LMFDB - Embedded newform 363.5.c.e.241.12

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [363,5,Mod(241,363)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(363, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1]))

N = Newforms(chi, 5, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("363.241");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 363 = 3 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	363.c (of order $2$, degree $1$, 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:	$37.5232965994$
Analytic rank:	$0$
Dimension:	$32$
Twist minimal:	no (minimal twist has level 33)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			241.12
Character	$\chi$	$=$	363.241
Dual form			363.5.c.e.241.21

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.73343i q^{2} -5.19615 q^{3} +12.9952 q^{4} +39.5641 q^{5} +9.00717i q^{6} +28.7171i q^{7} -50.2612i q^{8} +27.0000 q^{9} +O(q^{10})$	\(q-1.73343i q^{2} -5.19615 q^{3} +12.9952 q^{4} +39.5641 q^{5} +9.00717i q^{6} +28.7171i q^{7} -50.2612i q^{8} +27.0000 q^{9} -68.5817i q^{10} -67.5251 q^{12} -160.970i q^{13} +49.7791 q^{14} -205.581 q^{15} +120.799 q^{16} -241.372i q^{17} -46.8026i q^{18} -461.967i q^{19} +514.145 q^{20} -149.219i q^{21} -241.540 q^{23} +261.165i q^{24} +940.321 q^{25} -279.030 q^{26} -140.296 q^{27} +373.185i q^{28} +581.745i q^{29} +356.361i q^{30} -717.120 q^{31} -1013.58i q^{32} -418.402 q^{34} +1136.17i q^{35} +350.871 q^{36} +2452.32 q^{37} -800.787 q^{38} +836.424i q^{39} -1988.54i q^{40} -557.272i q^{41} -258.660 q^{42} +3168.88i q^{43} +1068.23 q^{45} +418.693i q^{46} +2516.10 q^{47} -627.691 q^{48} +1576.33 q^{49} -1629.98i q^{50} +1254.21i q^{51} -2091.84i q^{52} -3358.00 q^{53} +243.194i q^{54} +1443.36 q^{56} +2400.45i q^{57} +1008.41 q^{58} -2753.19 q^{59} -2671.57 q^{60} -6286.15i q^{61} +1243.08i q^{62} +775.362i q^{63} +175.825 q^{64} -6368.63i q^{65} +309.930 q^{67} -3136.68i q^{68} +1255.08 q^{69} +1969.47 q^{70} +5890.78 q^{71} -1357.05i q^{72} -2986.93i q^{73} -4250.92i q^{74} -4886.05 q^{75} -6003.36i q^{76} +1449.88 q^{78} +4620.07i q^{79} +4779.32 q^{80} +729.000 q^{81} -965.992 q^{82} -6902.64i q^{83} -1939.13i q^{84} -9549.68i q^{85} +5493.03 q^{86} -3022.84i q^{87} +6528.45 q^{89} -1851.71i q^{90} +4622.59 q^{91} -3138.87 q^{92} +3726.27 q^{93} -4361.48i q^{94} -18277.3i q^{95} +5266.70i q^{96} -6277.90 q^{97} -2732.45i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 32 q - 244 q^{4} + 36 q^{5} + 864 q^{9}+O(q^{10}) $ 32 * q - 244 * q^4 + 36 * q^5 + 864 * q^9	$ 32 q - 244 q^{4} + 36 q^{5} + 864 q^{9} + 360 q^{12} + 2220 q^{14} - 108 q^{15} + 3908 q^{16} + 468 q^{20} - 2196 q^{23} + 7280 q^{25} + 3564 q^{26} - 5872 q^{31} - 2320 q^{34} - 6588 q^{36} - 656 q^{37} - 2616 q^{38} - 1404 q^{42} + 972 q^{45} + 2640 q^{47} - 9936 q^{48} - 6988 q^{49} + 4560 q^{53} - 5604 q^{56} - 24644 q^{58} + 39612 q^{59} + 20592 q^{60} - 6232 q^{64} + 2796 q^{67} - 10476 q^{69} - 72692 q^{70} - 51828 q^{71} - 18072 q^{75} + 53640 q^{78} - 27624 q^{80} + 23328 q^{81} - 11548 q^{82} + 106284 q^{86} - 38748 q^{89} + 30672 q^{91} + 27000 q^{92} + 42624 q^{93} - 50544 q^{97}+O(q^{100}) $ 32 * q - 244 * q^4 + 36 * q^5 + 864 * q^9 + 360 * q^12 + 2220 * q^14 - 108 * q^15 + 3908 * q^16 + 468 * q^20 - 2196 * q^23 + 7280 * q^25 + 3564 * q^26 - 5872 * q^31 - 2320 * q^34 - 6588 * q^36 - 656 * q^37 - 2616 * q^38 - 1404 * q^42 + 972 * q^45 + 2640 * q^47 - 9936 * q^48 - 6988 * q^49 + 4560 * q^53 - 5604 * q^56 - 24644 * q^58 + 39612 * q^59 + 20592 * q^60 - 6232 * q^64 + 2796 * q^67 - 10476 * q^69 - 72692 * q^70 - 51828 * q^71 - 18072 * q^75 + 53640 * q^78 - 27624 * q^80 + 23328 * q^81 - 11548 * q^82 + 106284 * q^86 - 38748 * q^89 + 30672 * q^91 + 27000 * q^92 + 42624 * q^93 - 50544 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/363\mathbb{Z}\right)^\times$.

$n$	$122$	$244$
$\chi(n)$	$1$	$-1$

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 $

$2$	− 1.73343i	− 0.433358i	−0.976243	−	0.216679i	$-0.930478\pi$
$2$	− 1.73343i	− 0.433358i	0.976243	−	0.216679i	$-0.0695224\pi$
$3$	−5.19615	−0.577350
$3$	−5.19615	−0.577350
$4$	12.9952	0.812201
$5$	39.5641	1.58257	0.791283	−	0.611450i	$-0.209414\pi$
$5$	39.5641	1.58257	0.791283	+	0.611450i	$0.209414\pi$
$6$	9.00717i	0.250199i
$7$	28.7171i	0.586064i	0.956103	+	0.293032i	$0.0946642\pi$
$7$	28.7171i	0.586064i	−0.956103	+	0.293032i	$0.905336\pi$
$8$	− 50.2612i	− 0.785331i
$9$	27.0000	0.333333
$10$	− 68.5817i	− 0.685817i
$11$	0	0
$11$	0	0
$12$	−67.5251	−0.468925
$13$	− 160.970i	− 0.952484i	−0.879314	−	0.476242i	$-0.841999\pi$
$13$	− 160.970i	− 0.952484i	0.879314	−	0.476242i	$-0.158001\pi$
$14$	49.7791	0.253975
$15$	−205.581	−0.913695
$16$	120.799	0.471872
$17$	− 241.372i	− 0.835197i	−0.908632	−	0.417599i	$-0.862872\pi$
$17$	− 241.372i	− 0.835197i	0.908632	−	0.417599i	$-0.137128\pi$
$18$	− 46.8026i	− 0.144453i
$19$	− 461.967i	− 1.27969i	−0.768505	−	0.639843i	$-0.778999\pi$
$19$	− 461.967i	− 1.27969i	0.768505	−	0.639843i	$-0.221001\pi$
$20$	514.145	1.28536
$21$	− 149.219i	− 0.338364i
$22$	0	0
$23$	−241.540	−0.456598	−0.228299	−	0.973591i	$-0.573316\pi$
$23$	−241.540	−0.456598	−0.228299	+	0.973591i	$0.573316\pi$
$24$	261.165i	0.453411i
$25$	940.321	1.50451
$26$	−279.030	−0.412766
$27$	−140.296	−0.192450
$28$	373.185i	0.476002i
$29$	581.745i	0.691730i	0.938284	+	0.345865i	$0.112415\pi$
$29$	581.745i	0.691730i	−0.938284	+	0.345865i	$0.887585\pi$
$30$	356.361i	0.395956i
$31$	−717.120	−0.746223	−0.373111	−	0.927787i	$-0.621709\pi$
$31$	−717.120	−0.746223	−0.373111	+	0.927787i	$0.621709\pi$
$32$	− 1013.58i	− 0.989820i
$33$	0	0
$34$	−418.402	−0.361939
$35$	1136.17i	0.927484i
$36$	350.871	0.270734
$37$	2452.32	1.79132	0.895659	−	0.444741i	$-0.146704\pi$
$37$	2452.32	1.79132	0.895659	+	0.444741i	$0.146704\pi$
$38$	−800.787	−0.554562
$39$	836.424i	0.549917i
$40$	− 1988.54i	− 1.24284i
$41$	− 557.272i	− 0.331512i	−0.986167	−	0.165756i	$-0.946994\pi$
$41$	− 557.272i	− 0.331512i	0.986167	−	0.165756i	$-0.0530065\pi$
$42$	−258.660	−0.146633
$43$	3168.88i	1.71383i	0.515456	+	0.856916i	$0.327623\pi$
$43$	3168.88i	1.71383i	−0.515456	+	0.856916i	$0.672377\pi$
$44$	0	0
$45$	1068.23	0.527522
$46$	418.693i	0.197870i
$47$	2516.10	1.13902	0.569511	−	0.821984i	$-0.307133\pi$
$47$	2516.10	1.13902	0.569511	+	0.821984i	$0.307133\pi$
$48$	−627.691	−0.272435
$49$	1576.33	0.656529
$50$	− 1629.98i	− 0.651992i
$51$	1254.21i	0.482201i
$52$	− 2091.84i	− 0.773609i
$53$	−3358.00	−1.19544	−0.597722	−	0.801703i	$-0.703927\pi$
$53$	−3358.00	−1.19544	−0.597722	+	0.801703i	$0.703927\pi$
$54$	243.194i	0.0833997i
$55$	0	0
$56$	1443.36	0.460254
$57$	2400.45i	0.738827i
$58$	1008.41	0.299766
$59$	−2753.19	−0.790918	−0.395459	−	0.918484i	$-0.629414\pi$
$59$	−2753.19	−0.790918	−0.395459	+	0.918484i	$0.629414\pi$
$60$	−2671.57	−0.742104
$61$	− 6286.15i	− 1.68937i	−0.535263	−	0.844685i	$-0.679787\pi$
$61$	− 6286.15i	− 1.68937i	0.535263	−	0.844685i	$-0.320213\pi$
$62$	1243.08i	0.323381i
$63$	775.362i	0.195355i
$64$	175.825	0.0429260
$65$	− 6368.63i	− 1.50737i
$66$	0	0
$67$	309.930	0.0690421	0.0345210	−	0.999404i	$-0.489009\pi$
$67$	309.930	0.0690421	0.0345210	+	0.999404i	$0.489009\pi$
$68$	− 3136.68i	− 0.678348i
$69$	1255.08	0.263617
$70$	1969.47	0.401932
$71$	5890.78	1.16857	0.584287	−	0.811547i	$-0.301374\pi$
$71$	5890.78	1.16857	0.584287	+	0.811547i	$0.301374\pi$
$72$	− 1357.05i	− 0.261777i
$73$	− 2986.93i	− 0.560505i	−0.959926	−	0.280252i	$-0.909582\pi$
$73$	− 2986.93i	− 0.560505i	0.959926	−	0.280252i	$-0.0904182\pi$
$74$	− 4250.92i	− 0.776281i
$75$	−4886.05	−0.868631
$76$	− 6003.36i	− 1.03936i
$77$	0	0
$78$	1449.88	0.238311
$79$	4620.07i	0.740278i	0.928976	+	0.370139i	$0.120690\pi$
$79$	4620.07i	0.740278i	−0.928976	+	0.370139i	$0.879310\pi$
$80$	4779.32	0.746769
$81$	729.000	0.111111
$82$	−965.992	−0.143663
$83$	− 6902.64i	− 1.00198i	−0.865453	−	0.500990i	$-0.832969\pi$
$83$	− 6902.64i	− 1.00198i	0.865453	−	0.500990i	$-0.167031\pi$
$84$	− 1939.13i	− 0.274820i
$85$	− 9549.68i	− 1.32175i
$86$	5493.03	0.742702
$87$	− 3022.84i	− 0.399371i
$88$	0	0
$89$	6528.45	0.824196	0.412098	−	0.911140i	$-0.364796\pi$
$89$	6528.45	0.824196	0.412098	+	0.911140i	$0.364796\pi$
$90$	− 1851.71i	− 0.228606i
$91$	4622.59	0.558216
$92$	−3138.87	−0.370849
$93$	3726.27	0.430832
$94$	− 4361.48i	− 0.493604i
$95$	− 18277.3i	− 2.02519i
$96$	5266.70i	0.571473i
$97$	−6277.90	−0.667223	−0.333612	−	0.942711i	$-0.608267\pi$
$97$	−6277.90	−0.667223	−0.333612	+	0.942711i	$0.608267\pi$
$98$	− 2732.45i	− 0.284512i
$99$	0	0
$100$	12219.7	1.22197
$101$	2542.34i	0.249225i	0.992205	+	0.124612i	$0.0397688\pi$
$101$	2542.34i	0.249225i	−0.992205	+	0.124612i	$0.960231\pi$
$102$	2174.08	0.208966
$103$	−9208.11	−0.867953	−0.433976	−	0.900924i	$-0.642890\pi$
$103$	−9208.11	−0.867953	−0.433976	+	0.900924i	$0.642890\pi$
$104$	−8090.53	−0.748015
$105$	− 5903.70i	− 0.535483i
$106$	5820.86i	0.518055i
$107$	− 10994.9i	− 0.960338i	−0.877176	−	0.480169i	$-0.840575\pi$
$107$	− 10994.9i	− 0.960338i	0.877176	−	0.480169i	$-0.159425\pi$
$108$	−1823.18	−0.156308
$109$	15439.8i	1.29953i	0.760133	+	0.649767i	$0.225134\pi$
$109$	15439.8i	1.29953i	−0.760133	+	0.649767i	$0.774866\pi$
$110$	0	0
$111$	−12742.6	−1.03422
$112$	3469.01i	0.276547i
$113$	14562.0	1.14042	0.570210	−	0.821499i	$-0.306862\pi$
$113$	14562.0	1.14042	0.570210	+	0.821499i	$0.306862\pi$
$114$	4161.01	0.320176
$115$	−9556.33	−0.722596
$116$	7559.91i	0.561824i
$117$	− 4346.18i	− 0.317495i
$118$	4772.45i	0.342750i
$119$	6931.51	0.489479
$120$	10332.8i	0.717553i
$121$	0	0
$122$	−10896.6	−0.732102
$123$	2895.67i	0.191399i
$124$	−9319.13	−0.606083
$125$	12475.4	0.798426
$126$	1344.04	0.0846584
$127$	14846.5i	0.920487i	0.887793	+	0.460244i	$0.152238\pi$
$127$	14846.5i	0.920487i	−0.887793	+	0.460244i	$0.847762\pi$
$128$	− 16522.0i	− 1.00842i
$129$	− 16466.0i	− 0.989482i
$130$	−11039.6	−0.653229
$131$	7298.07i	0.425271i	0.977132	+	0.212635i	$0.0682047\pi$
$131$	7298.07i	0.425271i	−0.977132	+	0.212635i	$0.931795\pi$
$132$	0	0
$133$	13266.4	0.749978
$134$	− 537.242i	− 0.0299199i
$135$	−5550.69	−0.304565
$136$	−12131.6	−0.655907
$137$	−22747.2	−1.21195	−0.605977	−	0.795482i	$-0.707218\pi$
$137$	−22747.2	−1.21195	−0.605977	+	0.795482i	$0.707218\pi$
$138$	− 2175.59i	− 0.114240i
$139$	18543.1i	0.959740i	0.877340	+	0.479870i	$0.159316\pi$
$139$	18543.1i	0.959740i	−0.877340	+	0.479870i	$0.840684\pi$
$140$	14764.8i	0.753304i
$141$	−13074.0	−0.657615
$142$	− 10211.3i	− 0.506410i
$143$	0	0
$144$	3261.58	0.157291
$145$	23016.2i	1.09471i
$146$	−5177.63	−0.242899
$147$	−8190.83	−0.379047
$148$	31868.4	1.45491
$149$	21301.4i	0.959478i	0.877411	+	0.479739i	$0.159269\pi$
$149$	21301.4i	0.959478i	−0.877411	+	0.479739i	$0.840731\pi$
$150$	8469.63i	0.376428i
$151$	24729.2i	1.08457i	0.840195	+	0.542284i	$0.182440\pi$
$151$	24729.2i	1.08457i	−0.840195	+	0.542284i	$0.817560\pi$
$152$	−23219.0	−1.00498
$153$	− 6517.05i	− 0.278399i
$154$	0	0
$155$	−28372.2	−1.18095
$156$	10869.5i	0.446643i
$157$	3526.65	0.143075	0.0715373	−	0.997438i	$-0.477209\pi$
$157$	3526.65	0.143075	0.0715373	+	0.997438i	$0.477209\pi$
$158$	8008.58	0.320805
$159$	17448.7	0.690190
$160$	− 40101.3i	− 1.56646i
$161$	− 6936.34i	− 0.267596i
$162$	− 1263.67i	− 0.0481508i
$163$	21087.3	0.793682	0.396841	−	0.917887i	$-0.370106\pi$
$163$	21087.3	0.793682	0.396841	+	0.917887i	$0.370106\pi$
$164$	− 7241.87i	− 0.269255i
$165$	0	0
$166$	−11965.2	−0.434215
$167$	− 39552.6i	− 1.41821i	−0.705100	−	0.709107i	$-0.749098\pi$
$167$	− 39552.6i	− 1.41821i	0.705100	−	0.709107i	$-0.250902\pi$
$168$	−7499.90	−0.265728
$169$	2649.72	0.0927741
$170$	−16553.7	−0.572792
$171$	− 12473.1i	− 0.426562i
$172$	41180.2i	1.39198i
$173$	− 22652.3i	− 0.756869i	−0.925628	−	0.378434i	$-0.876463\pi$
$173$	− 22652.3i	− 0.756869i	0.925628	−	0.378434i	$-0.123537\pi$
$174$	−5239.88	−0.173070
$175$	27003.3i	0.881741i
$176$	0	0
$177$	14306.0	0.456637
$178$	− 11316.6i	− 0.357171i
$179$	−3209.13	−0.100157	−0.0500786	−	0.998745i	$-0.515947\pi$
$179$	−3209.13	−0.100157	−0.0500786	+	0.998745i	$0.515947\pi$
$180$	13881.9	0.428454
$181$	40082.1	1.22347	0.611736	−	0.791062i	$-0.290472\pi$
$181$	40082.1	1.22347	0.611736	+	0.791062i	$0.290472\pi$
$182$	− 8012.94i	− 0.241907i
$183$	32663.8i	0.975359i
$184$	12140.1i	0.358581i
$185$	97023.7	2.83488
$186$	− 6459.22i	− 0.186704i
$187$	0	0
$188$	32697.3	0.925115
$189$	− 4028.90i	− 0.112788i
$190$	−31682.5	−0.877630
$191$	−7033.03	−0.192786	−0.0963931	−	0.995343i	$-0.530731\pi$
$191$	−7033.03	−0.192786	−0.0963931	+	0.995343i	$0.530731\pi$
$192$	−913.613	−0.0247833
$193$	43421.3i	1.16570i	0.812579	+	0.582852i	$0.198063\pi$
$193$	43421.3i	1.16570i	−0.812579	+	0.582852i	$0.801937\pi$
$194$	10882.3i	0.289146i
$195$	33092.4i	0.870280i
$196$	20484.7	0.533234
$197$	− 41922.3i	− 1.08022i	−0.841594	−	0.540111i	$-0.818382\pi$
$197$	− 41922.3i	− 1.08022i	0.841594	−	0.540111i	$-0.181618\pi$
$198$	0	0
$199$	−23640.6	−0.596970	−0.298485	−	0.954414i	$-0.596481\pi$
$199$	−23640.6	−0.596970	−0.298485	+	0.954414i	$0.596481\pi$
$200$	− 47261.6i	− 1.18154i
$201$	−1610.44	−0.0398615
$202$	4406.98	0.108004
$203$	−16706.0	−0.405398
$204$	16298.7i	0.391645i
$205$	− 22048.0i	− 0.524640i
$206$	15961.6i	0.376134i
$207$	−6521.59	−0.152199
$208$	− 19445.0i	− 0.449451i
$209$	0	0
$210$	−10233.7	−0.232056
$211$	− 8467.94i	− 0.190201i	−0.995468	−	0.0951005i	$-0.969683\pi$
$211$	− 8467.94i	− 0.190201i	0.995468	−	0.0951005i	$-0.0303173\pi$
$212$	−43638.0	−0.970941
$213$	−30609.4	−0.674677
$214$	−19058.9	−0.416170
$215$	125374.i	2.71225i
$216$	7051.45i	0.151137i
$217$	− 20593.6i	− 0.437334i
$218$	26763.8	0.563163
$219$	15520.5i	0.323608i
$220$	0	0
$221$	−38853.6	−0.795512
$222$	22088.4i	0.448186i
$223$	−51541.6	−1.03645	−0.518225	−	0.855244i	$-0.673407\pi$
$223$	−51541.6	−1.03645	−0.518225	+	0.855244i	$0.673407\pi$
$224$	29107.0	0.580098
$225$	25388.7	0.501504
$226$	− 25242.3i	− 0.494210i
$227$	− 17412.1i	− 0.337910i	−0.985624	−	0.168955i	$-0.945961\pi$
$227$	− 17412.1i	− 0.337910i	0.985624	−	0.168955i	$-0.0540392\pi$
$228$	31194.4i	0.600077i
$229$	−69257.1	−1.32067	−0.660333	−	0.750973i	$-0.729585\pi$
$229$	−69257.1	−1.32067	−0.660333	+	0.750973i	$0.729585\pi$
$230$	16565.2i	0.313142i
$231$	0	0
$232$	29239.2	0.543237
$233$	− 31865.2i	− 0.586955i	−0.955966	−	0.293478i	$-0.905187\pi$
$233$	− 31865.2i	− 0.586955i	0.955966	−	0.293478i	$-0.0948126\pi$
$234$	−7533.81	−0.137589
$235$	99547.3	1.80258
$236$	−35778.3	−0.642385
$237$	− 24006.6i	− 0.427400i
$238$	− 12015.3i	− 0.212119i
$239$	110911.i	1.94169i	0.239702	+	0.970847i	$0.422950\pi$
$239$	110911.i	1.94169i	−0.239702	+	0.970847i	$0.577050\pi$
$240$	−24834.1	−0.431147
$241$	54085.5i	0.931207i	0.884993	+	0.465604i	$0.154163\pi$
$241$	54085.5i	0.931207i	−0.884993	+	0.465604i	$0.845837\pi$
$242$	0	0
$243$	−3788.00	−0.0641500
$244$	− 81689.9i	− 1.37211i
$245$	62366.0	1.03900
$246$	5019.44	0.0829441
$247$	−74362.7	−1.21888
$248$	36043.3i	0.586032i
$249$	35867.1i	0.578493i
$250$	− 21625.2i	− 0.346004i
$251$	87399.1	1.38727	0.693633	−	0.720329i	$-0.256009\pi$
$251$	87399.1	1.38727	0.693633	+	0.720329i	$0.256009\pi$
$252$	10076.0i	0.158667i
$253$	0	0
$254$	25735.4	0.398900
$255$	49621.6i	0.763115i
$256$	−25826.5	−0.394082
$257$	−87941.9	−1.33146	−0.665732	−	0.746190i	$-0.731881\pi$
$257$	−87941.9	−1.33146	−0.665732	+	0.746190i	$0.731881\pi$
$258$	−28542.6	−0.428799
$259$	70423.4i	1.04983i
$260$	− 82761.8i	− 1.22429i
$261$	15707.1i	0.230577i
$262$	12650.7	0.184294
$263$	− 19661.7i	− 0.284255i	−0.989848	−	0.142128i	$-0.954606\pi$
$263$	− 19661.7i	− 0.284255i	0.989848	−	0.142128i	$-0.0453943\pi$
$264$	0	0
$265$	−132857.	−1.89187
$266$	− 22996.3i	− 0.325009i
$267$	−33922.8	−0.475850
$268$	4027.61	0.0560761
$269$	3954.31	0.0546469	0.0273235	−	0.999627i	$-0.491302\pi$
$269$	3954.31	0.0546469	0.0273235	+	0.999627i	$0.491302\pi$
$270$	9621.74i	0.131985i
$271$	89380.3i	1.21704i	0.793540	+	0.608518i	$0.208236\pi$
$271$	89380.3i	1.21704i	−0.793540	+	0.608518i	$0.791764\pi$
$272$	− 29157.6i	− 0.394106i
$273$	−24019.7	−0.322286
$274$	39430.6i	0.525209i
$275$	0	0
$276$	16310.0	0.214110
$277$	110982.i	1.44642i	0.690627	+	0.723211i	$0.257334\pi$
$277$	110982.i	1.44642i	−0.690627	+	0.723211i	$0.742666\pi$
$278$	32143.2	0.415910
$279$	−19362.2	−0.248741
$280$	57105.2	0.728382
$281$	− 99631.7i	− 1.26178i	−0.775871	−	0.630892i	$-0.782689\pi$
$281$	− 99631.7i	− 1.26178i	0.775871	−	0.630892i	$-0.217311\pi$
$282$	22662.9i	0.284982i
$283$	34175.9i	0.426724i	0.976973	+	0.213362i	$0.0684414\pi$
$283$	34175.9i	0.426724i	−0.976973	+	0.213362i	$0.931559\pi$
$284$	76552.0	0.949117
$285$	94971.7i	1.16924i
$286$	0	0
$287$	16003.3	0.194287
$288$	− 27366.6i	− 0.329940i
$289$	25260.5	0.302445
$290$	39897.0	0.474400
$291$	32620.9	0.385221
$292$	− 38815.8i	− 0.455243i
$293$	48315.3i	0.562794i	0.959591	+	0.281397i	$0.0907977\pi$
$293$	48315.3i	0.562794i	−0.959591	+	0.281397i	$0.909202\pi$
$294$	14198.2i	0.164263i
$295$	−108927.	−1.25168
$296$	− 123256.i	− 1.40678i
$297$	0	0
$298$	36924.5	0.415797
$299$	38880.7i	0.434902i
$300$	−63495.3	−0.705503
$301$	−91001.0	−1.00442
$302$	42866.4	0.470006
$303$	− 13210.4i	− 0.143890i
$304$	− 55805.3i	− 0.603848i
$305$	− 248706.i	− 2.67354i
$306$	−11296.8	−0.120646
$307$	47042.3i	0.499128i	0.968358	+	0.249564i	$0.0802872\pi$
$307$	47042.3i	0.499128i	−0.968358	+	0.249564i	$0.919713\pi$
$308$	0	0
$309$	47846.7	0.501113
$310$	49181.3i	0.511772i
$311$	−135805.	−1.40409	−0.702046	−	0.712131i	$-0.747730\pi$
$311$	−135805.	−1.40409	−0.702046	+	0.712131i	$0.747730\pi$
$312$	42039.6	0.431867
$313$	116715.	1.19135	0.595675	−	0.803226i	$-0.296885\pi$
$313$	116715.	1.19135	0.595675	+	0.803226i	$0.296885\pi$
$314$	− 6113.20i	− 0.0620025i
$315$	30676.5i	0.309161i
$316$	60038.9i	0.601255i
$317$	−38815.5	−0.386266	−0.193133	−	0.981173i	$-0.561865\pi$
$317$	−38815.5	−0.386266	−0.193133	+	0.981173i	$0.561865\pi$
$318$	− 30246.1i	− 0.299099i
$319$	0	0
$320$	6956.36	0.0679332
$321$	57131.2i	0.554452i
$322$	−12023.7	−0.115965
$323$	−111506.	−1.06879
$324$	9473.52	0.0902446
$325$	− 151363.i	− 1.43303i
$326$	− 36553.4i	− 0.343948i
$327$	− 80227.4i	− 0.750287i
$328$	−28009.2	−0.260347
$329$	72255.2i	0.667540i
$330$	0	0
$331$	48824.2	0.445635	0.222818	−	0.974860i	$-0.428475\pi$
$331$	48824.2	0.445635	0.222818	+	0.974860i	$0.428475\pi$
$332$	− 89701.3i	− 0.813809i
$333$	66212.5	0.597106
$334$	−68561.7	−0.614594
$335$	12262.1	0.109264
$336$	− 18025.5i	− 0.159665i
$337$	26813.1i	0.236096i	0.993008	+	0.118048i	$0.0376636\pi$
$337$	26813.1i	0.236096i	−0.993008	+	0.118048i	$0.962336\pi$
$338$	− 4593.11i	− 0.0402044i
$339$	−75666.5	−0.658422
$340$	− 124100.i	− 1.07353i
$341$	0	0
$342$	−21621.3	−0.184854
$343$	114217.i	0.970832i
$344$	159271.	1.34593
$345$	49656.2	0.417191
$346$	−39266.2	−0.327995
$347$	121784.i	1.01142i	0.862705	+	0.505708i	$0.168769\pi$
$347$	121784.i	1.01142i	−0.862705	+	0.505708i	$0.831231\pi$
$348$	− 39282.4i	− 0.324369i
$349$	146055.i	1.19913i	0.800326	+	0.599565i	$0.204660\pi$
$349$	146055.i	1.19913i	−0.800326	+	0.599565i	$0.795340\pi$
$350$	46808.4	0.382109
$351$	22583.4i	0.183306i
$352$	0	0
$353$	−36019.5	−0.289061	−0.144530	−	0.989500i	$-0.546167\pi$
$353$	−36019.5	−0.289061	−0.144530	+	0.989500i	$0.546167\pi$
$354$	− 24798.4i	− 0.197887i
$355$	233064.	1.84934
$356$	84838.7	0.669413
$357$	−36017.2	−0.282601
$358$	5562.81i	0.0434038i
$359$	− 100539.i	− 0.780093i	−0.920795	−	0.390047i	$-0.872459\pi$
$359$	− 100539.i	− 0.780093i	0.920795	−	0.390047i	$-0.127541\pi$
$360$	− 53690.6i	− 0.414279i
$361$	−83092.4	−0.637598
$362$	− 69479.6i	− 0.530200i
$363$	0	0
$364$	60071.6	0.453384
$365$	− 118175.i	− 0.887036i
$366$	56620.4	0.422679
$367$	−130034.	−0.965441	−0.482720	−	0.875775i	$-0.660351\pi$
$367$	−130034.	−0.965441	−0.482720	+	0.875775i	$0.660351\pi$
$368$	−29177.9	−0.215456
$369$	− 15046.3i	− 0.110504i
$370$	− 168184.i	− 1.22852i
$371$	− 96432.2i	− 0.700607i
$372$	48423.6	0.349922
$373$	− 149908.i	− 1.07747i	−0.842474	−	0.538737i	$-0.818902\pi$
$373$	− 149908.i	− 1.07747i	0.842474	−	0.538737i	$-0.181098\pi$
$374$	0	0
$375$	−64824.1	−0.460971
$376$	− 126462.i	− 0.894510i
$377$	93643.4	0.658862
$378$	−6983.82	−0.0488775
$379$	−221325.	−1.54082	−0.770409	−	0.637550i	$-0.779948\pi$
$379$	−221325.	−1.54082	−0.770409	+	0.637550i	$0.779948\pi$
$380$	− 237518.i	− 1.64486i
$381$	− 77144.9i	− 0.531443i
$382$	12191.3i	0.0835453i
$383$	22226.4	0.151520	0.0757602	−	0.997126i	$-0.475862\pi$
$383$	22226.4	0.151520	0.0757602	+	0.997126i	$0.475862\pi$
$384$	85850.8i	0.582213i
$385$	0	0
$386$	75267.8	0.505166
$387$	85559.7i	0.571277i
$388$	−81582.7	−0.541920
$389$	173751.	1.14823	0.574114	−	0.818776i	$-0.305347\pi$
$389$	173751.	1.14823	0.574114	+	0.818776i	$0.305347\pi$
$390$	57363.3	0.377142
$391$	58301.1i	0.381349i
$392$	− 79228.1i	− 0.515593i
$393$	− 37921.9i	− 0.245530i
$394$	−72669.4	−0.468122
$395$	182789.i	1.17154i
$396$	0	0
$397$	61730.8	0.391671	0.195835	−	0.980637i	$-0.437258\pi$
$397$	61730.8	0.391671	0.195835	+	0.980637i	$0.437258\pi$
$398$	40979.4i	0.258701i
$399$	−68934.0	−0.433000
$400$	113590.	0.709938
$401$	−138269.	−0.859874	−0.429937	−	0.902859i	$-0.641464\pi$
$401$	−138269.	−0.859874	−0.429937	+	0.902859i	$0.641464\pi$
$402$	2791.59i	0.0172743i
$403$	115435.i	0.710765i
$404$	33038.3i	0.202421i
$405$	28842.3	0.175841
$406$	28958.8i	0.175682i
$407$	0	0
$408$	63037.9	0.378688
$409$	224986.i	1.34496i	0.740116	+	0.672479i	$0.234771\pi$
$409$	224986.i	1.34496i	−0.740116	+	0.672479i	$0.765229\pi$
$410$	−38218.7	−0.227357
$411$	118198.	0.699722
$412$	−119661.	−0.704952
$413$	− 79063.6i	− 0.463528i
$414$	11304.7i	0.0659567i
$415$	− 273097.i	− 1.58570i
$416$	−163155.	−0.942788
$417$	− 96352.9i	− 0.554106i
$418$	0	0
$419$	−86800.6	−0.494418	−0.247209	−	0.968962i	$-0.579513\pi$
$419$	−86800.6	−0.494418	−0.247209	+	0.968962i	$0.579513\pi$
$420$	− 76719.9i	− 0.434920i
$421$	166084.	0.937051	0.468526	−	0.883450i	$-0.344785\pi$
$421$	166084.	0.937051	0.468526	+	0.883450i	$0.344785\pi$
$422$	−14678.6	−0.0824251
$423$	67934.7	0.379674
$424$	168777.i	0.938820i
$425$	− 226967.i	− 1.25657i
$426$	53059.3i	0.292376i
$427$	180520.	0.990079
$428$	− 142881.i	− 0.779988i
$429$	0	0
$430$	217327.	1.17537
$431$	177448.i	0.955247i	0.878565	+	0.477623i	$0.158502\pi$
$431$	177448.i	0.955247i	−0.878565	+	0.477623i	$0.841498\pi$
$432$	−16947.7	−0.0908118
$433$	28492.8	0.151970	0.0759852	−	0.997109i	$-0.475790\pi$
$433$	28492.8	0.151970	0.0759852	+	0.997109i	$0.475790\pi$
$434$	−35697.6	−0.189522
$435$	− 119596.i	− 0.632030i
$436$	200643.i	1.05548i
$437$	111584.i	0.584302i
$438$	26903.8	0.140238
$439$	− 208514.i	− 1.08195i	−0.841039	−	0.540974i	$-0.818056\pi$
$439$	− 208514.i	− 1.08195i	0.841039	−	0.540974i	$-0.181944\pi$
$440$	0	0
$441$	42560.8	0.218843
$442$	67350.0i	0.344741i
$443$	213119.	1.08596	0.542980	−	0.839746i	$-0.317296\pi$
$443$	213119.	1.08596	0.542980	+	0.839746i	$0.317296\pi$
$444$	−165593.	−0.839993
$445$	258293.	1.30434
$446$	89343.8i	0.449154i
$447$	− 110685.i	− 0.553955i
$448$	5049.18i	0.0251574i
$449$	−62382.2	−0.309434	−0.154717	−	0.987959i	$-0.549447\pi$
$449$	−62382.2	−0.309434	−0.154717	+	0.987959i	$0.549447\pi$
$450$	− 44009.5i	− 0.217331i
$451$	0	0
$452$	189237.	0.926251
$453$	− 128497.i	− 0.626176i
$454$	−30182.7	−0.146436
$455$	182889.	0.883414
$456$	120649.	0.580224
$457$	− 341030.i	− 1.63290i	−0.577415	−	0.816451i	$-0.695938\pi$
$457$	− 341030.i	− 1.63290i	0.577415	−	0.816451i	$-0.304062\pi$
$458$	120052.i	0.572321i
$459$	33863.6i	0.160734i
$460$	−124187.	−0.586894
$461$	− 66165.6i	− 0.311337i	−0.987809	−	0.155668i	$-0.950247\pi$
$461$	− 66165.6i	− 0.311337i	0.987809	−	0.155668i	$-0.0497532\pi$
$462$	0	0
$463$	−255213.	−1.19053	−0.595265	−	0.803529i	$-0.702953\pi$
$463$	−255213.	−1.19053	−0.595265	+	0.803529i	$0.702953\pi$
$464$	70274.4i	0.326408i
$465$	147426.	0.681820
$466$	−55236.1	−0.254361
$467$	121217.	0.555816	0.277908	−	0.960608i	$-0.410359\pi$
$467$	121217.	0.555816	0.277908	+	0.960608i	$0.410359\pi$
$468$	− 56479.6i	− 0.257870i
$469$	8900.30i	0.0404631i
$470$	− 172558.i	− 0.781161i
$471$	−18325.0	−0.0826042
$472$	138378.i	0.621132i
$473$	0	0
$474$	−41613.8	−0.185217
$475$	− 434397.i	− 1.92531i
$476$	90076.5	0.397555
$477$	−90666.1	−0.398481
$478$	192257.	0.841447
$479$	59175.6i	0.257912i	0.991650	+	0.128956i	$0.0411626\pi$
$479$	59175.6i	0.257912i	−0.991650	+	0.128956i	$0.958837\pi$
$480$	208372.i	0.904394i
$481$	− 394749.i	− 1.70620i
$482$	93753.4	0.403546
$483$	36042.3i	0.154496i
$484$	0	0
$485$	−248380.	−1.05592
$486$	6566.22i	0.0277999i
$487$	108414.	0.457117	0.228558	−	0.973530i	$-0.426599\pi$
$487$	108414.	0.457117	0.228558	+	0.973530i	$0.426599\pi$
$488$	−315949.	−1.32672
$489$	−109573.	−0.458232
$490$	− 108107.i	− 0.450259i
$491$	440782.i	1.82835i	0.405314	+	0.914177i	$0.367162\pi$
$491$	440782.i	1.82835i	−0.405314	+	0.914177i	$0.632838\pi$
$492$	37629.9i	0.155454i
$493$	140417.	0.577731
$494$	128903.i	0.528211i
$495$	0	0
$496$	−86627.6	−0.352122
$497$	169166.i	0.684859i
$498$	62173.2	0.250694
$499$	−81740.4	−0.328273	−0.164137	−	0.986438i	$-0.552484\pi$
$499$	−81740.4	−0.328273	−0.164137	+	0.986438i	$0.552484\pi$
$500$	162121.	0.648482
$501$	205521.i	0.818807i
$502$	− 151500.i	− 0.601182i
$503$	451546.i	1.78470i	0.451340	+	0.892352i	$0.350946\pi$
$503$	451546.i	1.78470i	−0.451340	+	0.892352i	$0.649054\pi$
$504$	38970.6	0.153418
$505$	100586.i	0.394415i
$506$	0	0
$507$	−13768.4	−0.0535632
$508$	192934.i	0.747621i
$509$	−386890.	−1.49332	−0.746659	−	0.665207i	$-0.768343\pi$
$509$	−386890.	−1.49332	−0.746659	+	0.665207i	$0.768343\pi$
$510$	86015.5	0.330702
$511$	85776.0	0.328492
$512$	− 219583.i	− 0.837644i
$513$	64812.2i	0.246276i
$514$	152441.i	0.577000i
$515$	−364311.	−1.37359
$516$	− 213979.i	− 0.803658i
$517$	0	0
$518$	122074.	0.454950
$519$	117705.i	0.436978i
$520$	−320095.	−1.18378
$521$	−160763.	−0.592257	−0.296129	−	0.955148i	$-0.595696\pi$
$521$	−160763.	−0.592257	−0.296129	+	0.955148i	$0.595696\pi$
$522$	27227.2	0.0999222
$523$	− 27042.5i	− 0.0988651i	−0.998777	−	0.0494326i	$-0.984259\pi$
$523$	− 27042.5i	− 0.0988651i	0.998777	−	0.0494326i	$-0.0157413\pi$
$524$	94840.1i	0.345406i
$525$	− 140313.i	− 0.509073i
$526$	−34082.1	−0.123184
$527$	173093.i	0.623243i
$528$	0	0
$529$	−221499.	−0.791518
$530$	230297.i	0.819856i
$531$	−74336.0	−0.263639
$532$	172399.	0.609133
$533$	−89704.0	−0.315760
$534$	58802.9i	0.206213i
$535$	− 435004.i	− 1.51980i
$536$	− 15577.4i	− 0.0542209i
$537$	16675.2	0.0578257
$538$	− 6854.51i	− 0.0236817i
$539$	0	0
$540$	−72132.5	−0.247368
$541$	− 293862.i	− 1.00404i	−0.864857	−	0.502018i	$-0.832591\pi$
$541$	− 293862.i	− 1.00404i	0.864857	−	0.502018i	$-0.167409\pi$
$542$	154935.	0.527412
$543$	−208273.	−0.706371
$544$	−244649.	−0.826695
$545$	610861.i	2.05660i
$546$	41636.4i	0.139665i
$547$	− 85962.3i	− 0.287298i	−0.989629	−	0.143649i	$-0.954116\pi$
$547$	− 85962.3i	− 0.287298i	0.989629	−	0.143649i	$-0.0458837\pi$
$548$	−295604.	−0.984350
$549$	− 169726.i	− 0.563124i
$550$	0	0
$551$	268747.	0.885198
$552$	− 63081.8i	− 0.207027i
$553$	−132675.	−0.433850
$554$	192380.	0.626818
$555$	−504150.	−1.63672
$556$	240972.i	0.779502i
$557$	90499.1i	0.291698i	0.989307	+	0.145849i	$0.0465914\pi$
$557$	90499.1i	0.291698i	−0.989307	+	0.145849i	$0.953409\pi$
$558$	33563.1i	0.107794i
$559$	510093.	1.63240
$560$	137248.i	0.437654i
$561$	0	0
$562$	−172705.	−0.546804
$563$	− 78157.9i	− 0.246579i	−0.992371	−	0.123289i	$-0.960656\pi$
$563$	− 78157.9i	− 0.246579i	0.992371	−	0.123289i	$-0.0393444\pi$
$564$	−169900.	−0.534116
$565$	576134.	1.80479
$566$	59241.5	0.184924
$567$	20934.8i	0.0651182i
$568$	− 296078.i	− 0.917717i
$569$	43315.0i	0.133787i	0.997760	+	0.0668935i	$0.0213088\pi$
$569$	43315.0i	0.133787i	−0.997760	+	0.0668935i	$0.978691\pi$
$570$	164627.	0.506700
$571$	429216.i	1.31645i	0.752823	+	0.658223i	$0.228692\pi$
$571$	429216.i	1.31645i	−0.752823	+	0.658223i	$0.771308\pi$
$572$	0	0
$573$	36544.7	0.111305
$574$	− 27740.5i	− 0.0841959i
$575$	−227125.	−0.686958
$576$	4747.27	0.0143087
$577$	202595.	0.608523	0.304261	−	0.952589i	$-0.401590\pi$
$577$	202595.	0.608523	0.304261	+	0.952589i	$0.401590\pi$
$578$	− 43787.4i	− 0.131067i
$579$	− 225624.i	− 0.673019i
$580$	299101.i	0.889123i
$581$	198224.	0.587224
$582$	− 56546.1i	− 0.166939i
$583$	0	0
$584$	−150127.	−0.440182
$585$	− 171953.i	− 0.502456i
$586$	83751.2	0.243891
$587$	−89331.7	−0.259256	−0.129628	−	0.991563i	$-0.541378\pi$
$587$	−89331.7	−0.259256	−0.129628	+	0.991563i	$0.541378\pi$
$588$	−106442.	−0.307863
$589$	331286.i	0.954931i
$590$	188818.i	0.542425i
$591$	217835.i	0.623666i
$592$	296238.	0.845273
$593$	274181.i	0.779702i	0.920878	+	0.389851i	$0.127473\pi$
$593$	274181.i	0.779702i	−0.920878	+	0.389851i	$0.872527\pi$
$594$	0	0
$595$	274239.	0.774632
$596$	276816.i	0.779290i
$597$	122840.	0.344661
$598$	67397.0	0.188468
$599$	−170742.	−0.475867	−0.237933	−	0.971281i	$-0.576470\pi$
$599$	−170742.	−0.475867	−0.237933	+	0.971281i	$0.576470\pi$
$600$	245579.i	0.682163i
$601$	425813.i	1.17888i	0.807812	+	0.589441i	$0.200652\pi$
$601$	425813.i	1.17888i	−0.807812	+	0.589441i	$0.799348\pi$
$602$	157744.i	0.435271i
$603$	8368.11	0.0230140
$604$	321362.i	0.880887i
$605$	0	0
$606$	−22899.3	−0.0623559
$607$	− 108598.i	− 0.294744i	−0.989081	−	0.147372i	$-0.952919\pi$
$607$	− 108598.i	− 0.294744i	0.989081	−	0.147372i	$-0.0470815\pi$
$608$	−468239.	−1.26666
$609$	86807.2	0.234057
$610$	−431115.	−1.15860
$611$	− 405016.i	− 1.08490i
$612$	− 84690.4i	− 0.226116i
$613$	127106.i	0.338255i	0.985594	+	0.169127i	$0.0540949\pi$
$613$	127106.i	0.338255i	−0.985594	+	0.169127i	$0.945905\pi$
$614$	81544.5	0.216301
$615$	114565.i	0.302901i
$616$	0	0
$617$	−132931.	−0.349186	−0.174593	−	0.984641i	$-0.555861\pi$
$617$	−132931.	−0.349186	−0.174593	+	0.984641i	$0.555861\pi$
$618$	− 82939.0i	− 0.217161i
$619$	−161373.	−0.421162	−0.210581	−	0.977576i	$-0.567536\pi$
$619$	−161373.	−0.421162	−0.210581	+	0.977576i	$0.567536\pi$
$620$	−368703.	−0.959166
$621$	33887.2	0.0878723
$622$	235409.i	0.608474i
$623$	187478.i	0.483031i
$624$	101039.i	0.259490i
$625$	−94122.1	−0.240953
$626$	− 202318.i	− 0.516280i
$627$	0	0
$628$	45829.6	0.116205
$629$	− 591920.i	− 1.49610i
$630$	53175.6	0.133977
$631$	−225686.	−0.566822	−0.283411	−	0.958999i	$-0.591466\pi$
$631$	−225686.	−0.566822	−0.283411	+	0.958999i	$0.591466\pi$
$632$	232210.	0.581363
$633$	44000.7i	0.109813i
$634$	67284.0i	0.167391i
$635$	587390.i	1.45673i
$636$	226750.	0.560573
$637$	− 253741.i	− 0.625334i
$638$	0	0
$639$	159051.	0.389525
$640$	− 653679.i	− 1.59589i
$641$	−777622.	−1.89257	−0.946286	−	0.323329i	$-0.895198\pi$
$641$	−777622.	−1.89257	−0.946286	+	0.323329i	$0.895198\pi$
$642$	99033.0	0.240276
$643$	−81816.0	−0.197887	−0.0989433	−	0.995093i	$-0.531546\pi$
$643$	−81816.0	−0.197887	−0.0989433	+	0.995093i	$0.531546\pi$
$644$	− 90139.3i	− 0.217341i
$645$	− 651462.i	− 1.56592i
$646$	193288.i	0.463169i
$647$	−358698.	−0.856882	−0.428441	−	0.903570i	$-0.640937\pi$
$647$	−358698.	−0.856882	−0.428441	+	0.903570i	$0.640937\pi$
$648$	− 36640.4i	− 0.0872590i
$649$	0	0
$650$	−262378.	−0.621012
$651$	107008.i	0.252495i
$652$	274034.	0.644629
$653$	58654.3	0.137554	0.0687770	−	0.997632i	$-0.478090\pi$
$653$	58654.3	0.137554	0.0687770	+	0.997632i	$0.478090\pi$
$654$	−139069.	−0.325142
$655$	288742.i	0.673019i
$656$	− 67318.1i	− 0.156431i
$657$	− 80647.1i	− 0.186835i
$658$	125249.	0.289283
$659$	− 203272.i	− 0.468066i	−0.972229	−	0.234033i	$-0.924808\pi$
$659$	− 203272.i	− 0.468066i	0.972229	−	0.234033i	$-0.0751924\pi$
$660$	0	0
$661$	354885.	0.812241	0.406120	−	0.913820i	$-0.366881\pi$
$661$	354885.	0.812241	0.406120	+	0.913820i	$0.366881\pi$
$662$	− 84633.4i	− 0.193119i
$663$	201889.	0.459289
$664$	−346935.	−0.786886
$665$	524872.	1.18689
$666$	− 114775.i	− 0.258760i
$667$	− 140515.i	− 0.315843i
$668$	− 513995.i	− 1.15188i
$669$	267818.	0.598395
$670$	− 21255.5i	− 0.0473502i
$671$	0	0
$672$	−151244.	−0.334920
$673$	65783.5i	0.145240i	0.997360	+	0.0726201i	$0.0231361\pi$
$673$	65783.5i	0.145240i	−0.997360	+	0.0726201i	$0.976864\pi$
$674$	46478.7	0.102314
$675$	−131923.	−0.289544
$676$	34433.7	0.0753513
$677$	− 282456.i	− 0.616273i	−0.951342	−	0.308136i	$-0.900295\pi$
$677$	− 282456.i	− 0.616273i	0.951342	−	0.308136i	$-0.0997053\pi$
$678$	131163.i	0.285332i
$679$	− 180283.i	− 0.391035i
$680$	−479978.	−1.03801
$681$	90476.2i	0.195092i
$682$	0	0
$683$	−295282.	−0.632987	−0.316494	−	0.948595i	$-0.602506\pi$
$683$	−295282.	−0.632987	−0.316494	+	0.948595i	$0.602506\pi$
$684$	− 162091.i	− 0.346454i
$685$	−899972.	−1.91800
$686$	197988.	0.420717
$687$	359870.	0.762487
$688$	382798.i	0.808710i
$689$	540537.i	1.13864i
$690$	− 86075.5i	− 0.180793i
$691$	151414.	0.317109	0.158555	−	0.987350i	$-0.449317\pi$
$691$	151414.	0.317109	0.158555	+	0.987350i	$0.449317\pi$
$692$	− 294372.i	− 0.614730i
$693$	0	0
$694$	211103.	0.438305
$695$	733643.i	1.51885i
$696$	−151931.	−0.313638
$697$	−134510.	−0.276878
$698$	253177.	0.519652
$699$	165576.i	0.338879i
$700$	350914.i	0.716151i
$701$	66690.5i	0.135715i	0.997695	+	0.0678575i	$0.0216163\pi$
$701$	66690.5i	0.135715i	−0.997695	+	0.0678575i	$0.978384\pi$
$702$	39146.8	0.0794369
$703$	− 1.13289e6i	− 2.29233i
$704$	0	0
$705$	−517263.	−1.04072
$706$	62437.4i	0.125267i
$707$	−73008.8	−0.146062
$708$	185909.	0.370881
$709$	4407.88	0.00876874	0.00438437	−	0.999990i	$-0.498604\pi$
$709$	4407.88	0.00876874	0.00438437	+	0.999990i	$0.498604\pi$
$710$	− 404000.i	− 0.801428i
$711$	124742.i	0.246759i
$712$	− 328128.i	− 0.647266i
$713$	173213.	0.340724
$714$	62433.3i	0.122467i
$715$	0	0
$716$	−41703.4	−0.0813477
$717$	− 576313.i	− 1.12104i
$718$	−174278.	−0.338059
$719$	223520.	0.432374	0.216187	−	0.976352i	$-0.430638\pi$
$719$	223520.	0.432374	0.216187	+	0.976352i	$0.430638\pi$
$720$	129042.	0.248923
$721$	− 264430.i	− 0.508676i
$722$	144035.i	0.276308i
$723$	− 281036.i	− 0.537633i
$724$	520876.	0.993705
$725$	547027.i	1.04072i
$726$	0	0
$727$	774400.	1.46520	0.732599	−	0.680660i	$-0.238307\pi$
$727$	774400.	1.46520	0.732599	+	0.680660i	$0.238307\pi$
$728$	− 232337.i	− 0.438385i
$729$	19683.0	0.0370370
$730$	−204849.	−0.384404
$731$	764878.	1.43139
$732$	424473.i	0.792188i
$733$	368024.i	0.684965i	0.939524	+	0.342483i	$0.111268\pi$
$733$	368024.i	0.684965i	−0.939524	+	0.342483i	$0.888732\pi$
$734$	225405.i	0.418381i
$735$	−324063.	−0.599867
$736$	244819.i	0.451950i
$737$	0	0
$738$	−26081.8	−0.0478878
$739$	520127.i	0.952402i	0.879337	+	0.476201i	$0.157986\pi$
$739$	520127.i	0.952402i	−0.879337	+	0.476201i	$0.842014\pi$
$740$	1.26084e6	2.30249
$741$	386400.	0.703721
$742$	−167158.	−0.303613
$743$	− 766775.i	− 1.38896i	−0.719511	−	0.694481i	$-0.755634\pi$
$743$	− 766775.i	− 1.38896i	0.719511	−	0.694481i	$-0.244366\pi$
$744$	− 187287.i	− 0.338346i
$745$	842771.i	1.51844i
$746$	−259855.	−0.466931
$747$	− 186371.i	− 0.333993i
$748$	0	0
$749$	315742.	0.562819
$750$	112368.i	0.199765i
$751$	527356.	0.935027	0.467513	−	0.883986i	$-0.345150\pi$
$751$	527356.	0.935027	0.467513	+	0.883986i	$0.345150\pi$
$752$	303943.	0.537473
$753$	−454139.	−0.800938
$754$	− 162324.i	− 0.285523i
$755$	978391.i	1.71640i
$756$	− 52356.5i	− 0.0916066i
$757$	−458670.	−0.800403	−0.400201	−	0.916427i	$-0.631060\pi$
$757$	−458670.	−0.800403	−0.400201	+	0.916427i	$0.631060\pi$
$758$	383651.i	0.667725i
$759$	0	0
$760$	−918640.	−1.59044
$761$	609724.i	1.05284i	0.850223	+	0.526422i	$0.176467\pi$
$761$	609724.i	1.05284i	−0.850223	+	0.526422i	$0.823533\pi$
$762$	−133725.	−0.230305
$763$	−443386.	−0.761610
$764$	−91395.8	−0.156581
$765$	− 257841.i	− 0.440585i
$766$	− 38527.8i	− 0.0656625i
$767$	443180.i	0.753337i
$768$	134199.	0.227523
$769$	652888.i	1.10404i	0.833830	+	0.552022i	$0.186143\pi$
$769$	652888.i	1.10404i	−0.833830	+	0.552022i	$0.813857\pi$
$770$	0	0
$771$	456960.	0.768722
$772$	564269.i	0.946786i
$773$	−587326.	−0.982925	−0.491462	−	0.870899i	$-0.663537\pi$
$773$	−587326.	−0.982925	−0.491462	+	0.870899i	$0.663537\pi$
$774$	148312.	0.247567
$775$	−674323.	−1.12270
$776$	315535.i	0.523991i
$777$	− 365931.i	− 0.606118i
$778$	− 301185.i	− 0.497593i
$779$	−257441.	−0.424232
$780$	430043.i	0.706842i
$781$	0	0
$782$	101061.	0.165261
$783$	− 81616.6i	− 0.133124i
$784$	190419.	0.309798
$785$	139529.	0.226425
$786$	−65735.0	−0.106402
$787$	220596.i	0.356163i	0.984016	+	0.178081i	$0.0569890\pi$
$787$	220596.i	0.356163i	−0.984016	+	0.178081i	$0.943011\pi$
$788$	− 544790.i	− 0.877357i
$789$	102165.i	0.164115i
$790$	316852.	0.507695
$791$	418180.i	0.668359i
$792$	0	0
$793$	−1.01188e6	−1.60910
$794$	− 107006.i	− 0.169733i
$795$	690343.	1.09227
$796$	−307215.	−0.484860
$797$	−421386.	−0.663382	−0.331691	−	0.943388i	$-0.607619\pi$
$797$	−421386.	−0.663382	−0.331691	+	0.943388i	$0.607619\pi$
$798$	119492.i	0.187644i
$799$	− 607316.i	− 0.951309i
$800$	− 953087.i	− 1.48920i
$801$	176268.	0.274732
$802$	239679.i	0.372633i
$803$	0	0
$804$	−20928.1	−0.0323755
$805$	− 274430.i	− 0.423487i
$806$	200098.	0.308015
$807$	−20547.2	−0.0315504
$808$	127781.	0.195724
$809$	1.23170e6i	1.88195i	0.338474	+	0.940976i	$0.390089\pi$
$809$	1.23170e6i	1.88195i	−0.338474	+	0.940976i	$0.609911\pi$
$810$	− 49996.0i	− 0.0762019i
$811$	148145.i	0.225239i	0.993638	+	0.112620i	$0.0359242\pi$
$811$	148145.i	0.225239i	−0.993638	+	0.112620i	$0.964076\pi$
$812$	−217099.	−0.329265
$813$	− 464434.i	− 0.702656i
$814$	0	0
$815$	834302.	1.25605
$816$	151507.i	0.227537i
$817$	1.46392e6	2.19317
$818$	389998.	0.582848
$819$	124810.	0.186072
$820$	− 286518.i	− 0.426113i
$821$	− 636494.i	− 0.944295i	−0.881519	−	0.472148i	$-0.843479\pi$
$821$	− 636494.i	− 0.944295i	0.881519	−	0.472148i	$-0.156521\pi$
$822$	− 204887.i	− 0.303230i
$823$	4033.25	0.00595463	0.00297732	−	0.999996i	$-0.499052\pi$
$823$	4033.25	0.00595463	0.00297732	+	0.999996i	$0.499052\pi$
$824$	462811.i	0.681630i
$825$	0	0
$826$	−137051.	−0.200874
$827$	− 164298.i	− 0.240226i	−0.992760	−	0.120113i	$-0.961674\pi$
$827$	− 164298.i	− 0.240226i	0.992760	−	0.120113i	$-0.0383257\pi$
$828$	−84749.5	−0.123616
$829$	974718.	1.41831	0.709153	−	0.705054i	$-0.249077\pi$
$829$	974718.	1.41831	0.709153	+	0.705054i	$0.249077\pi$
$830$	−473394.	−0.687174
$831$	− 576682.i	− 0.835092i
$832$	− 28302.5i	− 0.0408863i
$833$	− 380481.i	− 0.548332i
$834$	−167021.	−0.240126
$835$	− 1.56486e6i	− 2.24442i
$836$	0	0
$837$	100609.	0.143611
$838$	150463.i	0.214260i
$839$	−542452.	−0.770615	−0.385308	−	0.922788i	$-0.625905\pi$
$839$	−542452.	−0.770615	−0.385308	+	0.922788i	$0.625905\pi$
$840$	−296727.	−0.420532
$841$	368854.	0.521509
$842$	− 287895.i	− 0.406078i
$843$	517702.i	0.728491i
$844$	− 110043.i	− 0.154482i
$845$	104834.	0.146821
$846$	− 117760.i	− 0.164535i
$847$	0	0
$848$	−405644.	−0.564097
$849$	− 177583.i	− 0.246369i
$850$	−393432.	−0.544542
$851$	−592333.	−0.817912
$852$	−397776.	−0.547973
$853$	− 196623.i	− 0.270232i	−0.990830	−	0.135116i	$-0.956859\pi$
$853$	− 196623.i	− 0.270232i	0.990830	−	0.135116i	$-0.0431407\pi$
$854$	− 312919.i	− 0.429058i
$855$	− 493488.i	− 0.675063i
$856$	−552617.	−0.754184
$857$	− 609696.i	− 0.830141i	−0.909789	−	0.415070i	$-0.863757\pi$
$857$	− 609696.i	− 0.830141i	0.909789	−	0.415070i	$-0.136243\pi$
$858$	0	0
$859$	364464.	0.493933	0.246967	−	0.969024i	$-0.420566\pi$
$859$	364464.	0.493933	0.246967	+	0.969024i	$0.420566\pi$
$860$	1.62926e6i	2.20289i
$861$	−83155.3	−0.112172
$862$	307593.	0.413963
$863$	−570445.	−0.765935	−0.382968	−	0.923762i	$-0.625098\pi$
$863$	−570445.	−0.765935	−0.382968	+	0.923762i	$0.625098\pi$
$864$	142201.i	0.190491i
$865$	− 896220.i	− 1.19779i
$866$	− 49390.2i	− 0.0658575i
$867$	−131258.	−0.174617
$868$	− 267619.i	− 0.355203i
$869$	0	0
$870$	−207311.	−0.273895
$871$	− 49889.4i	− 0.0657615i
$872$	776021.	1.02056
$873$	−169503.	−0.222408
$874$	193422.	0.253212
$875$	358258.i	0.467928i
$876$	201693.i	0.262835i
$877$	− 1.11668e6i	− 1.45188i	−0.687758	−	0.725940i	$-0.741405\pi$
$877$	− 1.11668e6i	− 1.45188i	0.687758	−	0.725940i	$-0.258595\pi$
$878$	−361445.	−0.468871
$879$	− 251054.i	− 0.324929i
$880$	0	0
$881$	−134195.	−0.172896	−0.0864480	−	0.996256i	$-0.527552\pi$
$881$	−134195.	−0.172896	−0.0864480	+	0.996256i	$0.527552\pi$
$882$	− 73776.2i	− 0.0948373i
$883$	1.40998e6	1.80838	0.904192	−	0.427126i	$-0.140474\pi$
$883$	1.40998e6	1.80838	0.904192	+	0.427126i	$0.140474\pi$
$884$	−504911.	−0.646116
$885$	566003.	0.722658
$886$	− 369426.i	− 0.470609i
$887$	− 386690.i	− 0.491491i	−0.969334	−	0.245746i	$-0.920967\pi$
$887$	− 386690.i	− 0.491491i	0.969334	−	0.245746i	$-0.0790328\pi$
$888$	640458.i	0.812204i
$889$	−426350.	−0.539464
$890$	− 447732.i	− 0.565247i
$891$	0	0
$892$	−669795.	−0.841806
$893$	− 1.16236e6i	− 1.45759i
$894$	−191865.	−0.240061
$895$	−126967.	−0.158505
$896$	474464.	0.591000
$897$	− 202030.i	− 0.251091i
$898$	108135.i	0.134096i
$899$	− 417181.i	− 0.516185i
$900$	329931.	0.407323
$901$	810528.i	0.998432i
$902$	0	0
$903$	472855.	0.579899
$904$	− 731905.i	− 0.895608i
$905$	1.58582e6	1.93622
$906$	−222740.	−0.271358
$907$	−231818.	−0.281795	−0.140898	−	0.990024i	$-0.544999\pi$
$907$	−231818.	−0.281795	−0.140898	+	0.990024i	$0.544999\pi$
$908$	− 226275.i	− 0.274451i
$909$	68643.3i	0.0830750i
$910$	− 317025.i	− 0.382834i
$911$	80338.9	0.0968031	0.0484016	−	0.998828i	$-0.484587\pi$
$911$	80338.9	0.0968031	0.0484016	+	0.998828i	$0.484587\pi$
$912$	289973.i	0.348632i
$913$	0	0
$914$	−591152.	−0.707630
$915$	1.29231e6i	1.54357i
$916$	−900011.	−1.07265
$917$	−209580.	−0.249236
$918$	58700.1	0.0696552
$919$	− 746583.i	− 0.883989i	−0.897018	−	0.441995i	$-0.854271\pi$
$919$	− 746583.i	− 0.883989i	0.897018	−	0.441995i	$-0.145729\pi$
$920$	480313.i	0.567477i
$921$	− 244439.i	− 0.288171i
$922$	−114694.	−0.134920
$923$	− 948238.i	− 1.11305i
$924$	0	0
$925$	2.30596e6	2.69506
$926$	442394.i	0.515925i
$927$	−248619.	−0.289318
$928$	589643.	0.684689
$929$	766392.	0.888014	0.444007	−	0.896023i	$-0.353557\pi$
$929$	766392.	0.888014	0.444007	+	0.896023i	$0.353557\pi$
$930$	− 255553.i	− 0.295472i
$931$	− 728211.i	− 0.840152i
$932$	− 414095.i	− 0.476726i
$933$	705665.	0.810653
$934$	− 210122.i	− 0.240867i
$935$	0	0
$936$	−218444.	−0.249338
$937$	− 992801.i	− 1.13079i	−0.824819	−	0.565397i	$-0.808723\pi$
$937$	− 992801.i	− 1.13079i	0.824819	−	0.565397i	$-0.191277\pi$
$938$	15428.0	0.0175350
$939$	−606471.	−0.687826
$940$	1.29364e6	1.46406
$941$	− 1.65199e6i	− 1.86565i	−0.360335	−	0.932823i	$-0.617338\pi$
$941$	− 1.65199e6i	− 1.86565i	0.360335	−	0.932823i	$-0.382662\pi$
$942$	31765.1i	0.0357972i
$943$	134604.i	0.151368i
$944$	−332583.	−0.373212
$945$	− 159400.i	− 0.178494i
$946$	0	0
$947$	1.33055e6	1.48365	0.741827	−	0.670592i	$-0.233960\pi$
$947$	1.33055e6	1.48365	0.741827	+	0.670592i	$0.233960\pi$
$948$	− 311971.i	− 0.347135i
$949$	−480806.	−0.533872
$950$	−752997.	−0.834346
$951$	201691.	0.223011
$952$	− 348386.i	− 0.384403i
$953$	− 1.00158e6i	− 1.10280i	−0.834240	−	0.551402i	$-0.814093\pi$
$953$	− 1.00158e6i	− 1.10280i	0.834240	−	0.551402i	$-0.185907\pi$
$954$	157163.i	0.172685i
$955$	−278256.	−0.305097
$956$	1.44132e6i	1.57705i
$957$	0	0
$958$	102577.	0.111768
$959$	− 653233.i	− 0.710282i
$960$	−36146.3	−0.0392212
$961$	−409260.	−0.443152
$962$	−684269.	−0.739396
$963$	− 296863.i	− 0.320113i
$964$	702852.i	0.756328i
$965$	1.71793e6i	1.84480i
$966$	62476.8	0.0669522
$967$	− 154811.i	− 0.165558i	−0.996568	−	0.0827788i	$-0.973621\pi$
$967$	− 154811.i	− 0.165558i	0.996568	−	0.0827788i	$-0.0263795\pi$
$968$	0	0
$969$	579402.	0.617067
$970$	430549.i	0.457593i
$971$	−1.45138e6	−1.53937	−0.769685	−	0.638424i	$-0.779587\pi$
$971$	−1.45138e6	−1.53937	−0.769685	+	0.638424i	$0.779587\pi$
$972$	−49225.8	−0.0521027
$973$	−532505.	−0.562469
$974$	− 187928.i	− 0.198095i
$975$	786507.i	0.827357i
$976$	− 759362.i	− 0.797167i
$977$	1.53715e6	1.61037	0.805186	−	0.593022i	$-0.202065\pi$
$977$	1.53715e6	1.61037	0.805186	+	0.593022i	$0.202065\pi$
$978$	189937.i	0.198578i
$979$	0	0
$980$	810460.	0.843878
$981$	416874.i	0.433178i
$982$	764064.	0.792331
$983$	120168.	0.124360	0.0621802	−	0.998065i	$-0.480195\pi$
$983$	120168.	0.124360	0.0621802	+	0.998065i	$0.480195\pi$
$984$	145540.	0.150311
$985$	− 1.65862e6i	− 1.70952i
$986$	− 243403.i	− 0.250364i
$987$	− 375449.i	− 0.385404i
$988$	−966360.	−0.989977
$989$	− 765411.i	− 0.782532i
$990$	0	0
$991$	−1.11523e6	−1.13558	−0.567791	−	0.823173i	$-0.692202\pi$
$991$	−1.11523e6	−1.13558	−0.567791	+	0.823173i	$0.692202\pi$
$992$	726856.i	0.738627i
$993$	−253698.	−0.257288
$994$	293238.	0.296789
$995$	−935320.	−0.944744
$996$	466101.i	0.469853i
$997$	− 746115.i	− 0.750612i	−0.926901	−	0.375306i	$-0.877538\pi$
$997$	− 746115.i	− 0.750612i	0.926901	−	0.375306i	$-0.122462\pi$
$998$	141691.i	0.142260i
$999$	−344050.	−0.344739

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	363.5.c.e.241.12		32
11.2	odd	10		33.5.g.a.7.6	✓	32
11.5	even	5		33.5.g.a.19.6	yes	32
11.10	odd	2	inner	363.5.c.e.241.21		32
33.2	even	10		99.5.k.c.73.3		32
33.5	odd	10		99.5.k.c.19.3		32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
33.5.g.a.7.6	✓	32	11.2	odd	10
33.5.g.a.19.6	yes	32	11.5	even	5
99.5.k.c.19.3		32	33.5	odd	10
99.5.k.c.73.3		32	33.2	even	10
363.5.c.e.241.12		32	1.1	even	1	trivial
363.5.c.e.241.21		32	11.10	odd	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:	\( N \)	\(=\)	\( 363 = 3 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	363.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.73343i q^{2} -5.19615 q^{3} +12.9952 q^{4} +39.5641 q^{5} +9.00717i q^{6} +28.7171i q^{7} -50.2612i q^{8} +27.0000 q^{9} +O(q^{10})\)	\(q-1.73343i q^{2} -5.19615 q^{3} +12.9952 q^{4} +39.5641 q^{5} +9.00717i q^{6} +28.7171i q^{7} -50.2612i q^{8} +27.0000 q^{9} -68.5817i q^{10} -67.5251 q^{12} -160.970i q^{13} +49.7791 q^{14} -205.581 q^{15} +120.799 q^{16} -241.372i q^{17} -46.8026i q^{18} -461.967i q^{19} +514.145 q^{20} -149.219i q^{21} -241.540 q^{23} +261.165i q^{24} +940.321 q^{25} -279.030 q^{26} -140.296 q^{27} +373.185i q^{28} +581.745i q^{29} +356.361i q^{30} -717.120 q^{31} -1013.58i q^{32} -418.402 q^{34} +1136.17i q^{35} +350.871 q^{36} +2452.32 q^{37} -800.787 q^{38} +836.424i q^{39} -1988.54i q^{40} -557.272i q^{41} -258.660 q^{42} +3168.88i q^{43} +1068.23 q^{45} +418.693i q^{46} +2516.10 q^{47} -627.691 q^{48} +1576.33 q^{49} -1629.98i q^{50} +1254.21i q^{51} -2091.84i q^{52} -3358.00 q^{53} +243.194i q^{54} +1443.36 q^{56} +2400.45i q^{57} +1008.41 q^{58} -2753.19 q^{59} -2671.57 q^{60} -6286.15i q^{61} +1243.08i q^{62} +775.362i q^{63} +175.825 q^{64} -6368.63i q^{65} +309.930 q^{67} -3136.68i q^{68} +1255.08 q^{69} +1969.47 q^{70} +5890.78 q^{71} -1357.05i q^{72} -2986.93i q^{73} -4250.92i q^{74} -4886.05 q^{75} -6003.36i q^{76} +1449.88 q^{78} +4620.07i q^{79} +4779.32 q^{80} +729.000 q^{81} -965.992 q^{82} -6902.64i q^{83} -1939.13i q^{84} -9549.68i q^{85} +5493.03 q^{86} -3022.84i q^{87} +6528.45 q^{89} -1851.71i q^{90} +4622.59 q^{91} -3138.87 q^{92} +3726.27 q^{93} -4361.48i q^{94} -18277.3i q^{95} +5266.70i q^{96} -6277.90 q^{97} -2732.45i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - 244 q^{4} + 36 q^{5} + 864 q^{9}+O(q^{10}) \) 32 * q - 244 * q^4 + 36 * q^5 + 864 * q^9	\( 32 q - 244 q^{4} + 36 q^{5} + 864 q^{9} + 360 q^{12} + 2220 q^{14} - 108 q^{15} + 3908 q^{16} + 468 q^{20} - 2196 q^{23} + 7280 q^{25} + 3564 q^{26} - 5872 q^{31} - 2320 q^{34} - 6588 q^{36} - 656 q^{37} - 2616 q^{38} - 1404 q^{42} + 972 q^{45} + 2640 q^{47} - 9936 q^{48} - 6988 q^{49} + 4560 q^{53} - 5604 q^{56} - 24644 q^{58} + 39612 q^{59} + 20592 q^{60} - 6232 q^{64} + 2796 q^{67} - 10476 q^{69} - 72692 q^{70} - 51828 q^{71} - 18072 q^{75} + 53640 q^{78} - 27624 q^{80} + 23328 q^{81} - 11548 q^{82} + 106284 q^{86} - 38748 q^{89} + 30672 q^{91} + 27000 q^{92} + 42624 q^{93} - 50544 q^{97}+O(q^{100}) \) 32 * q - 244 * q^4 + 36 * q^5 + 864 * q^9 + 360 * q^12 + 2220 * q^14 - 108 * q^15 + 3908 * q^16 + 468 * q^20 - 2196 * q^23 + 7280 * q^25 + 3564 * q^26 - 5872 * q^31 - 2320 * q^34 - 6588 * q^36 - 656 * q^37 - 2616 * q^38 - 1404 * q^42 + 972 * q^45 + 2640 * q^47 - 9936 * q^48 - 6988 * q^49 + 4560 * q^53 - 5604 * q^56 - 24644 * q^58 + 39612 * q^59 + 20592 * q^60 - 6232 * q^64 + 2796 * q^67 - 10476 * q^69 - 72692 * q^70 - 51828 * q^71 - 18072 * q^75 + 53640 * q^78 - 27624 * q^80 + 23328 * q^81 - 11548 * q^82 + 106284 * q^86 - 38748 * q^89 + 30672 * q^91 + 27000 * q^92 + 42624 * q^93 - 50544 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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