LMFDB - Embedded newform 5520.2.a.cb.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5520,2,Mod(1,5520)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5520, 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("5520.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5520 = 2^{4} \cdot 3 \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5520.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:	$44.0774219157$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.54764.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 9x^{2} + 3x + 2 $ x^4 - x^3 - 9x^2 + 3x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 2760)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-2.67673$ of defining polynomial
Character	$\chi$	$=$	5520.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} -1.00000 q^{5} +4.13277 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -1.00000 q^{5} +4.13277 q^{7} +1.00000 q^{9} -2.92955 q^{11} +2.42391 q^{13} -1.00000 q^{15} -4.15025 q^{17} -3.84163 q^{19} +4.13277 q^{21} -1.00000 q^{23} +1.00000 q^{25} +1.00000 q^{27} +5.64461 q^{29} +3.22069 q^{31} -2.92955 q^{33} -4.13277 q^{35} +5.55049 q^{37} +2.42391 q^{39} -5.06232 q^{41} +7.41772 q^{43} -1.00000 q^{45} +3.84163 q^{47} +10.0798 q^{49} -4.15025 q^{51} +8.48004 q^{53} +2.92955 q^{55} -3.84163 q^{57} -3.64461 q^{59} -1.51183 q^{61} +4.13277 q^{63} -2.42391 q^{65} +13.0448 q^{67} -1.00000 q^{69} +7.06232 q^{71} -1.57609 q^{73} +1.00000 q^{75} -12.1072 q^{77} -0.329796 q^{79} +1.00000 q^{81} +13.5680 q^{83} +4.15025 q^{85} +5.64461 q^{87} +1.08792 q^{89} +10.0175 q^{91} +3.22069 q^{93} +3.84163 q^{95} +0.441388 q^{97} -2.92955 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{3} - 4 q^{5} + 4 q^{9}+O(q^{10}) $ 4 * q + 4 * q^3 - 4 * q^5 + 4 * q^9	$ 4 q + 4 q^{3} - 4 q^{5} + 4 q^{9} - 2 q^{13} - 4 q^{15} + 2 q^{17} + 6 q^{19} - 4 q^{23} + 4 q^{25} + 4 q^{27} + 4 q^{29} + 6 q^{31} - 4 q^{37} - 2 q^{39} + 8 q^{41} + 20 q^{43} - 4 q^{45} - 6 q^{47} + 10 q^{49} + 2 q^{51} - 4 q^{53} + 6 q^{57} + 4 q^{59} - 4 q^{61} + 2 q^{65} + 26 q^{67} - 4 q^{69} - 18 q^{73} + 4 q^{75} + 6 q^{77} + 18 q^{79} + 4 q^{81} + 26 q^{83} - 2 q^{85} + 4 q^{87} + 14 q^{89} + 38 q^{91} + 6 q^{93} - 6 q^{95} - 12 q^{97}+O(q^{100}) $ 4 * q + 4 * q^3 - 4 * q^5 + 4 * q^9 - 2 * q^13 - 4 * q^15 + 2 * q^17 + 6 * q^19 - 4 * q^23 + 4 * q^25 + 4 * q^27 + 4 * q^29 + 6 * q^31 - 4 * q^37 - 2 * q^39 + 8 * q^41 + 20 * q^43 - 4 * q^45 - 6 * q^47 + 10 * q^49 + 2 * q^51 - 4 * q^53 + 6 * q^57 + 4 * q^59 - 4 * q^61 + 2 * q^65 + 26 * q^67 - 4 * q^69 - 18 * q^73 + 4 * q^75 + 6 * q^77 + 18 * q^79 + 4 * q^81 + 26 * q^83 - 2 * q^85 + 4 * q^87 + 14 * q^89 + 38 * q^91 + 6 * q^93 - 6 * q^95 - 12 * 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
$3$	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
$7$	4.13277	1.56204	0.781020	−	0.624506i	$-0.214699\pi$
$7$	4.13277	1.56204	0.781020	+	0.624506i	$0.214699\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−2.92955	−0.883294	−0.441647	−	0.897189i	$-0.645606\pi$
$11$	−2.92955	−0.883294	−0.441647	+	0.897189i	$0.645606\pi$
$12$	0	0
$13$	2.42391	0.672272	0.336136	−	0.941813i	$-0.390880\pi$
$13$	2.42391	0.672272	0.336136	+	0.941813i	$0.390880\pi$
$14$	0	0
$15$	−1.00000	−0.258199
$16$	0	0
$17$	−4.15025	−1.00658	−0.503291	−	0.864117i	$-0.667878\pi$
$17$	−4.15025	−1.00658	−0.503291	+	0.864117i	$0.667878\pi$
$18$	0	0
$19$	−3.84163	−0.881331	−0.440665	−	0.897672i	$-0.645257\pi$
$19$	−3.84163	−0.881331	−0.440665	+	0.897672i	$0.645257\pi$
$20$	0	0
$21$	4.13277	0.901845
$22$	0	0
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	5.64461	1.04818	0.524089	−	0.851664i	$-0.324406\pi$
$29$	5.64461	1.04818	0.524089	+	0.851664i	$0.324406\pi$
$30$	0	0
$31$	3.22069	0.578454	0.289227	−	0.957261i	$-0.406602\pi$
$31$	3.22069	0.578454	0.289227	+	0.957261i	$0.406602\pi$
$32$	0	0
$33$	−2.92955	−0.509970
$34$	0	0
$35$	−4.13277	−0.698566
$36$	0	0
$37$	5.55049	0.912495	0.456247	−	0.889853i	$-0.349193\pi$
$37$	5.55049	0.912495	0.456247	+	0.889853i	$0.349193\pi$
$38$	0	0
$39$	2.42391	0.388137
$40$	0	0
$41$	−5.06232	−0.790602	−0.395301	−	0.918552i	$-0.629360\pi$
$41$	−5.06232	−0.790602	−0.395301	+	0.918552i	$0.629360\pi$
$42$	0	0
$43$	7.41772	1.13119	0.565596	−	0.824683i	$-0.308646\pi$
$43$	7.41772	1.13119	0.565596	+	0.824683i	$0.308646\pi$
$44$	0	0
$45$	−1.00000	−0.149071
$46$	0	0
$47$	3.84163	0.560359	0.280180	−	0.959948i	$-0.409606\pi$
$47$	3.84163	0.560359	0.280180	+	0.959948i	$0.409606\pi$
$48$	0	0
$49$	10.0798	1.43997
$50$	0	0
$51$	−4.15025	−0.581151
$52$	0	0
$53$	8.48004	1.16482	0.582412	−	0.812894i	$-0.302109\pi$
$53$	8.48004	1.16482	0.582412	+	0.812894i	$0.302109\pi$
$54$	0	0
$55$	2.92955	0.395021
$56$	0	0
$57$	−3.84163	−0.508836
$58$	0	0
$59$	−3.64461	−0.474487	−0.237244	−	0.971450i	$-0.576244\pi$
$59$	−3.64461	−0.474487	−0.237244	+	0.971450i	$0.576244\pi$
$60$	0	0
$61$	−1.51183	−0.193571	−0.0967853	−	0.995305i	$-0.530856\pi$
$61$	−1.51183	−0.193571	−0.0967853	+	0.995305i	$0.530856\pi$
$62$	0	0
$63$	4.13277	0.520680
$64$	0	0
$65$	−2.42391	−0.300649
$66$	0	0
$67$	13.0448	1.59368	0.796841	−	0.604189i	$-0.206503\pi$
$67$	13.0448	1.59368	0.796841	+	0.604189i	$0.206503\pi$
$68$	0	0
$69$	−1.00000	−0.120386
$70$	0	0
$71$	7.06232	0.838144	0.419072	−	0.907953i	$-0.362356\pi$
$71$	7.06232	0.838144	0.419072	+	0.907953i	$0.362356\pi$
$72$	0	0
$73$	−1.57609	−0.184467	−0.0922336	−	0.995737i	$-0.529401\pi$
$73$	−1.57609	−0.184467	−0.0922336	+	0.995737i	$0.529401\pi$
$74$	0	0
$75$	1.00000	0.115470
$76$	0	0
$77$	−12.1072	−1.37974
$78$	0	0
$79$	−0.329796	−0.0371049	−0.0185525	−	0.999828i	$-0.505906\pi$
$79$	−0.329796	−0.0371049	−0.0185525	+	0.999828i	$0.505906\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	13.5680	1.48928	0.744639	−	0.667468i	$-0.232622\pi$
$83$	13.5680	1.48928	0.744639	+	0.667468i	$0.232622\pi$
$84$	0	0
$85$	4.15025	0.450157
$86$	0	0
$87$	5.64461	0.605165
$88$	0	0
$89$	1.08792	0.115320	0.0576598	−	0.998336i	$-0.481636\pi$
$89$	1.08792	0.115320	0.0576598	+	0.998336i	$0.481636\pi$
$90$	0	0
$91$	10.0175	1.05012
$92$	0	0
$93$	3.22069	0.333970
$94$	0	0
$95$	3.84163	0.394143
$96$	0	0
$97$	0.441388	0.0448161	0.0224081	−	0.999749i	$-0.492867\pi$
$97$	0.441388	0.0448161	0.0224081	+	0.999749i	$0.492867\pi$
$98$	0	0
$99$	−2.92955	−0.294431
$100$	0	0
$101$	4.21450	0.419358	0.209679	−	0.977770i	$-0.432758\pi$
$101$	4.21450	0.419358	0.209679	+	0.977770i	$0.432758\pi$
$102$	0	0
$103$	1.22882	0.121079	0.0605394	−	0.998166i	$-0.480718\pi$
$103$	1.22882	0.121079	0.0605394	+	0.998166i	$0.480718\pi$
$104$	0	0
$105$	−4.13277	−0.403317
$106$	0	0
$107$	0.291141	0.0281456	0.0140728	−	0.999901i	$-0.495520\pi$
$107$	0.291141	0.0281456	0.0140728	+	0.999901i	$0.495520\pi$
$108$	0	0
$109$	−0.929553	−0.0890350	−0.0445175	−	0.999009i	$-0.514175\pi$
$109$	−0.929553	−0.0890350	−0.0445175	+	0.999009i	$0.514175\pi$
$110$	0	0
$111$	5.55049	0.526829
$112$	0	0
$113$	−14.5567	−1.36938	−0.684689	−	0.728836i	$-0.740062\pi$
$113$	−14.5567	−1.36938	−0.684689	+	0.728836i	$0.740062\pi$
$114$	0	0
$115$	1.00000	0.0932505
$116$	0	0
$117$	2.42391	0.224091
$118$	0	0
$119$	−17.1520	−1.57232
$120$	0	0
$121$	−2.41772	−0.219793
$122$	0	0
$123$	−5.06232	−0.456454
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	14.2481	1.26431	0.632156	−	0.774841i	$-0.282170\pi$
$127$	14.2481	1.26431	0.632156	+	0.774841i	$0.282170\pi$
$128$	0	0
$129$	7.41772	0.653094
$130$	0	0
$131$	20.5311	1.79381	0.896905	−	0.442224i	$-0.145810\pi$
$131$	20.5311	1.79381	0.896905	+	0.442224i	$0.145810\pi$
$132$	0	0
$133$	−15.8766	−1.37667
$134$	0	0
$135$	−1.00000	−0.0860663
$136$	0	0
$137$	−2.94703	−0.251782	−0.125891	−	0.992044i	$-0.540179\pi$
$137$	−2.94703	−0.251782	−0.125891	+	0.992044i	$0.540179\pi$
$138$	0	0
$139$	14.0561	1.19223	0.596113	−	0.802901i	$-0.296711\pi$
$139$	14.0561	1.19223	0.596113	+	0.802901i	$0.296711\pi$
$140$	0	0
$141$	3.84163	0.323524
$142$	0	0
$143$	−7.10098	−0.593814
$144$	0	0
$145$	−5.64461	−0.468759
$146$	0	0
$147$	10.0798	0.831368
$148$	0	0
$149$	−17.1308	−1.40341	−0.701707	−	0.712466i	$-0.747578\pi$
$149$	−17.1308	−1.40341	−0.701707	+	0.712466i	$0.747578\pi$
$150$	0	0
$151$	−9.11337	−0.741635	−0.370818	−	0.928706i	$-0.620923\pi$
$151$	−9.11337	−0.741635	−0.370818	+	0.928706i	$0.620923\pi$
$152$	0	0
$153$	−4.15025	−0.335528
$154$	0	0
$155$	−3.22069	−0.258692
$156$	0	0
$157$	10.4333	0.832665	0.416333	−	0.909212i	$-0.363315\pi$
$157$	10.4333	0.832665	0.416333	+	0.909212i	$0.363315\pi$
$158$	0	0
$159$	8.48004	0.672511
$160$	0	0
$161$	−4.13277	−0.325708
$162$	0	0
$163$	18.4414	1.44444	0.722220	−	0.691663i	$-0.243122\pi$
$163$	18.4414	1.44444	0.722220	+	0.691663i	$0.243122\pi$
$164$	0	0
$165$	2.92955	0.228065
$166$	0	0
$167$	−20.3727	−1.57649	−0.788244	−	0.615363i	$-0.789010\pi$
$167$	−20.3727	−1.57649	−0.788244	+	0.615363i	$0.789010\pi$
$168$	0	0
$169$	−7.12465	−0.548050
$170$	0	0
$171$	−3.84163	−0.293777
$172$	0	0
$173$	−1.73446	−0.131868	−0.0659342	−	0.997824i	$-0.521003\pi$
$173$	−1.73446	−0.131868	−0.0659342	+	0.997824i	$0.521003\pi$
$174$	0	0
$175$	4.13277	0.312408
$176$	0	0
$177$	−3.64461	−0.273945
$178$	0	0
$179$	−5.24187	−0.391796	−0.195898	−	0.980624i	$-0.562762\pi$
$179$	−5.24187	−0.391796	−0.195898	+	0.980624i	$0.562762\pi$
$180$	0	0
$181$	−8.80613	−0.654555	−0.327277	−	0.944928i	$-0.606131\pi$
$181$	−8.80613	−0.654555	−0.327277	+	0.944928i	$0.606131\pi$
$182$	0	0
$183$	−1.51183	−0.111758
$184$	0	0
$185$	−5.55049	−0.408080
$186$	0	0
$187$	12.1584	0.889108
$188$	0	0
$189$	4.13277	0.300615
$190$	0	0
$191$	−16.7244	−1.21014	−0.605068	−	0.796174i	$-0.706854\pi$
$191$	−16.7244	−1.21014	−0.605068	+	0.796174i	$0.706854\pi$
$192$	0	0
$193$	6.30049	0.453519	0.226760	−	0.973951i	$-0.427187\pi$
$193$	6.30049	0.453519	0.226760	+	0.973951i	$0.427187\pi$
$194$	0	0
$195$	−2.42391	−0.173580
$196$	0	0
$197$	26.5660	1.89275	0.946376	−	0.323068i	$-0.104714\pi$
$197$	26.5660	1.89275	0.946376	+	0.323068i	$0.104714\pi$
$198$	0	0
$199$	7.74751	0.549207	0.274603	−	0.961558i	$-0.411453\pi$
$199$	7.74751	0.549207	0.274603	+	0.961558i	$0.411453\pi$
$200$	0	0
$201$	13.0448	0.920113
$202$	0	0
$203$	23.3279	1.63730
$204$	0	0
$205$	5.06232	0.353568
$206$	0	0
$207$	−1.00000	−0.0695048
$208$	0	0
$209$	11.2543	0.778474
$210$	0	0
$211$	−19.4700	−1.34037	−0.670185	−	0.742194i	$-0.733785\pi$
$211$	−19.4700	−1.34037	−0.670185	+	0.742194i	$0.733785\pi$
$212$	0	0
$213$	7.06232	0.483903
$214$	0	0
$215$	−7.41772	−0.505884
$216$	0	0
$217$	13.3104	0.903568
$218$	0	0
$219$	−1.57609	−0.106502
$220$	0	0
$221$	−10.0598	−0.676698
$222$	0	0
$223$	18.0897	1.21138	0.605688	−	0.795702i	$-0.292898\pi$
$223$	18.0897	1.21138	0.605688	+	0.795702i	$0.292898\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	−5.21257	−0.345971	−0.172985	−	0.984924i	$-0.555341\pi$
$227$	−5.21257	−0.345971	−0.172985	+	0.984924i	$0.555341\pi$
$228$	0	0
$229$	−25.3372	−1.67433	−0.837165	−	0.546950i	$-0.815789\pi$
$229$	−25.3372	−1.67433	−0.837165	+	0.546950i	$0.815789\pi$
$230$	0	0
$231$	−12.1072	−0.796594
$232$	0	0
$233$	−9.38277	−0.614686	−0.307343	−	0.951599i	$-0.599440\pi$
$233$	−9.38277	−0.614686	−0.307343	+	0.951599i	$0.599440\pi$
$234$	0	0
$235$	−3.84163	−0.250600
$236$	0	0
$237$	−0.329796	−0.0214226
$238$	0	0
$239$	−27.9451	−1.80762	−0.903809	−	0.427936i	$-0.859241\pi$
$239$	−27.9451	−1.80762	−0.903809	+	0.427936i	$0.859241\pi$
$240$	0	0
$241$	−25.9787	−1.67343	−0.836717	−	0.547636i	$-0.815528\pi$
$241$	−25.9787	−1.67343	−0.836717	+	0.547636i	$0.815528\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−10.0798	−0.643975
$246$	0	0
$247$	−9.31178	−0.592494
$248$	0	0
$249$	13.5680	0.859835
$250$	0	0
$251$	9.75990	0.616040	0.308020	−	0.951380i	$-0.400334\pi$
$251$	9.75990	0.616040	0.308020	+	0.951380i	$0.400334\pi$
$252$	0	0
$253$	2.92955	0.184179
$254$	0	0
$255$	4.15025	0.259899
$256$	0	0
$257$	14.1072	0.879981	0.439991	−	0.898002i	$-0.354982\pi$
$257$	14.1072	0.879981	0.439991	+	0.898002i	$0.354982\pi$
$258$	0	0
$259$	22.9389	1.42535
$260$	0	0
$261$	5.64461	0.349392
$262$	0	0
$263$	−3.60966	−0.222581	−0.111290	−	0.993788i	$-0.535498\pi$
$263$	−3.60966	−0.222581	−0.111290	+	0.993788i	$0.535498\pi$
$264$	0	0
$265$	−8.48004	−0.520925
$266$	0	0
$267$	1.08792	0.0665798
$268$	0	0
$269$	20.7456	1.26488	0.632440	−	0.774609i	$-0.282053\pi$
$269$	20.7456	1.26488	0.632440	+	0.774609i	$0.282053\pi$
$270$	0	0
$271$	20.7281	1.25914	0.629572	−	0.776943i	$-0.283230\pi$
$271$	20.7281	1.25914	0.629572	+	0.776943i	$0.283230\pi$
$272$	0	0
$273$	10.0175	0.606285
$274$	0	0
$275$	−2.92955	−0.176659
$276$	0	0
$277$	−10.5311	−0.632752	−0.316376	−	0.948634i	$-0.602466\pi$
$277$	−10.5311	−0.632752	−0.316376	+	0.948634i	$0.602466\pi$
$278$	0	0
$279$	3.22069	0.192818
$280$	0	0
$281$	27.5853	1.64560	0.822800	−	0.568331i	$-0.192411\pi$
$281$	27.5853	1.64560	0.822800	+	0.568331i	$0.192411\pi$
$282$	0	0
$283$	12.9163	0.767797	0.383898	−	0.923375i	$-0.374581\pi$
$283$	12.9163	0.767797	0.383898	+	0.923375i	$0.374581\pi$
$284$	0	0
$285$	3.84163	0.227559
$286$	0	0
$287$	−20.9214	−1.23495
$288$	0	0
$289$	0.224551	0.0132089
$290$	0	0
$291$	0.441388	0.0258746
$292$	0	0
$293$	16.1633	0.944270	0.472135	−	0.881526i	$-0.343483\pi$
$293$	16.1633	0.944270	0.472135	+	0.881526i	$0.343483\pi$
$294$	0	0
$295$	3.64461	0.212197
$296$	0	0
$297$	−2.92955	−0.169990
$298$	0	0
$299$	−2.42391	−0.140178
$300$	0	0
$301$	30.6557	1.76697
$302$	0	0
$303$	4.21450	0.242117
$304$	0	0
$305$	1.51183	0.0865674
$306$	0	0
$307$	32.4426	1.85160	0.925799	−	0.378016i	$-0.123394\pi$
$307$	32.4426	1.85160	0.925799	+	0.378016i	$0.123394\pi$
$308$	0	0
$309$	1.22882	0.0699049
$310$	0	0
$311$	24.6010	1.39499	0.697497	−	0.716588i	$-0.254297\pi$
$311$	24.6010	1.39499	0.697497	+	0.716588i	$0.254297\pi$
$312$	0	0
$313$	−17.0922	−0.966108	−0.483054	−	0.875591i	$-0.660472\pi$
$313$	−17.0922	−0.966108	−0.483054	+	0.875591i	$0.660472\pi$
$314$	0	0
$315$	−4.13277	−0.232855
$316$	0	0
$317$	−2.14212	−0.120314	−0.0601569	−	0.998189i	$-0.519160\pi$
$317$	−2.14212	−0.120314	−0.0601569	+	0.998189i	$0.519160\pi$
$318$	0	0
$319$	−16.5362	−0.925848
$320$	0	0
$321$	0.291141	0.0162499
$322$	0	0
$323$	15.9437	0.887132
$324$	0	0
$325$	2.42391	0.134454
$326$	0	0
$327$	−0.929553	−0.0514044
$328$	0	0
$329$	15.8766	0.875304
$330$	0	0
$331$	−13.3453	−0.733526	−0.366763	−	0.930314i	$-0.619534\pi$
$331$	−13.3453	−0.733526	−0.366763	+	0.930314i	$0.619534\pi$
$332$	0	0
$333$	5.55049	0.304165
$334$	0	0
$335$	−13.0448	−0.712716
$336$	0	0
$337$	0.175845	0.00957888	0.00478944	−	0.999989i	$-0.498475\pi$
$337$	0.175845	0.00957888	0.00478944	+	0.999989i	$0.498475\pi$
$338$	0	0
$339$	−14.5567	−0.790611
$340$	0	0
$341$	−9.43519	−0.510944
$342$	0	0
$343$	12.7281	0.687253
$344$	0	0
$345$	1.00000	0.0538382
$346$	0	0
$347$	1.85911	0.0998021	0.0499010	−	0.998754i	$-0.484109\pi$
$347$	1.85911	0.0998021	0.0499010	+	0.998754i	$0.484109\pi$
$348$	0	0
$349$	7.39654	0.395928	0.197964	−	0.980209i	$-0.436567\pi$
$349$	7.39654	0.395928	0.197964	+	0.980209i	$0.436567\pi$
$350$	0	0
$351$	2.42391	0.129379
$352$	0	0
$353$	9.16579	0.487846	0.243923	−	0.969795i	$-0.421566\pi$
$353$	9.16579	0.487846	0.243923	+	0.969795i	$0.421566\pi$
$354$	0	0
$355$	−7.06232	−0.374829
$356$	0	0
$357$	−17.1520	−0.907781
$358$	0	0
$359$	24.0909	1.27147	0.635735	−	0.771907i	$-0.280697\pi$
$359$	24.0909	1.27147	0.635735	+	0.771907i	$0.280697\pi$
$360$	0	0
$361$	−4.24187	−0.223257
$362$	0	0
$363$	−2.41772	−0.126897
$364$	0	0
$365$	1.57609	0.0824962
$366$	0	0
$367$	23.9795	1.25172	0.625860	−	0.779936i	$-0.284748\pi$
$367$	23.9795	1.25172	0.625860	+	0.779936i	$0.284748\pi$
$368$	0	0
$369$	−5.06232	−0.263534
$370$	0	0
$371$	35.0461	1.81950
$372$	0	0
$373$	−27.7949	−1.43916	−0.719581	−	0.694408i	$-0.755666\pi$
$373$	−27.7949	−1.43916	−0.719581	+	0.694408i	$0.755666\pi$
$374$	0	0
$375$	−1.00000	−0.0516398
$376$	0	0
$377$	13.6820	0.704660
$378$	0	0
$379$	−12.8828	−0.661744	−0.330872	−	0.943676i	$-0.607343\pi$
$379$	−12.8828	−0.661744	−0.330872	+	0.943676i	$0.607343\pi$
$380$	0	0
$381$	14.2481	0.729951
$382$	0	0
$383$	19.0461	0.973209	0.486605	−	0.873622i	$-0.338235\pi$
$383$	19.0461	0.973209	0.486605	+	0.873622i	$0.338235\pi$
$384$	0	0
$385$	12.1072	0.617039
$386$	0	0
$387$	7.41772	0.377064
$388$	0	0
$389$	−30.2842	−1.53547	−0.767736	−	0.640766i	$-0.778617\pi$
$389$	−30.2842	−1.53547	−0.767736	+	0.640766i	$0.778617\pi$
$390$	0	0
$391$	4.15025	0.209887
$392$	0	0
$393$	20.5311	1.03566
$394$	0	0
$395$	0.329796	0.0165938
$396$	0	0
$397$	−15.1483	−0.760272	−0.380136	−	0.924931i	$-0.624123\pi$
$397$	−15.1483	−0.760272	−0.380136	+	0.924931i	$0.624123\pi$
$398$	0	0
$399$	−15.8766	−0.794823
$400$	0	0
$401$	−23.0491	−1.15102	−0.575509	−	0.817795i	$-0.695196\pi$
$401$	−23.0491	−1.15102	−0.575509	+	0.817795i	$0.695196\pi$
$402$	0	0
$403$	7.80668	0.388878
$404$	0	0
$405$	−1.00000	−0.0496904
$406$	0	0
$407$	−16.2605	−0.806001
$408$	0	0
$409$	11.7906	0.583007	0.291504	−	0.956570i	$-0.405844\pi$
$409$	11.7906	0.583007	0.291504	+	0.956570i	$0.405844\pi$
$410$	0	0
$411$	−2.94703	−0.145366
$412$	0	0
$413$	−15.0623	−0.741169
$414$	0	0
$415$	−13.5680	−0.666025
$416$	0	0
$417$	14.0561	0.688332
$418$	0	0
$419$	35.9917	1.75831	0.879155	−	0.476535i	$-0.158108\pi$
$419$	35.9917	1.75831	0.879155	+	0.476535i	$0.158108\pi$
$420$	0	0
$421$	−1.58915	−0.0774502	−0.0387251	−	0.999250i	$-0.512330\pi$
$421$	−1.58915	−0.0774502	−0.0387251	+	0.999250i	$0.512330\pi$
$422$	0	0
$423$	3.84163	0.186786
$424$	0	0
$425$	−4.15025	−0.201317
$426$	0	0
$427$	−6.24807	−0.302365
$428$	0	0
$429$	−7.10098	−0.342839
$430$	0	0
$431$	−0.445245	−0.0214467	−0.0107233	−	0.999943i	$-0.503413\pi$
$431$	−0.445245	−0.0214467	−0.0107233	+	0.999943i	$0.503413\pi$
$432$	0	0
$433$	−9.34920	−0.449294	−0.224647	−	0.974440i	$-0.572123\pi$
$433$	−9.34920	−0.449294	−0.224647	+	0.974440i	$0.572123\pi$
$434$	0	0
$435$	−5.64461	−0.270638
$436$	0	0
$437$	3.84163	0.183770
$438$	0	0
$439$	−33.5261	−1.60011	−0.800057	−	0.599924i	$-0.795198\pi$
$439$	−33.5261	−1.60011	−0.800057	+	0.599924i	$0.795198\pi$
$440$	0	0
$441$	10.0798	0.479990
$442$	0	0
$443$	−13.8890	−0.659885	−0.329942	−	0.944001i	$-0.607029\pi$
$443$	−13.8890	−0.659885	−0.329942	+	0.944001i	$0.607029\pi$
$444$	0	0
$445$	−1.08792	−0.0515725
$446$	0	0
$447$	−17.1308	−0.810261
$448$	0	0
$449$	21.1644	0.998810	0.499405	−	0.866369i	$-0.333552\pi$
$449$	21.1644	0.998810	0.499405	+	0.866369i	$0.333552\pi$
$450$	0	0
$451$	14.8303	0.698334
$452$	0	0
$453$	−9.11337	−0.428183
$454$	0	0
$455$	−10.0175	−0.469626
$456$	0	0
$457$	31.3104	1.46464	0.732319	−	0.680962i	$-0.238438\pi$
$457$	31.3104	1.46464	0.732319	+	0.680962i	$0.238438\pi$
$458$	0	0
$459$	−4.15025	−0.193717
$460$	0	0
$461$	20.5187	0.955651	0.477826	−	0.878455i	$-0.341425\pi$
$461$	20.5187	0.955651	0.477826	+	0.878455i	$0.341425\pi$
$462$	0	0
$463$	−29.0323	−1.34925	−0.674623	−	0.738163i	$-0.735694\pi$
$463$	−29.0323	−1.34925	−0.674623	+	0.738163i	$0.735694\pi$
$464$	0	0
$465$	−3.22069	−0.149356
$466$	0	0
$467$	−10.6976	−0.495025	−0.247512	−	0.968885i	$-0.579613\pi$
$467$	−10.6976	−0.495025	−0.247512	+	0.968885i	$0.579613\pi$
$468$	0	0
$469$	53.9114	2.48940
$470$	0	0
$471$	10.4333	0.480740
$472$	0	0
$473$	−21.7306	−0.999174
$474$	0	0
$475$	−3.84163	−0.176266
$476$	0	0
$477$	8.48004	0.388274
$478$	0	0
$479$	−7.02876	−0.321152	−0.160576	−	0.987023i	$-0.551335\pi$
$479$	−7.02876	−0.321152	−0.160576	+	0.987023i	$0.551335\pi$
$480$	0	0
$481$	13.4539	0.613445
$482$	0	0
$483$	−4.13277	−0.188048
$484$	0	0
$485$	−0.441388	−0.0200424
$486$	0	0
$487$	−0.388961	−0.0176255	−0.00881276	−	0.999961i	$-0.502805\pi$
$487$	−0.388961	−0.0176255	−0.00881276	+	0.999961i	$0.502805\pi$
$488$	0	0
$489$	18.4414	0.833948
$490$	0	0
$491$	−2.92529	−0.132016	−0.0660082	−	0.997819i	$-0.521026\pi$
$491$	−2.92529	−0.132016	−0.0660082	+	0.997819i	$0.521026\pi$
$492$	0	0
$493$	−23.4265	−1.05508
$494$	0	0
$495$	2.92955	0.131674
$496$	0	0
$497$	29.1870	1.30921
$498$	0	0
$499$	−1.23308	−0.0552003	−0.0276002	−	0.999619i	$-0.508787\pi$
$499$	−1.23308	−0.0552003	−0.0276002	+	0.999619i	$0.508787\pi$
$500$	0	0
$501$	−20.3727	−0.910186
$502$	0	0
$503$	−23.7816	−1.06037	−0.530186	−	0.847882i	$-0.677878\pi$
$503$	−23.7816	−1.06037	−0.530186	+	0.847882i	$0.677878\pi$
$504$	0	0
$505$	−4.21450	−0.187543
$506$	0	0
$507$	−7.12465	−0.316417
$508$	0	0
$509$	−6.54733	−0.290205	−0.145103	−	0.989417i	$-0.546351\pi$
$509$	−6.54733	−0.290205	−0.145103	+	0.989417i	$0.546351\pi$
$510$	0	0
$511$	−6.51361	−0.288145
$512$	0	0
$513$	−3.84163	−0.169612
$514$	0	0
$515$	−1.22882	−0.0541481
$516$	0	0
$517$	−11.2543	−0.494962
$518$	0	0
$519$	−1.73446	−0.0761342
$520$	0	0
$521$	0.312320	0.0136830	0.00684150	−	0.999977i	$-0.497822\pi$
$521$	0.312320	0.0136830	0.00684150	+	0.999977i	$0.497822\pi$
$522$	0	0
$523$	−33.1907	−1.45133	−0.725664	−	0.688050i	$-0.758467\pi$
$523$	−33.1907	−1.45133	−0.725664	+	0.688050i	$0.758467\pi$
$524$	0	0
$525$	4.13277	0.180369
$526$	0	0
$527$	−13.3667	−0.582262
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	−3.64461	−0.158162
$532$	0	0
$533$	−12.2706	−0.531500
$534$	0	0
$535$	−0.291141	−0.0125871
$536$	0	0
$537$	−5.24187	−0.226204
$538$	0	0
$539$	−29.5293	−1.27192
$540$	0	0
$541$	11.9364	0.513187	0.256593	−	0.966519i	$-0.417400\pi$
$541$	11.9364	0.513187	0.256593	+	0.966519i	$0.417400\pi$
$542$	0	0
$543$	−8.80613	−0.377907
$544$	0	0
$545$	0.929553	0.0398177
$546$	0	0
$547$	13.1646	0.562876	0.281438	−	0.959579i	$-0.409189\pi$
$547$	13.1646	0.562876	0.281438	+	0.959579i	$0.409189\pi$
$548$	0	0
$549$	−1.51183	−0.0645235
$550$	0	0
$551$	−21.6845	−0.923790
$552$	0	0
$553$	−1.36297	−0.0579594
$554$	0	0
$555$	−5.55049	−0.235605
$556$	0	0
$557$	7.82045	0.331363	0.165682	−	0.986179i	$-0.447018\pi$
$557$	7.82045	0.331363	0.165682	+	0.986179i	$0.447018\pi$
$558$	0	0
$559$	17.9799	0.760469
$560$	0	0
$561$	12.1584	0.513327
$562$	0	0
$563$	1.00193	0.0422262	0.0211131	−	0.999777i	$-0.493279\pi$
$563$	1.00193	0.0422262	0.0211131	+	0.999777i	$0.493279\pi$
$564$	0	0
$565$	14.5567	0.612404
$566$	0	0
$567$	4.13277	0.173560
$568$	0	0
$569$	5.50675	0.230855	0.115427	−	0.993316i	$-0.463176\pi$
$569$	5.50675	0.230855	0.115427	+	0.993316i	$0.463176\pi$
$570$	0	0
$571$	28.7082	1.20140	0.600700	−	0.799475i	$-0.294889\pi$
$571$	28.7082	1.20140	0.600700	+	0.799475i	$0.294889\pi$
$572$	0	0
$573$	−16.7244	−0.698672
$574$	0	0
$575$	−1.00000	−0.0417029
$576$	0	0
$577$	14.9499	0.622373	0.311186	−	0.950349i	$-0.399274\pi$
$577$	14.9499	0.622373	0.311186	+	0.950349i	$0.399274\pi$
$578$	0	0
$579$	6.30049	0.261840
$580$	0	0
$581$	56.0733	2.32631
$582$	0	0
$583$	−24.8427	−1.02888
$584$	0	0
$585$	−2.42391	−0.100216
$586$	0	0
$587$	−31.3503	−1.29396	−0.646982	−	0.762505i	$-0.723969\pi$
$587$	−31.3503	−1.29396	−0.646982	+	0.762505i	$0.723969\pi$
$588$	0	0
$589$	−12.3727	−0.509809
$590$	0	0
$591$	26.5660	1.09278
$592$	0	0
$593$	8.11956	0.333430	0.166715	−	0.986005i	$-0.446684\pi$
$593$	8.11956	0.333430	0.166715	+	0.986005i	$0.446684\pi$
$594$	0	0
$595$	17.1520	0.703164
$596$	0	0
$597$	7.74751	0.317085
$598$	0	0
$599$	17.3789	0.710083	0.355042	−	0.934851i	$-0.384467\pi$
$599$	17.3789	0.710083	0.355042	+	0.934851i	$0.384467\pi$
$600$	0	0
$601$	3.13100	0.127716	0.0638580	−	0.997959i	$-0.479660\pi$
$601$	3.13100	0.127716	0.0638580	+	0.997959i	$0.479660\pi$
$602$	0	0
$603$	13.0448	0.531227
$604$	0	0
$605$	2.41772	0.0982942
$606$	0	0
$607$	10.8141	0.438931	0.219465	−	0.975620i	$-0.429569\pi$
$607$	10.8141	0.438931	0.219465	+	0.975620i	$0.429569\pi$
$608$	0	0
$609$	23.3279	0.945293
$610$	0	0
$611$	9.31178	0.376714
$612$	0	0
$613$	4.22385	0.170600	0.0852999	−	0.996355i	$-0.472815\pi$
$613$	4.22385	0.170600	0.0852999	+	0.996355i	$0.472815\pi$
$614$	0	0
$615$	5.06232	0.204133
$616$	0	0
$617$	9.72757	0.391617	0.195809	−	0.980642i	$-0.437267\pi$
$617$	9.72757	0.391617	0.195809	+	0.980642i	$0.437267\pi$
$618$	0	0
$619$	−29.3440	−1.17943	−0.589717	−	0.807610i	$-0.700761\pi$
$619$	−29.3440	−1.17943	−0.589717	+	0.807610i	$0.700761\pi$
$620$	0	0
$621$	−1.00000	−0.0401286
$622$	0	0
$623$	4.49613	0.180134
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	11.2543	0.449452
$628$	0	0
$629$	−23.0359	−0.918502
$630$	0	0
$631$	−20.8236	−0.828975	−0.414487	−	0.910055i	$-0.636039\pi$
$631$	−20.8236	−0.828975	−0.414487	+	0.910055i	$0.636039\pi$
$632$	0	0
$633$	−19.4700	−0.773863
$634$	0	0
$635$	−14.2481	−0.565417
$636$	0	0
$637$	24.4326	0.968053
$638$	0	0
$639$	7.06232	0.279381
$640$	0	0
$641$	−37.1277	−1.46645	−0.733227	−	0.679984i	$-0.761987\pi$
$641$	−37.1277	−1.46645	−0.733227	+	0.679984i	$0.761987\pi$
$642$	0	0
$643$	−18.0049	−0.710045	−0.355023	−	0.934858i	$-0.615527\pi$
$643$	−18.0049	−0.710045	−0.355023	+	0.934858i	$0.615527\pi$
$644$	0	0
$645$	−7.41772	−0.292072
$646$	0	0
$647$	−6.56481	−0.258089	−0.129045	−	0.991639i	$-0.541191\pi$
$647$	−6.56481	−0.258089	−0.129045	+	0.991639i	$0.541191\pi$
$648$	0	0
$649$	10.6771	0.419112
$650$	0	0
$651$	13.3104	0.521675
$652$	0	0
$653$	−39.8215	−1.55834	−0.779168	−	0.626815i	$-0.784358\pi$
$653$	−39.8215	−1.55834	−0.779168	+	0.626815i	$0.784358\pi$
$654$	0	0
$655$	−20.5311	−0.802216
$656$	0	0
$657$	−1.57609	−0.0614890
$658$	0	0
$659$	−44.3186	−1.72641	−0.863205	−	0.504854i	$-0.831546\pi$
$659$	−44.3186	−1.72641	−0.863205	+	0.504854i	$0.831546\pi$
$660$	0	0
$661$	10.3895	0.404106	0.202053	−	0.979375i	$-0.435239\pi$
$661$	10.3895	0.404106	0.202053	+	0.979375i	$0.435239\pi$
$662$	0	0
$663$	−10.0598	−0.390692
$664$	0	0
$665$	15.8766	0.615667
$666$	0	0
$667$	−5.64461	−0.218560
$668$	0	0
$669$	18.0897	0.699388
$670$	0	0
$671$	4.42900	0.170980
$672$	0	0
$673$	−11.5373	−0.444729	−0.222365	−	0.974964i	$-0.571378\pi$
$673$	−11.5373	−0.444729	−0.222365	+	0.974964i	$0.571378\pi$
$674$	0	0
$675$	1.00000	0.0384900
$676$	0	0
$677$	15.9677	0.613687	0.306844	−	0.951760i	$-0.400727\pi$
$677$	15.9677	0.613687	0.306844	+	0.951760i	$0.400727\pi$
$678$	0	0
$679$	1.82416	0.0700046
$680$	0	0
$681$	−5.21257	−0.199746
$682$	0	0
$683$	−32.1845	−1.23151	−0.615753	−	0.787940i	$-0.711148\pi$
$683$	−32.1845	−1.23151	−0.615753	+	0.787940i	$0.711148\pi$
$684$	0	0
$685$	2.94703	0.112600
$686$	0	0
$687$	−25.3372	−0.966675
$688$	0	0
$689$	20.5549	0.783079
$690$	0	0
$691$	40.9601	1.55820	0.779098	−	0.626903i	$-0.215678\pi$
$691$	40.9601	1.55820	0.779098	+	0.626903i	$0.215678\pi$
$692$	0	0
$693$	−12.1072	−0.459913
$694$	0	0
$695$	−14.0561	−0.533179
$696$	0	0
$697$	21.0099	0.795807
$698$	0	0
$699$	−9.38277	−0.354889
$700$	0	0
$701$	−25.4737	−0.962128	−0.481064	−	0.876685i	$-0.659750\pi$
$701$	−25.4737	−0.962128	−0.481064	+	0.876685i	$0.659750\pi$
$702$	0	0
$703$	−21.3229	−0.804210
$704$	0	0
$705$	−3.84163	−0.144684
$706$	0	0
$707$	17.4176	0.655055
$708$	0	0
$709$	4.46560	0.167709	0.0838546	−	0.996478i	$-0.473277\pi$
$709$	4.46560	0.167709	0.0838546	+	0.996478i	$0.473277\pi$
$710$	0	0
$711$	−0.329796	−0.0123683
$712$	0	0
$713$	−3.22069	−0.120616
$714$	0	0
$715$	7.10098	0.265562
$716$	0	0
$717$	−27.9451	−1.04363
$718$	0	0
$719$	−4.90272	−0.182841	−0.0914204	−	0.995812i	$-0.529141\pi$
$719$	−4.90272	−0.182841	−0.0914204	+	0.995812i	$0.529141\pi$
$720$	0	0
$721$	5.07842	0.189130
$722$	0	0
$723$	−25.9787	−0.966157
$724$	0	0
$725$	5.64461	0.209635
$726$	0	0
$727$	35.4220	1.31373	0.656864	−	0.754009i	$-0.271882\pi$
$727$	35.4220	1.31373	0.656864	+	0.754009i	$0.271882\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−30.7854	−1.13864
$732$	0	0
$733$	−51.9244	−1.91787	−0.958936	−	0.283621i	$-0.908464\pi$
$733$	−51.9244	−1.91787	−0.958936	+	0.283621i	$0.908464\pi$
$734$	0	0
$735$	−10.0798	−0.371799
$736$	0	0
$737$	−38.2156	−1.40769
$738$	0	0
$739$	13.7394	0.505412	0.252706	−	0.967543i	$-0.418680\pi$
$739$	13.7394	0.505412	0.252706	+	0.967543i	$0.418680\pi$
$740$	0	0
$741$	−9.31178	−0.342077
$742$	0	0
$743$	−15.8016	−0.579704	−0.289852	−	0.957071i	$-0.593606\pi$
$743$	−15.8016	−0.579704	−0.289852	+	0.957071i	$0.593606\pi$
$744$	0	0
$745$	17.1308	0.627626
$746$	0	0
$747$	13.5680	0.496426
$748$	0	0
$749$	1.20322	0.0439646
$750$	0	0
$751$	26.5616	0.969247	0.484624	−	0.874723i	$-0.338957\pi$
$751$	26.5616	0.969247	0.484624	+	0.874723i	$0.338957\pi$
$752$	0	0
$753$	9.75990	0.355671
$754$	0	0
$755$	9.11337	0.331669
$756$	0	0
$757$	−24.9682	−0.907485	−0.453742	−	0.891133i	$-0.649911\pi$
$757$	−24.9682	−0.907485	−0.453742	+	0.891133i	$0.649911\pi$
$758$	0	0
$759$	2.92955	0.106336
$760$	0	0
$761$	−29.3279	−1.06313	−0.531567	−	0.847016i	$-0.678397\pi$
$761$	−29.3279	−1.06313	−0.531567	+	0.847016i	$0.678397\pi$
$762$	0	0
$763$	−3.84163	−0.139076
$764$	0	0
$765$	4.15025	0.150052
$766$	0	0
$767$	−8.83421	−0.318985
$768$	0	0
$769$	3.09975	0.111780	0.0558899	−	0.998437i	$-0.482200\pi$
$769$	3.09975	0.111780	0.0558899	+	0.998437i	$0.482200\pi$
$770$	0	0
$771$	14.1072	0.508057
$772$	0	0
$773$	−7.50742	−0.270023	−0.135012	−	0.990844i	$-0.543107\pi$
$773$	−7.50742	−0.270023	−0.135012	+	0.990844i	$0.543107\pi$
$774$	0	0
$775$	3.22069	0.115691
$776$	0	0
$777$	22.9389	0.822929
$778$	0	0
$779$	19.4476	0.696782
$780$	0	0
$781$	−20.6895	−0.740327
$782$	0	0
$783$	5.64461	0.201722
$784$	0	0
$785$	−10.4333	−0.372379
$786$	0	0
$787$	43.5635	1.55287	0.776436	−	0.630196i	$-0.217025\pi$
$787$	43.5635	1.55287	0.776436	+	0.630196i	$0.217025\pi$
$788$	0	0
$789$	−3.60966	−0.128507
$790$	0	0
$791$	−60.1594	−2.13902
$792$	0	0
$793$	−3.66455	−0.130132
$794$	0	0
$795$	−8.48004	−0.300756
$796$	0	0
$797$	−16.3268	−0.578324	−0.289162	−	0.957280i	$-0.593377\pi$
$797$	−16.3268	−0.578324	−0.289162	+	0.957280i	$0.593377\pi$
$798$	0	0
$799$	−15.9437	−0.564048
$800$	0	0
$801$	1.08792	0.0384398
$802$	0	0
$803$	4.61723	0.162939
$804$	0	0
$805$	4.13277	0.145661
$806$	0	0
$807$	20.7456	0.730279
$808$	0	0
$809$	22.0736	0.776067	0.388033	−	0.921645i	$-0.373155\pi$
$809$	22.0736	0.776067	0.388033	+	0.921645i	$0.373155\pi$
$810$	0	0
$811$	−42.8365	−1.50419	−0.752097	−	0.659053i	$-0.770957\pi$
$811$	−42.8365	−1.50419	−0.752097	+	0.659053i	$0.770957\pi$
$812$	0	0
$813$	20.7281	0.726967
$814$	0	0
$815$	−18.4414	−0.645974
$816$	0	0
$817$	−28.4961	−0.996954
$818$	0	0
$819$	10.0175	0.350039
$820$	0	0
$821$	29.0424	1.01359	0.506793	−	0.862068i	$-0.330831\pi$
$821$	29.0424	1.01359	0.506793	+	0.862068i	$0.330831\pi$
$822$	0	0
$823$	−44.9089	−1.56543	−0.782713	−	0.622383i	$-0.786165\pi$
$823$	−44.9089	−1.56543	−0.782713	+	0.622383i	$0.786165\pi$
$824$	0	0
$825$	−2.92955	−0.101994
$826$	0	0
$827$	11.6739	0.405942	0.202971	−	0.979185i	$-0.434940\pi$
$827$	11.6739	0.405942	0.202971	+	0.979185i	$0.434940\pi$
$828$	0	0
$829$	27.3291	0.949179	0.474589	−	0.880207i	$-0.342597\pi$
$829$	27.3291	0.949179	0.474589	+	0.880207i	$0.342597\pi$
$830$	0	0
$831$	−10.5311	−0.365319
$832$	0	0
$833$	−41.8337	−1.44945
$834$	0	0
$835$	20.3727	0.705027
$836$	0	0
$837$	3.22069	0.111323
$838$	0	0
$839$	15.1010	0.521344	0.260672	−	0.965427i	$-0.416056\pi$
$839$	15.1010	0.521344	0.260672	+	0.965427i	$0.416056\pi$
$840$	0	0
$841$	2.86158	0.0986751
$842$	0	0
$843$	27.5853	0.950088
$844$	0	0
$845$	7.12465	0.245095
$846$	0	0
$847$	−9.99188	−0.343325
$848$	0	0
$849$	12.9163	0.443288
$850$	0	0
$851$	−5.55049	−0.190268
$852$	0	0
$853$	−14.3676	−0.491938	−0.245969	−	0.969278i	$-0.579106\pi$
$853$	−14.3676	−0.491938	−0.245969	+	0.969278i	$0.579106\pi$
$854$	0	0
$855$	3.84163	0.131381
$856$	0	0
$857$	25.5649	0.873281	0.436641	−	0.899636i	$-0.356168\pi$
$857$	25.5649	0.873281	0.436641	+	0.899636i	$0.356168\pi$
$858$	0	0
$859$	−50.3054	−1.71640	−0.858200	−	0.513316i	$-0.828417\pi$
$859$	−50.3054	−1.71640	−0.858200	+	0.513316i	$0.828417\pi$
$860$	0	0
$861$	−20.9214	−0.713000
$862$	0	0
$863$	−17.8429	−0.607378	−0.303689	−	0.952771i	$-0.598218\pi$
$863$	−17.8429	−0.607378	−0.303689	+	0.952771i	$0.598218\pi$
$864$	0	0
$865$	1.73446	0.0589733
$866$	0	0
$867$	0.224551	0.00762614
$868$	0	0
$869$	0.966155	0.0327746
$870$	0	0
$871$	31.6196	1.07139
$872$	0	0
$873$	0.441388	0.0149387
$874$	0	0
$875$	−4.13277	−0.139713
$876$	0	0
$877$	−51.3464	−1.73385	−0.866923	−	0.498443i	$-0.833905\pi$
$877$	−51.3464	−1.73385	−0.866923	+	0.498443i	$0.833905\pi$
$878$	0	0
$879$	16.1633	0.545175
$880$	0	0
$881$	−15.6365	−0.526807	−0.263403	−	0.964686i	$-0.584845\pi$
$881$	−15.6365	−0.526807	−0.263403	+	0.964686i	$0.584845\pi$
$882$	0	0
$883$	−20.5874	−0.692820	−0.346410	−	0.938083i	$-0.612599\pi$
$883$	−20.5874	−0.692820	−0.346410	+	0.938083i	$0.612599\pi$
$884$	0	0
$885$	3.64461	0.122512
$886$	0	0
$887$	6.05475	0.203299	0.101649	−	0.994820i	$-0.467588\pi$
$887$	6.05475	0.203299	0.101649	+	0.994820i	$0.467588\pi$
$888$	0	0
$889$	58.8840	1.97491
$890$	0	0
$891$	−2.92955	−0.0981437
$892$	0	0
$893$	−14.7581	−0.493862
$894$	0	0
$895$	5.24187	0.175217
$896$	0	0
$897$	−2.42391	−0.0809321
$898$	0	0
$899$	18.1795	0.606322
$900$	0	0
$901$	−35.1943	−1.17249
$902$	0	0
$903$	30.6557	1.02016
$904$	0	0
$905$	8.80613	0.292726
$906$	0	0
$907$	−2.19317	−0.0728229	−0.0364115	−	0.999337i	$-0.511593\pi$
$907$	−2.19317	−0.0728229	−0.0364115	+	0.999337i	$0.511593\pi$
$908$	0	0
$909$	4.21450	0.139786
$910$	0	0
$911$	−20.9089	−0.692742	−0.346371	−	0.938098i	$-0.612586\pi$
$911$	−20.9089	−0.692742	−0.346371	+	0.938098i	$0.612586\pi$
$912$	0	0
$913$	−39.7481	−1.31547
$914$	0	0
$915$	1.51183	0.0499797
$916$	0	0
$917$	84.8503	2.80200
$918$	0	0
$919$	−4.04801	−0.133531	−0.0667657	−	0.997769i	$-0.521268\pi$
$919$	−4.04801	−0.133531	−0.0667657	+	0.997769i	$0.521268\pi$
$920$	0	0
$921$	32.4426	1.06902
$922$	0	0
$923$	17.1185	0.563461
$924$	0	0
$925$	5.55049	0.182499
$926$	0	0
$927$	1.22882	0.0403596
$928$	0	0
$929$	−12.8155	−0.420463	−0.210231	−	0.977652i	$-0.567422\pi$
$929$	−12.8155	−0.420463	−0.210231	+	0.977652i	$0.567422\pi$
$930$	0	0
$931$	−38.7229	−1.26909
$932$	0	0
$933$	24.6010	0.805400
$934$	0	0
$935$	−12.1584	−0.397621
$936$	0	0
$937$	22.9499	0.749741	0.374870	−	0.927077i	$-0.377687\pi$
$937$	22.9499	0.749741	0.374870	+	0.927077i	$0.377687\pi$
$938$	0	0
$939$	−17.0922	−0.557782
$940$	0	0
$941$	−44.5323	−1.45171	−0.725856	−	0.687847i	$-0.758556\pi$
$941$	−44.5323	−1.45171	−0.725856	+	0.687847i	$0.758556\pi$
$942$	0	0
$943$	5.06232	0.164852
$944$	0	0
$945$	−4.13277	−0.134439
$946$	0	0
$947$	52.1557	1.69483	0.847417	−	0.530928i	$-0.178157\pi$
$947$	52.1557	1.69483	0.847417	+	0.530928i	$0.178157\pi$
$948$	0	0
$949$	−3.82030	−0.124012
$950$	0	0
$951$	−2.14212	−0.0694632
$952$	0	0
$953$	−42.9556	−1.39147	−0.695734	−	0.718300i	$-0.744921\pi$
$953$	−42.9556	−1.39147	−0.695734	+	0.718300i	$0.744921\pi$
$954$	0	0
$955$	16.7244	0.541189
$956$	0	0
$957$	−16.5362	−0.534539
$958$	0	0
$959$	−12.1794	−0.393293
$960$	0	0
$961$	−20.6271	−0.665391
$962$	0	0
$963$	0.291141	0.00938188
$964$	0	0
$965$	−6.30049	−0.202820
$966$	0	0
$967$	14.1196	0.454054	0.227027	−	0.973888i	$-0.427099\pi$
$967$	14.1196	0.454054	0.227027	+	0.973888i	$0.427099\pi$
$968$	0	0
$969$	15.9437	0.512186
$970$	0	0
$971$	−2.21368	−0.0710403	−0.0355201	−	0.999369i	$-0.511309\pi$
$971$	−2.21368	−0.0710403	−0.0355201	+	0.999369i	$0.511309\pi$
$972$	0	0
$973$	58.0908	1.86230
$974$	0	0
$975$	2.42391	0.0776273
$976$	0	0
$977$	−34.4034	−1.10066	−0.550331	−	0.834946i	$-0.685499\pi$
$977$	−34.4034	−1.10066	−0.550331	+	0.834946i	$0.685499\pi$
$978$	0	0
$979$	−3.18713	−0.101861
$980$	0	0
$981$	−0.929553	−0.0296783
$982$	0	0
$983$	2.05104	0.0654181	0.0327091	−	0.999465i	$-0.489587\pi$
$983$	2.05104	0.0654181	0.0327091	+	0.999465i	$0.489587\pi$
$984$	0	0
$985$	−26.5660	−0.846464
$986$	0	0
$987$	15.8766	0.505357
$988$	0	0
$989$	−7.41772	−0.235870
$990$	0	0
$991$	47.5371	1.51007	0.755033	−	0.655686i	$-0.227621\pi$
$991$	47.5371	1.51007	0.755033	+	0.655686i	$0.227621\pi$
$992$	0	0
$993$	−13.3453	−0.423502
$994$	0	0
$995$	−7.74751	−0.245613
$996$	0	0
$997$	−57.1868	−1.81112	−0.905562	−	0.424213i	$-0.860551\pi$
$997$	−57.1868	−1.81112	−0.905562	+	0.424213i	$0.860551\pi$
$998$	0	0
$999$	5.55049	0.175610

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5520.2.a.cb.1.4		4
4.3	odd	2		2760.2.a.v.1.1	✓	4
12.11	even	2		8280.2.a.bq.1.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2760.2.a.v.1.1	✓	4	4.3	odd	2
5520.2.a.cb.1.4		4	1.1	even	1	trivial
8280.2.a.bq.1.1		4	12.11	even	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:	\( N \)	\(=\)	\( 5520 = 2^{4} \cdot 3 \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5520.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -1.00000 q^{5} +4.13277 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -1.00000 q^{5} +4.13277 q^{7} +1.00000 q^{9} -2.92955 q^{11} +2.42391 q^{13} -1.00000 q^{15} -4.15025 q^{17} -3.84163 q^{19} +4.13277 q^{21} -1.00000 q^{23} +1.00000 q^{25} +1.00000 q^{27} +5.64461 q^{29} +3.22069 q^{31} -2.92955 q^{33} -4.13277 q^{35} +5.55049 q^{37} +2.42391 q^{39} -5.06232 q^{41} +7.41772 q^{43} -1.00000 q^{45} +3.84163 q^{47} +10.0798 q^{49} -4.15025 q^{51} +8.48004 q^{53} +2.92955 q^{55} -3.84163 q^{57} -3.64461 q^{59} -1.51183 q^{61} +4.13277 q^{63} -2.42391 q^{65} +13.0448 q^{67} -1.00000 q^{69} +7.06232 q^{71} -1.57609 q^{73} +1.00000 q^{75} -12.1072 q^{77} -0.329796 q^{79} +1.00000 q^{81} +13.5680 q^{83} +4.15025 q^{85} +5.64461 q^{87} +1.08792 q^{89} +10.0175 q^{91} +3.22069 q^{93} +3.84163 q^{95} +0.441388 q^{97} -2.92955 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{3} - 4 q^{5} + 4 q^{9}+O(q^{10}) \) 4 * q + 4 * q^3 - 4 * q^5 + 4 * q^9	\( 4 q + 4 q^{3} - 4 q^{5} + 4 q^{9} - 2 q^{13} - 4 q^{15} + 2 q^{17} + 6 q^{19} - 4 q^{23} + 4 q^{25} + 4 q^{27} + 4 q^{29} + 6 q^{31} - 4 q^{37} - 2 q^{39} + 8 q^{41} + 20 q^{43} - 4 q^{45} - 6 q^{47} + 10 q^{49} + 2 q^{51} - 4 q^{53} + 6 q^{57} + 4 q^{59} - 4 q^{61} + 2 q^{65} + 26 q^{67} - 4 q^{69} - 18 q^{73} + 4 q^{75} + 6 q^{77} + 18 q^{79} + 4 q^{81} + 26 q^{83} - 2 q^{85} + 4 q^{87} + 14 q^{89} + 38 q^{91} + 6 q^{93} - 6 q^{95} - 12 q^{97}+O(q^{100}) \) 4 * q + 4 * q^3 - 4 * q^5 + 4 * q^9 - 2 * q^13 - 4 * q^15 + 2 * q^17 + 6 * q^19 - 4 * q^23 + 4 * q^25 + 4 * q^27 + 4 * q^29 + 6 * q^31 - 4 * q^37 - 2 * q^39 + 8 * q^41 + 20 * q^43 - 4 * q^45 - 6 * q^47 + 10 * q^49 + 2 * q^51 - 4 * q^53 + 6 * q^57 + 4 * q^59 - 4 * q^61 + 2 * q^65 + 26 * q^67 - 4 * q^69 - 18 * q^73 + 4 * q^75 + 6 * q^77 + 18 * q^79 + 4 * q^81 + 26 * q^83 - 2 * q^85 + 4 * q^87 + 14 * q^89 + 38 * q^91 + 6 * q^93 - 6 * q^95 - 12 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more