LMFDB - Embedded newform 6480.2.a.cb.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6480,2,Mod(1,6480)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6480, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("6480.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6480 = 2^{4} \cdot 3^{4} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6480.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:	$51.7430605098$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.29268.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 9x^{2} + 5x + 16 $ x^4 - x^3 - 9x^2 + 5x + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 3 $
Twist minimal:	no (minimal twist has level 360)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.83719$ of defining polynomial
Character	$\chi$	$=$	6480.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{5} -0.867736 q^{7} +4.08667 q^{11} +1.21245 q^{13} +7.82214 q^{17} +4.51156 q^{19} -2.86774 q^{23} +1.00000 q^{25} +6.29912 q^{29} +2.52303 q^{31} -0.867736 q^{35} -0.523026 q^{37} -8.12126 q^{41} -3.82214 q^{43} +1.39076 q^{47} -6.24703 q^{49} +13.9088 q^{53} +4.08667 q^{55} -6.87423 q^{59} +9.08667 q^{61} +1.21245 q^{65} -3.37930 q^{67} -3.21245 q^{71} +8.60970 q^{73} -3.54615 q^{77} -16.2425 q^{79} -6.44284 q^{83} +7.82214 q^{85} -10.2079 q^{89} -1.05208 q^{91} +4.51156 q^{95} -8.77006 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{5} + q^{7} - q^{11} + 4 q^{13} + 5 q^{17} - q^{19} - 7 q^{23} + 4 q^{25} + 7 q^{29} + 2 q^{31} + q^{35} + 6 q^{37} + 12 q^{41} + 11 q^{43} - 7 q^{47} + 3 q^{49} + 12 q^{53} - q^{55} - 11 q^{59}+ \cdots + q^{97}+O(q^{100}) $ 4 * q + 4 * q^5 + q^7 - q^11 + 4 * q^13 + 5 * q^17 - q^19 - 7 * q^23 + 4 * q^25 + 7 * q^29 + 2 * q^31 + q^35 + 6 * q^37 + 12 * q^41 + 11 * q^43 - 7 * q^47 + 3 * q^49 + 12 * q^53 - q^55 - 11 * q^59 + 19 * q^61 + 4 * q^65 + 10 * q^67 - 12 * q^71 + 9 * q^73 + 32 * q^77 + 24 * q^79 - 23 * q^83 + 5 * q^85 + 21 * q^89 - 14 * q^91 - q^95 + q^97

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
$3$	0	0
$3$	0	0
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−0.867736	−0.327973	−0.163987	−	0.986463i	$-0.552435\pi$
$7$	−0.867736	−0.327973	−0.163987	+	0.986463i	$0.552435\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	4.08667	1.23218	0.616089	−	0.787677i	$-0.288716\pi$
$11$	4.08667	1.23218	0.616089	+	0.787677i	$0.288716\pi$
$12$	0	0
$13$	1.21245	0.336272	0.168136	−	0.985764i	$-0.446225\pi$
$13$	1.21245	0.336272	0.168136	+	0.985764i	$0.446225\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	7.82214	1.89715	0.948574	−	0.316555i	$-0.102526\pi$
$17$	7.82214	1.89715	0.948574	+	0.316555i	$0.102526\pi$
$18$	0	0
$19$	4.51156	1.03502	0.517512	−	0.855676i	$-0.326858\pi$
$19$	4.51156	1.03502	0.517512	+	0.855676i	$0.326858\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−2.86774	−0.597964	−0.298982	−	0.954259i	$-0.596647\pi$
$23$	−2.86774	−0.597964	−0.298982	+	0.954259i	$0.596647\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	6.29912	1.16972	0.584858	−	0.811135i	$-0.301150\pi$
$29$	6.29912	1.16972	0.584858	+	0.811135i	$0.301150\pi$
$30$	0	0
$31$	2.52303	0.453149	0.226574	−	0.973994i	$-0.427247\pi$
$31$	2.52303	0.453149	0.226574	+	0.973994i	$0.427247\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−0.867736	−0.146674
$36$	0	0
$37$	−0.523026	−0.0859850	−0.0429925	−	0.999075i	$-0.513689\pi$
$37$	−0.523026	−0.0859850	−0.0429925	+	0.999075i	$0.513689\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−8.12126	−1.26833	−0.634164	−	0.773199i	$-0.718656\pi$
$41$	−8.12126	−1.26833	−0.634164	+	0.773199i	$0.718656\pi$
$42$	0	0
$43$	−3.82214	−0.582871	−0.291436	−	0.956590i	$-0.594133\pi$
$43$	−3.82214	−0.582871	−0.291436	+	0.956590i	$0.594133\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	1.39076	0.202863	0.101432	−	0.994843i	$-0.467658\pi$
$47$	1.39076	0.202863	0.101432	+	0.994843i	$0.467658\pi$
$48$	0	0
$49$	−6.24703	−0.892433
$50$	0	0
$51$	0	0
$52$	0	0
$53$	13.9088	1.91052	0.955261	−	0.295763i	$-0.0955739\pi$
$53$	13.9088	1.91052	0.955261	+	0.295763i	$0.0955739\pi$
$54$	0	0
$55$	4.08667	0.551047
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−6.87423	−0.894948	−0.447474	−	0.894297i	$-0.647676\pi$
$59$	−6.87423	−0.894948	−0.447474	+	0.894297i	$0.647676\pi$
$60$	0	0
$61$	9.08667	1.16343	0.581715	−	0.813393i	$-0.302382\pi$
$61$	9.08667	1.16343	0.581715	+	0.813393i	$0.302382\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	1.21245	0.150385
$66$	0	0
$67$	−3.37930	−0.412847	−0.206424	−	0.978463i	$-0.566182\pi$
$67$	−3.37930	−0.412847	−0.206424	+	0.978463i	$0.566182\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−3.21245	−0.381247	−0.190624	−	0.981663i	$-0.561051\pi$
$71$	−3.21245	−0.381247	−0.190624	+	0.981663i	$0.561051\pi$
$72$	0	0
$73$	8.60970	1.00769	0.503844	−	0.863794i	$-0.331918\pi$
$73$	8.60970	1.00769	0.503844	+	0.863794i	$0.331918\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−3.54615	−0.404121
$78$	0	0
$79$	−16.2425	−1.82743	−0.913713	−	0.406360i	$-0.866798\pi$
$79$	−16.2425	−1.82743	−0.913713	+	0.406360i	$0.866798\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−6.44284	−0.707194	−0.353597	−	0.935398i	$-0.615042\pi$
$83$	−6.44284	−0.707194	−0.353597	+	0.935398i	$0.615042\pi$
$84$	0	0
$85$	7.82214	0.848431
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−10.2079	−1.08204	−0.541019	−	0.841010i	$-0.681961\pi$
$89$	−10.2079	−1.08204	−0.541019	+	0.841010i	$0.681961\pi$
$90$	0	0
$91$	−1.05208	−0.110288
$92$	0	0
$93$	0	0
$94$	0	0
$95$	4.51156	0.462877
$96$	0	0
$97$	−8.77006	−0.890465	−0.445232	−	0.895415i	$-0.646879\pi$
$97$	−8.77006	−0.890465	−0.445232	+	0.895415i	$0.646879\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	2.26453	0.225329	0.112664	−	0.993633i	$-0.464061\pi$
$101$	2.26453	0.225329	0.112664	+	0.993633i	$0.464061\pi$
$102$	0	0
$103$	15.3858	1.51601	0.758003	−	0.652251i	$-0.226175\pi$
$103$	15.3858	1.51601	0.758003	+	0.652251i	$0.226175\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−14.3402	−1.38632	−0.693160	−	0.720784i	$-0.743782\pi$
$107$	−14.3402	−1.38632	−0.693160	+	0.720784i	$0.743782\pi$
$108$	0	0
$109$	−5.35120	−0.512552	−0.256276	−	0.966604i	$-0.582496\pi$
$109$	−5.35120	−0.512552	−0.256276	+	0.966604i	$0.582496\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	2.68942	0.252999	0.126500	−	0.991967i	$-0.459626\pi$
$113$	2.68942	0.252999	0.126500	+	0.991967i	$0.459626\pi$
$114$	0	0
$115$	−2.86774	−0.267418
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−6.78755	−0.622214
$120$	0	0
$121$	5.70088	0.518262
$122$	0	0
$123$	0	0
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	21.5641	1.91350	0.956752	−	0.290903i	$-0.0939558\pi$
$127$	21.5641	1.91350	0.956752	+	0.290903i	$0.0939558\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−18.4941	−1.61583	−0.807917	−	0.589296i	$-0.799405\pi$
$131$	−18.4941	−1.61583	−0.807917	+	0.589296i	$0.799405\pi$
$132$	0	0
$133$	−3.91484	−0.339460
$134$	0	0
$135$	0	0
$136$	0	0
$137$	9.72401	0.830778	0.415389	−	0.909644i	$-0.363645\pi$
$137$	9.72401	0.830778	0.415389	+	0.909644i	$0.363645\pi$
$138$	0	0
$139$	16.0806	1.36394	0.681971	−	0.731379i	$-0.261123\pi$
$139$	16.0806	1.36394	0.681971	+	0.731379i	$0.261123\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	4.95487	0.414347
$144$	0	0
$145$	6.29912	0.523113
$146$	0	0
$147$	0	0
$148$	0	0
$149$	15.8453	1.29809	0.649047	−	0.760748i	$-0.275168\pi$
$149$	15.8453	1.29809	0.649047	+	0.760748i	$0.275168\pi$
$150$	0	0
$151$	15.1213	1.23055	0.615275	−	0.788312i	$-0.289045\pi$
$151$	15.1213	1.23055	0.615275	+	0.788312i	$0.289045\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	2.52303	0.202654
$156$	0	0
$157$	17.5531	1.40089	0.700445	−	0.713706i	$-0.252985\pi$
$157$	17.5531	1.40089	0.700445	+	0.713706i	$0.252985\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	2.48844	0.196116
$162$	0	0
$163$	−6.85673	−0.537061	−0.268530	−	0.963271i	$-0.586538\pi$
$163$	−6.85673	−0.537061	−0.268530	+	0.963271i	$0.586538\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−3.91982	−0.303325	−0.151662	−	0.988432i	$-0.548463\pi$
$167$	−3.91982	−0.303325	−0.151662	+	0.988432i	$0.548463\pi$
$168$	0	0
$169$	−11.5300	−0.886921
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−8.17334	−0.621408	−0.310704	−	0.950507i	$-0.600565\pi$
$173$	−8.17334	−0.621408	−0.310704	+	0.950507i	$0.600565\pi$
$174$	0	0
$175$	−0.867736	−0.0655947
$176$	0	0
$177$	0	0
$178$	0	0
$179$	6.85673	0.512496	0.256248	−	0.966611i	$-0.417514\pi$
$179$	6.85673	0.512496	0.256248	+	0.966611i	$0.417514\pi$
$180$	0	0
$181$	16.2991	1.21150	0.605752	−	0.795654i	$-0.292872\pi$
$181$	16.2991	1.21150	0.605752	+	0.795654i	$0.292872\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−0.523026	−0.0384536
$186$	0	0
$187$	31.9665	2.33762
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−11.8336	−0.856250	−0.428125	−	0.903720i	$-0.640826\pi$
$191$	−11.8336	−0.856250	−0.428125	+	0.903720i	$0.640826\pi$
$192$	0	0
$193$	−1.82214	−0.131161	−0.0655804	−	0.997847i	$-0.520890\pi$
$193$	−1.82214	−0.131161	−0.0655804	+	0.997847i	$0.520890\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	19.3798	1.38075	0.690375	−	0.723451i	$-0.257445\pi$
$197$	19.3798	1.38075	0.690375	+	0.723451i	$0.257445\pi$
$198$	0	0
$199$	−6.69637	−0.474693	−0.237347	−	0.971425i	$-0.576278\pi$
$199$	−6.69637	−0.474693	−0.237347	+	0.971425i	$0.576278\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−5.46597	−0.383636
$204$	0	0
$205$	−8.12126	−0.567213
$206$	0	0
$207$	0	0
$208$	0	0
$209$	18.4373	1.27533
$210$	0	0
$211$	−8.52303	−0.586749	−0.293375	−	0.955998i	$-0.594778\pi$
$211$	−8.52303	−0.586749	−0.293375	+	0.955998i	$0.594778\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−3.82214	−0.260668
$216$	0	0
$217$	−2.18932	−0.148621
$218$	0	0
$219$	0	0
$220$	0	0
$221$	9.48392	0.637958
$222$	0	0
$223$	10.8617	0.727354	0.363677	−	0.931525i	$-0.381521\pi$
$223$	10.8617	0.727354	0.363677	+	0.931525i	$0.381521\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	18.6849	1.24016	0.620080	−	0.784538i	$-0.287100\pi$
$227$	18.6849	1.24016	0.620080	+	0.784538i	$0.287100\pi$
$228$	0	0
$229$	9.59672	0.634169	0.317084	−	0.948397i	$-0.397296\pi$
$229$	9.59672	0.634169	0.317084	+	0.948397i	$0.397296\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−20.5807	−1.34829	−0.674145	−	0.738599i	$-0.735488\pi$
$233$	−20.5807	−1.34829	−0.674145	+	0.738599i	$0.735488\pi$
$234$	0	0
$235$	1.39076	0.0907233
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−10.1604	−0.657219	−0.328610	−	0.944466i	$-0.606580\pi$
$239$	−10.1604	−0.657219	−0.328610	+	0.944466i	$0.606580\pi$
$240$	0	0
$241$	14.2124	0.915504	0.457752	−	0.889080i	$-0.348655\pi$
$241$	14.2124	0.915504	0.457752	+	0.889080i	$0.348655\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−6.24703	−0.399108
$246$	0	0
$247$	5.47002	0.348049
$248$	0	0
$249$	0	0
$250$	0	0
$251$	0.247034	0.0155927	0.00779634	−	0.999970i	$-0.497518\pi$
$251$	0.247034	0.0155927	0.00779634	+	0.999970i	$0.497518\pi$
$252$	0	0
$253$	−11.7195	−0.736798
$254$	0	0
$255$	0	0
$256$	0	0
$257$	4.86125	0.303236	0.151618	−	0.988439i	$-0.451552\pi$
$257$	4.86125	0.303236	0.151618	+	0.988439i	$0.451552\pi$
$258$	0	0
$259$	0.453849	0.0282008
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−20.4318	−1.25988	−0.629941	−	0.776643i	$-0.716921\pi$
$263$	−20.4318	−1.25988	−0.629941	+	0.776643i	$0.716921\pi$
$264$	0	0
$265$	13.9088	0.854412
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−25.5125	−1.55552	−0.777762	−	0.628559i	$-0.783645\pi$
$269$	−25.5125	−1.55552	−0.777762	+	0.628559i	$0.783645\pi$
$270$	0	0
$271$	−5.12126	−0.311094	−0.155547	−	0.987828i	$-0.549714\pi$
$271$	−5.12126	−0.311094	−0.155547	+	0.987828i	$0.549714\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	4.08667	0.246436
$276$	0	0
$277$	11.4087	0.685483	0.342742	−	0.939430i	$-0.388644\pi$
$277$	11.4087	0.685483	0.342742	+	0.939430i	$0.388644\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	21.5937	1.28817	0.644087	−	0.764953i	$-0.277238\pi$
$281$	21.5937	1.28817	0.644087	+	0.764953i	$0.277238\pi$
$282$	0	0
$283$	−19.9268	−1.18452	−0.592262	−	0.805746i	$-0.701765\pi$
$283$	−19.9268	−1.18452	−0.592262	+	0.805746i	$0.701765\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	7.04711	0.415978
$288$	0	0
$289$	44.1859	2.59917
$290$	0	0
$291$	0	0
$292$	0	0
$293$	1.83361	0.107120	0.0535602	−	0.998565i	$-0.482943\pi$
$293$	1.83361	0.107120	0.0535602	+	0.998565i	$0.482943\pi$
$294$	0	0
$295$	−6.87423	−0.400233
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−3.47697	−0.201079
$300$	0	0
$301$	3.31661	0.191166
$302$	0	0
$303$	0	0
$304$	0	0
$305$	9.08667	0.520301
$306$	0	0
$307$	−9.02962	−0.515347	−0.257674	−	0.966232i	$-0.582956\pi$
$307$	−9.02962	−0.515347	−0.257674	+	0.966232i	$0.582956\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−2.96090	−0.167897	−0.0839485	−	0.996470i	$-0.526753\pi$
$311$	−2.96090	−0.167897	−0.0839485	+	0.996470i	$0.526753\pi$
$312$	0	0
$313$	19.2831	1.08995	0.544974	−	0.838453i	$-0.316540\pi$
$313$	19.2831	1.08995	0.544974	+	0.838453i	$0.316540\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−6.16731	−0.346391	−0.173195	−	0.984887i	$-0.555409\pi$
$317$	−6.16731	−0.346391	−0.173195	+	0.984887i	$0.555409\pi$
$318$	0	0
$319$	25.7424	1.44130
$320$	0	0
$321$	0	0
$322$	0	0
$323$	35.2901	1.96359
$324$	0	0
$325$	1.21245	0.0672544
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−1.20681	−0.0665338
$330$	0	0
$331$	−20.6894	−1.13719	−0.568597	−	0.822616i	$-0.692513\pi$
$331$	−20.6894	−1.13719	−0.568597	+	0.822616i	$0.692513\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−3.37930	−0.184631
$336$	0	0
$337$	−1.72401	−0.0939127	−0.0469564	−	0.998897i	$-0.514952\pi$
$337$	−1.72401	−0.0939127	−0.0469564	+	0.998897i	$0.514952\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	10.3108	0.558360
$342$	0	0
$343$	11.4949	0.620668
$344$	0	0
$345$	0	0
$346$	0	0
$347$	32.4896	1.74413	0.872065	−	0.489389i	$-0.162780\pi$
$347$	32.4896	1.74413	0.872065	+	0.489389i	$0.162780\pi$
$348$	0	0
$349$	5.18481	0.277536	0.138768	−	0.990325i	$-0.455686\pi$
$349$	5.18481	0.277536	0.138768	+	0.990325i	$0.455686\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−2.86125	−0.152289	−0.0761444	−	0.997097i	$-0.524261\pi$
$353$	−2.86125	−0.152289	−0.0761444	+	0.997097i	$0.524261\pi$
$354$	0	0
$355$	−3.21245	−0.170499
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−37.0301	−1.95437	−0.977186	−	0.212384i	$-0.931877\pi$
$359$	−37.0301	−1.95437	−0.977186	+	0.212384i	$0.931877\pi$
$360$	0	0
$361$	1.35420	0.0712736
$362$	0	0
$363$	0	0
$364$	0	0
$365$	8.60970	0.450652
$366$	0	0
$367$	36.4318	1.90173	0.950863	−	0.309611i	$-0.100199\pi$
$367$	36.4318	1.90173	0.950863	+	0.309611i	$0.100199\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−12.0692	−0.626600
$372$	0	0
$373$	−16.5300	−0.855890	−0.427945	−	0.903805i	$-0.640762\pi$
$373$	−16.5300	−0.855890	−0.427945	+	0.903805i	$0.640762\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	7.63734	0.393343
$378$	0	0
$379$	31.8062	1.63377	0.816886	−	0.576798i	$-0.195698\pi$
$379$	31.8062	1.63377	0.816886	+	0.576798i	$0.195698\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−20.2356	−1.03399	−0.516995	−	0.855989i	$-0.672949\pi$
$383$	−20.2356	−1.03399	−0.516995	+	0.855989i	$0.672949\pi$
$384$	0	0
$385$	−3.54615	−0.180729
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−24.0056	−1.21713	−0.608567	−	0.793503i	$-0.708255\pi$
$389$	−24.0056	−1.21713	−0.608567	+	0.793503i	$0.708255\pi$
$390$	0	0
$391$	−22.4318	−1.13443
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−16.2425	−0.817250
$396$	0	0
$397$	−35.8868	−1.80111	−0.900554	−	0.434745i	$-0.856839\pi$
$397$	−35.8868	−1.80111	−0.900554	+	0.434745i	$0.856839\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−3.21788	−0.160693	−0.0803466	−	0.996767i	$-0.525603\pi$
$401$	−3.21788	−0.160693	−0.0803466	+	0.996767i	$0.525603\pi$
$402$	0	0
$403$	3.05903	0.152381
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−2.13744	−0.105949
$408$	0	0
$409$	−26.2311	−1.29704	−0.648521	−	0.761197i	$-0.724612\pi$
$409$	−26.2311	−1.29704	−0.648521	+	0.761197i	$0.724612\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	5.96501	0.293519
$414$	0	0
$415$	−6.44284	−0.316267
$416$	0	0
$417$	0	0
$418$	0	0
$419$	22.7655	1.11217	0.556085	−	0.831126i	$-0.312303\pi$
$419$	22.7655	1.11217	0.556085	+	0.831126i	$0.312303\pi$
$420$	0	0
$421$	28.2425	1.37646	0.688228	−	0.725494i	$-0.258389\pi$
$421$	28.2425	1.37646	0.688228	+	0.725494i	$0.258389\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	7.82214	0.379430
$426$	0	0
$427$	−7.88483	−0.381574
$428$	0	0
$429$	0	0
$430$	0	0
$431$	21.4480	1.03311	0.516557	−	0.856253i	$-0.327213\pi$
$431$	21.4480	1.03311	0.516557	+	0.856253i	$0.327213\pi$
$432$	0	0
$433$	−0.498583	−0.0239604	−0.0119802	−	0.999928i	$-0.503814\pi$
$433$	−0.498583	−0.0239604	−0.0119802	+	0.999928i	$0.503814\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−12.9380	−0.618907
$438$	0	0
$439$	24.4149	1.16526	0.582631	−	0.812737i	$-0.302023\pi$
$439$	24.4149	1.16526	0.582631	+	0.812737i	$0.302023\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−16.1509	−0.767351	−0.383676	−	0.923468i	$-0.625342\pi$
$443$	−16.1509	−0.767351	−0.383676	+	0.923468i	$0.625342\pi$
$444$	0	0
$445$	−10.2079	−0.483902
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−30.8392	−1.45539	−0.727697	−	0.685899i	$-0.759409\pi$
$449$	−30.8392	−1.45539	−0.727697	+	0.685899i	$0.759409\pi$
$450$	0	0
$451$	−33.1889	−1.56281
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−1.05208	−0.0493224
$456$	0	0
$457$	−15.2369	−0.712752	−0.356376	−	0.934343i	$-0.615988\pi$
$457$	−15.2369	−0.712752	−0.356376	+	0.934343i	$0.615988\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−41.7610	−1.94501	−0.972503	−	0.232892i	$-0.925181\pi$
$461$	−41.7610	−1.94501	−0.972503	+	0.232892i	$0.925181\pi$
$462$	0	0
$463$	18.9479	0.880584	0.440292	−	0.897855i	$-0.354875\pi$
$463$	18.9479	0.880584	0.440292	+	0.897855i	$0.354875\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	1.90035	0.0879377	0.0439688	−	0.999033i	$-0.486000\pi$
$467$	1.90035	0.0879377	0.0439688	+	0.999033i	$0.486000\pi$
$468$	0	0
$469$	2.93234	0.135403
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−15.6198	−0.718201
$474$	0	0
$475$	4.51156	0.207005
$476$	0	0
$477$	0	0
$478$	0	0
$479$	29.8176	1.36240	0.681201	−	0.732096i	$-0.261458\pi$
$479$	29.8176	1.36240	0.681201	+	0.732096i	$0.261458\pi$
$480$	0	0
$481$	−0.634141	−0.0289143
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−8.77006	−0.398228
$486$	0	0
$487$	43.1213	1.95401	0.977005	−	0.213215i	$-0.0683933\pi$
$487$	43.1213	1.95401	0.977005	+	0.213215i	$0.0683933\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	11.7932	0.532219	0.266110	−	0.963943i	$-0.414262\pi$
$491$	11.7932	0.532219	0.266110	+	0.963943i	$0.414262\pi$
$492$	0	0
$493$	49.2726	2.21913
$494$	0	0
$495$	0	0
$496$	0	0
$497$	2.78755	0.125039
$498$	0	0
$499$	19.1167	0.855783	0.427892	−	0.903830i	$-0.359257\pi$
$499$	19.1167	0.855783	0.427892	+	0.903830i	$0.359257\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−21.1522	−0.943130	−0.471565	−	0.881831i	$-0.656311\pi$
$503$	−21.1522	−0.943130	−0.471565	+	0.881831i	$0.656311\pi$
$504$	0	0
$505$	2.26453	0.100770
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−1.96541	−0.0871153	−0.0435577	−	0.999051i	$-0.513869\pi$
$509$	−1.96541	−0.0871153	−0.0435577	+	0.999051i	$0.513869\pi$
$510$	0	0
$511$	−7.47094	−0.330495
$512$	0	0
$513$	0	0
$514$	0	0
$515$	15.3858	0.677979
$516$	0	0
$517$	5.68359	0.249964
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−14.1143	−0.618359	−0.309180	−	0.951004i	$-0.600054\pi$
$521$	−14.1143	−0.618359	−0.309180	+	0.951004i	$0.600054\pi$
$522$	0	0
$523$	−5.03413	−0.220127	−0.110064	−	0.993925i	$-0.535105\pi$
$523$	−5.03413	−0.220127	−0.110064	+	0.993925i	$0.535105\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	19.7355	0.859691
$528$	0	0
$529$	−14.7761	−0.642439
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−9.84659	−0.426503
$534$	0	0
$535$	−14.3402	−0.619981
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−25.5296	−1.09964
$540$	0	0
$541$	−9.30926	−0.400236	−0.200118	−	0.979772i	$-0.564133\pi$
$541$	−9.30926	−0.400236	−0.200118	+	0.979772i	$0.564133\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−5.35120	−0.229220
$546$	0	0
$547$	−30.2490	−1.29335	−0.646677	−	0.762764i	$-0.723842\pi$
$547$	−30.2490	−1.29335	−0.646677	+	0.762764i	$0.723842\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	28.4189	1.21068
$552$	0	0
$553$	14.0942	0.599347
$554$	0	0
$555$	0	0
$556$	0	0
$557$	14.6903	0.622450	0.311225	−	0.950336i	$-0.399261\pi$
$557$	14.6903	0.622450	0.311225	+	0.950336i	$0.399261\pi$
$558$	0	0
$559$	−4.63414	−0.196003
$560$	0	0
$561$	0	0
$562$	0	0
$563$	3.29714	0.138958	0.0694790	−	0.997583i	$-0.477866\pi$
$563$	3.29714	0.138958	0.0694790	+	0.997583i	$0.477866\pi$
$564$	0	0
$565$	2.68942	0.113145
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−23.9885	−1.00565	−0.502826	−	0.864388i	$-0.667706\pi$
$569$	−23.9885	−1.00565	−0.502826	+	0.864388i	$0.667706\pi$
$570$	0	0
$571$	−20.0806	−0.840349	−0.420174	−	0.907443i	$-0.638031\pi$
$571$	−20.0806	−0.840349	−0.420174	+	0.907443i	$0.638031\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−2.86774	−0.119593
$576$	0	0
$577$	22.8622	0.951764	0.475882	−	0.879509i	$-0.342129\pi$
$577$	22.8622	0.951764	0.475882	+	0.879509i	$0.342129\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	5.59069	0.231941
$582$	0	0
$583$	56.8408	2.35410
$584$	0	0
$585$	0	0
$586$	0	0
$587$	25.8893	1.06857	0.534284	−	0.845305i	$-0.320581\pi$
$587$	25.8893	1.06857	0.534284	+	0.845305i	$0.320581\pi$
$588$	0	0
$589$	11.3828	0.469020
$590$	0	0
$591$	0	0
$592$	0	0
$593$	39.7887	1.63392	0.816962	−	0.576691i	$-0.195656\pi$
$593$	39.7887	1.63392	0.816962	+	0.576691i	$0.195656\pi$
$594$	0	0
$595$	−6.78755	−0.278263
$596$	0	0
$597$	0	0
$598$	0	0
$599$	27.8868	1.13942	0.569712	−	0.821844i	$-0.307055\pi$
$599$	27.8868	1.13942	0.569712	+	0.821844i	$0.307055\pi$
$600$	0	0
$601$	27.1267	1.10652	0.553260	−	0.833008i	$-0.313383\pi$
$601$	27.1267	1.10652	0.553260	+	0.833008i	$0.313383\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	5.70088	0.231774
$606$	0	0
$607$	12.8617	0.522041	0.261020	−	0.965333i	$-0.415941\pi$
$607$	12.8617	0.522041	0.261020	+	0.965333i	$0.415941\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	1.68622	0.0682173
$612$	0	0
$613$	−29.2726	−1.18231	−0.591154	−	0.806558i	$-0.701328\pi$
$613$	−29.2726	−1.18231	−0.591154	+	0.806558i	$0.701328\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	34.7731	1.39991	0.699956	−	0.714186i	$-0.253203\pi$
$617$	34.7731	1.39991	0.699956	+	0.714186i	$0.253203\pi$
$618$	0	0
$619$	−35.4565	−1.42512	−0.712558	−	0.701613i	$-0.752464\pi$
$619$	−35.4565	−1.42512	−0.712558	+	0.701613i	$0.752464\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	8.85779	0.354880
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−4.09119	−0.163126
$630$	0	0
$631$	−6.68942	−0.266302	−0.133151	−	0.991096i	$-0.542509\pi$
$631$	−6.68942	−0.266302	−0.133151	+	0.991096i	$0.542509\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	21.5641	0.855745
$636$	0	0
$637$	−7.57419	−0.300100
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−4.93777	−0.195030	−0.0975151	−	0.995234i	$-0.531089\pi$
$641$	−4.93777	−0.195030	−0.0975151	+	0.995234i	$0.531089\pi$
$642$	0	0
$643$	15.2881	0.602904	0.301452	−	0.953481i	$-0.402529\pi$
$643$	15.2881	0.602904	0.301452	+	0.953481i	$0.402529\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	31.2304	1.22779	0.613897	−	0.789386i	$-0.289601\pi$
$647$	31.2304	1.22779	0.613897	+	0.789386i	$0.289601\pi$
$648$	0	0
$649$	−28.0927	−1.10273
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−7.65032	−0.299380	−0.149690	−	0.988733i	$-0.547828\pi$
$653$	−7.65032	−0.299380	−0.149690	+	0.988733i	$0.547828\pi$
$654$	0	0
$655$	−18.4941	−0.722623
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−41.9551	−1.63434	−0.817169	−	0.576399i	$-0.804458\pi$
$659$	−41.9551	−1.63434	−0.817169	+	0.576399i	$0.804458\pi$
$660$	0	0
$661$	40.5982	1.57909	0.789544	−	0.613694i	$-0.210317\pi$
$661$	40.5982	1.57909	0.789544	+	0.613694i	$0.210317\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−3.91484	−0.151811
$666$	0	0
$667$	−18.0642	−0.699449
$668$	0	0
$669$	0	0
$670$	0	0
$671$	37.1342	1.43355
$672$	0	0
$673$	14.8697	0.573185	0.286593	−	0.958053i	$-0.407477\pi$
$673$	14.8697	0.573185	0.286593	+	0.958053i	$0.407477\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−11.2254	−0.431428	−0.215714	−	0.976457i	$-0.569208\pi$
$677$	−11.2254	−0.431428	−0.215714	+	0.976457i	$0.569208\pi$
$678$	0	0
$679$	7.61010	0.292049
$680$	0	0
$681$	0	0
$682$	0	0
$683$	6.60275	0.252647	0.126324	−	0.991989i	$-0.459682\pi$
$683$	6.60275	0.252647	0.126324	+	0.991989i	$0.459682\pi$
$684$	0	0
$685$	9.72401	0.371535
$686$	0	0
$687$	0	0
$688$	0	0
$689$	16.8637	0.642455
$690$	0	0
$691$	−11.7485	−0.446932	−0.223466	−	0.974712i	$-0.571737\pi$
$691$	−11.7485	−0.446932	−0.223466	+	0.974712i	$0.571737\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	16.0806	0.609973
$696$	0	0
$697$	−63.5257	−2.40621
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−6.21396	−0.234698	−0.117349	−	0.993091i	$-0.537440\pi$
$701$	−6.21396	−0.234698	−0.117349	+	0.993091i	$0.537440\pi$
$702$	0	0
$703$	−2.35967	−0.0889965
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−1.96501	−0.0739019
$708$	0	0
$709$	−12.9895	−0.487829	−0.243915	−	0.969797i	$-0.578432\pi$
$709$	−12.9895	−0.487829	−0.243915	+	0.969797i	$0.578432\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−7.23537	−0.270967
$714$	0	0
$715$	4.95487	0.185302
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−15.4610	−0.576598	−0.288299	−	0.957540i	$-0.593090\pi$
$719$	−15.4610	−0.576598	−0.288299	+	0.957540i	$0.593090\pi$
$720$	0	0
$721$	−13.3508	−0.497210
$722$	0	0
$723$	0	0
$724$	0	0
$725$	6.29912	0.233943
$726$	0	0
$727$	−12.8518	−0.476645	−0.238323	−	0.971186i	$-0.576598\pi$
$727$	−12.8518	−0.476645	−0.238323	+	0.971186i	$0.576598\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−29.8974	−1.10579
$732$	0	0
$733$	−21.1273	−0.780354	−0.390177	−	0.920740i	$-0.627586\pi$
$733$	−21.1273	−0.780354	−0.390177	+	0.920740i	$0.627586\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−13.8101	−0.508701
$738$	0	0
$739$	18.8652	0.693968	0.346984	−	0.937871i	$-0.387206\pi$
$739$	18.8652	0.693968	0.346984	+	0.937871i	$0.387206\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	7.99503	0.293309	0.146655	−	0.989188i	$-0.453149\pi$
$743$	7.99503	0.293309	0.146655	+	0.989188i	$0.453149\pi$
$744$	0	0
$745$	15.8453	0.580526
$746$	0	0
$747$	0	0
$748$	0	0
$749$	12.4435	0.454676
$750$	0	0
$751$	30.4781	1.11216	0.556081	−	0.831128i	$-0.312305\pi$
$751$	30.4781	1.11216	0.556081	+	0.831128i	$0.312305\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	15.1213	0.550319
$756$	0	0
$757$	−30.4318	−1.10606	−0.553032	−	0.833160i	$-0.686529\pi$
$757$	−30.4318	−1.10606	−0.553032	+	0.833160i	$0.686529\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−46.7150	−1.69342	−0.846708	−	0.532057i	$-0.821419\pi$
$761$	−46.7150	−1.69342	−0.846708	+	0.532057i	$0.821419\pi$
$762$	0	0
$763$	4.64343	0.168103
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−8.33462	−0.300946
$768$	0	0
$769$	36.9895	1.33387	0.666937	−	0.745114i	$-0.267605\pi$
$769$	36.9895	1.33387	0.666937	+	0.745114i	$0.267605\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−33.7485	−1.21385	−0.606924	−	0.794760i	$-0.707597\pi$
$773$	−33.7485	−1.21385	−0.606924	+	0.794760i	$0.707597\pi$
$774$	0	0
$775$	2.52303	0.0906298
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−36.6396	−1.31275
$780$	0	0
$781$	−13.1282	−0.469764
$782$	0	0
$783$	0	0
$784$	0	0
$785$	17.5531	0.626497
$786$	0	0
$787$	41.9490	1.49532	0.747661	−	0.664081i	$-0.231177\pi$
$787$	41.9490	1.49532	0.747661	+	0.664081i	$0.231177\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−2.33371	−0.0829770
$792$	0	0
$793$	11.0171	0.391229
$794$	0	0
$795$	0	0
$796$	0	0
$797$	13.1333	0.465206	0.232603	−	0.972572i	$-0.425276\pi$
$797$	13.1333	0.465206	0.232603	+	0.972572i	$0.425276\pi$
$798$	0	0
$799$	10.8787	0.384862
$800$	0	0
$801$	0	0
$802$	0	0
$803$	35.1850	1.24165
$804$	0	0
$805$	2.48844	0.0877059
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−26.0015	−0.914165	−0.457082	−	0.889424i	$-0.651106\pi$
$809$	−26.0015	−0.914165	−0.457082	+	0.889424i	$0.651106\pi$
$810$	0	0
$811$	−14.9425	−0.524702	−0.262351	−	0.964973i	$-0.584498\pi$
$811$	−14.9425	−0.524702	−0.262351	+	0.964973i	$0.584498\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−6.85673	−0.240181
$816$	0	0
$817$	−17.2438	−0.603286
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−21.6620	−0.756008	−0.378004	−	0.925804i	$-0.623389\pi$
$821$	−21.6620	−0.756008	−0.378004	+	0.925804i	$0.623389\pi$
$822$	0	0
$823$	−39.9540	−1.39271	−0.696355	−	0.717698i	$-0.745196\pi$
$823$	−39.9540	−1.39271	−0.696355	+	0.717698i	$0.745196\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	7.04711	0.245052	0.122526	−	0.992465i	$-0.460901\pi$
$827$	7.04711	0.245052	0.122526	+	0.992465i	$0.460901\pi$
$828$	0	0
$829$	−7.21376	−0.250544	−0.125272	−	0.992122i	$-0.539980\pi$
$829$	−7.21376	−0.250544	−0.125272	+	0.992122i	$0.539980\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−48.8652	−1.69308
$834$	0	0
$835$	−3.91982	−0.135651
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−2.32676	−0.0803286	−0.0401643	−	0.999193i	$-0.512788\pi$
$839$	−2.32676	−0.0803286	−0.0401643	+	0.999193i	$0.512788\pi$
$840$	0	0
$841$	10.6789	0.368237
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−11.5300	−0.396643
$846$	0	0
$847$	−4.94686	−0.169976
$848$	0	0
$849$	0	0
$850$	0	0
$851$	1.49990	0.0514159
$852$	0	0
$853$	−36.7426	−1.25804	−0.629022	−	0.777388i	$-0.716544\pi$
$853$	−36.7426	−1.25804	−0.629022	+	0.777388i	$0.716544\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	8.91465	0.304519	0.152259	−	0.988341i	$-0.451345\pi$
$857$	8.91465	0.304519	0.152259	+	0.988341i	$0.451345\pi$
$858$	0	0
$859$	−38.9633	−1.32941	−0.664706	−	0.747105i	$-0.731443\pi$
$859$	−38.9633	−1.32941	−0.664706	+	0.747105i	$0.731443\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	15.2084	0.517700	0.258850	−	0.965918i	$-0.416657\pi$
$863$	15.2084	0.517700	0.258850	+	0.965918i	$0.416657\pi$
$864$	0	0
$865$	−8.17334	−0.277902
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−66.3778	−2.25171
$870$	0	0
$871$	−4.09722	−0.138829
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−0.867736	−0.0293348
$876$	0	0
$877$	−10.4318	−0.352258	−0.176129	−	0.984367i	$-0.556358\pi$
$877$	−10.4318	−0.352258	−0.176129	+	0.984367i	$0.556358\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	22.9272	0.772438	0.386219	−	0.922407i	$-0.373781\pi$
$881$	22.9272	0.772438	0.386219	+	0.922407i	$0.373781\pi$
$882$	0	0
$883$	−10.9773	−0.369417	−0.184708	−	0.982793i	$-0.559134\pi$
$883$	−10.9773	−0.369417	−0.184708	+	0.982793i	$0.559134\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	12.9699	0.435487	0.217744	−	0.976006i	$-0.430130\pi$
$887$	12.9699	0.435487	0.217744	+	0.976006i	$0.430130\pi$
$888$	0	0
$889$	−18.7119	−0.627579
$890$	0	0
$891$	0	0
$892$	0	0
$893$	6.27451	0.209968
$894$	0	0
$895$	6.85673	0.229195
$896$	0	0
$897$	0	0
$898$	0	0
$899$	15.8928	0.530056
$900$	0	0
$901$	108.797	3.62454
$902$	0	0
$903$	0	0
$904$	0	0
$905$	16.2991	0.541801
$906$	0	0
$907$	11.4835	0.381302	0.190651	−	0.981658i	$-0.438940\pi$
$907$	11.4835	0.381302	0.190651	+	0.981658i	$0.438940\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	15.2877	0.506503	0.253251	−	0.967401i	$-0.418500\pi$
$911$	15.2877	0.506503	0.253251	+	0.967401i	$0.418500\pi$
$912$	0	0
$913$	−26.3298	−0.871389
$914$	0	0
$915$	0	0
$916$	0	0
$917$	16.0480	0.529951
$918$	0	0
$919$	43.5890	1.43787	0.718934	−	0.695078i	$-0.244630\pi$
$919$	43.5890	1.43787	0.718934	+	0.695078i	$0.244630\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−3.89492	−0.128203
$924$	0	0
$925$	−0.523026	−0.0171970
$926$	0	0
$927$	0	0
$928$	0	0
$929$	1.48300	0.0486558	0.0243279	−	0.999704i	$-0.492255\pi$
$929$	1.48300	0.0486558	0.0243279	+	0.999704i	$0.492255\pi$
$930$	0	0
$931$	−28.1839	−0.923690
$932$	0	0
$933$	0	0
$934$	0	0
$935$	31.9665	1.04542
$936$	0	0
$937$	5.12821	0.167531	0.0837657	−	0.996485i	$-0.473305\pi$
$937$	5.12821	0.167531	0.0837657	+	0.996485i	$0.473305\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	46.8763	1.52812	0.764061	−	0.645144i	$-0.223203\pi$
$941$	46.8763	1.52812	0.764061	+	0.645144i	$0.223203\pi$
$942$	0	0
$943$	23.2896	0.758415
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−34.3701	−1.11688	−0.558439	−	0.829546i	$-0.688599\pi$
$947$	−34.3701	−1.11688	−0.558439	+	0.829546i	$0.688599\pi$
$948$	0	0
$949$	10.4388	0.338857
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−23.8092	−0.771254	−0.385627	−	0.922655i	$-0.626015\pi$
$953$	−23.8092	−0.771254	−0.385627	+	0.922655i	$0.626015\pi$
$954$	0	0
$955$	−11.8336	−0.382927
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−8.43787	−0.272473
$960$	0	0
$961$	−24.6343	−0.794656
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−1.82214	−0.0586569
$966$	0	0
$967$	−2.85779	−0.0919003	−0.0459501	−	0.998944i	$-0.514632\pi$
$967$	−2.85779	−0.0919003	−0.0459501	+	0.998944i	$0.514632\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−3.55821	−0.114188	−0.0570942	−	0.998369i	$-0.518184\pi$
$971$	−3.55821	−0.114188	−0.0570942	+	0.998369i	$0.518184\pi$
$972$	0	0
$973$	−13.9537	−0.447337
$974$	0	0
$975$	0	0
$976$	0	0
$977$	29.6960	0.950058	0.475029	−	0.879970i	$-0.342438\pi$
$977$	29.6960	0.950058	0.475029	+	0.879970i	$0.342438\pi$
$978$	0	0
$979$	−41.7165	−1.33326
$980$	0	0
$981$	0	0
$982$	0	0
$983$	58.1114	1.85346	0.926732	−	0.375722i	$-0.122605\pi$
$983$	58.1114	1.85346	0.926732	+	0.375722i	$0.122605\pi$
$984$	0	0
$985$	19.3798	0.617490
$986$	0	0
$987$	0	0
$988$	0	0
$989$	10.9609	0.348536
$990$	0	0
$991$	19.4489	0.617816	0.308908	−	0.951092i	$-0.400037\pi$
$991$	19.4489	0.617816	0.308908	+	0.951092i	$0.400037\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−6.69637	−0.212289
$996$	0	0
$997$	−8.50402	−0.269325	−0.134662	−	0.990892i	$-0.542995\pi$
$997$	−8.50402	−0.269325	−0.134662	+	0.990892i	$0.542995\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6480.2.a.cb.1.2		4
3.2	odd	2		6480.2.a.bz.1.2		4
4.3	odd	2		3240.2.a.u.1.3		4
9.2	odd	6		2160.2.q.l.1441.3		8
9.4	even	3		720.2.q.l.241.1		8
9.5	odd	6		2160.2.q.l.721.3		8
9.7	even	3		720.2.q.l.481.1		8
12.11	even	2		3240.2.a.s.1.3		4
36.7	odd	6		360.2.q.e.121.4	✓	8
36.11	even	6		1080.2.q.e.361.2		8
36.23	even	6		1080.2.q.e.721.2		8
36.31	odd	6		360.2.q.e.241.4	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
360.2.q.e.121.4	✓	8	36.7	odd	6
360.2.q.e.241.4	yes	8	36.31	odd	6
720.2.q.l.241.1		8	9.4	even	3
720.2.q.l.481.1		8	9.7	even	3
1080.2.q.e.361.2		8	36.11	even	6
1080.2.q.e.721.2		8	36.23	even	6
2160.2.q.l.721.3		8	9.5	odd	6
2160.2.q.l.1441.3		8	9.2	odd	6
3240.2.a.s.1.3		4	12.11	even	2
3240.2.a.u.1.3		4	4.3	odd	2
6480.2.a.bz.1.2		4	3.2	odd	2
6480.2.a.cb.1.2		4	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:	\( N \)	\(=\)	\( 6480 = 2^{4} \cdot 3^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6480.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{5} -0.867736 q^{7} +4.08667 q^{11} +1.21245 q^{13} +7.82214 q^{17} +4.51156 q^{19} -2.86774 q^{23} +1.00000 q^{25} +6.29912 q^{29} +2.52303 q^{31} -0.867736 q^{35} -0.523026 q^{37} -8.12126 q^{41} -3.82214 q^{43} +1.39076 q^{47} -6.24703 q^{49} +13.9088 q^{53} +4.08667 q^{55} -6.87423 q^{59} +9.08667 q^{61} +1.21245 q^{65} -3.37930 q^{67} -3.21245 q^{71} +8.60970 q^{73} -3.54615 q^{77} -16.2425 q^{79} -6.44284 q^{83} +7.82214 q^{85} -10.2079 q^{89} -1.05208 q^{91} +4.51156 q^{95} -8.77006 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{5} + q^{7} - q^{11} + 4 q^{13} + 5 q^{17} - q^{19} - 7 q^{23} + 4 q^{25} + 7 q^{29} + 2 q^{31} + q^{35} + 6 q^{37} + 12 q^{41} + 11 q^{43} - 7 q^{47} + 3 q^{49} + 12 q^{53} - q^{55} - 11 q^{59}+ \cdots + q^{97}+O(q^{100}) \) 4 * q + 4 * q^5 + q^7 - q^11 + 4 * q^13 + 5 * q^17 - q^19 - 7 * q^23 + 4 * q^25 + 7 * q^29 + 2 * q^31 + q^35 + 6 * q^37 + 12 * q^41 + 11 * q^43 - 7 * q^47 + 3 * q^49 + 12 * q^53 - q^55 - 11 * q^59 + 19 * q^61 + 4 * q^65 + 10 * q^67 - 12 * q^71 + 9 * q^73 + 32 * q^77 + 24 * q^79 - 23 * q^83 + 5 * q^85 + 21 * q^89 - 14 * q^91 - q^95 + q^97

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