LMFDB - Embedded newform 325.6.b.i.274.6

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			274.6
Character	$\chi$	$=$	325.274
Dual form			325.6.b.i.274.17

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-5.84001i q^{2} +12.3562i q^{3} -2.10573 q^{4} +72.1603 q^{6} -135.041i q^{7} -174.583i q^{8} +90.3246 q^{9} +O(q^{10})$	\(q-5.84001i q^{2} +12.3562i q^{3} -2.10573 q^{4} +72.1603 q^{6} -135.041i q^{7} -174.583i q^{8} +90.3246 q^{9} +564.387 q^{11} -26.0188i q^{12} -169.000i q^{13} -788.641 q^{14} -1086.95 q^{16} -1449.57i q^{17} -527.496i q^{18} +530.407 q^{19} +1668.59 q^{21} -3296.03i q^{22} +4715.23i q^{23} +2157.18 q^{24} -986.962 q^{26} +4118.62i q^{27} +284.360i q^{28} -3322.19 q^{29} +1863.25 q^{31} +761.144i q^{32} +6973.68i q^{33} -8465.52 q^{34} -190.199 q^{36} -10222.0i q^{37} -3097.58i q^{38} +2088.20 q^{39} -17960.2 q^{41} -9744.60i q^{42} +7561.88i q^{43} -1188.45 q^{44} +27537.0 q^{46} -24045.4i q^{47} -13430.6i q^{48} -1429.08 q^{49} +17911.2 q^{51} +355.868i q^{52} -5497.17i q^{53} +24052.8 q^{54} -23575.9 q^{56} +6553.80i q^{57} +19401.6i q^{58} +45424.9 q^{59} +29346.1 q^{61} -10881.4i q^{62} -12197.5i q^{63} -30337.3 q^{64} +40726.4 q^{66} -69953.5i q^{67} +3052.41i q^{68} -58262.3 q^{69} +49147.7 q^{71} -15769.1i q^{72} -41342.1i q^{73} -59696.3 q^{74} -1116.89 q^{76} -76215.5i q^{77} -12195.1i q^{78} -48493.3 q^{79} -28941.6 q^{81} +104888. i q^{82} -106062. i q^{83} -3513.61 q^{84} +44161.5 q^{86} -41049.6i q^{87} -98532.4i q^{88} +23361.1 q^{89} -22821.9 q^{91} -9929.00i q^{92} +23022.6i q^{93} -140425. q^{94} -9404.84 q^{96} +77533.3i q^{97} +8345.84i q^{98} +50978.0 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	− 5.84001i	− 1.03238i	−0.856475	−	0.516189i	$-0.827350\pi$
$2$	− 5.84001i	− 1.03238i	0.856475	−	0.516189i	$-0.172650\pi$
$3$	12.3562i	0.792650i	0.918110	+	0.396325i	$0.129715\pi$
$3$	12.3562i	0.792650i	−0.918110	+	0.396325i	$0.870285\pi$
$4$	−2.10573	−0.0658041
$5$	0	0
$5$	0	0
$6$	72.1603	0.818314
$7$	− 135.041i	− 1.04165i	−0.853664	−	0.520824i	$-0.825625\pi$
$7$	− 135.041i	− 1.04165i	0.853664	−	0.520824i	$-0.174375\pi$
$8$	− 174.583i	− 0.964443i
$9$	90.3246	0.371706
$10$	0	0
$11$	564.387	1.40636	0.703179	−	0.711013i	$-0.251763\pi$
$11$	564.387	1.40636	0.703179	+	0.711013i	$0.251763\pi$
$12$	− 26.0188i	− 0.0521596i
$13$	− 169.000i	− 0.277350i
$13$	− 169.000i	− 0.277350i
$14$	−788.641	−1.07537
$15$	0	0
$16$	−1086.95	−1.06147
$17$	− 1449.57i	− 1.21652i	−0.793740	−	0.608258i	$-0.791869\pi$
$17$	− 1449.57i	− 1.21652i	0.793740	−	0.608258i	$-0.208131\pi$
$18$	− 527.496i	− 0.383741i
$19$	530.407	0.337074	0.168537	−	0.985695i	$-0.446096\pi$
$19$	530.407	0.337074	0.168537	+	0.985695i	$0.446096\pi$
$20$	0	0
$21$	1668.59	0.825662
$22$	− 3296.03i	− 1.45189i
$23$	4715.23i	1.85859i	0.369341	+	0.929294i	$0.379584\pi$
$23$	4715.23i	1.85859i	−0.369341	+	0.929294i	$0.620416\pi$
$24$	2157.18	0.764466
$25$	0	0
$26$	−986.962	−0.286330
$27$	4118.62i	1.08728i
$28$	284.360i	0.0685446i
$29$	−3322.19	−0.733549	−0.366774	−	0.930310i	$-0.619538\pi$
$29$	−3322.19	−0.733549	−0.366774	+	0.930310i	$0.619538\pi$
$30$	0	0
$31$	1863.25	0.348230	0.174115	−	0.984725i	$-0.444294\pi$
$31$	1863.25	0.348230	0.174115	+	0.984725i	$0.444294\pi$
$32$	761.144i	0.131399i
$33$	6973.68i	1.11475i
$34$	−8465.52	−1.25590
$35$	0	0
$36$	−190.199	−0.0244598
$37$	− 10222.0i	− 1.22752i	−0.789491	−	0.613762i	$-0.789656\pi$
$37$	− 10222.0i	− 1.22752i	0.789491	−	0.613762i	$-0.210344\pi$
$38$	− 3097.58i	− 0.347987i
$39$	2088.20	0.219842
$40$	0	0
$41$	−17960.2	−1.66859	−0.834297	−	0.551315i	$-0.814126\pi$
$41$	−17960.2	−1.66859	−0.834297	+	0.551315i	$0.814126\pi$
$42$	− 9744.60i	− 0.852395i
$43$	7561.88i	0.623675i	0.950135	+	0.311838i	$0.100945\pi$
$43$	7561.88i	0.623675i	−0.950135	+	0.311838i	$0.899055\pi$
$44$	−1188.45	−0.0925440
$45$	0	0
$46$	27537.0	1.91877
$47$	− 24045.4i	− 1.58777i	−0.608069	−	0.793884i	$-0.708056\pi$
$47$	− 24045.4i	− 1.58777i	0.608069	−	0.793884i	$-0.291944\pi$
$48$	− 13430.6i	− 0.841377i
$49$	−1429.08	−0.0850288
$50$	0	0
$51$	17911.2	0.964271
$52$	355.868i	0.0182508i
$53$	− 5497.17i	− 0.268812i	−0.990926	−	0.134406i	$-0.957087\pi$
$53$	− 5497.17i	− 0.268812i	0.990926	−	0.134406i	$-0.0429127\pi$
$54$	24052.8	1.12249
$55$	0	0
$56$	−23575.9	−1.00461
$57$	6553.80i	0.267181i
$58$	19401.6i	0.757299i
$59$	45424.9	1.69889	0.849443	−	0.527680i	$-0.176938\pi$
$59$	45424.9	1.69889	0.849443	+	0.527680i	$0.176938\pi$
$60$	0	0
$61$	29346.1	1.00978	0.504889	−	0.863185i	$-0.331534\pi$
$61$	29346.1	1.00978	0.504889	+	0.863185i	$0.331534\pi$
$62$	− 10881.4i	− 0.359505i
$63$	− 12197.5i	− 0.387186i
$64$	−30337.3	−0.925821
$65$	0	0
$66$	40726.4	1.15084
$67$	− 69953.5i	− 1.90381i	−0.306399	−	0.951903i	$-0.599124\pi$
$67$	− 69953.5i	− 1.90381i	0.306399	−	0.951903i	$-0.400876\pi$
$68$	3052.41i	0.0800517i
$69$	−58262.3	−1.47321
$70$	0	0
$71$	49147.7	1.15706	0.578532	−	0.815659i	$-0.303626\pi$
$71$	49147.7	1.15706	0.578532	+	0.815659i	$0.303626\pi$
$72$	− 15769.1i	− 0.358489i
$73$	− 41342.1i	− 0.907999i	−0.891002	−	0.454000i	$-0.849997\pi$
$73$	− 41342.1i	− 0.907999i	0.891002	−	0.454000i	$-0.150003\pi$
$74$	−59696.3	−1.26727
$75$	0	0
$76$	−1116.89	−0.0221808
$77$	− 76215.5i	− 1.46493i
$78$	− 12195.1i	− 0.226960i
$79$	−48493.3	−0.874207	−0.437104	−	0.899411i	$-0.643996\pi$
$79$	−48493.3	−0.874207	−0.437104	+	0.899411i	$0.643996\pi$
$80$	0	0
$81$	−28941.6	−0.490129
$82$	104888.i	1.72262i
$83$	− 106062.i	− 1.68991i	−0.534834	−	0.844957i	$-0.679626\pi$
$83$	− 106062.i	− 1.68991i	0.534834	−	0.844957i	$-0.320374\pi$
$84$	−3513.61	−0.0543319
$85$	0	0
$86$	44161.5	0.643869
$87$	− 41049.6i	− 0.581447i
$88$	− 98532.4i	− 1.35635i
$89$	23361.1	0.312621	0.156311	−	0.987708i	$-0.450040\pi$
$89$	23361.1	0.312621	0.156311	+	0.987708i	$0.450040\pi$
$90$	0	0
$91$	−22821.9	−0.288901
$92$	− 9929.00i	− 0.122303i
$93$	23022.6i	0.276025i
$94$	−140425.	−1.63918
$95$	0	0
$96$	−9404.84	−0.104153
$97$	77533.3i	0.836679i	0.908291	+	0.418340i	$0.137388\pi$
$97$	77533.3i	0.836679i	−0.908291	+	0.418340i	$0.862612\pi$
$98$	8345.84i	0.0877818i
$99$	50978.0	0.522751
$100$	0	0
$101$	−90450.2	−0.882279	−0.441140	−	0.897438i	$-0.645426\pi$
$101$	−90450.2	−0.882279	−0.441140	+	0.897438i	$0.645426\pi$
$102$	− 104602.i	− 0.995492i
$103$	51829.2i	0.481373i	0.970603	+	0.240686i	$0.0773725\pi$
$103$	51829.2i	0.481373i	−0.970603	+	0.240686i	$0.922628\pi$
$104$	−29504.5	−0.267488
$105$	0	0
$106$	−32103.5	−0.277516
$107$	143602.i	1.21255i	0.795254	+	0.606277i	$0.207338\pi$
$107$	143602.i	1.21255i	−0.795254	+	0.606277i	$0.792662\pi$
$108$	− 8672.70i	− 0.0715476i
$109$	−67227.5	−0.541977	−0.270989	−	0.962583i	$-0.587351\pi$
$109$	−67227.5	−0.541977	−0.270989	+	0.962583i	$0.587351\pi$
$110$	0	0
$111$	126304.	0.972996
$112$	146783.i	1.10568i
$113$	− 8555.23i	− 0.0630283i	−0.999503	−	0.0315142i	$-0.989967\pi$
$113$	− 8555.23i	− 0.0630283i	0.999503	−	0.0315142i	$-0.0100329\pi$
$114$	38274.3	0.275832
$115$	0	0
$116$	6995.63	0.0482705
$117$	− 15264.9i	− 0.103093i
$118$	− 265282.i	− 1.75389i
$119$	−195752.	−1.26718
$120$	0	0
$121$	157482.	0.977841
$122$	− 171381.i	− 1.04247i
$123$	− 221919.i	− 1.32261i
$124$	−3923.50	−0.0229150
$125$	0	0
$126$	−71233.7	−0.399723
$127$	203972.i	1.12218i	0.827755	+	0.561089i	$0.189618\pi$
$127$	203972.i	1.12218i	−0.827755	+	0.561089i	$0.810382\pi$
$128$	201527.i	1.08720i
$129$	−93436.0	−0.494356
$130$	0	0
$131$	−121053.	−0.616308	−0.308154	−	0.951336i	$-0.599711\pi$
$131$	−121053.	−0.616308	−0.308154	+	0.951336i	$0.599711\pi$
$132$	− 14684.7i	− 0.0733550i
$133$	− 71626.6i	− 0.351112i
$134$	−408529.	−1.96545
$135$	0	0
$136$	−253071.	−1.17326
$137$	− 313663.i	− 1.42778i	−0.700257	−	0.713891i	$-0.746931\pi$
$137$	− 313663.i	− 1.42778i	0.700257	−	0.713891i	$-0.253069\pi$
$138$	340252.i	1.52091i
$139$	200259.	0.879132	0.439566	−	0.898210i	$-0.355132\pi$
$139$	200259.	0.879132	0.439566	+	0.898210i	$0.355132\pi$
$140$	0	0
$141$	297109.	1.25854
$142$	− 287023.i	− 1.19453i
$143$	− 95381.5i	− 0.390053i
$144$	−98178.2	−0.394556
$145$	0	0
$146$	−241438.	−0.937398
$147$	− 17658.0i	− 0.0673981i
$148$	21524.7i	0.0807760i
$149$	−256467.	−0.946379	−0.473190	−	0.880961i	$-0.656897\pi$
$149$	−256467.	−0.946379	−0.473190	+	0.880961i	$0.656897\pi$
$150$	0	0
$151$	−111385.	−0.397542	−0.198771	−	0.980046i	$-0.563695\pi$
$151$	−111385.	−0.397542	−0.198771	+	0.980046i	$0.563695\pi$
$152$	− 92599.9i	− 0.325088i
$153$	− 130932.i	− 0.452186i
$154$	−445099.	−1.51236
$155$	0	0
$156$	−4397.18	−0.0144665
$157$	− 104914.i	− 0.339691i	−0.985471	−	0.169845i	$-0.945673\pi$
$157$	− 104914.i	− 0.339691i	0.985471	−	0.169845i	$-0.0543268\pi$
$158$	283202.i	0.902512i
$159$	67924.1	0.213074
$160$	0	0
$161$	636749.	1.93599
$162$	169019.i	0.505998i
$163$	159530.i	0.470298i	0.971959	+	0.235149i	$0.0755578\pi$
$163$	159530.i	0.470298i	−0.971959	+	0.235149i	$0.924442\pi$
$164$	37819.3	0.109800
$165$	0	0
$166$	−619403.	−1.74463
$167$	687941.i	1.90880i	0.298533	+	0.954399i	$0.403503\pi$
$167$	687941.i	1.90880i	−0.298533	+	0.954399i	$0.596497\pi$
$168$	− 291308.i	− 0.796304i
$169$	−28561.0	−0.0769231
$170$	0	0
$171$	47908.7	0.125292
$172$	− 15923.3i	− 0.0410404i
$173$	− 195842.i	− 0.497497i	−0.968568	−	0.248748i	$-0.919981\pi$
$173$	− 195842.i	− 0.497497i	0.968568	−	0.248748i	$-0.0800192\pi$
$174$	−239730.	−0.600273
$175$	0	0
$176$	−613461.	−1.49281
$177$	561279.i	1.34662i
$178$	− 136429.i	− 0.322743i
$179$	444742.	1.03747	0.518735	−	0.854935i	$-0.326403\pi$
$179$	444742.	1.03747	0.518735	+	0.854935i	$0.326403\pi$
$180$	0	0
$181$	303087.	0.687654	0.343827	−	0.939033i	$-0.388277\pi$
$181$	303087.	0.687654	0.343827	+	0.939033i	$0.388277\pi$
$182$	133280.i	0.298255i
$183$	362606.i	0.800400i
$184$	823198.	1.79250
$185$	0	0
$186$	134453.	0.284962
$187$	− 818121.i	− 1.71086i
$188$	50633.1i	0.104482i
$189$	556183.	1.13256
$190$	0	0
$191$	161932.	0.321181	0.160590	−	0.987021i	$-0.448660\pi$
$191$	161932.	0.321181	0.160590	+	0.987021i	$0.448660\pi$
$192$	− 374853.i	− 0.733852i
$193$	− 37091.5i	− 0.0716773i	−0.999358	−	0.0358386i	$-0.988590\pi$
$193$	− 37091.5i	− 0.0716773i	0.999358	−	0.0358386i	$-0.0114102\pi$
$194$	452795.	0.863769
$195$	0	0
$196$	3009.25	0.00559524
$197$	− 643579.i	− 1.18151i	−0.806852	−	0.590753i	$-0.798831\pi$
$197$	− 643579.i	− 1.18151i	0.806852	−	0.590753i	$-0.201169\pi$
$198$	− 297712.i	− 0.539677i
$199$	344561.	0.616785	0.308392	−	0.951259i	$-0.400209\pi$
$199$	344561.	0.616785	0.308392	+	0.951259i	$0.400209\pi$
$200$	0	0
$201$	864359.	1.50905
$202$	528230.i	0.910846i
$203$	448631.i	0.764099i
$204$	−37716.1	−0.0634529
$205$	0	0
$206$	302683.	0.496958
$207$	425901.i	0.690848i
$208$	183694.i	0.294400i
$209$	299355.	0.474046
$210$	0	0
$211$	−680992.	−1.05302	−0.526509	−	0.850169i	$-0.676499\pi$
$211$	−680992.	−1.05302	−0.526509	+	0.850169i	$0.676499\pi$
$212$	11575.5i	0.0176889i
$213$	607279.i	0.917147i
$214$	838637.	1.25181
$215$	0	0
$216$	719041.	1.04862
$217$	− 251615.i	− 0.362733i
$218$	392609.i	0.559525i
$219$	510831.	0.719726
$220$	0	0
$221$	−244978.	−0.337401
$222$	− 737619.i	− 1.00450i
$223$	− 568334.i	− 0.765318i	−0.923890	−	0.382659i	$-0.875008\pi$
$223$	− 568334.i	− 0.765318i	0.923890	−	0.382659i	$-0.124992\pi$
$224$	102786.	0.136871
$225$	0	0
$226$	−49962.6	−0.0650690
$227$	997825.i	1.28526i	0.766178	+	0.642628i	$0.222156\pi$
$227$	997825.i	1.28526i	−0.766178	+	0.642628i	$0.777844\pi$
$228$	− 13800.5i	− 0.0175816i
$229$	164230.	0.206949	0.103475	−	0.994632i	$-0.467004\pi$
$229$	164230.	0.206949	0.103475	+	0.994632i	$0.467004\pi$
$230$	0	0
$231$	941733.	1.16118
$232$	579997.i	0.707466i
$233$	− 1.08853e6i	− 1.31356i	−0.754082	−	0.656781i	$-0.771918\pi$
$233$	− 1.08853e6i	− 1.31356i	0.754082	−	0.656781i	$-0.228082\pi$
$234$	−89146.9	−0.106431
$235$	0	0
$236$	−95652.6	−0.111794
$237$	− 599193.i	− 0.692940i
$238$	1.14319e6i	1.30821i
$239$	−230234.	−0.260720	−0.130360	−	0.991467i	$-0.541613\pi$
$239$	−230234.	−0.260720	−0.130360	+	0.991467i	$0.541613\pi$
$240$	0	0
$241$	−482240.	−0.534836	−0.267418	−	0.963581i	$-0.586170\pi$
$241$	−482240.	−0.534836	−0.267418	+	0.963581i	$0.586170\pi$
$242$	− 919698.i	− 1.00950i
$243$	643217.i	0.698782i
$244$	−61794.9	−0.0664474
$245$	0	0
$246$	−1.29601e6	−1.36543
$247$	− 89638.7i	− 0.0934874i
$248$	− 325291.i	− 0.335848i
$249$	1.31052e6	1.33951
$250$	0	0
$251$	1.08969e6	1.09174	0.545869	−	0.837871i	$-0.316200\pi$
$251$	1.08969e6	1.09174	0.545869	+	0.837871i	$0.316200\pi$
$252$	25684.7i	0.0254784i
$253$	2.66122e6i	2.61384i
$254$	1.19120e6	1.15851
$255$	0	0
$256$	206125.	0.196576
$257$	888841.i	0.839443i	0.907653	+	0.419721i	$0.137872\pi$
$257$	888841.i	0.839443i	−0.907653	+	0.419721i	$0.862128\pi$
$258$	545667.i	0.510363i
$259$	−1.38038e6	−1.27865
$260$	0	0
$261$	−300075.	−0.272664
$262$	706952.i	0.636263i
$263$	− 911106.i	− 0.812231i	−0.913822	−	0.406115i	$-0.866883\pi$
$263$	− 911106.i	− 0.812231i	0.913822	−	0.406115i	$-0.133117\pi$
$264$	1.21748e6	1.07511
$265$	0	0
$266$	−418300.	−0.362480
$267$	288654.i	0.247799i
$268$	147303.i	0.125278i
$269$	1.21729e6	1.02568	0.512841	−	0.858484i	$-0.328593\pi$
$269$	1.21729e6	1.02568	0.512841	+	0.858484i	$0.328593\pi$
$270$	0	0
$271$	−1.72452e6	−1.42642	−0.713208	−	0.700953i	$-0.752758\pi$
$271$	−1.72452e6	−1.42642	−0.713208	+	0.700953i	$0.752758\pi$
$272$	1.57561e6i	1.29130i
$273$	− 281992.i	− 0.228997i
$274$	−1.83180e6	−1.47401
$275$	0	0
$276$	122685.	0.0969432
$277$	− 1.14049e6i	− 0.893086i	−0.894762	−	0.446543i	$-0.852655\pi$
$277$	− 1.14049e6i	− 0.893086i	0.894762	−	0.446543i	$-0.147345\pi$
$278$	− 1.16951e6i	− 0.907596i
$279$	168297.	0.129439
$280$	0	0
$281$	−42346.3	−0.0319926	−0.0159963	−	0.999872i	$-0.505092\pi$
$281$	−42346.3	−0.0319926	−0.0159963	+	0.999872i	$0.505092\pi$
$282$	− 1.73512e6i	− 1.29929i
$283$	− 1.73212e6i	− 1.28562i	−0.766027	−	0.642808i	$-0.777769\pi$
$283$	− 1.73212e6i	− 1.28562i	0.766027	−	0.642808i	$-0.222231\pi$
$284$	−103492.	−0.0761396
$285$	0	0
$286$	−557029.	−0.402682
$287$	2.42536e6i	1.73809i
$288$	68750.0i	0.0488418i
$289$	−681404.	−0.479910
$290$	0	0
$291$	−958016.	−0.663194
$292$	87055.3i	0.0597500i
$293$	1.09014e6i	0.741842i	0.928664	+	0.370921i	$0.120958\pi$
$293$	1.09014e6i	0.741842i	−0.928664	+	0.370921i	$0.879042\pi$
$294$	−103123.	−0.0695803
$295$	0	0
$296$	−1.78458e6	−1.18388
$297$	2.32450e6i	1.52911i
$298$	1.49777e6i	0.977021i
$299$	796874.	0.515480
$300$	0	0
$301$	1.02116e6	0.649650
$302$	650488.i	0.410413i
$303$	− 1.11762e6i	− 0.699339i
$304$	−576525.	−0.357795
$305$	0	0
$306$	−764644.	−0.466827
$307$	1.05264e6i	0.637432i	0.947850	+	0.318716i	$0.103252\pi$
$307$	1.05264e6i	0.637432i	−0.947850	+	0.318716i	$0.896748\pi$
$308$	160489.i	0.0963982i
$309$	−640411.	−0.381560
$310$	0	0
$311$	1.49770e6	0.878058	0.439029	−	0.898473i	$-0.355323\pi$
$311$	1.49770e6	0.878058	0.439029	+	0.898473i	$0.355323\pi$
$312$	− 364563.i	− 0.212025i
$313$	1.99926e6i	1.15347i	0.816930	+	0.576737i	$0.195674\pi$
$313$	1.99926e6i	1.15347i	−0.816930	+	0.576737i	$0.804326\pi$
$314$	−612698.	−0.350689
$315$	0	0
$316$	102114.	0.0575264
$317$	1.56963e6i	0.877303i	0.898657	+	0.438652i	$0.144544\pi$
$317$	1.56963e6i	0.877303i	−0.898657	+	0.438652i	$0.855456\pi$
$318$	− 396677.i	− 0.219973i
$319$	−1.87500e6	−1.03163
$320$	0	0
$321$	−1.77437e6	−0.961131
$322$	− 3.71862e6i	− 1.99868i
$323$	− 768863.i	− 0.410055i
$324$	60943.2	0.0322525
$325$	0	0
$326$	931656.	0.485525
$327$	− 830676.i	− 0.429598i
$328$	3.13554e6i	1.60926i
$329$	−3.24711e6	−1.65389
$330$	0	0
$331$	1.49980e6	0.752425	0.376213	−	0.926533i	$-0.377226\pi$
$331$	1.49980e6	0.752425	0.376213	+	0.926533i	$0.377226\pi$
$332$	223338.i	0.111203i
$333$	− 923294.i	− 0.456278i
$334$	4.01758e6	1.97060
$335$	0	0
$336$	−1.81368e6	−0.876418
$337$	1.70350e6i	0.817084i	0.912740	+	0.408542i	$0.133963\pi$
$337$	1.70350e6i	0.817084i	−0.912740	+	0.408542i	$0.866037\pi$
$338$	166797.i	0.0794137i
$339$	105710.	0.0499594
$340$	0	0
$341$	1.05159e6	0.489736
$342$	− 279788.i	− 0.129349i
$343$	− 2.07665e6i	− 0.953077i
$344$	1.32017e6	0.601500
$345$	0	0
$346$	−1.14372e6	−0.513604
$347$	1.57481e6i	0.702109i	0.936355	+	0.351055i	$0.114177\pi$
$347$	1.57481e6i	0.702109i	−0.936355	+	0.351055i	$0.885823\pi$
$348$	86439.3i	0.0382616i
$349$	87651.2	0.0385207	0.0192604	−	0.999815i	$-0.493869\pi$
$349$	87651.2	0.0385207	0.0192604	+	0.999815i	$0.493869\pi$
$350$	0	0
$351$	696047.	0.301558
$352$	429580.i	0.184794i
$353$	3.19310e6i	1.36388i	0.731409	+	0.681939i	$0.238863\pi$
$353$	3.19310e6i	1.36388i	−0.731409	+	0.681939i	$0.761137\pi$
$354$	3.27788e6	1.39022
$355$	0	0
$356$	−49192.2	−0.0205717
$357$	− 2.41875e6i	− 1.00443i
$358$	− 2.59730e6i	− 1.07106i
$359$	−1.23310e6	−0.504965	−0.252482	−	0.967601i	$-0.581247\pi$
$359$	−1.23310e6	−0.504965	−0.252482	+	0.967601i	$0.581247\pi$
$360$	0	0
$361$	−2.19477e6	−0.886381
$362$	− 1.77003e6i	− 0.709919i
$363$	1.94588e6i	0.775085i
$364$	48056.8	0.0190109
$365$	0	0
$366$	2.11762e6	0.826315
$367$	2.30172e6i	0.892046i	0.895021	+	0.446023i	$0.147160\pi$
$367$	2.30172e6i	0.892046i	−0.895021	+	0.446023i	$0.852840\pi$
$368$	− 5.12521e6i	− 1.97284i
$369$	−1.62224e6	−0.620226
$370$	0	0
$371$	−742343.	−0.280008
$372$	− 48479.5i	− 0.0181635i
$373$	1.62545e6i	0.604925i	0.953161	+	0.302462i	$0.0978087\pi$
$373$	1.62545e6i	0.604925i	−0.953161	+	0.302462i	$0.902191\pi$
$374$	−4.77783e6	−1.76625
$375$	0	0
$376$	−4.19791e6	−1.53131
$377$	561449.i	0.203450i
$378$	− 3.24811e6i	− 1.16924i
$379$	−3.62459e6	−1.29617	−0.648083	−	0.761570i	$-0.724429\pi$
$379$	−3.62459e6	−1.29617	−0.648083	+	0.761570i	$0.724429\pi$
$380$	0	0
$381$	−2.52032e6	−0.889495
$382$	− 945685.i	− 0.331580i
$383$	− 271233.i	− 0.0944813i	−0.998884	−	0.0472407i	$-0.984957\pi$
$383$	− 271233.i	− 0.0944813i	0.998884	−	0.0472407i	$-0.0150428\pi$
$384$	−2.49010e6	−0.861766
$385$	0	0
$386$	−216615.	−0.0739980
$387$	683024.i	0.231824i
$388$	− 163264.i	− 0.0550569i
$389$	1.60205e6	0.536787	0.268394	−	0.963309i	$-0.413507\pi$
$389$	1.60205e6	0.536787	0.268394	+	0.963309i	$0.413507\pi$
$390$	0	0
$391$	6.83507e6	2.26100
$392$	249493.i	0.0820054i
$393$	− 1.49576e6i	− 0.488516i
$394$	−3.75851e6	−1.21976
$395$	0	0
$396$	−107346.	−0.0343992
$397$	57870.0i	0.0184280i	0.999958	+	0.00921398i	$0.00293294\pi$
$397$	57870.0i	0.0184280i	−0.999958	+	0.00921398i	$0.997067\pi$
$398$	− 2.01224e6i	− 0.636755i
$399$	885032.	0.278309
$400$	0	0
$401$	5.74549e6	1.78429	0.892147	−	0.451746i	$-0.149199\pi$
$401$	5.74549e6	1.78429	0.892147	+	0.451746i	$0.149199\pi$
$402$	− 5.04787e6i	− 1.55791i
$403$	− 314889.i	− 0.0965817i
$404$	190464.	0.0580576
$405$	0	0
$406$	2.62001e6	0.788839
$407$	− 5.76914e6i	− 1.72634i
$408$	− 3.12699e6i	− 0.929985i
$409$	346702.	0.102482	0.0512411	−	0.998686i	$-0.483682\pi$
$409$	346702.	0.102482	0.0512411	+	0.998686i	$0.483682\pi$
$410$	0	0
$411$	3.87568e6	1.13173
$412$	− 109138.i	− 0.0316763i
$413$	− 6.13423e6i	− 1.76964i
$414$	2.48727e6	0.713217
$415$	0	0
$416$	128633.	0.0364435
$417$	2.47443e6i	0.696844i
$418$	− 1.74824e6i	− 0.489395i
$419$	−5.22131e6	−1.45293	−0.726464	−	0.687205i	$-0.758838\pi$
$419$	−5.22131e6	−1.45293	−0.726464	+	0.687205i	$0.758838\pi$
$420$	0	0
$421$	790165.	0.217276	0.108638	−	0.994081i	$-0.465351\pi$
$421$	790165.	0.217276	0.108638	+	0.994081i	$0.465351\pi$
$422$	3.97700e6i	1.08711i
$423$	− 2.17189e6i	− 0.590183i
$424$	−959711.	−0.259254
$425$	0	0
$426$	3.54652e6	0.946843
$427$	− 3.96292e6i	− 1.05183i
$428$	− 302387.i	− 0.0797910i
$429$	1.17855e6	0.309176
$430$	0	0
$431$	6.34150e6	1.64437	0.822184	−	0.569221i	$-0.192755\pi$
$431$	6.34150e6	1.64437	0.822184	+	0.569221i	$0.192755\pi$
$432$	− 4.47673e6i	− 1.15412i
$433$	3.67068e6i	0.940864i	0.882436	+	0.470432i	$0.155902\pi$
$433$	3.67068e6i	0.940864i	−0.882436	+	0.470432i	$0.844098\pi$
$434$	−1.46943e6	−0.374478
$435$	0	0
$436$	141563.	0.0356643
$437$	2.50099e6i	0.626481i
$438$	− 2.98326e6i	− 0.743029i
$439$	−2.64288e6	−0.654510	−0.327255	−	0.944936i	$-0.606124\pi$
$439$	−2.64288e6	−0.654510	−0.327255	+	0.944936i	$0.606124\pi$
$440$	0	0
$441$	−129081.	−0.0316057
$442$	1.43067e6i	0.348325i
$443$	7.06558e6i	1.71056i	0.518165	+	0.855281i	$0.326615\pi$
$443$	7.06558e6i	1.71056i	−0.518165	+	0.855281i	$0.673385\pi$
$444$	−265963.	−0.0640271
$445$	0	0
$446$	−3.31908e6	−0.790097
$447$	− 3.16895e6i	− 0.750147i
$448$	4.09678e6i	0.964378i
$449$	6.28802e6	1.47197	0.735984	−	0.676999i	$-0.236720\pi$
$449$	6.28802e6	1.47197	0.735984	+	0.676999i	$0.236720\pi$
$450$	0	0
$451$	−1.01365e7	−2.34664
$452$	18015.0i	0.00414752i
$453$	− 1.37629e6i	− 0.315112i
$454$	5.82731e6	1.32687
$455$	0	0
$456$	1.14418e6	0.257681
$457$	3.10589e6i	0.695657i	0.937558	+	0.347829i	$0.113081\pi$
$457$	3.10589e6i	0.695657i	−0.937558	+	0.347829i	$0.886919\pi$
$458$	− 959104.i	− 0.213650i
$459$	5.97024e6	1.32270
$460$	0	0
$461$	2.98228e6	0.653575	0.326788	−	0.945098i	$-0.394034\pi$
$461$	2.98228e6	0.653575	0.326788	+	0.945098i	$0.394034\pi$
$462$	− 5.49973e6i	− 1.19877i
$463$	450492.i	0.0976641i	0.998807	+	0.0488320i	$0.0155499\pi$
$463$	450492.i	0.0976641i	−0.998807	+	0.0488320i	$0.984450\pi$
$464$	3.61105e6	0.778643
$465$	0	0
$466$	−6.35702e6	−1.35609
$467$	− 8.15375e6i	− 1.73008i	−0.501707	−	0.865038i	$-0.667294\pi$
$467$	− 8.15375e6i	− 1.73008i	0.501707	−	0.865038i	$-0.332706\pi$
$468$	32143.6i	0.00678392i
$469$	−9.44660e6	−1.98309
$470$	0	0
$471$	1.29633e6	0.269256
$472$	− 7.93042e6i	− 1.63848i
$473$	4.26783e6i	0.877111i
$474$	−3.49929e6	−0.715376
$475$	0	0
$476$	412200.	0.0833856
$477$	− 496529.i	− 0.0999192i
$478$	1.34457e6i	0.269161i
$479$	6.60072e6	1.31448	0.657238	−	0.753683i	$-0.271725\pi$
$479$	6.60072e6	1.31448	0.657238	+	0.753683i	$0.271725\pi$
$480$	0	0
$481$	−1.72751e6	−0.340454
$482$	2.81628e6i	0.552152i
$483$	7.86780e6i	1.53456i
$484$	−331615.	−0.0643459
$485$	0	0
$486$	3.75639e6	0.721407
$487$	7.74626e6i	1.48003i	0.672592	+	0.740014i	$0.265181\pi$
$487$	7.74626e6i	1.48003i	−0.672592	+	0.740014i	$0.734819\pi$
$488$	− 5.12332e6i	− 0.973873i
$489$	−1.97118e6	−0.372781
$490$	0	0
$491$	9.71168e6	1.81799	0.908994	−	0.416810i	$-0.136852\pi$
$491$	9.71168e6	1.81799	0.908994	+	0.416810i	$0.136852\pi$
$492$	467302.i	0.0870332i
$493$	4.81575e6i	0.892373i
$494$	−523491.	−0.0965143
$495$	0	0
$496$	−2.02526e6	−0.369637
$497$	− 6.63696e6i	− 1.20525i
$498$	− 7.65347e6i	− 1.38288i
$499$	−6.38167e6	−1.14732	−0.573658	−	0.819095i	$-0.694476\pi$
$499$	−6.38167e6	−1.14732	−0.573658	+	0.819095i	$0.694476\pi$
$500$	0	0
$501$	−8.50033e6	−1.51301
$502$	− 6.36380e6i	− 1.12709i
$503$	899505.i	0.158520i	0.996854	+	0.0792600i	$0.0252557\pi$
$503$	899505.i	0.158520i	−0.996854	+	0.0792600i	$0.974744\pi$
$504$	−2.12948e6	−0.373419
$505$	0	0
$506$	1.55415e7	2.69847
$507$	− 352905.i	− 0.0609731i
$508$	− 429511.i	− 0.0738439i
$509$	−5.90487e6	−1.01022	−0.505110	−	0.863055i	$-0.668548\pi$
$509$	−5.90487e6	−1.01022	−0.505110	+	0.863055i	$0.668548\pi$
$510$	0	0
$511$	−5.58288e6	−0.945815
$512$	5.24508e6i	0.884255i
$513$	2.18454e6i	0.366494i
$514$	5.19084e6	0.866622
$515$	0	0
$516$	196751.	0.0325307
$517$	− 1.35709e7i	− 2.23297i
$518$	8.06146e6i	1.32005i
$519$	2.41986e6	0.394341
$520$	0	0
$521$	3.84063e6	0.619880	0.309940	−	0.950756i	$-0.399691\pi$
$521$	3.84063e6	0.619880	0.309940	+	0.950756i	$0.399691\pi$
$522$	1.75244e6i	0.281493i
$523$	− 2.64332e6i	− 0.422567i	−0.977425	−	0.211283i	$-0.932236\pi$
$523$	− 2.64332e6i	− 0.422567i	0.977425	−	0.211283i	$-0.0677643\pi$
$524$	254905.	0.0405556
$525$	0	0
$526$	−5.32087e6	−0.838529
$527$	− 2.70091e6i	− 0.423627i
$528$	− 7.58004e6i	− 1.18328i
$529$	−1.57970e7	−2.45435
$530$	0	0
$531$	4.10299e6	0.631486
$532$	150826.i	0.0231046i
$533$	3.03527e6i	0.462785i
$534$	1.68574e6	0.255822
$535$	0	0
$536$	−1.22127e7	−1.83611
$537$	5.49532e6i	0.822351i
$538$	− 7.10898e6i	− 1.05889i
$539$	−806554.	−0.119581
$540$	0	0
$541$	−6.61133e6	−0.971171	−0.485585	−	0.874189i	$-0.661393\pi$
$541$	−6.61133e6	−0.971171	−0.485585	+	0.874189i	$0.661393\pi$
$542$	1.00712e7i	1.47260i
$543$	3.74499e6i	0.545069i
$544$	1.10333e6	0.159849
$545$	0	0
$546$	−1.64684e6	−0.236412
$547$	6.74890e6i	0.964416i	0.876057	+	0.482208i	$0.160165\pi$
$547$	6.74890e6i	0.964416i	−0.876057	+	0.482208i	$0.839835\pi$
$548$	660489.i	0.0939538i
$549$	2.65067e6	0.375340
$550$	0	0
$551$	−1.76211e6	−0.247260
$552$	1.01716e7i	1.42083i
$553$	6.54859e6i	0.910615i
$554$	−6.66049e6	−0.922002
$555$	0	0
$556$	−421690.	−0.0578504
$557$	2.65890e6i	0.363132i	0.983379	+	0.181566i	$0.0581165\pi$
$557$	2.65890e6i	0.363132i	−0.983379	+	0.181566i	$0.941883\pi$
$558$	− 982857.i	− 0.133630i
$559$	1.27796e6	0.172976
$560$	0	0
$561$	1.01089e7	1.35611
$562$	247303.i	0.0330284i
$563$	− 6.17659e6i	− 0.821254i	−0.911803	−	0.410627i	$-0.865310\pi$
$563$	− 6.17659e6i	− 0.821254i	0.911803	−	0.410627i	$-0.134690\pi$
$564$	−625632.	−0.0828173
$565$	0	0
$566$	−1.01156e7	−1.32724
$567$	3.90830e6i	0.510541i
$568$	− 8.58035e6i	− 1.11592i
$569$	5.90074e6	0.764056	0.382028	−	0.924151i	$-0.375226\pi$
$569$	5.90074e6	0.764056	0.382028	+	0.924151i	$0.375226\pi$
$570$	0	0
$571$	2.88636e6	0.370476	0.185238	−	0.982694i	$-0.440694\pi$
$571$	2.88636e6	0.370476	0.185238	+	0.982694i	$0.440694\pi$
$572$	200848.i	0.0256671i
$573$	2.00086e6i	0.254584i
$574$	1.41641e7	1.79436
$575$	0	0
$576$	−2.74020e6	−0.344133
$577$	− 6.83624e6i	− 0.854826i	−0.904057	−	0.427413i	$-0.859425\pi$
$577$	− 6.83624e6i	− 0.854826i	0.904057	−	0.427413i	$-0.140575\pi$
$578$	3.97941e6i	0.495449i
$579$	458310.	0.0568150
$580$	0	0
$581$	−1.43227e7	−1.76029
$582$	5.59483e6i	0.684667i
$583$	− 3.10253e6i	− 0.378046i
$584$	−7.21762e6	−0.875714
$585$	0	0
$586$	6.36640e6	0.765861
$587$	6.16627e6i	0.738630i	0.929304	+	0.369315i	$0.120408\pi$
$587$	6.16627e6i	0.738630i	−0.929304	+	0.369315i	$0.879592\pi$
$588$	37182.9i	0.00443507i
$589$	988279.	0.117379
$590$	0	0
$591$	7.95218e6	0.936521
$592$	1.11107e7i	1.30298i
$593$	1.14506e7i	1.33718i	0.743630	+	0.668591i	$0.233102\pi$
$593$	1.14506e7i	1.33718i	−0.743630	+	0.668591i	$0.766898\pi$
$594$	1.35751e7	1.57862
$595$	0	0
$596$	540049.	0.0622756
$597$	4.25746e6i	0.488894i
$598$	− 4.65375e6i	− 0.532170i
$599$	−1.97315e6	−0.224695	−0.112347	−	0.993669i	$-0.535837\pi$
$599$	−1.97315e6	−0.224695	−0.112347	+	0.993669i	$0.535837\pi$
$600$	0	0
$601$	1.47387e7	1.66446	0.832228	−	0.554433i	$-0.187065\pi$
$601$	1.47387e7	1.66446	0.832228	+	0.554433i	$0.187065\pi$
$602$	− 5.96361e6i	− 0.670684i
$603$	− 6.31852e6i	− 0.707656i
$604$	234546.	0.0261599
$605$	0	0
$606$	−6.52691e6	−0.721982
$607$	788080.i	0.0868158i	0.999057	+	0.0434079i	$0.0138215\pi$
$607$	788080.i	0.0868158i	−0.999057	+	0.0434079i	$0.986178\pi$
$608$	403716.i	0.0442911i
$609$	−5.54337e6	−0.605663
$610$	0	0
$611$	−4.06367e6	−0.440368
$612$	275707.i	0.0297557i
$613$	4.78326e6i	0.514130i	0.966394	+	0.257065i	$0.0827554\pi$
$613$	4.78326e6i	0.514130i	−0.966394	+	0.257065i	$0.917245\pi$
$614$	6.14743e6	0.658071
$615$	0	0
$616$	−1.33059e7	−1.41284
$617$	1.00768e7i	1.06563i	0.846230	+	0.532817i	$0.178867\pi$
$617$	1.00768e7i	1.06563i	−0.846230	+	0.532817i	$0.821133\pi$
$618$	3.74001e6i	0.393914i
$619$	7.36866e6	0.772968	0.386484	−	0.922296i	$-0.373689\pi$
$619$	7.36866e6	0.772968	0.386484	+	0.922296i	$0.373689\pi$
$620$	0	0
$621$	−1.94202e7	−2.02081
$622$	− 8.74657e6i	− 0.906487i
$623$	− 3.15471e6i	− 0.325641i
$624$	−2.26976e6	−0.233356
$625$	0	0
$626$	1.16757e7	1.19082
$627$	3.69889e6i	0.375753i
$628$	220920.i	0.0223530i
$629$	−1.48175e7	−1.49330
$630$	0	0
$631$	−1.06769e7	−1.06751	−0.533755	−	0.845639i	$-0.679219\pi$
$631$	−1.06769e7	−1.06751	−0.533755	+	0.845639i	$0.679219\pi$
$632$	8.46611e6i	0.843123i
$633$	− 8.41447e6i	− 0.834675i
$634$	9.16667e6	0.905708
$635$	0	0
$636$	−143030.	−0.0140211
$637$	241514.i	0.0235827i
$638$	1.09500e7i	1.06503i
$639$	4.43925e6	0.430088
$640$	0	0
$641$	9.72701e6	0.935049	0.467524	−	0.883980i	$-0.345146\pi$
$641$	9.72701e6	0.935049	0.467524	+	0.883980i	$0.345146\pi$
$642$	1.03624e7i	0.992250i
$643$	8.57219e6i	0.817644i	0.912614	+	0.408822i	$0.134060\pi$
$643$	8.57219e6i	0.817644i	−0.912614	+	0.408822i	$0.865940\pi$
$644$	−1.34082e6	−0.127396
$645$	0	0
$646$	−4.49017e6	−0.423332
$647$	5.95965e6i	0.559706i	0.960043	+	0.279853i	$0.0902857\pi$
$647$	5.95965e6i	0.559706i	−0.960043	+	0.279853i	$0.909714\pi$
$648$	5.05271e6i	0.472701i
$649$	2.56373e7	2.38924
$650$	0	0
$651$	3.10900e6	0.287520
$652$	− 335927.i	− 0.0309475i
$653$	1.27099e7i	1.16644i	0.812316	+	0.583218i	$0.198207\pi$
$653$	1.27099e7i	1.16644i	−0.812316	+	0.583218i	$0.801793\pi$
$654$	−4.85116e6	−0.443508
$655$	0	0
$656$	1.95218e7	1.77117
$657$	− 3.73421e6i	− 0.337509i
$658$	1.89632e7i	1.70744i
$659$	1.81505e7	1.62808	0.814038	−	0.580811i	$-0.197265\pi$
$659$	1.81505e7	1.62808	0.814038	+	0.580811i	$0.197265\pi$
$660$	0	0
$661$	−9.11641e6	−0.811559	−0.405779	−	0.913971i	$-0.633000\pi$
$661$	−9.11641e6	−0.811559	−0.405779	+	0.913971i	$0.633000\pi$
$662$	− 8.75885e6i	− 0.776787i
$663$	− 3.02699e6i	− 0.267441i
$664$	−1.85166e7	−1.62983
$665$	0	0
$666$	−5.39205e6	−0.471051
$667$	− 1.56649e7i	− 1.36336i
$668$	− 1.44862e6i	− 0.125607i
$669$	7.02245e6	0.606629
$670$	0	0
$671$	1.65626e7	1.42011
$672$	1.27004e6i	0.108491i
$673$	− 1.85348e7i	− 1.57743i	−0.614758	−	0.788716i	$-0.710746\pi$
$673$	− 1.85348e7i	− 1.57743i	0.614758	−	0.788716i	$-0.289254\pi$
$674$	9.94844e6	0.843539
$675$	0	0
$676$	60141.7	0.00506185
$677$	2.22101e7i	1.86242i	0.364481	+	0.931211i	$0.381247\pi$
$677$	2.22101e7i	1.86242i	−0.364481	+	0.931211i	$0.618753\pi$
$678$	− 617348.i	− 0.0515770i
$679$	1.04702e7	0.871524
$680$	0	0
$681$	−1.23293e7	−1.01876
$682$	− 6.14132e6i	− 0.505593i
$683$	− 1.86642e7i	− 1.53094i	−0.643473	−	0.765469i	$-0.722507\pi$
$683$	− 1.86642e7i	− 1.53094i	0.643473	−	0.765469i	$-0.277493\pi$
$684$	−100883.	−0.00824474
$685$	0	0
$686$	−1.21277e7	−0.983936
$687$	2.02926e6i	0.164038i
$688$	− 8.21938e6i	− 0.662015i
$689$	−929021.	−0.0745551
$690$	0	0
$691$	−2.75399e6	−0.219415	−0.109708	−	0.993964i	$-0.534991\pi$
$691$	−2.75399e6	−0.219415	−0.109708	+	0.993964i	$0.534991\pi$
$692$	412390.i	0.0327373i
$693$	− 6.88413e6i	− 0.544523i
$694$	9.19691e6	0.724842
$695$	0	0
$696$	−7.16655e6	−0.560773
$697$	2.60346e7i	2.02987i
$698$	− 511884.i	− 0.0397679i
$699$	1.34501e7	1.04119
$700$	0	0
$701$	−1.15791e7	−0.889976	−0.444988	−	0.895536i	$-0.646792\pi$
$701$	−1.15791e7	−0.889976	−0.444988	+	0.895536i	$0.646792\pi$
$702$	− 4.06492e6i	− 0.311322i
$703$	− 5.42179e6i	− 0.413766i
$704$	−1.71220e7	−1.30203
$705$	0	0
$706$	1.86477e7	1.40804
$707$	1.22145e7i	0.919024i
$708$	− 1.18190e6i	− 0.0886132i
$709$	−708197.	−0.0529101	−0.0264551	−	0.999650i	$-0.508422\pi$
$709$	−708197.	−0.0529101	−0.0264551	+	0.999650i	$0.508422\pi$
$710$	0	0
$711$	−4.38014e6	−0.324948
$712$	− 4.07845e6i	− 0.301505i
$713$	8.78564e6i	0.647217i
$714$	−1.41255e7	−1.03695
$715$	0	0
$716$	−936506.	−0.0682697
$717$	− 2.84481e6i	− 0.206659i
$718$	7.20130e6i	0.521315i
$719$	1.23580e7	0.891513	0.445757	−	0.895154i	$-0.352935\pi$
$719$	1.23580e7	0.891513	0.445757	+	0.895154i	$0.352935\pi$
$720$	0	0
$721$	6.99907e6	0.501420
$722$	1.28175e7i	0.915080i
$723$	− 5.95864e6i	− 0.423937i
$724$	−638218.	−0.0452504
$725$	0	0
$726$	1.13640e7	0.800181
$727$	− 1.87595e7i	− 1.31639i	−0.752847	−	0.658196i	$-0.771320\pi$
$727$	− 1.87595e7i	− 1.31639i	0.752847	−	0.658196i	$-0.228680\pi$
$728$	3.98432e6i	0.278629i
$729$	−1.49805e7	−1.04402
$730$	0	0
$731$	1.09615e7	0.758711
$732$	− 763550.i	− 0.0526696i
$733$	8.72767e6i	0.599982i	0.953942	+	0.299991i	$0.0969838\pi$
$733$	8.72767e6i	0.599982i	−0.953942	+	0.299991i	$0.903016\pi$
$734$	1.34421e7	0.920929
$735$	0	0
$736$	−3.58897e6	−0.244216
$737$	− 3.94809e7i	− 2.67743i
$738$	9.47392e6i	0.640308i
$739$	−1.00941e7	−0.679915	−0.339958	−	0.940441i	$-0.610413\pi$
$739$	−1.00941e7	−0.679915	−0.339958	+	0.940441i	$0.610413\pi$
$740$	0	0
$741$	1.10759e6	0.0741028
$742$	4.33529e6i	0.289074i
$743$	1.04314e7i	0.693220i	0.938009	+	0.346610i	$0.112667\pi$
$743$	1.04314e7i	0.693220i	−0.938009	+	0.346610i	$0.887333\pi$
$744$	4.01936e6	0.266210
$745$	0	0
$746$	9.49264e6	0.624511
$747$	− 9.58000e6i	− 0.628151i
$748$	1.72274e6i	0.112581i
$749$	1.93922e7	1.26305
$750$	0	0
$751$	−2.12033e7	−1.37184	−0.685920	−	0.727677i	$-0.740600\pi$
$751$	−2.12033e7	−1.37184	−0.685920	+	0.727677i	$0.740600\pi$
$752$	2.61361e7i	1.68537i
$753$	1.34644e7i	0.865366i
$754$	3.27887e6	0.210037
$755$	0	0
$756$	−1.17117e6	−0.0745274
$757$	− 1.08555e7i	− 0.688510i	−0.938876	−	0.344255i	$-0.888131\pi$
$757$	− 1.08555e7i	− 0.688510i	0.938876	−	0.344255i	$-0.111869\pi$
$758$	2.11676e7i	1.33813i
$759$	−3.28825e7	−2.07186
$760$	0	0
$761$	9.82464e6	0.614971	0.307486	−	0.951553i	$-0.400512\pi$
$761$	9.82464e6	0.614971	0.307486	+	0.951553i	$0.400512\pi$
$762$	1.47187e7i	0.918295i
$763$	9.07847e6i	0.564549i
$764$	−340985.	−0.0211350
$765$	0	0
$766$	−1.58401e6	−0.0975404
$767$	− 7.67681e6i	− 0.471186i
$768$	2.54692e6i	0.155816i
$769$	2.66376e7	1.62435	0.812175	−	0.583414i	$-0.198283\pi$
$769$	2.66376e7	1.62435	0.812175	+	0.583414i	$0.198283\pi$
$770$	0	0
$771$	−1.09827e7	−0.665384
$772$	78104.8i	0.00471666i
$773$	− 2.30179e6i	− 0.138553i	−0.997597	−	0.0692767i	$-0.977931\pi$
$773$	− 2.30179e6i	− 0.138553i	0.997597	−	0.0692767i	$-0.0220691\pi$
$774$	3.98886e6	0.239330
$775$	0	0
$776$	1.35360e7	0.806930
$777$	− 1.70563e7i	− 1.01352i
$778$	− 9.35600e6i	− 0.554168i
$779$	−9.52619e6	−0.562439
$780$	0	0
$781$	2.77384e7	1.62725
$782$	− 3.99169e7i	− 2.33421i
$783$	− 1.36828e7i	− 0.797575i
$784$	1.55334e6	0.0902558
$785$	0	0
$786$	−8.73523e6	−0.504334
$787$	4.31784e6i	0.248502i	0.992251	+	0.124251i	$0.0396528\pi$
$787$	4.31784e6i	0.248502i	−0.992251	+	0.124251i	$0.960347\pi$
$788$	1.35520e6i	0.0777479i
$789$	1.12578e7	0.643815
$790$	0	0
$791$	−1.15531e6	−0.0656533
$792$	− 8.89989e6i	− 0.504164i
$793$	− 4.95949e6i	− 0.280062i
$794$	337961.	0.0190246
$795$	0	0
$796$	−725553.	−0.0405869
$797$	855962.i	0.0477319i	0.999715	+	0.0238659i	$0.00759749\pi$
$797$	855962.i	0.0477319i	−0.999715	+	0.0238659i	$0.992403\pi$
$798$	− 5.16860e6i	− 0.287320i
$799$	−3.48555e7	−1.93154
$800$	0	0
$801$	2.11008e6	0.116203
$802$	− 3.35537e7i	− 1.84207i
$803$	− 2.33330e7i	− 1.27697i
$804$	−1.82011e6	−0.0993017
$805$	0	0
$806$	−1.83895e6	−0.0997088
$807$	1.50410e7i	0.813007i
$808$	1.57911e7i	0.850908i
$809$	−1.27474e7	−0.684780	−0.342390	−	0.939558i	$-0.611236\pi$
$809$	−1.27474e7	−0.684780	−0.342390	+	0.939558i	$0.611236\pi$
$810$	0	0
$811$	864775.	0.0461690	0.0230845	−	0.999734i	$-0.492651\pi$
$811$	864775.	0.0461690	0.0230845	+	0.999734i	$0.492651\pi$
$812$	− 944696.i	− 0.0502808i
$813$	− 2.13086e7i	− 1.13065i
$814$	−3.36919e7	−1.78223
$815$	0	0
$816$	−1.94686e7	−1.02355
$817$	4.01087e6i	0.210225i
$818$	− 2.02474e6i	− 0.105800i
$819$	−2.06138e6	−0.107386
$820$	0	0
$821$	1.81218e7	0.938304	0.469152	−	0.883118i	$-0.344560\pi$
$821$	1.81218e7	0.938304	0.469152	+	0.883118i	$0.344560\pi$
$822$	− 2.26340e7i	− 1.16837i
$823$	3.51363e7i	1.80824i	0.427277	+	0.904121i	$0.359473\pi$
$823$	3.51363e7i	1.80824i	−0.427277	+	0.904121i	$0.640527\pi$
$824$	9.04849e6	0.464256
$825$	0	0
$826$	−3.58240e7	−1.82694
$827$	− 6.09168e6i	− 0.309723i	−0.987936	−	0.154861i	$-0.950507\pi$
$827$	− 6.09168e6i	− 0.309723i	0.987936	−	0.154861i	$-0.0494931\pi$
$828$	− 896832.i	− 0.0454606i
$829$	1.41349e7	0.714341	0.357171	−	0.934039i	$-0.383741\pi$
$829$	1.41349e7	0.714341	0.357171	+	0.934039i	$0.383741\pi$
$830$	0	0
$831$	1.40922e7	0.707905
$832$	5.12700e6i	0.256776i
$833$	2.07155e6i	0.103439i
$834$	1.44507e7	0.719406
$835$	0	0
$836$	−630360.	−0.0311942
$837$	7.67401e6i	0.378625i
$838$	3.04925e7i	1.49997i
$839$	−8.32022e6	−0.408066	−0.204033	−	0.978964i	$-0.565405\pi$
$839$	−8.32022e6	−0.408066	−0.204033	+	0.978964i	$0.565405\pi$
$840$	0	0
$841$	−9.47423e6	−0.461906
$842$	− 4.61457e6i	− 0.224311i
$843$	− 523239.i	− 0.0253589i
$844$	1.43399e6	0.0692929
$845$	0	0
$846$	−1.26839e7	−0.609292
$847$	− 2.12666e7i	− 1.01856i
$848$	5.97514e6i	0.285337i
$849$	2.14024e7	1.01904
$850$	0	0
$851$	4.81989e7	2.28146
$852$	− 1.27877e6i	− 0.0603520i
$853$	3.61621e6i	0.170169i	0.996374	+	0.0850846i	$0.0271161\pi$
$853$	3.61621e6i	0.170169i	−0.996374	+	0.0850846i	$0.972884\pi$
$854$	−2.31435e7	−1.08589
$855$	0	0
$856$	2.50704e7	1.16944
$857$	1.82373e7i	0.848219i	0.905611	+	0.424110i	$0.139413\pi$
$857$	1.82373e7i	0.848219i	−0.905611	+	0.424110i	$0.860587\pi$
$858$	− 6.88276e6i	− 0.319186i
$859$	1.61982e7	0.749005	0.374502	−	0.927226i	$-0.377814\pi$
$859$	1.61982e7	0.749005	0.374502	+	0.927226i	$0.377814\pi$
$860$	0	0
$861$	−2.99682e7	−1.37769
$862$	− 3.70345e7i	− 1.69761i
$863$	− 3.38931e7i	− 1.54912i	−0.632502	−	0.774559i	$-0.717972\pi$
$863$	− 3.38931e7i	− 1.54912i	0.632502	−	0.774559i	$-0.282028\pi$
$864$	−3.13486e6	−0.142868
$865$	0	0
$866$	2.14368e7	0.971327
$867$	− 8.41955e6i	− 0.380401i
$868$	529833.i	0.0238693i
$869$	−2.73690e7	−1.22945
$870$	0	0
$871$	−1.18221e7	−0.528021
$872$	1.17368e7i	0.522706i
$873$	7.00316e6i	0.310999i
$874$	1.46058e7	0.646765
$875$	0	0
$876$	−1.07567e6	−0.0473609
$877$	2.80787e7i	1.23276i	0.787450	+	0.616379i	$0.211401\pi$
$877$	2.80787e7i	1.23276i	−0.787450	+	0.616379i	$0.788599\pi$
$878$	1.54345e7i	0.675702i
$879$	−1.34699e7	−0.588021
$880$	0	0
$881$	1.18418e7	0.514017	0.257009	−	0.966409i	$-0.417263\pi$
$881$	1.18418e7	0.514017	0.257009	+	0.966409i	$0.417263\pi$
$882$	753834.i	0.0326290i
$883$	− 1.69484e6i	− 0.0731523i	−0.999331	−	0.0365761i	$-0.988355\pi$
$883$	− 1.69484e6i	− 0.0731523i	0.999331	−	0.0365761i	$-0.0116451\pi$
$884$	515857.	0.0222023
$885$	0	0
$886$	4.12631e7	1.76595
$887$	− 2.51794e7i	− 1.07458i	−0.843399	−	0.537288i	$-0.819449\pi$
$887$	− 2.51794e7i	− 1.07458i	0.843399	−	0.537288i	$-0.180551\pi$
$888$	− 2.20506e7i	− 0.938400i
$889$	2.75446e7	1.16891
$890$	0	0
$891$	−1.63343e7	−0.689296
$892$	1.19676e6i	0.0503610i
$893$	− 1.27538e7i	− 0.535195i
$894$	−1.85067e7	−0.774436
$895$	0	0
$896$	2.72144e7	1.13247
$897$	9.84632e6i	0.408595i
$898$	− 3.67221e7i	− 1.51963i
$899$	−6.19006e6	−0.255444
$900$	0	0
$901$	−7.96854e6	−0.327014
$902$	5.91972e7i	2.42262i
$903$	1.26177e7i	0.514945i
$904$	−1.49360e6	−0.0607872
$905$	0	0
$906$	−8.03755e6	−0.325314
$907$	3.96582e7i	1.60072i	0.599522	+	0.800359i	$0.295358\pi$
$907$	3.96582e7i	1.60072i	−0.599522	+	0.800359i	$0.704642\pi$
$908$	− 2.10115e6i	− 0.0845751i
$909$	−8.16987e6	−0.327949
$910$	0	0
$911$	−1.52269e7	−0.607879	−0.303939	−	0.952691i	$-0.598302\pi$
$911$	−1.52269e7	−0.607879	−0.303939	+	0.952691i	$0.598302\pi$
$912$	− 7.12365e6i	− 0.283606i
$913$	− 5.98601e7i	− 2.37662i
$914$	1.81384e7	0.718181
$915$	0	0
$916$	−345824.	−0.0136181
$917$	1.63471e7i	0.641975i
$918$	− 3.48663e7i	− 1.36552i
$919$	−1.07946e7	−0.421615	−0.210808	−	0.977528i	$-0.567609\pi$
$919$	−1.07946e7	−0.421615	−0.210808	+	0.977528i	$0.567609\pi$
$920$	0	0
$921$	−1.30066e7	−0.505260
$922$	− 1.74165e7i	− 0.674737i
$923$	− 8.30597e6i	− 0.320912i
$924$	−1.98303e6	−0.0764100
$925$	0	0
$926$	2.63088e6	0.100826
$927$	4.68145e6i	0.178929i
$928$	− 2.52866e6i	− 0.0963875i
$929$	−2.43276e7	−0.924826	−0.462413	−	0.886665i	$-0.653016\pi$
$929$	−2.43276e7	−0.924826	−0.462413	+	0.886665i	$0.653016\pi$
$930$	0	0
$931$	−757993.	−0.0286610
$932$	2.29215e6i	0.0864376i
$933$	1.85058e7i	0.695993i
$934$	−4.76180e7	−1.78609
$935$	0	0
$936$	−2.66498e6	−0.0994270
$937$	1.68889e7i	0.628423i	0.949353	+	0.314211i	$0.101740\pi$
$937$	1.68889e7i	0.628423i	−0.949353	+	0.314211i	$0.898260\pi$
$938$	5.51682e7i	2.04730i
$939$	−2.47032e7	−0.914302
$940$	0	0
$941$	−3.14827e7	−1.15904	−0.579519	−	0.814959i	$-0.696759\pi$
$941$	−3.14827e7	−1.15904	−0.579519	+	0.814959i	$0.696759\pi$
$942$	− 7.57061e6i	− 0.277974i
$943$	− 8.46863e7i	− 3.10123i
$944$	−4.93746e7	−1.80332
$945$	0	0
$946$	2.49242e7	0.905510
$947$	8.67071e6i	0.314181i	0.987584	+	0.157090i	$0.0502114\pi$
$947$	8.67071e6i	0.314181i	−0.987584	+	0.157090i	$0.949789\pi$
$948$	1.26174e6i	0.0455983i
$949$	−6.98682e6	−0.251834
$950$	0	0
$951$	−1.93947e7	−0.695394
$952$	3.41749e7i	1.22212i
$953$	− 190552.i	− 0.00679645i	−0.999994	−	0.00339823i	$-0.998918\pi$
$953$	− 190552.i	− 0.00679645i	0.999994	−	0.00339823i	$-0.00108169\pi$
$954$	−2.89974e6	−0.103154
$955$	0	0
$956$	484810.	0.0171564
$957$	− 2.31679e7i	− 0.817723i
$958$	− 3.85483e7i	− 1.35704i
$959$	−4.23574e7	−1.48724
$960$	0	0
$961$	−2.51575e7	−0.878736
$962$	1.00887e7i	0.351477i
$963$	1.29708e7i	0.450714i
$964$	1.01547e6	0.0351943
$965$	0	0
$966$	4.59480e7	1.58425
$967$	− 4.15794e7i	− 1.42992i	−0.699166	−	0.714960i	$-0.746445\pi$
$967$	− 4.15794e7i	− 1.42992i	0.699166	−	0.714960i	$-0.253555\pi$
$968$	− 2.74937e7i	− 0.943072i
$969$	9.50022e6	0.325030
$970$	0	0
$971$	2.33633e7	0.795219	0.397609	−	0.917555i	$-0.369840\pi$
$971$	2.33633e7	0.795219	0.397609	+	0.917555i	$0.369840\pi$
$972$	− 1.35444e6i	− 0.0459827i
$973$	− 2.70431e7i	− 0.915745i
$974$	4.52383e7	1.52795
$975$	0	0
$976$	−3.18977e7	−1.07185
$977$	− 3.30710e7i	− 1.10844i	−0.832371	−	0.554219i	$-0.813017\pi$
$977$	− 3.30710e7i	− 1.10844i	0.832371	−	0.554219i	$-0.186983\pi$
$978$	1.15117e7i	0.384851i
$979$	1.31847e7	0.439657
$980$	0	0
$981$	−6.07230e6	−0.201456
$982$	− 5.67163e7i	− 1.87685i
$983$	− 2.20784e7i	− 0.728759i	−0.931251	−	0.364379i	$-0.881281\pi$
$983$	− 2.20784e7i	− 0.728759i	0.931251	−	0.364379i	$-0.118719\pi$
$984$	−3.87433e7	−1.27558
$985$	0	0
$986$	2.81240e7	0.921266
$987$	− 4.01220e7i	− 1.31096i
$988$	188755.i	0.00615185i
$989$	−3.56560e7	−1.15916
$990$	0	0
$991$	3.54322e6	0.114608	0.0573039	−	0.998357i	$-0.481750\pi$
$991$	3.54322e6	0.114608	0.0573039	+	0.998357i	$0.481750\pi$
$992$	1.41820e6i	0.0457571i
$993$	1.85318e7i	0.596410i
$994$	−3.87599e7	−1.24428
$995$	0	0
$996$	−2.75961e6	−0.0881452
$997$	2.82295e7i	0.899426i	0.893173	+	0.449713i	$0.148474\pi$
$997$	2.82295e7i	0.899426i	−0.893173	+	0.449713i	$0.851526\pi$
$998$	3.72690e7i	1.18446i
$999$	4.21004e7	1.33466

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	325.6.b.i.274.6		22
5.2	odd	4		325.6.a.j.1.9	✓	11
5.3	odd	4		325.6.a.k.1.3	yes	11
5.4	even	2	inner	325.6.b.i.274.17		22

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
325.6.a.j.1.9	✓	11	5.2	odd	4
325.6.a.k.1.3	yes	11	5.3	odd	4
325.6.b.i.274.6		22	1.1	even	1	trivial
325.6.b.i.274.17		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)

\(f(q)\)	\(=\)	\(q-5.84001i q^{2} +12.3562i q^{3} -2.10573 q^{4} +72.1603 q^{6} -135.041i q^{7} -174.583i q^{8} +90.3246 q^{9} +O(q^{10})\)	\(q-5.84001i q^{2} +12.3562i q^{3} -2.10573 q^{4} +72.1603 q^{6} -135.041i q^{7} -174.583i q^{8} +90.3246 q^{9} +564.387 q^{11} -26.0188i q^{12} -169.000i q^{13} -788.641 q^{14} -1086.95 q^{16} -1449.57i q^{17} -527.496i q^{18} +530.407 q^{19} +1668.59 q^{21} -3296.03i q^{22} +4715.23i q^{23} +2157.18 q^{24} -986.962 q^{26} +4118.62i q^{27} +284.360i q^{28} -3322.19 q^{29} +1863.25 q^{31} +761.144i q^{32} +6973.68i q^{33} -8465.52 q^{34} -190.199 q^{36} -10222.0i q^{37} -3097.58i q^{38} +2088.20 q^{39} -17960.2 q^{41} -9744.60i q^{42} +7561.88i q^{43} -1188.45 q^{44} +27537.0 q^{46} -24045.4i q^{47} -13430.6i q^{48} -1429.08 q^{49} +17911.2 q^{51} +355.868i q^{52} -5497.17i q^{53} +24052.8 q^{54} -23575.9 q^{56} +6553.80i q^{57} +19401.6i q^{58} +45424.9 q^{59} +29346.1 q^{61} -10881.4i q^{62} -12197.5i q^{63} -30337.3 q^{64} +40726.4 q^{66} -69953.5i q^{67} +3052.41i q^{68} -58262.3 q^{69} +49147.7 q^{71} -15769.1i q^{72} -41342.1i q^{73} -59696.3 q^{74} -1116.89 q^{76} -76215.5i q^{77} -12195.1i q^{78} -48493.3 q^{79} -28941.6 q^{81} +104888. i q^{82} -106062. i q^{83} -3513.61 q^{84} +44161.5 q^{86} -41049.6i q^{87} -98532.4i q^{88} +23361.1 q^{89} -22821.9 q^{91} -9929.00i q^{92} +23022.6i q^{93} -140425. q^{94} -9404.84 q^{96} +77533.3i q^{97} +8345.84i q^{98} +50978.0 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 22 q - 374 q^{4} + 702 q^{6} - 2744 q^{9}+O(q^{10}) \) 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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