LMFDB - Embedded newform 6930.2.a.ca.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6930,2,Mod(1,6930)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6930, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("6930.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6930 = 2 \cdot 3^{2} \cdot 5 \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6930.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:	$55.3363286007$
Analytic rank:	$0$
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:	$\Z[a_1, \ldots, a_{19}]$
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	$\chi$	$=$	6930.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{7} +1.00000 q^{8} +O(q^{10})$	$q+1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{7} +1.00000 q^{8} -1.00000 q^{10} -1.00000 q^{11} +5.46410 q^{13} +1.00000 q^{14} +1.00000 q^{16} +3.46410 q^{17} +0.732051 q^{19} -1.00000 q^{20} -1.00000 q^{22} -4.73205 q^{23} +1.00000 q^{25} +5.46410 q^{26} +1.00000 q^{28} +1.26795 q^{29} -4.92820 q^{31} +1.00000 q^{32} +3.46410 q^{34} -1.00000 q^{35} +6.73205 q^{37} +0.732051 q^{38} -1.00000 q^{40} +1.26795 q^{41} +8.92820 q^{43} -1.00000 q^{44} -4.73205 q^{46} +1.00000 q^{49} +1.00000 q^{50} +5.46410 q^{52} +1.26795 q^{53} +1.00000 q^{55} +1.00000 q^{56} +1.26795 q^{58} -13.8564 q^{59} +2.00000 q^{61} -4.92820 q^{62} +1.00000 q^{64} -5.46410 q^{65} +2.92820 q^{67} +3.46410 q^{68} -1.00000 q^{70} -2.53590 q^{71} +4.53590 q^{73} +6.73205 q^{74} +0.732051 q^{76} -1.00000 q^{77} +3.26795 q^{79} -1.00000 q^{80} +1.26795 q^{82} -16.3923 q^{83} -3.46410 q^{85} +8.92820 q^{86} -1.00000 q^{88} +8.53590 q^{89} +5.46410 q^{91} -4.73205 q^{92} -0.732051 q^{95} +16.1962 q^{97} +1.00000 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} + 2 q^{4} - 2 q^{5} + 2 q^{7} + 2 q^{8}+O(q^{10}) $ 2 * q + 2 * q^2 + 2 * q^4 - 2 * q^5 + 2 * q^7 + 2 * q^8	$ 2 q + 2 q^{2} + 2 q^{4} - 2 q^{5} + 2 q^{7} + 2 q^{8} - 2 q^{10} - 2 q^{11} + 4 q^{13} + 2 q^{14} + 2 q^{16} - 2 q^{19} - 2 q^{20} - 2 q^{22} - 6 q^{23} + 2 q^{25} + 4 q^{26} + 2 q^{28} + 6 q^{29} + 4 q^{31} + 2 q^{32} - 2 q^{35} + 10 q^{37} - 2 q^{38} - 2 q^{40} + 6 q^{41} + 4 q^{43} - 2 q^{44} - 6 q^{46} + 2 q^{49} + 2 q^{50} + 4 q^{52} + 6 q^{53} + 2 q^{55} + 2 q^{56} + 6 q^{58} + 4 q^{61} + 4 q^{62} + 2 q^{64} - 4 q^{65} - 8 q^{67} - 2 q^{70} - 12 q^{71} + 16 q^{73} + 10 q^{74} - 2 q^{76} - 2 q^{77} + 10 q^{79} - 2 q^{80} + 6 q^{82} - 12 q^{83} + 4 q^{86} - 2 q^{88} + 24 q^{89} + 4 q^{91} - 6 q^{92} + 2 q^{95} + 22 q^{97} + 2 q^{98}+O(q^{100}) $ 2 * q + 2 * q^2 + 2 * q^4 - 2 * q^5 + 2 * q^7 + 2 * q^8 - 2 * q^10 - 2 * q^11 + 4 * q^13 + 2 * q^14 + 2 * q^16 - 2 * q^19 - 2 * q^20 - 2 * q^22 - 6 * q^23 + 2 * q^25 + 4 * q^26 + 2 * q^28 + 6 * q^29 + 4 * q^31 + 2 * q^32 - 2 * q^35 + 10 * q^37 - 2 * q^38 - 2 * q^40 + 6 * q^41 + 4 * q^43 - 2 * q^44 - 6 * q^46 + 2 * q^49 + 2 * q^50 + 4 * q^52 + 6 * q^53 + 2 * q^55 + 2 * q^56 + 6 * q^58 + 4 * q^61 + 4 * q^62 + 2 * q^64 - 4 * q^65 - 8 * q^67 - 2 * q^70 - 12 * q^71 + 16 * q^73 + 10 * q^74 - 2 * q^76 - 2 * q^77 + 10 * q^79 - 2 * q^80 + 6 * q^82 - 12 * q^83 + 4 * q^86 - 2 * q^88 + 24 * q^89 + 4 * q^91 - 6 * q^92 + 2 * q^95 + 22 * q^97 + 2 * q^98

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 $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	1.00000	0.353553
$9$	0	0
$10$	−1.00000	−0.316228
$11$	−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	$0.226306\pi$
$14$	1.00000	0.267261
$15$	0	0
$16$	1.00000	0.250000
$17$	3.46410	0.840168	0.420084	−	0.907485i	$-0.362001\pi$
$17$	3.46410	0.840168	0.420084	+	0.907485i	$0.362001\pi$
$18$	0	0
$19$	0.732051	0.167944	0.0839720	−	0.996468i	$-0.473239\pi$
$19$	0.732051	0.167944	0.0839720	+	0.996468i	$0.473239\pi$
$20$	−1.00000	−0.223607
$21$	0	0
$22$	−1.00000	−0.213201
$23$	−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
$26$	5.46410	1.07160
$27$	0	0
$28$	1.00000	0.188982
$29$	1.26795	0.235452	0.117726	−	0.993046i	$-0.462440\pi$
$29$	1.26795	0.235452	0.117726	+	0.993046i	$0.462440\pi$
$30$	0	0
$31$	−4.92820	−0.885131	−0.442566	−	0.896736i	$-0.645932\pi$
$31$	−4.92820	−0.885131	−0.442566	+	0.896736i	$0.645932\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	3.46410	0.594089
$35$	−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	$0.313341\pi$
$38$	0.732051	0.118754
$39$	0	0
$40$	−1.00000	−0.158114
$41$	1.26795	0.198020	0.0990102	−	0.995086i	$-0.468432\pi$
$41$	1.26795	0.198020	0.0990102	+	0.995086i	$0.468432\pi$
$42$	0	0
$43$	8.92820	1.36154	0.680769	−	0.732498i	$-0.261646\pi$
$43$	8.92820	1.36154	0.680769	+	0.732498i	$0.261646\pi$
$44$	−1.00000	−0.150756
$45$	0	0
$46$	−4.73205	−0.697703
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	1.00000	0.141421
$51$	0	0
$52$	5.46410	0.757735
$53$	1.26795	0.174166	0.0870831	−	0.996201i	$-0.472245\pi$
$53$	1.26795	0.174166	0.0870831	+	0.996201i	$0.472245\pi$
$54$	0	0
$55$	1.00000	0.134840
$56$	1.00000	0.133631
$57$	0	0
$58$	1.26795	0.166490
$59$	−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	$0.459132\pi$
$62$	−4.92820	−0.625882
$63$	0	0
$64$	1.00000	0.125000
$65$	−5.46410	−0.677738
$66$	0	0
$67$	2.92820	0.357737	0.178868	−	0.983873i	$-0.442756\pi$
$67$	2.92820	0.357737	0.178868	+	0.983873i	$0.442756\pi$
$68$	3.46410	0.420084
$69$	0	0
$70$	−1.00000	−0.119523
$71$	−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	$0.414482\pi$
$74$	6.73205	0.782585
$75$	0	0
$76$	0.732051	0.0839720
$77$	−1.00000	−0.113961
$78$	0	0
$79$	3.26795	0.367673	0.183837	−	0.982957i	$-0.441148\pi$
$79$	3.26795	0.367673	0.183837	+	0.982957i	$0.441148\pi$
$80$	−1.00000	−0.111803
$81$	0	0
$82$	1.26795	0.140022
$83$	−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
$86$	8.92820	0.962753
$87$	0	0
$88$	−1.00000	−0.106600
$89$	8.53590	0.904803	0.452402	−	0.891814i	$-0.350567\pi$
$89$	8.53590	0.904803	0.452402	+	0.891814i	$0.350567\pi$
$90$	0	0
$91$	5.46410	0.572793
$92$	−4.73205	−0.493350
$93$	0	0
$94$	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	$0.192727\pi$
$98$	1.00000	0.101015
$99$	0	0
$100$	1.00000	0.100000
$101$	6.00000	0.597022	0.298511	−	0.954406i	$-0.403510\pi$
$101$	6.00000	0.597022	0.298511	+	0.954406i	$0.403510\pi$
$102$	0	0
$103$	17.4641	1.72079	0.860395	−	0.509629i	$-0.170217\pi$
$103$	17.4641	1.72079	0.860395	+	0.509629i	$0.170217\pi$
$104$	5.46410	0.535799
$105$	0	0
$106$	1.26795	0.123154
$107$	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	$0.698548\pi$
$110$	1.00000	0.0953463
$111$	0	0
$112$	1.00000	0.0944911
$113$	0.928203	0.0873180	0.0436590	−	0.999046i	$-0.486098\pi$
$113$	0.928203	0.0873180	0.0436590	+	0.999046i	$0.486098\pi$
$114$	0	0
$115$	4.73205	0.441266
$116$	1.26795	0.117726
$117$	0	0
$118$	−13.8564	−1.27559
$119$	3.46410	0.317554
$120$	0	0
$121$	1.00000	0.0909091
$122$	2.00000	0.181071
$123$	0	0
$124$	−4.92820	−0.442566
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−17.8564	−1.58450	−0.792250	−	0.610197i	$-0.791090\pi$
$127$	−17.8564	−1.58450	−0.792250	+	0.610197i	$0.791090\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	−5.46410	−0.479233
$131$	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
$134$	2.92820	0.252958
$135$	0	0
$136$	3.46410	0.297044
$137$	19.8564	1.69645	0.848224	−	0.529638i	$-0.177672\pi$
$137$	19.8564	1.69645	0.848224	+	0.529638i	$0.177672\pi$
$138$	0	0
$139$	−6.19615	−0.525551	−0.262775	−	0.964857i	$-0.584638\pi$
$139$	−6.19615	−0.525551	−0.262775	+	0.964857i	$0.584638\pi$
$140$	−1.00000	−0.0845154
$141$	0	0
$142$	−2.53590	−0.212808
$143$	−5.46410	−0.456931
$144$	0	0
$145$	−1.26795	−0.105297
$146$	4.53590	0.375394
$147$	0	0
$148$	6.73205	0.553371
$149$	10.7321	0.879204	0.439602	−	0.898193i	$-0.355120\pi$
$149$	10.7321	0.879204	0.439602	+	0.898193i	$0.355120\pi$
$150$	0	0
$151$	−18.1962	−1.48078	−0.740391	−	0.672177i	$-0.765360\pi$
$151$	−18.1962	−1.48078	−0.740391	+	0.672177i	$0.765360\pi$
$152$	0.732051	0.0593772
$153$	0	0
$154$	−1.00000	−0.0805823
$155$	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	$0.694732\pi$
$158$	3.26795	0.259984
$159$	0	0
$160$	−1.00000	−0.0790569
$161$	−4.73205	−0.372938
$162$	0	0
$163$	8.00000	0.626608	0.313304	−	0.949653i	$-0.398564\pi$
$163$	8.00000	0.626608	0.313304	+	0.949653i	$0.398564\pi$
$164$	1.26795	0.0990102
$165$	0	0
$166$	−16.3923	−1.27229
$167$	−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
$170$	−3.46410	−0.265684
$171$	0	0
$172$	8.92820	0.680769
$173$	12.9282	0.982913	0.491457	−	0.870902i	$-0.336465\pi$
$173$	12.9282	0.982913	0.491457	+	0.870902i	$0.336465\pi$
$174$	0	0
$175$	1.00000	0.0755929
$176$	−1.00000	−0.0753778
$177$	0	0
$178$	8.53590	0.639793
$179$	−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	$0.299404\pi$
$182$	5.46410	0.405026
$183$	0	0
$184$	−4.73205	−0.348851
$185$	−6.73205	−0.494950
$186$	0	0
$187$	−3.46410	−0.253320
$188$	0	0
$189$	0	0
$190$	−0.732051	−0.0531085
$191$	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	$0.732474\pi$
$194$	16.1962	1.16282
$195$	0	0
$196$	1.00000	0.0714286
$197$	1.60770	0.114544	0.0572718	−	0.998359i	$-0.481760\pi$
$197$	1.60770	0.114544	0.0572718	+	0.998359i	$0.481760\pi$
$198$	0	0
$199$	16.7846	1.18983	0.594915	−	0.803789i	$-0.297186\pi$
$199$	16.7846	1.18983	0.594915	+	0.803789i	$0.297186\pi$
$200$	1.00000	0.0707107
$201$	0	0
$202$	6.00000	0.422159
$203$	1.26795	0.0889926
$204$	0	0
$205$	−1.26795	−0.0885574
$206$	17.4641	1.21678
$207$	0	0
$208$	5.46410	0.378867
$209$	−0.732051	−0.0506370
$210$	0	0
$211$	8.00000	0.550743	0.275371	−	0.961338i	$-0.411199\pi$
$211$	8.00000	0.550743	0.275371	+	0.961338i	$0.411199\pi$
$212$	1.26795	0.0870831
$213$	0	0
$214$	12.9282	0.883754
$215$	−8.92820	−0.608898
$216$	0	0
$217$	−4.92820	−0.334548
$218$	−12.1962	−0.826028
$219$	0	0
$220$	1.00000	0.0674200
$221$	18.9282	1.27325
$222$	0	0
$223$	12.3923	0.829850	0.414925	−	0.909856i	$-0.363808\pi$
$223$	12.3923	0.829850	0.414925	+	0.909856i	$0.363808\pi$
$224$	1.00000	0.0668153
$225$	0	0
$226$	0.928203	0.0617432
$227$	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	$0.569286\pi$
$230$	4.73205	0.312022
$231$	0	0
$232$	1.26795	0.0832449
$233$	19.8564	1.30084	0.650418	−	0.759576i	$-0.274594\pi$
$233$	19.8564	1.30084	0.650418	+	0.759576i	$0.274594\pi$
$234$	0	0
$235$	0	0
$236$	−13.8564	−0.901975
$237$	0	0
$238$	3.46410	0.224544
$239$	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	$0.580873\pi$
$242$	1.00000	0.0642824
$243$	0	0
$244$	2.00000	0.128037
$245$	−1.00000	−0.0638877
$246$	0	0
$247$	4.00000	0.254514
$248$	−4.92820	−0.312941
$249$	0	0
$250$	−1.00000	−0.0632456
$251$	0	0		−	1.00000i	$-0.5\pi$
$251$	0	0			1.00000i	$0.5\pi$
$252$	0	0
$253$	4.73205	0.297501
$254$	−17.8564	−1.12041
$255$	0	0
$256$	1.00000	0.0625000
$257$	28.9808	1.80777	0.903885	−	0.427775i	$-0.140703\pi$
$257$	28.9808	1.80777	0.903885	+	0.427775i	$0.140703\pi$
$258$	0	0
$259$	6.73205	0.418309
$260$	−5.46410	−0.338869
$261$	0	0
$262$	0.339746	0.0209896
$263$	−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
$266$	0.732051	0.0448849
$267$	0	0
$268$	2.92820	0.178868
$269$	16.3923	0.999456	0.499728	−	0.866182i	$-0.333433\pi$
$269$	16.3923	0.999456	0.499728	+	0.866182i	$0.333433\pi$
$270$	0	0
$271$	3.60770	0.219152	0.109576	−	0.993978i	$-0.465051\pi$
$271$	3.60770	0.219152	0.109576	+	0.993978i	$0.465051\pi$
$272$	3.46410	0.210042
$273$	0	0
$274$	19.8564	1.19957
$275$	−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	$0.498627\pi$
$278$	−6.19615	−0.371621
$279$	0	0
$280$	−1.00000	−0.0597614
$281$	−10.3923	−0.619953	−0.309976	−	0.950744i	$-0.600321\pi$
$281$	−10.3923	−0.619953	−0.309976	+	0.950744i	$0.600321\pi$
$282$	0	0
$283$	2.92820	0.174064	0.0870318	−	0.996206i	$-0.472262\pi$
$283$	2.92820	0.174064	0.0870318	+	0.996206i	$0.472262\pi$
$284$	−2.53590	−0.150478
$285$	0	0
$286$	−5.46410	−0.323099
$287$	1.26795	0.0748447
$288$	0	0
$289$	−5.00000	−0.294118
$290$	−1.26795	−0.0744565
$291$	0	0
$292$	4.53590	0.265443
$293$	9.46410	0.552899	0.276449	−	0.961028i	$-0.410842\pi$
$293$	9.46410	0.552899	0.276449	+	0.961028i	$0.410842\pi$
$294$	0	0
$295$	13.8564	0.806751
$296$	6.73205	0.391293
$297$	0	0
$298$	10.7321	0.621691
$299$	−25.8564	−1.49531
$300$	0	0
$301$	8.92820	0.514613
$302$	−18.1962	−1.04707
$303$	0	0
$304$	0.732051	0.0419860
$305$	−2.00000	−0.114520
$306$	0	0
$307$	−3.32051	−0.189511	−0.0947557	−	0.995501i	$-0.530207\pi$
$307$	−3.32051	−0.189511	−0.0947557	+	0.995501i	$0.530207\pi$
$308$	−1.00000	−0.0569803
$309$	0	0
$310$	4.92820	0.279903
$311$	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	$0.165059\pi$
$314$	−14.3923	−0.812205
$315$	0	0
$316$	3.26795	0.183837
$317$	−26.4449	−1.48529	−0.742646	−	0.669684i	$-0.766429\pi$
$317$	−26.4449	−1.48529	−0.742646	+	0.669684i	$0.766429\pi$
$318$	0	0
$319$	−1.26795	−0.0709915
$320$	−1.00000	−0.0559017
$321$	0	0
$322$	−4.73205	−0.263707
$323$	2.53590	0.141101
$324$	0	0
$325$	5.46410	0.303094
$326$	8.00000	0.443079
$327$	0	0
$328$	1.26795	0.0700108
$329$	0	0
$330$	0	0
$331$	8.92820	0.490738	0.245369	−	0.969430i	$-0.421091\pi$
$331$	8.92820	0.490738	0.245369	+	0.969430i	$0.421091\pi$
$332$	−16.3923	−0.899645
$333$	0	0
$334$	−13.8564	−0.758189
$335$	−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	$0.405101\pi$
$338$	16.8564	0.916868
$339$	0	0
$340$	−3.46410	−0.187867
$341$	4.92820	0.266877
$342$	0	0
$343$	1.00000	0.0539949
$344$	8.92820	0.481376
$345$	0	0
$346$	12.9282	0.695025
$347$	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	$0.417219\pi$
$350$	1.00000	0.0534522
$351$	0	0
$352$	−1.00000	−0.0533002
$353$	−26.4449	−1.40752	−0.703759	−	0.710439i	$-0.748497\pi$
$353$	−26.4449	−1.40752	−0.703759	+	0.710439i	$0.748497\pi$
$354$	0	0
$355$	2.53590	0.134592
$356$	8.53590	0.452402
$357$	0	0
$358$	−7.85641	−0.415224
$359$	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
$362$	15.8564	0.833394
$363$	0	0
$364$	5.46410	0.286397
$365$	−4.53590	−0.237420
$366$	0	0
$367$	22.5359	1.17636	0.588182	−	0.808728i	$-0.299844\pi$
$367$	22.5359	1.17636	0.588182	+	0.808728i	$0.299844\pi$
$368$	−4.73205	−0.246675
$369$	0	0
$370$	−6.73205	−0.349983
$371$	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	$0.419986\pi$
$374$	−3.46410	−0.179124
$375$	0	0
$376$	0	0
$377$	6.92820	0.356821
$378$	0	0
$379$	−31.7128	−1.62898	−0.814489	−	0.580179i	$-0.802983\pi$
$379$	−31.7128	−1.62898	−0.814489	+	0.580179i	$0.802983\pi$
$380$	−0.732051	−0.0375534
$381$	0	0
$382$	6.92820	0.354478
$383$	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
$386$	−18.5359	−0.943452
$387$	0	0
$388$	16.1962	0.822235
$389$	−24.2487	−1.22946	−0.614729	−	0.788738i	$-0.710735\pi$
$389$	−24.2487	−1.22946	−0.614729	+	0.788738i	$0.710735\pi$
$390$	0	0
$391$	−16.3923	−0.828994
$392$	1.00000	0.0505076
$393$	0	0
$394$	1.60770	0.0809945
$395$	−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	$0.645679\pi$
$398$	16.7846	0.841336
$399$	0	0
$400$	1.00000	0.0500000
$401$	−37.1769	−1.85653	−0.928263	−	0.371924i	$-0.878698\pi$
$401$	−37.1769	−1.85653	−0.928263	+	0.371924i	$0.878698\pi$
$402$	0	0
$403$	−26.9282	−1.34139
$404$	6.00000	0.298511
$405$	0	0
$406$	1.26795	0.0629273
$407$	−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	$0.405735\pi$
$410$	−1.26795	−0.0626195
$411$	0	0
$412$	17.4641	0.860395
$413$	−13.8564	−0.681829
$414$	0	0
$415$	16.3923	0.804667
$416$	5.46410	0.267900
$417$	0	0
$418$	−0.732051	−0.0358058
$419$	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	$0.770031\pi$
$422$	8.00000	0.389434
$423$	0	0
$424$	1.26795	0.0615771
$425$	3.46410	0.168034
$426$	0	0
$427$	2.00000	0.0967868
$428$	12.9282	0.624908
$429$	0	0
$430$	−8.92820	−0.430556
$431$	−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	$0.520911\pi$
$434$	−4.92820	−0.236561
$435$	0	0
$436$	−12.1962	−0.584090
$437$	−3.46410	−0.165710
$438$	0	0
$439$	15.6077	0.744915	0.372457	−	0.928049i	$-0.378515\pi$
$439$	15.6077	0.744915	0.372457	+	0.928049i	$0.378515\pi$
$440$	1.00000	0.0476731
$441$	0	0
$442$	18.9282	0.900323
$443$	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
$446$	12.3923	0.586793
$447$	0	0
$448$	1.00000	0.0472456
$449$	0.679492	0.0320672	0.0160336	−	0.999871i	$-0.494896\pi$
$449$	0.679492	0.0320672	0.0160336	+	0.999871i	$0.494896\pi$
$450$	0	0
$451$	−1.26795	−0.0597054
$452$	0.928203	0.0436590
$453$	0	0
$454$	13.8564	0.650313
$455$	−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	$0.792648\pi$
$458$	−6.53590	−0.305402
$459$	0	0
$460$	4.73205	0.220633
$461$	5.32051	0.247801	0.123900	−	0.992295i	$-0.460460\pi$
$461$	5.32051	0.247801	0.123900	+	0.992295i	$0.460460\pi$
$462$	0	0
$463$	−38.9808	−1.81159	−0.905795	−	0.423717i	$-0.860725\pi$
$463$	−38.9808	−1.81159	−0.905795	+	0.423717i	$0.860725\pi$
$464$	1.26795	0.0588631
$465$	0	0
$466$	19.8564	0.919830
$467$	−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
$471$	0	0
$472$	−13.8564	−0.637793
$473$	−8.92820	−0.410519
$474$	0	0
$475$	0.732051	0.0335888
$476$	3.46410	0.158777
$477$	0	0
$478$	14.1962	0.649317
$479$	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
$482$	−7.80385	−0.355456
$483$	0	0
$484$	1.00000	0.0454545
$485$	−16.1962	−0.735429
$486$	0	0
$487$	29.1244	1.31975	0.659875	−	0.751375i	$-0.270609\pi$
$487$	29.1244	1.31975	0.659875	+	0.751375i	$0.270609\pi$
$488$	2.00000	0.0905357
$489$	0	0
$490$	−1.00000	−0.0451754
$491$	−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
$494$	4.00000	0.179969
$495$	0	0
$496$	−4.92820	−0.221283
$497$	−2.53590	−0.113751
$498$	0	0
$499$	2.00000	0.0895323	0.0447661	−	0.998997i	$-0.485746\pi$
$499$	2.00000	0.0895323	0.0447661	+	0.998997i	$0.485746\pi$
$500$	−1.00000	−0.0447214
$501$	0	0
$502$	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
$506$	4.73205	0.210365
$507$	0	0
$508$	−17.8564	−0.792250
$509$	38.7846	1.71910	0.859549	−	0.511054i	$-0.170745\pi$
$509$	38.7846	1.71910	0.859549	+	0.511054i	$0.170745\pi$
$510$	0	0
$511$	4.53590	0.200656
$512$	1.00000	0.0441942
$513$	0	0
$514$	28.9808	1.27829
$515$	−17.4641	−0.769560
$516$	0	0
$517$	0	0
$518$	6.73205	0.295789
$519$	0	0
$520$	−5.46410	−0.239617
$521$	10.3923	0.455295	0.227648	−	0.973744i	$-0.426897\pi$
$521$	10.3923	0.455295	0.227648	+	0.973744i	$0.426897\pi$
$522$	0	0
$523$	−15.3205	−0.669919	−0.334960	−	0.942233i	$-0.608723\pi$
$523$	−15.3205	−0.669919	−0.334960	+	0.942233i	$0.608723\pi$
$524$	0.339746	0.0148419
$525$	0	0
$526$	−5.07180	−0.221141
$527$	−17.0718	−0.743659
$528$	0	0
$529$	−0.607695	−0.0264215
$530$	−1.26795	−0.0550762
$531$	0	0
$532$	0.732051	0.0317384
$533$	6.92820	0.300094
$534$	0	0
$535$	−12.9282	−0.558935
$536$	2.92820	0.126479
$537$	0	0
$538$	16.3923	0.706722
$539$	−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	$0.548942\pi$
$542$	3.60770	0.154964
$543$	0	0
$544$	3.46410	0.148522
$545$	12.1962	0.522426
$546$	0	0
$547$	−36.7846	−1.57280	−0.786398	−	0.617720i	$-0.788057\pi$
$547$	−36.7846	−1.57280	−0.786398	+	0.617720i	$0.788057\pi$
$548$	19.8564	0.848224
$549$	0	0
$550$	−1.00000	−0.0426401
$551$	0.928203	0.0395428
$552$	0	0
$553$	3.26795	0.138967
$554$	0.143594	0.00610070
$555$	0	0
$556$	−6.19615	−0.262775
$557$	−36.2487	−1.53591	−0.767954	−	0.640505i	$-0.778725\pi$
$557$	−36.2487	−1.53591	−0.767954	+	0.640505i	$0.778725\pi$
$558$	0	0
$559$	48.7846	2.06337
$560$	−1.00000	−0.0422577
$561$	0	0
$562$	−10.3923	−0.438373
$563$	−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
$566$	2.92820	0.123082
$567$	0	0
$568$	−2.53590	−0.106404
$569$	30.0000	1.25767	0.628833	−	0.777541i	$-0.283533\pi$
$569$	30.0000	1.25767	0.628833	+	0.777541i	$0.283533\pi$
$570$	0	0
$571$	0.392305	0.0164174	0.00820872	−	0.999966i	$-0.497387\pi$
$571$	0.392305	0.0164174	0.00820872	+	0.999966i	$0.497387\pi$
$572$	−5.46410	−0.228466
$573$	0	0
$574$	1.26795	0.0529232
$575$	−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	$0.320641\pi$
$578$	−5.00000	−0.207973
$579$	0	0
$580$	−1.26795	−0.0526487
$581$	−16.3923	−0.680067
$582$	0	0
$583$	−1.26795	−0.0525131
$584$	4.53590	0.187697
$585$	0	0
$586$	9.46410	0.390958
$587$	−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
$590$	13.8564	0.570459
$591$	0	0
$592$	6.73205	0.276686
$593$	−39.4641	−1.62060	−0.810298	−	0.586018i	$-0.800695\pi$
$593$	−39.4641	−1.62060	−0.810298	+	0.586018i	$0.800695\pi$
$594$	0	0
$595$	−3.46410	−0.142014
$596$	10.7321	0.439602
$597$	0	0
$598$	−25.8564	−1.05735
$599$	−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	$0.713247\pi$
$602$	8.92820	0.363886
$603$	0	0
$604$	−18.1962	−0.740391
$605$	−1.00000	−0.0406558
$606$	0	0
$607$	−6.78461	−0.275379	−0.137689	−	0.990475i	$-0.543968\pi$
$607$	−6.78461	−0.275379	−0.137689	+	0.990475i	$0.543968\pi$
$608$	0.732051	0.0296886
$609$	0	0
$610$	−2.00000	−0.0809776
$611$	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	$0.751482\pi$
$614$	−3.32051	−0.134005
$615$	0	0
$616$	−1.00000	−0.0402911
$617$	−15.4641	−0.622561	−0.311281	−	0.950318i	$-0.600758\pi$
$617$	−15.4641	−0.622561	−0.311281	+	0.950318i	$0.600758\pi$
$618$	0	0
$619$	2.92820	0.117694	0.0588472	−	0.998267i	$-0.481258\pi$
$619$	2.92820	0.117694	0.0588472	+	0.998267i	$0.481258\pi$
$620$	4.92820	0.197921
$621$	0	0
$622$	19.8564	0.796169
$623$	8.53590	0.341984
$624$	0	0
$625$	1.00000	0.0400000
$626$	30.7321	1.22830
$627$	0	0
$628$	−14.3923	−0.574315
$629$	23.3205	0.929850
$630$	0	0
$631$	28.7846	1.14590	0.572949	−	0.819591i	$-0.305799\pi$
$631$	28.7846	1.14590	0.572949	+	0.819591i	$0.305799\pi$
$632$	3.26795	0.129992
$633$	0	0
$634$	−26.4449	−1.05026
$635$	17.8564	0.708610
$636$	0	0
$637$	5.46410	0.216496
$638$	−1.26795	−0.0501986
$639$	0	0
$640$	−1.00000	−0.0395285
$641$	7.85641	0.310309	0.155155	−	0.987890i	$-0.450412\pi$
$641$	7.85641	0.310309	0.155155	+	0.987890i	$0.450412\pi$
$642$	0	0
$643$	−11.2679	−0.444365	−0.222182	−	0.975005i	$-0.571318\pi$
$643$	−11.2679	−0.444365	−0.222182	+	0.975005i	$0.571318\pi$
$644$	−4.73205	−0.186469
$645$	0	0
$646$	2.53590	0.0997736
$647$	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
$650$	5.46410	0.214320
$651$	0	0
$652$	8.00000	0.313304
$653$	20.1962	0.790337	0.395168	−	0.918609i	$-0.370686\pi$
$653$	20.1962	0.790337	0.395168	+	0.918609i	$0.370686\pi$
$654$	0	0
$655$	−0.339746	−0.0132750
$656$	1.26795	0.0495051
$657$	0	0
$658$	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	$0.653569\pi$
$662$	8.92820	0.347004
$663$	0	0
$664$	−16.3923	−0.636145
$665$	−0.732051	−0.0283877
$666$	0	0
$667$	−6.00000	−0.232321
$668$	−13.8564	−0.536120
$669$	0	0
$670$	−2.92820	−0.113126
$671$	−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	$0.873746\pi$
$674$	10.7846	0.415408
$675$	0	0
$676$	16.8564	0.648323
$677$	7.85641	0.301946	0.150973	−	0.988538i	$-0.451759\pi$
$677$	7.85641	0.301946	0.150973	+	0.988538i	$0.451759\pi$
$678$	0	0
$679$	16.1962	0.621551
$680$	−3.46410	−0.132842
$681$	0	0
$682$	4.92820	0.188711
$683$	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
$686$	1.00000	0.0381802
$687$	0	0
$688$	8.92820	0.340385
$689$	6.92820	0.263944
$690$	0	0
$691$	−3.32051	−0.126318	−0.0631590	−	0.998003i	$-0.520118\pi$
$691$	−3.32051	−0.126318	−0.0631590	+	0.998003i	$0.520118\pi$
$692$	12.9282	0.491457
$693$	0	0
$694$	19.8564	0.753739
$695$	6.19615	0.235033
$696$	0	0
$697$	4.39230	0.166370
$698$	9.60770	0.363657
$699$	0	0
$700$	1.00000	0.0377964
$701$	−3.80385	−0.143669	−0.0718347	−	0.997417i	$-0.522885\pi$
$701$	−3.80385	−0.143669	−0.0718347	+	0.997417i	$0.522885\pi$
$702$	0	0
$703$	4.92820	0.185871
$704$	−1.00000	−0.0376889
$705$	0	0
$706$	−26.4449	−0.995266
$707$	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	$0.387754\pi$
$710$	2.53590	0.0951706
$711$	0	0
$712$	8.53590	0.319896
$713$	23.3205	0.873360
$714$	0	0
$715$	5.46410	0.204346
$716$	−7.85641	−0.293608
$717$	0	0
$718$	4.73205	0.176599
$719$	−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
$722$	−18.4641	−0.687163
$723$	0	0
$724$	15.8564	0.589299
$725$	1.26795	0.0470905
$726$	0	0
$727$	30.6410	1.13641	0.568206	−	0.822886i	$-0.307638\pi$
$727$	30.6410	1.13641	0.568206	+	0.822886i	$0.307638\pi$
$728$	5.46410	0.202513
$729$	0	0
$730$	−4.53590	−0.167881
$731$	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	$0.488240\pi$
$734$	22.5359	0.831815
$735$	0	0
$736$	−4.73205	−0.174426
$737$	−2.92820	−0.107862
$738$	0	0
$739$	45.8564	1.68686	0.843428	−	0.537243i	$-0.180534\pi$
$739$	45.8564	1.68686	0.843428	+	0.537243i	$0.180534\pi$
$740$	−6.73205	−0.247475
$741$	0	0
$742$	1.26795	0.0465479
$743$	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
$746$	9.60770	0.351763
$747$	0	0
$748$	−3.46410	−0.126660
$749$	12.9282	0.472386
$750$	0	0
$751$	−46.2487	−1.68764	−0.843820	−	0.536627i	$-0.819698\pi$
$751$	−46.2487	−1.68764	−0.843820	+	0.536627i	$0.819698\pi$
$752$	0	0
$753$	0	0
$754$	6.92820	0.252310
$755$	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	$0.854436\pi$
$758$	−31.7128	−1.15186
$759$	0	0
$760$	−0.732051	−0.0265543
$761$	−22.7321	−0.824036	−0.412018	−	0.911176i	$-0.635176\pi$
$761$	−22.7321	−0.824036	−0.412018	+	0.911176i	$0.635176\pi$
$762$	0	0
$763$	−12.1962	−0.441530
$764$	6.92820	0.250654
$765$	0	0
$766$	4.39230	0.158700
$767$	−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	$0.286704\pi$
$770$	1.00000	0.0360375
$771$	0	0
$772$	−18.5359	−0.667122
$773$	−30.0000	−1.07903	−0.539513	−	0.841978i	$-0.681391\pi$
$773$	−30.0000	−1.07903	−0.539513	+	0.841978i	$0.681391\pi$
$774$	0	0
$775$	−4.92820	−0.177026
$776$	16.1962	0.581408
$777$	0	0
$778$	−24.2487	−0.869358
$779$	0.928203	0.0332563
$780$	0	0
$781$	2.53590	0.0907416
$782$	−16.3923	−0.586188
$783$	0	0
$784$	1.00000	0.0357143
$785$	14.3923	0.513683
$786$	0	0
$787$	−28.0000	−0.998092	−0.499046	−	0.866575i	$-0.666316\pi$
$787$	−28.0000	−0.998092	−0.499046	+	0.866575i	$0.666316\pi$
$788$	1.60770	0.0572718
$789$	0	0
$790$	−3.26795	−0.116268
$791$	0.928203	0.0330031
$792$	0	0
$793$	10.9282	0.388072
$794$	−17.6077	−0.624874
$795$	0	0
$796$	16.7846	0.594915
$797$	−41.3205	−1.46365	−0.731824	−	0.681494i	$-0.761331\pi$
$797$	−41.3205	−1.46365	−0.731824	+	0.681494i	$0.761331\pi$
$798$	0	0
$799$	0	0
$800$	1.00000	0.0353553
$801$	0	0
$802$	−37.1769	−1.31276
$803$	−4.53590	−0.160068
$804$	0	0
$805$	4.73205	0.166783
$806$	−26.9282	−0.948506
$807$	0	0
$808$	6.00000	0.211079
$809$	29.3205	1.03085	0.515427	−	0.856933i	$-0.327633\pi$
$809$	29.3205	1.03085	0.515427	+	0.856933i	$0.327633\pi$
$810$	0	0
$811$	26.5885	0.933647	0.466824	−	0.884351i	$-0.345398\pi$
$811$	26.5885	0.933647	0.466824	+	0.884351i	$0.345398\pi$
$812$	1.26795	0.0444963
$813$	0	0
$814$	−6.73205	−0.235958
$815$	−8.00000	−0.280228
$816$	0	0
$817$	6.53590	0.228662
$818$	11.8038	0.412712
$819$	0	0
$820$	−1.26795	−0.0442787
$821$	−13.9474	−0.486769	−0.243385	−	0.969930i	$-0.578258\pi$
$821$	−13.9474	−0.486769	−0.243385	+	0.969930i	$0.578258\pi$
$822$	0	0
$823$	−13.8038	−0.481172	−0.240586	−	0.970628i	$-0.577340\pi$
$823$	−13.8038	−0.481172	−0.240586	+	0.970628i	$0.577340\pi$
$824$	17.4641	0.608391
$825$	0	0
$826$	−13.8564	−0.482126
$827$	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	$0.715714\pi$
$830$	16.3923	0.568985
$831$	0	0
$832$	5.46410	0.189434
$833$	3.46410	0.120024
$834$	0	0
$835$	13.8564	0.479521
$836$	−0.732051	−0.0253185
$837$	0	0
$838$	8.78461	0.303459
$839$	−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
$842$	−30.7846	−1.06091
$843$	0	0
$844$	8.00000	0.275371
$845$	−16.8564	−0.579878
$846$	0	0
$847$	1.00000	0.0343604
$848$	1.26795	0.0435416
$849$	0	0
$850$	3.46410	0.118818
$851$	−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	$0.461368\pi$
$854$	2.00000	0.0684386
$855$	0	0
$856$	12.9282	0.441877
$857$	−43.1769	−1.47490	−0.737448	−	0.675404i	$-0.763969\pi$
$857$	−43.1769	−1.47490	−0.737448	+	0.675404i	$0.763969\pi$
$858$	0	0
$859$	−0.784610	−0.0267705	−0.0133853	−	0.999910i	$-0.504261\pi$
$859$	−0.784610	−0.0267705	−0.0133853	+	0.999910i	$0.504261\pi$
$860$	−8.92820	−0.304449
$861$	0	0
$862$	−35.6603	−1.21459
$863$	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
$866$	−2.73205	−0.0928389
$867$	0	0
$868$	−4.92820	−0.167274
$869$	−3.26795	−0.110858
$870$	0	0
$871$	16.0000	0.542139
$872$	−12.1962	−0.413014
$873$	0	0
$874$	−3.46410	−0.117175
$875$	−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	$0.269939\pi$
$878$	15.6077	0.526734
$879$	0	0
$880$	1.00000	0.0337100
$881$	−11.0718	−0.373018	−0.186509	−	0.982453i	$-0.559717\pi$
$881$	−11.0718	−0.373018	−0.186509	+	0.982453i	$0.559717\pi$
$882$	0	0
$883$	21.1769	0.712660	0.356330	−	0.934360i	$-0.384028\pi$
$883$	21.1769	0.712660	0.356330	+	0.934360i	$0.384028\pi$
$884$	18.9282	0.636624
$885$	0	0
$886$	20.7846	0.698273
$887$	−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
$890$	−8.53590	−0.286124
$891$	0	0
$892$	12.3923	0.414925
$893$	0	0
$894$	0	0
$895$	7.85641	0.262611
$896$	1.00000	0.0334077
$897$	0	0
$898$	0.679492	0.0226749
$899$	−6.24871	−0.208406
$900$	0	0
$901$	4.39230	0.146329
$902$	−1.26795	−0.0422181
$903$	0	0
$904$	0.928203	0.0308716
$905$	−15.8564	−0.527085
$906$	0	0
$907$	19.3205	0.641527	0.320763	−	0.947159i	$-0.396061\pi$
$907$	19.3205	0.641527	0.320763	+	0.947159i	$0.396061\pi$
$908$	13.8564	0.459841
$909$	0	0
$910$	−5.46410	−0.181133
$911$	−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
$914$	−34.0000	−1.12462
$915$	0	0
$916$	−6.53590	−0.215952
$917$	0.339746	0.0112194
$918$	0	0
$919$	−26.9808	−0.890013	−0.445007	−	0.895527i	$-0.646799\pi$
$919$	−26.9808	−0.890013	−0.445007	+	0.895527i	$0.646799\pi$
$920$	4.73205	0.156011
$921$	0	0
$922$	5.32051	0.175222
$923$	−13.8564	−0.456089
$924$	0	0
$925$	6.73205	0.221348
$926$	−38.9808	−1.28099
$927$	0	0
$928$	1.26795	0.0416225
$929$	50.1051	1.64390	0.821948	−	0.569563i	$-0.192887\pi$
$929$	50.1051	1.64390	0.821948	+	0.569563i	$0.192887\pi$
$930$	0	0
$931$	0.732051	0.0239920
$932$	19.8564	0.650418
$933$	0	0
$934$	−0.339746	−0.0111168
$935$	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	$0.360383\pi$
$938$	2.92820	0.0956092
$939$	0	0
$940$	0	0
$941$	−47.5692	−1.55071	−0.775356	−	0.631524i	$-0.782430\pi$
$941$	−47.5692	−1.55071	−0.775356	+	0.631524i	$0.782430\pi$
$942$	0	0
$943$	−6.00000	−0.195387
$944$	−13.8564	−0.450988
$945$	0	0
$946$	−8.92820	−0.290281
$947$	−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
$950$	0.732051	0.0237509
$951$	0	0
$952$	3.46410	0.112272
$953$	−23.3205	−0.755425	−0.377713	−	0.925923i	$-0.623289\pi$
$953$	−23.3205	−0.755425	−0.377713	+	0.925923i	$0.623289\pi$
$954$	0	0
$955$	−6.92820	−0.224191
$956$	14.1962	0.459136
$957$	0	0
$958$	13.8564	0.447680
$959$	19.8564	0.641197
$960$	0	0
$961$	−6.71281	−0.216542
$962$	36.7846	1.18598
$963$	0	0
$964$	−7.80385	−0.251345
$965$	18.5359	0.596692
$966$	0	0
$967$	−53.8564	−1.73191	−0.865953	−	0.500126i	$-0.833287\pi$
$967$	−53.8564	−1.73191	−0.865953	+	0.500126i	$0.833287\pi$
$968$	1.00000	0.0321412
$969$	0	0
$970$	−16.1962	−0.520027
$971$	−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
$974$	29.1244	0.933205
$975$	0	0
$976$	2.00000	0.0640184
$977$	−12.9282	−0.413610	−0.206805	−	0.978382i	$-0.566307\pi$
$977$	−12.9282	−0.413610	−0.206805	+	0.978382i	$0.566307\pi$
$978$	0	0
$979$	−8.53590	−0.272808
$980$	−1.00000	−0.0319438
$981$	0	0
$982$	−29.0718	−0.927718
$983$	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
$986$	4.39230	0.139879
$987$	0	0
$988$	4.00000	0.127257
$989$	−42.2487	−1.34343
$990$	0	0
$991$	−9.07180	−0.288175	−0.144088	−	0.989565i	$-0.546025\pi$
$991$	−9.07180	−0.288175	−0.144088	+	0.989565i	$0.546025\pi$
$992$	−4.92820	−0.156471
$993$	0	0
$994$	−2.53590	−0.0804338
$995$	−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	$0.480550\pi$
$998$	2.00000	0.0633089
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6930.2.a.ca.1.2		2
3.2	odd	2		770.2.a.h.1.1	✓	2
12.11	even	2		6160.2.a.v.1.2		2
15.2	even	4		3850.2.c.s.1849.2		4
15.8	even	4		3850.2.c.s.1849.3		4
15.14	odd	2		3850.2.a.bm.1.2		2
21.20	even	2		5390.2.a.bk.1.2		2
33.32	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	3.2	odd	2
3850.2.a.bm.1.2		2	15.14	odd	2
3850.2.c.s.1849.2		4	15.2	even	4
3850.2.c.s.1849.3		4	15.8	even	4
5390.2.a.bk.1.2		2	21.20	even	2
6160.2.a.v.1.2		2	12.11	even	2
6930.2.a.ca.1.2		2	1.1	even	1	trivial
8470.2.a.ce.1.1		2	33.32	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:	\( N \)	\(=\)	\( 6930 = 2 \cdot 3^{2} \cdot 5 \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6930.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{7} +1.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{7} +1.00000 q^{8} -1.00000 q^{10} -1.00000 q^{11} +5.46410 q^{13} +1.00000 q^{14} +1.00000 q^{16} +3.46410 q^{17} +0.732051 q^{19} -1.00000 q^{20} -1.00000 q^{22} -4.73205 q^{23} +1.00000 q^{25} +5.46410 q^{26} +1.00000 q^{28} +1.26795 q^{29} -4.92820 q^{31} +1.00000 q^{32} +3.46410 q^{34} -1.00000 q^{35} +6.73205 q^{37} +0.732051 q^{38} -1.00000 q^{40} +1.26795 q^{41} +8.92820 q^{43} -1.00000 q^{44} -4.73205 q^{46} +1.00000 q^{49} +1.00000 q^{50} +5.46410 q^{52} +1.26795 q^{53} +1.00000 q^{55} +1.00000 q^{56} +1.26795 q^{58} -13.8564 q^{59} +2.00000 q^{61} -4.92820 q^{62} +1.00000 q^{64} -5.46410 q^{65} +2.92820 q^{67} +3.46410 q^{68} -1.00000 q^{70} -2.53590 q^{71} +4.53590 q^{73} +6.73205 q^{74} +0.732051 q^{76} -1.00000 q^{77} +3.26795 q^{79} -1.00000 q^{80} +1.26795 q^{82} -16.3923 q^{83} -3.46410 q^{85} +8.92820 q^{86} -1.00000 q^{88} +8.53590 q^{89} +5.46410 q^{91} -4.73205 q^{92} -0.732051 q^{95} +16.1962 q^{97} +1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} + 2 q^{4} - 2 q^{5} + 2 q^{7} + 2 q^{8}+O(q^{10}) \) 2 * q + 2 * q^2 + 2 * q^4 - 2 * q^5 + 2 * q^7 + 2 * q^8	\( 2 q + 2 q^{2} + 2 q^{4} - 2 q^{5} + 2 q^{7} + 2 q^{8} - 2 q^{10} - 2 q^{11} + 4 q^{13} + 2 q^{14} + 2 q^{16} - 2 q^{19} - 2 q^{20} - 2 q^{22} - 6 q^{23} + 2 q^{25} + 4 q^{26} + 2 q^{28} + 6 q^{29} + 4 q^{31} + 2 q^{32} - 2 q^{35} + 10 q^{37} - 2 q^{38} - 2 q^{40} + 6 q^{41} + 4 q^{43} - 2 q^{44} - 6 q^{46} + 2 q^{49} + 2 q^{50} + 4 q^{52} + 6 q^{53} + 2 q^{55} + 2 q^{56} + 6 q^{58} + 4 q^{61} + 4 q^{62} + 2 q^{64} - 4 q^{65} - 8 q^{67} - 2 q^{70} - 12 q^{71} + 16 q^{73} + 10 q^{74} - 2 q^{76} - 2 q^{77} + 10 q^{79} - 2 q^{80} + 6 q^{82} - 12 q^{83} + 4 q^{86} - 2 q^{88} + 24 q^{89} + 4 q^{91} - 6 q^{92} + 2 q^{95} + 22 q^{97} + 2 q^{98}+O(q^{100}) \) 2 * q + 2 * q^2 + 2 * q^4 - 2 * q^5 + 2 * q^7 + 2 * q^8 - 2 * q^10 - 2 * q^11 + 4 * q^13 + 2 * q^14 + 2 * q^16 - 2 * q^19 - 2 * q^20 - 2 * q^22 - 6 * q^23 + 2 * q^25 + 4 * q^26 + 2 * q^28 + 6 * q^29 + 4 * q^31 + 2 * q^32 - 2 * q^35 + 10 * q^37 - 2 * q^38 - 2 * q^40 + 6 * q^41 + 4 * q^43 - 2 * q^44 - 6 * q^46 + 2 * q^49 + 2 * q^50 + 4 * q^52 + 6 * q^53 + 2 * q^55 + 2 * q^56 + 6 * q^58 + 4 * q^61 + 4 * q^62 + 2 * q^64 - 4 * q^65 - 8 * q^67 - 2 * q^70 - 12 * q^71 + 16 * q^73 + 10 * q^74 - 2 * q^76 - 2 * q^77 + 10 * q^79 - 2 * q^80 + 6 * q^82 - 12 * q^83 + 4 * q^86 - 2 * q^88 + 24 * q^89 + 4 * q^91 - 6 * q^92 + 2 * q^95 + 22 * q^97 + 2 * q^98

Introduction

L-functions

Modular forms

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Database

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