LMFDB - Embedded newform 475.2.b.c.324.1

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			324.1
Root			$-1.24698i$ of defining polynomial
Character	$\chi$	$=$	475.324
Dual form			475.2.b.c.324.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.80194i q^{2} +0.554958i q^{3} -5.85086 q^{4} +1.55496 q^{6} +3.04892i q^{7} +10.7899i q^{8} +2.69202 q^{9} +O(q^{10})$	\(q-2.80194i q^{2} +0.554958i q^{3} -5.85086 q^{4} +1.55496 q^{6} +3.04892i q^{7} +10.7899i q^{8} +2.69202 q^{9} -2.93900 q^{11} -3.24698i q^{12} +3.24698i q^{13} +8.54288 q^{14} +18.5308 q^{16} +2.15883i q^{17} -7.54288i q^{18} +1.00000 q^{19} -1.69202 q^{21} +8.23490i q^{22} +1.19806i q^{23} -5.98792 q^{24} +9.09783 q^{26} +3.15883i q^{27} -17.8388i q^{28} +1.77479 q^{29} -9.34481 q^{31} -30.3424i q^{32} -1.63102i q^{33} +6.04892 q^{34} -15.7506 q^{36} +1.15883i q^{37} -2.80194i q^{38} -1.80194 q^{39} +8.57002 q^{41} +4.74094i q^{42} +5.27413i q^{43} +17.1957 q^{44} +3.35690 q^{46} -2.35690i q^{47} +10.2838i q^{48} -2.29590 q^{49} -1.19806 q^{51} -18.9976i q^{52} +8.82371i q^{53} +8.85086 q^{54} -32.8974 q^{56} +0.554958i q^{57} -4.97285i q^{58} +5.70171 q^{59} -9.96077 q^{61} +26.1836i q^{62} +8.20775i q^{63} -47.9560 q^{64} -4.57002 q^{66} -4.98254i q^{67} -12.6310i q^{68} -0.664874 q^{69} +2.70171 q^{71} +29.0465i q^{72} +13.7778i q^{73} +3.24698 q^{74} -5.85086 q^{76} -8.96077i q^{77} +5.04892i q^{78} -5.66487 q^{79} +6.32304 q^{81} -24.0127i q^{82} -3.00969i q^{83} +9.89977 q^{84} +14.7778 q^{86} +0.984935i q^{87} -31.7114i q^{88} +10.2838 q^{89} -9.89977 q^{91} -7.00969i q^{92} -5.18598i q^{93} -6.60388 q^{94} +16.8388 q^{96} -3.24698i q^{97} +6.43296i q^{98} -7.91185 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	− 2.80194i	− 1.98127i	−0.136540	−	0.990635i	$-0.543598\pi$
$2$	− 2.80194i	− 1.98127i	0.136540	−	0.990635i	$-0.456402\pi$
$3$	0.554958i	0.320405i	0.987084	+	0.160203i	$0.0512148\pi$
$3$	0.554958i	0.320405i	−0.987084	+	0.160203i	$0.948785\pi$
$4$	−5.85086	−2.92543
$5$	0	0
$5$	0	0
$6$	1.55496	0.634809
$7$	3.04892i	1.15238i	0.817315	+	0.576191i	$0.195462\pi$
$7$	3.04892i	1.15238i	−0.817315	+	0.576191i	$0.804538\pi$
$8$	10.7899i	3.81479i
$9$	2.69202	0.897340
$10$	0	0
$11$	−2.93900	−0.886142	−0.443071	−	0.896486i	$-0.646111\pi$
$11$	−2.93900	−0.886142	−0.443071	+	0.896486i	$0.646111\pi$
$12$	− 3.24698i	− 0.937322i
$13$	3.24698i	0.900550i	0.892890	+	0.450275i	$0.148674\pi$
$13$	3.24698i	0.900550i	−0.892890	+	0.450275i	$0.851326\pi$
$14$	8.54288	2.28318
$15$	0	0
$16$	18.5308	4.63270
$17$	2.15883i	0.523594i	0.965123	+	0.261797i	$0.0843151\pi$
$17$	2.15883i	0.523594i	−0.965123	+	0.261797i	$0.915685\pi$
$18$	− 7.54288i	− 1.77787i
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	−1.69202	−0.369229
$22$	8.23490i	1.75569i
$23$	1.19806i	0.249813i	0.992169	+	0.124907i	$0.0398631\pi$
$23$	1.19806i	0.249813i	−0.992169	+	0.124907i	$0.960137\pi$
$24$	−5.98792	−1.22228
$25$	0	0
$26$	9.09783	1.78423
$27$	3.15883i	0.607918i
$28$	− 17.8388i	− 3.37121i
$29$	1.77479	0.329570	0.164785	−	0.986329i	$-0.447307\pi$
$29$	1.77479	0.329570	0.164785	+	0.986329i	$0.447307\pi$
$30$	0	0
$31$	−9.34481	−1.67838	−0.839189	−	0.543840i	$-0.816970\pi$
$31$	−9.34481	−1.67838	−0.839189	+	0.543840i	$0.816970\pi$
$32$	− 30.3424i	− 5.36383i
$33$	− 1.63102i	− 0.283925i
$34$	6.04892	1.03738
$35$	0	0
$36$	−15.7506	−2.62510
$37$	1.15883i	0.190511i	0.995453	+	0.0952555i	$0.0303668\pi$
$37$	1.15883i	0.190511i	−0.995453	+	0.0952555i	$0.969633\pi$
$38$	− 2.80194i	− 0.454534i
$39$	−1.80194	−0.288541
$40$	0	0
$41$	8.57002	1.33841	0.669206	−	0.743077i	$-0.266634\pi$
$41$	8.57002	1.33841	0.669206	+	0.743077i	$0.266634\pi$
$42$	4.74094i	0.731543i
$43$	5.27413i	0.804297i	0.915575	+	0.402148i	$0.131736\pi$
$43$	5.27413i	0.804297i	−0.915575	+	0.402148i	$0.868264\pi$
$44$	17.1957	2.59234
$45$	0	0
$46$	3.35690	0.494947
$47$	− 2.35690i	− 0.343789i	−0.985115	−	0.171894i	$-0.945011\pi$
$47$	− 2.35690i	− 0.343789i	0.985115	−	0.171894i	$-0.0549888\pi$
$48$	10.2838i	1.48434i
$49$	−2.29590	−0.327985
$50$	0	0
$51$	−1.19806	−0.167762
$52$	− 18.9976i	− 2.63449i
$53$	8.82371i	1.21203i	0.795453	+	0.606015i	$0.207233\pi$
$53$	8.82371i	1.21203i	−0.795453	+	0.606015i	$0.792767\pi$
$54$	8.85086	1.20445
$55$	0	0
$56$	−32.8974	−4.39610
$57$	0.554958i	0.0735060i
$58$	− 4.97285i	− 0.652968i
$59$	5.70171	0.742299	0.371150	−	0.928573i	$-0.378964\pi$
$59$	5.70171	0.742299	0.371150	+	0.928573i	$0.378964\pi$
$60$	0	0
$61$	−9.96077	−1.27535	−0.637673	−	0.770307i	$-0.720103\pi$
$61$	−9.96077	−1.27535	−0.637673	+	0.770307i	$0.720103\pi$
$62$	26.1836i	3.32532i
$63$	8.20775i	1.03408i
$64$	−47.9560	−5.99450
$65$	0	0
$66$	−4.57002	−0.562531
$67$	− 4.98254i	− 0.608714i	−0.952558	−	0.304357i	$-0.901558\pi$
$67$	− 4.98254i	− 0.608714i	0.952558	−	0.304357i	$-0.0984416\pi$
$68$	− 12.6310i	− 1.53174i
$69$	−0.664874	−0.0800415
$70$	0	0
$71$	2.70171	0.320634	0.160317	−	0.987066i	$-0.448748\pi$
$71$	2.70171	0.320634	0.160317	+	0.987066i	$0.448748\pi$
$72$	29.0465i	3.42317i
$73$	13.7778i	1.61257i	0.591530	+	0.806283i	$0.298524\pi$
$73$	13.7778i	1.61257i	−0.591530	+	0.806283i	$0.701476\pi$
$74$	3.24698	0.377454
$75$	0	0
$76$	−5.85086	−0.671139
$77$	− 8.96077i	− 1.02117i
$78$	5.04892i	0.571677i
$79$	−5.66487	−0.637348	−0.318674	−	0.947864i	$-0.603238\pi$
$79$	−5.66487	−0.637348	−0.318674	+	0.947864i	$0.603238\pi$
$80$	0	0
$81$	6.32304	0.702560
$82$	− 24.0127i	− 2.65176i
$83$	− 3.00969i	− 0.330356i	−0.986264	−	0.165178i	$-0.947180\pi$
$83$	− 3.00969i	− 0.330356i	0.986264	−	0.165178i	$-0.0528199\pi$
$84$	9.89977	1.08015
$85$	0	0
$86$	14.7778	1.59353
$87$	0.984935i	0.105596i
$88$	− 31.7114i	− 3.38045i
$89$	10.2838	1.09008	0.545041	−	0.838409i	$-0.316514\pi$
$89$	10.2838	1.09008	0.545041	+	0.838409i	$0.316514\pi$
$90$	0	0
$91$	−9.89977	−1.03778
$92$	− 7.00969i	− 0.730811i
$93$	− 5.18598i	− 0.537761i
$94$	−6.60388	−0.681138
$95$	0	0
$96$	16.8388	1.71860
$97$	− 3.24698i	− 0.329681i	−0.986320	−	0.164840i	$-0.947289\pi$
$97$	− 3.24698i	− 0.329681i	0.986320	−	0.164840i	$-0.0527109\pi$
$98$	6.43296i	0.649827i
$99$	−7.91185	−0.795171
$100$	0	0
$101$	5.09246	0.506718	0.253359	−	0.967372i	$-0.418465\pi$
$101$	5.09246	0.506718	0.253359	+	0.967372i	$0.418465\pi$
$102$	3.35690i	0.332382i
$103$	− 14.3110i	− 1.41010i	−0.709157	−	0.705051i	$-0.750924\pi$
$103$	− 14.3110i	− 1.41010i	0.709157	−	0.705051i	$-0.249076\pi$
$104$	−35.0344	−3.43541
$105$	0	0
$106$	24.7235	2.40136
$107$	8.18598i	0.791369i	0.918387	+	0.395684i	$0.129493\pi$
$107$	8.18598i	0.791369i	−0.918387	+	0.395684i	$0.870507\pi$
$108$	− 18.4819i	− 1.77842i
$109$	11.3274	1.08496	0.542482	−	0.840067i	$-0.317485\pi$
$109$	11.3274	1.08496	0.542482	+	0.840067i	$0.317485\pi$
$110$	0	0
$111$	−0.643104	−0.0610407
$112$	56.4989i	5.33864i
$113$	3.43535i	0.323171i	0.986859	+	0.161585i	$0.0516607\pi$
$113$	3.43535i	0.323171i	−0.986859	+	0.161585i	$0.948339\pi$
$114$	1.55496	0.145635
$115$	0	0
$116$	−10.3840	−0.964134
$117$	8.74094i	0.808100i
$118$	− 15.9758i	− 1.47069i
$119$	−6.58211	−0.603381
$120$	0	0
$121$	−2.36227	−0.214752
$122$	27.9095i	2.52680i
$123$	4.75600i	0.428834i
$124$	54.6752	4.90997
$125$	0	0
$126$	22.9976	2.04879
$127$	− 1.46144i	− 0.129681i	−0.997896	−	0.0648407i	$-0.979346\pi$
$127$	− 1.46144i	− 0.129681i	0.997896	−	0.0648407i	$-0.0206539\pi$
$128$	73.6848i	6.51288i
$129$	−2.92692	−0.257701
$130$	0	0
$131$	−13.2295	−1.15587	−0.577934	−	0.816083i	$-0.696141\pi$
$131$	−13.2295	−1.15587	−0.577934	+	0.816083i	$0.696141\pi$
$132$	9.54288i	0.830601i
$133$	3.04892i	0.264375i
$134$	−13.9608	−1.20603
$135$	0	0
$136$	−23.2935	−1.99740
$137$	− 16.1739i	− 1.38183i	−0.722936	−	0.690915i	$-0.757208\pi$
$137$	− 16.1739i	− 1.38183i	0.722936	−	0.690915i	$-0.242792\pi$
$138$	1.86294i	0.158584i
$139$	−13.0978	−1.11094	−0.555472	−	0.831535i	$-0.687462\pi$
$139$	−13.0978	−1.11094	−0.555472	+	0.831535i	$0.687462\pi$
$140$	0	0
$141$	1.30798	0.110152
$142$	− 7.57002i	− 0.635262i
$143$	− 9.54288i	− 0.798015i
$144$	49.8853	4.15711
$145$	0	0
$146$	38.6045	3.19493
$147$	− 1.27413i	− 0.105088i
$148$	− 6.78017i	− 0.557326i
$149$	−11.0640	−0.906397	−0.453198	−	0.891410i	$-0.649717\pi$
$149$	−11.0640	−0.906397	−0.453198	+	0.891410i	$0.649717\pi$
$150$	0	0
$151$	10.4426	0.849811	0.424905	−	0.905238i	$-0.360307\pi$
$151$	10.4426	0.849811	0.424905	+	0.905238i	$0.360307\pi$
$152$	10.7899i	0.875173i
$153$	5.81163i	0.469842i
$154$	−25.1075	−2.02322
$155$	0	0
$156$	10.5429	0.844106
$157$	− 2.48427i	− 0.198266i	−0.995074	−	0.0991332i	$-0.968393\pi$
$157$	− 2.48427i	− 0.198266i	0.995074	−	0.0991332i	$-0.0316070\pi$
$158$	15.8726i	1.26276i
$159$	−4.89679	−0.388341
$160$	0	0
$161$	−3.65279	−0.287880
$162$	− 17.7168i	− 1.39196i
$163$	3.16852i	0.248178i	0.992271	+	0.124089i	$0.0396008\pi$
$163$	3.16852i	0.248178i	−0.992271	+	0.124089i	$0.960399\pi$
$164$	−50.1420	−3.91543
$165$	0	0
$166$	−8.43296	−0.654525
$167$	− 4.74632i	− 0.367281i	−0.982993	−	0.183640i	$-0.941212\pi$
$167$	− 4.74632i	− 0.367281i	0.982993	−	0.183640i	$-0.0587882\pi$
$168$	− 18.2567i	− 1.40853i
$169$	2.45712	0.189009
$170$	0	0
$171$	2.69202	0.205864
$172$	− 30.8582i	− 2.35291i
$173$	3.96316i	0.301314i	0.988586	+	0.150657i	$0.0481389\pi$
$173$	3.96316i	0.301314i	−0.988586	+	0.150657i	$0.951861\pi$
$174$	2.75973	0.209214
$175$	0	0
$176$	−54.4620	−4.10523
$177$	3.16421i	0.237837i
$178$	− 28.8146i	− 2.15975i
$179$	14.0858	1.05282	0.526409	−	0.850231i	$-0.323538\pi$
$179$	14.0858	1.05282	0.526409	+	0.850231i	$0.323538\pi$
$180$	0	0
$181$	−12.2513	−0.910631	−0.455316	−	0.890330i	$-0.650474\pi$
$181$	−12.2513	−0.910631	−0.455316	+	0.890330i	$0.650474\pi$
$182$	27.7385i	2.05612i
$183$	− 5.52781i	− 0.408628i
$184$	−12.9269	−0.952985
$185$	0	0
$186$	−14.5308	−1.06545
$187$	− 6.34481i	− 0.463979i
$188$	13.7899i	1.00573i
$189$	−9.63102	−0.700554
$190$	0	0
$191$	20.1468	1.45777	0.728884	−	0.684637i	$-0.240039\pi$
$191$	20.1468	1.45777	0.728884	+	0.684637i	$0.240039\pi$
$192$	− 26.6136i	− 1.92067i
$193$	− 9.52781i	− 0.685827i	−0.939367	−	0.342913i	$-0.888586\pi$
$193$	− 9.52781i	− 0.685827i	0.939367	−	0.342913i	$-0.111414\pi$
$194$	−9.09783	−0.653186
$195$	0	0
$196$	13.4330	0.959497
$197$	− 4.96316i	− 0.353611i	−0.984246	−	0.176805i	$-0.943424\pi$
$197$	− 4.96316i	− 0.353611i	0.984246	−	0.176805i	$-0.0565763\pi$
$198$	22.1685i	1.57545i
$199$	−5.42221	−0.384370	−0.192185	−	0.981359i	$-0.561557\pi$
$199$	−5.42221	−0.384370	−0.192185	+	0.981359i	$0.561557\pi$
$200$	0	0
$201$	2.76510	0.195035
$202$	− 14.2687i	− 1.00395i
$203$	5.41119i	0.379791i
$204$	7.00969	0.490776
$205$	0	0
$206$	−40.0984	−2.79379
$207$	3.22521i	0.224168i
$208$	60.1691i	4.17198i
$209$	−2.93900	−0.203295
$210$	0	0
$211$	15.9638	1.09899	0.549495	−	0.835497i	$-0.314820\pi$
$211$	15.9638	1.09899	0.549495	+	0.835497i	$0.314820\pi$
$212$	− 51.6262i	− 3.54570i
$213$	1.49934i	0.102733i
$214$	22.9366	1.56791
$215$	0	0
$216$	−34.0834	−2.31908
$217$	− 28.4916i	− 1.93413i
$218$	− 31.7385i	− 2.14961i
$219$	−7.64609	−0.516675
$220$	0	0
$221$	−7.00969	−0.471523
$222$	1.80194i	0.120938i
$223$	− 23.2150i	− 1.55459i	−0.629134	−	0.777297i	$-0.716590\pi$
$223$	− 23.2150i	− 1.55459i	0.629134	−	0.777297i	$-0.283410\pi$
$224$	92.5115	6.18119
$225$	0	0
$226$	9.62565	0.640288
$227$	17.4795i	1.16015i	0.814562	+	0.580077i	$0.196978\pi$
$227$	17.4795i	1.16015i	−0.814562	+	0.580077i	$0.803022\pi$
$228$	− 3.24698i	− 0.215036i
$229$	2.32544	0.153669	0.0768346	−	0.997044i	$-0.475519\pi$
$229$	2.32544	0.153669	0.0768346	+	0.997044i	$0.475519\pi$
$230$	0	0
$231$	4.97285	0.327190
$232$	19.1497i	1.25724i
$233$	− 18.9342i	− 1.24042i	−0.784435	−	0.620211i	$-0.787047\pi$
$233$	− 18.9342i	− 1.24042i	0.784435	−	0.620211i	$-0.212953\pi$
$234$	24.4916	1.60106
$235$	0	0
$236$	−33.3599	−2.17154
$237$	− 3.14377i	− 0.204210i
$238$	18.4426i	1.19546i
$239$	−11.1685	−0.722432	−0.361216	−	0.932482i	$-0.617638\pi$
$239$	−11.1685	−0.722432	−0.361216	+	0.932482i	$0.617638\pi$
$240$	0	0
$241$	7.42998	0.478607	0.239303	−	0.970945i	$-0.423081\pi$
$241$	7.42998	0.478607	0.239303	+	0.970945i	$0.423081\pi$
$242$	6.61894i	0.425482i
$243$	12.9855i	0.833022i
$244$	58.2790	3.73093
$245$	0	0
$246$	13.3260	0.849637
$247$	3.24698i	0.206600i
$248$	− 100.829i	− 6.40266i
$249$	1.67025	0.105848
$250$	0	0
$251$	−14.7409	−0.930440	−0.465220	−	0.885195i	$-0.654025\pi$
$251$	−14.7409	−0.930440	−0.465220	+	0.885195i	$0.654025\pi$
$252$	− 48.0224i	− 3.02512i
$253$	− 3.52111i	− 0.221370i
$254$	−4.09485	−0.256934
$255$	0	0
$256$	110.548	6.90927
$257$	− 3.34721i	− 0.208793i	−0.994536	−	0.104397i	$-0.966709\pi$
$257$	− 3.34721i	− 0.208793i	0.994536	−	0.104397i	$-0.0332911\pi$
$258$	8.20105i	0.510575i
$259$	−3.53319	−0.219542
$260$	0	0
$261$	4.77777	0.295737
$262$	37.0683i	2.29009i
$263$	− 16.6853i	− 1.02886i	−0.857532	−	0.514430i	$-0.828003\pi$
$263$	− 16.6853i	− 1.02886i	0.857532	−	0.514430i	$-0.171997\pi$
$264$	17.5985	1.08311
$265$	0	0
$266$	8.54288	0.523797
$267$	5.70709i	0.349268i
$268$	29.1521i	1.78075i
$269$	−15.5961	−0.950911	−0.475456	−	0.879740i	$-0.657717\pi$
$269$	−15.5961	−0.950911	−0.475456	+	0.879740i	$0.657717\pi$
$270$	0	0
$271$	−13.2131	−0.802640	−0.401320	−	0.915938i	$-0.631449\pi$
$271$	−13.2131	−0.802640	−0.401320	+	0.915938i	$0.631449\pi$
$272$	40.0049i	2.42565i
$273$	− 5.49396i	− 0.332510i
$274$	−45.3183	−2.73778
$275$	0	0
$276$	3.89008	0.234156
$277$	− 12.9758i	− 0.779642i	−0.920891	−	0.389821i	$-0.872537\pi$
$277$	− 12.9758i	− 0.779642i	0.920891	−	0.389821i	$-0.127463\pi$
$278$	36.6993i	2.20108i
$279$	−25.1564	−1.50608
$280$	0	0
$281$	24.8901	1.48482	0.742409	−	0.669947i	$-0.233683\pi$
$281$	24.8901	1.48482	0.742409	+	0.669947i	$0.233683\pi$
$282$	− 3.66487i	− 0.218240i
$283$	− 13.5821i	− 0.807372i	−0.914898	−	0.403686i	$-0.867729\pi$
$283$	− 13.5821i	− 0.807372i	0.914898	−	0.403686i	$-0.132271\pi$
$284$	−15.8073	−0.937992
$285$	0	0
$286$	−26.7385	−1.58108
$287$	26.1293i	1.54236i
$288$	− 81.6824i	− 4.81318i
$289$	12.3394	0.725849
$290$	0	0
$291$	1.80194	0.105631
$292$	− 80.6118i	− 4.71745i
$293$	1.27652i	0.0745751i	0.999305	+	0.0372875i	$0.0118717\pi$
$293$	1.27652i	0.0745751i	−0.999305	+	0.0372875i	$0.988128\pi$
$294$	−3.57002	−0.208208
$295$	0	0
$296$	−12.5036	−0.726760
$297$	− 9.28382i	− 0.538702i
$298$	31.0006i	1.79582i
$299$	−3.89008	−0.224969
$300$	0	0
$301$	−16.0804	−0.926857
$302$	− 29.2597i	− 1.68370i
$303$	2.82610i	0.162355i
$304$	18.5308	1.06281
$305$	0	0
$306$	16.2838	0.930884
$307$	32.2295i	1.83944i	0.392580	+	0.919718i	$0.371583\pi$
$307$	32.2295i	1.83944i	−0.392580	+	0.919718i	$0.628417\pi$
$308$	52.4282i	2.98737i
$309$	7.94198	0.451804
$310$	0	0
$311$	12.4983	0.708712	0.354356	−	0.935111i	$-0.384700\pi$
$311$	12.4983	0.708712	0.354356	+	0.935111i	$0.384700\pi$
$312$	− 19.4426i	− 1.10072i
$313$	− 20.9390i	− 1.18354i	−0.806106	−	0.591771i	$-0.798429\pi$
$313$	− 20.9390i	− 1.18354i	0.806106	−	0.591771i	$-0.201571\pi$
$314$	−6.96077	−0.392819
$315$	0	0
$316$	33.1444	1.86452
$317$	− 15.8562i	− 0.890575i	−0.895388	−	0.445287i	$-0.853102\pi$
$317$	− 15.8562i	− 0.890575i	0.895388	−	0.445287i	$-0.146898\pi$
$318$	13.7205i	0.769407i
$319$	−5.21611	−0.292046
$320$	0	0
$321$	−4.54288	−0.253559
$322$	10.2349i	0.570369i
$323$	2.15883i	0.120121i
$324$	−36.9952	−2.05529
$325$	0	0
$326$	8.87800	0.491707
$327$	6.28621i	0.347628i
$328$	92.4693i	5.10576i
$329$	7.18598	0.396176
$330$	0	0
$331$	−13.2349	−0.727456	−0.363728	−	0.931505i	$-0.618496\pi$
$331$	−13.2349	−0.727456	−0.363728	+	0.931505i	$0.618496\pi$
$332$	17.6093i	0.966433i
$333$	3.11960i	0.170953i
$334$	−13.2989	−0.727682
$335$	0	0
$336$	−31.3545	−1.71053
$337$	26.0780i	1.42056i	0.703920	+	0.710279i	$0.251431\pi$
$337$	26.0780i	1.42056i	−0.703920	+	0.710279i	$0.748569\pi$
$338$	− 6.88471i	− 0.374479i
$339$	−1.90648	−0.103546
$340$	0	0
$341$	27.4644	1.48728
$342$	− 7.54288i	− 0.407872i
$343$	14.3424i	0.774418i
$344$	−56.9071	−3.06822
$345$	0	0
$346$	11.1045	0.596984
$347$	− 8.98254i	− 0.482208i	−0.970499	−	0.241104i	$-0.922490\pi$
$347$	− 8.98254i	− 0.482208i	0.970499	−	0.241104i	$-0.0775095\pi$
$348$	− 5.76271i	− 0.308914i
$349$	−0.599564	−0.0320939	−0.0160470	−	0.999871i	$-0.505108\pi$
$349$	−0.599564	−0.0320939	−0.0160470	+	0.999871i	$0.505108\pi$
$350$	0	0
$351$	−10.2567	−0.547460
$352$	89.1764i	4.75312i
$353$	31.8896i	1.69731i	0.528945	+	0.848656i	$0.322588\pi$
$353$	31.8896i	1.69731i	−0.528945	+	0.848656i	$0.677412\pi$
$354$	8.86592	0.471218
$355$	0	0
$356$	−60.1691	−3.18896
$357$	− 3.65279i	− 0.193326i
$358$	− 39.4674i	− 2.08592i
$359$	18.3763	0.969863	0.484931	−	0.874552i	$-0.338845\pi$
$359$	18.3763	0.969863	0.484931	+	0.874552i	$0.338845\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	34.3274i	1.80421i
$363$	− 1.31096i	− 0.0688077i
$364$	57.9221	3.03594
$365$	0	0
$366$	−15.4886	−0.809601
$367$	14.6377i	0.764083i	0.924145	+	0.382042i	$0.124779\pi$
$367$	14.6377i	0.764083i	−0.924145	+	0.382042i	$0.875221\pi$
$368$	22.2010i	1.15731i
$369$	23.0707	1.20101
$370$	0	0
$371$	−26.9028	−1.39672
$372$	30.3424i	1.57318i
$373$	− 1.02715i	− 0.0531837i	−0.999646	−	0.0265918i	$-0.991535\pi$
$373$	− 1.02715i	− 0.0531837i	0.999646	−	0.0265918i	$-0.00846544\pi$
$374$	−17.7778	−0.919267
$375$	0	0
$376$	25.4306	1.31148
$377$	5.76271i	0.296795i
$378$	26.9855i	1.38799i
$379$	22.5284	1.15721	0.578603	−	0.815609i	$-0.303598\pi$
$379$	22.5284	1.15721	0.578603	+	0.815609i	$0.303598\pi$
$380$	0	0
$381$	0.811035	0.0415506
$382$	− 56.4499i	− 2.88823i
$383$	32.6698i	1.66935i	0.550745	+	0.834674i	$0.314344\pi$
$383$	32.6698i	1.66935i	−0.550745	+	0.834674i	$0.685656\pi$
$384$	−40.8920	−2.08676
$385$	0	0
$386$	−26.6963	−1.35881
$387$	14.1981i	0.721728i
$388$	18.9976i	0.964457i
$389$	8.16421	0.413942	0.206971	−	0.978347i	$-0.433639\pi$
$389$	8.16421	0.413942	0.206971	+	0.978347i	$0.433639\pi$
$390$	0	0
$391$	−2.58642	−0.130801
$392$	− 24.7724i	− 1.25120i
$393$	− 7.34183i	− 0.370346i
$394$	−13.9065	−0.700598
$395$	0	0
$396$	46.2911	2.32622
$397$	− 37.5478i	− 1.88447i	−0.334954	−	0.942235i	$-0.608721\pi$
$397$	− 37.5478i	− 1.88447i	0.334954	−	0.942235i	$-0.391279\pi$
$398$	15.1927i	0.761541i
$399$	−1.69202	−0.0847070
$400$	0	0
$401$	32.2519	1.61058	0.805291	−	0.592880i	$-0.202009\pi$
$401$	32.2519	1.61058	0.805291	+	0.592880i	$0.202009\pi$
$402$	− 7.74764i	− 0.386417i
$403$	− 30.3424i	− 1.51146i
$404$	−29.7952	−1.48237
$405$	0	0
$406$	15.1618	0.752468
$407$	− 3.40581i	− 0.168820i
$408$	− 12.9269i	− 0.639978i
$409$	31.5459	1.55984	0.779921	−	0.625878i	$-0.215259\pi$
$409$	31.5459	1.55984	0.779921	+	0.625878i	$0.215259\pi$
$410$	0	0
$411$	8.97584	0.442745
$412$	83.7314i	4.12515i
$413$	17.3840i	0.855413i
$414$	9.03684	0.444136
$415$	0	0
$416$	98.5212	4.83040
$417$	− 7.26875i	− 0.355952i
$418$	8.23490i	0.402782i
$419$	14.6866	0.717490	0.358745	−	0.933436i	$-0.383205\pi$
$419$	14.6866	0.717490	0.358745	+	0.933436i	$0.383205\pi$
$420$	0	0
$421$	15.1142	0.736622	0.368311	−	0.929703i	$-0.379936\pi$
$421$	15.1142	0.736622	0.368311	+	0.929703i	$0.379936\pi$
$422$	− 44.7294i	− 2.17740i
$423$	− 6.34481i	− 0.308495i
$424$	−95.2065	−4.62364
$425$	0	0
$426$	4.20105	0.203541
$427$	− 30.3696i	− 1.46969i
$428$	− 47.8950i	− 2.31509i
$429$	5.29590	0.255688
$430$	0	0
$431$	1.67696	0.0807761	0.0403881	−	0.999184i	$-0.487141\pi$
$431$	1.67696	0.0807761	0.0403881	+	0.999184i	$0.487141\pi$
$432$	58.5357i	2.81630i
$433$	− 13.1545i	− 0.632166i	−0.948732	−	0.316083i	$-0.897632\pi$
$433$	− 13.1545i	− 0.632166i	0.948732	−	0.316083i	$-0.102368\pi$
$434$	−79.8316	−3.83204
$435$	0	0
$436$	−66.2747	−3.17398
$437$	1.19806i	0.0573111i
$438$	21.4239i	1.02367i
$439$	−6.45580	−0.308118	−0.154059	−	0.988062i	$-0.549235\pi$
$439$	−6.45580	−0.308118	−0.154059	+	0.988062i	$0.549235\pi$
$440$	0	0
$441$	−6.18060	−0.294314
$442$	19.6407i	0.934213i
$443$	− 25.3709i	− 1.20541i	−0.797965	−	0.602704i	$-0.794090\pi$
$443$	− 25.3709i	− 1.20541i	0.797965	−	0.602704i	$-0.205910\pi$
$444$	3.76271	0.178570
$445$	0	0
$446$	−65.0471	−3.08007
$447$	− 6.14005i	− 0.290414i
$448$	− 146.214i	− 6.90795i
$449$	3.59956	0.169874	0.0849370	−	0.996386i	$-0.472931\pi$
$449$	3.59956	0.169874	0.0849370	+	0.996386i	$0.472931\pi$
$450$	0	0
$451$	−25.1873	−1.18602
$452$	− 20.0998i	− 0.945413i
$453$	5.79523i	0.272284i
$454$	48.9764	2.29858
$455$	0	0
$456$	−5.98792	−0.280410
$457$	12.2784i	0.574361i	0.957876	+	0.287181i	$0.0927180\pi$
$457$	12.2784i	0.574361i	−0.957876	+	0.287181i	$0.907282\pi$
$458$	− 6.51573i	− 0.304460i
$459$	−6.81940	−0.318302
$460$	0	0
$461$	39.4935	1.83939	0.919697	−	0.392628i	$-0.128434\pi$
$461$	39.4935	1.83939	0.919697	+	0.392628i	$0.128434\pi$
$462$	− 13.9336i	− 0.648251i
$463$	18.3991i	0.855079i	0.903997	+	0.427540i	$0.140620\pi$
$463$	18.3991i	0.855079i	−0.903997	+	0.427540i	$0.859380\pi$
$464$	32.8883	1.52680
$465$	0	0
$466$	−53.0525	−2.45761
$467$	36.2282i	1.67644i	0.545332	+	0.838220i	$0.316404\pi$
$467$	36.2282i	1.67644i	−0.545332	+	0.838220i	$0.683596\pi$
$468$	− 51.1420i	− 2.36404i
$469$	15.1914	0.701472
$470$	0	0
$471$	1.37867	0.0635256
$472$	61.5206i	2.83172i
$473$	− 15.5007i	− 0.712721i
$474$	−8.80864	−0.404594
$475$	0	0
$476$	38.5109	1.76515
$477$	23.7536i	1.08760i
$478$	31.2935i	1.43133i
$479$	34.3129	1.56780	0.783898	−	0.620890i	$-0.213229\pi$
$479$	34.3129	1.56780	0.783898	+	0.620890i	$0.213229\pi$
$480$	0	0
$481$	−3.76271	−0.171565
$482$	− 20.8183i	− 0.948249i
$483$	− 2.02715i	− 0.0922384i
$484$	13.8213	0.628242
$485$	0	0
$486$	36.3846	1.65044
$487$	24.4722i	1.10894i	0.832203	+	0.554470i	$0.187079\pi$
$487$	24.4722i	1.10894i	−0.832203	+	0.554470i	$0.812921\pi$
$488$	− 107.475i	− 4.86518i
$489$	−1.75840	−0.0795175
$490$	0	0
$491$	18.3067	0.826168	0.413084	−	0.910693i	$-0.364452\pi$
$491$	18.3067	0.826168	0.413084	+	0.910693i	$0.364452\pi$
$492$	− 27.8267i	− 1.25452i
$493$	3.83148i	0.172561i
$494$	9.09783	0.409331
$495$	0	0
$496$	−173.167	−7.77542
$497$	8.23729i	0.369493i
$498$	− 4.67994i	− 0.209713i
$499$	25.3424	1.13448	0.567241	−	0.823552i	$-0.308011\pi$
$499$	25.3424	1.13448	0.567241	+	0.823552i	$0.308011\pi$
$500$	0	0
$501$	2.63401	0.117679
$502$	41.3032i	1.84345i
$503$	14.1105i	0.629156i	0.949232	+	0.314578i	$0.101863\pi$
$503$	14.1105i	0.629156i	−0.949232	+	0.314578i	$0.898137\pi$
$504$	−88.5605	−3.94480
$505$	0	0
$506$	−9.86592	−0.438594
$507$	1.36360i	0.0605596i
$508$	8.55065i	0.379374i
$509$	17.8019	0.789057	0.394529	−	0.918884i	$-0.370908\pi$
$509$	17.8019	0.789057	0.394529	+	0.918884i	$0.370908\pi$
$510$	0	0
$511$	−42.0073	−1.85829
$512$	− 162.380i	− 7.17625i
$513$	3.15883i	0.139466i
$514$	−9.37867	−0.413675
$515$	0	0
$516$	17.1250	0.753885
$517$	6.92692i	0.304646i
$518$	9.89977i	0.434971i
$519$	−2.19939	−0.0965425
$520$	0	0
$521$	13.3948	0.586837	0.293418	−	0.955984i	$-0.405207\pi$
$521$	13.3948	0.586837	0.293418	+	0.955984i	$0.405207\pi$
$522$	− 13.3870i	− 0.585934i
$523$	− 1.46921i	− 0.0642439i	−0.999484	−	0.0321219i	$-0.989774\pi$
$523$	− 1.46921i	− 0.0642439i	0.999484	−	0.0321219i	$-0.0102265\pi$
$524$	77.4040	3.38141
$525$	0	0
$526$	−46.7512	−2.03845
$527$	− 20.1739i	− 0.878789i
$528$	− 30.2241i	− 1.31534i
$529$	21.5646	0.937593
$530$	0	0
$531$	15.3491	0.666095
$532$	− 17.8388i	− 0.773409i
$533$	27.8267i	1.20531i
$534$	15.9909	0.691994
$535$	0	0
$536$	53.7609	2.32212
$537$	7.81700i	0.337329i
$538$	43.6993i	1.88401i
$539$	6.74764	0.290642
$540$	0	0
$541$	−42.7144	−1.83643	−0.918217	−	0.396077i	$-0.870371\pi$
$541$	−42.7144	−1.83643	−0.918217	+	0.396077i	$0.870371\pi$
$542$	37.0224i	1.59025i
$543$	− 6.79895i	− 0.291771i
$544$	65.5042	2.80847
$545$	0	0
$546$	−15.3937	−0.658791
$547$	− 10.6866i	− 0.456928i	−0.973552	−	0.228464i	$-0.926630\pi$
$547$	− 10.6866i	− 0.456928i	0.973552	−	0.228464i	$-0.0733703\pi$
$548$	94.6311i	4.04244i
$549$	−26.8146	−1.14442
$550$	0	0
$551$	1.77479	0.0756086
$552$	− 7.17390i	− 0.305341i
$553$	− 17.2717i	− 0.734469i
$554$	−36.3575	−1.54468
$555$	0	0
$556$	76.6335	3.24999
$557$	34.2083i	1.44945i	0.689036	+	0.724727i	$0.258034\pi$
$557$	34.2083i	1.44945i	−0.689036	+	0.724727i	$0.741966\pi$
$558$	70.4868i	2.98394i
$559$	−17.1250	−0.724310
$560$	0	0
$561$	3.52111	0.148661
$562$	− 69.7405i	− 2.94182i
$563$	− 5.46788i	− 0.230444i	−0.993340	−	0.115222i	$-0.963242\pi$
$563$	− 5.46788i	− 0.230444i	0.993340	−	0.115222i	$-0.0367579\pi$
$564$	−7.65279	−0.322241
$565$	0	0
$566$	−38.0562	−1.59962
$567$	19.2784i	0.809618i
$568$	29.1511i	1.22315i
$569$	17.4819	0.732878	0.366439	−	0.930442i	$-0.380577\pi$
$569$	17.4819	0.732878	0.366439	+	0.930442i	$0.380577\pi$
$570$	0	0
$571$	−7.21877	−0.302096	−0.151048	−	0.988526i	$-0.548265\pi$
$571$	−7.21877	−0.302096	−0.151048	+	0.988526i	$0.548265\pi$
$572$	55.8340i	2.33454i
$573$	11.1806i	0.467076i
$574$	73.2127	3.05584
$575$	0	0
$576$	−129.099	−5.37911
$577$	− 1.80625i	− 0.0751952i	−0.999293	−	0.0375976i	$-0.988029\pi$
$577$	− 1.80625i	− 0.0751952i	0.999293	−	0.0375976i	$-0.0119705\pi$
$578$	− 34.5743i	− 1.43810i
$579$	5.28754	0.219743
$580$	0	0
$581$	9.17629	0.380697
$582$	− 5.04892i	− 0.209284i
$583$	− 25.9329i	− 1.07403i
$584$	−148.660	−6.15160
$585$	0	0
$586$	3.57673	0.147753
$587$	− 2.39075i	− 0.0986767i	−0.998782	−	0.0493384i	$-0.984289\pi$
$587$	− 2.39075i	− 0.0986767i	0.998782	−	0.0493384i	$-0.0157113\pi$
$588$	7.45473i	0.307428i
$589$	−9.34481	−0.385046
$590$	0	0
$591$	2.75435	0.113299
$592$	21.4741i	0.882580i
$593$	− 42.2650i	− 1.73562i	−0.496899	−	0.867808i	$-0.665528\pi$
$593$	− 42.2650i	− 1.73562i	0.496899	−	0.867808i	$-0.334472\pi$
$594$	−26.0127	−1.06731
$595$	0	0
$596$	64.7338	2.65160
$597$	− 3.00910i	− 0.123154i
$598$	10.8998i	0.445725i
$599$	−8.95838	−0.366029	−0.183015	−	0.983110i	$-0.558586\pi$
$599$	−8.95838	−0.366029	−0.183015	+	0.983110i	$0.558586\pi$
$600$	0	0
$601$	−19.8998	−0.811729	−0.405864	−	0.913933i	$-0.633029\pi$
$601$	−19.8998	−0.811729	−0.405864	+	0.913933i	$0.633029\pi$
$602$	45.0562i	1.83635i
$603$	− 13.4131i	− 0.546224i
$604$	−61.0984	−2.48606
$605$	0	0
$606$	7.91856	0.321669
$607$	− 26.0084i	− 1.05565i	−0.849354	−	0.527823i	$-0.823008\pi$
$607$	− 26.0084i	− 1.05565i	0.849354	−	0.527823i	$-0.176992\pi$
$608$	− 30.3424i	− 1.23055i
$609$	−3.00298	−0.121687
$610$	0	0
$611$	7.65279	0.309599
$612$	− 34.0030i	− 1.37449i
$613$	41.3763i	1.67117i	0.549360	+	0.835586i	$0.314872\pi$
$613$	41.3763i	1.67117i	−0.549360	+	0.835586i	$0.685128\pi$
$614$	90.3051	3.64442
$615$	0	0
$616$	96.6854	3.89557
$617$	− 43.9845i	− 1.77075i	−0.464880	−	0.885374i	$-0.653902\pi$
$617$	− 43.9845i	− 1.77075i	0.464880	−	0.885374i	$-0.346098\pi$
$618$	− 22.2529i	− 0.895145i
$619$	−23.4553	−0.942749	−0.471374	−	0.881933i	$-0.656242\pi$
$619$	−23.4553	−0.942749	−0.471374	+	0.881933i	$0.656242\pi$
$620$	0	0
$621$	−3.78448	−0.151866
$622$	− 35.0194i	− 1.40415i
$623$	31.3545i	1.25619i
$624$	−33.3913	−1.33672
$625$	0	0
$626$	−58.6698	−2.34492
$627$	− 1.63102i	− 0.0651368i
$628$	14.5351i	0.580014i
$629$	−2.50173	−0.0997505
$630$	0	0
$631$	−34.8877	−1.38886	−0.694429	−	0.719562i	$-0.744343\pi$
$631$	−34.8877	−1.38886	−0.694429	+	0.719562i	$0.744343\pi$
$632$	− 61.1232i	− 2.43135i
$633$	8.85922i	0.352122i
$634$	−44.4282	−1.76447
$635$	0	0
$636$	28.6504	1.13606
$637$	− 7.45473i	− 0.295367i
$638$	14.6152i	0.578622i
$639$	7.27306	0.287718
$640$	0	0
$641$	−38.5314	−1.52190	−0.760949	−	0.648812i	$-0.775266\pi$
$641$	−38.5314	−1.52190	−0.760949	+	0.648812i	$0.775266\pi$
$642$	12.7289i	0.502368i
$643$	− 18.6353i	− 0.734906i	−0.930042	−	0.367453i	$-0.880230\pi$
$643$	− 18.6353i	− 0.734906i	0.930042	−	0.367453i	$-0.119770\pi$
$644$	21.3720	0.842173
$645$	0	0
$646$	6.04892	0.237991
$647$	− 19.2282i	− 0.755938i	−0.925818	−	0.377969i	$-0.876623\pi$
$647$	− 19.2282i	− 0.755938i	0.925818	−	0.377969i	$-0.123377\pi$
$648$	68.2247i	2.68012i
$649$	−16.7573	−0.657783
$650$	0	0
$651$	15.8116	0.619706
$652$	− 18.5386i	− 0.726026i
$653$	− 3.98553i	− 0.155966i	−0.996955	−	0.0779828i	$-0.975152\pi$
$653$	− 3.98553i	− 0.155966i	0.996955	−	0.0779828i	$-0.0248479\pi$
$654$	17.6136	0.688745
$655$	0	0
$656$	158.809	6.20046
$657$	37.0901i	1.44702i
$658$	− 20.1347i	− 0.784931i
$659$	−33.5719	−1.30778	−0.653889	−	0.756591i	$-0.726864\pi$
$659$	−33.5719	−1.30778	−0.653889	+	0.756591i	$0.726864\pi$
$660$	0	0
$661$	36.9909	1.43878	0.719390	−	0.694607i	$-0.244422\pi$
$661$	36.9909	1.43878	0.719390	+	0.694607i	$0.244422\pi$
$662$	37.0834i	1.44129i
$663$	− 3.89008i	− 0.151078i
$664$	32.4741	1.26024
$665$	0	0
$666$	8.74094	0.338704
$667$	2.12631i	0.0823310i
$668$	27.7700i	1.07445i
$669$	12.8834	0.498100
$670$	0	0
$671$	29.2747	1.13014
$672$	51.3400i	1.98048i
$673$	12.0747i	0.465447i	0.972543	+	0.232723i	$0.0747637\pi$
$673$	12.0747i	0.465447i	−0.972543	+	0.232723i	$0.925236\pi$
$674$	73.0689	2.81451
$675$	0	0
$676$	−14.3763	−0.552934
$677$	− 41.9135i	− 1.61087i	−0.592686	−	0.805434i	$-0.701933\pi$
$677$	− 41.9135i	− 1.61087i	0.592686	−	0.805434i	$-0.298067\pi$
$678$	5.34183i	0.205152i
$679$	9.89977	0.379918
$680$	0	0
$681$	−9.70038	−0.371719
$682$	− 76.9536i	− 2.94671i
$683$	9.17092i	0.350915i	0.984487	+	0.175458i	$0.0561405\pi$
$683$	9.17092i	0.350915i	−0.984487	+	0.175458i	$0.943859\pi$
$684$	−15.7506	−0.602240
$685$	0	0
$686$	40.1866	1.53433
$687$	1.29052i	0.0492364i
$688$	97.7338i	3.72606i
$689$	−28.6504	−1.09149
$690$	0	0
$691$	−0.111244	−0.00423193	−0.00211596	−	0.999998i	$-0.500674\pi$
$691$	−0.111244	−0.00423193	−0.00211596	+	0.999998i	$0.500674\pi$
$692$	− 23.1879i	− 0.881472i
$693$	− 24.1226i	− 0.916341i
$694$	−25.1685	−0.955384
$695$	0	0
$696$	−10.6273	−0.402827
$697$	18.5013i	0.700785i
$698$	1.67994i	0.0635867i
$699$	10.5077	0.397438
$700$	0	0
$701$	−27.7832	−1.04936	−0.524678	−	0.851301i	$-0.675814\pi$
$701$	−27.7832	−1.04936	−0.524678	+	0.851301i	$0.675814\pi$
$702$	28.7385i	1.08467i
$703$	1.15883i	0.0437062i
$704$	140.943	5.31198
$705$	0	0
$706$	89.3527	3.36283
$707$	15.5265i	0.583933i
$708$	− 18.5133i	− 0.695774i
$709$	−37.4668	−1.40710	−0.703548	−	0.710648i	$-0.748402\pi$
$709$	−37.4668	−1.40710	−0.703548	+	0.710648i	$0.748402\pi$
$710$	0	0
$711$	−15.2500	−0.571918
$712$	110.961i	4.15844i
$713$	− 11.1957i	− 0.419281i
$714$	−10.2349	−0.383031
$715$	0	0
$716$	−82.4137	−3.07994
$717$	− 6.19806i	− 0.231471i
$718$	− 51.4892i	− 1.92156i
$719$	21.0513	0.785081	0.392541	−	0.919735i	$-0.371596\pi$
$719$	21.0513	0.785081	0.392541	+	0.919735i	$0.371596\pi$
$720$	0	0
$721$	43.6329	1.62498
$722$	− 2.80194i	− 0.104277i
$723$	4.12333i	0.153348i
$724$	71.6805	2.66399
$725$	0	0
$726$	−3.67324	−0.136327
$727$	44.7023i	1.65792i	0.559310	+	0.828958i	$0.311066\pi$
$727$	44.7023i	1.65792i	−0.559310	+	0.828958i	$0.688934\pi$
$728$	− 106.817i	− 3.95891i
$729$	11.7627	0.435656
$730$	0	0
$731$	−11.3860	−0.421125
$732$	32.3424i	1.19541i
$733$	25.0301i	0.924509i	0.886747	+	0.462254i	$0.152959\pi$
$733$	25.0301i	0.924509i	−0.886747	+	0.462254i	$0.847041\pi$
$734$	41.0140	1.51385
$735$	0	0
$736$	36.3521	1.33996
$737$	14.6437i	0.539407i
$738$	− 64.6426i	− 2.37953i
$739$	23.9982	0.882788	0.441394	−	0.897313i	$-0.354484\pi$
$739$	23.9982	0.882788	0.441394	+	0.897313i	$0.354484\pi$
$740$	0	0
$741$	−1.80194	−0.0661958
$742$	75.3798i	2.76728i
$743$	10.4601i	0.383744i	0.981420	+	0.191872i	$0.0614559\pi$
$743$	10.4601i	0.383744i	−0.981420	+	0.191872i	$0.938544\pi$
$744$	55.9560	2.05145
$745$	0	0
$746$	−2.87800	−0.105371
$747$	− 8.10215i	− 0.296442i
$748$	37.1226i	1.35734i
$749$	−24.9584	−0.911959
$750$	0	0
$751$	−29.9420	−1.09260	−0.546299	−	0.837590i	$-0.683964\pi$
$751$	−29.9420	−1.09260	−0.546299	+	0.837590i	$0.683964\pi$
$752$	− 43.6752i	− 1.59267i
$753$	− 8.18060i	− 0.298118i
$754$	16.1468	0.588030
$755$	0	0
$756$	56.3497	2.04942
$757$	− 27.2174i	− 0.989235i	−0.869111	−	0.494617i	$-0.835308\pi$
$757$	− 27.2174i	− 0.989235i	0.869111	−	0.494617i	$-0.164692\pi$
$758$	− 63.1232i	− 2.29274i
$759$	1.95407	0.0709281
$760$	0	0
$761$	18.6813	0.677195	0.338598	−	0.940931i	$-0.390047\pi$
$761$	18.6813	0.677195	0.338598	+	0.940931i	$0.390047\pi$
$762$	− 2.27247i	− 0.0823229i
$763$	34.5362i	1.25029i
$764$	−117.876	−4.26459
$765$	0	0
$766$	91.5387	3.30743
$767$	18.5133i	0.668478i
$768$	61.3497i	2.21377i
$769$	−6.34780	−0.228907	−0.114454	−	0.993429i	$-0.536512\pi$
$769$	−6.34780	−0.228907	−0.114454	+	0.993429i	$0.536512\pi$
$770$	0	0
$771$	1.85756	0.0668984
$772$	55.7458i	2.00634i
$773$	49.7851i	1.79064i	0.445419	+	0.895322i	$0.353055\pi$
$773$	49.7851i	1.79064i	−0.445419	+	0.895322i	$0.646945\pi$
$774$	39.7821	1.42994
$775$	0	0
$776$	35.0344	1.25766
$777$	− 1.96077i	− 0.0703423i
$778$	− 22.8756i	− 0.820130i
$779$	8.57002	0.307053
$780$	0	0
$781$	−7.94033	−0.284127
$782$	7.24698i	0.259151i
$783$	5.60627i	0.200352i
$784$	−42.5448	−1.51946
$785$	0	0
$786$	−20.5714	−0.733756
$787$	9.19865i	0.327897i	0.986469	+	0.163948i	$0.0524230\pi$
$787$	9.19865i	0.327897i	−0.986469	+	0.163948i	$0.947577\pi$
$788$	29.0388i	1.03446i
$789$	9.25965	0.329652
$790$	0	0
$791$	−10.4741	−0.372416
$792$	− 85.3678i	− 3.03341i
$793$	− 32.3424i	− 1.14851i
$794$	−105.207	−3.73364
$795$	0	0
$796$	31.7245	1.12445
$797$	− 13.5084i	− 0.478493i	−0.970959	−	0.239247i	$-0.923100\pi$
$797$	− 13.5084i	− 0.478493i	0.970959	−	0.239247i	$-0.0769004\pi$
$798$	4.74094i	0.167827i
$799$	5.08815	0.180006
$800$	0	0
$801$	27.6843	0.978175
$802$	− 90.3678i	− 3.19100i
$803$	− 40.4929i	− 1.42896i
$804$	−16.1782	−0.570562
$805$	0	0
$806$	−85.0176	−2.99462
$807$	− 8.65519i	− 0.304677i
$808$	54.9469i	1.93302i
$809$	−31.9560	−1.12351	−0.561756	−	0.827303i	$-0.689874\pi$
$809$	−31.9560	−1.12351	−0.561756	+	0.827303i	$0.689874\pi$
$810$	0	0
$811$	40.7012	1.42921	0.714607	−	0.699526i	$-0.246606\pi$
$811$	40.7012	1.42921	0.714607	+	0.699526i	$0.246606\pi$
$812$	− 31.6601i	− 1.11105i
$813$	− 7.33273i	− 0.257170i
$814$	−9.54288	−0.334478
$815$	0	0
$816$	−22.2010	−0.777192
$817$	5.27413i	0.184518i
$818$	− 88.3895i	− 3.09047i
$819$	−26.6504	−0.931240
$820$	0	0
$821$	−13.4300	−0.468709	−0.234355	−	0.972151i	$-0.575298\pi$
$821$	−13.4300	−0.468709	−0.234355	+	0.972151i	$0.575298\pi$
$822$	− 25.1497i	− 0.877198i
$823$	− 26.7275i	− 0.931663i	−0.884873	−	0.465832i	$-0.845755\pi$
$823$	− 26.7275i	− 0.931663i	0.884873	−	0.465832i	$-0.154245\pi$
$824$	154.413	5.37924
$825$	0	0
$826$	48.7090	1.69480
$827$	− 33.3435i	− 1.15947i	−0.814806	−	0.579733i	$-0.803157\pi$
$827$	− 33.3435i	− 1.15947i	0.814806	−	0.579733i	$-0.196843\pi$
$828$	− 18.8702i	− 0.655786i
$829$	−12.2849	−0.426672	−0.213336	−	0.976979i	$-0.568433\pi$
$829$	−12.2849	−0.426672	−0.213336	+	0.976979i	$0.568433\pi$
$830$	0	0
$831$	7.20105	0.249802
$832$	− 155.712i	− 5.39835i
$833$	− 4.95646i	− 0.171731i
$834$	−20.3666	−0.705237
$835$	0	0
$836$	17.1957	0.594725
$837$	− 29.5187i	− 1.02032i
$838$	− 41.1511i	− 1.42154i
$839$	−36.9922	−1.27711	−0.638557	−	0.769575i	$-0.720468\pi$
$839$	−36.9922	−1.27711	−0.638557	+	0.769575i	$0.720468\pi$
$840$	0	0
$841$	−25.8501	−0.891383
$842$	− 42.3491i	− 1.45945i
$843$	13.8130i	0.475743i
$844$	−93.4016	−3.21502
$845$	0	0
$846$	−17.7778	−0.611212
$847$	− 7.20237i	− 0.247477i
$848$	163.510i	5.61497i
$849$	7.53750	0.258686
$850$	0	0
$851$	−1.38835	−0.0475922
$852$	− 8.77240i	− 0.300537i
$853$	7.16959i	0.245482i	0.992439	+	0.122741i	$0.0391684\pi$
$853$	7.16959i	0.245482i	−0.992439	+	0.122741i	$0.960832\pi$
$854$	−85.0936	−2.91184
$855$	0	0
$856$	−88.3256	−3.01891
$857$	− 2.55124i	− 0.0871486i	−0.999050	−	0.0435743i	$-0.986125\pi$
$857$	− 2.55124i	− 0.0871486i	0.999050	−	0.0435743i	$-0.0138745\pi$
$858$	− 14.8388i	− 0.506587i
$859$	−53.9493	−1.84073	−0.920363	−	0.391065i	$-0.872107\pi$
$859$	−53.9493	−1.84073	−0.920363	+	0.391065i	$0.872107\pi$
$860$	0	0
$861$	−14.5007	−0.494181
$862$	− 4.69873i	− 0.160039i
$863$	21.4698i	0.730840i	0.930843	+	0.365420i	$0.119075\pi$
$863$	21.4698i	0.730840i	−0.930843	+	0.365420i	$0.880925\pi$
$864$	95.8467	3.26077
$865$	0	0
$866$	−36.8582	−1.25249
$867$	6.84787i	0.232566i
$868$	166.700i	5.65817i
$869$	16.6491	0.564781
$870$	0	0
$871$	16.1782	0.548178
$872$	122.221i	4.13891i
$873$	− 8.74094i	− 0.295836i
$874$	3.35690	0.113549
$875$	0	0
$876$	44.7362	1.51149
$877$	38.9638i	1.31571i	0.753144	+	0.657856i	$0.228537\pi$
$877$	38.9638i	1.31571i	−0.753144	+	0.657856i	$0.771463\pi$
$878$	18.0887i	0.610465i
$879$	−0.708415	−0.0238942
$880$	0	0
$881$	−13.4373	−0.452713	−0.226357	−	0.974045i	$-0.572681\pi$
$881$	−13.4373	−0.452713	−0.226357	+	0.974045i	$0.572681\pi$
$882$	17.3177i	0.583116i
$883$	− 27.8799i	− 0.938234i	−0.883136	−	0.469117i	$-0.844572\pi$
$883$	− 27.8799i	− 0.938234i	0.883136	−	0.469117i	$-0.155428\pi$
$884$	41.0127	1.37941
$885$	0	0
$886$	−71.0877	−2.38824
$887$	28.9135i	0.970821i	0.874286	+	0.485410i	$0.161330\pi$
$887$	28.9135i	0.970821i	−0.874286	+	0.485410i	$0.838670\pi$
$888$	− 6.93900i	− 0.232858i
$889$	4.45580	0.149443
$890$	0	0
$891$	−18.5834	−0.622568
$892$	135.828i	4.54785i
$893$	− 2.35690i	− 0.0788705i
$894$	−17.2040	−0.575389
$895$	0	0
$896$	−224.659	−7.50533
$897$	− 2.15883i	− 0.0720814i
$898$	− 10.0858i	− 0.336566i
$899$	−16.5851	−0.553144
$900$	0	0
$901$	−19.0489	−0.634611
$902$	70.5733i	2.34983i
$903$	− 8.92394i	− 0.296970i
$904$	−37.0670	−1.23283
$905$	0	0
$906$	16.2379	0.539467
$907$	− 30.7127i	− 1.01980i	−0.860234	−	0.509900i	$-0.829683\pi$
$907$	− 30.7127i	− 1.01980i	0.860234	−	0.509900i	$-0.170317\pi$
$908$	− 102.270i	− 3.39395i
$909$	13.7090	0.454699
$910$	0	0
$911$	39.3690	1.30435	0.652176	−	0.758067i	$-0.273856\pi$
$911$	39.3690	1.30435	0.652176	+	0.758067i	$0.273856\pi$
$912$	10.2838i	0.340531i
$913$	8.84548i	0.292743i
$914$	34.4034	1.13796
$915$	0	0
$916$	−13.6058	−0.449548
$917$	− 40.3357i	− 1.33200i
$918$	19.1075i	0.630642i
$919$	7.87023	0.259615	0.129808	−	0.991539i	$-0.458564\pi$
$919$	7.87023	0.259615	0.129808	+	0.991539i	$0.458564\pi$
$920$	0	0
$921$	−17.8860	−0.589365
$922$	− 110.658i	− 3.64434i
$923$	8.77240i	0.288747i
$924$	−29.0954	−0.957170
$925$	0	0
$926$	51.5532	1.69414
$927$	− 38.5254i	− 1.26534i
$928$	− 53.8514i	− 1.76776i
$929$	−39.2924	−1.28914	−0.644572	−	0.764544i	$-0.722964\pi$
$929$	−39.2924	−1.28914	−0.644572	+	0.764544i	$0.722964\pi$
$930$	0	0
$931$	−2.29590	−0.0752450
$932$	110.781i	3.62876i
$933$	6.93602i	0.227075i
$934$	101.509	3.32148
$935$	0	0
$936$	−94.3135	−3.08273
$937$	26.7554i	0.874061i	0.899447	+	0.437031i	$0.143970\pi$
$937$	26.7554i	0.874061i	−0.899447	+	0.437031i	$0.856030\pi$
$938$	− 42.5652i	− 1.38980i
$939$	11.6203	0.379213
$940$	0	0
$941$	−41.9191	−1.36653	−0.683263	−	0.730173i	$-0.739440\pi$
$941$	−41.9191	−1.36653	−0.683263	+	0.730173i	$0.739440\pi$
$942$	− 3.86294i	− 0.125861i
$943$	10.2674i	0.334353i
$944$	105.657	3.43885
$945$	0	0
$946$	−43.4319	−1.41209
$947$	− 55.4868i	− 1.80308i	−0.432698	−	0.901539i	$-0.642438\pi$
$947$	− 55.4868i	− 1.80308i	0.432698	−	0.901539i	$-0.357562\pi$
$948$	18.3937i	0.597401i
$949$	−44.7362	−1.45220
$950$	0	0
$951$	8.79954	0.285345
$952$	− 71.0200i	− 2.30177i
$953$	8.48081i	0.274720i	0.990521	+	0.137360i	$0.0438618\pi$
$953$	8.48081i	0.274720i	−0.990521	+	0.137360i	$0.956138\pi$
$954$	66.5561	2.15483
$955$	0	0
$956$	65.3454	2.11342
$957$	− 2.89472i	− 0.0935731i
$958$	− 96.1426i	− 3.10623i
$959$	49.3129	1.59240
$960$	0	0
$961$	56.3256	1.81695
$962$	10.5429i	0.339916i
$963$	22.0368i	0.710127i
$964$	−43.4717	−1.40013
$965$	0	0
$966$	−5.67994	−0.182749
$967$	7.63102i	0.245397i	0.992444	+	0.122699i	$0.0391548\pi$
$967$	7.63102i	0.245397i	−0.992444	+	0.122699i	$0.960845\pi$
$968$	− 25.4886i	− 0.819234i
$969$	−1.19806	−0.0384873
$970$	0	0
$971$	−19.3599	−0.621288	−0.310644	−	0.950526i	$-0.600545\pi$
$971$	−19.3599	−0.621288	−0.310644	+	0.950526i	$0.600545\pi$
$972$	− 75.9764i	− 2.43695i
$973$	− 39.9342i	− 1.28023i
$974$	68.5695	2.19711
$975$	0	0
$976$	−184.581	−5.90829
$977$	− 55.8883i	− 1.78802i	−0.448042	−	0.894012i	$-0.647879\pi$
$977$	− 55.8883i	− 1.78802i	0.448042	−	0.894012i	$-0.352121\pi$
$978$	4.92692i	0.157546i
$979$	−30.2241	−0.965968
$980$	0	0
$981$	30.4935	0.973582
$982$	− 51.2941i	− 1.63686i
$983$	− 4.58450i	− 0.146223i	−0.997324	−	0.0731114i	$-0.976707\pi$
$983$	− 4.58450i	− 0.146223i	0.997324	−	0.0731114i	$-0.0232929\pi$
$984$	−51.3166	−1.63591
$985$	0	0
$986$	10.7356	0.341890
$987$	3.98792i	0.126937i
$988$	− 18.9976i	− 0.604394i
$989$	−6.31873	−0.200924
$990$	0	0
$991$	27.5666	0.875681	0.437840	−	0.899053i	$-0.355743\pi$
$991$	27.5666	0.875681	0.437840	+	0.899053i	$0.355743\pi$
$992$	283.544i	9.00254i
$993$	− 7.34481i	− 0.233081i
$994$	23.0804	0.732065
$995$	0	0
$996$	−9.77240	−0.309650
$997$	− 15.9119i	− 0.503933i	−0.967736	−	0.251967i	$-0.918923\pi$
$997$	− 15.9119i	− 0.503933i	0.967736	−	0.251967i	$-0.0810774\pi$
$998$	− 71.0079i	− 2.24772i
$999$	−3.66056	−0.115815

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	475.2.b.c.324.1		6
5.2	odd	4		475.2.a.h.1.3	yes	3
5.3	odd	4		475.2.a.d.1.1	✓	3
5.4	even	2	inner	475.2.b.c.324.6		6
15.2	even	4		4275.2.a.z.1.1		3
15.8	even	4		4275.2.a.bn.1.3		3
20.3	even	4		7600.2.a.bw.1.2		3
20.7	even	4		7600.2.a.bn.1.2		3
95.18	even	4		9025.2.a.be.1.3		3
95.37	even	4		9025.2.a.w.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
475.2.a.d.1.1	✓	3	5.3	odd	4
475.2.a.h.1.3	yes	3	5.2	odd	4
475.2.b.c.324.1		6	1.1	even	1	trivial
475.2.b.c.324.6		6	5.4	even	2	inner
4275.2.a.z.1.1		3	15.2	even	4
4275.2.a.bn.1.3		3	15.8	even	4
7600.2.a.bn.1.2		3	20.7	even	4
7600.2.a.bw.1.2		3	20.3	even	4
9025.2.a.w.1.1		3	95.37	even	4
9025.2.a.be.1.3		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)

\(f(q)\)	\(=\)	\(q-2.80194i q^{2} +0.554958i q^{3} -5.85086 q^{4} +1.55496 q^{6} +3.04892i q^{7} +10.7899i q^{8} +2.69202 q^{9} +O(q^{10})\)	\(q-2.80194i q^{2} +0.554958i q^{3} -5.85086 q^{4} +1.55496 q^{6} +3.04892i q^{7} +10.7899i q^{8} +2.69202 q^{9} -2.93900 q^{11} -3.24698i q^{12} +3.24698i q^{13} +8.54288 q^{14} +18.5308 q^{16} +2.15883i q^{17} -7.54288i q^{18} +1.00000 q^{19} -1.69202 q^{21} +8.23490i q^{22} +1.19806i q^{23} -5.98792 q^{24} +9.09783 q^{26} +3.15883i q^{27} -17.8388i q^{28} +1.77479 q^{29} -9.34481 q^{31} -30.3424i q^{32} -1.63102i q^{33} +6.04892 q^{34} -15.7506 q^{36} +1.15883i q^{37} -2.80194i q^{38} -1.80194 q^{39} +8.57002 q^{41} +4.74094i q^{42} +5.27413i q^{43} +17.1957 q^{44} +3.35690 q^{46} -2.35690i q^{47} +10.2838i q^{48} -2.29590 q^{49} -1.19806 q^{51} -18.9976i q^{52} +8.82371i q^{53} +8.85086 q^{54} -32.8974 q^{56} +0.554958i q^{57} -4.97285i q^{58} +5.70171 q^{59} -9.96077 q^{61} +26.1836i q^{62} +8.20775i q^{63} -47.9560 q^{64} -4.57002 q^{66} -4.98254i q^{67} -12.6310i q^{68} -0.664874 q^{69} +2.70171 q^{71} +29.0465i q^{72} +13.7778i q^{73} +3.24698 q^{74} -5.85086 q^{76} -8.96077i q^{77} +5.04892i q^{78} -5.66487 q^{79} +6.32304 q^{81} -24.0127i q^{82} -3.00969i q^{83} +9.89977 q^{84} +14.7778 q^{86} +0.984935i q^{87} -31.7114i q^{88} +10.2838 q^{89} -9.89977 q^{91} -7.00969i q^{92} -5.18598i q^{93} -6.60388 q^{94} +16.8388 q^{96} -3.24698i q^{97} +6.43296i q^{98} -7.91185 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 8 q^{4} + 10 q^{6} + 6 q^{9}+O(q^{10}) \) 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

Related objects

Downloads

Learn more