LMFDB - Embedded newform 4620.2.a.t.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4620,2,Mod(1,4620)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4620, 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("4620.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4620 = 2^{2} \cdot 3 \cdot 5 \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4620.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:	$36.8908857338$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.1944.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 9x - 6 $ x^3 - 9*x - 6
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$3.28995$ of defining polynomial
Character	$\chi$	$=$	4620.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} -1.00000 q^{5} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} -1.00000 q^{5} +1.00000 q^{7} +1.00000 q^{9} +1.00000 q^{11} -1.53379 q^{13} +1.00000 q^{15} +1.75615 q^{17} -4.82374 q^{19} -1.00000 q^{21} +3.75615 q^{23} +1.00000 q^{25} -1.00000 q^{27} -7.86984 q^{29} +1.53379 q^{31} -1.00000 q^{33} -1.00000 q^{35} +6.11368 q^{37} +1.53379 q^{39} +12.1137 q^{41} +1.28995 q^{43} -1.00000 q^{45} +1.53379 q^{47} +1.00000 q^{49} -1.75615 q^{51} -8.33604 q^{53} -1.00000 q^{55} +4.82374 q^{57} -8.35753 q^{59} -5.75615 q^{61} +1.00000 q^{63} +1.53379 q^{65} +3.51230 q^{67} -3.75615 q^{69} +2.95390 q^{71} +3.06759 q^{73} -1.00000 q^{75} +1.00000 q^{77} +16.6936 q^{79} +1.00000 q^{81} +1.75615 q^{83} -1.75615 q^{85} +7.86984 q^{87} +3.28995 q^{89} -1.53379 q^{91} -1.53379 q^{93} +4.82374 q^{95} -16.9374 q^{97} +1.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{3} - 3 q^{5} + 3 q^{7} + 3 q^{9}+O(q^{10}) $ 3 * q - 3 * q^3 - 3 * q^5 + 3 * q^7 + 3 * q^9	$ 3 q - 3 q^{3} - 3 q^{5} + 3 q^{7} + 3 q^{9} + 3 q^{11} + 3 q^{15} - 3 q^{21} + 6 q^{23} + 3 q^{25} - 3 q^{27} + 6 q^{29} - 3 q^{33} - 3 q^{35} - 6 q^{37} + 12 q^{41} - 6 q^{43} - 3 q^{45} + 3 q^{49} - 3 q^{55} - 6 q^{59} - 12 q^{61} + 3 q^{63} - 6 q^{69} + 24 q^{71} - 3 q^{75} + 3 q^{77} + 6 q^{79} + 3 q^{81} - 6 q^{87} - 12 q^{97} + 3 q^{99}+O(q^{100}) $ 3 * q - 3 * q^3 - 3 * q^5 + 3 * q^7 + 3 * q^9 + 3 * q^11 + 3 * q^15 - 3 * q^21 + 6 * q^23 + 3 * q^25 - 3 * q^27 + 6 * q^29 - 3 * q^33 - 3 * q^35 - 6 * q^37 + 12 * q^41 - 6 * q^43 - 3 * q^45 + 3 * q^49 - 3 * q^55 - 6 * q^59 - 12 * q^61 + 3 * q^63 - 6 * q^69 + 24 * q^71 - 3 * q^75 + 3 * q^77 + 6 * q^79 + 3 * q^81 - 6 * q^87 - 12 * q^97 + 3 * q^99

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
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	−1.53379	−0.425398	−0.212699	−	0.977118i	$-0.568225\pi$
$13$	−1.53379	−0.425398	−0.212699	+	0.977118i	$0.568225\pi$
$14$	0	0
$15$	1.00000	0.258199
$16$	0	0
$17$	1.75615	0.425929	0.212965	−	0.977060i	$-0.431688\pi$
$17$	1.75615	0.425929	0.212965	+	0.977060i	$0.431688\pi$
$18$	0	0
$19$	−4.82374	−1.10664	−0.553321	−	0.832968i	$-0.686640\pi$
$19$	−4.82374	−1.10664	−0.553321	+	0.832968i	$0.686640\pi$
$20$	0	0
$21$	−1.00000	−0.218218
$22$	0	0
$23$	3.75615	0.783212	0.391606	−	0.920133i	$-0.371920\pi$
$23$	3.75615	0.783212	0.391606	+	0.920133i	$0.371920\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−7.86984	−1.46139	−0.730696	−	0.682703i	$-0.760804\pi$
$29$	−7.86984	−1.46139	−0.730696	+	0.682703i	$0.760804\pi$
$30$	0	0
$31$	1.53379	0.275477	0.137739	−	0.990469i	$-0.456017\pi$
$31$	1.53379	0.275477	0.137739	+	0.990469i	$0.456017\pi$
$32$	0	0
$33$	−1.00000	−0.174078
$34$	0	0
$35$	−1.00000	−0.169031
$36$	0	0
$37$	6.11368	1.00508	0.502542	−	0.864553i	$-0.332398\pi$
$37$	6.11368	1.00508	0.502542	+	0.864553i	$0.332398\pi$
$38$	0	0
$39$	1.53379	0.245604
$40$	0	0
$41$	12.1137	1.89184	0.945920	−	0.324401i	$-0.105163\pi$
$41$	12.1137	1.89184	0.945920	+	0.324401i	$0.105163\pi$
$42$	0	0
$43$	1.28995	0.196715	0.0983574	−	0.995151i	$-0.468641\pi$
$43$	1.28995	0.196715	0.0983574	+	0.995151i	$0.468641\pi$
$44$	0	0
$45$	−1.00000	−0.149071
$46$	0	0
$47$	1.53379	0.223727	0.111863	−	0.993724i	$-0.464318\pi$
$47$	1.53379	0.223727	0.111863	+	0.993724i	$0.464318\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−1.75615	−0.245910
$52$	0	0
$53$	−8.33604	−1.14504	−0.572522	−	0.819890i	$-0.694035\pi$
$53$	−8.33604	−1.14504	−0.572522	+	0.819890i	$0.694035\pi$
$54$	0	0
$55$	−1.00000	−0.134840
$56$	0	0
$57$	4.82374	0.638920
$58$	0	0
$59$	−8.35753	−1.08806	−0.544029	−	0.839066i	$-0.683102\pi$
$59$	−8.35753	−1.08806	−0.544029	+	0.839066i	$0.683102\pi$
$60$	0	0
$61$	−5.75615	−0.737000	−0.368500	−	0.929628i	$-0.620128\pi$
$61$	−5.75615	−0.737000	−0.368500	+	0.929628i	$0.620128\pi$
$62$	0	0
$63$	1.00000	0.125988
$64$	0	0
$65$	1.53379	0.190244
$66$	0	0
$67$	3.51230	0.429096	0.214548	−	0.976713i	$-0.431172\pi$
$67$	3.51230	0.429096	0.214548	+	0.976713i	$0.431172\pi$
$68$	0	0
$69$	−3.75615	−0.452188
$70$	0	0
$71$	2.95390	0.350564	0.175282	−	0.984518i	$-0.443916\pi$
$71$	2.95390	0.350564	0.175282	+	0.984518i	$0.443916\pi$
$72$	0	0
$73$	3.06759	0.359034	0.179517	−	0.983755i	$-0.442547\pi$
$73$	3.06759	0.359034	0.179517	+	0.983755i	$0.442547\pi$
$74$	0	0
$75$	−1.00000	−0.115470
$76$	0	0
$77$	1.00000	0.113961
$78$	0	0
$79$	16.6936	1.87817	0.939087	−	0.343680i	$-0.111673\pi$
$79$	16.6936	1.87817	0.939087	+	0.343680i	$0.111673\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	1.75615	0.192763	0.0963813	−	0.995344i	$-0.469273\pi$
$83$	1.75615	0.192763	0.0963813	+	0.995344i	$0.469273\pi$
$84$	0	0
$85$	−1.75615	−0.190481
$86$	0	0
$87$	7.86984	0.843735
$88$	0	0
$89$	3.28995	0.348733	0.174367	−	0.984681i	$-0.444212\pi$
$89$	3.28995	0.348733	0.174367	+	0.984681i	$0.444212\pi$
$90$	0	0
$91$	−1.53379	−0.160785
$92$	0	0
$93$	−1.53379	−0.159047
$94$	0	0
$95$	4.82374	0.494905
$96$	0	0
$97$	−16.9374	−1.71973	−0.859867	−	0.510518i	$-0.829454\pi$
$97$	−16.9374	−1.71973	−0.859867	+	0.510518i	$0.829454\pi$
$98$	0	0
$99$	1.00000	0.100504
$100$	0	0
$101$	7.51230	0.747502	0.373751	−	0.927529i	$-0.378071\pi$
$101$	7.51230	0.747502	0.373751	+	0.927529i	$0.378071\pi$
$102$	0	0
$103$	−11.8698	−1.16957	−0.584785	−	0.811188i	$-0.698821\pi$
$103$	−11.8698	−1.16957	−0.584785	+	0.811188i	$0.698821\pi$
$104$	0	0
$105$	1.00000	0.0975900
$106$	0	0
$107$	−15.2735	−1.47654	−0.738271	−	0.674504i	$-0.764357\pi$
$107$	−15.2735	−1.47654	−0.738271	+	0.674504i	$0.764357\pi$
$108$	0	0
$109$	−7.04610	−0.674894	−0.337447	−	0.941345i	$-0.609563\pi$
$109$	−7.04610	−0.674894	−0.337447	+	0.941345i	$0.609563\pi$
$110$	0	0
$111$	−6.11368	−0.580285
$112$	0	0
$113$	16.3360	1.53677	0.768383	−	0.639991i	$-0.221062\pi$
$113$	16.3360	1.53677	0.768383	+	0.639991i	$0.221062\pi$
$114$	0	0
$115$	−3.75615	−0.350263
$116$	0	0
$117$	−1.53379	−0.141799
$118$	0	0
$119$	1.75615	0.160986
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−12.1137	−1.09225
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−5.28995	−0.469407	−0.234703	−	0.972067i	$-0.575412\pi$
$127$	−5.28995	−0.469407	−0.234703	+	0.972067i	$0.575412\pi$
$128$	0	0
$129$	−1.28995	−0.113573
$130$	0	0
$131$	5.53379	0.483490	0.241745	−	0.970340i	$-0.422280\pi$
$131$	5.53379	0.483490	0.241745	+	0.970340i	$0.422280\pi$
$132$	0	0
$133$	−4.82374	−0.418271
$134$	0	0
$135$	1.00000	0.0860663
$136$	0	0
$137$	13.5123	1.15443	0.577217	−	0.816591i	$-0.304139\pi$
$137$	13.5123	1.15443	0.577217	+	0.816591i	$0.304139\pi$
$138$	0	0
$139$	11.1598	0.946560	0.473280	−	0.880912i	$-0.343070\pi$
$139$	11.1598	0.946560	0.473280	+	0.880912i	$0.343070\pi$
$140$	0	0
$141$	−1.53379	−0.129169
$142$	0	0
$143$	−1.53379	−0.128262
$144$	0	0
$145$	7.86984	0.653554
$146$	0	0
$147$	−1.00000	−0.0824786
$148$	0	0
$149$	6.57989	0.539046	0.269523	−	0.962994i	$-0.413134\pi$
$149$	6.57989	0.539046	0.269523	+	0.962994i	$0.413134\pi$
$150$	0	0
$151$	14.0000	1.13930	0.569652	−	0.821886i	$-0.307078\pi$
$151$	14.0000	1.13930	0.569652	+	0.821886i	$0.307078\pi$
$152$	0	0
$153$	1.75615	0.141976
$154$	0	0
$155$	−1.53379	−0.123197
$156$	0	0
$157$	−23.5173	−1.87689	−0.938443	−	0.345434i	$-0.887732\pi$
$157$	−23.5173	−1.87689	−0.938443	+	0.345434i	$0.887732\pi$
$158$	0	0
$159$	8.33604	0.661091
$160$	0	0
$161$	3.75615	0.296026
$162$	0	0
$163$	17.7612	1.39116	0.695581	−	0.718448i	$-0.255147\pi$
$163$	17.7612	1.39116	0.695581	+	0.718448i	$0.255147\pi$
$164$	0	0
$165$	1.00000	0.0778499
$166$	0	0
$167$	12.1352	0.939048	0.469524	−	0.882920i	$-0.344426\pi$
$167$	12.1352	0.939048	0.469524	+	0.882920i	$0.344426\pi$
$168$	0	0
$169$	−10.6475	−0.819037
$170$	0	0
$171$	−4.82374	−0.368881
$172$	0	0
$173$	20.8073	1.58195	0.790973	−	0.611851i	$-0.209575\pi$
$173$	20.8073	1.58195	0.790973	+	0.611851i	$0.209575\pi$
$174$	0	0
$175$	1.00000	0.0755929
$176$	0	0
$177$	8.35753	0.628191
$178$	0	0
$179$	26.2489	1.96193	0.980966	−	0.194180i	$-0.0622046\pi$
$179$	26.2489	1.96193	0.980966	+	0.194180i	$0.0622046\pi$
$180$	0	0
$181$	23.1598	1.72145	0.860726	−	0.509068i	$-0.170010\pi$
$181$	23.1598	1.72145	0.860726	+	0.509068i	$0.170010\pi$
$182$	0	0
$183$	5.75615	0.425507
$184$	0	0
$185$	−6.11368	−0.449487
$186$	0	0
$187$	1.75615	0.128423
$188$	0	0
$189$	−1.00000	−0.0727393
$190$	0	0
$191$	15.0246	1.08714	0.543571	−	0.839363i	$-0.317072\pi$
$191$	15.0246	1.08714	0.543571	+	0.839363i	$0.317072\pi$
$192$	0	0
$193$	12.7151	0.915250	0.457625	−	0.889145i	$-0.348700\pi$
$193$	12.7151	0.915250	0.457625	+	0.889145i	$0.348700\pi$
$194$	0	0
$195$	−1.53379	−0.109837
$196$	0	0
$197$	5.55528	0.395798	0.197899	−	0.980222i	$-0.436588\pi$
$197$	5.55528	0.395798	0.197899	+	0.980222i	$0.436588\pi$
$198$	0	0
$199$	−26.3196	−1.86574	−0.932872	−	0.360208i	$-0.882706\pi$
$199$	−26.3196	−1.86574	−0.932872	+	0.360208i	$0.882706\pi$
$200$	0	0
$201$	−3.51230	−0.247739
$202$	0	0
$203$	−7.86984	−0.552354
$204$	0	0
$205$	−12.1137	−0.846056
$206$	0	0
$207$	3.75615	0.261071
$208$	0	0
$209$	−4.82374	−0.333665
$210$	0	0
$211$	21.0676	1.45035	0.725176	−	0.688563i	$-0.241758\pi$
$211$	21.0676	1.45035	0.725176	+	0.688563i	$0.241758\pi$
$212$	0	0
$213$	−2.95390	−0.202398
$214$	0	0
$215$	−1.28995	−0.0879735
$216$	0	0
$217$	1.53379	0.104121
$218$	0	0
$219$	−3.06759	−0.207288
$220$	0	0
$221$	−2.69357	−0.181189
$222$	0	0
$223$	−9.51731	−0.637326	−0.318663	−	0.947868i	$-0.603234\pi$
$223$	−9.51731	−0.637326	−0.318663	+	0.947868i	$0.603234\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	1.31144	0.0870430	0.0435215	−	0.999052i	$-0.486142\pi$
$227$	1.31144	0.0870430	0.0435215	+	0.999052i	$0.486142\pi$
$228$	0	0
$229$	−8.95390	−0.591691	−0.295845	−	0.955236i	$-0.595601\pi$
$229$	−8.95390	−0.591691	−0.295845	+	0.955236i	$0.595601\pi$
$230$	0	0
$231$	−1.00000	−0.0657952
$232$	0	0
$233$	−22.7858	−1.49275	−0.746373	−	0.665528i	$-0.768206\pi$
$233$	−22.7858	−1.49275	−0.746373	+	0.665528i	$0.768206\pi$
$234$	0	0
$235$	−1.53379	−0.100054
$236$	0	0
$237$	−16.6936	−1.08436
$238$	0	0
$239$	27.8698	1.80275	0.901375	−	0.433040i	$-0.142559\pi$
$239$	27.8698	1.80275	0.901375	+	0.433040i	$0.142559\pi$
$240$	0	0
$241$	4.57989	0.295017	0.147508	−	0.989061i	$-0.452875\pi$
$241$	4.57989	0.295017	0.147508	+	0.989061i	$0.452875\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	−1.00000	−0.0638877
$246$	0	0
$247$	7.39862	0.470763
$248$	0	0
$249$	−1.75615	−0.111292
$250$	0	0
$251$	18.1352	1.14468	0.572341	−	0.820016i	$-0.306035\pi$
$251$	18.1352	1.14468	0.572341	+	0.820016i	$0.306035\pi$
$252$	0	0
$253$	3.75615	0.236147
$254$	0	0
$255$	1.75615	0.109974
$256$	0	0
$257$	−9.62599	−0.600453	−0.300226	−	0.953868i	$-0.597062\pi$
$257$	−9.62599	−0.600453	−0.300226	+	0.953868i	$0.597062\pi$
$258$	0	0
$259$	6.11368	0.379886
$260$	0	0
$261$	−7.86984	−0.487131
$262$	0	0
$263$	17.7397	1.09387	0.546937	−	0.837173i	$-0.315794\pi$
$263$	17.7397	1.09387	0.546937	+	0.837173i	$0.315794\pi$
$264$	0	0
$265$	8.33604	0.512079
$266$	0	0
$267$	−3.28995	−0.201341
$268$	0	0
$269$	31.0296	1.89191	0.945955	−	0.324299i	$-0.105128\pi$
$269$	31.0296	1.89191	0.945955	+	0.324299i	$0.105128\pi$
$270$	0	0
$271$	−9.98352	−0.606456	−0.303228	−	0.952918i	$-0.598064\pi$
$271$	−9.98352	−0.606456	−0.303228	+	0.952918i	$0.598064\pi$
$272$	0	0
$273$	1.53379	0.0928294
$274$	0	0
$275$	1.00000	0.0603023
$276$	0	0
$277$	−14.0922	−0.846718	−0.423359	−	0.905962i	$-0.639149\pi$
$277$	−14.0922	−0.846718	−0.423359	+	0.905962i	$0.639149\pi$
$278$	0	0
$279$	1.53379	0.0918258
$280$	0	0
$281$	30.0922	1.79515	0.897575	−	0.440862i	$-0.145327\pi$
$281$	30.0922	1.79515	0.897575	+	0.440862i	$0.145327\pi$
$282$	0	0
$283$	11.6690	0.693648	0.346824	−	0.937930i	$-0.387260\pi$
$283$	11.6690	0.693648	0.346824	+	0.937930i	$0.387260\pi$
$284$	0	0
$285$	−4.82374	−0.285734
$286$	0	0
$287$	12.1137	0.715048
$288$	0	0
$289$	−13.9159	−0.818584
$290$	0	0
$291$	16.9374	0.992889
$292$	0	0
$293$	−22.9159	−1.33876	−0.669382	−	0.742919i	$-0.733441\pi$
$293$	−22.9159	−1.33876	−0.669382	+	0.742919i	$0.733441\pi$
$294$	0	0
$295$	8.35753	0.486594
$296$	0	0
$297$	−1.00000	−0.0580259
$298$	0	0
$299$	−5.76116	−0.333177
$300$	0	0
$301$	1.28995	0.0743512
$302$	0	0
$303$	−7.51230	−0.431571
$304$	0	0
$305$	5.75615	0.329596
$306$	0	0
$307$	−4.71506	−0.269103	−0.134551	−	0.990907i	$-0.542959\pi$
$307$	−4.71506	−0.269103	−0.134551	+	0.990907i	$0.542959\pi$
$308$	0	0
$309$	11.8698	0.675251
$310$	0	0
$311$	32.2274	1.82745	0.913723	−	0.406337i	$-0.133194\pi$
$311$	32.2274	1.82745	0.913723	+	0.406337i	$0.133194\pi$
$312$	0	0
$313$	−2.84523	−0.160822	−0.0804109	−	0.996762i	$-0.525623\pi$
$313$	−2.84523	−0.160822	−0.0804109	+	0.996762i	$0.525623\pi$
$314$	0	0
$315$	−1.00000	−0.0563436
$316$	0	0
$317$	27.8748	1.56561	0.782804	−	0.622269i	$-0.213789\pi$
$317$	27.8748	1.56561	0.782804	+	0.622269i	$0.213789\pi$
$318$	0	0
$319$	−7.86984	−0.440626
$320$	0	0
$321$	15.2735	0.852482
$322$	0	0
$323$	−8.47122	−0.471351
$324$	0	0
$325$	−1.53379	−0.0850796
$326$	0	0
$327$	7.04610	0.389650
$328$	0	0
$329$	1.53379	0.0845608
$330$	0	0
$331$	−0.471216	−0.0259004	−0.0129502	−	0.999916i	$-0.504122\pi$
$331$	−0.471216	−0.0259004	−0.0129502	+	0.999916i	$0.504122\pi$
$332$	0	0
$333$	6.11368	0.335028
$334$	0	0
$335$	−3.51230	−0.191898
$336$	0	0
$337$	21.9620	1.19635	0.598174	−	0.801366i	$-0.295893\pi$
$337$	21.9620	1.19635	0.598174	+	0.801366i	$0.295893\pi$
$338$	0	0
$339$	−16.3360	−0.887252
$340$	0	0
$341$	1.53379	0.0830596
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	3.75615	0.202224
$346$	0	0
$347$	−2.71506	−0.145752	−0.0728761	−	0.997341i	$-0.523218\pi$
$347$	−2.71506	−0.145752	−0.0728761	+	0.997341i	$0.523218\pi$
$348$	0	0
$349$	36.0757	1.93109	0.965544	−	0.260239i	$-0.0838013\pi$
$349$	36.0757	1.93109	0.965544	+	0.260239i	$0.0838013\pi$
$350$	0	0
$351$	1.53379	0.0818678
$352$	0	0
$353$	−1.95702	−0.104162	−0.0520808	−	0.998643i	$-0.516585\pi$
$353$	−1.95702	−0.104162	−0.0520808	+	0.998643i	$0.516585\pi$
$354$	0	0
$355$	−2.95390	−0.156777
$356$	0	0
$357$	−1.75615	−0.0929454
$358$	0	0
$359$	−22.0050	−1.16138	−0.580690	−	0.814125i	$-0.697217\pi$
$359$	−22.0050	−1.16138	−0.580690	+	0.814125i	$0.697217\pi$
$360$	0	0
$361$	4.26845	0.224656
$362$	0	0
$363$	−1.00000	−0.0524864
$364$	0	0
$365$	−3.06759	−0.160565
$366$	0	0
$367$	15.1648	0.791596	0.395798	−	0.918338i	$-0.370468\pi$
$367$	15.1648	0.791596	0.395798	+	0.918338i	$0.370468\pi$
$368$	0	0
$369$	12.1137	0.630613
$370$	0	0
$371$	−8.33604	−0.432786
$372$	0	0
$373$	−28.8022	−1.49132	−0.745662	−	0.666324i	$-0.767867\pi$
$373$	−28.8022	−1.49132	−0.745662	+	0.666324i	$0.767867\pi$
$374$	0	0
$375$	1.00000	0.0516398
$376$	0	0
$377$	12.0707	0.621673
$378$	0	0
$379$	28.4712	1.46247	0.731234	−	0.682127i	$-0.238945\pi$
$379$	28.4712	1.46247	0.731234	+	0.682127i	$0.238945\pi$
$380$	0	0
$381$	5.28995	0.271012
$382$	0	0
$383$	−9.04610	−0.462234	−0.231117	−	0.972926i	$-0.574238\pi$
$383$	−9.04610	−0.462234	−0.231117	+	0.972926i	$0.574238\pi$
$384$	0	0
$385$	−1.00000	−0.0509647
$386$	0	0
$387$	1.28995	0.0655716
$388$	0	0
$389$	−4.02149	−0.203898	−0.101949	−	0.994790i	$-0.532508\pi$
$389$	−4.02149	−0.203898	−0.101949	+	0.994790i	$0.532508\pi$
$390$	0	0
$391$	6.59637	0.333593
$392$	0	0
$393$	−5.53379	−0.279143
$394$	0	0
$395$	−16.6936	−0.839945
$396$	0	0
$397$	9.73967	0.488820	0.244410	−	0.969672i	$-0.421406\pi$
$397$	9.73967	0.488820	0.244410	+	0.969672i	$0.421406\pi$
$398$	0	0
$399$	4.82374	0.241489
$400$	0	0
$401$	−28.9209	−1.44424	−0.722121	−	0.691766i	$-0.756833\pi$
$401$	−28.9209	−1.44424	−0.722121	+	0.691766i	$0.756833\pi$
$402$	0	0
$403$	−2.35252	−0.117188
$404$	0	0
$405$	−1.00000	−0.0496904
$406$	0	0
$407$	6.11368	0.303044
$408$	0	0
$409$	−31.8748	−1.57611	−0.788055	−	0.615605i	$-0.788912\pi$
$409$	−31.8748	−1.57611	−0.788055	+	0.615605i	$0.788912\pi$
$410$	0	0
$411$	−13.5123	−0.666513
$412$	0	0
$413$	−8.35753	−0.411247
$414$	0	0
$415$	−1.75615	−0.0862061
$416$	0	0
$417$	−11.1598	−0.546497
$418$	0	0
$419$	28.1402	1.37474	0.687369	−	0.726308i	$-0.258766\pi$
$419$	28.1402	1.37474	0.687369	+	0.726308i	$0.258766\pi$
$420$	0	0
$421$	−0.379023	−0.0184724	−0.00923622	−	0.999957i	$-0.502940\pi$
$421$	−0.379023	−0.0184724	−0.00923622	+	0.999957i	$0.502940\pi$
$422$	0	0
$423$	1.53379	0.0745756
$424$	0	0
$425$	1.75615	0.0851859
$426$	0	0
$427$	−5.75615	−0.278560
$428$	0	0
$429$	1.53379	0.0740523
$430$	0	0
$431$	−4.22737	−0.203625	−0.101813	−	0.994804i	$-0.532464\pi$
$431$	−4.22737	−0.203625	−0.101813	+	0.994804i	$0.532464\pi$
$432$	0	0
$433$	20.3625	0.978561	0.489281	−	0.872126i	$-0.337259\pi$
$433$	20.3625	0.978561	0.489281	+	0.872126i	$0.337259\pi$
$434$	0	0
$435$	−7.86984	−0.377330
$436$	0	0
$437$	−18.1187	−0.866735
$438$	0	0
$439$	8.37902	0.399909	0.199954	−	0.979805i	$-0.435921\pi$
$439$	8.37902	0.399909	0.199954	+	0.979805i	$0.435921\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	−6.62287	−0.314662	−0.157331	−	0.987546i	$-0.550289\pi$
$443$	−6.62287	−0.314662	−0.157331	+	0.987546i	$0.550289\pi$
$444$	0	0
$445$	−3.28995	−0.155958
$446$	0	0
$447$	−6.57989	−0.311218
$448$	0	0
$449$	−0.693574	−0.0327318	−0.0163659	−	0.999866i	$-0.505210\pi$
$449$	−0.693574	−0.0327318	−0.0163659	+	0.999866i	$0.505210\pi$
$450$	0	0
$451$	12.1137	0.570411
$452$	0	0
$453$	−14.0000	−0.657777
$454$	0	0
$455$	1.53379	0.0719054
$456$	0	0
$457$	−17.2470	−0.806779	−0.403389	−	0.915028i	$-0.632168\pi$
$457$	−17.2470	−0.806779	−0.403389	+	0.915028i	$0.632168\pi$
$458$	0	0
$459$	−1.75615	−0.0819701
$460$	0	0
$461$	−39.0346	−1.81802	−0.909012	−	0.416770i	$-0.863162\pi$
$461$	−39.0346	−1.81802	−0.909012	+	0.416770i	$0.863162\pi$
$462$	0	0
$463$	12.4447	0.578355	0.289177	−	0.957275i	$-0.406618\pi$
$463$	12.4447	0.578355	0.289177	+	0.957275i	$0.406618\pi$
$464$	0	0
$465$	1.53379	0.0711280
$466$	0	0
$467$	15.0676	0.697245	0.348622	−	0.937263i	$-0.386650\pi$
$467$	15.0676	0.697245	0.348622	+	0.937263i	$0.386650\pi$
$468$	0	0
$469$	3.51230	0.162183
$470$	0	0
$471$	23.5173	1.08362
$472$	0	0
$473$	1.28995	0.0593117
$474$	0	0
$475$	−4.82374	−0.221328
$476$	0	0
$477$	−8.33604	−0.381681
$478$	0	0
$479$	−3.44160	−0.157251	−0.0786254	−	0.996904i	$-0.525053\pi$
$479$	−3.44160	−0.157251	−0.0786254	+	0.996904i	$0.525053\pi$
$480$	0	0
$481$	−9.37713	−0.427560
$482$	0	0
$483$	−3.75615	−0.170911
$484$	0	0
$485$	16.9374	0.769089
$486$	0	0
$487$	13.4201	0.608123	0.304062	−	0.952652i	$-0.401657\pi$
$487$	13.4201	0.608123	0.304062	+	0.952652i	$0.401657\pi$
$488$	0	0
$489$	−17.7612	−0.803187
$490$	0	0
$491$	18.4927	0.834564	0.417282	−	0.908777i	$-0.362983\pi$
$491$	18.4927	0.834564	0.417282	+	0.908777i	$0.362983\pi$
$492$	0	0
$493$	−13.8206	−0.622450
$494$	0	0
$495$	−1.00000	−0.0449467
$496$	0	0
$497$	2.95390	0.132501
$498$	0	0
$499$	−29.4036	−1.31629	−0.658144	−	0.752892i	$-0.728658\pi$
$499$	−29.4036	−1.31629	−0.658144	+	0.752892i	$0.728658\pi$
$500$	0	0
$501$	−12.1352	−0.542160
$502$	0	0
$503$	3.62098	0.161451	0.0807257	−	0.996736i	$-0.474276\pi$
$503$	3.62098	0.161451	0.0807257	+	0.996736i	$0.474276\pi$
$504$	0	0
$505$	−7.51230	−0.334293
$506$	0	0
$507$	10.6475	0.472871
$508$	0	0
$509$	−15.2470	−0.675810	−0.337905	−	0.941180i	$-0.609718\pi$
$509$	−15.2470	−0.675810	−0.337905	+	0.941180i	$0.609718\pi$
$510$	0	0
$511$	3.06759	0.135702
$512$	0	0
$513$	4.82374	0.212973
$514$	0	0
$515$	11.8698	0.523047
$516$	0	0
$517$	1.53379	0.0674562
$518$	0	0
$519$	−20.8073	−0.913337
$520$	0	0
$521$	−18.8022	−0.823741	−0.411871	−	0.911242i	$-0.635124\pi$
$521$	−18.8022	−0.823741	−0.411871	+	0.911242i	$0.635124\pi$
$522$	0	0
$523$	−5.20276	−0.227501	−0.113750	−	0.993509i	$-0.536286\pi$
$523$	−5.20276	−0.227501	−0.113750	+	0.993509i	$0.536286\pi$
$524$	0	0
$525$	−1.00000	−0.0436436
$526$	0	0
$527$	2.69357	0.117334
$528$	0	0
$529$	−8.89133	−0.386579
$530$	0	0
$531$	−8.35753	−0.362686
$532$	0	0
$533$	−18.5799	−0.804784
$534$	0	0
$535$	15.2735	0.660329
$536$	0	0
$537$	−26.2489	−1.13272
$538$	0	0
$539$	1.00000	0.0430730
$540$	0	0
$541$	19.2735	0.828631	0.414315	−	0.910133i	$-0.364021\pi$
$541$	19.2735	0.828631	0.414315	+	0.910133i	$0.364021\pi$
$542$	0	0
$543$	−23.1598	−0.993881
$544$	0	0
$545$	7.04610	0.301822
$546$	0	0
$547$	10.2224	0.437076	0.218538	−	0.975828i	$-0.429871\pi$
$547$	10.2224	0.437076	0.218538	+	0.975828i	$0.429871\pi$
$548$	0	0
$549$	−5.75615	−0.245667
$550$	0	0
$551$	37.9620	1.61724
$552$	0	0
$553$	16.6936	0.709883
$554$	0	0
$555$	6.11368	0.259511
$556$	0	0
$557$	42.4118	1.79704	0.898522	−	0.438929i	$-0.144642\pi$
$557$	42.4118	1.79704	0.898522	+	0.438929i	$0.144642\pi$
$558$	0	0
$559$	−1.97851	−0.0836820
$560$	0	0
$561$	−1.75615	−0.0741448
$562$	0	0
$563$	32.5469	1.37169	0.685845	−	0.727748i	$-0.259433\pi$
$563$	32.5469	1.37169	0.685845	+	0.727748i	$0.259433\pi$
$564$	0	0
$565$	−16.3360	−0.687262
$566$	0	0
$567$	1.00000	0.0419961
$568$	0	0
$569$	17.9620	0.753007	0.376504	−	0.926415i	$-0.377126\pi$
$569$	17.9620	0.753007	0.376504	+	0.926415i	$0.377126\pi$
$570$	0	0
$571$	18.5154	0.774846	0.387423	−	0.921902i	$-0.373365\pi$
$571$	18.5154	0.774846	0.387423	+	0.921902i	$0.373365\pi$
$572$	0	0
$573$	−15.0246	−0.627662
$574$	0	0
$575$	3.75615	0.156642
$576$	0	0
$577$	31.4301	1.30845	0.654227	−	0.756299i	$-0.272994\pi$
$577$	31.4301	1.30845	0.654227	+	0.756299i	$0.272994\pi$
$578$	0	0
$579$	−12.7151	−0.528420
$580$	0	0
$581$	1.75615	0.0728574
$582$	0	0
$583$	−8.33604	−0.345244
$584$	0	0
$585$	1.53379	0.0634146
$586$	0	0
$587$	−6.95390	−0.287018	−0.143509	−	0.989649i	$-0.545839\pi$
$587$	−6.95390	−0.287018	−0.143509	+	0.989649i	$0.545839\pi$
$588$	0	0
$589$	−7.39862	−0.304855
$590$	0	0
$591$	−5.55528	−0.228514
$592$	0	0
$593$	−9.55528	−0.392388	−0.196194	−	0.980565i	$-0.562858\pi$
$593$	−9.55528	−0.392388	−0.196194	+	0.980565i	$0.562858\pi$
$594$	0	0
$595$	−1.75615	−0.0719952
$596$	0	0
$597$	26.3196	1.07719
$598$	0	0
$599$	−27.0346	−1.10460	−0.552302	−	0.833644i	$-0.686251\pi$
$599$	−27.0346	−1.10460	−0.552302	+	0.833644i	$0.686251\pi$
$600$	0	0
$601$	−28.5204	−1.16337	−0.581686	−	0.813413i	$-0.697607\pi$
$601$	−28.5204	−1.16337	−0.581686	+	0.813413i	$0.697607\pi$
$602$	0	0
$603$	3.51230	0.143032
$604$	0	0
$605$	−1.00000	−0.0406558
$606$	0	0
$607$	10.8502	0.440398	0.220199	−	0.975455i	$-0.429329\pi$
$607$	10.8502	0.440398	0.220199	+	0.975455i	$0.429329\pi$
$608$	0	0
$609$	7.86984	0.318902
$610$	0	0
$611$	−2.35252	−0.0951729
$612$	0	0
$613$	−37.8319	−1.52801	−0.764007	−	0.645208i	$-0.776771\pi$
$613$	−37.8319	−1.52801	−0.764007	+	0.645208i	$0.776771\pi$
$614$	0	0
$615$	12.1137	0.488471
$616$	0	0
$617$	−5.73967	−0.231070	−0.115535	−	0.993303i	$-0.536858\pi$
$617$	−5.73967	−0.231070	−0.115535	+	0.993303i	$0.536858\pi$
$618$	0	0
$619$	8.67208	0.348560	0.174280	−	0.984696i	$-0.444240\pi$
$619$	8.67208	0.348560	0.174280	+	0.984696i	$0.444240\pi$
$620$	0	0
$621$	−3.75615	−0.150729
$622$	0	0
$623$	3.28995	0.131809
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	4.82374	0.192642
$628$	0	0
$629$	10.7366	0.428094
$630$	0	0
$631$	−10.1187	−0.402819	−0.201409	−	0.979507i	$-0.564552\pi$
$631$	−10.1187	−0.402819	−0.201409	+	0.979507i	$0.564552\pi$
$632$	0	0
$633$	−21.0676	−0.837361
$634$	0	0
$635$	5.28995	0.209925
$636$	0	0
$637$	−1.53379	−0.0607711
$638$	0	0
$639$	2.95390	0.116855
$640$	0	0
$641$	−26.1844	−1.03422	−0.517111	−	0.855919i	$-0.672992\pi$
$641$	−26.1844	−1.03422	−0.517111	+	0.855919i	$0.672992\pi$
$642$	0	0
$643$	−28.5419	−1.12558	−0.562792	−	0.826599i	$-0.690273\pi$
$643$	−28.5419	−1.12558	−0.562792	+	0.826599i	$0.690273\pi$
$644$	0	0
$645$	1.28995	0.0507915
$646$	0	0
$647$	−20.2274	−0.795220	−0.397610	−	0.917555i	$-0.630160\pi$
$647$	−20.2274	−0.795220	−0.397610	+	0.917555i	$0.630160\pi$
$648$	0	0
$649$	−8.35753	−0.328062
$650$	0	0
$651$	−1.53379	−0.0601141
$652$	0	0
$653$	3.63100	0.142092	0.0710459	−	0.997473i	$-0.477366\pi$
$653$	3.63100	0.142092	0.0710459	+	0.997473i	$0.477366\pi$
$654$	0	0
$655$	−5.53379	−0.216223
$656$	0	0
$657$	3.06759	0.119678
$658$	0	0
$659$	−2.71005	−0.105569	−0.0527844	−	0.998606i	$-0.516810\pi$
$659$	−2.71005	−0.105569	−0.0527844	+	0.998606i	$0.516810\pi$
$660$	0	0
$661$	−12.9539	−0.503848	−0.251924	−	0.967747i	$-0.581063\pi$
$661$	−12.9539	−0.503848	−0.251924	+	0.967747i	$0.581063\pi$
$662$	0	0
$663$	2.69357	0.104610
$664$	0	0
$665$	4.82374	0.187057
$666$	0	0
$667$	−29.5603	−1.14458
$668$	0	0
$669$	9.51731	0.367961
$670$	0	0
$671$	−5.75615	−0.222214
$672$	0	0
$673$	−8.58490	−0.330924	−0.165462	−	0.986216i	$-0.552911\pi$
$673$	−8.58490	−0.330924	−0.165462	+	0.986216i	$0.552911\pi$
$674$	0	0
$675$	−1.00000	−0.0384900
$676$	0	0
$677$	−47.5880	−1.82896	−0.914478	−	0.404636i	$-0.867398\pi$
$677$	−47.5880	−1.82896	−0.914478	+	0.404636i	$0.867398\pi$
$678$	0	0
$679$	−16.9374	−0.649999
$680$	0	0
$681$	−1.31144	−0.0502543
$682$	0	0
$683$	18.8402	0.720901	0.360450	−	0.932778i	$-0.382623\pi$
$683$	18.8402	0.720901	0.360450	+	0.932778i	$0.382623\pi$
$684$	0	0
$685$	−13.5123	−0.516279
$686$	0	0
$687$	8.95390	0.341613
$688$	0	0
$689$	12.7858	0.487099
$690$	0	0
$691$	−13.6475	−0.519175	−0.259587	−	0.965720i	$-0.583587\pi$
$691$	−13.6475	−0.519175	−0.259587	+	0.965720i	$0.583587\pi$
$692$	0	0
$693$	1.00000	0.0379869
$694$	0	0
$695$	−11.1598	−0.423315
$696$	0	0
$697$	21.2735	0.805790
$698$	0	0
$699$	22.7858	0.861837
$700$	0	0
$701$	35.8698	1.35478	0.677392	−	0.735622i	$-0.263110\pi$
$701$	35.8698	1.35478	0.677392	+	0.735622i	$0.263110\pi$
$702$	0	0
$703$	−29.4908	−1.11227
$704$	0	0
$705$	1.53379	0.0577660
$706$	0	0
$707$	7.51230	0.282529
$708$	0	0
$709$	−39.1863	−1.47167	−0.735836	−	0.677160i	$-0.763210\pi$
$709$	−39.1863	−1.47167	−0.735836	+	0.677160i	$0.763210\pi$
$710$	0	0
$711$	16.6936	0.626058
$712$	0	0
$713$	5.76116	0.215757
$714$	0	0
$715$	1.53379	0.0573606
$716$	0	0
$717$	−27.8698	−1.04082
$718$	0	0
$719$	37.9620	1.41574	0.707872	−	0.706340i	$-0.249655\pi$
$719$	37.9620	1.41574	0.707872	+	0.706340i	$0.249655\pi$
$720$	0	0
$721$	−11.8698	−0.442056
$722$	0	0
$723$	−4.57989	−0.170328
$724$	0	0
$725$	−7.86984	−0.292278
$726$	0	0
$727$	−2.89444	−0.107349	−0.0536744	−	0.998558i	$-0.517093\pi$
$727$	−2.89444	−0.107349	−0.0536744	+	0.998558i	$0.517093\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	2.26534	0.0837866
$732$	0	0
$733$	−48.7151	−1.79933	−0.899666	−	0.436579i	$-0.856190\pi$
$733$	−48.7151	−1.79933	−0.899666	+	0.436579i	$0.856190\pi$
$734$	0	0
$735$	1.00000	0.0368856
$736$	0	0
$737$	3.51230	0.129377
$738$	0	0
$739$	45.5223	1.67457	0.837283	−	0.546770i	$-0.184143\pi$
$739$	45.5223	1.67457	0.837283	+	0.546770i	$0.184143\pi$
$740$	0	0
$741$	−7.39862	−0.271795
$742$	0	0
$743$	−14.9324	−0.547817	−0.273909	−	0.961756i	$-0.588317\pi$
$743$	−14.9324	−0.547817	−0.273909	+	0.961756i	$0.588317\pi$
$744$	0	0
$745$	−6.57989	−0.241069
$746$	0	0
$747$	1.75615	0.0642542
$748$	0	0
$749$	−15.2735	−0.558080
$750$	0	0
$751$	46.3031	1.68962	0.844812	−	0.535064i	$-0.179712\pi$
$751$	46.3031	1.68962	0.844812	+	0.535064i	$0.179712\pi$
$752$	0	0
$753$	−18.1352	−0.660882
$754$	0	0
$755$	−14.0000	−0.509512
$756$	0	0
$757$	−42.4118	−1.54148	−0.770741	−	0.637149i	$-0.780114\pi$
$757$	−42.4118	−1.54148	−0.770741	+	0.637149i	$0.780114\pi$
$758$	0	0
$759$	−3.75615	−0.136340
$760$	0	0
$761$	−9.64748	−0.349721	−0.174860	−	0.984593i	$-0.555947\pi$
$761$	−9.64748	−0.349721	−0.174860	+	0.984593i	$0.555947\pi$
$762$	0	0
$763$	−7.04610	−0.255086
$764$	0	0
$765$	−1.75615	−0.0634938
$766$	0	0
$767$	12.8187	0.462857
$768$	0	0
$769$	−5.26845	−0.189985	−0.0949927	−	0.995478i	$-0.530283\pi$
$769$	−5.26845	−0.189985	−0.0949927	+	0.995478i	$0.530283\pi$
$770$	0	0
$771$	9.62599	0.346671
$772$	0	0
$773$	−31.2028	−1.12229	−0.561143	−	0.827719i	$-0.689638\pi$
$773$	−31.2028	−1.12229	−0.561143	+	0.827719i	$0.689638\pi$
$774$	0	0
$775$	1.53379	0.0550955
$776$	0	0
$777$	−6.11368	−0.219327
$778$	0	0
$779$	−58.4332	−2.09359
$780$	0	0
$781$	2.95390	0.105699
$782$	0	0
$783$	7.86984	0.281245
$784$	0	0
$785$	23.5173	0.839369
$786$	0	0
$787$	−41.4301	−1.47682	−0.738412	−	0.674350i	$-0.764424\pi$
$787$	−41.4301	−1.47682	−0.738412	+	0.674350i	$0.764424\pi$
$788$	0	0
$789$	−17.7397	−0.631549
$790$	0	0
$791$	16.3360	0.580843
$792$	0	0
$793$	8.82875	0.313518
$794$	0	0
$795$	−8.33604	−0.295649
$796$	0	0
$797$	9.95702	0.352696	0.176348	−	0.984328i	$-0.443572\pi$
$797$	9.95702	0.352696	0.176348	+	0.984328i	$0.443572\pi$
$798$	0	0
$799$	2.69357	0.0952918
$800$	0	0
$801$	3.28995	0.116244
$802$	0	0
$803$	3.06759	0.108253
$804$	0	0
$805$	−3.75615	−0.132387
$806$	0	0
$807$	−31.0296	−1.09229
$808$	0	0
$809$	34.5799	1.21576	0.607882	−	0.794027i	$-0.292019\pi$
$809$	34.5799	1.21576	0.607882	+	0.794027i	$0.292019\pi$
$810$	0	0
$811$	21.5123	0.755399	0.377700	−	0.925928i	$-0.376715\pi$
$811$	21.5123	0.755399	0.377700	+	0.925928i	$0.376715\pi$
$812$	0	0
$813$	9.98352	0.350137
$814$	0	0
$815$	−17.7612	−0.622146
$816$	0	0
$817$	−6.22236	−0.217693
$818$	0	0
$819$	−1.53379	−0.0535951
$820$	0	0
$821$	27.6525	0.965078	0.482539	−	0.875874i	$-0.339715\pi$
$821$	27.6525	0.965078	0.482539	+	0.875874i	$0.339715\pi$
$822$	0	0
$823$	−42.0315	−1.46513	−0.732563	−	0.680699i	$-0.761676\pi$
$823$	−42.0315	−1.46513	−0.732563	+	0.680699i	$0.761676\pi$
$824$	0	0
$825$	−1.00000	−0.0348155
$826$	0	0
$827$	33.0346	1.14873	0.574363	−	0.818601i	$-0.305250\pi$
$827$	33.0346	1.14873	0.574363	+	0.818601i	$0.305250\pi$
$828$	0	0
$829$	−44.0922	−1.53139	−0.765693	−	0.643207i	$-0.777604\pi$
$829$	−44.0922	−1.53139	−0.765693	+	0.643207i	$0.777604\pi$
$830$	0	0
$831$	14.0922	0.488853
$832$	0	0
$833$	1.75615	0.0608470
$834$	0	0
$835$	−12.1352	−0.419955
$836$	0	0
$837$	−1.53379	−0.0530157
$838$	0	0
$839$	−20.3246	−0.701682	−0.350841	−	0.936435i	$-0.614104\pi$
$839$	−20.3246	−0.701682	−0.350841	+	0.936435i	$0.614104\pi$
$840$	0	0
$841$	32.9343	1.13567
$842$	0	0
$843$	−30.0922	−1.03643
$844$	0	0
$845$	10.6475	0.366284
$846$	0	0
$847$	1.00000	0.0343604
$848$	0	0
$849$	−11.6690	−0.400478
$850$	0	0
$851$	22.9639	0.787193
$852$	0	0
$853$	49.7282	1.70266	0.851331	−	0.524630i	$-0.175796\pi$
$853$	49.7282	1.70266	0.851331	+	0.524630i	$0.175796\pi$
$854$	0	0
$855$	4.82374	0.164968
$856$	0	0
$857$	6.97539	0.238275	0.119137	−	0.992878i	$-0.461987\pi$
$857$	6.97539	0.238275	0.119137	+	0.992878i	$0.461987\pi$
$858$	0	0
$859$	36.0807	1.23106	0.615529	−	0.788114i	$-0.288942\pi$
$859$	36.0807	1.23106	0.615529	+	0.788114i	$0.288942\pi$
$860$	0	0
$861$	−12.1137	−0.412833
$862$	0	0
$863$	−2.36900	−0.0806418	−0.0403209	−	0.999187i	$-0.512838\pi$
$863$	−2.36900	−0.0806418	−0.0403209	+	0.999187i	$0.512838\pi$
$864$	0	0
$865$	−20.8073	−0.707468
$866$	0	0
$867$	13.9159	0.472610
$868$	0	0
$869$	16.6936	0.566291
$870$	0	0
$871$	−5.38715	−0.182537
$872$	0	0
$873$	−16.9374	−0.573245
$874$	0	0
$875$	−1.00000	−0.0338062
$876$	0	0
$877$	−18.9374	−0.639471	−0.319736	−	0.947507i	$-0.603594\pi$
$877$	−18.9374	−0.639471	−0.319736	+	0.947507i	$0.603594\pi$
$878$	0	0
$879$	22.9159	0.772935
$880$	0	0
$881$	38.5419	1.29851	0.649255	−	0.760571i	$-0.275081\pi$
$881$	38.5419	1.29851	0.649255	+	0.760571i	$0.275081\pi$
$882$	0	0
$883$	18.3625	0.617949	0.308974	−	0.951070i	$-0.400014\pi$
$883$	18.3625	0.617949	0.308974	+	0.951070i	$0.400014\pi$
$884$	0	0
$885$	−8.35753	−0.280935
$886$	0	0
$887$	−33.9835	−1.14105	−0.570527	−	0.821279i	$-0.693261\pi$
$887$	−33.9835	−1.14105	−0.570527	+	0.821279i	$0.693261\pi$
$888$	0	0
$889$	−5.28995	−0.177419
$890$	0	0
$891$	1.00000	0.0335013
$892$	0	0
$893$	−7.39862	−0.247585
$894$	0	0
$895$	−26.2489	−0.877403
$896$	0	0
$897$	5.76116	0.192360
$898$	0	0
$899$	−12.0707	−0.402580
$900$	0	0
$901$	−14.6394	−0.487708
$902$	0	0
$903$	−1.28995	−0.0429267
$904$	0	0
$905$	−23.1598	−0.769857
$906$	0	0
$907$	20.6014	0.684058	0.342029	−	0.939689i	$-0.388886\pi$
$907$	20.6014	0.684058	0.342029	+	0.939689i	$0.388886\pi$
$908$	0	0
$909$	7.51230	0.249167
$910$	0	0
$911$	2.61285	0.0865677	0.0432838	−	0.999063i	$-0.486218\pi$
$911$	2.61285	0.0865677	0.0432838	+	0.999063i	$0.486218\pi$
$912$	0	0
$913$	1.75615	0.0581201
$914$	0	0
$915$	−5.75615	−0.190292
$916$	0	0
$917$	5.53379	0.182742
$918$	0	0
$919$	10.2703	0.338788	0.169394	−	0.985548i	$-0.445819\pi$
$919$	10.2703	0.338788	0.169394	+	0.985548i	$0.445819\pi$
$920$	0	0
$921$	4.71506	0.155367
$922$	0	0
$923$	−4.53068	−0.149129
$924$	0	0
$925$	6.11368	0.201017
$926$	0	0
$927$	−11.8698	−0.389857
$928$	0	0
$929$	17.5553	0.575970	0.287985	−	0.957635i	$-0.407015\pi$
$929$	17.5553	0.575970	0.287985	+	0.957635i	$0.407015\pi$
$930$	0	0
$931$	−4.82374	−0.158092
$932$	0	0
$933$	−32.2274	−1.05508
$934$	0	0
$935$	−1.75615	−0.0574323
$936$	0	0
$937$	48.2551	1.57643	0.788213	−	0.615403i	$-0.211007\pi$
$937$	48.2551	1.57643	0.788213	+	0.615403i	$0.211007\pi$
$938$	0	0
$939$	2.84523	0.0928505
$940$	0	0
$941$	8.60138	0.280397	0.140198	−	0.990123i	$-0.455226\pi$
$941$	8.60138	0.280397	0.140198	+	0.990123i	$0.455226\pi$
$942$	0	0
$943$	45.5008	1.48171
$944$	0	0
$945$	1.00000	0.0325300
$946$	0	0
$947$	16.6886	0.542305	0.271153	−	0.962536i	$-0.412595\pi$
$947$	16.6886	0.542305	0.271153	+	0.962536i	$0.412595\pi$
$948$	0	0
$949$	−4.70505	−0.152732
$950$	0	0
$951$	−27.8748	−0.903904
$952$	0	0
$953$	−20.0492	−0.649458	−0.324729	−	0.945807i	$-0.605273\pi$
$953$	−20.0492	−0.649458	−0.324729	+	0.945807i	$0.605273\pi$
$954$	0	0
$955$	−15.0246	−0.486185
$956$	0	0
$957$	7.86984	0.254396
$958$	0	0
$959$	13.5123	0.436335
$960$	0	0
$961$	−28.6475	−0.924112
$962$	0	0
$963$	−15.2735	−0.492181
$964$	0	0
$965$	−12.7151	−0.409312
$966$	0	0
$967$	−30.0050	−0.964896	−0.482448	−	0.875925i	$-0.660252\pi$
$967$	−30.0050	−0.964896	−0.482448	+	0.875925i	$0.660252\pi$
$968$	0	0
$969$	8.47122	0.272135
$970$	0	0
$971$	−10.9374	−0.350999	−0.175499	−	0.984480i	$-0.556154\pi$
$971$	−10.9374	−0.350999	−0.175499	+	0.984480i	$0.556154\pi$
$972$	0	0
$973$	11.1598	0.357766
$974$	0	0
$975$	1.53379	0.0491207
$976$	0	0
$977$	−8.42200	−0.269444	−0.134722	−	0.990883i	$-0.543014\pi$
$977$	−8.42200	−0.269444	−0.134722	+	0.990883i	$0.543014\pi$
$978$	0	0
$979$	3.28995	0.105147
$980$	0	0
$981$	−7.04610	−0.224965
$982$	0	0
$983$	−43.7397	−1.39508	−0.697539	−	0.716546i	$-0.745722\pi$
$983$	−43.7397	−1.39508	−0.697539	+	0.716546i	$0.745722\pi$
$984$	0	0
$985$	−5.55528	−0.177006
$986$	0	0
$987$	−1.53379	−0.0488212
$988$	0	0
$989$	4.84523	0.154069
$990$	0	0
$991$	58.5634	1.86033	0.930164	−	0.367144i	$-0.119664\pi$
$991$	58.5634	1.86033	0.930164	+	0.367144i	$0.119664\pi$
$992$	0	0
$993$	0.471216	0.0149536
$994$	0	0
$995$	26.3196	0.834386
$996$	0	0
$997$	3.51230	0.111236	0.0556179	−	0.998452i	$-0.482287\pi$
$997$	3.51230	0.111236	0.0556179	+	0.998452i	$0.482287\pi$
$998$	0	0
$999$	−6.11368	−0.193428

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4620.2.a.t.1.2	✓	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4620.2.a.t.1.2	✓	3	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 4620 = 2^{2} \cdot 3 \cdot 5 \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4620.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -1.00000 q^{5} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} -1.00000 q^{5} +1.00000 q^{7} +1.00000 q^{9} +1.00000 q^{11} -1.53379 q^{13} +1.00000 q^{15} +1.75615 q^{17} -4.82374 q^{19} -1.00000 q^{21} +3.75615 q^{23} +1.00000 q^{25} -1.00000 q^{27} -7.86984 q^{29} +1.53379 q^{31} -1.00000 q^{33} -1.00000 q^{35} +6.11368 q^{37} +1.53379 q^{39} +12.1137 q^{41} +1.28995 q^{43} -1.00000 q^{45} +1.53379 q^{47} +1.00000 q^{49} -1.75615 q^{51} -8.33604 q^{53} -1.00000 q^{55} +4.82374 q^{57} -8.35753 q^{59} -5.75615 q^{61} +1.00000 q^{63} +1.53379 q^{65} +3.51230 q^{67} -3.75615 q^{69} +2.95390 q^{71} +3.06759 q^{73} -1.00000 q^{75} +1.00000 q^{77} +16.6936 q^{79} +1.00000 q^{81} +1.75615 q^{83} -1.75615 q^{85} +7.86984 q^{87} +3.28995 q^{89} -1.53379 q^{91} -1.53379 q^{93} +4.82374 q^{95} -16.9374 q^{97} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{3} - 3 q^{5} + 3 q^{7} + 3 q^{9}+O(q^{10}) \) 3 * q - 3 * q^3 - 3 * q^5 + 3 * q^7 + 3 * q^9	\( 3 q - 3 q^{3} - 3 q^{5} + 3 q^{7} + 3 q^{9} + 3 q^{11} + 3 q^{15} - 3 q^{21} + 6 q^{23} + 3 q^{25} - 3 q^{27} + 6 q^{29} - 3 q^{33} - 3 q^{35} - 6 q^{37} + 12 q^{41} - 6 q^{43} - 3 q^{45} + 3 q^{49} - 3 q^{55} - 6 q^{59} - 12 q^{61} + 3 q^{63} - 6 q^{69} + 24 q^{71} - 3 q^{75} + 3 q^{77} + 6 q^{79} + 3 q^{81} - 6 q^{87} - 12 q^{97} + 3 q^{99}+O(q^{100}) \) 3 * q - 3 * q^3 - 3 * q^5 + 3 * q^7 + 3 * q^9 + 3 * q^11 + 3 * q^15 - 3 * q^21 + 6 * q^23 + 3 * q^25 - 3 * q^27 + 6 * q^29 - 3 * q^33 - 3 * q^35 - 6 * q^37 + 12 * q^41 - 6 * q^43 - 3 * q^45 + 3 * q^49 - 3 * q^55 - 6 * q^59 - 12 * q^61 + 3 * q^63 - 6 * q^69 + 24 * q^71 - 3 * q^75 + 3 * q^77 + 6 * q^79 + 3 * q^81 - 6 * q^87 - 12 * q^97 + 3 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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