LMFDB - Embedded newform 8450.2.a.cs.1.5

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [8450,2,Mod(1,8450)] 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("8450.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(8450, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 8450 = 2 \cdot 5^{2} \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8450.a (trivial)

Newform invariants

comment:select newform

sage:traces = [8,8,0,8,0,0,10,8,16,0,0,0,0,10,0,8,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)

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

Self dual:	yes
Analytic conductor:	$67.4735897080$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 20x^{6} + 132x^{4} - 332x^{2} + 256 $ x^8 - 20x^6 + 132x^4 - 332*x^2 + 256
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 130)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$1.17644$ of defining polynomial
Character	$\chi$	$=$	8450.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.00000 q^{2} +1.17644 q^{3} +1.00000 q^{4} +1.17644 q^{6} -2.08692 q^{7} +1.00000 q^{8} -1.61598 q^{9} +2.99224 q^{11} +1.17644 q^{12} -2.08692 q^{14} +1.00000 q^{16} +1.26018 q^{17} -1.61598 q^{18} +3.89166 q^{19} -2.45514 q^{21} +2.99224 q^{22} +0.899428 q^{23} +1.17644 q^{24} -5.43044 q^{27} -2.08692 q^{28} -0.470937 q^{29} +0.277015 q^{31} +1.00000 q^{32} +3.52020 q^{33} +1.26018 q^{34} -1.61598 q^{36} +7.03766 q^{37} +3.89166 q^{38} -11.1637 q^{41} -2.45514 q^{42} +7.78333 q^{43} +2.99224 q^{44} +0.899428 q^{46} -11.3189 q^{47} +1.17644 q^{48} -2.64477 q^{49} +1.48254 q^{51} +12.0148 q^{53} -5.43044 q^{54} -2.08692 q^{56} +4.57832 q^{57} -0.470937 q^{58} +12.3586 q^{59} +7.64204 q^{61} +0.277015 q^{62} +3.37242 q^{63} +1.00000 q^{64} +3.52020 q^{66} +4.00000 q^{67} +1.26018 q^{68} +1.05813 q^{69} +4.70577 q^{71} -1.61598 q^{72} +15.2984 q^{73} +7.03766 q^{74} +3.89166 q^{76} -6.24455 q^{77} +1.63645 q^{79} -1.54066 q^{81} -11.1637 q^{82} +11.1943 q^{83} -2.45514 q^{84} +7.78333 q^{86} -0.554031 q^{87} +2.99224 q^{88} +6.91127 q^{89} +0.899428 q^{92} +0.325893 q^{93} -11.3189 q^{94} +1.17644 q^{96} +2.57832 q^{97} -2.64477 q^{98} -4.83540 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 8 q^{2} + 8 q^{4} + 10 q^{7} + 8 q^{8} + 16 q^{9} + 10 q^{14} + 8 q^{16} + 16 q^{18} + 10 q^{28} - 6 q^{29} + 8 q^{32} + 20 q^{33} + 16 q^{36} + 40 q^{37} - 6 q^{47} + 30 q^{49} + 20 q^{51} + 10 q^{56}+ \cdots + 30 q^{98}+O(q^{100}) $ 8 * q + 8 * q^2 + 8 * q^4 + 10 * q^7 + 8 * q^8 + 16 * q^9 + 10 * q^14 + 8 * q^16 + 16 * q^18 + 10 * q^28 - 6 * q^29 + 8 * q^32 + 20 * q^33 + 16 * q^36 + 40 * q^37 - 6 * q^47 + 30 * q^49 + 20 * q^51 + 10 * q^56 + 24 * q^57 - 6 * q^58 + 10 * q^61 + 50 * q^63 + 8 * q^64 + 20 * q^66 + 32 * q^67 + 4 * q^69 + 16 * q^72 + 26 * q^73 + 40 * q^74 - 4 * q^79 - 16 * q^81 + 48 * q^83 + 36 * q^93 - 6 * q^94 + 8 * q^97 + 30 * 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
$3$	1.17644	0.679220	0.339610	−	0.940566i	$-0.389705\pi$
$3$	1.17644	0.679220	0.339610	+	0.940566i	$0.389705\pi$
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	1.17644	0.480281
$7$	−2.08692	−0.788781	−0.394390	−	0.918943i	$-0.629044\pi$
$7$	−2.08692	−0.788781	−0.394390	+	0.918943i	$0.629044\pi$
$8$	1.00000	0.353553
$9$	−1.61598	−0.538660
$10$	0	0
$11$	2.99224	0.902193	0.451096	−	0.892475i	$-0.351033\pi$
$11$	2.99224	0.902193	0.451096	+	0.892475i	$0.351033\pi$
$12$	1.17644	0.339610
$13$	0	0
$13$	0	0
$14$	−2.08692	−0.557752
$15$	0	0
$16$	1.00000	0.250000
$17$	1.26018	0.305640	0.152820	−	0.988254i	$-0.451165\pi$
$17$	1.26018	0.305640	0.152820	+	0.988254i	$0.451165\pi$
$18$	−1.61598	−0.380890
$19$	3.89166	0.892809	0.446404	−	0.894831i	$-0.352704\pi$
$19$	3.89166	0.892809	0.446404	+	0.894831i	$0.352704\pi$
$20$	0	0
$21$	−2.45514	−0.535756
$22$	2.99224	0.637947
$23$	0.899428	0.187544	0.0937719	−	0.995594i	$-0.470108\pi$
$23$	0.899428	0.187544	0.0937719	+	0.995594i	$0.470108\pi$
$24$	1.17644	0.240140
$25$	0	0
$26$	0	0
$27$	−5.43044	−1.04509
$28$	−2.08692	−0.394390
$29$	−0.470937	−0.0874508	−0.0437254	−	0.999044i	$-0.513923\pi$
$29$	−0.470937	−0.0874508	−0.0437254	+	0.999044i	$0.513923\pi$
$30$	0	0
$31$	0.277015	0.0497534	0.0248767	−	0.999691i	$-0.492081\pi$
$31$	0.277015	0.0497534	0.0248767	+	0.999691i	$0.492081\pi$
$32$	1.00000	0.176777
$33$	3.52020	0.612787
$34$	1.26018	0.216120
$35$	0	0
$36$	−1.61598	−0.269330
$37$	7.03766	1.15698	0.578492	−	0.815688i	$-0.303641\pi$
$37$	7.03766	1.15698	0.578492	+	0.815688i	$0.303641\pi$
$38$	3.89166	0.631311
$39$	0	0
$40$	0	0
$41$	−11.1637	−1.74348	−0.871738	−	0.489973i	$-0.837007\pi$
$41$	−11.1637	−1.74348	−0.871738	+	0.489973i	$0.837007\pi$
$42$	−2.45514	−0.378836
$43$	7.78333	1.18695	0.593473	−	0.804854i	$-0.297756\pi$
$43$	7.78333	1.18695	0.593473	+	0.804854i	$0.297756\pi$
$44$	2.99224	0.451096
$45$	0	0
$46$	0.899428	0.132613
$47$	−11.3189	−1.65103	−0.825514	−	0.564381i	$-0.809115\pi$
$47$	−11.3189	−1.65103	−0.825514	+	0.564381i	$0.809115\pi$
$48$	1.17644	0.169805
$49$	−2.64477	−0.377825
$50$	0	0
$51$	1.48254	0.207597
$52$	0	0
$53$	12.0148	1.65036	0.825181	−	0.564868i	$-0.191073\pi$
$53$	12.0148	1.65036	0.825181	+	0.564868i	$0.191073\pi$
$54$	−5.43044	−0.738989
$55$	0	0
$56$	−2.08692	−0.278876
$57$	4.57832	0.606413
$58$	−0.470937	−0.0618371
$59$	12.3586	1.60896	0.804479	−	0.593981i	$-0.202445\pi$
$59$	12.3586	1.60896	0.804479	+	0.593981i	$0.202445\pi$
$60$	0	0
$61$	7.64204	0.978463	0.489232	−	0.872154i	$-0.337277\pi$
$61$	7.64204	0.978463	0.489232	+	0.872154i	$0.337277\pi$
$62$	0.277015	0.0351810
$63$	3.37242	0.424885
$64$	1.00000	0.125000
$65$	0	0
$66$	3.52020	0.433306
$67$	4.00000	0.488678	0.244339	−	0.969690i	$-0.421429\pi$
$67$	4.00000	0.488678	0.244339	+	0.969690i	$0.421429\pi$
$68$	1.26018	0.152820
$69$	1.05813	0.127383
$70$	0	0
$71$	4.70577	0.558473	0.279236	−	0.960222i	$-0.409919\pi$
$71$	4.70577	0.558473	0.279236	+	0.960222i	$0.409919\pi$
$72$	−1.61598	−0.190445
$73$	15.2984	1.79054	0.895272	−	0.445520i	$-0.146981\pi$
$73$	15.2984	1.79054	0.895272	+	0.445520i	$0.146981\pi$
$74$	7.03766	0.818111
$75$	0	0
$76$	3.89166	0.446404
$77$	−6.24455	−0.711633
$78$	0	0
$79$	1.63645	0.184115	0.0920574	−	0.995754i	$-0.470656\pi$
$79$	1.63645	0.184115	0.0920574	+	0.995754i	$0.470656\pi$
$80$	0	0
$81$	−1.54066	−0.171185
$82$	−11.1637	−1.23282
$83$	11.1943	1.22873	0.614367	−	0.789020i	$-0.289411\pi$
$83$	11.1943	1.22873	0.614367	+	0.789020i	$0.289411\pi$
$84$	−2.45514	−0.267878
$85$	0	0
$86$	7.78333	0.839298
$87$	−0.554031	−0.0593983
$88$	2.99224	0.318973
$89$	6.91127	0.732593	0.366296	−	0.930498i	$-0.380626\pi$
$89$	6.91127	0.732593	0.366296	+	0.930498i	$0.380626\pi$
$90$	0	0
$91$	0	0
$92$	0.899428	0.0937719
$93$	0.325893	0.0337935
$94$	−11.3189	−1.16745
$95$	0	0
$96$	1.17644	0.120070
$97$	2.57832	0.261789	0.130894	−	0.991396i	$-0.458215\pi$
$97$	2.57832	0.261789	0.130894	+	0.991396i	$0.458215\pi$
$98$	−2.64477	−0.267162
$99$	−4.83540	−0.485976
$100$	0	0
$101$	−1.56672	−0.155895	−0.0779474	−	0.996957i	$-0.524837\pi$
$101$	−1.56672	−0.155895	−0.0779474	+	0.996957i	$0.524837\pi$
$102$	1.48254	0.146793
$103$	0.916364	0.0902920	0.0451460	−	0.998980i	$-0.485625\pi$
$103$	0.916364	0.0902920	0.0451460	+	0.998980i	$0.485625\pi$
$104$	0	0
$105$	0	0
$106$	12.0148	1.16698
$107$	−11.7024	−1.13131	−0.565655	−	0.824642i	$-0.691377\pi$
$107$	−11.7024	−1.13131	−0.565655	+	0.824642i	$0.691377\pi$
$108$	−5.43044	−0.522544
$109$	−15.9902	−1.53159	−0.765793	−	0.643087i	$-0.777653\pi$
$109$	−15.9902	−1.53159	−0.765793	+	0.643087i	$0.777653\pi$
$110$	0	0
$111$	8.27941	0.785847
$112$	−2.08692	−0.197195
$113$	−2.78203	−0.261711	−0.130855	−	0.991401i	$-0.541772\pi$
$113$	−2.78203	−0.261711	−0.130855	+	0.991401i	$0.541772\pi$
$114$	4.57832	0.428799
$115$	0	0
$116$	−0.470937	−0.0437254
$117$	0	0
$118$	12.3586	1.13771
$119$	−2.62990	−0.241083
$120$	0	0
$121$	−2.04653	−0.186048
$122$	7.64204	0.691878
$123$	−13.1334	−1.18420
$124$	0.277015	0.0248767
$125$	0	0
$126$	3.37242	0.300439
$127$	−8.69969	−0.771973	−0.385986	−	0.922504i	$-0.626139\pi$
$127$	−8.69969	−0.771973	−0.385986	+	0.922504i	$0.626139\pi$
$128$	1.00000	0.0883883
$129$	9.15664	0.806197
$130$	0	0
$131$	−15.8740	−1.38692	−0.693459	−	0.720496i	$-0.743914\pi$
$131$	−15.8740	−1.38692	−0.693459	+	0.720496i	$0.743914\pi$
$132$	3.52020	0.306394
$133$	−8.12158	−0.704231
$134$	4.00000	0.345547
$135$	0	0
$136$	1.26018	0.108060
$137$	7.22309	0.617111	0.308555	−	0.951206i	$-0.400155\pi$
$137$	7.22309	0.617111	0.308555	+	0.951206i	$0.400155\pi$
$138$	1.05813	0.0900737
$139$	19.8740	1.68569	0.842846	−	0.538156i	$-0.180879\pi$
$139$	19.8740	1.68569	0.842846	+	0.538156i	$0.180879\pi$
$140$	0	0
$141$	−13.3160	−1.12141
$142$	4.70577	0.394900
$143$	0	0
$144$	−1.61598	−0.134665
$145$	0	0
$146$	15.2984	1.26611
$147$	−3.11143	−0.256626
$148$	7.03766	0.578492
$149$	0.983169	0.0805444	0.0402722	−	0.999189i	$-0.487177\pi$
$149$	0.983169	0.0805444	0.0402722	+	0.999189i	$0.487177\pi$
$150$	0	0
$151$	8.06034	0.655941	0.327971	−	0.944688i	$-0.393635\pi$
$151$	8.06034	0.655941	0.327971	+	0.944688i	$0.393635\pi$
$152$	3.89166	0.315656
$153$	−2.03643	−0.164636
$154$	−6.24455	−0.503200
$155$	0	0
$156$	0	0
$157$	2.73373	0.218176	0.109088	−	0.994032i	$-0.465207\pi$
$157$	2.73373	0.218176	0.109088	+	0.994032i	$0.465207\pi$
$158$	1.63645	0.130189
$159$	14.1348	1.12096
$160$	0	0
$161$	−1.87703	−0.147931
$162$	−1.54066	−0.121046
$163$	24.5769	1.92501	0.962506	−	0.271261i	$-0.0874407\pi$
$163$	24.5769	1.92501	0.962506	+	0.271261i	$0.0874407\pi$
$164$	−11.1637	−0.871738
$165$	0	0
$166$	11.1943	0.868846
$167$	13.8363	1.07069	0.535344	−	0.844634i	$-0.320182\pi$
$167$	13.8363	1.07069	0.535344	+	0.844634i	$0.320182\pi$
$168$	−2.45514	−0.189418
$169$	0	0
$170$	0	0
$171$	−6.28885	−0.480921
$172$	7.78333	0.593473
$173$	−13.1728	−1.00151	−0.500753	−	0.865590i	$-0.666943\pi$
$173$	−13.1728	−1.00151	−0.500753	+	0.865590i	$0.666943\pi$
$174$	−0.554031	−0.0420010
$175$	0	0
$176$	2.99224	0.225548
$177$	14.5392	1.09284
$178$	6.91127	0.518021
$179$	3.11571	0.232879	0.116440	−	0.993198i	$-0.462852\pi$
$179$	3.11571	0.232879	0.116440	+	0.993198i	$0.462852\pi$
$180$	0	0
$181$	−7.91439	−0.588272	−0.294136	−	0.955764i	$-0.595032\pi$
$181$	−7.91439	−0.588272	−0.294136	+	0.955764i	$0.595032\pi$
$182$	0	0
$183$	8.99043	0.664592
$184$	0.899428	0.0663067
$185$	0	0
$186$	0.325893	0.0238956
$187$	3.77077	0.275746
$188$	−11.3189	−0.825514
$189$	11.3329	0.824346
$190$	0	0
$191$	13.9609	1.01018	0.505088	−	0.863068i	$-0.331460\pi$
$191$	13.9609	1.01018	0.505088	+	0.863068i	$0.331460\pi$
$192$	1.17644	0.0849025
$193$	13.6276	0.980935	0.490467	−	0.871460i	$-0.336826\pi$
$193$	13.6276	0.980935	0.490467	+	0.871460i	$0.336826\pi$
$194$	2.57832	0.185113
$195$	0	0
$196$	−2.64477	−0.188912
$197$	17.7219	1.26264	0.631318	−	0.775524i	$-0.282514\pi$
$197$	17.7219	1.26264	0.631318	+	0.775524i	$0.282514\pi$
$198$	−4.83540	−0.343637
$199$	−14.3654	−1.01834	−0.509168	−	0.860667i	$-0.670047\pi$
$199$	−14.3654	−1.01834	−0.509168	+	0.860667i	$0.670047\pi$
$200$	0	0
$201$	4.70577	0.331920
$202$	−1.56672	−0.110234
$203$	0.982807	0.0689795
$204$	1.48254	0.103798
$205$	0	0
$206$	0.916364	0.0638461
$207$	−1.45346	−0.101022
$208$	0	0
$209$	11.6448	0.805486
$210$	0	0
$211$	−20.2231	−1.39222	−0.696108	−	0.717937i	$-0.745086\pi$
$211$	−20.2231	−1.39222	−0.696108	+	0.717937i	$0.745086\pi$
$212$	12.0148	0.825181
$213$	5.53608	0.379326
$214$	−11.7024	−0.799957
$215$	0	0
$216$	−5.43044	−0.369495
$217$	−0.578108	−0.0392445
$218$	−15.9902	−1.08299
$219$	17.9977	1.21617
$220$	0	0
$221$	0	0
$222$	8.27941	0.555677
$223$	11.1450	0.746327	0.373164	−	0.927766i	$-0.378273\pi$
$223$	11.1450	0.746327	0.373164	+	0.927766i	$0.378273\pi$
$224$	−2.08692	−0.139438
$225$	0	0
$226$	−2.78203	−0.185058
$227$	−17.3073	−1.14872	−0.574362	−	0.818601i	$-0.694750\pi$
$227$	−17.3073	−1.14872	−0.574362	+	0.818601i	$0.694750\pi$
$228$	4.57832	0.303207
$229$	9.90350	0.654442	0.327221	−	0.944948i	$-0.393888\pi$
$229$	9.90350	0.654442	0.327221	+	0.944948i	$0.393888\pi$
$230$	0	0
$231$	−7.34636	−0.483355
$232$	−0.470937	−0.0309185
$233$	−19.1742	−1.25614	−0.628070	−	0.778156i	$-0.716155\pi$
$233$	−19.1742	−1.25614	−0.628070	+	0.778156i	$0.716155\pi$
$234$	0	0
$235$	0	0
$236$	12.3586	0.804479
$237$	1.92519	0.125054
$238$	−2.62990	−0.170471
$239$	−2.16448	−0.140009	−0.0700043	−	0.997547i	$-0.522301\pi$
$239$	−2.16448	−0.140009	−0.0700043	+	0.997547i	$0.522301\pi$
$240$	0	0
$241$	16.1013	1.03718	0.518589	−	0.855024i	$-0.326457\pi$
$241$	16.1013	1.03718	0.518589	+	0.855024i	$0.326457\pi$
$242$	−2.04653	−0.131556
$243$	14.4788	0.928817
$244$	7.64204	0.489232
$245$	0	0
$246$	−13.1334	−0.837358
$247$	0	0
$248$	0.277015	0.0175905
$249$	13.1695	0.834581
$250$	0	0
$251$	18.0888	1.14175	0.570877	−	0.821036i	$-0.306603\pi$
$251$	18.0888	1.14175	0.570877	+	0.821036i	$0.306603\pi$
$252$	3.37242	0.212442
$253$	2.69130	0.169201
$254$	−8.69969	−0.545867
$255$	0	0
$256$	1.00000	0.0625000
$257$	10.0452	0.626602	0.313301	−	0.949654i	$-0.398565\pi$
$257$	10.0452	0.626602	0.313301	+	0.949654i	$0.398565\pi$
$258$	9.15664	0.570068
$259$	−14.6870	−0.912607
$260$	0	0
$261$	0.761025	0.0471063
$262$	−15.8740	−0.980699
$263$	−15.8027	−0.974433	−0.487217	−	0.873281i	$-0.661988\pi$
$263$	−15.8027	−0.974433	−0.487217	+	0.873281i	$0.661988\pi$
$264$	3.52020	0.216653
$265$	0	0
$266$	−8.12158	−0.497966
$267$	8.13071	0.497592
$268$	4.00000	0.244339
$269$	−7.38587	−0.450325	−0.225162	−	0.974321i	$-0.572291\pi$
$269$	−7.38587	−0.450325	−0.225162	+	0.974321i	$0.572291\pi$
$270$	0	0
$271$	31.0029	1.88329	0.941645	−	0.336609i	$-0.109280\pi$
$271$	31.0029	1.88329	0.941645	+	0.336609i	$0.109280\pi$
$272$	1.26018	0.0764099
$273$	0	0
$274$	7.22309	0.436363
$275$	0	0
$276$	1.05813	0.0636917
$277$	−4.70735	−0.282837	−0.141419	−	0.989950i	$-0.545166\pi$
$277$	−4.70735	−0.282837	−0.141419	+	0.989950i	$0.545166\pi$
$278$	19.8740	1.19196
$279$	−0.447652	−0.0268002
$280$	0	0
$281$	−10.7771	−0.642910	−0.321455	−	0.946925i	$-0.604172\pi$
$281$	−10.7771	−0.642910	−0.321455	+	0.946925i	$0.604172\pi$
$282$	−13.3160	−0.792958
$283$	−29.1220	−1.73112	−0.865561	−	0.500804i	$-0.833038\pi$
$283$	−29.1220	−1.73112	−0.865561	+	0.500804i	$0.833038\pi$
$284$	4.70577	0.279236
$285$	0	0
$286$	0	0
$287$	23.2977	1.37522
$288$	−1.61598	−0.0952226
$289$	−15.4119	−0.906584
$290$	0	0
$291$	3.03325	0.177812
$292$	15.2984	0.895272
$293$	−25.4248	−1.48534	−0.742668	−	0.669660i	$-0.766440\pi$
$293$	−25.4248	−1.48534	−0.742668	+	0.669660i	$0.766440\pi$
$294$	−3.11143	−0.181462
$295$	0	0
$296$	7.03766	0.409056
$297$	−16.2492	−0.942872
$298$	0.983169	0.0569535
$299$	0	0
$300$	0	0
$301$	−16.2432	−0.936240
$302$	8.06034	0.463821
$303$	−1.84316	−0.105887
$304$	3.89166	0.223202
$305$	0	0
$306$	−2.03643	−0.116415
$307$	25.3305	1.44569	0.722843	−	0.691012i	$-0.242835\pi$
$307$	25.3305	1.44569	0.722843	+	0.691012i	$0.242835\pi$
$308$	−6.24455	−0.355816
$309$	1.07805	0.0613281
$310$	0	0
$311$	24.3495	1.38074	0.690368	−	0.723459i	$-0.257449\pi$
$311$	24.3495	1.38074	0.690368	+	0.723459i	$0.257449\pi$
$312$	0	0
$313$	15.6891	0.886802	0.443401	−	0.896323i	$-0.353772\pi$
$313$	15.6891	0.886802	0.443401	+	0.896323i	$0.353772\pi$
$314$	2.73373	0.154273
$315$	0	0
$316$	1.63645	0.0920574
$317$	−19.4607	−1.09302	−0.546510	−	0.837453i	$-0.684044\pi$
$317$	−19.4607	−1.09302	−0.546510	+	0.837453i	$0.684044\pi$
$318$	14.1348	0.792638
$319$	−1.40915	−0.0788975
$320$	0	0
$321$	−13.7672	−0.768408
$322$	−1.87703	−0.104603
$323$	4.90421	0.272878
$324$	−1.54066	−0.0855923
$325$	0	0
$326$	24.5769	1.36119
$327$	−18.8116	−1.04028
$328$	−11.1637	−0.616412
$329$	23.6216	1.30230
$330$	0	0
$331$	6.62711	0.364259	0.182129	−	0.983275i	$-0.441701\pi$
$331$	6.62711	0.364259	0.182129	+	0.983275i	$0.441701\pi$
$332$	11.1943	0.614367
$333$	−11.3727	−0.623221
$334$	13.8363	0.757091
$335$	0	0
$336$	−2.45514	−0.133939
$337$	−11.0614	−0.602555	−0.301277	−	0.953537i	$-0.597413\pi$
$337$	−11.0614	−0.602555	−0.301277	+	0.953537i	$0.597413\pi$
$338$	0	0
$339$	−3.27290	−0.177759
$340$	0	0
$341$	0.828895	0.0448872
$342$	−6.28885	−0.340062
$343$	20.1279	1.08680
$344$	7.78333	0.419649
$345$	0	0
$346$	−13.1728	−0.708172
$347$	17.7207	0.951297	0.475649	−	0.879635i	$-0.342213\pi$
$347$	17.7207	0.951297	0.475649	+	0.879635i	$0.342213\pi$
$348$	−0.554031	−0.0296992
$349$	−17.6868	−0.946754	−0.473377	−	0.880860i	$-0.656965\pi$
$349$	−17.6868	−0.946754	−0.473377	+	0.880860i	$0.656965\pi$
$350$	0	0
$351$	0	0
$352$	2.99224	0.159487
$353$	33.6229	1.78957	0.894783	−	0.446501i	$-0.147330\pi$
$353$	33.6229	1.78957	0.894783	+	0.446501i	$0.147330\pi$
$354$	14.5392	0.772752
$355$	0	0
$356$	6.91127	0.366296
$357$	−3.09393	−0.163748
$358$	3.11571	0.164670
$359$	−17.4880	−0.922981	−0.461490	−	0.887145i	$-0.652685\pi$
$359$	−17.4880	−0.922981	−0.461490	+	0.887145i	$0.652685\pi$
$360$	0	0
$361$	−3.85496	−0.202892
$362$	−7.91439	−0.415971
$363$	−2.40762	−0.126367
$364$	0	0
$365$	0	0
$366$	8.99043	0.469937
$367$	−7.35975	−0.384176	−0.192088	−	0.981378i	$-0.561526\pi$
$367$	−7.35975	−0.384176	−0.192088	+	0.981378i	$0.561526\pi$
$368$	0.899428	0.0468859
$369$	18.0403	0.939141
$370$	0	0
$371$	−25.0740	−1.30177
$372$	0.325893	0.0168968
$373$	29.2016	1.51200	0.756000	−	0.654571i	$-0.227151\pi$
$373$	29.2016	1.51200	0.756000	+	0.654571i	$0.227151\pi$
$374$	3.77077	0.194982
$375$	0	0
$376$	−11.3189	−0.583727
$377$	0	0
$378$	11.3329	0.582901
$379$	−15.3509	−0.788522	−0.394261	−	0.918999i	$-0.628999\pi$
$379$	−15.3509	−0.788522	−0.394261	+	0.918999i	$0.628999\pi$
$380$	0	0
$381$	−10.2347	−0.524339
$382$	13.9609	0.714303
$383$	−3.19245	−0.163126	−0.0815632	−	0.996668i	$-0.525991\pi$
$383$	−3.19245	−0.163126	−0.0815632	+	0.996668i	$0.525991\pi$
$384$	1.17644	0.0600351
$385$	0	0
$386$	13.6276	0.693626
$387$	−12.5777	−0.639361
$388$	2.57832	0.130894
$389$	28.9316	1.46689	0.733445	−	0.679749i	$-0.237911\pi$
$389$	28.9316	1.46689	0.733445	+	0.679749i	$0.237911\pi$
$390$	0	0
$391$	1.13345	0.0573208
$392$	−2.64477	−0.133581
$393$	−18.6749	−0.942022
$394$	17.7219	0.892819
$395$	0	0
$396$	−4.83540	−0.242988
$397$	4.99113	0.250498	0.125249	−	0.992125i	$-0.460027\pi$
$397$	4.99113	0.250498	0.125249	+	0.992125i	$0.460027\pi$
$398$	−14.3654	−0.720073
$399$	−9.55458	−0.478327
$400$	0	0
$401$	3.99563	0.199532	0.0997662	−	0.995011i	$-0.468191\pi$
$401$	3.99563	0.199532	0.0997662	+	0.995011i	$0.468191\pi$
$402$	4.70577	0.234703
$403$	0	0
$404$	−1.56672	−0.0779474
$405$	0	0
$406$	0.982807	0.0487759
$407$	21.0583	1.04382
$408$	1.48254	0.0733965
$409$	−2.92942	−0.144850	−0.0724252	−	0.997374i	$-0.523074\pi$
$409$	−2.92942	−0.144850	−0.0724252	+	0.997374i	$0.523074\pi$
$410$	0	0
$411$	8.49756	0.419154
$412$	0.916364	0.0451460
$413$	−25.7915	−1.26912
$414$	−1.45346	−0.0714336
$415$	0	0
$416$	0	0
$417$	23.3806	1.14495
$418$	11.6448	0.569564
$419$	−9.70849	−0.474291	−0.237145	−	0.971474i	$-0.576212\pi$
$419$	−9.70849	−0.474291	−0.237145	+	0.971474i	$0.576212\pi$
$420$	0	0
$421$	−22.8217	−1.11226	−0.556132	−	0.831094i	$-0.687715\pi$
$421$	−22.8217	−1.11226	−0.556132	+	0.831094i	$0.687715\pi$
$422$	−20.2231	−0.984445
$423$	18.2911	0.889344
$424$	12.0148	0.583491
$425$	0	0
$426$	5.53608	0.268224
$427$	−15.9483	−0.771793
$428$	−11.7024	−0.565655
$429$	0	0
$430$	0	0
$431$	−11.5615	−0.556896	−0.278448	−	0.960451i	$-0.589820\pi$
$431$	−11.5615	−0.556896	−0.278448	+	0.960451i	$0.589820\pi$
$432$	−5.43044	−0.261272
$433$	−12.5656	−0.603862	−0.301931	−	0.953330i	$-0.597631\pi$
$433$	−12.5656	−0.603862	−0.301931	+	0.953330i	$0.597631\pi$
$434$	−0.578108	−0.0277501
$435$	0	0
$436$	−15.9902	−0.765793
$437$	3.50027	0.167441
$438$	17.9977	0.859964
$439$	1.81028	0.0864001	0.0432001	−	0.999066i	$-0.486245\pi$
$439$	1.81028	0.0864001	0.0432001	+	0.999066i	$0.486245\pi$
$440$	0	0
$441$	4.27390	0.203519
$442$	0	0
$443$	16.7633	0.796450	0.398225	−	0.917288i	$-0.369626\pi$
$443$	16.7633	0.796450	0.398225	+	0.917288i	$0.369626\pi$
$444$	8.27941	0.392923
$445$	0	0
$446$	11.1450	0.527733
$447$	1.15664	0.0547073
$448$	−2.08692	−0.0985976
$449$	7.08190	0.334216	0.167108	−	0.985939i	$-0.446557\pi$
$449$	7.08190	0.334216	0.167108	+	0.985939i	$0.446557\pi$
$450$	0	0
$451$	−33.4044	−1.57295
$452$	−2.78203	−0.130855
$453$	9.48254	0.445528
$454$	−17.3073	−0.812271
$455$	0	0
$456$	4.57832	0.214400
$457$	−1.09019	−0.0509970	−0.0254985	−	0.999675i	$-0.508117\pi$
$457$	−1.09019	−0.0509970	−0.0254985	+	0.999675i	$0.508117\pi$
$458$	9.90350	0.462760
$459$	−6.84336	−0.319421
$460$	0	0
$461$	−15.5893	−0.726066	−0.363033	−	0.931776i	$-0.618259\pi$
$461$	−15.5893	−0.726066	−0.363033	+	0.931776i	$0.618259\pi$
$462$	−7.34636	−0.341784
$463$	−19.1896	−0.891817	−0.445908	−	0.895079i	$-0.647119\pi$
$463$	−19.1896	−0.891817	−0.445908	+	0.895079i	$0.647119\pi$
$464$	−0.470937	−0.0218627
$465$	0	0
$466$	−19.1742	−0.888226
$467$	23.1857	1.07290	0.536452	−	0.843931i	$-0.319764\pi$
$467$	23.1857	1.07290	0.536452	+	0.843931i	$0.319764\pi$
$468$	0	0
$469$	−8.34767	−0.385460
$470$	0	0
$471$	3.21608	0.148189
$472$	12.3586	0.568853
$473$	23.2895	1.07085
$474$	1.92519	0.0884268
$475$	0	0
$476$	−2.62990	−0.120541
$477$	−19.4157	−0.888985
$478$	−2.16448	−0.0990010
$479$	−3.48503	−0.159235	−0.0796174	−	0.996825i	$-0.525370\pi$
$479$	−3.48503	−0.159235	−0.0796174	+	0.996825i	$0.525370\pi$
$480$	0	0
$481$	0	0
$482$	16.1013	0.733396
$483$	−2.20822	−0.100478
$484$	−2.04653	−0.0930240
$485$	0	0
$486$	14.4788	0.656773
$487$	2.14046	0.0969934	0.0484967	−	0.998823i	$-0.484557\pi$
$487$	2.14046	0.0969934	0.0484967	+	0.998823i	$0.484557\pi$
$488$	7.64204	0.345939
$489$	28.9133	1.30751
$490$	0	0
$491$	−27.2403	−1.22934	−0.614668	−	0.788786i	$-0.710710\pi$
$491$	−27.2403	−1.22934	−0.614668	+	0.788786i	$0.710710\pi$
$492$	−13.1334	−0.592101
$493$	−0.593468	−0.0267284
$494$	0	0
$495$	0	0
$496$	0.277015	0.0124384
$497$	−9.82056	−0.440512
$498$	13.1695	0.590138
$499$	−22.7855	−1.02002	−0.510010	−	0.860169i	$-0.670358\pi$
$499$	−22.7855	−1.02002	−0.510010	+	0.860169i	$0.670358\pi$
$500$	0	0
$501$	16.2777	0.727233
$502$	18.0888	0.807341
$503$	9.60324	0.428187	0.214094	−	0.976813i	$-0.431320\pi$
$503$	9.60324	0.428187	0.214094	+	0.976813i	$0.431320\pi$
$504$	3.37242	0.150220
$505$	0	0
$506$	2.69130	0.119643
$507$	0	0
$508$	−8.69969	−0.385986
$509$	−15.6608	−0.694154	−0.347077	−	0.937837i	$-0.612826\pi$
$509$	−15.6608	−0.694154	−0.347077	+	0.937837i	$0.612826\pi$
$510$	0	0
$511$	−31.9265	−1.41235
$512$	1.00000	0.0441942
$513$	−21.1334	−0.933064
$514$	10.0452	0.443074
$515$	0	0
$516$	9.15664	0.403099
$517$	−33.8688	−1.48955
$518$	−14.6870	−0.645311
$519$	−15.4970	−0.680243
$520$	0	0
$521$	3.26689	0.143125	0.0715625	−	0.997436i	$-0.477201\pi$
$521$	3.26689	0.143125	0.0715625	+	0.997436i	$0.477201\pi$
$522$	0.761025	0.0333092
$523$	−15.9902	−0.699204	−0.349602	−	0.936898i	$-0.613683\pi$
$523$	−15.9902	−0.699204	−0.349602	+	0.936898i	$0.613683\pi$
$524$	−15.8740	−0.693459
$525$	0	0
$526$	−15.8027	−0.689029
$527$	0.349090	0.0152066
$528$	3.52020	0.153197
$529$	−22.1910	−0.964827
$530$	0	0
$531$	−19.9713	−0.866682
$532$	−8.12158	−0.352115
$533$	0	0
$534$	8.13071	0.351850
$535$	0	0
$536$	4.00000	0.172774
$537$	3.66546	0.158176
$538$	−7.38587	−0.318428
$539$	−7.91378	−0.340871
$540$	0	0
$541$	−7.38554	−0.317529	−0.158765	−	0.987316i	$-0.550751\pi$
$541$	−7.38554	−0.317529	−0.158765	+	0.987316i	$0.550751\pi$
$542$	31.0029	1.33169
$543$	−9.31084	−0.399566
$544$	1.26018	0.0540300
$545$	0	0
$546$	0	0
$547$	−11.4488	−0.489515	−0.244757	−	0.969584i	$-0.578708\pi$
$547$	−11.4488	−0.489515	−0.244757	+	0.969584i	$0.578708\pi$
$548$	7.22309	0.308555
$549$	−12.3494	−0.527059
$550$	0	0
$551$	−1.83273	−0.0780768
$552$	1.05813	0.0450368
$553$	−3.41513	−0.145226
$554$	−4.70735	−0.199996
$555$	0	0
$556$	19.8740	0.842846
$557$	19.4071	0.822306	0.411153	−	0.911566i	$-0.365126\pi$
$557$	19.4071	0.822306	0.411153	+	0.911566i	$0.365126\pi$
$558$	−0.447652	−0.0189506
$559$	0	0
$560$	0	0
$561$	4.43610	0.187292
$562$	−10.7771	−0.454606
$563$	−8.89770	−0.374993	−0.187497	−	0.982265i	$-0.560037\pi$
$563$	−8.89770	−0.374993	−0.187497	+	0.982265i	$0.560037\pi$
$564$	−13.3160	−0.560706
$565$	0	0
$566$	−29.1220	−1.22409
$567$	3.21524	0.135027
$568$	4.70577	0.197450
$569$	−6.89993	−0.289260	−0.144630	−	0.989486i	$-0.546199\pi$
$569$	−6.89993	−0.289260	−0.144630	+	0.989486i	$0.546199\pi$
$570$	0	0
$571$	32.4491	1.35795	0.678975	−	0.734161i	$-0.262424\pi$
$571$	32.4491	1.35795	0.678975	+	0.734161i	$0.262424\pi$
$572$	0	0
$573$	16.4242	0.686132
$574$	23.2977	0.972427
$575$	0	0
$576$	−1.61598	−0.0673325
$577$	25.3915	1.05706	0.528530	−	0.848914i	$-0.322743\pi$
$577$	25.3915	1.05706	0.528530	+	0.848914i	$0.322743\pi$
$578$	−15.4119	−0.641052
$579$	16.0321	0.666270
$580$	0	0
$581$	−23.3616	−0.969202
$582$	3.03325	0.125732
$583$	35.9512	1.48895
$584$	15.2984	0.633053
$585$	0	0
$586$	−25.4248	−1.05029
$587$	−2.46392	−0.101697	−0.0508485	−	0.998706i	$-0.516193\pi$
$587$	−2.46392	−0.101697	−0.0508485	+	0.998706i	$0.516193\pi$
$588$	−3.11143	−0.128313
$589$	1.07805	0.0444203
$590$	0	0
$591$	20.8489	0.857608
$592$	7.03766	0.289246
$593$	40.6651	1.66992	0.834958	−	0.550313i	$-0.185492\pi$
$593$	40.6651	1.66992	0.834958	+	0.550313i	$0.185492\pi$
$594$	−16.2492	−0.666711
$595$	0	0
$596$	0.983169	0.0402722
$597$	−16.9001	−0.691675
$598$	0	0
$599$	−26.1916	−1.07016	−0.535079	−	0.844802i	$-0.679718\pi$
$599$	−26.1916	−1.07016	−0.535079	+	0.844802i	$0.679718\pi$
$600$	0	0
$601$	−13.4983	−0.550608	−0.275304	−	0.961357i	$-0.588778\pi$
$601$	−13.4983	−0.550608	−0.275304	+	0.961357i	$0.588778\pi$
$602$	−16.2432	−0.662022
$603$	−6.46392	−0.263231
$604$	8.06034	0.327971
$605$	0	0
$606$	−1.84316	−0.0748733
$607$	9.82862	0.398931	0.199466	−	0.979905i	$-0.436079\pi$
$607$	9.82862	0.398931	0.199466	+	0.979905i	$0.436079\pi$
$608$	3.89166	0.157828
$609$	1.15622	0.0468523
$610$	0	0
$611$	0	0
$612$	−2.03643	−0.0823180
$613$	−20.6173	−0.832725	−0.416362	−	0.909199i	$-0.636695\pi$
$613$	−20.6173	−0.832725	−0.416362	+	0.909199i	$0.636695\pi$
$614$	25.3305	1.02225
$615$	0	0
$616$	−6.24455	−0.251600
$617$	−17.8192	−0.717372	−0.358686	−	0.933458i	$-0.616775\pi$
$617$	−17.8192	−0.717372	−0.358686	+	0.933458i	$0.616775\pi$
$618$	1.07805	0.0433655
$619$	−29.3377	−1.17918	−0.589592	−	0.807701i	$-0.700711\pi$
$619$	−29.3377	−1.17918	−0.589592	+	0.807701i	$0.700711\pi$
$620$	0	0
$621$	−4.88429	−0.196000
$622$	24.3495	0.976327
$623$	−14.4232	−0.577855
$624$	0	0
$625$	0	0
$626$	15.6891	0.627064
$627$	13.6994	0.547102
$628$	2.73373	0.109088
$629$	8.86875	0.353620
$630$	0	0
$631$	17.9743	0.715545	0.357772	−	0.933809i	$-0.383536\pi$
$631$	17.9743	0.715545	0.357772	+	0.933809i	$0.383536\pi$
$632$	1.63645	0.0650944
$633$	−23.7913	−0.945620
$634$	−19.4607	−0.772881
$635$	0	0
$636$	14.1348	0.560480
$637$	0	0
$638$	−1.40915	−0.0557890
$639$	−7.60444	−0.300827
$640$	0	0
$641$	−25.5610	−1.00960	−0.504800	−	0.863236i	$-0.668434\pi$
$641$	−25.5610	−1.00960	−0.504800	+	0.863236i	$0.668434\pi$
$642$	−13.7672	−0.543347
$643$	−21.9827	−0.866913	−0.433457	−	0.901174i	$-0.642706\pi$
$643$	−21.9827	−0.866913	−0.433457	+	0.901174i	$0.642706\pi$
$644$	−1.87703	−0.0739654
$645$	0	0
$646$	4.90421	0.192954
$647$	31.1502	1.22464	0.612321	−	0.790609i	$-0.290236\pi$
$647$	31.1502	1.22464	0.612321	+	0.790609i	$0.290236\pi$
$648$	−1.54066	−0.0605229
$649$	36.9800	1.45159
$650$	0	0
$651$	−0.680112	−0.0266557
$652$	24.5769	0.962506
$653$	17.8342	0.697907	0.348954	−	0.937140i	$-0.386537\pi$
$653$	17.8342	0.697907	0.348954	+	0.937140i	$0.386537\pi$
$654$	−18.8116	−0.735592
$655$	0	0
$656$	−11.1637	−0.435869
$657$	−24.7219	−0.964495
$658$	23.6216	0.920865
$659$	19.7893	0.770881	0.385440	−	0.922733i	$-0.374050\pi$
$659$	19.7893	0.770881	0.385440	+	0.922733i	$0.374050\pi$
$660$	0	0
$661$	−18.2321	−0.709148	−0.354574	−	0.935028i	$-0.615374\pi$
$661$	−18.2321	−0.709148	−0.354574	+	0.935028i	$0.615374\pi$
$662$	6.62711	0.257570
$663$	0	0
$664$	11.1943	0.434423
$665$	0	0
$666$	−11.3727	−0.440684
$667$	−0.423574	−0.0164008
$668$	13.8363	0.535344
$669$	13.1115	0.506920
$670$	0	0
$671$	22.8668	0.882763
$672$	−2.45514	−0.0947091
$673$	−19.8464	−0.765024	−0.382512	−	0.923950i	$-0.624941\pi$
$673$	−19.8464	−0.765024	−0.382512	+	0.923950i	$0.624941\pi$
$674$	−11.0614	−0.426070
$675$	0	0
$676$	0	0
$677$	4.02129	0.154551	0.0772753	−	0.997010i	$-0.475378\pi$
$677$	4.02129	0.154551	0.0772753	+	0.997010i	$0.475378\pi$
$678$	−3.27290	−0.125695
$679$	−5.38075	−0.206494
$680$	0	0
$681$	−20.3610	−0.780237
$682$	0.828895	0.0317400
$683$	−10.3886	−0.397509	−0.198754	−	0.980049i	$-0.563690\pi$
$683$	−10.3886	−0.397509	−0.198754	+	0.980049i	$0.563690\pi$
$684$	−6.28885	−0.240460
$685$	0	0
$686$	20.1279	0.768485
$687$	11.6509	0.444510
$688$	7.78333	0.296737
$689$	0	0
$690$	0	0
$691$	28.3899	1.08000	0.540001	−	0.841664i	$-0.318424\pi$
$691$	28.3899	1.08000	0.540001	+	0.841664i	$0.318424\pi$
$692$	−13.1728	−0.500753
$693$	10.0911	0.383328
$694$	17.7207	0.672669
$695$	0	0
$696$	−0.554031	−0.0210005
$697$	−14.0683	−0.532875
$698$	−17.6868	−0.669456
$699$	−22.5573	−0.853196
$700$	0	0
$701$	−16.7917	−0.634213	−0.317106	−	0.948390i	$-0.602711\pi$
$701$	−16.7917	−0.634213	−0.317106	+	0.948390i	$0.602711\pi$
$702$	0	0
$703$	27.3882	1.03297
$704$	2.99224	0.112774
$705$	0	0
$706$	33.6229	1.26541
$707$	3.26962	0.122967
$708$	14.5392	0.546418
$709$	−48.4796	−1.82069	−0.910345	−	0.413850i	$-0.864184\pi$
$709$	−48.4796	−1.82069	−0.910345	+	0.413850i	$0.864184\pi$
$710$	0	0
$711$	−2.64447	−0.0991753
$712$	6.91127	0.259011
$713$	0.249155	0.00933094
$714$	−3.09393	−0.115787
$715$	0	0
$716$	3.11571	0.116440
$717$	−2.54639	−0.0950966
$718$	−17.4880	−0.652646
$719$	−12.2956	−0.458547	−0.229273	−	0.973362i	$-0.573635\pi$
$719$	−12.2956	−0.458547	−0.229273	+	0.973362i	$0.573635\pi$
$720$	0	0
$721$	−1.91238	−0.0712206
$722$	−3.85496	−0.143467
$723$	18.9423	0.704472
$724$	−7.91439	−0.294136
$725$	0	0
$726$	−2.40762	−0.0893553
$727$	−36.1491	−1.34070	−0.670349	−	0.742046i	$-0.733855\pi$
$727$	−36.1491	−1.34070	−0.670349	+	0.742046i	$0.733855\pi$
$728$	0	0
$729$	21.6555	0.802055
$730$	0	0
$731$	9.80843	0.362778
$732$	8.99043	0.332296
$733$	22.4789	0.830278	0.415139	−	0.909758i	$-0.363733\pi$
$733$	22.4789	0.830278	0.415139	+	0.909758i	$0.363733\pi$
$734$	−7.35975	−0.271653
$735$	0	0
$736$	0.899428	0.0331534
$737$	11.9689	0.440882
$738$	18.0403	0.664073
$739$	−12.3515	−0.454357	−0.227179	−	0.973853i	$-0.572950\pi$
$739$	−12.3515	−0.454357	−0.227179	+	0.973853i	$0.572950\pi$
$740$	0	0
$741$	0	0
$742$	−25.0740	−0.920494
$743$	−22.9637	−0.842455	−0.421227	−	0.906955i	$-0.638401\pi$
$743$	−22.9637	−0.842455	−0.421227	+	0.906955i	$0.638401\pi$
$744$	0.325893	0.0119478
$745$	0	0
$746$	29.2016	1.06915
$747$	−18.0898	−0.661870
$748$	3.77077	0.137873
$749$	24.4219	0.892355
$750$	0	0
$751$	−32.8525	−1.19881	−0.599403	−	0.800447i	$-0.704595\pi$
$751$	−32.8525	−1.19881	−0.599403	+	0.800447i	$0.704595\pi$
$752$	−11.3189	−0.412757
$753$	21.2804	0.775501
$754$	0	0
$755$	0	0
$756$	11.3329	0.412173
$757$	33.0618	1.20165	0.600826	−	0.799380i	$-0.294839\pi$
$757$	33.0618	1.20165	0.600826	+	0.799380i	$0.294839\pi$
$758$	−15.3509	−0.557569
$759$	3.16616	0.114924
$760$	0	0
$761$	31.5634	1.14417	0.572086	−	0.820194i	$-0.306134\pi$
$761$	31.5634	1.14417	0.572086	+	0.820194i	$0.306134\pi$
$762$	−10.2347	−0.370764
$763$	33.3703	1.20809
$764$	13.9609	0.505088
$765$	0	0
$766$	−3.19245	−0.115348
$767$	0	0
$768$	1.17644	0.0424512
$769$	−42.5444	−1.53419	−0.767095	−	0.641534i	$-0.778298\pi$
$769$	−42.5444	−1.53419	−0.767095	+	0.641534i	$0.778298\pi$
$770$	0	0
$771$	11.8176	0.425601
$772$	13.6276	0.490467
$773$	1.06645	0.0383576	0.0191788	−	0.999816i	$-0.493895\pi$
$773$	1.06645	0.0383576	0.0191788	+	0.999816i	$0.493895\pi$
$774$	−12.5777	−0.452096
$775$	0	0
$776$	2.57832	0.0925563
$777$	−17.2784	−0.619861
$778$	28.9316	1.03725
$779$	−43.4453	−1.55659
$780$	0	0
$781$	14.0808	0.503850
$782$	1.13345	0.0405319
$783$	2.55739	0.0913938
$784$	−2.64477	−0.0944562
$785$	0	0
$786$	−18.6749	−0.666110
$787$	−0.0431219	−0.00153713	−0.000768565	−	1.00000i	$-0.500245\pi$
$787$	−0.0431219	−0.00153713	−0.000768565		1.00000i	$0.500245\pi$
$788$	17.7219	0.631318
$789$	−18.5909	−0.661855
$790$	0	0
$791$	5.80586	0.206433
$792$	−4.83540	−0.171818
$793$	0	0
$794$	4.99113	0.177129
$795$	0	0
$796$	−14.3654	−0.509168
$797$	−34.5901	−1.22525	−0.612623	−	0.790376i	$-0.709885\pi$
$797$	−34.5901	−1.22525	−0.612623	+	0.790376i	$0.709885\pi$
$798$	−9.55458	−0.338229
$799$	−14.2639	−0.504620
$800$	0	0
$801$	−11.1685	−0.394619
$802$	3.99563	0.141091
$803$	45.7765	1.61542
$804$	4.70577	0.165960
$805$	0	0
$806$	0	0
$807$	−8.68906	−0.305869
$808$	−1.56672	−0.0551171
$809$	−6.08091	−0.213794	−0.106897	−	0.994270i	$-0.534091\pi$
$809$	−6.08091	−0.213794	−0.106897	+	0.994270i	$0.534091\pi$
$810$	0	0
$811$	−5.47145	−0.192129	−0.0960644	−	0.995375i	$-0.530625\pi$
$811$	−5.47145	−0.192129	−0.0960644	+	0.995375i	$0.530625\pi$
$812$	0.982807	0.0344898
$813$	36.4731	1.27917
$814$	21.0583	0.738094
$815$	0	0
$816$	1.48254	0.0518991
$817$	30.2901	1.05972
$818$	−2.92942	−0.102425
$819$	0	0
$820$	0	0
$821$	−44.0662	−1.53792	−0.768961	−	0.639296i	$-0.779226\pi$
$821$	−44.0662	−1.53792	−0.768961	+	0.639296i	$0.779226\pi$
$822$	8.49756	0.296386
$823$	12.5165	0.436297	0.218148	−	0.975916i	$-0.429998\pi$
$823$	12.5165	0.436297	0.218148	+	0.975916i	$0.429998\pi$
$824$	0.916364	0.0319231
$825$	0	0
$826$	−25.7915	−0.897400
$827$	−51.7679	−1.80015	−0.900074	−	0.435738i	$-0.856488\pi$
$827$	−51.7679	−1.80015	−0.900074	+	0.435738i	$0.856488\pi$
$828$	−1.45346	−0.0505112
$829$	−13.9177	−0.483381	−0.241690	−	0.970353i	$-0.577702\pi$
$829$	−13.9177	−0.483381	−0.241690	+	0.970353i	$0.577702\pi$
$830$	0	0
$831$	−5.53793	−0.192109
$832$	0	0
$833$	−3.33290	−0.115478
$834$	23.3806	0.809605
$835$	0	0
$836$	11.6448	0.402743
$837$	−1.50432	−0.0519967
$838$	−9.70849	−0.335374
$839$	−50.0466	−1.72780	−0.863899	−	0.503664i	$-0.831985\pi$
$839$	−50.0466	−1.72780	−0.863899	+	0.503664i	$0.831985\pi$
$840$	0	0
$841$	−28.7782	−0.992352
$842$	−22.8217	−0.786489
$843$	−12.6787	−0.436677
$844$	−20.2231	−0.696108
$845$	0	0
$846$	18.2911	0.628861
$847$	4.27093	0.146751
$848$	12.0148	0.412591
$849$	−34.2603	−1.17581
$850$	0	0
$851$	6.32987	0.216985
$852$	5.53608	0.189663
$853$	12.7392	0.436183	0.218092	−	0.975928i	$-0.430017\pi$
$853$	12.7392	0.436183	0.218092	+	0.975928i	$0.430017\pi$
$854$	−15.9483	−0.545740
$855$	0	0
$856$	−11.7024	−0.399978
$857$	30.3306	1.03607	0.518036	−	0.855359i	$-0.326663\pi$
$857$	30.3306	1.03607	0.518036	+	0.855359i	$0.326663\pi$
$858$	0	0
$859$	−48.7446	−1.66314	−0.831572	−	0.555417i	$-0.812559\pi$
$859$	−48.7446	−1.66314	−0.831572	+	0.555417i	$0.812559\pi$
$860$	0	0
$861$	27.4084	0.934077
$862$	−11.5615	−0.393785
$863$	−17.6100	−0.599451	−0.299725	−	0.954026i	$-0.596895\pi$
$863$	−17.6100	−0.599451	−0.299725	+	0.954026i	$0.596895\pi$
$864$	−5.43044	−0.184747
$865$	0	0
$866$	−12.5656	−0.426995
$867$	−18.1313	−0.615770
$868$	−0.578108	−0.0196223
$869$	4.89664	0.166107
$870$	0	0
$871$	0	0
$872$	−15.9902	−0.541497
$873$	−4.16652	−0.141015
$874$	3.50027	0.118398
$875$	0	0
$876$	17.9977	0.608086
$877$	41.9289	1.41584	0.707918	−	0.706294i	$-0.249634\pi$
$877$	41.9289	1.41584	0.707918	+	0.706294i	$0.249634\pi$
$878$	1.81028	0.0610941
$879$	−29.9109	−1.00887
$880$	0	0
$881$	29.9390	1.00867	0.504336	−	0.863508i	$-0.331737\pi$
$881$	29.9390	1.00867	0.504336	+	0.863508i	$0.331737\pi$
$882$	4.27390	0.143910
$883$	35.3391	1.18926	0.594629	−	0.804001i	$-0.297299\pi$
$883$	35.3391	1.18926	0.594629	+	0.804001i	$0.297299\pi$
$884$	0	0
$885$	0	0
$886$	16.7633	0.563175
$887$	−42.5717	−1.42942	−0.714709	−	0.699421i	$-0.753441\pi$
$887$	−42.5717	−1.42942	−0.714709	+	0.699421i	$0.753441\pi$
$888$	8.27941	0.277839
$889$	18.1555	0.608917
$890$	0	0
$891$	−4.61002	−0.154442
$892$	11.1450	0.373164
$893$	−44.0493	−1.47405
$894$	1.15664	0.0386839
$895$	0	0
$896$	−2.08692	−0.0697190
$897$	0	0
$898$	7.08190	0.236326
$899$	−0.130457	−0.00435098
$900$	0	0
$901$	15.1409	0.504416
$902$	−33.4044	−1.11224
$903$	−19.1092	−0.635913
$904$	−2.78203	−0.0925288
$905$	0	0
$906$	9.48254	0.315036
$907$	−14.4449	−0.479637	−0.239818	−	0.970818i	$-0.577088\pi$
$907$	−14.4449	−0.479637	−0.239818	+	0.970818i	$0.577088\pi$
$908$	−17.3073	−0.574362
$909$	2.53179	0.0839743
$910$	0	0
$911$	−42.9419	−1.42273	−0.711364	−	0.702824i	$-0.751922\pi$
$911$	−42.9419	−1.42273	−0.711364	+	0.702824i	$0.751922\pi$
$912$	4.57832	0.151603
$913$	33.4960	1.10856
$914$	−1.09019	−0.0360603
$915$	0	0
$916$	9.90350	0.327221
$917$	33.1277	1.09397
$918$	−6.84336	−0.225864
$919$	−40.7540	−1.34435	−0.672175	−	0.740392i	$-0.734640\pi$
$919$	−40.7540	−1.34435	−0.672175	+	0.740392i	$0.734640\pi$
$920$	0	0
$921$	29.7999	0.981939
$922$	−15.5893	−0.513406
$923$	0	0
$924$	−7.34636	−0.241677
$925$	0	0
$926$	−19.1896	−0.630610
$927$	−1.48083	−0.0486367
$928$	−0.470937	−0.0154593
$929$	13.0857	0.429329	0.214664	−	0.976688i	$-0.431134\pi$
$929$	13.0857	0.429329	0.214664	+	0.976688i	$0.431134\pi$
$930$	0	0
$931$	−10.2926	−0.337325
$932$	−19.1742	−0.628070
$933$	28.6458	0.937823
$934$	23.1857	0.758658
$935$	0	0
$936$	0	0
$937$	−41.0546	−1.34120	−0.670598	−	0.741821i	$-0.733962\pi$
$937$	−41.0546	−1.34120	−0.670598	+	0.741821i	$0.733962\pi$
$938$	−8.34767	−0.272561
$939$	18.4574	0.602334
$940$	0	0
$941$	−24.4162	−0.795945	−0.397973	−	0.917397i	$-0.630286\pi$
$941$	−24.4162	−0.795945	−0.397973	+	0.917397i	$0.630286\pi$
$942$	3.21608	0.104786
$943$	−10.0409	−0.326978
$944$	12.3586	0.402240
$945$	0	0
$946$	23.2895	0.757208
$947$	−4.60866	−0.149761	−0.0748807	−	0.997192i	$-0.523858\pi$
$947$	−4.60866	−0.149761	−0.0748807	+	0.997192i	$0.523858\pi$
$948$	1.92519	0.0625272
$949$	0	0
$950$	0	0
$951$	−22.8944	−0.742400
$952$	−2.62990	−0.0852356
$953$	4.72985	0.153215	0.0766075	−	0.997061i	$-0.475591\pi$
$953$	4.72985	0.153215	0.0766075	+	0.997061i	$0.475591\pi$
$954$	−19.4157	−0.628607
$955$	0	0
$956$	−2.16448	−0.0700043
$957$	−1.65779	−0.0535887
$958$	−3.48503	−0.112596
$959$	−15.0740	−0.486765
$960$	0	0
$961$	−30.9233	−0.997525
$962$	0	0
$963$	18.9108	0.609392
$964$	16.1013	0.518589
$965$	0	0
$966$	−2.20822	−0.0710484
$967$	23.8186	0.765955	0.382977	−	0.923758i	$-0.374899\pi$
$967$	23.8186	0.765955	0.382977	+	0.923758i	$0.374899\pi$
$968$	−2.04653	−0.0657779
$969$	5.76953	0.185344
$970$	0	0
$971$	−31.6698	−1.01633	−0.508167	−	0.861259i	$-0.669677\pi$
$971$	−31.6698	−1.01633	−0.508167	+	0.861259i	$0.669677\pi$
$972$	14.4788	0.464408
$973$	−41.4754	−1.32964
$974$	2.14046	0.0685847
$975$	0	0
$976$	7.64204	0.244616
$977$	26.8707	0.859671	0.429835	−	0.902907i	$-0.358572\pi$
$977$	26.8707	0.859671	0.429835	+	0.902907i	$0.358572\pi$
$978$	28.9133	0.924546
$979$	20.6801	0.660940
$980$	0	0
$981$	25.8399	0.825005
$982$	−27.2403	−0.869273
$983$	8.22637	0.262380	0.131190	−	0.991357i	$-0.458120\pi$
$983$	8.22637	0.262380	0.131190	+	0.991357i	$0.458120\pi$
$984$	−13.1334	−0.418679
$985$	0	0
$986$	−0.593468	−0.0188999
$987$	27.7894	0.884548
$988$	0	0
$989$	7.00054	0.222604
$990$	0	0
$991$	−22.2314	−0.706204	−0.353102	−	0.935585i	$-0.614873\pi$
$991$	−22.2314	−0.706204	−0.353102	+	0.935585i	$0.614873\pi$
$992$	0.277015	0.00879525
$993$	7.79642	0.247412
$994$	−9.82056	−0.311489
$995$	0	0
$996$	13.1695	0.417290
$997$	23.5699	0.746467	0.373234	−	0.927737i	$-0.378249\pi$
$997$	23.5699	0.746467	0.373234	+	0.927737i	$0.378249\pi$
$998$	−22.7855	−0.721263
$999$	−38.2176	−1.20915

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8450.2.a.cs.1.5		8
5.2	odd	4		1690.2.b.e.339.12		16
5.3	odd	4		1690.2.b.e.339.5		16
5.4	even	2		8450.2.a.cr.1.4		8
13.6	odd	12		650.2.m.e.101.2		16
13.11	odd	12		650.2.m.e.251.2		16
13.12	even	2		8450.2.a.cr.1.5		8
65.8	even	4		1690.2.c.f.1689.5		8
65.12	odd	4		1690.2.b.e.339.4		16
65.18	even	4		1690.2.c.e.1689.5		8
65.19	odd	12		650.2.m.e.101.7		16
65.24	odd	12		650.2.m.e.251.7		16
65.32	even	12		130.2.m.a.49.2	✓	8
65.37	even	12		130.2.m.b.69.3	yes	8
65.38	odd	4		1690.2.b.e.339.13		16
65.47	even	4		1690.2.c.e.1689.4		8
65.57	even	4		1690.2.c.f.1689.4		8
65.58	even	12		130.2.m.b.49.3	yes	8
65.63	even	12		130.2.m.a.69.2	yes	8
65.64	even	2	inner	8450.2.a.cs.1.4		8
195.32	odd	12		1170.2.bj.b.829.3		8
195.128	odd	12		1170.2.bj.b.199.3		8
195.167	odd	12		1170.2.bj.a.199.2		8
195.188	odd	12		1170.2.bj.a.829.2		8
260.63	odd	12		1040.2.df.c.849.3		8
260.123	odd	12		1040.2.df.a.49.2		8
260.167	odd	12		1040.2.df.a.849.2		8
260.227	odd	12		1040.2.df.c.49.3		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
130.2.m.a.49.2	✓	8	65.32	even	12
130.2.m.a.69.2	yes	8	65.63	even	12
130.2.m.b.49.3	yes	8	65.58	even	12
130.2.m.b.69.3	yes	8	65.37	even	12
650.2.m.e.101.2		16	13.6	odd	12
650.2.m.e.101.7		16	65.19	odd	12
650.2.m.e.251.2		16	13.11	odd	12
650.2.m.e.251.7		16	65.24	odd	12
1040.2.df.a.49.2		8	260.123	odd	12
1040.2.df.a.849.2		8	260.167	odd	12
1040.2.df.c.49.3		8	260.227	odd	12
1040.2.df.c.849.3		8	260.63	odd	12
1170.2.bj.a.199.2		8	195.167	odd	12
1170.2.bj.a.829.2		8	195.188	odd	12
1170.2.bj.b.199.3		8	195.128	odd	12
1170.2.bj.b.829.3		8	195.32	odd	12
1690.2.b.e.339.4		16	65.12	odd	4
1690.2.b.e.339.5		16	5.3	odd	4
1690.2.b.e.339.12		16	5.2	odd	4
1690.2.b.e.339.13		16	65.38	odd	4
1690.2.c.e.1689.4		8	65.47	even	4
1690.2.c.e.1689.5		8	65.18	even	4
1690.2.c.f.1689.4		8	65.57	even	4
1690.2.c.f.1689.5		8	65.8	even	4
8450.2.a.cr.1.4		8	5.4	even	2
8450.2.a.cr.1.5		8	13.12	even	2
8450.2.a.cs.1.4		8	65.64	even	2	inner
8450.2.a.cs.1.5		8	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 8450 = 2 \cdot 5^{2} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8450.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.17644 q^{3} +1.00000 q^{4} +1.17644 q^{6} -2.08692 q^{7} +1.00000 q^{8} -1.61598 q^{9} +2.99224 q^{11} +1.17644 q^{12} -2.08692 q^{14} +1.00000 q^{16} +1.26018 q^{17} -1.61598 q^{18} +3.89166 q^{19} -2.45514 q^{21} +2.99224 q^{22} +0.899428 q^{23} +1.17644 q^{24} -5.43044 q^{27} -2.08692 q^{28} -0.470937 q^{29} +0.277015 q^{31} +1.00000 q^{32} +3.52020 q^{33} +1.26018 q^{34} -1.61598 q^{36} +7.03766 q^{37} +3.89166 q^{38} -11.1637 q^{41} -2.45514 q^{42} +7.78333 q^{43} +2.99224 q^{44} +0.899428 q^{46} -11.3189 q^{47} +1.17644 q^{48} -2.64477 q^{49} +1.48254 q^{51} +12.0148 q^{53} -5.43044 q^{54} -2.08692 q^{56} +4.57832 q^{57} -0.470937 q^{58} +12.3586 q^{59} +7.64204 q^{61} +0.277015 q^{62} +3.37242 q^{63} +1.00000 q^{64} +3.52020 q^{66} +4.00000 q^{67} +1.26018 q^{68} +1.05813 q^{69} +4.70577 q^{71} -1.61598 q^{72} +15.2984 q^{73} +7.03766 q^{74} +3.89166 q^{76} -6.24455 q^{77} +1.63645 q^{79} -1.54066 q^{81} -11.1637 q^{82} +11.1943 q^{83} -2.45514 q^{84} +7.78333 q^{86} -0.554031 q^{87} +2.99224 q^{88} +6.91127 q^{89} +0.899428 q^{92} +0.325893 q^{93} -11.3189 q^{94} +1.17644 q^{96} +2.57832 q^{97} -2.64477 q^{98} -4.83540 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{2} + 8 q^{4} + 10 q^{7} + 8 q^{8} + 16 q^{9} + 10 q^{14} + 8 q^{16} + 16 q^{18} + 10 q^{28} - 6 q^{29} + 8 q^{32} + 20 q^{33} + 16 q^{36} + 40 q^{37} - 6 q^{47} + 30 q^{49} + 20 q^{51} + 10 q^{56}+ \cdots + 30 q^{98}+O(q^{100}) \) 8 * q + 8 * q^2 + 8 * q^4 + 10 * q^7 + 8 * q^8 + 16 * q^9 + 10 * q^14 + 8 * q^16 + 16 * q^18 + 10 * q^28 - 6 * q^29 + 8 * q^32 + 20 * q^33 + 16 * q^36 + 40 * q^37 - 6 * q^47 + 30 * q^49 + 20 * q^51 + 10 * q^56 + 24 * q^57 - 6 * q^58 + 10 * q^61 + 50 * q^63 + 8 * q^64 + 20 * q^66 + 32 * q^67 + 4 * q^69 + 16 * q^72 + 26 * q^73 + 40 * q^74 - 4 * q^79 - 16 * q^81 + 48 * q^83 + 36 * q^93 - 6 * q^94 + 8 * q^97 + 30 * q^98

Introduction

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