LMFDB - Embedded newform 775.2.a.j.1.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [775,2,Mod(1,775)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("775.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(775, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 775 = 5^{2} \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	775.a (trivial)

Newform invariants

comment:select newform

sage:traces = [5,4,1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

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

Self dual:	yes
Analytic conductor:	$6.18840615665$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	5.5.205225.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 6x^{3} + 3x^{2} + 7x - 3 $ x^5 - x^4 - 6x^3 + 3x^2 + 7*x - 3
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-1.35347$ of defining polynomial
Character	$\chi$	$=$	775.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+2.35347 q^{2} +1.91723 q^{3} +3.53883 q^{4} +4.51216 q^{6} -0.845802 q^{7} +3.62160 q^{8} +0.675783 q^{9} -1.11393 q^{11} +6.78477 q^{12} +2.84580 q^{13} -1.99057 q^{14} +1.44567 q^{16} -3.11651 q^{17} +1.59044 q^{18} +4.69752 q^{19} -1.62160 q^{21} -2.62160 q^{22} +4.28204 q^{23} +6.94345 q^{24} +6.69752 q^{26} -4.45607 q^{27} -2.99315 q^{28} -2.93448 q^{29} +1.00000 q^{31} -3.84086 q^{32} -2.13566 q^{33} -7.33461 q^{34} +2.39148 q^{36} -11.4123 q^{37} +11.0555 q^{38} +5.45607 q^{39} -0.658536 q^{41} -3.81639 q^{42} -7.69784 q^{43} -3.94200 q^{44} +10.0777 q^{46} +6.10640 q^{47} +2.77168 q^{48} -6.28462 q^{49} -5.97507 q^{51} +10.0708 q^{52} +12.5433 q^{53} -10.4872 q^{54} -3.06315 q^{56} +9.00623 q^{57} -6.90622 q^{58} -11.2928 q^{59} -5.35748 q^{61} +2.35347 q^{62} -0.571579 q^{63} -11.9307 q^{64} -5.02622 q^{66} -0.847229 q^{67} -11.0288 q^{68} +8.20967 q^{69} +1.92315 q^{71} +2.44742 q^{72} +6.95030 q^{73} -26.8585 q^{74} +16.6237 q^{76} +0.942162 q^{77} +12.8407 q^{78} +3.73152 q^{79} -10.5707 q^{81} -1.54985 q^{82} +7.38112 q^{83} -5.73857 q^{84} -18.1166 q^{86} -5.62608 q^{87} -4.03420 q^{88} +9.22291 q^{89} -2.40698 q^{91} +15.1534 q^{92} +1.91723 q^{93} +14.3713 q^{94} -7.36382 q^{96} +16.9797 q^{97} -14.7907 q^{98} -0.752774 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 4 q^{2} + q^{3} + 6 q^{4} - q^{6} + 6 q^{7} + 15 q^{8} + 2 q^{9} - 11 q^{12} + 4 q^{13} + 2 q^{14} + 20 q^{16} + 11 q^{17} + 19 q^{18} - 4 q^{19} - 5 q^{21} - 10 q^{22} + 12 q^{23} - 26 q^{24} + 6 q^{26}+ \cdots + 5 q^{99}+O(q^{100}) $ 5 * q + 4 * q^2 + q^3 + 6 * q^4 - q^6 + 6 * q^7 + 15 * q^8 + 2 * q^9 - 11 * q^12 + 4 * q^13 + 2 * q^14 + 20 * q^16 + 11 * q^17 + 19 * q^18 - 4 * q^19 - 5 * q^21 - 10 * q^22 + 12 * q^23 - 26 * q^24 + 6 * q^26 - 2 * q^27 + 18 * q^28 - 6 * q^29 + 5 * q^31 + 29 * q^32 + 21 * q^33 - 5 * q^34 + 23 * q^36 - 2 * q^37 - 6 * q^38 + 7 * q^39 - 2 * q^41 - 24 * q^42 + 7 * q^43 - 28 * q^44 + 27 * q^46 + 8 * q^47 - 39 * q^48 - q^49 - 19 * q^51 - 6 * q^52 + 25 * q^53 - 18 * q^54 + 35 * q^56 + 20 * q^57 - q^58 + 4 * q^59 - 17 * q^61 + 4 * q^62 + 10 * q^63 + 27 * q^64 + 27 * q^66 - 13 * q^67 + 18 * q^68 - 10 * q^69 - 6 * q^71 + 26 * q^72 + 7 * q^73 + 6 * q^74 - 5 * q^76 + 7 * q^77 + 22 * q^78 + 12 * q^79 - 11 * q^81 - 21 * q^82 - 4 * q^83 - 63 * q^84 - 41 * q^86 + q^87 - 49 * q^88 - 3 * q^89 - 22 * q^91 + 34 * q^92 + q^93 - 34 * q^94 - 64 * q^96 + 25 * q^97 - 22 * q^98 + 5 * 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 $

$2$	2.35347	1.66416	0.832078	−	0.554658i	$-0.187151\pi$
$2$	2.35347	1.66416	0.832078	+	0.554658i	$0.187151\pi$
$3$	1.91723	1.10692	0.553458	−	0.832877i	$-0.313308\pi$
$3$	1.91723	1.10692	0.553458	+	0.832877i	$0.313308\pi$
$4$	3.53883	1.76942
$5$	0	0
$5$	0	0
$6$	4.51216	1.84208
$7$	−0.845802	−0.319683	−0.159841	−	0.987143i	$-0.551098\pi$
$7$	−0.845802	−0.319683	−0.159841	+	0.987143i	$0.551098\pi$
$8$	3.62160	1.28043
$9$	0.675783	0.225261
$10$	0	0
$11$	−1.11393	−0.335862	−0.167931	−	0.985799i	$-0.553709\pi$
$11$	−1.11393	−0.335862	−0.167931	+	0.985799i	$0.553709\pi$
$12$	6.78477	1.95859
$13$	2.84580	0.789283	0.394642	−	0.918835i	$-0.370869\pi$
$13$	2.84580	0.789283	0.394642	+	0.918835i	$0.370869\pi$
$14$	−1.99057	−0.532002
$15$	0	0
$16$	1.44567	0.361417
$17$	−3.11651	−0.755864	−0.377932	−	0.925833i	$-0.623365\pi$
$17$	−3.11651	−0.755864	−0.377932	+	0.925833i	$0.623365\pi$
$18$	1.59044	0.374870
$19$	4.69752	1.07768	0.538842	−	0.842407i	$-0.318862\pi$
$19$	4.69752	1.07768	0.538842	+	0.842407i	$0.318862\pi$
$20$	0	0
$21$	−1.62160	−0.353862
$22$	−2.62160	−0.558927
$23$	4.28204	0.892867	0.446434	−	0.894817i	$-0.352694\pi$
$23$	4.28204	0.892867	0.446434	+	0.894817i	$0.352694\pi$
$24$	6.94345	1.41733
$25$	0	0
$26$	6.69752	1.31349
$27$	−4.45607	−0.857570
$28$	−2.99315	−0.565652
$29$	−2.93448	−0.544919	−0.272460	−	0.962167i	$-0.587837\pi$
$29$	−2.93448	−0.544919	−0.272460	+	0.962167i	$0.587837\pi$
$30$	0	0
$31$	1.00000	0.179605
$31$	1.00000	0.179605
$32$	−3.84086	−0.678974
$33$	−2.13566	−0.371771
$34$	−7.33461	−1.25788
$35$	0	0
$36$	2.39148	0.398581
$37$	−11.4123	−1.87617	−0.938083	−	0.346409i	$-0.887401\pi$
$37$	−11.4123	−1.87617	−0.938083	+	0.346409i	$0.887401\pi$
$38$	11.0555	1.79343
$39$	5.45607	0.873670
$40$	0	0
$41$	−0.658536	−0.102846	−0.0514230	−	0.998677i	$-0.516376\pi$
$41$	−0.658536	−0.102846	−0.0514230	+	0.998677i	$0.516376\pi$
$42$	−3.81639	−0.588881
$43$	−7.69784	−1.17391	−0.586954	−	0.809620i	$-0.699673\pi$
$43$	−7.69784	−1.17391	−0.586954	+	0.809620i	$0.699673\pi$
$44$	−3.94200	−0.594280
$45$	0	0
$46$	10.0777	1.48587
$47$	6.10640	0.890711	0.445355	−	0.895354i	$-0.353077\pi$
$47$	6.10640	0.890711	0.445355	+	0.895354i	$0.353077\pi$
$48$	2.77168	0.400058
$49$	−6.28462	−0.897803
$50$	0	0
$51$	−5.97507	−0.836677
$52$	10.0708	1.39657
$53$	12.5433	1.72296	0.861479	−	0.507794i	$-0.169539\pi$
$53$	12.5433	1.72296	0.861479	+	0.507794i	$0.169539\pi$
$54$	−10.4872	−1.42713
$55$	0	0
$56$	−3.06315	−0.409331
$57$	9.00623	1.19290
$58$	−6.90622	−0.906831
$59$	−11.2928	−1.47019	−0.735096	−	0.677963i	$-0.762863\pi$
$59$	−11.2928	−1.47019	−0.735096	+	0.677963i	$0.762863\pi$
$60$	0	0
$61$	−5.35748	−0.685955	−0.342977	−	0.939344i	$-0.611435\pi$
$61$	−5.35748	−0.685955	−0.342977	+	0.939344i	$0.611435\pi$
$62$	2.35347	0.298891
$63$	−0.571579	−0.0720121
$64$	−11.9307	−1.49134
$65$	0	0
$66$	−5.02622	−0.618684
$67$	−0.847229	−0.103506	−0.0517528	−	0.998660i	$-0.516481\pi$
$67$	−0.847229	−0.103506	−0.0517528	+	0.998660i	$0.516481\pi$
$68$	−11.0288	−1.33744
$69$	8.20967	0.988328
$70$	0	0
$71$	1.92315	0.228235	0.114118	−	0.993467i	$-0.463596\pi$
$71$	1.92315	0.228235	0.114118	+	0.993467i	$0.463596\pi$
$72$	2.44742	0.288431
$73$	6.95030	0.813471	0.406736	−	0.913546i	$-0.366667\pi$
$73$	6.95030	0.813471	0.406736	+	0.913546i	$0.366667\pi$
$74$	−26.8585	−3.12223
$75$	0	0
$76$	16.6237	1.90687
$77$	0.942162	0.107369
$78$	12.8407	1.45392
$79$	3.73152	0.419829	0.209914	−	0.977720i	$-0.432681\pi$
$79$	3.73152	0.419829	0.209914	+	0.977720i	$0.432681\pi$
$80$	0	0
$81$	−10.5707	−1.17452
$82$	−1.54985	−0.171152
$83$	7.38112	0.810183	0.405091	−	0.914276i	$-0.367240\pi$
$83$	7.38112	0.810183	0.405091	+	0.914276i	$0.367240\pi$
$84$	−5.73857	−0.626129
$85$	0	0
$86$	−18.1166	−1.95357
$87$	−5.62608	−0.603179
$88$	−4.03420	−0.430047
$89$	9.22291	0.977627	0.488813	−	0.872388i	$-0.337430\pi$
$89$	9.22291	0.977627	0.488813	+	0.872388i	$0.337430\pi$
$90$	0	0
$91$	−2.40698	−0.252320
$92$	15.1534	1.57985
$93$	1.91723	0.198808
$94$	14.3713	1.48228
$95$	0	0
$96$	−7.36382	−0.751567
$97$	16.9797	1.72403	0.862013	−	0.506886i	$-0.169203\pi$
$97$	16.9797	1.72403	0.862013	+	0.506886i	$0.169203\pi$
$98$	−14.7907	−1.49408
$99$	−0.752774	−0.0756566
$100$	0	0
$101$	−6.13411	−0.610367	−0.305183	−	0.952294i	$-0.598718\pi$
$101$	−6.13411	−0.610367	−0.305183	+	0.952294i	$0.598718\pi$
$102$	−14.0622	−1.39236
$103$	−10.7904	−1.06321	−0.531604	−	0.846993i	$-0.678411\pi$
$103$	−10.7904	−1.06321	−0.531604	+	0.846993i	$0.678411\pi$
$104$	10.3064	1.01062
$105$	0	0
$106$	29.5203	2.86727
$107$	13.0938	1.26583	0.632915	−	0.774222i	$-0.281858\pi$
$107$	13.0938	1.26583	0.632915	+	0.774222i	$0.281858\pi$
$108$	−15.7693	−1.51740
$109$	3.30187	0.316261	0.158131	−	0.987418i	$-0.449453\pi$
$109$	3.30187	0.316261	0.158131	+	0.987418i	$0.449453\pi$
$110$	0	0
$111$	−21.8800	−2.07676
$112$	−1.22275	−0.115539
$113$	17.7370	1.66856	0.834279	−	0.551343i	$-0.185884\pi$
$113$	17.7370	1.66856	0.834279	+	0.551343i	$0.185884\pi$
$114$	21.1959	1.98518
$115$	0	0
$116$	−10.3846	−0.964189
$117$	1.92315	0.177795
$118$	−26.5772	−2.44663
$119$	2.63595	0.241637
$120$	0	0
$121$	−9.75916	−0.887197
$122$	−12.6087	−1.14154
$123$	−1.26257	−0.113842
$124$	3.53883	0.317797
$125$	0	0
$126$	−1.34519	−0.119839
$127$	15.5525	1.38006	0.690030	−	0.723781i	$-0.257597\pi$
$127$	15.5525	1.38006	0.690030	+	0.723781i	$0.257597\pi$
$128$	−20.3968	−1.80284
$129$	−14.7585	−1.29942
$130$	0	0
$131$	3.67449	0.321042	0.160521	−	0.987032i	$-0.448683\pi$
$131$	3.67449	0.321042	0.160521	+	0.987032i	$0.448683\pi$
$132$	−7.55774	−0.657817
$133$	−3.97317	−0.344517
$134$	−1.99393	−0.172249
$135$	0	0
$136$	−11.2867	−0.967830
$137$	0.609454	0.0520692	0.0260346	−	0.999661i	$-0.491712\pi$
$137$	0.609454	0.0520692	0.0260346	+	0.999661i	$0.491712\pi$
$138$	19.3212	1.64473
$139$	4.52126	0.383489	0.191744	−	0.981445i	$-0.438586\pi$
$139$	4.52126	0.383489	0.191744	+	0.981445i	$0.438586\pi$
$140$	0	0
$141$	11.7074	0.985941
$142$	4.52607	0.379819
$143$	−3.17002	−0.265090
$144$	0.976959	0.0814132
$145$	0	0
$146$	16.3573	1.35374
$147$	−12.0491	−0.993792
$148$	−40.3861	−3.31972
$149$	−0.407473	−0.0333815	−0.0166907	−	0.999861i	$-0.505313\pi$
$149$	−0.407473	−0.0333815	−0.0166907	+	0.999861i	$0.505313\pi$
$150$	0	0
$151$	−1.27003	−0.103354	−0.0516769	−	0.998664i	$-0.516457\pi$
$151$	−1.27003	−0.103354	−0.0516769	+	0.998664i	$0.516457\pi$
$152$	17.0125	1.37990
$153$	−2.10608	−0.170267
$154$	2.21735	0.178679
$155$	0	0
$156$	19.3081	1.54589
$157$	20.0229	1.59800	0.798999	−	0.601332i	$-0.205363\pi$
$157$	20.0229	1.59800	0.798999	+	0.601332i	$0.205363\pi$
$158$	8.78203	0.698661
$159$	24.0485	1.90717
$160$	0	0
$161$	−3.62176	−0.285434
$162$	−24.8778	−1.95458
$163$	−1.14858	−0.0899637	−0.0449819	−	0.998988i	$-0.514323\pi$
$163$	−1.14858	−0.0899637	−0.0449819	+	0.998988i	$0.514323\pi$
$164$	−2.33045	−0.181978
$165$	0	0
$166$	17.3713	1.34827
$167$	−10.8292	−0.837987	−0.418994	−	0.907989i	$-0.637617\pi$
$167$	−10.8292	−0.837987	−0.418994	+	0.907989i	$0.637617\pi$
$168$	−5.87278	−0.453095
$169$	−4.90141	−0.377032
$170$	0	0
$171$	3.17450	0.242760
$172$	−27.2413	−2.07713
$173$	−1.03183	−0.0784490	−0.0392245	−	0.999230i	$-0.512489\pi$
$173$	−1.03183	−0.0784490	−0.0392245	+	0.999230i	$0.512489\pi$
$174$	−13.2408	−1.00378
$175$	0	0
$176$	−1.61037	−0.121386
$177$	−21.6509	−1.62738
$178$	21.7059	1.62692
$179$	10.1389	0.757814	0.378907	−	0.925435i	$-0.376300\pi$
$179$	10.1389	0.757814	0.378907	+	0.925435i	$0.376300\pi$
$180$	0	0
$181$	7.01602	0.521496	0.260748	−	0.965407i	$-0.416031\pi$
$181$	7.01602	0.521496	0.260748	+	0.965407i	$0.416031\pi$
$182$	−5.66477	−0.419901
$183$	−10.2715	−0.759294
$184$	15.5078	1.14325
$185$	0	0
$186$	4.51216	0.330847
$187$	3.47156	0.253866
$188$	21.6095	1.57604
$189$	3.76895	0.274151
$190$	0	0
$191$	16.1486	1.16847	0.584236	−	0.811584i	$-0.301394\pi$
$191$	16.1486	1.16847	0.584236	+	0.811584i	$0.301394\pi$
$192$	−22.8739	−1.65078
$193$	11.5525	0.831566	0.415783	−	0.909464i	$-0.363508\pi$
$193$	11.5525	0.831566	0.415783	+	0.909464i	$0.363508\pi$
$194$	39.9612	2.86905
$195$	0	0
$196$	−22.2402	−1.58859
$197$	−23.1695	−1.65076	−0.825378	−	0.564581i	$-0.809038\pi$
$197$	−23.1695	−1.65076	−0.825378	+	0.564581i	$0.809038\pi$
$198$	−1.77163	−0.125904
$199$	−18.4173	−1.30557	−0.652783	−	0.757545i	$-0.726399\pi$
$199$	−18.4173	−1.30557	−0.652783	+	0.757545i	$0.726399\pi$
$200$	0	0
$201$	−1.62434	−0.114572
$202$	−14.4365	−1.01575
$203$	2.48199	0.174201
$204$	−21.1448	−1.48043
$205$	0	0
$206$	−25.3949	−1.76934
$207$	2.89373	0.201128
$208$	4.11409	0.285261
$209$	−5.23269	−0.361953
$210$	0	0
$211$	16.0087	1.10208	0.551041	−	0.834478i	$-0.314231\pi$
$211$	16.0087	1.10208	0.551041	+	0.834478i	$0.314231\pi$
$212$	44.3887	3.04863
$213$	3.68712	0.252637
$214$	30.8160	2.10654
$215$	0	0
$216$	−16.1381	−1.09806
$217$	−0.845802	−0.0574167
$218$	7.77085	0.526309
$219$	13.3253	0.900443
$220$	0	0
$221$	−8.86896	−0.596591
$222$	−51.4940	−3.45605
$223$	−21.4379	−1.43559	−0.717793	−	0.696256i	$-0.754848\pi$
$223$	−21.4379	−1.43559	−0.717793	+	0.696256i	$0.754848\pi$
$224$	3.24860	0.217056
$225$	0	0
$226$	41.7436	2.77674
$227$	24.5675	1.63060	0.815300	−	0.579039i	$-0.196572\pi$
$227$	24.5675	1.63060	0.815300	+	0.579039i	$0.196572\pi$
$228$	31.8715	2.11074
$229$	−28.1744	−1.86182	−0.930908	−	0.365253i	$-0.880983\pi$
$229$	−28.1744	−1.86182	−0.930908	+	0.365253i	$0.880983\pi$
$230$	0	0
$231$	1.80634	0.118849
$232$	−10.6275	−0.697730
$233$	22.2888	1.46019	0.730093	−	0.683347i	$-0.239477\pi$
$233$	22.2888	1.46019	0.730093	+	0.683347i	$0.239477\pi$
$234$	4.52607	0.295878
$235$	0	0
$236$	−39.9632	−2.60138
$237$	7.15419	0.464715
$238$	6.20363	0.402121
$239$	−22.5505	−1.45867	−0.729336	−	0.684156i	$-0.760171\pi$
$239$	−22.5505	−1.45867	−0.729336	+	0.684156i	$0.760171\pi$
$240$	0	0
$241$	−20.1781	−1.29978	−0.649892	−	0.760027i	$-0.725186\pi$
$241$	−20.1781	−1.29978	−0.649892	+	0.760027i	$0.725186\pi$
$242$	−22.9679	−1.47643
$243$	−6.89824	−0.442522
$244$	−18.9592	−1.21374
$245$	0	0
$246$	−2.97142	−0.189451
$247$	13.3682	0.850598
$248$	3.62160	0.229972
$249$	14.1513	0.896804
$250$	0	0
$251$	0.688730	0.0434723	0.0217361	−	0.999764i	$-0.493081\pi$
$251$	0.688730	0.0434723	0.0217361	+	0.999764i	$0.493081\pi$
$252$	−2.02272	−0.127419
$253$	−4.76989	−0.299880
$254$	36.6023	2.29663
$255$	0	0
$256$	−24.1420	−1.50887
$257$	3.58629	0.223707	0.111853	−	0.993725i	$-0.464321\pi$
$257$	3.58629	0.223707	0.111853	+	0.993725i	$0.464321\pi$
$258$	−34.7338	−2.16243
$259$	9.65252	0.599779
$260$	0	0
$261$	−1.98307	−0.122749
$262$	8.64782	0.534264
$263$	−16.1485	−0.995758	−0.497879	−	0.867247i	$-0.665888\pi$
$263$	−16.1485	−0.995758	−0.497879	+	0.867247i	$0.665888\pi$
$264$	−7.73450	−0.476026
$265$	0	0
$266$	−9.35074	−0.573330
$267$	17.6825	1.08215
$268$	−2.99820	−0.183144
$269$	17.6584	1.07665	0.538327	−	0.842736i	$-0.319057\pi$
$269$	17.6584	1.07665	0.538327	+	0.842736i	$0.319057\pi$
$270$	0	0
$271$	−6.72006	−0.408215	−0.204107	−	0.978949i	$-0.565429\pi$
$271$	−6.72006	−0.408215	−0.204107	+	0.978949i	$0.565429\pi$
$272$	−4.50544	−0.273182
$273$	−4.61475	−0.279297
$274$	1.43433	0.0866513
$275$	0	0
$276$	29.0526	1.74876
$277$	−12.3222	−0.740368	−0.370184	−	0.928959i	$-0.620705\pi$
$277$	−12.3222	−0.740368	−0.370184	+	0.928959i	$0.620705\pi$
$278$	10.6407	0.638185
$279$	0.675783	0.0404581
$280$	0	0
$281$	5.28442	0.315242	0.157621	−	0.987500i	$-0.449618\pi$
$281$	5.28442	0.315242	0.157621	+	0.987500i	$0.449618\pi$
$282$	27.5530	1.64076
$283$	2.35777	0.140155	0.0700775	−	0.997542i	$-0.477675\pi$
$283$	2.35777	0.140155	0.0700775	+	0.997542i	$0.477675\pi$
$284$	6.80569	0.403843
$285$	0	0
$286$	−7.46055	−0.441152
$287$	0.556991	0.0328781
$288$	−2.59559	−0.152946
$289$	−7.28738	−0.428670
$290$	0	0
$291$	32.5540	1.90835
$292$	24.5959	1.43937
$293$	−3.96455	−0.231611	−0.115806	−	0.993272i	$-0.536945\pi$
$293$	−3.96455	−0.231611	−0.115806	+	0.993272i	$0.536945\pi$
$294$	−28.3572	−1.65382
$295$	0	0
$296$	−41.3307	−2.40230
$297$	4.96374	0.288025
$298$	−0.958976	−0.0555520
$299$	12.1858	0.704725
$300$	0	0
$301$	6.51084	0.375279
$302$	−2.98899	−0.171997
$303$	−11.7605	−0.675624
$304$	6.79105	0.389493
$305$	0	0
$306$	−4.95661	−0.283351
$307$	−13.8678	−0.791479	−0.395739	−	0.918363i	$-0.629512\pi$
$307$	−13.8678	−0.791479	−0.395739	+	0.918363i	$0.629512\pi$
$308$	3.33415	0.189981
$309$	−20.6877	−1.17688
$310$	0	0
$311$	−27.7794	−1.57522	−0.787612	−	0.616172i	$-0.788683\pi$
$311$	−27.7794	−1.57522	−0.787612	+	0.616172i	$0.788683\pi$
$312$	19.7597	1.11867
$313$	5.97265	0.337594	0.168797	−	0.985651i	$-0.446012\pi$
$313$	5.97265	0.337594	0.168797	+	0.985651i	$0.446012\pi$
$314$	47.1233	2.65932
$315$	0	0
$316$	13.2052	0.742852
$317$	18.9483	1.06424	0.532120	−	0.846669i	$-0.321396\pi$
$317$	18.9483	1.06424	0.532120	+	0.846669i	$0.321396\pi$
$318$	56.5974	3.17382
$319$	3.26880	0.183018
$320$	0	0
$321$	25.1039	1.40117
$322$	−8.52370	−0.475007
$323$	−14.6398	−0.814582
$324$	−37.4078	−2.07821
$325$	0	0
$326$	−2.70315	−0.149714
$327$	6.33045	0.350075
$328$	−2.38495	−0.131687
$329$	−5.16481	−0.284745
$330$	0	0
$331$	18.1992	1.00032	0.500159	−	0.865934i	$-0.333275\pi$
$331$	18.1992	1.00032	0.500159	+	0.865934i	$0.333275\pi$
$332$	26.1205	1.43355
$333$	−7.71223	−0.422627
$334$	−25.4862	−1.39454
$335$	0	0
$336$	−2.34429	−0.127892
$337$	23.5886	1.28495	0.642475	−	0.766306i	$-0.277908\pi$
$337$	23.5886	1.28495	0.642475	+	0.766306i	$0.277908\pi$
$338$	−11.5353	−0.627440
$339$	34.0060	1.84695
$340$	0	0
$341$	−1.11393	−0.0603226
$342$	7.47110	0.403991
$343$	11.2362	0.606695
$344$	−27.8785	−1.50311
$345$	0	0
$346$	−2.42839	−0.130551
$347$	−28.8421	−1.54833	−0.774164	−	0.632985i	$-0.781829\pi$
$347$	−28.8421	−1.54833	−0.774164	+	0.632985i	$0.781829\pi$
$348$	−19.9098	−1.06728
$349$	13.5128	0.723324	0.361662	−	0.932309i	$-0.382209\pi$
$349$	13.5128	0.723324	0.361662	+	0.932309i	$0.382209\pi$
$350$	0	0
$351$	−12.6811	−0.676866
$352$	4.27844	0.228042
$353$	−6.16862	−0.328323	−0.164161	−	0.986433i	$-0.552492\pi$
$353$	−6.16862	−0.328323	−0.164161	+	0.986433i	$0.552492\pi$
$354$	−50.9547	−2.70821
$355$	0	0
$356$	32.6383	1.72983
$357$	5.05372	0.267471
$358$	23.8615	1.26112
$359$	−22.8460	−1.20577	−0.602883	−	0.797829i	$-0.705982\pi$
$359$	−22.8460	−1.20577	−0.602883	+	0.797829i	$0.705982\pi$
$360$	0	0
$361$	3.06665	0.161403
$362$	16.5120	0.867851
$363$	−18.7106	−0.982052
$364$	−8.51791	−0.446460
$365$	0	0
$366$	−24.1738	−1.26358
$367$	9.10378	0.475214	0.237607	−	0.971361i	$-0.423637\pi$
$367$	9.10378	0.475214	0.237607	+	0.971361i	$0.423637\pi$
$368$	6.19041	0.322698
$369$	−0.445028	−0.0231672
$370$	0	0
$371$	−10.6092	−0.550800
$372$	6.78477	0.351774
$373$	8.53974	0.442171	0.221086	−	0.975254i	$-0.429040\pi$
$373$	8.53974	0.442171	0.221086	+	0.975254i	$0.429040\pi$
$374$	8.17023	0.422473
$375$	0	0
$376$	22.1149	1.14049
$377$	−8.35095	−0.430096
$378$	8.87011	0.456229
$379$	−22.9091	−1.17676	−0.588380	−	0.808585i	$-0.700234\pi$
$379$	−22.9091	−1.17676	−0.588380	+	0.808585i	$0.700234\pi$
$380$	0	0
$381$	29.8177	1.52761
$382$	38.0053	1.94452
$383$	−13.7673	−0.703477	−0.351738	−	0.936098i	$-0.614409\pi$
$383$	−13.7673	−0.703477	−0.351738	+	0.936098i	$0.614409\pi$
$384$	−39.1055	−1.99559
$385$	0	0
$386$	27.1884	1.38386
$387$	−5.20207	−0.264436
$388$	60.0883	3.05052
$389$	−30.5457	−1.54873	−0.774365	−	0.632739i	$-0.781931\pi$
$389$	−30.5457	−1.54873	−0.774365	+	0.632739i	$0.781931\pi$
$390$	0	0
$391$	−13.3450	−0.674886
$392$	−22.7604	−1.14957
$393$	7.04486	0.355366
$394$	−54.5287	−2.74711
$395$	0	0
$396$	−2.66394	−0.133868
$397$	−35.4267	−1.77802	−0.889008	−	0.457891i	$-0.848605\pi$
$397$	−35.4267	−1.77802	−0.889008	+	0.457891i	$0.848605\pi$
$398$	−43.3446	−2.17267
$399$	−7.61749	−0.381351
$400$	0	0
$401$	10.9416	0.546396	0.273198	−	0.961958i	$-0.411919\pi$
$401$	10.9416	0.546396	0.273198	+	0.961958i	$0.411919\pi$
$402$	−3.82283	−0.190665
$403$	2.84580	0.141759
$404$	−21.7076	−1.07999
$405$	0	0
$406$	5.84129	0.289898
$407$	12.7125	0.630133
$408$	−21.6393	−1.07131
$409$	−25.5385	−1.26280	−0.631399	−	0.775458i	$-0.717519\pi$
$409$	−25.5385	−1.26280	−0.631399	+	0.775458i	$0.717519\pi$
$410$	0	0
$411$	1.16847	0.0576362
$412$	−38.1854	−1.88126
$413$	9.55143	0.469995
$414$	6.81032	0.334709
$415$	0	0
$416$	−10.9303	−0.535903
$417$	8.66832	0.424489
$418$	−12.3150	−0.602346
$419$	30.9321	1.51113	0.755565	−	0.655073i	$-0.227362\pi$
$419$	30.9321	1.51113	0.755565	+	0.655073i	$0.227362\pi$
$420$	0	0
$421$	36.0126	1.75514	0.877572	−	0.479444i	$-0.159162\pi$
$421$	36.0126	1.75514	0.877572	+	0.479444i	$0.159162\pi$
$422$	37.6760	1.83404
$423$	4.12661	0.200642
$424$	45.4269	2.20612
$425$	0	0
$426$	8.67753	0.420428
$427$	4.53136	0.219288
$428$	46.3369	2.23978
$429$	−6.07766	−0.293432
$430$	0	0
$431$	17.0184	0.819748	0.409874	−	0.912142i	$-0.365573\pi$
$431$	17.0184	0.819748	0.409874	+	0.912142i	$0.365573\pi$
$432$	−6.44199	−0.309941
$433$	−29.3623	−1.41106	−0.705531	−	0.708679i	$-0.749292\pi$
$433$	−29.3623	−1.41106	−0.705531	+	0.708679i	$0.749292\pi$
$434$	−1.99057	−0.0955504
$435$	0	0
$436$	11.6848	0.559598
$437$	20.1150	0.962229
$438$	31.3608	1.49848
$439$	31.4815	1.50253	0.751264	−	0.660002i	$-0.229444\pi$
$439$	31.4815	1.50253	0.751264	+	0.660002i	$0.229444\pi$
$440$	0	0
$441$	−4.24704	−0.202240
$442$	−20.8729	−0.992820
$443$	9.63758	0.457895	0.228948	−	0.973439i	$-0.426472\pi$
$443$	9.63758	0.457895	0.228948	+	0.973439i	$0.426472\pi$
$444$	−77.4296	−3.67465
$445$	0	0
$446$	−50.4535	−2.38904
$447$	−0.781220	−0.0369504
$448$	10.0910	0.476755
$449$	12.8898	0.608307	0.304154	−	0.952623i	$-0.401626\pi$
$449$	12.8898	0.608307	0.304154	+	0.952623i	$0.401626\pi$
$450$	0	0
$451$	0.733562	0.0345421
$452$	62.7683	2.95237
$453$	−2.43495	−0.114404
$454$	57.8189	2.71357
$455$	0	0
$456$	32.6170	1.52743
$457$	23.9973	1.12255	0.561274	−	0.827630i	$-0.310311\pi$
$457$	23.9973	1.12255	0.561274	+	0.827630i	$0.310311\pi$
$458$	−66.3077	−3.09835
$459$	13.8874	0.648206
$460$	0	0
$461$	21.0775	0.981677	0.490838	−	0.871251i	$-0.336691\pi$
$461$	21.0775	0.981677	0.490838	+	0.871251i	$0.336691\pi$
$462$	4.25118	0.197783
$463$	22.6584	1.05303	0.526513	−	0.850167i	$-0.323499\pi$
$463$	22.6584	1.05303	0.526513	+	0.850167i	$0.323499\pi$
$464$	−4.24229	−0.196943
$465$	0	0
$466$	52.4560	2.42998
$467$	−37.0281	−1.71346	−0.856728	−	0.515769i	$-0.827506\pi$
$467$	−37.0281	−1.71346	−0.856728	+	0.515769i	$0.827506\pi$
$468$	6.80569	0.314593
$469$	0.716588	0.0330889
$470$	0	0
$471$	38.3885	1.76885
$472$	−40.8978	−1.88248
$473$	8.57483	0.394271
$474$	16.8372	0.773358
$475$	0	0
$476$	9.32817	0.427556
$477$	8.47657	0.388115
$478$	−53.0720	−2.42746
$479$	12.8675	0.587932	0.293966	−	0.955816i	$-0.405025\pi$
$479$	12.8675	0.587932	0.293966	+	0.955816i	$0.405025\pi$
$480$	0	0
$481$	−32.4771	−1.48083
$482$	−47.4885	−2.16304
$483$	−6.94375	−0.315952
$484$	−34.5360	−1.56982
$485$	0	0
$486$	−16.2348	−0.736426
$487$	−10.5437	−0.477781	−0.238891	−	0.971047i	$-0.576784\pi$
$487$	−10.5437	−0.477781	−0.238891	+	0.971047i	$0.576784\pi$
$488$	−19.4026	−0.878316
$489$	−2.20210	−0.0995822
$490$	0	0
$491$	−29.1141	−1.31390	−0.656950	−	0.753934i	$-0.728154\pi$
$491$	−29.1141	−1.31390	−0.656950	+	0.753934i	$0.728154\pi$
$492$	−4.46801	−0.201434
$493$	9.14533	0.411885
$494$	31.4617	1.41553
$495$	0	0
$496$	1.44567	0.0649124
$497$	−1.62660	−0.0729630
$498$	33.3047	1.49242
$499$	27.2074	1.21797	0.608986	−	0.793181i	$-0.291576\pi$
$499$	27.2074	1.21797	0.608986	+	0.793181i	$0.291576\pi$
$500$	0	0
$501$	−20.7621	−0.927581
$502$	1.62091	0.0723446
$503$	0.458009	0.0204216	0.0102108	−	0.999948i	$-0.496750\pi$
$503$	0.458009	0.0204216	0.0102108	+	0.999948i	$0.496750\pi$
$504$	−2.07003	−0.0922064
$505$	0	0
$506$	−11.2258	−0.499047
$507$	−9.39715	−0.417342
$508$	55.0376	2.44190
$509$	−10.3329	−0.457996	−0.228998	−	0.973427i	$-0.573545\pi$
$509$	−10.3329	−0.457996	−0.228998	+	0.973427i	$0.573545\pi$
$510$	0	0
$511$	−5.87857	−0.260053
$512$	−16.0239	−0.708162
$513$	−20.9324	−0.924190
$514$	8.44025	0.372283
$515$	0	0
$516$	−52.2280	−2.29921
$517$	−6.80209	−0.299156
$518$	22.7169	0.998125
$519$	−1.97827	−0.0868363
$520$	0	0
$521$	31.0648	1.36097	0.680486	−	0.732761i	$-0.261769\pi$
$521$	31.0648	1.36097	0.680486	+	0.732761i	$0.261769\pi$
$522$	−4.66711	−0.204274
$523$	−32.8528	−1.43655	−0.718277	−	0.695757i	$-0.755069\pi$
$523$	−32.8528	−1.43655	−0.718277	+	0.695757i	$0.755069\pi$
$524$	13.0034	0.568057
$525$	0	0
$526$	−38.0050	−1.65710
$527$	−3.11651	−0.135757
$528$	−3.08746	−0.134364
$529$	−4.66413	−0.202788
$530$	0	0
$531$	−7.63146	−0.331177
$532$	−14.0604	−0.609594
$533$	−1.87406	−0.0811747
$534$	41.6152	1.80087
$535$	0	0
$536$	−3.06832	−0.132531
$537$	19.4386	0.838836
$538$	41.5586	1.79172
$539$	7.00061	0.301538
$540$	0	0
$541$	15.5263	0.667526	0.333763	−	0.942657i	$-0.391681\pi$
$541$	15.5263	0.667526	0.333763	+	0.942657i	$0.391681\pi$
$542$	−15.8155	−0.679333
$543$	13.4513	0.577252
$544$	11.9701	0.513212
$545$	0	0
$546$	−10.8607	−0.464794
$547$	−19.1811	−0.820125	−0.410063	−	0.912057i	$-0.634493\pi$
$547$	−19.1811	−0.820125	−0.410063	+	0.912057i	$0.634493\pi$
$548$	2.15676	0.0921320
$549$	−3.62050	−0.154519
$550$	0	0
$551$	−13.7848	−0.587251
$552$	29.7321	1.26548
$553$	−3.15613	−0.134212
$554$	−28.9999	−1.23209
$555$	0	0
$556$	16.0000	0.678551
$557$	19.4476	0.824021	0.412011	−	0.911179i	$-0.364827\pi$
$557$	19.4476	0.824021	0.412011	+	0.911179i	$0.364827\pi$
$558$	1.59044	0.0673286
$559$	−21.9065	−0.926547
$560$	0	0
$561$	6.65580	0.281008
$562$	12.4367	0.524612
$563$	28.3630	1.19536	0.597679	−	0.801736i	$-0.296090\pi$
$563$	28.3630	1.19536	0.597679	+	0.801736i	$0.296090\pi$
$564$	41.4305	1.74454
$565$	0	0
$566$	5.54895	0.233240
$567$	8.94069	0.375474
$568$	6.96486	0.292239
$569$	38.1347	1.59869	0.799344	−	0.600873i	$-0.205180\pi$
$569$	38.1347	1.59869	0.799344	+	0.600873i	$0.205180\pi$
$570$	0	0
$571$	−40.9488	−1.71365	−0.856826	−	0.515605i	$-0.827567\pi$
$571$	−40.9488	−1.71365	−0.856826	+	0.515605i	$0.827567\pi$
$572$	−11.2182	−0.469055
$573$	30.9607	1.29340
$574$	1.31086	0.0547144
$575$	0	0
$576$	−8.06256	−0.335940
$577$	5.75214	0.239465	0.119732	−	0.992806i	$-0.461796\pi$
$577$	5.75214	0.239465	0.119732	+	0.992806i	$0.461796\pi$
$578$	−17.1507	−0.713373
$579$	22.1488	0.920473
$580$	0	0
$581$	−6.24296	−0.259002
$582$	76.6150	3.17579
$583$	−13.9724	−0.578676
$584$	25.1712	1.04159
$585$	0	0
$586$	−9.33045	−0.385437
$587$	−18.8442	−0.777785	−0.388893	−	0.921283i	$-0.627142\pi$
$587$	−18.8442	−0.777785	−0.388893	+	0.921283i	$0.627142\pi$
$588$	−42.6397	−1.75843
$589$	4.69752	0.193558
$590$	0	0
$591$	−44.4213	−1.82725
$592$	−16.4984	−0.678079
$593$	38.2117	1.56916	0.784582	−	0.620025i	$-0.212877\pi$
$593$	38.2117	1.56916	0.784582	+	0.620025i	$0.212877\pi$
$594$	11.6820	0.479319
$595$	0	0
$596$	−1.44198	−0.0590657
$597$	−35.3102	−1.44515
$598$	28.6790	1.17277
$599$	−31.6403	−1.29279	−0.646394	−	0.763004i	$-0.723724\pi$
$599$	−31.6403	−1.29279	−0.646394	+	0.763004i	$0.723724\pi$
$600$	0	0
$601$	−11.9133	−0.485952	−0.242976	−	0.970032i	$-0.578124\pi$
$601$	−11.9133	−0.485952	−0.242976	+	0.970032i	$0.578124\pi$
$602$	15.3231	0.624522
$603$	−0.572543	−0.0233158
$604$	−4.49443	−0.182876
$605$	0	0
$606$	−27.6780	−1.12434
$607$	−29.7256	−1.20653	−0.603263	−	0.797543i	$-0.706133\pi$
$607$	−29.7256	−1.20653	−0.603263	+	0.797543i	$0.706133\pi$
$608$	−18.0425	−0.731719
$609$	4.75855	0.192826
$610$	0	0
$611$	17.3776	0.703023
$612$	−7.45308	−0.301273
$613$	49.0260	1.98014	0.990071	−	0.140569i	$-0.0448931\pi$
$613$	49.0260	1.98014	0.990071	+	0.140569i	$0.0448931\pi$
$614$	−32.6375	−1.31714
$615$	0	0
$616$	3.41213	0.137479
$617$	−3.03996	−0.122384	−0.0611921	−	0.998126i	$-0.519490\pi$
$617$	−3.03996	−0.122384	−0.0611921	+	0.998126i	$0.519490\pi$
$618$	−48.6879	−1.95851
$619$	−36.6166	−1.47175	−0.735873	−	0.677120i	$-0.763228\pi$
$619$	−36.6166	−1.47175	−0.735873	+	0.677120i	$0.763228\pi$
$620$	0	0
$621$	−19.0811	−0.765696
$622$	−65.3780	−2.62142
$623$	−7.80075	−0.312531
$624$	7.88766	0.315759
$625$	0	0
$626$	14.0565	0.561809
$627$	−10.0323	−0.400651
$628$	70.8575	2.82752
$629$	35.5664	1.41813
$630$	0	0
$631$	28.7537	1.14467	0.572334	−	0.820020i	$-0.306038\pi$
$631$	28.7537	1.14467	0.572334	+	0.820020i	$0.306038\pi$
$632$	13.5141	0.537561
$633$	30.6924	1.21991
$634$	44.5942	1.77106
$635$	0	0
$636$	85.1035	3.37457
$637$	−17.8848	−0.708621
$638$	7.69303	0.304570
$639$	1.29963	0.0514126
$640$	0	0
$641$	37.1115	1.46581	0.732907	−	0.680329i	$-0.238163\pi$
$641$	37.1115	1.46581	0.732907	+	0.680329i	$0.238163\pi$
$642$	59.0814	2.33176
$643$	−19.8629	−0.783314	−0.391657	−	0.920111i	$-0.628098\pi$
$643$	−19.8629	−0.783314	−0.391657	+	0.920111i	$0.628098\pi$
$644$	−12.8168	−0.505052
$645$	0	0
$646$	−34.4545	−1.35559
$647$	4.08298	0.160519	0.0802593	−	0.996774i	$-0.474425\pi$
$647$	4.08298	0.160519	0.0802593	+	0.996774i	$0.474425\pi$
$648$	−38.2827	−1.50389
$649$	12.5793	0.493781
$650$	0	0
$651$	−1.62160	−0.0635555
$652$	−4.06463	−0.159183
$653$	1.63252	0.0638856	0.0319428	−	0.999490i	$-0.489831\pi$
$653$	1.63252	0.0638856	0.0319428	+	0.999490i	$0.489831\pi$
$654$	14.8985	0.582579
$655$	0	0
$656$	−0.952025	−0.0371703
$657$	4.69690	0.183243
$658$	−12.1552	−0.473860
$659$	−18.1615	−0.707473	−0.353737	−	0.935345i	$-0.615089\pi$
$659$	−18.1615	−0.707473	−0.353737	+	0.935345i	$0.615089\pi$
$660$	0	0
$661$	−14.2028	−0.552424	−0.276212	−	0.961097i	$-0.589079\pi$
$661$	−14.2028	−0.552424	−0.276212	+	0.961097i	$0.589079\pi$
$662$	42.8313	1.66469
$663$	−17.0039	−0.660375
$664$	26.7314	1.03738
$665$	0	0
$666$	−18.1505	−0.703318
$667$	−12.5656	−0.486541
$668$	−38.3226	−1.48275
$669$	−41.1014	−1.58907
$670$	0	0
$671$	5.96785	0.230386
$672$	6.22833	0.240263
$673$	1.63580	0.0630554	0.0315277	−	0.999503i	$-0.489963\pi$
$673$	1.63580	0.0630554	0.0315277	+	0.999503i	$0.489963\pi$
$674$	55.5150	2.13836
$675$	0	0
$676$	−17.3453	−0.667126
$677$	−8.09313	−0.311044	−0.155522	−	0.987832i	$-0.549706\pi$
$677$	−8.09313	−0.311044	−0.155522	+	0.987832i	$0.549706\pi$
$678$	80.0321	3.07362
$679$	−14.3614	−0.551142
$680$	0	0
$681$	47.1016	1.80494
$682$	−2.62160	−0.100386
$683$	43.9387	1.68127	0.840634	−	0.541604i	$-0.182183\pi$
$683$	43.9387	1.68127	0.840634	+	0.541604i	$0.182183\pi$
$684$	11.2340	0.429544
$685$	0	0
$686$	26.4440	1.00964
$687$	−54.0169	−2.06087
$688$	−11.1285	−0.424271
$689$	35.6958	1.35990
$690$	0	0
$691$	−13.5532	−0.515588	−0.257794	−	0.966200i	$-0.582996\pi$
$691$	−13.5532	−0.515588	−0.257794	+	0.966200i	$0.582996\pi$
$692$	−3.65149	−0.138809
$693$	0.636697	0.0241861
$694$	−67.8792	−2.57666
$695$	0	0
$696$	−20.3754	−0.772328
$697$	2.05233	0.0777376
$698$	31.8020	1.20372
$699$	42.7328	1.61630
$700$	0	0
$701$	7.91834	0.299072	0.149536	−	0.988756i	$-0.452222\pi$
$701$	7.91834	0.299072	0.149536	+	0.988756i	$0.452222\pi$
$702$	−29.8446	−1.12641
$703$	−53.6093	−2.02191
$704$	13.2899	0.500883
$705$	0	0
$706$	−14.5177	−0.546380
$707$	5.18824	0.195124
$708$	−76.6187	−2.87951
$709$	−18.7213	−0.703093	−0.351546	−	0.936170i	$-0.614344\pi$
$709$	−18.7213	−0.703093	−0.351546	+	0.936170i	$0.614344\pi$
$710$	0	0
$711$	2.52170	0.0945711
$712$	33.4017	1.25178
$713$	4.28204	0.160364
$714$	11.8938	0.445114
$715$	0	0
$716$	35.8797	1.34089
$717$	−43.2346	−1.61463
$718$	−53.7675	−2.00658
$719$	−12.2009	−0.455019	−0.227509	−	0.973776i	$-0.573058\pi$
$719$	−12.2009	−0.455019	−0.227509	+	0.973776i	$0.573058\pi$
$720$	0	0
$721$	9.12652	0.339890
$722$	7.21728	0.268599
$723$	−38.6861	−1.43875
$724$	24.8285	0.922744
$725$	0	0
$726$	−44.0349	−1.63429
$727$	18.7571	0.695663	0.347832	−	0.937557i	$-0.386918\pi$
$727$	18.7571	0.695663	0.347832	+	0.937557i	$0.386918\pi$
$728$	−8.71713	−0.323078
$729$	18.4865	0.684684
$730$	0	0
$731$	23.9904	0.887316
$732$	−36.3492	−1.34351
$733$	−47.4991	−1.75442	−0.877209	−	0.480108i	$-0.840598\pi$
$733$	−47.4991	−1.75442	−0.877209	+	0.480108i	$0.840598\pi$
$734$	21.4255	0.790830
$735$	0	0
$736$	−16.4467	−0.606234
$737$	0.943752	0.0347636
$738$	−1.04736	−0.0385539
$739$	−29.4012	−1.08154	−0.540770	−	0.841171i	$-0.681867\pi$
$739$	−29.4012	−1.08154	−0.540770	+	0.841171i	$0.681867\pi$
$740$	0	0
$741$	25.6300	0.941540
$742$	−24.9684	−0.916617
$743$	5.32108	0.195211	0.0976057	−	0.995225i	$-0.468882\pi$
$743$	5.32108	0.195211	0.0976057	+	0.995225i	$0.468882\pi$
$744$	6.94345	0.254559
$745$	0	0
$746$	20.0981	0.735842
$747$	4.98804	0.182503
$748$	12.2853	0.449194
$749$	−11.0748	−0.404664
$750$	0	0
$751$	−43.6516	−1.59287	−0.796434	−	0.604725i	$-0.793283\pi$
$751$	−43.6516	−1.59287	−0.796434	+	0.604725i	$0.793283\pi$
$752$	8.82784	0.321918
$753$	1.32046	0.0481201
$754$	−19.6537	−0.715747
$755$	0	0
$756$	13.3377	0.485086
$757$	−21.4070	−0.778049	−0.389024	−	0.921227i	$-0.627188\pi$
$757$	−21.4070	−0.778049	−0.389024	+	0.921227i	$0.627188\pi$
$758$	−53.9159	−1.95831
$759$	−9.14498	−0.331942
$760$	0	0
$761$	2.43950	0.0884319	0.0442160	−	0.999022i	$-0.485921\pi$
$761$	2.43950	0.0884319	0.0442160	+	0.999022i	$0.485921\pi$
$762$	70.1752	2.54218
$763$	−2.79272	−0.101103
$764$	57.1472	2.06751
$765$	0	0
$766$	−32.4010	−1.17070
$767$	−32.1369	−1.16040
$768$	−46.2858	−1.67020
$769$	−43.7175	−1.57649	−0.788247	−	0.615358i	$-0.789011\pi$
$769$	−43.7175	−1.57649	−0.788247	+	0.615358i	$0.789011\pi$
$770$	0	0
$771$	6.87576	0.247625
$772$	40.8823	1.47139
$773$	−11.4467	−0.411709	−0.205855	−	0.978583i	$-0.565997\pi$
$773$	−11.4467	−0.411709	−0.205855	+	0.978583i	$0.565997\pi$
$774$	−12.2429	−0.440063
$775$	0	0
$776$	61.4936	2.20749
$777$	18.5061	0.663904
$778$	−71.8885	−2.57733
$779$	−3.09348	−0.110836
$780$	0	0
$781$	−2.14225	−0.0766556
$782$	−31.4071	−1.12312
$783$	13.0762	0.467307
$784$	−9.08548	−0.324481
$785$	0	0
$786$	16.5799	0.591385
$787$	0.723361	0.0257850	0.0128925	−	0.999917i	$-0.495896\pi$
$787$	0.723361	0.0257850	0.0128925	+	0.999917i	$0.495896\pi$
$788$	−81.9928	−2.92087
$789$	−30.9604	−1.10222
$790$	0	0
$791$	−15.0020	−0.533409
$792$	−2.72625	−0.0968729
$793$	−15.2463	−0.541413
$794$	−83.3758	−2.95890
$795$	0	0
$796$	−65.1757	−2.31009
$797$	−24.0082	−0.850413	−0.425206	−	0.905096i	$-0.639798\pi$
$797$	−24.0082	−0.850413	−0.425206	+	0.905096i	$0.639798\pi$
$798$	−17.9275	−0.634628
$799$	−19.0307	−0.673256
$800$	0	0
$801$	6.23269	0.220221
$802$	25.7507	0.909288
$803$	−7.74213	−0.273214
$804$	−5.74825	−0.202725
$805$	0	0
$806$	6.69752	0.235910
$807$	33.8553	1.19176
$808$	−22.2153	−0.781531
$809$	−22.5695	−0.793500	−0.396750	−	0.917927i	$-0.629862\pi$
$809$	−22.5695	−0.793500	−0.396750	+	0.917927i	$0.629862\pi$
$810$	0	0
$811$	−21.1890	−0.744047	−0.372023	−	0.928223i	$-0.621336\pi$
$811$	−21.1890	−0.744047	−0.372023	+	0.928223i	$0.621336\pi$
$812$	8.78334	0.308235
$813$	−12.8839	−0.451859
$814$	29.9184	1.04864
$815$	0	0
$816$	−8.63797	−0.302390
$817$	−36.1607	−1.26510
$818$	−60.1042	−2.10149
$819$	−1.62660	−0.0568380
$820$	0	0
$821$	−9.51769	−0.332170	−0.166085	−	0.986111i	$-0.553113\pi$
$821$	−9.51769	−0.332170	−0.166085	+	0.986111i	$0.553113\pi$
$822$	2.74995	0.0959156
$823$	−16.2177	−0.565313	−0.282656	−	0.959221i	$-0.591216\pi$
$823$	−16.2177	−0.565313	−0.282656	+	0.959221i	$0.591216\pi$
$824$	−39.0784	−1.36136
$825$	0	0
$826$	22.4790	0.782146
$827$	−47.7896	−1.66181	−0.830904	−	0.556416i	$-0.812176\pi$
$827$	−47.7896	−1.66181	−0.830904	+	0.556416i	$0.812176\pi$
$828$	10.2404	0.355880
$829$	−38.9021	−1.35112	−0.675562	−	0.737303i	$-0.736099\pi$
$829$	−38.9021	−1.35112	−0.675562	+	0.737303i	$0.736099\pi$
$830$	0	0
$831$	−23.6245	−0.819524
$832$	−33.9524	−1.17709
$833$	19.5861	0.678617
$834$	20.4006	0.706417
$835$	0	0
$836$	−18.5176	−0.640445
$837$	−4.45607	−0.154024
$838$	72.7978	2.51476
$839$	−47.3970	−1.63633	−0.818164	−	0.574985i	$-0.805008\pi$
$839$	−47.3970	−1.63633	−0.818164	+	0.574985i	$0.805008\pi$
$840$	0	0
$841$	−20.3888	−0.703063
$842$	84.7546	2.92084
$843$	10.1315	0.348946
$844$	56.6520	1.95004
$845$	0	0
$846$	9.71185	0.333900
$847$	8.25432	0.283622
$848$	18.1335	0.622706
$849$	4.52040	0.155140
$850$	0	0
$851$	−48.8678	−1.67517
$852$	13.0481	0.447020
$853$	−2.74048	−0.0938322	−0.0469161	−	0.998899i	$-0.514939\pi$
$853$	−2.74048	−0.0938322	−0.0469161	+	0.998899i	$0.514939\pi$
$854$	10.6644	0.364930
$855$	0	0
$856$	47.4206	1.62080
$857$	10.2298	0.349444	0.174722	−	0.984618i	$-0.444097\pi$
$857$	10.2298	0.349444	0.174722	+	0.984618i	$0.444097\pi$
$858$	−14.3036	−0.488317
$859$	21.6715	0.739421	0.369710	−	0.929147i	$-0.379457\pi$
$859$	21.6715	0.739421	0.369710	+	0.929147i	$0.379457\pi$
$860$	0	0
$861$	1.06788	0.0363933
$862$	40.0524	1.36419
$863$	−3.35079	−0.114062	−0.0570311	−	0.998372i	$-0.518163\pi$
$863$	−3.35079	−0.114062	−0.0570311	+	0.998372i	$0.518163\pi$
$864$	17.1151	0.582268
$865$	0	0
$866$	−69.1034	−2.34823
$867$	−13.9716	−0.474501
$868$	−2.99315	−0.101594
$869$	−4.15665	−0.141005
$870$	0	0
$871$	−2.41105	−0.0816952
$872$	11.9580	0.404950
$873$	11.4746	0.388356
$874$	47.3400	1.60130
$875$	0	0
$876$	47.1562	1.59326
$877$	40.6601	1.37299	0.686497	−	0.727133i	$-0.259147\pi$
$877$	40.6601	1.37299	0.686497	+	0.727133i	$0.259147\pi$
$878$	74.0908	2.50044
$879$	−7.60096	−0.256374
$880$	0	0
$881$	0.745750	0.0251250	0.0125625	−	0.999921i	$-0.496001\pi$
$881$	0.745750	0.0251250	0.0125625	+	0.999921i	$0.496001\pi$
$882$	−9.99530	−0.336559
$883$	3.94352	0.132710	0.0663550	−	0.997796i	$-0.478863\pi$
$883$	3.94352	0.132710	0.0663550	+	0.997796i	$0.478863\pi$
$884$	−31.3858	−1.05562
$885$	0	0
$886$	22.6818	0.762009
$887$	−49.4892	−1.66168	−0.830842	−	0.556509i	$-0.812140\pi$
$887$	−49.4892	−1.66168	−0.830842	+	0.556509i	$0.812140\pi$
$888$	−79.2406	−2.65914
$889$	−13.1543	−0.441181
$890$	0	0
$891$	11.7750	0.394476
$892$	−75.8651	−2.54015
$893$	28.6849	0.959904
$894$	−1.83858	−0.0614913
$895$	0	0
$896$	17.2517	0.576338
$897$	23.3631	0.780071
$898$	30.3358	1.01232
$899$	−2.93448	−0.0978704
$900$	0	0
$901$	−39.0913	−1.30232
$902$	1.72642	0.0574834
$903$	12.4828	0.415402
$904$	64.2363	2.13647
$905$	0	0
$906$	−5.73058	−0.190386
$907$	−11.2154	−0.372403	−0.186201	−	0.982512i	$-0.559618\pi$
$907$	−11.2154	−0.372403	−0.186201	+	0.982512i	$0.559618\pi$
$908$	86.9402	2.88521
$909$	−4.14533	−0.137492
$910$	0	0
$911$	−22.5579	−0.747375	−0.373688	−	0.927555i	$-0.621907\pi$
$911$	−22.5579	−0.747375	−0.373688	+	0.927555i	$0.621907\pi$
$912$	13.0200	0.431136
$913$	−8.22203	−0.272110
$914$	56.4771	1.86810
$915$	0	0
$916$	−99.7045	−3.29433
$917$	−3.10789	−0.102632
$918$	32.6835	1.07872
$919$	−15.7009	−0.517924	−0.258962	−	0.965888i	$-0.583380\pi$
$919$	−15.7009	−0.517924	−0.258962	+	0.965888i	$0.583380\pi$
$920$	0	0
$921$	−26.5879	−0.876100
$922$	49.6053	1.63366
$923$	5.47289	0.180142
$924$	6.39235	0.210293
$925$	0	0
$926$	53.3260	1.75240
$927$	−7.29196	−0.239499
$928$	11.2709	0.369986
$929$	−9.90563	−0.324993	−0.162497	−	0.986709i	$-0.551955\pi$
$929$	−9.90563	−0.324993	−0.162497	+	0.986709i	$0.551955\pi$
$930$	0	0
$931$	−29.5221	−0.967548
$932$	78.8763	2.58368
$933$	−53.2595	−1.74364
$934$	−87.1446	−2.85146
$935$	0	0
$936$	6.96486	0.227654
$937$	−7.99567	−0.261207	−0.130604	−	0.991435i	$-0.541691\pi$
$937$	−7.99567	−0.261207	−0.130604	+	0.991435i	$0.541691\pi$
$938$	1.68647	0.0550652
$939$	11.4510	0.373688
$940$	0	0
$941$	36.2677	1.18229	0.591147	−	0.806564i	$-0.298675\pi$
$941$	36.2677	1.18229	0.591147	+	0.806564i	$0.298675\pi$
$942$	90.3463	2.94364
$943$	−2.81988	−0.0918279
$944$	−16.3256	−0.531353
$945$	0	0
$946$	20.1806	0.656129
$947$	41.9574	1.36343	0.681716	−	0.731617i	$-0.261234\pi$
$947$	41.9574	1.36343	0.681716	+	0.731617i	$0.261234\pi$
$948$	25.3175	0.822274
$949$	19.7792	0.642059
$950$	0	0
$951$	36.3282	1.17802
$952$	9.54634	0.309399
$953$	5.18005	0.167798	0.0838991	−	0.996474i	$-0.473263\pi$
$953$	5.18005	0.167798	0.0838991	+	0.996474i	$0.473263\pi$
$954$	19.9494	0.645884
$955$	0	0
$956$	−79.8025	−2.58100
$957$	6.26705	0.202585
$958$	30.2833	0.978410
$959$	−0.515477	−0.0166456
$960$	0	0
$961$	1.00000	0.0322581
$962$	−76.4339	−2.46433
$963$	8.84860	0.285142
$964$	−71.4068	−2.29986
$965$	0	0
$966$	−16.3419	−0.525793
$967$	43.9028	1.41182	0.705909	−	0.708303i	$-0.250539\pi$
$967$	43.9028	1.41182	0.705909	+	0.708303i	$0.250539\pi$
$968$	−35.3438	−1.13599
$969$	−28.0680	−0.901674
$970$	0	0
$971$	48.4479	1.55477	0.777384	−	0.629026i	$-0.216546\pi$
$971$	48.4479	1.55477	0.777384	+	0.629026i	$0.216546\pi$
$972$	−24.4117	−0.783006
$973$	−3.82409	−0.122595
$974$	−24.8143	−0.795102
$975$	0	0
$976$	−7.74514	−0.247916
$977$	17.4324	0.557711	0.278855	−	0.960333i	$-0.410045\pi$
$977$	17.4324	0.557711	0.278855	+	0.960333i	$0.410045\pi$
$978$	−5.18257	−0.165720
$979$	−10.2737	−0.328348
$980$	0	0
$981$	2.23135	0.0712414
$982$	−68.5192	−2.18654
$983$	12.8471	0.409758	0.204879	−	0.978787i	$-0.434320\pi$
$983$	12.8471	0.409758	0.204879	+	0.978787i	$0.434320\pi$
$984$	−4.57251	−0.145766
$985$	0	0
$986$	21.5233	0.685441
$987$	−9.90214	−0.315189
$988$	47.3078	1.50506
$989$	−32.9624	−1.04814
$990$	0	0
$991$	−43.2787	−1.37479	−0.687396	−	0.726283i	$-0.741246\pi$
$991$	−43.2787	−1.37479	−0.687396	+	0.726283i	$0.741246\pi$
$992$	−3.84086	−0.121947
$993$	34.8921	1.10727
$994$	−3.82816	−0.121422
$995$	0	0
$996$	50.0792	1.58682
$997$	29.6158	0.937940	0.468970	−	0.883214i	$-0.344625\pi$
$997$	29.6158	0.937940	0.468970	+	0.883214i	$0.344625\pi$
$998$	64.0320	2.02690
$999$	50.8539	1.60894

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	775.2.a.j.1.4	yes	5
3.2	odd	2		6975.2.a.bq.1.2		5
5.2	odd	4		775.2.b.h.249.9		10
5.3	odd	4		775.2.b.h.249.2		10
5.4	even	2		775.2.a.i.1.2	✓	5
15.14	odd	2		6975.2.a.bx.1.4		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
775.2.a.i.1.2	✓	5	5.4	even	2
775.2.a.j.1.4	yes	5	1.1	even	1	trivial
775.2.b.h.249.2		10	5.3	odd	4
775.2.b.h.249.9		10	5.2	odd	4
6975.2.a.bq.1.2		5	3.2	odd	2
6975.2.a.bx.1.4		5	15.14	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:	\( N \)	\(=\)	\( 775 = 5^{2} \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	775.a (trivial)

\(f(q)\)	\(=\)	\(q+2.35347 q^{2} +1.91723 q^{3} +3.53883 q^{4} +4.51216 q^{6} -0.845802 q^{7} +3.62160 q^{8} +0.675783 q^{9} -1.11393 q^{11} +6.78477 q^{12} +2.84580 q^{13} -1.99057 q^{14} +1.44567 q^{16} -3.11651 q^{17} +1.59044 q^{18} +4.69752 q^{19} -1.62160 q^{21} -2.62160 q^{22} +4.28204 q^{23} +6.94345 q^{24} +6.69752 q^{26} -4.45607 q^{27} -2.99315 q^{28} -2.93448 q^{29} +1.00000 q^{31} -3.84086 q^{32} -2.13566 q^{33} -7.33461 q^{34} +2.39148 q^{36} -11.4123 q^{37} +11.0555 q^{38} +5.45607 q^{39} -0.658536 q^{41} -3.81639 q^{42} -7.69784 q^{43} -3.94200 q^{44} +10.0777 q^{46} +6.10640 q^{47} +2.77168 q^{48} -6.28462 q^{49} -5.97507 q^{51} +10.0708 q^{52} +12.5433 q^{53} -10.4872 q^{54} -3.06315 q^{56} +9.00623 q^{57} -6.90622 q^{58} -11.2928 q^{59} -5.35748 q^{61} +2.35347 q^{62} -0.571579 q^{63} -11.9307 q^{64} -5.02622 q^{66} -0.847229 q^{67} -11.0288 q^{68} +8.20967 q^{69} +1.92315 q^{71} +2.44742 q^{72} +6.95030 q^{73} -26.8585 q^{74} +16.6237 q^{76} +0.942162 q^{77} +12.8407 q^{78} +3.73152 q^{79} -10.5707 q^{81} -1.54985 q^{82} +7.38112 q^{83} -5.73857 q^{84} -18.1166 q^{86} -5.62608 q^{87} -4.03420 q^{88} +9.22291 q^{89} -2.40698 q^{91} +15.1534 q^{92} +1.91723 q^{93} +14.3713 q^{94} -7.36382 q^{96} +16.9797 q^{97} -14.7907 q^{98} -0.752774 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 4 q^{2} + q^{3} + 6 q^{4} - q^{6} + 6 q^{7} + 15 q^{8} + 2 q^{9} - 11 q^{12} + 4 q^{13} + 2 q^{14} + 20 q^{16} + 11 q^{17} + 19 q^{18} - 4 q^{19} - 5 q^{21} - 10 q^{22} + 12 q^{23} - 26 q^{24} + 6 q^{26}+ \cdots + 5 q^{99}+O(q^{100}) \) 5 * q + 4 * q^2 + q^3 + 6 * q^4 - q^6 + 6 * q^7 + 15 * q^8 + 2 * q^9 - 11 * q^12 + 4 * q^13 + 2 * q^14 + 20 * q^16 + 11 * q^17 + 19 * q^18 - 4 * q^19 - 5 * q^21 - 10 * q^22 + 12 * q^23 - 26 * q^24 + 6 * q^26 - 2 * q^27 + 18 * q^28 - 6 * q^29 + 5 * q^31 + 29 * q^32 + 21 * q^33 - 5 * q^34 + 23 * q^36 - 2 * q^37 - 6 * q^38 + 7 * q^39 - 2 * q^41 - 24 * q^42 + 7 * q^43 - 28 * q^44 + 27 * q^46 + 8 * q^47 - 39 * q^48 - q^49 - 19 * q^51 - 6 * q^52 + 25 * q^53 - 18 * q^54 + 35 * q^56 + 20 * q^57 - q^58 + 4 * q^59 - 17 * q^61 + 4 * q^62 + 10 * q^63 + 27 * q^64 + 27 * q^66 - 13 * q^67 + 18 * q^68 - 10 * q^69 - 6 * q^71 + 26 * q^72 + 7 * q^73 + 6 * q^74 - 5 * q^76 + 7 * q^77 + 22 * q^78 + 12 * q^79 - 11 * q^81 - 21 * q^82 - 4 * q^83 - 63 * q^84 - 41 * q^86 + q^87 - 49 * q^88 - 3 * q^89 - 22 * q^91 + 34 * q^92 + q^93 - 34 * q^94 - 64 * q^96 + 25 * q^97 - 22 * q^98 + 5 * q^99

Introduction

L-functions

Modular forms

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