LMFDB - Embedded newform 1050.4.g.a.799.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1050,4,Mod(799,1050)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1050, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("1050.799");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1050 = 2 \cdot 3 \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1050.g (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:	$61.9520055060$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 42)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			799.2
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	1050.799
Dual form			1050.4.g.a.799.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.00000i q^{2} +3.00000i q^{3} -4.00000 q^{4} -6.00000 q^{6} +7.00000i q^{7} -8.00000i q^{8} -9.00000 q^{9} +O(q^{10})$	\(q+2.00000i q^{2} +3.00000i q^{3} -4.00000 q^{4} -6.00000 q^{6} +7.00000i q^{7} -8.00000i q^{8} -9.00000 q^{9} -72.0000 q^{11} -12.0000i q^{12} +34.0000i q^{13} -14.0000 q^{14} +16.0000 q^{16} +6.00000i q^{17} -18.0000i q^{18} -92.0000 q^{19} -21.0000 q^{21} -144.000i q^{22} +180.000i q^{23} +24.0000 q^{24} -68.0000 q^{26} -27.0000i q^{27} -28.0000i q^{28} +114.000 q^{29} +56.0000 q^{31} +32.0000i q^{32} -216.000i q^{33} -12.0000 q^{34} +36.0000 q^{36} -34.0000i q^{37} -184.000i q^{38} -102.000 q^{39} +6.00000 q^{41} -42.0000i q^{42} -164.000i q^{43} +288.000 q^{44} -360.000 q^{46} +168.000i q^{47} +48.0000i q^{48} -49.0000 q^{49} -18.0000 q^{51} -136.000i q^{52} -654.000i q^{53} +54.0000 q^{54} +56.0000 q^{56} -276.000i q^{57} +228.000i q^{58} +492.000 q^{59} -250.000 q^{61} +112.000i q^{62} -63.0000i q^{63} -64.0000 q^{64} +432.000 q^{66} -124.000i q^{67} -24.0000i q^{68} -540.000 q^{69} +36.0000 q^{71} +72.0000i q^{72} -1010.00i q^{73} +68.0000 q^{74} +368.000 q^{76} -504.000i q^{77} -204.000i q^{78} -56.0000 q^{79} +81.0000 q^{81} +12.0000i q^{82} -228.000i q^{83} +84.0000 q^{84} +328.000 q^{86} +342.000i q^{87} +576.000i q^{88} -390.000 q^{89} -238.000 q^{91} -720.000i q^{92} +168.000i q^{93} -336.000 q^{94} -96.0000 q^{96} -70.0000i q^{97} -98.0000i q^{98} +648.000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 8 q^{4} - 12 q^{6} - 18 q^{9}+O(q^{10}) $ 2 * q - 8 * q^4 - 12 * q^6 - 18 * q^9	$ 2 q - 8 q^{4} - 12 q^{6} - 18 q^{9} - 144 q^{11} - 28 q^{14} + 32 q^{16} - 184 q^{19} - 42 q^{21} + 48 q^{24} - 136 q^{26} + 228 q^{29} + 112 q^{31} - 24 q^{34} + 72 q^{36} - 204 q^{39} + 12 q^{41} + 576 q^{44} - 720 q^{46} - 98 q^{49} - 36 q^{51} + 108 q^{54} + 112 q^{56} + 984 q^{59} - 500 q^{61} - 128 q^{64} + 864 q^{66} - 1080 q^{69} + 72 q^{71} + 136 q^{74} + 736 q^{76} - 112 q^{79} + 162 q^{81} + 168 q^{84} + 656 q^{86} - 780 q^{89} - 476 q^{91} - 672 q^{94} - 192 q^{96} + 1296 q^{99}+O(q^{100}) $ 2 * q - 8 * q^4 - 12 * q^6 - 18 * q^9 - 144 * q^11 - 28 * q^14 + 32 * q^16 - 184 * q^19 - 42 * q^21 + 48 * q^24 - 136 * q^26 + 228 * q^29 + 112 * q^31 - 24 * q^34 + 72 * q^36 - 204 * q^39 + 12 * q^41 + 576 * q^44 - 720 * q^46 - 98 * q^49 - 36 * q^51 + 108 * q^54 + 112 * q^56 + 984 * q^59 - 500 * q^61 - 128 * q^64 + 864 * q^66 - 1080 * q^69 + 72 * q^71 + 136 * q^74 + 736 * q^76 - 112 * q^79 + 162 * q^81 + 168 * q^84 + 656 * q^86 - 780 * q^89 - 476 * q^91 - 672 * q^94 - 192 * q^96 + 1296 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1050\mathbb{Z}\right)^\times$.

$n$	$127$	$451$	$701$
$\chi(n)$	$-1$	$1$	$1$

Coefficient data

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

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

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

$2$	2.00000i	0.707107i
$2$	2.00000i	0.707107i
$3$	3.00000i	0.577350i
$3$	3.00000i	0.577350i
$4$	−4.00000	−0.500000
$5$	0	0
$5$	0	0
$6$	−6.00000	−0.408248
$7$	7.00000i	0.377964i
$7$	7.00000i	0.377964i
$8$	− 8.00000i	− 0.353553i
$9$	−9.00000	−0.333333
$10$	0	0
$11$	−72.0000	−1.97353	−0.986764	−	0.162160i	$-0.948154\pi$
$11$	−72.0000	−1.97353	−0.986764	+	0.162160i	$0.948154\pi$
$12$	− 12.0000i	− 0.288675i
$13$	34.0000i	0.725377i	0.931910	+	0.362689i	$0.118141\pi$
$13$	34.0000i	0.725377i	−0.931910	+	0.362689i	$0.881859\pi$
$14$	−14.0000	−0.267261
$15$	0	0
$16$	16.0000	0.250000
$17$	6.00000i	0.0856008i	0.999084	+	0.0428004i	$0.0136280\pi$
$17$	6.00000i	0.0856008i	−0.999084	+	0.0428004i	$0.986372\pi$
$18$	− 18.0000i	− 0.235702i
$19$	−92.0000	−1.11086	−0.555428	−	0.831565i	$-0.687445\pi$
$19$	−92.0000	−1.11086	−0.555428	+	0.831565i	$0.687445\pi$
$20$	0	0
$21$	−21.0000	−0.218218
$22$	− 144.000i	− 1.39550i
$23$	180.000i	1.63185i	0.578156	+	0.815926i	$0.303772\pi$
$23$	180.000i	1.63185i	−0.578156	+	0.815926i	$0.696228\pi$
$24$	24.0000	0.204124
$25$	0	0
$26$	−68.0000	−0.512919
$27$	− 27.0000i	− 0.192450i
$28$	− 28.0000i	− 0.188982i
$29$	114.000	0.729975	0.364987	−	0.931012i	$-0.381073\pi$
$29$	114.000	0.729975	0.364987	+	0.931012i	$0.381073\pi$
$30$	0	0
$31$	56.0000	0.324448	0.162224	−	0.986754i	$-0.448133\pi$
$31$	56.0000	0.324448	0.162224	+	0.986754i	$0.448133\pi$
$32$	32.0000i	0.176777i
$33$	− 216.000i	− 1.13942i
$34$	−12.0000	−0.0605289
$35$	0	0
$36$	36.0000	0.166667
$37$	− 34.0000i	− 0.151069i	−0.997143	−	0.0755347i	$-0.975934\pi$
$37$	− 34.0000i	− 0.151069i	0.997143	−	0.0755347i	$-0.0240664\pi$
$38$	− 184.000i	− 0.785493i
$39$	−102.000	−0.418797
$40$	0	0
$41$	6.00000	0.0228547	0.0114273	−	0.999935i	$-0.496362\pi$
$41$	6.00000	0.0228547	0.0114273	+	0.999935i	$0.496362\pi$
$42$	− 42.0000i	− 0.154303i
$43$	− 164.000i	− 0.581622i	−0.956780	−	0.290811i	$-0.906075\pi$
$43$	− 164.000i	− 0.581622i	0.956780	−	0.290811i	$-0.0939252\pi$
$44$	288.000	0.986764
$45$	0	0
$46$	−360.000	−1.15389
$47$	168.000i	0.521390i	0.965421	+	0.260695i	$0.0839517\pi$
$47$	168.000i	0.521390i	−0.965421	+	0.260695i	$0.916048\pi$
$48$	48.0000i	0.144338i
$49$	−49.0000	−0.142857
$50$	0	0
$51$	−18.0000	−0.0494217
$52$	− 136.000i	− 0.362689i
$53$	− 654.000i	− 1.69498i	−0.530813	−	0.847489i	$-0.678113\pi$
$53$	− 654.000i	− 1.69498i	0.530813	−	0.847489i	$-0.321887\pi$
$54$	54.0000	0.136083
$55$	0	0
$56$	56.0000	0.133631
$57$	− 276.000i	− 0.641353i
$58$	228.000i	0.516170i
$59$	492.000	1.08564	0.542822	−	0.839848i	$-0.317356\pi$
$59$	492.000	1.08564	0.542822	+	0.839848i	$0.317356\pi$
$60$	0	0
$61$	−250.000	−0.524741	−0.262371	−	0.964967i	$-0.584504\pi$
$61$	−250.000	−0.524741	−0.262371	+	0.964967i	$0.584504\pi$
$62$	112.000i	0.229420i
$63$	− 63.0000i	− 0.125988i
$64$	−64.0000	−0.125000
$65$	0	0
$66$	432.000	0.805690
$67$	− 124.000i	− 0.226105i	−0.993589	−	0.113052i	$-0.963937\pi$
$67$	− 124.000i	− 0.226105i	0.993589	−	0.113052i	$-0.0360628\pi$
$68$	− 24.0000i	− 0.0428004i
$69$	−540.000	−0.942150
$70$	0	0
$71$	36.0000	0.0601748	0.0300874	−	0.999547i	$-0.490421\pi$
$71$	36.0000	0.0601748	0.0300874	+	0.999547i	$0.490421\pi$
$72$	72.0000i	0.117851i
$73$	− 1010.00i	− 1.61934i	−0.586888	−	0.809668i	$-0.699647\pi$
$73$	− 1010.00i	− 1.61934i	0.586888	−	0.809668i	$-0.300353\pi$
$74$	68.0000	0.106822
$75$	0	0
$76$	368.000	0.555428
$77$	− 504.000i	− 0.745924i
$78$	− 204.000i	− 0.296134i
$79$	−56.0000	−0.0797531	−0.0398765	−	0.999205i	$-0.512696\pi$
$79$	−56.0000	−0.0797531	−0.0398765	+	0.999205i	$0.512696\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	12.0000i	0.0161607i
$83$	− 228.000i	− 0.301521i	−0.988570	−	0.150761i	$-0.951828\pi$
$83$	− 228.000i	− 0.301521i	0.988570	−	0.150761i	$-0.0481722\pi$
$84$	84.0000	0.109109
$85$	0	0
$86$	328.000	0.411269
$87$	342.000i	0.421451i
$88$	576.000i	0.697748i
$89$	−390.000	−0.464493	−0.232247	−	0.972657i	$-0.574608\pi$
$89$	−390.000	−0.464493	−0.232247	+	0.972657i	$0.574608\pi$
$90$	0	0
$91$	−238.000	−0.274167
$92$	− 720.000i	− 0.815926i
$93$	168.000i	0.187320i
$94$	−336.000	−0.368678
$95$	0	0
$96$	−96.0000	−0.102062
$97$	− 70.0000i	− 0.0732724i	−0.999329	−	0.0366362i	$-0.988336\pi$
$97$	− 70.0000i	− 0.0732724i	0.999329	−	0.0366362i	$-0.0116643\pi$
$98$	− 98.0000i	− 0.101015i
$99$	648.000	0.657843
$100$	0	0
$101$	−1350.00	−1.33000	−0.665000	−	0.746843i	$-0.731569\pi$
$101$	−1350.00	−1.33000	−0.665000	+	0.746843i	$0.731569\pi$
$102$	− 36.0000i	− 0.0349464i
$103$	− 2000.00i	− 1.91326i	−0.291305	−	0.956630i	$-0.594089\pi$
$103$	− 2000.00i	− 1.91326i	0.291305	−	0.956630i	$-0.405911\pi$
$104$	272.000	0.256460
$105$	0	0
$106$	1308.00	1.19853
$107$	696.000i	0.628830i	0.949286	+	0.314415i	$0.101808\pi$
$107$	696.000i	0.628830i	−0.949286	+	0.314415i	$0.898192\pi$
$108$	108.000i	0.0962250i
$109$	1114.00	0.978916	0.489458	−	0.872027i	$-0.337195\pi$
$109$	1114.00	0.978916	0.489458	+	0.872027i	$0.337195\pi$
$110$	0	0
$111$	102.000	0.0872199
$112$	112.000i	0.0944911i
$113$	462.000i	0.384613i	0.981335	+	0.192307i	$0.0615968\pi$
$113$	462.000i	0.384613i	−0.981335	+	0.192307i	$0.938403\pi$
$114$	552.000	0.453505
$115$	0	0
$116$	−456.000	−0.364987
$117$	− 306.000i	− 0.241792i
$118$	984.000i	0.767666i
$119$	−42.0000	−0.0323541
$120$	0	0
$121$	3853.00	2.89482
$122$	− 500.000i	− 0.371048i
$123$	18.0000i	0.0131952i
$124$	−224.000	−0.162224
$125$	0	0
$126$	126.000	0.0890871
$127$	1064.00i	0.743423i	0.928348	+	0.371712i	$0.121229\pi$
$127$	1064.00i	0.743423i	−0.928348	+	0.371712i	$0.878771\pi$
$128$	− 128.000i	− 0.0883883i
$129$	492.000	0.335800
$130$	0	0
$131$	180.000	0.120051	0.0600255	−	0.998197i	$-0.480882\pi$
$131$	180.000	0.120051	0.0600255	+	0.998197i	$0.480882\pi$
$132$	864.000i	0.569709i
$133$	− 644.000i	− 0.419864i
$134$	248.000	0.159880
$135$	0	0
$136$	48.0000	0.0302645
$137$	− 2718.00i	− 1.69500i	−0.530799	−	0.847498i	$-0.678108\pi$
$137$	− 2718.00i	− 1.69500i	0.530799	−	0.847498i	$-0.321892\pi$
$138$	− 1080.00i	− 0.666201i
$139$	1348.00	0.822560	0.411280	−	0.911509i	$-0.365082\pi$
$139$	1348.00	0.822560	0.411280	+	0.911509i	$0.365082\pi$
$140$	0	0
$141$	−504.000	−0.301025
$142$	72.0000i	0.0425500i
$143$	− 2448.00i	− 1.43155i
$144$	−144.000	−0.0833333
$145$	0	0
$146$	2020.00	1.14504
$147$	− 147.000i	− 0.0824786i
$148$	136.000i	0.0755347i
$149$	−558.000	−0.306800	−0.153400	−	0.988164i	$-0.549022\pi$
$149$	−558.000	−0.306800	−0.153400	+	0.988164i	$0.549022\pi$
$150$	0	0
$151$	1928.00	1.03906	0.519531	−	0.854451i	$-0.326107\pi$
$151$	1928.00	1.03906	0.519531	+	0.854451i	$0.326107\pi$
$152$	736.000i	0.392747i
$153$	− 54.0000i	− 0.0285336i
$154$	1008.00	0.527448
$155$	0	0
$156$	408.000	0.209398
$157$	− 2410.00i	− 1.22509i	−0.790436	−	0.612544i	$-0.790146\pi$
$157$	− 2410.00i	− 1.22509i	0.790436	−	0.612544i	$-0.209854\pi$
$158$	− 112.000i	− 0.0563939i
$159$	1962.00	0.978596
$160$	0	0
$161$	−1260.00	−0.616782
$162$	162.000i	0.0785674i
$163$	− 740.000i	− 0.355591i	−0.984067	−	0.177795i	$-0.943104\pi$
$163$	− 740.000i	− 0.355591i	0.984067	−	0.177795i	$-0.0568965\pi$
$164$	−24.0000	−0.0114273
$165$	0	0
$166$	456.000	0.213208
$167$	3984.00i	1.84605i	0.384734	+	0.923027i	$0.374293\pi$
$167$	3984.00i	1.84605i	−0.384734	+	0.923027i	$0.625707\pi$
$168$	168.000i	0.0771517i
$169$	1041.00	0.473828
$170$	0	0
$171$	828.000	0.370285
$172$	656.000i	0.290811i
$173$	1038.00i	0.456172i	0.973641	+	0.228086i	$0.0732467\pi$
$173$	1038.00i	0.456172i	−0.973641	+	0.228086i	$0.926753\pi$
$174$	−684.000	−0.298011
$175$	0	0
$176$	−1152.00	−0.493382
$177$	1476.00i	0.626796i
$178$	− 780.000i	− 0.328446i
$179$	2568.00	1.07230	0.536149	−	0.844123i	$-0.319879\pi$
$179$	2568.00	1.07230	0.536149	+	0.844123i	$0.319879\pi$
$180$	0	0
$181$	−2698.00	−1.10796	−0.553980	−	0.832530i	$-0.686892\pi$
$181$	−2698.00	−1.10796	−0.553980	+	0.832530i	$0.686892\pi$
$182$	− 476.000i	− 0.193865i
$183$	− 750.000i	− 0.302960i
$184$	1440.00	0.576947
$185$	0	0
$186$	−336.000	−0.132455
$187$	− 432.000i	− 0.168936i
$188$	− 672.000i	− 0.260695i
$189$	189.000	0.0727393
$190$	0	0
$191$	−4116.00	−1.55928	−0.779642	−	0.626225i	$-0.784599\pi$
$191$	−4116.00	−1.55928	−0.779642	+	0.626225i	$0.784599\pi$
$192$	− 192.000i	− 0.0721688i
$193$	3310.00i	1.23450i	0.786766	+	0.617251i	$0.211754\pi$
$193$	3310.00i	1.23450i	−0.786766	+	0.617251i	$0.788246\pi$
$194$	140.000	0.0518114
$195$	0	0
$196$	196.000	0.0714286
$197$	1278.00i	0.462202i	0.972930	+	0.231101i	$0.0742327\pi$
$197$	1278.00i	0.462202i	−0.972930	+	0.231101i	$0.925767\pi$
$198$	1296.00i	0.465165i
$199$	−2936.00	−1.04587	−0.522933	−	0.852374i	$-0.675162\pi$
$199$	−2936.00	−1.04587	−0.522933	+	0.852374i	$0.675162\pi$
$200$	0	0
$201$	372.000	0.130542
$202$	− 2700.00i	− 0.940452i
$203$	798.000i	0.275905i
$204$	72.0000	0.0247108
$205$	0	0
$206$	4000.00	1.35288
$207$	− 1620.00i	− 0.543951i
$208$	544.000i	0.181344i
$209$	6624.00	2.19230
$210$	0	0
$211$	−3508.00	−1.14455	−0.572276	−	0.820061i	$-0.693940\pi$
$211$	−3508.00	−1.14455	−0.572276	+	0.820061i	$0.693940\pi$
$212$	2616.00i	0.847489i
$213$	108.000i	0.0347420i
$214$	−1392.00	−0.444650
$215$	0	0
$216$	−216.000	−0.0680414
$217$	392.000i	0.122630i
$218$	2228.00i	0.692198i
$219$	3030.00	0.934924
$220$	0	0
$221$	−204.000	−0.0620929
$222$	204.000i	0.0616738i
$223$	1888.00i	0.566950i	0.958980	+	0.283475i	$0.0914873\pi$
$223$	1888.00i	0.566950i	−0.958980	+	0.283475i	$0.908513\pi$
$224$	−224.000	−0.0668153
$225$	0	0
$226$	−924.000	−0.271963
$227$	3564.00i	1.04207i	0.853534	+	0.521037i	$0.174455\pi$
$227$	3564.00i	1.04207i	−0.853534	+	0.521037i	$0.825545\pi$
$228$	1104.00i	0.320676i
$229$	−1334.00	−0.384948	−0.192474	−	0.981302i	$-0.561651\pi$
$229$	−1334.00	−0.384948	−0.192474	+	0.981302i	$0.561651\pi$
$230$	0	0
$231$	1512.00	0.430659
$232$	− 912.000i	− 0.258085i
$233$	− 2658.00i	− 0.747345i	−0.927561	−	0.373672i	$-0.878098\pi$
$233$	− 2658.00i	− 0.747345i	0.927561	−	0.373672i	$-0.121902\pi$
$234$	612.000	0.170973
$235$	0	0
$236$	−1968.00	−0.542822
$237$	− 168.000i	− 0.0460455i
$238$	− 84.0000i	− 0.0228778i
$239$	588.000	0.159140	0.0795702	−	0.996829i	$-0.474645\pi$
$239$	588.000	0.159140	0.0795702	+	0.996829i	$0.474645\pi$
$240$	0	0
$241$	5690.00	1.52085	0.760426	−	0.649425i	$-0.224990\pi$
$241$	5690.00	1.52085	0.760426	+	0.649425i	$0.224990\pi$
$242$	7706.00i	2.04694i
$243$	243.000i	0.0641500i
$244$	1000.00	0.262371
$245$	0	0
$246$	−36.0000	−0.00933039
$247$	− 3128.00i	− 0.805789i
$248$	− 448.000i	− 0.114710i
$249$	684.000	0.174083
$250$	0	0
$251$	180.000	0.0452649	0.0226325	−	0.999744i	$-0.492795\pi$
$251$	180.000	0.0452649	0.0226325	+	0.999744i	$0.492795\pi$
$252$	252.000i	0.0629941i
$253$	− 12960.0i	− 3.22051i
$254$	−2128.00	−0.525680
$255$	0	0
$256$	256.000	0.0625000
$257$	5310.00i	1.28883i	0.764677	+	0.644414i	$0.222899\pi$
$257$	5310.00i	1.28883i	−0.764677	+	0.644414i	$0.777101\pi$
$258$	984.000i	0.237446i
$259$	238.000	0.0570988
$260$	0	0
$261$	−1026.00	−0.243325
$262$	360.000i	0.0848888i
$263$	− 828.000i	− 0.194132i	−0.995278	−	0.0970659i	$-0.969054\pi$
$263$	− 828.000i	− 0.194132i	0.995278	−	0.0970659i	$-0.0309458\pi$
$264$	−1728.00	−0.402845
$265$	0	0
$266$	1288.00	0.296889
$267$	− 1170.00i	− 0.268175i
$268$	496.000i	0.113052i
$269$	4134.00	0.937005	0.468503	−	0.883462i	$-0.344794\pi$
$269$	4134.00	0.937005	0.468503	+	0.883462i	$0.344794\pi$
$270$	0	0
$271$	−2968.00	−0.665288	−0.332644	−	0.943052i	$-0.607941\pi$
$271$	−2968.00	−0.665288	−0.332644	+	0.943052i	$0.607941\pi$
$272$	96.0000i	0.0214002i
$273$	− 714.000i	− 0.158290i
$274$	5436.00	1.19854
$275$	0	0
$276$	2160.00	0.471075
$277$	− 4786.00i	− 1.03813i	−0.854734	−	0.519067i	$-0.826280\pi$
$277$	− 4786.00i	− 1.03813i	0.854734	−	0.519067i	$-0.173720\pi$
$278$	2696.00i	0.581638i
$279$	−504.000	−0.108149
$280$	0	0
$281$	−4398.00	−0.933675	−0.466838	−	0.884343i	$-0.654607\pi$
$281$	−4398.00	−0.933675	−0.466838	+	0.884343i	$0.654607\pi$
$282$	− 1008.00i	− 0.212856i
$283$	− 4772.00i	− 1.00235i	−0.865345	−	0.501177i	$-0.832901\pi$
$283$	− 4772.00i	− 1.00235i	0.865345	−	0.501177i	$-0.167099\pi$
$284$	−144.000	−0.0300874
$285$	0	0
$286$	4896.00	1.01226
$287$	42.0000i	0.00863826i
$288$	− 288.000i	− 0.0589256i
$289$	4877.00	0.992673
$290$	0	0
$291$	210.000	0.0423038
$292$	4040.00i	0.809668i
$293$	− 6522.00i	− 1.30041i	−0.759760	−	0.650204i	$-0.774684\pi$
$293$	− 6522.00i	− 1.30041i	0.759760	−	0.650204i	$-0.225316\pi$
$294$	294.000	0.0583212
$295$	0	0
$296$	−272.000	−0.0534111
$297$	1944.00i	0.379806i
$298$	− 1116.00i	− 0.216940i
$299$	−6120.00	−1.18371
$300$	0	0
$301$	1148.00	0.219833
$302$	3856.00i	0.734728i
$303$	− 4050.00i	− 0.767876i
$304$	−1472.00	−0.277714
$305$	0	0
$306$	108.000	0.0201763
$307$	− 6244.00i	− 1.16079i	−0.814333	−	0.580397i	$-0.802897\pi$
$307$	− 6244.00i	− 1.16079i	0.814333	−	0.580397i	$-0.197103\pi$
$308$	2016.00i	0.372962i
$309$	6000.00	1.10462
$310$	0	0
$311$	−528.000	−0.0962705	−0.0481353	−	0.998841i	$-0.515328\pi$
$311$	−528.000	−0.0962705	−0.0481353	+	0.998841i	$0.515328\pi$
$312$	816.000i	0.148067i
$313$	5830.00i	1.05281i	0.850232	+	0.526407i	$0.176461\pi$
$313$	5830.00i	1.05281i	−0.850232	+	0.526407i	$0.823539\pi$
$314$	4820.00	0.866269
$315$	0	0
$316$	224.000	0.0398765
$317$	5046.00i	0.894043i	0.894523	+	0.447021i	$0.147515\pi$
$317$	5046.00i	0.894043i	−0.894523	+	0.447021i	$0.852485\pi$
$318$	3924.00i	0.691972i
$319$	−8208.00	−1.44063
$320$	0	0
$321$	−2088.00	−0.363055
$322$	− 2520.00i	− 0.436131i
$323$	− 552.000i	− 0.0950901i
$324$	−324.000	−0.0555556
$325$	0	0
$326$	1480.00	0.251441
$327$	3342.00i	0.565177i
$328$	− 48.0000i	− 0.00808036i
$329$	−1176.00	−0.197067
$330$	0	0
$331$	−5020.00	−0.833608	−0.416804	−	0.908996i	$-0.636850\pi$
$331$	−5020.00	−0.833608	−0.416804	+	0.908996i	$0.636850\pi$
$332$	912.000i	0.150761i
$333$	306.000i	0.0503564i
$334$	−7968.00	−1.30536
$335$	0	0
$336$	−336.000	−0.0545545
$337$	− 7486.00i	− 1.21005i	−0.796205	−	0.605027i	$-0.793162\pi$
$337$	− 7486.00i	− 1.21005i	0.796205	−	0.605027i	$-0.206838\pi$
$338$	2082.00i	0.335047i
$339$	−1386.00	−0.222057
$340$	0	0
$341$	−4032.00	−0.640308
$342$	1656.00i	0.261831i
$343$	− 343.000i	− 0.0539949i
$344$	−1312.00	−0.205635
$345$	0	0
$346$	−2076.00	−0.322562
$347$	− 10032.0i	− 1.55201i	−0.630729	−	0.776003i	$-0.717244\pi$
$347$	− 10032.0i	− 1.55201i	0.630729	−	0.776003i	$-0.282756\pi$
$348$	− 1368.00i	− 0.210726i
$349$	−5942.00	−0.911370	−0.455685	−	0.890141i	$-0.650606\pi$
$349$	−5942.00	−0.911370	−0.455685	+	0.890141i	$0.650606\pi$
$350$	0	0
$351$	918.000	0.139599
$352$	− 2304.00i	− 0.348874i
$353$	90.0000i	0.0135700i	0.999977	+	0.00678501i	$0.00215975\pi$
$353$	90.0000i	0.0135700i	−0.999977	+	0.00678501i	$0.997840\pi$
$354$	−2952.00	−0.443212
$355$	0	0
$356$	1560.00	0.232247
$357$	− 126.000i	− 0.0186796i
$358$	5136.00i	0.758229i
$359$	−10596.0	−1.55776	−0.778880	−	0.627174i	$-0.784212\pi$
$359$	−10596.0	−1.55776	−0.778880	+	0.627174i	$0.784212\pi$
$360$	0	0
$361$	1605.00	0.233999
$362$	− 5396.00i	− 0.783446i
$363$	11559.0i	1.67132i
$364$	952.000	0.137083
$365$	0	0
$366$	1500.00	0.214225
$367$	4016.00i	0.571208i	0.958348	+	0.285604i	$0.0921943\pi$
$367$	4016.00i	0.571208i	−0.958348	+	0.285604i	$0.907806\pi$
$368$	2880.00i	0.407963i
$369$	−54.0000	−0.00761823
$370$	0	0
$371$	4578.00	0.640641
$372$	− 672.000i	− 0.0936602i
$373$	− 3278.00i	− 0.455036i	−0.973774	−	0.227518i	$-0.926939\pi$
$373$	− 3278.00i	− 0.455036i	0.973774	−	0.227518i	$-0.0730610\pi$
$374$	864.000	0.119456
$375$	0	0
$376$	1344.00	0.184339
$377$	3876.00i	0.529507i
$378$	378.000i	0.0514344i
$379$	−4628.00	−0.627241	−0.313621	−	0.949548i	$-0.601542\pi$
$379$	−4628.00	−0.627241	−0.313621	+	0.949548i	$0.601542\pi$
$380$	0	0
$381$	−3192.00	−0.429216
$382$	− 8232.00i	− 1.10258i
$383$	2880.00i	0.384233i	0.981372	+	0.192116i	$0.0615351\pi$
$383$	2880.00i	0.384233i	−0.981372	+	0.192116i	$0.938465\pi$
$384$	384.000	0.0510310
$385$	0	0
$386$	−6620.00	−0.872925
$387$	1476.00i	0.193874i
$388$	280.000i	0.0366362i
$389$	−7974.00	−1.03933	−0.519663	−	0.854371i	$-0.673943\pi$
$389$	−7974.00	−1.03933	−0.519663	+	0.854371i	$0.673943\pi$
$390$	0	0
$391$	−1080.00	−0.139688
$392$	392.000i	0.0505076i
$393$	540.000i	0.0693114i
$394$	−2556.00	−0.326826
$395$	0	0
$396$	−2592.00	−0.328921
$397$	− 12346.0i	− 1.56078i	−0.625296	−	0.780388i	$-0.715022\pi$
$397$	− 12346.0i	− 1.56078i	0.625296	−	0.780388i	$-0.284978\pi$
$398$	− 5872.00i	− 0.739540i
$399$	1932.00	0.242408
$400$	0	0
$401$	9738.00	1.21270	0.606350	−	0.795198i	$-0.292633\pi$
$401$	9738.00	1.21270	0.606350	+	0.795198i	$0.292633\pi$
$402$	744.000i	0.0923068i
$403$	1904.00i	0.235347i
$404$	5400.00	0.665000
$405$	0	0
$406$	−1596.00	−0.195094
$407$	2448.00i	0.298140i
$408$	144.000i	0.0174732i
$409$	430.000	0.0519857	0.0259928	−	0.999662i	$-0.491725\pi$
$409$	430.000	0.0519857	0.0259928	+	0.999662i	$0.491725\pi$
$410$	0	0
$411$	8154.00	0.978606
$412$	8000.00i	0.956630i
$413$	3444.00i	0.410335i
$414$	3240.00	0.384631
$415$	0	0
$416$	−1088.00	−0.128230
$417$	4044.00i	0.474905i
$418$	13248.0i	1.55019i
$419$	1812.00	0.211270	0.105635	−	0.994405i	$-0.466313\pi$
$419$	1812.00	0.211270	0.105635	+	0.994405i	$0.466313\pi$
$420$	0	0
$421$	−10690.0	−1.23753	−0.618763	−	0.785577i	$-0.712366\pi$
$421$	−10690.0	−1.23753	−0.618763	+	0.785577i	$0.712366\pi$
$422$	− 7016.00i	− 0.809321i
$423$	− 1512.00i	− 0.173797i
$424$	−5232.00	−0.599265
$425$	0	0
$426$	−216.000	−0.0245663
$427$	− 1750.00i	− 0.198334i
$428$	− 2784.00i	− 0.314415i
$429$	7344.00	0.826507
$430$	0	0
$431$	−4116.00	−0.460002	−0.230001	−	0.973190i	$-0.573873\pi$
$431$	−4116.00	−0.460002	−0.230001	+	0.973190i	$0.573873\pi$
$432$	− 432.000i	− 0.0481125i
$433$	− 9938.00i	− 1.10298i	−0.834182	−	0.551489i	$-0.814060\pi$
$433$	− 9938.00i	− 1.10298i	0.834182	−	0.551489i	$-0.185940\pi$
$434$	−784.000	−0.0867125
$435$	0	0
$436$	−4456.00	−0.489458
$437$	− 16560.0i	− 1.81275i
$438$	6060.00i	0.661091i
$439$	−1784.00	−0.193954	−0.0969769	−	0.995287i	$-0.530917\pi$
$439$	−1784.00	−0.193954	−0.0969769	+	0.995287i	$0.530917\pi$
$440$	0	0
$441$	441.000	0.0476190
$442$	− 408.000i	− 0.0439063i
$443$	11712.0i	1.25610i	0.778172	+	0.628052i	$0.216147\pi$
$443$	11712.0i	1.25610i	−0.778172	+	0.628052i	$0.783853\pi$
$444$	−408.000	−0.0436100
$445$	0	0
$446$	−3776.00	−0.400894
$447$	− 1674.00i	− 0.177131i
$448$	− 448.000i	− 0.0472456i
$449$	−7650.00	−0.804066	−0.402033	−	0.915625i	$-0.631696\pi$
$449$	−7650.00	−0.804066	−0.402033	+	0.915625i	$0.631696\pi$
$450$	0	0
$451$	−432.000	−0.0451044
$452$	− 1848.00i	− 0.192307i
$453$	5784.00i	0.599903i
$454$	−7128.00	−0.736858
$455$	0	0
$456$	−2208.00	−0.226752
$457$	3674.00i	0.376067i	0.982163	+	0.188033i	$0.0602113\pi$
$457$	3674.00i	0.376067i	−0.982163	+	0.188033i	$0.939789\pi$
$458$	− 2668.00i	− 0.272200i
$459$	162.000	0.0164739
$460$	0	0
$461$	−3102.00	−0.313394	−0.156697	−	0.987647i	$-0.550085\pi$
$461$	−3102.00	−0.313394	−0.156697	+	0.987647i	$0.550085\pi$
$462$	3024.00i	0.304522i
$463$	− 8984.00i	− 0.901775i	−0.892581	−	0.450888i	$-0.851108\pi$
$463$	− 8984.00i	− 0.901775i	0.892581	−	0.450888i	$-0.148892\pi$
$464$	1824.00	0.182494
$465$	0	0
$466$	5316.00	0.528453
$467$	3612.00i	0.357909i	0.983857	+	0.178954i	$0.0572715\pi$
$467$	3612.00i	0.357909i	−0.983857	+	0.178954i	$0.942729\pi$
$468$	1224.00i	0.120896i
$469$	868.000	0.0854595
$470$	0	0
$471$	7230.00	0.707305
$472$	− 3936.00i	− 0.383833i
$473$	11808.0i	1.14785i
$474$	336.000	0.0325591
$475$	0	0
$476$	168.000	0.0161770
$477$	5886.00i	0.564993i
$478$	1176.00i	0.112529i
$479$	9288.00	0.885970	0.442985	−	0.896529i	$-0.353920\pi$
$479$	9288.00	0.885970	0.442985	+	0.896529i	$0.353920\pi$
$480$	0	0
$481$	1156.00	0.109582
$482$	11380.0i	1.07540i
$483$	− 3780.00i	− 0.356099i
$484$	−15412.0	−1.44741
$485$	0	0
$486$	−486.000	−0.0453609
$487$	− 5848.00i	− 0.544144i	−0.962277	−	0.272072i	$-0.912291\pi$
$487$	− 5848.00i	− 0.544144i	0.962277	−	0.272072i	$-0.0877089\pi$
$488$	2000.00i	0.185524i
$489$	2220.00	0.205300
$490$	0	0
$491$	−5952.00	−0.547067	−0.273534	−	0.961862i	$-0.588192\pi$
$491$	−5952.00	−0.547067	−0.273534	+	0.961862i	$0.588192\pi$
$492$	− 72.0000i	− 0.00659758i
$493$	684.000i	0.0624864i
$494$	6256.00	0.569779
$495$	0	0
$496$	896.000	0.0811121
$497$	252.000i	0.0227440i
$498$	1368.00i	0.123095i
$499$	−10748.0	−0.964222	−0.482111	−	0.876110i	$-0.660130\pi$
$499$	−10748.0	−0.964222	−0.482111	+	0.876110i	$0.660130\pi$
$500$	0	0
$501$	−11952.0	−1.06582
$502$	360.000i	0.0320071i
$503$	16488.0i	1.46156i	0.682614	+	0.730779i	$0.260843\pi$
$503$	16488.0i	1.46156i	−0.682614	+	0.730779i	$0.739157\pi$
$504$	−504.000	−0.0445435
$505$	0	0
$506$	25920.0	2.27724
$507$	3123.00i	0.273565i
$508$	− 4256.00i	− 0.371712i
$509$	−14058.0	−1.22418	−0.612092	−	0.790786i	$-0.709672\pi$
$509$	−14058.0	−1.22418	−0.612092	+	0.790786i	$0.709672\pi$
$510$	0	0
$511$	7070.00	0.612052
$512$	512.000i	0.0441942i
$513$	2484.00i	0.213784i
$514$	−10620.0	−0.911339
$515$	0	0
$516$	−1968.00	−0.167900
$517$	− 12096.0i	− 1.02898i
$518$	476.000i	0.0403750i
$519$	−3114.00	−0.263371
$520$	0	0
$521$	−14466.0	−1.21644	−0.608222	−	0.793767i	$-0.708117\pi$
$521$	−14466.0	−1.21644	−0.608222	+	0.793767i	$0.708117\pi$
$522$	− 2052.00i	− 0.172057i
$523$	− 18524.0i	− 1.54875i	−0.632725	−	0.774377i	$-0.718064\pi$
$523$	− 18524.0i	− 1.54875i	0.632725	−	0.774377i	$-0.281936\pi$
$524$	−720.000	−0.0600255
$525$	0	0
$526$	1656.00	0.137272
$527$	336.000i	0.0277730i
$528$	− 3456.00i	− 0.284854i
$529$	−20233.0	−1.66294
$530$	0	0
$531$	−4428.00	−0.361881
$532$	2576.00i	0.209932i
$533$	204.000i	0.0165783i
$534$	2340.00	0.189629
$535$	0	0
$536$	−992.000	−0.0799401
$537$	7704.00i	0.619092i
$538$	8268.00i	0.662563i
$539$	3528.00	0.281933
$540$	0	0
$541$	4358.00	0.346331	0.173165	−	0.984893i	$-0.444600\pi$
$541$	4358.00	0.346331	0.173165	+	0.984893i	$0.444600\pi$
$542$	− 5936.00i	− 0.470430i
$543$	− 8094.00i	− 0.639681i
$544$	−192.000	−0.0151322
$545$	0	0
$546$	1428.00	0.111928
$547$	− 2140.00i	− 0.167276i	−0.996496	−	0.0836378i	$-0.973346\pi$
$547$	− 2140.00i	− 0.167276i	0.996496	−	0.0836378i	$-0.0266539\pi$
$548$	10872.0i	0.847498i
$549$	2250.00	0.174914
$550$	0	0
$551$	−10488.0	−0.810896
$552$	4320.00i	0.333100i
$553$	− 392.000i	− 0.0301438i
$554$	9572.00	0.734071
$555$	0	0
$556$	−5392.00	−0.411280
$557$	2022.00i	0.153815i	0.997038	+	0.0769074i	$0.0245046\pi$
$557$	2022.00i	0.153815i	−0.997038	+	0.0769074i	$0.975495\pi$
$558$	− 1008.00i	− 0.0764732i
$559$	5576.00	0.421896
$560$	0	0
$561$	1296.00	0.0975350
$562$	− 8796.00i	− 0.660208i
$563$	− 7356.00i	− 0.550654i	−0.961351	−	0.275327i	$-0.911214\pi$
$563$	− 7356.00i	− 0.550654i	0.961351	−	0.275327i	$-0.0887862\pi$
$564$	2016.00	0.150512
$565$	0	0
$566$	9544.00	0.708771
$567$	567.000i	0.0419961i
$568$	− 288.000i	− 0.0212750i
$569$	−11202.0	−0.825329	−0.412665	−	0.910883i	$-0.635402\pi$
$569$	−11202.0	−0.825329	−0.412665	+	0.910883i	$0.635402\pi$
$570$	0	0
$571$	−10564.0	−0.774238	−0.387119	−	0.922030i	$-0.626530\pi$
$571$	−10564.0	−0.774238	−0.387119	+	0.922030i	$0.626530\pi$
$572$	9792.00i	0.715776i
$573$	− 12348.0i	− 0.900253i
$574$	−84.0000	−0.00610817
$575$	0	0
$576$	576.000	0.0416667
$577$	− 18574.0i	− 1.34011i	−0.742310	−	0.670057i	$-0.766270\pi$
$577$	− 18574.0i	− 1.34011i	0.742310	−	0.670057i	$-0.233730\pi$
$578$	9754.00i	0.701925i
$579$	−9930.00	−0.712740
$580$	0	0
$581$	1596.00	0.113964
$582$	420.000i	0.0299133i
$583$	47088.0i	3.34509i
$584$	−8080.00	−0.572522
$585$	0	0
$586$	13044.0	0.919527
$587$	13188.0i	0.927303i	0.886018	+	0.463652i	$0.153461\pi$
$587$	13188.0i	0.927303i	−0.886018	+	0.463652i	$0.846539\pi$
$588$	588.000i	0.0412393i
$589$	−5152.00	−0.360415
$590$	0	0
$591$	−3834.00	−0.266852
$592$	− 544.000i	− 0.0377673i
$593$	22506.0i	1.55853i	0.626692	+	0.779267i	$0.284408\pi$
$593$	22506.0i	1.55853i	−0.626692	+	0.779267i	$0.715592\pi$
$594$	−3888.00	−0.268563
$595$	0	0
$596$	2232.00	0.153400
$597$	− 8808.00i	− 0.603832i
$598$	− 12240.0i	− 0.837008i
$599$	−10596.0	−0.722773	−0.361386	−	0.932416i	$-0.617696\pi$
$599$	−10596.0	−0.722773	−0.361386	+	0.932416i	$0.617696\pi$
$600$	0	0
$601$	14618.0	0.992148	0.496074	−	0.868280i	$-0.334775\pi$
$601$	14618.0	0.992148	0.496074	+	0.868280i	$0.334775\pi$
$602$	2296.00i	0.155445i
$603$	1116.00i	0.0753682i
$604$	−7712.00	−0.519531
$605$	0	0
$606$	8100.00	0.542970
$607$	5168.00i	0.345573i	0.984959	+	0.172786i	$0.0552770\pi$
$607$	5168.00i	0.345573i	−0.984959	+	0.172786i	$0.944723\pi$
$608$	− 2944.00i	− 0.196373i
$609$	−2394.00	−0.159294
$610$	0	0
$611$	−5712.00	−0.378204
$612$	216.000i	0.0142668i
$613$	− 5726.00i	− 0.377277i	−0.982047	−	0.188639i	$-0.939593\pi$
$613$	− 5726.00i	− 0.377277i	0.982047	−	0.188639i	$-0.0604075\pi$
$614$	12488.0	0.820806
$615$	0	0
$616$	−4032.00	−0.263724
$617$	− 7806.00i	− 0.509332i	−0.967029	−	0.254666i	$-0.918035\pi$
$617$	− 7806.00i	− 0.509332i	0.967029	−	0.254666i	$-0.0819655\pi$
$618$	12000.0i	0.781085i
$619$	18052.0	1.17217	0.586083	−	0.810251i	$-0.300669\pi$
$619$	18052.0	1.17217	0.586083	+	0.810251i	$0.300669\pi$
$620$	0	0
$621$	4860.00	0.314050
$622$	− 1056.00i	− 0.0680735i
$623$	− 2730.00i	− 0.175562i
$624$	−1632.00	−0.104699
$625$	0	0
$626$	−11660.0	−0.744453
$627$	19872.0i	1.26573i
$628$	9640.00i	0.612544i
$629$	204.000	0.0129317
$630$	0	0
$631$	−6208.00	−0.391659	−0.195829	−	0.980638i	$-0.562740\pi$
$631$	−6208.00	−0.391659	−0.195829	+	0.980638i	$0.562740\pi$
$632$	448.000i	0.0281970i
$633$	− 10524.0i	− 0.660808i
$634$	−10092.0	−0.632184
$635$	0	0
$636$	−7848.00	−0.489298
$637$	− 1666.00i	− 0.103625i
$638$	− 16416.0i	− 1.01868i
$639$	−324.000	−0.0200583
$640$	0	0
$641$	−21510.0	−1.32542	−0.662710	−	0.748876i	$-0.730594\pi$
$641$	−21510.0	−1.32542	−0.662710	+	0.748876i	$0.730594\pi$
$642$	− 4176.00i	− 0.256719i
$643$	11140.0i	0.683233i	0.939839	+	0.341616i	$0.110974\pi$
$643$	11140.0i	0.683233i	−0.939839	+	0.341616i	$0.889026\pi$
$644$	5040.00	0.308391
$645$	0	0
$646$	1104.00	0.0672389
$647$	9312.00i	0.565831i	0.959145	+	0.282915i	$0.0913016\pi$
$647$	9312.00i	0.565831i	−0.959145	+	0.282915i	$0.908698\pi$
$648$	− 648.000i	− 0.0392837i
$649$	−35424.0	−2.14255
$650$	0	0
$651$	−1176.00	−0.0708004
$652$	2960.00i	0.177795i
$653$	− 4878.00i	− 0.292329i	−0.989260	−	0.146165i	$-0.953307\pi$
$653$	− 4878.00i	− 0.292329i	0.989260	−	0.146165i	$-0.0466929\pi$
$654$	−6684.00	−0.399641
$655$	0	0
$656$	96.0000	0.00571367
$657$	9090.00i	0.539779i
$658$	− 2352.00i	− 0.139347i
$659$	9744.00	0.575982	0.287991	−	0.957633i	$-0.407013\pi$
$659$	9744.00	0.575982	0.287991	+	0.957633i	$0.407013\pi$
$660$	0	0
$661$	2990.00	0.175942	0.0879709	−	0.996123i	$-0.471962\pi$
$661$	2990.00	0.175942	0.0879709	+	0.996123i	$0.471962\pi$
$662$	− 10040.0i	− 0.589450i
$663$	− 612.000i	− 0.0358493i
$664$	−1824.00	−0.106604
$665$	0	0
$666$	−612.000	−0.0356074
$667$	20520.0i	1.19121i
$668$	− 15936.0i	− 0.923027i
$669$	−5664.00	−0.327329
$670$	0	0
$671$	18000.0	1.03559
$672$	− 672.000i	− 0.0385758i
$673$	− 33266.0i	− 1.90536i	−0.303969	−	0.952682i	$-0.598312\pi$
$673$	− 33266.0i	− 1.90536i	0.303969	−	0.952682i	$-0.401688\pi$
$674$	14972.0	0.855638
$675$	0	0
$676$	−4164.00	−0.236914
$677$	5370.00i	0.304854i	0.988315	+	0.152427i	$0.0487088\pi$
$677$	5370.00i	0.304854i	−0.988315	+	0.152427i	$0.951291\pi$
$678$	− 2772.00i	− 0.157018i
$679$	490.000	0.0276944
$680$	0	0
$681$	−10692.0	−0.601642
$682$	− 8064.00i	− 0.452766i
$683$	− 384.000i	− 0.0215130i	−0.999942	−	0.0107565i	$-0.996576\pi$
$683$	− 384.000i	− 0.0215130i	0.999942	−	0.0107565i	$-0.00342396\pi$
$684$	−3312.00	−0.185143
$685$	0	0
$686$	686.000	0.0381802
$687$	− 4002.00i	− 0.222250i
$688$	− 2624.00i	− 0.145406i
$689$	22236.0	1.22950
$690$	0	0
$691$	−14524.0	−0.799593	−0.399797	−	0.916604i	$-0.630919\pi$
$691$	−14524.0	−0.799593	−0.399797	+	0.916604i	$0.630919\pi$
$692$	− 4152.00i	− 0.228086i
$693$	4536.00i	0.248641i
$694$	20064.0	1.09743
$695$	0	0
$696$	2736.00	0.149005
$697$	36.0000i	0.00195638i
$698$	− 11884.0i	− 0.644436i
$699$	7974.00	0.431480
$700$	0	0
$701$	24750.0	1.33352	0.666758	−	0.745274i	$-0.267682\pi$
$701$	24750.0	1.33352	0.666758	+	0.745274i	$0.267682\pi$
$702$	1836.00i	0.0987113i
$703$	3128.00i	0.167816i
$704$	4608.00	0.246691
$705$	0	0
$706$	−180.000	−0.00959545
$707$	− 9450.00i	− 0.502693i
$708$	− 5904.00i	− 0.313398i
$709$	1042.00	0.0551948	0.0275974	−	0.999619i	$-0.491214\pi$
$709$	1042.00	0.0551948	0.0275974	+	0.999619i	$0.491214\pi$
$710$	0	0
$711$	504.000	0.0265844
$712$	3120.00i	0.164223i
$713$	10080.0i	0.529452i
$714$	252.000	0.0132085
$715$	0	0
$716$	−10272.0	−0.536149
$717$	1764.00i	0.0918798i
$718$	− 21192.0i	− 1.10150i
$719$	36960.0	1.91707	0.958536	−	0.284970i	$-0.0919836\pi$
$719$	36960.0	1.91707	0.958536	+	0.284970i	$0.0919836\pi$
$720$	0	0
$721$	14000.0	0.723145
$722$	3210.00i	0.165462i
$723$	17070.0i	0.878064i
$724$	10792.0	0.553980
$725$	0	0
$726$	−23118.0	−1.18180
$727$	− 16288.0i	− 0.830933i	−0.909608	−	0.415467i	$-0.863618\pi$
$727$	− 16288.0i	− 0.830933i	0.909608	−	0.415467i	$-0.136382\pi$
$728$	1904.00i	0.0969326i
$729$	−729.000	−0.0370370
$730$	0	0
$731$	984.000	0.0497874
$732$	3000.00i	0.151480i
$733$	7810.00i	0.393546i	0.980449	+	0.196773i	$0.0630461\pi$
$733$	7810.00i	0.393546i	−0.980449	+	0.196773i	$0.936954\pi$
$734$	−8032.00	−0.403905
$735$	0	0
$736$	−5760.00	−0.288473
$737$	8928.00i	0.446224i
$738$	− 108.000i	− 0.00538690i
$739$	36700.0	1.82684	0.913418	−	0.407024i	$-0.133433\pi$
$739$	36700.0	1.82684	0.913418	+	0.407024i	$0.133433\pi$
$740$	0	0
$741$	9384.00	0.465222
$742$	9156.00i	0.453002i
$743$	− 29508.0i	− 1.45699i	−0.685051	−	0.728495i	$-0.740220\pi$
$743$	− 29508.0i	− 1.45699i	0.685051	−	0.728495i	$-0.259780\pi$
$744$	1344.00	0.0662277
$745$	0	0
$746$	6556.00	0.321759
$747$	2052.00i	0.100507i
$748$	1728.00i	0.0844678i
$749$	−4872.00	−0.237676
$750$	0	0
$751$	−15136.0	−0.735447	−0.367723	−	0.929935i	$-0.619863\pi$
$751$	−15136.0	−0.735447	−0.367723	+	0.929935i	$0.619863\pi$
$752$	2688.00i	0.130347i
$753$	540.000i	0.0261337i
$754$	−7752.00	−0.374418
$755$	0	0
$756$	−756.000	−0.0363696
$757$	3422.00i	0.164299i	0.996620	+	0.0821497i	$0.0261786\pi$
$757$	3422.00i	0.164299i	−0.996620	+	0.0821497i	$0.973821\pi$
$758$	− 9256.00i	− 0.443526i
$759$	38880.0	1.85936
$760$	0	0
$761$	31446.0	1.49792	0.748960	−	0.662616i	$-0.230554\pi$
$761$	31446.0	1.49792	0.748960	+	0.662616i	$0.230554\pi$
$762$	− 6384.00i	− 0.303501i
$763$	7798.00i	0.369995i
$764$	16464.0	0.779642
$765$	0	0
$766$	−5760.00	−0.271694
$767$	16728.0i	0.787501i
$768$	768.000i	0.0360844i
$769$	18718.0	0.877748	0.438874	−	0.898549i	$-0.355377\pi$
$769$	18718.0	0.877748	0.438874	+	0.898549i	$0.355377\pi$
$770$	0	0
$771$	−15930.0	−0.744105
$772$	− 13240.0i	− 0.617251i
$773$	1686.00i	0.0784492i	0.999230	+	0.0392246i	$0.0124888\pi$
$773$	1686.00i	0.0784492i	−0.999230	+	0.0392246i	$0.987511\pi$
$774$	−2952.00	−0.137090
$775$	0	0
$776$	−560.000	−0.0259057
$777$	714.000i	0.0329660i
$778$	− 15948.0i	− 0.734915i
$779$	−552.000	−0.0253883
$780$	0	0
$781$	−2592.00	−0.118757
$782$	− 2160.00i	− 0.0987742i
$783$	− 3078.00i	− 0.140484i
$784$	−784.000	−0.0357143
$785$	0	0
$786$	−1080.00	−0.0490106
$787$	5492.00i	0.248753i	0.992235	+	0.124377i	$0.0396931\pi$
$787$	5492.00i	0.248753i	−0.992235	+	0.124377i	$0.960307\pi$
$788$	− 5112.00i	− 0.231101i
$789$	2484.00	0.112082
$790$	0	0
$791$	−3234.00	−0.145370
$792$	− 5184.00i	− 0.232583i
$793$	− 8500.00i	− 0.380635i
$794$	24692.0	1.10364
$795$	0	0
$796$	11744.0	0.522933
$797$	− 17310.0i	− 0.769325i	−0.923057	−	0.384662i	$-0.874318\pi$
$797$	− 17310.0i	− 0.769325i	0.923057	−	0.384662i	$-0.125682\pi$
$798$	3864.00i	0.171409i
$799$	−1008.00	−0.0446314
$800$	0	0
$801$	3510.00	0.154831
$802$	19476.0i	0.857508i
$803$	72720.0i	3.19581i
$804$	−1488.00	−0.0652708
$805$	0	0
$806$	−3808.00	−0.166416
$807$	12402.0i	0.540980i
$808$	10800.0i	0.470226i
$809$	−35754.0	−1.55382	−0.776912	−	0.629609i	$-0.783215\pi$
$809$	−35754.0	−1.55382	−0.776912	+	0.629609i	$0.783215\pi$
$810$	0	0
$811$	33644.0	1.45672	0.728360	−	0.685194i	$-0.240283\pi$
$811$	33644.0	1.45672	0.728360	+	0.685194i	$0.240283\pi$
$812$	− 3192.00i	− 0.137952i
$813$	− 8904.00i	− 0.384104i
$814$	−4896.00	−0.210817
$815$	0	0
$816$	−288.000	−0.0123554
$817$	15088.0i	0.646098i
$818$	860.000i	0.0367594i
$819$	2142.00	0.0913889
$820$	0	0
$821$	28734.0	1.22147	0.610733	−	0.791837i	$-0.290875\pi$
$821$	28734.0	1.22147	0.610733	+	0.791837i	$0.290875\pi$
$822$	16308.0i	0.691979i
$823$	28672.0i	1.21439i	0.794553	+	0.607195i	$0.207705\pi$
$823$	28672.0i	1.21439i	−0.794553	+	0.607195i	$0.792295\pi$
$824$	−16000.0	−0.676440
$825$	0	0
$826$	−6888.00	−0.290150
$827$	− 15912.0i	− 0.669062i	−0.942385	−	0.334531i	$-0.891422\pi$
$827$	− 15912.0i	− 0.669062i	0.942385	−	0.334531i	$-0.108578\pi$
$828$	6480.00i	0.271975i
$829$	−17534.0	−0.734597	−0.367299	−	0.930103i	$-0.619717\pi$
$829$	−17534.0	−0.734597	−0.367299	+	0.930103i	$0.619717\pi$
$830$	0	0
$831$	14358.0	0.599366
$832$	− 2176.00i	− 0.0906721i
$833$	− 294.000i	− 0.0122287i
$834$	−8088.00	−0.335809
$835$	0	0
$836$	−26496.0	−1.09615
$837$	− 1512.00i	− 0.0624401i
$838$	3624.00i	0.149390i
$839$	−40656.0	−1.67295	−0.836473	−	0.548009i	$-0.815386\pi$
$839$	−40656.0	−1.67295	−0.836473	+	0.548009i	$0.815386\pi$
$840$	0	0
$841$	−11393.0	−0.467137
$842$	− 21380.0i	− 0.875063i
$843$	− 13194.0i	− 0.539058i
$844$	14032.0	0.572276
$845$	0	0
$846$	3024.00	0.122893
$847$	26971.0i	1.09414i
$848$	− 10464.0i	− 0.423744i
$849$	14316.0	0.578709
$850$	0	0
$851$	6120.00	0.246523
$852$	− 432.000i	− 0.0173710i
$853$	− 23870.0i	− 0.958140i	−0.877777	−	0.479070i	$-0.840974\pi$
$853$	− 23870.0i	− 0.958140i	0.877777	−	0.479070i	$-0.159026\pi$
$854$	3500.00	0.140243
$855$	0	0
$856$	5568.00	0.222325
$857$	− 29610.0i	− 1.18023i	−0.807319	−	0.590116i	$-0.799082\pi$
$857$	− 29610.0i	− 1.18023i	0.807319	−	0.590116i	$-0.200918\pi$
$858$	14688.0i	0.584429i
$859$	45484.0	1.80663	0.903314	−	0.428979i	$-0.141127\pi$
$859$	45484.0	1.80663	0.903314	+	0.428979i	$0.141127\pi$
$860$	0	0
$861$	−126.000	−0.00498730
$862$	− 8232.00i	− 0.325270i
$863$	− 46164.0i	− 1.82090i	−0.413614	−	0.910452i	$-0.635734\pi$
$863$	− 46164.0i	− 1.82090i	0.413614	−	0.910452i	$-0.364266\pi$
$864$	864.000	0.0340207
$865$	0	0
$866$	19876.0	0.779924
$867$	14631.0i	0.573120i
$868$	− 1568.00i	− 0.0613150i
$869$	4032.00	0.157395
$870$	0	0
$871$	4216.00	0.164011
$872$	− 8912.00i	− 0.346099i
$873$	630.000i	0.0244241i
$874$	33120.0	1.28181
$875$	0	0
$876$	−12120.0	−0.467462
$877$	− 2986.00i	− 0.114972i	−0.998346	−	0.0574858i	$-0.981692\pi$
$877$	− 2986.00i	− 0.114972i	0.998346	−	0.0574858i	$-0.0183084\pi$
$878$	− 3568.00i	− 0.137146i
$879$	19566.0	0.750790
$880$	0	0
$881$	6534.00	0.249871	0.124935	−	0.992165i	$-0.460128\pi$
$881$	6534.00	0.249871	0.124935	+	0.992165i	$0.460128\pi$
$882$	882.000i	0.0336718i
$883$	− 29756.0i	− 1.13405i	−0.823699	−	0.567027i	$-0.808094\pi$
$883$	− 29756.0i	− 1.13405i	0.823699	−	0.567027i	$-0.191906\pi$
$884$	816.000	0.0310464
$885$	0	0
$886$	−23424.0	−0.888199
$887$	29952.0i	1.13381i	0.823783	+	0.566905i	$0.191859\pi$
$887$	29952.0i	1.13381i	−0.823783	+	0.566905i	$0.808141\pi$
$888$	− 816.000i	− 0.0308369i
$889$	−7448.00	−0.280988
$890$	0	0
$891$	−5832.00	−0.219281
$892$	− 7552.00i	− 0.283475i
$893$	− 15456.0i	− 0.579188i
$894$	3348.00	0.125250
$895$	0	0
$896$	896.000	0.0334077
$897$	− 18360.0i	− 0.683414i
$898$	− 15300.0i	− 0.568561i
$899$	6384.00	0.236839
$900$	0	0
$901$	3924.00	0.145091
$902$	− 864.000i	− 0.0318936i
$903$	3444.00i	0.126920i
$904$	3696.00	0.135981
$905$	0	0
$906$	−11568.0	−0.424195
$907$	− 36268.0i	− 1.32774i	−0.747848	−	0.663869i	$-0.768913\pi$
$907$	− 36268.0i	− 1.32774i	0.747848	−	0.663869i	$-0.231087\pi$
$908$	− 14256.0i	− 0.521037i
$909$	12150.0	0.443333
$910$	0	0
$911$	−23604.0	−0.858436	−0.429218	−	0.903201i	$-0.641211\pi$
$911$	−23604.0	−0.858436	−0.429218	+	0.903201i	$0.641211\pi$
$912$	− 4416.00i	− 0.160338i
$913$	16416.0i	0.595061i
$914$	−7348.00	−0.265919
$915$	0	0
$916$	5336.00	0.192474
$917$	1260.00i	0.0453750i
$918$	324.000i	0.0116488i
$919$	−34184.0	−1.22701	−0.613507	−	0.789689i	$-0.710242\pi$
$919$	−34184.0	−1.22701	−0.613507	+	0.789689i	$0.710242\pi$
$920$	0	0
$921$	18732.0	0.670185
$922$	− 6204.00i	− 0.221603i
$923$	1224.00i	0.0436495i
$924$	−6048.00	−0.215330
$925$	0	0
$926$	17968.0	0.637651
$927$	18000.0i	0.637754i
$928$	3648.00i	0.129043i
$929$	53922.0	1.90433	0.952165	−	0.305583i	$-0.0988513\pi$
$929$	53922.0	1.90433	0.952165	+	0.305583i	$0.0988513\pi$
$930$	0	0
$931$	4508.00	0.158694
$932$	10632.0i	0.373672i
$933$	− 1584.00i	− 0.0555818i
$934$	−7224.00	−0.253080
$935$	0	0
$936$	−2448.00	−0.0854865
$937$	40538.0i	1.41336i	0.707533	+	0.706680i	$0.249808\pi$
$937$	40538.0i	1.41336i	−0.707533	+	0.706680i	$0.750192\pi$
$938$	1736.00i	0.0604290i
$939$	−17490.0	−0.607843
$940$	0	0
$941$	−3606.00	−0.124923	−0.0624613	−	0.998047i	$-0.519895\pi$
$941$	−3606.00	−0.124923	−0.0624613	+	0.998047i	$0.519895\pi$
$942$	14460.0i	0.500140i
$943$	1080.00i	0.0372955i
$944$	7872.00	0.271411
$945$	0	0
$946$	−23616.0	−0.811652
$947$	14064.0i	0.482596i	0.970451	+	0.241298i	$0.0775732\pi$
$947$	14064.0i	0.482596i	−0.970451	+	0.241298i	$0.922427\pi$
$948$	672.000i	0.0230227i
$949$	34340.0	1.17463
$950$	0	0
$951$	−15138.0	−0.516176
$952$	336.000i	0.0114389i
$953$	− 33066.0i	− 1.12394i	−0.827158	−	0.561969i	$-0.810044\pi$
$953$	− 33066.0i	− 1.12394i	0.827158	−	0.561969i	$-0.189956\pi$
$954$	−11772.0	−0.399510
$955$	0	0
$956$	−2352.00	−0.0795702
$957$	− 24624.0i	− 0.831746i
$958$	18576.0i	0.626475i
$959$	19026.0	0.640648
$960$	0	0
$961$	−26655.0	−0.894733
$962$	2312.00i	0.0774864i
$963$	− 6264.00i	− 0.209610i
$964$	−22760.0	−0.760426
$965$	0	0
$966$	7560.00	0.251800
$967$	− 26368.0i	− 0.876875i	−0.898762	−	0.438437i	$-0.855532\pi$
$967$	− 26368.0i	− 0.876875i	0.898762	−	0.438437i	$-0.144468\pi$
$968$	− 30824.0i	− 1.02347i
$969$	1656.00	0.0549003
$970$	0	0
$971$	55884.0	1.84696	0.923482	−	0.383641i	$-0.125330\pi$
$971$	55884.0	1.84696	0.923482	+	0.383641i	$0.125330\pi$
$972$	− 972.000i	− 0.0320750i
$973$	9436.00i	0.310899i
$974$	11696.0	0.384768
$975$	0	0
$976$	−4000.00	−0.131185
$977$	− 51126.0i	− 1.67417i	−0.547072	−	0.837086i	$-0.684257\pi$
$977$	− 51126.0i	− 1.67417i	0.547072	−	0.837086i	$-0.315743\pi$
$978$	4440.00i	0.145169i
$979$	28080.0	0.916691
$980$	0	0
$981$	−10026.0	−0.326305
$982$	− 11904.0i	− 0.386835i
$983$	14184.0i	0.460223i	0.973164	+	0.230112i	$0.0739091\pi$
$983$	14184.0i	0.460223i	−0.973164	+	0.230112i	$0.926091\pi$
$984$	144.000	0.00466520
$985$	0	0
$986$	−1368.00	−0.0441846
$987$	− 3528.00i	− 0.113777i
$988$	12512.0i	0.402894i
$989$	29520.0	0.949122
$990$	0	0
$991$	51680.0	1.65658	0.828289	−	0.560301i	$-0.189314\pi$
$991$	51680.0	1.65658	0.828289	+	0.560301i	$0.189314\pi$
$992$	1792.00i	0.0573549i
$993$	− 15060.0i	− 0.481284i
$994$	−504.000	−0.0160824
$995$	0	0
$996$	−2736.00	−0.0870416
$997$	52094.0i	1.65480i	0.561615	+	0.827399i	$0.310180\pi$
$997$	52094.0i	1.65480i	−0.561615	+	0.827399i	$0.689820\pi$
$998$	− 21496.0i	− 0.681808i
$999$	−918.000	−0.0290733

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1050.4.g.a.799.2		2
5.2	odd	4		1050.4.a.g.1.1		1
5.3	odd	4		42.4.a.a.1.1	✓	1
5.4	even	2	inner	1050.4.g.a.799.1		2
15.8	even	4		126.4.a.a.1.1		1
20.3	even	4		336.4.a.l.1.1		1
35.3	even	12		294.4.e.b.79.1		2
35.13	even	4		294.4.a.i.1.1		1
35.18	odd	12		294.4.e.c.79.1		2
35.23	odd	12		294.4.e.c.67.1		2
35.33	even	12		294.4.e.b.67.1		2
40.3	even	4		1344.4.a.a.1.1		1
40.13	odd	4		1344.4.a.o.1.1		1
60.23	odd	4		1008.4.a.b.1.1		1
105.23	even	12		882.4.g.w.361.1		2
105.38	odd	12		882.4.g.o.667.1		2
105.53	even	12		882.4.g.w.667.1		2
105.68	odd	12		882.4.g.o.361.1		2
105.83	odd	4		882.4.a.g.1.1		1
140.83	odd	4		2352.4.a.a.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
42.4.a.a.1.1	✓	1	5.3	odd	4
126.4.a.a.1.1		1	15.8	even	4
294.4.a.i.1.1		1	35.13	even	4
294.4.e.b.67.1		2	35.33	even	12
294.4.e.b.79.1		2	35.3	even	12
294.4.e.c.67.1		2	35.23	odd	12
294.4.e.c.79.1		2	35.18	odd	12
336.4.a.l.1.1		1	20.3	even	4
882.4.a.g.1.1		1	105.83	odd	4
882.4.g.o.361.1		2	105.68	odd	12
882.4.g.o.667.1		2	105.38	odd	12
882.4.g.w.361.1		2	105.23	even	12
882.4.g.w.667.1		2	105.53	even	12
1008.4.a.b.1.1		1	60.23	odd	4
1050.4.a.g.1.1		1	5.2	odd	4
1050.4.g.a.799.1		2	5.4	even	2	inner
1050.4.g.a.799.2		2	1.1	even	1	trivial
1344.4.a.a.1.1		1	40.3	even	4
1344.4.a.o.1.1		1	40.13	odd	4
2352.4.a.a.1.1		1	140.83	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 1050 = 2 \cdot 3 \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1050.g (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+2.00000i q^{2} +3.00000i q^{3} -4.00000 q^{4} -6.00000 q^{6} +7.00000i q^{7} -8.00000i q^{8} -9.00000 q^{9} +O(q^{10})\)	\(q+2.00000i q^{2} +3.00000i q^{3} -4.00000 q^{4} -6.00000 q^{6} +7.00000i q^{7} -8.00000i q^{8} -9.00000 q^{9} -72.0000 q^{11} -12.0000i q^{12} +34.0000i q^{13} -14.0000 q^{14} +16.0000 q^{16} +6.00000i q^{17} -18.0000i q^{18} -92.0000 q^{19} -21.0000 q^{21} -144.000i q^{22} +180.000i q^{23} +24.0000 q^{24} -68.0000 q^{26} -27.0000i q^{27} -28.0000i q^{28} +114.000 q^{29} +56.0000 q^{31} +32.0000i q^{32} -216.000i q^{33} -12.0000 q^{34} +36.0000 q^{36} -34.0000i q^{37} -184.000i q^{38} -102.000 q^{39} +6.00000 q^{41} -42.0000i q^{42} -164.000i q^{43} +288.000 q^{44} -360.000 q^{46} +168.000i q^{47} +48.0000i q^{48} -49.0000 q^{49} -18.0000 q^{51} -136.000i q^{52} -654.000i q^{53} +54.0000 q^{54} +56.0000 q^{56} -276.000i q^{57} +228.000i q^{58} +492.000 q^{59} -250.000 q^{61} +112.000i q^{62} -63.0000i q^{63} -64.0000 q^{64} +432.000 q^{66} -124.000i q^{67} -24.0000i q^{68} -540.000 q^{69} +36.0000 q^{71} +72.0000i q^{72} -1010.00i q^{73} +68.0000 q^{74} +368.000 q^{76} -504.000i q^{77} -204.000i q^{78} -56.0000 q^{79} +81.0000 q^{81} +12.0000i q^{82} -228.000i q^{83} +84.0000 q^{84} +328.000 q^{86} +342.000i q^{87} +576.000i q^{88} -390.000 q^{89} -238.000 q^{91} -720.000i q^{92} +168.000i q^{93} -336.000 q^{94} -96.0000 q^{96} -70.0000i q^{97} -98.0000i q^{98} +648.000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 8 q^{4} - 12 q^{6} - 18 q^{9}+O(q^{10}) \) 2 * q - 8 * q^4 - 12 * q^6 - 18 * q^9	\( 2 q - 8 q^{4} - 12 q^{6} - 18 q^{9} - 144 q^{11} - 28 q^{14} + 32 q^{16} - 184 q^{19} - 42 q^{21} + 48 q^{24} - 136 q^{26} + 228 q^{29} + 112 q^{31} - 24 q^{34} + 72 q^{36} - 204 q^{39} + 12 q^{41} + 576 q^{44} - 720 q^{46} - 98 q^{49} - 36 q^{51} + 108 q^{54} + 112 q^{56} + 984 q^{59} - 500 q^{61} - 128 q^{64} + 864 q^{66} - 1080 q^{69} + 72 q^{71} + 136 q^{74} + 736 q^{76} - 112 q^{79} + 162 q^{81} + 168 q^{84} + 656 q^{86} - 780 q^{89} - 476 q^{91} - 672 q^{94} - 192 q^{96} + 1296 q^{99}+O(q^{100}) \) 2 * q - 8 * q^4 - 12 * q^6 - 18 * q^9 - 144 * q^11 - 28 * q^14 + 32 * q^16 - 184 * q^19 - 42 * q^21 + 48 * q^24 - 136 * q^26 + 228 * q^29 + 112 * q^31 - 24 * q^34 + 72 * q^36 - 204 * q^39 + 12 * q^41 + 576 * q^44 - 720 * q^46 - 98 * q^49 - 36 * q^51 + 108 * q^54 + 112 * q^56 + 984 * q^59 - 500 * q^61 - 128 * q^64 + 864 * q^66 - 1080 * q^69 + 72 * q^71 + 136 * q^74 + 736 * q^76 - 112 * q^79 + 162 * q^81 + 168 * q^84 + 656 * q^86 - 780 * q^89 - 476 * q^91 - 672 * q^94 - 192 * q^96 + 1296 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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