LMFDB - Embedded newform 475.2.b.e.324.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [475,2,Mod(324,475)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(475, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("475.324");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 475 = 5^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	475.b (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:	$3.79289409601$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.2058981376.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 2x^{7} + 2x^{6} + 18x^{4} - 34x^{3} + 32x^{2} - 8x + 1 $ x^8 - 2x^7 + 2x^6 + 18x^4 - 34x^3 + 32x^2 - 8x + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	no (minimal twist has level 95)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			324.2
Root			$-1.43917 + 1.43917i$ of defining polynomial
Character	$\chi$	$=$	475.324
Dual form			475.2.b.e.324.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.14243i q^{2} +2.87834i q^{3} -2.59002 q^{4} +6.16666 q^{6} +3.10482i q^{7} +1.26409i q^{8} -5.28487 q^{9} +O(q^{10})$	\(q-2.14243i q^{2} +2.87834i q^{3} -2.59002 q^{4} +6.16666 q^{6} +3.10482i q^{7} +1.26409i q^{8} -5.28487 q^{9} -1.10482 q^{11} -7.45498i q^{12} -1.77353i q^{13} +6.65187 q^{14} -2.47182 q^{16} +7.75669i q^{17} +11.3225i q^{18} -1.00000 q^{19} -8.93674 q^{21} +2.36700i q^{22} +6.65187i q^{23} -3.63849 q^{24} -3.79966 q^{26} -6.57664i q^{27} -8.04156i q^{28} -7.75669 q^{29} +6.57664 q^{31} +7.82389i q^{32} -3.18005i q^{33} +16.6182 q^{34} +13.6879 q^{36} -1.40652i q^{37} +2.14243i q^{38} +5.10482 q^{39} +2.81995 q^{41} +19.1464i q^{42} -3.10482i q^{43} +2.86151 q^{44} +14.2512 q^{46} +1.46492i q^{47} -7.11475i q^{48} -2.63990 q^{49} -22.3264 q^{51} +4.59348i q^{52} +2.59348i q^{53} -14.0900 q^{54} -3.92477 q^{56} -2.87834i q^{57} +16.6182i q^{58} +5.38969 q^{59} +4.07523 q^{61} -14.0900i q^{62} -16.4086i q^{63} +11.8185 q^{64} -6.81305 q^{66} -15.8151i q^{67} -20.0900i q^{68} -19.1464 q^{69} +7.14638 q^{71} -6.68055i q^{72} -0.243310i q^{73} -3.01339 q^{74} +2.59002 q^{76} -3.43026i q^{77} -10.9367i q^{78} +9.38969 q^{79} +3.07523 q^{81} -6.04156i q^{82} -8.86151i q^{83} +23.1464 q^{84} -6.65187 q^{86} -22.3264i q^{87} -1.39659i q^{88} -0.813048 q^{89} +5.50648 q^{91} -17.2285i q^{92} +18.9298i q^{93} +3.13849 q^{94} -22.5199 q^{96} +3.19689i q^{97} +5.65581i q^{98} +5.83882 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 16 q^{4} - 16 q^{9}+O(q^{10}) $ 8 * q - 16 * q^4 - 16 * q^9	$ 8 q - 16 q^{4} - 16 q^{9} + 8 q^{11} + 16 q^{14} + 8 q^{16} - 8 q^{19} - 8 q^{21} + 48 q^{24} + 8 q^{26} - 8 q^{29} + 8 q^{31} + 8 q^{34} + 80 q^{36} + 24 q^{39} + 32 q^{41} - 48 q^{44} - 40 q^{49} - 72 q^{51} + 40 q^{54} - 24 q^{56} + 40 q^{61} + 8 q^{64} - 56 q^{66} - 56 q^{69} - 40 q^{71} - 64 q^{74} + 16 q^{76} + 32 q^{79} + 32 q^{81} + 88 q^{84} - 16 q^{86} - 8 q^{89} - 72 q^{91} + 96 q^{94} - 104 q^{96} + 8 q^{99}+O(q^{100}) $ 8 * q - 16 * q^4 - 16 * q^9 + 8 * q^11 + 16 * q^14 + 8 * q^16 - 8 * q^19 - 8 * q^21 + 48 * q^24 + 8 * q^26 - 8 * q^29 + 8 * q^31 + 8 * q^34 + 80 * q^36 + 24 * q^39 + 32 * q^41 - 48 * q^44 - 40 * q^49 - 72 * q^51 + 40 * q^54 - 24 * q^56 + 40 * q^61 + 8 * q^64 - 56 * q^66 - 56 * q^69 - 40 * q^71 - 64 * q^74 + 16 * q^76 + 32 * q^79 + 32 * q^81 + 88 * q^84 - 16 * q^86 - 8 * q^89 - 72 * q^91 + 96 * q^94 - 104 * q^96 + 8 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/475\mathbb{Z}\right)^\times$.

$n$	$77$	$401$
$\chi(n)$	$-1$	$1$

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 $

$2$	− 2.14243i	− 1.51493i	−0.652876	−	0.757465i	$-0.726438\pi$
$2$	− 2.14243i	− 1.51493i	0.652876	−	0.757465i	$-0.273562\pi$
$3$	2.87834i	1.66181i	0.556412	+	0.830907i	$0.312178\pi$
$3$	2.87834i	1.66181i	−0.556412	+	0.830907i	$0.687822\pi$
$4$	−2.59002	−1.29501
$5$	0	0
$5$	0	0
$6$	6.16666	2.51753
$7$	3.10482i	1.17351i	0.809764	+	0.586756i	$0.199595\pi$
$7$	3.10482i	1.17351i	−0.809764	+	0.586756i	$0.800405\pi$
$8$	1.26409i	0.446923i
$9$	−5.28487	−1.76162
$10$	0	0
$11$	−1.10482	−0.333115	−0.166558	−	0.986032i	$-0.553265\pi$
$11$	−1.10482	−0.333115	−0.166558	+	0.986032i	$0.553265\pi$
$12$	− 7.45498i	− 2.15207i
$13$	− 1.77353i	− 0.491888i	−0.969284	−	0.245944i	$-0.920902\pi$
$13$	− 1.77353i	− 0.491888i	0.969284	−	0.245944i	$-0.0790979\pi$
$14$	6.65187	1.77779
$15$	0	0
$16$	−2.47182	−0.617955
$17$	7.75669i	1.88127i	0.339415	+	0.940637i	$0.389771\pi$
$17$	7.75669i	1.88127i	−0.339415	+	0.940637i	$0.610229\pi$
$18$	11.3225i	2.66874i
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	−8.93674	−1.95016
$22$	2.36700i	0.504647i
$23$	6.65187i	1.38701i	0.720451	+	0.693505i	$0.243935\pi$
$23$	6.65187i	1.38701i	−0.720451	+	0.693505i	$0.756065\pi$
$24$	−3.63849	−0.742703
$25$	0	0
$26$	−3.79966	−0.745175
$27$	− 6.57664i	− 1.26567i
$28$	− 8.04156i	− 1.51971i
$29$	−7.75669	−1.44038	−0.720191	−	0.693776i	$-0.755946\pi$
$29$	−7.75669	−1.44038	−0.720191	+	0.693776i	$0.755946\pi$
$30$	0	0
$31$	6.57664	1.18120	0.590600	−	0.806965i	$-0.298891\pi$
$31$	6.57664	1.18120	0.590600	+	0.806965i	$0.298891\pi$
$32$	7.82389i	1.38308i
$33$	− 3.18005i	− 0.553576i
$34$	16.6182	2.85000
$35$	0	0
$36$	13.6879	2.28132
$37$	− 1.40652i	− 0.231231i	−0.993294	−	0.115616i	$-0.963116\pi$
$37$	− 1.40652i	− 0.231231i	0.993294	−	0.115616i	$-0.0368841\pi$
$38$	2.14243i	0.347549i
$39$	5.10482	0.817425
$40$	0	0
$41$	2.81995	0.440402	0.220201	−	0.975454i	$-0.429329\pi$
$41$	2.81995	0.440402	0.220201	+	0.975454i	$0.429329\pi$
$42$	19.1464i	2.95435i
$43$	− 3.10482i	− 0.473480i	−0.971573	−	0.236740i	$-0.923921\pi$
$43$	− 3.10482i	− 0.473480i	0.971573	−	0.236740i	$-0.0760790\pi$
$44$	2.86151	0.431389
$45$	0	0
$46$	14.2512	2.10122
$47$	1.46492i	0.213680i	0.994276	+	0.106840i	$0.0340733\pi$
$47$	1.46492i	0.213680i	−0.994276	+	0.106840i	$0.965927\pi$
$48$	− 7.11475i	− 1.02693i
$49$	−2.63990	−0.377129
$50$	0	0
$51$	−22.3264	−3.12633
$52$	4.59348i	0.637001i
$53$	2.59348i	0.356241i	0.984009	+	0.178121i	$0.0570017\pi$
$53$	2.59348i	0.356241i	−0.984009	+	0.178121i	$0.942998\pi$
$54$	−14.0900	−1.91741
$55$	0	0
$56$	−3.92477	−0.524469
$57$	− 2.87834i	− 0.381246i
$58$	16.6182i	2.18208i
$59$	5.38969	0.701678	0.350839	−	0.936436i	$-0.385897\pi$
$59$	5.38969	0.701678	0.350839	+	0.936436i	$0.385897\pi$
$60$	0	0
$61$	4.07523	0.521780	0.260890	−	0.965369i	$-0.415984\pi$
$61$	4.07523	0.521780	0.260890	+	0.965369i	$0.415984\pi$
$62$	− 14.0900i	− 1.78943i
$63$	− 16.4086i	− 2.06728i
$64$	11.8185	1.47732
$65$	0	0
$66$	−6.81305	−0.838628
$67$	− 15.8151i	− 1.93212i	−0.258318	−	0.966060i	$-0.583168\pi$
$67$	− 15.8151i	− 1.93212i	0.258318	−	0.966060i	$-0.416832\pi$
$68$	− 20.0900i	− 2.43627i
$69$	−19.1464	−2.30495
$70$	0	0
$71$	7.14638	0.848119	0.424059	−	0.905634i	$-0.360605\pi$
$71$	7.14638	0.848119	0.424059	+	0.905634i	$0.360605\pi$
$72$	− 6.68055i	− 0.787310i
$73$	− 0.243310i	− 0.0284773i	−0.999899	−	0.0142387i	$-0.995468\pi$
$73$	− 0.243310i	− 0.0284773i	0.999899	−	0.0142387i	$-0.00453246\pi$
$74$	−3.01339	−0.350299
$75$	0	0
$76$	2.59002	0.297096
$77$	− 3.43026i	− 0.390915i
$78$	− 10.9367i	− 1.23834i
$79$	9.38969	1.05642	0.528211	−	0.849113i	$-0.322863\pi$
$79$	9.38969	1.05642	0.528211	+	0.849113i	$0.322863\pi$
$80$	0	0
$81$	3.07523	0.341692
$82$	− 6.04156i	− 0.667178i
$83$	− 8.86151i	− 0.972677i	−0.873771	−	0.486338i	$-0.838332\pi$
$83$	− 8.86151i	− 0.972677i	0.873771	−	0.486338i	$-0.161668\pi$
$84$	23.1464	2.52548
$85$	0	0
$86$	−6.65187	−0.717290
$87$	− 22.3264i	− 2.39364i
$88$	− 1.39659i	− 0.148877i
$89$	−0.813048	−0.0861829	−0.0430914	−	0.999071i	$-0.513721\pi$
$89$	−0.813048	−0.0861829	−0.0430914	+	0.999071i	$0.513721\pi$
$90$	0	0
$91$	5.50648	0.577236
$92$	− 17.2285i	− 1.79620i
$93$	18.9298i	1.96293i
$94$	3.13849	0.323711
$95$	0	0
$96$	−22.5199	−2.29842
$97$	3.19689i	0.324595i	0.986742	+	0.162297i	$0.0518904\pi$
$97$	3.19689i	0.324595i	−0.986742	+	0.162297i	$0.948110\pi$
$98$	5.65581i	0.571323i
$99$	5.83882	0.586824
$100$	0	0
$101$	−3.01887	−0.300389	−0.150195	−	0.988656i	$-0.547990\pi$
$101$	−3.01887	−0.300389	−0.150195	+	0.988656i	$0.547990\pi$
$102$	47.8329i	4.73616i
$103$	− 6.05839i	− 0.596951i	−0.954417	−	0.298476i	$-0.903522\pi$
$103$	− 6.05839i	− 0.596951i	0.954417	−	0.298476i	$-0.0964782\pi$
$104$	2.24190	0.219836
$105$	0	0
$106$	5.55635	0.539681
$107$	20.3848i	1.97068i	0.170616	+	0.985338i	$0.445424\pi$
$107$	20.3848i	1.97068i	−0.170616	+	0.985338i	$0.554576\pi$
$108$	17.0337i	1.63906i
$109$	−4.72710	−0.452774	−0.226387	−	0.974037i	$-0.572691\pi$
$109$	−4.72710	−0.452774	−0.226387	+	0.974037i	$0.572691\pi$
$110$	0	0
$111$	4.04846	0.384263
$112$	− 7.67456i	− 0.725177i
$113$	7.98316i	0.750993i	0.926824	+	0.375496i	$0.122528\pi$
$113$	7.98316i	0.750993i	−0.926824	+	0.375496i	$0.877472\pi$
$114$	−6.16666	−0.577561
$115$	0	0
$116$	20.0900	1.86531
$117$	9.37285i	0.866520i
$118$	− 11.5471i	− 1.06299i
$119$	−24.0831	−2.20770
$120$	0	0
$121$	−9.77938	−0.889034
$122$	− 8.73091i	− 0.790460i
$123$	8.11679i	0.731866i
$124$	−17.0337	−1.52967
$125$	0	0
$126$	−35.1543	−3.13179
$127$	4.63503i	0.411293i	0.978626	+	0.205646i	$0.0659297\pi$
$127$	4.63503i	0.411293i	−0.978626	+	0.205646i	$0.934070\pi$
$128$	− 9.67265i	− 0.854950i
$129$	8.93674	0.786836
$130$	0	0
$131$	15.5134	1.35541	0.677705	−	0.735334i	$-0.262975\pi$
$131$	15.5134	1.35541	0.677705	+	0.735334i	$0.262975\pi$
$132$	8.23641i	0.716887i
$133$	− 3.10482i	− 0.269222i
$134$	−33.8828	−2.92703
$135$	0	0
$136$	−9.80515	−0.840785
$137$	− 0.813048i	− 0.0694634i	−0.999397	−	0.0347317i	$-0.988942\pi$
$137$	− 0.813048i	− 0.0694634i	0.999397	−	0.0347317i	$-0.0110577\pi$
$138$	41.0199i	3.49184i
$139$	12.7687	1.08302	0.541512	−	0.840693i	$-0.317852\pi$
$139$	12.7687	1.08302	0.541512	+	0.840693i	$0.317852\pi$
$140$	0	0
$141$	−4.21654	−0.355097
$142$	− 15.3106i	− 1.28484i
$143$	1.95942i	0.163855i
$144$	13.0632	1.08860
$145$	0	0
$146$	−0.521277	−0.0431412
$147$	− 7.59854i	− 0.626717i
$148$	3.64293i	0.299447i
$149$	13.7982	1.13040	0.565198	−	0.824955i	$-0.308800\pi$
$149$	13.7982	1.13040	0.565198	+	0.824955i	$0.308800\pi$
$150$	0	0
$151$	5.02269	0.408740	0.204370	−	0.978894i	$-0.434485\pi$
$151$	5.02269	0.408740	0.204370	+	0.978894i	$0.434485\pi$
$152$	− 1.26409i	− 0.102531i
$153$	− 40.9931i	− 3.31409i
$154$	−7.34911	−0.592208
$155$	0	0
$156$	−13.2216	−1.05858
$157$	7.18695i	0.573581i	0.957993	+	0.286791i	$0.0925884\pi$
$157$	7.18695i	0.573581i	−0.957993	+	0.286791i	$0.907412\pi$
$158$	− 20.1168i	− 1.60041i
$159$	−7.46492	−0.592007
$160$	0	0
$161$	−20.6529	−1.62767
$162$	− 6.58848i	− 0.517640i
$163$	− 7.25528i	− 0.568277i	−0.958783	−	0.284139i	$-0.908292\pi$
$163$	− 7.25528i	− 0.568277i	0.958783	−	0.284139i	$-0.0917076\pi$
$164$	−7.30374	−0.570326
$165$	0	0
$166$	−18.9852	−1.47354
$167$	− 7.33129i	− 0.567312i	−0.958926	−	0.283656i	$-0.908453\pi$
$167$	− 7.33129i	− 0.567312i	0.958926	−	0.283656i	$-0.0915475\pi$
$168$	− 11.2968i	− 0.871570i
$169$	9.85461	0.758047
$170$	0	0
$171$	5.28487	0.404144
$172$	8.04156i	0.613163i
$173$	5.77353i	0.438953i	0.975618	+	0.219477i	$0.0704349\pi$
$173$	5.77353i	0.438953i	−0.975618	+	0.219477i	$0.929565\pi$
$174$	−47.8329	−3.62620
$175$	0	0
$176$	2.73091	0.205850
$177$	15.5134i	1.16606i
$178$	1.74190i	0.130561i
$179$	4.48379	0.335134	0.167567	−	0.985861i	$-0.446409\pi$
$179$	4.48379	0.335134	0.167567	+	0.985861i	$0.446409\pi$
$180$	0	0
$181$	−2.73400	−0.203217	−0.101608	−	0.994824i	$-0.532399\pi$
$181$	−2.73400	−0.203217	−0.101608	+	0.994824i	$0.532399\pi$
$182$	− 11.7973i	− 0.874471i
$183$	11.7299i	0.867101i
$184$	−8.40856	−0.619887
$185$	0	0
$186$	40.5559	2.97371
$187$	− 8.56974i	− 0.626681i
$188$	− 3.79418i	− 0.276719i
$189$	20.4193	1.48528
$190$	0	0
$191$	−17.7230	−1.28239	−0.641196	−	0.767377i	$-0.721562\pi$
$191$	−17.7230	−1.28239	−0.641196	+	0.767377i	$0.721562\pi$
$192$	34.0178i	2.45502i
$193$	13.5371i	0.974423i	0.873284	+	0.487212i	$0.161986\pi$
$193$	13.5371i	0.974423i	−0.873284	+	0.487212i	$0.838014\pi$
$194$	6.84912	0.491738
$195$	0	0
$196$	6.83741	0.488386
$197$	− 21.5134i	− 1.53276i	−0.642385	−	0.766382i	$-0.722055\pi$
$197$	− 21.5134i	− 1.53276i	0.642385	−	0.766382i	$-0.277945\pi$
$198$	− 12.5093i	− 0.888997i
$199$	−7.09410	−0.502888	−0.251444	−	0.967872i	$-0.580905\pi$
$199$	−7.09410	−0.502888	−0.251444	+	0.967872i	$0.580905\pi$
$200$	0	0
$201$	45.5213	3.21082
$202$	6.46774i	0.455068i
$203$	− 24.0831i	− 1.69030i
$204$	57.8260	4.04863
$205$	0	0
$206$	−12.9797	−0.904339
$207$	− 35.1543i	− 2.44339i
$208$	4.38384i	0.303964i
$209$	1.10482	0.0764219
$210$	0	0
$211$	11.0604	0.761431	0.380716	−	0.924692i	$-0.375678\pi$
$211$	11.0604	0.761431	0.380716	+	0.924692i	$0.375678\pi$
$212$	− 6.71717i	− 0.461337i
$213$	20.5697i	1.40942i
$214$	43.6731	2.98543
$215$	0	0
$216$	8.31346	0.565659
$217$	20.4193i	1.38615i
$218$	10.1275i	0.685921i
$219$	0.700331	0.0473240
$220$	0	0
$221$	13.7567	0.925375
$222$	− 8.67356i	− 0.582131i
$223$	12.6350i	0.846104i	0.906105	+	0.423052i	$0.139041\pi$
$223$	12.6350i	0.846104i	−0.906105	+	0.423052i	$0.860959\pi$
$224$	−24.2918	−1.62306
$225$	0	0
$226$	17.1034	1.13770
$227$	− 6.62813i	− 0.439925i	−0.975508	−	0.219962i	$-0.929407\pi$
$227$	− 6.62813i	− 0.439925i	0.975508	−	0.219962i	$-0.0705934\pi$
$228$	7.45498i	0.493718i
$229$	0.0752308	0.00497139	0.00248570	−	0.999997i	$-0.499209\pi$
$229$	0.0752308	0.00497139	0.00248570	+	0.999997i	$0.499209\pi$
$230$	0	0
$231$	9.87348	0.649627
$232$	− 9.80515i	− 0.643740i
$233$	− 9.51338i	− 0.623242i	−0.950206	−	0.311621i	$-0.899128\pi$
$233$	− 9.51338i	− 0.623242i	0.950206	−	0.311621i	$-0.100872\pi$
$234$	20.0807	1.31272
$235$	0	0
$236$	−13.9594	−0.908681
$237$	27.0268i	1.75558i
$238$	51.5965i	3.34450i
$239$	2.92984	0.189515	0.0947577	−	0.995500i	$-0.469792\pi$
$239$	2.92984	0.189515	0.0947577	+	0.995500i	$0.469792\pi$
$240$	0	0
$241$	−9.42743	−0.607274	−0.303637	−	0.952788i	$-0.598201\pi$
$241$	−9.42743	−0.607274	−0.303637	+	0.952788i	$0.598201\pi$
$242$	20.9517i	1.34682i
$243$	− 10.8783i	− 0.697846i
$244$	−10.5549	−0.675711
$245$	0	0
$246$	17.3897	1.10873
$247$	1.77353i	0.112847i
$248$	8.31346i	0.527905i
$249$	25.5065	1.61641
$250$	0	0
$251$	−14.9298	−0.942363	−0.471181	−	0.882036i	$-0.656172\pi$
$251$	−14.9298	−0.942363	−0.471181	+	0.882036i	$0.656172\pi$
$252$	42.4986i	2.67716i
$253$	− 7.34911i	− 0.462035i
$254$	9.93026	0.623080
$255$	0	0
$256$	2.91405	0.182128
$257$	− 15.1295i	− 0.943755i	−0.881664	−	0.471877i	$-0.843576\pi$
$257$	− 15.1295i	− 0.943755i	0.881664	−	0.471877i	$-0.156424\pi$
$258$	− 19.1464i	− 1.19200i
$259$	4.36700	0.271352
$260$	0	0
$261$	40.9931	2.53741
$262$	− 33.2364i	− 2.05335i
$263$	15.2216i	0.938605i	0.883038	+	0.469302i	$0.155495\pi$
$263$	15.2216i	0.938605i	−0.883038	+	0.469302i	$0.844505\pi$
$264$	4.01987	0.247406
$265$	0	0
$266$	−6.65187	−0.407852
$267$	− 2.34023i	− 0.143220i
$268$	40.9615i	2.50212i
$269$	−11.5203	−0.702404	−0.351202	−	0.936300i	$-0.614227\pi$
$269$	−11.5203	−0.702404	−0.351202	+	0.936300i	$0.614227\pi$
$270$	0	0
$271$	2.16118	0.131282	0.0656411	−	0.997843i	$-0.479091\pi$
$271$	2.16118	0.131282	0.0656411	+	0.997843i	$0.479091\pi$
$272$	− 19.1731i	− 1.16254i
$273$	15.8495i	0.959258i
$274$	−1.74190	−0.105232
$275$	0	0
$276$	49.5896	2.98494
$277$	23.4205i	1.40720i	0.710595	+	0.703602i	$0.248426\pi$
$277$	23.4205i	1.40720i	−0.710595	+	0.703602i	$0.751574\pi$
$278$	− 27.3560i	− 1.64070i
$279$	−34.7567	−2.08083
$280$	0	0
$281$	25.6824	1.53209	0.766043	−	0.642789i	$-0.222223\pi$
$281$	25.6824	1.53209	0.766043	+	0.642789i	$0.222223\pi$
$282$	9.03366i	0.537947i
$283$	− 7.38587i	− 0.439045i	−0.975607	−	0.219522i	$-0.929550\pi$
$283$	− 7.38587i	− 0.439045i	0.975607	−	0.219522i	$-0.0704499\pi$
$284$	−18.5093	−1.09832
$285$	0	0
$286$	4.19794	0.248229
$287$	8.75543i	0.516817i
$288$	− 41.3482i	− 2.43647i
$289$	−43.1662	−2.53919
$290$	0	0
$291$	−9.20174	−0.539416
$292$	0.630180i	0.0368785i
$293$	24.1522i	1.41099i	0.708716	+	0.705494i	$0.249275\pi$
$293$	24.1522i	1.41099i	−0.708716	+	0.705494i	$0.750725\pi$
$294$	−16.2794	−0.949433
$295$	0	0
$296$	1.77797	0.103343
$297$	7.26600i	0.421616i
$298$	− 29.5618i	− 1.71247i
$299$	11.7973	0.682253
$300$	0	0
$301$	9.63990	0.555635
$302$	− 10.7608i	− 0.619213i
$303$	− 8.68936i	− 0.499190i
$304$	2.47182	0.141769
$305$	0	0
$306$	−87.8250	−5.02062
$307$	4.38482i	0.250255i	0.992141	+	0.125127i	$0.0399340\pi$
$307$	4.38482i	0.250255i	−0.992141	+	0.125127i	$0.960066\pi$
$308$	8.88447i	0.506239i
$309$	17.4381	0.992022
$310$	0	0
$311$	−15.7123	−0.890963	−0.445481	−	0.895291i	$-0.646967\pi$
$311$	−15.7123	−0.890963	−0.445481	+	0.895291i	$0.646967\pi$
$312$	6.45295i	0.365326i
$313$	− 24.7794i	− 1.40061i	−0.713842	−	0.700307i	$-0.753047\pi$
$313$	− 24.7794i	− 1.40061i	0.713842	−	0.700307i	$-0.246953\pi$
$314$	15.3976	0.868935
$315$	0	0
$316$	−24.3195	−1.36808
$317$	− 27.3798i	− 1.53780i	−0.639369	−	0.768900i	$-0.720804\pi$
$317$	− 27.3798i	− 1.53780i	0.639369	−	0.768900i	$-0.279196\pi$
$318$	15.9931i	0.896848i
$319$	8.56974	0.479813
$320$	0	0
$321$	−58.6745	−3.27489
$322$	44.2474i	2.46581i
$323$	− 7.75669i	− 0.431594i
$324$	−7.96492	−0.442496
$325$	0	0
$326$	−15.5440	−0.860900
$327$	− 13.6062i	− 0.752426i
$328$	3.56467i	0.196826i
$329$	−4.54831	−0.250756
$330$	0	0
$331$	4.70033	0.258354	0.129177	−	0.991622i	$-0.458767\pi$
$331$	4.70033	0.258354	0.129177	+	0.991622i	$0.458767\pi$
$332$	22.9515i	1.25963i
$333$	7.43329i	0.407342i
$334$	−15.7068	−0.859439
$335$	0	0
$336$	22.0900	1.20511
$337$	− 20.6360i	− 1.12412i	−0.827098	−	0.562058i	$-0.810010\pi$
$337$	− 20.6360i	− 1.12412i	0.827098	−	0.562058i	$-0.189990\pi$
$338$	− 21.1128i	− 1.14839i
$339$	−22.9783	−1.24801
$340$	0	0
$341$	−7.26600	−0.393476
$342$	− 11.3225i	− 0.612250i
$343$	13.5373i	0.730947i
$344$	3.92477	0.211609
$345$	0	0
$346$	12.3694	0.664983
$347$	30.0148i	1.61128i	0.592407	+	0.805639i	$0.298178\pi$
$347$	30.0148i	1.61128i	−0.592407	+	0.805639i	$0.701822\pi$
$348$	57.8260i	3.09980i
$349$	14.7340	0.788693	0.394347	−	0.918962i	$-0.370971\pi$
$349$	14.7340	0.788693	0.394347	+	0.918962i	$0.370971\pi$
$350$	0	0
$351$	−11.6638	−0.622570
$352$	− 8.64399i	− 0.460726i
$353$	29.6302i	1.57705i	0.615000	+	0.788527i	$0.289156\pi$
$353$	29.6302i	1.57705i	−0.615000	+	0.788527i	$0.710844\pi$
$354$	33.2364	1.76649
$355$	0	0
$356$	2.10581	0.111608
$357$	− 69.3195i	− 3.66878i
$358$	− 9.60623i	− 0.507705i
$359$	−21.7577	−1.14833	−0.574163	−	0.818741i	$-0.694672\pi$
$359$	−21.7577	−1.14833	−0.574163	+	0.818741i	$0.694672\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	5.85742i	0.307859i
$363$	− 28.1484i	− 1.47741i
$364$	−14.2619	−0.747527
$365$	0	0
$366$	25.1306	1.31360
$367$	30.0347i	1.56780i	0.620889	+	0.783898i	$0.286772\pi$
$367$	30.0347i	1.56780i	−0.620889	+	0.783898i	$0.713228\pi$
$368$	− 16.4422i	− 0.857111i
$369$	−14.9031	−0.775823
$370$	0	0
$371$	−8.05227	−0.418053
$372$	− 49.0287i	− 2.54202i
$373$	37.3055i	1.93161i	0.259277	+	0.965803i	$0.416516\pi$
$373$	37.3055i	1.93161i	−0.259277	+	0.965803i	$0.583484\pi$
$374$	−18.3601	−0.949378
$375$	0	0
$376$	−1.85179	−0.0954987
$377$	13.7567i	0.708506i
$378$	− 43.7470i	− 2.25010i
$379$	14.7794	0.759166	0.379583	−	0.925158i	$-0.376068\pi$
$379$	14.7794	0.759166	0.379583	+	0.925158i	$0.376068\pi$
$380$	0	0
$381$	−13.3412	−0.683492
$382$	37.9704i	1.94273i
$383$	29.0020i	1.48193i	0.671541	+	0.740967i	$0.265633\pi$
$383$	29.0020i	1.48193i	−0.671541	+	0.740967i	$0.734367\pi$
$384$	27.8412	1.42077
$385$	0	0
$386$	29.0024	1.47618
$387$	16.4086i	0.834094i
$388$	− 8.28001i	− 0.420354i
$389$	−21.1987	−1.07481	−0.537407	−	0.843323i	$-0.680596\pi$
$389$	−21.1987	−1.07481	−0.537407	+	0.843323i	$0.680596\pi$
$390$	0	0
$391$	−51.5965	−2.60935
$392$	− 3.33707i	− 0.168548i
$393$	44.6529i	2.25244i
$394$	−46.0910	−2.32203
$395$	0	0
$396$	−15.1227	−0.759944
$397$	10.3856i	0.521238i	0.965442	+	0.260619i	$0.0839267\pi$
$397$	10.3856i	0.521238i	−0.965442	+	0.260619i	$0.916073\pi$
$398$	15.1987i	0.761840i
$399$	8.93674	0.447397
$400$	0	0
$401$	25.6638	1.28159	0.640796	−	0.767712i	$-0.278605\pi$
$401$	25.6638	1.28159	0.640796	+	0.767712i	$0.278605\pi$
$402$	− 97.5263i	− 4.86417i
$403$	− 11.6638i	− 0.581017i
$404$	7.81896	0.389008
$405$	0	0
$406$	−51.5965	−2.56069
$407$	1.55395i	0.0770266i
$408$	− 28.2226i	− 1.39723i
$409$	−5.71611	−0.282644	−0.141322	−	0.989964i	$-0.545135\pi$
$409$	−5.71611	−0.282644	−0.141322	+	0.989964i	$0.545135\pi$
$410$	0	0
$411$	2.34023	0.115435
$412$	15.6914i	0.773059i
$413$	16.7340i	0.823427i
$414$	−75.3157	−3.70156
$415$	0	0
$416$	13.8759	0.680321
$417$	36.7526i	1.79978i
$418$	− 2.36700i	− 0.115774i
$419$	−15.4303	−0.753818	−0.376909	−	0.926250i	$-0.623013\pi$
$419$	−15.4303	−0.753818	−0.376909	+	0.926250i	$0.623013\pi$
$420$	0	0
$421$	−1.18005	−0.0575121	−0.0287561	−	0.999586i	$-0.509155\pi$
$421$	−1.18005	−0.0575121	−0.0287561	+	0.999586i	$0.509155\pi$
$422$	− 23.6962i	− 1.15352i
$423$	− 7.74190i	− 0.376424i
$424$	−3.27839	−0.159213
$425$	0	0
$426$	44.0693	2.13517
$427$	12.6529i	0.612315i
$428$	− 52.7972i	− 2.55205i
$429$	−5.63990	−0.272297
$430$	0	0
$431$	−14.0255	−0.675585	−0.337792	−	0.941221i	$-0.609680\pi$
$431$	−14.0255	−0.675585	−0.337792	+	0.941221i	$0.609680\pi$
$432$	16.2563i	0.782130i
$433$	− 25.5894i	− 1.22975i	−0.788625	−	0.614874i	$-0.789207\pi$
$433$	− 25.5894i	− 1.22975i	0.788625	−	0.614874i	$-0.210793\pi$
$434$	43.7470	2.09992
$435$	0	0
$436$	12.2433	0.586348
$437$	− 6.65187i	− 0.318202i
$438$	− 1.50041i	− 0.0716925i
$439$	21.9732	1.04873	0.524363	−	0.851495i	$-0.324304\pi$
$439$	21.9732	1.04873	0.524363	+	0.851495i	$0.324304\pi$
$440$	0	0
$441$	13.9515	0.664358
$442$	− 29.4728i	− 1.40188i
$443$	19.3721i	0.920395i	0.887817	+	0.460197i	$0.152221\pi$
$443$	19.3721i	0.920395i	−0.887817	+	0.460197i	$0.847779\pi$
$444$	−10.4856	−0.497625
$445$	0	0
$446$	27.0697	1.28179
$447$	39.7161i	1.87851i
$448$	36.6944i	1.73365i
$449$	19.1841	0.905355	0.452677	−	0.891674i	$-0.350469\pi$
$449$	19.1841	0.905355	0.452677	+	0.891674i	$0.350469\pi$
$450$	0	0
$451$	−3.11553	−0.146705
$452$	− 20.6766i	− 0.972545i
$453$	14.4570i	0.679250i
$454$	−14.2003	−0.666455
$455$	0	0
$456$	3.63849	0.170388
$457$	8.36010i	0.391069i	0.980697	+	0.195534i	$0.0626442\pi$
$457$	8.36010i	0.391069i	−0.980697	+	0.195534i	$0.937356\pi$
$458$	− 0.161177i	− 0.00753131i
$459$	51.0130	2.38108
$460$	0	0
$461$	−25.0268	−1.16561	−0.582806	−	0.812611i	$-0.698045\pi$
$461$	−25.0268	−1.16561	−0.582806	+	0.812611i	$0.698045\pi$
$462$	− 21.1533i	− 0.984140i
$463$	− 32.3749i	− 1.50459i	−0.658827	−	0.752294i	$-0.728947\pi$
$463$	− 32.3749i	− 1.50459i	0.658827	−	0.752294i	$-0.271053\pi$
$464$	19.1731	0.890091
$465$	0	0
$466$	−20.3818	−0.944168
$467$	15.2216i	0.704372i	0.935930	+	0.352186i	$0.114562\pi$
$467$	15.2216i	0.704372i	−0.935930	+	0.352186i	$0.885438\pi$
$468$	− 24.2759i	− 1.12215i
$469$	49.1030	2.26736
$470$	0	0
$471$	−20.6865	−0.953185
$472$	6.81305i	0.313596i
$473$	3.43026i	0.157724i
$474$	57.9031	2.65958
$475$	0	0
$476$	62.3759	2.85899
$477$	− 13.7062i	− 0.627563i
$478$	− 6.27698i	− 0.287103i
$479$	20.0485	0.916038	0.458019	−	0.888943i	$-0.348559\pi$
$479$	20.0485	0.916038	0.458019	+	0.888943i	$0.348559\pi$
$480$	0	0
$481$	−2.49451	−0.113740
$482$	20.1977i	0.919978i
$483$	− 59.4460i	− 2.70489i
$484$	25.3288	1.15131
$485$	0	0
$486$	−23.3061	−1.05719
$487$	− 31.6508i	− 1.43424i	−0.696952	−	0.717118i	$-0.745461\pi$
$487$	− 31.6508i	− 1.43424i	0.696952	−	0.717118i	$-0.254539\pi$
$488$	5.15146i	0.233195i
$489$	20.8832	0.944371
$490$	0	0
$491$	24.8033	1.11936	0.559679	−	0.828710i	$-0.310924\pi$
$491$	24.8033	1.11936	0.559679	+	0.828710i	$0.310924\pi$
$492$	− 21.0227i	− 0.947776i
$493$	− 60.1662i	− 2.70975i
$494$	3.79966	0.170955
$495$	0	0
$496$	−16.2563	−0.729928
$497$	22.1882i	0.995277i
$498$	− 54.6460i	− 2.44874i
$499$	33.0237	1.47834	0.739171	−	0.673518i	$-0.235217\pi$
$499$	33.0237	1.47834	0.739171	+	0.673518i	$0.235217\pi$
$500$	0	0
$501$	21.1020	0.942767
$502$	31.9862i	1.42761i
$503$	3.54396i	0.158017i	0.996874	+	0.0790087i	$0.0251755\pi$
$503$	3.54396i	0.158017i	−0.996874	+	0.0790087i	$0.974825\pi$
$504$	20.7419	0.923917
$505$	0	0
$506$	−15.7450	−0.699950
$507$	28.3650i	1.25973i
$508$	− 12.0049i	− 0.532629i
$509$	2.11679	0.0938250	0.0469125	−	0.998899i	$-0.485062\pi$
$509$	2.11679	0.0938250	0.0469125	+	0.998899i	$0.485062\pi$
$510$	0	0
$511$	0.755435	0.0334185
$512$	− 25.5885i	− 1.13086i
$513$	6.57664i	0.290366i
$514$	−32.4140	−1.42972
$515$	0	0
$516$	−23.1464	−1.01896
$517$	− 1.61847i	− 0.0711802i
$518$	− 9.35601i	− 0.411080i
$519$	−16.6182	−0.729458
$520$	0	0
$521$	−8.60748	−0.377101	−0.188550	−	0.982064i	$-0.560379\pi$
$521$	−8.60748	−0.377101	−0.188550	+	0.982064i	$0.560379\pi$
$522$	− 87.8250i	− 3.84400i
$523$	− 36.8349i	− 1.61068i	−0.592814	−	0.805340i	$-0.701983\pi$
$523$	− 36.8349i	− 1.61068i	0.592814	−	0.805340i	$-0.298017\pi$
$524$	−40.1800	−1.75527
$525$	0	0
$526$	32.6113	1.42192
$527$	51.0130i	2.22216i
$528$	7.86051i	0.342085i
$529$	−21.2474	−0.923799
$530$	0	0
$531$	−28.4838	−1.23609
$532$	8.04156i	0.348646i
$533$	− 5.00125i	− 0.216628i
$534$	−5.01379	−0.216968
$535$	0	0
$536$	19.9917	0.863509
$537$	12.9059i	0.556931i
$538$	24.6814i	1.06409i
$539$	2.91661	0.125627
$540$	0	0
$541$	−32.0079	−1.37613	−0.688063	−	0.725651i	$-0.741539\pi$
$541$	−32.0079	−1.37613	−0.688063	+	0.725651i	$0.741539\pi$
$542$	− 4.63018i	− 0.198883i
$543$	− 7.86941i	− 0.337709i
$544$	−60.6875	−2.60196
$545$	0	0
$546$	33.9566	1.45321
$547$	17.6240i	0.753550i	0.926305	+	0.376775i	$0.122967\pi$
$547$	17.6240i	0.753550i	−0.926305	+	0.376775i	$0.877033\pi$
$548$	2.10581i	0.0899559i
$549$	−21.5371	−0.919179
$550$	0	0
$551$	7.75669	0.330446
$552$	− 24.2027i	− 1.03014i
$553$	29.1533i	1.23972i
$554$	50.1769	2.13181
$555$	0	0
$556$	−33.0711	−1.40253
$557$	− 34.6865i	− 1.46972i	−0.678221	−	0.734858i	$-0.737249\pi$
$557$	− 34.6865i	− 1.46972i	0.678221	−	0.734858i	$-0.262751\pi$
$558$	74.4639i	3.15231i
$559$	−5.50648	−0.232899
$560$	0	0
$561$	24.6667	1.04143
$562$	− 55.0229i	− 2.32100i
$563$	− 13.5073i	− 0.569263i	−0.958637	−	0.284632i	$-0.908129\pi$
$563$	− 13.5073i	− 0.569263i	0.958637	−	0.284632i	$-0.0918712\pi$
$564$	10.9209	0.459855
$565$	0	0
$566$	−15.8238	−0.665122
$567$	9.54803i	0.400980i
$568$	9.03366i	0.379044i
$569$	−32.3588	−1.35655	−0.678276	−	0.734807i	$-0.737273\pi$
$569$	−32.3588	−1.35655	−0.678276	+	0.734807i	$0.737273\pi$
$570$	0	0
$571$	−8.93293	−0.373831	−0.186916	−	0.982376i	$-0.559849\pi$
$571$	−8.93293	−0.373831	−0.186916	+	0.982376i	$0.559849\pi$
$572$	− 5.07496i	− 0.212195i
$573$	− 51.0130i	− 2.13110i
$574$	18.7579	0.782941
$575$	0	0
$576$	−62.4594	−2.60248
$577$	14.9059i	0.620541i	0.950648	+	0.310270i	$0.100420\pi$
$577$	14.9059i	0.620541i	−0.950648	+	0.310270i	$0.899580\pi$
$578$	92.4808i	3.84669i
$579$	−38.9645	−1.61931
$580$	0	0
$581$	27.5134	1.14145
$582$	19.7141i	0.817177i
$583$	− 2.86532i	− 0.118669i
$584$	0.307566	0.0127272
$585$	0	0
$586$	51.7446	2.13755
$587$	7.37207i	0.304278i	0.988359	+	0.152139i	$0.0486161\pi$
$587$	7.37207i	0.304278i	−0.988359	+	0.152139i	$0.951384\pi$
$588$	19.6804i	0.811607i
$589$	−6.57664	−0.270986
$590$	0	0
$591$	61.9229	2.54717
$592$	3.47668i	0.142890i
$593$	17.6853i	0.726247i	0.931741	+	0.363124i	$0.118290\pi$
$593$	17.6853i	0.726247i	−0.931741	+	0.363124i	$0.881710\pi$
$594$	15.5669	0.638718
$595$	0	0
$596$	−35.7378	−1.46388
$597$	− 20.4193i	− 0.835705i
$598$	− 25.2749i	− 1.03357i
$599$	−5.30657	−0.216821	−0.108410	−	0.994106i	$-0.534576\pi$
$599$	−5.30657	−0.216821	−0.108410	+	0.994106i	$0.534576\pi$
$600$	0	0
$601$	43.7875	1.78613	0.893065	−	0.449927i	$-0.148550\pi$
$601$	43.7875	1.78613	0.893065	+	0.449927i	$0.148550\pi$
$602$	− 20.6529i	− 0.841747i
$603$	83.5806i	3.40367i
$604$	−13.0089	−0.529324
$605$	0	0
$606$	−18.6164	−0.756239
$607$	− 7.24535i	− 0.294080i	−0.989131	−	0.147040i	$-0.953025\pi$
$607$	− 7.24535i	− 0.294080i	0.989131	−	0.147040i	$-0.0469745\pi$
$608$	− 7.82389i	− 0.317301i
$609$	69.3195	2.80897
$610$	0	0
$611$	2.59807	0.105107
$612$	106.173i	4.29179i
$613$	− 20.8962i	− 0.843988i	−0.906599	−	0.421994i	$-0.861330\pi$
$613$	− 20.8962i	− 0.843988i	0.906599	−	0.421994i	$-0.138670\pi$
$614$	9.39419	0.379119
$615$	0	0
$616$	4.33616	0.174709
$617$	− 22.0494i	− 0.887677i	−0.896107	−	0.443839i	$-0.853616\pi$
$617$	− 22.0494i	− 0.887677i	0.896107	−	0.443839i	$-0.146384\pi$
$618$	− 37.3601i	− 1.50284i
$619$	0.0484607	0.00194780	0.000973900	−	1.00000i	$-0.499690\pi$
$619$	0.0484607	0.00194780	0.000973900		1.00000i	$0.499690\pi$
$620$	0	0
$621$	43.7470	1.75550
$622$	33.6626i	1.34975i
$623$	− 2.52437i	− 0.101137i
$624$	−12.6182	−0.505132
$625$	0	0
$626$	−53.0882	−2.12183
$627$	3.18005i	0.126999i
$628$	− 18.6144i	− 0.742795i
$629$	10.9100	0.435009
$630$	0	0
$631$	−32.6182	−1.29851	−0.649255	−	0.760571i	$-0.724919\pi$
$631$	−32.6182	−1.29851	−0.649255	+	0.760571i	$0.724919\pi$
$632$	11.8694i	0.472140i
$633$	31.8357i	1.26536i
$634$	−58.6593	−2.32966
$635$	0	0
$636$	19.3343	0.766656
$637$	4.68193i	0.185505i
$638$	− 18.3601i	− 0.726883i
$639$	−37.7677	−1.49407
$640$	0	0
$641$	−13.4136	−0.529806	−0.264903	−	0.964275i	$-0.585340\pi$
$641$	−13.4136	−0.529806	−0.264903	+	0.964275i	$0.585340\pi$
$642$	125.706i	4.96123i
$643$	− 23.8052i	− 0.938783i	−0.882990	−	0.469392i	$-0.844473\pi$
$643$	− 23.8052i	− 0.938783i	0.882990	−	0.469392i	$-0.155527\pi$
$644$	53.4914	2.10786
$645$	0	0
$646$	−16.6182	−0.653834
$647$	− 48.4580i	− 1.90508i	−0.304412	−	0.952540i	$-0.598460\pi$
$647$	− 48.4580i	− 1.90508i	0.304412	−	0.952540i	$-0.401540\pi$
$648$	3.88737i	0.152710i
$649$	−5.95463	−0.233740
$650$	0	0
$651$	−58.7737	−2.30352
$652$	18.7914i	0.735926i
$653$	− 22.2351i	− 0.870128i	−0.900399	−	0.435064i	$-0.856726\pi$
$653$	− 22.2351i	− 0.870128i	0.900399	−	0.435064i	$-0.143274\pi$
$654$	−29.1504	−1.13987
$655$	0	0
$656$	−6.97041	−0.272149
$657$	1.28586i	0.0501663i
$658$	9.74445i	0.379878i
$659$	10.7072	0.417095	0.208547	−	0.978012i	$-0.433126\pi$
$659$	10.7072	0.417095	0.208547	+	0.978012i	$0.433126\pi$
$660$	0	0
$661$	−44.7980	−1.74244	−0.871220	−	0.490893i	$-0.836670\pi$
$661$	−44.7980	−1.74244	−0.871220	+	0.490893i	$0.836670\pi$
$662$	− 10.0702i	− 0.391388i
$663$	39.5965i	1.53780i
$664$	11.2017	0.434712
$665$	0	0
$666$	15.9253	0.617095
$667$	− 51.5965i	− 1.99782i
$668$	18.9882i	0.734677i
$669$	−36.3680	−1.40607
$670$	0	0
$671$	−4.50239	−0.173813
$672$	− 69.9201i	− 2.69723i
$673$	43.2772i	1.66821i	0.551604	+	0.834106i	$0.314016\pi$
$673$	43.2772i	1.66821i	−0.551604	+	0.834106i	$0.685984\pi$
$674$	−44.2113	−1.70296
$675$	0	0
$676$	−25.5237	−0.981680
$677$	11.5330i	0.443251i	0.975132	+	0.221625i	$0.0711362\pi$
$677$	11.5330i	0.443251i	−0.975132	+	0.221625i	$0.928864\pi$
$678$	49.2295i	1.89065i
$679$	−9.92575	−0.380915
$680$	0	0
$681$	19.0780	0.731072
$682$	15.5669i	0.596088i
$683$	− 0.916090i	− 0.0350532i	−0.999846	−	0.0175266i	$-0.994421\pi$
$683$	− 0.916090i	− 0.0350532i	0.999846	−	0.0175266i	$-0.00557918\pi$
$684$	−13.6879	−0.523372
$685$	0	0
$686$	29.0028	1.10733
$687$	0.216540i	0.00826153i
$688$	7.67456i	0.292590i
$689$	4.59960	0.175231
$690$	0	0
$691$	14.8278	0.564077	0.282039	−	0.959403i	$-0.408989\pi$
$691$	14.8278	0.564077	0.282039	+	0.959403i	$0.408989\pi$
$692$	− 14.9536i	− 0.568450i
$693$	18.1285i	0.688644i
$694$	64.3047	2.44097
$695$	0	0
$696$	28.2226	1.06978
$697$	21.8735i	0.828517i
$698$	− 31.5666i	− 1.19481i
$699$	27.3828	1.03571
$700$	0	0
$701$	36.1722	1.36620	0.683102	−	0.730323i	$-0.260631\pi$
$701$	36.1722	1.36620	0.683102	+	0.730323i	$0.260631\pi$
$702$	24.9890i	0.943150i
$703$	1.40652i	0.0530481i
$704$	−13.0573	−0.492117
$705$	0	0
$706$	63.4807	2.38913
$707$	− 9.37305i	− 0.352510i
$708$	− 40.1800i	− 1.51006i
$709$	−10.4331	−0.391823	−0.195911	−	0.980622i	$-0.562766\pi$
$709$	−10.4331	−0.391823	−0.195911	+	0.980622i	$0.562766\pi$
$710$	0	0
$711$	−49.6233	−1.86102
$712$	− 1.02777i	− 0.0385171i
$713$	43.7470i	1.63834i
$714$	−148.513	−5.55794
$715$	0	0
$716$	−11.6131	−0.434003
$717$	8.43308i	0.314939i
$718$	46.6144i	1.73963i
$719$	−7.73373	−0.288420	−0.144210	−	0.989547i	$-0.546064\pi$
$719$	−7.73373	−0.288420	−0.144210	+	0.989547i	$0.546064\pi$
$720$	0	0
$721$	18.8102	0.700529
$722$	− 2.14243i	− 0.0797331i
$723$	− 27.1354i	− 1.00918i
$724$	7.08114	0.263168
$725$	0	0
$726$	−60.3061	−2.23817
$727$	− 39.5241i	− 1.46587i	−0.680300	−	0.732934i	$-0.738151\pi$
$727$	− 39.5241i	− 1.46587i	0.680300	−	0.732934i	$-0.261849\pi$
$728$	6.96068i	0.257980i
$729$	40.5373	1.50138
$730$	0	0
$731$	24.0831	0.890746
$732$	− 30.3808i	− 1.12291i
$733$	− 50.7498i	− 1.87449i	−0.348677	−	0.937243i	$-0.613369\pi$
$733$	− 50.7498i	− 1.87449i	0.348677	−	0.937243i	$-0.386631\pi$
$734$	64.3473	2.37510
$735$	0	0
$736$	−52.0435	−1.91835
$737$	17.4728i	0.643619i
$738$	31.9288i	1.17532i
$739$	−0.419276	−0.0154233	−0.00771165	−	0.999970i	$-0.502455\pi$
$739$	−0.419276	−0.0154233	−0.00771165	+	0.999970i	$0.502455\pi$
$740$	0	0
$741$	−5.10482	−0.187530
$742$	17.2515i	0.633321i
$743$	− 19.5511i	− 0.717259i	−0.933480	−	0.358630i	$-0.883244\pi$
$743$	− 19.5511i	− 0.717259i	0.933480	−	0.358630i	$-0.116756\pi$
$744$	−23.9290	−0.877280
$745$	0	0
$746$	79.9246	2.92625
$747$	46.8319i	1.71349i
$748$	22.1958i	0.811560i
$749$	−63.2912	−2.31261
$750$	0	0
$751$	8.76768	0.319937	0.159969	−	0.987122i	$-0.448861\pi$
$751$	8.76768	0.319937	0.159969	+	0.987122i	$0.448861\pi$
$752$	− 3.62102i	− 0.132045i
$753$	− 42.9732i	− 1.56603i
$754$	29.4728	1.07334
$755$	0	0
$756$	−52.8864	−1.92346
$757$	− 48.0535i	− 1.74653i	−0.487241	−	0.873267i	$-0.661997\pi$
$757$	− 48.0535i	− 1.74653i	0.487241	−	0.873267i	$-0.338003\pi$
$758$	− 31.6638i	− 1.15008i
$759$	21.1533	0.767815
$760$	0	0
$761$	29.3947	1.06556	0.532779	−	0.846254i	$-0.321148\pi$
$761$	29.3947	1.06556	0.532779	+	0.846254i	$0.321148\pi$
$762$	28.5827i	1.03544i
$763$	− 14.6768i	− 0.531336i
$764$	45.9031	1.66071
$765$	0	0
$766$	62.1350	2.24503
$767$	− 9.55875i	− 0.345146i
$768$	8.38765i	0.302663i
$769$	13.3790	0.482458	0.241229	−	0.970468i	$-0.422449\pi$
$769$	13.3790	0.482458	0.241229	+	0.970468i	$0.422449\pi$
$770$	0	0
$771$	43.5480	1.56834
$772$	− 35.0615i	− 1.26189i
$773$	− 0.133625i	− 0.00480617i	−0.999997	−	0.00240309i	$-0.999235\pi$
$773$	− 0.133625i	− 0.00480617i	0.999997	−	0.00240309i	$-0.000764927\pi$
$774$	35.1543	1.26359
$775$	0	0
$776$	−4.04115	−0.145069
$777$	12.5697i	0.450937i
$778$	45.4167i	1.62827i
$779$	−2.81995	−0.101035
$780$	0	0
$781$	−7.89545	−0.282522
$782$	110.542i	3.95298i
$783$	51.0130i	1.82305i
$784$	6.52536	0.233049
$785$	0	0
$786$	95.6658	3.41229
$787$	− 4.84060i	− 0.172549i	−0.996271	−	0.0862744i	$-0.972504\pi$
$787$	− 4.84060i	− 0.172549i	0.996271	−	0.0862744i	$-0.0274962\pi$
$788$	55.7202i	1.98495i
$789$	−43.8130	−1.55979
$790$	0	0
$791$	−24.7863	−0.881299
$792$	7.38079i	0.262265i
$793$	− 7.22753i	− 0.256657i
$794$	22.2505	0.789640
$795$	0	0
$796$	18.3739	0.651246
$797$	11.6993i	0.414410i	0.978298	+	0.207205i	$0.0664367\pi$
$797$	11.6993i	0.414410i	−0.978298	+	0.207205i	$0.933563\pi$
$798$	− 19.1464i	− 0.677774i
$799$	−11.3629	−0.401991
$800$	0	0
$801$	4.29685	0.151822
$802$	− 54.9831i	− 1.94152i
$803$	0.268814i	0.00948624i
$804$	−117.901	−4.15806
$805$	0	0
$806$	−24.9890	−0.880200
$807$	− 33.1593i	− 1.16726i
$808$	− 3.81613i	− 0.134251i
$809$	33.1987	1.16720	0.583601	−	0.812040i	$-0.301643\pi$
$809$	33.1987	1.16720	0.583601	+	0.812040i	$0.301643\pi$
$810$	0	0
$811$	−26.5766	−0.933232	−0.466616	−	0.884460i	$-0.654527\pi$
$811$	−26.5766	−0.933232	−0.466616	+	0.884460i	$0.654527\pi$
$812$	62.3759i	2.18896i
$813$	6.22061i	0.218166i
$814$	3.32924	0.116690
$815$	0	0
$816$	55.1869	1.93193
$817$	3.10482i	0.108624i
$818$	12.2464i	0.428185i
$819$	−29.1010	−1.01687
$820$	0	0
$821$	25.5807	0.892773	0.446387	−	0.894840i	$-0.352711\pi$
$821$	25.5807	0.892773	0.446387	+	0.894840i	$0.352711\pi$
$822$	− 5.01379i	− 0.174876i
$823$	− 12.9783i	− 0.452395i	−0.974081	−	0.226198i	$-0.927371\pi$
$823$	− 12.9783i	− 0.452395i	0.974081	−	0.226198i	$-0.0726295\pi$
$824$	7.65835	0.266791
$825$	0	0
$826$	35.8515	1.24743
$827$	− 12.5167i	− 0.435248i	−0.976033	−	0.217624i	$-0.930169\pi$
$827$	− 12.5167i	− 0.435248i	0.976033	−	0.217624i	$-0.0698306\pi$
$828$	91.0504i	3.16422i
$829$	19.4391	0.675149	0.337574	−	0.941299i	$-0.390394\pi$
$829$	19.4391	0.675149	0.337574	+	0.941299i	$0.390394\pi$
$830$	0	0
$831$	−67.4124	−2.33851
$832$	− 20.9605i	− 0.726674i
$833$	− 20.4769i	− 0.709482i
$834$	78.7400	2.72654
$835$	0	0
$836$	−2.86151	−0.0989673
$837$	− 43.2522i	− 1.49501i
$838$	33.0583i	1.14198i
$839$	43.9972	1.51895	0.759475	−	0.650536i	$-0.225456\pi$
$839$	43.9972	1.51895	0.759475	+	0.650536i	$0.225456\pi$
$840$	0	0
$841$	31.1662	1.07470
$842$	2.52818i	0.0871268i
$843$	73.9229i	2.54604i
$844$	−28.6468	−0.986063
$845$	0	0
$846$	−16.5865	−0.570256
$847$	− 30.3632i	− 1.04329i
$848$	− 6.41061i	− 0.220141i
$849$	21.2591	0.729610
$850$	0	0
$851$	9.35601	0.320720
$852$	− 53.2761i	− 1.82521i
$853$	− 3.30374i	− 0.113118i	−0.998399	−	0.0565590i	$-0.981987\pi$
$853$	− 3.30374i	− 0.113118i	0.998399	−	0.0565590i	$-0.0180129\pi$
$854$	27.1079	0.927614
$855$	0	0
$856$	−25.7682	−0.880740
$857$	6.02374i	0.205767i	0.994693	+	0.102883i	$0.0328069\pi$
$857$	6.02374i	0.205767i	−0.994693	+	0.102883i	$0.967193\pi$
$858$	12.0831i	0.412511i
$859$	36.0158	1.22884	0.614421	−	0.788978i	$-0.289390\pi$
$859$	36.0158	1.22884	0.614421	+	0.788978i	$0.289390\pi$
$860$	0	0
$861$	−25.2012	−0.858853
$862$	30.0487i	1.02346i
$863$	− 29.9250i	− 1.01866i	−0.860572	−	0.509329i	$-0.829894\pi$
$863$	− 29.9250i	− 1.01866i	0.860572	−	0.509329i	$-0.170106\pi$
$864$	51.4549	1.75053
$865$	0	0
$866$	−54.8236	−1.86298
$867$	− 124.247i	− 4.21966i
$868$	− 52.8864i	− 1.79508i
$869$	−10.3739	−0.351911
$870$	0	0
$871$	−28.0485	−0.950386
$872$	− 5.97548i	− 0.202355i
$873$	− 16.8951i	− 0.571813i
$874$	−14.2512	−0.482054
$875$	0	0
$876$	−1.81388	−0.0612852
$877$	0.618980i	0.0209015i	0.999945	+	0.0104507i	$0.00332663\pi$
$877$	0.618980i	0.0209015i	−0.999945	+	0.0104507i	$0.996673\pi$
$878$	− 47.0762i	− 1.58874i
$879$	−69.5184	−2.34480
$880$	0	0
$881$	−21.2423	−0.715672	−0.357836	−	0.933784i	$-0.616485\pi$
$881$	−21.2423	−0.715672	−0.357836	+	0.933784i	$0.616485\pi$
$882$	− 29.8902i	− 1.00646i
$883$	48.8971i	1.64552i	0.568389	+	0.822760i	$0.307567\pi$
$883$	48.8971i	1.64552i	−0.568389	+	0.822760i	$0.692433\pi$
$884$	−35.6302	−1.19837
$885$	0	0
$886$	41.5034	1.39433
$887$	58.2797i	1.95684i	0.206622	+	0.978421i	$0.433753\pi$
$887$	58.2797i	1.95684i	−0.206622	+	0.978421i	$0.566247\pi$
$888$	5.11762i	0.171736i
$889$	−14.3909	−0.482657
$890$	0	0
$891$	−3.39757	−0.113823
$892$	− 32.7251i	− 1.09572i
$893$	− 1.46492i	− 0.0490216i
$894$	85.0892	2.84581
$895$	0	0
$896$	30.0318	1.00329
$897$	33.9566i	1.13378i
$898$	− 41.1007i	− 1.37155i
$899$	−51.0130	−1.70138
$900$	0	0
$901$	−20.1168	−0.670187
$902$	6.67483i	0.222247i
$903$	27.7470i	0.923361i
$904$	−10.0914	−0.335636
$905$	0	0
$906$	30.9732	1.02902
$907$	− 36.5154i	− 1.21247i	−0.795284	−	0.606237i	$-0.792678\pi$
$907$	− 36.5154i	− 1.21247i	0.795284	−	0.606237i	$-0.207322\pi$
$908$	17.1670i	0.569708i
$909$	15.9543	0.529172
$910$	0	0
$911$	6.62735	0.219574	0.109787	−	0.993955i	$-0.464983\pi$
$911$	6.62735	0.219574	0.109787	+	0.993955i	$0.464983\pi$
$912$	7.11475i	0.235593i
$913$	9.79036i	0.324014i
$914$	17.9110	0.592442
$915$	0	0
$916$	−0.194850	−0.00643802
$917$	48.1662i	1.59059i
$918$	− 109.292i	− 3.60717i
$919$	−7.91688	−0.261154	−0.130577	−	0.991438i	$-0.541683\pi$
$919$	−7.91688	−0.261154	−0.130577	+	0.991438i	$0.541683\pi$
$920$	0	0
$921$	−12.6210	−0.415877
$922$	53.6182i	1.76582i
$923$	− 12.6743i	− 0.417179i
$924$	−25.5726	−0.841275
$925$	0	0
$926$	−69.3611	−2.27935
$927$	32.0178i	1.05160i
$928$	− 60.6875i	− 1.99217i
$929$	−35.8597	−1.17652	−0.588259	−	0.808673i	$-0.700186\pi$
$929$	−35.8597	−1.17652	−0.588259	+	0.808673i	$0.700186\pi$
$930$	0	0
$931$	2.63990	0.0865192
$932$	24.6399i	0.807106i
$933$	− 45.2254i	− 1.48061i
$934$	32.6113	1.06707
$935$	0	0
$936$	−11.8481	−0.387268
$937$	− 11.4420i	− 0.373793i	−0.982380	−	0.186896i	$-0.940157\pi$
$937$	− 11.4420i	− 0.373793i	0.982380	−	0.186896i	$-0.0598428\pi$
$938$	− 105.200i	− 3.43490i
$939$	71.3236	2.32756
$940$	0	0
$941$	−0.360100	−0.0117389	−0.00586945	−	0.999983i	$-0.501868\pi$
$941$	−0.360100	−0.0117389	−0.00586945	+	0.999983i	$0.501868\pi$
$942$	44.3195i	1.44401i
$943$	18.7579i	0.610843i
$944$	−13.3223	−0.433605
$945$	0	0
$946$	7.34911	0.238940
$947$	2.63807i	0.0857256i	0.999081	+	0.0428628i	$0.0136478\pi$
$947$	2.63807i	0.0857256i	−0.999081	+	0.0428628i	$0.986352\pi$
$948$	− 70.0000i	− 2.27349i
$949$	−0.431517	−0.0140076
$950$	0	0
$951$	78.8084	2.55554
$952$	− 30.4432i	− 0.986670i
$953$	− 10.4430i	− 0.338282i	−0.985592	−	0.169141i	$-0.945901\pi$
$953$	− 10.4430i	− 0.338282i	0.985592	−	0.169141i	$-0.0540994\pi$
$954$	−29.3646	−0.950714
$955$	0	0
$956$	−7.58835	−0.245425
$957$	24.6667i	0.797360i
$958$	− 42.9525i	− 1.38773i
$959$	2.52437	0.0815160
$960$	0	0
$961$	12.2522	0.395232
$962$	5.34432i	0.172308i
$963$	− 107.731i	− 3.47159i
$964$	24.4173	0.786428
$965$	0	0
$966$	−127.359	−4.09772
$967$	33.2732i	1.06999i	0.844854	+	0.534996i	$0.179687\pi$
$967$	33.2732i	1.06999i	−0.844854	+	0.534996i	$0.820313\pi$
$968$	− 12.3620i	− 0.397330i
$969$	22.3264	0.717228
$970$	0	0
$971$	23.5657	0.756258	0.378129	−	0.925753i	$-0.376568\pi$
$971$	23.5657	0.756258	0.378129	+	0.925753i	$0.376568\pi$
$972$	28.1752i	0.903719i
$973$	39.6444i	1.27094i
$974$	−67.8098	−2.17277
$975$	0	0
$976$	−10.0732	−0.322437
$977$	0.402439i	0.0128752i	0.999979	+	0.00643759i	$0.00204916\pi$
$977$	0.402439i	0.0128752i	−0.999979	+	0.00643759i	$0.997951\pi$
$978$	− 44.7409i	− 1.43066i
$979$	0.898271	0.0287089
$980$	0	0
$981$	24.9821	0.797617
$982$	− 53.1395i	− 1.69575i
$983$	− 11.4927i	− 0.366561i	−0.983061	−	0.183281i	$-0.941328\pi$
$983$	− 11.4927i	− 0.366561i	0.983061	−	0.183281i	$-0.0586717\pi$
$984$	−10.2603	−0.327088
$985$	0	0
$986$	−128.902	−4.10508
$987$	− 13.0916i	− 0.416710i
$988$	− 4.59348i	− 0.146138i
$989$	20.6529	0.656723
$990$	0	0
$991$	6.05761	0.192426	0.0962132	−	0.995361i	$-0.469327\pi$
$991$	6.05761	0.192426	0.0962132	+	0.995361i	$0.469327\pi$
$992$	51.4549i	1.63370i
$993$	13.5292i	0.429335i
$994$	47.5368	1.50777
$995$	0	0
$996$	−66.0624	−2.09327
$997$	− 48.1086i	− 1.52362i	−0.647803	−	0.761808i	$-0.724312\pi$
$997$	− 48.1086i	− 1.52362i	0.647803	−	0.761808i	$-0.275688\pi$
$998$	− 70.7510i	− 2.23959i
$999$	−9.25020	−0.292663

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	475.2.b.e.324.2		8
5.2	odd	4		475.2.a.i.1.3		4
5.3	odd	4		95.2.a.b.1.2	✓	4
5.4	even	2	inner	475.2.b.e.324.7		8
15.2	even	4		4275.2.a.bo.1.2		4
15.8	even	4		855.2.a.m.1.3		4
20.3	even	4		1520.2.a.t.1.4		4
20.7	even	4		7600.2.a.cf.1.1		4
35.13	even	4		4655.2.a.y.1.2		4
40.3	even	4		6080.2.a.ch.1.1		4
40.13	odd	4		6080.2.a.cc.1.4		4
95.18	even	4		1805.2.a.p.1.3		4
95.37	even	4		9025.2.a.bf.1.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
95.2.a.b.1.2	✓	4	5.3	odd	4
475.2.a.i.1.3		4	5.2	odd	4
475.2.b.e.324.2		8	1.1	even	1	trivial
475.2.b.e.324.7		8	5.4	even	2	inner
855.2.a.m.1.3		4	15.8	even	4
1520.2.a.t.1.4		4	20.3	even	4
1805.2.a.p.1.3		4	95.18	even	4
4275.2.a.bo.1.2		4	15.2	even	4
4655.2.a.y.1.2		4	35.13	even	4
6080.2.a.cc.1.4		4	40.13	odd	4
6080.2.a.ch.1.1		4	40.3	even	4
7600.2.a.cf.1.1		4	20.7	even	4
9025.2.a.bf.1.2		4	95.37	even	4

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:	\( N \)	\(=\)	\( 475 = 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	475.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-2.14243i q^{2} +2.87834i q^{3} -2.59002 q^{4} +6.16666 q^{6} +3.10482i q^{7} +1.26409i q^{8} -5.28487 q^{9} +O(q^{10})\)	\(q-2.14243i q^{2} +2.87834i q^{3} -2.59002 q^{4} +6.16666 q^{6} +3.10482i q^{7} +1.26409i q^{8} -5.28487 q^{9} -1.10482 q^{11} -7.45498i q^{12} -1.77353i q^{13} +6.65187 q^{14} -2.47182 q^{16} +7.75669i q^{17} +11.3225i q^{18} -1.00000 q^{19} -8.93674 q^{21} +2.36700i q^{22} +6.65187i q^{23} -3.63849 q^{24} -3.79966 q^{26} -6.57664i q^{27} -8.04156i q^{28} -7.75669 q^{29} +6.57664 q^{31} +7.82389i q^{32} -3.18005i q^{33} +16.6182 q^{34} +13.6879 q^{36} -1.40652i q^{37} +2.14243i q^{38} +5.10482 q^{39} +2.81995 q^{41} +19.1464i q^{42} -3.10482i q^{43} +2.86151 q^{44} +14.2512 q^{46} +1.46492i q^{47} -7.11475i q^{48} -2.63990 q^{49} -22.3264 q^{51} +4.59348i q^{52} +2.59348i q^{53} -14.0900 q^{54} -3.92477 q^{56} -2.87834i q^{57} +16.6182i q^{58} +5.38969 q^{59} +4.07523 q^{61} -14.0900i q^{62} -16.4086i q^{63} +11.8185 q^{64} -6.81305 q^{66} -15.8151i q^{67} -20.0900i q^{68} -19.1464 q^{69} +7.14638 q^{71} -6.68055i q^{72} -0.243310i q^{73} -3.01339 q^{74} +2.59002 q^{76} -3.43026i q^{77} -10.9367i q^{78} +9.38969 q^{79} +3.07523 q^{81} -6.04156i q^{82} -8.86151i q^{83} +23.1464 q^{84} -6.65187 q^{86} -22.3264i q^{87} -1.39659i q^{88} -0.813048 q^{89} +5.50648 q^{91} -17.2285i q^{92} +18.9298i q^{93} +3.13849 q^{94} -22.5199 q^{96} +3.19689i q^{97} +5.65581i q^{98} +5.83882 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 16 q^{4} - 16 q^{9}+O(q^{10}) \) 8 * q - 16 * q^4 - 16 * q^9	\( 8 q - 16 q^{4} - 16 q^{9} + 8 q^{11} + 16 q^{14} + 8 q^{16} - 8 q^{19} - 8 q^{21} + 48 q^{24} + 8 q^{26} - 8 q^{29} + 8 q^{31} + 8 q^{34} + 80 q^{36} + 24 q^{39} + 32 q^{41} - 48 q^{44} - 40 q^{49} - 72 q^{51} + 40 q^{54} - 24 q^{56} + 40 q^{61} + 8 q^{64} - 56 q^{66} - 56 q^{69} - 40 q^{71} - 64 q^{74} + 16 q^{76} + 32 q^{79} + 32 q^{81} + 88 q^{84} - 16 q^{86} - 8 q^{89} - 72 q^{91} + 96 q^{94} - 104 q^{96} + 8 q^{99}+O(q^{100}) \) 8 * q - 16 * q^4 - 16 * q^9 + 8 * q^11 + 16 * q^14 + 8 * q^16 - 8 * q^19 - 8 * q^21 + 48 * q^24 + 8 * q^26 - 8 * q^29 + 8 * q^31 + 8 * q^34 + 80 * q^36 + 24 * q^39 + 32 * q^41 - 48 * q^44 - 40 * q^49 - 72 * q^51 + 40 * q^54 - 24 * q^56 + 40 * q^61 + 8 * q^64 - 56 * q^66 - 56 * q^69 - 40 * q^71 - 64 * q^74 + 16 * q^76 + 32 * q^79 + 32 * q^81 + 88 * q^84 - 16 * q^86 - 8 * q^89 - 72 * q^91 + 96 * q^94 - 104 * q^96 + 8 * q^99

Introduction

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