LMFDB - Embedded newform 2156.2.a.e.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2156,2,Mod(1,2156)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2156, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("2156.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2156 = 2^{2} \cdot 7^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2156.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:	$17.2157466758$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.41421$ of defining polynomial
Character	$\chi$	$=$	2156.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.41421 q^{3} -1.00000 q^{9} -1.00000 q^{11} -1.41421 q^{13} -7.07107 q^{17} +2.82843 q^{19} -4.00000 q^{23} -5.00000 q^{25} -5.65685 q^{27} +2.00000 q^{29} -4.24264 q^{31} -1.41421 q^{33} -4.00000 q^{37} -2.00000 q^{39} +1.41421 q^{41} +2.00000 q^{43} +9.89949 q^{47} -10.0000 q^{51} -4.00000 q^{53} +4.00000 q^{57} +4.24264 q^{59} -12.7279 q^{61} -8.00000 q^{67} -5.65685 q^{69} +1.41421 q^{73} -7.07107 q^{75} -10.0000 q^{79} -5.00000 q^{81} +8.48528 q^{83} +2.82843 q^{87} +11.3137 q^{89} -6.00000 q^{93} +8.48528 q^{97} +1.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{9} - 2 q^{11} - 8 q^{23} - 10 q^{25} + 4 q^{29} - 8 q^{37} - 4 q^{39} + 4 q^{43} - 20 q^{51} - 8 q^{53} + 8 q^{57} - 16 q^{67} - 20 q^{79} - 10 q^{81} - 12 q^{93} + 2 q^{99}+O(q^{100}) $ 2 * q - 2 * q^9 - 2 * q^11 - 8 * q^23 - 10 * q^25 + 4 * q^29 - 8 * q^37 - 4 * q^39 + 4 * q^43 - 20 * q^51 - 8 * q^53 + 8 * q^57 - 16 * q^67 - 20 * q^79 - 10 * q^81 - 12 * q^93 + 2 * q^99

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
$3$	1.41421	0.816497	0.408248	−	0.912871i	$-0.366140\pi$
$3$	1.41421	0.816497	0.408248	+	0.912871i	$0.366140\pi$
$4$	0	0
$5$	0	0		−	1.00000i	$-0.5\pi$
$5$	0	0			1.00000i	$0.5\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	−1.00000	−0.333333
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	−1.41421	−0.392232	−0.196116	−	0.980581i	$-0.562833\pi$
$13$	−1.41421	−0.392232	−0.196116	+	0.980581i	$0.562833\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−7.07107	−1.71499	−0.857493	−	0.514496i	$-0.827979\pi$
$17$	−7.07107	−1.71499	−0.857493	+	0.514496i	$0.827979\pi$
$18$	0	0
$19$	2.82843	0.648886	0.324443	−	0.945905i	$-0.394823\pi$
$19$	2.82843	0.648886	0.324443	+	0.945905i	$0.394823\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−4.00000	−0.834058	−0.417029	−	0.908893i	$-0.636929\pi$
$23$	−4.00000	−0.834058	−0.417029	+	0.908893i	$0.636929\pi$
$24$	0	0
$25$	−5.00000	−1.00000
$26$	0	0
$27$	−5.65685	−1.08866
$28$	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
$31$	−4.24264	−0.762001	−0.381000	−	0.924575i	$-0.624420\pi$
$31$	−4.24264	−0.762001	−0.381000	+	0.924575i	$0.624420\pi$
$32$	0	0
$33$	−1.41421	−0.246183
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−4.00000	−0.657596	−0.328798	−	0.944400i	$-0.606644\pi$
$37$	−4.00000	−0.657596	−0.328798	+	0.944400i	$0.606644\pi$
$38$	0	0
$39$	−2.00000	−0.320256
$40$	0	0
$41$	1.41421	0.220863	0.110432	−	0.993884i	$-0.464777\pi$
$41$	1.41421	0.220863	0.110432	+	0.993884i	$0.464777\pi$
$42$	0	0
$43$	2.00000	0.304997	0.152499	−	0.988304i	$-0.451268\pi$
$43$	2.00000	0.304997	0.152499	+	0.988304i	$0.451268\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	9.89949	1.44399	0.721995	−	0.691898i	$-0.243225\pi$
$47$	9.89949	1.44399	0.721995	+	0.691898i	$0.243225\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	−10.0000	−1.40028
$52$	0	0
$53$	−4.00000	−0.549442	−0.274721	−	0.961524i	$-0.588586\pi$
$53$	−4.00000	−0.549442	−0.274721	+	0.961524i	$0.588586\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	4.00000	0.529813
$58$	0	0
$59$	4.24264	0.552345	0.276172	−	0.961108i	$-0.410934\pi$
$59$	4.24264	0.552345	0.276172	+	0.961108i	$0.410934\pi$
$60$	0	0
$61$	−12.7279	−1.62964	−0.814822	−	0.579712i	$-0.803165\pi$
$61$	−12.7279	−1.62964	−0.814822	+	0.579712i	$0.803165\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−8.00000	−0.977356	−0.488678	−	0.872464i	$-0.662521\pi$
$67$	−8.00000	−0.977356	−0.488678	+	0.872464i	$0.662521\pi$
$68$	0	0
$69$	−5.65685	−0.681005
$70$	0	0
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	0	0
$73$	1.41421	0.165521	0.0827606	−	0.996569i	$-0.473626\pi$
$73$	1.41421	0.165521	0.0827606	+	0.996569i	$0.473626\pi$
$74$	0	0
$75$	−7.07107	−0.816497
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−10.0000	−1.12509	−0.562544	−	0.826767i	$-0.690177\pi$
$79$	−10.0000	−1.12509	−0.562544	+	0.826767i	$0.690177\pi$
$80$	0	0
$81$	−5.00000	−0.555556
$82$	0	0
$83$	8.48528	0.931381	0.465690	−	0.884948i	$-0.345806\pi$
$83$	8.48528	0.931381	0.465690	+	0.884948i	$0.345806\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	2.82843	0.303239
$88$	0	0
$89$	11.3137	1.19925	0.599625	−	0.800281i	$-0.295316\pi$
$89$	11.3137	1.19925	0.599625	+	0.800281i	$0.295316\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−6.00000	−0.622171
$94$	0	0
$95$	0	0
$96$	0	0
$97$	8.48528	0.861550	0.430775	−	0.902459i	$-0.358240\pi$
$97$	8.48528	0.861550	0.430775	+	0.902459i	$0.358240\pi$
$98$	0	0
$99$	1.00000	0.100504
$100$	0	0
$101$	−1.41421	−0.140720	−0.0703598	−	0.997522i	$-0.522415\pi$
$101$	−1.41421	−0.140720	−0.0703598	+	0.997522i	$0.522415\pi$
$102$	0	0
$103$	7.07107	0.696733	0.348367	−	0.937358i	$-0.386736\pi$
$103$	7.07107	0.696733	0.348367	+	0.937358i	$0.386736\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−12.0000	−1.16008	−0.580042	−	0.814587i	$-0.696964\pi$
$107$	−12.0000	−1.16008	−0.580042	+	0.814587i	$0.696964\pi$
$108$	0	0
$109$	−10.0000	−0.957826	−0.478913	−	0.877862i	$-0.658969\pi$
$109$	−10.0000	−0.957826	−0.478913	+	0.877862i	$0.658969\pi$
$110$	0	0
$111$	−5.65685	−0.536925
$112$	0	0
$113$	14.0000	1.31701	0.658505	−	0.752577i	$-0.271189\pi$
$113$	14.0000	1.31701	0.658505	+	0.752577i	$0.271189\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	1.41421	0.130744
$118$	0	0
$119$	0	0
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	2.00000	0.180334
$124$	0	0
$125$	0	0
$126$	0	0
$127$	0	0		−	1.00000i	$-0.5\pi$
$127$	0	0			1.00000i	$0.5\pi$
$128$	0	0
$129$	2.82843	0.249029
$130$	0	0
$131$	−11.3137	−0.988483	−0.494242	−	0.869325i	$-0.664554\pi$
$131$	−11.3137	−0.988483	−0.494242	+	0.869325i	$0.664554\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−8.00000	−0.683486	−0.341743	−	0.939793i	$-0.611017\pi$
$137$	−8.00000	−0.683486	−0.341743	+	0.939793i	$0.611017\pi$
$138$	0	0
$139$	−8.48528	−0.719712	−0.359856	−	0.933008i	$-0.617174\pi$
$139$	−8.48528	−0.719712	−0.359856	+	0.933008i	$0.617174\pi$
$140$	0	0
$141$	14.0000	1.17901
$142$	0	0
$143$	1.41421	0.118262
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−6.00000	−0.491539	−0.245770	−	0.969328i	$-0.579041\pi$
$149$	−6.00000	−0.491539	−0.245770	+	0.969328i	$0.579041\pi$
$150$	0	0
$151$	−2.00000	−0.162758	−0.0813788	−	0.996683i	$-0.525932\pi$
$151$	−2.00000	−0.162758	−0.0813788	+	0.996683i	$0.525932\pi$
$152$	0	0
$153$	7.07107	0.571662
$154$	0	0
$155$	0	0
$156$	0	0
$157$	11.3137	0.902932	0.451466	−	0.892288i	$-0.350901\pi$
$157$	11.3137	0.902932	0.451466	+	0.892288i	$0.350901\pi$
$158$	0	0
$159$	−5.65685	−0.448618
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−4.00000	−0.313304	−0.156652	−	0.987654i	$-0.550070\pi$
$163$	−4.00000	−0.313304	−0.156652	+	0.987654i	$0.550070\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	11.3137	0.875481	0.437741	−	0.899101i	$-0.355779\pi$
$167$	11.3137	0.875481	0.437741	+	0.899101i	$0.355779\pi$
$168$	0	0
$169$	−11.0000	−0.846154
$170$	0	0
$171$	−2.82843	−0.216295
$172$	0	0
$173$	−15.5563	−1.18273	−0.591364	−	0.806405i	$-0.701410\pi$
$173$	−15.5563	−1.18273	−0.591364	+	0.806405i	$0.701410\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	6.00000	0.450988
$178$	0	0
$179$	−4.00000	−0.298974	−0.149487	−	0.988764i	$-0.547762\pi$
$179$	−4.00000	−0.298974	−0.149487	+	0.988764i	$0.547762\pi$
$180$	0	0
$181$	19.7990	1.47165	0.735824	−	0.677173i	$-0.236795\pi$
$181$	19.7990	1.47165	0.735824	+	0.677173i	$0.236795\pi$
$182$	0	0
$183$	−18.0000	−1.33060
$184$	0	0
$185$	0	0
$186$	0	0
$187$	7.07107	0.517088
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−8.00000	−0.578860	−0.289430	−	0.957199i	$-0.593466\pi$
$191$	−8.00000	−0.578860	−0.289430	+	0.957199i	$0.593466\pi$
$192$	0	0
$193$	−14.0000	−1.00774	−0.503871	−	0.863779i	$-0.668091\pi$
$193$	−14.0000	−1.00774	−0.503871	+	0.863779i	$0.668091\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	26.0000	1.85242	0.926212	−	0.377004i	$-0.123046\pi$
$197$	26.0000	1.85242	0.926212	+	0.377004i	$0.123046\pi$
$198$	0	0
$199$	24.0416	1.70427	0.852133	−	0.523325i	$-0.175309\pi$
$199$	24.0416	1.70427	0.852133	+	0.523325i	$0.175309\pi$
$200$	0	0
$201$	−11.3137	−0.798007
$202$	0	0
$203$	0	0
$204$	0	0
$205$	0	0
$206$	0	0
$207$	4.00000	0.278019
$208$	0	0
$209$	−2.82843	−0.195646
$210$	0	0
$211$	−20.0000	−1.37686	−0.688428	−	0.725304i	$-0.741699\pi$
$211$	−20.0000	−1.37686	−0.688428	+	0.725304i	$0.741699\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	0	0
$218$	0	0
$219$	2.00000	0.135147
$220$	0	0
$221$	10.0000	0.672673
$222$	0	0
$223$	7.07107	0.473514	0.236757	−	0.971569i	$-0.423916\pi$
$223$	7.07107	0.473514	0.236757	+	0.971569i	$0.423916\pi$
$224$	0	0
$225$	5.00000	0.333333
$226$	0	0
$227$	22.6274	1.50183	0.750917	−	0.660396i	$-0.229612\pi$
$227$	22.6274	1.50183	0.750917	+	0.660396i	$0.229612\pi$
$228$	0	0
$229$	14.1421	0.934539	0.467269	−	0.884115i	$-0.345238\pi$
$229$	14.1421	0.934539	0.467269	+	0.884115i	$0.345238\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−2.00000	−0.131024	−0.0655122	−	0.997852i	$-0.520868\pi$
$233$	−2.00000	−0.131024	−0.0655122	+	0.997852i	$0.520868\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−14.1421	−0.918630
$238$	0	0
$239$	−16.0000	−1.03495	−0.517477	−	0.855697i	$-0.673129\pi$
$239$	−16.0000	−1.03495	−0.517477	+	0.855697i	$0.673129\pi$
$240$	0	0
$241$	−26.8701	−1.73085	−0.865426	−	0.501036i	$-0.832952\pi$
$241$	−26.8701	−1.73085	−0.865426	+	0.501036i	$0.832952\pi$
$242$	0	0
$243$	9.89949	0.635053
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−4.00000	−0.254514
$248$	0	0
$249$	12.0000	0.760469
$250$	0	0
$251$	−15.5563	−0.981908	−0.490954	−	0.871185i	$-0.663352\pi$
$251$	−15.5563	−0.981908	−0.490954	+	0.871185i	$0.663352\pi$
$252$	0	0
$253$	4.00000	0.251478
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−5.65685	−0.352865	−0.176432	−	0.984313i	$-0.556456\pi$
$257$	−5.65685	−0.352865	−0.176432	+	0.984313i	$0.556456\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−2.00000	−0.123797
$262$	0	0
$263$	8.00000	0.493301	0.246651	−	0.969104i	$-0.420670\pi$
$263$	8.00000	0.493301	0.246651	+	0.969104i	$0.420670\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	16.0000	0.979184
$268$	0	0
$269$	−19.7990	−1.20717	−0.603583	−	0.797300i	$-0.706261\pi$
$269$	−19.7990	−1.20717	−0.603583	+	0.797300i	$0.706261\pi$
$270$	0	0
$271$	25.4558	1.54633	0.773166	−	0.634203i	$-0.218672\pi$
$271$	25.4558	1.54633	0.773166	+	0.634203i	$0.218672\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	5.00000	0.301511
$276$	0	0
$277$	−2.00000	−0.120168	−0.0600842	−	0.998193i	$-0.519137\pi$
$277$	−2.00000	−0.120168	−0.0600842	+	0.998193i	$0.519137\pi$
$278$	0	0
$279$	4.24264	0.254000
$280$	0	0
$281$	2.00000	0.119310	0.0596550	−	0.998219i	$-0.481000\pi$
$281$	2.00000	0.119310	0.0596550	+	0.998219i	$0.481000\pi$
$282$	0	0
$283$	−28.2843	−1.68133	−0.840663	−	0.541559i	$-0.817834\pi$
$283$	−28.2843	−1.68133	−0.840663	+	0.541559i	$0.817834\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	33.0000	1.94118
$290$	0	0
$291$	12.0000	0.703452
$292$	0	0
$293$	−1.41421	−0.0826192	−0.0413096	−	0.999146i	$-0.513153\pi$
$293$	−1.41421	−0.0826192	−0.0413096	+	0.999146i	$0.513153\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	5.65685	0.328244
$298$	0	0
$299$	5.65685	0.327144
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−2.00000	−0.114897
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−11.3137	−0.645707	−0.322854	−	0.946449i	$-0.604642\pi$
$307$	−11.3137	−0.645707	−0.322854	+	0.946449i	$0.604642\pi$
$308$	0	0
$309$	10.0000	0.568880
$310$	0	0
$311$	−15.5563	−0.882120	−0.441060	−	0.897478i	$-0.645397\pi$
$311$	−15.5563	−0.882120	−0.441060	+	0.897478i	$0.645397\pi$
$312$	0	0
$313$	−22.6274	−1.27898	−0.639489	−	0.768801i	$-0.720854\pi$
$313$	−22.6274	−1.27898	−0.639489	+	0.768801i	$0.720854\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	22.0000	1.23564	0.617822	−	0.786318i	$-0.288015\pi$
$317$	22.0000	1.23564	0.617822	+	0.786318i	$0.288015\pi$
$318$	0	0
$319$	−2.00000	−0.111979
$320$	0	0
$321$	−16.9706	−0.947204
$322$	0	0
$323$	−20.0000	−1.11283
$324$	0	0
$325$	7.07107	0.392232
$326$	0	0
$327$	−14.1421	−0.782062
$328$	0	0
$329$	0	0
$330$	0	0
$331$	8.00000	0.439720	0.219860	−	0.975531i	$-0.429440\pi$
$331$	8.00000	0.439720	0.219860	+	0.975531i	$0.429440\pi$
$332$	0	0
$333$	4.00000	0.219199
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−22.0000	−1.19842	−0.599208	−	0.800593i	$-0.704518\pi$
$337$	−22.0000	−1.19842	−0.599208	+	0.800593i	$0.704518\pi$
$338$	0	0
$339$	19.7990	1.07533
$340$	0	0
$341$	4.24264	0.229752
$342$	0	0
$343$	0	0
$344$	0	0
$345$	0	0
$346$	0	0
$347$	22.0000	1.18102	0.590511	−	0.807030i	$-0.298926\pi$
$347$	22.0000	1.18102	0.590511	+	0.807030i	$0.298926\pi$
$348$	0	0
$349$	4.24264	0.227103	0.113552	−	0.993532i	$-0.463777\pi$
$349$	4.24264	0.227103	0.113552	+	0.993532i	$0.463777\pi$
$350$	0	0
$351$	8.00000	0.427008
$352$	0	0
$353$	11.3137	0.602168	0.301084	−	0.953598i	$-0.402652\pi$
$353$	11.3137	0.602168	0.301084	+	0.953598i	$0.402652\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−22.0000	−1.16112	−0.580558	−	0.814219i	$-0.697165\pi$
$359$	−22.0000	−1.16112	−0.580558	+	0.814219i	$0.697165\pi$
$360$	0	0
$361$	−11.0000	−0.578947
$362$	0	0
$363$	1.41421	0.0742270
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−29.6985	−1.55025	−0.775124	−	0.631809i	$-0.782313\pi$
$367$	−29.6985	−1.55025	−0.775124	+	0.631809i	$0.782313\pi$
$368$	0	0
$369$	−1.41421	−0.0736210
$370$	0	0
$371$	0	0
$372$	0	0
$373$	38.0000	1.96757	0.983783	−	0.179364i	$-0.0574041\pi$
$373$	38.0000	1.96757	0.983783	+	0.179364i	$0.0574041\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−2.82843	−0.145671
$378$	0	0
$379$	−24.0000	−1.23280	−0.616399	−	0.787434i	$-0.711409\pi$
$379$	−24.0000	−1.23280	−0.616399	+	0.787434i	$0.711409\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−1.41421	−0.0722629	−0.0361315	−	0.999347i	$-0.511504\pi$
$383$	−1.41421	−0.0722629	−0.0361315	+	0.999347i	$0.511504\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−2.00000	−0.101666
$388$	0	0
$389$	26.0000	1.31825	0.659126	−	0.752032i	$-0.270926\pi$
$389$	26.0000	1.31825	0.659126	+	0.752032i	$0.270926\pi$
$390$	0	0
$391$	28.2843	1.43040
$392$	0	0
$393$	−16.0000	−0.807093
$394$	0	0
$395$	0	0
$396$	0	0
$397$	22.6274	1.13564	0.567819	−	0.823154i	$-0.307787\pi$
$397$	22.6274	1.13564	0.567819	+	0.823154i	$0.307787\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	18.0000	0.898877	0.449439	−	0.893311i	$-0.351624\pi$
$401$	18.0000	0.898877	0.449439	+	0.893311i	$0.351624\pi$
$402$	0	0
$403$	6.00000	0.298881
$404$	0	0
$405$	0	0
$406$	0	0
$407$	4.00000	0.198273
$408$	0	0
$409$	12.7279	0.629355	0.314678	−	0.949199i	$-0.398104\pi$
$409$	12.7279	0.629355	0.314678	+	0.949199i	$0.398104\pi$
$410$	0	0
$411$	−11.3137	−0.558064
$412$	0	0
$413$	0	0
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−12.0000	−0.587643
$418$	0	0
$419$	−21.2132	−1.03633	−0.518166	−	0.855280i	$-0.673385\pi$
$419$	−21.2132	−1.03633	−0.518166	+	0.855280i	$0.673385\pi$
$420$	0	0
$421$	32.0000	1.55958	0.779792	−	0.626038i	$-0.215325\pi$
$421$	32.0000	1.55958	0.779792	+	0.626038i	$0.215325\pi$
$422$	0	0
$423$	−9.89949	−0.481330
$424$	0	0
$425$	35.3553	1.71499
$426$	0	0
$427$	0	0
$428$	0	0
$429$	2.00000	0.0965609
$430$	0	0
$431$	30.0000	1.44505	0.722525	−	0.691345i	$-0.242982\pi$
$431$	30.0000	1.44505	0.722525	+	0.691345i	$0.242982\pi$
$432$	0	0
$433$	−25.4558	−1.22333	−0.611665	−	0.791117i	$-0.709500\pi$
$433$	−25.4558	−1.22333	−0.611665	+	0.791117i	$0.709500\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−11.3137	−0.541208
$438$	0	0
$439$	−8.48528	−0.404980	−0.202490	−	0.979284i	$-0.564903\pi$
$439$	−8.48528	−0.404980	−0.202490	+	0.979284i	$0.564903\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	24.0000	1.14027	0.570137	−	0.821549i	$-0.306890\pi$
$443$	24.0000	1.14027	0.570137	+	0.821549i	$0.306890\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−8.48528	−0.401340
$448$	0	0
$449$	4.00000	0.188772	0.0943858	−	0.995536i	$-0.469911\pi$
$449$	4.00000	0.188772	0.0943858	+	0.995536i	$0.469911\pi$
$450$	0	0
$451$	−1.41421	−0.0665927
$452$	0	0
$453$	−2.82843	−0.132891
$454$	0	0
$455$	0	0
$456$	0	0
$457$	38.0000	1.77757	0.888783	−	0.458329i	$-0.151552\pi$
$457$	38.0000	1.77757	0.888783	+	0.458329i	$0.151552\pi$
$458$	0	0
$459$	40.0000	1.86704
$460$	0	0
$461$	24.0416	1.11973	0.559865	−	0.828584i	$-0.310853\pi$
$461$	24.0416	1.11973	0.559865	+	0.828584i	$0.310853\pi$
$462$	0	0
$463$	−20.0000	−0.929479	−0.464739	−	0.885448i	$-0.653852\pi$
$463$	−20.0000	−0.929479	−0.464739	+	0.885448i	$0.653852\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−12.7279	−0.588978	−0.294489	−	0.955655i	$-0.595149\pi$
$467$	−12.7279	−0.588978	−0.294489	+	0.955655i	$0.595149\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	16.0000	0.737241
$472$	0	0
$473$	−2.00000	−0.0919601
$474$	0	0
$475$	−14.1421	−0.648886
$476$	0	0
$477$	4.00000	0.183147
$478$	0	0
$479$	−25.4558	−1.16311	−0.581554	−	0.813508i	$-0.697555\pi$
$479$	−25.4558	−1.16311	−0.581554	+	0.813508i	$0.697555\pi$
$480$	0	0
$481$	5.65685	0.257930
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−32.0000	−1.45006	−0.725029	−	0.688718i	$-0.758174\pi$
$487$	−32.0000	−1.45006	−0.725029	+	0.688718i	$0.758174\pi$
$488$	0	0
$489$	−5.65685	−0.255812
$490$	0	0
$491$	−34.0000	−1.53440	−0.767199	−	0.641409i	$-0.778350\pi$
$491$	−34.0000	−1.53440	−0.767199	+	0.641409i	$0.778350\pi$
$492$	0	0
$493$	−14.1421	−0.636930
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−12.0000	−0.537194	−0.268597	−	0.963253i	$-0.586560\pi$
$499$	−12.0000	−0.537194	−0.268597	+	0.963253i	$0.586560\pi$
$500$	0	0
$501$	16.0000	0.714827
$502$	0	0
$503$	19.7990	0.882793	0.441397	−	0.897312i	$-0.354483\pi$
$503$	19.7990	0.882793	0.441397	+	0.897312i	$0.354483\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−15.5563	−0.690882
$508$	0	0
$509$	25.4558	1.12831	0.564155	−	0.825669i	$-0.309202\pi$
$509$	25.4558	1.12831	0.564155	+	0.825669i	$0.309202\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−16.0000	−0.706417
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−9.89949	−0.435379
$518$	0	0
$519$	−22.0000	−0.965693
$520$	0	0
$521$	36.7696	1.61090	0.805452	−	0.592661i	$-0.201923\pi$
$521$	36.7696	1.61090	0.805452	+	0.592661i	$0.201923\pi$
$522$	0	0
$523$	−16.9706	−0.742071	−0.371035	−	0.928619i	$-0.620997\pi$
$523$	−16.9706	−0.742071	−0.371035	+	0.928619i	$0.620997\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	30.0000	1.30682
$528$	0	0
$529$	−7.00000	−0.304348
$530$	0	0
$531$	−4.24264	−0.184115
$532$	0	0
$533$	−2.00000	−0.0866296
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−5.65685	−0.244111
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−6.00000	−0.257960	−0.128980	−	0.991647i	$-0.541170\pi$
$541$	−6.00000	−0.257960	−0.128980	+	0.991647i	$0.541170\pi$
$542$	0	0
$543$	28.0000	1.20160
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−20.0000	−0.855138	−0.427569	−	0.903983i	$-0.640630\pi$
$547$	−20.0000	−0.855138	−0.427569	+	0.903983i	$0.640630\pi$
$548$	0	0
$549$	12.7279	0.543214
$550$	0	0
$551$	5.65685	0.240990
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	0	0
$557$	30.0000	1.27114	0.635570	−	0.772043i	$-0.280765\pi$
$557$	30.0000	1.27114	0.635570	+	0.772043i	$0.280765\pi$
$558$	0	0
$559$	−2.82843	−0.119630
$560$	0	0
$561$	10.0000	0.422200
$562$	0	0
$563$	−2.82843	−0.119204	−0.0596020	−	0.998222i	$-0.518983\pi$
$563$	−2.82843	−0.119204	−0.0596020	+	0.998222i	$0.518983\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−6.00000	−0.251533	−0.125767	−	0.992060i	$-0.540139\pi$
$569$	−6.00000	−0.251533	−0.125767	+	0.992060i	$0.540139\pi$
$570$	0	0
$571$	−20.0000	−0.836974	−0.418487	−	0.908223i	$-0.637439\pi$
$571$	−20.0000	−0.836974	−0.418487	+	0.908223i	$0.637439\pi$
$572$	0	0
$573$	−11.3137	−0.472637
$574$	0	0
$575$	20.0000	0.834058
$576$	0	0
$577$	19.7990	0.824243	0.412121	−	0.911129i	$-0.364788\pi$
$577$	19.7990	0.824243	0.412121	+	0.911129i	$0.364788\pi$
$578$	0	0
$579$	−19.7990	−0.822818
$580$	0	0
$581$	0	0
$582$	0	0
$583$	4.00000	0.165663
$584$	0	0
$585$	0	0
$586$	0	0
$587$	26.8701	1.10905	0.554523	−	0.832168i	$-0.312901\pi$
$587$	26.8701	1.10905	0.554523	+	0.832168i	$0.312901\pi$
$588$	0	0
$589$	−12.0000	−0.494451
$590$	0	0
$591$	36.7696	1.51250
$592$	0	0
$593$	−12.7279	−0.522673	−0.261337	−	0.965248i	$-0.584163\pi$
$593$	−12.7279	−0.522673	−0.261337	+	0.965248i	$0.584163\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	34.0000	1.39153
$598$	0	0
$599$	36.0000	1.47092	0.735460	−	0.677568i	$-0.236966\pi$
$599$	36.0000	1.47092	0.735460	+	0.677568i	$0.236966\pi$
$600$	0	0
$601$	12.7279	0.519183	0.259591	−	0.965719i	$-0.416412\pi$
$601$	12.7279	0.519183	0.259591	+	0.965719i	$0.416412\pi$
$602$	0	0
$603$	8.00000	0.325785
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−16.9706	−0.688814	−0.344407	−	0.938820i	$-0.611920\pi$
$607$	−16.9706	−0.688814	−0.344407	+	0.938820i	$0.611920\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−14.0000	−0.566379
$612$	0	0
$613$	10.0000	0.403896	0.201948	−	0.979396i	$-0.435273\pi$
$613$	10.0000	0.403896	0.201948	+	0.979396i	$0.435273\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	32.0000	1.28827	0.644136	−	0.764911i	$-0.277217\pi$
$617$	32.0000	1.28827	0.644136	+	0.764911i	$0.277217\pi$
$618$	0	0
$619$	32.5269	1.30737	0.653683	−	0.756768i	$-0.273223\pi$
$619$	32.5269	1.30737	0.653683	+	0.756768i	$0.273223\pi$
$620$	0	0
$621$	22.6274	0.908007
$622$	0	0
$623$	0	0
$624$	0	0
$625$	25.0000	1.00000
$626$	0	0
$627$	−4.00000	−0.159745
$628$	0	0
$629$	28.2843	1.12777
$630$	0	0
$631$	−36.0000	−1.43314	−0.716569	−	0.697517i	$-0.754288\pi$
$631$	−36.0000	−1.43314	−0.716569	+	0.697517i	$0.754288\pi$
$632$	0	0
$633$	−28.2843	−1.12420
$634$	0	0
$635$	0	0
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−20.0000	−0.789953	−0.394976	−	0.918691i	$-0.629247\pi$
$641$	−20.0000	−0.789953	−0.394976	+	0.918691i	$0.629247\pi$
$642$	0	0
$643$	12.7279	0.501940	0.250970	−	0.967995i	$-0.419250\pi$
$643$	12.7279	0.501940	0.250970	+	0.967995i	$0.419250\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−15.5563	−0.611583	−0.305792	−	0.952098i	$-0.598921\pi$
$647$	−15.5563	−0.611583	−0.305792	+	0.952098i	$0.598921\pi$
$648$	0	0
$649$	−4.24264	−0.166538
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−18.0000	−0.704394	−0.352197	−	0.935926i	$-0.614565\pi$
$653$	−18.0000	−0.704394	−0.352197	+	0.935926i	$0.614565\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−1.41421	−0.0551737
$658$	0	0
$659$	38.0000	1.48027	0.740135	−	0.672458i	$-0.234762\pi$
$659$	38.0000	1.48027	0.740135	+	0.672458i	$0.234762\pi$
$660$	0	0
$661$	33.9411	1.32016	0.660078	−	0.751197i	$-0.270523\pi$
$661$	33.9411	1.32016	0.660078	+	0.751197i	$0.270523\pi$
$662$	0	0
$663$	14.1421	0.549235
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−8.00000	−0.309761
$668$	0	0
$669$	10.0000	0.386622
$670$	0	0
$671$	12.7279	0.491356
$672$	0	0
$673$	−46.0000	−1.77317	−0.886585	−	0.462566i	$-0.846929\pi$
$673$	−46.0000	−1.77317	−0.886585	+	0.462566i	$0.846929\pi$
$674$	0	0
$675$	28.2843	1.08866
$676$	0	0
$677$	−35.3553	−1.35882	−0.679408	−	0.733761i	$-0.737763\pi$
$677$	−35.3553	−1.35882	−0.679408	+	0.733761i	$0.737763\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	32.0000	1.22624
$682$	0	0
$683$	4.00000	0.153056	0.0765279	−	0.997067i	$-0.475617\pi$
$683$	4.00000	0.153056	0.0765279	+	0.997067i	$0.475617\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	20.0000	0.763048
$688$	0	0
$689$	5.65685	0.215509
$690$	0	0
$691$	−38.1838	−1.45258	−0.726289	−	0.687389i	$-0.758757\pi$
$691$	−38.1838	−1.45258	−0.726289	+	0.687389i	$0.758757\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−10.0000	−0.378777
$698$	0	0
$699$	−2.82843	−0.106981
$700$	0	0
$701$	−18.0000	−0.679851	−0.339925	−	0.940452i	$-0.610402\pi$
$701$	−18.0000	−0.679851	−0.339925	+	0.940452i	$0.610402\pi$
$702$	0	0
$703$	−11.3137	−0.426705
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−10.0000	−0.375558	−0.187779	−	0.982211i	$-0.560129\pi$
$709$	−10.0000	−0.375558	−0.187779	+	0.982211i	$0.560129\pi$
$710$	0	0
$711$	10.0000	0.375029
$712$	0	0
$713$	16.9706	0.635553
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−22.6274	−0.845036
$718$	0	0
$719$	43.8406	1.63498	0.817490	−	0.575943i	$-0.195365\pi$
$719$	43.8406	1.63498	0.817490	+	0.575943i	$0.195365\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−38.0000	−1.41324
$724$	0	0
$725$	−10.0000	−0.371391
$726$	0	0
$727$	35.3553	1.31126	0.655628	−	0.755084i	$-0.272404\pi$
$727$	35.3553	1.31126	0.655628	+	0.755084i	$0.272404\pi$
$728$	0	0
$729$	29.0000	1.07407
$730$	0	0
$731$	−14.1421	−0.523066
$732$	0	0
$733$	−18.3848	−0.679057	−0.339529	−	0.940596i	$-0.610268\pi$
$733$	−18.3848	−0.679057	−0.339529	+	0.940596i	$0.610268\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	8.00000	0.294684
$738$	0	0
$739$	−20.0000	−0.735712	−0.367856	−	0.929883i	$-0.619908\pi$
$739$	−20.0000	−0.735712	−0.367856	+	0.929883i	$0.619908\pi$
$740$	0	0
$741$	−5.65685	−0.207810
$742$	0	0
$743$	16.0000	0.586983	0.293492	−	0.955962i	$-0.405183\pi$
$743$	16.0000	0.586983	0.293492	+	0.955962i	$0.405183\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−8.48528	−0.310460
$748$	0	0
$749$	0	0
$750$	0	0
$751$	40.0000	1.45962	0.729810	−	0.683650i	$-0.239608\pi$
$751$	40.0000	1.45962	0.729810	+	0.683650i	$0.239608\pi$
$752$	0	0
$753$	−22.0000	−0.801725
$754$	0	0
$755$	0	0
$756$	0	0
$757$	22.0000	0.799604	0.399802	−	0.916602i	$-0.369079\pi$
$757$	22.0000	0.799604	0.399802	+	0.916602i	$0.369079\pi$
$758$	0	0
$759$	5.65685	0.205331
$760$	0	0
$761$	12.7279	0.461387	0.230693	−	0.973026i	$-0.425901\pi$
$761$	12.7279	0.461387	0.230693	+	0.973026i	$0.425901\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−6.00000	−0.216647
$768$	0	0
$769$	−1.41421	−0.0509978	−0.0254989	−	0.999675i	$-0.508117\pi$
$769$	−1.41421	−0.0509978	−0.0254989	+	0.999675i	$0.508117\pi$
$770$	0	0
$771$	−8.00000	−0.288113
$772$	0	0
$773$	14.1421	0.508657	0.254329	−	0.967118i	$-0.418146\pi$
$773$	14.1421	0.508657	0.254329	+	0.967118i	$0.418146\pi$
$774$	0	0
$775$	21.2132	0.762001
$776$	0	0
$777$	0	0
$778$	0	0
$779$	4.00000	0.143315
$780$	0	0
$781$	0	0
$782$	0	0
$783$	−11.3137	−0.404319
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−2.82843	−0.100823	−0.0504113	−	0.998729i	$-0.516053\pi$
$787$	−2.82843	−0.100823	−0.0504113	+	0.998729i	$0.516053\pi$
$788$	0	0
$789$	11.3137	0.402779
$790$	0	0
$791$	0	0
$792$	0	0
$793$	18.0000	0.639199
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−39.5980	−1.40263	−0.701316	−	0.712850i	$-0.747404\pi$
$797$	−39.5980	−1.40263	−0.701316	+	0.712850i	$0.747404\pi$
$798$	0	0
$799$	−70.0000	−2.47642
$800$	0	0
$801$	−11.3137	−0.399750
$802$	0	0
$803$	−1.41421	−0.0499065
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−28.0000	−0.985647
$808$	0	0
$809$	−30.0000	−1.05474	−0.527372	−	0.849635i	$-0.676823\pi$
$809$	−30.0000	−1.05474	−0.527372	+	0.849635i	$0.676823\pi$
$810$	0	0
$811$	−45.2548	−1.58911	−0.794556	−	0.607191i	$-0.792296\pi$
$811$	−45.2548	−1.58911	−0.794556	+	0.607191i	$0.792296\pi$
$812$	0	0
$813$	36.0000	1.26258
$814$	0	0
$815$	0	0
$816$	0	0
$817$	5.65685	0.197908
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−54.0000	−1.88461	−0.942306	−	0.334751i	$-0.891348\pi$
$821$	−54.0000	−1.88461	−0.942306	+	0.334751i	$0.891348\pi$
$822$	0	0
$823$	−28.0000	−0.976019	−0.488009	−	0.872838i	$-0.662277\pi$
$823$	−28.0000	−0.976019	−0.488009	+	0.872838i	$0.662277\pi$
$824$	0	0
$825$	7.07107	0.246183
$826$	0	0
$827$	−6.00000	−0.208640	−0.104320	−	0.994544i	$-0.533267\pi$
$827$	−6.00000	−0.208640	−0.104320	+	0.994544i	$0.533267\pi$
$828$	0	0
$829$	−11.3137	−0.392941	−0.196471	−	0.980510i	$-0.562948\pi$
$829$	−11.3137	−0.392941	−0.196471	+	0.980510i	$0.562948\pi$
$830$	0	0
$831$	−2.82843	−0.0981170
$832$	0	0
$833$	0	0
$834$	0	0
$835$	0	0
$836$	0	0
$837$	24.0000	0.829561
$838$	0	0
$839$	−35.3553	−1.22060	−0.610301	−	0.792170i	$-0.708951\pi$
$839$	−35.3553	−1.22060	−0.610301	+	0.792170i	$0.708951\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	0	0
$843$	2.82843	0.0974162
$844$	0	0
$845$	0	0
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−40.0000	−1.37280
$850$	0	0
$851$	16.0000	0.548473
$852$	0	0
$853$	4.24264	0.145265	0.0726326	−	0.997359i	$-0.476860\pi$
$853$	4.24264	0.145265	0.0726326	+	0.997359i	$0.476860\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	29.6985	1.01448	0.507240	−	0.861805i	$-0.330666\pi$
$857$	29.6985	1.01448	0.507240	+	0.861805i	$0.330666\pi$
$858$	0	0
$859$	12.7279	0.434271	0.217136	−	0.976141i	$-0.430329\pi$
$859$	12.7279	0.434271	0.217136	+	0.976141i	$0.430329\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−16.0000	−0.544646	−0.272323	−	0.962206i	$-0.587792\pi$
$863$	−16.0000	−0.544646	−0.272323	+	0.962206i	$0.587792\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	46.6690	1.58496
$868$	0	0
$869$	10.0000	0.339227
$870$	0	0
$871$	11.3137	0.383350
$872$	0	0
$873$	−8.48528	−0.287183
$874$	0	0
$875$	0	0
$876$	0	0
$877$	2.00000	0.0675352	0.0337676	−	0.999430i	$-0.489249\pi$
$877$	2.00000	0.0675352	0.0337676	+	0.999430i	$0.489249\pi$
$878$	0	0
$879$	−2.00000	−0.0674583
$880$	0	0
$881$	−19.7990	−0.667045	−0.333522	−	0.942742i	$-0.608237\pi$
$881$	−19.7990	−0.667045	−0.333522	+	0.942742i	$0.608237\pi$
$882$	0	0
$883$	36.0000	1.21150	0.605748	−	0.795656i	$-0.292874\pi$
$883$	36.0000	1.21150	0.605748	+	0.795656i	$0.292874\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−36.7696	−1.23460	−0.617300	−	0.786728i	$-0.711774\pi$
$887$	−36.7696	−1.23460	−0.617300	+	0.786728i	$0.711774\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	5.00000	0.167506
$892$	0	0
$893$	28.0000	0.936984
$894$	0	0
$895$	0	0
$896$	0	0
$897$	8.00000	0.267112
$898$	0	0
$899$	−8.48528	−0.283000
$900$	0	0
$901$	28.2843	0.942286
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−32.0000	−1.06254	−0.531271	−	0.847202i	$-0.678286\pi$
$907$	−32.0000	−1.06254	−0.531271	+	0.847202i	$0.678286\pi$
$908$	0	0
$909$	1.41421	0.0469065
$910$	0	0
$911$	36.0000	1.19273	0.596367	−	0.802712i	$-0.296610\pi$
$911$	36.0000	1.19273	0.596367	+	0.802712i	$0.296610\pi$
$912$	0	0
$913$	−8.48528	−0.280822
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−40.0000	−1.31948	−0.659739	−	0.751495i	$-0.729333\pi$
$919$	−40.0000	−1.31948	−0.659739	+	0.751495i	$0.729333\pi$
$920$	0	0
$921$	−16.0000	−0.527218
$922$	0	0
$923$	0	0
$924$	0	0
$925$	20.0000	0.657596
$926$	0	0
$927$	−7.07107	−0.232244
$928$	0	0
$929$	2.82843	0.0927977	0.0463988	−	0.998923i	$-0.485225\pi$
$929$	2.82843	0.0927977	0.0463988	+	0.998923i	$0.485225\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−22.0000	−0.720248
$934$	0	0
$935$	0	0
$936$	0	0
$937$	55.1543	1.80181	0.900907	−	0.434013i	$-0.142903\pi$
$937$	55.1543	1.80181	0.900907	+	0.434013i	$0.142903\pi$
$938$	0	0
$939$	−32.0000	−1.04428
$940$	0	0
$941$	26.8701	0.875939	0.437969	−	0.898990i	$-0.355698\pi$
$941$	26.8701	0.875939	0.437969	+	0.898990i	$0.355698\pi$
$942$	0	0
$943$	−5.65685	−0.184213
$944$	0	0
$945$	0	0
$946$	0	0
$947$	20.0000	0.649913	0.324956	−	0.945729i	$-0.394650\pi$
$947$	20.0000	0.649913	0.324956	+	0.945729i	$0.394650\pi$
$948$	0	0
$949$	−2.00000	−0.0649227
$950$	0	0
$951$	31.1127	1.00890
$952$	0	0
$953$	−34.0000	−1.10137	−0.550684	−	0.834714i	$-0.685633\pi$
$953$	−34.0000	−1.10137	−0.550684	+	0.834714i	$0.685633\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−2.82843	−0.0914301
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−13.0000	−0.419355
$962$	0	0
$963$	12.0000	0.386695
$964$	0	0
$965$	0	0
$966$	0	0
$967$	58.0000	1.86515	0.932577	−	0.360971i	$-0.117555\pi$
$967$	58.0000	1.86515	0.932577	+	0.360971i	$0.117555\pi$
$968$	0	0
$969$	−28.2843	−0.908622
$970$	0	0
$971$	−18.3848	−0.589996	−0.294998	−	0.955498i	$-0.595319\pi$
$971$	−18.3848	−0.589996	−0.294998	+	0.955498i	$0.595319\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	10.0000	0.320256
$976$	0	0
$977$	−30.0000	−0.959785	−0.479893	−	0.877327i	$-0.659324\pi$
$977$	−30.0000	−0.959785	−0.479893	+	0.877327i	$0.659324\pi$
$978$	0	0
$979$	−11.3137	−0.361588
$980$	0	0
$981$	10.0000	0.319275
$982$	0	0
$983$	−24.0416	−0.766809	−0.383404	−	0.923581i	$-0.625248\pi$
$983$	−24.0416	−0.766809	−0.383404	+	0.923581i	$0.625248\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−8.00000	−0.254385
$990$	0	0
$991$	−40.0000	−1.27064	−0.635321	−	0.772248i	$-0.719132\pi$
$991$	−40.0000	−1.27064	−0.635321	+	0.772248i	$0.719132\pi$
$992$	0	0
$993$	11.3137	0.359030
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−4.24264	−0.134366	−0.0671829	−	0.997741i	$-0.521401\pi$
$997$	−4.24264	−0.134366	−0.0671829	+	0.997741i	$0.521401\pi$
$998$	0	0
$999$	22.6274	0.715900

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2156.2.a.e.1.2	yes	2
4.3	odd	2		8624.2.a.bu.1.1		2
7.2	even	3		2156.2.i.g.1145.1		4
7.3	odd	6		2156.2.i.g.177.2		4
7.4	even	3		2156.2.i.g.177.1		4
7.5	odd	6		2156.2.i.g.1145.2		4
7.6	odd	2	inner	2156.2.a.e.1.1	✓	2
28.27	even	2		8624.2.a.bu.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2156.2.a.e.1.1	✓	2	7.6	odd	2	inner
2156.2.a.e.1.2	yes	2	1.1	even	1	trivial
2156.2.i.g.177.1		4	7.4	even	3
2156.2.i.g.177.2		4	7.3	odd	6
2156.2.i.g.1145.1		4	7.2	even	3
2156.2.i.g.1145.2		4	7.5	odd	6
8624.2.a.bu.1.1		2	4.3	odd	2
8624.2.a.bu.1.2		2	28.27	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:	\( N \)	\(=\)	\( 2156 = 2^{2} \cdot 7^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2156.a (trivial)

\(f(q)\)	\(=\)	\(q+1.41421 q^{3} -1.00000 q^{9} -1.00000 q^{11} -1.41421 q^{13} -7.07107 q^{17} +2.82843 q^{19} -4.00000 q^{23} -5.00000 q^{25} -5.65685 q^{27} +2.00000 q^{29} -4.24264 q^{31} -1.41421 q^{33} -4.00000 q^{37} -2.00000 q^{39} +1.41421 q^{41} +2.00000 q^{43} +9.89949 q^{47} -10.0000 q^{51} -4.00000 q^{53} +4.00000 q^{57} +4.24264 q^{59} -12.7279 q^{61} -8.00000 q^{67} -5.65685 q^{69} +1.41421 q^{73} -7.07107 q^{75} -10.0000 q^{79} -5.00000 q^{81} +8.48528 q^{83} +2.82843 q^{87} +11.3137 q^{89} -6.00000 q^{93} +8.48528 q^{97} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{9} - 2 q^{11} - 8 q^{23} - 10 q^{25} + 4 q^{29} - 8 q^{37} - 4 q^{39} + 4 q^{43} - 20 q^{51} - 8 q^{53} + 8 q^{57} - 16 q^{67} - 20 q^{79} - 10 q^{81} - 12 q^{93} + 2 q^{99}+O(q^{100}) \) 2 * q - 2 * q^9 - 2 * q^11 - 8 * q^23 - 10 * q^25 + 4 * q^29 - 8 * q^37 - 4 * q^39 + 4 * q^43 - 20 * q^51 - 8 * q^53 + 8 * q^57 - 16 * q^67 - 20 * q^79 - 10 * q^81 - 12 * q^93 + 2 * q^99

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