LMFDB - Embedded newform 7440.2.a.bu.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7440,2,Mod(1,7440)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

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")

//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);

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: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$59.4086991038$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.404.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 5x - 1 $ x^3 - x^2 - 5*x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 3720)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-0.210756$ 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)

$f(q)$	$=$	$q+1.00000 q^{3} -1.00000 q^{5} -2.95558 q^{7} +1.00000 q^{9} -4.00000 q^{11} +4.95558 q^{13} -1.00000 q^{15} +0.421512 q^{17} +5.06814 q^{19} -2.95558 q^{21} -7.48965 q^{23} +1.00000 q^{25} +1.00000 q^{27} +8.44523 q^{29} +1.00000 q^{31} -4.00000 q^{33} +2.95558 q^{35} +4.11256 q^{37} +4.95558 q^{39} -12.1363 q^{41} -1.06814 q^{43} -1.00000 q^{45} -6.64663 q^{47} +1.73546 q^{49} +0.421512 q^{51} -3.57849 q^{53} +4.00000 q^{55} +5.06814 q^{57} +0.534070 q^{59} +3.91116 q^{61} -2.95558 q^{63} -4.95558 q^{65} +8.02372 q^{67} -7.48965 q^{69} -3.46593 q^{71} -8.11256 q^{73} +1.00000 q^{75} +11.8223 q^{77} +3.48965 q^{79} +1.00000 q^{81} +1.80361 q^{83} -0.421512 q^{85} +8.44523 q^{87} +7.60221 q^{89} -14.6466 q^{91} +1.00000 q^{93} -5.06814 q^{95} -18.0474 q^{97} -4.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{3} - 3 q^{5} + 2 q^{7} + 3 q^{9} - 12 q^{11} + 4 q^{13} - 3 q^{15} - 2 q^{17} + 2 q^{21} - 4 q^{23} + 3 q^{25} + 3 q^{27} - 4 q^{29} + 3 q^{31} - 12 q^{33} - 2 q^{35} + 8 q^{37} + 4 q^{39} - 6 q^{41}+ \cdots - 12 q^{99}+O(q^{100}) $ 3 * q + 3 * q^3 - 3 * q^5 + 2 * q^7 + 3 * q^9 - 12 * q^11 + 4 * q^13 - 3 * q^15 - 2 * q^17 + 2 * q^21 - 4 * q^23 + 3 * q^25 + 3 * q^27 - 4 * q^29 + 3 * q^31 - 12 * q^33 - 2 * q^35 + 8 * q^37 + 4 * q^39 - 6 * q^41 + 12 * q^43 - 3 * q^45 - 8 * q^47 + 15 * q^49 - 2 * q^51 - 14 * q^53 + 12 * q^55 - 6 * q^59 - 10 * q^61 + 2 * q^63 - 4 * q^65 - 2 * q^67 - 4 * q^69 - 18 * q^71 - 20 * q^73 + 3 * q^75 - 8 * q^77 - 8 * q^79 + 3 * q^81 + 2 * q^85 - 4 * q^87 - 32 * q^91 + 3 * q^93 - 2 * q^97 - 12 * 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
$7$	−2.95558	−1.11710	−0.558552	−	0.829469i	$-0.688643\pi$
$7$	−2.95558	−1.11710	−0.558552	+	0.829469i	$0.688643\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−4.00000	−1.20605	−0.603023	−	0.797724i	$-0.706037\pi$
$11$	−4.00000	−1.20605	−0.603023	+	0.797724i	$0.706037\pi$
$12$	0	0
$13$	4.95558	1.37443	0.687216	−	0.726454i	$-0.258833\pi$
$13$	4.95558	1.37443	0.687216	+	0.726454i	$0.258833\pi$
$14$	0	0
$15$	−1.00000	−0.258199
$16$	0	0
$17$	0.421512	0.102232	0.0511158	−	0.998693i	$-0.483722\pi$
$17$	0.421512	0.102232	0.0511158	+	0.998693i	$0.483722\pi$
$18$	0	0
$19$	5.06814	1.16271	0.581356	−	0.813650i	$-0.302523\pi$
$19$	5.06814	1.16271	0.581356	+	0.813650i	$0.302523\pi$
$20$	0	0
$21$	−2.95558	−0.644961
$22$	0	0
$23$	−7.48965	−1.56170	−0.780850	−	0.624718i	$-0.785214\pi$
$23$	−7.48965	−1.56170	−0.780850	+	0.624718i	$0.785214\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	8.44523	1.56824	0.784120	−	0.620609i	$-0.213114\pi$
$29$	8.44523	1.56824	0.784120	+	0.620609i	$0.213114\pi$
$30$	0	0
$31$	1.00000	0.179605
$31$	1.00000	0.179605
$32$	0	0
$33$	−4.00000	−0.696311
$34$	0	0
$35$	2.95558	0.499585
$36$	0	0
$37$	4.11256	0.676100	0.338050	−	0.941128i	$-0.390233\pi$
$37$	4.11256	0.676100	0.338050	+	0.941128i	$0.390233\pi$
$38$	0	0
$39$	4.95558	0.793528
$40$	0	0
$41$	−12.1363	−1.89537	−0.947684	−	0.319209i	$-0.896583\pi$
$41$	−12.1363	−1.89537	−0.947684	+	0.319209i	$0.896583\pi$
$42$	0	0
$43$	−1.06814	−0.162890	−0.0814449	−	0.996678i	$-0.525953\pi$
$43$	−1.06814	−0.162890	−0.0814449	+	0.996678i	$0.525953\pi$
$44$	0	0
$45$	−1.00000	−0.149071
$46$	0	0
$47$	−6.64663	−0.969510	−0.484755	−	0.874650i	$-0.661091\pi$
$47$	−6.64663	−0.969510	−0.484755	+	0.874650i	$0.661091\pi$
$48$	0	0
$49$	1.73546	0.247924
$50$	0	0
$51$	0.421512	0.0590235
$52$	0	0
$53$	−3.57849	−0.491543	−0.245772	−	0.969328i	$-0.579041\pi$
$53$	−3.57849	−0.491543	−0.245772	+	0.969328i	$0.579041\pi$
$54$	0	0
$55$	4.00000	0.539360
$56$	0	0
$57$	5.06814	0.671292
$58$	0	0
$59$	0.534070	0.0695300	0.0347650	−	0.999396i	$-0.488932\pi$
$59$	0.534070	0.0695300	0.0347650	+	0.999396i	$0.488932\pi$
$60$	0	0
$61$	3.91116	0.500773	0.250387	−	0.968146i	$-0.419442\pi$
$61$	3.91116	0.500773	0.250387	+	0.968146i	$0.419442\pi$
$62$	0	0
$63$	−2.95558	−0.372368
$64$	0	0
$65$	−4.95558	−0.614664
$66$	0	0
$67$	8.02372	0.980254	0.490127	−	0.871651i	$-0.336950\pi$
$67$	8.02372	0.980254	0.490127	+	0.871651i	$0.336950\pi$
$68$	0	0
$69$	−7.48965	−0.901648
$70$	0	0
$71$	−3.46593	−0.411330	−0.205665	−	0.978622i	$-0.565936\pi$
$71$	−3.46593	−0.411330	−0.205665	+	0.978622i	$0.565936\pi$
$72$	0	0
$73$	−8.11256	−0.949503	−0.474752	−	0.880120i	$-0.657462\pi$
$73$	−8.11256	−0.949503	−0.474752	+	0.880120i	$0.657462\pi$
$74$	0	0
$75$	1.00000	0.115470
$76$	0	0
$77$	11.8223	1.34728
$78$	0	0
$79$	3.48965	0.392617	0.196308	−	0.980542i	$-0.437105\pi$
$79$	3.48965	0.392617	0.196308	+	0.980542i	$0.437105\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	1.80361	0.197971	0.0989857	−	0.995089i	$-0.468440\pi$
$83$	1.80361	0.197971	0.0989857	+	0.995089i	$0.468440\pi$
$84$	0	0
$85$	−0.421512	−0.0457194
$86$	0	0
$87$	8.44523	0.905424
$88$	0	0
$89$	7.60221	0.805833	0.402916	−	0.915237i	$-0.367997\pi$
$89$	7.60221	0.805833	0.402916	+	0.915237i	$0.367997\pi$
$90$	0	0
$91$	−14.6466	−1.53538
$92$	0	0
$93$	1.00000	0.103695
$94$	0	0
$95$	−5.06814	−0.519980
$96$	0	0
$97$	−18.0474	−1.83244	−0.916220	−	0.400675i	$-0.868776\pi$
$97$	−18.0474	−1.83244	−0.916220	+	0.400675i	$0.868776\pi$
$98$	0	0
$99$	−4.00000	−0.402015
$100$	0	0
$101$	11.2933	1.12372	0.561861	−	0.827232i	$-0.310086\pi$
$101$	11.2933	1.12372	0.561861	+	0.827232i	$0.310086\pi$
$102$	0	0
$103$	17.0919	1.68411	0.842056	−	0.539391i	$-0.181345\pi$
$103$	17.0919	1.68411	0.842056	+	0.539391i	$0.181345\pi$
$104$	0	0
$105$	2.95558	0.288435
$106$	0	0
$107$	−15.4897	−1.49744	−0.748721	−	0.662886i	$-0.769332\pi$
$107$	−15.4897	−1.49744	−0.748721	+	0.662886i	$0.769332\pi$
$108$	0	0
$109$	0.421512	0.0403735	0.0201868	−	0.999796i	$-0.493574\pi$
$109$	0.421512	0.0403735	0.0201868	+	0.999796i	$0.493574\pi$
$110$	0	0
$111$	4.11256	0.390347
$112$	0	0
$113$	−16.9793	−1.59728	−0.798639	−	0.601810i	$-0.794446\pi$
$113$	−16.9793	−1.59728	−0.798639	+	0.601810i	$0.794446\pi$
$114$	0	0
$115$	7.48965	0.698414
$116$	0	0
$117$	4.95558	0.458144
$118$	0	0
$119$	−1.24581	−0.114203
$120$	0	0
$121$	5.00000	0.454545
$122$	0	0
$123$	−12.1363	−1.09429
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−8.22512	−0.729861	−0.364931	−	0.931035i	$-0.618907\pi$
$127$	−8.22512	−0.729861	−0.364931	+	0.931035i	$0.618907\pi$
$128$	0	0
$129$	−1.06814	−0.0940445
$130$	0	0
$131$	−15.4659	−1.35126	−0.675632	−	0.737239i	$-0.736129\pi$
$131$	−15.4659	−1.35126	−0.675632	+	0.737239i	$0.736129\pi$
$132$	0	0
$133$	−14.9793	−1.29887
$134$	0	0
$135$	−1.00000	−0.0860663
$136$	0	0
$137$	17.4897	1.49424	0.747121	−	0.664688i	$-0.231436\pi$
$137$	17.4897	1.49424	0.747121	+	0.664688i	$0.231436\pi$
$138$	0	0
$139$	2.97930	0.252701	0.126351	−	0.991986i	$-0.459674\pi$
$139$	2.97930	0.252701	0.126351	+	0.991986i	$0.459674\pi$
$140$	0	0
$141$	−6.64663	−0.559747
$142$	0	0
$143$	−19.8223	−1.65763
$144$	0	0
$145$	−8.44523	−0.701339
$146$	0	0
$147$	1.73546	0.143139
$148$	0	0
$149$	−16.9793	−1.39100	−0.695499	−	0.718527i	$-0.744817\pi$
$149$	−16.9793	−1.39100	−0.695499	+	0.718527i	$0.744817\pi$
$150$	0	0
$151$	−11.4897	−0.935015	−0.467507	−	0.883989i	$-0.654848\pi$
$151$	−11.4897	−0.935015	−0.467507	+	0.883989i	$0.654848\pi$
$152$	0	0
$153$	0.421512	0.0340772
$154$	0	0
$155$	−1.00000	−0.0803219
$156$	0	0
$157$	−6.22512	−0.496818	−0.248409	−	0.968655i	$-0.579908\pi$
$157$	−6.22512	−0.496818	−0.248409	+	0.968655i	$0.579908\pi$
$158$	0	0
$159$	−3.57849	−0.283793
$160$	0	0
$161$	22.1363	1.74458
$162$	0	0
$163$	−9.09186	−0.712130	−0.356065	−	0.934461i	$-0.615882\pi$
$163$	−9.09186	−0.712130	−0.356065	+	0.934461i	$0.615882\pi$
$164$	0	0
$165$	4.00000	0.311400
$166$	0	0
$167$	14.9793	1.15913	0.579567	−	0.814925i	$-0.303222\pi$
$167$	14.9793	1.15913	0.579567	+	0.814925i	$0.303222\pi$
$168$	0	0
$169$	11.5578	0.889061
$170$	0	0
$171$	5.06814	0.387570
$172$	0	0
$173$	5.15698	0.392078	0.196039	−	0.980596i	$-0.437192\pi$
$173$	5.15698	0.392078	0.196039	+	0.980596i	$0.437192\pi$
$174$	0	0
$175$	−2.95558	−0.223421
$176$	0	0
$177$	0.534070	0.0401432
$178$	0	0
$179$	−11.1570	−0.833912	−0.416956	−	0.908927i	$-0.636903\pi$
$179$	−11.1570	−0.833912	−0.416956	+	0.908927i	$0.636903\pi$
$180$	0	0
$181$	11.9112	0.885350	0.442675	−	0.896682i	$-0.354030\pi$
$181$	11.9112	0.885350	0.442675	+	0.896682i	$0.354030\pi$
$182$	0	0
$183$	3.91116	0.289122
$184$	0	0
$185$	−4.11256	−0.302361
$186$	0	0
$187$	−1.68605	−0.123296
$188$	0	0
$189$	−2.95558	−0.214987
$190$	0	0
$191$	−18.2201	−1.31836	−0.659181	−	0.751985i	$-0.729097\pi$
$191$	−18.2201	−1.31836	−0.659181	+	0.751985i	$0.729097\pi$
$192$	0	0
$193$	2.22512	0.160167	0.0800837	−	0.996788i	$-0.474481\pi$
$193$	2.22512	0.160167	0.0800837	+	0.996788i	$0.474481\pi$
$194$	0	0
$195$	−4.95558	−0.354877
$196$	0	0
$197$	20.4690	1.45835	0.729176	−	0.684326i	$-0.239903\pi$
$197$	20.4690	1.45835	0.729176	+	0.684326i	$0.239903\pi$
$198$	0	0
$199$	−25.4483	−1.80398	−0.901990	−	0.431758i	$-0.857894\pi$
$199$	−25.4483	−1.80398	−0.901990	+	0.431758i	$0.857894\pi$
$200$	0	0
$201$	8.02372	0.565950
$202$	0	0
$203$	−24.9606	−1.75189
$204$	0	0
$205$	12.1363	0.847635
$206$	0	0
$207$	−7.48965	−0.520567
$208$	0	0
$209$	−20.2726	−1.40228
$210$	0	0
$211$	−21.0681	−1.45039	−0.725195	−	0.688543i	$-0.758251\pi$
$211$	−21.0681	−1.45039	−0.725195	+	0.688543i	$0.758251\pi$
$212$	0	0
$213$	−3.46593	−0.237482
$214$	0	0
$215$	1.06814	0.0728466
$216$	0	0
$217$	−2.95558	−0.200638
$218$	0	0
$219$	−8.11256	−0.548196
$220$	0	0
$221$	2.08884	0.140510
$222$	0	0
$223$	−25.1156	−1.68186	−0.840932	−	0.541141i	$-0.817993\pi$
$223$	−25.1156	−1.68186	−0.840932	+	0.541141i	$0.817993\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	−14.8717	−0.987072	−0.493536	−	0.869725i	$-0.664296\pi$
$227$	−14.8717	−0.987072	−0.493536	+	0.869725i	$0.664296\pi$
$228$	0	0
$229$	−20.1363	−1.33064	−0.665321	−	0.746557i	$-0.731706\pi$
$229$	−20.1363	−1.33064	−0.665321	+	0.746557i	$0.731706\pi$
$230$	0	0
$231$	11.8223	0.777852
$232$	0	0
$233$	−5.26454	−0.344891	−0.172446	−	0.985019i	$-0.555167\pi$
$233$	−5.26454	−0.344891	−0.172446	+	0.985019i	$0.555167\pi$
$234$	0	0
$235$	6.64663	0.433578
$236$	0	0
$237$	3.48965	0.226677
$238$	0	0
$239$	18.3614	1.18770	0.593850	−	0.804576i	$-0.297607\pi$
$239$	18.3614	1.18770	0.593850	+	0.804576i	$0.297607\pi$
$240$	0	0
$241$	16.9793	1.09373	0.546867	−	0.837220i	$-0.315820\pi$
$241$	16.9793	1.09373	0.546867	+	0.837220i	$0.315820\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−1.73546	−0.110875
$246$	0	0
$247$	25.1156	1.59807
$248$	0	0
$249$	1.80361	0.114299
$250$	0	0
$251$	−5.86372	−0.370115	−0.185057	−	0.982728i	$-0.559247\pi$
$251$	−5.86372	−0.370115	−0.185057	+	0.982728i	$0.559247\pi$
$252$	0	0
$253$	29.9586	1.88348
$254$	0	0
$255$	−0.421512	−0.0263961
$256$	0	0
$257$	−20.2438	−1.26278	−0.631388	−	0.775467i	$-0.717514\pi$
$257$	−20.2438	−1.26278	−0.631388	+	0.775467i	$0.717514\pi$
$258$	0	0
$259$	−12.1550	−0.755275
$260$	0	0
$261$	8.44523	0.522747
$262$	0	0
$263$	−9.68605	−0.597267	−0.298634	−	0.954368i	$-0.596531\pi$
$263$	−9.68605	−0.597267	−0.298634	+	0.954368i	$0.596531\pi$
$264$	0	0
$265$	3.57849	0.219825
$266$	0	0
$267$	7.60221	0.465248
$268$	0	0
$269$	−18.3564	−1.11921	−0.559605	−	0.828760i	$-0.689047\pi$
$269$	−18.3564	−1.11921	−0.559605	+	0.828760i	$0.689047\pi$
$270$	0	0
$271$	3.15698	0.191773	0.0958863	−	0.995392i	$-0.469431\pi$
$271$	3.15698	0.191773	0.0958863	+	0.995392i	$0.469431\pi$
$272$	0	0
$273$	−14.6466	−0.886454
$274$	0	0
$275$	−4.00000	−0.241209
$276$	0	0
$277$	−18.9142	−1.13644	−0.568222	−	0.822875i	$-0.692368\pi$
$277$	−18.9142	−1.13644	−0.568222	+	0.822875i	$0.692368\pi$
$278$	0	0
$279$	1.00000	0.0598684
$280$	0	0
$281$	2.61791	0.156171	0.0780856	−	0.996947i	$-0.475119\pi$
$281$	2.61791	0.156171	0.0780856	+	0.996947i	$0.475119\pi$
$282$	0	0
$283$	23.8461	1.41750	0.708750	−	0.705459i	$-0.249259\pi$
$283$	23.8461	1.41750	0.708750	+	0.705459i	$0.249259\pi$
$284$	0	0
$285$	−5.06814	−0.300211
$286$	0	0
$287$	35.8698	2.11733
$288$	0	0
$289$	−16.8223	−0.989549
$290$	0	0
$291$	−18.0474	−1.05796
$292$	0	0
$293$	−4.42151	−0.258307	−0.129154	−	0.991625i	$-0.541226\pi$
$293$	−4.42151	−0.258307	−0.129154	+	0.991625i	$0.541226\pi$
$294$	0	0
$295$	−0.534070	−0.0310948
$296$	0	0
$297$	−4.00000	−0.232104
$298$	0	0
$299$	−37.1156	−2.14645
$300$	0	0
$301$	3.15698	0.181965
$302$	0	0
$303$	11.2933	0.648781
$304$	0	0
$305$	−3.91116	−0.223953
$306$	0	0
$307$	−19.0030	−1.08456	−0.542280	−	0.840198i	$-0.682439\pi$
$307$	−19.0030	−1.08456	−0.542280	+	0.840198i	$0.682439\pi$
$308$	0	0
$309$	17.0919	0.972322
$310$	0	0
$311$	−3.51337	−0.199225	−0.0996126	−	0.995026i	$-0.531760\pi$
$311$	−3.51337	−0.199225	−0.0996126	+	0.995026i	$0.531760\pi$
$312$	0	0
$313$	−28.7779	−1.62662	−0.813312	−	0.581828i	$-0.802338\pi$
$313$	−28.7779	−1.62662	−0.813312	+	0.581828i	$0.802338\pi$
$314$	0	0
$315$	2.95558	0.166528
$316$	0	0
$317$	−23.6734	−1.32963	−0.664815	−	0.747008i	$-0.731489\pi$
$317$	−23.6734	−1.32963	−0.664815	+	0.747008i	$0.731489\pi$
$318$	0	0
$319$	−33.7809	−1.89137
$320$	0	0
$321$	−15.4897	−0.864548
$322$	0	0
$323$	2.13628	0.118866
$324$	0	0
$325$	4.95558	0.274886
$326$	0	0
$327$	0.421512	0.0233097
$328$	0	0
$329$	19.6447	1.08304
$330$	0	0
$331$	−3.66732	−0.201574	−0.100787	−	0.994908i	$-0.532136\pi$
$331$	−3.66732	−0.201574	−0.100787	+	0.994908i	$0.532136\pi$
$332$	0	0
$333$	4.11256	0.225367
$334$	0	0
$335$	−8.02372	−0.438383
$336$	0	0
$337$	−11.8874	−0.647550	−0.323775	−	0.946134i	$-0.604952\pi$
$337$	−11.8874	−0.647550	−0.323775	+	0.946134i	$0.604952\pi$
$338$	0	0
$339$	−16.9793	−0.922189
$340$	0	0
$341$	−4.00000	−0.216612
$342$	0	0
$343$	15.5598	0.840148
$344$	0	0
$345$	7.48965	0.403229
$346$	0	0
$347$	−10.5291	−0.565230	−0.282615	−	0.959233i	$-0.591202\pi$
$347$	−10.5291	−0.565230	−0.282615	+	0.959233i	$0.591202\pi$
$348$	0	0
$349$	−30.5578	−1.63572	−0.817861	−	0.575416i	$-0.804840\pi$
$349$	−30.5578	−1.63572	−0.817861	+	0.575416i	$0.804840\pi$
$350$	0	0
$351$	4.95558	0.264509
$352$	0	0
$353$	2.55779	0.136138	0.0680688	−	0.997681i	$-0.478316\pi$
$353$	2.55779	0.136138	0.0680688	+	0.997681i	$0.478316\pi$
$354$	0	0
$355$	3.46593	0.183952
$356$	0	0
$357$	−1.24581	−0.0659354
$358$	0	0
$359$	29.4245	1.55297	0.776484	−	0.630137i	$-0.217001\pi$
$359$	29.4245	1.55297	0.776484	+	0.630137i	$0.217001\pi$
$360$	0	0
$361$	6.68605	0.351897
$362$	0	0
$363$	5.00000	0.262432
$364$	0	0
$365$	8.11256	0.424631
$366$	0	0
$367$	−10.9793	−0.573115	−0.286558	−	0.958063i	$-0.592511\pi$
$367$	−10.9793	−0.573115	−0.286558	+	0.958063i	$0.592511\pi$
$368$	0	0
$369$	−12.1363	−0.631790
$370$	0	0
$371$	10.5765	0.549105
$372$	0	0
$373$	−7.51837	−0.389287	−0.194643	−	0.980874i	$-0.562355\pi$
$373$	−7.51837	−0.389287	−0.194643	+	0.980874i	$0.562355\pi$
$374$	0	0
$375$	−1.00000	−0.0516398
$376$	0	0
$377$	41.8510	2.15544
$378$	0	0
$379$	−2.93186	−0.150600	−0.0752998	−	0.997161i	$-0.523991\pi$
$379$	−2.93186	−0.150600	−0.0752998	+	0.997161i	$0.523991\pi$
$380$	0	0
$381$	−8.22512	−0.421386
$382$	0	0
$383$	−12.7829	−0.653176	−0.326588	−	0.945167i	$-0.605899\pi$
$383$	−12.7829	−0.653176	−0.326588	+	0.945167i	$0.605899\pi$
$384$	0	0
$385$	−11.8223	−0.602522
$386$	0	0
$387$	−1.06814	−0.0542966
$388$	0	0
$389$	13.9162	0.705578	0.352789	−	0.935703i	$-0.385233\pi$
$389$	13.9162	0.705578	0.352789	+	0.935703i	$0.385233\pi$
$390$	0	0
$391$	−3.15698	−0.159655
$392$	0	0
$393$	−15.4659	−0.780153
$394$	0	0
$395$	−3.48965	−0.175583
$396$	0	0
$397$	34.0949	1.71117	0.855587	−	0.517659i	$-0.173197\pi$
$397$	34.0949	1.71117	0.855587	+	0.517659i	$0.173197\pi$
$398$	0	0
$399$	−14.9793	−0.749903
$400$	0	0
$401$	17.6910	0.883449	0.441724	−	0.897151i	$-0.354367\pi$
$401$	17.6910	0.883449	0.441724	+	0.897151i	$0.354367\pi$
$402$	0	0
$403$	4.95558	0.246855
$404$	0	0
$405$	−1.00000	−0.0496904
$406$	0	0
$407$	−16.4502	−0.815408
$408$	0	0
$409$	9.59721	0.474552	0.237276	−	0.971442i	$-0.423746\pi$
$409$	9.59721	0.474552	0.237276	+	0.971442i	$0.423746\pi$
$410$	0	0
$411$	17.4897	0.862701
$412$	0	0
$413$	−1.57849	−0.0776723
$414$	0	0
$415$	−1.80361	−0.0885355
$416$	0	0
$417$	2.97930	0.145897
$418$	0	0
$419$	8.75919	0.427914	0.213957	−	0.976843i	$-0.431365\pi$
$419$	8.75919	0.427914	0.213957	+	0.976843i	$0.431365\pi$
$420$	0	0
$421$	2.10756	0.102716	0.0513581	−	0.998680i	$-0.483645\pi$
$421$	2.10756	0.102716	0.0513581	+	0.998680i	$0.483645\pi$
$422$	0	0
$423$	−6.64663	−0.323170
$424$	0	0
$425$	0.421512	0.0204463
$426$	0	0
$427$	−11.5598	−0.559416
$428$	0	0
$429$	−19.8223	−0.957031
$430$	0	0
$431$	−16.8066	−0.809547	−0.404773	−	0.914417i	$-0.632650\pi$
$431$	−16.8066	−0.809547	−0.404773	+	0.914417i	$0.632650\pi$
$432$	0	0
$433$	−4.95558	−0.238150	−0.119075	−	0.992885i	$-0.537993\pi$
$433$	−4.95558	−0.238150	−0.119075	+	0.992885i	$0.537993\pi$
$434$	0	0
$435$	−8.44523	−0.404918
$436$	0	0
$437$	−37.9586	−1.81581
$438$	0	0
$439$	10.1363	0.483778	0.241889	−	0.970304i	$-0.422233\pi$
$439$	10.1363	0.483778	0.241889	+	0.970304i	$0.422233\pi$
$440$	0	0
$441$	1.73546	0.0826412
$442$	0	0
$443$	−1.12825	−0.0536050	−0.0268025	−	0.999641i	$-0.508533\pi$
$443$	−1.12825	−0.0536050	−0.0268025	+	0.999641i	$0.508533\pi$
$444$	0	0
$445$	−7.60221	−0.360379
$446$	0	0
$447$	−16.9793	−0.803094
$448$	0	0
$449$	33.7859	1.59446	0.797228	−	0.603678i	$-0.206299\pi$
$449$	33.7859	1.59446	0.797228	+	0.603678i	$0.206299\pi$
$450$	0	0
$451$	48.5451	2.28590
$452$	0	0
$453$	−11.4897	−0.539831
$454$	0	0
$455$	14.6466	0.686645
$456$	0	0
$457$	−19.9349	−0.932515	−0.466257	−	0.884649i	$-0.654398\pi$
$457$	−19.9349	−0.932515	−0.466257	+	0.884649i	$0.654398\pi$
$458$	0	0
$459$	0.421512	0.0196745
$460$	0	0
$461$	38.3564	1.78644	0.893218	−	0.449624i	$-0.148442\pi$
$461$	38.3564	1.78644	0.893218	+	0.449624i	$0.148442\pi$
$462$	0	0
$463$	−14.9319	−0.693942	−0.346971	−	0.937876i	$-0.612790\pi$
$463$	−14.9319	−0.693942	−0.346971	+	0.937876i	$0.612790\pi$
$464$	0	0
$465$	−1.00000	−0.0463739
$466$	0	0
$467$	8.96058	0.414646	0.207323	−	0.978273i	$-0.433525\pi$
$467$	8.96058	0.414646	0.207323	+	0.978273i	$0.433525\pi$
$468$	0	0
$469$	−23.7148	−1.09505
$470$	0	0
$471$	−6.22512	−0.286838
$472$	0	0
$473$	4.27256	0.196453
$474$	0	0
$475$	5.06814	0.232542
$476$	0	0
$477$	−3.57849	−0.163848
$478$	0	0
$479$	20.6290	0.942561	0.471281	−	0.881983i	$-0.343792\pi$
$479$	20.6290	0.942561	0.471281	+	0.881983i	$0.343792\pi$
$480$	0	0
$481$	20.3801	0.929254
$482$	0	0
$483$	22.1363	1.00724
$484$	0	0
$485$	18.0474	0.819492
$486$	0	0
$487$	11.5972	0.525520	0.262760	−	0.964861i	$-0.415367\pi$
$487$	11.5972	0.525520	0.262760	+	0.964861i	$0.415367\pi$
$488$	0	0
$489$	−9.09186	−0.411148
$490$	0	0
$491$	−20.8905	−0.942774	−0.471387	−	0.881927i	$-0.656246\pi$
$491$	−20.8905	−0.942774	−0.471387	+	0.881927i	$0.656246\pi$
$492$	0	0
$493$	3.55977	0.160324
$494$	0	0
$495$	4.00000	0.179787
$496$	0	0
$497$	10.2438	0.459499
$498$	0	0
$499$	7.37209	0.330020	0.165010	−	0.986292i	$-0.447234\pi$
$499$	7.37209	0.330020	0.165010	+	0.986292i	$0.447234\pi$
$500$	0	0
$501$	14.9793	0.669226
$502$	0	0
$503$	15.3120	0.682727	0.341364	−	0.939931i	$-0.389111\pi$
$503$	15.3120	0.682727	0.341364	+	0.939931i	$0.389111\pi$
$504$	0	0
$505$	−11.2933	−0.502543
$506$	0	0
$507$	11.5578	0.513300
$508$	0	0
$509$	23.1994	1.02830	0.514148	−	0.857701i	$-0.328108\pi$
$509$	23.1994	1.02830	0.514148	+	0.857701i	$0.328108\pi$
$510$	0	0
$511$	23.9773	1.06069
$512$	0	0
$513$	5.06814	0.223764
$514$	0	0
$515$	−17.0919	−0.753157
$516$	0	0
$517$	26.5865	1.16927
$518$	0	0
$519$	5.15698	0.226366
$520$	0	0
$521$	−45.2519	−1.98252	−0.991260	−	0.131922i	$-0.957885\pi$
$521$	−45.2519	−1.98252	−0.991260	+	0.131922i	$0.957885\pi$
$522$	0	0
$523$	32.8905	1.43820	0.719100	−	0.694907i	$-0.244554\pi$
$523$	32.8905	1.43820	0.719100	+	0.694907i	$0.244554\pi$
$524$	0	0
$525$	−2.95558	−0.128992
$526$	0	0
$527$	0.421512	0.0183613
$528$	0	0
$529$	33.0949	1.43891
$530$	0	0
$531$	0.534070	0.0231767
$532$	0	0
$533$	−60.1423	−2.60505
$534$	0	0
$535$	15.4897	0.669676
$536$	0	0
$537$	−11.1570	−0.481459
$538$	0	0
$539$	−6.94186	−0.299007
$540$	0	0
$541$	16.8717	0.725373	0.362686	−	0.931911i	$-0.381860\pi$
$541$	16.8717	0.725373	0.362686	+	0.931911i	$0.381860\pi$
$542$	0	0
$543$	11.9112	0.511157
$544$	0	0
$545$	−0.421512	−0.0180556
$546$	0	0
$547$	11.8461	0.506501	0.253250	−	0.967401i	$-0.418500\pi$
$547$	11.8461	0.506501	0.253250	+	0.967401i	$0.418500\pi$
$548$	0	0
$549$	3.91116	0.166924
$550$	0	0
$551$	42.8016	1.82341
$552$	0	0
$553$	−10.3140	−0.438594
$554$	0	0
$555$	−4.11256	−0.174568
$556$	0	0
$557$	40.9092	1.73338	0.866689	−	0.498849i	$-0.166244\pi$
$557$	40.9092	1.73338	0.866689	+	0.498849i	$0.166244\pi$
$558$	0	0
$559$	−5.29326	−0.223881
$560$	0	0
$561$	−1.68605	−0.0711850
$562$	0	0
$563$	−4.33268	−0.182601	−0.0913003	−	0.995823i	$-0.529102\pi$
$563$	−4.33268	−0.182601	−0.0913003	+	0.995823i	$0.529102\pi$
$564$	0	0
$565$	16.9793	0.714325
$566$	0	0
$567$	−2.95558	−0.124123
$568$	0	0
$569$	−28.8955	−1.21136	−0.605681	−	0.795708i	$-0.707099\pi$
$569$	−28.8955	−1.21136	−0.605681	+	0.795708i	$0.707099\pi$
$570$	0	0
$571$	30.8016	1.28901	0.644504	−	0.764601i	$-0.277064\pi$
$571$	30.8016	1.28901	0.644504	+	0.764601i	$0.277064\pi$
$572$	0	0
$573$	−18.2201	−0.761156
$574$	0	0
$575$	−7.48965	−0.312340
$576$	0	0
$577$	−1.54977	−0.0645176	−0.0322588	−	0.999480i	$-0.510270\pi$
$577$	−1.54977	−0.0645176	−0.0322588	+	0.999480i	$0.510270\pi$
$578$	0	0
$579$	2.22512	0.0924727
$580$	0	0
$581$	−5.33070	−0.221155
$582$	0	0
$583$	14.3140	0.592823
$584$	0	0
$585$	−4.95558	−0.204888
$586$	0	0
$587$	−28.0000	−1.15568	−0.577842	−	0.816149i	$-0.696105\pi$
$587$	−28.0000	−1.15568	−0.577842	+	0.816149i	$0.696105\pi$
$588$	0	0
$589$	5.06814	0.208829
$590$	0	0
$591$	20.4690	0.841980
$592$	0	0
$593$	−47.9586	−1.96942	−0.984712	−	0.174190i	$-0.944269\pi$
$593$	−47.9586	−1.96942	−0.984712	+	0.174190i	$0.944269\pi$
$594$	0	0
$595$	1.24581	0.0510733
$596$	0	0
$597$	−25.4483	−1.04153
$598$	0	0
$599$	20.1313	0.822542	0.411271	−	0.911513i	$-0.365085\pi$
$599$	20.1313	0.822542	0.411271	+	0.911513i	$0.365085\pi$
$600$	0	0
$601$	13.7749	0.561889	0.280945	−	0.959724i	$-0.409352\pi$
$601$	13.7749	0.561889	0.280945	+	0.959724i	$0.409352\pi$
$602$	0	0
$603$	8.02372	0.326751
$604$	0	0
$605$	−5.00000	−0.203279
$606$	0	0
$607$	33.5421	1.36143	0.680716	−	0.732548i	$-0.261669\pi$
$607$	33.5421	1.36143	0.680716	+	0.732548i	$0.261669\pi$
$608$	0	0
$609$	−24.9606	−1.01145
$610$	0	0
$611$	−32.9379	−1.33253
$612$	0	0
$613$	43.5321	1.75824	0.879122	−	0.476596i	$-0.158130\pi$
$613$	43.5321	1.75824	0.879122	+	0.476596i	$0.158130\pi$
$614$	0	0
$615$	12.1363	0.489382
$616$	0	0
$617$	−7.86372	−0.316581	−0.158291	−	0.987393i	$-0.550598\pi$
$617$	−7.86372	−0.316581	−0.158291	+	0.987393i	$0.550598\pi$
$618$	0	0
$619$	−24.1550	−0.970872	−0.485436	−	0.874272i	$-0.661339\pi$
$619$	−24.1550	−0.970872	−0.485436	+	0.874272i	$0.661339\pi$
$620$	0	0
$621$	−7.48965	−0.300549
$622$	0	0
$623$	−22.4690	−0.900200
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−20.2726	−0.809608
$628$	0	0
$629$	1.73349	0.0691188
$630$	0	0
$631$	21.8637	0.870381	0.435190	−	0.900338i	$-0.356681\pi$
$631$	21.8637	0.870381	0.435190	+	0.900338i	$0.356681\pi$
$632$	0	0
$633$	−21.0681	−0.837383
$634$	0	0
$635$	8.22512	0.326404
$636$	0	0
$637$	8.60024	0.340754
$638$	0	0
$639$	−3.46593	−0.137110
$640$	0	0
$641$	−9.73849	−0.384647	−0.192324	−	0.981332i	$-0.561602\pi$
$641$	−9.73849	−0.384647	−0.192324	+	0.981332i	$0.561602\pi$
$642$	0	0
$643$	5.63860	0.222365	0.111182	−	0.993800i	$-0.464536\pi$
$643$	5.63860	0.222365	0.111182	+	0.993800i	$0.464536\pi$
$644$	0	0
$645$	1.06814	0.0420580
$646$	0	0
$647$	3.60721	0.141814	0.0709070	−	0.997483i	$-0.477411\pi$
$647$	3.60721	0.141814	0.0709070	+	0.997483i	$0.477411\pi$
$648$	0	0
$649$	−2.13628	−0.0838564
$650$	0	0
$651$	−2.95558	−0.115838
$652$	0	0
$653$	8.87175	0.347178	0.173589	−	0.984818i	$-0.444464\pi$
$653$	8.87175	0.347178	0.173589	+	0.984818i	$0.444464\pi$
$654$	0	0
$655$	15.4659	0.604304
$656$	0	0
$657$	−8.11256	−0.316501
$658$	0	0
$659$	−20.5341	−0.799894	−0.399947	−	0.916538i	$-0.630971\pi$
$659$	−20.5341	−0.799894	−0.399947	+	0.916538i	$0.630971\pi$
$660$	0	0
$661$	22.0000	0.855701	0.427850	−	0.903850i	$-0.359271\pi$
$661$	22.0000	0.855701	0.427850	+	0.903850i	$0.359271\pi$
$662$	0	0
$663$	2.08884	0.0811237
$664$	0	0
$665$	14.9793	0.580873
$666$	0	0
$667$	−63.2519	−2.44912
$668$	0	0
$669$	−25.1156	−0.971025
$670$	0	0
$671$	−15.6447	−0.603955
$672$	0	0
$673$	−37.1807	−1.43321	−0.716605	−	0.697479i	$-0.754305\pi$
$673$	−37.1807	−1.43321	−0.716605	+	0.697479i	$0.754305\pi$
$674$	0	0
$675$	1.00000	0.0384900
$676$	0	0
$677$	−39.8510	−1.53160	−0.765800	−	0.643079i	$-0.777657\pi$
$677$	−39.8510	−1.53160	−0.765800	+	0.643079i	$0.777657\pi$
$678$	0	0
$679$	53.3407	2.04703
$680$	0	0
$681$	−14.8717	−0.569887
$682$	0	0
$683$	32.1076	1.22856	0.614281	−	0.789088i	$-0.289446\pi$
$683$	32.1076	1.22856	0.614281	+	0.789088i	$0.289446\pi$
$684$	0	0
$685$	−17.4897	−0.668245
$686$	0	0
$687$	−20.1363	−0.768247
$688$	0	0
$689$	−17.7335	−0.675592
$690$	0	0
$691$	−25.5084	−0.970384	−0.485192	−	0.874408i	$-0.661250\pi$
$691$	−25.5084	−0.970384	−0.485192	+	0.874408i	$0.661250\pi$
$692$	0	0
$693$	11.8223	0.449093
$694$	0	0
$695$	−2.97930	−0.113011
$696$	0	0
$697$	−5.11559	−0.193767
$698$	0	0
$699$	−5.26454	−0.199123
$700$	0	0
$701$	−9.02675	−0.340936	−0.170468	−	0.985363i	$-0.554528\pi$
$701$	−9.02675	−0.340936	−0.170468	+	0.985363i	$0.554528\pi$
$702$	0	0
$703$	20.8430	0.786110
$704$	0	0
$705$	6.64663	0.250327
$706$	0	0
$707$	−33.3781	−1.25531
$708$	0	0
$709$	−23.8637	−0.896221	−0.448110	−	0.893978i	$-0.647903\pi$
$709$	−23.8637	−0.896221	−0.448110	+	0.893978i	$0.647903\pi$
$710$	0	0
$711$	3.48965	0.130872
$712$	0	0
$713$	−7.48965	−0.280490
$714$	0	0
$715$	19.8223	0.741313
$716$	0	0
$717$	18.3614	0.685719
$718$	0	0
$719$	−12.4502	−0.464315	−0.232158	−	0.972678i	$-0.574579\pi$
$719$	−12.4502	−0.464315	−0.232158	+	0.972678i	$0.574579\pi$
$720$	0	0
$721$	−50.5164	−1.88133
$722$	0	0
$723$	16.9793	0.631467
$724$	0	0
$725$	8.44523	0.313648
$726$	0	0
$727$	−35.8935	−1.33122	−0.665608	−	0.746302i	$-0.731828\pi$
$727$	−35.8935	−1.33122	−0.665608	+	0.746302i	$0.731828\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−0.450234	−0.0166525
$732$	0	0
$733$	35.5658	1.31365	0.656827	−	0.754042i	$-0.271898\pi$
$733$	35.5658	1.31365	0.656827	+	0.754042i	$0.271898\pi$
$734$	0	0
$735$	−1.73546	−0.0640136
$736$	0	0
$737$	−32.0949	−1.18223
$738$	0	0
$739$	38.4690	1.41510	0.707552	−	0.706662i	$-0.249800\pi$
$739$	38.4690	1.41510	0.707552	+	0.706662i	$0.249800\pi$
$740$	0	0
$741$	25.1156	0.922644
$742$	0	0
$743$	3.72744	0.136746	0.0683732	−	0.997660i	$-0.478219\pi$
$743$	3.72744	0.136746	0.0683732	+	0.997660i	$0.478219\pi$
$744$	0	0
$745$	16.9793	0.622074
$746$	0	0
$747$	1.80361	0.0659905
$748$	0	0
$749$	45.7809	1.67280
$750$	0	0
$751$	−28.8430	−1.05250	−0.526248	−	0.850331i	$-0.676402\pi$
$751$	−28.8430	−1.05250	−0.526248	+	0.850331i	$0.676402\pi$
$752$	0	0
$753$	−5.86372	−0.213686
$754$	0	0
$755$	11.4897	0.418151
$756$	0	0
$757$	−7.31698	−0.265940	−0.132970	−	0.991120i	$-0.542451\pi$
$757$	−7.31698	−0.265940	−0.132970	+	0.991120i	$0.542451\pi$
$758$	0	0
$759$	29.9586	1.08743
$760$	0	0
$761$	17.5134	0.634859	0.317430	−	0.948282i	$-0.397180\pi$
$761$	17.5134	0.634859	0.317430	+	0.948282i	$0.397180\pi$
$762$	0	0
$763$	−1.24581	−0.0451014
$764$	0	0
$765$	−0.421512	−0.0152398
$766$	0	0
$767$	2.64663	0.0955642
$768$	0	0
$769$	16.6941	0.602004	0.301002	−	0.953624i	$-0.402679\pi$
$769$	16.6941	0.602004	0.301002	+	0.953624i	$0.402679\pi$
$770$	0	0
$771$	−20.2438	−0.729064
$772$	0	0
$773$	−13.2645	−0.477092	−0.238546	−	0.971131i	$-0.576671\pi$
$773$	−13.2645	−0.477092	−0.238546	+	0.971131i	$0.576671\pi$
$774$	0	0
$775$	1.00000	0.0359211
$776$	0	0
$777$	−12.1550	−0.436058
$778$	0	0
$779$	−61.5084	−2.20377
$780$	0	0
$781$	13.8637	0.496083
$782$	0	0
$783$	8.44523	0.301808
$784$	0	0
$785$	6.22512	0.222184
$786$	0	0
$787$	46.6240	1.66196	0.830982	−	0.556299i	$-0.187779\pi$
$787$	46.6240	1.66196	0.830982	+	0.556299i	$0.187779\pi$
$788$	0	0
$789$	−9.68605	−0.344832
$790$	0	0
$791$	50.1837	1.78433
$792$	0	0
$793$	19.3821	0.688278
$794$	0	0
$795$	3.57849	0.126916
$796$	0	0
$797$	−35.8411	−1.26956	−0.634778	−	0.772695i	$-0.718908\pi$
$797$	−35.8411	−1.26956	−0.634778	+	0.772695i	$0.718908\pi$
$798$	0	0
$799$	−2.80163	−0.0991146
$800$	0	0
$801$	7.60221	0.268611
$802$	0	0
$803$	32.4502	1.14514
$804$	0	0
$805$	−22.1363	−0.780201
$806$	0	0
$807$	−18.3564	−0.646176
$808$	0	0
$809$	39.8648	1.40157	0.700785	−	0.713372i	$-0.252833\pi$
$809$	39.8648	1.40157	0.700785	+	0.713372i	$0.252833\pi$
$810$	0	0
$811$	−7.86977	−0.276345	−0.138173	−	0.990408i	$-0.544123\pi$
$811$	−7.86977	−0.276345	−0.138173	+	0.990408i	$0.544123\pi$
$812$	0	0
$813$	3.15698	0.110720
$814$	0	0
$815$	9.09186	0.318474
$816$	0	0
$817$	−5.41349	−0.189394
$818$	0	0
$819$	−14.6466	−0.511795
$820$	0	0
$821$	8.62291	0.300942	0.150471	−	0.988614i	$-0.451921\pi$
$821$	8.62291	0.300942	0.150471	+	0.988614i	$0.451921\pi$
$822$	0	0
$823$	−1.06814	−0.0372330	−0.0186165	−	0.999827i	$-0.505926\pi$
$823$	−1.06814	−0.0372330	−0.0186165	+	0.999827i	$0.505926\pi$
$824$	0	0
$825$	−4.00000	−0.139262
$826$	0	0
$827$	−30.6466	−1.06569	−0.532844	−	0.846214i	$-0.678877\pi$
$827$	−30.6466	−1.06569	−0.532844	+	0.846214i	$0.678877\pi$
$828$	0	0
$829$	−10.6179	−0.368775	−0.184388	−	0.982854i	$-0.559030\pi$
$829$	−10.6179	−0.368775	−0.184388	+	0.982854i	$0.559030\pi$
$830$	0	0
$831$	−18.9142	−0.656126
$832$	0	0
$833$	0.731519	0.0253456
$834$	0	0
$835$	−14.9793	−0.518380
$836$	0	0
$837$	1.00000	0.0345651
$838$	0	0
$839$	1.15198	0.0397707	0.0198853	−	0.999802i	$-0.493670\pi$
$839$	1.15198	0.0397707	0.0198853	+	0.999802i	$0.493670\pi$
$840$	0	0
$841$	42.3220	1.45938
$842$	0	0
$843$	2.61791	0.0901655
$844$	0	0
$845$	−11.5578	−0.397600
$846$	0	0
$847$	−14.7779	−0.507775
$848$	0	0
$849$	23.8461	0.818394
$850$	0	0
$851$	−30.8016	−1.05587
$852$	0	0
$853$	36.9793	1.26615	0.633074	−	0.774092i	$-0.281793\pi$
$853$	36.9793	1.26615	0.633074	+	0.774092i	$0.281793\pi$
$854$	0	0
$855$	−5.06814	−0.173327
$856$	0	0
$857$	1.15698	0.0395216	0.0197608	−	0.999805i	$-0.493710\pi$
$857$	1.15698	0.0395216	0.0197608	+	0.999805i	$0.493710\pi$
$858$	0	0
$859$	8.17767	0.279019	0.139509	−	0.990221i	$-0.455447\pi$
$859$	8.17767	0.279019	0.139509	+	0.990221i	$0.455447\pi$
$860$	0	0
$861$	35.8698	1.22244
$862$	0	0
$863$	54.4088	1.85210	0.926049	−	0.377403i	$-0.123183\pi$
$863$	54.4088	1.85210	0.926049	+	0.377403i	$0.123183\pi$
$864$	0	0
$865$	−5.15698	−0.175342
$866$	0	0
$867$	−16.8223	−0.571316
$868$	0	0
$869$	−13.9586	−0.473513
$870$	0	0
$871$	39.7622	1.34729
$872$	0	0
$873$	−18.0474	−0.610813
$874$	0	0
$875$	2.95558	0.0999169
$876$	0	0
$877$	−36.1363	−1.22024	−0.610118	−	0.792311i	$-0.708878\pi$
$877$	−36.1363	−1.22024	−0.610118	+	0.792311i	$0.708878\pi$
$878$	0	0
$879$	−4.42151	−0.149134
$880$	0	0
$881$	47.6971	1.60696	0.803478	−	0.595334i	$-0.202980\pi$
$881$	47.6971	1.60696	0.803478	+	0.595334i	$0.202980\pi$
$882$	0	0
$883$	42.8016	1.44039	0.720195	−	0.693772i	$-0.244052\pi$
$883$	42.8016	1.44039	0.720195	+	0.693772i	$0.244052\pi$
$884$	0	0
$885$	−0.534070	−0.0179526
$886$	0	0
$887$	−53.6734	−1.80218	−0.901088	−	0.433637i	$-0.857230\pi$
$887$	−53.6734	−1.80218	−0.901088	+	0.433637i	$0.857230\pi$
$888$	0	0
$889$	24.3100	0.815331
$890$	0	0
$891$	−4.00000	−0.134005
$892$	0	0
$893$	−33.6860	−1.12726
$894$	0	0
$895$	11.1570	0.372937
$896$	0	0
$897$	−37.1156	−1.23925
$898$	0	0
$899$	8.44523	0.281664
$900$	0	0
$901$	−1.50837	−0.0502513
$902$	0	0
$903$	3.15698	0.105058
$904$	0	0
$905$	−11.9112	−0.395940
$906$	0	0
$907$	45.1868	1.50040	0.750201	−	0.661210i	$-0.229957\pi$
$907$	45.1868	1.50040	0.750201	+	0.661210i	$0.229957\pi$
$908$	0	0
$909$	11.2933	0.374574
$910$	0	0
$911$	24.7228	0.819103	0.409551	−	0.912287i	$-0.365685\pi$
$911$	24.7228	0.819103	0.409551	+	0.912287i	$0.365685\pi$
$912$	0	0
$913$	−7.21442	−0.238762
$914$	0	0
$915$	−3.91116	−0.129299
$916$	0	0
$917$	45.7108	1.50950
$918$	0	0
$919$	30.3614	1.00153	0.500765	−	0.865583i	$-0.333052\pi$
$919$	30.3614	1.00153	0.500765	+	0.865583i	$0.333052\pi$
$920$	0	0
$921$	−19.0030	−0.626171
$922$	0	0
$923$	−17.1757	−0.565345
$924$	0	0
$925$	4.11256	0.135220
$926$	0	0
$927$	17.0919	0.561370
$928$	0	0
$929$	24.4452	0.802022	0.401011	−	0.916073i	$-0.368659\pi$
$929$	24.4452	0.802022	0.401011	+	0.916073i	$0.368659\pi$
$930$	0	0
$931$	8.79558	0.288263
$932$	0	0
$933$	−3.51337	−0.115023
$934$	0	0
$935$	1.68605	0.0551396
$936$	0	0
$937$	10.7956	0.352676	0.176338	−	0.984330i	$-0.443575\pi$
$937$	10.7956	0.352676	0.176338	+	0.984330i	$0.443575\pi$
$938$	0	0
$939$	−28.7779	−0.939132
$940$	0	0
$941$	−20.6704	−0.673834	−0.336917	−	0.941534i	$-0.609384\pi$
$941$	−20.6704	−0.673834	−0.336917	+	0.941534i	$0.609384\pi$
$942$	0	0
$943$	90.8965	2.96000
$944$	0	0
$945$	2.95558	0.0961451
$946$	0	0
$947$	−39.7622	−1.29210	−0.646049	−	0.763296i	$-0.723580\pi$
$947$	−39.7622	−1.29210	−0.646049	+	0.763296i	$0.723580\pi$
$948$	0	0
$949$	−40.2024	−1.30503
$950$	0	0
$951$	−23.6734	−0.767662
$952$	0	0
$953$	−41.3120	−1.33823	−0.669113	−	0.743161i	$-0.733326\pi$
$953$	−41.3120	−1.33823	−0.669113	+	0.743161i	$0.733326\pi$
$954$	0	0
$955$	18.2201	0.589589
$956$	0	0
$957$	−33.7809	−1.09198
$958$	0	0
$959$	−51.6921	−1.66922
$960$	0	0
$961$	1.00000	0.0322581
$962$	0	0
$963$	−15.4897	−0.499147
$964$	0	0
$965$	−2.22512	−0.0716290
$966$	0	0
$967$	20.4877	0.658839	0.329420	−	0.944184i	$-0.393147\pi$
$967$	20.4877	0.658839	0.329420	+	0.944184i	$0.393147\pi$
$968$	0	0
$969$	2.13628	0.0686272
$970$	0	0
$971$	−8.35640	−0.268170	−0.134085	−	0.990970i	$-0.542809\pi$
$971$	−8.35640	−0.268170	−0.134085	+	0.990970i	$0.542809\pi$
$972$	0	0
$973$	−8.80558	−0.282294
$974$	0	0
$975$	4.95558	0.158706
$976$	0	0
$977$	−8.40884	−0.269023	−0.134511	−	0.990912i	$-0.542946\pi$
$977$	−8.40884	−0.269023	−0.134511	+	0.990912i	$0.542946\pi$
$978$	0	0
$979$	−30.4088	−0.971871
$980$	0	0
$981$	0.421512	0.0134578
$982$	0	0
$983$	−21.7435	−0.693510	−0.346755	−	0.937956i	$-0.612716\pi$
$983$	−21.7435	−0.693510	−0.346755	+	0.937956i	$0.612716\pi$
$984$	0	0
$985$	−20.4690	−0.652195
$986$	0	0
$987$	19.6447	0.625296
$988$	0	0
$989$	8.00000	0.254385
$990$	0	0
$991$	30.5291	0.969788	0.484894	−	0.874573i	$-0.338858\pi$
$991$	30.5291	0.969788	0.484894	+	0.874573i	$0.338858\pi$
$992$	0	0
$993$	−3.66732	−0.116379
$994$	0	0
$995$	25.4483	0.806764
$996$	0	0
$997$	−23.5184	−0.744834	−0.372417	−	0.928065i	$-0.621471\pi$
$997$	−23.5184	−0.744834	−0.372417	+	0.928065i	$0.621471\pi$
$998$	0	0
$999$	4.11256	0.130116

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7440.2.a.bu.1.1		3
4.3	odd	2		3720.2.a.l.1.3	✓	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3720.2.a.l.1.3	✓	3	4.3	odd	2
7440.2.a.bu.1.1		3	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 $\F_{q}$

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

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -1.00000 q^{5} -2.95558 q^{7} +1.00000 q^{9} -4.00000 q^{11} +4.95558 q^{13} -1.00000 q^{15} +0.421512 q^{17} +5.06814 q^{19} -2.95558 q^{21} -7.48965 q^{23} +1.00000 q^{25} +1.00000 q^{27} +8.44523 q^{29} +1.00000 q^{31} -4.00000 q^{33} +2.95558 q^{35} +4.11256 q^{37} +4.95558 q^{39} -12.1363 q^{41} -1.06814 q^{43} -1.00000 q^{45} -6.64663 q^{47} +1.73546 q^{49} +0.421512 q^{51} -3.57849 q^{53} +4.00000 q^{55} +5.06814 q^{57} +0.534070 q^{59} +3.91116 q^{61} -2.95558 q^{63} -4.95558 q^{65} +8.02372 q^{67} -7.48965 q^{69} -3.46593 q^{71} -8.11256 q^{73} +1.00000 q^{75} +11.8223 q^{77} +3.48965 q^{79} +1.00000 q^{81} +1.80361 q^{83} -0.421512 q^{85} +8.44523 q^{87} +7.60221 q^{89} -14.6466 q^{91} +1.00000 q^{93} -5.06814 q^{95} -18.0474 q^{97} -4.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{3} - 3 q^{5} + 2 q^{7} + 3 q^{9} - 12 q^{11} + 4 q^{13} - 3 q^{15} - 2 q^{17} + 2 q^{21} - 4 q^{23} + 3 q^{25} + 3 q^{27} - 4 q^{29} + 3 q^{31} - 12 q^{33} - 2 q^{35} + 8 q^{37} + 4 q^{39} - 6 q^{41}+ \cdots - 12 q^{99}+O(q^{100}) \) 3 * q + 3 * q^3 - 3 * q^5 + 2 * q^7 + 3 * q^9 - 12 * q^11 + 4 * q^13 - 3 * q^15 - 2 * q^17 + 2 * q^21 - 4 * q^23 + 3 * q^25 + 3 * q^27 - 4 * q^29 + 3 * q^31 - 12 * q^33 - 2 * q^35 + 8 * q^37 + 4 * q^39 - 6 * q^41 + 12 * q^43 - 3 * q^45 - 8 * q^47 + 15 * q^49 - 2 * q^51 - 14 * q^53 + 12 * q^55 - 6 * q^59 - 10 * q^61 + 2 * q^63 - 4 * q^65 - 2 * q^67 - 4 * q^69 - 18 * q^71 - 20 * q^73 + 3 * q^75 - 8 * q^77 - 8 * q^79 + 3 * q^81 + 2 * q^85 - 4 * q^87 - 32 * q^91 + 3 * q^93 - 2 * q^97 - 12 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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