LMFDB - Embedded newform 325.6.b.i.274.2

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			274.2
Character	$\chi$	$=$	325.274
Dual form			325.6.b.i.274.21

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-10.1023i q^{2} +24.5176i q^{3} -70.0572 q^{4} +247.685 q^{6} -72.1186i q^{7} +384.466i q^{8} -358.110 q^{9} +O(q^{10})$	\(q-10.1023i q^{2} +24.5176i q^{3} -70.0572 q^{4} +247.685 q^{6} -72.1186i q^{7} +384.466i q^{8} -358.110 q^{9} +127.428 q^{11} -1717.63i q^{12} -169.000i q^{13} -728.566 q^{14} +1642.18 q^{16} +2152.48i q^{17} +3617.75i q^{18} -2726.20 q^{19} +1768.17 q^{21} -1287.32i q^{22} -2658.77i q^{23} -9426.17 q^{24} -1707.29 q^{26} -2822.22i q^{27} +5052.43i q^{28} +5885.60 q^{29} +1366.14 q^{31} -4286.91i q^{32} +3124.23i q^{33} +21745.0 q^{34} +25088.2 q^{36} -481.670i q^{37} +27541.0i q^{38} +4143.47 q^{39} +7385.44 q^{41} -17862.7i q^{42} -10167.0i q^{43} -8927.26 q^{44} -26859.8 q^{46} -16361.7i q^{47} +40262.2i q^{48} +11605.9 q^{49} -52773.4 q^{51} +11839.7i q^{52} +9061.97i q^{53} -28511.0 q^{54} +27727.2 q^{56} -66839.8i q^{57} -59458.3i q^{58} -29585.4 q^{59} +16410.0 q^{61} -13801.2i q^{62} +25826.4i q^{63} +9241.89 q^{64} +31562.0 q^{66} -61557.7i q^{67} -150796. i q^{68} +65186.5 q^{69} -5229.46 q^{71} -137681. i q^{72} -67851.0i q^{73} -4865.99 q^{74} +190990. q^{76} -9189.95i q^{77} -41858.7i q^{78} +89505.0 q^{79} -17826.8 q^{81} -74610.2i q^{82} -78989.9i q^{83} -123873. q^{84} -102711. q^{86} +144300. i q^{87} +48991.9i q^{88} -111252. q^{89} -12188.0 q^{91} +186266. i q^{92} +33494.5i q^{93} -165292. q^{94} +105105. q^{96} -71553.7i q^{97} -117247. i q^{98} -45633.3 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	− 10.1023i	− 1.78586i	−0.450198	−	0.892929i	$-0.648647\pi$
$2$	− 10.1023i	− 1.78586i	0.450198	−	0.892929i	$-0.351353\pi$
$3$	24.5176i	1.57280i	0.617717	+	0.786401i	$0.288058\pi$
$3$	24.5176i	1.57280i	−0.617717	+	0.786401i	$0.711942\pi$
$4$	−70.0572	−2.18929
$5$	0	0
$5$	0	0
$6$	247.685	2.80880
$7$	− 72.1186i	− 0.556291i	−0.960539	−	0.278146i	$-0.910280\pi$
$7$	− 72.1186i	− 0.556291i	0.960539	−	0.278146i	$-0.0897198\pi$
$8$	384.466i	2.12390i
$9$	−358.110	−1.47370
$10$	0	0
$11$	127.428	0.317529	0.158765	−	0.987316i	$-0.449249\pi$
$11$	127.428	0.317529	0.158765	+	0.987316i	$0.449249\pi$
$12$	− 1717.63i	− 3.44331i
$13$	− 169.000i	− 0.277350i
$13$	− 169.000i	− 0.277350i
$14$	−728.566	−0.993457
$15$	0	0
$16$	1642.18	1.60369
$17$	2152.48i	1.80641i	0.429211	+	0.903204i	$0.358792\pi$
$17$	2152.48i	1.80641i	−0.429211	+	0.903204i	$0.641208\pi$
$18$	3617.75i	2.63183i
$19$	−2726.20	−1.73250	−0.866252	−	0.499608i	$-0.833477\pi$
$19$	−2726.20	−1.73250	−0.866252	+	0.499608i	$0.833477\pi$
$20$	0	0
$21$	1768.17	0.874936
$22$	− 1287.32i	− 0.567062i
$23$	− 2658.77i	− 1.04800i	−0.851719	−	0.524000i	$-0.824439\pi$
$23$	− 2658.77i	− 1.04800i	0.851719	−	0.524000i	$-0.175561\pi$
$24$	−9426.17	−3.34047
$25$	0	0
$26$	−1707.29	−0.495308
$27$	− 2822.22i	− 0.745044i
$28$	5052.43i	1.21788i
$29$	5885.60	1.29956	0.649779	−	0.760123i	$-0.274861\pi$
$29$	5885.60	1.29956	0.649779	+	0.760123i	$0.274861\pi$
$30$	0	0
$31$	1366.14	0.255324	0.127662	−	0.991818i	$-0.459253\pi$
$31$	1366.14	0.255324	0.127662	+	0.991818i	$0.459253\pi$
$32$	− 4286.91i	− 0.740064i
$33$	3124.23i	0.499411i
$34$	21745.0	3.22599
$35$	0	0
$36$	25088.2	3.22636
$37$	− 481.670i	− 0.0578423i	−0.999582	−	0.0289211i	$-0.990793\pi$
$37$	− 481.670i	− 0.0578423i	0.999582	−	0.0289211i	$-0.00920717\pi$
$38$	27541.0i	3.09400i
$39$	4143.47	0.436217
$40$	0	0
$41$	7385.44	0.686146	0.343073	−	0.939309i	$-0.388532\pi$
$41$	7385.44	0.686146	0.343073	+	0.939309i	$0.388532\pi$
$42$	− 17862.7i	− 1.56251i
$43$	− 10167.0i	− 0.838538i	−0.907862	−	0.419269i	$-0.862286\pi$
$43$	− 10167.0i	− 0.838538i	0.907862	−	0.419269i	$-0.137714\pi$
$44$	−8927.26	−0.695163
$45$	0	0
$46$	−26859.8	−1.87158
$47$	− 16361.7i	− 1.08040i	−0.841537	−	0.540200i	$-0.818349\pi$
$47$	− 16361.7i	− 1.08040i	0.841537	−	0.540200i	$-0.181651\pi$
$48$	40262.2i	2.52229i
$49$	11605.9	0.690540
$50$	0	0
$51$	−52773.4	−2.84112
$52$	11839.7i	0.607199i
$53$	9061.97i	0.443132i	0.975145	+	0.221566i	$0.0711168\pi$
$53$	9061.97i	0.443132i	−0.975145	+	0.221566i	$0.928883\pi$
$54$	−28511.0	−1.33054
$55$	0	0
$56$	27727.2	1.18151
$57$	− 66839.8i	− 2.72488i
$58$	− 59458.3i	− 2.32082i
$59$	−29585.4	−1.10649	−0.553245	−	0.833019i	$-0.686611\pi$
$59$	−29585.4	−1.10649	−0.553245	+	0.833019i	$0.686611\pi$
$60$	0	0
$61$	16410.0	0.564656	0.282328	−	0.959318i	$-0.408893\pi$
$61$	16410.0	0.564656	0.282328	+	0.959318i	$0.408893\pi$
$62$	− 13801.2i	− 0.455972i
$63$	25826.4i	0.819809i
$64$	9241.89	0.282040
$65$	0	0
$66$	31562.0	0.891876
$67$	− 61557.7i	− 1.67531i	−0.546198	−	0.837656i	$-0.683926\pi$
$67$	− 61557.7i	− 1.67531i	0.546198	−	0.837656i	$-0.316074\pi$
$68$	− 150796.i	− 3.95475i
$69$	65186.5	1.64829
$70$	0	0
$71$	−5229.46	−0.123115	−0.0615575	−	0.998104i	$-0.519607\pi$
$71$	−5229.46	−0.123115	−0.0615575	+	0.998104i	$0.519607\pi$
$72$	− 137681.i	− 3.13000i
$73$	− 67851.0i	− 1.49021i	−0.666944	−	0.745107i	$-0.732398\pi$
$73$	− 67851.0i	− 1.49021i	0.666944	−	0.745107i	$-0.267602\pi$
$74$	−4865.99	−0.103298
$75$	0	0
$76$	190990.	3.79295
$77$	− 9189.95i	− 0.176639i
$78$	− 41858.7i	− 0.779021i
$79$	89505.0	1.61354	0.806769	−	0.590867i	$-0.201214\pi$
$79$	89505.0	1.61354	0.806769	+	0.590867i	$0.201214\pi$
$80$	0	0
$81$	−17826.8	−0.301899
$82$	− 74610.2i	− 1.22536i
$83$	− 78989.9i	− 1.25857i	−0.777176	−	0.629283i	$-0.783348\pi$
$83$	− 78989.9i	− 1.25857i	0.777176	−	0.629283i	$-0.216652\pi$
$84$	−123873.	−1.91549
$85$	0	0
$86$	−102711.	−1.49751
$87$	144300.i	2.04395i
$88$	48991.9i	0.674399i
$89$	−111252.	−1.48879	−0.744394	−	0.667741i	$-0.767262\pi$
$89$	−111252.	−1.48879	−0.744394	+	0.667741i	$0.767262\pi$
$90$	0	0
$91$	−12188.0	−0.154287
$92$	186266.i	2.29437i
$93$	33494.5i	0.401574i
$94$	−165292.	−1.92944
$95$	0	0
$96$	105105.	1.16397
$97$	− 71553.7i	− 0.772151i	−0.922467	−	0.386076i	$-0.873830\pi$
$97$	− 71553.7i	− 0.772151i	0.922467	−	0.386076i	$-0.126170\pi$
$98$	− 117247.i	− 1.23321i
$99$	−45633.3	−0.467945
$100$	0	0
$101$	150676.	1.46974	0.734868	−	0.678210i	$-0.237244\pi$
$101$	150676.	1.46974	0.734868	+	0.678210i	$0.237244\pi$
$102$	533135.i	5.07384i
$103$	− 160191.i	− 1.48781i	−0.668288	−	0.743903i	$-0.732973\pi$
$103$	− 160191.i	− 1.48781i	0.668288	−	0.743903i	$-0.267027\pi$
$104$	64974.8	0.589063
$105$	0	0
$106$	91547.1	0.791371
$107$	78017.7i	0.658769i	0.944196	+	0.329385i	$0.106841\pi$
$107$	78017.7i	0.658769i	−0.944196	+	0.329385i	$0.893159\pi$
$108$	197717.i	1.63111i
$109$	219872.	1.77257	0.886284	−	0.463141i	$-0.153278\pi$
$109$	219872.	1.77257	0.886284	+	0.463141i	$0.153278\pi$
$110$	0	0
$111$	11809.4	0.0909745
$112$	− 118432.i	− 0.892118i
$113$	171300.i	1.26201i	0.775780	+	0.631004i	$0.217357\pi$
$113$	171300.i	1.26201i	−0.775780	+	0.631004i	$0.782643\pi$
$114$	−675238.	−4.86625
$115$	0	0
$116$	−412328.	−2.84510
$117$	60520.6i	0.408732i
$118$	298882.i	1.97603i
$119$	155234.	1.00489
$120$	0	0
$121$	−144813.	−0.899175
$122$	− 165779.i	− 1.00839i
$123$	181073.i	1.07917i
$124$	−95708.0	−0.558977
$125$	0	0
$126$	260907.	1.46406
$127$	− 98906.0i	− 0.544143i	−0.962277	−	0.272072i	$-0.912291\pi$
$127$	− 98906.0i	− 0.544143i	0.962277	−	0.272072i	$-0.0877088\pi$
$128$	− 230546.i	− 1.24375i
$129$	249271.	1.31885
$130$	0	0
$131$	266465.	1.35663	0.678317	−	0.734769i	$-0.262710\pi$
$131$	266465.	1.35663	0.678317	+	0.734769i	$0.262710\pi$
$132$	− 218875.i	− 1.09335i
$133$	196610.i	0.963777i
$134$	−621877.	−2.99187
$135$	0	0
$136$	−827554.	−3.83662
$137$	35452.7i	0.161379i	0.996739	+	0.0806896i	$0.0257122\pi$
$137$	35452.7i	0.161379i	−0.996739	+	0.0806896i	$0.974288\pi$
$138$	− 658536.i	− 2.94362i
$139$	63698.3	0.279635	0.139817	−	0.990177i	$-0.455348\pi$
$139$	63698.3	0.279635	0.139817	+	0.990177i	$0.455348\pi$
$140$	0	0
$141$	401150.	1.69926
$142$	52829.7i	0.219866i
$143$	− 21535.4i	− 0.0880668i
$144$	−588081.	−2.36336
$145$	0	0
$146$	−685453.	−2.66131
$147$	284548.i	1.08608i
$148$	33744.5i	0.126633i
$149$	26631.1	0.0982707	0.0491353	−	0.998792i	$-0.484353\pi$
$149$	26631.1	0.0982707	0.0491353	+	0.998792i	$0.484353\pi$
$150$	0	0
$151$	448794.	1.60179	0.800893	−	0.598808i	$-0.204359\pi$
$151$	448794.	1.60179	0.800893	+	0.598808i	$0.204359\pi$
$152$	− 1.04813e6i	− 3.67966i
$153$	− 770824.i	− 2.66211i
$154$	−92839.9	−0.315452
$155$	0	0
$156$	−290280.	−0.955003
$157$	− 16497.9i	− 0.0534170i	−0.999643	−	0.0267085i	$-0.991497\pi$
$157$	− 16497.9i	− 0.0534170i	0.999643	−	0.0267085i	$-0.00850260\pi$
$158$	− 904209.i	− 2.88155i
$159$	−222177.	−0.696959
$160$	0	0
$161$	−191747.	−0.582993
$162$	180093.i	0.539149i
$163$	− 83354.3i	− 0.245731i	−0.992423	−	0.122865i	$-0.960792\pi$
$163$	− 83354.3i	− 0.245731i	0.992423	−	0.122865i	$-0.0392083\pi$
$164$	−517403.	−1.50217
$165$	0	0
$166$	−797982.	−2.24762
$167$	215768.i	0.598680i	0.954146	+	0.299340i	$0.0967665\pi$
$167$	215768.i	0.598680i	−0.954146	+	0.299340i	$0.903233\pi$
$168$	679802.i	1.85827i
$169$	−28561.0	−0.0769231
$170$	0	0
$171$	976281.	2.55320
$172$	712273.i	1.83580i
$173$	− 42375.0i	− 0.107645i	−0.998551	−	0.0538225i	$-0.982859\pi$
$173$	− 42375.0i	− 0.107645i	0.998551	−	0.0538225i	$-0.0171405\pi$
$174$	1.45777e6	3.65020
$175$	0	0
$176$	209260.	0.509218
$177$	− 725362.i	− 1.74029i
$178$	1.12391e6i	2.65876i
$179$	−147708.	−0.344566	−0.172283	−	0.985047i	$-0.555114\pi$
$179$	−147708.	−0.344566	−0.172283	+	0.985047i	$0.555114\pi$
$180$	0	0
$181$	−742092.	−1.68369	−0.841843	−	0.539722i	$-0.818529\pi$
$181$	−742092.	−1.68369	−0.841843	+	0.539722i	$0.818529\pi$
$182$	123128.i	0.275535i
$183$	402333.i	0.888092i
$184$	1.02221e6	2.22584
$185$	0	0
$186$	338372.	0.717154
$187$	274286.i	0.573588i
$188$	1.14626e6i	2.36531i
$189$	−203535.	−0.414461
$190$	0	0
$191$	698156.	1.38474	0.692372	−	0.721541i	$-0.256566\pi$
$191$	698156.	1.38474	0.692372	+	0.721541i	$0.256566\pi$
$192$	226589.i	0.443593i
$193$	− 190607.i	− 0.368338i	−0.982895	−	0.184169i	$-0.941041\pi$
$193$	− 190607.i	− 0.368338i	0.982895	−	0.184169i	$-0.0589594\pi$
$194$	−722859.	−1.37895
$195$	0	0
$196$	−813077.	−1.51179
$197$	− 817552.i	− 1.50089i	−0.660931	−	0.750447i	$-0.729838\pi$
$197$	− 817552.i	− 1.50089i	0.660931	−	0.750447i	$-0.270162\pi$
$198$	461003.i	0.835682i
$199$	288709.	0.516806	0.258403	−	0.966037i	$-0.416804\pi$
$199$	288709.	0.516806	0.258403	+	0.966037i	$0.416804\pi$
$200$	0	0
$201$	1.50924e6	2.63493
$202$	− 1.52218e6i	− 2.62474i
$203$	− 424461.i	− 0.722933i
$204$	3.69716e6	6.22003
$205$	0	0
$206$	−1.61831e6	−2.65701
$207$	952132.i	1.54444i
$208$	− 277528.i	− 0.444783i
$209$	−347395.	−0.550121
$210$	0	0
$211$	134806.	0.208451	0.104225	−	0.994554i	$-0.466764\pi$
$211$	134806.	0.208451	0.104225	+	0.994554i	$0.466764\pi$
$212$	− 634856.i	− 0.970143i
$213$	− 128213.i	− 0.193635i
$214$	788161.	1.17647
$215$	0	0
$216$	1.08505e6	1.58240
$217$	− 98524.3i	− 0.142035i
$218$	− 2.22122e6i	− 3.16556i
$219$	1.66354e6	2.34381
$220$	0	0
$221$	363768.	0.501007
$222$	− 119302.i	− 0.162467i
$223$	320971.i	0.432218i	0.976369	+	0.216109i	$0.0693367\pi$
$223$	320971.i	0.432218i	−0.976369	+	0.216109i	$0.930663\pi$
$224$	−309166.	−0.411691
$225$	0	0
$226$	1.73053e6	2.25377
$227$	28722.2i	0.0369958i	0.999829	+	0.0184979i	$0.00588841\pi$
$227$	28722.2i	0.0369958i	−0.999829	+	0.0184979i	$0.994112\pi$
$228$	4.68261e6i	5.96555i
$229$	−850808.	−1.07212	−0.536059	−	0.844180i	$-0.680088\pi$
$229$	−850808.	−1.07212	−0.536059	+	0.844180i	$0.680088\pi$
$230$	0	0
$231$	225315.	0.277818
$232$	2.26281e6i	2.76013i
$233$	432423.i	0.521818i	0.965363	+	0.260909i	$0.0840222\pi$
$233$	432423.i	0.521818i	−0.965363	+	0.260909i	$0.915978\pi$
$234$	611400.	0.729937
$235$	0	0
$236$	2.07267e6	2.42242
$237$	2.19444e6i	2.53778i
$238$	− 1.56822e6i	− 1.79459i
$239$	−530968.	−0.601276	−0.300638	−	0.953738i	$-0.597200\pi$
$239$	−530968.	−0.601276	−0.300638	+	0.953738i	$0.597200\pi$
$240$	0	0
$241$	−741085.	−0.821912	−0.410956	−	0.911655i	$-0.634805\pi$
$241$	−741085.	−0.821912	−0.410956	+	0.911655i	$0.634805\pi$
$242$	1.46295e6i	1.60580i
$243$	− 1.12287e6i	− 1.21987i
$244$	−1.14964e6	−1.23619
$245$	0	0
$246$	1.82926e6	1.92725
$247$	460728.i	0.480510i
$248$	525236.i	0.542282i
$249$	1.93664e6	1.97948
$250$	0	0
$251$	1.09761e6	1.09968	0.549839	−	0.835271i	$-0.314689\pi$
$251$	1.09761e6	1.09968	0.549839	+	0.835271i	$0.314689\pi$
$252$	− 1.80933e6i	− 1.79480i
$253$	− 338802.i	− 0.332770i
$254$	−999182.	−0.971762
$255$	0	0
$256$	−2.03331e6	−1.93912
$257$	− 379916.i	− 0.358802i	−0.983776	−	0.179401i	$-0.942584\pi$
$257$	− 379916.i	− 0.358802i	0.983776	−	0.179401i	$-0.0574160\pi$
$258$	− 2.51822e6i	− 2.35529i
$259$	−34737.4	−0.0321772
$260$	0	0
$261$	−2.10769e6	−1.91516
$262$	− 2.69192e6i	− 2.42275i
$263$	− 261845.i	− 0.233429i	−0.993166	−	0.116714i	$-0.962764\pi$
$263$	− 261845.i	− 0.233429i	0.993166	−	0.116714i	$-0.0372362\pi$
$264$	−1.20116e6	−1.06070
$265$	0	0
$266$	1.98622e6	1.72117
$267$	− 2.72763e6i	− 2.34157i
$268$	4.31256e6i	3.66774i
$269$	892153.	0.751724	0.375862	−	0.926676i	$-0.377347\pi$
$269$	892153.	0.751724	0.375862	+	0.926676i	$0.377347\pi$
$270$	0	0
$271$	−1.54998e6	−1.28204	−0.641021	−	0.767524i	$-0.721489\pi$
$271$	−1.54998e6	−1.28204	−0.641021	+	0.767524i	$0.721489\pi$
$272$	3.53475e6i	2.89692i
$273$	− 298821.i	− 0.242664i
$274$	358155.	0.288200
$275$	0	0
$276$	−4.56678e6	−3.60859
$277$	797920.i	0.624828i	0.949946	+	0.312414i	$0.101138\pi$
$277$	797920.i	0.624828i	−0.949946	+	0.312414i	$0.898862\pi$
$278$	− 643502.i	− 0.499388i
$279$	−489229.	−0.376272
$280$	0	0
$281$	1.16864e6	0.882907	0.441453	−	0.897284i	$-0.354463\pi$
$281$	1.16864e6	0.882907	0.441453	+	0.897284i	$0.354463\pi$
$282$	− 4.05255e6i	− 3.03463i
$283$	− 1.09370e6i	− 0.811771i	−0.913924	−	0.405886i	$-0.866963\pi$
$283$	− 1.09370e6i	− 0.811771i	0.913924	−	0.405886i	$-0.133037\pi$
$284$	366361.	0.269534
$285$	0	0
$286$	−217557.	−0.157275
$287$	− 532628.i	− 0.381697i
$288$	1.53519e6i	1.09064i
$289$	−3.21329e6	−2.26311
$290$	0	0
$291$	1.75432e6	1.21444
$292$	4.75345e6i	3.26251i
$293$	− 1.02050e6i	− 0.694451i	−0.937782	−	0.347226i	$-0.887124\pi$
$293$	− 1.02050e6i	− 0.694451i	0.937782	−	0.347226i	$-0.112876\pi$
$294$	2.87460e6	1.93959
$295$	0	0
$296$	185186.	0.122851
$297$	− 359631.i	− 0.236573i
$298$	− 269037.i	− 0.175497i
$299$	−449332.	−0.290663
$300$	0	0
$301$	−733232.	−0.466472
$302$	− 4.53386e6i	− 2.86056i
$303$	3.69420e6i	2.31160i
$304$	−4.47691e6	−2.77840
$305$	0	0
$306$	−7.78712e6	−4.75415
$307$	3.21266e6i	1.94544i	0.231975	+	0.972722i	$0.425481\pi$
$307$	3.21266e6i	1.94544i	−0.231975	+	0.972722i	$0.574519\pi$
$308$	643822.i	0.386713i
$309$	3.92750e6	2.34002
$310$	0	0
$311$	−2.57875e6	−1.51185	−0.755924	−	0.654659i	$-0.772812\pi$
$311$	−2.57875e6	−1.51185	−0.755924	+	0.654659i	$0.772812\pi$
$312$	1.59302e6i	0.926479i
$313$	2.26878e6i	1.30898i	0.756072	+	0.654488i	$0.227116\pi$
$313$	2.26878e6i	1.30898i	−0.756072	+	0.654488i	$0.772884\pi$
$314$	−166667.	−0.0953952
$315$	0	0
$316$	−6.27046e6	−3.53250
$317$	− 200774.i	− 0.112217i	−0.998425	−	0.0561086i	$-0.982131\pi$
$317$	− 200774.i	− 0.112217i	0.998425	−	0.0561086i	$-0.0178693\pi$
$318$	2.24451e6i	1.24467i
$319$	749991.	0.412648
$320$	0	0
$321$	−1.91280e6	−1.03611
$322$	1.93709e6i	1.04114i
$323$	− 5.86808e6i	− 3.12961i
$324$	1.24890e6	0.660943
$325$	0	0
$326$	−842073.	−0.438840
$327$	5.39072e6i	2.78790i
$328$	2.83945e6i	1.45730i
$329$	−1.17999e6	−0.601017
$330$	0	0
$331$	−1.31065e6	−0.657530	−0.328765	−	0.944412i	$-0.606632\pi$
$331$	−1.31065e6	−0.657530	−0.328765	+	0.944412i	$0.606632\pi$
$332$	5.53381e6i	2.75536i
$333$	172491.i	0.0852425i
$334$	2.17976e6	1.06916
$335$	0	0
$336$	2.90365e6	1.40313
$337$	1.35360e6i	0.649255i	0.945842	+	0.324628i	$0.105239\pi$
$337$	1.35360e6i	0.649255i	−0.945842	+	0.324628i	$0.894761\pi$
$338$	288533.i	0.137374i
$339$	−4.19986e6	−1.98489
$340$	0	0
$341$	174085.	0.0810729
$342$	− 9.86272e6i	− 4.55965i
$343$	− 2.04910e6i	− 0.940433i
$344$	3.90888e6	1.78097
$345$	0	0
$346$	−428086.	−0.192239
$347$	− 231416.i	− 0.103174i	−0.998669	−	0.0515869i	$-0.983572\pi$
$347$	− 231416.i	− 0.103174i	0.998669	−	0.0515869i	$-0.0164279\pi$
$348$	− 1.01093e7i	− 4.47478i
$349$	−112062.	−0.0492486	−0.0246243	−	0.999697i	$-0.507839\pi$
$349$	−112062.	−0.0492486	−0.0246243	+	0.999697i	$0.507839\pi$
$350$	0	0
$351$	−476955.	−0.206638
$352$	− 546273.i	− 0.234992i
$353$	1.64207e6i	0.701383i	0.936491	+	0.350692i	$0.114053\pi$
$353$	1.64207e6i	0.701383i	−0.936491	+	0.350692i	$0.885947\pi$
$354$	−7.32785e6	−3.10791
$355$	0	0
$356$	7.79400e6	3.25938
$357$	3.80595e6i	1.58049i
$358$	1.49220e6i	0.615346i
$359$	1.14212e6	0.467708	0.233854	−	0.972272i	$-0.424866\pi$
$359$	1.14212e6	0.467708	0.233854	+	0.972272i	$0.424866\pi$
$360$	0	0
$361$	4.95608e6	2.00157
$362$	7.49686e6i	3.00682i
$363$	− 3.55046e6i	− 1.41422i
$364$	853860.	0.337779
$365$	0	0
$366$	4.06450e6	1.58600
$367$	− 3.01075e6i	− 1.16683i	−0.812173	−	0.583417i	$-0.801715\pi$
$367$	− 3.01075e6i	− 1.16683i	0.812173	−	0.583417i	$-0.198285\pi$
$368$	− 4.36617e6i	− 1.68067i
$369$	−2.64480e6	−1.01118
$370$	0	0
$371$	653537.	0.246510
$372$	− 2.34653e6i	− 0.879161i
$373$	2.87230e6i	1.06895i	0.845183	+	0.534476i	$0.179491\pi$
$373$	2.87230e6i	1.06895i	−0.845183	+	0.534476i	$0.820509\pi$
$374$	2.77093e6	1.02435
$375$	0	0
$376$	6.29054e6	2.29466
$377$	− 994666.i	− 0.360432i
$378$	2.05618e6i	0.740169i
$379$	−382184.	−0.136670	−0.0683352	−	0.997662i	$-0.521769\pi$
$379$	−382184.	−0.136670	−0.0683352	+	0.997662i	$0.521769\pi$
$380$	0	0
$381$	2.42493e6	0.855830
$382$	− 7.05301e6i	− 2.47295i
$383$	789967.i	0.275177i	0.990489	+	0.137588i	$0.0439351\pi$
$383$	789967.i	0.275177i	−0.990489	+	0.137588i	$0.956065\pi$
$384$	5.65242e6	1.95617
$385$	0	0
$386$	−1.92558e6	−0.657799
$387$	3.64092e6i	1.23576i
$388$	5.01285e6i	1.69046i
$389$	−4.32009e6	−1.44750	−0.723751	−	0.690061i	$-0.757584\pi$
$389$	−4.32009e6	−1.44750	−0.723751	+	0.690061i	$0.757584\pi$
$390$	0	0
$391$	5.72293e6	1.89311
$392$	4.46208e6i	1.46664i
$393$	6.53308e6i	2.13372i
$394$	−8.25919e6	−2.68038
$395$	0	0
$396$	3.19694e6	1.02446
$397$	− 3.54302e6i	− 1.12823i	−0.825696	−	0.564115i	$-0.809217\pi$
$397$	− 3.54302e6i	− 1.12823i	0.825696	−	0.564115i	$-0.190783\pi$
$398$	− 2.91664e6i	− 0.922943i
$399$	−4.82039e6	−1.51583
$400$	0	0
$401$	3.78071e6	1.17412	0.587060	−	0.809543i	$-0.300285\pi$
$401$	3.78071e6	1.17412	0.587060	+	0.809543i	$0.300285\pi$
$402$	− 1.52469e7i	− 4.70561i
$403$	− 230878.i	− 0.0708141i
$404$	−1.05559e7	−3.21768
$405$	0	0
$406$	−4.28805e6	−1.29105
$407$	− 61378.4i	− 0.0183666i
$408$	− 2.02896e7i	− 6.03425i
$409$	−5.36133e6	−1.58476	−0.792381	−	0.610026i	$-0.791159\pi$
$409$	−5.36133e6	−1.58476	−0.792381	+	0.610026i	$0.791159\pi$
$410$	0	0
$411$	−869212.	−0.253817
$412$	1.12226e7i	3.25723i
$413$	2.13366e6i	0.615531i
$414$	9.61876e6	2.75815
$415$	0	0
$416$	−724488.	−0.205257
$417$	1.56173e6i	0.439810i
$418$	3.50950e6i	0.982437i
$419$	4.19124e6	1.16629	0.583146	−	0.812368i	$-0.301822\pi$
$419$	4.19124e6	1.16629	0.583146	+	0.812368i	$0.301822\pi$
$420$	0	0
$421$	−1.98109e6	−0.544752	−0.272376	−	0.962191i	$-0.587809\pi$
$421$	−1.98109e6	−0.544752	−0.272376	+	0.962191i	$0.587809\pi$
$422$	− 1.36186e6i	− 0.372264i
$423$	5.85930e6i	1.59219i
$424$	−3.48402e6	−0.941166
$425$	0	0
$426$	−1.29526e6	−0.345805
$427$	− 1.18347e6i	− 0.314113i
$428$	− 5.46570e6i	− 1.44224i
$429$	527994.	0.138512
$430$	0	0
$431$	4.49072e6	1.16446	0.582228	−	0.813026i	$-0.302181\pi$
$431$	4.49072e6	1.16446	0.582228	+	0.813026i	$0.302181\pi$
$432$	− 4.63459e6i	− 1.19482i
$433$	1.75380e6i	0.449531i	0.974413	+	0.224765i	$0.0721616\pi$
$433$	1.75380e6i	0.449531i	−0.974413	+	0.224765i	$0.927838\pi$
$434$	−995325.	−0.253653
$435$	0	0
$436$	−1.54036e7	−3.88066
$437$	7.24834e6i	1.81566i
$438$	− 1.68056e7i	− 4.18571i
$439$	5.04733e6	1.24997	0.624987	−	0.780635i	$-0.285104\pi$
$439$	5.04733e6	1.24997	0.624987	+	0.780635i	$0.285104\pi$
$440$	0	0
$441$	−4.15619e6	−1.01765
$442$	− 3.67491e6i	− 0.894728i
$443$	3.47296e6i	0.840795i	0.907340	+	0.420398i	$0.138109\pi$
$443$	3.47296e6i	0.840795i	−0.907340	+	0.420398i	$0.861891\pi$
$444$	−827331.	−0.199169
$445$	0	0
$446$	3.24255e6	0.771880
$447$	652930.i	0.154560i
$448$	− 666513.i	− 0.156897i
$449$	−298830.	−0.0699533	−0.0349767	−	0.999388i	$-0.511136\pi$
$449$	−298830.	−0.0699533	−0.0349767	+	0.999388i	$0.511136\pi$
$450$	0	0
$451$	941114.	0.217872
$452$	− 1.20008e7i	− 2.76290i
$453$	1.10033e7i	2.51929i
$454$	290161.	0.0660693
$455$	0	0
$456$	2.56977e7	5.78737
$457$	− 4.62298e6i	− 1.03546i	−0.855545	−	0.517728i	$-0.826778\pi$
$457$	− 4.62298e6i	− 1.03546i	0.855545	−	0.517728i	$-0.173222\pi$
$458$	8.59515e6i	1.91465i
$459$	6.07476e6	1.34585
$460$	0	0
$461$	1.73228e6	0.379634	0.189817	−	0.981820i	$-0.439211\pi$
$461$	1.73228e6	0.379634	0.189817	+	0.981820i	$0.439211\pi$
$462$	− 2.27621e6i	− 0.496143i
$463$	− 3.98725e6i	− 0.864412i	−0.901775	−	0.432206i	$-0.857735\pi$
$463$	− 3.98725e6i	− 0.864412i	0.901775	−	0.432206i	$-0.142265\pi$
$464$	9.66520e6	2.08409
$465$	0	0
$466$	4.36848e6	0.931892
$467$	− 4.73552e6i	− 1.00479i	−0.864638	−	0.502395i	$-0.832452\pi$
$467$	− 4.73552e6i	− 1.00479i	0.864638	−	0.502395i	$-0.167548\pi$
$468$	− 4.23990e6i	− 0.894832i
$469$	−4.43946e6	−0.931961
$470$	0	0
$471$	404488.	0.0840144
$472$	− 1.13746e7i	− 2.35007i
$473$	− 1.29557e6i	− 0.266261i
$474$	2.21690e7	4.53211
$475$	0	0
$476$	−1.08752e7	−2.19999
$477$	− 3.24519e6i	− 0.653046i
$478$	5.36402e6i	1.07379i
$479$	−974320.	−0.194027	−0.0970137	−	0.995283i	$-0.530929\pi$
$479$	−974320.	−0.194027	−0.0970137	+	0.995283i	$0.530929\pi$
$480$	0	0
$481$	−81402.3	−0.0160426
$482$	7.48669e6i	1.46782i
$483$	− 4.70116e6i	− 0.916932i
$484$	1.01452e7	1.96855
$485$	0	0
$486$	−1.13436e7	−2.17852
$487$	834741.i	0.159489i	0.996815	+	0.0797443i	$0.0254104\pi$
$487$	834741.i	0.159489i	−0.996815	+	0.0797443i	$0.974590\pi$
$488$	6.30909e6i	1.19927i
$489$	2.04364e6	0.386485
$490$	0	0
$491$	829646.	0.155306	0.0776532	−	0.996980i	$-0.475257\pi$
$491$	829646.	0.155306	0.0776532	+	0.996980i	$0.475257\pi$
$492$	− 1.26855e7i	− 2.36262i
$493$	1.26686e7i	2.34753i
$494$	4.65443e6	0.858122
$495$	0	0
$496$	2.24345e6	0.409460
$497$	377141.i	0.0684878i
$498$	− 1.95646e7i	− 3.53506i
$499$	−8.32784e6	−1.49720	−0.748602	−	0.663019i	$-0.769275\pi$
$499$	−8.32784e6	−1.49720	−0.748602	+	0.663019i	$0.769275\pi$
$500$	0	0
$501$	−5.29009e6	−0.941605
$502$	− 1.10885e7i	− 1.96387i
$503$	6.37207e6i	1.12295i	0.827493	+	0.561475i	$0.189766\pi$
$503$	6.37207e6i	1.12295i	−0.827493	+	0.561475i	$0.810234\pi$
$504$	−9.92939e6	−1.74119
$505$	0	0
$506$	−3.42269e6	−0.594281
$507$	− 700246.i	− 0.120985i
$508$	6.92907e6i	1.19129i
$509$	−3.38691e6	−0.579441	−0.289721	−	0.957111i	$-0.593562\pi$
$509$	−3.38691e6	−0.579441	−0.289721	+	0.957111i	$0.593562\pi$
$510$	0	0
$511$	−4.89332e6	−0.828994
$512$	1.31637e7i	2.21924i
$513$	7.69395e6i	1.29079i
$514$	−3.83804e6	−0.640769
$515$	0	0
$516$	−1.74632e7	−2.88735
$517$	− 2.08495e6i	− 0.343059i
$518$	350929.i	0.0574638i
$519$	1.03893e6	0.169304
$520$	0	0
$521$	8.84049e6	1.42686	0.713431	−	0.700725i	$-0.247140\pi$
$521$	8.84049e6	1.42686	0.713431	+	0.700725i	$0.247140\pi$
$522$	2.12926e7i	3.42021i
$523$	975493.i	0.155944i	0.996956	+	0.0779722i	$0.0248446\pi$
$523$	975493.i	0.155944i	−0.996956	+	0.0779722i	$0.975155\pi$
$524$	−1.86678e7	−2.97006
$525$	0	0
$526$	−2.64524e6	−0.416870
$527$	2.94059e6i	0.461219i
$528$	5.13054e6i	0.800900i
$529$	−632708.	−0.0983024
$530$	0	0
$531$	1.05948e7	1.63064
$532$	− 1.37739e7i	− 2.10998i
$533$	− 1.24814e6i	− 0.190303i
$534$	−2.75554e7	−4.18171
$535$	0	0
$536$	2.36669e7	3.55819
$537$	− 3.62145e6i	− 0.541934i
$538$	− 9.01283e6i	− 1.34247i
$539$	1.47892e6	0.219267
$540$	0	0
$541$	−2.36388e6	−0.347242	−0.173621	−	0.984813i	$-0.555547\pi$
$541$	−2.36388e6	−0.347242	−0.173621	+	0.984813i	$0.555547\pi$
$542$	1.56584e7i	2.28954i
$543$	− 1.81943e7i	− 2.64811i
$544$	9.22747e6	1.33686
$545$	0	0
$546$	−3.01879e6	−0.433362
$547$	− 4.75103e6i	− 0.678922i	−0.940620	−	0.339461i	$-0.889755\pi$
$547$	− 4.75103e6i	− 0.678922i	0.940620	−	0.339461i	$-0.110245\pi$
$548$	− 2.48371e6i	− 0.353305i
$549$	−5.87659e6	−0.832136
$550$	0	0
$551$	−1.60453e7	−2.25149
$552$	2.50620e7i	3.50081i
$553$	− 6.45497e6i	− 0.897597i
$554$	8.06086e6	1.11585
$555$	0	0
$556$	−4.46252e6	−0.612200
$557$	− 8.56354e6i	− 1.16954i	−0.811199	−	0.584770i	$-0.801185\pi$
$557$	− 8.56354e6i	− 1.16954i	0.811199	−	0.584770i	$-0.198815\pi$
$558$	4.94236e6i	0.671968i
$559$	−1.71823e6	−0.232569
$560$	0	0
$561$	−6.72482e6	−0.902139
$562$	− 1.18060e7i	− 1.57675i
$563$	− 3.06139e6i	− 0.407049i	−0.979070	−	0.203525i	$-0.934760\pi$
$563$	− 3.06139e6i	− 0.407049i	0.979070	−	0.203525i	$-0.0652397\pi$
$564$	−2.81034e7	−3.72016
$565$	0	0
$566$	−1.10490e7	−1.44971
$567$	1.28565e6i	0.167944i
$568$	− 2.01055e6i	− 0.261483i
$569$	−1.01121e6	−0.130937	−0.0654685	−	0.997855i	$-0.520854\pi$
$569$	−1.01121e6	−0.130937	−0.0654685	+	0.997855i	$0.520854\pi$
$570$	0	0
$571$	4.63373e6	0.594759	0.297379	−	0.954759i	$-0.403887\pi$
$571$	4.63373e6	0.594759	0.297379	+	0.954759i	$0.403887\pi$
$572$	1.50871e6i	0.192803i
$573$	1.71171e7i	2.17793i
$574$	−5.38078e6	−0.681657
$575$	0	0
$576$	−3.30962e6	−0.415644
$577$	− 328589.i	− 0.0410879i	−0.999789	−	0.0205439i	$-0.993460\pi$
$577$	− 328589.i	− 0.0410879i	0.999789	−	0.0205439i	$-0.00653980\pi$
$578$	3.24618e7i	4.04159i
$579$	4.67323e6	0.579323
$580$	0	0
$581$	−5.69664e6	−0.700130
$582$	− 1.77227e7i	− 2.16882i
$583$	1.15475e6i	0.140707i
$584$	2.60864e7	3.16506
$585$	0	0
$586$	−1.03094e7	−1.24019
$587$	1.02498e7i	1.22778i	0.789393	+	0.613888i	$0.210395\pi$
$587$	1.02498e7i	1.22778i	−0.789393	+	0.613888i	$0.789605\pi$
$588$	− 1.99347e7i	− 2.37775i
$589$	−3.72438e6	−0.442350
$590$	0	0
$591$	2.00444e7	2.36061
$592$	− 790988.i	− 0.0927611i
$593$	− 2.82390e6i	− 0.329771i	−0.986313	−	0.164886i	$-0.947275\pi$
$593$	− 2.82390e6i	− 0.329771i	0.986313	−	0.164886i	$-0.0527255\pi$
$594$	−3.63311e6	−0.422486
$595$	0	0
$596$	−1.86570e6	−0.215143
$597$	7.07844e6i	0.812834i
$598$	4.53930e6i	0.519082i
$599$	1.52502e7	1.73664	0.868318	−	0.496007i	$-0.165201\pi$
$599$	1.52502e7	1.73664	0.868318	+	0.496007i	$0.165201\pi$
$600$	0	0
$601$	−6.06769e6	−0.685231	−0.342616	−	0.939476i	$-0.611313\pi$
$601$	−6.06769e6	−0.685231	−0.342616	+	0.939476i	$0.611313\pi$
$602$	7.40735e6i	0.833052i
$603$	2.20445e7i	2.46891i
$604$	−3.14412e7	−3.50677
$605$	0	0
$606$	3.73200e7	4.12820
$607$	− 2.77430e6i	− 0.305620i	−0.988256	−	0.152810i	$-0.951168\pi$
$607$	− 2.77430e6i	− 0.305620i	0.988256	−	0.152810i	$-0.0488323\pi$
$608$	1.16870e7i	1.28216i
$609$	1.04067e7	1.13703
$610$	0	0
$611$	−2.76513e6	−0.299649
$612$	5.40017e7i	5.82813i
$613$	− 694666.i	− 0.0746664i	−0.999303	−	0.0373332i	$-0.988114\pi$
$613$	− 694666.i	− 0.0746664i	0.999303	−	0.0373332i	$-0.0118863\pi$
$614$	3.24554e7	3.47428
$615$	0	0
$616$	3.53322e6	0.375163
$617$	− 6.78058e6i	− 0.717057i	−0.933519	−	0.358529i	$-0.883279\pi$
$617$	− 6.78058e6i	− 0.717057i	0.933519	−	0.358529i	$-0.116721\pi$
$618$	− 3.96769e7i	− 4.17895i
$619$	2.60889e6	0.273671	0.136835	−	0.990594i	$-0.456307\pi$
$619$	2.60889e6	0.273671	0.136835	+	0.990594i	$0.456307\pi$
$620$	0	0
$621$	−7.50363e6	−0.780805
$622$	2.60514e7i	2.69994i
$623$	8.02334e6i	0.828200i
$624$	6.80431e6	0.699556
$625$	0	0
$626$	2.29200e7	2.33765
$627$	− 8.51728e6i	− 0.865231i
$628$	1.15580e6i	0.116945i
$629$	1.03678e6	0.104487
$630$	0	0
$631$	1.81425e7	1.81395	0.906973	−	0.421190i	$-0.138387\pi$
$631$	1.81425e7	1.81395	0.906973	+	0.421190i	$0.138387\pi$
$632$	3.44116e7i	3.42699i
$633$	3.30512e6i	0.327852i
$634$	−2.02829e6	−0.200404
$635$	0	0
$636$	1.55651e7	1.52584
$637$	− 1.96140e6i	− 0.191521i
$638$	− 7.57666e6i	− 0.736930i
$639$	1.87272e6	0.181435
$640$	0	0
$641$	−2.04257e7	−1.96350	−0.981750	−	0.190174i	$-0.939095\pi$
$641$	−2.04257e7	−1.96350	−0.981750	+	0.190174i	$0.939095\pi$
$642$	1.93238e7i	1.85035i
$643$	− 1.26717e7i	− 1.20867i	−0.796730	−	0.604335i	$-0.793439\pi$
$643$	− 1.26717e7i	− 1.20867i	0.796730	−	0.604335i	$-0.206561\pi$
$644$	1.34332e7	1.27634
$645$	0	0
$646$	−5.92813e7	−5.58903
$647$	− 8.01621e6i	− 0.752850i	−0.926447	−	0.376425i	$-0.877153\pi$
$647$	− 8.01621e6i	− 0.752850i	0.926447	−	0.376425i	$-0.122847\pi$
$648$	− 6.85382e6i	− 0.641202i
$649$	−3.77002e6	−0.351343
$650$	0	0
$651$	2.41557e6	0.223392
$652$	5.83957e6i	0.537975i
$653$	− 1.98662e7i	− 1.82319i	−0.411086	−	0.911597i	$-0.634850\pi$
$653$	− 1.98662e7i	− 1.82319i	0.411086	−	0.911597i	$-0.365150\pi$
$654$	5.44588e7	4.97879
$655$	0	0
$656$	1.21282e7	1.10037
$657$	2.42981e7i	2.19614i
$658$	1.19206e7i	1.07333i
$659$	−1.33714e6	−0.119940	−0.0599699	−	0.998200i	$-0.519100\pi$
$659$	−1.33714e6	−0.119940	−0.0599699	+	0.998200i	$0.519100\pi$
$660$	0	0
$661$	−9.97249e6	−0.887769	−0.443884	−	0.896084i	$-0.646400\pi$
$661$	−9.97249e6	−0.887769	−0.443884	+	0.896084i	$0.646400\pi$
$662$	1.32406e7i	1.17426i
$663$	8.91871e6i	0.787985i
$664$	3.03690e7	2.67307
$665$	0	0
$666$	1.74256e6	0.152231
$667$	− 1.56484e7i	− 1.36194i
$668$	− 1.51161e7i	− 1.31068i
$669$	−7.86941e6	−0.679793
$670$	0	0
$671$	2.09110e6	0.179295
$672$	− 7.57999e6i	− 0.647509i
$673$	− 1.31355e7i	− 1.11792i	−0.829196	−	0.558958i	$-0.811201\pi$
$673$	− 1.31355e7i	− 1.11792i	0.829196	−	0.558958i	$-0.188799\pi$
$674$	1.36745e7	1.15948
$675$	0	0
$676$	2.00090e6	0.168407
$677$	− 1.58935e7i	− 1.33275i	−0.745616	−	0.666376i	$-0.767845\pi$
$677$	− 1.58935e7i	− 1.33275i	0.745616	−	0.666376i	$-0.232155\pi$
$678$	4.24284e7i	3.54473i
$679$	−5.16035e6	−0.429541
$680$	0	0
$681$	−704198.	−0.0581871
$682$	− 1.75867e6i	− 0.144785i
$683$	− 1.90939e7i	− 1.56618i	−0.621905	−	0.783092i	$-0.713641\pi$
$683$	− 1.90939e7i	− 1.56618i	0.621905	−	0.783092i	$-0.286359\pi$
$684$	−6.83955e7	−5.58968
$685$	0	0
$686$	−2.07007e7	−1.67948
$687$	− 2.08597e7i	− 1.68623i
$688$	− 1.66961e7i	− 1.34475i
$689$	1.53147e6	0.122903
$690$	0	0
$691$	2.12588e7	1.69373	0.846863	−	0.531810i	$-0.178488\pi$
$691$	2.12588e7	1.69373	0.846863	+	0.531810i	$0.178488\pi$
$692$	2.96867e6i	0.235666i
$693$	3.29101e6i	0.260313i
$694$	−2.33784e6	−0.184254
$695$	0	0
$696$	−5.54787e7	−4.34113
$697$	1.58970e7i	1.23946i
$698$	1.13209e6i	0.0879510i
$699$	−1.06020e7	−0.820716
$700$	0	0
$701$	9.34437e6	0.718216	0.359108	−	0.933296i	$-0.383081\pi$
$701$	9.34437e6	0.718216	0.359108	+	0.933296i	$0.383081\pi$
$702$	4.81836e6i	0.369026i
$703$	1.31313e6i	0.100212i
$704$	1.17768e6	0.0895560
$705$	0	0
$706$	1.65888e7	1.25257
$707$	− 1.08665e7i	− 0.817602i
$708$	5.08168e7i	3.80999i
$709$	−2.84238e6	−0.212357	−0.106178	−	0.994347i	$-0.533861\pi$
$709$	−2.84238e6	−0.212357	−0.106178	+	0.994347i	$0.533861\pi$
$710$	0	0
$711$	−3.20526e7	−2.37788
$712$	− 4.27727e7i	− 3.16203i
$713$	− 3.63226e6i	− 0.267579i
$714$	3.84489e7	2.82253
$715$	0	0
$716$	1.03480e7	0.754354
$717$	− 1.30180e7i	− 0.945687i
$718$	− 1.15381e7i	− 0.835260i
$719$	−1.63490e6	−0.117942	−0.0589712	−	0.998260i	$-0.518782\pi$
$719$	−1.63490e6	−0.117942	−0.0589712	+	0.998260i	$0.518782\pi$
$720$	0	0
$721$	−1.15528e7	−0.827653
$722$	− 5.00680e7i	− 3.57451i
$723$	− 1.81696e7i	− 1.29270i
$724$	5.19889e7	3.68607
$725$	0	0
$726$	−3.58679e7	−2.52560
$727$	1.60243e7i	1.12446i	0.826981	+	0.562230i	$0.190057\pi$
$727$	1.60243e7i	1.12446i	−0.826981	+	0.562230i	$0.809943\pi$
$728$	− 4.68589e6i	− 0.327691i
$729$	2.31981e7	1.61672
$730$	0	0
$731$	2.18843e7	1.51474
$732$	− 2.81863e7i	− 1.94429i
$733$	1.85018e7i	1.27190i	0.771730	+	0.635951i	$0.219392\pi$
$733$	1.85018e7i	1.27190i	−0.771730	+	0.635951i	$0.780608\pi$
$734$	−3.04156e7	−2.08380
$735$	0	0
$736$	−1.13979e7	−0.775587
$737$	− 7.84419e6i	− 0.531961i
$738$	2.67187e7i	1.80582i
$739$	−8.93389e6	−0.601769	−0.300884	−	0.953661i	$-0.597282\pi$
$739$	−8.93389e6	−0.601769	−0.300884	+	0.953661i	$0.597282\pi$
$740$	0	0
$741$	−1.12959e7	−0.755747
$742$	− 6.60225e6i	− 0.440233i
$743$	− 1.81525e7i	− 1.20633i	−0.797617	−	0.603164i	$-0.793907\pi$
$743$	− 1.81525e7i	− 1.20633i	0.797617	−	0.603164i	$-0.206093\pi$
$744$	−1.28775e7	−0.852901
$745$	0	0
$746$	2.90170e7	1.90900
$747$	2.82871e7i	1.85476i
$748$	− 1.92157e7i	− 1.25575i
$749$	5.62653e6	0.366468
$750$	0	0
$751$	−4.74200e6	−0.306805	−0.153402	−	0.988164i	$-0.549023\pi$
$751$	−4.74200e6	−0.306805	−0.153402	+	0.988164i	$0.549023\pi$
$752$	− 2.68689e7i	− 1.73263i
$753$	2.69108e7i	1.72958i
$754$	−1.00484e7	−0.643681
$755$	0	0
$756$	1.42591e7	0.907375
$757$	1.70743e6i	0.108294i	0.998533	+	0.0541469i	$0.0172439\pi$
$757$	1.70743e6i	0.108294i	−0.998533	+	0.0541469i	$0.982756\pi$
$758$	3.86095e6i	0.244074i
$759$	8.30660e6	0.523382
$760$	0	0
$761$	1.53942e7	0.963596	0.481798	−	0.876282i	$-0.339984\pi$
$761$	1.53942e7	0.963596	0.481798	+	0.876282i	$0.339984\pi$
$762$	− 2.44975e7i	− 1.52839i
$763$	− 1.58568e7i	− 0.986065i
$764$	−4.89109e7	−3.03160
$765$	0	0
$766$	7.98051e6	0.491427
$767$	4.99993e6i	0.306885i
$768$	− 4.98518e7i	− 3.04984i
$769$	−2.79442e7	−1.70402	−0.852012	−	0.523522i	$-0.824618\pi$
$769$	−2.79442e7	−1.70402	−0.852012	+	0.523522i	$0.824618\pi$
$770$	0	0
$771$	9.31461e6	0.564325
$772$	1.33534e7i	0.806398i
$773$	− 2.09934e6i	− 0.126367i	−0.998002	−	0.0631834i	$-0.979875\pi$
$773$	− 2.09934e6i	− 0.126367i	0.998002	−	0.0631834i	$-0.0201253\pi$
$774$	3.67818e7	2.20689
$775$	0	0
$776$	2.75100e7	1.63997
$777$	− 851676.i	− 0.0506083i
$778$	4.36430e7i	2.58503i
$779$	−2.01342e7	−1.18875
$780$	0	0
$781$	−666380.	−0.0390926
$782$	− 5.78150e7i	− 3.38083i
$783$	− 1.66105e7i	− 0.968227i
$784$	1.90590e7	1.10741
$785$	0	0
$786$	6.59994e7	3.81051
$787$	2.97406e7i	1.71164i	0.517274	+	0.855820i	$0.326947\pi$
$787$	2.97406e7i	1.71164i	−0.517274	+	0.855820i	$0.673053\pi$
$788$	5.72754e7i	3.28589i
$789$	6.41979e6	0.367137
$790$	0	0
$791$	1.23539e7	0.702044
$792$	− 1.75445e7i	− 0.993866i
$793$	− 2.77329e6i	− 0.156607i
$794$	−3.57928e7	−2.01486
$795$	0	0
$796$	−2.02261e7	−1.13144
$797$	1.67730e7i	0.935329i	0.883906	+	0.467664i	$0.154904\pi$
$797$	1.67730e7i	0.935329i	−0.883906	+	0.467664i	$0.845096\pi$
$798$	4.86972e7i	2.70705i
$799$	3.52182e7	1.95164
$800$	0	0
$801$	3.98405e7	2.19403
$802$	− 3.81940e7i	− 2.09681i
$803$	− 8.64613e6i	− 0.473187i
$804$	−1.05733e8	−5.76862
$805$	0	0
$806$	−2.33241e6	−0.126464
$807$	2.18734e7i	1.18231i
$808$	5.79297e7i	3.12157i
$809$	2.01463e7	1.08224	0.541120	−	0.840945i	$-0.318000\pi$
$809$	2.01463e7	1.08224	0.541120	+	0.840945i	$0.318000\pi$
$810$	0	0
$811$	3.28200e6	0.175221	0.0876105	−	0.996155i	$-0.472077\pi$
$811$	3.28200e6	0.175221	0.0876105	+	0.996155i	$0.472077\pi$
$812$	2.97365e7i	1.58271i
$813$	− 3.80016e7i	− 2.01640i
$814$	−620065.	−0.0328002
$815$	0	0
$816$	−8.66634e7	−4.55628
$817$	2.77174e7i	1.45277i
$818$	5.41619e7i	2.83016i
$819$	4.36466e6	0.227374
$820$	0	0
$821$	698069.	0.0361444	0.0180722	−	0.999837i	$-0.494247\pi$
$821$	698069.	0.0361444	0.0180722	+	0.999837i	$0.494247\pi$
$822$	8.78107e6i	0.453282i
$823$	− 2.24014e7i	− 1.15286i	−0.817148	−	0.576428i	$-0.804446\pi$
$823$	− 2.24014e7i	− 1.15286i	0.817148	−	0.576428i	$-0.195554\pi$
$824$	6.15882e7	3.15994
$825$	0	0
$826$	2.15549e7	1.09925
$827$	1.30401e7i	0.663005i	0.943454	+	0.331503i	$0.107556\pi$
$827$	1.30401e7i	0.663005i	−0.943454	+	0.331503i	$0.892444\pi$
$828$	− 6.67037e7i	− 3.38123i
$829$	3.65981e7	1.84958	0.924788	−	0.380483i	$-0.124242\pi$
$829$	3.65981e7	1.84958	0.924788	+	0.380483i	$0.124242\pi$
$830$	0	0
$831$	−1.95631e7	−0.982730
$832$	− 1.56188e6i	− 0.0782239i
$833$	2.49814e7i	1.24740i
$834$	1.57771e7	0.785438
$835$	0	0
$836$	2.43375e7	1.20437
$837$	− 3.85556e6i	− 0.190228i
$838$	− 4.23413e7i	− 2.08283i
$839$	−2.56393e7	−1.25748	−0.628740	−	0.777616i	$-0.716429\pi$
$839$	−2.56393e7	−1.25748	−0.628740	+	0.777616i	$0.716429\pi$
$840$	0	0
$841$	1.41291e7	0.688850
$842$	2.00136e7i	0.972849i
$843$	2.86522e7i	1.38864i
$844$	−9.44414e6	−0.456359
$845$	0	0
$846$	5.91927e7	2.84343
$847$	1.04437e7i	0.500203i
$848$	1.48814e7i	0.710646i
$849$	2.68149e7	1.27675
$850$	0	0
$851$	−1.28065e6	−0.0606187
$852$	8.98227e6i	0.423923i
$853$	− 1.27542e7i	− 0.600178i	−0.953911	−	0.300089i	$-0.902984\pi$
$853$	− 1.27542e7i	− 0.600178i	0.953911	−	0.300089i	$-0.0970164\pi$
$854$	−1.19558e7	−0.560961
$855$	0	0
$856$	−2.99952e7	−1.39916
$857$	− 3.05245e7i	− 1.41970i	−0.704353	−	0.709850i	$-0.748763\pi$
$857$	− 3.05245e7i	− 1.41970i	0.704353	−	0.709850i	$-0.251237\pi$
$858$	− 5.33398e6i	− 0.247362i
$859$	−3.68731e7	−1.70501	−0.852504	−	0.522721i	$-0.824917\pi$
$859$	−3.68731e7	−1.70501	−0.852504	+	0.522721i	$0.824917\pi$
$860$	0	0
$861$	1.30587e7	0.600334
$862$	− 4.53668e7i	− 2.07955i
$863$	2.27261e7i	1.03872i	0.854555	+	0.519360i	$0.173830\pi$
$863$	2.27261e7i	1.03872i	−0.854555	+	0.519360i	$0.826170\pi$
$864$	−1.20986e7	−0.551380
$865$	0	0
$866$	1.77174e7	0.802798
$867$	− 7.87821e7i	− 3.55942i
$868$	6.90233e6i	0.310954i
$869$	1.14055e7	0.512346
$870$	0	0
$871$	−1.04033e7	−0.464648
$872$	8.45333e7i	3.76475i
$873$	2.56241e7i	1.13792i
$874$	7.32252e7	3.24251
$875$	0	0
$876$	−1.16543e8	−5.13128
$877$	− 471311.i	− 0.0206923i	−0.999946	−	0.0103461i	$-0.996707\pi$
$877$	− 471311.i	− 0.0206923i	0.999946	−	0.0103461i	$-0.00329334\pi$
$878$	− 5.09899e7i	− 2.23227i
$879$	2.50200e7	1.09223
$880$	0	0
$881$	3.32241e7	1.44216	0.721080	−	0.692852i	$-0.243646\pi$
$881$	3.32241e7	1.44216	0.721080	+	0.692852i	$0.243646\pi$
$882$	4.19873e7i	1.81738i
$883$	− 1.34686e6i	− 0.0581328i	−0.999577	−	0.0290664i	$-0.990747\pi$
$883$	− 1.34686e6i	− 0.0581328i	0.999577	−	0.0290664i	$-0.00925343\pi$
$884$	−2.54846e7	−1.09685
$885$	0	0
$886$	3.50850e7	1.50154
$887$	4.53882e7i	1.93702i	0.248980	+	0.968509i	$0.419905\pi$
$887$	4.53882e7i	1.93702i	−0.248980	+	0.968509i	$0.580095\pi$
$888$	4.54031e6i	0.193220i
$889$	−7.13296e6	−0.302702
$890$	0	0
$891$	−2.27164e6	−0.0958618
$892$	− 2.24863e7i	− 0.946249i
$893$	4.46054e7i	1.87180i
$894$	6.59612e6	0.276023
$895$	0	0
$896$	−1.66266e7	−0.691886
$897$	− 1.10165e7i	− 0.457155i
$898$	3.01888e6i	0.124927i
$899$	8.04056e6	0.331808
$900$	0	0
$901$	−1.95057e7	−0.800477
$902$	− 9.50744e6i	− 0.389088i
$903$	− 1.79771e7i	− 0.733667i
$904$	−6.58592e7	−2.68037
$905$	0	0
$906$	1.11159e8	4.49909
$907$	5.01481e6i	0.202412i	0.994865	+	0.101206i	$0.0322701\pi$
$907$	5.01481e6i	0.202412i	−0.994865	+	0.101206i	$0.967730\pi$
$908$	− 2.01219e6i	− 0.0809945i
$909$	−5.39585e7	−2.16596
$910$	0	0
$911$	2.48880e7	0.993560	0.496780	−	0.867877i	$-0.334516\pi$
$911$	2.48880e7	0.993560	0.496780	+	0.867877i	$0.334516\pi$
$912$	− 1.09763e8i	− 4.36987i
$913$	− 1.00655e7i	− 0.399632i
$914$	−4.67029e7	−1.84918
$915$	0	0
$916$	5.96052e7	2.34717
$917$	− 1.92171e7i	− 0.754684i
$918$	− 6.13693e7i	− 2.40350i
$919$	3.35944e7	1.31213	0.656067	−	0.754703i	$-0.272219\pi$
$919$	3.35944e7	1.31213	0.656067	+	0.754703i	$0.272219\pi$
$920$	0	0
$921$	−7.87665e7	−3.05980
$922$	− 1.75000e7i	− 0.677972i
$923$	883778.i	0.0341459i
$924$	−1.57849e7	−0.608223
$925$	0	0
$926$	−4.02805e7	−1.54372
$927$	5.73662e7i	2.19259i
$928$	− 2.52310e7i	− 0.961756i
$929$	−2.73986e7	−1.04157	−0.520785	−	0.853688i	$-0.674361\pi$
$929$	−2.73986e7	−1.04157	−0.520785	+	0.853688i	$0.674361\pi$
$930$	0	0
$931$	−3.16400e7	−1.19636
$932$	− 3.02943e7i	− 1.14241i
$933$	− 6.32246e7i	− 2.37784i
$934$	−4.78398e7	−1.79441
$935$	0	0
$936$	−2.32681e7	−0.868105
$937$	4.73279e7i	1.76104i	0.474011	+	0.880519i	$0.342806\pi$
$937$	4.73279e7i	1.76104i	−0.474011	+	0.880519i	$0.657194\pi$
$938$	4.48489e7i	1.66435i
$939$	−5.56250e7	−2.05876
$940$	0	0
$941$	−2.81830e7	−1.03756	−0.518780	−	0.854908i	$-0.673614\pi$
$941$	−2.81830e7	−1.03756	−0.518780	+	0.854908i	$0.673614\pi$
$942$	− 4.08628e6i	− 0.150038i
$943$	− 1.96362e7i	− 0.719081i
$944$	−4.85845e7	−1.77447
$945$	0	0
$946$	−1.30882e7	−0.475503
$947$	− 1.35455e7i	− 0.490817i	−0.969420	−	0.245409i	$-0.921078\pi$
$947$	− 1.35455e7i	− 0.490817i	0.969420	−	0.245409i	$-0.0789221\pi$
$948$	− 1.53736e8i	− 5.55592i
$949$	−1.14668e7	−0.413311
$950$	0	0
$951$	4.92249e6	0.176495
$952$	5.96821e7i	2.13428i
$953$	− 2.85179e7i	− 1.01715i	−0.861017	−	0.508576i	$-0.830172\pi$
$953$	− 2.85179e7i	− 1.01715i	0.861017	−	0.508576i	$-0.169828\pi$
$954$	−3.27840e7	−1.16625
$955$	0	0
$956$	3.71981e7	1.31637
$957$	1.83879e7i	0.649013i
$958$	9.84291e6i	0.346505i
$959$	2.55680e6	0.0897738
$960$	0	0
$961$	−2.67628e7	−0.934810
$962$	822353.i	0.0286497i
$963$	− 2.79389e7i	− 0.970832i
$964$	5.19183e7	1.79940
$965$	0	0
$966$	−4.74927e7	−1.63751
$967$	− 2.08677e7i	− 0.717644i	−0.933406	−	0.358822i	$-0.883179\pi$
$967$	− 2.08677e7i	− 0.717644i	0.933406	−	0.358822i	$-0.116821\pi$
$968$	− 5.56757e7i	− 1.90975i
$969$	1.43871e8	4.92225
$970$	0	0
$971$	−3.67948e7	−1.25239	−0.626194	−	0.779667i	$-0.715388\pi$
$971$	−3.67948e7	−1.25239	−0.626194	+	0.779667i	$0.715388\pi$
$972$	7.86651e7i	2.67065i
$973$	− 4.59383e6i	− 0.155558i
$974$	8.43284e6	0.284824
$975$	0	0
$976$	2.69481e7	0.905533
$977$	− 1.31397e7i	− 0.440403i	−0.975454	−	0.220202i	$-0.929328\pi$
$977$	− 1.31397e7i	− 0.440403i	0.975454	−	0.220202i	$-0.0706715\pi$
$978$	− 2.06456e7i	− 0.690208i
$979$	−1.41766e7	−0.472734
$980$	0	0
$981$	−7.87383e7	−2.61224
$982$	− 8.38137e6i	− 0.277355i
$983$	− 2.66889e7i	− 0.880941i	−0.897767	−	0.440470i	$-0.854812\pi$
$983$	− 2.66889e7i	− 0.880941i	0.897767	−	0.440470i	$-0.145188\pi$
$984$	−6.96164e7	−2.29205
$985$	0	0
$986$	1.27982e8	4.19236
$987$	− 2.89304e7i	− 0.945281i
$988$	− 3.22773e7i	− 1.05197i
$989$	−2.70318e7	−0.878787
$990$	0	0
$991$	−2.39103e7	−0.773394	−0.386697	−	0.922207i	$-0.626384\pi$
$991$	−2.39103e7	−0.773394	−0.386697	+	0.922207i	$0.626384\pi$
$992$	− 5.85653e6i	− 0.188956i
$993$	− 3.21339e7i	− 1.03416i
$994$	3.81001e6	0.122309
$995$	0	0
$996$	−1.35675e8	−4.33364
$997$	3.32793e7i	1.06032i	0.847898	+	0.530160i	$0.177868\pi$
$997$	3.32793e7i	1.06032i	−0.847898	+	0.530160i	$0.822132\pi$
$998$	8.41307e7i	2.67379i
$999$	−1.35938e6	−0.0430950

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	325.6.b.i.274.2		22
5.2	odd	4		325.6.a.j.1.11	✓	11
5.3	odd	4		325.6.a.k.1.1	yes	11
5.4	even	2	inner	325.6.b.i.274.21		22

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
325.6.a.j.1.11	✓	11	5.2	odd	4
325.6.a.k.1.1	yes	11	5.3	odd	4
325.6.b.i.274.2		22	1.1	even	1	trivial
325.6.b.i.274.21		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)

\(f(q)\)	\(=\)	\(q-10.1023i q^{2} +24.5176i q^{3} -70.0572 q^{4} +247.685 q^{6} -72.1186i q^{7} +384.466i q^{8} -358.110 q^{9} +O(q^{10})\)	\(q-10.1023i q^{2} +24.5176i q^{3} -70.0572 q^{4} +247.685 q^{6} -72.1186i q^{7} +384.466i q^{8} -358.110 q^{9} +127.428 q^{11} -1717.63i q^{12} -169.000i q^{13} -728.566 q^{14} +1642.18 q^{16} +2152.48i q^{17} +3617.75i q^{18} -2726.20 q^{19} +1768.17 q^{21} -1287.32i q^{22} -2658.77i q^{23} -9426.17 q^{24} -1707.29 q^{26} -2822.22i q^{27} +5052.43i q^{28} +5885.60 q^{29} +1366.14 q^{31} -4286.91i q^{32} +3124.23i q^{33} +21745.0 q^{34} +25088.2 q^{36} -481.670i q^{37} +27541.0i q^{38} +4143.47 q^{39} +7385.44 q^{41} -17862.7i q^{42} -10167.0i q^{43} -8927.26 q^{44} -26859.8 q^{46} -16361.7i q^{47} +40262.2i q^{48} +11605.9 q^{49} -52773.4 q^{51} +11839.7i q^{52} +9061.97i q^{53} -28511.0 q^{54} +27727.2 q^{56} -66839.8i q^{57} -59458.3i q^{58} -29585.4 q^{59} +16410.0 q^{61} -13801.2i q^{62} +25826.4i q^{63} +9241.89 q^{64} +31562.0 q^{66} -61557.7i q^{67} -150796. i q^{68} +65186.5 q^{69} -5229.46 q^{71} -137681. i q^{72} -67851.0i q^{73} -4865.99 q^{74} +190990. q^{76} -9189.95i q^{77} -41858.7i q^{78} +89505.0 q^{79} -17826.8 q^{81} -74610.2i q^{82} -78989.9i q^{83} -123873. q^{84} -102711. q^{86} +144300. i q^{87} +48991.9i q^{88} -111252. q^{89} -12188.0 q^{91} +186266. i q^{92} +33494.5i q^{93} -165292. q^{94} +105105. q^{96} -71553.7i q^{97} -117247. i q^{98} -45633.3 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 22 q - 374 q^{4} + 702 q^{6} - 2744 q^{9}+O(q^{10}) \) 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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