LMFDB - Embedded newform 1050.4.g.f.799.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1050,4,Mod(799,1050)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://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);

sage: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")

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:traces = [2,0,0,-8,0,-12,0,0,-18,0,32,0,0,28,0,32,0,0,-128] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(19)] == traces)

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 210)
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.f.799.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+2.00000i q^{2} +3.00000i q^{3} -4.00000 q^{4} -6.00000 q^{6} -7.00000i q^{7} -8.00000i q^{8} -9.00000 q^{9} +16.0000 q^{11} -12.0000i q^{12} -58.0000i q^{13} +14.0000 q^{14} +16.0000 q^{16} +34.0000i q^{17} -18.0000i q^{18} -64.0000 q^{19} +21.0000 q^{21} +32.0000i q^{22} +16.0000i q^{23} +24.0000 q^{24} +116.000 q^{26} -27.0000i q^{27} +28.0000i q^{28} -62.0000 q^{29} +60.0000 q^{31} +32.0000i q^{32} +48.0000i q^{33} -68.0000 q^{34} +36.0000 q^{36} +150.000i q^{37} -128.000i q^{38} +174.000 q^{39} +474.000 q^{41} +42.0000i q^{42} +292.000i q^{43} -64.0000 q^{44} -32.0000 q^{46} +240.000i q^{47} +48.0000i q^{48} -49.0000 q^{49} -102.000 q^{51} +232.000i q^{52} +662.000i q^{53} +54.0000 q^{54} -56.0000 q^{56} -192.000i q^{57} -124.000i q^{58} +324.000 q^{59} -514.000 q^{61} +120.000i q^{62} +63.0000i q^{63} -64.0000 q^{64} -96.0000 q^{66} -372.000i q^{67} -136.000i q^{68} -48.0000 q^{69} -412.000 q^{71} +72.0000i q^{72} +770.000i q^{73} -300.000 q^{74} +256.000 q^{76} -112.000i q^{77} +348.000i q^{78} +560.000 q^{79} +81.0000 q^{81} +948.000i q^{82} +852.000i q^{83} -84.0000 q^{84} -584.000 q^{86} -186.000i q^{87} -128.000i q^{88} -1466.00 q^{89} -406.000 q^{91} -64.0000i q^{92} +180.000i q^{93} -480.000 q^{94} -96.0000 q^{96} -178.000i q^{97} -98.0000i q^{98} -144.000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 8 q^{4} - 12 q^{6} - 18 q^{9} + 32 q^{11} + 28 q^{14} + 32 q^{16} - 128 q^{19} + 42 q^{21} + 48 q^{24} + 232 q^{26} - 124 q^{29} + 120 q^{31} - 136 q^{34} + 72 q^{36} + 348 q^{39} + 948 q^{41} - 128 q^{44}+ \cdots - 288 q^{99}+O(q^{100}) $ 2 * q - 8 * q^4 - 12 * q^6 - 18 * q^9 + 32 * q^11 + 28 * q^14 + 32 * q^16 - 128 * q^19 + 42 * q^21 + 48 * q^24 + 232 * q^26 - 124 * q^29 + 120 * q^31 - 136 * q^34 + 72 * q^36 + 348 * q^39 + 948 * q^41 - 128 * q^44 - 64 * q^46 - 98 * q^49 - 204 * q^51 + 108 * q^54 - 112 * q^56 + 648 * q^59 - 1028 * q^61 - 128 * q^64 - 192 * q^66 - 96 * q^69 - 824 * q^71 - 600 * q^74 + 512 * q^76 + 1120 * q^79 + 162 * q^81 - 168 * q^84 - 1168 * q^86 - 2932 * q^89 - 812 * q^91 - 960 * q^94 - 192 * q^96 - 288 * q^99

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
$11$	16.0000	0.438562	0.219281	−	0.975662i	$-0.429629\pi$
$11$	16.0000	0.438562	0.219281	+	0.975662i	$0.429629\pi$
$12$	− 12.0000i	− 0.288675i
$13$	− 58.0000i	− 1.23741i	−0.785624	−	0.618704i	$-0.787658\pi$
$13$	− 58.0000i	− 1.23741i	0.785624	−	0.618704i	$-0.212342\pi$
$14$	14.0000	0.267261
$15$	0	0
$16$	16.0000	0.250000
$17$	34.0000i	0.485071i	0.970143	+	0.242536i	$0.0779791\pi$
$17$	34.0000i	0.485071i	−0.970143	+	0.242536i	$0.922021\pi$
$18$	− 18.0000i	− 0.235702i
$19$	−64.0000	−0.772769	−0.386384	−	0.922338i	$-0.626276\pi$
$19$	−64.0000	−0.772769	−0.386384	+	0.922338i	$0.626276\pi$
$20$	0	0
$21$	21.0000	0.218218
$22$	32.0000i	0.310110i
$23$	16.0000i	0.145054i	0.997366	+	0.0725268i	$0.0231063\pi$
$23$	16.0000i	0.145054i	−0.997366	+	0.0725268i	$0.976894\pi$
$24$	24.0000	0.204124
$25$	0	0
$26$	116.000	0.874980
$27$	− 27.0000i	− 0.192450i
$28$	28.0000i	0.188982i
$29$	−62.0000	−0.397004	−0.198502	−	0.980101i	$-0.563608\pi$
$29$	−62.0000	−0.397004	−0.198502	+	0.980101i	$0.563608\pi$
$30$	0	0
$31$	60.0000	0.347623	0.173812	−	0.984779i	$-0.444392\pi$
$31$	60.0000	0.347623	0.173812	+	0.984779i	$0.444392\pi$
$32$	32.0000i	0.176777i
$33$	48.0000i	0.253204i
$34$	−68.0000	−0.342997
$35$	0	0
$36$	36.0000	0.166667
$37$	150.000i	0.666482i	0.942842	+	0.333241i	$0.108142\pi$
$37$	150.000i	0.666482i	−0.942842	+	0.333241i	$0.891858\pi$
$38$	− 128.000i	− 0.546430i
$39$	174.000	0.714418
$40$	0	0
$41$	474.000	1.80552	0.902761	−	0.430144i	$-0.141537\pi$
$41$	474.000	1.80552	0.902761	+	0.430144i	$0.141537\pi$
$42$	42.0000i	0.154303i
$43$	292.000i	1.03557i	0.855510	+	0.517786i	$0.173244\pi$
$43$	292.000i	1.03557i	−0.855510	+	0.517786i	$0.826756\pi$
$44$	−64.0000	−0.219281
$45$	0	0
$46$	−32.0000	−0.102568
$47$	240.000i	0.744843i	0.928064	+	0.372421i	$0.121472\pi$
$47$	240.000i	0.744843i	−0.928064	+	0.372421i	$0.878528\pi$
$48$	48.0000i	0.144338i
$49$	−49.0000	−0.142857
$50$	0	0
$51$	−102.000	−0.280056
$52$	232.000i	0.618704i
$53$	662.000i	1.71571i	0.513891	+	0.857856i	$0.328204\pi$
$53$	662.000i	1.71571i	−0.513891	+	0.857856i	$0.671796\pi$
$54$	54.0000	0.136083
$55$	0	0
$56$	−56.0000	−0.133631
$57$	− 192.000i	− 0.446158i
$58$	− 124.000i	− 0.280724i
$59$	324.000	0.714936	0.357468	−	0.933925i	$-0.383640\pi$
$59$	324.000	0.714936	0.357468	+	0.933925i	$0.383640\pi$
$60$	0	0
$61$	−514.000	−1.07887	−0.539434	−	0.842028i	$-0.681362\pi$
$61$	−514.000	−1.07887	−0.539434	+	0.842028i	$0.681362\pi$
$62$	120.000i	0.245807i
$63$	63.0000i	0.125988i
$64$	−64.0000	−0.125000
$65$	0	0
$66$	−96.0000	−0.179042
$67$	− 372.000i	− 0.678314i	−0.940730	−	0.339157i	$-0.889858\pi$
$67$	− 372.000i	− 0.678314i	0.940730	−	0.339157i	$-0.110142\pi$
$68$	− 136.000i	− 0.242536i
$69$	−48.0000	−0.0837467
$70$	0	0
$71$	−412.000	−0.688668	−0.344334	−	0.938847i	$-0.611895\pi$
$71$	−412.000	−0.688668	−0.344334	+	0.938847i	$0.611895\pi$
$72$	72.0000i	0.117851i
$73$	770.000i	1.23454i	0.786750	+	0.617272i	$0.211762\pi$
$73$	770.000i	1.23454i	−0.786750	+	0.617272i	$0.788238\pi$
$74$	−300.000	−0.471274
$75$	0	0
$76$	256.000	0.386384
$77$	− 112.000i	− 0.165761i
$78$	348.000i	0.505170i
$79$	560.000	0.797531	0.398765	−	0.917053i	$-0.369439\pi$
$79$	560.000	0.797531	0.398765	+	0.917053i	$0.369439\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	948.000i	1.27670i
$83$	852.000i	1.12674i	0.826206	+	0.563368i	$0.190495\pi$
$83$	852.000i	1.12674i	−0.826206	+	0.563368i	$0.809505\pi$
$84$	−84.0000	−0.109109
$85$	0	0
$86$	−584.000	−0.732260
$87$	− 186.000i	− 0.229210i
$88$	− 128.000i	− 0.155055i
$89$	−1466.00	−1.74602	−0.873009	−	0.487703i	$-0.837835\pi$
$89$	−1466.00	−1.74602	−0.873009	+	0.487703i	$0.837835\pi$
$90$	0	0
$91$	−406.000	−0.467696
$92$	− 64.0000i	− 0.0725268i
$93$	180.000i	0.200700i
$94$	−480.000	−0.526683
$95$	0	0
$96$	−96.0000	−0.102062
$97$	− 178.000i	− 0.186321i	−0.995651	−	0.0931606i	$-0.970303\pi$
$97$	− 178.000i	− 0.186321i	0.995651	−	0.0931606i	$-0.0296970\pi$
$98$	− 98.0000i	− 0.101015i
$99$	−144.000	−0.146187
$100$	0	0
$101$	1278.00	1.25907	0.629533	−	0.776973i	$-0.283246\pi$
$101$	1278.00	1.25907	0.629533	+	0.776973i	$0.283246\pi$
$102$	− 204.000i	− 0.198030i
$103$	296.000i	0.283163i	0.989927	+	0.141581i	$0.0452187\pi$
$103$	296.000i	0.283163i	−0.989927	+	0.141581i	$0.954781\pi$
$104$	−464.000	−0.437490
$105$	0	0
$106$	−1324.00	−1.21319
$107$	1164.00i	1.05166i	0.850588	+	0.525832i	$0.176246\pi$
$107$	1164.00i	1.05166i	−0.850588	+	0.525832i	$0.823754\pi$
$108$	108.000i	0.0962250i
$109$	778.000	0.683659	0.341830	−	0.939762i	$-0.388953\pi$
$109$	778.000	0.683659	0.341830	+	0.939762i	$0.388953\pi$
$110$	0	0
$111$	−450.000	−0.384794
$112$	− 112.000i	− 0.0944911i
$113$	42.0000i	0.0349648i	0.999847	+	0.0174824i	$0.00556511\pi$
$113$	42.0000i	0.0349648i	−0.999847	+	0.0174824i	$0.994435\pi$
$114$	384.000	0.315482
$115$	0	0
$116$	248.000	0.198502
$117$	522.000i	0.412469i
$118$	648.000i	0.505536i
$119$	238.000	0.183340
$120$	0	0
$121$	−1075.00	−0.807663
$122$	− 1028.00i	− 0.762875i
$123$	1422.00i	1.04242i
$124$	−240.000	−0.173812
$125$	0	0
$126$	−126.000	−0.0890871
$127$	− 280.000i	− 0.195638i	−0.995204	−	0.0978188i	$-0.968813\pi$
$127$	− 280.000i	− 0.195638i	0.995204	−	0.0978188i	$-0.0311866\pi$
$128$	− 128.000i	− 0.0883883i
$129$	−876.000	−0.597888
$130$	0	0
$131$	1180.00	0.787001	0.393500	−	0.919324i	$-0.371264\pi$
$131$	1180.00	0.787001	0.393500	+	0.919324i	$0.371264\pi$
$132$	− 192.000i	− 0.126602i
$133$	448.000i	0.292079i
$134$	744.000	0.479640
$135$	0	0
$136$	272.000	0.171499
$137$	− 730.000i	− 0.455242i	−0.973750	−	0.227621i	$-0.926905\pi$
$137$	− 730.000i	− 0.455242i	0.973750	−	0.227621i	$-0.0730947\pi$
$138$	− 96.0000i	− 0.0592178i
$139$	664.000	0.405178	0.202589	−	0.979264i	$-0.435064\pi$
$139$	664.000	0.405178	0.202589	+	0.979264i	$0.435064\pi$
$140$	0	0
$141$	−720.000	−0.430035
$142$	− 824.000i	− 0.486962i
$143$	− 928.000i	− 0.542680i
$144$	−144.000	−0.0833333
$145$	0	0
$146$	−1540.00	−0.872954
$147$	− 147.000i	− 0.0824786i
$148$	− 600.000i	− 0.333241i
$149$	−2974.00	−1.63516	−0.817582	−	0.575812i	$-0.804686\pi$
$149$	−2974.00	−1.63516	−0.817582	+	0.575812i	$0.804686\pi$
$150$	0	0
$151$	−2768.00	−1.49177	−0.745883	−	0.666077i	$-0.767972\pi$
$151$	−2768.00	−1.49177	−0.745883	+	0.666077i	$0.767972\pi$
$152$	512.000i	0.273215i
$153$	− 306.000i	− 0.161690i
$154$	224.000	0.117211
$155$	0	0
$156$	−696.000	−0.357209
$157$	1890.00i	0.960754i	0.877062	+	0.480377i	$0.159500\pi$
$157$	1890.00i	0.960754i	−0.877062	+	0.480377i	$0.840500\pi$
$158$	1120.00i	0.563939i
$159$	−1986.00	−0.990566
$160$	0	0
$161$	112.000	0.0548251
$162$	162.000i	0.0785674i
$163$	1180.00i	0.567023i	0.958969	+	0.283511i	$0.0914994\pi$
$163$	1180.00i	0.567023i	−0.958969	+	0.283511i	$0.908501\pi$
$164$	−1896.00	−0.902761
$165$	0	0
$166$	−1704.00	−0.796723
$167$	− 1216.00i	− 0.563455i	−0.959495	−	0.281727i	$-0.909093\pi$
$167$	− 1216.00i	− 0.563455i	0.959495	−	0.281727i	$-0.0909073\pi$
$168$	− 168.000i	− 0.0771517i
$169$	−1167.00	−0.531179
$170$	0	0
$171$	576.000	0.257590
$172$	− 1168.00i	− 0.517786i
$173$	− 126.000i	− 0.0553734i	−0.999617	−	0.0276867i	$-0.991186\pi$
$173$	− 126.000i	− 0.0553734i	0.999617	−	0.0276867i	$-0.00881408\pi$
$174$	372.000	0.162076
$175$	0	0
$176$	256.000	0.109640
$177$	972.000i	0.412768i
$178$	− 2932.00i	− 1.23462i
$179$	−872.000	−0.364114	−0.182057	−	0.983288i	$-0.558275\pi$
$179$	−872.000	−0.364114	−0.182057	+	0.983288i	$0.558275\pi$
$180$	0	0
$181$	−18.0000	−0.00739188	−0.00369594	−	0.999993i	$-0.501176\pi$
$181$	−18.0000	−0.00739188	−0.00369594	+	0.999993i	$0.501176\pi$
$182$	− 812.000i	− 0.330711i
$183$	− 1542.00i	− 0.622885i
$184$	128.000	0.0512842
$185$	0	0
$186$	−360.000	−0.141917
$187$	544.000i	0.212734i
$188$	− 960.000i	− 0.372421i
$189$	−189.000	−0.0727393
$190$	0	0
$191$	4420.00	1.67445	0.837225	−	0.546858i	$-0.184176\pi$
$191$	4420.00	1.67445	0.837225	+	0.546858i	$0.184176\pi$
$192$	− 192.000i	− 0.0721688i
$193$	2254.00i	0.840655i	0.907372	+	0.420328i	$0.138085\pi$
$193$	2254.00i	0.840655i	−0.907372	+	0.420328i	$0.861915\pi$
$194$	356.000	0.131749
$195$	0	0
$196$	196.000	0.0714286
$197$	− 750.000i	− 0.271245i	−0.990761	−	0.135623i	$-0.956697\pi$
$197$	− 750.000i	− 0.271245i	0.990761	−	0.135623i	$-0.0433034\pi$
$198$	− 288.000i	− 0.103370i
$199$	−3732.00	−1.32942	−0.664710	−	0.747102i	$-0.731445\pi$
$199$	−3732.00	−1.32942	−0.664710	+	0.747102i	$0.731445\pi$
$200$	0	0
$201$	1116.00	0.391625
$202$	2556.00i	0.890295i
$203$	434.000i	0.150053i
$204$	408.000	0.140028
$205$	0	0
$206$	−592.000	−0.200226
$207$	− 144.000i	− 0.0483512i
$208$	− 928.000i	− 0.309352i
$209$	−1024.00	−0.338907
$210$	0	0
$211$	1980.00	0.646013	0.323007	−	0.946397i	$-0.395306\pi$
$211$	1980.00	0.646013	0.323007	+	0.946397i	$0.395306\pi$
$212$	− 2648.00i	− 0.857856i
$213$	− 1236.00i	− 0.397602i
$214$	−2328.00	−0.743639
$215$	0	0
$216$	−216.000	−0.0680414
$217$	− 420.000i	− 0.131389i
$218$	1556.00i	0.483420i
$219$	−2310.00	−0.712764
$220$	0	0
$221$	1972.00	0.600231
$222$	− 900.000i	− 0.272090i
$223$	6328.00i	1.90024i	0.311880	+	0.950122i	$0.399041\pi$
$223$	6328.00i	1.90024i	−0.311880	+	0.950122i	$0.600959\pi$
$224$	224.000	0.0668153
$225$	0	0
$226$	−84.0000	−0.0247239
$227$	− 2596.00i	− 0.759042i	−0.925183	−	0.379521i	$-0.876089\pi$
$227$	− 2596.00i	− 0.759042i	0.925183	−	0.379521i	$-0.123911\pi$
$228$	768.000i	0.223079i
$229$	−4742.00	−1.36839	−0.684193	−	0.729301i	$-0.739845\pi$
$229$	−4742.00	−1.36839	−0.684193	+	0.729301i	$0.739845\pi$
$230$	0	0
$231$	336.000	0.0957021
$232$	496.000i	0.140362i
$233$	− 1294.00i	− 0.363832i	−0.983314	−	0.181916i	$-0.941770\pi$
$233$	− 1294.00i	− 0.363832i	0.983314	−	0.181916i	$-0.0582298\pi$
$234$	−1044.00	−0.291660
$235$	0	0
$236$	−1296.00	−0.357468
$237$	1680.00i	0.460455i
$238$	476.000i	0.129641i
$239$	2340.00	0.633314	0.316657	−	0.948540i	$-0.397440\pi$
$239$	2340.00	0.633314	0.316657	+	0.948540i	$0.397440\pi$
$240$	0	0
$241$	5962.00	1.59355	0.796776	−	0.604274i	$-0.206537\pi$
$241$	5962.00	1.59355	0.796776	+	0.604274i	$0.206537\pi$
$242$	− 2150.00i	− 0.571104i
$243$	243.000i	0.0641500i
$244$	2056.00	0.539434
$245$	0	0
$246$	−2844.00	−0.737101
$247$	3712.00i	0.956230i
$248$	− 480.000i	− 0.122903i
$249$	−2556.00	−0.650522
$250$	0	0
$251$	−1572.00	−0.395314	−0.197657	−	0.980271i	$-0.563333\pi$
$251$	−1572.00	−0.395314	−0.197657	+	0.980271i	$0.563333\pi$
$252$	− 252.000i	− 0.0629941i
$253$	256.000i	0.0636149i
$254$	560.000	0.138337
$255$	0	0
$256$	256.000	0.0625000
$257$	− 3910.00i	− 0.949024i	−0.880249	−	0.474512i	$-0.842625\pi$
$257$	− 3910.00i	− 0.949024i	0.880249	−	0.474512i	$-0.157375\pi$
$258$	− 1752.00i	− 0.422770i
$259$	1050.00	0.251907
$260$	0	0
$261$	558.000	0.132335
$262$	2360.00i	0.556493i
$263$	5672.00i	1.32985i	0.746910	+	0.664925i	$0.231536\pi$
$263$	5672.00i	1.32985i	−0.746910	+	0.664925i	$0.768464\pi$
$264$	384.000	0.0895211
$265$	0	0
$266$	−896.000	−0.206531
$267$	− 4398.00i	− 1.00806i
$268$	1488.00i	0.339157i
$269$	1002.00	0.227112	0.113556	−	0.993532i	$-0.463776\pi$
$269$	1002.00	0.227112	0.113556	+	0.993532i	$0.463776\pi$
$270$	0	0
$271$	−6140.00	−1.37630	−0.688152	−	0.725566i	$-0.741578\pi$
$271$	−6140.00	−1.37630	−0.688152	+	0.725566i	$0.741578\pi$
$272$	544.000i	0.121268i
$273$	− 1218.00i	− 0.270025i
$274$	1460.00	0.321904
$275$	0	0
$276$	192.000	0.0418733
$277$	70.0000i	0.0151837i	0.999971	+	0.00759186i	$0.00241659\pi$
$277$	70.0000i	0.0151837i	−0.999971	+	0.00759186i	$0.997583\pi$
$278$	1328.00i	0.286504i
$279$	−540.000	−0.115874
$280$	0	0
$281$	−3294.00	−0.699301	−0.349650	−	0.936880i	$-0.613700\pi$
$281$	−3294.00	−0.699301	−0.349650	+	0.936880i	$0.613700\pi$
$282$	− 1440.00i	− 0.304081i
$283$	− 1852.00i	− 0.389011i	−0.980901	−	0.194505i	$-0.937690\pi$
$283$	− 1852.00i	− 0.389011i	0.980901	−	0.194505i	$-0.0623101\pi$
$284$	1648.00	0.344334
$285$	0	0
$286$	1856.00	0.383733
$287$	− 3318.00i	− 0.682423i
$288$	− 288.000i	− 0.0589256i
$289$	3757.00	0.764706
$290$	0	0
$291$	534.000	0.107573
$292$	− 3080.00i	− 0.617272i
$293$	5130.00i	1.02286i	0.859325	+	0.511430i	$0.170884\pi$
$293$	5130.00i	1.02286i	−0.859325	+	0.511430i	$0.829116\pi$
$294$	294.000	0.0583212
$295$	0	0
$296$	1200.00	0.235637
$297$	− 432.000i	− 0.0844013i
$298$	− 5948.00i	− 1.15624i
$299$	928.000	0.179490
$300$	0	0
$301$	2044.00	0.391409
$302$	− 5536.00i	− 1.05484i
$303$	3834.00i	0.726923i
$304$	−1024.00	−0.193192
$305$	0	0
$306$	612.000	0.114332
$307$	956.000i	0.177726i	0.996044	+	0.0888629i	$0.0283233\pi$
$307$	956.000i	0.177726i	−0.996044	+	0.0888629i	$0.971677\pi$
$308$	448.000i	0.0828804i
$309$	−888.000	−0.163484
$310$	0	0
$311$	−3448.00	−0.628676	−0.314338	−	0.949311i	$-0.601782\pi$
$311$	−3448.00	−0.628676	−0.314338	+	0.949311i	$0.601782\pi$
$312$	− 1392.00i	− 0.252585i
$313$	5850.00i	1.05643i	0.849112	+	0.528213i	$0.177138\pi$
$313$	5850.00i	1.05643i	−0.849112	+	0.528213i	$0.822862\pi$
$314$	−3780.00	−0.679356
$315$	0	0
$316$	−2240.00	−0.398765
$317$	3794.00i	0.672215i	0.941824	+	0.336108i	$0.109111\pi$
$317$	3794.00i	0.672215i	−0.941824	+	0.336108i	$0.890889\pi$
$318$	− 3972.00i	− 0.700436i
$319$	−992.000	−0.174111
$320$	0	0
$321$	−3492.00	−0.607179
$322$	224.000i	0.0387672i
$323$	− 2176.00i	− 0.374848i
$324$	−324.000	−0.0555556
$325$	0	0
$326$	−2360.00	−0.400946
$327$	2334.00i	0.394711i
$328$	− 3792.00i	− 0.638348i
$329$	1680.00	0.281524
$330$	0	0
$331$	4116.00	0.683492	0.341746	−	0.939792i	$-0.388982\pi$
$331$	4116.00	0.683492	0.341746	+	0.939792i	$0.388982\pi$
$332$	− 3408.00i	− 0.563368i
$333$	− 1350.00i	− 0.222161i
$334$	2432.00	0.398423
$335$	0	0
$336$	336.000	0.0545545
$337$	7506.00i	1.21329i	0.794974	+	0.606644i	$0.207485\pi$
$337$	7506.00i	1.21329i	−0.794974	+	0.606644i	$0.792515\pi$
$338$	− 2334.00i	− 0.375600i
$339$	−126.000	−0.0201870
$340$	0	0
$341$	960.000	0.152454
$342$	1152.00i	0.182143i
$343$	343.000i	0.0539949i
$344$	2336.00	0.366130
$345$	0	0
$346$	252.000	0.0391549
$347$	− 11516.0i	− 1.78159i	−0.454407	−	0.890794i	$-0.650149\pi$
$347$	− 11516.0i	− 1.78159i	0.454407	−	0.890794i	$-0.349851\pi$
$348$	744.000i	0.114605i
$349$	11362.0	1.74268	0.871338	−	0.490683i	$-0.163253\pi$
$349$	11362.0	1.74268	0.871338	+	0.490683i	$0.163253\pi$
$350$	0	0
$351$	−1566.00	−0.238139
$352$	512.000i	0.0775275i
$353$	− 3890.00i	− 0.586526i	−0.956032	−	0.293263i	$-0.905259\pi$
$353$	− 3890.00i	− 0.586526i	0.956032	−	0.293263i	$-0.0947412\pi$
$354$	−1944.00	−0.291871
$355$	0	0
$356$	5864.00	0.873009
$357$	714.000i	0.105851i
$358$	− 1744.00i	− 0.257467i
$359$	1332.00	0.195822	0.0979112	−	0.995195i	$-0.468784\pi$
$359$	1332.00	0.195822	0.0979112	+	0.995195i	$0.468784\pi$
$360$	0	0
$361$	−2763.00	−0.402828
$362$	− 36.0000i	− 0.00522685i
$363$	− 3225.00i	− 0.466305i
$364$	1624.00	0.233848
$365$	0	0
$366$	3084.00	0.440446
$367$	10264.0i	1.45988i	0.683511	+	0.729941i	$0.260452\pi$
$367$	10264.0i	1.45988i	−0.683511	+	0.729941i	$0.739548\pi$
$368$	256.000i	0.0362634i
$369$	−4266.00	−0.601840
$370$	0	0
$371$	4634.00	0.648478
$372$	− 720.000i	− 0.100350i
$373$	7714.00i	1.07082i	0.844592	+	0.535410i	$0.179843\pi$
$373$	7714.00i	1.07082i	−0.844592	+	0.535410i	$0.820157\pi$
$374$	−1088.00	−0.150426
$375$	0	0
$376$	1920.00	0.263342
$377$	3596.00i	0.491256i
$378$	− 378.000i	− 0.0514344i
$379$	6020.00	0.815901	0.407951	−	0.913004i	$-0.366244\pi$
$379$	6020.00	0.815901	0.407951	+	0.913004i	$0.366244\pi$
$380$	0	0
$381$	840.000	0.112951
$382$	8840.00i	1.18402i
$383$	5368.00i	0.716167i	0.933690	+	0.358084i	$0.116570\pi$
$383$	5368.00i	0.716167i	−0.933690	+	0.358084i	$0.883430\pi$
$384$	384.000	0.0510310
$385$	0	0
$386$	−4508.00	−0.594433
$387$	− 2628.00i	− 0.345191i
$388$	712.000i	0.0931606i
$389$	−10526.0	−1.37195	−0.685976	−	0.727624i	$-0.740625\pi$
$389$	−10526.0	−1.37195	−0.685976	+	0.727624i	$0.740625\pi$
$390$	0	0
$391$	−544.000	−0.0703613
$392$	392.000i	0.0505076i
$393$	3540.00i	0.454375i
$394$	1500.00	0.191799
$395$	0	0
$396$	576.000	0.0730937
$397$	15642.0i	1.97745i	0.149728	+	0.988727i	$0.452160\pi$
$397$	15642.0i	1.97745i	−0.149728	+	0.988727i	$0.547840\pi$
$398$	− 7464.00i	− 0.940041i
$399$	−1344.00	−0.168632
$400$	0	0
$401$	14498.0	1.80548	0.902738	−	0.430192i	$-0.141554\pi$
$401$	14498.0	1.80548	0.902738	+	0.430192i	$0.141554\pi$
$402$	2232.00i	0.276921i
$403$	− 3480.00i	− 0.430152i
$404$	−5112.00	−0.629533
$405$	0	0
$406$	−868.000	−0.106104
$407$	2400.00i	0.292294i
$408$	816.000i	0.0990148i
$409$	13718.0	1.65846	0.829232	−	0.558905i	$-0.188778\pi$
$409$	13718.0	1.65846	0.829232	+	0.558905i	$0.188778\pi$
$410$	0	0
$411$	2190.00	0.262834
$412$	− 1184.00i	− 0.141581i
$413$	− 2268.00i	− 0.270220i
$414$	288.000	0.0341894
$415$	0	0
$416$	1856.00	0.218745
$417$	1992.00i	0.233930i
$418$	− 2048.00i	− 0.239643i
$419$	10484.0	1.22238	0.611190	−	0.791484i	$-0.290691\pi$
$419$	10484.0	1.22238	0.611190	+	0.791484i	$0.290691\pi$
$420$	0	0
$421$	−8594.00	−0.994883	−0.497442	−	0.867497i	$-0.665727\pi$
$421$	−8594.00	−0.994883	−0.497442	+	0.867497i	$0.665727\pi$
$422$	3960.00i	0.456800i
$423$	− 2160.00i	− 0.248281i
$424$	5296.00	0.606596
$425$	0	0
$426$	2472.00	0.281147
$427$	3598.00i	0.407774i
$428$	− 4656.00i	− 0.525832i
$429$	2784.00	0.313317
$430$	0	0
$431$	−420.000	−0.0469390	−0.0234695	−	0.999725i	$-0.507471\pi$
$431$	−420.000	−0.0469390	−0.0234695	+	0.999725i	$0.507471\pi$
$432$	− 432.000i	− 0.0481125i
$433$	9794.00i	1.08700i	0.839410	+	0.543498i	$0.182901\pi$
$433$	9794.00i	1.08700i	−0.839410	+	0.543498i	$0.817099\pi$
$434$	840.000	0.0929062
$435$	0	0
$436$	−3112.00	−0.341830
$437$	− 1024.00i	− 0.112093i
$438$	− 4620.00i	− 0.504000i
$439$	1436.00	0.156120	0.0780598	−	0.996949i	$-0.475127\pi$
$439$	1436.00	0.156120	0.0780598	+	0.996949i	$0.475127\pi$
$440$	0	0
$441$	441.000	0.0476190
$442$	3944.00i	0.424427i
$443$	− 12228.0i	− 1.31144i	−0.755002	−	0.655722i	$-0.772364\pi$
$443$	− 12228.0i	− 1.31144i	0.755002	−	0.655722i	$-0.227636\pi$
$444$	1800.00	0.192397
$445$	0	0
$446$	−12656.0	−1.34367
$447$	− 8922.00i	− 0.944063i
$448$	448.000i	0.0472456i
$449$	6734.00	0.707789	0.353894	−	0.935285i	$-0.384857\pi$
$449$	6734.00	0.707789	0.353894	+	0.935285i	$0.384857\pi$
$450$	0	0
$451$	7584.00	0.791833
$452$	− 168.000i	− 0.0174824i
$453$	− 8304.00i	− 0.861271i
$454$	5192.00	0.536724
$455$	0	0
$456$	−1536.00	−0.157741
$457$	10690.0i	1.09422i	0.837062	+	0.547108i	$0.184271\pi$
$457$	10690.0i	1.09422i	−0.837062	+	0.547108i	$0.815729\pi$
$458$	− 9484.00i	− 0.967594i
$459$	918.000	0.0933520
$460$	0	0
$461$	14902.0	1.50554	0.752772	−	0.658282i	$-0.228717\pi$
$461$	14902.0	1.50554	0.752772	+	0.658282i	$0.228717\pi$
$462$	672.000i	0.0676716i
$463$	− 17064.0i	− 1.71281i	−0.516304	−	0.856405i	$-0.672693\pi$
$463$	− 17064.0i	− 1.71281i	0.516304	−	0.856405i	$-0.327307\pi$
$464$	−992.000	−0.0992510
$465$	0	0
$466$	2588.00	0.257268
$467$	3036.00i	0.300834i	0.988623	+	0.150417i	$0.0480616\pi$
$467$	3036.00i	0.300834i	−0.988623	+	0.150417i	$0.951938\pi$
$468$	− 2088.00i	− 0.206235i
$469$	−2604.00	−0.256379
$470$	0	0
$471$	−5670.00	−0.554692
$472$	− 2592.00i	− 0.252768i
$473$	4672.00i	0.454162i
$474$	−3360.00	−0.325591
$475$	0	0
$476$	−952.000	−0.0916698
$477$	− 5958.00i	− 0.571904i
$478$	4680.00i	0.447821i
$479$	13664.0	1.30339	0.651695	−	0.758481i	$-0.274058\pi$
$479$	13664.0	1.30339	0.651695	+	0.758481i	$0.274058\pi$
$480$	0	0
$481$	8700.00	0.824711
$482$	11924.0i	1.12681i
$483$	336.000i	0.0316533i
$484$	4300.00	0.403832
$485$	0	0
$486$	−486.000	−0.0453609
$487$	− 11744.0i	− 1.09275i	−0.837539	−	0.546377i	$-0.816006\pi$
$487$	− 11744.0i	− 1.09275i	0.837539	−	0.546377i	$-0.183994\pi$
$488$	4112.00i	0.381437i
$489$	−3540.00	−0.327371
$490$	0	0
$491$	−48.0000	−0.00441183	−0.00220592	−	0.999998i	$-0.500702\pi$
$491$	−48.0000	−0.00441183	−0.00220592	+	0.999998i	$0.500702\pi$
$492$	− 5688.00i	− 0.521209i
$493$	− 2108.00i	− 0.192575i
$494$	−7424.00	−0.676157
$495$	0	0
$496$	960.000	0.0869058
$497$	2884.00i	0.260292i
$498$	− 5112.00i	− 0.459988i
$499$	−1044.00	−0.0936590	−0.0468295	−	0.998903i	$-0.514912\pi$
$499$	−1044.00	−0.0936590	−0.0468295	+	0.998903i	$0.514912\pi$
$500$	0	0
$501$	3648.00	0.325311
$502$	− 3144.00i	− 0.279529i
$503$	− 14432.0i	− 1.27931i	−0.768664	−	0.639653i	$-0.779078\pi$
$503$	− 14432.0i	− 1.27931i	0.768664	−	0.639653i	$-0.220922\pi$
$504$	504.000	0.0445435
$505$	0	0
$506$	−512.000	−0.0449826
$507$	− 3501.00i	− 0.306676i
$508$	1120.00i	0.0978188i
$509$	6426.00	0.559582	0.279791	−	0.960061i	$-0.409735\pi$
$509$	6426.00	0.559582	0.279791	+	0.960061i	$0.409735\pi$
$510$	0	0
$511$	5390.00	0.466614
$512$	512.000i	0.0441942i
$513$	1728.00i	0.148719i
$514$	7820.00	0.671061
$515$	0	0
$516$	3504.00	0.298944
$517$	3840.00i	0.326660i
$518$	2100.00i	0.178125i
$519$	378.000	0.0319699
$520$	0	0
$521$	−11766.0	−0.989401	−0.494700	−	0.869064i	$-0.664722\pi$
$521$	−11766.0	−0.989401	−0.494700	+	0.869064i	$0.664722\pi$
$522$	1116.00i	0.0935747i
$523$	11900.0i	0.994934i	0.867483	+	0.497467i	$0.165737\pi$
$523$	11900.0i	0.994934i	−0.867483	+	0.497467i	$0.834263\pi$
$524$	−4720.00	−0.393500
$525$	0	0
$526$	−11344.0	−0.940346
$527$	2040.00i	0.168622i
$528$	768.000i	0.0633010i
$529$	11911.0	0.978959
$530$	0	0
$531$	−2916.00	−0.238312
$532$	− 1792.00i	− 0.146040i
$533$	− 27492.0i	− 2.23417i
$534$	8796.00	0.712809
$535$	0	0
$536$	−2976.00	−0.239820
$537$	− 2616.00i	− 0.210221i
$538$	2004.00i	0.160592i
$539$	−784.000	−0.0626517
$540$	0	0
$541$	−22330.0	−1.77457	−0.887284	−	0.461223i	$-0.847411\pi$
$541$	−22330.0	−1.77457	−0.887284	+	0.461223i	$0.847411\pi$
$542$	− 12280.0i	− 0.973194i
$543$	− 54.0000i	− 0.00426770i
$544$	−1088.00	−0.0857493
$545$	0	0
$546$	2436.00	0.190936
$547$	18396.0i	1.43795i	0.695038	+	0.718973i	$0.255387\pi$
$547$	18396.0i	1.43795i	−0.695038	+	0.718973i	$0.744613\pi$
$548$	2920.00i	0.227621i
$549$	4626.00	0.359623
$550$	0	0
$551$	3968.00	0.306792
$552$	384.000i	0.0296089i
$553$	− 3920.00i	− 0.301438i
$554$	−140.000	−0.0107365
$555$	0	0
$556$	−2656.00	−0.202589
$557$	− 3774.00i	− 0.287091i	−0.989644	−	0.143545i	$-0.954150\pi$
$557$	− 3774.00i	− 0.287091i	0.989644	−	0.143545i	$-0.0458503\pi$
$558$	− 1080.00i	− 0.0819356i
$559$	16936.0	1.28142
$560$	0	0
$561$	−1632.00	−0.122822
$562$	− 6588.00i	− 0.494480i
$563$	− 9412.00i	− 0.704562i	−0.935894	−	0.352281i	$-0.885406\pi$
$563$	− 9412.00i	− 0.704562i	0.935894	−	0.352281i	$-0.114594\pi$
$564$	2880.00	0.215018
$565$	0	0
$566$	3704.00	0.275072
$567$	− 567.000i	− 0.0419961i
$568$	3296.00i	0.243481i
$569$	−11146.0	−0.821203	−0.410602	−	0.911815i	$-0.634681\pi$
$569$	−11146.0	−0.821203	−0.410602	+	0.911815i	$0.634681\pi$
$570$	0	0
$571$	15292.0	1.12075	0.560377	−	0.828238i	$-0.310656\pi$
$571$	15292.0	1.12075	0.560377	+	0.828238i	$0.310656\pi$
$572$	3712.00i	0.271340i
$573$	13260.0i	0.966744i
$574$	6636.00	0.482546
$575$	0	0
$576$	576.000	0.0416667
$577$	− 22322.0i	− 1.61053i	−0.592914	−	0.805266i	$-0.702022\pi$
$577$	− 22322.0i	− 1.61053i	0.592914	−	0.805266i	$-0.297978\pi$
$578$	7514.00i	0.540729i
$579$	−6762.00	−0.485353
$580$	0	0
$581$	5964.00	0.425866
$582$	1068.00i	0.0760653i
$583$	10592.0i	0.752446i
$584$	6160.00	0.436477
$585$	0	0
$586$	−10260.0	−0.723271
$587$	684.000i	0.0480949i	0.999711	+	0.0240474i	$0.00765528\pi$
$587$	684.000i	0.0480949i	−0.999711	+	0.0240474i	$0.992345\pi$
$588$	588.000i	0.0412393i
$589$	−3840.00	−0.268632
$590$	0	0
$591$	2250.00	0.156603
$592$	2400.00i	0.166621i
$593$	7806.00i	0.540563i	0.962781	+	0.270282i	$0.0871168\pi$
$593$	7806.00i	0.540563i	−0.962781	+	0.270282i	$0.912883\pi$
$594$	864.000	0.0596807
$595$	0	0
$596$	11896.0	0.817582
$597$	− 11196.0i	− 0.767541i
$598$	1856.00i	0.126919i
$599$	1284.00	0.0875840	0.0437920	−	0.999041i	$-0.486056\pi$
$599$	1284.00	0.0875840	0.0437920	+	0.999041i	$0.486056\pi$
$600$	0	0
$601$	12706.0	0.862377	0.431188	−	0.902262i	$-0.358094\pi$
$601$	12706.0	0.862377	0.431188	+	0.902262i	$0.358094\pi$
$602$	4088.00i	0.276768i
$603$	3348.00i	0.226105i
$604$	11072.0	0.745883
$605$	0	0
$606$	−7668.00	−0.514012
$607$	9016.00i	0.602880i	0.953485	+	0.301440i	$0.0974673\pi$
$607$	9016.00i	0.602880i	−0.953485	+	0.301440i	$0.902533\pi$
$608$	− 2048.00i	− 0.136608i
$609$	−1302.00	−0.0866333
$610$	0	0
$611$	13920.0	0.921674
$612$	1224.00i	0.0808452i
$613$	19386.0i	1.27731i	0.769492	+	0.638657i	$0.220510\pi$
$613$	19386.0i	1.27731i	−0.769492	+	0.638657i	$0.779490\pi$
$614$	−1912.00	−0.125671
$615$	0	0
$616$	−896.000	−0.0586053
$617$	14206.0i	0.926924i	0.886117	+	0.463462i	$0.153393\pi$
$617$	14206.0i	0.926924i	−0.886117	+	0.463462i	$0.846607\pi$
$618$	− 1776.00i	− 0.115601i
$619$	−344.000	−0.0223369	−0.0111684	−	0.999938i	$-0.503555\pi$
$619$	−344.000	−0.0223369	−0.0111684	+	0.999938i	$0.503555\pi$
$620$	0	0
$621$	432.000	0.0279156
$622$	− 6896.00i	− 0.444541i
$623$	10262.0i	0.659933i
$624$	2784.00	0.178604
$625$	0	0
$626$	−11700.0	−0.747006
$627$	− 3072.00i	− 0.195668i
$628$	− 7560.00i	− 0.480377i
$629$	−5100.00	−0.323291
$630$	0	0
$631$	3608.00	0.227626	0.113813	−	0.993502i	$-0.463693\pi$
$631$	3608.00	0.227626	0.113813	+	0.993502i	$0.463693\pi$
$632$	− 4480.00i	− 0.281970i
$633$	5940.00i	0.372976i
$634$	−7588.00	−0.475328
$635$	0	0
$636$	7944.00	0.495283
$637$	2842.00i	0.176773i
$638$	− 1984.00i	− 0.123115i
$639$	3708.00	0.229556
$640$	0	0
$641$	−18838.0	−1.16077	−0.580387	−	0.814341i	$-0.697099\pi$
$641$	−18838.0	−1.16077	−0.580387	+	0.814341i	$0.697099\pi$
$642$	− 6984.00i	− 0.429340i
$643$	− 27068.0i	− 1.66012i	−0.557673	−	0.830060i	$-0.688306\pi$
$643$	− 27068.0i	− 1.66012i	0.557673	−	0.830060i	$-0.311694\pi$
$644$	−448.000	−0.0274125
$645$	0	0
$646$	4352.00	0.265058
$647$	− 18912.0i	− 1.14916i	−0.818448	−	0.574581i	$-0.805165\pi$
$647$	− 18912.0i	− 1.14916i	0.818448	−	0.574581i	$-0.194835\pi$
$648$	− 648.000i	− 0.0392837i
$649$	5184.00	0.313544
$650$	0	0
$651$	1260.00	0.0758576
$652$	− 4720.00i	− 0.283511i
$653$	614.000i	0.0367958i	0.999831	+	0.0183979i	$0.00585657\pi$
$653$	614.000i	0.0367958i	−0.999831	+	0.0183979i	$0.994143\pi$
$654$	−4668.00	−0.279103
$655$	0	0
$656$	7584.00	0.451380
$657$	− 6930.00i	− 0.411515i
$658$	3360.00i	0.199068i
$659$	−31248.0	−1.84712	−0.923558	−	0.383459i	$-0.874733\pi$
$659$	−31248.0	−1.84712	−0.923558	+	0.383459i	$0.874733\pi$
$660$	0	0
$661$	−19882.0	−1.16992	−0.584962	−	0.811060i	$-0.698891\pi$
$661$	−19882.0	−1.16992	−0.584962	+	0.811060i	$0.698891\pi$
$662$	8232.00i	0.483302i
$663$	5916.00i	0.346544i
$664$	6816.00	0.398362
$665$	0	0
$666$	2700.00	0.157091
$667$	− 992.000i	− 0.0575868i
$668$	4864.00i	0.281727i
$669$	−18984.0	−1.09711
$670$	0	0
$671$	−8224.00	−0.473151
$672$	672.000i	0.0385758i
$673$	− 31866.0i	− 1.82518i	−0.408879	−	0.912588i	$-0.634080\pi$
$673$	− 31866.0i	− 1.82518i	0.408879	−	0.912588i	$-0.365920\pi$
$674$	−15012.0	−0.857924
$675$	0	0
$676$	4668.00	0.265589
$677$	19574.0i	1.11121i	0.831446	+	0.555606i	$0.187514\pi$
$677$	19574.0i	1.11121i	−0.831446	+	0.555606i	$0.812486\pi$
$678$	− 252.000i	− 0.0142743i
$679$	−1246.00	−0.0704228
$680$	0	0
$681$	7788.00	0.438233
$682$	1920.00i	0.107801i
$683$	− 24036.0i	− 1.34658i	−0.739380	−	0.673288i	$-0.764881\pi$
$683$	− 24036.0i	− 1.34658i	0.739380	−	0.673288i	$-0.235119\pi$
$684$	−2304.00	−0.128795
$685$	0	0
$686$	−686.000	−0.0381802
$687$	− 14226.0i	− 0.790037i
$688$	4672.00i	0.258893i
$689$	38396.0	2.12303
$690$	0	0
$691$	29496.0	1.62385	0.811925	−	0.583761i	$-0.198420\pi$
$691$	29496.0	1.62385	0.811925	+	0.583761i	$0.198420\pi$
$692$	504.000i	0.0276867i
$693$	1008.00i	0.0552536i
$694$	23032.0	1.25977
$695$	0	0
$696$	−1488.00	−0.0810381
$697$	16116.0i	0.875806i
$698$	22724.0i	1.23226i
$699$	3882.00	0.210058
$700$	0	0
$701$	−10242.0	−0.551833	−0.275917	−	0.961182i	$-0.588981\pi$
$701$	−10242.0	−0.551833	−0.275917	+	0.961182i	$0.588981\pi$
$702$	− 3132.00i	− 0.168390i
$703$	− 9600.00i	− 0.515037i
$704$	−1024.00	−0.0548202
$705$	0	0
$706$	7780.00	0.414737
$707$	− 8946.00i	− 0.475883i
$708$	− 3888.00i	− 0.206384i
$709$	13730.0	0.727279	0.363640	−	0.931540i	$-0.381534\pi$
$709$	13730.0	0.727279	0.363640	+	0.931540i	$0.381534\pi$
$710$	0	0
$711$	−5040.00	−0.265844
$712$	11728.0i	0.617311i
$713$	960.000i	0.0504240i
$714$	−1428.00	−0.0748481
$715$	0	0
$716$	3488.00	0.182057
$717$	7020.00i	0.365644i
$718$	2664.00i	0.138467i
$719$	14840.0	0.769734	0.384867	−	0.922972i	$-0.374247\pi$
$719$	14840.0	0.769734	0.384867	+	0.922972i	$0.374247\pi$
$720$	0	0
$721$	2072.00	0.107025
$722$	− 5526.00i	− 0.284843i
$723$	17886.0i	0.920038i
$724$	72.0000	0.00369594
$725$	0	0
$726$	6450.00	0.329727
$727$	− 3616.00i	− 0.184470i	−0.995737	−	0.0922352i	$-0.970599\pi$
$727$	− 3616.00i	− 0.184470i	0.995737	−	0.0922352i	$-0.0294012\pi$
$728$	3248.00i	0.165356i
$729$	−729.000	−0.0370370
$730$	0	0
$731$	−9928.00	−0.502326
$732$	6168.00i	0.311442i
$733$	24182.0i	1.21853i	0.792967	+	0.609265i	$0.208535\pi$
$733$	24182.0i	1.21853i	−0.792967	+	0.609265i	$0.791465\pi$
$734$	−20528.0	−1.03229
$735$	0	0
$736$	−512.000	−0.0256421
$737$	− 5952.00i	− 0.297483i
$738$	− 8532.00i	− 0.425565i
$739$	−34052.0	−1.69502	−0.847512	−	0.530776i	$-0.821901\pi$
$739$	−34052.0	−1.69502	−0.847512	+	0.530776i	$0.821901\pi$
$740$	0	0
$741$	−11136.0	−0.552080
$742$	9268.00i	0.458543i
$743$	− 21192.0i	− 1.04638i	−0.852217	−	0.523189i	$-0.824742\pi$
$743$	− 21192.0i	− 1.04638i	0.852217	−	0.523189i	$-0.175258\pi$
$744$	1440.00	0.0709583
$745$	0	0
$746$	−15428.0	−0.757184
$747$	− 7668.00i	− 0.375579i
$748$	− 2176.00i	− 0.106367i
$749$	8148.00	0.397492
$750$	0	0
$751$	−2192.00	−0.106508	−0.0532538	−	0.998581i	$-0.516959\pi$
$751$	−2192.00	−0.106508	−0.0532538	+	0.998581i	$0.516959\pi$
$752$	3840.00i	0.186211i
$753$	− 4716.00i	− 0.228235i
$754$	−7192.00	−0.347370
$755$	0	0
$756$	756.000	0.0363696
$757$	− 39458.0i	− 1.89449i	−0.320517	−	0.947243i	$-0.603857\pi$
$757$	− 39458.0i	− 1.89449i	0.320517	−	0.947243i	$-0.396143\pi$
$758$	12040.0i	0.576929i
$759$	−768.000	−0.0367281
$760$	0	0
$761$	458.000	0.0218167	0.0109083	−	0.999941i	$-0.496528\pi$
$761$	458.000	0.0218167	0.0109083	+	0.999941i	$0.496528\pi$
$762$	1680.00i	0.0798687i
$763$	− 5446.00i	− 0.258399i
$764$	−17680.0	−0.837225
$765$	0	0
$766$	−10736.0	−0.506407
$767$	− 18792.0i	− 0.884667i
$768$	768.000i	0.0360844i
$769$	−25354.0	−1.18893	−0.594466	−	0.804121i	$-0.702636\pi$
$769$	−25354.0	−1.18893	−0.594466	+	0.804121i	$0.702636\pi$
$770$	0	0
$771$	11730.0	0.547919
$772$	− 9016.00i	− 0.420328i
$773$	25306.0i	1.17748i	0.808322	+	0.588741i	$0.200376\pi$
$773$	25306.0i	1.17748i	−0.808322	+	0.588741i	$0.799624\pi$
$774$	5256.00	0.244087
$775$	0	0
$776$	−1424.00	−0.0658745
$777$	3150.00i	0.145438i
$778$	− 21052.0i	− 0.970117i
$779$	−30336.0	−1.39525
$780$	0	0
$781$	−6592.00	−0.302023
$782$	− 1088.00i	− 0.0497529i
$783$	1674.00i	0.0764034i
$784$	−784.000	−0.0357143
$785$	0	0
$786$	−7080.00	−0.321292
$787$	26588.0i	1.20427i	0.798395	+	0.602135i	$0.205683\pi$
$787$	26588.0i	1.20427i	−0.798395	+	0.602135i	$0.794317\pi$
$788$	3000.00i	0.135623i
$789$	−17016.0	−0.767789
$790$	0	0
$791$	294.000	0.0132155
$792$	1152.00i	0.0516850i
$793$	29812.0i	1.33500i
$794$	−31284.0	−1.39827
$795$	0	0
$796$	14928.0	0.664710
$797$	38862.0i	1.72718i	0.504194	+	0.863590i	$0.331789\pi$
$797$	38862.0i	1.72718i	−0.504194	+	0.863590i	$0.668211\pi$
$798$	− 2688.00i	− 0.119241i
$799$	−8160.00	−0.361302
$800$	0	0
$801$	13194.0	0.582006
$802$	28996.0i	1.27666i
$803$	12320.0i	0.541424i
$804$	−4464.00	−0.195812
$805$	0	0
$806$	6960.00	0.304163
$807$	3006.00i	0.131123i
$808$	− 10224.0i	− 0.445147i
$809$	−6610.00	−0.287262	−0.143631	−	0.989631i	$-0.545878\pi$
$809$	−6610.00	−0.287262	−0.143631	+	0.989631i	$0.545878\pi$
$810$	0	0
$811$	4696.00	0.203328	0.101664	−	0.994819i	$-0.467583\pi$
$811$	4696.00	0.203328	0.101664	+	0.994819i	$0.467583\pi$
$812$	− 1736.00i	− 0.0750267i
$813$	− 18420.0i	− 0.794610i
$814$	−4800.00	−0.206683
$815$	0	0
$816$	−1632.00	−0.0700140
$817$	− 18688.0i	− 0.800257i
$818$	27436.0i	1.17271i
$819$	3654.00	0.155899
$820$	0	0
$821$	−20362.0	−0.865577	−0.432788	−	0.901495i	$-0.642470\pi$
$821$	−20362.0	−0.865577	−0.432788	+	0.901495i	$0.642470\pi$
$822$	4380.00i	0.185852i
$823$	9888.00i	0.418802i	0.977830	+	0.209401i	$0.0671514\pi$
$823$	9888.00i	0.418802i	−0.977830	+	0.209401i	$0.932849\pi$
$824$	2368.00	0.100113
$825$	0	0
$826$	4536.00	0.191075
$827$	− 19252.0i	− 0.809501i	−0.914427	−	0.404751i	$-0.867358\pi$
$827$	− 19252.0i	− 0.809501i	0.914427	−	0.404751i	$-0.132642\pi$
$828$	576.000i	0.0241756i
$829$	12954.0	0.542715	0.271358	−	0.962479i	$-0.412527\pi$
$829$	12954.0	0.542715	0.271358	+	0.962479i	$0.412527\pi$
$830$	0	0
$831$	−210.000	−0.00876633
$832$	3712.00i	0.154676i
$833$	− 1666.00i	− 0.0692959i
$834$	−3984.00	−0.165413
$835$	0	0
$836$	4096.00	0.169453
$837$	− 1620.00i	− 0.0669001i
$838$	20968.0i	0.864353i
$839$	−18784.0	−0.772939	−0.386469	−	0.922302i	$-0.626306\pi$
$839$	−18784.0	−0.772939	−0.386469	+	0.922302i	$0.626306\pi$
$840$	0	0
$841$	−20545.0	−0.842388
$842$	− 17188.0i	− 0.703489i
$843$	− 9882.00i	− 0.403742i
$844$	−7920.00	−0.323007
$845$	0	0
$846$	4320.00	0.175561
$847$	7525.00i	0.305268i
$848$	10592.0i	0.428928i
$849$	5556.00	0.224595
$850$	0	0
$851$	−2400.00	−0.0966756
$852$	4944.00i	0.198801i
$853$	4958.00i	0.199014i	0.995037	+	0.0995069i	$0.0317265\pi$
$853$	4958.00i	0.199014i	−0.995037	+	0.0995069i	$0.968273\pi$
$854$	−7196.00	−0.288340
$855$	0	0
$856$	9312.00	0.371820
$857$	− 15326.0i	− 0.610882i	−0.952211	−	0.305441i	$-0.901196\pi$
$857$	− 15326.0i	− 0.610882i	0.952211	−	0.305441i	$-0.0988039\pi$
$858$	5568.00i	0.221548i
$859$	−49840.0	−1.97965	−0.989825	−	0.142292i	$-0.954553\pi$
$859$	−49840.0	−1.97965	−0.989825	+	0.142292i	$0.954553\pi$
$860$	0	0
$861$	9954.00	0.393997
$862$	− 840.000i	− 0.0331909i
$863$	− 13384.0i	− 0.527922i	−0.964533	−	0.263961i	$-0.914971\pi$
$863$	− 13384.0i	− 0.527922i	0.964533	−	0.263961i	$-0.0850290\pi$
$864$	864.000	0.0340207
$865$	0	0
$866$	−19588.0	−0.768623
$867$	11271.0i	0.441503i
$868$	1680.00i	0.0656946i
$869$	8960.00	0.349767
$870$	0	0
$871$	−21576.0	−0.839351
$872$	− 6224.00i	− 0.241710i
$873$	1602.00i	0.0621071i
$874$	2048.00	0.0792616
$875$	0	0
$876$	9240.00	0.356382
$877$	5006.00i	0.192749i	0.995345	+	0.0963743i	$0.0307246\pi$
$877$	5006.00i	0.192749i	−0.995345	+	0.0963743i	$0.969275\pi$
$878$	2872.00i	0.110393i
$879$	−15390.0	−0.590548
$880$	0	0
$881$	14098.0	0.539130	0.269565	−	0.962982i	$-0.413120\pi$
$881$	14098.0	0.539130	0.269565	+	0.962982i	$0.413120\pi$
$882$	882.000i	0.0336718i
$883$	− 13580.0i	− 0.517558i	−0.965937	−	0.258779i	$-0.916680\pi$
$883$	− 13580.0i	− 0.517558i	0.965937	−	0.258779i	$-0.0833201\pi$
$884$	−7888.00	−0.300116
$885$	0	0
$886$	24456.0	0.927331
$887$	− 14648.0i	− 0.554489i	−0.960799	−	0.277244i	$-0.910579\pi$
$887$	− 14648.0i	− 0.554489i	0.960799	−	0.277244i	$-0.0894212\pi$
$888$	3600.00i	0.136045i
$889$	−1960.00	−0.0739441
$890$	0	0
$891$	1296.00	0.0487291
$892$	− 25312.0i	− 0.950122i
$893$	− 15360.0i	− 0.575591i
$894$	17844.0	0.667553
$895$	0	0
$896$	−896.000	−0.0334077
$897$	2784.00i	0.103629i
$898$	13468.0i	0.500482i
$899$	−3720.00	−0.138008
$900$	0	0
$901$	−22508.0	−0.832242
$902$	15168.0i	0.559910i
$903$	6132.00i	0.225980i
$904$	336.000	0.0123619
$905$	0	0
$906$	16608.0	0.609011
$907$	− 17996.0i	− 0.658817i	−0.944187	−	0.329409i	$-0.893151\pi$
$907$	− 17996.0i	− 0.658817i	0.944187	−	0.329409i	$-0.106849\pi$
$908$	10384.0i	0.379521i
$909$	−11502.0	−0.419689
$910$	0	0
$911$	−41420.0	−1.50637	−0.753187	−	0.657807i	$-0.771484\pi$
$911$	−41420.0	−1.50637	−0.753187	+	0.657807i	$0.771484\pi$
$912$	− 3072.00i	− 0.111540i
$913$	13632.0i	0.494144i
$914$	−21380.0	−0.773728
$915$	0	0
$916$	18968.0	0.684193
$917$	− 8260.00i	− 0.297458i
$918$	1836.00i	0.0660098i
$919$	−33640.0	−1.20749	−0.603744	−	0.797178i	$-0.706325\pi$
$919$	−33640.0	−1.20749	−0.603744	+	0.797178i	$0.706325\pi$
$920$	0	0
$921$	−2868.00	−0.102610
$922$	29804.0i	1.06458i
$923$	23896.0i	0.852163i
$924$	−1344.00	−0.0478510
$925$	0	0
$926$	34128.0	1.21114
$927$	− 2664.00i	− 0.0943875i
$928$	− 1984.00i	− 0.0701810i
$929$	37918.0	1.33913	0.669564	−	0.742755i	$-0.266481\pi$
$929$	37918.0	1.33913	0.669564	+	0.742755i	$0.266481\pi$
$930$	0	0
$931$	3136.00	0.110396
$932$	5176.00i	0.181916i
$933$	− 10344.0i	− 0.362966i
$934$	−6072.00	−0.212722
$935$	0	0
$936$	4176.00	0.145830
$937$	− 5954.00i	− 0.207587i	−0.994599	−	0.103793i	$-0.966902\pi$
$937$	− 5954.00i	− 0.207587i	0.994599	−	0.103793i	$-0.0330981\pi$
$938$	− 5208.00i	− 0.181287i
$939$	−17550.0	−0.609928
$940$	0	0
$941$	−33066.0	−1.14551	−0.572753	−	0.819728i	$-0.694125\pi$
$941$	−33066.0	−1.14551	−0.572753	+	0.819728i	$0.694125\pi$
$942$	− 11340.0i	− 0.392226i
$943$	7584.00i	0.261897i
$944$	5184.00	0.178734
$945$	0	0
$946$	−9344.00	−0.321141
$947$	28508.0i	0.978232i	0.872219	+	0.489116i	$0.162680\pi$
$947$	28508.0i	0.978232i	−0.872219	+	0.489116i	$0.837320\pi$
$948$	− 6720.00i	− 0.230227i
$949$	44660.0	1.52763
$950$	0	0
$951$	−11382.0	−0.388104
$952$	− 1904.00i	− 0.0648204i
$953$	− 24718.0i	− 0.840183i	−0.907482	−	0.420092i	$-0.861998\pi$
$953$	− 24718.0i	− 0.840183i	0.907482	−	0.420092i	$-0.138002\pi$
$954$	11916.0	0.404397
$955$	0	0
$956$	−9360.00	−0.316657
$957$	− 2976.00i	− 0.100523i
$958$	27328.0i	0.921636i
$959$	−5110.00	−0.172065
$960$	0	0
$961$	−26191.0	−0.879158
$962$	17400.0i	0.583159i
$963$	− 10476.0i	− 0.350555i
$964$	−23848.0	−0.796776
$965$	0	0
$966$	−672.000	−0.0223822
$967$	− 52424.0i	− 1.74337i	−0.490063	−	0.871687i	$-0.663026\pi$
$967$	− 52424.0i	− 1.74337i	0.490063	−	0.871687i	$-0.336974\pi$
$968$	8600.00i	0.285552i
$969$	6528.00	0.216419
$970$	0	0
$971$	10988.0	0.363153	0.181577	−	0.983377i	$-0.441880\pi$
$971$	10988.0	0.363153	0.181577	+	0.983377i	$0.441880\pi$
$972$	− 972.000i	− 0.0320750i
$973$	− 4648.00i	− 0.153143i
$974$	23488.0	0.772694
$975$	0	0
$976$	−8224.00	−0.269717
$977$	31446.0i	1.02973i	0.857271	+	0.514865i	$0.172158\pi$
$977$	31446.0i	1.02973i	−0.857271	+	0.514865i	$0.827842\pi$
$978$	− 7080.00i	− 0.231486i
$979$	−23456.0	−0.765737
$980$	0	0
$981$	−7002.00	−0.227886
$982$	− 96.0000i	− 0.00311964i
$983$	− 33528.0i	− 1.08787i	−0.839127	−	0.543935i	$-0.816934\pi$
$983$	− 33528.0i	− 1.08787i	0.839127	−	0.543935i	$-0.183066\pi$
$984$	11376.0	0.368550
$985$	0	0
$986$	4216.00	0.136171
$987$	5040.00i	0.162538i
$988$	− 14848.0i	− 0.478115i
$989$	−4672.00	−0.150213
$990$	0	0
$991$	−49856.0	−1.59811	−0.799056	−	0.601257i	$-0.794667\pi$
$991$	−49856.0	−1.59811	−0.799056	+	0.601257i	$0.794667\pi$
$992$	1920.00i	0.0614517i
$993$	12348.0i	0.394614i
$994$	−5768.00	−0.184054
$995$	0	0
$996$	10224.0	0.325261
$997$	9298.00i	0.295357i	0.989035	+	0.147678i	$0.0471800\pi$
$997$	9298.00i	0.295357i	−0.989035	+	0.147678i	$0.952820\pi$
$998$	− 2088.00i	− 0.0662269i
$999$	4050.00	0.128265

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1050.4.g.f.799.2		2
5.2	odd	4		1050.4.a.l.1.1		1
5.3	odd	4		210.4.a.i.1.1	✓	1
5.4	even	2	inner	1050.4.g.f.799.1		2
15.8	even	4		630.4.a.a.1.1		1
20.3	even	4		1680.4.a.w.1.1		1
35.13	even	4		1470.4.a.z.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
210.4.a.i.1.1	✓	1	5.3	odd	4
630.4.a.a.1.1		1	15.8	even	4
1050.4.a.l.1.1		1	5.2	odd	4
1050.4.g.f.799.1		2	5.4	even	2	inner
1050.4.g.f.799.2		2	1.1	even	1	trivial
1470.4.a.z.1.1		1	35.13	even	4
1680.4.a.w.1.1		1	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 $\F_{q}$
Belyi maps

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)

\(f(q)\)	\(=\)	\(q+2.00000i q^{2} +3.00000i q^{3} -4.00000 q^{4} -6.00000 q^{6} -7.00000i q^{7} -8.00000i q^{8} -9.00000 q^{9} +16.0000 q^{11} -12.0000i q^{12} -58.0000i q^{13} +14.0000 q^{14} +16.0000 q^{16} +34.0000i q^{17} -18.0000i q^{18} -64.0000 q^{19} +21.0000 q^{21} +32.0000i q^{22} +16.0000i q^{23} +24.0000 q^{24} +116.000 q^{26} -27.0000i q^{27} +28.0000i q^{28} -62.0000 q^{29} +60.0000 q^{31} +32.0000i q^{32} +48.0000i q^{33} -68.0000 q^{34} +36.0000 q^{36} +150.000i q^{37} -128.000i q^{38} +174.000 q^{39} +474.000 q^{41} +42.0000i q^{42} +292.000i q^{43} -64.0000 q^{44} -32.0000 q^{46} +240.000i q^{47} +48.0000i q^{48} -49.0000 q^{49} -102.000 q^{51} +232.000i q^{52} +662.000i q^{53} +54.0000 q^{54} -56.0000 q^{56} -192.000i q^{57} -124.000i q^{58} +324.000 q^{59} -514.000 q^{61} +120.000i q^{62} +63.0000i q^{63} -64.0000 q^{64} -96.0000 q^{66} -372.000i q^{67} -136.000i q^{68} -48.0000 q^{69} -412.000 q^{71} +72.0000i q^{72} +770.000i q^{73} -300.000 q^{74} +256.000 q^{76} -112.000i q^{77} +348.000i q^{78} +560.000 q^{79} +81.0000 q^{81} +948.000i q^{82} +852.000i q^{83} -84.0000 q^{84} -584.000 q^{86} -186.000i q^{87} -128.000i q^{88} -1466.00 q^{89} -406.000 q^{91} -64.0000i q^{92} +180.000i q^{93} -480.000 q^{94} -96.0000 q^{96} -178.000i q^{97} -98.0000i q^{98} -144.000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 8 q^{4} - 12 q^{6} - 18 q^{9} + 32 q^{11} + 28 q^{14} + 32 q^{16} - 128 q^{19} + 42 q^{21} + 48 q^{24} + 232 q^{26} - 124 q^{29} + 120 q^{31} - 136 q^{34} + 72 q^{36} + 348 q^{39} + 948 q^{41} - 128 q^{44}+ \cdots - 288 q^{99}+O(q^{100}) \) 2 * q - 8 * q^4 - 12 * q^6 - 18 * q^9 + 32 * q^11 + 28 * q^14 + 32 * q^16 - 128 * q^19 + 42 * q^21 + 48 * q^24 + 232 * q^26 - 124 * q^29 + 120 * q^31 - 136 * q^34 + 72 * q^36 + 348 * q^39 + 948 * q^41 - 128 * q^44 - 64 * q^46 - 98 * q^49 - 204 * q^51 + 108 * q^54 - 112 * q^56 + 648 * q^59 - 1028 * q^61 - 128 * q^64 - 192 * q^66 - 96 * q^69 - 824 * q^71 - 600 * q^74 + 512 * q^76 + 1120 * q^79 + 162 * q^81 - 168 * q^84 - 1168 * q^86 - 2932 * q^89 - 812 * q^91 - 960 * q^94 - 192 * q^96 - 288 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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