LMFDB - Embedded newform 6160.2.a.bs.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6160,2,Mod(1,6160)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6160, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("6160.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6160 = 2^{4} \cdot 5 \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6160.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$49.1878476451$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.111028.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 10x^{2} - 2x + 4 $ x^4 - 10x^2 - 2x + 4
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 1540)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$0.547280$ of defining polynomial
Character	$\chi$	$=$	6160.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.547280 q^{3} +1.00000 q^{5} +1.00000 q^{7} -2.70049 q^{9} +O(q^{10})$	$q+0.547280 q^{3} +1.00000 q^{5} +1.00000 q^{7} -2.70049 q^{9} -1.00000 q^{11} +6.76160 q^{13} +0.547280 q^{15} +1.45272 q^{17} +5.30888 q^{19} +0.547280 q^{21} +2.51383 q^{23} +1.00000 q^{25} -3.11976 q^{27} +2.00000 q^{29} -6.24776 q^{31} -0.547280 q^{33} +1.00000 q^{35} +1.48617 q^{37} +3.70049 q^{39} +7.15321 q^{41} +1.60839 q^{43} -2.70049 q^{45} -3.66704 q^{47} +1.00000 q^{49} +0.795044 q^{51} +4.70295 q^{53} -1.00000 q^{55} +2.90544 q^{57} -7.34232 q^{59} -5.55664 q^{61} -2.70049 q^{63} +6.76160 q^{65} +0.391607 q^{67} +1.37577 q^{69} -12.4286 q^{71} +6.54728 q^{73} +0.547280 q^{75} -1.00000 q^{77} -12.0118 q^{79} +6.39407 q^{81} +15.6153 q^{83} +1.45272 q^{85} +1.09456 q^{87} -5.18665 q^{89} +6.76160 q^{91} -3.41927 q^{93} +5.30888 q^{95} -6.21432 q^{97} +2.70049 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{5} + 4 q^{7} + 8 q^{9}+O(q^{10}) $ 4 * q + 4 * q^5 + 4 * q^7 + 8 * q^9	$ 4 q + 4 q^{5} + 4 q^{7} + 8 q^{9} - 4 q^{11} + 2 q^{13} + 8 q^{17} - 6 q^{19} + 6 q^{23} + 4 q^{25} + 6 q^{27} + 8 q^{29} - 4 q^{31} + 4 q^{35} + 10 q^{37} - 4 q^{39} + 12 q^{41} - 2 q^{43} + 8 q^{45} + 6 q^{47} + 4 q^{49} - 20 q^{51} + 6 q^{53} - 4 q^{55} + 16 q^{57} - 4 q^{59} + 26 q^{61} + 8 q^{63} + 2 q^{65} + 10 q^{67} - 18 q^{69} - 4 q^{71} + 24 q^{73} - 4 q^{77} - 8 q^{79} + 40 q^{81} + 2 q^{83} + 8 q^{85} - 6 q^{89} + 2 q^{91} - 14 q^{93} - 6 q^{95} - 2 q^{97} - 8 q^{99}+O(q^{100}) $ 4 * q + 4 * q^5 + 4 * q^7 + 8 * q^9 - 4 * q^11 + 2 * q^13 + 8 * q^17 - 6 * q^19 + 6 * q^23 + 4 * q^25 + 6 * q^27 + 8 * q^29 - 4 * q^31 + 4 * q^35 + 10 * q^37 - 4 * q^39 + 12 * q^41 - 2 * q^43 + 8 * q^45 + 6 * q^47 + 4 * q^49 - 20 * q^51 + 6 * q^53 - 4 * q^55 + 16 * q^57 - 4 * q^59 + 26 * q^61 + 8 * q^63 + 2 * q^65 + 10 * q^67 - 18 * q^69 - 4 * q^71 + 24 * q^73 - 4 * q^77 - 8 * q^79 + 40 * q^81 + 2 * q^83 + 8 * q^85 - 6 * q^89 + 2 * q^91 - 14 * q^93 - 6 * q^95 - 2 * q^97 - 8 * q^99

Display 100 coefficients 10 coefficients

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
$3$	0.547280	0.315972	0.157986	−	0.987441i	$-0.449500\pi$
$3$	0.547280	0.315972	0.157986	+	0.987441i	$0.449500\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	−2.70049	−0.900162
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	6.76160	1.87533	0.937665	−	0.347541i	$-0.112983\pi$
$13$	6.76160	1.87533	0.937665	+	0.347541i	$0.112983\pi$
$14$	0	0
$15$	0.547280	0.141307
$16$	0	0
$17$	1.45272	0.352336	0.176168	−	0.984360i	$-0.443630\pi$
$17$	1.45272	0.352336	0.176168	+	0.984360i	$0.443630\pi$
$18$	0	0
$19$	5.30888	1.21794	0.608970	−	0.793193i	$-0.291583\pi$
$19$	5.30888	1.21794	0.608970	+	0.793193i	$0.291583\pi$
$20$	0	0
$21$	0.547280	0.119426
$22$	0	0
$23$	2.51383	0.524171	0.262085	−	0.965045i	$-0.415590\pi$
$23$	2.51383	0.524171	0.262085	+	0.965045i	$0.415590\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−3.11976	−0.600398
$28$	0	0
$29$	2.00000	0.371391	0.185695	−	0.982607i	$-0.440546\pi$
$29$	2.00000	0.371391	0.185695	+	0.982607i	$0.440546\pi$
$30$	0	0
$31$	−6.24776	−1.12213	−0.561066	−	0.827771i	$-0.689609\pi$
$31$	−6.24776	−1.12213	−0.561066	+	0.827771i	$0.689609\pi$
$32$	0	0
$33$	−0.547280	−0.0952692
$34$	0	0
$35$	1.00000	0.169031
$36$	0	0
$37$	1.48617	0.244324	0.122162	−	0.992510i	$-0.461017\pi$
$37$	1.48617	0.244324	0.122162	+	0.992510i	$0.461017\pi$
$38$	0	0
$39$	3.70049	0.592552
$40$	0	0
$41$	7.15321	1.11714	0.558571	−	0.829456i	$-0.311350\pi$
$41$	7.15321	1.11714	0.558571	+	0.829456i	$0.311350\pi$
$42$	0	0
$43$	1.60839	0.245278	0.122639	−	0.992451i	$-0.460864\pi$
$43$	1.60839	0.245278	0.122639	+	0.992451i	$0.460864\pi$
$44$	0	0
$45$	−2.70049	−0.402565
$46$	0	0
$47$	−3.66704	−0.534893	−0.267446	−	0.963573i	$-0.586180\pi$
$47$	−3.66704	−0.534893	−0.267446	+	0.963573i	$0.586180\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0.795044	0.111328
$52$	0	0
$53$	4.70295	0.646000	0.323000	−	0.946399i	$-0.395309\pi$
$53$	4.70295	0.646000	0.323000	+	0.946399i	$0.395309\pi$
$54$	0	0
$55$	−1.00000	−0.134840
$56$	0	0
$57$	2.90544	0.384835
$58$	0	0
$59$	−7.34232	−0.955889	−0.477945	−	0.878390i	$-0.658618\pi$
$59$	−7.34232	−0.955889	−0.477945	+	0.878390i	$0.658618\pi$
$60$	0	0
$61$	−5.55664	−0.711455	−0.355728	−	0.934590i	$-0.615767\pi$
$61$	−5.55664	−0.711455	−0.355728	+	0.934590i	$0.615767\pi$
$62$	0	0
$63$	−2.70049	−0.340229
$64$	0	0
$65$	6.76160	0.838673
$66$	0	0
$67$	0.391607	0.0478424	0.0239212	−	0.999714i	$-0.492385\pi$
$67$	0.391607	0.0478424	0.0239212	+	0.999714i	$0.492385\pi$
$68$	0	0
$69$	1.37577	0.165623
$70$	0	0
$71$	−12.4286	−1.47501	−0.737504	−	0.675343i	$-0.763996\pi$
$71$	−12.4286	−1.47501	−0.737504	+	0.675343i	$0.763996\pi$
$72$	0	0
$73$	6.54728	0.766301	0.383151	−	0.923686i	$-0.374839\pi$
$73$	6.54728	0.766301	0.383151	+	0.923686i	$0.374839\pi$
$74$	0	0
$75$	0.547280	0.0631944
$76$	0	0
$77$	−1.00000	−0.113961
$78$	0	0
$79$	−12.0118	−1.35144	−0.675718	−	0.737160i	$-0.736166\pi$
$79$	−12.0118	−1.35144	−0.675718	+	0.737160i	$0.736166\pi$
$80$	0	0
$81$	6.39407	0.710453
$82$	0	0
$83$	15.6153	1.71400	0.857000	−	0.515316i	$-0.172325\pi$
$83$	15.6153	1.71400	0.857000	+	0.515316i	$0.172325\pi$
$84$	0	0
$85$	1.45272	0.157570
$86$	0	0
$87$	1.09456	0.117349
$88$	0	0
$89$	−5.18665	−0.549784	−0.274892	−	0.961475i	$-0.588642\pi$
$89$	−5.18665	−0.549784	−0.274892	+	0.961475i	$0.588642\pi$
$90$	0	0
$91$	6.76160	0.708808
$92$	0	0
$93$	−3.41927	−0.354562
$94$	0	0
$95$	5.30888	0.544679
$96$	0	0
$97$	−6.21432	−0.630968	−0.315484	−	0.948931i	$-0.602167\pi$
$97$	−6.21432	−0.630968	−0.315484	+	0.948931i	$0.602167\pi$
$98$	0	0
$99$	2.70049	0.271409
$100$	0	0
$101$	14.4369	1.43652	0.718262	−	0.695773i	$-0.244938\pi$
$101$	14.4369	1.43652	0.718262	+	0.695773i	$0.244938\pi$
$102$	0	0
$103$	−10.0705	−0.992273	−0.496137	−	0.868244i	$-0.665249\pi$
$103$	−10.0705	−0.992273	−0.496137	+	0.868244i	$0.665249\pi$
$104$	0	0
$105$	0.547280	0.0534090
$106$	0	0
$107$	7.40097	0.715479	0.357739	−	0.933821i	$-0.383548\pi$
$107$	7.40097	0.715479	0.357739	+	0.933821i	$0.383548\pi$
$108$	0	0
$109$	3.09456	0.296405	0.148203	−	0.988957i	$-0.452651\pi$
$109$	3.09456	0.296405	0.148203	+	0.988957i	$0.452651\pi$
$110$	0	0
$111$	0.813349	0.0771996
$112$	0	0
$113$	2.06689	0.194437	0.0972184	−	0.995263i	$-0.469005\pi$
$113$	2.06689	0.194437	0.0972184	+	0.995263i	$0.469005\pi$
$114$	0	0
$115$	2.51383	0.234416
$116$	0	0
$117$	−18.2596	−1.68810
$118$	0	0
$119$	1.45272	0.133171
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	3.91480	0.352986
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	17.5232	1.55493	0.777466	−	0.628925i	$-0.216505\pi$
$127$	17.5232	1.55493	0.777466	+	0.628925i	$0.216505\pi$
$128$	0	0
$129$	0.880241	0.0775009
$130$	0	0
$131$	−4.00000	−0.349482	−0.174741	−	0.984614i	$-0.555909\pi$
$131$	−4.00000	−0.349482	−0.174741	+	0.984614i	$0.555909\pi$
$132$	0	0
$133$	5.30888	0.460338
$134$	0	0
$135$	−3.11976	−0.268506
$136$	0	0
$137$	−4.70295	−0.401800	−0.200900	−	0.979612i	$-0.564387\pi$
$137$	−4.70295	−0.401800	−0.200900	+	0.979612i	$0.564387\pi$
$138$	0	0
$139$	−9.52320	−0.807747	−0.403873	−	0.914815i	$-0.632336\pi$
$139$	−9.52320	−0.807747	−0.403873	+	0.914815i	$0.632336\pi$
$140$	0	0
$141$	−2.00690	−0.169011
$142$	0	0
$143$	−6.76160	−0.565433
$144$	0	0
$145$	2.00000	0.166091
$146$	0	0
$147$	0.547280	0.0451389
$148$	0	0
$149$	21.1133	1.72967	0.864834	−	0.502058i	$-0.167424\pi$
$149$	21.1133	1.72967	0.864834	+	0.502058i	$0.167424\pi$
$150$	0	0
$151$	−21.2906	−1.73260	−0.866301	−	0.499522i	$-0.833509\pi$
$151$	−21.2906	−1.73260	−0.866301	+	0.499522i	$0.833509\pi$
$152$	0	0
$153$	−3.92305	−0.317160
$154$	0	0
$155$	−6.24776	−0.501833
$156$	0	0
$157$	15.7375	1.25599	0.627995	−	0.778217i	$-0.283876\pi$
$157$	15.7375	1.25599	0.627995	+	0.778217i	$0.283876\pi$
$158$	0	0
$159$	2.57383	0.204118
$160$	0	0
$161$	2.51383	0.198118
$162$	0	0
$163$	7.79258	0.610362	0.305181	−	0.952294i	$-0.401283\pi$
$163$	7.79258	0.610362	0.305181	+	0.952294i	$0.401283\pi$
$164$	0	0
$165$	−0.547280	−0.0426057
$166$	0	0
$167$	23.9266	1.85150	0.925749	−	0.378138i	$-0.123436\pi$
$167$	23.9266	1.85150	0.925749	+	0.378138i	$0.123436\pi$
$168$	0	0
$169$	32.7192	2.51686
$170$	0	0
$171$	−14.3365	−1.09634
$172$	0	0
$173$	16.5660	1.25949	0.629745	−	0.776802i	$-0.283159\pi$
$173$	16.5660	1.25949	0.629745	+	0.776802i	$0.283159\pi$
$174$	0	0
$175$	1.00000	0.0755929
$176$	0	0
$177$	−4.01830	−0.302034
$178$	0	0
$179$	19.6454	1.46837	0.734184	−	0.678951i	$-0.237565\pi$
$179$	19.6454	1.46837	0.734184	+	0.678951i	$0.237565\pi$
$180$	0	0
$181$	−17.4980	−1.30062	−0.650308	−	0.759671i	$-0.725360\pi$
$181$	−17.4980	−1.30062	−0.650308	+	0.759671i	$0.725360\pi$
$182$	0	0
$183$	−3.04104	−0.224800
$184$	0	0
$185$	1.48617	0.109265
$186$	0	0
$187$	−1.45272	−0.106233
$188$	0	0
$189$	−3.11976	−0.226929
$190$	0	0
$191$	3.33408	0.241245	0.120623	−	0.992698i	$-0.461511\pi$
$191$	3.33408	0.241245	0.120623	+	0.992698i	$0.461511\pi$
$192$	0	0
$193$	3.79751	0.273351	0.136675	−	0.990616i	$-0.456358\pi$
$193$	3.79751	0.273351	0.136675	+	0.990616i	$0.456358\pi$
$194$	0	0
$195$	3.70049	0.264997
$196$	0	0
$197$	26.0370	1.85506	0.927531	−	0.373747i	$-0.121927\pi$
$197$	26.0370	1.85506	0.927531	+	0.373747i	$0.121927\pi$
$198$	0	0
$199$	13.3675	0.947599	0.473799	−	0.880633i	$-0.342882\pi$
$199$	13.3675	0.947599	0.473799	+	0.880633i	$0.342882\pi$
$200$	0	0
$201$	0.214319	0.0151169
$202$	0	0
$203$	2.00000	0.140372
$204$	0	0
$205$	7.15321	0.499602
$206$	0	0
$207$	−6.78857	−0.471838
$208$	0	0
$209$	−5.30888	−0.367223
$210$	0	0
$211$	7.33408	0.504899	0.252449	−	0.967610i	$-0.418764\pi$
$211$	7.33408	0.504899	0.252449	+	0.967610i	$0.418764\pi$
$212$	0	0
$213$	−6.80194	−0.466061
$214$	0	0
$215$	1.60839	0.109691
$216$	0	0
$217$	−6.24776	−0.424126
$218$	0	0
$219$	3.58319	0.242130
$220$	0	0
$221$	9.82271	0.660747
$222$	0	0
$223$	19.3492	1.29572	0.647860	−	0.761760i	$-0.275664\pi$
$223$	19.3492	1.29572	0.647860	+	0.761760i	$0.275664\pi$
$224$	0	0
$225$	−2.70049	−0.180032
$226$	0	0
$227$	12.4286	0.824918	0.412459	−	0.910976i	$-0.364670\pi$
$227$	12.4286	0.824918	0.412459	+	0.910976i	$0.364670\pi$
$228$	0	0
$229$	−26.7400	−1.76703	−0.883514	−	0.468405i	$-0.844829\pi$
$229$	−26.7400	−1.76703	−0.883514	+	0.468405i	$0.844829\pi$
$230$	0	0
$231$	−0.547280	−0.0360084
$232$	0	0
$233$	9.07625	0.594605	0.297303	−	0.954783i	$-0.403913\pi$
$233$	9.07625	0.594605	0.297303	+	0.954783i	$0.403913\pi$
$234$	0	0
$235$	−3.66704	−0.239211
$236$	0	0
$237$	−6.57383	−0.427016
$238$	0	0
$239$	1.27874	0.0827150	0.0413575	−	0.999144i	$-0.486832\pi$
$239$	1.27874	0.0827150	0.0413575	+	0.999144i	$0.486832\pi$
$240$	0	0
$241$	−21.4833	−1.38386	−0.691930	−	0.721965i	$-0.743239\pi$
$241$	−21.4833	−1.38386	−0.691930	+	0.721965i	$0.743239\pi$
$242$	0	0
$243$	12.8586	0.824881
$244$	0	0
$245$	1.00000	0.0638877
$246$	0	0
$247$	35.8965	2.28404
$248$	0	0
$249$	8.54593	0.541576
$250$	0	0
$251$	−9.46455	−0.597397	−0.298699	−	0.954348i	$-0.596553\pi$
$251$	−9.46455	−0.597397	−0.298699	+	0.954348i	$0.596553\pi$
$252$	0	0
$253$	−2.51383	−0.158043
$254$	0	0
$255$	0.795044	0.0497876
$256$	0	0
$257$	1.71879	0.107215	0.0536076	−	0.998562i	$-0.482928\pi$
$257$	1.71879	0.107215	0.0536076	+	0.998562i	$0.482928\pi$
$258$	0	0
$259$	1.48617	0.0923459
$260$	0	0
$261$	−5.40097	−0.334312
$262$	0	0
$263$	1.52320	0.0939243	0.0469622	−	0.998897i	$-0.485046\pi$
$263$	1.52320	0.0939243	0.0469622	+	0.998897i	$0.485046\pi$
$264$	0	0
$265$	4.70295	0.288900
$266$	0	0
$267$	−2.83855	−0.173716
$268$	0	0
$269$	−25.3276	−1.54425	−0.772126	−	0.635470i	$-0.780806\pi$
$269$	−25.3276	−1.54425	−0.772126	+	0.635470i	$0.780806\pi$
$270$	0	0
$271$	7.28368	0.442452	0.221226	−	0.975223i	$-0.428994\pi$
$271$	7.28368	0.442452	0.221226	+	0.975223i	$0.428994\pi$
$272$	0	0
$273$	3.70049	0.223964
$274$	0	0
$275$	−1.00000	−0.0603023
$276$	0	0
$277$	−18.7030	−1.12375	−0.561876	−	0.827222i	$-0.689920\pi$
$277$	−18.7030	−1.12375	−0.561876	+	0.827222i	$0.689920\pi$
$278$	0	0
$279$	16.8720	1.01010
$280$	0	0
$281$	17.0227	1.01549	0.507746	−	0.861507i	$-0.330479\pi$
$281$	17.0227	1.01549	0.507746	+	0.861507i	$0.330479\pi$
$282$	0	0
$283$	−22.0187	−1.30888	−0.654439	−	0.756115i	$-0.727095\pi$
$283$	−22.0187	−1.30888	−0.654439	+	0.756115i	$0.727095\pi$
$284$	0	0
$285$	2.90544	0.172103
$286$	0	0
$287$	7.15321	0.422240
$288$	0	0
$289$	−14.8896	−0.875859
$290$	0	0
$291$	−3.40097	−0.199368
$292$	0	0
$293$	−20.5660	−1.20148	−0.600739	−	0.799445i	$-0.705127\pi$
$293$	−20.5660	−1.20148	−0.600739	+	0.799445i	$0.705127\pi$
$294$	0	0
$295$	−7.34232	−0.427487
$296$	0	0
$297$	3.11976	0.181027
$298$	0	0
$299$	16.9975	0.982993
$300$	0	0
$301$	1.60839	0.0927062
$302$	0	0
$303$	7.90101	0.453901
$304$	0	0
$305$	−5.55664	−0.318172
$306$	0	0
$307$	−10.8990	−0.622037	−0.311018	−	0.950404i	$-0.600670\pi$
$307$	−10.8990	−0.622037	−0.311018	+	0.950404i	$0.600670\pi$
$308$	0	0
$309$	−5.51137	−0.313531
$310$	0	0
$311$	−7.48975	−0.424705	−0.212352	−	0.977193i	$-0.568112\pi$
$311$	−7.48975	−0.424705	−0.212352	+	0.977193i	$0.568112\pi$
$312$	0	0
$313$	−29.4010	−1.66184	−0.830921	−	0.556391i	$-0.812186\pi$
$313$	−29.4010	−1.66184	−0.830921	+	0.556391i	$0.812186\pi$
$314$	0	0
$315$	−2.70049	−0.152155
$316$	0	0
$317$	−6.18418	−0.347338	−0.173669	−	0.984804i	$-0.555562\pi$
$317$	−6.18418	−0.347338	−0.173669	+	0.984804i	$0.555562\pi$
$318$	0	0
$319$	−2.00000	−0.111979
$320$	0	0
$321$	4.05040	0.226071
$322$	0	0
$323$	7.71232	0.429125
$324$	0	0
$325$	6.76160	0.375066
$326$	0	0
$327$	1.69359	0.0936557
$328$	0	0
$329$	−3.66704	−0.202170
$330$	0	0
$331$	−2.68465	−0.147562	−0.0737808	−	0.997274i	$-0.523507\pi$
$331$	−2.68465	−0.147562	−0.0737808	+	0.997274i	$0.523507\pi$
$332$	0	0
$333$	−4.01337	−0.219931
$334$	0	0
$335$	0.391607	0.0213958
$336$	0	0
$337$	−1.17976	−0.0642654	−0.0321327	−	0.999484i	$-0.510230\pi$
$337$	−1.17976	−0.0642654	−0.0321327	+	0.999484i	$0.510230\pi$
$338$	0	0
$339$	1.13117	0.0614366
$340$	0	0
$341$	6.24776	0.338335
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	1.37577	0.0740690
$346$	0	0
$347$	−7.19848	−0.386435	−0.193217	−	0.981156i	$-0.561892\pi$
$347$	−7.19848	−0.386435	−0.193217	+	0.981156i	$0.561892\pi$
$348$	0	0
$349$	−0.150738	−0.00806883	−0.00403442	−	0.999992i	$-0.501284\pi$
$349$	−0.150738	−0.00806883	−0.00403442	+	0.999992i	$0.501284\pi$
$350$	0	0
$351$	−21.0946	−1.12594
$352$	0	0
$353$	−24.0187	−1.27839	−0.639194	−	0.769046i	$-0.720732\pi$
$353$	−24.0187	−1.27839	−0.639194	+	0.769046i	$0.720732\pi$
$354$	0	0
$355$	−12.4286	−0.659644
$356$	0	0
$357$	0.795044	0.0420782
$358$	0	0
$359$	1.39407	0.0735764	0.0367882	−	0.999323i	$-0.488287\pi$
$359$	1.39407	0.0735764	0.0367882	+	0.999323i	$0.488287\pi$
$360$	0	0
$361$	9.18418	0.483378
$362$	0	0
$363$	0.547280	0.0287247
$364$	0	0
$365$	6.54728	0.342700
$366$	0	0
$367$	17.9734	0.938206	0.469103	−	0.883143i	$-0.344577\pi$
$367$	17.9734	0.938206	0.469103	+	0.883143i	$0.344577\pi$
$368$	0	0
$369$	−19.3171	−1.00561
$370$	0	0
$371$	4.70295	0.244165
$372$	0	0
$373$	15.3827	0.796484	0.398242	−	0.917280i	$-0.369620\pi$
$373$	15.3827	0.796484	0.398242	+	0.917280i	$0.369620\pi$
$374$	0	0
$375$	0.547280	0.0282614
$376$	0	0
$377$	13.5232	0.696480
$378$	0	0
$379$	31.0464	1.59475	0.797373	−	0.603487i	$-0.206222\pi$
$379$	31.0464	1.59475	0.797373	+	0.603487i	$0.206222\pi$
$380$	0	0
$381$	9.59009	0.491315
$382$	0	0
$383$	−3.94825	−0.201746	−0.100873	−	0.994899i	$-0.532164\pi$
$383$	−3.94825	−0.201746	−0.100873	+	0.994899i	$0.532164\pi$
$384$	0	0
$385$	−1.00000	−0.0509647
$386$	0	0
$387$	−4.34344	−0.220790
$388$	0	0
$389$	26.1173	1.32420	0.662100	−	0.749416i	$-0.269666\pi$
$389$	26.1173	1.32420	0.662100	+	0.749416i	$0.269666\pi$
$390$	0	0
$391$	3.65190	0.184684
$392$	0	0
$393$	−2.18912	−0.110426
$394$	0	0
$395$	−12.0118	−0.604381
$396$	0	0
$397$	−11.7139	−0.587902	−0.293951	−	0.955821i	$-0.594970\pi$
$397$	−11.7139	−0.587902	−0.293951	+	0.955821i	$0.594970\pi$
$398$	0	0
$399$	2.90544	0.145454
$400$	0	0
$401$	−19.4197	−0.969773	−0.484887	−	0.874577i	$-0.661139\pi$
$401$	−19.4197	−0.969773	−0.484887	+	0.874577i	$0.661139\pi$
$402$	0	0
$403$	−42.2449	−2.10437
$404$	0	0
$405$	6.39407	0.317724
$406$	0	0
$407$	−1.48617	−0.0736665
$408$	0	0
$409$	29.8011	1.47357	0.736785	−	0.676127i	$-0.236343\pi$
$409$	29.8011	1.47357	0.736785	+	0.676127i	$0.236343\pi$
$410$	0	0
$411$	−2.57383	−0.126958
$412$	0	0
$413$	−7.34232	−0.361292
$414$	0	0
$415$	15.6153	0.766524
$416$	0	0
$417$	−5.21185	−0.255225
$418$	0	0
$419$	1.72210	0.0841301	0.0420651	−	0.999115i	$-0.486606\pi$
$419$	1.72210	0.0841301	0.0420651	+	0.999115i	$0.486606\pi$
$420$	0	0
$421$	−12.8138	−0.624505	−0.312252	−	0.949999i	$-0.601083\pi$
$421$	−12.8138	−0.624505	−0.312252	+	0.949999i	$0.601083\pi$
$422$	0	0
$423$	9.90278	0.481490
$424$	0	0
$425$	1.45272	0.0704673
$426$	0	0
$427$	−5.55664	−0.268905
$428$	0	0
$429$	−3.70049	−0.178661
$430$	0	0
$431$	4.41681	0.212750	0.106375	−	0.994326i	$-0.466076\pi$
$431$	4.41681	0.212750	0.106375	+	0.994326i	$0.466076\pi$
$432$	0	0
$433$	−23.5232	−1.13045	−0.565226	−	0.824936i	$-0.691211\pi$
$433$	−23.5232	−1.13045	−0.565226	+	0.824936i	$0.691211\pi$
$434$	0	0
$435$	1.09456	0.0524801
$436$	0	0
$437$	13.3456	0.638408
$438$	0	0
$439$	33.7310	1.60989	0.804947	−	0.593346i	$-0.202193\pi$
$439$	33.7310	1.60989	0.804947	+	0.593346i	$0.202193\pi$
$440$	0	0
$441$	−2.70049	−0.128595
$442$	0	0
$443$	15.5049	0.736660	0.368330	−	0.929695i	$-0.379930\pi$
$443$	15.5049	0.736660	0.368330	+	0.929695i	$0.379930\pi$
$444$	0	0
$445$	−5.18665	−0.245871
$446$	0	0
$447$	11.5549	0.546527
$448$	0	0
$449$	30.8454	1.45569	0.727843	−	0.685744i	$-0.240523\pi$
$449$	30.8454	1.45569	0.727843	+	0.685744i	$0.240523\pi$
$450$	0	0
$451$	−7.15321	−0.336831
$452$	0	0
$453$	−11.6519	−0.547454
$454$	0	0
$455$	6.76160	0.316989
$456$	0	0
$457$	−31.1985	−1.45940	−0.729702	−	0.683766i	$-0.760341\pi$
$457$	−31.1985	−1.45940	−0.729702	+	0.683766i	$0.760341\pi$
$458$	0	0
$459$	−4.53214	−0.211542
$460$	0	0
$461$	35.7660	1.66579	0.832895	−	0.553431i	$-0.186682\pi$
$461$	35.7660	1.66579	0.832895	+	0.553431i	$0.186682\pi$
$462$	0	0
$463$	−29.4567	−1.36897	−0.684485	−	0.729027i	$-0.739973\pi$
$463$	−29.4567	−1.36897	−0.684485	+	0.729027i	$0.739973\pi$
$464$	0	0
$465$	−3.41927	−0.158565
$466$	0	0
$467$	−16.9255	−0.783220	−0.391610	−	0.920131i	$-0.628082\pi$
$467$	−16.9255	−0.783220	−0.391610	+	0.920131i	$0.628082\pi$
$468$	0	0
$469$	0.391607	0.0180827
$470$	0	0
$471$	8.61282	0.396858
$472$	0	0
$473$	−1.60839	−0.0739540
$474$	0	0
$475$	5.30888	0.243588
$476$	0	0
$477$	−12.7003	−0.581505
$478$	0	0
$479$	17.7139	0.809367	0.404683	−	0.914457i	$-0.367382\pi$
$479$	17.7139	0.809367	0.404683	+	0.914457i	$0.367382\pi$
$480$	0	0
$481$	10.0489	0.458189
$482$	0	0
$483$	1.37577	0.0625997
$484$	0	0
$485$	−6.21432	−0.282178
$486$	0	0
$487$	34.1226	1.54624	0.773122	−	0.634257i	$-0.218694\pi$
$487$	34.1226	1.54624	0.773122	+	0.634257i	$0.218694\pi$
$488$	0	0
$489$	4.26472	0.192857
$490$	0	0
$491$	−20.3664	−0.919123	−0.459562	−	0.888146i	$-0.651993\pi$
$491$	−20.3664	−0.919123	−0.459562	+	0.888146i	$0.651993\pi$
$492$	0	0
$493$	2.90544	0.130854
$494$	0	0
$495$	2.70049	0.121378
$496$	0	0
$497$	−12.4286	−0.557501
$498$	0	0
$499$	−9.21185	−0.412379	−0.206190	−	0.978512i	$-0.566106\pi$
$499$	−9.21185	−0.412379	−0.206190	+	0.978512i	$0.566106\pi$
$500$	0	0
$501$	13.0946	0.585022
$502$	0	0
$503$	−15.9935	−0.713116	−0.356558	−	0.934273i	$-0.616050\pi$
$503$	−15.9935	−0.713116	−0.356558	+	0.934273i	$0.616050\pi$
$504$	0	0
$505$	14.4369	0.642433
$506$	0	0
$507$	17.9066	0.795258
$508$	0	0
$509$	7.90143	0.350225	0.175112	−	0.984548i	$-0.443971\pi$
$509$	7.90143	0.350225	0.175112	+	0.984548i	$0.443971\pi$
$510$	0	0
$511$	6.54728	0.289635
$512$	0	0
$513$	−16.5624	−0.731249
$514$	0	0
$515$	−10.0705	−0.443758
$516$	0	0
$517$	3.66704	0.161276
$518$	0	0
$519$	9.06624	0.397964
$520$	0	0
$521$	33.7326	1.47785	0.738926	−	0.673787i	$-0.235333\pi$
$521$	33.7326	1.47785	0.738926	+	0.673787i	$0.235333\pi$
$522$	0	0
$523$	20.3316	0.889039	0.444520	−	0.895769i	$-0.353374\pi$
$523$	20.3316	0.889039	0.444520	+	0.895769i	$0.353374\pi$
$524$	0	0
$525$	0.547280	0.0238852
$526$	0	0
$527$	−9.07625	−0.395368
$528$	0	0
$529$	−16.6806	−0.725245
$530$	0	0
$531$	19.8278	0.860455
$532$	0	0
$533$	48.3671	2.09501
$534$	0	0
$535$	7.40097	0.319972
$536$	0	0
$537$	10.7515	0.463963
$538$	0	0
$539$	−1.00000	−0.0430730
$540$	0	0
$541$	10.9242	0.469667	0.234833	−	0.972036i	$-0.424546\pi$
$541$	10.9242	0.469667	0.234833	+	0.972036i	$0.424546\pi$
$542$	0	0
$543$	−9.57630	−0.410958
$544$	0	0
$545$	3.09456	0.132556
$546$	0	0
$547$	−9.49954	−0.406171	−0.203085	−	0.979161i	$-0.565097\pi$
$547$	−9.49954	−0.406171	−0.203085	+	0.979161i	$0.565097\pi$
$548$	0	0
$549$	15.0056	0.640425
$550$	0	0
$551$	10.6178	0.452332
$552$	0	0
$553$	−12.0118	−0.510795
$554$	0	0
$555$	0.813349	0.0345247
$556$	0	0
$557$	−39.0950	−1.65651	−0.828254	−	0.560353i	$-0.810665\pi$
$557$	−39.0950	−1.65651	−0.828254	+	0.560353i	$0.810665\pi$
$558$	0	0
$559$	10.8753	0.459976
$560$	0	0
$561$	−0.795044	−0.0335668
$562$	0	0
$563$	−24.7400	−1.04267	−0.521333	−	0.853353i	$-0.674565\pi$
$563$	−24.7400	−1.04267	−0.521333	+	0.853353i	$0.674565\pi$
$564$	0	0
$565$	2.06689	0.0869548
$566$	0	0
$567$	6.39407	0.268526
$568$	0	0
$569$	−10.3251	−0.432852	−0.216426	−	0.976299i	$-0.569440\pi$
$569$	−10.3251	−0.432852	−0.216426	+	0.976299i	$0.569440\pi$
$570$	0	0
$571$	−27.5785	−1.15413	−0.577063	−	0.816700i	$-0.695801\pi$
$571$	−27.5785	−1.15413	−0.577063	+	0.816700i	$0.695801\pi$
$572$	0	0
$573$	1.82467	0.0762268
$574$	0	0
$575$	2.51383	0.104834
$576$	0	0
$577$	6.42216	0.267358	0.133679	−	0.991025i	$-0.457321\pi$
$577$	6.42216	0.267358	0.133679	+	0.991025i	$0.457321\pi$
$578$	0	0
$579$	2.07830	0.0863712
$580$	0	0
$581$	15.6153	0.647831
$582$	0	0
$583$	−4.70295	−0.194776
$584$	0	0
$585$	−18.2596	−0.754941
$586$	0	0
$587$	−34.6164	−1.42877	−0.714386	−	0.699752i	$-0.753294\pi$
$587$	−34.6164	−1.42877	−0.714386	+	0.699752i	$0.753294\pi$
$588$	0	0
$589$	−33.1686	−1.36669
$590$	0	0
$591$	14.2495	0.586148
$592$	0	0
$593$	14.5774	0.598623	0.299311	−	0.954156i	$-0.403243\pi$
$593$	14.5774	0.598623	0.299311	+	0.954156i	$0.403243\pi$
$594$	0	0
$595$	1.45272	0.0595557
$596$	0	0
$597$	7.31577	0.299415
$598$	0	0
$599$	−9.07807	−0.370920	−0.185460	−	0.982652i	$-0.559377\pi$
$599$	−9.07807	−0.370920	−0.185460	+	0.982652i	$0.559377\pi$
$600$	0	0
$601$	22.4204	0.914547	0.457273	−	0.889326i	$-0.348826\pi$
$601$	22.4204	0.914547	0.457273	+	0.889326i	$0.348826\pi$
$602$	0	0
$603$	−1.05753	−0.0430659
$604$	0	0
$605$	1.00000	0.0406558
$606$	0	0
$607$	15.3039	0.621168	0.310584	−	0.950546i	$-0.399475\pi$
$607$	15.3039	0.621168	0.310584	+	0.950546i	$0.399475\pi$
$608$	0	0
$609$	1.09456	0.0443538
$610$	0	0
$611$	−24.7950	−1.00310
$612$	0	0
$613$	−22.3988	−0.904678	−0.452339	−	0.891846i	$-0.649410\pi$
$613$	−22.3988	−0.904678	−0.452339	+	0.891846i	$0.649410\pi$
$614$	0	0
$615$	3.91480	0.157860
$616$	0	0
$617$	36.4822	1.46872	0.734358	−	0.678762i	$-0.237483\pi$
$617$	36.4822	1.46872	0.734358	+	0.678762i	$0.237483\pi$
$618$	0	0
$619$	−32.1191	−1.29097	−0.645487	−	0.763771i	$-0.723346\pi$
$619$	−32.1191	−1.29097	−0.645487	+	0.763771i	$0.723346\pi$
$620$	0	0
$621$	−7.84256	−0.314711
$622$	0	0
$623$	−5.18665	−0.207799
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−2.90544	−0.116032
$628$	0	0
$629$	2.15898	0.0860843
$630$	0	0
$631$	−14.4840	−0.576598	−0.288299	−	0.957540i	$-0.593090\pi$
$631$	−14.4840	−0.576598	−0.288299	+	0.957540i	$0.593090\pi$
$632$	0	0
$633$	4.01379	0.159534
$634$	0	0
$635$	17.5232	0.695387
$636$	0	0
$637$	6.76160	0.267904
$638$	0	0
$639$	33.5633	1.32775
$640$	0	0
$641$	−31.2237	−1.23326	−0.616631	−	0.787253i	$-0.711503\pi$
$641$	−31.2237	−1.23326	−0.616631	+	0.787253i	$0.711503\pi$
$642$	0	0
$643$	48.2531	1.90292	0.951459	−	0.307777i	$-0.0995849\pi$
$643$	48.2531	1.90292	0.951459	+	0.307777i	$0.0995849\pi$
$644$	0	0
$645$	0.880241	0.0346594
$646$	0	0
$647$	18.6394	0.732789	0.366395	−	0.930460i	$-0.380592\pi$
$647$	18.6394	0.732789	0.366395	+	0.930460i	$0.380592\pi$
$648$	0	0
$649$	7.34232	0.288211
$650$	0	0
$651$	−3.41927	−0.134012
$652$	0	0
$653$	44.9291	1.75821	0.879106	−	0.476627i	$-0.158141\pi$
$653$	44.9291	1.75821	0.879106	+	0.476627i	$0.158141\pi$
$654$	0	0
$655$	−4.00000	−0.156293
$656$	0	0
$657$	−17.6808	−0.689795
$658$	0	0
$659$	−44.7419	−1.74290	−0.871449	−	0.490486i	$-0.836819\pi$
$659$	−44.7419	−1.74290	−0.871449	+	0.490486i	$0.836819\pi$
$660$	0	0
$661$	−6.03013	−0.234545	−0.117273	−	0.993100i	$-0.537415\pi$
$661$	−6.03013	−0.234545	−0.117273	+	0.993100i	$0.537415\pi$
$662$	0	0
$663$	5.37577	0.208778
$664$	0	0
$665$	5.30888	0.205869
$666$	0	0
$667$	5.02767	0.194672
$668$	0	0
$669$	10.5894	0.409411
$670$	0	0
$671$	5.55664	0.214512
$672$	0	0
$673$	17.0094	0.655663	0.327831	−	0.944736i	$-0.393682\pi$
$673$	17.0094	0.655663	0.327831	+	0.944736i	$0.393682\pi$
$674$	0	0
$675$	−3.11976	−0.120080
$676$	0	0
$677$	−7.20673	−0.276977	−0.138489	−	0.990364i	$-0.544224\pi$
$677$	−7.20673	−0.276977	−0.138489	+	0.990364i	$0.544224\pi$
$678$	0	0
$679$	−6.21432	−0.238484
$680$	0	0
$681$	6.80194	0.260651
$682$	0	0
$683$	−25.9817	−0.994162	−0.497081	−	0.867704i	$-0.665595\pi$
$683$	−25.9817	−0.994162	−0.497081	+	0.867704i	$0.665595\pi$
$684$	0	0
$685$	−4.70295	−0.179691
$686$	0	0
$687$	−14.6342	−0.558331
$688$	0	0
$689$	31.7995	1.21146
$690$	0	0
$691$	−8.87199	−0.337507	−0.168753	−	0.985658i	$-0.553974\pi$
$691$	−8.87199	−0.337507	−0.168753	+	0.985658i	$0.553974\pi$
$692$	0	0
$693$	2.70049	0.102583
$694$	0	0
$695$	−9.52320	−0.361235
$696$	0	0
$697$	10.3916	0.393610
$698$	0	0
$699$	4.96725	0.187879
$700$	0	0
$701$	2.25108	0.0850220	0.0425110	−	0.999096i	$-0.486464\pi$
$701$	2.25108	0.0850220	0.0425110	+	0.999096i	$0.486464\pi$
$702$	0	0
$703$	7.88988	0.297572
$704$	0	0
$705$	−2.00690	−0.0755841
$706$	0	0
$707$	14.4369	0.542955
$708$	0	0
$709$	16.9960	0.638298	0.319149	−	0.947705i	$-0.396603\pi$
$709$	16.9960	0.638298	0.319149	+	0.947705i	$0.396603\pi$
$710$	0	0
$711$	32.4378	1.21651
$712$	0	0
$713$	−15.7058	−0.588188
$714$	0	0
$715$	−6.76160	−0.252869
$716$	0	0
$717$	0.699830	0.0261356
$718$	0	0
$719$	16.2226	0.605000	0.302500	−	0.953149i	$-0.402179\pi$
$719$	16.2226	0.605000	0.302500	+	0.953149i	$0.402179\pi$
$720$	0	0
$721$	−10.0705	−0.375044
$722$	0	0
$723$	−11.7574	−0.437261
$724$	0	0
$725$	2.00000	0.0742781
$726$	0	0
$727$	1.64831	0.0611326	0.0305663	−	0.999533i	$-0.490269\pi$
$727$	1.64831	0.0611326	0.0305663	+	0.999533i	$0.490269\pi$
$728$	0	0
$729$	−12.1450	−0.449813
$730$	0	0
$731$	2.33655	0.0864202
$732$	0	0
$733$	−49.3996	−1.82462	−0.912308	−	0.409504i	$-0.865702\pi$
$733$	−49.3996	−1.82462	−0.912308	+	0.409504i	$0.865702\pi$
$734$	0	0
$735$	0.547280	0.0201867
$736$	0	0
$737$	−0.391607	−0.0144250
$738$	0	0
$739$	21.6137	0.795075	0.397537	−	0.917586i	$-0.369865\pi$
$739$	21.6137	0.795075	0.397537	+	0.917586i	$0.369865\pi$
$740$	0	0
$741$	19.6454	0.721693
$742$	0	0
$743$	9.34564	0.342858	0.171429	−	0.985196i	$-0.445162\pi$
$743$	9.34564	0.342858	0.171429	+	0.985196i	$0.445162\pi$
$744$	0	0
$745$	21.1133	0.773531
$746$	0	0
$747$	−42.1689	−1.54288
$748$	0	0
$749$	7.40097	0.270426
$750$	0	0
$751$	6.47680	0.236342	0.118171	−	0.992993i	$-0.462297\pi$
$751$	6.47680	0.236342	0.118171	+	0.992993i	$0.462297\pi$
$752$	0	0
$753$	−5.17976	−0.188761
$754$	0	0
$755$	−21.2906	−0.774843
$756$	0	0
$757$	13.1499	0.477941	0.238971	−	0.971027i	$-0.423190\pi$
$757$	13.1499	0.477941	0.238971	+	0.971027i	$0.423190\pi$
$758$	0	0
$759$	−1.37577	−0.0499373
$760$	0	0
$761$	−5.84433	−0.211857	−0.105928	−	0.994374i	$-0.533781\pi$
$761$	−5.84433	−0.211857	−0.105928	+	0.994374i	$0.533781\pi$
$762$	0	0
$763$	3.09456	0.112031
$764$	0	0
$765$	−3.92305	−0.141838
$766$	0	0
$767$	−49.6458	−1.79261
$768$	0	0
$769$	8.00331	0.288607	0.144303	−	0.989533i	$-0.453906\pi$
$769$	8.00331	0.288607	0.144303	+	0.989533i	$0.453906\pi$
$770$	0	0
$771$	0.940658	0.0338770
$772$	0	0
$773$	−0.587621	−0.0211353	−0.0105676	−	0.999944i	$-0.503364\pi$
$773$	−0.587621	−0.0211353	−0.0105676	+	0.999944i	$0.503364\pi$
$774$	0	0
$775$	−6.24776	−0.224426
$776$	0	0
$777$	0.813349	0.0291787
$778$	0	0
$779$	37.9755	1.36061
$780$	0	0
$781$	12.4286	0.444732
$782$	0	0
$783$	−6.23952	−0.222982
$784$	0	0
$785$	15.7375	0.561696
$786$	0	0
$787$	25.1084	0.895016	0.447508	−	0.894280i	$-0.352312\pi$
$787$	25.1084	0.895016	0.447508	+	0.894280i	$0.352312\pi$
$788$	0	0
$789$	0.833614	0.0296775
$790$	0	0
$791$	2.06689	0.0734902
$792$	0	0
$793$	−37.5718	−1.33421
$794$	0	0
$795$	2.57383	0.0912843
$796$	0	0
$797$	−32.5444	−1.15278	−0.576391	−	0.817174i	$-0.695539\pi$
$797$	−32.5444	−1.15278	−0.576391	+	0.817174i	$0.695539\pi$
$798$	0	0
$799$	−5.32718	−0.188462
$800$	0	0
$801$	14.0065	0.494894
$802$	0	0
$803$	−6.54728	−0.231048
$804$	0	0
$805$	2.51383	0.0886010
$806$	0	0
$807$	−13.8613	−0.487940
$808$	0	0
$809$	−13.6454	−0.479748	−0.239874	−	0.970804i	$-0.577106\pi$
$809$	−13.6454	−0.479748	−0.239874	+	0.970804i	$0.577106\pi$
$810$	0	0
$811$	21.8548	0.767426	0.383713	−	0.923452i	$-0.374645\pi$
$811$	21.8548	0.767426	0.383713	+	0.923452i	$0.374645\pi$
$812$	0	0
$813$	3.98621	0.139802
$814$	0	0
$815$	7.79258	0.272962
$816$	0	0
$817$	8.53876	0.298733
$818$	0	0
$819$	−18.2596	−0.638042
$820$	0	0
$821$	25.3845	0.885924	0.442962	−	0.896540i	$-0.353928\pi$
$821$	25.3845	0.885924	0.442962	+	0.896540i	$0.353928\pi$
$822$	0	0
$823$	22.2261	0.774755	0.387377	−	0.921921i	$-0.373381\pi$
$823$	22.2261	0.774755	0.387377	+	0.921921i	$0.373381\pi$
$824$	0	0
$825$	−0.547280	−0.0190538
$826$	0	0
$827$	−11.2252	−0.390339	−0.195170	−	0.980770i	$-0.562526\pi$
$827$	−11.2252	−0.390339	−0.195170	+	0.980770i	$0.562526\pi$
$828$	0	0
$829$	46.8573	1.62742	0.813710	−	0.581271i	$-0.197444\pi$
$829$	46.8573	1.62742	0.813710	+	0.581271i	$0.197444\pi$
$830$	0	0
$831$	−10.2357	−0.355074
$832$	0	0
$833$	1.45272	0.0503338
$834$	0	0
$835$	23.9266	0.828015
$836$	0	0
$837$	19.4915	0.673726
$838$	0	0
$839$	14.2176	0.490847	0.245424	−	0.969416i	$-0.421073\pi$
$839$	14.2176	0.490847	0.245424	+	0.969416i	$0.421073\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	0	0
$843$	9.31620	0.320867
$844$	0	0
$845$	32.7192	1.12558
$846$	0	0
$847$	1.00000	0.0343604
$848$	0	0
$849$	−12.0504	−0.413569
$850$	0	0
$851$	3.73598	0.128068
$852$	0	0
$853$	3.19024	0.109232	0.0546158	−	0.998507i	$-0.482607\pi$
$853$	3.19024	0.109232	0.0546158	+	0.998507i	$0.482607\pi$
$854$	0	0
$855$	−14.3365	−0.490299
$856$	0	0
$857$	−1.28233	−0.0438035	−0.0219017	−	0.999760i	$-0.506972\pi$
$857$	−1.28233	−0.0438035	−0.0219017	+	0.999760i	$0.506972\pi$
$858$	0	0
$859$	−14.6764	−0.500752	−0.250376	−	0.968149i	$-0.580554\pi$
$859$	−14.6764	−0.500752	−0.250376	+	0.968149i	$0.580554\pi$
$860$	0	0
$861$	3.91480	0.133416
$862$	0	0
$863$	−28.1039	−0.956669	−0.478334	−	0.878178i	$-0.658759\pi$
$863$	−28.1039	−0.956669	−0.478334	+	0.878178i	$0.658759\pi$
$864$	0	0
$865$	16.5660	0.563261
$866$	0	0
$867$	−8.14878	−0.276747
$868$	0	0
$869$	12.0118	0.407473
$870$	0	0
$871$	2.64789	0.0897203
$872$	0	0
$873$	16.7817	0.567974
$874$	0	0
$875$	1.00000	0.0338062
$876$	0	0
$877$	−14.1039	−0.476256	−0.238128	−	0.971234i	$-0.576534\pi$
$877$	−14.1039	−0.476256	−0.238128	+	0.971234i	$0.576534\pi$
$878$	0	0
$879$	−11.2554	−0.379634
$880$	0	0
$881$	−26.5395	−0.894137	−0.447068	−	0.894500i	$-0.647532\pi$
$881$	−26.5395	−0.894137	−0.447068	+	0.894500i	$0.647532\pi$
$882$	0	0
$883$	−27.4884	−0.925058	−0.462529	−	0.886604i	$-0.653058\pi$
$883$	−27.4884	−0.925058	−0.462529	+	0.886604i	$0.653058\pi$
$884$	0	0
$885$	−4.01830	−0.135074
$886$	0	0
$887$	−44.1410	−1.48211	−0.741054	−	0.671445i	$-0.765674\pi$
$887$	−44.1410	−1.48211	−0.741054	+	0.671445i	$0.765674\pi$
$888$	0	0
$889$	17.5232	0.587709
$890$	0	0
$891$	−6.39407	−0.214210
$892$	0	0
$893$	−19.4679	−0.651467
$894$	0	0
$895$	19.6454	0.656674
$896$	0	0
$897$	9.30240	0.310598
$898$	0	0
$899$	−12.4955	−0.416749
$900$	0	0
$901$	6.83207	0.227609
$902$	0	0
$903$	0.880241	0.0292926
$904$	0	0
$905$	−17.4980	−0.581653
$906$	0	0
$907$	−19.6753	−0.653307	−0.326654	−	0.945144i	$-0.605921\pi$
$907$	−19.6753	−0.653307	−0.326654	+	0.945144i	$0.605921\pi$
$908$	0	0
$909$	−38.9866	−1.29310
$910$	0	0
$911$	−26.7515	−0.886318	−0.443159	−	0.896443i	$-0.646142\pi$
$911$	−26.7515	−0.886318	−0.443159	+	0.896443i	$0.646142\pi$
$912$	0	0
$913$	−15.6153	−0.516791
$914$	0	0
$915$	−3.04104	−0.100534
$916$	0	0
$917$	−4.00000	−0.132092
$918$	0	0
$919$	−11.3341	−0.373877	−0.186938	−	0.982372i	$-0.559856\pi$
$919$	−11.3341	−0.373877	−0.186938	+	0.982372i	$0.559856\pi$
$920$	0	0
$921$	−5.96478	−0.196546
$922$	0	0
$923$	−84.0375	−2.76613
$924$	0	0
$925$	1.48617	0.0488649
$926$	0	0
$927$	27.1952	0.893207
$928$	0	0
$929$	−0.691122	−0.0226750	−0.0113375	−	0.999936i	$-0.503609\pi$
$929$	−0.691122	−0.0226750	−0.0113375	+	0.999936i	$0.503609\pi$
$930$	0	0
$931$	5.30888	0.173991
$932$	0	0
$933$	−4.09899	−0.134195
$934$	0	0
$935$	−1.45272	−0.0475090
$936$	0	0
$937$	7.76561	0.253691	0.126846	−	0.991922i	$-0.459515\pi$
$937$	7.76561	0.253691	0.126846	+	0.991922i	$0.459515\pi$
$938$	0	0
$939$	−16.0906	−0.525095
$940$	0	0
$941$	−49.3496	−1.60875	−0.804376	−	0.594120i	$-0.797500\pi$
$941$	−49.3496	−1.60875	−0.804376	+	0.594120i	$0.797500\pi$
$942$	0	0
$943$	17.9820	0.585573
$944$	0	0
$945$	−3.11976	−0.101486
$946$	0	0
$947$	−48.3118	−1.56992	−0.784961	−	0.619546i	$-0.787317\pi$
$947$	−48.3118	−1.56992	−0.784961	+	0.619546i	$0.787317\pi$
$948$	0	0
$949$	44.2701	1.43707
$950$	0	0
$951$	−3.38448	−0.109749
$952$	0	0
$953$	−10.5692	−0.342369	−0.171184	−	0.985239i	$-0.554759\pi$
$953$	−10.5692	−0.342369	−0.171184	+	0.985239i	$0.554759\pi$
$954$	0	0
$955$	3.33408	0.107888
$956$	0	0
$957$	−1.09456	−0.0353821
$958$	0	0
$959$	−4.70295	−0.151866
$960$	0	0
$961$	8.03456	0.259179
$962$	0	0
$963$	−19.9862	−0.644047
$964$	0	0
$965$	3.79751	0.122246
$966$	0	0
$967$	−24.8153	−0.798007	−0.399003	−	0.916949i	$-0.630644\pi$
$967$	−24.8153	−0.798007	−0.399003	+	0.916949i	$0.630644\pi$
$968$	0	0
$969$	4.22079	0.135591
$970$	0	0
$971$	28.4670	0.913550	0.456775	−	0.889582i	$-0.349004\pi$
$971$	28.4670	0.913550	0.456775	+	0.889582i	$0.349004\pi$
$972$	0	0
$973$	−9.52320	−0.305300
$974$	0	0
$975$	3.70049	0.118510
$976$	0	0
$977$	−51.3024	−1.64131	−0.820655	−	0.571424i	$-0.806391\pi$
$977$	−51.3024	−1.64131	−0.820655	+	0.571424i	$0.806391\pi$
$978$	0	0
$979$	5.18665	0.165766
$980$	0	0
$981$	−8.35681	−0.266812
$982$	0	0
$983$	29.3843	0.937213	0.468607	−	0.883407i	$-0.344756\pi$
$983$	29.3843	0.937213	0.468607	+	0.883407i	$0.344756\pi$
$984$	0	0
$985$	26.0370	0.829609
$986$	0	0
$987$	−2.00690	−0.0638802
$988$	0	0
$989$	4.04323	0.128567
$990$	0	0
$991$	−35.3577	−1.12318	−0.561588	−	0.827417i	$-0.689809\pi$
$991$	−35.3577	−1.12318	−0.561588	+	0.827417i	$0.689809\pi$
$992$	0	0
$993$	−1.46925	−0.0466253
$994$	0	0
$995$	13.3675	0.423779
$996$	0	0
$997$	29.6441	0.938837	0.469419	−	0.882976i	$-0.344464\pi$
$997$	29.6441	0.938837	0.469419	+	0.882976i	$0.344464\pi$
$998$	0	0
$999$	−4.63648	−0.146692

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6160.2.a.bs.1.3		4
4.3	odd	2		1540.2.a.i.1.2	✓	4
20.3	even	4		7700.2.e.t.1849.4		8
20.7	even	4		7700.2.e.t.1849.5		8
20.19	odd	2		7700.2.a.bb.1.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1540.2.a.i.1.2	✓	4	4.3	odd	2
6160.2.a.bs.1.3		4	1.1	even	1	trivial
7700.2.a.bb.1.3		4	20.19	odd	2
7700.2.e.t.1849.4		8	20.3	even	4
7700.2.e.t.1849.5		8	20.7	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 6160 = 2^{4} \cdot 5 \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6160.a (trivial)

\(f(q)\)	\(=\)	\(q+0.547280 q^{3} +1.00000 q^{5} +1.00000 q^{7} -2.70049 q^{9} +O(q^{10})\)	\(q+0.547280 q^{3} +1.00000 q^{5} +1.00000 q^{7} -2.70049 q^{9} -1.00000 q^{11} +6.76160 q^{13} +0.547280 q^{15} +1.45272 q^{17} +5.30888 q^{19} +0.547280 q^{21} +2.51383 q^{23} +1.00000 q^{25} -3.11976 q^{27} +2.00000 q^{29} -6.24776 q^{31} -0.547280 q^{33} +1.00000 q^{35} +1.48617 q^{37} +3.70049 q^{39} +7.15321 q^{41} +1.60839 q^{43} -2.70049 q^{45} -3.66704 q^{47} +1.00000 q^{49} +0.795044 q^{51} +4.70295 q^{53} -1.00000 q^{55} +2.90544 q^{57} -7.34232 q^{59} -5.55664 q^{61} -2.70049 q^{63} +6.76160 q^{65} +0.391607 q^{67} +1.37577 q^{69} -12.4286 q^{71} +6.54728 q^{73} +0.547280 q^{75} -1.00000 q^{77} -12.0118 q^{79} +6.39407 q^{81} +15.6153 q^{83} +1.45272 q^{85} +1.09456 q^{87} -5.18665 q^{89} +6.76160 q^{91} -3.41927 q^{93} +5.30888 q^{95} -6.21432 q^{97} +2.70049 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{5} + 4 q^{7} + 8 q^{9}+O(q^{10}) \) 4 * q + 4 * q^5 + 4 * q^7 + 8 * q^9	\( 4 q + 4 q^{5} + 4 q^{7} + 8 q^{9} - 4 q^{11} + 2 q^{13} + 8 q^{17} - 6 q^{19} + 6 q^{23} + 4 q^{25} + 6 q^{27} + 8 q^{29} - 4 q^{31} + 4 q^{35} + 10 q^{37} - 4 q^{39} + 12 q^{41} - 2 q^{43} + 8 q^{45} + 6 q^{47} + 4 q^{49} - 20 q^{51} + 6 q^{53} - 4 q^{55} + 16 q^{57} - 4 q^{59} + 26 q^{61} + 8 q^{63} + 2 q^{65} + 10 q^{67} - 18 q^{69} - 4 q^{71} + 24 q^{73} - 4 q^{77} - 8 q^{79} + 40 q^{81} + 2 q^{83} + 8 q^{85} - 6 q^{89} + 2 q^{91} - 14 q^{93} - 6 q^{95} - 2 q^{97} - 8 q^{99}+O(q^{100}) \) 4 * q + 4 * q^5 + 4 * q^7 + 8 * q^9 - 4 * q^11 + 2 * q^13 + 8 * q^17 - 6 * q^19 + 6 * q^23 + 4 * q^25 + 6 * q^27 + 8 * q^29 - 4 * q^31 + 4 * q^35 + 10 * q^37 - 4 * q^39 + 12 * q^41 - 2 * q^43 + 8 * q^45 + 6 * q^47 + 4 * q^49 - 20 * q^51 + 6 * q^53 - 4 * q^55 + 16 * q^57 - 4 * q^59 + 26 * q^61 + 8 * q^63 + 2 * q^65 + 10 * q^67 - 18 * q^69 - 4 * q^71 + 24 * q^73 - 4 * q^77 - 8 * q^79 + 40 * q^81 + 2 * q^83 + 8 * q^85 - 6 * q^89 + 2 * q^91 - 14 * q^93 - 6 * q^95 - 2 * q^97 - 8 * q^99

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