LMFDB - Embedded newform 1617.2.c.b.538.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1617,2,Mod(538,1617)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1617, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("1617.538");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1617 = 3 \cdot 7^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1617.c (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:	$12.9118100068$
Analytic rank:	$0$
Dimension:	$48$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			538.8
Character	$\chi$	$=$	1617.538
Dual form			1617.2.c.b.538.41

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.14374i q^{2} +1.00000i q^{3} -2.59563 q^{4} -1.55945i q^{5} +2.14374 q^{6} +1.27689i q^{8} -1.00000 q^{9} +O(q^{10})$	\(q-2.14374i q^{2} +1.00000i q^{3} -2.59563 q^{4} -1.55945i q^{5} +2.14374 q^{6} +1.27689i q^{8} -1.00000 q^{9} -3.34306 q^{10} +(1.68920 + 2.85422i) q^{11} -2.59563i q^{12} +4.23323 q^{13} +1.55945 q^{15} -2.45395 q^{16} +0.829168 q^{17} +2.14374i q^{18} +4.75731 q^{19} +4.04777i q^{20} +(6.11872 - 3.62122i) q^{22} -1.60908 q^{23} -1.27689 q^{24} +2.56811 q^{25} -9.07497i q^{26} -1.00000i q^{27} -0.853870i q^{29} -3.34306i q^{30} -9.33663i q^{31} +7.81442i q^{32} +(-2.85422 + 1.68920i) q^{33} -1.77752i q^{34} +2.59563 q^{36} +3.41775 q^{37} -10.1985i q^{38} +4.23323i q^{39} +1.99124 q^{40} +0.308307 q^{41} +6.72552i q^{43} +(-4.38455 - 7.40852i) q^{44} +1.55945i q^{45} +3.44945i q^{46} -5.93972i q^{47} -2.45395i q^{48} -5.50537i q^{50} +0.829168i q^{51} -10.9879 q^{52} -4.16866 q^{53} -2.14374 q^{54} +(4.45102 - 2.63423i) q^{55} +4.75731i q^{57} -1.83048 q^{58} -8.37253i q^{59} -4.04777 q^{60} +7.95518 q^{61} -20.0153 q^{62} +11.8442 q^{64} -6.60153i q^{65} +(3.62122 + 6.11872i) q^{66} +12.9150 q^{67} -2.15222 q^{68} -1.60908i q^{69} -8.47875 q^{71} -1.27689i q^{72} -9.63338 q^{73} -7.32677i q^{74} +2.56811i q^{75} -12.3482 q^{76} +9.07497 q^{78} -7.07971i q^{79} +3.82682i q^{80} +1.00000 q^{81} -0.660930i q^{82} +1.82395 q^{83} -1.29305i q^{85} +14.4178 q^{86} +0.853870 q^{87} +(-3.64452 + 2.15692i) q^{88} +1.92740i q^{89} +3.34306 q^{90} +4.17658 q^{92} +9.33663 q^{93} -12.7332 q^{94} -7.41880i q^{95} -7.81442 q^{96} -17.0963i q^{97} +(-1.68920 - 2.85422i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 48 q - 64 q^{4} - 48 q^{9}+O(q^{10}) $ 48 * q - 64 * q^4 - 48 * q^9	$ 48 q - 64 q^{4} - 48 q^{9} - 16 q^{11} + 64 q^{16} + 16 q^{22} + 32 q^{23} - 80 q^{25} + 64 q^{36} - 96 q^{37} - 32 q^{44} + 64 q^{53} + 48 q^{58} - 48 q^{60} - 240 q^{64} + 96 q^{67} - 32 q^{71} + 48 q^{78} + 48 q^{81} - 96 q^{86} - 48 q^{88} - 32 q^{92} + 96 q^{93} + 16 q^{99}+O(q^{100}) $ 48 * q - 64 * q^4 - 48 * q^9 - 16 * q^11 + 64 * q^16 + 16 * q^22 + 32 * q^23 - 80 * q^25 + 64 * q^36 - 96 * q^37 - 32 * q^44 + 64 * q^53 + 48 * q^58 - 48 * q^60 - 240 * q^64 + 96 * q^67 - 32 * q^71 + 48 * q^78 + 48 * q^81 - 96 * q^86 - 48 * q^88 - 32 * q^92 + 96 * q^93 + 16 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1617\mathbb{Z}\right)^\times$.

$n$	$199$	$442$	$1079$
$\chi(n)$	$-1$	$-1$	$1$

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 $

$2$		−	2.14374i		−	1.51586i	−0.652339	−	0.757928i	$-0.726212\pi$
$2$		−	2.14374i		−	1.51586i	0.652339	−	0.757928i	$-0.273788\pi$
$3$			1.00000i			0.577350i
$3$			1.00000i			0.577350i
$4$	−2.59563			−1.29782
$5$		−	1.55945i		−	0.697408i	−0.937233	−	0.348704i	$-0.886622\pi$
$5$		−	1.55945i		−	0.697408i	0.937233	−	0.348704i	$-0.113378\pi$
$6$	2.14374			0.875179
$7$		0			0
$7$		0			0
$8$			1.27689i			0.451447i
$9$	−1.00000			−0.333333
$10$	−3.34306			−1.05717
$11$	1.68920	+	2.85422i	0.509314	+	0.860581i
$11$	1.68920	+	2.85422i	0.509314	+	0.860581i
$12$		−	2.59563i		−	0.749295i
$13$	4.23323			1.17409			0.587044	−	0.809555i	$-0.300292\pi$
$13$	4.23323			1.17409			0.587044	+	0.809555i	$0.300292\pi$
$14$		0			0
$15$	1.55945			0.402649
$16$	−2.45395			−0.613488
$17$	0.829168			0.201103			0.100551	−	0.994932i	$-0.467939\pi$
$17$	0.829168			0.201103			0.100551	+	0.994932i	$0.467939\pi$
$18$			2.14374i			0.505285i
$19$	4.75731			1.09140			0.545701	−	0.837980i	$-0.316263\pi$
$19$	4.75731			1.09140			0.545701	+	0.837980i	$0.316263\pi$
$20$			4.04777i			0.905108i
$21$		0			0
$22$	6.11872	−	3.62122i	1.30452	−	0.772046i
$23$	−1.60908			−0.335516			−0.167758	−	0.985828i	$-0.553653\pi$
$23$	−1.60908			−0.335516			−0.167758	+	0.985828i	$0.553653\pi$
$24$	−1.27689			−0.260643
$25$	2.56811			0.513622
$26$		−	9.07497i		−	1.77975i
$27$		−	1.00000i		−	0.192450i
$28$		0			0
$29$		−	0.853870i		−	0.158560i	−0.996852	−	0.0792799i	$-0.974738\pi$
$29$		−	0.853870i		−	0.158560i	0.996852	−	0.0792799i	$-0.0252621\pi$
$30$		−	3.34306i		−	0.610357i
$31$		−	9.33663i		−	1.67691i	−0.544972	−	0.838454i	$-0.683460\pi$
$31$		−	9.33663i		−	1.67691i	0.544972	−	0.838454i	$-0.316540\pi$
$32$			7.81442i			1.38141i
$33$	−2.85422	+	1.68920i	−0.496857	+	0.294053i
$34$		−	1.77752i		−	0.304843i
$35$		0			0
$36$	2.59563			0.432606
$37$	3.41775			0.561874			0.280937	−	0.959726i	$-0.409355\pi$
$37$	3.41775			0.561874			0.280937	+	0.959726i	$0.409355\pi$
$38$		−	10.1985i		−	1.65441i
$39$			4.23323i			0.677860i
$40$	1.99124			0.314843
$41$	0.308307			0.0481494			0.0240747	−	0.999710i	$-0.492336\pi$
$41$	0.308307			0.0481494			0.0240747	+	0.999710i	$0.492336\pi$
$42$		0			0
$43$			6.72552i			1.02563i	0.858498	+	0.512816i	$0.171398\pi$
$43$			6.72552i			1.02563i	−0.858498	+	0.512816i	$0.828602\pi$
$44$	−4.38455	−	7.40852i	−0.660996	−	1.11688i
$45$			1.55945i			0.232469i
$46$			3.44945i			0.508593i
$47$		−	5.93972i		−	0.866398i	−0.901298	−	0.433199i	$-0.857385\pi$
$47$		−	5.93972i		−	0.866398i	0.901298	−	0.433199i	$-0.142615\pi$
$48$		−	2.45395i		−	0.354198i
$49$		0			0
$50$		−	5.50537i		−	0.778576i
$51$			0.829168i			0.116107i
$52$	−10.9879			−1.52375
$53$	−4.16866			−0.572610			−0.286305	−	0.958139i	$-0.592427\pi$
$53$	−4.16866			−0.572610			−0.286305	+	0.958139i	$0.592427\pi$
$54$	−2.14374			−0.291726
$55$	4.45102	−	2.63423i	0.600176	−	0.355200i
$56$		0			0
$57$			4.75731i			0.630121i
$58$	−1.83048			−0.240354
$59$		−	8.37253i		−	1.09001i	−0.838433	−	0.545005i	$-0.816528\pi$
$59$		−	8.37253i		−	1.09001i	0.838433	−	0.545005i	$-0.183472\pi$
$60$	−4.04777			−0.522564
$61$	7.95518			1.01856			0.509278	−	0.860602i	$-0.329912\pi$
$61$	7.95518			1.01856			0.509278	+	0.860602i	$0.329912\pi$
$62$	−20.0153			−2.54195
$63$		0			0
$64$	11.8442			1.48052
$65$		−	6.60153i		−	0.818819i
$66$	3.62122	+	6.11872i	0.445741	+	0.753163i
$67$	12.9150			1.57782			0.788909	−	0.614511i	$-0.210647\pi$
$67$	12.9150			1.57782			0.788909	+	0.614511i	$0.210647\pi$
$68$	−2.15222			−0.260994
$69$		−	1.60908i		−	0.193710i
$70$		0			0
$71$	−8.47875			−1.00624			−0.503121	−	0.864216i	$-0.667815\pi$
$71$	−8.47875			−1.00624			−0.503121	+	0.864216i	$0.667815\pi$
$72$		−	1.27689i		−	0.150482i
$73$	−9.63338			−1.12750			−0.563751	−	0.825945i	$-0.690642\pi$
$73$	−9.63338			−1.12750			−0.563751	+	0.825945i	$0.690642\pi$
$74$		−	7.32677i		−	0.851719i
$75$			2.56811i			0.296540i
$76$	−12.3482			−1.41644
$77$		0			0
$78$	9.07497			1.02754
$79$		−	7.07971i		−	0.796530i	−0.917270	−	0.398265i	$-0.869612\pi$
$79$		−	7.07971i		−	0.796530i	0.917270	−	0.398265i	$-0.130388\pi$
$80$			3.82682i			0.427852i
$81$	1.00000			0.111111
$82$		−	0.660930i		−	0.0729875i
$83$	1.82395			0.200205			0.100102	−	0.994977i	$-0.468083\pi$
$83$	1.82395			0.200205			0.100102	+	0.994977i	$0.468083\pi$
$84$		0			0
$85$		−	1.29305i		−	0.140251i
$86$	14.4178			1.55471
$87$	0.853870			0.0915445
$88$	−3.64452	+	2.15692i	−0.388507	+	0.229928i
$89$			1.92740i			0.204304i	0.994769	+	0.102152i	$0.0325728\pi$
$89$			1.92740i			0.204304i	−0.994769	+	0.102152i	$0.967427\pi$
$90$	3.34306			0.352390
$91$		0			0
$92$	4.17658			0.435438
$93$	9.33663			0.968164
$94$	−12.7332			−1.31333
$95$		−	7.41880i		−	0.761153i
$96$	−7.81442			−0.797555
$97$		−	17.0963i		−	1.73586i	−0.496684	−	0.867931i	$-0.665449\pi$
$97$		−	17.0963i		−	1.73586i	0.496684	−	0.867931i	$-0.334551\pi$
$98$		0			0
$99$	−1.68920	−	2.85422i	−0.169771	−	0.286860i
$100$	−6.66587			−0.666587
$101$	3.88039			0.386113			0.193057	−	0.981188i	$-0.438160\pi$
$101$	3.88039			0.386113			0.193057	+	0.981188i	$0.438160\pi$
$102$	1.77752			0.176001
$103$			6.79735i			0.669763i	0.942260	+	0.334881i	$0.108696\pi$
$103$			6.79735i			0.669763i	−0.942260	+	0.334881i	$0.891304\pi$
$104$			5.40536i			0.530039i
$105$		0			0
$106$			8.93654i			0.867993i
$107$		−	12.6312i		−	1.22110i	−0.791977	−	0.610551i	$-0.790948\pi$
$107$		−	12.6312i		−	1.22110i	0.791977	−	0.610551i	$-0.209052\pi$
$108$			2.59563i			0.249765i
$109$			10.7289i			1.02764i	0.857899	+	0.513819i	$0.171770\pi$
$109$			10.7289i			1.02764i	−0.857899	+	0.513819i	$0.828230\pi$
$110$	−5.64712	−	9.54185i	−0.538431	−	0.909780i
$111$			3.41775i			0.324398i
$112$		0			0
$113$	−20.4465			−1.92344			−0.961720	−	0.274033i	$-0.911642\pi$
$113$	−20.4465			−1.92344			−0.961720	+	0.274033i	$0.911642\pi$
$114$	10.1985			0.955173
$115$			2.50928i			0.233992i
$116$			2.21633i			0.205782i
$117$	−4.23323			−0.391363
$118$	−17.9486			−1.65230
$119$		0			0
$120$			1.99124i			0.181775i
$121$	−5.29318	+	9.64273i	−0.481198	+	0.876612i
$122$		−	17.0539i		−	1.54398i
$123$			0.308307i			0.0277991i
$124$			24.2345i			2.17632i
$125$		−	11.8021i		−	1.05561i
$126$		0			0
$127$		−	17.1488i		−	1.52171i	−0.648924	−	0.760853i	$-0.724781\pi$
$127$		−	17.1488i		−	1.52171i	0.648924	−	0.760853i	$-0.275219\pi$
$128$		−	9.76207i		−	0.862853i
$129$	−6.72552			−0.592149
$130$	−14.1520			−1.24121
$131$	11.0205			0.962864			0.481432	−	0.876483i	$-0.340117\pi$
$131$	11.0205			0.962864			0.481432	+	0.876483i	$0.340117\pi$
$132$	7.40852	−	4.38455i	0.644829	−	0.381626i
$133$		0			0
$134$		−	27.6864i		−	2.39174i
$135$	−1.55945			−0.134216
$136$			1.05875i			0.0907872i
$137$	−10.0337			−0.857234			−0.428617	−	0.903486i	$-0.640999\pi$
$137$	−10.0337			−0.857234			−0.428617	+	0.903486i	$0.640999\pi$
$138$	−3.44945			−0.293637
$139$	−10.9473			−0.928540			−0.464270	−	0.885694i	$-0.653683\pi$
$139$	−10.9473			−0.928540			−0.464270	+	0.885694i	$0.653683\pi$
$140$		0			0
$141$	5.93972			0.500215
$142$			18.1763i			1.52532i
$143$	7.15080	+	12.0826i	0.597980	+	1.01040i
$144$	2.45395			0.204496
$145$	−1.33157			−0.110581
$146$			20.6515i			1.70913i
$147$		0			0
$148$	−8.87121			−0.729209
$149$			10.3292i			0.846197i	0.906084	+	0.423099i	$0.139058\pi$
$149$			10.3292i			0.846197i	−0.906084	+	0.423099i	$0.860942\pi$
$150$	5.50537			0.449511
$151$		−	8.39206i		−	0.682936i	−0.939893	−	0.341468i	$-0.889076\pi$
$151$		−	8.39206i		−	0.682936i	0.939893	−	0.341468i	$-0.110924\pi$
$152$			6.07454i			0.492711i
$153$	−0.829168			−0.0670342
$154$		0			0
$155$	−14.5600			−1.16949
$156$		−	10.9879i		−	0.879738i
$157$			12.4498i			0.993606i	0.867863	+	0.496803i	$0.165493\pi$
$157$			12.4498i			0.993606i	−0.867863	+	0.496803i	$0.834507\pi$
$158$	−15.1771			−1.20742
$159$		−	4.16866i		−	0.330596i
$160$	12.1862			0.963404
$161$		0			0
$162$		−	2.14374i		−	0.168428i
$163$	21.2931			1.66780			0.833902	−	0.551913i	$-0.186102\pi$
$163$	21.2931			1.66780			0.833902	+	0.551913i	$0.186102\pi$
$164$	−0.800251			−0.0624891
$165$	2.63423	+	4.45102i	0.205075	+	0.346512i
$166$		−	3.91008i		−	0.303481i
$167$	25.4765			1.97143			0.985715	−	0.168419i	$-0.0538662\pi$
$167$	25.4765			1.97143			0.985715	+	0.168419i	$0.0538662\pi$
$168$		0			0
$169$	4.92028			0.378483
$170$	−2.77196			−0.212600
$171$	−4.75731			−0.363801
$172$		−	17.4570i		−	1.33108i
$173$	7.24911			0.551140			0.275570	−	0.961281i	$-0.411133\pi$
$173$	7.24911			0.551140			0.275570	+	0.961281i	$0.411133\pi$
$174$		−	1.83048i		−	0.138768i
$175$		0			0
$176$	−4.14523	−	7.00413i	−0.312458	−	0.527956i
$177$	8.37253			0.629318
$178$	4.13186			0.309696
$179$	−0.284605			−0.0212724			−0.0106362	−	0.999943i	$-0.503386\pi$
$179$	−0.284605			−0.0212724			−0.0106362	+	0.999943i	$0.503386\pi$
$180$		−	4.04777i		−	0.301703i
$181$			8.32716i			0.618953i	0.950907	+	0.309477i	$0.100154\pi$
$181$			8.32716i			0.618953i	−0.950907	+	0.309477i	$0.899846\pi$
$182$		0			0
$183$			7.95518i			0.588064i
$184$		−	2.05461i		−	0.151468i
$185$		−	5.32981i		−	0.391855i
$186$		−	20.0153i		−	1.46760i
$187$	1.40063	+	2.36663i	0.102424	+	0.173065i
$188$			15.4173i			1.12443i
$189$		0			0
$190$	−15.9040			−1.15380
$191$	5.13110			0.371273			0.185637	−	0.982618i	$-0.440565\pi$
$191$	5.13110			0.371273			0.185637	+	0.982618i	$0.440565\pi$
$192$			11.8442i			0.854781i
$193$			22.1879i			1.59712i	0.601914	+	0.798561i	$0.294405\pi$
$193$			22.1879i			1.59712i	−0.601914	+	0.798561i	$0.705595\pi$
$194$	−36.6500			−2.63132
$195$	6.60153			0.472745
$196$		0			0
$197$		−	21.9462i		−	1.56360i	−0.623526	−	0.781802i	$-0.714301\pi$
$197$		−	21.9462i		−	1.56360i	0.623526	−	0.781802i	$-0.285699\pi$
$198$	−6.11872	+	3.62122i	−0.434839	+	0.257349i
$199$			4.71207i			0.334030i	0.985954	+	0.167015i	$0.0534128\pi$
$199$			4.71207i			0.334030i	−0.985954	+	0.167015i	$0.946587\pi$
$200$			3.27918i			0.231873i
$201$			12.9150i			0.910953i
$202$		−	8.31856i		−	0.585292i
$203$		0			0
$204$		−	2.15222i		−	0.150685i
$205$		−	0.480789i		−	0.0335798i
$206$	14.5718			1.01526
$207$	1.60908			0.111839
$208$	−10.3882			−0.720289
$209$	8.03607	+	13.5784i	0.555867	+	0.939240i
$210$		0			0
$211$			8.05258i			0.554362i	0.960818	+	0.277181i	$0.0894002\pi$
$211$			8.05258i			0.554362i	−0.960818	+	0.277181i	$0.910600\pi$
$212$	10.8203			0.743142
$213$		−	8.47875i		−	0.580954i
$214$	−27.0780			−1.85101
$215$	10.4881			0.715285
$216$	1.27689			0.0868811
$217$		0			0
$218$	22.9999			1.55775
$219$		−	9.63338i		−	0.650963i
$220$	−11.5532	+	6.83750i	−0.778919	+	0.460984i
$221$	3.51006			0.236112
$222$	7.32677			0.491740
$223$		−	13.4410i		−	0.900078i	−0.893009	−	0.450039i	$-0.851410\pi$
$223$		−	13.4410i		−	0.900078i	0.893009	−	0.450039i	$-0.148590\pi$
$224$		0			0
$225$	−2.56811			−0.171207
$226$			43.8319i			2.91566i
$227$	−8.69091			−0.576836			−0.288418	−	0.957505i	$-0.593129\pi$
$227$	−8.69091			−0.576836			−0.288418	+	0.957505i	$0.593129\pi$
$228$		−	12.3482i		−	0.817782i
$229$			18.1025i			1.19625i	0.801403	+	0.598125i	$0.204087\pi$
$229$			18.1025i			1.19625i	−0.801403	+	0.598125i	$0.795913\pi$
$230$	5.37925			0.354697
$231$		0			0
$232$	1.09029			0.0715814
$233$			17.0200i			1.11501i	0.830172	+	0.557507i	$0.188242\pi$
$233$			17.0200i			1.11501i	−0.830172	+	0.557507i	$0.811758\pi$
$234$			9.07497i			0.593249i
$235$	−9.26271			−0.604233
$236$			21.7320i			1.41463i
$237$	7.07971			0.459877
$238$		0			0
$239$			24.1941i			1.56499i	0.622657	+	0.782495i	$0.286053\pi$
$239$			24.1941i			1.56499i	−0.622657	+	0.782495i	$0.713947\pi$
$240$	−3.82682			−0.247020
$241$	10.8139			0.696584			0.348292	−	0.937386i	$-0.386762\pi$
$241$	10.8139			0.696584			0.348292	+	0.937386i	$0.386762\pi$
$242$	20.6715	+	11.3472i	1.32882	+	0.729427i
$243$			1.00000i			0.0641500i
$244$	−20.6487			−1.32190
$245$		0			0
$246$	0.660930			0.0421394
$247$	20.1388			1.28140
$248$	11.9218			0.757036
$249$			1.82395i			0.115588i
$250$	−25.3007			−1.60016
$251$			10.8787i			0.686659i	0.939215	+	0.343330i	$0.111555\pi$
$251$			10.8787i			0.686659i	−0.939215	+	0.343330i	$0.888445\pi$
$252$		0			0
$253$	−2.71806	−	4.59267i	−0.170883	−	0.288738i
$254$	−36.7625			−2.30669
$255$	1.29305			0.0809737
$256$	2.76101			0.172563
$257$		−	7.05171i		−	0.439874i	−0.975514	−	0.219937i	$-0.929415\pi$
$257$		−	7.05171i		−	0.439874i	0.975514	−	0.219937i	$-0.0705851\pi$
$258$			14.4178i			0.897613i
$259$		0			0
$260$			17.1351i			1.06268i
$261$			0.853870i			0.0528532i
$262$		−	23.6251i		−	1.45956i
$263$		−	11.3740i		−	0.701352i	−0.936497	−	0.350676i	$-0.885952\pi$
$263$		−	11.3740i		−	0.701352i	0.936497	−	0.350676i	$-0.114048\pi$
$264$	−2.15692	−	3.64452i	−0.132749	−	0.224305i
$265$			6.50083i			0.399343i
$266$		0			0
$267$	−1.92740			−0.117955
$268$	−33.5226			−2.04772
$269$			3.54905i			0.216389i	0.994130	+	0.108195i	$0.0345070\pi$
$269$			3.54905i			0.216389i	−0.994130	+	0.108195i	$0.965493\pi$
$270$			3.34306i			0.203452i
$271$	19.5850			1.18970			0.594851	−	0.803836i	$-0.297211\pi$
$271$	19.5850			1.18970			0.594851	+	0.803836i	$0.297211\pi$
$272$	−2.03474			−0.123374
$273$		0			0
$274$			21.5096i			1.29944i
$275$	4.33806	+	7.32996i	0.261595	+	0.442013i
$276$			4.17658i			0.251400i
$277$			28.2941i			1.70003i	0.526759	+	0.850015i	$0.323407\pi$
$277$			28.2941i			1.70003i	−0.526759	+	0.850015i	$0.676593\pi$
$278$			23.4683i			1.40753i
$279$			9.33663i			0.558969i
$280$		0			0
$281$			0.263523i			0.0157205i	0.999969	+	0.00786023i	$0.00250202\pi$
$281$			0.263523i			0.0157205i	−0.999969	+	0.00786023i	$0.997498\pi$
$282$		−	12.7332i		−	0.758253i
$283$	26.7033			1.58735			0.793673	−	0.608345i	$-0.208166\pi$
$283$	26.7033			1.58735			0.793673	+	0.608345i	$0.208166\pi$
$284$	22.0077			1.30592
$285$	7.41880			0.439452
$286$	25.9020	−	15.3295i	1.53162	−	0.906450i
$287$		0			0
$288$		−	7.81442i		−	0.460469i
$289$	−16.3125			−0.959558
$290$			2.85454i			0.167625i
$291$	17.0963			1.00220
$292$	25.0047			1.46329
$293$	−14.7712			−0.862941			−0.431470	−	0.902127i	$-0.642005\pi$
$293$	−14.7712			−0.862941			−0.431470	+	0.902127i	$0.642005\pi$
$294$		0			0
$295$	−13.0566			−0.760182
$296$			4.36407i			0.253656i
$297$	2.85422	−	1.68920i	0.165619	−	0.0980175i
$298$	22.1430			1.28271
$299$	−6.81160			−0.393925
$300$		−	6.66587i		−	0.384854i
$301$		0			0
$302$	−17.9904			−1.03523
$303$			3.88039i			0.222923i
$304$	−11.6742			−0.669562
$305$		−	12.4057i		−	0.710350i
$306$			1.77752i			0.101614i
$307$	−20.3711			−1.16264			−0.581320	−	0.813675i	$-0.697464\pi$
$307$	−20.3711			−1.16264			−0.581320	+	0.813675i	$0.697464\pi$
$308$		0			0
$309$	−6.79735			−0.386688
$310$			31.2130i			1.77278i
$311$			30.2672i			1.71629i	0.513403	+	0.858147i	$0.328384\pi$
$311$			30.2672i			1.71629i	−0.513403	+	0.858147i	$0.671616\pi$
$312$	−5.40536			−0.306018
$313$			3.97342i			0.224591i	0.993675	+	0.112296i	$0.0358203\pi$
$313$			3.97342i			0.224591i	−0.993675	+	0.112296i	$0.964180\pi$
$314$	26.6893			1.50616
$315$		0			0
$316$			18.3763i			1.03375i
$317$	31.6997			1.78044			0.890218	−	0.455536i	$-0.150552\pi$
$317$	31.6997			1.78044			0.890218	+	0.455536i	$0.150552\pi$
$318$	−8.93654			−0.501136
$319$	2.43714	−	1.44236i	0.136453	−	0.0807567i
$320$		−	18.4704i		−	1.03253i
$321$	12.6312			0.705004
$322$		0			0
$323$	3.94461			0.219484
$324$	−2.59563			−0.144202
$325$	10.8714			0.603037
$326$		−	45.6469i		−	2.52815i
$327$	−10.7289			−0.593307
$328$			0.393672i			0.0217369i
$329$		0			0
$330$	9.54185	−	5.64712i	0.525262	−	0.310864i
$331$	−14.4947			−0.796699			−0.398349	−	0.917234i	$-0.630417\pi$
$331$	−14.4947			−0.796699			−0.398349	+	0.917234i	$0.630417\pi$
$332$	−4.73431			−0.259829
$333$	−3.41775			−0.187291
$334$		−	54.6151i		−	2.98840i
$335$		−	20.1403i		−	1.10038i
$336$		0			0
$337$			23.1999i			1.26378i	0.775058	+	0.631890i	$0.217720\pi$
$337$			23.1999i			1.26378i	−0.775058	+	0.631890i	$0.782280\pi$
$338$		−	10.5478i		−	0.573725i
$339$		−	20.4465i		−	1.11050i
$340$			3.35628i			0.182020i
$341$	26.6488	−	15.7715i	1.44312	−	0.854073i
$342$			10.1985i			0.551469i
$343$		0			0
$344$	−8.58772			−0.463019
$345$	−2.50928			−0.135095
$346$		−	15.5402i		−	0.835448i
$347$			29.2674i			1.57116i	0.618761	+	0.785579i	$0.287635\pi$
$347$			29.2674i			1.57116i	−0.618761	+	0.785579i	$0.712365\pi$
$348$	−2.21633			−0.118808
$349$	−27.4077			−1.46710			−0.733550	−	0.679635i	$-0.762138\pi$
$349$	−27.4077			−1.46710			−0.733550	+	0.679635i	$0.762138\pi$
$350$		0			0
$351$		−	4.23323i		−	0.225953i
$352$	−22.3041	+	13.2001i	−1.18881	+	0.703570i
$353$		−	2.33213i		−	0.124126i	−0.998072	−	0.0620632i	$-0.980232\pi$
$353$		−	2.33213i		−	0.124126i	0.998072	−	0.0620632i	$-0.0197680\pi$
$354$		−	17.9486i		−	0.953955i
$355$			13.2222i			0.701762i
$356$		−	5.00283i		−	0.265149i
$357$		0			0
$358$			0.610121i			0.0322459i
$359$		−	26.0920i		−	1.37709i	−0.725196	−	0.688543i	$-0.758251\pi$
$359$		−	26.0920i		−	1.37709i	0.725196	−	0.688543i	$-0.241749\pi$
$360$	−1.99124			−0.104948
$361$	3.63202			0.191159
$362$	17.8513			0.938243
$363$	−9.64273	−	5.29318i	−0.506112	−	0.277820i
$364$		0			0
$365$			15.0228i			0.786329i
$366$	17.0539			0.891420
$367$		−	10.3989i		−	0.542818i	−0.962464	−	0.271409i	$-0.912510\pi$
$367$		−	10.3989i		−	0.542818i	0.962464	−	0.271409i	$-0.0874896\pi$
$368$	3.94860			0.205835
$369$	−0.308307			−0.0160498
$370$	−11.4257			−0.593996
$371$		0			0
$372$	−24.2345			−1.25650
$373$		−	13.6459i		−	0.706559i	−0.935518	−	0.353279i	$-0.885066\pi$
$373$		−	13.6459i		−	0.706559i	0.935518	−	0.353279i	$-0.114934\pi$
$374$	5.07345	−	3.00260i	0.262342	−	0.155261i
$375$	11.8021			0.609458
$376$	7.58435			0.391133
$377$		−	3.61463i		−	0.186163i
$378$		0			0
$379$	5.24267			0.269298			0.134649	−	0.990893i	$-0.457009\pi$
$379$	5.24267			0.269298			0.134649	+	0.990893i	$0.457009\pi$
$380$			19.2565i			0.987837i
$381$	17.1488			0.878557
$382$		−	10.9998i		−	0.562797i
$383$		−	29.9485i		−	1.53030i	−0.643854	−	0.765149i	$-0.722666\pi$
$383$		−	29.9485i		−	1.53030i	0.643854	−	0.765149i	$-0.277334\pi$
$384$	9.76207			0.498169
$385$		0			0
$386$	47.5652			2.42101
$387$		−	6.72552i		−	0.341878i
$388$			44.3756i			2.25283i
$389$	−9.99564			−0.506799			−0.253399	−	0.967362i	$-0.581549\pi$
$389$	−9.99564			−0.506799			−0.253399	+	0.967362i	$0.581549\pi$
$390$		−	14.1520i		−	0.716613i
$391$	−1.33420			−0.0674731
$392$		0			0
$393$			11.0205i			0.555910i
$394$	−47.0471			−2.37020
$395$	−11.0405			−0.555506
$396$	4.38455	+	7.40852i	0.220332	+	0.372292i
$397$		−	11.8819i		−	0.596335i	−0.954514	−	0.298167i	$-0.903625\pi$
$397$		−	11.8819i		−	0.596335i	0.954514	−	0.298167i	$-0.0963753\pi$
$398$	10.1015			0.506341
$399$		0			0
$400$	−6.30202			−0.315101
$401$	−36.5693			−1.82618			−0.913092	−	0.407754i	$-0.866312\pi$
$401$	−36.5693			−1.82618			−0.913092	+	0.407754i	$0.866312\pi$
$402$	27.6864			1.38087
$403$		−	39.5242i		−	1.96884i
$404$	−10.0721			−0.501104
$405$		−	1.55945i		−	0.0774898i
$406$		0			0
$407$	5.77327	+	9.75501i	0.286170	+	0.483538i
$408$	−1.05875			−0.0524160
$409$	28.1191			1.39040			0.695200	−	0.718816i	$-0.255316\pi$
$409$	28.1191			1.39040			0.695200	+	0.718816i	$0.255316\pi$
$410$	−1.03069			−0.0509021
$411$		−	10.0337i		−	0.494924i
$412$		−	17.6434i		−	0.869229i
$413$		0			0
$414$		−	3.44945i		−	0.169531i
$415$		−	2.84436i		−	0.139624i
$416$			33.0803i			1.62189i
$417$		−	10.9473i		−	0.536093i
$418$	29.1087	−	17.2273i	1.42375	−	0.842613i
$419$		−	32.7688i		−	1.60086i	−0.599427	−	0.800429i	$-0.704605\pi$
$419$		−	32.7688i		−	1.60086i	0.599427	−	0.800429i	$-0.295395\pi$
$420$		0			0
$421$	−3.80183			−0.185290			−0.0926450	−	0.995699i	$-0.529532\pi$
$421$	−3.80183			−0.185290			−0.0926450	+	0.995699i	$0.529532\pi$
$422$	17.2627			0.840333
$423$			5.93972i			0.288799i
$424$		−	5.32290i		−	0.258503i
$425$	2.12939			0.103291
$426$	−18.1763			−0.880642
$427$		0			0
$428$			32.7859i			1.58477i
$429$	−12.0826	+	7.15080i	−0.583353	+	0.345244i
$430$		−	22.4839i		−	1.08427i
$431$		−	25.8407i		−	1.24470i	−0.782737	−	0.622352i	$-0.786177\pi$
$431$		−	25.8407i		−	1.24470i	0.782737	−	0.622352i	$-0.213823\pi$
$432$			2.45395i			0.118066i
$433$			15.8416i			0.761300i	0.924719	+	0.380650i	$0.124300\pi$
$433$			15.8416i			0.761300i	−0.924719	+	0.380650i	$0.875700\pi$
$434$		0			0
$435$		−	1.33157i		−	0.0638439i
$436$		−	27.8482i		−	1.33369i
$437$	−7.65488			−0.366183
$438$	−20.6515			−0.986766
$439$	−30.1923			−1.44100			−0.720500	−	0.693455i	$-0.756087\pi$
$439$	−30.1923			−1.44100			−0.720500	+	0.693455i	$0.756087\pi$
$440$	3.36361	+	5.68345i	0.160354	+	0.270948i
$441$		0			0
$442$		−	7.52467i		−	0.357912i
$443$	16.7535			0.795981			0.397991	−	0.917389i	$-0.369708\pi$
$443$	16.7535			0.795981			0.397991	+	0.917389i	$0.369708\pi$
$444$		−	8.87121i		−	0.421009i
$445$	3.00569			0.142483
$446$	−28.8141			−1.36439
$447$	−10.3292			−0.488552
$448$		0			0
$449$	30.8320			1.45505			0.727527	−	0.686079i	$-0.240670\pi$
$449$	30.8320			1.45505			0.727527	+	0.686079i	$0.240670\pi$
$450$			5.50537i			0.259525i
$451$	0.520793	+	0.879976i	0.0245232	+	0.0414365i
$452$	53.0715			2.49627
$453$	8.39206			0.394293
$454$			18.6311i			0.874400i
$455$		0			0
$456$	−6.07454			−0.284467
$457$		−	27.2671i		−	1.27550i	−0.770243	−	0.637750i	$-0.779865\pi$
$457$		−	27.2671i		−	1.27550i	0.770243	−	0.637750i	$-0.220135\pi$
$458$	38.8072			1.81334
$459$		−	0.829168i		−	0.0387022i
$460$		−	6.51317i		−	0.303678i
$461$	33.9242			1.58001			0.790004	−	0.613102i	$-0.210079\pi$
$461$	33.9242			1.58001			0.790004	+	0.613102i	$0.210079\pi$
$462$		0			0
$463$	20.8947			0.971058			0.485529	−	0.874221i	$-0.338627\pi$
$463$	20.8947			0.971058			0.485529	+	0.874221i	$0.338627\pi$
$464$			2.09536i			0.0972745i
$465$		−	14.5600i		−	0.675205i
$466$	36.4864			1.69020
$467$			32.8901i			1.52197i	0.648769	+	0.760985i	$0.275284\pi$
$467$			32.8901i			1.52197i	−0.648769	+	0.760985i	$0.724716\pi$
$468$	10.9879			0.507917
$469$		0			0
$470$			19.8569i			0.915929i
$471$	−12.4498			−0.573659
$472$	10.6908			0.492082
$473$	−19.1961	+	11.3608i	−0.882640	+	0.522369i
$474$		−	15.1771i		−	0.697107i
$475$	12.2173			0.560568
$476$		0			0
$477$	4.16866			0.190870
$478$	51.8660			2.37230
$479$	−8.46142			−0.386612			−0.193306	−	0.981139i	$-0.561921\pi$
$479$	−8.46142			−0.386612			−0.193306	+	0.981139i	$0.561921\pi$
$480$			12.1862i			0.556222i
$481$	14.4681			0.659689
$482$		−	23.1822i		−	1.05592i
$483$		0			0
$484$	13.7392	−	25.0290i	0.624507	−	1.13768i
$485$	−26.6608			−1.21060
$486$	2.14374			0.0972422
$487$	7.17070			0.324936			0.162468	−	0.986714i	$-0.448055\pi$
$487$	7.17070			0.324936			0.162468	+	0.986714i	$0.448055\pi$
$488$			10.1579i			0.459825i
$489$			21.2931i			0.962907i
$490$		0			0
$491$			41.4512i			1.87067i	0.353770	+	0.935333i	$0.384900\pi$
$491$			41.4512i			1.87067i	−0.353770	+	0.935333i	$0.615100\pi$
$492$		−	0.800251i		−	0.0360781i
$493$		−	0.708002i		−	0.0318868i
$494$		−	43.1725i		−	1.94242i
$495$	−4.45102	+	2.63423i	−0.200059	+	0.118400i
$496$			22.9117i			1.02876i
$497$		0			0
$498$	3.91008			0.175215
$499$	−8.32544			−0.372698			−0.186349	−	0.982484i	$-0.559665\pi$
$499$	−8.32544			−0.372698			−0.186349	+	0.982484i	$0.559665\pi$
$500$			30.6339i			1.36999i
$501$			25.4765i			1.13821i
$502$	23.3212			1.04088
$503$	−30.2705			−1.34969			−0.674847	−	0.737957i	$-0.735791\pi$
$503$	−30.2705			−1.34969			−0.674847	+	0.737957i	$0.735791\pi$
$504$		0			0
$505$		−	6.05128i		−	0.269278i
$506$	−9.84550	+	5.82682i	−0.437686	+	0.259034i
$507$			4.92028i			0.218517i
$508$			44.5119i			1.97490i
$509$		−	1.58196i		−	0.0701194i	−0.999385	−	0.0350597i	$-0.988838\pi$
$509$		−	1.58196i		−	0.0701194i	0.999385	−	0.0350597i	$-0.0111621\pi$
$510$		−	2.77196i		−	0.122744i
$511$		0			0
$512$		−	25.4430i		−	1.12443i
$513$		−	4.75731i		−	0.210040i
$514$	−15.1171			−0.666785
$515$	10.6001			0.467098
$516$	17.4570			0.768501
$517$	16.9533	−	10.0334i	0.745605	−	0.441269i
$518$		0			0
$519$			7.24911i			0.318201i
$520$	8.42939			0.369653
$521$			21.8500i			0.957268i	0.878015	+	0.478634i	$0.158868\pi$
$521$			21.8500i			0.957268i	−0.878015	+	0.478634i	$0.841132\pi$
$522$	1.83048			0.0801179
$523$	18.0592			0.789672			0.394836	−	0.918752i	$-0.370801\pi$
$523$	18.0592			0.789672			0.394836	+	0.918752i	$0.370801\pi$
$524$	−28.6051			−1.24962
$525$		0			0
$526$	−24.3830			−1.06315
$527$		−	7.74163i		−	0.337231i
$528$	7.00413	−	4.14523i	0.304816	−	0.180398i
$529$	−20.4109			−0.887429
$530$	13.9361			0.605346
$531$			8.37253i			0.363337i
$532$		0			0
$533$	1.30513			0.0565317
$534$			4.13186i			0.178803i
$535$	−19.6977			−0.851607
$536$			16.4910i			0.712301i
$537$		−	0.284605i		−	0.0122816i
$538$	7.60825			0.328015
$539$		0			0
$540$	4.04777			0.174188
$541$			31.7429i			1.36473i	0.731011	+	0.682366i	$0.239049\pi$
$541$			31.7429i			1.36473i	−0.731011	+	0.682366i	$0.760951\pi$
$542$		−	41.9851i		−	1.80342i
$543$	−8.32716			−0.357353
$544$			6.47946i			0.277805i
$545$	16.7311			0.716683
$546$		0			0
$547$		−	11.5925i		−	0.495660i	−0.968804	−	0.247830i	$-0.920283\pi$
$547$		−	11.5925i		−	0.495660i	0.968804	−	0.247830i	$-0.0797174\pi$
$548$	26.0437			1.11253
$549$	−7.95518			−0.339519
$550$	15.7135	−	9.29969i	0.670028	−	0.396540i
$551$		−	4.06213i		−	0.173052i
$552$	2.05461			0.0874499
$553$		0			0
$554$	60.6554			2.57700
$555$	5.32981			0.226238
$556$	28.4153			1.20508
$557$			28.5977i			1.21172i	0.795570	+	0.605862i	$0.207172\pi$
$557$			28.5977i			1.21172i	−0.795570	+	0.605862i	$0.792828\pi$
$558$	20.0153			0.847317
$559$			28.4707i			1.20418i
$560$		0			0
$561$	−2.36663	+	1.40063i	−0.0999192	+	0.0591348i
$562$	0.564926			0.0238299
$563$	−18.8906			−0.796144			−0.398072	−	0.917354i	$-0.630321\pi$
$563$	−18.8906			−0.796144			−0.398072	+	0.917354i	$0.630321\pi$
$564$	−15.4173			−0.649187
$565$			31.8853i			1.34142i
$566$		−	57.2450i		−	2.40619i
$567$		0			0
$568$		−	10.8264i		−	0.454265i
$569$		−	7.33803i		−	0.307626i	−0.988100	−	0.153813i	$-0.950845\pi$
$569$		−	7.33803i		−	0.307626i	0.988100	−	0.153813i	$-0.0491554\pi$
$570$		−	15.9040i		−	0.666145i
$571$			22.8369i			0.955695i	0.878443	+	0.477847i	$0.158583\pi$
$571$			22.8369i			0.955695i	−0.878443	+	0.477847i	$0.841417\pi$
$572$	−18.5608	−	31.3620i	−0.776068	−	1.31131i
$573$			5.13110i			0.214355i
$574$		0			0
$575$	−4.13229			−0.172328
$576$	−11.8442			−0.493508
$577$		−	0.0447835i		−	0.00186436i	−1.00000		0.000932180i	$-0.999703\pi$
$577$		−	0.0447835i		−	0.00186436i	1.00000		0.000932180i	$-0.000296722\pi$
$578$			34.9698i			1.45455i
$579$	−22.1879			−0.922099
$580$	3.45627			0.143514
$581$		0			0
$582$		−	36.6500i		−	1.51919i
$583$	−7.04172	−	11.8983i	−0.291638	−	0.492777i
$584$		−	12.3007i		−	0.509008i
$585$			6.60153i			0.272940i
$586$			31.6656i			1.30809i
$587$		−	8.99242i		−	0.371157i	−0.982629	−	0.185578i	$-0.940584\pi$
$587$		−	8.99242i		−	0.371157i	0.982629	−	0.185578i	$-0.0594159\pi$
$588$		0			0
$589$		−	44.4173i		−	1.83018i
$590$			27.9899i			1.15233i
$591$	21.9462			0.902748
$592$	−8.38699			−0.344703
$593$	−10.1326			−0.416097			−0.208049	−	0.978118i	$-0.566711\pi$
$593$	−10.1326			−0.416097			−0.208049	+	0.978118i	$0.566711\pi$
$594$	−3.62122	−	6.11872i	−0.148580	−	0.251054i
$595$		0			0
$596$		−	26.8107i		−	1.09821i
$597$	−4.71207			−0.192852
$598$			14.6023i			0.597134i
$599$	−32.8384			−1.34174			−0.670871	−	0.741574i	$-0.734079\pi$
$599$	−32.8384			−1.34174			−0.670871	+	0.741574i	$0.734079\pi$
$600$	−3.27918			−0.133872
$601$	−13.8713			−0.565823			−0.282912	−	0.959146i	$-0.591300\pi$
$601$	−13.8713			−0.565823			−0.282912	+	0.959146i	$0.591300\pi$
$602$		0			0
$603$	−12.9150			−0.525939
$604$			21.7827i			0.886326i
$605$	15.0374	+	8.25446i	0.611356	+	0.335592i
$606$	8.31856			0.337918
$607$	−4.94782			−0.200826			−0.100413	−	0.994946i	$-0.532016\pi$
$607$	−4.94782			−0.200826			−0.100413	+	0.994946i	$0.532016\pi$
$608$			37.1756i			1.50767i
$609$		0			0
$610$	−26.5947			−1.07679
$611$		−	25.1442i		−	1.01723i
$612$	2.15222			0.0869981
$613$			2.78451i			0.112465i	0.998418	+	0.0562327i	$0.0179089\pi$
$613$			2.78451i			0.112465i	−0.998418	+	0.0562327i	$0.982091\pi$
$614$			43.6704i			1.76240i
$615$	0.480789			0.0193873
$616$		0			0
$617$	−12.0230			−0.484028			−0.242014	−	0.970273i	$-0.577808\pi$
$617$	−12.0230			−0.484028			−0.242014	+	0.970273i	$0.577808\pi$
$618$			14.5718i			0.586163i
$619$		−	25.9305i		−	1.04224i	−0.853485	−	0.521118i	$-0.825515\pi$
$619$		−	25.9305i		−	1.04224i	0.853485	−	0.521118i	$-0.174485\pi$
$620$	37.7925			1.51778
$621$			1.60908i			0.0645701i
$622$	64.8851			2.60165
$623$		0			0
$624$		−	10.3882i		−	0.415859i
$625$	−5.56427			−0.222571
$626$	8.51799			0.340448
$627$	−13.5784	+	8.03607i	−0.542270	+	0.320930i
$628$		−	32.3152i		−	1.28952i
$629$	2.83388			0.112994
$630$		0			0
$631$	−9.42416			−0.375170			−0.187585	−	0.982248i	$-0.560066\pi$
$631$	−9.42416			−0.375170			−0.187585	+	0.982248i	$0.560066\pi$
$632$	9.03998			0.359591
$633$	−8.05258			−0.320061
$634$		−	67.9561i		−	2.69888i
$635$	−26.7427			−1.06125
$636$			10.8203i			0.429053i
$637$		0			0
$638$	−3.09205	−	5.22459i	−0.122415	−	0.206844i
$639$	8.47875			0.335414
$640$	−15.2235			−0.601761
$641$	−40.1886			−1.58735			−0.793676	−	0.608341i	$-0.791836\pi$
$641$	−40.1886			−1.58735			−0.793676	+	0.608341i	$0.791836\pi$
$642$		−	27.0780i		−	1.06868i
$643$			34.1769i			1.34781i	0.738820	+	0.673903i	$0.235384\pi$
$643$			34.1769i			1.34781i	−0.738820	+	0.673903i	$0.764616\pi$
$644$		0			0
$645$			10.4881i			0.412970i
$646$		−	8.45623i		−	0.332706i
$647$			43.4863i			1.70962i	0.518940	+	0.854811i	$0.326327\pi$
$647$			43.4863i			1.70962i	−0.518940	+	0.854811i	$0.673673\pi$
$648$			1.27689i			0.0501608i
$649$	23.8971	−	14.1429i	0.938042	−	0.555158i
$650$		−	23.3055i		−	0.914117i
$651$		0			0
$652$	−55.2690			−2.16450
$653$	−38.4628			−1.50516			−0.752582	−	0.658498i	$-0.771192\pi$
$653$	−38.4628			−1.50516			−0.752582	+	0.658498i	$0.771192\pi$
$654$			22.9999i			0.899367i
$655$		−	17.1859i		−	0.671509i
$656$	−0.756570			−0.0295391
$657$	9.63338			0.375834
$658$		0			0
$659$			0.500879i			0.0195115i	0.999952	+	0.00975574i	$0.00310540\pi$
$659$			0.500879i			0.0195115i	−0.999952	+	0.00975574i	$0.996895\pi$
$660$	−6.83750	−	11.5532i	−0.266149	−	0.449709i
$661$			3.07700i			0.119681i	0.998208	+	0.0598407i	$0.0190593\pi$
$661$			3.07700i			0.119681i	−0.998208	+	0.0598407i	$0.980941\pi$
$662$			31.0728i			1.20768i
$663$			3.51006i			0.136319i
$664$			2.32898i			0.0903818i
$665$		0			0
$666$			7.32677i			0.283906i
$667$			1.37394i			0.0531993i
$668$	−66.1277			−2.55856
$669$	13.4410			0.519660
$670$	−43.1756			−1.66802
$671$	13.4379	+	22.7059i	0.518765	+	0.876550i
$672$		0			0
$673$			5.04021i			0.194286i	0.995270	+	0.0971429i	$0.0309704\pi$
$673$			5.04021i			0.194286i	−0.995270	+	0.0971429i	$0.969030\pi$
$674$	49.7346			1.91571
$675$		−	2.56811i		−	0.0988466i
$676$	−12.7712			−0.491202
$677$	−9.43692			−0.362690			−0.181345	−	0.983420i	$-0.558045\pi$
$677$	−9.43692			−0.362690			−0.181345	+	0.983420i	$0.558045\pi$
$678$	−43.8319			−1.68336
$679$		0			0
$680$	1.65107			0.0633158
$681$		−	8.69091i		−	0.333036i
$682$	−33.8100	−	57.1282i	−1.29465	−	2.18755i
$683$	−25.7818			−0.986511			−0.493256	−	0.869884i	$-0.664193\pi$
$683$	−25.7818			−0.986511			−0.493256	+	0.869884i	$0.664193\pi$
$684$	12.3482			0.472147
$685$			15.6470i			0.597842i
$686$		0			0
$687$	−18.1025			−0.690655
$688$		−	16.5041i		−	0.629214i
$689$	−17.6469			−0.672294
$690$			5.37925i			0.204785i
$691$			37.6575i			1.43256i	0.697814	+	0.716279i	$0.254156\pi$
$691$			37.6575i			1.43256i	−0.697814	+	0.716279i	$0.745844\pi$
$692$	−18.8160			−0.715279
$693$		0			0
$694$	62.7419			2.38165
$695$			17.0718i			0.647572i
$696$			1.09029i			0.0413275i
$697$	0.255638			0.00968298
$698$			58.7551i			2.22391i
$699$	−17.0200			−0.643754
$700$		0			0
$701$		−	3.99611i		−	0.150931i	−0.997148	−	0.0754655i	$-0.975956\pi$
$701$		−	3.99611i		−	0.150931i	0.997148	−	0.0754655i	$-0.0240443\pi$
$702$	−9.07497			−0.342513
$703$	16.2593			0.613230
$704$	20.0073	+	33.8060i	0.754052	+	1.27411i
$705$		−	9.26271i		−	0.348854i
$706$	−4.99948			−0.188158
$707$		0			0
$708$	−21.7320			−0.816739
$709$	12.1335			0.455681			0.227841	−	0.973698i	$-0.426833\pi$
$709$	12.1335			0.455681			0.227841	+	0.973698i	$0.426833\pi$
$710$	28.3450			1.06377
$711$			7.07971i			0.265510i
$712$	−2.46107			−0.0922326
$713$			15.0234i			0.562629i
$714$		0			0
$715$	18.8422	−	11.1513i	0.704660	−	0.417036i
$716$	0.738732			0.0276077
$717$	−24.1941			−0.903547
$718$	−55.9346			−2.08746
$719$			26.4236i			0.985436i	0.870189	+	0.492718i	$0.163997\pi$
$719$			26.4236i			0.985436i	−0.870189	+	0.492718i	$0.836003\pi$
$720$		−	3.82682i		−	0.142617i
$721$		0			0
$722$		−	7.78611i		−	0.289769i
$723$			10.8139i			0.402173i
$724$		−	21.6143i		−	0.803288i
$725$		−	2.19283i		−	0.0814398i
$726$	−11.3472	+	20.6715i	−0.421135	+	0.767193i
$727$		−	42.8586i		−	1.58954i	−0.606912	−	0.794769i	$-0.707592\pi$
$727$		−	42.8586i		−	1.58954i	0.606912	−	0.794769i	$-0.292408\pi$
$728$		0			0
$729$	−1.00000			−0.0370370
$730$	32.2050			1.19196
$731$			5.57659i			0.206257i
$732$		−	20.6487i		−	0.763199i
$733$	5.89765			0.217834			0.108917	−	0.994051i	$-0.465262\pi$
$733$	5.89765			0.217834			0.108917	+	0.994051i	$0.465262\pi$
$734$	−22.2926			−0.822833
$735$		0			0
$736$		−	12.5740i		−	0.463484i
$737$	21.8160	+	36.8623i	0.803604	+	1.35784i
$738$			0.660930i			0.0243292i
$739$		−	28.6728i		−	1.05475i	−0.849634	−	0.527373i	$-0.823177\pi$
$739$		−	28.6728i		−	1.05475i	0.849634	−	0.527373i	$-0.176823\pi$
$740$			13.8342i			0.508557i
$741$			20.1388i			0.739818i
$742$		0			0
$743$			10.4915i			0.384895i	0.981307	+	0.192448i	$0.0616425\pi$
$743$			10.4915i			0.384895i	−0.981307	+	0.192448i	$0.938357\pi$
$744$			11.9218i			0.437075i
$745$	16.1078			0.590145
$746$	−29.2533			−1.07104
$747$	−1.82395			−0.0667349
$748$	−3.63553	−	6.14290i	−0.132928	−	0.224607i
$749$		0			0
$750$		−	25.3007i		−	0.923850i
$751$	27.0539			0.987210			0.493605	−	0.869686i	$-0.335679\pi$
$751$	27.0539			0.987210			0.493605	+	0.869686i	$0.335679\pi$
$752$			14.5758i			0.531525i
$753$	−10.8787			−0.396443
$754$	−7.74885			−0.282196
$755$	−13.0870			−0.476285
$756$		0			0
$757$	−7.04600			−0.256091			−0.128046	−	0.991768i	$-0.540870\pi$
$757$	−7.04600			−0.256091			−0.128046	+	0.991768i	$0.540870\pi$
$758$		−	11.2389i		−	0.408217i
$759$	4.59267	−	2.71806i	0.166703	−	0.0986593i
$760$	9.47296			0.343620
$761$	6.18771			0.224304			0.112152	−	0.993691i	$-0.464226\pi$
$761$	6.18771			0.224304			0.112152	+	0.993691i	$0.464226\pi$
$762$		−	36.7625i		−	1.33177i
$763$		0			0
$764$	−13.3185			−0.481845
$765$			1.29305i			0.0467502i
$766$	−64.2019			−2.31971
$767$		−	35.4429i		−	1.27977i
$768$			2.76101i			0.0996294i
$769$	−37.5046			−1.35245			−0.676225	−	0.736695i	$-0.736385\pi$
$769$	−37.5046			−1.35245			−0.676225	+	0.736695i	$0.736385\pi$
$770$		0			0
$771$	7.05171			0.253961
$772$		−	57.5917i		−	2.07277i
$773$		−	26.6942i		−	0.960124i	−0.877235	−	0.480062i	$-0.840614\pi$
$773$		−	26.6942i		−	0.960124i	0.877235	−	0.480062i	$-0.159386\pi$
$774$	−14.4178			−0.518237
$775$		−	23.9775i		−	0.861297i
$776$	21.8300			0.783651
$777$		0			0
$778$			21.4281i			0.768234i
$779$	1.46671			0.0525504
$780$	−17.1351			−0.613537
$781$	−14.3223	−	24.2002i	−0.512493	−	0.865953i
$782$			2.86017i			0.102280i
$783$	−0.853870			−0.0305148
$784$		0			0
$785$	19.4149			0.692949
$786$	23.6251			0.842679
$787$	−52.9058			−1.88589			−0.942945	−	0.332949i	$-0.891956\pi$
$787$	−52.9058			−1.88589			−0.942945	+	0.332949i	$0.891956\pi$
$788$			56.9644i			2.02927i
$789$	11.3740			0.404926
$790$			23.6679i			0.842067i
$791$		0			0
$792$	3.64452	−	2.15692i	0.129502	−	0.0766428i
$793$	33.6762			1.19588
$794$	−25.4717			−0.903957
$795$	−6.50083			−0.230561
$796$		−	12.2308i		−	0.433510i
$797$			8.46959i			0.300008i	0.988685	+	0.150004i	$0.0479287\pi$
$797$			8.46959i			0.300008i	−0.988685	+	0.150004i	$0.952071\pi$
$798$		0			0
$799$		−	4.92503i		−	0.174235i
$800$			20.0683i			0.709521i
$801$		−	1.92740i		−	0.0681014i
$802$			78.3952i			2.76823i
$803$	−16.2727	−	27.4958i	−0.574252	−	0.970306i
$804$		−	33.5226i		−	1.18225i
$805$		0			0
$806$	−84.7296			−2.98447
$807$	−3.54905			−0.124932
$808$			4.95481i			0.174310i
$809$		−	10.3210i		−	0.362866i	−0.983403	−	0.181433i	$-0.941927\pi$
$809$		−	10.3210i		−	0.362866i	0.983403	−	0.181433i	$-0.0580735\pi$
$810$	−3.34306			−0.117463
$811$	−39.1772			−1.37570			−0.687849	−	0.725854i	$-0.741445\pi$
$811$	−39.1772			−1.37570			−0.687849	+	0.725854i	$0.741445\pi$
$812$		0			0
$813$			19.5850i			0.686875i
$814$	20.9122	−	12.3764i	0.732973	−	0.433793i
$815$		−	33.2055i		−	1.16314i
$816$		−	2.03474i		−	0.0712301i
$817$			31.9954i			1.11938i
$818$		−	60.2801i		−	2.10764i
$819$		0			0
$820$			1.24795i			0.0435804i
$821$		−	3.25008i		−	0.113429i	−0.998390	−	0.0567143i	$-0.981938\pi$
$821$		−	3.25008i		−	0.113429i	0.998390	−	0.0567143i	$-0.0180624\pi$
$822$	−21.5096			−0.750234
$823$	11.9143			0.415306			0.207653	−	0.978203i	$-0.433418\pi$
$823$	11.9143			0.415306			0.207653	+	0.978203i	$0.433418\pi$
$824$	−8.67944			−0.302363
$825$	−7.32996	+	4.33806i	−0.255196	+	0.151032i
$826$		0			0
$827$			8.56076i			0.297687i	0.988861	+	0.148843i	$0.0475550\pi$
$827$			8.56076i			0.297687i	−0.988861	+	0.148843i	$0.952445\pi$
$828$	−4.17658			−0.145146
$829$			28.5276i			0.990805i	0.868664	+	0.495402i	$0.164979\pi$
$829$			28.5276i			0.990805i	−0.868664	+	0.495402i	$0.835021\pi$
$830$	−6.09758			−0.211650
$831$	−28.2941			−0.981513
$832$	50.1392			1.73827
$833$		0			0
$834$	−23.4683			−0.812639
$835$		−	39.7294i		−	1.37489i
$836$	−20.8587	−	35.2446i	−0.721413	−	1.21896i
$837$	−9.33663			−0.322721
$838$	−70.2478			−2.42667
$839$			8.11044i			0.280003i	0.990151	+	0.140002i	$0.0447108\pi$
$839$			8.11044i			0.280003i	−0.990151	+	0.140002i	$0.955289\pi$
$840$		0			0
$841$	28.2709			0.974859
$842$			8.15016i			0.280873i
$843$	−0.263523			−0.00907621
$844$		−	20.9015i		−	0.719461i
$845$		−	7.67294i		−	0.263957i
$846$	12.7332			0.437778
$847$		0			0
$848$	10.2297			0.351289
$849$			26.7033i			0.916454i
$850$		−	4.56487i		−	0.156574i
$851$	−5.49942			−0.188518
$852$			22.0077i			0.753972i
$853$	48.6237			1.66484			0.832421	−	0.554143i	$-0.186954\pi$
$853$	48.6237			1.66484			0.832421	+	0.554143i	$0.186954\pi$
$854$		0			0
$855$			7.41880i			0.253718i
$856$	16.1286			0.551263
$857$	−47.0669			−1.60778			−0.803888	−	0.594781i	$-0.797239\pi$
$857$	−47.0669			−1.60778			−0.803888	+	0.594781i	$0.797239\pi$
$858$	15.3295	+	25.9020i	0.523339	+	0.884279i
$859$		−	16.5396i		−	0.564322i	−0.959367	−	0.282161i	$-0.908949\pi$
$859$		−	16.5396i		−	0.564322i	0.959367	−	0.282161i	$-0.0910513\pi$
$860$	−27.2233			−0.928308
$861$		0			0
$862$	−55.3959			−1.88679
$863$	8.22143			0.279861			0.139930	−	0.990161i	$-0.455312\pi$
$863$	8.22143			0.279861			0.139930	+	0.990161i	$0.455312\pi$
$864$	7.81442			0.265852
$865$		−	11.3046i		−	0.384369i
$866$	33.9604			1.15402
$867$		−	16.3125i		−	0.554001i
$868$		0			0
$869$	20.2071	−	11.9591i	0.685478	−	0.405684i
$870$	−2.85454			−0.0967781
$871$	54.6722			1.85250
$872$	−13.6995			−0.463924
$873$			17.0963i			0.578621i
$874$			16.4101i			0.555080i
$875$		0			0
$876$			25.0047i			0.844831i
$877$			30.6808i			1.03602i	0.855375	+	0.518009i	$0.173327\pi$
$877$			30.6808i			1.03602i	−0.855375	+	0.518009i	$0.826673\pi$
$878$			64.7245i			2.18435i
$879$		−	14.7712i		−	0.498219i
$880$	−10.9226	+	6.46428i	−0.368201	+	0.217911i
$881$		−	2.60718i		−	0.0878382i	−0.999035	−	0.0439191i	$-0.986016\pi$
$881$		−	2.60718i		−	0.0878382i	0.999035	−	0.0439191i	$-0.0139844\pi$
$882$		0			0
$883$	13.7800			0.463735			0.231868	−	0.972747i	$-0.425516\pi$
$883$	13.7800			0.463735			0.231868	+	0.972747i	$0.425516\pi$
$884$	−9.11083			−0.306430
$885$		−	13.0566i		−	0.438891i
$886$		−	35.9151i		−	1.20659i
$887$	−35.3358			−1.18646			−0.593230	−	0.805033i	$-0.702147\pi$
$887$	−35.3358			−1.18646			−0.593230	+	0.805033i	$0.702147\pi$
$888$	−4.36407			−0.146449
$889$		0			0
$890$		−	6.44343i		−	0.215984i
$891$	1.68920	+	2.85422i	0.0565905	+	0.0956201i
$892$			34.8880i			1.16814i
$893$		−	28.2571i		−	0.945588i
$894$			22.1430i			0.740574i
$895$			0.443829i			0.0148356i
$896$		0			0
$897$		−	6.81160i		−	0.227433i
$898$		−	66.0960i		−	2.20565i
$899$	−7.97227			−0.265890
$900$	6.66587			0.222196
$901$	−3.45652			−0.115153
$902$	1.88644	−	1.11645i	0.0628117	−	0.0371736i
$903$		0			0
$904$		−	26.1078i		−	0.868332i
$905$	12.9858			0.431663
$906$		−	17.9904i		−	0.597691i
$907$	31.0255			1.03018			0.515092	−	0.857135i	$-0.327758\pi$
$907$	31.0255			1.03018			0.515092	+	0.857135i	$0.327758\pi$
$908$	22.5584			0.748627
$909$	−3.88039			−0.128704
$910$		0			0
$911$	−54.7340			−1.81342			−0.906709	−	0.421757i	$-0.861414\pi$
$911$	−54.7340			−1.81342			−0.906709	+	0.421757i	$0.861414\pi$
$912$		−	11.6742i		−	0.386572i
$913$	3.08102	+	5.20596i	0.101967	+	0.172292i
$914$	−58.4536			−1.93347
$915$	12.4057			0.410121
$916$		−	46.9876i		−	1.55251i
$917$		0			0
$918$	−1.77752			−0.0586670
$919$		−	26.2740i		−	0.866699i	−0.901226	−	0.433349i	$-0.857332\pi$
$919$		−	26.2740i		−	0.866699i	0.901226	−	0.433349i	$-0.142668\pi$
$920$	−3.20406			−0.105635
$921$		−	20.3711i		−	0.671251i
$922$		−	72.7247i		−	2.39506i
$923$	−35.8925			−1.18142
$924$		0			0
$925$	8.77714			0.288591
$926$		−	44.7928i		−	1.47198i
$927$		−	6.79735i		−	0.223254i
$928$	6.67250			0.219035
$929$			17.3111i			0.567958i	0.958830	+	0.283979i	$0.0916547\pi$
$929$			17.3111i			0.567958i	−0.958830	+	0.283979i	$0.908345\pi$
$930$	−31.2130			−1.02351
$931$		0			0
$932$		−	44.1776i		−	1.44708i
$933$	−30.2672			−0.990903
$934$	70.5078			2.30709
$935$	3.69065	−	2.18422i	0.120697	−	0.0714316i
$936$		−	5.40536i		−	0.176680i
$937$	−19.7352			−0.644720			−0.322360	−	0.946617i	$-0.604476\pi$
$937$	−19.7352			−0.644720			−0.322360	+	0.946617i	$0.604476\pi$
$938$		0			0
$939$	−3.97342			−0.129668
$940$	24.0426			0.784183
$941$	−41.6578			−1.35801			−0.679003	−	0.734136i	$-0.737588\pi$
$941$	−41.6578			−1.35801			−0.679003	+	0.734136i	$0.737588\pi$
$942$			26.6893i			0.869583i
$943$	−0.496089			−0.0161549
$944$			20.5458i			0.668709i
$945$		0			0
$946$	24.3546	+	41.1516i	0.791836	+	1.33795i
$947$	11.1262			0.361553			0.180776	−	0.983524i	$-0.442139\pi$
$947$	11.1262			0.361553			0.180776	+	0.983524i	$0.442139\pi$
$948$	−18.3763			−0.596836
$949$	−40.7804			−1.32379
$950$		−	26.1907i		−	0.849740i
$951$			31.6997i			1.02793i
$952$		0			0
$953$		−	22.5363i		−	0.730023i	−0.931003	−	0.365011i	$-0.881065\pi$
$953$		−	22.5363i		−	0.730023i	0.931003	−	0.365011i	$-0.118935\pi$
$954$		−	8.93654i		−	0.289331i
$955$		−	8.00171i		−	0.258929i
$956$		−	62.7991i		−	2.03107i
$957$	1.44236	+	2.43714i	0.0466249	+	0.0787814i
$958$			18.1391i			0.586048i
$959$		0			0
$960$	18.4704			0.596131
$961$	−56.1727			−1.81202
$962$		−	31.0159i		−	0.999994i
$963$			12.6312i			0.407034i
$964$	−28.0689			−0.904038
$965$	34.6010			1.11385
$966$		0			0
$967$			4.33211i			0.139311i	0.997571	+	0.0696557i	$0.0221901\pi$
$967$			4.33211i			0.139311i	−0.997571	+	0.0696557i	$0.977810\pi$
$968$	−12.3127	−	6.75879i	−0.395744	−	0.217236i
$969$			3.94461i			0.126719i
$970$			57.1539i			1.83510i
$971$		−	31.7702i		−	1.01955i	−0.860307	−	0.509777i	$-0.829728\pi$
$971$		−	31.7702i		−	1.01955i	0.860307	−	0.509777i	$-0.170272\pi$
$972$		−	2.59563i		−	0.0832550i
$973$		0			0
$974$		−	15.3721i		−	0.492555i
$975$			10.8714i			0.348164i
$976$	−19.5216			−0.624873
$977$	40.4975			1.29563			0.647814	−	0.761798i	$-0.275683\pi$
$977$	40.4975			1.29563			0.647814	+	0.761798i	$0.275683\pi$
$978$	45.6469			1.45963
$979$	−5.50124	+	3.25578i	−0.175820	+	0.104055i
$980$		0			0
$981$		−	10.7289i		−	0.342546i
$982$	88.8607			2.83566
$983$		−	35.6734i		−	1.13781i	−0.822405	−	0.568903i	$-0.807368\pi$
$983$		−	35.6734i		−	1.13781i	0.822405	−	0.568903i	$-0.192632\pi$
$984$	−0.393672			−0.0125498
$985$	−34.2241			−1.09047
$986$	−1.51777			−0.0483358
$987$		0			0
$988$	−52.2730			−1.66303
$989$		−	10.8219i		−	0.344116i
$990$	5.64712	+	9.54185i	0.179477	+	0.303260i
$991$	62.3315			1.98003			0.990013	−	0.140973i	$-0.0450232\pi$
$991$	62.3315			1.98003			0.990013	+	0.140973i	$0.0450232\pi$
$992$	72.9603			2.31649
$993$		−	14.4947i		−	0.459974i
$994$		0			0
$995$	7.34825			0.232955
$996$		−	4.73431i		−	0.150012i
$997$	14.7933			0.468508			0.234254	−	0.972175i	$-0.424735\pi$
$997$	14.7933			0.468508			0.234254	+	0.972175i	$0.424735\pi$
$998$			17.8476i			0.564956i
$999$		−	3.41775i		−	0.108133i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1617.2.c.b.538.8	yes	48
7.6	odd	2	inner	1617.2.c.b.538.7	✓	48
11.10	odd	2	inner	1617.2.c.b.538.42	yes	48
77.76	even	2	inner	1617.2.c.b.538.41	yes	48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1617.2.c.b.538.7	✓	48	7.6	odd	2	inner
1617.2.c.b.538.8	yes	48	1.1	even	1	trivial
1617.2.c.b.538.41	yes	48	77.76	even	2	inner
1617.2.c.b.538.42	yes	48	11.10	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:	\( N \)	\(=\)	\( 1617 = 3 \cdot 7^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1617.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-2.14374i q^{2} +1.00000i q^{3} -2.59563 q^{4} -1.55945i q^{5} +2.14374 q^{6} +1.27689i q^{8} -1.00000 q^{9} +O(q^{10})\)	\(q-2.14374i q^{2} +1.00000i q^{3} -2.59563 q^{4} -1.55945i q^{5} +2.14374 q^{6} +1.27689i q^{8} -1.00000 q^{9} -3.34306 q^{10} +(1.68920 + 2.85422i) q^{11} -2.59563i q^{12} +4.23323 q^{13} +1.55945 q^{15} -2.45395 q^{16} +0.829168 q^{17} +2.14374i q^{18} +4.75731 q^{19} +4.04777i q^{20} +(6.11872 - 3.62122i) q^{22} -1.60908 q^{23} -1.27689 q^{24} +2.56811 q^{25} -9.07497i q^{26} -1.00000i q^{27} -0.853870i q^{29} -3.34306i q^{30} -9.33663i q^{31} +7.81442i q^{32} +(-2.85422 + 1.68920i) q^{33} -1.77752i q^{34} +2.59563 q^{36} +3.41775 q^{37} -10.1985i q^{38} +4.23323i q^{39} +1.99124 q^{40} +0.308307 q^{41} +6.72552i q^{43} +(-4.38455 - 7.40852i) q^{44} +1.55945i q^{45} +3.44945i q^{46} -5.93972i q^{47} -2.45395i q^{48} -5.50537i q^{50} +0.829168i q^{51} -10.9879 q^{52} -4.16866 q^{53} -2.14374 q^{54} +(4.45102 - 2.63423i) q^{55} +4.75731i q^{57} -1.83048 q^{58} -8.37253i q^{59} -4.04777 q^{60} +7.95518 q^{61} -20.0153 q^{62} +11.8442 q^{64} -6.60153i q^{65} +(3.62122 + 6.11872i) q^{66} +12.9150 q^{67} -2.15222 q^{68} -1.60908i q^{69} -8.47875 q^{71} -1.27689i q^{72} -9.63338 q^{73} -7.32677i q^{74} +2.56811i q^{75} -12.3482 q^{76} +9.07497 q^{78} -7.07971i q^{79} +3.82682i q^{80} +1.00000 q^{81} -0.660930i q^{82} +1.82395 q^{83} -1.29305i q^{85} +14.4178 q^{86} +0.853870 q^{87} +(-3.64452 + 2.15692i) q^{88} +1.92740i q^{89} +3.34306 q^{90} +4.17658 q^{92} +9.33663 q^{93} -12.7332 q^{94} -7.41880i q^{95} -7.81442 q^{96} -17.0963i q^{97} +(-1.68920 - 2.85422i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q - 64 q^{4} - 48 q^{9}+O(q^{10}) \) 48 * q - 64 * q^4 - 48 * q^9	\( 48 q - 64 q^{4} - 48 q^{9} - 16 q^{11} + 64 q^{16} + 16 q^{22} + 32 q^{23} - 80 q^{25} + 64 q^{36} - 96 q^{37} - 32 q^{44} + 64 q^{53} + 48 q^{58} - 48 q^{60} - 240 q^{64} + 96 q^{67} - 32 q^{71} + 48 q^{78} + 48 q^{81} - 96 q^{86} - 48 q^{88} - 32 q^{92} + 96 q^{93} + 16 q^{99}+O(q^{100}) \) 48 * q - 64 * q^4 - 48 * q^9 - 16 * q^11 + 64 * q^16 + 16 * q^22 + 32 * q^23 - 80 * q^25 + 64 * q^36 - 96 * q^37 - 32 * q^44 + 64 * q^53 + 48 * q^58 - 48 * q^60 - 240 * q^64 + 96 * q^67 - 32 * q^71 + 48 * q^78 + 48 * q^81 - 96 * q^86 - 48 * q^88 - 32 * q^92 + 96 * q^93 + 16 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more