LMFDB - Embedded newform 7728.2.a.bi.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7728,2,Mod(1,7728)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7728, 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("7728.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7728 = 2^{4} \cdot 3 \cdot 7 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7728.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:	$61.7083906820$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{41}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 10 $ x^2 - x - 10
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 3864)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$3.70156$ of defining polynomial
Character	$\chi$	$=$	7728.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -1.00000 q^{7} +1.00000 q^{9} +3.70156 q^{11} +2.00000 q^{13} +5.40312 q^{17} +7.70156 q^{19} -1.00000 q^{21} -1.00000 q^{23} -5.00000 q^{25} +1.00000 q^{27} +2.00000 q^{29} -3.40312 q^{31} +3.70156 q^{33} +3.40312 q^{37} +2.00000 q^{39} -3.70156 q^{41} +1.40312 q^{43} +6.29844 q^{47} +1.00000 q^{49} +5.40312 q^{51} -9.10469 q^{53} +7.70156 q^{57} -5.10469 q^{59} +6.29844 q^{61} -1.00000 q^{63} +13.4031 q^{67} -1.00000 q^{69} -5.40312 q^{73} -5.00000 q^{75} -3.70156 q^{77} -11.4031 q^{79} +1.00000 q^{81} +10.0000 q^{83} +2.00000 q^{87} -2.00000 q^{89} -2.00000 q^{91} -3.40312 q^{93} +6.00000 q^{97} +3.70156 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} - 2 q^{7} + 2 q^{9}+O(q^{10}) $ 2 * q + 2 * q^3 - 2 * q^7 + 2 * q^9	$ 2 q + 2 q^{3} - 2 q^{7} + 2 q^{9} + q^{11} + 4 q^{13} - 2 q^{17} + 9 q^{19} - 2 q^{21} - 2 q^{23} - 10 q^{25} + 2 q^{27} + 4 q^{29} + 6 q^{31} + q^{33} - 6 q^{37} + 4 q^{39} - q^{41} - 10 q^{43} + 19 q^{47} + 2 q^{49} - 2 q^{51} + q^{53} + 9 q^{57} + 9 q^{59} + 19 q^{61} - 2 q^{63} + 14 q^{67} - 2 q^{69} + 2 q^{73} - 10 q^{75} - q^{77} - 10 q^{79} + 2 q^{81} + 20 q^{83} + 4 q^{87} - 4 q^{89} - 4 q^{91} + 6 q^{93} + 12 q^{97} + q^{99}+O(q^{100}) $ 2 * q + 2 * q^3 - 2 * q^7 + 2 * q^9 + q^11 + 4 * q^13 - 2 * q^17 + 9 * q^19 - 2 * q^21 - 2 * q^23 - 10 * q^25 + 2 * q^27 + 4 * q^29 + 6 * q^31 + q^33 - 6 * q^37 + 4 * q^39 - q^41 - 10 * q^43 + 19 * q^47 + 2 * q^49 - 2 * q^51 + q^53 + 9 * q^57 + 9 * q^59 + 19 * q^61 - 2 * q^63 + 14 * q^67 - 2 * q^69 + 2 * q^73 - 10 * q^75 - q^77 - 10 * q^79 + 2 * q^81 + 20 * q^83 + 4 * q^87 - 4 * q^89 - 4 * q^91 + 6 * q^93 + 12 * q^97 + 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
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	0	0		−	1.00000i	$-0.5\pi$
$5$	0	0			1.00000i	$0.5\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	3.70156	1.11606	0.558031	−	0.829820i	$-0.311557\pi$
$11$	3.70156	1.11606	0.558031	+	0.829820i	$0.311557\pi$
$12$	0	0
$13$	2.00000	0.554700	0.277350	−	0.960769i	$-0.410544\pi$
$13$	2.00000	0.554700	0.277350	+	0.960769i	$0.410544\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	5.40312	1.31045	0.655225	−	0.755434i	$-0.272574\pi$
$17$	5.40312	1.31045	0.655225	+	0.755434i	$0.272574\pi$
$18$	0	0
$19$	7.70156	1.76686	0.883430	−	0.468564i	$-0.155228\pi$
$19$	7.70156	1.76686	0.883430	+	0.468564i	$0.155228\pi$
$20$	0	0
$21$	−1.00000	−0.218218
$22$	0	0
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	−5.00000	−1.00000
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	2.00000	0.371391	0.185695	−	0.982607i	$-0.440546\pi$
$29$	2.00000	0.371391	0.185695	+	0.982607i	$0.440546\pi$
$30$	0	0
$31$	−3.40312	−0.611219	−0.305610	−	0.952157i	$-0.598860\pi$
$31$	−3.40312	−0.611219	−0.305610	+	0.952157i	$0.598860\pi$
$32$	0	0
$33$	3.70156	0.644359
$34$	0	0
$35$	0	0
$36$	0	0
$37$	3.40312	0.559470	0.279735	−	0.960077i	$-0.409753\pi$
$37$	3.40312	0.559470	0.279735	+	0.960077i	$0.409753\pi$
$38$	0	0
$39$	2.00000	0.320256
$40$	0	0
$41$	−3.70156	−0.578087	−0.289043	−	0.957316i	$-0.593337\pi$
$41$	−3.70156	−0.578087	−0.289043	+	0.957316i	$0.593337\pi$
$42$	0	0
$43$	1.40312	0.213974	0.106987	−	0.994260i	$-0.465880\pi$
$43$	1.40312	0.213974	0.106987	+	0.994260i	$0.465880\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	6.29844	0.918722	0.459361	−	0.888250i	$-0.348079\pi$
$47$	6.29844	0.918722	0.459361	+	0.888250i	$0.348079\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	5.40312	0.756589
$52$	0	0
$53$	−9.10469	−1.25062	−0.625312	−	0.780375i	$-0.715028\pi$
$53$	−9.10469	−1.25062	−0.625312	+	0.780375i	$0.715028\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	7.70156	1.02010
$58$	0	0
$59$	−5.10469	−0.664574	−0.332287	−	0.943178i	$-0.607820\pi$
$59$	−5.10469	−0.664574	−0.332287	+	0.943178i	$0.607820\pi$
$60$	0	0
$61$	6.29844	0.806432	0.403216	−	0.915105i	$-0.367892\pi$
$61$	6.29844	0.806432	0.403216	+	0.915105i	$0.367892\pi$
$62$	0	0
$63$	−1.00000	−0.125988
$64$	0	0
$65$	0	0
$66$	0	0
$67$	13.4031	1.63745	0.818726	−	0.574184i	$-0.194681\pi$
$67$	13.4031	1.63745	0.818726	+	0.574184i	$0.194681\pi$
$68$	0	0
$69$	−1.00000	−0.120386
$70$	0	0
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	0	0
$73$	−5.40312	−0.632388	−0.316194	−	0.948695i	$-0.602405\pi$
$73$	−5.40312	−0.632388	−0.316194	+	0.948695i	$0.602405\pi$
$74$	0	0
$75$	−5.00000	−0.577350
$76$	0	0
$77$	−3.70156	−0.421832
$78$	0	0
$79$	−11.4031	−1.28295	−0.641476	−	0.767143i	$-0.721678\pi$
$79$	−11.4031	−1.28295	−0.641476	+	0.767143i	$0.721678\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	10.0000	1.09764	0.548821	−	0.835940i	$-0.315077\pi$
$83$	10.0000	1.09764	0.548821	+	0.835940i	$0.315077\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	2.00000	0.214423
$88$	0	0
$89$	−2.00000	−0.212000	−0.106000	−	0.994366i	$-0.533804\pi$
$89$	−2.00000	−0.212000	−0.106000	+	0.994366i	$0.533804\pi$
$90$	0	0
$91$	−2.00000	−0.209657
$92$	0	0
$93$	−3.40312	−0.352888
$94$	0	0
$95$	0	0
$96$	0	0
$97$	6.00000	0.609208	0.304604	−	0.952479i	$-0.401476\pi$
$97$	6.00000	0.609208	0.304604	+	0.952479i	$0.401476\pi$
$98$	0	0
$99$	3.70156	0.372021
$100$	0	0
$101$	−7.70156	−0.766334	−0.383167	−	0.923679i	$-0.625167\pi$
$101$	−7.70156	−0.766334	−0.383167	+	0.923679i	$0.625167\pi$
$102$	0	0
$103$	−2.29844	−0.226472	−0.113236	−	0.993568i	$-0.536122\pi$
$103$	−2.29844	−0.226472	−0.113236	+	0.993568i	$0.536122\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	2.59688	0.251049	0.125525	−	0.992090i	$-0.459939\pi$
$107$	2.59688	0.251049	0.125525	+	0.992090i	$0.459939\pi$
$108$	0	0
$109$	10.8062	1.03505	0.517525	−	0.855668i	$-0.326853\pi$
$109$	10.8062	1.03505	0.517525	+	0.855668i	$0.326853\pi$
$110$	0	0
$111$	3.40312	0.323010
$112$	0	0
$113$	−2.00000	−0.188144	−0.0940721	−	0.995565i	$-0.529988\pi$
$113$	−2.00000	−0.188144	−0.0940721	+	0.995565i	$0.529988\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	2.00000	0.184900
$118$	0	0
$119$	−5.40312	−0.495304
$120$	0	0
$121$	2.70156	0.245597
$122$	0	0
$123$	−3.70156	−0.333759
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−13.1047	−1.16285	−0.581426	−	0.813599i	$-0.697505\pi$
$127$	−13.1047	−1.16285	−0.581426	+	0.813599i	$0.697505\pi$
$128$	0	0
$129$	1.40312	0.123538
$130$	0	0
$131$	14.2984	1.24926	0.624630	−	0.780921i	$-0.285250\pi$
$131$	14.2984	1.24926	0.624630	+	0.780921i	$0.285250\pi$
$132$	0	0
$133$	−7.70156	−0.667810
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−14.5078	−1.23949	−0.619743	−	0.784805i	$-0.712763\pi$
$137$	−14.5078	−1.23949	−0.619743	+	0.784805i	$0.712763\pi$
$138$	0	0
$139$	8.00000	0.678551	0.339276	−	0.940687i	$-0.389818\pi$
$139$	8.00000	0.678551	0.339276	+	0.940687i	$0.389818\pi$
$140$	0	0
$141$	6.29844	0.530424
$142$	0	0
$143$	7.40312	0.619080
$144$	0	0
$145$	0	0
$146$	0	0
$147$	1.00000	0.0824786
$148$	0	0
$149$	−13.7016	−1.12248	−0.561238	−	0.827655i	$-0.689675\pi$
$149$	−13.7016	−1.12248	−0.561238	+	0.827655i	$0.689675\pi$
$150$	0	0
$151$	−6.29844	−0.512560	−0.256280	−	0.966603i	$-0.582497\pi$
$151$	−6.29844	−0.512560	−0.256280	+	0.966603i	$0.582497\pi$
$152$	0	0
$153$	5.40312	0.436817
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−5.70156	−0.455034	−0.227517	−	0.973774i	$-0.573061\pi$
$157$	−5.70156	−0.455034	−0.227517	+	0.973774i	$0.573061\pi$
$158$	0	0
$159$	−9.10469	−0.722049
$160$	0	0
$161$	1.00000	0.0788110
$162$	0	0
$163$	−5.10469	−0.399830	−0.199915	−	0.979813i	$-0.564067\pi$
$163$	−5.10469	−0.399830	−0.199915	+	0.979813i	$0.564067\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	5.10469	0.395012	0.197506	−	0.980302i	$-0.436716\pi$
$167$	5.10469	0.395012	0.197506	+	0.980302i	$0.436716\pi$
$168$	0	0
$169$	−9.00000	−0.692308
$170$	0	0
$171$	7.70156	0.588953
$172$	0	0
$173$	−16.8062	−1.27776	−0.638878	−	0.769308i	$-0.720601\pi$
$173$	−16.8062	−1.27776	−0.638878	+	0.769308i	$0.720601\pi$
$174$	0	0
$175$	5.00000	0.377964
$176$	0	0
$177$	−5.10469	−0.383692
$178$	0	0
$179$	16.0000	1.19590	0.597948	−	0.801535i	$-0.295983\pi$
$179$	16.0000	1.19590	0.597948	+	0.801535i	$0.295983\pi$
$180$	0	0
$181$	3.40312	0.252952	0.126476	−	0.991970i	$-0.459633\pi$
$181$	3.40312	0.252952	0.126476	+	0.991970i	$0.459633\pi$
$182$	0	0
$183$	6.29844	0.465594
$184$	0	0
$185$	0	0
$186$	0	0
$187$	20.0000	1.46254
$188$	0	0
$189$	−1.00000	−0.0727393
$190$	0	0
$191$	21.7016	1.57027	0.785135	−	0.619325i	$-0.212594\pi$
$191$	21.7016	1.57027	0.785135	+	0.619325i	$0.212594\pi$
$192$	0	0
$193$	0.298438	0.0214820	0.0107410	−	0.999942i	$-0.496581\pi$
$193$	0.298438	0.0214820	0.0107410	+	0.999942i	$0.496581\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	8.80625	0.627419	0.313710	−	0.949519i	$-0.398428\pi$
$197$	8.80625	0.627419	0.313710	+	0.949519i	$0.398428\pi$
$198$	0	0
$199$	13.1047	0.928967	0.464483	−	0.885582i	$-0.346240\pi$
$199$	13.1047	0.928967	0.464483	+	0.885582i	$0.346240\pi$
$200$	0	0
$201$	13.4031	0.945383
$202$	0	0
$203$	−2.00000	−0.140372
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−1.00000	−0.0695048
$208$	0	0
$209$	28.5078	1.97193
$210$	0	0
$211$	5.70156	0.392512	0.196256	−	0.980553i	$-0.437122\pi$
$211$	5.70156	0.392512	0.196256	+	0.980553i	$0.437122\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	3.40312	0.231019
$218$	0	0
$219$	−5.40312	−0.365109
$220$	0	0
$221$	10.8062	0.726907
$222$	0	0
$223$	−10.2094	−0.683670	−0.341835	−	0.939760i	$-0.611048\pi$
$223$	−10.2094	−0.683670	−0.341835	+	0.939760i	$0.611048\pi$
$224$	0	0
$225$	−5.00000	−0.333333
$226$	0	0
$227$	20.8062	1.38096	0.690480	−	0.723352i	$-0.257400\pi$
$227$	20.8062	1.38096	0.690480	+	0.723352i	$0.257400\pi$
$228$	0	0
$229$	−9.70156	−0.641097	−0.320549	−	0.947232i	$-0.603867\pi$
$229$	−9.70156	−0.641097	−0.320549	+	0.947232i	$0.603867\pi$
$230$	0	0
$231$	−3.70156	−0.243545
$232$	0	0
$233$	6.59688	0.432176	0.216088	−	0.976374i	$-0.430670\pi$
$233$	6.59688	0.432176	0.216088	+	0.976374i	$0.430670\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−11.4031	−0.740713
$238$	0	0
$239$	−8.00000	−0.517477	−0.258738	−	0.965947i	$-0.583307\pi$
$239$	−8.00000	−0.517477	−0.258738	+	0.965947i	$0.583307\pi$
$240$	0	0
$241$	3.10469	0.199990	0.0999952	−	0.994988i	$-0.468117\pi$
$241$	3.10469	0.199990	0.0999952	+	0.994988i	$0.468117\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	15.4031	0.980077
$248$	0	0
$249$	10.0000	0.633724
$250$	0	0
$251$	−6.00000	−0.378717	−0.189358	−	0.981908i	$-0.560641\pi$
$251$	−6.00000	−0.378717	−0.189358	+	0.981908i	$0.560641\pi$
$252$	0	0
$253$	−3.70156	−0.232715
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−4.89531	−0.305361	−0.152681	−	0.988276i	$-0.548791\pi$
$257$	−4.89531	−0.305361	−0.152681	+	0.988276i	$0.548791\pi$
$258$	0	0
$259$	−3.40312	−0.211460
$260$	0	0
$261$	2.00000	0.123797
$262$	0	0
$263$	21.1047	1.30137	0.650685	−	0.759347i	$-0.274482\pi$
$263$	21.1047	1.30137	0.650685	+	0.759347i	$0.274482\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−2.00000	−0.122398
$268$	0	0
$269$	19.6125	1.19580	0.597898	−	0.801573i	$-0.296003\pi$
$269$	19.6125	1.19580	0.597898	+	0.801573i	$0.296003\pi$
$270$	0	0
$271$	14.8062	0.899416	0.449708	−	0.893176i	$-0.351528\pi$
$271$	14.8062	0.899416	0.449708	+	0.893176i	$0.351528\pi$
$272$	0	0
$273$	−2.00000	−0.121046
$274$	0	0
$275$	−18.5078	−1.11606
$276$	0	0
$277$	−3.10469	−0.186543	−0.0932713	−	0.995641i	$-0.529732\pi$
$277$	−3.10469	−0.186543	−0.0932713	+	0.995641i	$0.529732\pi$
$278$	0	0
$279$	−3.40312	−0.203740
$280$	0	0
$281$	20.8062	1.24120	0.620598	−	0.784129i	$-0.286890\pi$
$281$	20.8062	1.24120	0.620598	+	0.784129i	$0.286890\pi$
$282$	0	0
$283$	29.4031	1.74783	0.873917	−	0.486075i	$-0.161572\pi$
$283$	29.4031	1.74783	0.873917	+	0.486075i	$0.161572\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	3.70156	0.218496
$288$	0	0
$289$	12.1938	0.717280
$290$	0	0
$291$	6.00000	0.351726
$292$	0	0
$293$	21.6125	1.26262	0.631308	−	0.775532i	$-0.282518\pi$
$293$	21.6125	1.26262	0.631308	+	0.775532i	$0.282518\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	3.70156	0.214786
$298$	0	0
$299$	−2.00000	−0.115663
$300$	0	0
$301$	−1.40312	−0.0808747
$302$	0	0
$303$	−7.70156	−0.442443
$304$	0	0
$305$	0	0
$306$	0	0
$307$	2.80625	0.160161	0.0800805	−	0.996788i	$-0.474482\pi$
$307$	2.80625	0.160161	0.0800805	+	0.996788i	$0.474482\pi$
$308$	0	0
$309$	−2.29844	−0.130754
$310$	0	0
$311$	17.7016	1.00376	0.501882	−	0.864936i	$-0.332641\pi$
$311$	17.7016	1.00376	0.501882	+	0.864936i	$0.332641\pi$
$312$	0	0
$313$	−21.9109	−1.23848	−0.619240	−	0.785202i	$-0.712559\pi$
$313$	−21.9109	−1.23848	−0.619240	+	0.785202i	$0.712559\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−19.6125	−1.10155	−0.550774	−	0.834655i	$-0.685667\pi$
$317$	−19.6125	−1.10155	−0.550774	+	0.834655i	$0.685667\pi$
$318$	0	0
$319$	7.40312	0.414495
$320$	0	0
$321$	2.59688	0.144943
$322$	0	0
$323$	41.6125	2.31538
$324$	0	0
$325$	−10.0000	−0.554700
$326$	0	0
$327$	10.8062	0.597587
$328$	0	0
$329$	−6.29844	−0.347244
$330$	0	0
$331$	−2.29844	−0.126334	−0.0631668	−	0.998003i	$-0.520120\pi$
$331$	−2.29844	−0.126334	−0.0631668	+	0.998003i	$0.520120\pi$
$332$	0	0
$333$	3.40312	0.186490
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−12.8062	−0.697601	−0.348800	−	0.937197i	$-0.613411\pi$
$337$	−12.8062	−0.697601	−0.348800	+	0.937197i	$0.613411\pi$
$338$	0	0
$339$	−2.00000	−0.108625
$340$	0	0
$341$	−12.5969	−0.682159
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−10.2094	−0.548068	−0.274034	−	0.961720i	$-0.588358\pi$
$347$	−10.2094	−0.548068	−0.274034	+	0.961720i	$0.588358\pi$
$348$	0	0
$349$	−23.6125	−1.26395	−0.631974	−	0.774990i	$-0.717755\pi$
$349$	−23.6125	−1.26395	−0.631974	+	0.774990i	$0.717755\pi$
$350$	0	0
$351$	2.00000	0.106752
$352$	0	0
$353$	−27.6125	−1.46966	−0.734832	−	0.678249i	$-0.762739\pi$
$353$	−27.6125	−1.46966	−0.734832	+	0.678249i	$0.762739\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−5.40312	−0.285964
$358$	0	0
$359$	18.8062	0.992556	0.496278	−	0.868164i	$-0.334700\pi$
$359$	18.8062	0.992556	0.496278	+	0.868164i	$0.334700\pi$
$360$	0	0
$361$	40.3141	2.12179
$362$	0	0
$363$	2.70156	0.141795
$364$	0	0
$365$	0	0
$366$	0	0
$367$	29.1047	1.51925	0.759626	−	0.650360i	$-0.225382\pi$
$367$	29.1047	1.51925	0.759626	+	0.650360i	$0.225382\pi$
$368$	0	0
$369$	−3.70156	−0.192696
$370$	0	0
$371$	9.10469	0.472692
$372$	0	0
$373$	27.4031	1.41888	0.709440	−	0.704766i	$-0.248948\pi$
$373$	27.4031	1.41888	0.709440	+	0.704766i	$0.248948\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	4.00000	0.206010
$378$	0	0
$379$	−22.0000	−1.13006	−0.565032	−	0.825069i	$-0.691136\pi$
$379$	−22.0000	−1.13006	−0.565032	+	0.825069i	$0.691136\pi$
$380$	0	0
$381$	−13.1047	−0.671373
$382$	0	0
$383$	−10.2094	−0.521675	−0.260837	−	0.965383i	$-0.583999\pi$
$383$	−10.2094	−0.521675	−0.260837	+	0.965383i	$0.583999\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	1.40312	0.0713248
$388$	0	0
$389$	18.2094	0.923252	0.461626	−	0.887075i	$-0.347266\pi$
$389$	18.2094	0.923252	0.461626	+	0.887075i	$0.347266\pi$
$390$	0	0
$391$	−5.40312	−0.273248
$392$	0	0
$393$	14.2984	0.721261
$394$	0	0
$395$	0	0
$396$	0	0
$397$	16.8062	0.843481	0.421741	−	0.906716i	$-0.361419\pi$
$397$	16.8062	0.843481	0.421741	+	0.906716i	$0.361419\pi$
$398$	0	0
$399$	−7.70156	−0.385560
$400$	0	0
$401$	−19.1047	−0.954043	−0.477021	−	0.878892i	$-0.658284\pi$
$401$	−19.1047	−0.954043	−0.477021	+	0.878892i	$0.658284\pi$
$402$	0	0
$403$	−6.80625	−0.339043
$404$	0	0
$405$	0	0
$406$	0	0
$407$	12.5969	0.624404
$408$	0	0
$409$	10.0000	0.494468	0.247234	−	0.968956i	$-0.420478\pi$
$409$	10.0000	0.494468	0.247234	+	0.968956i	$0.420478\pi$
$410$	0	0
$411$	−14.5078	−0.715618
$412$	0	0
$413$	5.10469	0.251185
$414$	0	0
$415$	0	0
$416$	0	0
$417$	8.00000	0.391762
$418$	0	0
$419$	−4.80625	−0.234801	−0.117400	−	0.993085i	$-0.537456\pi$
$419$	−4.80625	−0.234801	−0.117400	+	0.993085i	$0.537456\pi$
$420$	0	0
$421$	−16.0000	−0.779792	−0.389896	−	0.920859i	$-0.627489\pi$
$421$	−16.0000	−0.779792	−0.389896	+	0.920859i	$0.627489\pi$
$422$	0	0
$423$	6.29844	0.306241
$424$	0	0
$425$	−27.0156	−1.31045
$426$	0	0
$427$	−6.29844	−0.304803
$428$	0	0
$429$	7.40312	0.357426
$430$	0	0
$431$	3.49219	0.168213	0.0841064	−	0.996457i	$-0.473196\pi$
$431$	3.49219	0.168213	0.0841064	+	0.996457i	$0.473196\pi$
$432$	0	0
$433$	−17.3141	−0.832061	−0.416030	−	0.909351i	$-0.636579\pi$
$433$	−17.3141	−0.832061	−0.416030	+	0.909351i	$0.636579\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−7.70156	−0.368416
$438$	0	0
$439$	−9.19375	−0.438794	−0.219397	−	0.975636i	$-0.570409\pi$
$439$	−9.19375	−0.438794	−0.219397	+	0.975636i	$0.570409\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	8.00000	0.380091	0.190046	−	0.981775i	$-0.439136\pi$
$443$	8.00000	0.380091	0.190046	+	0.981775i	$0.439136\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−13.7016	−0.648062
$448$	0	0
$449$	28.8062	1.35945	0.679725	−	0.733467i	$-0.262099\pi$
$449$	28.8062	1.35945	0.679725	+	0.733467i	$0.262099\pi$
$450$	0	0
$451$	−13.7016	−0.645181
$452$	0	0
$453$	−6.29844	−0.295926
$454$	0	0
$455$	0	0
$456$	0	0
$457$	11.6125	0.543210	0.271605	−	0.962409i	$-0.412446\pi$
$457$	11.6125	0.543210	0.271605	+	0.962409i	$0.412446\pi$
$458$	0	0
$459$	5.40312	0.252196
$460$	0	0
$461$	−0.806248	−0.0375507	−0.0187754	−	0.999824i	$-0.505977\pi$
$461$	−0.806248	−0.0375507	−0.0187754	+	0.999824i	$0.505977\pi$
$462$	0	0
$463$	−11.9109	−0.553548	−0.276774	−	0.960935i	$-0.589265\pi$
$463$	−11.9109	−0.553548	−0.276774	+	0.960935i	$0.589265\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	11.1938	0.517985	0.258993	−	0.965879i	$-0.416609\pi$
$467$	11.1938	0.517985	0.258993	+	0.965879i	$0.416609\pi$
$468$	0	0
$469$	−13.4031	−0.618899
$470$	0	0
$471$	−5.70156	−0.262714
$472$	0	0
$473$	5.19375	0.238809
$474$	0	0
$475$	−38.5078	−1.76686
$476$	0	0
$477$	−9.10469	−0.416875
$478$	0	0
$479$	−8.00000	−0.365529	−0.182765	−	0.983157i	$-0.558505\pi$
$479$	−8.00000	−0.365529	−0.182765	+	0.983157i	$0.558505\pi$
$480$	0	0
$481$	6.80625	0.310338
$482$	0	0
$483$	1.00000	0.0455016
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−22.8062	−1.03345	−0.516725	−	0.856152i	$-0.672849\pi$
$487$	−22.8062	−1.03345	−0.516725	+	0.856152i	$0.672849\pi$
$488$	0	0
$489$	−5.10469	−0.230842
$490$	0	0
$491$	−4.59688	−0.207454	−0.103727	−	0.994606i	$-0.533077\pi$
$491$	−4.59688	−0.207454	−0.103727	+	0.994606i	$0.533077\pi$
$492$	0	0
$493$	10.8062	0.486689
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−17.1938	−0.769698	−0.384849	−	0.922980i	$-0.625746\pi$
$499$	−17.1938	−0.769698	−0.384849	+	0.922980i	$0.625746\pi$
$500$	0	0
$501$	5.10469	0.228061
$502$	0	0
$503$	10.8062	0.481827	0.240913	−	0.970547i	$-0.422553\pi$
$503$	10.8062	0.481827	0.240913	+	0.970547i	$0.422553\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−9.00000	−0.399704
$508$	0	0
$509$	−20.8953	−0.926168	−0.463084	−	0.886314i	$-0.653257\pi$
$509$	−20.8953	−0.926168	−0.463084	+	0.886314i	$0.653257\pi$
$510$	0	0
$511$	5.40312	0.239020
$512$	0	0
$513$	7.70156	0.340032
$514$	0	0
$515$	0	0
$516$	0	0
$517$	23.3141	1.02535
$518$	0	0
$519$	−16.8062	−0.737712
$520$	0	0
$521$	−30.0000	−1.31432	−0.657162	−	0.753749i	$-0.728243\pi$
$521$	−30.0000	−1.31432	−0.657162	+	0.753749i	$0.728243\pi$
$522$	0	0
$523$	31.7016	1.38621	0.693106	−	0.720835i	$-0.256242\pi$
$523$	31.7016	1.38621	0.693106	+	0.720835i	$0.256242\pi$
$524$	0	0
$525$	5.00000	0.218218
$526$	0	0
$527$	−18.3875	−0.800972
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	−5.10469	−0.221525
$532$	0	0
$533$	−7.40312	−0.320665
$534$	0	0
$535$	0	0
$536$	0	0
$537$	16.0000	0.690451
$538$	0	0
$539$	3.70156	0.159438
$540$	0	0
$541$	17.3141	0.744390	0.372195	−	0.928155i	$-0.378605\pi$
$541$	17.3141	0.744390	0.372195	+	0.928155i	$0.378605\pi$
$542$	0	0
$543$	3.40312	0.146042
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−34.8062	−1.48821	−0.744104	−	0.668064i	$-0.767123\pi$
$547$	−34.8062	−1.48821	−0.744104	+	0.668064i	$0.767123\pi$
$548$	0	0
$549$	6.29844	0.268811
$550$	0	0
$551$	15.4031	0.656195
$552$	0	0
$553$	11.4031	0.484910
$554$	0	0
$555$	0	0
$556$	0	0
$557$	42.2094	1.78847	0.894234	−	0.447599i	$-0.147721\pi$
$557$	42.2094	1.78847	0.894234	+	0.447599i	$0.147721\pi$
$558$	0	0
$559$	2.80625	0.118692
$560$	0	0
$561$	20.0000	0.844401
$562$	0	0
$563$	−5.40312	−0.227715	−0.113857	−	0.993497i	$-0.536321\pi$
$563$	−5.40312	−0.227715	−0.113857	+	0.993497i	$0.536321\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−1.00000	−0.0419961
$568$	0	0
$569$	19.7016	0.825932	0.412966	−	0.910746i	$-0.364493\pi$
$569$	19.7016	0.825932	0.412966	+	0.910746i	$0.364493\pi$
$570$	0	0
$571$	−2.59688	−0.108676	−0.0543379	−	0.998523i	$-0.517305\pi$
$571$	−2.59688	−0.108676	−0.0543379	+	0.998523i	$0.517305\pi$
$572$	0	0
$573$	21.7016	0.906596
$574$	0	0
$575$	5.00000	0.208514
$576$	0	0
$577$	−16.8062	−0.699653	−0.349827	−	0.936814i	$-0.613760\pi$
$577$	−16.8062	−0.699653	−0.349827	+	0.936814i	$0.613760\pi$
$578$	0	0
$579$	0.298438	0.0124027
$580$	0	0
$581$	−10.0000	−0.414870
$582$	0	0
$583$	−33.7016	−1.39578
$584$	0	0
$585$	0	0
$586$	0	0
$587$	5.10469	0.210693	0.105346	−	0.994436i	$-0.466405\pi$
$587$	5.10469	0.210693	0.105346	+	0.994436i	$0.466405\pi$
$588$	0	0
$589$	−26.2094	−1.07994
$590$	0	0
$591$	8.80625	0.362241
$592$	0	0
$593$	14.0000	0.574911	0.287456	−	0.957794i	$-0.407191\pi$
$593$	14.0000	0.574911	0.287456	+	0.957794i	$0.407191\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	13.1047	0.536339
$598$	0	0
$599$	20.5969	0.841566	0.420783	−	0.907161i	$-0.361755\pi$
$599$	20.5969	0.841566	0.420783	+	0.907161i	$0.361755\pi$
$600$	0	0
$601$	8.20937	0.334867	0.167434	−	0.985883i	$-0.446452\pi$
$601$	8.20937	0.334867	0.167434	+	0.985883i	$0.446452\pi$
$602$	0	0
$603$	13.4031	0.545817
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−38.8062	−1.57510	−0.787549	−	0.616253i	$-0.788650\pi$
$607$	−38.8062	−1.57510	−0.787549	+	0.616253i	$0.788650\pi$
$608$	0	0
$609$	−2.00000	−0.0810441
$610$	0	0
$611$	12.5969	0.509615
$612$	0	0
$613$	1.79063	0.0723228	0.0361614	−	0.999346i	$-0.488487\pi$
$613$	1.79063	0.0723228	0.0361614	+	0.999346i	$0.488487\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	20.8062	0.837628	0.418814	−	0.908072i	$-0.362446\pi$
$617$	20.8062	0.837628	0.418814	+	0.908072i	$0.362446\pi$
$618$	0	0
$619$	−45.8219	−1.84174	−0.920868	−	0.389874i	$-0.872519\pi$
$619$	−45.8219	−1.84174	−0.920868	+	0.389874i	$0.872519\pi$
$620$	0	0
$621$	−1.00000	−0.0401286
$622$	0	0
$623$	2.00000	0.0801283
$624$	0	0
$625$	25.0000	1.00000
$626$	0	0
$627$	28.5078	1.13849
$628$	0	0
$629$	18.3875	0.733158
$630$	0	0
$631$	−39.4031	−1.56861	−0.784307	−	0.620373i	$-0.786981\pi$
$631$	−39.4031	−1.56861	−0.784307	+	0.620373i	$0.786981\pi$
$632$	0	0
$633$	5.70156	0.226617
$634$	0	0
$635$	0	0
$636$	0	0
$637$	2.00000	0.0792429
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−25.9109	−1.02342	−0.511710	−	0.859158i	$-0.670988\pi$
$641$	−25.9109	−1.02342	−0.511710	+	0.859158i	$0.670988\pi$
$642$	0	0
$643$	−16.8953	−0.666286	−0.333143	−	0.942876i	$-0.608109\pi$
$643$	−16.8953	−0.666286	−0.333143	+	0.942876i	$0.608109\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−1.19375	−0.0469312	−0.0234656	−	0.999725i	$-0.507470\pi$
$647$	−1.19375	−0.0469312	−0.0234656	+	0.999725i	$0.507470\pi$
$648$	0	0
$649$	−18.8953	−0.741706
$650$	0	0
$651$	3.40312	0.133379
$652$	0	0
$653$	8.20937	0.321258	0.160629	−	0.987015i	$-0.448648\pi$
$653$	8.20937	0.321258	0.160629	+	0.987015i	$0.448648\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−5.40312	−0.210796
$658$	0	0
$659$	47.0156	1.83147	0.915734	−	0.401784i	$-0.131610\pi$
$659$	47.0156	1.83147	0.915734	+	0.401784i	$0.131610\pi$
$660$	0	0
$661$	−39.3141	−1.52914	−0.764570	−	0.644541i	$-0.777049\pi$
$661$	−39.3141	−1.52914	−0.764570	+	0.644541i	$0.777049\pi$
$662$	0	0
$663$	10.8062	0.419680
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−2.00000	−0.0774403
$668$	0	0
$669$	−10.2094	−0.394717
$670$	0	0
$671$	23.3141	0.900029
$672$	0	0
$673$	49.9109	1.92393	0.961963	−	0.273181i	$-0.0880759\pi$
$673$	49.9109	1.92393	0.961963	+	0.273181i	$0.0880759\pi$
$674$	0	0
$675$	−5.00000	−0.192450
$676$	0	0
$677$	8.00000	0.307465	0.153732	−	0.988113i	$-0.450871\pi$
$677$	8.00000	0.307465	0.153732	+	0.988113i	$0.450871\pi$
$678$	0	0
$679$	−6.00000	−0.230259
$680$	0	0
$681$	20.8062	0.797297
$682$	0	0
$683$	9.61250	0.367812	0.183906	−	0.982944i	$-0.441126\pi$
$683$	9.61250	0.367812	0.183906	+	0.982944i	$0.441126\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−9.70156	−0.370138
$688$	0	0
$689$	−18.2094	−0.693722
$690$	0	0
$691$	−6.80625	−0.258922	−0.129461	−	0.991585i	$-0.541325\pi$
$691$	−6.80625	−0.258922	−0.129461	+	0.991585i	$0.541325\pi$
$692$	0	0
$693$	−3.70156	−0.140611
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−20.0000	−0.757554
$698$	0	0
$699$	6.59688	0.249517
$700$	0	0
$701$	−35.9109	−1.35634	−0.678169	−	0.734906i	$-0.737226\pi$
$701$	−35.9109	−1.35634	−0.678169	+	0.734906i	$0.737226\pi$
$702$	0	0
$703$	26.2094	0.988505
$704$	0	0
$705$	0	0
$706$	0	0
$707$	7.70156	0.289647
$708$	0	0
$709$	3.40312	0.127807	0.0639035	−	0.997956i	$-0.479645\pi$
$709$	3.40312	0.127807	0.0639035	+	0.997956i	$0.479645\pi$
$710$	0	0
$711$	−11.4031	−0.427651
$712$	0	0
$713$	3.40312	0.127448
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−8.00000	−0.298765
$718$	0	0
$719$	−29.6125	−1.10436	−0.552180	−	0.833725i	$-0.686204\pi$
$719$	−29.6125	−1.10436	−0.552180	+	0.833725i	$0.686204\pi$
$720$	0	0
$721$	2.29844	0.0855983
$722$	0	0
$723$	3.10469	0.115465
$724$	0	0
$725$	−10.0000	−0.371391
$726$	0	0
$727$	−11.9109	−0.441752	−0.220876	−	0.975302i	$-0.570892\pi$
$727$	−11.9109	−0.441752	−0.220876	+	0.975302i	$0.570892\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	7.58125	0.280403
$732$	0	0
$733$	8.59688	0.317533	0.158766	−	0.987316i	$-0.449248\pi$
$733$	8.59688	0.317533	0.158766	+	0.987316i	$0.449248\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	49.6125	1.82750
$738$	0	0
$739$	38.8062	1.42751	0.713755	−	0.700395i	$-0.246993\pi$
$739$	38.8062	1.42751	0.713755	+	0.700395i	$0.246993\pi$
$740$	0	0
$741$	15.4031	0.565848
$742$	0	0
$743$	−18.8953	−0.693202	−0.346601	−	0.938013i	$-0.612664\pi$
$743$	−18.8953	−0.693202	−0.346601	+	0.938013i	$0.612664\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	10.0000	0.365881
$748$	0	0
$749$	−2.59688	−0.0948878
$750$	0	0
$751$	−49.6125	−1.81039	−0.905193	−	0.425001i	$-0.860274\pi$
$751$	−49.6125	−1.81039	−0.905193	+	0.425001i	$0.860274\pi$
$752$	0	0
$753$	−6.00000	−0.218652
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−2.80625	−0.101995	−0.0509974	−	0.998699i	$-0.516240\pi$
$757$	−2.80625	−0.101995	−0.0509974	+	0.998699i	$0.516240\pi$
$758$	0	0
$759$	−3.70156	−0.134358
$760$	0	0
$761$	19.7016	0.714181	0.357091	−	0.934070i	$-0.383769\pi$
$761$	19.7016	0.714181	0.357091	+	0.934070i	$0.383769\pi$
$762$	0	0
$763$	−10.8062	−0.391212
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−10.2094	−0.368639
$768$	0	0
$769$	−23.6125	−0.851488	−0.425744	−	0.904844i	$-0.639988\pi$
$769$	−23.6125	−0.851488	−0.425744	+	0.904844i	$0.639988\pi$
$770$	0	0
$771$	−4.89531	−0.176300
$772$	0	0
$773$	2.80625	0.100934	0.0504669	−	0.998726i	$-0.483929\pi$
$773$	2.80625	0.100934	0.0504669	+	0.998726i	$0.483929\pi$
$774$	0	0
$775$	17.0156	0.611219
$776$	0	0
$777$	−3.40312	−0.122086
$778$	0	0
$779$	−28.5078	−1.02140
$780$	0	0
$781$	0	0
$782$	0	0
$783$	2.00000	0.0714742
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−46.5078	−1.65782	−0.828912	−	0.559379i	$-0.811040\pi$
$787$	−46.5078	−1.65782	−0.828912	+	0.559379i	$0.811040\pi$
$788$	0	0
$789$	21.1047	0.751347
$790$	0	0
$791$	2.00000	0.0711118
$792$	0	0
$793$	12.5969	0.447328
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−14.2094	−0.503322	−0.251661	−	0.967815i	$-0.580977\pi$
$797$	−14.2094	−0.503322	−0.251661	+	0.967815i	$0.580977\pi$
$798$	0	0
$799$	34.0312	1.20394
$800$	0	0
$801$	−2.00000	−0.0706665
$802$	0	0
$803$	−20.0000	−0.705785
$804$	0	0
$805$	0	0
$806$	0	0
$807$	19.6125	0.690393
$808$	0	0
$809$	19.1938	0.674816	0.337408	−	0.941358i	$-0.390450\pi$
$809$	19.1938	0.674816	0.337408	+	0.941358i	$0.390450\pi$
$810$	0	0
$811$	2.20937	0.0775816	0.0387908	−	0.999247i	$-0.487649\pi$
$811$	2.20937	0.0775816	0.0387908	+	0.999247i	$0.487649\pi$
$812$	0	0
$813$	14.8062	0.519278
$814$	0	0
$815$	0	0
$816$	0	0
$817$	10.8062	0.378063
$818$	0	0
$819$	−2.00000	−0.0698857
$820$	0	0
$821$	−12.2094	−0.426110	−0.213055	−	0.977040i	$-0.568341\pi$
$821$	−12.2094	−0.426110	−0.213055	+	0.977040i	$0.568341\pi$
$822$	0	0
$823$	−7.31406	−0.254952	−0.127476	−	0.991842i	$-0.540688\pi$
$823$	−7.31406	−0.254952	−0.127476	+	0.991842i	$0.540688\pi$
$824$	0	0
$825$	−18.5078	−0.644359
$826$	0	0
$827$	−28.7172	−0.998594	−0.499297	−	0.866431i	$-0.666408\pi$
$827$	−28.7172	−0.998594	−0.499297	+	0.866431i	$0.666408\pi$
$828$	0	0
$829$	−10.0000	−0.347314	−0.173657	−	0.984806i	$-0.555558\pi$
$829$	−10.0000	−0.347314	−0.173657	+	0.984806i	$0.555558\pi$
$830$	0	0
$831$	−3.10469	−0.107700
$832$	0	0
$833$	5.40312	0.187207
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−3.40312	−0.117629
$838$	0	0
$839$	−26.2094	−0.904848	−0.452424	−	0.891803i	$-0.649441\pi$
$839$	−26.2094	−0.904848	−0.452424	+	0.891803i	$0.649441\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	0	0
$843$	20.8062	0.716605
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−2.70156	−0.0928268
$848$	0	0
$849$	29.4031	1.00911
$850$	0	0
$851$	−3.40312	−0.116658
$852$	0	0
$853$	−39.0156	−1.33587	−0.667935	−	0.744220i	$-0.732821\pi$
$853$	−39.0156	−1.33587	−0.667935	+	0.744220i	$0.732821\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−31.7016	−1.08290	−0.541452	−	0.840731i	$-0.682125\pi$
$857$	−31.7016	−1.08290	−0.541452	+	0.840731i	$0.682125\pi$
$858$	0	0
$859$	−13.1938	−0.450165	−0.225082	−	0.974340i	$-0.572265\pi$
$859$	−13.1938	−0.450165	−0.225082	+	0.974340i	$0.572265\pi$
$860$	0	0
$861$	3.70156	0.126149
$862$	0	0
$863$	11.4031	0.388167	0.194083	−	0.980985i	$-0.437827\pi$
$863$	11.4031	0.388167	0.194083	+	0.980985i	$0.437827\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	12.1938	0.414122
$868$	0	0
$869$	−42.2094	−1.43185
$870$	0	0
$871$	26.8062	0.908295
$872$	0	0
$873$	6.00000	0.203069
$874$	0	0
$875$	0	0
$876$	0	0
$877$	4.89531	0.165303	0.0826515	−	0.996579i	$-0.473661\pi$
$877$	4.89531	0.165303	0.0826515	+	0.996579i	$0.473661\pi$
$878$	0	0
$879$	21.6125	0.728971
$880$	0	0
$881$	11.6125	0.391235	0.195617	−	0.980680i	$-0.437329\pi$
$881$	11.6125	0.391235	0.195617	+	0.980680i	$0.437329\pi$
$882$	0	0
$883$	−42.8062	−1.44054	−0.720272	−	0.693691i	$-0.755983\pi$
$883$	−42.8062	−1.44054	−0.720272	+	0.693691i	$0.755983\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−8.00000	−0.268614	−0.134307	−	0.990940i	$-0.542881\pi$
$887$	−8.00000	−0.268614	−0.134307	+	0.990940i	$0.542881\pi$
$888$	0	0
$889$	13.1047	0.439517
$890$	0	0
$891$	3.70156	0.124007
$892$	0	0
$893$	48.5078	1.62325
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−2.00000	−0.0667781
$898$	0	0
$899$	−6.80625	−0.227001
$900$	0	0
$901$	−49.1938	−1.63888
$902$	0	0
$903$	−1.40312	−0.0466931
$904$	0	0
$905$	0	0
$906$	0	0
$907$	40.2094	1.33513	0.667565	−	0.744551i	$-0.267337\pi$
$907$	40.2094	1.33513	0.667565	+	0.744551i	$0.267337\pi$
$908$	0	0
$909$	−7.70156	−0.255445
$910$	0	0
$911$	−2.80625	−0.0929752	−0.0464876	−	0.998919i	$-0.514803\pi$
$911$	−2.80625	−0.0929752	−0.0464876	+	0.998919i	$0.514803\pi$
$912$	0	0
$913$	37.0156	1.22504
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−14.2984	−0.472176
$918$	0	0
$919$	3.40312	0.112259	0.0561294	−	0.998424i	$-0.482124\pi$
$919$	3.40312	0.112259	0.0561294	+	0.998424i	$0.482124\pi$
$920$	0	0
$921$	2.80625	0.0924690
$922$	0	0
$923$	0	0
$924$	0	0
$925$	−17.0156	−0.559470
$926$	0	0
$927$	−2.29844	−0.0754906
$928$	0	0
$929$	50.4187	1.65419	0.827093	−	0.562066i	$-0.189993\pi$
$929$	50.4187	1.65419	0.827093	+	0.562066i	$0.189993\pi$
$930$	0	0
$931$	7.70156	0.252409
$932$	0	0
$933$	17.7016	0.579523
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−0.895314	−0.0292486	−0.0146243	−	0.999893i	$-0.504655\pi$
$937$	−0.895314	−0.0292486	−0.0146243	+	0.999893i	$0.504655\pi$
$938$	0	0
$939$	−21.9109	−0.715036
$940$	0	0
$941$	−52.4187	−1.70880	−0.854401	−	0.519614i	$-0.826076\pi$
$941$	−52.4187	−1.70880	−0.854401	+	0.519614i	$0.826076\pi$
$942$	0	0
$943$	3.70156	0.120539
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−46.8062	−1.52100	−0.760499	−	0.649339i	$-0.775046\pi$
$947$	−46.8062	−1.52100	−0.760499	+	0.649339i	$0.775046\pi$
$948$	0	0
$949$	−10.8062	−0.350786
$950$	0	0
$951$	−19.6125	−0.635979
$952$	0	0
$953$	31.1047	1.00758	0.503790	−	0.863826i	$-0.331939\pi$
$953$	31.1047	1.00758	0.503790	+	0.863826i	$0.331939\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	7.40312	0.239309
$958$	0	0
$959$	14.5078	0.468482
$960$	0	0
$961$	−19.4187	−0.626411
$962$	0	0
$963$	2.59688	0.0836832
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−38.8062	−1.24792	−0.623962	−	0.781454i	$-0.714478\pi$
$967$	−38.8062	−1.24792	−0.623962	+	0.781454i	$0.714478\pi$
$968$	0	0
$969$	41.6125	1.33679
$970$	0	0
$971$	36.8062	1.18117	0.590584	−	0.806976i	$-0.298897\pi$
$971$	36.8062	1.18117	0.590584	+	0.806976i	$0.298897\pi$
$972$	0	0
$973$	−8.00000	−0.256468
$974$	0	0
$975$	−10.0000	−0.320256
$976$	0	0
$977$	24.2984	0.777376	0.388688	−	0.921369i	$-0.372928\pi$
$977$	24.2984	0.777376	0.388688	+	0.921369i	$0.372928\pi$
$978$	0	0
$979$	−7.40312	−0.236605
$980$	0	0
$981$	10.8062	0.345017
$982$	0	0
$983$	16.5969	0.529358	0.264679	−	0.964337i	$-0.414734\pi$
$983$	16.5969	0.529358	0.264679	+	0.964337i	$0.414734\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−6.29844	−0.200481
$988$	0	0
$989$	−1.40312	−0.0446168
$990$	0	0
$991$	12.0891	0.384022	0.192011	−	0.981393i	$-0.438499\pi$
$991$	12.0891	0.384022	0.192011	+	0.981393i	$0.438499\pi$
$992$	0	0
$993$	−2.29844	−0.0729387
$994$	0	0
$995$	0	0
$996$	0	0
$997$	15.0156	0.475549	0.237775	−	0.971320i	$-0.423582\pi$
$997$	15.0156	0.475549	0.237775	+	0.971320i	$0.423582\pi$
$998$	0	0
$999$	3.40312	0.107670

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7728.2.a.bi.1.2		2
4.3	odd	2		3864.2.a.g.1.1	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3864.2.a.g.1.1	✓	2	4.3	odd	2
7728.2.a.bi.1.2		2	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 7728 = 2^{4} \cdot 3 \cdot 7 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7728.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -1.00000 q^{7} +1.00000 q^{9} +3.70156 q^{11} +2.00000 q^{13} +5.40312 q^{17} +7.70156 q^{19} -1.00000 q^{21} -1.00000 q^{23} -5.00000 q^{25} +1.00000 q^{27} +2.00000 q^{29} -3.40312 q^{31} +3.70156 q^{33} +3.40312 q^{37} +2.00000 q^{39} -3.70156 q^{41} +1.40312 q^{43} +6.29844 q^{47} +1.00000 q^{49} +5.40312 q^{51} -9.10469 q^{53} +7.70156 q^{57} -5.10469 q^{59} +6.29844 q^{61} -1.00000 q^{63} +13.4031 q^{67} -1.00000 q^{69} -5.40312 q^{73} -5.00000 q^{75} -3.70156 q^{77} -11.4031 q^{79} +1.00000 q^{81} +10.0000 q^{83} +2.00000 q^{87} -2.00000 q^{89} -2.00000 q^{91} -3.40312 q^{93} +6.00000 q^{97} +3.70156 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} - 2 q^{7} + 2 q^{9}+O(q^{10}) \) 2 * q + 2 * q^3 - 2 * q^7 + 2 * q^9	\( 2 q + 2 q^{3} - 2 q^{7} + 2 q^{9} + q^{11} + 4 q^{13} - 2 q^{17} + 9 q^{19} - 2 q^{21} - 2 q^{23} - 10 q^{25} + 2 q^{27} + 4 q^{29} + 6 q^{31} + q^{33} - 6 q^{37} + 4 q^{39} - q^{41} - 10 q^{43} + 19 q^{47} + 2 q^{49} - 2 q^{51} + q^{53} + 9 q^{57} + 9 q^{59} + 19 q^{61} - 2 q^{63} + 14 q^{67} - 2 q^{69} + 2 q^{73} - 10 q^{75} - q^{77} - 10 q^{79} + 2 q^{81} + 20 q^{83} + 4 q^{87} - 4 q^{89} - 4 q^{91} + 6 q^{93} + 12 q^{97} + q^{99}+O(q^{100}) \) 2 * q + 2 * q^3 - 2 * q^7 + 2 * q^9 + q^11 + 4 * q^13 - 2 * q^17 + 9 * q^19 - 2 * q^21 - 2 * q^23 - 10 * q^25 + 2 * q^27 + 4 * q^29 + 6 * q^31 + q^33 - 6 * q^37 + 4 * q^39 - q^41 - 10 * q^43 + 19 * q^47 + 2 * q^49 - 2 * q^51 + q^53 + 9 * q^57 + 9 * q^59 + 19 * q^61 - 2 * q^63 + 14 * q^67 - 2 * q^69 + 2 * q^73 - 10 * q^75 - q^77 - 10 * q^79 + 2 * q^81 + 20 * q^83 + 4 * q^87 - 4 * q^89 - 4 * q^91 + 6 * q^93 + 12 * q^97 + q^99

Introduction

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