LMFDB - Embedded newform 525.6.d.c.274.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [525,6,Mod(274,525)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(525, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("525.274");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 525 = 3 \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	525.d (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:	$84.2015054018$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 21)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			274.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	525.274
Dual form			525.6.d.c.274.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+5.00000i q^{2} -9.00000i q^{3} +7.00000 q^{4} +45.0000 q^{6} -49.0000i q^{7} +195.000i q^{8} -81.0000 q^{9} +52.0000 q^{11} -63.0000i q^{12} +770.000i q^{13} +245.000 q^{14} -751.000 q^{16} -2022.00i q^{17} -405.000i q^{18} -1732.00 q^{19} -441.000 q^{21} +260.000i q^{22} +576.000i q^{23} +1755.00 q^{24} -3850.00 q^{26} +729.000i q^{27} -343.000i q^{28} -5518.00 q^{29} +6336.00 q^{31} +2485.00i q^{32} -468.000i q^{33} +10110.0 q^{34} -567.000 q^{36} -7338.00i q^{37} -8660.00i q^{38} +6930.00 q^{39} -3262.00 q^{41} -2205.00i q^{42} -5420.00i q^{43} +364.000 q^{44} -2880.00 q^{46} +864.000i q^{47} +6759.00i q^{48} -2401.00 q^{49} -18198.0 q^{51} +5390.00i q^{52} -4182.00i q^{53} -3645.00 q^{54} +9555.00 q^{56} +15588.0i q^{57} -27590.0i q^{58} +11220.0 q^{59} -45602.0 q^{61} +31680.0i q^{62} +3969.00i q^{63} -36457.0 q^{64} +2340.00 q^{66} +1396.00i q^{67} -14154.0i q^{68} +5184.00 q^{69} +18720.0 q^{71} -15795.0i q^{72} -46362.0i q^{73} +36690.0 q^{74} -12124.0 q^{76} -2548.00i q^{77} +34650.0i q^{78} -97424.0 q^{79} +6561.00 q^{81} -16310.0i q^{82} +81228.0i q^{83} -3087.00 q^{84} +27100.0 q^{86} +49662.0i q^{87} +10140.0i q^{88} +3182.00 q^{89} +37730.0 q^{91} +4032.00i q^{92} -57024.0i q^{93} -4320.00 q^{94} +22365.0 q^{96} +4914.00i q^{97} -12005.0i q^{98} -4212.00 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 14 q^{4} + 90 q^{6} - 162 q^{9} + 104 q^{11} + 490 q^{14} - 1502 q^{16} - 3464 q^{19} - 882 q^{21} + 3510 q^{24} - 7700 q^{26} - 11036 q^{29} + 12672 q^{31} + 20220 q^{34} - 1134 q^{36} + 13860 q^{39}+ \cdots - 8424 q^{99}+O(q^{100}) $ 2 * q + 14 * q^4 + 90 * q^6 - 162 * q^9 + 104 * q^11 + 490 * q^14 - 1502 * q^16 - 3464 * q^19 - 882 * q^21 + 3510 * q^24 - 7700 * q^26 - 11036 * q^29 + 12672 * q^31 + 20220 * q^34 - 1134 * q^36 + 13860 * q^39 - 6524 * q^41 + 728 * q^44 - 5760 * q^46 - 4802 * q^49 - 36396 * q^51 - 7290 * q^54 + 19110 * q^56 + 22440 * q^59 - 91204 * q^61 - 72914 * q^64 + 4680 * q^66 + 10368 * q^69 + 37440 * q^71 + 73380 * q^74 - 24248 * q^76 - 194848 * q^79 + 13122 * q^81 - 6174 * q^84 + 54200 * q^86 + 6364 * q^89 + 75460 * q^91 - 8640 * q^94 + 44730 * q^96 - 8424 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/525\mathbb{Z}\right)^\times$.

$n$	$127$	$176$	$451$
$\chi(n)$	$-1$	$1$	$1$

Coefficient data

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

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

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

$2$	5.00000i	0.883883i	0.897044	+	0.441942i	$0.145710\pi$
$2$	5.00000i	0.883883i	−0.897044	+	0.441942i	$0.854290\pi$
$3$	− 9.00000i	− 0.577350i
$3$	− 9.00000i	− 0.577350i
$4$	7.00000	0.218750
$5$	0	0
$5$	0	0
$6$	45.0000	0.510310
$7$	− 49.0000i	− 0.377964i
$7$	− 49.0000i	− 0.377964i
$8$	195.000i	1.07723i
$9$	−81.0000	−0.333333
$10$	0	0
$11$	52.0000	0.129575	0.0647876	−	0.997899i	$-0.479363\pi$
$11$	52.0000	0.129575	0.0647876	+	0.997899i	$0.479363\pi$
$12$	− 63.0000i	− 0.126295i
$13$	770.000i	1.26367i	0.775104	+	0.631833i	$0.217697\pi$
$13$	770.000i	1.26367i	−0.775104	+	0.631833i	$0.782303\pi$
$14$	245.000	0.334077
$15$	0	0
$16$	−751.000	−0.733398
$17$	− 2022.00i	− 1.69691i	−0.529267	−	0.848455i	$-0.677533\pi$
$17$	− 2022.00i	− 1.69691i	0.529267	−	0.848455i	$-0.322467\pi$
$18$	− 405.000i	− 0.294628i
$19$	−1732.00	−1.10069	−0.550344	−	0.834938i	$-0.685503\pi$
$19$	−1732.00	−1.10069	−0.550344	+	0.834938i	$0.685503\pi$
$20$	0	0
$21$	−441.000	−0.218218
$22$	260.000i	0.114529i
$23$	576.000i	0.227040i	0.993536	+	0.113520i	$0.0362126\pi$
$23$	576.000i	0.227040i	−0.993536	+	0.113520i	$0.963787\pi$
$24$	1755.00	0.621941
$25$	0	0
$26$	−3850.00	−1.11693
$27$	729.000i	0.192450i
$28$	− 343.000i	− 0.0826797i
$29$	−5518.00	−1.21839	−0.609196	−	0.793020i	$-0.708508\pi$
$29$	−5518.00	−1.21839	−0.609196	+	0.793020i	$0.708508\pi$
$30$	0	0
$31$	6336.00	1.18416	0.592081	−	0.805879i	$-0.298307\pi$
$31$	6336.00	1.18416	0.592081	+	0.805879i	$0.298307\pi$
$32$	2485.00i	0.428994i
$33$	− 468.000i	− 0.0748102i
$34$	10110.0	1.49987
$35$	0	0
$36$	−567.000	−0.0729167
$37$	− 7338.00i	− 0.881198i	−0.897704	−	0.440599i	$-0.854766\pi$
$37$	− 7338.00i	− 0.881198i	0.897704	−	0.440599i	$-0.145234\pi$
$38$	− 8660.00i	− 0.972879i
$39$	6930.00	0.729578
$40$	0	0
$41$	−3262.00	−0.303057	−0.151528	−	0.988453i	$-0.548420\pi$
$41$	−3262.00	−0.303057	−0.151528	+	0.988453i	$0.548420\pi$
$42$	− 2205.00i	− 0.192879i
$43$	− 5420.00i	− 0.447021i	−0.974701	−	0.223511i	$-0.928248\pi$
$43$	− 5420.00i	− 0.447021i	0.974701	−	0.223511i	$-0.0717517\pi$
$44$	364.000	0.0283446
$45$	0	0
$46$	−2880.00	−0.200677
$47$	864.000i	0.0570518i	0.999593	+	0.0285259i	$0.00908130\pi$
$47$	864.000i	0.0570518i	−0.999593	+	0.0285259i	$0.990919\pi$
$48$	6759.00i	0.423428i
$49$	−2401.00	−0.142857
$50$	0	0
$51$	−18198.0	−0.979712
$52$	5390.00i	0.276427i
$53$	− 4182.00i	− 0.204500i	−0.994759	−	0.102250i	$-0.967396\pi$
$53$	− 4182.00i	− 0.204500i	0.994759	−	0.102250i	$-0.0326042\pi$
$54$	−3645.00	−0.170103
$55$	0	0
$56$	9555.00	0.407156
$57$	15588.0i	0.635482i
$58$	− 27590.0i	− 1.07692i
$59$	11220.0	0.419626	0.209813	−	0.977741i	$-0.432714\pi$
$59$	11220.0	0.419626	0.209813	+	0.977741i	$0.432714\pi$
$60$	0	0
$61$	−45602.0	−1.56913	−0.784566	−	0.620046i	$-0.787114\pi$
$61$	−45602.0	−1.56913	−0.784566	+	0.620046i	$0.787114\pi$
$62$	31680.0i	1.04666i
$63$	3969.00i	0.125988i
$64$	−36457.0	−1.11258
$65$	0	0
$66$	2340.00	0.0661235
$67$	1396.00i	0.0379925i	0.999820	+	0.0189963i	$0.00604707\pi$
$67$	1396.00i	0.0379925i	−0.999820	+	0.0189963i	$0.993953\pi$
$68$	− 14154.0i	− 0.371199i
$69$	5184.00	0.131082
$70$	0	0
$71$	18720.0	0.440717	0.220359	−	0.975419i	$-0.429277\pi$
$71$	18720.0	0.440717	0.220359	+	0.975419i	$0.429277\pi$
$72$	− 15795.0i	− 0.359078i
$73$	− 46362.0i	− 1.01825i	−0.860692	−	0.509126i	$-0.829969\pi$
$73$	− 46362.0i	− 1.01825i	0.860692	−	0.509126i	$-0.170031\pi$
$74$	36690.0	0.778876
$75$	0	0
$76$	−12124.0	−0.240775
$77$	− 2548.00i	− 0.0489748i
$78$	34650.0i	0.644862i
$79$	−97424.0	−1.75630	−0.878149	−	0.478387i	$-0.841222\pi$
$79$	−97424.0	−1.75630	−0.878149	+	0.478387i	$0.841222\pi$
$80$	0	0
$81$	6561.00	0.111111
$82$	− 16310.0i	− 0.267867i
$83$	81228.0i	1.29423i	0.762394	+	0.647114i	$0.224024\pi$
$83$	81228.0i	1.29423i	−0.762394	+	0.647114i	$0.775976\pi$
$84$	−3087.00	−0.0477352
$85$	0	0
$86$	27100.0	0.395115
$87$	49662.0i	0.703438i
$88$	10140.0i	0.139583i
$89$	3182.00	0.0425819	0.0212910	−	0.999773i	$-0.493222\pi$
$89$	3182.00	0.0425819	0.0212910	+	0.999773i	$0.493222\pi$
$90$	0	0
$91$	37730.0	0.477621
$92$	4032.00i	0.0496651i
$93$	− 57024.0i	− 0.683676i
$94$	−4320.00	−0.0504271
$95$	0	0
$96$	22365.0	0.247680
$97$	4914.00i	0.0530281i	0.999648	+	0.0265140i	$0.00844067\pi$
$97$	4914.00i	0.0530281i	−0.999648	+	0.0265140i	$0.991559\pi$
$98$	− 12005.0i	− 0.126269i
$99$	−4212.00	−0.0431917
$100$	0	0
$101$	−166354.	−1.62267	−0.811334	−	0.584583i	$-0.801258\pi$
$101$	−166354.	−1.62267	−0.811334	+	0.584583i	$0.801258\pi$
$102$	− 90990.0i	− 0.865951i
$103$	− 157160.i	− 1.45965i	−0.683634	−	0.729825i	$-0.739601\pi$
$103$	− 157160.i	− 1.45965i	0.683634	−	0.729825i	$-0.260399\pi$
$104$	−150150.	−1.36126
$105$	0	0
$106$	20910.0	0.180755
$107$	− 6764.00i	− 0.0571142i	−0.999592	−	0.0285571i	$-0.990909\pi$
$107$	− 6764.00i	− 0.0571142i	0.999592	−	0.0285571i	$-0.00909124\pi$
$108$	5103.00i	0.0420985i
$109$	−178398.	−1.43821	−0.719107	−	0.694899i	$-0.755449\pi$
$109$	−178398.	−1.43821	−0.719107	+	0.694899i	$0.755449\pi$
$110$	0	0
$111$	−66042.0	−0.508760
$112$	36799.0i	0.277199i
$113$	45134.0i	0.332512i	0.986083	+	0.166256i	$0.0531679\pi$
$113$	45134.0i	0.332512i	−0.986083	+	0.166256i	$0.946832\pi$
$114$	−77940.0	−0.561692
$115$	0	0
$116$	−38626.0	−0.266523
$117$	− 62370.0i	− 0.421222i
$118$	56100.0i	0.370901i
$119$	−99078.0	−0.641372
$120$	0	0
$121$	−158347.	−0.983210
$122$	− 228010.i	− 1.38693i
$123$	29358.0i	0.174970i
$124$	44352.0	0.259035
$125$	0	0
$126$	−19845.0	−0.111359
$127$	− 205056.i	− 1.12814i	−0.825727	−	0.564070i	$-0.809235\pi$
$127$	− 205056.i	− 1.12814i	0.825727	−	0.564070i	$-0.190765\pi$
$128$	− 102765.i	− 0.554396i
$129$	−48780.0	−0.258088
$130$	0	0
$131$	72964.0	0.371476	0.185738	−	0.982599i	$-0.440532\pi$
$131$	72964.0	0.371476	0.185738	+	0.982599i	$0.440532\pi$
$132$	− 3276.00i	− 0.0163647i
$133$	84868.0i	0.416021i
$134$	−6980.00	−0.0335810
$135$	0	0
$136$	394290.	1.82797
$137$	− 94182.0i	− 0.428713i	−0.976756	−	0.214356i	$-0.931235\pi$
$137$	− 94182.0i	− 0.428713i	0.976756	−	0.214356i	$-0.0687654\pi$
$138$	25920.0i	0.115861i
$139$	47796.0	0.209824	0.104912	−	0.994482i	$-0.466544\pi$
$139$	47796.0	0.209824	0.104912	+	0.994482i	$0.466544\pi$
$140$	0	0
$141$	7776.00	0.0329389
$142$	93600.0i	0.389543i
$143$	40040.0i	0.163740i
$144$	60831.0	0.244466
$145$	0	0
$146$	231810.	0.900016
$147$	21609.0i	0.0824786i
$148$	− 51366.0i	− 0.192762i
$149$	124266.	0.458550	0.229275	−	0.973362i	$-0.426364\pi$
$149$	124266.	0.458550	0.229275	+	0.973362i	$0.426364\pi$
$150$	0	0
$151$	−446296.	−1.59287	−0.796436	−	0.604723i	$-0.793284\pi$
$151$	−446296.	−1.59287	−0.796436	+	0.604723i	$0.793284\pi$
$152$	− 337740.i	− 1.18570i
$153$	163782.i	0.565637i
$154$	12740.0	0.0432880
$155$	0	0
$156$	48510.0	0.159595
$157$	− 159746.i	− 0.517227i	−0.965981	−	0.258613i	$-0.916734\pi$
$157$	− 159746.i	− 0.517227i	0.965981	−	0.258613i	$-0.0832655\pi$
$158$	− 487120.i	− 1.55236i
$159$	−37638.0	−0.118068
$160$	0	0
$161$	28224.0	0.0858132
$162$	32805.0i	0.0982093i
$163$	− 247252.i	− 0.728905i	−0.931222	−	0.364452i	$-0.881256\pi$
$163$	− 247252.i	− 0.728905i	0.931222	−	0.364452i	$-0.118744\pi$
$164$	−22834.0	−0.0662937
$165$	0	0
$166$	−406140.	−1.14395
$167$	− 684488.i	− 1.89922i	−0.313438	−	0.949609i	$-0.601481\pi$
$167$	− 684488.i	− 1.89922i	0.313438	−	0.949609i	$-0.398519\pi$
$168$	− 85995.0i	− 0.235072i
$169$	−221607.	−0.596852
$170$	0	0
$171$	140292.	0.366896
$172$	− 37940.0i	− 0.0977859i
$173$	610474.i	1.55079i	0.631479	+	0.775393i	$0.282448\pi$
$173$	610474.i	1.55079i	−0.631479	+	0.775393i	$0.717552\pi$
$174$	−248310.	−0.621758
$175$	0	0
$176$	−39052.0	−0.0950302
$177$	− 100980.i	− 0.242271i
$178$	15910.0i	0.0376374i
$179$	−662252.	−1.54487	−0.772433	−	0.635097i	$-0.780960\pi$
$179$	−662252.	−1.54487	−0.772433	+	0.635097i	$0.780960\pi$
$180$	0	0
$181$	154630.	0.350830	0.175415	−	0.984495i	$-0.443873\pi$
$181$	154630.	0.350830	0.175415	+	0.984495i	$0.443873\pi$
$182$	188650.i	0.422161i
$183$	410418.i	0.905938i
$184$	−112320.	−0.244575
$185$	0	0
$186$	285120.	0.604290
$187$	− 105144.i	− 0.219877i
$188$	6048.00i	0.0124801i
$189$	35721.0	0.0727393
$190$	0	0
$191$	486904.	0.965739	0.482870	−	0.875692i	$-0.339594\pi$
$191$	486904.	0.965739	0.482870	+	0.875692i	$0.339594\pi$
$192$	328113.i	0.642348i
$193$	− 620546.i	− 1.19917i	−0.800311	−	0.599585i	$-0.795332\pi$
$193$	− 620546.i	− 1.19917i	0.800311	−	0.599585i	$-0.204668\pi$
$194$	−24570.0	−0.0468706
$195$	0	0
$196$	−16807.0	−0.0312500
$197$	− 236570.i	− 0.434304i	−0.976138	−	0.217152i	$-0.930323\pi$
$197$	− 236570.i	− 0.434304i	0.976138	−	0.217152i	$-0.0696768\pi$
$198$	− 21060.0i	− 0.0381764i
$199$	−82104.0	−0.146971	−0.0734855	−	0.997296i	$-0.523412\pi$
$199$	−82104.0	−0.146971	−0.0734855	+	0.997296i	$0.523412\pi$
$200$	0	0
$201$	12564.0	0.0219350
$202$	− 831770.i	− 1.43425i
$203$	270382.i	0.460509i
$204$	−127386.	−0.214312
$205$	0	0
$206$	785800.	1.29016
$207$	− 46656.0i	− 0.0756801i
$208$	− 578270.i	− 0.926771i
$209$	−90064.0	−0.142622
$210$	0	0
$211$	99892.0	0.154463	0.0772315	−	0.997013i	$-0.475392\pi$
$211$	99892.0	0.154463	0.0772315	+	0.997013i	$0.475392\pi$
$212$	− 29274.0i	− 0.0447345i
$213$	− 168480.i	− 0.254448i
$214$	33820.0	0.0504823
$215$	0	0
$216$	−142155.	−0.207314
$217$	− 310464.i	− 0.447571i
$218$	− 891990.i	− 1.27121i
$219$	−417258.	−0.587888
$220$	0	0
$221$	1.55694e6	2.14433
$222$	− 330210.i	− 0.449684i
$223$	186704.i	0.251415i	0.992067	+	0.125708i	$0.0401201\pi$
$223$	186704.i	0.251415i	−0.992067	+	0.125708i	$0.959880\pi$
$224$	121765.	0.162145
$225$	0	0
$226$	−225670.	−0.293902
$227$	336372.i	0.433267i	0.976253	+	0.216633i	$0.0695076\pi$
$227$	336372.i	0.433267i	−0.976253	+	0.216633i	$0.930492\pi$
$228$	109116.i	0.139012i
$229$	926314.	1.16727	0.583633	−	0.812018i	$-0.301631\pi$
$229$	926314.	1.16727	0.583633	+	0.812018i	$0.301631\pi$
$230$	0	0
$231$	−22932.0	−0.0282756
$232$	− 1.07601e6i	− 1.31249i
$233$	− 1.25711e6i	− 1.51700i	−0.651675	−	0.758499i	$-0.725933\pi$
$233$	− 1.25711e6i	− 1.51700i	0.651675	−	0.758499i	$-0.274067\pi$
$234$	311850.	0.372311
$235$	0	0
$236$	78540.0	0.0917933
$237$	876816.i	1.01400i
$238$	− 495390.i	− 0.566898i
$239$	347016.	0.392966	0.196483	−	0.980507i	$-0.437048\pi$
$239$	347016.	0.392966	0.196483	+	0.980507i	$0.437048\pi$
$240$	0	0
$241$	99170.0	0.109986	0.0549930	−	0.998487i	$-0.482486\pi$
$241$	99170.0	0.109986	0.0549930	+	0.998487i	$0.482486\pi$
$242$	− 791735.i	− 0.869043i
$243$	− 59049.0i	− 0.0641500i
$244$	−319214.	−0.343247
$245$	0	0
$246$	−146790.	−0.154653
$247$	− 1.33364e6i	− 1.39090i
$248$	1.23552e6i	1.27562i
$249$	731052.	0.747222
$250$	0	0
$251$	344428.	0.345076	0.172538	−	0.985003i	$-0.444803\pi$
$251$	344428.	0.345076	0.172538	+	0.985003i	$0.444803\pi$
$252$	27783.0i	0.0275599i
$253$	29952.0i	0.0294188i
$254$	1.02528e6	0.997145
$255$	0	0
$256$	−652799.	−0.622558
$257$	295130.i	0.278728i	0.990241	+	0.139364i	$0.0445058\pi$
$257$	295130.i	0.278728i	−0.990241	+	0.139364i	$0.955494\pi$
$258$	− 243900.i	− 0.228120i
$259$	−359562.	−0.333061
$260$	0	0
$261$	446958.	0.406130
$262$	364820.i	0.328341i
$263$	1.27246e6i	1.13437i	0.823589	+	0.567187i	$0.191968\pi$
$263$	1.27246e6i	1.13437i	−0.823589	+	0.567187i	$0.808032\pi$
$264$	91260.0	0.0805881
$265$	0	0
$266$	−424340.	−0.367714
$267$	− 28638.0i	− 0.0245847i
$268$	9772.00i	0.00831087i
$269$	−276774.	−0.233209	−0.116604	−	0.993178i	$-0.537201\pi$
$269$	−276774.	−0.233209	−0.116604	+	0.993178i	$0.537201\pi$
$270$	0	0
$271$	−1.28994e6	−1.06695	−0.533476	−	0.845815i	$-0.679115\pi$
$271$	−1.28994e6	−1.06695	−0.533476	+	0.845815i	$0.679115\pi$
$272$	1.51852e6i	1.24451i
$273$	− 339570.i	− 0.275755i
$274$	470910.	0.378932
$275$	0	0
$276$	36288.0	0.0286741
$277$	1.71655e6i	1.34418i	0.740470	+	0.672089i	$0.234603\pi$
$277$	1.71655e6i	1.34418i	−0.740470	+	0.672089i	$0.765397\pi$
$278$	238980.i	0.185460i
$279$	−513216.	−0.394720
$280$	0	0
$281$	−1.47218e6	−1.11223	−0.556116	−	0.831104i	$-0.687709\pi$
$281$	−1.47218e6	−1.11223	−0.556116	+	0.831104i	$0.687709\pi$
$282$	38880.0i	0.0291141i
$283$	− 1.02881e6i	− 0.763607i	−0.924244	−	0.381804i	$-0.875303\pi$
$283$	− 1.02881e6i	− 0.763607i	0.924244	−	0.381804i	$-0.124697\pi$
$284$	131040.	0.0964069
$285$	0	0
$286$	−200200.	−0.144727
$287$	159838.i	0.114545i
$288$	− 201285.i	− 0.142998i
$289$	−2.66863e6	−1.87950
$290$	0	0
$291$	44226.0	0.0306158
$292$	− 324534.i	− 0.222742i
$293$	1.18607e6i	0.807123i	0.914952	+	0.403562i	$0.132228\pi$
$293$	1.18607e6i	0.807123i	−0.914952	+	0.403562i	$0.867772\pi$
$294$	−108045.	−0.0729015
$295$	0	0
$296$	1.43091e6	0.949255
$297$	37908.0i	0.0249367i
$298$	621330.i	0.405305i
$299$	−443520.	−0.286903
$300$	0	0
$301$	−265580.	−0.168958
$302$	− 2.23148e6i	− 1.40791i
$303$	1.49719e6i	0.936848i
$304$	1.30073e6	0.807242
$305$	0	0
$306$	−818910.	−0.499957
$307$	1.51892e6i	0.919788i	0.887974	+	0.459894i	$0.152113\pi$
$307$	1.51892e6i	0.919788i	−0.887974	+	0.459894i	$0.847887\pi$
$308$	− 17836.0i	− 0.0107132i
$309$	−1.41444e6	−0.842730
$310$	0	0
$311$	212808.	0.124763	0.0623817	−	0.998052i	$-0.480130\pi$
$311$	212808.	0.124763	0.0623817	+	0.998052i	$0.480130\pi$
$312$	1.35135e6i	0.785925i
$313$	1894.00i	0.00109275i	1.00000		0.000546373i	$0.000173916\pi$
$313$	1894.00i	0.00109275i	−1.00000		0.000546373i	$0.999826\pi$
$314$	798730.	0.457168
$315$	0	0
$316$	−681968.	−0.384190
$317$	− 1.57898e6i	− 0.882527i	−0.897378	−	0.441263i	$-0.854530\pi$
$317$	− 1.57898e6i	− 0.882527i	0.897378	−	0.441263i	$-0.145470\pi$
$318$	− 188190.i	− 0.104359i
$319$	−286936.	−0.157873
$320$	0	0
$321$	−60876.0	−0.0329749
$322$	141120.i	0.0758488i
$323$	3.50210e6i	1.86777i
$324$	45927.0	0.0243056
$325$	0	0
$326$	1.23626e6	0.644267
$327$	1.60558e6i	0.830354i
$328$	− 636090.i	− 0.326463i
$329$	42336.0	0.0215635
$330$	0	0
$331$	−3.39471e6	−1.70307	−0.851535	−	0.524298i	$-0.824328\pi$
$331$	−3.39471e6	−1.70307	−0.851535	+	0.524298i	$0.824328\pi$
$332$	568596.i	0.283112i
$333$	594378.i	0.293733i
$334$	3.42244e6	1.67869
$335$	0	0
$336$	331191.	0.160041
$337$	2.02731e6i	0.972403i	0.873847	+	0.486201i	$0.161618\pi$
$337$	2.02731e6i	0.972403i	−0.873847	+	0.486201i	$0.838382\pi$
$338$	− 1.10804e6i	− 0.527548i
$339$	406206.	0.191976
$340$	0	0
$341$	329472.	0.153438
$342$	701460.i	0.324293i
$343$	117649.i	0.0539949i
$344$	1.05690e6	0.481546
$345$	0	0
$346$	−3.05237e6	−1.37071
$347$	3.48885e6i	1.55546i	0.628598	+	0.777730i	$0.283629\pi$
$347$	3.48885e6i	1.55546i	−0.628598	+	0.777730i	$0.716371\pi$
$348$	347634.i	0.153877i
$349$	−965566.	−0.424344	−0.212172	−	0.977232i	$-0.568054\pi$
$349$	−965566.	−0.424344	−0.212172	+	0.977232i	$0.568054\pi$
$350$	0	0
$351$	−561330.	−0.243193
$352$	129220.i	0.0555870i
$353$	− 1.15393e6i	− 0.492882i	−0.969158	−	0.246441i	$-0.920739\pi$
$353$	− 1.15393e6i	− 0.492882i	0.969158	−	0.246441i	$-0.0792611\pi$
$354$	504900.	0.214140
$355$	0	0
$356$	22274.0	0.00931479
$357$	891702.i	0.370296i
$358$	− 3.31126e6i	− 1.36548i
$359$	−1.61110e6	−0.659762	−0.329881	−	0.944022i	$-0.607009\pi$
$359$	−1.61110e6	−0.659762	−0.329881	+	0.944022i	$0.607009\pi$
$360$	0	0
$361$	523725.	0.211512
$362$	773150.i	0.310093i
$363$	1.42512e6i	0.567657i
$364$	264110.	0.104480
$365$	0	0
$366$	−2.05209e6	−0.800744
$367$	3.67747e6i	1.42523i	0.701557	+	0.712614i	$0.252489\pi$
$367$	3.67747e6i	1.42523i	−0.701557	+	0.712614i	$0.747511\pi$
$368$	− 432576.i	− 0.166511i
$369$	264222.	0.101019
$370$	0	0
$371$	−204918.	−0.0772939
$372$	− 399168.i	− 0.149554i
$373$	− 649766.i	− 0.241816i	−0.992664	−	0.120908i	$-0.961419\pi$
$373$	− 649766.i	− 0.241816i	0.992664	−	0.120908i	$-0.0385806\pi$
$374$	525720.	0.194346
$375$	0	0
$376$	−168480.	−0.0614580
$377$	− 4.24886e6i	− 1.53964i
$378$	178605.i	0.0642931i
$379$	−320700.	−0.114683	−0.0573417	−	0.998355i	$-0.518262\pi$
$379$	−320700.	−0.114683	−0.0573417	+	0.998355i	$0.518262\pi$
$380$	0	0
$381$	−1.84550e6	−0.651332
$382$	2.43452e6i	0.853601i
$383$	2.36189e6i	0.822740i	0.911469	+	0.411370i	$0.134950\pi$
$383$	2.36189e6i	0.822740i	−0.911469	+	0.411370i	$0.865050\pi$
$384$	−924885.	−0.320081
$385$	0	0
$386$	3.10273e6	1.05993
$387$	439020.i	0.149007i
$388$	34398.0i	0.0115999i
$389$	3.53390e6	1.18408	0.592039	−	0.805910i	$-0.298323\pi$
$389$	3.53390e6	1.18408	0.592039	+	0.805910i	$0.298323\pi$
$390$	0	0
$391$	1.16467e6	0.385267
$392$	− 468195.i	− 0.153890i
$393$	− 656676.i	− 0.214472i
$394$	1.18285e6	0.383874
$395$	0	0
$396$	−29484.0	−0.00944819
$397$	4.04811e6i	1.28907i	0.764575	+	0.644534i	$0.222949\pi$
$397$	4.04811e6i	1.28907i	−0.764575	+	0.644534i	$0.777051\pi$
$398$	− 410520.i	− 0.129905i
$399$	763812.	0.240190
$400$	0	0
$401$	2.07645e6	0.644853	0.322426	−	0.946595i	$-0.395502\pi$
$401$	2.07645e6	0.644853	0.322426	+	0.946595i	$0.395502\pi$
$402$	62820.0i	0.0193880i
$403$	4.87872e6i	1.49638i
$404$	−1.16448e6	−0.354959
$405$	0	0
$406$	−1.35191e6	−0.407036
$407$	− 381576.i	− 0.114181i
$408$	− 3.54861e6i	− 1.05538i
$409$	−2.57431e6	−0.760945	−0.380472	−	0.924792i	$-0.624239\pi$
$409$	−2.57431e6	−0.760945	−0.380472	+	0.924792i	$0.624239\pi$
$410$	0	0
$411$	−847638.	−0.247517
$412$	− 1.10012e6i	− 0.319299i
$413$	− 549780.i	− 0.158604i
$414$	233280.	0.0668924
$415$	0	0
$416$	−1.91345e6	−0.542105
$417$	− 430164.i	− 0.121142i
$418$	− 450320.i	− 0.126061i
$419$	−848148.	−0.236013	−0.118007	−	0.993013i	$-0.537650\pi$
$419$	−848148.	−0.236013	−0.118007	+	0.993013i	$0.537650\pi$
$420$	0	0
$421$	1.43682e6	0.395092	0.197546	−	0.980294i	$-0.436703\pi$
$421$	1.43682e6	0.395092	0.197546	+	0.980294i	$0.436703\pi$
$422$	499460.i	0.136527i
$423$	− 69984.0i	− 0.0190173i
$424$	815490.	0.220295
$425$	0	0
$426$	842400.	0.224903
$427$	2.23450e6i	0.593076i
$428$	− 47348.0i	− 0.0124937i
$429$	360360.	0.0945352
$430$	0	0
$431$	2.35438e6	0.610496	0.305248	−	0.952273i	$-0.401261\pi$
$431$	2.35438e6	0.610496	0.305248	+	0.952273i	$0.401261\pi$
$432$	− 547479.i	− 0.141143i
$433$	3.78808e6i	0.970955i	0.874249	+	0.485478i	$0.161354\pi$
$433$	3.78808e6i	0.970955i	−0.874249	+	0.485478i	$0.838646\pi$
$434$	1.55232e6	0.395601
$435$	0	0
$436$	−1.24879e6	−0.314609
$437$	− 997632.i	− 0.249900i
$438$	− 2.08629e6i	− 0.519624i
$439$	3.64322e6	0.902245	0.451123	−	0.892462i	$-0.351024\pi$
$439$	3.64322e6	0.902245	0.451123	+	0.892462i	$0.351024\pi$
$440$	0	0
$441$	194481.	0.0476190
$442$	7.78470e6i	1.89534i
$443$	− 2.48389e6i	− 0.601345i	−0.953728	−	0.300672i	$-0.902789\pi$
$443$	− 2.48389e6i	− 0.601345i	0.953728	−	0.300672i	$-0.0972110\pi$
$444$	−462294.	−0.111291
$445$	0	0
$446$	−933520.	−0.222222
$447$	− 1.11839e6i	− 0.264744i
$448$	1.78639e6i	0.420515i
$449$	2.63177e6	0.616074	0.308037	−	0.951374i	$-0.400328\pi$
$449$	2.63177e6	0.616074	0.308037	+	0.951374i	$0.400328\pi$
$450$	0	0
$451$	−169624.	−0.0392686
$452$	315938.i	0.0727371i
$453$	4.01666e6i	0.919645i
$454$	−1.68186e6	−0.382957
$455$	0	0
$456$	−3.03966e6	−0.684562
$457$	− 1.16130e6i	− 0.260109i	−0.991507	−	0.130054i	$-0.958485\pi$
$457$	− 1.16130e6i	− 0.260109i	0.991507	−	0.130054i	$-0.0415152\pi$
$458$	4.63157e6i	1.03173i
$459$	1.47404e6	0.326571
$460$	0	0
$461$	2.81385e6	0.616663	0.308332	−	0.951279i	$-0.400229\pi$
$461$	2.81385e6	0.616663	0.308332	+	0.951279i	$0.400229\pi$
$462$	− 114660.i	− 0.0249923i
$463$	− 6.84299e6i	− 1.48352i	−0.670665	−	0.741760i	$-0.733991\pi$
$463$	− 6.84299e6i	− 1.48352i	0.670665	−	0.741760i	$-0.266009\pi$
$464$	4.14402e6	0.893566
$465$	0	0
$466$	6.28557e6	1.34085
$467$	3.34314e6i	0.709353i	0.934989	+	0.354676i	$0.115409\pi$
$467$	3.34314e6i	0.709353i	−0.934989	+	0.354676i	$0.884591\pi$
$468$	− 436590.i	− 0.0921423i
$469$	68404.0	0.0143598
$470$	0	0
$471$	−1.43771e6	−0.298621
$472$	2.18790e6i	0.452035i
$473$	− 281840.i	− 0.0579228i
$474$	−4.38408e6	−0.896257
$475$	0	0
$476$	−693546.	−0.140300
$477$	338742.i	0.0681668i
$478$	1.73508e6i	0.347336i
$479$	4.28248e6	0.852818	0.426409	−	0.904530i	$-0.359778\pi$
$479$	4.28248e6	0.852818	0.426409	+	0.904530i	$0.359778\pi$
$480$	0	0
$481$	5.65026e6	1.11354
$482$	495850.i	0.0972149i
$483$	− 254016.i	− 0.0495442i
$484$	−1.10843e6	−0.215077
$485$	0	0
$486$	295245.	0.0567012
$487$	− 8.93175e6i	− 1.70653i	−0.521477	−	0.853266i	$-0.674619\pi$
$487$	− 8.93175e6i	− 1.70653i	0.521477	−	0.853266i	$-0.325381\pi$
$488$	− 8.89239e6i	− 1.69032i
$489$	−2.22527e6	−0.420833
$490$	0	0
$491$	2.75306e6	0.515361	0.257681	−	0.966230i	$-0.417042\pi$
$491$	2.75306e6	0.515361	0.257681	+	0.966230i	$0.417042\pi$
$492$	205506.i	0.0382747i
$493$	1.11574e7i	2.06750i
$494$	6.66820e6	1.22939
$495$	0	0
$496$	−4.75834e6	−0.868462
$497$	− 917280.i	− 0.166575i
$498$	3.65526e6i	0.660458i
$499$	−4.80408e6	−0.863693	−0.431846	−	0.901947i	$-0.642138\pi$
$499$	−4.80408e6	−0.863693	−0.431846	+	0.901947i	$0.642138\pi$
$500$	0	0
$501$	−6.16039e6	−1.09651
$502$	1.72214e6i	0.305007i
$503$	6.02465e6i	1.06172i	0.847458	+	0.530862i	$0.178132\pi$
$503$	6.02465e6i	1.06172i	−0.847458	+	0.530862i	$0.821868\pi$
$504$	−773955.	−0.135719
$505$	0	0
$506$	−149760.	−0.0260028
$507$	1.99446e6i	0.344593i
$508$	− 1.43539e6i	− 0.246781i
$509$	8.42987e6	1.44220	0.721101	−	0.692830i	$-0.243636\pi$
$509$	8.42987e6	1.44220	0.721101	+	0.692830i	$0.243636\pi$
$510$	0	0
$511$	−2.27174e6	−0.384863
$512$	− 6.55248e6i	− 1.10466i
$513$	− 1.26263e6i	− 0.211827i
$514$	−1.47565e6	−0.246363
$515$	0	0
$516$	−341460.	−0.0564567
$517$	44928.0i	0.00739249i
$518$	− 1.79781e6i	− 0.294388i
$519$	5.49427e6	0.895347
$520$	0	0
$521$	9.25058e6	1.49305	0.746525	−	0.665357i	$-0.231721\pi$
$521$	9.25058e6	1.49305	0.746525	+	0.665357i	$0.231721\pi$
$522$	2.23479e6i	0.358972i
$523$	− 5.84494e6i	− 0.934385i	−0.884156	−	0.467192i	$-0.845266\pi$
$523$	− 5.84494e6i	− 0.934385i	0.884156	−	0.467192i	$-0.154734\pi$
$524$	510748.	0.0812603
$525$	0	0
$526$	−6.36232e6	−1.00265
$527$	− 1.28114e7i	− 2.00942i
$528$	351468.i	0.0548657i
$529$	6.10457e6	0.948453
$530$	0	0
$531$	−908820.	−0.139875
$532$	594076.i	0.0910045i
$533$	− 2.51174e6i	− 0.382963i
$534$	143190.	0.0217300
$535$	0	0
$536$	−272220.	−0.0409268
$537$	5.96027e6i	0.891929i
$538$	− 1.38387e6i	− 0.206129i
$539$	−124852.	−0.0185107
$540$	0	0
$541$	9.22533e6	1.35515	0.677577	−	0.735452i	$-0.263030\pi$
$541$	9.22533e6	1.35515	0.677577	+	0.735452i	$0.263030\pi$
$542$	− 6.44968e6i	− 0.943061i
$543$	− 1.39167e6i	− 0.202552i
$544$	5.02467e6	0.727965
$545$	0	0
$546$	1.69785e6	0.243735
$547$	− 6.44337e6i	− 0.920757i	−0.887723	−	0.460378i	$-0.847714\pi$
$547$	− 6.44337e6i	− 0.920757i	0.887723	−	0.460378i	$-0.152286\pi$
$548$	− 659274.i	− 0.0937809i
$549$	3.69376e6	0.523044
$550$	0	0
$551$	9.55718e6	1.34107
$552$	1.01088e6i	0.141206i
$553$	4.77378e6i	0.663818i
$554$	−8.58275e6	−1.18810
$555$	0	0
$556$	334572.	0.0458989
$557$	3.74213e6i	0.511070i	0.966800	+	0.255535i	$0.0822516\pi$
$557$	3.74213e6i	0.511070i	−0.966800	+	0.255535i	$0.917748\pi$
$558$	− 2.56608e6i	− 0.348887i
$559$	4.17340e6	0.564886
$560$	0	0
$561$	−946296.	−0.126946
$562$	− 7.36091e6i	− 0.983084i
$563$	1.46384e7i	1.94635i	0.230060	+	0.973176i	$0.426108\pi$
$563$	1.46384e7i	1.94635i	−0.230060	+	0.973176i	$0.573892\pi$
$564$	54432.0	0.00720537
$565$	0	0
$566$	5.14406e6	0.674940
$567$	− 321489.i	− 0.0419961i
$568$	3.65040e6i	0.474755i
$569$	−1.41805e7	−1.83616	−0.918078	−	0.396400i	$-0.870259\pi$
$569$	−1.41805e7	−1.83616	−0.918078	+	0.396400i	$0.870259\pi$
$570$	0	0
$571$	−1.25160e6	−0.160648	−0.0803242	−	0.996769i	$-0.525596\pi$
$571$	−1.25160e6	−0.160648	−0.0803242	+	0.996769i	$0.525596\pi$
$572$	280280.i	0.0358181i
$573$	− 4.38214e6i	− 0.557570i
$574$	−799190.	−0.101244
$575$	0	0
$576$	2.95302e6	0.370860
$577$	5.94378e6i	0.743230i	0.928387	+	0.371615i	$0.121196\pi$
$577$	5.94378e6i	0.743230i	−0.928387	+	0.371615i	$0.878804\pi$
$578$	− 1.33431e7i	− 1.66126i
$579$	−5.58491e6	−0.692341
$580$	0	0
$581$	3.98017e6	0.489172
$582$	221130.i	0.0270608i
$583$	− 217464.i	− 0.0264982i
$584$	9.04059e6	1.09689
$585$	0	0
$586$	−5.93033e6	−0.713403
$587$	− 6.46192e6i	− 0.774046i	−0.922070	−	0.387023i	$-0.873503\pi$
$587$	− 6.46192e6i	− 0.774046i	0.922070	−	0.387023i	$-0.126497\pi$
$588$	151263.i	0.0180422i
$589$	−1.09740e7	−1.30339
$590$	0	0
$591$	−2.12913e6	−0.250746
$592$	5.51084e6i	0.646269i
$593$	2.34605e6i	0.273969i	0.990573	+	0.136984i	$0.0437410\pi$
$593$	2.34605e6i	0.273969i	−0.990573	+	0.136984i	$0.956259\pi$
$594$	−189540.	−0.0220412
$595$	0	0
$596$	869862.	0.100308
$597$	738936.i	0.0848537i
$598$	− 2.21760e6i	− 0.253589i
$599$	1.34959e7	1.53686	0.768432	−	0.639931i	$-0.221037\pi$
$599$	1.34959e7	1.53686	0.768432	+	0.639931i	$0.221037\pi$
$600$	0	0
$601$	3.87849e6	0.438002	0.219001	−	0.975725i	$-0.429720\pi$
$601$	3.87849e6	0.438002	0.219001	+	0.975725i	$0.429720\pi$
$602$	− 1.32790e6i	− 0.149339i
$603$	− 113076.i	− 0.0126642i
$604$	−3.12407e6	−0.348441
$605$	0	0
$606$	−7.48593e6	−0.828065
$607$	− 533488.i	− 0.0587696i	−0.999568	−	0.0293848i	$-0.990645\pi$
$607$	− 533488.i	− 0.0587696i	0.999568	−	0.0293848i	$-0.00935482\pi$
$608$	− 4.30402e6i	− 0.472188i
$609$	2.43344e6	0.265875
$610$	0	0
$611$	−665280.	−0.0720944
$612$	1.14647e6i	0.123733i
$613$	− 5.14610e6i	− 0.553130i	−0.960995	−	0.276565i	$-0.910804\pi$
$613$	− 5.14610e6i	− 0.553130i	0.960995	−	0.276565i	$-0.0891961\pi$
$614$	−7.59458e6	−0.812986
$615$	0	0
$616$	496860.	0.0527573
$617$	− 2.37860e6i	− 0.251541i	−0.992059	−	0.125770i	$-0.959860\pi$
$617$	− 2.37860e6i	− 0.251541i	0.992059	−	0.125770i	$-0.0401402\pi$
$618$	− 7.07220e6i	− 0.744875i
$619$	−1.60023e7	−1.67863	−0.839317	−	0.543642i	$-0.817045\pi$
$619$	−1.60023e7	−1.67863	−0.839317	+	0.543642i	$0.817045\pi$
$620$	0	0
$621$	−419904.	−0.0436939
$622$	1.06404e6i	0.110276i
$623$	− 155918.i	− 0.0160944i
$624$	−5.20443e6	−0.535071
$625$	0	0
$626$	−9470.00	−0.000965860 0
$627$	810576.i	0.0823427i
$628$	− 1.11822e6i	− 0.113143i
$629$	−1.48374e7	−1.49531
$630$	0	0
$631$	1.23459e7	1.23439	0.617193	−	0.786812i	$-0.288270\pi$
$631$	1.23459e7	1.23439	0.617193	+	0.786812i	$0.288270\pi$
$632$	− 1.89977e7i	− 1.89194i
$633$	− 899028.i	− 0.0891793i
$634$	7.89489e6	0.780051
$635$	0	0
$636$	−263466.	−0.0258275
$637$	− 1.84877e6i	− 0.180524i
$638$	− 1.43468e6i	− 0.139541i
$639$	−1.51632e6	−0.146906
$640$	0	0
$641$	−3.43755e6	−0.330449	−0.165224	−	0.986256i	$-0.552835\pi$
$641$	−3.43755e6	−0.330449	−0.165224	+	0.986256i	$0.552835\pi$
$642$	− 304380.i	− 0.0291460i
$643$	1.62191e7i	1.54703i	0.633778	+	0.773515i	$0.281503\pi$
$643$	1.62191e7i	1.54703i	−0.633778	+	0.773515i	$0.718497\pi$
$644$	197568.	0.0187716
$645$	0	0
$646$	−1.75105e7	−1.65089
$647$	1.19929e7i	1.12632i	0.826348	+	0.563160i	$0.190415\pi$
$647$	1.19929e7i	1.12632i	−0.826348	+	0.563160i	$0.809585\pi$
$648$	1.27940e6i	0.119693i
$649$	583440.	0.0543731
$650$	0	0
$651$	−2.79418e6	−0.258405
$652$	− 1.73076e6i	− 0.159448i
$653$	− 1.58009e6i	− 0.145011i	−0.997368	−	0.0725053i	$-0.976901\pi$
$653$	− 1.58009e6i	− 0.145011i	0.997368	−	0.0725053i	$-0.0230994\pi$
$654$	−8.02791e6	−0.733936
$655$	0	0
$656$	2.44976e6	0.222262
$657$	3.75532e6i	0.339417i
$658$	211680.i	0.0190597i
$659$	−6.98358e6	−0.626419	−0.313209	−	0.949684i	$-0.601404\pi$
$659$	−6.98358e6	−0.626419	−0.313209	+	0.949684i	$0.601404\pi$
$660$	0	0
$661$	3.69602e6	0.329027	0.164513	−	0.986375i	$-0.447395\pi$
$661$	3.69602e6	0.329027	0.164513	+	0.986375i	$0.447395\pi$
$662$	− 1.69735e7i	− 1.50532i
$663$	− 1.40125e7i	− 1.23803i
$664$	−1.58395e7	−1.39418
$665$	0	0
$666$	−2.97189e6	−0.259625
$667$	− 3.17837e6i	− 0.276624i
$668$	− 4.79142e6i	− 0.415454i
$669$	1.68034e6	0.145155
$670$	0	0
$671$	−2.37130e6	−0.203320
$672$	− 1.09588e6i	− 0.0936142i
$673$	− 1.84688e6i	− 0.157182i	−0.996907	−	0.0785908i	$-0.974958\pi$
$673$	− 1.84688e6i	− 0.157182i	0.996907	−	0.0785908i	$-0.0250420\pi$
$674$	−1.01366e7	−0.859491
$675$	0	0
$676$	−1.55125e6	−0.130561
$677$	− 7.68501e6i	− 0.644426i	−0.946667	−	0.322213i	$-0.895573\pi$
$677$	− 7.68501e6i	− 0.644426i	0.946667	−	0.322213i	$-0.104427\pi$
$678$	2.03103e6i	0.169684i
$679$	240786.	0.0200427
$680$	0	0
$681$	3.02735e6	0.250147
$682$	1.64736e6i	0.135621i
$683$	− 7.12180e6i	− 0.584168i	−0.956393	−	0.292084i	$-0.905651\pi$
$683$	− 7.12180e6i	− 0.584168i	0.956393	−	0.292084i	$-0.0943487\pi$
$684$	982044.	0.0802584
$685$	0	0
$686$	−588245.	−0.0477252
$687$	− 8.33683e6i	− 0.673921i
$688$	4.07042e6i	0.327845i
$689$	3.22014e6	0.258420
$690$	0	0
$691$	−3.23787e6	−0.257967	−0.128983	−	0.991647i	$-0.541171\pi$
$691$	−3.23787e6	−0.257967	−0.128983	+	0.991647i	$0.541171\pi$
$692$	4.27332e6i	0.339234i
$693$	206388.i	0.0163249i
$694$	−1.74443e7	−1.37485
$695$	0	0
$696$	−9.68409e6	−0.757767
$697$	6.59576e6i	0.514260i
$698$	− 4.82783e6i	− 0.375071i
$699$	−1.13140e7	−0.875839
$700$	0	0
$701$	−7.39163e6	−0.568127	−0.284063	−	0.958805i	$-0.591683\pi$
$701$	−7.39163e6	−0.568127	−0.284063	+	0.958805i	$0.591683\pi$
$702$	− 2.80665e6i	− 0.214954i
$703$	1.27094e7i	0.969923i
$704$	−1.89576e6	−0.144163
$705$	0	0
$706$	5.76965e6	0.435650
$707$	8.15135e6i	0.613311i
$708$	− 706860.i	− 0.0529969i
$709$	5.33361e6	0.398479	0.199240	−	0.979951i	$-0.436153\pi$
$709$	5.33361e6	0.398479	0.199240	+	0.979951i	$0.436153\pi$
$710$	0	0
$711$	7.89134e6	0.585433
$712$	620490.i	0.0458706i
$713$	3.64954e6i	0.268852i
$714$	−4.45851e6	−0.327299
$715$	0	0
$716$	−4.63576e6	−0.337939
$717$	− 3.12314e6i	− 0.226879i
$718$	− 8.05552e6i	− 0.583153i
$719$	1.14564e7	0.826468	0.413234	−	0.910625i	$-0.364399\pi$
$719$	1.14564e7	0.826468	0.413234	+	0.910625i	$0.364399\pi$
$720$	0	0
$721$	−7.70084e6	−0.551696
$722$	2.61862e6i	0.186952i
$723$	− 892530.i	− 0.0635005i
$724$	1.08241e6	0.0767442
$725$	0	0
$726$	−7.12562e6	−0.501742
$727$	− 2.49540e7i	− 1.75107i	−0.483153	−	0.875536i	$-0.660508\pi$
$727$	− 2.49540e7i	− 1.75107i	0.483153	−	0.875536i	$-0.339492\pi$
$728$	7.35735e6i	0.514509i
$729$	−531441.	−0.0370370
$730$	0	0
$731$	−1.09592e7	−0.758555
$732$	2.87293e6i	0.198174i
$733$	1.43398e7i	0.985789i	0.870089	+	0.492894i	$0.164061\pi$
$733$	1.43398e7i	0.985789i	−0.870089	+	0.492894i	$0.835939\pi$
$734$	−1.83874e7	−1.25974
$735$	0	0
$736$	−1.43136e6	−0.0973990
$737$	72592.0i	0.00492289i
$738$	1.32111e6i	0.0892890i
$739$	−922932.	−0.0621668	−0.0310834	−	0.999517i	$-0.509896\pi$
$739$	−922932.	−0.0621668	−0.0310834	+	0.999517i	$0.509896\pi$
$740$	0	0
$741$	−1.20028e7	−0.803037
$742$	− 1.02459e6i	− 0.0683188i
$743$	9.38995e6i	0.624010i	0.950081	+	0.312005i	$0.101001\pi$
$743$	9.38995e6i	0.624010i	−0.950081	+	0.312005i	$0.898999\pi$
$744$	1.11197e7	0.736478
$745$	0	0
$746$	3.24883e6	0.213737
$747$	− 6.57947e6i	− 0.431409i
$748$	− 736008.i	− 0.0480982i
$749$	−331436.	−0.0215871
$750$	0	0
$751$	−408032.	−0.0263994	−0.0131997	−	0.999913i	$-0.504202\pi$
$751$	−408032.	−0.0263994	−0.0131997	+	0.999913i	$0.504202\pi$
$752$	− 648864.i	− 0.0418417i
$753$	− 3.09985e6i	− 0.199229i
$754$	2.12443e7	1.36086
$755$	0	0
$756$	250047.	0.0159117
$757$	2.59605e7i	1.64654i	0.567649	+	0.823271i	$0.307853\pi$
$757$	2.59605e7i	1.64654i	−0.567649	+	0.823271i	$0.692147\pi$
$758$	− 1.60350e6i	− 0.101367i
$759$	269568.	0.0169849
$760$	0	0
$761$	1.83554e7	1.14895	0.574477	−	0.818521i	$-0.305206\pi$
$761$	1.83554e7	1.14895	0.574477	+	0.818521i	$0.305206\pi$
$762$	− 9.22752e6i	− 0.575702i
$763$	8.74150e6i	0.543594i
$764$	3.40833e6	0.211255
$765$	0	0
$766$	−1.18094e7	−0.727206
$767$	8.63940e6i	0.530268i
$768$	5.87519e6i	0.359434i
$769$	747166.	0.0455618	0.0227809	−	0.999740i	$-0.492748\pi$
$769$	747166.	0.0455618	0.0227809	+	0.999740i	$0.492748\pi$
$770$	0	0
$771$	2.65617e6	0.160924
$772$	− 4.34382e6i	− 0.262318i
$773$	− 2.02692e7i	− 1.22008i	−0.792372	−	0.610038i	$-0.791154\pi$
$773$	− 2.02692e7i	− 1.22008i	0.792372	−	0.610038i	$-0.208846\pi$
$774$	−2.19510e6	−0.131705
$775$	0	0
$776$	−958230.	−0.0571236
$777$	3.23606e6i	0.192293i
$778$	1.76695e7i	1.04659i
$779$	5.64978e6	0.333571
$780$	0	0
$781$	973440.	0.0571060
$782$	5.82336e6i	0.340531i
$783$	− 4.02262e6i	− 0.234479i
$784$	1.80315e6	0.104771
$785$	0	0
$786$	3.28338e6	0.189568
$787$	− 4.69982e6i	− 0.270486i	−0.990812	−	0.135243i	$-0.956819\pi$
$787$	− 4.69982e6i	− 0.270486i	0.990812	−	0.135243i	$-0.0431815\pi$
$788$	− 1.65599e6i	− 0.0950041i
$789$	1.14522e7	0.654931
$790$	0	0
$791$	2.21157e6	0.125678
$792$	− 821340.i	− 0.0465275i
$793$	− 3.51135e7i	− 1.98286i
$794$	−2.02406e7	−1.13939
$795$	0	0
$796$	−574728.	−0.0321499
$797$	584710.i	0.0326058i	0.999867	+	0.0163029i	$0.00518960\pi$
$797$	584710.i	0.0326058i	−0.999867	+	0.0163029i	$0.994810\pi$
$798$	3.81906e6i	0.212300i
$799$	1.74701e6	0.0968117
$800$	0	0
$801$	−257742.	−0.0141940
$802$	1.03822e7i	0.569975i
$803$	− 2.41082e6i	− 0.131940i
$804$	87948.0	0.00479828
$805$	0	0
$806$	−2.43936e7	−1.32263
$807$	2.49097e6i	0.134643i
$808$	− 3.24390e7i	− 1.74799i
$809$	1.64013e7	0.881061	0.440531	−	0.897738i	$-0.354790\pi$
$809$	1.64013e7	0.881061	0.440531	+	0.897738i	$0.354790\pi$
$810$	0	0
$811$	−304948.	−0.0162807	−0.00814036	−	0.999967i	$-0.502591\pi$
$811$	−304948.	−0.0162807	−0.00814036	+	0.999967i	$0.502591\pi$
$812$	1.89267e6i	0.100736i
$813$	1.16094e7i	0.616005i
$814$	1.90788e6	0.100923
$815$	0	0
$816$	1.36667e7	0.718519
$817$	9.38744e6i	0.492031i
$818$	− 1.28716e7i	− 0.672587i
$819$	−3.05613e6	−0.159207
$820$	0	0
$821$	3.43428e7	1.77819	0.889095	−	0.457722i	$-0.151335\pi$
$821$	3.43428e7	1.77819	0.889095	+	0.457722i	$0.151335\pi$
$822$	− 4.23819e6i	− 0.218777i
$823$	− 1.56684e7i	− 0.806351i	−0.915123	−	0.403176i	$-0.867906\pi$
$823$	− 1.56684e7i	− 0.806351i	0.915123	−	0.403176i	$-0.132094\pi$
$824$	3.06462e7	1.57238
$825$	0	0
$826$	2.74890e6	0.140187
$827$	− 2.96886e7i	− 1.50948i	−0.656026	−	0.754738i	$-0.727764\pi$
$827$	− 2.96886e7i	− 1.50948i	0.656026	−	0.754738i	$-0.272236\pi$
$828$	− 326592.i	− 0.0165550i
$829$	2.30708e7	1.16594	0.582970	−	0.812494i	$-0.301890\pi$
$829$	2.30708e7	1.16594	0.582970	+	0.812494i	$0.301890\pi$
$830$	0	0
$831$	1.54490e7	0.776062
$832$	− 2.80719e7i	− 1.40593i
$833$	4.85482e6i	0.242416i
$834$	2.15082e6	0.107075
$835$	0	0
$836$	−630448.	−0.0311985
$837$	4.61894e6i	0.227892i
$838$	− 4.24074e6i	− 0.208608i
$839$	−2.32642e7	−1.14100	−0.570498	−	0.821299i	$-0.693250\pi$
$839$	−2.32642e7	−1.14100	−0.570498	+	0.821299i	$0.693250\pi$
$840$	0	0
$841$	9.93718e6	0.484477
$842$	7.18411e6i	0.349215i
$843$	1.32496e7i	0.642148i
$844$	699244.	0.0337888
$845$	0	0
$846$	349920.	0.0168090
$847$	7.75900e6i	0.371619i
$848$	3.14068e6i	0.149980i
$849$	−9.25931e6	−0.440869
$850$	0	0
$851$	4.22669e6	0.200067
$852$	− 1.17936e6i	− 0.0556605i
$853$	− 1.91515e7i	− 0.901219i	−0.892721	−	0.450610i	$-0.851207\pi$
$853$	− 1.91515e7i	− 0.901219i	0.892721	−	0.450610i	$-0.148793\pi$
$854$	−1.11725e7	−0.524210
$855$	0	0
$856$	1.31898e6	0.0615253
$857$	− 5.34683e6i	− 0.248682i	−0.992240	−	0.124341i	$-0.960318\pi$
$857$	− 5.34683e6i	− 0.248682i	0.992240	−	0.124341i	$-0.0396817\pi$
$858$	1.80180e6i	0.0835581i
$859$	−3.95858e7	−1.83045	−0.915223	−	0.402948i	$-0.867986\pi$
$859$	−3.95858e7	−1.83045	−0.915223	+	0.402948i	$0.867986\pi$
$860$	0	0
$861$	1.43854e6	0.0661325
$862$	1.17719e7i	0.539607i
$863$	2.50284e7i	1.14395i	0.820272	+	0.571973i	$0.193822\pi$
$863$	2.50284e7i	1.14395i	−0.820272	+	0.571973i	$0.806178\pi$
$864$	−1.81156e6	−0.0825600
$865$	0	0
$866$	−1.89404e7	−0.858211
$867$	2.40176e7i	1.08513i
$868$	− 2.17325e6i	− 0.0979062i
$869$	−5.06605e6	−0.227573
$870$	0	0
$871$	−1.07492e6	−0.0480099
$872$	− 3.47876e7i	− 1.54929i
$873$	− 398034.i	− 0.0176760i
$874$	4.98816e6	0.220883
$875$	0	0
$876$	−2.92081e6	−0.128600
$877$	− 5.02589e6i	− 0.220655i	−0.993895	−	0.110328i	$-0.964810\pi$
$877$	− 5.02589e6i	− 0.220655i	0.993895	−	0.110328i	$-0.0351900\pi$
$878$	1.82161e7i	0.797480i
$879$	1.06746e7	0.465993
$880$	0	0
$881$	−2.60490e7	−1.13071	−0.565356	−	0.824847i	$-0.691261\pi$
$881$	−2.60490e7	−1.13071	−0.565356	+	0.824847i	$0.691261\pi$
$882$	972405.i	0.0420897i
$883$	6.82462e6i	0.294562i	0.989095	+	0.147281i	$0.0470522\pi$
$883$	6.82462e6i	0.294562i	−0.989095	+	0.147281i	$0.952948\pi$
$884$	1.08986e7	0.469072
$885$	0	0
$886$	1.24195e7	0.531519
$887$	2.33835e7i	0.997931i	0.866622	+	0.498965i	$0.166287\pi$
$887$	2.33835e7i	0.997931i	−0.866622	+	0.498965i	$0.833713\pi$
$888$	− 1.28782e7i	− 0.548053i
$889$	−1.00477e7	−0.426397
$890$	0	0
$891$	341172.	0.0143972
$892$	1.30693e6i	0.0549971i
$893$	− 1.49645e6i	− 0.0627961i
$894$	5.59197e6	0.234003
$895$	0	0
$896$	−5.03548e6	−0.209542
$897$	3.99168e6i	0.165644i
$898$	1.31589e7i	0.544537i
$899$	−3.49620e7	−1.44277
$900$	0	0
$901$	−8.45600e6	−0.347019
$902$	− 848120.i	− 0.0347089i
$903$	2.39022e6i	0.0975480i
$904$	−8.80113e6	−0.358193
$905$	0	0
$906$	−2.00833e7	−0.812859
$907$	3.95959e7i	1.59820i	0.601196	+	0.799102i	$0.294691\pi$
$907$	3.95959e7i	1.59820i	−0.601196	+	0.799102i	$0.705309\pi$
$908$	2.35460e6i	0.0947771i
$909$	1.34747e7	0.540890
$910$	0	0
$911$	−4.67570e6	−0.186660	−0.0933300	−	0.995635i	$-0.529751\pi$
$911$	−4.67570e6	−0.186660	−0.0933300	+	0.995635i	$0.529751\pi$
$912$	− 1.17066e7i	− 0.466062i
$913$	4.22386e6i	0.167700i
$914$	5.80651e6	0.229906
$915$	0	0
$916$	6.48420e6	0.255339
$917$	− 3.57524e6i	− 0.140405i
$918$	7.37019e6i	0.288650i
$919$	4.92594e6	0.192398	0.0961990	−	0.995362i	$-0.469331\pi$
$919$	4.92594e6	0.192398	0.0961990	+	0.995362i	$0.469331\pi$
$920$	0	0
$921$	1.36702e7	0.531040
$922$	1.40692e7i	0.545058i
$923$	1.44144e7i	0.556919i
$924$	−160524.	−0.00618529
$925$	0	0
$926$	3.42150e7	1.31126
$927$	1.27300e7i	0.486550i
$928$	− 1.37122e7i	− 0.522683i
$929$	3.23688e7	1.23052	0.615258	−	0.788326i	$-0.289052\pi$
$929$	3.23688e7	1.23052	0.615258	+	0.788326i	$0.289052\pi$
$930$	0	0
$931$	4.15853e6	0.157241
$932$	− 8.79980e6i	− 0.331843i
$933$	− 1.91527e6i	− 0.0720321i
$934$	−1.67157e7	−0.626985
$935$	0	0
$936$	1.21622e7	0.453754
$937$	− 3.32337e7i	− 1.23660i	−0.785941	−	0.618301i	$-0.787821\pi$
$937$	− 3.32337e7i	− 1.23660i	0.785941	−	0.618301i	$-0.212179\pi$
$938$	342020.i	0.0126924i
$939$	17046.0	0.000630897 0
$940$	0	0
$941$	−2.66426e7	−0.980852	−0.490426	−	0.871483i	$-0.663159\pi$
$941$	−2.66426e7	−0.980852	−0.490426	+	0.871483i	$0.663159\pi$
$942$	− 7.18857e6i	− 0.263946i
$943$	− 1.87891e6i	− 0.0688061i
$944$	−8.42622e6	−0.307753
$945$	0	0
$946$	1.40920e6	0.0511970
$947$	3.14663e7i	1.14017i	0.821585	+	0.570086i	$0.193090\pi$
$947$	3.14663e7i	1.14017i	−0.821585	+	0.570086i	$0.806910\pi$
$948$	6.13771e6i	0.221812i
$949$	3.56987e7	1.28673
$950$	0	0
$951$	−1.42108e7	−0.509527
$952$	− 1.93202e7i	− 0.690907i
$953$	1.34516e7i	0.479779i	0.970800	+	0.239890i	$0.0771112\pi$
$953$	1.34516e7i	0.479779i	−0.970800	+	0.239890i	$0.922889\pi$
$954$	−1.69371e6	−0.0602515
$955$	0	0
$956$	2.42911e6	0.0859613
$957$	2.58242e6i	0.0911481i
$958$	2.14124e7i	0.753792i
$959$	−4.61492e6	−0.162038
$960$	0	0
$961$	1.15157e7	0.402238
$962$	2.82513e7i	0.984239i
$963$	547884.i	0.0190381i
$964$	694190.	0.0240595
$965$	0	0
$966$	1.27008e6	0.0437913
$967$	− 2.84963e7i	− 0.979992i	−0.871725	−	0.489996i	$-0.836998\pi$
$967$	− 2.84963e7i	− 0.979992i	0.871725	−	0.489996i	$-0.163002\pi$
$968$	− 3.08777e7i	− 1.05915i
$969$	3.15189e7	1.07836
$970$	0	0
$971$	1.81858e7	0.618990	0.309495	−	0.950901i	$-0.399840\pi$
$971$	1.81858e7	0.618990	0.309495	+	0.950901i	$0.399840\pi$
$972$	− 413343.i	− 0.0140328i
$973$	− 2.34200e6i	− 0.0793059i
$974$	4.46588e7	1.50837
$975$	0	0
$976$	3.42471e7	1.15080
$977$	3.20941e7i	1.07569i	0.843042	+	0.537847i	$0.180762\pi$
$977$	3.20941e7i	1.07569i	−0.843042	+	0.537847i	$0.819238\pi$
$978$	− 1.11263e7i	− 0.371968i
$979$	165464.	0.00551756
$980$	0	0
$981$	1.44502e7	0.479405
$982$	1.37653e7i	0.455519i
$983$	− 1.56154e7i	− 0.515429i	−0.966221	−	0.257715i	$-0.917031\pi$
$983$	− 1.56154e7i	− 0.515429i	0.966221	−	0.257715i	$-0.0829694\pi$
$984$	−5.72481e6	−0.188483
$985$	0	0
$986$	−5.57870e7	−1.82743
$987$	− 381024.i	− 0.0124497i
$988$	− 9.33548e6i	− 0.304260i
$989$	3.12192e6	0.101492
$990$	0	0
$991$	4.84499e7	1.56714	0.783572	−	0.621301i	$-0.213395\pi$
$991$	4.84499e7	1.56714	0.783572	+	0.621301i	$0.213395\pi$
$992$	1.57450e7i	0.507998i
$993$	3.05524e7i	0.983268i
$994$	4.58640e6	0.147233
$995$	0	0
$996$	5.11736e6	0.163455
$997$	− 4.54336e7i	− 1.44757i	−0.690027	−	0.723784i	$-0.742401\pi$
$997$	− 4.54336e7i	− 1.44757i	0.690027	−	0.723784i	$-0.257599\pi$
$998$	− 2.40204e7i	− 0.763404i
$999$	5.34940e6	0.169587

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	525.6.d.c.274.2		2
5.2	odd	4		525.6.a.b.1.1		1
5.3	odd	4		21.6.a.c.1.1	✓	1
5.4	even	2	inner	525.6.d.c.274.1		2
15.8	even	4		63.6.a.b.1.1		1
20.3	even	4		336.6.a.i.1.1		1
35.3	even	12		147.6.e.d.79.1		2
35.13	even	4		147.6.a.f.1.1		1
35.18	odd	12		147.6.e.c.79.1		2
35.23	odd	12		147.6.e.c.67.1		2
35.33	even	12		147.6.e.d.67.1		2
60.23	odd	4		1008.6.a.a.1.1		1
105.83	odd	4		441.6.a.c.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
21.6.a.c.1.1	✓	1	5.3	odd	4
63.6.a.b.1.1		1	15.8	even	4
147.6.a.f.1.1		1	35.13	even	4
147.6.e.c.67.1		2	35.23	odd	12
147.6.e.c.79.1		2	35.18	odd	12
147.6.e.d.67.1		2	35.33	even	12
147.6.e.d.79.1		2	35.3	even	12
336.6.a.i.1.1		1	20.3	even	4
441.6.a.c.1.1		1	105.83	odd	4
525.6.a.b.1.1		1	5.2	odd	4
525.6.d.c.274.1		2	5.4	even	2	inner
525.6.d.c.274.2		2	1.1	even	1	trivial
1008.6.a.a.1.1		1	60.23	odd	4

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:	\( N \)	\(=\)	\( 525 = 3 \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	525.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+5.00000i q^{2} -9.00000i q^{3} +7.00000 q^{4} +45.0000 q^{6} -49.0000i q^{7} +195.000i q^{8} -81.0000 q^{9} +52.0000 q^{11} -63.0000i q^{12} +770.000i q^{13} +245.000 q^{14} -751.000 q^{16} -2022.00i q^{17} -405.000i q^{18} -1732.00 q^{19} -441.000 q^{21} +260.000i q^{22} +576.000i q^{23} +1755.00 q^{24} -3850.00 q^{26} +729.000i q^{27} -343.000i q^{28} -5518.00 q^{29} +6336.00 q^{31} +2485.00i q^{32} -468.000i q^{33} +10110.0 q^{34} -567.000 q^{36} -7338.00i q^{37} -8660.00i q^{38} +6930.00 q^{39} -3262.00 q^{41} -2205.00i q^{42} -5420.00i q^{43} +364.000 q^{44} -2880.00 q^{46} +864.000i q^{47} +6759.00i q^{48} -2401.00 q^{49} -18198.0 q^{51} +5390.00i q^{52} -4182.00i q^{53} -3645.00 q^{54} +9555.00 q^{56} +15588.0i q^{57} -27590.0i q^{58} +11220.0 q^{59} -45602.0 q^{61} +31680.0i q^{62} +3969.00i q^{63} -36457.0 q^{64} +2340.00 q^{66} +1396.00i q^{67} -14154.0i q^{68} +5184.00 q^{69} +18720.0 q^{71} -15795.0i q^{72} -46362.0i q^{73} +36690.0 q^{74} -12124.0 q^{76} -2548.00i q^{77} +34650.0i q^{78} -97424.0 q^{79} +6561.00 q^{81} -16310.0i q^{82} +81228.0i q^{83} -3087.00 q^{84} +27100.0 q^{86} +49662.0i q^{87} +10140.0i q^{88} +3182.00 q^{89} +37730.0 q^{91} +4032.00i q^{92} -57024.0i q^{93} -4320.00 q^{94} +22365.0 q^{96} +4914.00i q^{97} -12005.0i q^{98} -4212.00 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 14 q^{4} + 90 q^{6} - 162 q^{9} + 104 q^{11} + 490 q^{14} - 1502 q^{16} - 3464 q^{19} - 882 q^{21} + 3510 q^{24} - 7700 q^{26} - 11036 q^{29} + 12672 q^{31} + 20220 q^{34} - 1134 q^{36} + 13860 q^{39}+ \cdots - 8424 q^{99}+O(q^{100}) \) 2 * q + 14 * q^4 + 90 * q^6 - 162 * q^9 + 104 * q^11 + 490 * q^14 - 1502 * q^16 - 3464 * q^19 - 882 * q^21 + 3510 * q^24 - 7700 * q^26 - 11036 * q^29 + 12672 * q^31 + 20220 * q^34 - 1134 * q^36 + 13860 * q^39 - 6524 * q^41 + 728 * q^44 - 5760 * q^46 - 4802 * q^49 - 36396 * q^51 - 7290 * q^54 + 19110 * q^56 + 22440 * q^59 - 91204 * q^61 - 72914 * q^64 + 4680 * q^66 + 10368 * q^69 + 37440 * q^71 + 73380 * q^74 - 24248 * q^76 - 194848 * q^79 + 13122 * q^81 - 6174 * q^84 + 54200 * q^86 + 6364 * q^89 + 75460 * q^91 - 8640 * q^94 + 44730 * q^96 - 8424 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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