LMFDB - Embedded newform 750.6.c.c.499.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [750,6,Mod(499,750)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(750, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("750.499");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 750 = 2 \cdot 3 \cdot 5^{3} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	750.c (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:	$120.287864860$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.289444000000.3
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 39x^{6} + 541x^{4} + 3084x^{2} + 5776 $ x^8 + 39x^6 + 541x^4 + 3084*x^2 + 5776
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{4}\cdot 5^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			499.7
Root			$-3.67526i$ of defining polynomial
Character	$\chi$	$=$	750.499
Dual form			750.6.c.c.499.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+4.00000i q^{2} -9.00000i q^{3} -16.0000 q^{4} +36.0000 q^{6} +89.0490i q^{7} -64.0000i q^{8} -81.0000 q^{9} +O(q^{10})$	\(q+4.00000i q^{2} -9.00000i q^{3} -16.0000 q^{4} +36.0000 q^{6} +89.0490i q^{7} -64.0000i q^{8} -81.0000 q^{9} -129.203 q^{11} +144.000i q^{12} +1041.54i q^{13} -356.196 q^{14} +256.000 q^{16} -1180.79i q^{17} -324.000i q^{18} -764.251 q^{19} +801.441 q^{21} -516.814i q^{22} +1202.47i q^{23} -576.000 q^{24} -4166.18 q^{26} +729.000i q^{27} -1424.78i q^{28} +1709.80 q^{29} -9126.51 q^{31} +1024.00i q^{32} +1162.83i q^{33} +4723.17 q^{34} +1296.00 q^{36} +3759.79i q^{37} -3057.01i q^{38} +9373.90 q^{39} +7742.13 q^{41} +3205.76i q^{42} +16352.6i q^{43} +2067.26 q^{44} -4809.87 q^{46} -26640.4i q^{47} -2304.00i q^{48} +8877.27 q^{49} -10627.1 q^{51} -16664.7i q^{52} -3501.44i q^{53} -2916.00 q^{54} +5699.14 q^{56} +6878.26i q^{57} +6839.20i q^{58} +16267.2 q^{59} +5248.23 q^{61} -36506.1i q^{62} -7212.97i q^{63} -4096.00 q^{64} -4651.32 q^{66} +65574.7i q^{67} +18892.7i q^{68} +10822.2 q^{69} -16219.1 q^{71} +5184.00i q^{72} +10482.2i q^{73} -15039.2 q^{74} +12228.0 q^{76} -11505.4i q^{77} +37495.6i q^{78} -54607.9 q^{79} +6561.00 q^{81} +30968.5i q^{82} -81509.4i q^{83} -12823.1 q^{84} -65410.4 q^{86} -15388.2i q^{87} +8269.02i q^{88} -16837.3 q^{89} -92748.5 q^{91} -19239.5i q^{92} +82138.6i q^{93} +106562. q^{94} +9216.00 q^{96} -145703. i q^{97} +35509.1i q^{98} +10465.5 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 128 q^{4} + 288 q^{6} - 648 q^{9}+O(q^{10}) $ 8 * q - 128 * q^4 + 288 * q^6 - 648 * q^9	$ 8 q - 128 q^{4} + 288 q^{6} - 648 q^{9} - 804 q^{11} + 336 q^{14} + 2048 q^{16} + 3320 q^{19} - 756 q^{21} - 4608 q^{24} - 10128 q^{26} + 11400 q^{29} - 26004 q^{31} + 17296 q^{34} + 10368 q^{36} + 22788 q^{39} + 26316 q^{41} + 12864 q^{44} - 64048 q^{46} + 16064 q^{49} - 38916 q^{51} - 23328 q^{54} - 5376 q^{56} + 21660 q^{59} - 97324 q^{61} - 32768 q^{64} - 28944 q^{66} + 144108 q^{69} - 138444 q^{71} + 223776 q^{74} - 53120 q^{76} + 346560 q^{79} + 52488 q^{81} + 12096 q^{84} - 58848 q^{86} - 123600 q^{89} - 283544 q^{91} + 190336 q^{94} + 73728 q^{96} + 65124 q^{99}+O(q^{100}) $ 8 * q - 128 * q^4 + 288 * q^6 - 648 * q^9 - 804 * q^11 + 336 * q^14 + 2048 * q^16 + 3320 * q^19 - 756 * q^21 - 4608 * q^24 - 10128 * q^26 + 11400 * q^29 - 26004 * q^31 + 17296 * q^34 + 10368 * q^36 + 22788 * q^39 + 26316 * q^41 + 12864 * q^44 - 64048 * q^46 + 16064 * q^49 - 38916 * q^51 - 23328 * q^54 - 5376 * q^56 + 21660 * q^59 - 97324 * q^61 - 32768 * q^64 - 28944 * q^66 + 144108 * q^69 - 138444 * q^71 + 223776 * q^74 - 53120 * q^76 + 346560 * q^79 + 52488 * q^81 + 12096 * q^84 - 58848 * q^86 - 123600 * q^89 - 283544 * q^91 + 190336 * q^94 + 73728 * q^96 + 65124 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/750\mathbb{Z}\right)^\times$.

$n$	$127$	$251$
$\chi(n)$	$-1$	$1$

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 $

$2$	4.00000i	0.707107i
$2$	4.00000i	0.707107i
$3$	− 9.00000i	− 0.577350i
$3$	− 9.00000i	− 0.577350i
$4$	−16.0000	−0.500000
$5$	0	0
$5$	0	0
$6$	36.0000	0.408248
$7$	89.0490i	0.686885i	0.939174	+	0.343442i	$0.111593\pi$
$7$	89.0490i	0.686885i	−0.939174	+	0.343442i	$0.888407\pi$
$8$	− 64.0000i	− 0.353553i
$9$	−81.0000	−0.333333
$10$	0	0
$11$	−129.203	−0.321953	−0.160976	−	0.986958i	$-0.551464\pi$
$11$	−129.203	−0.321953	−0.160976	+	0.986958i	$0.551464\pi$
$12$	144.000i	0.288675i
$13$	1041.54i	1.70931i	0.519200	+	0.854653i	$0.326230\pi$
$13$	1041.54i	1.70931i	−0.519200	+	0.854653i	$0.673770\pi$
$14$	−356.196	−0.485701
$15$	0	0
$16$	256.000	0.250000
$17$	− 1180.79i	− 0.990949i	−0.868623	−	0.495474i	$-0.834994\pi$
$17$	− 1180.79i	− 0.990949i	0.868623	−	0.495474i	$-0.165006\pi$
$18$	− 324.000i	− 0.235702i
$19$	−764.251	−0.485682	−0.242841	−	0.970066i	$-0.578079\pi$
$19$	−764.251	−0.485682	−0.242841	+	0.970066i	$0.578079\pi$
$20$	0	0
$21$	801.441	0.396573
$22$	− 516.814i	− 0.227655i
$23$	1202.47i	0.473973i	0.971513	+	0.236987i	$0.0761598\pi$
$23$	1202.47i	0.473973i	−0.971513	+	0.236987i	$0.923840\pi$
$24$	−576.000	−0.204124
$25$	0	0
$26$	−4166.18	−1.20866
$27$	729.000i	0.192450i
$28$	− 1424.78i	− 0.343442i
$29$	1709.80	0.377529	0.188764	−	0.982022i	$-0.439552\pi$
$29$	1709.80	0.377529	0.188764	+	0.982022i	$0.439552\pi$
$30$	0	0
$31$	−9126.51	−1.70569	−0.852846	−	0.522162i	$-0.825126\pi$
$31$	−9126.51	−1.70569	−0.852846	+	0.522162i	$0.825126\pi$
$32$	1024.00i	0.176777i
$33$	1162.83i	0.185880i
$34$	4723.17	0.700707
$35$	0	0
$36$	1296.00	0.166667
$37$	3759.79i	0.451501i	0.974185	+	0.225751i	$0.0724835\pi$
$37$	3759.79i	0.451501i	−0.974185	+	0.225751i	$0.927517\pi$
$38$	− 3057.01i	− 0.343429i
$39$	9373.90	0.986868
$40$	0	0
$41$	7742.13	0.719285	0.359642	−	0.933090i	$-0.382899\pi$
$41$	7742.13	0.719285	0.359642	+	0.933090i	$0.382899\pi$
$42$	3205.76i	0.280420i
$43$	16352.6i	1.34870i	0.738411	+	0.674350i	$0.235576\pi$
$43$	16352.6i	1.34870i	−0.738411	+	0.674350i	$0.764424\pi$
$44$	2067.26	0.160976
$45$	0	0
$46$	−4809.87	−0.335150
$47$	− 26640.4i	− 1.75912i	−0.475786	−	0.879561i	$-0.657837\pi$
$47$	− 26640.4i	− 1.75912i	0.475786	−	0.879561i	$-0.342163\pi$
$48$	− 2304.00i	− 0.144338i
$49$	8877.27	0.528189
$50$	0	0
$51$	−10627.1	−0.572125
$52$	− 16664.7i	− 0.854653i
$53$	− 3501.44i	− 0.171221i	−0.996329	−	0.0856106i	$-0.972716\pi$
$53$	− 3501.44i	− 0.171221i	0.996329	−	0.0856106i	$-0.0272841\pi$
$54$	−2916.00	−0.136083
$55$	0	0
$56$	5699.14	0.242850
$57$	6878.26i	0.280409i
$58$	6839.20i	0.266953i
$59$	16267.2	0.608390	0.304195	−	0.952610i	$-0.401612\pi$
$59$	16267.2	0.608390	0.304195	+	0.952610i	$0.401612\pi$
$60$	0	0
$61$	5248.23	0.180588	0.0902939	−	0.995915i	$-0.471219\pi$
$61$	5248.23	0.180588	0.0902939	+	0.995915i	$0.471219\pi$
$62$	− 36506.1i	− 1.20611i
$63$	− 7212.97i	− 0.228962i
$64$	−4096.00	−0.125000
$65$	0	0
$66$	−4651.32	−0.131437
$67$	65574.7i	1.78464i	0.451408	+	0.892318i	$0.350922\pi$
$67$	65574.7i	1.78464i	−0.451408	+	0.892318i	$0.649078\pi$
$68$	18892.7i	0.495474i
$69$	10822.2	0.273649
$70$	0	0
$71$	−16219.1	−0.381840	−0.190920	−	0.981606i	$-0.561147\pi$
$71$	−16219.1	−0.381840	−0.190920	+	0.981606i	$0.561147\pi$
$72$	5184.00i	0.117851i
$73$	10482.2i	0.230221i	0.993353	+	0.115110i	$0.0367222\pi$
$73$	10482.2i	0.230221i	−0.993353	+	0.115110i	$0.963278\pi$
$74$	−15039.2	−0.319260
$75$	0	0
$76$	12228.0	0.242841
$77$	− 11505.4i	− 0.221145i
$78$	37495.6i	0.697821i
$79$	−54607.9	−0.984436	−0.492218	−	0.870472i	$-0.663814\pi$
$79$	−54607.9	−0.984436	−0.492218	+	0.870472i	$0.663814\pi$
$80$	0	0
$81$	6561.00	0.111111
$82$	30968.5i	0.508611i
$83$	− 81509.4i	− 1.29871i	−0.760485	−	0.649356i	$-0.775039\pi$
$83$	− 81509.4i	− 1.29871i	0.760485	−	0.649356i	$-0.224961\pi$
$84$	−12823.1	−0.198287
$85$	0	0
$86$	−65410.4	−0.953676
$87$	− 15388.2i	− 0.217966i
$88$	8269.02i	0.113828i
$89$	−16837.3	−0.225319	−0.112659	−	0.993634i	$-0.535937\pi$
$89$	−16837.3	−0.225319	−0.112659	+	0.993634i	$0.535937\pi$
$90$	0	0
$91$	−92748.5	−1.17410
$92$	− 19239.5i	− 0.236987i
$93$	82138.6i	0.984782i
$94$	106562.	1.24389
$95$	0	0
$96$	9216.00	0.102062
$97$	− 145703.i	− 1.57231i	−0.618027	−	0.786157i	$-0.712068\pi$
$97$	− 145703.i	− 1.57231i	0.618027	−	0.786157i	$-0.287932\pi$
$98$	35509.1i	0.373486i
$99$	10465.5	0.107318
$100$	0	0
$101$	−24354.9	−0.237566	−0.118783	−	0.992920i	$-0.537899\pi$
$101$	−24354.9	−0.237566	−0.118783	+	0.992920i	$0.537899\pi$
$102$	− 42508.5i	− 0.404553i
$103$	− 134559.i	− 1.24974i	−0.780728	−	0.624871i	$-0.785151\pi$
$103$	− 134559.i	− 1.24974i	0.780728	−	0.624871i	$-0.214849\pi$
$104$	66658.9	0.604331
$105$	0	0
$106$	14005.8	0.121072
$107$	146012.i	1.23291i	0.787391	+	0.616454i	$0.211431\pi$
$107$	146012.i	1.23291i	−0.787391	+	0.616454i	$0.788569\pi$
$108$	− 11664.0i	− 0.0962250i
$109$	97863.8	0.788962	0.394481	−	0.918904i	$-0.370924\pi$
$109$	97863.8	0.788962	0.394481	+	0.918904i	$0.370924\pi$
$110$	0	0
$111$	33838.1	0.260674
$112$	22796.5i	0.171721i
$113$	28667.9i	0.211203i	0.994409	+	0.105602i	$0.0336768\pi$
$113$	28667.9i	0.211203i	−0.994409	+	0.105602i	$0.966323\pi$
$114$	−27513.0	−0.198279
$115$	0	0
$116$	−27356.8	−0.188764
$117$	− 84365.1i	− 0.569768i
$118$	65068.7i	0.430197i
$119$	105148.	0.680668
$120$	0	0
$121$	−144357.	−0.896346
$122$	20992.9i	0.127695i
$123$	− 69679.2i	− 0.415279i
$124$	146024.	0.852846
$125$	0	0
$126$	28851.9	0.161900
$127$	− 211473.i	− 1.16344i	−0.813389	−	0.581721i	$-0.802380\pi$
$127$	− 211473.i	− 1.16344i	0.813389	−	0.581721i	$-0.197620\pi$
$128$	− 16384.0i	− 0.0883883i
$129$	147173.	0.778673
$130$	0	0
$131$	−30184.8	−0.153678	−0.0768388	−	0.997044i	$-0.524483\pi$
$131$	−30184.8	−0.153678	−0.0768388	+	0.997044i	$0.524483\pi$
$132$	− 18605.3i	− 0.0929398i
$133$	− 68055.8i	− 0.333608i
$134$	−262299.	−1.26193
$135$	0	0
$136$	−75570.7	−0.350353
$137$	− 289701.i	− 1.31871i	−0.751834	−	0.659353i	$-0.770830\pi$
$137$	− 289701.i	− 1.31871i	0.751834	−	0.659353i	$-0.229170\pi$
$138$	43288.8i	0.193499i
$139$	−79187.2	−0.347630	−0.173815	−	0.984778i	$-0.555610\pi$
$139$	−79187.2	−0.347630	−0.173815	+	0.984778i	$0.555610\pi$
$140$	0	0
$141$	−239763.	−1.01563
$142$	− 64876.4i	− 0.270001i
$143$	− 134571.i	− 0.550316i
$144$	−20736.0	−0.0833333
$145$	0	0
$146$	−41928.7	−0.162791
$147$	− 79895.5i	− 0.304950i
$148$	− 60156.6i	− 0.225751i
$149$	−270884.	−0.999579	−0.499790	−	0.866147i	$-0.666589\pi$
$149$	−270884.	−0.999579	−0.499790	+	0.866147i	$0.666589\pi$
$150$	0	0
$151$	−236755.	−0.845001	−0.422501	−	0.906363i	$-0.638848\pi$
$151$	−236755.	−0.845001	−0.422501	+	0.906363i	$0.638848\pi$
$152$	48912.1i	0.171715i
$153$	95644.2i	0.330316i
$154$	46021.8	0.156373
$155$	0	0
$156$	−149982.	−0.493434
$157$	− 465383.i	− 1.50682i	−0.657551	−	0.753410i	$-0.728408\pi$
$157$	− 465383.i	− 1.50682i	0.657551	−	0.753410i	$-0.271592\pi$
$158$	− 218432.i	− 0.696101i
$159$	−31513.0	−0.0988546
$160$	0	0
$161$	−107079.	−0.325565
$162$	26244.0i	0.0785674i
$163$	127374.i	0.375501i	0.982217	+	0.187750i	$0.0601196\pi$
$163$	127374.i	0.375501i	−0.982217	+	0.187750i	$0.939880\pi$
$164$	−123874.	−0.359642
$165$	0	0
$166$	326038.	0.918327
$167$	− 213512.i	− 0.592423i	−0.955122	−	0.296211i	$-0.904277\pi$
$167$	− 213512.i	− 0.592423i	0.955122	−	0.296211i	$-0.0957233\pi$
$168$	− 51292.2i	− 0.140210i
$169$	−713523.	−1.92172
$170$	0	0
$171$	61904.4	0.161894
$172$	− 261642.i	− 0.674350i
$173$	− 35158.7i	− 0.0893135i	−0.999002	−	0.0446567i	$-0.985781\pi$
$173$	− 35158.7i	− 0.0893135i	0.999002	−	0.0446567i	$-0.0142194\pi$
$174$	61552.8	0.154126
$175$	0	0
$176$	−33076.1	−0.0804882
$177$	− 146405.i	− 0.351254i
$178$	− 67349.3i	− 0.159325i
$179$	−667332.	−1.55672	−0.778358	−	0.627820i	$-0.783947\pi$
$179$	−667332.	−1.55672	−0.778358	+	0.627820i	$0.783947\pi$
$180$	0	0
$181$	−40248.8	−0.0913180	−0.0456590	−	0.998957i	$-0.514539\pi$
$181$	−40248.8	−0.0913180	−0.0456590	+	0.998957i	$0.514539\pi$
$182$	− 370994.i	− 0.830211i
$183$	− 47234.1i	− 0.104262i
$184$	76957.9	0.167575
$185$	0	0
$186$	−328555.	−0.696346
$187$	152562.i	0.319039i
$188$	426246.i	0.879561i
$189$	−64916.7	−0.132191
$190$	0	0
$191$	−322518.	−0.639692	−0.319846	−	0.947469i	$-0.603631\pi$
$191$	−322518.	−0.639692	−0.319846	+	0.947469i	$0.603631\pi$
$192$	36864.0i	0.0721688i
$193$	− 661970.i	− 1.27922i	−0.768700	−	0.639609i	$-0.779096\pi$
$193$	− 661970.i	− 1.27922i	0.768700	−	0.639609i	$-0.220904\pi$
$194$	582812.	1.11179
$195$	0	0
$196$	−142036.	−0.264095
$197$	− 698123.i	− 1.28164i	−0.767691	−	0.640820i	$-0.778594\pi$
$197$	− 698123.i	− 1.28164i	0.767691	−	0.640820i	$-0.221406\pi$
$198$	41861.9i	0.0758850i
$199$	815553.	1.45989	0.729944	−	0.683507i	$-0.239546\pi$
$199$	815553.	1.45989	0.729944	+	0.683507i	$0.239546\pi$
$200$	0	0
$201$	590173.	1.03036
$202$	− 97419.8i	− 0.167984i
$203$	152256.i	0.259319i
$204$	170034.	0.286062
$205$	0	0
$206$	538237.	0.883701
$207$	− 97399.9i	− 0.157991i
$208$	266636.i	0.427326i
$209$	98743.9	0.156367
$210$	0	0
$211$	846709.	1.30927	0.654633	−	0.755947i	$-0.272823\pi$
$211$	846709.	1.30927	0.654633	+	0.755947i	$0.272823\pi$
$212$	56023.1i	0.0856106i
$213$	145972.i	0.220455i
$214$	−584050.	−0.871797
$215$	0	0
$216$	46656.0	0.0680414
$217$	− 812707.i	− 1.17161i
$218$	391455.i	0.557880i
$219$	94339.6	0.132918
$220$	0	0
$221$	1.22985e6	1.69383
$222$	135352.i	0.184325i
$223$	− 643352.i	− 0.866336i	−0.901313	−	0.433168i	$-0.857396\pi$
$223$	− 643352.i	− 0.866336i	0.901313	−	0.433168i	$-0.142604\pi$
$224$	−91186.2	−0.121425
$225$	0	0
$226$	−114672.	−0.149343
$227$	− 832367.i	− 1.07214i	−0.844175	−	0.536068i	$-0.819909\pi$
$227$	− 832367.i	− 1.07214i	0.844175	−	0.536068i	$-0.180091\pi$
$228$	− 110052.i	− 0.140204i
$229$	−51116.3	−0.0644126	−0.0322063	−	0.999481i	$-0.510253\pi$
$229$	−51116.3	−0.0644126	−0.0322063	+	0.999481i	$0.510253\pi$
$230$	0	0
$231$	−103549.	−0.127678
$232$	− 109427.i	− 0.133477i
$233$	769657.i	0.928769i	0.885634	+	0.464384i	$0.153724\pi$
$233$	769657.i	0.928769i	−0.885634	+	0.464384i	$0.846276\pi$
$234$	337461.	0.402887
$235$	0	0
$236$	−260275.	−0.304195
$237$	491471.i	0.568364i
$238$	420594.i	0.481305i
$239$	−636498.	−0.720780	−0.360390	−	0.932802i	$-0.617356\pi$
$239$	−636498.	−0.720780	−0.360390	+	0.932802i	$0.617356\pi$
$240$	0	0
$241$	−637791.	−0.707353	−0.353676	−	0.935368i	$-0.615069\pi$
$241$	−637791.	−0.707353	−0.353676	+	0.935368i	$0.615069\pi$
$242$	− 577430.i	− 0.633813i
$243$	− 59049.0i	− 0.0641500i
$244$	−83971.7	−0.0902939
$245$	0	0
$246$	278717.	0.293647
$247$	− 796002.i	− 0.830179i
$248$	584097.i	0.603053i
$249$	−733585.	−0.749811
$250$	0	0
$251$	692324.	0.693626	0.346813	−	0.937934i	$-0.387264\pi$
$251$	692324.	0.693626	0.346813	+	0.937934i	$0.387264\pi$
$252$	115408.i	0.114481i
$253$	− 155363.i	− 0.152597i
$254$	845890.	0.822678
$255$	0	0
$256$	65536.0	0.0625000
$257$	− 1.78221e6i	− 1.68316i	−0.540134	−	0.841579i	$-0.681626\pi$
$257$	− 1.78221e6i	− 1.68316i	0.540134	−	0.841579i	$-0.318374\pi$
$258$	588694.i	0.550605i
$259$	−334805.	−0.310130
$260$	0	0
$261$	−138494.	−0.125843
$262$	− 120739.i	− 0.108667i
$263$	225723.i	0.201227i	0.994926	+	0.100614i	$0.0320806\pi$
$263$	225723.i	0.201227i	−0.994926	+	0.100614i	$0.967919\pi$
$264$	74421.2	0.0657184
$265$	0	0
$266$	272223.	0.235896
$267$	151536.i	0.130088i
$268$	− 1.04920e6i	− 0.892318i
$269$	−898113.	−0.756746	−0.378373	−	0.925653i	$-0.623516\pi$
$269$	−898113.	−0.756746	−0.378373	+	0.925653i	$0.623516\pi$
$270$	0	0
$271$	−455573.	−0.376821	−0.188410	−	0.982090i	$-0.560333\pi$
$271$	−455573.	−0.376821	−0.188410	+	0.982090i	$0.560333\pi$
$272$	− 302283.i	− 0.247737i
$273$	834737.i	0.677865i
$274$	1.15880e6	0.932466
$275$	0	0
$276$	−173155.	−0.136824
$277$	− 168150.i	− 0.131673i	−0.997830	−	0.0658366i	$-0.979028\pi$
$277$	− 168150.i	− 0.131673i	0.997830	−	0.0658366i	$-0.0209716\pi$
$278$	− 316749.i	− 0.245812i
$279$	739248.	0.568564
$280$	0	0
$281$	1.62527e6	1.22789	0.613944	−	0.789349i	$-0.289582\pi$
$281$	1.62527e6	1.22789	0.613944	+	0.789349i	$0.289582\pi$
$282$	− 959054.i	− 0.718158i
$283$	− 1.03227e6i	− 0.766177i	−0.923712	−	0.383089i	$-0.874860\pi$
$283$	− 1.03227e6i	− 0.766177i	0.923712	−	0.383089i	$-0.125140\pi$
$284$	259506.	0.190920
$285$	0	0
$286$	538285.	0.389132
$287$	689429.i	0.494066i
$288$	− 82944.0i	− 0.0589256i
$289$	25586.5	0.0180205
$290$	0	0
$291$	−1.31133e6	−0.907776
$292$	− 167715.i	− 0.115110i
$293$	2.53642e6i	1.72605i	0.505165	+	0.863023i	$0.331431\pi$
$293$	2.53642e6i	1.72605i	−0.505165	+	0.863023i	$0.668569\pi$
$294$	319582.	0.215632
$295$	0	0
$296$	240626.	0.159630
$297$	− 94189.3i	− 0.0619599i
$298$	− 1.08353e6i	− 0.706809i
$299$	−1.25242e6	−0.810165
$300$	0	0
$301$	−1.45618e6	−0.926402
$302$	− 947021.i	− 0.597506i
$303$	219195.i	0.137159i
$304$	−195648.	−0.121421
$305$	0	0
$306$	−382577.	−0.233569
$307$	− 1.58046e6i	− 0.957055i	−0.878073	−	0.478527i	$-0.841171\pi$
$307$	− 1.58046e6i	− 0.957055i	0.878073	−	0.478527i	$-0.158829\pi$
$308$	184087.i	0.110572i
$309$	−1.21103e6	−0.721539
$310$	0	0
$311$	170550.	0.0999887	0.0499944	−	0.998750i	$-0.484080\pi$
$311$	170550.	0.0999887	0.0499944	+	0.998750i	$0.484080\pi$
$312$	− 599930.i	− 0.348910i
$313$	− 2.10998e6i	− 1.21736i	−0.793417	−	0.608678i	$-0.791700\pi$
$313$	− 2.10998e6i	− 1.21736i	0.793417	−	0.608678i	$-0.208300\pi$
$314$	1.86153e6	1.06548
$315$	0	0
$316$	873726.	0.492218
$317$	2.43889e6i	1.36315i	0.731748	+	0.681575i	$0.238705\pi$
$317$	2.43889e6i	1.36315i	−0.731748	+	0.681575i	$0.761295\pi$
$318$	− 126052.i	− 0.0699008i
$319$	−220912.	−0.121547
$320$	0	0
$321$	1.31411e6	0.711819
$322$	− 428314.i	− 0.230209i
$323$	902422.i	0.481286i
$324$	−104976.	−0.0555556
$325$	0	0
$326$	−509495.	−0.265519
$327$	− 880774.i	− 0.455507i
$328$	− 495496.i	− 0.254306i
$329$	2.37230e6	1.20831
$330$	0	0
$331$	2.91169e6	1.46075	0.730373	−	0.683049i	$-0.239346\pi$
$331$	2.91169e6	1.46075	0.730373	+	0.683049i	$0.239346\pi$
$332$	1.30415e6i	0.649356i
$333$	− 304543.i	− 0.150500i
$334$	854049.	0.418906
$335$	0	0
$336$	205169.	0.0991433
$337$	− 1.39716e6i	− 0.670149i	−0.942192	−	0.335074i	$-0.891239\pi$
$337$	− 1.39716e6i	− 0.670149i	0.942192	−	0.335074i	$-0.108761\pi$
$338$	− 2.85409e6i	− 1.35886i
$339$	258011.	0.121938
$340$	0	0
$341$	1.17918e6	0.549153
$342$	247617.i	0.114476i
$343$	2.28716e6i	1.04969i
$344$	1.04657e6	0.476838
$345$	0	0
$346$	140635.	0.0631542
$347$	− 3.34489e6i	− 1.49127i	−0.666352	−	0.745637i	$-0.732145\pi$
$347$	− 3.34489e6i	− 1.49127i	0.666352	−	0.745637i	$-0.267855\pi$
$348$	246211.i	0.108983i
$349$	1.75860e6	0.772864	0.386432	−	0.922318i	$-0.373707\pi$
$349$	1.75860e6	0.772864	0.386432	+	0.922318i	$0.373707\pi$
$350$	0	0
$351$	−759286.	−0.328956
$352$	− 132304.i	− 0.0569138i
$353$	− 3.54037e6i	− 1.51221i	−0.654450	−	0.756105i	$-0.727100\pi$
$353$	− 3.54037e6i	− 1.51221i	0.654450	−	0.756105i	$-0.272900\pi$
$354$	585619.	0.248374
$355$	0	0
$356$	269397.	0.112659
$357$	− 946335.i	− 0.392984i
$358$	− 2.66933e6i	− 1.10076i
$359$	−699728.	−0.286545	−0.143273	−	0.989683i	$-0.545763\pi$
$359$	−699728.	−0.286545	−0.143273	+	0.989683i	$0.545763\pi$
$360$	0	0
$361$	−1.89202e6	−0.764113
$362$	− 160995.i	− 0.0645716i
$363$	1.29922e6i	0.517506i
$364$	1.48398e6	0.587048
$365$	0	0
$366$	188936.	0.0737247
$367$	502286.i	0.194664i	0.995252	+	0.0973321i	$0.0310309\pi$
$367$	502286.i	0.194664i	−0.995252	+	0.0973321i	$0.968969\pi$
$368$	307832.i	0.118493i
$369$	−627113.	−0.239762
$370$	0	0
$371$	311800.	0.117609
$372$	− 1.31422e6i	− 0.492391i
$373$	3.84750e6i	1.43188i	0.698162	+	0.715940i	$0.254001\pi$
$373$	3.84750e6i	1.43188i	−0.698162	+	0.715940i	$0.745999\pi$
$374$	−610250.	−0.225595
$375$	0	0
$376$	−1.70498e6	−0.621943
$377$	1.78083e6i	0.645312i
$378$	− 259667.i	− 0.0934732i
$379$	−1.96748e6	−0.703577	−0.351789	−	0.936079i	$-0.614426\pi$
$379$	−1.96748e6	−0.703577	−0.351789	+	0.936079i	$0.614426\pi$
$380$	0	0
$381$	−1.90325e6	−0.671713
$382$	− 1.29007e6i	− 0.452331i
$383$	418398.i	0.145745i	0.997341	+	0.0728724i	$0.0232166\pi$
$383$	418398.i	0.145745i	−0.997341	+	0.0728724i	$0.976783\pi$
$384$	−147456.	−0.0510310
$385$	0	0
$386$	2.64788e6	0.904544
$387$	− 1.32456e6i	− 0.449567i
$388$	2.33125e6i	0.786157i
$389$	−1.55706e6	−0.521713	−0.260857	−	0.965378i	$-0.584005\pi$
$389$	−1.55706e6	−0.521713	−0.260857	+	0.965378i	$0.584005\pi$
$390$	0	0
$391$	1.41986e6	0.469683
$392$	− 568146.i	− 0.186743i
$393$	271664.i	0.0887258i
$394$	2.79249e6	0.906257
$395$	0	0
$396$	−167448.	−0.0536588
$397$	2.57150e6i	0.818860i	0.912342	+	0.409430i	$0.134272\pi$
$397$	2.57150e6i	0.818860i	−0.912342	+	0.409430i	$0.865728\pi$
$398$	3.26221e6i	1.03230i
$399$	−612502.	−0.192609
$400$	0	0
$401$	3.48615e6	1.08264	0.541322	−	0.840816i	$-0.317924\pi$
$401$	3.48615e6	1.08264	0.541322	+	0.840816i	$0.317924\pi$
$402$	2.36069e6i	0.728574i
$403$	− 9.50567e6i	− 2.91555i
$404$	389679.	0.118783
$405$	0	0
$406$	−609024.	−0.183366
$407$	− 485778.i	− 0.145362i
$408$	680136.i	0.202277i
$409$	−5.49587e6	−1.62453	−0.812266	−	0.583287i	$-0.801766\pi$
$409$	−5.49587e6	−1.62453	−0.812266	+	0.583287i	$0.801766\pi$
$410$	0	0
$411$	−2.60731e6	−0.761355
$412$	2.15295e6i	0.624871i
$413$	1.44858e6i	0.417894i
$414$	389600.	0.111717
$415$	0	0
$416$	−1.06654e6	−0.302165
$417$	712684.i	0.200704i
$418$	394976.i	0.110568i
$419$	−490754.	−0.136562	−0.0682808	−	0.997666i	$-0.521751\pi$
$419$	−490754.	−0.136562	−0.0682808	+	0.997666i	$0.521751\pi$
$420$	0	0
$421$	−4.49723e6	−1.23663	−0.618315	−	0.785930i	$-0.712184\pi$
$421$	−4.49723e6	−1.23663	−0.618315	+	0.785930i	$0.712184\pi$
$422$	3.38683e6i	0.925791i
$423$	2.15787e6i	0.586374i
$424$	−224092.	−0.0605358
$425$	0	0
$426$	−583888.	−0.155885
$427$	467350.i	0.124043i
$428$	− 2.33620e6i	− 0.616454i
$429$	−1.21114e6	−0.317725
$430$	0	0
$431$	3.65565e6	0.947918	0.473959	−	0.880547i	$-0.342824\pi$
$431$	3.65565e6	0.947918	0.473959	+	0.880547i	$0.342824\pi$
$432$	186624.i	0.0481125i
$433$	1.90074e6i	0.487195i	0.969876	+	0.243597i	$0.0783276\pi$
$433$	1.90074e6i	0.487195i	−0.969876	+	0.243597i	$0.921672\pi$
$434$	3.25083e6	0.828456
$435$	0	0
$436$	−1.56582e6	−0.394481
$437$	− 918988.i	− 0.230200i
$438$	377358.i	0.0939872i
$439$	−6.31273e6	−1.56335	−0.781675	−	0.623686i	$-0.785634\pi$
$439$	−6.31273e6	−1.56335	−0.781675	+	0.623686i	$0.785634\pi$
$440$	0	0
$441$	−719059.	−0.176063
$442$	4.91939e6i	1.19772i
$443$	− 1.11593e6i	− 0.270165i	−0.990834	−	0.135082i	$-0.956870\pi$
$443$	− 1.11593e6i	− 0.270165i	0.990834	−	0.135082i	$-0.0431299\pi$
$444$	−541410.	−0.130337
$445$	0	0
$446$	2.57341e6	0.612592
$447$	2.43795e6i	0.577107i
$448$	− 364745.i	− 0.0858606i
$449$	2.18307e6	0.511037	0.255518	−	0.966804i	$-0.417754\pi$
$449$	2.18307e6	0.511037	0.255518	+	0.966804i	$0.417754\pi$
$450$	0	0
$451$	−1.00031e6	−0.231576
$452$	− 458687.i	− 0.105602i
$453$	2.13080e6i	0.487862i
$454$	3.32947e6	0.758115
$455$	0	0
$456$	440209.	0.0991395
$457$	− 2.61536e6i	− 0.585788i	−0.956145	−	0.292894i	$-0.905382\pi$
$457$	− 2.61536e6i	− 0.585788i	0.956145	−	0.292894i	$-0.0946183\pi$
$458$	− 204465.i	− 0.0455466i
$459$	860798.	0.190708
$460$	0	0
$461$	5.00945e6	1.09784	0.548918	−	0.835876i	$-0.315040\pi$
$461$	5.00945e6	1.09784	0.548918	+	0.835876i	$0.315040\pi$
$462$	− 414196.i	− 0.0902819i
$463$	− 2.46856e6i	− 0.535170i	−0.963534	−	0.267585i	$-0.913774\pi$
$463$	− 2.46856e6i	− 0.535170i	0.963534	−	0.267585i	$-0.0862256\pi$
$464$	437709.	0.0943822
$465$	0	0
$466$	−3.07863e6	−0.656739
$467$	4.08810e6i	0.867420i	0.901052	+	0.433710i	$0.142796\pi$
$467$	4.08810e6i	0.867420i	−0.901052	+	0.433710i	$0.857204\pi$
$468$	1.34984e6i	0.284884i
$469$	−5.83937e6	−1.22584
$470$	0	0
$471$	−4.18845e6	−0.869963
$472$	− 1.04110e6i	− 0.215099i
$473$	− 2.11281e6i	− 0.434218i
$474$	−1.96588e6	−0.401894
$475$	0	0
$476$	−1.68237e6	−0.340334
$477$	283617.i	0.0570737i
$478$	− 2.54599e6i	− 0.509668i
$479$	2.59311e6	0.516394	0.258197	−	0.966092i	$-0.416872\pi$
$479$	2.59311e6	0.516394	0.258197	+	0.966092i	$0.416872\pi$
$480$	0	0
$481$	−3.91599e6	−0.771754
$482$	− 2.55117e6i	− 0.500174i
$483$	963707.i	0.187965i
$484$	2.30972e6	0.448173
$485$	0	0
$486$	236196.	0.0453609
$487$	7.43245e6i	1.42007i	0.704167	+	0.710035i	$0.251321\pi$
$487$	7.43245e6i	1.42007i	−0.704167	+	0.710035i	$0.748679\pi$
$488$	− 335887.i	− 0.0638474i
$489$	1.14636e6	0.216795
$490$	0	0
$491$	−6.97457e6	−1.30561	−0.652805	−	0.757526i	$-0.726408\pi$
$491$	−6.97457e6	−1.30561	−0.652805	+	0.757526i	$0.726408\pi$
$492$	1.11487e6i	0.207640i
$493$	− 2.01892e6i	− 0.374112i
$494$	3.18401e6	0.587025
$495$	0	0
$496$	−2.33639e6	−0.426423
$497$	− 1.44430e6i	− 0.262280i
$498$	− 2.93434e6i	− 0.530197i
$499$	−2.96304e6	−0.532704	−0.266352	−	0.963876i	$-0.585818\pi$
$499$	−2.96304e6	−0.532704	−0.266352	+	0.963876i	$0.585818\pi$
$500$	0	0
$501$	−1.92161e6	−0.342035
$502$	2.76930e6i	0.490467i
$503$	3.83854e6i	0.676465i	0.941063	+	0.338233i	$0.109829\pi$
$503$	3.83854e6i	0.676465i	−0.941063	+	0.338233i	$0.890171\pi$
$504$	−461630.	−0.0809502
$505$	0	0
$506$	621452.	0.107902
$507$	6.42171e6i	1.10951i
$508$	3.38356e6i	0.581721i
$509$	−6.44622e6	−1.10283	−0.551417	−	0.834229i	$-0.685913\pi$
$509$	−6.44622e6	−1.10283	−0.551417	+	0.834229i	$0.685913\pi$
$510$	0	0
$511$	−933427.	−0.158135
$512$	262144.i	0.0441942i
$513$	− 557139.i	− 0.0934696i
$514$	7.12882e6	1.19017
$515$	0	0
$516$	−2.35477e6	−0.389336
$517$	3.44203e6i	0.566354i
$518$	− 1.33922e6i	− 0.219295i
$519$	−316428.	−0.0515652
$520$	0	0
$521$	−4.89694e6	−0.790369	−0.395185	−	0.918602i	$-0.629319\pi$
$521$	−4.89694e6	−0.790369	−0.395185	+	0.918602i	$0.629319\pi$
$522$	− 553975.i	− 0.0889844i
$523$	5.24383e6i	0.838291i	0.907919	+	0.419145i	$0.137670\pi$
$523$	5.24383e6i	0.838291i	−0.907919	+	0.419145i	$0.862330\pi$
$524$	482958.	0.0768388
$525$	0	0
$526$	−902892.	−0.142289
$527$	1.07765e7i	1.69025i
$528$	297685.i	0.0464699i
$529$	4.99041e6	0.775349
$530$	0	0
$531$	−1.31764e6	−0.202797
$532$	1.08889e6i	0.166804i
$533$	8.06378e6i	1.22948i
$534$	−606143.	−0.0919861
$535$	0	0
$536$	4.19678e6	0.630964
$537$	6.00599e6i	0.898771i
$538$	− 3.59245e6i	− 0.535100i
$539$	−1.14697e6	−0.170052
$540$	0	0
$541$	−75006.1	−0.0110180	−0.00550901	−	0.999985i	$-0.501754\pi$
$541$	−75006.1	−0.0110180	−0.00550901	+	0.999985i	$0.501754\pi$
$542$	− 1.82229e6i	− 0.266452i
$543$	362239.i	0.0527225i
$544$	1.20913e6	0.175177
$545$	0	0
$546$	−3.33895e6	−0.479323
$547$	8.56356e6i	1.22373i	0.790962	+	0.611865i	$0.209581\pi$
$547$	8.56356e6i	1.22373i	−0.790962	+	0.611865i	$0.790419\pi$
$548$	4.63521e6i	0.659353i
$549$	−425107.	−0.0601959
$550$	0	0
$551$	−1.30672e6	−0.183359
$552$	− 692621.i	− 0.0967494i
$553$	− 4.86278e6i	− 0.676194i
$554$	672600.	0.0931070
$555$	0	0
$556$	1.26699e6	0.173815
$557$	− 1.35579e6i	− 0.185163i	−0.995705	−	0.0925816i	$-0.970488\pi$
$557$	− 1.35579e6i	− 0.185163i	0.995705	−	0.0925816i	$-0.0295119\pi$
$558$	2.95699e6i	0.402036i
$559$	−1.70320e7	−2.30534
$560$	0	0
$561$	1.37306e6	0.184197
$562$	6.50107e6i	0.868248i
$563$	6.73716e6i	0.895789i	0.894086	+	0.447895i	$0.147826\pi$
$563$	6.73716e6i	0.895789i	−0.894086	+	0.447895i	$0.852174\pi$
$564$	3.83622e6	0.507815
$565$	0	0
$566$	4.12910e6	0.541769
$567$	584251.i	0.0763205i
$568$	1.03802e6i	0.135001i
$569$	−1.15693e7	−1.49805	−0.749023	−	0.662544i	$-0.769477\pi$
$569$	−1.15693e7	−1.49805	−0.749023	+	0.662544i	$0.769477\pi$
$570$	0	0
$571$	−7.49457e6	−0.961958	−0.480979	−	0.876732i	$-0.659719\pi$
$571$	−7.49457e6	−0.961958	−0.480979	+	0.876732i	$0.659719\pi$
$572$	2.15314e6i	0.275158i
$573$	2.90267e6i	0.369327i
$574$	−2.75772e6	−0.349357
$575$	0	0
$576$	331776.	0.0416667
$577$	− 1.51983e7i	− 1.90044i	−0.311574	−	0.950222i	$-0.600856\pi$
$577$	− 1.51983e7i	− 1.90044i	0.311574	−	0.950222i	$-0.399144\pi$
$578$	102346.i	0.0127424i
$579$	−5.95773e6	−0.738557
$580$	0	0
$581$	7.25833e6	0.892065
$582$	− 5.24531e6i	− 0.641894i
$583$	452399.i	0.0551252i
$584$	670859.	0.0813953
$585$	0	0
$586$	−1.01457e7	−1.22050
$587$	1.28621e7i	1.54070i	0.637621	+	0.770350i	$0.279919\pi$
$587$	1.28621e7i	1.54070i	−0.637621	+	0.770350i	$0.720081\pi$
$588$	1.27833e6i	0.152475i
$589$	6.97495e6	0.828424
$590$	0	0
$591$	−6.28310e6	−0.739955
$592$	962506.i	0.112875i
$593$	− 1.12822e7i	− 1.31752i	−0.752353	−	0.658760i	$-0.771081\pi$
$593$	− 1.12822e7i	− 1.31752i	0.752353	−	0.658760i	$-0.228919\pi$
$594$	376757.	0.0438122
$595$	0	0
$596$	4.33414e6	0.499790
$597$	− 7.33998e6i	− 0.842867i
$598$	− 5.00970e6i	− 0.572873i
$599$	−1.61733e7	−1.84175	−0.920877	−	0.389854i	$-0.872525\pi$
$599$	−1.61733e7	−1.84175	−0.920877	+	0.389854i	$0.872525\pi$
$600$	0	0
$601$	4.28796e6	0.484244	0.242122	−	0.970246i	$-0.422157\pi$
$601$	4.28796e6	0.484244	0.242122	+	0.970246i	$0.422157\pi$
$602$	− 5.82473e6i	− 0.655065i
$603$	− 5.31155e6i	− 0.594879i
$604$	3.78808e6	0.422501
$605$	0	0
$606$	−876778.	−0.0969858
$607$	− 887110.i	− 0.0977251i	−0.998806	−	0.0488625i	$-0.984440\pi$
$607$	− 887110.i	− 0.0977251i	0.998806	−	0.0488625i	$-0.0155596\pi$
$608$	− 782593.i	− 0.0858573i
$609$	1.37030e6	0.149718
$610$	0	0
$611$	2.77472e7	3.00688
$612$	− 1.53031e6i	− 0.165158i
$613$	8.04073e6i	0.864260i	0.901811	+	0.432130i	$0.142238\pi$
$613$	8.04073e6i	0.864260i	−0.901811	+	0.432130i	$0.857762\pi$
$614$	6.32183e6	0.676740
$615$	0	0
$616$	−736348.	−0.0781864
$617$	1.44062e7i	1.52348i	0.647881	+	0.761742i	$0.275655\pi$
$617$	1.44062e7i	1.52348i	−0.647881	+	0.761742i	$0.724345\pi$
$618$	− 4.84413e6i	− 0.510205i
$619$	1.58481e7	1.66246	0.831231	−	0.555927i	$-0.187637\pi$
$619$	1.58481e7	1.66246	0.831231	+	0.555927i	$0.187637\pi$
$620$	0	0
$621$	−876599.	−0.0912162
$622$	682200.i	0.0707027i
$623$	− 1.49935e6i	− 0.154768i
$624$	2.39972e6	0.246717
$625$	0	0
$626$	8.43993e6	0.860801
$627$	− 888695.i	− 0.0902784i
$628$	7.44613e6i	0.753410i
$629$	4.43953e6	0.447415
$630$	0	0
$631$	−4.34095e6	−0.434022	−0.217011	−	0.976169i	$-0.569631\pi$
$631$	−4.34095e6	−0.434022	−0.217011	+	0.976169i	$0.569631\pi$
$632$	3.49490e6i	0.348051i
$633$	− 7.62038e6i	− 0.755905i
$634$	−9.75555e6	−0.963893
$635$	0	0
$636$	504208.	0.0494273
$637$	9.24608e6i	0.902836i
$638$	− 883648.i	− 0.0859464i
$639$	1.31375e6	0.127280
$640$	0	0
$641$	1.46690e7	1.41012	0.705058	−	0.709149i	$-0.250921\pi$
$641$	1.46690e7	1.41012	0.705058	+	0.709149i	$0.250921\pi$
$642$	5.25645e6i	0.503332i
$643$	6.54309e6i	0.624102i	0.950065	+	0.312051i	$0.101016\pi$
$643$	6.54309e6i	0.624102i	−0.950065	+	0.312051i	$0.898984\pi$
$644$	1.71326e6	0.162783
$645$	0	0
$646$	−3.60969e6	−0.340321
$647$	1.39086e7i	1.30624i	0.757254	+	0.653121i	$0.226541\pi$
$647$	1.39086e7i	1.30624i	−0.757254	+	0.653121i	$0.773459\pi$
$648$	− 419904.i	− 0.0392837i
$649$	−2.10178e6	−0.195873
$650$	0	0
$651$	−7.31436e6	−0.676432
$652$	− 2.03798e6i	− 0.187750i
$653$	− 8.71155e6i	− 0.799489i	−0.916627	−	0.399744i	$-0.869099\pi$
$653$	− 8.71155e6i	− 0.799489i	0.916627	−	0.399744i	$-0.130901\pi$
$654$	3.52310e6	0.322092
$655$	0	0
$656$	1.98199e6	0.179821
$657$	− 849056.i	− 0.0767402i
$658$	9.48920e6i	0.854407i
$659$	1.49770e7	1.34342	0.671709	−	0.740815i	$-0.265560\pi$
$659$	1.49770e7	1.34342	0.671709	+	0.740815i	$0.265560\pi$
$660$	0	0
$661$	−1.30247e7	−1.15948	−0.579742	−	0.814800i	$-0.696847\pi$
$661$	−1.30247e7	−1.15948	−0.579742	+	0.814800i	$0.696847\pi$
$662$	1.16467e7i	1.03290i
$663$	− 1.10686e7i	− 0.977936i
$664$	−5.21660e6	−0.459164
$665$	0	0
$666$	1.21817e6	0.106420
$667$	2.05598e6i	0.178939i
$668$	3.41620e6i	0.296211i
$669$	−5.79016e6	−0.500179
$670$	0	0
$671$	−678090.	−0.0581408
$672$	820676.i	0.0701049i
$673$	− 1.35401e7i	− 1.15235i	−0.817326	−	0.576175i	$-0.804545\pi$
$673$	− 1.35401e7i	− 1.15235i	0.817326	−	0.576175i	$-0.195455\pi$
$674$	5.58864e6	0.473867
$675$	0	0
$676$	1.14164e7	0.960862
$677$	− 3.95478e6i	− 0.331627i	−0.986157	−	0.165814i	$-0.946975\pi$
$677$	− 3.95478e6i	− 0.331627i	0.986157	−	0.165814i	$-0.0530250\pi$
$678$	1.03205e6i	0.0862233i
$679$	1.29747e7	1.08000
$680$	0	0
$681$	−7.49130e6	−0.618998
$682$	4.71671e6i	0.388310i
$683$	2.95710e6i	0.242557i	0.992619	+	0.121279i	$0.0386994\pi$
$683$	2.95710e6i	0.242557i	−0.992619	+	0.121279i	$0.961301\pi$
$684$	−990470.	−0.0809470
$685$	0	0
$686$	−9.14864e6	−0.742243
$687$	460047.i	0.0371886i
$688$	4.18627e6i	0.337175i
$689$	3.64691e6	0.292669
$690$	0	0
$691$	−1.77351e7	−1.41299	−0.706493	−	0.707720i	$-0.749724\pi$
$691$	−1.77351e7	−1.41299	−0.706493	+	0.707720i	$0.749724\pi$
$692$	562538.i	0.0446567i
$693$	931941.i	0.0737149i
$694$	1.33795e7	1.05449
$695$	0	0
$696$	−984845.	−0.0770628
$697$	− 9.14185e6i	− 0.712774i
$698$	7.03439e6i	0.546497i
$699$	6.92692e6	0.536225
$700$	0	0
$701$	1.97596e7	1.51874	0.759368	−	0.650661i	$-0.225508\pi$
$701$	1.97596e7	1.51874	0.759368	+	0.650661i	$0.225508\pi$
$702$	− 3.03715e6i	− 0.232607i
$703$	− 2.87342e6i	− 0.219286i
$704$	529217.	0.0402441
$705$	0	0
$706$	1.41615e7	1.06929
$707$	− 2.16878e6i	− 0.163180i
$708$	2.34247e6i	0.175627i
$709$	5.55927e6	0.415338	0.207669	−	0.978199i	$-0.433412\pi$
$709$	5.55927e6	0.415338	0.207669	+	0.978199i	$0.433412\pi$
$710$	0	0
$711$	4.42324e6	0.328145
$712$	1.07759e6i	0.0796623i
$713$	− 1.09743e7i	− 0.808453i
$714$	3.78534e6	0.277881
$715$	0	0
$716$	1.06773e7	0.778358
$717$	5.72848e6i	0.416142i
$718$	− 2.79891e6i	− 0.202618i
$719$	2.23562e6	0.161278	0.0806390	−	0.996743i	$-0.474304\pi$
$719$	2.23562e6	0.161278	0.0806390	+	0.996743i	$0.474304\pi$
$720$	0	0
$721$	1.19824e7	0.858429
$722$	− 7.56808e6i	− 0.540309i
$723$	5.74012e6i	0.408390i
$724$	643981.	0.0456590
$725$	0	0
$726$	−5.19687e6	−0.365932
$727$	4.26363e6i	0.299187i	0.988748	+	0.149594i	$0.0477965\pi$
$727$	4.26363e6i	0.299187i	−0.988748	+	0.149594i	$0.952203\pi$
$728$	5.93591e6i	0.415106i
$729$	−531441.	−0.0370370
$730$	0	0
$731$	1.93090e7	1.33649
$732$	755745.i	0.0521312i
$733$	− 1.03774e7i	− 0.713396i	−0.934220	−	0.356698i	$-0.883903\pi$
$733$	− 1.03774e7i	− 0.713396i	0.934220	−	0.356698i	$-0.116097\pi$
$734$	−2.00914e6	−0.137648
$735$	0	0
$736$	−1.23133e6	−0.0837874
$737$	− 8.47248e6i	− 0.574569i
$738$	− 2.50845e6i	− 0.169537i
$739$	599439.	0.0403770	0.0201885	−	0.999796i	$-0.493573\pi$
$739$	599439.	0.0403770	0.0201885	+	0.999796i	$0.493573\pi$
$740$	0	0
$741$	−7.16402e6	−0.479304
$742$	1.24720e6i	0.0831623i
$743$	− 7.62440e6i	− 0.506680i	−0.967377	−	0.253340i	$-0.918471\pi$
$743$	− 7.62440e6i	− 0.506680i	0.967377	−	0.253340i	$-0.0815291\pi$
$744$	5.25687e6	0.348173
$745$	0	0
$746$	−1.53900e7	−1.01249
$747$	6.60226e6i	0.432904i
$748$	− 2.44100e6i	− 0.159519i
$749$	−1.30023e7	−0.846865
$750$	0	0
$751$	1.90191e7	1.23053	0.615263	−	0.788322i	$-0.289050\pi$
$751$	1.90191e7	1.23053	0.615263	+	0.788322i	$0.289050\pi$
$752$	− 6.81994e6i	− 0.439780i
$753$	− 6.23092e6i	− 0.400465i
$754$	−7.12333e6	−0.456305
$755$	0	0
$756$	1.03867e6	0.0660955
$757$	1.90005e7i	1.20511i	0.798078	+	0.602554i	$0.205850\pi$
$757$	1.90005e7i	1.20511i	−0.798078	+	0.602554i	$0.794150\pi$
$758$	− 7.86991e6i	− 0.497504i
$759$	−1.39827e6	−0.0881020
$760$	0	0
$761$	3.07129e6	0.192247	0.0961233	−	0.995369i	$-0.469356\pi$
$761$	3.07129e6	0.192247	0.0961233	+	0.995369i	$0.469356\pi$
$762$	− 7.61301e6i	− 0.474973i
$763$	8.71467e6i	0.541926i
$764$	5.16029e6	0.319846
$765$	0	0
$766$	−1.67359e6	−0.103057
$767$	1.69430e7i	1.03993i
$768$	− 589824.i	− 0.0360844i
$769$	−7.49158e6	−0.456833	−0.228417	−	0.973563i	$-0.573355\pi$
$769$	−7.49158e6	−0.456833	−0.228417	+	0.973563i	$0.573355\pi$
$770$	0	0
$771$	−1.60398e7	−0.971772
$772$	1.05915e7i	0.639609i
$773$	1.12771e7i	0.678813i	0.940640	+	0.339406i	$0.110226\pi$
$773$	1.12771e7i	0.678813i	−0.940640	+	0.339406i	$0.889774\pi$
$774$	5.29824e6	0.317892
$775$	0	0
$776$	−9.32499e6	−0.555897
$777$	3.01325e6i	0.179053i
$778$	− 6.22825e6i	− 0.368907i
$779$	−5.91693e6	−0.349344
$780$	0	0
$781$	2.09556e6	0.122934
$782$	5.67946e6i	0.332116i
$783$	1.24644e6i	0.0726555i
$784$	2.27258e6	0.132047
$785$	0	0
$786$	−1.08665e6	−0.0627386
$787$	− 2.06176e7i	− 1.18659i	−0.804984	−	0.593296i	$-0.797826\pi$
$787$	− 2.06176e7i	− 1.18659i	0.804984	−	0.593296i	$-0.202174\pi$
$788$	1.11700e7i	0.640820i
$789$	2.03151e6	0.116178
$790$	0	0
$791$	−2.55285e6	−0.145072
$792$	− 669791.i	− 0.0379425i
$793$	5.46627e6i	0.308680i
$794$	−1.02860e7	−0.579021
$795$	0	0
$796$	−1.30488e7	−0.729944
$797$	− 1.96946e7i	− 1.09825i	−0.835739	−	0.549126i	$-0.814961\pi$
$797$	− 1.96946e7i	− 1.09825i	0.835739	−	0.549126i	$-0.185039\pi$
$798$	− 2.45001e6i	− 0.136195i
$799$	−3.14568e7	−1.74320
$800$	0	0
$801$	1.36382e6	0.0751063
$802$	1.39446e7i	0.765544i
$803$	− 1.35433e6i	− 0.0741202i
$804$	−9.44276e6	−0.515180
$805$	0	0
$806$	3.80227e7	2.06160
$807$	8.08302e6i	0.436908i
$808$	1.55872e6i	0.0839922i
$809$	−2.78799e7	−1.49768	−0.748840	−	0.662750i	$-0.769389\pi$
$809$	−2.78799e7	−1.49768	−0.748840	+	0.662750i	$0.769389\pi$
$810$	0	0
$811$	7.28842e6	0.389118	0.194559	−	0.980891i	$-0.437672\pi$
$811$	7.28842e6	0.389118	0.194559	+	0.980891i	$0.437672\pi$
$812$	− 2.43610e6i	− 0.129659i
$813$	4.10016e6i	0.217557i
$814$	1.94311e6	0.102787
$815$	0	0
$816$	−2.72055e6	−0.143031
$817$	− 1.24975e7i	− 0.655040i
$818$	− 2.19835e7i	− 1.14872i
$819$	7.51263e6	0.391365
$820$	0	0
$821$	3.42734e7	1.77459	0.887297	−	0.461198i	$-0.152580\pi$
$821$	3.42734e7	1.77459	0.887297	+	0.461198i	$0.152580\pi$
$822$	− 1.04292e7i	− 0.538359i
$823$	3.60257e7i	1.85402i	0.375042	+	0.927008i	$0.377628\pi$
$823$	3.60257e7i	1.85402i	−0.375042	+	0.927008i	$0.622372\pi$
$824$	−8.61179e6	−0.441851
$825$	0	0
$826$	−5.79431e6	−0.295496
$827$	− 3.86553e7i	− 1.96537i	−0.185272	−	0.982687i	$-0.559316\pi$
$827$	− 3.86553e7i	− 1.96537i	0.185272	−	0.982687i	$-0.440684\pi$
$828$	1.55840e6i	0.0789955i
$829$	3.76786e7	1.90418	0.952091	−	0.305816i	$-0.0989293\pi$
$829$	3.76786e7	1.90418	0.952091	+	0.305816i	$0.0989293\pi$
$830$	0	0
$831$	−1.51335e6	−0.0760215
$832$	− 4.26617e6i	− 0.213663i
$833$	− 1.04822e7i	− 0.523408i
$834$	−2.85074e6	−0.141920
$835$	0	0
$836$	−1.57990e6	−0.0781834
$837$	− 6.65323e6i	− 0.328261i
$838$	− 1.96302e6i	− 0.0965636i
$839$	2.44851e7	1.20087	0.600437	−	0.799672i	$-0.294993\pi$
$839$	2.44851e7	1.20087	0.600437	+	0.799672i	$0.294993\pi$
$840$	0	0
$841$	−1.75877e7	−0.857472
$842$	− 1.79889e7i	− 0.874430i
$843$	− 1.46274e7i	− 0.708922i
$844$	−1.35473e7	−0.654633
$845$	0	0
$846$	−8.63149e6	−0.414629
$847$	− 1.28549e7i	− 0.615687i
$848$	− 896370.i	− 0.0428053i
$849$	−9.29047e6	−0.442353
$850$	0	0
$851$	−4.52102e6	−0.214000
$852$	− 2.33555e6i	− 0.110228i
$853$	1.36982e7i	0.644602i	0.946637	+	0.322301i	$0.104456\pi$
$853$	1.36982e7i	0.644602i	−0.946637	+	0.322301i	$0.895544\pi$
$854$	−1.86940e6	−0.0877117
$855$	0	0
$856$	9.34479e6	0.435898
$857$	− 2.74878e6i	− 0.127846i	−0.997955	−	0.0639231i	$-0.979639\pi$
$857$	− 2.74878e6i	− 0.127846i	0.997955	−	0.0639231i	$-0.0203612\pi$
$858$	− 4.84456e6i	− 0.224666i
$859$	−1.24721e7	−0.576708	−0.288354	−	0.957524i	$-0.593108\pi$
$859$	−1.24721e7	−0.576708	−0.288354	+	0.957524i	$0.593108\pi$
$860$	0	0
$861$	6.20486e6	0.285249
$862$	1.46226e7i	0.670279i
$863$	− 1.14518e7i	− 0.523417i	−0.965147	−	0.261708i	$-0.915714\pi$
$863$	− 1.14518e7i	− 0.523417i	0.965147	−	0.261708i	$-0.0842858\pi$
$864$	−746496.	−0.0340207
$865$	0	0
$866$	−7.60295e6	−0.344499
$867$	− 230279.i	− 0.0104041i
$868$	1.30033e7i	0.585807i
$869$	7.05553e6	0.316942
$870$	0	0
$871$	−6.82990e7	−3.05049
$872$	− 6.26328e6i	− 0.278940i
$873$	1.18019e7i	0.524105i
$874$	3.67595e6	0.162776
$875$	0	0
$876$	−1.50943e6	−0.0664590
$877$	− 3.26366e7i	− 1.43287i	−0.697656	−	0.716433i	$-0.745774\pi$
$877$	− 3.26366e7i	− 1.43287i	0.697656	−	0.716433i	$-0.254226\pi$
$878$	− 2.52509e7i	− 1.10546i
$879$	2.28278e7	0.996533
$880$	0	0
$881$	4.07451e7	1.76862	0.884312	−	0.466895i	$-0.154628\pi$
$881$	4.07451e7	1.76862	0.884312	+	0.466895i	$0.154628\pi$
$882$	− 2.87624e6i	− 0.124495i
$883$	1.94427e7i	0.839179i	0.907714	+	0.419589i	$0.137826\pi$
$883$	1.94427e7i	0.839179i	−0.907714	+	0.419589i	$0.862174\pi$
$884$	−1.96776e7	−0.846917
$885$	0	0
$886$	4.46373e6	0.191035
$887$	− 2.53539e7i	− 1.08202i	−0.841016	−	0.541010i	$-0.818042\pi$
$887$	− 2.53539e7i	− 1.08202i	0.841016	−	0.541010i	$-0.181958\pi$
$888$	− 2.16564e6i	− 0.0921623i
$889$	1.88314e7	0.799151
$890$	0	0
$891$	−847704.	−0.0357726
$892$	1.02936e7i	0.433168i
$893$	2.03599e7i	0.854374i
$894$	−9.75181e6	−0.408077
$895$	0	0
$896$	1.45898e6	0.0607126
$897$	1.12718e7i	0.467749i
$898$	8.73228e6i	0.361357i
$899$	−1.56045e7	−0.643948
$900$	0	0
$901$	−4.13448e6	−0.169671
$902$	− 4.00124e6i	− 0.163749i
$903$	1.31056e7i	0.534859i
$904$	1.83475e6	0.0746716
$905$	0	0
$906$	−8.52319e6	−0.344970
$907$	− 2.09341e7i	− 0.844958i	−0.906373	−	0.422479i	$-0.861160\pi$
$907$	− 2.09341e7i	− 0.844958i	0.906373	−	0.422479i	$-0.138840\pi$
$908$	1.33179e7i	0.536068i
$909$	1.97275e6	0.0791886
$910$	0	0
$911$	1.75143e7	0.699191	0.349596	−	0.936901i	$-0.386319\pi$
$911$	1.75143e7	0.699191	0.349596	+	0.936901i	$0.386319\pi$
$912$	1.76083e6i	0.0701022i
$913$	1.05313e7i	0.418124i
$914$	1.04614e7	0.414215
$915$	0	0
$916$	817861.	0.0322063
$917$	− 2.68793e6i	− 0.105559i
$918$	3.44319e6i	0.134851i
$919$	2.89296e6	0.112993	0.0564967	−	0.998403i	$-0.482007\pi$
$919$	2.89296e6	0.112993	0.0564967	+	0.998403i	$0.482007\pi$
$920$	0	0
$921$	−1.42241e7	−0.552556
$922$	2.00378e7i	0.776288i
$923$	− 1.68929e7i	− 0.652681i
$924$	1.65678e6	0.0638390
$925$	0	0
$926$	9.87425e6	0.378422
$927$	1.08993e7i	0.416581i
$928$	1.75083e6i	0.0667383i
$929$	3.00755e7	1.14334	0.571668	−	0.820485i	$-0.306297\pi$
$929$	3.00755e7	1.14334	0.571668	+	0.820485i	$0.306297\pi$
$930$	0	0
$931$	−6.78447e6	−0.256532
$932$	− 1.23145e7i	− 0.464384i
$933$	− 1.53495e6i	− 0.0577285i
$934$	−1.63524e7	−0.613359
$935$	0	0
$936$	−5.39937e6	−0.201444
$937$	− 8.55143e6i	− 0.318192i	−0.987263	−	0.159096i	$-0.949142\pi$
$937$	− 8.55143e6i	− 0.318192i	0.987263	−	0.159096i	$-0.0508580\pi$
$938$	− 2.33575e7i	− 0.866799i
$939$	−1.89898e7	−0.702841
$940$	0	0
$941$	−1.69719e7	−0.624823	−0.312412	−	0.949947i	$-0.601137\pi$
$941$	−1.69719e7	−0.624823	−0.312412	+	0.949947i	$0.601137\pi$
$942$	− 1.67538e7i	− 0.615157i
$943$	9.30966e6i	0.340922i
$944$	4.16440e6	0.152098
$945$	0	0
$946$	8.45125e6	0.307039
$947$	1.06112e7i	0.384493i	0.981347	+	0.192246i	$0.0615772\pi$
$947$	1.06112e7i	0.384493i	−0.981347	+	0.192246i	$0.938423\pi$
$948$	− 7.86353e6i	− 0.284182i
$949$	−1.09177e7	−0.393517
$950$	0	0
$951$	2.19500e7	0.787015
$952$	− 6.72950e6i	− 0.240652i
$953$	3.28727e7i	1.17247i	0.810140	+	0.586237i	$0.199391\pi$
$953$	3.28727e7i	1.17247i	−0.810140	+	0.586237i	$0.800609\pi$
$954$	−1.13447e6	−0.0403572
$955$	0	0
$956$	1.01840e7	0.360390
$957$	1.98821e6i	0.0701749i
$958$	1.03724e7i	0.365146i
$959$	2.57975e7	0.905799
$960$	0	0
$961$	5.46641e7	1.90939
$962$	− 1.56640e7i	− 0.545712i
$963$	− 1.18270e7i	− 0.410969i
$964$	1.02047e7	0.353676
$965$	0	0
$966$	−3.85483e6	−0.132911
$967$	− 3.99873e7i	− 1.37517i	−0.726105	−	0.687584i	$-0.758671\pi$
$967$	− 3.99873e7i	− 1.37517i	0.726105	−	0.687584i	$-0.241329\pi$
$968$	9.23888e6i	0.316906i
$969$	8.12180e6	0.277871
$970$	0	0
$971$	−1.82273e7	−0.620403	−0.310201	−	0.950671i	$-0.600396\pi$
$971$	−1.82273e7	−0.620403	−0.310201	+	0.950671i	$0.600396\pi$
$972$	944784.i	0.0320750i
$973$	− 7.05154e6i	− 0.238782i
$974$	−2.97298e7	−1.00414
$975$	0	0
$976$	1.34355e6	0.0451469
$977$	2.80863e7i	0.941364i	0.882303	+	0.470682i	$0.155992\pi$
$977$	2.80863e7i	0.941364i	−0.882303	+	0.470682i	$0.844008\pi$
$978$	4.58545e6i	0.153297i
$979$	2.17544e6	0.0725421
$980$	0	0
$981$	−7.92697e6	−0.262987
$982$	− 2.78983e7i	− 0.923206i
$983$	− 1.94351e7i	− 0.641510i	−0.947162	−	0.320755i	$-0.896063\pi$
$983$	− 1.94351e7i	− 0.641510i	0.947162	−	0.320755i	$-0.103937\pi$
$984$	−4.45947e6	−0.146823
$985$	0	0
$986$	8.07567e6	0.264537
$987$	− 2.13507e7i	− 0.697620i
$988$	1.27360e7i	0.415090i
$989$	−1.96635e7	−0.639248
$990$	0	0
$991$	−3.58851e7	−1.16073	−0.580363	−	0.814358i	$-0.697089\pi$
$991$	−3.58851e7	−1.16073	−0.580363	+	0.814358i	$0.697089\pi$
$992$	− 9.34555e6i	− 0.301527i
$993$	− 2.62052e7i	− 0.843362i
$994$	5.77718e6	0.185460
$995$	0	0
$996$	1.17374e7	0.374906
$997$	− 2.54523e7i	− 0.810941i	−0.914108	−	0.405471i	$-0.867108\pi$
$997$	− 2.54523e7i	− 0.810941i	0.914108	−	0.405471i	$-0.132892\pi$
$998$	− 1.18522e7i	− 0.376679i
$999$	−2.74089e6	−0.0868915

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	750.6.c.c.499.7		8
5.2	odd	4		750.6.a.a.1.2	✓	4
5.3	odd	4		750.6.a.h.1.3	yes	4
5.4	even	2	inner	750.6.c.c.499.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
750.6.a.a.1.2	✓	4	5.2	odd	4
750.6.a.h.1.3	yes	4	5.3	odd	4
750.6.c.c.499.2		8	5.4	even	2	inner
750.6.c.c.499.7		8	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 750 = 2 \cdot 3 \cdot 5^{3} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	750.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+4.00000i q^{2} -9.00000i q^{3} -16.0000 q^{4} +36.0000 q^{6} +89.0490i q^{7} -64.0000i q^{8} -81.0000 q^{9} +O(q^{10})\)	\(q+4.00000i q^{2} -9.00000i q^{3} -16.0000 q^{4} +36.0000 q^{6} +89.0490i q^{7} -64.0000i q^{8} -81.0000 q^{9} -129.203 q^{11} +144.000i q^{12} +1041.54i q^{13} -356.196 q^{14} +256.000 q^{16} -1180.79i q^{17} -324.000i q^{18} -764.251 q^{19} +801.441 q^{21} -516.814i q^{22} +1202.47i q^{23} -576.000 q^{24} -4166.18 q^{26} +729.000i q^{27} -1424.78i q^{28} +1709.80 q^{29} -9126.51 q^{31} +1024.00i q^{32} +1162.83i q^{33} +4723.17 q^{34} +1296.00 q^{36} +3759.79i q^{37} -3057.01i q^{38} +9373.90 q^{39} +7742.13 q^{41} +3205.76i q^{42} +16352.6i q^{43} +2067.26 q^{44} -4809.87 q^{46} -26640.4i q^{47} -2304.00i q^{48} +8877.27 q^{49} -10627.1 q^{51} -16664.7i q^{52} -3501.44i q^{53} -2916.00 q^{54} +5699.14 q^{56} +6878.26i q^{57} +6839.20i q^{58} +16267.2 q^{59} +5248.23 q^{61} -36506.1i q^{62} -7212.97i q^{63} -4096.00 q^{64} -4651.32 q^{66} +65574.7i q^{67} +18892.7i q^{68} +10822.2 q^{69} -16219.1 q^{71} +5184.00i q^{72} +10482.2i q^{73} -15039.2 q^{74} +12228.0 q^{76} -11505.4i q^{77} +37495.6i q^{78} -54607.9 q^{79} +6561.00 q^{81} +30968.5i q^{82} -81509.4i q^{83} -12823.1 q^{84} -65410.4 q^{86} -15388.2i q^{87} +8269.02i q^{88} -16837.3 q^{89} -92748.5 q^{91} -19239.5i q^{92} +82138.6i q^{93} +106562. q^{94} +9216.00 q^{96} -145703. i q^{97} +35509.1i q^{98} +10465.5 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 128 q^{4} + 288 q^{6} - 648 q^{9}+O(q^{10}) \) 8 * q - 128 * q^4 + 288 * q^6 - 648 * q^9	\( 8 q - 128 q^{4} + 288 q^{6} - 648 q^{9} - 804 q^{11} + 336 q^{14} + 2048 q^{16} + 3320 q^{19} - 756 q^{21} - 4608 q^{24} - 10128 q^{26} + 11400 q^{29} - 26004 q^{31} + 17296 q^{34} + 10368 q^{36} + 22788 q^{39} + 26316 q^{41} + 12864 q^{44} - 64048 q^{46} + 16064 q^{49} - 38916 q^{51} - 23328 q^{54} - 5376 q^{56} + 21660 q^{59} - 97324 q^{61} - 32768 q^{64} - 28944 q^{66} + 144108 q^{69} - 138444 q^{71} + 223776 q^{74} - 53120 q^{76} + 346560 q^{79} + 52488 q^{81} + 12096 q^{84} - 58848 q^{86} - 123600 q^{89} - 283544 q^{91} + 190336 q^{94} + 73728 q^{96} + 65124 q^{99}+O(q^{100}) \) 8 * q - 128 * q^4 + 288 * q^6 - 648 * q^9 - 804 * q^11 + 336 * q^14 + 2048 * q^16 + 3320 * q^19 - 756 * q^21 - 4608 * q^24 - 10128 * q^26 + 11400 * q^29 - 26004 * q^31 + 17296 * q^34 + 10368 * q^36 + 22788 * q^39 + 26316 * q^41 + 12864 * q^44 - 64048 * q^46 + 16064 * q^49 - 38916 * q^51 - 23328 * q^54 - 5376 * q^56 + 21660 * q^59 - 97324 * q^61 - 32768 * q^64 - 28944 * q^66 + 144108 * q^69 - 138444 * q^71 + 223776 * q^74 - 53120 * q^76 + 346560 * q^79 + 52488 * q^81 + 12096 * q^84 - 58848 * q^86 - 123600 * q^89 - 283544 * q^91 + 190336 * q^94 + 73728 * q^96 + 65124 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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