LMFDB - Embedded newform 475.2.b.d.324.3

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:	6.0.350464.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 4x^{2} - 4x + 2 $ x^6 - 2x^5 + 2x^4 + 2x^3 + 4x^2 - 4*x + 2
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 95)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			324.3
Root			$1.45161 + 1.45161i$ of defining polynomial
Character	$\chi$	$=$	475.324
Dual form			475.2.b.d.324.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.311108i q^{2} +2.90321i q^{3} +1.90321 q^{4} +0.903212 q^{6} +4.42864i q^{7} -1.21432i q^{8} -5.42864 q^{9} +O(q^{10})$	\(q-0.311108i q^{2} +2.90321i q^{3} +1.90321 q^{4} +0.903212 q^{6} +4.42864i q^{7} -1.21432i q^{8} -5.42864 q^{9} -2.62222 q^{11} +5.52543i q^{12} +0.474572i q^{13} +1.37778 q^{14} +3.42864 q^{16} -5.05086i q^{17} +1.68889i q^{18} +1.00000 q^{19} -12.8573 q^{21} +0.815792i q^{22} -1.37778i q^{23} +3.52543 q^{24} +0.147643 q^{26} -7.05086i q^{27} +8.42864i q^{28} +7.80642 q^{29} +1.24443 q^{31} -3.49532i q^{32} -7.61285i q^{33} -1.57136 q^{34} -10.3319 q^{36} -4.47457i q^{37} -0.311108i q^{38} -1.37778 q^{39} -5.05086 q^{41} +4.00000i q^{42} +12.0415i q^{43} -4.99063 q^{44} -0.428639 q^{46} +4.42864i q^{47} +9.95407i q^{48} -12.6128 q^{49} +14.6637 q^{51} +0.903212i q^{52} +7.52543i q^{53} -2.19358 q^{54} +5.37778 q^{56} +2.90321i q^{57} -2.42864i q^{58} +2.19358 q^{59} +3.67307 q^{61} -0.387152i q^{62} -24.0415i q^{63} +5.76986 q^{64} -2.36842 q^{66} +1.65878i q^{67} -9.61285i q^{68} +4.00000 q^{69} +7.61285 q^{71} +6.59210i q^{72} -3.80642i q^{73} -1.39207 q^{74} +1.90321 q^{76} -11.6128i q^{77} +0.428639i q^{78} +13.4193 q^{79} +4.18421 q^{81} +1.57136i q^{82} -10.6222i q^{83} -24.4701 q^{84} +3.74620 q^{86} +22.6637i q^{87} +3.18421i q^{88} +12.6637 q^{89} -2.10171 q^{91} -2.62222i q^{92} +3.61285i q^{93} +1.37778 q^{94} +10.1476 q^{96} -17.8938i q^{97} +3.92396i q^{98} +14.2351 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 2 q^{4} - 8 q^{6} - 6 q^{9}+O(q^{10}) $ 6 * q - 2 * q^4 - 8 * q^6 - 6 * q^9	$ 6 q - 2 q^{4} - 8 q^{6} - 6 q^{9} - 16 q^{11} + 8 q^{14} - 6 q^{16} + 6 q^{19} - 24 q^{21} + 8 q^{24} - 12 q^{26} + 20 q^{29} + 8 q^{31} - 36 q^{34} - 22 q^{36} - 8 q^{39} - 4 q^{41} + 24 q^{44} + 24 q^{46} - 22 q^{49} + 8 q^{51} - 40 q^{54} + 32 q^{56} + 40 q^{59} - 4 q^{61} + 22 q^{64} + 40 q^{66} + 24 q^{69} - 8 q^{71} + 4 q^{74} - 2 q^{76} - 2 q^{81} - 40 q^{84} - 32 q^{86} - 4 q^{89} + 40 q^{91} + 8 q^{94} + 48 q^{96} + 32 q^{99}+O(q^{100}) $ 6 * q - 2 * q^4 - 8 * q^6 - 6 * q^9 - 16 * q^11 + 8 * q^14 - 6 * q^16 + 6 * q^19 - 24 * q^21 + 8 * q^24 - 12 * q^26 + 20 * q^29 + 8 * q^31 - 36 * q^34 - 22 * q^36 - 8 * q^39 - 4 * q^41 + 24 * q^44 + 24 * q^46 - 22 * q^49 + 8 * q^51 - 40 * q^54 + 32 * q^56 + 40 * q^59 - 4 * q^61 + 22 * q^64 + 40 * q^66 + 24 * q^69 - 8 * q^71 + 4 * q^74 - 2 * q^76 - 2 * q^81 - 40 * q^84 - 32 * q^86 - 4 * q^89 + 40 * q^91 + 8 * q^94 + 48 * q^96 + 32 * 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 $

$2$	− 0.311108i	− 0.219986i	−0.993932	−	0.109993i	$-0.964917\pi$
$2$	− 0.311108i	− 0.219986i	0.993932	−	0.109993i	$-0.0350829\pi$
$3$	2.90321i	1.67617i	0.545540	+	0.838085i	$0.316325\pi$
$3$	2.90321i	1.67617i	−0.545540	+	0.838085i	$0.683675\pi$
$4$	1.90321	0.951606
$5$	0	0
$5$	0	0
$6$	0.903212	0.368735
$7$	4.42864i	1.67387i	0.547304	+	0.836934i	$0.315654\pi$
$7$	4.42864i	1.67387i	−0.547304	+	0.836934i	$0.684346\pi$
$8$	− 1.21432i	− 0.429327i
$9$	−5.42864	−1.80955
$10$	0	0
$11$	−2.62222	−0.790628	−0.395314	−	0.918546i	$-0.629364\pi$
$11$	−2.62222	−0.790628	−0.395314	+	0.918546i	$0.629364\pi$
$12$	5.52543i	1.59505i
$13$	0.474572i	0.131623i	0.997832	+	0.0658114i	$0.0209636\pi$
$13$	0.474572i	0.131623i	−0.997832	+	0.0658114i	$0.979036\pi$
$14$	1.37778	0.368228
$15$	0	0
$16$	3.42864	0.857160
$17$	− 5.05086i	− 1.22501i	−0.790466	−	0.612506i	$-0.790161\pi$
$17$	− 5.05086i	− 1.22501i	0.790466	−	0.612506i	$-0.209839\pi$
$18$	1.68889i	0.398076i
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	−12.8573	−2.80569
$22$	0.815792i	0.173927i
$23$	− 1.37778i	− 0.287288i	−0.989629	−	0.143644i	$-0.954118\pi$
$23$	− 1.37778i	− 0.287288i	0.989629	−	0.143644i	$-0.0458820\pi$
$24$	3.52543	0.719625
$25$	0	0
$26$	0.147643	0.0289552
$27$	− 7.05086i	− 1.35694i
$28$	8.42864i	1.59286i
$29$	7.80642	1.44962	0.724808	−	0.688951i	$-0.241928\pi$
$29$	7.80642	1.44962	0.724808	+	0.688951i	$0.241928\pi$
$30$	0	0
$31$	1.24443	0.223506	0.111753	−	0.993736i	$-0.464353\pi$
$31$	1.24443	0.223506	0.111753	+	0.993736i	$0.464353\pi$
$32$	− 3.49532i	− 0.617890i
$33$	− 7.61285i	− 1.32523i
$34$	−1.57136	−0.269486
$35$	0	0
$36$	−10.3319	−1.72198
$37$	− 4.47457i	− 0.735615i	−0.929902	−	0.367808i	$-0.880109\pi$
$37$	− 4.47457i	− 0.735615i	0.929902	−	0.367808i	$-0.119891\pi$
$38$	− 0.311108i	− 0.0504684i
$39$	−1.37778	−0.220622
$40$	0	0
$41$	−5.05086	−0.788811	−0.394406	−	0.918936i	$-0.629049\pi$
$41$	−5.05086	−0.788811	−0.394406	+	0.918936i	$0.629049\pi$
$42$	4.00000i	0.617213i
$43$	12.0415i	1.83631i	0.396222	+	0.918155i	$0.370321\pi$
$43$	12.0415i	1.83631i	−0.396222	+	0.918155i	$0.629679\pi$
$44$	−4.99063	−0.752366
$45$	0	0
$46$	−0.428639	−0.0631994
$47$	4.42864i	0.645983i	0.946402	+	0.322992i	$0.104689\pi$
$47$	4.42864i	0.645983i	−0.946402	+	0.322992i	$0.895311\pi$
$48$	9.95407i	1.43675i
$49$	−12.6128	−1.80184
$50$	0	0
$51$	14.6637	2.05333
$52$	0.903212i	0.125253i
$53$	7.52543i	1.03370i	0.856077	+	0.516848i	$0.172895\pi$
$53$	7.52543i	1.03370i	−0.856077	+	0.516848i	$0.827105\pi$
$54$	−2.19358	−0.298508
$55$	0	0
$56$	5.37778	0.718637
$57$	2.90321i	0.384540i
$58$	− 2.42864i	− 0.318896i
$59$	2.19358	0.285579	0.142790	−	0.989753i	$-0.454393\pi$
$59$	2.19358	0.285579	0.142790	+	0.989753i	$0.454393\pi$
$60$	0	0
$61$	3.67307	0.470289	0.235144	−	0.971960i	$-0.424444\pi$
$61$	3.67307	0.470289	0.235144	+	0.971960i	$0.424444\pi$
$62$	− 0.387152i	− 0.0491684i
$63$	− 24.0415i	− 3.02894i
$64$	5.76986	0.721232
$65$	0	0
$66$	−2.36842	−0.291532
$67$	1.65878i	0.202652i	0.994853	+	0.101326i	$0.0323086\pi$
$67$	1.65878i	0.202652i	−0.994853	+	0.101326i	$0.967691\pi$
$68$	− 9.61285i	− 1.16573i
$69$	4.00000	0.481543
$70$	0	0
$71$	7.61285	0.903479	0.451739	−	0.892150i	$-0.350804\pi$
$71$	7.61285	0.903479	0.451739	+	0.892150i	$0.350804\pi$
$72$	6.59210i	0.776887i
$73$	− 3.80642i	− 0.445508i	−0.974875	−	0.222754i	$-0.928495\pi$
$73$	− 3.80642i	− 0.445508i	0.974875	−	0.222754i	$-0.0715047\pi$
$74$	−1.39207	−0.161825
$75$	0	0
$76$	1.90321	0.218313
$77$	− 11.6128i	− 1.32341i
$78$	0.428639i	0.0485339i
$79$	13.4193	1.50979	0.754893	−	0.655848i	$-0.227689\pi$
$79$	13.4193	1.50979	0.754893	+	0.655848i	$0.227689\pi$
$80$	0	0
$81$	4.18421	0.464912
$82$	1.57136i	0.173528i
$83$	− 10.6222i	− 1.16594i	−0.812494	−	0.582970i	$-0.801891\pi$
$83$	− 10.6222i	− 1.16594i	0.812494	−	0.582970i	$-0.198109\pi$
$84$	−24.4701	−2.66991
$85$	0	0
$86$	3.74620	0.403963
$87$	22.6637i	2.42980i
$88$	3.18421i	0.339438i
$89$	12.6637	1.34235	0.671175	−	0.741299i	$-0.265790\pi$
$89$	12.6637	1.34235	0.671175	+	0.741299i	$0.265790\pi$
$90$	0	0
$91$	−2.10171	−0.220319
$92$	− 2.62222i	− 0.273385i
$93$	3.61285i	0.374635i
$94$	1.37778	0.142108
$95$	0	0
$96$	10.1476	1.03569
$97$	− 17.8938i	− 1.81684i	−0.418054	−	0.908422i	$-0.637288\pi$
$97$	− 17.8938i	− 1.81684i	0.418054	−	0.908422i	$-0.362712\pi$
$98$	3.92396i	0.396379i
$99$	14.2351	1.43068
$100$	0	0
$101$	−10.4286	−1.03769	−0.518844	−	0.854869i	$-0.673638\pi$
$101$	−10.4286	−1.03769	−0.518844	+	0.854869i	$0.673638\pi$
$102$	− 4.56199i	− 0.451705i
$103$	− 5.65878i	− 0.557576i	−0.960353	−	0.278788i	$-0.910067\pi$
$103$	− 5.65878i	− 0.557576i	0.960353	−	0.278788i	$-0.0899327\pi$
$104$	0.576283	0.0565092
$105$	0	0
$106$	2.34122	0.227399
$107$	− 6.90321i	− 0.667359i	−0.942687	−	0.333679i	$-0.891710\pi$
$107$	− 6.90321i	− 0.667359i	0.942687	−	0.333679i	$-0.108290\pi$
$108$	− 13.4193i	− 1.29127i
$109$	−5.61285	−0.537613	−0.268807	−	0.963194i	$-0.586629\pi$
$109$	−5.61285	−0.537613	−0.268807	+	0.963194i	$0.586629\pi$
$110$	0	0
$111$	12.9906	1.23302
$112$	15.1842i	1.43477i
$113$	13.8938i	1.30702i	0.756917	+	0.653511i	$0.226705\pi$
$113$	13.8938i	1.30702i	−0.756917	+	0.653511i	$0.773295\pi$
$114$	0.903212	0.0845935
$115$	0	0
$116$	14.8573	1.37946
$117$	− 2.57628i	− 0.238177i
$118$	− 0.682439i	− 0.0628236i
$119$	22.3684	2.05051
$120$	0	0
$121$	−4.12399	−0.374908
$122$	− 1.14272i	− 0.103457i
$123$	− 14.6637i	− 1.32218i
$124$	2.36842	0.212690
$125$	0	0
$126$	−7.47949	−0.666326
$127$	7.19850i	0.638763i	0.947626	+	0.319382i	$0.103475\pi$
$127$	7.19850i	0.638763i	−0.947626	+	0.319382i	$0.896525\pi$
$128$	− 8.78568i	− 0.776552i
$129$	−34.9590	−3.07797
$130$	0	0
$131$	−2.10171	−0.183627	−0.0918136	−	0.995776i	$-0.529266\pi$
$131$	−2.10171	−0.183627	−0.0918136	+	0.995776i	$0.529266\pi$
$132$	− 14.4889i	− 1.26109i
$133$	4.42864i	0.384012i
$134$	0.516060	0.0445808
$135$	0	0
$136$	−6.13335	−0.525931
$137$	− 1.70471i	− 0.145644i	−0.997345	−	0.0728218i	$-0.976800\pi$
$137$	− 1.70471i	− 0.145644i	0.997345	−	0.0728218i	$-0.0232004\pi$
$138$	− 1.24443i	− 0.105933i
$139$	−8.72393	−0.739954	−0.369977	−	0.929041i	$-0.620634\pi$
$139$	−8.72393	−0.739954	−0.369977	+	0.929041i	$0.620634\pi$
$140$	0	0
$141$	−12.8573	−1.08278
$142$	− 2.36842i	− 0.198753i
$143$	− 1.24443i	− 0.104065i
$144$	−18.6128	−1.55107
$145$	0	0
$146$	−1.18421	−0.0980058
$147$	− 36.6178i	− 3.02018i
$148$	− 8.51606i	− 0.700016i
$149$	6.81579	0.558371	0.279186	−	0.960237i	$-0.409935\pi$
$149$	6.81579	0.558371	0.279186	+	0.960237i	$0.409935\pi$
$150$	0	0
$151$	−5.80642	−0.472520	−0.236260	−	0.971690i	$-0.575922\pi$
$151$	−5.80642	−0.472520	−0.236260	+	0.971690i	$0.575922\pi$
$152$	− 1.21432i	− 0.0984943i
$153$	27.4193i	2.21672i
$154$	−3.61285	−0.291132
$155$	0	0
$156$	−2.62222	−0.209945
$157$	− 0.193576i	− 0.0154491i	−0.999970	−	0.00772453i	$-0.997541\pi$
$157$	− 0.193576i	− 0.0154491i	0.999970	−	0.00772453i	$-0.00245882\pi$
$158$	− 4.17484i	− 0.332132i
$159$	−21.8479	−1.73265
$160$	0	0
$161$	6.10171	0.480882
$162$	− 1.30174i	− 0.102274i
$163$	− 8.42864i	− 0.660182i	−0.943949	−	0.330091i	$-0.892921\pi$
$163$	− 8.42864i	− 0.660182i	0.943949	−	0.330091i	$-0.107079\pi$
$164$	−9.61285	−0.750637
$165$	0	0
$166$	−3.30465	−0.256491
$167$	− 12.4429i	− 0.962863i	−0.876484	−	0.481431i	$-0.840117\pi$
$167$	− 12.4429i	− 0.962863i	0.876484	−	0.481431i	$-0.159883\pi$
$168$	15.6128i	1.20456i
$169$	12.7748	0.982675
$170$	0	0
$171$	−5.42864	−0.415138
$172$	22.9175i	1.74744i
$173$	− 22.1891i	− 1.68701i	−0.537123	−	0.843504i	$-0.680489\pi$
$173$	− 22.1891i	− 1.68701i	0.537123	−	0.843504i	$-0.319511\pi$
$174$	7.05086	0.534524
$175$	0	0
$176$	−8.99063	−0.677694
$177$	6.36842i	0.478679i
$178$	− 3.93978i	− 0.295299i
$179$	11.9081	0.890056	0.445028	−	0.895517i	$-0.353194\pi$
$179$	11.9081	0.890056	0.445028	+	0.895517i	$0.353194\pi$
$180$	0	0
$181$	17.6128	1.30915	0.654576	−	0.755996i	$-0.272847\pi$
$181$	17.6128	1.30915	0.654576	+	0.755996i	$0.272847\pi$
$182$	0.653858i	0.0484672i
$183$	10.6637i	0.788284i
$184$	−1.67307	−0.123340
$185$	0	0
$186$	1.12399	0.0824146
$187$	13.2444i	0.968529i
$188$	8.42864i	0.614722i
$189$	31.2257	2.27134
$190$	0	0
$191$	−0.266706	−0.0192982	−0.00964909	−	0.999953i	$-0.503071\pi$
$191$	−0.266706	−0.0192982	−0.00964909	+	0.999953i	$0.503071\pi$
$192$	16.7511i	1.20891i
$193$	2.66815i	0.192058i	0.995379	+	0.0960288i	$0.0306141\pi$
$193$	2.66815i	0.192058i	−0.995379	+	0.0960288i	$0.969386\pi$
$194$	−5.56691	−0.399681
$195$	0	0
$196$	−24.0049	−1.71464
$197$	5.34614i	0.380897i	0.981697	+	0.190448i	$0.0609942\pi$
$197$	5.34614i	0.380897i	−0.981697	+	0.190448i	$0.939006\pi$
$198$	− 4.42864i	− 0.314730i
$199$	−17.1240	−1.21389	−0.606944	−	0.794745i	$-0.707605\pi$
$199$	−17.1240	−1.21389	−0.606944	+	0.794745i	$0.707605\pi$
$200$	0	0
$201$	−4.81579	−0.339680
$202$	3.24443i	0.228277i
$203$	34.5718i	2.42647i
$204$	27.9081	1.95396
$205$	0	0
$206$	−1.76049	−0.122659
$207$	7.47949i	0.519861i
$208$	1.62714i	0.112822i
$209$	−2.62222	−0.181382
$210$	0	0
$211$	13.1526	0.905460	0.452730	−	0.891648i	$-0.350450\pi$
$211$	13.1526	0.905460	0.452730	+	0.891648i	$0.350450\pi$
$212$	14.3225i	0.983672i
$213$	22.1017i	1.51438i
$214$	−2.14764	−0.146810
$215$	0	0
$216$	−8.56199	−0.582570
$217$	5.51114i	0.374120i
$218$	1.74620i	0.118268i
$219$	11.0509	0.746748
$220$	0	0
$221$	2.39700	0.161239
$222$	− 4.04149i	− 0.271247i
$223$	− 10.5161i	− 0.704207i	−0.935961	−	0.352104i	$-0.885466\pi$
$223$	− 10.5161i	− 0.704207i	0.935961	−	0.352104i	$-0.114534\pi$
$224$	15.4795	1.03427
$225$	0	0
$226$	4.32248	0.287527
$227$	− 6.90321i	− 0.458182i	−0.973405	−	0.229091i	$-0.926425\pi$
$227$	− 6.90321i	− 0.458182i	0.973405	−	0.229091i	$-0.0735754\pi$
$228$	5.52543i	0.365930i
$229$	−18.0415	−1.19222	−0.596108	−	0.802905i	$-0.703287\pi$
$229$	−18.0415	−1.19222	−0.596108	+	0.802905i	$0.703287\pi$
$230$	0	0
$231$	33.7146	2.21826
$232$	− 9.47949i	− 0.622359i
$233$	− 12.3684i	− 0.810282i	−0.914254	−	0.405141i	$-0.867222\pi$
$233$	− 12.3684i	− 0.810282i	0.914254	−	0.405141i	$-0.132778\pi$
$234$	−0.801502	−0.0523958
$235$	0	0
$236$	4.17484	0.271759
$237$	38.9590i	2.53066i
$238$	− 6.95899i	− 0.451084i
$239$	−24.8573	−1.60788	−0.803942	−	0.594708i	$-0.797268\pi$
$239$	−24.8573	−1.60788	−0.803942	+	0.594708i	$0.797268\pi$
$240$	0	0
$241$	9.05086	0.583017	0.291508	−	0.956568i	$-0.405843\pi$
$241$	9.05086	0.583017	0.291508	+	0.956568i	$0.405843\pi$
$242$	1.28300i	0.0824746i
$243$	− 9.00492i	− 0.577666i
$244$	6.99063	0.447529
$245$	0	0
$246$	−4.56199	−0.290862
$247$	0.474572i	0.0301963i
$248$	− 1.51114i	− 0.0959573i
$249$	30.8385	1.95431
$250$	0	0
$251$	−22.5718	−1.42472	−0.712361	−	0.701813i	$-0.752374\pi$
$251$	−22.5718	−1.42472	−0.712361	+	0.701813i	$0.752374\pi$
$252$	− 45.7560i	− 2.88236i
$253$	3.61285i	0.227138i
$254$	2.23951	0.140519
$255$	0	0
$256$	8.80642	0.550401
$257$	− 4.94470i	− 0.308442i	−0.988036	−	0.154221i	$-0.950713\pi$
$257$	− 4.94470i	− 0.308442i	0.988036	−	0.154221i	$-0.0492868\pi$
$258$	10.8760i	0.677111i
$259$	19.8163	1.23132
$260$	0	0
$261$	−42.3783	−2.62315
$262$	0.653858i	0.0403955i
$263$	9.37778i	0.578259i	0.957290	+	0.289129i	$0.0933658\pi$
$263$	9.37778i	0.578259i	−0.957290	+	0.289129i	$0.906634\pi$
$264$	−9.24443	−0.568955
$265$	0	0
$266$	1.37778	0.0844774
$267$	36.7654i	2.25001i
$268$	3.15701i	0.192845i
$269$	−19.7146	−1.20202	−0.601009	−	0.799242i	$-0.705234\pi$
$269$	−19.7146	−1.20202	−0.601009	+	0.799242i	$0.705234\pi$
$270$	0	0
$271$	−1.11108	−0.0674932	−0.0337466	−	0.999430i	$-0.510744\pi$
$271$	−1.11108	−0.0674932	−0.0337466	+	0.999430i	$0.510744\pi$
$272$	− 17.3176i	− 1.05003i
$273$	− 6.10171i	− 0.369292i
$274$	−0.530350	−0.0320396
$275$	0	0
$276$	7.61285	0.458240
$277$	− 5.52098i	− 0.331724i	−0.986149	−	0.165862i	$-0.946959\pi$
$277$	− 5.52098i	− 0.331724i	0.986149	−	0.165862i	$-0.0530406\pi$
$278$	2.71408i	0.162780i
$279$	−6.75557	−0.404445
$280$	0	0
$281$	−15.8064	−0.942932	−0.471466	−	0.881884i	$-0.656275\pi$
$281$	−15.8064	−0.942932	−0.471466	+	0.881884i	$0.656275\pi$
$282$	4.00000i	0.238197i
$283$	14.2351i	0.846187i	0.906086	+	0.423093i	$0.139056\pi$
$283$	14.2351i	0.846187i	−0.906086	+	0.423093i	$0.860944\pi$
$284$	14.4889	0.859756
$285$	0	0
$286$	−0.387152	−0.0228928
$287$	− 22.3684i	− 1.32037i
$288$	18.9748i	1.11810i
$289$	−8.51114	−0.500655
$290$	0	0
$291$	51.9496	3.04534
$292$	− 7.24443i	− 0.423948i
$293$	7.52543i	0.439640i	0.975540	+	0.219820i	$0.0705470\pi$
$293$	7.52543i	0.439640i	−0.975540	+	0.219820i	$0.929453\pi$
$294$	−11.3921	−0.664399
$295$	0	0
$296$	−5.43356	−0.315819
$297$	18.4889i	1.07283i
$298$	− 2.12045i	− 0.122834i
$299$	0.653858	0.0378136
$300$	0	0
$301$	−53.3274	−3.07374
$302$	1.80642i	0.103948i
$303$	− 30.2766i	− 1.73934i
$304$	3.42864	0.196646
$305$	0	0
$306$	8.53035	0.487648
$307$	2.81135i	0.160452i	0.996777	+	0.0802260i	$0.0255642\pi$
$307$	2.81135i	0.160452i	−0.996777	+	0.0802260i	$0.974436\pi$
$308$	− 22.1017i	− 1.25936i
$309$	16.4286	0.934593
$310$	0	0
$311$	−13.8479	−0.785243	−0.392621	−	0.919700i	$-0.628432\pi$
$311$	−13.8479	−0.785243	−0.392621	+	0.919700i	$0.628432\pi$
$312$	1.67307i	0.0947190i
$313$	23.2444i	1.31385i	0.753955	+	0.656926i	$0.228144\pi$
$313$	23.2444i	1.31385i	−0.753955	+	0.656926i	$0.771856\pi$
$314$	−0.0602231	−0.00339858
$315$	0	0
$316$	25.5397	1.43672
$317$	− 2.96343i	− 0.166443i	−0.996531	−	0.0832215i	$-0.973479\pi$
$317$	− 2.96343i	− 0.166443i	0.996531	−	0.0832215i	$-0.0265209\pi$
$318$	6.79706i	0.381160i
$319$	−20.4701	−1.14611
$320$	0	0
$321$	20.0415	1.11861
$322$	− 1.89829i	− 0.105788i
$323$	− 5.05086i	− 0.281037i
$324$	7.96343	0.442413
$325$	0	0
$326$	−2.62222	−0.145231
$327$	− 16.2953i	− 0.901131i
$328$	6.13335i	0.338658i
$329$	−19.6128	−1.08129
$330$	0	0
$331$	−0.949145	−0.0521697	−0.0260849	−	0.999660i	$-0.508304\pi$
$331$	−0.949145	−0.0521697	−0.0260849	+	0.999660i	$0.508304\pi$
$332$	− 20.2163i	− 1.10952i
$333$	24.2908i	1.33113i
$334$	−3.87109	−0.211817
$335$	0	0
$336$	−44.0830	−2.40492
$337$	− 2.28100i	− 0.124254i	−0.998068	−	0.0621269i	$-0.980212\pi$
$337$	− 2.28100i	− 0.124254i	0.998068	−	0.0621269i	$-0.0197883\pi$
$338$	− 3.97433i	− 0.216175i
$339$	−40.3368	−2.19079
$340$	0	0
$341$	−3.26317	−0.176710
$342$	1.68889i	0.0913248i
$343$	− 24.8573i	− 1.34217i
$344$	14.6222	0.788377
$345$	0	0
$346$	−6.90321	−0.371119
$347$	− 12.3368i	− 0.662273i	−0.943583	−	0.331136i	$-0.892568\pi$
$347$	− 12.3368i	− 0.662273i	0.943583	−	0.331136i	$-0.107432\pi$
$348$	43.1338i	2.31222i
$349$	10.0000	0.535288	0.267644	−	0.963518i	$-0.413755\pi$
$349$	10.0000	0.535288	0.267644	+	0.963518i	$0.413755\pi$
$350$	0	0
$351$	3.34614	0.178604
$352$	9.16547i	0.488521i
$353$	2.56199i	0.136361i	0.997673	+	0.0681806i	$0.0217194\pi$
$353$	2.56199i	0.136361i	−0.997673	+	0.0681806i	$0.978281\pi$
$354$	1.98126	0.105303
$355$	0	0
$356$	24.1017	1.27739
$357$	64.9403i	3.43700i
$358$	− 3.70471i	− 0.195800i
$359$	−24.3368	−1.28445	−0.642223	−	0.766518i	$-0.721988\pi$
$359$	−24.3368	−1.28445	−0.642223	+	0.766518i	$0.721988\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	− 5.47949i	− 0.287996i
$363$	− 11.9728i	− 0.628409i
$364$	−4.00000	−0.209657
$365$	0	0
$366$	3.31756	0.173412
$367$	4.42864i	0.231173i	0.993297	+	0.115587i	$0.0368748\pi$
$367$	4.42864i	0.231173i	−0.993297	+	0.115587i	$0.963125\pi$
$368$	− 4.72393i	− 0.246252i
$369$	27.4193	1.42739
$370$	0	0
$371$	−33.3274	−1.73027
$372$	6.87601i	0.356505i
$373$	− 23.7003i	− 1.22715i	−0.789635	−	0.613577i	$-0.789730\pi$
$373$	− 23.7003i	− 1.22715i	0.789635	−	0.613577i	$-0.210270\pi$
$374$	4.12045	0.213063
$375$	0	0
$376$	5.37778	0.277338
$377$	3.70471i	0.190802i
$378$	− 9.71456i	− 0.499663i
$379$	−8.20342	−0.421381	−0.210691	−	0.977553i	$-0.567571\pi$
$379$	−8.20342	−0.421381	−0.210691	+	0.977553i	$0.567571\pi$
$380$	0	0
$381$	−20.8988	−1.07068
$382$	0.0829744i	0.00424534i
$383$	− 20.9131i	− 1.06861i	−0.845293	−	0.534304i	$-0.820574\pi$
$383$	− 20.9131i	− 1.06861i	0.845293	−	0.534304i	$-0.179426\pi$
$384$	25.5067	1.30163
$385$	0	0
$386$	0.830082	0.0422501
$387$	− 65.3689i	− 3.32289i
$388$	− 34.0558i	− 1.72892i
$389$	24.1017	1.22201	0.611003	−	0.791629i	$-0.290766\pi$
$389$	24.1017	1.22201	0.611003	+	0.791629i	$0.290766\pi$
$390$	0	0
$391$	−6.95899	−0.351931
$392$	15.3160i	0.773576i
$393$	− 6.10171i	− 0.307791i
$394$	1.66323	0.0837921
$395$	0	0
$396$	27.0923	1.36144
$397$	− 7.92687i	− 0.397838i	−0.980016	−	0.198919i	$-0.936257\pi$
$397$	− 7.92687i	− 0.397838i	0.980016	−	0.198919i	$-0.0637431\pi$
$398$	5.32741i	0.267039i
$399$	−12.8573	−0.643669
$400$	0	0
$401$	−32.5718	−1.62656	−0.813280	−	0.581873i	$-0.802320\pi$
$401$	−32.5718	−1.62656	−0.813280	+	0.581873i	$0.802320\pi$
$402$	1.49823i	0.0747249i
$403$	0.590573i	0.0294185i
$404$	−19.8479	−0.987470
$405$	0	0
$406$	10.7556	0.533790
$407$	11.7333i	0.581598i
$408$	− 17.8064i	− 0.881549i
$409$	−36.3684	−1.79830	−0.899151	−	0.437638i	$-0.855815\pi$
$409$	−36.3684	−1.79830	−0.899151	+	0.437638i	$0.855815\pi$
$410$	0	0
$411$	4.94914	0.244123
$412$	− 10.7699i	− 0.530593i
$413$	9.71456i	0.478022i
$414$	2.32693	0.114362
$415$	0	0
$416$	1.65878	0.0813284
$417$	− 25.3274i	− 1.24029i
$418$	0.815792i	0.0399017i
$419$	31.6958	1.54844	0.774221	−	0.632915i	$-0.218142\pi$
$419$	31.6958	1.54844	0.774221	+	0.632915i	$0.218142\pi$
$420$	0	0
$421$	37.4005	1.82279	0.911395	−	0.411532i	$-0.135006\pi$
$421$	37.4005	1.82279	0.911395	+	0.411532i	$0.135006\pi$
$422$	− 4.09187i	− 0.199189i
$423$	− 24.0415i	− 1.16894i
$424$	9.13828	0.443794
$425$	0	0
$426$	6.87601	0.333144
$427$	16.2667i	0.787201i
$428$	− 13.1383i	− 0.635063i
$429$	3.61285	0.174430
$430$	0	0
$431$	4.94914	0.238392	0.119196	−	0.992871i	$-0.461968\pi$
$431$	4.94914	0.238392	0.119196	+	0.992871i	$0.461968\pi$
$432$	− 24.1748i	− 1.16311i
$433$	32.3827i	1.55621i	0.628132	+	0.778107i	$0.283820\pi$
$433$	32.3827i	1.55621i	−0.628132	+	0.778107i	$0.716180\pi$
$434$	1.71456	0.0823014
$435$	0	0
$436$	−10.6824	−0.511596
$437$	− 1.37778i	− 0.0659084i
$438$	− 3.43801i	− 0.164274i
$439$	−10.0731	−0.480764	−0.240382	−	0.970678i	$-0.577273\pi$
$439$	−10.0731	−0.480764	−0.240382	+	0.970678i	$0.577273\pi$
$440$	0	0
$441$	68.4706	3.26050
$442$	− 0.745724i	− 0.0354705i
$443$	− 13.9684i	− 0.663657i	−0.943340	−	0.331828i	$-0.892335\pi$
$443$	− 13.9684i	− 0.663657i	0.943340	−	0.331828i	$-0.107665\pi$
$444$	24.7239	1.17335
$445$	0	0
$446$	−3.27163	−0.154916
$447$	19.7877i	0.935926i
$448$	25.5526i	1.20725i
$449$	−24.5718	−1.15962	−0.579808	−	0.814753i	$-0.696873\pi$
$449$	−24.5718	−1.15962	−0.579808	+	0.814753i	$0.696873\pi$
$450$	0	0
$451$	13.2444	0.623656
$452$	26.4429i	1.24377i
$453$	− 16.8573i	− 0.792024i
$454$	−2.14764	−0.100794
$455$	0	0
$456$	3.52543	0.165093
$457$	3.51114i	0.164244i	0.996622	+	0.0821220i	$0.0261697\pi$
$457$	3.51114i	0.164244i	−0.996622	+	0.0821220i	$0.973830\pi$
$458$	5.61285i	0.262271i
$459$	−35.6128	−1.66227
$460$	0	0
$461$	10.2034	0.475221	0.237610	−	0.971361i	$-0.423636\pi$
$461$	10.2034	0.475221	0.237610	+	0.971361i	$0.423636\pi$
$462$	− 10.4889i	− 0.487986i
$463$	8.33677i	0.387443i	0.981057	+	0.193721i	$0.0620558\pi$
$463$	8.33677i	0.387443i	−0.981057	+	0.193721i	$0.937944\pi$
$464$	26.7654	1.24255
$465$	0	0
$466$	−3.84791	−0.178251
$467$	42.7052i	1.97616i	0.153940	+	0.988080i	$0.450804\pi$
$467$	42.7052i	1.97616i	−0.153940	+	0.988080i	$0.549196\pi$
$468$	− 4.90321i	− 0.226651i
$469$	−7.34614	−0.339213
$470$	0	0
$471$	0.561993	0.0258953
$472$	− 2.66370i	− 0.122607i
$473$	− 31.5754i	− 1.45184i
$474$	12.1204	0.556711
$475$	0	0
$476$	42.5718	1.95128
$477$	− 40.8528i	− 1.87052i
$478$	7.73329i	0.353713i
$479$	−41.4608	−1.89439	−0.947195	−	0.320658i	$-0.896096\pi$
$479$	−41.4608	−1.89439	−0.947195	+	0.320658i	$0.896096\pi$
$480$	0	0
$481$	2.12351	0.0968237
$482$	− 2.81579i	− 0.128256i
$483$	17.7146i	0.806040i
$484$	−7.84882	−0.356764
$485$	0	0
$486$	−2.80150	−0.127079
$487$	− 30.0370i	− 1.36111i	−0.732698	−	0.680554i	$-0.761739\pi$
$487$	− 30.0370i	− 1.36111i	0.732698	−	0.680554i	$-0.238261\pi$
$488$	− 4.46028i	− 0.201907i
$489$	24.4701	1.10658
$490$	0	0
$491$	15.3461	0.692562	0.346281	−	0.938131i	$-0.387444\pi$
$491$	15.3461	0.692562	0.346281	+	0.938131i	$0.387444\pi$
$492$	− 27.9081i	− 1.25820i
$493$	− 39.4291i	− 1.77580i
$494$	0.147643	0.00664278
$495$	0	0
$496$	4.26671	0.191581
$497$	33.7146i	1.51230i
$498$	− 9.59411i	− 0.429922i
$499$	25.8479	1.15711	0.578556	−	0.815643i	$-0.303617\pi$
$499$	25.8479	1.15711	0.578556	+	0.815643i	$0.303617\pi$
$500$	0	0
$501$	36.1245	1.61392
$502$	7.02227i	0.313419i
$503$	− 4.40006i	− 0.196189i	−0.995177	−	0.0980945i	$-0.968725\pi$
$503$	− 4.40006i	− 0.196189i	0.995177	−	0.0980945i	$-0.0312747\pi$
$504$	−29.1941	−1.30041
$505$	0	0
$506$	1.12399	0.0499672
$507$	37.0879i	1.64713i
$508$	13.7003i	0.607851i
$509$	27.2355	1.20719	0.603597	−	0.797290i	$-0.293734\pi$
$509$	27.2355	1.20719	0.603597	+	0.797290i	$0.293734\pi$
$510$	0	0
$511$	16.8573	0.745722
$512$	− 20.3111i	− 0.897633i
$513$	− 7.05086i	− 0.311303i
$514$	−1.53833	−0.0678530
$515$	0	0
$516$	−66.5344	−2.92901
$517$	− 11.6128i	− 0.510732i
$518$	− 6.16500i	− 0.270874i
$519$	64.4197	2.82771
$520$	0	0
$521$	38.5531	1.68904	0.844521	−	0.535522i	$-0.179885\pi$
$521$	38.5531	1.68904	0.844521	+	0.535522i	$0.179885\pi$
$522$	13.1842i	0.577057i
$523$	18.1575i	0.793971i	0.917825	+	0.396986i	$0.129944\pi$
$523$	18.1575i	0.793971i	−0.917825	+	0.396986i	$0.870056\pi$
$524$	−4.00000	−0.174741
$525$	0	0
$526$	2.91750	0.127209
$527$	− 6.28544i	− 0.273798i
$528$	− 26.1017i	− 1.13593i
$529$	21.1017	0.917466
$530$	0	0
$531$	−11.9081	−0.516769
$532$	8.42864i	0.365428i
$533$	− 2.39700i	− 0.103825i
$534$	11.4380	0.494971
$535$	0	0
$536$	2.01429	0.0870041
$537$	34.5718i	1.49188i
$538$	6.13335i	0.264428i
$539$	33.0736	1.42458
$540$	0	0
$541$	−13.7748	−0.592224	−0.296112	−	0.955153i	$-0.595690\pi$
$541$	−13.7748	−0.592224	−0.296112	+	0.955153i	$0.595690\pi$
$542$	0.345665i	0.0148476i
$543$	51.1338i	2.19436i
$544$	−17.6543	−0.756923
$545$	0	0
$546$	−1.89829	−0.0812393
$547$	− 42.9862i	− 1.83796i	−0.394308	−	0.918978i	$-0.629016\pi$
$547$	− 42.9862i	− 1.83796i	0.394308	−	0.918978i	$-0.370984\pi$
$548$	− 3.24443i	− 0.138595i
$549$	−19.9398	−0.851009
$550$	0	0
$551$	7.80642	0.332565
$552$	− 4.85728i	− 0.206740i
$553$	59.4291i	2.52718i
$554$	−1.71762	−0.0729747
$555$	0	0
$556$	−16.6035	−0.704144
$557$	− 14.2953i	− 0.605711i	−0.953037	−	0.302855i	$-0.902060\pi$
$557$	− 14.2953i	− 0.605711i	0.953037	−	0.302855i	$-0.0979399\pi$
$558$	2.10171i	0.0889725i
$559$	−5.71456	−0.241700
$560$	0	0
$561$	−38.4514	−1.62342
$562$	4.91750i	0.207432i
$563$	29.9541i	1.26241i	0.775615	+	0.631207i	$0.217440\pi$
$563$	29.9541i	1.26241i	−0.775615	+	0.631207i	$0.782560\pi$
$564$	−24.4701	−1.03038
$565$	0	0
$566$	4.42864	0.186150
$567$	18.5303i	0.778202i
$568$	− 9.24443i	− 0.387888i
$569$	10.0000	0.419222	0.209611	−	0.977785i	$-0.432780\pi$
$569$	10.0000	0.419222	0.209611	+	0.977785i	$0.432780\pi$
$570$	0	0
$571$	−38.2351	−1.60009	−0.800044	−	0.599942i	$-0.795190\pi$
$571$	−38.2351	−1.60009	−0.800044	+	0.599942i	$0.795190\pi$
$572$	− 2.36842i	− 0.0990285i
$573$	− 0.774305i	− 0.0323470i
$574$	−6.95899	−0.290463
$575$	0	0
$576$	−31.3225	−1.30510
$577$	16.5718i	0.689895i	0.938622	+	0.344947i	$0.112103\pi$
$577$	16.5718i	0.689895i	−0.938622	+	0.344947i	$0.887897\pi$
$578$	2.64788i	0.110137i
$579$	−7.74620	−0.321921
$580$	0	0
$581$	47.0420	1.95163
$582$	− 16.1619i	− 0.669934i
$583$	− 19.7333i	− 0.817270i
$584$	−4.62222	−0.191269
$585$	0	0
$586$	2.34122	0.0967149
$587$	− 7.94962i	− 0.328116i	−0.986451	−	0.164058i	$-0.947542\pi$
$587$	− 7.94962i	− 0.328116i	0.986451	−	0.164058i	$-0.0524584\pi$
$588$	− 69.6914i	− 2.87402i
$589$	1.24443	0.0512759
$590$	0	0
$591$	−15.5210	−0.638448
$592$	− 15.3417i	− 0.630540i
$593$	− 17.0794i	− 0.701368i	−0.936494	−	0.350684i	$-0.885949\pi$
$593$	− 17.0794i	− 0.701368i	0.936494	−	0.350684i	$-0.114051\pi$
$594$	5.75203	0.236009
$595$	0	0
$596$	12.9719	0.531350
$597$	− 49.7146i	− 2.03468i
$598$	− 0.203420i	− 0.00831848i
$599$	−5.68598	−0.232323	−0.116161	−	0.993230i	$-0.537059\pi$
$599$	−5.68598	−0.232323	−0.116161	+	0.993230i	$0.537059\pi$
$600$	0	0
$601$	−33.2543	−1.35647	−0.678235	−	0.734845i	$-0.737255\pi$
$601$	−33.2543	−1.35647	−0.678235	+	0.734845i	$0.737255\pi$
$602$	16.5906i	0.676181i
$603$	− 9.00492i	− 0.366709i
$604$	−11.0509	−0.449653
$605$	0	0
$606$	−9.41927	−0.382632
$607$	10.9032i	0.442548i	0.975212	+	0.221274i	$0.0710215\pi$
$607$	10.9032i	0.442548i	−0.975212	+	0.221274i	$0.928979\pi$
$608$	− 3.49532i	− 0.141754i
$609$	−100.369	−4.06717
$610$	0	0
$611$	−2.10171	−0.0850261
$612$	52.1847i	2.10944i
$613$	− 47.6227i	− 1.92346i	−0.273995	−	0.961731i	$-0.588345\pi$
$613$	− 47.6227i	− 1.92346i	0.273995	−	0.961731i	$-0.411655\pi$
$614$	0.874632	0.0352973
$615$	0	0
$616$	−14.1017	−0.568174
$617$	46.6450i	1.87786i	0.344114	+	0.938928i	$0.388179\pi$
$617$	46.6450i	1.87786i	−0.344114	+	0.938928i	$0.611821\pi$
$618$	− 5.11108i	− 0.205598i
$619$	32.2163	1.29488	0.647442	−	0.762115i	$-0.275839\pi$
$619$	32.2163	1.29488	0.647442	+	0.762115i	$0.275839\pi$
$620$	0	0
$621$	−9.71456	−0.389832
$622$	4.30819i	0.172743i
$623$	56.0830i	2.24692i
$624$	−4.72393	−0.189108
$625$	0	0
$626$	7.23152	0.289030
$627$	− 7.61285i	− 0.304028i
$628$	− 0.368416i	− 0.0147014i
$629$	−22.6004	−0.901138
$630$	0	0
$631$	−30.9719	−1.23297	−0.616486	−	0.787366i	$-0.711444\pi$
$631$	−30.9719	−1.23297	−0.616486	+	0.787366i	$0.711444\pi$
$632$	− 16.2953i	− 0.648192i
$633$	38.1847i	1.51770i
$634$	−0.921948	−0.0366152
$635$	0	0
$636$	−41.5812	−1.64880
$637$	− 5.98571i	− 0.237162i
$638$	6.36842i	0.252128i
$639$	−41.3274	−1.63489
$640$	0	0
$641$	−22.1748	−0.875854	−0.437927	−	0.899011i	$-0.644287\pi$
$641$	−22.1748	−0.875854	−0.437927	+	0.899011i	$0.644287\pi$
$642$	− 6.23506i	− 0.246078i
$643$	− 6.23506i	− 0.245887i	−0.992414	−	0.122943i	$-0.960767\pi$
$643$	− 6.23506i	− 0.245887i	0.992414	−	0.122943i	$-0.0392334\pi$
$644$	11.6128	0.457610
$645$	0	0
$646$	−1.57136	−0.0618244
$647$	1.29481i	0.0509042i	0.999676	+	0.0254521i	$0.00810254\pi$
$647$	1.29481i	0.0509042i	−0.999676	+	0.0254521i	$0.991897\pi$
$648$	− 5.08097i	− 0.199599i
$649$	−5.75203	−0.225787
$650$	0	0
$651$	−16.0000	−0.627089
$652$	− 16.0415i	− 0.628233i
$653$	30.2953i	1.18555i	0.805370	+	0.592773i	$0.201967\pi$
$653$	30.2953i	1.18555i	−0.805370	+	0.592773i	$0.798033\pi$
$654$	−5.06959	−0.198237
$655$	0	0
$656$	−17.3176	−0.676137
$657$	20.6637i	0.806168i
$658$	6.10171i	0.237869i
$659$	4.17484	0.162629	0.0813143	−	0.996689i	$-0.474088\pi$
$659$	4.17484	0.162629	0.0813143	+	0.996689i	$0.474088\pi$
$660$	0	0
$661$	−6.56199	−0.255232	−0.127616	−	0.991824i	$-0.540732\pi$
$661$	−6.56199	−0.255232	−0.127616	+	0.991824i	$0.540732\pi$
$662$	0.295286i	0.0114766i
$663$	6.95899i	0.270265i
$664$	−12.8988	−0.500569
$665$	0	0
$666$	7.55707	0.292831
$667$	− 10.7556i	− 0.416457i
$668$	− 23.6815i	− 0.916266i
$669$	30.5303	1.18037
$670$	0	0
$671$	−9.63158	−0.371823
$672$	44.9403i	1.73361i
$673$	12.7413i	0.491140i	0.969379	+	0.245570i	$0.0789752\pi$
$673$	12.7413i	0.491140i	−0.969379	+	0.245570i	$0.921025\pi$
$674$	−0.709636	−0.0273341
$675$	0	0
$676$	24.3131	0.935120
$677$	30.9260i	1.18858i	0.804250	+	0.594291i	$0.202567\pi$
$677$	30.9260i	1.18858i	−0.804250	+	0.594291i	$0.797433\pi$
$678$	12.5491i	0.481945i
$679$	79.2454	3.04116
$680$	0	0
$681$	20.0415	0.767991
$682$	1.01520i	0.0388739i
$683$	34.3412i	1.31403i	0.753877	+	0.657015i	$0.228181\pi$
$683$	34.3412i	1.31403i	−0.753877	+	0.657015i	$0.771819\pi$
$684$	−10.3319	−0.395048
$685$	0	0
$686$	−7.73329	−0.295259
$687$	− 52.3783i	− 1.99836i
$688$	41.2859i	1.57401i
$689$	−3.57136	−0.136058
$690$	0	0
$691$	24.2163	0.921233	0.460616	−	0.887599i	$-0.347628\pi$
$691$	24.2163	0.921233	0.460616	+	0.887599i	$0.347628\pi$
$692$	− 42.2306i	− 1.60537i
$693$	63.0420i	2.39477i
$694$	−3.83807	−0.145691
$695$	0	0
$696$	27.5210	1.04318
$697$	25.5111i	0.966303i
$698$	− 3.11108i	− 0.117756i
$699$	35.9081	1.35817
$700$	0	0
$701$	11.4064	0.430812	0.215406	−	0.976525i	$-0.430892\pi$
$701$	11.4064	0.430812	0.215406	+	0.976525i	$0.430892\pi$
$702$	− 1.04101i	− 0.0392904i
$703$	− 4.47457i	− 0.168762i
$704$	−15.1298	−0.570226
$705$	0	0
$706$	0.797056	0.0299976
$707$	− 46.1847i	− 1.73695i
$708$	12.1204i	0.455514i
$709$	13.0223	0.489062	0.244531	−	0.969642i	$-0.421366\pi$
$709$	13.0223	0.489062	0.244531	+	0.969642i	$0.421366\pi$
$710$	0	0
$711$	−72.8484	−2.73203
$712$	− 15.3778i	− 0.576307i
$713$	− 1.71456i	− 0.0642107i
$714$	20.2034	0.756094
$715$	0	0
$716$	22.6637	0.846982
$717$	− 72.1659i	− 2.69509i
$718$	7.57136i	0.282561i
$719$	−52.2163	−1.94734	−0.973670	−	0.227961i	$-0.926794\pi$
$719$	−52.2163	−1.94734	−0.973670	+	0.227961i	$0.926794\pi$
$720$	0	0
$721$	25.0607	0.933309
$722$	− 0.311108i	− 0.0115782i
$723$	26.2766i	0.977235i
$724$	33.5210	1.24580
$725$	0	0
$726$	−3.72483	−0.138242
$727$	− 14.0602i	− 0.521465i	−0.965411	−	0.260732i	$-0.916036\pi$
$727$	− 14.0602i	− 0.521465i	0.965411	−	0.260732i	$-0.0839640\pi$
$728$	2.55215i	0.0945889i
$729$	38.6958	1.43318
$730$	0	0
$731$	60.8198	2.24950
$732$	20.2953i	0.750135i
$733$	− 4.48886i	− 0.165800i	−0.996558	−	0.0829000i	$-0.973582\pi$
$733$	− 4.48886i	− 0.165800i	0.996558	−	0.0829000i	$-0.0264182\pi$
$734$	1.37778	0.0508549
$735$	0	0
$736$	−4.81579	−0.177512
$737$	− 4.34968i	− 0.160223i
$738$	− 8.53035i	− 0.314007i
$739$	20.0000	0.735712	0.367856	−	0.929883i	$-0.380092\pi$
$739$	20.0000	0.735712	0.367856	+	0.929883i	$0.380092\pi$
$740$	0	0
$741$	−1.37778	−0.0506142
$742$	10.3684i	0.380637i
$743$	− 37.5669i	− 1.37820i	−0.724668	−	0.689098i	$-0.758007\pi$
$743$	− 37.5669i	− 1.37820i	0.724668	−	0.689098i	$-0.241993\pi$
$744$	4.38715	0.160841
$745$	0	0
$746$	−7.37334	−0.269957
$747$	57.6642i	2.10982i
$748$	25.2070i	0.921658i
$749$	30.5718	1.11707
$750$	0	0
$751$	52.5817	1.91873	0.959366	−	0.282163i	$-0.0910520\pi$
$751$	52.5817	1.91873	0.959366	+	0.282163i	$0.0910520\pi$
$752$	15.1842i	0.553711i
$753$	− 65.5308i	− 2.38808i
$754$	1.15257	0.0419740
$755$	0	0
$756$	59.4291	2.16142
$757$	− 5.73329i	− 0.208380i	−0.994557	−	0.104190i	$-0.966775\pi$
$757$	− 5.73329i	− 0.208380i	0.994557	−	0.104190i	$-0.0332250\pi$
$758$	2.55215i	0.0926982i
$759$	−10.4889	−0.380722
$760$	0	0
$761$	−32.7338	−1.18660	−0.593299	−	0.804982i	$-0.702175\pi$
$761$	−32.7338	−1.18660	−0.593299	+	0.804982i	$0.702175\pi$
$762$	6.50177i	0.235534i
$763$	− 24.8573i	− 0.899894i
$764$	−0.507598	−0.0183643
$765$	0	0
$766$	−6.50622	−0.235079
$767$	1.04101i	0.0375887i
$768$	25.5669i	0.922567i
$769$	−10.1619	−0.366449	−0.183224	−	0.983071i	$-0.558653\pi$
$769$	−10.1619	−0.366449	−0.183224	+	0.983071i	$0.558653\pi$
$770$	0	0
$771$	14.3555	0.517001
$772$	5.07805i	0.182763i
$773$	− 36.0785i	− 1.29765i	−0.760936	−	0.648827i	$-0.775260\pi$
$773$	− 36.0785i	− 1.29765i	0.760936	−	0.648827i	$-0.224740\pi$
$774$	−20.3368	−0.730990
$775$	0	0
$776$	−21.7288	−0.780020
$777$	57.5308i	2.06391i
$778$	− 7.49823i	− 0.268825i
$779$	−5.05086	−0.180966
$780$	0	0
$781$	−19.9625	−0.714315
$782$	2.16500i	0.0774201i
$783$	− 55.0420i	− 1.96704i
$784$	−43.2449	−1.54446
$785$	0	0
$786$	−1.89829	−0.0677098
$787$	30.5446i	1.08880i	0.838826	+	0.544399i	$0.183242\pi$
$787$	30.5446i	1.08880i	−0.838826	+	0.544399i	$0.816758\pi$
$788$	10.1748i	0.362464i
$789$	−27.2257	−0.969260
$790$	0	0
$791$	−61.5308	−2.18778
$792$	− 17.2859i	− 0.614228i
$793$	1.74314i	0.0619007i
$794$	−2.46611	−0.0875190
$795$	0	0
$796$	−32.5906	−1.15514
$797$	27.9037i	0.988399i	0.869348	+	0.494200i	$0.164539\pi$
$797$	27.9037i	0.988399i	−0.869348	+	0.494200i	$0.835461\pi$
$798$	4.00000i	0.141598i
$799$	22.3684	0.791338
$800$	0	0
$801$	−68.7467	−2.42904
$802$	10.1334i	0.357821i
$803$	9.98126i	0.352231i
$804$	−9.16547	−0.323241
$805$	0	0
$806$	0.183732	0.00647168
$807$	− 57.2355i	− 2.01479i
$808$	12.6637i	0.445508i
$809$	25.6128	0.900500	0.450250	−	0.892903i	$-0.351335\pi$
$809$	25.6128	0.900500	0.450250	+	0.892903i	$0.351335\pi$
$810$	0	0
$811$	−6.01874	−0.211346	−0.105673	−	0.994401i	$-0.533700\pi$
$811$	−6.01874	−0.211346	−0.105673	+	0.994401i	$0.533700\pi$
$812$	65.7975i	2.30904i
$813$	− 3.22570i	− 0.113130i
$814$	3.65032	0.127944
$815$	0	0
$816$	50.2766	1.76003
$817$	12.0415i	0.421278i
$818$	11.3145i	0.395602i
$819$	11.4094	0.398678
$820$	0	0
$821$	−6.20342	−0.216501	−0.108250	−	0.994124i	$-0.534525\pi$
$821$	−6.20342	−0.216501	−0.108250	+	0.994124i	$0.534525\pi$
$822$	− 1.53972i	− 0.0537038i
$823$	1.75605i	0.0612119i	0.999532	+	0.0306059i	$0.00974370\pi$
$823$	1.75605i	0.0612119i	−0.999532	+	0.0306059i	$0.990256\pi$
$824$	−6.87157	−0.239382
$825$	0	0
$826$	3.02227	0.105158
$827$	53.2083i	1.85024i	0.379680	+	0.925118i	$0.376034\pi$
$827$	53.2083i	1.85024i	−0.379680	+	0.925118i	$0.623966\pi$
$828$	14.2351i	0.494703i
$829$	26.9777	0.936975	0.468488	−	0.883470i	$-0.344799\pi$
$829$	26.9777	0.936975	0.468488	+	0.883470i	$0.344799\pi$
$830$	0	0
$831$	16.0286	0.556025
$832$	2.73822i	0.0949306i
$833$	63.7057i	2.20727i
$834$	−7.87955	−0.272847
$835$	0	0
$836$	−4.99063	−0.172605
$837$	− 8.77430i	− 0.303284i
$838$	− 9.86082i	− 0.340636i
$839$	−46.9501	−1.62090	−0.810449	−	0.585810i	$-0.800777\pi$
$839$	−46.9501	−1.62090	−0.810449	+	0.585810i	$0.800777\pi$
$840$	0	0
$841$	31.9403	1.10139
$842$	− 11.6356i	− 0.400989i
$843$	− 45.8894i	− 1.58051i
$844$	25.0321	0.861641
$845$	0	0
$846$	−7.47949	−0.257150
$847$	− 18.2636i	− 0.627546i
$848$	25.8020i	0.886044i
$849$	−41.3274	−1.41835
$850$	0	0
$851$	−6.16500	−0.211333
$852$	42.0642i	1.44110i
$853$	− 46.4701i	− 1.59111i	−0.605883	−	0.795553i	$-0.707180\pi$
$853$	− 46.4701i	− 1.59111i	0.605883	−	0.795553i	$-0.292820\pi$
$854$	5.06070	0.173174
$855$	0	0
$856$	−8.38271	−0.286515
$857$	7.79213i	0.266174i	0.991104	+	0.133087i	$0.0424890\pi$
$857$	7.79213i	0.266174i	−0.991104	+	0.133087i	$0.957511\pi$
$858$	− 1.12399i	− 0.0383722i
$859$	37.4479	1.27770	0.638852	−	0.769330i	$-0.279410\pi$
$859$	37.4479	1.27770	0.638852	+	0.769330i	$0.279410\pi$
$860$	0	0
$861$	64.9403	2.21316
$862$	− 1.53972i	− 0.0524430i
$863$	47.7605i	1.62579i	0.582413	+	0.812893i	$0.302109\pi$
$863$	47.7605i	1.62579i	−0.582413	+	0.812893i	$0.697891\pi$
$864$	−24.6450	−0.838439
$865$	0	0
$866$	10.0745	0.342346
$867$	− 24.7096i	− 0.839183i
$868$	10.4889i	0.356015i
$869$	−35.1882	−1.19368
$870$	0	0
$871$	−0.787212	−0.0266736
$872$	6.81579i	0.230812i
$873$	97.1392i	3.28766i
$874$	−0.428639	−0.0144989
$875$	0	0
$876$	21.0321	0.710609
$877$	22.5763i	0.762347i	0.924504	+	0.381173i	$0.124480\pi$
$877$	22.5763i	0.762347i	−0.924504	+	0.381173i	$0.875520\pi$
$878$	3.13383i	0.105762i
$879$	−21.8479	−0.736912
$880$	0	0
$881$	−10.8988	−0.367189	−0.183594	−	0.983002i	$-0.558773\pi$
$881$	−10.8988	−0.367189	−0.183594	+	0.983002i	$0.558773\pi$
$882$	− 21.3017i	− 0.717267i
$883$	− 39.9782i	− 1.34537i	−0.739927	−	0.672687i	$-0.765140\pi$
$883$	− 39.9782i	− 1.34537i	0.739927	−	0.672687i	$-0.234860\pi$
$884$	4.56199	0.153436
$885$	0	0
$886$	−4.34567	−0.145995
$887$	− 49.6785i	− 1.66804i	−0.551734	−	0.834020i	$-0.686034\pi$
$887$	− 49.6785i	− 1.66804i	0.551734	−	0.834020i	$-0.313966\pi$
$888$	− 15.7748i	− 0.529367i
$889$	−31.8796	−1.06921
$890$	0	0
$891$	−10.9719	−0.367572
$892$	− 20.0143i	− 0.670128i
$893$	4.42864i	0.148199i
$894$	6.15610	0.205891
$895$	0	0
$896$	38.9086	1.29985
$897$	1.89829i	0.0633821i
$898$	7.64449i	0.255100i
$899$	9.71456	0.323999
$900$	0	0
$901$	38.0098	1.26629
$902$	− 4.12045i	− 0.137196i
$903$	− 154.821i	− 5.15211i
$904$	16.8716	0.561140
$905$	0	0
$906$	−5.24443	−0.174235
$907$	18.2779i	0.606909i	0.952846	+	0.303454i	$0.0981400\pi$
$907$	18.2779i	0.606909i	−0.952846	+	0.303454i	$0.901860\pi$
$908$	− 13.1383i	− 0.436009i
$909$	56.6133	1.87775
$910$	0	0
$911$	−44.7654	−1.48314	−0.741572	−	0.670873i	$-0.765920\pi$
$911$	−44.7654	−1.48314	−0.741572	+	0.670873i	$0.765920\pi$
$912$	9.95407i	0.329612i
$913$	27.8537i	0.921824i
$914$	1.09234	0.0361315
$915$	0	0
$916$	−34.3368	−1.13452
$917$	− 9.30772i	− 0.307368i
$918$	11.0794i	0.365676i
$919$	33.6316	1.10940	0.554702	−	0.832049i	$-0.312832\pi$
$919$	33.6316	1.10940	0.554702	+	0.832049i	$0.312832\pi$
$920$	0	0
$921$	−8.16193	−0.268945
$922$	− 3.17436i	− 0.104542i
$923$	3.61285i	0.118918i
$924$	64.1659	2.11090
$925$	0	0
$926$	2.59364	0.0852321
$927$	30.7195i	1.00896i
$928$	− 27.2859i	− 0.895704i
$929$	−13.0223	−0.427247	−0.213623	−	0.976916i	$-0.568527\pi$
$929$	−13.0223	−0.427247	−0.213623	+	0.976916i	$0.568527\pi$
$930$	0	0
$931$	−12.6128	−0.413369
$932$	− 23.5397i	− 0.771069i
$933$	− 40.2034i	− 1.31620i
$934$	13.2859	0.434729
$935$	0	0
$936$	−3.12843	−0.102256
$937$	− 28.5433i	− 0.932468i	−0.884662	−	0.466234i	$-0.845611\pi$
$937$	− 28.5433i	− 0.932468i	0.884662	−	0.466234i	$-0.154389\pi$
$938$	2.28544i	0.0746223i
$939$	−67.4835	−2.20224
$940$	0	0
$941$	13.2257	0.431145	0.215573	−	0.976488i	$-0.430838\pi$
$941$	13.2257	0.431145	0.215573	+	0.976488i	$0.430838\pi$
$942$	− 0.174840i	− 0.00569660i
$943$	6.95899i	0.226616i
$944$	7.52098	0.244787
$945$	0	0
$946$	−9.82335	−0.319385
$947$	− 3.66323i	− 0.119039i	−0.998227	−	0.0595194i	$-0.981043\pi$
$947$	− 3.66323i	− 0.119039i	0.998227	−	0.0595194i	$-0.0189568\pi$
$948$	74.1472i	2.40819i
$949$	1.80642	0.0586390
$950$	0	0
$951$	8.60348	0.278987
$952$	− 27.1624i	− 0.880339i
$953$	12.1704i	0.394238i	0.980380	+	0.197119i	$0.0631585\pi$
$953$	12.1704i	0.394238i	−0.980380	+	0.197119i	$0.936842\pi$
$954$	−12.7096	−0.411490
$955$	0	0
$956$	−47.3087	−1.53007
$957$	− 59.4291i	− 1.92107i
$958$	12.8988i	0.416740i
$959$	7.54956	0.243788
$960$	0	0
$961$	−29.4514	−0.950045
$962$	− 0.660640i	− 0.0212999i
$963$	37.4750i	1.20762i
$964$	17.2257	0.554802
$965$	0	0
$966$	5.51114	0.177318
$967$	− 8.52051i	− 0.274001i	−0.990571	−	0.137000i	$-0.956254\pi$
$967$	− 8.52051i	− 0.274001i	0.990571	−	0.137000i	$-0.0437462\pi$
$968$	5.00784i	0.160958i
$969$	14.6637	0.471066
$970$	0	0
$971$	−2.67259	−0.0857676	−0.0428838	−	0.999080i	$-0.513655\pi$
$971$	−2.67259	−0.0857676	−0.0428838	+	0.999080i	$0.513655\pi$
$972$	− 17.1383i	− 0.549710i
$973$	− 38.6351i	− 1.23859i
$974$	−9.34476	−0.299425
$975$	0	0
$976$	12.5936	0.403112
$977$	20.3827i	0.652101i	0.945352	+	0.326050i	$0.105718\pi$
$977$	20.3827i	0.652101i	−0.945352	+	0.326050i	$0.894282\pi$
$978$	− 7.61285i	− 0.243432i
$979$	−33.2070	−1.06130
$980$	0	0
$981$	30.4701	0.972836
$982$	− 4.77430i	− 0.152354i
$983$	10.3126i	0.328922i	0.986384	+	0.164461i	$0.0525885\pi$
$983$	10.3126i	0.328922i	−0.986384	+	0.164461i	$0.947412\pi$
$984$	−17.8064	−0.567648
$985$	0	0
$986$	−12.2667	−0.390652
$987$	− 56.9403i	− 1.81243i
$988$	0.903212i	0.0287350i
$989$	16.5906	0.527550
$990$	0	0
$991$	20.0919	0.638239	0.319120	−	0.947714i	$-0.396613\pi$
$991$	20.0919	0.638239	0.319120	+	0.947714i	$0.396613\pi$
$992$	− 4.34968i	− 0.138102i
$993$	− 2.75557i	− 0.0874453i
$994$	10.4889	0.332687
$995$	0	0
$996$	58.6923	1.85974
$997$	− 25.0509i	− 0.793369i	−0.917955	−	0.396684i	$-0.870161\pi$
$997$	− 25.0509i	− 0.793369i	0.917955	−	0.396684i	$-0.129839\pi$
$998$	− 8.04149i	− 0.254549i
$999$	−31.5496	−0.998184

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	475.2.b.d.324.3		6
5.2	odd	4		95.2.a.a.1.2	✓	3
5.3	odd	4		475.2.a.f.1.2		3
5.4	even	2	inner	475.2.b.d.324.4		6
15.2	even	4		855.2.a.i.1.2		3
15.8	even	4		4275.2.a.bk.1.2		3
20.3	even	4		7600.2.a.bx.1.3		3
20.7	even	4		1520.2.a.p.1.1		3
35.27	even	4		4655.2.a.u.1.2		3
40.27	even	4		6080.2.a.by.1.3		3
40.37	odd	4		6080.2.a.bo.1.1		3
95.18	even	4		9025.2.a.bb.1.2		3
95.37	even	4		1805.2.a.f.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
95.2.a.a.1.2	✓	3	5.2	odd	4
475.2.a.f.1.2		3	5.3	odd	4
475.2.b.d.324.3		6	1.1	even	1	trivial
475.2.b.d.324.4		6	5.4	even	2	inner
855.2.a.i.1.2		3	15.2	even	4
1520.2.a.p.1.1		3	20.7	even	4
1805.2.a.f.1.2		3	95.37	even	4
4275.2.a.bk.1.2		3	15.8	even	4
4655.2.a.u.1.2		3	35.27	even	4
6080.2.a.bo.1.1		3	40.37	odd	4
6080.2.a.by.1.3		3	40.27	even	4
7600.2.a.bx.1.3		3	20.3	even	4
9025.2.a.bb.1.2		3	95.18	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)

\(f(q)\)	\(=\)	\(q-0.311108i q^{2} +2.90321i q^{3} +1.90321 q^{4} +0.903212 q^{6} +4.42864i q^{7} -1.21432i q^{8} -5.42864 q^{9} +O(q^{10})\)	\(q-0.311108i q^{2} +2.90321i q^{3} +1.90321 q^{4} +0.903212 q^{6} +4.42864i q^{7} -1.21432i q^{8} -5.42864 q^{9} -2.62222 q^{11} +5.52543i q^{12} +0.474572i q^{13} +1.37778 q^{14} +3.42864 q^{16} -5.05086i q^{17} +1.68889i q^{18} +1.00000 q^{19} -12.8573 q^{21} +0.815792i q^{22} -1.37778i q^{23} +3.52543 q^{24} +0.147643 q^{26} -7.05086i q^{27} +8.42864i q^{28} +7.80642 q^{29} +1.24443 q^{31} -3.49532i q^{32} -7.61285i q^{33} -1.57136 q^{34} -10.3319 q^{36} -4.47457i q^{37} -0.311108i q^{38} -1.37778 q^{39} -5.05086 q^{41} +4.00000i q^{42} +12.0415i q^{43} -4.99063 q^{44} -0.428639 q^{46} +4.42864i q^{47} +9.95407i q^{48} -12.6128 q^{49} +14.6637 q^{51} +0.903212i q^{52} +7.52543i q^{53} -2.19358 q^{54} +5.37778 q^{56} +2.90321i q^{57} -2.42864i q^{58} +2.19358 q^{59} +3.67307 q^{61} -0.387152i q^{62} -24.0415i q^{63} +5.76986 q^{64} -2.36842 q^{66} +1.65878i q^{67} -9.61285i q^{68} +4.00000 q^{69} +7.61285 q^{71} +6.59210i q^{72} -3.80642i q^{73} -1.39207 q^{74} +1.90321 q^{76} -11.6128i q^{77} +0.428639i q^{78} +13.4193 q^{79} +4.18421 q^{81} +1.57136i q^{82} -10.6222i q^{83} -24.4701 q^{84} +3.74620 q^{86} +22.6637i q^{87} +3.18421i q^{88} +12.6637 q^{89} -2.10171 q^{91} -2.62222i q^{92} +3.61285i q^{93} +1.37778 q^{94} +10.1476 q^{96} -17.8938i q^{97} +3.92396i q^{98} +14.2351 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 2 q^{4} - 8 q^{6} - 6 q^{9}+O(q^{10}) \) 6 * q - 2 * q^4 - 8 * q^6 - 6 * q^9	\( 6 q - 2 q^{4} - 8 q^{6} - 6 q^{9} - 16 q^{11} + 8 q^{14} - 6 q^{16} + 6 q^{19} - 24 q^{21} + 8 q^{24} - 12 q^{26} + 20 q^{29} + 8 q^{31} - 36 q^{34} - 22 q^{36} - 8 q^{39} - 4 q^{41} + 24 q^{44} + 24 q^{46} - 22 q^{49} + 8 q^{51} - 40 q^{54} + 32 q^{56} + 40 q^{59} - 4 q^{61} + 22 q^{64} + 40 q^{66} + 24 q^{69} - 8 q^{71} + 4 q^{74} - 2 q^{76} - 2 q^{81} - 40 q^{84} - 32 q^{86} - 4 q^{89} + 40 q^{91} + 8 q^{94} + 48 q^{96} + 32 q^{99}+O(q^{100}) \) 6 * q - 2 * q^4 - 8 * q^6 - 6 * q^9 - 16 * q^11 + 8 * q^14 - 6 * q^16 + 6 * q^19 - 24 * q^21 + 8 * q^24 - 12 * q^26 + 20 * q^29 + 8 * q^31 - 36 * q^34 - 22 * q^36 - 8 * q^39 - 4 * q^41 + 24 * q^44 + 24 * q^46 - 22 * q^49 + 8 * q^51 - 40 * q^54 + 32 * q^56 + 40 * q^59 - 4 * q^61 + 22 * q^64 + 40 * q^66 + 24 * q^69 - 8 * q^71 + 4 * q^74 - 2 * q^76 - 2 * q^81 - 40 * q^84 - 32 * q^86 - 4 * q^89 + 40 * q^91 + 8 * q^94 + 48 * q^96 + 32 * q^99

Introduction

L-functions

Modular forms

Varieties

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Representations

Groups

Database

Properties

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