LMFDB - Embedded newform 5776.2.a.cd.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [5776,2,Mod(1,5776)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(5776, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("5776.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 5776 = 2^{4} \cdot 19^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5776.a (trivial)

Newform invariants

comment:select newform

sage:traces = [9,0,-3,0,3,0,-9,0,6,0,-3,0,-6,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(15)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$46.1215922075$
Analytic rank:	$1$
Dimension:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - 3x^{8} - 12x^{7} + 35x^{6} + 45x^{5} - 117x^{4} - 55x^{3} + 96x^{2} - 6x - 1 $ x^9 - 3x^8 - 12x^7 + 35x^6 + 45x^5 - 117x^4 - 55x^3 + 96x^2 - 6x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 152)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$2.36188$ of defining polynomial
Character	$\chi$	$=$	5776.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-2.36188 q^{3} -2.34290 q^{5} -4.15573 q^{7} +2.57847 q^{9} -5.36729 q^{11} -2.95483 q^{13} +5.53365 q^{15} +5.54437 q^{17} +9.81534 q^{21} -1.79384 q^{23} +0.489197 q^{25} +0.995607 q^{27} -0.215600 q^{29} +2.40651 q^{31} +12.6769 q^{33} +9.73649 q^{35} +6.54964 q^{37} +6.97896 q^{39} -1.74834 q^{41} +1.64951 q^{43} -6.04110 q^{45} +6.85481 q^{47} +10.2701 q^{49} -13.0951 q^{51} -6.33010 q^{53} +12.5750 q^{55} -10.2413 q^{59} +9.99468 q^{61} -10.7154 q^{63} +6.92289 q^{65} -8.94912 q^{67} +4.23684 q^{69} +10.7435 q^{71} +12.4701 q^{73} -1.15542 q^{75} +22.3050 q^{77} +5.60302 q^{79} -10.0869 q^{81} +7.31227 q^{83} -12.9899 q^{85} +0.509222 q^{87} +5.42611 q^{89} +12.2795 q^{91} -5.68388 q^{93} -17.8445 q^{97} -13.8394 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q - 3 q^{3} + 3 q^{5} - 9 q^{7} + 6 q^{9} - 3 q^{11} - 6 q^{13} - 3 q^{17} + 15 q^{21} - 24 q^{23} + 30 q^{25} - 12 q^{27} - 15 q^{29} - 6 q^{31} - 18 q^{33} - 15 q^{35} + 24 q^{37} - 6 q^{39} - 12 q^{41}+ \cdots - 48 q^{99}+O(q^{100}) $ 9 * q - 3 * q^3 + 3 * q^5 - 9 * q^7 + 6 * q^9 - 3 * q^11 - 6 * q^13 - 3 * q^17 + 15 * q^21 - 24 * q^23 + 30 * q^25 - 12 * q^27 - 15 * q^29 - 6 * q^31 - 18 * q^33 - 15 * q^35 + 24 * q^37 - 6 * q^39 - 12 * q^41 - 9 * q^43 + 42 * q^45 + 12 * q^47 + 18 * q^49 - 12 * q^51 - 18 * q^53 - 21 * q^55 - 57 * q^59 + 15 * q^61 - 30 * q^63 - 27 * q^65 - 6 * q^67 + 3 * q^69 + 36 * q^73 - 45 * q^75 + 30 * q^77 + 3 * q^79 - 15 * q^81 - 27 * q^83 + 18 * q^85 + 18 * q^87 - 30 * q^89 + 21 * q^91 - 24 * q^93 + 3 * q^97 - 48 * 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
$3$	−2.36188	−1.36363	−0.681815	−	0.731524i	$-0.738809\pi$
$3$	−2.36188	−1.36363	−0.681815	+	0.731524i	$0.738809\pi$
$4$	0	0
$5$	−2.34290	−1.04778	−0.523889	−	0.851786i	$-0.675519\pi$
$5$	−2.34290	−1.04778	−0.523889	+	0.851786i	$0.675519\pi$
$6$	0	0
$7$	−4.15573	−1.57072	−0.785360	−	0.619039i	$-0.787522\pi$
$7$	−4.15573	−1.57072	−0.785360	+	0.619039i	$0.787522\pi$
$8$	0	0
$9$	2.57847	0.859489
$10$	0	0
$11$	−5.36729	−1.61830	−0.809149	−	0.587604i	$-0.800071\pi$
$11$	−5.36729	−1.61830	−0.809149	+	0.587604i	$0.800071\pi$
$12$	0	0
$13$	−2.95483	−0.819524	−0.409762	−	0.912193i	$-0.634388\pi$
$13$	−2.95483	−0.819524	−0.409762	+	0.912193i	$0.634388\pi$
$14$	0	0
$15$	5.53365	1.42878
$16$	0	0
$17$	5.54437	1.34471	0.672354	−	0.740230i	$-0.265283\pi$
$17$	5.54437	1.34471	0.672354	+	0.740230i	$0.265283\pi$
$18$	0	0
$19$	0	0
$19$	0	0
$20$	0	0
$21$	9.81534	2.14188
$22$	0	0
$23$	−1.79384	−0.374042	−0.187021	−	0.982356i	$-0.559883\pi$
$23$	−1.79384	−0.374042	−0.187021	+	0.982356i	$0.559883\pi$
$24$	0	0
$25$	0.489197	0.0978395
$26$	0	0
$27$	0.995607	0.191605
$28$	0	0
$29$	−0.215600	−0.0400360	−0.0200180	−	0.999800i	$-0.506372\pi$
$29$	−0.215600	−0.0400360	−0.0200180	+	0.999800i	$0.506372\pi$
$30$	0	0
$31$	2.40651	0.432221	0.216111	−	0.976369i	$-0.430663\pi$
$31$	2.40651	0.432221	0.216111	+	0.976369i	$0.430663\pi$
$32$	0	0
$33$	12.6769	2.20676
$34$	0	0
$35$	9.73649	1.64577
$36$	0	0
$37$	6.54964	1.07676	0.538378	−	0.842704i	$-0.319037\pi$
$37$	6.54964	1.07676	0.538378	+	0.842704i	$0.319037\pi$
$38$	0	0
$39$	6.97896	1.11753
$40$	0	0
$41$	−1.74834	−0.273044	−0.136522	−	0.990637i	$-0.543592\pi$
$41$	−1.74834	−0.273044	−0.136522	+	0.990637i	$0.543592\pi$
$42$	0	0
$43$	1.64951	0.251548	0.125774	−	0.992059i	$-0.459859\pi$
$43$	1.64951	0.251548	0.125774	+	0.992059i	$0.459859\pi$
$44$	0	0
$45$	−6.04110	−0.900554
$46$	0	0
$47$	6.85481	0.999876	0.499938	−	0.866061i	$-0.333356\pi$
$47$	6.85481	0.999876	0.499938	+	0.866061i	$0.333356\pi$
$48$	0	0
$49$	10.2701	1.46716
$50$	0	0
$51$	−13.0951	−1.83368
$52$	0	0
$53$	−6.33010	−0.869506	−0.434753	−	0.900550i	$-0.643164\pi$
$53$	−6.33010	−0.869506	−0.434753	+	0.900550i	$0.643164\pi$
$54$	0	0
$55$	12.5750	1.69562
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−10.2413	−1.33330	−0.666649	−	0.745371i	$-0.732272\pi$
$59$	−10.2413	−1.33330	−0.666649	+	0.745371i	$0.732272\pi$
$60$	0	0
$61$	9.99468	1.27969	0.639844	−	0.768505i	$-0.278999\pi$
$61$	9.99468	1.27969	0.639844	+	0.768505i	$0.278999\pi$
$62$	0	0
$63$	−10.7154	−1.35002
$64$	0	0
$65$	6.92289	0.858679
$66$	0	0
$67$	−8.94912	−1.09331	−0.546654	−	0.837358i	$-0.684099\pi$
$67$	−8.94912	−1.09331	−0.546654	+	0.837358i	$0.684099\pi$
$68$	0	0
$69$	4.23684	0.510056
$70$	0	0
$71$	10.7435	1.27502	0.637509	−	0.770443i	$-0.279965\pi$
$71$	10.7435	1.27502	0.637509	+	0.770443i	$0.279965\pi$
$72$	0	0
$73$	12.4701	1.45952	0.729760	−	0.683704i	$-0.239632\pi$
$73$	12.4701	1.45952	0.729760	+	0.683704i	$0.239632\pi$
$74$	0	0
$75$	−1.15542	−0.133417
$76$	0	0
$77$	22.3050	2.54189
$78$	0	0
$79$	5.60302	0.630389	0.315194	−	0.949027i	$-0.397930\pi$
$79$	5.60302	0.630389	0.315194	+	0.949027i	$0.397930\pi$
$80$	0	0
$81$	−10.0869	−1.12077
$82$	0	0
$83$	7.31227	0.802626	0.401313	−	0.915941i	$-0.368554\pi$
$83$	7.31227	0.802626	0.401313	+	0.915941i	$0.368554\pi$
$84$	0	0
$85$	−12.9899	−1.40896
$86$	0	0
$87$	0.509222	0.0545943
$88$	0	0
$89$	5.42611	0.575166	0.287583	−	0.957756i	$-0.407148\pi$
$89$	5.42611	0.575166	0.287583	+	0.957756i	$0.407148\pi$
$90$	0	0
$91$	12.2795	1.28724
$92$	0	0
$93$	−5.68388	−0.589391
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−17.8445	−1.81184	−0.905919	−	0.423451i	$-0.860819\pi$
$97$	−17.8445	−1.81184	−0.905919	+	0.423451i	$0.860819\pi$
$98$	0	0
$99$	−13.8394	−1.39091
$100$	0	0
$101$	−10.1207	−1.00704	−0.503521	−	0.863983i	$-0.667962\pi$
$101$	−10.1207	−1.00704	−0.503521	+	0.863983i	$0.667962\pi$
$102$	0	0
$103$	−4.72949	−0.466010	−0.233005	−	0.972475i	$-0.574856\pi$
$103$	−4.72949	−0.466010	−0.233005	+	0.972475i	$0.574856\pi$
$104$	0	0
$105$	−22.9964	−2.24422
$106$	0	0
$107$	−9.04533	−0.874446	−0.437223	−	0.899353i	$-0.644038\pi$
$107$	−9.04533	−0.874446	−0.437223	+	0.899353i	$0.644038\pi$
$108$	0	0
$109$	8.04468	0.770541	0.385270	−	0.922804i	$-0.374108\pi$
$109$	8.04468	0.770541	0.385270	+	0.922804i	$0.374108\pi$
$110$	0	0
$111$	−15.4695	−1.46830
$112$	0	0
$113$	−4.20523	−0.395595	−0.197797	−	0.980243i	$-0.563379\pi$
$113$	−4.20523	−0.395595	−0.197797	+	0.980243i	$0.563379\pi$
$114$	0	0
$115$	4.20280	0.391913
$116$	0	0
$117$	−7.61895	−0.704372
$118$	0	0
$119$	−23.0409	−2.11216
$120$	0	0
$121$	17.8078	1.61889
$122$	0	0
$123$	4.12935	0.372331
$124$	0	0
$125$	10.5684	0.945264
$126$	0	0
$127$	−5.30569	−0.470804	−0.235402	−	0.971898i	$-0.575641\pi$
$127$	−5.30569	−0.470804	−0.235402	+	0.971898i	$0.575641\pi$
$128$	0	0
$129$	−3.89595	−0.343019
$130$	0	0
$131$	2.46781	0.215614	0.107807	−	0.994172i	$-0.465617\pi$
$131$	2.46781	0.215614	0.107807	+	0.994172i	$0.465617\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−2.33261	−0.200759
$136$	0	0
$137$	1.14001	0.0973973	0.0486987	−	0.998814i	$-0.484493\pi$
$137$	1.14001	0.0973973	0.0486987	+	0.998814i	$0.484493\pi$
$138$	0	0
$139$	−16.7713	−1.42252	−0.711260	−	0.702929i	$-0.751875\pi$
$139$	−16.7713	−1.42252	−0.711260	+	0.702929i	$0.751875\pi$
$140$	0	0
$141$	−16.1902	−1.36346
$142$	0	0
$143$	15.8594	1.32623
$144$	0	0
$145$	0.505131	0.0419488
$146$	0	0
$147$	−24.2568	−2.00067
$148$	0	0
$149$	6.44232	0.527776	0.263888	−	0.964553i	$-0.414995\pi$
$149$	6.44232	0.527776	0.263888	+	0.964553i	$0.414995\pi$
$150$	0	0
$151$	9.15755	0.745231	0.372616	−	0.927986i	$-0.378461\pi$
$151$	9.15755	0.745231	0.372616	+	0.927986i	$0.378461\pi$
$152$	0	0
$153$	14.2960	1.15576
$154$	0	0
$155$	−5.63821	−0.452872
$156$	0	0
$157$	18.1429	1.44796	0.723982	−	0.689819i	$-0.242310\pi$
$157$	18.1429	1.44796	0.723982	+	0.689819i	$0.242310\pi$
$158$	0	0
$159$	14.9509	1.18568
$160$	0	0
$161$	7.45474	0.587516
$162$	0	0
$163$	7.75111	0.607114	0.303557	−	0.952813i	$-0.401826\pi$
$163$	7.75111	0.607114	0.303557	+	0.952813i	$0.401826\pi$
$164$	0	0
$165$	−29.7007	−2.31220
$166$	0	0
$167$	4.70588	0.364152	0.182076	−	0.983285i	$-0.441718\pi$
$167$	4.70588	0.364152	0.182076	+	0.983285i	$0.441718\pi$
$168$	0	0
$169$	−4.26895	−0.328381
$170$	0	0
$171$	0	0
$172$	0	0
$173$	21.5283	1.63677	0.818383	−	0.574673i	$-0.194871\pi$
$173$	21.5283	1.63677	0.818383	+	0.574673i	$0.194871\pi$
$174$	0	0
$175$	−2.03297	−0.153678
$176$	0	0
$177$	24.1886	1.81813
$178$	0	0
$179$	−11.1355	−0.832305	−0.416153	−	0.909295i	$-0.636622\pi$
$179$	−11.1355	−0.832305	−0.416153	+	0.909295i	$0.636622\pi$
$180$	0	0
$181$	−19.5795	−1.45533	−0.727666	−	0.685931i	$-0.759395\pi$
$181$	−19.5795	−1.45533	−0.727666	+	0.685931i	$0.759395\pi$
$182$	0	0
$183$	−23.6062	−1.74502
$184$	0	0
$185$	−15.3452	−1.12820
$186$	0	0
$187$	−29.7582	−2.17614
$188$	0	0
$189$	−4.13748	−0.300957
$190$	0	0
$191$	14.0015	1.01312	0.506558	−	0.862206i	$-0.330918\pi$
$191$	14.0015	1.01312	0.506558	+	0.862206i	$0.330918\pi$
$192$	0	0
$193$	−0.130767	−0.00941284	−0.00470642	−	0.999989i	$-0.501498\pi$
$193$	−0.130767	−0.00941284	−0.00470642	+	0.999989i	$0.501498\pi$
$194$	0	0
$195$	−16.3510	−1.17092
$196$	0	0
$197$	−12.1046	−0.862420	−0.431210	−	0.902251i	$-0.641913\pi$
$197$	−12.1046	−0.862420	−0.431210	+	0.902251i	$0.641913\pi$
$198$	0	0
$199$	−11.6872	−0.828482	−0.414241	−	0.910167i	$-0.635953\pi$
$199$	−11.6872	−0.828482	−0.414241	+	0.910167i	$0.635953\pi$
$200$	0	0
$201$	21.1367	1.49087
$202$	0	0
$203$	0.895978	0.0628853
$204$	0	0
$205$	4.09618	0.286090
$206$	0	0
$207$	−4.62537	−0.321485
$208$	0	0
$209$	0	0
$210$	0	0
$211$	1.64762	0.113427	0.0567133	−	0.998391i	$-0.481938\pi$
$211$	1.64762	0.113427	0.0567133	+	0.998391i	$0.481938\pi$
$212$	0	0
$213$	−25.3748	−1.73865
$214$	0	0
$215$	−3.86465	−0.263567
$216$	0	0
$217$	−10.0008	−0.678899
$218$	0	0
$219$	−29.4530	−1.99025
$220$	0	0
$221$	−16.3827	−1.10202
$222$	0	0
$223$	18.4561	1.23591	0.617957	−	0.786212i	$-0.287961\pi$
$223$	18.4561	1.23591	0.617957	+	0.786212i	$0.287961\pi$
$224$	0	0
$225$	1.26138	0.0840920
$226$	0	0
$227$	−3.16564	−0.210111	−0.105055	−	0.994466i	$-0.533502\pi$
$227$	−3.16564	−0.210111	−0.105055	+	0.994466i	$0.533502\pi$
$228$	0	0
$229$	−12.8379	−0.848353	−0.424177	−	0.905580i	$-0.639436\pi$
$229$	−12.8379	−0.848353	−0.424177	+	0.905580i	$0.639436\pi$
$230$	0	0
$231$	−52.6817	−3.46620
$232$	0	0
$233$	5.78280	0.378844	0.189422	−	0.981896i	$-0.439339\pi$
$233$	5.78280	0.378844	0.189422	+	0.981896i	$0.439339\pi$
$234$	0	0
$235$	−16.0602	−1.04765
$236$	0	0
$237$	−13.2336	−0.859618
$238$	0	0
$239$	13.0527	0.844308	0.422154	−	0.906524i	$-0.361274\pi$
$239$	13.0527	0.844308	0.422154	+	0.906524i	$0.361274\pi$
$240$	0	0
$241$	10.8508	0.698962	0.349481	−	0.936943i	$-0.386358\pi$
$241$	10.8508	0.698962	0.349481	+	0.936943i	$0.386358\pi$
$242$	0	0
$243$	20.8372	1.33671
$244$	0	0
$245$	−24.0619	−1.53726
$246$	0	0
$247$	0	0
$248$	0	0
$249$	−17.2707	−1.09449
$250$	0	0
$251$	11.2491	0.710039	0.355020	−	0.934859i	$-0.384474\pi$
$251$	11.2491	0.710039	0.355020	+	0.934859i	$0.384474\pi$
$252$	0	0
$253$	9.62807	0.605312
$254$	0	0
$255$	30.6806	1.92130
$256$	0	0
$257$	−20.4744	−1.27716	−0.638580	−	0.769556i	$-0.720478\pi$
$257$	−20.4744	−1.27716	−0.638580	+	0.769556i	$0.720478\pi$
$258$	0	0
$259$	−27.2186	−1.69128
$260$	0	0
$261$	−0.555918	−0.0344105
$262$	0	0
$263$	27.8919	1.71989	0.859946	−	0.510386i	$-0.170497\pi$
$263$	27.8919	1.71989	0.859946	+	0.510386i	$0.170497\pi$
$264$	0	0
$265$	14.8308	0.911049
$266$	0	0
$267$	−12.8158	−0.784314
$268$	0	0
$269$	−5.23607	−0.319249	−0.159625	−	0.987178i	$-0.551028\pi$
$269$	−5.23607	−0.319249	−0.159625	+	0.987178i	$0.551028\pi$
$270$	0	0
$271$	−5.39188	−0.327533	−0.163767	−	0.986499i	$-0.552364\pi$
$271$	−5.39188	−0.327533	−0.163767	+	0.986499i	$0.552364\pi$
$272$	0	0
$273$	−29.0027	−1.75532
$274$	0	0
$275$	−2.62566	−0.158333
$276$	0	0
$277$	25.4903	1.53156	0.765781	−	0.643102i	$-0.222353\pi$
$277$	25.4903	1.53156	0.765781	+	0.643102i	$0.222353\pi$
$278$	0	0
$279$	6.20510	0.371490
$280$	0	0
$281$	−18.4077	−1.09811	−0.549056	−	0.835786i	$-0.685012\pi$
$281$	−18.4077	−1.09811	−0.549056	+	0.835786i	$0.685012\pi$
$282$	0	0
$283$	−19.7747	−1.17549	−0.587743	−	0.809047i	$-0.699983\pi$
$283$	−19.7747	−1.17549	−0.587743	+	0.809047i	$0.699983\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	7.26562	0.428876
$288$	0	0
$289$	13.7401	0.808239
$290$	0	0
$291$	42.1466	2.47068
$292$	0	0
$293$	−1.69621	−0.0990936	−0.0495468	−	0.998772i	$-0.515778\pi$
$293$	−1.69621	−0.0990936	−0.0495468	+	0.998772i	$0.515778\pi$
$294$	0	0
$295$	23.9943	1.39700
$296$	0	0
$297$	−5.34371	−0.310073
$298$	0	0
$299$	5.30051	0.306537
$300$	0	0
$301$	−6.85493	−0.395112
$302$	0	0
$303$	23.9037	1.37323
$304$	0	0
$305$	−23.4166	−1.34083
$306$	0	0
$307$	22.6157	1.29075	0.645373	−	0.763868i	$-0.276702\pi$
$307$	22.6157	1.29075	0.645373	+	0.763868i	$0.276702\pi$
$308$	0	0
$309$	11.1705	0.635466
$310$	0	0
$311$	−1.29863	−0.0736384	−0.0368192	−	0.999322i	$-0.511723\pi$
$311$	−1.29863	−0.0736384	−0.0368192	+	0.999322i	$0.511723\pi$
$312$	0	0
$313$	−8.08418	−0.456945	−0.228473	−	0.973550i	$-0.573373\pi$
$313$	−8.08418	−0.456945	−0.228473	+	0.973550i	$0.573373\pi$
$314$	0	0
$315$	25.1052	1.41452
$316$	0	0
$317$	−7.14759	−0.401449	−0.200724	−	0.979648i	$-0.564330\pi$
$317$	−7.14759	−0.401449	−0.200724	+	0.979648i	$0.564330\pi$
$318$	0	0
$319$	1.15719	0.0647901
$320$	0	0
$321$	21.3640	1.19242
$322$	0	0
$323$	0	0
$324$	0	0
$325$	−1.44550	−0.0801818
$326$	0	0
$327$	−19.0006	−1.05073
$328$	0	0
$329$	−28.4868	−1.57053
$330$	0	0
$331$	−27.2644	−1.49858	−0.749292	−	0.662239i	$-0.769606\pi$
$331$	−27.2644	−1.49858	−0.749292	+	0.662239i	$0.769606\pi$
$332$	0	0
$333$	16.8880	0.925459
$334$	0	0
$335$	20.9669	1.14554
$336$	0	0
$337$	−24.4792	−1.33347	−0.666733	−	0.745297i	$-0.732308\pi$
$337$	−24.4792	−1.33347	−0.666733	+	0.745297i	$0.732308\pi$
$338$	0	0
$339$	9.93224	0.539445
$340$	0	0
$341$	−12.9164	−0.699463
$342$	0	0
$343$	−13.5898	−0.733780
$344$	0	0
$345$	−9.92651	−0.534425
$346$	0	0
$347$	22.8285	1.22550	0.612750	−	0.790277i	$-0.290063\pi$
$347$	22.8285	1.22550	0.612750	+	0.790277i	$0.290063\pi$
$348$	0	0
$349$	2.58715	0.138487	0.0692435	−	0.997600i	$-0.477941\pi$
$349$	2.58715	0.138487	0.0692435	+	0.997600i	$0.477941\pi$
$350$	0	0
$351$	−2.94186	−0.157025
$352$	0	0
$353$	−10.9610	−0.583394	−0.291697	−	0.956511i	$-0.594220\pi$
$353$	−10.9610	−0.583394	−0.291697	+	0.956511i	$0.594220\pi$
$354$	0	0
$355$	−25.1710	−1.33594
$356$	0	0
$357$	54.4199	2.88021
$358$	0	0
$359$	23.5184	1.24126	0.620628	−	0.784105i	$-0.286878\pi$
$359$	23.5184	1.24126	0.620628	+	0.784105i	$0.286878\pi$
$360$	0	0
$361$	0	0
$362$	0	0
$363$	−42.0598	−2.20756
$364$	0	0
$365$	−29.2163	−1.52925
$366$	0	0
$367$	8.01699	0.418484	0.209242	−	0.977864i	$-0.432900\pi$
$367$	8.01699	0.418484	0.209242	+	0.977864i	$0.432900\pi$
$368$	0	0
$369$	−4.50803	−0.234678
$370$	0	0
$371$	26.3062	1.36575
$372$	0	0
$373$	−16.9610	−0.878209	−0.439104	−	0.898436i	$-0.644704\pi$
$373$	−16.9610	−0.878209	−0.439104	+	0.898436i	$0.644704\pi$
$374$	0	0
$375$	−24.9612	−1.28899
$376$	0	0
$377$	0.637063	0.0328104
$378$	0	0
$379$	22.6828	1.16513	0.582567	−	0.812782i	$-0.302048\pi$
$379$	22.6828	1.16513	0.582567	+	0.812782i	$0.302048\pi$
$380$	0	0
$381$	12.5314	0.642002
$382$	0	0
$383$	−12.3594	−0.631536	−0.315768	−	0.948837i	$-0.602262\pi$
$383$	−12.3594	−0.631536	−0.315768	+	0.948837i	$0.602262\pi$
$384$	0	0
$385$	−52.2585	−2.66334
$386$	0	0
$387$	4.25321	0.216203
$388$	0	0
$389$	−19.3892	−0.983073	−0.491536	−	0.870857i	$-0.663565\pi$
$389$	−19.3892	−0.983073	−0.491536	+	0.870857i	$0.663565\pi$
$390$	0	0
$391$	−9.94574	−0.502978
$392$	0	0
$393$	−5.82867	−0.294017
$394$	0	0
$395$	−13.1273	−0.660508
$396$	0	0
$397$	5.50592	0.276334	0.138167	−	0.990409i	$-0.455879\pi$
$397$	5.50592	0.276334	0.138167	+	0.990409i	$0.455879\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−7.04316	−0.351719	−0.175859	−	0.984415i	$-0.556270\pi$
$401$	−7.04316	−0.351719	−0.175859	+	0.984415i	$0.556270\pi$
$402$	0	0
$403$	−7.11083	−0.354216
$404$	0	0
$405$	23.6327	1.17432
$406$	0	0
$407$	−35.1538	−1.74251
$408$	0	0
$409$	38.1564	1.88671	0.943356	−	0.331782i	$-0.107650\pi$
$409$	38.1564	1.88671	0.943356	+	0.331782i	$0.107650\pi$
$410$	0	0
$411$	−2.69256	−0.132814
$412$	0	0
$413$	42.5600	2.09424
$414$	0	0
$415$	−17.1319	−0.840974
$416$	0	0
$417$	39.6117	1.93979
$418$	0	0
$419$	−8.28410	−0.404705	−0.202352	−	0.979313i	$-0.564859\pi$
$419$	−8.28410	−0.404705	−0.202352	+	0.979313i	$0.564859\pi$
$420$	0	0
$421$	−0.0790672	−0.00385350	−0.00192675	−	0.999998i	$-0.500613\pi$
$421$	−0.0790672	−0.00385350	−0.00192675	+	0.999998i	$0.500613\pi$
$422$	0	0
$423$	17.6749	0.859383
$424$	0	0
$425$	2.71229	0.131565
$426$	0	0
$427$	−41.5352	−2.01003
$428$	0	0
$429$	−37.4581	−1.80849
$430$	0	0
$431$	−0.887413	−0.0427452	−0.0213726	−	0.999772i	$-0.506804\pi$
$431$	−0.887413	−0.0427452	−0.0213726	+	0.999772i	$0.506804\pi$
$432$	0	0
$433$	14.5212	0.697845	0.348922	−	0.937152i	$-0.386548\pi$
$433$	14.5212	0.697845	0.348922	+	0.937152i	$0.386548\pi$
$434$	0	0
$435$	−1.19306	−0.0572027
$436$	0	0
$437$	0	0
$438$	0	0
$439$	11.6694	0.556950	0.278475	−	0.960443i	$-0.410171\pi$
$439$	11.6694	0.556950	0.278475	+	0.960443i	$0.410171\pi$
$440$	0	0
$441$	26.4812	1.26101
$442$	0	0
$443$	10.9049	0.518109	0.259055	−	0.965863i	$-0.416589\pi$
$443$	10.9049	0.518109	0.259055	+	0.965863i	$0.416589\pi$
$444$	0	0
$445$	−12.7128	−0.602646
$446$	0	0
$447$	−15.2160	−0.719691
$448$	0	0
$449$	2.81252	0.132731	0.0663655	−	0.997795i	$-0.478860\pi$
$449$	2.81252	0.132731	0.0663655	+	0.997795i	$0.478860\pi$
$450$	0	0
$451$	9.38381	0.441867
$452$	0	0
$453$	−21.6290	−1.01622
$454$	0	0
$455$	−28.7697	−1.34874
$456$	0	0
$457$	4.19534	0.196250	0.0981248	−	0.995174i	$-0.468716\pi$
$457$	4.19534	0.196250	0.0981248	+	0.995174i	$0.468716\pi$
$458$	0	0
$459$	5.52002	0.257652
$460$	0	0
$461$	−32.3784	−1.50801	−0.754006	−	0.656867i	$-0.771881\pi$
$461$	−32.3784	−1.50801	−0.754006	+	0.656867i	$0.771881\pi$
$462$	0	0
$463$	−22.0392	−1.02425	−0.512124	−	0.858911i	$-0.671141\pi$
$463$	−22.0392	−1.02425	−0.512124	+	0.858911i	$0.671141\pi$
$464$	0	0
$465$	13.3168	0.617551
$466$	0	0
$467$	−15.3581	−0.710688	−0.355344	−	0.934736i	$-0.615636\pi$
$467$	−15.3581	−0.710688	−0.355344	+	0.934736i	$0.615636\pi$
$468$	0	0
$469$	37.1901	1.71728
$470$	0	0
$471$	−42.8514	−1.97449
$472$	0	0
$473$	−8.85340	−0.407080
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−16.3220	−0.747331
$478$	0	0
$479$	−20.2977	−0.927427	−0.463714	−	0.885985i	$-0.653483\pi$
$479$	−20.2977	−0.927427	−0.463714	+	0.885985i	$0.653483\pi$
$480$	0	0
$481$	−19.3531	−0.882426
$482$	0	0
$483$	−17.6072	−0.801155
$484$	0	0
$485$	41.8080	1.89841
$486$	0	0
$487$	29.8902	1.35445	0.677227	−	0.735774i	$-0.263182\pi$
$487$	29.8902	1.35445	0.677227	+	0.735774i	$0.263182\pi$
$488$	0	0
$489$	−18.3072	−0.827880
$490$	0	0
$491$	−7.48271	−0.337690	−0.168845	−	0.985643i	$-0.554004\pi$
$491$	−7.48271	−0.337690	−0.168845	+	0.985643i	$0.554004\pi$
$492$	0	0
$493$	−1.19537	−0.0538367
$494$	0	0
$495$	32.4243	1.45736
$496$	0	0
$497$	−44.6471	−2.00270
$498$	0	0
$499$	9.96687	0.446178	0.223089	−	0.974798i	$-0.428386\pi$
$499$	9.96687	0.446178	0.223089	+	0.974798i	$0.428386\pi$
$500$	0	0
$501$	−11.1147	−0.496568
$502$	0	0
$503$	7.42061	0.330869	0.165434	−	0.986221i	$-0.447097\pi$
$503$	7.42061	0.330869	0.165434	+	0.986221i	$0.447097\pi$
$504$	0	0
$505$	23.7117	1.05516
$506$	0	0
$507$	10.0827	0.447790
$508$	0	0
$509$	−41.7515	−1.85060	−0.925301	−	0.379232i	$-0.876188\pi$
$509$	−41.7515	−1.85060	−0.925301	+	0.379232i	$0.876188\pi$
$510$	0	0
$511$	−51.8226	−2.29250
$512$	0	0
$513$	0	0
$514$	0	0
$515$	11.0807	0.488276
$516$	0	0
$517$	−36.7917	−1.61810
$518$	0	0
$519$	−50.8472	−2.23195
$520$	0	0
$521$	−33.2868	−1.45832	−0.729160	−	0.684343i	$-0.760089\pi$
$521$	−33.2868	−1.45832	−0.729160	+	0.684343i	$0.760089\pi$
$522$	0	0
$523$	−30.5185	−1.33448	−0.667239	−	0.744843i	$-0.732524\pi$
$523$	−30.5185	−1.33448	−0.667239	+	0.744843i	$0.732524\pi$
$524$	0	0
$525$	4.80164	0.209561
$526$	0	0
$527$	13.3426	0.581211
$528$	0	0
$529$	−19.7821	−0.860092
$530$	0	0
$531$	−26.4068	−1.14596
$532$	0	0
$533$	5.16604	0.223766
$534$	0	0
$535$	21.1923	0.916225
$536$	0	0
$537$	26.3007	1.13496
$538$	0	0
$539$	−55.1227	−2.37430
$540$	0	0
$541$	2.49043	0.107072	0.0535360	−	0.998566i	$-0.482951\pi$
$541$	2.49043	0.107072	0.0535360	+	0.998566i	$0.482951\pi$
$542$	0	0
$543$	46.2444	1.98454
$544$	0	0
$545$	−18.8479	−0.807356
$546$	0	0
$547$	14.1882	0.606644	0.303322	−	0.952888i	$-0.401904\pi$
$547$	14.1882	0.606644	0.303322	+	0.952888i	$0.401904\pi$
$548$	0	0
$549$	25.7709	1.09988
$550$	0	0
$551$	0	0
$552$	0	0
$553$	−23.2847	−0.990164
$554$	0	0
$555$	36.2435	1.53845
$556$	0	0
$557$	6.40556	0.271412	0.135706	−	0.990749i	$-0.456670\pi$
$557$	6.40556	0.271412	0.135706	+	0.990749i	$0.456670\pi$
$558$	0	0
$559$	−4.87404	−0.206150
$560$	0	0
$561$	70.2853	2.96745
$562$	0	0
$563$	28.3494	1.19479	0.597393	−	0.801949i	$-0.296203\pi$
$563$	28.3494	1.19479	0.597393	+	0.801949i	$0.296203\pi$
$564$	0	0
$565$	9.85244	0.414495
$566$	0	0
$567$	41.9185	1.76041
$568$	0	0
$569$	−16.4200	−0.688362	−0.344181	−	0.938903i	$-0.611843\pi$
$569$	−16.4200	−0.688362	−0.344181	+	0.938903i	$0.611843\pi$
$570$	0	0
$571$	36.9283	1.54540	0.772701	−	0.634771i	$-0.218905\pi$
$571$	36.9283	1.54540	0.772701	+	0.634771i	$0.218905\pi$
$572$	0	0
$573$	−33.0700	−1.38152
$574$	0	0
$575$	−0.877544	−0.0365961
$576$	0	0
$577$	41.1638	1.71367	0.856835	−	0.515591i	$-0.172428\pi$
$577$	41.1638	1.71367	0.856835	+	0.515591i	$0.172428\pi$
$578$	0	0
$579$	0.308857	0.0128356
$580$	0	0
$581$	−30.3878	−1.26070
$582$	0	0
$583$	33.9754	1.40712
$584$	0	0
$585$	17.8505	0.738026
$586$	0	0
$587$	−7.31112	−0.301762	−0.150881	−	0.988552i	$-0.548211\pi$
$587$	−7.31112	−0.301762	−0.150881	+	0.988552i	$0.548211\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	28.5897	1.17602
$592$	0	0
$593$	0.185394	0.00761323	0.00380662	−	0.999993i	$-0.498788\pi$
$593$	0.185394	0.00761323	0.00380662	+	0.999993i	$0.498788\pi$
$594$	0	0
$595$	53.9827	2.21307
$596$	0	0
$597$	27.6037	1.12974
$598$	0	0
$599$	−23.3882	−0.955617	−0.477809	−	0.878464i	$-0.658569\pi$
$599$	−23.3882	−0.955617	−0.477809	+	0.878464i	$0.658569\pi$
$600$	0	0
$601$	26.6578	1.08739	0.543697	−	0.839282i	$-0.317024\pi$
$601$	26.6578	1.08739	0.543697	+	0.839282i	$0.317024\pi$
$602$	0	0
$603$	−23.0750	−0.939687
$604$	0	0
$605$	−41.7219	−1.69623
$606$	0	0
$607$	−20.2286	−0.821052	−0.410526	−	0.911849i	$-0.634655\pi$
$607$	−20.2286	−0.821052	−0.410526	+	0.911849i	$0.634655\pi$
$608$	0	0
$609$	−2.11619	−0.0857523
$610$	0	0
$611$	−20.2548	−0.819422
$612$	0	0
$613$	24.2426	0.979151	0.489576	−	0.871961i	$-0.337152\pi$
$613$	24.2426	0.979151	0.489576	+	0.871961i	$0.337152\pi$
$614$	0	0
$615$	−9.67468	−0.390121
$616$	0	0
$617$	−22.1691	−0.892493	−0.446246	−	0.894910i	$-0.647239\pi$
$617$	−22.1691	−0.892493	−0.446246	+	0.894910i	$0.647239\pi$
$618$	0	0
$619$	−10.6658	−0.428697	−0.214348	−	0.976757i	$-0.568763\pi$
$619$	−10.6658	−0.428697	−0.214348	+	0.976757i	$0.568763\pi$
$620$	0	0
$621$	−1.78596	−0.0716683
$622$	0	0
$623$	−22.5495	−0.903425
$624$	0	0
$625$	−27.2067	−1.08827
$626$	0	0
$627$	0	0
$628$	0	0
$629$	36.3137	1.44792
$630$	0	0
$631$	−39.0376	−1.55406	−0.777031	−	0.629463i	$-0.783275\pi$
$631$	−39.0376	−1.55406	−0.777031	+	0.629463i	$0.783275\pi$
$632$	0	0
$633$	−3.89147	−0.154672
$634$	0	0
$635$	12.4307	0.493298
$636$	0	0
$637$	−30.3465	−1.20237
$638$	0	0
$639$	27.7017	1.09586
$640$	0	0
$641$	15.8845	0.627401	0.313701	−	0.949522i	$-0.398431\pi$
$641$	15.8845	0.627401	0.313701	+	0.949522i	$0.398431\pi$
$642$	0	0
$643$	26.3254	1.03817	0.519086	−	0.854722i	$-0.326273\pi$
$643$	26.3254	1.03817	0.519086	+	0.854722i	$0.326273\pi$
$644$	0	0
$645$	9.12783	0.359408
$646$	0	0
$647$	−27.2044	−1.06952	−0.534758	−	0.845005i	$-0.679597\pi$
$647$	−27.2044	−1.06952	−0.534758	+	0.845005i	$0.679597\pi$
$648$	0	0
$649$	54.9678	2.15767
$650$	0	0
$651$	23.6207	0.925768
$652$	0	0
$653$	30.4067	1.18990	0.594952	−	0.803761i	$-0.297171\pi$
$653$	30.4067	1.18990	0.594952	+	0.803761i	$0.297171\pi$
$654$	0	0
$655$	−5.78184	−0.225915
$656$	0	0
$657$	32.1539	1.25444
$658$	0	0
$659$	−15.1628	−0.590659	−0.295330	−	0.955395i	$-0.595429\pi$
$659$	−15.1628	−0.590659	−0.295330	+	0.955395i	$0.595429\pi$
$660$	0	0
$661$	−9.40611	−0.365855	−0.182928	−	0.983126i	$-0.558557\pi$
$661$	−9.40611	−0.365855	−0.182928	+	0.983126i	$0.558557\pi$
$662$	0	0
$663$	38.6939	1.50275
$664$	0	0
$665$	0	0
$666$	0	0
$667$	0.386753	0.0149751
$668$	0	0
$669$	−43.5911	−1.68533
$670$	0	0
$671$	−53.6443	−2.07091
$672$	0	0
$673$	41.6071	1.60383	0.801917	−	0.597435i	$-0.203814\pi$
$673$	41.6071	1.60383	0.801917	+	0.597435i	$0.203814\pi$
$674$	0	0
$675$	0.487048	0.0187465
$676$	0	0
$677$	−32.9902	−1.26792	−0.633959	−	0.773367i	$-0.718571\pi$
$677$	−32.9902	−1.26792	−0.633959	+	0.773367i	$0.718571\pi$
$678$	0	0
$679$	74.1572	2.84589
$680$	0	0
$681$	7.47686	0.286514
$682$	0	0
$683$	42.6980	1.63379	0.816896	−	0.576785i	$-0.195693\pi$
$683$	42.6980	1.63379	0.816896	+	0.576785i	$0.195693\pi$
$684$	0	0
$685$	−2.67093	−0.102051
$686$	0	0
$687$	30.3216	1.15684
$688$	0	0
$689$	18.7044	0.712581
$690$	0	0
$691$	34.6814	1.31934	0.659671	−	0.751555i	$-0.270696\pi$
$691$	34.6814	1.31934	0.659671	+	0.751555i	$0.270696\pi$
$692$	0	0
$693$	57.5128	2.18473
$694$	0	0
$695$	39.2934	1.49049
$696$	0	0
$697$	−9.69342	−0.367164
$698$	0	0
$699$	−13.6583	−0.516603
$700$	0	0
$701$	−12.8129	−0.483936	−0.241968	−	0.970284i	$-0.577793\pi$
$701$	−12.8129	−0.483936	−0.241968	+	0.970284i	$0.577793\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	37.9321	1.42861
$706$	0	0
$707$	42.0587	1.58178
$708$	0	0
$709$	−28.8302	−1.08274	−0.541370	−	0.840784i	$-0.682094\pi$
$709$	−28.8302	−1.08274	−0.541370	+	0.840784i	$0.682094\pi$
$710$	0	0
$711$	14.4472	0.541812
$712$	0	0
$713$	−4.31690	−0.161669
$714$	0	0
$715$	−37.1571	−1.38960
$716$	0	0
$717$	−30.8289	−1.15133
$718$	0	0
$719$	2.29122	0.0854480	0.0427240	−	0.999087i	$-0.486396\pi$
$719$	2.29122	0.0854480	0.0427240	+	0.999087i	$0.486396\pi$
$720$	0	0
$721$	19.6545	0.731972
$722$	0	0
$723$	−25.6283	−0.953126
$724$	0	0
$725$	−0.105471	−0.00391710
$726$	0	0
$727$	39.1971	1.45374	0.726870	−	0.686775i	$-0.240974\pi$
$727$	39.1971	1.45374	0.726870	+	0.686775i	$0.240974\pi$
$728$	0	0
$729$	−18.9543	−0.702010
$730$	0	0
$731$	9.14551	0.338259
$732$	0	0
$733$	−17.5243	−0.647275	−0.323638	−	0.946181i	$-0.604906\pi$
$733$	−17.5243	−0.647275	−0.323638	+	0.946181i	$0.604906\pi$
$734$	0	0
$735$	56.8313	2.09626
$736$	0	0
$737$	48.0325	1.76930
$738$	0	0
$739$	−33.6463	−1.23770	−0.618850	−	0.785509i	$-0.712401\pi$
$739$	−33.6463	−1.23770	−0.618850	+	0.785509i	$0.712401\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−6.26736	−0.229927	−0.114964	−	0.993370i	$-0.536675\pi$
$743$	−6.26736	−0.229927	−0.114964	+	0.993370i	$0.536675\pi$
$744$	0	0
$745$	−15.0937	−0.552992
$746$	0	0
$747$	18.8544	0.689848
$748$	0	0
$749$	37.5900	1.37351
$750$	0	0
$751$	3.50442	0.127878	0.0639391	−	0.997954i	$-0.479634\pi$
$751$	3.50442	0.127878	0.0639391	+	0.997954i	$0.479634\pi$
$752$	0	0
$753$	−26.5691	−0.968232
$754$	0	0
$755$	−21.4553	−0.780837
$756$	0	0
$757$	18.1851	0.660948	0.330474	−	0.943815i	$-0.392791\pi$
$757$	18.1851	0.660948	0.330474	+	0.943815i	$0.392791\pi$
$758$	0	0
$759$	−22.7403	−0.825422
$760$	0	0
$761$	−30.9777	−1.12294	−0.561469	−	0.827497i	$-0.689764\pi$
$761$	−30.9777	−1.12294	−0.561469	+	0.827497i	$0.689764\pi$
$762$	0	0
$763$	−33.4316	−1.21030
$764$	0	0
$765$	−33.4941	−1.21098
$766$	0	0
$767$	30.2612	1.09267
$768$	0	0
$769$	44.0854	1.58976	0.794880	−	0.606767i	$-0.207534\pi$
$769$	44.0854	1.58976	0.794880	+	0.606767i	$0.207534\pi$
$770$	0	0
$771$	48.3581	1.74157
$772$	0	0
$773$	7.31289	0.263027	0.131513	−	0.991314i	$-0.458016\pi$
$773$	7.31289	0.263027	0.131513	+	0.991314i	$0.458016\pi$
$774$	0	0
$775$	1.17726	0.0422883
$776$	0	0
$777$	64.2870	2.30628
$778$	0	0
$779$	0	0
$780$	0	0
$781$	−57.6634	−2.06336
$782$	0	0
$783$	−0.214653	−0.00767108
$784$	0	0
$785$	−42.5072	−1.51715
$786$	0	0
$787$	38.8126	1.38352	0.691760	−	0.722128i	$-0.256836\pi$
$787$	38.8126	1.38352	0.691760	+	0.722128i	$0.256836\pi$
$788$	0	0
$789$	−65.8774	−2.34530
$790$	0	0
$791$	17.4758	0.621368
$792$	0	0
$793$	−29.5326	−1.04873
$794$	0	0
$795$	−35.0286	−1.24233
$796$	0	0
$797$	−29.8075	−1.05583	−0.527917	−	0.849296i	$-0.677027\pi$
$797$	−29.8075	−1.05583	−0.527917	+	0.849296i	$0.677027\pi$
$798$	0	0
$799$	38.0056	1.34454
$800$	0	0
$801$	13.9910	0.494349
$802$	0	0
$803$	−66.9308	−2.36194
$804$	0	0
$805$	−17.4657	−0.615586
$806$	0	0
$807$	12.3670	0.435338
$808$	0	0
$809$	−33.8617	−1.19051	−0.595257	−	0.803536i	$-0.702950\pi$
$809$	−33.8617	−1.19051	−0.595257	+	0.803536i	$0.702950\pi$
$810$	0	0
$811$	−3.32445	−0.116737	−0.0583686	−	0.998295i	$-0.518590\pi$
$811$	−3.32445	−0.116737	−0.0583686	+	0.998295i	$0.518590\pi$
$812$	0	0
$813$	12.7350	0.446635
$814$	0	0
$815$	−18.1601	−0.636121
$816$	0	0
$817$	0	0
$818$	0	0
$819$	31.6623	1.10637
$820$	0	0
$821$	55.2790	1.92925	0.964624	−	0.263628i	$-0.0849191\pi$
$821$	55.2790	1.92925	0.964624	+	0.263628i	$0.0849191\pi$
$822$	0	0
$823$	38.2602	1.33367	0.666833	−	0.745207i	$-0.267649\pi$
$823$	38.2602	1.33367	0.666833	+	0.745207i	$0.267649\pi$
$824$	0	0
$825$	6.20149	0.215908
$826$	0	0
$827$	−0.615969	−0.0214194	−0.0107097	−	0.999943i	$-0.503409\pi$
$827$	−0.615969	−0.0214194	−0.0107097	+	0.999943i	$0.503409\pi$
$828$	0	0
$829$	0.302667	0.0105121	0.00525604	−	0.999986i	$-0.498327\pi$
$829$	0.302667	0.0105121	0.00525604	+	0.999986i	$0.498327\pi$
$830$	0	0
$831$	−60.2049	−2.08848
$832$	0	0
$833$	56.9414	1.97290
$834$	0	0
$835$	−11.0254	−0.381550
$836$	0	0
$837$	2.39594	0.0828157
$838$	0	0
$839$	−19.3281	−0.667279	−0.333640	−	0.942701i	$-0.608277\pi$
$839$	−19.3281	−0.667279	−0.333640	+	0.942701i	$0.608277\pi$
$840$	0	0
$841$	−28.9535	−0.998397
$842$	0	0
$843$	43.4767	1.49742
$844$	0	0
$845$	10.0017	0.344070
$846$	0	0
$847$	−74.0043	−2.54282
$848$	0	0
$849$	46.7055	1.60293
$850$	0	0
$851$	−11.7490	−0.402752
$852$	0	0
$853$	14.1179	0.483386	0.241693	−	0.970353i	$-0.422297\pi$
$853$	14.1179	0.483386	0.241693	+	0.970353i	$0.422297\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−17.4528	−0.596175	−0.298088	−	0.954539i	$-0.596349\pi$
$857$	−17.4528	−0.596175	−0.298088	+	0.954539i	$0.596349\pi$
$858$	0	0
$859$	−3.11876	−0.106411	−0.0532054	−	0.998584i	$-0.516944\pi$
$859$	−3.11876	−0.106411	−0.0532054	+	0.998584i	$0.516944\pi$
$860$	0	0
$861$	−17.1605	−0.584828
$862$	0	0
$863$	53.2436	1.81243	0.906217	−	0.422812i	$-0.138957\pi$
$863$	53.2436	1.81243	0.906217	+	0.422812i	$0.138957\pi$
$864$	0	0
$865$	−50.4387	−1.71497
$866$	0	0
$867$	−32.4523	−1.10214
$868$	0	0
$869$	−30.0730	−1.02016
$870$	0	0
$871$	26.4432	0.895992
$872$	0	0
$873$	−46.0116	−1.55726
$874$	0	0
$875$	−43.9194	−1.48475
$876$	0	0
$877$	28.7134	0.969582	0.484791	−	0.874630i	$-0.338896\pi$
$877$	28.7134	0.969582	0.484791	+	0.874630i	$0.338896\pi$
$878$	0	0
$879$	4.00624	0.135127
$880$	0	0
$881$	6.65371	0.224169	0.112085	−	0.993699i	$-0.464247\pi$
$881$	6.65371	0.224169	0.112085	+	0.993699i	$0.464247\pi$
$882$	0	0
$883$	13.1273	0.441770	0.220885	−	0.975300i	$-0.429105\pi$
$883$	13.1273	0.441770	0.220885	+	0.975300i	$0.429105\pi$
$884$	0	0
$885$	−56.6716	−1.90499
$886$	0	0
$887$	−10.5914	−0.355625	−0.177812	−	0.984064i	$-0.556902\pi$
$887$	−10.5914	−0.355625	−0.177812	+	0.984064i	$0.556902\pi$
$888$	0	0
$889$	22.0490	0.739501
$890$	0	0
$891$	54.1393	1.81374
$892$	0	0
$893$	0	0
$894$	0	0
$895$	26.0894	0.872072
$896$	0	0
$897$	−12.5192	−0.418003
$898$	0	0
$899$	−0.518844	−0.0173044
$900$	0	0
$901$	−35.0964	−1.16923
$902$	0	0
$903$	16.1905	0.538787
$904$	0	0
$905$	45.8729	1.52487
$906$	0	0
$907$	37.0777	1.23114	0.615572	−	0.788081i	$-0.288925\pi$
$907$	37.0777	1.23114	0.615572	+	0.788081i	$0.288925\pi$
$908$	0	0
$909$	−26.0958	−0.865542
$910$	0	0
$911$	29.8600	0.989307	0.494653	−	0.869090i	$-0.335295\pi$
$911$	29.8600	0.989307	0.494653	+	0.869090i	$0.335295\pi$
$912$	0	0
$913$	−39.2470	−1.29889
$914$	0	0
$915$	55.3071	1.82840
$916$	0	0
$917$	−10.2556	−0.338669
$918$	0	0
$919$	29.2234	0.963991	0.481996	−	0.876174i	$-0.339912\pi$
$919$	29.2234	0.963991	0.481996	+	0.876174i	$0.339912\pi$
$920$	0	0
$921$	−53.4155	−1.76010
$922$	0	0
$923$	−31.7452	−1.04491
$924$	0	0
$925$	3.20407	0.105349
$926$	0	0
$927$	−12.1948	−0.400531
$928$	0	0
$929$	10.0629	0.330152	0.165076	−	0.986281i	$-0.447213\pi$
$929$	10.0629	0.330152	0.165076	+	0.986281i	$0.447213\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	3.06720	0.100416
$934$	0	0
$935$	69.7207	2.28011
$936$	0	0
$937$	−7.11210	−0.232342	−0.116171	−	0.993229i	$-0.537062\pi$
$937$	−7.11210	−0.232342	−0.116171	+	0.993229i	$0.537062\pi$
$938$	0	0
$939$	19.0939	0.623104
$940$	0	0
$941$	32.6177	1.06331	0.531653	−	0.846962i	$-0.321571\pi$
$941$	32.6177	1.06331	0.531653	+	0.846962i	$0.321571\pi$
$942$	0	0
$943$	3.13624	0.102130
$944$	0	0
$945$	9.69372	0.315337
$946$	0	0
$947$	2.09984	0.0682357	0.0341178	−	0.999418i	$-0.489138\pi$
$947$	2.09984	0.0682357	0.0341178	+	0.999418i	$0.489138\pi$
$948$	0	0
$949$	−36.8472	−1.19611
$950$	0	0
$951$	16.8817	0.547428
$952$	0	0
$953$	16.0496	0.519897	0.259948	−	0.965623i	$-0.416294\pi$
$953$	16.0496	0.519897	0.259948	+	0.965623i	$0.416294\pi$
$954$	0	0
$955$	−32.8043	−1.06152
$956$	0	0
$957$	−2.73314	−0.0883498
$958$	0	0
$959$	−4.73756	−0.152984
$960$	0	0
$961$	−25.2087	−0.813185
$962$	0	0
$963$	−23.3231	−0.751577
$964$	0	0
$965$	0.306375	0.00986257
$966$	0	0
$967$	29.8722	0.960626	0.480313	−	0.877097i	$-0.340523\pi$
$967$	29.8722	0.960626	0.480313	+	0.877097i	$0.340523\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	19.5209	0.626455	0.313227	−	0.949678i	$-0.398590\pi$
$971$	19.5209	0.626455	0.313227	+	0.949678i	$0.398590\pi$
$972$	0	0
$973$	69.6969	2.23438
$974$	0	0
$975$	3.41409	0.109338
$976$	0	0
$977$	−54.1172	−1.73136	−0.865681	−	0.500596i	$-0.833114\pi$
$977$	−54.1172	−1.73136	−0.865681	+	0.500596i	$0.833114\pi$
$978$	0	0
$979$	−29.1235	−0.930790
$980$	0	0
$981$	20.7430	0.662272
$982$	0	0
$983$	26.1274	0.833335	0.416668	−	0.909059i	$-0.363198\pi$
$983$	26.1274	0.833335	0.416668	+	0.909059i	$0.363198\pi$
$984$	0	0
$985$	28.3600	0.903626
$986$	0	0
$987$	67.2822	2.14162
$988$	0	0
$989$	−2.95897	−0.0940897
$990$	0	0
$991$	−24.5391	−0.779509	−0.389754	−	0.920919i	$-0.627440\pi$
$991$	−24.5391	−0.779509	−0.389754	+	0.920919i	$0.627440\pi$
$992$	0	0
$993$	64.3951	2.04352
$994$	0	0
$995$	27.3819	0.868065
$996$	0	0
$997$	11.3393	0.359118	0.179559	−	0.983747i	$-0.442533\pi$
$997$	11.3393	0.359118	0.179559	+	0.983747i	$0.442533\pi$
$998$	0	0
$999$	6.52087	0.206311

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5776.2.a.cd.1.3		9
4.3	odd	2		2888.2.a.y.1.7		9
19.6	even	9		304.2.u.f.17.1		18
19.16	even	9		304.2.u.f.161.1		18
19.18	odd	2		5776.2.a.ce.1.7		9
76.35	odd	18		152.2.q.c.9.3	✓	18
76.63	odd	18		152.2.q.c.17.3	yes	18
76.75	even	2		2888.2.a.x.1.3		9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
152.2.q.c.9.3	✓	18	76.35	odd	18
152.2.q.c.17.3	yes	18	76.63	odd	18
304.2.u.f.17.1		18	19.6	even	9
304.2.u.f.161.1		18	19.16	even	9
2888.2.a.x.1.3		9	76.75	even	2
2888.2.a.y.1.7		9	4.3	odd	2
5776.2.a.cd.1.3		9	1.1	even	1	trivial
5776.2.a.ce.1.7		9	19.18	odd	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 5776 = 2^{4} \cdot 19^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5776.a (trivial)

\(f(q)\)	\(=\)	\(q-2.36188 q^{3} -2.34290 q^{5} -4.15573 q^{7} +2.57847 q^{9} -5.36729 q^{11} -2.95483 q^{13} +5.53365 q^{15} +5.54437 q^{17} +9.81534 q^{21} -1.79384 q^{23} +0.489197 q^{25} +0.995607 q^{27} -0.215600 q^{29} +2.40651 q^{31} +12.6769 q^{33} +9.73649 q^{35} +6.54964 q^{37} +6.97896 q^{39} -1.74834 q^{41} +1.64951 q^{43} -6.04110 q^{45} +6.85481 q^{47} +10.2701 q^{49} -13.0951 q^{51} -6.33010 q^{53} +12.5750 q^{55} -10.2413 q^{59} +9.99468 q^{61} -10.7154 q^{63} +6.92289 q^{65} -8.94912 q^{67} +4.23684 q^{69} +10.7435 q^{71} +12.4701 q^{73} -1.15542 q^{75} +22.3050 q^{77} +5.60302 q^{79} -10.0869 q^{81} +7.31227 q^{83} -12.9899 q^{85} +0.509222 q^{87} +5.42611 q^{89} +12.2795 q^{91} -5.68388 q^{93} -17.8445 q^{97} -13.8394 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q - 3 q^{3} + 3 q^{5} - 9 q^{7} + 6 q^{9} - 3 q^{11} - 6 q^{13} - 3 q^{17} + 15 q^{21} - 24 q^{23} + 30 q^{25} - 12 q^{27} - 15 q^{29} - 6 q^{31} - 18 q^{33} - 15 q^{35} + 24 q^{37} - 6 q^{39} - 12 q^{41}+ \cdots - 48 q^{99}+O(q^{100}) \) 9 * q - 3 * q^3 + 3 * q^5 - 9 * q^7 + 6 * q^9 - 3 * q^11 - 6 * q^13 - 3 * q^17 + 15 * q^21 - 24 * q^23 + 30 * q^25 - 12 * q^27 - 15 * q^29 - 6 * q^31 - 18 * q^33 - 15 * q^35 + 24 * q^37 - 6 * q^39 - 12 * q^41 - 9 * q^43 + 42 * q^45 + 12 * q^47 + 18 * q^49 - 12 * q^51 - 18 * q^53 - 21 * q^55 - 57 * q^59 + 15 * q^61 - 30 * q^63 - 27 * q^65 - 6 * q^67 + 3 * q^69 + 36 * q^73 - 45 * q^75 + 30 * q^77 + 3 * q^79 - 15 * q^81 - 27 * q^83 + 18 * q^85 + 18 * q^87 - 30 * q^89 + 21 * q^91 - 24 * q^93 + 3 * q^97 - 48 * q^99

Introduction

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