LMFDB - Embedded newform 650.6.b.j.599.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [650,6,Mod(599,650)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(650, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("650.599");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 650 = 2 \cdot 5^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	650.b (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$104.249482878$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} + 2x^{4} - 390x^{3} + 32400x^{2} - 135000x + 281250 $ x^6 - 2x^5 + 2x^4 - 390x^3 + 32400x^2 - 135000*x + 281250
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 130)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			599.3
Root			$-9.94525 + 9.94525i$ of defining polynomial
Character	$\chi$	$=$	650.599
Dual form			650.6.b.j.599.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-4.00000i q^{2} +27.8905i q^{3} -16.0000 q^{4} +111.562 q^{6} -240.426i q^{7} +64.0000i q^{8} -534.880 q^{9} +544.151 q^{11} -446.248i q^{12} -169.000i q^{13} -961.702 q^{14} +256.000 q^{16} +1629.30i q^{17} +2139.52i q^{18} +805.920 q^{19} +6705.59 q^{21} -2176.61i q^{22} -373.152i q^{23} -1784.99 q^{24} -676.000 q^{26} -8140.67i q^{27} +3846.81i q^{28} -1503.62 q^{29} -2200.08 q^{31} -1024.00i q^{32} +15176.7i q^{33} +6517.20 q^{34} +8558.08 q^{36} -13109.1i q^{37} -3223.68i q^{38} +4713.49 q^{39} -17099.9 q^{41} -26822.4i q^{42} +8935.58i q^{43} -8706.42 q^{44} -1492.61 q^{46} +15749.7i q^{47} +7139.97i q^{48} -40997.5 q^{49} -45442.0 q^{51} +2704.00i q^{52} -40379.7i q^{53} -32562.7 q^{54} +15387.2 q^{56} +22477.5i q^{57} +6014.46i q^{58} +47562.1 q^{59} -30280.0 q^{61} +8800.31i q^{62} +128599. i q^{63} -4096.00 q^{64} +60706.6 q^{66} +38769.4i q^{67} -26068.8i q^{68} +10407.4 q^{69} -10519.7 q^{71} -34232.3i q^{72} -1582.12i q^{73} -52436.5 q^{74} -12894.7 q^{76} -130828. i q^{77} -18854.0i q^{78} +6191.23 q^{79} +97071.7 q^{81} +68399.8i q^{82} -37849.2i q^{83} -107289. q^{84} +35742.3 q^{86} -41936.6i q^{87} +34825.7i q^{88} -49151.0 q^{89} -40631.9 q^{91} +5970.43i q^{92} -61361.3i q^{93} +62999.0 q^{94} +28559.9 q^{96} -15654.2i q^{97} +163990. i q^{98} -291056. q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 96 q^{4} + 176 q^{6} - 310 q^{9} + 96 q^{11} - 1872 q^{14} + 1536 q^{16} + 720 q^{19} + 13808 q^{21} - 2816 q^{24} - 4056 q^{26} + 6156 q^{29} - 10776 q^{31} + 12048 q^{34} + 4960 q^{36} + 7436 q^{39}+ \cdots - 700680 q^{99}+O(q^{100}) $ 6 * q - 96 * q^4 + 176 * q^6 - 310 * q^9 + 96 * q^11 - 1872 * q^14 + 1536 * q^16 + 720 * q^19 + 13808 * q^21 - 2816 * q^24 - 4056 * q^26 + 6156 * q^29 - 10776 * q^31 + 12048 * q^34 + 4960 * q^36 + 7436 * q^39 - 31812 * q^41 - 1536 * q^44 + 18960 * q^46 - 15534 * q^49 - 67944 * q^51 - 81344 * q^54 + 29952 * q^56 + 155520 * q^59 + 35964 * q^61 - 24576 * q^64 + 180576 * q^66 + 63720 * q^69 - 9888 * q^71 - 202896 * q^74 - 11520 * q^76 - 84240 * q^79 + 293038 * q^81 - 220928 * q^84 + 314448 * q^86 - 39228 * q^89 - 79092 * q^91 - 142224 * q^94 + 45056 * q^96 - 700680 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/650\mathbb{Z}\right)^\times$.

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

Coefficient data

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

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

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

$2$	− 4.00000i	− 0.707107i
$2$	− 4.00000i	− 0.707107i
$3$	27.8905i	1.78918i	0.446892	+	0.894588i	$0.352531\pi$
$3$	27.8905i	1.78918i	−0.446892	+	0.894588i	$0.647469\pi$
$4$	−16.0000	−0.500000
$5$	0	0
$5$	0	0
$6$	111.562	1.26514
$7$	− 240.426i	− 1.85454i	−0.374397	−	0.927269i	$-0.622150\pi$
$7$	− 240.426i	− 1.85454i	0.374397	−	0.927269i	$-0.377850\pi$
$8$	64.0000i	0.353553i
$9$	−534.880	−2.20115
$10$	0	0
$11$	544.151	1.35593	0.677966	−	0.735093i	$-0.262862\pi$
$11$	544.151	1.35593	0.677966	+	0.735093i	$0.262862\pi$
$12$	− 446.248i	− 0.894588i
$13$	− 169.000i	− 0.277350i
$13$	− 169.000i	− 0.277350i
$14$	−961.702	−1.31136
$15$	0	0
$16$	256.000	0.250000
$17$	1629.30i	1.36735i	0.729788	+	0.683673i	$0.239619\pi$
$17$	1629.30i	1.36735i	−0.729788	+	0.683673i	$0.760381\pi$
$18$	2139.52i	1.55645i
$19$	805.920	0.512162	0.256081	−	0.966655i	$-0.417569\pi$
$19$	805.920	0.512162	0.256081	+	0.966655i	$0.417569\pi$
$20$	0	0
$21$	6705.59	3.31809
$22$	− 2176.61i	− 0.958789i
$23$	− 373.152i	− 0.147084i	−0.997292	−	0.0735421i	$-0.976570\pi$
$23$	− 373.152i	− 0.147084i	0.997292	−	0.0735421i	$-0.0234303\pi$
$24$	−1784.99	−0.632569
$25$	0	0
$26$	−676.000	−0.196116
$27$	− 8140.67i	− 2.14907i
$28$	3846.81i	0.927269i
$29$	−1503.62	−0.332003	−0.166001	−	0.986126i	$-0.553086\pi$
$29$	−1503.62	−0.332003	−0.166001	+	0.986126i	$0.553086\pi$
$30$	0	0
$31$	−2200.08	−0.411182	−0.205591	−	0.978638i	$-0.565912\pi$
$31$	−2200.08	−0.411182	−0.205591	+	0.978638i	$0.565912\pi$
$32$	− 1024.00i	− 0.176777i
$33$	15176.7i	2.42600i
$34$	6517.20	0.966860
$35$	0	0
$36$	8558.08	1.10058
$37$	− 13109.1i	− 1.57423i	−0.616804	−	0.787117i	$-0.711573\pi$
$37$	− 13109.1i	− 1.57423i	0.616804	−	0.787117i	$-0.288427\pi$
$38$	− 3223.68i	− 0.362154i
$39$	4713.49	0.496228
$40$	0	0
$41$	−17099.9	−1.58867	−0.794337	−	0.607477i	$-0.792182\pi$
$41$	−17099.9	−1.58867	−0.794337	+	0.607477i	$0.792182\pi$
$42$	− 26822.4i	− 2.34625i
$43$	8935.58i	0.736973i	0.929633	+	0.368487i	$0.120124\pi$
$43$	8935.58i	0.736973i	−0.929633	+	0.368487i	$0.879876\pi$
$44$	−8706.42	−0.677966
$45$	0	0
$46$	−1492.61	−0.104004
$47$	15749.7i	1.03999i	0.854170	+	0.519995i	$0.174066\pi$
$47$	15749.7i	1.03999i	−0.854170	+	0.519995i	$0.825934\pi$
$48$	7139.97i	0.447294i
$49$	−40997.5	−2.43931
$50$	0	0
$51$	−45442.0	−2.44642
$52$	2704.00i	0.138675i
$53$	− 40379.7i	− 1.97457i	−0.158951	−	0.987286i	$-0.550811\pi$
$53$	− 40379.7i	− 1.97457i	0.158951	−	0.987286i	$-0.449189\pi$
$54$	−32562.7	−1.51962
$55$	0	0
$56$	15387.2	0.655678
$57$	22477.5i	0.916349i
$58$	6014.46i	0.234761i
$59$	47562.1	1.77881	0.889407	−	0.457116i	$-0.151118\pi$
$59$	47562.1	1.77881	0.889407	+	0.457116i	$0.151118\pi$
$60$	0	0
$61$	−30280.0	−1.04191	−0.520956	−	0.853584i	$-0.674424\pi$
$61$	−30280.0	−1.04191	−0.520956	+	0.853584i	$0.674424\pi$
$62$	8800.31i	0.290749i
$63$	128599.i	4.08212i
$64$	−4096.00	−0.125000
$65$	0	0
$66$	60706.6	1.71544
$67$	38769.4i	1.05512i	0.849517	+	0.527561i	$0.176893\pi$
$67$	38769.4i	1.05512i	−0.849517	+	0.527561i	$0.823107\pi$
$68$	− 26068.8i	− 0.683673i
$69$	10407.4	0.263160
$70$	0	0
$71$	−10519.7	−0.247660	−0.123830	−	0.992303i	$-0.539518\pi$
$71$	−10519.7	−0.247660	−0.123830	+	0.992303i	$0.539518\pi$
$72$	− 34232.3i	− 0.778225i
$73$	− 1582.12i	− 0.0347483i	−0.999849	−	0.0173742i	$-0.994469\pi$
$73$	− 1582.12i	− 0.0347483i	0.999849	−	0.0173742i	$-0.00553064\pi$
$74$	−52436.5	−1.11315
$75$	0	0
$76$	−12894.7	−0.256081
$77$	− 130828.i	− 2.51463i
$78$	− 18854.0i	− 0.350886i
$79$	6191.23	0.111612	0.0558058	−	0.998442i	$-0.482227\pi$
$79$	6191.23	0.111612	0.0558058	+	0.998442i	$0.482227\pi$
$80$	0	0
$81$	97071.7	1.64392
$82$	68399.8i	1.12336i
$83$	− 37849.2i	− 0.603062i	−0.953456	−	0.301531i	$-0.902502\pi$
$83$	− 37849.2i	− 0.603062i	0.953456	−	0.301531i	$-0.0974977\pi$
$84$	−107289.	−1.65905
$85$	0	0
$86$	35742.3	0.521119
$87$	− 41936.6i	− 0.594011i
$88$	34825.7i	0.479394i
$89$	−49151.0	−0.657744	−0.328872	−	0.944374i	$-0.606669\pi$
$89$	−49151.0	−0.657744	−0.328872	+	0.944374i	$0.606669\pi$
$90$	0	0
$91$	−40631.9	−0.514356
$92$	5970.43i	0.0735421i
$93$	− 61361.3i	− 0.735677i
$94$	62999.0	0.735383
$95$	0	0
$96$	28559.9	0.316285
$97$	− 15654.2i	− 0.168928i	−0.996427	−	0.0844641i	$-0.973082\pi$
$97$	− 15654.2i	− 0.168928i	0.996427	−	0.0844641i	$-0.0269178\pi$
$98$	163990.i	1.72485i
$99$	−291056.	−2.98461
$100$	0	0
$101$	−138108.	−1.34714	−0.673572	−	0.739122i	$-0.735241\pi$
$101$	−138108.	−1.34714	−0.673572	+	0.739122i	$0.735241\pi$
$102$	181768.i	1.72988i
$103$	47642.1i	0.442484i	0.975219	+	0.221242i	$0.0710111\pi$
$103$	47642.1i	0.442484i	−0.975219	+	0.221242i	$0.928989\pi$
$104$	10816.0	0.0980581
$105$	0	0
$106$	−161519.	−1.39623
$107$	− 186527.i	− 1.57500i	−0.616313	−	0.787501i	$-0.711374\pi$
$107$	− 186527.i	− 1.57500i	0.616313	−	0.787501i	$-0.288626\pi$
$108$	130251.i	1.07454i
$109$	−14407.1	−0.116147	−0.0580736	−	0.998312i	$-0.518496\pi$
$109$	−14407.1	−0.116147	−0.0580736	+	0.998312i	$0.518496\pi$
$110$	0	0
$111$	365620.	2.81658
$112$	− 61548.9i	− 0.463634i
$113$	50802.0i	0.374270i	0.982334	+	0.187135i	$0.0599201\pi$
$113$	50802.0i	0.374270i	−0.982334	+	0.187135i	$0.940080\pi$
$114$	89910.0	0.647957
$115$	0	0
$116$	24057.8	0.166001
$117$	90394.7i	0.610490i
$118$	− 190248.i	− 1.25781i
$119$	391725.	2.53580
$120$	0	0
$121$	135050.	0.838552
$122$	121120.i	0.736742i
$123$	− 476926.i	− 2.84242i
$124$	35201.3	0.205591
$125$	0	0
$126$	514395.	2.88649
$127$	− 182278.i	− 1.00282i	−0.865209	−	0.501411i	$-0.832814\pi$
$127$	− 182278.i	− 1.00282i	0.865209	−	0.501411i	$-0.167186\pi$
$128$	16384.0i	0.0883883i
$129$	−249218.	−1.31858
$130$	0	0
$131$	199714.	1.01679	0.508394	−	0.861124i	$-0.330239\pi$
$131$	199714.	1.01679	0.508394	+	0.861124i	$0.330239\pi$
$132$	− 242826.i	− 1.21300i
$133$	− 193764.i	− 0.949824i
$134$	155078.	0.746083
$135$	0	0
$136$	−104275.	−0.483430
$137$	− 345418.i	− 1.57233i	−0.618017	−	0.786165i	$-0.712064\pi$
$137$	− 345418.i	− 1.57233i	0.618017	−	0.786165i	$-0.287936\pi$
$138$	− 41629.6i	− 0.186082i
$139$	−183088.	−0.803754	−0.401877	−	0.915694i	$-0.631642\pi$
$139$	−183088.	−0.803754	−0.401877	+	0.915694i	$0.631642\pi$
$140$	0	0
$141$	−439268.	−1.86072
$142$	42078.7i	0.175122i
$143$	− 91961.6i	− 0.376068i
$144$	−136929.	−0.550288
$145$	0	0
$146$	−6328.50	−0.0245708
$147$	− 1.14344e6i	− 4.36435i
$148$	209746.i	0.787117i
$149$	187082.	0.690344	0.345172	−	0.938540i	$-0.387821\pi$
$149$	187082.	0.690344	0.345172	+	0.938540i	$0.387821\pi$
$150$	0	0
$151$	−10616.3	−0.0378907	−0.0189454	−	0.999821i	$-0.506031\pi$
$151$	−10616.3	−0.0378907	−0.0189454	+	0.999821i	$0.506031\pi$
$152$	51578.9i	0.181077i
$153$	− 871480.i	− 3.00974i
$154$	−523312.	−1.77811
$155$	0	0
$156$	−75415.9	−0.248114
$157$	− 335884.i	− 1.08753i	−0.839238	−	0.543764i	$-0.816999\pi$
$157$	− 335884.i	− 1.08753i	0.839238	−	0.543764i	$-0.183001\pi$
$158$	− 24764.9i	− 0.0789213i
$159$	1.12621e6	3.53286
$160$	0	0
$161$	−89715.3	−0.272773
$162$	− 388287.i	− 1.16242i
$163$	− 201881.i	− 0.595150i	−0.954698	−	0.297575i	$-0.903822\pi$
$163$	− 201881.i	− 0.595150i	0.954698	−	0.297575i	$-0.0961778\pi$
$164$	273599.	0.794337
$165$	0	0
$166$	−151397.	−0.426429
$167$	− 446050.i	− 1.23764i	−0.785535	−	0.618818i	$-0.787612\pi$
$167$	− 446050.i	− 1.23764i	0.785535	−	0.618818i	$-0.212388\pi$
$168$	429158.i	1.17312i
$169$	−28561.0	−0.0769231
$170$	0	0
$171$	−431070.	−1.12735
$172$	− 142969.i	− 0.368487i
$173$	− 57408.8i	− 0.145835i	−0.997338	−	0.0729177i	$-0.976769\pi$
$173$	− 57408.8i	− 0.145835i	0.997338	−	0.0729177i	$-0.0232311\pi$
$174$	−167746.	−0.420030
$175$	0	0
$176$	139303.	0.338983
$177$	1.32653e6i	3.18261i
$178$	196604.i	0.465095i
$179$	−87439.4	−0.203974	−0.101987	−	0.994786i	$-0.532520\pi$
$179$	−87439.4	−0.203974	−0.101987	+	0.994786i	$0.532520\pi$
$180$	0	0
$181$	21341.6	0.0484206	0.0242103	−	0.999707i	$-0.492293\pi$
$181$	21341.6	0.0484206	0.0242103	+	0.999707i	$0.492293\pi$
$182$	162528.i	0.363705i
$183$	− 844523.i	− 1.86416i
$184$	23881.7	0.0520021
$185$	0	0
$186$	−245445.	−0.520202
$187$	886586.i	1.85403i
$188$	− 251996.i	− 0.519995i
$189$	−1.95723e6	−3.98553
$190$	0	0
$191$	307517.	0.609938	0.304969	−	0.952362i	$-0.401354\pi$
$191$	307517.	0.609938	0.304969	+	0.952362i	$0.401354\pi$
$192$	− 114239.i	− 0.223647i
$193$	− 182409.i	− 0.352496i	−0.984346	−	0.176248i	$-0.943604\pi$
$193$	− 182409.i	− 0.352496i	0.984346	−	0.176248i	$-0.0563960\pi$
$194$	−62616.9	−0.119450
$195$	0	0
$196$	655959.	1.21965
$197$	− 152508.i	− 0.279980i	−0.990153	−	0.139990i	$-0.955293\pi$
$197$	− 152508.i	− 0.279980i	0.990153	−	0.139990i	$-0.0447070\pi$
$198$	1.16422e6i	2.11044i
$199$	586606.	1.05006	0.525029	−	0.851084i	$-0.324054\pi$
$199$	586606.	1.05006	0.525029	+	0.851084i	$0.324054\pi$
$200$	0	0
$201$	−1.08130e6	−1.88780
$202$	552430.i	0.952574i
$203$	361508.i	0.615711i
$204$	727072.	1.22321
$205$	0	0
$206$	190568.	0.312884
$207$	199592.i	0.323755i
$208$	− 43264.0i	− 0.0693375i
$209$	438542.	0.694458
$210$	0	0
$211$	−207041.	−0.320147	−0.160073	−	0.987105i	$-0.551173\pi$
$211$	−207041.	−0.320147	−0.160073	+	0.987105i	$0.551173\pi$
$212$	646075.i	0.987286i
$213$	− 293399.i	− 0.443108i
$214$	−746106.	−1.11369
$215$	0	0
$216$	521003.	0.759812
$217$	528955.i	0.762552i
$218$	57628.2i	0.0821285i
$219$	44126.3	0.0621708
$220$	0	0
$221$	275352.	0.379234
$222$	− 1.46248e6i	− 1.99162i
$223$	1.04889e6i	1.41243i	0.707998	+	0.706214i	$0.249599\pi$
$223$	1.04889e6i	1.41243i	−0.707998	+	0.706214i	$0.750401\pi$
$224$	−246196.	−0.327839
$225$	0	0
$226$	203208.	0.264649
$227$	− 100709.i	− 0.129719i	−0.997894	−	0.0648593i	$-0.979340\pi$
$227$	− 100709.i	− 0.129719i	0.997894	−	0.0648593i	$-0.0206599\pi$
$228$	− 359640.i	− 0.458174i
$229$	−487174.	−0.613896	−0.306948	−	0.951726i	$-0.599308\pi$
$229$	−487174.	−0.613896	−0.306948	+	0.951726i	$0.599308\pi$
$230$	0	0
$231$	3.64886e6	4.49911
$232$	− 96231.4i	− 0.117381i
$233$	832844.i	1.00502i	0.864572	+	0.502509i	$0.167590\pi$
$233$	832844.i	1.00502i	−0.864572	+	0.502509i	$0.832410\pi$
$234$	361579.	0.431681
$235$	0	0
$236$	−760993.	−0.889407
$237$	172677.i	0.199693i
$238$	− 1.56690e6i	− 1.79308i
$239$	−1.34919e6	−1.52785	−0.763923	−	0.645308i	$-0.776729\pi$
$239$	−1.34919e6	−1.52785	−0.763923	+	0.645308i	$0.776729\pi$
$240$	0	0
$241$	−1.66283e6	−1.84419	−0.922096	−	0.386962i	$-0.873524\pi$
$241$	−1.66283e6	−1.84419	−0.922096	+	0.386962i	$0.873524\pi$
$242$	− 540199.i	− 0.592946i
$243$	729193.i	0.792185i
$244$	484479.	0.520956
$245$	0	0
$246$	−1.90770e6	−2.00989
$247$	− 136200.i	− 0.142048i
$248$	− 140805.i	− 0.145375i
$249$	1.05563e6	1.07898
$250$	0	0
$251$	−291273.	−0.291820	−0.145910	−	0.989298i	$-0.546611\pi$
$251$	−291273.	−0.291820	−0.145910	+	0.989298i	$0.546611\pi$
$252$	− 2.05758e6i	− 2.04106i
$253$	− 203051.i	− 0.199436i
$254$	−729110.	−0.709102
$255$	0	0
$256$	65536.0	0.0625000
$257$	− 548204.i	− 0.517737i	−0.965913	−	0.258868i	$-0.916650\pi$
$257$	− 548204.i	− 0.517737i	0.965913	−	0.258868i	$-0.0833496\pi$
$258$	996872.i	0.932374i
$259$	−3.15177e6	−2.91947
$260$	0	0
$261$	804253.	0.730788
$262$	− 798857.i	− 0.718978i
$263$	− 818412.i	− 0.729596i	−0.931087	−	0.364798i	$-0.881138\pi$
$263$	− 818412.i	− 0.729596i	0.931087	−	0.364798i	$-0.118862\pi$
$264$	−971306.	−0.857721
$265$	0	0
$266$	−775055.	−0.671627
$267$	− 1.37085e6i	− 1.17682i
$268$	− 620311.i	− 0.527561i
$269$	939455.	0.791581	0.395790	−	0.918341i	$-0.370471\pi$
$269$	939455.	0.791581	0.395790	+	0.918341i	$0.370471\pi$
$270$	0	0
$271$	−1.52416e6	−1.26069	−0.630345	−	0.776315i	$-0.717087\pi$
$271$	−1.52416e6	−1.26069	−0.630345	+	0.776315i	$0.717087\pi$
$272$	417101.i	0.341837i
$273$	− 1.13324e6i	− 0.920274i
$274$	−1.38167e6	−1.11180
$275$	0	0
$276$	−166518.	−0.131580
$277$	− 439704.i	− 0.344319i	−0.985069	−	0.172159i	$-0.944926\pi$
$277$	− 439704.i	− 0.344319i	0.985069	−	0.172159i	$-0.0550744\pi$
$278$	732352.i	0.568340i
$279$	1.17678e6	0.905074
$280$	0	0
$281$	−592033.	−0.447280	−0.223640	−	0.974672i	$-0.571794\pi$
$281$	−592033.	−0.447280	−0.223640	+	0.974672i	$0.571794\pi$
$282$	1.75707e6i	1.31573i
$283$	777226.i	0.576875i	0.957499	+	0.288437i	$0.0931357\pi$
$283$	777226.i	0.576875i	−0.957499	+	0.288437i	$0.906864\pi$
$284$	168315.	0.123830
$285$	0	0
$286$	−367846.	−0.265920
$287$	4.11126e6i	2.94626i
$288$	547717.i	0.389112i
$289$	−1.23476e6	−0.869637
$290$	0	0
$291$	436604.	0.302242
$292$	25314.0i	0.0173742i
$293$	− 609025.i	− 0.414444i	−0.978294	−	0.207222i	$-0.933558\pi$
$293$	− 609025.i	− 0.414444i	0.978294	−	0.207222i	$-0.0664423\pi$
$294$	−4.57376e6	−3.08606
$295$	0	0
$296$	838984.	0.556576
$297$	− 4.42976e6i	− 2.91400i
$298$	− 748326.i	− 0.488147i
$299$	−63062.7	−0.0407938
$300$	0	0
$301$	2.14834e6	1.36674
$302$	42465.4i	0.0267928i
$303$	− 3.85189e6i	− 2.41028i
$304$	206315.	0.128041
$305$	0	0
$306$	−3.48592e6	−2.12821
$307$	− 2.66761e6i	− 1.61538i	−0.589604	−	0.807692i	$-0.700716\pi$
$307$	− 2.66761e6i	− 1.61538i	0.589604	−	0.807692i	$-0.299284\pi$
$308$	2.09325e6i	1.25731i
$309$	−1.32876e6	−0.791682
$310$	0	0
$311$	−1.62320e6	−0.951634	−0.475817	−	0.879544i	$-0.657847\pi$
$311$	−1.62320e6	−0.951634	−0.475817	+	0.879544i	$0.657847\pi$
$312$	301664.i	0.175443i
$313$	− 592115.i	− 0.341622i	−0.985304	−	0.170811i	$-0.945361\pi$
$313$	− 592115.i	− 0.341622i	0.985304	−	0.170811i	$-0.0546387\pi$
$314$	−1.34354e6	−0.768998
$315$	0	0
$316$	−99059.7	−0.0558058
$317$	− 2.61036e6i	− 1.45899i	−0.683987	−	0.729494i	$-0.739756\pi$
$317$	− 2.61036e6i	− 1.45899i	0.683987	−	0.729494i	$-0.260244\pi$
$318$	− 4.50484e6i	− 2.49811i
$319$	−818194.	−0.450173
$320$	0	0
$321$	5.20232e6	2.81796
$322$	358861.i	0.192880i
$323$	1.31308e6i	0.700304i
$324$	−1.55315e6	−0.821959
$325$	0	0
$326$	−807524.	−0.420834
$327$	− 401820.i	− 0.207808i
$328$	− 1.09440e6i	− 0.561681i
$329$	3.78664e6	1.92870
$330$	0	0
$331$	818785.	0.410771	0.205386	−	0.978681i	$-0.434155\pi$
$331$	818785.	0.410771	0.205386	+	0.978681i	$0.434155\pi$
$332$	605588.i	0.301531i
$333$	7.01180e6i	3.46513i
$334$	−1.78420e6	−0.875140
$335$	0	0
$336$	1.71663e6	0.829523
$337$	3.28910e6i	1.57762i	0.614638	+	0.788809i	$0.289302\pi$
$337$	3.28910e6i	1.57762i	−0.614638	+	0.788809i	$0.710698\pi$
$338$	114244.i	0.0543928i
$339$	−1.41689e6	−0.669634
$340$	0	0
$341$	−1.19718e6	−0.557535
$342$	1.72428e6i	0.797155i
$343$	5.81600e6i	2.66925i
$344$	−571877.	−0.260559
$345$	0	0
$346$	−229635.	−0.103121
$347$	139108.i	0.0620195i	0.999519	+	0.0310098i	$0.00987229\pi$
$347$	139108.i	0.0620195i	−0.999519	+	0.0310098i	$0.990128\pi$
$348$	670985.i	0.297006i
$349$	−1.41781e6	−0.623097	−0.311549	−	0.950230i	$-0.600848\pi$
$349$	−1.41781e6	−0.623097	−0.311549	+	0.950230i	$0.600848\pi$
$350$	0	0
$351$	−1.37577e6	−0.596045
$352$	− 557211.i	− 0.239697i
$353$	− 2.45675e6i	− 1.04936i	−0.851300	−	0.524679i	$-0.824185\pi$
$353$	− 2.45675e6i	− 1.04936i	0.851300	−	0.524679i	$-0.175815\pi$
$354$	5.30612e6	2.25045
$355$	0	0
$356$	786416.	0.328872
$357$	1.09254e7i	4.53698i
$358$	349758.i	0.144231i
$359$	2.88601e6	1.18185	0.590924	−	0.806727i	$-0.298763\pi$
$359$	2.88601e6	1.18185	0.590924	+	0.806727i	$0.298763\pi$
$360$	0	0
$361$	−1.82659e6	−0.737690
$362$	− 85366.4i	− 0.0342386i
$363$	3.76660e6i	1.50032i
$364$	650111.	0.257178
$365$	0	0
$366$	−3.37809e6	−1.31816
$367$	− 2.83587e6i	− 1.09906i	−0.835474	−	0.549530i	$-0.814807\pi$
$367$	− 2.83587e6i	− 1.09906i	0.835474	−	0.549530i	$-0.185193\pi$
$368$	− 95526.9i	− 0.0367711i
$369$	9.14641e6	3.49691
$370$	0	0
$371$	−9.70831e6	−3.66192
$372$	981781.i	0.367838i
$373$	− 5.02691e6i	− 1.87081i	−0.353583	−	0.935403i	$-0.615037\pi$
$373$	− 5.02691e6i	− 1.87081i	0.353583	−	0.935403i	$-0.384963\pi$
$374$	3.54634e6	1.31100
$375$	0	0
$376$	−1.00798e6	−0.367692
$377$	254111.i	0.0920810i
$378$	7.82891e6i	2.81820i
$379$	−2.14892e6	−0.768462	−0.384231	−	0.923237i	$-0.625533\pi$
$379$	−2.14892e6	−0.768462	−0.384231	+	0.923237i	$0.625533\pi$
$380$	0	0
$381$	5.08381e6	1.79423
$382$	− 1.23007e6i	− 0.431291i
$383$	− 832971.i	− 0.290157i	−0.989420	−	0.145078i	$-0.953657\pi$
$383$	− 832971.i	− 0.290157i	0.989420	−	0.145078i	$-0.0463434\pi$
$384$	−456958.	−0.158142
$385$	0	0
$386$	−729637.	−0.249252
$387$	− 4.77946e6i	− 1.62219i
$388$	250468.i	0.0844641i
$389$	725243.	0.243002	0.121501	−	0.992591i	$-0.461229\pi$
$389$	725243.	0.243002	0.121501	+	0.992591i	$0.461229\pi$
$390$	0	0
$391$	607976.	0.201115
$392$	− 2.62384e6i	− 0.862426i
$393$	5.57013e6i	1.81921i
$394$	−610032.	−0.197976
$395$	0	0
$396$	4.65689e6	1.49231
$397$	2.42654e6i	0.772701i	0.922352	+	0.386351i	$0.126265\pi$
$397$	2.42654e6i	0.772701i	−0.922352	+	0.386351i	$0.873735\pi$
$398$	− 2.34642e6i	− 0.742504i
$399$	5.40417e6	1.69940
$400$	0	0
$401$	−2.14376e6	−0.665758	−0.332879	−	0.942970i	$-0.608020\pi$
$401$	−2.14376e6	−0.665758	−0.332879	+	0.942970i	$0.608020\pi$
$402$	4.32519e6i	1.33487i
$403$	371813.i	0.114041i
$404$	2.20972e6	0.673572
$405$	0	0
$406$	1.44603e6	0.435374
$407$	− 7.13335e6i	− 2.13455i
$408$	− 2.90829e6i	− 0.864942i
$409$	3.66144e6	1.08229	0.541145	−	0.840929i	$-0.317991\pi$
$409$	3.66144e6	1.08229	0.541145	+	0.840929i	$0.317991\pi$
$410$	0	0
$411$	9.63388e6	2.81317
$412$	− 762273.i	− 0.221242i
$413$	− 1.14351e7i	− 3.29888i
$414$	798366.	0.228929
$415$	0	0
$416$	−173056.	−0.0490290
$417$	− 5.10642e6i	− 1.43806i
$418$	− 1.75417e6i	− 0.491056i
$419$	−3.03786e6	−0.845341	−0.422671	−	0.906283i	$-0.638907\pi$
$419$	−3.03786e6	−0.845341	−0.422671	+	0.906283i	$0.638907\pi$
$420$	0	0
$421$	−1.88663e6	−0.518777	−0.259389	−	0.965773i	$-0.583521\pi$
$421$	−1.88663e6	−0.518777	−0.259389	+	0.965773i	$0.583521\pi$
$422$	828162.i	0.226378i
$423$	− 8.42422e6i	− 2.28917i
$424$	2.58430e6	0.698117
$425$	0	0
$426$	−1.17360e6	−0.313325
$427$	7.28008e6i	1.93226i
$428$	2.98442e6i	0.787501i
$429$	2.56485e6	0.672852
$430$	0	0
$431$	717193.	0.185970	0.0929850	−	0.995668i	$-0.470359\pi$
$431$	717193.	0.185970	0.0929850	+	0.995668i	$0.470359\pi$
$432$	− 2.08401e6i	− 0.537268i
$433$	− 2.64569e6i	− 0.678140i	−0.940761	−	0.339070i	$-0.889888\pi$
$433$	− 2.64569e6i	− 0.678140i	0.940761	−	0.339070i	$-0.110112\pi$
$434$	2.11582e6	0.539206
$435$	0	0
$436$	230513.	0.0580736
$437$	− 300731.i	− 0.0753310i
$438$	− 176505.i	− 0.0439614i
$439$	−4.47973e6	−1.10941	−0.554704	−	0.832048i	$-0.687168\pi$
$439$	−4.47973e6	−1.10941	−0.554704	+	0.832048i	$0.687168\pi$
$440$	0	0
$441$	2.19287e7	5.36929
$442$	− 1.10141e6i	− 0.268159i
$443$	− 4.30731e6i	− 1.04279i	−0.853315	−	0.521395i	$-0.825412\pi$
$443$	− 4.30731e6i	− 1.04279i	0.853315	−	0.521395i	$-0.174588\pi$
$444$	−5.84992e6	−1.40829
$445$	0	0
$446$	4.19555e6	0.998738
$447$	5.21780e6i	1.23515i
$448$	984783.i	0.231817i
$449$	25272.9	0.00591615	0.00295808	−	0.999996i	$-0.499058\pi$
$449$	25272.9	0.00591615	0.00295808	+	0.999996i	$0.499058\pi$
$450$	0	0
$451$	−9.30495e6	−2.15413
$452$	− 812832.i	− 0.187135i
$453$	− 296095.i	− 0.0677932i
$454$	−402835.	−0.0917250
$455$	0	0
$456$	−1.43856e6	−0.323978
$457$	1.14360e6i	0.256143i	0.991765	+	0.128072i	$0.0408787\pi$
$457$	1.14360e6i	0.256143i	−0.991765	+	0.128072i	$0.959121\pi$
$458$	1.94869e6i	0.434090i
$459$	1.32636e7	2.93853
$460$	0	0
$461$	−8.34911e6	−1.82973	−0.914867	−	0.403755i	$-0.867705\pi$
$461$	−8.34911e6	−1.82973	−0.914867	+	0.403755i	$0.867705\pi$
$462$	− 1.45954e7i	− 3.18135i
$463$	3.41596e6i	0.740561i	0.928920	+	0.370280i	$0.120738\pi$
$463$	3.41596e6i	0.740561i	−0.928920	+	0.370280i	$0.879262\pi$
$464$	−384925.	−0.0830007
$465$	0	0
$466$	3.33138e6	0.710655
$467$	5.67105e6i	1.20329i	0.798763	+	0.601646i	$0.205488\pi$
$467$	5.67105e6i	1.20329i	−0.798763	+	0.601646i	$0.794512\pi$
$468$	− 1.44632e6i	− 0.305245i
$469$	9.32116e6	1.95676
$470$	0	0
$471$	9.36797e6	1.94578
$472$	3.04397e6i	0.628906i
$473$	4.86231e6i	0.999286i
$474$	690706.	0.141204
$475$	0	0
$476$	−6.26760e6	−1.26790
$477$	2.15983e7i	4.34633i
$478$	5.39677e6i	1.08035i
$479$	6.16340e6	1.22739	0.613694	−	0.789544i	$-0.289683\pi$
$479$	6.16340e6	1.22739	0.613694	+	0.789544i	$0.289683\pi$
$480$	0	0
$481$	−2.21544e6	−0.436614
$482$	6.65133e6i	1.30404i
$483$	− 2.50220e6i	− 0.488039i
$484$	−2.16080e6	−0.419276
$485$	0	0
$486$	2.91677e6	0.560160
$487$	− 7.32701e6i	− 1.39992i	−0.714180	−	0.699962i	$-0.753200\pi$
$487$	− 7.32701e6i	− 1.39992i	0.714180	−	0.699962i	$-0.246800\pi$
$488$	− 1.93792e6i	− 0.368371i
$489$	5.63056e6	1.06483
$490$	0	0
$491$	8.89739e6	1.66555	0.832777	−	0.553608i	$-0.186750\pi$
$491$	8.89739e6	1.66555	0.832777	+	0.553608i	$0.186750\pi$
$492$	7.63081e6i	1.42121i
$493$	− 2.44984e6i	− 0.453963i
$494$	−544802.	−0.100443
$495$	0	0
$496$	−563220.	−0.102795
$497$	2.52920e6i	0.459295i
$498$	− 4.22254e6i	− 0.762957i
$499$	−3.02237e6	−0.543371	−0.271686	−	0.962386i	$-0.587581\pi$
$499$	−3.02237e6	−0.543371	−0.271686	+	0.962386i	$0.587581\pi$
$500$	0	0
$501$	1.24406e7	2.21435
$502$	1.16509e6i	0.206348i
$503$	7.48799e6i	1.31961i	0.751437	+	0.659805i	$0.229361\pi$
$503$	7.48799e6i	1.31961i	−0.751437	+	0.659805i	$0.770639\pi$
$504$	−8.23032e6	−1.44325
$505$	0	0
$506$	−812205.	−0.141023
$507$	− 796581.i	− 0.137629i
$508$	2.91644e6i	0.501411i
$509$	−4.87997e6	−0.834878	−0.417439	−	0.908705i	$-0.637072\pi$
$509$	−4.87997e6	−0.834878	−0.417439	+	0.908705i	$0.637072\pi$
$510$	0	0
$511$	−380383.	−0.0644420
$512$	− 262144.i	− 0.0441942i
$513$	− 6.56073e6i	− 1.10067i
$514$	−2.19281e6	−0.366095
$515$	0	0
$516$	3.98749e6	0.659288
$517$	8.57024e6i	1.41015i
$518$	1.26071e7i	2.06438i
$519$	1.60116e6	0.260925
$520$	0	0
$521$	−3.79679e6	−0.612805	−0.306403	−	0.951902i	$-0.599125\pi$
$521$	−3.79679e6	−0.612805	−0.306403	+	0.951902i	$0.599125\pi$
$522$	− 3.21701e6i	− 0.516745i
$523$	5.66643e6i	0.905848i	0.891549	+	0.452924i	$0.149619\pi$
$523$	5.66643e6i	0.905848i	−0.891549	+	0.452924i	$0.850381\pi$
$524$	−3.19543e6	−0.508394
$525$	0	0
$526$	−3.27365e6	−0.515902
$527$	− 3.58459e6i	− 0.562228i
$528$	3.88522e6i	0.606500i
$529$	6.29710e6	0.978366
$530$	0	0
$531$	−2.54400e7	−3.91544
$532$	3.10022e6i	0.474912i
$533$	2.88989e6i	0.440619i
$534$	−5.48338e6	−0.832138
$535$	0	0
$536$	−2.48124e6	−0.373042
$537$	− 2.43873e6i	− 0.364945i
$538$	− 3.75782e6i	− 0.559732i
$539$	−2.23088e7	−3.30754
$540$	0	0
$541$	1.15648e7	1.69881	0.849403	−	0.527745i	$-0.176962\pi$
$541$	1.15648e7	1.69881	0.849403	+	0.527745i	$0.176962\pi$
$542$	6.09665e6i	0.891442i
$543$	595228.i	0.0866330i
$544$	1.66840e6	0.241715
$545$	0	0
$546$	−4.53298e6	−0.650732
$547$	− 1.28104e7i	− 1.83060i	−0.402776	−	0.915299i	$-0.631955\pi$
$547$	− 1.28104e7i	− 1.83060i	0.402776	−	0.915299i	$-0.368045\pi$
$548$	5.52669e6i	0.786165i
$549$	1.61961e7	2.29340
$550$	0	0
$551$	−1.21179e6	−0.170039
$552$	666073.i	0.0930410i
$553$	− 1.48853e6i	− 0.206988i
$554$	−1.75881e6	−0.243470
$555$	0	0
$556$	2.92941e6	0.401877
$557$	3.51737e6i	0.480374i	0.970727	+	0.240187i	$0.0772088\pi$
$557$	3.51737e6i	0.480374i	−0.970727	+	0.240187i	$0.922791\pi$
$558$	− 4.70711e6i	− 0.639984i
$559$	1.51011e6	0.204400
$560$	0	0
$561$	−2.47273e7	−3.31719
$562$	2.36813e6i	0.316275i
$563$	− 9.45258e6i	− 1.25684i	−0.777875	−	0.628419i	$-0.783702\pi$
$563$	− 9.45258e6i	− 1.25684i	0.777875	−	0.628419i	$-0.216298\pi$
$564$	7.02829e6	0.930362
$565$	0	0
$566$	3.10891e6	0.407912
$567$	− 2.33385e7i	− 3.04871i
$568$	− 673259.i	− 0.0875611i
$569$	−1.11315e7	−1.44136	−0.720679	−	0.693269i	$-0.756170\pi$
$569$	−1.11315e7	−1.44136	−0.720679	+	0.693269i	$0.756170\pi$
$570$	0	0
$571$	4.92935e6	0.632702	0.316351	−	0.948642i	$-0.397542\pi$
$571$	4.92935e6	0.632702	0.316351	+	0.948642i	$0.397542\pi$
$572$	1.47139e6i	0.188034i
$573$	8.57680e6i	1.09129i
$574$	1.64450e7	2.08332
$575$	0	0
$576$	2.19087e6	0.275144
$577$	1.09437e7i	1.36843i	0.729279	+	0.684217i	$0.239856\pi$
$577$	1.09437e7i	1.36843i	−0.729279	+	0.684217i	$0.760144\pi$
$578$	4.93904e6i	0.614926i
$579$	5.08749e6	0.630677
$580$	0	0
$581$	−9.09993e6	−1.11840
$582$	− 1.74642e6i	− 0.213718i
$583$	− 2.19727e7i	− 2.67739i
$584$	101256.	0.0122854
$585$	0	0
$586$	−2.43610e6	−0.293056
$587$	7.96444e6i	0.954025i	0.878896	+	0.477013i	$0.158280\pi$
$587$	7.96444e6i	0.954025i	−0.878896	+	0.477013i	$0.841720\pi$
$588$	1.82950e7i	2.18218i
$589$	−1.77309e6	−0.210592
$590$	0	0
$591$	4.25352e6	0.500934
$592$	− 3.35593e6i	− 0.393558i
$593$	1.19793e7i	1.39892i	0.714671	+	0.699460i	$0.246576\pi$
$593$	1.19793e7i	1.39892i	−0.714671	+	0.699460i	$0.753424\pi$
$594$	−1.77190e7	−2.06051
$595$	0	0
$596$	−2.99330e6	−0.345172
$597$	1.63607e7i	1.87874i
$598$	252251.i	0.0288456i
$599$	−7.45663e6	−0.849132	−0.424566	−	0.905397i	$-0.639573\pi$
$599$	−7.45663e6	−0.849132	−0.424566	+	0.905397i	$0.639573\pi$
$600$	0	0
$601$	−1.04196e7	−1.17670	−0.588351	−	0.808606i	$-0.700223\pi$
$601$	−1.04196e7	−1.17670	−0.588351	+	0.808606i	$0.700223\pi$
$602$	− 8.59337e6i	− 0.966434i
$603$	− 2.07370e7i	− 2.32248i
$604$	169862.	0.0189454
$605$	0	0
$606$	−1.54075e7	−1.70432
$607$	571939.i	0.0630054i	0.999504	+	0.0315027i	$0.0100293\pi$
$607$	571939.i	0.0630054i	−0.999504	+	0.0315027i	$0.989971\pi$
$608$	− 825262.i	− 0.0905384i
$609$	−1.00826e7	−1.10162
$610$	0	0
$611$	2.66171e6	0.288441
$612$	1.39437e7i	1.50487i
$613$	1.61323e7i	1.73398i	0.498323	+	0.866991i	$0.333949\pi$
$613$	1.61323e7i	1.73398i	−0.498323	+	0.866991i	$0.666051\pi$
$614$	−1.06704e7	−1.14225
$615$	0	0
$616$	8.37299e6	0.889055
$617$	− 6.39133e6i	− 0.675894i	−0.941165	−	0.337947i	$-0.890268\pi$
$617$	− 6.39133e6i	− 0.675894i	0.941165	−	0.337947i	$-0.109732\pi$
$618$	5.31505e6i	0.559804i
$619$	−1.27062e7	−1.33287	−0.666435	−	0.745563i	$-0.732181\pi$
$619$	−1.27062e7	−1.33287	−0.666435	+	0.745563i	$0.732181\pi$
$620$	0	0
$621$	−3.03771e6	−0.316095
$622$	6.49278e6i	0.672907i
$623$	1.18172e7i	1.21981i
$624$	1.20665e6	0.124057
$625$	0	0
$626$	−2.36846e6	−0.241563
$627$	1.22312e7i	1.24251i
$628$	5.37414e6i	0.543764i
$629$	2.13587e7	2.15252
$630$	0	0
$631$	−694354.	−0.0694237	−0.0347118	−	0.999397i	$-0.511051\pi$
$631$	−694354.	−0.0694237	−0.0347118	+	0.999397i	$0.511051\pi$
$632$	396239.i	0.0394607i
$633$	− 5.77447e6i	− 0.572799i
$634$	−1.04414e7	−1.03166
$635$	0	0
$636$	−1.80193e7	−1.76643
$637$	6.92857e6i	0.676542i
$638$	3.27278e6i	0.318321i
$639$	5.62676e6	0.545138
$640$	0	0
$641$	−8.71863e6	−0.838114	−0.419057	−	0.907960i	$-0.637639\pi$
$641$	−8.71863e6	−0.838114	−0.419057	+	0.907960i	$0.637639\pi$
$642$	− 2.08093e7i	− 1.99260i
$643$	− 5.98737e6i	− 0.571096i	−0.958364	−	0.285548i	$-0.907824\pi$
$643$	− 5.98737e6i	− 0.571096i	0.958364	−	0.285548i	$-0.0921756\pi$
$644$	1.43544e6	0.136387
$645$	0	0
$646$	5.25234e6	0.495189
$647$	− 1.09581e7i	− 1.02914i	−0.857448	−	0.514571i	$-0.827951\pi$
$647$	− 1.09581e7i	− 1.02914i	0.857448	−	0.514571i	$-0.172049\pi$
$648$	6.21259e6i	0.581212i
$649$	2.58810e7	2.41195
$650$	0	0
$651$	−1.47528e7	−1.36434
$652$	3.23009e6i	0.297575i
$653$	− 6.47503e6i	− 0.594236i	−0.954841	−	0.297118i	$-0.903975\pi$
$653$	− 6.47503e6i	− 0.594236i	0.954841	−	0.297118i	$-0.0960255\pi$
$654$	−1.60728e6	−0.146942
$655$	0	0
$656$	−4.37758e6	−0.397169
$657$	846247.i	0.0764863i
$658$	− 1.51466e7i	− 1.36380i
$659$	−501723.	−0.0450039	−0.0225020	−	0.999747i	$-0.507163\pi$
$659$	−501723.	−0.0450039	−0.0225020	+	0.999747i	$0.507163\pi$
$660$	0	0
$661$	4.00659e6	0.356674	0.178337	−	0.983969i	$-0.442928\pi$
$661$	4.00659e6	0.356674	0.178337	+	0.983969i	$0.442928\pi$
$662$	− 3.27514e6i	− 0.290459i
$663$	7.67969e6i	0.678516i
$664$	2.42235e6	0.213215
$665$	0	0
$666$	2.80472e7	2.45021
$667$	561077.i	0.0488324i
$668$	7.13681e6i	0.618818i
$669$	−2.92540e7	−2.52708
$670$	0	0
$671$	−1.64769e7	−1.41276
$672$	− 6.86652e6i	− 0.586562i
$673$	− 8.17473e6i	− 0.695722i	−0.937546	−	0.347861i	$-0.886908\pi$
$673$	− 8.17473e6i	− 0.695722i	0.937546	−	0.347861i	$-0.113092\pi$
$674$	1.31564e7	1.11554
$675$	0	0
$676$	456976.	0.0384615
$677$	1.00935e6i	0.0846392i	0.999104	+	0.0423196i	$0.0134748\pi$
$677$	1.00935e6i	0.0846392i	−0.999104	+	0.0423196i	$0.986525\pi$
$678$	5.66757e6i	0.473503i
$679$	−3.76368e6	−0.313284
$680$	0	0
$681$	2.80882e6	0.232090
$682$	4.78870e6i	0.394237i
$683$	− 1.99189e7i	− 1.63386i	−0.576739	−	0.816928i	$-0.695675\pi$
$683$	− 1.99189e7i	− 1.63386i	0.576739	−	0.816928i	$-0.304325\pi$
$684$	6.89712e6	0.563674
$685$	0	0
$686$	2.32640e7	1.88745
$687$	− 1.35875e7i	− 1.09837i
$688$	2.28751e6i	0.184243i
$689$	−6.82417e6	−0.547648
$690$	0	0
$691$	−677407.	−0.0539703	−0.0269851	−	0.999636i	$-0.508591\pi$
$691$	−677407.	−0.0539703	−0.0269851	+	0.999636i	$0.508591\pi$
$692$	918541.i	0.0729177i
$693$	6.99772e7i	5.53508i
$694$	556432.	0.0438544
$695$	0	0
$696$	2.68394e6	0.210015
$697$	− 2.78609e7i	− 2.17227i
$698$	5.67126e6i	0.440596i
$699$	−2.32284e7	−1.79815
$700$	0	0
$701$	1.04542e7	0.803518	0.401759	−	0.915745i	$-0.368399\pi$
$701$	1.04542e7	0.803518	0.401759	+	0.915745i	$0.368399\pi$
$702$	5.50310e6i	0.421468i
$703$	− 1.05649e7i	− 0.806263i
$704$	−2.22884e6	−0.169492
$705$	0	0
$706$	−9.82700e6	−0.742009
$707$	3.32046e7i	2.49833i
$708$	− 2.12245e7i	− 1.59131i
$709$	2.25125e7	1.68193	0.840964	−	0.541091i	$-0.181989\pi$
$709$	2.25125e7	1.68193	0.840964	+	0.541091i	$0.181989\pi$
$710$	0	0
$711$	−3.31157e6	−0.245674
$712$	− 3.14566e6i	− 0.232548i
$713$	820964.i	0.0604784i
$714$	4.37017e7	3.20813
$715$	0	0
$716$	1.39903e6	0.101987
$717$	− 3.76297e7i	− 2.73358i
$718$	− 1.15440e7i	− 0.835693i
$719$	2.77678e6	0.200317	0.100159	−	0.994971i	$-0.468065\pi$
$719$	2.77678e6	0.200317	0.100159	+	0.994971i	$0.468065\pi$
$720$	0	0
$721$	1.14544e7	0.820603
$722$	7.30637e6i	0.521625i
$723$	− 4.63772e7i	− 3.29958i
$724$	−341466.	−0.0242103
$725$	0	0
$726$	1.50664e7	1.06089
$727$	2.76872e6i	0.194287i	0.995270	+	0.0971434i	$0.0309705\pi$
$727$	2.76872e6i	0.194287i	−0.995270	+	0.0971434i	$0.969029\pi$
$728$	− 2.60044e6i	− 0.181852i
$729$	3.25086e6	0.226558
$730$	0	0
$731$	−1.45587e7	−1.00770
$732$	1.35124e7i	0.932081i
$733$	1.80998e7i	1.24427i	0.782911	+	0.622133i	$0.213734\pi$
$733$	1.80998e7i	1.24427i	−0.782911	+	0.622133i	$0.786266\pi$
$734$	−1.13435e7	−0.777152
$735$	0	0
$736$	−382108.	−0.0260011
$737$	2.10964e7i	1.43067i
$738$	− 3.65857e7i	− 2.47269i
$739$	3.57016e6	0.240479	0.120239	−	0.992745i	$-0.461634\pi$
$739$	3.57016e6	0.240479	0.120239	+	0.992745i	$0.461634\pi$
$740$	0	0
$741$	3.79870e6	0.254149
$742$	3.88332e7i	2.58937i
$743$	1.73285e7i	1.15157i	0.817603	+	0.575783i	$0.195303\pi$
$743$	1.73285e7i	1.15157i	−0.817603	+	0.575783i	$0.804697\pi$
$744$	3.92712e6	0.260101
$745$	0	0
$746$	−2.01076e7	−1.32286
$747$	2.02448e7i	1.32743i
$748$	− 1.41854e7i	− 0.927015i
$749$	−4.48457e7	−2.92090
$750$	0	0
$751$	3.04634e6	0.197096	0.0985480	−	0.995132i	$-0.468580\pi$
$751$	3.04634e6	0.197096	0.0985480	+	0.995132i	$0.468580\pi$
$752$	4.03193e6i	0.259997i
$753$	− 8.12374e6i	− 0.522118i
$754$	1.01644e6	0.0651111
$755$	0	0
$756$	3.13156e7	1.99277
$757$	− 8.98398e6i	− 0.569808i	−0.958556	−	0.284904i	$-0.908038\pi$
$757$	− 8.98398e6i	− 0.569808i	0.958556	−	0.284904i	$-0.0919618\pi$
$758$	8.59568e6i	0.543384i
$759$	5.66320e6	0.356827
$760$	0	0
$761$	1.31757e7	0.824731	0.412366	−	0.911018i	$-0.364703\pi$
$761$	1.31757e7	0.824731	0.412366	+	0.911018i	$0.364703\pi$
$762$	− 2.03352e7i	− 1.26871i
$763$	3.46382e6i	0.215399i
$764$	−4.92027e6	−0.304969
$765$	0	0
$766$	−3.33188e6	−0.205172
$767$	− 8.03799e6i	− 0.493354i
$768$	1.82783e6i	0.111824i
$769$	−3.53661e6	−0.215661	−0.107830	−	0.994169i	$-0.534390\pi$
$769$	−3.53661e6	−0.215661	−0.107830	+	0.994169i	$0.534390\pi$
$770$	0	0
$771$	1.52897e7	0.926322
$772$	2.91855e6i	0.176248i
$773$	3.14397e7i	1.89247i	0.323479	+	0.946235i	$0.395147\pi$
$773$	3.14397e7i	1.89247i	−0.323479	+	0.946235i	$0.604853\pi$
$774$	−1.91179e7	−1.14706
$775$	0	0
$776$	1.00187e6	0.0597251
$777$	− 8.79044e7i	− 5.22346i
$778$	− 2.90097e6i	− 0.171828i
$779$	−1.37812e7	−0.813659
$780$	0	0
$781$	−5.72429e6	−0.335810
$782$	− 2.43191e6i	− 0.142210i
$783$	1.22404e7i	0.713498i
$784$	−1.04953e7	−0.609827
$785$	0	0
$786$	2.22805e7	1.28638
$787$	1.27213e7i	0.732139i	0.930588	+	0.366069i	$0.119297\pi$
$787$	1.27213e7i	0.732139i	−0.930588	+	0.366069i	$0.880703\pi$
$788$	2.44013e6i	0.139990i
$789$	2.28259e7	1.30538
$790$	0	0
$791$	1.22141e7	0.694097
$792$	− 1.86276e7i	− 1.05522i
$793$	5.11731e6i	0.288974i
$794$	9.70617e6	0.546382
$795$	0	0
$796$	−9.38569e6	−0.525029
$797$	1.84237e7i	1.02738i	0.857977	+	0.513689i	$0.171721\pi$
$797$	1.84237e7i	1.02738i	−0.857977	+	0.513689i	$0.828279\pi$
$798$	− 2.16167e7i	− 1.20166i
$799$	−2.56610e7	−1.42203
$800$	0	0
$801$	2.62899e7	1.44780
$802$	8.57506e6i	0.470762i
$803$	− 860915.i	− 0.0471164i
$804$	1.73008e7	0.943899
$805$	0	0
$806$	1.48725e6	0.0806394
$807$	2.62019e7i	1.41628i
$808$	− 8.83888e6i	− 0.476287i
$809$	−2.10360e7	−1.13004	−0.565018	−	0.825079i	$-0.691131\pi$
$809$	−2.10360e7	−1.13004	−0.565018	+	0.825079i	$0.691131\pi$
$810$	0	0
$811$	3.53571e7	1.88767	0.943833	−	0.330424i	$-0.107192\pi$
$811$	3.53571e7	1.88767	0.943833	+	0.330424i	$0.107192\pi$
$812$	− 5.78412e6i	− 0.307856i
$813$	− 4.25097e7i	− 2.25560i
$814$	−2.85334e7	−1.50936
$815$	0	0
$816$	−1.16331e7	−0.611606
$817$	7.20136e6i	0.377450i
$818$	− 1.46458e7i	− 0.765295i
$819$	2.17332e7	1.13218
$820$	0	0
$821$	1.72038e7	0.890770	0.445385	−	0.895339i	$-0.353067\pi$
$821$	1.72038e7	0.890770	0.445385	+	0.895339i	$0.353067\pi$
$822$	− 3.85355e7i	− 1.98921i
$823$	947098.i	0.0487411i	0.999703	+	0.0243705i	$0.00775815\pi$
$823$	947098.i	0.0487411i	−0.999703	+	0.0243705i	$0.992242\pi$
$824$	−3.04909e6	−0.156442
$825$	0	0
$826$	−4.57405e7	−2.33266
$827$	− 3.05385e7i	− 1.55269i	−0.630308	−	0.776345i	$-0.717072\pi$
$827$	− 3.05385e7i	− 1.55269i	0.630308	−	0.776345i	$-0.282928\pi$
$828$	− 3.19346e6i	− 0.161877i
$829$	1.57415e7	0.795538	0.397769	−	0.917486i	$-0.369785\pi$
$829$	1.57415e7	0.795538	0.397769	+	0.917486i	$0.369785\pi$
$830$	0	0
$831$	1.22636e7	0.616047
$832$	692224.i	0.0346688i
$833$	− 6.67971e7i	− 3.33538i
$834$	−2.04257e7	−1.01686
$835$	0	0
$836$	−7.01668e6	−0.347229
$837$	1.79101e7i	0.883659i
$838$	1.21514e7i	0.597746i
$839$	−2.07700e7	−1.01867	−0.509333	−	0.860570i	$-0.670108\pi$
$839$	−2.07700e7	−1.01867	−0.509333	+	0.860570i	$0.670108\pi$
$840$	0	0
$841$	−1.82503e7	−0.889774
$842$	7.54651e6i	0.366831i
$843$	− 1.65121e7i	− 0.800263i
$844$	3.31265e6	0.160073
$845$	0	0
$846$	−3.36969e7	−1.61869
$847$	− 3.24694e7i	− 1.55513i
$848$	− 1.03372e7i	− 0.493643i
$849$	−2.16772e7	−1.03213
$850$	0	0
$851$	−4.89169e6	−0.231545
$852$	4.69438e6i	0.221554i
$853$	1.91838e7i	0.902740i	0.892337	+	0.451370i	$0.149064\pi$
$853$	1.91838e7i	0.902740i	−0.892337	+	0.451370i	$0.850936\pi$
$854$	2.91203e7	1.36632
$855$	0	0
$856$	1.19377e7	0.556847
$857$	2.64211e7i	1.22885i	0.788975	+	0.614425i	$0.210612\pi$
$857$	2.64211e7i	1.22885i	−0.788975	+	0.614425i	$0.789388\pi$
$858$	− 1.02594e7i	− 0.475778i
$859$	3.30576e7	1.52858	0.764289	−	0.644873i	$-0.223090\pi$
$859$	3.30576e7	1.52858	0.764289	+	0.644873i	$0.223090\pi$
$860$	0	0
$861$	−1.14665e8	−5.27137
$862$	− 2.86877e6i	− 0.131501i
$863$	− 1.19324e7i	− 0.545380i	−0.962102	−	0.272690i	$-0.912087\pi$
$863$	− 1.19324e7i	− 0.545380i	0.962102	−	0.272690i	$-0.0879133\pi$
$864$	−8.33605e6	−0.379906
$865$	0	0
$866$	−1.05828e7	−0.479518
$867$	− 3.44381e7i	− 1.55593i
$868$	− 8.46328e6i	− 0.381276i
$869$	3.36897e6	0.151338
$870$	0	0
$871$	6.55203e6	0.292638
$872$	− 922051.i	− 0.0410643i
$873$	8.37313e6i	0.371837i
$874$	−1.20292e6	−0.0532671
$875$	0	0
$876$	−706020.	−0.0310854
$877$	3.88434e6i	0.170537i	0.996358	+	0.0852684i	$0.0271748\pi$
$877$	3.88434e6i	0.170537i	−0.996358	+	0.0852684i	$0.972825\pi$
$878$	1.79189e7i	0.784469i
$879$	1.69860e7	0.741513
$880$	0	0
$881$	−8.93397e6	−0.387797	−0.193899	−	0.981022i	$-0.562113\pi$
$881$	−8.93397e6	−0.387797	−0.193899	+	0.981022i	$0.562113\pi$
$882$	− 8.77148e7i	− 3.79666i
$883$	4.32552e6i	0.186697i	0.995634	+	0.0933484i	$0.0297570\pi$
$883$	4.32552e6i	0.186697i	−0.995634	+	0.0933484i	$0.970243\pi$
$884$	−4.40563e6	−0.189617
$885$	0	0
$886$	−1.72292e7	−0.737364
$887$	3.55552e7i	1.51738i	0.651453	+	0.758689i	$0.274160\pi$
$887$	3.55552e7i	1.51738i	−0.651453	+	0.758689i	$0.725840\pi$
$888$	2.33997e7i	0.995812i
$889$	−4.38242e7	−1.85977
$890$	0	0
$891$	5.28217e7	2.22904
$892$	− 1.67822e7i	− 0.706214i
$893$	1.26930e7i	0.532643i
$894$	2.08712e7	0.873380
$895$	0	0
$896$	3.93913e6	0.163919
$897$	− 1.75885e6i	− 0.0729874i
$898$	− 101092.i	− 0.00418335i
$899$	3.30807e6	0.136514
$900$	0	0
$901$	6.57906e7	2.69993
$902$	3.72198e7i	1.52320i
$903$	5.99184e7i	2.44535i
$904$	−3.25133e6	−0.132324
$905$	0	0
$906$	−1.18438e6	−0.0479370
$907$	2.44157e7i	0.985486i	0.870175	+	0.492743i	$0.164006\pi$
$907$	2.44157e7i	0.985486i	−0.870175	+	0.492743i	$0.835994\pi$
$908$	1.61134e6i	0.0648593i
$909$	7.38709e7	2.96527
$910$	0	0
$911$	3.00267e7	1.19870	0.599352	−	0.800486i	$-0.295425\pi$
$911$	3.00267e7	1.19870	0.599352	+	0.800486i	$0.295425\pi$
$912$	5.75424e6i	0.229087i
$913$	− 2.05957e7i	− 0.817711i
$914$	4.57439e6	0.181120
$915$	0	0
$916$	7.79478e6	0.306948
$917$	− 4.80164e7i	− 1.88567i
$918$	− 5.30544e7i	− 2.07785i
$919$	1.30118e7	0.508216	0.254108	−	0.967176i	$-0.418218\pi$
$919$	1.30118e7	0.508216	0.254108	+	0.967176i	$0.418218\pi$
$920$	0	0
$921$	7.44009e7	2.89021
$922$	3.33964e7i	1.29382i
$923$	1.77782e6i	0.0686886i
$924$	−5.83817e7	−2.24956
$925$	0	0
$926$	1.36639e7	0.523656
$927$	− 2.54828e7i	− 0.973975i
$928$	1.53970e6i	0.0586904i
$929$	5.16678e7	1.96418	0.982089	−	0.188419i	$-0.0603362\pi$
$929$	5.16678e7	1.96418	0.982089	+	0.188419i	$0.0603362\pi$
$930$	0	0
$931$	−3.30407e7	−1.24932
$932$	− 1.33255e7i	− 0.502509i
$933$	− 4.52717e7i	− 1.70264i
$934$	2.26842e7	0.850857
$935$	0	0
$936$	−5.78526e6	−0.215841
$937$	− 4.04703e6i	− 0.150587i	−0.997161	−	0.0752936i	$-0.976011\pi$
$937$	− 4.04703e6i	− 0.150587i	0.997161	−	0.0752936i	$-0.0239894\pi$
$938$	− 3.72846e7i	− 1.38364i
$939$	1.65144e7	0.611221
$940$	0	0
$941$	−4.43008e7	−1.63094	−0.815470	−	0.578799i	$-0.803521\pi$
$941$	−4.43008e7	−1.63094	−0.815470	+	0.578799i	$0.803521\pi$
$942$	− 3.74719e7i	− 1.37587i
$943$	6.38088e6i	0.233669i
$944$	1.21759e7	0.444704
$945$	0	0
$946$	1.94492e7	0.706602
$947$	− 3.60463e7i	− 1.30613i	−0.757303	−	0.653064i	$-0.773483\pi$
$947$	− 3.60463e7i	− 1.30613i	0.757303	−	0.653064i	$-0.226517\pi$
$948$	− 2.76283e6i	− 0.0998464i
$949$	−267379.	−0.00963745
$950$	0	0
$951$	7.28042e7	2.61039
$952$	2.50704e7i	0.896539i
$953$	1.31248e7i	0.468123i	0.972222	+	0.234061i	$0.0752017\pi$
$953$	1.31248e7i	0.468123i	−0.972222	+	0.234061i	$0.924798\pi$
$954$	8.63931e7	3.07332
$955$	0	0
$956$	2.15871e7	0.763923
$957$	− 2.28198e7i	− 0.805439i
$958$	− 2.46536e7i	− 0.867894i
$959$	−8.30473e7	−2.91594
$960$	0	0
$961$	−2.37888e7	−0.830929
$962$	8.86176e6i	0.308733i
$963$	9.97693e7i	3.46682i
$964$	2.66053e7	0.922096
$965$	0	0
$966$	−1.00088e7	−0.345096
$967$	2.48150e7i	0.853390i	0.904396	+	0.426695i	$0.140322\pi$
$967$	2.48150e7i	0.853390i	−0.904396	+	0.426695i	$0.859678\pi$
$968$	8.64318e6i	0.296473i
$969$	−3.66226e7	−1.25297
$970$	0	0
$971$	−3.54227e6	−0.120569	−0.0602843	−	0.998181i	$-0.519201\pi$
$971$	−3.54227e6	−0.120569	−0.0602843	+	0.998181i	$0.519201\pi$
$972$	− 1.16671e7i	− 0.396093i
$973$	4.40191e7i	1.49059i
$974$	−2.93080e7	−0.989896
$975$	0	0
$976$	−7.75167e6	−0.260478
$977$	9.16916e6i	0.307322i	0.988124	+	0.153661i	$0.0491063\pi$
$977$	9.16916e6i	0.307322i	−0.988124	+	0.153661i	$0.950894\pi$
$978$	− 2.25222e7i	− 0.752947i
$979$	−2.67456e7	−0.891857
$980$	0	0
$981$	7.70604e6	0.255658
$982$	− 3.55895e7i	− 1.17772i
$983$	3.05912e7i	1.00975i	0.863193	+	0.504874i	$0.168461\pi$
$983$	3.05912e7i	1.00975i	−0.863193	+	0.504874i	$0.831539\pi$
$984$	3.05233e7	1.00495
$985$	0	0
$986$	−9.79936e6	−0.321000
$987$	1.05611e8i	3.45078i
$988$	2.17921e6i	0.0710242i
$989$	3.33433e6	0.108397
$990$	0	0
$991$	3.47852e7	1.12515	0.562575	−	0.826746i	$-0.309811\pi$
$991$	3.47852e7	1.12515	0.562575	+	0.826746i	$0.309811\pi$
$992$	2.25288e6i	0.0726874i
$993$	2.28363e7i	0.734942i
$994$	1.01168e7	0.324771
$995$	0	0
$996$	−1.68902e7	−0.539492
$997$	− 4.21812e7i	− 1.34394i	−0.740577	−	0.671972i	$-0.765448\pi$
$997$	− 4.21812e7i	− 1.34394i	0.740577	−	0.671972i	$-0.234552\pi$
$998$	1.20895e7i	0.384222i
$999$	−1.06717e8	−3.38314

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	650.6.b.j.599.3		6
5.2	odd	4		650.6.a.j.1.3		3
5.3	odd	4		130.6.a.f.1.1	✓	3
5.4	even	2	inner	650.6.b.j.599.4		6
20.3	even	4		1040.6.a.l.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
130.6.a.f.1.1	✓	3	5.3	odd	4
650.6.a.j.1.3		3	5.2	odd	4
650.6.b.j.599.3		6	1.1	even	1	trivial
650.6.b.j.599.4		6	5.4	even	2	inner
1040.6.a.l.1.3		3	20.3	even	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:	\( N \)	\(=\)	\( 650 = 2 \cdot 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	650.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-4.00000i q^{2} +27.8905i q^{3} -16.0000 q^{4} +111.562 q^{6} -240.426i q^{7} +64.0000i q^{8} -534.880 q^{9} +544.151 q^{11} -446.248i q^{12} -169.000i q^{13} -961.702 q^{14} +256.000 q^{16} +1629.30i q^{17} +2139.52i q^{18} +805.920 q^{19} +6705.59 q^{21} -2176.61i q^{22} -373.152i q^{23} -1784.99 q^{24} -676.000 q^{26} -8140.67i q^{27} +3846.81i q^{28} -1503.62 q^{29} -2200.08 q^{31} -1024.00i q^{32} +15176.7i q^{33} +6517.20 q^{34} +8558.08 q^{36} -13109.1i q^{37} -3223.68i q^{38} +4713.49 q^{39} -17099.9 q^{41} -26822.4i q^{42} +8935.58i q^{43} -8706.42 q^{44} -1492.61 q^{46} +15749.7i q^{47} +7139.97i q^{48} -40997.5 q^{49} -45442.0 q^{51} +2704.00i q^{52} -40379.7i q^{53} -32562.7 q^{54} +15387.2 q^{56} +22477.5i q^{57} +6014.46i q^{58} +47562.1 q^{59} -30280.0 q^{61} +8800.31i q^{62} +128599. i q^{63} -4096.00 q^{64} +60706.6 q^{66} +38769.4i q^{67} -26068.8i q^{68} +10407.4 q^{69} -10519.7 q^{71} -34232.3i q^{72} -1582.12i q^{73} -52436.5 q^{74} -12894.7 q^{76} -130828. i q^{77} -18854.0i q^{78} +6191.23 q^{79} +97071.7 q^{81} +68399.8i q^{82} -37849.2i q^{83} -107289. q^{84} +35742.3 q^{86} -41936.6i q^{87} +34825.7i q^{88} -49151.0 q^{89} -40631.9 q^{91} +5970.43i q^{92} -61361.3i q^{93} +62999.0 q^{94} +28559.9 q^{96} -15654.2i q^{97} +163990. i q^{98} -291056. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 96 q^{4} + 176 q^{6} - 310 q^{9} + 96 q^{11} - 1872 q^{14} + 1536 q^{16} + 720 q^{19} + 13808 q^{21} - 2816 q^{24} - 4056 q^{26} + 6156 q^{29} - 10776 q^{31} + 12048 q^{34} + 4960 q^{36} + 7436 q^{39}+ \cdots - 700680 q^{99}+O(q^{100}) \) 6 * q - 96 * q^4 + 176 * q^6 - 310 * q^9 + 96 * q^11 - 1872 * q^14 + 1536 * q^16 + 720 * q^19 + 13808 * q^21 - 2816 * q^24 - 4056 * q^26 + 6156 * q^29 - 10776 * q^31 + 12048 * q^34 + 4960 * q^36 + 7436 * q^39 - 31812 * q^41 - 1536 * q^44 + 18960 * q^46 - 15534 * q^49 - 67944 * q^51 - 81344 * q^54 + 29952 * q^56 + 155520 * q^59 + 35964 * q^61 - 24576 * q^64 + 180576 * q^66 + 63720 * q^69 - 9888 * q^71 - 202896 * q^74 - 11520 * q^76 - 84240 * q^79 + 293038 * q^81 - 220928 * q^84 + 314448 * q^86 - 39228 * q^89 - 79092 * q^91 - 142224 * q^94 + 45056 * q^96 - 700680 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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