LMFDB - Embedded newform 4114.2.a.bc.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [4114,2,Mod(1,4114)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4114, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("4114.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 4114 = 2 \cdot 11^{2} \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4114.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4,4,1,4,5,1,-1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	yes
Analytic conductor:	$32.8504553916$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.55585.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 10x^{2} + 9x + 16 $ x^4 - x^3 - 10x^2 + 9x + 16
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 374)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$2.22696$ of defining polynomial
Character	$\chi$	$=$	4114.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.00000 q^{2} +2.22696 q^{3} +1.00000 q^{4} +0.959363 q^{5} +2.22696 q^{6} -4.62570 q^{7} +1.00000 q^{8} +1.95936 q^{9} +0.959363 q^{10} +2.22696 q^{12} +6.94316 q^{13} -4.62570 q^{14} +2.13646 q^{15} +1.00000 q^{16} +1.00000 q^{17} +1.95936 q^{18} -2.17177 q^{19} +0.959363 q^{20} -10.3013 q^{21} -2.45393 q^{23} +2.22696 q^{24} -4.07962 q^{25} +6.94316 q^{26} -2.31746 q^{27} -4.62570 q^{28} +7.17012 q^{29} +2.13646 q^{30} +9.17012 q^{31} +1.00000 q^{32} +1.00000 q^{34} -4.43772 q^{35} +1.95936 q^{36} +10.6257 q^{37} -2.17177 q^{38} +15.4622 q^{39} +0.959363 q^{40} +10.5834 q^{41} -10.3013 q^{42} +1.75683 q^{43} +1.87974 q^{45} -2.45393 q^{46} -0.324441 q^{47} +2.22696 q^{48} +14.3971 q^{49} -4.07962 q^{50} +2.22696 q^{51} +6.94316 q^{52} +4.71620 q^{53} -2.31746 q^{54} -4.62570 q^{56} -4.83646 q^{57} +7.17012 q^{58} +13.7053 q^{59} +2.13646 q^{60} +0.453925 q^{61} +9.17012 q^{62} -9.06342 q^{63} +1.00000 q^{64} +6.66101 q^{65} -0.453925 q^{67} +1.00000 q^{68} -5.46480 q^{69} -4.43772 q^{70} -13.8050 q^{71} +1.95936 q^{72} -0.861884 q^{73} +10.6257 q^{74} -9.08517 q^{75} -2.17177 q^{76} +15.4622 q^{78} +3.10571 q^{79} +0.959363 q^{80} -11.0390 q^{81} +10.5834 q^{82} -3.26062 q^{83} -10.3013 q^{84} +0.959363 q^{85} +1.75683 q^{86} +15.9676 q^{87} -9.26062 q^{89} +1.87974 q^{90} -32.1170 q^{91} -2.45393 q^{92} +20.4215 q^{93} -0.324441 q^{94} -2.08352 q^{95} +2.22696 q^{96} -13.6240 q^{97} +14.3971 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{2} + q^{3} + 4 q^{4} + 5 q^{5} + q^{6} - q^{7} + 4 q^{8} + 9 q^{9} + 5 q^{10} + q^{12} + 3 q^{13} - q^{14} + 4 q^{16} + 4 q^{17} + 9 q^{18} - 7 q^{19} + 5 q^{20} - 8 q^{21} + 6 q^{23} + q^{24}+ \cdots + 17 q^{98}+O(q^{100}) $ 4 * q + 4 * q^2 + q^3 + 4 * q^4 + 5 * q^5 + q^6 - q^7 + 4 * q^8 + 9 * q^9 + 5 * q^10 + q^12 + 3 * q^13 - q^14 + 4 * q^16 + 4 * q^17 + 9 * q^18 - 7 * q^19 + 5 * q^20 - 8 * q^21 + 6 * q^23 + q^24 + 17 * q^25 + 3 * q^26 - 2 * q^27 - q^28 - 4 * q^29 + 4 * q^31 + 4 * q^32 + 4 * q^34 + 24 * q^35 + 9 * q^36 + 25 * q^37 - 7 * q^38 - 7 * q^39 + 5 * q^40 - 5 * q^41 - 8 * q^42 - 11 * q^43 + 42 * q^45 + 6 * q^46 - 17 * q^47 + q^48 + 17 * q^49 + 17 * q^50 + q^51 + 3 * q^52 + 2 * q^53 - 2 * q^54 - q^56 + 32 * q^57 - 4 * q^58 + 4 * q^59 - 14 * q^61 + 4 * q^62 + 23 * q^63 + 4 * q^64 + 16 * q^65 + 14 * q^67 + 4 * q^68 - 40 * q^69 + 24 * q^70 - 8 * q^71 + 9 * q^72 + 19 * q^73 + 25 * q^74 - 45 * q^75 - 7 * q^76 - 7 * q^78 - 17 * q^79 + 5 * q^80 - 12 * q^81 - 5 * q^82 + 19 * q^83 - 8 * q^84 + 5 * q^85 - 11 * q^86 + 12 * q^87 - 5 * q^89 + 42 * q^90 - 32 * q^91 + 6 * q^92 + 14 * q^93 - 17 * q^94 + 14 * q^95 + q^96 - 6 * q^97 + 17 * q^98

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 $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	2.22696	1.28574	0.642869	−	0.765976i	$-0.277744\pi$
$3$	2.22696	1.28574	0.642869	+	0.765976i	$0.277744\pi$
$4$	1.00000	0.500000
$5$	0.959363	0.429040	0.214520	−	0.976720i	$-0.431181\pi$
$5$	0.959363	0.429040	0.214520	+	0.976720i	$0.431181\pi$
$6$	2.22696	0.909154
$7$	−4.62570	−1.74835	−0.874175	−	0.485611i	$-0.838597\pi$
$7$	−4.62570	−1.74835	−0.874175	+	0.485611i	$0.838597\pi$
$8$	1.00000	0.353553
$9$	1.95936	0.653121
$10$	0.959363	0.303377
$11$	0	0
$11$	0	0
$12$	2.22696	0.642869
$13$	6.94316	1.92569	0.962843	−	0.270062i	$-0.0870442\pi$
$13$	6.94316	1.92569	0.962843	+	0.270062i	$0.0870442\pi$
$14$	−4.62570	−1.23627
$15$	2.13646	0.551633
$16$	1.00000	0.250000
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	1.95936	0.461826
$19$	−2.17177	−0.498239	−0.249119	−	0.968473i	$-0.580141\pi$
$19$	−2.17177	−0.498239	−0.249119	+	0.968473i	$0.580141\pi$
$20$	0.959363	0.214520
$21$	−10.3013	−2.24792
$22$	0	0
$23$	−2.45393	−0.511679	−0.255839	−	0.966719i	$-0.582352\pi$
$23$	−2.45393	−0.511679	−0.255839	+	0.966719i	$0.582352\pi$
$24$	2.22696	0.454577
$25$	−4.07962	−0.815925
$26$	6.94316	1.36167
$27$	−2.31746	−0.445996
$28$	−4.62570	−0.874175
$29$	7.17012	1.33146	0.665729	−	0.746194i	$-0.268121\pi$
$29$	7.17012	1.33146	0.665729	+	0.746194i	$0.268121\pi$
$30$	2.13646	0.390063
$31$	9.17012	1.64700	0.823501	−	0.567314i	$-0.192018\pi$
$31$	9.17012	1.64700	0.823501	+	0.567314i	$0.192018\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	1.00000	0.171499
$35$	−4.43772	−0.750112
$36$	1.95936	0.326560
$37$	10.6257	1.74685	0.873427	−	0.486955i	$-0.161892\pi$
$37$	10.6257	1.74685	0.873427	+	0.486955i	$0.161892\pi$
$38$	−2.17177	−0.352308
$39$	15.4622	2.47593
$40$	0.959363	0.151689
$41$	10.5834	1.65285	0.826425	−	0.563046i	$-0.190371\pi$
$41$	10.5834	1.65285	0.826425	+	0.563046i	$0.190371\pi$
$42$	−10.3013	−1.58952
$43$	1.75683	0.267915	0.133957	−	0.990987i	$-0.457231\pi$
$43$	1.75683	0.267915	0.133957	+	0.990987i	$0.457231\pi$
$44$	0	0
$45$	1.87974	0.280215
$46$	−2.45393	−0.361812
$47$	−0.324441	−0.0473246	−0.0236623	−	0.999720i	$-0.507533\pi$
$47$	−0.324441	−0.0473246	−0.0236623	+	0.999720i	$0.507533\pi$
$48$	2.22696	0.321434
$49$	14.3971	2.05673
$50$	−4.07962	−0.576946
$51$	2.22696	0.311837
$52$	6.94316	0.962843
$53$	4.71620	0.647819	0.323910	−	0.946088i	$-0.395003\pi$
$53$	4.71620	0.647819	0.323910	+	0.946088i	$0.395003\pi$
$54$	−2.31746	−0.315366
$55$	0	0
$56$	−4.62570	−0.618135
$57$	−4.83646	−0.640604
$58$	7.17012	0.941483
$59$	13.7053	1.78428	0.892140	−	0.451758i	$-0.149203\pi$
$59$	13.7053	1.78428	0.892140	+	0.451758i	$0.149203\pi$
$60$	2.13646	0.275816
$61$	0.453925	0.0581192	0.0290596	−	0.999578i	$-0.490749\pi$
$61$	0.453925	0.0581192	0.0290596	+	0.999578i	$0.490749\pi$
$62$	9.17012	1.16461
$63$	−9.06342	−1.14188
$64$	1.00000	0.125000
$65$	6.66101	0.826196
$66$	0	0
$67$	−0.453925	−0.0554558	−0.0277279	−	0.999616i	$-0.508827\pi$
$67$	−0.453925	−0.0554558	−0.0277279	+	0.999616i	$0.508827\pi$
$68$	1.00000	0.121268
$69$	−5.46480	−0.657885
$70$	−4.43772	−0.530409
$71$	−13.8050	−1.63836	−0.819179	−	0.573539i	$-0.805570\pi$
$71$	−13.8050	−1.63836	−0.819179	+	0.573539i	$0.805570\pi$
$72$	1.95936	0.230913
$73$	−0.861884	−0.100876	−0.0504379	−	0.998727i	$-0.516062\pi$
$73$	−0.861884	−0.100876	−0.0504379	+	0.998727i	$0.516062\pi$
$74$	10.6257	1.23521
$75$	−9.08517	−1.04906
$76$	−2.17177	−0.249119
$77$	0	0
$78$	15.4622	1.75074
$79$	3.10571	0.349420	0.174710	−	0.984620i	$-0.444101\pi$
$79$	3.10571	0.349420	0.174710	+	0.984620i	$0.444101\pi$
$80$	0.959363	0.107260
$81$	−11.0390	−1.22655
$82$	10.5834	1.16874
$83$	−3.26062	−0.357900	−0.178950	−	0.983858i	$-0.557270\pi$
$83$	−3.26062	−0.357900	−0.178950	+	0.983858i	$0.557270\pi$
$84$	−10.3013	−1.12396
$85$	0.959363	0.104057
$86$	1.75683	0.189444
$87$	15.9676	1.71191
$88$	0	0
$89$	−9.26062	−0.981624	−0.490812	−	0.871266i	$-0.663300\pi$
$89$	−9.26062	−0.981624	−0.490812	+	0.871266i	$0.663300\pi$
$90$	1.87974	0.198142
$91$	−32.1170	−3.36677
$92$	−2.45393	−0.255839
$93$	20.4215	2.11761
$94$	−0.324441	−0.0334635
$95$	−2.08352	−0.213764
$96$	2.22696	0.227288
$97$	−13.6240	−1.38331	−0.691656	−	0.722227i	$-0.743119\pi$
$97$	−13.6240	−1.38331	−0.691656	+	0.722227i	$0.743119\pi$
$98$	14.3971	1.45433
$99$	0	0
$100$	−4.07962	−0.407962
$101$	6.95936	0.692482	0.346241	−	0.938146i	$-0.387458\pi$
$101$	6.95936	0.692482	0.346241	+	0.938146i	$0.387458\pi$
$102$	2.22696	0.220502
$103$	−13.2867	−1.30918	−0.654589	−	0.755985i	$-0.727158\pi$
$103$	−13.2867	−1.30918	−0.654589	+	0.755985i	$0.727158\pi$
$104$	6.94316	0.680833
$105$	−9.88264	−0.964447
$106$	4.71620	0.458077
$107$	1.28380	0.124110	0.0620550	−	0.998073i	$-0.480235\pi$
$107$	1.28380	0.124110	0.0620550	+	0.998073i	$0.480235\pi$
$108$	−2.31746	−0.222998
$109$	−11.6240	−1.11338	−0.556691	−	0.830720i	$-0.687929\pi$
$109$	−11.6240	−1.11338	−0.556691	+	0.830720i	$0.687929\pi$
$110$	0	0
$111$	23.6630	2.24600
$112$	−4.62570	−0.437087
$113$	2.79747	0.263164	0.131582	−	0.991305i	$-0.457994\pi$
$113$	2.79747	0.263164	0.131582	+	0.991305i	$0.457994\pi$
$114$	−4.83646	−0.452976
$115$	−2.35420	−0.219531
$116$	7.17012	0.665729
$117$	13.6042	1.25771
$118$	13.7053	1.26168
$119$	−4.62570	−0.424037
$120$	2.13646	0.195032
$121$	0	0
$122$	0.453925	0.0410965
$123$	23.5689	2.12513
$124$	9.17012	0.823501
$125$	−8.71065	−0.779104
$126$	−9.06342	−0.807434
$127$	−16.4539	−1.46005	−0.730025	−	0.683421i	$-0.760492\pi$
$127$	−16.4539	−1.46005	−0.730025	+	0.683421i	$0.760492\pi$
$128$	1.00000	0.0883883
$129$	3.91240	0.344468
$130$	6.66101	0.584209
$131$	−9.43239	−0.824112	−0.412056	−	0.911159i	$-0.635189\pi$
$131$	−9.43239	−0.824112	−0.412056	+	0.911159i	$0.635189\pi$
$132$	0	0
$133$	10.0460	0.871096
$134$	−0.453925	−0.0392132
$135$	−2.22329	−0.191350
$136$	1.00000	0.0857493
$137$	−6.48923	−0.554413	−0.277206	−	0.960810i	$-0.589409\pi$
$137$	−6.48923	−0.554413	−0.277206	+	0.960810i	$0.589409\pi$
$138$	−5.46480	−0.465195
$139$	4.29138	0.363990	0.181995	−	0.983299i	$-0.441745\pi$
$139$	4.29138	0.363990	0.181995	+	0.983299i	$0.441745\pi$
$140$	−4.43772	−0.375056
$141$	−0.722518	−0.0608470
$142$	−13.8050	−1.15849
$143$	0	0
$144$	1.95936	0.163280
$145$	6.87875	0.571249
$146$	−0.861884	−0.0713300
$147$	32.0618	2.64441
$148$	10.6257	0.873427
$149$	−10.1272	−0.829656	−0.414828	−	0.909900i	$-0.636158\pi$
$149$	−10.1272	−0.829656	−0.414828	+	0.909900i	$0.636158\pi$
$150$	−9.08517	−0.741801
$151$	−6.71620	−0.546556	−0.273278	−	0.961935i	$-0.588108\pi$
$151$	−6.71620	−0.546556	−0.273278	+	0.961935i	$0.588108\pi$
$152$	−2.17177	−0.176154
$153$	1.95936	0.158405
$154$	0	0
$155$	8.79747	0.706630
$156$	15.4622	1.23796
$157$	0.424821	0.0339044	0.0169522	−	0.999856i	$-0.494604\pi$
$157$	0.424821	0.0339044	0.0169522	+	0.999856i	$0.494604\pi$
$158$	3.10571	0.247077
$159$	10.5028	0.832926
$160$	0.959363	0.0758443
$161$	11.3511	0.894593
$162$	−11.0390	−0.867305
$163$	−17.3809	−1.36138	−0.680688	−	0.732574i	$-0.738319\pi$
$163$	−17.3809	−1.36138	−0.680688	+	0.732574i	$0.738319\pi$
$164$	10.5834	0.826425
$165$	0	0
$166$	−3.26062	−0.253073
$167$	−1.83811	−0.142237	−0.0711186	−	0.997468i	$-0.522657\pi$
$167$	−1.83811	−0.142237	−0.0711186	+	0.997468i	$0.522657\pi$
$168$	−10.3013	−0.794759
$169$	35.2075	2.70827
$170$	0.959363	0.0735797
$171$	−4.25529	−0.325410
$172$	1.75683	0.133957
$173$	−16.9079	−1.28548	−0.642740	−	0.766084i	$-0.722202\pi$
$173$	−16.9079	−1.28548	−0.642740	+	0.766084i	$0.722202\pi$
$174$	15.9676	1.21050
$175$	18.8711	1.42652
$176$	0	0
$177$	30.5212	2.29412
$178$	−9.26062	−0.694113
$179$	0.343546	0.0256778	0.0128389	−	0.999918i	$-0.495913\pi$
$179$	0.343546	0.0256778	0.0128389	+	0.999918i	$0.495913\pi$
$180$	1.87974	0.140107
$181$	16.3082	1.21218	0.606091	−	0.795395i	$-0.292737\pi$
$181$	16.3082	1.21218	0.606091	+	0.795395i	$0.292737\pi$
$182$	−32.1170	−2.38067
$183$	1.01087	0.0747260
$184$	−2.45393	−0.180906
$185$	10.1939	0.749470
$186$	20.4215	1.49738
$187$	0	0
$188$	−0.324441	−0.0236623
$189$	10.7199	0.779756
$190$	−2.08352	−0.151154
$191$	−5.74063	−0.415377	−0.207689	−	0.978195i	$-0.566594\pi$
$191$	−5.74063	−0.415377	−0.207689	+	0.978195i	$0.566594\pi$
$192$	2.22696	0.160717
$193$	12.4080	0.893144	0.446572	−	0.894748i	$-0.352645\pi$
$193$	12.4080	0.893144	0.446572	+	0.894748i	$0.352645\pi$
$194$	−13.6240	−0.978150
$195$	14.8338	1.06227
$196$	14.3971	1.02836
$197$	−6.74860	−0.480818	−0.240409	−	0.970672i	$-0.577282\pi$
$197$	−6.74860	−0.480818	−0.240409	+	0.970672i	$0.577282\pi$
$198$	0	0
$199$	8.53520	0.605044	0.302522	−	0.953142i	$-0.402171\pi$
$199$	8.53520	0.605044	0.302522	+	0.953142i	$0.402171\pi$
$200$	−4.07962	−0.288473
$201$	−1.01087	−0.0713016
$202$	6.95936	0.489659
$203$	−33.1668	−2.32785
$204$	2.22696	0.155919
$205$	10.1533	0.709139
$206$	−13.2867	−0.925729
$207$	−4.80813	−0.334188
$208$	6.94316	0.481421
$209$	0	0
$210$	−9.88264	−0.681967
$211$	0.162550	0.0111904	0.00559519	−	0.999984i	$-0.498219\pi$
$211$	0.162550	0.0111904	0.00559519	+	0.999984i	$0.498219\pi$
$212$	4.71620	0.323910
$213$	−30.7433	−2.10650
$214$	1.28380	0.0877590
$215$	1.68544	0.114946
$216$	−2.31746	−0.157683
$217$	−42.4182	−2.87954
$218$	−11.6240	−0.787280
$219$	−1.91938	−0.129700
$220$	0	0
$221$	6.94316	0.467047
$222$	23.6630	1.58816
$223$	−10.1718	−0.681152	−0.340576	−	0.940217i	$-0.610622\pi$
$223$	−10.1718	−0.681152	−0.340576	+	0.940217i	$0.610622\pi$
$224$	−4.62570	−0.309067
$225$	−7.99346	−0.532897
$226$	2.79747	0.186085
$227$	29.0380	1.92732	0.963660	−	0.267133i	$-0.0860763\pi$
$227$	29.0380	1.92732	0.963660	+	0.267133i	$0.0860763\pi$
$228$	−4.83646	−0.320302
$229$	15.9967	1.05709	0.528546	−	0.848905i	$-0.322738\pi$
$229$	15.9967	1.05709	0.528546	+	0.848905i	$0.322738\pi$
$230$	−2.35420	−0.155232
$231$	0	0
$232$	7.17012	0.470742
$233$	−5.93228	−0.388637	−0.194318	−	0.980939i	$-0.562250\pi$
$233$	−5.93228	−0.388637	−0.194318	+	0.980939i	$0.562250\pi$
$234$	13.6042	0.889332
$235$	−0.311257	−0.0203041
$236$	13.7053	0.892140
$237$	6.91630	0.449262
$238$	−4.62570	−0.299839
$239$	−21.8157	−1.41114	−0.705570	−	0.708640i	$-0.749309\pi$
$239$	−21.8157	−1.41114	−0.705570	+	0.708640i	$0.749309\pi$
$240$	2.13646	0.137908
$241$	−6.90785	−0.444974	−0.222487	−	0.974936i	$-0.571417\pi$
$241$	−6.90785	−0.444974	−0.222487	+	0.974936i	$0.571417\pi$
$242$	0	0
$243$	−17.6310	−1.13103
$244$	0.453925	0.0290596
$245$	13.8120	0.882418
$246$	23.5689	1.50270
$247$	−15.0790	−0.959452
$248$	9.17012	0.582303
$249$	−7.26128	−0.460165
$250$	−8.71065	−0.550910
$251$	1.57518	0.0994244	0.0497122	−	0.998764i	$-0.484170\pi$
$251$	1.57518	0.0994244	0.0497122	+	0.998764i	$0.484170\pi$
$252$	−9.06342	−0.570942
$253$	0	0
$254$	−16.4539	−1.03241
$255$	2.13646	0.133791
$256$	1.00000	0.0625000
$257$	4.73608	0.295428	0.147714	−	0.989030i	$-0.452808\pi$
$257$	4.73608	0.295428	0.147714	+	0.989030i	$0.452808\pi$
$258$	3.91240	0.243576
$259$	−49.1513	−3.05411
$260$	6.66101	0.413098
$261$	14.0489	0.869603
$262$	−9.43239	−0.582735
$263$	−14.6838	−0.905441	−0.452721	−	0.891652i	$-0.649547\pi$
$263$	−14.6838	−0.905441	−0.452721	+	0.891652i	$0.649547\pi$
$264$	0	0
$265$	4.52454	0.277940
$266$	10.0460	0.615958
$267$	−20.6231	−1.26211
$268$	−0.453925	−0.0277279
$269$	9.37430	0.571561	0.285781	−	0.958295i	$-0.407747\pi$
$269$	9.37430	0.571561	0.285781	+	0.958295i	$0.407747\pi$
$270$	−2.22329	−0.135305
$271$	−2.87875	−0.174871	−0.0874357	−	0.996170i	$-0.527867\pi$
$271$	−2.87875	−0.174871	−0.0874357	+	0.996170i	$0.527867\pi$
$272$	1.00000	0.0606339
$273$	−71.5233	−4.32878
$274$	−6.48923	−0.392029
$275$	0	0
$276$	−5.46480	−0.328942
$277$	14.4506	0.868254	0.434127	−	0.900852i	$-0.357057\pi$
$277$	14.4506	0.868254	0.434127	+	0.900852i	$0.357057\pi$
$278$	4.29138	0.257380
$279$	17.9676	1.07569
$280$	−4.43772	−0.265205
$281$	14.5319	0.866900	0.433450	−	0.901178i	$-0.357296\pi$
$281$	14.5319	0.866900	0.433450	+	0.901178i	$0.357296\pi$
$282$	−0.722518	−0.0430253
$283$	5.26984	0.313260	0.156630	−	0.987657i	$-0.449937\pi$
$283$	5.26984	0.313260	0.156630	+	0.987657i	$0.449937\pi$
$284$	−13.8050	−0.819179
$285$	−4.63992	−0.274845
$286$	0	0
$287$	−48.9557	−2.88976
$288$	1.95936	0.115457
$289$	1.00000	0.0588235
$290$	6.87875	0.403934
$291$	−30.3402	−1.77858
$292$	−0.861884	−0.0504379
$293$	−16.1262	−0.942102	−0.471051	−	0.882106i	$-0.656125\pi$
$293$	−16.1262	−0.942102	−0.471051	+	0.882106i	$0.656125\pi$
$294$	32.0618	1.86988
$295$	13.1484	0.765528
$296$	10.6257	0.617606
$297$	0	0
$298$	−10.1272	−0.586655
$299$	−17.0380	−0.985333
$300$	−9.08517	−0.524532
$301$	−8.12658	−0.468408
$302$	−6.71620	−0.386474
$303$	15.4982	0.890351
$304$	−2.17177	−0.124560
$305$	0.435479	0.0249354
$306$	1.95936	0.112009
$307$	−2.03201	−0.115973	−0.0579863	−	0.998317i	$-0.518468\pi$
$307$	−2.03201	−0.115973	−0.0579863	+	0.998317i	$0.518468\pi$
$308$	0	0
$309$	−29.5890	−1.68326
$310$	8.79747	0.499663
$311$	14.7453	0.836129	0.418065	−	0.908417i	$-0.362709\pi$
$311$	14.7453	0.836129	0.418065	+	0.908417i	$0.362709\pi$
$312$	15.4622	0.875372
$313$	−25.4628	−1.43924	−0.719622	−	0.694366i	$-0.755685\pi$
$313$	−25.4628	−1.43924	−0.719622	+	0.694366i	$0.755685\pi$
$314$	0.424821	0.0239740
$315$	−8.69511	−0.489914
$316$	3.10571	0.174710
$317$	−3.62405	−0.203547	−0.101773	−	0.994808i	$-0.532452\pi$
$317$	−3.62405	−0.203547	−0.101773	+	0.994808i	$0.532452\pi$
$318$	10.5028	0.588967
$319$	0	0
$320$	0.959363	0.0536300
$321$	2.85898	0.159573
$322$	11.3511	0.632573
$323$	−2.17177	−0.120841
$324$	−11.0390	−0.613277
$325$	−28.3255	−1.57121
$326$	−17.3809	−0.962638
$327$	−25.8863	−1.43152
$328$	10.5834	0.584371
$329$	1.50077	0.0827399
$330$	0	0
$331$	2.40506	0.132194	0.0660970	−	0.997813i	$-0.478945\pi$
$331$	2.40506	0.132194	0.0660970	+	0.997813i	$0.478945\pi$
$332$	−3.26062	−0.178950
$333$	20.8196	1.14091
$334$	−1.83811	−0.100577
$335$	−0.435479	−0.0237928
$336$	−10.3013	−0.561980
$337$	−4.24317	−0.231140	−0.115570	−	0.993299i	$-0.536869\pi$
$337$	−4.24317	−0.231140	−0.115570	+	0.993299i	$0.536869\pi$
$338$	35.2075	1.91503
$339$	6.22986	0.338360
$340$	0.959363	0.0520287
$341$	0	0
$342$	−4.25529	−0.230100
$343$	−34.2167	−1.84753
$344$	1.75683	0.0947221
$345$	−5.24272	−0.282259
$346$	−16.9079	−0.908972
$347$	10.7162	0.575276	0.287638	−	0.957739i	$-0.407130\pi$
$347$	10.7162	0.575276	0.287638	+	0.957739i	$0.407130\pi$
$348$	15.9676	0.855953
$349$	13.8375	0.740702	0.370351	−	0.928892i	$-0.379237\pi$
$349$	13.8375	0.740702	0.370351	+	0.928892i	$0.379237\pi$
$350$	18.8711	1.00870
$351$	−16.0905	−0.858847
$352$	0	0
$353$	9.29961	0.494968	0.247484	−	0.968892i	$-0.420396\pi$
$353$	9.29961	0.494968	0.247484	+	0.968892i	$0.420396\pi$
$354$	30.5212	1.62219
$355$	−13.2440	−0.702921
$356$	−9.26062	−0.490812
$357$	−10.3013	−0.545200
$358$	0.343546	0.0181570
$359$	−3.54607	−0.187155	−0.0935773	−	0.995612i	$-0.529830\pi$
$359$	−3.54607	−0.187155	−0.0935773	+	0.995612i	$0.529830\pi$
$360$	1.87974	0.0990709
$361$	−14.2834	−0.751758
$362$	16.3082	0.857142
$363$	0	0
$364$	−32.1170	−1.68339
$365$	−0.826859	−0.0432798
$366$	1.01087	0.0528393
$367$	18.6165	0.971772	0.485886	−	0.874022i	$-0.338497\pi$
$367$	18.6165	0.971772	0.485886	+	0.874022i	$0.338497\pi$
$368$	−2.45393	−0.127920
$369$	20.7367	1.07951
$370$	10.1939	0.529955
$371$	−21.8157	−1.13261
$372$	20.4215	1.05881
$373$	16.8986	0.874978	0.437489	−	0.899224i	$-0.355868\pi$
$373$	16.8986	0.874978	0.437489	+	0.899224i	$0.355868\pi$
$374$	0	0
$375$	−19.3983	−1.00172
$376$	−0.324441	−0.0167318
$377$	49.7833	2.56397
$378$	10.7199	0.551371
$379$	17.0558	0.876097	0.438048	−	0.898951i	$-0.355670\pi$
$379$	17.0558	0.876097	0.438048	+	0.898951i	$0.355670\pi$
$380$	−2.08352	−0.106882
$381$	−36.6423	−1.87724
$382$	−5.74063	−0.293716
$383$	−7.09215	−0.362392	−0.181196	−	0.983447i	$-0.557997\pi$
$383$	−7.09215	−0.362392	−0.181196	+	0.983447i	$0.557997\pi$
$384$	2.22696	0.113644
$385$	0	0
$386$	12.4080	0.631548
$387$	3.44227	0.174981
$388$	−13.6240	−0.691656
$389$	0.826576	0.0419090	0.0209545	−	0.999780i	$-0.493329\pi$
$389$	0.826576	0.0419090	0.0209545	+	0.999780i	$0.493329\pi$
$390$	14.8338	0.751139
$391$	−2.45393	−0.124100
$392$	14.3971	0.727163
$393$	−21.0056	−1.05959
$394$	−6.74860	−0.339990
$395$	2.97950	0.149915
$396$	0	0
$397$	30.2173	1.51656	0.758282	−	0.651926i	$-0.226039\pi$
$397$	30.2173	1.51656	0.758282	+	0.651926i	$0.226039\pi$
$398$	8.53520	0.427831
$399$	22.3720	1.12000
$400$	−4.07962	−0.203981
$401$	18.1301	0.905376	0.452688	−	0.891669i	$-0.350465\pi$
$401$	18.1301	0.905376	0.452688	+	0.891669i	$0.350465\pi$
$402$	−1.01087	−0.0504178
$403$	63.6696	3.17161
$404$	6.95936	0.346241
$405$	−10.5904	−0.526241
$406$	−33.1668	−1.64604
$407$	0	0
$408$	2.22696	0.110251
$409$	9.78660	0.483916	0.241958	−	0.970287i	$-0.422210\pi$
$409$	9.78660	0.483916	0.241958	+	0.970287i	$0.422210\pi$
$410$	10.1533	0.501437
$411$	−14.4513	−0.712829
$412$	−13.2867	−0.654589
$413$	−63.3967	−3.11955
$414$	−4.80813	−0.236307
$415$	−3.12812	−0.153553
$416$	6.94316	0.340416
$417$	9.55673	0.467995
$418$	0	0
$419$	25.5811	1.24972	0.624859	−	0.780737i	$-0.285156\pi$
$419$	25.5811	1.24972	0.624859	+	0.780737i	$0.285156\pi$
$420$	−9.88264	−0.482223
$421$	−33.5592	−1.63558	−0.817788	−	0.575519i	$-0.804800\pi$
$421$	−33.5592	−1.63558	−0.817788	+	0.575519i	$0.804800\pi$
$422$	0.162550	0.00791280
$423$	−0.635698	−0.0309087
$424$	4.71620	0.229039
$425$	−4.07962	−0.197891
$426$	−30.7433	−1.48952
$427$	−2.09972	−0.101613
$428$	1.28380	0.0620550
$429$	0	0
$430$	1.68544	0.0812792
$431$	−4.10116	−0.197546	−0.0987729	−	0.995110i	$-0.531492\pi$
$431$	−4.10116	−0.197546	−0.0987729	+	0.995110i	$0.531492\pi$
$432$	−2.31746	−0.111499
$433$	−28.9758	−1.39249	−0.696245	−	0.717805i	$-0.745147\pi$
$433$	−28.9758	−1.39249	−0.696245	+	0.717805i	$0.745147\pi$
$434$	−42.4182	−2.03614
$435$	15.3187	0.734476
$436$	−11.6240	−0.556691
$437$	5.32937	0.254938
$438$	−1.91938	−0.0917117
$439$	23.1675	1.10572	0.552862	−	0.833273i	$-0.313536\pi$
$439$	23.1675	1.10572	0.552862	+	0.833273i	$0.313536\pi$
$440$	0	0
$441$	28.2091	1.34329
$442$	6.94316	0.330252
$443$	−12.3693	−0.587685	−0.293843	−	0.955854i	$-0.594934\pi$
$443$	−12.3693	−0.587685	−0.293843	+	0.955854i	$0.594934\pi$
$444$	23.6630	1.12300
$445$	−8.88429	−0.421156
$446$	−10.1718	−0.481647
$447$	−22.5530	−1.06672
$448$	−4.62570	−0.218544
$449$	16.2114	0.765064	0.382532	−	0.923942i	$-0.375052\pi$
$449$	16.2114	0.765064	0.382532	+	0.923942i	$0.375052\pi$
$450$	−7.99346	−0.376815
$451$	0	0
$452$	2.79747	0.131582
$453$	−14.9567	−0.702728
$454$	29.0380	1.36282
$455$	−30.8118	−1.44448
$456$	−4.83646	−0.226488
$457$	18.8754	0.882956	0.441478	−	0.897272i	$-0.354454\pi$
$457$	18.8754	0.882956	0.441478	+	0.897272i	$0.354454\pi$
$458$	15.9967	0.747477
$459$	−2.31746	−0.108170
$460$	−2.35420	−0.109765
$461$	−24.9583	−1.16242	−0.581212	−	0.813752i	$-0.697421\pi$
$461$	−24.9583	−1.16242	−0.581212	+	0.813752i	$0.697421\pi$
$462$	0	0
$463$	14.5197	0.674786	0.337393	−	0.941364i	$-0.390455\pi$
$463$	14.5197	0.674786	0.337393	+	0.941364i	$0.390455\pi$
$464$	7.17012	0.332865
$465$	19.5916	0.908541
$466$	−5.93228	−0.274808
$467$	25.8355	1.19552	0.597761	−	0.801674i	$-0.296057\pi$
$467$	25.8355	1.19552	0.597761	+	0.801674i	$0.296057\pi$
$468$	13.6042	0.628853
$469$	2.09972	0.0969561
$470$	−0.311257	−0.0143572
$471$	0.946060	0.0435922
$472$	13.7053	0.630839
$473$	0	0
$474$	6.91630	0.317676
$475$	8.86002	0.406525
$476$	−4.62570	−0.212019
$477$	9.24074	0.423104
$478$	−21.8157	−0.997827
$479$	−4.01356	−0.183384	−0.0916921	−	0.995787i	$-0.529228\pi$
$479$	−4.01356	−0.183384	−0.0916921	+	0.995787i	$0.529228\pi$
$480$	2.13646	0.0975158
$481$	73.7759	3.36389
$482$	−6.90785	−0.314644
$483$	25.2785	1.15021
$484$	0	0
$485$	−13.0704	−0.593496
$486$	−17.6310	−0.799760
$487$	−11.7193	−0.531051	−0.265526	−	0.964104i	$-0.585546\pi$
$487$	−11.7193	−0.531051	−0.265526	+	0.964104i	$0.585546\pi$
$488$	0.453925	0.0205482
$489$	−38.7066	−1.75037
$490$	13.8120	0.623964
$491$	−17.6538	−0.796705	−0.398353	−	0.917232i	$-0.630418\pi$
$491$	−17.6538	−0.796705	−0.398353	+	0.917232i	$0.630418\pi$
$492$	23.5689	1.06257
$493$	7.17012	0.322926
$494$	−15.0790	−0.678435
$495$	0	0
$496$	9.17012	0.411751
$497$	63.8580	2.86442
$498$	−7.26128	−0.325386
$499$	3.36798	0.150771	0.0753857	−	0.997154i	$-0.475981\pi$
$499$	3.36798	0.150771	0.0753857	+	0.997154i	$0.475981\pi$
$500$	−8.71065	−0.389552
$501$	−4.09340	−0.182880
$502$	1.57518	0.0703037
$503$	−26.9475	−1.20153	−0.600765	−	0.799426i	$-0.705137\pi$
$503$	−26.9475	−1.20153	−0.600765	+	0.799426i	$0.705137\pi$
$504$	−9.06342	−0.403717
$505$	6.67655	0.297103
$506$	0	0
$507$	78.4057	3.48212
$508$	−16.4539	−0.730025
$509$	−12.9045	−0.571984	−0.285992	−	0.958232i	$-0.592323\pi$
$509$	−12.9045	−0.571984	−0.285992	+	0.958232i	$0.592323\pi$
$510$	2.13646	0.0946042
$511$	3.98682	0.176366
$512$	1.00000	0.0441942
$513$	5.03300	0.222212
$514$	4.73608	0.208899
$515$	−12.7468	−0.561690
$516$	3.91240	0.172234
$517$	0	0
$518$	−49.1513	−2.15958
$519$	−37.6532	−1.65279
$520$	6.66101	0.292104
$521$	−42.4076	−1.85791	−0.928954	−	0.370194i	$-0.879291\pi$
$521$	−42.4076	−1.85791	−0.928954	+	0.370194i	$0.879291\pi$
$522$	14.0489	0.614902
$523$	8.84898	0.386939	0.193470	−	0.981106i	$-0.438026\pi$
$523$	8.84898	0.386939	0.193470	+	0.981106i	$0.438026\pi$
$524$	−9.43239	−0.412056
$525$	42.0253	1.83413
$526$	−14.6838	−0.640244
$527$	9.17012	0.399457
$528$	0	0
$529$	−16.9783	−0.738185
$530$	4.52454	0.196534
$531$	26.8537	1.16535
$532$	10.0460	0.435548
$533$	73.4823	3.18287
$534$	−20.6231	−0.892447
$535$	1.23163	0.0532481
$536$	−0.453925	−0.0196066
$537$	0.765064	0.0330149
$538$	9.37430	0.404155
$539$	0	0
$540$	−2.22329	−0.0956750
$541$	−43.4291	−1.86716	−0.933581	−	0.358366i	$-0.883334\pi$
$541$	−43.4291	−1.86716	−0.933581	+	0.358366i	$0.883334\pi$
$542$	−2.87875	−0.123653
$543$	36.3178	1.55855
$544$	1.00000	0.0428746
$545$	−11.1517	−0.477685
$546$	−71.5233	−3.06091
$547$	−23.6240	−1.01009	−0.505046	−	0.863093i	$-0.668524\pi$
$547$	−23.6240	−1.01009	−0.505046	+	0.863093i	$0.668524\pi$
$548$	−6.48923	−0.277206
$549$	0.889404	0.0379588
$550$	0	0
$551$	−15.5719	−0.663384
$552$	−5.46480	−0.232597
$553$	−14.3661	−0.610908
$554$	14.4506	0.613948
$555$	22.7014	0.963622
$556$	4.29138	0.181995
$557$	−0.819879	−0.0347394	−0.0173697	−	0.999849i	$-0.505529\pi$
$557$	−0.819879	−0.0347394	−0.0173697	+	0.999849i	$0.505529\pi$
$558$	17.9676	0.760629
$559$	12.1980	0.515919
$560$	−4.43772	−0.187528
$561$	0	0
$562$	14.5319	0.612991
$563$	−7.15102	−0.301379	−0.150690	−	0.988581i	$-0.548149\pi$
$563$	−7.15102	−0.301379	−0.150690	+	0.988581i	$0.548149\pi$
$564$	−0.722518	−0.0304235
$565$	2.68379	0.112908
$566$	5.26984	0.221508
$567$	51.0630	2.14445
$568$	−13.8050	−0.579247
$569$	1.39749	0.0585857	0.0292928	−	0.999571i	$-0.490674\pi$
$569$	1.39749	0.0585857	0.0292928	+	0.999571i	$0.490674\pi$
$570$	−4.63992	−0.194345
$571$	−26.9045	−1.12592	−0.562960	−	0.826484i	$-0.690338\pi$
$571$	−26.9045	−1.12592	−0.562960	+	0.826484i	$0.690338\pi$
$572$	0	0
$573$	−12.7842	−0.534066
$574$	−48.9557	−2.04337
$575$	10.0111	0.417491
$576$	1.95936	0.0816401
$577$	20.6713	0.860556	0.430278	−	0.902696i	$-0.358416\pi$
$577$	20.6713	0.860556	0.430278	+	0.902696i	$0.358416\pi$
$578$	1.00000	0.0415945
$579$	27.6321	1.14835
$580$	6.87875	0.285624
$581$	15.0826	0.625733
$582$	−30.3402	−1.25764
$583$	0	0
$584$	−0.861884	−0.0356650
$585$	13.0513	0.539606
$586$	−16.1262	−0.666166
$587$	−17.9003	−0.738824	−0.369412	−	0.929266i	$-0.620441\pi$
$587$	−17.9003	−0.738824	−0.369412	+	0.929266i	$0.620441\pi$
$588$	32.0618	1.32221
$589$	−19.9154	−0.820601
$590$	13.1484	0.541310
$591$	−15.0289	−0.618206
$592$	10.6257	0.436714
$593$	13.1701	0.540832	0.270416	−	0.962744i	$-0.412839\pi$
$593$	13.1701	0.540832	0.270416	+	0.962744i	$0.412839\pi$
$594$	0	0
$595$	−4.43772	−0.181929
$596$	−10.1272	−0.414828
$597$	19.0076	0.777928
$598$	−17.0380	−0.696735
$599$	21.8157	0.891365	0.445683	−	0.895191i	$-0.352961\pi$
$599$	21.8157	0.891365	0.445683	+	0.895191i	$0.352961\pi$
$600$	−9.08517	−0.370900
$601$	10.0721	0.410848	0.205424	−	0.978673i	$-0.434143\pi$
$601$	10.0721	0.410848	0.205424	+	0.978673i	$0.434143\pi$
$602$	−8.12658	−0.331215
$603$	−0.889404	−0.0362193
$604$	−6.71620	−0.273278
$605$	0	0
$606$	15.4982	0.629573
$607$	22.0971	0.896893	0.448446	−	0.893810i	$-0.351978\pi$
$607$	22.0971	0.896893	0.448446	+	0.893810i	$0.351978\pi$
$608$	−2.17177	−0.0880770
$609$	−73.8613	−2.99301
$610$	0.435479	0.0176320
$611$	−2.25265	−0.0911323
$612$	1.95936	0.0792025
$613$	−36.8523	−1.48845	−0.744224	−	0.667930i	$-0.767181\pi$
$613$	−36.8523	−1.48845	−0.744224	+	0.667930i	$0.767181\pi$
$614$	−2.03201	−0.0820051
$615$	22.6111	0.911767
$616$	0	0
$617$	−28.5286	−1.14852	−0.574259	−	0.818674i	$-0.694710\pi$
$617$	−28.5286	−1.14852	−0.574259	+	0.818674i	$0.694710\pi$
$618$	−29.5890	−1.19024
$619$	11.8219	0.475162	0.237581	−	0.971368i	$-0.423645\pi$
$619$	11.8219	0.475162	0.237581	+	0.971368i	$0.423645\pi$
$620$	8.79747	0.353315
$621$	5.68688	0.228206
$622$	14.7453	0.591233
$623$	42.8368	1.71622
$624$	15.4622	0.618982
$625$	12.0414	0.481658
$626$	−25.4628	−1.01770
$627$	0	0
$628$	0.424821	0.0169522
$629$	10.6257	0.423674
$630$	−8.69511	−0.346421
$631$	−2.06573	−0.0822354	−0.0411177	−	0.999154i	$-0.513092\pi$
$631$	−2.06573	−0.0822354	−0.0411177	+	0.999154i	$0.513092\pi$
$632$	3.10571	0.123538
$633$	0.361992	0.0143879
$634$	−3.62405	−0.143929
$635$	−15.7853	−0.626420
$636$	10.5028	0.416463
$637$	99.9612	3.96061
$638$	0	0
$639$	−27.0491	−1.07005
$640$	0.959363	0.0379221
$641$	−8.63162	−0.340928	−0.170464	−	0.985364i	$-0.554527\pi$
$641$	−8.63162	−0.340928	−0.170464	+	0.985364i	$0.554527\pi$
$642$	2.85898	0.112835
$643$	−29.0704	−1.14642	−0.573212	−	0.819407i	$-0.694303\pi$
$643$	−29.0704	−1.14642	−0.573212	+	0.819407i	$0.694303\pi$
$644$	11.3511	0.447297
$645$	3.75341	0.147790
$646$	−2.17177	−0.0854473
$647$	36.4060	1.43127	0.715633	−	0.698476i	$-0.246138\pi$
$647$	36.4060	1.43127	0.715633	+	0.698476i	$0.246138\pi$
$648$	−11.0390	−0.433652
$649$	0	0
$650$	−28.3255	−1.11102
$651$	−94.4638	−3.70233
$652$	−17.3809	−0.680688
$653$	−47.1436	−1.84487	−0.922436	−	0.386149i	$-0.873805\pi$
$653$	−47.1436	−1.84487	−0.922436	+	0.386149i	$0.873805\pi$
$654$	−25.8863	−1.01224
$655$	−9.04908	−0.353577
$656$	10.5834	0.413213
$657$	−1.68874	−0.0658842
$658$	1.50077	0.0585060
$659$	−12.4936	−0.486680	−0.243340	−	0.969941i	$-0.578243\pi$
$659$	−12.4936	−0.486680	−0.243340	+	0.969941i	$0.578243\pi$
$660$	0	0
$661$	2.97847	0.115849	0.0579245	−	0.998321i	$-0.481552\pi$
$661$	2.97847	0.115849	0.0579245	+	0.998321i	$0.481552\pi$
$662$	2.40506	0.0934752
$663$	15.4622	0.600500
$664$	−3.26062	−0.126537
$665$	9.63772	0.373735
$666$	20.8196	0.806743
$667$	−17.5949	−0.681279
$668$	−1.83811	−0.0711186
$669$	−22.6522	−0.875783
$670$	−0.435479	−0.0168240
$671$	0	0
$672$	−10.3013	−0.397380
$673$	−26.7460	−1.03098	−0.515490	−	0.856895i	$-0.672390\pi$
$673$	−26.7460	−1.03098	−0.515490	+	0.856895i	$0.672390\pi$
$674$	−4.24317	−0.163441
$675$	9.45437	0.363899
$676$	35.2075	1.35413
$677$	−26.7096	−1.02653	−0.513266	−	0.858229i	$-0.671565\pi$
$677$	−26.7096	−1.02653	−0.513266	+	0.858229i	$0.671565\pi$
$678$	6.22986	0.239256
$679$	63.0207	2.41851
$680$	0.959363	0.0367899
$681$	64.6665	2.47803
$682$	0	0
$683$	13.8024	0.528134	0.264067	−	0.964504i	$-0.414936\pi$
$683$	13.8024	0.528134	0.264067	+	0.964504i	$0.414936\pi$
$684$	−4.25529	−0.162705
$685$	−6.22553	−0.237865
$686$	−34.2167	−1.30640
$687$	35.6240	1.35914
$688$	1.75683	0.0669787
$689$	32.7453	1.24750
$690$	−5.24272	−0.199587
$691$	30.8140	1.17222	0.586111	−	0.810231i	$-0.300658\pi$
$691$	30.8140	1.17222	0.586111	+	0.810231i	$0.300658\pi$
$692$	−16.9079	−0.642740
$693$	0	0
$694$	10.7162	0.406781
$695$	4.11698	0.156166
$696$	15.9676	0.605250
$697$	10.5834	0.400875
$698$	13.8375	0.523755
$699$	−13.2110	−0.499685
$700$	18.8711	0.713261
$701$	−35.0181	−1.32262	−0.661308	−	0.750115i	$-0.729998\pi$
$701$	−35.0181	−1.32262	−0.661308	+	0.750115i	$0.729998\pi$
$702$	−16.0905	−0.607297
$703$	−23.0766	−0.870351
$704$	0	0
$705$	−0.693157	−0.0261058
$706$	9.29961	0.349995
$707$	−32.1919	−1.21070
$708$	30.5212	1.14706
$709$	20.0979	0.754791	0.377395	−	0.926052i	$-0.376820\pi$
$709$	20.0979	0.754791	0.377395	+	0.926052i	$0.376820\pi$
$710$	−13.2440	−0.497040
$711$	6.08521	0.228213
$712$	−9.26062	−0.347056
$713$	−22.5028	−0.842736
$714$	−10.3013	−0.385515
$715$	0	0
$716$	0.343546	0.0128389
$717$	−48.5828	−1.81436
$718$	−3.54607	−0.132338
$719$	22.3369	0.833027	0.416514	−	0.909129i	$-0.363252\pi$
$719$	22.3369	0.833027	0.416514	+	0.909129i	$0.363252\pi$
$720$	1.87974	0.0700537
$721$	61.4603	2.28890
$722$	−14.2834	−0.531573
$723$	−15.3835	−0.572119
$724$	16.3082	0.606091
$725$	−29.2514	−1.08637
$726$	0	0
$727$	−0.0168621	−0.000625380 0	−0.000312690	−	1.00000i	$-0.500100\pi$
$727$	−0.0168621	−0.000625380 0	−0.000312690		1.00000i	$0.500100\pi$
$728$	−32.1170	−1.19033
$729$	−6.14668	−0.227655
$730$	−0.826859	−0.0306034
$731$	1.75683	0.0649788
$732$	1.01087	0.0373630
$733$	17.2032	0.635414	0.317707	−	0.948189i	$-0.397087\pi$
$733$	17.2032	0.635414	0.317707	+	0.948189i	$0.397087\pi$
$734$	18.6165	0.687147
$735$	30.7589	1.13456
$736$	−2.45393	−0.0904529
$737$	0	0
$738$	20.7367	0.763330
$739$	−32.1457	−1.18250	−0.591249	−	0.806489i	$-0.701365\pi$
$739$	−32.1457	−1.18250	−0.591249	+	0.806489i	$0.701365\pi$
$740$	10.1939	0.374735
$741$	−33.5803	−1.23360
$742$	−21.8157	−0.800880
$743$	50.2596	1.84385	0.921923	−	0.387372i	$-0.126617\pi$
$743$	50.2596	1.84385	0.921923	+	0.387372i	$0.126617\pi$
$744$	20.4215	0.748689
$745$	−9.71570	−0.355956
$746$	16.8986	0.618703
$747$	−6.38874	−0.233752
$748$	0	0
$749$	−5.93849	−0.216988
$750$	−19.3983	−0.708326
$751$	−5.55343	−0.202648	−0.101324	−	0.994854i	$-0.532308\pi$
$751$	−5.55343	−0.202648	−0.101324	+	0.994854i	$0.532308\pi$
$752$	−0.324441	−0.0118312
$753$	3.50787	0.127834
$754$	49.7833	1.81300
$755$	−6.44327	−0.234495
$756$	10.7199	0.389878
$757$	−42.7400	−1.55341	−0.776706	−	0.629863i	$-0.783111\pi$
$757$	−42.7400	−1.55341	−0.776706	+	0.629863i	$0.783111\pi$
$758$	17.0558	0.619494
$759$	0	0
$760$	−2.08352	−0.0755771
$761$	1.18408	0.0429230	0.0214615	−	0.999770i	$-0.493168\pi$
$761$	1.18408	0.0429230	0.0214615	+	0.999770i	$0.493168\pi$
$762$	−36.6423	−1.32741
$763$	53.7693	1.94658
$764$	−5.74063	−0.207689
$765$	1.87974	0.0679621
$766$	−7.09215	−0.256250
$767$	95.1582	3.43596
$768$	2.22696	0.0803586
$769$	−18.3369	−0.661247	−0.330623	−	0.943763i	$-0.607259\pi$
$769$	−18.3369	−0.661247	−0.330623	+	0.943763i	$0.607259\pi$
$770$	0	0
$771$	10.5471	0.379843
$772$	12.4080	0.446572
$773$	7.08436	0.254807	0.127403	−	0.991851i	$-0.459336\pi$
$773$	7.08436	0.254807	0.127403	+	0.991851i	$0.459336\pi$
$774$	3.44227	0.123730
$775$	−37.4106	−1.34383
$776$	−13.6240	−0.489075
$777$	−109.458	−3.92679
$778$	0.826576	0.0296342
$779$	−22.9848	−0.823515
$780$	14.8338	0.531136
$781$	0	0
$782$	−2.45393	−0.0877522
$783$	−16.6165	−0.593824
$784$	14.3971	0.514182
$785$	0.407557	0.0145463
$786$	−21.0056	−0.749244
$787$	35.9352	1.28095	0.640476	−	0.767979i	$-0.278737\pi$
$787$	35.9352	1.28095	0.640476	+	0.767979i	$0.278737\pi$
$788$	−6.74860	−0.240409
$789$	−32.7002	−1.16416
$790$	2.97950	0.106006
$791$	−12.9403	−0.460103
$792$	0	0
$793$	3.15168	0.111919
$794$	30.2173	1.07237
$795$	10.0760	0.357358
$796$	8.53520	0.302522
$797$	−24.1777	−0.856418	−0.428209	−	0.903680i	$-0.640855\pi$
$797$	−24.1777	−0.856418	−0.428209	+	0.903680i	$0.640855\pi$
$798$	22.3720	0.791960
$799$	−0.324441	−0.0114779
$800$	−4.07962	−0.144236
$801$	−18.1449	−0.641119
$802$	18.1301	0.640198
$803$	0	0
$804$	−1.01087	−0.0356508
$805$	10.8898	0.383816
$806$	63.6696	2.24267
$807$	20.8762	0.734878
$808$	6.95936	0.244830
$809$	35.1681	1.23645	0.618223	−	0.786003i	$-0.287853\pi$
$809$	35.1681	1.23645	0.618223	+	0.786003i	$0.287853\pi$
$810$	−10.5904	−0.372108
$811$	−23.3967	−0.821569	−0.410784	−	0.911733i	$-0.634745\pi$
$811$	−23.3967	−0.821569	−0.410784	+	0.911733i	$0.634745\pi$
$812$	−33.1668	−1.16393
$813$	−6.41086	−0.224839
$814$	0	0
$815$	−16.6746	−0.584085
$816$	2.22696	0.0779593
$817$	−3.81544	−0.133485
$818$	9.78660	0.342180
$819$	−62.9288	−2.19891
$820$	10.1533	0.354570
$821$	−18.7744	−0.655231	−0.327616	−	0.944811i	$-0.606245\pi$
$821$	−18.7744	−0.655231	−0.327616	+	0.944811i	$0.606245\pi$
$822$	−14.4513	−0.504046
$823$	29.3047	1.02150	0.510750	−	0.859729i	$-0.329368\pi$
$823$	29.3047	1.02150	0.510750	+	0.859729i	$0.329368\pi$
$824$	−13.2867	−0.462864
$825$	0	0
$826$	−63.3967	−2.20585
$827$	23.4431	0.815195	0.407597	−	0.913162i	$-0.366367\pi$
$827$	23.4431	0.815195	0.407597	+	0.913162i	$0.366367\pi$
$828$	−4.80813	−0.167094
$829$	−54.4658	−1.89167	−0.945837	−	0.324642i	$-0.894756\pi$
$829$	−54.4658	−1.89167	−0.945837	+	0.324642i	$0.894756\pi$
$830$	−3.12812	−0.108579
$831$	32.1810	1.11635
$832$	6.94316	0.240711
$833$	14.3971	0.498829
$834$	9.55673	0.330923
$835$	−1.76341	−0.0610254
$836$	0	0
$837$	−21.2514	−0.734556
$838$	25.5811	0.883684
$839$	−35.5288	−1.22659	−0.613295	−	0.789854i	$-0.710156\pi$
$839$	−35.5288	−1.22659	−0.613295	+	0.789854i	$0.710156\pi$
$840$	−9.88264	−0.340983
$841$	22.4106	0.772781
$842$	−33.5592	−1.15653
$843$	32.3620	1.11461
$844$	0.162550	0.00559519
$845$	33.7767	1.16195
$846$	−0.635698	−0.0218557
$847$	0	0
$848$	4.71620	0.161955
$849$	11.7357	0.402770
$850$	−4.07962	−0.139930
$851$	−26.0747	−0.893828
$852$	−30.7433	−1.05325
$853$	25.4184	0.870311	0.435155	−	0.900355i	$-0.356693\pi$
$853$	25.4184	0.870311	0.435155	+	0.900355i	$0.356693\pi$
$854$	−2.09972	−0.0718510
$855$	−4.08237	−0.139614
$856$	1.28380	0.0438795
$857$	−47.4794	−1.62187	−0.810933	−	0.585140i	$-0.801040\pi$
$857$	−47.4794	−1.62187	−0.810933	+	0.585140i	$0.801040\pi$
$858$	0	0
$859$	−19.6762	−0.671344	−0.335672	−	0.941979i	$-0.608963\pi$
$859$	−19.6762	−0.671344	−0.335672	+	0.941979i	$0.608963\pi$
$860$	1.68544	0.0574730
$861$	−109.022	−3.71547
$862$	−4.10116	−0.139686
$863$	−33.2389	−1.13146	−0.565732	−	0.824589i	$-0.691406\pi$
$863$	−33.2389	−1.13146	−0.565732	+	0.824589i	$0.691406\pi$
$864$	−2.31746	−0.0788416
$865$	−16.2208	−0.551522
$866$	−28.9758	−0.984639
$867$	2.22696	0.0756316
$868$	−42.4182	−1.43977
$869$	0	0
$870$	15.3187	0.519353
$871$	−3.15168	−0.106790
$872$	−11.6240	−0.393640
$873$	−26.6944	−0.903470
$874$	5.32937	0.180269
$875$	40.2928	1.36215
$876$	−1.91938	−0.0648500
$877$	−11.3327	−0.382677	−0.191339	−	0.981524i	$-0.561283\pi$
$877$	−11.3327	−0.382677	−0.191339	+	0.981524i	$0.561283\pi$
$878$	23.1675	0.781864
$879$	−35.9124	−1.21130
$880$	0	0
$881$	36.9707	1.24557	0.622787	−	0.782392i	$-0.286000\pi$
$881$	36.9707	1.24557	0.622787	+	0.782392i	$0.286000\pi$
$882$	28.2091	0.949850
$883$	−7.18857	−0.241915	−0.120957	−	0.992658i	$-0.538596\pi$
$883$	−7.18857	−0.241915	−0.120957	+	0.992658i	$0.538596\pi$
$884$	6.94316	0.233524
$885$	29.2809	0.984268
$886$	−12.3693	−0.415556
$887$	17.7145	0.594796	0.297398	−	0.954754i	$-0.403881\pi$
$887$	17.7145	0.594796	0.297398	+	0.954754i	$0.403881\pi$
$888$	23.6630	0.794079
$889$	76.1109	2.55268
$890$	−8.88429	−0.297802
$891$	0	0
$892$	−10.1718	−0.340576
$893$	0.704612	0.0235790
$894$	−22.5530	−0.754285
$895$	0.329585	0.0110168
$896$	−4.62570	−0.154534
$897$	−37.9430	−1.26688
$898$	16.2114	0.540982
$899$	65.7509	2.19291
$900$	−7.99346	−0.266449
$901$	4.71620	0.157119
$902$	0	0
$903$	−18.0976	−0.602250
$904$	2.79747	0.0930425
$905$	15.6455	0.520074
$906$	−14.9567	−0.496904
$907$	−12.6378	−0.419632	−0.209816	−	0.977741i	$-0.567287\pi$
$907$	−12.6378	−0.419632	−0.209816	+	0.977741i	$0.567287\pi$
$908$	29.0380	0.963660
$909$	13.6359	0.452275
$910$	−30.8118	−1.02140
$911$	−25.0921	−0.831340	−0.415670	−	0.909516i	$-0.636453\pi$
$911$	−25.0921	−0.831340	−0.415670	+	0.909516i	$0.636453\pi$
$912$	−4.83646	−0.160151
$913$	0	0
$914$	18.8754	0.624344
$915$	0.969795	0.0320604
$916$	15.9967	0.528546
$917$	43.6314	1.44084
$918$	−2.31746	−0.0764876
$919$	−11.6380	−0.383902	−0.191951	−	0.981404i	$-0.561482\pi$
$919$	−11.6380	−0.383902	−0.191951	+	0.981404i	$0.561482\pi$
$920$	−2.35420	−0.0776158
$921$	−4.52520	−0.149110
$922$	−24.9583	−0.821958
$923$	−95.8506	−3.15496
$924$	0	0
$925$	−43.3488	−1.42530
$926$	14.5197	0.477145
$927$	−26.0335	−0.855051
$928$	7.17012	0.235371
$929$	53.0023	1.73895	0.869474	−	0.493978i	$-0.164458\pi$
$929$	53.0023	1.73895	0.869474	+	0.493978i	$0.164458\pi$
$930$	19.5916	0.642435
$931$	−31.2672	−1.02474
$932$	−5.93228	−0.194318
$933$	32.8372	1.07504
$934$	25.8355	0.845362
$935$	0	0
$936$	13.6042	0.444666
$937$	−4.65799	−0.152170	−0.0760849	−	0.997101i	$-0.524242\pi$
$937$	−4.65799	−0.152170	−0.0760849	+	0.997101i	$0.524242\pi$
$938$	2.09972	0.0685583
$939$	−56.7047	−1.85049
$940$	−0.311257	−0.0101521
$941$	−19.4997	−0.635672	−0.317836	−	0.948146i	$-0.602956\pi$
$941$	−19.4997	−0.635672	−0.317836	+	0.948146i	$0.602956\pi$
$942$	0.946060	0.0308243
$943$	−25.9709	−0.845729
$944$	13.7053	0.446070
$945$	10.2842	0.334547
$946$	0	0
$947$	56.3634	1.83156	0.915782	−	0.401675i	$-0.131572\pi$
$947$	56.3634	1.83156	0.915782	+	0.401675i	$0.131572\pi$
$948$	6.91630	0.224631
$949$	−5.98420	−0.194255
$950$	8.86002	0.287457
$951$	−8.07062	−0.261708
$952$	−4.62570	−0.149920
$953$	−36.0944	−1.16921	−0.584607	−	0.811317i	$-0.698751\pi$
$953$	−36.0944	−1.16921	−0.584607	+	0.811317i	$0.698751\pi$
$954$	9.24074	0.299180
$955$	−5.50735	−0.178214
$956$	−21.8157	−0.705570
$957$	0	0
$958$	−4.01356	−0.129672
$959$	30.0172	0.969307
$960$	2.13646	0.0689541
$961$	53.0911	1.71262
$962$	73.7759	2.37863
$963$	2.51544	0.0810588
$964$	−6.90785	−0.222487
$965$	11.9037	0.383195
$966$	25.2785	0.813323
$967$	−13.8605	−0.445724	−0.222862	−	0.974850i	$-0.571540\pi$
$967$	−13.8605	−0.445724	−0.222862	+	0.974850i	$0.571540\pi$
$968$	0	0
$969$	−4.83646	−0.155369
$970$	−13.0704	−0.419665
$971$	36.2603	1.16365	0.581824	−	0.813315i	$-0.302339\pi$
$971$	36.2603	1.16365	0.581824	+	0.813315i	$0.302339\pi$
$972$	−17.6310	−0.565515
$973$	−19.8506	−0.636381
$974$	−11.7193	−0.375510
$975$	−63.0798	−2.02017
$976$	0.453925	0.0145298
$977$	−24.3549	−0.779181	−0.389591	−	0.920988i	$-0.627383\pi$
$977$	−24.3549	−0.779181	−0.389591	+	0.920988i	$0.627383\pi$
$978$	−38.7066	−1.23770
$979$	0	0
$980$	13.8120	0.441209
$981$	−22.7757	−0.727173
$982$	−17.6538	−0.563356
$983$	−53.3076	−1.70025	−0.850125	−	0.526582i	$-0.823473\pi$
$983$	−53.3076	−1.70025	−0.850125	+	0.526582i	$0.823473\pi$
$984$	23.5689	0.751348
$985$	−6.47436	−0.206290
$986$	7.17012	0.228343
$987$	3.34215	0.106382
$988$	−15.0790	−0.479726
$989$	−4.31114	−0.137086
$990$	0	0
$991$	−59.7833	−1.89908	−0.949539	−	0.313648i	$-0.898449\pi$
$991$	−59.7833	−1.89908	−0.949539	+	0.313648i	$0.898449\pi$
$992$	9.17012	0.291152
$993$	5.35597	0.169967
$994$	63.8580	2.02545
$995$	8.18835	0.259588
$996$	−7.26128	−0.230082
$997$	28.4473	0.900936	0.450468	−	0.892793i	$-0.351257\pi$
$997$	28.4473	0.900936	0.450468	+	0.892793i	$0.351257\pi$
$998$	3.36798	0.106612
$999$	−24.6246	−0.779089

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4114.2.a.bc.1.3		4
11.10	odd	2		374.2.a.d.1.3	✓	4
33.32	even	2		3366.2.a.bg.1.3		4
44.43	even	2		2992.2.a.w.1.2		4
55.54	odd	2		9350.2.a.cl.1.2		4
187.186	odd	2		6358.2.a.t.1.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
374.2.a.d.1.3	✓	4	11.10	odd	2
2992.2.a.w.1.2		4	44.43	even	2
3366.2.a.bg.1.3		4	33.32	even	2
4114.2.a.bc.1.3		4	1.1	even	1	trivial
6358.2.a.t.1.2		4	187.186	odd	2
9350.2.a.cl.1.2		4	55.54	odd	2

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:	\( N \)	\(=\)	\( 4114 = 2 \cdot 11^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4114.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +2.22696 q^{3} +1.00000 q^{4} +0.959363 q^{5} +2.22696 q^{6} -4.62570 q^{7} +1.00000 q^{8} +1.95936 q^{9} +0.959363 q^{10} +2.22696 q^{12} +6.94316 q^{13} -4.62570 q^{14} +2.13646 q^{15} +1.00000 q^{16} +1.00000 q^{17} +1.95936 q^{18} -2.17177 q^{19} +0.959363 q^{20} -10.3013 q^{21} -2.45393 q^{23} +2.22696 q^{24} -4.07962 q^{25} +6.94316 q^{26} -2.31746 q^{27} -4.62570 q^{28} +7.17012 q^{29} +2.13646 q^{30} +9.17012 q^{31} +1.00000 q^{32} +1.00000 q^{34} -4.43772 q^{35} +1.95936 q^{36} +10.6257 q^{37} -2.17177 q^{38} +15.4622 q^{39} +0.959363 q^{40} +10.5834 q^{41} -10.3013 q^{42} +1.75683 q^{43} +1.87974 q^{45} -2.45393 q^{46} -0.324441 q^{47} +2.22696 q^{48} +14.3971 q^{49} -4.07962 q^{50} +2.22696 q^{51} +6.94316 q^{52} +4.71620 q^{53} -2.31746 q^{54} -4.62570 q^{56} -4.83646 q^{57} +7.17012 q^{58} +13.7053 q^{59} +2.13646 q^{60} +0.453925 q^{61} +9.17012 q^{62} -9.06342 q^{63} +1.00000 q^{64} +6.66101 q^{65} -0.453925 q^{67} +1.00000 q^{68} -5.46480 q^{69} -4.43772 q^{70} -13.8050 q^{71} +1.95936 q^{72} -0.861884 q^{73} +10.6257 q^{74} -9.08517 q^{75} -2.17177 q^{76} +15.4622 q^{78} +3.10571 q^{79} +0.959363 q^{80} -11.0390 q^{81} +10.5834 q^{82} -3.26062 q^{83} -10.3013 q^{84} +0.959363 q^{85} +1.75683 q^{86} +15.9676 q^{87} -9.26062 q^{89} +1.87974 q^{90} -32.1170 q^{91} -2.45393 q^{92} +20.4215 q^{93} -0.324441 q^{94} -2.08352 q^{95} +2.22696 q^{96} -13.6240 q^{97} +14.3971 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{2} + q^{3} + 4 q^{4} + 5 q^{5} + q^{6} - q^{7} + 4 q^{8} + 9 q^{9} + 5 q^{10} + q^{12} + 3 q^{13} - q^{14} + 4 q^{16} + 4 q^{17} + 9 q^{18} - 7 q^{19} + 5 q^{20} - 8 q^{21} + 6 q^{23} + q^{24}+ \cdots + 17 q^{98}+O(q^{100}) \) 4 * q + 4 * q^2 + q^3 + 4 * q^4 + 5 * q^5 + q^6 - q^7 + 4 * q^8 + 9 * q^9 + 5 * q^10 + q^12 + 3 * q^13 - q^14 + 4 * q^16 + 4 * q^17 + 9 * q^18 - 7 * q^19 + 5 * q^20 - 8 * q^21 + 6 * q^23 + q^24 + 17 * q^25 + 3 * q^26 - 2 * q^27 - q^28 - 4 * q^29 + 4 * q^31 + 4 * q^32 + 4 * q^34 + 24 * q^35 + 9 * q^36 + 25 * q^37 - 7 * q^38 - 7 * q^39 + 5 * q^40 - 5 * q^41 - 8 * q^42 - 11 * q^43 + 42 * q^45 + 6 * q^46 - 17 * q^47 + q^48 + 17 * q^49 + 17 * q^50 + q^51 + 3 * q^52 + 2 * q^53 - 2 * q^54 - q^56 + 32 * q^57 - 4 * q^58 + 4 * q^59 - 14 * q^61 + 4 * q^62 + 23 * q^63 + 4 * q^64 + 16 * q^65 + 14 * q^67 + 4 * q^68 - 40 * q^69 + 24 * q^70 - 8 * q^71 + 9 * q^72 + 19 * q^73 + 25 * q^74 - 45 * q^75 - 7 * q^76 - 7 * q^78 - 17 * q^79 + 5 * q^80 - 12 * q^81 - 5 * q^82 + 19 * q^83 - 8 * q^84 + 5 * q^85 - 11 * q^86 + 12 * q^87 - 5 * q^89 + 42 * q^90 - 32 * q^91 + 6 * q^92 + 14 * q^93 - 17 * q^94 + 14 * q^95 + q^96 - 6 * q^97 + 17 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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