LMFDB - Embedded newform 3380.2.f.h.3041.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3380,2,Mod(3041,3380)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3380, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 1]))

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

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

chi := DirichletCharacter("3380.3041");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3380 = 2^{2} \cdot 5 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3380.f (of order $2$, degree $1$, not minimal)

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:	no
Analytic conductor:	$26.9894358832$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.5089536.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 16x^{2} - 24x + 18 $ x^6 - 2x^5 + 2x^4 + 2x^3 + 16x^2 - 24*x + 18
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 260)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			3041.5
Root			$1.66044 - 1.66044i$ of defining polynomial
Character	$\chi$	$=$	3380.3041
Dual form			3380.2.f.h.3041.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.32088 q^{3} -1.00000i q^{5} -5.02827i q^{7} +8.02827 q^{9} +O(q^{10})$	$q+3.32088 q^{3} -1.00000i q^{5} -5.02827i q^{7} +8.02827 q^{9} +1.70739i q^{11} -3.32088i q^{15} +4.64177 q^{17} +4.34916i q^{19} -16.6983i q^{21} +0.679116 q^{23} -1.00000 q^{25} +16.6983 q^{27} -1.02827 q^{29} +2.29261i q^{31} +5.67004i q^{33} -5.02827 q^{35} -1.61350i q^{37} -4.64177i q^{41} +3.32088 q^{43} -8.02827i q^{45} +1.02827i q^{47} -18.2835 q^{49} +15.4148 q^{51} -9.41478 q^{53} +1.70739 q^{55} +14.4431i q^{57} -8.93438i q^{59} +9.02827 q^{61} -40.3684i q^{63} -5.61350i q^{67} +2.25526 q^{69} -1.70739i q^{71} +12.4431i q^{73} -3.32088 q^{75} +8.58522 q^{77} -2.64177 q^{79} +31.3684 q^{81} +8.25526i q^{83} -4.64177i q^{85} -3.41478 q^{87} +1.22699i q^{89} +7.61350i q^{93} +4.34916 q^{95} +0.0565477i q^{97} +13.7074i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 4 q^{3} + 22 q^{9}+O(q^{10}) $ 6 * q + 4 * q^3 + 22 * q^9	$ 6 q + 4 q^{3} + 22 q^{9} - 4 q^{17} + 20 q^{23} - 6 q^{25} + 16 q^{27} + 20 q^{29} - 4 q^{35} + 4 q^{43} - 46 q^{49} + 72 q^{51} - 36 q^{53} + 28 q^{61} - 24 q^{69} - 4 q^{75} + 72 q^{77} + 16 q^{79} + 46 q^{81} - 16 q^{95}+O(q^{100}) $ 6 * q + 4 * q^3 + 22 * q^9 - 4 * q^17 + 20 * q^23 - 6 * q^25 + 16 * q^27 + 20 * q^29 - 4 * q^35 + 4 * q^43 - 46 * q^49 + 72 * q^51 - 36 * q^53 + 28 * q^61 - 24 * q^69 - 4 * q^75 + 72 * q^77 + 16 * q^79 + 46 * q^81 - 16 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3380\mathbb{Z}\right)^\times$.

$n$	$677$	$1691$	$1861$
$\chi(n)$	$1$	$1$	$-1$

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
$3$	3.32088	1.91731	0.958657	−	0.284565i	$-0.0918491\pi$
$3$	3.32088	1.91731	0.958657	+	0.284565i	$0.0918491\pi$
$4$	0	0
$5$	− 1.00000i	− 0.447214i
$5$	− 1.00000i	− 0.447214i
$6$	0	0
$7$	− 5.02827i	− 1.90051i	−0.311475	−	0.950254i	$-0.600823\pi$
$7$	− 5.02827i	− 1.90051i	0.311475	−	0.950254i	$-0.399177\pi$
$8$	0	0
$9$	8.02827	2.67609
$10$	0	0
$11$	1.70739i	0.514797i	0.966305	+	0.257399i	$0.0828653\pi$
$11$	1.70739i	0.514797i	−0.966305	+	0.257399i	$0.917135\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	0	0
$15$	− 3.32088i	− 0.857449i
$16$	0	0
$17$	4.64177	1.12579	0.562897	−	0.826527i	$-0.309687\pi$
$17$	4.64177	1.12579	0.562897	+	0.826527i	$0.309687\pi$
$18$	0	0
$19$	4.34916i	0.997765i	0.866670	+	0.498883i	$0.166256\pi$
$19$	4.34916i	0.997765i	−0.866670	+	0.498883i	$0.833744\pi$
$20$	0	0
$21$	− 16.6983i	− 3.64387i
$22$	0	0
$23$	0.679116	0.141605	0.0708027	−	0.997490i	$-0.477444\pi$
$23$	0.679116	0.141605	0.0708027	+	0.997490i	$0.477444\pi$
$24$	0	0
$25$	−1.00000	−0.200000
$26$	0	0
$27$	16.6983	3.21359
$28$	0	0
$29$	−1.02827	−0.190946	−0.0954728	−	0.995432i	$-0.530436\pi$
$29$	−1.02827	−0.190946	−0.0954728	+	0.995432i	$0.530436\pi$
$30$	0	0
$31$	2.29261i	0.411765i	0.978577	+	0.205883i	$0.0660065\pi$
$31$	2.29261i	0.411765i	−0.978577	+	0.205883i	$0.933994\pi$
$32$	0	0
$33$	5.67004i	0.987028i
$34$	0	0
$35$	−5.02827	−0.849933
$36$	0	0
$37$	− 1.61350i	− 0.265257i	−0.991166	−	0.132628i	$-0.957658\pi$
$37$	− 1.61350i	− 0.265257i	0.991166	−	0.132628i	$-0.0423417\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 4.64177i	− 0.724923i	−0.931999	−	0.362461i	$-0.881937\pi$
$41$	− 4.64177i	− 0.724923i	0.931999	−	0.362461i	$-0.118063\pi$
$42$	0	0
$43$	3.32088	0.506430	0.253215	−	0.967410i	$-0.418512\pi$
$43$	3.32088	0.506430	0.253215	+	0.967410i	$0.418512\pi$
$44$	0	0
$45$	− 8.02827i	− 1.19678i
$46$	0	0
$47$	1.02827i	0.149989i	0.997184	+	0.0749946i	$0.0238939\pi$
$47$	1.02827i	0.149989i	−0.997184	+	0.0749946i	$0.976106\pi$
$48$	0	0
$49$	−18.2835	−2.61193
$50$	0	0
$51$	15.4148	2.15850
$52$	0	0
$53$	−9.41478	−1.29322	−0.646610	−	0.762821i	$-0.723814\pi$
$53$	−9.41478	−1.29322	−0.646610	+	0.762821i	$0.723814\pi$
$54$	0	0
$55$	1.70739	0.230224
$56$	0	0
$57$	14.4431i	1.91303i
$58$	0	0
$59$	− 8.93438i	− 1.16316i	−0.813490	−	0.581579i	$-0.802435\pi$
$59$	− 8.93438i	− 1.16316i	0.813490	−	0.581579i	$-0.197565\pi$
$60$	0	0
$61$	9.02827	1.15595	0.577976	−	0.816054i	$-0.303843\pi$
$61$	9.02827	1.15595	0.577976	+	0.816054i	$0.303843\pi$
$62$	0	0
$63$	− 40.3684i	− 5.08594i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 5.61350i	− 0.685798i	−0.939372	−	0.342899i	$-0.888591\pi$
$67$	− 5.61350i	− 0.685798i	0.939372	−	0.342899i	$-0.111409\pi$
$68$	0	0
$69$	2.25526	0.271502
$70$	0	0
$71$	− 1.70739i	− 0.202630i	−0.994854	−	0.101315i	$-0.967695\pi$
$71$	− 1.70739i	− 0.202630i	0.994854	−	0.101315i	$-0.0323050\pi$
$72$	0	0
$73$	12.4431i	1.45635i	0.685392	+	0.728175i	$0.259631\pi$
$73$	12.4431i	1.45635i	−0.685392	+	0.728175i	$0.740369\pi$
$74$	0	0
$75$	−3.32088	−0.383463
$76$	0	0
$77$	8.58522	0.978377
$78$	0	0
$79$	−2.64177	−0.297222	−0.148611	−	0.988896i	$-0.547480\pi$
$79$	−2.64177	−0.297222	−0.148611	+	0.988896i	$0.547480\pi$
$80$	0	0
$81$	31.3684	3.48537
$82$	0	0
$83$	8.25526i	0.906133i	0.891477	+	0.453066i	$0.149670\pi$
$83$	8.25526i	0.906133i	−0.891477	+	0.453066i	$0.850330\pi$
$84$	0	0
$85$	− 4.64177i	− 0.503471i
$86$	0	0
$87$	−3.41478	−0.366103
$88$	0	0
$89$	1.22699i	0.130061i	0.997883	+	0.0650304i	$0.0207144\pi$
$89$	1.22699i	0.130061i	−0.997883	+	0.0650304i	$0.979286\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	7.61350i	0.789483i
$94$	0	0
$95$	4.34916	0.446214
$96$	0	0
$97$	0.0565477i	0.00574155i	0.999996	+	0.00287078i	$0.000913797\pi$
$97$	0.0565477i	0.00574155i	−0.999996	+	0.00287078i	$0.999086\pi$
$98$	0	0
$99$	13.7074i	1.37764i
$100$	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
$103$	−9.37743	−0.923986	−0.461993	−	0.886884i	$-0.652865\pi$
$103$	−9.37743	−0.923986	−0.461993	+	0.886884i	$0.652865\pi$
$104$	0	0
$105$	−16.6983	−1.62959
$106$	0	0
$107$	−13.3774	−1.29325	−0.646623	−	0.762810i	$-0.723819\pi$
$107$	−13.3774	−1.29325	−0.646623	+	0.762810i	$0.723819\pi$
$108$	0	0
$109$	− 11.2835i	− 1.08077i	−0.841419	−	0.540383i	$-0.818279\pi$
$109$	− 11.2835i	− 1.08077i	0.841419	−	0.540383i	$-0.181721\pi$
$110$	0	0
$111$	− 5.35823i	− 0.508581i
$112$	0	0
$113$	−18.6983	−1.75899	−0.879495	−	0.475908i	$-0.842119\pi$
$113$	−18.6983	−1.75899	−0.879495	+	0.475908i	$0.842119\pi$
$114$	0	0
$115$	− 0.679116i	− 0.0633278i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	− 23.3401i	− 2.13958i
$120$	0	0
$121$	8.08482	0.734984
$122$	0	0
$123$	− 15.4148i	− 1.38990i
$124$	0	0
$125$	1.00000i	0.0894427i
$126$	0	0
$127$	−10.7357	−0.952636	−0.476318	−	0.879273i	$-0.658029\pi$
$127$	−10.7357	−0.952636	−0.476318	+	0.879273i	$0.658029\pi$
$128$	0	0
$129$	11.0283	0.970985
$130$	0	0
$131$	7.22699	0.631425	0.315713	−	0.948855i	$-0.397756\pi$
$131$	7.22699	0.631425	0.315713	+	0.948855i	$0.397756\pi$
$132$	0	0
$133$	21.8688	1.89626
$134$	0	0
$135$	− 16.6983i	− 1.43716i
$136$	0	0
$137$	17.3401i	1.48146i	0.671801	+	0.740732i	$0.265521\pi$
$137$	17.3401i	1.48146i	−0.671801	+	0.740732i	$0.734479\pi$
$138$	0	0
$139$	9.35823	0.793755	0.396877	−	0.917872i	$-0.370094\pi$
$139$	9.35823	0.793755	0.396877	+	0.917872i	$0.370094\pi$
$140$	0	0
$141$	3.41478i	0.287576i
$142$	0	0
$143$	0	0
$144$	0	0
$145$	1.02827i	0.0853935i
$146$	0	0
$147$	−60.7175	−5.00790
$148$	0	0
$149$	− 17.3401i	− 1.42056i	−0.703922	−	0.710278i	$-0.748569\pi$
$149$	− 17.3401i	− 1.42056i	0.703922	−	0.710278i	$-0.251431\pi$
$150$	0	0
$151$	3.57615i	0.291023i	0.989357	+	0.145511i	$0.0464828\pi$
$151$	3.57615i	0.291023i	−0.989357	+	0.145511i	$0.953517\pi$
$152$	0	0
$153$	37.2654	3.01273
$154$	0	0
$155$	2.29261	0.184147
$156$	0	0
$157$	−7.28354	−0.581290	−0.290645	−	0.956831i	$-0.593870\pi$
$157$	−7.28354	−0.581290	−0.290645	+	0.956831i	$0.593870\pi$
$158$	0	0
$159$	−31.2654	−2.47951
$160$	0	0
$161$	− 3.41478i	− 0.269122i
$162$	0	0
$163$	0.840485i	0.0658319i	0.999458	+	0.0329159i	$0.0104794\pi$
$163$	0.840485i	0.0658319i	−0.999458	+	0.0329159i	$0.989521\pi$
$164$	0	0
$165$	5.67004	0.441412
$166$	0	0
$167$	− 13.0283i	− 1.00816i	−0.863658	−	0.504079i	$-0.831832\pi$
$167$	− 13.0283i	− 1.00816i	0.863658	−	0.504079i	$-0.168168\pi$
$168$	0	0
$169$	0	0
$170$	0	0
$171$	34.9162i	2.67011i
$172$	0	0
$173$	−25.9253	−1.97106	−0.985532	−	0.169488i	$-0.945789\pi$
$173$	−25.9253	−1.97106	−0.985532	+	0.169488i	$0.945789\pi$
$174$	0	0
$175$	5.02827i	0.380102i
$176$	0	0
$177$	− 29.6700i	− 2.23014i
$178$	0	0
$179$	4.77301	0.356751	0.178376	−	0.983962i	$-0.442916\pi$
$179$	4.77301	0.356751	0.178376	+	0.983962i	$0.442916\pi$
$180$	0	0
$181$	14.3118	1.06379	0.531894	−	0.846811i	$-0.321480\pi$
$181$	14.3118	1.06379	0.531894	+	0.846811i	$0.321480\pi$
$182$	0	0
$183$	29.9819	2.21632
$184$	0	0
$185$	−1.61350	−0.118627
$186$	0	0
$187$	7.92531i	0.579556i
$188$	0	0
$189$	− 83.9637i	− 6.10746i
$190$	0	0
$191$	12.0000	0.868290	0.434145	−	0.900843i	$-0.357051\pi$
$191$	12.0000	0.868290	0.434145	+	0.900843i	$0.357051\pi$
$192$	0	0
$193$	4.05655i	0.291997i	0.989285	+	0.145998i	$0.0466394\pi$
$193$	4.05655i	0.291997i	−0.989285	+	0.145998i	$0.953361\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	15.2835i	1.08891i	0.838791	+	0.544453i	$0.183263\pi$
$197$	15.2835i	1.08891i	−0.838791	+	0.544453i	$0.816737\pi$
$198$	0	0
$199$	8.77301	0.621902	0.310951	−	0.950426i	$-0.399352\pi$
$199$	8.77301	0.621902	0.310951	+	0.950426i	$0.399352\pi$
$200$	0	0
$201$	− 18.6418i	− 1.31489i
$202$	0	0
$203$	5.17044i	0.362894i
$204$	0	0
$205$	−4.64177	−0.324195
$206$	0	0
$207$	5.45213	0.378949
$208$	0	0
$209$	−7.42571	−0.513647
$210$	0	0
$211$	0.773010	0.0532162	0.0266081	−	0.999646i	$-0.491529\pi$
$211$	0.773010	0.0532162	0.0266081	+	0.999646i	$0.491529\pi$
$212$	0	0
$213$	− 5.67004i	− 0.388505i
$214$	0	0
$215$	− 3.32088i	− 0.226482i
$216$	0	0
$217$	11.5279	0.782563
$218$	0	0
$219$	41.3219i	2.79228i
$220$	0	0
$221$	0	0
$222$	0	0
$223$	14.3118i	0.958390i	0.877709	+	0.479195i	$0.159071\pi$
$223$	14.3118i	0.958390i	−0.877709	+	0.479195i	$0.840929\pi$
$224$	0	0
$225$	−8.02827	−0.535218
$226$	0	0
$227$	13.7266i	0.911066i	0.890219	+	0.455533i	$0.150551\pi$
$227$	13.7266i	0.911066i	−0.890219	+	0.455533i	$0.849449\pi$
$228$	0	0
$229$	21.9253i	1.44887i	0.689346	+	0.724433i	$0.257898\pi$
$229$	21.9253i	1.44887i	−0.689346	+	0.724433i	$0.742102\pi$
$230$	0	0
$231$	28.5105	1.87586
$232$	0	0
$233$	6.00000	0.393073	0.196537	−	0.980497i	$-0.437031\pi$
$233$	6.00000	0.393073	0.196537	+	0.980497i	$0.437031\pi$
$234$	0	0
$235$	1.02827	0.0670772
$236$	0	0
$237$	−8.77301	−0.569868
$238$	0	0
$239$	15.7639i	1.01968i	0.860268	+	0.509842i	$0.170296\pi$
$239$	15.7639i	1.01968i	−0.860268	+	0.509842i	$0.829704\pi$
$240$	0	0
$241$	9.92531i	0.639345i	0.947528	+	0.319673i	$0.103573\pi$
$241$	9.92531i	0.639345i	−0.947528	+	0.319673i	$0.896427\pi$
$242$	0	0
$243$	54.0757	3.46896
$244$	0	0
$245$	18.2835i	1.16809i
$246$	0	0
$247$	0	0
$248$	0	0
$249$	27.4148i	1.73734i
$250$	0	0
$251$	−24.6983	−1.55894	−0.779472	−	0.626437i	$-0.784513\pi$
$251$	−24.6983	−1.55894	−0.779472	+	0.626437i	$0.784513\pi$
$252$	0	0
$253$	1.15951i	0.0728981i
$254$	0	0
$255$	− 15.4148i	− 0.965311i
$256$	0	0
$257$	20.0565	1.25109	0.625547	−	0.780187i	$-0.284876\pi$
$257$	20.0565	1.25109	0.625547	+	0.780187i	$0.284876\pi$
$258$	0	0
$259$	−8.11310	−0.504123
$260$	0	0
$261$	−8.25526	−0.510988
$262$	0	0
$263$	26.0757	1.60790	0.803950	−	0.594697i	$-0.202728\pi$
$263$	26.0757	1.60790	0.803950	+	0.594697i	$0.202728\pi$
$264$	0	0
$265$	9.41478i	0.578345i
$266$	0	0
$267$	4.07469i	0.249367i
$268$	0	0
$269$	−12.8296	−0.782232	−0.391116	−	0.920341i	$-0.627911\pi$
$269$	−12.8296	−0.782232	−0.391116	+	0.920341i	$0.627911\pi$
$270$	0	0
$271$	− 17.7074i	− 1.07565i	−0.843057	−	0.537824i	$-0.819247\pi$
$271$	− 17.7074i	− 1.07565i	0.843057	−	0.537824i	$-0.180753\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 1.70739i	− 0.102959i
$276$	0	0
$277$	15.4713	0.929582	0.464791	−	0.885420i	$-0.346129\pi$
$277$	15.4713	0.929582	0.464791	+	0.885420i	$0.346129\pi$
$278$	0	0
$279$	18.4057i	1.10192i
$280$	0	0
$281$	7.35823i	0.438955i	0.975618	+	0.219478i	$0.0704353\pi$
$281$	7.35823i	0.438955i	−0.975618	+	0.219478i	$0.929565\pi$
$282$	0	0
$283$	0.604422	0.0359292	0.0179646	−	0.999839i	$-0.494281\pi$
$283$	0.604422	0.0359292	0.0179646	+	0.999839i	$0.494281\pi$
$284$	0	0
$285$	14.4431	0.855533
$286$	0	0
$287$	−23.3401	−1.37772
$288$	0	0
$289$	4.54602	0.267413
$290$	0	0
$291$	0.187788i	0.0110084i
$292$	0	0
$293$	23.0101i	1.34427i	0.740430	+	0.672133i	$0.234622\pi$
$293$	23.0101i	1.34427i	−0.740430	+	0.672133i	$0.765378\pi$
$294$	0	0
$295$	−8.93438	−0.520180
$296$	0	0
$297$	28.5105i	1.65435i
$298$	0	0
$299$	0	0
$300$	0	0
$301$	− 16.6983i	− 0.962475i
$302$	0	0
$303$	−19.9253	−1.14468
$304$	0	0
$305$	− 9.02827i	− 0.516957i
$306$	0	0
$307$	20.3684i	1.16248i	0.813731	+	0.581242i	$0.197433\pi$
$307$	20.3684i	1.16248i	−0.813731	+	0.581242i	$0.802567\pi$
$308$	0	0
$309$	−31.1414	−1.77157
$310$	0	0
$311$	19.9253	1.12986	0.564930	−	0.825139i	$-0.308903\pi$
$311$	19.9253	1.12986	0.564930	+	0.825139i	$0.308903\pi$
$312$	0	0
$313$	31.4713	1.77886	0.889432	−	0.457067i	$-0.151100\pi$
$313$	31.4713	1.77886	0.889432	+	0.457067i	$0.151100\pi$
$314$	0	0
$315$	−40.3684	−2.27450
$316$	0	0
$317$	− 1.72659i	− 0.0969750i	−0.998824	−	0.0484875i	$-0.984560\pi$
$317$	− 1.72659i	− 0.0969750i	0.998824	−	0.0484875i	$-0.0154401\pi$
$318$	0	0
$319$	− 1.75566i	− 0.0982983i
$320$	0	0
$321$	−44.4249	−2.47956
$322$	0	0
$323$	20.1878i	1.12328i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 37.4713i	− 2.07217i
$328$	0	0
$329$	5.17044	0.285056
$330$	0	0
$331$	− 10.4057i	− 0.571949i	−0.958237	−	0.285975i	$-0.907683\pi$
$331$	− 10.4057i	− 0.571949i	0.958237	−	0.285975i	$-0.0923173\pi$
$332$	0	0
$333$	− 12.9536i	− 0.709852i
$334$	0	0
$335$	−5.61350	−0.306698
$336$	0	0
$337$	10.6983	0.582774	0.291387	−	0.956605i	$-0.405883\pi$
$337$	10.6983	0.582774	0.291387	+	0.956605i	$0.405883\pi$
$338$	0	0
$339$	−62.0950	−3.37253
$340$	0	0
$341$	−3.91438	−0.211976
$342$	0	0
$343$	56.7367i	3.06349i
$344$	0	0
$345$	− 2.25526i	− 0.121419i
$346$	0	0
$347$	−32.6044	−1.75030	−0.875149	−	0.483854i	$-0.839236\pi$
$347$	−32.6044	−1.75030	−0.875149	+	0.483854i	$0.839236\pi$
$348$	0	0
$349$	− 13.4148i	− 0.718077i	−0.933323	−	0.359038i	$-0.883105\pi$
$349$	− 13.4148i	− 0.718077i	0.933323	−	0.359038i	$-0.116895\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	35.0101i	1.86340i	0.363227	+	0.931701i	$0.381675\pi$
$353$	35.0101i	1.86340i	−0.363227	+	0.931701i	$0.618325\pi$
$354$	0	0
$355$	−1.70739	−0.0906188
$356$	0	0
$357$	− 77.5097i	− 4.10225i
$358$	0	0
$359$	20.9344i	1.10487i	0.833555	+	0.552437i	$0.186302\pi$
$359$	20.9344i	1.10487i	−0.833555	+	0.552437i	$0.813698\pi$
$360$	0	0
$361$	0.0848216	0.00446429
$362$	0	0
$363$	26.8488	1.40919
$364$	0	0
$365$	12.4431	0.651299
$366$	0	0
$367$	11.4340	0.596849	0.298424	−	0.954433i	$-0.403539\pi$
$367$	11.4340	0.596849	0.298424	+	0.954433i	$0.403539\pi$
$368$	0	0
$369$	− 37.2654i	− 1.93996i
$370$	0	0
$371$	47.3401i	2.45777i
$372$	0	0
$373$	−14.1131	−0.730748	−0.365374	−	0.930861i	$-0.619059\pi$
$373$	−14.1131	−0.730748	−0.365374	+	0.930861i	$0.619059\pi$
$374$	0	0
$375$	3.32088i	0.171490i
$376$	0	0
$377$	0	0
$378$	0	0
$379$	1.89518i	0.0973487i	0.998815	+	0.0486744i	$0.0154997\pi$
$379$	1.89518i	0.0973487i	−0.998815	+	0.0486744i	$0.984500\pi$
$380$	0	0
$381$	−35.6519	−1.82650
$382$	0	0
$383$	22.3118i	1.14008i	0.821617	+	0.570040i	$0.193072\pi$
$383$	22.3118i	1.14008i	−0.821617	+	0.570040i	$0.806928\pi$
$384$	0	0
$385$	− 8.58522i	− 0.437543i
$386$	0	0
$387$	26.6610	1.35525
$388$	0	0
$389$	28.6802	1.45414	0.727071	−	0.686562i	$-0.240881\pi$
$389$	28.6802	1.45414	0.727071	+	0.686562i	$0.240881\pi$
$390$	0	0
$391$	3.15230	0.159419
$392$	0	0
$393$	24.0000	1.21064
$394$	0	0
$395$	2.64177i	0.132922i
$396$	0	0
$397$	15.5569i	0.780781i	0.920649	+	0.390390i	$0.127660\pi$
$397$	15.5569i	0.780781i	−0.920649	+	0.390390i	$0.872340\pi$
$398$	0	0
$399$	72.6236	3.63573
$400$	0	0
$401$	− 24.1696i	− 1.20697i	−0.797373	−	0.603487i	$-0.793777\pi$
$401$	− 24.1696i	− 1.20697i	0.797373	−	0.603487i	$-0.206223\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	− 31.3684i	− 1.55871i
$406$	0	0
$407$	2.75486	0.136554
$408$	0	0
$409$	29.9253i	1.47971i	0.672766	+	0.739856i	$0.265106\pi$
$409$	29.9253i	1.47971i	−0.672766	+	0.739856i	$0.734894\pi$
$410$	0	0
$411$	57.5844i	2.84043i
$412$	0	0
$413$	−44.9245	−2.21059
$414$	0	0
$415$	8.25526	0.405235
$416$	0	0
$417$	31.0776	1.52188
$418$	0	0
$419$	−8.58522	−0.419416	−0.209708	−	0.977764i	$-0.567251\pi$
$419$	−8.58522	−0.419416	−0.209708	+	0.977764i	$0.567251\pi$
$420$	0	0
$421$	21.7375i	1.05942i	0.848178	+	0.529711i	$0.177700\pi$
$421$	21.7375i	1.05942i	−0.848178	+	0.529711i	$0.822300\pi$
$422$	0	0
$423$	8.25526i	0.401385i
$424$	0	0
$425$	−4.64177	−0.225159
$426$	0	0
$427$	− 45.3966i	− 2.19690i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	− 13.7074i	− 0.660262i	−0.943935	−	0.330131i	$-0.892907\pi$
$431$	− 13.7074i	− 0.660262i	0.943935	−	0.330131i	$-0.107093\pi$
$432$	0	0
$433$	40.5671	1.94953	0.974765	−	0.223235i	$-0.0716618\pi$
$433$	40.5671	1.94953	0.974765	+	0.223235i	$0.0716618\pi$
$434$	0	0
$435$	3.41478i	0.163726i
$436$	0	0
$437$	2.95358i	0.141289i
$438$	0	0
$439$	−26.5671	−1.26798	−0.633989	−	0.773342i	$-0.718583\pi$
$439$	−26.5671	−1.26798	−0.633989	+	0.773342i	$0.718583\pi$
$440$	0	0
$441$	−146.785	−6.98977
$442$	0	0
$443$	−16.7922	−0.797822	−0.398911	−	0.916990i	$-0.630612\pi$
$443$	−16.7922	−0.797822	−0.398911	+	0.916990i	$0.630612\pi$
$444$	0	0
$445$	1.22699	0.0581649
$446$	0	0
$447$	− 57.5844i	− 2.72365i
$448$	0	0
$449$	− 16.2443i	− 0.766618i	−0.923620	−	0.383309i	$-0.874785\pi$
$449$	− 16.2443i	− 0.766618i	0.923620	−	0.383309i	$-0.125215\pi$
$450$	0	0
$451$	7.92531	0.373188
$452$	0	0
$453$	11.8760i	0.557982i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 6.77301i	− 0.316828i	−0.987373	−	0.158414i	$-0.949362\pi$
$457$	− 6.77301i	− 0.316828i	0.987373	−	0.158414i	$-0.0506381\pi$
$458$	0	0
$459$	77.5097	3.61784
$460$	0	0
$461$	22.8114i	1.06243i	0.847236	+	0.531217i	$0.178265\pi$
$461$	22.8114i	1.06243i	−0.847236	+	0.531217i	$0.821735\pi$
$462$	0	0
$463$	− 16.3300i	− 0.758917i	−0.925209	−	0.379459i	$-0.876110\pi$
$463$	− 16.3300i	− 0.758917i	0.925209	−	0.379459i	$-0.123890\pi$
$464$	0	0
$465$	7.61350	0.353067
$466$	0	0
$467$	−10.6610	−0.493331	−0.246665	−	0.969101i	$-0.579335\pi$
$467$	−10.6610	−0.493331	−0.246665	+	0.969101i	$0.579335\pi$
$468$	0	0
$469$	−28.2262	−1.30336
$470$	0	0
$471$	−24.1878	−1.11451
$472$	0	0
$473$	5.67004i	0.260709i
$474$	0	0
$475$	− 4.34916i	− 0.199553i
$476$	0	0
$477$	−75.5844	−3.46077
$478$	0	0
$479$	6.48040i	0.296097i	0.988980	+	0.148048i	$0.0472992\pi$
$479$	6.48040i	0.296097i	−0.988980	+	0.148048i	$0.952701\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	− 11.3401i	− 0.515992i
$484$	0	0
$485$	0.0565477	0.00256770
$486$	0	0
$487$	− 23.7831i	− 1.07772i	−0.842396	−	0.538858i	$-0.818856\pi$
$487$	− 23.7831i	− 1.07772i	0.842396	−	0.538858i	$-0.181144\pi$
$488$	0	0
$489$	2.79116i	0.126220i
$490$	0	0
$491$	6.13124	0.276699	0.138350	−	0.990383i	$-0.455820\pi$
$491$	6.13124	0.276699	0.138350	+	0.990383i	$0.455820\pi$
$492$	0	0
$493$	−4.77301	−0.214966
$494$	0	0
$495$	13.7074	0.616101
$496$	0	0
$497$	−8.58522	−0.385100
$498$	0	0
$499$	− 21.0475i	− 0.942214i	−0.882076	−	0.471107i	$-0.843854\pi$
$499$	− 21.0475i	− 0.942214i	0.882076	−	0.471107i	$-0.156146\pi$
$500$	0	0
$501$	− 43.2654i	− 1.93296i
$502$	0	0
$503$	33.5652	1.49660	0.748300	−	0.663361i	$-0.230871\pi$
$503$	33.5652	1.49660	0.748300	+	0.663361i	$0.230871\pi$
$504$	0	0
$505$	6.00000i	0.266996i
$506$	0	0
$507$	0	0
$508$	0	0
$509$	14.5852i	0.646479i	0.946317	+	0.323239i	$0.104772\pi$
$509$	14.5852i	0.646479i	−0.946317	+	0.323239i	$0.895228\pi$
$510$	0	0
$511$	62.5671	2.76780
$512$	0	0
$513$	72.6236i	3.20641i
$514$	0	0
$515$	9.37743i	0.413219i
$516$	0	0
$517$	−1.75566	−0.0772140
$518$	0	0
$519$	−86.0950	−3.77915
$520$	0	0
$521$	−3.48225	−0.152560	−0.0762802	−	0.997086i	$-0.524304\pi$
$521$	−3.48225	−0.152560	−0.0762802	+	0.997086i	$0.524304\pi$
$522$	0	0
$523$	−11.5087	−0.503239	−0.251620	−	0.967826i	$-0.580963\pi$
$523$	−11.5087	−0.503239	−0.251620	+	0.967826i	$0.580963\pi$
$524$	0	0
$525$	16.6983i	0.728774i
$526$	0	0
$527$	10.6418i	0.463563i
$528$	0	0
$529$	−22.5388	−0.979948
$530$	0	0
$531$	− 71.7276i	− 3.11271i
$532$	0	0
$533$	0	0
$534$	0	0
$535$	13.3774i	0.578357i
$536$	0	0
$537$	15.8506	0.684004
$538$	0	0
$539$	− 31.2171i	− 1.34462i
$540$	0	0
$541$	− 13.8122i	− 0.593833i	−0.954903	−	0.296917i	$-0.904042\pi$
$541$	− 13.8122i	− 0.593833i	0.954903	−	0.296917i	$-0.0959584\pi$
$542$	0	0
$543$	47.5279	2.03962
$544$	0	0
$545$	−11.2835	−0.483334
$546$	0	0
$547$	−5.37743	−0.229922	−0.114961	−	0.993370i	$-0.536674\pi$
$547$	−5.37743	−0.229922	−0.114961	+	0.993370i	$0.536674\pi$
$548$	0	0
$549$	72.4815	3.09343
$550$	0	0
$551$	− 4.47213i	− 0.190519i
$552$	0	0
$553$	13.2835i	0.564873i
$554$	0	0
$555$	−5.35823	−0.227444
$556$	0	0
$557$	− 0.0674757i	− 0.00285904i	−0.999999	−	0.00142952i	$-0.999545\pi$
$557$	− 0.0674757i	− 0.00285904i	0.999999	−	0.00142952i	$-0.000455030\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	26.3190i	1.11119i
$562$	0	0
$563$	7.20779	0.303772	0.151886	−	0.988398i	$-0.451465\pi$
$563$	7.20779	0.303772	0.151886	+	0.988398i	$0.451465\pi$
$564$	0	0
$565$	18.6983i	0.786644i
$566$	0	0
$567$	− 157.729i	− 6.62398i
$568$	0	0
$569$	7.85783	0.329417	0.164709	−	0.986342i	$-0.447332\pi$
$569$	7.85783	0.329417	0.164709	+	0.986342i	$0.447332\pi$
$570$	0	0
$571$	23.9253	1.00124	0.500621	−	0.865666i	$-0.333105\pi$
$571$	23.9253	1.00124	0.500621	+	0.865666i	$0.333105\pi$
$572$	0	0
$573$	39.8506	1.66478
$574$	0	0
$575$	−0.679116	−0.0283211
$576$	0	0
$577$	− 31.0101i	− 1.29097i	−0.763774	−	0.645484i	$-0.776656\pi$
$577$	− 31.0101i	− 1.29097i	0.763774	−	0.645484i	$-0.223344\pi$
$578$	0	0
$579$	13.4713i	0.559849i
$580$	0	0
$581$	41.5097	1.72211
$582$	0	0
$583$	− 16.0747i	− 0.665746i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 47.4076i	− 1.95672i	−0.206911	−	0.978360i	$-0.566341\pi$
$587$	− 47.4076i	− 1.95672i	0.206911	−	0.978360i	$-0.433659\pi$
$588$	0	0
$589$	−9.97093	−0.410845
$590$	0	0
$591$	50.7549i	2.08778i
$592$	0	0
$593$	− 14.8861i	− 0.611299i	−0.952144	−	0.305650i	$-0.901126\pi$
$593$	− 14.8861i	− 0.611299i	0.952144	−	0.305650i	$-0.0988736\pi$
$594$	0	0
$595$	−23.3401	−0.956850
$596$	0	0
$597$	29.1342	1.19238
$598$	0	0
$599$	42.8680	1.75154	0.875769	−	0.482731i	$-0.160355\pi$
$599$	42.8680	1.75154	0.875769	+	0.482731i	$0.160355\pi$
$600$	0	0
$601$	−10.0000	−0.407909	−0.203954	−	0.978980i	$-0.565379\pi$
$601$	−10.0000	−0.407909	−0.203954	+	0.978980i	$0.565379\pi$
$602$	0	0
$603$	− 45.0667i	− 1.83526i
$604$	0	0
$605$	− 8.08482i	− 0.328695i
$606$	0	0
$607$	−18.7357	−0.760457	−0.380229	−	0.924893i	$-0.624155\pi$
$607$	−18.7357	−0.760457	−0.380229	+	0.924893i	$0.624155\pi$
$608$	0	0
$609$	17.1704i	0.695781i
$610$	0	0
$611$	0	0
$612$	0	0
$613$	− 36.6802i	− 1.48150i	−0.671782	−	0.740749i	$-0.734471\pi$
$613$	− 36.6802i	− 1.48150i	0.671782	−	0.740749i	$-0.265529\pi$
$614$	0	0
$615$	−15.4148	−0.621584
$616$	0	0
$617$	8.05655i	0.324344i	0.986762	+	0.162172i	$0.0518500\pi$
$617$	8.05655i	0.324344i	−0.986762	+	0.162172i	$0.948150\pi$
$618$	0	0
$619$	− 33.1606i	− 1.33284i	−0.745578	−	0.666418i	$-0.767827\pi$
$619$	− 33.1606i	− 1.33284i	0.745578	−	0.666418i	$-0.232173\pi$
$620$	0	0
$621$	11.3401	0.455062
$622$	0	0
$623$	6.16964	0.247182
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−24.6599	−0.984822
$628$	0	0
$629$	− 7.48947i	− 0.298625i
$630$	0	0
$631$	− 33.8205i	− 1.34637i	−0.739473	−	0.673186i	$-0.764925\pi$
$631$	− 33.8205i	− 1.34637i	0.739473	−	0.673186i	$-0.235075\pi$
$632$	0	0
$633$	2.56708	0.102032
$634$	0	0
$635$	10.7357i	0.426032i
$636$	0	0
$637$	0	0
$638$	0	0
$639$	− 13.7074i	− 0.542256i
$640$	0	0
$641$	−12.8296	−0.506737	−0.253369	−	0.967370i	$-0.581539\pi$
$641$	−12.8296	−0.506737	−0.253369	+	0.967370i	$0.581539\pi$
$642$	0	0
$643$	− 43.7084i	− 1.72369i	−0.507169	−	0.861846i	$-0.669308\pi$
$643$	− 43.7084i	− 1.72369i	0.507169	−	0.861846i	$-0.330692\pi$
$644$	0	0
$645$	− 11.0283i	− 0.434238i
$646$	0	0
$647$	−25.4158	−0.999200	−0.499600	−	0.866256i	$-0.666520\pi$
$647$	−25.4158	−0.999200	−0.499600	+	0.866256i	$0.666520\pi$
$648$	0	0
$649$	15.2545	0.598790
$650$	0	0
$651$	38.2827	1.50042
$652$	0	0
$653$	−3.94345	−0.154319	−0.0771596	−	0.997019i	$-0.524585\pi$
$653$	−3.94345	−0.154319	−0.0771596	+	0.997019i	$0.524585\pi$
$654$	0	0
$655$	− 7.22699i	− 0.282382i
$656$	0	0
$657$	99.8962i	3.89732i
$658$	0	0
$659$	−38.0950	−1.48397	−0.741984	−	0.670417i	$-0.766115\pi$
$659$	−38.0950	−1.48397	−0.741984	+	0.670417i	$0.766115\pi$
$660$	0	0
$661$	18.1131i	0.704518i	0.935903	+	0.352259i	$0.114586\pi$
$661$	18.1131i	0.704518i	−0.935903	+	0.352259i	$0.885414\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	− 21.8688i	− 0.848034i
$666$	0	0
$667$	−0.698317	−0.0270389
$668$	0	0
$669$	47.5279i	1.83753i
$670$	0	0
$671$	15.4148i	0.595081i
$672$	0	0
$673$	−43.2088	−1.66558	−0.832789	−	0.553590i	$-0.813257\pi$
$673$	−43.2088	−1.66558	−0.832789	+	0.553590i	$0.813257\pi$
$674$	0	0
$675$	−16.6983	−0.642719
$676$	0	0
$677$	−30.6983	−1.17983	−0.589916	−	0.807465i	$-0.700839\pi$
$677$	−30.6983	−1.17983	−0.589916	+	0.807465i	$0.700839\pi$
$678$	0	0
$679$	0.284337	0.0109119
$680$	0	0
$681$	45.5844i	1.74680i
$682$	0	0
$683$	− 32.9536i	− 1.26093i	−0.776216	−	0.630467i	$-0.782863\pi$
$683$	− 32.9536i	− 1.26093i	0.776216	−	0.630467i	$-0.217137\pi$
$684$	0	0
$685$	17.3401	0.662531
$686$	0	0
$687$	72.8114i	2.77793i
$688$	0	0
$689$	0	0
$690$	0	0
$691$	− 37.1222i	− 1.41219i	−0.708115	−	0.706097i	$-0.750454\pi$
$691$	− 37.1222i	− 1.41219i	0.708115	−	0.706097i	$-0.249546\pi$
$692$	0	0
$693$	68.9245	2.61823
$694$	0	0
$695$	− 9.35823i	− 0.354978i
$696$	0	0
$697$	− 21.5460i	− 0.816114i
$698$	0	0
$699$	19.9253	0.753644
$700$	0	0
$701$	7.39663	0.279367	0.139683	−	0.990196i	$-0.455391\pi$
$701$	7.39663	0.279367	0.139683	+	0.990196i	$0.455391\pi$
$702$	0	0
$703$	7.01735	0.264664
$704$	0	0
$705$	3.41478	0.128608
$706$	0	0
$707$	30.1696i	1.13465i
$708$	0	0
$709$	28.7549i	1.07991i	0.841693	+	0.539956i	$0.181559\pi$
$709$	28.7549i	1.07991i	−0.841693	+	0.539956i	$0.818441\pi$
$710$	0	0
$711$	−21.2088	−0.795394
$712$	0	0
$713$	1.55695i	0.0583081i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	52.3502i	1.95505i
$718$	0	0
$719$	39.4532	1.47136	0.735678	−	0.677332i	$-0.236864\pi$
$719$	39.4532	1.47136	0.735678	+	0.677332i	$0.236864\pi$
$720$	0	0
$721$	47.1523i	1.75604i
$722$	0	0
$723$	32.9608i	1.22583i
$724$	0	0
$725$	1.02827	0.0381891
$726$	0	0
$727$	0.866904	0.0321517	0.0160758	−	0.999871i	$-0.494883\pi$
$727$	0.866904	0.0321517	0.0160758	+	0.999871i	$0.494883\pi$
$728$	0	0
$729$	85.4742	3.16571
$730$	0	0
$731$	15.4148	0.570136
$732$	0	0
$733$	10.0000i	0.369358i	0.982799	+	0.184679i	$0.0591246\pi$
$733$	10.0000i	0.369358i	−0.982799	+	0.184679i	$0.940875\pi$
$734$	0	0
$735$	60.7175i	2.23960i
$736$	0	0
$737$	9.58442	0.353047
$738$	0	0
$739$	− 31.5015i	− 1.15880i	−0.815043	−	0.579400i	$-0.803287\pi$
$739$	− 31.5015i	− 1.15880i	0.815043	−	0.579400i	$-0.196713\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	31.8578i	1.16875i	0.811484	+	0.584375i	$0.198660\pi$
$743$	31.8578i	1.16875i	−0.811484	+	0.584375i	$0.801340\pi$
$744$	0	0
$745$	−17.3401	−0.635292
$746$	0	0
$747$	66.2755i	2.42489i
$748$	0	0
$749$	67.2654i	2.45782i
$750$	0	0
$751$	−46.0950	−1.68203	−0.841014	−	0.541013i	$-0.818041\pi$
$751$	−46.0950	−1.68203	−0.841014	+	0.541013i	$0.818041\pi$
$752$	0	0
$753$	−82.0203	−2.98898
$754$	0	0
$755$	3.57615	0.130149
$756$	0	0
$757$	5.01735	0.182359	0.0911793	−	0.995834i	$-0.470936\pi$
$757$	5.01735	0.182359	0.0911793	+	0.995834i	$0.470936\pi$
$758$	0	0
$759$	3.85061i	0.139768i
$760$	0	0
$761$	24.8296i	0.900071i	0.893011	+	0.450035i	$0.148589\pi$
$761$	24.8296i	0.900071i	−0.893011	+	0.450035i	$0.851411\pi$
$762$	0	0
$763$	−56.7367	−2.05401
$764$	0	0
$765$	− 37.2654i	− 1.34733i
$766$	0	0
$767$	0	0
$768$	0	0
$769$	− 36.9427i	− 1.33219i	−0.745869	−	0.666093i	$-0.767965\pi$
$769$	− 36.9427i	− 1.33219i	0.745869	−	0.666093i	$-0.232035\pi$
$770$	0	0
$771$	66.6055	2.39874
$772$	0	0
$773$	16.4431i	0.591415i	0.955278	+	0.295708i	$0.0955555\pi$
$773$	16.4431i	0.591415i	−0.955278	+	0.295708i	$0.904445\pi$
$774$	0	0
$775$	− 2.29261i	− 0.0823530i
$776$	0	0
$777$	−26.9427	−0.966562
$778$	0	0
$779$	20.1878	0.723303
$780$	0	0
$781$	2.91518	0.104313
$782$	0	0
$783$	−17.1704	−0.613622
$784$	0	0
$785$	7.28354i	0.259961i
$786$	0	0
$787$	40.2553i	1.43495i	0.696587	+	0.717473i	$0.254701\pi$
$787$	40.2553i	1.43495i	−0.696587	+	0.717473i	$0.745299\pi$
$788$	0	0
$789$	86.5946	3.08285
$790$	0	0
$791$	94.0203i	3.34298i
$792$	0	0
$793$	0	0
$794$	0	0
$795$	31.2654i	1.10887i
$796$	0	0
$797$	−27.2835	−0.966432	−0.483216	−	0.875501i	$-0.660532\pi$
$797$	−27.2835	−0.966432	−0.483216	+	0.875501i	$0.660532\pi$
$798$	0	0
$799$	4.77301i	0.168857i
$800$	0	0
$801$	9.85061i	0.348054i
$802$	0	0
$803$	−21.2451	−0.749725
$804$	0	0
$805$	−3.41478	−0.120355
$806$	0	0
$807$	−42.6055	−1.49978
$808$	0	0
$809$	−28.4815	−1.00135	−0.500677	−	0.865634i	$-0.666916\pi$
$809$	−28.4815	−1.00135	−0.500677	+	0.865634i	$0.666916\pi$
$810$	0	0
$811$	− 31.6892i	− 1.11276i	−0.830928	−	0.556380i	$-0.812190\pi$
$811$	− 31.6892i	− 1.11276i	0.830928	−	0.556380i	$-0.187810\pi$
$812$	0	0
$813$	− 58.8042i	− 2.06235i
$814$	0	0
$815$	0.840485	0.0294409
$816$	0	0
$817$	14.4431i	0.505298i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	6.65991i	0.232433i	0.993224	+	0.116216i	$0.0370766\pi$
$821$	6.65991i	0.232433i	−0.993224	+	0.116216i	$0.962923\pi$
$822$	0	0
$823$	7.79301	0.271647	0.135824	−	0.990733i	$-0.456632\pi$
$823$	7.79301	0.271647	0.135824	+	0.990733i	$0.456632\pi$
$824$	0	0
$825$	− 5.67004i	− 0.197406i
$826$	0	0
$827$	26.4249i	0.918884i	0.888208	+	0.459442i	$0.151951\pi$
$827$	26.4249i	0.918884i	−0.888208	+	0.459442i	$0.848049\pi$
$828$	0	0
$829$	−11.7447	−0.407912	−0.203956	−	0.978980i	$-0.565380\pi$
$829$	−11.7447	−0.407912	−0.203956	+	0.978980i	$0.565380\pi$
$830$	0	0
$831$	51.3785	1.78230
$832$	0	0
$833$	−84.8680	−2.94050
$834$	0	0
$835$	−13.0283	−0.450862
$836$	0	0
$837$	38.2827i	1.32325i
$838$	0	0
$839$	− 44.5753i	− 1.53891i	−0.638700	−	0.769456i	$-0.720527\pi$
$839$	− 44.5753i	− 1.53891i	0.638700	−	0.769456i	$-0.279473\pi$
$840$	0	0
$841$	−27.9427	−0.963540
$842$	0	0
$843$	24.4358i	0.841615i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 40.6527i	− 1.39684i
$848$	0	0
$849$	2.00722	0.0688875
$850$	0	0
$851$	− 1.09575i	− 0.0375618i
$852$	0	0
$853$	29.6135i	1.01395i	0.861962	+	0.506973i	$0.169236\pi$
$853$	29.6135i	1.01395i	−0.861962	+	0.506973i	$0.830764\pi$
$854$	0	0
$855$	34.9162	1.19411
$856$	0	0
$857$	−22.5105	−0.768945	−0.384472	−	0.923136i	$-0.625617\pi$
$857$	−22.5105	−0.768945	−0.384472	+	0.923136i	$0.625617\pi$
$858$	0	0
$859$	−0.585221	−0.0199675	−0.00998375	−	0.999950i	$-0.503178\pi$
$859$	−0.585221	−0.0199675	−0.00998375	+	0.999950i	$0.503178\pi$
$860$	0	0
$861$	−77.5097	−2.64152
$862$	0	0
$863$	− 45.6519i	− 1.55401i	−0.629495	−	0.777004i	$-0.716738\pi$
$863$	− 45.6519i	− 1.55401i	0.629495	−	0.777004i	$-0.283262\pi$
$864$	0	0
$865$	25.9253i	0.881487i
$866$	0	0
$867$	15.0968	0.512714
$868$	0	0
$869$	− 4.51053i	− 0.153009i
$870$	0	0
$871$	0	0
$872$	0	0
$873$	0.453981i	0.0153649i
$874$	0	0
$875$	5.02827	0.169987
$876$	0	0
$877$	10.0000i	0.337676i	0.985644	+	0.168838i	$0.0540015\pi$
$877$	10.0000i	0.337676i	−0.985644	+	0.168838i	$0.945999\pi$
$878$	0	0
$879$	76.4140i	2.57738i
$880$	0	0
$881$	−42.5380	−1.43314	−0.716571	−	0.697514i	$-0.754289\pi$
$881$	−42.5380	−1.43314	−0.716571	+	0.697514i	$0.754289\pi$
$882$	0	0
$883$	−6.22513	−0.209492	−0.104746	−	0.994499i	$-0.533403\pi$
$883$	−6.22513	−0.209492	−0.104746	+	0.994499i	$0.533403\pi$
$884$	0	0
$885$	−29.6700	−0.997348
$886$	0	0
$887$	22.0011	0.738723	0.369362	−	0.929286i	$-0.379576\pi$
$887$	22.0011	0.738723	0.369362	+	0.929286i	$0.379576\pi$
$888$	0	0
$889$	53.9819i	1.81049i
$890$	0	0
$891$	53.5580i	1.79426i
$892$	0	0
$893$	−4.47213	−0.149654
$894$	0	0
$895$	− 4.77301i	− 0.159544i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 2.35743i	− 0.0786247i
$900$	0	0
$901$	−43.7012	−1.45590
$902$	0	0
$903$	− 55.4532i	− 1.84537i
$904$	0	0
$905$	− 14.3118i	− 0.475741i
$906$	0	0
$907$	−28.2070	−0.936598	−0.468299	−	0.883570i	$-0.655133\pi$
$907$	−28.2070	−0.936598	−0.468299	+	0.883570i	$0.655133\pi$
$908$	0	0
$909$	−48.1696	−1.59769
$910$	0	0
$911$	40.5105	1.34217	0.671087	−	0.741379i	$-0.265828\pi$
$911$	40.5105	1.34217	0.671087	+	0.741379i	$0.265828\pi$
$912$	0	0
$913$	−14.0950	−0.466475
$914$	0	0
$915$	− 29.9819i	− 0.991170i
$916$	0	0
$917$	− 36.3393i	− 1.20003i
$918$	0	0
$919$	0.0746930	0.00246389	0.00123195	−	0.999999i	$-0.499608\pi$
$919$	0.0746930	0.00246389	0.00123195	+	0.999999i	$0.499608\pi$
$920$	0	0
$921$	67.6410i	2.22885i
$922$	0	0
$923$	0	0
$924$	0	0
$925$	1.61350i	0.0530514i
$926$	0	0
$927$	−75.2846	−2.47267
$928$	0	0
$929$	− 53.6410i	− 1.75990i	−0.475063	−	0.879952i	$-0.657575\pi$
$929$	− 53.6410i	− 1.75990i	0.475063	−	0.879952i	$-0.342425\pi$
$930$	0	0
$931$	− 79.5180i	− 2.60610i
$932$	0	0
$933$	66.1696	2.16630
$934$	0	0
$935$	7.92531	0.259185
$936$	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
$939$	104.513	3.41064
$940$	0	0
$941$	49.9253i	1.62752i	0.581202	+	0.813759i	$0.302583\pi$
$941$	49.9253i	1.62752i	−0.581202	+	0.813759i	$0.697417\pi$
$942$	0	0
$943$	− 3.15230i	− 0.102653i
$944$	0	0
$945$	−83.9637	−2.73134
$946$	0	0
$947$	− 5.80128i	− 0.188516i	−0.995548	−	0.0942582i	$-0.969952\pi$
$947$	− 5.80128i	− 0.188516i	0.995548	−	0.0942582i	$-0.0300479\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	− 5.73381i	− 0.185931i
$952$	0	0
$953$	13.2270	0.428464	0.214232	−	0.976783i	$-0.431275\pi$
$953$	13.2270	0.428464	0.214232	+	0.976783i	$0.431275\pi$
$954$	0	0
$955$	− 12.0000i	− 0.388311i
$956$	0	0
$957$	− 5.83036i	− 0.188469i
$958$	0	0
$959$	87.1907	2.81553
$960$	0	0
$961$	25.7439	0.830450
$962$	0	0
$963$	−107.398	−3.46084
$964$	0	0
$965$	4.05655	0.130585
$966$	0	0
$967$	1.61350i	0.0518865i	0.999663	+	0.0259433i	$0.00825892\pi$
$967$	1.61350i	0.0518865i	−0.999663	+	0.0259433i	$0.991741\pi$
$968$	0	0
$969$	67.0413i	2.15368i
$970$	0	0
$971$	16.7730	0.538271	0.269136	−	0.963102i	$-0.413262\pi$
$971$	16.7730	0.538271	0.269136	+	0.963102i	$0.413262\pi$
$972$	0	0
$973$	− 47.0557i	− 1.50854i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	47.6700i	1.52510i	0.646929	+	0.762550i	$0.276053\pi$
$977$	47.6700i	1.52510i	−0.646929	+	0.762550i	$0.723947\pi$
$978$	0	0
$979$	−2.09495	−0.0669549
$980$	0	0
$981$	− 90.5873i	− 2.89223i
$982$	0	0
$983$	30.4996i	0.972786i	0.873740	+	0.486393i	$0.161688\pi$
$983$	30.4996i	0.972786i	−0.873740	+	0.486393i	$0.838312\pi$
$984$	0	0
$985$	15.2835	0.486974
$986$	0	0
$987$	17.1704	0.546541
$988$	0	0
$989$	2.25526	0.0717132
$990$	0	0
$991$	−19.8506	−0.630576	−0.315288	−	0.948996i	$-0.602101\pi$
$991$	−19.8506	−0.630576	−0.315288	+	0.948996i	$0.602101\pi$
$992$	0	0
$993$	− 34.5561i	− 1.09661i
$994$	0	0
$995$	− 8.77301i	− 0.278123i
$996$	0	0
$997$	−33.6410	−1.06542	−0.532710	−	0.846298i	$-0.678826\pi$
$997$	−33.6410	−1.06542	−0.532710	+	0.846298i	$0.678826\pi$
$998$	0	0
$999$	− 26.9427i	− 0.852428i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3380.2.f.h.3041.5		6
13.5	odd	4		3380.2.a.o.1.3		3
13.8	odd	4		260.2.a.b.1.3	✓	3
13.12	even	2	inner	3380.2.f.h.3041.6		6
39.8	even	4		2340.2.a.n.1.1		3
52.47	even	4		1040.2.a.o.1.1		3
65.8	even	4		1300.2.c.f.1249.6		6
65.34	odd	4		1300.2.a.i.1.1		3
65.47	even	4		1300.2.c.f.1249.1		6
104.21	odd	4		4160.2.a.bo.1.1		3
104.99	even	4		4160.2.a.br.1.3		3
156.47	odd	4		9360.2.a.da.1.3		3
260.99	even	4		5200.2.a.ci.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
260.2.a.b.1.3	✓	3	13.8	odd	4
1040.2.a.o.1.1		3	52.47	even	4
1300.2.a.i.1.1		3	65.34	odd	4
1300.2.c.f.1249.1		6	65.47	even	4
1300.2.c.f.1249.6		6	65.8	even	4
2340.2.a.n.1.1		3	39.8	even	4
3380.2.a.o.1.3		3	13.5	odd	4
3380.2.f.h.3041.5		6	1.1	even	1	trivial
3380.2.f.h.3041.6		6	13.12	even	2	inner
4160.2.a.bo.1.1		3	104.21	odd	4
4160.2.a.br.1.3		3	104.99	even	4
5200.2.a.ci.1.3		3	260.99	even	4
9360.2.a.da.1.3		3	156.47	odd	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:	\( N \)	\(=\)	\( 3380 = 2^{2} \cdot 5 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3380.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+3.32088 q^{3} -1.00000i q^{5} -5.02827i q^{7} +8.02827 q^{9} +O(q^{10})\)	\(q+3.32088 q^{3} -1.00000i q^{5} -5.02827i q^{7} +8.02827 q^{9} +1.70739i q^{11} -3.32088i q^{15} +4.64177 q^{17} +4.34916i q^{19} -16.6983i q^{21} +0.679116 q^{23} -1.00000 q^{25} +16.6983 q^{27} -1.02827 q^{29} +2.29261i q^{31} +5.67004i q^{33} -5.02827 q^{35} -1.61350i q^{37} -4.64177i q^{41} +3.32088 q^{43} -8.02827i q^{45} +1.02827i q^{47} -18.2835 q^{49} +15.4148 q^{51} -9.41478 q^{53} +1.70739 q^{55} +14.4431i q^{57} -8.93438i q^{59} +9.02827 q^{61} -40.3684i q^{63} -5.61350i q^{67} +2.25526 q^{69} -1.70739i q^{71} +12.4431i q^{73} -3.32088 q^{75} +8.58522 q^{77} -2.64177 q^{79} +31.3684 q^{81} +8.25526i q^{83} -4.64177i q^{85} -3.41478 q^{87} +1.22699i q^{89} +7.61350i q^{93} +4.34916 q^{95} +0.0565477i q^{97} +13.7074i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 4 q^{3} + 22 q^{9}+O(q^{10}) \) 6 * q + 4 * q^3 + 22 * q^9	\( 6 q + 4 q^{3} + 22 q^{9} - 4 q^{17} + 20 q^{23} - 6 q^{25} + 16 q^{27} + 20 q^{29} - 4 q^{35} + 4 q^{43} - 46 q^{49} + 72 q^{51} - 36 q^{53} + 28 q^{61} - 24 q^{69} - 4 q^{75} + 72 q^{77} + 16 q^{79} + 46 q^{81} - 16 q^{95}+O(q^{100}) \) 6 * q + 4 * q^3 + 22 * q^9 - 4 * q^17 + 20 * q^23 - 6 * q^25 + 16 * q^27 + 20 * q^29 - 4 * q^35 + 4 * q^43 - 46 * q^49 + 72 * q^51 - 36 * q^53 + 28 * q^61 - 24 * q^69 - 4 * q^75 + 72 * q^77 + 16 * q^79 + 46 * q^81 - 16 * q^95

Introduction

L-functions

Modular forms

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Database

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