LMFDB - Embedded newform 288.8.f.a.143.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [288,8,Mod(143,288)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(288, 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("288.143");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 288 = 2^{5} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	288.f (of order $2$, degree $1$, not 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:	$89.9668873394$
Analytic rank:	$0$
Dimension:	$28$
Twist minimal:	no (minimal twist has level 72)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			143.9
Character	$\chi$	$=$	288.143
Dual form			288.8.f.a.143.10

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-168.010 q^{5} -1488.58i q^{7} +O(q^{10})$	$q-168.010 q^{5} -1488.58i q^{7} +6118.18i q^{11} -7316.03i q^{13} +22822.8i q^{17} +47190.9 q^{19} +54570.3 q^{23} -49897.5 q^{25} +97385.2 q^{29} -98959.4i q^{31} +250097. i q^{35} +361266. i q^{37} -404129. i q^{41} -2896.98 q^{43} -889688. q^{47} -1.39232e6 q^{49} +527514. q^{53} -1.02792e6i q^{55} -2.05569e6i q^{59} +276924. i q^{61} +1.22917e6i q^{65} +1.14980e6 q^{67} -1.07344e6 q^{71} +5.20577e6 q^{73} +9.10739e6 q^{77} -7.25453e6i q^{79} -2.79942e6i q^{83} -3.83447e6i q^{85} +4.23875e6i q^{89} -1.08905e7 q^{91} -7.92856e6 q^{95} -1.66999e7 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 28 q+O(q^{10}) $ 28 * q	$ 28 q + 121168 q^{19} + 437500 q^{25} - 1505696 q^{43} - 2272076 q^{49} + 776272 q^{67} - 2534128 q^{73} + 3406992 q^{91} - 26311456 q^{97}+O(q^{100}) $ 28 * q + 121168 * q^19 + 437500 * q^25 - 1505696 * q^43 - 2272076 * q^49 + 776272 * q^67 - 2534128 * q^73 + 3406992 * q^91 - 26311456 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/288\mathbb{Z}\right)^\times$.

$n$	$37$	$65$	$127$
$\chi(n)$	$-1$	$-1$	$-1$

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 $

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	−168.010	−0.601092	−0.300546	−	0.953767i	$-0.597169\pi$
$5$	−168.010	−0.601092	−0.300546	+	0.953767i	$0.597169\pi$
$6$	0	0
$7$	− 1488.58i	− 1.64032i	−0.572135	−	0.820160i	$-0.693885\pi$
$7$	− 1488.58i	− 1.64032i	0.572135	−	0.820160i	$-0.306115\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	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	$-0.152792\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	22822.8i	1.12667i	0.826228	+	0.563336i	$0.190482\pi$
$17$	22822.8i	1.12667i	−0.826228	+	0.563336i	$0.809518\pi$
$18$	0	0
$19$	47190.9	1.57841	0.789206	−	0.614128i	$-0.210492\pi$
$19$	47190.9	1.57841	0.789206	+	0.614128i	$0.210492\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	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
$26$	0	0
$27$	0	0
$28$	0	0
$29$	97385.2	0.741481	0.370740	−	0.928737i	$-0.379104\pi$
$29$	97385.2	0.741481	0.370740	+	0.928737i	$0.379104\pi$
$30$	0	0
$31$	− 98959.4i	− 0.596611i	−0.954471	−	0.298305i	$-0.903579\pi$
$31$	− 98959.4i	− 0.596611i	0.954471	−	0.298305i	$-0.0964214\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	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	$0.800600\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 404129.i	− 0.915749i	−0.889017	−	0.457874i	$-0.848611\pi$
$41$	− 404129.i	− 0.915749i	0.889017	−	0.457874i	$-0.151389\pi$
$42$	0	0
$43$	−2896.98	−0.00555656	−0.00277828	−	0.999996i	$-0.500884\pi$
$43$	−2896.98	−0.00555656	−0.00277828	+	0.999996i	$0.500884\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−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
$50$	0	0
$51$	0	0
$52$	0	0
$53$	527514.	0.486708	0.243354	−	0.969938i	$-0.421752\pi$
$53$	527514.	0.486708	0.243354	+	0.969938i	$0.421752\pi$
$54$	0	0
$55$	− 1.02792e6i	− 0.833085i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	− 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	$0.975113\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	1.22917e6i	0.555156i
$66$	0	0
$67$	1.14980e6	0.467049	0.233524	−	0.972351i	$-0.424974\pi$
$67$	1.14980e6	0.467049	0.233524	+	0.972351i	$0.424974\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−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	$0.213630\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	9.10739e6	2.27340
$78$	0	0
$79$	− 7.25453e6i	− 1.65544i	−0.561138	−	0.827722i	$-0.689636\pi$
$79$	− 7.25453e6i	− 1.65544i	0.561138	−	0.827722i	$-0.310364\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	− 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
$86$	0	0
$87$	0	0
$88$	0	0
$89$	4.23875e6i	0.637343i	0.947865	+	0.318671i	$0.103237\pi$
$89$	4.23875e6i	0.637343i	−0.947865	+	0.318671i	$0.896763\pi$
$90$	0	0
$91$	−1.08905e7	−1.51496
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−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	$0.879269\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	3.86611e6	0.373379	0.186689	−	0.982419i	$-0.440224\pi$
$101$	3.86611e6	0.373379	0.186689	+	0.982419i	$0.440224\pi$
$102$	0	0
$103$	− 4.50297e6i	− 0.406040i	−0.979175	−	0.203020i	$-0.934924\pi$
$103$	− 4.50297e6i	− 0.406040i	0.979175	−	0.203020i	$-0.0650757\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 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	$0.894622\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 2.64197e7i	− 1.72248i	−0.508201	−	0.861238i	$-0.669689\pi$
$113$	− 2.64197e7i	− 1.72248i	0.508201	−	0.861238i	$-0.330311\pi$
$114$	0	0
$115$	−9.16838e6	−0.562147
$116$	0	0
$117$	0	0
$118$	0	0
$119$	3.39735e7	1.84810
$120$	0	0
$121$	−1.79450e7	−0.920861
$122$	0	0
$123$	0	0
$124$	0	0
$125$	2.15091e7	0.985003
$126$	0	0
$127$	− 1.69006e7i	− 0.732133i	−0.930589	−	0.366067i	$-0.880704\pi$
$127$	− 1.69006e7i	− 0.732133i	0.930589	−	0.366067i	$-0.119296\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	− 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
$134$	0	0
$135$	0	0
$136$	0	0
$137$	8.09631e6i	0.269008i	0.990913	+	0.134504i	$0.0429441\pi$
$137$	8.09631e6i	0.269008i	−0.990913	+	0.134504i	$0.957056\pi$
$138$	0	0
$139$	2.33396e7	0.737126	0.368563	−	0.929603i	$-0.379850\pi$
$139$	2.33396e7	0.737126	0.368563	+	0.929603i	$0.379850\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	4.47608e7	1.28004
$144$	0	0
$145$	−1.63617e7	−0.445698
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−6.65657e7	−1.64854	−0.824269	−	0.566199i	$-0.808413\pi$
$149$	−6.65657e7	−1.64854	−0.824269	+	0.566199i	$0.808413\pi$
$150$	0	0
$151$	− 6.91146e7i	− 1.63362i	−0.576907	−	0.816810i	$-0.695741\pi$
$151$	− 6.91146e7i	− 1.63362i	0.576907	−	0.816810i	$-0.304259\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	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	$-0.175702\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	− 8.12322e7i	− 1.53404i
$162$	0	0
$163$	−8.87701e7	−1.60550	−0.802749	−	0.596317i	$-0.796630\pi$
$163$	−8.87701e7	−1.60550	−0.802749	+	0.596317i	$0.796630\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	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
$170$	0	0
$171$	0	0
$172$	0	0
$173$	1.26544e8	1.85814	0.929072	−	0.369899i	$-0.120608\pi$
$173$	1.26544e8	1.85814	0.929072	+	0.369899i	$0.120608\pi$
$174$	0	0
$175$	7.42763e7i	1.04765i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	− 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	$-0.216996\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	− 6.06964e7i	− 0.704793i
$186$	0	0
$187$	−1.39634e8	−1.56151
$188$	0	0
$189$	0	0
$190$	0	0
$191$	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	$0.548231\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−1.38163e8	−1.28754	−0.643768	−	0.765221i	$-0.722630\pi$
$197$	−1.38163e8	−1.28754	−0.643768	+	0.765221i	$0.722630\pi$
$198$	0	0
$199$	− 3.24443e7i	− 0.291846i	−0.989296	−	0.145923i	$-0.953385\pi$
$199$	− 3.24443e7i	− 0.291846i	0.989296	−	0.145923i	$-0.0466151\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	− 1.44965e8i	− 1.21627i
$204$	0	0
$205$	6.78979e7i	0.550450i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	2.88722e8i	2.18760i
$210$	0	0
$211$	1.93728e6	0.0141973	0.00709863	−	0.999975i	$-0.497740\pi$
$211$	1.93728e6	0.0141973	0.00709863	+	0.999975i	$0.497740\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	486722.	0.00334000
$216$	0	0
$217$	−1.47309e8	−0.978632
$218$	0	0
$219$	0	0
$220$	0	0
$221$	1.66972e8	1.04057
$222$	0	0
$223$	− 5.78083e7i	− 0.349078i	−0.984650	−	0.174539i	$-0.944156\pi$
$223$	− 5.78083e7i	− 0.349078i	0.984650	−	0.174539i	$-0.0558436\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	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	$-0.107423\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 2.64384e8i	− 1.36927i	−0.728886	−	0.684636i	$-0.759961\pi$
$233$	− 2.64384e8i	− 1.36927i	0.728886	−	0.684636i	$-0.240039\pi$
$234$	0	0
$235$	1.49477e8	0.751340
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−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	$0.318892\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	2.33924e8	1.01623
$246$	0	0
$247$	− 3.45250e8i	− 1.45779i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	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
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 2.09860e7i	− 0.0771195i	−0.999256	−	0.0385597i	$-0.987723\pi$
$257$	− 2.09860e7i	− 0.0771195i	0.999256	−	0.0385597i	$-0.0122770\pi$
$258$	0	0
$259$	5.37772e8	1.92331
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−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
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−1.66474e8	−0.521451	−0.260725	−	0.965413i	$-0.583962\pi$
$269$	−1.66474e8	−0.521451	−0.260725	+	0.965413i	$0.583962\pi$
$270$	0	0
$271$	2.83438e8i	0.865097i	0.901611	+	0.432549i	$0.142386\pi$
$271$	2.83438e8i	0.865097i	−0.901611	+	0.432549i	$0.857614\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 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	$0.634455\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	2.64934e8i	0.712305i	0.934428	+	0.356153i	$0.115912\pi$
$281$	2.64934e8i	0.712305i	−0.934428	+	0.356153i	$0.884088\pi$
$282$	0	0
$283$	−7.23578e7	−0.189772	−0.0948861	−	0.995488i	$-0.530249\pi$
$283$	−7.23578e7	−0.189772	−0.0948861	+	0.995488i	$0.530249\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−6.01577e8	−1.50212
$288$	0	0
$289$	−1.10542e8	−0.269392
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−1.41789e8	−0.329310	−0.164655	−	0.986351i	$-0.552651\pi$
$293$	−1.41789e8	−0.329310	−0.164655	+	0.986351i	$0.552651\pi$
$294$	0	0
$295$	3.45378e8i	0.783281i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	− 3.99238e8i	− 0.863740i
$300$	0	0
$301$	4.31238e6i	0.00911452i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	− 4.65261e7i	− 0.0938961i
$306$	0	0
$307$	−5.38374e8	−1.06194	−0.530970	−	0.847391i	$-0.678172\pi$
$307$	−5.38374e8	−1.06194	−0.530970	+	0.847391i	$0.678172\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−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	$0.410116\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−7.70255e8	−1.35809	−0.679043	−	0.734099i	$-0.737605\pi$
$317$	−7.70255e8	−1.35809	−0.679043	+	0.734099i	$0.737605\pi$
$318$	0	0
$319$	5.95820e8i	1.02766i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	1.07703e9i	1.77835i
$324$	0	0
$325$	3.65052e8i	0.589879i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	1.32437e9i	2.05033i
$330$	0	0
$331$	8.90041e8	1.34900	0.674500	−	0.738274i	$-0.264359\pi$
$331$	8.90041e8	1.34900	0.674500	+	0.738274i	$0.264359\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−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	$0.186168\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	6.05451e8	0.826873
$342$	0	0
$343$	8.46670e8i	1.13288i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	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	$-0.113754\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 1.97472e8i	− 0.238942i	−0.992838	−	0.119471i	$-0.961880\pi$
$353$	− 1.97472e8i	− 0.238942i	0.992838	−	0.119471i	$-0.0381199\pi$
$354$	0	0
$355$	1.80348e8	0.213950
$356$	0	0
$357$	0	0
$358$	0	0
$359$	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
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−8.74624e8	−0.941448
$366$	0	0
$367$	− 1.80227e8i	− 0.190321i	−0.995462	−	0.0951607i	$-0.969664\pi$
$367$	− 1.80227e8i	− 0.190321i	0.995462	−	0.0951607i	$-0.0303365\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	− 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	$0.863190\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 7.12473e8i	− 0.684816i
$378$	0	0
$379$	8.71747e7	0.0822532	0.0411266	−	0.999154i	$-0.486905\pi$
$379$	8.71747e7	0.0822532	0.0411266	+	0.999154i	$0.486905\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	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
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−1.05419e9	−0.908021	−0.454010	−	0.890996i	$-0.650007\pi$
$389$	−1.05419e9	−0.908021	−0.454010	+	0.890996i	$0.650007\pi$
$390$	0	0
$391$	1.24545e9i	1.05368i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	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	$-0.142132\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	− 5.27042e8i	− 0.408169i	−0.978953	−	0.204085i	$-0.934578\pi$
$401$	− 5.27042e8i	− 0.408169i	0.978953	−	0.204085i	$-0.0654218\pi$
$402$	0	0
$403$	−7.23990e8	−0.551017
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−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	$0.488283\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−3.06006e9	−2.13749
$414$	0	0
$415$	4.70332e8i	0.323025i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	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	$0.928579\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	− 1.13880e9i	− 0.719593i
$426$	0	0
$427$	4.12223e8	0.256233
$428$	0	0
$429$	0	0
$430$	0	0
$431$	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	$0.769743\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	2.57522e9	1.47615
$438$	0	0
$439$	− 2.76692e9i	− 1.56088i	−0.625229	−	0.780441i	$-0.714994\pi$
$439$	− 2.76692e9i	− 1.56088i	0.625229	−	0.780441i	$-0.285006\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 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
$446$	0	0
$447$	0	0
$448$	0	0
$449$	4.35223e8i	0.226908i	0.993543	+	0.113454i	$0.0361915\pi$
$449$	4.35223e8i	0.226908i	−0.993543	+	0.113454i	$0.963809\pi$
$450$	0	0
$451$	2.47253e9	1.26918
$452$	0	0
$453$	0	0
$454$	0	0
$455$	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	$0.531567\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	2.95816e9	1.40627	0.703134	−	0.711058i	$-0.251784\pi$
$461$	2.95816e9	1.40627	0.703134	+	0.711058i	$0.251784\pi$
$462$	0	0
$463$	− 7.95716e7i	− 0.0372584i	−0.999826	−	0.0186292i	$-0.994070\pi$
$463$	− 7.95716e7i	− 0.0372584i	0.999826	−	0.0186292i	$-0.00593021\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	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
$470$	0	0
$471$	0	0
$472$	0	0
$473$	− 1.77242e7i	− 0.00770112i
$474$	0	0
$475$	−2.35471e9	−1.00811
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−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
$482$	0	0
$483$	0	0
$484$	0	0
$485$	2.80575e9	1.11674
$486$	0	0
$487$	− 1.30663e9i	− 0.512627i	−0.966594	−	0.256314i	$-0.917492\pi$
$487$	− 1.30663e9i	− 0.512627i	0.966594	−	0.256314i	$-0.0825080\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	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
$494$	0	0
$495$	0	0
$496$	0	0
$497$	1.59789e9i	0.583849i
$498$	0	0
$499$	−5.46162e9	−1.96775	−0.983874	−	0.178861i	$-0.942759\pi$
$499$	−5.46162e9	−1.96775	−0.983874	+	0.178861i	$0.942759\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−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
$506$	0	0
$507$	0	0
$508$	0	0
$509$	5.19906e9	1.74748	0.873741	−	0.486392i	$-0.161687\pi$
$509$	5.19906e9	1.74748	0.873741	+	0.486392i	$0.161687\pi$
$510$	0	0
$511$	− 7.74920e9i	− 2.56912i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	7.56547e8i	0.244068i
$516$	0	0
$517$	− 5.44327e9i	− 1.73238i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	9.03089e8i	0.279768i	0.990168	+	0.139884i	$0.0446730\pi$
$521$	9.03089e8i	0.279768i	−0.990168	+	0.139884i	$0.955327\pi$
$522$	0	0
$523$	−1.94736e9	−0.595238	−0.297619	−	0.954685i	$-0.596192\pi$
$523$	−1.94736e9	−0.595238	−0.297619	+	0.954685i	$0.596192\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	2.25853e9	0.672185
$528$	0	0
$529$	−4.26905e8	−0.125382
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−2.95662e9	−0.845766
$534$	0	0
$535$	8.02605e8i	0.226602i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	− 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	$-0.00394551\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	− 1.47672e9i	− 0.390760i
$546$	0	0
$547$	7.28778e8	0.190388	0.0951940	−	0.995459i	$-0.469653\pi$
$547$	7.28778e8	0.190388	0.0951940	+	0.995459i	$0.469653\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	4.59569e9	1.17036
$552$	0	0
$553$	−1.07989e10	−2.71546
$554$	0	0
$555$	0	0
$556$	0	0
$557$	4.48763e9	1.10033	0.550166	−	0.835055i	$-0.314564\pi$
$557$	4.48763e9	1.10033	0.550166	+	0.835055i	$0.314564\pi$
$558$	0	0
$559$	2.11944e7i	0.00513192i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 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
$566$	0	0
$567$	0	0
$568$	0	0
$569$	− 2.51985e9i	− 0.573431i	−0.958016	−	0.286716i	$-0.907436\pi$
$569$	− 2.51985e9i	− 0.573431i	0.958016	−	0.286716i	$-0.0925635\pi$
$570$	0	0
$571$	4.52309e9	1.01674	0.508368	−	0.861140i	$-0.330249\pi$
$571$	4.52309e9	1.01674	0.508368	+	0.861140i	$0.330249\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−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	$0.346253\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−4.16716e9	−0.881502
$582$	0	0
$583$	3.22743e9i	0.674554i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	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
$590$	0	0
$591$	0	0
$592$	0	0
$593$	− 4.94870e9i	− 0.974539i	−0.873252	−	0.487270i	$-0.837993\pi$
$593$	− 4.94870e9i	− 0.974539i	0.873252	−	0.487270i	$-0.162007\pi$
$594$	0	0
$595$	−5.70790e9	−1.11088
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−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	$0.558956\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	3.01494e9	0.553523
$606$	0	0
$607$	− 2.96338e9i	− 0.537808i	−0.963167	−	0.268904i	$-0.913339\pi$
$607$	− 2.96338e9i	− 0.537808i	0.963167	−	0.268904i	$-0.0866614\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	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	$-0.207478\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 6.20892e9i	− 1.06419i	−0.846685	−	0.532094i	$-0.821405\pi$
$617$	− 6.20892e9i	− 1.06419i	0.846685	−	0.532094i	$-0.178595\pi$
$618$	0	0
$619$	5.22520e9	0.885494	0.442747	−	0.896647i	$-0.354004\pi$
$619$	5.22520e9	0.885494	0.442747	+	0.896647i	$0.354004\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	6.30971e9	1.04545
$624$	0	0
$625$	2.84489e8	0.0466107
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−8.24509e9	−1.32105
$630$	0	0
$631$	− 8.31627e9i	− 1.31773i	−0.752262	−	0.658864i	$-0.771037\pi$
$631$	− 8.31627e9i	− 1.31773i	0.752262	−	0.658864i	$-0.228963\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	2.83948e9i	0.440080i
$636$	0	0
$637$	1.01863e10i	1.56145i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	7.19590e9i	1.07915i	0.841937	+	0.539575i	$0.181415\pi$
$641$	7.19590e9i	1.07915i	−0.841937	+	0.539575i	$0.818585\pi$
$642$	0	0
$643$	−4.49943e8	−0.0667451	−0.0333725	−	0.999443i	$-0.510625\pi$
$643$	−4.49943e8	−0.0667451	−0.0333725	+	0.999443i	$0.510625\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	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
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−2.41085e9	−0.338824	−0.169412	−	0.985545i	$-0.554187\pi$
$653$	−2.41085e9	−0.338824	−0.169412	+	0.985545i	$0.554187\pi$
$654$	0	0
$655$	1.57237e9i	0.218631i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	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	$-0.241198\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	1.18023e10i	1.55629i
$666$	0	0
$667$	5.31434e9	0.693440
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−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	$0.677122\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−7.79709e8	−0.0965767	−0.0482883	−	0.998833i	$-0.515377\pi$
$677$	−7.79709e8	−0.0965767	−0.0482883	+	0.998833i	$0.515377\pi$
$678$	0	0
$679$	2.48591e10i	3.04748i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 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
$686$	0	0
$687$	0	0
$688$	0	0
$689$	− 3.85931e9i	− 0.449513i
$690$	0	0
$691$	5.92970e8	0.0683690	0.0341845	−	0.999416i	$-0.489117\pi$
$691$	5.92970e8	0.0683690	0.0341845	+	0.999416i	$0.489117\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−3.92130e9	−0.443081
$696$	0	0
$697$	9.22336e9	1.03175
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−9.18280e9	−1.00684	−0.503422	−	0.864041i	$-0.667926\pi$
$701$	−9.18280e9	−1.00684	−0.503422	+	0.864041i	$0.667926\pi$
$702$	0	0
$703$	1.70484e10i	1.85072i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 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	$0.730290\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	− 5.40024e9i	− 0.557956i
$714$	0	0
$715$	−7.52028e9	−0.769419
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−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
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−4.85928e9	−0.473575
$726$	0	0
$727$	− 1.18128e10i	− 1.14020i	−0.821575	−	0.570100i	$-0.806904\pi$
$727$	− 1.18128e10i	− 1.14020i	0.821575	−	0.570100i	$-0.193096\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	− 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	$0.843096\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	7.03471e9i	0.647307i
$738$	0	0
$739$	−1.06274e10	−0.968661	−0.484331	−	0.874885i	$-0.660937\pi$
$739$	−1.06274e10	−0.968661	−0.484331	+	0.874885i	$0.660937\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	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
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−7.11111e9	−0.618374
$750$	0	0
$751$	1.15775e10i	0.997411i	0.866772	+	0.498705i	$0.166191\pi$
$751$	1.15775e10i	0.997411i	−0.866772	+	0.498705i	$0.833809\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	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	$0.933994\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	2.53368e9i	0.208404i	0.994556	+	0.104202i	$0.0332288\pi$
$761$	2.53368e9i	0.208404i	−0.994556	+	0.104202i	$0.966771\pi$
$762$	0	0
$763$	1.30838e10	1.06635
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−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	$0.227226\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	6.81224e9	0.530471	0.265236	−	0.964184i	$-0.414550\pi$
$773$	6.81224e9	0.530471	0.265236	+	0.964184i	$0.414550\pi$
$774$	0	0
$775$	4.93783e9i	0.381048i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 1.90712e10i	− 1.44543i
$780$	0	0
$781$	− 6.56748e9i	− 0.493310i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	8.54394e9i	0.630397i
$786$	0	0
$787$	1.56544e10	1.14479	0.572393	−	0.819979i	$-0.306015\pi$
$787$	1.56544e10	1.14479	0.572393	+	0.819979i	$0.306015\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−3.93278e10	−2.82541
$792$	0	0
$793$	2.02599e9	0.144271
$794$	0	0
$795$	0	0
$796$	0	0
$797$	1.50517e9	0.105313	0.0526565	−	0.998613i	$-0.483231\pi$
$797$	1.50517e9	0.105313	0.0526565	+	0.998613i	$0.483231\pi$
$798$	0	0
$799$	− 2.03052e10i	− 1.40829i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	3.18499e10i	2.17072i
$804$	0	0
$805$	1.36478e10i	0.922101i
$806$	0	0
$807$	0	0
$808$	0	0
$809$	2.50669e10i	1.66449i	0.554412	+	0.832243i	$0.312943\pi$
$809$	2.50669e10i	1.66449i	−0.554412	+	0.832243i	$0.687057\pi$
$810$	0	0
$811$	2.89387e10	1.90505	0.952525	−	0.304459i	$-0.0984757\pi$
$811$	2.89387e10	1.90505	0.952525	+	0.304459i	$0.0984757\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	1.49143e10	0.965053
$816$	0	0
$817$	−1.36711e8	−0.00877053
$818$	0	0
$819$	0	0
$820$	0	0
$821$	9.16791e9	0.578188	0.289094	−	0.957301i	$-0.406646\pi$
$821$	9.16791e9	0.578188	0.289094	+	0.957301i	$0.406646\pi$
$822$	0	0
$823$	− 2.14215e10i	− 1.33952i	−0.742576	−	0.669762i	$-0.766396\pi$
$823$	− 2.14215e10i	− 1.33952i	0.742576	−	0.669762i	$-0.233604\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 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	$0.969106\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	− 3.17767e10i	− 1.90481i
$834$	0	0
$835$	−7.70727e9	−0.458140
$836$	0	0
$837$	0	0
$838$	0	0
$839$	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
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−1.54976e9	−0.0883620
$846$	0	0
$847$	2.67125e10i	1.51051i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	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	$0.664644\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 1.20759e10i	− 0.655371i	−0.944787	−	0.327685i	$-0.893731\pi$
$857$	− 1.20759e10i	− 0.655371i	0.944787	−	0.327685i	$-0.106269\pi$
$858$	0	0
$859$	−7.65080e8	−0.0411842	−0.0205921	−	0.999788i	$-0.506555\pi$
$859$	−7.65080e8	−0.0411842	−0.0205921	+	0.999788i	$0.506555\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−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
$866$	0	0
$867$	0	0
$868$	0	0
$869$	4.43846e10	2.29437
$870$	0	0
$871$	− 8.41201e9i	− 0.431356i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	− 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	$-0.0934005\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	7.56814e9i	0.372884i	0.982466	+	0.186442i	$0.0596956\pi$
$881$	7.56814e9i	0.372884i	−0.982466	+	0.186442i	$0.940304\pi$
$882$	0	0
$883$	4.48773e9	0.219364	0.109682	−	0.993967i	$-0.465017\pi$
$883$	4.48773e9	0.219364	0.109682	+	0.993967i	$0.465017\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	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
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−4.19852e10	−1.97295
$894$	0	0
$895$	1.01629e10i	0.473848i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 9.63718e9i	− 0.442375i
$900$	0	0
$901$	1.20394e10i	0.548361i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	1.68915e10i	0.757526i
$906$	0	0
$907$	7.51806e9	0.334565	0.167282	−	0.985909i	$-0.446501\pi$
$907$	7.51806e9	0.334565	0.167282	+	0.985909i	$0.446501\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−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
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−1.39313e10	−0.596621
$918$	0	0
$919$	− 5.05031e9i	− 0.214642i	−0.994224	−	0.107321i	$-0.965773\pi$
$919$	− 5.05031e9i	− 0.214642i	0.994224	−	0.107321i	$-0.0342272\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	7.85329e9i	0.328735i
$924$	0	0
$925$	− 1.80263e10i	− 0.748875i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	1.97133e10i	0.806685i	0.915049	+	0.403343i	$0.132152\pi$
$929$	1.97133e10i	0.806685i	−0.915049	+	0.403343i	$0.867848\pi$
$930$	0	0
$931$	−6.57048e10	−2.66854
$932$	0	0
$933$	0	0
$934$	0	0
$935$	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	$0.657809\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	2.89908e10	1.13422	0.567109	−	0.823643i	$-0.308062\pi$
$941$	2.89908e10	1.13422	0.567109	+	0.823643i	$0.308062\pi$
$942$	0	0
$943$	− 2.20534e10i	− 0.856417i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 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
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 2.75031e10i	− 1.02933i	−0.857390	−	0.514667i	$-0.827916\pi$
$953$	− 2.75031e10i	− 1.02933i	0.857390	−	0.514667i	$-0.172084\pi$
$954$	0	0
$955$	−1.41543e10	−0.525867
$956$	0	0
$957$	0	0
$958$	0	0
$959$	1.20520e10	0.441259
$960$	0	0
$961$	1.77197e10	0.644056
$962$	0	0
$963$	0	0
$964$	0	0
$965$	5.06560e9	0.181462
$966$	0	0
$967$	− 4.75920e9i	− 0.169255i	−0.996413	−	0.0846274i	$-0.973030\pi$
$967$	− 4.75920e9i	− 0.169255i	0.996413	−	0.0846274i	$-0.0269700\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	− 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
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 6.45020e9i	− 0.221280i	−0.993861	−	0.110640i	$-0.964710\pi$
$977$	− 6.45020e9i	− 0.221280i	0.993861	−	0.110640i	$-0.0352900\pi$
$978$	0	0
$979$	−2.59335e10	−0.883326
$980$	0	0
$981$	0	0
$982$	0	0
$983$	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
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−1.58089e8	−0.00519655
$990$	0	0
$991$	− 1.68634e10i	− 0.550412i	−0.961385	−	0.275206i	$-0.911254\pi$
$991$	− 1.68634e10i	− 0.550412i	0.961385	−	0.275206i	$-0.0887460\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	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	$-0.159086\pi$
$998$	0	0
$999$	0	0

Twists

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

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

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:	\( N \)	\(=\)	\( 288 = 2^{5} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	288.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-168.010 q^{5} -1488.58i q^{7} +O(q^{10})\)	\(q-168.010 q^{5} -1488.58i q^{7} +6118.18i q^{11} -7316.03i q^{13} +22822.8i q^{17} +47190.9 q^{19} +54570.3 q^{23} -49897.5 q^{25} +97385.2 q^{29} -98959.4i q^{31} +250097. i q^{35} +361266. i q^{37} -404129. i q^{41} -2896.98 q^{43} -889688. q^{47} -1.39232e6 q^{49} +527514. q^{53} -1.02792e6i q^{55} -2.05569e6i q^{59} +276924. i q^{61} +1.22917e6i q^{65} +1.14980e6 q^{67} -1.07344e6 q^{71} +5.20577e6 q^{73} +9.10739e6 q^{77} -7.25453e6i q^{79} -2.79942e6i q^{83} -3.83447e6i q^{85} +4.23875e6i q^{89} -1.08905e7 q^{91} -7.92856e6 q^{95} -1.66999e7 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q+O(q^{10}) \) 28 * q	\( 28 q + 121168 q^{19} + 437500 q^{25} - 1505696 q^{43} - 2272076 q^{49} + 776272 q^{67} - 2534128 q^{73} + 3406992 q^{91} - 26311456 q^{97}+O(q^{100}) \) 28 * q + 121168 * q^19 + 437500 * q^25 - 1505696 * q^43 - 2272076 * q^49 + 776272 * q^67 - 2534128 * q^73 + 3406992 * q^91 - 26311456 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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