LMFDB - Embedded newform 325.6.b.i.274.20

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:	$22$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			274.20
Character	$\chi$	$=$	325.274
Dual form			325.6.b.i.274.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+9.60154i q^{2} +25.9222i q^{3} -60.1895 q^{4} -248.893 q^{6} -181.554i q^{7} -270.662i q^{8} -428.963 q^{9} +O(q^{10})$	\(q+9.60154i q^{2} +25.9222i q^{3} -60.1895 q^{4} -248.893 q^{6} -181.554i q^{7} -270.662i q^{8} -428.963 q^{9} +512.944 q^{11} -1560.25i q^{12} +169.000i q^{13} +1743.19 q^{14} +672.710 q^{16} -1668.15i q^{17} -4118.70i q^{18} +464.071 q^{19} +4706.28 q^{21} +4925.05i q^{22} -2095.04i q^{23} +7016.17 q^{24} -1622.66 q^{26} -4820.57i q^{27} +10927.6i q^{28} -5935.75 q^{29} +8397.76 q^{31} -2202.14i q^{32} +13296.7i q^{33} +16016.8 q^{34} +25819.0 q^{36} +8909.35i q^{37} +4455.79i q^{38} -4380.86 q^{39} +3256.85 q^{41} +45187.5i q^{42} -5951.12i q^{43} -30873.8 q^{44} +20115.6 q^{46} +10741.1i q^{47} +17438.1i q^{48} -16154.7 q^{49} +43242.3 q^{51} -10172.0i q^{52} -39344.1i q^{53} +46284.9 q^{54} -49139.7 q^{56} +12029.8i q^{57} -56992.4i q^{58} +3940.20 q^{59} -41477.5 q^{61} +80631.4i q^{62} +77879.7i q^{63} +42670.7 q^{64} -127668. q^{66} -18589.7i q^{67} +100405. i q^{68} +54308.1 q^{69} +70348.8 q^{71} +116104. i q^{72} -78602.6i q^{73} -85543.4 q^{74} -27932.2 q^{76} -93126.8i q^{77} -42063.0i q^{78} +71703.3 q^{79} +20722.1 q^{81} +31270.8i q^{82} +208.877i q^{83} -283268. q^{84} +57139.9 q^{86} -153868. i q^{87} -138835. i q^{88} +43537.5 q^{89} +30682.6 q^{91} +126099. i q^{92} +217689. i q^{93} -103131. q^{94} +57084.5 q^{96} -15298.0i q^{97} -155110. i q^{98} -220034. q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 22 q - 374 q^{4} + 702 q^{6} - 2744 q^{9}+O(q^{10}) $ 22 * q - 374 * q^4 + 702 * q^6 - 2744 * q^9	$ 22 q - 374 q^{4} + 702 q^{6} - 2744 q^{9} + 2552 q^{11} - 1156 q^{14} + 11414 q^{16} - 7040 q^{19} + 3412 q^{21} - 15446 q^{24} + 1690 q^{26} - 36850 q^{29} + 18136 q^{31} + 28910 q^{34} + 75368 q^{36} - 3718 q^{39} - 20020 q^{41} - 121302 q^{44} - 19716 q^{46} - 146626 q^{49} + 73212 q^{51} - 96026 q^{54} - 2524 q^{56} - 263284 q^{59} - 86730 q^{61} - 360142 q^{64} - 117214 q^{66} - 118690 q^{69} + 136840 q^{71} - 742792 q^{74} + 45034 q^{76} - 104478 q^{79} + 215014 q^{81} - 1705432 q^{84} + 404492 q^{86} - 655424 q^{89} + 70304 q^{91} - 1299948 q^{94} + 1799326 q^{96} - 853396 q^{99}+O(q^{100}) $ 22 * q - 374 * q^4 + 702 * q^6 - 2744 * q^9 + 2552 * q^11 - 1156 * q^14 + 11414 * q^16 - 7040 * q^19 + 3412 * q^21 - 15446 * q^24 + 1690 * q^26 - 36850 * q^29 + 18136 * q^31 + 28910 * q^34 + 75368 * q^36 - 3718 * q^39 - 20020 * q^41 - 121302 * q^44 - 19716 * q^46 - 146626 * q^49 + 73212 * q^51 - 96026 * q^54 - 2524 * q^56 - 263284 * q^59 - 86730 * q^61 - 360142 * q^64 - 117214 * q^66 - 118690 * q^69 + 136840 * q^71 - 742792 * q^74 + 45034 * q^76 - 104478 * q^79 + 215014 * q^81 - 1705432 * q^84 + 404492 * q^86 - 655424 * q^89 + 70304 * q^91 - 1299948 * q^94 + 1799326 * q^96 - 853396 * q^99

Display 100 coefficients 10 coefficients

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 $

$2$	9.60154i	1.69733i	0.528933	+	0.848664i	$0.322592\pi$
$2$	9.60154i	1.69733i	−0.528933	+	0.848664i	$0.677408\pi$
$3$	25.9222i	1.66291i	0.555590	+	0.831456i	$0.312492\pi$
$3$	25.9222i	1.66291i	−0.555590	+	0.831456i	$0.687508\pi$
$4$	−60.1895	−1.88092
$5$	0	0
$5$	0	0
$6$	−248.893	−2.82251
$7$	− 181.554i	− 1.40042i	−0.713935	−	0.700212i	$-0.753089\pi$
$7$	− 181.554i	− 1.40042i	0.713935	−	0.700212i	$-0.246911\pi$
$8$	− 270.662i	− 1.49521i
$9$	−428.963	−1.76528
$10$	0	0
$11$	512.944	1.27817	0.639084	−	0.769137i	$-0.279313\pi$
$11$	512.944	1.27817	0.639084	+	0.769137i	$0.279313\pi$
$12$	− 1560.25i	− 3.12781i
$13$	169.000i	0.277350i
$13$	169.000i	0.277350i
$14$	1743.19	2.37698
$15$	0	0
$16$	672.710	0.656943
$17$	− 1668.15i	− 1.39995i	−0.714165	−	0.699977i	$-0.753193\pi$
$17$	− 1668.15i	− 1.39995i	0.714165	−	0.699977i	$-0.246807\pi$
$18$	− 4118.70i	− 2.99626i
$19$	464.071	0.294917	0.147459	−	0.989068i	$-0.452891\pi$
$19$	464.071	0.294917	0.147459	+	0.989068i	$0.452891\pi$
$20$	0	0
$21$	4706.28	2.32878
$22$	4925.05i	2.16947i
$23$	− 2095.04i	− 0.825795i	−0.910778	−	0.412897i	$-0.864517\pi$
$23$	− 2095.04i	− 0.825795i	0.910778	−	0.412897i	$-0.135483\pi$
$24$	7016.17	2.48641
$25$	0	0
$26$	−1622.66	−0.470754
$27$	− 4820.57i	− 1.27259i
$28$	10927.6i	2.63409i
$29$	−5935.75	−1.31063	−0.655316	−	0.755355i	$-0.727465\pi$
$29$	−5935.75	−1.31063	−0.655316	+	0.755355i	$0.727465\pi$
$30$	0	0
$31$	8397.76	1.56949	0.784746	−	0.619817i	$-0.212793\pi$
$31$	8397.76	1.56949	0.784746	+	0.619817i	$0.212793\pi$
$32$	− 2202.14i	− 0.380164i
$33$	13296.7i	2.12548i
$34$	16016.8	2.37618
$35$	0	0
$36$	25819.0	3.32035
$37$	8909.35i	1.06990i	0.844885	+	0.534948i	$0.179669\pi$
$37$	8909.35i	1.06990i	−0.844885	+	0.534948i	$0.820331\pi$
$38$	4455.79i	0.500571i
$39$	−4380.86	−0.461209
$40$	0	0
$41$	3256.85	0.302579	0.151289	−	0.988490i	$-0.451657\pi$
$41$	3256.85	0.302579	0.151289	+	0.988490i	$0.451657\pi$
$42$	45187.5i	3.95271i
$43$	− 5951.12i	− 0.490826i	−0.969419	−	0.245413i	$-0.921076\pi$
$43$	− 5951.12i	− 0.490826i	0.969419	−	0.245413i	$-0.0789236\pi$
$44$	−30873.8	−2.40413
$45$	0	0
$46$	20115.6	1.40164
$47$	10741.1i	0.709255i	0.935008	+	0.354628i	$0.115392\pi$
$47$	10741.1i	0.709255i	−0.935008	+	0.354628i	$0.884608\pi$
$48$	17438.1i	1.09244i
$49$	−16154.7	−0.961190
$50$	0	0
$51$	43242.3	2.32800
$52$	− 10172.0i	− 0.521674i
$53$	− 39344.1i	− 1.92393i	−0.273167	−	0.961967i	$-0.588071\pi$
$53$	− 39344.1i	− 1.92393i	0.273167	−	0.961967i	$-0.411929\pi$
$54$	46284.9	2.16001
$55$	0	0
$56$	−49139.7	−2.09393
$57$	12029.8i	0.490422i
$58$	− 56992.4i	− 2.22457i
$59$	3940.20	0.147363	0.0736815	−	0.997282i	$-0.476525\pi$
$59$	3940.20	0.147363	0.0736815	+	0.997282i	$0.476525\pi$
$60$	0	0
$61$	−41477.5	−1.42721	−0.713605	−	0.700548i	$-0.752939\pi$
$61$	−41477.5	−1.42721	−0.713605	+	0.700548i	$0.752939\pi$
$62$	80631.4i	2.66394i
$63$	77879.7i	2.47214i
$64$	42670.7	1.30221
$65$	0	0
$66$	−127668.	−3.60764
$67$	− 18589.7i	− 0.505923i	−0.967476	−	0.252961i	$-0.918595\pi$
$67$	− 18589.7i	− 0.505923i	0.967476	−	0.252961i	$-0.0814046\pi$
$68$	100405.i	2.63320i
$69$	54308.1	1.37322
$70$	0	0
$71$	70348.8	1.65619	0.828096	−	0.560586i	$-0.189424\pi$
$71$	70348.8	1.65619	0.828096	+	0.560586i	$0.189424\pi$
$72$	116104.i	2.63947i
$73$	− 78602.6i	− 1.72635i	−0.504902	−	0.863176i	$-0.668471\pi$
$73$	− 78602.6i	− 1.72635i	0.504902	−	0.863176i	$-0.331529\pi$
$74$	−85543.4	−1.81596
$75$	0	0
$76$	−27932.2	−0.554716
$77$	− 93126.8i	− 1.78998i
$78$	− 42063.0i	− 0.782823i
$79$	71703.3	1.29262	0.646310	−	0.763075i	$-0.276311\pi$
$79$	71703.3	1.29262	0.646310	+	0.763075i	$0.276311\pi$
$80$	0	0
$81$	20722.1	0.350931
$82$	31270.8i	0.513575i
$83$	208.877i	0.00332808i	0.999999	+	0.00166404i	$0.000529681\pi$
$83$	208.877i	0.00332808i	−0.999999	+	0.00166404i	$0.999470\pi$
$84$	−283268.	−4.38026
$85$	0	0
$86$	57139.9	0.833093
$87$	− 153868.i	− 2.17947i
$88$	− 138835.i	− 1.91113i
$89$	43537.5	0.582624	0.291312	−	0.956628i	$-0.405908\pi$
$89$	43537.5	0.582624	0.291312	+	0.956628i	$0.405908\pi$
$90$	0	0
$91$	30682.6	0.388408
$92$	126099.i	1.55325i
$93$	217689.i	2.60993i
$94$	−103131.	−1.20384
$95$	0	0
$96$	57084.5	0.632179
$97$	− 15298.0i	− 0.165084i	−0.996588	−	0.0825419i	$-0.973696\pi$
$97$	− 15298.0i	− 0.165084i	0.996588	−	0.0825419i	$-0.0263038\pi$
$98$	− 155110.i	− 1.63145i
$99$	−220034.	−2.25632
$100$	0	0
$101$	2600.25	0.0253636	0.0126818	−	0.999920i	$-0.495963\pi$
$101$	2600.25	0.0253636	0.0126818	+	0.999920i	$0.495963\pi$
$102$	415193.i	3.95138i
$103$	− 210878.i	− 1.95857i	−0.202497	−	0.979283i	$-0.564906\pi$
$103$	− 210878.i	− 1.95857i	0.202497	−	0.979283i	$-0.435094\pi$
$104$	45741.9	0.414697
$105$	0	0
$106$	377764.	3.26555
$107$	− 107005.i	− 0.903531i	−0.892137	−	0.451765i	$-0.850794\pi$
$107$	− 107005.i	− 0.903531i	0.892137	−	0.451765i	$-0.149206\pi$
$108$	290148.i	2.39365i
$109$	−151212.	−1.21904	−0.609522	−	0.792769i	$-0.708638\pi$
$109$	−151212.	−1.21904	−0.609522	+	0.792769i	$0.708638\pi$
$110$	0	0
$111$	−230950.	−1.77914
$112$	− 122133.i	− 0.920000i
$113$	176779.i	1.30237i	0.758918	+	0.651186i	$0.225728\pi$
$113$	176779.i	1.30237i	−0.758918	+	0.651186i	$0.774272\pi$
$114$	−115504.	−0.832407
$115$	0	0
$116$	357270.	2.46520
$117$	− 72494.7i	− 0.489600i
$118$	37832.0i	0.250123i
$119$	−302860.	−1.96053
$120$	0	0
$121$	102060.	0.633715
$122$	− 398248.i	− 2.42244i
$123$	84424.9i	0.503162i
$124$	−505457.	−2.95209
$125$	0	0
$126$	−747765.	−4.19603
$127$	3379.52i	0.0185929i	0.999957	+	0.00929643i	$0.00295919\pi$
$127$	3379.52i	0.0185929i	−0.999957	+	0.00929643i	$0.997041\pi$
$128$	339235.i	1.83011i
$129$	154266.	0.816201
$130$	0	0
$131$	−165563.	−0.842915	−0.421458	−	0.906848i	$-0.638481\pi$
$131$	−165563.	−0.842915	−0.421458	+	0.906848i	$0.638481\pi$
$132$	− 800319.i	− 3.99787i
$133$	− 84253.8i	− 0.413010i
$134$	178489.	0.858717
$135$	0	0
$136$	−451507.	−2.09323
$137$	− 54105.0i	− 0.246284i	−0.992389	−	0.123142i	$-0.960703\pi$
$137$	− 54105.0i	− 0.246284i	0.992389	−	0.123142i	$-0.0392970\pi$
$138$	521441.i	2.33081i
$139$	248178.	1.08950	0.544749	−	0.838599i	$-0.316625\pi$
$139$	248178.	1.08950	0.544749	+	0.838599i	$0.316625\pi$
$140$	0	0
$141$	−278432.	−1.17943
$142$	675457.i	2.81110i
$143$	86687.5i	0.354500i
$144$	−288567.	−1.15969
$145$	0	0
$146$	754705.	2.93019
$147$	− 418767.i	− 1.59838i
$148$	− 536249.i	− 2.01239i
$149$	−102637.	−0.378736	−0.189368	−	0.981906i	$-0.560644\pi$
$149$	−102637.	−0.378736	−0.189368	+	0.981906i	$0.560644\pi$
$150$	0	0
$151$	513812.	1.83384	0.916921	−	0.399068i	$-0.130666\pi$
$151$	513812.	1.83384	0.916921	+	0.399068i	$0.130666\pi$
$152$	− 125606.i	− 0.440964i
$153$	715576.i	2.47131i
$154$	894161.	3.03818
$155$	0	0
$156$	263682.	0.867498
$157$	276514.i	0.895300i	0.894209	+	0.447650i	$0.147739\pi$
$157$	276514.i	0.895300i	−0.894209	+	0.447650i	$0.852261\pi$
$158$	688461.i	2.19400i
$159$	1.01989e6	3.19933
$160$	0	0
$161$	−380362.	−1.15646
$162$	198964.i	0.595644i
$163$	278424.i	0.820801i	0.911905	+	0.410401i	$0.134611\pi$
$163$	278424.i	0.820801i	−0.911905	+	0.410401i	$0.865389\pi$
$164$	−196028.	−0.569127
$165$	0	0
$166$	−2005.54	−0.00564885
$167$	407177.i	1.12978i	0.825167	+	0.564888i	$0.191081\pi$
$167$	407177.i	1.12978i	−0.825167	+	0.564888i	$0.808919\pi$
$168$	− 1.27381e6i	− 3.48203i
$169$	−28561.0	−0.0769231
$170$	0	0
$171$	−199069.	−0.520611
$172$	358195.i	0.923206i
$173$	49123.8i	0.124789i	0.998052	+	0.0623946i	$0.0198737\pi$
$173$	49123.8i	0.124789i	−0.998052	+	0.0623946i	$0.980126\pi$
$174$	1.47737e6	3.69927
$175$	0	0
$176$	345062.	0.839684
$177$	102139.i	0.245052i
$178$	418026.i	0.988903i
$179$	−386317.	−0.901179	−0.450590	−	0.892731i	$-0.648786\pi$
$179$	−386317.	−0.901179	−0.450590	+	0.892731i	$0.648786\pi$
$180$	0	0
$181$	−218966.	−0.496799	−0.248400	−	0.968658i	$-0.579905\pi$
$181$	−218966.	−0.496799	−0.248400	+	0.968658i	$0.579905\pi$
$182$	294600.i	0.659256i
$183$	− 1.07519e6i	− 2.37333i
$184$	−567047.	−1.23474
$185$	0	0
$186$	−2.09015e6	−4.42991
$187$	− 855670.i	− 1.78938i
$188$	− 646499.i	− 1.33405i
$189$	−875192.	−1.78217
$190$	0	0
$191$	277244.	0.549894	0.274947	−	0.961459i	$-0.411340\pi$
$191$	277244.	0.549894	0.274947	+	0.961459i	$0.411340\pi$
$192$	1.10612e6i	2.16545i
$193$	310407.i	0.599844i	0.953964	+	0.299922i	$0.0969606\pi$
$193$	310407.i	0.599844i	−0.953964	+	0.299922i	$0.903039\pi$
$194$	146884.	0.280201
$195$	0	0
$196$	972344.	1.80792
$197$	− 516120.i	− 0.947513i	−0.880656	−	0.473757i	$-0.842898\pi$
$197$	− 516120.i	− 0.947513i	0.880656	−	0.473757i	$-0.157102\pi$
$198$	− 2.11266e6i	− 3.82972i
$199$	−647020.	−1.15820	−0.579102	−	0.815255i	$-0.696597\pi$
$199$	−647020.	−1.15820	−0.579102	+	0.815255i	$0.696597\pi$
$200$	0	0
$201$	481886.	0.841306
$202$	24966.4i	0.0430504i
$203$	1.07766e6i	1.83544i
$204$	−2.60273e6	−4.37879
$205$	0	0
$206$	2.02475e6	3.32433
$207$	898693.i	1.45776i
$208$	113688.i	0.182203i
$209$	238042.	0.376954
$210$	0	0
$211$	132130.	0.204312	0.102156	−	0.994768i	$-0.467426\pi$
$211$	132130.	0.204312	0.102156	+	0.994768i	$0.467426\pi$
$212$	2.36810e6i	3.61877i
$213$	1.82360e6i	2.75410i
$214$	1.02741e6	1.53359
$215$	0	0
$216$	−1.30475e6	−1.90279
$217$	− 1.52464e6i	− 2.19796i
$218$	− 1.45186e6i	− 2.06912i
$219$	2.03756e6	2.87077
$220$	0	0
$221$	281918.	0.388278
$222$	− 2.21748e6i	− 3.01979i
$223$	− 919984.i	− 1.23885i	−0.785057	−	0.619424i	$-0.787366\pi$
$223$	− 919984.i	− 1.23885i	0.785057	−	0.619424i	$-0.212634\pi$
$224$	−399807.	−0.532391
$225$	0	0
$226$	−1.69735e6	−2.21055
$227$	30962.8i	0.0398819i	0.999801	+	0.0199409i	$0.00634781\pi$
$227$	30962.8i	0.0398819i	−0.999801	+	0.0199409i	$0.993652\pi$
$228$	− 724065.i	− 0.922445i
$229$	−163086.	−0.205508	−0.102754	−	0.994707i	$-0.532765\pi$
$229$	−163086.	−0.205508	−0.102754	+	0.994707i	$0.532765\pi$
$230$	0	0
$231$	2.41406e6	2.97658
$232$	1.60658e6i	1.95967i
$233$	1.19712e6i	1.44460i	0.691582	+	0.722298i	$0.256914\pi$
$233$	1.19712e6i	1.44460i	−0.691582	+	0.722298i	$0.743086\pi$
$234$	696060.	0.831012
$235$	0	0
$236$	−237159.	−0.277178
$237$	1.85871e6i	2.14952i
$238$	− 2.90792e6i	− 3.32766i
$239$	769561.	0.871462	0.435731	−	0.900077i	$-0.356490\pi$
$239$	769561.	0.871462	0.435731	+	0.900077i	$0.356490\pi$
$240$	0	0
$241$	−850451.	−0.943206	−0.471603	−	0.881811i	$-0.656324\pi$
$241$	−850451.	−0.943206	−0.471603	+	0.881811i	$0.656324\pi$
$242$	979937.i	1.07562i
$243$	− 634236.i	− 0.689025i
$244$	2.49651e6	2.68447
$245$	0	0
$246$	−810609.	−0.854031
$247$	78428.0i	0.0817954i
$248$	− 2.27296e6i	− 2.34672i
$249$	−5414.55	−0.00553431
$250$	0	0
$251$	992979.	0.994846	0.497423	−	0.867508i	$-0.334280\pi$
$251$	992979.	0.994846	0.497423	+	0.867508i	$0.334280\pi$
$252$	− 4.68754e6i	− 4.64990i
$253$	− 1.07464e6i	− 1.05550i
$254$	−32448.6	−0.0315582
$255$	0	0
$256$	−1.89172e6	−1.80408
$257$	− 1.29318e6i	− 1.22131i	−0.791897	−	0.610655i	$-0.790906\pi$
$257$	− 1.29318e6i	− 1.22131i	0.791897	−	0.610655i	$-0.209094\pi$
$258$	1.48120e6i	1.38536i
$259$	1.61752e6	1.49831
$260$	0	0
$261$	2.54622e6	2.31363
$262$	− 1.58965e6i	− 1.43070i
$263$	− 1.69375e6i	− 1.50994i	−0.655757	−	0.754972i	$-0.727650\pi$
$263$	− 1.69375e6i	− 1.50994i	0.655757	−	0.754972i	$-0.272350\pi$
$264$	3.59890e6	3.17805
$265$	0	0
$266$	808966.	0.701013
$267$	1.12859e6i	0.968852i
$268$	1.11890e6i	0.951601i
$269$	857080.	0.722172	0.361086	−	0.932532i	$-0.382406\pi$
$269$	857080.	0.722172	0.361086	+	0.932532i	$0.382406\pi$
$270$	0	0
$271$	1.66147e6	1.37426	0.687130	−	0.726534i	$-0.258870\pi$
$271$	1.66147e6	1.37426	0.687130	+	0.726534i	$0.258870\pi$
$272$	− 1.12218e6i	− 0.919691i
$273$	795361.i	0.645889i
$274$	519491.	0.418024
$275$	0	0
$276$	−3.26877e6	−2.58293
$277$	− 2.15375e6i	− 1.68654i	−0.537491	−	0.843269i	$-0.680628\pi$
$277$	− 2.15375e6i	− 1.68654i	0.537491	−	0.843269i	$-0.319372\pi$
$278$	2.38289e6i	1.84924i
$279$	−3.60233e6	−2.77059
$280$	0	0
$281$	−547631.	−0.413735	−0.206868	−	0.978369i	$-0.566327\pi$
$281$	−547631.	−0.413735	−0.206868	+	0.978369i	$0.566327\pi$
$282$	− 2.67338e6i	− 2.00188i
$283$	− 609080.i	− 0.452073i	−0.974119	−	0.226036i	$-0.927423\pi$
$283$	− 609080.i	− 0.452073i	0.974119	−	0.226036i	$-0.0725768\pi$
$284$	−4.23426e6	−3.11517
$285$	0	0
$286$	−832333.	−0.601703
$287$	− 591293.i	− 0.423739i
$288$	944638.i	0.671095i
$289$	−1.36288e6	−0.959874
$290$	0	0
$291$	396558.	0.274520
$292$	4.73105e6i	3.24713i
$293$	927787.i	0.631363i	0.948865	+	0.315682i	$0.102233\pi$
$293$	927787.i	0.631363i	−0.948865	+	0.315682i	$0.897767\pi$
$294$	4.02080e6	2.71297
$295$	0	0
$296$	2.41142e6	1.59972
$297$	− 2.47268e6i	− 1.62659i
$298$	− 985469.i	− 0.642839i
$299$	354061.	0.229034
$300$	0	0
$301$	−1.08045e6	−0.687365
$302$	4.93339e6i	3.11263i
$303$	67404.3i	0.0421775i
$304$	312185.	0.193744
$305$	0	0
$306$	−6.87063e6	−4.19462
$307$	2.39073e6i	1.44772i	0.689948	+	0.723859i	$0.257633\pi$
$307$	2.39073e6i	1.44772i	−0.689948	+	0.723859i	$0.742367\pi$
$308$	5.60526e6i	3.36681i
$309$	5.46643e6	3.25692
$310$	0	0
$311$	−1.27080e6	−0.745032	−0.372516	−	0.928026i	$-0.621505\pi$
$311$	−1.27080e6	−0.745032	−0.372516	+	0.928026i	$0.621505\pi$
$312$	1.18573e6i	0.689605i
$313$	− 1.69293e6i	− 0.976738i	−0.872637	−	0.488369i	$-0.837592\pi$
$313$	− 1.69293e6i	− 0.976738i	0.872637	−	0.488369i	$-0.162408\pi$
$314$	−2.65496e6	−1.51962
$315$	0	0
$316$	−4.31578e6	−2.43132
$317$	958135.i	0.535523i	0.963485	+	0.267762i	$0.0862840\pi$
$317$	958135.i	0.535523i	−0.963485	+	0.267762i	$0.913716\pi$
$318$	9.79249e6i	5.43032i
$319$	−3.04471e6	−1.67521
$320$	0	0
$321$	2.77380e6	1.50249
$322$	− 3.65205e6i	− 1.96290i
$323$	− 774142.i	− 0.412871i
$324$	−1.24725e6	−0.660073
$325$	0	0
$326$	−2.67330e6	−1.39317
$327$	− 3.91975e6i	− 2.02716i
$328$	− 881507.i	− 0.452419i
$329$	1.95008e6	0.993259
$330$	0	0
$331$	2.29139e6	1.14955	0.574777	−	0.818310i	$-0.305089\pi$
$331$	2.29139e6	1.14955	0.574777	+	0.818310i	$0.305089\pi$
$332$	− 12572.2i	− 0.00625986i
$333$	− 3.82178e6i	− 1.88866i
$334$	−3.90953e6	−1.91760
$335$	0	0
$336$	3.16596e6	1.52988
$337$	1.20806e6i	0.579446i	0.957110	+	0.289723i	$0.0935632\pi$
$337$	1.20806e6i	0.579446i	−0.957110	+	0.289723i	$0.906437\pi$
$338$	− 274229.i	− 0.130564i
$339$	−4.58251e6	−2.16573
$340$	0	0
$341$	4.30758e6	2.00608
$342$	− 1.91137e6i	− 0.883648i
$343$	− 118424.i	− 0.0543505i
$344$	−1.61074e6	−0.733889
$345$	0	0
$346$	−471664.	−0.211808
$347$	4.27369e6i	1.90537i	0.303962	+	0.952684i	$0.401690\pi$
$347$	4.27369e6i	1.90537i	−0.303962	+	0.952684i	$0.598310\pi$
$348$	9.26124e6i	4.09941i
$349$	1.49638e6	0.657625	0.328812	−	0.944395i	$-0.393352\pi$
$349$	1.49638e6	0.657625	0.328812	+	0.944395i	$0.393352\pi$
$350$	0	0
$351$	814677.	0.352954
$352$	− 1.12958e6i	− 0.485914i
$353$	730249.i	0.311914i	0.987764	+	0.155957i	$0.0498461\pi$
$353$	730249.i	0.311914i	−0.987764	+	0.155957i	$0.950154\pi$
$354$	−980690.	−0.415933
$355$	0	0
$356$	−2.62050e6	−1.09587
$357$	− 7.85080e6i	− 3.26019i
$358$	− 3.70924e6i	− 1.52960i
$359$	−1.04399e6	−0.427522	−0.213761	−	0.976886i	$-0.568571\pi$
$359$	−1.04399e6	−0.427522	−0.213761	+	0.976886i	$0.568571\pi$
$360$	0	0
$361$	−2.26074e6	−0.913024
$362$	− 2.10241e6i	− 0.843231i
$363$	2.64564e6i	1.05381i
$364$	−1.84677e6	−0.730565
$365$	0	0
$366$	1.03235e7	4.02831
$367$	75157.4i	0.0291277i	0.999894	+	0.0145639i	$0.00463598\pi$
$367$	75157.4i	0.0291277i	−0.999894	+	0.0145639i	$0.995364\pi$
$368$	− 1.40935e6i	− 0.542500i
$369$	−1.39707e6	−0.534136
$370$	0	0
$371$	−7.14307e6	−2.69432
$372$	− 1.31026e7i	− 4.90907i
$373$	− 1.44619e6i	− 0.538212i	−0.963111	−	0.269106i	$-0.913272\pi$
$373$	− 1.44619e6i	− 0.538212i	0.963111	−	0.269106i	$-0.0867283\pi$
$374$	8.21574e6	3.03716
$375$	0	0
$376$	2.90720e6	1.06049
$377$	− 1.00314e6i	− 0.363504i
$378$	− 8.40319e6i	− 3.02493i
$379$	−291357.	−0.104190	−0.0520951	−	0.998642i	$-0.516590\pi$
$379$	−291357.	−0.104190	−0.0520951	+	0.998642i	$0.516590\pi$
$380$	0	0
$381$	−87604.8	−0.0309183
$382$	2.66197e6i	0.933350i
$383$	− 3.40227e6i	− 1.18515i	−0.805517	−	0.592573i	$-0.798112\pi$
$383$	− 3.40227e6i	− 1.18515i	0.805517	−	0.592573i	$-0.201888\pi$
$384$	−8.79374e6	−3.04331
$385$	0	0
$386$	−2.98038e6	−1.01813
$387$	2.55281e6i	0.866445i
$388$	920777.i	0.310510i
$389$	−1.64309e6	−0.550537	−0.275268	−	0.961367i	$-0.588767\pi$
$389$	−1.64309e6	−0.550537	−0.275268	+	0.961367i	$0.588767\pi$
$390$	0	0
$391$	−3.49485e6	−1.15608
$392$	4.37247e6i	1.43718i
$393$	− 4.29175e6i	− 1.40169i
$394$	4.95555e6	1.60824
$395$	0	0
$396$	1.32437e7	4.24397
$397$	2.47328e6i	0.787585i	0.919199	+	0.393792i	$0.128837\pi$
$397$	2.47328e6i	0.787585i	−0.919199	+	0.393792i	$0.871163\pi$
$398$	− 6.21239e6i	− 1.96585i
$399$	2.18405e6	0.686799
$400$	0	0
$401$	4.41469e6	1.37100	0.685502	−	0.728071i	$-0.259583\pi$
$401$	4.41469e6	1.37100	0.685502	+	0.728071i	$0.259583\pi$
$402$	4.62684e6i	1.42797i
$403$	1.41922e6i	0.435299i
$404$	−156508.	−0.0477070
$405$	0	0
$406$	−1.03472e7	−3.11535
$407$	4.56999e6i	1.36751i
$408$	− 1.17041e7i	− 3.48086i
$409$	2.67344e6	0.790244	0.395122	−	0.918629i	$-0.370702\pi$
$409$	2.67344e6	0.790244	0.395122	+	0.918629i	$0.370702\pi$
$410$	0	0
$411$	1.40252e6	0.409549
$412$	1.26926e7i	3.68391i
$413$	− 715358.i	− 0.206371i
$414$	−8.62883e6	−2.47429
$415$	0	0
$416$	372162.	0.105439
$417$	6.43334e6i	1.81174i
$418$	2.28557e6i	0.639815i
$419$	−2.27577e6	−0.633277	−0.316639	−	0.948546i	$-0.602554\pi$
$419$	−2.27577e6	−0.633277	−0.316639	+	0.948546i	$0.602554\pi$
$420$	0	0
$421$	4.79147e6	1.31754	0.658770	−	0.752345i	$-0.271077\pi$
$421$	4.79147e6	1.31754	0.658770	+	0.752345i	$0.271077\pi$
$422$	1.26865e6i	0.346785i
$423$	− 4.60751e6i	− 1.25203i
$424$	−1.06490e7	−2.87669
$425$	0	0
$426$	−1.75093e7	−4.67462
$427$	7.53039e6i	1.99870i
$428$	6.44055e6i	1.69947i
$429$	−2.24714e6	−0.589503
$430$	0	0
$431$	−2.09885e6	−0.544238	−0.272119	−	0.962264i	$-0.587725\pi$
$431$	−2.09885e6	−0.544238	−0.272119	+	0.962264i	$0.587725\pi$
$432$	− 3.24285e6i	− 0.836021i
$433$	− 2.53735e6i	− 0.650370i	−0.945650	−	0.325185i	$-0.894573\pi$
$433$	− 2.53735e6i	− 0.650370i	0.945650	−	0.325185i	$-0.105427\pi$
$434$	1.46389e7	3.73065
$435$	0	0
$436$	9.10135e6	2.29292
$437$	− 972246.i	− 0.243541i
$438$	1.95637e7i	4.87264i
$439$	−3.67042e6	−0.908979	−0.454490	−	0.890752i	$-0.650178\pi$
$439$	−3.67042e6	−0.908979	−0.454490	+	0.890752i	$0.650178\pi$
$440$	0	0
$441$	6.92977e6	1.69677
$442$	2.70685e6i	0.659034i
$443$	2.32270e6i	0.562321i	0.959661	+	0.281161i	$0.0907193\pi$
$443$	2.32270e6i	0.562321i	−0.959661	+	0.281161i	$0.909281\pi$
$444$	1.39008e7	3.34643
$445$	0	0
$446$	8.83325e6	2.10273
$447$	− 2.66057e6i	− 0.629805i
$448$	− 7.74702e6i	− 1.82364i
$449$	4.30759e6	1.00837	0.504183	−	0.863597i	$-0.331794\pi$
$449$	4.30759e6	1.00837	0.504183	+	0.863597i	$0.331794\pi$
$450$	0	0
$451$	1.67058e6	0.386747
$452$	− 1.06403e7i	− 2.44966i
$453$	1.33192e7i	3.04952i
$454$	−297290.	−0.0676926
$455$	0	0
$456$	3.25600e6	0.733285
$457$	1.14361e6i	0.256145i	0.991765	+	0.128073i	$0.0408791\pi$
$457$	1.14361e6i	0.256145i	−0.991765	+	0.128073i	$0.959121\pi$
$458$	− 1.56588e6i	− 0.348814i
$459$	−8.04146e6	−1.78157
$460$	0	0
$461$	−3.73288e6	−0.818071	−0.409036	−	0.912518i	$-0.634135\pi$
$461$	−3.73288e6	−0.818071	−0.409036	+	0.912518i	$0.634135\pi$
$462$	2.31786e7i	5.05223i
$463$	− 4.46756e6i	− 0.968540i	−0.874919	−	0.484270i	$-0.839085\pi$
$463$	− 4.46756e6i	− 0.968540i	0.874919	−	0.484270i	$-0.160915\pi$
$464$	−3.99304e6	−0.861011
$465$	0	0
$466$	−1.14942e7	−2.45195
$467$	− 5.73689e6i	− 1.21726i	−0.793453	−	0.608632i	$-0.791719\pi$
$467$	− 5.73689e6i	− 1.21726i	0.793453	−	0.608632i	$-0.208281\pi$
$468$	4.36342e6i	0.920900i
$469$	−3.37502e6	−0.708507
$470$	0	0
$471$	−7.16787e6	−1.48881
$472$	− 1.06646e6i	− 0.220339i
$473$	− 3.05259e6i	− 0.627359i
$474$	−1.78465e7	−3.64843
$475$	0	0
$476$	1.82290e7	3.68761
$477$	1.68772e7i	3.39628i
$478$	7.38897e6i	1.47916i
$479$	8.44282e6	1.68131	0.840657	−	0.541567i	$-0.182169\pi$
$479$	8.44282e6	1.68131	0.840657	+	0.541567i	$0.182169\pi$
$480$	0	0
$481$	−1.50568e6	−0.296736
$482$	− 8.16563e6i	− 1.60093i
$483$	− 9.85982e6i	− 1.92310i
$484$	−6.14297e6	−1.19197
$485$	0	0
$486$	6.08964e6	1.16950
$487$	− 2.30080e6i	− 0.439599i	−0.975545	−	0.219799i	$-0.929460\pi$
$487$	− 2.30080e6i	− 0.439599i	0.975545	−	0.219799i	$-0.0705403\pi$
$488$	1.12264e7i	2.13398i
$489$	−7.21738e6	−1.36492
$490$	0	0
$491$	−1.83374e6	−0.343269	−0.171635	−	0.985161i	$-0.554905\pi$
$491$	−1.83374e6	−0.343269	−0.171635	+	0.985161i	$0.554905\pi$
$492$	− 5.08149e6i	− 0.946408i
$493$	9.90176e6i	1.83483i
$494$	−753029.	−0.138834
$495$	0	0
$496$	5.64926e6	1.03107
$497$	− 1.27721e7i	− 2.31937i
$498$	− 51988.0i	− 0.00939354i
$499$	1.65708e6	0.297916	0.148958	−	0.988844i	$-0.452408\pi$
$499$	1.65708e6	0.297916	0.148958	+	0.988844i	$0.452408\pi$
$500$	0	0
$501$	−1.05550e7	−1.87872
$502$	9.53412e6i	1.68858i
$503$	− 674839.i	− 0.118927i	−0.998230	−	0.0594634i	$-0.981061\pi$
$503$	− 674839.i	− 0.118927i	0.998230	−	0.0594634i	$-0.0189390\pi$
$504$	2.10791e7	3.69637
$505$	0	0
$506$	1.03182e7	1.79154
$507$	− 740365.i	− 0.127916i
$508$	− 203412.i	− 0.0349717i
$509$	−214603.	−0.0367148	−0.0183574	−	0.999831i	$-0.505844\pi$
$509$	−214603.	−0.0367148	−0.0183574	+	0.999831i	$0.505844\pi$
$510$	0	0
$511$	−1.42706e7	−2.41763
$512$	− 7.30787e6i	− 1.23202i
$513$	− 2.23709e6i	− 0.375310i
$514$	1.24165e7	2.07296
$515$	0	0
$516$	−9.28522e6	−1.53521
$517$	5.50956e6i	0.906548i
$518$	1.55307e7i	2.54312i
$519$	−1.27340e6	−0.207513
$520$	0	0
$521$	9.75675e6	1.57475	0.787373	−	0.616476i	$-0.211440\pi$
$521$	9.75675e6	1.57475	0.787373	+	0.616476i	$0.211440\pi$
$522$	2.44476e7i	3.92699i
$523$	1.41439e6i	0.226107i	0.993589	+	0.113054i	$0.0360632\pi$
$523$	1.41439e6i	0.226107i	−0.993589	+	0.113054i	$0.963937\pi$
$524$	9.96512e6	1.58546
$525$	0	0
$526$	1.62626e7	2.56287
$527$	− 1.40088e7i	− 2.19722i
$528$	8.94479e6i	1.39632i
$529$	2.04716e6	0.318063
$530$	0	0
$531$	−1.69020e6	−0.260137
$532$	5.07119e6i	0.776839i
$533$	550408.i	0.0839203i
$534$	−1.08362e7	−1.64446
$535$	0	0
$536$	−5.03152e6	−0.756462
$537$	− 1.00142e7i	− 1.49858i
$538$	8.22929e6i	1.22576i
$539$	−8.28647e6	−1.22856
$540$	0	0
$541$	6.38226e6	0.937522	0.468761	−	0.883325i	$-0.344700\pi$
$541$	6.38226e6	0.937522	0.468761	+	0.883325i	$0.344700\pi$
$542$	1.59527e7i	2.33257i
$543$	− 5.67610e6i	− 0.826134i
$544$	−3.67352e6	−0.532212
$545$	0	0
$546$	−7.63669e6	−1.09628
$547$	− 6.98550e6i	− 0.998227i	−0.866537	−	0.499113i	$-0.833659\pi$
$547$	− 6.98550e6i	− 0.998227i	0.866537	−	0.499113i	$-0.166341\pi$
$548$	3.25655e6i	0.463241i
$549$	1.77923e7	2.51943
$550$	0	0
$551$	−2.75461e6	−0.386528
$552$	− 1.46991e7i	− 2.05326i
$553$	− 1.30180e7i	− 1.81022i
$554$	2.06793e7	2.86261
$555$	0	0
$556$	−1.49377e7	−2.04926
$557$	− 8.50622e6i	− 1.16171i	−0.814006	−	0.580856i	$-0.802718\pi$
$557$	− 8.50622e6i	− 1.16171i	0.814006	−	0.580856i	$-0.197282\pi$
$558$	− 3.45879e7i	− 4.70260i
$559$	1.00574e6	0.136131
$560$	0	0
$561$	2.21809e7	2.97558
$562$	− 5.25810e6i	− 0.702244i
$563$	− 5.22994e6i	− 0.695386i	−0.937608	−	0.347693i	$-0.886965\pi$
$563$	− 5.22994e6i	− 0.695386i	0.937608	−	0.347693i	$-0.113035\pi$
$564$	1.67587e7	2.21841
$565$	0	0
$566$	5.84810e6	0.767315
$567$	− 3.76217e6i	− 0.491452i
$568$	− 1.90408e7i	− 2.47636i
$569$	−1.12412e7	−1.45556	−0.727781	−	0.685809i	$-0.759448\pi$
$569$	−1.12412e7	−1.45556	−0.727781	+	0.685809i	$0.759448\pi$
$570$	0	0
$571$	−3.42903e6	−0.440130	−0.220065	−	0.975485i	$-0.570627\pi$
$571$	−3.42903e6	−0.440130	−0.220065	+	0.975485i	$0.570627\pi$
$572$	− 5.21768e6i	− 0.666787i
$573$	7.18679e6i	0.914426i
$574$	5.67732e6	0.719224
$575$	0	0
$576$	−1.83041e7	−2.29876
$577$	− 1.39888e7i	− 1.74920i	−0.484845	−	0.874600i	$-0.661124\pi$
$577$	− 1.39888e7i	− 1.74920i	0.484845	−	0.874600i	$-0.338876\pi$
$578$	− 1.30858e7i	− 1.62922i
$579$	−8.04645e6	−0.997488
$580$	0	0
$581$	37922.3	0.00466073
$582$	3.80756e6i	0.465950i
$583$	− 2.01813e7i	− 2.45911i
$584$	−2.12747e7	−2.58126
$585$	0	0
$586$	−8.90818e6	−1.07163
$587$	2.19504e6i	0.262934i	0.991321	+	0.131467i	$0.0419687\pi$
$587$	2.19504e6i	0.262934i	−0.991321	+	0.131467i	$0.958031\pi$
$588$	2.52053e7i	3.00642i
$589$	3.89716e6	0.462871
$590$	0	0
$591$	1.33790e7	1.57563
$592$	5.99340e6i	0.702861i
$593$	− 1.41425e6i	− 0.165154i	−0.996585	−	0.0825772i	$-0.973685\pi$
$593$	− 1.41425e6i	− 0.165154i	0.996585	−	0.0825772i	$-0.0263151\pi$
$594$	2.37416e7	2.76085
$595$	0	0
$596$	6.17765e6	0.712373
$597$	− 1.67722e7i	− 1.92599i
$598$	3.39953e6i	0.388746i
$599$	9.66661e6	1.10080	0.550398	−	0.834902i	$-0.314476\pi$
$599$	9.66661e6	1.10080	0.550398	+	0.834902i	$0.314476\pi$
$600$	0	0
$601$	−8.23263e6	−0.929721	−0.464860	−	0.885384i	$-0.653895\pi$
$601$	−8.23263e6	−0.929721	−0.464860	+	0.885384i	$0.653895\pi$
$602$	− 1.03740e7i	− 1.16668i
$603$	7.97427e6i	0.893095i
$604$	−3.09261e7	−3.44931
$605$	0	0
$606$	−647185.	−0.0715891
$607$	7.61203e6i	0.838549i	0.907859	+	0.419275i	$0.137716\pi$
$607$	7.61203e6i	0.838549i	−0.907859	+	0.419275i	$0.862284\pi$
$608$	− 1.02195e6i	− 0.112117i
$609$	−2.79353e7	−3.05218
$610$	0	0
$611$	−1.81524e6	−0.196712
$612$	− 4.30702e7i	− 4.64834i
$613$	− 1.04544e7i	− 1.12369i	−0.827243	−	0.561844i	$-0.810092\pi$
$613$	− 1.04544e7i	− 1.12369i	0.827243	−	0.561844i	$-0.189908\pi$
$614$	−2.29546e7	−2.45725
$615$	0	0
$616$	−2.52059e7	−2.67640
$617$	− 8.78600e6i	− 0.929134i	−0.885538	−	0.464567i	$-0.846210\pi$
$617$	− 8.78600e6i	− 0.929134i	0.885538	−	0.464567i	$-0.153790\pi$
$618$	5.24861e7i	5.52807i
$619$	−8.05994e6	−0.845484	−0.422742	−	0.906250i	$-0.638932\pi$
$619$	−8.05994e6	−0.845484	−0.422742	+	0.906250i	$0.638932\pi$
$620$	0	0
$621$	−1.00993e7	−1.05090
$622$	− 1.22016e7i	− 1.26456i
$623$	− 7.90438e6i	− 0.815921i
$624$	−2.94705e6	−0.302988
$625$	0	0
$626$	1.62547e7	1.65784
$627$	6.17059e6i	0.626842i
$628$	− 1.66433e7i	− 1.68399i
$629$	1.48622e7	1.49781
$630$	0	0
$631$	1.42505e6	0.142481	0.0712404	−	0.997459i	$-0.477304\pi$
$631$	1.42505e6	0.142481	0.0712404	+	0.997459i	$0.477304\pi$
$632$	− 1.94074e7i	− 1.93274i
$633$	3.42510e6i	0.339754i
$634$	−9.19956e6	−0.908959
$635$	0	0
$636$	−6.13865e7	−6.01769
$637$	− 2.73015e6i	− 0.266586i
$638$	− 2.92339e7i	− 2.84338i
$639$	−3.01770e7	−2.92364
$640$	0	0
$641$	1.11250e6	0.106944	0.0534718	−	0.998569i	$-0.482971\pi$
$641$	1.11250e6	0.106944	0.0534718	+	0.998569i	$0.482971\pi$
$642$	2.66327e7i	2.55022i
$643$	9.07113e6i	0.865235i	0.901578	+	0.432617i	$0.142410\pi$
$643$	9.07113e6i	0.865235i	−0.901578	+	0.432617i	$0.857590\pi$
$644$	2.28938e7	2.17522
$645$	0	0
$646$	7.43295e6	0.700777
$647$	2.19842e6i	0.206467i	0.994657	+	0.103234i	$0.0329189\pi$
$647$	2.19842e6i	0.206467i	−0.994657	+	0.103234i	$0.967081\pi$
$648$	− 5.60869e6i	− 0.524716i
$649$	2.02110e6	0.188355
$650$	0	0
$651$	3.95222e7	3.65501
$652$	− 1.67582e7i	− 1.54386i
$653$	1.42857e7i	1.31105i	0.755173	+	0.655525i	$0.227553\pi$
$653$	1.42857e7i	1.31105i	−0.755173	+	0.655525i	$0.772447\pi$
$654$	3.76356e7	3.44076
$655$	0	0
$656$	2.19092e6	0.198777
$657$	3.37176e7i	3.04749i
$658$	1.87237e7i	1.68589i
$659$	−1.54265e7	−1.38374	−0.691869	−	0.722023i	$-0.743213\pi$
$659$	−1.54265e7	−1.38374	−0.691869	+	0.722023i	$0.743213\pi$
$660$	0	0
$661$	1.65514e7	1.47344	0.736719	−	0.676199i	$-0.236374\pi$
$661$	1.65514e7	1.47344	0.736719	+	0.676199i	$0.236374\pi$
$662$	2.20009e7i	1.95117i
$663$	7.30795e6i	0.645672i
$664$	56535.0	0.00497619
$665$	0	0
$666$	3.66949e7	3.20568
$667$	1.24356e7i	1.08231i
$668$	− 2.45078e7i	− 2.12502i
$669$	2.38480e7	2.06010
$670$	0	0
$671$	−2.12756e7	−1.82422
$672$	− 1.03639e7i	− 0.885320i
$673$	− 8.91603e6i	− 0.758811i	−0.925230	−	0.379406i	$-0.876129\pi$
$673$	− 8.91603e6i	− 0.758811i	0.925230	−	0.379406i	$-0.123871\pi$
$674$	−1.15992e7	−0.983510
$675$	0	0
$676$	1.71907e6	0.144686
$677$	− 8.15442e6i	− 0.683788i	−0.939739	−	0.341894i	$-0.888932\pi$
$677$	− 8.15442e6i	− 0.683788i	0.939739	−	0.341894i	$-0.111068\pi$
$678$	− 4.39992e7i	− 3.67596i
$679$	−2.77740e6	−0.231187
$680$	0	0
$681$	−802625.	−0.0663200
$682$	4.13594e7i	3.40497i
$683$	− 2.02314e6i	− 0.165948i	−0.996552	−	0.0829742i	$-0.973558\pi$
$683$	− 2.02314e6i	− 0.165948i	0.996552	−	0.0829742i	$-0.0264419\pi$
$684$	1.19819e7	0.979229
$685$	0	0
$686$	1.13705e6	0.0922507
$687$	− 4.22756e6i	− 0.341742i
$688$	− 4.00338e6i	− 0.322445i
$689$	6.64915e6	0.533603
$690$	0	0
$691$	−5.61163e6	−0.447089	−0.223545	−	0.974694i	$-0.571763\pi$
$691$	−5.61163e6	−0.447089	−0.223545	+	0.974694i	$0.571763\pi$
$692$	− 2.95674e6i	− 0.234719i
$693$	3.99479e7i	3.15981i
$694$	−4.10339e7	−3.23403
$695$	0	0
$696$	−4.16463e7	−3.25877
$697$	− 5.43293e6i	− 0.423597i
$698$	1.43675e7i	1.11620i
$699$	−3.10319e7	−2.40224
$700$	0	0
$701$	−5.09127e6	−0.391319	−0.195660	−	0.980672i	$-0.562685\pi$
$701$	−5.09127e6	−0.391319	−0.195660	+	0.980672i	$0.562685\pi$
$702$	7.82215e6i	0.599078i
$703$	4.13457e6i	0.315531i
$704$	2.18877e7	1.66444
$705$	0	0
$706$	−7.01152e6	−0.529420
$707$	− 472085.i	− 0.0355199i
$708$	− 6.14769e6i	− 0.460923i
$709$	−1.54906e7	−1.15732	−0.578658	−	0.815570i	$-0.696423\pi$
$709$	−1.54906e7	−1.15732	−0.578658	+	0.815570i	$0.696423\pi$
$710$	0	0
$711$	−3.07580e7	−2.28184
$712$	− 1.17839e7i	− 0.871146i
$713$	− 1.75936e7i	− 1.29608i
$714$	7.53797e7	5.53362
$715$	0	0
$716$	2.32522e7	1.69505
$717$	1.99488e7i	1.44916i
$718$	− 1.00239e7i	− 0.725645i
$719$	1.12604e7	0.812328	0.406164	−	0.913800i	$-0.366866\pi$
$719$	1.12604e7	0.812328	0.406164	+	0.913800i	$0.366866\pi$
$720$	0	0
$721$	−3.82857e7	−2.74282
$722$	− 2.17065e7i	− 1.54970i
$723$	− 2.20456e7i	− 1.56847i
$724$	1.31795e7	0.934440
$725$	0	0
$726$	−2.54022e7	−1.78867
$727$	− 8.14136e6i	− 0.571296i	−0.958335	−	0.285648i	$-0.907791\pi$
$727$	− 8.14136e6i	− 0.571296i	0.958335	−	0.285648i	$-0.0922088\pi$
$728$	− 8.30461e6i	− 0.580752i
$729$	2.14763e7	1.49672
$730$	0	0
$731$	−9.92740e6	−0.687135
$732$	6.47151e7i	4.46404i
$733$	− 1.30341e7i	− 0.896025i	−0.894027	−	0.448013i	$-0.852132\pi$
$733$	− 1.30341e7i	− 0.896025i	0.894027	−	0.448013i	$-0.147868\pi$
$734$	−721626.	−0.0494393
$735$	0	0
$736$	−4.61357e6	−0.313937
$737$	− 9.53545e6i	− 0.646655i
$738$	− 1.34140e7i	− 0.906604i
$739$	1.95666e7	1.31796	0.658981	−	0.752159i	$-0.270988\pi$
$739$	1.95666e7	1.31796	0.658981	+	0.752159i	$0.270988\pi$
$740$	0	0
$741$	−2.03303e6	−0.136019
$742$	− 6.85844e7i	− 4.57315i
$743$	− 1.78971e6i	− 0.118935i	−0.998230	−	0.0594675i	$-0.981060\pi$
$743$	− 1.78971e6i	− 0.118935i	0.998230	−	0.0594675i	$-0.0189403\pi$
$744$	5.89201e7	3.90240
$745$	0	0
$746$	1.38857e7	0.913523
$747$	− 89600.2i	− 0.00587500i
$748$	5.15023e7i	3.36568i
$749$	−1.94271e7	−1.26533
$750$	0	0
$751$	−2.13528e7	−1.38151	−0.690755	−	0.723089i	$-0.742722\pi$
$751$	−2.13528e7	−1.38151	−0.690755	+	0.723089i	$0.742722\pi$
$752$	7.22562e6i	0.465940i
$753$	2.57402e7i	1.65434i
$754$	9.63171e6	0.616985
$755$	0	0
$756$	5.26774e7	3.35212
$757$	7.21618e6i	0.457686i	0.973463	+	0.228843i	$0.0734942\pi$
$757$	7.21618e6i	0.457686i	−0.973463	+	0.228843i	$0.926506\pi$
$758$	− 2.79747e6i	− 0.176845i
$759$	2.78570e7	1.75521
$760$	0	0
$761$	−1.18696e7	−0.742974	−0.371487	−	0.928438i	$-0.621152\pi$
$761$	−1.18696e7	−0.742974	−0.371487	+	0.928438i	$0.621152\pi$
$762$	− 841141.i	− 0.0524785i
$763$	2.74530e7i	1.70718i
$764$	−1.66872e7	−1.03431
$765$	0	0
$766$	3.26670e7	2.01158
$767$	665894.i	0.0408711i
$768$	− 4.90376e7i	− 3.00003i
$769$	2.02111e7	1.23247	0.616233	−	0.787564i	$-0.288658\pi$
$769$	2.02111e7	1.23247	0.616233	+	0.787564i	$0.288658\pi$
$770$	0	0
$771$	3.35221e7	2.03093
$772$	− 1.86832e7i	− 1.12826i
$773$	− 4.38847e6i	− 0.264158i	−0.991239	−	0.132079i	$-0.957835\pi$
$773$	− 4.38847e6i	− 0.264158i	0.991239	−	0.132079i	$-0.0421653\pi$
$774$	−2.45109e7	−1.47064
$775$	0	0
$776$	−4.14058e6	−0.246835
$777$	4.19299e7i	2.49156i
$778$	− 1.57762e7i	− 0.934442i
$779$	1.51141e6	0.0892357
$780$	0	0
$781$	3.60850e7	2.11689
$782$	− 3.35559e7i	− 1.96224i
$783$	2.86137e7i	1.66790i
$784$	−1.08674e7	−0.631447
$785$	0	0
$786$	4.12074e7	2.37913
$787$	− 4.56154e6i	− 0.262527i	−0.991347	−	0.131264i	$-0.958097\pi$
$787$	− 4.56154e6i	− 0.262527i	0.991347	−	0.131264i	$-0.0419034\pi$
$788$	3.10650e7i	1.78220i
$789$	4.39059e7	2.51090
$790$	0	0
$791$	3.20949e7	1.82387
$792$	5.95549e7i	3.37368i
$793$	− 7.00970e6i	− 0.395837i
$794$	−2.37473e7	−1.33679
$795$	0	0
$796$	3.89438e7	2.17849
$797$	1.86077e7i	1.03764i	0.854883	+	0.518820i	$0.173629\pi$
$797$	1.86077e7i	1.03764i	−0.854883	+	0.518820i	$0.826371\pi$
$798$	2.09702e7i	1.16572i
$799$	1.79177e7	0.992925
$800$	0	0
$801$	−1.86759e7	−1.02849
$802$	4.23878e7i	2.32704i
$803$	− 4.03187e7i	− 2.20657i
$804$	−2.90044e7	−1.58243
$805$	0	0
$806$	−1.36267e7	−0.738845
$807$	2.22175e7i	1.20091i
$808$	− 703789.i	− 0.0379240i
$809$	1.95576e7	1.05062	0.525309	−	0.850911i	$-0.323950\pi$
$809$	1.95576e7	1.05062	0.525309	+	0.850911i	$0.323950\pi$
$810$	0	0
$811$	−1.88885e7	−1.00843	−0.504214	−	0.863579i	$-0.668218\pi$
$811$	−1.88885e7	−1.00843	−0.504214	+	0.863579i	$0.668218\pi$
$812$	− 6.48637e7i	− 3.45232i
$813$	4.30690e7i	2.28528i
$814$	−4.38790e7	−2.32111
$815$	0	0
$816$	2.90895e7	1.52937
$817$	− 2.76174e6i	− 0.144753i
$818$	2.56691e7i	1.34130i
$819$	−1.31617e7	−0.685648
$820$	0	0
$821$	7.00926e6	0.362923	0.181461	−	0.983398i	$-0.441917\pi$
$821$	7.00926e6	0.362923	0.181461	+	0.983398i	$0.441917\pi$
$822$	1.34664e7i	0.695138i
$823$	2.33685e7i	1.20263i	0.799014	+	0.601313i	$0.205355\pi$
$823$	2.33685e7i	1.20263i	−0.799014	+	0.601313i	$0.794645\pi$
$824$	−5.70767e7	−2.92847
$825$	0	0
$826$	6.86853e6	0.350279
$827$	2.84662e7i	1.44732i	0.690155	+	0.723662i	$0.257542\pi$
$827$	2.84662e7i	1.44732i	−0.690155	+	0.723662i	$0.742458\pi$
$828$	− 5.40918e7i	− 2.74193i
$829$	1.37621e7	0.695503	0.347752	−	0.937587i	$-0.386945\pi$
$829$	1.37621e7	0.695503	0.347752	+	0.937587i	$0.386945\pi$
$830$	0	0
$831$	5.58301e7	2.80457
$832$	7.21135e6i	0.361167i
$833$	2.69486e7i	1.34562i
$834$	−6.17699e7	−3.07512
$835$	0	0
$836$	−1.43276e7	−0.709021
$837$	− 4.04820e7i	− 1.99732i
$838$	− 2.18509e7i	− 1.07488i
$839$	3.18392e7	1.56155	0.780776	−	0.624811i	$-0.214824\pi$
$839$	3.18392e7	1.56155	0.780776	+	0.624811i	$0.214824\pi$
$840$	0	0
$841$	1.47220e7	0.717757
$842$	4.60055e7i	2.23630i
$843$	− 1.41958e7i	− 0.688006i
$844$	−7.95282e6	−0.384295
$845$	0	0
$846$	4.42392e7	2.12511
$847$	− 1.85294e7i	− 0.887471i
$848$	− 2.64672e7i	− 1.26392i
$849$	1.57887e7	0.751757
$850$	0	0
$851$	1.86654e7	0.883514
$852$	− 1.09761e8i	− 5.18025i
$853$	− 2.97955e6i	− 0.140210i	−0.997540	−	0.0701049i	$-0.977667\pi$
$853$	− 2.97955e6i	− 0.140210i	0.997540	−	0.0701049i	$-0.0223334\pi$
$854$	−7.23033e7	−3.39245
$855$	0	0
$856$	−2.89621e7	−1.35097
$857$	− 1.00599e7i	− 0.467886i	−0.972250	−	0.233943i	$-0.924837\pi$
$857$	− 1.00599e7i	− 0.467886i	0.972250	−	0.233943i	$-0.0751630\pi$
$858$	− 2.15759e7i	− 1.00058i
$859$	−1.59873e7	−0.739252	−0.369626	−	0.929181i	$-0.620514\pi$
$859$	−1.59873e7	−0.739252	−0.369626	+	0.929181i	$0.620514\pi$
$860$	0	0
$861$	1.53277e7	0.704641
$862$	− 2.01522e7i	− 0.923751i
$863$	− 9.59580e6i	− 0.438586i	−0.975659	−	0.219293i	$-0.929625\pi$
$863$	− 9.59580e6i	− 0.438586i	0.975659	−	0.219293i	$-0.0703750\pi$
$864$	−1.06156e7	−0.483794
$865$	0	0
$866$	2.43624e7	1.10389
$867$	− 3.53290e7i	− 1.59619i
$868$	9.17675e7i	4.13418i
$869$	3.67797e7	1.65219
$870$	0	0
$871$	3.14165e6	0.140318
$872$	4.09273e7i	1.82273i
$873$	6.56226e6i	0.291419i
$874$	9.33505e6	0.413369
$875$	0	0
$876$	−1.22639e8	−5.39970
$877$	1.09080e7i	0.478902i	0.970908	+	0.239451i	$0.0769675\pi$
$877$	1.09080e7i	0.478902i	−0.970908	+	0.239451i	$0.923033\pi$
$878$	− 3.52416e7i	− 1.54284i
$879$	−2.40503e7	−1.04990
$880$	0	0
$881$	−897407.	−0.0389538	−0.0194769	−	0.999810i	$-0.506200\pi$
$881$	−897407.	−0.0389538	−0.0194769	+	0.999810i	$0.506200\pi$
$882$	6.65365e7i	2.87997i
$883$	3.37890e7i	1.45839i	0.684306	+	0.729195i	$0.260105\pi$
$883$	3.37890e7i	1.45839i	−0.684306	+	0.729195i	$0.739895\pi$
$884$	−1.69685e7	−0.730320
$885$	0	0
$886$	−2.23015e7	−0.954443
$887$	2.26350e7i	0.965989i	0.875623	+	0.482994i	$0.160451\pi$
$887$	2.26350e7i	0.965989i	−0.875623	+	0.482994i	$0.839549\pi$
$888$	6.25095e7i	2.66020i
$889$	613565.	0.0260379
$890$	0	0
$891$	1.06293e7	0.448549
$892$	5.53733e7i	2.33017i
$893$	4.98461e6i	0.209172i
$894$	2.55456e7	1.06899
$895$	0	0
$896$	6.15894e7	2.56293
$897$	9.17806e6i	0.380864i
$898$	4.13594e7i	1.71153i
$899$	−4.98470e7	−2.05703
$900$	0	0
$901$	−6.56321e7	−2.69342
$902$	1.60402e7i	0.656436i
$903$	− 2.80076e7i	− 1.14303i
$904$	4.78475e7	1.94732
$905$	0	0
$906$	−1.27884e8	−5.17604
$907$	− 3.13826e7i	− 1.26669i	−0.773870	−	0.633345i	$-0.781681\pi$
$907$	− 3.13826e7i	− 1.26669i	0.773870	−	0.633345i	$-0.218319\pi$
$908$	− 1.86363e6i	− 0.0750146i
$909$	−1.11541e6	−0.0447739
$910$	0	0
$911$	−2.54672e7	−1.01668	−0.508341	−	0.861156i	$-0.669741\pi$
$911$	−2.54672e7	−1.01668	−0.508341	+	0.861156i	$0.669741\pi$
$912$	8.09254e6i	0.322179i
$913$	107142.i	0.00425385i
$914$	−1.09804e7	−0.434763
$915$	0	0
$916$	9.81608e6	0.386544
$917$	3.00585e7i	1.18044i
$918$	− 7.72104e7i	− 3.02391i
$919$	2.19820e7	0.858577	0.429288	−	0.903167i	$-0.358764\pi$
$919$	2.19820e7	0.858577	0.429288	+	0.903167i	$0.358764\pi$
$920$	0	0
$921$	−6.19730e7	−2.40743
$922$	− 3.58413e7i	− 1.38854i
$923$	1.18889e7i	0.459345i
$924$	−1.45301e8	−5.59871
$925$	0	0
$926$	4.28954e7	1.64393
$927$	9.04588e7i	3.45741i
$928$	1.30714e7i	0.498255i
$929$	−2.75340e7	−1.04672	−0.523359	−	0.852112i	$-0.675321\pi$
$929$	−2.75340e7	−1.04672	−0.523359	+	0.852112i	$0.675321\pi$
$930$	0	0
$931$	−7.49694e6	−0.283472
$932$	− 7.20538e7i	− 2.71717i
$933$	− 3.29419e7i	− 1.23892i
$934$	5.50830e7	2.06610
$935$	0	0
$936$	−1.96216e7	−0.732056
$937$	− 117742.i	− 0.00438111i	−0.999998	−	0.00219055i	$-0.999303\pi$
$937$	− 117742.i	− 0.00438111i	0.999998	−	0.00219055i	$-0.000697275\pi$
$938$	− 3.24054e7i	− 1.20257i
$939$	4.38845e7	1.62423
$940$	0	0
$941$	1.19282e7	0.439136	0.219568	−	0.975597i	$-0.429535\pi$
$941$	1.19282e7	0.439136	0.219568	+	0.975597i	$0.429535\pi$
$942$	− 6.88226e7i	− 2.52699i
$943$	− 6.82323e6i	− 0.249868i
$944$	2.65061e6	0.0968091
$945$	0	0
$946$	2.93096e7	1.06483
$947$	4.67856e7i	1.69526i	0.530585	+	0.847632i	$0.321972\pi$
$947$	4.67856e7i	1.69526i	−0.530585	+	0.847632i	$0.678028\pi$
$948$	− 1.11875e8i	− 4.04307i
$949$	1.32838e7	0.478804
$950$	0	0
$951$	−2.48370e7	−0.890529
$952$	8.19727e7i	2.93141i
$953$	− 1.84653e6i	− 0.0658603i	−0.999458	−	0.0329301i	$-0.989516\pi$
$953$	− 1.84653e6i	− 0.0658603i	0.999458	−	0.0329301i	$-0.0104839\pi$
$954$	−1.62047e8	−5.76460
$955$	0	0
$956$	−4.63195e7	−1.63915
$957$	− 7.89257e7i	− 2.78573i
$958$	8.10641e7i	2.85374i
$959$	−9.82296e6	−0.344902
$960$	0	0
$961$	4.18932e7	1.46331
$962$	− 1.44568e7i	− 0.503658i
$963$	4.59010e7i	1.59498i
$964$	5.11882e7	1.77410
$965$	0	0
$966$	9.46694e7	3.26413
$967$	− 3.47395e7i	− 1.19470i	−0.801982	−	0.597349i	$-0.796221\pi$
$967$	− 3.47395e7i	− 1.19470i	0.801982	−	0.597349i	$-0.203779\pi$
$968$	− 2.76239e7i	− 0.947538i
$969$	2.00675e7	0.686569
$970$	0	0
$971$	1.02513e7	0.348923	0.174462	−	0.984664i	$-0.444182\pi$
$971$	1.02513e7	0.348923	0.174462	+	0.984664i	$0.444182\pi$
$972$	3.81743e7i	1.29600i
$973$	− 4.50577e7i	− 1.52576i
$974$	2.20912e7	0.746143
$975$	0	0
$976$	−2.79023e7	−0.937597
$977$	1.54408e6i	0.0517528i	0.999665	+	0.0258764i	$0.00823763\pi$
$977$	1.54408e6i	0.0517528i	−0.999665	+	0.0258764i	$0.991762\pi$
$978$	− 6.92980e7i	− 2.31672i
$979$	2.23323e7	0.744691
$980$	0	0
$981$	6.48642e7	2.15195
$982$	− 1.76068e7i	− 0.582640i
$983$	3.24347e7i	1.07060i	0.844663	+	0.535299i	$0.179801\pi$
$983$	3.24347e7i	1.07060i	−0.844663	+	0.535299i	$0.820199\pi$
$984$	2.28506e7	0.752334
$985$	0	0
$986$	−9.50721e7	−3.11430
$987$	5.05504e7i	1.65170i
$988$	− 4.72054e6i	− 0.153851i
$989$	−1.24678e7	−0.405322
$990$	0	0
$991$	−3.76595e7	−1.21812	−0.609060	−	0.793124i	$-0.708453\pi$
$991$	−3.76595e7	−1.21812	−0.609060	+	0.793124i	$0.708453\pi$
$992$	− 1.84931e7i	− 0.596664i
$993$	5.93980e7i	1.91161i
$994$	1.22632e8	3.93674
$995$	0	0
$996$	325899.	0.0104096
$997$	− 3.72285e7i	− 1.18614i	−0.805149	−	0.593072i	$-0.797915\pi$
$997$	− 3.72285e7i	− 1.18614i	0.805149	−	0.593072i	$-0.202085\pi$
$998$	1.59106e7i	0.505661i
$999$	4.29481e7	1.36154

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	325.6.b.i.274.20		22
5.2	odd	4		325.6.a.k.1.2	yes	11
5.3	odd	4		325.6.a.j.1.10	✓	11
5.4	even	2	inner	325.6.b.i.274.3		22

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
325.6.a.j.1.10	✓	11	5.3	odd	4
325.6.a.k.1.2	yes	11	5.2	odd	4
325.6.b.i.274.3		22	5.4	even	2	inner
325.6.b.i.274.20		22	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:	\( N \)	\(=\)	\( 325 = 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	325.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+9.60154i q^{2} +25.9222i q^{3} -60.1895 q^{4} -248.893 q^{6} -181.554i q^{7} -270.662i q^{8} -428.963 q^{9} +O(q^{10})\)	\(q+9.60154i q^{2} +25.9222i q^{3} -60.1895 q^{4} -248.893 q^{6} -181.554i q^{7} -270.662i q^{8} -428.963 q^{9} +512.944 q^{11} -1560.25i q^{12} +169.000i q^{13} +1743.19 q^{14} +672.710 q^{16} -1668.15i q^{17} -4118.70i q^{18} +464.071 q^{19} +4706.28 q^{21} +4925.05i q^{22} -2095.04i q^{23} +7016.17 q^{24} -1622.66 q^{26} -4820.57i q^{27} +10927.6i q^{28} -5935.75 q^{29} +8397.76 q^{31} -2202.14i q^{32} +13296.7i q^{33} +16016.8 q^{34} +25819.0 q^{36} +8909.35i q^{37} +4455.79i q^{38} -4380.86 q^{39} +3256.85 q^{41} +45187.5i q^{42} -5951.12i q^{43} -30873.8 q^{44} +20115.6 q^{46} +10741.1i q^{47} +17438.1i q^{48} -16154.7 q^{49} +43242.3 q^{51} -10172.0i q^{52} -39344.1i q^{53} +46284.9 q^{54} -49139.7 q^{56} +12029.8i q^{57} -56992.4i q^{58} +3940.20 q^{59} -41477.5 q^{61} +80631.4i q^{62} +77879.7i q^{63} +42670.7 q^{64} -127668. q^{66} -18589.7i q^{67} +100405. i q^{68} +54308.1 q^{69} +70348.8 q^{71} +116104. i q^{72} -78602.6i q^{73} -85543.4 q^{74} -27932.2 q^{76} -93126.8i q^{77} -42063.0i q^{78} +71703.3 q^{79} +20722.1 q^{81} +31270.8i q^{82} +208.877i q^{83} -283268. q^{84} +57139.9 q^{86} -153868. i q^{87} -138835. i q^{88} +43537.5 q^{89} +30682.6 q^{91} +126099. i q^{92} +217689. i q^{93} -103131. q^{94} +57084.5 q^{96} -15298.0i q^{97} -155110. i q^{98} -220034. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 22 q - 374 q^{4} + 702 q^{6} - 2744 q^{9}+O(q^{10}) \) 22 * q - 374 * q^4 + 702 * q^6 - 2744 * q^9	\( 22 q - 374 q^{4} + 702 q^{6} - 2744 q^{9} + 2552 q^{11} - 1156 q^{14} + 11414 q^{16} - 7040 q^{19} + 3412 q^{21} - 15446 q^{24} + 1690 q^{26} - 36850 q^{29} + 18136 q^{31} + 28910 q^{34} + 75368 q^{36} - 3718 q^{39} - 20020 q^{41} - 121302 q^{44} - 19716 q^{46} - 146626 q^{49} + 73212 q^{51} - 96026 q^{54} - 2524 q^{56} - 263284 q^{59} - 86730 q^{61} - 360142 q^{64} - 117214 q^{66} - 118690 q^{69} + 136840 q^{71} - 742792 q^{74} + 45034 q^{76} - 104478 q^{79} + 215014 q^{81} - 1705432 q^{84} + 404492 q^{86} - 655424 q^{89} + 70304 q^{91} - 1299948 q^{94} + 1799326 q^{96} - 853396 q^{99}+O(q^{100}) \) 22 * q - 374 * q^4 + 702 * q^6 - 2744 * q^9 + 2552 * q^11 - 1156 * q^14 + 11414 * q^16 - 7040 * q^19 + 3412 * q^21 - 15446 * q^24 + 1690 * q^26 - 36850 * q^29 + 18136 * q^31 + 28910 * q^34 + 75368 * q^36 - 3718 * q^39 - 20020 * q^41 - 121302 * q^44 - 19716 * q^46 - 146626 * q^49 + 73212 * q^51 - 96026 * q^54 - 2524 * q^56 - 263284 * q^59 - 86730 * q^61 - 360142 * q^64 - 117214 * q^66 - 118690 * q^69 + 136840 * q^71 - 742792 * q^74 + 45034 * q^76 - 104478 * q^79 + 215014 * q^81 - 1705432 * q^84 + 404492 * q^86 - 655424 * q^89 + 70304 * q^91 - 1299948 * q^94 + 1799326 * q^96 - 853396 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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