LMFDB - Embedded newform 8028.2.a.l.1.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8028,2,Mod(1,8028)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8028, 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("8028.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8028 = 2^{2} \cdot 3^{2} \cdot 223 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8028.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:	$64.1039027427$
Analytic rank:	$1$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{7} - 13x^{6} + 10x^{5} + 48x^{4} - 23x^{3} - 44x^{2} + 20x - 2 $ x^8 - x^7 - 13x^6 + 10x^5 + 48x^4 - 23x^3 - 44x^2 + 20x - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 2676)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$0.270821$ of defining polynomial
Character	$\chi$	$=$	8028.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.48691 q^{5} +0.673508 q^{7} +O(q^{10})$	$q+3.48691 q^{5} +0.673508 q^{7} -3.16525 q^{11} -3.20261 q^{13} +1.26740 q^{17} -3.85497 q^{19} -6.82609 q^{23} +7.15852 q^{25} -0.126794 q^{29} -7.36909 q^{31} +2.34846 q^{35} +7.86985 q^{37} +3.51837 q^{41} -2.65783 q^{43} +12.9825 q^{47} -6.54639 q^{49} +4.87891 q^{53} -11.0369 q^{55} -7.94095 q^{59} -9.84128 q^{61} -11.1672 q^{65} -4.02109 q^{67} +1.45708 q^{71} +7.06561 q^{73} -2.13182 q^{77} -12.6020 q^{79} +11.7168 q^{83} +4.41930 q^{85} -11.6717 q^{89} -2.15698 q^{91} -13.4419 q^{95} -7.04958 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 5 q^{5} - 13 q^{7}+O(q^{10}) $ 8 * q + 5 * q^5 - 13 * q^7	$ 8 q + 5 q^{5} - 13 q^{7} + q^{11} - 10 q^{13} + 10 q^{17} - 5 q^{19} - q^{23} + 5 q^{25} + 14 q^{29} - 21 q^{31} - 21 q^{37} + 31 q^{41} - 18 q^{43} + 10 q^{47} + q^{49} + 12 q^{53} - 31 q^{55} - 2 q^{59} - 32 q^{61} + 26 q^{65} - 25 q^{67} + 11 q^{71} - 14 q^{73} + 12 q^{77} - 9 q^{79} - 25 q^{83} - 45 q^{85} + 23 q^{89} - 4 q^{91} + 6 q^{95} - 11 q^{97}+O(q^{100}) $ 8 * q + 5 * q^5 - 13 * q^7 + q^11 - 10 * q^13 + 10 * q^17 - 5 * q^19 - q^23 + 5 * q^25 + 14 * q^29 - 21 * q^31 - 21 * q^37 + 31 * q^41 - 18 * q^43 + 10 * q^47 + q^49 + 12 * q^53 - 31 * q^55 - 2 * q^59 - 32 * q^61 + 26 * q^65 - 25 * q^67 + 11 * q^71 - 14 * q^73 + 12 * q^77 - 9 * q^79 - 25 * q^83 - 45 * q^85 + 23 * q^89 - 4 * q^91 + 6 * q^95 - 11 * q^97

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	3.48691	1.55939	0.779696	−	0.626158i	$-0.215374\pi$
$5$	3.48691	1.55939	0.779696	+	0.626158i	$0.215374\pi$
$6$	0	0
$7$	0.673508	0.254562	0.127281	−	0.991867i	$-0.459375\pi$
$7$	0.673508	0.254562	0.127281	+	0.991867i	$0.459375\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−3.16525	−0.954359	−0.477180	−	0.878806i	$-0.658341\pi$
$11$	−3.16525	−0.954359	−0.477180	+	0.878806i	$0.658341\pi$
$12$	0	0
$13$	−3.20261	−0.888245	−0.444122	−	0.895966i	$-0.646484\pi$
$13$	−3.20261	−0.888245	−0.444122	+	0.895966i	$0.646484\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	1.26740	0.307389	0.153695	−	0.988118i	$-0.450883\pi$
$17$	1.26740	0.307389	0.153695	+	0.988118i	$0.450883\pi$
$18$	0	0
$19$	−3.85497	−0.884391	−0.442196	−	0.896919i	$-0.645800\pi$
$19$	−3.85497	−0.884391	−0.442196	+	0.896919i	$0.645800\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−6.82609	−1.42334	−0.711670	−	0.702514i	$-0.752061\pi$
$23$	−6.82609	−1.42334	−0.711670	+	0.702514i	$0.752061\pi$
$24$	0	0
$25$	7.15852	1.43170
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−0.126794	−0.0235451	−0.0117725	−	0.999931i	$-0.503747\pi$
$29$	−0.126794	−0.0235451	−0.0117725	+	0.999931i	$0.503747\pi$
$30$	0	0
$31$	−7.36909	−1.32353	−0.661764	−	0.749712i	$-0.730192\pi$
$31$	−7.36909	−1.32353	−0.661764	+	0.749712i	$0.730192\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	2.34846	0.396962
$36$	0	0
$37$	7.86985	1.29380	0.646898	−	0.762577i	$-0.276066\pi$
$37$	7.86985	1.29380	0.646898	+	0.762577i	$0.276066\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	3.51837	0.549478	0.274739	−	0.961519i	$-0.411409\pi$
$41$	3.51837	0.549478	0.274739	+	0.961519i	$0.411409\pi$
$42$	0	0
$43$	−2.65783	−0.405316	−0.202658	−	0.979250i	$-0.564958\pi$
$43$	−2.65783	−0.405316	−0.202658	+	0.979250i	$0.564958\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	12.9825	1.89369	0.946843	−	0.321696i	$-0.104253\pi$
$47$	12.9825	1.89369	0.946843	+	0.321696i	$0.104253\pi$
$48$	0	0
$49$	−6.54639	−0.935198
$50$	0	0
$51$	0	0
$52$	0	0
$53$	4.87891	0.670169	0.335085	−	0.942188i	$-0.391235\pi$
$53$	4.87891	0.670169	0.335085	+	0.942188i	$0.391235\pi$
$54$	0	0
$55$	−11.0369	−1.48822
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−7.94095	−1.03382	−0.516912	−	0.856039i	$-0.672918\pi$
$59$	−7.94095	−1.03382	−0.516912	+	0.856039i	$0.672918\pi$
$60$	0	0
$61$	−9.84128	−1.26005	−0.630023	−	0.776576i	$-0.716955\pi$
$61$	−9.84128	−1.26005	−0.630023	+	0.776576i	$0.716955\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−11.1672	−1.38512
$66$	0	0
$67$	−4.02109	−0.491255	−0.245627	−	0.969364i	$-0.578994\pi$
$67$	−4.02109	−0.491255	−0.245627	+	0.969364i	$0.578994\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	1.45708	0.172923	0.0864616	−	0.996255i	$-0.472444\pi$
$71$	1.45708	0.172923	0.0864616	+	0.996255i	$0.472444\pi$
$72$	0	0
$73$	7.06561	0.826967	0.413483	−	0.910512i	$-0.364312\pi$
$73$	7.06561	0.826967	0.413483	+	0.910512i	$0.364312\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−2.13182	−0.242944
$78$	0	0
$79$	−12.6020	−1.41784	−0.708921	−	0.705288i	$-0.750818\pi$
$79$	−12.6020	−1.41784	−0.708921	+	0.705288i	$0.750818\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	11.7168	1.28609	0.643045	−	0.765829i	$-0.277671\pi$
$83$	11.7168	1.28609	0.643045	+	0.765829i	$0.277671\pi$
$84$	0	0
$85$	4.41930	0.479340
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−11.6717	−1.23720	−0.618598	−	0.785708i	$-0.712299\pi$
$89$	−11.6717	−1.23720	−0.618598	+	0.785708i	$0.712299\pi$
$90$	0	0
$91$	−2.15698	−0.226113
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−13.4419	−1.37911
$96$	0	0
$97$	−7.04958	−0.715776	−0.357888	−	0.933764i	$-0.616503\pi$
$97$	−7.04958	−0.715776	−0.357888	+	0.933764i	$0.616503\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−7.83166	−0.779280	−0.389640	−	0.920967i	$-0.627400\pi$
$101$	−7.83166	−0.779280	−0.389640	+	0.920967i	$0.627400\pi$
$102$	0	0
$103$	−16.1302	−1.58936	−0.794679	−	0.607030i	$-0.792361\pi$
$103$	−16.1302	−1.58936	−0.794679	+	0.607030i	$0.792361\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−7.70196	−0.744577	−0.372288	−	0.928117i	$-0.621427\pi$
$107$	−7.70196	−0.744577	−0.372288	+	0.928117i	$0.621427\pi$
$108$	0	0
$109$	0.274466	0.0262891	0.0131445	−	0.999914i	$-0.495816\pi$
$109$	0.274466	0.0262891	0.0131445	+	0.999914i	$0.495816\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	18.4991	1.74025	0.870127	−	0.492828i	$-0.164037\pi$
$113$	18.4991	1.74025	0.870127	+	0.492828i	$0.164037\pi$
$114$	0	0
$115$	−23.8020	−2.21954
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0.853602	0.0782496
$120$	0	0
$121$	−0.981182	−0.0891983
$122$	0	0
$123$	0	0
$124$	0	0
$125$	7.52656	0.673196
$126$	0	0
$127$	−13.1510	−1.16696	−0.583481	−	0.812127i	$-0.698310\pi$
$127$	−13.1510	−1.16696	−0.583481	+	0.812127i	$0.698310\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−10.4817	−0.915795	−0.457897	−	0.889005i	$-0.651397\pi$
$131$	−10.4817	−0.915795	−0.457897	+	0.889005i	$0.651397\pi$
$132$	0	0
$133$	−2.59635	−0.225132
$134$	0	0
$135$	0	0
$136$	0	0
$137$	1.31702	0.112520	0.0562601	−	0.998416i	$-0.482082\pi$
$137$	1.31702	0.112520	0.0562601	+	0.998416i	$0.482082\pi$
$138$	0	0
$139$	−11.0451	−0.936836	−0.468418	−	0.883507i	$-0.655176\pi$
$139$	−11.0451	−0.936836	−0.468418	+	0.883507i	$0.655176\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	10.1371	0.847705
$144$	0	0
$145$	−0.442119	−0.0367160
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−9.00785	−0.737952	−0.368976	−	0.929439i	$-0.620291\pi$
$149$	−9.00785	−0.737952	−0.368976	+	0.929439i	$0.620291\pi$
$150$	0	0
$151$	−15.6021	−1.26968	−0.634841	−	0.772643i	$-0.718934\pi$
$151$	−15.6021	−1.26968	−0.634841	+	0.772643i	$0.718934\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−25.6953	−2.06390
$156$	0	0
$157$	−0.399326	−0.0318697	−0.0159348	−	0.999873i	$-0.505072\pi$
$157$	−0.399326	−0.0318697	−0.0159348	+	0.999873i	$0.505072\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−4.59743	−0.362328
$162$	0	0
$163$	24.0039	1.88013	0.940064	−	0.340999i	$-0.110765\pi$
$163$	24.0039	1.88013	0.940064	+	0.340999i	$0.110765\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	3.94556	0.305317	0.152658	−	0.988279i	$-0.451217\pi$
$167$	3.94556	0.305317	0.152658	+	0.988279i	$0.451217\pi$
$168$	0	0
$169$	−2.74327	−0.211021
$170$	0	0
$171$	0	0
$172$	0	0
$173$	21.4199	1.62852	0.814262	−	0.580497i	$-0.197142\pi$
$173$	21.4199	1.62852	0.814262	+	0.580497i	$0.197142\pi$
$174$	0	0
$175$	4.82132	0.364457
$176$	0	0
$177$	0	0
$178$	0	0
$179$	5.46276	0.408306	0.204153	−	0.978939i	$-0.434556\pi$
$179$	5.46276	0.408306	0.204153	+	0.978939i	$0.434556\pi$
$180$	0	0
$181$	−5.33411	−0.396481	−0.198241	−	0.980153i	$-0.563523\pi$
$181$	−5.33411	−0.396481	−0.198241	+	0.980153i	$0.563523\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	27.4414	2.01753
$186$	0	0
$187$	−4.01163	−0.293360
$188$	0	0
$189$	0	0
$190$	0	0
$191$	5.90512	0.427280	0.213640	−	0.976912i	$-0.431468\pi$
$191$	5.90512	0.427280	0.213640	+	0.976912i	$0.431468\pi$
$192$	0	0
$193$	−19.6638	−1.41543	−0.707717	−	0.706496i	$-0.750275\pi$
$193$	−19.6638	−1.41543	−0.707717	+	0.706496i	$0.750275\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−20.6283	−1.46970	−0.734852	−	0.678228i	$-0.762748\pi$
$197$	−20.6283	−1.46970	−0.734852	+	0.678228i	$0.762748\pi$
$198$	0	0
$199$	18.7303	1.32776	0.663878	−	0.747841i	$-0.268910\pi$
$199$	18.7303	1.32776	0.663878	+	0.747841i	$0.268910\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−0.0853968	−0.00599368
$204$	0	0
$205$	12.2682	0.856851
$206$	0	0
$207$	0	0
$208$	0	0
$209$	12.2020	0.844027
$210$	0	0
$211$	−0.808891	−0.0556864	−0.0278432	−	0.999612i	$-0.508864\pi$
$211$	−0.808891	−0.0556864	−0.0278432	+	0.999612i	$0.508864\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−9.26762	−0.632046
$216$	0	0
$217$	−4.96314	−0.336920
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−4.05899	−0.273037
$222$	0	0
$223$	−1.00000	−0.0669650
$223$	−1.00000	−0.0669650
$224$	0	0
$225$	0	0
$226$	0	0
$227$	11.2931	0.749546	0.374773	−	0.927117i	$-0.377721\pi$
$227$	11.2931	0.749546	0.374773	+	0.927117i	$0.377721\pi$
$228$	0	0
$229$	−18.3361	−1.21169	−0.605843	−	0.795584i	$-0.707164\pi$
$229$	−18.3361	−1.21169	−0.605843	+	0.795584i	$0.707164\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−26.7413	−1.75188	−0.875942	−	0.482417i	$-0.839759\pi$
$233$	−26.7413	−1.75188	−0.875942	+	0.482417i	$0.839759\pi$
$234$	0	0
$235$	45.2686	2.95300
$236$	0	0
$237$	0	0
$238$	0	0
$239$	0.803219	0.0519559	0.0259780	−	0.999663i	$-0.491730\pi$
$239$	0.803219	0.0519559	0.0259780	+	0.999663i	$0.491730\pi$
$240$	0	0
$241$	11.6621	0.751220	0.375610	−	0.926778i	$-0.377433\pi$
$241$	11.6621	0.751220	0.375610	+	0.926778i	$0.377433\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−22.8266	−1.45834
$246$	0	0
$247$	12.3460	0.785556
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−24.9833	−1.57693	−0.788466	−	0.615078i	$-0.789124\pi$
$251$	−24.9833	−1.57693	−0.788466	+	0.615078i	$0.789124\pi$
$252$	0	0
$253$	21.6063	1.35838
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−15.4011	−0.960693	−0.480346	−	0.877079i	$-0.659489\pi$
$257$	−15.4011	−0.960693	−0.480346	+	0.877079i	$0.659489\pi$
$258$	0	0
$259$	5.30040	0.329351
$260$	0	0
$261$	0	0
$262$	0	0
$263$	9.98930	0.615966	0.307983	−	0.951392i	$-0.400346\pi$
$263$	9.98930	0.615966	0.307983	+	0.951392i	$0.400346\pi$
$264$	0	0
$265$	17.0123	1.04506
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−5.41452	−0.330129	−0.165065	−	0.986283i	$-0.552783\pi$
$269$	−5.41452	−0.330129	−0.165065	+	0.986283i	$0.552783\pi$
$270$	0	0
$271$	−3.52162	−0.213923	−0.106962	−	0.994263i	$-0.534112\pi$
$271$	−3.52162	−0.213923	−0.106962	+	0.994263i	$0.534112\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−22.6585	−1.36636
$276$	0	0
$277$	8.01814	0.481763	0.240882	−	0.970555i	$-0.422563\pi$
$277$	8.01814	0.481763	0.240882	+	0.970555i	$0.422563\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	6.92486	0.413103	0.206551	−	0.978436i	$-0.433776\pi$
$281$	6.92486	0.413103	0.206551	+	0.978436i	$0.433776\pi$
$282$	0	0
$283$	19.4749	1.15766	0.578832	−	0.815447i	$-0.303509\pi$
$283$	19.4749	1.15766	0.578832	+	0.815447i	$0.303509\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	2.36965	0.139876
$288$	0	0
$289$	−15.3937	−0.905512
$290$	0	0
$291$	0	0
$292$	0	0
$293$	20.3050	1.18623	0.593116	−	0.805117i	$-0.297898\pi$
$293$	20.3050	1.18623	0.593116	+	0.805117i	$0.297898\pi$
$294$	0	0
$295$	−27.6893	−1.61214
$296$	0	0
$297$	0	0
$298$	0	0
$299$	21.8613	1.26427
$300$	0	0
$301$	−1.79007	−0.103178
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−34.3156	−1.96491
$306$	0	0
$307$	−31.3013	−1.78646	−0.893229	−	0.449601i	$-0.851566\pi$
$307$	−31.3013	−1.78646	−0.893229	+	0.449601i	$0.851566\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−12.4775	−0.707534	−0.353767	−	0.935334i	$-0.615099\pi$
$311$	−12.4775	−0.707534	−0.353767	+	0.935334i	$0.615099\pi$
$312$	0	0
$313$	2.43778	0.137791	0.0688956	−	0.997624i	$-0.478052\pi$
$313$	2.43778	0.137791	0.0688956	+	0.997624i	$0.478052\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−20.0966	−1.12874	−0.564368	−	0.825524i	$-0.690880\pi$
$317$	−20.0966	−1.12874	−0.564368	+	0.825524i	$0.690880\pi$
$318$	0	0
$319$	0.401335	0.0224705
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−4.88579	−0.271852
$324$	0	0
$325$	−22.9260	−1.27170
$326$	0	0
$327$	0	0
$328$	0	0
$329$	8.74378	0.482060
$330$	0	0
$331$	18.6846	1.02700	0.513498	−	0.858091i	$-0.328349\pi$
$331$	18.6846	1.02700	0.513498	+	0.858091i	$0.328349\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−14.0212	−0.766059
$336$	0	0
$337$	29.6887	1.61725	0.808624	−	0.588326i	$-0.200213\pi$
$337$	29.6887	1.61725	0.808624	+	0.588326i	$0.200213\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	23.3250	1.26312
$342$	0	0
$343$	−9.12360	−0.492628
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−32.2512	−1.73134	−0.865669	−	0.500617i	$-0.833106\pi$
$347$	−32.2512	−1.73134	−0.865669	+	0.500617i	$0.833106\pi$
$348$	0	0
$349$	−9.14754	−0.489656	−0.244828	−	0.969566i	$-0.578732\pi$
$349$	−9.14754	−0.489656	−0.244828	+	0.969566i	$0.578732\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	27.2823	1.45209	0.726046	−	0.687646i	$-0.241356\pi$
$353$	27.2823	1.45209	0.726046	+	0.687646i	$0.241356\pi$
$354$	0	0
$355$	5.08069	0.269655
$356$	0	0
$357$	0	0
$358$	0	0
$359$	16.9281	0.893432	0.446716	−	0.894676i	$-0.352593\pi$
$359$	16.9281	0.893432	0.446716	+	0.894676i	$0.352593\pi$
$360$	0	0
$361$	−4.13918	−0.217852
$362$	0	0
$363$	0	0
$364$	0	0
$365$	24.6371	1.28957
$366$	0	0
$367$	21.1056	1.10170	0.550851	−	0.834604i	$-0.314303\pi$
$367$	21.1056	1.10170	0.550851	+	0.834604i	$0.314303\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	3.28598	0.170600
$372$	0	0
$373$	−1.13230	−0.0586282	−0.0293141	−	0.999570i	$-0.509332\pi$
$373$	−1.13230	−0.0586282	−0.0293141	+	0.999570i	$0.509332\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	0.406072	0.0209138
$378$	0	0
$379$	21.9241	1.12616	0.563082	−	0.826401i	$-0.309616\pi$
$379$	21.9241	1.12616	0.563082	+	0.826401i	$0.309616\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	37.1005	1.89575	0.947873	−	0.318648i	$-0.103229\pi$
$383$	37.1005	1.89575	0.947873	+	0.318648i	$0.103229\pi$
$384$	0	0
$385$	−7.43346	−0.378844
$386$	0	0
$387$	0	0
$388$	0	0
$389$	24.7355	1.25414	0.627071	−	0.778962i	$-0.284254\pi$
$389$	24.7355	1.25414	0.627071	+	0.778962i	$0.284254\pi$
$390$	0	0
$391$	−8.65138	−0.437519
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−43.9422	−2.21097
$396$	0	0
$397$	−8.78391	−0.440852	−0.220426	−	0.975404i	$-0.570745\pi$
$397$	−8.78391	−0.440852	−0.220426	+	0.975404i	$0.570745\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	11.8162	0.590073	0.295036	−	0.955486i	$-0.404668\pi$
$401$	11.8162	0.590073	0.295036	+	0.955486i	$0.404668\pi$
$402$	0	0
$403$	23.6003	1.17562
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−24.9101	−1.23475
$408$	0	0
$409$	33.8137	1.67198	0.835991	−	0.548744i	$-0.184894\pi$
$409$	33.8137	1.67198	0.835991	+	0.548744i	$0.184894\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−5.34829	−0.263172
$414$	0	0
$415$	40.8555	2.00552
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−10.9954	−0.537159	−0.268579	−	0.963258i	$-0.586554\pi$
$419$	−10.9954	−0.537159	−0.268579	+	0.963258i	$0.586554\pi$
$420$	0	0
$421$	12.0449	0.587033	0.293516	−	0.955954i	$-0.405174\pi$
$421$	12.0449	0.587033	0.293516	+	0.955954i	$0.405174\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	9.07270	0.440090
$426$	0	0
$427$	−6.62818	−0.320760
$428$	0	0
$429$	0	0
$430$	0	0
$431$	4.66999	0.224946	0.112473	−	0.993655i	$-0.464123\pi$
$431$	4.66999	0.224946	0.112473	+	0.993655i	$0.464123\pi$
$432$	0	0
$433$	16.2458	0.780724	0.390362	−	0.920662i	$-0.372350\pi$
$433$	16.2458	0.780724	0.390362	+	0.920662i	$0.372350\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	26.3144	1.25879
$438$	0	0
$439$	−5.96330	−0.284613	−0.142306	−	0.989823i	$-0.545452\pi$
$439$	−5.96330	−0.284613	−0.142306	+	0.989823i	$0.545452\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−16.5367	−0.785681	−0.392841	−	0.919607i	$-0.628508\pi$
$443$	−16.5367	−0.785681	−0.392841	+	0.919607i	$0.628508\pi$
$444$	0	0
$445$	−40.6981	−1.92927
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−2.11620	−0.0998694	−0.0499347	−	0.998752i	$-0.515901\pi$
$449$	−2.11620	−0.0998694	−0.0499347	+	0.998752i	$0.515901\pi$
$450$	0	0
$451$	−11.1365	−0.524399
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−7.52120	−0.352599
$456$	0	0
$457$	31.3873	1.46824	0.734118	−	0.679022i	$-0.237596\pi$
$457$	31.3873	1.46824	0.734118	+	0.679022i	$0.237596\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−17.1267	−0.797668	−0.398834	−	0.917023i	$-0.630585\pi$
$461$	−17.1267	−0.797668	−0.398834	+	0.917023i	$0.630585\pi$
$462$	0	0
$463$	34.5679	1.60651	0.803253	−	0.595638i	$-0.203101\pi$
$463$	34.5679	1.60651	0.803253	+	0.595638i	$0.203101\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−36.0820	−1.66967	−0.834837	−	0.550497i	$-0.814438\pi$
$467$	−36.0820	−1.66967	−0.834837	+	0.550497i	$0.814438\pi$
$468$	0	0
$469$	−2.70824	−0.125055
$470$	0	0
$471$	0	0
$472$	0	0
$473$	8.41271	0.386817
$474$	0	0
$475$	−27.5959	−1.26619
$476$	0	0
$477$	0	0
$478$	0	0
$479$	14.2307	0.650217	0.325109	−	0.945677i	$-0.394599\pi$
$479$	14.2307	0.650217	0.325109	+	0.945677i	$0.394599\pi$
$480$	0	0
$481$	−25.2041	−1.14921
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−24.5812	−1.11618
$486$	0	0
$487$	−30.0161	−1.36016	−0.680079	−	0.733139i	$-0.738054\pi$
$487$	−30.0161	−1.36016	−0.680079	+	0.733139i	$0.738054\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−3.87677	−0.174956	−0.0874780	−	0.996166i	$-0.527881\pi$
$491$	−3.87677	−0.174956	−0.0874780	+	0.996166i	$0.527881\pi$
$492$	0	0
$493$	−0.160699	−0.00723750
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0.981352	0.0440197
$498$	0	0
$499$	−18.9056	−0.846331	−0.423166	−	0.906052i	$-0.639081\pi$
$499$	−18.9056	−0.846331	−0.423166	+	0.906052i	$0.639081\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−17.9727	−0.801363	−0.400682	−	0.916217i	$-0.631227\pi$
$503$	−17.9727	−0.801363	−0.400682	+	0.916217i	$0.631227\pi$
$504$	0	0
$505$	−27.3083	−1.21520
$506$	0	0
$507$	0	0
$508$	0	0
$509$	28.0966	1.24536	0.622680	−	0.782477i	$-0.286044\pi$
$509$	28.0966	1.24536	0.622680	+	0.782477i	$0.286044\pi$
$510$	0	0
$511$	4.75874	0.210514
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−56.2446	−2.47843
$516$	0	0
$517$	−41.0927	−1.80726
$518$	0	0
$519$	0	0
$520$	0	0
$521$	10.5394	0.461741	0.230870	−	0.972985i	$-0.425843\pi$
$521$	10.5394	0.461741	0.230870	+	0.972985i	$0.425843\pi$
$522$	0	0
$523$	18.0571	0.789584	0.394792	−	0.918771i	$-0.370817\pi$
$523$	18.0571	0.789584	0.394792	+	0.918771i	$0.370817\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−9.33957	−0.406838
$528$	0	0
$529$	23.5956	1.02589
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−11.2680	−0.488071
$534$	0	0
$535$	−26.8560	−1.16109
$536$	0	0
$537$	0	0
$538$	0	0
$539$	20.7210	0.892515
$540$	0	0
$541$	−14.7059	−0.632258	−0.316129	−	0.948716i	$-0.602383\pi$
$541$	−14.7059	−0.632258	−0.316129	+	0.948716i	$0.602383\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	0.957038	0.0409950
$546$	0	0
$547$	−18.1233	−0.774895	−0.387448	−	0.921892i	$-0.626643\pi$
$547$	−18.1233	−0.774895	−0.387448	+	0.921892i	$0.626643\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	0.488788	0.0208231
$552$	0	0
$553$	−8.48758	−0.360928
$554$	0	0
$555$	0	0
$556$	0	0
$557$	1.59030	0.0673832	0.0336916	−	0.999432i	$-0.489274\pi$
$557$	1.59030	0.0673832	0.0336916	+	0.999432i	$0.489274\pi$
$558$	0	0
$559$	8.51201	0.360020
$560$	0	0
$561$	0	0
$562$	0	0
$563$	15.1498	0.638489	0.319244	−	0.947672i	$-0.396571\pi$
$563$	15.1498	0.638489	0.319244	+	0.947672i	$0.396571\pi$
$564$	0	0
$565$	64.5048	2.71374
$566$	0	0
$567$	0	0
$568$	0	0
$569$	3.49601	0.146560	0.0732802	−	0.997311i	$-0.476653\pi$
$569$	3.49601	0.146560	0.0732802	+	0.997311i	$0.476653\pi$
$570$	0	0
$571$	−6.25042	−0.261572	−0.130786	−	0.991411i	$-0.541750\pi$
$571$	−6.25042	−0.261572	−0.130786	+	0.991411i	$0.541750\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−48.8647	−2.03780
$576$	0	0
$577$	−14.0355	−0.584304	−0.292152	−	0.956372i	$-0.594371\pi$
$577$	−14.0355	−0.584304	−0.292152	+	0.956372i	$0.594371\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	7.89138	0.327389
$582$	0	0
$583$	−15.4430	−0.639582
$584$	0	0
$585$	0	0
$586$	0	0
$587$	24.7922	1.02328	0.511642	−	0.859199i	$-0.329037\pi$
$587$	24.7922	1.02328	0.511642	+	0.859199i	$0.329037\pi$
$588$	0	0
$589$	28.4076	1.17052
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−16.9857	−0.697520	−0.348760	−	0.937212i	$-0.613397\pi$
$593$	−16.9857	−0.697520	−0.348760	+	0.937212i	$0.613397\pi$
$594$	0	0
$595$	2.97643	0.122022
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−26.3290	−1.07577	−0.537887	−	0.843017i	$-0.680777\pi$
$599$	−26.3290	−1.07577	−0.537887	+	0.843017i	$0.680777\pi$
$600$	0	0
$601$	−37.1753	−1.51641	−0.758206	−	0.652016i	$-0.773924\pi$
$601$	−37.1753	−1.51641	−0.758206	+	0.652016i	$0.773924\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−3.42129	−0.139095
$606$	0	0
$607$	35.5467	1.44280	0.721398	−	0.692521i	$-0.243500\pi$
$607$	35.5467	1.44280	0.721398	+	0.692521i	$0.243500\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−41.5778	−1.68206
$612$	0	0
$613$	−18.0429	−0.728744	−0.364372	−	0.931254i	$-0.618716\pi$
$613$	−18.0429	−0.728744	−0.364372	+	0.931254i	$0.618716\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	20.1412	0.810853	0.405426	−	0.914128i	$-0.367123\pi$
$617$	20.1412	0.810853	0.405426	+	0.914128i	$0.367123\pi$
$618$	0	0
$619$	37.6199	1.51207	0.756037	−	0.654529i	$-0.227133\pi$
$619$	37.6199	1.51207	0.756037	+	0.654529i	$0.227133\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−7.86097	−0.314943
$624$	0	0
$625$	−9.54820	−0.381928
$626$	0	0
$627$	0	0
$628$	0	0
$629$	9.97423	0.397699
$630$	0	0
$631$	3.90613	0.155500	0.0777502	−	0.996973i	$-0.475226\pi$
$631$	3.90613	0.155500	0.0777502	+	0.996973i	$0.475226\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−45.8563	−1.81975
$636$	0	0
$637$	20.9655	0.830685
$638$	0	0
$639$	0	0
$640$	0	0
$641$	5.92278	0.233936	0.116968	−	0.993136i	$-0.462683\pi$
$641$	5.92278	0.233936	0.116968	+	0.993136i	$0.462683\pi$
$642$	0	0
$643$	0.0958707	0.00378077	0.00189039	−	0.999998i	$-0.499398\pi$
$643$	0.0958707	0.00378077	0.00189039	+	0.999998i	$0.499398\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−28.5343	−1.12180	−0.560899	−	0.827884i	$-0.689544\pi$
$647$	−28.5343	−1.12180	−0.560899	+	0.827884i	$0.689544\pi$
$648$	0	0
$649$	25.1351	0.986639
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−34.4756	−1.34913	−0.674567	−	0.738214i	$-0.735670\pi$
$653$	−34.4756	−1.34913	−0.674567	+	0.738214i	$0.735670\pi$
$654$	0	0
$655$	−36.5489	−1.42808
$656$	0	0
$657$	0	0
$658$	0	0
$659$	20.8366	0.811678	0.405839	−	0.913945i	$-0.366979\pi$
$659$	20.8366	0.811678	0.405839	+	0.913945i	$0.366979\pi$
$660$	0	0
$661$	1.84083	0.0715999	0.0358000	−	0.999359i	$-0.488602\pi$
$661$	1.84083	0.0715999	0.0358000	+	0.999359i	$0.488602\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−9.05324	−0.351070
$666$	0	0
$667$	0.865508	0.0335126
$668$	0	0
$669$	0	0
$670$	0	0
$671$	31.1501	1.20254
$672$	0	0
$673$	−25.5117	−0.983405	−0.491702	−	0.870763i	$-0.663625\pi$
$673$	−25.5117	−0.983405	−0.491702	+	0.870763i	$0.663625\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−25.2937	−0.972115	−0.486058	−	0.873927i	$-0.661566\pi$
$677$	−25.2937	−0.972115	−0.486058	+	0.873927i	$0.661566\pi$
$678$	0	0
$679$	−4.74794	−0.182209
$680$	0	0
$681$	0	0
$682$	0	0
$683$	15.3313	0.586637	0.293319	−	0.956015i	$-0.405240\pi$
$683$	15.3313	0.586637	0.293319	+	0.956015i	$0.405240\pi$
$684$	0	0
$685$	4.59231	0.175463
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−15.6253	−0.595275
$690$	0	0
$691$	34.0751	1.29628	0.648139	−	0.761522i	$-0.275547\pi$
$691$	34.0751	1.29628	0.648139	+	0.761522i	$0.275547\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−38.5133	−1.46089
$696$	0	0
$697$	4.45918	0.168904
$698$	0	0
$699$	0	0
$700$	0	0
$701$	12.2257	0.461759	0.230879	−	0.972982i	$-0.425840\pi$
$701$	12.2257	0.461759	0.230879	+	0.972982i	$0.425840\pi$
$702$	0	0
$703$	−30.3381	−1.14422
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−5.27469	−0.198375
$708$	0	0
$709$	10.9021	0.409438	0.204719	−	0.978821i	$-0.434372\pi$
$709$	10.9021	0.409438	0.204719	+	0.978821i	$0.434372\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	50.3021	1.88383
$714$	0	0
$715$	35.3470	1.32190
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−39.2123	−1.46237	−0.731185	−	0.682179i	$-0.761033\pi$
$719$	−39.2123	−1.46237	−0.731185	+	0.682179i	$0.761033\pi$
$720$	0	0
$721$	−10.8638	−0.404590
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−0.907658	−0.0337096
$726$	0	0
$727$	25.4121	0.942481	0.471241	−	0.882005i	$-0.343806\pi$
$727$	25.4121	0.942481	0.471241	+	0.882005i	$0.343806\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−3.36853	−0.124590
$732$	0	0
$733$	−6.82055	−0.251923	−0.125961	−	0.992035i	$-0.540202\pi$
$733$	−6.82055	−0.251923	−0.125961	+	0.992035i	$0.540202\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	12.7278	0.468833
$738$	0	0
$739$	−46.2469	−1.70122	−0.850609	−	0.525798i	$-0.823767\pi$
$739$	−46.2469	−1.70122	−0.850609	+	0.525798i	$0.823767\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−23.8491	−0.874938	−0.437469	−	0.899234i	$-0.644125\pi$
$743$	−23.8491	−0.874938	−0.437469	+	0.899234i	$0.644125\pi$
$744$	0	0
$745$	−31.4095	−1.15076
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−5.18733	−0.189541
$750$	0	0
$751$	9.12574	0.333003	0.166501	−	0.986041i	$-0.446753\pi$
$751$	9.12574	0.333003	0.166501	+	0.986041i	$0.446753\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−54.4031	−1.97993
$756$	0	0
$757$	28.9969	1.05391	0.526955	−	0.849893i	$-0.323334\pi$
$757$	28.9969	1.05391	0.526955	+	0.849893i	$0.323334\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	13.0536	0.473191	0.236595	−	0.971608i	$-0.423968\pi$
$761$	13.0536	0.473191	0.236595	+	0.971608i	$0.423968\pi$
$762$	0	0
$763$	0.184855	0.00669220
$764$	0	0
$765$	0	0
$766$	0	0
$767$	25.4318	0.918288
$768$	0	0
$769$	−33.9453	−1.22410	−0.612049	−	0.790820i	$-0.709654\pi$
$769$	−33.9453	−1.22410	−0.612049	+	0.790820i	$0.709654\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	20.6514	0.742781	0.371390	−	0.928477i	$-0.378881\pi$
$773$	20.6514	0.742781	0.371390	+	0.928477i	$0.378881\pi$
$774$	0	0
$775$	−52.7518	−1.89490
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−13.5632	−0.485953
$780$	0	0
$781$	−4.61201	−0.165031
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−1.39241	−0.0496973
$786$	0	0
$787$	−35.8313	−1.27725	−0.638625	−	0.769518i	$-0.720496\pi$
$787$	−35.8313	−1.27725	−0.638625	+	0.769518i	$0.720496\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	12.4593	0.443002
$792$	0	0
$793$	31.5178	1.11923
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−21.8185	−0.772851	−0.386425	−	0.922321i	$-0.626290\pi$
$797$	−21.8185	−0.772851	−0.386425	+	0.922321i	$0.626290\pi$
$798$	0	0
$799$	16.4539	0.582099
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−22.3644	−0.789223
$804$	0	0
$805$	−16.0308	−0.565011
$806$	0	0
$807$	0	0
$808$	0	0
$809$	7.89212	0.277472	0.138736	−	0.990329i	$-0.455696\pi$
$809$	7.89212	0.277472	0.138736	+	0.990329i	$0.455696\pi$
$810$	0	0
$811$	−12.8991	−0.452949	−0.226474	−	0.974017i	$-0.572720\pi$
$811$	−12.8991	−0.452949	−0.226474	+	0.974017i	$0.572720\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	83.6992	2.93186
$816$	0	0
$817$	10.2459	0.358458
$818$	0	0
$819$	0	0
$820$	0	0
$821$	45.7640	1.59718	0.798588	−	0.601878i	$-0.205581\pi$
$821$	45.7640	1.59718	0.798588	+	0.601878i	$0.205581\pi$
$822$	0	0
$823$	−32.3284	−1.12690	−0.563449	−	0.826151i	$-0.690526\pi$
$823$	−32.3284	−1.12690	−0.563449	+	0.826151i	$0.690526\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−41.9294	−1.45803	−0.729013	−	0.684499i	$-0.760021\pi$
$827$	−41.9294	−1.45803	−0.729013	+	0.684499i	$0.760021\pi$
$828$	0	0
$829$	−27.0415	−0.939191	−0.469596	−	0.882882i	$-0.655600\pi$
$829$	−27.0415	−0.939191	−0.469596	+	0.882882i	$0.655600\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−8.29688	−0.287470
$834$	0	0
$835$	13.7578	0.476108
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−32.6654	−1.12774	−0.563868	−	0.825865i	$-0.690687\pi$
$839$	−32.6654	−1.12774	−0.563868	+	0.825865i	$0.690687\pi$
$840$	0	0
$841$	−28.9839	−0.999446
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−9.56554	−0.329064
$846$	0	0
$847$	−0.660833	−0.0227065
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−53.7203	−1.84151
$852$	0	0
$853$	19.9755	0.683947	0.341974	−	0.939710i	$-0.388905\pi$
$853$	19.9755	0.683947	0.341974	+	0.939710i	$0.388905\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−41.8526	−1.42966	−0.714830	−	0.699299i	$-0.753496\pi$
$857$	−41.8526	−1.42966	−0.714830	+	0.699299i	$0.753496\pi$
$858$	0	0
$859$	18.9635	0.647025	0.323513	−	0.946224i	$-0.395136\pi$
$859$	18.9635	0.647025	0.323513	+	0.946224i	$0.395136\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	4.35696	0.148313	0.0741563	−	0.997247i	$-0.476374\pi$
$863$	4.35696	0.148313	0.0741563	+	0.997247i	$0.476374\pi$
$864$	0	0
$865$	74.6892	2.53951
$866$	0	0
$867$	0	0
$868$	0	0
$869$	39.8887	1.35313
$870$	0	0
$871$	12.8780	0.436354
$872$	0	0
$873$	0	0
$874$	0	0
$875$	5.06919	0.171370
$876$	0	0
$877$	1.52476	0.0514874	0.0257437	−	0.999669i	$-0.491805\pi$
$877$	1.52476	0.0514874	0.0257437	+	0.999669i	$0.491805\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−25.5225	−0.859874	−0.429937	−	0.902859i	$-0.641464\pi$
$881$	−25.5225	−0.859874	−0.429937	+	0.902859i	$0.641464\pi$
$882$	0	0
$883$	−2.99519	−0.100796	−0.0503980	−	0.998729i	$-0.516049\pi$
$883$	−2.99519	−0.100796	−0.0503980	+	0.998729i	$0.516049\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	20.9048	0.701915	0.350957	−	0.936391i	$-0.385856\pi$
$887$	20.9048	0.701915	0.350957	+	0.936391i	$0.385856\pi$
$888$	0	0
$889$	−8.85730	−0.297064
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−50.0470	−1.67476
$894$	0	0
$895$	19.0481	0.636709
$896$	0	0
$897$	0	0
$898$	0	0
$899$	0.934357	0.0311625
$900$	0	0
$901$	6.18352	0.206003
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−18.5996	−0.618270
$906$	0	0
$907$	12.0101	0.398790	0.199395	−	0.979919i	$-0.436102\pi$
$907$	12.0101	0.398790	0.199395	+	0.979919i	$0.436102\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	13.4836	0.446731	0.223366	−	0.974735i	$-0.428296\pi$
$911$	13.4836	0.446731	0.223366	+	0.974735i	$0.428296\pi$
$912$	0	0
$913$	−37.0867	−1.22739
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−7.05954	−0.233126
$918$	0	0
$919$	−7.87535	−0.259784	−0.129892	−	0.991528i	$-0.541463\pi$
$919$	−7.87535	−0.259784	−0.129892	+	0.991528i	$0.541463\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−4.66645	−0.153598
$924$	0	0
$925$	56.3365	1.85233
$926$	0	0
$927$	0	0
$928$	0	0
$929$	18.0643	0.592672	0.296336	−	0.955084i	$-0.404235\pi$
$929$	18.0643	0.592672	0.296336	+	0.955084i	$0.404235\pi$
$930$	0	0
$931$	25.2361	0.827081
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−13.9882	−0.457463
$936$	0	0
$937$	42.1732	1.37774	0.688870	−	0.724885i	$-0.258107\pi$
$937$	42.1732	1.37774	0.688870	+	0.724885i	$0.258107\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	29.1276	0.949533	0.474767	−	0.880112i	$-0.342532\pi$
$941$	29.1276	0.949533	0.474767	+	0.880112i	$0.342532\pi$
$942$	0	0
$943$	−24.0168	−0.782093
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−10.5529	−0.342922	−0.171461	−	0.985191i	$-0.554849\pi$
$947$	−10.5529	−0.342922	−0.171461	+	0.985191i	$0.554849\pi$
$948$	0	0
$949$	−22.6284	−0.734549
$950$	0	0
$951$	0	0
$952$	0	0
$953$	45.2917	1.46714	0.733572	−	0.679612i	$-0.237852\pi$
$953$	45.2917	1.46714	0.733572	+	0.679612i	$0.237852\pi$
$954$	0	0
$955$	20.5906	0.666297
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0.887020	0.0286434
$960$	0	0
$961$	23.3035	0.751726
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−68.5659	−2.20722
$966$	0	0
$967$	−20.2984	−0.652751	−0.326376	−	0.945240i	$-0.605827\pi$
$967$	−20.2984	−0.652751	−0.326376	+	0.945240i	$0.605827\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−37.5363	−1.20460	−0.602298	−	0.798271i	$-0.705748\pi$
$971$	−37.5363	−1.20460	−0.602298	+	0.798271i	$0.705748\pi$
$972$	0	0
$973$	−7.43898	−0.238483
$974$	0	0
$975$	0	0
$976$	0	0
$977$	19.9861	0.639412	0.319706	−	0.947517i	$-0.396416\pi$
$977$	19.9861	0.639412	0.319706	+	0.947517i	$0.396416\pi$
$978$	0	0
$979$	36.9438	1.18073
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−17.6745	−0.563728	−0.281864	−	0.959454i	$-0.590953\pi$
$983$	−17.6745	−0.563728	−0.281864	+	0.959454i	$0.590953\pi$
$984$	0	0
$985$	−71.9289	−2.29184
$986$	0	0
$987$	0	0
$988$	0	0
$989$	18.1426	0.576902
$990$	0	0
$991$	25.9500	0.824328	0.412164	−	0.911110i	$-0.364773\pi$
$991$	25.9500	0.824328	0.412164	+	0.911110i	$0.364773\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	65.3108	2.07049
$996$	0	0
$997$	−28.6571	−0.907579	−0.453789	−	0.891109i	$-0.649928\pi$
$997$	−28.6571	−0.907579	−0.453789	+	0.891109i	$0.649928\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8028.2.a.l.1.7		8
3.2	odd	2		2676.2.a.d.1.2	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2676.2.a.d.1.2	✓	8	3.2	odd	2
8028.2.a.l.1.7		8	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:	\( N \)	\(=\)	\( 8028 = 2^{2} \cdot 3^{2} \cdot 223 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8028.a (trivial)

\(f(q)\)	\(=\)	\(q+3.48691 q^{5} +0.673508 q^{7} +O(q^{10})\)	\(q+3.48691 q^{5} +0.673508 q^{7} -3.16525 q^{11} -3.20261 q^{13} +1.26740 q^{17} -3.85497 q^{19} -6.82609 q^{23} +7.15852 q^{25} -0.126794 q^{29} -7.36909 q^{31} +2.34846 q^{35} +7.86985 q^{37} +3.51837 q^{41} -2.65783 q^{43} +12.9825 q^{47} -6.54639 q^{49} +4.87891 q^{53} -11.0369 q^{55} -7.94095 q^{59} -9.84128 q^{61} -11.1672 q^{65} -4.02109 q^{67} +1.45708 q^{71} +7.06561 q^{73} -2.13182 q^{77} -12.6020 q^{79} +11.7168 q^{83} +4.41930 q^{85} -11.6717 q^{89} -2.15698 q^{91} -13.4419 q^{95} -7.04958 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 5 q^{5} - 13 q^{7}+O(q^{10}) \) 8 * q + 5 * q^5 - 13 * q^7	\( 8 q + 5 q^{5} - 13 q^{7} + q^{11} - 10 q^{13} + 10 q^{17} - 5 q^{19} - q^{23} + 5 q^{25} + 14 q^{29} - 21 q^{31} - 21 q^{37} + 31 q^{41} - 18 q^{43} + 10 q^{47} + q^{49} + 12 q^{53} - 31 q^{55} - 2 q^{59} - 32 q^{61} + 26 q^{65} - 25 q^{67} + 11 q^{71} - 14 q^{73} + 12 q^{77} - 9 q^{79} - 25 q^{83} - 45 q^{85} + 23 q^{89} - 4 q^{91} + 6 q^{95} - 11 q^{97}+O(q^{100}) \) 8 * q + 5 * q^5 - 13 * q^7 + q^11 - 10 * q^13 + 10 * q^17 - 5 * q^19 - q^23 + 5 * q^25 + 14 * q^29 - 21 * q^31 - 21 * q^37 + 31 * q^41 - 18 * q^43 + 10 * q^47 + q^49 + 12 * q^53 - 31 * q^55 - 2 * q^59 - 32 * q^61 + 26 * q^65 - 25 * q^67 + 11 * q^71 - 14 * q^73 + 12 * q^77 - 9 * q^79 - 25 * q^83 - 45 * q^85 + 23 * q^89 - 4 * q^91 + 6 * q^95 - 11 * q^97

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