LMFDB - Embedded newform 325.6.b.d.274.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [325,6,Mod(274,325)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(325, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("325.274");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 325 = 5^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	325.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:	$52.1247414392$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} + 85x^{4} + 1668x^{2} + 2304 $ x^6 + 85x^4 + 1668x^2 + 2304
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 65)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			274.5
Root			$5.25457i$ of defining polynomial
Character	$\chi$	$=$	325.274
Dual form			325.6.b.d.274.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+6.25457i q^{2} -21.3742i q^{3} -7.11967 q^{4} +133.687 q^{6} -116.319i q^{7} +155.616i q^{8} -213.858 q^{9} -259.239 q^{11} +152.178i q^{12} +169.000i q^{13} +727.523 q^{14} -1201.14 q^{16} -876.601i q^{17} -1337.59i q^{18} +921.477 q^{19} -2486.22 q^{21} -1621.43i q^{22} -2050.72i q^{23} +3326.17 q^{24} -1057.02 q^{26} -622.884i q^{27} +828.150i q^{28} -781.288 q^{29} -5802.03 q^{31} -2532.91i q^{32} +5541.03i q^{33} +5482.76 q^{34} +1522.60 q^{36} +2818.28i q^{37} +5763.44i q^{38} +3612.25 q^{39} -5408.91 q^{41} -15550.3i q^{42} +5189.97i q^{43} +1845.69 q^{44} +12826.4 q^{46} +9661.60i q^{47} +25673.5i q^{48} +3276.98 q^{49} -18736.7 q^{51} -1203.22i q^{52} -763.033i q^{53} +3895.87 q^{54} +18101.0 q^{56} -19695.9i q^{57} -4886.62i q^{58} -44477.7 q^{59} -50834.3 q^{61} -36289.2i q^{62} +24875.7i q^{63} -22594.2 q^{64} -34656.8 q^{66} +67343.8i q^{67} +6241.11i q^{68} -43832.6 q^{69} -79516.3 q^{71} -33279.7i q^{72} -55128.4i q^{73} -17627.2 q^{74} -6560.61 q^{76} +30154.3i q^{77} +22593.1i q^{78} +66467.3 q^{79} -65281.2 q^{81} -33830.4i q^{82} +39771.5i q^{83} +17701.1 q^{84} -32461.0 q^{86} +16699.4i q^{87} -40341.6i q^{88} -59807.6 q^{89} +19657.8 q^{91} +14600.4i q^{92} +124014. i q^{93} -60429.2 q^{94} -54139.0 q^{96} -41624.9i q^{97} +20496.1i q^{98} +55440.3 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 20 q^{4} + 44 q^{6} - 286 q^{9} - 1232 q^{11} + 2568 q^{14} - 3740 q^{16} + 6720 q^{19} - 7584 q^{21} - 1044 q^{24} - 676 q^{26} - 4612 q^{29} - 12760 q^{31} + 5720 q^{34} - 1372 q^{36} + 5408 q^{39}+ \cdots + 138896 q^{99}+O(q^{100}) $ 6 * q + 20 * q^4 + 44 * q^6 - 286 * q^9 - 1232 * q^11 + 2568 * q^14 - 3740 * q^16 + 6720 * q^19 - 7584 * q^21 - 1044 * q^24 - 676 * q^26 - 4612 * q^29 - 12760 * q^31 + 5720 * q^34 - 1372 * q^36 + 5408 * q^39 + 16572 * q^41 - 15380 * q^44 + 28372 * q^46 - 32342 * q^49 - 86656 * q^51 - 60008 * q^54 + 31560 * q^56 - 9504 * q^59 - 173788 * q^61 - 67236 * q^64 - 42376 * q^66 + 17776 * q^69 - 246504 * q^71 - 187776 * q^74 - 51820 * q^76 + 221392 * q^79 - 304154 * q^81 - 159888 * q^84 - 237932 * q^86 + 224420 * q^89 - 70304 * q^91 - 255032 * q^94 - 212876 * q^96 + 138896 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/325\mathbb{Z}\right)^\times$.

$n$	$27$	$301$
$\chi(n)$	$-1$	$1$

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 $

$2$	6.25457i	1.10566i	0.833293	+	0.552831i	$0.186453\pi$
$2$	6.25457i	1.10566i	−0.833293	+	0.552831i	$0.813547\pi$
$3$	− 21.3742i	− 1.37116i	−0.727998	−	0.685579i	$-0.759549\pi$
$3$	− 21.3742i	− 1.37116i	0.727998	−	0.685579i	$-0.240451\pi$
$4$	−7.11967	−0.222490
$5$	0	0
$5$	0	0
$6$	133.687	1.51604
$7$	− 116.319i	− 0.897231i	−0.893725	−	0.448615i	$-0.851917\pi$
$7$	− 116.319i	− 0.897231i	0.893725	−	0.448615i	$-0.148083\pi$
$8$	155.616i	0.859664i
$9$	−213.858	−0.880075
$10$	0	0
$11$	−259.239	−0.645978	−0.322989	−	0.946403i	$-0.604688\pi$
$11$	−259.239	−0.645978	−0.322989	+	0.946403i	$0.604688\pi$
$12$	152.178i	0.305069i
$13$	169.000i	0.277350i
$13$	169.000i	0.277350i
$14$	727.523	0.992034
$15$	0	0
$16$	−1201.14	−1.17299
$17$	− 876.601i	− 0.735664i	−0.929892	−	0.367832i	$-0.880100\pi$
$17$	− 876.601i	− 0.735664i	0.929892	−	0.367832i	$-0.119900\pi$
$18$	− 1337.59i	− 0.973066i
$19$	921.477	0.585599	0.292800	−	0.956174i	$-0.405413\pi$
$19$	921.477	0.585599	0.292800	+	0.956174i	$0.405413\pi$
$20$	0	0
$21$	−2486.22	−1.23025
$22$	− 1621.43i	− 0.714234i
$23$	− 2050.72i	− 0.808326i	−0.914687	−	0.404163i	$-0.867563\pi$
$23$	− 2050.72i	− 0.808326i	0.914687	−	0.404163i	$-0.132437\pi$
$24$	3326.17	1.17874
$25$	0	0
$26$	−1057.02	−0.306656
$27$	− 622.884i	− 0.164436i
$28$	828.150i	0.199625i
$29$	−781.288	−0.172511	−0.0862554	−	0.996273i	$-0.527490\pi$
$29$	−781.288	−0.172511	−0.0862554	+	0.996273i	$0.527490\pi$
$30$	0	0
$31$	−5802.03	−1.08437	−0.542183	−	0.840260i	$-0.682402\pi$
$31$	−5802.03	−1.08437	−0.542183	+	0.840260i	$0.682402\pi$
$32$	− 2532.91i	− 0.437265i
$33$	5541.03i	0.885738i
$34$	5482.76	0.813396
$35$	0	0
$36$	1522.60	0.195808
$37$	2818.28i	0.338439i	0.985578	+	0.169220i	$0.0541247\pi$
$37$	2818.28i	0.338439i	−0.985578	+	0.169220i	$0.945875\pi$
$38$	5763.44i	0.647475i
$39$	3612.25	0.380291
$40$	0	0
$41$	−5408.91	−0.502516	−0.251258	−	0.967920i	$-0.580844\pi$
$41$	−5408.91	−0.502516	−0.251258	+	0.967920i	$0.580844\pi$
$42$	− 15550.3i	− 1.36024i
$43$	5189.97i	0.428049i	0.976828	+	0.214025i	$0.0686573\pi$
$43$	5189.97i	0.428049i	−0.976828	+	0.214025i	$0.931343\pi$
$44$	1845.69	0.143723
$45$	0	0
$46$	12826.4	0.893736
$47$	9661.60i	0.637976i	0.947759	+	0.318988i	$0.103343\pi$
$47$	9661.60i	0.637976i	−0.947759	+	0.318988i	$0.896657\pi$
$48$	25673.5i	1.60835i
$49$	3276.98	0.194977
$50$	0	0
$51$	−18736.7	−1.00871
$52$	− 1203.22i	− 0.0617075i
$53$	− 763.033i	− 0.0373125i	−0.999826	−	0.0186562i	$-0.994061\pi$
$53$	− 763.033i	− 0.0373125i	0.999826	−	0.0186562i	$-0.00593881\pi$
$54$	3895.87	0.181811
$55$	0	0
$56$	18101.0	0.771317
$57$	− 19695.9i	− 0.802949i
$58$	− 4886.62i	− 0.190739i
$59$	−44477.7	−1.66346	−0.831731	−	0.555179i	$-0.812650\pi$
$59$	−44477.7	−1.66346	−0.831731	+	0.555179i	$0.812650\pi$
$60$	0	0
$61$	−50834.3	−1.74917	−0.874585	−	0.484872i	$-0.838866\pi$
$61$	−50834.3	−1.74917	−0.874585	+	0.484872i	$0.838866\pi$
$62$	− 36289.2i	− 1.19894i
$63$	24875.7i	0.789630i
$64$	−22594.2	−0.689521
$65$	0	0
$66$	−34656.8	−0.979328
$67$	67343.8i	1.83278i	0.400285	+	0.916391i	$0.368911\pi$
$67$	67343.8i	1.83278i	−0.400285	+	0.916391i	$0.631089\pi$
$68$	6241.11i	0.163678i
$69$	−43832.6	−1.10834
$70$	0	0
$71$	−79516.3	−1.87202	−0.936009	−	0.351976i	$-0.885510\pi$
$71$	−79516.3	−1.87202	−0.936009	+	0.351976i	$0.885510\pi$
$72$	− 33279.7i	− 0.756569i
$73$	− 55128.4i	− 1.21079i	−0.795926	−	0.605394i	$-0.793016\pi$
$73$	− 55128.4i	− 1.21079i	0.795926	−	0.605394i	$-0.206984\pi$
$74$	−17627.2	−0.374199
$75$	0	0
$76$	−6560.61	−0.130290
$77$	30154.3i	0.579591i
$78$	22593.1i	0.420473i
$79$	66467.3	1.19823	0.599115	−	0.800663i	$-0.295519\pi$
$79$	66467.3	1.19823	0.599115	+	0.800663i	$0.295519\pi$
$80$	0	0
$81$	−65281.2	−1.10554
$82$	− 33830.4i	− 0.555614i
$83$	39771.5i	0.633690i	0.948477	+	0.316845i	$0.102624\pi$
$83$	39771.5i	0.633690i	−0.948477	+	0.316845i	$0.897376\pi$
$84$	17701.1	0.273717
$85$	0	0
$86$	−32461.0	−0.473278
$87$	16699.4i	0.236540i
$88$	− 40341.6i	− 0.555324i
$89$	−59807.6	−0.800352	−0.400176	−	0.916438i	$-0.631051\pi$
$89$	−59807.6	−0.800352	−0.400176	+	0.916438i	$0.631051\pi$
$90$	0	0
$91$	19657.8	0.248847
$92$	14600.4i	0.179844i
$93$	124014.i	1.48684i
$94$	−60429.2	−0.705386
$95$	0	0
$96$	−54139.0	−0.599559
$97$	− 41624.9i	− 0.449183i	−0.974453	−	0.224592i	$-0.927895\pi$
$97$	− 41624.9i	− 0.449183i	0.974453	−	0.224592i	$-0.0721048\pi$
$98$	20496.1i	0.215579i
$99$	55440.3	0.568509
$100$	0	0
$101$	160951.	1.56996	0.784982	−	0.619519i	$-0.212672\pi$
$101$	160951.	1.56996	0.784982	+	0.619519i	$0.212672\pi$
$102$	− 117190.i	− 1.11529i
$103$	− 167554.i	− 1.55618i	−0.628150	−	0.778092i	$-0.716188\pi$
$103$	− 167554.i	− 1.55618i	0.628150	−	0.778092i	$-0.283812\pi$
$104$	−26299.1	−0.238428
$105$	0	0
$106$	4772.45	0.0412550
$107$	58033.6i	0.490027i	0.969520	+	0.245013i	$0.0787924\pi$
$107$	58033.6i	0.490027i	−0.969520	+	0.245013i	$0.921208\pi$
$108$	4434.73i	0.0365854i
$109$	−133082.	−1.07288	−0.536442	−	0.843937i	$-0.680232\pi$
$109$	−133082.	−1.07288	−0.536442	+	0.843937i	$0.680232\pi$
$110$	0	0
$111$	60238.7	0.464053
$112$	139715.i	1.05244i
$113$	148757.i	1.09593i	0.836503	+	0.547963i	$0.184596\pi$
$113$	148757.i	1.09593i	−0.836503	+	0.547963i	$0.815404\pi$
$114$	123189.	0.887791
$115$	0	0
$116$	5562.51	0.0383819
$117$	− 36142.0i	− 0.244089i
$118$	− 278189.i	− 1.83923i
$119$	−101965.	−0.660060
$120$	0	0
$121$	−93846.4	−0.582712
$122$	− 317947.i	− 1.93399i
$123$	115611.i	0.689029i
$124$	41308.6	0.241260
$125$	0	0
$126$	−155587.	−0.873064
$127$	− 133292.i	− 0.733324i	−0.930354	−	0.366662i	$-0.880500\pi$
$127$	− 133292.i	− 0.733324i	0.930354	−	0.366662i	$-0.119500\pi$
$128$	− 222370.i	− 1.19964i
$129$	110932.	0.586923
$130$	0	0
$131$	−140851.	−0.717102	−0.358551	−	0.933510i	$-0.616729\pi$
$131$	−140851.	−0.717102	−0.358551	+	0.933510i	$0.616729\pi$
$132$	− 39450.3i	− 0.197068i
$133$	− 107185.i	− 0.525417i
$134$	−421207.	−2.02644
$135$	0	0
$136$	136413.	0.632424
$137$	− 251353.i	− 1.14415i	−0.820202	−	0.572074i	$-0.806139\pi$
$137$	− 251353.i	− 1.14415i	0.820202	−	0.572074i	$-0.193861\pi$
$138$	− 274154.i	− 1.22545i
$139$	284722.	1.24992	0.624962	−	0.780655i	$-0.285115\pi$
$139$	284722.	1.24992	0.624962	+	0.780655i	$0.285115\pi$
$140$	0	0
$141$	206509.	0.874766
$142$	− 497340.i	− 2.06982i
$143$	− 43811.3i	− 0.179162i
$144$	256874.	1.03232
$145$	0	0
$146$	344804.	1.33872
$147$	− 70043.0i	− 0.267345i
$148$	− 20065.3i	− 0.0752992i
$149$	−34484.7	−0.127251	−0.0636254	−	0.997974i	$-0.520266\pi$
$149$	−34484.7	−0.127251	−0.0636254	+	0.997974i	$0.520266\pi$
$150$	0	0
$151$	−330461.	−1.17945	−0.589723	−	0.807606i	$-0.700763\pi$
$151$	−330461.	−1.17945	−0.589723	+	0.807606i	$0.700763\pi$
$152$	143396.i	0.503418i
$153$	187468.i	0.647440i
$154$	−188602.	−0.640832
$155$	0	0
$156$	−25718.0	−0.0846108
$157$	− 384612.i	− 1.24530i	−0.782501	−	0.622650i	$-0.786056\pi$
$157$	− 384612.i	− 1.24530i	0.782501	−	0.622650i	$-0.213944\pi$
$158$	415724.i	1.32484i
$159$	−16309.3	−0.0511613
$160$	0	0
$161$	−238537.	−0.725255
$162$	− 408306.i	− 1.22236i
$163$	190142.i	0.560543i	0.959921	+	0.280272i	$0.0904246\pi$
$163$	190142.i	0.560543i	−0.959921	+	0.280272i	$0.909575\pi$
$164$	38509.7	0.111805
$165$	0	0
$166$	−248754.	−0.700648
$167$	327019.i	0.907366i	0.891163	+	0.453683i	$0.149890\pi$
$167$	327019.i	0.907366i	−0.891163	+	0.453683i	$0.850110\pi$
$168$	− 386895.i	− 1.05760i
$169$	−28561.0	−0.0769231
$170$	0	0
$171$	−197065.	−0.515371
$172$	− 36950.9i	− 0.0952365i
$173$	− 104123.i	− 0.264504i	−0.991216	−	0.132252i	$-0.957779\pi$
$173$	− 104123.i	− 0.264504i	0.991216	−	0.132252i	$-0.0422209\pi$
$174$	−104448.	−0.261533
$175$	0	0
$176$	311382.	0.757725
$177$	950678.i	2.28087i
$178$	− 374071.i	− 0.884919i
$179$	216503.	0.505047	0.252523	−	0.967591i	$-0.418740\pi$
$179$	216503.	0.505047	0.252523	+	0.967591i	$0.418740\pi$
$180$	0	0
$181$	−46482.5	−0.105461	−0.0527306	−	0.998609i	$-0.516792\pi$
$181$	−46482.5	−0.105461	−0.0527306	+	0.998609i	$0.516792\pi$
$182$	122951.i	0.275141i
$183$	1.08654e6i	2.39839i
$184$	319124.	0.694889
$185$	0	0
$186$	−775655.	−1.64394
$187$	227249.i	0.475223i
$188$	− 68787.4i	− 0.141943i
$189$	−72453.0	−0.147537
$190$	0	0
$191$	−854719.	−1.69527	−0.847637	−	0.530576i	$-0.821975\pi$
$191$	−854719.	−1.69527	−0.847637	+	0.530576i	$0.821975\pi$
$192$	482934.i	0.945442i
$193$	− 869527.i	− 1.68031i	−0.542345	−	0.840156i	$-0.682464\pi$
$193$	− 869527.i	− 1.68031i	0.542345	−	0.840156i	$-0.317536\pi$
$194$	260346.	0.496645
$195$	0	0
$196$	−23331.0	−0.0433804
$197$	674594.i	1.23845i	0.785215	+	0.619223i	$0.212552\pi$
$197$	674594.i	1.23845i	−0.785215	+	0.619223i	$0.787448\pi$
$198$	346755.i	0.628579i
$199$	630277.	1.12823	0.564117	−	0.825695i	$-0.309217\pi$
$199$	630277.	1.12823	0.564117	+	0.825695i	$0.309217\pi$
$200$	0	0
$201$	1.43942e6	2.51303
$202$	1.00668e6i	1.73585i
$203$	90878.3i	0.154782i
$204$	133399.	0.224428
$205$	0	0
$206$	1.04798e6	1.72062
$207$	438563.i	0.711388i
$208$	− 202993.i	− 0.325328i
$209$	−238882.	−0.378284
$210$	0	0
$211$	1.19496e6	1.84777	0.923885	−	0.382671i	$-0.124996\pi$
$211$	1.19496e6	1.84777	0.923885	+	0.382671i	$0.124996\pi$
$212$	5432.55i	0.00830164i
$213$	1.69960e6i	2.56683i
$214$	−362975.	−0.541804
$215$	0	0
$216$	96930.6	0.141360
$217$	674885.i	0.972927i
$218$	− 832371.i	− 1.18625i
$219$	−1.17833e6	−1.66018
$220$	0	0
$221$	148146.	0.204037
$222$	376767.i	0.513087i
$223$	1.26371e6i	1.70171i	0.525403	+	0.850854i	$0.323915\pi$
$223$	1.26371e6i	1.70171i	−0.525403	+	0.850854i	$0.676085\pi$
$224$	−294624.	−0.392327
$225$	0	0
$226$	−930411.	−1.21172
$227$	389798.i	0.502082i	0.967976	+	0.251041i	$0.0807729\pi$
$227$	389798.i	0.502082i	−0.967976	+	0.251041i	$0.919227\pi$
$228$	140228.i	0.178648i
$229$	67870.2	0.0855245	0.0427623	−	0.999085i	$-0.486384\pi$
$229$	67870.2	0.0855245	0.0427623	+	0.999085i	$0.486384\pi$
$230$	0	0
$231$	644524.	0.794711
$232$	− 121581.i	− 0.148301i
$233$	− 204180.i	− 0.246390i	−0.992382	−	0.123195i	$-0.960686\pi$
$233$	− 204180.i	− 0.246390i	0.992382	−	0.123195i	$-0.0393140\pi$
$234$	226053.	0.269880
$235$	0	0
$236$	316667.	0.370103
$237$	− 1.42069e6i	− 1.64296i
$238$	− 637747.i	− 0.729804i
$239$	−1.35649e6	−1.53611	−0.768056	−	0.640383i	$-0.778776\pi$
$239$	−1.35649e6	−1.53611	−0.768056	+	0.640383i	$0.778776\pi$
$240$	0	0
$241$	230905.	0.256089	0.128044	−	0.991768i	$-0.459130\pi$
$241$	230905.	0.256089	0.128044	+	0.991768i	$0.459130\pi$
$242$	− 586969.i	− 0.644283i
$243$	1.24398e6i	1.35144i
$244$	361923.	0.389172
$245$	0	0
$246$	−723100.	−0.761834
$247$	155730.i	0.162416i
$248$	− 902888.i	− 0.932191i
$249$	850086.	0.868890
$250$	0	0
$251$	−1.20774e6	−1.21001	−0.605003	−	0.796223i	$-0.706828\pi$
$251$	−1.20774e6	−1.21001	−0.605003	+	0.796223i	$0.706828\pi$
$252$	− 177107.i	− 0.175685i
$253$	531626.i	0.522161i
$254$	833687.	0.810809
$255$	0	0
$256$	667816.	0.636879
$257$	− 675820.i	− 0.638261i	−0.947711	−	0.319131i	$-0.896609\pi$
$257$	− 675820.i	− 0.638261i	0.947711	−	0.319131i	$-0.103391\pi$
$258$	693830.i	0.648939i
$259$	327819.	0.303658
$260$	0	0
$261$	167085.	0.151822
$262$	− 880961.i	− 0.792873i
$263$	− 377744.i	− 0.336751i	−0.985723	−	0.168375i	$-0.946148\pi$
$263$	− 377744.i	− 0.336751i	0.985723	−	0.168375i	$-0.0538521\pi$
$264$	−862271.	−0.761437
$265$	0	0
$266$	670396.	0.580934
$267$	1.27834e6i	1.09741i
$268$	− 479466.i	− 0.407775i
$269$	−82864.5	−0.0698213	−0.0349106	−	0.999390i	$-0.511115\pi$
$269$	−82864.5	−0.0698213	−0.0349106	+	0.999390i	$0.511115\pi$
$270$	0	0
$271$	315805.	0.261214	0.130607	−	0.991434i	$-0.458307\pi$
$271$	315805.	0.261214	0.130607	+	0.991434i	$0.458307\pi$
$272$	1.05292e6i	0.862925i
$273$	− 420172.i	− 0.341209i
$274$	1.57210e6	1.26504
$275$	0	0
$276$	312074.	0.246595
$277$	73722.7i	0.0577300i	0.999583	+	0.0288650i	$0.00918929\pi$
$277$	73722.7i	0.0577300i	−0.999583	+	0.0288650i	$0.990811\pi$
$278$	1.78081e6i	1.38199i
$279$	1.24081e6	0.954324
$280$	0	0
$281$	983448.	0.742995	0.371497	−	0.928434i	$-0.378844\pi$
$281$	983448.	0.742995	0.371497	+	0.928434i	$0.378844\pi$
$282$	1.29163e6i	0.967196i
$283$	− 689549.i	− 0.511798i	−0.966703	−	0.255899i	$-0.917628\pi$
$283$	− 689549.i	− 0.511798i	0.966703	−	0.255899i	$-0.0823715\pi$
$284$	566130.	0.416505
$285$	0	0
$286$	274021.	0.198093
$287$	629157.i	0.450873i
$288$	541683.i	0.384826i
$289$	651428.	0.458798
$290$	0	0
$291$	−889700.	−0.615902
$292$	392496.i	0.269388i
$293$	− 2.26444e6i	− 1.54096i	−0.637462	−	0.770482i	$-0.720016\pi$
$293$	− 2.26444e6i	− 1.54096i	0.637462	−	0.770482i	$-0.279984\pi$
$294$	438089.	0.295593
$295$	0	0
$296$	−438570.	−0.290944
$297$	161475.i	0.106222i
$298$	− 215687.i	− 0.140696i
$299$	346572.	0.224189
$300$	0	0
$301$	603690.	0.384059
$302$	− 2.06689e6i	− 1.30407i
$303$	− 3.44020e6i	− 2.15267i
$304$	−1.10682e6	−0.686901
$305$	0	0
$306$	−1.17253e6	−0.715850
$307$	133545.i	0.0808690i	0.999182	+	0.0404345i	$0.0128742\pi$
$307$	133545.i	0.0808690i	−0.999182	+	0.0404345i	$0.987126\pi$
$308$	− 214688.i	− 0.128953i
$309$	−3.58133e6	−2.13378
$310$	0	0
$311$	2.40605e6	1.41060	0.705299	−	0.708910i	$-0.250813\pi$
$311$	2.40605e6	1.41060	0.705299	+	0.708910i	$0.250813\pi$
$312$	562123.i	0.326922i
$313$	− 1.83885e6i	− 1.06093i	−0.847708	−	0.530463i	$-0.822018\pi$
$313$	− 1.83885e6i	− 1.06093i	0.847708	−	0.530463i	$-0.177982\pi$
$314$	2.40558e6	1.37688
$315$	0	0
$316$	−473225.	−0.266594
$317$	− 3.10514e6i	− 1.73553i	−0.496972	−	0.867767i	$-0.665555\pi$
$317$	− 3.10514e6i	− 1.73553i	0.496972	−	0.867767i	$-0.334445\pi$
$318$	− 102007.i	− 0.0565671i
$319$	202540.	0.111438
$320$	0	0
$321$	1.24042e6	0.671904
$322$	− 1.49195e6i	− 0.801887i
$323$	− 807767.i	− 0.430804i
$324$	464781.	0.245972
$325$	0	0
$326$	−1.18926e6	−0.619772
$327$	2.84453e6i	1.47109i
$328$	− 841712.i	− 0.431995i
$329$	1.12382e6	0.572412
$330$	0	0
$331$	2.20989e6	1.10866	0.554332	−	0.832295i	$-0.312974\pi$
$331$	2.20989e6	1.10866	0.554332	+	0.832295i	$0.312974\pi$
$332$	− 283160.i	− 0.140990i
$333$	− 602713.i	− 0.297852i
$334$	−2.04537e6	−1.00324
$335$	0	0
$336$	2.98630e6	1.44306
$337$	− 2.47122e6i	− 1.18532i	−0.805452	−	0.592661i	$-0.798077\pi$
$337$	− 2.47122e6i	− 1.18532i	0.805452	−	0.592661i	$-0.201923\pi$
$338$	− 178637.i	− 0.0850510i
$339$	3.17957e6	1.50269
$340$	0	0
$341$	1.50411e6	0.700477
$342$	− 1.23256e6i	− 0.569826i
$343$	− 2.33614e6i	− 1.07217i
$344$	−807641.	−0.367978
$345$	0	0
$346$	651247.	0.292453
$347$	1.49298e6i	0.665626i	0.942993	+	0.332813i	$0.107998\pi$
$347$	1.49298e6i	0.665626i	−0.942993	+	0.332813i	$0.892002\pi$
$348$	− 118895.i	− 0.0526276i
$349$	4.00494e6	1.76008	0.880040	−	0.474900i	$-0.157516\pi$
$349$	4.00494e6	1.76008	0.880040	+	0.474900i	$0.157516\pi$
$350$	0	0
$351$	105267.	0.0456064
$352$	656628.i	0.282464i
$353$	− 548627.i	− 0.234337i	−0.993112	−	0.117168i	$-0.962618\pi$
$353$	− 548627.i	− 0.234337i	0.993112	−	0.117168i	$-0.0373817\pi$
$354$	−5.94608e6	−2.52187
$355$	0	0
$356$	425810.	0.178070
$357$	2.17942e6i	0.905047i
$358$	1.35413e6i	0.558412i
$359$	2.00206e6	0.819862	0.409931	−	0.912117i	$-0.365553\pi$
$359$	2.00206e6	0.819862	0.409931	+	0.912117i	$0.365553\pi$
$360$	0	0
$361$	−1.62698e6	−0.657074
$362$	− 290728.i	− 0.116604i
$363$	2.00590e6i	0.798991i
$364$	−139957.	−0.0553659
$365$	0	0
$366$	−6.79587e6	−2.65181
$367$	− 2.30086e6i	− 0.891712i	−0.895105	−	0.445856i	$-0.852899\pi$
$367$	− 2.30086e6i	− 0.891712i	0.895105	−	0.445856i	$-0.147101\pi$
$368$	2.46320e6i	0.948157i
$369$	1.15674e6	0.442252
$370$	0	0
$371$	−88755.0	−0.0334779
$372$	− 882939.i	− 0.330806i
$373$	1.36153e6i	0.506704i	0.967374	+	0.253352i	$0.0815330\pi$
$373$	1.36153e6i	0.506704i	−0.967374	+	0.253352i	$0.918467\pi$
$374$	−1.42134e6	−0.525436
$375$	0	0
$376$	−1.50350e6	−0.548445
$377$	− 132038.i	− 0.0478459i
$378$	− 453162.i	− 0.163126i
$379$	−913278.	−0.326591	−0.163296	−	0.986577i	$-0.552212\pi$
$379$	−913278.	−0.326591	−0.163296	+	0.986577i	$0.552212\pi$
$380$	0	0
$381$	−2.84902e6	−1.00550
$382$	− 5.34590e6i	− 1.87440i
$383$	3.16295e6i	1.10178i	0.834577	+	0.550891i	$0.185712\pi$
$383$	3.16295e6i	1.10178i	−0.834577	+	0.550891i	$0.814288\pi$
$384$	−4.75299e6	−1.64490
$385$	0	0
$386$	5.43852e6	1.85786
$387$	− 1.10992e6i	− 0.376715i
$388$	296355.i	0.0999387i
$389$	−1.11256e6	−0.372778	−0.186389	−	0.982476i	$-0.559678\pi$
$389$	−1.11256e6	−0.372778	−0.186389	+	0.982476i	$0.559678\pi$
$390$	0	0
$391$	−1.79766e6	−0.594657
$392$	509950.i	0.167615i
$393$	3.01058e6i	0.983261i
$394$	−4.21930e6	−1.36930
$395$	0	0
$396$	−394717.	−0.126487
$397$	− 5.30834e6i	− 1.69037i	−0.534471	−	0.845187i	$-0.679489\pi$
$397$	− 5.30834e6i	− 1.69037i	0.534471	−	0.845187i	$-0.320511\pi$
$398$	3.94211e6i	1.24745i
$399$	−2.29100e6	−0.720430
$400$	0	0
$401$	−2.83298e6	−0.879796	−0.439898	−	0.898048i	$-0.644985\pi$
$401$	−2.83298e6	−0.879796	−0.439898	+	0.898048i	$0.644985\pi$
$402$	9.00297e6i	2.77857i
$403$	− 980544.i	− 0.300749i
$404$	−1.14592e6	−0.349301
$405$	0	0
$406$	−568405.	−0.171137
$407$	− 730608.i	− 0.218624i
$408$	− 2.91572e6i	− 0.867153i
$409$	5.48711e6	1.62194	0.810971	−	0.585086i	$-0.198939\pi$
$409$	5.48711e6	1.62194	0.810971	+	0.585086i	$0.198939\pi$
$410$	0	0
$411$	−5.37247e6	−1.56881
$412$	1.19293e6i	0.346235i
$413$	5.17359e6i	1.49251i
$414$	−2.74303e6	−0.786555
$415$	0	0
$416$	428062.	0.121275
$417$	− 6.08571e6i	− 1.71384i
$418$	− 1.49411e6i	− 0.418255i
$419$	2.60460e6	0.724780	0.362390	−	0.932027i	$-0.381961\pi$
$419$	2.60460e6	0.724780	0.362390	+	0.932027i	$0.381961\pi$
$420$	0	0
$421$	−154318.	−0.0424338	−0.0212169	−	0.999775i	$-0.506754\pi$
$421$	−154318.	−0.0424338	−0.0212169	+	0.999775i	$0.506754\pi$
$422$	7.47397e6i	2.04301i
$423$	− 2.06621e6i	− 0.561467i
$424$	118740.	0.0320762
$425$	0	0
$426$	−1.06303e7	−2.83805
$427$	5.91297e6i	1.56941i
$428$	− 413180.i	− 0.109026i
$429$	−936434.	−0.245660
$430$	0	0
$431$	4.30539e6	1.11640	0.558199	−	0.829707i	$-0.311492\pi$
$431$	4.30539e6	1.11640	0.558199	+	0.829707i	$0.311492\pi$
$432$	748170.i	0.192882i
$433$	3.05395e6i	0.782784i	0.920224	+	0.391392i	$0.128006\pi$
$433$	3.05395e6i	0.782784i	−0.920224	+	0.391392i	$0.871994\pi$
$434$	−4.22111e6	−1.07573
$435$	0	0
$436$	947500.	0.238706
$437$	− 1.88969e6i	− 0.473355i
$438$	− 7.36993e6i	− 1.83560i
$439$	−6.28386e6	−1.55620	−0.778100	−	0.628141i	$-0.783816\pi$
$439$	−6.28386e6	−1.55620	−0.778100	+	0.628141i	$0.783816\pi$
$440$	0	0
$441$	−700810.	−0.171595
$442$	926587.i	0.225596i
$443$	5.38851e6i	1.30455i	0.757984	+	0.652273i	$0.226184\pi$
$443$	5.38851e6i	1.30455i	−0.757984	+	0.652273i	$0.773816\pi$
$444$	−428880.	−0.103247
$445$	0	0
$446$	−7.90396e6	−1.88151
$447$	737084.i	0.174481i
$448$	2.62813e6i	0.618659i
$449$	285189.	0.0667602	0.0333801	−	0.999443i	$-0.489373\pi$
$449$	285189.	0.0667602	0.0333801	+	0.999443i	$0.489373\pi$
$450$	0	0
$451$	1.40220e6	0.324615
$452$	− 1.05910e6i	− 0.243832i
$453$	7.06335e6i	1.61721i
$454$	−2.43802e6	−0.555134
$455$	0	0
$456$	3.06499e6	0.690266
$457$	− 5.26002e6i	− 1.17814i	−0.808082	−	0.589070i	$-0.799494\pi$
$457$	− 5.26002e6i	− 1.17814i	0.808082	−	0.589070i	$-0.200506\pi$
$458$	424499.i	0.0945613i
$459$	−546020.	−0.120970
$460$	0	0
$461$	6.93938e6	1.52079	0.760394	−	0.649463i	$-0.225006\pi$
$461$	6.93938e6	1.52079	0.760394	+	0.649463i	$0.225006\pi$
$462$	4.03122e6i	0.878683i
$463$	− 113379.i	− 0.0245798i	−0.999924	−	0.0122899i	$-0.996088\pi$
$463$	− 113379.i	− 0.0245798i	0.999924	−	0.0122899i	$-0.00391210\pi$
$464$	938436.	0.202353
$465$	0	0
$466$	1.27706e6	0.272424
$467$	− 2.78525e6i	− 0.590980i	−0.955346	−	0.295490i	$-0.904517\pi$
$467$	− 2.78525e6i	− 0.590980i	0.955346	−	0.295490i	$-0.0954829\pi$
$468$	257319.i	0.0543073i
$469$	7.83334e6	1.64443
$470$	0	0
$471$	−8.22079e6	−1.70750
$472$	− 6.92144e6i	− 1.43002i
$473$	− 1.34544e6i	− 0.276510i
$474$	8.88579e6	1.81656
$475$	0	0
$476$	725957.	0.146857
$477$	163181.i	0.0328378i
$478$	− 8.48428e6i	− 1.69842i
$479$	6.54147e6	1.30268	0.651339	−	0.758787i	$-0.274208\pi$
$479$	6.54147e6	1.30268	0.651339	+	0.758787i	$0.274208\pi$
$480$	0	0
$481$	−476290.	−0.0938661
$482$	1.44421e6i	0.283147i
$483$	5.09854e6i	0.994440i
$484$	668155.	0.129647
$485$	0	0
$486$	−7.78054e6	−1.49423
$487$	− 4.65514e6i	− 0.889427i	−0.895673	−	0.444714i	$-0.853305\pi$
$487$	− 4.65514e6i	− 0.889427i	0.895673	−	0.444714i	$-0.146695\pi$
$488$	− 7.91062e6i	− 1.50370i
$489$	4.06414e6	0.768594
$490$	0	0
$491$	−6.54709e6	−1.22559	−0.612794	−	0.790243i	$-0.709955\pi$
$491$	−6.54709e6	−1.22559	−0.612794	+	0.790243i	$0.709955\pi$
$492$	− 823115.i	− 0.153302i
$493$	684878.i	0.126910i
$494$	−974022.	−0.179577
$495$	0	0
$496$	6.96905e6	1.27195
$497$	9.24922e6i	1.67963i
$498$	5.31693e6i	0.960699i
$499$	−187544.	−0.0337173	−0.0168586	−	0.999858i	$-0.505367\pi$
$499$	−187544.	−0.0337173	−0.0168586	+	0.999858i	$0.505367\pi$
$500$	0	0
$501$	6.98979e6	1.24414
$502$	− 7.55387e6i	− 1.33786i
$503$	− 7.63064e6i	− 1.34475i	−0.740212	−	0.672374i	$-0.765275\pi$
$503$	− 7.63064e6i	− 1.34475i	0.740212	−	0.672374i	$-0.234725\pi$
$504$	−3.87105e6	−0.678817
$505$	0	0
$506$	−3.32509e6	−0.577334
$507$	610470.i	0.105474i
$508$	948998.i	0.163157i
$509$	1.64426e6	0.281304	0.140652	−	0.990059i	$-0.455080\pi$
$509$	1.64426e6	0.281304	0.140652	+	0.990059i	$0.455080\pi$
$510$	0	0
$511$	−6.41246e6	−1.08636
$512$	− 2.93895e6i	− 0.495469i
$513$	− 573973.i	− 0.0962937i
$514$	4.22697e6	0.705702
$515$	0	0
$516$	−789797.	−0.130584
$517$	− 2.50466e6i	− 0.412119i
$518$	2.05037e6i	0.335743i
$519$	−2.22556e6	−0.362678
$520$	0	0
$521$	−2.37187e6	−0.382823	−0.191411	−	0.981510i	$-0.561306\pi$
$521$	−2.37187e6	−0.382823	−0.191411	+	0.981510i	$0.561306\pi$
$522$	1.04504e6i	0.167864i
$523$	2.16500e6i	0.346101i	0.984913	+	0.173051i	$0.0553624\pi$
$523$	2.16500e6i	0.346101i	−0.984913	+	0.173051i	$0.944638\pi$
$524$	1.00281e6	0.159548
$525$	0	0
$526$	2.36263e6	0.372333
$527$	5.08607e6i	0.797729i
$528$	− 6.65555e6i	− 1.03896i
$529$	2.23089e6	0.346608
$530$	0	0
$531$	9.51193e6	1.46397
$532$	763121.i	0.116900i
$533$	− 914106.i	− 0.139373i
$534$	−7.99548e6	−1.21336
$535$	0	0
$536$	−1.04798e7	−1.57558
$537$	− 4.62759e6i	− 0.692499i
$538$	− 518282.i	− 0.0771988i
$539$	−849520.	−0.125951
$540$	0	0
$541$	2.41746e6	0.355112	0.177556	−	0.984111i	$-0.443181\pi$
$541$	2.41746e6	0.355112	0.177556	+	0.984111i	$0.443181\pi$
$542$	1.97523e6i	0.288814i
$543$	993527.i	0.144604i
$544$	−2.22035e6	−0.321680
$545$	0	0
$546$	2.62799e6	0.377262
$547$	− 9.98124e6i	− 1.42632i	−0.701002	−	0.713159i	$-0.747264\pi$
$547$	− 9.98124e6i	− 1.42632i	0.701002	−	0.713159i	$-0.252736\pi$
$548$	1.78955e6i	0.254561i
$549$	1.08713e7	1.53940
$550$	0	0
$551$	−719939.	−0.101022
$552$	− 6.82104e6i	− 0.952803i
$553$	− 7.73138e6i	− 1.07509i
$554$	−461104.	−0.0638299
$555$	0	0
$556$	−2.02712e6	−0.278095
$557$	139762.i	0.0190875i	0.999954	+	0.00954377i	$0.00303792\pi$
$557$	139762.i	0.0190875i	−0.999954	+	0.00954377i	$0.996962\pi$
$558$	7.76075e6i	1.05516i
$559$	−877105.	−0.118719
$560$	0	0
$561$	4.85727e6	0.651606
$562$	6.15105e6i	0.821502i
$563$	2.73423e6i	0.363550i	0.983340	+	0.181775i	$0.0581843\pi$
$563$	2.73423e6i	0.363550i	−0.983340	+	0.181775i	$0.941816\pi$
$564$	−1.47028e6	−0.194626
$565$	0	0
$566$	4.31283e6	0.565876
$567$	7.59342e6i	0.991927i
$568$	− 1.23740e7i	− 1.60931i
$569$	−8.14061e6	−1.05409	−0.527043	−	0.849838i	$-0.676699\pi$
$569$	−8.14061e6	−1.05409	−0.527043	+	0.849838i	$0.676699\pi$
$570$	0	0
$571$	−6.32458e6	−0.811785	−0.405893	−	0.913921i	$-0.633039\pi$
$571$	−6.32458e6	−0.811785	−0.405893	+	0.913921i	$0.633039\pi$
$572$	311922.i	0.0398617i
$573$	1.82690e7i	2.32449i
$574$	−3.93511e6	−0.498513
$575$	0	0
$576$	4.83196e6	0.606830
$577$	− 2.57561e6i	− 0.322063i	−0.986949	−	0.161032i	$-0.948518\pi$
$577$	− 2.57561e6i	− 0.322063i	0.986949	−	0.161032i	$-0.0514821\pi$
$578$	4.07440e6i	0.507276i
$579$	−1.85855e7	−2.30397
$580$	0	0
$581$	4.62617e6	0.568566
$582$	− 5.56470e6i	− 0.680979i
$583$	197808.i	0.0241030i
$584$	8.57885e6	1.04087
$585$	0	0
$586$	1.41631e7	1.70379
$587$	− 3.08620e6i	− 0.369682i	−0.982768	−	0.184841i	$-0.940823\pi$
$587$	− 3.08620e6i	− 0.369682i	0.982768	−	0.184841i	$-0.0591771\pi$
$588$	498683.i	0.0594814i
$589$	−5.34644e6	−0.635004
$590$	0	0
$591$	1.44189e7	1.69810
$592$	− 3.38515e6i	− 0.396985i
$593$	7.24899e6i	0.846527i	0.906007	+	0.423263i	$0.139115\pi$
$593$	7.24899e6i	0.846527i	−0.906007	+	0.423263i	$0.860885\pi$
$594$	−1.00996e6	−0.117446
$595$	0	0
$596$	245520.	0.0283120
$597$	− 1.34717e7i	− 1.54699i
$598$	2.16766e6i	0.247878i
$599$	8.78328e6	1.00021	0.500104	−	0.865966i	$-0.333295\pi$
$599$	8.78328e6	1.00021	0.500104	+	0.865966i	$0.333295\pi$
$600$	0	0
$601$	−1.04096e7	−1.17557	−0.587783	−	0.809019i	$-0.699999\pi$
$601$	−1.04096e7	−1.17557	−0.587783	+	0.809019i	$0.699999\pi$
$602$	3.77582e6i	0.424639i
$603$	− 1.44020e7i	− 1.61298i
$604$	2.35277e6	0.262414
$605$	0	0
$606$	2.15170e7	2.38012
$607$	1.02228e7i	1.12616i	0.826403	+	0.563079i	$0.190383\pi$
$607$	1.02228e7i	1.12616i	−0.826403	+	0.563079i	$0.809617\pi$
$608$	− 2.33402e6i	− 0.256062i
$609$	1.94246e6	0.212231
$610$	0	0
$611$	−1.63281e6	−0.176943
$612$	− 1.33471e6i	− 0.144049i
$613$	− 1.24077e6i	− 0.133364i	−0.997774	−	0.0666822i	$-0.978759\pi$
$613$	− 1.24077e6i	− 0.133364i	0.997774	−	0.0666822i	$-0.0212414\pi$
$614$	−835268.	−0.0894139
$615$	0	0
$616$	−4.69248e6	−0.498254
$617$	7.95105e6i	0.840836i	0.907330	+	0.420418i	$0.138117\pi$
$617$	7.95105e6i	0.840836i	−0.907330	+	0.420418i	$0.861883\pi$
$618$	− 2.23997e7i	− 2.35924i
$619$	−4.21032e6	−0.441661	−0.220830	−	0.975312i	$-0.570877\pi$
$619$	−4.21032e6	−0.441661	−0.220830	+	0.975312i	$0.570877\pi$
$620$	0	0
$621$	−1.27736e6	−0.132918
$622$	1.50488e7i	1.55965i
$623$	6.95673e6i	0.718100i
$624$	−4.33881e6	−0.446077
$625$	0	0
$626$	1.15012e7	1.17303
$627$	5.10593e6i	0.518687i
$628$	2.73831e6i	0.277066i
$629$	2.47051e6	0.248977
$630$	0	0
$631$	1.10594e7	1.10575	0.552877	−	0.833263i	$-0.313530\pi$
$631$	1.10594e7	1.10575	0.552877	+	0.833263i	$0.313530\pi$
$632$	1.03434e7i	1.03007i
$633$	− 2.55414e7i	− 2.53358i
$634$	1.94213e7	1.91891
$635$	0	0
$636$	116117.	0.0113829
$637$	553810.i	0.0540770i
$638$	1.26680e6i	0.123213i
$639$	1.70052e7	1.64752
$640$	0	0
$641$	−5.22205e6	−0.501991	−0.250995	−	0.967988i	$-0.580758\pi$
$641$	−5.22205e6	−0.501991	−0.250995	+	0.967988i	$0.580758\pi$
$642$	7.75832e6i	0.742900i
$643$	8.62447e6i	0.822631i	0.911493	+	0.411316i	$0.134931\pi$
$643$	8.62447e6i	0.822631i	−0.911493	+	0.411316i	$0.865069\pi$
$644$	1.69830e6	0.161362
$645$	0	0
$646$	5.05224e6	0.476324
$647$	1.74948e7i	1.64304i	0.570182	+	0.821518i	$0.306873\pi$
$647$	1.74948e7i	1.64304i	−0.570182	+	0.821518i	$0.693127\pi$
$648$	− 1.01588e7i	− 0.950396i
$649$	1.15303e7	1.07456
$650$	0	0
$651$	1.44251e7	1.33404
$652$	− 1.35375e6i	− 0.124715i
$653$	− 1.54315e7i	− 1.41620i	−0.706113	−	0.708099i	$-0.749553\pi$
$653$	− 1.54315e7i	− 1.41620i	0.706113	−	0.708099i	$-0.250447\pi$
$654$	−1.77913e7	−1.62653
$655$	0	0
$656$	6.49686e6	0.589446
$657$	1.17897e7i	1.06558i
$658$	7.02904e6i	0.632894i
$659$	−2.04187e7	−1.83154	−0.915768	−	0.401708i	$-0.868417\pi$
$659$	−2.04187e7	−1.83154	−0.915768	+	0.401708i	$0.868417\pi$
$660$	0	0
$661$	−1.40718e7	−1.25270	−0.626349	−	0.779543i	$-0.715451\pi$
$661$	−1.40718e7	−1.25270	−0.626349	+	0.779543i	$0.715451\pi$
$662$	1.38219e7i	1.22581i
$663$	− 3.16650e6i	− 0.279766i
$664$	−6.18908e6	−0.544761
$665$	0	0
$666$	3.76971e6	0.329323
$667$	1.60220e6i	0.139445i
$668$	− 2.32827e6i	− 0.201880i
$669$	2.70108e7	2.33331
$670$	0	0
$671$	1.31782e7	1.12993
$672$	6.29737e6i	0.537943i
$673$	− 1.50655e7i	− 1.28217i	−0.767469	−	0.641086i	$-0.778484\pi$
$673$	− 1.50655e7i	− 1.28217i	0.767469	−	0.641086i	$-0.221516\pi$
$674$	1.54564e7	1.31057
$675$	0	0
$676$	203345.	0.0171146
$677$	− 4.13338e6i	− 0.346604i	−0.984869	−	0.173302i	$-0.944556\pi$
$677$	− 4.13338e6i	− 0.346604i	0.984869	−	0.173302i	$-0.0554437\pi$
$678$	1.98868e7i	1.66147i
$679$	−4.84175e6	−0.403021
$680$	0	0
$681$	8.33163e6	0.688434
$682$	9.40757e6i	0.774491i
$683$	1.67899e7i	1.37720i	0.725142	+	0.688599i	$0.241774\pi$
$683$	1.67899e7i	1.37720i	−0.725142	+	0.688599i	$0.758226\pi$
$684$	1.40304e6	0.114665
$685$	0	0
$686$	1.46116e7	1.18546
$687$	− 1.45067e6i	− 0.117268i
$688$	− 6.23388e6i	− 0.502096i
$689$	128953.	0.0103486
$690$	0	0
$691$	−1.85418e7	−1.47726	−0.738631	−	0.674110i	$-0.764527\pi$
$691$	−1.85418e7	−1.47726	−0.738631	+	0.674110i	$0.764527\pi$
$692$	741324.i	0.0588495i
$693$	− 6.44874e6i	− 0.510084i
$694$	−9.33795e6	−0.735958
$695$	0	0
$696$	−2.59870e6	−0.203345
$697$	4.74146e6i	0.369683i
$698$	2.50492e7i	1.94605i
$699$	−4.36419e6	−0.337840
$700$	0	0
$701$	4.81303e6	0.369933	0.184967	−	0.982745i	$-0.440782\pi$
$701$	4.81303e6	0.369933	0.184967	+	0.982745i	$0.440782\pi$
$702$	658402.i	0.0504253i
$703$	2.59698e6i	0.198190i
$704$	5.85729e6	0.445415
$705$	0	0
$706$	3.43142e6	0.259097
$707$	− 1.87216e7i	− 1.40862i
$708$	− 6.76851e6i	− 0.507470i
$709$	−2.07738e7	−1.55203	−0.776015	−	0.630714i	$-0.782762\pi$
$709$	−2.07738e7	−1.55203	−0.776015	+	0.630714i	$0.782762\pi$
$710$	0	0
$711$	−1.42146e7	−1.05453
$712$	− 9.30700e6i	− 0.688034i
$713$	1.18983e7i	0.876522i
$714$	−1.36314e7	−1.00068
$715$	0	0
$716$	−1.54143e6	−0.112368
$717$	2.89940e7i	2.10625i
$718$	1.25220e7i	0.906491i
$719$	−1.17074e7	−0.844573	−0.422287	−	0.906462i	$-0.638772\pi$
$719$	−1.17074e7	−0.844573	−0.422287	+	0.906462i	$0.638772\pi$
$720$	0	0
$721$	−1.94896e7	−1.39626
$722$	− 1.01761e7i	− 0.726502i
$723$	− 4.93541e6i	− 0.351138i
$724$	330940.	0.0234640
$725$	0	0
$726$	−1.25460e7	−0.883414
$727$	− 1.45991e7i	− 1.02445i	−0.858851	−	0.512226i	$-0.828821\pi$
$727$	− 1.45991e7i	− 1.02445i	0.858851	−	0.512226i	$-0.171179\pi$
$728$	3.05907e6i	0.213925i
$729$	1.07257e7	0.747493
$730$	0	0
$731$	4.54953e6	0.314900
$732$	− 7.73584e6i	− 0.533617i
$733$	− 2.44712e7i	− 1.68227i	−0.540826	−	0.841135i	$-0.681888\pi$
$733$	− 2.44712e7i	− 1.68227i	0.540826	−	0.841135i	$-0.318112\pi$
$734$	1.43909e7	0.985932
$735$	0	0
$736$	−5.19429e6	−0.353453
$737$	− 1.74581e7i	− 1.18394i
$738$	7.23491e6i	0.488982i
$739$	−1.42425e7	−0.959343	−0.479671	−	0.877448i	$-0.659244\pi$
$739$	−1.42425e7	−0.959343	−0.479671	+	0.877448i	$0.659244\pi$
$740$	0	0
$741$	3.32860e6	0.222698
$742$	− 555124.i	− 0.0370152i
$743$	2.54417e7i	1.69073i	0.534191	+	0.845364i	$0.320616\pi$
$743$	2.54417e7i	1.69073i	−0.534191	+	0.845364i	$0.679384\pi$
$744$	−1.92986e7	−1.27818
$745$	0	0
$746$	−8.51576e6	−0.560243
$747$	− 8.50547e6i	− 0.557695i
$748$	− 1.61794e6i	− 0.105732i
$749$	6.75039e6	0.439667
$750$	0	0
$751$	−6.95129e6	−0.449744	−0.224872	−	0.974388i	$-0.572196\pi$
$751$	−6.95129e6	−0.449744	−0.224872	+	0.974388i	$0.572196\pi$
$752$	− 1.16049e7i	− 0.748338i
$753$	2.58144e7i	1.65911i
$754$	825839.	0.0529014
$755$	0	0
$756$	515841.	0.0328255
$757$	2.30272e7i	1.46050i	0.683181	+	0.730249i	$0.260596\pi$
$757$	2.30272e7i	1.46050i	−0.683181	+	0.730249i	$0.739404\pi$
$758$	− 5.71216e6i	− 0.361100i
$759$	1.13631e7	0.715966
$760$	0	0
$761$	−1.15710e7	−0.724287	−0.362144	−	0.932122i	$-0.617955\pi$
$761$	−1.15710e7	−0.724287	−0.362144	+	0.932122i	$0.617955\pi$
$762$	− 1.78194e7i	− 1.11175i
$763$	1.54799e7i	0.962625i
$764$	6.08532e6	0.377181
$765$	0	0
$766$	−1.97829e7	−1.21820
$767$	− 7.51674e6i	− 0.461361i
$768$	− 1.42741e7i	− 0.873261i
$769$	1.20721e7	0.736153	0.368076	−	0.929796i	$-0.380017\pi$
$769$	1.20721e7	0.736153	0.368076	+	0.929796i	$0.380017\pi$
$770$	0	0
$771$	−1.44451e7	−0.875157
$772$	6.19074e6i	0.373852i
$773$	− 7.33162e6i	− 0.441318i	−0.975351	−	0.220659i	$-0.929179\pi$
$773$	− 7.33162e6i	− 0.441318i	0.975351	−	0.220659i	$-0.0708208\pi$
$774$	6.94206e6	0.416520
$775$	0	0
$776$	6.47749e6	0.386147
$777$	− 7.00688e6i	− 0.416363i
$778$	− 6.95860e6i	− 0.412167i
$779$	−4.98419e6	−0.294273
$780$	0	0
$781$	2.06137e7	1.20928
$782$	− 1.12436e7i	− 0.657490i
$783$	486652.i	0.0283670i
$784$	−3.93611e6	−0.228706
$785$	0	0
$786$	−1.88299e7	−1.08715
$787$	1.65857e7i	0.954546i	0.878755	+	0.477273i	$0.158375\pi$
$787$	1.65857e7i	0.954546i	−0.878755	+	0.477273i	$0.841625\pi$
$788$	− 4.80289e6i	− 0.275541i
$789$	−8.07400e6	−0.461738
$790$	0	0
$791$	1.73032e7	0.983298
$792$	8.62739e6i	0.488727i
$793$	− 8.59099e6i	− 0.485133i
$794$	3.32014e7	1.86898
$795$	0	0
$796$	−4.48737e6	−0.251020
$797$	6.27622e6i	0.349987i	0.984570	+	0.174994i	$0.0559905\pi$
$797$	6.27622e6i	0.349987i	−0.984570	+	0.174994i	$0.944010\pi$
$798$	− 1.43292e7i	− 0.796553i
$799$	8.46937e6	0.469336
$800$	0	0
$801$	1.27903e7	0.704370
$802$	− 1.77191e7i	− 0.972758i
$803$	1.42914e7i	0.782142i
$804$	−1.02482e7	−0.559124
$805$	0	0
$806$	6.13288e6	0.332527
$807$	1.77117e6i	0.0957360i
$808$	2.50465e7i	1.34964i
$809$	2.64760e7	1.42227	0.711134	−	0.703057i	$-0.248182\pi$
$809$	2.64760e7	1.42227	0.711134	+	0.703057i	$0.248182\pi$
$810$	0	0
$811$	1.73949e7	0.928688	0.464344	−	0.885655i	$-0.346290\pi$
$811$	1.73949e7	0.928688	0.464344	+	0.885655i	$0.346290\pi$
$812$	− 647024.i	− 0.0344374i
$813$	− 6.75010e6i	− 0.358166i
$814$	4.56964e6	0.241725
$815$	0	0
$816$	2.25054e7	1.18321
$817$	4.78244e6i	0.250665i
$818$	3.43195e7i	1.79332i
$819$	−4.20399e6	−0.219004
$820$	0	0
$821$	−2.24599e7	−1.16292	−0.581460	−	0.813575i	$-0.697518\pi$
$821$	−2.24599e7	−1.16292	−0.581460	+	0.813575i	$0.697518\pi$
$822$	− 3.36025e7i	− 1.73457i
$823$	− 2.29479e7i	− 1.18098i	−0.807043	−	0.590492i	$-0.798934\pi$
$823$	− 2.29479e7i	− 1.18098i	0.807043	−	0.590492i	$-0.201066\pi$
$824$	2.60740e7	1.33780
$825$	0	0
$826$	−3.23586e7	−1.65021
$827$	− 2.60201e7i	− 1.32295i	−0.749965	−	0.661477i	$-0.769930\pi$
$827$	− 2.60201e7i	− 1.32295i	0.749965	−	0.661477i	$-0.230070\pi$
$828$	− 3.12243e6i	− 0.158276i
$829$	−2.59397e7	−1.31093	−0.655463	−	0.755227i	$-0.727527\pi$
$829$	−2.59397e7	−1.31093	−0.655463	+	0.755227i	$0.727527\pi$
$830$	0	0
$831$	1.57577e6	0.0791570
$832$	− 3.81842e6i	− 0.191239i
$833$	− 2.87261e6i	− 0.143438i
$834$	3.80635e7	1.89493
$835$	0	0
$836$	1.70076e6	0.0841643
$837$	3.61399e6i	0.178309i
$838$	1.62907e7i	0.801362i
$839$	7.59928e6	0.372707	0.186353	−	0.982483i	$-0.440333\pi$
$839$	7.59928e6	0.372707	0.186353	+	0.982483i	$0.440333\pi$
$840$	0	0
$841$	−1.99007e7	−0.970240
$842$	− 965195.i	− 0.0469175i
$843$	− 2.10205e7i	− 1.01876i
$844$	−8.50773e6	−0.411110
$845$	0	0
$846$	1.29233e7	0.620793
$847$	1.09161e7i	0.522827i
$848$	916510.i	0.0437671i
$849$	−1.47386e7	−0.701756
$850$	0	0
$851$	5.77951e6	0.273569
$852$	− 1.21006e7i	− 0.571094i
$853$	4.38692e6i	0.206437i	0.994659	+	0.103218i	$0.0329140\pi$
$853$	4.38692e6i	0.206437i	−0.994659	+	0.103218i	$0.967086\pi$
$854$	−3.69831e7	−1.73524
$855$	0	0
$856$	−9.03094e6	−0.421259
$857$	3.98007e6i	0.185114i	0.995707	+	0.0925569i	$0.0295040\pi$
$857$	3.98007e6i	0.185114i	−0.995707	+	0.0925569i	$0.970496\pi$
$858$	− 5.85699e6i	− 0.271617i
$859$	1.81700e7	0.840178	0.420089	−	0.907483i	$-0.361999\pi$
$859$	1.81700e7	0.840178	0.420089	+	0.907483i	$0.361999\pi$
$860$	0	0
$861$	1.34478e7	0.618218
$862$	2.69284e7i	1.23436i
$863$	3.33461e7i	1.52412i	0.647509	+	0.762058i	$0.275811\pi$
$863$	3.33461e7i	1.52412i	−0.647509	+	0.762058i	$0.724189\pi$
$864$	−1.57771e6	−0.0719022
$865$	0	0
$866$	−1.91011e7	−0.865495
$867$	− 1.39238e7i	− 0.629085i
$868$	− 4.80495e6i	− 0.216466i
$869$	−1.72309e7	−0.774030
$870$	0	0
$871$	−1.13811e7	−0.508322
$872$	− 2.07097e7i	− 0.922320i
$873$	8.90182e6i	0.395315i
$874$	1.18192e7	0.523371
$875$	0	0
$876$	8.38930e6	0.369373
$877$	2.27510e7i	0.998853i	0.866356	+	0.499426i	$0.166456\pi$
$877$	2.27510e7i	0.998853i	−0.866356	+	0.499426i	$0.833544\pi$
$878$	− 3.93029e7i	− 1.72063i
$879$	−4.84008e7	−2.11291
$880$	0	0
$881$	2.66503e7	1.15681	0.578405	−	0.815750i	$-0.303676\pi$
$881$	2.66503e7	1.15681	0.578405	+	0.815750i	$0.303676\pi$
$882$	− 4.38326e6i	− 0.189726i
$883$	− 2.54804e7i	− 1.09978i	−0.835238	−	0.549888i	$-0.814670\pi$
$883$	− 2.54804e7i	− 1.09978i	0.835238	−	0.549888i	$-0.185330\pi$
$884$	−1.05475e6	−0.0453960
$885$	0	0
$886$	−3.37028e7	−1.44239
$887$	4.94339e6i	0.210968i	0.994421	+	0.105484i	$0.0336392\pi$
$887$	4.94339e6i	0.210968i	−0.994421	+	0.105484i	$0.966361\pi$
$888$	9.37409e6i	0.398930i
$889$	−1.55044e7	−0.657961
$890$	0	0
$891$	1.69234e7	0.714157
$892$	− 8.99719e6i	− 0.378612i
$893$	8.90294e6i	0.373598i
$894$	−4.61015e6	−0.192917
$895$	0	0
$896$	−2.58658e7	−1.07636
$897$	− 7.40771e6i	− 0.307399i
$898$	1.78374e6i	0.0738142i
$899$	4.53306e6	0.187065
$900$	0	0
$901$	−668876.	−0.0274494
$902$	8.77015e6i	0.358914i
$903$	− 1.29034e7i	− 0.526605i
$904$	−2.31489e7	−0.942128
$905$	0	0
$906$	−4.41783e7	−1.78808
$907$	1.72075e7i	0.694542i	0.937765	+	0.347271i	$0.112892\pi$
$907$	1.72075e7i	0.694542i	−0.937765	+	0.347271i	$0.887108\pi$
$908$	− 2.77523e6i	− 0.111708i
$909$	−3.44206e7	−1.38169
$910$	0	0
$911$	−3.34765e7	−1.33642	−0.668212	−	0.743971i	$-0.732940\pi$
$911$	−3.34765e7	−1.33642	−0.668212	+	0.743971i	$0.732940\pi$
$912$	2.36575e7i	0.941849i
$913$	− 1.03103e7i	− 0.409350i
$914$	3.28992e7	1.30262
$915$	0	0
$916$	−483214.	−0.0190283
$917$	1.63836e7i	0.643406i
$918$	− 3.41512e6i	− 0.133752i
$919$	−7.59930e6	−0.296814	−0.148407	−	0.988926i	$-0.547415\pi$
$919$	−7.59930e6	−0.296814	−0.148407	+	0.988926i	$0.547415\pi$
$920$	0	0
$921$	2.85443e6	0.110884
$922$	4.34029e7i	1.68148i
$923$	− 1.34382e7i	− 0.519204i
$924$	−4.58880e6	−0.176815
$925$	0	0
$926$	709135.	0.0271770
$927$	3.58327e7i	1.36956i
$928$	1.97893e6i	0.0754329i
$929$	−1.98974e7	−0.756411	−0.378205	−	0.925722i	$-0.623459\pi$
$929$	−1.98974e7	−0.756411	−0.378205	+	0.925722i	$0.623459\pi$
$930$	0	0
$931$	3.01966e6	0.114179
$932$	1.45369e6i	0.0548192i
$933$	− 5.14275e7i	− 1.93415i
$934$	1.74206e7	0.653424
$935$	0	0
$936$	5.62427e6	0.209834
$937$	− 1.39414e7i	− 0.518749i	−0.965777	−	0.259375i	$-0.916484\pi$
$937$	− 1.39414e7i	− 0.518749i	0.965777	−	0.259375i	$-0.0835164\pi$
$938$	4.89942e7i	1.81818i
$939$	−3.93040e7	−1.45470
$940$	0	0
$941$	3.02780e7	1.11469	0.557345	−	0.830281i	$-0.311820\pi$
$941$	3.02780e7	1.11469	0.557345	+	0.830281i	$0.311820\pi$
$942$	− 5.14175e7i	− 1.88792i
$943$	1.10922e7i	0.406197i
$944$	5.34240e7	1.95122
$945$	0	0
$946$	8.41515e6	0.305727
$947$	− 5.48834e7i	− 1.98868i	−0.106223	−	0.994342i	$-0.533876\pi$
$947$	− 5.48834e7i	− 1.98868i	0.106223	−	0.994342i	$-0.466124\pi$
$948$	1.01148e7i	0.365542i
$949$	9.31670e6	0.335812
$950$	0	0
$951$	−6.63700e7	−2.37969
$952$	− 1.58674e7i	− 0.567430i
$953$	3.67850e7i	1.31201i	0.754755	+	0.656007i	$0.227756\pi$
$953$	3.67850e7i	1.31201i	−0.754755	+	0.656007i	$0.772244\pi$
$954$	−1.02063e6	−0.0363075
$955$	0	0
$956$	9.65778e6	0.341769
$957$	− 4.32914e6i	− 0.152799i
$958$	4.09141e7i	1.44032i
$959$	−2.92370e7	−1.02656
$960$	0	0
$961$	5.03445e6	0.175851
$962$	− 2.97899e6i	− 0.103784i
$963$	− 1.24110e7i	− 0.431260i
$964$	−1.64396e6	−0.0569770
$965$	0	0
$966$	−3.18892e7	−1.09951
$967$	2.94486e6i	0.101274i	0.998717	+	0.0506371i	$0.0161252\pi$
$967$	2.94486e6i	0.101274i	−0.998717	+	0.0506371i	$0.983875\pi$
$968$	− 1.46040e7i	− 0.500937i
$969$	−1.72654e7	−0.590701
$970$	0	0
$971$	−3.29113e7	−1.12020	−0.560102	−	0.828424i	$-0.689238\pi$
$971$	−3.29113e7	−1.12020	−0.560102	+	0.828424i	$0.689238\pi$
$972$	− 8.85670e6i	− 0.300681i
$973$	− 3.31184e7i	− 1.12147i
$974$	2.91159e7	0.983407
$975$	0	0
$976$	6.10591e7	2.05176
$977$	− 3.00384e7i	− 1.00679i	−0.864056	−	0.503396i	$-0.832084\pi$
$977$	− 3.00384e7i	− 1.00679i	0.864056	−	0.503396i	$-0.167916\pi$
$978$	2.54195e7i	0.849805i
$979$	1.55044e7	0.517010
$980$	0	0
$981$	2.84607e7	0.944219
$982$	− 4.09492e7i	− 1.35509i
$983$	− 4.02782e7i	− 1.32949i	−0.747069	−	0.664747i	$-0.768540\pi$
$983$	− 4.02782e7i	− 1.32949i	0.747069	−	0.664747i	$-0.231460\pi$
$984$	−1.79910e7	−0.592334
$985$	0	0
$986$	−4.28362e6	−0.140320
$987$	− 2.40209e7i	− 0.784867i
$988$	− 1.10874e6i	− 0.0361359i
$989$	1.06432e7	0.346003
$990$	0	0
$991$	4.26432e7	1.37932	0.689661	−	0.724132i	$-0.257760\pi$
$991$	4.26432e7	1.37932	0.689661	+	0.724132i	$0.257760\pi$
$992$	1.46960e7i	0.474155i
$993$	− 4.72347e7i	− 1.52015i
$994$	−5.78499e7	−1.85711
$995$	0	0
$996$	−6.05233e6	−0.193319
$997$	1.22251e7i	0.389508i	0.980852	+	0.194754i	$0.0623908\pi$
$997$	1.22251e7i	0.389508i	−0.980852	+	0.194754i	$0.937609\pi$
$998$	− 1.17301e6i	− 0.0372799i
$999$	1.75546e6	0.0556516

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	325.6.b.d.274.5		6
5.2	odd	4		65.6.a.b.1.1	✓	3
5.3	odd	4		325.6.a.d.1.3		3
5.4	even	2	inner	325.6.b.d.274.2		6
15.2	even	4		585.6.a.c.1.3		3
20.7	even	4		1040.6.a.k.1.3		3
65.12	odd	4		845.6.a.c.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
65.6.a.b.1.1	✓	3	5.2	odd	4
325.6.a.d.1.3		3	5.3	odd	4
325.6.b.d.274.2		6	5.4	even	2	inner
325.6.b.d.274.5		6	1.1	even	1	trivial
585.6.a.c.1.3		3	15.2	even	4
845.6.a.c.1.3		3	65.12	odd	4
1040.6.a.k.1.3		3	20.7	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:	\( N \)	\(=\)	\( 325 = 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	325.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+6.25457i q^{2} -21.3742i q^{3} -7.11967 q^{4} +133.687 q^{6} -116.319i q^{7} +155.616i q^{8} -213.858 q^{9} -259.239 q^{11} +152.178i q^{12} +169.000i q^{13} +727.523 q^{14} -1201.14 q^{16} -876.601i q^{17} -1337.59i q^{18} +921.477 q^{19} -2486.22 q^{21} -1621.43i q^{22} -2050.72i q^{23} +3326.17 q^{24} -1057.02 q^{26} -622.884i q^{27} +828.150i q^{28} -781.288 q^{29} -5802.03 q^{31} -2532.91i q^{32} +5541.03i q^{33} +5482.76 q^{34} +1522.60 q^{36} +2818.28i q^{37} +5763.44i q^{38} +3612.25 q^{39} -5408.91 q^{41} -15550.3i q^{42} +5189.97i q^{43} +1845.69 q^{44} +12826.4 q^{46} +9661.60i q^{47} +25673.5i q^{48} +3276.98 q^{49} -18736.7 q^{51} -1203.22i q^{52} -763.033i q^{53} +3895.87 q^{54} +18101.0 q^{56} -19695.9i q^{57} -4886.62i q^{58} -44477.7 q^{59} -50834.3 q^{61} -36289.2i q^{62} +24875.7i q^{63} -22594.2 q^{64} -34656.8 q^{66} +67343.8i q^{67} +6241.11i q^{68} -43832.6 q^{69} -79516.3 q^{71} -33279.7i q^{72} -55128.4i q^{73} -17627.2 q^{74} -6560.61 q^{76} +30154.3i q^{77} +22593.1i q^{78} +66467.3 q^{79} -65281.2 q^{81} -33830.4i q^{82} +39771.5i q^{83} +17701.1 q^{84} -32461.0 q^{86} +16699.4i q^{87} -40341.6i q^{88} -59807.6 q^{89} +19657.8 q^{91} +14600.4i q^{92} +124014. i q^{93} -60429.2 q^{94} -54139.0 q^{96} -41624.9i q^{97} +20496.1i q^{98} +55440.3 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 20 q^{4} + 44 q^{6} - 286 q^{9} - 1232 q^{11} + 2568 q^{14} - 3740 q^{16} + 6720 q^{19} - 7584 q^{21} - 1044 q^{24} - 676 q^{26} - 4612 q^{29} - 12760 q^{31} + 5720 q^{34} - 1372 q^{36} + 5408 q^{39}+ \cdots + 138896 q^{99}+O(q^{100}) \) 6 * q + 20 * q^4 + 44 * q^6 - 286 * q^9 - 1232 * q^11 + 2568 * q^14 - 3740 * q^16 + 6720 * q^19 - 7584 * q^21 - 1044 * q^24 - 676 * q^26 - 4612 * q^29 - 12760 * q^31 + 5720 * q^34 - 1372 * q^36 + 5408 * q^39 + 16572 * q^41 - 15380 * q^44 + 28372 * q^46 - 32342 * q^49 - 86656 * q^51 - 60008 * q^54 + 31560 * q^56 - 9504 * q^59 - 173788 * q^61 - 67236 * q^64 - 42376 * q^66 + 17776 * q^69 - 246504 * q^71 - 187776 * q^74 - 51820 * q^76 + 221392 * q^79 - 304154 * q^81 - 159888 * q^84 - 237932 * q^86 + 224420 * q^89 - 70304 * q^91 - 255032 * q^94 - 212876 * q^96 + 138896 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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