LMFDB - Embedded newform 72.8.f.a.35.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [72,8,Mod(35,72)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(72, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 1, 1]))

N = Newforms(chi, 8, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("72.35");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 72 = 2^{3} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	72.f (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$22.4917218349$
Analytic rank:	$0$
Dimension:	$28$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			35.5
Character	$\chi$	$=$	72.35
Dual form			72.8.f.a.35.6

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-9.08380 - 6.74423i) q^{2} +(37.0308 + 122.526i) q^{4} +168.010 q^{5} +1488.58i q^{7} +(489.966 - 1362.75i) q^{8} +O(q^{10})$	\(q+(-9.08380 - 6.74423i) q^{2} +(37.0308 + 122.526i) q^{4} +168.010 q^{5} +1488.58i q^{7} +(489.966 - 1362.75i) q^{8} +(-1526.17 - 1133.10i) q^{10} +6118.18i q^{11} -7316.03i q^{13} +(10039.3 - 13521.9i) q^{14} +(-13641.4 + 9074.50i) q^{16} -22822.8i q^{17} -47190.9 q^{19} +(6221.56 + 20585.7i) q^{20} +(41262.4 - 55576.3i) q^{22} +54570.3 q^{23} -49897.5 q^{25} +(-49341.0 + 66457.4i) q^{26} +(-182390. + 55123.2i) q^{28} -97385.2 q^{29} +98959.4i q^{31} +(185117. + 9570.02i) q^{32} +(-153922. + 207318. i) q^{34} +250097. i q^{35} +361266. i q^{37} +(428672. + 318266. i) q^{38} +(82319.3 - 228956. i) q^{40} +404129. i q^{41} +2896.98 q^{43} +(-749639. + 226561. i) q^{44} +(-495706. - 368035. i) q^{46} -889688. q^{47} -1.39232e6 q^{49} +(453259. + 336520. i) q^{50} +(896407. - 270919. i) q^{52} -527514. q^{53} +1.02792e6i q^{55} +(2.02856e6 + 729352. i) q^{56} +(884627. + 656788. i) q^{58} -2.05569e6i q^{59} +276924. i q^{61} +(667404. - 898927. i) q^{62} +(-1.61702e6 - 1.33540e6i) q^{64} -1.22917e6i q^{65} -1.14980e6 q^{67} +(2.79640e6 - 845147. i) q^{68} +(1.68671e6 - 2.27183e6i) q^{70} -1.07344e6 q^{71} +5.20577e6 q^{73} +(2.43646e6 - 3.28166e6i) q^{74} +(-1.74752e6 - 5.78213e6i) q^{76} -9.10739e6 q^{77} +7.25453e6i q^{79} +(-2.29190e6 + 1.52461e6i) q^{80} +(2.72554e6 - 3.67103e6i) q^{82} -2.79942e6i q^{83} -3.83447e6i q^{85} +(-26315.6 - 19537.9i) q^{86} +(8.33755e6 + 2.99770e6i) q^{88} -4.23875e6i q^{89} +1.08905e7 q^{91} +(2.02078e6 + 6.68631e6i) q^{92} +(8.08175e6 + 6.00026e6i) q^{94} -7.92856e6 q^{95} -1.66999e7 q^{97} +(1.26476e7 + 9.39013e6i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 28 q + 52 q^{4}+O(q^{10}) $ 28 * q + 52 * q^4	$ 28 q + 52 q^{4} + 10092 q^{10} - 1928 q^{16} - 121168 q^{19} + 59576 q^{22} + 437500 q^{25} + 46872 q^{28} - 114748 q^{34} + 1054752 q^{40} + 1505696 q^{43} - 476184 q^{46} - 2272076 q^{49} + 1468392 q^{52} + 3054996 q^{58} - 4186016 q^{64} - 776272 q^{67} + 3238872 q^{70} - 2534128 q^{73} - 21642832 q^{76} + 10334372 q^{82} + 10834016 q^{88} - 3406992 q^{91} - 22555944 q^{94} - 26311456 q^{97}+O(q^{100}) $ 28 * q + 52 * q^4 + 10092 * q^10 - 1928 * q^16 - 121168 * q^19 + 59576 * q^22 + 437500 * q^25 + 46872 * q^28 - 114748 * q^34 + 1054752 * q^40 + 1505696 * q^43 - 476184 * q^46 - 2272076 * q^49 + 1468392 * q^52 + 3054996 * q^58 - 4186016 * q^64 - 776272 * q^67 + 3238872 * q^70 - 2534128 * q^73 - 21642832 * q^76 + 10334372 * q^82 + 10834016 * q^88 - 3406992 * q^91 - 22555944 * q^94 - 26311456 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/72\mathbb{Z}\right)^\times$.

$n$	$37$	$55$	$65$
$\chi(n)$	$-1$	$-1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$	−9.08380	−	6.74423i	−0.802902	−	0.596111i
$2$	−9.08380	−	6.74423i	−0.802902	−	0.596111i
$3$		0			0
$3$		0			0
$4$	37.0308	+	122.526i	0.289303	+	0.957238i
$5$	168.010			0.601092			0.300546	−	0.953767i	$-0.402831\pi$
$5$	168.010			0.601092			0.300546	+	0.953767i	$0.402831\pi$
$6$		0			0
$7$			1488.58i			1.64032i	0.572135	+	0.820160i	$0.306115\pi$
$7$			1488.58i			1.64032i	−0.572135	+	0.820160i	$0.693885\pi$
$8$	489.966	−	1362.75i	0.338338	−	0.941025i
$9$		0			0
$10$	−1526.17	−	1133.10i	−0.482618	−	0.358318i
$11$			6118.18i			1.38595i	0.720961	+	0.692976i	$0.243701\pi$
$11$			6118.18i			1.38595i	−0.720961	+	0.692976i	$0.756299\pi$
$12$		0			0
$13$		−	7316.03i		−	0.923579i	−0.886990	−	0.461789i	$-0.847208\pi$
$13$		−	7316.03i		−	0.923579i	0.886990	−	0.461789i	$-0.152792\pi$
$14$	10039.3	−	13521.9i	0.977813	−	1.31702i
$15$		0			0
$16$	−13641.4	+	9074.50i	−0.832607	+	0.553864i
$17$		−	22822.8i		−	1.12667i	−0.826228	−	0.563336i	$-0.809518\pi$
$17$		−	22822.8i		−	1.12667i	0.826228	−	0.563336i	$-0.190482\pi$
$18$		0			0
$19$	−47190.9			−1.57841			−0.789206	−	0.614128i	$-0.789508\pi$
$19$	−47190.9			−1.57841			−0.789206	+	0.614128i	$0.789508\pi$
$20$	6221.56	+	20585.7i	0.173898	+	0.575388i
$21$		0			0
$22$	41262.4	−	55576.3i	0.826181	−	1.11278i
$23$	54570.3			0.935210			0.467605	−	0.883938i	$-0.345117\pi$
$23$	54570.3			0.935210			0.467605	+	0.883938i	$0.345117\pi$
$24$		0			0
$25$	−49897.5			−0.638688
$26$	−49341.0	+	66457.4i	−0.550556	+	0.741543i
$27$		0			0
$28$	−182390.	+	55123.2i	−1.57018	+	0.474550i
$29$	−97385.2			−0.741481			−0.370740	−	0.928737i	$-0.620896\pi$
$29$	−97385.2			−0.741481			−0.370740	+	0.928737i	$0.620896\pi$
$30$		0			0
$31$			98959.4i			0.596611i	0.954471	+	0.298305i	$0.0964214\pi$
$31$			98959.4i			0.596611i	−0.954471	+	0.298305i	$0.903579\pi$
$32$	185117.	+	9570.02i	0.998666	+	0.0516283i
$33$		0			0
$34$	−153922.	+	207318.i	−0.671622	+	0.904608i
$35$			250097.i			0.985983i
$36$		0			0
$37$			361266.i			1.17252i	0.810123	+	0.586260i	$0.199400\pi$
$37$			361266.i			1.17252i	−0.810123	+	0.586260i	$0.800600\pi$
$38$	428672.	+	318266.i	1.26731	+	0.940909i
$39$		0			0
$40$	82319.3	−	228956.i	0.203372	−	0.565643i
$41$			404129.i			0.915749i	0.889017	+	0.457874i	$0.151389\pi$
$41$			404129.i			0.915749i	−0.889017	+	0.457874i	$0.848611\pi$
$42$		0			0
$43$	2896.98			0.00555656			0.00277828	−	0.999996i	$-0.499116\pi$
$43$	2896.98			0.00555656			0.00277828	+	0.999996i	$0.499116\pi$
$44$	−749639.	+	226561.i	−1.32668	+	0.400960i
$45$		0			0
$46$	−495706.	−	368035.i	−0.750882	−	0.557489i
$47$	−889688.			−1.24996			−0.624979	−	0.780642i	$-0.714892\pi$
$47$	−889688.			−1.24996			−0.624979	+	0.780642i	$0.714892\pi$
$48$		0			0
$49$	−1.39232e6			−1.69065
$50$	453259.	+	336520.i	0.512804	+	0.380729i
$51$		0			0
$52$	896407.	−	270919.i	0.884084	−	0.267194i
$53$	−527514.			−0.486708			−0.243354	−	0.969938i	$-0.578248\pi$
$53$	−527514.			−0.486708			−0.243354	+	0.969938i	$0.578248\pi$
$54$		0			0
$55$			1.02792e6i			0.833085i
$56$	2.02856e6	+	729352.i	1.54358	+	0.554982i
$57$		0			0
$58$	884627.	+	656788.i	0.595336	+	0.442005i
$59$		−	2.05569e6i		−	1.30310i	−0.758608	−	0.651548i	$-0.774120\pi$
$59$		−	2.05569e6i		−	1.30310i	0.758608	−	0.651548i	$-0.225880\pi$
$60$		0			0
$61$			276924.i			0.156209i	0.996945	+	0.0781046i	$0.0248868\pi$
$61$			276924.i			0.156209i	−0.996945	+	0.0781046i	$0.975113\pi$
$62$	667404.	−	898927.i	0.355646	−	0.479020i
$63$		0			0
$64$	−1.61702e6	−	1.33540e6i	−0.771055	−	0.636769i
$65$		−	1.22917e6i		−	0.555156i
$66$		0			0
$67$	−1.14980e6			−0.467049			−0.233524	−	0.972351i	$-0.575026\pi$
$67$	−1.14980e6			−0.467049			−0.233524	+	0.972351i	$0.575026\pi$
$68$	2.79640e6	−	845147.i	1.07849	−	0.325950i
$69$		0			0
$70$	1.68671e6	−	2.27183e6i	0.587755	−	0.791648i
$71$	−1.07344e6			−0.355936			−0.177968	−	0.984036i	$-0.556952\pi$
$71$	−1.07344e6			−0.355936			−0.177968	+	0.984036i	$0.556952\pi$
$72$		0			0
$73$	5.20577e6			1.56623			0.783115	−	0.621877i	$-0.213630\pi$
$73$	5.20577e6			1.56623			0.783115	+	0.621877i	$0.213630\pi$
$74$	2.43646e6	−	3.28166e6i	0.698953	−	0.941419i
$75$		0			0
$76$	−1.74752e6	−	5.78213e6i	−0.456640	−	1.51092i
$77$	−9.10739e6			−2.27340
$78$		0			0
$79$			7.25453e6i			1.65544i	0.561138	+	0.827722i	$0.310364\pi$
$79$			7.25453e6i			1.65544i	−0.561138	+	0.827722i	$0.689636\pi$
$80$	−2.29190e6	+	1.52461e6i	−0.500474	+	0.332923i
$81$		0			0
$82$	2.72554e6	−	3.67103e6i	0.545888	−	0.735257i
$83$		−	2.79942e6i		−	0.537396i	−0.963224	−	0.268698i	$-0.913407\pi$
$83$		−	2.79942e6i		−	0.537396i	0.963224	−	0.268698i	$-0.0865934\pi$
$84$		0			0
$85$		−	3.83447e6i		−	0.677234i
$86$	−26315.6	−	19537.9i	−0.00446137	−	0.00331232i
$87$		0			0
$88$	8.33755e6	+	2.99770e6i	1.30421	+	0.468920i
$89$		−	4.23875e6i		−	0.637343i	−0.947865	−	0.318671i	$-0.896763\pi$
$89$		−	4.23875e6i		−	0.637343i	0.947865	−	0.318671i	$-0.103237\pi$
$90$		0			0
$91$	1.08905e7			1.51496
$92$	2.02078e6	+	6.68631e6i	0.270559	+	0.895218i
$93$		0			0
$94$	8.08175e6	+	6.00026e6i	1.00359	+	0.745114i
$95$	−7.92856e6			−0.948771
$96$		0			0
$97$	−1.66999e7			−1.85786			−0.928929	−	0.370259i	$-0.879269\pi$
$97$	−1.66999e7			−1.85786			−0.928929	+	0.370259i	$0.879269\pi$
$98$	1.26476e7	+	9.39013e6i	1.35742	+	1.00781i
$99$		0			0
$100$	−1.84774e6	−	6.11376e6i	−0.184774	−	0.611376i
$101$	−3.86611e6			−0.373379			−0.186689	−	0.982419i	$-0.559776\pi$
$101$	−3.86611e6			−0.373379			−0.186689	+	0.982419i	$0.559776\pi$
$102$		0			0
$103$			4.50297e6i			0.406040i	0.979175	+	0.203020i	$0.0650757\pi$
$103$			4.50297e6i			0.406040i	−0.979175	+	0.203020i	$0.934924\pi$
$104$	−9.96992e6	−	3.58460e6i	−0.869110	−	0.312482i
$105$		0			0
$106$	4.79183e6	+	3.55768e6i	0.390779	+	0.290132i
$107$		−	4.77712e6i		−	0.376984i	−0.982075	−	0.188492i	$-0.939640\pi$
$107$		−	4.77712e6i		−	0.376984i	0.982075	−	0.188492i	$-0.0603599\pi$
$108$		0			0
$109$			8.78946e6i			0.650084i	0.945700	+	0.325042i	$0.105378\pi$
$109$			8.78946e6i			0.650084i	−0.945700	+	0.325042i	$0.894622\pi$
$110$	6.93251e6	−	9.33740e6i	0.496611	−	0.668885i
$111$		0			0
$112$	−1.35081e7	−	2.03063e7i	−0.908513	−	1.36574i
$113$			2.64197e7i			1.72248i	0.508201	+	0.861238i	$0.330311\pi$
$113$			2.64197e7i			1.72248i	−0.508201	+	0.861238i	$0.669689\pi$
$114$		0			0
$115$	9.16838e6			0.562147
$116$	−3.60625e6	−	1.19323e7i	−0.214513	−	0.709773i
$117$		0			0
$118$	−1.38641e7	+	1.86735e7i	−0.776790	+	1.04626i
$119$	3.39735e7			1.84810
$120$		0			0
$121$	−1.79450e7			−0.920861
$122$	1.86764e6	−	2.51552e6i	0.0931180	−	0.125421i
$123$		0			0
$124$	−1.21251e7	+	3.66454e6i	−0.571098	+	0.172601i
$125$	−2.15091e7			−0.985003
$126$		0			0
$127$			1.69006e7i			0.732133i	0.930589	+	0.366067i	$0.119296\pi$
$127$			1.69006e7i			0.732133i	−0.930589	+	0.366067i	$0.880704\pi$
$128$	5.68244e6	+	2.30361e7i	0.239497	+	0.970897i
$129$		0			0
$130$	−8.28980e6	+	1.11655e7i	−0.330935	+	0.445736i
$131$		−	9.35878e6i		−	0.363722i	−0.983324	−	0.181861i	$-0.941788\pi$
$131$		−	9.35878e6i		−	0.363722i	0.983324	−	0.181861i	$-0.0582121\pi$
$132$		0			0
$133$		−	7.02473e7i		−	2.58910i
$134$	1.04446e7	+	7.75454e6i	0.374994	+	0.278413i
$135$		0			0
$136$	−3.11018e7	−	1.11824e7i	−1.06023	−	0.381196i
$137$		−	8.09631e6i		−	0.269008i	−0.990913	−	0.134504i	$-0.957056\pi$
$137$		−	8.09631e6i		−	0.269008i	0.990913	−	0.134504i	$-0.0429441\pi$
$138$		0			0
$139$	−2.33396e7			−0.737126			−0.368563	−	0.929603i	$-0.620150\pi$
$139$	−2.33396e7			−0.737126			−0.368563	+	0.929603i	$0.620150\pi$
$140$	−3.06434e7	+	9.26127e6i	−0.943820	+	0.285248i
$141$		0			0
$142$	9.75088e6	+	7.23950e6i	0.285782	+	0.212177i
$143$	4.47608e7			1.28004
$144$		0			0
$145$	−1.63617e7			−0.445698
$146$	−4.72882e7	−	3.51089e7i	−1.25753	−	0.933647i
$147$		0			0
$148$	−4.42646e7	+	1.33780e7i	−1.12238	+	0.339214i
$149$	6.65657e7			1.64854			0.824269	−	0.566199i	$-0.191587\pi$
$149$	6.65657e7			1.64854			0.824269	+	0.566199i	$0.191587\pi$
$150$		0			0
$151$			6.91146e7i			1.63362i	0.576907	+	0.816810i	$0.304259\pi$
$151$			6.91146e7i			1.63362i	−0.576907	+	0.816810i	$0.695741\pi$
$152$	−2.31219e7	+	6.43093e7i	−0.534037	+	1.48532i
$153$		0			0
$154$	8.27297e7	+	6.14223e7i	1.82532	+	1.35520i
$155$			1.66262e7i			0.358618i
$156$		0			0
$157$		−	5.08536e7i		−	1.04875i	−0.851487	−	0.524377i	$-0.824298\pi$
$157$		−	5.08536e7i		−	1.04875i	0.851487	−	0.524377i	$-0.175702\pi$
$158$	4.89262e7	−	6.58987e7i	0.986829	−	1.32916i
$159$		0			0
$160$	3.11015e7	+	1.60786e6i	0.600291	+	0.0310334i
$161$			8.12322e7i			1.53404i
$162$		0			0
$163$	8.87701e7			1.60550			0.802749	−	0.596317i	$-0.203370\pi$
$163$	8.87701e7			1.60550			0.802749	+	0.596317i	$0.203370\pi$
$164$	−4.95165e7	+	1.49652e7i	−0.876589	+	0.264929i
$165$		0			0
$166$	−1.88799e7	+	2.54294e7i	−0.320348	+	0.431477i
$167$	4.58738e7			0.762179			0.381089	−	0.924538i	$-0.375549\pi$
$167$	4.58738e7			0.762179			0.381089	+	0.924538i	$0.375549\pi$
$168$		0			0
$169$	9.22418e6			0.147002
$170$	−2.58605e7	+	3.48315e7i	−0.403707	+	0.543753i
$171$		0			0
$172$	107277.	+	354956.i	0.00160753	+	0.00531894i
$173$	−1.26544e8			−1.85814			−0.929072	−	0.369899i	$-0.879392\pi$
$173$	−1.26544e8			−1.85814			−0.929072	+	0.369899i	$0.879392\pi$
$174$		0			0
$175$		−	7.42763e7i		−	1.04765i
$176$	−5.55195e7	−	8.34608e7i	−0.767628	−	1.15395i
$177$		0			0
$178$	−2.85871e7	+	3.85040e7i	−0.379927	+	0.511724i
$179$		−	6.04899e7i		−	0.788311i	−0.919044	−	0.394155i	$-0.871037\pi$
$179$		−	6.04899e7i		−	0.788311i	0.919044	−	0.394155i	$-0.128963\pi$
$180$		0			0
$181$		−	1.00538e8i		−	1.26025i	−0.776494	−	0.630125i	$-0.783004\pi$
$181$		−	1.00538e8i		−	1.26025i	0.776494	−	0.630125i	$-0.216996\pi$
$182$	−9.89270e7	−	7.34479e7i	−1.21637	−	0.903087i
$183$		0			0
$184$	2.67376e7	−	7.43657e7i	0.316417	−	0.880056i
$185$			6.06964e7i			0.704793i
$186$		0			0
$187$	1.39634e8			1.56151
$188$	−3.29459e7	−	1.09010e8i	−0.361617	−	1.19651i
$189$		0			0
$190$	7.20214e7	+	5.34720e7i	0.761770	+	0.565573i
$191$	8.42465e7			0.874853			0.437426	−	0.899254i	$-0.355890\pi$
$191$	8.42465e7			0.874853			0.437426	+	0.899254i	$0.355890\pi$
$192$		0			0
$193$	−3.01505e7			−0.301887			−0.150943	−	0.988542i	$-0.548231\pi$
$193$	−3.01505e7			−0.301887			−0.150943	+	0.988542i	$0.548231\pi$
$194$	1.51698e8	+	1.12628e8i	1.49168	+	1.10749i
$195$		0			0
$196$	−5.15588e7	−	1.70596e8i	−0.489110	−	1.61835i
$197$	1.38163e8			1.28754			0.643768	−	0.765221i	$-0.277370\pi$
$197$	1.38163e8			1.28754			0.643768	+	0.765221i	$0.277370\pi$
$198$		0			0
$199$			3.24443e7i			0.291846i	0.989296	+	0.145923i	$0.0466151\pi$
$199$			3.24443e7i			0.291846i	−0.989296	+	0.145923i	$0.953385\pi$
$200$	−2.44481e7	+	6.79978e7i	−0.216092	+	0.601021i
$201$		0			0
$202$	3.51190e7	+	2.60739e7i	0.299786	+	0.222575i
$203$		−	1.44965e8i		−	1.21627i
$204$		0			0
$205$			6.78979e7i			0.550450i
$206$	3.03691e7	−	4.09041e7i	0.242045	−	0.326011i
$207$		0			0
$208$	6.63894e7	+	9.98012e7i	0.511537	+	0.768978i
$209$		−	2.88722e8i		−	2.18760i
$210$		0			0
$211$	−1.93728e6			−0.0141973			−0.00709863	−	0.999975i	$-0.502260\pi$
$211$	−1.93728e6			−0.0141973			−0.00709863	+	0.999975i	$0.502260\pi$
$212$	−1.95343e7	−	6.46344e7i	−0.140806	−	0.465895i
$213$		0			0
$214$	−3.22180e7	+	4.33944e7i	−0.224724	+	0.302681i
$215$	486722.			0.00334000
$216$		0			0
$217$	−1.47309e8			−0.978632
$218$	5.92781e7	−	7.98417e7i	0.387522	−	0.521954i
$219$		0			0
$220$	−1.25947e8	+	3.80646e7i	−0.797460	+	0.241014i
$221$	−1.66972e8			−1.04057
$222$		0			0
$223$			5.78083e7i			0.349078i	0.984650	+	0.174539i	$0.0558436\pi$
$223$			5.78083e7i			0.349078i	−0.984650	+	0.174539i	$0.944156\pi$
$224$	−1.42457e7	+	2.75560e8i	−0.0846869	+	1.63813i
$225$		0			0
$226$	1.78180e8	−	2.39991e8i	1.02679	−	1.38298i
$227$			1.79786e8i			1.02015i	0.860129	+	0.510076i	$0.170383\pi$
$227$			1.79786e8i			1.02015i	−0.860129	+	0.510076i	$0.829617\pi$
$228$		0			0
$229$		−	1.20344e8i		−	0.662218i	−0.943593	−	0.331109i	$-0.892577\pi$
$229$		−	1.20344e8i		−	0.662218i	0.943593	−	0.331109i	$-0.107423\pi$
$230$	−8.32837e7	−	6.18336e7i	−0.451349	−	0.335102i
$231$		0			0
$232$	−4.77154e7	+	1.32712e8i	−0.250871	+	0.697752i
$233$			2.64384e8i			1.36927i	0.728886	+	0.684636i	$0.240039\pi$
$233$			2.64384e8i			1.36927i	−0.728886	+	0.684636i	$0.759961\pi$
$234$		0			0
$235$	−1.49477e8			−0.751340
$236$	2.51877e8	−	7.61240e7i	1.24737	−	0.376990i
$237$		0			0
$238$	−3.08609e8	−	2.29125e8i	−1.48385	−	1.10167i
$239$	−2.38126e8			−1.12827			−0.564137	−	0.825681i	$-0.690791\pi$
$239$	−2.38126e8			−1.12827			−0.564137	+	0.825681i	$0.690791\pi$
$240$		0			0
$241$	2.34147e8			1.07753			0.538763	−	0.842457i	$-0.318892\pi$
$241$	2.34147e8			1.07753			0.538763	+	0.842457i	$0.318892\pi$
$242$	1.63009e8	+	1.21025e8i	0.739361	+	0.548936i
$243$		0			0
$244$	−3.39305e7	+	1.02547e7i	−0.149529	+	0.0451918i
$245$	−2.33924e8			−1.01623
$246$		0			0
$247$			3.45250e8i			1.45779i
$248$	1.34857e8	+	4.84867e7i	0.561425	+	0.201856i
$249$		0			0
$250$	1.95384e8	+	1.45062e8i	0.790861	+	0.587171i
$251$			2.51967e8i			1.00574i	0.864362	+	0.502871i	$0.167723\pi$
$251$			2.51967e8i			1.00574i	−0.864362	+	0.502871i	$0.832277\pi$
$252$		0			0
$253$			3.33871e8i			1.29616i
$254$	1.13982e8	−	1.53522e8i	0.436433	−	0.587831i
$255$		0			0
$256$	1.03742e8	−	2.47579e8i	0.386470	−	0.922302i
$257$			2.09860e7i			0.0771195i	0.999256	+	0.0385597i	$0.0122770\pi$
$257$			2.09860e7i			0.0771195i	−0.999256	+	0.0385597i	$0.987723\pi$
$258$		0			0
$259$	−5.37772e8			−1.92331
$260$	1.50606e8	−	4.55171e7i	0.531416	−	0.160608i
$261$		0			0
$262$	−6.31178e7	+	8.50133e7i	−0.216819	+	0.292033i
$263$	−1.44706e8			−0.490503			−0.245251	−	0.969460i	$-0.578870\pi$
$263$	−1.44706e8			−0.490503			−0.245251	+	0.969460i	$0.578870\pi$
$264$		0			0
$265$	−8.86279e7			−0.292556
$266$	−4.73764e8	+	6.38112e8i	−1.54339	+	2.07879i
$267$		0			0
$268$	−4.25782e7	−	1.40881e8i	−0.135119	−	0.447076i
$269$	1.66474e8			0.521451			0.260725	−	0.965413i	$-0.416038\pi$
$269$	1.66474e8			0.521451			0.260725	+	0.965413i	$0.416038\pi$
$270$		0			0
$271$		−	2.83438e8i		−	0.865097i	−0.901611	−	0.432549i	$-0.857614\pi$
$271$		−	2.83438e8i		−	0.865097i	0.901611	−	0.432549i	$-0.142386\pi$
$272$	2.07106e8	+	3.11336e8i	0.624023	+	0.938076i
$273$		0			0
$274$	−5.46034e7	+	7.35453e7i	−0.160359	+	0.215987i
$275$		−	3.05282e8i		−	0.885191i
$276$		0			0
$277$			6.45289e8i			1.82421i	0.409954	+	0.912106i	$0.365545\pi$
$277$			6.45289e8i			1.82421i	−0.409954	+	0.912106i	$0.634455\pi$
$278$	2.12012e8	+	1.57408e8i	0.591840	+	0.439409i
$279$		0			0
$280$	3.40819e8	+	1.22539e8i	0.927834	+	0.333595i
$281$		−	2.64934e8i		−	0.712305i	−0.934428	−	0.356153i	$-0.884088\pi$
$281$		−	2.64934e8i		−	0.712305i	0.934428	−	0.356153i	$-0.115912\pi$
$282$		0			0
$283$	7.23578e7			0.189772			0.0948861	−	0.995488i	$-0.469751\pi$
$283$	7.23578e7			0.189772			0.0948861	+	0.995488i	$0.469751\pi$
$284$	−3.97502e7	−	1.31524e8i	−0.102973	−	0.340715i
$285$		0			0
$286$	−4.06598e8	−	3.01877e8i	−1.02774	−	0.763043i
$287$	−6.01577e8			−1.50212
$288$		0			0
$289$	−1.10542e8			−0.269392
$290$	1.48627e8	+	1.10347e8i	0.357852	+	0.265686i
$291$		0			0
$292$	1.92774e8	+	6.37845e8i	0.453115	+	1.49925i
$293$	1.41789e8			0.329310			0.164655	−	0.986351i	$-0.447349\pi$
$293$	1.41789e8			0.329310			0.164655	+	0.986351i	$0.447349\pi$
$294$		0			0
$295$		−	3.45378e8i		−	0.783281i
$296$	4.92314e8	+	1.77008e8i	1.10337	+	0.396708i
$297$		0			0
$298$	−6.04670e8	−	4.48934e8i	−1.32361	−	0.982711i
$299$		−	3.99238e8i		−	0.863740i
$300$		0			0
$301$			4.31238e6i			0.00911452i
$302$	4.66125e8	−	6.27824e8i	0.973818	−	1.31164i
$303$		0			0
$304$	6.43751e8	−	4.28234e8i	1.31420	−	0.874225i
$305$			4.65261e7i			0.0938961i
$306$		0			0
$307$	5.38374e8			1.06194			0.530970	−	0.847391i	$-0.321828\pi$
$307$	5.38374e8			1.06194			0.530970	+	0.847391i	$0.321828\pi$
$308$	−3.37254e8	−	1.11590e9i	−0.657703	−	2.17619i
$309$		0			0
$310$	1.12131e8	−	1.51029e8i	0.213776	−	0.287935i
$311$	−8.28423e7			−0.156168			−0.0780838	−	0.996947i	$-0.524880\pi$
$311$	−8.28423e7			−0.156168			−0.0780838	+	0.996947i	$0.524880\pi$
$312$		0			0
$313$	3.02329e8			0.557282			0.278641	−	0.960395i	$-0.410116\pi$
$313$	3.02329e8			0.557282			0.278641	+	0.960395i	$0.410116\pi$
$314$	−3.42968e8	+	4.61944e8i	−0.625173	+	0.842046i
$315$		0			0
$316$	−8.88872e8	+	2.68641e8i	−1.58465	+	0.478925i
$317$	7.70255e8			1.35809			0.679043	−	0.734099i	$-0.262395\pi$
$317$	7.70255e8			1.35809			0.679043	+	0.734099i	$0.262395\pi$
$318$		0			0
$319$		−	5.95820e8i		−	1.02766i
$320$	−2.71676e8	−	2.24361e8i	−0.463475	−	0.382757i
$321$		0			0
$322$	5.47848e8	−	7.37897e8i	0.914460	−	1.23169i
$323$			1.07703e9i			1.77835i
$324$		0			0
$325$			3.65052e8i			0.589879i
$326$	−8.06370e8	−	5.98686e8i	−1.28906	−	0.957056i
$327$		0			0
$328$	5.50726e8	+	1.98009e8i	0.861742	+	0.309833i
$329$		−	1.32437e9i		−	2.05033i
$330$		0			0
$331$	−8.90041e8			−1.34900			−0.674500	−	0.738274i	$-0.735641\pi$
$331$	−8.90041e8			−1.34900			−0.674500	+	0.738274i	$0.735641\pi$
$332$	3.43003e8	−	1.03665e8i	0.514416	−	0.155470i
$333$		0			0
$334$	−4.16708e8	−	3.09383e8i	−0.611955	−	0.454343i
$335$	−1.93179e8			−0.280739
$336$		0			0
$337$	1.17163e9			1.66757			0.833787	−	0.552087i	$-0.186168\pi$
$337$	1.17163e9			1.66757			0.833787	+	0.552087i	$0.186168\pi$
$338$	−8.37906e7	−	6.22100e7i	−0.118028	−	0.0876297i
$339$		0			0
$340$	4.69824e8	−	1.41993e8i	0.648274	−	0.195926i
$341$	−6.05451e8			−0.826873
$342$		0			0
$343$		−	8.46670e8i		−	1.13288i
$344$	1.41942e6	−	3.94786e6i	0.00187999	−	0.00522886i
$345$		0			0
$346$	1.14950e9	+	8.53440e8i	1.49191	+	1.10766i
$347$			5.95493e8i			0.765109i	0.923933	+	0.382555i	$0.124956\pi$
$347$			5.95493e8i			0.765109i	−0.923933	+	0.382555i	$0.875044\pi$
$348$		0			0
$349$		−	5.55585e8i		−	0.699619i	−0.936821	−	0.349810i	$-0.886246\pi$
$349$		−	5.55585e8i		−	0.699619i	0.936821	−	0.349810i	$-0.113754\pi$
$350$	−5.00936e8	+	6.74711e8i	−0.624517	+	0.841162i
$351$		0			0
$352$	−5.85511e7	+	1.13258e9i	−0.0715543	+	1.38410i
$353$			1.97472e8i			0.238942i	0.992838	+	0.119471i	$0.0381199\pi$
$353$			1.97472e8i			0.238942i	−0.992838	+	0.119471i	$0.961880\pi$
$354$		0			0
$355$	−1.80348e8			−0.213950
$356$	5.19359e8	−	1.56964e8i	0.610088	−	0.184385i
$357$		0			0
$358$	−4.07958e8	+	5.49478e8i	−0.469921	+	0.632936i
$359$	1.31954e9			1.50519			0.752594	−	0.658485i	$-0.228802\pi$
$359$	1.31954e9			1.50519			0.752594	+	0.658485i	$0.228802\pi$
$360$		0			0
$361$	1.33311e9			1.49138
$362$	−6.78053e8	+	9.13270e8i	−0.751249	+	1.01186i
$363$		0			0
$364$	4.03283e8	+	1.33437e9i	0.438284	+	1.45018i
$365$	8.74624e8			0.941448
$366$		0			0
$367$			1.80227e8i			0.190321i	0.995462	+	0.0951607i	$0.0303365\pi$
$367$			1.80227e8i			0.190321i	−0.995462	+	0.0951607i	$0.969664\pi$
$368$	−7.44418e8	+	4.95198e8i	−0.778663	+	0.517979i
$369$		0			0
$370$	4.09350e8	−	5.51354e8i	0.420135	−	0.565880i
$371$		−	7.85246e8i		−	0.798357i
$372$		0			0
$373$			8.35266e8i			0.833382i	0.909048	+	0.416691i	$0.136810\pi$
$373$			8.35266e8i			0.833382i	−0.909048	+	0.416691i	$0.863190\pi$
$374$	−1.26841e9	−	9.41724e8i	−1.25374	−	0.930836i
$375$		0			0
$376$	−4.35917e8	+	1.21242e9i	−0.422908	+	1.17624i
$377$			7.12473e8i			0.684816i
$378$		0			0
$379$	−8.71747e7			−0.0822532			−0.0411266	−	0.999154i	$-0.513095\pi$
$379$	−8.71747e7			−0.0822532			−0.0411266	+	0.999154i	$0.513095\pi$
$380$	−2.93601e8	−	9.71457e8i	−0.274482	−	0.908199i
$381$		0			0
$382$	−7.65278e8	−	5.68177e8i	−0.702421	−	0.521509i
$383$	2.24075e8			0.203797			0.101899	−	0.994795i	$-0.467508\pi$
$383$	2.24075e8			0.203797			0.101899	+	0.994795i	$0.467508\pi$
$384$		0			0
$385$	−1.53014e9			−1.36652
$386$	2.73881e8	+	2.03342e8i	0.242386	+	0.179958i
$387$		0			0
$388$	−6.18410e8	−	2.04618e9i	−0.537484	−	1.77841i
$389$	1.05419e9			0.908021			0.454010	−	0.890996i	$-0.349993\pi$
$389$	1.05419e9			0.908021			0.454010	+	0.890996i	$0.349993\pi$
$390$		0			0
$391$		−	1.24545e9i		−	1.05368i
$392$	−6.82189e8	+	1.89738e9i	−0.572010	+	1.59094i
$393$		0			0
$394$	−1.25504e9	−	9.31801e8i	−1.03376	−	0.767514i
$395$			1.21884e9i			0.995075i
$396$		0			0
$397$		−	1.07674e9i		−	0.863662i	−0.901955	−	0.431831i	$-0.857868\pi$
$397$		−	1.07674e9i		−	0.863662i	0.901955	−	0.431831i	$-0.142132\pi$
$398$	2.18812e8	−	2.94718e8i	0.173972	−	0.234323i
$399$		0			0
$400$	6.80674e8	−	4.52795e8i	0.531776	−	0.353746i
$401$			5.27042e8i			0.408169i	0.978953	+	0.204085i	$0.0654218\pi$
$401$			5.27042e8i			0.408169i	−0.978953	+	0.204085i	$0.934578\pi$
$402$		0			0
$403$	7.23990e8			0.551017
$404$	−1.43165e8	−	4.73701e8i	−0.108020	−	0.357412i
$405$		0			0
$406$	−9.77680e8	+	1.31684e9i	−0.725029	+	0.976542i
$407$	−2.21029e9			−1.62506
$408$		0			0
$409$	1.01839e8			0.0736006			0.0368003	−	0.999323i	$-0.488283\pi$
$409$	1.01839e8			0.0736006			0.0368003	+	0.999323i	$0.488283\pi$
$410$	4.57919e8	−	6.16770e8i	0.328129	−	0.441957i
$411$		0			0
$412$	−5.51733e8	+	1.66749e8i	−0.388677	+	0.117469i
$413$	3.06006e9			2.13749
$414$		0			0
$415$		−	4.70332e8i		−	0.323025i
$416$	7.00146e7	−	1.35432e9i	0.0476828	−	0.922347i
$417$		0			0
$418$	−1.94721e9	+	2.62270e9i	−1.30405	+	1.75643i
$419$			3.15851e8i			0.209765i	0.994485	+	0.104883i	$0.0334467\pi$
$419$			3.15851e8i			0.209765i	−0.994485	+	0.104883i	$0.966553\pi$
$420$		0			0
$421$			6.81309e8i			0.444997i	0.974933	+	0.222498i	$0.0714212\pi$
$421$			6.81309e8i			0.444997i	−0.974933	+	0.222498i	$0.928579\pi$
$422$	1.75979e7	+	1.30655e7i	0.0113990	+	0.00846314i
$423$		0			0
$424$	−2.58464e8	+	7.18870e8i	−0.164672	+	0.458004i
$425$			1.13880e9i			0.719593i
$426$		0			0
$427$	−4.12223e8			−0.256233
$428$	5.85323e8	−	1.76900e8i	0.360863	−	0.109063i
$429$		0			0
$430$	−4.42129e6	−	3.28257e6i	−0.00268169	−	0.00199101i
$431$	2.39495e9			1.44088			0.720438	−	0.693520i	$-0.243941\pi$
$431$	2.39495e9			1.44088			0.720438	+	0.693520i	$0.243941\pi$
$432$		0			0
$433$	−2.53253e9			−1.49915			−0.749577	−	0.661917i	$-0.769743\pi$
$433$	−2.53253e9			−1.49915			−0.749577	+	0.661917i	$0.769743\pi$
$434$	1.33812e9	+	9.93483e8i	0.785745	+	0.583373i
$435$		0			0
$436$	−1.07694e9	+	3.25481e8i	−0.622285	+	0.188071i
$437$	−2.57522e9			−1.47615
$438$		0			0
$439$			2.76692e9i			1.56088i	0.625229	+	0.780441i	$0.285006\pi$
$439$			2.76692e9i			1.56088i	−0.625229	+	0.780441i	$0.714994\pi$
$440$	1.40079e9	+	5.03644e8i	0.783953	+	0.281864i
$441$		0			0
$442$	1.51674e9	+	1.12610e9i	0.835477	+	0.620296i
$443$		−	3.23885e9i		−	1.77002i	−0.465575	−	0.885008i	$-0.654153\pi$
$443$		−	3.23885e9i		−	1.77002i	0.465575	−	0.885008i	$-0.345847\pi$
$444$		0			0
$445$		−	7.12154e8i		−	0.383102i
$446$	3.89872e8	−	5.25119e8i	0.208089	−	0.280276i
$447$		0			0
$448$	1.98785e9	−	2.40706e9i	1.04450	−	1.26478i
$449$		−	4.35223e8i		−	0.226908i	−0.993543	−	0.113454i	$-0.963809\pi$
$449$		−	4.35223e8i		−	0.226908i	0.993543	−	0.113454i	$-0.0361915\pi$
$450$		0			0
$451$	−2.47253e9			−1.26918
$452$	−3.23711e9	+	9.78343e8i	−1.64882	+	0.498318i
$453$		0			0
$454$	1.21252e9	−	1.63314e9i	0.608124	−	0.819082i
$455$	1.82971e9			0.910633
$456$		0			0
$457$	−4.04027e8			−0.198018			−0.0990088	−	0.995087i	$-0.531567\pi$
$457$	−4.04027e8			−0.198018			−0.0990088	+	0.995087i	$0.531567\pi$
$458$	−8.11628e8	+	1.09318e9i	−0.394755	+	0.531696i
$459$		0			0
$460$	3.39512e8	+	1.12337e9i	0.162631	+	0.538109i
$461$	−2.95816e9			−1.40627			−0.703134	−	0.711058i	$-0.748216\pi$
$461$	−2.95816e9			−1.40627			−0.703134	+	0.711058i	$0.748216\pi$
$462$		0			0
$463$			7.95716e7i			0.0372584i	0.999826	+	0.0186292i	$0.00593021\pi$
$463$			7.95716e7i			0.0372584i	−0.999826	+	0.0186292i	$0.994070\pi$
$464$	1.32847e9	−	8.83722e8i	0.617362	−	0.410679i
$465$		0			0
$466$	1.78307e9	−	2.40161e9i	0.816238	−	1.09939i
$467$			1.51432e9i			0.688033i	0.938964	+	0.344017i	$0.111788\pi$
$467$			1.51432e9i			0.688033i	−0.938964	+	0.344017i	$0.888212\pi$
$468$		0			0
$469$		−	1.71157e9i		−	0.766109i
$470$	1.35782e9	+	1.00811e9i	0.603252	+	0.447882i
$471$		0			0
$472$	−2.80139e9	−	1.00722e9i	−1.22625	−	0.440887i
$473$			1.77242e7i			0.00770112i
$474$		0			0
$475$	2.35471e9			1.00811
$476$	1.25807e9	+	4.16265e9i	0.534662	+	1.76907i
$477$		0			0
$478$	2.16309e9	+	1.60598e9i	0.905894	+	0.672577i
$479$	−5.30880e8			−0.220710			−0.110355	−	0.993892i	$-0.535199\pi$
$479$	−5.30880e8			−0.220710			−0.110355	+	0.993892i	$0.535199\pi$
$480$		0			0
$481$	2.64303e9			1.08292
$482$	−2.12694e9	−	1.57914e9i	−0.865148	−	0.642326i
$483$		0			0
$484$	−6.64517e8	−	2.19873e9i	−0.266408	−	0.881483i
$485$	−2.80575e9			−1.11674
$486$		0			0
$487$			1.30663e9i			0.512627i	0.966594	+	0.256314i	$0.0825080\pi$
$487$			1.30663e9i			0.512627i	−0.966594	+	0.256314i	$0.917492\pi$
$488$	3.77378e8	+	1.35683e8i	0.146997	+	0.0528515i
$489$		0			0
$490$	2.12492e9	+	1.57764e9i	0.815937	+	0.605789i
$491$			1.03337e8i			0.0393976i	0.999806	+	0.0196988i	$0.00627073\pi$
$491$			1.03337e8i			0.0393976i	−0.999806	+	0.0196988i	$0.993729\pi$
$492$		0			0
$493$			2.22260e9i			0.835406i
$494$	2.32844e9	−	3.13618e9i	0.869004	−	1.17046i
$495$		0			0
$496$	−8.98007e8	−	1.34995e9i	−0.330441	−	0.496742i
$497$		−	1.59789e9i		−	0.583849i
$498$		0			0
$499$	5.46162e9			1.96775			0.983874	−	0.178861i	$-0.0572411\pi$
$499$	5.46162e9			1.96775			0.983874	+	0.178861i	$0.0572411\pi$
$500$	−7.96500e8	−	2.63543e9i	−0.284964	−	0.942882i
$501$		0			0
$502$	1.69933e9	−	2.28882e9i	0.599534	−	0.807512i
$503$	−5.34875e9			−1.87398			−0.936989	−	0.349359i	$-0.886399\pi$
$503$	−5.34875e9			−1.87398			−0.936989	+	0.349359i	$0.886399\pi$
$504$		0			0
$505$	−6.49547e8			−0.224435
$506$	2.25170e9	−	3.03282e9i	0.772653	−	1.04069i
$507$		0			0
$508$	−2.07077e9	+	6.25844e8i	−0.700825	+	0.211808i
$509$	−5.19906e9			−1.74748			−0.873741	−	0.486392i	$-0.838313\pi$
$509$	−5.19906e9			−1.74748			−0.873741	+	0.486392i	$0.838313\pi$
$510$		0			0
$511$			7.74920e9i			2.56912i
$512$	−2.61210e9	+	1.54929e9i	−0.860092	+	0.510139i
$513$		0			0
$514$	1.41534e8	−	1.90633e8i	0.0459718	−	0.0619194i
$515$			7.56547e8i			0.244068i
$516$		0			0
$517$		−	5.44327e9i		−	1.73238i
$518$	4.88501e9	+	3.62686e9i	1.54423	+	1.14651i
$519$		0			0
$520$	−1.67505e9	−	6.02251e8i	−0.522415	−	0.187830i
$521$		−	9.03089e8i		−	0.279768i	−0.990168	−	0.139884i	$-0.955327\pi$
$521$		−	9.03089e8i		−	0.279768i	0.990168	−	0.139884i	$-0.0446730\pi$
$522$		0			0
$523$	1.94736e9			0.595238			0.297619	−	0.954685i	$-0.403808\pi$
$523$	1.94736e9			0.595238			0.297619	+	0.954685i	$0.403808\pi$
$524$	1.14670e9	−	3.46563e8i	0.348169	−	0.105226i
$525$		0			0
$526$	1.31448e9	+	9.75930e8i	0.393826	+	0.292394i
$527$	2.25853e9			0.672185
$528$		0			0
$529$	−4.26905e8			−0.125382
$530$	8.05078e8	+	5.97726e8i	0.234894	+	0.174396i
$531$		0			0
$532$	8.60715e9	−	2.60131e9i	2.47838	−	0.749035i
$533$	2.95662e9			0.845766
$534$		0			0
$535$		−	8.02605e8i		−	0.226602i
$536$	−5.63365e8	+	1.56690e9i	−0.158020	+	0.439504i
$537$		0			0
$538$	−1.51222e9	−	1.12274e9i	−0.418674	−	0.310843i
$539$		−	8.51847e9i		−	2.34316i
$540$		0			0
$541$		−	9.12981e7i		−	0.0247897i	−0.999923	−	0.0123949i	$-0.996054\pi$
$541$		−	9.12981e7i		−	0.0247897i	0.999923	−	0.0123949i	$-0.00394551\pi$
$542$	−1.91157e9	+	2.57469e9i	−0.515694	+	0.694588i
$543$		0			0
$544$	2.18415e8	−	4.22488e9i	0.0581682	−	1.12517i
$545$			1.47672e9i			0.390760i
$546$		0			0
$547$	−7.28778e8			−0.190388			−0.0951940	−	0.995459i	$-0.530347\pi$
$547$	−7.28778e8			−0.190388			−0.0951940	+	0.995459i	$0.530347\pi$
$548$	9.92012e8	−	2.99813e8i	0.257505	−	0.0778249i
$549$		0			0
$550$	−2.05889e9	+	2.77312e9i	−0.527672	+	0.710721i
$551$	4.59569e9			1.17036
$552$		0			0
$553$	−1.07989e10			−2.71546
$554$	4.35198e9	−	5.86168e9i	1.08743	−	1.46466i
$555$		0			0
$556$	−8.64284e8	−	2.85972e9i	−0.213253	−	0.705605i
$557$	−4.48763e9			−1.10033			−0.550166	−	0.835055i	$-0.685436\pi$
$557$	−4.48763e9			−1.10033			−0.550166	+	0.835055i	$0.685436\pi$
$558$		0			0
$559$		−	2.11944e7i		−	0.00513192i
$560$	−2.26950e9	−	3.41168e9i	−0.546100	−	0.820937i
$561$		0			0
$562$	−1.78678e9	+	2.40661e9i	−0.424613	+	0.571911i
$563$		−	5.25800e9i		−	1.24177i	−0.783902	−	0.620885i	$-0.786774\pi$
$563$		−	5.25800e9i		−	1.24177i	0.783902	−	0.620885i	$-0.213226\pi$
$564$		0			0
$565$			4.43878e9i			1.03537i
$566$	−6.57283e8	−	4.87997e8i	−0.152368	−	0.113125i
$567$		0			0
$568$	−5.25947e8	+	1.46282e9i	−0.120427	+	0.334945i
$569$			2.51985e9i			0.573431i	0.958016	+	0.286716i	$0.0925635\pi$
$569$			2.51985e9i			0.573431i	−0.958016	+	0.286716i	$0.907436\pi$
$570$		0			0
$571$	−4.52309e9			−1.01674			−0.508368	−	0.861140i	$-0.669751\pi$
$571$	−4.52309e9			−1.01674			−0.508368	+	0.861140i	$0.669751\pi$
$572$	1.65753e9	+	5.48438e9i	0.370318	+	1.22530i
$573$		0			0
$574$	5.46461e9	+	4.05717e9i	1.20606	+	0.895431i
$575$	−2.72292e9			−0.597307
$576$		0			0
$577$	4.28629e9			0.928893			0.464447	−	0.885601i	$-0.346253\pi$
$577$	4.28629e9			0.928893			0.464447	+	0.885601i	$0.346253\pi$
$578$	1.00414e9	+	7.45519e8i	0.216295	+	0.160587i
$579$		0			0
$580$	−6.05888e8	−	2.00474e9i	−0.128942	−	0.426639i
$581$	4.16716e9			0.881502
$582$		0			0
$583$		−	3.22743e9i		−	0.674554i
$584$	2.55065e9	−	7.09417e9i	0.529915	−	1.47386i
$585$		0			0
$586$	−1.28798e9	−	9.56255e8i	−0.264404	−	0.196305i
$587$			4.91633e9i			1.00325i	0.865086	+	0.501623i	$0.167264\pi$
$587$			4.91633e9i			1.00325i	−0.865086	+	0.501623i	$0.832736\pi$
$588$		0			0
$589$		−	4.66998e9i		−	0.941697i
$590$	−2.32931e9	+	3.13734e9i	−0.466922	+	0.628898i
$591$		0			0
$592$	−3.27831e9	−	4.92818e9i	−0.649417	−	0.976249i
$593$			4.94870e9i			0.974539i	0.873252	+	0.487270i	$0.162007\pi$
$593$			4.94870e9i			0.974539i	−0.873252	+	0.487270i	$0.837993\pi$
$594$		0			0
$595$	5.70790e9			1.11088
$596$	2.46498e9	+	8.15606e9i	0.476927	+	1.57804i
$597$		0			0
$598$	−2.69255e9	+	3.62660e9i	−0.514885	+	0.693499i
$599$	−2.07676e9			−0.394813			−0.197407	−	0.980322i	$-0.563252\pi$
$599$	−2.07676e9			−0.394813			−0.197407	+	0.980322i	$0.563252\pi$
$600$		0			0
$601$	−1.96011e9			−0.368315			−0.184158	−	0.982897i	$-0.558956\pi$
$601$	−1.96011e9			−0.368315			−0.184158	+	0.982897i	$0.558956\pi$
$602$	2.90837e7	−	3.91728e7i	0.00543327	−	0.00731807i
$603$		0			0
$604$	−8.46837e9	+	2.55937e9i	−1.56376	+	0.472611i
$605$	−3.01494e9			−0.553523
$606$		0			0
$607$			2.96338e9i			0.537808i	0.963167	+	0.268904i	$0.0866614\pi$
$607$			2.96338e9i			0.537808i	−0.963167	+	0.268904i	$0.913339\pi$
$608$	−8.73581e9	−	4.51618e8i	−1.57631	−	0.0814907i
$609$		0			0
$610$	3.13783e8	−	4.22634e8i	0.0559725	−	0.0753894i
$611$			6.50899e9i			1.15443i
$612$		0			0
$613$		−	6.91931e9i		−	1.21325i	−0.794987	−	0.606626i	$-0.792522\pi$
$613$		−	6.91931e9i		−	1.21325i	0.794987	−	0.606626i	$-0.207478\pi$
$614$	−4.89048e9	−	3.63092e9i	−0.852634	−	0.633034i
$615$		0			0
$616$	−4.46231e9	+	1.24111e10i	−0.769178	+	2.13933i
$617$			6.20892e9i			1.06419i	0.846685	+	0.532094i	$0.178595\pi$
$617$			6.20892e9i			1.06419i	−0.846685	+	0.532094i	$0.821405\pi$
$618$		0			0
$619$	−5.22520e9			−0.885494			−0.442747	−	0.896647i	$-0.645996\pi$
$619$	−5.22520e9			−0.885494			−0.442747	+	0.896647i	$0.645996\pi$
$620$	−2.03715e9	+	6.15682e8i	−0.343283	+	0.103749i
$621$		0			0
$622$	7.52523e8	+	5.58707e8i	0.125387	+	0.0930932i
$623$	6.30971e9			1.04545
$624$		0			0
$625$	2.84489e8			0.0466107
$626$	−2.74630e9	−	2.03898e9i	−0.447443	−	0.332202i
$627$		0			0
$628$	6.23091e9	−	1.88315e9i	1.00391	−	0.303408i
$629$	8.24509e9			1.32105
$630$		0			0
$631$			8.31627e9i			1.31773i	0.752262	+	0.658864i	$0.228963\pi$
$631$			8.31627e9i			1.31773i	−0.752262	+	0.658864i	$0.771037\pi$
$632$	9.88611e9	+	3.55447e9i	1.55781	+	0.560100i
$633$		0			0
$634$	−6.99684e9	−	5.19477e9i	−1.09041	−	0.809570i
$635$			2.83948e9i			0.440080i
$636$		0			0
$637$			1.01863e10i			1.56145i
$638$	−4.01835e9	+	5.41231e9i	−0.612597	+	0.825107i
$639$		0			0
$640$	9.54708e8	+	3.87030e9i	0.143960	+	0.583599i
$641$		−	7.19590e9i		−	1.07915i	−0.841937	−	0.539575i	$-0.818585\pi$
$641$		−	7.19590e9i		−	1.07915i	0.841937	−	0.539575i	$-0.181415\pi$
$642$		0			0
$643$	4.49943e8			0.0667451			0.0333725	−	0.999443i	$-0.489375\pi$
$643$	4.49943e8			0.0667451			0.0333725	+	0.999443i	$0.489375\pi$
$644$	−9.95309e9	+	3.00809e9i	−1.46844	+	0.443803i
$645$		0			0
$646$	7.26372e9	−	9.78351e9i	1.06010	−	1.42784i
$647$	9.91493e9			1.43921			0.719605	−	0.694383i	$-0.244323\pi$
$647$	9.91493e9			1.43921			0.719605	+	0.694383i	$0.244323\pi$
$648$		0			0
$649$	1.25771e10			1.80603
$650$	2.46199e9	−	3.31606e9i	0.351633	−	0.473615i
$651$		0			0
$652$	3.28723e9	+	1.08767e10i	0.464476	+	1.53684i
$653$	2.41085e9			0.338824			0.169412	−	0.985545i	$-0.445813\pi$
$653$	2.41085e9			0.338824			0.169412	+	0.985545i	$0.445813\pi$
$654$		0			0
$655$		−	1.57237e9i		−	0.218631i
$656$	−3.66727e9	−	5.51290e9i	−0.507200	−	0.762459i
$657$		0			0
$658$	−8.93185e9	+	1.20303e10i	−1.22222	+	1.64621i
$659$			1.19216e10i			1.62268i	0.584571	+	0.811342i	$0.301263\pi$
$659$			1.19216e10i			1.62268i	−0.584571	+	0.811342i	$0.698737\pi$
$660$		0			0
$661$		−	1.02064e10i		−	1.37457i	−0.726387	−	0.687286i	$-0.758802\pi$
$661$		−	1.02064e10i		−	1.37457i	0.726387	−	0.687286i	$-0.241198\pi$
$662$	8.08496e9	+	6.00264e9i	1.08312	+	0.804154i
$663$		0			0
$664$	−3.81491e9	−	1.37162e9i	−0.505703	−	0.181822i
$665$		−	1.18023e10i		−	1.55629i
$666$		0			0
$667$	−5.31434e9			−0.693440
$668$	1.69874e9	+	5.62075e9i	0.220501	+	0.729586i
$669$		0			0
$670$	1.75480e9	+	1.30284e9i	0.225406	+	0.167352i
$671$	−1.69427e9			−0.216498
$672$		0			0
$673$	−8.35328e9			−1.05634			−0.528170	−	0.849138i	$-0.677122\pi$
$673$	−8.35328e9			−1.05634			−0.528170	+	0.849138i	$0.677122\pi$
$674$	−1.06428e10	−	7.90172e9i	−1.33890	−	0.994059i
$675$		0			0
$676$	3.41579e8	+	1.13021e9i	0.0425282	+	0.140716i
$677$	7.79709e8			0.0965767			0.0482883	−	0.998833i	$-0.484623\pi$
$677$	7.79709e8			0.0965767			0.0482883	+	0.998833i	$0.484623\pi$
$678$		0			0
$679$		−	2.48591e10i		−	3.04748i
$680$	−5.22542e9	−	1.87876e9i	−0.637294	−	0.229134i
$681$		0			0
$682$	5.49980e9	+	4.08330e9i	0.663898	+	0.492908i
$683$		−	9.64107e9i		−	1.15785i	−0.815381	−	0.578925i	$-0.803472\pi$
$683$		−	9.64107e9i		−	1.15785i	0.815381	−	0.578925i	$-0.196528\pi$
$684$		0			0
$685$		−	1.36026e9i		−	0.161699i
$686$	−5.71014e9	+	7.69098e9i	−0.675324	+	0.909593i
$687$		0			0
$688$	−3.95190e7	+	2.62886e7i	−0.00462643	+	0.00307757i
$689$			3.85931e9i			0.449513i
$690$		0			0
$691$	−5.92970e8			−0.0683690			−0.0341845	−	0.999416i	$-0.510883\pi$
$691$	−5.92970e8			−0.0683690			−0.0341845	+	0.999416i	$0.510883\pi$
$692$	−4.68602e9	−	1.55050e10i	−0.537567	−	1.77869i
$693$		0			0
$694$	4.01614e9	−	5.40934e9i	0.456090	−	0.614308i
$695$	−3.92130e9			−0.443081
$696$		0			0
$697$	9.22336e9			1.03175
$698$	−3.74699e9	+	5.04683e9i	−0.417051	+	0.561726i
$699$		0			0
$700$	9.10081e9	−	2.75051e9i	1.00285	−	0.303089i
$701$	9.18280e9			1.00684			0.503422	−	0.864041i	$-0.332074\pi$
$701$	9.18280e9			1.00684			0.503422	+	0.864041i	$0.332074\pi$
$702$		0			0
$703$		−	1.70484e10i		−	1.85072i
$704$	8.17022e9	−	9.89322e9i	0.882530	−	1.06864i
$705$		0			0
$706$	1.33179e9	−	1.79379e9i	0.142436	−	0.191847i
$707$		−	5.75501e9i		−	0.612460i
$708$		0			0
$709$			1.42255e10i			1.49902i	0.661994	+	0.749509i	$0.269710\pi$
$709$			1.42255e10i			1.49902i	−0.661994	+	0.749509i	$0.730290\pi$
$710$	1.63825e9	+	1.21631e9i	0.171781	+	0.127538i
$711$		0			0
$712$	−5.77636e9	−	2.07684e9i	−0.599755	−	0.215637i
$713$			5.40024e9i			0.557956i
$714$		0			0
$715$	7.52028e9			0.769419
$716$	7.41162e9	−	2.23999e9i	0.754601	−	0.228061i
$717$		0			0
$718$	−1.19864e10	−	8.89925e9i	−1.20852	−	0.897259i
$719$	−8.99661e9			−0.902667			−0.451334	−	0.892355i	$-0.649052\pi$
$719$	−8.99661e9			−0.902667			−0.451334	+	0.892355i	$0.649052\pi$
$720$		0			0
$721$	−6.70303e9			−0.666036
$722$	−1.21097e10	−	8.99077e9i	−1.19744	−	0.889031i
$723$		0			0
$724$	1.23186e10	−	3.72302e9i	1.20636	−	0.364594i
$725$	4.85928e9			0.473575
$726$		0			0
$727$			1.18128e10i			1.14020i	0.821575	+	0.570100i	$0.193096\pi$
$727$			1.18128e10i			1.14020i	−0.821575	+	0.570100i	$0.806904\pi$
$728$	5.33596e9	−	1.48410e10i	0.512570	−	1.42562i
$729$		0			0
$730$	−7.94491e9	−	5.89866e9i	−0.755891	−	0.561208i
$731$		−	6.61172e7i		−	0.00626042i
$732$		0			0
$733$			1.00912e10i			0.946414i	0.880951	+	0.473207i	$0.156904\pi$
$733$			1.00912e10i			0.946414i	−0.880951	+	0.473207i	$0.843096\pi$
$734$	1.21549e9	−	1.63714e9i	0.113453	−	0.152809i
$735$		0			0
$736$	1.01019e10	+	5.22239e8i	0.933963	+	0.0482833i
$737$		−	7.03471e9i		−	0.647307i
$738$		0			0
$739$	1.06274e10			0.968661			0.484331	−	0.874885i	$-0.339063\pi$
$739$	1.06274e10			0.968661			0.484331	+	0.874885i	$0.339063\pi$
$740$	−7.43691e9	+	2.24763e9i	−0.674654	+	0.203899i
$741$		0			0
$742$	−5.29588e9	+	7.13302e9i	−0.475909	+	0.641002i
$743$	7.94701e9			0.710792			0.355396	−	0.934716i	$-0.384346\pi$
$743$	7.94701e9			0.710792			0.355396	+	0.934716i	$0.384346\pi$
$744$		0			0
$745$	1.11837e10			0.990923
$746$	5.63322e9	−	7.58739e9i	0.496788	−	0.669124i
$747$		0			0
$748$	5.17076e9	+	1.71089e10i	0.451751	+	1.49474i
$749$	7.11111e9			0.618374
$750$		0			0
$751$		−	1.15775e10i		−	0.997411i	−0.866772	−	0.498705i	$-0.833809\pi$
$751$		−	1.15775e10i		−	0.997411i	0.866772	−	0.498705i	$-0.166191\pi$
$752$	1.21366e10	−	8.07348e9i	1.04072	−	0.692306i
$753$		0			0
$754$	4.80508e9	−	6.47196e9i	0.408226	−	0.549840i
$755$			1.16120e10i			0.981956i
$756$		0			0
$757$			4.91453e9i			0.411762i	0.978577	+	0.205881i	$0.0660061\pi$
$757$			4.91453e9i			0.411762i	−0.978577	+	0.205881i	$0.933994\pi$
$758$	7.91877e8	+	5.87926e8i	0.0660413	+	0.0490321i
$759$		0			0
$760$	−3.88472e9	+	1.08046e10i	−0.321005	+	0.892817i
$761$		−	2.53368e9i		−	0.208404i	−0.994556	−	0.104202i	$-0.966771\pi$
$761$		−	2.53368e9i		−	0.208404i	0.994556	−	0.104202i	$-0.0332288\pi$
$762$		0			0
$763$	−1.30838e10			−1.06635
$764$	3.11971e9	+	1.03224e10i	0.253098	+	0.837442i
$765$		0			0
$766$	−2.03545e9	−	1.51121e9i	−0.163629	−	0.121486i
$767$	−1.50395e10			−1.20351
$768$		0			0
$769$	1.90636e10			1.51169			0.755845	−	0.654750i	$-0.227226\pi$
$769$	1.90636e10			1.51169			0.755845	+	0.654750i	$0.227226\pi$
$770$	1.38994e10	+	1.03196e10i	1.09719	+	0.814601i
$771$		0			0
$772$	−1.11650e9	−	3.69424e9i	−0.0873368	−	0.288977i
$773$	−6.81224e9			−0.530471			−0.265236	−	0.964184i	$-0.585450\pi$
$773$	−6.81224e9			−0.530471			−0.265236	+	0.964184i	$0.585450\pi$
$774$		0			0
$775$		−	4.93783e9i		−	0.381048i
$776$	−8.18237e9	+	2.27578e10i	−0.628583	+	1.74829i
$777$		0			0
$778$	−9.57606e9	−	7.10971e9i	−0.729052	−	0.541281i
$779$		−	1.90712e10i		−	1.44543i
$780$		0			0
$781$		−	6.56748e9i		−	0.493310i
$782$	−8.39958e9	+	1.13134e10i	−0.628108	+	0.845998i
$783$		0			0
$784$	1.89933e10	−	1.26346e10i	1.40765	−	0.936388i
$785$		−	8.54394e9i		−	0.630397i
$786$		0			0
$787$	−1.56544e10			−1.14479			−0.572393	−	0.819979i	$-0.693985\pi$
$787$	−1.56544e10			−1.14479			−0.572393	+	0.819979i	$0.693985\pi$
$788$	5.11628e9	+	1.69286e10i	0.372488	+	1.23248i
$789$		0			0
$790$	8.22011e9	−	1.10717e10i	0.593175	−	0.798948i
$791$	−3.93278e10			−2.82541
$792$		0			0
$793$	2.02599e9			0.144271
$794$	−7.26177e9	+	9.78088e9i	−0.514838	+	0.693436i
$795$		0			0
$796$	−3.97529e9	+	1.20144e9i	−0.279366	+	0.0844318i
$797$	−1.50517e9			−0.105313			−0.0526565	−	0.998613i	$-0.516769\pi$
$797$	−1.50517e9			−0.105313			−0.0526565	+	0.998613i	$0.516769\pi$
$798$		0			0
$799$			2.03052e10i			1.40829i
$800$	−9.23686e9	−	4.77520e8i	−0.637836	−	0.0329744i
$801$		0			0
$802$	3.55449e9	−	4.78755e9i	0.243314	−	0.327720i
$803$			3.18499e10i			2.17072i
$804$		0			0
$805$			1.36478e10i			0.922101i
$806$	−6.57658e9	−	4.88275e9i	−0.442413	−	0.328467i
$807$		0			0
$808$	−1.89426e9	+	5.26854e9i	−0.126328	+	0.351359i
$809$		−	2.50669e10i		−	1.66449i	−0.554412	−	0.832243i	$-0.687057\pi$
$809$		−	2.50669e10i		−	1.66449i	0.554412	−	0.832243i	$-0.312943\pi$
$810$		0			0
$811$	−2.89387e10			−1.90505			−0.952525	−	0.304459i	$-0.901524\pi$
$811$	−2.89387e10			−1.90505			−0.952525	+	0.304459i	$0.901524\pi$
$812$	1.77621e10	−	5.36819e9i	1.16425	−	0.351869i
$813$		0			0
$814$	2.00778e10	+	1.49067e10i	1.30476	+	0.968714i
$815$	1.49143e10			0.965053
$816$		0			0
$817$	−1.36711e8			−0.00877053
$818$	−9.25082e8	−	6.86823e8i	−0.0590940	−	0.0438741i
$819$		0			0
$820$	−8.31928e9	+	2.51431e9i	−0.526911	+	0.159247i
$821$	−9.16791e9			−0.578188			−0.289094	−	0.957301i	$-0.593354\pi$
$821$	−9.16791e9			−0.578188			−0.289094	+	0.957301i	$0.593354\pi$
$822$		0			0
$823$			2.14215e10i			1.33952i	0.742576	+	0.669762i	$0.233604\pi$
$823$			2.14215e10i			1.33952i	−0.742576	+	0.669762i	$0.766396\pi$
$824$	6.13643e9	+	2.20630e9i	0.382094	+	0.137379i
$825$		0			0
$826$	−2.77970e10	−	2.06377e10i	−1.71620	−	1.27418i
$827$		−	5.43716e9i		−	0.334274i	−0.985934	−	0.167137i	$-0.946548\pi$
$827$		−	5.43716e9i		−	0.334274i	0.985934	−	0.167137i	$-0.0534523\pi$
$828$		0			0
$829$			3.17916e9i			0.193808i	0.995294	+	0.0969040i	$0.0308940\pi$
$829$			3.17916e9i			0.193808i	−0.995294	+	0.0969040i	$0.969106\pi$
$830$	−3.17202e9	+	4.27240e9i	−0.192559	+	0.259357i
$831$		0			0
$832$	−9.76983e9	+	1.18302e10i	−0.588106	+	0.712130i
$833$			3.17767e10i			1.90481i
$834$		0			0
$835$	7.70727e9			0.458140
$836$	3.53761e10	−	1.06916e10i	2.09406	−	0.632880i
$837$		0			0
$838$	2.13017e9	−	2.86913e9i	0.125043	−	0.168421i
$839$	5.66513e9			0.331164			0.165582	−	0.986196i	$-0.447050\pi$
$839$	5.66513e9			0.331164			0.165582	+	0.986196i	$0.447050\pi$
$840$		0			0
$841$	−7.76600e9			−0.450206
$842$	4.59490e9	−	6.18887e9i	0.265267	−	0.357289i
$843$		0			0
$844$	−7.17391e7	−	2.37368e8i	−0.00410731	−	0.0135901i
$845$	1.54976e9			0.0883620
$846$		0			0
$847$		−	2.67125e10i		−	1.51051i
$848$	7.19605e9	−	4.78693e9i	0.405237	−	0.269570i
$849$		0			0
$850$	7.68033e9	−	1.03446e10i	0.428957	−	0.577762i
$851$			1.97144e10i			1.09655i
$852$		0			0
$853$			3.15111e10i			1.73837i	0.494487	+	0.869185i	$0.335356\pi$
$853$			3.15111e10i			1.73837i	−0.494487	+	0.869185i	$0.664644\pi$
$854$	3.74455e9	+	2.78013e9i	0.205730	+	0.152743i
$855$		0			0
$856$	−6.51001e9	−	2.34062e9i	−0.354751	−	0.127548i
$857$			1.20759e10i			0.655371i	0.944787	+	0.327685i	$0.106269\pi$
$857$			1.20759e10i			0.655371i	−0.944787	+	0.327685i	$0.893731\pi$
$858$		0			0
$859$	7.65080e8			0.0411842			0.0205921	−	0.999788i	$-0.493445\pi$
$859$	7.65080e8			0.0411842			0.0205921	+	0.999788i	$0.493445\pi$
$860$	1.80237e7	+	5.96363e7i	0.000966273	+	0.00319718i
$861$		0			0
$862$	−2.17553e10	−	1.61521e10i	−1.15688	−	0.858922i
$863$	−4.39561e9			−0.232799			−0.116399	−	0.993202i	$-0.537135\pi$
$863$	−4.39561e9			−0.232799			−0.116399	+	0.993202i	$0.537135\pi$
$864$		0			0
$865$	−2.12607e10			−1.11692
$866$	2.30050e10	+	1.70799e10i	1.20367	+	0.893663i
$867$		0			0
$868$	−5.45496e9	−	1.80492e10i	−0.283121	−	0.936783i
$869$	−4.43846e10			−2.29437
$870$		0			0
$871$			8.41201e9i			0.431356i
$872$	1.19778e10	+	4.30653e9i	0.611745	+	0.219948i
$873$		0			0
$874$	2.33928e10	+	1.73679e10i	1.18520	+	0.879947i
$875$		−	3.20180e10i		−	1.61572i
$876$		0			0
$877$		−	1.15552e10i		−	0.578468i	−0.957258	−	0.289234i	$-0.906599\pi$
$877$		−	1.15552e10i		−	0.578468i	0.957258	−	0.289234i	$-0.0934005\pi$
$878$	1.86607e10	−	2.51341e10i	0.930459	−	1.25324i
$879$		0			0
$880$	−9.32784e9	−	1.40223e10i	−0.461415	−	0.693632i
$881$		−	7.56814e9i		−	0.372884i	−0.982466	−	0.186442i	$-0.940304\pi$
$881$		−	7.56814e9i		−	0.372884i	0.982466	−	0.186442i	$-0.0596956\pi$
$882$		0			0
$883$	−4.48773e9			−0.219364			−0.109682	−	0.993967i	$-0.534983\pi$
$883$	−4.48773e9			−0.219364			−0.109682	+	0.993967i	$0.534983\pi$
$884$	−6.18312e9	−	2.04585e10i	−0.301040	−	0.996074i
$885$		0			0
$886$	−2.18435e10	+	2.94210e10i	−1.05513	+	1.42115i
$887$	5.36413e9			0.258087			0.129044	−	0.991639i	$-0.458809\pi$
$887$	5.36413e9			0.258087			0.129044	+	0.991639i	$0.458809\pi$
$888$		0			0
$889$	−2.51579e10			−1.20093
$890$	−4.80293e9	+	6.46907e9i	−0.228371	+	0.307593i
$891$		0			0
$892$	−7.08304e9	+	2.14069e9i	−0.334151	+	0.100989i
$893$	4.19852e10			1.97295
$894$		0			0
$895$		−	1.01629e10i		−	0.473848i
$896$	−3.42910e10	+	8.45875e9i	−1.59258	+	0.392851i
$897$		0			0
$898$	−2.93524e9	+	3.95348e9i	−0.135262	+	0.182185i
$899$		−	9.63718e9i		−	0.442375i
$900$		0			0
$901$			1.20394e10i			0.548361i
$902$	2.24600e10	+	1.66753e10i	1.01903	+	0.756574i
$903$		0			0
$904$	3.60034e10	+	1.29447e10i	1.62089	+	0.582779i
$905$		−	1.68915e10i		−	0.757526i
$906$		0			0
$907$	−7.51806e9			−0.334565			−0.167282	−	0.985909i	$-0.553499\pi$
$907$	−7.51806e9			−0.334565			−0.167282	+	0.985909i	$0.553499\pi$
$908$	−2.20285e10	+	6.65761e9i	−0.976528	+	0.295133i
$909$		0			0
$910$	−1.66208e10	−	1.23400e10i	−0.731149	−	0.542838i
$911$	−2.19518e10			−0.961959			−0.480979	−	0.876732i	$-0.659719\pi$
$911$	−2.19518e10			−0.961959			−0.480979	+	0.876732i	$0.659719\pi$
$912$		0			0
$913$	1.71274e10			0.744805
$914$	3.67010e9	+	2.72485e9i	0.158989	+	0.118040i
$915$		0			0
$916$	1.47453e10	−	4.45644e9i	0.633900	−	0.191582i
$917$	1.39313e10			0.596621
$918$		0			0
$919$			5.05031e9i			0.214642i	0.994224	+	0.107321i	$0.0342272\pi$
$919$			5.05031e9i			0.214642i	−0.994224	+	0.107321i	$0.965773\pi$
$920$	4.49219e9	−	1.24942e10i	0.190196	−	0.528995i
$921$		0			0
$922$	2.68713e10	+	1.99505e10i	1.12909	+	0.838292i
$923$			7.85329e9i			0.328735i
$924$		0			0
$925$		−	1.80263e10i		−	0.748875i
$926$	5.36649e8	−	7.22812e8i	0.0222102	−	0.0299149i
$927$		0			0
$928$	−1.80276e10	−	9.31978e8i	−0.740492	−	0.0382814i
$929$		−	1.97133e10i		−	0.806685i	−0.915049	−	0.403343i	$-0.867848\pi$
$929$		−	1.97133e10i		−	0.806685i	0.915049	−	0.403343i	$-0.132152\pi$
$930$		0			0
$931$	6.57048e10			2.66854
$932$	−3.23940e10	+	9.79035e9i	−1.31072	+	0.396134i
$933$		0			0
$934$	1.02129e10	−	1.37558e10i	0.410144	−	0.552423i
$935$	2.34600e10			0.938614
$936$		0			0
$937$	−2.39586e10			−0.951422			−0.475711	−	0.879602i	$-0.657809\pi$
$937$	−2.39586e10			−0.951422			−0.475711	+	0.879602i	$0.657809\pi$
$938$	−1.15432e10	+	1.55476e10i	−0.456686	+	0.615110i
$939$		0			0
$940$	−5.53525e9	−	1.83149e10i	−0.217365	−	0.719211i
$941$	−2.89908e10			−1.13422			−0.567109	−	0.823643i	$-0.691938\pi$
$941$	−2.89908e10			−1.13422			−0.567109	+	0.823643i	$0.691938\pi$
$942$		0			0
$943$			2.20534e10i			0.856417i
$944$	1.86544e10	+	2.80426e10i	0.721737	+	1.08497i
$945$		0			0
$946$	1.19536e8	−	1.61003e8i	0.00459072	−	0.00618324i
$947$		−	4.26639e9i		−	0.163244i	−0.996663	−	0.0816218i	$-0.973990\pi$
$947$		−	4.26639e9i		−	0.163244i	0.996663	−	0.0816218i	$-0.0260100\pi$
$948$		0			0
$949$		−	3.80856e10i		−	1.44654i
$950$	−2.13897e10	−	1.58807e10i	−0.809416	−	0.600947i
$951$		0			0
$952$	1.66459e10	−	4.62974e10i	0.625283	−	1.73911i
$953$			2.75031e10i			1.02933i	0.857390	+	0.514667i	$0.172084\pi$
$953$			2.75031e10i			1.02933i	−0.857390	+	0.514667i	$0.827916\pi$
$954$		0			0
$955$	1.41543e10			0.525867
$956$	−8.81801e9	−	2.91768e10i	−0.326413	−	1.08003i
$957$		0			0
$958$	4.82241e9	+	3.58038e9i	0.177209	+	0.131568i
$959$	1.20520e10			0.441259
$960$		0			0
$961$	1.77197e10			0.644056
$962$	−2.40088e10	−	1.78252e10i	−0.869475	−	0.645538i
$963$		0			0
$964$	8.67063e9	+	2.86891e10i	0.311732	+	1.03145i
$965$	−5.06560e9			−0.181462
$966$		0			0
$967$			4.75920e9i			0.169255i	0.996413	+	0.0846274i	$0.0269700\pi$
$967$			4.75920e9i			0.169255i	−0.996413	+	0.0846274i	$0.973030\pi$
$968$	−8.79242e9	+	2.44545e10i	−0.311562	+	0.866553i
$969$		0			0
$970$	2.54869e10	+	1.89226e10i	0.896636	+	0.665703i
$971$		−	1.02554e10i		−	0.359488i	−0.983713	−	0.179744i	$-0.942473\pi$
$971$		−	1.02554e10i		−	0.359488i	0.983713	−	0.179744i	$-0.0575270\pi$
$972$		0			0
$973$		−	3.47428e10i		−	1.20912i
$974$	8.81222e9	−	1.18692e10i	0.305583	−	0.411589i
$975$		0			0
$976$	−2.51295e9	−	3.77764e9i	−0.0865186	−	0.130061i
$977$			6.45020e9i			0.221280i	0.993861	+	0.110640i	$0.0352900\pi$
$977$			6.45020e9i			0.221280i	−0.993861	+	0.110640i	$0.964710\pi$
$978$		0			0
$979$	2.59335e10			0.883326
$980$	−8.66241e9	−	2.86619e10i	−0.294000	−	0.972778i
$981$		0			0
$982$	6.96927e8	−	9.38691e8i	0.0234853	−	0.0316324i
$983$	2.23515e10			0.750531			0.375265	−	0.926917i	$-0.377552\pi$
$983$	2.23515e10			0.750531			0.375265	+	0.926917i	$0.377552\pi$
$984$		0			0
$985$	2.32128e10			0.773927
$986$	1.49897e10	−	2.01897e10i	0.497995	−	0.670749i
$987$		0			0
$988$	−4.23022e10	+	1.27849e10i	−1.39545	+	0.421743i
$989$	1.58089e8			0.00519655
$990$		0			0
$991$			1.68634e10i			0.550412i	0.961385	+	0.275206i	$0.0887460\pi$
$991$			1.68634e10i			0.550412i	−0.961385	+	0.275206i	$0.911254\pi$
$992$	−9.47043e8	+	1.83190e10i	−0.0308020	+	0.595815i
$993$		0			0
$994$	−1.07766e10	+	1.45149e10i	−0.348039	+	0.468773i
$995$			5.45098e9i			0.175426i
$996$		0			0
$997$		−	2.99924e10i		−	0.958468i	−0.877687	−	0.479234i	$-0.840914\pi$
$997$		−	2.99924e10i		−	0.958468i	0.877687	−	0.479234i	$-0.159086\pi$
$998$	−4.96123e10	−	3.68344e10i	−1.57991	−	1.17300i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	72.8.f.a.35.5	✓	28
3.2	odd	2	inner	72.8.f.a.35.24	yes	28
4.3	odd	2		288.8.f.a.143.19		28
8.3	odd	2	inner	72.8.f.a.35.23	yes	28
8.5	even	2		288.8.f.a.143.10		28
12.11	even	2		288.8.f.a.143.9		28
24.5	odd	2		288.8.f.a.143.20		28
24.11	even	2	inner	72.8.f.a.35.6	yes	28

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
72.8.f.a.35.5	✓	28	1.1	even	1	trivial
72.8.f.a.35.6	yes	28	24.11	even	2	inner
72.8.f.a.35.23	yes	28	8.3	odd	2	inner
72.8.f.a.35.24	yes	28	3.2	odd	2	inner
288.8.f.a.143.9		28	12.11	even	2
288.8.f.a.143.10		28	8.5	even	2
288.8.f.a.143.19		28	4.3	odd	2
288.8.f.a.143.20		28	24.5	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 72 = 2^{3} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	72.f (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(-9.08380 - 6.74423i) q^{2} +(37.0308 + 122.526i) q^{4} +168.010 q^{5} +1488.58i q^{7} +(489.966 - 1362.75i) q^{8} +O(q^{10})\)	\(q+(-9.08380 - 6.74423i) q^{2} +(37.0308 + 122.526i) q^{4} +168.010 q^{5} +1488.58i q^{7} +(489.966 - 1362.75i) q^{8} +(-1526.17 - 1133.10i) q^{10} +6118.18i q^{11} -7316.03i q^{13} +(10039.3 - 13521.9i) q^{14} +(-13641.4 + 9074.50i) q^{16} -22822.8i q^{17} -47190.9 q^{19} +(6221.56 + 20585.7i) q^{20} +(41262.4 - 55576.3i) q^{22} +54570.3 q^{23} -49897.5 q^{25} +(-49341.0 + 66457.4i) q^{26} +(-182390. + 55123.2i) q^{28} -97385.2 q^{29} +98959.4i q^{31} +(185117. + 9570.02i) q^{32} +(-153922. + 207318. i) q^{34} +250097. i q^{35} +361266. i q^{37} +(428672. + 318266. i) q^{38} +(82319.3 - 228956. i) q^{40} +404129. i q^{41} +2896.98 q^{43} +(-749639. + 226561. i) q^{44} +(-495706. - 368035. i) q^{46} -889688. q^{47} -1.39232e6 q^{49} +(453259. + 336520. i) q^{50} +(896407. - 270919. i) q^{52} -527514. q^{53} +1.02792e6i q^{55} +(2.02856e6 + 729352. i) q^{56} +(884627. + 656788. i) q^{58} -2.05569e6i q^{59} +276924. i q^{61} +(667404. - 898927. i) q^{62} +(-1.61702e6 - 1.33540e6i) q^{64} -1.22917e6i q^{65} -1.14980e6 q^{67} +(2.79640e6 - 845147. i) q^{68} +(1.68671e6 - 2.27183e6i) q^{70} -1.07344e6 q^{71} +5.20577e6 q^{73} +(2.43646e6 - 3.28166e6i) q^{74} +(-1.74752e6 - 5.78213e6i) q^{76} -9.10739e6 q^{77} +7.25453e6i q^{79} +(-2.29190e6 + 1.52461e6i) q^{80} +(2.72554e6 - 3.67103e6i) q^{82} -2.79942e6i q^{83} -3.83447e6i q^{85} +(-26315.6 - 19537.9i) q^{86} +(8.33755e6 + 2.99770e6i) q^{88} -4.23875e6i q^{89} +1.08905e7 q^{91} +(2.02078e6 + 6.68631e6i) q^{92} +(8.08175e6 + 6.00026e6i) q^{94} -7.92856e6 q^{95} -1.66999e7 q^{97} +(1.26476e7 + 9.39013e6i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q + 52 q^{4}+O(q^{10}) \) 28 * q + 52 * q^4	\( 28 q + 52 q^{4} + 10092 q^{10} - 1928 q^{16} - 121168 q^{19} + 59576 q^{22} + 437500 q^{25} + 46872 q^{28} - 114748 q^{34} + 1054752 q^{40} + 1505696 q^{43} - 476184 q^{46} - 2272076 q^{49} + 1468392 q^{52} + 3054996 q^{58} - 4186016 q^{64} - 776272 q^{67} + 3238872 q^{70} - 2534128 q^{73} - 21642832 q^{76} + 10334372 q^{82} + 10834016 q^{88} - 3406992 q^{91} - 22555944 q^{94} - 26311456 q^{97}+O(q^{100}) \) 28 * q + 52 * q^4 + 10092 * q^10 - 1928 * q^16 - 121168 * q^19 + 59576 * q^22 + 437500 * q^25 + 46872 * q^28 - 114748 * q^34 + 1054752 * q^40 + 1505696 * q^43 - 476184 * q^46 - 2272076 * q^49 + 1468392 * q^52 + 3054996 * q^58 - 4186016 * q^64 - 776272 * q^67 + 3238872 * q^70 - 2534128 * q^73 - 21642832 * q^76 + 10334372 * q^82 + 10834016 * q^88 - 3406992 * q^91 - 22555944 * q^94 - 26311456 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more