LMFDB - Embedded newform 475.2.b.c.324.5

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:	$6$
Coefficient field:	6.0.153664.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} + 5x^{4} + 6x^{2} + 1 $ x^6 + 5x^4 + 6x^2 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			324.5
Root			$-1.80194i$ of defining polynomial
Character	$\chi$	$=$	475.324
Dual form			475.2.b.c.324.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.44504i q^{2} -2.24698i q^{3} -0.0881460 q^{4} +3.24698 q^{6} +1.35690i q^{7} +2.76271i q^{8} -2.04892 q^{9} +O(q^{10})$	\(q+1.44504i q^{2} -2.24698i q^{3} -0.0881460 q^{4} +3.24698 q^{6} +1.35690i q^{7} +2.76271i q^{8} -2.04892 q^{9} +4.85086 q^{11} +0.198062i q^{12} -0.198062i q^{13} -1.96077 q^{14} -4.16852 q^{16} -1.13706i q^{17} -2.96077i q^{18} +1.00000 q^{19} +3.04892 q^{21} +7.00969i q^{22} -2.55496i q^{23} +6.20775 q^{24} +0.286208 q^{26} -2.13706i q^{27} -0.119605i q^{28} +10.2349 q^{29} +2.51573 q^{31} -0.498271i q^{32} -10.8998i q^{33} +1.64310 q^{34} +0.180604 q^{36} -0.137063i q^{37} +1.44504i q^{38} -0.445042 q^{39} -11.7506 q^{41} +4.40581i q^{42} +7.59179i q^{43} -0.427583 q^{44} +3.69202 q^{46} +2.69202i q^{47} +9.36658i q^{48} +5.15883 q^{49} -2.55496 q^{51} +0.0174584i q^{52} -12.8780i q^{53} +3.08815 q^{54} -3.74871 q^{56} -2.24698i q^{57} +14.7899i q^{58} -5.82371 q^{59} -7.58211 q^{61} +3.63533i q^{62} -2.78017i q^{63} -7.61702 q^{64} +15.7506 q^{66} +8.01507i q^{67} +0.100228i q^{68} -5.74094 q^{69} -8.82371 q^{71} -5.66056i q^{72} +11.9705i q^{73} +0.198062 q^{74} -0.0881460 q^{76} +6.58211i q^{77} -0.643104i q^{78} -10.7409 q^{79} -10.9487 q^{81} -16.9801i q^{82} -3.77479i q^{83} -0.268750 q^{84} -10.9705 q^{86} -22.9976i q^{87} +13.4015i q^{88} -9.36658 q^{89} +0.268750 q^{91} +0.225209i q^{92} -5.65279i q^{93} -3.89008 q^{94} -1.11960 q^{96} +0.198062i q^{97} +7.45473i q^{98} -9.93900 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 8 q^{4} + 10 q^{6} + 6 q^{9}+O(q^{10}) $ 6 * q - 8 * q^4 + 10 * q^6 + 6 * q^9	$ 6 q - 8 q^{4} + 10 q^{6} + 6 q^{9} + 2 q^{11} + 14 q^{14} + 36 q^{16} + 6 q^{19} + 2 q^{24} + 18 q^{26} + 14 q^{29} - 10 q^{31} + 18 q^{34} - 22 q^{36} - 2 q^{39} + 2 q^{41} + 30 q^{44} + 12 q^{46} + 14 q^{49} - 16 q^{51} + 26 q^{54} - 70 q^{56} - 20 q^{59} - 34 q^{61} - 98 q^{64} + 22 q^{66} - 6 q^{69} - 38 q^{71} + 10 q^{74} - 8 q^{76} - 36 q^{79} - 2 q^{81} + 14 q^{84} + 4 q^{86} - 4 q^{89} - 14 q^{91} - 22 q^{94} + 36 q^{96} - 40 q^{99}+O(q^{100}) $ 6 * q - 8 * q^4 + 10 * q^6 + 6 * q^9 + 2 * q^11 + 14 * q^14 + 36 * q^16 + 6 * q^19 + 2 * q^24 + 18 * q^26 + 14 * q^29 - 10 * q^31 + 18 * q^34 - 22 * q^36 - 2 * q^39 + 2 * q^41 + 30 * q^44 + 12 * q^46 + 14 * q^49 - 16 * q^51 + 26 * q^54 - 70 * q^56 - 20 * q^59 - 34 * q^61 - 98 * q^64 + 22 * q^66 - 6 * q^69 - 38 * q^71 + 10 * q^74 - 8 * q^76 - 36 * q^79 - 2 * q^81 + 14 * q^84 + 4 * q^86 - 4 * q^89 - 14 * q^91 - 22 * q^94 + 36 * q^96 - 40 * 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 $

$2$	1.44504i	1.02180i	0.859640	+	0.510899i	$0.170688\pi$
$2$	1.44504i	1.02180i	−0.859640	+	0.510899i	$0.829312\pi$
$3$	− 2.24698i	− 1.29729i	−0.761089	−	0.648647i	$-0.775335\pi$
$3$	− 2.24698i	− 1.29729i	0.761089	−	0.648647i	$-0.224665\pi$
$4$	−0.0881460	−0.0440730
$5$	0	0
$5$	0	0
$6$	3.24698	1.32557
$7$	1.35690i	0.512858i	0.966563	+	0.256429i	$0.0825461\pi$
$7$	1.35690i	0.512858i	−0.966563	+	0.256429i	$0.917454\pi$
$8$	2.76271i	0.976765i
$9$	−2.04892	−0.682972
$10$	0	0
$11$	4.85086	1.46259	0.731294	−	0.682062i	$-0.238917\pi$
$11$	4.85086	1.46259	0.731294	+	0.682062i	$0.238917\pi$
$12$	0.198062i	0.0571757i
$13$	− 0.198062i	− 0.0549326i	−0.999623	−	0.0274663i	$-0.991256\pi$
$13$	− 0.198062i	− 0.0549326i	0.999623	−	0.0274663i	$-0.00874389\pi$
$14$	−1.96077	−0.524038
$15$	0	0
$16$	−4.16852	−1.04213
$17$	− 1.13706i	− 0.275778i	−0.990448	−	0.137889i	$-0.955968\pi$
$17$	− 1.13706i	− 0.275778i	0.990448	−	0.137889i	$-0.0440318\pi$
$18$	− 2.96077i	− 0.697860i
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	3.04892	0.665328
$22$	7.00969i	1.49447i
$23$	− 2.55496i	− 0.532746i	−0.963870	−	0.266373i	$-0.914175\pi$
$23$	− 2.55496i	− 0.532746i	0.963870	−	0.266373i	$-0.0858252\pi$
$24$	6.20775	1.26715
$25$	0	0
$26$	0.286208	0.0561301
$27$	− 2.13706i	− 0.411278i
$28$	− 0.119605i	− 0.0226032i
$29$	10.2349	1.90057	0.950286	−	0.311377i	$-0.100790\pi$
$29$	10.2349	1.90057	0.950286	+	0.311377i	$0.100790\pi$
$30$	0	0
$31$	2.51573	0.451838	0.225919	−	0.974146i	$-0.427461\pi$
$31$	2.51573	0.451838	0.225919	+	0.974146i	$0.427461\pi$
$32$	− 0.498271i	− 0.0880827i
$33$	− 10.8998i	− 1.89741i
$34$	1.64310	0.281790
$35$	0	0
$36$	0.180604	0.0301006
$37$	− 0.137063i	− 0.0225331i	−0.999937	−	0.0112665i	$-0.996414\pi$
$37$	− 0.137063i	− 0.0225331i	0.999937	−	0.0112665i	$-0.00358633\pi$
$38$	1.44504i	0.234417i
$39$	−0.445042	−0.0712637
$40$	0	0
$41$	−11.7506	−1.83514	−0.917570	−	0.397575i	$-0.869852\pi$
$41$	−11.7506	−1.83514	−0.917570	+	0.397575i	$0.869852\pi$
$42$	4.40581i	0.679832i
$43$	7.59179i	1.15774i	0.815421	+	0.578869i	$0.196506\pi$
$43$	7.59179i	1.15774i	−0.815421	+	0.578869i	$0.803494\pi$
$44$	−0.427583	−0.0644606
$45$	0	0
$46$	3.69202	0.544359
$47$	2.69202i	0.392672i	0.980537	+	0.196336i	$0.0629043\pi$
$47$	2.69202i	0.392672i	−0.980537	+	0.196336i	$0.937096\pi$
$48$	9.36658i	1.35195i
$49$	5.15883	0.736976
$50$	0	0
$51$	−2.55496	−0.357766
$52$	0.0174584i	0.00242104i
$53$	− 12.8780i	− 1.76893i	−0.466607	−	0.884465i	$-0.654524\pi$
$53$	− 12.8780i	− 1.76893i	0.466607	−	0.884465i	$-0.345476\pi$
$54$	3.08815	0.420243
$55$	0	0
$56$	−3.74871	−0.500942
$57$	− 2.24698i	− 0.297620i
$58$	14.7899i	1.94200i
$59$	−5.82371	−0.758182	−0.379091	−	0.925359i	$-0.623763\pi$
$59$	−5.82371	−0.758182	−0.379091	+	0.925359i	$0.623763\pi$
$60$	0	0
$61$	−7.58211	−0.970789	−0.485395	−	0.874295i	$-0.661324\pi$
$61$	−7.58211	−0.970789	−0.485395	+	0.874295i	$0.661324\pi$
$62$	3.63533i	0.461688i
$63$	− 2.78017i	− 0.350268i
$64$	−7.61702	−0.952128
$65$	0	0
$66$	15.7506	1.93877
$67$	8.01507i	0.979196i	0.871948	+	0.489598i	$0.162856\pi$
$67$	8.01507i	0.979196i	−0.871948	+	0.489598i	$0.837144\pi$
$68$	0.100228i	0.0121544i
$69$	−5.74094	−0.691128
$70$	0	0
$71$	−8.82371	−1.04718	−0.523591	−	0.851970i	$-0.675408\pi$
$71$	−8.82371	−1.04718	−0.523591	+	0.851970i	$0.675408\pi$
$72$	− 5.66056i	− 0.667104i
$73$	11.9705i	1.40104i	0.713635	+	0.700518i	$0.247048\pi$
$73$	11.9705i	1.40104i	−0.713635	+	0.700518i	$0.752952\pi$
$74$	0.198062	0.0230243
$75$	0	0
$76$	−0.0881460	−0.0101110
$77$	6.58211i	0.750101i
$78$	− 0.643104i	− 0.0728172i
$79$	−10.7409	−1.20845	−0.604225	−	0.796814i	$-0.706517\pi$
$79$	−10.7409	−1.20845	−0.604225	+	0.796814i	$0.706517\pi$
$80$	0	0
$81$	−10.9487	−1.21652
$82$	− 16.9801i	− 1.87514i
$83$	− 3.77479i	− 0.414337i	−0.978305	−	0.207169i	$-0.933575\pi$
$83$	− 3.77479i	− 0.414337i	0.978305	−	0.207169i	$-0.0664249\pi$
$84$	−0.268750	−0.0293230
$85$	0	0
$86$	−10.9705	−1.18298
$87$	− 22.9976i	− 2.46560i
$88$	13.4015i	1.42860i
$89$	−9.36658	−0.992856	−0.496428	−	0.868078i	$-0.665355\pi$
$89$	−9.36658	−0.992856	−0.496428	+	0.868078i	$0.665355\pi$
$90$	0	0
$91$	0.268750	0.0281726
$92$	0.225209i	0.0234797i
$93$	− 5.65279i	− 0.586167i
$94$	−3.89008	−0.401232
$95$	0	0
$96$	−1.11960	−0.114269
$97$	0.198062i	0.0201102i	0.999949	+	0.0100551i	$0.00320069\pi$
$97$	0.198062i	0.0201102i	−0.999949	+	0.0100551i	$0.996799\pi$
$98$	7.45473i	0.753042i
$99$	−9.93900	−0.998907
$100$	0	0
$101$	11.5090	1.14519	0.572595	−	0.819838i	$-0.305937\pi$
$101$	11.5090	1.14519	0.572595	+	0.819838i	$0.305937\pi$
$102$	− 3.69202i	− 0.365565i
$103$	− 15.1564i	− 1.49341i	−0.665156	−	0.746704i	$-0.731635\pi$
$103$	− 15.1564i	− 1.49341i	0.665156	−	0.746704i	$-0.268365\pi$
$104$	0.547188	0.0536562
$105$	0	0
$106$	18.6093	1.80749
$107$	2.65279i	0.256455i	0.991745	+	0.128228i	$0.0409288\pi$
$107$	2.65279i	0.256455i	−0.991745	+	0.128228i	$0.959071\pi$
$108$	0.188374i	0.0181263i
$109$	2.49934	0.239393	0.119696	−	0.992811i	$-0.461808\pi$
$109$	2.49934	0.239393	0.119696	+	0.992811i	$0.461808\pi$
$110$	0	0
$111$	−0.307979	−0.0292320
$112$	− 5.65625i	− 0.534465i
$113$	− 8.52781i	− 0.802229i	−0.916028	−	0.401114i	$-0.868623\pi$
$113$	− 8.52781i	− 0.802229i	0.916028	−	0.401114i	$-0.131377\pi$
$114$	3.24698	0.304108
$115$	0	0
$116$	−0.902165	−0.0837639
$117$	0.405813i	0.0375174i
$118$	− 8.41550i	− 0.774710i
$119$	1.54288	0.141435
$120$	0	0
$121$	12.5308	1.13916
$122$	− 10.9565i	− 0.991951i
$123$	26.4034i	2.38072i
$124$	−0.221751	−0.0199139
$125$	0	0
$126$	4.01746	0.357904
$127$	20.4088i	1.81099i	0.424359	+	0.905494i	$0.360499\pi$
$127$	20.4088i	1.81099i	−0.424359	+	0.905494i	$0.639501\pi$
$128$	− 12.0035i	− 1.06097i
$129$	17.0586	1.50193
$130$	0	0
$131$	−13.2131	−1.15444	−0.577218	−	0.816590i	$-0.695862\pi$
$131$	−13.2131	−1.15444	−0.577218	+	0.816590i	$0.695862\pi$
$132$	0.960771i	0.0836244i
$133$	1.35690i	0.117658i
$134$	−11.5821	−1.00054
$135$	0	0
$136$	3.14138	0.269371
$137$	− 6.86054i	− 0.586136i	−0.956092	−	0.293068i	$-0.905324\pi$
$137$	− 6.86054i	− 0.586136i	0.956092	−	0.293068i	$-0.0946762\pi$
$138$	− 8.29590i	− 0.706194i
$139$	−4.28621	−0.363551	−0.181776	−	0.983340i	$-0.558184\pi$
$139$	−4.28621	−0.363551	−0.181776	+	0.983340i	$0.558184\pi$
$140$	0	0
$141$	6.04892	0.509411
$142$	− 12.7506i	− 1.07001i
$143$	− 0.960771i	− 0.0803437i
$144$	8.54096	0.711746
$145$	0	0
$146$	−17.2978	−1.43158
$147$	− 11.5918i	− 0.956075i
$148$	0.0120816i	0 0.000993100i
$149$	15.3545	1.25789	0.628945	−	0.777450i	$-0.283487\pi$
$149$	15.3545	1.25789	0.628945	+	0.777450i	$0.283487\pi$
$150$	0	0
$151$	−10.2295	−0.832467	−0.416233	−	0.909258i	$-0.636650\pi$
$151$	−10.2295	−0.832467	−0.416233	+	0.909258i	$0.636650\pi$
$152$	2.76271i	0.224085i
$153$	2.32975i	0.188349i
$154$	−9.51142	−0.766452
$155$	0	0
$156$	0.0392287	0.00314081
$157$	3.17092i	0.253067i	0.991962	+	0.126533i	$0.0403851\pi$
$157$	3.17092i	0.253067i	−0.991962	+	0.126533i	$0.959615\pi$
$158$	− 15.5211i	− 1.23479i
$159$	−28.9366	−2.29482
$160$	0	0
$161$	3.46681	0.273223
$162$	− 15.8213i	− 1.24304i
$163$	4.63773i	0.363255i	0.983367	+	0.181627i	$0.0581365\pi$
$163$	4.63773i	0.363255i	−0.983367	+	0.181627i	$0.941864\pi$
$164$	1.03577	0.0808801
$165$	0	0
$166$	5.45473	0.423369
$167$	− 19.6286i	− 1.51891i	−0.650560	−	0.759454i	$-0.725466\pi$
$167$	− 19.6286i	− 1.51891i	0.650560	−	0.759454i	$-0.274534\pi$
$168$	8.42327i	0.649870i
$169$	12.9608	0.996982
$170$	0	0
$171$	−2.04892	−0.156685
$172$	− 0.669186i	− 0.0510250i
$173$	− 20.5646i	− 1.56350i	−0.623591	−	0.781751i	$-0.714327\pi$
$173$	− 20.5646i	− 1.56350i	0.623591	−	0.781751i	$-0.285673\pi$
$174$	33.2325	2.51935
$175$	0	0
$176$	−20.2209	−1.52421
$177$	13.0858i	0.983585i
$178$	− 13.5351i	− 1.01450i
$179$	−6.92154	−0.517340	−0.258670	−	0.965966i	$-0.583284\pi$
$179$	−6.92154	−0.517340	−0.258670	+	0.965966i	$0.583284\pi$
$180$	0	0
$181$	−17.6461	−1.31162	−0.655812	−	0.754925i	$-0.727673\pi$
$181$	−17.6461	−1.31162	−0.655812	+	0.754925i	$0.727673\pi$
$182$	0.388355i	0.0287868i
$183$	17.0368i	1.25940i
$184$	7.05861	0.520367
$185$	0	0
$186$	8.16852	0.598945
$187$	− 5.51573i	− 0.403350i
$188$	− 0.237291i	− 0.0173062i
$189$	2.89977	0.210927
$190$	0	0
$191$	6.92931	0.501387	0.250694	−	0.968066i	$-0.419341\pi$
$191$	6.92931	0.501387	0.250694	+	0.968066i	$0.419341\pi$
$192$	17.1153i	1.23519i
$193$	21.0368i	1.51426i	0.653262	+	0.757132i	$0.273400\pi$
$193$	21.0368i	1.51426i	−0.653262	+	0.757132i	$0.726600\pi$
$194$	−0.286208	−0.0205486
$195$	0	0
$196$	−0.454731	−0.0324808
$197$	21.5646i	1.53642i	0.640199	+	0.768209i	$0.278852\pi$
$197$	21.5646i	1.53642i	−0.640199	+	0.768209i	$0.721148\pi$
$198$	− 14.3623i	− 1.02068i
$199$	−21.9909	−1.55889	−0.779447	−	0.626468i	$-0.784500\pi$
$199$	−21.9909	−1.55889	−0.779447	+	0.626468i	$0.784500\pi$
$200$	0	0
$201$	18.0097	1.27031
$202$	16.6310i	1.17015i
$203$	13.8877i	0.974725i
$204$	0.225209	0.0157678
$205$	0	0
$206$	21.9017	1.52596
$207$	5.23490i	0.363851i
$208$	0.825627i	0.0572469i
$209$	4.85086	0.335541
$210$	0	0
$211$	−20.6233	−1.41976	−0.709882	−	0.704321i	$-0.751252\pi$
$211$	−20.6233	−1.41976	−0.709882	+	0.704321i	$0.751252\pi$
$212$	1.13514i	0.0779620i
$213$	19.8267i	1.35850i
$214$	−3.83340	−0.262046
$215$	0	0
$216$	5.90408	0.401722
$217$	3.41358i	0.231729i
$218$	3.61165i	0.244611i
$219$	26.8974	1.81756
$220$	0	0
$221$	−0.225209	−0.0151492
$222$	− 0.445042i	− 0.0298693i
$223$	− 7.97716i	− 0.534190i	−0.963670	−	0.267095i	$-0.913936\pi$
$223$	− 7.97716i	− 0.534190i	0.963670	−	0.267095i	$-0.0860638\pi$
$224$	0.676102	0.0451740
$225$	0	0
$226$	12.3230	0.819717
$227$	19.7942i	1.31379i	0.753984	+	0.656893i	$0.228129\pi$
$227$	19.7942i	1.31379i	−0.753984	+	0.656893i	$0.771871\pi$
$228$	0.198062i	0.0131170i
$229$	4.03385	0.266564	0.133282	−	0.991078i	$-0.457448\pi$
$229$	4.03385	0.266564	0.133282	+	0.991078i	$0.457448\pi$
$230$	0	0
$231$	14.7899	0.973101
$232$	28.2760i	1.85641i
$233$	− 26.8159i	− 1.75677i	−0.477953	−	0.878385i	$-0.658621\pi$
$233$	− 26.8159i	− 1.75677i	0.477953	−	0.878385i	$-0.341379\pi$
$234$	−0.586417	−0.0383353
$235$	0	0
$236$	0.513337	0.0334154
$237$	24.1347i	1.56772i
$238$	2.22952i	0.144518i
$239$	−3.36227	−0.217487	−0.108744	−	0.994070i	$-0.534683\pi$
$239$	−3.36227	−0.217487	−0.108744	+	0.994070i	$0.534683\pi$
$240$	0	0
$241$	27.7506	1.78758	0.893788	−	0.448491i	$-0.148038\pi$
$241$	27.7506	1.78758	0.893788	+	0.448491i	$0.148038\pi$
$242$	18.1075i	1.16400i
$243$	18.1903i	1.16691i
$244$	0.668332	0.0427856
$245$	0	0
$246$	−38.1540	−2.43261
$247$	− 0.198062i	− 0.0126024i
$248$	6.95023i	0.441340i
$249$	−8.48188	−0.537517
$250$	0	0
$251$	−5.59419	−0.353102	−0.176551	−	0.984291i	$-0.556494\pi$
$251$	−5.59419	−0.353102	−0.176551	+	0.984291i	$0.556494\pi$
$252$	0.245061i	0.0154374i
$253$	− 12.3937i	− 0.779187i
$254$	−29.4916	−1.85047
$255$	0	0
$256$	2.11146	0.131966
$257$	10.4668i	0.652902i	0.945214	+	0.326451i	$0.105853\pi$
$257$	10.4668i	0.652902i	−0.945214	+	0.326451i	$0.894147\pi$
$258$	24.6504i	1.53467i
$259$	0.185981	0.0115563
$260$	0	0
$261$	−20.9705	−1.29804
$262$	− 19.0935i	− 1.17960i
$263$	− 15.4795i	− 0.954506i	−0.878766	−	0.477253i	$-0.841633\pi$
$263$	− 15.4795i	− 0.954506i	0.878766	−	0.477253i	$-0.158367\pi$
$264$	30.1129	1.85332
$265$	0	0
$266$	−1.96077	−0.120223
$267$	21.0465i	1.28803i
$268$	− 0.706496i	− 0.0431561i
$269$	−9.13036	−0.556688	−0.278344	−	0.960481i	$-0.589785\pi$
$269$	−9.13036	−0.556688	−0.278344	+	0.960481i	$0.589785\pi$
$270$	0	0
$271$	7.44265	0.452109	0.226054	−	0.974115i	$-0.427417\pi$
$271$	7.44265	0.452109	0.226054	+	0.974115i	$0.427417\pi$
$272$	4.73987i	0.287397i
$273$	− 0.603875i	− 0.0365482i
$274$	9.91377	0.598913
$275$	0	0
$276$	0.506041	0.0304601
$277$	− 11.4155i	− 0.685891i	−0.939355	−	0.342946i	$-0.888575\pi$
$277$	− 11.4155i	− 0.685891i	0.939355	−	0.342946i	$-0.111425\pi$
$278$	− 6.19375i	− 0.371476i
$279$	−5.15452	−0.308593
$280$	0	0
$281$	21.5060	1.28294	0.641471	−	0.767147i	$-0.278324\pi$
$281$	21.5060	1.28294	0.641471	+	0.767147i	$0.278324\pi$
$282$	8.74094i	0.520515i
$283$	5.45712i	0.324392i	0.986759	+	0.162196i	$0.0518577\pi$
$283$	5.45712i	0.324392i	−0.986759	+	0.162196i	$0.948142\pi$
$284$	0.777775	0.0461524
$285$	0	0
$286$	1.38835	0.0820951
$287$	− 15.9444i	− 0.941167i
$288$	1.02092i	0.0601581i
$289$	15.7071	0.923946
$290$	0	0
$291$	0.445042	0.0260888
$292$	− 1.05515i	− 0.0617479i
$293$	− 7.39075i	− 0.431772i	−0.976419	−	0.215886i	$-0.930736\pi$
$293$	− 7.39075i	− 0.431772i	0.976419	−	0.215886i	$-0.0692640\pi$
$294$	16.7506	0.976916
$295$	0	0
$296$	0.378666	0.0220095
$297$	− 10.3666i	− 0.601530i
$298$	22.1879i	1.28531i
$299$	−0.506041	−0.0292651
$300$	0	0
$301$	−10.3013	−0.593756
$302$	− 14.7821i	− 0.850613i
$303$	− 25.8605i	− 1.48565i
$304$	−4.16852	−0.239081
$305$	0	0
$306$	−3.36658	−0.192455
$307$	− 32.2131i	− 1.83850i	−0.393674	−	0.919250i	$-0.628796\pi$
$307$	− 32.2131i	− 1.83850i	0.393674	−	0.919250i	$-0.371204\pi$
$308$	− 0.580186i	− 0.0330592i
$309$	−34.0562	−1.93739
$310$	0	0
$311$	14.8442	0.841735	0.420867	−	0.907122i	$-0.361726\pi$
$311$	14.8442	0.841735	0.420867	+	0.907122i	$0.361726\pi$
$312$	− 1.22952i	− 0.0696079i
$313$	13.1491i	0.743234i	0.928386	+	0.371617i	$0.121196\pi$
$313$	13.1491i	0.743234i	−0.928386	+	0.371617i	$0.878804\pi$
$314$	−4.58211	−0.258583
$315$	0	0
$316$	0.946771	0.0532600
$317$	− 5.13467i	− 0.288392i	−0.989549	−	0.144196i	$-0.953940\pi$
$317$	− 5.13467i	− 0.288392i	0.989549	−	0.144196i	$-0.0460595\pi$
$318$	− 41.8146i	− 2.34485i
$319$	49.6480	2.77975
$320$	0	0
$321$	5.96077	0.332698
$322$	5.00969i	0.279179i
$323$	− 1.13706i	− 0.0632679i
$324$	0.965083	0.0536157
$325$	0	0
$326$	−6.70171	−0.371173
$327$	− 5.61596i	− 0.310563i
$328$	− 32.4636i	− 1.79250i
$329$	−3.65279	−0.201385
$330$	0	0
$331$	2.00969	0.110462	0.0552312	−	0.998474i	$-0.482410\pi$
$331$	2.00969	0.110462	0.0552312	+	0.998474i	$0.482410\pi$
$332$	0.332733i	0.0182611i
$333$	0.280831i	0.0153895i
$334$	28.3642	1.55202
$335$	0	0
$336$	−12.7095	−0.693359
$337$	− 1.31873i	− 0.0718359i	−0.999355	−	0.0359180i	$-0.988564\pi$
$337$	− 1.31873i	− 0.0718359i	0.999355	−	0.0359180i	$-0.0114355\pi$
$338$	18.7289i	1.01872i
$339$	−19.1618	−1.04073
$340$	0	0
$341$	12.2034	0.660853
$342$	− 2.96077i	− 0.160100i
$343$	16.4983i	0.890823i
$344$	−20.9739	−1.13084
$345$	0	0
$346$	29.7168	1.59758
$347$	12.0151i	0.645003i	0.946569	+	0.322501i	$0.104524\pi$
$347$	12.0151i	0.645003i	−0.946569	+	0.322501i	$0.895476\pi$
$348$	2.02715i	0.108666i
$349$	10.5579	0.565154	0.282577	−	0.959245i	$-0.408811\pi$
$349$	10.5579	0.565154	0.282577	+	0.959245i	$0.408811\pi$
$350$	0	0
$351$	−0.423272	−0.0225926
$352$	− 2.41704i	− 0.128829i
$353$	1.01102i	0.0538110i	0.999638	+	0.0269055i	$0.00856532\pi$
$353$	1.01102i	0.0538110i	−0.999638	+	0.0269055i	$0.991435\pi$
$354$	−18.9095	−1.00503
$355$	0	0
$356$	0.825627	0.0437581
$357$	− 3.46681i	− 0.183483i
$358$	− 10.0019i	− 0.528618i
$359$	5.14244	0.271408	0.135704	−	0.990749i	$-0.456670\pi$
$359$	5.14244	0.271408	0.135704	+	0.990749i	$0.456670\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	− 25.4993i	− 1.34022i
$363$	− 28.1564i	− 1.47783i
$364$	−0.0236892	−0.00124165
$365$	0	0
$366$	−24.6189	−1.28685
$367$	− 29.5308i	− 1.54149i	−0.637141	−	0.770747i	$-0.719883\pi$
$367$	− 29.5308i	− 1.54149i	0.637141	−	0.770747i	$-0.280117\pi$
$368$	10.6504i	0.555190i
$369$	24.0761	1.25335
$370$	0	0
$371$	17.4741	0.907210
$372$	0.498271i	0.0258342i
$373$	− 8.78986i	− 0.455121i	−0.973764	−	0.227561i	$-0.926925\pi$
$373$	− 8.78986i	− 0.455121i	0.973764	−	0.227561i	$-0.0730750\pi$
$374$	7.97046	0.412143
$375$	0	0
$376$	−7.43727	−0.383548
$377$	− 2.02715i	− 0.104403i
$378$	4.19029i	0.215525i
$379$	−19.1511	−0.983724	−0.491862	−	0.870673i	$-0.663684\pi$
$379$	−19.1511	−0.983724	−0.491862	+	0.870673i	$0.663684\pi$
$380$	0	0
$381$	45.8582	2.34938
$382$	10.0131i	0.512317i
$383$	6.99894i	0.357629i	0.983883	+	0.178814i	$0.0572262\pi$
$383$	6.99894i	0.357629i	−0.983883	+	0.178814i	$0.942774\pi$
$384$	−26.9715	−1.37638
$385$	0	0
$386$	−30.3991	−1.54727
$387$	− 15.5550i	− 0.790703i
$388$	− 0.0174584i	0 0.000886316i
$389$	−8.08575	−0.409964	−0.204982	−	0.978766i	$-0.565714\pi$
$389$	−8.08575	−0.409964	−0.204982	+	0.978766i	$0.565714\pi$
$390$	0	0
$391$	−2.90515	−0.146920
$392$	14.2524i	0.719853i
$393$	29.6896i	1.49764i
$394$	−31.1618	−1.56991
$395$	0	0
$396$	0.876083	0.0440248
$397$	− 17.7006i	− 0.888370i	−0.895935	−	0.444185i	$-0.853493\pi$
$397$	− 17.7006i	− 0.888370i	0.895935	−	0.444185i	$-0.146507\pi$
$398$	− 31.7778i	− 1.59288i
$399$	3.04892	0.152637
$400$	0	0
$401$	−15.5418	−0.776121	−0.388061	−	0.921634i	$-0.626855\pi$
$401$	−15.5418	−0.776121	−0.388061	+	0.921634i	$0.626855\pi$
$402$	26.0248i	1.29800i
$403$	− 0.498271i	− 0.0248207i
$404$	−1.01447	−0.0504720
$405$	0	0
$406$	−20.0683	−0.995973
$407$	− 0.664874i	− 0.0329566i
$408$	− 7.05861i	− 0.349453i
$409$	−13.1661	−0.651023	−0.325512	−	0.945538i	$-0.605537\pi$
$409$	−13.1661	−0.651023	−0.325512	+	0.945538i	$0.605537\pi$
$410$	0	0
$411$	−15.4155	−0.760391
$412$	1.33598i	0.0658190i
$413$	− 7.90217i	− 0.388840i
$414$	−7.56465	−0.371782
$415$	0	0
$416$	−0.0986887	−0.00483861
$417$	9.63102i	0.471633i
$418$	7.00969i	0.342855i
$419$	25.1739	1.22983	0.614913	−	0.788595i	$-0.289191\pi$
$419$	25.1739	1.22983	0.614913	+	0.788595i	$0.289191\pi$
$420$	0	0
$421$	26.9420	1.31307	0.656536	−	0.754295i	$-0.272021\pi$
$421$	26.9420	1.31307	0.656536	+	0.754295i	$0.272021\pi$
$422$	− 29.8015i	− 1.45071i
$423$	− 5.51573i	− 0.268184i
$424$	35.5782	1.72783
$425$	0	0
$426$	−28.6504	−1.38812
$427$	− 10.2881i	− 0.497877i
$428$	− 0.233833i	− 0.0113027i
$429$	−2.15883	−0.104229
$430$	0	0
$431$	18.9487	0.912726	0.456363	−	0.889794i	$-0.349152\pi$
$431$	18.9487	0.912726	0.456363	+	0.889794i	$0.349152\pi$
$432$	8.90840i	0.428605i
$433$	3.68904i	0.177284i	0.996064	+	0.0886419i	$0.0282527\pi$
$433$	3.68904i	0.177284i	−0.996064	+	0.0886419i	$0.971747\pi$
$434$	−4.93277	−0.236781
$435$	0	0
$436$	−0.220306	−0.0105508
$437$	− 2.55496i	− 0.122220i
$438$	38.8678i	1.85718i
$439$	25.6926	1.22624	0.613121	−	0.789989i	$-0.289914\pi$
$439$	25.6926	1.22624	0.613121	+	0.789989i	$0.289914\pi$
$440$	0	0
$441$	−10.5700	−0.503334
$442$	− 0.325437i	− 0.0154795i
$443$	27.3653i	1.30016i	0.759865	+	0.650081i	$0.225265\pi$
$443$	27.3653i	1.30016i	−0.759865	+	0.650081i	$0.774735\pi$
$444$	0.0271471	0.00128834
$445$	0	0
$446$	11.5273	0.545835
$447$	− 34.5013i	− 1.63185i
$448$	− 10.3355i	− 0.488307i
$449$	−7.55794	−0.356681	−0.178341	−	0.983969i	$-0.557073\pi$
$449$	−7.55794	−0.356681	−0.178341	+	0.983969i	$0.557073\pi$
$450$	0	0
$451$	−57.0006	−2.68405
$452$	0.751692i	0.0353566i
$453$	22.9855i	1.07995i
$454$	−28.6034	−1.34242
$455$	0	0
$456$	6.20775	0.290705
$457$	− 7.85623i	− 0.367499i	−0.982973	−	0.183750i	$-0.941176\pi$
$457$	− 7.85623i	− 0.367499i	0.982973	−	0.183750i	$-0.0588235\pi$
$458$	5.82908i	0.272375i
$459$	−2.42998	−0.113422
$460$	0	0
$461$	3.87907	0.180666	0.0903331	−	0.995912i	$-0.471207\pi$
$461$	3.87907	0.180666	0.0903331	+	0.995912i	$0.471207\pi$
$462$	21.3720i	0.994314i
$463$	13.0954i	0.608597i	0.952577	+	0.304298i	$0.0984220\pi$
$463$	13.0954i	0.608597i	−0.952577	+	0.304298i	$0.901578\pi$
$464$	−42.6644	−1.98065
$465$	0	0
$466$	38.7502	1.79507
$467$	6.44026i	0.298020i	0.988836	+	0.149010i	$0.0476086\pi$
$467$	6.44026i	0.298020i	−0.988836	+	0.149010i	$0.952391\pi$
$468$	− 0.0357708i	− 0.00165351i
$469$	−10.8756	−0.502189
$470$	0	0
$471$	7.12498	0.328302
$472$	− 16.0892i	− 0.740566i
$473$	36.8267i	1.69329i
$474$	−34.8756	−1.60189
$475$	0	0
$476$	−0.135998	−0.00623348
$477$	26.3860i	1.20813i
$478$	− 4.85862i	− 0.222228i
$479$	−5.69096	−0.260026	−0.130013	−	0.991512i	$-0.541502\pi$
$479$	−5.69096	−0.260026	−0.130013	+	0.991512i	$0.541502\pi$
$480$	0	0
$481$	−0.0271471	−0.00123780
$482$	40.1008i	1.82654i
$483$	− 7.78986i	− 0.354451i
$484$	−1.10454	−0.0502063
$485$	0	0
$486$	−26.2857	−1.19235
$487$	− 12.9632i	− 0.587417i	−0.955895	−	0.293709i	$-0.905110\pi$
$487$	− 12.9632i	− 0.587417i	0.955895	−	0.293709i	$-0.0948895\pi$
$488$	− 20.9472i	− 0.948233i
$489$	10.4209	0.471248
$490$	0	0
$491$	−19.6045	−0.884737	−0.442369	−	0.896833i	$-0.645862\pi$
$491$	−19.6045	−0.884737	−0.442369	+	0.896833i	$0.645862\pi$
$492$	− 2.32736i	− 0.104925i
$493$	− 11.6377i	− 0.524137i
$494$	0.286208	0.0128771
$495$	0	0
$496$	−10.4869	−0.470875
$497$	− 11.9729i	− 0.537056i
$498$	− 12.2567i	− 0.549234i
$499$	−5.49827	−0.246136	−0.123068	−	0.992398i	$-0.539273\pi$
$499$	−5.49827	−0.246136	−0.123068	+	0.992398i	$0.539273\pi$
$500$	0	0
$501$	−44.1051	−1.97047
$502$	− 8.08383i	− 0.360799i
$503$	35.6939i	1.59151i	0.605616	+	0.795757i	$0.292927\pi$
$503$	35.6939i	1.59151i	−0.605616	+	0.795757i	$0.707073\pi$
$504$	7.68079	0.342130
$505$	0	0
$506$	17.9095	0.796173
$507$	− 29.1226i	− 1.29338i
$508$	− 1.79895i	− 0.0798157i
$509$	16.4450	0.728914	0.364457	−	0.931220i	$-0.381255\pi$
$509$	16.4450	0.728914	0.364457	+	0.931220i	$0.381255\pi$
$510$	0	0
$511$	−16.2427	−0.718533
$512$	− 20.9558i	− 0.926123i
$513$	− 2.13706i	− 0.0943537i
$514$	−15.1250	−0.667134
$515$	0	0
$516$	−1.50365	−0.0661944
$517$	13.0586i	0.574317i
$518$	0.268750i	0.0118082i
$519$	−46.2083	−2.02832
$520$	0	0
$521$	−26.5435	−1.16289	−0.581445	−	0.813586i	$-0.697513\pi$
$521$	−26.5435	−1.16289	−0.581445	+	0.813586i	$0.697513\pi$
$522$	− 30.3032i	− 1.32633i
$523$	24.1685i	1.05682i	0.848991	+	0.528408i	$0.177211\pi$
$523$	24.1685i	1.05682i	−0.848991	+	0.528408i	$0.822789\pi$
$524$	1.16468	0.0508795
$525$	0	0
$526$	22.3685	0.975313
$527$	− 2.86054i	− 0.124607i
$528$	45.4359i	1.97735i
$529$	16.4722	0.716182
$530$	0	0
$531$	11.9323	0.517818
$532$	− 0.119605i	− 0.00518553i
$533$	2.32736i	0.100809i
$534$	−30.4131	−1.31610
$535$	0	0
$536$	−22.1433	−0.956445
$537$	15.5526i	0.671143i
$538$	− 13.1938i	− 0.568823i
$539$	25.0248	1.07789
$540$	0	0
$541$	9.80386	0.421501	0.210750	−	0.977540i	$-0.432409\pi$
$541$	9.80386	0.421501	0.210750	+	0.977540i	$0.432409\pi$
$542$	10.7549i	0.461964i
$543$	39.6504i	1.70156i
$544$	−0.566566	−0.0242913
$545$	0	0
$546$	0.872625	0.0373449
$547$	21.1739i	0.905331i	0.891681	+	0.452665i	$0.149527\pi$
$547$	21.1739i	0.905331i	−0.891681	+	0.452665i	$0.850473\pi$
$548$	0.604729i	0.0258328i
$549$	15.5351	0.663022
$550$	0	0
$551$	10.2349	0.436021
$552$	− 15.8605i	− 0.675070i
$553$	− 14.5743i	− 0.619764i
$554$	16.4959	0.700843
$555$	0	0
$556$	0.377812	0.0160228
$557$	24.4077i	1.03419i	0.855928	+	0.517094i	$0.172986\pi$
$557$	24.4077i	1.03419i	−0.855928	+	0.517094i	$0.827014\pi$
$558$	− 7.44850i	− 0.315320i
$559$	1.50365	0.0635975
$560$	0	0
$561$	−12.3937	−0.523264
$562$	31.0771i	1.31091i
$563$	− 14.4849i	− 0.610464i	−0.952278	−	0.305232i	$-0.901266\pi$
$563$	− 14.4849i	− 0.610464i	0.952278	−	0.305232i	$-0.0987340\pi$
$564$	−0.533188	−0.0224513
$565$	0	0
$566$	−7.88577	−0.331464
$567$	− 14.8562i	− 0.623903i
$568$	− 24.3773i	− 1.02285i
$569$	−0.811626	−0.0340251	−0.0170126	−	0.999855i	$-0.505416\pi$
$569$	−0.811626	−0.0340251	−0.0170126	+	0.999855i	$0.505416\pi$
$570$	0	0
$571$	−37.6588	−1.57597	−0.787985	−	0.615694i	$-0.788876\pi$
$571$	−37.6588	−1.57597	−0.787985	+	0.615694i	$0.788876\pi$
$572$	0.0846882i	0.00354099i
$573$	− 15.5700i	− 0.650447i
$574$	23.0403	0.961683
$575$	0	0
$576$	15.6066	0.650277
$577$	8.89307i	0.370223i	0.982718	+	0.185112i	$0.0592647\pi$
$577$	8.89307i	0.370223i	−0.982718	+	0.185112i	$0.940735\pi$
$578$	22.6974i	0.944087i
$579$	47.2693	1.96445
$580$	0	0
$581$	5.12200	0.212496
$582$	0.643104i	0.0266575i
$583$	− 62.4693i	− 2.58721i
$584$	−33.0709	−1.36848
$585$	0	0
$586$	10.6799	0.441184
$587$	20.3327i	0.839222i	0.907704	+	0.419611i	$0.137833\pi$
$587$	20.3327i	0.839222i	−0.907704	+	0.419611i	$0.862167\pi$
$588$	1.02177i	0.0421371i
$589$	2.51573	0.103659
$590$	0	0
$591$	48.4553	1.99319
$592$	0.571352i	0.0234824i
$593$	− 17.0049i	− 0.698308i	−0.937065	−	0.349154i	$-0.886469\pi$
$593$	− 17.0049i	− 0.698308i	0.937065	−	0.349154i	$-0.113531\pi$
$594$	14.9801	0.614643
$595$	0	0
$596$	−1.35344	−0.0554390
$597$	49.4131i	2.02234i
$598$	− 0.731250i	− 0.0299030i
$599$	12.4004	0.506668	0.253334	−	0.967379i	$-0.418473\pi$
$599$	12.4004	0.506668	0.253334	+	0.967379i	$0.418473\pi$
$600$	0	0
$601$	−9.73125	−0.396946	−0.198473	−	0.980106i	$-0.563598\pi$
$601$	−9.73125	−0.396946	−0.198473	+	0.980106i	$0.563598\pi$
$602$	− 14.8858i	− 0.606699i
$603$	− 16.4222i	− 0.668764i
$604$	0.901691	0.0366893
$605$	0	0
$606$	37.3696	1.51803
$607$	− 23.4282i	− 0.950920i	−0.879737	−	0.475460i	$-0.842282\pi$
$607$	− 23.4282i	− 0.950920i	0.879737	−	0.475460i	$-0.157718\pi$
$608$	− 0.498271i	− 0.0202076i
$609$	31.2054	1.26450
$610$	0	0
$611$	0.533188	0.0215705
$612$	− 0.205358i	− 0.00830111i
$613$	− 28.1424i	− 1.13666i	−0.822800	−	0.568331i	$-0.807589\pi$
$613$	− 28.1424i	− 1.13666i	0.822800	−	0.568331i	$-0.192411\pi$
$614$	46.5493	1.87858
$615$	0	0
$616$	−18.1844	−0.732672
$617$	36.4805i	1.46865i	0.678797	+	0.734326i	$0.262502\pi$
$617$	36.4805i	1.46865i	−0.678797	+	0.734326i	$0.737498\pi$
$618$	− 49.2127i	− 1.97962i
$619$	38.2097	1.53578	0.767888	−	0.640584i	$-0.221308\pi$
$619$	38.2097	1.53578	0.767888	+	0.640584i	$0.221308\pi$
$620$	0	0
$621$	−5.46011	−0.219107
$622$	21.4504i	0.860083i
$623$	− 12.7095i	− 0.509195i
$624$	1.85517	0.0742661
$625$	0	0
$626$	−19.0011	−0.759435
$627$	− 10.8998i	− 0.435295i
$628$	− 0.279503i	− 0.0111534i
$629$	−0.155850	−0.00621413
$630$	0	0
$631$	−12.5235	−0.498553	−0.249276	−	0.968432i	$-0.580193\pi$
$631$	−12.5235	−0.498553	−0.249276	+	0.968432i	$0.580193\pi$
$632$	− 29.6741i	− 1.18037i
$633$	46.3400i	1.84185i
$634$	7.41981	0.294678
$635$	0	0
$636$	2.55065	0.101140
$637$	− 1.02177i	− 0.0404840i
$638$	71.7434i	2.84035i
$639$	18.0790	0.715196
$640$	0	0
$641$	37.3564	1.47549	0.737745	−	0.675080i	$-0.235891\pi$
$641$	37.3564	1.47549	0.737745	+	0.675080i	$0.235891\pi$
$642$	8.61356i	0.339950i
$643$	14.5483i	0.573727i	0.957971	+	0.286864i	$0.0926126\pi$
$643$	14.5483i	0.573727i	−0.957971	+	0.286864i	$0.907387\pi$
$644$	−0.305586	−0.0120418
$645$	0	0
$646$	1.64310	0.0646471
$647$	− 23.4403i	− 0.921532i	−0.887522	−	0.460766i	$-0.847575\pi$
$647$	− 23.4403i	− 0.921532i	0.887522	−	0.460766i	$-0.152425\pi$
$648$	− 30.2480i	− 1.18826i
$649$	−28.2500	−1.10891
$650$	0	0
$651$	7.67025	0.300621
$652$	− 0.408797i	− 0.0160097i
$653$	− 27.1903i	− 1.06404i	−0.846732	−	0.532019i	$-0.821433\pi$
$653$	− 27.1903i	− 1.06404i	0.846732	−	0.532019i	$-0.178567\pi$
$654$	8.11529	0.317333
$655$	0	0
$656$	48.9828	1.91246
$657$	− 24.5265i	− 0.956869i
$658$	− 5.27844i	− 0.205775i
$659$	−2.71486	−0.105756	−0.0528779	−	0.998601i	$-0.516839\pi$
$659$	−2.71486	−0.105756	−0.0528779	+	0.998601i	$0.516839\pi$
$660$	0	0
$661$	−9.41311	−0.366128	−0.183064	−	0.983101i	$-0.558601\pi$
$661$	−9.41311	−0.366128	−0.183064	+	0.983101i	$0.558601\pi$
$662$	2.90408i	0.112870i
$663$	0.506041i	0.0196530i
$664$	10.4286	0.404710
$665$	0	0
$666$	−0.405813	−0.0157249
$667$	− 26.1497i	− 1.01252i
$668$	1.73019i	0.0669429i
$669$	−17.9245	−0.693002
$670$	0	0
$671$	−36.7797	−1.41986
$672$	− 1.51919i	− 0.0586039i
$673$	44.8001i	1.72692i	0.504419	+	0.863459i	$0.331707\pi$
$673$	44.8001i	1.72692i	−0.504419	+	0.863459i	$0.668293\pi$
$674$	1.90562	0.0734019
$675$	0	0
$676$	−1.14244	−0.0439400
$677$	− 32.9197i	− 1.26521i	−0.774475	−	0.632604i	$-0.781986\pi$
$677$	− 32.9197i	− 1.26521i	0.774475	−	0.632604i	$-0.218014\pi$
$678$	− 27.6896i	− 1.06341i
$679$	−0.268750	−0.0103137
$680$	0	0
$681$	44.4771	1.70437
$682$	17.6345i	0.675259i
$683$	− 20.3448i	− 0.778473i	−0.921138	−	0.389236i	$-0.872739\pi$
$683$	− 20.3448i	− 0.778473i	0.921138	−	0.389236i	$-0.127261\pi$
$684$	0.180604	0.00690556
$685$	0	0
$686$	−23.8407	−0.910242
$687$	− 9.06398i	− 0.345813i
$688$	− 31.6466i	− 1.20651i
$689$	−2.55065	−0.0971719
$690$	0	0
$691$	−46.1473	−1.75553	−0.877764	−	0.479094i	$-0.840965\pi$
$691$	−46.1473	−1.75553	−0.877764	+	0.479094i	$0.840965\pi$
$692$	1.81269i	0.0689082i
$693$	− 13.4862i	− 0.512298i
$694$	−17.3623	−0.659063
$695$	0	0
$696$	63.5357	2.40831
$697$	13.3612i	0.506092i
$698$	15.2567i	0.577473i
$699$	−60.2549	−2.27905
$700$	0	0
$701$	13.1933	0.498303	0.249152	−	0.968464i	$-0.419848\pi$
$701$	13.1933	0.498303	0.249152	+	0.968464i	$0.419848\pi$
$702$	− 0.611645i	− 0.0230851i
$703$	− 0.137063i	− 0.00516944i
$704$	−36.9491	−1.39257
$705$	0	0
$706$	−1.46096	−0.0549840
$707$	15.6165i	0.587321i
$708$	− 1.15346i	− 0.0433496i
$709$	−41.1860	−1.54677	−0.773386	−	0.633935i	$-0.781438\pi$
$709$	−41.1860	−1.54677	−0.773386	+	0.633935i	$0.781438\pi$
$710$	0	0
$711$	22.0073	0.825338
$712$	− 25.8771i	− 0.969787i
$713$	− 6.42758i	− 0.240715i
$714$	5.00969	0.187483
$715$	0	0
$716$	0.610106	0.0228007
$717$	7.55496i	0.282145i
$718$	7.43104i	0.277324i
$719$	35.6256	1.32861	0.664306	−	0.747461i	$-0.268727\pi$
$719$	35.6256	1.32861	0.664306	+	0.747461i	$0.268727\pi$
$720$	0	0
$721$	20.5657	0.765907
$722$	1.44504i	0.0537789i
$723$	− 62.3551i	− 2.31901i
$724$	1.55543	0.0578072
$725$	0	0
$726$	40.6872	1.51004
$727$	20.0116i	0.742189i	0.928595	+	0.371095i	$0.121017\pi$
$727$	20.0116i	0.742189i	−0.928595	+	0.371095i	$0.878983\pi$
$728$	0.742478i	0.0275181i
$729$	8.02715	0.297302
$730$	0	0
$731$	8.63235	0.319279
$732$	− 1.50173i	− 0.0555055i
$733$	18.9952i	0.701604i	0.936450	+	0.350802i	$0.114091\pi$
$733$	18.9952i	0.701604i	−0.936450	+	0.350802i	$0.885909\pi$
$734$	42.6732	1.57510
$735$	0	0
$736$	−1.27306	−0.0469257
$737$	38.8799i	1.43216i
$738$	34.7909i	1.28067i
$739$	−48.1704	−1.77198	−0.885989	−	0.463706i	$-0.846519\pi$
$739$	−48.1704	−1.77198	−0.885989	+	0.463706i	$0.846519\pi$
$740$	0	0
$741$	−0.445042	−0.0163490
$742$	25.2508i	0.926987i
$743$	13.2446i	0.485897i	0.970039	+	0.242948i	$0.0781146\pi$
$743$	13.2446i	0.485897i	−0.970039	+	0.242948i	$0.921885\pi$
$744$	15.6170	0.572548
$745$	0	0
$746$	12.7017	0.465043
$747$	7.73423i	0.282981i
$748$	0.486189i	0.0177768i
$749$	−3.59956	−0.131525
$750$	0	0
$751$	12.0562	0.439937	0.219969	−	0.975507i	$-0.429404\pi$
$751$	12.0562	0.439937	0.219969	+	0.975507i	$0.429404\pi$
$752$	− 11.2218i	− 0.409215i
$753$	12.5700i	0.458077i
$754$	2.92931	0.106679
$755$	0	0
$756$	−0.255603	−0.00929620
$757$	15.0054i	0.545380i	0.962102	+	0.272690i	$0.0879133\pi$
$757$	15.0054i	0.545380i	−0.962102	+	0.272690i	$0.912087\pi$
$758$	− 27.6741i	− 1.00517i
$759$	−27.8485	−1.01084
$760$	0	0
$761$	44.3967	1.60938	0.804690	−	0.593695i	$-0.202332\pi$
$761$	44.3967	1.60938	0.804690	+	0.593695i	$0.202332\pi$
$762$	66.2669i	2.40060i
$763$	3.39134i	0.122775i
$764$	−0.610791	−0.0220976
$765$	0	0
$766$	−10.1138	−0.365425
$767$	1.15346i	0.0416489i
$768$	− 4.74440i	− 0.171199i
$769$	39.7211	1.43238	0.716190	−	0.697906i	$-0.245885\pi$
$769$	39.7211	1.43238	0.716190	+	0.697906i	$0.245885\pi$
$770$	0	0
$771$	23.5187	0.847006
$772$	− 1.85431i	− 0.0667382i
$773$	1.72779i	0.0621444i	0.999517	+	0.0310722i	$0.00989218\pi$
$773$	1.72779i	0.0621444i	−0.999517	+	0.0310722i	$0.990108\pi$
$774$	22.4776	0.807939
$775$	0	0
$776$	−0.547188	−0.0196429
$777$	− 0.417895i	− 0.0149919i
$778$	− 11.6843i	− 0.418901i
$779$	−11.7506	−0.421010
$780$	0	0
$781$	−42.8025	−1.53159
$782$	− 4.19806i	− 0.150122i
$783$	− 21.8726i	− 0.781664i
$784$	−21.5047	−0.768025
$785$	0	0
$786$	−42.9028	−1.53029
$787$	42.6329i	1.51970i	0.650098	+	0.759850i	$0.274728\pi$
$787$	42.6329i	1.51970i	−0.650098	+	0.759850i	$0.725272\pi$
$788$	− 1.90084i	− 0.0677145i
$789$	−34.7821	−1.23828
$790$	0	0
$791$	11.5714	0.411430
$792$	− 27.4586i	− 0.975698i
$793$	1.50173i	0.0533280i
$794$	25.5782	0.907735
$795$	0	0
$796$	1.93841	0.0687051
$797$	38.5864i	1.36680i	0.730044	+	0.683401i	$0.239500\pi$
$797$	38.5864i	1.36680i	−0.730044	+	0.683401i	$0.760500\pi$
$798$	4.40581i	0.155964i
$799$	3.06100	0.108290
$800$	0	0
$801$	19.1914	0.678093
$802$	− 22.4586i	− 0.793040i
$803$	58.0670i	2.04914i
$804$	−1.58748	−0.0559862
$805$	0	0
$806$	0.720023	0.0253617
$807$	20.5157i	0.722188i
$808$	31.7961i	1.11858i
$809$	8.38298	0.294730	0.147365	−	0.989082i	$-0.452921\pi$
$809$	8.38298	0.294730	0.147365	+	0.989082i	$0.452921\pi$
$810$	0	0
$811$	−0.340765	−0.0119659	−0.00598295	−	0.999982i	$-0.501904\pi$
$811$	−0.340765	−0.0119659	−0.00598295	+	0.999982i	$0.501904\pi$
$812$	− 1.22414i	− 0.0429590i
$813$	− 16.7235i	− 0.586518i
$814$	0.960771	0.0336750
$815$	0	0
$816$	10.6504	0.372839
$817$	7.59179i	0.265603i
$818$	− 19.0256i	− 0.665215i
$819$	−0.550646	−0.0192411
$820$	0	0
$821$	−33.7506	−1.17791	−0.588953	−	0.808168i	$-0.700460\pi$
$821$	−33.7506	−1.17791	−0.588953	+	0.808168i	$0.700460\pi$
$822$	− 22.2760i	− 0.776966i
$823$	− 37.2669i	− 1.29904i	−0.760343	−	0.649522i	$-0.774969\pi$
$823$	− 37.2669i	− 1.29904i	0.760343	−	0.649522i	$-0.225031\pi$
$824$	41.8728	1.45871
$825$	0	0
$826$	11.4190	0.397316
$827$	− 21.1691i	− 0.736122i	−0.929802	−	0.368061i	$-0.880022\pi$
$827$	− 21.1691i	− 0.736122i	0.929802	−	0.368061i	$-0.119978\pi$
$828$	− 0.461435i	− 0.0160360i
$829$	31.0374	1.07797	0.538987	−	0.842314i	$-0.318807\pi$
$829$	31.0374	1.07797	0.538987	+	0.842314i	$0.318807\pi$
$830$	0	0
$831$	−25.6504	−0.889803
$832$	1.50864i	0.0523028i
$833$	− 5.86592i	− 0.203242i
$834$	−13.9172	−0.481914
$835$	0	0
$836$	−0.427583	−0.0147883
$837$	− 5.37627i	− 0.185831i
$838$	36.3773i	1.25663i
$839$	−33.2403	−1.14758	−0.573791	−	0.819002i	$-0.694528\pi$
$839$	−33.2403	−1.14758	−0.573791	+	0.819002i	$0.694528\pi$
$840$	0	0
$841$	75.7531	2.61218
$842$	38.9323i	1.34170i
$843$	− 48.3236i	− 1.66435i
$844$	1.81786	0.0625732
$845$	0	0
$846$	7.97046	0.274030
$847$	17.0030i	0.584229i
$848$	53.6822i	1.84346i
$849$	12.2620	0.420832
$850$	0	0
$851$	−0.350191	−0.0120044
$852$	− 1.74764i	− 0.0598733i
$853$	24.3086i	0.832310i	0.909294	+	0.416155i	$0.136623\pi$
$853$	24.3086i	0.832310i	−0.909294	+	0.416155i	$0.863377\pi$
$854$	14.8668	0.508731
$855$	0	0
$856$	−7.32889	−0.250496
$857$	− 57.3889i	− 1.96037i	−0.198087	−	0.980185i	$-0.563473\pi$
$857$	− 57.3889i	− 1.96037i	0.198087	−	0.980185i	$-0.436527\pi$
$858$	− 3.11960i	− 0.106502i
$859$	13.8135	0.471312	0.235656	−	0.971837i	$-0.424276\pi$
$859$	13.8135	0.471312	0.235656	+	0.971837i	$0.424276\pi$
$860$	0	0
$861$	−35.8267	−1.22097
$862$	27.3817i	0.932623i
$863$	9.01938i	0.307023i	0.988147	+	0.153512i	$0.0490582\pi$
$863$	9.01938i	0.307023i	−0.988147	+	0.153512i	$0.950942\pi$
$864$	−1.06484	−0.0362265
$865$	0	0
$866$	−5.33081	−0.181148
$867$	− 35.2935i	− 1.19863i
$868$	− 0.300894i	− 0.0102130i
$869$	−52.1027	−1.76746
$870$	0	0
$871$	1.58748	0.0537898
$872$	6.90494i	0.233831i
$873$	− 0.405813i	− 0.0137347i
$874$	3.69202	0.124884
$875$	0	0
$876$	−2.37090	−0.0801052
$877$	− 2.37675i	− 0.0802570i	−0.999195	−	0.0401285i	$-0.987223\pi$
$877$	− 2.37675i	− 0.0802570i	0.999195	−	0.0401285i	$-0.0127767\pi$
$878$	37.1269i	1.25297i
$879$	−16.6069	−0.560135
$880$	0	0
$881$	−7.99330	−0.269301	−0.134650	−	0.990893i	$-0.542991\pi$
$881$	−7.99330	−0.269301	−0.134650	+	0.990893i	$0.542991\pi$
$882$	− 15.2741i	− 0.514307i
$883$	1.76377i	0.0593557i	0.999560	+	0.0296779i	$0.00944814\pi$
$883$	1.76377i	0.0593557i	−0.999560	+	0.0296779i	$0.990552\pi$
$884$	0.0198513	0.000667672 0
$885$	0	0
$886$	−39.5439	−1.32850
$887$	45.9197i	1.54183i	0.636936	+	0.770917i	$0.280202\pi$
$887$	45.9197i	1.54183i	−0.636936	+	0.770917i	$0.719798\pi$
$888$	− 0.850855i	− 0.0285528i
$889$	−27.6926	−0.928780
$890$	0	0
$891$	−53.1105	−1.77927
$892$	0.703155i	0.0235434i
$893$	2.69202i	0.0900851i
$894$	49.8558	1.66743
$895$	0	0
$896$	16.2874	0.544125
$897$	1.13706i	0.0379654i
$898$	− 10.9215i	− 0.364457i
$899$	25.7482	0.858752
$900$	0	0
$901$	−14.6431	−0.487833
$902$	− 82.3682i	− 2.74256i
$903$	23.1468i	0.770276i
$904$	23.5599	0.783589
$905$	0	0
$906$	−33.2150	−1.10350
$907$	55.0549i	1.82807i	0.405638	+	0.914034i	$0.367049\pi$
$907$	55.0549i	1.82807i	−0.405638	+	0.914034i	$0.632951\pi$
$908$	− 1.74478i	− 0.0579024i
$909$	−23.5810	−0.782134
$910$	0	0
$911$	51.8998	1.71952	0.859758	−	0.510702i	$-0.170614\pi$
$911$	51.8998	1.71952	0.859758	+	0.510702i	$0.170614\pi$
$912$	9.36658i	0.310159i
$913$	− 18.3110i	− 0.606004i
$914$	11.3526	0.375510
$915$	0	0
$916$	−0.355568	−0.0117483
$917$	− 17.9288i	− 0.592062i
$918$	− 3.51142i	− 0.115894i
$919$	−11.4614	−0.378078	−0.189039	−	0.981970i	$-0.560537\pi$
$919$	−11.4614	−0.378078	−0.189039	+	0.981970i	$0.560537\pi$
$920$	0	0
$921$	−72.3822	−2.38508
$922$	5.60541i	0.184604i
$923$	1.74764i	0.0575244i
$924$	−1.30367	−0.0428875
$925$	0	0
$926$	−18.9235	−0.621864
$927$	31.0543i	1.01996i
$928$	− 5.09975i	− 0.167408i
$929$	−36.5295	−1.19849	−0.599246	−	0.800565i	$-0.704533\pi$
$929$	−36.5295	−1.19849	−0.599246	+	0.800565i	$0.704533\pi$
$930$	0	0
$931$	5.15883	0.169074
$932$	2.36372i	0.0774261i
$933$	− 33.3545i	− 1.09198i
$934$	−9.30644	−0.304516
$935$	0	0
$936$	−1.12114	−0.0366457
$937$	− 48.7845i	− 1.59372i	−0.604164	−	0.796860i	$-0.706493\pi$
$937$	− 48.7845i	− 1.59372i	0.604164	−	0.796860i	$-0.293507\pi$
$938$	− 15.7157i	− 0.513136i
$939$	29.5459	0.964193
$940$	0	0
$941$	−18.1817	−0.592705	−0.296353	−	0.955079i	$-0.595770\pi$
$941$	−18.1817	−0.592705	−0.296353	+	0.955079i	$0.595770\pi$
$942$	10.2959i	0.335458i
$943$	30.0224i	0.977663i
$944$	24.2763	0.790125
$945$	0	0
$946$	−53.2161	−1.73021
$947$	− 7.55150i	− 0.245391i	−0.992444	−	0.122695i	$-0.960846\pi$
$947$	− 7.55150i	− 0.245391i	0.992444	−	0.122695i	$-0.0391538\pi$
$948$	− 2.12737i	− 0.0690939i
$949$	2.37090	0.0769626
$950$	0	0
$951$	−11.5375	−0.374129
$952$	4.26252i	0.138149i
$953$	− 13.8592i	− 0.448944i	−0.974481	−	0.224472i	$-0.927934\pi$
$953$	− 13.8592i	− 0.448944i	0.974481	−	0.224472i	$-0.0720657\pi$
$954$	−38.1288	−1.23447
$955$	0	0
$956$	0.296371	0.00958532
$957$	− 111.558i	− 3.60616i
$958$	− 8.22367i	− 0.265695i
$959$	9.30904	0.300605
$960$	0	0
$961$	−24.6711	−0.795842
$962$	− 0.0392287i	− 0.00126478i
$963$	− 5.43535i	− 0.175152i
$964$	−2.44611	−0.0787838
$965$	0	0
$966$	11.2567	0.362177
$967$	4.89977i	0.157566i	0.996892	+	0.0787830i	$0.0251034\pi$
$967$	4.89977i	0.157566i	−0.996892	+	0.0787830i	$0.974897\pi$
$968$	34.6189i	1.11269i
$969$	−2.55496	−0.0820771
$970$	0	0
$971$	14.5133	0.465755	0.232878	−	0.972506i	$-0.425186\pi$
$971$	14.5133	0.465755	0.232878	+	0.972506i	$0.425186\pi$
$972$	− 1.60340i	− 0.0514291i
$973$	− 5.81594i	− 0.186450i
$974$	18.7323	0.600222
$975$	0	0
$976$	31.6062	1.01169
$977$	− 19.6644i	− 0.629120i	−0.949238	−	0.314560i	$-0.898143\pi$
$977$	− 19.6644i	− 0.629120i	0.949238	−	0.314560i	$-0.101857\pi$
$978$	15.0586i	0.481521i
$979$	−45.4359	−1.45214
$980$	0	0
$981$	−5.12093	−0.163499
$982$	− 28.3293i	− 0.904023i
$983$	15.4397i	0.492449i	0.969213	+	0.246224i	$0.0791900\pi$
$983$	15.4397i	0.492449i	−0.969213	+	0.246224i	$0.920810\pi$
$984$	−72.9450	−2.32540
$985$	0	0
$986$	16.8170	0.535562
$987$	8.20775i	0.261256i
$988$	0.0174584i	0 0.000555426i
$989$	19.3967	0.616780
$990$	0	0
$991$	11.9377	0.379213	0.189606	−	0.981860i	$-0.439279\pi$
$991$	11.9377	0.379213	0.189606	+	0.981860i	$0.439279\pi$
$992$	− 1.25352i	− 0.0397991i
$993$	− 4.51573i	− 0.143302i
$994$	17.3013	0.548763
$995$	0	0
$996$	0.747644	0.0236900
$997$	17.9390i	0.568134i	0.958804	+	0.284067i	$0.0916838\pi$
$997$	17.9390i	0.568134i	−0.958804	+	0.284067i	$0.908316\pi$
$998$	− 7.94523i	− 0.251502i
$999$	−0.292913	−0.00926736

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	475.2.b.c.324.5		6
5.2	odd	4		475.2.a.d.1.2	✓	3
5.3	odd	4		475.2.a.h.1.2	yes	3
5.4	even	2	inner	475.2.b.c.324.2		6
15.2	even	4		4275.2.a.bn.1.2		3
15.8	even	4		4275.2.a.z.1.2		3
20.3	even	4		7600.2.a.bn.1.1		3
20.7	even	4		7600.2.a.bw.1.3		3
95.18	even	4		9025.2.a.w.1.2		3
95.37	even	4		9025.2.a.be.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
475.2.a.d.1.2	✓	3	5.2	odd	4
475.2.a.h.1.2	yes	3	5.3	odd	4
475.2.b.c.324.2		6	5.4	even	2	inner
475.2.b.c.324.5		6	1.1	even	1	trivial
4275.2.a.z.1.2		3	15.8	even	4
4275.2.a.bn.1.2		3	15.2	even	4
7600.2.a.bn.1.1		3	20.3	even	4
7600.2.a.bw.1.3		3	20.7	even	4
9025.2.a.w.1.2		3	95.18	even	4
9025.2.a.be.1.2		3	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)

\(f(q)\)	\(=\)	\(q+1.44504i q^{2} -2.24698i q^{3} -0.0881460 q^{4} +3.24698 q^{6} +1.35690i q^{7} +2.76271i q^{8} -2.04892 q^{9} +O(q^{10})\)	\(q+1.44504i q^{2} -2.24698i q^{3} -0.0881460 q^{4} +3.24698 q^{6} +1.35690i q^{7} +2.76271i q^{8} -2.04892 q^{9} +4.85086 q^{11} +0.198062i q^{12} -0.198062i q^{13} -1.96077 q^{14} -4.16852 q^{16} -1.13706i q^{17} -2.96077i q^{18} +1.00000 q^{19} +3.04892 q^{21} +7.00969i q^{22} -2.55496i q^{23} +6.20775 q^{24} +0.286208 q^{26} -2.13706i q^{27} -0.119605i q^{28} +10.2349 q^{29} +2.51573 q^{31} -0.498271i q^{32} -10.8998i q^{33} +1.64310 q^{34} +0.180604 q^{36} -0.137063i q^{37} +1.44504i q^{38} -0.445042 q^{39} -11.7506 q^{41} +4.40581i q^{42} +7.59179i q^{43} -0.427583 q^{44} +3.69202 q^{46} +2.69202i q^{47} +9.36658i q^{48} +5.15883 q^{49} -2.55496 q^{51} +0.0174584i q^{52} -12.8780i q^{53} +3.08815 q^{54} -3.74871 q^{56} -2.24698i q^{57} +14.7899i q^{58} -5.82371 q^{59} -7.58211 q^{61} +3.63533i q^{62} -2.78017i q^{63} -7.61702 q^{64} +15.7506 q^{66} +8.01507i q^{67} +0.100228i q^{68} -5.74094 q^{69} -8.82371 q^{71} -5.66056i q^{72} +11.9705i q^{73} +0.198062 q^{74} -0.0881460 q^{76} +6.58211i q^{77} -0.643104i q^{78} -10.7409 q^{79} -10.9487 q^{81} -16.9801i q^{82} -3.77479i q^{83} -0.268750 q^{84} -10.9705 q^{86} -22.9976i q^{87} +13.4015i q^{88} -9.36658 q^{89} +0.268750 q^{91} +0.225209i q^{92} -5.65279i q^{93} -3.89008 q^{94} -1.11960 q^{96} +0.198062i q^{97} +7.45473i q^{98} -9.93900 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 8 q^{4} + 10 q^{6} + 6 q^{9}+O(q^{10}) \) 6 * q - 8 * q^4 + 10 * q^6 + 6 * q^9	\( 6 q - 8 q^{4} + 10 q^{6} + 6 q^{9} + 2 q^{11} + 14 q^{14} + 36 q^{16} + 6 q^{19} + 2 q^{24} + 18 q^{26} + 14 q^{29} - 10 q^{31} + 18 q^{34} - 22 q^{36} - 2 q^{39} + 2 q^{41} + 30 q^{44} + 12 q^{46} + 14 q^{49} - 16 q^{51} + 26 q^{54} - 70 q^{56} - 20 q^{59} - 34 q^{61} - 98 q^{64} + 22 q^{66} - 6 q^{69} - 38 q^{71} + 10 q^{74} - 8 q^{76} - 36 q^{79} - 2 q^{81} + 14 q^{84} + 4 q^{86} - 4 q^{89} - 14 q^{91} - 22 q^{94} + 36 q^{96} - 40 q^{99}+O(q^{100}) \) 6 * q - 8 * q^4 + 10 * q^6 + 6 * q^9 + 2 * q^11 + 14 * q^14 + 36 * q^16 + 6 * q^19 + 2 * q^24 + 18 * q^26 + 14 * q^29 - 10 * q^31 + 18 * q^34 - 22 * q^36 - 2 * q^39 + 2 * q^41 + 30 * q^44 + 12 * q^46 + 14 * q^49 - 16 * q^51 + 26 * q^54 - 70 * q^56 - 20 * q^59 - 34 * q^61 - 98 * q^64 + 22 * q^66 - 6 * q^69 - 38 * q^71 + 10 * q^74 - 8 * q^76 - 36 * q^79 - 2 * q^81 + 14 * q^84 + 4 * q^86 - 4 * q^89 - 14 * q^91 - 22 * q^94 + 36 * q^96 - 40 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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