LMFDB - Embedded newform 7440.2.a.bg.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [7440,2,Mod(1,7440)] 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("7440.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(7440, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 7440 = 2^{4} \cdot 3 \cdot 5 \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7440.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,0,-2,0,2,0,-1,0,2,0,-7,0,-4,0,-2,0,2] 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:	$59.4086991038$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{33}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 8 $ x^2 - x - 8
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 930)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$3.37228$ of defining polynomial
Character	$\chi$	$=$	7440.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} +1.00000 q^{5} -3.37228 q^{7} +1.00000 q^{9} -0.627719 q^{11} -2.00000 q^{13} -1.00000 q^{15} -4.74456 q^{17} +0.627719 q^{19} +3.37228 q^{21} -3.37228 q^{23} +1.00000 q^{25} -1.00000 q^{27} -8.74456 q^{29} +1.00000 q^{31} +0.627719 q^{33} -3.37228 q^{35} -0.744563 q^{37} +2.00000 q^{39} +0.744563 q^{41} -0.627719 q^{43} +1.00000 q^{45} +6.74456 q^{47} +4.37228 q^{49} +4.74456 q^{51} +1.37228 q^{53} -0.627719 q^{55} -0.627719 q^{57} -2.74456 q^{59} +11.4891 q^{61} -3.37228 q^{63} -2.00000 q^{65} -10.7446 q^{67} +3.37228 q^{69} -3.37228 q^{71} -8.11684 q^{73} -1.00000 q^{75} +2.11684 q^{77} +4.62772 q^{79} +1.00000 q^{81} +12.0000 q^{83} -4.74456 q^{85} +8.74456 q^{87} -1.37228 q^{89} +6.74456 q^{91} -1.00000 q^{93} +0.627719 q^{95} +2.00000 q^{97} -0.627719 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} + 2 q^{5} - q^{7} + 2 q^{9} - 7 q^{11} - 4 q^{13} - 2 q^{15} + 2 q^{17} + 7 q^{19} + q^{21} - q^{23} + 2 q^{25} - 2 q^{27} - 6 q^{29} + 2 q^{31} + 7 q^{33} - q^{35} + 10 q^{37} + 4 q^{39}+ \cdots - 7 q^{99}+O(q^{100}) $ 2 * q - 2 * q^3 + 2 * q^5 - q^7 + 2 * q^9 - 7 * q^11 - 4 * q^13 - 2 * q^15 + 2 * q^17 + 7 * q^19 + q^21 - q^23 + 2 * q^25 - 2 * q^27 - 6 * q^29 + 2 * q^31 + 7 * q^33 - q^35 + 10 * q^37 + 4 * q^39 - 10 * q^41 - 7 * q^43 + 2 * q^45 + 2 * q^47 + 3 * q^49 - 2 * q^51 - 3 * q^53 - 7 * q^55 - 7 * q^57 + 6 * q^59 - q^63 - 4 * q^65 - 10 * q^67 + q^69 - q^71 + q^73 - 2 * q^75 - 13 * q^77 + 15 * q^79 + 2 * q^81 + 24 * q^83 + 2 * q^85 + 6 * q^87 + 3 * q^89 + 2 * q^91 - 2 * q^93 + 7 * q^95 + 4 * q^97 - 7 * q^99

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−3.37228	−1.27460	−0.637301	−	0.770615i	$-0.719949\pi$
$7$	−3.37228	−1.27460	−0.637301	+	0.770615i	$0.719949\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−0.627719	−0.189264	−0.0946322	−	0.995512i	$-0.530167\pi$
$11$	−0.627719	−0.189264	−0.0946322	+	0.995512i	$0.530167\pi$
$12$	0	0
$13$	−2.00000	−0.554700	−0.277350	−	0.960769i	$-0.589456\pi$
$13$	−2.00000	−0.554700	−0.277350	+	0.960769i	$0.589456\pi$
$14$	0	0
$15$	−1.00000	−0.258199
$16$	0	0
$17$	−4.74456	−1.15073	−0.575363	−	0.817898i	$-0.695139\pi$
$17$	−4.74456	−1.15073	−0.575363	+	0.817898i	$0.695139\pi$
$18$	0	0
$19$	0.627719	0.144009	0.0720043	−	0.997404i	$-0.477060\pi$
$19$	0.627719	0.144009	0.0720043	+	0.997404i	$0.477060\pi$
$20$	0	0
$21$	3.37228	0.735892
$22$	0	0
$23$	−3.37228	−0.703169	−0.351585	−	0.936156i	$-0.614357\pi$
$23$	−3.37228	−0.703169	−0.351585	+	0.936156i	$0.614357\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−8.74456	−1.62382	−0.811912	−	0.583779i	$-0.801573\pi$
$29$	−8.74456	−1.62382	−0.811912	+	0.583779i	$0.801573\pi$
$30$	0	0
$31$	1.00000	0.179605
$31$	1.00000	0.179605
$32$	0	0
$33$	0.627719	0.109272
$34$	0	0
$35$	−3.37228	−0.570020
$36$	0	0
$37$	−0.744563	−0.122405	−0.0612027	−	0.998125i	$-0.519494\pi$
$37$	−0.744563	−0.122405	−0.0612027	+	0.998125i	$0.519494\pi$
$38$	0	0
$39$	2.00000	0.320256
$40$	0	0
$41$	0.744563	0.116281	0.0581406	−	0.998308i	$-0.481483\pi$
$41$	0.744563	0.116281	0.0581406	+	0.998308i	$0.481483\pi$
$42$	0	0
$43$	−0.627719	−0.0957262	−0.0478631	−	0.998854i	$-0.515241\pi$
$43$	−0.627719	−0.0957262	−0.0478631	+	0.998854i	$0.515241\pi$
$44$	0	0
$45$	1.00000	0.149071
$46$	0	0
$47$	6.74456	0.983796	0.491898	−	0.870653i	$-0.336303\pi$
$47$	6.74456	0.983796	0.491898	+	0.870653i	$0.336303\pi$
$48$	0	0
$49$	4.37228	0.624612
$50$	0	0
$51$	4.74456	0.664372
$52$	0	0
$53$	1.37228	0.188497	0.0942487	−	0.995549i	$-0.469955\pi$
$53$	1.37228	0.188497	0.0942487	+	0.995549i	$0.469955\pi$
$54$	0	0
$55$	−0.627719	−0.0846416
$56$	0	0
$57$	−0.627719	−0.0831434
$58$	0	0
$59$	−2.74456	−0.357312	−0.178656	−	0.983912i	$-0.557175\pi$
$59$	−2.74456	−0.357312	−0.178656	+	0.983912i	$0.557175\pi$
$60$	0	0
$61$	11.4891	1.47103	0.735516	−	0.677507i	$-0.236940\pi$
$61$	11.4891	1.47103	0.735516	+	0.677507i	$0.236940\pi$
$62$	0	0
$63$	−3.37228	−0.424868
$64$	0	0
$65$	−2.00000	−0.248069
$66$	0	0
$67$	−10.7446	−1.31266	−0.656329	−	0.754475i	$-0.727892\pi$
$67$	−10.7446	−1.31266	−0.656329	+	0.754475i	$0.727892\pi$
$68$	0	0
$69$	3.37228	0.405975
$70$	0	0
$71$	−3.37228	−0.400216	−0.200108	−	0.979774i	$-0.564129\pi$
$71$	−3.37228	−0.400216	−0.200108	+	0.979774i	$0.564129\pi$
$72$	0	0
$73$	−8.11684	−0.950005	−0.475002	−	0.879985i	$-0.657553\pi$
$73$	−8.11684	−0.950005	−0.475002	+	0.879985i	$0.657553\pi$
$74$	0	0
$75$	−1.00000	−0.115470
$76$	0	0
$77$	2.11684	0.241237
$78$	0	0
$79$	4.62772	0.520659	0.260330	−	0.965520i	$-0.416169\pi$
$79$	4.62772	0.520659	0.260330	+	0.965520i	$0.416169\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	12.0000	1.31717	0.658586	−	0.752506i	$-0.271155\pi$
$83$	12.0000	1.31717	0.658586	+	0.752506i	$0.271155\pi$
$84$	0	0
$85$	−4.74456	−0.514620
$86$	0	0
$87$	8.74456	0.937516
$88$	0	0
$89$	−1.37228	−0.145462	−0.0727308	−	0.997352i	$-0.523171\pi$
$89$	−1.37228	−0.145462	−0.0727308	+	0.997352i	$0.523171\pi$
$90$	0	0
$91$	6.74456	0.707022
$92$	0	0
$93$	−1.00000	−0.103695
$94$	0	0
$95$	0.627719	0.0644026
$96$	0	0
$97$	2.00000	0.203069	0.101535	−	0.994832i	$-0.467625\pi$
$97$	2.00000	0.203069	0.101535	+	0.994832i	$0.467625\pi$
$98$	0	0
$99$	−0.627719	−0.0630881
$100$	0	0
$101$	8.11684	0.807656	0.403828	−	0.914835i	$-0.367679\pi$
$101$	8.11684	0.807656	0.403828	+	0.914835i	$0.367679\pi$
$102$	0	0
$103$	8.00000	0.788263	0.394132	−	0.919054i	$-0.371045\pi$
$103$	8.00000	0.788263	0.394132	+	0.919054i	$0.371045\pi$
$104$	0	0
$105$	3.37228	0.329101
$106$	0	0
$107$	−0.627719	−0.0606839	−0.0303419	−	0.999540i	$-0.509660\pi$
$107$	−0.627719	−0.0606839	−0.0303419	+	0.999540i	$0.509660\pi$
$108$	0	0
$109$	4.74456	0.454447	0.227223	−	0.973843i	$-0.427035\pi$
$109$	4.74456	0.454447	0.227223	+	0.973843i	$0.427035\pi$
$110$	0	0
$111$	0.744563	0.0706708
$112$	0	0
$113$	−2.62772	−0.247195	−0.123597	−	0.992332i	$-0.539443\pi$
$113$	−2.62772	−0.247195	−0.123597	+	0.992332i	$0.539443\pi$
$114$	0	0
$115$	−3.37228	−0.314467
$116$	0	0
$117$	−2.00000	−0.184900
$118$	0	0
$119$	16.0000	1.46672
$120$	0	0
$121$	−10.6060	−0.964179
$122$	0	0
$123$	−0.744563	−0.0671350
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−13.4891	−1.19697	−0.598483	−	0.801135i	$-0.704230\pi$
$127$	−13.4891	−1.19697	−0.598483	+	0.801135i	$0.704230\pi$
$128$	0	0
$129$	0.627719	0.0552675
$130$	0	0
$131$	16.2337	1.41834	0.709172	−	0.705036i	$-0.249069\pi$
$131$	16.2337	1.41834	0.709172	+	0.705036i	$0.249069\pi$
$132$	0	0
$133$	−2.11684	−0.183554
$134$	0	0
$135$	−1.00000	−0.0860663
$136$	0	0
$137$	−3.48913	−0.298096	−0.149048	−	0.988830i	$-0.547621\pi$
$137$	−3.48913	−0.298096	−0.149048	+	0.988830i	$0.547621\pi$
$138$	0	0
$139$	10.7446	0.911342	0.455671	−	0.890148i	$-0.349399\pi$
$139$	10.7446	0.911342	0.455671	+	0.890148i	$0.349399\pi$
$140$	0	0
$141$	−6.74456	−0.567995
$142$	0	0
$143$	1.25544	0.104985
$144$	0	0
$145$	−8.74456	−0.726196
$146$	0	0
$147$	−4.37228	−0.360620
$148$	0	0
$149$	−12.1168	−0.992651	−0.496325	−	0.868137i	$-0.665318\pi$
$149$	−12.1168	−0.992651	−0.496325	+	0.868137i	$0.665318\pi$
$150$	0	0
$151$	8.00000	0.651031	0.325515	−	0.945537i	$-0.394462\pi$
$151$	8.00000	0.651031	0.325515	+	0.945537i	$0.394462\pi$
$152$	0	0
$153$	−4.74456	−0.383575
$154$	0	0
$155$	1.00000	0.0803219
$156$	0	0
$157$	9.37228	0.747989	0.373995	−	0.927431i	$-0.377988\pi$
$157$	9.37228	0.747989	0.373995	+	0.927431i	$0.377988\pi$
$158$	0	0
$159$	−1.37228	−0.108829
$160$	0	0
$161$	11.3723	0.896261
$162$	0	0
$163$	−24.2337	−1.89813	−0.949064	−	0.315082i	$-0.897968\pi$
$163$	−24.2337	−1.89813	−0.949064	+	0.315082i	$0.897968\pi$
$164$	0	0
$165$	0.627719	0.0488678
$166$	0	0
$167$	12.6277	0.977162	0.488581	−	0.872518i	$-0.337515\pi$
$167$	12.6277	0.977162	0.488581	+	0.872518i	$0.337515\pi$
$168$	0	0
$169$	−9.00000	−0.692308
$170$	0	0
$171$	0.627719	0.0480028
$172$	0	0
$173$	−18.0000	−1.36851	−0.684257	−	0.729241i	$-0.739873\pi$
$173$	−18.0000	−1.36851	−0.684257	+	0.729241i	$0.739873\pi$
$174$	0	0
$175$	−3.37228	−0.254921
$176$	0	0
$177$	2.74456	0.206294
$178$	0	0
$179$	12.0000	0.896922	0.448461	−	0.893802i	$-0.351972\pi$
$179$	12.0000	0.896922	0.448461	+	0.893802i	$0.351972\pi$
$180$	0	0
$181$	14.8614	1.10464	0.552320	−	0.833632i	$-0.313743\pi$
$181$	14.8614	1.10464	0.552320	+	0.833632i	$0.313743\pi$
$182$	0	0
$183$	−11.4891	−0.849301
$184$	0	0
$185$	−0.744563	−0.0547413
$186$	0	0
$187$	2.97825	0.217791
$188$	0	0
$189$	3.37228	0.245297
$190$	0	0
$191$	−16.0000	−1.15772	−0.578860	−	0.815427i	$-0.696502\pi$
$191$	−16.0000	−1.15772	−0.578860	+	0.815427i	$0.696502\pi$
$192$	0	0
$193$	15.4891	1.11493	0.557466	−	0.830200i	$-0.311774\pi$
$193$	15.4891	1.11493	0.557466	+	0.830200i	$0.311774\pi$
$194$	0	0
$195$	2.00000	0.143223
$196$	0	0
$197$	19.4891	1.38854	0.694271	−	0.719713i	$-0.255727\pi$
$197$	19.4891	1.38854	0.694271	+	0.719713i	$0.255727\pi$
$198$	0	0
$199$	12.6277	0.895155	0.447578	−	0.894245i	$-0.352287\pi$
$199$	12.6277	0.895155	0.447578	+	0.894245i	$0.352287\pi$
$200$	0	0
$201$	10.7446	0.757863
$202$	0	0
$203$	29.4891	2.06973
$204$	0	0
$205$	0.744563	0.0520025
$206$	0	0
$207$	−3.37228	−0.234390
$208$	0	0
$209$	−0.394031	−0.0272557
$210$	0	0
$211$	0.627719	0.0432139	0.0216070	−	0.999767i	$-0.493122\pi$
$211$	0.627719	0.0432139	0.0216070	+	0.999767i	$0.493122\pi$
$212$	0	0
$213$	3.37228	0.231065
$214$	0	0
$215$	−0.627719	−0.0428101
$216$	0	0
$217$	−3.37228	−0.228925
$218$	0	0
$219$	8.11684	0.548485
$220$	0	0
$221$	9.48913	0.638308
$222$	0	0
$223$	−9.25544	−0.619790	−0.309895	−	0.950771i	$-0.600294\pi$
$223$	−9.25544	−0.619790	−0.309895	+	0.950771i	$0.600294\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	−6.11684	−0.405989	−0.202995	−	0.979180i	$-0.565067\pi$
$227$	−6.11684	−0.405989	−0.202995	+	0.979180i	$0.565067\pi$
$228$	0	0
$229$	3.88316	0.256606	0.128303	−	0.991735i	$-0.459047\pi$
$229$	3.88316	0.256606	0.128303	+	0.991735i	$0.459047\pi$
$230$	0	0
$231$	−2.11684	−0.139278
$232$	0	0
$233$	−14.8614	−0.973603	−0.486802	−	0.873513i	$-0.661837\pi$
$233$	−14.8614	−0.973603	−0.486802	+	0.873513i	$0.661837\pi$
$234$	0	0
$235$	6.74456	0.439967
$236$	0	0
$237$	−4.62772	−0.300603
$238$	0	0
$239$	29.4891	1.90749	0.953746	−	0.300612i	$-0.0971910\pi$
$239$	29.4891	1.90749	0.953746	+	0.300612i	$0.0971910\pi$
$240$	0	0
$241$	4.51087	0.290571	0.145285	−	0.989390i	$-0.453590\pi$
$241$	4.51087	0.290571	0.145285	+	0.989390i	$0.453590\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	4.37228	0.279335
$246$	0	0
$247$	−1.25544	−0.0798816
$248$	0	0
$249$	−12.0000	−0.760469
$250$	0	0
$251$	4.00000	0.252478	0.126239	−	0.992000i	$-0.459709\pi$
$251$	4.00000	0.252478	0.126239	+	0.992000i	$0.459709\pi$
$252$	0	0
$253$	2.11684	0.133085
$254$	0	0
$255$	4.74456	0.297116
$256$	0	0
$257$	13.3723	0.834140	0.417070	−	0.908874i	$-0.363057\pi$
$257$	13.3723	0.834140	0.417070	+	0.908874i	$0.363057\pi$
$258$	0	0
$259$	2.51087	0.156018
$260$	0	0
$261$	−8.74456	−0.541275
$262$	0	0
$263$	18.9783	1.17025	0.585125	−	0.810943i	$-0.301046\pi$
$263$	18.9783	1.17025	0.585125	+	0.810943i	$0.301046\pi$
$264$	0	0
$265$	1.37228	0.0842986
$266$	0	0
$267$	1.37228	0.0839823
$268$	0	0
$269$	−2.00000	−0.121942	−0.0609711	−	0.998140i	$-0.519420\pi$
$269$	−2.00000	−0.121942	−0.0609711	+	0.998140i	$0.519420\pi$
$270$	0	0
$271$	13.8832	0.843342	0.421671	−	0.906749i	$-0.361444\pi$
$271$	13.8832	0.843342	0.421671	+	0.906749i	$0.361444\pi$
$272$	0	0
$273$	−6.74456	−0.408200
$274$	0	0
$275$	−0.627719	−0.0378529
$276$	0	0
$277$	15.2554	0.916610	0.458305	−	0.888795i	$-0.348457\pi$
$277$	15.2554	0.916610	0.458305	+	0.888795i	$0.348457\pi$
$278$	0	0
$279$	1.00000	0.0598684
$280$	0	0
$281$	16.7446	0.998897	0.499448	−	0.866344i	$-0.333536\pi$
$281$	16.7446	0.998897	0.499448	+	0.866344i	$0.333536\pi$
$282$	0	0
$283$	−12.0000	−0.713326	−0.356663	−	0.934233i	$-0.616086\pi$
$283$	−12.0000	−0.713326	−0.356663	+	0.934233i	$0.616086\pi$
$284$	0	0
$285$	−0.627719	−0.0371828
$286$	0	0
$287$	−2.51087	−0.148212
$288$	0	0
$289$	5.51087	0.324169
$290$	0	0
$291$	−2.00000	−0.117242
$292$	0	0
$293$	−7.48913	−0.437519	−0.218760	−	0.975779i	$-0.570201\pi$
$293$	−7.48913	−0.437519	−0.218760	+	0.975779i	$0.570201\pi$
$294$	0	0
$295$	−2.74456	−0.159795
$296$	0	0
$297$	0.627719	0.0364239
$298$	0	0
$299$	6.74456	0.390048
$300$	0	0
$301$	2.11684	0.122013
$302$	0	0
$303$	−8.11684	−0.466301
$304$	0	0
$305$	11.4891	0.657865
$306$	0	0
$307$	−30.9783	−1.76802	−0.884011	−	0.467466i	$-0.845167\pi$
$307$	−30.9783	−1.76802	−0.884011	+	0.467466i	$0.845167\pi$
$308$	0	0
$309$	−8.00000	−0.455104
$310$	0	0
$311$	−8.00000	−0.453638	−0.226819	−	0.973937i	$-0.572833\pi$
$311$	−8.00000	−0.453638	−0.226819	+	0.973937i	$0.572833\pi$
$312$	0	0
$313$	10.0000	0.565233	0.282617	−	0.959233i	$-0.408798\pi$
$313$	10.0000	0.565233	0.282617	+	0.959233i	$0.408798\pi$
$314$	0	0
$315$	−3.37228	−0.190007
$316$	0	0
$317$	−24.7446	−1.38979	−0.694897	−	0.719110i	$-0.744550\pi$
$317$	−24.7446	−1.38979	−0.694897	+	0.719110i	$0.744550\pi$
$318$	0	0
$319$	5.48913	0.307332
$320$	0	0
$321$	0.627719	0.0350358
$322$	0	0
$323$	−2.97825	−0.165714
$324$	0	0
$325$	−2.00000	−0.110940
$326$	0	0
$327$	−4.74456	−0.262375
$328$	0	0
$329$	−22.7446	−1.25395
$330$	0	0
$331$	1.48913	0.0818497	0.0409249	−	0.999162i	$-0.486970\pi$
$331$	1.48913	0.0818497	0.0409249	+	0.999162i	$0.486970\pi$
$332$	0	0
$333$	−0.744563	−0.0408018
$334$	0	0
$335$	−10.7446	−0.587038
$336$	0	0
$337$	28.9783	1.57855	0.789273	−	0.614043i	$-0.210458\pi$
$337$	28.9783	1.57855	0.789273	+	0.614043i	$0.210458\pi$
$338$	0	0
$339$	2.62772	0.142718
$340$	0	0
$341$	−0.627719	−0.0339929
$342$	0	0
$343$	8.86141	0.478471
$344$	0	0
$345$	3.37228	0.181558
$346$	0	0
$347$	−36.4674	−1.95767	−0.978836	−	0.204648i	$-0.934395\pi$
$347$	−36.4674	−1.95767	−0.978836	+	0.204648i	$0.934395\pi$
$348$	0	0
$349$	7.25544	0.388375	0.194187	−	0.980964i	$-0.437793\pi$
$349$	7.25544	0.388375	0.194187	+	0.980964i	$0.437793\pi$
$350$	0	0
$351$	2.00000	0.106752
$352$	0	0
$353$	15.4891	0.824403	0.412201	−	0.911093i	$-0.364760\pi$
$353$	15.4891	0.824403	0.412201	+	0.911093i	$0.364760\pi$
$354$	0	0
$355$	−3.37228	−0.178982
$356$	0	0
$357$	−16.0000	−0.846810
$358$	0	0
$359$	−3.37228	−0.177982	−0.0889911	−	0.996032i	$-0.528364\pi$
$359$	−3.37228	−0.177982	−0.0889911	+	0.996032i	$0.528364\pi$
$360$	0	0
$361$	−18.6060	−0.979262
$362$	0	0
$363$	10.6060	0.556669
$364$	0	0
$365$	−8.11684	−0.424855
$366$	0	0
$367$	20.2337	1.05619	0.528095	−	0.849185i	$-0.322907\pi$
$367$	20.2337	1.05619	0.528095	+	0.849185i	$0.322907\pi$
$368$	0	0
$369$	0.744563	0.0387604
$370$	0	0
$371$	−4.62772	−0.240259
$372$	0	0
$373$	14.8614	0.769494	0.384747	−	0.923022i	$-0.374289\pi$
$373$	14.8614	0.769494	0.384747	+	0.923022i	$0.374289\pi$
$374$	0	0
$375$	−1.00000	−0.0516398
$376$	0	0
$377$	17.4891	0.900736
$378$	0	0
$379$	28.8614	1.48251	0.741255	−	0.671223i	$-0.234231\pi$
$379$	28.8614	1.48251	0.741255	+	0.671223i	$0.234231\pi$
$380$	0	0
$381$	13.4891	0.691069
$382$	0	0
$383$	−13.4891	−0.689262	−0.344631	−	0.938738i	$-0.611996\pi$
$383$	−13.4891	−0.689262	−0.344631	+	0.938738i	$0.611996\pi$
$384$	0	0
$385$	2.11684	0.107884
$386$	0	0
$387$	−0.627719	−0.0319087
$388$	0	0
$389$	16.9783	0.860831	0.430416	−	0.902631i	$-0.358367\pi$
$389$	16.9783	0.860831	0.430416	+	0.902631i	$0.358367\pi$
$390$	0	0
$391$	16.0000	0.809155
$392$	0	0
$393$	−16.2337	−0.818881
$394$	0	0
$395$	4.62772	0.232846
$396$	0	0
$397$	−1.60597	−0.0806013	−0.0403006	−	0.999188i	$-0.512832\pi$
$397$	−1.60597	−0.0806013	−0.0403006	+	0.999188i	$0.512832\pi$
$398$	0	0
$399$	2.11684	0.105975
$400$	0	0
$401$	22.6277	1.12997	0.564987	−	0.825100i	$-0.308881\pi$
$401$	22.6277	1.12997	0.564987	+	0.825100i	$0.308881\pi$
$402$	0	0
$403$	−2.00000	−0.0996271
$404$	0	0
$405$	1.00000	0.0496904
$406$	0	0
$407$	0.467376	0.0231670
$408$	0	0
$409$	−10.2337	−0.506023	−0.253012	−	0.967463i	$-0.581421\pi$
$409$	−10.2337	−0.506023	−0.253012	+	0.967463i	$0.581421\pi$
$410$	0	0
$411$	3.48913	0.172106
$412$	0	0
$413$	9.25544	0.455430
$414$	0	0
$415$	12.0000	0.589057
$416$	0	0
$417$	−10.7446	−0.526163
$418$	0	0
$419$	12.0000	0.586238	0.293119	−	0.956076i	$-0.405307\pi$
$419$	12.0000	0.586238	0.293119	+	0.956076i	$0.405307\pi$
$420$	0	0
$421$	19.4891	0.949842	0.474921	−	0.880028i	$-0.342477\pi$
$421$	19.4891	0.949842	0.474921	+	0.880028i	$0.342477\pi$
$422$	0	0
$423$	6.74456	0.327932
$424$	0	0
$425$	−4.74456	−0.230145
$426$	0	0
$427$	−38.7446	−1.87498
$428$	0	0
$429$	−1.25544	−0.0606131
$430$	0	0
$431$	26.9783	1.29950	0.649748	−	0.760149i	$-0.274874\pi$
$431$	26.9783	1.29950	0.649748	+	0.760149i	$0.274874\pi$
$432$	0	0
$433$	36.1168	1.73566	0.867832	−	0.496857i	$-0.165513\pi$
$433$	36.1168	1.73566	0.867832	+	0.496857i	$0.165513\pi$
$434$	0	0
$435$	8.74456	0.419270
$436$	0	0
$437$	−2.11684	−0.101262
$438$	0	0
$439$	−14.7446	−0.703720	−0.351860	−	0.936053i	$-0.614451\pi$
$439$	−14.7446	−0.703720	−0.351860	+	0.936053i	$0.614451\pi$
$440$	0	0
$441$	4.37228	0.208204
$442$	0	0
$443$	−34.3505	−1.63204	−0.816022	−	0.578022i	$-0.803825\pi$
$443$	−34.3505	−1.63204	−0.816022	+	0.578022i	$0.803825\pi$
$444$	0	0
$445$	−1.37228	−0.0650524
$446$	0	0
$447$	12.1168	0.573107
$448$	0	0
$449$	−8.97825	−0.423710	−0.211855	−	0.977301i	$-0.567950\pi$
$449$	−8.97825	−0.423710	−0.211855	+	0.977301i	$0.567950\pi$
$450$	0	0
$451$	−0.467376	−0.0220079
$452$	0	0
$453$	−8.00000	−0.375873
$454$	0	0
$455$	6.74456	0.316190
$456$	0	0
$457$	34.4674	1.61232	0.806158	−	0.591700i	$-0.201543\pi$
$457$	34.4674	1.61232	0.806158	+	0.591700i	$0.201543\pi$
$458$	0	0
$459$	4.74456	0.221457
$460$	0	0
$461$	−19.7228	−0.918583	−0.459291	−	0.888286i	$-0.651897\pi$
$461$	−19.7228	−0.918583	−0.459291	+	0.888286i	$0.651897\pi$
$462$	0	0
$463$	−2.51087	−0.116690	−0.0583451	−	0.998296i	$-0.518582\pi$
$463$	−2.51087	−0.116690	−0.0583451	+	0.998296i	$0.518582\pi$
$464$	0	0
$465$	−1.00000	−0.0463739
$466$	0	0
$467$	−6.51087	−0.301287	−0.150644	−	0.988588i	$-0.548135\pi$
$467$	−6.51087	−0.301287	−0.150644	+	0.988588i	$0.548135\pi$
$468$	0	0
$469$	36.2337	1.67312
$470$	0	0
$471$	−9.37228	−0.431852
$472$	0	0
$473$	0.394031	0.0181176
$474$	0	0
$475$	0.627719	0.0288017
$476$	0	0
$477$	1.37228	0.0628324
$478$	0	0
$479$	−8.86141	−0.404888	−0.202444	−	0.979294i	$-0.564888\pi$
$479$	−8.86141	−0.404888	−0.202444	+	0.979294i	$0.564888\pi$
$480$	0	0
$481$	1.48913	0.0678983
$482$	0	0
$483$	−11.3723	−0.517457
$484$	0	0
$485$	2.00000	0.0908153
$486$	0	0
$487$	37.4891	1.69879	0.849397	−	0.527754i	$-0.176966\pi$
$487$	37.4891	1.69879	0.849397	+	0.527754i	$0.176966\pi$
$488$	0	0
$489$	24.2337	1.09589
$490$	0	0
$491$	−27.6060	−1.24584	−0.622920	−	0.782286i	$-0.714054\pi$
$491$	−27.6060	−1.24584	−0.622920	+	0.782286i	$0.714054\pi$
$492$	0	0
$493$	41.4891	1.86858
$494$	0	0
$495$	−0.627719	−0.0282139
$496$	0	0
$497$	11.3723	0.510117
$498$	0	0
$499$	−33.4891	−1.49918	−0.749590	−	0.661903i	$-0.769749\pi$
$499$	−33.4891	−1.49918	−0.749590	+	0.661903i	$0.769749\pi$
$500$	0	0
$501$	−12.6277	−0.564165
$502$	0	0
$503$	−5.48913	−0.244748	−0.122374	−	0.992484i	$-0.539051\pi$
$503$	−5.48913	−0.244748	−0.122374	+	0.992484i	$0.539051\pi$
$504$	0	0
$505$	8.11684	0.361195
$506$	0	0
$507$	9.00000	0.399704
$508$	0	0
$509$	−27.2554	−1.20808	−0.604038	−	0.796956i	$-0.706442\pi$
$509$	−27.2554	−1.20808	−0.604038	+	0.796956i	$0.706442\pi$
$510$	0	0
$511$	27.3723	1.21088
$512$	0	0
$513$	−0.627719	−0.0277145
$514$	0	0
$515$	8.00000	0.352522
$516$	0	0
$517$	−4.23369	−0.186197
$518$	0	0
$519$	18.0000	0.790112
$520$	0	0
$521$	36.9783	1.62005	0.810023	−	0.586398i	$-0.199454\pi$
$521$	36.9783	1.62005	0.810023	+	0.586398i	$0.199454\pi$
$522$	0	0
$523$	12.8614	0.562390	0.281195	−	0.959651i	$-0.409269\pi$
$523$	12.8614	0.562390	0.281195	+	0.959651i	$0.409269\pi$
$524$	0	0
$525$	3.37228	0.147178
$526$	0	0
$527$	−4.74456	−0.206676
$528$	0	0
$529$	−11.6277	−0.505553
$530$	0	0
$531$	−2.74456	−0.119104
$532$	0	0
$533$	−1.48913	−0.0645012
$534$	0	0
$535$	−0.627719	−0.0271386
$536$	0	0
$537$	−12.0000	−0.517838
$538$	0	0
$539$	−2.74456	−0.118217
$540$	0	0
$541$	−15.4891	−0.665930	−0.332965	−	0.942939i	$-0.608049\pi$
$541$	−15.4891	−0.665930	−0.332965	+	0.942939i	$0.608049\pi$
$542$	0	0
$543$	−14.8614	−0.637764
$544$	0	0
$545$	4.74456	0.203235
$546$	0	0
$547$	2.74456	0.117349	0.0586745	−	0.998277i	$-0.481313\pi$
$547$	2.74456	0.117349	0.0586745	+	0.998277i	$0.481313\pi$
$548$	0	0
$549$	11.4891	0.490344
$550$	0	0
$551$	−5.48913	−0.233845
$552$	0	0
$553$	−15.6060	−0.663633
$554$	0	0
$555$	0.744563	0.0316049
$556$	0	0
$557$	11.8832	0.503505	0.251753	−	0.967792i	$-0.418993\pi$
$557$	11.8832	0.503505	0.251753	+	0.967792i	$0.418993\pi$
$558$	0	0
$559$	1.25544	0.0530993
$560$	0	0
$561$	−2.97825	−0.125742
$562$	0	0
$563$	6.97825	0.294098	0.147049	−	0.989129i	$-0.453022\pi$
$563$	6.97825	0.294098	0.147049	+	0.989129i	$0.453022\pi$
$564$	0	0
$565$	−2.62772	−0.110549
$566$	0	0
$567$	−3.37228	−0.141623
$568$	0	0
$569$	−1.37228	−0.0575290	−0.0287645	−	0.999586i	$-0.509157\pi$
$569$	−1.37228	−0.0575290	−0.0287645	+	0.999586i	$0.509157\pi$
$570$	0	0
$571$	26.7446	1.11923	0.559613	−	0.828754i	$-0.310950\pi$
$571$	26.7446	1.11923	0.559613	+	0.828754i	$0.310950\pi$
$572$	0	0
$573$	16.0000	0.668410
$574$	0	0
$575$	−3.37228	−0.140634
$576$	0	0
$577$	−2.23369	−0.0929896	−0.0464948	−	0.998919i	$-0.514805\pi$
$577$	−2.23369	−0.0929896	−0.0464948	+	0.998919i	$0.514805\pi$
$578$	0	0
$579$	−15.4891	−0.643706
$580$	0	0
$581$	−40.4674	−1.67887
$582$	0	0
$583$	−0.861407	−0.0356758
$584$	0	0
$585$	−2.00000	−0.0826898
$586$	0	0
$587$	−12.0000	−0.495293	−0.247647	−	0.968850i	$-0.579657\pi$
$587$	−12.0000	−0.495293	−0.247647	+	0.968850i	$0.579657\pi$
$588$	0	0
$589$	0.627719	0.0258647
$590$	0	0
$591$	−19.4891	−0.801675
$592$	0	0
$593$	44.9783	1.84704	0.923518	−	0.383556i	$-0.125301\pi$
$593$	44.9783	1.84704	0.923518	+	0.383556i	$0.125301\pi$
$594$	0	0
$595$	16.0000	0.655936
$596$	0	0
$597$	−12.6277	−0.516818
$598$	0	0
$599$	−10.1168	−0.413363	−0.206682	−	0.978408i	$-0.566266\pi$
$599$	−10.1168	−0.413363	−0.206682	+	0.978408i	$0.566266\pi$
$600$	0	0
$601$	12.5109	0.510329	0.255165	−	0.966898i	$-0.417870\pi$
$601$	12.5109	0.510329	0.255165	+	0.966898i	$0.417870\pi$
$602$	0	0
$603$	−10.7446	−0.437552
$604$	0	0
$605$	−10.6060	−0.431194
$606$	0	0
$607$	4.62772	0.187833	0.0939167	−	0.995580i	$-0.470061\pi$
$607$	4.62772	0.187833	0.0939167	+	0.995580i	$0.470061\pi$
$608$	0	0
$609$	−29.4891	−1.19496
$610$	0	0
$611$	−13.4891	−0.545712
$612$	0	0
$613$	35.4891	1.43339	0.716696	−	0.697386i	$-0.245653\pi$
$613$	35.4891	1.43339	0.716696	+	0.697386i	$0.245653\pi$
$614$	0	0
$615$	−0.744563	−0.0300237
$616$	0	0
$617$	7.88316	0.317364	0.158682	−	0.987330i	$-0.449276\pi$
$617$	7.88316	0.317364	0.158682	+	0.987330i	$0.449276\pi$
$618$	0	0
$619$	−16.2337	−0.652487	−0.326244	−	0.945286i	$-0.605783\pi$
$619$	−16.2337	−0.652487	−0.326244	+	0.945286i	$0.605783\pi$
$620$	0	0
$621$	3.37228	0.135325
$622$	0	0
$623$	4.62772	0.185406
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0.394031	0.0157361
$628$	0	0
$629$	3.53262	0.140855
$630$	0	0
$631$	46.3505	1.84519	0.922593	−	0.385775i	$-0.126066\pi$
$631$	46.3505	1.84519	0.922593	+	0.385775i	$0.126066\pi$
$632$	0	0
$633$	−0.627719	−0.0249496
$634$	0	0
$635$	−13.4891	−0.535300
$636$	0	0
$637$	−8.74456	−0.346472
$638$	0	0
$639$	−3.37228	−0.133405
$640$	0	0
$641$	18.0000	0.710957	0.355479	−	0.934684i	$-0.384318\pi$
$641$	18.0000	0.710957	0.355479	+	0.934684i	$0.384318\pi$
$642$	0	0
$643$	25.0951	0.989654	0.494827	−	0.868992i	$-0.335231\pi$
$643$	25.0951	0.989654	0.494827	+	0.868992i	$0.335231\pi$
$644$	0	0
$645$	0.627719	0.0247164
$646$	0	0
$647$	−50.5842	−1.98867	−0.994335	−	0.106287i	$-0.966104\pi$
$647$	−50.5842	−1.98867	−0.994335	+	0.106287i	$0.966104\pi$
$648$	0	0
$649$	1.72281	0.0676263
$650$	0	0
$651$	3.37228	0.132170
$652$	0	0
$653$	40.9783	1.60360	0.801801	−	0.597591i	$-0.203875\pi$
$653$	40.9783	1.60360	0.801801	+	0.597591i	$0.203875\pi$
$654$	0	0
$655$	16.2337	0.634303
$656$	0	0
$657$	−8.11684	−0.316668
$658$	0	0
$659$	−36.0000	−1.40236	−0.701180	−	0.712984i	$-0.747343\pi$
$659$	−36.0000	−1.40236	−0.701180	+	0.712984i	$0.747343\pi$
$660$	0	0
$661$	−4.97825	−0.193632	−0.0968158	−	0.995302i	$-0.530866\pi$
$661$	−4.97825	−0.193632	−0.0968158	+	0.995302i	$0.530866\pi$
$662$	0	0
$663$	−9.48913	−0.368527
$664$	0	0
$665$	−2.11684	−0.0820877
$666$	0	0
$667$	29.4891	1.14182
$668$	0	0
$669$	9.25544	0.357836
$670$	0	0
$671$	−7.21194	−0.278414
$672$	0	0
$673$	−0.510875	−0.0196928	−0.00984639	−	0.999952i	$-0.503134\pi$
$673$	−0.510875	−0.0196928	−0.00984639	+	0.999952i	$0.503134\pi$
$674$	0	0
$675$	−1.00000	−0.0384900
$676$	0	0
$677$	−41.6060	−1.59905	−0.799524	−	0.600635i	$-0.794915\pi$
$677$	−41.6060	−1.59905	−0.799524	+	0.600635i	$0.794915\pi$
$678$	0	0
$679$	−6.74456	−0.258833
$680$	0	0
$681$	6.11684	0.234398
$682$	0	0
$683$	12.8614	0.492128	0.246064	−	0.969254i	$-0.420863\pi$
$683$	12.8614	0.492128	0.246064	+	0.969254i	$0.420863\pi$
$684$	0	0
$685$	−3.48913	−0.133313
$686$	0	0
$687$	−3.88316	−0.148152
$688$	0	0
$689$	−2.74456	−0.104560
$690$	0	0
$691$	19.1386	0.728066	0.364033	−	0.931386i	$-0.381399\pi$
$691$	19.1386	0.728066	0.364033	+	0.931386i	$0.381399\pi$
$692$	0	0
$693$	2.11684	0.0804123
$694$	0	0
$695$	10.7446	0.407564
$696$	0	0
$697$	−3.53262	−0.133808
$698$	0	0
$699$	14.8614	0.562110
$700$	0	0
$701$	45.6060	1.72251	0.861257	−	0.508170i	$-0.169678\pi$
$701$	45.6060	1.72251	0.861257	+	0.508170i	$0.169678\pi$
$702$	0	0
$703$	−0.467376	−0.0176274
$704$	0	0
$705$	−6.74456	−0.254015
$706$	0	0
$707$	−27.3723	−1.02944
$708$	0	0
$709$	40.1168	1.50662	0.753310	−	0.657666i	$-0.228456\pi$
$709$	40.1168	1.50662	0.753310	+	0.657666i	$0.228456\pi$
$710$	0	0
$711$	4.62772	0.173553
$712$	0	0
$713$	−3.37228	−0.126293
$714$	0	0
$715$	1.25544	0.0469507
$716$	0	0
$717$	−29.4891	−1.10129
$718$	0	0
$719$	−38.7446	−1.44493	−0.722464	−	0.691408i	$-0.756991\pi$
$719$	−38.7446	−1.44493	−0.722464	+	0.691408i	$0.756991\pi$
$720$	0	0
$721$	−26.9783	−1.00472
$722$	0	0
$723$	−4.51087	−0.167761
$724$	0	0
$725$	−8.74456	−0.324765
$726$	0	0
$727$	−19.3723	−0.718478	−0.359239	−	0.933246i	$-0.616964\pi$
$727$	−19.3723	−0.718478	−0.359239	+	0.933246i	$0.616964\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	2.97825	0.110155
$732$	0	0
$733$	−34.0000	−1.25582	−0.627909	−	0.778287i	$-0.716089\pi$
$733$	−34.0000	−1.25582	−0.627909	+	0.778287i	$0.716089\pi$
$734$	0	0
$735$	−4.37228	−0.161274
$736$	0	0
$737$	6.74456	0.248439
$738$	0	0
$739$	22.9783	0.845269	0.422634	−	0.906300i	$-0.361105\pi$
$739$	22.9783	0.845269	0.422634	+	0.906300i	$0.361105\pi$
$740$	0	0
$741$	1.25544	0.0461196
$742$	0	0
$743$	35.3723	1.29768	0.648842	−	0.760924i	$-0.275254\pi$
$743$	35.3723	1.29768	0.648842	+	0.760924i	$0.275254\pi$
$744$	0	0
$745$	−12.1168	−0.443927
$746$	0	0
$747$	12.0000	0.439057
$748$	0	0
$749$	2.11684	0.0773478
$750$	0	0
$751$	38.7446	1.41381	0.706905	−	0.707309i	$-0.250091\pi$
$751$	38.7446	1.41381	0.706905	+	0.707309i	$0.250091\pi$
$752$	0	0
$753$	−4.00000	−0.145768
$754$	0	0
$755$	8.00000	0.291150
$756$	0	0
$757$	−26.0000	−0.944986	−0.472493	−	0.881334i	$-0.656646\pi$
$757$	−26.0000	−0.944986	−0.472493	+	0.881334i	$0.656646\pi$
$758$	0	0
$759$	−2.11684	−0.0768366
$760$	0	0
$761$	−46.8614	−1.69872	−0.849362	−	0.527810i	$-0.823013\pi$
$761$	−46.8614	−1.69872	−0.849362	+	0.527810i	$0.823013\pi$
$762$	0	0
$763$	−16.0000	−0.579239
$764$	0	0
$765$	−4.74456	−0.171540
$766$	0	0
$767$	5.48913	0.198201
$768$	0	0
$769$	−9.37228	−0.337973	−0.168987	−	0.985618i	$-0.554049\pi$
$769$	−9.37228	−0.337973	−0.168987	+	0.985618i	$0.554049\pi$
$770$	0	0
$771$	−13.3723	−0.481591
$772$	0	0
$773$	21.6060	0.777113	0.388556	−	0.921425i	$-0.372974\pi$
$773$	21.6060	0.777113	0.388556	+	0.921425i	$0.372974\pi$
$774$	0	0
$775$	1.00000	0.0359211
$776$	0	0
$777$	−2.51087	−0.0900771
$778$	0	0
$779$	0.467376	0.0167455
$780$	0	0
$781$	2.11684	0.0757466
$782$	0	0
$783$	8.74456	0.312505
$784$	0	0
$785$	9.37228	0.334511
$786$	0	0
$787$	9.88316	0.352296	0.176148	−	0.984364i	$-0.443636\pi$
$787$	9.88316	0.352296	0.176148	+	0.984364i	$0.443636\pi$
$788$	0	0
$789$	−18.9783	−0.675644
$790$	0	0
$791$	8.86141	0.315075
$792$	0	0
$793$	−22.9783	−0.815982
$794$	0	0
$795$	−1.37228	−0.0486698
$796$	0	0
$797$	−20.5109	−0.726532	−0.363266	−	0.931685i	$-0.618338\pi$
$797$	−20.5109	−0.726532	−0.363266	+	0.931685i	$0.618338\pi$
$798$	0	0
$799$	−32.0000	−1.13208
$800$	0	0
$801$	−1.37228	−0.0484872
$802$	0	0
$803$	5.09509	0.179802
$804$	0	0
$805$	11.3723	0.400820
$806$	0	0
$807$	2.00000	0.0704033
$808$	0	0
$809$	9.60597	0.337728	0.168864	−	0.985639i	$-0.445990\pi$
$809$	9.60597	0.337728	0.168864	+	0.985639i	$0.445990\pi$
$810$	0	0
$811$	40.6277	1.42663	0.713316	−	0.700842i	$-0.247192\pi$
$811$	40.6277	1.42663	0.713316	+	0.700842i	$0.247192\pi$
$812$	0	0
$813$	−13.8832	−0.486904
$814$	0	0
$815$	−24.2337	−0.848869
$816$	0	0
$817$	−0.394031	−0.0137854
$818$	0	0
$819$	6.74456	0.235674
$820$	0	0
$821$	15.2554	0.532418	0.266209	−	0.963915i	$-0.414229\pi$
$821$	15.2554	0.532418	0.266209	+	0.963915i	$0.414229\pi$
$822$	0	0
$823$	17.2554	0.601487	0.300743	−	0.953705i	$-0.402765\pi$
$823$	17.2554	0.601487	0.300743	+	0.953705i	$0.402765\pi$
$824$	0	0
$825$	0.627719	0.0218544
$826$	0	0
$827$	17.4891	0.608156	0.304078	−	0.952647i	$-0.401652\pi$
$827$	17.4891	0.608156	0.304078	+	0.952647i	$0.401652\pi$
$828$	0	0
$829$	−31.0951	−1.07998	−0.539989	−	0.841672i	$-0.681571\pi$
$829$	−31.0951	−1.07998	−0.539989	+	0.841672i	$0.681571\pi$
$830$	0	0
$831$	−15.2554	−0.529205
$832$	0	0
$833$	−20.7446	−0.718756
$834$	0	0
$835$	12.6277	0.437000
$836$	0	0
$837$	−1.00000	−0.0345651
$838$	0	0
$839$	−16.8614	−0.582120	−0.291060	−	0.956705i	$-0.594008\pi$
$839$	−16.8614	−0.582120	−0.291060	+	0.956705i	$0.594008\pi$
$840$	0	0
$841$	47.4674	1.63681
$842$	0	0
$843$	−16.7446	−0.576713
$844$	0	0
$845$	−9.00000	−0.309609
$846$	0	0
$847$	35.7663	1.22895
$848$	0	0
$849$	12.0000	0.411839
$850$	0	0
$851$	2.51087	0.0860717
$852$	0	0
$853$	−39.0951	−1.33859	−0.669295	−	0.742997i	$-0.733404\pi$
$853$	−39.0951	−1.33859	−0.669295	+	0.742997i	$0.733404\pi$
$854$	0	0
$855$	0.627719	0.0214675
$856$	0	0
$857$	7.48913	0.255824	0.127912	−	0.991786i	$-0.459173\pi$
$857$	7.48913	0.255824	0.127912	+	0.991786i	$0.459173\pi$
$858$	0	0
$859$	17.4891	0.596721	0.298361	−	0.954453i	$-0.403560\pi$
$859$	17.4891	0.596721	0.298361	+	0.954453i	$0.403560\pi$
$860$	0	0
$861$	2.51087	0.0855704
$862$	0	0
$863$	6.35053	0.216175	0.108087	−	0.994141i	$-0.465527\pi$
$863$	6.35053	0.216175	0.108087	+	0.994141i	$0.465527\pi$
$864$	0	0
$865$	−18.0000	−0.612018
$866$	0	0
$867$	−5.51087	−0.187159
$868$	0	0
$869$	−2.90491	−0.0985422
$870$	0	0
$871$	21.4891	0.728131
$872$	0	0
$873$	2.00000	0.0676897
$874$	0	0
$875$	−3.37228	−0.114004
$876$	0	0
$877$	−44.9783	−1.51881	−0.759404	−	0.650620i	$-0.774509\pi$
$877$	−44.9783	−1.51881	−0.759404	+	0.650620i	$0.774509\pi$
$878$	0	0
$879$	7.48913	0.252602
$880$	0	0
$881$	−3.02175	−0.101805	−0.0509027	−	0.998704i	$-0.516210\pi$
$881$	−3.02175	−0.101805	−0.0509027	+	0.998704i	$0.516210\pi$
$882$	0	0
$883$	−44.8614	−1.50971	−0.754853	−	0.655894i	$-0.772292\pi$
$883$	−44.8614	−1.50971	−0.754853	+	0.655894i	$0.772292\pi$
$884$	0	0
$885$	2.74456	0.0922575
$886$	0	0
$887$	30.7446	1.03230	0.516151	−	0.856498i	$-0.327364\pi$
$887$	30.7446	1.03230	0.516151	+	0.856498i	$0.327364\pi$
$888$	0	0
$889$	45.4891	1.52566
$890$	0	0
$891$	−0.627719	−0.0210294
$892$	0	0
$893$	4.23369	0.141675
$894$	0	0
$895$	12.0000	0.401116
$896$	0	0
$897$	−6.74456	−0.225194
$898$	0	0
$899$	−8.74456	−0.291647
$900$	0	0
$901$	−6.51087	−0.216909
$902$	0	0
$903$	−2.11684	−0.0704442
$904$	0	0
$905$	14.8614	0.494010
$906$	0	0
$907$	17.4891	0.580717	0.290358	−	0.956918i	$-0.406225\pi$
$907$	17.4891	0.580717	0.290358	+	0.956918i	$0.406225\pi$
$908$	0	0
$909$	8.11684	0.269219
$910$	0	0
$911$	2.51087	0.0831890	0.0415945	−	0.999135i	$-0.486756\pi$
$911$	2.51087	0.0831890	0.0415945	+	0.999135i	$0.486756\pi$
$912$	0	0
$913$	−7.53262	−0.249293
$914$	0	0
$915$	−11.4891	−0.379819
$916$	0	0
$917$	−54.7446	−1.80782
$918$	0	0
$919$	12.2337	0.403552	0.201776	−	0.979432i	$-0.435329\pi$
$919$	12.2337	0.403552	0.201776	+	0.979432i	$0.435329\pi$
$920$	0	0
$921$	30.9783	1.02077
$922$	0	0
$923$	6.74456	0.222000
$924$	0	0
$925$	−0.744563	−0.0244811
$926$	0	0
$927$	8.00000	0.262754
$928$	0	0
$929$	−16.1168	−0.528776	−0.264388	−	0.964416i	$-0.585170\pi$
$929$	−16.1168	−0.528776	−0.264388	+	0.964416i	$0.585170\pi$
$930$	0	0
$931$	2.74456	0.0899494
$932$	0	0
$933$	8.00000	0.261908
$934$	0	0
$935$	2.97825	0.0973992
$936$	0	0
$937$	−31.2554	−1.02107	−0.510535	−	0.859857i	$-0.670553\pi$
$937$	−31.2554	−1.02107	−0.510535	+	0.859857i	$0.670553\pi$
$938$	0	0
$939$	−10.0000	−0.326338
$940$	0	0
$941$	−19.7228	−0.642945	−0.321473	−	0.946919i	$-0.604178\pi$
$941$	−19.7228	−0.642945	−0.321473	+	0.946919i	$0.604178\pi$
$942$	0	0
$943$	−2.51087	−0.0817653
$944$	0	0
$945$	3.37228	0.109700
$946$	0	0
$947$	2.74456	0.0891863	0.0445932	−	0.999005i	$-0.485801\pi$
$947$	2.74456	0.0891863	0.0445932	+	0.999005i	$0.485801\pi$
$948$	0	0
$949$	16.2337	0.526968
$950$	0	0
$951$	24.7446	0.802397
$952$	0	0
$953$	11.7228	0.379739	0.189870	−	0.981809i	$-0.439193\pi$
$953$	11.7228	0.379739	0.189870	+	0.981809i	$0.439193\pi$
$954$	0	0
$955$	−16.0000	−0.517748
$956$	0	0
$957$	−5.48913	−0.177438
$958$	0	0
$959$	11.7663	0.379954
$960$	0	0
$961$	1.00000	0.0322581
$962$	0	0
$963$	−0.627719	−0.0202280
$964$	0	0
$965$	15.4891	0.498613
$966$	0	0
$967$	−8.00000	−0.257263	−0.128631	−	0.991692i	$-0.541058\pi$
$967$	−8.00000	−0.257263	−0.128631	+	0.991692i	$0.541058\pi$
$968$	0	0
$969$	2.97825	0.0956752
$970$	0	0
$971$	48.7011	1.56289	0.781446	−	0.623973i	$-0.214483\pi$
$971$	48.7011	1.56289	0.781446	+	0.623973i	$0.214483\pi$
$972$	0	0
$973$	−36.2337	−1.16160
$974$	0	0
$975$	2.00000	0.0640513
$976$	0	0
$977$	−46.0000	−1.47167	−0.735835	−	0.677161i	$-0.763210\pi$
$977$	−46.0000	−1.47167	−0.735835	+	0.677161i	$0.763210\pi$
$978$	0	0
$979$	0.861407	0.0275307
$980$	0	0
$981$	4.74456	0.151482
$982$	0	0
$983$	2.97825	0.0949914	0.0474957	−	0.998871i	$-0.484876\pi$
$983$	2.97825	0.0949914	0.0474957	+	0.998871i	$0.484876\pi$
$984$	0	0
$985$	19.4891	0.620975
$986$	0	0
$987$	22.7446	0.723967
$988$	0	0
$989$	2.11684	0.0673117
$990$	0	0
$991$	47.6060	1.51225	0.756127	−	0.654425i	$-0.227089\pi$
$991$	47.6060	1.51225	0.756127	+	0.654425i	$0.227089\pi$
$992$	0	0
$993$	−1.48913	−0.0472560
$994$	0	0
$995$	12.6277	0.400326
$996$	0	0
$997$	6.00000	0.190022	0.0950110	−	0.995476i	$-0.469711\pi$
$997$	6.00000	0.190022	0.0950110	+	0.995476i	$0.469711\pi$
$998$	0	0
$999$	0.744563	0.0235569

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7440.2.a.bg.1.1		2
4.3	odd	2		930.2.a.r.1.2	✓	2
12.11	even	2		2790.2.a.bd.1.2		2
20.3	even	4		4650.2.d.bh.3349.1		4
20.7	even	4		4650.2.d.bh.3349.4		4
20.19	odd	2		4650.2.a.by.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
930.2.a.r.1.2	✓	2	4.3	odd	2
2790.2.a.bd.1.2		2	12.11	even	2
4650.2.a.by.1.1		2	20.19	odd	2
4650.2.d.bh.3349.1		4	20.3	even	4
4650.2.d.bh.3349.4		4	20.7	even	4
7440.2.a.bg.1.1		2	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:	\( N \)	\(=\)	\( 7440 = 2^{4} \cdot 3 \cdot 5 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7440.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +1.00000 q^{5} -3.37228 q^{7} +1.00000 q^{9} -0.627719 q^{11} -2.00000 q^{13} -1.00000 q^{15} -4.74456 q^{17} +0.627719 q^{19} +3.37228 q^{21} -3.37228 q^{23} +1.00000 q^{25} -1.00000 q^{27} -8.74456 q^{29} +1.00000 q^{31} +0.627719 q^{33} -3.37228 q^{35} -0.744563 q^{37} +2.00000 q^{39} +0.744563 q^{41} -0.627719 q^{43} +1.00000 q^{45} +6.74456 q^{47} +4.37228 q^{49} +4.74456 q^{51} +1.37228 q^{53} -0.627719 q^{55} -0.627719 q^{57} -2.74456 q^{59} +11.4891 q^{61} -3.37228 q^{63} -2.00000 q^{65} -10.7446 q^{67} +3.37228 q^{69} -3.37228 q^{71} -8.11684 q^{73} -1.00000 q^{75} +2.11684 q^{77} +4.62772 q^{79} +1.00000 q^{81} +12.0000 q^{83} -4.74456 q^{85} +8.74456 q^{87} -1.37228 q^{89} +6.74456 q^{91} -1.00000 q^{93} +0.627719 q^{95} +2.00000 q^{97} -0.627719 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} + 2 q^{5} - q^{7} + 2 q^{9} - 7 q^{11} - 4 q^{13} - 2 q^{15} + 2 q^{17} + 7 q^{19} + q^{21} - q^{23} + 2 q^{25} - 2 q^{27} - 6 q^{29} + 2 q^{31} + 7 q^{33} - q^{35} + 10 q^{37} + 4 q^{39}+ \cdots - 7 q^{99}+O(q^{100}) \) 2 * q - 2 * q^3 + 2 * q^5 - q^7 + 2 * q^9 - 7 * q^11 - 4 * q^13 - 2 * q^15 + 2 * q^17 + 7 * q^19 + q^21 - q^23 + 2 * q^25 - 2 * q^27 - 6 * q^29 + 2 * q^31 + 7 * q^33 - q^35 + 10 * q^37 + 4 * q^39 - 10 * q^41 - 7 * q^43 + 2 * q^45 + 2 * q^47 + 3 * q^49 - 2 * q^51 - 3 * q^53 - 7 * q^55 - 7 * q^57 + 6 * q^59 - q^63 - 4 * q^65 - 10 * q^67 + q^69 - q^71 + q^73 - 2 * q^75 - 13 * q^77 + 15 * q^79 + 2 * q^81 + 24 * q^83 + 2 * q^85 + 6 * q^87 + 3 * q^89 + 2 * q^91 - 2 * q^93 + 7 * q^95 + 4 * q^97 - 7 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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