LMFDB - Embedded newform 336.9.f.d.97.19

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [336,9,Mod(97,336)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(336, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 1]))

N = Newforms(chi, 9, names="a")

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

chi := DirichletCharacter("336.97");

S:= CuspForms(chi, 9);

N := Newforms(S);

Level:	$ N $	$=$	$ 336 = 2^{4} \cdot 3 \cdot 7 $
Weight:	$ k $	$=$	$ 9 $
Character orbit:	$[\chi]$	$=$	336.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:	$136.879212981$
Analytic rank:	$0$
Dimension:	$32$
Twist minimal:	no (minimal twist has level 168)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			97.19
Character	$\chi$	$=$	336.97
Dual form			336.9.f.d.97.14

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+46.7654i q^{3} -806.986i q^{5} +(-973.349 - 2194.86i) q^{7} -2187.00 q^{9} +O(q^{10})$	$q+46.7654i q^{3} -806.986i q^{5} +(-973.349 - 2194.86i) q^{7} -2187.00 q^{9} -2997.96 q^{11} +21481.3i q^{13} +37739.0 q^{15} -81540.7i q^{17} +182506. i q^{19} +(102643. - 45519.0i) q^{21} -449570. q^{23} -260601. q^{25} -102276. i q^{27} -474493. q^{29} +1.72422e6i q^{31} -140201. i q^{33} +(-1.77122e6 + 785479. i) q^{35} +437687. q^{37} -1.00458e6 q^{39} +2.12303e6i q^{41} +4.67720e6 q^{43} +1.76488e6i q^{45} -546959. i q^{47} +(-3.86998e6 + 4.27272e6i) q^{49} +3.81328e6 q^{51} +828142. q^{53} +2.41931e6i q^{55} -8.53496e6 q^{57} -9.00218e6i q^{59} -2.38193e7i q^{61} +(2.12872e6 + 4.80015e6i) q^{63} +1.73351e7 q^{65} +1.60886e7 q^{67} -2.10243e7i q^{69} +2.27822e6 q^{71} -7.58400e6i q^{73} -1.21871e7i q^{75} +(2.91806e6 + 6.58009e6i) q^{77} +2.67369e7 q^{79} +4.78297e6 q^{81} +2.05175e7i q^{83} -6.58022e7 q^{85} -2.21898e7i q^{87} -3.22886e7i q^{89} +(4.71483e7 - 2.09088e7i) q^{91} -8.06337e7 q^{93} +1.47280e8 q^{95} +1.23844e7i q^{97} +6.55654e6 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 32 q - 1424 q^{7} - 69984 q^{9}+O(q^{10}) $ 32 * q - 1424 * q^7 - 69984 * q^9	$ 32 q - 1424 q^{7} - 69984 q^{9} - 23616 q^{11} + 59616 q^{15} + 60912 q^{21} - 455040 q^{23} - 3404928 q^{25} + 632064 q^{29} + 1543200 q^{35} + 4150496 q^{37} + 4162752 q^{39} - 6028000 q^{43} + 15115072 q^{49} + 1340064 q^{51} - 37728576 q^{53} + 7286112 q^{57} + 3114288 q^{63} + 29977536 q^{65} - 68431648 q^{67} + 19788096 q^{71} + 87499392 q^{77} + 86954656 q^{79} + 153055008 q^{81} - 31326496 q^{85} + 24540288 q^{91} - 38833344 q^{93} - 329546688 q^{95} + 51648192 q^{99}+O(q^{100}) $ 32 * q - 1424 * q^7 - 69984 * q^9 - 23616 * q^11 + 59616 * q^15 + 60912 * q^21 - 455040 * q^23 - 3404928 * q^25 + 632064 * q^29 + 1543200 * q^35 + 4150496 * q^37 + 4162752 * q^39 - 6028000 * q^43 + 15115072 * q^49 + 1340064 * q^51 - 37728576 * q^53 + 7286112 * q^57 + 3114288 * q^63 + 29977536 * q^65 - 68431648 * q^67 + 19788096 * q^71 + 87499392 * q^77 + 86954656 * q^79 + 153055008 * q^81 - 31326496 * q^85 + 24540288 * q^91 - 38833344 * q^93 - 329546688 * q^95 + 51648192 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/336\mathbb{Z}\right)^\times$.

$n$	$85$	$113$	$127$	$241$
$\chi(n)$	$1$	$1$	$1$	$-1$

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
$3$			46.7654i			0.577350i
$3$			46.7654i			0.577350i
$4$		0			0
$5$		−	806.986i		−	1.29118i	−0.763685	−	0.645589i	$-0.776612\pi$
$5$		−	806.986i		−	1.29118i	0.763685	−	0.645589i	$-0.223388\pi$
$6$		0			0
$7$	−973.349	−	2194.86i	−0.405393	−	0.914142i
$7$	−973.349	−	2194.86i	−0.405393	−	0.914142i
$8$		0			0
$9$	−2187.00			−0.333333
$10$		0			0
$11$	−2997.96			−0.204765			−0.102382	−	0.994745i	$-0.532647\pi$
$11$	−2997.96			−0.204765			−0.102382	+	0.994745i	$0.532647\pi$
$12$		0			0
$13$			21481.3i			0.752119i	0.926596	+	0.376060i	$0.122721\pi$
$13$			21481.3i			0.752119i	−0.926596	+	0.376060i	$0.877279\pi$
$14$		0			0
$15$	37739.0			0.745461
$16$		0			0
$17$		−	81540.7i		−	0.976290i	−0.872763	−	0.488145i	$-0.837674\pi$
$17$		−	81540.7i		−	0.976290i	0.872763	−	0.488145i	$-0.162326\pi$
$18$		0			0
$19$			182506.i			1.40043i	0.713930	+	0.700217i	$0.246913\pi$
$19$			182506.i			1.40043i	−0.713930	+	0.700217i	$0.753087\pi$
$20$		0			0
$21$	102643.	−	45519.0i	0.527780	−	0.234054i
$22$		0			0
$23$	−449570.			−1.60652			−0.803259	−	0.595629i	$-0.796903\pi$
$23$	−449570.			−1.60652			−0.803259	+	0.595629i	$0.796903\pi$
$24$		0			0
$25$	−260601.			−0.667138
$26$		0			0
$27$		−	102276.i		−	0.192450i
$28$		0			0
$29$	−474493.			−0.670869			−0.335434	−	0.942064i	$-0.608883\pi$
$29$	−474493.			−0.670869			−0.335434	+	0.942064i	$0.608883\pi$
$30$		0			0
$31$			1.72422e6i			1.86701i	0.358570	+	0.933503i	$0.383264\pi$
$31$			1.72422e6i			1.86701i	−0.358570	+	0.933503i	$0.616736\pi$
$32$		0			0
$33$		−	140201.i		−	0.118221i
$34$		0			0
$35$	−1.77122e6	+	785479.i	−1.18032	+	0.523435i
$36$		0			0
$37$	437687.			0.233538			0.116769	−	0.993159i	$-0.462746\pi$
$37$	437687.			0.233538			0.116769	+	0.993159i	$0.462746\pi$
$38$		0			0
$39$	−1.00458e6			−0.434236
$40$		0			0
$41$			2.12303e6i			0.751311i	0.926759	+	0.375656i	$0.122582\pi$
$41$			2.12303e6i			0.751311i	−0.926759	+	0.375656i	$0.877418\pi$
$42$		0			0
$43$	4.67720e6			1.36808			0.684042	−	0.729443i	$-0.260221\pi$
$43$	4.67720e6			1.36808			0.684042	+	0.729443i	$0.260221\pi$
$44$		0			0
$45$			1.76488e6i			0.430392i
$46$		0			0
$47$		−	546959.i		−	0.112089i	−0.998428	−	0.0560446i	$-0.982151\pi$
$47$		−	546959.i		−	0.112089i	0.998428	−	0.0560446i	$-0.0178489\pi$
$48$		0			0
$49$	−3.86998e6	+	4.27272e6i	−0.671313	+	0.741174i
$50$		0			0
$51$	3.81328e6			0.563661
$52$		0			0
$53$	828142.			0.104955			0.0524773	−	0.998622i	$-0.483288\pi$
$53$	828142.			0.104955			0.0524773	+	0.998622i	$0.483288\pi$
$54$		0			0
$55$			2.41931e6i			0.264388i
$56$		0			0
$57$	−8.53496e6			−0.808541
$58$		0			0
$59$		−	9.00218e6i		−	0.742916i	−0.928450	−	0.371458i	$-0.878858\pi$
$59$		−	9.00218e6i		−	0.742916i	0.928450	−	0.371458i	$-0.121142\pi$
$60$		0			0
$61$		−	2.38193e7i		−	1.72032i	−0.510026	−	0.860159i	$-0.670364\pi$
$61$		−	2.38193e7i		−	1.72032i	0.510026	−	0.860159i	$-0.329636\pi$
$62$		0			0
$63$	2.12872e6	+	4.80015e6i	0.135131	+	0.304714i
$64$		0			0
$65$	1.73351e7			0.971119
$66$		0			0
$67$	1.60886e7			0.798399			0.399199	−	0.916864i	$-0.369288\pi$
$67$	1.60886e7			0.798399			0.399199	+	0.916864i	$0.369288\pi$
$68$		0			0
$69$		−	2.10243e7i		−	0.927524i
$70$		0			0
$71$	2.27822e6			0.0896525			0.0448263	−	0.998995i	$-0.485727\pi$
$71$	2.27822e6			0.0896525			0.0448263	+	0.998995i	$0.485727\pi$
$72$		0			0
$73$		−	7.58400e6i		−	0.267059i	−0.991045	−	0.133529i	$-0.957369\pi$
$73$		−	7.58400e6i		−	0.267059i	0.991045	−	0.133529i	$-0.0426310\pi$
$74$		0			0
$75$		−	1.21871e7i		−	0.385172i
$76$		0			0
$77$	2.91806e6	+	6.58009e6i	0.0830103	+	0.187184i
$78$		0			0
$79$	2.67369e7			0.686440			0.343220	−	0.939255i	$-0.388482\pi$
$79$	2.67369e7			0.686440			0.343220	+	0.939255i	$0.388482\pi$
$80$		0			0
$81$	4.78297e6			0.111111
$82$		0			0
$83$			2.05175e7i			0.432326i	0.976357	+	0.216163i	$0.0693543\pi$
$83$			2.05175e7i			0.432326i	−0.976357	+	0.216163i	$0.930646\pi$
$84$		0			0
$85$	−6.58022e7			−1.26056
$86$		0			0
$87$		−	2.21898e7i		−	0.387326i
$88$		0			0
$89$		−	3.22886e7i		−	0.514623i	−0.966328	−	0.257312i	$-0.917163\pi$
$89$		−	3.22886e7i		−	0.514623i	0.966328	−	0.257312i	$-0.0828367\pi$
$90$		0			0
$91$	4.71483e7	−	2.09088e7i	0.687544	−	0.304904i
$92$		0			0
$93$	−8.06337e7			−1.07792
$94$		0			0
$95$	1.47280e8			1.80821
$96$		0			0
$97$			1.23844e7i			0.139890i	0.997551	+	0.0699450i	$0.0222824\pi$
$97$			1.23844e7i			0.139890i	−0.997551	+	0.0699450i	$0.977718\pi$
$98$		0			0
$99$	6.55654e6			0.0682549
$100$		0			0
$101$			3.21497e6i			0.0308952i	0.999881	+	0.0154476i	$0.00491732\pi$
$101$			3.21497e6i			0.0308952i	−0.999881	+	0.0154476i	$0.995083\pi$
$102$		0			0
$103$		−	1.18290e8i		−	1.05099i	−0.850796	−	0.525495i	$-0.823880\pi$
$103$		−	1.18290e8i		−	1.05099i	0.850796	−	0.525495i	$-0.176120\pi$
$104$		0			0
$105$	−3.67332e7	−	8.28316e7i	−0.302205	−	0.681458i
$106$		0			0
$107$	1.60896e8			1.22747			0.613735	−	0.789512i	$-0.289666\pi$
$107$	1.60896e8			1.22747			0.613735	+	0.789512i	$0.289666\pi$
$108$		0			0
$109$	2.49441e8			1.76710			0.883552	−	0.468333i	$-0.155145\pi$
$109$	2.49441e8			1.76710			0.883552	+	0.468333i	$0.155145\pi$
$110$		0			0
$111$			2.04686e7i			0.134833i
$112$		0			0
$113$	−2.50123e8			−1.53405			−0.767025	−	0.641618i	$-0.778264\pi$
$113$	−2.50123e8			−1.53405			−0.767025	+	0.641618i	$0.778264\pi$
$114$		0			0
$115$			3.62796e8i			2.07430i
$116$		0			0
$117$		−	4.69795e7i		−	0.250706i
$118$		0			0
$119$	−1.78970e8	+	7.93676e7i	−0.892468	+	0.395781i
$120$		0			0
$121$	−2.05371e8			−0.958071
$122$		0			0
$123$	−9.92841e7			−0.433770
$124$		0			0
$125$		−	1.04928e8i		−	0.429784i
$126$		0			0
$127$	4.93103e8			1.89549			0.947747	−	0.319023i	$-0.103355\pi$
$127$	4.93103e8			1.89549			0.947747	+	0.319023i	$0.103355\pi$
$128$		0			0
$129$			2.18731e8i			0.789863i
$130$		0			0
$131$			4.85932e8i			1.65002i	0.565116	+	0.825012i	$0.308832\pi$
$131$			4.85932e8i			1.65002i	−0.565116	+	0.825012i	$0.691168\pi$
$132$		0			0
$133$	4.00574e8	−	1.77642e8i	1.28020	−	0.567727i
$134$		0			0
$135$	−8.25352e7			−0.248487
$136$		0			0
$137$	−2.56721e8			−0.728752			−0.364376	−	0.931252i	$-0.618718\pi$
$137$	−2.56721e8			−0.728752			−0.364376	+	0.931252i	$0.618718\pi$
$138$		0			0
$139$			5.09792e8i			1.36563i	0.730591	+	0.682816i	$0.239245\pi$
$139$			5.09792e8i			1.36563i	−0.730591	+	0.682816i	$0.760755\pi$
$140$		0			0
$141$	2.55788e7			0.0647147
$142$		0			0
$143$		−	6.44000e7i		−	0.154007i
$144$		0			0
$145$			3.82909e8i			0.866210i
$146$		0			0
$147$	−1.99815e8	−	1.80981e8i	−0.427917	−	0.387582i
$148$		0			0
$149$	−2.07052e8			−0.420081			−0.210041	−	0.977693i	$-0.567360\pi$
$149$	−2.07052e8			−0.420081			−0.210041	+	0.977693i	$0.567360\pi$
$150$		0			0
$151$	6.17799e8			1.18834			0.594169	−	0.804340i	$-0.297481\pi$
$151$	6.17799e8			1.18834			0.594169	+	0.804340i	$0.297481\pi$
$152$		0			0
$153$			1.78329e8i			0.325430i
$154$		0			0
$155$	1.39142e9			2.41063
$156$		0			0
$157$		−	8.54184e7i		−	0.140589i	−0.997526	−	0.0702947i	$-0.977606\pi$
$157$		−	8.54184e7i		−	0.140589i	0.997526	−	0.0702947i	$-0.0223940\pi$
$158$		0			0
$159$			3.87284e7i			0.0605956i
$160$		0			0
$161$	4.37588e8	+	9.86741e8i	0.651272	+	1.46859i
$162$		0			0
$163$	−3.84481e8			−0.544658			−0.272329	−	0.962204i	$-0.587794\pi$
$163$	−3.84481e8			−0.544658			−0.272329	+	0.962204i	$0.587794\pi$
$164$		0			0
$165$	−1.13140e8			−0.152644
$166$		0			0
$167$		−	1.05353e9i		−	1.35451i	−0.735750	−	0.677253i	$-0.763170\pi$
$167$		−	1.05353e9i		−	1.35451i	0.735750	−	0.677253i	$-0.236830\pi$
$168$		0			0
$169$	3.54286e8			0.434317
$170$		0			0
$171$		−	3.99141e8i		−	0.466811i
$172$		0			0
$173$		−	5.89006e7i		−	0.0657560i	−0.999459	−	0.0328780i	$-0.989533\pi$
$173$		−	5.89006e7i		−	0.0657560i	0.999459	−	0.0328780i	$-0.0104673\pi$
$174$		0			0
$175$	2.53656e8	+	5.71981e8i	0.270453	+	0.609859i
$176$		0			0
$177$	4.20990e8			0.428923
$178$		0			0
$179$	−1.76161e9			−1.71593			−0.857963	−	0.513711i	$-0.828270\pi$
$179$	−1.76161e9			−1.71593			−0.857963	+	0.513711i	$0.828270\pi$
$180$		0			0
$181$			1.01292e9i			0.943757i	0.881663	+	0.471879i	$0.156424\pi$
$181$			1.01292e9i			0.943757i	−0.881663	+	0.471879i	$0.843576\pi$
$182$		0			0
$183$	1.11392e9			0.993226
$184$		0			0
$185$		−	3.53207e8i		−	0.301538i
$186$		0			0
$187$			2.44456e8i			0.199910i
$188$		0			0
$189$	−2.24481e8	+	9.95502e7i	−0.175927	+	0.0780180i
$190$		0			0
$191$	−3.53684e8			−0.265755			−0.132877	−	0.991132i	$-0.542422\pi$
$191$	−3.53684e8			−0.265755			−0.132877	+	0.991132i	$0.542422\pi$
$192$		0			0
$193$	1.13560e9			0.818459			0.409230	−	0.912431i	$-0.365798\pi$
$193$	1.13560e9			0.818459			0.409230	+	0.912431i	$0.365798\pi$
$194$		0			0
$195$			8.10681e8i			0.560676i
$196$		0			0
$197$	2.26680e9			1.50504			0.752522	−	0.658567i	$-0.228837\pi$
$197$	2.26680e9			1.50504			0.752522	+	0.658567i	$0.228837\pi$
$198$		0			0
$199$			9.55717e8i			0.609420i	0.952445	+	0.304710i	$0.0985596\pi$
$199$			9.55717e8i			0.609420i	−0.952445	+	0.304710i	$0.901440\pi$
$200$		0			0
$201$			7.52391e8i			0.460956i
$202$		0			0
$203$	4.61847e8	+	1.04144e9i	0.271966	+	0.613270i
$204$		0			0
$205$	1.71325e9			0.970076
$206$		0			0
$207$	9.83209e8			0.535506
$208$		0			0
$209$		−	5.47146e8i		−	0.286760i
$210$		0			0
$211$	4.63063e8			0.233620			0.116810	−	0.993154i	$-0.462733\pi$
$211$	4.63063e8			0.233620			0.116810	+	0.993154i	$0.462733\pi$
$212$		0			0
$213$			1.06542e8i			0.0517609i
$214$		0			0
$215$		−	3.77444e9i		−	1.76644i
$216$		0			0
$217$	3.78441e9	−	1.67827e9i	1.70671	−	0.756872i
$218$		0			0
$219$	3.54668e8			0.154186
$220$		0			0
$221$	1.75160e9			0.734286
$222$		0			0
$223$		−	1.30073e8i		−	0.0525978i	−0.999654	−	0.0262989i	$-0.991628\pi$
$223$		−	1.30073e8i		−	0.0525978i	0.999654	−	0.0262989i	$-0.00837217\pi$
$224$		0			0
$225$	5.69934e8			0.222379
$226$		0			0
$227$		−	1.22680e9i		−	0.462030i	−0.972950	−	0.231015i	$-0.925795\pi$
$227$		−	1.22680e9i		−	0.462030i	0.972950	−	0.231015i	$-0.0742046\pi$
$228$		0			0
$229$		−	3.62581e9i		−	1.31845i	−0.751947	−	0.659224i	$-0.770885\pi$
$229$		−	3.62581e9i		−	1.31845i	0.751947	−	0.659224i	$-0.229115\pi$
$230$		0			0
$231$	−3.07720e8	+	1.36464e8i	−0.108071	+	0.0479260i
$232$		0			0
$233$	2.12035e9			0.719422			0.359711	−	0.933064i	$-0.382875\pi$
$233$	2.12035e9			0.719422			0.359711	+	0.933064i	$0.382875\pi$
$234$		0			0
$235$	−4.41388e8			−0.144727
$236$		0			0
$237$			1.25036e9i			0.396317i
$238$		0			0
$239$	8.23654e8			0.252437			0.126219	−	0.992002i	$-0.459716\pi$
$239$	8.23654e8			0.252437			0.126219	+	0.992002i	$0.459716\pi$
$240$		0			0
$241$			1.89634e9i			0.562146i	0.959686	+	0.281073i	$0.0906902\pi$
$241$			1.89634e9i			0.562146i	−0.959686	+	0.281073i	$0.909310\pi$
$242$		0			0
$243$			2.23677e8i			0.0641500i
$244$		0			0
$245$	3.44803e9	+	3.12302e9i	0.956987	+	0.866783i
$246$		0			0
$247$	−3.92046e9			−1.05329
$248$		0			0
$249$	−9.59507e8			−0.249604
$250$		0			0
$251$			3.57957e9i			0.901853i	0.892561	+	0.450927i	$0.148906\pi$
$251$			3.57957e9i			0.901853i	−0.892561	+	0.450927i	$0.851094\pi$
$252$		0			0
$253$	1.34779e9			0.328958
$254$		0			0
$255$		−	3.07726e9i		−	0.727786i
$256$		0			0
$257$			6.19783e9i			1.42072i	0.703841	+	0.710358i	$0.251467\pi$
$257$			6.19783e9i			1.42072i	−0.703841	+	0.710358i	$0.748533\pi$
$258$		0			0
$259$	−4.26023e8	−	9.60660e8i	−0.0946746	−	0.213487i
$260$		0			0
$261$	1.03772e9			0.223623
$262$		0			0
$263$	8.15860e9			1.70527			0.852634	−	0.522509i	$-0.175004\pi$
$263$	8.15860e9			1.70527			0.852634	+	0.522509i	$0.175004\pi$
$264$		0			0
$265$		−	6.68299e8i		−	0.135515i
$266$		0			0
$267$	1.50999e9			0.297118
$268$		0			0
$269$			2.73424e9i			0.522188i	0.965313	+	0.261094i	$0.0840833\pi$
$269$			2.73424e9i			0.522188i	−0.965313	+	0.261094i	$0.915917\pi$
$270$		0			0
$271$			9.86740e9i			1.82947i	0.404053	+	0.914736i	$0.367601\pi$
$271$			9.86740e9i			1.82947i	−0.404053	+	0.914736i	$0.632399\pi$
$272$		0			0
$273$	9.77807e8	+	2.20491e9i	0.176036	+	0.396954i
$274$		0			0
$275$	7.81271e8			0.136606
$276$		0			0
$277$	−3.65105e9			−0.620152			−0.310076	−	0.950712i	$-0.600354\pi$
$277$	−3.65105e9			−0.620152			−0.310076	+	0.950712i	$0.600354\pi$
$278$		0			0
$279$		−	3.77087e9i		−	0.622335i
$280$		0			0
$281$	8.60756e9			1.38056			0.690279	−	0.723543i	$-0.257488\pi$
$281$	8.60756e9			1.38056			0.690279	+	0.723543i	$0.257488\pi$
$282$		0			0
$283$		−	6.14091e8i		−	0.0957386i	−0.998854	−	0.0478693i	$-0.984757\pi$
$283$		−	6.14091e8i		−	0.0957386i	0.998854	−	0.0478693i	$-0.0152431\pi$
$284$		0			0
$285$			6.88759e9i			1.04397i
$286$		0			0
$287$	4.65973e9	−	2.06645e9i	0.686805	−	0.304576i
$288$		0			0
$289$	3.26873e8			0.0468584
$290$		0			0
$291$	−5.79159e8			−0.0807655
$292$		0			0
$293$		−	1.02177e9i		−	0.138638i	−0.997595	−	0.0693191i	$-0.977917\pi$
$293$		−	1.02177e9i		−	0.138638i	0.997595	−	0.0693191i	$-0.0220827\pi$
$294$		0			0
$295$	−7.26463e9			−0.959236
$296$		0			0
$297$			3.06619e8i			0.0394070i
$298$		0			0
$299$		−	9.65733e9i		−	1.20829i
$300$		0			0
$301$	−4.55255e9	−	1.02658e10i	−0.554612	−	1.25062i
$302$		0			0
$303$	−1.50349e8			−0.0178374
$304$		0			0
$305$	−1.92218e10			−2.22124
$306$		0			0
$307$			1.30959e10i			1.47429i	0.675736	+	0.737144i	$0.263826\pi$
$307$			1.30959e10i			1.47429i	−0.675736	+	0.737144i	$0.736174\pi$
$308$		0			0
$309$	5.53187e9			0.606790
$310$		0			0
$311$			1.83197e10i			1.95829i	0.203162	+	0.979145i	$0.434878\pi$
$311$			1.83197e10i			1.95829i	−0.203162	+	0.979145i	$0.565122\pi$
$312$		0			0
$313$		−	1.49868e9i		−	0.156147i	−0.996948	−	0.0780733i	$-0.975123\pi$
$313$		−	1.49868e9i		−	0.156147i	0.996948	−	0.0780733i	$-0.0248768\pi$
$314$		0			0
$315$	3.87365e9	−	1.71784e9i	0.393440	−	0.174478i
$316$		0			0
$317$	−4.69050e9			−0.464496			−0.232248	−	0.972657i	$-0.574608\pi$
$317$	−4.69050e9			−0.464496			−0.232248	+	0.972657i	$0.574608\pi$
$318$		0			0
$319$	1.42251e9			0.137370
$320$		0			0
$321$			7.52438e9i			0.708681i
$322$		0			0
$323$	1.48817e10			1.36723
$324$		0			0
$325$		−	5.59804e9i		−	0.501767i
$326$		0			0
$327$			1.16652e10i			1.02024i
$328$		0			0
$329$	−1.20050e9	+	5.32383e8i	−0.102465	+	0.0454402i
$330$		0			0
$331$	−1.40276e10			−1.16861			−0.584307	−	0.811533i	$-0.698633\pi$
$331$	−1.40276e10			−1.16861			−0.584307	+	0.811533i	$0.698633\pi$
$332$		0			0
$333$	−9.57222e8			−0.0778459
$334$		0			0
$335$		−	1.29833e10i		−	1.03087i
$336$		0			0
$337$	1.91233e10			1.48267			0.741333	−	0.671137i	$-0.234194\pi$
$337$	1.91233e10			1.48267			0.741333	+	0.671137i	$0.234194\pi$
$338$		0			0
$339$		−	1.16971e10i		−	0.885684i
$340$		0			0
$341$		−	5.16914e9i		−	0.382297i
$342$		0			0
$343$	1.31449e10	+	4.33520e9i	0.949685	+	0.313208i
$344$		0			0
$345$	−1.69663e10			−1.19760
$346$		0			0
$347$	1.49699e10			1.03253			0.516265	−	0.856429i	$-0.327322\pi$
$347$	1.49699e10			1.03253			0.516265	+	0.856429i	$0.327322\pi$
$348$		0			0
$349$		−	1.97205e10i		−	1.32928i	−0.747163	−	0.664641i	$-0.768584\pi$
$349$		−	1.97205e10i		−	1.32928i	0.747163	−	0.664641i	$-0.231416\pi$
$350$		0			0
$351$	2.19702e9			0.144745
$352$		0			0
$353$		−	5.14618e9i		−	0.331426i	−0.986174	−	0.165713i	$-0.947008\pi$
$353$		−	5.14618e9i		−	0.331426i	0.986174	−	0.165713i	$-0.0529925\pi$
$354$		0			0
$355$		−	1.83849e9i		−	0.115757i
$356$		0			0
$357$	−3.71165e9	−	8.36960e9i	−0.228504	−	0.515267i
$358$		0			0
$359$	−1.91319e10			−1.15181			−0.575904	−	0.817518i	$-0.695350\pi$
$359$	−1.91319e10			−1.15181			−0.575904	+	0.817518i	$0.695350\pi$
$360$		0			0
$361$	−1.63249e10			−0.961216
$362$		0			0
$363$		−	9.60426e9i		−	0.553143i
$364$		0			0
$365$	−6.12018e9			−0.344820
$366$		0			0
$367$			1.43239e10i			0.789582i	0.918771	+	0.394791i	$0.129183\pi$
$367$			1.43239e10i			0.789582i	−0.918771	+	0.394791i	$0.870817\pi$
$368$		0			0
$369$		−	4.64306e9i		−	0.250437i
$370$		0			0
$371$	−8.06072e8	−	1.81765e9i	−0.0425479	−	0.0959434i
$372$		0			0
$373$	−1.52674e10			−0.788732			−0.394366	−	0.918953i	$-0.629036\pi$
$373$	−1.52674e10			−0.788732			−0.394366	+	0.918953i	$0.629036\pi$
$374$		0			0
$375$	4.90698e9			0.248136
$376$		0			0
$377$		−	1.01927e10i		−	0.504573i
$378$		0			0
$379$	3.12545e10			1.51480			0.757400	−	0.652951i	$-0.226469\pi$
$379$	3.12545e10			1.51480			0.757400	+	0.652951i	$0.226469\pi$
$380$		0			0
$381$			2.30601e10i			1.09436i
$382$		0			0
$383$		−	3.65163e10i		−	1.69704i	−0.529166	−	0.848518i	$-0.677495\pi$
$383$		−	3.65163e10i		−	1.69704i	0.529166	−	0.848518i	$-0.322505\pi$
$384$		0			0
$385$	5.31004e9	−	2.35483e9i	0.241688	−	0.107181i
$386$		0			0
$387$	−1.02290e10			−0.456028
$388$		0			0
$389$	2.32602e10			1.01582			0.507908	−	0.861411i	$-0.330419\pi$
$389$	2.32602e10			1.01582			0.507908	+	0.861411i	$0.330419\pi$
$390$		0			0
$391$			3.66582e10i			1.56843i
$392$		0			0
$393$	−2.27248e10			−0.952641
$394$		0			0
$395$		−	2.15763e10i		−	0.886316i
$396$		0			0
$397$		−	8.57494e8i		−	0.0345198i	−0.999851	−	0.0172599i	$-0.994506\pi$
$397$		−	8.57494e8i		−	0.0345198i	0.999851	−	0.0172599i	$-0.00549428\pi$
$398$		0			0
$399$	8.30750e9	+	1.87330e10i	0.327777	+	0.739122i
$400$		0			0
$401$	−3.45424e10			−1.33590			−0.667952	−	0.744204i	$-0.732829\pi$
$401$	−3.45424e10			−1.33590			−0.667952	+	0.744204i	$0.732829\pi$
$402$		0			0
$403$	−3.70384e10			−1.40421
$404$		0			0
$405$		−	3.85979e9i		−	0.143464i
$406$		0			0
$407$	−1.31217e9			−0.0478203
$408$		0			0
$409$			1.12874e10i			0.403367i	0.979451	+	0.201683i	$0.0646412\pi$
$409$			1.12874e10i			0.403367i	−0.979451	+	0.201683i	$0.935359\pi$
$410$		0			0
$411$		−	1.20057e10i		−	0.420745i
$412$		0			0
$413$	−1.97585e10	+	8.76227e9i	−0.679131	+	0.301173i
$414$		0			0
$415$	1.65573e10			0.558210
$416$		0			0
$417$	−2.38406e10			−0.788448
$418$		0			0
$419$			4.67170e9i			0.151572i	0.997124	+	0.0757860i	$0.0241466\pi$
$419$			4.67170e9i			0.151572i	−0.997124	+	0.0757860i	$0.975853\pi$
$420$		0			0
$421$	1.78464e10			0.568098			0.284049	−	0.958810i	$-0.408322\pi$
$421$	1.78464e10			0.568098			0.284049	+	0.958810i	$0.408322\pi$
$422$		0			0
$423$			1.19620e9i			0.0373631i
$424$		0			0
$425$			2.12496e10i			0.651320i
$426$		0			0
$427$	−5.22798e10	+	2.31845e10i	−1.57262	+	0.697406i
$428$		0			0
$429$	3.01169e9			0.0889163
$430$		0			0
$431$	−4.99765e10			−1.44829			−0.724147	−	0.689646i	$-0.757766\pi$
$431$	−4.99765e10			−1.44829			−0.724147	+	0.689646i	$0.757766\pi$
$432$		0			0
$433$		−	2.18390e9i		−	0.0621270i	−0.999517	−	0.0310635i	$-0.990111\pi$
$433$		−	2.18390e9i		−	0.0621270i	0.999517	−	0.0310635i	$-0.00988942\pi$
$434$		0			0
$435$	−1.79069e10			−0.500107
$436$		0			0
$437$		−	8.20492e10i		−	2.24982i
$438$		0			0
$439$			2.15525e10i			0.580282i	0.956984	+	0.290141i	$0.0937022\pi$
$439$			2.15525e10i			0.580282i	−0.956984	+	0.290141i	$0.906298\pi$
$440$		0			0
$441$	8.46365e9	−	9.34445e9i	0.223771	−	0.247058i
$442$		0			0
$443$	−1.62040e10			−0.420735			−0.210367	−	0.977622i	$-0.567466\pi$
$443$	−1.62040e10			−0.420735			−0.210367	+	0.977622i	$0.567466\pi$
$444$		0			0
$445$	−2.60565e10			−0.664470
$446$		0			0
$447$		−	9.68284e9i		−	0.242534i
$448$		0			0
$449$	1.14427e10			0.281541			0.140771	−	0.990042i	$-0.455042\pi$
$449$	1.14427e10			0.281541			0.140771	+	0.990042i	$0.455042\pi$
$450$		0			0
$451$		−	6.36475e9i		−	0.153842i
$452$		0			0
$453$			2.88916e10i			0.686087i
$454$		0			0
$455$	−1.68731e10	−	3.80480e10i	−0.393685	−	0.887741i
$456$		0			0
$457$	1.90088e10			0.435804			0.217902	−	0.975971i	$-0.430079\pi$
$457$	1.90088e10			0.435804			0.217902	+	0.975971i	$0.430079\pi$
$458$		0			0
$459$	−8.33965e9			−0.187887
$460$		0			0
$461$			4.08620e9i			0.0904724i	0.998976	+	0.0452362i	$0.0144040\pi$
$461$			4.08620e9i			0.0904724i	−0.998976	+	0.0452362i	$0.985596\pi$
$462$		0			0
$463$	2.32777e10			0.506543			0.253271	−	0.967395i	$-0.418493\pi$
$463$	2.32777e10			0.506543			0.253271	+	0.967395i	$0.418493\pi$
$464$		0			0
$465$			6.50703e10i			1.39178i
$466$		0			0
$467$		−	7.81851e10i		−	1.64383i	−0.569611	−	0.821915i	$-0.692906\pi$
$467$		−	7.81851e10i		−	1.64383i	0.569611	−	0.821915i	$-0.307094\pi$
$468$		0			0
$469$	−1.56599e10	−	3.53122e10i	−0.323665	−	0.729850i
$470$		0			0
$471$	3.99462e9			0.0811693
$472$		0			0
$473$	−1.40221e10			−0.280135
$474$		0			0
$475$		−	4.75612e10i		−	0.934283i
$476$		0			0
$477$	−1.81115e9			−0.0349849
$478$		0			0
$479$		−	5.51766e10i		−	1.04812i	−0.851680	−	0.524062i	$-0.824416\pi$
$479$		−	5.51766e10i		−	1.04812i	0.851680	−	0.524062i	$-0.175584\pi$
$480$		0			0
$481$			9.40208e9i			0.175648i
$482$		0			0
$483$	−4.61453e10	+	2.04640e10i	−0.847889	+	0.376012i
$484$		0			0
$485$	9.99400e9			0.180623
$486$		0			0
$487$	−6.12116e10			−1.08822			−0.544112	−	0.839013i	$-0.683133\pi$
$487$	−6.12116e10			−1.08822			−0.544112	+	0.839013i	$0.683133\pi$
$488$		0			0
$489$		−	1.79804e10i		−	0.314458i
$490$		0			0
$491$	−1.12307e10			−0.193232			−0.0966160	−	0.995322i	$-0.530802\pi$
$491$	−1.12307e10			−0.193232			−0.0966160	+	0.995322i	$0.530802\pi$
$492$		0			0
$493$			3.86905e10i			0.654962i
$494$		0			0
$495$		−	5.29103e9i		−	0.0881292i
$496$		0			0
$497$	−2.21751e9	−	5.00037e9i	−0.0363445	−	0.0819552i
$498$		0			0
$499$	−8.83273e8			−0.0142460			−0.00712300	−	0.999975i	$-0.502267\pi$
$499$	−8.83273e8			−0.0142460			−0.00712300	+	0.999975i	$0.502267\pi$
$500$		0			0
$501$	4.92687e10			0.782025
$502$		0			0
$503$			2.14670e10i			0.335351i	0.985842	+	0.167676i	$0.0536261\pi$
$503$			2.14670e10i			0.335351i	−0.985842	+	0.167676i	$0.946374\pi$
$504$		0			0
$505$	2.59443e9			0.0398912
$506$		0			0
$507$			1.65683e10i			0.250753i
$508$		0			0
$509$			2.14265e10i			0.319212i	0.987181	+	0.159606i	$0.0510224\pi$
$509$			2.14265e10i			0.319212i	−0.987181	+	0.159606i	$0.948978\pi$
$510$		0			0
$511$	−1.66458e10	+	7.38188e9i	−0.244130	+	0.108264i
$512$		0			0
$513$	1.86660e10			0.269514
$514$		0			0
$515$	−9.54583e10			−1.35702
$516$		0			0
$517$			1.63976e9i			0.0229519i
$518$		0			0
$519$	2.75451e9			0.0379642
$520$		0			0
$521$		−	1.08758e11i		−	1.47608i	−0.674758	−	0.738039i	$-0.735752\pi$
$521$		−	1.08758e11i		−	1.47608i	0.674758	−	0.738039i	$-0.264248\pi$
$522$		0			0
$523$		−	1.89511e10i		−	0.253295i	−0.991948	−	0.126648i	$-0.959578\pi$
$523$		−	1.89511e10i		−	0.253295i	0.991948	−	0.126648i	$-0.0404217\pi$
$524$		0			0
$525$	−2.67489e10	+	1.18623e10i	−0.352102	+	0.156146i
$526$		0			0
$527$	1.40594e11			1.82274
$528$		0			0
$529$	1.23802e11			1.58090
$530$		0			0
$531$			1.96878e10i			0.247639i
$532$		0			0
$533$	−4.56053e10			−0.565075
$534$		0			0
$535$		−	1.29841e11i		−	1.58488i
$536$		0			0
$537$		−	8.23825e10i		−	0.990690i
$538$		0			0
$539$	1.16021e10	−	1.28095e10i	0.137461	−	0.151766i
$540$		0			0
$541$	1.25973e10			0.147057			0.0735287	−	0.997293i	$-0.476574\pi$
$541$	1.25973e10			0.147057			0.0735287	+	0.997293i	$0.476574\pi$
$542$		0			0
$543$	−4.73695e10			−0.544879
$544$		0			0
$545$		−	2.01295e11i		−	2.28164i
$546$		0			0
$547$	−1.37917e11			−1.54052			−0.770260	−	0.637730i	$-0.779873\pi$
$547$	−1.37917e11			−1.54052			−0.770260	+	0.637730i	$0.779873\pi$
$548$		0			0
$549$			5.20927e10i			0.573439i
$550$		0			0
$551$		−	8.65978e10i		−	0.939508i
$552$		0			0
$553$	−2.60244e10	−	5.86837e10i	−0.278278	−	0.627504i
$554$		0			0
$555$	1.65179e10			0.174093
$556$		0			0
$557$	1.16550e11			1.21085			0.605424	−	0.795903i	$-0.293004\pi$
$557$	1.16550e11			1.21085			0.605424	+	0.795903i	$0.293004\pi$
$558$		0			0
$559$			1.00472e11i			1.02896i
$560$		0			0
$561$	−1.14321e10			−0.115418
$562$		0			0
$563$			1.45751e11i			1.45070i	0.688381	+	0.725349i	$0.258322\pi$
$563$			1.45751e11i			1.45070i	−0.688381	+	0.725349i	$0.741678\pi$
$564$		0			0
$565$			2.01845e11i			1.98073i
$566$		0			0
$567$	−4.65550e9	−	1.04979e10i	−0.0450437	−	0.101571i
$568$		0			0
$569$	1.34221e11			1.28047			0.640236	−	0.768178i	$-0.278837\pi$
$569$	1.34221e11			1.28047			0.640236	+	0.768178i	$0.278837\pi$
$570$		0			0
$571$	−1.21915e11			−1.14687			−0.573433	−	0.819252i	$-0.694389\pi$
$571$	−1.21915e11			−1.14687			−0.573433	+	0.819252i	$0.694389\pi$
$572$		0			0
$573$		−	1.65401e10i		−	0.153434i
$574$		0			0
$575$	1.17158e11			1.07177
$576$		0			0
$577$			1.24482e11i			1.12306i	0.827457	+	0.561529i	$0.189787\pi$
$577$			1.24482e11i			1.12306i	−0.827457	+	0.561529i	$0.810213\pi$
$578$		0			0
$579$			5.31069e10i			0.472538i
$580$		0			0
$581$	4.50329e10	−	1.99707e10i	0.395208	−	0.175262i
$582$		0			0
$583$	−2.48274e9			−0.0214910
$584$		0			0
$585$	−3.79118e10			−0.323706
$586$		0			0
$587$			9.41980e10i			0.793395i	0.917949	+	0.396697i	$0.129844\pi$
$587$			9.41980e10i			0.793395i	−0.917949	+	0.396697i	$0.870156\pi$
$588$		0			0
$589$	−3.14680e11			−2.61462
$590$		0			0
$591$			1.06008e11i			0.868938i
$592$		0			0
$593$			1.56581e11i			1.26626i	0.774047	+	0.633128i	$0.218229\pi$
$593$			1.56581e11i			1.26626i	−0.774047	+	0.633128i	$0.781771\pi$
$594$		0			0
$595$	6.40485e10	+	1.44426e11i	0.511024	+	1.15233i
$596$		0			0
$597$	−4.46945e10			−0.351849
$598$		0			0
$599$	1.85009e11			1.43709			0.718547	−	0.695478i	$-0.244807\pi$
$599$	1.85009e11			1.43709			0.718547	+	0.695478i	$0.244807\pi$
$600$		0			0
$601$			1.69387e11i			1.29832i	0.760651	+	0.649161i	$0.224880\pi$
$601$			1.69387e11i			1.29832i	−0.760651	+	0.649161i	$0.775120\pi$
$602$		0			0
$603$	−3.51858e10			−0.266133
$604$		0			0
$605$			1.65732e11i			1.23704i
$606$		0			0
$607$		−	1.51101e11i		−	1.11305i	−0.830832	−	0.556523i	$-0.812135\pi$
$607$		−	1.51101e11i		−	1.11305i	0.830832	−	0.556523i	$-0.187865\pi$
$608$		0			0
$609$	−4.87035e10	+	2.15985e10i	−0.354071	+	0.157019i
$610$		0			0
$611$	1.17494e10			0.0843044
$612$		0			0
$613$	−2.47572e11			−1.75331			−0.876655	−	0.481119i	$-0.840231\pi$
$613$	−2.47572e11			−1.75331			−0.876655	+	0.481119i	$0.840231\pi$
$614$		0			0
$615$			8.01208e10i			0.560073i
$616$		0			0
$617$	1.87149e11			1.29136			0.645680	−	0.763608i	$-0.276574\pi$
$617$	1.87149e11			1.29136			0.645680	+	0.763608i	$0.276574\pi$
$618$		0			0
$619$		−	1.24121e11i		−	0.845441i	−0.906260	−	0.422721i	$-0.861075\pi$
$619$		−	1.24121e11i		−	0.845441i	0.906260	−	0.422721i	$-0.138925\pi$
$620$		0			0
$621$			4.59801e10i			0.309175i
$622$		0			0
$623$	−7.08689e10	+	3.14281e10i	−0.470439	+	0.208625i
$624$		0			0
$625$	−1.86472e11			−1.22206
$626$		0			0
$627$	2.55875e10			0.165561
$628$		0			0
$629$		−	3.56893e10i		−	0.228000i
$630$		0			0
$631$	1.40315e11			0.885091			0.442546	−	0.896746i	$-0.354075\pi$
$631$	1.40315e11			0.885091			0.442546	+	0.896746i	$0.354075\pi$
$632$		0			0
$633$			2.16553e10i			0.134881i
$634$		0			0
$635$		−	3.97927e11i		−	2.44742i
$636$		0			0
$637$	−9.17835e10	−	8.31322e10i	−0.557451	−	0.504907i
$638$		0			0
$639$	−4.98247e9			−0.0298842
$640$		0			0
$641$	−1.77074e11			−1.04887			−0.524435	−	0.851450i	$-0.675724\pi$
$641$	−1.77074e11			−1.04887			−0.524435	+	0.851450i	$0.675724\pi$
$642$		0			0
$643$			1.16691e11i			0.682643i	0.939947	+	0.341322i	$0.110875\pi$
$643$			1.16691e11i			0.682643i	−0.939947	+	0.341322i	$0.889125\pi$
$644$		0			0
$645$	1.76513e11			1.01985
$646$		0			0
$647$			1.82722e11i			1.04273i	0.853333	+	0.521366i	$0.174577\pi$
$647$			1.82722e11i			1.04273i	−0.853333	+	0.521366i	$0.825423\pi$
$648$		0			0
$649$			2.69882e10i			0.152123i
$650$		0			0
$651$	7.84848e10	+	1.76979e11i	0.436980	+	0.985369i
$652$		0			0
$653$	2.74748e11			1.51106			0.755530	−	0.655114i	$-0.227379\pi$
$653$	2.74748e11			1.51106			0.755530	+	0.655114i	$0.227379\pi$
$654$		0			0
$655$	3.92140e11			2.13047
$656$		0			0
$657$			1.65862e10i			0.0890196i
$658$		0			0
$659$	−1.30992e11			−0.694550			−0.347275	−	0.937763i	$-0.612893\pi$
$659$	−1.30992e11			−0.694550			−0.347275	+	0.937763i	$0.612893\pi$
$660$		0			0
$661$		−	1.50121e11i		−	0.786388i	−0.919456	−	0.393194i	$-0.871370\pi$
$661$		−	1.50121e11i		−	0.786388i	0.919456	−	0.393194i	$-0.128630\pi$
$662$		0			0
$663$			8.19141e10i			0.423940i
$664$		0			0
$665$	−1.43355e11	−	3.23258e11i	−0.733036	−	1.65296i
$666$		0			0
$667$	2.13318e11			1.07776
$668$		0			0
$669$	6.08291e9			0.0303674
$670$		0			0
$671$			7.14092e10i			0.352261i
$672$		0			0
$673$	3.70666e11			1.80685			0.903425	−	0.428746i	$-0.141045\pi$
$673$	3.70666e11			1.80685			0.903425	+	0.428746i	$0.141045\pi$
$674$		0			0
$675$			2.66532e10i			0.128391i
$676$		0			0
$677$		−	3.45722e11i		−	1.64578i	−0.568200	−	0.822891i	$-0.692360\pi$
$677$		−	3.45722e11i		−	1.64578i	0.568200	−	0.822891i	$-0.307640\pi$
$678$		0			0
$679$	2.71819e10	−	1.20543e10i	0.127879	−	0.0567105i
$680$		0			0
$681$	5.73717e10			0.266753
$682$		0			0
$683$	9.59098e10			0.440738			0.220369	−	0.975417i	$-0.429274\pi$
$683$	9.59098e10			0.440738			0.220369	+	0.975417i	$0.429274\pi$
$684$		0			0
$685$			2.07170e11i			0.940948i
$686$		0			0
$687$	1.69562e11			0.761206
$688$		0			0
$689$			1.77896e10i			0.0789384i
$690$		0			0
$691$		−	7.73267e10i		−	0.339170i	−0.985516	−	0.169585i	$-0.945757\pi$
$691$		−	7.73267e10i		−	0.339170i	0.985516	−	0.169585i	$-0.0542427\pi$
$692$		0			0
$693$	−6.38180e9	−	1.43907e10i	−0.0276701	−	0.0623947i
$694$		0			0
$695$	4.11394e11			1.76327
$696$		0			0
$697$	1.73113e11			0.733497
$698$		0			0
$699$			9.91589e10i			0.415358i
$700$		0			0
$701$	2.60432e11			1.07850			0.539252	−	0.842144i	$-0.318707\pi$
$701$	2.60432e11			1.07850			0.539252	+	0.842144i	$0.318707\pi$
$702$		0			0
$703$			7.98805e10i			0.327054i
$704$		0			0
$705$		−	2.06417e10i		−	0.0835582i
$706$		0			0
$707$	7.05639e9	−	3.12929e9i	0.0282426	−	0.0125247i
$708$		0			0
$709$	2.43073e11			0.961948			0.480974	−	0.876735i	$-0.340283\pi$
$709$	2.43073e11			0.961948			0.480974	+	0.876735i	$0.340283\pi$
$710$		0			0
$711$	−5.84736e10			−0.228813
$712$		0			0
$713$		−	7.75157e11i		−	2.99938i
$714$		0			0
$715$	−5.19699e10			−0.198851
$716$		0			0
$717$			3.85185e10i			0.145745i
$718$		0			0
$719$			2.65280e9i			0.00992633i	0.999988	+	0.00496317i	$0.00157983\pi$
$719$			2.65280e9i			0.00992633i	−0.999988	+	0.00496317i	$0.998420\pi$
$720$		0			0
$721$	−2.59629e11	+	1.15137e11i	−0.960755	+	0.426065i
$722$		0			0
$723$	−8.86832e10			−0.324555
$724$		0			0
$725$	1.23653e11			0.447562
$726$		0			0
$727$			5.08987e11i			1.82209i	0.412309	+	0.911044i	$0.364722\pi$
$727$			5.08987e11i			1.82209i	−0.412309	+	0.911044i	$0.635278\pi$
$728$		0			0
$729$	−1.04604e10			−0.0370370
$730$		0			0
$731$		−	3.81382e11i		−	1.33565i
$732$		0			0
$733$			5.60084e11i			1.94016i	0.242787	+	0.970080i	$0.421938\pi$
$733$			5.60084e11i			1.94016i	−0.242787	+	0.970080i	$0.578062\pi$
$734$		0			0
$735$	−1.46049e11	+	1.61248e11i	−0.500438	+	0.552517i
$736$		0			0
$737$	−4.82331e10			−0.163484
$738$		0			0
$739$	2.96802e11			0.995151			0.497576	−	0.867421i	$-0.334224\pi$
$739$	2.96802e11			0.995151			0.497576	+	0.867421i	$0.334224\pi$
$740$		0			0
$741$		−	1.83342e11i		−	0.608119i
$742$		0			0
$743$	2.79452e11			0.916964			0.458482	−	0.888704i	$-0.348393\pi$
$743$	2.79452e11			0.916964			0.458482	+	0.888704i	$0.348393\pi$
$744$		0			0
$745$			1.67088e11i			0.542399i
$746$		0			0
$747$		−	4.48717e10i		−	0.144109i
$748$		0			0
$749$	−1.56608e11	−	3.53144e11i	−0.497608	−	1.12208i
$750$		0			0
$751$	3.71025e11			1.16639			0.583195	−	0.812332i	$-0.301802\pi$
$751$	3.71025e11			1.16639			0.583195	+	0.812332i	$0.301802\pi$
$752$		0			0
$753$	−1.67400e11			−0.520685
$754$		0			0
$755$		−	4.98555e11i		−	1.53435i
$756$		0			0
$757$	−1.46122e11			−0.444972			−0.222486	−	0.974936i	$-0.571417\pi$
$757$	−1.46122e11			−0.444972			−0.222486	+	0.974936i	$0.571417\pi$
$758$		0			0
$759$			6.30300e10i			0.189924i
$760$		0			0
$761$			4.54125e11i			1.35406i	0.735958	+	0.677028i	$0.236732\pi$
$761$			4.54125e11i			1.35406i	−0.735958	+	0.677028i	$0.763268\pi$
$762$		0			0
$763$	−2.42793e11	−	5.47487e11i	−0.716372	−	1.61538i
$764$		0			0
$765$	1.43909e11			0.420188
$766$		0			0
$767$	1.93378e11			0.558761
$768$		0			0
$769$			2.07798e11i			0.594204i	0.954846	+	0.297102i	$0.0960202\pi$
$769$			2.07798e11i			0.594204i	−0.954846	+	0.297102i	$0.903980\pi$
$770$		0			0
$771$	−2.89844e11			−0.820251
$772$		0			0
$773$			1.03896e11i			0.290993i	0.989359	+	0.145496i	$0.0464779\pi$
$773$			1.03896e11i			0.290993i	−0.989359	+	0.145496i	$0.953522\pi$
$774$		0			0
$775$		−	4.49333e11i		−	1.24555i
$776$		0			0
$777$	4.49256e10	−	1.99231e10i	0.123257	−	0.0546604i
$778$		0			0
$779$	−3.87465e11			−1.05216
$780$		0			0
$781$	−6.83002e9			−0.0183577
$782$		0			0
$783$			4.85292e10i			0.129109i
$784$		0			0
$785$	−6.89314e10			−0.181526
$786$		0			0
$787$		−	1.87638e11i		−	0.489126i	−0.969633	−	0.244563i	$-0.921355\pi$
$787$		−	1.87638e11i		−	0.489126i	0.969633	−	0.244563i	$-0.0786445\pi$
$788$		0			0
$789$			3.81540e11i			0.984537i
$790$		0			0
$791$	2.43457e11	+	5.48983e11i	0.621893	+	1.40234i
$792$		0			0
$793$	5.11668e11			1.29388
$794$		0			0
$795$	3.12532e10			0.0782396
$796$		0			0
$797$		−	3.01628e11i		−	0.747546i	−0.927520	−	0.373773i	$-0.878064\pi$
$797$		−	3.01628e11i		−	0.747546i	0.927520	−	0.373773i	$-0.121936\pi$
$798$		0			0
$799$	−4.45995e10			−0.109432
$800$		0			0
$801$			7.06152e10i			0.171541i
$802$		0			0
$803$			2.27365e10i			0.0546842i
$804$		0			0
$805$	7.96286e11	−	3.53128e11i	1.89621	−	0.840907i
$806$		0			0
$807$	−1.27868e11			−0.301486
$808$		0			0
$809$	1.91427e11			0.446899			0.223450	−	0.974715i	$-0.428268\pi$
$809$	1.91427e11			0.446899			0.223450	+	0.974715i	$0.428268\pi$
$810$		0			0
$811$			6.32845e11i			1.46290i	0.681896	+	0.731449i	$0.261156\pi$
$811$			6.32845e11i			1.46290i	−0.681896	+	0.731449i	$0.738844\pi$
$812$		0			0
$813$	−4.61453e11			−1.05625
$814$		0			0
$815$			3.10270e11i			0.703250i
$816$		0			0
$817$			8.53618e11i			1.91591i
$818$		0			0
$819$	−1.03113e11	+	4.57275e10i	−0.229181	+	0.101635i
$820$		0			0
$821$	−2.41717e11			−0.532027			−0.266014	−	0.963969i	$-0.585707\pi$
$821$	−2.41717e11			−0.532027			−0.266014	+	0.963969i	$0.585707\pi$
$822$		0			0
$823$	−1.89207e11			−0.412418			−0.206209	−	0.978508i	$-0.566113\pi$
$823$	−1.89207e11			−0.412418			−0.206209	+	0.978508i	$0.566113\pi$
$824$		0			0
$825$			3.65364e10i			0.0788697i
$826$		0			0
$827$	−3.98214e11			−0.851324			−0.425662	−	0.904882i	$-0.639959\pi$
$827$	−3.98214e11			−0.851324			−0.425662	+	0.904882i	$0.639959\pi$
$828$		0			0
$829$			5.85111e11i			1.23885i	0.785054	+	0.619427i	$0.212635\pi$
$829$			5.85111e11i			1.23885i	−0.785054	+	0.619427i	$0.787365\pi$
$830$		0			0
$831$		−	1.70742e11i		−	0.358045i
$832$		0			0
$833$	3.48401e11	+	3.15561e11i	0.723601	+	0.655396i
$834$		0			0
$835$	−8.50184e11			−1.74891
$836$		0			0
$837$	1.76346e11			0.359305
$838$		0			0
$839$		−	4.29088e11i		−	0.865961i	−0.901403	−	0.432981i	$-0.857462\pi$
$839$		−	4.29088e11i		−	0.865961i	0.901403	−	0.432981i	$-0.142538\pi$
$840$		0			0
$841$	−2.75103e11			−0.549935
$842$		0			0
$843$			4.02536e11i			0.797066i
$844$		0			0
$845$		−	2.85903e11i		−	0.560780i
$846$		0			0
$847$	1.99898e11	+	4.50760e11i	0.388396	+	0.875814i
$848$		0			0
$849$	2.87182e10			0.0552747
$850$		0			0
$851$	−1.96771e11			−0.375183
$852$		0			0
$853$		−	1.72826e11i		−	0.326447i	−0.986589	−	0.163224i	$-0.947811\pi$
$853$		−	1.72826e11i		−	0.326447i	0.986589	−	0.163224i	$-0.0521892\pi$
$854$		0			0
$855$	−3.22101e11			−0.602736
$856$		0			0
$857$			6.08384e11i			1.12786i	0.825823	+	0.563929i	$0.190711\pi$
$857$			6.08384e11i			1.12786i	−0.825823	+	0.563929i	$0.809289\pi$
$858$		0			0
$859$		−	4.25905e11i		−	0.782240i	−0.920340	−	0.391120i	$-0.872088\pi$
$859$		−	4.25905e11i		−	0.782240i	0.920340	−	0.391120i	$-0.127912\pi$
$860$		0			0
$861$	9.66381e10	+	2.17914e11i	0.175847	+	0.396527i
$862$		0			0
$863$	−2.13399e11			−0.384724			−0.192362	−	0.981324i	$-0.561615\pi$
$863$	−2.13399e11			−0.384724			−0.192362	+	0.981324i	$0.561615\pi$
$864$		0			0
$865$	−4.75319e10			−0.0849026
$866$		0			0
$867$			1.52863e10i			0.0270537i
$868$		0			0
$869$	−8.01562e10			−0.140559
$870$		0			0
$871$			3.45604e11i			0.600491i
$872$		0			0
$873$		−	2.70846e10i		−	0.0466300i
$874$		0			0
$875$	−2.30301e11	+	1.02131e11i	−0.392883	+	0.174231i
$876$		0			0
$877$	−1.79590e11			−0.303587			−0.151794	−	0.988412i	$-0.548505\pi$
$877$	−1.79590e11			−0.303587			−0.151794	+	0.988412i	$0.548505\pi$
$878$		0			0
$879$	4.77835e10			0.0800428
$880$		0			0
$881$			1.74627e10i			0.0289873i	0.999895	+	0.0144937i	$0.00461363\pi$
$881$			1.74627e10i			0.0289873i	−0.999895	+	0.0144937i	$0.995386\pi$
$882$		0			0
$883$	4.08817e11			0.672491			0.336245	−	0.941774i	$-0.390843\pi$
$883$	4.08817e11			0.672491			0.336245	+	0.941774i	$0.390843\pi$
$884$		0			0
$885$		−	3.39733e11i		−	0.553815i
$886$		0			0
$887$		−	6.05445e11i		−	0.978093i	−0.872258	−	0.489046i	$-0.837345\pi$
$887$		−	6.05445e11i		−	0.978093i	0.872258	−	0.489046i	$-0.162655\pi$
$888$		0			0
$889$	−4.79961e11	−	1.08229e12i	−0.768421	−	1.73275i
$890$		0			0
$891$	−1.43392e10			−0.0227516
$892$		0			0
$893$	9.98234e10			0.156974
$894$		0			0
$895$			1.42160e12i			2.21556i
$896$		0			0
$897$	4.51629e11			0.697608
$898$		0			0
$899$		−	8.18129e11i		−	1.25252i
$900$		0			0
$901$		−	6.75273e10i		−	0.102466i
$902$		0			0
$903$	4.80083e11	−	2.12902e11i	0.722047	−	0.320205i
$904$		0			0
$905$	8.17411e11			1.21856
$906$		0			0
$907$	−7.97484e11			−1.17840			−0.589200	−	0.807987i	$-0.700557\pi$
$907$	−7.97484e11			−1.17840			−0.589200	+	0.807987i	$0.700557\pi$
$908$		0			0
$909$		−	7.03114e9i		−	0.0102984i
$910$		0			0
$911$	−1.70007e10			−0.0246827			−0.0123414	−	0.999924i	$-0.503928\pi$
$911$	−1.70007e10			−0.0246827			−0.0123414	+	0.999924i	$0.503928\pi$
$912$		0			0
$913$		−	6.15106e10i		−	0.0885252i
$914$		0			0
$915$		−	8.98914e11i		−	1.28243i
$916$		0			0
$917$	1.06655e12	−	4.72981e11i	1.50836	−	0.668908i
$918$		0			0
$919$	−2.72358e11			−0.381837			−0.190919	−	0.981606i	$-0.561147\pi$
$919$	−2.72358e11			−0.381837			−0.190919	+	0.981606i	$0.561147\pi$
$920$		0			0
$921$	−6.12435e11			−0.851180
$922$		0			0
$923$			4.89391e10i			0.0674294i
$924$		0			0
$925$	−1.14062e11			−0.155802
$926$		0			0
$927$			2.58700e11i			0.350330i
$928$		0			0
$929$		−	2.32000e11i		−	0.311476i	−0.987798	−	0.155738i	$-0.950224\pi$
$929$		−	2.32000e11i		−	0.311476i	0.987798	−	0.155738i	$-0.0497756\pi$
$930$		0			0
$931$	−7.79798e11	−	7.06295e11i	−1.03797	−	0.940129i
$932$		0			0
$933$	−8.56728e11			−1.13062
$934$		0			0
$935$	1.97272e11			0.258119
$936$		0			0
$937$		−	1.32500e11i		−	0.171893i	−0.996300	−	0.0859465i	$-0.972609\pi$
$937$		−	1.32500e11i		−	0.171893i	0.996300	−	0.0859465i	$-0.0273914\pi$
$938$		0			0
$939$	7.00865e10			0.0901513
$940$		0			0
$941$			1.30889e12i			1.66934i	0.550752	+	0.834669i	$0.314341\pi$
$941$			1.30889e12i			1.66934i	−0.550752	+	0.834669i	$0.685659\pi$
$942$		0			0
$943$		−	9.54448e11i		−	1.20700i
$944$		0			0
$945$	8.03355e10	+	1.81153e11i	0.100735	+	0.227153i
$946$		0			0
$947$	−1.26480e12			−1.57261			−0.786307	−	0.617836i	$-0.788010\pi$
$947$	−1.26480e12			−1.57261			−0.786307	+	0.617836i	$0.788010\pi$
$948$		0			0
$949$	1.62914e11			0.200860
$950$		0			0
$951$		−	2.19353e11i		−	0.268177i
$952$		0			0
$953$	−3.22636e11			−0.391148			−0.195574	−	0.980689i	$-0.562657\pi$
$953$	−3.22636e11			−0.391148			−0.195574	+	0.980689i	$0.562657\pi$
$954$		0			0
$955$			2.85418e11i			0.343137i
$956$		0			0
$957$			6.65242e10i			0.0793108i
$958$		0			0
$959$	2.49880e11	+	5.63466e11i	0.295431	+	0.666183i
$960$		0			0
$961$	−2.12004e12			−2.48571
$962$		0			0
$963$	−3.51880e11			−0.409157
$964$		0			0
$965$		−	9.16415e11i		−	1.05678i
$966$		0			0
$967$	1.09225e12			1.24915			0.624577	−	0.780963i	$-0.285271\pi$
$967$	1.09225e12			1.24915			0.624577	+	0.780963i	$0.285271\pi$
$968$		0			0
$969$			6.95947e11i			0.789370i
$970$		0			0
$971$		−	9.80303e11i		−	1.10277i	−0.834252	−	0.551383i	$-0.814100\pi$
$971$		−	9.80303e11i		−	1.10277i	0.834252	−	0.551383i	$-0.185900\pi$
$972$		0			0
$973$	1.11892e12	−	4.96205e11i	1.24838	−	0.553618i
$974$		0			0
$975$	2.61794e11			0.289695
$976$		0			0
$977$	−3.43312e11			−0.376800			−0.188400	−	0.982092i	$-0.560330\pi$
$977$	−3.43312e11			−0.376800			−0.188400	+	0.982092i	$0.560330\pi$
$978$		0			0
$979$			9.68000e10i			0.105377i
$980$		0			0
$981$	−5.45528e11			−0.589035
$982$		0			0
$983$		−	4.45649e11i		−	0.477286i	−0.971107	−	0.238643i	$-0.923297\pi$
$983$		−	4.45649e11i		−	0.477286i	0.971107	−	0.238643i	$-0.0767026\pi$
$984$		0			0
$985$		−	1.82928e12i		−	1.94328i
$986$		0			0
$987$	−2.48971e10	−	5.61417e10i	−0.0262349	−	0.0591585i
$988$		0			0
$989$	−2.10273e12			−2.19785
$990$		0			0
$991$	−1.12963e12			−1.17123			−0.585614	−	0.810590i	$-0.699147\pi$
$991$	−1.12963e12			−1.17123			−0.585614	+	0.810590i	$0.699147\pi$
$992$		0			0
$993$		−	6.56005e11i		−	0.674699i
$994$		0			0
$995$	7.71250e11			0.786870
$996$		0			0
$997$		−	1.70368e12i		−	1.72428i	−0.506672	−	0.862139i	$-0.669125\pi$
$997$		−	1.70368e12i		−	1.72428i	0.506672	−	0.862139i	$-0.330875\pi$
$998$		0			0
$999$		−	4.47648e10i		−	0.0449443i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	336.9.f.d.97.19		32
4.3	odd	2		168.9.f.a.97.3	✓	32
7.6	odd	2	inner	336.9.f.d.97.14		32
28.27	even	2		168.9.f.a.97.30	yes	32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
168.9.f.a.97.3	✓	32	4.3	odd	2
168.9.f.a.97.30	yes	32	28.27	even	2
336.9.f.d.97.14		32	7.6	odd	2	inner
336.9.f.d.97.19		32	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 336 = 2^{4} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 9 \)
Character orbit:	\([\chi]\)	\(=\)	336.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+46.7654i q^{3} -806.986i q^{5} +(-973.349 - 2194.86i) q^{7} -2187.00 q^{9} +O(q^{10})\)	\(q+46.7654i q^{3} -806.986i q^{5} +(-973.349 - 2194.86i) q^{7} -2187.00 q^{9} -2997.96 q^{11} +21481.3i q^{13} +37739.0 q^{15} -81540.7i q^{17} +182506. i q^{19} +(102643. - 45519.0i) q^{21} -449570. q^{23} -260601. q^{25} -102276. i q^{27} -474493. q^{29} +1.72422e6i q^{31} -140201. i q^{33} +(-1.77122e6 + 785479. i) q^{35} +437687. q^{37} -1.00458e6 q^{39} +2.12303e6i q^{41} +4.67720e6 q^{43} +1.76488e6i q^{45} -546959. i q^{47} +(-3.86998e6 + 4.27272e6i) q^{49} +3.81328e6 q^{51} +828142. q^{53} +2.41931e6i q^{55} -8.53496e6 q^{57} -9.00218e6i q^{59} -2.38193e7i q^{61} +(2.12872e6 + 4.80015e6i) q^{63} +1.73351e7 q^{65} +1.60886e7 q^{67} -2.10243e7i q^{69} +2.27822e6 q^{71} -7.58400e6i q^{73} -1.21871e7i q^{75} +(2.91806e6 + 6.58009e6i) q^{77} +2.67369e7 q^{79} +4.78297e6 q^{81} +2.05175e7i q^{83} -6.58022e7 q^{85} -2.21898e7i q^{87} -3.22886e7i q^{89} +(4.71483e7 - 2.09088e7i) q^{91} -8.06337e7 q^{93} +1.47280e8 q^{95} +1.23844e7i q^{97} +6.55654e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - 1424 q^{7} - 69984 q^{9}+O(q^{10}) \) 32 * q - 1424 * q^7 - 69984 * q^9	\( 32 q - 1424 q^{7} - 69984 q^{9} - 23616 q^{11} + 59616 q^{15} + 60912 q^{21} - 455040 q^{23} - 3404928 q^{25} + 632064 q^{29} + 1543200 q^{35} + 4150496 q^{37} + 4162752 q^{39} - 6028000 q^{43} + 15115072 q^{49} + 1340064 q^{51} - 37728576 q^{53} + 7286112 q^{57} + 3114288 q^{63} + 29977536 q^{65} - 68431648 q^{67} + 19788096 q^{71} + 87499392 q^{77} + 86954656 q^{79} + 153055008 q^{81} - 31326496 q^{85} + 24540288 q^{91} - 38833344 q^{93} - 329546688 q^{95} + 51648192 q^{99}+O(q^{100}) \) 32 * q - 1424 * q^7 - 69984 * q^9 - 23616 * q^11 + 59616 * q^15 + 60912 * q^21 - 455040 * q^23 - 3404928 * q^25 + 632064 * q^29 + 1543200 * q^35 + 4150496 * q^37 + 4162752 * q^39 - 6028000 * q^43 + 15115072 * q^49 + 1340064 * q^51 - 37728576 * q^53 + 7286112 * q^57 + 3114288 * q^63 + 29977536 * q^65 - 68431648 * q^67 + 19788096 * q^71 + 87499392 * q^77 + 86954656 * q^79 + 153055008 * q^81 - 31326496 * q^85 + 24540288 * q^91 - 38833344 * q^93 - 329546688 * q^95 + 51648192 * q^99

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