LMFDB - Embedded newform 783.2.a.i.1.5

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [783,2,Mod(1,783)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(783, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([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("783.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 783 = 3^{3} \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	783.a (trivial)

Newform invariants

comment:select newform

sage:traces = [8,-1,0,11] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(4)] == traces)

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

Self dual:	yes
Analytic conductor:	$6.25228647827$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{7} - 13x^{6} + 11x^{5} + 49x^{4} - 37x^{3} - 43x^{2} + 35x - 3 $ x^8 - x^7 - 13x^6 + 11x^5 + 49x^4 - 37x^3 - 43x^2 + 35x - 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			1.5
Root			$0.0985138$ of defining polynomial
Character	$\chi$	$=$	783.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-0.0985138 q^{2} -1.99030 q^{4} -4.03464 q^{5} -4.56635 q^{7} +0.393099 q^{8} +0.397468 q^{10} +0.732700 q^{11} +2.33523 q^{13} +0.449848 q^{14} +3.94186 q^{16} -5.93885 q^{17} +5.44014 q^{19} +8.03012 q^{20} -0.0721810 q^{22} -1.94595 q^{23} +11.2783 q^{25} -0.230053 q^{26} +9.08837 q^{28} +1.00000 q^{29} -5.73786 q^{31} -1.17453 q^{32} +0.585059 q^{34} +18.4236 q^{35} +2.29226 q^{37} -0.535929 q^{38} -1.58601 q^{40} -8.10257 q^{41} +6.23167 q^{43} -1.45829 q^{44} +0.191703 q^{46} +6.42814 q^{47} +13.8515 q^{49} -1.11107 q^{50} -4.64780 q^{52} +7.93489 q^{53} -2.95618 q^{55} -1.79503 q^{56} -0.0985138 q^{58} -12.5382 q^{59} -12.5422 q^{61} +0.565258 q^{62} -7.76802 q^{64} -9.42182 q^{65} +6.87092 q^{67} +11.8201 q^{68} -1.81497 q^{70} -3.01388 q^{71} +4.76256 q^{73} -0.225819 q^{74} -10.8275 q^{76} -3.34576 q^{77} -0.854385 q^{79} -15.9040 q^{80} +0.798215 q^{82} -6.55761 q^{83} +23.9611 q^{85} -0.613905 q^{86} +0.288024 q^{88} +1.41902 q^{89} -10.6635 q^{91} +3.87302 q^{92} -0.633261 q^{94} -21.9490 q^{95} +14.7352 q^{97} -1.36456 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - q^{2} + 11 q^{4} + 2 q^{5} + 5 q^{7} - 3 q^{8} + 7 q^{10} + 2 q^{11} + 11 q^{13} - 8 q^{14} + 21 q^{16} - 6 q^{17} + 5 q^{19} + 27 q^{20} + 17 q^{22} + 4 q^{23} + 30 q^{25} - 16 q^{26} + 2 q^{28}+ \cdots + 11 q^{98}+O(q^{100}) $ 8 * q - q^2 + 11 * q^4 + 2 * q^5 + 5 * q^7 - 3 * q^8 + 7 * q^10 + 2 * q^11 + 11 * q^13 - 8 * q^14 + 21 * q^16 - 6 * q^17 + 5 * q^19 + 27 * q^20 + 17 * q^22 + 4 * q^23 + 30 * q^25 - 16 * q^26 + 2 * q^28 + 8 * q^29 - q^31 - 37 * q^32 - 9 * q^34 - q^35 + 26 * q^37 + 14 * q^38 + 2 * q^40 - 19 * q^41 + 16 * q^43 + q^44 - 15 * q^46 + 18 * q^47 + 27 * q^49 - 4 * q^50 + 7 * q^52 + 19 * q^53 + 6 * q^55 + 27 * q^56 - q^58 - 13 * q^59 + 2 * q^61 - 7 * q^62 + 13 * q^64 + 19 * q^65 + 25 * q^67 - 22 * q^68 - 22 * q^70 + 7 * q^71 + 11 * q^73 + 9 * q^74 - 31 * q^76 - 26 * q^77 + 16 * q^79 + 19 * q^80 + 7 * q^82 - 19 * q^83 + 10 * q^85 - 63 * q^86 + 32 * q^88 + 16 * q^89 + 6 * q^91 + 101 * q^92 - 34 * q^94 - 43 * q^95 + 31 * q^97 + 11 * 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 $

$2$	−0.0985138	−0.0696598	−0.0348299	−	0.999393i	$-0.511089\pi$
$2$	−0.0985138	−0.0696598	−0.0348299	+	0.999393i	$0.511089\pi$
$3$	0	0
$3$	0	0
$4$	−1.99030	−0.995148
$5$	−4.03464	−1.80435	−0.902173	−	0.431375i	$-0.858028\pi$
$5$	−4.03464	−1.80435	−0.902173	+	0.431375i	$0.858028\pi$
$6$	0	0
$7$	−4.56635	−1.72592	−0.862958	−	0.505275i	$-0.831391\pi$
$7$	−4.56635	−1.72592	−0.862958	+	0.505275i	$0.831391\pi$
$8$	0.393099	0.138982
$9$	0	0
$10$	0.397468	0.125690
$11$	0.732700	0.220917	0.110459	−	0.993881i	$-0.464768\pi$
$11$	0.732700	0.220917	0.110459	+	0.993881i	$0.464768\pi$
$12$	0	0
$13$	2.33523	0.647677	0.323838	−	0.946112i	$-0.395027\pi$
$13$	2.33523	0.647677	0.323838	+	0.946112i	$0.395027\pi$
$14$	0.449848	0.120227
$15$	0	0
$16$	3.94186	0.985466
$17$	−5.93885	−1.44038	−0.720192	−	0.693775i	$-0.755946\pi$
$17$	−5.93885	−1.44038	−0.720192	+	0.693775i	$0.755946\pi$
$18$	0	0
$19$	5.44014	1.24805	0.624027	−	0.781403i	$-0.285495\pi$
$19$	5.44014	1.24805	0.624027	+	0.781403i	$0.285495\pi$
$20$	8.03012	1.79559
$21$	0	0
$22$	−0.0721810	−0.0153890
$23$	−1.94595	−0.405759	−0.202879	−	0.979204i	$-0.565030\pi$
$23$	−1.94595	−0.405759	−0.202879	+	0.979204i	$0.565030\pi$
$24$	0	0
$25$	11.2783	2.25566
$26$	−0.230053	−0.0451170
$27$	0	0
$28$	9.08837	1.71754
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	0	0
$31$	−5.73786	−1.03055	−0.515275	−	0.857025i	$-0.672310\pi$
$31$	−5.73786	−1.03055	−0.515275	+	0.857025i	$0.672310\pi$
$32$	−1.17453	−0.207629
$33$	0	0
$34$	0.585059	0.100337
$35$	18.4236	3.11415
$36$	0	0
$37$	2.29226	0.376845	0.188423	−	0.982088i	$-0.439662\pi$
$37$	2.29226	0.376845	0.188423	+	0.982088i	$0.439662\pi$
$38$	−0.535929	−0.0869392
$39$	0	0
$40$	−1.58601	−0.250771
$41$	−8.10257	−1.26541	−0.632704	−	0.774393i	$-0.718055\pi$
$41$	−8.10257	−1.26541	−0.632704	+	0.774393i	$0.718055\pi$
$42$	0	0
$43$	6.23167	0.950320	0.475160	−	0.879899i	$-0.342390\pi$
$43$	6.23167	0.950320	0.475160	+	0.879899i	$0.342390\pi$
$44$	−1.45829	−0.219845
$45$	0	0
$46$	0.191703	0.0282651
$47$	6.42814	0.937641	0.468820	−	0.883294i	$-0.344679\pi$
$47$	6.42814	0.937641	0.468820	+	0.883294i	$0.344679\pi$
$48$	0	0
$49$	13.8515	1.97879
$50$	−1.11107	−0.157129
$51$	0	0
$52$	−4.64780	−0.644534
$53$	7.93489	1.08994	0.544970	−	0.838455i	$-0.316541\pi$
$53$	7.93489	1.08994	0.544970	+	0.838455i	$0.316541\pi$
$54$	0	0
$55$	−2.95618	−0.398611
$56$	−1.79503	−0.239871
$57$	0	0
$58$	−0.0985138	−0.0129355
$59$	−12.5382	−1.63233	−0.816167	−	0.577816i	$-0.803905\pi$
$59$	−12.5382	−1.63233	−0.816167	+	0.577816i	$0.803905\pi$
$60$	0	0
$61$	−12.5422	−1.60586	−0.802931	−	0.596073i	$-0.796727\pi$
$61$	−12.5422	−1.60586	−0.802931	+	0.596073i	$0.796727\pi$
$62$	0.565258	0.0717879
$63$	0	0
$64$	−7.76802	−0.971003
$65$	−9.42182	−1.16863
$66$	0	0
$67$	6.87092	0.839416	0.419708	−	0.907659i	$-0.362132\pi$
$67$	6.87092	0.839416	0.419708	+	0.907659i	$0.362132\pi$
$68$	11.8201	1.43339
$69$	0	0
$70$	−1.81497	−0.216931
$71$	−3.01388	−0.357682	−0.178841	−	0.983878i	$-0.557235\pi$
$71$	−3.01388	−0.357682	−0.178841	+	0.983878i	$0.557235\pi$
$72$	0	0
$73$	4.76256	0.557416	0.278708	−	0.960376i	$-0.410094\pi$
$73$	4.76256	0.557416	0.278708	+	0.960376i	$0.410094\pi$
$74$	−0.225819	−0.0262510
$75$	0	0
$76$	−10.8275	−1.24200
$77$	−3.34576	−0.381285
$78$	0	0
$79$	−0.854385	−0.0961258	−0.0480629	−	0.998844i	$-0.515305\pi$
$79$	−0.854385	−0.0961258	−0.0480629	+	0.998844i	$0.515305\pi$
$80$	−15.9040	−1.77812
$81$	0	0
$82$	0.798215	0.0881481
$83$	−6.55761	−0.719791	−0.359895	−	0.932993i	$-0.617188\pi$
$83$	−6.55761	−0.719791	−0.359895	+	0.932993i	$0.617188\pi$
$84$	0	0
$85$	23.9611	2.59895
$86$	−0.613905	−0.0661991
$87$	0	0
$88$	0.288024	0.0307034
$89$	1.41902	0.150415	0.0752077	−	0.997168i	$-0.476038\pi$
$89$	1.41902	0.150415	0.0752077	+	0.997168i	$0.476038\pi$
$90$	0	0
$91$	−10.6635	−1.11784
$92$	3.87302	0.403790
$93$	0	0
$94$	−0.633261	−0.0653158
$95$	−21.9490	−2.25192
$96$	0	0
$97$	14.7352	1.49614	0.748068	−	0.663623i	$-0.230982\pi$
$97$	14.7352	1.49614	0.748068	+	0.663623i	$0.230982\pi$
$98$	−1.36456	−0.137842
$99$	0	0
$100$	−22.4472	−2.24472
$101$	6.16888	0.613826	0.306913	−	0.951738i	$-0.400704\pi$
$101$	6.16888	0.613826	0.306913	+	0.951738i	$0.400704\pi$
$102$	0	0
$103$	−0.991263	−0.0976721	−0.0488360	−	0.998807i	$-0.515551\pi$
$103$	−0.991263	−0.0976721	−0.0488360	+	0.998807i	$0.515551\pi$
$104$	0.917978	0.0900151
$105$	0	0
$106$	−0.781696	−0.0759250
$107$	18.8298	1.82034	0.910171	−	0.414234i	$-0.135950\pi$
$107$	18.8298	1.82034	0.910171	+	0.414234i	$0.135950\pi$
$108$	0	0
$109$	−1.57376	−0.150738	−0.0753692	−	0.997156i	$-0.524014\pi$
$109$	−1.57376	−0.150738	−0.0753692	+	0.997156i	$0.524014\pi$
$110$	0.291224	0.0277672
$111$	0	0
$112$	−17.9999	−1.70083
$113$	18.5165	1.74188	0.870941	−	0.491388i	$-0.163510\pi$
$113$	18.5165	1.74188	0.870941	+	0.491388i	$0.163510\pi$
$114$	0	0
$115$	7.85121	0.732129
$116$	−1.99030	−0.184794
$117$	0	0
$118$	1.23519	0.113708
$119$	27.1188	2.48598
$120$	0	0
$121$	−10.4632	−0.951196
$122$	1.23558	0.111864
$123$	0	0
$124$	11.4200	1.02555
$125$	−25.3307	−2.26565
$126$	0	0
$127$	−4.86735	−0.431907	−0.215954	−	0.976404i	$-0.569286\pi$
$127$	−4.86735	−0.431907	−0.215954	+	0.976404i	$0.569286\pi$
$128$	3.11431	0.275269
$129$	0	0
$130$	0.928179	0.0814067
$131$	12.1065	1.05775	0.528875	−	0.848700i	$-0.322614\pi$
$131$	12.1065	1.05775	0.528875	+	0.848700i	$0.322614\pi$
$132$	0	0
$133$	−24.8416	−2.15404
$134$	−0.676880	−0.0584736
$135$	0	0
$136$	−2.33456	−0.200187
$137$	−8.28144	−0.707531	−0.353766	−	0.935334i	$-0.615099\pi$
$137$	−8.28144	−0.707531	−0.353766	+	0.935334i	$0.615099\pi$
$138$	0	0
$139$	15.6455	1.32704	0.663519	−	0.748160i	$-0.269062\pi$
$139$	15.6455	1.32704	0.663519	+	0.748160i	$0.269062\pi$
$140$	−36.6683	−3.09904
$141$	0	0
$142$	0.296909	0.0249161
$143$	1.71102	0.143083
$144$	0	0
$145$	−4.03464	−0.335058
$146$	−0.469178	−0.0388295
$147$	0	0
$148$	−4.56228	−0.375017
$149$	11.0933	0.908798	0.454399	−	0.890798i	$-0.349854\pi$
$149$	11.0933	0.908798	0.454399	+	0.890798i	$0.349854\pi$
$150$	0	0
$151$	−21.6652	−1.76309	−0.881544	−	0.472103i	$-0.843495\pi$
$151$	−21.6652	−1.76309	−0.881544	+	0.472103i	$0.843495\pi$
$152$	2.13852	0.173457
$153$	0	0
$154$	0.329604	0.0265602
$155$	23.1502	1.85947
$156$	0	0
$157$	14.6471	1.16897	0.584483	−	0.811406i	$-0.301297\pi$
$157$	14.6471	1.16897	0.584483	+	0.811406i	$0.301297\pi$
$158$	0.0841687	0.00669610
$159$	0	0
$160$	4.73879	0.374634
$161$	8.88589	0.700306
$162$	0	0
$163$	7.44147	0.582861	0.291431	−	0.956592i	$-0.405869\pi$
$163$	7.44147	0.582861	0.291431	+	0.956592i	$0.405869\pi$
$164$	16.1265	1.25927
$165$	0	0
$166$	0.646015	0.0501405
$167$	7.84306	0.606915	0.303457	−	0.952845i	$-0.401859\pi$
$167$	7.84306	0.606915	0.303457	+	0.952845i	$0.401859\pi$
$168$	0	0
$169$	−7.54669	−0.580515
$170$	−2.36050	−0.181042
$171$	0	0
$172$	−12.4029	−0.945709
$173$	−4.93433	−0.375150	−0.187575	−	0.982250i	$-0.560063\pi$
$173$	−4.93433	−0.375150	−0.187575	+	0.982250i	$0.560063\pi$
$174$	0	0
$175$	−51.5006	−3.89308
$176$	2.88820	0.217706
$177$	0	0
$178$	−0.139793	−0.0104779
$179$	−3.58843	−0.268212	−0.134106	−	0.990967i	$-0.542816\pi$
$179$	−3.58843	−0.268212	−0.134106	+	0.990967i	$0.542816\pi$
$180$	0	0
$181$	11.3273	0.841951	0.420975	−	0.907072i	$-0.361688\pi$
$181$	11.3273	0.841951	0.420975	+	0.907072i	$0.361688\pi$
$182$	1.05050	0.0778682
$183$	0	0
$184$	−0.764952	−0.0563930
$185$	−9.24845	−0.679959
$186$	0	0
$187$	−4.35140	−0.318206
$188$	−12.7939	−0.933091
$189$	0	0
$190$	2.16228	0.156868
$191$	10.2667	0.742872	0.371436	−	0.928459i	$-0.378866\pi$
$191$	10.2667	0.742872	0.371436	+	0.928459i	$0.378866\pi$
$192$	0	0
$193$	3.41822	0.246049	0.123025	−	0.992404i	$-0.460741\pi$
$193$	3.41822	0.246049	0.123025	+	0.992404i	$0.460741\pi$
$194$	−1.45162	−0.104220
$195$	0	0
$196$	−27.5686	−1.96919
$197$	4.34099	0.309282	0.154641	−	0.987971i	$-0.450578\pi$
$197$	4.34099	0.309282	0.154641	+	0.987971i	$0.450578\pi$
$198$	0	0
$199$	−1.93557	−0.137209	−0.0686045	−	0.997644i	$-0.521855\pi$
$199$	−1.93557	−0.137209	−0.0686045	+	0.997644i	$0.521855\pi$
$200$	4.43349	0.313495
$201$	0	0
$202$	−0.607720	−0.0427590
$203$	−4.56635	−0.320495
$204$	0	0
$205$	32.6909	2.28323
$206$	0.0976531	0.00680381
$207$	0	0
$208$	9.20517	0.638264
$209$	3.98599	0.275717
$210$	0	0
$211$	0.533900	0.0367552	0.0183776	−	0.999831i	$-0.494150\pi$
$211$	0.533900	0.0367552	0.0183776	+	0.999831i	$0.494150\pi$
$212$	−15.7928	−1.08465
$213$	0	0
$214$	−1.85499	−0.126805
$215$	−25.1425	−1.71471
$216$	0	0
$217$	26.2010	1.77864
$218$	0.155037	0.0105004
$219$	0	0
$220$	5.88367	0.396677
$221$	−13.8686	−0.932903
$222$	0	0
$223$	11.8987	0.796796	0.398398	−	0.917213i	$-0.369566\pi$
$223$	11.8987	0.796796	0.398398	+	0.917213i	$0.369566\pi$
$224$	5.36329	0.358350
$225$	0	0
$226$	−1.82413	−0.121339
$227$	−18.8867	−1.25355	−0.626777	−	0.779199i	$-0.715626\pi$
$227$	−18.8867	−1.25355	−0.626777	+	0.779199i	$0.715626\pi$
$228$	0	0
$229$	−4.58382	−0.302907	−0.151454	−	0.988464i	$-0.548395\pi$
$229$	−4.58382	−0.302907	−0.151454	+	0.988464i	$0.548395\pi$
$230$	−0.773453	−0.0510000
$231$	0	0
$232$	0.393099	0.0258082
$233$	9.87450	0.646900	0.323450	−	0.946245i	$-0.395157\pi$
$233$	9.87450	0.646900	0.323450	+	0.946245i	$0.395157\pi$
$234$	0	0
$235$	−25.9352	−1.69183
$236$	24.9547	1.62441
$237$	0	0
$238$	−2.67158	−0.173173
$239$	−10.5588	−0.682992	−0.341496	−	0.939883i	$-0.610934\pi$
$239$	−10.5588	−0.682992	−0.341496	+	0.939883i	$0.610934\pi$
$240$	0	0
$241$	−6.76815	−0.435975	−0.217987	−	0.975952i	$-0.569949\pi$
$241$	−6.76815	−0.435975	−0.217987	+	0.975952i	$0.569949\pi$
$242$	1.03076	0.0662601
$243$	0	0
$244$	24.9626	1.59807
$245$	−55.8858	−3.57041
$246$	0	0
$247$	12.7040	0.808336
$248$	−2.25555	−0.143227
$249$	0	0
$250$	2.49542	0.157824
$251$	−4.61752	−0.291455	−0.145728	−	0.989325i	$-0.546552\pi$
$251$	−4.61752	−0.291455	−0.145728	+	0.989325i	$0.546552\pi$
$252$	0	0
$253$	−1.42580	−0.0896392
$254$	0.479501	0.0300866
$255$	0	0
$256$	15.2292	0.951828
$257$	16.2482	1.01354	0.506768	−	0.862082i	$-0.330840\pi$
$257$	16.2482	1.01354	0.506768	+	0.862082i	$0.330840\pi$
$258$	0	0
$259$	−10.4673	−0.650404
$260$	18.7522	1.16296
$261$	0	0
$262$	−1.19266	−0.0736826
$263$	−21.7865	−1.34341	−0.671707	−	0.740817i	$-0.734439\pi$
$263$	−21.7865	−1.34341	−0.671707	+	0.740817i	$0.734439\pi$
$264$	0	0
$265$	−32.0144	−1.96663
$266$	2.44724	0.150050
$267$	0	0
$268$	−13.6752	−0.835343
$269$	−13.0841	−0.797753	−0.398876	−	0.917005i	$-0.630600\pi$
$269$	−13.0841	−0.797753	−0.398876	+	0.917005i	$0.630600\pi$
$270$	0	0
$271$	−15.5239	−0.943007	−0.471504	−	0.881864i	$-0.656289\pi$
$271$	−15.5239	−0.943007	−0.471504	+	0.881864i	$0.656289\pi$
$272$	−23.4101	−1.41945
$273$	0	0
$274$	0.815836	0.0492865
$275$	8.26361	0.498315
$276$	0	0
$277$	−22.2494	−1.33684	−0.668419	−	0.743785i	$-0.733029\pi$
$277$	−22.2494	−1.33684	−0.668419	+	0.743785i	$0.733029\pi$
$278$	−1.54130	−0.0924411
$279$	0	0
$280$	7.24228	0.432809
$281$	21.0658	1.25668	0.628341	−	0.777938i	$-0.283734\pi$
$281$	21.0658	1.25668	0.628341	+	0.777938i	$0.283734\pi$
$282$	0	0
$283$	2.90593	0.172739	0.0863697	−	0.996263i	$-0.472473\pi$
$283$	2.90593	0.172739	0.0863697	+	0.996263i	$0.472473\pi$
$284$	5.99852	0.355947
$285$	0	0
$286$	−0.168559	−0.00996713
$287$	36.9991	2.18399
$288$	0	0
$289$	18.2700	1.07470
$290$	0.397468	0.0233401
$291$	0	0
$292$	−9.47891	−0.554711
$293$	12.9581	0.757022	0.378511	−	0.925597i	$-0.376436\pi$
$293$	12.9581	0.757022	0.378511	+	0.925597i	$0.376436\pi$
$294$	0	0
$295$	50.5871	2.94529
$296$	0.901086	0.0523746
$297$	0	0
$298$	−1.09284	−0.0633067
$299$	−4.54425	−0.262801
$300$	0	0
$301$	−28.4559	−1.64017
$302$	2.13432	0.122816
$303$	0	0
$304$	21.4443	1.22992
$305$	50.6031	2.89753
$306$	0	0
$307$	5.07988	0.289924	0.144962	−	0.989437i	$-0.453694\pi$
$307$	5.07988	0.289924	0.144962	+	0.989437i	$0.453694\pi$
$308$	6.65905	0.379435
$309$	0	0
$310$	−2.28061	−0.129530
$311$	5.07042	0.287517	0.143759	−	0.989613i	$-0.454081\pi$
$311$	5.07042	0.287517	0.143759	+	0.989613i	$0.454081\pi$
$312$	0	0
$313$	−20.6528	−1.16737	−0.583683	−	0.811982i	$-0.698389\pi$
$313$	−20.6528	−1.16737	−0.583683	+	0.811982i	$0.698389\pi$
$314$	−1.44294	−0.0814299
$315$	0	0
$316$	1.70048	0.0956593
$317$	0.986895	0.0554296	0.0277148	−	0.999616i	$-0.491177\pi$
$317$	0.986895	0.0554296	0.0277148	+	0.999616i	$0.491177\pi$
$318$	0	0
$319$	0.732700	0.0410233
$320$	31.3412	1.75202
$321$	0	0
$322$	−0.875383	−0.0487832
$323$	−32.3082	−1.79768
$324$	0	0
$325$	26.3375	1.46094
$326$	−0.733088	−0.0406020
$327$	0	0
$328$	−3.18511	−0.175868
$329$	−29.3531	−1.61829
$330$	0	0
$331$	11.2757	0.619770	0.309885	−	0.950774i	$-0.399709\pi$
$331$	11.2757	0.619770	0.309885	+	0.950774i	$0.399709\pi$
$332$	13.0516	0.716298
$333$	0	0
$334$	−0.772650	−0.0422775
$335$	−27.7217	−1.51460
$336$	0	0
$337$	−8.49874	−0.462956	−0.231478	−	0.972840i	$-0.574356\pi$
$337$	−8.49874	−0.462956	−0.231478	+	0.972840i	$0.574356\pi$
$338$	0.743453	0.0404385
$339$	0	0
$340$	−47.6897	−2.58634
$341$	−4.20413	−0.227666
$342$	0	0
$343$	−31.2864	−1.68930
$344$	2.44966	0.132077
$345$	0	0
$346$	0.486100	0.0261329
$347$	5.21157	0.279772	0.139886	−	0.990168i	$-0.455326\pi$
$347$	5.21157	0.279772	0.139886	+	0.990168i	$0.455326\pi$
$348$	0	0
$349$	6.71439	0.359413	0.179707	−	0.983720i	$-0.442485\pi$
$349$	6.71439	0.359413	0.179707	+	0.983720i	$0.442485\pi$
$350$	5.07352	0.271191
$351$	0	0
$352$	−0.860575	−0.0458688
$353$	5.29679	0.281920	0.140960	−	0.990015i	$-0.454981\pi$
$353$	5.29679	0.281920	0.140960	+	0.990015i	$0.454981\pi$
$354$	0	0
$355$	12.1599	0.645382
$356$	−2.82426	−0.149686
$357$	0	0
$358$	0.353510	0.0186836
$359$	1.18134	0.0623486	0.0311743	−	0.999514i	$-0.490075\pi$
$359$	1.18134	0.0623486	0.0311743	+	0.999514i	$0.490075\pi$
$360$	0	0
$361$	10.5952	0.557640
$362$	−1.11589	−0.0586501
$363$	0	0
$364$	21.2235	1.11241
$365$	−19.2152	−1.00577
$366$	0	0
$367$	17.7121	0.924567	0.462283	−	0.886732i	$-0.347030\pi$
$367$	17.7121	0.924567	0.462283	+	0.886732i	$0.347030\pi$
$368$	−7.67068	−0.399862
$369$	0	0
$370$	0.911100	0.0473658
$371$	−36.2334	−1.88115
$372$	0	0
$373$	24.9512	1.29193	0.645963	−	0.763369i	$-0.276456\pi$
$373$	24.9512	1.29193	0.645963	+	0.763369i	$0.276456\pi$
$374$	0.428672	0.0221661
$375$	0	0
$376$	2.52690	0.130315
$377$	2.33523	0.120271
$378$	0	0
$379$	−4.01067	−0.206014	−0.103007	−	0.994681i	$-0.532846\pi$
$379$	−4.01067	−0.206014	−0.103007	+	0.994681i	$0.532846\pi$
$380$	43.6850	2.24099
$381$	0	0
$382$	−1.01141	−0.0517483
$383$	31.6739	1.61846	0.809230	−	0.587492i	$-0.199885\pi$
$383$	31.6739	1.61846	0.809230	+	0.587492i	$0.199885\pi$
$384$	0	0
$385$	13.4989	0.687969
$386$	−0.336742	−0.0171397
$387$	0	0
$388$	−29.3274	−1.48888
$389$	35.1264	1.78098	0.890489	−	0.455005i	$-0.150362\pi$
$389$	35.1264	1.78098	0.890489	+	0.455005i	$0.150362\pi$
$390$	0	0
$391$	11.5567	0.584448
$392$	5.44502	0.275015
$393$	0	0
$394$	−0.427647	−0.0215445
$395$	3.44713	0.173444
$396$	0	0
$397$	15.0595	0.755814	0.377907	−	0.925844i	$-0.376644\pi$
$397$	15.0595	0.755814	0.377907	+	0.925844i	$0.376644\pi$
$398$	0.190681	0.00955795
$399$	0	0
$400$	44.4576	2.22288
$401$	37.4374	1.86953	0.934766	−	0.355263i	$-0.115609\pi$
$401$	37.4374	1.86953	0.934766	+	0.355263i	$0.115609\pi$
$402$	0	0
$403$	−13.3992	−0.667463
$404$	−12.2779	−0.610848
$405$	0	0
$406$	0.449848	0.0223256
$407$	1.67954	0.0832517
$408$	0	0
$409$	30.4089	1.50362	0.751811	−	0.659378i	$-0.229180\pi$
$409$	30.4089	1.50362	0.751811	+	0.659378i	$0.229180\pi$
$410$	−3.22051	−0.159050
$411$	0	0
$412$	1.97291	0.0971981
$413$	57.2537	2.81727
$414$	0	0
$415$	26.4576	1.29875
$416$	−2.74279	−0.134476
$417$	0	0
$418$	−0.392675	−0.0192064
$419$	−24.6235	−1.20293	−0.601467	−	0.798897i	$-0.705417\pi$
$419$	−24.6235	−1.20293	−0.601467	+	0.798897i	$0.705417\pi$
$420$	0	0
$421$	24.1809	1.17851	0.589253	−	0.807949i	$-0.299422\pi$
$421$	24.1809	1.17851	0.589253	+	0.807949i	$0.299422\pi$
$422$	−0.0525965	−0.00256036
$423$	0	0
$424$	3.11920	0.151482
$425$	−66.9802	−3.24902
$426$	0	0
$427$	57.2719	2.77158
$428$	−37.4768	−1.81151
$429$	0	0
$430$	2.47689	0.119446
$431$	−21.9830	−1.05889	−0.529443	−	0.848346i	$-0.677599\pi$
$431$	−21.9830	−1.05889	−0.529443	+	0.848346i	$0.677599\pi$
$432$	0	0
$433$	15.0884	0.725104	0.362552	−	0.931964i	$-0.381906\pi$
$433$	15.0884	0.725104	0.362552	+	0.931964i	$0.381906\pi$
$434$	−2.58116	−0.123900
$435$	0	0
$436$	3.13224	0.150007
$437$	−10.5863	−0.506409
$438$	0	0
$439$	−5.15949	−0.246249	−0.123125	−	0.992391i	$-0.539291\pi$
$439$	−5.15949	−0.246249	−0.123125	+	0.992391i	$0.539291\pi$
$440$	−1.16207	−0.0553996
$441$	0	0
$442$	1.36625	0.0649858
$443$	5.15492	0.244918	0.122459	−	0.992474i	$-0.460922\pi$
$443$	5.15492	0.244918	0.122459	+	0.992474i	$0.460922\pi$
$444$	0	0
$445$	−5.72522	−0.271401
$446$	−1.17219	−0.0555046
$447$	0	0
$448$	35.4715	1.67587
$449$	5.28749	0.249532	0.124766	−	0.992186i	$-0.460182\pi$
$449$	5.28749	0.249532	0.124766	+	0.992186i	$0.460182\pi$
$450$	0	0
$451$	−5.93675	−0.279551
$452$	−36.8532	−1.73343
$453$	0	0
$454$	1.86060	0.0873223
$455$	43.0233	2.01696
$456$	0	0
$457$	13.1977	0.617363	0.308681	−	0.951165i	$-0.400112\pi$
$457$	13.1977	0.617363	0.308681	+	0.951165i	$0.400112\pi$
$458$	0.451570	0.0211005
$459$	0	0
$460$	−15.6262	−0.728577
$461$	−35.8725	−1.67075	−0.835374	−	0.549683i	$-0.814749\pi$
$461$	−35.8725	−1.67075	−0.835374	+	0.549683i	$0.814749\pi$
$462$	0	0
$463$	−26.8663	−1.24858	−0.624291	−	0.781192i	$-0.714612\pi$
$463$	−26.8663	−1.24858	−0.624291	+	0.781192i	$0.714612\pi$
$464$	3.94186	0.182996
$465$	0	0
$466$	−0.972775	−0.0450629
$467$	−34.4414	−1.59376	−0.796880	−	0.604138i	$-0.793518\pi$
$467$	−34.4414	−1.59376	−0.796880	+	0.604138i	$0.793518\pi$
$468$	0	0
$469$	−31.3750	−1.44876
$470$	2.55498	0.117852
$471$	0	0
$472$	−4.92875	−0.226864
$473$	4.56594	0.209942
$474$	0	0
$475$	61.3556	2.81519
$476$	−53.9745	−2.47392
$477$	0	0
$478$	1.04019	0.0475771
$479$	24.8968	1.13756	0.568782	−	0.822489i	$-0.307415\pi$
$479$	24.8968	1.13756	0.568782	+	0.822489i	$0.307415\pi$
$480$	0	0
$481$	5.35296	0.244074
$482$	0.666756	0.0303699
$483$	0	0
$484$	20.8248	0.946580
$485$	−59.4513	−2.69954
$486$	0	0
$487$	−15.5650	−0.705317	−0.352659	−	0.935752i	$-0.614722\pi$
$487$	−15.5650	−0.705317	−0.352659	+	0.935752i	$0.614722\pi$
$488$	−4.93032	−0.223185
$489$	0	0
$490$	5.50553	0.248714
$491$	43.4099	1.95906	0.979531	−	0.201292i	$-0.0645139\pi$
$491$	43.4099	1.95906	0.979531	+	0.201292i	$0.0645139\pi$
$492$	0	0
$493$	−5.93885	−0.267472
$494$	−1.25152	−0.0563085
$495$	0	0
$496$	−22.6179	−1.01557
$497$	13.7624	0.617330
$498$	0	0
$499$	−8.64901	−0.387183	−0.193591	−	0.981082i	$-0.562014\pi$
$499$	−8.64901	−0.387183	−0.193591	+	0.981082i	$0.562014\pi$
$500$	50.4156	2.25465
$501$	0	0
$502$	0.454889	0.0203027
$503$	−19.1178	−0.852419	−0.426209	−	0.904625i	$-0.640151\pi$
$503$	−19.1178	−0.852419	−0.426209	+	0.904625i	$0.640151\pi$
$504$	0	0
$505$	−24.8892	−1.10755
$506$	0.140461	0.00624425
$507$	0	0
$508$	9.68746	0.429811
$509$	−7.37422	−0.326856	−0.163428	−	0.986555i	$-0.552255\pi$
$509$	−7.37422	−0.326856	−0.163428	+	0.986555i	$0.552255\pi$
$510$	0	0
$511$	−21.7475	−0.962053
$512$	−7.72891	−0.341573
$513$	0	0
$514$	−1.60067	−0.0706027
$515$	3.99939	0.176234
$516$	0	0
$517$	4.70990	0.207141
$518$	1.03117	0.0453070
$519$	0	0
$520$	−3.70371	−0.162418
$521$	1.31697	0.0576973	0.0288487	−	0.999584i	$-0.490816\pi$
$521$	1.31697	0.0576973	0.0288487	+	0.999584i	$0.490816\pi$
$522$	0	0
$523$	16.3118	0.713267	0.356633	−	0.934244i	$-0.383925\pi$
$523$	16.3118	0.713267	0.356633	+	0.934244i	$0.383925\pi$
$524$	−24.0955	−1.05262
$525$	0	0
$526$	2.14627	0.0935820
$527$	34.0763	1.48439
$528$	0	0
$529$	−19.2133	−0.835360
$530$	3.15386	0.136995
$531$	0	0
$532$	49.4421	2.14359
$533$	−18.9214	−0.819576
$534$	0	0
$535$	−75.9713	−3.28452
$536$	2.70095	0.116663
$537$	0	0
$538$	1.28897	0.0555713
$539$	10.1490	0.437148
$540$	0	0
$541$	−17.3788	−0.747175	−0.373587	−	0.927595i	$-0.621872\pi$
$541$	−17.3788	−0.747175	−0.373587	+	0.927595i	$0.621872\pi$
$542$	1.52931	0.0656897
$543$	0	0
$544$	6.97534	0.299065
$545$	6.34954	0.271984
$546$	0	0
$547$	1.61387	0.0690041	0.0345020	−	0.999405i	$-0.489015\pi$
$547$	1.61387	0.0690041	0.0345020	+	0.999405i	$0.489015\pi$
$548$	16.4825	0.704098
$549$	0	0
$550$	−0.814080	−0.0347125
$551$	5.44014	0.231758
$552$	0	0
$553$	3.90142	0.165905
$554$	2.19188	0.0931239
$555$	0	0
$556$	−31.1392	−1.32060
$557$	−11.8184	−0.500763	−0.250382	−	0.968147i	$-0.580556\pi$
$557$	−11.8184	−0.500763	−0.250382	+	0.968147i	$0.580556\pi$
$558$	0	0
$559$	14.5524	0.615500
$560$	72.6231	3.06889
$561$	0	0
$562$	−2.07528	−0.0875402
$563$	−22.5227	−0.949218	−0.474609	−	0.880197i	$-0.657411\pi$
$563$	−22.5227	−0.949218	−0.474609	+	0.880197i	$0.657411\pi$
$564$	0	0
$565$	−74.7072	−3.14296
$566$	−0.286274	−0.0120330
$567$	0	0
$568$	−1.18476	−0.0497112
$569$	41.2742	1.73030	0.865151	−	0.501511i	$-0.167222\pi$
$569$	41.2742	1.73030	0.865151	+	0.501511i	$0.167222\pi$
$570$	0	0
$571$	−32.4617	−1.35848	−0.679240	−	0.733916i	$-0.737691\pi$
$571$	−32.4617	−1.35848	−0.679240	+	0.733916i	$0.737691\pi$
$572$	−3.40544	−0.142389
$573$	0	0
$574$	−3.64493	−0.152136
$575$	−21.9470	−0.915255
$576$	0	0
$577$	33.3635	1.38894	0.694470	−	0.719522i	$-0.255639\pi$
$577$	33.3635	1.38894	0.694470	+	0.719522i	$0.255639\pi$
$578$	−1.79984	−0.0748636
$579$	0	0
$580$	8.03012	0.333433
$581$	29.9443	1.24230
$582$	0	0
$583$	5.81389	0.240787
$584$	1.87216	0.0774705
$585$	0	0
$586$	−1.27656	−0.0527340
$587$	−14.6927	−0.606434	−0.303217	−	0.952922i	$-0.598061\pi$
$587$	−14.6927	−0.606434	−0.303217	+	0.952922i	$0.598061\pi$
$588$	0	0
$589$	−31.2148	−1.28618
$590$	−4.98353	−0.205169
$591$	0	0
$592$	9.03578	0.371368
$593$	27.7605	1.13999	0.569994	−	0.821649i	$-0.306945\pi$
$593$	27.7605	1.13999	0.569994	+	0.821649i	$0.306945\pi$
$594$	0	0
$595$	−109.415	−4.48557
$596$	−22.0789	−0.904388
$597$	0	0
$598$	0.447671	0.0183066
$599$	−32.4749	−1.32689	−0.663445	−	0.748225i	$-0.730906\pi$
$599$	−32.4749	−1.32689	−0.663445	+	0.748225i	$0.730906\pi$
$600$	0	0
$601$	13.9953	0.570880	0.285440	−	0.958397i	$-0.407860\pi$
$601$	13.9953	0.570880	0.285440	+	0.958397i	$0.407860\pi$
$602$	2.80330	0.114254
$603$	0	0
$604$	43.1201	1.75453
$605$	42.2150	1.71629
$606$	0	0
$607$	30.0331	1.21901	0.609503	−	0.792783i	$-0.291369\pi$
$607$	30.0331	1.21901	0.609503	+	0.792783i	$0.291369\pi$
$608$	−6.38959	−0.259132
$609$	0	0
$610$	−4.98511	−0.201841
$611$	15.0112	0.607288
$612$	0	0
$613$	1.02509	0.0414029	0.0207015	−	0.999786i	$-0.493410\pi$
$613$	1.02509	0.0414029	0.0207015	+	0.999786i	$0.493410\pi$
$614$	−0.500438	−0.0201960
$615$	0	0
$616$	−1.31522	−0.0529915
$617$	45.0591	1.81401	0.907005	−	0.421120i	$-0.138363\pi$
$617$	45.0591	1.81401	0.907005	+	0.421120i	$0.138363\pi$
$618$	0	0
$619$	−17.0441	−0.685059	−0.342529	−	0.939507i	$-0.611284\pi$
$619$	−17.0441	−0.685059	−0.342529	+	0.939507i	$0.611284\pi$
$620$	−46.0757	−1.85044
$621$	0	0
$622$	−0.499506	−0.0200284
$623$	−6.47972	−0.259604
$624$	0	0
$625$	45.8087	1.83235
$626$	2.03459	0.0813184
$627$	0	0
$628$	−29.1521	−1.16329
$629$	−13.6134	−0.542802
$630$	0	0
$631$	36.8117	1.46545	0.732726	−	0.680523i	$-0.238248\pi$
$631$	36.8117	1.46545	0.732726	+	0.680523i	$0.238248\pi$
$632$	−0.335858	−0.0133597
$633$	0	0
$634$	−0.0972228	−0.00386121
$635$	19.6380	0.779310
$636$	0	0
$637$	32.3465	1.28161
$638$	−0.0721810	−0.00285767
$639$	0	0
$640$	−12.5651	−0.496680
$641$	−29.3025	−1.15738	−0.578689	−	0.815548i	$-0.696436\pi$
$641$	−29.3025	−1.15738	−0.578689	+	0.815548i	$0.696436\pi$
$642$	0	0
$643$	−38.8912	−1.53372	−0.766859	−	0.641815i	$-0.778182\pi$
$643$	−38.8912	−1.53372	−0.766859	+	0.641815i	$0.778182\pi$
$644$	−17.6855	−0.696908
$645$	0	0
$646$	3.18280	0.125226
$647$	−26.7727	−1.05254	−0.526271	−	0.850317i	$-0.676410\pi$
$647$	−26.7727	−1.05254	−0.526271	+	0.850317i	$0.676410\pi$
$648$	0	0
$649$	−9.18673	−0.360611
$650$	−2.59460	−0.101769
$651$	0	0
$652$	−14.8107	−0.580033
$653$	11.8552	0.463928	0.231964	−	0.972724i	$-0.425485\pi$
$653$	11.8552	0.463928	0.231964	+	0.972724i	$0.425485\pi$
$654$	0	0
$655$	−48.8453	−1.90855
$656$	−31.9392	−1.24702
$657$	0	0
$658$	2.89169	0.112730
$659$	−5.61121	−0.218582	−0.109291	−	0.994010i	$-0.534858\pi$
$659$	−5.61121	−0.218582	−0.109291	+	0.994010i	$0.534858\pi$
$660$	0	0
$661$	−20.4821	−0.796662	−0.398331	−	0.917242i	$-0.630410\pi$
$661$	−20.4821	−0.796662	−0.398331	+	0.917242i	$0.630410\pi$
$662$	−1.11081	−0.0431730
$663$	0	0
$664$	−2.57779	−0.100038
$665$	100.227	3.88663
$666$	0	0
$667$	−1.94595	−0.0753476
$668$	−15.6100	−0.603970
$669$	0	0
$670$	2.73097	0.105506
$671$	−9.18965	−0.354762
$672$	0	0
$673$	−7.81724	−0.301332	−0.150666	−	0.988585i	$-0.548142\pi$
$673$	−7.81724	−0.301332	−0.150666	+	0.988585i	$0.548142\pi$
$674$	0.837243	0.0322494
$675$	0	0
$676$	15.0201	0.577698
$677$	−8.26231	−0.317546	−0.158773	−	0.987315i	$-0.550754\pi$
$677$	−8.26231	−0.317546	−0.158773	+	0.987315i	$0.550754\pi$
$678$	0	0
$679$	−67.2861	−2.58220
$680$	9.41910	0.361206
$681$	0	0
$682$	0.414165	0.0158592
$683$	19.1396	0.732358	0.366179	−	0.930544i	$-0.380666\pi$
$683$	19.1396	0.732358	0.366179	+	0.930544i	$0.380666\pi$
$684$	0	0
$685$	33.4126	1.27663
$686$	3.08214	0.117677
$687$	0	0
$688$	24.5644	0.936508
$689$	18.5298	0.705929
$690$	0	0
$691$	−26.4595	−1.00657	−0.503283	−	0.864122i	$-0.667875\pi$
$691$	−26.4595	−1.00657	−0.503283	+	0.864122i	$0.667875\pi$
$692$	9.82078	0.373330
$693$	0	0
$694$	−0.513412	−0.0194889
$695$	−63.1241	−2.39443
$696$	0	0
$697$	48.1200	1.82267
$698$	−0.661460	−0.0250366
$699$	0	0
$700$	102.501	3.87419
$701$	40.1101	1.51494	0.757468	−	0.652872i	$-0.226436\pi$
$701$	40.1101	1.51494	0.757468	+	0.652872i	$0.226436\pi$
$702$	0	0
$703$	12.4702	0.470324
$704$	−5.69163	−0.214511
$705$	0	0
$706$	−0.521807	−0.0196385
$707$	−28.1692	−1.05941
$708$	0	0
$709$	31.9477	1.19982	0.599911	−	0.800067i	$-0.295203\pi$
$709$	31.9477	1.19982	0.599911	+	0.800067i	$0.295203\pi$
$710$	−1.19792	−0.0449572
$711$	0	0
$712$	0.557814	0.0209050
$713$	11.1656	0.418155
$714$	0	0
$715$	−6.90336	−0.258171
$716$	7.14204	0.266910
$717$	0	0
$718$	−0.116378	−0.00434319
$719$	15.8652	0.591672	0.295836	−	0.955239i	$-0.404402\pi$
$719$	15.8652	0.591672	0.295836	+	0.955239i	$0.404402\pi$
$720$	0	0
$721$	4.52645	0.168574
$722$	−1.04377	−0.0388451
$723$	0	0
$724$	−22.5446	−0.837865
$725$	11.2783	0.418866
$726$	0	0
$727$	−21.1870	−0.785782	−0.392891	−	0.919585i	$-0.628525\pi$
$727$	−21.1870	−0.785782	−0.392891	+	0.919585i	$0.628525\pi$
$728$	−4.19180	−0.155359
$729$	0	0
$730$	1.89296	0.0700618
$731$	−37.0089	−1.36883
$732$	0	0
$733$	−18.5954	−0.686836	−0.343418	−	0.939183i	$-0.611585\pi$
$733$	−18.5954	−0.686836	−0.343418	+	0.939183i	$0.611585\pi$
$734$	−1.74489	−0.0644051
$735$	0	0
$736$	2.28557	0.0842473
$737$	5.03432	0.185442
$738$	0	0
$739$	−8.18908	−0.301240	−0.150620	−	0.988592i	$-0.548127\pi$
$739$	−8.18908	−0.301240	−0.150620	+	0.988592i	$0.548127\pi$
$740$	18.4071	0.676660
$741$	0	0
$742$	3.56949	0.131040
$743$	24.7742	0.908878	0.454439	−	0.890778i	$-0.349840\pi$
$743$	24.7742	0.908878	0.454439	+	0.890778i	$0.349840\pi$
$744$	0	0
$745$	−44.7574	−1.63979
$746$	−2.45804	−0.0899952
$747$	0	0
$748$	8.66056	0.316661
$749$	−85.9832	−3.14176
$750$	0	0
$751$	−36.3697	−1.32715	−0.663575	−	0.748110i	$-0.730962\pi$
$751$	−36.3697	−1.32715	−0.663575	+	0.748110i	$0.730962\pi$
$752$	25.3389	0.924013
$753$	0	0
$754$	−0.230053	−0.00837802
$755$	87.4112	3.18122
$756$	0	0
$757$	21.3314	0.775305	0.387652	−	0.921806i	$-0.373286\pi$
$757$	21.3314	0.775305	0.387652	+	0.921806i	$0.373286\pi$
$758$	0.395107	0.0143509
$759$	0	0
$760$	−8.62814	−0.312975
$761$	16.3820	0.593848	0.296924	−	0.954901i	$-0.404039\pi$
$761$	16.3820	0.593848	0.296924	+	0.954901i	$0.404039\pi$
$762$	0	0
$763$	7.18631	0.260162
$764$	−20.4337	−0.739267
$765$	0	0
$766$	−3.12032	−0.112742
$767$	−29.2796	−1.05722
$768$	0	0
$769$	20.9892	0.756891	0.378445	−	0.925624i	$-0.376459\pi$
$769$	20.9892	0.756891	0.378445	+	0.925624i	$0.376459\pi$
$770$	−1.32983	−0.0479238
$771$	0	0
$772$	−6.80327	−0.244855
$773$	−4.99970	−0.179827	−0.0899134	−	0.995950i	$-0.528659\pi$
$773$	−4.99970	−0.179827	−0.0899134	+	0.995950i	$0.528659\pi$
$774$	0	0
$775$	−64.7133	−2.32457
$776$	5.79240	0.207935
$777$	0	0
$778$	−3.46043	−0.124063
$779$	−44.0791	−1.57930
$780$	0	0
$781$	−2.20827	−0.0790182
$782$	−1.13850	−0.0407125
$783$	0	0
$784$	54.6008	1.95003
$785$	−59.0958	−2.10922
$786$	0	0
$787$	−20.0672	−0.715320	−0.357660	−	0.933852i	$-0.616425\pi$
$787$	−20.0672	−0.715320	−0.357660	+	0.933852i	$0.616425\pi$
$788$	−8.63984	−0.307782
$789$	0	0
$790$	−0.339590	−0.0120821
$791$	−84.5526	−3.00634
$792$	0	0
$793$	−29.2889	−1.04008
$794$	−1.48357	−0.0526498
$795$	0	0
$796$	3.85236	0.136543
$797$	−46.1551	−1.63490	−0.817448	−	0.576002i	$-0.804612\pi$
$797$	−46.1551	−1.63490	−0.817448	+	0.576002i	$0.804612\pi$
$798$	0	0
$799$	−38.1758	−1.35056
$800$	−13.2467	−0.468340
$801$	0	0
$802$	−3.68810	−0.130231
$803$	3.48953	0.123143
$804$	0	0
$805$	−35.8513	−1.26359
$806$	1.32001	0.0464953
$807$	0	0
$808$	2.42498	0.0853105
$809$	44.1213	1.55122	0.775611	−	0.631211i	$-0.217442\pi$
$809$	44.1213	1.55122	0.775611	+	0.631211i	$0.217442\pi$
$810$	0	0
$811$	−48.1501	−1.69078	−0.845390	−	0.534150i	$-0.820632\pi$
$811$	−48.1501	−1.69078	−0.845390	+	0.534150i	$0.820632\pi$
$812$	9.08837	0.318939
$813$	0	0
$814$	−0.165458	−0.00579929
$815$	−30.0236	−1.05168
$816$	0	0
$817$	33.9012	1.18605
$818$	−2.99569	−0.104742
$819$	0	0
$820$	−65.0646	−2.27215
$821$	39.0794	1.36388	0.681941	−	0.731407i	$-0.261136\pi$
$821$	39.0794	1.36388	0.681941	+	0.731407i	$0.261136\pi$
$822$	0	0
$823$	41.2966	1.43951	0.719754	−	0.694229i	$-0.244254\pi$
$823$	41.2966	1.43951	0.719754	+	0.694229i	$0.244254\pi$
$824$	−0.389665	−0.0135746
$825$	0	0
$826$	−5.64028	−0.196251
$827$	−17.8844	−0.621900	−0.310950	−	0.950426i	$-0.600647\pi$
$827$	−17.8844	−0.621900	−0.310950	+	0.950426i	$0.600647\pi$
$828$	0	0
$829$	50.9416	1.76928	0.884638	−	0.466279i	$-0.154406\pi$
$829$	50.9416	1.76928	0.884638	+	0.466279i	$0.154406\pi$
$830$	−2.60644	−0.0904707
$831$	0	0
$832$	−18.1401	−0.628896
$833$	−82.2621	−2.85021
$834$	0	0
$835$	−31.6439	−1.09508
$836$	−7.93330	−0.274379
$837$	0	0
$838$	2.42575	0.0837961
$839$	35.6168	1.22963	0.614815	−	0.788671i	$-0.289231\pi$
$839$	35.6168	1.22963	0.614815	+	0.788671i	$0.289231\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	−2.38215	−0.0820945
$843$	0	0
$844$	−1.06262	−0.0365768
$845$	30.4482	1.04745
$846$	0	0
$847$	47.7784	1.64168
$848$	31.2782	1.07410
$849$	0	0
$850$	6.59847	0.226326
$851$	−4.46063	−0.152908
$852$	0	0
$853$	−35.8276	−1.22671	−0.613356	−	0.789806i	$-0.710181\pi$
$853$	−35.8276	−1.22671	−0.613356	+	0.789806i	$0.710181\pi$
$854$	−5.64207	−0.193068
$855$	0	0
$856$	7.40196	0.252994
$857$	2.07261	0.0707991	0.0353995	−	0.999373i	$-0.488730\pi$
$857$	2.07261	0.0707991	0.0353995	+	0.999373i	$0.488730\pi$
$858$	0	0
$859$	8.93407	0.304827	0.152413	−	0.988317i	$-0.451296\pi$
$859$	8.93407	0.304827	0.152413	+	0.988317i	$0.451296\pi$
$860$	50.0410	1.70638
$861$	0	0
$862$	2.16563	0.0737617
$863$	−39.6911	−1.35110	−0.675550	−	0.737315i	$-0.736094\pi$
$863$	−39.6911	−1.35110	−0.675550	+	0.737315i	$0.736094\pi$
$864$	0	0
$865$	19.9083	0.676901
$866$	−1.48642	−0.0505106
$867$	0	0
$868$	−52.1478	−1.77001
$869$	−0.626007	−0.0212358
$870$	0	0
$871$	16.0452	0.543671
$872$	−0.618642	−0.0209499
$873$	0	0
$874$	1.04289	0.0352764
$875$	115.669	3.91032
$876$	0	0
$877$	−19.9324	−0.673069	−0.336535	−	0.941671i	$-0.609255\pi$
$877$	−19.9324	−0.673069	−0.336535	+	0.941671i	$0.609255\pi$
$878$	0.508281	0.0171537
$879$	0	0
$880$	−11.6529	−0.392818
$881$	28.6600	0.965580	0.482790	−	0.875736i	$-0.339623\pi$
$881$	28.6600	0.965580	0.482790	+	0.875736i	$0.339623\pi$
$882$	0	0
$883$	29.8489	1.00450	0.502248	−	0.864724i	$-0.332507\pi$
$883$	29.8489	1.00450	0.502248	+	0.864724i	$0.332507\pi$
$884$	27.6026	0.928376
$885$	0	0
$886$	−0.507831	−0.0170609
$887$	39.8861	1.33924	0.669622	−	0.742702i	$-0.266456\pi$
$887$	39.8861	1.33924	0.669622	+	0.742702i	$0.266456\pi$
$888$	0	0
$889$	22.2260	0.745436
$890$	0.564013	0.0189058
$891$	0	0
$892$	−23.6819	−0.792929
$893$	34.9700	1.17023
$894$	0	0
$895$	14.4780	0.483947
$896$	−14.2210	−0.475091
$897$	0	0
$898$	−0.520890	−0.0173823
$899$	−5.73786	−0.191368
$900$	0	0
$901$	−47.1241	−1.56993
$902$	0.584852	0.0194734
$903$	0	0
$904$	7.27880	0.242089
$905$	−45.7015	−1.51917
$906$	0	0
$907$	−25.2967	−0.839962	−0.419981	−	0.907533i	$-0.637963\pi$
$907$	−25.2967	−0.839962	−0.419981	+	0.907533i	$0.637963\pi$
$908$	37.5901	1.24747
$909$	0	0
$910$	−4.23839	−0.140501
$911$	−1.31582	−0.0435949	−0.0217975	−	0.999762i	$-0.506939\pi$
$911$	−1.31582	−0.0435949	−0.0217975	+	0.999762i	$0.506939\pi$
$912$	0	0
$913$	−4.80476	−0.159014
$914$	−1.30016	−0.0430054
$915$	0	0
$916$	9.12316	0.301438
$917$	−55.2824	−1.82559
$918$	0	0
$919$	4.52035	0.149112	0.0745562	−	0.997217i	$-0.476246\pi$
$919$	4.52035	0.149112	0.0745562	+	0.997217i	$0.476246\pi$
$920$	3.08630	0.101752
$921$	0	0
$922$	3.53393	0.116384
$923$	−7.03812	−0.231662
$924$	0	0
$925$	25.8528	0.850036
$926$	2.64670	0.0869759
$927$	0	0
$928$	−1.17453	−0.0385557
$929$	7.26202	0.238259	0.119130	−	0.992879i	$-0.461990\pi$
$929$	7.26202	0.238259	0.119130	+	0.992879i	$0.461990\pi$
$930$	0	0
$931$	75.3542	2.46963
$932$	−19.6532	−0.643761
$933$	0	0
$934$	3.39296	0.111021
$935$	17.5563	0.574153
$936$	0	0
$937$	−35.8159	−1.17005	−0.585027	−	0.811014i	$-0.698916\pi$
$937$	−35.8159	−1.17005	−0.585027	+	0.811014i	$0.698916\pi$
$938$	3.09087	0.100920
$939$	0	0
$940$	51.6187	1.68362
$941$	−20.2410	−0.659838	−0.329919	−	0.944009i	$-0.607021\pi$
$941$	−20.2410	−0.659838	−0.329919	+	0.944009i	$0.607021\pi$
$942$	0	0
$943$	15.7672	0.513451
$944$	−49.4239	−1.60861
$945$	0	0
$946$	−0.449808	−0.0146245
$947$	−16.1762	−0.525655	−0.262828	−	0.964843i	$-0.584655\pi$
$947$	−16.1762	−0.525655	−0.262828	+	0.964843i	$0.584655\pi$
$948$	0	0
$949$	11.1217	0.361025
$950$	−6.04437	−0.196105
$951$	0	0
$952$	10.6604	0.345505
$953$	36.6035	1.18570	0.592851	−	0.805312i	$-0.298002\pi$
$953$	36.6035	1.18570	0.592851	+	0.805312i	$0.298002\pi$
$954$	0	0
$955$	−41.4224	−1.34040
$956$	21.0151	0.679678
$957$	0	0
$958$	−2.45268	−0.0792424
$959$	37.8159	1.22114
$960$	0	0
$961$	1.92303	0.0620331
$962$	−0.527341	−0.0170021
$963$	0	0
$964$	13.4706	0.433859
$965$	−13.7913	−0.443957
$966$	0	0
$967$	−42.6228	−1.37066	−0.685328	−	0.728234i	$-0.740341\pi$
$967$	−42.6228	−1.37066	−0.685328	+	0.728234i	$0.740341\pi$
$968$	−4.11306	−0.132199
$969$	0	0
$970$	5.85677	0.188050
$971$	30.2446	0.970595	0.485298	−	0.874349i	$-0.338711\pi$
$971$	30.2446	0.970595	0.485298	+	0.874349i	$0.338711\pi$
$972$	0	0
$973$	−71.4429	−2.29036
$974$	1.53337	0.0491323
$975$	0	0
$976$	−49.4396	−1.58252
$977$	5.14443	0.164585	0.0822925	−	0.996608i	$-0.473776\pi$
$977$	5.14443	0.164585	0.0822925	+	0.996608i	$0.473776\pi$
$978$	0	0
$979$	1.03971	0.0332294
$980$	111.229	3.55309
$981$	0	0
$982$	−4.27648	−0.136468
$983$	−35.7294	−1.13959	−0.569795	−	0.821787i	$-0.692977\pi$
$983$	−35.7294	−1.13959	−0.569795	+	0.821787i	$0.692977\pi$
$984$	0	0
$985$	−17.5143	−0.558052
$986$	0.585059	0.0186321
$987$	0	0
$988$	−25.2847	−0.804414
$989$	−12.1265	−0.385601
$990$	0	0
$991$	30.7868	0.977973	0.488987	−	0.872291i	$-0.337367\pi$
$991$	30.7868	0.977973	0.488987	+	0.872291i	$0.337367\pi$
$992$	6.73927	0.213972
$993$	0	0
$994$	−1.35579	−0.0430030
$995$	7.80933	0.247573
$996$	0	0
$997$	28.6023	0.905844	0.452922	−	0.891550i	$-0.350382\pi$
$997$	28.6023	0.905844	0.452922	+	0.891550i	$0.350382\pi$
$998$	0.852047	0.0269711
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	783.2.a.i.1.5	✓	8
3.2	odd	2		783.2.a.j.1.4	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
783.2.a.i.1.5	✓	8	1.1	even	1	trivial
783.2.a.j.1.4	yes	8	3.2	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:	\( N \)	\(=\)	\( 783 = 3^{3} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	783.a (trivial)

\(f(q)\)	\(=\)	\(q-0.0985138 q^{2} -1.99030 q^{4} -4.03464 q^{5} -4.56635 q^{7} +0.393099 q^{8} +0.397468 q^{10} +0.732700 q^{11} +2.33523 q^{13} +0.449848 q^{14} +3.94186 q^{16} -5.93885 q^{17} +5.44014 q^{19} +8.03012 q^{20} -0.0721810 q^{22} -1.94595 q^{23} +11.2783 q^{25} -0.230053 q^{26} +9.08837 q^{28} +1.00000 q^{29} -5.73786 q^{31} -1.17453 q^{32} +0.585059 q^{34} +18.4236 q^{35} +2.29226 q^{37} -0.535929 q^{38} -1.58601 q^{40} -8.10257 q^{41} +6.23167 q^{43} -1.45829 q^{44} +0.191703 q^{46} +6.42814 q^{47} +13.8515 q^{49} -1.11107 q^{50} -4.64780 q^{52} +7.93489 q^{53} -2.95618 q^{55} -1.79503 q^{56} -0.0985138 q^{58} -12.5382 q^{59} -12.5422 q^{61} +0.565258 q^{62} -7.76802 q^{64} -9.42182 q^{65} +6.87092 q^{67} +11.8201 q^{68} -1.81497 q^{70} -3.01388 q^{71} +4.76256 q^{73} -0.225819 q^{74} -10.8275 q^{76} -3.34576 q^{77} -0.854385 q^{79} -15.9040 q^{80} +0.798215 q^{82} -6.55761 q^{83} +23.9611 q^{85} -0.613905 q^{86} +0.288024 q^{88} +1.41902 q^{89} -10.6635 q^{91} +3.87302 q^{92} -0.633261 q^{94} -21.9490 q^{95} +14.7352 q^{97} -1.36456 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - q^{2} + 11 q^{4} + 2 q^{5} + 5 q^{7} - 3 q^{8} + 7 q^{10} + 2 q^{11} + 11 q^{13} - 8 q^{14} + 21 q^{16} - 6 q^{17} + 5 q^{19} + 27 q^{20} + 17 q^{22} + 4 q^{23} + 30 q^{25} - 16 q^{26} + 2 q^{28}+ \cdots + 11 q^{98}+O(q^{100}) \) 8 * q - q^2 + 11 * q^4 + 2 * q^5 + 5 * q^7 - 3 * q^8 + 7 * q^10 + 2 * q^11 + 11 * q^13 - 8 * q^14 + 21 * q^16 - 6 * q^17 + 5 * q^19 + 27 * q^20 + 17 * q^22 + 4 * q^23 + 30 * q^25 - 16 * q^26 + 2 * q^28 + 8 * q^29 - q^31 - 37 * q^32 - 9 * q^34 - q^35 + 26 * q^37 + 14 * q^38 + 2 * q^40 - 19 * q^41 + 16 * q^43 + q^44 - 15 * q^46 + 18 * q^47 + 27 * q^49 - 4 * q^50 + 7 * q^52 + 19 * q^53 + 6 * q^55 + 27 * q^56 - q^58 - 13 * q^59 + 2 * q^61 - 7 * q^62 + 13 * q^64 + 19 * q^65 + 25 * q^67 - 22 * q^68 - 22 * q^70 + 7 * q^71 + 11 * q^73 + 9 * q^74 - 31 * q^76 - 26 * q^77 + 16 * q^79 + 19 * q^80 + 7 * q^82 - 19 * q^83 + 10 * q^85 - 63 * q^86 + 32 * q^88 + 16 * q^89 + 6 * q^91 + 101 * q^92 - 34 * q^94 - 43 * q^95 + 31 * q^97 + 11 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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