LMFDB - Embedded newform 4680.2.a.bh.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4680,2,Mod(1,4680)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4680, 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("4680.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4680 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4680.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:	$37.3699881460$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.940.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 7x - 4 $ x^3 - 7*x - 4
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 1560)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2.29240$ of defining polynomial
Character	$\chi$	$=$	4680.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{5} -4.83991 q^{7} +O(q^{10})$	$q-1.00000 q^{5} -4.83991 q^{7} -2.25511 q^{11} -1.00000 q^{13} -6.32970 q^{17} -2.58480 q^{19} -4.83991 q^{23} +1.00000 q^{25} -9.09501 q^{29} -0.510210 q^{31} +4.83991 q^{35} +4.25511 q^{37} -0.255105 q^{41} +9.16961 q^{43} +4.51021 q^{47} +16.4247 q^{49} +9.93492 q^{53} +2.25511 q^{55} -14.1900 q^{59} +9.93492 q^{61} +1.00000 q^{65} -13.1696 q^{67} +5.74489 q^{71} +2.90499 q^{73} +10.9145 q^{77} +5.74489 q^{79} -10.1900 q^{83} +6.32970 q^{85} -3.74489 q^{89} +4.83991 q^{91} +2.58480 q^{95} -12.5197 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{5} + q^{7}+O(q^{10}) $ 3 * q - 3 * q^5 + q^7	$ 3 q - 3 q^{5} + q^{7} - 5 q^{11} - 3 q^{13} - 7 q^{17} + 6 q^{19} + q^{23} + 3 q^{25} - 10 q^{29} + 2 q^{31} - q^{35} + 11 q^{37} + q^{41} + 10 q^{47} + 20 q^{49} - 3 q^{53} + 5 q^{55} - 8 q^{59} - 3 q^{61} + 3 q^{65} - 12 q^{67} + 19 q^{71} + 26 q^{73} + 7 q^{77} + 19 q^{79} + 4 q^{83} + 7 q^{85} - 13 q^{89} - q^{91} - 6 q^{95} + 9 q^{97}+O(q^{100}) $ 3 * q - 3 * q^5 + q^7 - 5 * q^11 - 3 * q^13 - 7 * q^17 + 6 * q^19 + q^23 + 3 * q^25 - 10 * q^29 + 2 * q^31 - q^35 + 11 * q^37 + q^41 + 10 * q^47 + 20 * q^49 - 3 * q^53 + 5 * q^55 - 8 * q^59 - 3 * q^61 + 3 * q^65 - 12 * q^67 + 19 * q^71 + 26 * q^73 + 7 * q^77 + 19 * q^79 + 4 * q^83 + 7 * q^85 - 13 * q^89 - q^91 - 6 * q^95 + 9 * q^97

Display 100 coefficients 10 coefficients

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
$7$	−4.83991	−1.82931	−0.914657	−	0.404232i	$-0.867539\pi$
$7$	−4.83991	−1.82931	−0.914657	+	0.404232i	$0.867539\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−2.25511	−0.679940	−0.339970	−	0.940436i	$-0.610417\pi$
$11$	−2.25511	−0.679940	−0.339970	+	0.940436i	$0.610417\pi$
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−6.32970	−1.53518	−0.767589	−	0.640943i	$-0.778544\pi$
$17$	−6.32970	−1.53518	−0.767589	+	0.640943i	$0.778544\pi$
$18$	0	0
$19$	−2.58480	−0.592995	−0.296497	−	0.955034i	$-0.595819\pi$
$19$	−2.58480	−0.592995	−0.296497	+	0.955034i	$0.595819\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−4.83991	−1.00919	−0.504595	−	0.863356i	$-0.668358\pi$
$23$	−4.83991	−1.00919	−0.504595	+	0.863356i	$0.668358\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−9.09501	−1.68890	−0.844451	−	0.535633i	$-0.820073\pi$
$29$	−9.09501	−1.68890	−0.844451	+	0.535633i	$0.820073\pi$
$30$	0	0
$31$	−0.510210	−0.0916364	−0.0458182	−	0.998950i	$-0.514589\pi$
$31$	−0.510210	−0.0916364	−0.0458182	+	0.998950i	$0.514589\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	4.83991	0.818094
$36$	0	0
$37$	4.25511	0.699535	0.349767	−	0.936837i	$-0.386261\pi$
$37$	4.25511	0.699535	0.349767	+	0.936837i	$0.386261\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−0.255105	−0.0398407	−0.0199204	−	0.999802i	$-0.506341\pi$
$41$	−0.255105	−0.0398407	−0.0199204	+	0.999802i	$0.506341\pi$
$42$	0	0
$43$	9.16961	1.39835	0.699176	−	0.714950i	$-0.253550\pi$
$43$	9.16961	1.39835	0.699176	+	0.714950i	$0.253550\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	4.51021	0.657882	0.328941	−	0.944351i	$-0.393308\pi$
$47$	4.51021	0.657882	0.328941	+	0.944351i	$0.393308\pi$
$48$	0	0
$49$	16.4247	2.34639
$50$	0	0
$51$	0	0
$52$	0	0
$53$	9.93492	1.36467	0.682333	−	0.731041i	$-0.260965\pi$
$53$	9.93492	1.36467	0.682333	+	0.731041i	$0.260965\pi$
$54$	0	0
$55$	2.25511	0.304078
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−14.1900	−1.84738	−0.923692	−	0.383136i	$-0.874844\pi$
$59$	−14.1900	−1.84738	−0.923692	+	0.383136i	$0.874844\pi$
$60$	0	0
$61$	9.93492	1.27204	0.636018	−	0.771674i	$-0.280580\pi$
$61$	9.93492	1.27204	0.636018	+	0.771674i	$0.280580\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	1.00000	0.124035
$66$	0	0
$67$	−13.1696	−1.60892	−0.804462	−	0.594004i	$-0.797546\pi$
$67$	−13.1696	−1.60892	−0.804462	+	0.594004i	$0.797546\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	5.74489	0.681794	0.340897	−	0.940101i	$-0.389269\pi$
$71$	5.74489	0.681794	0.340897	+	0.940101i	$0.389269\pi$
$72$	0	0
$73$	2.90499	0.340003	0.170001	−	0.985444i	$-0.445623\pi$
$73$	2.90499	0.340003	0.170001	+	0.985444i	$0.445623\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	10.9145	1.24382
$78$	0	0
$79$	5.74489	0.646351	0.323176	−	0.946339i	$-0.395250\pi$
$79$	5.74489	0.646351	0.323176	+	0.946339i	$0.395250\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−10.1900	−1.11850	−0.559250	−	0.828999i	$-0.688911\pi$
$83$	−10.1900	−1.11850	−0.559250	+	0.828999i	$0.688911\pi$
$84$	0	0
$85$	6.32970	0.686552
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−3.74489	−0.396958	−0.198479	−	0.980105i	$-0.563600\pi$
$89$	−3.74489	−0.396958	−0.198479	+	0.980105i	$0.563600\pi$
$90$	0	0
$91$	4.83991	0.507360
$92$	0	0
$93$	0	0
$94$	0	0
$95$	2.58480	0.265195
$96$	0	0
$97$	−12.5197	−1.27119	−0.635593	−	0.772025i	$-0.719244\pi$
$97$	−12.5197	−1.27119	−0.635593	+	0.772025i	$0.719244\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	17.6052	1.75179	0.875893	−	0.482506i	$-0.160273\pi$
$101$	17.6052	1.75179	0.875893	+	0.482506i	$0.160273\pi$
$102$	0	0
$103$	−17.1696	−1.69177	−0.845886	−	0.533364i	$-0.820928\pi$
$103$	−17.1696	−1.69177	−0.845886	+	0.533364i	$0.820928\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−6.91450	−0.668450	−0.334225	−	0.942493i	$-0.608475\pi$
$107$	−6.91450	−0.668450	−0.334225	+	0.942493i	$0.608475\pi$
$108$	0	0
$109$	3.41520	0.327117	0.163558	−	0.986534i	$-0.447703\pi$
$109$	3.41520	0.327117	0.163558	+	0.986534i	$0.447703\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−5.09501	−0.479299	−0.239649	−	0.970860i	$-0.577032\pi$
$113$	−5.09501	−0.479299	−0.239649	+	0.970860i	$0.577032\pi$
$114$	0	0
$115$	4.83991	0.451324
$116$	0	0
$117$	0	0
$118$	0	0
$119$	30.6352	2.80832
$120$	0	0
$121$	−5.91450	−0.537682
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−12.6594	−1.12334	−0.561670	−	0.827361i	$-0.689841\pi$
$127$	−12.6594	−1.12334	−0.561670	+	0.827361i	$0.689841\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	16.7748	1.46562	0.732812	−	0.680431i	$-0.238208\pi$
$131$	16.7748	1.46562	0.732812	+	0.680431i	$0.238208\pi$
$132$	0	0
$133$	12.5102	1.08477
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−2.00000	−0.170872	−0.0854358	−	0.996344i	$-0.527228\pi$
$137$	−2.00000	−0.170872	−0.0854358	+	0.996344i	$0.527228\pi$
$138$	0	0
$139$	5.23468	0.444000	0.222000	−	0.975047i	$-0.428741\pi$
$139$	5.23468	0.444000	0.222000	+	0.975047i	$0.428741\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	2.25511	0.188581
$144$	0	0
$145$	9.09501	0.755300
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−3.74489	−0.306794	−0.153397	−	0.988165i	$-0.549021\pi$
$149$	−3.74489	−0.306794	−0.153397	+	0.988165i	$0.549021\pi$
$150$	0	0
$151$	15.3596	1.24995	0.624975	−	0.780645i	$-0.285109\pi$
$151$	15.3596	1.24995	0.624975	+	0.780645i	$0.285109\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0.510210	0.0409811
$156$	0	0
$157$	0.979580	0.0781790	0.0390895	−	0.999236i	$-0.487554\pi$
$157$	0.979580	0.0781790	0.0390895	+	0.999236i	$0.487554\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	23.4247	1.84613
$162$	0	0
$163$	7.42471	0.581548	0.290774	−	0.956792i	$-0.406087\pi$
$163$	7.42471	0.581548	0.290774	+	0.956792i	$0.406087\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	21.1696	1.63815	0.819077	−	0.573684i	$-0.194486\pi$
$167$	21.1696	1.63815	0.819077	+	0.573684i	$0.194486\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	0	0
$172$	0	0
$173$	6.00000	0.456172	0.228086	−	0.973641i	$-0.426753\pi$
$173$	6.00000	0.456172	0.228086	+	0.973641i	$0.426753\pi$
$174$	0	0
$175$	−4.83991	−0.365863
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−8.77483	−0.655862	−0.327931	−	0.944702i	$-0.606351\pi$
$179$	−8.77483	−0.655862	−0.327931	+	0.944702i	$0.606351\pi$
$180$	0	0
$181$	23.1045	1.71735	0.858673	−	0.512524i	$-0.171289\pi$
$181$	23.1045	1.71735	0.858673	+	0.512524i	$0.171289\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−4.25511	−0.312842
$186$	0	0
$187$	14.2741	1.04383
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−19.3596	−1.40081	−0.700407	−	0.713744i	$-0.746998\pi$
$191$	−19.3596	−1.40081	−0.700407	+	0.713744i	$0.746998\pi$
$192$	0	0
$193$	6.18051	0.444883	0.222441	−	0.974946i	$-0.428597\pi$
$193$	6.18051	0.444883	0.222441	+	0.974946i	$0.428597\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−21.8698	−1.55816	−0.779081	−	0.626923i	$-0.784314\pi$
$197$	−21.8698	−1.55816	−0.779081	+	0.626923i	$0.784314\pi$
$198$	0	0
$199$	2.83039	0.200641	0.100321	−	0.994955i	$-0.468013\pi$
$199$	2.83039	0.200641	0.100321	+	0.994955i	$0.468013\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	44.0190	3.08953
$204$	0	0
$205$	0.255105	0.0178173
$206$	0	0
$207$	0	0
$208$	0	0
$209$	5.82900	0.403201
$210$	0	0
$211$	−22.3392	−1.53789	−0.768947	−	0.639312i	$-0.779219\pi$
$211$	−22.3392	−1.53789	−0.768947	+	0.639312i	$0.779219\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−9.16961	−0.625362
$216$	0	0
$217$	2.46937	0.167632
$218$	0	0
$219$	0	0
$220$	0	0
$221$	6.32970	0.425782
$222$	0	0
$223$	−23.0950	−1.54656	−0.773278	−	0.634067i	$-0.781384\pi$
$223$	−23.0950	−1.54656	−0.773278	+	0.634067i	$0.781384\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	0.510210	0.0338638	0.0169319	−	0.999857i	$-0.494610\pi$
$227$	0.510210	0.0338638	0.0169319	+	0.999857i	$0.494610\pi$
$228$	0	0
$229$	−1.09501	−0.0723605	−0.0361803	−	0.999345i	$-0.511519\pi$
$229$	−1.09501	−0.0723605	−0.0361803	+	0.999345i	$0.511519\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−1.81949	−0.119199	−0.0595993	−	0.998222i	$-0.518982\pi$
$233$	−1.81949	−0.119199	−0.0595993	+	0.998222i	$0.518982\pi$
$234$	0	0
$235$	−4.51021	−0.294214
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−8.44513	−0.546270	−0.273135	−	0.961976i	$-0.588061\pi$
$239$	−8.44513	−0.546270	−0.273135	+	0.961976i	$0.588061\pi$
$240$	0	0
$241$	20.3392	1.31016	0.655082	−	0.755558i	$-0.272634\pi$
$241$	20.3392	1.31016	0.655082	+	0.755558i	$0.272634\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−16.4247	−1.04934
$246$	0	0
$247$	2.58480	0.164467
$248$	0	0
$249$	0	0
$250$	0	0
$251$	17.4342	1.10044	0.550219	−	0.835020i	$-0.314544\pi$
$251$	17.4342	1.10044	0.550219	+	0.835020i	$0.314544\pi$
$252$	0	0
$253$	10.9145	0.686189
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−15.4342	−0.962761	−0.481380	−	0.876512i	$-0.659864\pi$
$257$	−15.4342	−0.962761	−0.481380	+	0.876512i	$0.659864\pi$
$258$	0	0
$259$	−20.5943	−1.27967
$260$	0	0
$261$	0	0
$262$	0	0
$263$	8.90499	0.549105	0.274553	−	0.961572i	$-0.411470\pi$
$263$	8.90499	0.549105	0.274553	+	0.961572i	$0.411470\pi$
$264$	0	0
$265$	−9.93492	−0.610297
$266$	0	0
$267$	0	0
$268$	0	0
$269$	16.5848	1.01119	0.505597	−	0.862770i	$-0.331272\pi$
$269$	16.5848	1.01119	0.505597	+	0.862770i	$0.331272\pi$
$270$	0	0
$271$	17.8290	1.08303	0.541517	−	0.840690i	$-0.317850\pi$
$271$	17.8290	1.08303	0.541517	+	0.840690i	$0.317850\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−2.25511	−0.135988
$276$	0	0
$277$	22.3801	1.34469	0.672344	−	0.740239i	$-0.265288\pi$
$277$	22.3801	1.34469	0.672344	+	0.740239i	$0.265288\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	24.1900	1.44306	0.721528	−	0.692385i	$-0.243440\pi$
$281$	24.1900	1.44306	0.721528	+	0.692385i	$0.243440\pi$
$282$	0	0
$283$	−26.1900	−1.55684	−0.778418	−	0.627747i	$-0.783977\pi$
$283$	−26.1900	−1.55684	−0.778418	+	0.627747i	$0.783977\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	1.23468	0.0728811
$288$	0	0
$289$	23.0651	1.35677
$290$	0	0
$291$	0	0
$292$	0	0
$293$	5.48979	0.320717	0.160358	−	0.987059i	$-0.448735\pi$
$293$	5.48979	0.320717	0.160358	+	0.987059i	$0.448735\pi$
$294$	0	0
$295$	14.1900	0.826175
$296$	0	0
$297$	0	0
$298$	0	0
$299$	4.83991	0.279899
$300$	0	0
$301$	−44.3801	−2.55802
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−9.93492	−0.568872
$306$	0	0
$307$	−6.76532	−0.386117	−0.193058	−	0.981187i	$-0.561841\pi$
$307$	−6.76532	−0.386117	−0.193058	+	0.981187i	$0.561841\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	17.0204	0.965139	0.482570	−	0.875858i	$-0.339703\pi$
$311$	17.0204	0.965139	0.482570	+	0.875858i	$0.339703\pi$
$312$	0	0
$313$	12.8304	0.725217	0.362608	−	0.931942i	$-0.381886\pi$
$313$	12.8304	0.725217	0.362608	+	0.931942i	$0.381886\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	13.4898	0.757662	0.378831	−	0.925466i	$-0.376326\pi$
$317$	13.4898	0.757662	0.378831	+	0.925466i	$0.376326\pi$
$318$	0	0
$319$	20.5102	1.14835
$320$	0	0
$321$	0	0
$322$	0	0
$323$	16.3610	0.910352
$324$	0	0
$325$	−1.00000	−0.0554700
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−21.8290	−1.20347
$330$	0	0
$331$	−14.0746	−0.773610	−0.386805	−	0.922162i	$-0.626421\pi$
$331$	−14.0746	−0.773610	−0.386805	+	0.922162i	$0.626421\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	13.1696	0.719532
$336$	0	0
$337$	−22.6594	−1.23434	−0.617168	−	0.786831i	$-0.711720\pi$
$337$	−22.6594	−1.23434	−0.617168	+	0.786831i	$0.711720\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	1.15058	0.0623073
$342$	0	0
$343$	−45.6147	−2.46296
$344$	0	0
$345$	0	0
$346$	0	0
$347$	8.59432	0.461367	0.230684	−	0.973029i	$-0.425904\pi$
$347$	8.59432	0.461367	0.230684	+	0.973029i	$0.425904\pi$
$348$	0	0
$349$	−23.9444	−1.28172	−0.640858	−	0.767659i	$-0.721421\pi$
$349$	−23.9444	−1.28172	−0.640858	+	0.767659i	$0.721421\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	14.0000	0.745145	0.372572	−	0.928003i	$-0.378476\pi$
$353$	14.0000	0.745145	0.372572	+	0.928003i	$0.378476\pi$
$354$	0	0
$355$	−5.74489	−0.304907
$356$	0	0
$357$	0	0
$358$	0	0
$359$	1.80997	0.0955268	0.0477634	−	0.998859i	$-0.484791\pi$
$359$	1.80997	0.0955268	0.0477634	+	0.998859i	$0.484791\pi$
$360$	0	0
$361$	−12.3188	−0.648358
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−2.90499	−0.152054
$366$	0	0
$367$	9.16961	0.478650	0.239325	−	0.970940i	$-0.423074\pi$
$367$	9.16961	0.478650	0.239325	+	0.970940i	$0.423074\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−48.0841	−2.49640
$372$	0	0
$373$	−28.8494	−1.49377	−0.746883	−	0.664955i	$-0.768451\pi$
$373$	−28.8494	−1.49377	−0.746883	+	0.664955i	$0.768451\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	9.09501	0.468417
$378$	0	0
$379$	21.9444	1.12721	0.563605	−	0.826044i	$-0.309414\pi$
$379$	21.9444	1.12721	0.563605	+	0.826044i	$0.309414\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	6.32018	0.322946	0.161473	−	0.986877i	$-0.448375\pi$
$383$	6.32018	0.322946	0.161473	+	0.986877i	$0.448375\pi$
$384$	0	0
$385$	−10.9145	−0.556254
$386$	0	0
$387$	0	0
$388$	0	0
$389$	17.6052	0.892620	0.446310	−	0.894878i	$-0.352738\pi$
$389$	17.6052	0.892620	0.446310	+	0.894878i	$0.352738\pi$
$390$	0	0
$391$	30.6352	1.54929
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−5.74489	−0.289057
$396$	0	0
$397$	−30.0841	−1.50988	−0.754939	−	0.655795i	$-0.772334\pi$
$397$	−30.0841	−1.50988	−0.754939	+	0.655795i	$0.772334\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−22.5292	−1.12506	−0.562528	−	0.826778i	$-0.690171\pi$
$401$	−22.5292	−1.12506	−0.562528	+	0.826778i	$0.690171\pi$
$402$	0	0
$403$	0.510210	0.0254154
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−9.59571	−0.475642
$408$	0	0
$409$	23.1696	1.14566	0.572832	−	0.819673i	$-0.305845\pi$
$409$	23.1696	1.14566	0.572832	+	0.819673i	$0.305845\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	68.6784	3.37944
$414$	0	0
$415$	10.1900	0.500209
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−9.56438	−0.467251	−0.233625	−	0.972327i	$-0.575059\pi$
$419$	−9.56438	−0.467251	−0.233625	+	0.972327i	$0.575059\pi$
$420$	0	0
$421$	−35.7952	−1.74455	−0.872277	−	0.489012i	$-0.837357\pi$
$421$	−35.7952	−1.74455	−0.872277	+	0.489012i	$0.837357\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−6.32970	−0.307035
$426$	0	0
$427$	−48.0841	−2.32695
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−28.3801	−1.36702	−0.683510	−	0.729942i	$-0.739547\pi$
$431$	−28.3801	−1.36702	−0.683510	+	0.729942i	$0.739547\pi$
$432$	0	0
$433$	19.6798	0.945752	0.472876	−	0.881129i	$-0.343216\pi$
$433$	19.6798	0.945752	0.472876	+	0.881129i	$0.343216\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	12.5102	0.598445
$438$	0	0
$439$	33.7639	1.61146	0.805732	−	0.592280i	$-0.201772\pi$
$439$	33.7639	1.61146	0.805732	+	0.592280i	$0.201772\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−29.1045	−1.38280	−0.691399	−	0.722473i	$-0.743005\pi$
$443$	−29.1045	−1.38280	−0.691399	+	0.722473i	$0.743005\pi$
$444$	0	0
$445$	3.74489	0.177525
$446$	0	0
$447$	0	0
$448$	0	0
$449$	22.4642	1.06015	0.530075	−	0.847951i	$-0.322164\pi$
$449$	22.4642	1.06015	0.530075	+	0.847951i	$0.322164\pi$
$450$	0	0
$451$	0.575289	0.0270893
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−4.83991	−0.226898
$456$	0	0
$457$	−11.4993	−0.537915	−0.268957	−	0.963152i	$-0.586679\pi$
$457$	−11.4993	−0.537915	−0.268957	+	0.963152i	$0.586679\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−16.7843	−0.781725	−0.390862	−	0.920449i	$-0.627823\pi$
$461$	−16.7843	−0.781725	−0.390862	+	0.920449i	$0.627823\pi$
$462$	0	0
$463$	19.6893	0.915041	0.457520	−	0.889199i	$-0.348738\pi$
$463$	19.6893	0.915041	0.457520	+	0.889199i	$0.348738\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	1.38387	0.0640379	0.0320190	−	0.999487i	$-0.489806\pi$
$467$	1.38387	0.0640379	0.0320190	+	0.999487i	$0.489806\pi$
$468$	0	0
$469$	63.7397	2.94323
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−20.6784	−0.950795
$474$	0	0
$475$	−2.58480	−0.118599
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−28.9553	−1.32300	−0.661502	−	0.749944i	$-0.730081\pi$
$479$	−28.9553	−1.32300	−0.661502	+	0.749944i	$0.730081\pi$
$480$	0	0
$481$	−4.25511	−0.194016
$482$	0	0
$483$	0	0
$484$	0	0
$485$	12.5197	0.568491
$486$	0	0
$487$	−26.3705	−1.19496	−0.597482	−	0.801883i	$-0.703832\pi$
$487$	−26.3705	−1.19496	−0.597482	+	0.801883i	$0.703832\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−5.05417	−0.228092	−0.114046	−	0.993475i	$-0.536381\pi$
$491$	−5.05417	−0.228092	−0.114046	+	0.993475i	$0.536381\pi$
$492$	0	0
$493$	57.5687	2.59276
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−27.8048	−1.24721
$498$	0	0
$499$	−15.7544	−0.705264	−0.352632	−	0.935762i	$-0.614713\pi$
$499$	−15.7544	−0.705264	−0.352632	+	0.935762i	$0.614713\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	9.92541	0.442552	0.221276	−	0.975211i	$-0.428978\pi$
$503$	9.92541	0.442552	0.221276	+	0.975211i	$0.428978\pi$
$504$	0	0
$505$	−17.6052	−0.783422
$506$	0	0
$507$	0	0
$508$	0	0
$509$	25.4247	1.12693	0.563465	−	0.826140i	$-0.309468\pi$
$509$	25.4247	1.12693	0.563465	+	0.826140i	$0.309468\pi$
$510$	0	0
$511$	−14.0599	−0.621972
$512$	0	0
$513$	0	0
$514$	0	0
$515$	17.1696	0.756583
$516$	0	0
$517$	−10.1710	−0.447320
$518$	0	0
$519$	0	0
$520$	0	0
$521$	14.5292	0.636538	0.318269	−	0.948001i	$-0.396899\pi$
$521$	14.5292	0.636538	0.318269	+	0.948001i	$0.396899\pi$
$522$	0	0
$523$	−4.00000	−0.174908	−0.0874539	−	0.996169i	$-0.527873\pi$
$523$	−4.00000	−0.174908	−0.0874539	+	0.996169i	$0.527873\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	3.22948	0.140678
$528$	0	0
$529$	0.424711	0.0184657
$530$	0	0
$531$	0	0
$532$	0	0
$533$	0.255105	0.0110498
$534$	0	0
$535$	6.91450	0.298940
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−37.0394	−1.59540
$540$	0	0
$541$	6.11543	0.262923	0.131462	−	0.991321i	$-0.458033\pi$
$541$	6.11543	0.262923	0.131462	+	0.991321i	$0.458033\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−3.41520	−0.146291
$546$	0	0
$547$	15.3596	0.656730	0.328365	−	0.944551i	$-0.393502\pi$
$547$	15.3596	0.656730	0.328365	+	0.944551i	$0.393502\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	23.5088	1.00151
$552$	0	0
$553$	−27.8048	−1.18238
$554$	0	0
$555$	0	0
$556$	0	0
$557$	22.3801	0.948273	0.474137	−	0.880451i	$-0.342760\pi$
$557$	22.3801	0.948273	0.474137	+	0.880451i	$0.342760\pi$
$558$	0	0
$559$	−9.16961	−0.387833
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−17.7449	−0.747858	−0.373929	−	0.927457i	$-0.621990\pi$
$563$	−17.7449	−0.747858	−0.373929	+	0.927457i	$0.621990\pi$
$564$	0	0
$565$	5.09501	0.214349
$566$	0	0
$567$	0	0
$568$	0	0
$569$	10.0190	0.420020	0.210010	−	0.977699i	$-0.432650\pi$
$569$	10.0190	0.420020	0.210010	+	0.977699i	$0.432650\pi$
$570$	0	0
$571$	−17.6147	−0.737154	−0.368577	−	0.929597i	$-0.620155\pi$
$571$	−17.6147	−0.737154	−0.368577	+	0.929597i	$0.620155\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−4.83991	−0.201838
$576$	0	0
$577$	−15.3501	−0.639034	−0.319517	−	0.947581i	$-0.603521\pi$
$577$	−15.3501	−0.639034	−0.319517	+	0.947581i	$0.603521\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	49.3188	2.04609
$582$	0	0
$583$	−22.4043	−0.927891
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−23.4898	−0.969527	−0.484764	−	0.874645i	$-0.661094\pi$
$587$	−23.4898	−0.969527	−0.484764	+	0.874645i	$0.661094\pi$
$588$	0	0
$589$	1.31879	0.0543399
$590$	0	0
$591$	0	0
$592$	0	0
$593$	5.63898	0.231565	0.115782	−	0.993275i	$-0.463062\pi$
$593$	5.63898	0.231565	0.115782	+	0.993275i	$0.463062\pi$
$594$	0	0
$595$	−30.6352	−1.25592
$596$	0	0
$597$	0	0
$598$	0	0
$599$	10.0408	0.410258	0.205129	−	0.978735i	$-0.434239\pi$
$599$	10.0408	0.410258	0.205129	+	0.978735i	$0.434239\pi$
$600$	0	0
$601$	5.04466	0.205776	0.102888	−	0.994693i	$-0.467192\pi$
$601$	5.04466	0.205776	0.102888	+	0.994693i	$0.467192\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	5.91450	0.240459
$606$	0	0
$607$	−2.19003	−0.0888904	−0.0444452	−	0.999012i	$-0.514152\pi$
$607$	−2.19003	−0.0888904	−0.0444452	+	0.999012i	$0.514152\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−4.51021	−0.182464
$612$	0	0
$613$	45.4437	1.83546	0.917728	−	0.397210i	$-0.130022\pi$
$613$	45.4437	1.83546	0.917728	+	0.397210i	$0.130022\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	7.31879	0.294643	0.147322	−	0.989089i	$-0.452935\pi$
$617$	7.31879	0.294643	0.147322	+	0.989089i	$0.452935\pi$
$618$	0	0
$619$	20.6256	0.829015	0.414507	−	0.910046i	$-0.363954\pi$
$619$	20.6256	0.829015	0.414507	+	0.910046i	$0.363954\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	18.1249	0.726161
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−26.9335	−1.07391
$630$	0	0
$631$	14.8304	0.590389	0.295194	−	0.955437i	$-0.404616\pi$
$631$	14.8304	0.590389	0.295194	+	0.955437i	$0.404616\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	12.6594	0.502373
$636$	0	0
$637$	−16.4247	−0.650771
$638$	0	0
$639$	0	0
$640$	0	0
$641$	28.8494	1.13948	0.569742	−	0.821824i	$-0.307043\pi$
$641$	28.8494	1.13948	0.569742	+	0.821824i	$0.307043\pi$
$642$	0	0
$643$	−29.7449	−1.17302	−0.586512	−	0.809940i	$-0.699499\pi$
$643$	−29.7449	−1.17302	−0.586512	+	0.809940i	$0.699499\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−2.13967	−0.0841192	−0.0420596	−	0.999115i	$-0.513392\pi$
$647$	−2.13967	−0.0841192	−0.0420596	+	0.999115i	$0.513392\pi$
$648$	0	0
$649$	32.0000	1.25611
$650$	0	0
$651$	0	0
$652$	0	0
$653$	13.2104	0.516965	0.258482	−	0.966016i	$-0.416778\pi$
$653$	13.2104	0.516965	0.258482	+	0.966016i	$0.416778\pi$
$654$	0	0
$655$	−16.7748	−0.655447
$656$	0	0
$657$	0	0
$658$	0	0
$659$	48.6447	1.89493	0.947464	−	0.319863i	$-0.103637\pi$
$659$	48.6447	1.89493	0.947464	+	0.319863i	$0.103637\pi$
$660$	0	0
$661$	−24.7340	−0.962041	−0.481020	−	0.876709i	$-0.659734\pi$
$661$	−24.7340	−0.962041	−0.481020	+	0.876709i	$0.659734\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−12.5102	−0.485125
$666$	0	0
$667$	44.0190	1.70442
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−22.4043	−0.864908
$672$	0	0
$673$	39.6988	1.53028	0.765139	−	0.643865i	$-0.222670\pi$
$673$	39.6988	1.53028	0.765139	+	0.643865i	$0.222670\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−28.5535	−1.09740	−0.548700	−	0.836020i	$-0.684877\pi$
$677$	−28.5535	−1.09740	−0.548700	+	0.836020i	$0.684877\pi$
$678$	0	0
$679$	60.5943	2.32540
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−15.8508	−0.606515	−0.303257	−	0.952909i	$-0.598074\pi$
$683$	−15.8508	−0.606515	−0.303257	+	0.952909i	$0.598074\pi$
$684$	0	0
$685$	2.00000	0.0764161
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−9.93492	−0.378490
$690$	0	0
$691$	−28.6256	−1.08897	−0.544485	−	0.838770i	$-0.683275\pi$
$691$	−28.6256	−1.08897	−0.544485	+	0.838770i	$0.683275\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−5.23468	−0.198563
$696$	0	0
$697$	1.61474	0.0611626
$698$	0	0
$699$	0	0
$700$	0	0
$701$	8.58480	0.324244	0.162122	−	0.986771i	$-0.448166\pi$
$701$	8.58480	0.324244	0.162122	+	0.986771i	$0.448166\pi$
$702$	0	0
$703$	−10.9986	−0.414820
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−85.2077	−3.20456
$708$	0	0
$709$	−3.56438	−0.133863	−0.0669316	−	0.997758i	$-0.521321\pi$
$709$	−3.56438	−0.133863	−0.0669316	+	0.997758i	$0.521321\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	2.46937	0.0924786
$714$	0	0
$715$	−2.25511	−0.0843361
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−22.1900	−0.827548	−0.413774	−	0.910380i	$-0.635790\pi$
$719$	−22.1900	−0.827548	−0.413774	+	0.910380i	$0.635790\pi$
$720$	0	0
$721$	83.0993	3.09478
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−9.09501	−0.337780
$726$	0	0
$727$	9.82900	0.364538	0.182269	−	0.983249i	$-0.441656\pi$
$727$	9.82900	0.364538	0.182269	+	0.983249i	$0.441656\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−58.0408	−2.14672
$732$	0	0
$733$	−29.0637	−1.07349	−0.536746	−	0.843744i	$-0.680347\pi$
$733$	−29.0637	−1.07349	−0.536746	+	0.843744i	$0.680347\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	29.6988	1.09397
$738$	0	0
$739$	28.9240	1.06399	0.531994	−	0.846748i	$-0.321443\pi$
$739$	28.9240	1.06399	0.531994	+	0.846748i	$0.321443\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	20.0190	0.734427	0.367213	−	0.930137i	$-0.380312\pi$
$743$	20.0190	0.734427	0.367213	+	0.930137i	$0.380312\pi$
$744$	0	0
$745$	3.74489	0.137202
$746$	0	0
$747$	0	0
$748$	0	0
$749$	33.4656	1.22280
$750$	0	0
$751$	−22.2741	−0.812795	−0.406397	−	0.913696i	$-0.633215\pi$
$751$	−22.2741	−0.812795	−0.406397	+	0.913696i	$0.633215\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−15.3596	−0.558994
$756$	0	0
$757$	−23.1886	−0.842805	−0.421403	−	0.906874i	$-0.638462\pi$
$757$	−23.1886	−0.842805	−0.421403	+	0.906874i	$0.638462\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−47.5497	−1.72367	−0.861837	−	0.507186i	$-0.830686\pi$
$761$	−47.5497	−1.72367	−0.861837	+	0.507186i	$0.830686\pi$
$762$	0	0
$763$	−16.5292	−0.598399
$764$	0	0
$765$	0	0
$766$	0	0
$767$	14.1900	0.512372
$768$	0	0
$769$	−21.8698	−0.788647	−0.394323	−	0.918972i	$-0.629021\pi$
$769$	−21.8698	−0.788647	−0.394323	+	0.918972i	$0.629021\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	8.84942	0.318292	0.159146	−	0.987255i	$-0.449126\pi$
$773$	8.84942	0.318292	0.159146	+	0.987255i	$0.449126\pi$
$774$	0	0
$775$	−0.510210	−0.0183273
$776$	0	0
$777$	0	0
$778$	0	0
$779$	0.659396	0.0236253
$780$	0	0
$781$	−12.9553	−0.463579
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−0.979580	−0.0349627
$786$	0	0
$787$	9.02042	0.321543	0.160772	−	0.986992i	$-0.448602\pi$
$787$	9.02042	0.321543	0.160772	+	0.986992i	$0.448602\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	24.6594	0.876787
$792$	0	0
$793$	−9.93492	−0.352799
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−9.55487	−0.338451	−0.169225	−	0.985577i	$-0.554127\pi$
$797$	−9.55487	−0.338451	−0.169225	+	0.985577i	$0.554127\pi$
$798$	0	0
$799$	−28.5483	−1.00997
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−6.55105	−0.231182
$804$	0	0
$805$	−23.4247	−0.825613
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−53.3596	−1.87602	−0.938012	−	0.346602i	$-0.887336\pi$
$809$	−53.3596	−1.87602	−0.938012	+	0.346602i	$0.887336\pi$
$810$	0	0
$811$	10.9458	0.384360	0.192180	−	0.981360i	$-0.438444\pi$
$811$	10.9458	0.384360	0.192180	+	0.981360i	$0.438444\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−7.42471	−0.260076
$816$	0	0
$817$	−23.7016	−0.829215
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−35.6147	−1.24296	−0.621481	−	0.783429i	$-0.713469\pi$
$821$	−35.6147	−1.24296	−0.621481	+	0.783429i	$0.713469\pi$
$822$	0	0
$823$	10.1900	0.355202	0.177601	−	0.984103i	$-0.443166\pi$
$823$	10.1900	0.355202	0.177601	+	0.984103i	$0.443166\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	25.0394	0.870707	0.435353	−	0.900260i	$-0.356623\pi$
$827$	25.0394	0.870707	0.435353	+	0.900260i	$0.356623\pi$
$828$	0	0
$829$	−36.3392	−1.26211	−0.631057	−	0.775737i	$-0.717378\pi$
$829$	−36.3392	−1.26211	−0.631057	+	0.775737i	$0.717378\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−103.963	−3.60212
$834$	0	0
$835$	−21.1696	−0.732604
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−38.4043	−1.32586	−0.662932	−	0.748680i	$-0.730688\pi$
$839$	−38.4043	−1.32586	−0.662932	+	0.748680i	$0.730688\pi$
$840$	0	0
$841$	53.7193	1.85239
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−1.00000	−0.0344010
$846$	0	0
$847$	28.6256	0.983589
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−20.5943	−0.705964
$852$	0	0
$853$	22.7245	0.778071	0.389036	−	0.921223i	$-0.372808\pi$
$853$	22.7245	0.778071	0.389036	+	0.921223i	$0.372808\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	10.3297	0.352856	0.176428	−	0.984314i	$-0.443546\pi$
$857$	10.3297	0.352856	0.176428	+	0.984314i	$0.443546\pi$
$858$	0	0
$859$	2.40429	0.0820334	0.0410167	−	0.999158i	$-0.486940\pi$
$859$	2.40429	0.0820334	0.0410167	+	0.999158i	$0.486940\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−17.6798	−0.601828	−0.300914	−	0.953651i	$-0.597292\pi$
$863$	−17.6798	−0.601828	−0.300914	+	0.953651i	$0.597292\pi$
$864$	0	0
$865$	−6.00000	−0.204006
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−12.9553	−0.439480
$870$	0	0
$871$	13.1696	0.446235
$872$	0	0
$873$	0	0
$874$	0	0
$875$	4.83991	0.163619
$876$	0	0
$877$	50.7601	1.71405	0.857023	−	0.515277i	$-0.172311\pi$
$877$	50.7601	1.71405	0.857023	+	0.515277i	$0.172311\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	49.5905	1.67075	0.835373	−	0.549683i	$-0.185252\pi$
$881$	49.5905	1.67075	0.835373	+	0.549683i	$0.185252\pi$
$882$	0	0
$883$	21.3188	0.717434	0.358717	−	0.933446i	$-0.383214\pi$
$883$	21.3188	0.717434	0.358717	+	0.933446i	$0.383214\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	16.9891	0.570438	0.285219	−	0.958462i	$-0.407934\pi$
$887$	16.9891	0.570438	0.285219	+	0.958462i	$0.407934\pi$
$888$	0	0
$889$	61.2703	2.05494
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−11.6580	−0.390120
$894$	0	0
$895$	8.77483	0.293310
$896$	0	0
$897$	0	0
$898$	0	0
$899$	4.64037	0.154765
$900$	0	0
$901$	−62.8851	−2.09500
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−23.1045	−0.768020
$906$	0	0
$907$	−36.6594	−1.21726	−0.608628	−	0.793456i	$-0.708280\pi$
$907$	−36.6594	−1.21726	−0.608628	+	0.793456i	$0.708280\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−22.1900	−0.735188	−0.367594	−	0.929986i	$-0.619818\pi$
$911$	−22.1900	−0.735188	−0.367594	+	0.929986i	$0.619818\pi$
$912$	0	0
$913$	22.9796	0.760513
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−81.1886	−2.68108
$918$	0	0
$919$	25.5957	0.844325	0.422162	−	0.906520i	$-0.361271\pi$
$919$	25.5957	0.844325	0.422162	+	0.906520i	$0.361271\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−5.74489	−0.189096
$924$	0	0
$925$	4.25511	0.139907
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−9.40568	−0.308590	−0.154295	−	0.988025i	$-0.549311\pi$
$929$	−9.40568	−0.308590	−0.154295	+	0.988025i	$0.549311\pi$
$930$	0	0
$931$	−42.4546	−1.39139
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−14.2741	−0.466814
$936$	0	0
$937$	50.7601	1.65826	0.829130	−	0.559056i	$-0.188836\pi$
$937$	50.7601	1.65826	0.829130	+	0.559056i	$0.188836\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	30.5943	0.997346	0.498673	−	0.866790i	$-0.333821\pi$
$941$	30.5943	0.997346	0.498673	+	0.866790i	$0.333821\pi$
$942$	0	0
$943$	1.23468	0.0402069
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−7.12877	−0.231654	−0.115827	−	0.993269i	$-0.536952\pi$
$947$	−7.12877	−0.231654	−0.115827	+	0.993269i	$0.536952\pi$
$948$	0	0
$949$	−2.90499	−0.0942999
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−3.99049	−0.129265	−0.0646323	−	0.997909i	$-0.520587\pi$
$953$	−3.99049	−0.129265	−0.0646323	+	0.997909i	$0.520587\pi$
$954$	0	0
$955$	19.3596	0.626463
$956$	0	0
$957$	0	0
$958$	0	0
$959$	9.67982	0.312578
$960$	0	0
$961$	−30.7397	−0.991603
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−6.18051	−0.198958
$966$	0	0
$967$	27.4751	0.883539	0.441769	−	0.897129i	$-0.354351\pi$
$967$	27.4751	0.883539	0.441769	+	0.897129i	$0.354351\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−29.5834	−0.949377	−0.474688	−	0.880154i	$-0.657439\pi$
$971$	−29.5834	−0.949377	−0.474688	+	0.880154i	$0.657439\pi$
$972$	0	0
$973$	−25.3354	−0.812215
$974$	0	0
$975$	0	0
$976$	0	0
$977$	1.72066	0.0550487	0.0275243	−	0.999621i	$-0.491238\pi$
$977$	1.72066	0.0550487	0.0275243	+	0.999621i	$0.491238\pi$
$978$	0	0
$979$	8.44513	0.269908
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−27.8889	−0.889517	−0.444758	−	0.895651i	$-0.646710\pi$
$983$	−27.8889	−0.889517	−0.444758	+	0.895651i	$0.646710\pi$
$984$	0	0
$985$	21.8698	0.696831
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−44.3801	−1.41120
$990$	0	0
$991$	45.6147	1.44900	0.724500	−	0.689275i	$-0.242071\pi$
$991$	45.6147	1.44900	0.724500	+	0.689275i	$0.242071\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−2.83039	−0.0897295
$996$	0	0
$997$	−19.8290	−0.627991	−0.313995	−	0.949425i	$-0.601668\pi$
$997$	−19.8290	−0.627991	−0.313995	+	0.949425i	$0.601668\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4680.2.a.bh.1.1		3
3.2	odd	2		1560.2.a.q.1.1	✓	3
4.3	odd	2		9360.2.a.cy.1.3		3
12.11	even	2		3120.2.a.bi.1.3		3
15.14	odd	2		7800.2.a.bi.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1560.2.a.q.1.1	✓	3	3.2	odd	2
3120.2.a.bi.1.3		3	12.11	even	2
4680.2.a.bh.1.1		3	1.1	even	1	trivial
7800.2.a.bi.1.3		3	15.14	odd	2
9360.2.a.cy.1.3		3	4.3	odd	2

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:	\( N \)	\(=\)	\( 4680 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4680.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{5} -4.83991 q^{7} +O(q^{10})\)	\(q-1.00000 q^{5} -4.83991 q^{7} -2.25511 q^{11} -1.00000 q^{13} -6.32970 q^{17} -2.58480 q^{19} -4.83991 q^{23} +1.00000 q^{25} -9.09501 q^{29} -0.510210 q^{31} +4.83991 q^{35} +4.25511 q^{37} -0.255105 q^{41} +9.16961 q^{43} +4.51021 q^{47} +16.4247 q^{49} +9.93492 q^{53} +2.25511 q^{55} -14.1900 q^{59} +9.93492 q^{61} +1.00000 q^{65} -13.1696 q^{67} +5.74489 q^{71} +2.90499 q^{73} +10.9145 q^{77} +5.74489 q^{79} -10.1900 q^{83} +6.32970 q^{85} -3.74489 q^{89} +4.83991 q^{91} +2.58480 q^{95} -12.5197 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{5} + q^{7}+O(q^{10}) \) 3 * q - 3 * q^5 + q^7	\( 3 q - 3 q^{5} + q^{7} - 5 q^{11} - 3 q^{13} - 7 q^{17} + 6 q^{19} + q^{23} + 3 q^{25} - 10 q^{29} + 2 q^{31} - q^{35} + 11 q^{37} + q^{41} + 10 q^{47} + 20 q^{49} - 3 q^{53} + 5 q^{55} - 8 q^{59} - 3 q^{61} + 3 q^{65} - 12 q^{67} + 19 q^{71} + 26 q^{73} + 7 q^{77} + 19 q^{79} + 4 q^{83} + 7 q^{85} - 13 q^{89} - q^{91} - 6 q^{95} + 9 q^{97}+O(q^{100}) \) 3 * q - 3 * q^5 + q^7 - 5 * q^11 - 3 * q^13 - 7 * q^17 + 6 * q^19 + q^23 + 3 * q^25 - 10 * q^29 + 2 * q^31 - q^35 + 11 * q^37 + q^41 + 10 * q^47 + 20 * q^49 - 3 * q^53 + 5 * q^55 - 8 * q^59 - 3 * q^61 + 3 * q^65 - 12 * q^67 + 19 * q^71 + 26 * q^73 + 7 * q^77 + 19 * q^79 + 4 * q^83 + 7 * q^85 - 13 * q^89 - q^91 - 6 * q^95 + 9 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more