LMFDB - Embedded newform 6160.2.a.v.1.2

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:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{12})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 3 $ x^2 - 3
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 770)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.73205$ 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)

$f(q)$	$=$	$q+0.732051 q^{3} +1.00000 q^{5} -1.00000 q^{7} -2.46410 q^{9} +O(q^{10})$	$q+0.732051 q^{3} +1.00000 q^{5} -1.00000 q^{7} -2.46410 q^{9} -1.00000 q^{11} +5.46410 q^{13} +0.732051 q^{15} -3.46410 q^{17} -0.732051 q^{19} -0.732051 q^{21} -4.73205 q^{23} +1.00000 q^{25} -4.00000 q^{27} -1.26795 q^{29} +4.92820 q^{31} -0.732051 q^{33} -1.00000 q^{35} +6.73205 q^{37} +4.00000 q^{39} -1.26795 q^{41} -8.92820 q^{43} -2.46410 q^{45} +1.00000 q^{49} -2.53590 q^{51} -1.26795 q^{53} -1.00000 q^{55} -0.535898 q^{57} -13.8564 q^{59} +2.00000 q^{61} +2.46410 q^{63} +5.46410 q^{65} -2.92820 q^{67} -3.46410 q^{69} -2.53590 q^{71} +4.53590 q^{73} +0.732051 q^{75} +1.00000 q^{77} -3.26795 q^{79} +4.46410 q^{81} -16.3923 q^{83} -3.46410 q^{85} -0.928203 q^{87} -8.53590 q^{89} -5.46410 q^{91} +3.60770 q^{93} -0.732051 q^{95} +16.1962 q^{97} +2.46410 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} + 2 q^{5} - 2 q^{7} + 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^3 + 2 * q^5 - 2 * q^7 + 2 * q^9	$ 2 q - 2 q^{3} + 2 q^{5} - 2 q^{7} + 2 q^{9} - 2 q^{11} + 4 q^{13} - 2 q^{15} + 2 q^{19} + 2 q^{21} - 6 q^{23} + 2 q^{25} - 8 q^{27} - 6 q^{29} - 4 q^{31} + 2 q^{33} - 2 q^{35} + 10 q^{37} + 8 q^{39} - 6 q^{41} - 4 q^{43} + 2 q^{45} + 2 q^{49} - 12 q^{51} - 6 q^{53} - 2 q^{55} - 8 q^{57} + 4 q^{61} - 2 q^{63} + 4 q^{65} + 8 q^{67} - 12 q^{71} + 16 q^{73} - 2 q^{75} + 2 q^{77} - 10 q^{79} + 2 q^{81} - 12 q^{83} + 12 q^{87} - 24 q^{89} - 4 q^{91} + 28 q^{93} + 2 q^{95} + 22 q^{97} - 2 q^{99}+O(q^{100}) $ 2 * q - 2 * q^3 + 2 * q^5 - 2 * q^7 + 2 * q^9 - 2 * q^11 + 4 * q^13 - 2 * q^15 + 2 * q^19 + 2 * q^21 - 6 * q^23 + 2 * q^25 - 8 * q^27 - 6 * q^29 - 4 * q^31 + 2 * q^33 - 2 * q^35 + 10 * q^37 + 8 * q^39 - 6 * q^41 - 4 * q^43 + 2 * q^45 + 2 * q^49 - 12 * q^51 - 6 * q^53 - 2 * q^55 - 8 * q^57 + 4 * q^61 - 2 * q^63 + 4 * q^65 + 8 * q^67 - 12 * q^71 + 16 * q^73 - 2 * q^75 + 2 * q^77 - 10 * q^79 + 2 * q^81 - 12 * q^83 + 12 * q^87 - 24 * q^89 - 4 * q^91 + 28 * q^93 + 2 * q^95 + 22 * q^97 - 2 * 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
$3$	0.732051	0.422650	0.211325	−	0.977416i	$-0.432222\pi$
$3$	0.732051	0.422650	0.211325	+	0.977416i	$0.432222\pi$
$4$	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
$9$	−2.46410	−0.821367
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	5.46410	1.51547	0.757735	−	0.652563i	$-0.226306\pi$
$13$	5.46410	1.51547	0.757735	+	0.652563i	$0.226306\pi$
$14$	0	0
$15$	0.732051	0.189015
$16$	0	0
$17$	−3.46410	−0.840168	−0.420084	−	0.907485i	$-0.637999\pi$
$17$	−3.46410	−0.840168	−0.420084	+	0.907485i	$0.637999\pi$
$18$	0	0
$19$	−0.732051	−0.167944	−0.0839720	−	0.996468i	$-0.526761\pi$
$19$	−0.732051	−0.167944	−0.0839720	+	0.996468i	$0.526761\pi$
$20$	0	0
$21$	−0.732051	−0.159747
$22$	0	0
$23$	−4.73205	−0.986701	−0.493350	−	0.869831i	$-0.664228\pi$
$23$	−4.73205	−0.986701	−0.493350	+	0.869831i	$0.664228\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−4.00000	−0.769800
$28$	0	0
$29$	−1.26795	−0.235452	−0.117726	−	0.993046i	$-0.537560\pi$
$29$	−1.26795	−0.235452	−0.117726	+	0.993046i	$0.537560\pi$
$30$	0	0
$31$	4.92820	0.885131	0.442566	−	0.896736i	$-0.354068\pi$
$31$	4.92820	0.885131	0.442566	+	0.896736i	$0.354068\pi$
$32$	0	0
$33$	−0.732051	−0.127434
$34$	0	0
$35$	−1.00000	−0.169031
$36$	0	0
$37$	6.73205	1.10674	0.553371	−	0.832935i	$-0.313341\pi$
$37$	6.73205	1.10674	0.553371	+	0.832935i	$0.313341\pi$
$38$	0	0
$39$	4.00000	0.640513
$40$	0	0
$41$	−1.26795	−0.198020	−0.0990102	−	0.995086i	$-0.531568\pi$
$41$	−1.26795	−0.198020	−0.0990102	+	0.995086i	$0.531568\pi$
$42$	0	0
$43$	−8.92820	−1.36154	−0.680769	−	0.732498i	$-0.738354\pi$
$43$	−8.92820	−1.36154	−0.680769	+	0.732498i	$0.738354\pi$
$44$	0	0
$45$	−2.46410	−0.367327
$46$	0	0
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−2.53590	−0.355097
$52$	0	0
$53$	−1.26795	−0.174166	−0.0870831	−	0.996201i	$-0.527755\pi$
$53$	−1.26795	−0.174166	−0.0870831	+	0.996201i	$0.527755\pi$
$54$	0	0
$55$	−1.00000	−0.134840
$56$	0	0
$57$	−0.535898	−0.0709815
$58$	0	0
$59$	−13.8564	−1.80395	−0.901975	−	0.431788i	$-0.857883\pi$
$59$	−13.8564	−1.80395	−0.901975	+	0.431788i	$0.857883\pi$
$60$	0	0
$61$	2.00000	0.256074	0.128037	−	0.991769i	$-0.459132\pi$
$61$	2.00000	0.256074	0.128037	+	0.991769i	$0.459132\pi$
$62$	0	0
$63$	2.46410	0.310448
$64$	0	0
$65$	5.46410	0.677738
$66$	0	0
$67$	−2.92820	−0.357737	−0.178868	−	0.983873i	$-0.557244\pi$
$67$	−2.92820	−0.357737	−0.178868	+	0.983873i	$0.557244\pi$
$68$	0	0
$69$	−3.46410	−0.417029
$70$	0	0
$71$	−2.53590	−0.300956	−0.150478	−	0.988613i	$-0.548081\pi$
$71$	−2.53590	−0.300956	−0.150478	+	0.988613i	$0.548081\pi$
$72$	0	0
$73$	4.53590	0.530887	0.265443	−	0.964126i	$-0.414482\pi$
$73$	4.53590	0.530887	0.265443	+	0.964126i	$0.414482\pi$
$74$	0	0
$75$	0.732051	0.0845299
$76$	0	0
$77$	1.00000	0.113961
$78$	0	0
$79$	−3.26795	−0.367673	−0.183837	−	0.982957i	$-0.558852\pi$
$79$	−3.26795	−0.367673	−0.183837	+	0.982957i	$0.558852\pi$
$80$	0	0
$81$	4.46410	0.496011
$82$	0	0
$83$	−16.3923	−1.79929	−0.899645	−	0.436623i	$-0.856174\pi$
$83$	−16.3923	−1.79929	−0.899645	+	0.436623i	$0.856174\pi$
$84$	0	0
$85$	−3.46410	−0.375735
$86$	0	0
$87$	−0.928203	−0.0995138
$88$	0	0
$89$	−8.53590	−0.904803	−0.452402	−	0.891814i	$-0.649433\pi$
$89$	−8.53590	−0.904803	−0.452402	+	0.891814i	$0.649433\pi$
$90$	0	0
$91$	−5.46410	−0.572793
$92$	0	0
$93$	3.60770	0.374101
$94$	0	0
$95$	−0.732051	−0.0751068
$96$	0	0
$97$	16.1962	1.64447	0.822235	−	0.569148i	$-0.192727\pi$
$97$	16.1962	1.64447	0.822235	+	0.569148i	$0.192727\pi$
$98$	0	0
$99$	2.46410	0.247652
$100$	0	0
$101$	−6.00000	−0.597022	−0.298511	−	0.954406i	$-0.596490\pi$
$101$	−6.00000	−0.597022	−0.298511	+	0.954406i	$0.596490\pi$
$102$	0	0
$103$	−17.4641	−1.72079	−0.860395	−	0.509629i	$-0.829783\pi$
$103$	−17.4641	−1.72079	−0.860395	+	0.509629i	$0.829783\pi$
$104$	0	0
$105$	−0.732051	−0.0714408
$106$	0	0
$107$	12.9282	1.24982	0.624908	−	0.780698i	$-0.285136\pi$
$107$	12.9282	1.24982	0.624908	+	0.780698i	$0.285136\pi$
$108$	0	0
$109$	−12.1962	−1.16818	−0.584090	−	0.811689i	$-0.698548\pi$
$109$	−12.1962	−1.16818	−0.584090	+	0.811689i	$0.698548\pi$
$110$	0	0
$111$	4.92820	0.467764
$112$	0	0
$113$	−0.928203	−0.0873180	−0.0436590	−	0.999046i	$-0.513902\pi$
$113$	−0.928203	−0.0873180	−0.0436590	+	0.999046i	$0.513902\pi$
$114$	0	0
$115$	−4.73205	−0.441266
$116$	0	0
$117$	−13.4641	−1.24476
$118$	0	0
$119$	3.46410	0.317554
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−0.928203	−0.0836933
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	17.8564	1.58450	0.792250	−	0.610197i	$-0.208910\pi$
$127$	17.8564	1.58450	0.792250	+	0.610197i	$0.208910\pi$
$128$	0	0
$129$	−6.53590	−0.575454
$130$	0	0
$131$	0.339746	0.0296837	0.0148419	−	0.999890i	$-0.495276\pi$
$131$	0.339746	0.0296837	0.0148419	+	0.999890i	$0.495276\pi$
$132$	0	0
$133$	0.732051	0.0634769
$134$	0	0
$135$	−4.00000	−0.344265
$136$	0	0
$137$	−19.8564	−1.69645	−0.848224	−	0.529638i	$-0.822328\pi$
$137$	−19.8564	−1.69645	−0.848224	+	0.529638i	$0.822328\pi$
$138$	0	0
$139$	6.19615	0.525551	0.262775	−	0.964857i	$-0.415362\pi$
$139$	6.19615	0.525551	0.262775	+	0.964857i	$0.415362\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−5.46410	−0.456931
$144$	0	0
$145$	−1.26795	−0.105297
$146$	0	0
$147$	0.732051	0.0603785
$148$	0	0
$149$	−10.7321	−0.879204	−0.439602	−	0.898193i	$-0.644880\pi$
$149$	−10.7321	−0.879204	−0.439602	+	0.898193i	$0.644880\pi$
$150$	0	0
$151$	18.1962	1.48078	0.740391	−	0.672177i	$-0.234640\pi$
$151$	18.1962	1.48078	0.740391	+	0.672177i	$0.234640\pi$
$152$	0	0
$153$	8.53590	0.690086
$154$	0	0
$155$	4.92820	0.395843
$156$	0	0
$157$	−14.3923	−1.14863	−0.574315	−	0.818634i	$-0.694732\pi$
$157$	−14.3923	−1.14863	−0.574315	+	0.818634i	$0.694732\pi$
$158$	0	0
$159$	−0.928203	−0.0736113
$160$	0	0
$161$	4.73205	0.372938
$162$	0	0
$163$	−8.00000	−0.626608	−0.313304	−	0.949653i	$-0.601436\pi$
$163$	−8.00000	−0.626608	−0.313304	+	0.949653i	$0.601436\pi$
$164$	0	0
$165$	−0.732051	−0.0569901
$166$	0	0
$167$	−13.8564	−1.07224	−0.536120	−	0.844141i	$-0.680111\pi$
$167$	−13.8564	−1.07224	−0.536120	+	0.844141i	$0.680111\pi$
$168$	0	0
$169$	16.8564	1.29665
$170$	0	0
$171$	1.80385	0.137944
$172$	0	0
$173$	−12.9282	−0.982913	−0.491457	−	0.870902i	$-0.663535\pi$
$173$	−12.9282	−0.982913	−0.491457	+	0.870902i	$0.663535\pi$
$174$	0	0
$175$	−1.00000	−0.0755929
$176$	0	0
$177$	−10.1436	−0.762439
$178$	0	0
$179$	−7.85641	−0.587215	−0.293608	−	0.955926i	$-0.594856\pi$
$179$	−7.85641	−0.587215	−0.293608	+	0.955926i	$0.594856\pi$
$180$	0	0
$181$	15.8564	1.17860	0.589299	−	0.807915i	$-0.299404\pi$
$181$	15.8564	1.17860	0.589299	+	0.807915i	$0.299404\pi$
$182$	0	0
$183$	1.46410	0.108230
$184$	0	0
$185$	6.73205	0.494950
$186$	0	0
$187$	3.46410	0.253320
$188$	0	0
$189$	4.00000	0.290957
$190$	0	0
$191$	6.92820	0.501307	0.250654	−	0.968077i	$-0.419354\pi$
$191$	6.92820	0.501307	0.250654	+	0.968077i	$0.419354\pi$
$192$	0	0
$193$	−18.5359	−1.33424	−0.667122	−	0.744949i	$-0.732474\pi$
$193$	−18.5359	−1.33424	−0.667122	+	0.744949i	$0.732474\pi$
$194$	0	0
$195$	4.00000	0.286446
$196$	0	0
$197$	−1.60770	−0.114544	−0.0572718	−	0.998359i	$-0.518240\pi$
$197$	−1.60770	−0.114544	−0.0572718	+	0.998359i	$0.518240\pi$
$198$	0	0
$199$	−16.7846	−1.18983	−0.594915	−	0.803789i	$-0.702814\pi$
$199$	−16.7846	−1.18983	−0.594915	+	0.803789i	$0.702814\pi$
$200$	0	0
$201$	−2.14359	−0.151197
$202$	0	0
$203$	1.26795	0.0889926
$204$	0	0
$205$	−1.26795	−0.0885574
$206$	0	0
$207$	11.6603	0.810444
$208$	0	0
$209$	0.732051	0.0506370
$210$	0	0
$211$	−8.00000	−0.550743	−0.275371	−	0.961338i	$-0.588801\pi$
$211$	−8.00000	−0.550743	−0.275371	+	0.961338i	$0.588801\pi$
$212$	0	0
$213$	−1.85641	−0.127199
$214$	0	0
$215$	−8.92820	−0.608898
$216$	0	0
$217$	−4.92820	−0.334548
$218$	0	0
$219$	3.32051	0.224379
$220$	0	0
$221$	−18.9282	−1.27325
$222$	0	0
$223$	−12.3923	−0.829850	−0.414925	−	0.909856i	$-0.636192\pi$
$223$	−12.3923	−0.829850	−0.414925	+	0.909856i	$0.636192\pi$
$224$	0	0
$225$	−2.46410	−0.164273
$226$	0	0
$227$	13.8564	0.919682	0.459841	−	0.888001i	$-0.347906\pi$
$227$	13.8564	0.919682	0.459841	+	0.888001i	$0.347906\pi$
$228$	0	0
$229$	−6.53590	−0.431904	−0.215952	−	0.976404i	$-0.569286\pi$
$229$	−6.53590	−0.431904	−0.215952	+	0.976404i	$0.569286\pi$
$230$	0	0
$231$	0.732051	0.0481654
$232$	0	0
$233$	−19.8564	−1.30084	−0.650418	−	0.759576i	$-0.725406\pi$
$233$	−19.8564	−1.30084	−0.650418	+	0.759576i	$0.725406\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−2.39230	−0.155397
$238$	0	0
$239$	14.1962	0.918273	0.459136	−	0.888366i	$-0.348159\pi$
$239$	14.1962	0.918273	0.459136	+	0.888366i	$0.348159\pi$
$240$	0	0
$241$	−7.80385	−0.502690	−0.251345	−	0.967898i	$-0.580873\pi$
$241$	−7.80385	−0.502690	−0.251345	+	0.967898i	$0.580873\pi$
$242$	0	0
$243$	15.2679	0.979439
$244$	0	0
$245$	1.00000	0.0638877
$246$	0	0
$247$	−4.00000	−0.254514
$248$	0	0
$249$	−12.0000	−0.760469
$250$	0	0
$251$	0	0		−	1.00000i	$-0.5\pi$
$251$	0	0			1.00000i	$0.5\pi$
$252$	0	0
$253$	4.73205	0.297501
$254$	0	0
$255$	−2.53590	−0.158804
$256$	0	0
$257$	−28.9808	−1.80777	−0.903885	−	0.427775i	$-0.859297\pi$
$257$	−28.9808	−1.80777	−0.903885	+	0.427775i	$0.859297\pi$
$258$	0	0
$259$	−6.73205	−0.418309
$260$	0	0
$261$	3.12436	0.193393
$262$	0	0
$263$	−5.07180	−0.312740	−0.156370	−	0.987699i	$-0.549979\pi$
$263$	−5.07180	−0.312740	−0.156370	+	0.987699i	$0.549979\pi$
$264$	0	0
$265$	−1.26795	−0.0778895
$266$	0	0
$267$	−6.24871	−0.382415
$268$	0	0
$269$	−16.3923	−0.999456	−0.499728	−	0.866182i	$-0.666567\pi$
$269$	−16.3923	−0.999456	−0.499728	+	0.866182i	$0.666567\pi$
$270$	0	0
$271$	−3.60770	−0.219152	−0.109576	−	0.993978i	$-0.534949\pi$
$271$	−3.60770	−0.219152	−0.109576	+	0.993978i	$0.534949\pi$
$272$	0	0
$273$	−4.00000	−0.242091
$274$	0	0
$275$	−1.00000	−0.0603023
$276$	0	0
$277$	0.143594	0.00862770	0.00431385	−	0.999991i	$-0.498627\pi$
$277$	0.143594	0.00862770	0.00431385	+	0.999991i	$0.498627\pi$
$278$	0	0
$279$	−12.1436	−0.727018
$280$	0	0
$281$	10.3923	0.619953	0.309976	−	0.950744i	$-0.399679\pi$
$281$	10.3923	0.619953	0.309976	+	0.950744i	$0.399679\pi$
$282$	0	0
$283$	−2.92820	−0.174064	−0.0870318	−	0.996206i	$-0.527738\pi$
$283$	−2.92820	−0.174064	−0.0870318	+	0.996206i	$0.527738\pi$
$284$	0	0
$285$	−0.535898	−0.0317439
$286$	0	0
$287$	1.26795	0.0748447
$288$	0	0
$289$	−5.00000	−0.294118
$290$	0	0
$291$	11.8564	0.695035
$292$	0	0
$293$	−9.46410	−0.552899	−0.276449	−	0.961028i	$-0.589158\pi$
$293$	−9.46410	−0.552899	−0.276449	+	0.961028i	$0.589158\pi$
$294$	0	0
$295$	−13.8564	−0.806751
$296$	0	0
$297$	4.00000	0.232104
$298$	0	0
$299$	−25.8564	−1.49531
$300$	0	0
$301$	8.92820	0.514613
$302$	0	0
$303$	−4.39230	−0.252331
$304$	0	0
$305$	2.00000	0.114520
$306$	0	0
$307$	3.32051	0.189511	0.0947557	−	0.995501i	$-0.469793\pi$
$307$	3.32051	0.189511	0.0947557	+	0.995501i	$0.469793\pi$
$308$	0	0
$309$	−12.7846	−0.727291
$310$	0	0
$311$	19.8564	1.12595	0.562977	−	0.826473i	$-0.309656\pi$
$311$	19.8564	1.12595	0.562977	+	0.826473i	$0.309656\pi$
$312$	0	0
$313$	30.7321	1.73708	0.868539	−	0.495621i	$-0.165059\pi$
$313$	30.7321	1.73708	0.868539	+	0.495621i	$0.165059\pi$
$314$	0	0
$315$	2.46410	0.138836
$316$	0	0
$317$	26.4449	1.48529	0.742646	−	0.669684i	$-0.233571\pi$
$317$	26.4449	1.48529	0.742646	+	0.669684i	$0.233571\pi$
$318$	0	0
$319$	1.26795	0.0709915
$320$	0	0
$321$	9.46410	0.528235
$322$	0	0
$323$	2.53590	0.141101
$324$	0	0
$325$	5.46410	0.303094
$326$	0	0
$327$	−8.92820	−0.493731
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−8.92820	−0.490738	−0.245369	−	0.969430i	$-0.578909\pi$
$331$	−8.92820	−0.490738	−0.245369	+	0.969430i	$0.578909\pi$
$332$	0	0
$333$	−16.5885	−0.909042
$334$	0	0
$335$	−2.92820	−0.159985
$336$	0	0
$337$	10.7846	0.587475	0.293738	−	0.955886i	$-0.405101\pi$
$337$	10.7846	0.587475	0.293738	+	0.955886i	$0.405101\pi$
$338$	0	0
$339$	−0.679492	−0.0369049
$340$	0	0
$341$	−4.92820	−0.266877
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	−3.46410	−0.186501
$346$	0	0
$347$	19.8564	1.06595	0.532974	−	0.846132i	$-0.321074\pi$
$347$	19.8564	1.06595	0.532974	+	0.846132i	$0.321074\pi$
$348$	0	0
$349$	9.60770	0.514288	0.257144	−	0.966373i	$-0.417219\pi$
$349$	9.60770	0.514288	0.257144	+	0.966373i	$0.417219\pi$
$350$	0	0
$351$	−21.8564	−1.16661
$352$	0	0
$353$	26.4449	1.40752	0.703759	−	0.710439i	$-0.251503\pi$
$353$	26.4449	1.40752	0.703759	+	0.710439i	$0.251503\pi$
$354$	0	0
$355$	−2.53590	−0.134592
$356$	0	0
$357$	2.53590	0.134214
$358$	0	0
$359$	4.73205	0.249748	0.124874	−	0.992173i	$-0.460147\pi$
$359$	4.73205	0.249748	0.124874	+	0.992173i	$0.460147\pi$
$360$	0	0
$361$	−18.4641	−0.971795
$362$	0	0
$363$	0.732051	0.0384227
$364$	0	0
$365$	4.53590	0.237420
$366$	0	0
$367$	−22.5359	−1.17636	−0.588182	−	0.808728i	$-0.700156\pi$
$367$	−22.5359	−1.17636	−0.588182	+	0.808728i	$0.700156\pi$
$368$	0	0
$369$	3.12436	0.162647
$370$	0	0
$371$	1.26795	0.0658286
$372$	0	0
$373$	9.60770	0.497468	0.248734	−	0.968572i	$-0.419986\pi$
$373$	9.60770	0.497468	0.248734	+	0.968572i	$0.419986\pi$
$374$	0	0
$375$	0.732051	0.0378029
$376$	0	0
$377$	−6.92820	−0.356821
$378$	0	0
$379$	31.7128	1.62898	0.814489	−	0.580179i	$-0.197017\pi$
$379$	31.7128	1.62898	0.814489	+	0.580179i	$0.197017\pi$
$380$	0	0
$381$	13.0718	0.669688
$382$	0	0
$383$	4.39230	0.224436	0.112218	−	0.993684i	$-0.464204\pi$
$383$	4.39230	0.224436	0.112218	+	0.993684i	$0.464204\pi$
$384$	0	0
$385$	1.00000	0.0509647
$386$	0	0
$387$	22.0000	1.11832
$388$	0	0
$389$	24.2487	1.22946	0.614729	−	0.788738i	$-0.289265\pi$
$389$	24.2487	1.22946	0.614729	+	0.788738i	$0.289265\pi$
$390$	0	0
$391$	16.3923	0.828994
$392$	0	0
$393$	0.248711	0.0125458
$394$	0	0
$395$	−3.26795	−0.164428
$396$	0	0
$397$	−17.6077	−0.883705	−0.441852	−	0.897088i	$-0.645679\pi$
$397$	−17.6077	−0.883705	−0.441852	+	0.897088i	$0.645679\pi$
$398$	0	0
$399$	0.535898	0.0268285
$400$	0	0
$401$	37.1769	1.85653	0.928263	−	0.371924i	$-0.121302\pi$
$401$	37.1769	1.85653	0.928263	+	0.371924i	$0.121302\pi$
$402$	0	0
$403$	26.9282	1.34139
$404$	0	0
$405$	4.46410	0.221823
$406$	0	0
$407$	−6.73205	−0.333695
$408$	0	0
$409$	11.8038	0.583663	0.291831	−	0.956470i	$-0.405735\pi$
$409$	11.8038	0.583663	0.291831	+	0.956470i	$0.405735\pi$
$410$	0	0
$411$	−14.5359	−0.717003
$412$	0	0
$413$	13.8564	0.681829
$414$	0	0
$415$	−16.3923	−0.804667
$416$	0	0
$417$	4.53590	0.222124
$418$	0	0
$419$	8.78461	0.429156	0.214578	−	0.976707i	$-0.431162\pi$
$419$	8.78461	0.429156	0.214578	+	0.976707i	$0.431162\pi$
$420$	0	0
$421$	−30.7846	−1.50035	−0.750175	−	0.661239i	$-0.770031\pi$
$421$	−30.7846	−1.50035	−0.750175	+	0.661239i	$0.770031\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−3.46410	−0.168034
$426$	0	0
$427$	−2.00000	−0.0967868
$428$	0	0
$429$	−4.00000	−0.193122
$430$	0	0
$431$	−35.6603	−1.71769	−0.858847	−	0.512232i	$-0.828819\pi$
$431$	−35.6603	−1.71769	−0.858847	+	0.512232i	$0.828819\pi$
$432$	0	0
$433$	−2.73205	−0.131294	−0.0656470	−	0.997843i	$-0.520911\pi$
$433$	−2.73205	−0.131294	−0.0656470	+	0.997843i	$0.520911\pi$
$434$	0	0
$435$	−0.928203	−0.0445039
$436$	0	0
$437$	3.46410	0.165710
$438$	0	0
$439$	−15.6077	−0.744915	−0.372457	−	0.928049i	$-0.621485\pi$
$439$	−15.6077	−0.744915	−0.372457	+	0.928049i	$0.621485\pi$
$440$	0	0
$441$	−2.46410	−0.117338
$442$	0	0
$443$	20.7846	0.987507	0.493753	−	0.869602i	$-0.335625\pi$
$443$	20.7846	0.987507	0.493753	+	0.869602i	$0.335625\pi$
$444$	0	0
$445$	−8.53590	−0.404640
$446$	0	0
$447$	−7.85641	−0.371595
$448$	0	0
$449$	−0.679492	−0.0320672	−0.0160336	−	0.999871i	$-0.505104\pi$
$449$	−0.679492	−0.0320672	−0.0160336	+	0.999871i	$0.505104\pi$
$450$	0	0
$451$	1.26795	0.0597054
$452$	0	0
$453$	13.3205	0.625852
$454$	0	0
$455$	−5.46410	−0.256161
$456$	0	0
$457$	−34.0000	−1.59045	−0.795226	−	0.606313i	$-0.792648\pi$
$457$	−34.0000	−1.59045	−0.795226	+	0.606313i	$0.792648\pi$
$458$	0	0
$459$	13.8564	0.646762
$460$	0	0
$461$	−5.32051	−0.247801	−0.123900	−	0.992295i	$-0.539540\pi$
$461$	−5.32051	−0.247801	−0.123900	+	0.992295i	$0.539540\pi$
$462$	0	0
$463$	38.9808	1.81159	0.905795	−	0.423717i	$-0.139275\pi$
$463$	38.9808	1.81159	0.905795	+	0.423717i	$0.139275\pi$
$464$	0	0
$465$	3.60770	0.167303
$466$	0	0
$467$	−0.339746	−0.0157216	−0.00786078	−	0.999969i	$-0.502502\pi$
$467$	−0.339746	−0.0157216	−0.00786078	+	0.999969i	$0.502502\pi$
$468$	0	0
$469$	2.92820	0.135212
$470$	0	0
$471$	−10.5359	−0.485469
$472$	0	0
$473$	8.92820	0.410519
$474$	0	0
$475$	−0.732051	−0.0335888
$476$	0	0
$477$	3.12436	0.143054
$478$	0	0
$479$	13.8564	0.633115	0.316558	−	0.948573i	$-0.397473\pi$
$479$	13.8564	0.633115	0.316558	+	0.948573i	$0.397473\pi$
$480$	0	0
$481$	36.7846	1.67723
$482$	0	0
$483$	3.46410	0.157622
$484$	0	0
$485$	16.1962	0.735429
$486$	0	0
$487$	−29.1244	−1.31975	−0.659875	−	0.751375i	$-0.729391\pi$
$487$	−29.1244	−1.31975	−0.659875	+	0.751375i	$0.729391\pi$
$488$	0	0
$489$	−5.85641	−0.264836
$490$	0	0
$491$	−29.0718	−1.31199	−0.655996	−	0.754764i	$-0.727751\pi$
$491$	−29.0718	−1.31199	−0.655996	+	0.754764i	$0.727751\pi$
$492$	0	0
$493$	4.39230	0.197819
$494$	0	0
$495$	2.46410	0.110753
$496$	0	0
$497$	2.53590	0.113751
$498$	0	0
$499$	−2.00000	−0.0895323	−0.0447661	−	0.998997i	$-0.514254\pi$
$499$	−2.00000	−0.0895323	−0.0447661	+	0.998997i	$0.514254\pi$
$500$	0	0
$501$	−10.1436	−0.453182
$502$	0	0
$503$	5.07180	0.226140	0.113070	−	0.993587i	$-0.463932\pi$
$503$	5.07180	0.226140	0.113070	+	0.993587i	$0.463932\pi$
$504$	0	0
$505$	−6.00000	−0.266996
$506$	0	0
$507$	12.3397	0.548027
$508$	0	0
$509$	−38.7846	−1.71910	−0.859549	−	0.511054i	$-0.829255\pi$
$509$	−38.7846	−1.71910	−0.859549	+	0.511054i	$0.829255\pi$
$510$	0	0
$511$	−4.53590	−0.200656
$512$	0	0
$513$	2.92820	0.129283
$514$	0	0
$515$	−17.4641	−0.769560
$516$	0	0
$517$	0	0
$518$	0	0
$519$	−9.46410	−0.415428
$520$	0	0
$521$	−10.3923	−0.455295	−0.227648	−	0.973744i	$-0.573103\pi$
$521$	−10.3923	−0.455295	−0.227648	+	0.973744i	$0.573103\pi$
$522$	0	0
$523$	15.3205	0.669919	0.334960	−	0.942233i	$-0.391277\pi$
$523$	15.3205	0.669919	0.334960	+	0.942233i	$0.391277\pi$
$524$	0	0
$525$	−0.732051	−0.0319493
$526$	0	0
$527$	−17.0718	−0.743659
$528$	0	0
$529$	−0.607695	−0.0264215
$530$	0	0
$531$	34.1436	1.48171
$532$	0	0
$533$	−6.92820	−0.300094
$534$	0	0
$535$	12.9282	0.558935
$536$	0	0
$537$	−5.75129	−0.248186
$538$	0	0
$539$	−1.00000	−0.0430730
$540$	0	0
$541$	−7.12436	−0.306300	−0.153150	−	0.988203i	$-0.548942\pi$
$541$	−7.12436	−0.306300	−0.153150	+	0.988203i	$0.548942\pi$
$542$	0	0
$543$	11.6077	0.498134
$544$	0	0
$545$	−12.1962	−0.522426
$546$	0	0
$547$	36.7846	1.57280	0.786398	−	0.617720i	$-0.211943\pi$
$547$	36.7846	1.57280	0.786398	+	0.617720i	$0.211943\pi$
$548$	0	0
$549$	−4.92820	−0.210331
$550$	0	0
$551$	0.928203	0.0395428
$552$	0	0
$553$	3.26795	0.138967
$554$	0	0
$555$	4.92820	0.209191
$556$	0	0
$557$	36.2487	1.53591	0.767954	−	0.640505i	$-0.221275\pi$
$557$	36.2487	1.53591	0.767954	+	0.640505i	$0.221275\pi$
$558$	0	0
$559$	−48.7846	−2.06337
$560$	0	0
$561$	2.53590	0.107066
$562$	0	0
$563$	−20.7846	−0.875967	−0.437983	−	0.898983i	$-0.644307\pi$
$563$	−20.7846	−0.875967	−0.437983	+	0.898983i	$0.644307\pi$
$564$	0	0
$565$	−0.928203	−0.0390498
$566$	0	0
$567$	−4.46410	−0.187475
$568$	0	0
$569$	−30.0000	−1.25767	−0.628833	−	0.777541i	$-0.716467\pi$
$569$	−30.0000	−1.25767	−0.628833	+	0.777541i	$0.716467\pi$
$570$	0	0
$571$	−0.392305	−0.0164174	−0.00820872	−	0.999966i	$-0.502613\pi$
$571$	−0.392305	−0.0164174	−0.00820872	+	0.999966i	$0.502613\pi$
$572$	0	0
$573$	5.07180	0.211877
$574$	0	0
$575$	−4.73205	−0.197340
$576$	0	0
$577$	25.6603	1.06825	0.534125	−	0.845405i	$-0.320641\pi$
$577$	25.6603	1.06825	0.534125	+	0.845405i	$0.320641\pi$
$578$	0	0
$579$	−13.5692	−0.563918
$580$	0	0
$581$	16.3923	0.680067
$582$	0	0
$583$	1.26795	0.0525131
$584$	0	0
$585$	−13.4641	−0.556672
$586$	0	0
$587$	−22.9808	−0.948518	−0.474259	−	0.880385i	$-0.657284\pi$
$587$	−22.9808	−0.948518	−0.474259	+	0.880385i	$0.657284\pi$
$588$	0	0
$589$	−3.60770	−0.148652
$590$	0	0
$591$	−1.17691	−0.0484118
$592$	0	0
$593$	39.4641	1.62060	0.810298	−	0.586018i	$-0.199305\pi$
$593$	39.4641	1.62060	0.810298	+	0.586018i	$0.199305\pi$
$594$	0	0
$595$	3.46410	0.142014
$596$	0	0
$597$	−12.2872	−0.502881
$598$	0	0
$599$	−44.1051	−1.80209	−0.901043	−	0.433730i	$-0.857197\pi$
$599$	−44.1051	−1.80209	−0.901043	+	0.433730i	$0.857197\pi$
$600$	0	0
$601$	−30.4449	−1.24187	−0.620936	−	0.783861i	$-0.713247\pi$
$601$	−30.4449	−1.24187	−0.620936	+	0.783861i	$0.713247\pi$
$602$	0	0
$603$	7.21539	0.293833
$604$	0	0
$605$	1.00000	0.0406558
$606$	0	0
$607$	6.78461	0.275379	0.137689	−	0.990475i	$-0.456032\pi$
$607$	6.78461	0.275379	0.137689	+	0.990475i	$0.456032\pi$
$608$	0	0
$609$	0.928203	0.0376127
$610$	0	0
$611$	0	0
$612$	0	0
$613$	−35.1769	−1.42078	−0.710391	−	0.703807i	$-0.751482\pi$
$613$	−35.1769	−1.42078	−0.710391	+	0.703807i	$0.751482\pi$
$614$	0	0
$615$	−0.928203	−0.0374288
$616$	0	0
$617$	15.4641	0.622561	0.311281	−	0.950318i	$-0.399242\pi$
$617$	15.4641	0.622561	0.311281	+	0.950318i	$0.399242\pi$
$618$	0	0
$619$	−2.92820	−0.117694	−0.0588472	−	0.998267i	$-0.518742\pi$
$619$	−2.92820	−0.117694	−0.0588472	+	0.998267i	$0.518742\pi$
$620$	0	0
$621$	18.9282	0.759563
$622$	0	0
$623$	8.53590	0.341984
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0.535898	0.0214017
$628$	0	0
$629$	−23.3205	−0.929850
$630$	0	0
$631$	−28.7846	−1.14590	−0.572949	−	0.819591i	$-0.694201\pi$
$631$	−28.7846	−1.14590	−0.572949	+	0.819591i	$0.694201\pi$
$632$	0	0
$633$	−5.85641	−0.232771
$634$	0	0
$635$	17.8564	0.708610
$636$	0	0
$637$	5.46410	0.216496
$638$	0	0
$639$	6.24871	0.247195
$640$	0	0
$641$	−7.85641	−0.310309	−0.155155	−	0.987890i	$-0.549588\pi$
$641$	−7.85641	−0.310309	−0.155155	+	0.987890i	$0.549588\pi$
$642$	0	0
$643$	11.2679	0.444365	0.222182	−	0.975005i	$-0.428682\pi$
$643$	11.2679	0.444365	0.222182	+	0.975005i	$0.428682\pi$
$644$	0	0
$645$	−6.53590	−0.257351
$646$	0	0
$647$	25.1769	0.989807	0.494903	−	0.868948i	$-0.335203\pi$
$647$	25.1769	0.989807	0.494903	+	0.868948i	$0.335203\pi$
$648$	0	0
$649$	13.8564	0.543912
$650$	0	0
$651$	−3.60770	−0.141397
$652$	0	0
$653$	−20.1962	−0.790337	−0.395168	−	0.918609i	$-0.629314\pi$
$653$	−20.1962	−0.790337	−0.395168	+	0.918609i	$0.629314\pi$
$654$	0	0
$655$	0.339746	0.0132750
$656$	0	0
$657$	−11.1769	−0.436053
$658$	0	0
$659$	42.2487	1.64578	0.822888	−	0.568204i	$-0.192361\pi$
$659$	42.2487	1.64578	0.822888	+	0.568204i	$0.192361\pi$
$660$	0	0
$661$	−23.8564	−0.927907	−0.463953	−	0.885860i	$-0.653569\pi$
$661$	−23.8564	−0.927907	−0.463953	+	0.885860i	$0.653569\pi$
$662$	0	0
$663$	−13.8564	−0.538138
$664$	0	0
$665$	0.732051	0.0283877
$666$	0	0
$667$	6.00000	0.232321
$668$	0	0
$669$	−9.07180	−0.350736
$670$	0	0
$671$	−2.00000	−0.0772091
$672$	0	0
$673$	−47.8564	−1.84473	−0.922364	−	0.386321i	$-0.873746\pi$
$673$	−47.8564	−1.84473	−0.922364	+	0.386321i	$0.873746\pi$
$674$	0	0
$675$	−4.00000	−0.153960
$676$	0	0
$677$	−7.85641	−0.301946	−0.150973	−	0.988538i	$-0.548241\pi$
$677$	−7.85641	−0.301946	−0.150973	+	0.988538i	$0.548241\pi$
$678$	0	0
$679$	−16.1962	−0.621551
$680$	0	0
$681$	10.1436	0.388703
$682$	0	0
$683$	6.24871	0.239100	0.119550	−	0.992828i	$-0.461855\pi$
$683$	6.24871	0.239100	0.119550	+	0.992828i	$0.461855\pi$
$684$	0	0
$685$	−19.8564	−0.758674
$686$	0	0
$687$	−4.78461	−0.182544
$688$	0	0
$689$	−6.92820	−0.263944
$690$	0	0
$691$	3.32051	0.126318	0.0631590	−	0.998003i	$-0.479882\pi$
$691$	3.32051	0.126318	0.0631590	+	0.998003i	$0.479882\pi$
$692$	0	0
$693$	−2.46410	−0.0936035
$694$	0	0
$695$	6.19615	0.235033
$696$	0	0
$697$	4.39230	0.166370
$698$	0	0
$699$	−14.5359	−0.549798
$700$	0	0
$701$	3.80385	0.143669	0.0718347	−	0.997417i	$-0.477115\pi$
$701$	3.80385	0.143669	0.0718347	+	0.997417i	$0.477115\pi$
$702$	0	0
$703$	−4.92820	−0.185871
$704$	0	0
$705$	0	0
$706$	0	0
$707$	6.00000	0.225653
$708$	0	0
$709$	18.3923	0.690738	0.345369	−	0.938467i	$-0.387754\pi$
$709$	18.3923	0.690738	0.345369	+	0.938467i	$0.387754\pi$
$710$	0	0
$711$	8.05256	0.301995
$712$	0	0
$713$	−23.3205	−0.873360
$714$	0	0
$715$	−5.46410	−0.204346
$716$	0	0
$717$	10.3923	0.388108
$718$	0	0
$719$	−24.0000	−0.895049	−0.447524	−	0.894272i	$-0.647694\pi$
$719$	−24.0000	−0.895049	−0.447524	+	0.894272i	$0.647694\pi$
$720$	0	0
$721$	17.4641	0.650397
$722$	0	0
$723$	−5.71281	−0.212462
$724$	0	0
$725$	−1.26795	−0.0470905
$726$	0	0
$727$	−30.6410	−1.13641	−0.568206	−	0.822886i	$-0.692362\pi$
$727$	−30.6410	−1.13641	−0.568206	+	0.822886i	$0.692362\pi$
$728$	0	0
$729$	−2.21539	−0.0820515
$730$	0	0
$731$	30.9282	1.14392
$732$	0	0
$733$	2.00000	0.0738717	0.0369358	−	0.999318i	$-0.488240\pi$
$733$	2.00000	0.0738717	0.0369358	+	0.999318i	$0.488240\pi$
$734$	0	0
$735$	0.732051	0.0270021
$736$	0	0
$737$	2.92820	0.107862
$738$	0	0
$739$	−45.8564	−1.68686	−0.843428	−	0.537243i	$-0.819466\pi$
$739$	−45.8564	−1.68686	−0.843428	+	0.537243i	$0.819466\pi$
$740$	0	0
$741$	−2.92820	−0.107570
$742$	0	0
$743$	51.7128	1.89716	0.948580	−	0.316539i	$-0.102521\pi$
$743$	51.7128	1.89716	0.948580	+	0.316539i	$0.102521\pi$
$744$	0	0
$745$	−10.7321	−0.393192
$746$	0	0
$747$	40.3923	1.47788
$748$	0	0
$749$	−12.9282	−0.472386
$750$	0	0
$751$	46.2487	1.68764	0.843820	−	0.536627i	$-0.180302\pi$
$751$	46.2487	1.68764	0.843820	+	0.536627i	$0.180302\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	18.1962	0.662226
$756$	0	0
$757$	−49.3731	−1.79449	−0.897247	−	0.441528i	$-0.854436\pi$
$757$	−49.3731	−1.79449	−0.897247	+	0.441528i	$0.854436\pi$
$758$	0	0
$759$	3.46410	0.125739
$760$	0	0
$761$	22.7321	0.824036	0.412018	−	0.911176i	$-0.364824\pi$
$761$	22.7321	0.824036	0.412018	+	0.911176i	$0.364824\pi$
$762$	0	0
$763$	12.1962	0.441530
$764$	0	0
$765$	8.53590	0.308616
$766$	0	0
$767$	−75.7128	−2.73383
$768$	0	0
$769$	34.4449	1.24211	0.621057	−	0.783766i	$-0.286704\pi$
$769$	34.4449	1.24211	0.621057	+	0.783766i	$0.286704\pi$
$770$	0	0
$771$	−21.2154	−0.764054
$772$	0	0
$773$	30.0000	1.07903	0.539513	−	0.841978i	$-0.318609\pi$
$773$	30.0000	1.07903	0.539513	+	0.841978i	$0.318609\pi$
$774$	0	0
$775$	4.92820	0.177026
$776$	0	0
$777$	−4.92820	−0.176798
$778$	0	0
$779$	0.928203	0.0332563
$780$	0	0
$781$	2.53590	0.0907416
$782$	0	0
$783$	5.07180	0.181251
$784$	0	0
$785$	−14.3923	−0.513683
$786$	0	0
$787$	28.0000	0.998092	0.499046	−	0.866575i	$-0.333684\pi$
$787$	28.0000	0.998092	0.499046	+	0.866575i	$0.333684\pi$
$788$	0	0
$789$	−3.71281	−0.132180
$790$	0	0
$791$	0.928203	0.0330031
$792$	0	0
$793$	10.9282	0.388072
$794$	0	0
$795$	−0.928203	−0.0329200
$796$	0	0
$797$	41.3205	1.46365	0.731824	−	0.681494i	$-0.238669\pi$
$797$	41.3205	1.46365	0.731824	+	0.681494i	$0.238669\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	21.0333	0.743176
$802$	0	0
$803$	−4.53590	−0.160068
$804$	0	0
$805$	4.73205	0.166783
$806$	0	0
$807$	−12.0000	−0.422420
$808$	0	0
$809$	−29.3205	−1.03085	−0.515427	−	0.856933i	$-0.672367\pi$
$809$	−29.3205	−1.03085	−0.515427	+	0.856933i	$0.672367\pi$
$810$	0	0
$811$	−26.5885	−0.933647	−0.466824	−	0.884351i	$-0.654602\pi$
$811$	−26.5885	−0.933647	−0.466824	+	0.884351i	$0.654602\pi$
$812$	0	0
$813$	−2.64102	−0.0926245
$814$	0	0
$815$	−8.00000	−0.280228
$816$	0	0
$817$	6.53590	0.228662
$818$	0	0
$819$	13.4641	0.470474
$820$	0	0
$821$	13.9474	0.486769	0.243385	−	0.969930i	$-0.421742\pi$
$821$	13.9474	0.486769	0.243385	+	0.969930i	$0.421742\pi$
$822$	0	0
$823$	13.8038	0.481172	0.240586	−	0.970628i	$-0.422660\pi$
$823$	13.8038	0.481172	0.240586	+	0.970628i	$0.422660\pi$
$824$	0	0
$825$	−0.732051	−0.0254867
$826$	0	0
$827$	29.5692	1.02822	0.514111	−	0.857724i	$-0.328122\pi$
$827$	29.5692	1.02822	0.514111	+	0.857724i	$0.328122\pi$
$828$	0	0
$829$	−36.1051	−1.25398	−0.626991	−	0.779026i	$-0.715714\pi$
$829$	−36.1051	−1.25398	−0.626991	+	0.779026i	$0.715714\pi$
$830$	0	0
$831$	0.105118	0.00364649
$832$	0	0
$833$	−3.46410	−0.120024
$834$	0	0
$835$	−13.8564	−0.479521
$836$	0	0
$837$	−19.7128	−0.681374
$838$	0	0
$839$	−2.78461	−0.0961354	−0.0480677	−	0.998844i	$-0.515306\pi$
$839$	−2.78461	−0.0961354	−0.0480677	+	0.998844i	$0.515306\pi$
$840$	0	0
$841$	−27.3923	−0.944562
$842$	0	0
$843$	7.60770	0.262023
$844$	0	0
$845$	16.8564	0.579878
$846$	0	0
$847$	−1.00000	−0.0343604
$848$	0	0
$849$	−2.14359	−0.0735679
$850$	0	0
$851$	−31.8564	−1.09202
$852$	0	0
$853$	7.07180	0.242134	0.121067	−	0.992644i	$-0.461368\pi$
$853$	7.07180	0.242134	0.121067	+	0.992644i	$0.461368\pi$
$854$	0	0
$855$	1.80385	0.0616903
$856$	0	0
$857$	43.1769	1.47490	0.737448	−	0.675404i	$-0.236031\pi$
$857$	43.1769	1.47490	0.737448	+	0.675404i	$0.236031\pi$
$858$	0	0
$859$	0.784610	0.0267705	0.0133853	−	0.999910i	$-0.495739\pi$
$859$	0.784610	0.0267705	0.0133853	+	0.999910i	$0.495739\pi$
$860$	0	0
$861$	0.928203	0.0316331
$862$	0	0
$863$	6.58846	0.224274	0.112137	−	0.993693i	$-0.464230\pi$
$863$	6.58846	0.224274	0.112137	+	0.993693i	$0.464230\pi$
$864$	0	0
$865$	−12.9282	−0.439572
$866$	0	0
$867$	−3.66025	−0.124309
$868$	0	0
$869$	3.26795	0.110858
$870$	0	0
$871$	−16.0000	−0.542139
$872$	0	0
$873$	−39.9090	−1.35071
$874$	0	0
$875$	−1.00000	−0.0338062
$876$	0	0
$877$	39.1769	1.32291	0.661455	−	0.749985i	$-0.269939\pi$
$877$	39.1769	1.32291	0.661455	+	0.749985i	$0.269939\pi$
$878$	0	0
$879$	−6.92820	−0.233682
$880$	0	0
$881$	11.0718	0.373018	0.186509	−	0.982453i	$-0.440283\pi$
$881$	11.0718	0.373018	0.186509	+	0.982453i	$0.440283\pi$
$882$	0	0
$883$	−21.1769	−0.712660	−0.356330	−	0.934360i	$-0.615972\pi$
$883$	−21.1769	−0.712660	−0.356330	+	0.934360i	$0.615972\pi$
$884$	0	0
$885$	−10.1436	−0.340973
$886$	0	0
$887$	−24.0000	−0.805841	−0.402921	−	0.915235i	$-0.632005\pi$
$887$	−24.0000	−0.805841	−0.402921	+	0.915235i	$0.632005\pi$
$888$	0	0
$889$	−17.8564	−0.598885
$890$	0	0
$891$	−4.46410	−0.149553
$892$	0	0
$893$	0	0
$894$	0	0
$895$	−7.85641	−0.262611
$896$	0	0
$897$	−18.9282	−0.631994
$898$	0	0
$899$	−6.24871	−0.208406
$900$	0	0
$901$	4.39230	0.146329
$902$	0	0
$903$	6.53590	0.217501
$904$	0	0
$905$	15.8564	0.527085
$906$	0	0
$907$	−19.3205	−0.641527	−0.320763	−	0.947159i	$-0.603939\pi$
$907$	−19.3205	−0.641527	−0.320763	+	0.947159i	$0.603939\pi$
$908$	0	0
$909$	14.7846	0.490375
$910$	0	0
$911$	−21.4641	−0.711137	−0.355569	−	0.934650i	$-0.615713\pi$
$911$	−21.4641	−0.711137	−0.355569	+	0.934650i	$0.615713\pi$
$912$	0	0
$913$	16.3923	0.542506
$914$	0	0
$915$	1.46410	0.0484017
$916$	0	0
$917$	−0.339746	−0.0112194
$918$	0	0
$919$	26.9808	0.890013	0.445007	−	0.895527i	$-0.353201\pi$
$919$	26.9808	0.890013	0.445007	+	0.895527i	$0.353201\pi$
$920$	0	0
$921$	2.43078	0.0800969
$922$	0	0
$923$	−13.8564	−0.456089
$924$	0	0
$925$	6.73205	0.221348
$926$	0	0
$927$	43.0333	1.41340
$928$	0	0
$929$	−50.1051	−1.64390	−0.821948	−	0.569563i	$-0.807113\pi$
$929$	−50.1051	−1.64390	−0.821948	+	0.569563i	$0.807113\pi$
$930$	0	0
$931$	−0.732051	−0.0239920
$932$	0	0
$933$	14.5359	0.475884
$934$	0	0
$935$	3.46410	0.113288
$936$	0	0
$937$	26.0000	0.849383	0.424691	−	0.905338i	$-0.360383\pi$
$937$	26.0000	0.849383	0.424691	+	0.905338i	$0.360383\pi$
$938$	0	0
$939$	22.4974	0.734176
$940$	0	0
$941$	47.5692	1.55071	0.775356	−	0.631524i	$-0.217570\pi$
$941$	47.5692	1.55071	0.775356	+	0.631524i	$0.217570\pi$
$942$	0	0
$943$	6.00000	0.195387
$944$	0	0
$945$	4.00000	0.130120
$946$	0	0
$947$	−37.1769	−1.20809	−0.604044	−	0.796951i	$-0.706445\pi$
$947$	−37.1769	−1.20809	−0.604044	+	0.796951i	$0.706445\pi$
$948$	0	0
$949$	24.7846	0.804542
$950$	0	0
$951$	19.3590	0.627758
$952$	0	0
$953$	23.3205	0.755425	0.377713	−	0.925923i	$-0.376711\pi$
$953$	23.3205	0.755425	0.377713	+	0.925923i	$0.376711\pi$
$954$	0	0
$955$	6.92820	0.224191
$956$	0	0
$957$	0.928203	0.0300045
$958$	0	0
$959$	19.8564	0.641197
$960$	0	0
$961$	−6.71281	−0.216542
$962$	0	0
$963$	−31.8564	−1.02656
$964$	0	0
$965$	−18.5359	−0.596692
$966$	0	0
$967$	53.8564	1.73191	0.865953	−	0.500126i	$-0.166713\pi$
$967$	53.8564	1.73191	0.865953	+	0.500126i	$0.166713\pi$
$968$	0	0
$969$	1.85641	0.0596364
$970$	0	0
$971$	−22.1436	−0.710622	−0.355311	−	0.934748i	$-0.615625\pi$
$971$	−22.1436	−0.710622	−0.355311	+	0.934748i	$0.615625\pi$
$972$	0	0
$973$	−6.19615	−0.198640
$974$	0	0
$975$	4.00000	0.128103
$976$	0	0
$977$	12.9282	0.413610	0.206805	−	0.978382i	$-0.433693\pi$
$977$	12.9282	0.413610	0.206805	+	0.978382i	$0.433693\pi$
$978$	0	0
$979$	8.53590	0.272808
$980$	0	0
$981$	30.0526	0.959504
$982$	0	0
$983$	28.3923	0.905574	0.452787	−	0.891619i	$-0.350430\pi$
$983$	28.3923	0.905574	0.452787	+	0.891619i	$0.350430\pi$
$984$	0	0
$985$	−1.60770	−0.0512254
$986$	0	0
$987$	0	0
$988$	0	0
$989$	42.2487	1.34343
$990$	0	0
$991$	9.07180	0.288175	0.144088	−	0.989565i	$-0.453975\pi$
$991$	9.07180	0.288175	0.144088	+	0.989565i	$0.453975\pi$
$992$	0	0
$993$	−6.53590	−0.207410
$994$	0	0
$995$	−16.7846	−0.532108
$996$	0	0
$997$	3.85641	0.122134	0.0610668	−	0.998134i	$-0.480550\pi$
$997$	3.85641	0.122134	0.0610668	+	0.998134i	$0.480550\pi$
$998$	0	0
$999$	−26.9282	−0.851971

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6160.2.a.v.1.2		2
4.3	odd	2		770.2.a.h.1.1	✓	2
12.11	even	2		6930.2.a.ca.1.2		2
20.3	even	4		3850.2.c.s.1849.3		4
20.7	even	4		3850.2.c.s.1849.2		4
20.19	odd	2		3850.2.a.bm.1.2		2
28.27	even	2		5390.2.a.bk.1.2		2
44.43	even	2		8470.2.a.ce.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
770.2.a.h.1.1	✓	2	4.3	odd	2
3850.2.a.bm.1.2		2	20.19	odd	2
3850.2.c.s.1849.2		4	20.7	even	4
3850.2.c.s.1849.3		4	20.3	even	4
5390.2.a.bk.1.2		2	28.27	even	2
6160.2.a.v.1.2		2	1.1	even	1	trivial
6930.2.a.ca.1.2		2	12.11	even	2
8470.2.a.ce.1.1		2	44.43	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 $\F_{q}$

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

\(f(q)\)	\(=\)	\(q+0.732051 q^{3} +1.00000 q^{5} -1.00000 q^{7} -2.46410 q^{9} +O(q^{10})\)	\(q+0.732051 q^{3} +1.00000 q^{5} -1.00000 q^{7} -2.46410 q^{9} -1.00000 q^{11} +5.46410 q^{13} +0.732051 q^{15} -3.46410 q^{17} -0.732051 q^{19} -0.732051 q^{21} -4.73205 q^{23} +1.00000 q^{25} -4.00000 q^{27} -1.26795 q^{29} +4.92820 q^{31} -0.732051 q^{33} -1.00000 q^{35} +6.73205 q^{37} +4.00000 q^{39} -1.26795 q^{41} -8.92820 q^{43} -2.46410 q^{45} +1.00000 q^{49} -2.53590 q^{51} -1.26795 q^{53} -1.00000 q^{55} -0.535898 q^{57} -13.8564 q^{59} +2.00000 q^{61} +2.46410 q^{63} +5.46410 q^{65} -2.92820 q^{67} -3.46410 q^{69} -2.53590 q^{71} +4.53590 q^{73} +0.732051 q^{75} +1.00000 q^{77} -3.26795 q^{79} +4.46410 q^{81} -16.3923 q^{83} -3.46410 q^{85} -0.928203 q^{87} -8.53590 q^{89} -5.46410 q^{91} +3.60770 q^{93} -0.732051 q^{95} +16.1962 q^{97} +2.46410 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} + 2 q^{5} - 2 q^{7} + 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^3 + 2 * q^5 - 2 * q^7 + 2 * q^9	\( 2 q - 2 q^{3} + 2 q^{5} - 2 q^{7} + 2 q^{9} - 2 q^{11} + 4 q^{13} - 2 q^{15} + 2 q^{19} + 2 q^{21} - 6 q^{23} + 2 q^{25} - 8 q^{27} - 6 q^{29} - 4 q^{31} + 2 q^{33} - 2 q^{35} + 10 q^{37} + 8 q^{39} - 6 q^{41} - 4 q^{43} + 2 q^{45} + 2 q^{49} - 12 q^{51} - 6 q^{53} - 2 q^{55} - 8 q^{57} + 4 q^{61} - 2 q^{63} + 4 q^{65} + 8 q^{67} - 12 q^{71} + 16 q^{73} - 2 q^{75} + 2 q^{77} - 10 q^{79} + 2 q^{81} - 12 q^{83} + 12 q^{87} - 24 q^{89} - 4 q^{91} + 28 q^{93} + 2 q^{95} + 22 q^{97} - 2 q^{99}+O(q^{100}) \) 2 * q - 2 * q^3 + 2 * q^5 - 2 * q^7 + 2 * q^9 - 2 * q^11 + 4 * q^13 - 2 * q^15 + 2 * q^19 + 2 * q^21 - 6 * q^23 + 2 * q^25 - 8 * q^27 - 6 * q^29 - 4 * q^31 + 2 * q^33 - 2 * q^35 + 10 * q^37 + 8 * q^39 - 6 * q^41 - 4 * q^43 + 2 * q^45 + 2 * q^49 - 12 * q^51 - 6 * q^53 - 2 * q^55 - 8 * q^57 + 4 * q^61 - 2 * q^63 + 4 * q^65 + 8 * q^67 - 12 * q^71 + 16 * q^73 - 2 * q^75 + 2 * q^77 - 10 * q^79 + 2 * q^81 - 12 * q^83 + 12 * q^87 - 24 * q^89 - 4 * q^91 + 28 * q^93 + 2 * q^95 + 22 * q^97 - 2 * q^99

Introduction

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