LMFDB - Embedded newform 50.6.b.a.49.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [50,6,Mod(49,50)] 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("50.49"); S:= CuspForms(chi, 6); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(50, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1])) N = Newforms(chi, 6, names="a")

Level:	$ N $	$=$	$ 50 = 2 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	50.b (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,0,-32,0,-192] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(6)] == traces)

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

Self dual:	no
Analytic conductor:	$8.01919099065$
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:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 10)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			49.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	50.49
Dual form			50.6.b.a.49.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+4.00000i q^{2} +24.0000i q^{3} -16.0000 q^{4} -96.0000 q^{6} +172.000i q^{7} -64.0000i q^{8} -333.000 q^{9} +132.000 q^{11} -384.000i q^{12} -946.000i q^{13} -688.000 q^{14} +256.000 q^{16} +222.000i q^{17} -1332.00i q^{18} -500.000 q^{19} -4128.00 q^{21} +528.000i q^{22} +3564.00i q^{23} +1536.00 q^{24} +3784.00 q^{26} -2160.00i q^{27} -2752.00i q^{28} -2190.00 q^{29} +2312.00 q^{31} +1024.00i q^{32} +3168.00i q^{33} -888.000 q^{34} +5328.00 q^{36} +11242.0i q^{37} -2000.00i q^{38} +22704.0 q^{39} +1242.00 q^{41} -16512.0i q^{42} +20624.0i q^{43} -2112.00 q^{44} -14256.0 q^{46} -6588.00i q^{47} +6144.00i q^{48} -12777.0 q^{49} -5328.00 q^{51} +15136.0i q^{52} -21066.0i q^{53} +8640.00 q^{54} +11008.0 q^{56} -12000.0i q^{57} -8760.00i q^{58} -7980.00 q^{59} +16622.0 q^{61} +9248.00i q^{62} -57276.0i q^{63} -4096.00 q^{64} -12672.0 q^{66} -1808.00i q^{67} -3552.00i q^{68} -85536.0 q^{69} -24528.0 q^{71} +21312.0i q^{72} +20474.0i q^{73} -44968.0 q^{74} +8000.00 q^{76} +22704.0i q^{77} +90816.0i q^{78} +46240.0 q^{79} -29079.0 q^{81} +4968.00i q^{82} -51576.0i q^{83} +66048.0 q^{84} -82496.0 q^{86} -52560.0i q^{87} -8448.00i q^{88} +110310. q^{89} +162712. q^{91} -57024.0i q^{92} +55488.0i q^{93} +26352.0 q^{94} -24576.0 q^{96} +78382.0i q^{97} -51108.0i q^{98} -43956.0 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 32 q^{4} - 192 q^{6} - 666 q^{9} + 264 q^{11} - 1376 q^{14} + 512 q^{16} - 1000 q^{19} - 8256 q^{21} + 3072 q^{24} + 7568 q^{26} - 4380 q^{29} + 4624 q^{31} - 1776 q^{34} + 10656 q^{36} + 45408 q^{39}+ \cdots - 87912 q^{99}+O(q^{100}) $ 2 * q - 32 * q^4 - 192 * q^6 - 666 * q^9 + 264 * q^11 - 1376 * q^14 + 512 * q^16 - 1000 * q^19 - 8256 * q^21 + 3072 * q^24 + 7568 * q^26 - 4380 * q^29 + 4624 * q^31 - 1776 * q^34 + 10656 * q^36 + 45408 * q^39 + 2484 * q^41 - 4224 * q^44 - 28512 * q^46 - 25554 * q^49 - 10656 * q^51 + 17280 * q^54 + 22016 * q^56 - 15960 * q^59 + 33244 * q^61 - 8192 * q^64 - 25344 * q^66 - 171072 * q^69 - 49056 * q^71 - 89936 * q^74 + 16000 * q^76 + 92480 * q^79 - 58158 * q^81 + 132096 * q^84 - 164992 * q^86 + 220620 * q^89 + 325424 * q^91 + 52704 * q^94 - 49152 * q^96 - 87912 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/50\mathbb{Z}\right)^\times$.

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

Coefficient data

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

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

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

$2$	4.00000i	0.707107i
$2$	4.00000i	0.707107i
$3$	24.0000i	1.53960i	0.638285	+	0.769800i	$0.279644\pi$
$3$	24.0000i	1.53960i	−0.638285	+	0.769800i	$0.720356\pi$
$4$	−16.0000	−0.500000
$5$	0	0
$5$	0	0
$6$	−96.0000	−1.08866
$7$	172.000i	1.32673i	0.748295	+	0.663366i	$0.230873\pi$
$7$	172.000i	1.32673i	−0.748295	+	0.663366i	$0.769127\pi$
$8$	− 64.0000i	− 0.353553i
$9$	−333.000	−1.37037
$10$	0	0
$11$	132.000	0.328921	0.164461	−	0.986384i	$-0.447412\pi$
$11$	132.000	0.328921	0.164461	+	0.986384i	$0.447412\pi$
$12$	− 384.000i	− 0.769800i
$13$	− 946.000i	− 1.55250i	−0.630423	−	0.776252i	$-0.717118\pi$
$13$	− 946.000i	− 1.55250i	0.630423	−	0.776252i	$-0.282882\pi$
$14$	−688.000	−0.938142
$15$	0	0
$16$	256.000	0.250000
$17$	222.000i	0.186308i	0.995652	+	0.0931538i	$0.0296948\pi$
$17$	222.000i	0.186308i	−0.995652	+	0.0931538i	$0.970305\pi$
$18$	− 1332.00i	− 0.968998i
$19$	−500.000	−0.317750	−0.158875	−	0.987299i	$-0.550787\pi$
$19$	−500.000	−0.317750	−0.158875	+	0.987299i	$0.550787\pi$
$20$	0	0
$21$	−4128.00	−2.04264
$22$	528.000i	0.232583i
$23$	3564.00i	1.40481i	0.711777	+	0.702406i	$0.247891\pi$
$23$	3564.00i	1.40481i	−0.711777	+	0.702406i	$0.752109\pi$
$24$	1536.00	0.544331
$25$	0	0
$26$	3784.00	1.09779
$27$	− 2160.00i	− 0.570222i
$28$	− 2752.00i	− 0.663366i
$29$	−2190.00	−0.483559	−0.241779	−	0.970331i	$-0.577731\pi$
$29$	−2190.00	−0.483559	−0.241779	+	0.970331i	$0.577731\pi$
$30$	0	0
$31$	2312.00	0.432099	0.216050	−	0.976382i	$-0.430683\pi$
$31$	2312.00	0.432099	0.216050	+	0.976382i	$0.430683\pi$
$32$	1024.00i	0.176777i
$33$	3168.00i	0.506408i
$34$	−888.000	−0.131739
$35$	0	0
$36$	5328.00	0.685185
$37$	11242.0i	1.35002i	0.737810	+	0.675009i	$0.235860\pi$
$37$	11242.0i	1.35002i	−0.737810	+	0.675009i	$0.764140\pi$
$38$	− 2000.00i	− 0.224683i
$39$	22704.0	2.39024
$40$	0	0
$41$	1242.00	0.115388	0.0576942	−	0.998334i	$-0.481625\pi$
$41$	1242.00	0.115388	0.0576942	+	0.998334i	$0.481625\pi$
$42$	− 16512.0i	− 1.44436i
$43$	20624.0i	1.70099i	0.525983	+	0.850495i	$0.323697\pi$
$43$	20624.0i	1.70099i	−0.525983	+	0.850495i	$0.676303\pi$
$44$	−2112.00	−0.164461
$45$	0	0
$46$	−14256.0	−0.993352
$47$	− 6588.00i	− 0.435020i	−0.976058	−	0.217510i	$-0.930207\pi$
$47$	− 6588.00i	− 0.435020i	0.976058	−	0.217510i	$-0.0697934\pi$
$48$	6144.00i	0.384900i
$49$	−12777.0	−0.760219
$50$	0	0
$51$	−5328.00	−0.286839
$52$	15136.0i	0.776252i
$53$	− 21066.0i	− 1.03013i	−0.857151	−	0.515065i	$-0.827768\pi$
$53$	− 21066.0i	− 1.03013i	0.857151	−	0.515065i	$-0.172232\pi$
$54$	8640.00	0.403208
$55$	0	0
$56$	11008.0	0.469071
$57$	− 12000.0i	− 0.489209i
$58$	− 8760.00i	− 0.341928i
$59$	−7980.00	−0.298451	−0.149225	−	0.988803i	$-0.547678\pi$
$59$	−7980.00	−0.298451	−0.149225	+	0.988803i	$0.547678\pi$
$60$	0	0
$61$	16622.0	0.571951	0.285975	−	0.958237i	$-0.407682\pi$
$61$	16622.0	0.571951	0.285975	+	0.958237i	$0.407682\pi$
$62$	9248.00i	0.305540i
$63$	− 57276.0i	− 1.81811i
$64$	−4096.00	−0.125000
$65$	0	0
$66$	−12672.0	−0.358084
$67$	− 1808.00i	− 0.0492052i	−0.999697	−	0.0246026i	$-0.992168\pi$
$67$	− 1808.00i	− 0.0492052i	0.999697	−	0.0246026i	$-0.00783205\pi$
$68$	− 3552.00i	− 0.0931538i
$69$	−85536.0	−2.16285
$70$	0	0
$71$	−24528.0	−0.577452	−0.288726	−	0.957412i	$-0.593232\pi$
$71$	−24528.0	−0.577452	−0.288726	+	0.957412i	$0.593232\pi$
$72$	21312.0i	0.484499i
$73$	20474.0i	0.449672i	0.974397	+	0.224836i	$0.0721846\pi$
$73$	20474.0i	0.449672i	−0.974397	+	0.224836i	$0.927815\pi$
$74$	−44968.0	−0.954606
$75$	0	0
$76$	8000.00	0.158875
$77$	22704.0i	0.436391i
$78$	90816.0i	1.69015i
$79$	46240.0	0.833585	0.416793	−	0.909002i	$-0.363154\pi$
$79$	46240.0	0.833585	0.416793	+	0.909002i	$0.363154\pi$
$80$	0	0
$81$	−29079.0	−0.492455
$82$	4968.00i	0.0815919i
$83$	− 51576.0i	− 0.821774i	−0.911686	−	0.410887i	$-0.865219\pi$
$83$	− 51576.0i	− 0.821774i	0.911686	−	0.410887i	$-0.134781\pi$
$84$	66048.0	1.02132
$85$	0	0
$86$	−82496.0	−1.20278
$87$	− 52560.0i	− 0.744487i
$88$	− 8448.00i	− 0.116291i
$89$	110310.	1.47618	0.738091	−	0.674701i	$-0.235728\pi$
$89$	110310.	1.47618	0.738091	+	0.674701i	$0.235728\pi$
$90$	0	0
$91$	162712.	2.05976
$92$	− 57024.0i	− 0.702406i
$93$	55488.0i	0.665260i
$94$	26352.0	0.307605
$95$	0	0
$96$	−24576.0	−0.272166
$97$	78382.0i	0.845838i	0.906168	+	0.422919i	$0.138994\pi$
$97$	78382.0i	0.845838i	−0.906168	+	0.422919i	$0.861006\pi$
$98$	− 51108.0i	− 0.537556i
$99$	−43956.0	−0.450744
$100$	0	0
$101$	141942.	1.38455	0.692273	−	0.721636i	$-0.256609\pi$
$101$	141942.	1.38455	0.692273	+	0.721636i	$0.256609\pi$
$102$	− 21312.0i	− 0.202826i
$103$	− 436.000i	− 0.00404943i	−0.999998	−	0.00202471i	$-0.999356\pi$
$103$	− 436.000i	− 0.00404943i	0.999998	−	0.00202471i	$-0.000644487\pi$
$104$	−60544.0	−0.548893
$105$	0	0
$106$	84264.0	0.728413
$107$	− 232968.i	− 1.96715i	−0.180508	−	0.983574i	$-0.557774\pi$
$107$	− 232968.i	− 1.96715i	0.180508	−	0.983574i	$-0.442226\pi$
$108$	34560.0i	0.285111i
$109$	174850.	1.40961	0.704806	−	0.709400i	$-0.251034\pi$
$109$	174850.	1.40961	0.704806	+	0.709400i	$0.251034\pi$
$110$	0	0
$111$	−269808.	−2.07849
$112$	44032.0i	0.331683i
$113$	182994.i	1.34816i	0.738659	+	0.674079i	$0.235459\pi$
$113$	182994.i	1.34816i	−0.738659	+	0.674079i	$0.764541\pi$
$114$	48000.0	0.345923
$115$	0	0
$116$	35040.0	0.241779
$117$	315018.i	2.12751i
$118$	− 31920.0i	− 0.211037i
$119$	−38184.0	−0.247180
$120$	0	0
$121$	−143627.	−0.891811
$122$	66488.0i	0.404430i
$123$	29808.0i	0.177652i
$124$	−36992.0	−0.216050
$125$	0	0
$126$	229104.	1.28560
$127$	122452.i	0.673685i	0.941561	+	0.336842i	$0.109359\pi$
$127$	122452.i	0.673685i	−0.941561	+	0.336842i	$0.890641\pi$
$128$	− 16384.0i	− 0.0883883i
$129$	−494976.	−2.61885
$130$	0	0
$131$	−241908.	−1.23161	−0.615803	−	0.787900i	$-0.711168\pi$
$131$	−241908.	−1.23161	−0.615803	+	0.787900i	$0.711168\pi$
$132$	− 50688.0i	− 0.253204i
$133$	− 86000.0i	− 0.421570i
$134$	7232.00	0.0347934
$135$	0	0
$136$	14208.0	0.0658697
$137$	− 277098.i	− 1.26134i	−0.776051	−	0.630670i	$-0.782780\pi$
$137$	− 277098.i	− 1.26134i	0.776051	−	0.630670i	$-0.217220\pi$
$138$	− 342144.i	− 1.52937i
$139$	193540.	0.849638	0.424819	−	0.905278i	$-0.360338\pi$
$139$	193540.	0.849638	0.424819	+	0.905278i	$0.360338\pi$
$140$	0	0
$141$	158112.	0.669757
$142$	− 98112.0i	− 0.408321i
$143$	− 124872.i	− 0.510652i
$144$	−85248.0	−0.342593
$145$	0	0
$146$	−81896.0	−0.317966
$147$	− 306648.i	− 1.17043i
$148$	− 179872.i	− 0.675009i
$149$	−140550.	−0.518639	−0.259320	−	0.965792i	$-0.583498\pi$
$149$	−140550.	−0.518639	−0.259320	+	0.965792i	$0.583498\pi$
$150$	0	0
$151$	433952.	1.54881	0.774407	−	0.632688i	$-0.218048\pi$
$151$	433952.	1.54881	0.774407	+	0.632688i	$0.218048\pi$
$152$	32000.0i	0.112342i
$153$	− 73926.0i	− 0.255310i
$154$	−90816.0	−0.308575
$155$	0	0
$156$	−363264.	−1.19512
$157$	555922.i	1.79997i	0.435923	+	0.899984i	$0.356422\pi$
$157$	555922.i	1.79997i	−0.435923	+	0.899984i	$0.643578\pi$
$158$	184960.i	0.589434i
$159$	505584.	1.58599
$160$	0	0
$161$	−613008.	−1.86381
$162$	− 116316.i	− 0.348219i
$163$	− 66616.0i	− 0.196386i	−0.995167	−	0.0981928i	$-0.968694\pi$
$163$	− 66616.0i	− 0.196386i	0.995167	−	0.0981928i	$-0.0313062\pi$
$164$	−19872.0	−0.0576942
$165$	0	0
$166$	206304.	0.581082
$167$	205692.i	0.570724i	0.958420	+	0.285362i	$0.0921138\pi$
$167$	205692.i	0.570724i	−0.958420	+	0.285362i	$0.907886\pi$
$168$	264192.i	0.722182i
$169$	−523623.	−1.41027
$170$	0	0
$171$	166500.	0.435436
$172$	− 329984.i	− 0.850495i
$173$	433854.i	1.10212i	0.834466	+	0.551059i	$0.185776\pi$
$173$	433854.i	1.10212i	−0.834466	+	0.551059i	$0.814224\pi$
$174$	210240.	0.526432
$175$	0	0
$176$	33792.0	0.0822304
$177$	− 191520.i	− 0.459495i
$178$	441240.i	1.04382i
$179$	489180.	1.14113	0.570566	−	0.821252i	$-0.306724\pi$
$179$	489180.	1.14113	0.570566	+	0.821252i	$0.306724\pi$
$180$	0	0
$181$	719462.	1.63234	0.816172	−	0.577810i	$-0.196092\pi$
$181$	719462.	1.63234	0.816172	+	0.577810i	$0.196092\pi$
$182$	650848.i	1.45647i
$183$	398928.i	0.880576i
$184$	228096.	0.496676
$185$	0	0
$186$	−221952.	−0.470410
$187$	29304.0i	0.0612806i
$188$	105408.i	0.217510i
$189$	371520.	0.756533
$190$	0	0
$191$	−185928.	−0.368775	−0.184387	−	0.982854i	$-0.559030\pi$
$191$	−185928.	−0.368775	−0.184387	+	0.982854i	$0.559030\pi$
$192$	− 98304.0i	− 0.192450i
$193$	− 591406.i	− 1.14286i	−0.820651	−	0.571429i	$-0.806389\pi$
$193$	− 591406.i	− 1.14286i	0.820651	−	0.571429i	$-0.193611\pi$
$194$	−313528.	−0.598098
$195$	0	0
$196$	204432.	0.380109
$197$	− 449478.i	− 0.825169i	−0.910919	−	0.412584i	$-0.864626\pi$
$197$	− 449478.i	− 0.825169i	0.910919	−	0.412584i	$-0.135374\pi$
$198$	− 175824.i	− 0.318724i
$199$	−157160.	−0.281326	−0.140663	−	0.990058i	$-0.544923\pi$
$199$	−157160.	−0.281326	−0.140663	+	0.990058i	$0.544923\pi$
$200$	0	0
$201$	43392.0	0.0757564
$202$	567768.i	0.979022i
$203$	− 376680.i	− 0.641553i
$204$	85248.0	0.143420
$205$	0	0
$206$	1744.00	0.00286338
$207$	− 1.18681e6i	− 1.92511i
$208$	− 242176.i	− 0.388126i
$209$	−66000.0	−0.104515
$210$	0	0
$211$	253052.	0.391294	0.195647	−	0.980674i	$-0.437319\pi$
$211$	253052.	0.391294	0.195647	+	0.980674i	$0.437319\pi$
$212$	337056.i	0.515065i
$213$	− 588672.i	− 0.889046i
$214$	931872.	1.39098
$215$	0	0
$216$	−138240.	−0.201604
$217$	397664.i	0.573280i
$218$	699400.i	0.996746i
$219$	−491376.	−0.692315
$220$	0	0
$221$	210012.	0.289243
$222$	− 1.07923e6i	− 1.46971i
$223$	1.07344e6i	1.44550i	0.691111	+	0.722749i	$0.257122\pi$
$223$	1.07344e6i	1.44550i	−0.691111	+	0.722749i	$0.742878\pi$
$224$	−176128.	−0.234535
$225$	0	0
$226$	−731976.	−0.953292
$227$	626832.i	0.807396i	0.914892	+	0.403698i	$0.132275\pi$
$227$	626832.i	0.807396i	−0.914892	+	0.403698i	$0.867725\pi$
$228$	192000.i	0.244604i
$229$	116650.	0.146993	0.0734964	−	0.997295i	$-0.476584\pi$
$229$	116650.	0.146993	0.0734964	+	0.997295i	$0.476584\pi$
$230$	0	0
$231$	−544896.	−0.671868
$232$	140160.i	0.170964i
$233$	− 743046.i	− 0.896656i	−0.893869	−	0.448328i	$-0.852020\pi$
$233$	− 743046.i	− 0.896656i	0.893869	−	0.448328i	$-0.147980\pi$
$234$	−1.26007e6	−1.50437
$235$	0	0
$236$	127680.	0.149225
$237$	1.10976e6i	1.28339i
$238$	− 152736.i	− 0.174783i
$239$	−978720.	−1.10832	−0.554158	−	0.832411i	$-0.686960\pi$
$239$	−978720.	−1.10832	−0.554158	+	0.832411i	$0.686960\pi$
$240$	0	0
$241$	−1.13280e6	−1.25635	−0.628174	−	0.778073i	$-0.716197\pi$
$241$	−1.13280e6	−1.25635	−0.628174	+	0.778073i	$0.716197\pi$
$242$	− 574508.i	− 0.630605i
$243$	− 1.22278e6i	− 1.32841i
$244$	−265952.	−0.285975
$245$	0	0
$246$	−119232.	−0.125619
$247$	473000.i	0.493309i
$248$	− 147968.i	− 0.152770i
$249$	1.23782e6	1.26520
$250$	0	0
$251$	905652.	0.907355	0.453677	−	0.891166i	$-0.350112\pi$
$251$	905652.	0.907355	0.453677	+	0.891166i	$0.350112\pi$
$252$	916416.i	0.909057i
$253$	470448.i	0.462073i
$254$	−489808.	−0.476367
$255$	0	0
$256$	65536.0	0.0625000
$257$	− 1.93994e6i	− 1.83212i	−0.401036	−	0.916062i	$-0.631350\pi$
$257$	− 1.93994e6i	− 1.83212i	0.401036	−	0.916062i	$-0.368650\pi$
$258$	− 1.97990e6i	− 1.85180i
$259$	−1.93362e6	−1.79111
$260$	0	0
$261$	729270.	0.662654
$262$	− 967632.i	− 0.870877i
$263$	− 805476.i	− 0.718064i	−0.933325	−	0.359032i	$-0.883107\pi$
$263$	− 805476.i	− 0.718064i	0.933325	−	0.359032i	$-0.116893\pi$
$264$	202752.	0.179042
$265$	0	0
$266$	344000.	0.298095
$267$	2.64744e6i	2.27273i
$268$	28928.0i	0.0246026i
$269$	858690.	0.723529	0.361764	−	0.932270i	$-0.382175\pi$
$269$	858690.	0.723529	0.361764	+	0.932270i	$0.382175\pi$
$270$	0	0
$271$	−383608.	−0.317296	−0.158648	−	0.987335i	$-0.550713\pi$
$271$	−383608.	−0.317296	−0.158648	+	0.987335i	$0.550713\pi$
$272$	56832.0i	0.0465769i
$273$	3.90509e6i	3.17120i
$274$	1.10839e6	0.891902
$275$	0	0
$276$	1.36858e6	1.08142
$277$	− 2.01076e6i	− 1.57456i	−0.616593	−	0.787282i	$-0.711488\pi$
$277$	− 2.01076e6i	− 1.57456i	0.616593	−	0.787282i	$-0.288512\pi$
$278$	774160.i	0.600785i
$279$	−769896.	−0.592136
$280$	0	0
$281$	202602.	0.153066	0.0765329	−	0.997067i	$-0.475615\pi$
$281$	202602.	0.153066	0.0765329	+	0.997067i	$0.475615\pi$
$282$	632448.i	0.473589i
$283$	− 221536.i	− 0.164429i	−0.996615	−	0.0822145i	$-0.973801\pi$
$283$	− 221536.i	− 0.164429i	0.996615	−	0.0822145i	$-0.0261992\pi$
$284$	392448.	0.288726
$285$	0	0
$286$	499488.	0.361085
$287$	213624.i	0.153089i
$288$	− 340992.i	− 0.242250i
$289$	1.37057e6	0.965289
$290$	0	0
$291$	−1.88117e6	−1.30225
$292$	− 327584.i	− 0.224836i
$293$	− 322506.i	− 0.219467i	−0.993961	−	0.109733i	$-0.965000\pi$
$293$	− 322506.i	− 0.219467i	0.993961	−	0.109733i	$-0.0349997\pi$
$294$	1.22659e6	0.827622
$295$	0	0
$296$	719488.	0.477303
$297$	− 285120.i	− 0.187558i
$298$	− 562200.i	− 0.366733i
$299$	3.37154e6	2.18098
$300$	0	0
$301$	−3.54733e6	−2.25676
$302$	1.73581e6i	1.09518i
$303$	3.40661e6i	2.13165i
$304$	−128000.	−0.0794376
$305$	0	0
$306$	295704.	0.180532
$307$	− 1.44301e6i	− 0.873822i	−0.899505	−	0.436911i	$-0.856073\pi$
$307$	− 1.44301e6i	− 0.873822i	0.899505	−	0.436911i	$-0.143927\pi$
$308$	− 363264.i	− 0.218195i
$309$	10464.0	0.00623450
$310$	0	0
$311$	171312.	0.100435	0.0502177	−	0.998738i	$-0.484008\pi$
$311$	171312.	0.100435	0.0502177	+	0.998738i	$0.484008\pi$
$312$	− 1.45306e6i	− 0.845076i
$313$	− 1.02689e6i	− 0.592463i	−0.955116	−	0.296232i	$-0.904270\pi$
$313$	− 1.02689e6i	− 0.592463i	0.955116	−	0.296232i	$-0.0957300\pi$
$314$	−2.22369e6	−1.27277
$315$	0	0
$316$	−739840.	−0.416793
$317$	− 752958.i	− 0.420845i	−0.977610	−	0.210423i	$-0.932516\pi$
$317$	− 752958.i	− 0.420845i	0.977610	−	0.210423i	$-0.0674840\pi$
$318$	2.02234e6i	1.12146i
$319$	−289080.	−0.159053
$320$	0	0
$321$	5.59123e6	3.02862
$322$	− 2.45203e6i	− 1.31791i
$323$	− 111000.i	− 0.0591993i
$324$	465264.	0.246228
$325$	0	0
$326$	266464.	0.138866
$327$	4.19640e6i	2.17024i
$328$	− 79488.0i	− 0.0407959i
$329$	1.13314e6	0.577155
$330$	0	0
$331$	1.99413e6	1.00042	0.500212	−	0.865903i	$-0.333255\pi$
$331$	1.99413e6	1.00042	0.500212	+	0.865903i	$0.333255\pi$
$332$	825216.i	0.410887i
$333$	− 3.74359e6i	− 1.85002i
$334$	−822768.	−0.403563
$335$	0	0
$336$	−1.05677e6	−0.510660
$337$	987022.i	0.473426i	0.971580	+	0.236713i	$0.0760701\pi$
$337$	987022.i	0.473426i	−0.971580	+	0.236713i	$0.923930\pi$
$338$	− 2.09449e6i	− 0.997211i
$339$	−4.39186e6	−2.07562
$340$	0	0
$341$	305184.	0.142127
$342$	666000.i	0.307899i
$343$	693160.i	0.318125i
$344$	1.31994e6	0.601391
$345$	0	0
$346$	−1.73542e6	−0.779316
$347$	− 2.20601e6i	− 0.983520i	−0.870731	−	0.491760i	$-0.836354\pi$
$347$	− 2.20601e6i	− 0.983520i	0.870731	−	0.491760i	$-0.163646\pi$
$348$	840960.i	0.372244i
$349$	−2.74187e6	−1.20499	−0.602495	−	0.798123i	$-0.705827\pi$
$349$	−2.74187e6	−1.20499	−0.602495	+	0.798123i	$0.705827\pi$
$350$	0	0
$351$	−2.04336e6	−0.885273
$352$	135168.i	0.0581456i
$353$	− 2.38957e6i	− 1.02066i	−0.859978	−	0.510331i	$-0.829523\pi$
$353$	− 2.38957e6i	− 1.02066i	0.859978	−	0.510331i	$-0.170477\pi$
$354$	766080.	0.324912
$355$	0	0
$356$	−1.76496e6	−0.738091
$357$	− 916416.i	− 0.380559i
$358$	1.95672e6i	0.806903i
$359$	279480.	0.114450	0.0572248	−	0.998361i	$-0.481775\pi$
$359$	279480.	0.114450	0.0572248	+	0.998361i	$0.481775\pi$
$360$	0	0
$361$	−2.22610e6	−0.899035
$362$	2.87785e6i	1.15424i
$363$	− 3.44705e6i	− 1.37303i
$364$	−2.60339e6	−1.02988
$365$	0	0
$366$	−1.59571e6	−0.622661
$367$	2.47637e6i	0.959734i	0.877341	+	0.479867i	$0.159315\pi$
$367$	2.47637e6i	0.959734i	−0.877341	+	0.479867i	$0.840685\pi$
$368$	912384.i	0.351203i
$369$	−413586.	−0.158125
$370$	0	0
$371$	3.62335e6	1.36671
$372$	− 887808.i	− 0.332630i
$373$	2.74525e6i	1.02167i	0.859679	+	0.510835i	$0.170664\pi$
$373$	2.74525e6i	1.02167i	−0.859679	+	0.510835i	$0.829336\pi$
$374$	−117216.	−0.0433319
$375$	0	0
$376$	−421632.	−0.153803
$377$	2.07174e6i	0.750727i
$378$	1.48608e6i	0.534949i
$379$	1.18906e6	0.425212	0.212606	−	0.977138i	$-0.431805\pi$
$379$	1.18906e6	0.425212	0.212606	+	0.977138i	$0.431805\pi$
$380$	0	0
$381$	−2.93885e6	−1.03721
$382$	− 743712.i	− 0.260763i
$383$	3.25760e6i	1.13475i	0.823458	+	0.567377i	$0.192042\pi$
$383$	3.25760e6i	1.13475i	−0.823458	+	0.567377i	$0.807958\pi$
$384$	393216.	0.136083
$385$	0	0
$386$	2.36562e6	0.808123
$387$	− 6.86779e6i	− 2.33099i
$388$	− 1.25411e6i	− 0.422919i
$389$	−1.98351e6	−0.664600	−0.332300	−	0.943174i	$-0.607825\pi$
$389$	−1.98351e6	−0.664600	−0.332300	+	0.943174i	$0.607825\pi$
$390$	0	0
$391$	−791208.	−0.261727
$392$	817728.i	0.268778i
$393$	− 5.80579e6i	− 1.89618i
$394$	1.79791e6	0.583483
$395$	0	0
$396$	703296.	0.225372
$397$	− 4.97416e6i	− 1.58396i	−0.610549	−	0.791978i	$-0.709051\pi$
$397$	− 4.97416e6i	− 1.58396i	0.610549	−	0.791978i	$-0.290949\pi$
$398$	− 628640.i	− 0.198927i
$399$	2.06400e6	0.649049
$400$	0	0
$401$	−1.34264e6	−0.416963	−0.208482	−	0.978026i	$-0.566852\pi$
$401$	−1.34264e6	−0.416963	−0.208482	+	0.978026i	$0.566852\pi$
$402$	173568.i	0.0535679i
$403$	− 2.18715e6i	− 0.670836i
$404$	−2.27107e6	−0.692273
$405$	0	0
$406$	1.50672e6	0.453646
$407$	1.48394e6i	0.444050i
$408$	340992.i	0.101413i
$409$	1.09423e6	0.323445	0.161722	−	0.986836i	$-0.448295\pi$
$409$	1.09423e6	0.323445	0.161722	+	0.986836i	$0.448295\pi$
$410$	0	0
$411$	6.65035e6	1.94196
$412$	6976.00i	0.00202471i
$413$	− 1.37256e6i	− 0.395964i
$414$	4.74725e6	1.36126
$415$	0	0
$416$	968704.	0.274447
$417$	4.64496e6i	1.30810i
$418$	− 264000.i	− 0.0739032i
$419$	954060.	0.265485	0.132743	−	0.991151i	$-0.457622\pi$
$419$	954060.	0.265485	0.132743	+	0.991151i	$0.457622\pi$
$420$	0	0
$421$	−1.59390e6	−0.438284	−0.219142	−	0.975693i	$-0.570326\pi$
$421$	−1.59390e6	−0.438284	−0.219142	+	0.975693i	$0.570326\pi$
$422$	1.01221e6i	0.276687i
$423$	2.19380e6i	0.596138i
$424$	−1.34822e6	−0.364206
$425$	0	0
$426$	2.35469e6	0.628651
$427$	2.85898e6i	0.758826i
$428$	3.72749e6i	0.983574i
$429$	2.99693e6	0.786200
$430$	0	0
$431$	−2.64665e6	−0.686283	−0.343141	−	0.939284i	$-0.611491\pi$
$431$	−2.64665e6	−0.686283	−0.343141	+	0.939284i	$0.611491\pi$
$432$	− 552960.i	− 0.142556i
$433$	3.72355e6i	0.954416i	0.878790	+	0.477208i	$0.158351\pi$
$433$	3.72355e6i	0.954416i	−0.878790	+	0.477208i	$0.841649\pi$
$434$	−1.59066e6	−0.405370
$435$	0	0
$436$	−2.79760e6	−0.704806
$437$	− 1.78200e6i	− 0.446379i
$438$	− 1.96550e6i	− 0.489541i
$439$	2.58340e6	0.639780	0.319890	−	0.947455i	$-0.396354\pi$
$439$	2.58340e6	0.639780	0.319890	+	0.947455i	$0.396354\pi$
$440$	0	0
$441$	4.25474e6	1.04178
$442$	840048.i	0.204526i
$443$	7.56206e6i	1.83076i	0.402593	+	0.915379i	$0.368109\pi$
$443$	7.56206e6i	1.83076i	−0.402593	+	0.915379i	$0.631891\pi$
$444$	4.31693e6	1.03924
$445$	0	0
$446$	−4.29378e6	−1.02212
$447$	− 3.37320e6i	− 0.798497i
$448$	− 704512.i	− 0.165842i
$449$	−4.30773e6	−1.00840	−0.504200	−	0.863587i	$-0.668212\pi$
$449$	−4.30773e6	−1.00840	−0.504200	+	0.863587i	$0.668212\pi$
$450$	0	0
$451$	163944.	0.0379537
$452$	− 2.92790e6i	− 0.674079i
$453$	1.04148e7i	2.38456i
$454$	−2.50733e6	−0.570915
$455$	0	0
$456$	−768000.	−0.172961
$457$	2.24354e6i	0.502509i	0.967921	+	0.251254i	$0.0808431\pi$
$457$	2.24354e6i	0.502509i	−0.967921	+	0.251254i	$0.919157\pi$
$458$	466600.i	0.103940i
$459$	479520.	0.106237
$460$	0	0
$461$	1.65670e6	0.363071	0.181536	−	0.983384i	$-0.441893\pi$
$461$	1.65670e6	0.363071	0.181536	+	0.983384i	$0.441893\pi$
$462$	− 2.17958e6i	− 0.475082i
$463$	− 2.89160e6i	− 0.626881i	−0.949608	−	0.313441i	$-0.898518\pi$
$463$	− 2.89160e6i	− 0.626881i	0.949608	−	0.313441i	$-0.101482\pi$
$464$	−560640.	−0.120890
$465$	0	0
$466$	2.97218e6	0.634032
$467$	6.52699e6i	1.38491i	0.721462	+	0.692454i	$0.243470\pi$
$467$	6.52699e6i	1.38491i	−0.721462	+	0.692454i	$0.756530\pi$
$468$	− 5.04029e6i	− 1.06375i
$469$	310976.	0.0652822
$470$	0	0
$471$	−1.33421e7	−2.77123
$472$	510720.i	0.105518i
$473$	2.72237e6i	0.559492i
$474$	−4.43904e6	−0.907493
$475$	0	0
$476$	610944.	0.123590
$477$	7.01498e6i	1.41166i
$478$	− 3.91488e6i	− 0.783698i
$479$	5.96232e6	1.18734	0.593672	−	0.804707i	$-0.297678\pi$
$479$	5.96232e6	1.18734	0.593672	+	0.804707i	$0.297678\pi$
$480$	0	0
$481$	1.06349e7	2.09591
$482$	− 4.53119e6i	− 0.888372i
$483$	− 1.47122e7i	− 2.86952i
$484$	2.29803e6	0.445905
$485$	0	0
$486$	4.89110e6	0.939326
$487$	− 2.99191e6i	− 0.571644i	−0.958283	−	0.285822i	$-0.907733\pi$
$487$	− 2.99191e6i	− 0.571644i	0.958283	−	0.285822i	$-0.0922666\pi$
$488$	− 1.06381e6i	− 0.202215i
$489$	1.59878e6	0.302355
$490$	0	0
$491$	−1.20419e6	−0.225419	−0.112710	−	0.993628i	$-0.535953\pi$
$491$	−1.20419e6	−0.225419	−0.112710	+	0.993628i	$0.535953\pi$
$492$	− 476928.i	− 0.0888260i
$493$	− 486180.i	− 0.0900907i
$494$	−1.89200e6	−0.348822
$495$	0	0
$496$	591872.	0.108025
$497$	− 4.21882e6i	− 0.766125i
$498$	4.95130e6i	0.894634i
$499$	−9.20546e6	−1.65499	−0.827493	−	0.561477i	$-0.810233\pi$
$499$	−9.20546e6	−1.65499	−0.827493	+	0.561477i	$0.810233\pi$
$500$	0	0
$501$	−4.93661e6	−0.878687
$502$	3.62261e6i	0.641597i
$503$	− 3.35956e6i	− 0.592055i	−0.955179	−	0.296027i	$-0.904338\pi$
$503$	− 3.35956e6i	− 0.592055i	0.955179	−	0.296027i	$-0.0956620\pi$
$504$	−3.66566e6	−0.642801
$505$	0	0
$506$	−1.88179e6	−0.326735
$507$	− 1.25670e7i	− 2.17125i
$508$	− 1.95923e6i	− 0.336842i
$509$	2.53701e6	0.434038	0.217019	−	0.976167i	$-0.430367\pi$
$509$	2.53701e6	0.434038	0.217019	+	0.976167i	$0.430367\pi$
$510$	0	0
$511$	−3.52153e6	−0.596594
$512$	262144.i	0.0441942i
$513$	1.08000e6i	0.181188i
$514$	7.75975e6	1.29551
$515$	0	0
$516$	7.91962e6	1.30942
$517$	− 869616.i	− 0.143087i
$518$	− 7.73450e6i	− 1.26651i
$519$	−1.04125e7	−1.69682
$520$	0	0
$521$	−9.31580e6	−1.50358	−0.751789	−	0.659404i	$-0.770809\pi$
$521$	−9.31580e6	−1.50358	−0.751789	+	0.659404i	$0.770809\pi$
$522$	2.91708e6i	0.468567i
$523$	− 5.02802e6i	− 0.803790i	−0.915686	−	0.401895i	$-0.868352\pi$
$523$	− 5.02802e6i	− 0.803790i	0.915686	−	0.401895i	$-0.131648\pi$
$524$	3.87053e6	0.615803
$525$	0	0
$526$	3.22190e6	0.507748
$527$	513264.i	0.0805034i
$528$	811008.i	0.126602i
$529$	−6.26575e6	−0.973496
$530$	0	0
$531$	2.65734e6	0.408988
$532$	1.37600e6i	0.210785i
$533$	− 1.17493e6i	− 0.179141i
$534$	−1.05898e7	−1.60706
$535$	0	0
$536$	−115712.	−0.0173967
$537$	1.17403e7i	1.75689i
$538$	3.43476e6i	0.511612i
$539$	−1.68656e6	−0.250052
$540$	0	0
$541$	134222.	0.0197165	0.00985827	−	0.999951i	$-0.496862\pi$
$541$	134222.	0.0197165	0.00985827	+	0.999951i	$0.496862\pi$
$542$	− 1.53443e6i	− 0.224362i
$543$	1.72671e7i	2.51316i
$544$	−227328.	−0.0329348
$545$	0	0
$546$	−1.56204e7	−2.24238
$547$	− 605648.i	− 0.0865470i	−0.999063	−	0.0432735i	$-0.986221\pi$
$547$	− 605648.i	− 0.0865470i	0.999063	−	0.0432735i	$-0.0137787\pi$
$548$	4.43357e6i	0.630670i
$549$	−5.53513e6	−0.783784
$550$	0	0
$551$	1.09500e6	0.153651
$552$	5.47430e6i	0.764683i
$553$	7.95328e6i	1.10594i
$554$	8.04303e6	1.11339
$555$	0	0
$556$	−3.09664e6	−0.424819
$557$	7.06240e6i	0.964527i	0.876026	+	0.482264i	$0.160185\pi$
$557$	7.06240e6i	0.964527i	−0.876026	+	0.482264i	$0.839815\pi$
$558$	− 3.07958e6i	− 0.418703i
$559$	1.95103e7	2.64079
$560$	0	0
$561$	−703296.	−0.0943476
$562$	810408.i	0.108234i
$563$	− 1.03029e7i	− 1.36990i	−0.728588	−	0.684952i	$-0.759823\pi$
$563$	− 1.03029e7i	− 1.36990i	0.728588	−	0.684952i	$-0.240177\pi$
$564$	−2.52979e6	−0.334878
$565$	0	0
$566$	886144.	0.116269
$567$	− 5.00159e6i	− 0.653357i
$568$	1.56979e6i	0.204160i
$569$	−1.04769e6	−0.135660	−0.0678300	−	0.997697i	$-0.521608\pi$
$569$	−1.04769e6	−0.135660	−0.0678300	+	0.997697i	$0.521608\pi$
$570$	0	0
$571$	1.40765e7	1.80677	0.903385	−	0.428830i	$-0.141074\pi$
$571$	1.40765e7	1.80677	0.903385	+	0.428830i	$0.141074\pi$
$572$	1.99795e6i	0.255326i
$573$	− 4.46227e6i	− 0.567766i
$574$	−854496.	−0.108251
$575$	0	0
$576$	1.36397e6	0.171296
$577$	− 1.62682e6i	− 0.203423i	−0.994814	−	0.101711i	$-0.967568\pi$
$577$	− 1.62682e6i	− 0.203423i	0.994814	−	0.101711i	$-0.0324318\pi$
$578$	5.48229e6i	0.682563i
$579$	1.41937e7	1.75955
$580$	0	0
$581$	8.87107e6	1.09027
$582$	− 7.52467e6i	− 0.920831i
$583$	− 2.78071e6i	− 0.338832i
$584$	1.31034e6	0.158983
$585$	0	0
$586$	1.29002e6	0.155186
$587$	− 6.96089e6i	− 0.833814i	−0.908949	−	0.416907i	$-0.863114\pi$
$587$	− 6.96089e6i	− 0.833814i	0.908949	−	0.416907i	$-0.136886\pi$
$588$	4.90637e6i	0.585217i
$589$	−1.15600e6	−0.137300
$590$	0	0
$591$	1.07875e7	1.27043
$592$	2.87795e6i	0.337504i
$593$	− 1.13639e7i	− 1.32706i	−0.748150	−	0.663529i	$-0.769058\pi$
$593$	− 1.13639e7i	− 1.32706i	0.748150	−	0.663529i	$-0.230942\pi$
$594$	1.14048e6	0.132624
$595$	0	0
$596$	2.24880e6	0.259320
$597$	− 3.77184e6i	− 0.433129i
$598$	1.34862e7i	1.54218i
$599$	−1.48688e7	−1.69321	−0.846603	−	0.532224i	$-0.821356\pi$
$599$	−1.48688e7	−1.69321	−0.846603	+	0.532224i	$0.821356\pi$
$600$	0	0
$601$	−1.23612e6	−0.139596	−0.0697981	−	0.997561i	$-0.522236\pi$
$601$	−1.23612e6	−0.139596	−0.0697981	+	0.997561i	$0.522236\pi$
$602$	− 1.41893e7i	− 1.59577i
$603$	602064.i	0.0674294i
$604$	−6.94323e6	−0.774407
$605$	0	0
$606$	−1.36264e7	−1.50730
$607$	1.24498e7i	1.37149i	0.727844	+	0.685743i	$0.240522\pi$
$607$	1.24498e7i	1.37149i	−0.727844	+	0.685743i	$0.759478\pi$
$608$	− 512000.i	− 0.0561709i
$609$	9.04032e6	0.987735
$610$	0	0
$611$	−6.23225e6	−0.675370
$612$	1.18282e6i	0.127655i
$613$	− 8.73491e6i	− 0.938873i	−0.882966	−	0.469437i	$-0.844457\pi$
$613$	− 8.73491e6i	− 0.938873i	0.882966	−	0.469437i	$-0.155543\pi$
$614$	5.77203e6	0.617885
$615$	0	0
$616$	1.45306e6	0.154287
$617$	− 1.25495e7i	− 1.32713i	−0.748119	−	0.663565i	$-0.769043\pi$
$617$	− 1.25495e7i	− 1.32713i	0.748119	−	0.663565i	$-0.230957\pi$
$618$	41856.0i	0.00440846i
$619$	1.46658e7	1.53843	0.769216	−	0.638988i	$-0.220647\pi$
$619$	1.46658e7	1.53843	0.769216	+	0.638988i	$0.220647\pi$
$620$	0	0
$621$	7.69824e6	0.801055
$622$	685248.i	0.0710186i
$623$	1.89733e7i	1.95850i
$624$	5.81222e6	0.597559
$625$	0	0
$626$	4.10754e6	0.418935
$627$	− 1.58400e6i	− 0.160911i
$628$	− 8.89475e6i	− 0.899984i
$629$	−2.49572e6	−0.251519
$630$	0	0
$631$	−196288.	−0.0196255	−0.00981274	−	0.999952i	$-0.503124\pi$
$631$	−196288.	−0.0196255	−0.00981274	+	0.999952i	$0.503124\pi$
$632$	− 2.95936e6i	− 0.294717i
$633$	6.07325e6i	0.602437i
$634$	3.01183e6	0.297583
$635$	0	0
$636$	−8.08934e6	−0.792995
$637$	1.20870e7i	1.18024i
$638$	− 1.15632e6i	− 0.112467i
$639$	8.16782e6	0.791324
$640$	0	0
$641$	−1.11596e7	−1.07276	−0.536381	−	0.843976i	$-0.680209\pi$
$641$	−1.11596e7	−1.07276	−0.536381	+	0.843976i	$0.680209\pi$
$642$	2.23649e7i	2.14156i
$643$	− 2.25158e6i	− 0.214763i	−0.994218	−	0.107381i	$-0.965753\pi$
$643$	− 2.25158e6i	− 0.214763i	0.994218	−	0.107381i	$-0.0342466\pi$
$644$	9.80813e6	0.931905
$645$	0	0
$646$	444000.	0.0418602
$647$	− 8.05319e6i	− 0.756323i	−0.925740	−	0.378161i	$-0.876556\pi$
$647$	− 8.05319e6i	− 0.756323i	0.925740	−	0.378161i	$-0.123444\pi$
$648$	1.86106e6i	0.174109i
$649$	−1.05336e6	−0.0981669
$650$	0	0
$651$	−9.54394e6	−0.882623
$652$	1.06586e6i	0.0981928i
$653$	− 416466.i	− 0.0382205i	−0.999817	−	0.0191103i	$-0.993917\pi$
$653$	− 416466.i	− 0.0382205i	0.999817	−	0.0191103i	$-0.00608336\pi$
$654$	−1.67856e7	−1.53459
$655$	0	0
$656$	317952.	0.0288471
$657$	− 6.81784e6i	− 0.616217i
$658$	4.53254e6i	0.408110i
$659$	−1.31721e7	−1.18152	−0.590761	−	0.806847i	$-0.701172\pi$
$659$	−1.31721e7	−1.18152	−0.590761	+	0.806847i	$0.701172\pi$
$660$	0	0
$661$	−1.69494e6	−0.150886	−0.0754432	−	0.997150i	$-0.524037\pi$
$661$	−1.69494e6	−0.150886	−0.0754432	+	0.997150i	$0.524037\pi$
$662$	7.97653e6i	0.707406i
$663$	5.04029e6i	0.445319i
$664$	−3.30086e6	−0.290541
$665$	0	0
$666$	1.49743e7	1.30816
$667$	− 7.80516e6i	− 0.679309i
$668$	− 3.29107e6i	− 0.285362i
$669$	−2.57627e7	−2.22549
$670$	0	0
$671$	2.19410e6	0.188127
$672$	− 4.22707e6i	− 0.361091i
$673$	− 8.91605e6i	− 0.758813i	−0.925230	−	0.379406i	$-0.876128\pi$
$673$	− 8.91605e6i	− 0.758813i	0.925230	−	0.379406i	$-0.123872\pi$
$674$	−3.94809e6	−0.334763
$675$	0	0
$676$	8.37797e6	0.705134
$677$	1.42894e7i	1.19824i	0.800661	+	0.599118i	$0.204482\pi$
$677$	1.42894e7i	1.19824i	−0.800661	+	0.599118i	$0.795518\pi$
$678$	− 1.75674e7i	− 1.46769i
$679$	−1.34817e7	−1.12220
$680$	0	0
$681$	−1.50440e7	−1.24307
$682$	1.22074e6i	0.100499i
$683$	− 5.33314e6i	− 0.437452i	−0.975786	−	0.218726i	$-0.929810\pi$
$683$	− 5.33314e6i	− 0.437452i	0.975786	−	0.218726i	$-0.0701902\pi$
$684$	−2.66400e6	−0.217718
$685$	0	0
$686$	−2.77264e6	−0.224949
$687$	2.79960e6i	0.226310i
$688$	5.27974e6i	0.425248i
$689$	−1.99284e7	−1.59928
$690$	0	0
$691$	698252.	0.0556310	0.0278155	−	0.999613i	$-0.491145\pi$
$691$	698252.	0.0556310	0.0278155	+	0.999613i	$0.491145\pi$
$692$	− 6.94166e6i	− 0.551059i
$693$	− 7.56043e6i	− 0.598017i
$694$	8.82403e6	0.695454
$695$	0	0
$696$	−3.36384e6	−0.263216
$697$	275724.i	0.0214977i
$698$	− 1.09675e7i	− 0.852056i
$699$	1.78331e7	1.38049
$700$	0	0
$701$	1.79880e7	1.38257	0.691285	−	0.722582i	$-0.257045\pi$
$701$	1.79880e7	1.38257	0.691285	+	0.722582i	$0.257045\pi$
$702$	− 8.17344e6i	− 0.625982i
$703$	− 5.62100e6i	− 0.428968i
$704$	−540672.	−0.0411152
$705$	0	0
$706$	9.55826e6	0.721718
$707$	2.44140e7i	1.83692i
$708$	3.06432e6i	0.229748i
$709$	1.39464e7	1.04195	0.520975	−	0.853572i	$-0.325568\pi$
$709$	1.39464e7	1.04195	0.520975	+	0.853572i	$0.325568\pi$
$710$	0	0
$711$	−1.53979e7	−1.14232
$712$	− 7.05984e6i	− 0.521909i
$713$	8.23997e6i	0.607018i
$714$	3.66566e6	0.269096
$715$	0	0
$716$	−7.82688e6	−0.570566
$717$	− 2.34893e7i	− 1.70636i
$718$	1.11792e6i	0.0809282i
$719$	−6.22272e6	−0.448909	−0.224454	−	0.974485i	$-0.572060\pi$
$719$	−6.22272e6	−0.448909	−0.224454	+	0.974485i	$0.572060\pi$
$720$	0	0
$721$	74992.0	0.00537250
$722$	− 8.90440e6i	− 0.635714i
$723$	− 2.71872e7i	− 1.93427i
$724$	−1.15114e7	−0.816172
$725$	0	0
$726$	1.37882e7	0.970880
$727$	7.76729e6i	0.545047i	0.962149	+	0.272523i	$0.0878582\pi$
$727$	7.76729e6i	0.545047i	−0.962149	+	0.272523i	$0.912142\pi$
$728$	− 1.04136e7i	− 0.728234i
$729$	2.22804e7	1.55276
$730$	0	0
$731$	−4.57853e6	−0.316907
$732$	− 6.38285e6i	− 0.440288i
$733$	2.42083e7i	1.66420i	0.554627	+	0.832099i	$0.312861\pi$
$733$	2.42083e7i	1.66420i	−0.554627	+	0.832099i	$0.687139\pi$
$734$	−9.90549e6	−0.678634
$735$	0	0
$736$	−3.64954e6	−0.248338
$737$	− 238656.i	− 0.0161847i
$738$	− 1.65434e6i	− 0.111811i
$739$	−1.26850e7	−0.854434	−0.427217	−	0.904149i	$-0.640506\pi$
$739$	−1.26850e7	−0.854434	−0.427217	+	0.904149i	$0.640506\pi$
$740$	0	0
$741$	−1.13520e7	−0.759498
$742$	1.44934e7i	0.966409i
$743$	1.97632e7i	1.31337i	0.754166	+	0.656684i	$0.228041\pi$
$743$	1.97632e7i	1.31337i	−0.754166	+	0.656684i	$0.771959\pi$
$744$	3.55123e6	0.235205
$745$	0	0
$746$	−1.09810e7	−0.722429
$747$	1.71748e7i	1.12613i
$748$	− 468864.i	− 0.0306403i
$749$	4.00705e7	2.60988
$750$	0	0
$751$	−9.01761e6	−0.583434	−0.291717	−	0.956505i	$-0.594226\pi$
$751$	−9.01761e6	−0.583434	−0.291717	+	0.956505i	$0.594226\pi$
$752$	− 1.68653e6i	− 0.108755i
$753$	2.17356e7i	1.39696i
$754$	−8.28696e6	−0.530844
$755$	0	0
$756$	−5.94432e6	−0.378266
$757$	1.12556e6i	0.0713887i	0.999363	+	0.0356944i	$0.0113643\pi$
$757$	1.12556e6i	0.0713887i	−0.999363	+	0.0356944i	$0.988636\pi$
$758$	4.75624e6i	0.300670i
$759$	−1.12908e7	−0.711407
$760$	0	0
$761$	2.25747e7	1.41306	0.706529	−	0.707684i	$-0.250260\pi$
$761$	2.25747e7	1.41306	0.706529	+	0.707684i	$0.250260\pi$
$762$	− 1.17554e7i	− 0.733415i
$763$	3.00742e7i	1.87018i
$764$	2.97485e6	0.184387
$765$	0	0
$766$	−1.30304e7	−0.802392
$767$	7.54908e6i	0.463346i
$768$	1.57286e6i	0.0962250i
$769$	632350.	0.0385604	0.0192802	−	0.999814i	$-0.493863\pi$
$769$	632350.	0.0385604	0.0192802	+	0.999814i	$0.493863\pi$
$770$	0	0
$771$	4.65585e7	2.82074
$772$	9.46250e6i	0.571429i
$773$	− 1.25867e7i	− 0.757643i	−0.925470	−	0.378822i	$-0.876329\pi$
$773$	− 1.25867e7i	− 0.757643i	0.925470	−	0.378822i	$-0.123671\pi$
$774$	2.74712e7	1.64826
$775$	0	0
$776$	5.01645e6	0.299049
$777$	− 4.64070e7i	− 2.75760i
$778$	− 7.93404e6i	− 0.469943i
$779$	−621000.	−0.0366647
$780$	0	0
$781$	−3.23770e6	−0.189937
$782$	− 3.16483e6i	− 0.185069i
$783$	4.73040e6i	0.275736i
$784$	−3.27091e6	−0.190055
$785$	0	0
$786$	2.32232e7	1.34080
$787$	− 2.15792e7i	− 1.24194i	−0.783836	−	0.620968i	$-0.786740\pi$
$787$	− 2.15792e7i	− 1.24194i	0.783836	−	0.620968i	$-0.213260\pi$
$788$	7.19165e6i	0.412584i
$789$	1.93314e7	1.10553
$790$	0	0
$791$	−3.14750e7	−1.78864
$792$	2.81318e6i	0.159362i
$793$	− 1.57244e7i	− 0.887956i
$794$	1.98966e7	1.12003
$795$	0	0
$796$	2.51456e6	0.140663
$797$	3.09760e7i	1.72735i	0.504052	+	0.863673i	$0.331842\pi$
$797$	3.09760e7i	1.72735i	−0.504052	+	0.863673i	$0.668158\pi$
$798$	8.25600e6i	0.458947i
$799$	1.46254e6	0.0810475
$800$	0	0
$801$	−3.67332e7	−2.02292
$802$	− 5.37055e6i	− 0.294838i
$803$	2.70257e6i	0.147907i
$804$	−694272.	−0.0378782
$805$	0	0
$806$	8.74861e6	0.474353
$807$	2.06086e7i	1.11395i
$808$	− 9.08429e6i	− 0.489511i
$809$	−4.24929e6	−0.228268	−0.114134	−	0.993465i	$-0.536409\pi$
$809$	−4.24929e6	−0.228268	−0.114134	+	0.993465i	$0.536409\pi$
$810$	0	0
$811$	3.42333e6	0.182767	0.0913833	−	0.995816i	$-0.470871\pi$
$811$	3.42333e6	0.182767	0.0913833	+	0.995816i	$0.470871\pi$
$812$	6.02688e6i	0.320776i
$813$	− 9.20659e6i	− 0.488509i
$814$	−5.93578e6	−0.313990
$815$	0	0
$816$	−1.36397e6	−0.0717098
$817$	− 1.03120e7i	− 0.540490i
$818$	4.37692e6i	0.228710i
$819$	−5.41831e7	−2.82263
$820$	0	0
$821$	3.10571e7	1.60806	0.804030	−	0.594588i	$-0.202685\pi$
$821$	3.10571e7	1.60806	0.804030	+	0.594588i	$0.202685\pi$
$822$	2.66014e7i	1.37317i
$823$	− 3.11904e7i	− 1.60517i	−0.596538	−	0.802584i	$-0.703458\pi$
$823$	− 3.11904e7i	− 1.60517i	0.596538	−	0.802584i	$-0.296542\pi$
$824$	−27904.0	−0.00143169
$825$	0	0
$826$	5.49024e6	0.279989
$827$	8.28487e6i	0.421233i	0.977569	+	0.210616i	$0.0675471\pi$
$827$	8.28487e6i	0.421233i	−0.977569	+	0.210616i	$0.932453\pi$
$828$	1.89890e7i	0.962556i
$829$	1.81688e7	0.918208	0.459104	−	0.888383i	$-0.348171\pi$
$829$	1.81688e7	0.918208	0.459104	+	0.888383i	$0.348171\pi$
$830$	0	0
$831$	4.82582e7	2.42420
$832$	3.87482e6i	0.194063i
$833$	− 2.83649e6i	− 0.141635i
$834$	−1.85798e7	−0.924968
$835$	0	0
$836$	1.05600e6	0.0522575
$837$	− 4.99392e6i	− 0.246393i
$838$	3.81624e6i	0.187727i
$839$	1.02743e7	0.503902	0.251951	−	0.967740i	$-0.418928\pi$
$839$	1.02743e7	0.503902	0.251951	+	0.967740i	$0.418928\pi$
$840$	0	0
$841$	−1.57150e7	−0.766171
$842$	− 6.37559e6i	− 0.309913i
$843$	4.86245e6i	0.235660i
$844$	−4.04883e6	−0.195647
$845$	0	0
$846$	−8.77522e6	−0.421533
$847$	− 2.47038e7i	− 1.18319i
$848$	− 5.39290e6i	− 0.257533i
$849$	5.31686e6	0.253155
$850$	0	0
$851$	−4.00665e7	−1.89652
$852$	9.41875e6i	0.444523i
$853$	6.28597e6i	0.295801i	0.989002	+	0.147901i	$0.0472516\pi$
$853$	6.28597e6i	0.295801i	−0.989002	+	0.147901i	$0.952748\pi$
$854$	−1.14359e7	−0.536571
$855$	0	0
$856$	−1.49100e7	−0.695492
$857$	− 1.54050e7i	− 0.716490i	−0.933628	−	0.358245i	$-0.883375\pi$
$857$	− 1.54050e7i	− 0.716490i	0.933628	−	0.358245i	$-0.116625\pi$
$858$	1.19877e7i	0.555927i
$859$	−1.43526e7	−0.663664	−0.331832	−	0.943338i	$-0.607667\pi$
$859$	−1.43526e7	−0.663664	−0.331832	+	0.943338i	$0.607667\pi$
$860$	0	0
$861$	−5.12698e6	−0.235697
$862$	− 1.05866e7i	− 0.485275i
$863$	1.33278e7i	0.609158i	0.952487	+	0.304579i	$0.0985158\pi$
$863$	1.33278e7i	0.609158i	−0.952487	+	0.304579i	$0.901484\pi$
$864$	2.21184e6	0.100802
$865$	0	0
$866$	−1.48942e7	−0.674874
$867$	3.28938e7i	1.48616i
$868$	− 6.36262e6i	− 0.286640i
$869$	6.10368e6	0.274184
$870$	0	0
$871$	−1.71037e6	−0.0763913
$872$	− 1.11904e7i	− 0.498373i
$873$	− 2.61012e7i	− 1.15911i
$874$	7.12800e6	0.315638
$875$	0	0
$876$	7.86202e6	0.346157
$877$	− 3.24846e7i	− 1.42620i	−0.701065	−	0.713098i	$-0.747292\pi$
$877$	− 3.24846e7i	− 1.42620i	0.701065	−	0.713098i	$-0.252708\pi$
$878$	1.03336e7i	0.452392i
$879$	7.74014e6	0.337891
$880$	0	0
$881$	1.54600e7	0.671073	0.335537	−	0.942027i	$-0.391082\pi$
$881$	1.54600e7	0.671073	0.335537	+	0.942027i	$0.391082\pi$
$882$	1.70190e7i	0.736651i
$883$	− 1.69478e6i	− 0.0731494i	−0.999331	−	0.0365747i	$-0.988355\pi$
$883$	− 1.69478e6i	− 0.0731494i	0.999331	−	0.0365747i	$-0.0116447\pi$
$884$	−3.36019e6	−0.144622
$885$	0	0
$886$	−3.02483e7	−1.29454
$887$	2.87257e6i	0.122592i	0.998120	+	0.0612960i	$0.0195233\pi$
$887$	2.87257e6i	0.122592i	−0.998120	+	0.0612960i	$0.980477\pi$
$888$	1.72677e7i	0.734856i
$889$	−2.10617e7	−0.893799
$890$	0	0
$891$	−3.83843e6	−0.161979
$892$	− 1.71751e7i	− 0.722749i
$893$	3.29400e6i	0.138228i
$894$	1.34928e7	0.564623
$895$	0	0
$896$	2.81805e6	0.117268
$897$	8.09171e7i	3.35783i
$898$	− 1.72309e7i	− 0.713046i
$899$	−5.06328e6	−0.208945
$900$	0	0
$901$	4.67665e6	0.191921
$902$	655776.i	0.0268373i
$903$	− 8.51359e7i	− 3.47451i
$904$	1.17116e7	0.476646
$905$	0	0
$906$	−4.16594e7	−1.68614
$907$	− 3.95422e7i	− 1.59603i	−0.602635	−	0.798017i	$-0.705882\pi$
$907$	− 3.95422e7i	− 1.59603i	0.602635	−	0.798017i	$-0.294118\pi$
$908$	− 1.00293e7i	− 0.403698i
$909$	−4.72667e7	−1.89734
$910$	0	0
$911$	1.13178e7	0.451819	0.225909	−	0.974148i	$-0.427465\pi$
$911$	1.13178e7	0.451819	0.225909	+	0.974148i	$0.427465\pi$
$912$	− 3.07200e6i	− 0.122302i
$913$	− 6.80803e6i	− 0.270299i
$914$	−8.97417e6	−0.355327
$915$	0	0
$916$	−1.86640e6	−0.0734964
$917$	− 4.16082e7i	− 1.63401i
$918$	1.91808e6i	0.0751208i
$919$	−8.51348e6	−0.332520	−0.166260	−	0.986082i	$-0.553169\pi$
$919$	−8.51348e6	−0.332520	−0.166260	+	0.986082i	$0.553169\pi$
$920$	0	0
$921$	3.46322e7	1.34534
$922$	6.62681e6i	0.256730i
$923$	2.32035e7i	0.896497i
$924$	8.71834e6	0.335934
$925$	0	0
$926$	1.15664e7	0.443272
$927$	145188.i	0.00554921i
$928$	− 2.24256e6i	− 0.0854819i
$929$	7.54587e6	0.286860	0.143430	−	0.989660i	$-0.454187\pi$
$929$	7.54587e6	0.286860	0.143430	+	0.989660i	$0.454187\pi$
$930$	0	0
$931$	6.38850e6	0.241560
$932$	1.18887e7i	0.448328i
$933$	4.11149e6i	0.154630i
$934$	−2.61080e7	−0.979278
$935$	0	0
$936$	2.01612e7	0.752187
$937$	1.84500e7i	0.686512i	0.939242	+	0.343256i	$0.111530\pi$
$937$	1.84500e7i	0.686512i	−0.939242	+	0.343256i	$0.888470\pi$
$938$	1.24390e6i	0.0461615i
$939$	2.46453e7	0.912157
$940$	0	0
$941$	6.75046e6	0.248519	0.124259	−	0.992250i	$-0.460344\pi$
$941$	6.75046e6	0.248519	0.124259	+	0.992250i	$0.460344\pi$
$942$	− 5.33685e7i	− 1.95956i
$943$	4.42649e6i	0.162099i
$944$	−2.04288e6	−0.0746127
$945$	0	0
$946$	−1.08895e7	−0.395621
$947$	− 6.45677e6i	− 0.233959i	−0.993134	−	0.116980i	$-0.962679\pi$
$947$	− 6.45677e6i	− 0.233959i	0.993134	−	0.116980i	$-0.0373212\pi$
$948$	− 1.77562e7i	− 0.641694i
$949$	1.93684e7	0.698117
$950$	0	0
$951$	1.80710e7	0.647934
$952$	2.44378e6i	0.0873915i
$953$	− 3.96648e7i	− 1.41473i	−0.706849	−	0.707364i	$-0.749884\pi$
$953$	− 3.96648e7i	− 1.41473i	0.706849	−	0.707364i	$-0.250116\pi$
$954$	−2.80599e7	−0.998195
$955$	0	0
$956$	1.56595e7	0.554158
$957$	− 6.93792e6i	− 0.244878i
$958$	2.38493e7i	0.839579i
$959$	4.76609e7	1.67346
$960$	0	0
$961$	−2.32838e7	−0.813290
$962$	4.25397e7i	1.48203i
$963$	7.75783e7i	2.69572i
$964$	1.81248e7	0.628174
$965$	0	0
$966$	5.88488e7	2.02906
$967$	3.43015e7i	1.17963i	0.807538	+	0.589816i	$0.200800\pi$
$967$	3.43015e7i	1.17963i	−0.807538	+	0.589816i	$0.799200\pi$
$968$	9.19213e6i	0.315303i
$969$	2.66400e6	0.0911433
$970$	0	0
$971$	−5.77115e6	−0.196433	−0.0982164	−	0.995165i	$-0.531314\pi$
$971$	−5.77115e6	−0.196433	−0.0982164	+	0.995165i	$0.531314\pi$
$972$	1.95644e7i	0.664204i
$973$	3.32889e7i	1.12724i
$974$	1.19676e7	0.404214
$975$	0	0
$976$	4.25523e6	0.142988
$977$	− 7.08746e6i	− 0.237549i	−0.992921	−	0.118775i	$-0.962103\pi$
$977$	− 7.08746e6i	− 0.237549i	0.992921	−	0.118775i	$-0.0378966\pi$
$978$	6.39514e6i	0.213798i
$979$	1.45609e7	0.485548
$980$	0	0
$981$	−5.82250e7	−1.93169
$982$	− 4.81675e6i	− 0.159395i
$983$	4.59362e7i	1.51625i	0.652108	+	0.758126i	$0.273885\pi$
$983$	4.59362e7i	1.51625i	−0.652108	+	0.758126i	$0.726115\pi$
$984$	1.90771e6	0.0628095
$985$	0	0
$986$	1.94472e6	0.0637037
$987$	2.71953e7i	0.888588i
$988$	− 7.56800e6i	− 0.246654i
$989$	−7.35039e7	−2.38957
$990$	0	0
$991$	−4.50298e7	−1.45652	−0.728260	−	0.685301i	$-0.759671\pi$
$991$	−4.50298e7	−1.45652	−0.728260	+	0.685301i	$0.759671\pi$
$992$	2.36749e6i	0.0763851i
$993$	4.78592e7i	1.54025i
$994$	1.68753e7	0.541732
$995$	0	0
$996$	−1.98052e7	−0.632602
$997$	2.37364e7i	0.756271i	0.925750	+	0.378136i	$0.123435\pi$
$997$	2.37364e7i	0.756271i	−0.925750	+	0.378136i	$0.876565\pi$
$998$	− 3.68218e7i	− 1.17025i
$999$	2.42827e7	0.769810

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	50.6.b.a.49.2		2
3.2	odd	2		450.6.c.h.199.1		2
4.3	odd	2		400.6.c.b.49.1		2
5.2	odd	4		10.6.a.b.1.1	✓	1
5.3	odd	4		50.6.a.d.1.1		1
5.4	even	2	inner	50.6.b.a.49.1		2
15.2	even	4		90.6.a.d.1.1		1
15.8	even	4		450.6.a.l.1.1		1
15.14	odd	2		450.6.c.h.199.2		2
20.3	even	4		400.6.a.n.1.1		1
20.7	even	4		80.6.a.a.1.1		1
20.19	odd	2		400.6.c.b.49.2		2
35.27	even	4		490.6.a.a.1.1		1
40.27	even	4		320.6.a.o.1.1		1
40.37	odd	4		320.6.a.b.1.1		1
60.47	odd	4		720.6.a.j.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
10.6.a.b.1.1	✓	1	5.2	odd	4
50.6.a.d.1.1		1	5.3	odd	4
50.6.b.a.49.1		2	5.4	even	2	inner
50.6.b.a.49.2		2	1.1	even	1	trivial
80.6.a.a.1.1		1	20.7	even	4
90.6.a.d.1.1		1	15.2	even	4
320.6.a.b.1.1		1	40.37	odd	4
320.6.a.o.1.1		1	40.27	even	4
400.6.a.n.1.1		1	20.3	even	4
400.6.c.b.49.1		2	4.3	odd	2
400.6.c.b.49.2		2	20.19	odd	2
450.6.a.l.1.1		1	15.8	even	4
450.6.c.h.199.1		2	3.2	odd	2
450.6.c.h.199.2		2	15.14	odd	2
490.6.a.a.1.1		1	35.27	even	4
720.6.a.j.1.1		1	60.47	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 50 = 2 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	50.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+4.00000i q^{2} +24.0000i q^{3} -16.0000 q^{4} -96.0000 q^{6} +172.000i q^{7} -64.0000i q^{8} -333.000 q^{9} +132.000 q^{11} -384.000i q^{12} -946.000i q^{13} -688.000 q^{14} +256.000 q^{16} +222.000i q^{17} -1332.00i q^{18} -500.000 q^{19} -4128.00 q^{21} +528.000i q^{22} +3564.00i q^{23} +1536.00 q^{24} +3784.00 q^{26} -2160.00i q^{27} -2752.00i q^{28} -2190.00 q^{29} +2312.00 q^{31} +1024.00i q^{32} +3168.00i q^{33} -888.000 q^{34} +5328.00 q^{36} +11242.0i q^{37} -2000.00i q^{38} +22704.0 q^{39} +1242.00 q^{41} -16512.0i q^{42} +20624.0i q^{43} -2112.00 q^{44} -14256.0 q^{46} -6588.00i q^{47} +6144.00i q^{48} -12777.0 q^{49} -5328.00 q^{51} +15136.0i q^{52} -21066.0i q^{53} +8640.00 q^{54} +11008.0 q^{56} -12000.0i q^{57} -8760.00i q^{58} -7980.00 q^{59} +16622.0 q^{61} +9248.00i q^{62} -57276.0i q^{63} -4096.00 q^{64} -12672.0 q^{66} -1808.00i q^{67} -3552.00i q^{68} -85536.0 q^{69} -24528.0 q^{71} +21312.0i q^{72} +20474.0i q^{73} -44968.0 q^{74} +8000.00 q^{76} +22704.0i q^{77} +90816.0i q^{78} +46240.0 q^{79} -29079.0 q^{81} +4968.00i q^{82} -51576.0i q^{83} +66048.0 q^{84} -82496.0 q^{86} -52560.0i q^{87} -8448.00i q^{88} +110310. q^{89} +162712. q^{91} -57024.0i q^{92} +55488.0i q^{93} +26352.0 q^{94} -24576.0 q^{96} +78382.0i q^{97} -51108.0i q^{98} -43956.0 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 32 q^{4} - 192 q^{6} - 666 q^{9} + 264 q^{11} - 1376 q^{14} + 512 q^{16} - 1000 q^{19} - 8256 q^{21} + 3072 q^{24} + 7568 q^{26} - 4380 q^{29} + 4624 q^{31} - 1776 q^{34} + 10656 q^{36} + 45408 q^{39}+ \cdots - 87912 q^{99}+O(q^{100}) \) 2 * q - 32 * q^4 - 192 * q^6 - 666 * q^9 + 264 * q^11 - 1376 * q^14 + 512 * q^16 - 1000 * q^19 - 8256 * q^21 + 3072 * q^24 + 7568 * q^26 - 4380 * q^29 + 4624 * q^31 - 1776 * q^34 + 10656 * q^36 + 45408 * q^39 + 2484 * q^41 - 4224 * q^44 - 28512 * q^46 - 25554 * q^49 - 10656 * q^51 + 17280 * q^54 + 22016 * q^56 - 15960 * q^59 + 33244 * q^61 - 8192 * q^64 - 25344 * q^66 - 171072 * q^69 - 49056 * q^71 - 89936 * q^74 + 16000 * q^76 + 92480 * q^79 - 58158 * q^81 + 132096 * q^84 - 164992 * q^86 + 220620 * q^89 + 325424 * q^91 + 52704 * q^94 - 49152 * q^96 - 87912 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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