LMFDB - Embedded newform 7440.2.a.bm.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:	3.3.148.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 3x + 1 $ x^3 - x^2 - 3*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 465)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.17009$ 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)

$f(q)$	$=$	$q-1.00000 q^{3} -1.00000 q^{5} -4.87936 q^{7} +1.00000 q^{9} -4.34017 q^{11} -2.53919 q^{13} +1.00000 q^{15} +2.63090 q^{17} +7.41855 q^{19} +4.87936 q^{21} -2.29072 q^{23} +1.00000 q^{25} -1.00000 q^{27} +6.09171 q^{29} +1.00000 q^{31} +4.34017 q^{33} +4.87936 q^{35} -5.80098 q^{37} +2.53919 q^{39} -0.183417 q^{41} +6.49693 q^{43} -1.00000 q^{45} +9.80817 q^{47} +16.8082 q^{49} -2.63090 q^{51} -1.86603 q^{53} +4.34017 q^{55} -7.41855 q^{57} +7.90829 q^{59} -8.15676 q^{61} -4.87936 q^{63} +2.53919 q^{65} +10.4813 q^{67} +2.29072 q^{69} -3.17009 q^{71} -15.5597 q^{73} -1.00000 q^{75} +21.1773 q^{77} +6.23287 q^{79} +1.00000 q^{81} -6.38962 q^{83} -2.63090 q^{85} -6.09171 q^{87} -7.51026 q^{89} +12.3896 q^{91} -1.00000 q^{93} -7.41855 q^{95} +16.2823 q^{97} -4.34017 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{3} - 3 q^{5} - 2 q^{7} + 3 q^{9} - 2 q^{11} - 6 q^{13} + 3 q^{15} + 4 q^{17} + 8 q^{19} + 2 q^{21} - 14 q^{23} + 3 q^{25} - 3 q^{27} + 16 q^{29} + 3 q^{31} + 2 q^{33} + 2 q^{35} - 8 q^{37} + 6 q^{39}+ \cdots - 2 q^{99}+O(q^{100}) $ 3 * q - 3 * q^3 - 3 * q^5 - 2 * q^7 + 3 * q^9 - 2 * q^11 - 6 * q^13 + 3 * q^15 + 4 * q^17 + 8 * q^19 + 2 * q^21 - 14 * q^23 + 3 * q^25 - 3 * q^27 + 16 * q^29 + 3 * q^31 + 2 * q^33 + 2 * q^35 - 8 * q^37 + 6 * q^39 + 4 * q^41 + 2 * q^43 - 3 * q^45 - 14 * q^47 + 7 * q^49 - 4 * q^51 + 8 * q^53 + 2 * q^55 - 8 * q^57 + 26 * q^59 - 18 * q^61 - 2 * q^63 + 6 * q^65 + 14 * q^69 - 4 * q^71 - 12 * q^73 - 3 * q^75 + 24 * q^77 - 4 * q^79 + 3 * q^81 + 10 * q^83 - 4 * q^85 - 16 * q^87 - 6 * q^89 + 8 * q^91 - 3 * q^93 - 8 * q^95 + 8 * 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
$7$	−4.87936	−1.84423	−0.922113	−	0.386921i	$-0.873538\pi$
$7$	−4.87936	−1.84423	−0.922113	+	0.386921i	$0.873538\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−4.34017	−1.30861	−0.654306	−	0.756230i	$-0.727039\pi$
$11$	−4.34017	−1.30861	−0.654306	+	0.756230i	$0.727039\pi$
$12$	0	0
$13$	−2.53919	−0.704244	−0.352122	−	0.935954i	$-0.614540\pi$
$13$	−2.53919	−0.704244	−0.352122	+	0.935954i	$0.614540\pi$
$14$	0	0
$15$	1.00000	0.258199
$16$	0	0
$17$	2.63090	0.638086	0.319043	−	0.947740i	$-0.396638\pi$
$17$	2.63090	0.638086	0.319043	+	0.947740i	$0.396638\pi$
$18$	0	0
$19$	7.41855	1.70193	0.850966	−	0.525221i	$-0.176017\pi$
$19$	7.41855	1.70193	0.850966	+	0.525221i	$0.176017\pi$
$20$	0	0
$21$	4.87936	1.06476
$22$	0	0
$23$	−2.29072	−0.477649	−0.238825	−	0.971063i	$-0.576762\pi$
$23$	−2.29072	−0.477649	−0.238825	+	0.971063i	$0.576762\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	6.09171	1.13120	0.565601	−	0.824679i	$-0.308644\pi$
$29$	6.09171	1.13120	0.565601	+	0.824679i	$0.308644\pi$
$30$	0	0
$31$	1.00000	0.179605
$31$	1.00000	0.179605
$32$	0	0
$33$	4.34017	0.755527
$34$	0	0
$35$	4.87936	0.824763
$36$	0	0
$37$	−5.80098	−0.953676	−0.476838	−	0.878991i	$-0.658217\pi$
$37$	−5.80098	−0.953676	−0.476838	+	0.878991i	$0.658217\pi$
$38$	0	0
$39$	2.53919	0.406596
$40$	0	0
$41$	−0.183417	−0.0286450	−0.0143225	−	0.999897i	$-0.504559\pi$
$41$	−0.183417	−0.0286450	−0.0143225	+	0.999897i	$0.504559\pi$
$42$	0	0
$43$	6.49693	0.990772	0.495386	−	0.868673i	$-0.335027\pi$
$43$	6.49693	0.990772	0.495386	+	0.868673i	$0.335027\pi$
$44$	0	0
$45$	−1.00000	−0.149071
$46$	0	0
$47$	9.80817	1.43067	0.715334	−	0.698782i	$-0.246274\pi$
$47$	9.80817	1.43067	0.715334	+	0.698782i	$0.246274\pi$
$48$	0	0
$49$	16.8082	2.40117
$50$	0	0
$51$	−2.63090	−0.368399
$52$	0	0
$53$	−1.86603	−0.256319	−0.128160	−	0.991754i	$-0.540907\pi$
$53$	−1.86603	−0.256319	−0.128160	+	0.991754i	$0.540907\pi$
$54$	0	0
$55$	4.34017	0.585229
$56$	0	0
$57$	−7.41855	−0.982611
$58$	0	0
$59$	7.90829	1.02957	0.514786	−	0.857319i	$-0.327871\pi$
$59$	7.90829	1.02957	0.514786	+	0.857319i	$0.327871\pi$
$60$	0	0
$61$	−8.15676	−1.04437	−0.522183	−	0.852834i	$-0.674882\pi$
$61$	−8.15676	−1.04437	−0.522183	+	0.852834i	$0.674882\pi$
$62$	0	0
$63$	−4.87936	−0.614742
$64$	0	0
$65$	2.53919	0.314948
$66$	0	0
$67$	10.4813	1.28050	0.640249	−	0.768167i	$-0.278831\pi$
$67$	10.4813	1.28050	0.640249	+	0.768167i	$0.278831\pi$
$68$	0	0
$69$	2.29072	0.275771
$70$	0	0
$71$	−3.17009	−0.376220	−0.188110	−	0.982148i	$-0.560236\pi$
$71$	−3.17009	−0.376220	−0.188110	+	0.982148i	$0.560236\pi$
$72$	0	0
$73$	−15.5597	−1.82113	−0.910563	−	0.413370i	$-0.864352\pi$
$73$	−15.5597	−1.82113	−0.910563	+	0.413370i	$0.864352\pi$
$74$	0	0
$75$	−1.00000	−0.115470
$76$	0	0
$77$	21.1773	2.41337
$78$	0	0
$79$	6.23287	0.701252	0.350626	−	0.936516i	$-0.385969\pi$
$79$	6.23287	0.701252	0.350626	+	0.936516i	$0.385969\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−6.38962	−0.701352	−0.350676	−	0.936497i	$-0.614048\pi$
$83$	−6.38962	−0.701352	−0.350676	+	0.936497i	$0.614048\pi$
$84$	0	0
$85$	−2.63090	−0.285361
$86$	0	0
$87$	−6.09171	−0.653100
$88$	0	0
$89$	−7.51026	−0.796086	−0.398043	−	0.917367i	$-0.630311\pi$
$89$	−7.51026	−0.796086	−0.398043	+	0.917367i	$0.630311\pi$
$90$	0	0
$91$	12.3896	1.29879
$92$	0	0
$93$	−1.00000	−0.103695
$94$	0	0
$95$	−7.41855	−0.761127
$96$	0	0
$97$	16.2823	1.65322	0.826609	−	0.562776i	$-0.190267\pi$
$97$	16.2823	1.65322	0.826609	+	0.562776i	$0.190267\pi$
$98$	0	0
$99$	−4.34017	−0.436204
$100$	0	0
$101$	7.23513	0.719923	0.359961	−	0.932967i	$-0.382790\pi$
$101$	7.23513	0.719923	0.359961	+	0.932967i	$0.382790\pi$
$102$	0	0
$103$	−17.8999	−1.76373	−0.881864	−	0.471504i	$-0.843711\pi$
$103$	−17.8999	−1.76373	−0.881864	+	0.471504i	$0.843711\pi$
$104$	0	0
$105$	−4.87936	−0.476177
$106$	0	0
$107$	−12.9444	−1.25138	−0.625692	−	0.780071i	$-0.715183\pi$
$107$	−12.9444	−1.25138	−0.625692	+	0.780071i	$0.715183\pi$
$108$	0	0
$109$	10.2062	0.977577	0.488789	−	0.872402i	$-0.337439\pi$
$109$	10.2062	0.977577	0.488789	+	0.872402i	$0.337439\pi$
$110$	0	0
$111$	5.80098	0.550605
$112$	0	0
$113$	9.60197	0.903277	0.451639	−	0.892201i	$-0.350840\pi$
$113$	9.60197	0.903277	0.451639	+	0.892201i	$0.350840\pi$
$114$	0	0
$115$	2.29072	0.213611
$116$	0	0
$117$	−2.53919	−0.234748
$118$	0	0
$119$	−12.8371	−1.17678
$120$	0	0
$121$	7.83710	0.712464
$122$	0	0
$123$	0.183417	0.0165382
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−15.6020	−1.38445	−0.692225	−	0.721681i	$-0.743370\pi$
$127$	−15.6020	−1.38445	−0.692225	+	0.721681i	$0.743370\pi$
$128$	0	0
$129$	−6.49693	−0.572023
$130$	0	0
$131$	9.53692	0.833245	0.416622	−	0.909080i	$-0.363214\pi$
$131$	9.53692	0.833245	0.416622	+	0.909080i	$0.363214\pi$
$132$	0	0
$133$	−36.1978	−3.13875
$134$	0	0
$135$	1.00000	0.0860663
$136$	0	0
$137$	2.94441	0.251558	0.125779	−	0.992058i	$-0.459857\pi$
$137$	2.94441	0.251558	0.125779	+	0.992058i	$0.459857\pi$
$138$	0	0
$139$	−7.02052	−0.595473	−0.297736	−	0.954648i	$-0.596232\pi$
$139$	−7.02052	−0.595473	−0.297736	+	0.954648i	$0.596232\pi$
$140$	0	0
$141$	−9.80817	−0.825997
$142$	0	0
$143$	11.0205	0.921582
$144$	0	0
$145$	−6.09171	−0.505889
$146$	0	0
$147$	−16.8082	−1.38631
$148$	0	0
$149$	−8.49693	−0.696096	−0.348048	−	0.937477i	$-0.613155\pi$
$149$	−8.49693	−0.696096	−0.348048	+	0.937477i	$0.613155\pi$
$150$	0	0
$151$	3.52586	0.286930	0.143465	−	0.989655i	$-0.454176\pi$
$151$	3.52586	0.286930	0.143465	+	0.989655i	$0.454176\pi$
$152$	0	0
$153$	2.63090	0.212695
$154$	0	0
$155$	−1.00000	−0.0803219
$156$	0	0
$157$	4.49693	0.358894	0.179447	−	0.983768i	$-0.442569\pi$
$157$	4.49693	0.358894	0.179447	+	0.983768i	$0.442569\pi$
$158$	0	0
$159$	1.86603	0.147986
$160$	0	0
$161$	11.1773	0.880893
$162$	0	0
$163$	15.3763	1.20436	0.602182	−	0.798359i	$-0.294298\pi$
$163$	15.3763	1.20436	0.602182	+	0.798359i	$0.294298\pi$
$164$	0	0
$165$	−4.34017	−0.337882
$166$	0	0
$167$	−10.2413	−0.792494	−0.396247	−	0.918144i	$-0.629688\pi$
$167$	−10.2413	−0.792494	−0.396247	+	0.918144i	$0.629688\pi$
$168$	0	0
$169$	−6.55252	−0.504040
$170$	0	0
$171$	7.41855	0.567311
$172$	0	0
$173$	7.07838	0.538159	0.269080	−	0.963118i	$-0.413281\pi$
$173$	7.07838	0.538159	0.269080	+	0.963118i	$0.413281\pi$
$174$	0	0
$175$	−4.87936	−0.368845
$176$	0	0
$177$	−7.90829	−0.594424
$178$	0	0
$179$	−14.6225	−1.09294	−0.546468	−	0.837480i	$-0.684028\pi$
$179$	−14.6225	−1.09294	−0.546468	+	0.837480i	$0.684028\pi$
$180$	0	0
$181$	2.00000	0.148659	0.0743294	−	0.997234i	$-0.476318\pi$
$181$	2.00000	0.148659	0.0743294	+	0.997234i	$0.476318\pi$
$182$	0	0
$183$	8.15676	0.602965
$184$	0	0
$185$	5.80098	0.426497
$186$	0	0
$187$	−11.4186	−0.835007
$188$	0	0
$189$	4.87936	0.354921
$190$	0	0
$191$	−1.75154	−0.126737	−0.0633683	−	0.997990i	$-0.520184\pi$
$191$	−1.75154	−0.126737	−0.0633683	+	0.997990i	$0.520184\pi$
$192$	0	0
$193$	−4.86376	−0.350101	−0.175051	−	0.984559i	$-0.556009\pi$
$193$	−4.86376	−0.350101	−0.175051	+	0.984559i	$0.556009\pi$
$194$	0	0
$195$	−2.53919	−0.181835
$196$	0	0
$197$	−20.3051	−1.44668	−0.723339	−	0.690493i	$-0.757394\pi$
$197$	−20.3051	−1.44668	−0.723339	+	0.690493i	$0.757394\pi$
$198$	0	0
$199$	2.04945	0.145282	0.0726408	−	0.997358i	$-0.476857\pi$
$199$	2.04945	0.145282	0.0726408	+	0.997358i	$0.476857\pi$
$200$	0	0
$201$	−10.4813	−0.739296
$202$	0	0
$203$	−29.7237	−2.08619
$204$	0	0
$205$	0.183417	0.0128104
$206$	0	0
$207$	−2.29072	−0.159216
$208$	0	0
$209$	−32.1978	−2.22717
$210$	0	0
$211$	−12.1834	−0.838741	−0.419371	−	0.907815i	$-0.637749\pi$
$211$	−12.1834	−0.838741	−0.419371	+	0.907815i	$0.637749\pi$
$212$	0	0
$213$	3.17009	0.217211
$214$	0	0
$215$	−6.49693	−0.443087
$216$	0	0
$217$	−4.87936	−0.331233
$218$	0	0
$219$	15.5597	1.05143
$220$	0	0
$221$	−6.68035	−0.449369
$222$	0	0
$223$	−12.2557	−0.820699	−0.410350	−	0.911928i	$-0.634593\pi$
$223$	−12.2557	−0.820699	−0.410350	+	0.911928i	$0.634593\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	−12.1483	−0.806314	−0.403157	−	0.915131i	$-0.632087\pi$
$227$	−12.1483	−0.806314	−0.403157	+	0.915131i	$0.632087\pi$
$228$	0	0
$229$	6.28231	0.415147	0.207574	−	0.978219i	$-0.433443\pi$
$229$	6.28231	0.415147	0.207574	+	0.978219i	$0.433443\pi$
$230$	0	0
$231$	−21.1773	−1.39336
$232$	0	0
$233$	15.3112	1.00307	0.501536	−	0.865137i	$-0.332768\pi$
$233$	15.3112	1.00307	0.501536	+	0.865137i	$0.332768\pi$
$234$	0	0
$235$	−9.80817	−0.639815
$236$	0	0
$237$	−6.23287	−0.404868
$238$	0	0
$239$	30.1978	1.95333	0.976666	−	0.214762i	$-0.0688976\pi$
$239$	30.1978	1.95333	0.976666	+	0.214762i	$0.0688976\pi$
$240$	0	0
$241$	−25.6475	−1.65210	−0.826052	−	0.563594i	$-0.809418\pi$
$241$	−25.6475	−1.65210	−0.826052	+	0.563594i	$0.809418\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	−16.8082	−1.07383
$246$	0	0
$247$	−18.8371	−1.19858
$248$	0	0
$249$	6.38962	0.404926
$250$	0	0
$251$	6.25565	0.394853	0.197427	−	0.980318i	$-0.436742\pi$
$251$	6.25565	0.394853	0.197427	+	0.980318i	$0.436742\pi$
$252$	0	0
$253$	9.94214	0.625057
$254$	0	0
$255$	2.63090	0.164753
$256$	0	0
$257$	−13.1545	−0.820554	−0.410277	−	0.911961i	$-0.634568\pi$
$257$	−13.1545	−0.820554	−0.410277	+	0.911961i	$0.634568\pi$
$258$	0	0
$259$	28.3051	1.75879
$260$	0	0
$261$	6.09171	0.377067
$262$	0	0
$263$	−29.4908	−1.81848	−0.909240	−	0.416273i	$-0.863336\pi$
$263$	−29.4908	−1.81848	−0.909240	+	0.416273i	$0.863336\pi$
$264$	0	0
$265$	1.86603	0.114629
$266$	0	0
$267$	7.51026	0.459620
$268$	0	0
$269$	−20.7454	−1.26487	−0.632434	−	0.774614i	$-0.717944\pi$
$269$	−20.7454	−1.26487	−0.632434	+	0.774614i	$0.717944\pi$
$270$	0	0
$271$	−0.979481	−0.0594992	−0.0297496	−	0.999557i	$-0.509471\pi$
$271$	−0.979481	−0.0594992	−0.0297496	+	0.999557i	$0.509471\pi$
$272$	0	0
$273$	−12.3896	−0.749854
$274$	0	0
$275$	−4.34017	−0.261722
$276$	0	0
$277$	16.3246	0.980849	0.490424	−	0.871484i	$-0.336842\pi$
$277$	16.3246	0.980849	0.490424	+	0.871484i	$0.336842\pi$
$278$	0	0
$279$	1.00000	0.0598684
$280$	0	0
$281$	22.2823	1.32925	0.664626	−	0.747176i	$-0.268591\pi$
$281$	22.2823	1.32925	0.664626	+	0.747176i	$0.268591\pi$
$282$	0	0
$283$	2.05664	0.122254	0.0611272	−	0.998130i	$-0.480530\pi$
$283$	2.05664	0.122254	0.0611272	+	0.998130i	$0.480530\pi$
$284$	0	0
$285$	7.41855	0.439437
$286$	0	0
$287$	0.894960	0.0528278
$288$	0	0
$289$	−10.0784	−0.592846
$290$	0	0
$291$	−16.2823	−0.954486
$292$	0	0
$293$	22.3630	1.30646	0.653229	−	0.757160i	$-0.273414\pi$
$293$	22.3630	1.30646	0.653229	+	0.757160i	$0.273414\pi$
$294$	0	0
$295$	−7.90829	−0.460439
$296$	0	0
$297$	4.34017	0.251842
$298$	0	0
$299$	5.81658	0.336382
$300$	0	0
$301$	−31.7009	−1.82721
$302$	0	0
$303$	−7.23513	−0.415648
$304$	0	0
$305$	8.15676	0.467054
$306$	0	0
$307$	9.24620	0.527708	0.263854	−	0.964563i	$-0.415006\pi$
$307$	9.24620	0.527708	0.263854	+	0.964563i	$0.415006\pi$
$308$	0	0
$309$	17.8999	1.01829
$310$	0	0
$311$	−9.24232	−0.524084	−0.262042	−	0.965056i	$-0.584396\pi$
$311$	−9.24232	−0.524084	−0.262042	+	0.965056i	$0.584396\pi$
$312$	0	0
$313$	3.89988	0.220434	0.110217	−	0.993908i	$-0.464845\pi$
$313$	3.89988	0.220434	0.110217	+	0.993908i	$0.464845\pi$
$314$	0	0
$315$	4.87936	0.274921
$316$	0	0
$317$	3.62475	0.203586	0.101793	−	0.994806i	$-0.467542\pi$
$317$	3.62475	0.203586	0.101793	+	0.994806i	$0.467542\pi$
$318$	0	0
$319$	−26.4391	−1.48030
$320$	0	0
$321$	12.9444	0.722486
$322$	0	0
$323$	19.5174	1.08598
$324$	0	0
$325$	−2.53919	−0.140849
$326$	0	0
$327$	−10.2062	−0.564404
$328$	0	0
$329$	−47.8576	−2.63848
$330$	0	0
$331$	24.1217	1.32585	0.662924	−	0.748687i	$-0.269315\pi$
$331$	24.1217	1.32585	0.662924	+	0.748687i	$0.269315\pi$
$332$	0	0
$333$	−5.80098	−0.317892
$334$	0	0
$335$	−10.4813	−0.572656
$336$	0	0
$337$	3.19287	0.173927	0.0869634	−	0.996212i	$-0.472284\pi$
$337$	3.19287	0.173927	0.0869634	+	0.996212i	$0.472284\pi$
$338$	0	0
$339$	−9.60197	−0.521507
$340$	0	0
$341$	−4.34017	−0.235034
$342$	0	0
$343$	−47.8576	−2.58407
$344$	0	0
$345$	−2.29072	−0.123328
$346$	0	0
$347$	34.6369	1.85940	0.929702	−	0.368312i	$-0.120064\pi$
$347$	34.6369	1.85940	0.929702	+	0.368312i	$0.120064\pi$
$348$	0	0
$349$	5.31124	0.284304	0.142152	−	0.989845i	$-0.454598\pi$
$349$	5.31124	0.284304	0.142152	+	0.989845i	$0.454598\pi$
$350$	0	0
$351$	2.53919	0.135532
$352$	0	0
$353$	33.3523	1.77516	0.887581	−	0.460651i	$-0.152384\pi$
$353$	33.3523	1.77516	0.887581	+	0.460651i	$0.152384\pi$
$354$	0	0
$355$	3.17009	0.168251
$356$	0	0
$357$	12.8371	0.679411
$358$	0	0
$359$	−2.33299	−0.123130	−0.0615651	−	0.998103i	$-0.519609\pi$
$359$	−2.33299	−0.123130	−0.0615651	+	0.998103i	$0.519609\pi$
$360$	0	0
$361$	36.0349	1.89657
$362$	0	0
$363$	−7.83710	−0.411341
$364$	0	0
$365$	15.5597	0.814432
$366$	0	0
$367$	−12.3135	−0.642760	−0.321380	−	0.946950i	$-0.604147\pi$
$367$	−12.3135	−0.642760	−0.321380	+	0.946950i	$0.604147\pi$
$368$	0	0
$369$	−0.183417	−0.00954833
$370$	0	0
$371$	9.10504	0.472710
$372$	0	0
$373$	−5.86991	−0.303932	−0.151966	−	0.988386i	$-0.548560\pi$
$373$	−5.86991	−0.303932	−0.151966	+	0.988386i	$0.548560\pi$
$374$	0	0
$375$	1.00000	0.0516398
$376$	0	0
$377$	−15.4680	−0.796642
$378$	0	0
$379$	2.42469	0.124548	0.0622741	−	0.998059i	$-0.480165\pi$
$379$	2.42469	0.124548	0.0622741	+	0.998059i	$0.480165\pi$
$380$	0	0
$381$	15.6020	0.799313
$382$	0	0
$383$	32.9588	1.68412	0.842058	−	0.539388i	$-0.181344\pi$
$383$	32.9588	1.68412	0.842058	+	0.539388i	$0.181344\pi$
$384$	0	0
$385$	−21.1773	−1.07929
$386$	0	0
$387$	6.49693	0.330257
$388$	0	0
$389$	19.0589	0.966325	0.483162	−	0.875531i	$-0.339488\pi$
$389$	19.0589	0.966325	0.483162	+	0.875531i	$0.339488\pi$
$390$	0	0
$391$	−6.02666	−0.304781
$392$	0	0
$393$	−9.53692	−0.481074
$394$	0	0
$395$	−6.23287	−0.313610
$396$	0	0
$397$	−27.2618	−1.36823	−0.684115	−	0.729374i	$-0.739811\pi$
$397$	−27.2618	−1.36823	−0.684115	+	0.729374i	$0.739811\pi$
$398$	0	0
$399$	36.1978	1.81216
$400$	0	0
$401$	−23.6404	−1.18054	−0.590271	−	0.807205i	$-0.700979\pi$
$401$	−23.6404	−1.18054	−0.590271	+	0.807205i	$0.700979\pi$
$402$	0	0
$403$	−2.53919	−0.126486
$404$	0	0
$405$	−1.00000	−0.0496904
$406$	0	0
$407$	25.1773	1.24799
$408$	0	0
$409$	−7.16290	−0.354183	−0.177091	−	0.984194i	$-0.556669\pi$
$409$	−7.16290	−0.354183	−0.177091	+	0.984194i	$0.556669\pi$
$410$	0	0
$411$	−2.94441	−0.145237
$412$	0	0
$413$	−38.5874	−1.89876
$414$	0	0
$415$	6.38962	0.313654
$416$	0	0
$417$	7.02052	0.343796
$418$	0	0
$419$	4.43188	0.216512	0.108256	−	0.994123i	$-0.465473\pi$
$419$	4.43188	0.216512	0.108256	+	0.994123i	$0.465473\pi$
$420$	0	0
$421$	34.4885	1.68087	0.840434	−	0.541914i	$-0.182300\pi$
$421$	34.4885	1.68087	0.840434	+	0.541914i	$0.182300\pi$
$422$	0	0
$423$	9.80817	0.476890
$424$	0	0
$425$	2.63090	0.127617
$426$	0	0
$427$	39.7998	1.92605
$428$	0	0
$429$	−11.0205	−0.532076
$430$	0	0
$431$	−17.6092	−0.848203	−0.424102	−	0.905615i	$-0.639410\pi$
$431$	−17.6092	−0.848203	−0.424102	+	0.905615i	$0.639410\pi$
$432$	0	0
$433$	−36.5536	−1.75665	−0.878326	−	0.478062i	$-0.841339\pi$
$433$	−36.5536	−1.75665	−0.878326	+	0.478062i	$0.841339\pi$
$434$	0	0
$435$	6.09171	0.292075
$436$	0	0
$437$	−16.9939	−0.812926
$438$	0	0
$439$	12.3135	0.587692	0.293846	−	0.955853i	$-0.405065\pi$
$439$	12.3135	0.587692	0.293846	+	0.955853i	$0.405065\pi$
$440$	0	0
$441$	16.8082	0.800389
$442$	0	0
$443$	4.57757	0.217487	0.108744	−	0.994070i	$-0.465317\pi$
$443$	4.57757	0.217487	0.108744	+	0.994070i	$0.465317\pi$
$444$	0	0
$445$	7.51026	0.356020
$446$	0	0
$447$	8.49693	0.401891
$448$	0	0
$449$	−17.7971	−0.839897	−0.419949	−	0.907548i	$-0.637952\pi$
$449$	−17.7971	−0.839897	−0.419949	+	0.907548i	$0.637952\pi$
$450$	0	0
$451$	0.796064	0.0374852
$452$	0	0
$453$	−3.52586	−0.165659
$454$	0	0
$455$	−12.3896	−0.580834
$456$	0	0
$457$	−1.78661	−0.0835740	−0.0417870	−	0.999127i	$-0.513305\pi$
$457$	−1.78661	−0.0835740	−0.0417870	+	0.999127i	$0.513305\pi$
$458$	0	0
$459$	−2.63090	−0.122800
$460$	0	0
$461$	−33.9227	−1.57994	−0.789968	−	0.613148i	$-0.789903\pi$
$461$	−33.9227	−1.57994	−0.789968	+	0.613148i	$0.789903\pi$
$462$	0	0
$463$	−41.1194	−1.91098	−0.955491	−	0.295021i	$-0.904673\pi$
$463$	−41.1194	−1.91098	−0.955491	+	0.295021i	$0.904673\pi$
$464$	0	0
$465$	1.00000	0.0463739
$466$	0	0
$467$	−22.4885	−1.04064	−0.520322	−	0.853970i	$-0.674188\pi$
$467$	−22.4885	−1.04064	−0.520322	+	0.853970i	$0.674188\pi$
$468$	0	0
$469$	−51.1422	−2.36153
$470$	0	0
$471$	−4.49693	−0.207208
$472$	0	0
$473$	−28.1978	−1.29654
$474$	0	0
$475$	7.41855	0.340386
$476$	0	0
$477$	−1.86603	−0.0854397
$478$	0	0
$479$	32.2485	1.47347	0.736735	−	0.676182i	$-0.236367\pi$
$479$	32.2485	1.47347	0.736735	+	0.676182i	$0.236367\pi$
$480$	0	0
$481$	14.7298	0.671621
$482$	0	0
$483$	−11.1773	−0.508584
$484$	0	0
$485$	−16.2823	−0.739342
$486$	0	0
$487$	−23.3340	−1.05737	−0.528683	−	0.848819i	$-0.677314\pi$
$487$	−23.3340	−1.05737	−0.528683	+	0.848819i	$0.677314\pi$
$488$	0	0
$489$	−15.3763	−0.695340
$490$	0	0
$491$	−23.3340	−1.05305	−0.526525	−	0.850160i	$-0.676505\pi$
$491$	−23.3340	−1.05305	−0.526525	+	0.850160i	$0.676505\pi$
$492$	0	0
$493$	16.0267	0.721805
$494$	0	0
$495$	4.34017	0.195076
$496$	0	0
$497$	15.4680	0.693835
$498$	0	0
$499$	3.20394	0.143428	0.0717139	−	0.997425i	$-0.477153\pi$
$499$	3.20394	0.143428	0.0717139	+	0.997425i	$0.477153\pi$
$500$	0	0
$501$	10.2413	0.457546
$502$	0	0
$503$	−7.55252	−0.336750	−0.168375	−	0.985723i	$-0.553852\pi$
$503$	−7.55252	−0.336750	−0.168375	+	0.985723i	$0.553852\pi$
$504$	0	0
$505$	−7.23513	−0.321959
$506$	0	0
$507$	6.55252	0.291008
$508$	0	0
$509$	27.4401	1.21626	0.608131	−	0.793837i	$-0.291920\pi$
$509$	27.4401	1.21626	0.608131	+	0.793837i	$0.291920\pi$
$510$	0	0
$511$	75.9214	3.35857
$512$	0	0
$513$	−7.41855	−0.327537
$514$	0	0
$515$	17.8999	0.788763
$516$	0	0
$517$	−42.5692	−1.87219
$518$	0	0
$519$	−7.07838	−0.310706
$520$	0	0
$521$	28.0144	1.22733	0.613666	−	0.789566i	$-0.289694\pi$
$521$	28.0144	1.22733	0.613666	+	0.789566i	$0.289694\pi$
$522$	0	0
$523$	−3.33403	−0.145787	−0.0728935	−	0.997340i	$-0.523223\pi$
$523$	−3.33403	−0.145787	−0.0728935	+	0.997340i	$0.523223\pi$
$524$	0	0
$525$	4.87936	0.212953
$526$	0	0
$527$	2.63090	0.114604
$528$	0	0
$529$	−17.7526	−0.771851
$530$	0	0
$531$	7.90829	0.343191
$532$	0	0
$533$	0.465732	0.0201731
$534$	0	0
$535$	12.9444	0.559636
$536$	0	0
$537$	14.6225	0.631007
$538$	0	0
$539$	−72.9504	−3.14220
$540$	0	0
$541$	−7.55252	−0.324708	−0.162354	−	0.986733i	$-0.551909\pi$
$541$	−7.55252	−0.324708	−0.162354	+	0.986733i	$0.551909\pi$
$542$	0	0
$543$	−2.00000	−0.0858282
$544$	0	0
$545$	−10.2062	−0.437186
$546$	0	0
$547$	−32.4547	−1.38766	−0.693831	−	0.720138i	$-0.744078\pi$
$547$	−32.4547	−1.38766	−0.693831	+	0.720138i	$0.744078\pi$
$548$	0	0
$549$	−8.15676	−0.348122
$550$	0	0
$551$	45.1917	1.92523
$552$	0	0
$553$	−30.4124	−1.29327
$554$	0	0
$555$	−5.80098	−0.246238
$556$	0	0
$557$	23.0433	0.976376	0.488188	−	0.872738i	$-0.337658\pi$
$557$	23.0433	0.976376	0.488188	+	0.872738i	$0.337658\pi$
$558$	0	0
$559$	−16.4969	−0.697746
$560$	0	0
$561$	11.4186	0.482092
$562$	0	0
$563$	39.9793	1.68493	0.842463	−	0.538754i	$-0.181105\pi$
$563$	39.9793	1.68493	0.842463	+	0.538754i	$0.181105\pi$
$564$	0	0
$565$	−9.60197	−0.403958
$566$	0	0
$567$	−4.87936	−0.204914
$568$	0	0
$569$	−19.6670	−0.824484	−0.412242	−	0.911074i	$-0.635254\pi$
$569$	−19.6670	−0.824484	−0.412242	+	0.911074i	$0.635254\pi$
$570$	0	0
$571$	−11.8166	−0.494509	−0.247254	−	0.968951i	$-0.579528\pi$
$571$	−11.8166	−0.494509	−0.247254	+	0.968951i	$0.579528\pi$
$572$	0	0
$573$	1.75154	0.0731715
$574$	0	0
$575$	−2.29072	−0.0955298
$576$	0	0
$577$	−38.4079	−1.59894	−0.799470	−	0.600706i	$-0.794886\pi$
$577$	−38.4079	−1.59894	−0.799470	+	0.600706i	$0.794886\pi$
$578$	0	0
$579$	4.86376	0.202131
$580$	0	0
$581$	31.1773	1.29345
$582$	0	0
$583$	8.09890	0.335422
$584$	0	0
$585$	2.53919	0.104983
$586$	0	0
$587$	−11.3874	−0.470006	−0.235003	−	0.971995i	$-0.575510\pi$
$587$	−11.3874	−0.470006	−0.235003	+	0.971995i	$0.575510\pi$
$588$	0	0
$589$	7.41855	0.305676
$590$	0	0
$591$	20.3051	0.835240
$592$	0	0
$593$	31.5441	1.29536	0.647681	−	0.761912i	$-0.275739\pi$
$593$	31.5441	1.29536	0.647681	+	0.761912i	$0.275739\pi$
$594$	0	0
$595$	12.8371	0.526270
$596$	0	0
$597$	−2.04945	−0.0838783
$598$	0	0
$599$	−28.9744	−1.18386	−0.591931	−	0.805989i	$-0.701634\pi$
$599$	−28.9744	−1.18386	−0.591931	+	0.805989i	$0.701634\pi$
$600$	0	0
$601$	−23.9565	−0.977207	−0.488603	−	0.872506i	$-0.662493\pi$
$601$	−23.9565	−0.977207	−0.488603	+	0.872506i	$0.662493\pi$
$602$	0	0
$603$	10.4813	0.426833
$604$	0	0
$605$	−7.83710	−0.318623
$606$	0	0
$607$	−20.1724	−0.818771	−0.409385	−	0.912362i	$-0.634257\pi$
$607$	−20.1724	−0.818771	−0.409385	+	0.912362i	$0.634257\pi$
$608$	0	0
$609$	29.7237	1.20446
$610$	0	0
$611$	−24.9048	−1.00754
$612$	0	0
$613$	−4.46695	−0.180419	−0.0902093	−	0.995923i	$-0.528754\pi$
$613$	−4.46695	−0.180419	−0.0902093	+	0.995923i	$0.528754\pi$
$614$	0	0
$615$	−0.183417	−0.00739611
$616$	0	0
$617$	46.0821	1.85519	0.927597	−	0.373582i	$-0.121870\pi$
$617$	46.0821	1.85519	0.927597	+	0.373582i	$0.121870\pi$
$618$	0	0
$619$	9.55252	0.383948	0.191974	−	0.981400i	$-0.438511\pi$
$619$	9.55252	0.383948	0.191974	+	0.981400i	$0.438511\pi$
$620$	0	0
$621$	2.29072	0.0919236
$622$	0	0
$623$	36.6453	1.46816
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	32.1978	1.28586
$628$	0	0
$629$	−15.2618	−0.608528
$630$	0	0
$631$	−15.8310	−0.630221	−0.315110	−	0.949055i	$-0.602041\pi$
$631$	−15.8310	−0.630221	−0.315110	+	0.949055i	$0.602041\pi$
$632$	0	0
$633$	12.1834	0.484247
$634$	0	0
$635$	15.6020	0.619145
$636$	0	0
$637$	−42.6791	−1.69101
$638$	0	0
$639$	−3.17009	−0.125407
$640$	0	0
$641$	−5.87709	−0.232131	−0.116066	−	0.993242i	$-0.537028\pi$
$641$	−5.87709	−0.232131	−0.116066	+	0.993242i	$0.537028\pi$
$642$	0	0
$643$	−19.8166	−0.781490	−0.390745	−	0.920499i	$-0.627783\pi$
$643$	−19.8166	−0.781490	−0.390745	+	0.920499i	$0.627783\pi$
$644$	0	0
$645$	6.49693	0.255816
$646$	0	0
$647$	−10.7915	−0.424259	−0.212129	−	0.977242i	$-0.568040\pi$
$647$	−10.7915	−0.424259	−0.212129	+	0.977242i	$0.568040\pi$
$648$	0	0
$649$	−34.3234	−1.34731
$650$	0	0
$651$	4.87936	0.191237
$652$	0	0
$653$	−8.93987	−0.349844	−0.174922	−	0.984582i	$-0.555967\pi$
$653$	−8.93987	−0.349844	−0.174922	+	0.984582i	$0.555967\pi$
$654$	0	0
$655$	−9.53692	−0.372638
$656$	0	0
$657$	−15.5597	−0.607042
$658$	0	0
$659$	12.3330	0.480425	0.240212	−	0.970720i	$-0.422783\pi$
$659$	12.3330	0.480425	0.240212	+	0.970720i	$0.422783\pi$
$660$	0	0
$661$	−22.3980	−0.871182	−0.435591	−	0.900145i	$-0.643461\pi$
$661$	−22.3980	−0.871182	−0.435591	+	0.900145i	$0.643461\pi$
$662$	0	0
$663$	6.68035	0.259443
$664$	0	0
$665$	36.1978	1.40369
$666$	0	0
$667$	−13.9544	−0.540318
$668$	0	0
$669$	12.2557	0.473831
$670$	0	0
$671$	35.4017	1.36667
$672$	0	0
$673$	−15.4875	−0.596998	−0.298499	−	0.954410i	$-0.596486\pi$
$673$	−15.4875	−0.596998	−0.298499	+	0.954410i	$0.596486\pi$
$674$	0	0
$675$	−1.00000	−0.0384900
$676$	0	0
$677$	−18.5997	−0.714845	−0.357422	−	0.933943i	$-0.616344\pi$
$677$	−18.5997	−0.714845	−0.357422	+	0.933943i	$0.616344\pi$
$678$	0	0
$679$	−79.4473	−3.04891
$680$	0	0
$681$	12.1483	0.465526
$682$	0	0
$683$	−14.4040	−0.551154	−0.275577	−	0.961279i	$-0.588869\pi$
$683$	−14.4040	−0.551154	−0.275577	+	0.961279i	$0.588869\pi$
$684$	0	0
$685$	−2.94441	−0.112500
$686$	0	0
$687$	−6.28231	−0.239685
$688$	0	0
$689$	4.73820	0.180511
$690$	0	0
$691$	−33.9565	−1.29177	−0.645883	−	0.763436i	$-0.723511\pi$
$691$	−33.9565	−1.29177	−0.645883	+	0.763436i	$0.723511\pi$
$692$	0	0
$693$	21.1773	0.804458
$694$	0	0
$695$	7.02052	0.266303
$696$	0	0
$697$	−0.482553	−0.0182780
$698$	0	0
$699$	−15.3112	−0.579124
$700$	0	0
$701$	0.267938	0.0101199	0.00505994	−	0.999987i	$-0.498389\pi$
$701$	0.267938	0.0101199	0.00505994	+	0.999987i	$0.498389\pi$
$702$	0	0
$703$	−43.0349	−1.62309
$704$	0	0
$705$	9.80817	0.369397
$706$	0	0
$707$	−35.3028	−1.32770
$708$	0	0
$709$	7.36069	0.276437	0.138218	−	0.990402i	$-0.455862\pi$
$709$	7.36069	0.276437	0.138218	+	0.990402i	$0.455862\pi$
$710$	0	0
$711$	6.23287	0.233751
$712$	0	0
$713$	−2.29072	−0.0857883
$714$	0	0
$715$	−11.0205	−0.412144
$716$	0	0
$717$	−30.1978	−1.12776
$718$	0	0
$719$	30.3279	1.13104	0.565520	−	0.824735i	$-0.308676\pi$
$719$	30.3279	1.13104	0.565520	+	0.824735i	$0.308676\pi$
$720$	0	0
$721$	87.3400	3.25271
$722$	0	0
$723$	25.6475	0.953842
$724$	0	0
$725$	6.09171	0.226240
$726$	0	0
$727$	9.43415	0.349893	0.174947	−	0.984578i	$-0.444025\pi$
$727$	9.43415	0.349893	0.174947	+	0.984578i	$0.444025\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	17.0928	0.632198
$732$	0	0
$733$	27.9877	1.03375	0.516875	−	0.856061i	$-0.327095\pi$
$733$	27.9877	1.03375	0.516875	+	0.856061i	$0.327095\pi$
$734$	0	0
$735$	16.8082	0.619979
$736$	0	0
$737$	−45.4908	−1.67567
$738$	0	0
$739$	6.41628	0.236027	0.118013	−	0.993012i	$-0.462347\pi$
$739$	6.41628	0.236027	0.118013	+	0.993012i	$0.462347\pi$
$740$	0	0
$741$	18.8371	0.691998
$742$	0	0
$743$	−16.5503	−0.607170	−0.303585	−	0.952804i	$-0.598184\pi$
$743$	−16.5503	−0.607170	−0.303585	+	0.952804i	$0.598184\pi$
$744$	0	0
$745$	8.49693	0.311303
$746$	0	0
$747$	−6.38962	−0.233784
$748$	0	0
$749$	63.1605	2.30783
$750$	0	0
$751$	−38.1711	−1.39288	−0.696442	−	0.717613i	$-0.745235\pi$
$751$	−38.1711	−1.39288	−0.696442	+	0.717613i	$0.745235\pi$
$752$	0	0
$753$	−6.25565	−0.227969
$754$	0	0
$755$	−3.52586	−0.128319
$756$	0	0
$757$	0.512527	0.0186281	0.00931405	−	0.999957i	$-0.497035\pi$
$757$	0.512527	0.0186281	0.00931405	+	0.999957i	$0.497035\pi$
$758$	0	0
$759$	−9.94214	−0.360877
$760$	0	0
$761$	−1.92267	−0.0696966	−0.0348483	−	0.999393i	$-0.511095\pi$
$761$	−1.92267	−0.0696966	−0.0348483	+	0.999393i	$0.511095\pi$
$762$	0	0
$763$	−49.7998	−1.80287
$764$	0	0
$765$	−2.63090	−0.0951203
$766$	0	0
$767$	−20.0806	−0.725070
$768$	0	0
$769$	−50.8164	−1.83249	−0.916243	−	0.400622i	$-0.868794\pi$
$769$	−50.8164	−1.83249	−0.916243	+	0.400622i	$0.868794\pi$
$770$	0	0
$771$	13.1545	0.473747
$772$	0	0
$773$	17.9506	0.645636	0.322818	−	0.946461i	$-0.395370\pi$
$773$	17.9506	0.645636	0.322818	+	0.946461i	$0.395370\pi$
$774$	0	0
$775$	1.00000	0.0359211
$776$	0	0
$777$	−28.3051	−1.01544
$778$	0	0
$779$	−1.36069	−0.0487518
$780$	0	0
$781$	13.7587	0.492326
$782$	0	0
$783$	−6.09171	−0.217700
$784$	0	0
$785$	−4.49693	−0.160502
$786$	0	0
$787$	−6.68035	−0.238129	−0.119064	−	0.992887i	$-0.537989\pi$
$787$	−6.68035	−0.238129	−0.119064	+	0.992887i	$0.537989\pi$
$788$	0	0
$789$	29.4908	1.04990
$790$	0	0
$791$	−46.8515	−1.66585
$792$	0	0
$793$	20.7115	0.735488
$794$	0	0
$795$	−1.86603	−0.0661813
$796$	0	0
$797$	−24.8865	−0.881527	−0.440763	−	0.897623i	$-0.645292\pi$
$797$	−24.8865	−0.881527	−0.440763	+	0.897623i	$0.645292\pi$
$798$	0	0
$799$	25.8043	0.912890
$800$	0	0
$801$	−7.51026	−0.265362
$802$	0	0
$803$	67.5318	2.38315
$804$	0	0
$805$	−11.1773	−0.393947
$806$	0	0
$807$	20.7454	0.730272
$808$	0	0
$809$	−18.0027	−0.632940	−0.316470	−	0.948603i	$-0.602498\pi$
$809$	−18.0027	−0.632940	−0.316470	+	0.948603i	$0.602498\pi$
$810$	0	0
$811$	−30.2700	−1.06292	−0.531462	−	0.847082i	$-0.678357\pi$
$811$	−30.2700	−1.06292	−0.531462	+	0.847082i	$0.678357\pi$
$812$	0	0
$813$	0.979481	0.0343519
$814$	0	0
$815$	−15.3763	−0.538608
$816$	0	0
$817$	48.1978	1.68623
$818$	0	0
$819$	12.3896	0.432928
$820$	0	0
$821$	35.6235	1.24327	0.621635	−	0.783307i	$-0.286469\pi$
$821$	35.6235	1.24327	0.621635	+	0.783307i	$0.286469\pi$
$822$	0	0
$823$	53.9976	1.88224	0.941118	−	0.338078i	$-0.109777\pi$
$823$	53.9976	1.88224	0.941118	+	0.338078i	$0.109777\pi$
$824$	0	0
$825$	4.34017	0.151105
$826$	0	0
$827$	4.64527	0.161532	0.0807660	−	0.996733i	$-0.474263\pi$
$827$	4.64527	0.161532	0.0807660	+	0.996733i	$0.474263\pi$
$828$	0	0
$829$	44.6537	1.55089	0.775443	−	0.631417i	$-0.217526\pi$
$829$	44.6537	1.55089	0.775443	+	0.631417i	$0.217526\pi$
$830$	0	0
$831$	−16.3246	−0.566293
$832$	0	0
$833$	44.2206	1.53215
$834$	0	0
$835$	10.2413	0.354414
$836$	0	0
$837$	−1.00000	−0.0345651
$838$	0	0
$839$	−17.6814	−0.610429	−0.305215	−	0.952284i	$-0.598728\pi$
$839$	−17.6814	−0.610429	−0.305215	+	0.952284i	$0.598728\pi$
$840$	0	0
$841$	8.10892	0.279618
$842$	0	0
$843$	−22.2823	−0.767444
$844$	0	0
$845$	6.55252	0.225414
$846$	0	0
$847$	−38.2401	−1.31394
$848$	0	0
$849$	−2.05664	−0.0705836
$850$	0	0
$851$	13.2885	0.455522
$852$	0	0
$853$	−15.2495	−0.522133	−0.261067	−	0.965321i	$-0.584074\pi$
$853$	−15.2495	−0.522133	−0.261067	+	0.965321i	$0.584074\pi$
$854$	0	0
$855$	−7.41855	−0.253709
$856$	0	0
$857$	26.7649	0.914270	0.457135	−	0.889397i	$-0.348876\pi$
$857$	26.7649	0.914270	0.457135	+	0.889397i	$0.348876\pi$
$858$	0	0
$859$	42.7747	1.45945	0.729727	−	0.683739i	$-0.239647\pi$
$859$	42.7747	1.45945	0.729727	+	0.683739i	$0.239647\pi$
$860$	0	0
$861$	−0.894960	−0.0305002
$862$	0	0
$863$	−34.6681	−1.18011	−0.590057	−	0.807361i	$-0.700895\pi$
$863$	−34.6681	−1.18011	−0.590057	+	0.807361i	$0.700895\pi$
$864$	0	0
$865$	−7.07838	−0.240672
$866$	0	0
$867$	10.0784	0.342280
$868$	0	0
$869$	−27.0517	−0.917667
$870$	0	0
$871$	−26.6141	−0.901784
$872$	0	0
$873$	16.2823	0.551073
$874$	0	0
$875$	4.87936	0.164953
$876$	0	0
$877$	−24.0000	−0.810422	−0.405211	−	0.914223i	$-0.632802\pi$
$877$	−24.0000	−0.810422	−0.405211	+	0.914223i	$0.632802\pi$
$878$	0	0
$879$	−22.3630	−0.754284
$880$	0	0
$881$	−12.8254	−0.432098	−0.216049	−	0.976383i	$-0.569317\pi$
$881$	−12.8254	−0.432098	−0.216049	+	0.976383i	$0.569317\pi$
$882$	0	0
$883$	−33.5174	−1.12795	−0.563976	−	0.825791i	$-0.690729\pi$
$883$	−33.5174	−1.12795	−0.563976	+	0.825791i	$0.690729\pi$
$884$	0	0
$885$	7.90829	0.265834
$886$	0	0
$887$	−7.55252	−0.253589	−0.126794	−	0.991929i	$-0.540469\pi$
$887$	−7.55252	−0.253589	−0.126794	+	0.991929i	$0.540469\pi$
$888$	0	0
$889$	76.1276	2.55324
$890$	0	0
$891$	−4.34017	−0.145401
$892$	0	0
$893$	72.7624	2.43490
$894$	0	0
$895$	14.6225	0.488776
$896$	0	0
$897$	−5.81658	−0.194210
$898$	0	0
$899$	6.09171	0.203170
$900$	0	0
$901$	−4.90934	−0.163554
$902$	0	0
$903$	31.7009	1.05494
$904$	0	0
$905$	−2.00000	−0.0664822
$906$	0	0
$907$	1.97212	0.0654830	0.0327415	−	0.999464i	$-0.489576\pi$
$907$	1.97212	0.0654830	0.0327415	+	0.999464i	$0.489576\pi$
$908$	0	0
$909$	7.23513	0.239974
$910$	0	0
$911$	44.3689	1.47001	0.735004	−	0.678063i	$-0.237180\pi$
$911$	44.3689	1.47001	0.735004	+	0.678063i	$0.237180\pi$
$912$	0	0
$913$	27.7321	0.917797
$914$	0	0
$915$	−8.15676	−0.269654
$916$	0	0
$917$	−46.5341	−1.53669
$918$	0	0
$919$	−12.8494	−0.423862	−0.211931	−	0.977285i	$-0.567975\pi$
$919$	−12.8494	−0.423862	−0.211931	+	0.977285i	$0.567975\pi$
$920$	0	0
$921$	−9.24620	−0.304673
$922$	0	0
$923$	8.04945	0.264951
$924$	0	0
$925$	−5.80098	−0.190735
$926$	0	0
$927$	−17.8999	−0.587909
$928$	0	0
$929$	19.2013	0.629974	0.314987	−	0.949096i	$-0.398000\pi$
$929$	19.2013	0.629974	0.314987	+	0.949096i	$0.398000\pi$
$930$	0	0
$931$	124.692	4.08662
$932$	0	0
$933$	9.24232	0.302580
$934$	0	0
$935$	11.4186	0.373427
$936$	0	0
$937$	−59.0037	−1.92757	−0.963783	−	0.266686i	$-0.914071\pi$
$937$	−59.0037	−1.92757	−0.963783	+	0.266686i	$0.914071\pi$
$938$	0	0
$939$	−3.89988	−0.127268
$940$	0	0
$941$	−10.7142	−0.349273	−0.174636	−	0.984633i	$-0.555875\pi$
$941$	−10.7142	−0.349273	−0.174636	+	0.984633i	$0.555875\pi$
$942$	0	0
$943$	0.420159	0.0136823
$944$	0	0
$945$	−4.87936	−0.158726
$946$	0	0
$947$	7.23901	0.235236	0.117618	−	0.993059i	$-0.462474\pi$
$947$	7.23901	0.235236	0.117618	+	0.993059i	$0.462474\pi$
$948$	0	0
$949$	39.5090	1.28252
$950$	0	0
$951$	−3.62475	−0.117541
$952$	0	0
$953$	−44.1361	−1.42971	−0.714854	−	0.699274i	$-0.753507\pi$
$953$	−44.1361	−1.42971	−0.714854	+	0.699274i	$0.753507\pi$
$954$	0	0
$955$	1.75154	0.0566784
$956$	0	0
$957$	26.4391	0.854654
$958$	0	0
$959$	−14.3668	−0.463929
$960$	0	0
$961$	1.00000	0.0322581
$962$	0	0
$963$	−12.9444	−0.417128
$964$	0	0
$965$	4.86376	0.156570
$966$	0	0
$967$	10.3837	0.333916	0.166958	−	0.985964i	$-0.446606\pi$
$967$	10.3837	0.333916	0.166958	+	0.985964i	$0.446606\pi$
$968$	0	0
$969$	−19.5174	−0.626991
$970$	0	0
$971$	49.9448	1.60280	0.801402	−	0.598126i	$-0.204088\pi$
$971$	49.9448	1.60280	0.801402	+	0.598126i	$0.204088\pi$
$972$	0	0
$973$	34.2557	1.09819
$974$	0	0
$975$	2.53919	0.0813191
$976$	0	0
$977$	32.7214	1.04685	0.523425	−	0.852072i	$-0.324654\pi$
$977$	32.7214	1.04685	0.523425	+	0.852072i	$0.324654\pi$
$978$	0	0
$979$	32.5958	1.04177
$980$	0	0
$981$	10.2062	0.325859
$982$	0	0
$983$	40.3500	1.28697	0.643483	−	0.765461i	$-0.277489\pi$
$983$	40.3500	1.28697	0.643483	+	0.765461i	$0.277489\pi$
$984$	0	0
$985$	20.3051	0.646974
$986$	0	0
$987$	47.8576	1.52332
$988$	0	0
$989$	−14.8827	−0.473242
$990$	0	0
$991$	−7.81658	−0.248302	−0.124151	−	0.992263i	$-0.539621\pi$
$991$	−7.81658	−0.248302	−0.124151	+	0.992263i	$0.539621\pi$
$992$	0	0
$993$	−24.1217	−0.765478
$994$	0	0
$995$	−2.04945	−0.0649719
$996$	0	0
$997$	−40.2245	−1.27392	−0.636961	−	0.770896i	$-0.719809\pi$
$997$	−40.2245	−1.27392	−0.636961	+	0.770896i	$0.719809\pi$
$998$	0	0
$999$	5.80098	0.183535

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7440.2.a.bm.1.1		3
4.3	odd	2		465.2.a.g.1.2	✓	3
12.11	even	2		1395.2.a.h.1.2		3
20.3	even	4		2325.2.c.l.1024.2		6
20.7	even	4		2325.2.c.l.1024.5		6
20.19	odd	2		2325.2.a.p.1.2		3
60.59	even	2		6975.2.a.bi.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
465.2.a.g.1.2	✓	3	4.3	odd	2
1395.2.a.h.1.2		3	12.11	even	2
2325.2.a.p.1.2		3	20.19	odd	2
2325.2.c.l.1024.2		6	20.3	even	4
2325.2.c.l.1024.5		6	20.7	even	4
6975.2.a.bi.1.2		3	60.59	even	2
7440.2.a.bm.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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -1.00000 q^{5} -4.87936 q^{7} +1.00000 q^{9} -4.34017 q^{11} -2.53919 q^{13} +1.00000 q^{15} +2.63090 q^{17} +7.41855 q^{19} +4.87936 q^{21} -2.29072 q^{23} +1.00000 q^{25} -1.00000 q^{27} +6.09171 q^{29} +1.00000 q^{31} +4.34017 q^{33} +4.87936 q^{35} -5.80098 q^{37} +2.53919 q^{39} -0.183417 q^{41} +6.49693 q^{43} -1.00000 q^{45} +9.80817 q^{47} +16.8082 q^{49} -2.63090 q^{51} -1.86603 q^{53} +4.34017 q^{55} -7.41855 q^{57} +7.90829 q^{59} -8.15676 q^{61} -4.87936 q^{63} +2.53919 q^{65} +10.4813 q^{67} +2.29072 q^{69} -3.17009 q^{71} -15.5597 q^{73} -1.00000 q^{75} +21.1773 q^{77} +6.23287 q^{79} +1.00000 q^{81} -6.38962 q^{83} -2.63090 q^{85} -6.09171 q^{87} -7.51026 q^{89} +12.3896 q^{91} -1.00000 q^{93} -7.41855 q^{95} +16.2823 q^{97} -4.34017 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{3} - 3 q^{5} - 2 q^{7} + 3 q^{9} - 2 q^{11} - 6 q^{13} + 3 q^{15} + 4 q^{17} + 8 q^{19} + 2 q^{21} - 14 q^{23} + 3 q^{25} - 3 q^{27} + 16 q^{29} + 3 q^{31} + 2 q^{33} + 2 q^{35} - 8 q^{37} + 6 q^{39}+ \cdots - 2 q^{99}+O(q^{100}) \) 3 * q - 3 * q^3 - 3 * q^5 - 2 * q^7 + 3 * q^9 - 2 * q^11 - 6 * q^13 + 3 * q^15 + 4 * q^17 + 8 * q^19 + 2 * q^21 - 14 * q^23 + 3 * q^25 - 3 * q^27 + 16 * q^29 + 3 * q^31 + 2 * q^33 + 2 * q^35 - 8 * q^37 + 6 * q^39 + 4 * q^41 + 2 * q^43 - 3 * q^45 - 14 * q^47 + 7 * q^49 - 4 * q^51 + 8 * q^53 + 2 * q^55 - 8 * q^57 + 26 * q^59 - 18 * q^61 - 2 * q^63 + 6 * q^65 + 14 * q^69 - 4 * q^71 - 12 * q^73 - 3 * q^75 + 24 * q^77 - 4 * q^79 + 3 * q^81 + 10 * q^83 - 4 * q^85 - 16 * q^87 - 6 * q^89 + 8 * q^91 - 3 * q^93 - 8 * q^95 + 8 * q^97 - 2 * q^99

Introduction

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