LMFDB - Embedded newform 7440.2.a.bt.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 = [3,0,3,0,-3,0,-4,0,3,0,2,0,4,0,-3,0,-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:	$3$
Coefficient field:	3.3.564.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 5x + 3 $ x^3 - x^2 - 5*x + 3
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1860)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2.08613$ 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)

$f(q)$	$=$	$q+1.00000 q^{3} -1.00000 q^{5} -4.43807 q^{7} +1.00000 q^{9} -4.17226 q^{11} +4.43807 q^{13} -1.00000 q^{15} -1.35194 q^{17} -2.70388 q^{19} -4.43807 q^{21} +1.35194 q^{23} +1.00000 q^{25} +1.00000 q^{27} -7.25839 q^{29} +1.00000 q^{31} -4.17226 q^{33} +4.43807 q^{35} -10.7826 q^{37} +4.43807 q^{39} +2.17226 q^{41} -9.46838 q^{43} -1.00000 q^{45} +4.11644 q^{47} +12.6965 q^{49} -1.35194 q^{51} +12.8203 q^{53} +4.17226 q^{55} -2.70388 q^{57} -5.25839 q^{59} +6.00000 q^{61} -4.43807 q^{63} -4.43807 q^{65} +1.14195 q^{67} +1.35194 q^{69} +10.7268 q^{71} +10.4381 q^{73} +1.00000 q^{75} +18.5168 q^{77} +2.05582 q^{79} +1.00000 q^{81} +6.99258 q^{83} +1.35194 q^{85} -7.25839 q^{87} -2.55451 q^{89} -19.6965 q^{91} +1.00000 q^{93} +2.70388 q^{95} -15.9245 q^{97} -4.17226 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{3} - 3 q^{5} - 4 q^{7} + 3 q^{9} + 2 q^{11} + 4 q^{13} - 3 q^{15} - 2 q^{17} - 4 q^{19} - 4 q^{21} + 2 q^{23} + 3 q^{25} + 3 q^{27} + 3 q^{31} + 2 q^{33} + 4 q^{35} + 6 q^{37} + 4 q^{39} - 8 q^{41}+ \cdots + 2 q^{99}+O(q^{100}) $ 3 * q + 3 * q^3 - 3 * q^5 - 4 * q^7 + 3 * q^9 + 2 * q^11 + 4 * q^13 - 3 * q^15 - 2 * q^17 - 4 * q^19 - 4 * q^21 + 2 * q^23 + 3 * q^25 + 3 * q^27 + 3 * q^31 + 2 * q^33 + 4 * q^35 + 6 * q^37 + 4 * q^39 - 8 * q^41 - 18 * q^43 - 3 * q^45 + 4 * q^47 + 7 * q^49 - 2 * q^51 + 26 * q^53 - 2 * q^55 - 4 * q^57 + 6 * q^59 + 18 * q^61 - 4 * q^63 - 4 * q^65 - 10 * q^67 + 2 * q^69 + 22 * q^73 + 3 * q^75 + 12 * q^77 + 3 * q^81 - 6 * q^83 + 2 * q^85 + 10 * q^89 - 28 * q^91 + 3 * q^93 + 4 * q^95 + 4 * q^97 + 2 * 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
$7$	−4.43807	−1.67743	−0.838716	−	0.544569i	$-0.816693\pi$
$7$	−4.43807	−1.67743	−0.838716	+	0.544569i	$0.816693\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−4.17226	−1.25798	−0.628992	−	0.777412i	$-0.716532\pi$
$11$	−4.17226	−1.25798	−0.628992	+	0.777412i	$0.716532\pi$
$12$	0	0
$13$	4.43807	1.23090	0.615449	−	0.788176i	$-0.288975\pi$
$13$	4.43807	1.23090	0.615449	+	0.788176i	$0.288975\pi$
$14$	0	0
$15$	−1.00000	−0.258199
$16$	0	0
$17$	−1.35194	−0.327893	−0.163947	−	0.986469i	$-0.552423\pi$
$17$	−1.35194	−0.327893	−0.163947	+	0.986469i	$0.552423\pi$
$18$	0	0
$19$	−2.70388	−0.620312	−0.310156	−	0.950686i	$-0.600381\pi$
$19$	−2.70388	−0.620312	−0.310156	+	0.950686i	$0.600381\pi$
$20$	0	0
$21$	−4.43807	−0.968466
$22$	0	0
$23$	1.35194	0.281899	0.140949	−	0.990017i	$-0.454985\pi$
$23$	1.35194	0.281899	0.140949	+	0.990017i	$0.454985\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−7.25839	−1.34785	−0.673925	−	0.738800i	$-0.735393\pi$
$29$	−7.25839	−1.34785	−0.673925	+	0.738800i	$0.735393\pi$
$30$	0	0
$31$	1.00000	0.179605
$31$	1.00000	0.179605
$32$	0	0
$33$	−4.17226	−0.726297
$34$	0	0
$35$	4.43807	0.750171
$36$	0	0
$37$	−10.7826	−1.77265	−0.886323	−	0.463067i	$-0.846749\pi$
$37$	−10.7826	−1.77265	−0.886323	+	0.463067i	$0.846749\pi$
$38$	0	0
$39$	4.43807	0.710660
$40$	0	0
$41$	2.17226	0.339250	0.169625	−	0.985509i	$-0.445744\pi$
$41$	2.17226	0.339250	0.169625	+	0.985509i	$0.445744\pi$
$42$	0	0
$43$	−9.46838	−1.44391	−0.721957	−	0.691938i	$-0.756757\pi$
$43$	−9.46838	−1.44391	−0.721957	+	0.691938i	$0.756757\pi$
$44$	0	0
$45$	−1.00000	−0.149071
$46$	0	0
$47$	4.11644	0.600445	0.300222	−	0.953869i	$-0.402939\pi$
$47$	4.11644	0.600445	0.300222	+	0.953869i	$0.402939\pi$
$48$	0	0
$49$	12.6965	1.81378
$50$	0	0
$51$	−1.35194	−0.189309
$52$	0	0
$53$	12.8203	1.76101	0.880503	−	0.474040i	$-0.157205\pi$
$53$	12.8203	1.76101	0.880503	+	0.474040i	$0.157205\pi$
$54$	0	0
$55$	4.17226	0.562587
$56$	0	0
$57$	−2.70388	−0.358137
$58$	0	0
$59$	−5.25839	−0.684584	−0.342292	−	0.939594i	$-0.611203\pi$
$59$	−5.25839	−0.684584	−0.342292	+	0.939594i	$0.611203\pi$
$60$	0	0
$61$	6.00000	0.768221	0.384111	−	0.923287i	$-0.374508\pi$
$61$	6.00000	0.768221	0.384111	+	0.923287i	$0.374508\pi$
$62$	0	0
$63$	−4.43807	−0.559144
$64$	0	0
$65$	−4.43807	−0.550475
$66$	0	0
$67$	1.14195	0.139511	0.0697556	−	0.997564i	$-0.477778\pi$
$67$	1.14195	0.139511	0.0697556	+	0.997564i	$0.477778\pi$
$68$	0	0
$69$	1.35194	0.162754
$70$	0	0
$71$	10.7268	1.27303	0.636517	−	0.771263i	$-0.280375\pi$
$71$	10.7268	1.27303	0.636517	+	0.771263i	$0.280375\pi$
$72$	0	0
$73$	10.4381	1.22168	0.610842	−	0.791753i	$-0.290831\pi$
$73$	10.4381	1.22168	0.610842	+	0.791753i	$0.290831\pi$
$74$	0	0
$75$	1.00000	0.115470
$76$	0	0
$77$	18.5168	2.11018
$78$	0	0
$79$	2.05582	0.231298	0.115649	−	0.993290i	$-0.463105\pi$
$79$	2.05582	0.231298	0.115649	+	0.993290i	$0.463105\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	6.99258	0.767536	0.383768	−	0.923430i	$-0.374626\pi$
$83$	6.99258	0.767536	0.383768	+	0.923430i	$0.374626\pi$
$84$	0	0
$85$	1.35194	0.146638
$86$	0	0
$87$	−7.25839	−0.778181
$88$	0	0
$89$	−2.55451	−0.270778	−0.135389	−	0.990793i	$-0.543228\pi$
$89$	−2.55451	−0.270778	−0.135389	+	0.990793i	$0.543228\pi$
$90$	0	0
$91$	−19.6965	−2.06475
$92$	0	0
$93$	1.00000	0.103695
$94$	0	0
$95$	2.70388	0.277412
$96$	0	0
$97$	−15.9245	−1.61689	−0.808446	−	0.588570i	$-0.799691\pi$
$97$	−15.9245	−1.61689	−0.808446	+	0.588570i	$0.799691\pi$
$98$	0	0
$99$	−4.17226	−0.419328
$100$	0	0
$101$	−0.876139	−0.0871791	−0.0435895	−	0.999050i	$-0.513879\pi$
$101$	−0.876139	−0.0871791	−0.0435895	+	0.999050i	$0.513879\pi$
$102$	0	0
$103$	−6.43807	−0.634362	−0.317181	−	0.948365i	$-0.602736\pi$
$103$	−6.43807	−0.634362	−0.317181	+	0.948365i	$0.602736\pi$
$104$	0	0
$105$	4.43807	0.433111
$106$	0	0
$107$	−11.6965	−1.13074	−0.565370	−	0.824838i	$-0.691267\pi$
$107$	−11.6965	−1.13074	−0.565370	+	0.824838i	$0.691267\pi$
$108$	0	0
$109$	11.5242	1.10382	0.551909	−	0.833904i	$-0.313900\pi$
$109$	11.5242	1.10382	0.551909	+	0.833904i	$0.313900\pi$
$110$	0	0
$111$	−10.7826	−1.02344
$112$	0	0
$113$	17.5800	1.65379	0.826894	−	0.562357i	$-0.190105\pi$
$113$	17.5800	1.65379	0.826894	+	0.562357i	$0.190105\pi$
$114$	0	0
$115$	−1.35194	−0.126069
$116$	0	0
$117$	4.43807	0.410300
$118$	0	0
$119$	6.00000	0.550019
$120$	0	0
$121$	6.40776	0.582523
$122$	0	0
$123$	2.17226	0.195866
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−15.8129	−1.40317	−0.701584	−	0.712587i	$-0.747524\pi$
$127$	−15.8129	−1.40317	−0.701584	+	0.712587i	$0.747524\pi$
$128$	0	0
$129$	−9.46838	−0.833645
$130$	0	0
$131$	10.1345	0.885458	0.442729	−	0.896655i	$-0.354010\pi$
$131$	10.1345	0.885458	0.442729	+	0.896655i	$0.354010\pi$
$132$	0	0
$133$	12.0000	1.04053
$134$	0	0
$135$	−1.00000	−0.0860663
$136$	0	0
$137$	−15.1042	−1.29044	−0.645220	−	0.763997i	$-0.723234\pi$
$137$	−15.1042	−1.29044	−0.645220	+	0.763997i	$0.723234\pi$
$138$	0	0
$139$	17.2207	1.46064	0.730319	−	0.683107i	$-0.239372\pi$
$139$	17.2207	1.46064	0.730319	+	0.683107i	$0.239372\pi$
$140$	0	0
$141$	4.11644	0.346667
$142$	0	0
$143$	−18.5168	−1.54845
$144$	0	0
$145$	7.25839	0.602777
$146$	0	0
$147$	12.6965	1.04719
$148$	0	0
$149$	1.82774	0.149734	0.0748671	−	0.997194i	$-0.476147\pi$
$149$	1.82774	0.149734	0.0748671	+	0.997194i	$0.476147\pi$
$150$	0	0
$151$	16.4610	1.33957	0.669787	−	0.742553i	$-0.266385\pi$
$151$	16.4610	1.33957	0.669787	+	0.742553i	$0.266385\pi$
$152$	0	0
$153$	−1.35194	−0.109298
$154$	0	0
$155$	−1.00000	−0.0803219
$156$	0	0
$157$	4.53162	0.361662	0.180831	−	0.983514i	$-0.442121\pi$
$157$	4.53162	0.361662	0.180831	+	0.983514i	$0.442121\pi$
$158$	0	0
$159$	12.8203	1.01672
$160$	0	0
$161$	−6.00000	−0.472866
$162$	0	0
$163$	2.78259	0.217949	0.108975	−	0.994045i	$-0.465243\pi$
$163$	2.78259	0.217949	0.108975	+	0.994045i	$0.465243\pi$
$164$	0	0
$165$	4.17226	0.324810
$166$	0	0
$167$	12.7645	0.987747	0.493873	−	0.869534i	$-0.335581\pi$
$167$	12.7645	0.987747	0.493873	+	0.869534i	$0.335581\pi$
$168$	0	0
$169$	6.69646	0.515112
$170$	0	0
$171$	−2.70388	−0.206771
$172$	0	0
$173$	25.5800	1.94481	0.972406	−	0.233294i	$-0.0749503\pi$
$173$	25.5800	1.94481	0.972406	+	0.233294i	$0.0749503\pi$
$174$	0	0
$175$	−4.43807	−0.335487
$176$	0	0
$177$	−5.25839	−0.395245
$178$	0	0
$179$	−5.64064	−0.421601	−0.210801	−	0.977529i	$-0.567607\pi$
$179$	−5.64064	−0.421601	−0.210801	+	0.977529i	$0.567607\pi$
$180$	0	0
$181$	8.93676	0.664264	0.332132	−	0.943233i	$-0.392232\pi$
$181$	8.93676	0.664264	0.332132	+	0.943233i	$0.392232\pi$
$182$	0	0
$183$	6.00000	0.443533
$184$	0	0
$185$	10.7826	0.792752
$186$	0	0
$187$	5.64064	0.412485
$188$	0	0
$189$	−4.43807	−0.322822
$190$	0	0
$191$	2.32163	0.167987	0.0839935	−	0.996466i	$-0.473233\pi$
$191$	2.32163	0.167987	0.0839935	+	0.996466i	$0.473233\pi$
$192$	0	0
$193$	−5.22066	−0.375791	−0.187896	−	0.982189i	$-0.560167\pi$
$193$	−5.22066	−0.375791	−0.187896	+	0.982189i	$0.560167\pi$
$194$	0	0
$195$	−4.43807	−0.317817
$196$	0	0
$197$	−9.58482	−0.682890	−0.341445	−	0.939902i	$-0.610916\pi$
$197$	−9.58482	−0.682890	−0.341445	+	0.939902i	$0.610916\pi$
$198$	0	0
$199$	−12.9171	−0.915670	−0.457835	−	0.889037i	$-0.651375\pi$
$199$	−12.9171	−0.915670	−0.457835	+	0.889037i	$0.651375\pi$
$200$	0	0
$201$	1.14195	0.0805468
$202$	0	0
$203$	32.2132	2.26093
$204$	0	0
$205$	−2.17226	−0.151717
$206$	0	0
$207$	1.35194	0.0939663
$208$	0	0
$209$	11.2813	0.780343
$210$	0	0
$211$	−5.82774	−0.401198	−0.200599	−	0.979673i	$-0.564289\pi$
$211$	−5.82774	−0.401198	−0.200599	+	0.979673i	$0.564289\pi$
$212$	0	0
$213$	10.7268	0.734986
$214$	0	0
$215$	9.46838	0.645738
$216$	0	0
$217$	−4.43807	−0.301276
$218$	0	0
$219$	10.4381	0.705339
$220$	0	0
$221$	−6.00000	−0.403604
$222$	0	0
$223$	2.00000	0.133930	0.0669650	−	0.997755i	$-0.478668\pi$
$223$	2.00000	0.133930	0.0669650	+	0.997755i	$0.478668\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	−25.7933	−1.71196	−0.855979	−	0.517010i	$-0.827045\pi$
$227$	−25.7933	−1.71196	−0.855979	+	0.517010i	$0.827045\pi$
$228$	0	0
$229$	−7.93937	−0.524649	−0.262324	−	0.964980i	$-0.584489\pi$
$229$	−7.93937	−0.524649	−0.262324	+	0.964980i	$0.584489\pi$
$230$	0	0
$231$	18.5168	1.21831
$232$	0	0
$233$	24.4610	1.60249	0.801246	−	0.598336i	$-0.204171\pi$
$233$	24.4610	1.60249	0.801246	+	0.598336i	$0.204171\pi$
$234$	0	0
$235$	−4.11644	−0.268527
$236$	0	0
$237$	2.05582	0.133540
$238$	0	0
$239$	18.0000	1.16432	0.582162	−	0.813073i	$-0.302207\pi$
$239$	18.0000	1.16432	0.582162	+	0.813073i	$0.302207\pi$
$240$	0	0
$241$	−2.64325	−0.170267	−0.0851335	−	0.996370i	$-0.527132\pi$
$241$	−2.64325	−0.170267	−0.0851335	+	0.996370i	$0.527132\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−12.6965	−0.811147
$246$	0	0
$247$	−12.0000	−0.763542
$248$	0	0
$249$	6.99258	0.443137
$250$	0	0
$251$	16.4562	1.03870	0.519352	−	0.854561i	$-0.326173\pi$
$251$	16.4562	1.03870	0.519352	+	0.854561i	$0.326173\pi$
$252$	0	0
$253$	−5.64064	−0.354624
$254$	0	0
$255$	1.35194	0.0846617
$256$	0	0
$257$	7.16484	0.446931	0.223465	−	0.974712i	$-0.428263\pi$
$257$	7.16484	0.446931	0.223465	+	0.974712i	$0.428263\pi$
$258$	0	0
$259$	47.8539	2.97350
$260$	0	0
$261$	−7.25839	−0.449283
$262$	0	0
$263$	8.87614	0.547326	0.273663	−	0.961826i	$-0.411765\pi$
$263$	8.87614	0.547326	0.273663	+	0.961826i	$0.411765\pi$
$264$	0	0
$265$	−12.8203	−0.787546
$266$	0	0
$267$	−2.55451	−0.156334
$268$	0	0
$269$	3.90164	0.237887	0.118944	−	0.992901i	$-0.462049\pi$
$269$	3.90164	0.237887	0.118944	+	0.992901i	$0.462049\pi$
$270$	0	0
$271$	4.87614	0.296204	0.148102	−	0.988972i	$-0.452684\pi$
$271$	4.87614	0.296204	0.148102	+	0.988972i	$0.452684\pi$
$272$	0	0
$273$	−19.6965	−1.19208
$274$	0	0
$275$	−4.17226	−0.251597
$276$	0	0
$277$	29.5981	1.77838	0.889189	−	0.457540i	$-0.151269\pi$
$277$	29.5981	1.77838	0.889189	+	0.457540i	$0.151269\pi$
$278$	0	0
$279$	1.00000	0.0598684
$280$	0	0
$281$	−15.2207	−0.907988	−0.453994	−	0.891005i	$-0.650001\pi$
$281$	−15.2207	−0.907988	−0.453994	+	0.891005i	$0.650001\pi$
$282$	0	0
$283$	20.0181	1.18995	0.594976	−	0.803744i	$-0.297162\pi$
$283$	20.0181	1.18995	0.594976	+	0.803744i	$0.297162\pi$
$284$	0	0
$285$	2.70388	0.160164
$286$	0	0
$287$	−9.64064	−0.569069
$288$	0	0
$289$	−15.1723	−0.892486
$290$	0	0
$291$	−15.9245	−0.933513
$292$	0	0
$293$	−8.46096	−0.494295	−0.247147	−	0.968978i	$-0.579493\pi$
$293$	−8.46096	−0.494295	−0.247147	+	0.968978i	$0.579493\pi$
$294$	0	0
$295$	5.25839	0.306155
$296$	0	0
$297$	−4.17226	−0.242099
$298$	0	0
$299$	6.00000	0.346989
$300$	0	0
$301$	42.0213	2.42207
$302$	0	0
$303$	−0.876139	−0.0503329
$304$	0	0
$305$	−6.00000	−0.343559
$306$	0	0
$307$	8.43807	0.481586	0.240793	−	0.970576i	$-0.422592\pi$
$307$	8.43807	0.481586	0.240793	+	0.970576i	$0.422592\pi$
$308$	0	0
$309$	−6.43807	−0.366249
$310$	0	0
$311$	20.6661	1.17187	0.585935	−	0.810358i	$-0.300727\pi$
$311$	20.6661	1.17187	0.585935	+	0.810358i	$0.300727\pi$
$312$	0	0
$313$	21.3142	1.20475	0.602375	−	0.798213i	$-0.294221\pi$
$313$	21.3142	1.20475	0.602375	+	0.798213i	$0.294221\pi$
$314$	0	0
$315$	4.43807	0.250057
$316$	0	0
$317$	13.5242	0.759595	0.379797	−	0.925070i	$-0.375994\pi$
$317$	13.5242	0.759595	0.379797	+	0.925070i	$0.375994\pi$
$318$	0	0
$319$	30.2839	1.69557
$320$	0	0
$321$	−11.6965	−0.652833
$322$	0	0
$323$	3.65548	0.203396
$324$	0	0
$325$	4.43807	0.246180
$326$	0	0
$327$	11.5242	0.637290
$328$	0	0
$329$	−18.2691	−1.00721
$330$	0	0
$331$	−12.9926	−0.714137	−0.357068	−	0.934078i	$-0.616224\pi$
$331$	−12.9926	−0.714137	−0.357068	+	0.934078i	$0.616224\pi$
$332$	0	0
$333$	−10.7826	−0.590882
$334$	0	0
$335$	−1.14195	−0.0623913
$336$	0	0
$337$	22.5955	1.23085	0.615427	−	0.788194i	$-0.288983\pi$
$337$	22.5955	1.23085	0.615427	+	0.788194i	$0.288983\pi$
$338$	0	0
$339$	17.5800	0.954815
$340$	0	0
$341$	−4.17226	−0.225941
$342$	0	0
$343$	−25.2813	−1.36506
$344$	0	0
$345$	−1.35194	−0.0727860
$346$	0	0
$347$	0.419983	0.0225459	0.0112729	−	0.999936i	$-0.496412\pi$
$347$	0.419983	0.0225459	0.0112729	+	0.999936i	$0.496412\pi$
$348$	0	0
$349$	−11.8687	−0.635318	−0.317659	−	0.948205i	$-0.602897\pi$
$349$	−11.8687	−0.635318	−0.317659	+	0.948205i	$0.602897\pi$
$350$	0	0
$351$	4.43807	0.236887
$352$	0	0
$353$	−1.69646	−0.0902935	−0.0451467	−	0.998980i	$-0.514376\pi$
$353$	−1.69646	−0.0902935	−0.0451467	+	0.998980i	$0.514376\pi$
$354$	0	0
$355$	−10.7268	−0.569318
$356$	0	0
$357$	6.00000	0.317554
$358$	0	0
$359$	4.80223	0.253452	0.126726	−	0.991938i	$-0.459553\pi$
$359$	4.80223	0.253452	0.126726	+	0.991938i	$0.459553\pi$
$360$	0	0
$361$	−11.6890	−0.615213
$362$	0	0
$363$	6.40776	0.336320
$364$	0	0
$365$	−10.4381	−0.546354
$366$	0	0
$367$	31.6258	1.65085	0.825427	−	0.564509i	$-0.190935\pi$
$367$	31.6258	1.65085	0.825427	+	0.564509i	$0.190935\pi$
$368$	0	0
$369$	2.17226	0.113083
$370$	0	0
$371$	−56.8975	−2.95397
$372$	0	0
$373$	17.9245	0.928097	0.464049	−	0.885810i	$-0.346396\pi$
$373$	17.9245	0.928097	0.464049	+	0.885810i	$0.346396\pi$
$374$	0	0
$375$	−1.00000	−0.0516398
$376$	0	0
$377$	−32.2132	−1.65907
$378$	0	0
$379$	−32.8007	−1.68486	−0.842429	−	0.538808i	$-0.818875\pi$
$379$	−32.8007	−1.68486	−0.842429	+	0.538808i	$0.818875\pi$
$380$	0	0
$381$	−15.8129	−0.810120
$382$	0	0
$383$	29.8081	1.52312	0.761561	−	0.648093i	$-0.224433\pi$
$383$	29.8081	1.52312	0.761561	+	0.648093i	$0.224433\pi$
$384$	0	0
$385$	−18.5168	−0.943703
$386$	0	0
$387$	−9.46838	−0.481305
$388$	0	0
$389$	−32.7629	−1.66115	−0.830574	−	0.556909i	$-0.811987\pi$
$389$	−32.7629	−1.66115	−0.830574	+	0.556909i	$0.811987\pi$
$390$	0	0
$391$	−1.82774	−0.0924328
$392$	0	0
$393$	10.1345	0.511219
$394$	0	0
$395$	−2.05582	−0.103439
$396$	0	0
$397$	34.4562	1.72930	0.864652	−	0.502371i	$-0.167539\pi$
$397$	34.4562	1.72930	0.864652	+	0.502371i	$0.167539\pi$
$398$	0	0
$399$	12.0000	0.600751
$400$	0	0
$401$	29.7145	1.48387	0.741937	−	0.670470i	$-0.233907\pi$
$401$	29.7145	1.48387	0.741937	+	0.670470i	$0.233907\pi$
$402$	0	0
$403$	4.43807	0.221076
$404$	0	0
$405$	−1.00000	−0.0496904
$406$	0	0
$407$	44.9878	2.22996
$408$	0	0
$409$	−28.0968	−1.38930	−0.694649	−	0.719349i	$-0.744440\pi$
$409$	−28.0968	−1.38930	−0.694649	+	0.719349i	$0.744440\pi$
$410$	0	0
$411$	−15.1042	−0.745036
$412$	0	0
$413$	23.3371	1.14834
$414$	0	0
$415$	−6.99258	−0.343252
$416$	0	0
$417$	17.2207	0.843299
$418$	0	0
$419$	−6.83841	−0.334078	−0.167039	−	0.985950i	$-0.553421\pi$
$419$	−6.83841	−0.334078	−0.167039	+	0.985950i	$0.553421\pi$
$420$	0	0
$421$	−5.24030	−0.255397	−0.127698	−	0.991813i	$-0.540759\pi$
$421$	−5.24030	−0.255397	−0.127698	+	0.991813i	$0.540759\pi$
$422$	0	0
$423$	4.11644	0.200148
$424$	0	0
$425$	−1.35194	−0.0655787
$426$	0	0
$427$	−26.6284	−1.28864
$428$	0	0
$429$	−18.5168	−0.893999
$430$	0	0
$431$	−6.83841	−0.329395	−0.164697	−	0.986344i	$-0.552665\pi$
$431$	−6.83841	−0.329395	−0.164697	+	0.986344i	$0.552665\pi$
$432$	0	0
$433$	−11.3142	−0.543726	−0.271863	−	0.962336i	$-0.587640\pi$
$433$	−11.3142	−0.543726	−0.271863	+	0.962336i	$0.587640\pi$
$434$	0	0
$435$	7.25839	0.348013
$436$	0	0
$437$	−3.65548	−0.174865
$438$	0	0
$439$	10.8155	0.516196	0.258098	−	0.966119i	$-0.416904\pi$
$439$	10.8155	0.516196	0.258098	+	0.966119i	$0.416904\pi$
$440$	0	0
$441$	12.6965	0.604593
$442$	0	0
$443$	17.4487	0.829015	0.414507	−	0.910046i	$-0.363954\pi$
$443$	17.4487	0.829015	0.414507	+	0.910046i	$0.363954\pi$
$444$	0	0
$445$	2.55451	0.121095
$446$	0	0
$447$	1.82774	0.0864491
$448$	0	0
$449$	3.96227	0.186991	0.0934955	−	0.995620i	$-0.470196\pi$
$449$	3.96227	0.186991	0.0934955	+	0.995620i	$0.470196\pi$
$450$	0	0
$451$	−9.06324	−0.426771
$452$	0	0
$453$	16.4610	0.773404
$454$	0	0
$455$	19.6965	0.923384
$456$	0	0
$457$	33.8310	1.58255	0.791273	−	0.611462i	$-0.209418\pi$
$457$	33.8310	1.58255	0.791273	+	0.611462i	$0.209418\pi$
$458$	0	0
$459$	−1.35194	−0.0631031
$460$	0	0
$461$	−7.55712	−0.351970	−0.175985	−	0.984393i	$-0.556311\pi$
$461$	−7.55712	−0.351970	−0.175985	+	0.984393i	$0.556311\pi$
$462$	0	0
$463$	−28.7401	−1.33566	−0.667832	−	0.744312i	$-0.732777\pi$
$463$	−28.7401	−1.33566	−0.667832	+	0.744312i	$0.732777\pi$
$464$	0	0
$465$	−1.00000	−0.0463739
$466$	0	0
$467$	7.16484	0.331549	0.165775	−	0.986164i	$-0.446988\pi$
$467$	7.16484	0.331549	0.165775	+	0.986164i	$0.446988\pi$
$468$	0	0
$469$	−5.06804	−0.234021
$470$	0	0
$471$	4.53162	0.208806
$472$	0	0
$473$	39.5046	1.81642
$474$	0	0
$475$	−2.70388	−0.124062
$476$	0	0
$477$	12.8203	0.587002
$478$	0	0
$479$	30.8384	1.40904	0.704521	−	0.709683i	$-0.251162\pi$
$479$	30.8384	1.40904	0.704521	+	0.709683i	$0.251162\pi$
$480$	0	0
$481$	−47.8539	−2.18195
$482$	0	0
$483$	−6.00000	−0.273009
$484$	0	0
$485$	15.9245	0.723096
$486$	0	0
$487$	6.17226	0.279692	0.139846	−	0.990173i	$-0.455339\pi$
$487$	6.17226	0.279692	0.139846	+	0.990173i	$0.455339\pi$
$488$	0	0
$489$	2.78259	0.125833
$490$	0	0
$491$	−1.10902	−0.0500496	−0.0250248	−	0.999687i	$-0.507966\pi$
$491$	−1.10902	−0.0500496	−0.0250248	+	0.999687i	$0.507966\pi$
$492$	0	0
$493$	9.81290	0.441951
$494$	0	0
$495$	4.17226	0.187529
$496$	0	0
$497$	−47.6062	−2.13543
$498$	0	0
$499$	30.2180	1.35275	0.676373	−	0.736559i	$-0.263551\pi$
$499$	30.2180	1.35275	0.676373	+	0.736559i	$0.263551\pi$
$500$	0	0
$501$	12.7645	0.570276
$502$	0	0
$503$	15.2765	0.681144	0.340572	−	0.940218i	$-0.389379\pi$
$503$	15.2765	0.681144	0.340572	+	0.940218i	$0.389379\pi$
$504$	0	0
$505$	0.876139	0.0389877
$506$	0	0
$507$	6.69646	0.297400
$508$	0	0
$509$	−7.08613	−0.314087	−0.157044	−	0.987592i	$-0.550196\pi$
$509$	−7.08613	−0.314087	−0.157044	+	0.987592i	$0.550196\pi$
$510$	0	0
$511$	−46.3249	−2.04929
$512$	0	0
$513$	−2.70388	−0.119379
$514$	0	0
$515$	6.43807	0.283695
$516$	0	0
$517$	−17.1749	−0.755350
$518$	0	0
$519$	25.5800	1.12284
$520$	0	0
$521$	16.7645	0.734466	0.367233	−	0.930129i	$-0.380305\pi$
$521$	16.7645	0.734466	0.367233	+	0.930129i	$0.380305\pi$
$522$	0	0
$523$	20.9878	0.917731	0.458866	−	0.888506i	$-0.348256\pi$
$523$	20.9878	0.917731	0.458866	+	0.888506i	$0.348256\pi$
$524$	0	0
$525$	−4.43807	−0.193693
$526$	0	0
$527$	−1.35194	−0.0588914
$528$	0	0
$529$	−21.1723	−0.920533
$530$	0	0
$531$	−5.25839	−0.228195
$532$	0	0
$533$	9.64064	0.417583
$534$	0	0
$535$	11.6965	0.505682
$536$	0	0
$537$	−5.64064	−0.243412
$538$	0	0
$539$	−52.9729	−2.28171
$540$	0	0
$541$	−24.4003	−1.04905	−0.524526	−	0.851394i	$-0.675758\pi$
$541$	−24.4003	−1.04905	−0.524526	+	0.851394i	$0.675758\pi$
$542$	0	0
$543$	8.93676	0.383513
$544$	0	0
$545$	−11.5242	−0.493642
$546$	0	0
$547$	−23.4355	−1.00203	−0.501014	−	0.865439i	$-0.667039\pi$
$547$	−23.4355	−1.00203	−0.501014	+	0.865439i	$0.667039\pi$
$548$	0	0
$549$	6.00000	0.256074
$550$	0	0
$551$	19.6258	0.836087
$552$	0	0
$553$	−9.12386	−0.387986
$554$	0	0
$555$	10.7826	0.457695
$556$	0	0
$557$	−26.1526	−1.10812	−0.554061	−	0.832476i	$-0.686923\pi$
$557$	−26.1526	−1.10812	−0.554061	+	0.832476i	$0.686923\pi$
$558$	0	0
$559$	−42.0213	−1.77731
$560$	0	0
$561$	5.64064	0.238148
$562$	0	0
$563$	−37.9049	−1.59750	−0.798750	−	0.601663i	$-0.794505\pi$
$563$	−37.9049	−1.59750	−0.798750	+	0.601663i	$0.794505\pi$
$564$	0	0
$565$	−17.5800	−0.739597
$566$	0	0
$567$	−4.43807	−0.186381
$568$	0	0
$569$	−9.19777	−0.385590	−0.192795	−	0.981239i	$-0.561755\pi$
$569$	−9.19777	−0.385590	−0.192795	+	0.981239i	$0.561755\pi$
$570$	0	0
$571$	−29.2207	−1.22285	−0.611423	−	0.791304i	$-0.709403\pi$
$571$	−29.2207	−1.22285	−0.611423	+	0.791304i	$0.709403\pi$
$572$	0	0
$573$	2.32163	0.0969873
$574$	0	0
$575$	1.35194	0.0563798
$576$	0	0
$577$	−18.2084	−0.758027	−0.379014	−	0.925391i	$-0.623737\pi$
$577$	−18.2084	−0.758027	−0.379014	+	0.925391i	$0.623737\pi$
$578$	0	0
$579$	−5.22066	−0.216963
$580$	0	0
$581$	−31.0336	−1.28749
$582$	0	0
$583$	−53.4897	−2.21532
$584$	0	0
$585$	−4.43807	−0.183492
$586$	0	0
$587$	40.3807	1.66669	0.833345	−	0.552754i	$-0.186423\pi$
$587$	40.3807	1.66669	0.833345	+	0.552754i	$0.186423\pi$
$588$	0	0
$589$	−2.70388	−0.111411
$590$	0	0
$591$	−9.58482	−0.394267
$592$	0	0
$593$	−36.7497	−1.50913	−0.754564	−	0.656226i	$-0.772152\pi$
$593$	−36.7497	−1.50913	−0.754564	+	0.656226i	$0.772152\pi$
$594$	0	0
$595$	−6.00000	−0.245976
$596$	0	0
$597$	−12.9171	−0.528663
$598$	0	0
$599$	12.5545	0.512963	0.256482	−	0.966549i	$-0.417437\pi$
$599$	12.5545	0.512963	0.256482	+	0.966549i	$0.417437\pi$
$600$	0	0
$601$	−30.0362	−1.22520	−0.612601	−	0.790393i	$-0.709877\pi$
$601$	−30.0362	−1.22520	−0.612601	+	0.790393i	$0.709877\pi$
$602$	0	0
$603$	1.14195	0.0465037
$604$	0	0
$605$	−6.40776	−0.260512
$606$	0	0
$607$	10.1297	0.411153	0.205576	−	0.978641i	$-0.434093\pi$
$607$	10.1297	0.411153	0.205576	+	0.978641i	$0.434093\pi$
$608$	0	0
$609$	32.2132	1.30535
$610$	0	0
$611$	18.2691	0.739087
$612$	0	0
$613$	−38.8794	−1.57032	−0.785162	−	0.619291i	$-0.787420\pi$
$613$	−38.8794	−1.57032	−0.785162	+	0.619291i	$0.787420\pi$
$614$	0	0
$615$	−2.17226	−0.0875940
$616$	0	0
$617$	−27.4078	−1.10339	−0.551697	−	0.834044i	$-0.686020\pi$
$617$	−27.4078	−1.10339	−0.551697	+	0.834044i	$0.686020\pi$
$618$	0	0
$619$	−26.2887	−1.05663	−0.528316	−	0.849048i	$-0.677176\pi$
$619$	−26.2887	−1.05663	−0.528316	+	0.849048i	$0.677176\pi$
$620$	0	0
$621$	1.35194	0.0542515
$622$	0	0
$623$	11.3371	0.454211
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	11.2813	0.450531
$628$	0	0
$629$	14.5774	0.581239
$630$	0	0
$631$	3.87875	0.154411	0.0772053	−	0.997015i	$-0.475400\pi$
$631$	3.87875	0.154411	0.0772053	+	0.997015i	$0.475400\pi$
$632$	0	0
$633$	−5.82774	−0.231632
$634$	0	0
$635$	15.8129	0.627516
$636$	0	0
$637$	56.3478	2.23258
$638$	0	0
$639$	10.7268	0.424345
$640$	0	0
$641$	4.08874	0.161496	0.0807478	−	0.996735i	$-0.474269\pi$
$641$	4.08874	0.161496	0.0807478	+	0.996735i	$0.474269\pi$
$642$	0	0
$643$	−15.2355	−0.600829	−0.300415	−	0.953809i	$-0.597125\pi$
$643$	−15.2355	−0.600829	−0.300415	+	0.953809i	$0.597125\pi$
$644$	0	0
$645$	9.46838	0.372817
$646$	0	0
$647$	21.6406	0.850781	0.425391	−	0.905010i	$-0.360137\pi$
$647$	21.6406	0.850781	0.425391	+	0.905010i	$0.360137\pi$
$648$	0	0
$649$	21.9394	0.861196
$650$	0	0
$651$	−4.43807	−0.173942
$652$	0	0
$653$	−0.228078	−0.00892538	−0.00446269	−	0.999990i	$-0.501421\pi$
$653$	−0.228078	−0.00892538	−0.00446269	+	0.999990i	$0.501421\pi$
$654$	0	0
$655$	−10.1345	−0.395989
$656$	0	0
$657$	10.4381	0.407228
$658$	0	0
$659$	−50.9500	−1.98473	−0.992366	−	0.123328i	$-0.960643\pi$
$659$	−50.9500	−1.98473	−0.992366	+	0.123328i	$0.960643\pi$
$660$	0	0
$661$	−7.82774	−0.304464	−0.152232	−	0.988345i	$-0.548646\pi$
$661$	−7.82774	−0.304464	−0.152232	+	0.988345i	$0.548646\pi$
$662$	0	0
$663$	−6.00000	−0.233021
$664$	0	0
$665$	−12.0000	−0.465340
$666$	0	0
$667$	−9.81290	−0.379957
$668$	0	0
$669$	2.00000	0.0773245
$670$	0	0
$671$	−25.0336	−0.966410
$672$	0	0
$673$	−45.3962	−1.74989	−0.874947	−	0.484219i	$-0.839104\pi$
$673$	−45.3962	−1.74989	−0.874947	+	0.484219i	$0.839104\pi$
$674$	0	0
$675$	1.00000	0.0384900
$676$	0	0
$677$	51.9507	1.99663	0.998313	−	0.0580539i	$-0.0184895\pi$
$677$	51.9507	1.99663	0.998313	+	0.0580539i	$0.0184895\pi$
$678$	0	0
$679$	70.6742	2.71223
$680$	0	0
$681$	−25.7933	−0.988400
$682$	0	0
$683$	−47.0894	−1.80183	−0.900913	−	0.434001i	$-0.857102\pi$
$683$	−47.0894	−1.80183	−0.900913	+	0.434001i	$0.857102\pi$
$684$	0	0
$685$	15.1042	0.577102
$686$	0	0
$687$	−7.93937	−0.302906
$688$	0	0
$689$	56.8975	2.16762
$690$	0	0
$691$	29.9097	1.13782	0.568909	−	0.822400i	$-0.307366\pi$
$691$	29.9097	1.13782	0.568909	+	0.822400i	$0.307366\pi$
$692$	0	0
$693$	18.5168	0.703394
$694$	0	0
$695$	−17.2207	−0.653217
$696$	0	0
$697$	−2.93676	−0.111238
$698$	0	0
$699$	24.4610	0.925199
$700$	0	0
$701$	40.4562	1.52801	0.764004	−	0.645212i	$-0.223231\pi$
$701$	40.4562	1.52801	0.764004	+	0.645212i	$0.223231\pi$
$702$	0	0
$703$	29.1548	1.09959
$704$	0	0
$705$	−4.11644	−0.155034
$706$	0	0
$707$	3.88836	0.146237
$708$	0	0
$709$	−52.7858	−1.98241	−0.991207	−	0.132321i	$-0.957757\pi$
$709$	−52.7858	−1.98241	−0.991207	+	0.132321i	$0.957757\pi$
$710$	0	0
$711$	2.05582	0.0770992
$712$	0	0
$713$	1.35194	0.0506305
$714$	0	0
$715$	18.5168	0.692488
$716$	0	0
$717$	18.0000	0.672222
$718$	0	0
$719$	−3.82774	−0.142751	−0.0713753	−	0.997450i	$-0.522739\pi$
$719$	−3.82774	−0.142751	−0.0713753	+	0.997450i	$0.522739\pi$
$720$	0	0
$721$	28.5726	1.06410
$722$	0	0
$723$	−2.64325	−0.0983036
$724$	0	0
$725$	−7.25839	−0.269570
$726$	0	0
$727$	11.8554	0.439694	0.219847	−	0.975534i	$-0.429444\pi$
$727$	11.8554	0.439694	0.219847	+	0.975534i	$0.429444\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	12.8007	0.473450
$732$	0	0
$733$	−5.16003	−0.190590	−0.0952951	−	0.995449i	$-0.530379\pi$
$733$	−5.16003	−0.190590	−0.0952951	+	0.995449i	$0.530379\pi$
$734$	0	0
$735$	−12.6965	−0.468316
$736$	0	0
$737$	−4.76450	−0.175503
$738$	0	0
$739$	9.52420	0.350353	0.175177	−	0.984537i	$-0.443950\pi$
$739$	9.52420	0.350353	0.175177	+	0.984537i	$0.443950\pi$
$740$	0	0
$741$	−12.0000	−0.440831
$742$	0	0
$743$	33.7885	1.23958	0.619789	−	0.784768i	$-0.287218\pi$
$743$	33.7885	1.23958	0.619789	+	0.784768i	$0.287218\pi$
$744$	0	0
$745$	−1.82774	−0.0669632
$746$	0	0
$747$	6.99258	0.255845
$748$	0	0
$749$	51.9097	1.89674
$750$	0	0
$751$	−41.1090	−1.50009	−0.750045	−	0.661387i	$-0.769968\pi$
$751$	−41.1090	−1.50009	−0.750045	+	0.661387i	$0.769968\pi$
$752$	0	0
$753$	16.4562	0.599696
$754$	0	0
$755$	−16.4610	−0.599076
$756$	0	0
$757$	−24.3626	−0.885474	−0.442737	−	0.896652i	$-0.645992\pi$
$757$	−24.3626	−0.885474	−0.442737	+	0.896652i	$0.645992\pi$
$758$	0	0
$759$	−5.64064	−0.204742
$760$	0	0
$761$	26.4184	0.957667	0.478834	−	0.877906i	$-0.341060\pi$
$761$	26.4184	0.957667	0.478834	+	0.877906i	$0.341060\pi$
$762$	0	0
$763$	−51.1452	−1.85158
$764$	0	0
$765$	1.35194	0.0488795
$766$	0	0
$767$	−23.3371	−0.842654
$768$	0	0
$769$	32.9023	1.18649	0.593243	−	0.805023i	$-0.297847\pi$
$769$	32.9023	1.18649	0.593243	+	0.805023i	$0.297847\pi$
$770$	0	0
$771$	7.16484	0.258035
$772$	0	0
$773$	44.8661	1.61372	0.806861	−	0.590741i	$-0.201165\pi$
$773$	44.8661	1.61372	0.806861	+	0.590741i	$0.201165\pi$
$774$	0	0
$775$	1.00000	0.0359211
$776$	0	0
$777$	47.8539	1.71675
$778$	0	0
$779$	−5.87353	−0.210441
$780$	0	0
$781$	−44.7549	−1.60146
$782$	0	0
$783$	−7.25839	−0.259394
$784$	0	0
$785$	−4.53162	−0.161740
$786$	0	0
$787$	42.4265	1.51234	0.756170	−	0.654375i	$-0.227068\pi$
$787$	42.4265	1.51234	0.756170	+	0.654375i	$0.227068\pi$
$788$	0	0
$789$	8.87614	0.315999
$790$	0	0
$791$	−78.0213	−2.77412
$792$	0	0
$793$	26.6284	0.945603
$794$	0	0
$795$	−12.8203	−0.454690
$796$	0	0
$797$	28.9681	1.02610	0.513052	−	0.858358i	$-0.328515\pi$
$797$	28.9681	1.02610	0.513052	+	0.858358i	$0.328515\pi$
$798$	0	0
$799$	−5.56518	−0.196882
$800$	0	0
$801$	−2.55451	−0.0902593
$802$	0	0
$803$	−43.5503	−1.53686
$804$	0	0
$805$	6.00000	0.211472
$806$	0	0
$807$	3.90164	0.137344
$808$	0	0
$809$	26.4429	0.929682	0.464841	−	0.885394i	$-0.346111\pi$
$809$	26.4429	0.929682	0.464841	+	0.885394i	$0.346111\pi$
$810$	0	0
$811$	26.2542	0.921910	0.460955	−	0.887423i	$-0.347507\pi$
$811$	26.2542	0.921910	0.460955	+	0.887423i	$0.347507\pi$
$812$	0	0
$813$	4.87614	0.171014
$814$	0	0
$815$	−2.78259	−0.0974699
$816$	0	0
$817$	25.6014	0.895678
$818$	0	0
$819$	−19.6965	−0.688250
$820$	0	0
$821$	13.5571	0.473147	0.236573	−	0.971614i	$-0.423976\pi$
$821$	13.5571	0.473147	0.236573	+	0.971614i	$0.423976\pi$
$822$	0	0
$823$	−44.8613	−1.56377	−0.781883	−	0.623425i	$-0.785741\pi$
$823$	−44.8613	−1.56377	−0.781883	+	0.623425i	$0.785741\pi$
$824$	0	0
$825$	−4.17226	−0.145259
$826$	0	0
$827$	29.8081	1.03653	0.518265	−	0.855220i	$-0.326578\pi$
$827$	29.8081	1.03653	0.518265	+	0.855220i	$0.326578\pi$
$828$	0	0
$829$	−43.4535	−1.50920	−0.754602	−	0.656183i	$-0.772170\pi$
$829$	−43.4535	−1.50920	−0.754602	+	0.656183i	$0.772170\pi$
$830$	0	0
$831$	29.5981	1.02675
$832$	0	0
$833$	−17.1648	−0.594727
$834$	0	0
$835$	−12.7645	−0.441734
$836$	0	0
$837$	1.00000	0.0345651
$838$	0	0
$839$	12.9958	0.448666	0.224333	−	0.974513i	$-0.427980\pi$
$839$	12.9958	0.448666	0.224333	+	0.974513i	$0.427980\pi$
$840$	0	0
$841$	23.6842	0.816698
$842$	0	0
$843$	−15.2207	−0.524227
$844$	0	0
$845$	−6.69646	−0.230365
$846$	0	0
$847$	−28.4381	−0.977144
$848$	0	0
$849$	20.0181	0.687019
$850$	0	0
$851$	−14.5774	−0.499707
$852$	0	0
$853$	−49.7523	−1.70349	−0.851743	−	0.523960i	$-0.824454\pi$
$853$	−49.7523	−1.70349	−0.851743	+	0.523960i	$0.824454\pi$
$854$	0	0
$855$	2.70388	0.0924707
$856$	0	0
$857$	−21.2058	−0.724377	−0.362189	−	0.932105i	$-0.617970\pi$
$857$	−21.2058	−0.724377	−0.362189	+	0.932105i	$0.617970\pi$
$858$	0	0
$859$	10.4710	0.357266	0.178633	−	0.983916i	$-0.442833\pi$
$859$	10.4710	0.357266	0.178633	+	0.983916i	$0.442833\pi$
$860$	0	0
$861$	−9.64064	−0.328552
$862$	0	0
$863$	4.83997	0.164754	0.0823772	−	0.996601i	$-0.473749\pi$
$863$	4.83997	0.164754	0.0823772	+	0.996601i	$0.473749\pi$
$864$	0	0
$865$	−25.5800	−0.869747
$866$	0	0
$867$	−15.1723	−0.515277
$868$	0	0
$869$	−8.57741	−0.290969
$870$	0	0
$871$	5.06804	0.171724
$872$	0	0
$873$	−15.9245	−0.538964
$874$	0	0
$875$	4.43807	0.150034
$876$	0	0
$877$	−36.8007	−1.24267	−0.621335	−	0.783545i	$-0.713409\pi$
$877$	−36.8007	−1.24267	−0.621335	+	0.783545i	$0.713409\pi$
$878$	0	0
$879$	−8.46096	−0.285381
$880$	0	0
$881$	−39.1319	−1.31839	−0.659194	−	0.751973i	$-0.729102\pi$
$881$	−39.1319	−1.31839	−0.659194	+	0.751973i	$0.729102\pi$
$882$	0	0
$883$	17.7671	0.597911	0.298956	−	0.954267i	$-0.403362\pi$
$883$	17.7671	0.597911	0.298956	+	0.954267i	$0.403362\pi$
$884$	0	0
$885$	5.25839	0.176759
$886$	0	0
$887$	57.2520	1.92233	0.961167	−	0.275966i	$-0.0889977\pi$
$887$	57.2520	1.92233	0.961167	+	0.275966i	$0.0889977\pi$
$888$	0	0
$889$	70.1788	2.35372
$890$	0	0
$891$	−4.17226	−0.139776
$892$	0	0
$893$	−11.1304	−0.372463
$894$	0	0
$895$	5.64064	0.188546
$896$	0	0
$897$	6.00000	0.200334
$898$	0	0
$899$	−7.25839	−0.242081
$900$	0	0
$901$	−17.3323	−0.577422
$902$	0	0
$903$	42.0213	1.39838
$904$	0	0
$905$	−8.93676	−0.297068
$906$	0	0
$907$	−16.8581	−0.559762	−0.279881	−	0.960035i	$-0.590295\pi$
$907$	−16.8581	−0.559762	−0.279881	+	0.960035i	$0.590295\pi$
$908$	0	0
$909$	−0.876139	−0.0290597
$910$	0	0
$911$	−45.5800	−1.51013	−0.755067	−	0.655648i	$-0.772396\pi$
$911$	−45.5800	−1.51013	−0.755067	+	0.655648i	$0.772396\pi$
$912$	0	0
$913$	−29.1749	−0.965547
$914$	0	0
$915$	−6.00000	−0.198354
$916$	0	0
$917$	−44.9777	−1.48530
$918$	0	0
$919$	35.8491	1.18255	0.591276	−	0.806469i	$-0.298624\pi$
$919$	35.8491	1.18255	0.591276	+	0.806469i	$0.298624\pi$
$920$	0	0
$921$	8.43807	0.278044
$922$	0	0
$923$	47.6062	1.56698
$924$	0	0
$925$	−10.7826	−0.354529
$926$	0	0
$927$	−6.43807	−0.211454
$928$	0	0
$929$	45.7752	1.50183	0.750917	−	0.660396i	$-0.229612\pi$
$929$	45.7752	1.50183	0.750917	+	0.660396i	$0.229612\pi$
$930$	0	0
$931$	−34.3297	−1.12511
$932$	0	0
$933$	20.6661	0.676579
$934$	0	0
$935$	−5.64064	−0.184469
$936$	0	0
$937$	25.8277	0.843755	0.421878	−	0.906653i	$-0.361371\pi$
$937$	25.8277	0.843755	0.421878	+	0.906653i	$0.361371\pi$
$938$	0	0
$939$	21.3142	0.695563
$940$	0	0
$941$	−1.52095	−0.0495815	−0.0247908	−	0.999693i	$-0.507892\pi$
$941$	−1.52095	−0.0495815	−0.0247908	+	0.999693i	$0.507892\pi$
$942$	0	0
$943$	2.93676	0.0956342
$944$	0	0
$945$	4.43807	0.144370
$946$	0	0
$947$	−18.6481	−0.605981	−0.302990	−	0.952994i	$-0.597985\pi$
$947$	−18.6481	−0.605981	−0.302990	+	0.952994i	$0.597985\pi$
$948$	0	0
$949$	46.3249	1.50377
$950$	0	0
$951$	13.5242	0.438552
$952$	0	0
$953$	33.3519	1.08038	0.540188	−	0.841545i	$-0.318353\pi$
$953$	33.3519	1.08038	0.540188	+	0.841545i	$0.318353\pi$
$954$	0	0
$955$	−2.32163	−0.0751261
$956$	0	0
$957$	30.2839	0.978939
$958$	0	0
$959$	67.0336	2.16463
$960$	0	0
$961$	1.00000	0.0322581
$962$	0	0
$963$	−11.6965	−0.376913
$964$	0	0
$965$	5.22066	0.168059
$966$	0	0
$967$	2.00000	0.0643157	0.0321578	−	0.999483i	$-0.489762\pi$
$967$	2.00000	0.0643157	0.0321578	+	0.999483i	$0.489762\pi$
$968$	0	0
$969$	3.65548	0.117431
$970$	0	0
$971$	−21.6932	−0.696168	−0.348084	−	0.937463i	$-0.613168\pi$
$971$	−21.6932	−0.696168	−0.348084	+	0.937463i	$0.613168\pi$
$972$	0	0
$973$	−76.4265	−2.45012
$974$	0	0
$975$	4.43807	0.142132
$976$	0	0
$977$	17.2813	0.552877	0.276439	−	0.961032i	$-0.410846\pi$
$977$	17.2813	0.552877	0.276439	+	0.961032i	$0.410846\pi$
$978$	0	0
$979$	10.6581	0.340634
$980$	0	0
$981$	11.5242	0.367939
$982$	0	0
$983$	−29.1452	−0.929587	−0.464794	−	0.885419i	$-0.653872\pi$
$983$	−29.1452	−0.929587	−0.464794	+	0.885419i	$0.653872\pi$
$984$	0	0
$985$	9.58482	0.305398
$986$	0	0
$987$	−18.2691	−0.581511
$988$	0	0
$989$	−12.8007	−0.407038
$990$	0	0
$991$	5.34191	0.169691	0.0848457	−	0.996394i	$-0.472960\pi$
$991$	5.34191	0.169691	0.0848457	+	0.996394i	$0.472960\pi$
$992$	0	0
$993$	−12.9926	−0.412307
$994$	0	0
$995$	12.9171	0.409500
$996$	0	0
$997$	1.23550	0.0391286	0.0195643	−	0.999809i	$-0.493772\pi$
$997$	1.23550	0.0391286	0.0195643	+	0.999809i	$0.493772\pi$
$998$	0	0
$999$	−10.7826	−0.341146

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7440.2.a.bt.1.1		3
4.3	odd	2		1860.2.a.f.1.3	✓	3
12.11	even	2		5580.2.a.l.1.3		3
20.3	even	4		9300.2.g.p.3349.1		6
20.7	even	4		9300.2.g.p.3349.6		6
20.19	odd	2		9300.2.a.v.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1860.2.a.f.1.3	✓	3	4.3	odd	2
5580.2.a.l.1.3		3	12.11	even	2
7440.2.a.bt.1.1		3	1.1	even	1	trivial
9300.2.a.v.1.1		3	20.19	odd	2
9300.2.g.p.3349.1		6	20.3	even	4
9300.2.g.p.3349.6		6	20.7	even	4

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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -1.00000 q^{5} -4.43807 q^{7} +1.00000 q^{9} -4.17226 q^{11} +4.43807 q^{13} -1.00000 q^{15} -1.35194 q^{17} -2.70388 q^{19} -4.43807 q^{21} +1.35194 q^{23} +1.00000 q^{25} +1.00000 q^{27} -7.25839 q^{29} +1.00000 q^{31} -4.17226 q^{33} +4.43807 q^{35} -10.7826 q^{37} +4.43807 q^{39} +2.17226 q^{41} -9.46838 q^{43} -1.00000 q^{45} +4.11644 q^{47} +12.6965 q^{49} -1.35194 q^{51} +12.8203 q^{53} +4.17226 q^{55} -2.70388 q^{57} -5.25839 q^{59} +6.00000 q^{61} -4.43807 q^{63} -4.43807 q^{65} +1.14195 q^{67} +1.35194 q^{69} +10.7268 q^{71} +10.4381 q^{73} +1.00000 q^{75} +18.5168 q^{77} +2.05582 q^{79} +1.00000 q^{81} +6.99258 q^{83} +1.35194 q^{85} -7.25839 q^{87} -2.55451 q^{89} -19.6965 q^{91} +1.00000 q^{93} +2.70388 q^{95} -15.9245 q^{97} -4.17226 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{3} - 3 q^{5} - 4 q^{7} + 3 q^{9} + 2 q^{11} + 4 q^{13} - 3 q^{15} - 2 q^{17} - 4 q^{19} - 4 q^{21} + 2 q^{23} + 3 q^{25} + 3 q^{27} + 3 q^{31} + 2 q^{33} + 4 q^{35} + 6 q^{37} + 4 q^{39} - 8 q^{41}+ \cdots + 2 q^{99}+O(q^{100}) \) 3 * q + 3 * q^3 - 3 * q^5 - 4 * q^7 + 3 * q^9 + 2 * q^11 + 4 * q^13 - 3 * q^15 - 2 * q^17 - 4 * q^19 - 4 * q^21 + 2 * q^23 + 3 * q^25 + 3 * q^27 + 3 * q^31 + 2 * q^33 + 4 * q^35 + 6 * q^37 + 4 * q^39 - 8 * q^41 - 18 * q^43 - 3 * q^45 + 4 * q^47 + 7 * q^49 - 2 * q^51 + 26 * q^53 - 2 * q^55 - 4 * q^57 + 6 * q^59 + 18 * q^61 - 4 * q^63 - 4 * q^65 - 10 * q^67 + 2 * q^69 + 22 * q^73 + 3 * q^75 + 12 * q^77 + 3 * q^81 - 6 * q^83 + 2 * q^85 + 10 * q^89 - 28 * q^91 + 3 * q^93 + 4 * q^95 + 4 * q^97 + 2 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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