LMFDB - Embedded newform 8450.2.a.bc.1.2

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)

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")

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);

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 = [2,-2,0,2,0,0,2,-2,14,0,-2,0,0,-2,0,2,4] 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:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{10}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 10 $ x^2 - 10
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 130)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$3.16228$ 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)

$f(q)$	$=$	\(q-1.00000 q^{2} +3.16228 q^{3} +1.00000 q^{4} -3.16228 q^{6} +1.00000 q^{7} -1.00000 q^{8} +7.00000 q^{9} -4.16228 q^{11} +3.16228 q^{12} -1.00000 q^{14} +1.00000 q^{16} -1.16228 q^{17} -7.00000 q^{18} -8.16228 q^{19} +3.16228 q^{21} +4.16228 q^{22} -6.00000 q^{23} -3.16228 q^{24} +12.6491 q^{27} +1.00000 q^{28} -2.32456 q^{29} +0.837722 q^{31} -1.00000 q^{32} -13.1623 q^{33} +1.16228 q^{34} +7.00000 q^{36} -6.16228 q^{37} +8.16228 q^{38} +2.32456 q^{41} -3.16228 q^{42} -2.00000 q^{43} -4.16228 q^{44} +6.00000 q^{46} +3.00000 q^{47} +3.16228 q^{48} -6.00000 q^{49} -3.67544 q^{51} -4.16228 q^{53} -12.6491 q^{54} -1.00000 q^{56} -25.8114 q^{57} +2.32456 q^{58} +2.32456 q^{59} +11.4868 q^{61} -0.837722 q^{62} +7.00000 q^{63} +1.00000 q^{64} +13.1623 q^{66} -10.3246 q^{67} -1.16228 q^{68} -18.9737 q^{69} +8.32456 q^{71} -7.00000 q^{72} -9.16228 q^{73} +6.16228 q^{74} -8.16228 q^{76} -4.16228 q^{77} +5.48683 q^{79} +19.0000 q^{81} -2.32456 q^{82} +9.48683 q^{83} +3.16228 q^{84} +2.00000 q^{86} -7.35089 q^{87} +4.16228 q^{88} +5.32456 q^{89} -6.00000 q^{92} +2.64911 q^{93} -3.00000 q^{94} -3.16228 q^{96} -11.4868 q^{97} +6.00000 q^{98} -29.1359 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} + 2 q^{4} + 2 q^{7} - 2 q^{8} + 14 q^{9} - 2 q^{11} - 2 q^{14} + 2 q^{16} + 4 q^{17} - 14 q^{18} - 10 q^{19} + 2 q^{22} - 12 q^{23} + 2 q^{28} + 8 q^{29} + 8 q^{31} - 2 q^{32} - 20 q^{33}+ \cdots - 14 q^{99}+O(q^{100}) $ 2 * q - 2 * q^2 + 2 * q^4 + 2 * q^7 - 2 * q^8 + 14 * q^9 - 2 * q^11 - 2 * q^14 + 2 * q^16 + 4 * q^17 - 14 * q^18 - 10 * q^19 + 2 * q^22 - 12 * q^23 + 2 * q^28 + 8 * q^29 + 8 * q^31 - 2 * q^32 - 20 * q^33 - 4 * q^34 + 14 * q^36 - 6 * q^37 + 10 * q^38 - 8 * q^41 - 4 * q^43 - 2 * q^44 + 12 * q^46 + 6 * q^47 - 12 * q^49 - 20 * q^51 - 2 * q^53 - 2 * q^56 - 20 * q^57 - 8 * q^58 - 8 * q^59 + 4 * q^61 - 8 * q^62 + 14 * q^63 + 2 * q^64 + 20 * q^66 - 8 * q^67 + 4 * q^68 + 4 * q^71 - 14 * q^72 - 12 * q^73 + 6 * q^74 - 10 * q^76 - 2 * q^77 - 8 * q^79 + 38 * q^81 + 8 * q^82 + 4 * q^86 - 40 * q^87 + 2 * q^88 - 2 * q^89 - 12 * q^92 - 20 * q^93 - 6 * q^94 - 4 * q^97 + 12 * q^98 - 14 * q^99

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
$3$	3.16228	1.82574	0.912871	−	0.408248i	$-0.133860\pi$
$3$	3.16228	1.82574	0.912871	+	0.408248i	$0.133860\pi$
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	−3.16228	−1.29099
$7$	1.00000	0.377964	0.188982	−	0.981981i	$-0.439481\pi$
$7$	1.00000	0.377964	0.188982	+	0.981981i	$0.439481\pi$
$8$	−1.00000	−0.353553
$9$	7.00000	2.33333
$10$	0	0
$11$	−4.16228	−1.25497	−0.627487	−	0.778627i	$-0.715916\pi$
$11$	−4.16228	−1.25497	−0.627487	+	0.778627i	$0.715916\pi$
$12$	3.16228	0.912871
$13$	0	0
$13$	0	0
$14$	−1.00000	−0.267261
$15$	0	0
$16$	1.00000	0.250000
$17$	−1.16228	−0.281894	−0.140947	−	0.990017i	$-0.545015\pi$
$17$	−1.16228	−0.281894	−0.140947	+	0.990017i	$0.545015\pi$
$18$	−7.00000	−1.64992
$19$	−8.16228	−1.87255	−0.936277	−	0.351261i	$-0.885753\pi$
$19$	−8.16228	−1.87255	−0.936277	+	0.351261i	$0.885753\pi$
$20$	0	0
$21$	3.16228	0.690066
$22$	4.16228	0.887401
$23$	−6.00000	−1.25109	−0.625543	−	0.780189i	$-0.715123\pi$
$23$	−6.00000	−1.25109	−0.625543	+	0.780189i	$0.715123\pi$
$24$	−3.16228	−0.645497
$25$	0	0
$26$	0	0
$27$	12.6491	2.43432
$28$	1.00000	0.188982
$29$	−2.32456	−0.431659	−0.215830	−	0.976431i	$-0.569246\pi$
$29$	−2.32456	−0.431659	−0.215830	+	0.976431i	$0.569246\pi$
$30$	0	0
$31$	0.837722	0.150459	0.0752297	−	0.997166i	$-0.476031\pi$
$31$	0.837722	0.150459	0.0752297	+	0.997166i	$0.476031\pi$
$32$	−1.00000	−0.176777
$33$	−13.1623	−2.29126
$34$	1.16228	0.199329
$35$	0	0
$36$	7.00000	1.16667
$37$	−6.16228	−1.01307	−0.506536	−	0.862219i	$-0.669074\pi$
$37$	−6.16228	−1.01307	−0.506536	+	0.862219i	$0.669074\pi$
$38$	8.16228	1.32410
$39$	0	0
$40$	0	0
$41$	2.32456	0.363035	0.181517	−	0.983388i	$-0.441899\pi$
$41$	2.32456	0.363035	0.181517	+	0.983388i	$0.441899\pi$
$42$	−3.16228	−0.487950
$43$	−2.00000	−0.304997	−0.152499	−	0.988304i	$-0.548732\pi$
$43$	−2.00000	−0.304997	−0.152499	+	0.988304i	$0.548732\pi$
$44$	−4.16228	−0.627487
$45$	0	0
$46$	6.00000	0.884652
$47$	3.00000	0.437595	0.218797	−	0.975770i	$-0.429787\pi$
$47$	3.00000	0.437595	0.218797	+	0.975770i	$0.429787\pi$
$48$	3.16228	0.456435
$49$	−6.00000	−0.857143
$50$	0	0
$51$	−3.67544	−0.514665
$52$	0	0
$53$	−4.16228	−0.571733	−0.285866	−	0.958269i	$-0.592281\pi$
$53$	−4.16228	−0.571733	−0.285866	+	0.958269i	$0.592281\pi$
$54$	−12.6491	−1.72133
$55$	0	0
$56$	−1.00000	−0.133631
$57$	−25.8114	−3.41880
$58$	2.32456	0.305229
$59$	2.32456	0.302631	0.151316	−	0.988485i	$-0.451649\pi$
$59$	2.32456	0.302631	0.151316	+	0.988485i	$0.451649\pi$
$60$	0	0
$61$	11.4868	1.47074	0.735369	−	0.677667i	$-0.237009\pi$
$61$	11.4868	1.47074	0.735369	+	0.677667i	$0.237009\pi$
$62$	−0.837722	−0.106391
$63$	7.00000	0.881917
$64$	1.00000	0.125000
$65$	0	0
$66$	13.1623	1.62016
$67$	−10.3246	−1.26135	−0.630673	−	0.776049i	$-0.717221\pi$
$67$	−10.3246	−1.26135	−0.630673	+	0.776049i	$0.717221\pi$
$68$	−1.16228	−0.140947
$69$	−18.9737	−2.28416
$70$	0	0
$71$	8.32456	0.987943	0.493971	−	0.869478i	$-0.335545\pi$
$71$	8.32456	0.987943	0.493971	+	0.869478i	$0.335545\pi$
$72$	−7.00000	−0.824958
$73$	−9.16228	−1.07236	−0.536182	−	0.844103i	$-0.680134\pi$
$73$	−9.16228	−1.07236	−0.536182	+	0.844103i	$0.680134\pi$
$74$	6.16228	0.716350
$75$	0	0
$76$	−8.16228	−0.936277
$77$	−4.16228	−0.474336
$78$	0	0
$79$	5.48683	0.617317	0.308658	−	0.951173i	$-0.400120\pi$
$79$	5.48683	0.617317	0.308658	+	0.951173i	$0.400120\pi$
$80$	0	0
$81$	19.0000	2.11111
$82$	−2.32456	−0.256704
$83$	9.48683	1.04132	0.520658	−	0.853766i	$-0.325687\pi$
$83$	9.48683	1.04132	0.520658	+	0.853766i	$0.325687\pi$
$84$	3.16228	0.345033
$85$	0	0
$86$	2.00000	0.215666
$87$	−7.35089	−0.788098
$88$	4.16228	0.443700
$89$	5.32456	0.564402	0.282201	−	0.959355i	$-0.408936\pi$
$89$	5.32456	0.564402	0.282201	+	0.959355i	$0.408936\pi$
$90$	0	0
$91$	0	0
$92$	−6.00000	−0.625543
$93$	2.64911	0.274700
$94$	−3.00000	−0.309426
$95$	0	0
$96$	−3.16228	−0.322749
$97$	−11.4868	−1.16631	−0.583156	−	0.812360i	$-0.698182\pi$
$97$	−11.4868	−1.16631	−0.583156	+	0.812360i	$0.698182\pi$
$98$	6.00000	0.606092
$99$	−29.1359	−2.92827
$100$	0	0
$101$	−4.83772	−0.481371	−0.240686	−	0.970603i	$-0.577372\pi$
$101$	−4.83772	−0.481371	−0.240686	+	0.970603i	$0.577372\pi$
$102$	3.67544	0.363923
$103$	−13.3246	−1.31291	−0.656454	−	0.754366i	$-0.727944\pi$
$103$	−13.3246	−1.31291	−0.656454	+	0.754366i	$0.727944\pi$
$104$	0	0
$105$	0	0
$106$	4.16228	0.404276
$107$	−15.4868	−1.49717	−0.748584	−	0.663040i	$-0.769266\pi$
$107$	−15.4868	−1.49717	−0.748584	+	0.663040i	$0.769266\pi$
$108$	12.6491	1.21716
$109$	−14.6491	−1.40313	−0.701565	−	0.712605i	$-0.747515\pi$
$109$	−14.6491	−1.40313	−0.701565	+	0.712605i	$0.747515\pi$
$110$	0	0
$111$	−19.4868	−1.84961
$112$	1.00000	0.0944911
$113$	−16.6491	−1.56622	−0.783108	−	0.621885i	$-0.786367\pi$
$113$	−16.6491	−1.56622	−0.783108	+	0.621885i	$0.786367\pi$
$114$	25.8114	2.41746
$115$	0	0
$116$	−2.32456	−0.215830
$117$	0	0
$118$	−2.32456	−0.213993
$119$	−1.16228	−0.106546
$120$	0	0
$121$	6.32456	0.574960
$122$	−11.4868	−1.03997
$123$	7.35089	0.662807
$124$	0.837722	0.0752297
$125$	0	0
$126$	−7.00000	−0.623610
$127$	9.32456	0.827420	0.413710	−	0.910409i	$-0.364233\pi$
$127$	9.32456	0.827420	0.413710	+	0.910409i	$0.364233\pi$
$128$	−1.00000	−0.0883883
$129$	−6.32456	−0.556846
$130$	0	0
$131$	20.8114	1.81830	0.909150	−	0.416469i	$-0.136733\pi$
$131$	20.8114	1.81830	0.909150	+	0.416469i	$0.136733\pi$
$132$	−13.1623	−1.14563
$133$	−8.16228	−0.707759
$134$	10.3246	0.891906
$135$	0	0
$136$	1.16228	0.0996645
$137$	3.48683	0.297900	0.148950	−	0.988845i	$-0.452411\pi$
$137$	3.48683	0.297900	0.148950	+	0.988845i	$0.452411\pi$
$138$	18.9737	1.61515
$139$	−5.83772	−0.495149	−0.247575	−	0.968869i	$-0.579634\pi$
$139$	−5.83772	−0.495149	−0.247575	+	0.968869i	$0.579634\pi$
$140$	0	0
$141$	9.48683	0.798935
$142$	−8.32456	−0.698581
$143$	0	0
$144$	7.00000	0.583333
$145$	0	0
$146$	9.16228	0.758275
$147$	−18.9737	−1.56492
$148$	−6.16228	−0.506536
$149$	3.67544	0.301104	0.150552	−	0.988602i	$-0.451895\pi$
$149$	3.67544	0.301104	0.150552	+	0.988602i	$0.451895\pi$
$150$	0	0
$151$	−11.1623	−0.908373	−0.454187	−	0.890907i	$-0.650070\pi$
$151$	−11.1623	−0.908373	−0.454187	+	0.890907i	$0.650070\pi$
$152$	8.16228	0.662048
$153$	−8.13594	−0.657752
$154$	4.16228	0.335406
$155$	0	0
$156$	0	0
$157$	5.83772	0.465901	0.232950	−	0.972489i	$-0.425162\pi$
$157$	5.83772	0.465901	0.232950	+	0.972489i	$0.425162\pi$
$158$	−5.48683	−0.436509
$159$	−13.1623	−1.04384
$160$	0	0
$161$	−6.00000	−0.472866
$162$	−19.0000	−1.49278
$163$	1.48683	0.116458	0.0582289	−	0.998303i	$-0.481455\pi$
$163$	1.48683	0.116458	0.0582289	+	0.998303i	$0.481455\pi$
$164$	2.32456	0.181517
$165$	0	0
$166$	−9.48683	−0.736321
$167$	−0.675445	−0.0522675	−0.0261337	−	0.999658i	$-0.508320\pi$
$167$	−0.675445	−0.0522675	−0.0261337	+	0.999658i	$0.508320\pi$
$168$	−3.16228	−0.243975
$169$	0	0
$170$	0	0
$171$	−57.1359	−4.36929
$172$	−2.00000	−0.152499
$173$	4.16228	0.316452	0.158226	−	0.987403i	$-0.449423\pi$
$173$	4.16228	0.316452	0.158226	+	0.987403i	$0.449423\pi$
$174$	7.35089	0.557269
$175$	0	0
$176$	−4.16228	−0.313743
$177$	7.35089	0.552527
$178$	−5.32456	−0.399092
$179$	9.67544	0.723177	0.361588	−	0.932338i	$-0.382235\pi$
$179$	9.67544	0.723177	0.361588	+	0.932338i	$0.382235\pi$
$180$	0	0
$181$	−21.8114	−1.62123	−0.810614	−	0.585581i	$-0.800866\pi$
$181$	−21.8114	−1.62123	−0.810614	+	0.585581i	$0.800866\pi$
$182$	0	0
$183$	36.3246	2.68519
$184$	6.00000	0.442326
$185$	0	0
$186$	−2.64911	−0.194242
$187$	4.83772	0.353769
$188$	3.00000	0.218797
$189$	12.6491	0.920087
$190$	0	0
$191$	−9.48683	−0.686443	−0.343222	−	0.939254i	$-0.611518\pi$
$191$	−9.48683	−0.686443	−0.343222	+	0.939254i	$0.611518\pi$
$192$	3.16228	0.228218
$193$	4.00000	0.287926	0.143963	−	0.989583i	$-0.454015\pi$
$193$	4.00000	0.287926	0.143963	+	0.989583i	$0.454015\pi$
$194$	11.4868	0.824707
$195$	0	0
$196$	−6.00000	−0.428571
$197$	−0.486833	−0.0346854	−0.0173427	−	0.999850i	$-0.505521\pi$
$197$	−0.486833	−0.0346854	−0.0173427	+	0.999850i	$0.505521\pi$
$198$	29.1359	2.07060
$199$	−26.6491	−1.88911	−0.944553	−	0.328360i	$-0.893504\pi$
$199$	−26.6491	−1.88911	−0.944553	+	0.328360i	$0.893504\pi$
$200$	0	0
$201$	−32.6491	−2.30289
$202$	4.83772	0.340381
$203$	−2.32456	−0.163152
$204$	−3.67544	−0.257333
$205$	0	0
$206$	13.3246	0.928366
$207$	−42.0000	−2.91920
$208$	0	0
$209$	33.9737	2.35001
$210$	0	0
$211$	18.1623	1.25034	0.625171	−	0.780488i	$-0.285029\pi$
$211$	18.1623	1.25034	0.625171	+	0.780488i	$0.285029\pi$
$212$	−4.16228	−0.285866
$213$	26.3246	1.80373
$214$	15.4868	1.05866
$215$	0	0
$216$	−12.6491	−0.860663
$217$	0.837722	0.0568683
$218$	14.6491	0.992163
$219$	−28.9737	−1.95786
$220$	0	0
$221$	0	0
$222$	19.4868	1.30787
$223$	9.32456	0.624418	0.312209	−	0.950013i	$-0.398931\pi$
$223$	9.32456	0.624418	0.312209	+	0.950013i	$0.398931\pi$
$224$	−1.00000	−0.0668153
$225$	0	0
$226$	16.6491	1.10748
$227$	2.32456	0.154286	0.0771431	−	0.997020i	$-0.475420\pi$
$227$	2.32456	0.154286	0.0771431	+	0.997020i	$0.475420\pi$
$228$	−25.8114	−1.70940
$229$	9.16228	0.605460	0.302730	−	0.953076i	$-0.402102\pi$
$229$	9.16228	0.605460	0.302730	+	0.953076i	$0.402102\pi$
$230$	0	0
$231$	−13.1623	−0.866014
$232$	2.32456	0.152615
$233$	9.48683	0.621503	0.310752	−	0.950491i	$-0.399419\pi$
$233$	9.48683	0.621503	0.310752	+	0.950491i	$0.399419\pi$
$234$	0	0
$235$	0	0
$236$	2.32456	0.151316
$237$	17.3509	1.12706
$238$	1.16228	0.0753393
$239$	4.83772	0.312926	0.156463	−	0.987684i	$-0.449991\pi$
$239$	4.83772	0.312926	0.156463	+	0.987684i	$0.449991\pi$
$240$	0	0
$241$	11.9737	0.771292	0.385646	−	0.922647i	$-0.373979\pi$
$241$	11.9737	0.771292	0.385646	+	0.922647i	$0.373979\pi$
$242$	−6.32456	−0.406558
$243$	22.1359	1.42002
$244$	11.4868	0.735369
$245$	0	0
$246$	−7.35089	−0.468676
$247$	0	0
$248$	−0.837722	−0.0531954
$249$	30.0000	1.90117
$250$	0	0
$251$	−6.48683	−0.409445	−0.204723	−	0.978820i	$-0.565629\pi$
$251$	−6.48683	−0.409445	−0.204723	+	0.978820i	$0.565629\pi$
$252$	7.00000	0.440959
$253$	24.9737	1.57008
$254$	−9.32456	−0.585075
$255$	0	0
$256$	1.00000	0.0625000
$257$	0	0		−	1.00000i	$-0.5\pi$
$257$	0	0			1.00000i	$0.5\pi$
$258$	6.32456	0.393750
$259$	−6.16228	−0.382905
$260$	0	0
$261$	−16.2719	−1.00720
$262$	−20.8114	−1.28573
$263$	−13.6491	−0.841640	−0.420820	−	0.907144i	$-0.638258\pi$
$263$	−13.6491	−0.841640	−0.420820	+	0.907144i	$0.638258\pi$
$264$	13.1623	0.810082
$265$	0	0
$266$	8.16228	0.500461
$267$	16.8377	1.03045
$268$	−10.3246	−0.630673
$269$	21.4868	1.31008	0.655038	−	0.755596i	$-0.272653\pi$
$269$	21.4868	1.31008	0.655038	+	0.755596i	$0.272653\pi$
$270$	0	0
$271$	−6.32456	−0.384189	−0.192095	−	0.981376i	$-0.561528\pi$
$271$	−6.32456	−0.384189	−0.192095	+	0.981376i	$0.561528\pi$
$272$	−1.16228	−0.0704734
$273$	0	0
$274$	−3.48683	−0.210647
$275$	0	0
$276$	−18.9737	−1.14208
$277$	−20.4868	−1.23093	−0.615467	−	0.788162i	$-0.711033\pi$
$277$	−20.4868	−1.23093	−0.615467	+	0.788162i	$0.711033\pi$
$278$	5.83772	0.350123
$279$	5.86406	0.351072
$280$	0	0
$281$	18.9737	1.13187	0.565937	−	0.824448i	$-0.308515\pi$
$281$	18.9737	1.13187	0.565937	+	0.824448i	$0.308515\pi$
$282$	−9.48683	−0.564933
$283$	22.0000	1.30776	0.653882	−	0.756596i	$-0.273139\pi$
$283$	22.0000	1.30776	0.653882	+	0.756596i	$0.273139\pi$
$284$	8.32456	0.493971
$285$	0	0
$286$	0	0
$287$	2.32456	0.137214
$288$	−7.00000	−0.412479
$289$	−15.6491	−0.920536
$290$	0	0
$291$	−36.3246	−2.12938
$292$	−9.16228	−0.536182
$293$	28.1623	1.64526	0.822629	−	0.568579i	$-0.192506\pi$
$293$	28.1623	1.64526	0.822629	+	0.568579i	$0.192506\pi$
$294$	18.9737	1.10657
$295$	0	0
$296$	6.16228	0.358175
$297$	−52.6491	−3.05501
$298$	−3.67544	−0.212913
$299$	0	0
$300$	0	0
$301$	−2.00000	−0.115278
$302$	11.1623	0.642317
$303$	−15.2982	−0.878860
$304$	−8.16228	−0.468139
$305$	0	0
$306$	8.13594	0.465101
$307$	−28.3246	−1.61657	−0.808284	−	0.588793i	$-0.799603\pi$
$307$	−28.3246	−1.61657	−0.808284	+	0.588793i	$0.799603\pi$
$308$	−4.16228	−0.237168
$309$	−42.1359	−2.39703
$310$	0	0
$311$	15.4868	0.878178	0.439089	−	0.898444i	$-0.355301\pi$
$311$	15.4868	0.878178	0.439089	+	0.898444i	$0.355301\pi$
$312$	0	0
$313$	−4.32456	−0.244438	−0.122219	−	0.992503i	$-0.539001\pi$
$313$	−4.32456	−0.244438	−0.122219	+	0.992503i	$0.539001\pi$
$314$	−5.83772	−0.329442
$315$	0	0
$316$	5.48683	0.308658
$317$	6.48683	0.364337	0.182168	−	0.983267i	$-0.441688\pi$
$317$	6.48683	0.364337	0.182168	+	0.983267i	$0.441688\pi$
$318$	13.1623	0.738104
$319$	9.67544	0.541721
$320$	0	0
$321$	−48.9737	−2.73344
$322$	6.00000	0.334367
$323$	9.48683	0.527862
$324$	19.0000	1.05556
$325$	0	0
$326$	−1.48683	−0.0823481
$327$	−46.3246	−2.56175
$328$	−2.32456	−0.128352
$329$	3.00000	0.165395
$330$	0	0
$331$	0.649111	0.0356783	0.0178392	−	0.999841i	$-0.494321\pi$
$331$	0.649111	0.0356783	0.0178392	+	0.999841i	$0.494321\pi$
$332$	9.48683	0.520658
$333$	−43.1359	−2.36384
$334$	0.675445	0.0369587
$335$	0	0
$336$	3.16228	0.172516
$337$	1.48683	0.0809930	0.0404965	−	0.999180i	$-0.487106\pi$
$337$	1.48683	0.0809930	0.0404965	+	0.999180i	$0.487106\pi$
$338$	0	0
$339$	−52.6491	−2.85951
$340$	0	0
$341$	−3.48683	−0.188823
$342$	57.1359	3.08956
$343$	−13.0000	−0.701934
$344$	2.00000	0.107833
$345$	0	0
$346$	−4.16228	−0.223765
$347$	−21.4868	−1.15347	−0.576737	−	0.816930i	$-0.695674\pi$
$347$	−21.4868	−1.15347	−0.576737	+	0.816930i	$0.695674\pi$
$348$	−7.35089	−0.394049
$349$	−1.48683	−0.0795883	−0.0397942	−	0.999208i	$-0.512670\pi$
$349$	−1.48683	−0.0795883	−0.0397942	+	0.999208i	$0.512670\pi$
$350$	0	0
$351$	0	0
$352$	4.16228	0.221850
$353$	1.16228	0.0618618	0.0309309	−	0.999522i	$-0.490153\pi$
$353$	1.16228	0.0618618	0.0309309	+	0.999522i	$0.490153\pi$
$354$	−7.35089	−0.390695
$355$	0	0
$356$	5.32456	0.282201
$357$	−3.67544	−0.194525
$358$	−9.67544	−0.511363
$359$	−6.00000	−0.316668	−0.158334	−	0.987386i	$-0.550612\pi$
$359$	−6.00000	−0.316668	−0.158334	+	0.987386i	$0.550612\pi$
$360$	0	0
$361$	47.6228	2.50646
$362$	21.8114	1.14638
$363$	20.0000	1.04973
$364$	0	0
$365$	0	0
$366$	−36.3246	−1.89871
$367$	20.6491	1.07787	0.538937	−	0.842346i	$-0.318826\pi$
$367$	20.6491	1.07787	0.538937	+	0.842346i	$0.318826\pi$
$368$	−6.00000	−0.312772
$369$	16.2719	0.847081
$370$	0	0
$371$	−4.16228	−0.216095
$372$	2.64911	0.137350
$373$	5.35089	0.277059	0.138529	−	0.990358i	$-0.455762\pi$
$373$	5.35089	0.277059	0.138529	+	0.990358i	$0.455762\pi$
$374$	−4.83772	−0.250153
$375$	0	0
$376$	−3.00000	−0.154713
$377$	0	0
$378$	−12.6491	−0.650600
$379$	−7.18861	−0.369254	−0.184627	−	0.982809i	$-0.559108\pi$
$379$	−7.18861	−0.369254	−0.184627	+	0.982809i	$0.559108\pi$
$380$	0	0
$381$	29.4868	1.51066
$382$	9.48683	0.485389
$383$	−30.0000	−1.53293	−0.766464	−	0.642287i	$-0.777986\pi$
$383$	−30.0000	−1.53293	−0.766464	+	0.642287i	$0.777986\pi$
$384$	−3.16228	−0.161374
$385$	0	0
$386$	−4.00000	−0.203595
$387$	−14.0000	−0.711660
$388$	−11.4868	−0.583156
$389$	19.1623	0.971566	0.485783	−	0.874079i	$-0.338535\pi$
$389$	19.1623	0.971566	0.485783	+	0.874079i	$0.338535\pi$
$390$	0	0
$391$	6.97367	0.352673
$392$	6.00000	0.303046
$393$	65.8114	3.31975
$394$	0.486833	0.0245263
$395$	0	0
$396$	−29.1359	−1.46414
$397$	−19.5132	−0.979338	−0.489669	−	0.871908i	$-0.662882\pi$
$397$	−19.5132	−0.979338	−0.489669	+	0.871908i	$0.662882\pi$
$398$	26.6491	1.33580
$399$	−25.8114	−1.29219
$400$	0	0
$401$	−15.9737	−0.797687	−0.398843	−	0.917019i	$-0.630588\pi$
$401$	−15.9737	−0.797687	−0.398843	+	0.917019i	$0.630588\pi$
$402$	32.6491	1.62839
$403$	0	0
$404$	−4.83772	−0.240686
$405$	0	0
$406$	2.32456	0.115366
$407$	25.6491	1.27138
$408$	3.67544	0.181962
$409$	3.64911	0.180437	0.0902185	−	0.995922i	$-0.471243\pi$
$409$	3.64911	0.180437	0.0902185	+	0.995922i	$0.471243\pi$
$410$	0	0
$411$	11.0263	0.543889
$412$	−13.3246	−0.656454
$413$	2.32456	0.114384
$414$	42.0000	2.06419
$415$	0	0
$416$	0	0
$417$	−18.4605	−0.904015
$418$	−33.9737	−1.66171
$419$	−10.6491	−0.520243	−0.260122	−	0.965576i	$-0.583763\pi$
$419$	−10.6491	−0.520243	−0.260122	+	0.965576i	$0.583763\pi$
$420$	0	0
$421$	3.16228	0.154120	0.0770600	−	0.997026i	$-0.475447\pi$
$421$	3.16228	0.154120	0.0770600	+	0.997026i	$0.475447\pi$
$422$	−18.1623	−0.884126
$423$	21.0000	1.02105
$424$	4.16228	0.202138
$425$	0	0
$426$	−26.3246	−1.27543
$427$	11.4868	0.555887
$428$	−15.4868	−0.748584
$429$	0	0
$430$	0	0
$431$	−3.48683	−0.167955	−0.0839774	−	0.996468i	$-0.526762\pi$
$431$	−3.48683	−0.167955	−0.0839774	+	0.996468i	$0.526762\pi$
$432$	12.6491	0.608581
$433$	6.32456	0.303939	0.151969	−	0.988385i	$-0.451438\pi$
$433$	6.32456	0.303939	0.151969	+	0.988385i	$0.451438\pi$
$434$	−0.837722	−0.0402120
$435$	0	0
$436$	−14.6491	−0.701565
$437$	48.9737	2.34273
$438$	28.9737	1.38442
$439$	−27.8114	−1.32737	−0.663683	−	0.748014i	$-0.731007\pi$
$439$	−27.8114	−1.32737	−0.663683	+	0.748014i	$0.731007\pi$
$440$	0	0
$441$	−42.0000	−2.00000
$442$	0	0
$443$	−1.35089	−0.0641827	−0.0320913	−	0.999485i	$-0.510217\pi$
$443$	−1.35089	−0.0641827	−0.0320913	+	0.999485i	$0.510217\pi$
$444$	−19.4868	−0.924804
$445$	0	0
$446$	−9.32456	−0.441530
$447$	11.6228	0.549738
$448$	1.00000	0.0472456
$449$	−39.9737	−1.88647	−0.943237	−	0.332120i	$-0.892236\pi$
$449$	−39.9737	−1.88647	−0.943237	+	0.332120i	$0.892236\pi$
$450$	0	0
$451$	−9.67544	−0.455599
$452$	−16.6491	−0.783108
$453$	−35.2982	−1.65846
$454$	−2.32456	−0.109097
$455$	0	0
$456$	25.8114	1.20873
$457$	32.4605	1.51844	0.759219	−	0.650835i	$-0.225581\pi$
$457$	32.4605	1.51844	0.759219	+	0.650835i	$0.225581\pi$
$458$	−9.16228	−0.428125
$459$	−14.7018	−0.686220
$460$	0	0
$461$	2.51317	0.117050	0.0585249	−	0.998286i	$-0.481360\pi$
$461$	2.51317	0.117050	0.0585249	+	0.998286i	$0.481360\pi$
$462$	13.1623	0.612365
$463$	0.324555	0.0150834	0.00754168	−	0.999972i	$-0.497599\pi$
$463$	0.324555	0.0150834	0.00754168	+	0.999972i	$0.497599\pi$
$464$	−2.32456	−0.107915
$465$	0	0
$466$	−9.48683	−0.439469
$467$	−7.35089	−0.340159	−0.170079	−	0.985430i	$-0.554402\pi$
$467$	−7.35089	−0.340159	−0.170079	+	0.985430i	$0.554402\pi$
$468$	0	0
$469$	−10.3246	−0.476744
$470$	0	0
$471$	18.4605	0.850615
$472$	−2.32456	−0.106996
$473$	8.32456	0.382763
$474$	−17.3509	−0.796953
$475$	0	0
$476$	−1.16228	−0.0532729
$477$	−29.1359	−1.33404
$478$	−4.83772	−0.221272
$479$	32.1359	1.46833	0.734164	−	0.678972i	$-0.237574\pi$
$479$	32.1359	1.46833	0.734164	+	0.678972i	$0.237574\pi$
$480$	0	0
$481$	0	0
$482$	−11.9737	−0.545386
$483$	−18.9737	−0.863332
$484$	6.32456	0.287480
$485$	0	0
$486$	−22.1359	−1.00411
$487$	1.00000	0.0453143	0.0226572	−	0.999743i	$-0.492787\pi$
$487$	1.00000	0.0453143	0.0226572	+	0.999743i	$0.492787\pi$
$488$	−11.4868	−0.519984
$489$	4.70178	0.212622
$490$	0	0
$491$	36.4868	1.64663	0.823314	−	0.567586i	$-0.192123\pi$
$491$	36.4868	1.64663	0.823314	+	0.567586i	$0.192123\pi$
$492$	7.35089	0.331404
$493$	2.70178	0.121682
$494$	0	0
$495$	0	0
$496$	0.837722	0.0376148
$497$	8.32456	0.373407
$498$	−30.0000	−1.34433
$499$	−22.0000	−0.984855	−0.492428	−	0.870353i	$-0.663890\pi$
$499$	−22.0000	−0.984855	−0.492428	+	0.870353i	$0.663890\pi$
$500$	0	0
$501$	−2.13594	−0.0954269
$502$	6.48683	0.289522
$503$	−36.2982	−1.61846	−0.809229	−	0.587494i	$-0.800115\pi$
$503$	−36.2982	−1.61846	−0.809229	+	0.587494i	$0.800115\pi$
$504$	−7.00000	−0.311805
$505$	0	0
$506$	−24.9737	−1.11021
$507$	0	0
$508$	9.32456	0.413710
$509$	36.9737	1.63883	0.819414	−	0.573201i	$-0.194299\pi$
$509$	36.9737	1.63883	0.819414	+	0.573201i	$0.194299\pi$
$510$	0	0
$511$	−9.16228	−0.405315
$512$	−1.00000	−0.0441942
$513$	−103.246	−4.55840
$514$	0	0
$515$	0	0
$516$	−6.32456	−0.278423
$517$	−12.4868	−0.549170
$518$	6.16228	0.270755
$519$	13.1623	0.577760
$520$	0	0
$521$	35.3246	1.54760	0.773798	−	0.633432i	$-0.218354\pi$
$521$	35.3246	1.54760	0.773798	+	0.633432i	$0.218354\pi$
$522$	16.2719	0.712201
$523$	−11.2982	−0.494037	−0.247018	−	0.969011i	$-0.579451\pi$
$523$	−11.2982	−0.494037	−0.247018	+	0.969011i	$0.579451\pi$
$524$	20.8114	0.909150
$525$	0	0
$526$	13.6491	0.595130
$527$	−0.973666	−0.0424136
$528$	−13.1623	−0.572815
$529$	13.0000	0.565217
$530$	0	0
$531$	16.2719	0.706140
$532$	−8.16228	−0.353880
$533$	0	0
$534$	−16.8377	−0.728640
$535$	0	0
$536$	10.3246	0.445953
$537$	30.5964	1.32033
$538$	−21.4868	−0.926363
$539$	24.9737	1.07569
$540$	0	0
$541$	33.1623	1.42576	0.712879	−	0.701287i	$-0.247391\pi$
$541$	33.1623	1.42576	0.712879	+	0.701287i	$0.247391\pi$
$542$	6.32456	0.271663
$543$	−68.9737	−2.95994
$544$	1.16228	0.0498322
$545$	0	0
$546$	0	0
$547$	−18.6491	−0.797378	−0.398689	−	0.917086i	$-0.630535\pi$
$547$	−18.6491	−0.797378	−0.398689	+	0.917086i	$0.630535\pi$
$548$	3.48683	0.148950
$549$	80.4078	3.43172
$550$	0	0
$551$	18.9737	0.808305
$552$	18.9737	0.807573
$553$	5.48683	0.233324
$554$	20.4868	0.870402
$555$	0	0
$556$	−5.83772	−0.247575
$557$	−22.1623	−0.939046	−0.469523	−	0.882920i	$-0.655574\pi$
$557$	−22.1623	−0.939046	−0.469523	+	0.882920i	$0.655574\pi$
$558$	−5.86406	−0.248245
$559$	0	0
$560$	0	0
$561$	15.2982	0.645891
$562$	−18.9737	−0.800356
$563$	24.0000	1.01148	0.505740	−	0.862686i	$-0.331220\pi$
$563$	24.0000	1.01148	0.505740	+	0.862686i	$0.331220\pi$
$564$	9.48683	0.399468
$565$	0	0
$566$	−22.0000	−0.924729
$567$	19.0000	0.797925
$568$	−8.32456	−0.349291
$569$	3.97367	0.166585	0.0832924	−	0.996525i	$-0.473456\pi$
$569$	3.97367	0.166585	0.0832924	+	0.996525i	$0.473456\pi$
$570$	0	0
$571$	19.1359	0.800814	0.400407	−	0.916337i	$-0.368869\pi$
$571$	19.1359	0.800814	0.400407	+	0.916337i	$0.368869\pi$
$572$	0	0
$573$	−30.0000	−1.25327
$574$	−2.32456	−0.0970251
$575$	0	0
$576$	7.00000	0.291667
$577$	−0.837722	−0.0348748	−0.0174374	−	0.999848i	$-0.505551\pi$
$577$	−0.837722	−0.0348748	−0.0174374	+	0.999848i	$0.505551\pi$
$578$	15.6491	0.650917
$579$	12.6491	0.525679
$580$	0	0
$581$	9.48683	0.393580
$582$	36.3246	1.50570
$583$	17.3246	0.717510
$584$	9.16228	0.379138
$585$	0	0
$586$	−28.1623	−1.16337
$587$	24.9737	1.03077	0.515387	−	0.856958i	$-0.327648\pi$
$587$	24.9737	1.03077	0.515387	+	0.856958i	$0.327648\pi$
$588$	−18.9737	−0.782461
$589$	−6.83772	−0.281743
$590$	0	0
$591$	−1.53950	−0.0633266
$592$	−6.16228	−0.253268
$593$	28.6491	1.17648	0.588239	−	0.808687i	$-0.299821\pi$
$593$	28.6491	1.17648	0.588239	+	0.808687i	$0.299821\pi$
$594$	52.6491	2.16022
$595$	0	0
$596$	3.67544	0.150552
$597$	−84.2719	−3.44902
$598$	0	0
$599$	−31.9473	−1.30533	−0.652666	−	0.757646i	$-0.726350\pi$
$599$	−31.9473	−1.30533	−0.652666	+	0.757646i	$0.726350\pi$
$600$	0	0
$601$	−16.2982	−0.664818	−0.332409	−	0.943135i	$-0.607861\pi$
$601$	−16.2982	−0.664818	−0.332409	+	0.943135i	$0.607861\pi$
$602$	2.00000	0.0815139
$603$	−72.2719	−2.94314
$604$	−11.1623	−0.454187
$605$	0	0
$606$	15.2982	0.621448
$607$	−5.00000	−0.202944	−0.101472	−	0.994838i	$-0.532355\pi$
$607$	−5.00000	−0.202944	−0.101472	+	0.994838i	$0.532355\pi$
$608$	8.16228	0.331024
$609$	−7.35089	−0.297873
$610$	0	0
$611$	0	0
$612$	−8.13594	−0.328876
$613$	−1.51317	−0.0611162	−0.0305581	−	0.999533i	$-0.509728\pi$
$613$	−1.51317	−0.0611162	−0.0305581	+	0.999533i	$0.509728\pi$
$614$	28.3246	1.14309
$615$	0	0
$616$	4.16228	0.167703
$617$	−43.9473	−1.76925	−0.884626	−	0.466300i	$-0.845587\pi$
$617$	−43.9473	−1.76925	−0.884626	+	0.466300i	$0.845587\pi$
$618$	42.1359	1.69496
$619$	27.4605	1.10373	0.551865	−	0.833933i	$-0.313916\pi$
$619$	27.4605	1.10373	0.551865	+	0.833933i	$0.313916\pi$
$620$	0	0
$621$	−75.8947	−3.04555
$622$	−15.4868	−0.620965
$623$	5.32456	0.213324
$624$	0	0
$625$	0	0
$626$	4.32456	0.172844
$627$	107.434	4.29051
$628$	5.83772	0.232950
$629$	7.16228	0.285579
$630$	0	0
$631$	−1.67544	−0.0666984	−0.0333492	−	0.999444i	$-0.510617\pi$
$631$	−1.67544	−0.0666984	−0.0333492	+	0.999444i	$0.510617\pi$
$632$	−5.48683	−0.218254
$633$	57.4342	2.28280
$634$	−6.48683	−0.257625
$635$	0	0
$636$	−13.1623	−0.521918
$637$	0	0
$638$	−9.67544	−0.383055
$639$	58.2719	2.30520
$640$	0	0
$641$	27.9737	1.10489	0.552447	−	0.833548i	$-0.313694\pi$
$641$	27.9737	1.10489	0.552447	+	0.833548i	$0.313694\pi$
$642$	48.9737	1.93284
$643$	44.4605	1.75335	0.876675	−	0.481082i	$-0.159756\pi$
$643$	44.4605	1.75335	0.876675	+	0.481082i	$0.159756\pi$
$644$	−6.00000	−0.236433
$645$	0	0
$646$	−9.48683	−0.373254
$647$	−21.9737	−0.863874	−0.431937	−	0.901904i	$-0.642170\pi$
$647$	−21.9737	−0.863874	−0.431937	+	0.901904i	$0.642170\pi$
$648$	−19.0000	−0.746390
$649$	−9.67544	−0.379794
$650$	0	0
$651$	2.64911	0.103827
$652$	1.48683	0.0582289
$653$	−36.4868	−1.42784	−0.713920	−	0.700227i	$-0.753082\pi$
$653$	−36.4868	−1.42784	−0.713920	+	0.700227i	$0.753082\pi$
$654$	46.3246	1.81143
$655$	0	0
$656$	2.32456	0.0907586
$657$	−64.1359	−2.50218
$658$	−3.00000	−0.116952
$659$	−27.2982	−1.06339	−0.531694	−	0.846937i	$-0.678444\pi$
$659$	−27.2982	−1.06339	−0.531694	+	0.846937i	$0.678444\pi$
$660$	0	0
$661$	−17.1623	−0.667535	−0.333768	−	0.942655i	$-0.608320\pi$
$661$	−17.1623	−0.667535	−0.333768	+	0.942655i	$0.608320\pi$
$662$	−0.649111	−0.0252284
$663$	0	0
$664$	−9.48683	−0.368161
$665$	0	0
$666$	43.1359	1.67148
$667$	13.9473	0.540043
$668$	−0.675445	−0.0261337
$669$	29.4868	1.14003
$670$	0	0
$671$	−47.8114	−1.84574
$672$	−3.16228	−0.121988
$673$	22.9737	0.885570	0.442785	−	0.896628i	$-0.353991\pi$
$673$	22.9737	0.885570	0.442785	+	0.896628i	$0.353991\pi$
$674$	−1.48683	−0.0572707
$675$	0	0
$676$	0	0
$677$	−24.9737	−0.959816	−0.479908	−	0.877319i	$-0.659330\pi$
$677$	−24.9737	−0.959816	−0.479908	+	0.877319i	$0.659330\pi$
$678$	52.6491	2.02198
$679$	−11.4868	−0.440824
$680$	0	0
$681$	7.35089	0.281687
$682$	3.48683	0.133518
$683$	−32.3246	−1.23686	−0.618432	−	0.785838i	$-0.712232\pi$
$683$	−32.3246	−1.23686	−0.618432	+	0.785838i	$0.712232\pi$
$684$	−57.1359	−2.18465
$685$	0	0
$686$	13.0000	0.496342
$687$	28.9737	1.10541
$688$	−2.00000	−0.0762493
$689$	0	0
$690$	0	0
$691$	7.13594	0.271464	0.135732	−	0.990746i	$-0.456661\pi$
$691$	7.13594	0.271464	0.135732	+	0.990746i	$0.456661\pi$
$692$	4.16228	0.158226
$693$	−29.1359	−1.10678
$694$	21.4868	0.815629
$695$	0	0
$696$	7.35089	0.278635
$697$	−2.70178	−0.102337
$698$	1.48683	0.0562775
$699$	30.0000	1.13470
$700$	0	0
$701$	−16.8377	−0.635952	−0.317976	−	0.948099i	$-0.603003\pi$
$701$	−16.8377	−0.635952	−0.317976	+	0.948099i	$0.603003\pi$
$702$	0	0
$703$	50.2982	1.89703
$704$	−4.16228	−0.156872
$705$	0	0
$706$	−1.16228	−0.0437429
$707$	−4.83772	−0.181941
$708$	7.35089	0.276263
$709$	22.5132	0.845500	0.422750	−	0.906246i	$-0.361065\pi$
$709$	22.5132	0.845500	0.422750	+	0.906246i	$0.361065\pi$
$710$	0	0
$711$	38.4078	1.44041
$712$	−5.32456	−0.199546
$713$	−5.02633	−0.188238
$714$	3.67544	0.137550
$715$	0	0
$716$	9.67544	0.361588
$717$	15.2982	0.571323
$718$	6.00000	0.223918
$719$	12.0000	0.447524	0.223762	−	0.974644i	$-0.428166\pi$
$719$	12.0000	0.447524	0.223762	+	0.974644i	$0.428166\pi$
$720$	0	0
$721$	−13.3246	−0.496232
$722$	−47.6228	−1.77234
$723$	37.8641	1.40818
$724$	−21.8114	−0.810614
$725$	0	0
$726$	−20.0000	−0.742270
$727$	33.3246	1.23594	0.617970	−	0.786202i	$-0.287955\pi$
$727$	33.3246	1.23594	0.617970	+	0.786202i	$0.287955\pi$
$728$	0	0
$729$	13.0000	0.481481
$730$	0	0
$731$	2.32456	0.0859768
$732$	36.3246	1.34259
$733$	10.4868	0.387340	0.193670	−	0.981067i	$-0.437961\pi$
$733$	10.4868	0.387340	0.193670	+	0.981067i	$0.437961\pi$
$734$	−20.6491	−0.762173
$735$	0	0
$736$	6.00000	0.221163
$737$	42.9737	1.58296
$738$	−16.2719	−0.598976
$739$	−13.1886	−0.485151	−0.242575	−	0.970133i	$-0.577992\pi$
$739$	−13.1886	−0.485151	−0.242575	+	0.970133i	$0.577992\pi$
$740$	0	0
$741$	0	0
$742$	4.16228	0.152802
$743$	18.0000	0.660356	0.330178	−	0.943919i	$-0.392891\pi$
$743$	18.0000	0.660356	0.330178	+	0.943919i	$0.392891\pi$
$744$	−2.64911	−0.0971211
$745$	0	0
$746$	−5.35089	−0.195910
$747$	66.4078	2.42974
$748$	4.83772	0.176885
$749$	−15.4868	−0.565877
$750$	0	0
$751$	2.97367	0.108511	0.0542553	−	0.998527i	$-0.482722\pi$
$751$	2.97367	0.108511	0.0542553	+	0.998527i	$0.482722\pi$
$752$	3.00000	0.109399
$753$	−20.5132	−0.747541
$754$	0	0
$755$	0	0
$756$	12.6491	0.460044
$757$	−27.8377	−1.01178	−0.505890	−	0.862598i	$-0.668836\pi$
$757$	−27.8377	−1.01178	−0.505890	+	0.862598i	$0.668836\pi$
$758$	7.18861	0.261102
$759$	78.9737	2.86656
$760$	0	0
$761$	42.2982	1.53331	0.766655	−	0.642060i	$-0.221920\pi$
$761$	42.2982	1.53331	0.766655	+	0.642060i	$0.221920\pi$
$762$	−29.4868	−1.06820
$763$	−14.6491	−0.530333
$764$	−9.48683	−0.343222
$765$	0	0
$766$	30.0000	1.08394
$767$	0	0
$768$	3.16228	0.114109
$769$	−6.32456	−0.228069	−0.114035	−	0.993477i	$-0.536377\pi$
$769$	−6.32456	−0.228069	−0.114035	+	0.993477i	$0.536377\pi$
$770$	0	0
$771$	0	0
$772$	4.00000	0.143963
$773$	21.7851	0.783554	0.391777	−	0.920060i	$-0.371860\pi$
$773$	21.7851	0.783554	0.391777	+	0.920060i	$0.371860\pi$
$774$	14.0000	0.503220
$775$	0	0
$776$	11.4868	0.412353
$777$	−19.4868	−0.699086
$778$	−19.1623	−0.687001
$779$	−18.9737	−0.679802
$780$	0	0
$781$	−34.6491	−1.23984
$782$	−6.97367	−0.249378
$783$	−29.4036	−1.05080
$784$	−6.00000	−0.214286
$785$	0	0
$786$	−65.8114	−2.34741
$787$	1.86406	0.0664464	0.0332232	−	0.999448i	$-0.489423\pi$
$787$	1.86406	0.0664464	0.0332232	+	0.999448i	$0.489423\pi$
$788$	−0.486833	−0.0173427
$789$	−43.1623	−1.53662
$790$	0	0
$791$	−16.6491	−0.591974
$792$	29.1359	1.03530
$793$	0	0
$794$	19.5132	0.692496
$795$	0	0
$796$	−26.6491	−0.944553
$797$	26.3246	0.932464	0.466232	−	0.884663i	$-0.345611\pi$
$797$	26.3246	0.932464	0.466232	+	0.884663i	$0.345611\pi$
$798$	25.8114	0.913713
$799$	−3.48683	−0.123355
$800$	0	0
$801$	37.2719	1.31694
$802$	15.9737	0.564050
$803$	38.1359	1.34579
$804$	−32.6491	−1.15145
$805$	0	0
$806$	0	0
$807$	67.9473	2.39186
$808$	4.83772	0.170190
$809$	−2.32456	−0.0817270	−0.0408635	−	0.999165i	$-0.513011\pi$
$809$	−2.32456	−0.0817270	−0.0408635	+	0.999165i	$0.513011\pi$
$810$	0	0
$811$	0.162278	0.00569834	0.00284917	−	0.999996i	$-0.499093\pi$
$811$	0.162278	0.00569834	0.00284917	+	0.999996i	$0.499093\pi$
$812$	−2.32456	−0.0815759
$813$	−20.0000	−0.701431
$814$	−25.6491	−0.899001
$815$	0	0
$816$	−3.67544	−0.128666
$817$	16.3246	0.571124
$818$	−3.64911	−0.127588
$819$	0	0
$820$	0	0
$821$	1.16228	0.0405638	0.0202819	−	0.999794i	$-0.493544\pi$
$821$	1.16228	0.0405638	0.0202819	+	0.999794i	$0.493544\pi$
$822$	−11.0263	−0.384588
$823$	−47.0000	−1.63832	−0.819159	−	0.573567i	$-0.805559\pi$
$823$	−47.0000	−1.63832	−0.819159	+	0.573567i	$0.805559\pi$
$824$	13.3246	0.464183
$825$	0	0
$826$	−2.32456	−0.0808816
$827$	32.5132	1.13059	0.565297	−	0.824888i	$-0.308762\pi$
$827$	32.5132	1.13059	0.565297	+	0.824888i	$0.308762\pi$
$828$	−42.0000	−1.45960
$829$	32.0000	1.11141	0.555703	−	0.831381i	$-0.312449\pi$
$829$	32.0000	1.11141	0.555703	+	0.831381i	$0.312449\pi$
$830$	0	0
$831$	−64.7851	−2.24737
$832$	0	0
$833$	6.97367	0.241623
$834$	18.4605	0.639235
$835$	0	0
$836$	33.9737	1.17500
$837$	10.5964	0.366267
$838$	10.6491	0.367867
$839$	−16.8377	−0.581303	−0.290651	−	0.956829i	$-0.593872\pi$
$839$	−16.8377	−0.581303	−0.290651	+	0.956829i	$0.593872\pi$
$840$	0	0
$841$	−23.5964	−0.813670
$842$	−3.16228	−0.108979
$843$	60.0000	2.06651
$844$	18.1623	0.625171
$845$	0	0
$846$	−21.0000	−0.721995
$847$	6.32456	0.217314
$848$	−4.16228	−0.142933
$849$	69.5701	2.38764
$850$	0	0
$851$	36.9737	1.26744
$852$	26.3246	0.901864
$853$	32.2719	1.10497	0.552484	−	0.833523i	$-0.313680\pi$
$853$	32.2719	1.10497	0.552484	+	0.833523i	$0.313680\pi$
$854$	−11.4868	−0.393071
$855$	0	0
$856$	15.4868	0.529329
$857$	14.1359	0.482875	0.241437	−	0.970416i	$-0.422381\pi$
$857$	14.1359	0.482875	0.241437	+	0.970416i	$0.422381\pi$
$858$	0	0
$859$	2.86406	0.0977203	0.0488602	−	0.998806i	$-0.484441\pi$
$859$	2.86406	0.0977203	0.0488602	+	0.998806i	$0.484441\pi$
$860$	0	0
$861$	7.35089	0.250518
$862$	3.48683	0.118762
$863$	18.0000	0.612727	0.306364	−	0.951915i	$-0.400888\pi$
$863$	18.0000	0.612727	0.306364	+	0.951915i	$0.400888\pi$
$864$	−12.6491	−0.430331
$865$	0	0
$866$	−6.32456	−0.214917
$867$	−49.4868	−1.68066
$868$	0.837722	0.0284341
$869$	−22.8377	−0.774717
$870$	0	0
$871$	0	0
$872$	14.6491	0.496081
$873$	−80.4078	−2.72139
$874$	−48.9737	−1.65656
$875$	0	0
$876$	−28.9737	−0.978929
$877$	36.3246	1.22659	0.613297	−	0.789853i	$-0.289843\pi$
$877$	36.3246	1.22659	0.613297	+	0.789853i	$0.289843\pi$
$878$	27.8114	0.938589
$879$	89.0569	3.00382
$880$	0	0
$881$	−20.0263	−0.674704	−0.337352	−	0.941379i	$-0.609531\pi$
$881$	−20.0263	−0.674704	−0.337352	+	0.941379i	$0.609531\pi$
$882$	42.0000	1.41421
$883$	−16.5132	−0.555712	−0.277856	−	0.960623i	$-0.589624\pi$
$883$	−16.5132	−0.555712	−0.277856	+	0.960623i	$0.589624\pi$
$884$	0	0
$885$	0	0
$886$	1.35089	0.0453840
$887$	26.6228	0.893905	0.446953	−	0.894558i	$-0.352509\pi$
$887$	26.6228	0.893905	0.446953	+	0.894558i	$0.352509\pi$
$888$	19.4868	0.653935
$889$	9.32456	0.312736
$890$	0	0
$891$	−79.0833	−2.64939
$892$	9.32456	0.312209
$893$	−24.4868	−0.819421
$894$	−11.6228	−0.388724
$895$	0	0
$896$	−1.00000	−0.0334077
$897$	0	0
$898$	39.9737	1.33394
$899$	−1.94733	−0.0649472
$900$	0	0
$901$	4.83772	0.161168
$902$	9.67544	0.322157
$903$	−6.32456	−0.210468
$904$	16.6491	0.553741
$905$	0	0
$906$	35.2982	1.17270
$907$	−28.1359	−0.934239	−0.467119	−	0.884194i	$-0.654708\pi$
$907$	−28.1359	−0.934239	−0.467119	+	0.884194i	$0.654708\pi$
$908$	2.32456	0.0771431
$909$	−33.8641	−1.12320
$910$	0	0
$911$	33.2982	1.10322	0.551610	−	0.834102i	$-0.314014\pi$
$911$	33.2982	1.10322	0.551610	+	0.834102i	$0.314014\pi$
$912$	−25.8114	−0.854700
$913$	−39.4868	−1.30682
$914$	−32.4605	−1.07370
$915$	0	0
$916$	9.16228	0.302730
$917$	20.8114	0.687253
$918$	14.7018	0.485231
$919$	−24.3246	−0.802393	−0.401197	−	0.915992i	$-0.631406\pi$
$919$	−24.3246	−0.802393	−0.401197	+	0.915992i	$0.631406\pi$
$920$	0	0
$921$	−89.5701	−2.95144
$922$	−2.51317	−0.0827667
$923$	0	0
$924$	−13.1623	−0.433007
$925$	0	0
$926$	−0.324555	−0.0106655
$927$	−93.2719	−3.06345
$928$	2.32456	0.0763073
$929$	54.0000	1.77168	0.885841	−	0.463988i	$-0.153582\pi$
$929$	54.0000	1.77168	0.885841	+	0.463988i	$0.153582\pi$
$930$	0	0
$931$	48.9737	1.60505
$932$	9.48683	0.310752
$933$	48.9737	1.60333
$934$	7.35089	0.240528
$935$	0	0
$936$	0	0
$937$	−40.3246	−1.31735	−0.658673	−	0.752429i	$-0.728882\pi$
$937$	−40.3246	−1.31735	−0.658673	+	0.752429i	$0.728882\pi$
$938$	10.3246	0.337109
$939$	−13.6754	−0.446281
$940$	0	0
$941$	9.67544	0.315410	0.157705	−	0.987486i	$-0.449590\pi$
$941$	9.67544	0.315410	0.157705	+	0.987486i	$0.449590\pi$
$942$	−18.4605	−0.601476
$943$	−13.9473	−0.454188
$944$	2.32456	0.0756578
$945$	0	0
$946$	−8.32456	−0.270655
$947$	−29.8114	−0.968740	−0.484370	−	0.874863i	$-0.660951\pi$
$947$	−29.8114	−0.968740	−0.484370	+	0.874863i	$0.660951\pi$
$948$	17.3509	0.563531
$949$	0	0
$950$	0	0
$951$	20.5132	0.665185
$952$	1.16228	0.0376696
$953$	21.4868	0.696027	0.348013	−	0.937490i	$-0.386856\pi$
$953$	21.4868	0.696027	0.348013	+	0.937490i	$0.386856\pi$
$954$	29.1359	0.943311
$955$	0	0
$956$	4.83772	0.156463
$957$	30.5964	0.989043
$958$	−32.1359	−1.03827
$959$	3.48683	0.112596
$960$	0	0
$961$	−30.2982	−0.977362
$962$	0	0
$963$	−108.408	−3.49339
$964$	11.9737	0.385646
$965$	0	0
$966$	18.9737	0.610468
$967$	−46.6228	−1.49929	−0.749644	−	0.661842i	$-0.769775\pi$
$967$	−46.6228	−1.49929	−0.749644	+	0.661842i	$0.769775\pi$
$968$	−6.32456	−0.203279
$969$	30.0000	0.963739
$970$	0	0
$971$	−9.78505	−0.314017	−0.157009	−	0.987597i	$-0.550185\pi$
$971$	−9.78505	−0.314017	−0.157009	+	0.987597i	$0.550185\pi$
$972$	22.1359	0.710011
$973$	−5.83772	−0.187149
$974$	−1.00000	−0.0320421
$975$	0	0
$976$	11.4868	0.367685
$977$	18.9737	0.607021	0.303511	−	0.952828i	$-0.401841\pi$
$977$	18.9737	0.607021	0.303511	+	0.952828i	$0.401841\pi$
$978$	−4.70178	−0.150346
$979$	−22.1623	−0.708309
$980$	0	0
$981$	−102.544	−3.27397
$982$	−36.4868	−1.16434
$983$	−55.6491	−1.77493	−0.887465	−	0.460874i	$-0.847536\pi$
$983$	−55.6491	−1.77493	−0.887465	+	0.460874i	$0.847536\pi$
$984$	−7.35089	−0.234338
$985$	0	0
$986$	−2.70178	−0.0860422
$987$	9.48683	0.301969
$988$	0	0
$989$	12.0000	0.381578
$990$	0	0
$991$	11.2982	0.358900	0.179450	−	0.983767i	$-0.442568\pi$
$991$	11.2982	0.358900	0.179450	+	0.983767i	$0.442568\pi$
$992$	−0.837722	−0.0265977
$993$	2.05267	0.0651395
$994$	−8.32456	−0.264039
$995$	0	0
$996$	30.0000	0.950586
$997$	−18.1623	−0.575205	−0.287603	−	0.957750i	$-0.592858\pi$
$997$	−18.1623	−0.575205	−0.287603	+	0.957750i	$0.592858\pi$
$998$	22.0000	0.696398
$999$	−77.9473	−2.46614

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8450.2.a.bc.1.2		2
5.4	even	2		1690.2.a.n.1.1		2
13.3	even	3		650.2.e.h.451.1		4
13.9	even	3		650.2.e.h.601.1		4
13.12	even	2		8450.2.a.bj.1.2		2
65.3	odd	12		650.2.o.g.399.1		8
65.4	even	6		1690.2.e.m.991.2		4
65.9	even	6		130.2.e.c.81.2	yes	4
65.19	odd	12		1690.2.l.k.361.2		8
65.22	odd	12		650.2.o.g.549.1		8
65.24	odd	12		1690.2.l.k.1161.2		8
65.29	even	6		130.2.e.c.61.2	✓	4
65.34	odd	4		1690.2.d.g.1351.3		4
65.42	odd	12		650.2.o.g.399.4		8
65.44	odd	4		1690.2.d.g.1351.1		4
65.48	odd	12		650.2.o.g.549.4		8
65.49	even	6		1690.2.e.m.191.2		4
65.54	odd	12		1690.2.l.k.1161.4		8
65.59	odd	12		1690.2.l.k.361.4		8
65.64	even	2		1690.2.a.k.1.1		2
195.29	odd	6		1170.2.i.q.451.1		4
195.74	odd	6		1170.2.i.q.991.1		4
260.139	odd	6		1040.2.q.m.81.1		4
260.159	odd	6		1040.2.q.m.321.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
130.2.e.c.61.2	✓	4	65.29	even	6
130.2.e.c.81.2	yes	4	65.9	even	6
650.2.e.h.451.1		4	13.3	even	3
650.2.e.h.601.1		4	13.9	even	3
650.2.o.g.399.1		8	65.3	odd	12
650.2.o.g.399.4		8	65.42	odd	12
650.2.o.g.549.1		8	65.22	odd	12
650.2.o.g.549.4		8	65.48	odd	12
1040.2.q.m.81.1		4	260.139	odd	6
1040.2.q.m.321.1		4	260.159	odd	6
1170.2.i.q.451.1		4	195.29	odd	6
1170.2.i.q.991.1		4	195.74	odd	6
1690.2.a.k.1.1		2	65.64	even	2
1690.2.a.n.1.1		2	5.4	even	2
1690.2.d.g.1351.1		4	65.44	odd	4
1690.2.d.g.1351.3		4	65.34	odd	4
1690.2.e.m.191.2		4	65.49	even	6
1690.2.e.m.991.2		4	65.4	even	6
1690.2.l.k.361.2		8	65.19	odd	12
1690.2.l.k.361.4		8	65.59	odd	12
1690.2.l.k.1161.2		8	65.24	odd	12
1690.2.l.k.1161.4		8	65.54	odd	12
8450.2.a.bc.1.2		2	1.1	even	1	trivial
8450.2.a.bj.1.2		2	13.12	even	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:	\( N \)	\(=\)	\( 8450 = 2 \cdot 5^{2} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8450.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +3.16228 q^{3} +1.00000 q^{4} -3.16228 q^{6} +1.00000 q^{7} -1.00000 q^{8} +7.00000 q^{9} -4.16228 q^{11} +3.16228 q^{12} -1.00000 q^{14} +1.00000 q^{16} -1.16228 q^{17} -7.00000 q^{18} -8.16228 q^{19} +3.16228 q^{21} +4.16228 q^{22} -6.00000 q^{23} -3.16228 q^{24} +12.6491 q^{27} +1.00000 q^{28} -2.32456 q^{29} +0.837722 q^{31} -1.00000 q^{32} -13.1623 q^{33} +1.16228 q^{34} +7.00000 q^{36} -6.16228 q^{37} +8.16228 q^{38} +2.32456 q^{41} -3.16228 q^{42} -2.00000 q^{43} -4.16228 q^{44} +6.00000 q^{46} +3.00000 q^{47} +3.16228 q^{48} -6.00000 q^{49} -3.67544 q^{51} -4.16228 q^{53} -12.6491 q^{54} -1.00000 q^{56} -25.8114 q^{57} +2.32456 q^{58} +2.32456 q^{59} +11.4868 q^{61} -0.837722 q^{62} +7.00000 q^{63} +1.00000 q^{64} +13.1623 q^{66} -10.3246 q^{67} -1.16228 q^{68} -18.9737 q^{69} +8.32456 q^{71} -7.00000 q^{72} -9.16228 q^{73} +6.16228 q^{74} -8.16228 q^{76} -4.16228 q^{77} +5.48683 q^{79} +19.0000 q^{81} -2.32456 q^{82} +9.48683 q^{83} +3.16228 q^{84} +2.00000 q^{86} -7.35089 q^{87} +4.16228 q^{88} +5.32456 q^{89} -6.00000 q^{92} +2.64911 q^{93} -3.00000 q^{94} -3.16228 q^{96} -11.4868 q^{97} +6.00000 q^{98} -29.1359 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 2 q^{4} + 2 q^{7} - 2 q^{8} + 14 q^{9} - 2 q^{11} - 2 q^{14} + 2 q^{16} + 4 q^{17} - 14 q^{18} - 10 q^{19} + 2 q^{22} - 12 q^{23} + 2 q^{28} + 8 q^{29} + 8 q^{31} - 2 q^{32} - 20 q^{33}+ \cdots - 14 q^{99}+O(q^{100}) \) 2 * q - 2 * q^2 + 2 * q^4 + 2 * q^7 - 2 * q^8 + 14 * q^9 - 2 * q^11 - 2 * q^14 + 2 * q^16 + 4 * q^17 - 14 * q^18 - 10 * q^19 + 2 * q^22 - 12 * q^23 + 2 * q^28 + 8 * q^29 + 8 * q^31 - 2 * q^32 - 20 * q^33 - 4 * q^34 + 14 * q^36 - 6 * q^37 + 10 * q^38 - 8 * q^41 - 4 * q^43 - 2 * q^44 + 12 * q^46 + 6 * q^47 - 12 * q^49 - 20 * q^51 - 2 * q^53 - 2 * q^56 - 20 * q^57 - 8 * q^58 - 8 * q^59 + 4 * q^61 - 8 * q^62 + 14 * q^63 + 2 * q^64 + 20 * q^66 - 8 * q^67 + 4 * q^68 + 4 * q^71 - 14 * q^72 - 12 * q^73 + 6 * q^74 - 10 * q^76 - 2 * q^77 - 8 * q^79 + 38 * q^81 + 8 * q^82 + 4 * q^86 - 40 * q^87 + 2 * q^88 - 2 * q^89 - 12 * q^92 - 20 * q^93 - 6 * q^94 - 4 * q^97 + 12 * q^98 - 14 * q^99

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