LMFDB - Embedded newform 325.6.b.i.274.11

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			274.11
Character	$\chi$	$=$	325.274
Dual form			325.6.b.i.274.12

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.979695i q^{2} +20.7822i q^{3} +31.0402 q^{4} +20.3602 q^{6} +233.896i q^{7} -61.7601i q^{8} -188.898 q^{9} +O(q^{10})$	\(q-0.979695i q^{2} +20.7822i q^{3} +31.0402 q^{4} +20.3602 q^{6} +233.896i q^{7} -61.7601i q^{8} -188.898 q^{9} +621.155 q^{11} +645.082i q^{12} +169.000i q^{13} +229.147 q^{14} +932.780 q^{16} -287.693i q^{17} +185.063i q^{18} +2632.19 q^{19} -4860.87 q^{21} -608.542i q^{22} +2164.84i q^{23} +1283.51 q^{24} +165.568 q^{26} +1124.35i q^{27} +7260.18i q^{28} -2475.24 q^{29} +7148.94 q^{31} -2890.16i q^{32} +12908.9i q^{33} -281.851 q^{34} -5863.44 q^{36} -2727.11i q^{37} -2578.74i q^{38} -3512.19 q^{39} +6104.22 q^{41} +4762.17i q^{42} -2663.13i q^{43} +19280.8 q^{44} +2120.88 q^{46} -29613.8i q^{47} +19385.2i q^{48} -37900.4 q^{49} +5978.87 q^{51} +5245.79i q^{52} -11712.3i q^{53} +1101.52 q^{54} +14445.5 q^{56} +54702.5i q^{57} +2424.98i q^{58} -28664.9 q^{59} +30545.9 q^{61} -7003.78i q^{62} -44182.6i q^{63} +27017.5 q^{64} +12646.8 q^{66} -22127.6i q^{67} -8930.03i q^{68} -44990.0 q^{69} -62434.6 q^{71} +11666.4i q^{72} +40511.4i q^{73} -2671.73 q^{74} +81703.6 q^{76} +145286. i q^{77} +3440.87i q^{78} +1895.66 q^{79} -69268.7 q^{81} -5980.27i q^{82} -6639.87i q^{83} -150882. q^{84} -2609.05 q^{86} -51440.9i q^{87} -38362.6i q^{88} -139830. q^{89} -39528.5 q^{91} +67197.1i q^{92} +148570. i q^{93} -29012.5 q^{94} +60063.9 q^{96} -51749.0i q^{97} +37130.8i q^{98} -117335. q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	− 0.979695i	− 0.173187i	−0.996244	−	0.0865936i	$-0.972402\pi$
$2$	− 0.979695i	− 0.173187i	0.996244	−	0.0865936i	$-0.0275982\pi$
$3$	20.7822i	1.33318i	0.745426	+	0.666588i	$0.232246\pi$
$3$	20.7822i	1.33318i	−0.745426	+	0.666588i	$0.767754\pi$
$4$	31.0402	0.970006
$5$	0	0
$5$	0	0
$6$	20.3602	0.230889
$7$	233.896i	1.80417i	0.431556	+	0.902086i	$0.357965\pi$
$7$	233.896i	1.80417i	−0.431556	+	0.902086i	$0.642035\pi$
$8$	− 61.7601i	− 0.341180i
$9$	−188.898	−0.777359
$10$	0	0
$11$	621.155	1.54781	0.773906	−	0.633300i	$-0.218300\pi$
$11$	621.155	1.54781	0.773906	+	0.633300i	$0.218300\pi$
$12$	645.082i	1.29319i
$13$	169.000i	0.277350i
$13$	169.000i	0.277350i
$14$	229.147	0.312459
$15$	0	0
$16$	932.780	0.910918
$17$	− 287.693i	− 0.241438i	−0.992687	−	0.120719i	$-0.961480\pi$
$17$	− 287.693i	− 0.241438i	0.992687	−	0.120719i	$-0.0385201\pi$
$18$	185.063i	0.134629i
$19$	2632.19	1.67276	0.836378	−	0.548153i	$-0.184669\pi$
$19$	2632.19	1.67276	0.836378	+	0.548153i	$0.184669\pi$
$20$	0	0
$21$	−4860.87	−2.40528
$22$	− 608.542i	− 0.268061i
$23$	2164.84i	0.853309i	0.904415	+	0.426654i	$0.140308\pi$
$23$	2164.84i	0.853309i	−0.904415	+	0.426654i	$0.859692\pi$
$24$	1283.51	0.454853
$25$	0	0
$26$	165.568	0.0480335
$27$	1124.35i	0.296820i
$28$	7260.18i	1.75006i
$29$	−2475.24	−0.546541	−0.273271	−	0.961937i	$-0.588105\pi$
$29$	−2475.24	−0.546541	−0.273271	+	0.961937i	$0.588105\pi$
$30$	0	0
$31$	7148.94	1.33610	0.668048	−	0.744119i	$-0.267130\pi$
$31$	7148.94	1.33610	0.668048	+	0.744119i	$0.267130\pi$
$32$	− 2890.16i	− 0.498939i
$33$	12908.9i	2.06351i
$34$	−281.851	−0.0418140
$35$	0	0
$36$	−5863.44	−0.754043
$37$	− 2727.11i	− 0.327490i	−0.986503	−	0.163745i	$-0.947643\pi$
$37$	− 2727.11i	− 0.327490i	0.986503	−	0.163745i	$-0.0523574\pi$
$38$	− 2578.74i	− 0.289700i
$39$	−3512.19	−0.369757
$40$	0	0
$41$	6104.22	0.567114	0.283557	−	0.958955i	$-0.408486\pi$
$41$	6104.22	0.567114	0.283557	+	0.958955i	$0.408486\pi$
$42$	4762.17i	0.416564i
$43$	− 2663.13i	− 0.219645i	−0.993951	−	0.109822i	$-0.964972\pi$
$43$	− 2663.13i	− 0.219645i	0.993951	−	0.109822i	$-0.0350282\pi$
$44$	19280.8	1.50139
$45$	0	0
$46$	2120.88	0.147782
$47$	− 29613.8i	− 1.95546i	−0.209862	−	0.977731i	$-0.567301\pi$
$47$	− 29613.8i	− 1.95546i	0.209862	−	0.977731i	$-0.432699\pi$
$48$	19385.2i	1.21441i
$49$	−37900.4	−2.25504
$50$	0	0
$51$	5978.87	0.321880
$52$	5245.79i	0.269031i
$53$	− 11712.3i	− 0.572736i	−0.958120	−	0.286368i	$-0.907552\pi$
$53$	− 11712.3i	− 0.572736i	0.958120	−	0.286368i	$-0.0924479\pi$
$54$	1101.52	0.0514054
$55$	0	0
$56$	14445.5	0.615547
$57$	54702.5i	2.23008i
$58$	2424.98i	0.0946539i
$59$	−28664.9	−1.07206	−0.536031	−	0.844198i	$-0.680077\pi$
$59$	−28664.9	−1.07206	−0.536031	+	0.844198i	$0.680077\pi$
$60$	0	0
$61$	30545.9	1.05106	0.525531	−	0.850775i	$-0.323867\pi$
$61$	30545.9	1.05106	0.525531	+	0.850775i	$0.323867\pi$
$62$	− 7003.78i	− 0.231395i
$63$	− 44182.6i	− 1.40249i
$64$	27017.5	0.824508
$65$	0	0
$66$	12646.8	0.357373
$67$	− 22127.6i	− 0.602210i	−0.953591	−	0.301105i	$-0.902645\pi$
$67$	− 22127.6i	− 0.602210i	0.953591	−	0.301105i	$-0.0973554\pi$
$68$	− 8930.03i	− 0.234197i
$69$	−44990.0	−1.13761
$70$	0	0
$71$	−62434.6	−1.46987	−0.734936	−	0.678137i	$-0.762788\pi$
$71$	−62434.6	−1.46987	−0.734936	+	0.678137i	$0.762788\pi$
$72$	11666.4i	0.265219i
$73$	40511.4i	0.889754i	0.895592	+	0.444877i	$0.146753\pi$
$73$	40511.4i	0.889754i	−0.895592	+	0.444877i	$0.853247\pi$
$74$	−2671.73	−0.0567170
$75$	0	0
$76$	81703.6	1.62258
$77$	145286.i	2.79252i
$78$	3440.87i	0.0640371i
$79$	1895.66	0.0341737	0.0170869	−	0.999854i	$-0.494561\pi$
$79$	1895.66	0.0341737	0.0170869	+	0.999854i	$0.494561\pi$
$80$	0	0
$81$	−69268.7	−1.17307
$82$	− 5980.27i	− 0.0982168i
$83$	− 6639.87i	− 0.105795i	−0.998600	−	0.0528974i	$-0.983154\pi$
$83$	− 6639.87i	− 0.105795i	0.998600	−	0.0528974i	$-0.0168456\pi$
$84$	−150882.	−2.33314
$85$	0	0
$86$	−2609.05	−0.0380396
$87$	− 51440.9i	− 0.728636i
$88$	− 38362.6i	− 0.528082i
$89$	−139830.	−1.87123	−0.935613	−	0.353028i	$-0.885152\pi$
$89$	−139830.	−1.87123	−0.935613	+	0.353028i	$0.885152\pi$
$90$	0	0
$91$	−39528.5	−0.500387
$92$	67197.1i	0.827715i
$93$	148570.i	1.78125i
$94$	−29012.5	−0.338661
$95$	0	0
$96$	60063.9	0.665174
$97$	− 51749.0i	− 0.558436i	−0.960228	−	0.279218i	$-0.909925\pi$
$97$	− 51749.0i	− 0.558436i	0.960228	−	0.279218i	$-0.0900751\pi$
$98$	37130.8i	0.390544i
$99$	−117335.	−1.20321
$100$	0	0
$101$	−127703.	−1.24565	−0.622826	−	0.782360i	$-0.714016\pi$
$101$	−127703.	−1.24565	−0.622826	+	0.782360i	$0.714016\pi$
$102$	− 5857.47i	− 0.0557455i
$103$	− 47897.1i	− 0.444853i	−0.974949	−	0.222426i	$-0.928602\pi$
$103$	− 47897.1i	− 0.444853i	0.974949	−	0.222426i	$-0.0713977\pi$
$104$	10437.5	0.0946263
$105$	0	0
$106$	−11474.5	−0.0991905
$107$	− 65909.7i	− 0.556532i	−0.960504	−	0.278266i	$-0.910240\pi$
$107$	− 65909.7i	− 0.556532i	0.960504	−	0.278266i	$-0.0897596\pi$
$108$	34900.1i	0.287917i
$109$	48669.5	0.392365	0.196183	−	0.980567i	$-0.437145\pi$
$109$	48669.5	0.392365	0.196183	+	0.980567i	$0.437145\pi$
$110$	0	0
$111$	56675.2	0.436602
$112$	218174.i	1.64345i
$113$	− 76457.5i	− 0.563279i	−0.959520	−	0.281640i	$-0.909122\pi$
$113$	− 76457.5i	− 0.563279i	0.959520	−	0.281640i	$-0.0908783\pi$
$114$	53591.7	0.386221
$115$	0	0
$116$	−76832.1	−0.530148
$117$	− 31923.8i	− 0.215601i
$118$	28082.8i	0.185667i
$119$	67290.2	0.435596
$120$	0	0
$121$	224783.	1.39572
$122$	− 29925.6i	− 0.182030i
$123$	126859.i	0.756063i
$124$	221905.	1.29602
$125$	0	0
$126$	−43285.4	−0.242893
$127$	242910.i	1.33640i	0.743982	+	0.668199i	$0.232935\pi$
$127$	242910.i	1.33640i	−0.743982	+	0.668199i	$0.767065\pi$
$128$	− 118954.i	− 0.641733i
$129$	55345.5	0.292825
$130$	0	0
$131$	80066.9	0.407638	0.203819	−	0.979009i	$-0.434665\pi$
$131$	80066.9	0.407638	0.203819	+	0.979009i	$0.434665\pi$
$132$	400696.i	2.00161i
$133$	615658.i	3.01794i
$134$	−21678.3	−0.104295
$135$	0	0
$136$	−17767.9	−0.0823739
$137$	394611.i	1.79625i	0.439736	+	0.898127i	$0.355072\pi$
$137$	394611.i	1.79625i	−0.439736	+	0.898127i	$0.644928\pi$
$138$	44076.5i	0.197020i
$139$	−420116.	−1.84430	−0.922151	−	0.386830i	$-0.873570\pi$
$139$	−420116.	−1.84430	−0.922151	+	0.386830i	$0.873570\pi$
$140$	0	0
$141$	615439.	2.60698
$142$	61166.8i	0.254563i
$143$	104975.i	0.429286i
$144$	−176201.	−0.708110
$145$	0	0
$146$	39688.8	0.154094
$147$	− 787653.i	− 3.00636i
$148$	− 84649.9i	− 0.317667i
$149$	−164639.	−0.607528	−0.303764	−	0.952747i	$-0.598243\pi$
$149$	−164639.	−0.607528	−0.303764	+	0.952747i	$0.598243\pi$
$150$	0	0
$151$	475633.	1.69758	0.848789	−	0.528732i	$-0.177332\pi$
$151$	475633.	1.69758	0.848789	+	0.528732i	$0.177332\pi$
$152$	− 162564.i	− 0.570710i
$153$	54344.6i	0.187684i
$154$	142336.	0.483629
$155$	0	0
$156$	−109019.	−0.358666
$157$	− 314374.i	− 1.01788i	−0.860802	−	0.508940i	$-0.830037\pi$
$157$	− 314374.i	− 1.01788i	0.860802	−	0.508940i	$-0.169963\pi$
$158$	− 1857.17i	− 0.00591845i
$159$	243408.	0.763557
$160$	0	0
$161$	−506348.	−1.53952
$162$	67862.2i	0.203161i
$163$	− 118428.i	− 0.349130i	−0.984646	−	0.174565i	$-0.944148\pi$
$163$	− 118428.i	− 0.349130i	0.984646	−	0.174565i	$-0.0558518\pi$
$164$	189476.	0.550104
$165$	0	0
$166$	−6505.05	−0.0183223
$167$	− 311247.i	− 0.863603i	−0.901969	−	0.431802i	$-0.857878\pi$
$167$	− 311247.i	− 0.863603i	0.901969	−	0.431802i	$-0.142122\pi$
$168$	300208.i	0.820633i
$169$	−28561.0	−0.0769231
$170$	0	0
$171$	−497215.	−1.30033
$172$	− 82664.0i	− 0.213057i
$173$	− 658492.i	− 1.67277i	−0.548145	−	0.836383i	$-0.684666\pi$
$173$	− 658492.i	− 1.67277i	0.548145	−	0.836383i	$-0.315334\pi$
$174$	−50396.4	−0.126190
$175$	0	0
$176$	579401.	1.40993
$177$	− 595718.i	− 1.42925i
$178$	136991.i	0.324072i
$179$	−240265.	−0.560478	−0.280239	−	0.959930i	$-0.590414\pi$
$179$	−240265.	−0.560478	−0.280239	+	0.959930i	$0.590414\pi$
$180$	0	0
$181$	−125020.	−0.283651	−0.141825	−	0.989892i	$-0.545297\pi$
$181$	−125020.	−0.283651	−0.141825	+	0.989892i	$0.545297\pi$
$182$	38725.8i	0.0866607i
$183$	634810.i	1.40125i
$184$	133701.	0.291132
$185$	0	0
$186$	145554.	0.308490
$187$	− 178702.i	− 0.373701i
$188$	− 919218.i	− 1.89681i
$189$	−262982.	−0.535514
$190$	0	0
$191$	−175558.	−0.348207	−0.174103	−	0.984727i	$-0.555703\pi$
$191$	−175558.	−0.348207	−0.174103	+	0.984727i	$0.555703\pi$
$192$	561482.i	1.09921i
$193$	− 394803.i	− 0.762935i	−0.924382	−	0.381467i	$-0.875419\pi$
$193$	− 394803.i	− 0.762935i	0.924382	−	0.381467i	$-0.124581\pi$
$194$	−50698.3	−0.0967139
$195$	0	0
$196$	−1.17644e6	−2.18740
$197$	371003.i	0.681102i	0.940226	+	0.340551i	$0.110614\pi$
$197$	371003.i	0.681102i	−0.940226	+	0.340551i	$0.889386\pi$
$198$	114953.i	0.208380i
$199$	−54437.1	−0.0974456	−0.0487228	−	0.998812i	$-0.515515\pi$
$199$	−54437.1	−0.0974456	−0.0487228	+	0.998812i	$0.515515\pi$
$200$	0	0
$201$	459860.	0.802852
$202$	125110.i	0.215731i
$203$	− 578950.i	− 0.986055i
$204$	185585.	0.312225
$205$	0	0
$206$	−46924.6	−0.0770428
$207$	− 408934.i	− 0.663327i
$208$	157640.i	0.252643i
$209$	1.63500e6	2.58911
$210$	0	0
$211$	69737.6	0.107835	0.0539176	−	0.998545i	$-0.482829\pi$
$211$	69737.6	0.107835	0.0539176	+	0.998545i	$0.482829\pi$
$212$	− 363554.i	− 0.555557i
$213$	− 1.29753e6i	− 1.95960i
$214$	−64571.4	−0.0963841
$215$	0	0
$216$	69440.1	0.101269
$217$	1.67211e6i	2.41055i
$218$	− 47681.2i	− 0.0679526i
$219$	−841915.	−1.18620
$220$	0	0
$221$	48620.0	0.0669630
$222$	− 55524.4i	− 0.0756138i
$223$	− 175963.i	− 0.236951i	−0.992957	−	0.118476i	$-0.962199\pi$
$223$	− 175963.i	− 0.236951i	0.992957	−	0.118476i	$-0.0378007\pi$
$224$	675998.	0.900172
$225$	0	0
$226$	−74905.0	−0.0975528
$227$	113441.i	0.146118i	0.997328	+	0.0730591i	$0.0232762\pi$
$227$	113441.i	0.146118i	−0.997328	+	0.0730591i	$0.976724\pi$
$228$	1.69798e6i	2.16319i
$229$	−597110.	−0.752430	−0.376215	−	0.926532i	$-0.622775\pi$
$229$	−597110.	−0.752430	−0.376215	+	0.926532i	$0.622775\pi$
$230$	0	0
$231$	−3.01935e6	−3.72292
$232$	152871.i	0.186469i
$233$	− 1.04890e6i	− 1.26574i	−0.774259	−	0.632868i	$-0.781877\pi$
$233$	− 1.04890e6i	− 1.26574i	0.774259	−	0.632868i	$-0.218123\pi$
$234$	−31275.6	−0.0373393
$235$	0	0
$236$	−889764.	−1.03991
$237$	39395.9i	0.0455596i
$238$	− 65923.8i	− 0.0754397i
$239$	887699.	1.00524	0.502622	−	0.864507i	$-0.332369\pi$
$239$	887699.	1.00524	0.502622	+	0.864507i	$0.332369\pi$
$240$	0	0
$241$	−186313.	−0.206633	−0.103317	−	0.994649i	$-0.532946\pi$
$241$	−186313.	−0.206633	−0.103317	+	0.994649i	$0.532946\pi$
$242$	− 220218.i	− 0.241721i
$243$	− 1.16634e6i	− 1.26709i
$244$	948150.	1.01954
$245$	0	0
$246$	124283.	0.130940
$247$	444839.i	0.463939i
$248$	− 441520.i	− 0.455849i
$249$	137991.	0.141043
$250$	0	0
$251$	1.51057e6	1.51341	0.756703	−	0.653758i	$-0.226809\pi$
$251$	1.51057e6	1.51341	0.756703	+	0.653758i	$0.226809\pi$
$252$	− 1.37144e6i	− 1.36042i
$253$	1.34470e6i	1.32076i
$254$	237978.	0.231447
$255$	0	0
$256$	748021.	0.713368
$257$	− 1.81310e6i	− 1.71233i	−0.516700	−	0.856166i	$-0.672840\pi$
$257$	− 1.81310e6i	− 1.71233i	0.516700	−	0.856166i	$-0.327160\pi$
$258$	− 54221.7i	− 0.0507135i
$259$	637860.	0.590848
$260$	0	0
$261$	467569.	0.424859
$262$	− 78441.1i	− 0.0705977i
$263$	132233.i	0.117883i	0.998261	+	0.0589414i	$0.0187725\pi$
$263$	132233.i	0.117883i	−0.998261	+	0.0589414i	$0.981227\pi$
$264$	797258.	0.704027
$265$	0	0
$266$	603157.	0.522668
$267$	− 2.90598e6i	− 2.49467i
$268$	− 686846.i	− 0.584147i
$269$	971922.	0.818937	0.409469	−	0.912324i	$-0.365714\pi$
$269$	971922.	0.818937	0.409469	+	0.912324i	$0.365714\pi$
$270$	0	0
$271$	−919603.	−0.760637	−0.380318	−	0.924856i	$-0.624186\pi$
$271$	−919603.	−0.760637	−0.380318	+	0.924856i	$0.624186\pi$
$272$	− 268354.i	− 0.219931i
$273$	− 821487.i	− 0.667105i
$274$	386598.	0.311088
$275$	0	0
$276$	−1.39650e6	−1.10349
$277$	165694.i	0.129750i	0.997893	+	0.0648748i	$0.0206648\pi$
$277$	165694.i	0.129750i	−0.997893	+	0.0648748i	$0.979335\pi$
$278$	411585.i	0.319409i
$279$	−1.35042e6	−1.03863
$280$	0	0
$281$	949022.	0.716986	0.358493	−	0.933533i	$-0.383291\pi$
$281$	949022.	0.716986	0.358493	+	0.933533i	$0.383291\pi$
$282$	− 602942.i	− 0.451495i
$283$	− 1.59906e6i	− 1.18686i	−0.804886	−	0.593429i	$-0.797774\pi$
$283$	− 1.59906e6i	− 1.18686i	0.804886	−	0.593429i	$-0.202226\pi$
$284$	−1.93798e6	−1.42578
$285$	0	0
$286$	102844.	0.0743468
$287$	1.42775e6i	1.02317i
$288$	545947.i	0.387855i
$289$	1.33709e6	0.941708
$290$	0	0
$291$	1.07546e6	0.744493
$292$	1.25748e6i	0.863067i
$293$	1.67468e6i	1.13963i	0.821774	+	0.569814i	$0.192985\pi$
$293$	1.67468e6i	1.13963i	−0.821774	+	0.569814i	$0.807015\pi$
$294$	−771659.	−0.520663
$295$	0	0
$296$	−168427.	−0.111733
$297$	698397.i	0.459421i
$298$	161296.i	0.105216i
$299$	−365858.	−0.236665
$300$	0	0
$301$	622895.	0.396277
$302$	− 465975.i	− 0.293999i
$303$	− 2.65394e6i	− 1.66067i
$304$	2.45525e6	1.52374
$305$	0	0
$306$	53241.1	0.0325045
$307$	− 2.14086e6i	− 1.29641i	−0.761467	−	0.648203i	$-0.775521\pi$
$307$	− 2.14086e6i	− 1.29641i	0.761467	−	0.648203i	$-0.224479\pi$
$308$	4.50970e6i	2.70876i
$309$	995406.	0.593067
$310$	0	0
$311$	573238.	0.336073	0.168037	−	0.985781i	$-0.446257\pi$
$311$	573238.	0.336073	0.168037	+	0.985781i	$0.446257\pi$
$312$	216913.i	0.126153i
$313$	2.90875e6i	1.67821i	0.543970	+	0.839104i	$0.316920\pi$
$313$	2.90875e6i	1.67821i	−0.543970	+	0.839104i	$0.683080\pi$
$314$	−307990.	−0.176284
$315$	0	0
$316$	58841.6	0.0331487
$317$	− 739814.i	− 0.413499i	−0.978394	−	0.206749i	$-0.933711\pi$
$317$	− 739814.i	− 0.413499i	0.978394	−	0.206749i	$-0.0662885\pi$
$318$	− 238465.i	− 0.132238i
$319$	−1.53751e6	−0.845943
$320$	0	0
$321$	1.36975e6	0.741955
$322$	496066.i	0.266624i
$323$	− 757260.i	− 0.403867i
$324$	−2.15012e6	−1.13789
$325$	0	0
$326$	−116024.	−0.0604648
$327$	1.01146e6i	0.523092i
$328$	− 376997.i	− 0.193488i
$329$	6.92655e6	3.52799
$330$	0	0
$331$	2.93930e6	1.47460	0.737299	−	0.675566i	$-0.236101\pi$
$331$	2.93930e6	1.47460	0.737299	+	0.675566i	$0.236101\pi$
$332$	− 206103.i	− 0.102622i
$333$	515146.i	0.254577i
$334$	−304927.	−0.149565
$335$	0	0
$336$	−4.53412e6	−2.19101
$337$	− 982760.i	− 0.471382i	−0.971828	−	0.235691i	$-0.924265\pi$
$337$	− 982760.i	− 0.471382i	0.971828	−	0.235691i	$-0.0757352\pi$
$338$	27981.1i	0.0133221i
$339$	1.58895e6	0.750951
$340$	0	0
$341$	4.44060e6	2.06803
$342$	487119.i	0.225201i
$343$	− 4.93367e6i	− 2.26430i
$344$	−164475.	−0.0749383
$345$	0	0
$346$	−645121.	−0.289702
$347$	2.67215e6i	1.19135i	0.803227	+	0.595673i	$0.203115\pi$
$347$	2.67215e6i	1.19135i	−0.803227	+	0.595673i	$0.796885\pi$
$348$	− 1.59674e6i	− 0.706781i
$349$	−2.71620e6	−1.19371	−0.596854	−	0.802350i	$-0.703583\pi$
$349$	−2.71620e6	−1.19371	−0.596854	+	0.802350i	$0.703583\pi$
$350$	0	0
$351$	−190015.	−0.0823230
$352$	− 1.79524e6i	− 0.772264i
$353$	− 2.78459e6i	− 1.18939i	−0.803951	−	0.594696i	$-0.797272\pi$
$353$	− 2.78459e6i	− 1.18939i	0.803951	−	0.594696i	$-0.202728\pi$
$354$	−583622.	−0.247527
$355$	0	0
$356$	−4.34036e6	−1.81510
$357$	1.39844e6i	0.580727i
$358$	235387.i	0.0970676i
$359$	1.18365e6	0.484714	0.242357	−	0.970187i	$-0.422079\pi$
$359$	1.18365e6	0.484714	0.242357	+	0.970187i	$0.422079\pi$
$360$	0	0
$361$	4.45230e6	1.79811
$362$	122482.i	0.0491247i
$363$	4.67147e6i	1.86075i
$364$	−1.22697e6	−0.485379
$365$	0	0
$366$	621920.	0.242679
$367$	1.42155e6i	0.550932i	0.961311	+	0.275466i	$0.0888322\pi$
$367$	1.42155e6i	0.550932i	−0.961311	+	0.275466i	$0.911168\pi$
$368$	2.01932e6i	0.777294i
$369$	−1.15308e6	−0.440851
$370$	0	0
$371$	2.73947e6	1.03331
$372$	4.61166e6i	1.72782i
$373$	− 488425.i	− 0.181772i	−0.995861	−	0.0908858i	$-0.971030\pi$
$373$	− 488425.i	− 0.181772i	0.995861	−	0.0908858i	$-0.0289698\pi$
$374$	−175073.	−0.0647203
$375$	0	0
$376$	−1.82895e6	−0.667164
$377$	− 418316.i	− 0.151583i
$378$	257642.i	0.0927442i
$379$	−3.64996e6	−1.30524	−0.652619	−	0.757686i	$-0.726330\pi$
$379$	−3.64996e6	−1.30524	−0.652619	+	0.757686i	$0.726330\pi$
$380$	0	0
$381$	−5.04819e6	−1.78165
$382$	171993.i	0.0603049i
$383$	4.54415e6i	1.58291i	0.611228	+	0.791455i	$0.290676\pi$
$383$	4.54415e6i	1.58291i	−0.611228	+	0.791455i	$0.709324\pi$
$384$	2.47212e6	0.855544
$385$	0	0
$386$	−386786.	−0.132130
$387$	503060.i	0.170743i
$388$	− 1.60630e6i	− 0.541686i
$389$	−360319.	−0.120729	−0.0603646	−	0.998176i	$-0.519226\pi$
$389$	−360319.	−0.120729	−0.0603646	+	0.998176i	$0.519226\pi$
$390$	0	0
$391$	622808.	0.206021
$392$	2.34074e6i	0.769373i
$393$	1.66396e6i	0.543453i
$394$	363470.	0.117958
$395$	0	0
$396$	−3.64210e6	−1.16712
$397$	3.31026e6i	1.05411i	0.849831	+	0.527055i	$0.176704\pi$
$397$	3.31026e6i	1.05411i	−0.849831	+	0.527055i	$0.823296\pi$
$398$	53331.7i	0.0168763i
$399$	−1.27947e7	−4.02344
$400$	0	0
$401$	3.77349e6	1.17188	0.585938	−	0.810356i	$-0.300726\pi$
$401$	3.77349e6	1.17188	0.585938	+	0.810356i	$0.300726\pi$
$402$	− 450522.i	− 0.139044i
$403$	1.20817e6i	0.370566i
$404$	−3.96392e6	−1.20829
$405$	0	0
$406$	−567194.	−0.170772
$407$	− 1.69396e6i	− 0.506893i
$408$	− 369256.i	− 0.109819i
$409$	−2.46939e6	−0.729932	−0.364966	−	0.931021i	$-0.618919\pi$
$409$	−2.46939e6	−0.729932	−0.364966	+	0.931021i	$0.618919\pi$
$410$	0	0
$411$	−8.20087e6	−2.39472
$412$	− 1.48674e6i	− 0.431510i
$413$	− 6.70461e6i	− 1.93419i
$414$	−400631.	−0.114880
$415$	0	0
$416$	488438.	0.138381
$417$	− 8.73092e6i	− 2.45878i
$418$	− 1.60180e6i	− 0.448401i
$419$	−2.12907e6	−0.592455	−0.296227	−	0.955117i	$-0.595729\pi$
$419$	−2.12907e6	−0.592455	−0.296227	+	0.955117i	$0.595729\pi$
$420$	0	0
$421$	−4.89905e6	−1.34712	−0.673560	−	0.739133i	$-0.735236\pi$
$421$	−4.89905e6	−1.34712	−0.673560	+	0.739133i	$0.735236\pi$
$422$	− 68321.5i	− 0.0186757i
$423$	5.59399e6i	1.52010i
$424$	−723356.	−0.195406
$425$	0	0
$426$	−1.27118e6	−0.339377
$427$	7.14457e6i	1.89630i
$428$	− 2.04585e6i	− 0.539839i
$429$	−2.18161e6	−0.572314
$430$	0	0
$431$	519640.	0.134744	0.0673720	−	0.997728i	$-0.478539\pi$
$431$	519640.	0.134744	0.0673720	+	0.997728i	$0.478539\pi$
$432$	1.04877e6i	0.270379i
$433$	2.08313e6i	0.533946i	0.963704	+	0.266973i	$0.0860234\pi$
$433$	2.08313e6i	0.533946i	−0.963704	+	0.266973i	$0.913977\pi$
$434$	1.63816e6	0.417476
$435$	0	0
$436$	1.51071e6	0.380597
$437$	5.69826e6i	1.42738i
$438$	824819.i	0.205435i
$439$	5.11967e6	1.26789	0.633944	−	0.773379i	$-0.281435\pi$
$439$	5.11967e6	1.26789	0.633944	+	0.773379i	$0.281435\pi$
$440$	0	0
$441$	7.15932e6	1.75297
$442$	− 47632.8i	− 0.0115971i
$443$	− 8.00110e6i	− 1.93705i	−0.248922	−	0.968523i	$-0.580076\pi$
$443$	− 8.00110e6i	− 1.93705i	0.248922	−	0.968523i	$-0.419924\pi$
$444$	1.75921e6	0.423506
$445$	0	0
$446$	−172390.	−0.0410369
$447$	− 3.42155e6i	− 0.809942i
$448$	6.31929e6i	1.48756i
$449$	−6.98258e6	−1.63456	−0.817279	−	0.576242i	$-0.804519\pi$
$449$	−6.98258e6	−1.63456	−0.817279	+	0.576242i	$0.804519\pi$
$450$	0	0
$451$	3.79166e6	0.877786
$452$	− 2.37326e6i	− 0.546384i
$453$	9.88469e6i	2.26317i
$454$	111137.	0.0253058
$455$	0	0
$456$	3.37843e6	0.760858
$457$	− 1.37626e6i	− 0.308256i	−0.988051	−	0.154128i	$-0.950743\pi$
$457$	− 1.37626e6i	− 0.308256i	0.988051	−	0.154128i	$-0.0492568\pi$
$458$	584986.i	0.130311i
$459$	323468.	0.0716637
$460$	0	0
$461$	−2.50047e6	−0.547987	−0.273993	−	0.961732i	$-0.588345\pi$
$461$	−2.50047e6	−0.547987	−0.273993	+	0.961732i	$0.588345\pi$
$462$	2.95804e6i	0.644762i
$463$	865000.i	0.187527i	0.995595	+	0.0937635i	$0.0298897\pi$
$463$	865000.i	0.187527i	−0.995595	+	0.0937635i	$0.970110\pi$
$464$	−2.30886e6	−0.497854
$465$	0	0
$466$	−1.02760e6	−0.219209
$467$	4.99073e6i	1.05894i	0.848328	+	0.529470i	$0.177609\pi$
$467$	4.99073e6i	1.05894i	−0.848328	+	0.529470i	$0.822391\pi$
$468$	− 990921.i	− 0.209134i
$469$	5.17557e6	1.08649
$470$	0	0
$471$	6.53336e6	1.35701
$472$	1.77035e6i	0.365766i
$473$	− 1.65421e6i	− 0.339969i
$474$	38595.9	0.00789033
$475$	0	0
$476$	2.08870e6	0.422531
$477$	2.21244e6i	0.445221i
$478$	− 869674.i	− 0.174095i
$479$	−2.74347e6	−0.546337	−0.273169	−	0.961966i	$-0.588072\pi$
$479$	−2.74347e6	−0.546337	−0.273169	+	0.961966i	$0.588072\pi$
$480$	0	0
$481$	460881.	0.0908293
$482$	182530.i	0.0357862i
$483$	− 1.05230e7i	− 2.05245i
$484$	6.97730e6	1.35386
$485$	0	0
$486$	−1.14265e6	−0.219444
$487$	7.22585e6i	1.38060i	0.723525	+	0.690298i	$0.242521\pi$
$487$	7.22585e6i	1.38060i	−0.723525	+	0.690298i	$0.757479\pi$
$488$	− 1.88652e6i	− 0.358601i
$489$	2.46120e6	0.465451
$490$	0	0
$491$	7.90576e6	1.47993	0.739963	−	0.672647i	$-0.234843\pi$
$491$	7.90576e6	1.47993	0.739963	+	0.672647i	$0.234843\pi$
$492$	3.93772e6i	0.733385i
$493$	712109.i	0.131956i
$494$	435807.	0.0803483
$495$	0	0
$496$	6.66839e6	1.21707
$497$	− 1.46032e7i	− 2.65190i
$498$	− 135189.i	− 0.0244269i
$499$	−505183.	−0.0908234	−0.0454117	−	0.998968i	$-0.514460\pi$
$499$	−505183.	−0.0908234	−0.0454117	+	0.998968i	$0.514460\pi$
$500$	0	0
$501$	6.46839e6	1.15134
$502$	− 1.47989e6i	− 0.262103i
$503$	− 2.15590e6i	− 0.379934i	−0.981790	−	0.189967i	$-0.939162\pi$
$503$	− 2.15590e6i	− 0.379934i	0.981790	−	0.189967i	$-0.0608381\pi$
$504$	−2.72872e6	−0.478501
$505$	0	0
$506$	1.31740e6	0.228739
$507$	− 593559.i	− 0.102552i
$508$	7.53997e6i	1.29631i
$509$	−3.81217e6	−0.652195	−0.326097	−	0.945336i	$-0.605734\pi$
$509$	−3.81217e6	−0.652195	−0.326097	+	0.945336i	$0.605734\pi$
$510$	0	0
$511$	−9.47546e6	−1.60527
$512$	− 4.53936e6i	− 0.765280i
$513$	2.95950e6i	0.496507i
$514$	−1.77628e6	−0.296554
$515$	0	0
$516$	1.71794e6	0.284042
$517$	− 1.83948e7i	− 3.02669i
$518$	− 624908.i	− 0.102327i
$519$	1.36849e7	2.23009
$520$	0	0
$521$	2.81958e6	0.455083	0.227541	−	0.973768i	$-0.426931\pi$
$521$	2.81958e6	0.455083	0.227541	+	0.973768i	$0.426931\pi$
$522$	− 458075.i	− 0.0735801i
$523$	6.85609e6i	1.09603i	0.836468	+	0.548015i	$0.184616\pi$
$523$	6.85609e6i	1.09603i	−0.836468	+	0.548015i	$0.815384\pi$
$524$	2.48529e6	0.395411
$525$	0	0
$526$	129548.	0.0204158
$527$	− 2.05670e6i	− 0.322585i
$528$	1.20412e7i	1.87969i
$529$	1.74981e6	0.271864
$530$	0	0
$531$	5.41474e6	0.833377
$532$	1.91101e7i	2.92742i
$533$	1.03161e6i	0.157289i
$534$	−2.84697e6	−0.432046
$535$	0	0
$536$	−1.36661e6	−0.205462
$537$	− 4.99323e6i	− 0.747216i
$538$	− 952187.i	− 0.141829i
$539$	−2.35420e7	−3.49038
$540$	0	0
$541$	−9.25845e6	−1.36002	−0.680010	−	0.733203i	$-0.738025\pi$
$541$	−9.25845e6	−1.36002	−0.680010	+	0.733203i	$0.738025\pi$
$542$	900930.i	0.131733i
$543$	− 2.59819e6i	− 0.378157i
$544$	−831479.	−0.120463
$545$	0	0
$546$	−804806.	−0.115534
$547$	− 311761.i	− 0.0445506i	−0.999752	−	0.0222753i	$-0.992909\pi$
$547$	− 311761.i	− 0.0445506i	0.999752	−	0.0222753i	$-0.00709103\pi$
$548$	1.22488e7i	1.74238i
$549$	−5.77006e6	−0.817052
$550$	0	0
$551$	−6.51530e6	−0.914230
$552$	2.77859e6i	0.388130i
$553$	443387.i	0.0616552i
$554$	162329.	0.0224710
$555$	0	0
$556$	−1.30405e7	−1.78898
$557$	6.50976e6i	0.889051i	0.895766	+	0.444526i	$0.146628\pi$
$557$	6.50976e6i	0.889051i	−0.895766	+	0.444526i	$0.853372\pi$
$558$	1.32300e6i	0.179877i
$559$	450068.	0.0609185
$560$	0	0
$561$	3.71381e6	0.498210
$562$	− 929752.i	− 0.124173i
$563$	− 1.11564e6i	− 0.148338i	−0.997246	−	0.0741691i	$-0.976370\pi$
$563$	− 1.11564e6i	− 0.148338i	0.997246	−	0.0741691i	$-0.0236305\pi$
$564$	1.91033e7	2.52878
$565$	0	0
$566$	−1.56659e6	−0.205549
$567$	− 1.62017e7i	− 2.11642i
$568$	3.85597e6i	0.501491i
$569$	−3.26296e6	−0.422504	−0.211252	−	0.977432i	$-0.567754\pi$
$569$	−3.26296e6	−0.422504	−0.211252	+	0.977432i	$0.567754\pi$
$570$	0	0
$571$	−1.19763e6	−0.153721	−0.0768605	−	0.997042i	$-0.524490\pi$
$571$	−1.19763e6	−0.153721	−0.0768605	+	0.997042i	$0.524490\pi$
$572$	3.25845e6i	0.416410i
$573$	− 3.64847e6i	− 0.464221i
$574$	1.39876e6	0.177200
$575$	0	0
$576$	−5.10356e6	−0.640939
$577$	− 6.91253e6i	− 0.864365i	−0.901786	−	0.432183i	$-0.857744\pi$
$577$	− 6.91253e6i	− 0.864365i	0.901786	−	0.432183i	$-0.142256\pi$
$578$	− 1.30994e6i	− 0.163092i
$579$	8.20486e6	1.01713
$580$	0	0
$581$	1.55304e6	0.190872
$582$	− 1.05362e6i	− 0.128937i
$583$	− 7.27518e6i	− 0.886487i
$584$	2.50199e6	0.303566
$585$	0	0
$586$	1.64068e6	0.197369
$587$	9.02641e6i	1.08123i	0.841269	+	0.540617i	$0.181809\pi$
$587$	9.02641e6i	1.08123i	−0.841269	+	0.540617i	$0.818191\pi$
$588$	− 2.44489e7i	− 2.91619i
$589$	1.88173e7	2.23496
$590$	0	0
$591$	−7.71025e6	−0.908029
$592$	− 2.54379e6i	− 0.298316i
$593$	− 1.37184e6i	− 0.160202i	−0.996787	−	0.0801010i	$-0.974476\pi$
$593$	− 1.37184e6i	− 0.160202i	0.996787	−	0.0801010i	$-0.0255243\pi$
$594$	684216.	0.0795659
$595$	0	0
$596$	−5.11042e6	−0.589306
$597$	− 1.13132e6i	− 0.129912i
$598$	358429.i	0.0409874i
$599$	−1.17573e6	−0.133887	−0.0669436	−	0.997757i	$-0.521325\pi$
$599$	−1.17573e6	−0.133887	−0.0669436	+	0.997757i	$0.521325\pi$
$600$	0	0
$601$	−4.81704e6	−0.543994	−0.271997	−	0.962298i	$-0.587684\pi$
$601$	−4.81704e6	−0.543994	−0.271997	+	0.962298i	$0.587684\pi$
$602$	− 610247.i	− 0.0686301i
$603$	4.17987e6i	0.468133i
$604$	1.47637e7	1.64666
$605$	0	0
$606$	−2.60005e6	−0.287608
$607$	− 4.41344e6i	− 0.486189i	−0.970003	−	0.243094i	$-0.921838\pi$
$607$	− 4.41344e6i	− 0.486189i	0.970003	−	0.243094i	$-0.0781625\pi$
$608$	− 7.60745e6i	− 0.834603i
$609$	1.20318e7	1.31458
$610$	0	0
$611$	5.00473e6	0.542348
$612$	1.68687e6i	0.182055i
$613$	1.06623e7i	1.14604i	0.819542	+	0.573020i	$0.194228\pi$
$613$	1.06623e7i	1.14604i	−0.819542	+	0.573020i	$0.805772\pi$
$614$	−2.09738e6	−0.224521
$615$	0	0
$616$	8.97287e6	0.952752
$617$	− 3.10171e6i	− 0.328011i	−0.986459	−	0.164005i	$-0.947559\pi$
$617$	− 3.10171e6i	− 0.328011i	0.986459	−	0.164005i	$-0.0524414\pi$
$618$	− 975194.i	− 0.102712i
$619$	−2.68711e6	−0.281876	−0.140938	−	0.990018i	$-0.545012\pi$
$619$	−2.68711e6	−0.281876	−0.140938	+	0.990018i	$0.545012\pi$
$620$	0	0
$621$	−2.43404e6	−0.253279
$622$	− 561598.i	− 0.0582036i
$623$	− 3.27058e7i	− 3.37601i
$624$	−3.27610e6	−0.336818
$625$	0	0
$626$	2.84969e6	0.290644
$627$	3.39787e7i	3.45174i
$628$	− 9.75822e6i	− 0.987351i
$629$	−784568.	−0.0790686
$630$	0	0
$631$	−9.93210e6	−0.993042	−0.496521	−	0.868025i	$-0.665389\pi$
$631$	−9.93210e6	−0.993042	−0.496521	+	0.868025i	$0.665389\pi$
$632$	− 117076.i	− 0.0116594i
$633$	1.44930e6i	0.143763i
$634$	−724792.	−0.0716127
$635$	0	0
$636$	7.55543e6	0.740656
$637$	− 6.40517e6i	− 0.625435i
$638$	1.50629e6i	0.146507i
$639$	1.17938e7	1.14262
$640$	0	0
$641$	1.72516e7	1.65838	0.829188	−	0.558969i	$-0.188803\pi$
$641$	1.72516e7	1.65838	0.829188	+	0.558969i	$0.188803\pi$
$642$	− 1.34193e6i	− 0.128497i
$643$	− 3.88223e6i	− 0.370300i	−0.982710	−	0.185150i	$-0.940723\pi$
$643$	− 3.88223e6i	− 0.370300i	0.982710	−	0.185150i	$-0.0592771\pi$
$644$	−1.57171e7	−1.49334
$645$	0	0
$646$	−741884.	−0.0699447
$647$	1.46226e7i	1.37330i	0.726990	+	0.686648i	$0.240919\pi$
$647$	1.46226e7i	1.37330i	−0.726990	+	0.686648i	$0.759081\pi$
$648$	4.27805e6i	0.400229i
$649$	−1.78053e7	−1.65935
$650$	0	0
$651$	−3.47501e7	−3.21368
$652$	− 3.67604e6i	− 0.338658i
$653$	8.45344e6i	0.775801i	0.921701	+	0.387901i	$0.126800\pi$
$653$	8.45344e6i	0.775801i	−0.921701	+	0.387901i	$0.873200\pi$
$654$	990919.	0.0905928
$655$	0	0
$656$	5.69389e6	0.516594
$657$	− 7.65253e6i	− 0.691659i
$658$	− 6.78591e6i	− 0.611003i
$659$	3.95720e6	0.354956	0.177478	−	0.984125i	$-0.443206\pi$
$659$	3.95720e6	0.354956	0.177478	+	0.984125i	$0.443206\pi$
$660$	0	0
$661$	−1.71259e6	−0.152458	−0.0762290	−	0.997090i	$-0.524288\pi$
$661$	−1.71259e6	−0.152458	−0.0762290	+	0.997090i	$0.524288\pi$
$662$	− 2.87962e6i	− 0.255382i
$663$	1.01043e6i	0.0892734i
$664$	−410080.	−0.0360951
$665$	0	0
$666$	504685.	0.0440895
$667$	− 5.35851e6i	− 0.466368i
$668$	− 9.66117e6i	− 0.837700i
$669$	3.65689e6	0.315898
$670$	0	0
$671$	1.89737e7	1.62685
$672$	1.40487e7i	1.20009i
$673$	1.02167e7i	0.869510i	0.900549	+	0.434755i	$0.143165\pi$
$673$	1.02167e7i	0.869510i	−0.900549	+	0.434755i	$0.856835\pi$
$674$	−962804.	−0.0816372
$675$	0	0
$676$	−886539.	−0.0746159
$677$	− 1.08748e7i	− 0.911905i	−0.890004	−	0.455952i	$-0.849299\pi$
$677$	− 1.08748e7i	− 0.911905i	0.890004	−	0.455952i	$-0.150701\pi$
$678$	− 1.55669e6i	− 0.130055i
$679$	1.21039e7	1.00751
$680$	0	0
$681$	−2.35754e6	−0.194801
$682$	− 4.35043e6i	− 0.358155i
$683$	− 1.05416e6i	− 0.0864675i	−0.999065	−	0.0432338i	$-0.986234\pi$
$683$	− 1.05416e6i	− 0.0864675i	0.999065	−	0.0432338i	$-0.0137660\pi$
$684$	−1.54337e7	−1.26133
$685$	0	0
$686$	−4.83349e6	−0.392148
$687$	− 1.24092e7i	− 1.00312i
$688$	− 2.48411e6i	− 0.200078i
$689$	1.97939e6	0.158848
$690$	0	0
$691$	−2.62440e6	−0.209091	−0.104545	−	0.994520i	$-0.533339\pi$
$691$	−2.62440e6	−0.209091	−0.104545	+	0.994520i	$0.533339\pi$
$692$	− 2.04397e7i	− 1.62259i
$693$	− 2.74442e7i	− 2.17079i
$694$	2.61790e6	0.206326
$695$	0	0
$696$	−3.17700e6	−0.248596
$697$	− 1.75614e6i	− 0.136923i
$698$	2.66105e6i	0.206735i
$699$	2.17984e7	1.68745
$700$	0	0
$701$	−1.03468e7	−0.795262	−0.397631	−	0.917545i	$-0.630168\pi$
$701$	−1.03468e7	−0.795262	−0.397631	+	0.917545i	$0.630168\pi$
$702$	186157.i	0.0142573i
$703$	− 7.17825e6i	− 0.547810i
$704$	1.67821e7	1.27618
$705$	0	0
$706$	−2.72805e6	−0.205988
$707$	− 2.98692e7i	− 2.24737i
$708$	− 1.84912e7i	− 1.38638i
$709$	−1.53654e7	−1.14796	−0.573982	−	0.818868i	$-0.694602\pi$
$709$	−1.53654e7	−1.14796	−0.573982	+	0.818868i	$0.694602\pi$
$710$	0	0
$711$	−358086.	−0.0265652
$712$	8.63594e6i	0.638424i
$713$	1.54763e7i	1.14010i
$714$	1.37004e6	0.100574
$715$	0	0
$716$	−7.45789e6	−0.543667
$717$	1.84483e7i	1.34017i
$718$	− 1.15961e6i	− 0.0839463i
$719$	−1.67683e6	−0.120967	−0.0604835	−	0.998169i	$-0.519264\pi$
$719$	−1.67683e6	−0.120967	−0.0604835	+	0.998169i	$0.519264\pi$
$720$	0	0
$721$	1.12030e7	0.802591
$722$	− 4.36190e6i	− 0.311410i
$723$	− 3.87199e6i	− 0.275479i
$724$	−3.88066e6	−0.275143
$725$	0	0
$726$	4.57661e6	0.322257
$727$	− 2.10416e7i	− 1.47653i	−0.674508	−	0.738267i	$-0.735644\pi$
$727$	− 2.10416e7i	− 1.47653i	0.674508	−	0.738267i	$-0.264356\pi$
$728$	2.44128e6i	0.170722i
$729$	7.40669e6	0.516185
$730$	0	0
$731$	−766162.	−0.0530306
$732$	1.97046e7i	1.35922i
$733$	1.39822e7i	0.961205i	0.876939	+	0.480602i	$0.159582\pi$
$733$	1.39822e7i	0.961205i	−0.876939	+	0.480602i	$0.840418\pi$
$734$	1.39269e6	0.0954144
$735$	0	0
$736$	6.25674e6	0.425749
$737$	− 1.37447e7i	− 0.932108i
$738$	1.12966e6i	0.0763497i
$739$	−1.81147e7	−1.22017	−0.610084	−	0.792337i	$-0.708864\pi$
$739$	−1.81147e7	−1.22017	−0.610084	+	0.792337i	$0.708864\pi$
$740$	0	0
$741$	−9.24472e6	−0.618512
$742$	− 2.68385e6i	− 0.178957i
$743$	− 1.12917e7i	− 0.750391i	−0.926946	−	0.375196i	$-0.877576\pi$
$743$	− 1.12917e7i	− 0.750391i	0.926946	−	0.375196i	$-0.122424\pi$
$744$	9.17573e6	0.607727
$745$	0	0
$746$	−478507.	−0.0314805
$747$	1.25426e6i	0.0822406i
$748$	− 5.54694e6i	− 0.362493i
$749$	1.54160e7	1.00408
$750$	0	0
$751$	1.94291e7	1.25705	0.628525	−	0.777789i	$-0.283659\pi$
$751$	1.94291e7	1.25705	0.628525	+	0.777789i	$0.283659\pi$
$752$	− 2.76232e7i	− 1.78127i
$753$	3.13928e7i	2.01764i
$754$	−409822.	−0.0262523
$755$	0	0
$756$	−8.16300e6	−0.519452
$757$	2.19245e7i	1.39056i	0.718740	+	0.695279i	$0.244719\pi$
$757$	2.19245e7i	1.39056i	−0.718740	+	0.695279i	$0.755281\pi$
$758$	3.57584e6i	0.226051i
$759$	−2.79458e7	−1.76081
$760$	0	0
$761$	−1.29865e7	−0.812888	−0.406444	−	0.913676i	$-0.633231\pi$
$761$	−1.29865e7	−0.812888	−0.406444	+	0.913676i	$0.633231\pi$
$762$	4.94569e6i	0.308560i
$763$	1.13836e7i	0.707894i
$764$	−5.44935e6	−0.337763
$765$	0	0
$766$	4.45188e6	0.274140
$767$	− 4.84436e6i	− 0.297337i
$768$	1.55455e7i	0.951046i
$769$	3.08811e7	1.88311	0.941556	−	0.336856i	$-0.109364\pi$
$769$	3.08811e7	1.88311	0.941556	+	0.336856i	$0.109364\pi$
$770$	0	0
$771$	3.76801e7	2.28284
$772$	− 1.22548e7i	− 0.740051i
$773$	8.07818e6i	0.486256i	0.969994	+	0.243128i	$0.0781735\pi$
$773$	8.07818e6i	0.486256i	−0.969994	+	0.243128i	$0.921827\pi$
$774$	492845.	0.0295704
$775$	0	0
$776$	−3.19603e6	−0.190527
$777$	1.32561e7i	0.787705i
$778$	353002.i	0.0209088i
$779$	1.60674e7	0.948643
$780$	0	0
$781$	−3.87816e7	−2.27509
$782$	− 610162.i	− 0.0356803i
$783$	− 2.78305e6i	− 0.162224i
$784$	−3.53528e7	−2.05415
$785$	0	0
$786$	1.63018e6	0.0941191
$787$	5.74061e6i	0.330386i	0.986261	+	0.165193i	$0.0528246\pi$
$787$	5.74061e6i	0.330386i	−0.986261	+	0.165193i	$0.947175\pi$
$788$	1.15160e7i	0.660673i
$789$	−2.74809e6	−0.157159
$790$	0	0
$791$	1.78831e7	1.01625
$792$	7.24663e6i	0.410510i
$793$	5.16225e6i	0.291512i
$794$	3.24305e6	0.182558
$795$	0	0
$796$	−1.68974e6	−0.0945228
$797$	2.83035e7i	1.57832i	0.614188	+	0.789160i	$0.289484\pi$
$797$	2.83035e7i	1.57832i	−0.614188	+	0.789160i	$0.710516\pi$
$798$	1.25349e7i	0.696809i
$799$	−8.51967e6	−0.472124
$800$	0	0
$801$	2.64137e7	1.45461
$802$	− 3.69686e6i	− 0.202954i
$803$	2.51639e7i	1.37717i
$804$	1.42741e7	0.778771
$805$	0	0
$806$	1.18364e6	0.0641773
$807$	2.01986e7i	1.09179i
$808$	7.88694e6i	0.424992i
$809$	−9.64902e6	−0.518336	−0.259168	−	0.965832i	$-0.583448\pi$
$809$	−9.64902e6	−0.518336	−0.259168	+	0.965832i	$0.583448\pi$
$810$	0	0
$811$	−8.69726e6	−0.464334	−0.232167	−	0.972676i	$-0.574582\pi$
$811$	−8.69726e6	−0.464334	−0.232167	+	0.972676i	$0.574582\pi$
$812$	− 1.79707e7i	− 0.956479i
$813$	− 1.91113e7i	− 1.01406i
$814$	−1.65956e6	−0.0877874
$815$	0	0
$816$	5.57697e6	0.293206
$817$	− 7.00984e6i	− 0.367412i
$818$	2.41925e6i	0.126415i
$819$	7.46685e6	0.388981
$820$	0	0
$821$	1.35245e7	0.700269	0.350135	−	0.936699i	$-0.386136\pi$
$821$	1.35245e7	0.700269	0.350135	+	0.936699i	$0.386136\pi$
$822$	8.03435e6i	0.414735i
$823$	− 4.97088e6i	− 0.255820i	−0.991786	−	0.127910i	$-0.959173\pi$
$823$	− 4.97088e6i	− 0.255820i	0.991786	−	0.127910i	$-0.0408268\pi$
$824$	−2.95813e6	−0.151775
$825$	0	0
$826$	−6.56847e6	−0.334976
$827$	− 2.39690e7i	− 1.21867i	−0.792913	−	0.609335i	$-0.791436\pi$
$827$	− 2.39690e7i	− 1.21867i	0.792913	−	0.609335i	$-0.208564\pi$
$828$	− 1.26934e7i	− 0.643431i
$829$	2.55170e7	1.28956	0.644782	−	0.764367i	$-0.276948\pi$
$829$	2.55170e7	1.28956	0.644782	+	0.764367i	$0.276948\pi$
$830$	0	0
$831$	−3.44347e6	−0.172979
$832$	4.56596e6i	0.228677i
$833$	1.09037e7i	0.544453i
$834$	−8.55363e6	−0.425829
$835$	0	0
$836$	5.07506e7	2.51145
$837$	8.03792e6i	0.396580i
$838$	2.08584e6i	0.102606i
$839$	−2.32753e7	−1.14154	−0.570768	−	0.821111i	$-0.693354\pi$
$839$	−2.32753e7	−1.14154	−0.570768	+	0.821111i	$0.693354\pi$
$840$	0	0
$841$	−1.43843e7	−0.701293
$842$	4.79957e6i	0.233304i
$843$	1.97227e7i	0.955868i
$844$	2.16467e6	0.104601
$845$	0	0
$846$	5.48040e6	0.263261
$847$	5.25758e7i	2.51813i
$848$	− 1.09250e7i	− 0.521715i
$849$	3.32319e7	1.58229
$850$	0	0
$851$	5.90375e6	0.279450
$852$	− 4.02755e7i	− 1.90082i
$853$	− 2.63162e7i	− 1.23837i	−0.785245	−	0.619186i	$-0.787463\pi$
$853$	− 2.63162e7i	− 1.23837i	0.785245	−	0.619186i	$-0.212537\pi$
$854$	6.99949e6	0.328414
$855$	0	0
$856$	−4.07059e6	−0.189877
$857$	1.31408e7i	0.611179i	0.952163	+	0.305589i	$0.0988535\pi$
$857$	1.31408e7i	0.611179i	−0.952163	+	0.305589i	$0.901146\pi$
$858$	2.13731e6i	0.0991174i
$859$	−2.09626e7	−0.969311	−0.484655	−	0.874705i	$-0.661055\pi$
$859$	−2.09626e7	−0.969311	−0.484655	+	0.874705i	$0.661055\pi$
$860$	0	0
$861$	−2.96718e7	−1.36407
$862$	− 509088.i	− 0.0233359i
$863$	1.62144e7i	0.741095i	0.928813	+	0.370548i	$0.120830\pi$
$863$	1.62144e7i	0.741095i	−0.928813	+	0.370548i	$0.879170\pi$
$864$	3.24956e6	0.148095
$865$	0	0
$866$	2.04083e6	0.0924726
$867$	2.77876e7i	1.25546i
$868$	5.19026e7i	2.33824i
$869$	1.17750e6	0.0528945
$870$	0	0
$871$	3.73957e6	0.167023
$872$	− 3.00583e6i	− 0.133867i
$873$	9.77530e6i	0.434105i
$874$	5.58255e6	0.247203
$875$	0	0
$876$	−2.61332e7	−1.15062
$877$	1.83511e7i	0.805682i	0.915270	+	0.402841i	$0.131977\pi$
$877$	1.83511e7i	0.805682i	−0.915270	+	0.402841i	$0.868023\pi$
$878$	− 5.01571e6i	− 0.219582i
$879$	−3.48035e7	−1.51933
$880$	0	0
$881$	−2.38315e7	−1.03445	−0.517227	−	0.855848i	$-0.673036\pi$
$881$	−2.38315e7	−1.03445	−0.517227	+	0.855848i	$0.673036\pi$
$882$	− 7.01395e6i	− 0.303593i
$883$	− 2.82191e6i	− 0.121798i	−0.998144	−	0.0608992i	$-0.980603\pi$
$883$	− 2.82191e6i	− 0.121798i	0.998144	−	0.0608992i	$-0.0193968\pi$
$884$	1.50918e6	0.0649545
$885$	0	0
$886$	−7.83863e6	−0.335472
$887$	2.51901e7i	1.07503i	0.843253	+	0.537516i	$0.180637\pi$
$887$	2.51901e7i	1.07503i	−0.843253	+	0.537516i	$0.819363\pi$
$888$	− 3.50027e6i	− 0.148960i
$889$	−5.68157e7	−2.41109
$890$	0	0
$891$	−4.30266e7	−1.81570
$892$	− 5.46192e6i	− 0.229844i
$893$	− 7.79490e7i	− 3.27101i
$894$	−3.35207e6	−0.140272
$895$	0	0
$896$	2.78229e7	1.15780
$897$	− 7.60332e6i	− 0.315516i
$898$	6.84080e6i	0.283085i
$899$	−1.76954e7	−0.730231
$900$	0	0
$901$	−3.36955e6	−0.138280
$902$	− 3.71467e6i	− 0.152021i
$903$	1.29451e7i	0.528307i
$904$	−4.72202e6	−0.192180
$905$	0	0
$906$	9.68397e6	0.391952
$907$	− 1.27183e7i	− 0.513346i	−0.966498	−	0.256673i	$-0.917374\pi$
$907$	− 1.27183e7i	− 0.513346i	0.966498	−	0.256673i	$-0.0826264\pi$
$908$	3.52122e6i	0.141735i
$909$	2.41228e7	0.968319
$910$	0	0
$911$	1.25714e7	0.501864	0.250932	−	0.968005i	$-0.419263\pi$
$911$	1.25714e7	0.501864	0.250932	+	0.968005i	$0.419263\pi$
$912$	5.10254e7i	2.03142i
$913$	− 4.12439e6i	− 0.163751i
$914$	−1.34832e6	−0.0533860
$915$	0	0
$916$	−1.85344e7	−0.729861
$917$	1.87273e7i	0.735449i
$918$	− 316900.i	− 0.0124112i
$919$	4.77489e7	1.86498	0.932491	−	0.361194i	$-0.117631\pi$
$919$	4.77489e7	1.86498	0.932491	+	0.361194i	$0.117631\pi$
$920$	0	0
$921$	4.44916e7	1.72834
$922$	2.44970e6i	0.0949042i
$923$	− 1.05514e7i	− 0.407669i
$924$	−9.37213e7	−3.61126
$925$	0	0
$926$	847436.	0.0324773
$927$	9.04768e6i	0.345810i
$928$	7.15386e6i	0.272691i
$929$	4.02091e7	1.52857	0.764284	−	0.644880i	$-0.223093\pi$
$929$	4.02091e7	1.52857	0.764284	+	0.644880i	$0.223093\pi$
$930$	0	0
$931$	−9.97609e7	−3.77213
$932$	− 3.25580e7i	− 1.22777i
$933$	1.19131e7i	0.448045i
$934$	4.88939e6	0.183395
$935$	0	0
$936$	−1.97162e6	−0.0735586
$937$	− 2.41476e7i	− 0.898514i	−0.893403	−	0.449257i	$-0.851689\pi$
$937$	− 2.41476e7i	− 0.898514i	0.893403	−	0.449257i	$-0.148311\pi$
$938$	− 5.07047e6i	− 0.188166i
$939$	−6.04502e7	−2.23735
$940$	0	0
$941$	5.26445e7	1.93811	0.969055	−	0.246843i	$-0.0793932\pi$
$941$	5.26445e7	1.93811	0.969055	+	0.246843i	$0.0793932\pi$
$942$	− 6.40070e6i	− 0.235017i
$943$	1.32146e7i	0.483923i
$944$	−2.67380e7	−0.976561
$945$	0	0
$946$	−1.62063e6	−0.0588782
$947$	− 3.68602e7i	− 1.33562i	−0.744331	−	0.667811i	$-0.767232\pi$
$947$	− 3.68602e7i	− 1.33562i	0.744331	−	0.667811i	$-0.232768\pi$
$948$	1.22286e6i	0.0441931i
$949$	−6.84643e6	−0.246773
$950$	0	0
$951$	1.53749e7	0.551267
$952$	− 4.15585e6i	− 0.148617i
$953$	− 1.39818e7i	− 0.498691i	−0.968415	−	0.249346i	$-0.919784\pi$
$953$	− 1.39818e7i	− 0.498691i	0.968415	−	0.249346i	$-0.0802155\pi$
$954$	2.16752e6	0.0771066
$955$	0	0
$956$	2.75544e7	0.975092
$957$	− 3.19528e7i	− 1.12779i
$958$	2.68776e6i	0.0946186i
$959$	−9.22980e7	−3.24075
$960$	0	0
$961$	2.24782e7	0.785150
$962$	− 451523.i	− 0.0157305i
$963$	1.24502e7i	0.432625i
$964$	−5.78319e6	−0.200436
$965$	0	0
$966$	−1.03093e7	−0.355457
$967$	4.58056e7i	1.57526i	0.616147	+	0.787631i	$0.288693\pi$
$967$	4.58056e7i	1.57526i	−0.616147	+	0.787631i	$0.711307\pi$
$968$	− 1.38826e7i	− 0.476193i
$969$	1.57375e7	0.538426
$970$	0	0
$971$	4.00170e7	1.36206	0.681030	−	0.732256i	$-0.261532\pi$
$971$	4.00170e7	1.36206	0.681030	+	0.732256i	$0.261532\pi$
$972$	− 3.62033e7i	− 1.22909i
$973$	− 9.82635e7i	− 3.32744i
$974$	7.07913e6	0.239102
$975$	0	0
$976$	2.84926e7	0.957431
$977$	− 1.23889e7i	− 0.415236i	−0.978210	−	0.207618i	$-0.933429\pi$
$977$	− 1.23889e7i	− 0.415236i	0.978210	−	0.207618i	$-0.0665711\pi$
$978$	− 2.41122e6i	− 0.0806102i
$979$	−8.68563e7	−2.89631
$980$	0	0
$981$	−9.19358e6	−0.305008
$982$	− 7.74523e6i	− 0.256304i
$983$	4.61240e7i	1.52245i	0.648487	+	0.761226i	$0.275402\pi$
$983$	4.61240e7i	1.52245i	−0.648487	+	0.761226i	$0.724598\pi$
$984$	7.83482e6	0.257953
$985$	0	0
$986$	697649.	0.0228531
$987$	1.43949e8i	4.70343i
$988$	1.38079e7i	0.450024i
$989$	5.76524e6	0.187425
$990$	0	0
$991$	3.28504e7	1.06257	0.531284	−	0.847194i	$-0.321710\pi$
$991$	3.28504e7	1.06257	0.531284	+	0.847194i	$0.321710\pi$
$992$	− 2.06616e7i	− 0.666630i
$993$	6.10850e7i	1.96590i
$994$	−1.43067e7	−0.459275
$995$	0	0
$996$	4.28327e6	0.136813
$997$	1.53268e7i	0.488329i	0.969734	+	0.244164i	$0.0785136\pi$
$997$	1.53268e7i	0.488329i	−0.969734	+	0.244164i	$0.921486\pi$
$998$	494925.i	0.0157294i
$999$	3.06623e6	0.0972055

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	325.6.b.i.274.11		22
5.2	odd	4		325.6.a.k.1.6	yes	11
5.3	odd	4		325.6.a.j.1.6	✓	11
5.4	even	2	inner	325.6.b.i.274.12		22

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
325.6.a.j.1.6	✓	11	5.3	odd	4
325.6.a.k.1.6	yes	11	5.2	odd	4
325.6.b.i.274.11		22	1.1	even	1	trivial
325.6.b.i.274.12		22	5.4	even	2	inner

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)

\(f(q)\)	\(=\)	\(q-0.979695i q^{2} +20.7822i q^{3} +31.0402 q^{4} +20.3602 q^{6} +233.896i q^{7} -61.7601i q^{8} -188.898 q^{9} +O(q^{10})\)	\(q-0.979695i q^{2} +20.7822i q^{3} +31.0402 q^{4} +20.3602 q^{6} +233.896i q^{7} -61.7601i q^{8} -188.898 q^{9} +621.155 q^{11} +645.082i q^{12} +169.000i q^{13} +229.147 q^{14} +932.780 q^{16} -287.693i q^{17} +185.063i q^{18} +2632.19 q^{19} -4860.87 q^{21} -608.542i q^{22} +2164.84i q^{23} +1283.51 q^{24} +165.568 q^{26} +1124.35i q^{27} +7260.18i q^{28} -2475.24 q^{29} +7148.94 q^{31} -2890.16i q^{32} +12908.9i q^{33} -281.851 q^{34} -5863.44 q^{36} -2727.11i q^{37} -2578.74i q^{38} -3512.19 q^{39} +6104.22 q^{41} +4762.17i q^{42} -2663.13i q^{43} +19280.8 q^{44} +2120.88 q^{46} -29613.8i q^{47} +19385.2i q^{48} -37900.4 q^{49} +5978.87 q^{51} +5245.79i q^{52} -11712.3i q^{53} +1101.52 q^{54} +14445.5 q^{56} +54702.5i q^{57} +2424.98i q^{58} -28664.9 q^{59} +30545.9 q^{61} -7003.78i q^{62} -44182.6i q^{63} +27017.5 q^{64} +12646.8 q^{66} -22127.6i q^{67} -8930.03i q^{68} -44990.0 q^{69} -62434.6 q^{71} +11666.4i q^{72} +40511.4i q^{73} -2671.73 q^{74} +81703.6 q^{76} +145286. i q^{77} +3440.87i q^{78} +1895.66 q^{79} -69268.7 q^{81} -5980.27i q^{82} -6639.87i q^{83} -150882. q^{84} -2609.05 q^{86} -51440.9i q^{87} -38362.6i q^{88} -139830. q^{89} -39528.5 q^{91} +67197.1i q^{92} +148570. i q^{93} -29012.5 q^{94} +60063.9 q^{96} -51749.0i q^{97} +37130.8i q^{98} -117335. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 22 q - 374 q^{4} + 702 q^{6} - 2744 q^{9}+O(q^{10}) \) 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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