LMFDB - Embedded newform 5120.2.a.u.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [5120,2,Mod(1,5120)] 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("5120.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(5120, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 5120 = 2^{10} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5120.a (trivial)

Newform invariants

comment:select newform

sage:traces = [8,0,4,0,-8,0,-4,0,8,0,8,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

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

Self dual:	yes
Analytic conductor:	$40.8834058349$
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} - 12x^{6} - 8x^{5} + 21x^{4} + 12x^{3} - 10x^{2} - 4x + 1 $ x^8 - 12x^6 - 8x^5 + 21x^4 + 12x^3 - 10x^2 - 4x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 80)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-0.833171$ of defining polynomial
Character	$\chi$	$=$	5120.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-2.35278 q^{3} -1.00000 q^{5} -2.89402 q^{7} +2.53555 q^{9} +2.60869 q^{11} -4.35593 q^{13} +2.35278 q^{15} -7.29875 q^{17} +1.74787 q^{19} +6.80899 q^{21} -4.60490 q^{23} +1.00000 q^{25} +1.09274 q^{27} -6.00588 q^{29} -2.06299 q^{31} -6.13767 q^{33} +2.89402 q^{35} -1.66734 q^{37} +10.2485 q^{39} -4.61484 q^{41} +4.28533 q^{43} -2.53555 q^{45} -11.7111 q^{47} +1.37537 q^{49} +17.1723 q^{51} -3.86149 q^{53} -2.60869 q^{55} -4.11235 q^{57} -4.40253 q^{59} -3.32287 q^{61} -7.33795 q^{63} +4.35593 q^{65} +11.6575 q^{67} +10.8343 q^{69} +3.25937 q^{71} -12.6877 q^{73} -2.35278 q^{75} -7.54962 q^{77} -0.113885 q^{79} -10.1776 q^{81} -13.8142 q^{83} +7.29875 q^{85} +14.1305 q^{87} -3.74593 q^{89} +12.6062 q^{91} +4.85375 q^{93} -1.74787 q^{95} -13.9853 q^{97} +6.61448 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 4 q^{3} - 8 q^{5} - 4 q^{7} + 8 q^{9} + 8 q^{11} - 4 q^{15} + 16 q^{19} - 12 q^{23} + 8 q^{25} + 16 q^{27} + 4 q^{35} + 28 q^{43} - 8 q^{45} - 20 q^{47} + 8 q^{49} + 24 q^{51} - 8 q^{55} + 16 q^{59}+ \cdots + 40 q^{99}+O(q^{100}) $ 8 * q + 4 * q^3 - 8 * q^5 - 4 * q^7 + 8 * q^9 + 8 * q^11 - 4 * q^15 + 16 * q^19 - 12 * q^23 + 8 * q^25 + 16 * q^27 + 4 * q^35 + 28 * q^43 - 8 * q^45 - 20 * q^47 + 8 * q^49 + 24 * q^51 - 8 * q^55 + 16 * q^59 - 20 * q^63 + 36 * q^67 + 16 * q^69 + 8 * q^71 + 4 * q^75 + 8 * q^79 + 8 * q^81 + 20 * q^83 + 40 * q^91 - 16 * q^95 + 40 * 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 $

$2$	0	0
$2$	0	0
$3$	−2.35278	−1.35838	−0.679188	−	0.733965i	$-0.737668\pi$
$3$	−2.35278	−1.35838	−0.679188	+	0.733965i	$0.737668\pi$
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−2.89402	−1.09384	−0.546919	−	0.837186i	$-0.684199\pi$
$7$	−2.89402	−1.09384	−0.546919	+	0.837186i	$0.684199\pi$
$8$	0	0
$9$	2.53555	0.845184
$10$	0	0
$11$	2.60869	0.786551	0.393275	−	0.919421i	$-0.371342\pi$
$11$	2.60869	0.786551	0.393275	+	0.919421i	$0.371342\pi$
$12$	0	0
$13$	−4.35593	−1.20812	−0.604058	−	0.796940i	$-0.706451\pi$
$13$	−4.35593	−1.20812	−0.604058	+	0.796940i	$0.706451\pi$
$14$	0	0
$15$	2.35278	0.607484
$16$	0	0
$17$	−7.29875	−1.77021	−0.885104	−	0.465393i	$-0.845913\pi$
$17$	−7.29875	−1.77021	−0.885104	+	0.465393i	$0.845913\pi$
$18$	0	0
$19$	1.74787	0.400989	0.200495	−	0.979695i	$-0.435745\pi$
$19$	1.74787	0.400989	0.200495	+	0.979695i	$0.435745\pi$
$20$	0	0
$21$	6.80899	1.48584
$22$	0	0
$23$	−4.60490	−0.960189	−0.480094	−	0.877217i	$-0.659398\pi$
$23$	−4.60490	−0.960189	−0.480094	+	0.877217i	$0.659398\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	1.09274	0.210298
$28$	0	0
$29$	−6.00588	−1.11526	−0.557632	−	0.830088i	$-0.688290\pi$
$29$	−6.00588	−1.11526	−0.557632	+	0.830088i	$0.688290\pi$
$30$	0	0
$31$	−2.06299	−0.370524	−0.185262	−	0.982689i	$-0.559313\pi$
$31$	−2.06299	−0.370524	−0.185262	+	0.982689i	$0.559313\pi$
$32$	0	0
$33$	−6.13767	−1.06843
$34$	0	0
$35$	2.89402	0.489179
$36$	0	0
$37$	−1.66734	−0.274109	−0.137055	−	0.990563i	$-0.543764\pi$
$37$	−1.66734	−0.274109	−0.137055	+	0.990563i	$0.543764\pi$
$38$	0	0
$39$	10.2485	1.64108
$40$	0	0
$41$	−4.61484	−0.720717	−0.360359	−	0.932814i	$-0.617346\pi$
$41$	−4.61484	−0.720717	−0.360359	+	0.932814i	$0.617346\pi$
$42$	0	0
$43$	4.28533	0.653507	0.326753	−	0.945110i	$-0.394045\pi$
$43$	4.28533	0.653507	0.326753	+	0.945110i	$0.394045\pi$
$44$	0	0
$45$	−2.53555	−0.377978
$46$	0	0
$47$	−11.7111	−1.70823	−0.854117	−	0.520081i	$-0.825902\pi$
$47$	−11.7111	−1.70823	−0.854117	+	0.520081i	$0.825902\pi$
$48$	0	0
$49$	1.37537	0.196481
$50$	0	0
$51$	17.1723	2.40461
$52$	0	0
$53$	−3.86149	−0.530416	−0.265208	−	0.964191i	$-0.585441\pi$
$53$	−3.86149	−0.530416	−0.265208	+	0.964191i	$0.585441\pi$
$54$	0	0
$55$	−2.60869	−0.351756
$56$	0	0
$57$	−4.11235	−0.544694
$58$	0	0
$59$	−4.40253	−0.573160	−0.286580	−	0.958056i	$-0.592518\pi$
$59$	−4.40253	−0.573160	−0.286580	+	0.958056i	$0.592518\pi$
$60$	0	0
$61$	−3.32287	−0.425450	−0.212725	−	0.977112i	$-0.568234\pi$
$61$	−3.32287	−0.425450	−0.212725	+	0.977112i	$0.568234\pi$
$62$	0	0
$63$	−7.33795	−0.924495
$64$	0	0
$65$	4.35593	0.540286
$66$	0	0
$67$	11.6575	1.42419	0.712097	−	0.702082i	$-0.247746\pi$
$67$	11.6575	1.42419	0.712097	+	0.702082i	$0.247746\pi$
$68$	0	0
$69$	10.8343	1.30430
$70$	0	0
$71$	3.25937	0.386816	0.193408	−	0.981118i	$-0.438046\pi$
$71$	3.25937	0.386816	0.193408	+	0.981118i	$0.438046\pi$
$72$	0	0
$73$	−12.6877	−1.48499	−0.742494	−	0.669853i	$-0.766357\pi$
$73$	−12.6877	−1.48499	−0.742494	+	0.669853i	$0.766357\pi$
$74$	0	0
$75$	−2.35278	−0.271675
$76$	0	0
$77$	−7.54962	−0.860359
$78$	0	0
$79$	−0.113885	−0.0128130	−0.00640652	−	0.999979i	$-0.502039\pi$
$79$	−0.113885	−0.0128130	−0.00640652	+	0.999979i	$0.502039\pi$
$80$	0	0
$81$	−10.1776	−1.13085
$82$	0	0
$83$	−13.8142	−1.51631	−0.758154	−	0.652075i	$-0.773899\pi$
$83$	−13.8142	−1.51631	−0.758154	+	0.652075i	$0.773899\pi$
$84$	0	0
$85$	7.29875	0.791661
$86$	0	0
$87$	14.1305	1.51495
$88$	0	0
$89$	−3.74593	−0.397068	−0.198534	−	0.980094i	$-0.563618\pi$
$89$	−3.74593	−0.397068	−0.198534	+	0.980094i	$0.563618\pi$
$90$	0	0
$91$	12.6062	1.32148
$92$	0	0
$93$	4.85375	0.503310
$94$	0	0
$95$	−1.74787	−0.179328
$96$	0	0
$97$	−13.9853	−1.41999	−0.709995	−	0.704206i	$-0.751303\pi$
$97$	−13.9853	−1.41999	−0.709995	+	0.704206i	$0.751303\pi$
$98$	0	0
$99$	6.61448	0.664780
$100$	0	0
$101$	4.98126	0.495654	0.247827	−	0.968804i	$-0.420284\pi$
$101$	4.98126	0.495654	0.247827	+	0.968804i	$0.420284\pi$
$102$	0	0
$103$	−0.150216	−0.0148013	−0.00740063	−	0.999973i	$-0.502356\pi$
$103$	−0.150216	−0.0148013	−0.00740063	+	0.999973i	$0.502356\pi$
$104$	0	0
$105$	−6.80899	−0.664489
$106$	0	0
$107$	−3.88996	−0.376056	−0.188028	−	0.982164i	$-0.560210\pi$
$107$	−3.88996	−0.376056	−0.188028	+	0.982164i	$0.560210\pi$
$108$	0	0
$109$	−9.76908	−0.935708	−0.467854	−	0.883806i	$-0.654973\pi$
$109$	−9.76908	−0.935708	−0.467854	+	0.883806i	$0.654973\pi$
$110$	0	0
$111$	3.92288	0.372344
$112$	0	0
$113$	3.49507	0.328788	0.164394	−	0.986395i	$-0.447433\pi$
$113$	3.49507	0.328788	0.164394	+	0.986395i	$0.447433\pi$
$114$	0	0
$115$	4.60490	0.429410
$116$	0	0
$117$	−11.0447	−1.02108
$118$	0	0
$119$	21.1228	1.93632
$120$	0	0
$121$	−4.19472	−0.381338
$122$	0	0
$123$	10.8577	0.979005
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−6.25357	−0.554915	−0.277458	−	0.960738i	$-0.589492\pi$
$127$	−6.25357	−0.554915	−0.277458	+	0.960738i	$0.589492\pi$
$128$	0	0
$129$	−10.0824	−0.887708
$130$	0	0
$131$	−7.30427	−0.638177	−0.319089	−	0.947725i	$-0.603377\pi$
$131$	−7.30427	−0.638177	−0.319089	+	0.947725i	$0.603377\pi$
$132$	0	0
$133$	−5.05838	−0.438617
$134$	0	0
$135$	−1.09274	−0.0940480
$136$	0	0
$137$	18.9408	1.61823	0.809113	−	0.587654i	$-0.199948\pi$
$137$	18.9408	1.61823	0.809113	+	0.587654i	$0.199948\pi$
$138$	0	0
$139$	3.94646	0.334735	0.167367	−	0.985895i	$-0.446473\pi$
$139$	3.94646	0.334735	0.167367	+	0.985895i	$0.446473\pi$
$140$	0	0
$141$	27.5535	2.32042
$142$	0	0
$143$	−11.3633	−0.950245
$144$	0	0
$145$	6.00588	0.498761
$146$	0	0
$147$	−3.23593	−0.266895
$148$	0	0
$149$	2.26800	0.185802	0.0929009	−	0.995675i	$-0.470386\pi$
$149$	2.26800	0.185802	0.0929009	+	0.995675i	$0.470386\pi$
$150$	0	0
$151$	2.53754	0.206502	0.103251	−	0.994655i	$-0.467076\pi$
$151$	2.53754	0.206502	0.103251	+	0.994655i	$0.467076\pi$
$152$	0	0
$153$	−18.5064	−1.49615
$154$	0	0
$155$	2.06299	0.165703
$156$	0	0
$157$	−14.4822	−1.15581	−0.577903	−	0.816105i	$-0.696129\pi$
$157$	−14.4822	−1.15581	−0.577903	+	0.816105i	$0.696129\pi$
$158$	0	0
$159$	9.08521	0.720504
$160$	0	0
$161$	13.3267	1.05029
$162$	0	0
$163$	11.3506	0.889046	0.444523	−	0.895768i	$-0.353373\pi$
$163$	11.3506	0.889046	0.444523	+	0.895768i	$0.353373\pi$
$164$	0	0
$165$	6.13767	0.477817
$166$	0	0
$167$	6.82611	0.528221	0.264110	−	0.964492i	$-0.414922\pi$
$167$	6.82611	0.528221	0.264110	+	0.964492i	$0.414922\pi$
$168$	0	0
$169$	5.97411	0.459547
$170$	0	0
$171$	4.43182	0.338910
$172$	0	0
$173$	7.19695	0.547174	0.273587	−	0.961847i	$-0.411790\pi$
$173$	7.19695	0.547174	0.273587	+	0.961847i	$0.411790\pi$
$174$	0	0
$175$	−2.89402	−0.218768
$176$	0	0
$177$	10.3582	0.778567
$178$	0	0
$179$	−2.31644	−0.173139	−0.0865696	−	0.996246i	$-0.527590\pi$
$179$	−2.31644	−0.173139	−0.0865696	+	0.996246i	$0.527590\pi$
$180$	0	0
$181$	23.7244	1.76342	0.881709	−	0.471793i	$-0.156393\pi$
$181$	23.7244	1.76342	0.881709	+	0.471793i	$0.156393\pi$
$182$	0	0
$183$	7.81797	0.577921
$184$	0	0
$185$	1.66734	0.122585
$186$	0	0
$187$	−19.0402	−1.39236
$188$	0	0
$189$	−3.16241	−0.230032
$190$	0	0
$191$	−5.85815	−0.423881	−0.211940	−	0.977283i	$-0.567978\pi$
$191$	−5.85815	−0.423881	−0.211940	+	0.977283i	$0.567978\pi$
$192$	0	0
$193$	−0.0241155	−0.00173587	−0.000867935	−	1.00000i	$-0.500276\pi$
$193$	−0.0241155	−0.00173587	−0.000867935		1.00000i	$0.500276\pi$
$194$	0	0
$195$	−10.2485	−0.733912
$196$	0	0
$197$	21.0839	1.50216	0.751082	−	0.660208i	$-0.229532\pi$
$197$	21.0839	1.50216	0.751082	+	0.660208i	$0.229532\pi$
$198$	0	0
$199$	−13.6525	−0.967801	−0.483900	−	0.875123i	$-0.660780\pi$
$199$	−13.6525	−0.967801	−0.483900	+	0.875123i	$0.660780\pi$
$200$	0	0
$201$	−27.4275	−1.93459
$202$	0	0
$203$	17.3812	1.21992
$204$	0	0
$205$	4.61484	0.322315
$206$	0	0
$207$	−11.6760	−0.811537
$208$	0	0
$209$	4.55966	0.315398
$210$	0	0
$211$	3.46628	0.238629	0.119314	−	0.992857i	$-0.461930\pi$
$211$	3.46628	0.238629	0.119314	+	0.992857i	$0.461930\pi$
$212$	0	0
$213$	−7.66856	−0.525441
$214$	0	0
$215$	−4.28533	−0.292257
$216$	0	0
$217$	5.97033	0.405293
$218$	0	0
$219$	29.8514	2.01717
$220$	0	0
$221$	31.7928	2.13862
$222$	0	0
$223$	−13.9483	−0.934045	−0.467023	−	0.884245i	$-0.654673\pi$
$223$	−13.9483	−0.934045	−0.467023	+	0.884245i	$0.654673\pi$
$224$	0	0
$225$	2.53555	0.169037
$226$	0	0
$227$	6.27746	0.416650	0.208325	−	0.978060i	$-0.433199\pi$
$227$	6.27746	0.416650	0.208325	+	0.978060i	$0.433199\pi$
$228$	0	0
$229$	−7.56701	−0.500042	−0.250021	−	0.968240i	$-0.580438\pi$
$229$	−7.56701	−0.500042	−0.250021	+	0.968240i	$0.580438\pi$
$230$	0	0
$231$	17.7626	1.16869
$232$	0	0
$233$	11.9370	0.782019	0.391010	−	0.920387i	$-0.372126\pi$
$233$	11.9370	0.782019	0.391010	+	0.920387i	$0.372126\pi$
$234$	0	0
$235$	11.7111	0.763945
$236$	0	0
$237$	0.267945	0.0174049
$238$	0	0
$239$	−16.7720	−1.08489	−0.542445	−	0.840091i	$-0.682501\pi$
$239$	−16.7720	−1.08489	−0.542445	+	0.840091i	$0.682501\pi$
$240$	0	0
$241$	22.0294	1.41904	0.709519	−	0.704686i	$-0.248912\pi$
$241$	22.0294	1.41904	0.709519	+	0.704686i	$0.248912\pi$
$242$	0	0
$243$	20.6675	1.32582
$244$	0	0
$245$	−1.37537	−0.0878691
$246$	0	0
$247$	−7.61360	−0.484442
$248$	0	0
$249$	32.5018	2.05972
$250$	0	0
$251$	−9.38932	−0.592649	−0.296324	−	0.955087i	$-0.595761\pi$
$251$	−9.38932	−0.592649	−0.296324	+	0.955087i	$0.595761\pi$
$252$	0	0
$253$	−12.0128	−0.755237
$254$	0	0
$255$	−17.1723	−1.07537
$256$	0	0
$257$	−7.25821	−0.452755	−0.226377	−	0.974040i	$-0.572688\pi$
$257$	−7.25821	−0.452755	−0.226377	+	0.974040i	$0.572688\pi$
$258$	0	0
$259$	4.82533	0.299831
$260$	0	0
$261$	−15.2282	−0.942604
$262$	0	0
$263$	9.27431	0.571878	0.285939	−	0.958248i	$-0.407694\pi$
$263$	9.27431	0.571878	0.285939	+	0.958248i	$0.407694\pi$
$264$	0	0
$265$	3.86149	0.237209
$266$	0	0
$267$	8.81334	0.539368
$268$	0	0
$269$	−18.9780	−1.15711	−0.578554	−	0.815644i	$-0.696383\pi$
$269$	−18.9780	−1.15711	−0.578554	+	0.815644i	$0.696383\pi$
$270$	0	0
$271$	22.5999	1.37285	0.686423	−	0.727202i	$-0.259180\pi$
$271$	22.5999	1.37285	0.686423	+	0.727202i	$0.259180\pi$
$272$	0	0
$273$	−29.6595	−1.79507
$274$	0	0
$275$	2.60869	0.157310
$276$	0	0
$277$	−22.9123	−1.37667	−0.688334	−	0.725394i	$-0.741658\pi$
$277$	−22.9123	−1.37667	−0.688334	+	0.725394i	$0.741658\pi$
$278$	0	0
$279$	−5.23082	−0.313161
$280$	0	0
$281$	−8.84793	−0.527824	−0.263912	−	0.964547i	$-0.585013\pi$
$281$	−8.84793	−0.527824	−0.263912	+	0.964547i	$0.585013\pi$
$282$	0	0
$283$	28.7173	1.70707	0.853533	−	0.521039i	$-0.174455\pi$
$283$	28.7173	1.70707	0.853533	+	0.521039i	$0.174455\pi$
$284$	0	0
$285$	4.11235	0.243595
$286$	0	0
$287$	13.3555	0.788348
$288$	0	0
$289$	36.2718	2.13364
$290$	0	0
$291$	32.9042	1.92888
$292$	0	0
$293$	−10.1390	−0.592327	−0.296164	−	0.955137i	$-0.595707\pi$
$293$	−10.1390	−0.592327	−0.296164	+	0.955137i	$0.595707\pi$
$294$	0	0
$295$	4.40253	0.256325
$296$	0	0
$297$	2.85062	0.165410
$298$	0	0
$299$	20.0586	1.16002
$300$	0	0
$301$	−12.4018	−0.714830
$302$	0	0
$303$	−11.7198	−0.673284
$304$	0	0
$305$	3.32287	0.190267
$306$	0	0
$307$	26.0651	1.48761	0.743807	−	0.668395i	$-0.233018\pi$
$307$	26.0651	1.48761	0.743807	+	0.668395i	$0.233018\pi$
$308$	0	0
$309$	0.353425	0.0201057
$310$	0	0
$311$	7.08961	0.402015	0.201007	−	0.979590i	$-0.435578\pi$
$311$	7.08961	0.402015	0.201007	+	0.979590i	$0.435578\pi$
$312$	0	0
$313$	22.0477	1.24621	0.623104	−	0.782139i	$-0.285871\pi$
$313$	22.0477	1.24621	0.623104	+	0.782139i	$0.285871\pi$
$314$	0	0
$315$	7.33795	0.413447
$316$	0	0
$317$	8.76346	0.492205	0.246103	−	0.969244i	$-0.420850\pi$
$317$	8.76346	0.492205	0.246103	+	0.969244i	$0.420850\pi$
$318$	0	0
$319$	−15.6675	−0.877211
$320$	0	0
$321$	9.15220	0.510826
$322$	0	0
$323$	−12.7573	−0.709834
$324$	0	0
$325$	−4.35593	−0.241623
$326$	0	0
$327$	22.9845	1.27104
$328$	0	0
$329$	33.8921	1.86853
$330$	0	0
$331$	−26.3290	−1.44717	−0.723585	−	0.690235i	$-0.757507\pi$
$331$	−26.3290	−1.44717	−0.723585	+	0.690235i	$0.757507\pi$
$332$	0	0
$333$	−4.22764	−0.231673
$334$	0	0
$335$	−11.6575	−0.636919
$336$	0	0
$337$	−14.2577	−0.776666	−0.388333	−	0.921519i	$-0.626949\pi$
$337$	−14.2577	−0.776666	−0.388333	+	0.921519i	$0.626949\pi$
$338$	0	0
$339$	−8.22311	−0.446618
$340$	0	0
$341$	−5.38170	−0.291436
$342$	0	0
$343$	16.2778	0.878919
$344$	0	0
$345$	−10.8343	−0.583299
$346$	0	0
$347$	33.2899	1.78709	0.893547	−	0.448970i	$-0.148209\pi$
$347$	33.2899	1.78709	0.893547	+	0.448970i	$0.148209\pi$
$348$	0	0
$349$	2.21582	0.118610	0.0593051	−	0.998240i	$-0.481112\pi$
$349$	2.21582	0.118610	0.0593051	+	0.998240i	$0.481112\pi$
$350$	0	0
$351$	−4.75989	−0.254064
$352$	0	0
$353$	9.44678	0.502801	0.251401	−	0.967883i	$-0.419109\pi$
$353$	9.44678	0.502801	0.251401	+	0.967883i	$0.419109\pi$
$354$	0	0
$355$	−3.25937	−0.172989
$356$	0	0
$357$	−49.6971	−2.63025
$358$	0	0
$359$	18.0452	0.952392	0.476196	−	0.879339i	$-0.342015\pi$
$359$	18.0452	0.952392	0.476196	+	0.879339i	$0.342015\pi$
$360$	0	0
$361$	−15.9449	−0.839208
$362$	0	0
$363$	9.86924	0.518001
$364$	0	0
$365$	12.6877	0.664107
$366$	0	0
$367$	29.1329	1.52073	0.760363	−	0.649498i	$-0.225021\pi$
$367$	29.1329	1.52073	0.760363	+	0.649498i	$0.225021\pi$
$368$	0	0
$369$	−11.7012	−0.609139
$370$	0	0
$371$	11.1752	0.580189
$372$	0	0
$373$	−4.74607	−0.245742	−0.122871	−	0.992423i	$-0.539210\pi$
$373$	−4.74607	−0.245742	−0.122871	+	0.992423i	$0.539210\pi$
$374$	0	0
$375$	2.35278	0.121497
$376$	0	0
$377$	26.1612	1.34737
$378$	0	0
$379$	−16.4766	−0.846346	−0.423173	−	0.906049i	$-0.639084\pi$
$379$	−16.4766	−0.846346	−0.423173	+	0.906049i	$0.639084\pi$
$380$	0	0
$381$	14.7133	0.753783
$382$	0	0
$383$	21.8044	1.11415	0.557077	−	0.830461i	$-0.311923\pi$
$383$	21.8044	1.11415	0.557077	+	0.830461i	$0.311923\pi$
$384$	0	0
$385$	7.54962	0.384764
$386$	0	0
$387$	10.8657	0.552334
$388$	0	0
$389$	−16.8149	−0.852548	−0.426274	−	0.904594i	$-0.640174\pi$
$389$	−16.8149	−0.852548	−0.426274	+	0.904594i	$0.640174\pi$
$390$	0	0
$391$	33.6101	1.69973
$392$	0	0
$393$	17.1853	0.866884
$394$	0	0
$395$	0.113885	0.00573017
$396$	0	0
$397$	−13.0660	−0.655763	−0.327881	−	0.944719i	$-0.606335\pi$
$397$	−13.0660	−0.655763	−0.327881	+	0.944719i	$0.606335\pi$
$398$	0	0
$399$	11.9012	0.595807
$400$	0	0
$401$	14.4744	0.722818	0.361409	−	0.932407i	$-0.382296\pi$
$401$	14.4744	0.722818	0.361409	+	0.932407i	$0.382296\pi$
$402$	0	0
$403$	8.98623	0.447636
$404$	0	0
$405$	10.1776	0.505730
$406$	0	0
$407$	−4.34958	−0.215601
$408$	0	0
$409$	−9.54117	−0.471781	−0.235890	−	0.971780i	$-0.575801\pi$
$409$	−9.54117	−0.471781	−0.235890	+	0.971780i	$0.575801\pi$
$410$	0	0
$411$	−44.5636	−2.19816
$412$	0	0
$413$	12.7410	0.626944
$414$	0	0
$415$	13.8142	0.678114
$416$	0	0
$417$	−9.28514	−0.454695
$418$	0	0
$419$	1.18464	0.0578734	0.0289367	−	0.999581i	$-0.490788\pi$
$419$	1.18464	0.0578734	0.0289367	+	0.999581i	$0.490788\pi$
$420$	0	0
$421$	25.4104	1.23843	0.619215	−	0.785222i	$-0.287451\pi$
$421$	25.4104	1.23843	0.619215	+	0.785222i	$0.287451\pi$
$422$	0	0
$423$	−29.6940	−1.44377
$424$	0	0
$425$	−7.29875	−0.354042
$426$	0	0
$427$	9.61646	0.465373
$428$	0	0
$429$	26.7352	1.29079
$430$	0	0
$431$	−3.85473	−0.185676	−0.0928380	−	0.995681i	$-0.529594\pi$
$431$	−3.85473	−0.185676	−0.0928380	+	0.995681i	$0.529594\pi$
$432$	0	0
$433$	25.5651	1.22858	0.614289	−	0.789081i	$-0.289443\pi$
$433$	25.5651	1.22858	0.614289	+	0.789081i	$0.289443\pi$
$434$	0	0
$435$	−14.1305	−0.677505
$436$	0	0
$437$	−8.04878	−0.385025
$438$	0	0
$439$	−30.1311	−1.43808	−0.719039	−	0.694970i	$-0.755418\pi$
$439$	−30.1311	−1.43808	−0.719039	+	0.694970i	$0.755418\pi$
$440$	0	0
$441$	3.48732	0.166063
$442$	0	0
$443$	−28.5140	−1.35474	−0.677372	−	0.735641i	$-0.736881\pi$
$443$	−28.5140	−1.35474	−0.677372	+	0.735641i	$0.736881\pi$
$444$	0	0
$445$	3.74593	0.177574
$446$	0	0
$447$	−5.33610	−0.252389
$448$	0	0
$449$	−36.5827	−1.72644	−0.863221	−	0.504826i	$-0.831557\pi$
$449$	−36.5827	−1.72644	−0.863221	+	0.504826i	$0.831557\pi$
$450$	0	0
$451$	−12.0387	−0.566881
$452$	0	0
$453$	−5.97026	−0.280507
$454$	0	0
$455$	−12.6062	−0.590986
$456$	0	0
$457$	16.7340	0.782785	0.391392	−	0.920224i	$-0.371994\pi$
$457$	16.7340	0.782785	0.391392	+	0.920224i	$0.371994\pi$
$458$	0	0
$459$	−7.97563	−0.372271
$460$	0	0
$461$	16.7410	0.779706	0.389853	−	0.920877i	$-0.372526\pi$
$461$	16.7410	0.779706	0.389853	+	0.920877i	$0.372526\pi$
$462$	0	0
$463$	−32.2711	−1.49976	−0.749882	−	0.661572i	$-0.769890\pi$
$463$	−32.2711	−1.49976	−0.749882	+	0.661572i	$0.769890\pi$
$464$	0	0
$465$	−4.85375	−0.225087
$466$	0	0
$467$	−1.73333	−0.0802090	−0.0401045	−	0.999195i	$-0.512769\pi$
$467$	−1.73333	−0.0802090	−0.0401045	+	0.999195i	$0.512769\pi$
$468$	0	0
$469$	−33.7371	−1.55784
$470$	0	0
$471$	34.0734	1.57002
$472$	0	0
$473$	11.1791	0.514016
$474$	0	0
$475$	1.74787	0.0801978
$476$	0	0
$477$	−9.79100	−0.448299
$478$	0	0
$479$	−28.8399	−1.31773	−0.658865	−	0.752261i	$-0.728963\pi$
$479$	−28.8399	−1.31773	−0.658865	+	0.752261i	$0.728963\pi$
$480$	0	0
$481$	7.26282	0.331156
$482$	0	0
$483$	−31.3547	−1.42669
$484$	0	0
$485$	13.9853	0.635039
$486$	0	0
$487$	−32.1668	−1.45762	−0.728808	−	0.684718i	$-0.759925\pi$
$487$	−32.1668	−1.45762	−0.728808	+	0.684718i	$0.759925\pi$
$488$	0	0
$489$	−26.7054	−1.20766
$490$	0	0
$491$	−7.68776	−0.346944	−0.173472	−	0.984839i	$-0.555499\pi$
$491$	−7.68776	−0.346944	−0.173472	+	0.984839i	$0.555499\pi$
$492$	0	0
$493$	43.8354	1.97425
$494$	0	0
$495$	−6.61448	−0.297299
$496$	0	0
$497$	−9.43268	−0.423114
$498$	0	0
$499$	24.2229	1.08437	0.542183	−	0.840261i	$-0.317598\pi$
$499$	24.2229	1.08437	0.542183	+	0.840261i	$0.317598\pi$
$500$	0	0
$501$	−16.0603	−0.717522
$502$	0	0
$503$	−23.5180	−1.04862	−0.524308	−	0.851529i	$-0.675676\pi$
$503$	−23.5180	−1.04862	−0.524308	+	0.851529i	$0.675676\pi$
$504$	0	0
$505$	−4.98126	−0.221663
$506$	0	0
$507$	−14.0557	−0.624237
$508$	0	0
$509$	−28.7294	−1.27341	−0.636703	−	0.771109i	$-0.719702\pi$
$509$	−28.7294	−1.27341	−0.636703	+	0.771109i	$0.719702\pi$
$510$	0	0
$511$	36.7186	1.62434
$512$	0	0
$513$	1.90997	0.0843271
$514$	0	0
$515$	0.150216	0.00661932
$516$	0	0
$517$	−30.5506	−1.34361
$518$	0	0
$519$	−16.9328	−0.743268
$520$	0	0
$521$	35.5082	1.55564	0.777820	−	0.628487i	$-0.216325\pi$
$521$	35.5082	1.55564	0.777820	+	0.628487i	$0.216325\pi$
$522$	0	0
$523$	0.958506	0.0419125	0.0209563	−	0.999780i	$-0.493329\pi$
$523$	0.958506	0.0419125	0.0209563	+	0.999780i	$0.493329\pi$
$524$	0	0
$525$	6.80899	0.297169
$526$	0	0
$527$	15.0572	0.655904
$528$	0	0
$529$	−1.79485	−0.0780371
$530$	0	0
$531$	−11.1628	−0.484426
$532$	0	0
$533$	20.1019	0.870711
$534$	0	0
$535$	3.88996	0.168178
$536$	0	0
$537$	5.45007	0.235188
$538$	0	0
$539$	3.58791	0.154542
$540$	0	0
$541$	−7.59573	−0.326566	−0.163283	−	0.986579i	$-0.552208\pi$
$541$	−7.59573	−0.326566	−0.163283	+	0.986579i	$0.552208\pi$
$542$	0	0
$543$	−55.8181	−2.39539
$544$	0	0
$545$	9.76908	0.418461
$546$	0	0
$547$	12.5410	0.536214	0.268107	−	0.963389i	$-0.413602\pi$
$547$	12.5410	0.536214	0.268107	+	0.963389i	$0.413602\pi$
$548$	0	0
$549$	−8.42531	−0.359584
$550$	0	0
$551$	−10.4975	−0.447209
$552$	0	0
$553$	0.329585	0.0140154
$554$	0	0
$555$	−3.92288	−0.166517
$556$	0	0
$557$	−32.2567	−1.36676	−0.683381	−	0.730062i	$-0.739491\pi$
$557$	−32.2567	−1.36676	−0.683381	+	0.730062i	$0.739491\pi$
$558$	0	0
$559$	−18.6666	−0.789513
$560$	0	0
$561$	44.7973	1.89135
$562$	0	0
$563$	−29.6576	−1.24992	−0.624959	−	0.780657i	$-0.714884\pi$
$563$	−29.6576	−1.24992	−0.624959	+	0.780657i	$0.714884\pi$
$564$	0	0
$565$	−3.49507	−0.147039
$566$	0	0
$567$	29.4543	1.23696
$568$	0	0
$569$	−8.05295	−0.337597	−0.168799	−	0.985651i	$-0.553989\pi$
$569$	−8.05295	−0.337597	−0.168799	+	0.985651i	$0.553989\pi$
$570$	0	0
$571$	31.8254	1.33185	0.665926	−	0.746018i	$-0.268037\pi$
$571$	31.8254	1.33185	0.665926	+	0.746018i	$0.268037\pi$
$572$	0	0
$573$	13.7829	0.575789
$574$	0	0
$575$	−4.60490	−0.192038
$576$	0	0
$577$	15.9819	0.665334	0.332667	−	0.943044i	$-0.392051\pi$
$577$	15.9819	0.665334	0.332667	+	0.943044i	$0.392051\pi$
$578$	0	0
$579$	0.0567383	0.00235796
$580$	0	0
$581$	39.9787	1.65860
$582$	0	0
$583$	−10.0734	−0.417199
$584$	0	0
$585$	11.0447	0.456642
$586$	0	0
$587$	−7.43526	−0.306886	−0.153443	−	0.988157i	$-0.549036\pi$
$587$	−7.43526	−0.306886	−0.153443	+	0.988157i	$0.549036\pi$
$588$	0	0
$589$	−3.60584	−0.148576
$590$	0	0
$591$	−49.6057	−2.04050
$592$	0	0
$593$	−3.96571	−0.162852	−0.0814260	−	0.996679i	$-0.525947\pi$
$593$	−3.96571	−0.162852	−0.0814260	+	0.996679i	$0.525947\pi$
$594$	0	0
$595$	−21.1228	−0.865949
$596$	0	0
$597$	32.1213	1.31464
$598$	0	0
$599$	8.31600	0.339783	0.169891	−	0.985463i	$-0.445658\pi$
$599$	8.31600	0.339783	0.169891	+	0.985463i	$0.445658\pi$
$600$	0	0
$601$	−46.0550	−1.87862	−0.939310	−	0.343068i	$-0.888534\pi$
$601$	−46.0550	−1.87862	−0.939310	+	0.343068i	$0.888534\pi$
$602$	0	0
$603$	29.5583	1.20371
$604$	0	0
$605$	4.19472	0.170540
$606$	0	0
$607$	5.05760	0.205282	0.102641	−	0.994718i	$-0.467271\pi$
$607$	5.05760	0.205282	0.102641	+	0.994718i	$0.467271\pi$
$608$	0	0
$609$	−40.8940	−1.65711
$610$	0	0
$611$	51.0125	2.06375
$612$	0	0
$613$	44.1235	1.78213	0.891066	−	0.453874i	$-0.149958\pi$
$613$	44.1235	1.78213	0.891066	+	0.453874i	$0.149958\pi$
$614$	0	0
$615$	−10.8577	−0.437824
$616$	0	0
$617$	−30.7412	−1.23759	−0.618796	−	0.785551i	$-0.712379\pi$
$617$	−30.7412	−1.23759	−0.618796	+	0.785551i	$0.712379\pi$
$618$	0	0
$619$	−23.8672	−0.959303	−0.479651	−	0.877459i	$-0.659237\pi$
$619$	−23.8672	−0.959303	−0.479651	+	0.877459i	$0.659237\pi$
$620$	0	0
$621$	−5.03196	−0.201926
$622$	0	0
$623$	10.8408	0.434328
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−10.7279	−0.428429
$628$	0	0
$629$	12.1695	0.485231
$630$	0	0
$631$	30.7318	1.22342	0.611708	−	0.791084i	$-0.290483\pi$
$631$	30.7318	1.22342	0.611708	+	0.791084i	$0.290483\pi$
$632$	0	0
$633$	−8.15539	−0.324147
$634$	0	0
$635$	6.25357	0.248166
$636$	0	0
$637$	−5.99101	−0.237372
$638$	0	0
$639$	8.26430	0.326931
$640$	0	0
$641$	22.1658	0.875496	0.437748	−	0.899098i	$-0.355776\pi$
$641$	22.1658	0.875496	0.437748	+	0.899098i	$0.355776\pi$
$642$	0	0
$643$	1.37995	0.0544200	0.0272100	−	0.999630i	$-0.491338\pi$
$643$	1.37995	0.0544200	0.0272100	+	0.999630i	$0.491338\pi$
$644$	0	0
$645$	10.0824	0.396995
$646$	0	0
$647$	23.2610	0.914484	0.457242	−	0.889342i	$-0.348837\pi$
$647$	23.2610	0.914484	0.457242	+	0.889342i	$0.348837\pi$
$648$	0	0
$649$	−11.4848	−0.450819
$650$	0	0
$651$	−14.0469	−0.550540
$652$	0	0
$653$	−33.8523	−1.32474	−0.662372	−	0.749175i	$-0.730450\pi$
$653$	−33.8523	−1.32474	−0.662372	+	0.749175i	$0.730450\pi$
$654$	0	0
$655$	7.30427	0.285401
$656$	0	0
$657$	−32.1704	−1.25509
$658$	0	0
$659$	19.9495	0.777122	0.388561	−	0.921423i	$-0.372972\pi$
$659$	19.9495	0.777122	0.388561	+	0.921423i	$0.372972\pi$
$660$	0	0
$661$	4.30093	0.167287	0.0836433	−	0.996496i	$-0.473344\pi$
$661$	4.30093	0.167287	0.0836433	+	0.996496i	$0.473344\pi$
$662$	0	0
$663$	−74.8014	−2.90505
$664$	0	0
$665$	5.05838	0.196156
$666$	0	0
$667$	27.6565	1.07086
$668$	0	0
$669$	32.8171	1.26878
$670$	0	0
$671$	−8.66835	−0.334638
$672$	0	0
$673$	−25.3628	−0.977662	−0.488831	−	0.872378i	$-0.662577\pi$
$673$	−25.3628	−0.977662	−0.488831	+	0.872378i	$0.662577\pi$
$674$	0	0
$675$	1.09274	0.0420595
$676$	0	0
$677$	13.2445	0.509027	0.254513	−	0.967069i	$-0.418085\pi$
$677$	13.2445	0.509027	0.254513	+	0.967069i	$0.418085\pi$
$678$	0	0
$679$	40.4737	1.55324
$680$	0	0
$681$	−14.7695	−0.565967
$682$	0	0
$683$	5.94719	0.227563	0.113781	−	0.993506i	$-0.463704\pi$
$683$	5.94719	0.227563	0.113781	+	0.993506i	$0.463704\pi$
$684$	0	0
$685$	−18.9408	−0.723692
$686$	0	0
$687$	17.8035	0.679245
$688$	0	0
$689$	16.8204	0.640804
$690$	0	0
$691$	−8.19246	−0.311656	−0.155828	−	0.987784i	$-0.549805\pi$
$691$	−8.19246	−0.311656	−0.155828	+	0.987784i	$0.549805\pi$
$692$	0	0
$693$	−19.1425	−0.727162
$694$	0	0
$695$	−3.94646	−0.149698
$696$	0	0
$697$	33.6826	1.27582
$698$	0	0
$699$	−28.0851	−1.06228
$700$	0	0
$701$	0.366269	0.0138338	0.00691689	−	0.999976i	$-0.497798\pi$
$701$	0.366269	0.0138338	0.00691689	+	0.999976i	$0.497798\pi$
$702$	0	0
$703$	−2.91430	−0.109915
$704$	0	0
$705$	−27.5535	−1.03772
$706$	0	0
$707$	−14.4159	−0.542165
$708$	0	0
$709$	−1.06303	−0.0399229	−0.0199614	−	0.999801i	$-0.506354\pi$
$709$	−1.06303	−0.0399229	−0.0199614	+	0.999801i	$0.506354\pi$
$710$	0	0
$711$	−0.288761	−0.0108294
$712$	0	0
$713$	9.49986	0.355773
$714$	0	0
$715$	11.3633	0.424963
$716$	0	0
$717$	39.4607	1.47369
$718$	0	0
$719$	39.6557	1.47891	0.739455	−	0.673206i	$-0.235083\pi$
$719$	39.6557	1.47891	0.739455	+	0.673206i	$0.235083\pi$
$720$	0	0
$721$	0.434730	0.0161902
$722$	0	0
$723$	−51.8302	−1.92759
$724$	0	0
$725$	−6.00588	−0.223053
$726$	0	0
$727$	22.2952	0.826881	0.413441	−	0.910531i	$-0.364327\pi$
$727$	22.2952	0.826881	0.413441	+	0.910531i	$0.364327\pi$
$728$	0	0
$729$	−18.0930	−0.670112
$730$	0	0
$731$	−31.2776	−1.15684
$732$	0	0
$733$	−39.9245	−1.47464	−0.737322	−	0.675542i	$-0.763910\pi$
$733$	−39.9245	−1.47464	−0.737322	+	0.675542i	$0.763910\pi$
$734$	0	0
$735$	3.23593	0.119359
$736$	0	0
$737$	30.4109	1.12020
$738$	0	0
$739$	−7.70944	−0.283596	−0.141798	−	0.989896i	$-0.545288\pi$
$739$	−7.70944	−0.283596	−0.141798	+	0.989896i	$0.545288\pi$
$740$	0	0
$741$	17.9131	0.658054
$742$	0	0
$743$	−52.5667	−1.92849	−0.964243	−	0.265020i	$-0.914621\pi$
$743$	−52.5667	−1.92849	−0.964243	+	0.265020i	$0.914621\pi$
$744$	0	0
$745$	−2.26800	−0.0830931
$746$	0	0
$747$	−35.0267	−1.28156
$748$	0	0
$749$	11.2576	0.411345
$750$	0	0
$751$	31.0189	1.13190	0.565948	−	0.824441i	$-0.308510\pi$
$751$	31.0189	1.13190	0.565948	+	0.824441i	$0.308510\pi$
$752$	0	0
$753$	22.0910	0.805040
$754$	0	0
$755$	−2.53754	−0.0923505
$756$	0	0
$757$	3.49861	0.127159	0.0635796	−	0.997977i	$-0.479748\pi$
$757$	3.49861	0.127159	0.0635796	+	0.997977i	$0.479748\pi$
$758$	0	0
$759$	28.2634	1.02590
$760$	0	0
$761$	−2.48375	−0.0900358	−0.0450179	−	0.998986i	$-0.514334\pi$
$761$	−2.48375	−0.0900358	−0.0450179	+	0.998986i	$0.514334\pi$
$762$	0	0
$763$	28.2719	1.02351
$764$	0	0
$765$	18.5064	0.669100
$766$	0	0
$767$	19.1771	0.692444
$768$	0	0
$769$	43.4690	1.56753	0.783767	−	0.621055i	$-0.213296\pi$
$769$	43.4690	1.56753	0.783767	+	0.621055i	$0.213296\pi$
$770$	0	0
$771$	17.0769	0.615011
$772$	0	0
$773$	0.420058	0.0151084	0.00755421	−	0.999971i	$-0.497595\pi$
$773$	0.420058	0.0151084	0.00755421	+	0.999971i	$0.497595\pi$
$774$	0	0
$775$	−2.06299	−0.0741047
$776$	0	0
$777$	−11.3529	−0.407283
$778$	0	0
$779$	−8.06615	−0.289000
$780$	0	0
$781$	8.50269	0.304250
$782$	0	0
$783$	−6.56286	−0.234537
$784$	0	0
$785$	14.4822	0.516892
$786$	0	0
$787$	33.5071	1.19440	0.597199	−	0.802093i	$-0.296280\pi$
$787$	33.5071	1.19440	0.597199	+	0.802093i	$0.296280\pi$
$788$	0	0
$789$	−21.8204	−0.776825
$790$	0	0
$791$	−10.1148	−0.359641
$792$	0	0
$793$	14.4742	0.513993
$794$	0	0
$795$	−9.08521	−0.322219
$796$	0	0
$797$	−54.0643	−1.91506	−0.957528	−	0.288342i	$-0.906896\pi$
$797$	−54.0643	−1.91506	−0.957528	+	0.288342i	$0.906896\pi$
$798$	0	0
$799$	85.4762	3.02393
$800$	0	0
$801$	−9.49801	−0.335596
$802$	0	0
$803$	−33.0984	−1.16802
$804$	0	0
$805$	−13.3267	−0.469704
$806$	0	0
$807$	44.6509	1.57179
$808$	0	0
$809$	−53.8310	−1.89260	−0.946298	−	0.323296i	$-0.895209\pi$
$809$	−53.8310	−1.89260	−0.946298	+	0.323296i	$0.895209\pi$
$810$	0	0
$811$	38.2614	1.34354	0.671769	−	0.740761i	$-0.265535\pi$
$811$	38.2614	1.34354	0.671769	+	0.740761i	$0.265535\pi$
$812$	0	0
$813$	−53.1725	−1.86484
$814$	0	0
$815$	−11.3506	−0.397593
$816$	0	0
$817$	7.49020	0.262049
$818$	0	0
$819$	31.9636	1.11690
$820$	0	0
$821$	34.2480	1.19526	0.597632	−	0.801770i	$-0.296108\pi$
$821$	34.2480	1.19526	0.597632	+	0.801770i	$0.296108\pi$
$822$	0	0
$823$	−41.3013	−1.43967	−0.719836	−	0.694144i	$-0.755783\pi$
$823$	−41.3013	−1.43967	−0.719836	+	0.694144i	$0.755783\pi$
$824$	0	0
$825$	−6.13767	−0.213686
$826$	0	0
$827$	−22.2405	−0.773379	−0.386690	−	0.922210i	$-0.626382\pi$
$827$	−22.2405	−0.773379	−0.386690	+	0.922210i	$0.626382\pi$
$828$	0	0
$829$	29.3199	1.01832	0.509160	−	0.860672i	$-0.329956\pi$
$829$	29.3199	1.01832	0.509160	+	0.860672i	$0.329956\pi$
$830$	0	0
$831$	53.9075	1.87003
$832$	0	0
$833$	−10.0385	−0.347813
$834$	0	0
$835$	−6.82611	−0.236227
$836$	0	0
$837$	−2.25431	−0.0779203
$838$	0	0
$839$	−43.6919	−1.50841	−0.754206	−	0.656638i	$-0.771978\pi$
$839$	−43.6919	−1.50841	−0.754206	+	0.656638i	$0.771978\pi$
$840$	0	0
$841$	7.07060	0.243814
$842$	0	0
$843$	20.8172	0.716983
$844$	0	0
$845$	−5.97411	−0.205516
$846$	0	0
$847$	12.1396	0.417122
$848$	0	0
$849$	−67.5654	−2.31884
$850$	0	0
$851$	7.67795	0.263197
$852$	0	0
$853$	49.5837	1.69771	0.848856	−	0.528623i	$-0.177292\pi$
$853$	49.5837	1.69771	0.848856	+	0.528623i	$0.177292\pi$
$854$	0	0
$855$	−4.43182	−0.151565
$856$	0	0
$857$	−45.3397	−1.54878	−0.774388	−	0.632711i	$-0.781942\pi$
$857$	−45.3397	−1.54878	−0.774388	+	0.632711i	$0.781942\pi$
$858$	0	0
$859$	−45.4286	−1.55000	−0.775002	−	0.631958i	$-0.782251\pi$
$859$	−45.4286	−1.55000	−0.775002	+	0.631958i	$0.782251\pi$
$860$	0	0
$861$	−31.4224	−1.07087
$862$	0	0
$863$	−36.9142	−1.25657	−0.628287	−	0.777981i	$-0.716244\pi$
$863$	−36.9142	−1.25657	−0.628287	+	0.777981i	$0.716244\pi$
$864$	0	0
$865$	−7.19695	−0.244704
$866$	0	0
$867$	−85.3394	−2.89828
$868$	0	0
$869$	−0.297090	−0.0100781
$870$	0	0
$871$	−50.7793	−1.72059
$872$	0	0
$873$	−35.4604	−1.20015
$874$	0	0
$875$	2.89402	0.0978358
$876$	0	0
$877$	−22.2283	−0.750597	−0.375299	−	0.926904i	$-0.622460\pi$
$877$	−22.2283	−0.750597	−0.375299	+	0.926904i	$0.622460\pi$
$878$	0	0
$879$	23.8548	0.804603
$880$	0	0
$881$	−1.16748	−0.0393335	−0.0196667	−	0.999807i	$-0.506261\pi$
$881$	−1.16748	−0.0393335	−0.0196667	+	0.999807i	$0.506261\pi$
$882$	0	0
$883$	−45.5957	−1.53442	−0.767209	−	0.641397i	$-0.778355\pi$
$883$	−45.5957	−1.53442	−0.767209	+	0.641397i	$0.778355\pi$
$884$	0	0
$885$	−10.3582	−0.348186
$886$	0	0
$887$	−42.7282	−1.43467	−0.717336	−	0.696728i	$-0.754639\pi$
$887$	−42.7282	−1.43467	−0.717336	+	0.696728i	$0.754639\pi$
$888$	0	0
$889$	18.0980	0.606987
$890$	0	0
$891$	−26.5503	−0.889469
$892$	0	0
$893$	−20.4694	−0.684983
$894$	0	0
$895$	2.31644	0.0774302
$896$	0	0
$897$	−47.1935	−1.57574
$898$	0	0
$899$	12.3901	0.413232
$900$	0	0
$901$	28.1840	0.938946
$902$	0	0
$903$	29.1788	0.971008
$904$	0	0
$905$	−23.7244	−0.788625
$906$	0	0
$907$	1.74422	0.0579160	0.0289580	−	0.999581i	$-0.490781\pi$
$907$	1.74422	0.0579160	0.0289580	+	0.999581i	$0.490781\pi$
$908$	0	0
$909$	12.6302	0.418919
$910$	0	0
$911$	23.9284	0.792785	0.396392	−	0.918081i	$-0.370262\pi$
$911$	23.9284	0.792785	0.396392	+	0.918081i	$0.370262\pi$
$912$	0	0
$913$	−36.0371	−1.19265
$914$	0	0
$915$	−7.81797	−0.258454
$916$	0	0
$917$	21.1387	0.698062
$918$	0	0
$919$	−45.3844	−1.49709	−0.748546	−	0.663082i	$-0.769248\pi$
$919$	−45.3844	−1.49709	−0.748546	+	0.663082i	$0.769248\pi$
$920$	0	0
$921$	−61.3253	−2.02074
$922$	0	0
$923$	−14.1976	−0.467319
$924$	0	0
$925$	−1.66734	−0.0548219
$926$	0	0
$927$	−0.380882	−0.0125098
$928$	0	0
$929$	−6.51036	−0.213598	−0.106799	−	0.994281i	$-0.534060\pi$
$929$	−6.51036	−0.213598	−0.106799	+	0.994281i	$0.534060\pi$
$930$	0	0
$931$	2.40397	0.0787868
$932$	0	0
$933$	−16.6803	−0.546087
$934$	0	0
$935$	19.0402	0.622681
$936$	0	0
$937$	40.2986	1.31650	0.658248	−	0.752801i	$-0.271298\pi$
$937$	40.2986	1.31650	0.658248	+	0.752801i	$0.271298\pi$
$938$	0	0
$939$	−51.8732	−1.69282
$940$	0	0
$941$	1.56481	0.0510114	0.0255057	−	0.999675i	$-0.491880\pi$
$941$	1.56481	0.0510114	0.0255057	+	0.999675i	$0.491880\pi$
$942$	0	0
$943$	21.2509	0.692025
$944$	0	0
$945$	3.16241	0.102873
$946$	0	0
$947$	12.4993	0.406172	0.203086	−	0.979161i	$-0.434903\pi$
$947$	12.4993	0.406172	0.203086	+	0.979161i	$0.434903\pi$
$948$	0	0
$949$	55.2669	1.79404
$950$	0	0
$951$	−20.6185	−0.668600
$952$	0	0
$953$	14.9610	0.484636	0.242318	−	0.970197i	$-0.422092\pi$
$953$	14.9610	0.484636	0.242318	+	0.970197i	$0.422092\pi$
$954$	0	0
$955$	5.85815	0.189565
$956$	0	0
$957$	36.8621	1.19158
$958$	0	0
$959$	−54.8152	−1.77008
$960$	0	0
$961$	−26.7441	−0.862712
$962$	0	0
$963$	−9.86320	−0.317837
$964$	0	0
$965$	0.0241155	0.000776304 0
$966$	0	0
$967$	3.95287	0.127116	0.0635578	−	0.997978i	$-0.479755\pi$
$967$	3.95287	0.127116	0.0635578	+	0.997978i	$0.479755\pi$
$968$	0	0
$969$	30.0150	0.964221
$970$	0	0
$971$	−41.0883	−1.31858	−0.659292	−	0.751887i	$-0.729144\pi$
$971$	−41.0883	−1.31858	−0.659292	+	0.751887i	$0.729144\pi$
$972$	0	0
$973$	−11.4212	−0.366145
$974$	0	0
$975$	10.2485	0.328215
$976$	0	0
$977$	−25.8962	−0.828494	−0.414247	−	0.910164i	$-0.635955\pi$
$977$	−25.8962	−0.828494	−0.414247	+	0.910164i	$0.635955\pi$
$978$	0	0
$979$	−9.77199	−0.312314
$980$	0	0
$981$	−24.7700	−0.790846
$982$	0	0
$983$	22.0151	0.702173	0.351087	−	0.936343i	$-0.385812\pi$
$983$	22.0151	0.702173	0.351087	+	0.936343i	$0.385812\pi$
$984$	0	0
$985$	−21.0839	−0.671789
$986$	0	0
$987$	−79.7405	−2.53817
$988$	0	0
$989$	−19.7335	−0.627490
$990$	0	0
$991$	54.3207	1.72556	0.862778	−	0.505583i	$-0.168723\pi$
$991$	54.3207	1.72556	0.862778	+	0.505583i	$0.168723\pi$
$992$	0	0
$993$	61.9461	1.96580
$994$	0	0
$995$	13.6525	0.432814
$996$	0	0
$997$	11.5174	0.364761	0.182380	−	0.983228i	$-0.441620\pi$
$997$	11.5174	0.364761	0.182380	+	0.983228i	$0.441620\pi$
$998$	0	0
$999$	−1.82197	−0.0576446

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5120.2.a.u.1.1		8
4.3	odd	2		5120.2.a.s.1.8		8
8.3	odd	2		5120.2.a.v.1.1		8
8.5	even	2		5120.2.a.t.1.8		8
32.3	odd	8		640.2.l.b.161.7		16
32.5	even	8		320.2.l.a.241.7		16
32.11	odd	8		640.2.l.b.481.7		16
32.13	even	8		320.2.l.a.81.7		16
32.19	odd	8		80.2.l.a.61.4	yes	16
32.21	even	8		640.2.l.a.481.2		16
32.27	odd	8		80.2.l.a.21.4	✓	16
32.29	even	8		640.2.l.a.161.2		16
96.5	odd	8		2880.2.t.c.2161.6		16
96.59	even	8		720.2.t.c.181.5		16
96.77	odd	8		2880.2.t.c.721.7		16
96.83	even	8		720.2.t.c.541.5		16
160.13	odd	8		1600.2.q.h.849.2		16
160.19	odd	8		400.2.l.h.301.5		16
160.27	even	8		400.2.q.g.149.2		16
160.37	odd	8		1600.2.q.h.49.2		16
160.59	odd	8		400.2.l.h.101.5		16
160.69	even	8		1600.2.l.i.1201.2		16
160.77	odd	8		1600.2.q.g.849.7		16
160.83	even	8		400.2.q.g.349.2		16
160.109	even	8		1600.2.l.i.401.2		16
160.123	even	8		400.2.q.h.149.7		16
160.133	odd	8		1600.2.q.g.49.7		16
160.147	even	8		400.2.q.h.349.7		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
80.2.l.a.21.4	✓	16	32.27	odd	8
80.2.l.a.61.4	yes	16	32.19	odd	8
320.2.l.a.81.7		16	32.13	even	8
320.2.l.a.241.7		16	32.5	even	8
400.2.l.h.101.5		16	160.59	odd	8
400.2.l.h.301.5		16	160.19	odd	8
400.2.q.g.149.2		16	160.27	even	8
400.2.q.g.349.2		16	160.83	even	8
400.2.q.h.149.7		16	160.123	even	8
400.2.q.h.349.7		16	160.147	even	8
640.2.l.a.161.2		16	32.29	even	8
640.2.l.a.481.2		16	32.21	even	8
640.2.l.b.161.7		16	32.3	odd	8
640.2.l.b.481.7		16	32.11	odd	8
720.2.t.c.181.5		16	96.59	even	8
720.2.t.c.541.5		16	96.83	even	8
1600.2.l.i.401.2		16	160.109	even	8
1600.2.l.i.1201.2		16	160.69	even	8
1600.2.q.g.49.7		16	160.133	odd	8
1600.2.q.g.849.7		16	160.77	odd	8
1600.2.q.h.49.2		16	160.37	odd	8
1600.2.q.h.849.2		16	160.13	odd	8
2880.2.t.c.721.7		16	96.77	odd	8
2880.2.t.c.2161.6		16	96.5	odd	8
5120.2.a.s.1.8		8	4.3	odd	2
5120.2.a.t.1.8		8	8.5	even	2
5120.2.a.u.1.1		8	1.1	even	1	trivial
5120.2.a.v.1.1		8	8.3	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:	\( N \)	\(=\)	\( 5120 = 2^{10} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5120.a (trivial)

\(f(q)\)	\(=\)	\(q-2.35278 q^{3} -1.00000 q^{5} -2.89402 q^{7} +2.53555 q^{9} +2.60869 q^{11} -4.35593 q^{13} +2.35278 q^{15} -7.29875 q^{17} +1.74787 q^{19} +6.80899 q^{21} -4.60490 q^{23} +1.00000 q^{25} +1.09274 q^{27} -6.00588 q^{29} -2.06299 q^{31} -6.13767 q^{33} +2.89402 q^{35} -1.66734 q^{37} +10.2485 q^{39} -4.61484 q^{41} +4.28533 q^{43} -2.53555 q^{45} -11.7111 q^{47} +1.37537 q^{49} +17.1723 q^{51} -3.86149 q^{53} -2.60869 q^{55} -4.11235 q^{57} -4.40253 q^{59} -3.32287 q^{61} -7.33795 q^{63} +4.35593 q^{65} +11.6575 q^{67} +10.8343 q^{69} +3.25937 q^{71} -12.6877 q^{73} -2.35278 q^{75} -7.54962 q^{77} -0.113885 q^{79} -10.1776 q^{81} -13.8142 q^{83} +7.29875 q^{85} +14.1305 q^{87} -3.74593 q^{89} +12.6062 q^{91} +4.85375 q^{93} -1.74787 q^{95} -13.9853 q^{97} +6.61448 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 4 q^{3} - 8 q^{5} - 4 q^{7} + 8 q^{9} + 8 q^{11} - 4 q^{15} + 16 q^{19} - 12 q^{23} + 8 q^{25} + 16 q^{27} + 4 q^{35} + 28 q^{43} - 8 q^{45} - 20 q^{47} + 8 q^{49} + 24 q^{51} - 8 q^{55} + 16 q^{59}+ \cdots + 40 q^{99}+O(q^{100}) \) 8 * q + 4 * q^3 - 8 * q^5 - 4 * q^7 + 8 * q^9 + 8 * q^11 - 4 * q^15 + 16 * q^19 - 12 * q^23 + 8 * q^25 + 16 * q^27 + 4 * q^35 + 28 * q^43 - 8 * q^45 - 20 * q^47 + 8 * q^49 + 24 * q^51 - 8 * q^55 + 16 * q^59 - 20 * q^63 + 36 * q^67 + 16 * q^69 + 8 * q^71 + 4 * q^75 + 8 * q^79 + 8 * q^81 + 20 * q^83 + 40 * q^91 - 16 * q^95 + 40 * q^99

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