LMFDB - Embedded newform 6975.2.a.bx.1.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [6975,2,Mod(1,6975)] 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("6975.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(6975, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 6975 = 3^{2} \cdot 5^{2} \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6975.a (trivial)

Newform invariants

comment:select newform

sage:traces = [5,4,0,6,0,0,-6,15,0,0,0,0,-4,-2,0,20,11] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)

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

Self dual:	yes
Analytic conductor:	$55.6956554098$
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:	no (minimal twist has level 775)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-1.35347$ of defining polynomial
Character	$\chi$	$=$	6975.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+2.35347 q^{2} +3.53883 q^{4} +0.845802 q^{7} +3.62160 q^{8} +1.11393 q^{11} -2.84580 q^{13} +1.99057 q^{14} +1.44567 q^{16} -3.11651 q^{17} +4.69752 q^{19} +2.62160 q^{22} +4.28204 q^{23} -6.69752 q^{26} +2.99315 q^{28} +2.93448 q^{29} +1.00000 q^{31} -3.84086 q^{32} -7.33461 q^{34} +11.4123 q^{37} +11.0555 q^{38} +0.658536 q^{41} +7.69784 q^{43} +3.94200 q^{44} +10.0777 q^{46} +6.10640 q^{47} -6.28462 q^{49} -10.0708 q^{52} +12.5433 q^{53} +3.06315 q^{56} +6.90622 q^{58} +11.2928 q^{59} -5.35748 q^{61} +2.35347 q^{62} -11.9307 q^{64} +0.847229 q^{67} -11.0288 q^{68} -1.92315 q^{71} -6.95030 q^{73} +26.8585 q^{74} +16.6237 q^{76} +0.942162 q^{77} +3.73152 q^{79} +1.54985 q^{82} +7.38112 q^{83} +18.1166 q^{86} +4.03420 q^{88} -9.22291 q^{89} -2.40698 q^{91} +15.1534 q^{92} +14.3713 q^{94} -16.9797 q^{97} -14.7907 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 4 q^{2} + 6 q^{4} - 6 q^{7} + 15 q^{8} - 4 q^{13} - 2 q^{14} + 20 q^{16} + 11 q^{17} - 4 q^{19} + 10 q^{22} + 12 q^{23} - 6 q^{26} - 18 q^{28} + 6 q^{29} + 5 q^{31} + 29 q^{32} - 5 q^{34} + 2 q^{37}+ \cdots - 22 q^{98}+O(q^{100}) $ 5 * q + 4 * q^2 + 6 * q^4 - 6 * q^7 + 15 * q^8 - 4 * q^13 - 2 * q^14 + 20 * q^16 + 11 * q^17 - 4 * q^19 + 10 * q^22 + 12 * q^23 - 6 * q^26 - 18 * q^28 + 6 * q^29 + 5 * q^31 + 29 * q^32 - 5 * q^34 + 2 * q^37 - 6 * q^38 + 2 * q^41 - 7 * q^43 + 28 * q^44 + 27 * q^46 + 8 * q^47 - q^49 + 6 * q^52 + 25 * q^53 - 35 * q^56 + q^58 - 4 * q^59 - 17 * q^61 + 4 * q^62 + 27 * q^64 + 13 * q^67 + 18 * q^68 + 6 * q^71 - 7 * q^73 - 6 * q^74 - 5 * q^76 + 7 * q^77 + 12 * q^79 + 21 * q^82 - 4 * q^83 + 41 * q^86 + 49 * q^88 + 3 * q^89 - 22 * q^91 + 34 * q^92 - 34 * q^94 - 25 * q^97 - 22 * q^98

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	$0.187151\pi$
$3$	0	0
$3$	0	0
$4$	3.53883	1.76942
$5$	0	0
$5$	0	0
$6$	0	0
$7$	0.845802	0.319683	0.159841	−	0.987143i	$-0.448902\pi$
$7$	0.845802	0.319683	0.159841	+	0.987143i	$0.448902\pi$
$8$	3.62160	1.28043
$9$	0	0
$10$	0	0
$11$	1.11393	0.335862	0.167931	−	0.985799i	$-0.446291\pi$
$11$	1.11393	0.335862	0.167931	+	0.985799i	$0.446291\pi$
$12$	0	0
$13$	−2.84580	−0.789283	−0.394642	−	0.918835i	$-0.629131\pi$
$13$	−2.84580	−0.789283	−0.394642	+	0.918835i	$0.629131\pi$
$14$	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	$0.623365\pi$
$18$	0	0
$19$	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
$21$	0	0
$22$	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	$0.352694\pi$
$24$	0	0
$25$	0	0
$26$	−6.69752	−1.31349
$27$	0	0
$28$	2.99315	0.565652
$29$	2.93448	0.544919	0.272460	−	0.962167i	$-0.412163\pi$
$29$	2.93448	0.544919	0.272460	+	0.962167i	$0.412163\pi$
$30$	0	0
$31$	1.00000	0.179605
$31$	1.00000	0.179605
$32$	−3.84086	−0.678974
$33$	0	0
$34$	−7.33461	−1.25788
$35$	0	0
$36$	0	0
$37$	11.4123	1.87617	0.938083	−	0.346409i	$-0.112599\pi$
$37$	11.4123	1.87617	0.938083	+	0.346409i	$0.112599\pi$
$38$	11.0555	1.79343
$39$	0	0
$40$	0	0
$41$	0.658536	0.102846	0.0514230	−	0.998677i	$-0.483624\pi$
$41$	0.658536	0.102846	0.0514230	+	0.998677i	$0.483624\pi$
$42$	0	0
$43$	7.69784	1.17391	0.586954	−	0.809620i	$-0.300327\pi$
$43$	7.69784	1.17391	0.586954	+	0.809620i	$0.300327\pi$
$44$	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	$0.353077\pi$
$48$	0	0
$49$	−6.28462	−0.897803
$50$	0	0
$51$	0	0
$52$	−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	$0.169539\pi$
$54$	0	0
$55$	0	0
$56$	3.06315	0.409331
$57$	0	0
$58$	6.90622	0.906831
$59$	11.2928	1.47019	0.735096	−	0.677963i	$-0.237137\pi$
$59$	11.2928	1.47019	0.735096	+	0.677963i	$0.237137\pi$
$60$	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	0
$64$	−11.9307	−1.49134
$65$	0	0
$66$	0	0
$67$	0.847229	0.103506	0.0517528	−	0.998660i	$-0.483519\pi$
$67$	0.847229	0.103506	0.0517528	+	0.998660i	$0.483519\pi$
$68$	−11.0288	−1.33744
$69$	0	0
$70$	0	0
$71$	−1.92315	−0.228235	−0.114118	−	0.993467i	$-0.536404\pi$
$71$	−1.92315	−0.228235	−0.114118	+	0.993467i	$0.536404\pi$
$72$	0	0
$73$	−6.95030	−0.813471	−0.406736	−	0.913546i	$-0.633333\pi$
$73$	−6.95030	−0.813471	−0.406736	+	0.913546i	$0.633333\pi$
$74$	26.8585	3.12223
$75$	0	0
$76$	16.6237	1.90687
$77$	0.942162	0.107369
$78$	0	0
$79$	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
$81$	0	0
$82$	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	$0.367240\pi$
$84$	0	0
$85$	0	0
$86$	18.1166	1.95357
$87$	0	0
$88$	4.03420	0.430047
$89$	−9.22291	−0.977627	−0.488813	−	0.872388i	$-0.662570\pi$
$89$	−9.22291	−0.977627	−0.488813	+	0.872388i	$0.662570\pi$
$90$	0	0
$91$	−2.40698	−0.252320
$92$	15.1534	1.57985
$93$	0	0
$94$	14.3713	1.48228
$95$	0	0
$96$	0	0
$97$	−16.9797	−1.72403	−0.862013	−	0.506886i	$-0.830797\pi$
$97$	−16.9797	−1.72403	−0.862013	+	0.506886i	$0.830797\pi$
$98$	−14.7907	−1.49408
$99$	0	0
$100$	0	0
$101$	6.13411	0.610367	0.305183	−	0.952294i	$-0.401282\pi$
$101$	6.13411	0.610367	0.305183	+	0.952294i	$0.401282\pi$
$102$	0	0
$103$	10.7904	1.06321	0.531604	−	0.846993i	$-0.321589\pi$
$103$	10.7904	1.06321	0.531604	+	0.846993i	$0.321589\pi$
$104$	−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	$0.281858\pi$
$108$	0	0
$109$	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
$111$	0	0
$112$	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	$0.185884\pi$
$114$	0	0
$115$	0	0
$116$	10.3846	0.964189
$117$	0	0
$118$	26.5772	2.44663
$119$	−2.63595	−0.241637
$120$	0	0
$121$	−9.75916	−0.887197
$122$	−12.6087	−1.14154
$123$	0	0
$124$	3.53883	0.317797
$125$	0	0
$126$	0	0
$127$	−15.5525	−1.38006	−0.690030	−	0.723781i	$-0.742403\pi$
$127$	−15.5525	−1.38006	−0.690030	+	0.723781i	$0.742403\pi$
$128$	−20.3968	−1.80284
$129$	0	0
$130$	0	0
$131$	−3.67449	−0.321042	−0.160521	−	0.987032i	$-0.551317\pi$
$131$	−3.67449	−0.321042	−0.160521	+	0.987032i	$0.551317\pi$
$132$	0	0
$133$	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	$0.491712\pi$
$138$	0	0
$139$	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
$141$	0	0
$142$	−4.52607	−0.379819
$143$	−3.17002	−0.265090
$144$	0	0
$145$	0	0
$146$	−16.3573	−1.35374
$147$	0	0
$148$	40.3861	3.31972
$149$	0.407473	0.0333815	0.0166907	−	0.999861i	$-0.494687\pi$
$149$	0.407473	0.0333815	0.0166907	+	0.999861i	$0.494687\pi$
$150$	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
$153$	0	0
$154$	2.21735	0.178679
$155$	0	0
$156$	0	0
$157$	−20.0229	−1.59800	−0.798999	−	0.601332i	$-0.794637\pi$
$157$	−20.0229	−1.59800	−0.798999	+	0.601332i	$0.794637\pi$
$158$	8.78203	0.698661
$159$	0	0
$160$	0	0
$161$	3.62176	0.285434
$162$	0	0
$163$	1.14858	0.0899637	0.0449819	−	0.998988i	$-0.485677\pi$
$163$	1.14858	0.0899637	0.0449819	+	0.998988i	$0.485677\pi$
$164$	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	$0.637617\pi$
$168$	0	0
$169$	−4.90141	−0.377032
$170$	0	0
$171$	0	0
$172$	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	$0.512489\pi$
$174$	0	0
$175$	0	0
$176$	1.61037	0.121386
$177$	0	0
$178$	−21.7059	−1.62692
$179$	−10.1389	−0.757814	−0.378907	−	0.925435i	$-0.623700\pi$
$179$	−10.1389	−0.757814	−0.378907	+	0.925435i	$0.623700\pi$
$180$	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
$183$	0	0
$184$	15.5078	1.14325
$185$	0	0
$186$	0	0
$187$	−3.47156	−0.253866
$188$	21.6095	1.57604
$189$	0	0
$190$	0	0
$191$	−16.1486	−1.16847	−0.584236	−	0.811584i	$-0.698606\pi$
$191$	−16.1486	−1.16847	−0.584236	+	0.811584i	$0.698606\pi$
$192$	0	0
$193$	−11.5525	−0.831566	−0.415783	−	0.909464i	$-0.636492\pi$
$193$	−11.5525	−0.831566	−0.415783	+	0.909464i	$0.636492\pi$
$194$	−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	$0.809038\pi$
$198$	0	0
$199$	−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
$201$	0	0
$202$	14.4365	1.01575
$203$	2.48199	0.174201
$204$	0	0
$205$	0	0
$206$	25.3949	1.76934
$207$	0	0
$208$	−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
$213$	0	0
$214$	30.8160	2.10654
$215$	0	0
$216$	0	0
$217$	0.845802	0.0574167
$218$	7.77085	0.526309
$219$	0	0
$220$	0	0
$221$	8.86896	0.596591
$222$	0	0
$223$	21.4379	1.43559	0.717793	−	0.696256i	$-0.245152\pi$
$223$	21.4379	1.43559	0.717793	+	0.696256i	$0.245152\pi$
$224$	−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	$0.196572\pi$
$228$	0	0
$229$	−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
$231$	0	0
$232$	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	$0.239477\pi$
$234$	0	0
$235$	0	0
$236$	39.9632	2.60138
$237$	0	0
$238$	−6.20363	−0.402121
$239$	22.5505	1.45867	0.729336	−	0.684156i	$-0.239829\pi$
$239$	22.5505	1.45867	0.729336	+	0.684156i	$0.239829\pi$
$240$	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
$243$	0	0
$244$	−18.9592	−1.21374
$245$	0	0
$246$	0	0
$247$	−13.3682	−0.850598
$248$	3.62160	0.229972
$249$	0	0
$250$	0	0
$251$	−0.688730	−0.0434723	−0.0217361	−	0.999764i	$-0.506919\pi$
$251$	−0.688730	−0.0434723	−0.0217361	+	0.999764i	$0.506919\pi$
$252$	0	0
$253$	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	$0.464321\pi$
$258$	0	0
$259$	9.65252	0.599779
$260$	0	0
$261$	0	0
$262$	−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	$0.665888\pi$
$264$	0	0
$265$	0	0
$266$	9.35074	0.573330
$267$	0	0
$268$	2.99820	0.183144
$269$	−17.6584	−1.07665	−0.538327	−	0.842736i	$-0.680943\pi$
$269$	−17.6584	−1.07665	−0.538327	+	0.842736i	$0.680943\pi$
$270$	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
$273$	0	0
$274$	1.43433	0.0866513
$275$	0	0
$276$	0	0
$277$	12.3222	0.740368	0.370184	−	0.928959i	$-0.379295\pi$
$277$	12.3222	0.740368	0.370184	+	0.928959i	$0.379295\pi$
$278$	10.6407	0.638185
$279$	0	0
$280$	0	0
$281$	−5.28442	−0.315242	−0.157621	−	0.987500i	$-0.550382\pi$
$281$	−5.28442	−0.315242	−0.157621	+	0.987500i	$0.550382\pi$
$282$	0	0
$283$	−2.35777	−0.140155	−0.0700775	−	0.997542i	$-0.522325\pi$
$283$	−2.35777	−0.140155	−0.0700775	+	0.997542i	$0.522325\pi$
$284$	−6.80569	−0.403843
$285$	0	0
$286$	−7.46055	−0.441152
$287$	0.556991	0.0328781
$288$	0	0
$289$	−7.28738	−0.428670
$290$	0	0
$291$	0	0
$292$	−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	$0.536945\pi$
$294$	0	0
$295$	0	0
$296$	41.3307	2.40230
$297$	0	0
$298$	0.958976	0.0555520
$299$	−12.1858	−0.704725
$300$	0	0
$301$	6.51084	0.375279
$302$	−2.98899	−0.171997
$303$	0	0
$304$	6.79105	0.389493
$305$	0	0
$306$	0	0
$307$	13.8678	0.791479	0.395739	−	0.918363i	$-0.370488\pi$
$307$	13.8678	0.791479	0.395739	+	0.918363i	$0.370488\pi$
$308$	3.33415	0.189981
$309$	0	0
$310$	0	0
$311$	27.7794	1.57522	0.787612	−	0.616172i	$-0.211317\pi$
$311$	27.7794	1.57522	0.787612	+	0.616172i	$0.211317\pi$
$312$	0	0
$313$	−5.97265	−0.337594	−0.168797	−	0.985651i	$-0.553988\pi$
$313$	−5.97265	−0.337594	−0.168797	+	0.985651i	$0.553988\pi$
$314$	−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	$0.321396\pi$
$318$	0	0
$319$	3.26880	0.183018
$320$	0	0
$321$	0	0
$322$	8.52370	0.475007
$323$	−14.6398	−0.814582
$324$	0	0
$325$	0	0
$326$	2.70315	0.149714
$327$	0	0
$328$	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
$333$	0	0
$334$	−25.4862	−1.39454
$335$	0	0
$336$	0	0
$337$	−23.5886	−1.28495	−0.642475	−	0.766306i	$-0.722092\pi$
$337$	−23.5886	−1.28495	−0.642475	+	0.766306i	$0.722092\pi$
$338$	−11.5353	−0.627440
$339$	0	0
$340$	0	0
$341$	1.11393	0.0603226
$342$	0	0
$343$	−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	$0.781829\pi$
$348$	0	0
$349$	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
$351$	0	0
$352$	−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	$0.552492\pi$
$354$	0	0
$355$	0	0
$356$	−32.6383	−1.72983
$357$	0	0
$358$	−23.8615	−1.26112
$359$	22.8460	1.20577	0.602883	−	0.797829i	$-0.294018\pi$
$359$	22.8460	1.20577	0.602883	+	0.797829i	$0.294018\pi$
$360$	0	0
$361$	3.06665	0.161403
$362$	16.5120	0.867851
$363$	0	0
$364$	−8.51791	−0.446460
$365$	0	0
$366$	0	0
$367$	−9.10378	−0.475214	−0.237607	−	0.971361i	$-0.576363\pi$
$367$	−9.10378	−0.475214	−0.237607	+	0.971361i	$0.576363\pi$
$368$	6.19041	0.322698
$369$	0	0
$370$	0	0
$371$	10.6092	0.550800
$372$	0	0
$373$	−8.53974	−0.442171	−0.221086	−	0.975254i	$-0.570960\pi$
$373$	−8.53974	−0.442171	−0.221086	+	0.975254i	$0.570960\pi$
$374$	−8.17023	−0.422473
$375$	0	0
$376$	22.1149	1.14049
$377$	−8.35095	−0.430096
$378$	0	0
$379$	−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
$381$	0	0
$382$	−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	$0.614409\pi$
$384$	0	0
$385$	0	0
$386$	−27.1884	−1.38386
$387$	0	0
$388$	−60.0883	−3.05052
$389$	30.5457	1.54873	0.774365	−	0.632739i	$-0.218069\pi$
$389$	30.5457	1.54873	0.774365	+	0.632739i	$0.218069\pi$
$390$	0	0
$391$	−13.3450	−0.674886
$392$	−22.7604	−1.14957
$393$	0	0
$394$	−54.5287	−2.74711
$395$	0	0
$396$	0	0
$397$	35.4267	1.77802	0.889008	−	0.457891i	$-0.151395\pi$
$397$	35.4267	1.77802	0.889008	+	0.457891i	$0.151395\pi$
$398$	−43.3446	−2.17267
$399$	0	0
$400$	0	0
$401$	−10.9416	−0.546396	−0.273198	−	0.961958i	$-0.588081\pi$
$401$	−10.9416	−0.546396	−0.273198	+	0.961958i	$0.588081\pi$
$402$	0	0
$403$	−2.84580	−0.141759
$404$	21.7076	1.07999
$405$	0	0
$406$	5.84129	0.289898
$407$	12.7125	0.630133
$408$	0	0
$409$	−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
$411$	0	0
$412$	38.1854	1.88126
$413$	9.55143	0.469995
$414$	0	0
$415$	0	0
$416$	10.9303	0.535903
$417$	0	0
$418$	12.3150	0.602346
$419$	−30.9321	−1.51113	−0.755565	−	0.655073i	$-0.772638\pi$
$419$	−30.9321	−1.51113	−0.755565	+	0.655073i	$0.772638\pi$
$420$	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
$423$	0	0
$424$	45.4269	2.20612
$425$	0	0
$426$	0	0
$427$	−4.53136	−0.219288
$428$	46.3369	2.23978
$429$	0	0
$430$	0	0
$431$	−17.0184	−0.819748	−0.409874	−	0.912142i	$-0.634427\pi$
$431$	−17.0184	−0.819748	−0.409874	+	0.912142i	$0.634427\pi$
$432$	0	0
$433$	29.3623	1.41106	0.705531	−	0.708679i	$-0.250708\pi$
$433$	29.3623	1.41106	0.705531	+	0.708679i	$0.250708\pi$
$434$	1.99057	0.0955504
$435$	0	0
$436$	11.6848	0.559598
$437$	20.1150	0.962229
$438$	0	0
$439$	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
$441$	0	0
$442$	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	$0.426472\pi$
$444$	0	0
$445$	0	0
$446$	50.4535	2.38904
$447$	0	0
$448$	−10.0910	−0.476755
$449$	−12.8898	−0.608307	−0.304154	−	0.952623i	$-0.598374\pi$
$449$	−12.8898	−0.608307	−0.304154	+	0.952623i	$0.598374\pi$
$450$	0	0
$451$	0.733562	0.0345421
$452$	62.7683	2.95237
$453$	0	0
$454$	57.8189	2.71357
$455$	0	0
$456$	0	0
$457$	−23.9973	−1.12255	−0.561274	−	0.827630i	$-0.689689\pi$
$457$	−23.9973	−1.12255	−0.561274	+	0.827630i	$0.689689\pi$
$458$	−66.3077	−3.09835
$459$	0	0
$460$	0	0
$461$	−21.0775	−0.981677	−0.490838	−	0.871251i	$-0.663309\pi$
$461$	−21.0775	−0.981677	−0.490838	+	0.871251i	$0.663309\pi$
$462$	0	0
$463$	−22.6584	−1.05303	−0.526513	−	0.850167i	$-0.676501\pi$
$463$	−22.6584	−1.05303	−0.526513	+	0.850167i	$0.676501\pi$
$464$	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	$0.827506\pi$
$468$	0	0
$469$	0.716588	0.0330889
$470$	0	0
$471$	0	0
$472$	40.8978	1.88248
$473$	8.57483	0.394271
$474$	0	0
$475$	0	0
$476$	−9.32817	−0.427556
$477$	0	0
$478$	53.0720	2.42746
$479$	−12.8675	−0.587932	−0.293966	−	0.955816i	$-0.594975\pi$
$479$	−12.8675	−0.587932	−0.293966	+	0.955816i	$0.594975\pi$
$480$	0	0
$481$	−32.4771	−1.48083
$482$	−47.4885	−2.16304
$483$	0	0
$484$	−34.5360	−1.56982
$485$	0	0
$486$	0	0
$487$	10.5437	0.477781	0.238891	−	0.971047i	$-0.423216\pi$
$487$	10.5437	0.477781	0.238891	+	0.971047i	$0.423216\pi$
$488$	−19.4026	−0.878316
$489$	0	0
$490$	0	0
$491$	29.1141	1.31390	0.656950	−	0.753934i	$-0.271846\pi$
$491$	29.1141	1.31390	0.656950	+	0.753934i	$0.271846\pi$
$492$	0	0
$493$	−9.14533	−0.411885
$494$	−31.4617	−1.41553
$495$	0	0
$496$	1.44567	0.0649124
$497$	−1.62660	−0.0729630
$498$	0	0
$499$	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
$501$	0	0
$502$	−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	$0.496750\pi$
$504$	0	0
$505$	0	0
$506$	11.2258	0.499047
$507$	0	0
$508$	−55.0376	−2.44190
$509$	10.3329	0.457996	0.228998	−	0.973427i	$-0.426455\pi$
$509$	10.3329	0.457996	0.228998	+	0.973427i	$0.426455\pi$
$510$	0	0
$511$	−5.87857	−0.260053
$512$	−16.0239	−0.708162
$513$	0	0
$514$	8.44025	0.372283
$515$	0	0
$516$	0	0
$517$	6.80209	0.299156
$518$	22.7169	0.998125
$519$	0	0
$520$	0	0
$521$	−31.0648	−1.36097	−0.680486	−	0.732761i	$-0.738231\pi$
$521$	−31.0648	−1.36097	−0.680486	+	0.732761i	$0.738231\pi$
$522$	0	0
$523$	32.8528	1.43655	0.718277	−	0.695757i	$-0.244931\pi$
$523$	32.8528	1.43655	0.718277	+	0.695757i	$0.244931\pi$
$524$	−13.0034	−0.568057
$525$	0	0
$526$	−38.0050	−1.65710
$527$	−3.11651	−0.135757
$528$	0	0
$529$	−4.66413	−0.202788
$530$	0	0
$531$	0	0
$532$	14.0604	0.609594
$533$	−1.87406	−0.0811747
$534$	0	0
$535$	0	0
$536$	3.06832	0.132531
$537$	0	0
$538$	−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
$543$	0	0
$544$	11.9701	0.513212
$545$	0	0
$546$	0	0
$547$	19.1811	0.820125	0.410063	−	0.912057i	$-0.365507\pi$
$547$	19.1811	0.820125	0.410063	+	0.912057i	$0.365507\pi$
$548$	2.15676	0.0921320
$549$	0	0
$550$	0	0
$551$	13.7848	0.587251
$552$	0	0
$553$	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	$0.364827\pi$
$558$	0	0
$559$	−21.9065	−0.926547
$560$	0	0
$561$	0	0
$562$	−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	$0.296090\pi$
$564$	0	0
$565$	0	0
$566$	−5.54895	−0.233240
$567$	0	0
$568$	−6.96486	−0.292239
$569$	−38.1347	−1.59869	−0.799344	−	0.600873i	$-0.794820\pi$
$569$	−38.1347	−1.59869	−0.799344	+	0.600873i	$0.794820\pi$
$570$	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
$573$	0	0
$574$	1.31086	0.0547144
$575$	0	0
$576$	0	0
$577$	−5.75214	−0.239465	−0.119732	−	0.992806i	$-0.538204\pi$
$577$	−5.75214	−0.239465	−0.119732	+	0.992806i	$0.538204\pi$
$578$	−17.1507	−0.713373
$579$	0	0
$580$	0	0
$581$	6.24296	0.259002
$582$	0	0
$583$	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	$0.627142\pi$
$588$	0	0
$589$	4.69752	0.193558
$590$	0	0
$591$	0	0
$592$	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	$0.212877\pi$
$594$	0	0
$595$	0	0
$596$	1.44198	0.0590657
$597$	0	0
$598$	−28.6790	−1.17277
$599$	31.6403	1.29279	0.646394	−	0.763004i	$-0.276276\pi$
$599$	31.6403	1.29279	0.646394	+	0.763004i	$0.276276\pi$
$600$	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	0
$604$	−4.49443	−0.182876
$605$	0	0
$606$	0	0
$607$	29.7256	1.20653	0.603263	−	0.797543i	$-0.293867\pi$
$607$	29.7256	1.20653	0.603263	+	0.797543i	$0.293867\pi$
$608$	−18.0425	−0.731719
$609$	0	0
$610$	0	0
$611$	−17.3776	−0.703023
$612$	0	0
$613$	−49.0260	−1.98014	−0.990071	−	0.140569i	$-0.955107\pi$
$613$	−49.0260	−1.98014	−0.990071	+	0.140569i	$0.955107\pi$
$614$	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	$0.519490\pi$
$618$	0	0
$619$	−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
$621$	0	0
$622$	65.3780	2.62142
$623$	−7.80075	−0.312531
$624$	0	0
$625$	0	0
$626$	−14.0565	−0.561809
$627$	0	0
$628$	−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
$633$	0	0
$634$	44.5942	1.77106
$635$	0	0
$636$	0	0
$637$	17.8848	0.708621
$638$	7.69303	0.304570
$639$	0	0
$640$	0	0
$641$	−37.1115	−1.46581	−0.732907	−	0.680329i	$-0.761837\pi$
$641$	−37.1115	−1.46581	−0.732907	+	0.680329i	$0.761837\pi$
$642$	0	0
$643$	19.8629	0.783314	0.391657	−	0.920111i	$-0.371902\pi$
$643$	19.8629	0.783314	0.391657	+	0.920111i	$0.371902\pi$
$644$	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	$0.474425\pi$
$648$	0	0
$649$	12.5793	0.493781
$650$	0	0
$651$	0	0
$652$	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	$0.489831\pi$
$654$	0	0
$655$	0	0
$656$	0.952025	0.0371703
$657$	0	0
$658$	12.1552	0.473860
$659$	18.1615	0.707473	0.353737	−	0.935345i	$-0.384911\pi$
$659$	18.1615	0.707473	0.353737	+	0.935345i	$0.384911\pi$
$660$	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
$663$	0	0
$664$	26.7314	1.03738
$665$	0	0
$666$	0	0
$667$	12.5656	0.486541
$668$	−38.3226	−1.48275
$669$	0	0
$670$	0	0
$671$	−5.96785	−0.230386
$672$	0	0
$673$	−1.63580	−0.0630554	−0.0315277	−	0.999503i	$-0.510037\pi$
$673$	−1.63580	−0.0630554	−0.0315277	+	0.999503i	$0.510037\pi$
$674$	−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	$0.549706\pi$
$678$	0	0
$679$	−14.3614	−0.551142
$680$	0	0
$681$	0	0
$682$	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	$0.182183\pi$
$684$	0	0
$685$	0	0
$686$	−26.4440	−1.00964
$687$	0	0
$688$	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	0
$694$	−67.8792	−2.57666
$695$	0	0
$696$	0	0
$697$	−2.05233	−0.0777376
$698$	31.8020	1.20372
$699$	0	0
$700$	0	0
$701$	−7.91834	−0.299072	−0.149536	−	0.988756i	$-0.547778\pi$
$701$	−7.91834	−0.299072	−0.149536	+	0.988756i	$0.547778\pi$
$702$	0	0
$703$	53.6093	2.02191
$704$	−13.2899	−0.500883
$705$	0	0
$706$	−14.5177	−0.546380
$707$	5.18824	0.195124
$708$	0	0
$709$	−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
$711$	0	0
$712$	−33.4017	−1.25178
$713$	4.28204	0.160364
$714$	0	0
$715$	0	0
$716$	−35.8797	−1.34089
$717$	0	0
$718$	53.7675	2.00658
$719$	12.2009	0.455019	0.227509	−	0.973776i	$-0.426942\pi$
$719$	12.2009	0.455019	0.227509	+	0.973776i	$0.426942\pi$
$720$	0	0
$721$	9.12652	0.339890
$722$	7.21728	0.268599
$723$	0	0
$724$	24.8285	0.922744
$725$	0	0
$726$	0	0
$727$	−18.7571	−0.695663	−0.347832	−	0.937557i	$-0.613082\pi$
$727$	−18.7571	−0.695663	−0.347832	+	0.937557i	$0.613082\pi$
$728$	−8.71713	−0.323078
$729$	0	0
$730$	0	0
$731$	−23.9904	−0.887316
$732$	0	0
$733$	47.4991	1.75442	0.877209	−	0.480108i	$-0.159402\pi$
$733$	47.4991	1.75442	0.877209	+	0.480108i	$0.159402\pi$
$734$	−21.4255	−0.790830
$735$	0	0
$736$	−16.4467	−0.606234
$737$	0.943752	0.0347636
$738$	0	0
$739$	−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
$741$	0	0
$742$	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	$0.468882\pi$
$744$	0	0
$745$	0	0
$746$	−20.0981	−0.735842
$747$	0	0
$748$	−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
$753$	0	0
$754$	−19.6537	−0.715747
$755$	0	0
$756$	0	0
$757$	21.4070	0.778049	0.389024	−	0.921227i	$-0.372812\pi$
$757$	21.4070	0.778049	0.389024	+	0.921227i	$0.372812\pi$
$758$	−53.9159	−1.95831
$759$	0	0
$760$	0	0
$761$	−2.43950	−0.0884319	−0.0442160	−	0.999022i	$-0.514079\pi$
$761$	−2.43950	−0.0884319	−0.0442160	+	0.999022i	$0.514079\pi$
$762$	0	0
$763$	2.79272	0.101103
$764$	−57.1472	−2.06751
$765$	0	0
$766$	−32.4010	−1.17070
$767$	−32.1369	−1.16040
$768$	0	0
$769$	−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
$771$	0	0
$772$	−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	$0.565997\pi$
$774$	0	0
$775$	0	0
$776$	−61.4936	−2.20749
$777$	0	0
$778$	71.8885	2.57733
$779$	3.09348	0.110836
$780$	0	0
$781$	−2.14225	−0.0766556
$782$	−31.4071	−1.12312
$783$	0	0
$784$	−9.08548	−0.324481
$785$	0	0
$786$	0	0
$787$	−0.723361	−0.0257850	−0.0128925	−	0.999917i	$-0.504104\pi$
$787$	−0.723361	−0.0257850	−0.0128925	+	0.999917i	$0.504104\pi$
$788$	−81.9928	−2.92087
$789$	0	0
$790$	0	0
$791$	15.0020	0.533409
$792$	0	0
$793$	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	$0.639798\pi$
$798$	0	0
$799$	−19.0307	−0.673256
$800$	0	0
$801$	0	0
$802$	−25.7507	−0.909288
$803$	−7.74213	−0.273214
$804$	0	0
$805$	0	0
$806$	−6.69752	−0.235910
$807$	0	0
$808$	22.2153	0.781531
$809$	22.5695	0.793500	0.396750	−	0.917927i	$-0.370138\pi$
$809$	22.5695	0.793500	0.396750	+	0.917927i	$0.370138\pi$
$810$	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
$813$	0	0
$814$	29.9184	1.04864
$815$	0	0
$816$	0	0
$817$	36.1607	1.26510
$818$	−60.1042	−2.10149
$819$	0	0
$820$	0	0
$821$	9.51769	0.332170	0.166085	−	0.986111i	$-0.446887\pi$
$821$	9.51769	0.332170	0.166085	+	0.986111i	$0.446887\pi$
$822$	0	0
$823$	16.2177	0.565313	0.282656	−	0.959221i	$-0.408784\pi$
$823$	16.2177	0.565313	0.282656	+	0.959221i	$0.408784\pi$
$824$	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	$0.812176\pi$
$828$	0	0
$829$	−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
$831$	0	0
$832$	33.9524	1.17709
$833$	19.5861	0.678617
$834$	0	0
$835$	0	0
$836$	18.5176	0.640445
$837$	0	0
$838$	−72.7978	−2.51476
$839$	47.3970	1.63633	0.818164	−	0.574985i	$-0.194992\pi$
$839$	47.3970	1.63633	0.818164	+	0.574985i	$0.194992\pi$
$840$	0	0
$841$	−20.3888	−0.703063
$842$	84.7546	2.92084
$843$	0	0
$844$	56.6520	1.95004
$845$	0	0
$846$	0	0
$847$	−8.25432	−0.283622
$848$	18.1335	0.622706
$849$	0	0
$850$	0	0
$851$	48.8678	1.67517
$852$	0	0
$853$	2.74048	0.0938322	0.0469161	−	0.998899i	$-0.485061\pi$
$853$	2.74048	0.0938322	0.0469161	+	0.998899i	$0.485061\pi$
$854$	−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	$0.444097\pi$
$858$	0	0
$859$	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
$861$	0	0
$862$	−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	$0.518163\pi$
$864$	0	0
$865$	0	0
$866$	69.1034	2.34823
$867$	0	0
$868$	2.99315	0.101594
$869$	4.15665	0.141005
$870$	0	0
$871$	−2.41105	−0.0816952
$872$	11.9580	0.404950
$873$	0	0
$874$	47.3400	1.60130
$875$	0	0
$876$	0	0
$877$	−40.6601	−1.37299	−0.686497	−	0.727133i	$-0.740853\pi$
$877$	−40.6601	−1.37299	−0.686497	+	0.727133i	$0.740853\pi$
$878$	74.0908	2.50044
$879$	0	0
$880$	0	0
$881$	−0.745750	−0.0251250	−0.0125625	−	0.999921i	$-0.503999\pi$
$881$	−0.745750	−0.0251250	−0.0125625	+	0.999921i	$0.503999\pi$
$882$	0	0
$883$	−3.94352	−0.132710	−0.0663550	−	0.997796i	$-0.521137\pi$
$883$	−3.94352	−0.132710	−0.0663550	+	0.997796i	$0.521137\pi$
$884$	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	$0.812140\pi$
$888$	0	0
$889$	−13.1543	−0.441181
$890$	0	0
$891$	0	0
$892$	75.8651	2.54015
$893$	28.6849	0.959904
$894$	0	0
$895$	0	0
$896$	−17.2517	−0.576338
$897$	0	0
$898$	−30.3358	−1.01232
$899$	2.93448	0.0978704
$900$	0	0
$901$	−39.0913	−1.30232
$902$	1.72642	0.0574834
$903$	0	0
$904$	64.2363	2.13647
$905$	0	0
$906$	0	0
$907$	11.2154	0.372403	0.186201	−	0.982512i	$-0.440382\pi$
$907$	11.2154	0.372403	0.186201	+	0.982512i	$0.440382\pi$
$908$	86.9402	2.88521
$909$	0	0
$910$	0	0
$911$	22.5579	0.747375	0.373688	−	0.927555i	$-0.378093\pi$
$911$	22.5579	0.747375	0.373688	+	0.927555i	$0.378093\pi$
$912$	0	0
$913$	8.22203	0.272110
$914$	−56.4771	−1.86810
$915$	0	0
$916$	−99.7045	−3.29433
$917$	−3.10789	−0.102632
$918$	0	0
$919$	−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
$921$	0	0
$922$	−49.6053	−1.63366
$923$	5.47289	0.180142
$924$	0	0
$925$	0	0
$926$	−53.3260	−1.75240
$927$	0	0
$928$	−11.2709	−0.369986
$929$	9.90563	0.324993	0.162497	−	0.986709i	$-0.448045\pi$
$929$	9.90563	0.324993	0.162497	+	0.986709i	$0.448045\pi$
$930$	0	0
$931$	−29.5221	−0.967548
$932$	78.8763	2.58368
$933$	0	0
$934$	−87.1446	−2.85146
$935$	0	0
$936$	0	0
$937$	7.99567	0.261207	0.130604	−	0.991435i	$-0.458309\pi$
$937$	7.99567	0.261207	0.130604	+	0.991435i	$0.458309\pi$
$938$	1.68647	0.0550652
$939$	0	0
$940$	0	0
$941$	−36.2677	−1.18229	−0.591147	−	0.806564i	$-0.701325\pi$
$941$	−36.2677	−1.18229	−0.591147	+	0.806564i	$0.701325\pi$
$942$	0	0
$943$	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	$0.261234\pi$
$948$	0	0
$949$	19.7792	0.642059
$950$	0	0
$951$	0	0
$952$	−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	$0.473263\pi$
$954$	0	0
$955$	0	0
$956$	79.8025	2.58100
$957$	0	0
$958$	−30.2833	−0.978410
$959$	0.515477	0.0166456
$960$	0	0
$961$	1.00000	0.0322581
$962$	−76.4339	−2.46433
$963$	0	0
$964$	−71.4068	−2.29986
$965$	0	0
$966$	0	0
$967$	−43.9028	−1.41182	−0.705909	−	0.708303i	$-0.749461\pi$
$967$	−43.9028	−1.41182	−0.705909	+	0.708303i	$0.749461\pi$
$968$	−35.3438	−1.13599
$969$	0	0
$970$	0	0
$971$	−48.4479	−1.55477	−0.777384	−	0.629026i	$-0.783454\pi$
$971$	−48.4479	−1.55477	−0.777384	+	0.629026i	$0.783454\pi$
$972$	0	0
$973$	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	$0.410045\pi$
$978$	0	0
$979$	−10.2737	−0.328348
$980$	0	0
$981$	0	0
$982$	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	$0.434320\pi$
$984$	0	0
$985$	0	0
$986$	−21.5233	−0.685441
$987$	0	0
$988$	−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
$993$	0	0
$994$	−3.82816	−0.121422
$995$	0	0
$996$	0	0
$997$	−29.6158	−0.937940	−0.468970	−	0.883214i	$-0.655375\pi$
$997$	−29.6158	−0.937940	−0.468970	+	0.883214i	$0.655375\pi$
$998$	64.0320	2.02690
$999$	0	0

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
775.2.a.i.1.2	✓	5	3.2	odd	2
775.2.a.j.1.4	yes	5	15.14	odd	2
775.2.b.h.249.2		10	15.2	even	4
775.2.b.h.249.9		10	15.8	even	4
6975.2.a.bq.1.2		5	5.4	even	2
6975.2.a.bx.1.4		5	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 6975 = 3^{2} \cdot 5^{2} \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6975.a (trivial)

\(f(q)\)	\(=\)	\(q+2.35347 q^{2} +3.53883 q^{4} +0.845802 q^{7} +3.62160 q^{8} +1.11393 q^{11} -2.84580 q^{13} +1.99057 q^{14} +1.44567 q^{16} -3.11651 q^{17} +4.69752 q^{19} +2.62160 q^{22} +4.28204 q^{23} -6.69752 q^{26} +2.99315 q^{28} +2.93448 q^{29} +1.00000 q^{31} -3.84086 q^{32} -7.33461 q^{34} +11.4123 q^{37} +11.0555 q^{38} +0.658536 q^{41} +7.69784 q^{43} +3.94200 q^{44} +10.0777 q^{46} +6.10640 q^{47} -6.28462 q^{49} -10.0708 q^{52} +12.5433 q^{53} +3.06315 q^{56} +6.90622 q^{58} +11.2928 q^{59} -5.35748 q^{61} +2.35347 q^{62} -11.9307 q^{64} +0.847229 q^{67} -11.0288 q^{68} -1.92315 q^{71} -6.95030 q^{73} +26.8585 q^{74} +16.6237 q^{76} +0.942162 q^{77} +3.73152 q^{79} +1.54985 q^{82} +7.38112 q^{83} +18.1166 q^{86} +4.03420 q^{88} -9.22291 q^{89} -2.40698 q^{91} +15.1534 q^{92} +14.3713 q^{94} -16.9797 q^{97} -14.7907 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 4 q^{2} + 6 q^{4} - 6 q^{7} + 15 q^{8} - 4 q^{13} - 2 q^{14} + 20 q^{16} + 11 q^{17} - 4 q^{19} + 10 q^{22} + 12 q^{23} - 6 q^{26} - 18 q^{28} + 6 q^{29} + 5 q^{31} + 29 q^{32} - 5 q^{34} + 2 q^{37}+ \cdots - 22 q^{98}+O(q^{100}) \) 5 * q + 4 * q^2 + 6 * q^4 - 6 * q^7 + 15 * q^8 - 4 * q^13 - 2 * q^14 + 20 * q^16 + 11 * q^17 - 4 * q^19 + 10 * q^22 + 12 * q^23 - 6 * q^26 - 18 * q^28 + 6 * q^29 + 5 * q^31 + 29 * q^32 - 5 * q^34 + 2 * q^37 - 6 * q^38 + 2 * q^41 - 7 * q^43 + 28 * q^44 + 27 * q^46 + 8 * q^47 - q^49 + 6 * q^52 + 25 * q^53 - 35 * q^56 + q^58 - 4 * q^59 - 17 * q^61 + 4 * q^62 + 27 * q^64 + 13 * q^67 + 18 * q^68 + 6 * q^71 - 7 * q^73 - 6 * q^74 - 5 * q^76 + 7 * q^77 + 12 * q^79 + 21 * q^82 - 4 * q^83 + 41 * q^86 + 49 * q^88 + 3 * q^89 - 22 * q^91 + 34 * q^92 - 34 * q^94 - 25 * q^97 - 22 * q^98

Introduction

L-functions

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Database

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