LMFDB - Embedded newform 1470.3.f.a.391.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1470,3,Mod(391,1470)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1470, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("1470.391");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1470 = 2 \cdot 3 \cdot 5 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1470.f (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$40.0545988610$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.3317760000.3
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 4x^{6} + 7x^{4} - 36x^{2} + 81 $ x^8 - 4x^6 + 7x^4 - 36*x^2 + 81
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	no (minimal twist has level 210)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			391.8
Root			$-1.01575 - 1.40294i$ of defining polynomial
Character	$\chi$	$=$	1470.391
Dual form			1470.3.f.a.391.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.41421 q^{2} +1.73205i q^{3} +2.00000 q^{4} +2.23607i q^{5} +2.44949i q^{6} +2.82843 q^{8} -3.00000 q^{9} +O(q^{10})$	\(q+1.41421 q^{2} +1.73205i q^{3} +2.00000 q^{4} +2.23607i q^{5} +2.44949i q^{6} +2.82843 q^{8} -3.00000 q^{9} +3.16228i q^{10} -18.0701 q^{11} +3.46410i q^{12} +18.6604i q^{13} -3.87298 q^{15} +4.00000 q^{16} +1.54109i q^{17} -4.24264 q^{18} -34.0447i q^{19} +4.47214i q^{20} -25.5549 q^{22} -26.9876 q^{23} +4.89898i q^{24} -5.00000 q^{25} +26.3898i q^{26} -5.19615i q^{27} -16.4662 q^{29} -5.47723 q^{30} -27.9423i q^{31} +5.65685 q^{32} -31.2983i q^{33} +2.17942i q^{34} -6.00000 q^{36} +51.6706 q^{37} -48.1465i q^{38} -32.3208 q^{39} +6.32456i q^{40} -37.4818i q^{41} -63.6947 q^{43} -36.1402 q^{44} -6.70820i q^{45} -38.1663 q^{46} +28.2408i q^{47} +6.92820i q^{48} -7.07107 q^{50} -2.66924 q^{51} +37.3208i q^{52} +0.443681 q^{53} -7.34847i q^{54} -40.4059i q^{55} +58.9672 q^{57} -23.2868 q^{58} +74.1618i q^{59} -7.74597 q^{60} -105.129i q^{61} -39.5164i q^{62} +8.00000 q^{64} -41.7259 q^{65} -44.2625i q^{66} -12.7088 q^{67} +3.08217i q^{68} -46.7439i q^{69} -45.7647 q^{71} -8.48528 q^{72} +36.4265i q^{73} +73.0732 q^{74} -8.66025i q^{75} -68.0895i q^{76} -45.7085 q^{78} -133.198 q^{79} +8.94427i q^{80} +9.00000 q^{81} -53.0073i q^{82} +49.9265i q^{83} -3.44597 q^{85} -90.0779 q^{86} -28.5204i q^{87} -51.1099 q^{88} -99.2041i q^{89} -9.48683i q^{90} -53.9752 q^{92} +48.3975 q^{93} +39.9385i q^{94} +76.1264 q^{95} +9.79796i q^{96} +150.376i q^{97} +54.2102 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 16 q^{4} - 24 q^{9}+O(q^{10}) $ 8 * q + 16 * q^4 - 24 * q^9	$ 8 q + 16 q^{4} - 24 q^{9} + 8 q^{11} + 32 q^{16} - 48 q^{22} - 24 q^{23} - 40 q^{25} + 72 q^{29} - 48 q^{36} + 192 q^{37} - 48 q^{39} - 112 q^{43} + 16 q^{44} - 16 q^{46} - 168 q^{51} - 64 q^{53} + 216 q^{57} - 208 q^{58} + 64 q^{64} - 40 q^{65} + 240 q^{67} + 8 q^{71} + 32 q^{74} - 192 q^{78} - 24 q^{79} + 72 q^{81} + 120 q^{85} + 80 q^{86} - 96 q^{88} - 48 q^{92} + 264 q^{93} + 80 q^{95} - 24 q^{99}+O(q^{100}) $ 8 * q + 16 * q^4 - 24 * q^9 + 8 * q^11 + 32 * q^16 - 48 * q^22 - 24 * q^23 - 40 * q^25 + 72 * q^29 - 48 * q^36 + 192 * q^37 - 48 * q^39 - 112 * q^43 + 16 * q^44 - 16 * q^46 - 168 * q^51 - 64 * q^53 + 216 * q^57 - 208 * q^58 + 64 * q^64 - 40 * q^65 + 240 * q^67 + 8 * q^71 + 32 * q^74 - 192 * q^78 - 24 * q^79 + 72 * q^81 + 120 * q^85 + 80 * q^86 - 96 * q^88 - 48 * q^92 + 264 * q^93 + 80 * q^95 - 24 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1470\mathbb{Z}\right)^\times$.

$n$	$491$	$1081$	$1177$
$\chi(n)$	$1$	$-1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	1.41421	0.707107
$2$	1.41421	0.707107
$3$	1.73205i	0.577350i
$3$	1.73205i	0.577350i
$4$	2.00000	0.500000
$5$	2.23607i	0.447214i
$5$	2.23607i	0.447214i
$6$	2.44949i	0.408248i
$7$	0	0
$7$	0	0
$8$	2.82843	0.353553
$9$	−3.00000	−0.333333
$10$	3.16228i	0.316228i
$11$	−18.0701	−1.64273	−0.821367	−	0.570400i	$-0.806788\pi$
$11$	−18.0701	−1.64273	−0.821367	+	0.570400i	$0.806788\pi$
$12$	3.46410i	0.288675i
$13$	18.6604i	1.43542i	0.696344	+	0.717708i	$0.254809\pi$
$13$	18.6604i	1.43542i	−0.696344	+	0.717708i	$0.745191\pi$
$14$	0	0
$15$	−3.87298	−0.258199
$16$	4.00000	0.250000
$17$	1.54109i	0.0906521i	0.998972	+	0.0453261i	$0.0144327\pi$
$17$	1.54109i	0.0906521i	−0.998972	+	0.0453261i	$0.985567\pi$
$18$	−4.24264	−0.235702
$19$	− 34.0447i	− 1.79183i	−0.444227	−	0.895914i	$-0.646522\pi$
$19$	− 34.0447i	− 1.79183i	0.444227	−	0.895914i	$-0.353478\pi$
$20$	4.47214i	0.223607i
$21$	0	0
$22$	−25.5549	−1.16159
$23$	−26.9876	−1.17337	−0.586687	−	0.809814i	$-0.699568\pi$
$23$	−26.9876	−1.17337	−0.586687	+	0.809814i	$0.699568\pi$
$24$	4.89898i	0.204124i
$25$	−5.00000	−0.200000
$26$	26.3898i	1.01499i
$27$	− 5.19615i	− 0.192450i
$28$	0	0
$29$	−16.4662	−0.567802	−0.283901	−	0.958854i	$-0.591629\pi$
$29$	−16.4662	−0.567802	−0.283901	+	0.958854i	$0.591629\pi$
$30$	−5.47723	−0.182574
$31$	− 27.9423i	− 0.901365i	−0.892684	−	0.450683i	$-0.851181\pi$
$31$	− 27.9423i	− 0.901365i	0.892684	−	0.450683i	$-0.148819\pi$
$32$	5.65685	0.176777
$33$	− 31.2983i	− 0.948433i
$34$	2.17942i	0.0641007i
$35$	0	0
$36$	−6.00000	−0.166667
$37$	51.6706	1.39650	0.698251	−	0.715853i	$-0.253962\pi$
$37$	51.6706	1.39650	0.698251	+	0.715853i	$0.253962\pi$
$38$	− 48.1465i	− 1.26701i
$39$	−32.3208	−0.828738
$40$	6.32456i	0.158114i
$41$	− 37.4818i	− 0.914191i	−0.889418	−	0.457095i	$-0.848890\pi$
$41$	− 37.4818i	− 0.914191i	0.889418	−	0.457095i	$-0.151110\pi$
$42$	0	0
$43$	−63.6947	−1.48127	−0.740636	−	0.671907i	$-0.765476\pi$
$43$	−63.6947	−1.48127	−0.740636	+	0.671907i	$0.765476\pi$
$44$	−36.1402	−0.821367
$45$	− 6.70820i	− 0.149071i
$46$	−38.1663	−0.829701
$47$	28.2408i	0.600868i	0.953803	+	0.300434i	$0.0971315\pi$
$47$	28.2408i	0.600868i	−0.953803	+	0.300434i	$0.902868\pi$
$48$	6.92820i	0.144338i
$49$	0	0
$50$	−7.07107	−0.141421
$51$	−2.66924	−0.0523380
$52$	37.3208i	0.717708i
$53$	0.443681	0.00837135	0.00418567	−	0.999991i	$-0.498668\pi$
$53$	0.443681	0.00837135	0.00418567	+	0.999991i	$0.498668\pi$
$54$	− 7.34847i	− 0.136083i
$55$	− 40.4059i	− 0.734653i
$56$	0	0
$57$	58.9672	1.03451
$58$	−23.2868	−0.401496
$59$	74.1618i	1.25698i	0.777818	+	0.628490i	$0.216327\pi$
$59$	74.1618i	1.25698i	−0.777818	+	0.628490i	$0.783673\pi$
$60$	−7.74597	−0.129099
$61$	− 105.129i	− 1.72342i	−0.507397	−	0.861712i	$-0.669392\pi$
$61$	− 105.129i	− 1.72342i	0.507397	−	0.861712i	$-0.330608\pi$
$62$	− 39.5164i	− 0.637361i
$63$	0	0
$64$	8.00000	0.125000
$65$	−41.7259	−0.641937
$66$	− 44.2625i	− 0.670643i
$67$	−12.7088	−0.189684	−0.0948420	−	0.995492i	$-0.530235\pi$
$67$	−12.7088	−0.189684	−0.0948420	+	0.995492i	$0.530235\pi$
$68$	3.08217i	0.0453261i
$69$	− 46.7439i	− 0.677448i
$70$	0	0
$71$	−45.7647	−0.644573	−0.322286	−	0.946642i	$-0.604451\pi$
$71$	−45.7647	−0.644573	−0.322286	+	0.946642i	$0.604451\pi$
$72$	−8.48528	−0.117851
$73$	36.4265i	0.498994i	0.968376	+	0.249497i	$0.0802652\pi$
$73$	36.4265i	0.498994i	−0.968376	+	0.249497i	$0.919735\pi$
$74$	73.0732	0.987476
$75$	− 8.66025i	− 0.115470i
$76$	− 68.0895i	− 0.895914i
$77$	0	0
$78$	−45.7085	−0.586006
$79$	−133.198	−1.68605	−0.843025	−	0.537874i	$-0.819228\pi$
$79$	−133.198	−1.68605	−0.843025	+	0.537874i	$0.819228\pi$
$80$	8.94427i	0.111803i
$81$	9.00000	0.111111
$82$	− 53.0073i	− 0.646431i
$83$	49.9265i	0.601524i	0.953699	+	0.300762i	$0.0972409\pi$
$83$	49.9265i	0.601524i	−0.953699	+	0.300762i	$0.902759\pi$
$84$	0	0
$85$	−3.44597	−0.0405409
$86$	−90.0779	−1.04742
$87$	− 28.5204i	− 0.327820i
$88$	−51.1099	−0.580794
$89$	− 99.2041i	− 1.11465i	−0.830293	−	0.557326i	$-0.811827\pi$
$89$	− 99.2041i	− 1.11465i	0.830293	−	0.557326i	$-0.188173\pi$
$90$	− 9.48683i	− 0.105409i
$91$	0	0
$92$	−53.9752	−0.586687
$93$	48.3975	0.520403
$94$	39.9385i	0.424878i
$95$	76.1264	0.801330
$96$	9.79796i	0.102062i
$97$	150.376i	1.55027i	0.631796	+	0.775134i	$0.282318\pi$
$97$	150.376i	1.55027i	−0.631796	+	0.775134i	$0.717682\pi$
$98$	0	0
$99$	54.2102	0.547578
$100$	−10.0000	−0.100000
$101$	47.0801i	0.466140i	0.972460	+	0.233070i	$0.0748771\pi$
$101$	47.0801i	0.466140i	−0.972460	+	0.233070i	$0.925123\pi$
$102$	−3.77487	−0.0370086
$103$	75.9456i	0.737336i	0.929561	+	0.368668i	$0.120186\pi$
$103$	75.9456i	0.737336i	−0.929561	+	0.368668i	$0.879814\pi$
$104$	52.7796i	0.507496i
$105$	0	0
$106$	0.627460	0.00591944
$107$	−107.225	−1.00210	−0.501052	−	0.865417i	$-0.667053\pi$
$107$	−107.225	−1.00210	−0.501052	+	0.865417i	$0.667053\pi$
$108$	− 10.3923i	− 0.0962250i
$109$	80.8905	0.742114	0.371057	−	0.928610i	$-0.378995\pi$
$109$	80.8905	0.742114	0.371057	+	0.928610i	$0.378995\pi$
$110$	− 57.1426i	− 0.519478i
$111$	89.4961i	0.806271i
$112$	0	0
$113$	−79.9061	−0.707134	−0.353567	−	0.935409i	$-0.615031\pi$
$113$	−79.9061	−0.707134	−0.353567	+	0.935409i	$0.615031\pi$
$114$	83.3923	0.731511
$115$	− 60.3461i	− 0.524749i
$116$	−32.9325	−0.283901
$117$	− 55.9812i	− 0.478472i
$118$	104.881i	0.888819i
$119$	0	0
$120$	−10.9545	−0.0912871
$121$	205.528	1.69858
$122$	− 148.675i	− 1.21865i
$123$	64.9204	0.527808
$124$	− 55.8846i	− 0.450683i
$125$	− 11.1803i	− 0.0894427i
$126$	0	0
$127$	99.1937	0.781053	0.390526	−	0.920592i	$-0.372293\pi$
$127$	99.1937	0.781053	0.390526	+	0.920592i	$0.372293\pi$
$128$	11.3137	0.0883883
$129$	− 110.322i	− 0.855213i
$130$	−59.0094	−0.453918
$131$	− 166.634i	− 1.27202i	−0.771682	−	0.636008i	$-0.780585\pi$
$131$	− 166.634i	− 1.27202i	0.771682	−	0.636008i	$-0.219415\pi$
$132$	− 62.5966i	− 0.474216i
$133$	0	0
$134$	−17.9730	−0.134127
$135$	11.6190	0.0860663
$136$	4.35885i	0.0320504i
$137$	−151.517	−1.10597	−0.552983	−	0.833192i	$-0.686511\pi$
$137$	−151.517	−1.10597	−0.552983	+	0.833192i	$0.686511\pi$
$138$	− 66.1059i	− 0.479028i
$139$	− 151.816i	− 1.09220i	−0.837719	−	0.546101i	$-0.816111\pi$
$139$	− 151.816i	− 1.09220i	0.837719	−	0.546101i	$-0.183889\pi$
$140$	0	0
$141$	−48.9145	−0.346911
$142$	−64.7210	−0.455782
$143$	− 337.195i	− 2.35801i
$144$	−12.0000	−0.0833333
$145$	− 36.8196i	− 0.253929i
$146$	51.5149i	0.352842i
$147$	0	0
$148$	103.341	0.698251
$149$	33.1939	0.222778	0.111389	−	0.993777i	$-0.464470\pi$
$149$	33.1939	0.222778	0.111389	+	0.993777i	$0.464470\pi$
$150$	− 12.2474i	− 0.0816497i
$151$	−9.30317	−0.0616104	−0.0308052	−	0.999525i	$-0.509807\pi$
$151$	−9.30317	−0.0616104	−0.0308052	+	0.999525i	$0.509807\pi$
$152$	− 96.2931i	− 0.633507i
$153$	− 4.62326i	− 0.0302174i
$154$	0	0
$155$	62.4809	0.403103
$156$	−64.6415	−0.414369
$157$	− 23.9884i	− 0.152793i	−0.997078	−	0.0763963i	$-0.975659\pi$
$157$	− 23.9884i	− 0.152793i	0.997078	−	0.0763963i	$-0.0243414\pi$
$158$	−188.370	−1.19222
$159$	0.768479i	0.00483320i
$160$	12.6491i	0.0790569i
$161$	0	0
$162$	12.7279	0.0785674
$163$	−253.127	−1.55293	−0.776464	−	0.630161i	$-0.782989\pi$
$163$	−253.127	−1.55293	−0.776464	+	0.630161i	$0.782989\pi$
$164$	− 74.9636i	− 0.457095i
$165$	69.9851	0.424152
$166$	70.6067i	0.425341i
$167$	85.7259i	0.513329i	0.966501	+	0.256664i	$0.0826235\pi$
$167$	85.7259i	0.513329i	−0.966501	+	0.256664i	$0.917376\pi$
$168$	0	0
$169$	−179.211	−1.06042
$170$	−4.87334	−0.0286667
$171$	102.134i	0.597276i
$172$	−127.389	−0.740636
$173$	19.4765i	0.112581i	0.998414	+	0.0562905i	$0.0179273\pi$
$173$	19.4765i	0.112581i	−0.998414	+	0.0562905i	$0.982073\pi$
$174$	− 40.3339i	− 0.231804i
$175$	0	0
$176$	−72.2803	−0.410684
$177$	−128.452	−0.725717
$178$	− 140.296i	− 0.788179i
$179$	252.834	1.41248	0.706241	−	0.707972i	$-0.250390\pi$
$179$	252.834	1.41248	0.706241	+	0.707972i	$0.250390\pi$
$180$	− 13.4164i	− 0.0745356i
$181$	144.224i	0.796820i	0.917207	+	0.398410i	$0.130438\pi$
$181$	144.224i	0.796820i	−0.917207	+	0.398410i	$0.869562\pi$
$182$	0	0
$183$	182.089	0.995020
$184$	−76.3325	−0.414851
$185$	115.539i	0.624535i
$186$	68.4444	0.367981
$187$	− 27.8475i	− 0.148917i
$188$	56.4816i	0.300434i
$189$	0	0
$190$	107.659	0.566626
$191$	−314.098	−1.64449	−0.822246	−	0.569132i	$-0.807279\pi$
$191$	−314.098	−1.64449	−0.822246	+	0.569132i	$0.807279\pi$
$192$	13.8564i	0.0721688i
$193$	52.8284	0.273722	0.136861	−	0.990590i	$-0.456299\pi$
$193$	52.8284	0.273722	0.136861	+	0.990590i	$0.456299\pi$
$194$	212.664i	1.09621i
$195$	− 72.2714i	− 0.370623i
$196$	0	0
$197$	54.2005	0.275129	0.137565	−	0.990493i	$-0.456073\pi$
$197$	54.2005	0.275129	0.137565	+	0.990493i	$0.456073\pi$
$198$	76.6648	0.387196
$199$	22.9710i	0.115432i	0.998333	+	0.0577160i	$0.0183818\pi$
$199$	22.9710i	0.115432i	−0.998333	+	0.0577160i	$0.981618\pi$
$200$	−14.1421	−0.0707107
$201$	− 22.0123i	− 0.109514i
$202$	66.5813i	0.329611i
$203$	0	0
$204$	−5.33848	−0.0261690
$205$	83.8119	0.408839
$206$	107.403i	0.521375i
$207$	80.9628	0.391125
$208$	74.6416i	0.358854i
$209$	615.191i	2.94350i
$210$	0	0
$211$	−292.203	−1.38485	−0.692425	−	0.721490i	$-0.743457\pi$
$211$	−292.203	−1.38485	−0.692425	+	0.721490i	$0.743457\pi$
$212$	0.887363	0.00418567
$213$	− 79.2667i	− 0.372144i
$214$	−151.639	−0.708594
$215$	− 142.426i	− 0.662445i
$216$	− 14.6969i	− 0.0680414i
$217$	0	0
$218$	114.396	0.524754
$219$	−63.0926	−0.288094
$220$	− 80.8118i	− 0.367327i
$221$	−28.7573	−0.130123
$222$	126.567i	0.570120i
$223$	− 271.305i	− 1.21661i	−0.793702	−	0.608306i	$-0.791849\pi$
$223$	− 271.305i	− 1.21661i	0.793702	−	0.608306i	$-0.208151\pi$
$224$	0	0
$225$	15.0000	0.0666667
$226$	−113.004	−0.500019
$227$	148.609i	0.654664i	0.944909	+	0.327332i	$0.106150\pi$
$227$	148.609i	0.654664i	−0.944909	+	0.327332i	$0.893850\pi$
$228$	117.934	0.517256
$229$	− 25.3335i	− 0.110626i	−0.998469	−	0.0553132i	$-0.982384\pi$
$229$	− 25.3335i	− 0.110626i	0.998469	−	0.0553132i	$-0.0176157\pi$
$230$	− 85.3423i	− 0.371054i
$231$	0	0
$232$	−46.5736	−0.200748
$233$	−408.886	−1.75488	−0.877438	−	0.479691i	$-0.840749\pi$
$233$	−408.886	−1.75488	−0.877438	+	0.479691i	$0.840749\pi$
$234$	− 79.1694i	− 0.338331i
$235$	−63.1483	−0.268716
$236$	148.324i	0.628490i
$237$	− 230.706i	− 0.973442i
$238$	0	0
$239$	67.0352	0.280482	0.140241	−	0.990117i	$-0.455212\pi$
$239$	67.0352	0.280482	0.140241	+	0.990117i	$0.455212\pi$
$240$	−15.4919	−0.0645497
$241$	236.879i	0.982901i	0.870905	+	0.491451i	$0.163533\pi$
$241$	236.879i	0.982901i	−0.870905	+	0.491451i	$0.836467\pi$
$242$	290.660	1.20107
$243$	15.5885i	0.0641500i
$244$	− 210.258i	− 0.861712i
$245$	0	0
$246$	91.8113	0.373217
$247$	635.289	2.57202
$248$	− 79.0328i	− 0.318681i
$249$	−86.4752	−0.347290
$250$	− 15.8114i	− 0.0632456i
$251$	483.382i	1.92582i	0.269815	+	0.962912i	$0.413037\pi$
$251$	483.382i	1.92582i	−0.269815	+	0.962912i	$0.586963\pi$
$252$	0	0
$253$	487.668	1.92754
$254$	140.281	0.552288
$255$	− 5.96860i	− 0.0234063i
$256$	16.0000	0.0625000
$257$	324.583i	1.26297i	0.775389	+	0.631484i	$0.217554\pi$
$257$	324.583i	1.26297i	−0.775389	+	0.631484i	$0.782446\pi$
$258$	− 156.019i	− 0.604727i
$259$	0	0
$260$	−83.4519	−0.320969
$261$	49.3987	0.189267
$262$	− 235.656i	− 0.899451i
$263$	307.372	1.16872	0.584358	−	0.811496i	$-0.301347\pi$
$263$	307.372	1.16872	0.584358	+	0.811496i	$0.301347\pi$
$264$	− 88.5249i	− 0.335322i
$265$	0.992102i	0.00374378i
$266$	0	0
$267$	171.827	0.643545
$268$	−25.4177	−0.0948420
$269$	− 75.8484i	− 0.281964i	−0.990012	−	0.140982i	$-0.954974\pi$
$269$	− 75.8484i	− 0.281964i	0.990012	−	0.140982i	$-0.0450260\pi$
$270$	16.4317	0.0608581
$271$	21.7258i	0.0801689i	0.999196	+	0.0400844i	$0.0127627\pi$
$271$	21.7258i	0.0801689i	−0.999196	+	0.0400844i	$0.987237\pi$
$272$	6.16434i	0.0226630i
$273$	0	0
$274$	−214.278	−0.782036
$275$	90.3504	0.328547
$276$	− 93.4878i	− 0.338724i
$277$	−111.732	−0.403364	−0.201682	−	0.979451i	$-0.564641\pi$
$277$	−111.732	−0.403364	−0.201682	+	0.979451i	$0.564641\pi$
$278$	− 214.700i	− 0.772304i
$279$	83.8270i	0.300455i
$280$	0	0
$281$	−188.298	−0.670101	−0.335051	−	0.942200i	$-0.608753\pi$
$281$	−188.298	−0.670101	−0.335051	+	0.942200i	$0.608753\pi$
$282$	−69.1755	−0.245303
$283$	216.817i	0.766137i	0.923720	+	0.383069i	$0.125133\pi$
$283$	216.817i	0.766137i	−0.923720	+	0.383069i	$0.874867\pi$
$284$	−91.5293	−0.322286
$285$	131.855i	0.462648i
$286$	− 476.866i	− 1.66736i
$287$	0	0
$288$	−16.9706	−0.0589256
$289$	286.625	0.991782
$290$	− 52.0708i	− 0.179555i
$291$	−260.459	−0.895048
$292$	72.8531i	0.249497i
$293$	57.3776i	0.195828i	0.995195	+	0.0979140i	$0.0312170\pi$
$293$	57.3776i	0.195828i	−0.995195	+	0.0979140i	$0.968783\pi$
$294$	0	0
$295$	−165.831	−0.562138
$296$	146.146	0.493738
$297$	93.8949i	0.316144i
$298$	46.9433	0.157528
$299$	− 503.600i	− 1.68428i
$300$	− 17.3205i	− 0.0577350i
$301$	0	0
$302$	−13.1567	−0.0435651
$303$	−81.5451	−0.269126
$304$	− 136.179i	− 0.447957i
$305$	235.075	0.770739
$306$	− 6.53827i	− 0.0213669i
$307$	291.273i	0.948773i	0.880317	+	0.474386i	$0.157330\pi$
$307$	291.273i	0.948773i	−0.880317	+	0.474386i	$0.842670\pi$
$308$	0	0
$309$	−131.542	−0.425701
$310$	88.3614	0.285037
$311$	249.840i	0.803345i	0.915783	+	0.401673i	$0.131571\pi$
$311$	249.840i	0.803345i	−0.915783	+	0.401673i	$0.868429\pi$
$312$	−91.4169	−0.293003
$313$	3.78548i	0.0120942i	0.999982	+	0.00604709i	$0.00192486\pi$
$313$	3.78548i	0.0120942i	−0.999982	+	0.00604709i	$0.998075\pi$
$314$	− 33.9248i	− 0.108041i
$315$	0	0
$316$	−266.396	−0.843025
$317$	−71.3610	−0.225114	−0.112557	−	0.993645i	$-0.535904\pi$
$317$	−71.3610	−0.225114	−0.112557	+	0.993645i	$0.535904\pi$
$318$	1.08679i	0.00341759i
$319$	297.546	0.932747
$320$	17.8885i	0.0559017i
$321$	− 185.719i	− 0.578565i
$322$	0	0
$323$	52.4659	0.162433
$324$	18.0000	0.0555556
$325$	− 93.3020i	− 0.287083i
$326$	−357.976	−1.09809
$327$	140.106i	0.428460i
$328$	− 106.015i	− 0.323215i
$329$	0	0
$330$	98.9739	0.299921
$331$	−341.966	−1.03313	−0.516566	−	0.856248i	$-0.672790\pi$
$331$	−341.966	−1.03313	−0.516566	+	0.856248i	$0.672790\pi$
$332$	99.8529i	0.300762i
$333$	−155.012	−0.465501
$334$	121.235i	0.362978i
$335$	− 28.4178i	− 0.0848293i
$336$	0	0
$337$	246.396	0.731145	0.365573	−	0.930783i	$-0.380873\pi$
$337$	246.396	0.731145	0.365573	+	0.930783i	$0.380873\pi$
$338$	−253.442	−0.749829
$339$	− 138.401i	− 0.408264i
$340$	−6.89195	−0.0202704
$341$	504.920i	1.48070i
$342$	144.440i	0.422338i
$343$	0	0
$344$	−180.156	−0.523709
$345$	104.523	0.302964
$346$	27.5440i	0.0796068i
$347$	138.352	0.398708	0.199354	−	0.979928i	$-0.436116\pi$
$347$	138.352	0.398708	0.199354	+	0.979928i	$0.436116\pi$
$348$	− 57.0407i	− 0.163910i
$349$	115.858i	0.331971i	0.986128	+	0.165986i	$0.0530805\pi$
$349$	115.858i	0.331971i	−0.986128	+	0.165986i	$0.946919\pi$
$350$	0	0
$351$	96.9623	0.276246
$352$	−102.220	−0.290397
$353$	187.219i	0.530364i	0.964198	+	0.265182i	$0.0854321\pi$
$353$	187.219i	0.530364i	−0.964198	+	0.265182i	$0.914568\pi$
$354$	−181.659	−0.513160
$355$	− 102.333i	− 0.288262i
$356$	− 198.408i	− 0.557326i
$357$	0	0
$358$	357.562	0.998775
$359$	637.495	1.77575	0.887876	−	0.460082i	$-0.152180\pi$
$359$	637.495	1.77575	0.887876	+	0.460082i	$0.152180\pi$
$360$	− 18.9737i	− 0.0527046i
$361$	−798.045	−2.21065
$362$	203.964i	0.563437i
$363$	355.984i	0.980673i
$364$	0	0
$365$	−81.4522	−0.223157
$366$	257.512	0.703585
$367$	55.0504i	0.150001i	0.997183	+	0.0750005i	$0.0238959\pi$
$367$	55.0504i	0.150001i	−0.997183	+	0.0750005i	$0.976104\pi$
$368$	−107.950	−0.293344
$369$	112.445i	0.304730i
$370$	163.397i	0.441613i
$371$	0	0
$372$	96.7950	0.260202
$373$	269.157	0.721600	0.360800	−	0.932643i	$-0.382504\pi$
$373$	269.157	0.721600	0.360800	+	0.932643i	$0.382504\pi$
$374$	− 39.3824i	− 0.105300i
$375$	19.3649	0.0516398
$376$	79.8770i	0.212439i
$377$	− 307.267i	− 0.815031i
$378$	0	0
$379$	469.785	1.23954	0.619769	−	0.784784i	$-0.287226\pi$
$379$	469.785	1.23954	0.619769	+	0.784784i	$0.287226\pi$
$380$	152.253	0.400665
$381$	171.809i	0.450941i
$382$	−444.202	−1.16283
$383$	402.210i	1.05016i	0.851054	+	0.525078i	$0.175964\pi$
$383$	402.210i	1.05016i	−0.851054	+	0.525078i	$0.824036\pi$
$384$	19.5959i	0.0510310i
$385$	0	0
$386$	74.7107	0.193551
$387$	191.084	0.493757
$388$	300.752i	0.775134i
$389$	−110.761	−0.284732	−0.142366	−	0.989814i	$-0.545471\pi$
$389$	−110.761	−0.284732	−0.142366	+	0.989814i	$0.545471\pi$
$390$	− 102.207i	− 0.262070i
$391$	− 41.5902i	− 0.106369i
$392$	0	0
$393$	288.619	0.734399
$394$	76.6510	0.194546
$395$	− 297.840i	− 0.754025i
$396$	108.420	0.273789
$397$	− 419.814i	− 1.05747i	−0.848788	−	0.528733i	$-0.822667\pi$
$397$	− 419.814i	− 1.05747i	0.848788	−	0.528733i	$-0.177333\pi$
$398$	32.4858i	0.0816227i
$399$	0	0
$400$	−20.0000	−0.0500000
$401$	314.067	0.783210	0.391605	−	0.920133i	$-0.371920\pi$
$401$	314.067	0.783210	0.391605	+	0.920133i	$0.371920\pi$
$402$	− 31.1302i	− 0.0774382i
$403$	521.415	1.29383
$404$	94.1602i	0.233070i
$405$	20.1246i	0.0496904i
$406$	0	0
$407$	−933.691	−2.29408
$408$	−7.54975	−0.0185043
$409$	121.828i	0.297868i	0.988847	+	0.148934i	$0.0475842\pi$
$409$	121.828i	0.297868i	−0.988847	+	0.148934i	$0.952416\pi$
$410$	118.528	0.289093
$411$	− 262.436i	− 0.638530i
$412$	151.891i	0.368668i
$413$	0	0
$414$	114.499	0.276567
$415$	−111.639	−0.269010
$416$	105.559i	0.253748i
$417$	262.953	0.630583
$418$	870.012i	2.08137i
$419$	43.0872i	0.102833i	0.998677	+	0.0514167i	$0.0163737\pi$
$419$	43.0872i	0.102833i	−0.998677	+	0.0514167i	$0.983626\pi$
$420$	0	0
$421$	135.571	0.322022	0.161011	−	0.986953i	$-0.448525\pi$
$421$	135.571	0.322022	0.161011	+	0.986953i	$0.448525\pi$
$422$	−413.238	−0.979236
$423$	− 84.7224i	− 0.200289i
$424$	1.25492	0.00295972
$425$	− 7.70543i	− 0.0181304i
$426$	− 112.100i	− 0.263146i
$427$	0	0
$428$	−214.450	−0.501052
$429$	584.039	1.36140
$430$	− 201.420i	− 0.468419i
$431$	191.226	0.443681	0.221840	−	0.975083i	$-0.428794\pi$
$431$	191.226	0.443681	0.221840	+	0.975083i	$0.428794\pi$
$432$	− 20.7846i	− 0.0481125i
$433$	− 339.825i	− 0.784815i	−0.919791	−	0.392408i	$-0.871642\pi$
$433$	− 339.825i	− 0.784815i	0.919791	−	0.392408i	$-0.128358\pi$
$434$	0	0
$435$	63.7735	0.146606
$436$	161.781	0.371057
$437$	918.786i	2.10249i
$438$	−89.2264	−0.203713
$439$	− 330.262i	− 0.752305i	−0.926558	−	0.376153i	$-0.877247\pi$
$439$	− 330.262i	− 0.752305i	0.926558	−	0.376153i	$-0.122753\pi$
$440$	− 114.285i	− 0.259739i
$441$	0	0
$442$	−40.6690	−0.0920112
$443$	−561.662	−1.26786	−0.633931	−	0.773390i	$-0.718560\pi$
$443$	−561.662	−1.26786	−0.633931	+	0.773390i	$0.718560\pi$
$444$	178.992i	0.403135i
$445$	221.827	0.498488
$446$	− 383.683i	− 0.860275i
$447$	57.4936i	0.128621i
$448$	0	0
$449$	−386.250	−0.860244	−0.430122	−	0.902771i	$-0.641529\pi$
$449$	−386.250	−0.860244	−0.430122	+	0.902771i	$0.641529\pi$
$450$	21.2132	0.0471405
$451$	677.299i	1.50177i
$452$	−159.812	−0.353567
$453$	− 16.1136i	− 0.0355708i
$454$	210.165i	0.462918i
$455$	0	0
$456$	166.785	0.365755
$457$	499.066	1.09205	0.546024	−	0.837770i	$-0.316141\pi$
$457$	499.066	1.09205	0.546024	+	0.837770i	$0.316141\pi$
$458$	− 35.8269i	− 0.0782247i
$459$	8.00772	0.0174460
$460$	− 120.692i	− 0.262375i
$461$	− 618.290i	− 1.34119i	−0.741823	−	0.670596i	$-0.766038\pi$
$461$	− 618.290i	− 1.34119i	0.741823	−	0.670596i	$-0.233962\pi$
$462$	0	0
$463$	−194.433	−0.419941	−0.209971	−	0.977708i	$-0.567337\pi$
$463$	−194.433	−0.419941	−0.209971	+	0.977708i	$0.567337\pi$
$464$	−65.8650	−0.141950
$465$	108.220i	0.232731i
$466$	−578.252	−1.24088
$467$	− 151.785i	− 0.325021i	−0.986707	−	0.162510i	$-0.948041\pi$
$467$	− 151.785i	− 0.325021i	0.986707	−	0.162510i	$-0.0519591\pi$
$468$	− 111.962i	− 0.239236i
$469$	0	0
$470$	−89.3052	−0.190011
$471$	41.5492	0.0882149
$472$	209.761i	0.444409i
$473$	1150.97	2.43334
$474$	− 326.267i	− 0.688327i
$475$	170.224i	0.358366i
$476$	0	0
$477$	−1.33104	−0.00279045
$478$	94.8020	0.198331
$479$	224.770i	0.469248i	0.972086	+	0.234624i	$0.0753859\pi$
$479$	224.770i	0.469248i	−0.972086	+	0.234624i	$0.924614\pi$
$480$	−21.9089	−0.0456435
$481$	964.194i	2.00456i
$482$	334.998i	0.695016i
$483$	0	0
$484$	411.055	0.849288
$485$	−336.251	−0.693301
$486$	22.0454i	0.0453609i
$487$	69.5628	0.142840	0.0714198	−	0.997446i	$-0.477247\pi$
$487$	69.5628	0.142840	0.0714198	+	0.997446i	$0.477247\pi$
$488$	− 297.349i	− 0.609323i
$489$	− 438.429i	− 0.896584i
$490$	0	0
$491$	−750.454	−1.52842	−0.764210	−	0.644967i	$-0.776871\pi$
$491$	−750.454	−1.52842	−0.764210	+	0.644967i	$0.776871\pi$
$492$	129.841	0.263904
$493$	− 25.3759i	− 0.0514724i
$494$	898.434	1.81869
$495$	121.218i	0.244884i
$496$	− 111.769i	− 0.225341i
$497$	0	0
$498$	−122.294	−0.245571
$499$	160.170	0.320981	0.160491	−	0.987037i	$-0.448692\pi$
$499$	160.170	0.320981	0.160491	+	0.987037i	$0.448692\pi$
$500$	− 22.3607i	− 0.0447214i
$501$	−148.482	−0.296371
$502$	683.605i	1.36176i
$503$	724.412i	1.44018i	0.693879	+	0.720092i	$0.255900\pi$
$503$	724.412i	1.44018i	−0.693879	+	0.720092i	$0.744100\pi$
$504$	0	0
$505$	−105.274	−0.208464
$506$	689.667	1.36298
$507$	− 310.402i	− 0.612233i
$508$	198.387	0.390526
$509$	822.082i	1.61509i	0.589804	+	0.807547i	$0.299205\pi$
$509$	822.082i	1.61509i	−0.589804	+	0.807547i	$0.700795\pi$
$510$	− 8.44088i	− 0.0165507i
$511$	0	0
$512$	22.6274	0.0441942
$513$	−176.902	−0.344838
$514$	459.029i	0.893053i
$515$	−169.820	−0.329747
$516$	− 220.645i	− 0.427606i
$517$	− 510.313i	− 0.987066i
$518$	0	0
$519$	−33.7343	−0.0649987
$520$	−118.019	−0.226959
$521$	452.812i	0.869122i	0.900642	+	0.434561i	$0.143096\pi$
$521$	452.812i	0.869122i	−0.900642	+	0.434561i	$0.856904\pi$
$522$	69.8604	0.133832
$523$	− 974.102i	− 1.86253i	−0.364345	−	0.931264i	$-0.618707\pi$
$523$	− 974.102i	− 1.86253i	0.364345	−	0.931264i	$-0.381293\pi$
$524$	− 333.268i	− 0.636008i
$525$	0	0
$526$	434.690	0.826406
$527$	43.0615	0.0817107
$528$	− 125.193i	− 0.237108i
$529$	199.331	0.376808
$530$	1.40304i	0.00264725i
$531$	− 222.485i	− 0.418993i
$532$	0	0
$533$	699.426	1.31224
$534$	242.999	0.455055
$535$	− 239.763i	− 0.448154i
$536$	−35.9460	−0.0670634
$537$	437.922i	0.815497i
$538$	− 107.266i	− 0.199379i
$539$	0	0
$540$	23.2379	0.0430331
$541$	276.156	0.510455	0.255228	−	0.966881i	$-0.417850\pi$
$541$	276.156	0.510455	0.255228	+	0.966881i	$0.417850\pi$
$542$	30.7249i	0.0566880i
$543$	−249.804	−0.460044
$544$	8.71770i	0.0160252i
$545$	180.877i	0.331884i
$546$	0	0
$547$	426.436	0.779591	0.389795	−	0.920901i	$-0.372546\pi$
$547$	426.436	0.779591	0.389795	+	0.920901i	$0.372546\pi$
$548$	−303.035	−0.552983
$549$	315.387i	0.574475i
$550$	127.775	0.232318
$551$	560.589i	1.01740i
$552$	− 132.212i	− 0.239514i
$553$	0	0
$554$	−158.013	−0.285222
$555$	−200.119	−0.360575
$556$	− 303.632i	− 0.546101i
$557$	−121.956	−0.218952	−0.109476	−	0.993989i	$-0.534917\pi$
$557$	−121.956	−0.218952	−0.109476	+	0.993989i	$0.534917\pi$
$558$	118.549i	0.212454i
$559$	− 1188.57i	− 2.12624i
$560$	0	0
$561$	48.2334	0.0859775
$562$	−266.294	−0.473833
$563$	− 434.782i	− 0.772259i	−0.922445	−	0.386130i	$-0.873812\pi$
$563$	− 434.782i	− 0.772259i	0.922445	−	0.386130i	$-0.126188\pi$
$564$	−97.8290	−0.173456
$565$	− 178.675i	− 0.316240i
$566$	306.625i	0.541741i
$567$	0	0
$568$	−129.442	−0.227891
$569$	−296.277	−0.520697	−0.260349	−	0.965515i	$-0.583838\pi$
$569$	−296.277	−0.520697	−0.260349	+	0.965515i	$0.583838\pi$
$570$	186.471i	0.327142i
$571$	59.9156	0.104931	0.0524655	−	0.998623i	$-0.483292\pi$
$571$	59.9156	0.104931	0.0524655	+	0.998623i	$0.483292\pi$
$572$	− 674.390i	− 1.17900i
$573$	− 544.034i	− 0.949448i
$574$	0	0
$575$	134.938	0.234675
$576$	−24.0000	−0.0416667
$577$	332.701i	0.576605i	0.957539	+	0.288303i	$0.0930909\pi$
$577$	332.701i	0.576605i	−0.957539	+	0.288303i	$0.906909\pi$
$578$	405.349	0.701296
$579$	91.5015i	0.158034i
$580$	− 73.6393i	− 0.126964i
$581$	0	0
$582$	−368.345	−0.632895
$583$	−8.01736	−0.0137519
$584$	103.030i	0.176421i
$585$	125.178	0.213979
$586$	81.1442i	0.138471i
$587$	− 656.221i	− 1.11792i	−0.829194	−	0.558961i	$-0.811200\pi$
$587$	− 656.221i	− 1.11792i	0.829194	−	0.558961i	$-0.188800\pi$
$588$	0	0
$589$	−951.289	−1.61509
$590$	−234.520	−0.397492
$591$	93.8780i	0.158846i
$592$	206.682	0.349125
$593$	408.542i	0.688941i	0.938797	+	0.344471i	$0.111942\pi$
$593$	408.542i	0.688941i	−0.938797	+	0.344471i	$0.888058\pi$
$594$	132.787i	0.223548i
$595$	0	0
$596$	66.3879	0.111389
$597$	−39.7869	−0.0666447
$598$	− 712.198i	− 1.19097i
$599$	−1177.29	−1.96542	−0.982709	−	0.185157i	$-0.940721\pi$
$599$	−1177.29	−1.96542	−0.982709	+	0.185157i	$0.940721\pi$
$600$	− 24.4949i	− 0.0408248i
$601$	162.592i	0.270536i	0.990809	+	0.135268i	$0.0431896\pi$
$601$	162.592i	0.270536i	−0.990809	+	0.135268i	$0.956810\pi$
$602$	0	0
$603$	38.1265	0.0632280
$604$	−18.6063	−0.0308052
$605$	459.574i	0.759626i
$606$	−115.322	−0.190301
$607$	367.329i	0.605154i	0.953125	+	0.302577i	$0.0978470\pi$
$607$	367.329i	0.605154i	−0.953125	+	0.302577i	$0.902153\pi$
$608$	− 192.586i	− 0.316754i
$609$	0	0
$610$	332.447	0.544995
$611$	−526.985	−0.862495
$612$	− 9.24652i	− 0.0151087i
$613$	365.272	0.595877	0.297938	−	0.954585i	$-0.403701\pi$
$613$	365.272	0.595877	0.297938	+	0.954585i	$0.403701\pi$
$614$	411.923i	0.670884i
$615$	145.166i	0.236043i
$616$	0	0
$617$	−64.2245	−0.104092	−0.0520458	−	0.998645i	$-0.516574\pi$
$617$	−64.2245	−0.104092	−0.0520458	+	0.998645i	$0.516574\pi$
$618$	−186.028	−0.301016
$619$	− 1143.90i	− 1.84798i	−0.382418	−	0.923989i	$-0.624909\pi$
$619$	− 1143.90i	− 1.84798i	0.382418	−	0.923989i	$-0.375091\pi$
$620$	124.962	0.201551
$621$	140.232i	0.225816i
$622$	353.328i	0.568051i
$623$	0	0
$624$	−129.283	−0.207184
$625$	25.0000	0.0400000
$626$	5.35347i	0.00855188i
$627$	−1065.54	−1.69943
$628$	− 47.9769i	− 0.0763963i
$629$	79.6288i	0.126596i
$630$	0	0
$631$	257.367	0.407872	0.203936	−	0.978984i	$-0.434627\pi$
$631$	257.367	0.407872	0.203936	+	0.978984i	$0.434627\pi$
$632$	−376.741	−0.596109
$633$	− 506.111i	− 0.799543i
$634$	−100.920	−0.159179
$635$	221.804i	0.349298i
$636$	1.53696i	0.00241660i
$637$	0	0
$638$	420.794	0.659552
$639$	137.294	0.214858
$640$	25.2982i	0.0395285i
$641$	198.972	0.310409	0.155204	−	0.987882i	$-0.450396\pi$
$641$	198.972	0.310409	0.155204	+	0.987882i	$0.450396\pi$
$642$	− 262.647i	− 0.409107i
$643$	− 708.223i	− 1.10144i	−0.834692	−	0.550718i	$-0.814354\pi$
$643$	− 708.223i	− 1.10144i	0.834692	−	0.550718i	$-0.185646\pi$
$644$	0	0
$645$	246.688	0.382463
$646$	74.1980	0.114858
$647$	− 162.575i	− 0.251276i	−0.992076	−	0.125638i	$-0.959902\pi$
$647$	− 162.575i	− 0.251276i	0.992076	−	0.125638i	$-0.0400977\pi$
$648$	25.4558	0.0392837
$649$	− 1340.11i	− 2.06488i
$650$	− 131.949i	− 0.202998i
$651$	0	0
$652$	−506.255	−0.776464
$653$	995.760	1.52490	0.762450	−	0.647047i	$-0.223996\pi$
$653$	995.760	1.52490	0.762450	+	0.647047i	$0.223996\pi$
$654$	198.140i	0.302967i
$655$	372.605	0.568863
$656$	− 149.927i	− 0.228548i
$657$	− 109.280i	− 0.166331i
$658$	0	0
$659$	897.542	1.36198	0.680988	−	0.732295i	$-0.261551\pi$
$659$	897.542	1.36198	0.680988	+	0.732295i	$0.261551\pi$
$660$	139.970	0.212076
$661$	− 86.2616i	− 0.130502i	−0.997869	−	0.0652508i	$-0.979215\pi$
$661$	− 86.2616i	− 0.130502i	0.997869	−	0.0652508i	$-0.0207848\pi$
$662$	−483.614	−0.730534
$663$	− 49.8091i	− 0.0751268i
$664$	141.213i	0.212671i
$665$	0	0
$666$	−219.220	−0.329159
$667$	444.385	0.666244
$668$	171.452i	0.256664i
$669$	469.913	0.702412
$670$	− 40.1889i	− 0.0599834i
$671$	1899.69i	2.83113i
$672$	0	0
$673$	486.598	0.723028	0.361514	−	0.932367i	$-0.382260\pi$
$673$	486.598	0.723028	0.361514	+	0.932367i	$0.382260\pi$
$674$	348.456	0.516998
$675$	25.9808i	0.0384900i
$676$	−358.422	−0.530209
$677$	− 54.9254i	− 0.0811306i	−0.999177	−	0.0405653i	$-0.987084\pi$
$677$	− 54.9254i	− 0.0811306i	0.999177	−	0.0405653i	$-0.0129159\pi$
$678$	− 195.729i	− 0.288686i
$679$	0	0
$680$	−9.74668	−0.0143334
$681$	−257.398	−0.377971
$682$	714.064i	1.04702i
$683$	−954.370	−1.39732	−0.698661	−	0.715453i	$-0.746220\pi$
$683$	−954.370	−1.39732	−0.698661	+	0.715453i	$0.746220\pi$
$684$	204.268i	0.298638i
$685$	− 338.803i	− 0.494603i
$686$	0	0
$687$	43.8788	0.0638702
$688$	−254.779	−0.370318
$689$	8.27928i	0.0120164i
$690$	147.817	0.214228
$691$	25.8492i	0.0374084i	0.999825	+	0.0187042i	$0.00595407\pi$
$691$	25.8492i	0.0374084i	−0.999825	+	0.0187042i	$0.994046\pi$
$692$	38.9530i	0.0562905i
$693$	0	0
$694$	195.659	0.281929
$695$	339.471	0.488448
$696$	− 80.6678i	− 0.115902i
$697$	57.7627	0.0828733
$698$	163.848i	0.234739i
$699$	− 708.211i	− 1.01318i
$700$	0	0
$701$	−942.060	−1.34388	−0.671940	−	0.740606i	$-0.734539\pi$
$701$	−942.060	−1.34388	−0.671940	+	0.740606i	$0.734539\pi$
$702$	137.125	0.195335
$703$	− 1759.11i	− 2.50229i
$704$	−144.561	−0.205342
$705$	− 109.376i	− 0.155143i
$706$	264.767i	0.375024i
$707$	0	0
$708$	−256.904	−0.362859
$709$	−336.564	−0.474703	−0.237351	−	0.971424i	$-0.576279\pi$
$709$	−336.564	−0.474703	−0.237351	+	0.971424i	$0.576279\pi$
$710$	− 144.721i	− 0.203832i
$711$	399.594	0.562017
$712$	− 280.592i	− 0.394089i
$713$	754.097i	1.05764i
$714$	0	0
$715$	753.991	1.05453
$716$	505.668	0.706241
$717$	116.108i	0.161936i
$718$	901.555	1.25565
$719$	128.688i	0.178981i	0.995988	+	0.0894907i	$0.0285239\pi$
$719$	128.688i	0.178981i	−0.995988	+	0.0894907i	$0.971476\pi$
$720$	− 26.8328i	− 0.0372678i
$721$	0	0
$722$	−1128.61	−1.56317
$723$	−410.287	−0.567478
$724$	288.449i	0.398410i
$725$	82.3312	0.113560
$726$	503.438i	0.693440i
$727$	684.683i	0.941792i	0.882189	+	0.470896i	$0.156069\pi$
$727$	684.683i	0.941792i	−0.882189	+	0.470896i	$0.843931\pi$
$728$	0	0
$729$	−27.0000	−0.0370370
$730$	−115.191	−0.157796
$731$	− 98.1590i	− 0.134280i
$732$	364.177	0.497510
$733$	− 591.823i	− 0.807398i	−0.914892	−	0.403699i	$-0.867724\pi$
$733$	− 591.823i	− 0.807398i	0.914892	−	0.403699i	$-0.132276\pi$
$734$	77.8530i	0.106067i
$735$	0	0
$736$	−152.665	−0.207425
$737$	229.650	0.311600
$738$	159.022i	0.215477i
$739$	504.210	0.682287	0.341144	−	0.940011i	$-0.389186\pi$
$739$	504.210	0.682287	0.341144	+	0.940011i	$0.389186\pi$
$740$	231.078i	0.312267i
$741$	1100.35i	1.48496i
$742$	0	0
$743$	983.540	1.32374	0.661871	−	0.749618i	$-0.269763\pi$
$743$	983.540	1.32374	0.661871	+	0.749618i	$0.269763\pi$
$744$	136.889	0.183990
$745$	74.2239i	0.0996294i
$746$	380.645	0.510249
$747$	− 149.779i	− 0.200508i
$748$	− 55.6951i	− 0.0744587i
$749$	0	0
$750$	27.3861	0.0365148
$751$	916.925	1.22094	0.610469	−	0.792040i	$-0.290981\pi$
$751$	916.925	1.22094	0.610469	+	0.792040i	$0.290981\pi$
$752$	112.963i	0.150217i
$753$	−837.242	−1.11188
$754$	− 434.541i	− 0.576314i
$755$	− 20.8025i	− 0.0275530i
$756$	0	0
$757$	82.2419	0.108642	0.0543209	−	0.998524i	$-0.482701\pi$
$757$	82.2419	0.108642	0.0543209	+	0.998524i	$0.482701\pi$
$758$	664.377	0.876486
$759$	844.666i	1.11287i
$760$	215.318	0.283313
$761$	− 350.217i	− 0.460206i	−0.973166	−	0.230103i	$-0.926094\pi$
$761$	− 350.217i	− 0.460206i	0.973166	−	0.230103i	$-0.0739063\pi$
$762$	242.974i	0.318864i
$763$	0	0
$764$	−628.196	−0.822246
$765$	10.3379	0.0135136
$766$	568.811i	0.742573i
$767$	−1383.89	−1.80429
$768$	27.7128i	0.0360844i
$769$	319.560i	0.415553i	0.978176	+	0.207777i	$0.0666227\pi$
$769$	319.560i	0.415553i	−0.978176	+	0.207777i	$0.933377\pi$
$770$	0	0
$771$	−562.194	−0.729175
$772$	105.657	0.136861
$773$	− 1001.01i	− 1.29497i	−0.762077	−	0.647487i	$-0.775820\pi$
$773$	− 1001.01i	− 1.29497i	0.762077	−	0.647487i	$-0.224180\pi$
$774$	270.234	0.349139
$775$	139.712i	0.180273i
$776$	425.328i	0.548103i
$777$	0	0
$778$	−156.639	−0.201336
$779$	−1276.06	−1.63807
$780$	− 144.543i	− 0.185311i
$781$	826.971	1.05886
$782$	− 58.8175i	− 0.0752142i
$783$	85.5611i	0.109273i
$784$	0	0
$785$	53.6398	0.0683309
$786$	408.168	0.519298
$787$	129.701i	0.164804i	0.996599	+	0.0824022i	$0.0262592\pi$
$787$	129.701i	0.164804i	−0.996599	+	0.0824022i	$0.973741\pi$
$788$	108.401	0.137565
$789$	532.384i	0.674758i
$790$	− 421.209i	− 0.533176i
$791$	0	0
$792$	153.330	0.193598
$793$	1961.75	2.47383
$794$	− 593.706i	− 0.747741i
$795$	−1.71837	−0.00216147
$796$	45.9419i	0.0577160i
$797$	− 420.926i	− 0.528138i	−0.964504	−	0.264069i	$-0.914935\pi$
$797$	− 420.926i	− 0.528138i	0.964504	−	0.264069i	$-0.0850647\pi$
$798$	0	0
$799$	−43.5215	−0.0544700
$800$	−28.2843	−0.0353553
$801$	297.612i	0.371551i
$802$	444.158	0.553813
$803$	− 658.230i	− 0.819714i
$804$	− 44.0247i	− 0.0547571i
$805$	0	0
$806$	737.392	0.914879
$807$	131.373	0.162792
$808$	133.163i	0.164805i
$809$	−1164.33	−1.43922	−0.719612	−	0.694377i	$-0.755680\pi$
$809$	−1164.33	−1.43922	−0.719612	+	0.694377i	$0.755680\pi$
$810$	28.4605i	0.0351364i
$811$	− 753.691i	− 0.929335i	−0.885485	−	0.464668i	$-0.846174\pi$
$811$	− 753.691i	− 0.929335i	0.885485	−	0.464668i	$-0.153826\pi$
$812$	0	0
$813$	−37.6301	−0.0462855
$814$	−1320.44	−1.62216
$815$	− 566.010i	− 0.694491i
$816$	−10.6770	−0.0130845
$817$	2168.47i	2.65419i
$818$	172.291i	0.210625i
$819$	0	0
$820$	167.624	0.204419
$821$	−368.797	−0.449204	−0.224602	−	0.974451i	$-0.572108\pi$
$821$	−368.797	−0.449204	−0.224602	+	0.974451i	$0.572108\pi$
$822$	− 371.140i	− 0.451509i
$823$	−516.245	−0.627273	−0.313636	−	0.949543i	$-0.601547\pi$
$823$	−516.245	−0.627273	−0.313636	+	0.949543i	$0.601547\pi$
$824$	214.807i	0.260688i
$825$	156.491i	0.189687i
$826$	0	0
$827$	−534.206	−0.645957	−0.322978	−	0.946406i	$-0.604684\pi$
$827$	−534.206	−0.645957	−0.322978	+	0.946406i	$0.604684\pi$
$828$	161.926	0.195562
$829$	799.884i	0.964879i	0.875929	+	0.482439i	$0.160249\pi$
$829$	799.884i	0.964879i	−0.875929	+	0.482439i	$0.839751\pi$
$830$	−157.881	−0.190218
$831$	− 193.525i	− 0.232882i
$832$	149.283i	0.179427i
$833$	0	0
$834$	371.872	0.445890
$835$	−191.689	−0.229568
$836$	1230.38i	1.47175i
$837$	−145.193	−0.173468
$838$	60.9345i	0.0727142i
$839$	− 1250.09i	− 1.48998i	−0.667078	−	0.744988i	$-0.732455\pi$
$839$	− 1250.09i	− 1.48998i	0.667078	−	0.744988i	$-0.267545\pi$
$840$	0	0
$841$	−569.863	−0.677601
$842$	191.727	0.227704
$843$	− 326.142i	− 0.386883i
$844$	−584.406	−0.692425
$845$	− 400.727i	− 0.474234i
$846$	− 119.816i	− 0.141626i
$847$	0	0
$848$	1.77473	0.00209284
$849$	−375.538	−0.442330
$850$	− 10.8971i	− 0.0128201i
$851$	−1394.47	−1.63862
$852$	− 158.533i	− 0.186072i
$853$	− 427.261i	− 0.500893i	−0.968130	−	0.250446i	$-0.919423\pi$
$853$	− 427.261i	− 0.500893i	0.968130	−	0.250446i	$-0.0805774\pi$
$854$	0	0
$855$	−228.379	−0.267110
$856$	−303.278	−0.354297
$857$	421.205i	0.491488i	0.969335	+	0.245744i	$0.0790322\pi$
$857$	421.205i	0.491488i	−0.969335	+	0.245744i	$0.920968\pi$
$858$	825.956	0.962652
$859$	− 769.636i	− 0.895967i	−0.894042	−	0.447983i	$-0.852142\pi$
$859$	− 769.636i	− 0.895967i	0.894042	−	0.447983i	$-0.147858\pi$
$860$	− 284.851i	− 0.331222i
$861$	0	0
$862$	270.435	0.313730
$863$	−1518.41	−1.75945	−0.879726	−	0.475481i	$-0.842274\pi$
$863$	−1518.41	−1.75945	−0.879726	+	0.475481i	$0.842274\pi$
$864$	− 29.3939i	− 0.0340207i
$865$	−43.5508	−0.0503478
$866$	− 480.585i	− 0.554948i
$867$	496.449i	0.572606i
$868$	0	0
$869$	2406.90	2.76973
$870$	90.1893	0.103666
$871$	− 237.152i	− 0.272276i
$872$	228.793	0.262377
$873$	− 451.128i	− 0.516756i
$874$	1299.36i	1.48668i
$875$	0	0
$876$	−126.185	−0.144047
$877$	−1605.83	−1.83105	−0.915525	−	0.402261i	$-0.868225\pi$
$877$	−1605.83	−1.83105	−0.915525	+	0.402261i	$0.868225\pi$
$878$	− 467.061i	− 0.531960i
$879$	−99.3809	−0.113061
$880$	− 161.624i	− 0.183663i
$881$	538.120i	0.610806i	0.952223	+	0.305403i	$0.0987912\pi$
$881$	538.120i	0.610806i	−0.952223	+	0.305403i	$0.901209\pi$
$882$	0	0
$883$	880.262	0.996899	0.498450	−	0.866919i	$-0.333903\pi$
$883$	880.262	0.996899	0.498450	+	0.866919i	$0.333903\pi$
$884$	−57.5146	−0.0650617
$885$	− 287.227i	− 0.324551i
$886$	−794.311	−0.896513
$887$	− 804.812i	− 0.907342i	−0.891169	−	0.453671i	$-0.850114\pi$
$887$	− 804.812i	− 0.907342i	0.891169	−	0.453671i	$-0.149886\pi$
$888$	253.133i	0.285060i
$889$	0	0
$890$	313.711	0.352484
$891$	−162.631	−0.182526
$892$	− 542.609i	− 0.608306i
$893$	961.451	1.07665
$894$	81.3082i	0.0909488i
$895$	565.354i	0.631681i
$896$	0	0
$897$	872.261	0.972420
$898$	−546.239	−0.608284
$899$	460.105i	0.511797i
$900$	30.0000	0.0333333
$901$	0.683751i	0 0.000758880i
$902$	957.846i	1.06191i
$903$	0	0
$904$	−226.009	−0.250009
$905$	−322.496	−0.356349
$906$	− 22.7880i	− 0.0251523i
$907$	1066.33	1.17567	0.587836	−	0.808980i	$-0.299980\pi$
$907$	1066.33	1.17567	0.587836	+	0.808980i	$0.299980\pi$
$908$	297.218i	0.327332i
$909$	− 141.240i	− 0.155380i
$910$	0	0
$911$	−1052.95	−1.15582	−0.577911	−	0.816100i	$-0.696132\pi$
$911$	−1052.95	−1.15582	−0.577911	+	0.816100i	$0.696132\pi$
$912$	235.869	0.258628
$913$	− 902.175i	− 0.988143i
$914$	705.785	0.772194
$915$	407.163i	0.444986i
$916$	− 50.6669i	− 0.0553132i
$917$	0	0
$918$	11.3246	0.0123362
$919$	−1428.70	−1.55462	−0.777312	−	0.629115i	$-0.783417\pi$
$919$	−1428.70	−1.55462	−0.777312	+	0.629115i	$0.783417\pi$
$920$	− 170.685i	− 0.185527i
$921$	−504.500	−0.547774
$922$	− 874.393i	− 0.948366i
$923$	− 853.987i	− 0.925230i
$924$	0	0
$925$	−258.353	−0.279300
$926$	−274.970	−0.296943
$927$	− 227.837i	− 0.245779i
$928$	−93.1471	−0.100374
$929$	1759.33i	1.89379i	0.321548	+	0.946893i	$0.395797\pi$
$929$	1759.33i	1.89379i	−0.321548	+	0.946893i	$0.604203\pi$
$930$	153.046i	0.164566i
$931$	0	0
$932$	−817.772	−0.877438
$933$	−432.736	−0.463812
$934$	− 214.656i	− 0.229824i
$935$	62.2690	0.0665979
$936$	− 158.339i	− 0.169165i
$937$	− 1264.00i	− 1.34898i	−0.738284	−	0.674490i	$-0.764363\pi$
$937$	− 1264.00i	− 1.34898i	0.738284	−	0.674490i	$-0.235637\pi$
$938$	0	0
$939$	−6.55664	−0.00698258
$940$	−126.297	−0.134358
$941$	264.006i	0.280559i	0.990112	+	0.140280i	$0.0448002\pi$
$941$	264.006i	0.280559i	−0.990112	+	0.140280i	$0.955200\pi$
$942$	58.7594	0.0623773
$943$	1011.55i	1.07269i
$944$	296.647i	0.314245i
$945$	0	0
$946$	1627.71	1.72063
$947$	−69.9848	−0.0739016	−0.0369508	−	0.999317i	$-0.511764\pi$
$947$	−69.9848	−0.0739016	−0.0369508	+	0.999317i	$0.511764\pi$
$948$	− 461.412i	− 0.486721i
$949$	−679.734	−0.716263
$950$	240.733i	0.253403i
$951$	− 123.601i	− 0.129969i
$952$	0	0
$953$	−1410.53	−1.48009	−0.740047	−	0.672555i	$-0.765197\pi$
$953$	−1410.53	−1.48009	−0.740047	+	0.672555i	$0.765197\pi$
$954$	−1.88238	−0.00197315
$955$	− 702.344i	− 0.735439i
$956$	134.070	0.140241
$957$	515.365i	0.538522i
$958$	317.872i	0.331808i
$959$	0	0
$960$	−30.9839	−0.0322749
$961$	180.227	0.187541
$962$	1363.58i	1.41744i
$963$	321.675	0.334035
$964$	473.758i	0.491451i
$965$	118.128i	0.122412i
$966$	0	0
$967$	−1581.56	−1.63554	−0.817768	−	0.575548i	$-0.804789\pi$
$967$	−1581.56	−1.63554	−0.817768	+	0.575548i	$0.804789\pi$
$968$	581.320	0.600537
$969$	90.8736i	0.0937808i
$970$	−475.531	−0.490238
$971$	598.490i	0.616365i	0.951327	+	0.308182i	$0.0997207\pi$
$971$	598.490i	0.616365i	−0.951327	+	0.308182i	$0.900279\pi$
$972$	31.1769i	0.0320750i
$973$	0	0
$974$	98.3767	0.101003
$975$	161.604	0.165748
$976$	− 420.516i	− 0.430856i
$977$	−172.779	−0.176847	−0.0884235	−	0.996083i	$-0.528183\pi$
$977$	−172.779	−0.176847	−0.0884235	+	0.996083i	$0.528183\pi$
$978$	− 620.033i	− 0.633980i
$979$	1792.63i	1.83108i
$980$	0	0
$981$	−242.671	−0.247371
$982$	−1061.30	−1.08076
$983$	− 216.413i	− 0.220155i	−0.993923	−	0.110078i	$-0.964890\pi$
$983$	− 216.413i	− 0.220155i	0.993923	−	0.110078i	$-0.0351100\pi$
$984$	183.623	0.186608
$985$	121.196i	0.123042i
$986$	− 35.8869i	− 0.0363965i
$987$	0	0
$988$	1270.58	1.28601
$989$	1718.97	1.73809
$990$	171.428i	0.173159i
$991$	−321.987	−0.324911	−0.162455	−	0.986716i	$-0.551941\pi$
$991$	−321.987	−0.324911	−0.162455	+	0.986716i	$0.551941\pi$
$992$	− 158.066i	− 0.159340i
$993$	− 592.303i	− 0.596479i
$994$	0	0
$995$	−51.3646	−0.0516227
$996$	−172.950	−0.173645
$997$	− 46.8126i	− 0.0469535i	−0.999724	−	0.0234767i	$-0.992526\pi$
$997$	− 46.8126i	− 0.0469535i	0.999724	−	0.0234767i	$-0.00747356\pi$
$998$	226.514	0.226968
$999$	− 268.488i	− 0.268757i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1470.3.f.a.391.8		8
7.4	even	3		210.3.o.a.61.1	yes	8
7.5	odd	6		210.3.o.a.31.1	✓	8
7.6	odd	2	inner	1470.3.f.a.391.5		8
21.5	even	6		630.3.v.b.451.4		8
21.11	odd	6		630.3.v.b.271.4		8
35.4	even	6		1050.3.p.b.901.4		8
35.12	even	12		1050.3.q.c.199.8		16
35.18	odd	12		1050.3.q.c.649.8		16
35.19	odd	6		1050.3.p.b.451.4		8
35.32	odd	12		1050.3.q.c.649.1		16
35.33	even	12		1050.3.q.c.199.1		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
210.3.o.a.31.1	✓	8	7.5	odd	6
210.3.o.a.61.1	yes	8	7.4	even	3
630.3.v.b.271.4		8	21.11	odd	6
630.3.v.b.451.4		8	21.5	even	6
1050.3.p.b.451.4		8	35.19	odd	6
1050.3.p.b.901.4		8	35.4	even	6
1050.3.q.c.199.1		16	35.33	even	12
1050.3.q.c.199.8		16	35.12	even	12
1050.3.q.c.649.1		16	35.32	odd	12
1050.3.q.c.649.8		16	35.18	odd	12
1470.3.f.a.391.5		8	7.6	odd	2	inner
1470.3.f.a.391.8		8	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 1470 = 2 \cdot 3 \cdot 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1470.f (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.41421 q^{2} +1.73205i q^{3} +2.00000 q^{4} +2.23607i q^{5} +2.44949i q^{6} +2.82843 q^{8} -3.00000 q^{9} +O(q^{10})\)	\(q+1.41421 q^{2} +1.73205i q^{3} +2.00000 q^{4} +2.23607i q^{5} +2.44949i q^{6} +2.82843 q^{8} -3.00000 q^{9} +3.16228i q^{10} -18.0701 q^{11} +3.46410i q^{12} +18.6604i q^{13} -3.87298 q^{15} +4.00000 q^{16} +1.54109i q^{17} -4.24264 q^{18} -34.0447i q^{19} +4.47214i q^{20} -25.5549 q^{22} -26.9876 q^{23} +4.89898i q^{24} -5.00000 q^{25} +26.3898i q^{26} -5.19615i q^{27} -16.4662 q^{29} -5.47723 q^{30} -27.9423i q^{31} +5.65685 q^{32} -31.2983i q^{33} +2.17942i q^{34} -6.00000 q^{36} +51.6706 q^{37} -48.1465i q^{38} -32.3208 q^{39} +6.32456i q^{40} -37.4818i q^{41} -63.6947 q^{43} -36.1402 q^{44} -6.70820i q^{45} -38.1663 q^{46} +28.2408i q^{47} +6.92820i q^{48} -7.07107 q^{50} -2.66924 q^{51} +37.3208i q^{52} +0.443681 q^{53} -7.34847i q^{54} -40.4059i q^{55} +58.9672 q^{57} -23.2868 q^{58} +74.1618i q^{59} -7.74597 q^{60} -105.129i q^{61} -39.5164i q^{62} +8.00000 q^{64} -41.7259 q^{65} -44.2625i q^{66} -12.7088 q^{67} +3.08217i q^{68} -46.7439i q^{69} -45.7647 q^{71} -8.48528 q^{72} +36.4265i q^{73} +73.0732 q^{74} -8.66025i q^{75} -68.0895i q^{76} -45.7085 q^{78} -133.198 q^{79} +8.94427i q^{80} +9.00000 q^{81} -53.0073i q^{82} +49.9265i q^{83} -3.44597 q^{85} -90.0779 q^{86} -28.5204i q^{87} -51.1099 q^{88} -99.2041i q^{89} -9.48683i q^{90} -53.9752 q^{92} +48.3975 q^{93} +39.9385i q^{94} +76.1264 q^{95} +9.79796i q^{96} +150.376i q^{97} +54.2102 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 16 q^{4} - 24 q^{9}+O(q^{10}) \) 8 * q + 16 * q^4 - 24 * q^9	\( 8 q + 16 q^{4} - 24 q^{9} + 8 q^{11} + 32 q^{16} - 48 q^{22} - 24 q^{23} - 40 q^{25} + 72 q^{29} - 48 q^{36} + 192 q^{37} - 48 q^{39} - 112 q^{43} + 16 q^{44} - 16 q^{46} - 168 q^{51} - 64 q^{53} + 216 q^{57} - 208 q^{58} + 64 q^{64} - 40 q^{65} + 240 q^{67} + 8 q^{71} + 32 q^{74} - 192 q^{78} - 24 q^{79} + 72 q^{81} + 120 q^{85} + 80 q^{86} - 96 q^{88} - 48 q^{92} + 264 q^{93} + 80 q^{95} - 24 q^{99}+O(q^{100}) \) 8 * q + 16 * q^4 - 24 * q^9 + 8 * q^11 + 32 * q^16 - 48 * q^22 - 24 * q^23 - 40 * q^25 + 72 * q^29 - 48 * q^36 + 192 * q^37 - 48 * q^39 - 112 * q^43 + 16 * q^44 - 16 * q^46 - 168 * q^51 - 64 * q^53 + 216 * q^57 - 208 * q^58 + 64 * q^64 - 40 * q^65 + 240 * q^67 + 8 * q^71 + 32 * q^74 - 192 * q^78 - 24 * q^79 + 72 * q^81 + 120 * q^85 + 80 * q^86 - 96 * q^88 - 48 * q^92 + 264 * q^93 + 80 * q^95 - 24 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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