LMFDB - Embedded newform 325.6.b.i.274.4

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			274.4
Character	$\chi$	$=$	325.274
Dual form			325.6.b.i.274.19

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-9.27946i q^{2} -5.85765i q^{3} -54.1083 q^{4} -54.3558 q^{6} +90.4001i q^{7} +205.153i q^{8} +208.688 q^{9} +O(q^{10})$	\(q-9.27946i q^{2} -5.85765i q^{3} -54.1083 q^{4} -54.3558 q^{6} +90.4001i q^{7} +205.153i q^{8} +208.688 q^{9} +597.487 q^{11} +316.948i q^{12} +169.000i q^{13} +838.864 q^{14} +172.246 q^{16} -1347.74i q^{17} -1936.51i q^{18} -2986.90 q^{19} +529.532 q^{21} -5544.36i q^{22} -1354.91i q^{23} +1201.72 q^{24} +1568.23 q^{26} -2645.83i q^{27} -4891.40i q^{28} -6786.00 q^{29} +661.963 q^{31} +4966.56i q^{32} -3499.87i q^{33} -12506.3 q^{34} -11291.8 q^{36} -9682.46i q^{37} +27716.8i q^{38} +989.943 q^{39} +6933.76 q^{41} -4913.77i q^{42} -2511.55i q^{43} -32329.0 q^{44} -12572.8 q^{46} -10294.9i q^{47} -1008.96i q^{48} +8634.82 q^{49} -7894.59 q^{51} -9144.31i q^{52} -1060.19i q^{53} -24551.9 q^{54} -18545.9 q^{56} +17496.2i q^{57} +62970.4i q^{58} +7026.83 q^{59} -31222.1 q^{61} -6142.65i q^{62} +18865.4i q^{63} +51598.9 q^{64} -32476.9 q^{66} -27843.5i q^{67} +72924.0i q^{68} -7936.60 q^{69} -59047.0 q^{71} +42813.1i q^{72} -69099.5i q^{73} -89848.0 q^{74} +161616. q^{76} +54012.9i q^{77} -9186.13i q^{78} -59896.7 q^{79} +35212.8 q^{81} -64341.5i q^{82} -35435.2i q^{83} -28652.1 q^{84} -23305.8 q^{86} +39750.0i q^{87} +122577. i q^{88} -79177.3 q^{89} -15277.6 q^{91} +73312.0i q^{92} -3877.55i q^{93} -95531.5 q^{94} +29092.4 q^{96} -59235.2i q^{97} -80126.5i q^{98} +124688. q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 22 q - 374 q^{4} + 702 q^{6} - 2744 q^{9}+O(q^{10}) $ 22 * q - 374 * q^4 + 702 * q^6 - 2744 * q^9	$ 22 q - 374 q^{4} + 702 q^{6} - 2744 q^{9} + 2552 q^{11} - 1156 q^{14} + 11414 q^{16} - 7040 q^{19} + 3412 q^{21} - 15446 q^{24} + 1690 q^{26} - 36850 q^{29} + 18136 q^{31} + 28910 q^{34} + 75368 q^{36} - 3718 q^{39} - 20020 q^{41} - 121302 q^{44} - 19716 q^{46} - 146626 q^{49} + 73212 q^{51} - 96026 q^{54} - 2524 q^{56} - 263284 q^{59} - 86730 q^{61} - 360142 q^{64} - 117214 q^{66} - 118690 q^{69} + 136840 q^{71} - 742792 q^{74} + 45034 q^{76} - 104478 q^{79} + 215014 q^{81} - 1705432 q^{84} + 404492 q^{86} - 655424 q^{89} + 70304 q^{91} - 1299948 q^{94} + 1799326 q^{96} - 853396 q^{99}+O(q^{100}) $ 22 * q - 374 * q^4 + 702 * q^6 - 2744 * q^9 + 2552 * q^11 - 1156 * q^14 + 11414 * q^16 - 7040 * q^19 + 3412 * q^21 - 15446 * q^24 + 1690 * q^26 - 36850 * q^29 + 18136 * q^31 + 28910 * q^34 + 75368 * q^36 - 3718 * q^39 - 20020 * q^41 - 121302 * q^44 - 19716 * q^46 - 146626 * q^49 + 73212 * q^51 - 96026 * q^54 - 2524 * q^56 - 263284 * q^59 - 86730 * q^61 - 360142 * q^64 - 117214 * q^66 - 118690 * q^69 + 136840 * q^71 - 742792 * q^74 + 45034 * q^76 - 104478 * q^79 + 215014 * q^81 - 1705432 * q^84 + 404492 * q^86 - 655424 * q^89 + 70304 * q^91 - 1299948 * q^94 + 1799326 * q^96 - 853396 * q^99

Display 100 coefficients 10 coefficients

Character values

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

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

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	− 9.27946i	− 1.64039i	−0.572083	−	0.820196i	$-0.693864\pi$
$2$	− 9.27946i	− 1.64039i	0.572083	−	0.820196i	$-0.306136\pi$
$3$	− 5.85765i	− 0.375768i	−0.982191	−	0.187884i	$-0.939837\pi$
$3$	− 5.85765i	− 0.375768i	0.982191	−	0.187884i	$-0.0601630\pi$
$4$	−54.1083	−1.69089
$5$	0	0
$5$	0	0
$6$	−54.3558	−0.616408
$7$	90.4001i	0.697307i	0.937252	+	0.348653i	$0.113361\pi$
$7$	90.4001i	0.697307i	−0.937252	+	0.348653i	$0.886639\pi$
$8$	205.153i	1.13332i
$9$	208.688	0.858798
$10$	0	0
$11$	597.487	1.48884	0.744418	−	0.667714i	$-0.232727\pi$
$11$	597.487	1.48884	0.744418	+	0.667714i	$0.232727\pi$
$12$	316.948i	0.635382i
$13$	169.000i	0.277350i
$13$	169.000i	0.277350i
$14$	838.864	1.14386
$15$	0	0
$16$	172.246	0.168209
$17$	− 1347.74i	− 1.13106i	−0.824729	−	0.565528i	$-0.808673\pi$
$17$	− 1347.74i	− 1.13106i	0.824729	−	0.565528i	$-0.191327\pi$
$18$	− 1936.51i	− 1.40877i
$19$	−2986.90	−1.89818	−0.949089	−	0.315008i	$-0.897993\pi$
$19$	−2986.90	−1.89818	−0.949089	+	0.315008i	$0.897993\pi$
$20$	0	0
$21$	529.532	0.262026
$22$	− 5544.36i	− 2.44227i
$23$	− 1354.91i	− 0.534061i	−0.963688	−	0.267031i	$-0.913957\pi$
$23$	− 1354.91i	− 0.534061i	0.963688	−	0.267031i	$-0.0860425\pi$
$24$	1201.72	0.425867
$25$	0	0
$26$	1568.23	0.454963
$27$	− 2645.83i	− 0.698478i
$28$	− 4891.40i	− 1.17907i
$29$	−6786.00	−1.49837	−0.749184	−	0.662362i	$-0.769554\pi$
$29$	−6786.00	−1.49837	−0.749184	+	0.662362i	$0.769554\pi$
$30$	0	0
$31$	661.963	0.123717	0.0618585	−	0.998085i	$-0.480297\pi$
$31$	661.963	0.123717	0.0618585	+	0.998085i	$0.480297\pi$
$32$	4966.56i	0.857395i
$33$	− 3499.87i	− 0.559458i
$34$	−12506.3	−1.85537
$35$	0	0
$36$	−11291.8	−1.45213
$37$	− 9682.46i	− 1.16274i	−0.813640	−	0.581368i	$-0.802518\pi$
$37$	− 9682.46i	− 1.16274i	0.813640	−	0.581368i	$-0.197482\pi$
$38$	27716.8i	3.11376i
$39$	989.943	0.104219
$40$	0	0
$41$	6933.76	0.644183	0.322091	−	0.946709i	$-0.395614\pi$
$41$	6933.76	0.644183	0.322091	+	0.946709i	$0.395614\pi$
$42$	− 4913.77i	− 0.429825i
$43$	− 2511.55i	− 0.207143i	−0.994622	−	0.103572i	$-0.966973\pi$
$43$	− 2511.55i	− 0.207143i	0.994622	−	0.103572i	$-0.0330271\pi$
$44$	−32329.0	−2.51745
$45$	0	0
$46$	−12572.8	−0.876070
$47$	− 10294.9i	− 0.679797i	−0.940462	−	0.339899i	$-0.889607\pi$
$47$	− 10294.9i	− 0.679797i	0.940462	−	0.339899i	$-0.110393\pi$
$48$	− 1008.96i	− 0.0632076i
$49$	8634.82	0.513763
$50$	0	0
$51$	−7894.59	−0.425015
$52$	− 9144.31i	− 0.468967i
$53$	− 1060.19i	− 0.0518436i	−0.999664	−	0.0259218i	$-0.991748\pi$
$53$	− 1060.19i	− 0.0518436i	0.999664	−	0.0259218i	$-0.00825209\pi$
$54$	−24551.9	−1.14578
$55$	0	0
$56$	−18545.9	−0.790274
$57$	17496.2i	0.713275i
$58$	62970.4i	2.45791i
$59$	7026.83	0.262802	0.131401	−	0.991329i	$-0.458052\pi$
$59$	7026.83	0.262802	0.131401	+	0.991329i	$0.458052\pi$
$60$	0	0
$61$	−31222.1	−1.07433	−0.537164	−	0.843478i	$-0.680505\pi$
$61$	−31222.1	−1.07433	−0.537164	+	0.843478i	$0.680505\pi$
$62$	− 6142.65i	− 0.202944i
$63$	18865.4i	0.598846i
$64$	51598.9	1.57467
$65$	0	0
$66$	−32476.9	−0.917730
$67$	− 27843.5i	− 0.757768i	−0.925444	−	0.378884i	$-0.876308\pi$
$67$	− 27843.5i	− 0.757768i	0.925444	−	0.378884i	$-0.123692\pi$
$68$	72924.0i	1.91249i
$69$	−7936.60	−0.200683
$70$	0	0
$71$	−59047.0	−1.39012	−0.695060	−	0.718952i	$-0.744622\pi$
$71$	−59047.0	−1.39012	−0.695060	+	0.718952i	$0.744622\pi$
$72$	42813.1i	0.973296i
$73$	− 69099.5i	− 1.51764i	−0.651302	−	0.758818i	$-0.725777\pi$
$73$	− 69099.5i	− 1.51764i	0.651302	−	0.758818i	$-0.274223\pi$
$74$	−89848.0	−1.90734
$75$	0	0
$76$	161616.	3.20960
$77$	54012.9i	1.03817i
$78$	− 9186.13i	− 0.170961i
$79$	−59896.7	−1.07978	−0.539890	−	0.841736i	$-0.681534\pi$
$79$	−59896.7	−1.07978	−0.539890	+	0.841736i	$0.681534\pi$
$80$	0	0
$81$	35212.8	0.596332
$82$	− 64341.5i	− 1.05671i
$83$	− 35435.2i	− 0.564599i	−0.959326	−	0.282300i	$-0.908903\pi$
$83$	− 35435.2i	− 0.564599i	0.959326	−	0.282300i	$-0.0910972\pi$
$84$	−28652.1	−0.443056
$85$	0	0
$86$	−23305.8	−0.339796
$87$	39750.0i	0.563040i
$88$	122577.i	1.68733i
$89$	−79177.3	−1.05956	−0.529780	−	0.848135i	$-0.677726\pi$
$89$	−79177.3	−1.05956	−0.529780	+	0.848135i	$0.677726\pi$
$90$	0	0
$91$	−15277.6	−0.193398
$92$	73312.0i	0.903037i
$93$	− 3877.55i	− 0.0464889i
$94$	−95531.5	−1.11513
$95$	0	0
$96$	29092.4	0.322182
$97$	− 59235.2i	− 0.639221i	−0.947549	−	0.319610i	$-0.896448\pi$
$97$	− 59235.2i	− 0.639221i	0.947549	−	0.319610i	$-0.103552\pi$
$98$	− 80126.5i	− 0.842773i
$99$	124688.	1.27861
$100$	0	0
$101$	190448.	1.85769	0.928844	−	0.370471i	$-0.120804\pi$
$101$	190448.	1.85769	0.928844	+	0.370471i	$0.120804\pi$
$102$	73257.5i	0.697191i
$103$	60755.7i	0.564279i	0.959373	+	0.282140i	$0.0910441\pi$
$103$	60755.7i	0.564279i	−0.959373	+	0.282140i	$0.908956\pi$
$104$	−34670.9	−0.314327
$105$	0	0
$106$	−9838.01	−0.0850438
$107$	131827.i	1.11313i	0.830805	+	0.556563i	$0.187880\pi$
$107$	131827.i	1.11313i	−0.830805	+	0.556563i	$0.812120\pi$
$108$	143161.i	1.18105i
$109$	−148041.	−1.19348	−0.596742	−	0.802433i	$-0.703539\pi$
$109$	−148041.	−1.19348	−0.596742	+	0.802433i	$0.703539\pi$
$110$	0	0
$111$	−56716.5	−0.436920
$112$	15571.1i	0.117293i
$113$	47163.1i	0.347461i	0.984793	+	0.173731i	$0.0555822\pi$
$113$	47163.1i	0.347461i	−0.984793	+	0.173731i	$0.944418\pi$
$114$	162356.	1.17005
$115$	0	0
$116$	367179.	2.53357
$117$	35268.3i	0.238188i
$118$	− 65205.1i	− 0.431099i
$119$	121836.	0.788692
$120$	0	0
$121$	195940.	1.21663
$122$	289724.i	1.76232i
$123$	− 40615.5i	− 0.242064i
$124$	−35817.7	−0.209191
$125$	0	0
$126$	175061.	0.982341
$127$	− 130485.i	− 0.717881i	−0.933360	−	0.358941i	$-0.883138\pi$
$127$	− 130485.i	− 0.717881i	0.933360	−	0.358941i	$-0.116862\pi$
$128$	− 319880.i	− 1.72569i
$129$	−14711.8	−0.0778379
$130$	0	0
$131$	39838.7	0.202828	0.101414	−	0.994844i	$-0.467663\pi$
$131$	39838.7	0.202828	0.101414	+	0.994844i	$0.467663\pi$
$132$	189372.i	0.945979i
$133$	− 270016.i	− 1.32361i
$134$	−258372.	−1.24304
$135$	0	0
$136$	276494.	1.28185
$137$	405016.i	1.84362i	0.387644	+	0.921809i	$0.373289\pi$
$137$	405016.i	1.84362i	−0.387644	+	0.921809i	$0.626711\pi$
$138$	73647.3i	0.329199i
$139$	−146603.	−0.643584	−0.321792	−	0.946810i	$-0.604285\pi$
$139$	−146603.	−0.643584	−0.321792	+	0.946810i	$0.604285\pi$
$140$	0	0
$141$	−60304.2	−0.255446
$142$	547924.i	2.28034i
$143$	100975.i	0.412929i
$144$	35945.7	0.144458
$145$	0	0
$146$	−641206.	−2.48952
$147$	− 50579.8i	− 0.193056i
$148$	523902.i	1.96606i
$149$	−45802.9	−0.169016	−0.0845079	−	0.996423i	$-0.526932\pi$
$149$	−45802.9	−0.169016	−0.0845079	+	0.996423i	$0.526932\pi$
$150$	0	0
$151$	−299685.	−1.06960	−0.534802	−	0.844977i	$-0.679614\pi$
$151$	−299685.	−1.06960	−0.534802	+	0.844977i	$0.679614\pi$
$152$	− 612773.i	− 2.15125i
$153$	− 281257.i	− 0.971348i
$154$	501210.	1.70301
$155$	0	0
$156$	−53564.2	−0.176223
$157$	248510.i	0.804626i	0.915502	+	0.402313i	$0.131794\pi$
$157$	248510.i	0.804626i	−0.915502	+	0.402313i	$0.868206\pi$
$158$	555809.i	1.77126i
$159$	−6210.24	−0.0194812
$160$	0	0
$161$	122484.	0.372405
$162$	− 326756.i	− 0.978218i
$163$	− 610706.i	− 1.80038i	−0.435501	−	0.900188i	$-0.643429\pi$
$163$	− 610706.i	− 1.80038i	0.435501	−	0.900188i	$-0.356571\pi$
$164$	−375174.	−1.08924
$165$	0	0
$166$	−328820.	−0.926164
$167$	383764.i	1.06481i	0.846489	+	0.532406i	$0.178712\pi$
$167$	383764.i	1.06481i	−0.846489	+	0.532406i	$0.821288\pi$
$168$	108635.i	0.296960i
$169$	−28561.0	−0.0769231
$170$	0	0
$171$	−623330.	−1.63015
$172$	135896.i	0.350255i
$173$	− 544383.i	− 1.38290i	−0.722426	−	0.691448i	$-0.756973\pi$
$173$	− 544383.i	− 1.38290i	0.722426	−	0.691448i	$-0.243027\pi$
$174$	368858.	0.923606
$175$	0	0
$176$	102915.	0.250435
$177$	− 41160.7i	− 0.0987528i
$178$	734723.i	1.73809i
$179$	459644.	1.07223	0.536117	−	0.844144i	$-0.319891\pi$
$179$	459644.	1.07223	0.536117	+	0.844144i	$0.319891\pi$
$180$	0	0
$181$	−209506.	−0.475334	−0.237667	−	0.971347i	$-0.576383\pi$
$181$	−209506.	−0.475334	−0.237667	+	0.971347i	$0.576383\pi$
$182$	141768.i	0.317249i
$183$	182888.i	0.403699i
$184$	277965.	0.605264
$185$	0	0
$186$	−35981.5	−0.0762601
$187$	− 805257.i	− 1.68396i
$188$	557042.i	1.14946i
$189$	239183.	0.487053
$190$	0	0
$191$	311387.	0.617614	0.308807	−	0.951125i	$-0.400070\pi$
$191$	311387.	0.617614	0.308807	+	0.951125i	$0.400070\pi$
$192$	− 302248.i	− 0.591712i
$193$	775484.i	1.49858i	0.662243	+	0.749289i	$0.269605\pi$
$193$	775484.i	1.49858i	−0.662243	+	0.749289i	$0.730395\pi$
$194$	−549671.	−1.04857
$195$	0	0
$196$	−467216.	−0.868715
$197$	− 492489.i	− 0.904130i	−0.891985	−	0.452065i	$-0.850687\pi$
$197$	− 492489.i	− 0.904130i	0.891985	−	0.452065i	$-0.149313\pi$
$198$	− 1.15704e6i	− 2.09742i
$199$	−805242.	−1.44143	−0.720716	−	0.693231i	$-0.756187\pi$
$199$	−805242.	−1.44143	−0.720716	+	0.693231i	$0.756187\pi$
$200$	0	0
$201$	−163097.	−0.284745
$202$	− 1.76725e6i	− 3.04734i
$203$	− 613455.i	− 1.04482i
$204$	427163.	0.718652
$205$	0	0
$206$	563780.	0.925639
$207$	− 282754.i	− 0.458651i
$208$	29109.6i	0.0466528i
$209$	−1.78463e6	−2.82607
$210$	0	0
$211$	112124.	0.173377	0.0866885	−	0.996235i	$-0.472372\pi$
$211$	112124.	0.173377	0.0866885	+	0.996235i	$0.472372\pi$
$212$	57365.2i	0.0876616i
$213$	345877.i	0.522363i
$214$	1.22328e6	1.82596
$215$	0	0
$216$	542801.	0.791601
$217$	59841.5i	0.0862686i
$218$	1.37374e6i	1.95778i
$219$	−404761.	−0.570280
$220$	0	0
$221$	227768.	0.313698
$222$	526298.i	0.716720i
$223$	− 309032.i	− 0.416142i	−0.978114	−	0.208071i	$-0.933281\pi$
$223$	− 309032.i	− 0.416142i	0.978114	−	0.208071i	$-0.0667186\pi$
$224$	−448978.	−0.597867
$225$	0	0
$226$	437648.	0.569973
$227$	− 229817.i	− 0.296018i	−0.988986	−	0.148009i	$-0.952714\pi$
$227$	− 229817.i	− 0.296018i	0.988986	−	0.148009i	$-0.0472865\pi$
$228$	− 946692.i	− 1.20607i
$229$	1.48633e6	1.87295	0.936475	−	0.350735i	$-0.114068\pi$
$229$	1.48633e6	1.87295	0.936475	+	0.350735i	$0.114068\pi$
$230$	0	0
$231$	316389.	0.390113
$232$	− 1.39217e6i	− 1.69814i
$233$	− 706867.i	− 0.852997i	−0.904488	−	0.426499i	$-0.859747\pi$
$233$	− 706867.i	− 0.852997i	0.904488	−	0.426499i	$-0.140253\pi$
$234$	327270.	0.390721
$235$	0	0
$236$	−380210.	−0.444369
$237$	350854.i	0.405747i
$238$	− 1.13057e6i	− 1.29376i
$239$	−657958.	−0.745080	−0.372540	−	0.928016i	$-0.621513\pi$
$239$	−657958.	−0.745080	−0.372540	+	0.928016i	$0.621513\pi$
$240$	0	0
$241$	−790694.	−0.876932	−0.438466	−	0.898748i	$-0.644478\pi$
$241$	−790694.	−0.876932	−0.438466	+	0.898748i	$0.644478\pi$
$242$	− 1.81821e6i	− 1.99575i
$243$	− 849201.i	− 0.922561i
$244$	1.68938e6	1.81657
$245$	0	0
$246$	−376890.	−0.397079
$247$	− 504786.i	− 0.526460i
$248$	135804.i	0.140211i
$249$	−207567.	−0.212159
$250$	0	0
$251$	−129798.	−0.130042	−0.0650212	−	0.997884i	$-0.520711\pi$
$251$	−129798.	−0.130042	−0.0650212	+	0.997884i	$0.520711\pi$
$252$	− 1.02078e6i	− 1.01258i
$253$	− 809542.i	− 0.795130i
$254$	−1.21083e6	−1.17761
$255$	0	0
$256$	−1.31715e6	−1.25613
$257$	475029.i	0.448629i	0.974517	+	0.224314i	$0.0720142\pi$
$257$	475029.i	0.448629i	−0.974517	+	0.224314i	$0.927986\pi$
$258$	136517.i	0.127685i
$259$	875295.	0.810784
$260$	0	0
$261$	−1.41616e6	−1.28680
$262$	− 369682.i	− 0.332717i
$263$	− 1.81430e6i	− 1.61741i	−0.588214	−	0.808705i	$-0.700169\pi$
$263$	− 1.81430e6i	− 1.61741i	0.588214	−	0.808705i	$-0.299831\pi$
$264$	718011.	0.634046
$265$	0	0
$266$	−2.50560e6	−2.17124
$267$	463793.i	0.398149i
$268$	1.50656e6i	1.28130i
$269$	−1.77036e6	−1.49170	−0.745849	−	0.666115i	$-0.767956\pi$
$269$	−1.77036e6	−1.49170	−0.745849	+	0.666115i	$0.767956\pi$
$270$	0	0
$271$	1.11442e6	0.921777	0.460888	−	0.887458i	$-0.347531\pi$
$271$	1.11442e6	0.921777	0.460888	+	0.887458i	$0.347531\pi$
$272$	− 232143.i	− 0.190254i
$273$	89490.9i	0.0726729i
$274$	3.75833e6	3.02426
$275$	0	0
$276$	429436.	0.339333
$277$	− 508146.i	− 0.397914i	−0.980008	−	0.198957i	$-0.936245\pi$
$277$	− 508146.i	− 0.397914i	0.980008	−	0.198957i	$-0.0637554\pi$
$278$	1.36039e6i	1.05573i
$279$	138144.	0.106248
$280$	0	0
$281$	1.60415e6	1.21193	0.605966	−	0.795491i	$-0.292787\pi$
$281$	1.60415e6	1.21193	0.605966	+	0.795491i	$0.292787\pi$
$282$	559590.i	0.419032i
$283$	1.80648e6i	1.34081i	0.741997	+	0.670403i	$0.233879\pi$
$283$	1.80648e6i	1.34081i	−0.741997	+	0.670403i	$0.766121\pi$
$284$	3.19494e6	2.35053
$285$	0	0
$286$	936996.	0.677365
$287$	626812.i	0.449193i
$288$	1.03646e6i	0.736329i
$289$	−396546.	−0.279286
$290$	0	0
$291$	−346979.	−0.240199
$292$	3.73886e6i	2.56615i
$293$	− 2.20331e6i	− 1.49936i	−0.661798	−	0.749682i	$-0.730206\pi$
$293$	− 2.20331e6i	− 1.49936i	0.661798	−	0.749682i	$-0.269794\pi$
$294$	−469353.	−0.316688
$295$	0	0
$296$	1.98639e6	1.31776
$297$	− 1.58085e6i	− 1.03992i
$298$	425026.i	0.277252i
$299$	228980.	0.148122
$300$	0	0
$301$	227044.	0.144442
$302$	2.78092e6i	1.75457i
$303$	− 1.11558e6i	− 0.698061i
$304$	−514482.	−0.319290
$305$	0	0
$306$	−2.60991e6	−1.59339
$307$	2.20792e6i	1.33702i	0.743704	+	0.668509i	$0.233067\pi$
$307$	2.20792e6i	1.33702i	−0.743704	+	0.668509i	$0.766933\pi$
$308$	− 2.92255e6i	− 1.75544i
$309$	355886.	0.212038
$310$	0	0
$311$	−1.09149e6	−0.639910	−0.319955	−	0.947433i	$-0.603668\pi$
$311$	−1.09149e6	−0.639910	−0.319955	+	0.947433i	$0.603668\pi$
$312$	203090.i	0.118114i
$313$	− 1.38817e6i	− 0.800905i	−0.916318	−	0.400452i	$-0.868853\pi$
$313$	− 1.38817e6i	− 0.800905i	0.916318	−	0.400452i	$-0.131147\pi$
$314$	2.30604e6	1.31990
$315$	0	0
$316$	3.24091e6	1.82578
$317$	1.04832e6i	0.585931i	0.956123	+	0.292965i	$0.0946421\pi$
$317$	1.04832e6i	0.585931i	−0.956123	+	0.292965i	$0.905358\pi$
$318$	57627.6i	0.0319568i
$319$	−4.05454e6	−2.23082
$320$	0	0
$321$	772196.	0.418278
$322$	− 1.13659e6i	− 0.610889i
$323$	4.02557e6i	2.14694i
$324$	−1.90531e6	−1.00833
$325$	0	0
$326$	−5.66702e6	−2.95332
$327$	867175.i	0.448474i
$328$	1.42248e6i	0.730067i
$329$	930664.	0.474027
$330$	0	0
$331$	3.48155e6	1.74664	0.873320	−	0.487148i	$-0.161963\pi$
$331$	3.48155e6	1.74664	0.873320	+	0.487148i	$0.161963\pi$
$332$	1.91734e6i	0.954673i
$333$	− 2.02061e6i	− 0.998556i
$334$	3.56112e6	1.74671
$335$	0	0
$336$	91209.8	0.0440751
$337$	− 1.92271e6i	− 0.922231i	−0.887340	−	0.461115i	$-0.847449\pi$
$337$	− 1.92271e6i	− 0.922231i	0.887340	−	0.461115i	$-0.152551\pi$
$338$	265031.i	0.126184i
$339$	276265.	0.130565
$340$	0	0
$341$	395514.	0.184194
$342$	5.78417e6i	2.67409i
$343$	2.29994e6i	1.05556i
$344$	515253.	0.234760
$345$	0	0
$346$	−5.05158e6	−2.26849
$347$	− 3.52602e6i	− 1.57203i	−0.618207	−	0.786016i	$-0.712140\pi$
$347$	− 3.52602e6i	− 1.57203i	0.618207	−	0.786016i	$-0.287860\pi$
$348$	− 2.15081e6i	− 0.952036i
$349$	1.36701e6	0.600771	0.300385	−	0.953818i	$-0.402885\pi$
$349$	1.36701e6	0.600771	0.300385	+	0.953818i	$0.402885\pi$
$350$	0	0
$351$	447145.	0.193723
$352$	2.96746e6i	1.27652i
$353$	829000.i	0.354094i	0.984202	+	0.177047i	$0.0566544\pi$
$353$	829000.i	0.354094i	−0.984202	+	0.177047i	$0.943346\pi$
$354$	−381949.	−0.161993
$355$	0	0
$356$	4.28415e6	1.79160
$357$	− 713672.i	− 0.296366i
$358$	− 4.26525e6i	− 1.75888i
$359$	−663764.	−0.271818	−0.135909	−	0.990721i	$-0.543395\pi$
$359$	−663764.	−0.271818	−0.135909	+	0.990721i	$0.543395\pi$
$360$	0	0
$361$	6.44548e6	2.60308
$362$	1.94410e6i	0.779734i
$363$	− 1.14775e6i	− 0.457172i
$364$	826647.	0.327014
$365$	0	0
$366$	1.69710e6	0.662224
$367$	193910.i	0.0751511i	0.999294	+	0.0375755i	$0.0119635\pi$
$367$	193910.i	0.0751511i	−0.999294	+	0.0375755i	$0.988037\pi$
$368$	− 233378.i	− 0.0898339i
$369$	1.44699e6	0.553223
$370$	0	0
$371$	95841.5	0.0361509
$372$	209808.i	0.0786075i
$373$	− 1.83984e6i	− 0.684712i	−0.939570	−	0.342356i	$-0.888775\pi$
$373$	− 1.83984e6i	− 0.684712i	0.939570	−	0.342356i	$-0.111225\pi$
$374$	−7.47235e6	−2.76235
$375$	0	0
$376$	2.11204e6	0.770430
$377$	− 1.14683e6i	− 0.415573i
$378$	− 2.21949e6i	− 0.798958i
$379$	1.64249e6	0.587359	0.293680	−	0.955904i	$-0.405120\pi$
$379$	1.64249e6	0.587359	0.293680	+	0.955904i	$0.405120\pi$
$380$	0	0
$381$	−764338.	−0.269757
$382$	− 2.88951e6i	− 1.01313i
$383$	126562.i	0.0440865i	0.999757	+	0.0220432i	$0.00701715\pi$
$383$	126562.i	0.0440865i	−0.999757	+	0.0220432i	$0.992983\pi$
$384$	−1.87374e6	−0.648458
$385$	0	0
$386$	7.19607e6	2.45826
$387$	− 524130.i	− 0.177894i
$388$	3.20512e6i	1.08085i
$389$	5.20026e6	1.74241	0.871206	−	0.490917i	$-0.163338\pi$
$389$	5.20026e6	1.74241	0.871206	+	0.490917i	$0.163338\pi$
$390$	0	0
$391$	−1.82607e6	−0.604053
$392$	1.77146e6i	0.582260i
$393$	− 233361.i	− 0.0762163i
$394$	−4.57003e6	−1.48313
$395$	0	0
$396$	−6.74668e6	−2.16198
$397$	− 2.24899e6i	− 0.716163i	−0.933690	−	0.358081i	$-0.883431\pi$
$397$	− 2.24899e6i	− 0.716163i	0.933690	−	0.358081i	$-0.116569\pi$
$398$	7.47221e6i	2.36451i
$399$	−1.58166e6	−0.497372
$400$	0	0
$401$	2.11199e6	0.655889	0.327945	−	0.944697i	$-0.393644\pi$
$401$	2.11199e6	0.655889	0.327945	+	0.944697i	$0.393644\pi$
$402$	1.51345e6i	0.467094i
$403$	111872.i	0.0343129i
$404$	−1.03048e7	−3.14114
$405$	0	0
$406$	−5.69253e6	−1.71392
$407$	− 5.78514e6i	− 1.73112i
$408$	− 1.61960e6i	− 0.481679i
$409$	−1.01144e6	−0.298974	−0.149487	−	0.988764i	$-0.547762\pi$
$409$	−1.01144e6	−0.298974	−0.149487	+	0.988764i	$0.547762\pi$
$410$	0	0
$411$	2.37244e6	0.692774
$412$	− 3.28739e6i	− 0.954132i
$413$	635226.i	0.183254i
$414$	−2.62380e6	−0.752367
$415$	0	0
$416$	−839349.	−0.237799
$417$	858748.i	0.241839i
$418$	1.65604e7i	4.63587i
$419$	84677.5	0.0235631	0.0117816	−	0.999931i	$-0.496250\pi$
$419$	84677.5	0.0235631	0.0117816	+	0.999931i	$0.496250\pi$
$420$	0	0
$421$	−2.58658e6	−0.711247	−0.355624	−	0.934629i	$-0.615731\pi$
$421$	−2.58658e6	−0.711247	−0.355624	+	0.934629i	$0.615731\pi$
$422$	− 1.04045e6i	− 0.284406i
$423$	− 2.14843e6i	− 0.583808i
$424$	217502.	0.0587555
$425$	0	0
$426$	3.20955e6	0.856880
$427$	− 2.82248e6i	− 0.749137i
$428$	− 7.13294e6i	− 1.88217i
$429$	591478.	0.155166
$430$	0	0
$431$	63405.6	0.0164412	0.00822062	−	0.999966i	$-0.497383\pi$
$431$	63405.6	0.0164412	0.00822062	+	0.999966i	$0.497383\pi$
$432$	− 455734.i	− 0.117490i
$433$	916925.i	0.235025i	0.993071	+	0.117512i	$0.0374920\pi$
$433$	916925.i	0.235025i	−0.993071	+	0.117512i	$0.962508\pi$
$434$	555296.	0.141514
$435$	0	0
$436$	8.01027e6	2.01805
$437$	4.04699e6i	1.01374i
$438$	3.75596e6i	0.935483i
$439$	−1.74762e6	−0.432799	−0.216399	−	0.976305i	$-0.569431\pi$
$439$	−1.74762e6	−0.432799	−0.216399	+	0.976305i	$0.569431\pi$
$440$	0	0
$441$	1.80198e6	0.441219
$442$	− 2.11356e6i	− 0.514588i
$443$	3.55036e6i	0.859533i	0.902940	+	0.429766i	$0.141404\pi$
$443$	3.55036e6i	0.859533i	−0.902940	+	0.429766i	$0.858596\pi$
$444$	3.06883e6	0.738782
$445$	0	0
$446$	−2.86765e6	−0.682637
$447$	268297.i	0.0635108i
$448$	4.66454e6i	1.09803i
$449$	−7.03537e6	−1.64691	−0.823457	−	0.567378i	$-0.807958\pi$
$449$	−7.03537e6	−1.64691	−0.823457	+	0.567378i	$0.807958\pi$
$450$	0	0
$451$	4.14283e6	0.959082
$452$	− 2.55192e6i	− 0.587517i
$453$	1.75545e6i	0.401924i
$454$	−2.13258e6	−0.485586
$455$	0	0
$456$	−3.58941e6	−0.808372
$457$	1.68083e6i	0.376472i	0.982124	+	0.188236i	$0.0602770\pi$
$457$	1.68083e6i	0.376472i	−0.982124	+	0.188236i	$0.939723\pi$
$458$	− 1.37923e7i	− 3.07237i
$459$	−3.56589e6	−0.790017
$460$	0	0
$461$	−8.85997e6	−1.94169	−0.970845	−	0.239709i	$-0.922948\pi$
$461$	−8.85997e6	−1.94169	−0.970845	+	0.239709i	$0.922948\pi$
$462$	− 2.93592e6i	− 0.639939i
$463$	7.32234e6i	1.58744i	0.608284	+	0.793720i	$0.291858\pi$
$463$	7.32234e6i	1.58744i	−0.608284	+	0.793720i	$0.708142\pi$
$464$	−1.16886e6	−0.252039
$465$	0	0
$466$	−6.55934e6	−1.39925
$467$	840696.i	0.178380i	0.996015	+	0.0891901i	$0.0284279\pi$
$467$	840696.i	0.178380i	−0.996015	+	0.0891901i	$0.971572\pi$
$468$	− 1.90831e6i	− 0.402748i
$469$	2.51705e6	0.528396
$470$	0	0
$471$	1.45568e6	0.302353
$472$	1.44158e6i	0.297840i
$473$	− 1.50062e6i	− 0.308402i
$474$	3.25574e6	0.665585
$475$	0	0
$476$	−6.59233e6	−1.33359
$477$	− 221249.i	− 0.0445232i
$478$	6.10549e6i	1.22222i
$479$	4.72739e6	0.941418	0.470709	−	0.882288i	$-0.343998\pi$
$479$	4.72739e6	0.941418	0.470709	+	0.882288i	$0.343998\pi$
$480$	0	0
$481$	1.63634e6	0.322485
$482$	7.33722e6i	1.43851i
$483$	− 717469.i	− 0.139938i
$484$	−1.06020e7	−2.05718
$485$	0	0
$486$	−7.88013e6	−1.51336
$487$	5.60717e6i	1.07133i	0.844432	+	0.535663i	$0.179938\pi$
$487$	5.60717e6i	1.07133i	−0.844432	+	0.535663i	$0.820062\pi$
$488$	− 6.40532e6i	− 1.21756i
$489$	−3.57730e6	−0.676525
$490$	0	0
$491$	117889.	0.0220683	0.0110341	−	0.999939i	$-0.496488\pi$
$491$	117889.	0.0220683	0.0110341	+	0.999939i	$0.496488\pi$
$492$	2.19764e6i	0.409302i
$493$	9.14576e6i	1.69474i
$494$	−4.68414e6	−0.863600
$495$	0	0
$496$	114020.	0.0208103
$497$	− 5.33786e6i	− 0.969340i
$498$	1.92611e6i	0.348023i
$499$	314105.	0.0564707	0.0282354	−	0.999601i	$-0.491011\pi$
$499$	314105.	0.0564707	0.0282354	+	0.999601i	$0.491011\pi$
$500$	0	0
$501$	2.24796e6	0.400123
$502$	1.20446e6i	0.213320i
$503$	− 8.44792e6i	− 1.48878i	−0.667747	−	0.744389i	$-0.732741\pi$
$503$	− 8.44792e6i	− 1.48878i	0.667747	−	0.744389i	$-0.267259\pi$
$504$	−3.87030e6	−0.678686
$505$	0	0
$506$	−7.51211e6	−1.30432
$507$	167300.i	0.0289053i
$508$	7.06035e6i	1.21385i
$509$	4.29988e6	0.735635	0.367817	−	0.929898i	$-0.380105\pi$
$509$	4.29988e6	0.735635	0.367817	+	0.929898i	$0.380105\pi$
$510$	0	0
$511$	6.24660e6	1.05826
$512$	1.98625e6i	0.334857i
$513$	7.90283e6i	1.32583i
$514$	4.40801e6	0.735927
$515$	0	0
$516$	796030.	0.131615
$517$	− 6.15110e6i	− 1.01211i
$518$	− 8.12227e6i	− 1.33000i
$519$	−3.18881e6	−0.519649
$520$	0	0
$521$	−7.31158e6	−1.18009	−0.590047	−	0.807369i	$-0.700891\pi$
$521$	−7.31158e6	−1.18009	−0.590047	+	0.807369i	$0.700891\pi$
$522$	1.31412e7i	2.11085i
$523$	3.72278e6i	0.595131i	0.954701	+	0.297566i	$0.0961747\pi$
$523$	3.72278e6i	0.595131i	−0.954701	+	0.297566i	$0.903825\pi$
$524$	−2.15561e6	−0.342959
$525$	0	0
$526$	−1.68357e7	−2.65319
$527$	− 892153.i	− 0.139931i
$528$	− 602839.i	− 0.0941058i
$529$	4.60056e6	0.714778
$530$	0	0
$531$	1.46641e6	0.225694
$532$	1.46101e7i	2.23808i
$533$	1.17181e6i	0.178664i
$534$	4.30375e6	0.653121
$535$	0	0
$536$	5.71218e6	0.858796
$537$	− 2.69244e6i	− 0.402912i
$538$	1.64280e7i	2.44697i
$539$	5.15919e6	0.764909
$540$	0	0
$541$	612758.	0.0900111	0.0450056	−	0.998987i	$-0.485669\pi$
$541$	612758.	0.0900111	0.0450056	+	0.998987i	$0.485669\pi$
$542$	− 1.03412e7i	− 1.51208i
$543$	1.22721e6i	0.178616i
$544$	6.69363e6	0.969761
$545$	0	0
$546$	830428.	0.119212
$547$	− 1.31870e7i	− 1.88442i	−0.335020	−	0.942211i	$-0.608743\pi$
$547$	− 1.31870e7i	− 1.88442i	0.335020	−	0.942211i	$-0.391257\pi$
$548$	− 2.19148e7i	− 3.11735i
$549$	−6.51567e6	−0.922632
$550$	0	0
$551$	2.02691e7	2.84417
$552$	− 1.62822e6i	− 0.227439i
$553$	− 5.41467e6i	− 0.752938i
$554$	−4.71532e6	−0.652735
$555$	0	0
$556$	7.93244e6	1.08823
$557$	− 2.49720e6i	− 0.341048i	−0.985354	−	0.170524i	$-0.945454\pi$
$557$	− 2.49720e6i	− 0.341048i	0.985354	−	0.170524i	$-0.0545460\pi$
$558$	− 1.28190e6i	− 0.174288i
$559$	424452.	0.0574512
$560$	0	0
$561$	−4.71692e6	−0.632777
$562$	− 1.48856e7i	− 1.98804i
$563$	− 8.39064e6i	− 1.11564i	−0.829962	−	0.557820i	$-0.811638\pi$
$563$	− 8.39064e6i	− 1.11564i	0.829962	−	0.557820i	$-0.188362\pi$
$564$	3.26296e6	0.431931
$565$	0	0
$566$	1.67631e7	2.19945
$567$	3.18324e6i	0.415826i
$568$	− 1.21137e7i	− 1.57546i
$569$	2.18516e6	0.282945	0.141473	−	0.989942i	$-0.454816\pi$
$569$	2.18516e6	0.282945	0.141473	+	0.989942i	$0.454816\pi$
$570$	0	0
$571$	1.46108e7	1.87535	0.937676	−	0.347510i	$-0.112973\pi$
$571$	1.46108e7	1.87535	0.937676	+	0.347510i	$0.112973\pi$
$572$	− 5.46361e6i	− 0.698215i
$573$	− 1.82400e6i	− 0.232080i
$574$	5.81648e6	0.736852
$575$	0	0
$576$	1.07681e7	1.35233
$577$	756371.i	0.0945791i	0.998881	+	0.0472896i	$0.0150583\pi$
$577$	756371.i	0.0945791i	−0.998881	+	0.0472896i	$0.984942\pi$
$578$	3.67973e6i	0.458138i
$579$	4.54252e6	0.563119
$580$	0	0
$581$	3.20335e6	0.393699
$582$	3.21978e6i	0.394020i
$583$	− 633451.i	− 0.0771866i
$584$	1.41760e7	1.71997
$585$	0	0
$586$	−2.04456e7	−2.45955
$587$	797341.i	0.0955100i	0.998859	+	0.0477550i	$0.0152067\pi$
$587$	797341.i	0.0955100i	−0.998859	+	0.0477550i	$0.984793\pi$
$588$	2.73679e6i	0.326436i
$589$	−1.97722e6	−0.234837
$590$	0	0
$591$	−2.88483e6	−0.339744
$592$	− 1.66776e6i	− 0.195583i
$593$	− 9.00116e6i	− 1.05114i	−0.850750	−	0.525571i	$-0.823852\pi$
$593$	− 9.00116e6i	− 1.05114i	0.850750	−	0.525571i	$-0.176148\pi$
$594$	−1.46694e7	−1.70587
$595$	0	0
$596$	2.47832e6	0.285786
$597$	4.71683e6i	0.541644i
$598$	− 2.12481e6i	− 0.242978i
$599$	1.40266e7	1.59730	0.798648	−	0.601799i	$-0.205549\pi$
$599$	1.40266e7	1.59730	0.798648	+	0.601799i	$0.205549\pi$
$600$	0	0
$601$	1.55864e7	1.76019	0.880094	−	0.474800i	$-0.157480\pi$
$601$	1.55864e7	1.76019	0.880094	+	0.474800i	$0.157480\pi$
$602$	− 2.10685e6i	− 0.236942i
$603$	− 5.81059e6i	− 0.650769i
$604$	1.62155e7	1.80858
$605$	0	0
$606$	−1.03520e7	−1.14509
$607$	6.45850e6i	0.711475i	0.934586	+	0.355738i	$0.115770\pi$
$607$	6.45850e6i	0.711475i	−0.934586	+	0.355738i	$0.884230\pi$
$608$	− 1.48346e7i	− 1.62749i
$609$	−3.59340e6	−0.392611
$610$	0	0
$611$	1.73985e6	0.188542
$612$	1.52184e7i	1.64244i
$613$	− 8.15464e6i	− 0.876503i	−0.898852	−	0.438252i	$-0.855598\pi$
$613$	− 8.15464e6i	− 0.876503i	0.898852	−	0.438252i	$-0.144402\pi$
$614$	2.04883e7	2.19323
$615$	0	0
$616$	−1.10809e7	−1.17659
$617$	1.30591e7i	1.38102i	0.723323	+	0.690510i	$0.242614\pi$
$617$	1.30591e7i	1.38102i	−0.723323	+	0.690510i	$0.757386\pi$
$618$	− 3.30243e6i	− 0.347826i
$619$	1.16624e7	1.22338	0.611690	−	0.791098i	$-0.290490\pi$
$619$	1.16624e7	1.22338	0.611690	+	0.791098i	$0.290490\pi$
$620$	0	0
$621$	−3.58486e6	−0.373030
$622$	1.01284e7i	1.04970i
$623$	− 7.15764e6i	− 0.738839i
$624$	170514.	0.0175306
$625$	0	0
$626$	−1.28814e7	−1.31380
$627$	1.04538e7i	1.06195i
$628$	− 1.34465e7i	− 1.36053i
$629$	−1.30494e7	−1.31512
$630$	0	0
$631$	6.56162e6	0.656051	0.328026	−	0.944669i	$-0.393617\pi$
$631$	6.56162e6	0.656051	0.328026	+	0.944669i	$0.393617\pi$
$632$	− 1.22880e7i	− 1.22374i
$633$	− 656782.i	− 0.0651496i
$634$	9.72786e6	0.961156
$635$	0	0
$636$	336026.	0.0329405
$637$	1.45928e6i	0.142492i
$638$	3.76240e7i	3.65943i
$639$	−1.23224e7	−1.19383
$640$	0	0
$641$	8.91450e6	0.856943	0.428472	−	0.903555i	$-0.359052\pi$
$641$	8.91450e6	0.856943	0.428472	+	0.903555i	$0.359052\pi$
$642$	− 7.16556e6i	− 0.686140i
$643$	1.65259e7i	1.57629i	0.615489	+	0.788146i	$0.288959\pi$
$643$	1.65259e7i	1.57629i	−0.615489	+	0.788146i	$0.711041\pi$
$644$	−6.62741e6	−0.629694
$645$	0	0
$646$	3.73551e7	3.52183
$647$	− 1.36339e7i	− 1.28044i	−0.768192	−	0.640220i	$-0.778843\pi$
$647$	− 1.36339e7i	− 1.28044i	0.768192	−	0.640220i	$-0.221157\pi$
$648$	7.22403e6i	0.675837i
$649$	4.19844e6	0.391269
$650$	0	0
$651$	350530.	0.0324170
$652$	3.30443e7i	3.04423i
$653$	9.93610e6i	0.911870i	0.890013	+	0.455935i	$0.150695\pi$
$653$	9.93610e6i	0.911870i	−0.890013	+	0.455935i	$0.849305\pi$
$654$	8.04691e6	0.735673
$655$	0	0
$656$	1.19431e6	0.108357
$657$	− 1.44202e7i	− 1.30334i
$658$	− 8.63606e6i	− 0.777590i
$659$	1.18157e6	0.105985	0.0529926	−	0.998595i	$-0.483124\pi$
$659$	1.18157e6	0.105985	0.0529926	+	0.998595i	$0.483124\pi$
$660$	0	0
$661$	−1.40076e7	−1.24698	−0.623490	−	0.781831i	$-0.714286\pi$
$661$	−1.40076e7	−1.24698	−0.623490	+	0.781831i	$0.714286\pi$
$662$	− 3.23069e7i	− 2.86517i
$663$	− 1.33419e6i	− 0.117878i
$664$	7.26966e6	0.639873
$665$	0	0
$666$	−1.87502e7	−1.63802
$667$	9.19442e6i	0.800221i
$668$	− 2.07648e7i	− 1.80048i
$669$	−1.81020e6	−0.156373
$670$	0	0
$671$	−1.86548e7	−1.59950
$672$	2.62995e6i	0.224660i
$673$	− 1.45325e7i	− 1.23681i	−0.785860	−	0.618404i	$-0.787779\pi$
$673$	− 1.45325e7i	− 1.23681i	0.785860	−	0.618404i	$-0.212221\pi$
$674$	−1.78417e7	−1.51282
$675$	0	0
$676$	1.54539e6	0.130068
$677$	6.27094e6i	0.525849i	0.964816	+	0.262925i	$0.0846870\pi$
$677$	6.27094e6i	0.525849i	−0.964816	+	0.262925i	$0.915313\pi$
$678$	− 2.56359e6i	− 0.214178i
$679$	5.35487e6	0.445733
$680$	0	0
$681$	−1.34619e6	−0.111234
$682$	− 3.67016e6i	− 0.302151i
$683$	− 188702.i	− 0.0154784i	−0.999970	−	0.00773919i	$-0.997537\pi$
$683$	− 188702.i	− 0.0154784i	0.999970	−	0.00773919i	$-0.00246348\pi$
$684$	3.37274e7	2.75640
$685$	0	0
$686$	2.13422e7	1.73153
$687$	− 8.70639e6i	− 0.703796i
$688$	− 432604.i	− 0.0348433i
$689$	179172.	0.0143788
$690$	0	0
$691$	−2.49501e7	−1.98782	−0.993910	−	0.110195i	$-0.964852\pi$
$691$	−2.49501e7	−1.98782	−0.993910	+	0.110195i	$0.964852\pi$
$692$	2.94557e7i	2.33832i
$693$	1.12718e7i	0.891583i
$694$	−3.27196e7	−2.57875
$695$	0	0
$696$	−8.15485e6	−0.638106
$697$	− 9.34490e6i	− 0.728606i
$698$	− 1.26851e7i	− 0.985500i
$699$	−4.14058e6	−0.320530
$700$	0	0
$701$	1.11757e7	0.858975	0.429488	−	0.903073i	$-0.358694\pi$
$701$	1.11757e7	0.858975	0.429488	+	0.903073i	$0.358694\pi$
$702$	− 4.14927e6i	− 0.317781i
$703$	2.89206e7i	2.20708i
$704$	3.08297e7	2.34443
$705$	0	0
$706$	7.69267e6	0.580852
$707$	1.72165e7i	1.29538i
$708$	2.22714e6i	0.166980i
$709$	1.91144e7	1.42806	0.714028	−	0.700117i	$-0.246869\pi$
$709$	1.91144e7	1.42806	0.714028	+	0.700117i	$0.246869\pi$
$710$	0	0
$711$	−1.24997e7	−0.927313
$712$	− 1.62435e7i	− 1.20082i
$713$	− 896900.i	− 0.0660724i
$714$	−6.62249e6	−0.486156
$715$	0	0
$716$	−2.48706e7	−1.81302
$717$	3.85409e6i	0.279978i
$718$	6.15937e6i	0.445887i
$719$	9.47384e6	0.683446	0.341723	−	0.939801i	$-0.388990\pi$
$719$	9.47384e6	0.683446	0.341723	+	0.939801i	$0.388990\pi$
$720$	0	0
$721$	−5.49232e6	−0.393476
$722$	− 5.98106e7i	− 4.27007i
$723$	4.63161e6i	0.329524i
$724$	1.13360e7	0.803736
$725$	0	0
$726$	−1.06505e7	−0.749941
$727$	2.00091e7i	1.40408i	0.712139	+	0.702039i	$0.247727\pi$
$727$	2.00091e7i	1.40408i	−0.712139	+	0.702039i	$0.752273\pi$
$728$	− 3.13426e6i	− 0.219183i
$729$	3.58239e6	0.249663
$730$	0	0
$731$	−3.38492e6	−0.234290
$732$	− 9.89577e6i	− 0.682609i
$733$	− 8.14639e6i	− 0.560023i	−0.959997	−	0.280011i	$-0.909662\pi$
$733$	− 8.14639e6i	− 0.560023i	0.959997	−	0.280011i	$-0.0903382\pi$
$734$	1.79938e6	0.123277
$735$	0	0
$736$	6.72925e6	0.457901
$737$	− 1.66361e7i	− 1.12819i
$738$	− 1.34273e7i	− 0.907502i
$739$	−9.22447e6	−0.621342	−0.310671	−	0.950518i	$-0.600554\pi$
$739$	−9.22447e6	−0.621342	−0.310671	+	0.950518i	$0.600554\pi$
$740$	0	0
$741$	−2.95686e6	−0.197827
$742$	− 889357.i	− 0.0593016i
$743$	1.27277e7i	0.845820i	0.906172	+	0.422910i	$0.138991\pi$
$743$	1.27277e7i	0.845820i	−0.906172	+	0.422910i	$0.861009\pi$
$744$	795492.	0.0526870
$745$	0	0
$746$	−1.70727e7	−1.12320
$747$	− 7.39491e6i	− 0.484877i
$748$	4.35711e7i	2.84738i
$749$	−1.19172e7	−0.776191
$750$	0	0
$751$	−1.45313e7	−0.940169	−0.470084	−	0.882622i	$-0.655776\pi$
$751$	−1.45313e7	−0.940169	−0.470084	+	0.882622i	$0.655776\pi$
$752$	− 1.77326e6i	− 0.114348i
$753$	760313.i	0.0488658i
$754$	−1.06420e7	−0.681702
$755$	0	0
$756$	−1.29418e7	−0.823551
$757$	− 9.46969e6i	− 0.600615i	−0.953842	−	0.300307i	$-0.902911\pi$
$757$	− 9.46969e6i	− 0.600615i	0.953842	−	0.300307i	$-0.0970893\pi$
$758$	− 1.52414e7i	− 0.963499i
$759$	−4.74201e6	−0.298785
$760$	0	0
$761$	2.61063e7	1.63412	0.817059	−	0.576554i	$-0.195603\pi$
$761$	2.61063e7	1.63412	0.817059	+	0.576554i	$0.195603\pi$
$762$	7.09264e6i	0.442507i
$763$	− 1.33830e7i	− 0.832225i
$764$	−1.68486e7	−1.04432
$765$	0	0
$766$	1.17442e6	0.0723191
$767$	1.18753e6i	0.0728883i
$768$	7.71538e6i	0.472013i
$769$	5.26481e6	0.321045	0.160523	−	0.987032i	$-0.448682\pi$
$769$	5.26481e6	0.321045	0.160523	+	0.987032i	$0.448682\pi$
$770$	0	0
$771$	2.78255e6	0.168580
$772$	− 4.19602e7i	− 2.53393i
$773$	− 6.66639e6i	− 0.401275i	−0.979666	−	0.200638i	$-0.935699\pi$
$773$	− 6.66639e6i	− 0.401275i	0.979666	−	0.200638i	$-0.0643014\pi$
$774$	−4.86364e6	−0.291816
$775$	0	0
$776$	1.21523e7	0.724444
$777$	− 5.12717e6i	− 0.304667i
$778$	− 4.82556e7i	− 2.85824i
$779$	−2.07105e7	−1.22277
$780$	0	0
$781$	−3.52798e7	−2.06966
$782$	1.69449e7i	0.990883i
$783$	1.79546e7i	1.04658i
$784$	1.48731e6	0.0864196
$785$	0	0
$786$	−2.16547e6	−0.125025
$787$	− 1.01573e7i	− 0.584577i	−0.956330	−	0.292288i	$-0.905583\pi$
$787$	− 1.01573e7i	− 0.584577i	0.956330	−	0.292288i	$-0.0944167\pi$
$788$	2.66478e7i	1.52878i
$789$	−1.06275e7	−0.607772
$790$	0	0
$791$	−4.26355e6	−0.242287
$792$	2.55802e7i	1.44908i
$793$	− 5.27653e6i	− 0.297965i
$794$	−2.08694e7	−1.17479
$795$	0	0
$796$	4.35703e7	2.43730
$797$	9.12058e6i	0.508600i	0.967125	+	0.254300i	$0.0818451\pi$
$797$	9.12058e6i	0.508600i	−0.967125	+	0.254300i	$0.918155\pi$
$798$	1.46770e7i	0.815885i
$799$	−1.38749e7	−0.768888
$800$	0	0
$801$	−1.65233e7	−0.909948
$802$	− 1.95981e7i	− 1.07592i
$803$	− 4.12861e7i	− 2.25951i
$804$	8.82492e6	0.481472
$805$	0	0
$806$	1.03811e6	0.0562866
$807$	1.03702e7i	0.560533i
$808$	3.90710e7i	2.10536i
$809$	1.59436e7	0.856475	0.428238	−	0.903666i	$-0.359135\pi$
$809$	1.59436e7	0.856475	0.428238	+	0.903666i	$0.359135\pi$
$810$	0	0
$811$	−1.92860e7	−1.02965	−0.514825	−	0.857295i	$-0.672143\pi$
$811$	−1.92860e7	−1.02965	−0.514825	+	0.857295i	$0.672143\pi$
$812$	3.31930e7i	1.76668i
$813$	− 6.52789e6i	− 0.346375i
$814$	−5.36830e7	−2.83972
$815$	0	0
$816$	−1.35981e6	−0.0714913
$817$	7.50175e6i	0.393195i
$818$	9.38565e6i	0.490435i
$819$	−3.18825e6	−0.166090
$820$	0	0
$821$	−3.67112e6	−0.190082	−0.0950408	−	0.995473i	$-0.530298\pi$
$821$	−3.67112e6	−0.190082	−0.0950408	+	0.995473i	$0.530298\pi$
$822$	− 2.20150e7i	− 1.13642i
$823$	− 7.63657e6i	− 0.393006i	−0.980503	−	0.196503i	$-0.937042\pi$
$823$	− 7.63657e6i	− 0.393006i	0.980503	−	0.196503i	$-0.0629585\pi$
$824$	−1.24642e7	−0.639511
$825$	0	0
$826$	5.89455e6	0.300608
$827$	− 1.92879e7i	− 0.980667i	−0.871535	−	0.490334i	$-0.836875\pi$
$827$	− 1.92879e7i	− 0.980667i	0.871535	−	0.490334i	$-0.163125\pi$
$828$	1.52993e7i	0.775526i
$829$	−1.14847e7	−0.580407	−0.290203	−	0.956965i	$-0.593723\pi$
$829$	−1.14847e7	−0.580407	−0.290203	+	0.956965i	$0.593723\pi$
$830$	0	0
$831$	−2.97654e6	−0.149524
$832$	8.72021e6i	0.436736i
$833$	− 1.16375e7i	− 0.581095i
$834$	7.96872e6	0.396710
$835$	0	0
$836$	9.65636e7	4.77857
$837$	− 1.75144e6i	− 0.0864135i
$838$	− 785761.i	− 0.0386528i
$839$	−3.21966e7	−1.57908	−0.789541	−	0.613698i	$-0.789681\pi$
$839$	−3.21966e7	−1.57908	−0.789541	+	0.613698i	$0.789681\pi$
$840$	0	0
$841$	2.55386e7	1.24511
$842$	2.40020e7i	1.16672i
$843$	− 9.39653e6i	− 0.455406i
$844$	−6.06683e6	−0.293161
$845$	0	0
$846$	−1.99363e7	−0.957675
$847$	1.77130e7i	0.848365i
$848$	− 182614.i	− 0.00872055i
$849$	1.05817e7	0.503833
$850$	0	0
$851$	−1.31189e7	−0.620973
$852$	− 1.87148e7i	− 0.883256i
$853$	− 3.91351e7i	− 1.84160i	−0.390040	−	0.920798i	$-0.627539\pi$
$853$	− 3.91351e7i	− 1.84160i	0.390040	−	0.920798i	$-0.372461\pi$
$854$	−2.61911e7	−1.22888
$855$	0	0
$856$	−2.70448e7	−1.26153
$857$	− 2.45430e7i	− 1.14150i	−0.821124	−	0.570750i	$-0.806653\pi$
$857$	− 2.45430e7i	− 1.14150i	0.821124	−	0.570750i	$-0.193347\pi$
$858$	− 5.48860e6i	− 0.254532i
$859$	3.17559e7	1.46839	0.734196	−	0.678937i	$-0.237559\pi$
$859$	3.17559e7	1.46839	0.734196	+	0.678937i	$0.237559\pi$
$860$	0	0
$861$	3.67165e6	0.168793
$862$	− 588370.i	− 0.0269701i
$863$	2.85461e7i	1.30473i	0.757906	+	0.652363i	$0.226222\pi$
$863$	2.85461e7i	1.30473i	−0.757906	+	0.652363i	$0.773778\pi$
$864$	1.31407e7	0.598871
$865$	0	0
$866$	8.50857e6	0.385533
$867$	2.32283e6i	0.104947i
$868$	− 3.23792e6i	− 0.145870i
$869$	−3.57875e7	−1.60761
$870$	0	0
$871$	4.70554e6	0.210167
$872$	− 3.03712e7i	− 1.35260i
$873$	− 1.23617e7i	− 0.548961i
$874$	3.75538e7	1.66294
$875$	0	0
$876$	2.19009e7	0.964278
$877$	3.56134e7i	1.56356i	0.623556	+	0.781779i	$0.285688\pi$
$877$	3.56134e7i	1.56356i	−0.623556	+	0.781779i	$0.714312\pi$
$878$	1.62170e7i	0.709959i
$879$	−1.29062e7	−0.563414
$880$	0	0
$881$	−2.79621e7	−1.21375	−0.606877	−	0.794796i	$-0.707578\pi$
$881$	−2.79621e7	−1.21375	−0.606877	+	0.794796i	$0.707578\pi$
$882$	− 1.67214e7i	− 0.723772i
$883$	1.65853e7i	0.715849i	0.933751	+	0.357924i	$0.116515\pi$
$883$	1.65853e7i	0.715849i	−0.933751	+	0.357924i	$0.883485\pi$
$884$	−1.23242e7	−0.530428
$885$	0	0
$886$	3.29454e7	1.40997
$887$	376818.i	0.0160813i	0.999968	+	0.00804067i	$0.00255945\pi$
$887$	376818.i	0.0160813i	−0.999968	+	0.00804067i	$0.997441\pi$
$888$	− 1.16356e7i	− 0.495172i
$889$	1.17959e7	0.500583
$890$	0	0
$891$	2.10392e7	0.887840
$892$	1.67212e7i	0.703649i
$893$	3.07500e7i	1.29038i
$894$	2.48966e6	0.104183
$895$	0	0
$896$	2.89171e7	1.20333
$897$	− 1.34128e6i	− 0.0556596i
$898$	6.52844e7i	2.70159i
$899$	−4.49208e6	−0.185374
$900$	0	0
$901$	−1.42886e6	−0.0586379
$902$	− 3.84432e7i	− 1.57327i
$903$	− 1.32995e6i	− 0.0542769i
$904$	−9.67568e6	−0.393786
$905$	0	0
$906$	1.62896e7	0.659312
$907$	2.16129e7i	0.872360i	0.899860	+	0.436180i	$0.143669\pi$
$907$	2.16129e7i	0.872360i	−0.899860	+	0.436180i	$0.856331\pi$
$908$	1.24350e7i	0.500533i
$909$	3.97442e7	1.59538
$910$	0	0
$911$	2.44743e7	0.977043	0.488521	−	0.872552i	$-0.337536\pi$
$911$	2.44743e7	0.977043	0.488521	+	0.872552i	$0.337536\pi$
$912$	3.01365e6i	0.119979i
$913$	− 2.11721e7i	− 0.840595i
$914$	1.55972e7	0.617561
$915$	0	0
$916$	−8.04228e7	−3.16694
$917$	3.60143e6i	0.141433i
$918$	3.30895e7i	1.29594i
$919$	−3.34269e7	−1.30559	−0.652796	−	0.757534i	$-0.726404\pi$
$919$	−3.34269e7	−1.30559	−0.652796	+	0.757534i	$0.726404\pi$
$920$	0	0
$921$	1.29332e7	0.502409
$922$	8.22157e7i	3.18513i
$923$	− 9.97895e6i	− 0.385550i
$924$	−1.71193e7	−0.659637
$925$	0	0
$926$	6.79473e7	2.60402
$927$	1.26790e7i	0.484602i
$928$	− 3.37031e7i	− 1.28469i
$929$	2.68705e7	1.02149	0.510747	−	0.859731i	$-0.329369\pi$
$929$	2.68705e7	1.02149	0.510747	+	0.859731i	$0.329369\pi$
$930$	0	0
$931$	−2.57914e7	−0.975214
$932$	3.82474e7i	1.44232i
$933$	6.39357e6i	0.240458i
$934$	7.80120e6	0.292613
$935$	0	0
$936$	−7.23541e6	−0.269944
$937$	1.17808e7i	0.438353i	0.975685	+	0.219177i	$0.0703371\pi$
$937$	1.17808e7i	0.438353i	−0.975685	+	0.219177i	$0.929663\pi$
$938$	− 2.33569e7i	− 0.866777i
$939$	−8.13140e6	−0.300955
$940$	0	0
$941$	−4.37398e7	−1.61029	−0.805143	−	0.593081i	$-0.797911\pi$
$941$	−4.37398e7	−1.61029	−0.805143	+	0.593081i	$0.797911\pi$
$942$	− 1.35080e7i	− 0.495978i
$943$	− 9.39463e6i	− 0.344033i
$944$	1.21034e6	0.0442057
$945$	0	0
$946$	−1.39249e7	−0.505900
$947$	5.14429e7i	1.86402i	0.362435	+	0.932009i	$0.381946\pi$
$947$	5.14429e7i	1.86402i	−0.362435	+	0.932009i	$0.618054\pi$
$948$	− 1.89841e7i	− 0.686072i
$949$	1.16778e7	0.420917
$950$	0	0
$951$	6.14070e6	0.220174
$952$	2.49950e7i	0.893843i
$953$	5.27803e7i	1.88252i	0.337682	+	0.941260i	$0.390357\pi$
$953$	5.27803e7i	1.88252i	−0.337682	+	0.941260i	$0.609643\pi$
$954$	−2.05307e6	−0.0730354
$955$	0	0
$956$	3.56010e7	1.25985
$957$	2.37501e7i	0.838274i
$958$	− 4.38676e7i	− 1.54429i
$959$	−3.66135e7	−1.28557
$960$	0	0
$961$	−2.81910e7	−0.984694
$962$	− 1.51843e7i	− 0.529002i
$963$	2.75107e7i	0.955951i
$964$	4.27832e7	1.48279
$965$	0	0
$966$	−6.65772e6	−0.229553
$967$	− 2.62374e6i	− 0.0902307i	−0.998982	−	0.0451154i	$-0.985634\pi$
$967$	− 2.62374e6i	− 0.0902307i	0.998982	−	0.0451154i	$-0.0143655\pi$
$968$	4.01977e7i	1.37884i
$969$	2.35804e7	0.806754
$970$	0	0
$971$	3.08316e6	0.104942	0.0524708	−	0.998622i	$-0.483290\pi$
$971$	3.08316e6	0.104942	0.0524708	+	0.998622i	$0.483290\pi$
$972$	4.59489e7i	1.55994i
$973$	− 1.32529e7i	− 0.448775i
$974$	5.20315e7	1.75739
$975$	0	0
$976$	−5.37788e6	−0.180712
$977$	3.86878e7i	1.29669i	0.761345	+	0.648347i	$0.224539\pi$
$977$	3.86878e7i	1.29669i	−0.761345	+	0.648347i	$0.775461\pi$
$978$	3.31954e7i	1.10977i
$979$	−4.73074e7	−1.57751
$980$	0	0
$981$	−3.08944e7	−1.02496
$982$	− 1.09394e6i	− 0.0362006i
$983$	4.32293e7i	1.42690i	0.700705	+	0.713451i	$0.252869\pi$
$983$	4.32293e7i	1.42690i	−0.700705	+	0.713451i	$0.747131\pi$
$984$	8.33242e6	0.274336
$985$	0	0
$986$	8.48677e7	2.78003
$987$	− 5.45150e6i	− 0.178124i
$988$	2.73132e7i	0.890183i
$989$	−3.40293e6	−0.110627
$990$	0	0
$991$	−1.44224e7	−0.466501	−0.233251	−	0.972417i	$-0.574936\pi$
$991$	−1.44224e7	−0.466501	−0.233251	+	0.972417i	$0.574936\pi$
$992$	3.28768e6i	0.106074i
$993$	− 2.03937e7i	− 0.656332i
$994$	−4.95324e7	−1.59010
$995$	0	0
$996$	1.12311e7	0.358736
$997$	− 4.29304e7i	− 1.36781i	−0.729569	−	0.683907i	$-0.760279\pi$
$997$	− 4.29304e7i	− 1.36781i	0.729569	−	0.683907i	$-0.239721\pi$
$998$	− 2.91472e6i	− 0.0926341i
$999$	−2.56181e7	−0.812146

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	325.6.b.i.274.4		22
5.2	odd	4		325.6.a.k.1.10	yes	11
5.3	odd	4		325.6.a.j.1.2	✓	11
5.4	even	2	inner	325.6.b.i.274.19		22

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
325.6.a.j.1.2	✓	11	5.3	odd	4
325.6.a.k.1.10	yes	11	5.2	odd	4
325.6.b.i.274.4		22	1.1	even	1	trivial
325.6.b.i.274.19		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)

\(f(q)\)	\(=\)	\(q-9.27946i q^{2} -5.85765i q^{3} -54.1083 q^{4} -54.3558 q^{6} +90.4001i q^{7} +205.153i q^{8} +208.688 q^{9} +O(q^{10})\)	\(q-9.27946i q^{2} -5.85765i q^{3} -54.1083 q^{4} -54.3558 q^{6} +90.4001i q^{7} +205.153i q^{8} +208.688 q^{9} +597.487 q^{11} +316.948i q^{12} +169.000i q^{13} +838.864 q^{14} +172.246 q^{16} -1347.74i q^{17} -1936.51i q^{18} -2986.90 q^{19} +529.532 q^{21} -5544.36i q^{22} -1354.91i q^{23} +1201.72 q^{24} +1568.23 q^{26} -2645.83i q^{27} -4891.40i q^{28} -6786.00 q^{29} +661.963 q^{31} +4966.56i q^{32} -3499.87i q^{33} -12506.3 q^{34} -11291.8 q^{36} -9682.46i q^{37} +27716.8i q^{38} +989.943 q^{39} +6933.76 q^{41} -4913.77i q^{42} -2511.55i q^{43} -32329.0 q^{44} -12572.8 q^{46} -10294.9i q^{47} -1008.96i q^{48} +8634.82 q^{49} -7894.59 q^{51} -9144.31i q^{52} -1060.19i q^{53} -24551.9 q^{54} -18545.9 q^{56} +17496.2i q^{57} +62970.4i q^{58} +7026.83 q^{59} -31222.1 q^{61} -6142.65i q^{62} +18865.4i q^{63} +51598.9 q^{64} -32476.9 q^{66} -27843.5i q^{67} +72924.0i q^{68} -7936.60 q^{69} -59047.0 q^{71} +42813.1i q^{72} -69099.5i q^{73} -89848.0 q^{74} +161616. q^{76} +54012.9i q^{77} -9186.13i q^{78} -59896.7 q^{79} +35212.8 q^{81} -64341.5i q^{82} -35435.2i q^{83} -28652.1 q^{84} -23305.8 q^{86} +39750.0i q^{87} +122577. i q^{88} -79177.3 q^{89} -15277.6 q^{91} +73312.0i q^{92} -3877.55i q^{93} -95531.5 q^{94} +29092.4 q^{96} -59235.2i q^{97} -80126.5i q^{98} +124688. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 22 q - 374 q^{4} + 702 q^{6} - 2744 q^{9}+O(q^{10}) \) 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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