LMFDB - Embedded newform 1620.3.o.g.701.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1620,3,Mod(701,1620)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1620, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 5, 0]))

N = Newforms(chi, 3, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1620.701");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1620 = 2^{2} \cdot 3^{4} \cdot 5 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1620.o (of order $6$, degree $2$, not minimal)

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:	no
Analytic conductor:	$44.1418028264$
Analytic rank:	$0$
Dimension:	$32$
Relative dimension:	$16$ over $\Q(\zeta_{6})$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			701.3
Character	$\chi$	$=$	1620.701
Dual form			1620.3.o.g.1241.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.93649 - 1.11803i) q^{5} +(-0.973755 - 1.68659i) q^{7} +O(q^{10})$	$q+(-1.93649 - 1.11803i) q^{5} +(-0.973755 - 1.68659i) q^{7} +(-13.6991 + 7.90918i) q^{11} +(11.4700 - 19.8666i) q^{13} +21.9502i q^{17} +16.8264 q^{19} +(24.2401 + 13.9950i) q^{23} +(2.50000 + 4.33013i) q^{25} +(36.0462 - 20.8113i) q^{29} +(-25.9962 + 45.0268i) q^{31} +4.35476i q^{35} -49.8407 q^{37} +(-52.0551 - 30.0541i) q^{41} +(-25.1531 - 43.5665i) q^{43} +(61.6553 - 35.5967i) q^{47} +(22.6036 - 39.1506i) q^{49} -21.7336i q^{53} +35.3709 q^{55} +(-34.4517 - 19.8907i) q^{59} +(-53.7423 - 93.0844i) q^{61} +(-44.4230 + 25.6476i) q^{65} +(40.9371 - 70.9052i) q^{67} +62.1499i q^{71} +46.9217 q^{73} +(26.6791 + 15.4032i) q^{77} +(-0.00315546 - 0.00546542i) q^{79} +(-55.5608 + 32.0780i) q^{83} +(24.5410 - 42.5063i) q^{85} +51.0401i q^{89} -44.6757 q^{91} +(-32.5842 - 18.8125i) q^{95} +(-22.3365 - 38.6879i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 32 q - 8 q^{7}+O(q^{10}) $ 32 * q - 8 * q^7	$ 32 q - 8 q^{7} + 40 q^{13} + 112 q^{19} + 80 q^{25} + 64 q^{31} - 176 q^{37} - 128 q^{43} - 216 q^{49} - 8 q^{61} + 40 q^{67} + 112 q^{73} + 136 q^{79} - 784 q^{91} - 128 q^{97}+O(q^{100}) $ 32 * q - 8 * q^7 + 40 * q^13 + 112 * q^19 + 80 * q^25 + 64 * q^31 - 176 * q^37 - 128 * q^43 - 216 * q^49 - 8 * q^61 + 40 * q^67 + 112 * q^73 + 136 * q^79 - 784 * q^91 - 128 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1620\mathbb{Z}\right)^\times$.

$n$	$811$	$1297$	$1541$
$\chi(n)$	$1$	$1$	$e\left(\frac{5}{6}\right)$

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
$5$	−1.93649	−	1.11803i	−0.387298	−	0.223607i
$5$	−1.93649	−	1.11803i	−0.387298	−	0.223607i
$6$		0			0
$7$	−0.973755	−	1.68659i	−0.139108	−	0.240942i	0.788051	−	0.615610i	$-0.211090\pi$
$7$	−0.973755	−	1.68659i	−0.139108	−	0.240942i	−0.927159	+	0.374668i	$0.877757\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−13.6991	+	7.90918i	−1.24537	+	0.719016i	−0.970183	−	0.242374i	$-0.922074\pi$
$11$	−13.6991	+	7.90918i	−1.24537	+	0.719016i	−0.275190	+	0.961390i	$0.588741\pi$
$12$		0			0
$13$	11.4700	−	19.8666i	0.882305	−	1.52820i	0.0335334	−	0.999438i	$-0.489324\pi$
$13$	11.4700	−	19.8666i	0.882305	−	1.52820i	0.848772	−	0.528760i	$-0.177343\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$			21.9502i			1.29119i	0.763682	+	0.645593i	$0.223390\pi$
$17$			21.9502i			1.29119i	−0.763682	+	0.645593i	$0.776610\pi$
$18$		0			0
$19$	16.8264			0.885600			0.442800	−	0.896621i	$-0.353985\pi$
$19$	16.8264			0.885600			0.442800	+	0.896621i	$0.353985\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	24.2401	+	13.9950i	1.05392	+	0.608478i	0.923743	−	0.383013i	$-0.125114\pi$
$23$	24.2401	+	13.9950i	1.05392	+	0.608478i	0.130172	+	0.991491i	$0.458447\pi$
$24$		0			0
$25$	2.50000	+	4.33013i	0.100000	+	0.173205i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	36.0462	−	20.8113i	1.24297	−	0.717630i	0.273274	−	0.961936i	$-0.411893\pi$
$29$	36.0462	−	20.8113i	1.24297	−	0.717630i	0.969698	+	0.244306i	$0.0785601\pi$
$30$		0			0
$31$	−25.9962	+	45.0268i	−0.838588	+	1.45248i	0.0524872	+	0.998622i	$0.483285\pi$
$31$	−25.9962	+	45.0268i	−0.838588	+	1.45248i	−0.891075	+	0.453856i	$0.850048\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$			4.35476i			0.124422i
$36$		0			0
$37$	−49.8407			−1.34705			−0.673523	−	0.739166i	$-0.735220\pi$
$37$	−49.8407			−1.34705			−0.673523	+	0.739166i	$0.735220\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−52.0551	−	30.0541i	−1.26964	−	0.733026i	−0.294718	−	0.955584i	$-0.595226\pi$
$41$	−52.0551	−	30.0541i	−1.26964	−	0.733026i	−0.974919	+	0.222558i	$0.928559\pi$
$42$		0			0
$43$	−25.1531	−	43.5665i	−0.584956	−	1.01317i	−0.994881	−	0.101055i	$-0.967778\pi$
$43$	−25.1531	−	43.5665i	−0.584956	−	1.01317i	0.409925	−	0.912119i	$-0.365555\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	61.6553	−	35.5967i	1.31181	−	0.757377i	0.329418	−	0.944184i	$-0.393147\pi$
$47$	61.6553	−	35.5967i	1.31181	−	0.757377i	0.982397	+	0.186807i	$0.0598141\pi$
$48$		0			0
$49$	22.6036	−	39.1506i	0.461298	−	0.798992i
$50$		0			0
$51$		0			0
$52$		0			0
$53$		−	21.7336i		−	0.410068i	−0.978755	−	0.205034i	$-0.934270\pi$
$53$		−	21.7336i		−	0.410068i	0.978755	−	0.205034i	$-0.0657304\pi$
$54$		0			0
$55$	35.3709			0.643108
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−34.4517	−	19.8907i	−0.583927	−	0.337131i	0.178765	−	0.983892i	$-0.442790\pi$
$59$	−34.4517	−	19.8907i	−0.583927	−	0.337131i	−0.762693	+	0.646761i	$0.776123\pi$
$60$		0			0
$61$	−53.7423	−	93.0844i	−0.881021	−	1.52597i	−0.850207	−	0.526448i	$-0.823523\pi$
$61$	−53.7423	−	93.0844i	−0.881021	−	1.52597i	−0.0308138	−	0.999525i	$-0.509810\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−44.4230	+	25.6476i	−0.683431	+	0.394579i
$66$		0			0
$67$	40.9371	−	70.9052i	0.611002	−	1.05829i	−0.380070	−	0.924958i	$-0.624100\pi$
$67$	40.9371	−	70.9052i	0.611002	−	1.05829i	0.991072	−	0.133328i	$-0.0425665\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			62.1499i			0.875351i	0.899133	+	0.437676i	$0.144198\pi$
$71$			62.1499i			0.875351i	−0.899133	+	0.437676i	$0.855802\pi$
$72$		0			0
$73$	46.9217			0.642763			0.321382	−	0.946950i	$-0.395853\pi$
$73$	46.9217			0.642763			0.321382	+	0.946950i	$0.395853\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	26.6791	+	15.4032i	0.346482	+	0.200042i
$78$		0			0
$79$	−0.00315546	−	0.00546542i	−3.99426e−5	−	6.91826e-5i	0.866005	−	0.500035i	$-0.166679\pi$
$79$	−0.00315546	−	0.00546542i	−3.99426e−5	−	6.91826e-5i	−0.866045	+	0.499965i	$0.833346\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−55.5608	+	32.0780i	−0.669407	+	0.386482i	−0.795852	−	0.605491i	$-0.792977\pi$
$83$	−55.5608	+	32.0780i	−0.669407	+	0.386482i	0.126445	+	0.991974i	$0.459643\pi$
$84$		0			0
$85$	24.5410	−	42.5063i	0.288718	−	0.500074i
$86$		0			0
$87$		0			0
$88$		0			0
$89$			51.0401i			0.573485i	0.958008	+	0.286742i	$0.0925723\pi$
$89$			51.0401i			0.573485i	−0.958008	+	0.286742i	$0.907428\pi$
$90$		0			0
$91$	−44.6757			−0.490942
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−32.5842	−	18.8125i	−0.342991	−	0.198026i
$96$		0			0
$97$	−22.3365	−	38.6879i	−0.230273	−	0.398844i	0.727616	−	0.685985i	$-0.240629\pi$
$97$	−22.3365	−	38.6879i	−0.230273	−	0.398844i	−0.957888	+	0.287141i	$0.907295\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	109.929	−	63.4673i	1.08840	−	0.628389i	0.155251	−	0.987875i	$-0.450381\pi$
$101$	109.929	−	63.4673i	1.08840	−	0.628389i	0.933151	+	0.359486i	$0.117048\pi$
$102$		0			0
$103$	93.7094	−	162.309i	0.909800	−	1.57582i	0.0954582	−	0.995433i	$-0.469568\pi$
$103$	93.7094	−	162.309i	0.909800	−	1.57582i	0.814342	−	0.580386i	$-0.197098\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	23.9616i		−	0.223940i	−0.993712	−	0.111970i	$-0.964284\pi$
$107$		−	23.9616i		−	0.223940i	0.993712	−	0.111970i	$-0.0357161\pi$
$108$		0			0
$109$	−161.184			−1.47875			−0.739375	−	0.673294i	$-0.764879\pi$
$109$	−161.184			−1.47875			−0.739375	+	0.673294i	$0.764879\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−51.8987	−	29.9638i	−0.459281	−	0.265166i	0.252461	−	0.967607i	$-0.418760\pi$
$113$	−51.8987	−	29.9638i	−0.459281	−	0.265166i	−0.711742	+	0.702441i	$0.752093\pi$
$114$		0			0
$115$	−31.2938	−	54.2024i	−0.272120	−	0.471325i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	37.0210	−	21.3741i	0.311101	−	0.179614i
$120$		0			0
$121$	64.6103	−	111.908i	0.533969	−	0.924861i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		−	11.1803i		−	0.0894427i
$126$		0			0
$127$	95.3125			0.750492			0.375246	−	0.926925i	$-0.377558\pi$
$127$	95.3125			0.750492			0.375246	+	0.926925i	$0.377558\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−111.927	−	64.6208i	−0.854401	−	0.493289i	0.00773223	−	0.999970i	$-0.497539\pi$
$131$	−111.927	−	64.6208i	−0.854401	−	0.493289i	−0.862133	+	0.506681i	$0.830872\pi$
$132$		0			0
$133$	−16.3848	−	28.3793i	−0.123194	−	0.213378i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	45.7808	−	26.4315i	0.334166	−	0.192931i	−0.323523	−	0.946220i	$-0.604867\pi$
$137$	45.7808	−	26.4315i	0.334166	−	0.192931i	0.657689	+	0.753289i	$0.271534\pi$
$138$		0			0
$139$	27.4479	−	47.5411i	0.197467	−	0.342023i	−0.750240	−	0.661166i	$-0.770062\pi$
$139$	27.4479	−	47.5411i	0.197467	−	0.342023i	0.947706	+	0.319144i	$0.103395\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$			362.872i			2.53757i
$144$		0			0
$145$	−93.0709			−0.641868
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−63.0312	−	36.3911i	−0.423028	−	0.244235i	0.273344	−	0.961916i	$-0.411870\pi$
$149$	−63.0312	−	36.3911i	−0.423028	−	0.244235i	−0.696372	+	0.717681i	$0.745204\pi$
$150$		0			0
$151$	−28.6242	−	49.5785i	−0.189564	−	0.328335i	0.755541	−	0.655101i	$-0.227374\pi$
$151$	−28.6242	−	49.5785i	−0.189564	−	0.328335i	−0.945105	+	0.326767i	$0.894041\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	100.683	−	58.1293i	0.649568	−	0.375028i
$156$		0			0
$157$	−74.1524	+	128.436i	−0.472308	+	0.818062i	−0.999498	−	0.0316858i	$-0.989912\pi$
$157$	−74.1524	+	128.436i	−0.472308	+	0.818062i	0.527190	+	0.849748i	$0.323246\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		−	54.5108i		−	0.338576i
$162$		0			0
$163$	−81.1571			−0.497897			−0.248948	−	0.968517i	$-0.580085\pi$
$163$	−81.1571			−0.497897			−0.248948	+	0.968517i	$0.580085\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−248.322	−	143.368i	−1.48696	−	0.858494i	−0.487066	−	0.873365i	$-0.661933\pi$
$167$	−248.322	−	143.368i	−1.48696	−	0.858494i	−0.999889	+	0.0148713i	$0.995266\pi$
$168$		0			0
$169$	−178.620	−	309.379i	−1.05692	−	1.83065i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	140.719	−	81.2442i	0.813405	−	0.469620i	−0.0347320	−	0.999397i	$-0.511058\pi$
$173$	140.719	−	81.2442i	0.813405	−	0.469620i	0.848137	+	0.529777i	$0.177724\pi$
$174$		0			0
$175$	4.86877	−	8.43297i	0.0278216	−	0.0481884i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		−	130.340i		−	0.728157i	−0.931368	−	0.364078i	$-0.881384\pi$
$179$		−	130.340i		−	0.728157i	0.931368	−	0.364078i	$-0.118616\pi$
$180$		0			0
$181$	187.368			1.03519			0.517593	−	0.855627i	$-0.326828\pi$
$181$	187.368			1.03519			0.517593	+	0.855627i	$0.326828\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	96.5161	+	55.7236i	0.521708	+	0.301209i
$186$		0			0
$187$	−173.608	−	300.697i	−0.928383	−	1.60801i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	235.130	−	135.753i	1.23105	−	0.710747i	0.263801	−	0.964577i	$-0.415024\pi$
$191$	235.130	−	135.753i	1.23105	−	0.710747i	0.967249	+	0.253831i	$0.0816906\pi$
$192$		0			0
$193$	14.4873	−	25.0928i	0.0750638	−	0.130014i	−0.826050	−	0.563597i	$-0.809417\pi$
$193$	14.4873	−	25.0928i	0.0750638	−	0.130014i	0.901114	+	0.433582i	$0.142751\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			29.6598i			0.150558i	0.997163	+	0.0752788i	$0.0239847\pi$
$197$			29.6598i			0.150558i	−0.997163	+	0.0752788i	$0.976015\pi$
$198$		0			0
$199$	270.976			1.36169			0.680843	−	0.732429i	$-0.261614\pi$
$199$	270.976			1.36169			0.680843	+	0.732429i	$0.261614\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−70.2003	−	40.5302i	−0.345814	−	0.199656i
$204$		0			0
$205$	67.2029	+	116.399i	0.327819	+	0.567799i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−230.506	+	133.083i	−1.10290	+	0.636761i
$210$		0			0
$211$	92.5134	−	160.238i	0.438452	−	0.759421i	−0.559118	−	0.829088i	$-0.688860\pi$
$211$	92.5134	−	160.238i	0.438452	−	0.759421i	0.997570	+	0.0696667i	$0.0221936\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$			112.488i			0.523201i
$216$		0			0
$217$	101.256			0.466617
$218$		0			0
$219$		0			0
$220$		0			0
$221$	436.074	+	251.767i	1.97319	+	1.13922i
$222$		0			0
$223$	109.188	+	189.120i	0.489633	+	0.848070i	0.999929	−	0.0119293i	$-0.00379729\pi$
$223$	109.188	+	189.120i	0.489633	+	0.848070i	−0.510295	+	0.859999i	$0.670464\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−38.2709	+	22.0957i	−0.168594	+	0.0973379i	−0.581923	−	0.813244i	$-0.697699\pi$
$227$	−38.2709	+	22.0957i	−0.168594	+	0.0973379i	0.413329	+	0.910582i	$0.364366\pi$
$228$		0			0
$229$	142.091	−	246.109i	0.620484	−	1.07471i	−0.368911	−	0.929465i	$-0.620269\pi$
$229$	142.091	−	246.109i	0.620484	−	1.07471i	0.989396	−	0.145246i	$-0.0463973\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			141.768i			0.608446i	0.952601	+	0.304223i	$0.0983968\pi$
$233$			141.768i			0.608446i	−0.952601	+	0.304223i	$0.901603\pi$
$234$		0			0
$235$	−159.193			−0.677418
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−16.0331	−	9.25672i	−0.0670841	−	0.0387310i	0.466083	−	0.884741i	$-0.345665\pi$
$239$	−16.0331	−	9.25672i	−0.0670841	−	0.0387310i	−0.533167	+	0.846010i	$0.678998\pi$
$240$		0			0
$241$	−70.1268	−	121.463i	−0.290983	−	0.503997i	0.683060	−	0.730363i	$-0.260649\pi$
$241$	−70.1268	−	121.463i	−0.290983	−	0.503997i	−0.974042	+	0.226366i	$0.927316\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−87.5434	+	50.5432i	−0.357320	+	0.206299i
$246$		0			0
$247$	192.998	−	334.283i	0.781369	−	1.35337i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		−	277.396i		−	1.10516i	−0.833459	−	0.552582i	$-0.813643\pi$
$251$		−	277.396i		−	1.10516i	0.833459	−	0.552582i	$-0.186357\pi$
$252$		0			0
$253$	−442.756			−1.75002
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−43.3050	−	25.0021i	−0.168502	−	0.0972845i	0.413377	−	0.910560i	$-0.364349\pi$
$257$	−43.3050	−	25.0021i	−0.168502	−	0.0972845i	−0.581879	+	0.813275i	$0.697682\pi$
$258$		0			0
$259$	48.5326	+	84.0609i	0.187385	+	0.324560i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−280.519	+	161.958i	−1.06661	+	0.615809i	−0.927254	−	0.374433i	$-0.877837\pi$
$263$	−280.519	+	161.958i	−1.06661	+	0.615809i	−0.139358	+	0.990242i	$0.544504\pi$
$264$		0			0
$265$	−24.2989	+	42.0869i	−0.0916939	+	0.158819i
$266$		0			0
$267$		0			0
$268$		0			0
$269$			357.041i			1.32729i	0.748048	+	0.663645i	$0.230991\pi$
$269$			357.041i			1.32729i	−0.748048	+	0.663645i	$0.769009\pi$
$270$		0			0
$271$	124.084			0.457875			0.228938	−	0.973441i	$-0.426475\pi$
$271$	124.084			0.457875			0.228938	+	0.973441i	$0.426475\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−68.4955	−	39.5459i	−0.249075	−	0.143803i
$276$		0			0
$277$	−53.2553	−	92.2408i	−0.192257	−	0.332999i	0.753741	−	0.657172i	$-0.228247\pi$
$277$	−53.2553	−	92.2408i	−0.192257	−	0.332999i	−0.945998	+	0.324173i	$0.894914\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	147.325	−	85.0579i	0.524287	−	0.302697i	−0.214400	−	0.976746i	$-0.568780\pi$
$281$	147.325	−	85.0579i	0.524287	−	0.302697i	0.738687	+	0.674049i	$0.235446\pi$
$282$		0			0
$283$	−137.277	+	237.770i	−0.485077	+	0.840178i	−0.999853	−	0.0171467i	$-0.994542\pi$
$283$	−137.277	+	237.770i	−0.485077	+	0.840178i	0.514776	+	0.857325i	$0.327875\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$			117.061i			0.407879i
$288$		0			0
$289$	−192.809			−0.667160
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−161.982	−	93.5202i	−0.552839	−	0.319182i	0.197427	−	0.980318i	$-0.436741\pi$
$293$	−161.982	−	93.5202i	−0.552839	−	0.319182i	−0.750266	+	0.661136i	$0.770075\pi$
$294$		0			0
$295$	44.4770	+	77.0364i	0.150769	+	0.261140i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	556.065	−	321.044i	1.85975	−	1.07373i
$300$		0			0
$301$	−48.9859	+	84.8461i	−0.162744	+	0.281881i
$302$		0			0
$303$		0			0
$304$		0			0
$305$			240.343i			0.788009i
$306$		0			0
$307$	110.783			0.360857			0.180428	−	0.983588i	$-0.442252\pi$
$307$	110.783			0.360857			0.180428	+	0.983588i	$0.442252\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−150.770	−	87.0474i	−0.484793	−	0.279895i	0.237619	−	0.971358i	$-0.423633\pi$
$311$	−150.770	−	87.0474i	−0.484793	−	0.279895i	−0.722412	+	0.691463i	$0.756966\pi$
$312$		0			0
$313$	−220.004	−	381.058i	−0.702887	−	1.21744i	−0.967449	−	0.253068i	$-0.918560\pi$
$313$	−220.004	−	381.058i	−0.702887	−	1.21744i	0.264561	−	0.964369i	$-0.414773\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−142.068	+	82.0228i	−0.448163	+	0.258747i	−0.707054	−	0.707160i	$-0.749976\pi$
$317$	−142.068	+	82.0228i	−0.448163	+	0.258747i	0.258891	+	0.965907i	$0.416643\pi$
$318$		0			0
$319$	−329.200	+	570.192i	−1.03198	+	1.78743i
$320$		0			0
$321$		0			0
$322$		0			0
$323$			369.342i			1.14347i
$324$		0			0
$325$	114.700			0.352922
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−120.074	−	69.3249i	−0.364967	−	0.210714i
$330$		0			0
$331$	51.7350	+	89.6077i	0.156299	+	0.270718i	0.933531	−	0.358496i	$-0.116710\pi$
$331$	51.7350	+	89.6077i	0.156299	+	0.270718i	−0.777232	+	0.629214i	$0.783377\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−158.549	+	91.5382i	−0.473280	+	0.273248i
$336$		0			0
$337$	−137.233	+	237.694i	−0.407218	+	0.705323i	−0.994577	−	0.104004i	$-0.966835\pi$
$337$	−137.233	+	237.694i	−0.407218	+	0.705323i	0.587359	+	0.809327i	$0.300168\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		−	822.435i		−	2.41183i
$342$		0			0
$343$	−183.469			−0.534896
$344$		0			0
$345$		0			0
$346$		0			0
$347$	414.256	+	239.171i	1.19382	+	0.689253i	0.959171	−	0.282827i	$-0.0912721\pi$
$347$	414.256	+	239.171i	1.19382	+	0.689253i	0.234650	+	0.972080i	$0.424605\pi$
$348$		0			0
$349$	129.612	+	224.495i	0.371382	+	0.643253i	0.989778	−	0.142614i	$-0.0455507\pi$
$349$	129.612	+	224.495i	0.371382	+	0.643253i	−0.618396	+	0.785866i	$0.712217\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−465.135	+	268.546i	−1.31766	+	0.760753i	−0.983352	−	0.181710i	$-0.941837\pi$
$353$	−465.135	+	268.546i	−1.31766	+	0.760753i	−0.334310	+	0.942463i	$0.608503\pi$
$354$		0			0
$355$	69.4858	−	120.353i	0.195735	−	0.339022i
$356$		0			0
$357$		0			0
$358$		0			0
$359$		−	340.713i		−	0.949062i	−0.880239	−	0.474531i	$-0.842618\pi$
$359$		−	340.713i		−	0.949062i	0.880239	−	0.474531i	$-0.157382\pi$
$360$		0			0
$361$	−77.8726			−0.215713
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−90.8635	−	52.4601i	−0.248941	−	0.143726i
$366$		0			0
$367$	289.579	+	501.566i	0.789044	+	1.36667i	0.926553	+	0.376164i	$0.122757\pi$
$367$	289.579	+	501.566i	0.789044	+	1.36667i	−0.137508	+	0.990501i	$0.543909\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−36.6557	+	21.1632i	−0.0988025	+	0.0570436i
$372$		0			0
$373$	274.217	−	474.958i	0.735166	−	1.27335i	−0.219484	−	0.975616i	$-0.570437\pi$
$373$	274.217	−	474.958i	0.735166	−	1.27335i	0.954650	−	0.297729i	$-0.0962292\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	954.819i		−	2.53268i
$378$		0			0
$379$	564.649			1.48984			0.744920	−	0.667154i	$-0.232488\pi$
$379$	564.649			1.48984			0.744920	+	0.667154i	$0.232488\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	201.735	+	116.472i	0.526724	+	0.304104i	0.739681	−	0.672957i	$-0.234976\pi$
$383$	201.735	+	116.472i	0.526724	+	0.304104i	−0.212958	+	0.977061i	$0.568310\pi$
$384$		0			0
$385$	−34.4426	−	59.6564i	−0.0894613	−	0.154952i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	12.9315	−	7.46602i	0.0332430	−	0.0191928i	−0.483286	−	0.875462i	$-0.660557\pi$
$389$	12.9315	−	7.46602i	0.0332430	−	0.0191928i	0.516529	+	0.856269i	$0.327224\pi$
$390$		0			0
$391$	−307.192	+	532.073i	−0.785658	+	1.36080i
$392$		0			0
$393$		0			0
$394$		0			0
$395$			0.0141117i			3.57257e-5i
$396$		0			0
$397$	−111.461			−0.280759			−0.140379	−	0.990098i	$-0.544832\pi$
$397$	−111.461			−0.280759			−0.140379	+	0.990098i	$0.544832\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−556.258	−	321.155i	−1.38718	−	0.800886i	−0.394180	−	0.919033i	$-0.628971\pi$
$401$	−556.258	−	321.155i	−1.38718	−	0.800886i	−0.992996	+	0.118147i	$0.962305\pi$
$402$		0			0
$403$	596.352	+	1032.91i	1.47978	+	2.56306i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	682.773	−	394.199i	1.67757	−	0.968548i
$408$		0			0
$409$	45.1673	−	78.2320i	0.110433	−	0.191276i	−0.805512	−	0.592580i	$-0.798109\pi$
$409$	45.1673	−	78.2320i	0.110433	−	0.191276i	0.915945	+	0.401304i	$0.131443\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$			77.4747i			0.187590i
$414$		0			0
$415$	143.457			0.345680
$416$		0			0
$417$		0			0
$418$		0			0
$419$	445.690	+	257.319i	1.06370	+	0.614127i	0.926453	−	0.376411i	$-0.122842\pi$
$419$	445.690	+	257.319i	1.06370	+	0.614127i	0.137245	+	0.990537i	$0.456175\pi$
$420$		0			0
$421$	−359.722	−	623.057i	−0.854448	−	1.47995i	−0.877157	−	0.480204i	$-0.840562\pi$
$421$	−359.722	−	623.057i	−0.854448	−	1.47995i	0.0227091	−	0.999742i	$-0.492771\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−95.0469	+	54.8754i	−0.223640	+	0.129119i
$426$		0			0
$427$	−104.664	+	181.283i	−0.245114	+	0.424550i
$428$		0			0
$429$		0			0
$430$		0			0
$431$			346.715i			0.804443i	0.915542	+	0.402222i	$0.131762\pi$
$431$			346.715i			0.804443i	−0.915542	+	0.402222i	$0.868238\pi$
$432$		0			0
$433$	−464.421			−1.07257			−0.536283	−	0.844038i	$-0.680172\pi$
$433$	−464.421			−1.07257			−0.536283	+	0.844038i	$0.680172\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	407.873	+	235.485i	0.933347	+	0.538868i
$438$		0			0
$439$	134.229	+	232.492i	0.305761	+	0.529594i	0.977431	−	0.211257i	$-0.0677557\pi$
$439$	134.229	+	232.492i	0.305761	+	0.529594i	−0.671669	+	0.740851i	$0.734422\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−409.784	+	236.589i	−0.925021	+	0.534061i	−0.885234	−	0.465147i	$-0.846002\pi$
$443$	−409.784	+	236.589i	−0.925021	+	0.534061i	−0.0397878	+	0.999208i	$0.512668\pi$
$444$		0			0
$445$	57.0646	−	98.8388i	0.128235	−	0.222110i
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	469.742i		−	1.04620i	−0.852272	−	0.523098i	$-0.824776\pi$
$449$		−	469.742i		−	1.04620i	0.852272	−	0.523098i	$-0.175224\pi$
$450$		0			0
$451$	950.812			2.10823
$452$		0			0
$453$		0			0
$454$		0			0
$455$	86.5142	+	49.9490i	0.190141	+	0.109778i
$456$		0			0
$457$	91.3917	+	158.295i	0.199982	+	0.346379i	0.948522	−	0.316711i	$-0.102578\pi$
$457$	91.3917	+	158.295i	0.199982	+	0.346379i	−0.748540	+	0.663089i	$0.769245\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−114.319	+	66.0024i	−0.247982	+	0.143172i	−0.618840	−	0.785517i	$-0.712397\pi$
$461$	−114.319	+	66.0024i	−0.247982	+	0.143172i	0.370858	+	0.928690i	$0.379064\pi$
$462$		0			0
$463$	316.849	−	548.798i	0.684338	−	1.18531i	−0.289306	−	0.957237i	$-0.593425\pi$
$463$	316.849	−	548.798i	0.684338	−	1.18531i	0.973644	−	0.228072i	$-0.0732422\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$			140.531i			0.300923i	0.988616	+	0.150461i	$0.0480759\pi$
$467$			140.531i			0.300923i	−0.988616	+	0.150461i	$0.951924\pi$
$468$		0			0
$469$	−159.451			−0.339981
$470$		0			0
$471$		0			0
$472$		0			0
$473$	689.150	+	397.881i	1.45698	+	0.841186i
$474$		0			0
$475$	42.0660	+	72.8604i	0.0885600	+	0.153390i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−685.976	+	396.048i	−1.43210	+	0.826823i	−0.997281	−	0.0736949i	$-0.976521\pi$
$479$	−685.976	+	396.048i	−1.43210	+	0.826823i	−0.434819	+	0.900518i	$0.643188\pi$
$480$		0			0
$481$	−571.671	+	990.163i	−1.18851	+	2.05855i
$482$		0			0
$483$		0			0
$484$		0			0
$485$			99.8917i			0.205962i
$486$		0			0
$487$	−242.911			−0.498790			−0.249395	−	0.968402i	$-0.580232\pi$
$487$	−242.911			−0.498790			−0.249395	+	0.968402i	$0.580232\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−266.368	−	153.787i	−0.542500	−	0.313213i	0.203591	−	0.979056i	$-0.434739\pi$
$491$	−266.368	−	153.787i	−0.542500	−	0.313213i	−0.746092	+	0.665843i	$0.768072\pi$
$492$		0			0
$493$	456.811	+	791.219i	0.926594	+	1.60491i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	104.822	−	60.5188i	0.210909	−	0.121768i
$498$		0			0
$499$	65.1507	−	112.844i	0.130563	−	0.226141i	−0.793331	−	0.608790i	$-0.791655\pi$
$499$	65.1507	−	112.844i	0.130563	−	0.226141i	0.923894	+	0.382650i	$0.124988\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$			142.999i			0.284292i	0.989846	+	0.142146i	$0.0454003\pi$
$503$			142.999i			0.284292i	−0.989846	+	0.142146i	$0.954600\pi$
$504$		0			0
$505$	−283.834			−0.562048
$506$		0			0
$507$		0			0
$508$		0			0
$509$	412.864	+	238.367i	0.811129	+	0.468305i	0.847348	−	0.531039i	$-0.178198\pi$
$509$	412.864	+	238.367i	0.811129	+	0.468305i	−0.0362190	+	0.999344i	$0.511531\pi$
$510$		0			0
$511$	−45.6903	−	79.1379i	−0.0894134	−	0.154869i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−362.935	+	209.541i	−0.704728	+	0.406875i
$516$		0			0
$517$	−563.081	+	975.286i	−1.08913	+	1.88643i
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	384.813i		−	0.738605i	−0.929309	−	0.369303i	$-0.879597\pi$
$521$		−	384.813i		−	0.738605i	0.929309	−	0.369303i	$-0.120403\pi$
$522$		0			0
$523$	710.188			1.35791			0.678956	−	0.734179i	$-0.262433\pi$
$523$	710.188			1.35791			0.678956	+	0.734179i	$0.262433\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−988.345	−	570.621i	−1.87542	−	1.08277i
$528$		0			0
$529$	127.220	+	220.352i	0.240492	+	0.416544i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−1194.14	+	689.438i	−2.24042	+	1.29350i
$534$		0			0
$535$	−26.7899	+	46.4015i	−0.0500746	+	0.0867317i
$536$		0			0
$537$		0			0
$538$		0			0
$539$			715.104i			1.32672i
$540$		0			0
$541$	−554.714			−1.02535			−0.512675	−	0.858583i	$-0.671345\pi$
$541$	−554.714			−1.02535			−0.512675	+	0.858583i	$0.671345\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	312.131	+	180.209i	0.572717	+	0.330658i
$546$		0			0
$547$	72.5065	+	125.585i	0.132553	+	0.229589i	0.924660	−	0.380794i	$-0.124349\pi$
$547$	72.5065	+	125.585i	0.132553	+	0.229589i	−0.792107	+	0.610382i	$0.791016\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	606.527	−	350.179i	1.10078	−	0.635533i
$552$		0			0
$553$	−0.00614530	+	0.0106440i	−1.11127e−5	+	1.92477e-5i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		−	260.152i		−	0.467060i	−0.972350	−	0.233530i	$-0.924972\pi$
$557$		−	260.152i		−	0.467060i	0.972350	−	0.233530i	$-0.0750276\pi$
$558$		0			0
$559$	−1154.02			−2.06444
$560$		0			0
$561$		0			0
$562$		0			0
$563$	165.527	+	95.5670i	0.294009	+	0.169746i	0.639748	−	0.768585i	$-0.279039\pi$
$563$	165.527	+	95.5670i	0.294009	+	0.169746i	−0.345740	+	0.938330i	$0.612372\pi$
$564$		0			0
$565$	67.0010	+	116.049i	0.118586	+	0.205397i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	52.6219	−	30.3813i	0.0924814	−	0.0533942i	−0.453046	−	0.891487i	$-0.649663\pi$
$569$	52.6219	−	30.3813i	0.0924814	−	0.0533942i	0.545527	+	0.838093i	$0.316329\pi$
$570$		0			0
$571$	−337.159	+	583.976i	−0.590470	+	1.02272i	0.403699	+	0.914892i	$0.367724\pi$
$571$	−337.159	+	583.976i	−0.590470	+	1.02272i	−0.994169	+	0.107833i	$0.965609\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$			139.950i			0.243391i
$576$		0			0
$577$	173.574			0.300821			0.150411	−	0.988624i	$-0.451940\pi$
$577$	173.574			0.300821			0.150411	+	0.988624i	$0.451940\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	108.205	+	62.4723i	0.186239	+	0.107525i
$582$		0			0
$583$	171.895	+	297.731i	0.294845	+	0.510687i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	445.578	−	257.254i	0.759076	−	0.438253i	−0.0698878	−	0.997555i	$-0.522264\pi$
$587$	445.578	−	257.254i	0.759076	−	0.438253i	0.828964	+	0.559302i	$0.188931\pi$
$588$		0			0
$589$	−437.423	+	757.638i	−0.742653	+	1.28631i
$590$		0			0
$591$		0			0
$592$		0			0
$593$			894.385i			1.50824i	0.656738	+	0.754119i	$0.271936\pi$
$593$			894.385i			1.50824i	−0.656738	+	0.754119i	$0.728064\pi$
$594$		0			0
$595$	−95.5877			−0.160652
$596$		0			0
$597$		0			0
$598$		0			0
$599$	74.8641	+	43.2228i	0.124982	+	0.0721583i	0.561187	−	0.827689i	$-0.310345\pi$
$599$	74.8641	+	43.2228i	0.124982	+	0.0721583i	−0.436206	+	0.899847i	$0.643678\pi$
$600$		0			0
$601$	498.997	+	864.288i	0.830278	+	1.43808i	0.897818	+	0.440367i	$0.145152\pi$
$601$	498.997	+	864.288i	0.830278	+	1.43808i	−0.0675402	+	0.997717i	$0.521515\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−250.234	+	144.473i	−0.413611	+	0.238798i
$606$		0			0
$607$	−267.168	+	462.749i	−0.440145	+	0.762354i	−0.997700	−	0.0677865i	$-0.978406\pi$
$607$	−267.168	+	462.749i	−0.440145	+	0.762354i	0.557555	+	0.830140i	$0.311740\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		−	1633.17i		−	2.67295i
$612$		0			0
$613$	283.029			0.461712			0.230856	−	0.972988i	$-0.425847\pi$
$613$	283.029			0.461712			0.230856	+	0.972988i	$0.425847\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	1039.61	+	600.221i	1.68495	+	0.972806i	0.958286	+	0.285812i	$0.0922632\pi$
$617$	1039.61	+	600.221i	1.68495	+	0.972806i	0.726663	+	0.686994i	$0.241070\pi$
$618$		0			0
$619$	−159.866	−	276.895i	−0.258264	−	0.447327i	0.707513	−	0.706701i	$-0.249817\pi$
$619$	−159.866	−	276.895i	−0.258264	−	0.447327i	−0.965777	+	0.259374i	$0.916484\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	86.0839	−	49.7006i	0.138176	−	0.0797762i
$624$		0			0
$625$	−12.5000	+	21.6506i	−0.0200000	+	0.0346410i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		−	1094.01i		−	1.73929i
$630$		0			0
$631$	679.823			1.07737			0.538687	−	0.842506i	$-0.318921\pi$
$631$	679.823			1.07737			0.538687	+	0.842506i	$0.318921\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−184.572	−	106.563i	−0.290664	−	0.167815i
$636$		0			0
$637$	−518.525	−	898.112i	−0.814011	−	1.40991i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	893.212	−	515.696i	1.39347	−	0.804518i	0.399769	−	0.916616i	$-0.369090\pi$
$641$	893.212	−	515.696i	1.39347	−	0.804518i	0.993697	+	0.112098i	$0.0357570\pi$
$642$		0			0
$643$	273.461	−	473.648i	0.425289	−	0.736623i	−0.571158	−	0.820840i	$-0.693506\pi$
$643$	273.461	−	473.648i	0.425289	−	0.736623i	0.996447	+	0.0842174i	$0.0268390\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	1073.99i		−	1.65995i	−0.557802	−	0.829974i	$-0.688355\pi$
$647$		−	1073.99i		−	1.65995i	0.557802	−	0.829974i	$-0.311645\pi$
$648$		0			0
$649$	629.277			0.969609
$650$		0			0
$651$		0			0
$652$		0			0
$653$	15.8347	+	9.14215i	0.0242491	+	0.0140002i	0.512076	−	0.858940i	$-0.328877\pi$
$653$	15.8347	+	9.14215i	0.0242491	+	0.0140002i	−0.487826	+	0.872941i	$0.662210\pi$
$654$		0			0
$655$	144.497	+	250.275i	0.220605	+	0.382100i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	127.107	−	73.3854i	0.192879	−	0.111359i	−0.400451	−	0.916318i	$-0.631146\pi$
$659$	127.107	−	73.3854i	0.192879	−	0.111359i	0.593330	+	0.804960i	$0.297813\pi$
$660$		0			0
$661$	−103.053	+	178.492i	−0.155904	+	0.270034i	−0.933388	−	0.358869i	$-0.883162\pi$
$661$	−103.053	+	178.492i	−0.155904	+	0.270034i	0.777484	+	0.628903i	$0.216496\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$			73.2750i			0.110188i
$666$		0			0
$667$	1165.02			1.74665
$668$		0			0
$669$		0			0
$670$		0			0
$671$	1472.44	+	850.115i	2.19440	+	1.26694i
$672$		0			0
$673$	−41.3730	−	71.6602i	−0.0614755	−	0.106479i	0.833650	−	0.552294i	$-0.186247\pi$
$673$	−41.3730	−	71.6602i	−0.0614755	−	0.106479i	−0.895125	+	0.445815i	$0.852914\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−635.500	+	366.906i	−0.938700	+	0.541959i	−0.889552	−	0.456833i	$-0.848984\pi$
$677$	−635.500	+	366.906i	−0.938700	+	0.541959i	−0.0491473	+	0.998792i	$0.515650\pi$
$678$		0			0
$679$	−43.5005	+	75.3450i	−0.0640655	+	0.110965i
$680$		0			0
$681$		0			0
$682$		0			0
$683$			920.831i			1.34821i	0.738633	+	0.674107i	$0.235471\pi$
$683$			920.831i			1.34821i	−0.738633	+	0.674107i	$0.764529\pi$
$684$		0			0
$685$	−118.205			−0.172563
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−431.772	−	249.283i	−0.626664	−	0.361805i
$690$		0			0
$691$	262.535	+	454.724i	0.379935	+	0.658067i	0.991052	−	0.133473i	$-0.0426130\pi$
$691$	262.535	+	454.724i	0.379935	+	0.658067i	−0.611117	+	0.791540i	$0.709280\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−106.305	+	61.3753i	−0.152957	+	0.0883098i
$696$		0			0
$697$	659.691	−	1142.62i	0.946472	−	1.63934i
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	644.204i		−	0.918979i	−0.888183	−	0.459490i	$-0.848032\pi$
$701$		−	644.204i		−	0.918979i	0.888183	−	0.459490i	$-0.151968\pi$
$702$		0			0
$703$	−838.639			−1.19294
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−214.087	−	123.603i	−0.302811	−	0.174828i
$708$		0			0
$709$	−415.976	−	720.492i	−0.586708	−	1.01621i	−0.994660	−	0.103205i	$-0.967090\pi$
$709$	−415.976	−	720.492i	−0.586708	−	1.01621i	0.407952	−	0.913003i	$-0.366243\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−1260.30	+	727.635i	−1.76760	+	1.02053i
$714$		0			0
$715$	405.703	−	702.699i	0.567417	−	0.982795i
$716$		0			0
$717$		0			0
$718$		0			0
$719$			127.368i			0.177146i	0.996070	+	0.0885730i	$0.0282307\pi$
$719$			127.368i			0.177146i	−0.996070	+	0.0885730i	$0.971769\pi$
$720$		0			0
$721$	−365.000			−0.506241
$722$		0			0
$723$		0			0
$724$		0			0
$725$	180.231	+	104.056i	0.248594	+	0.143526i
$726$		0			0
$727$	355.972	+	616.562i	0.489645	+	0.848091i	0.999929	−	0.0119156i	$-0.00379293\pi$
$727$	355.972	+	616.562i	0.489645	+	0.848091i	−0.510284	+	0.860006i	$0.670460\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	956.291	−	552.115i	1.30820	−	0.755287i
$732$		0			0
$733$	−228.941	+	396.538i	−0.312335	+	0.540979i	−0.978867	−	0.204496i	$-0.934444\pi$
$733$	−228.941	+	396.538i	−0.312335	+	0.540979i	0.666533	+	0.745476i	$0.267778\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			1295.12i			1.75728i
$738$		0			0
$739$	1100.81			1.48959			0.744794	−	0.667295i	$-0.232548\pi$
$739$	1100.81			1.48959			0.744794	+	0.667295i	$0.232548\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−1193.39	−	689.004i	−1.60618	−	0.927327i	−0.990215	−	0.139551i	$-0.955434\pi$
$743$	−1193.39	−	689.004i	−1.60618	−	0.927327i	−0.615962	−	0.787776i	$-0.711233\pi$
$744$		0			0
$745$	81.3729	+	140.942i	0.109225	+	0.189184i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−40.4135	+	23.3327i	−0.0539566	+	0.0311519i
$750$		0			0
$751$	138.984	−	240.727i	0.185065	−	0.320542i	−0.758533	−	0.651634i	$-0.774084\pi$
$751$	138.984	−	240.727i	0.185065	−	0.320542i	0.943598	+	0.331092i	$0.107417\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$			128.011i			0.169551i
$756$		0			0
$757$	71.0992			0.0939224			0.0469612	−	0.998897i	$-0.485046\pi$
$757$	71.0992			0.0939224			0.0469612	+	0.998897i	$0.485046\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	449.264	+	259.383i	0.590360	+	0.340845i	0.765240	−	0.643745i	$-0.222620\pi$
$761$	449.264	+	259.383i	0.590360	+	0.340845i	−0.174880	+	0.984590i	$0.555954\pi$
$762$		0			0
$763$	156.953	+	271.851i	0.205706	+	0.356293i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−790.320	+	456.291i	−1.03040	+	0.594904i
$768$		0			0
$769$	−240.525	+	416.601i	−0.312776	+	0.541744i	−0.978962	−	0.204041i	$-0.934592\pi$
$769$	−240.525	+	416.601i	−0.312776	+	0.541744i	0.666186	+	0.745786i	$0.267926\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$			794.504i			1.02782i	0.857845	+	0.513909i	$0.171803\pi$
$773$			794.504i			1.02782i	−0.857845	+	0.513909i	$0.828197\pi$
$774$		0			0
$775$	−259.962			−0.335435
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−875.900	−	505.701i	−1.12439	−	0.649167i
$780$		0			0
$781$	−491.555	−	851.398i	−0.629392	−	1.09014i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	287.191	−	165.810i	0.365848	−	0.211223i
$786$		0			0
$787$	−520.719	+	901.912i	−0.661651	+	1.14601i	0.318531	+	0.947912i	$0.396810\pi$
$787$	−520.719	+	901.912i	−0.661651	+	1.14601i	−0.980182	+	0.198100i	$0.936523\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$			116.709i			0.147547i
$792$		0			0
$793$	−2465.69			−3.10932
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−1181.27	−	682.004i	−1.48214	−	0.855714i	−0.482346	−	0.875981i	$-0.660215\pi$
$797$	−1181.27	−	682.004i	−1.48214	−	0.855714i	−0.999795	+	0.0202662i	$0.993549\pi$
$798$		0			0
$799$	781.353	+	1353.34i	0.977914	+	1.69380i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−642.785	+	371.112i	−0.800480	+	0.462157i
$804$		0			0
$805$	−60.9449	+	105.560i	−0.0757080	+	0.131130i
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	1274.65i		−	1.57559i	−0.615940	−	0.787793i	$-0.711224\pi$
$809$		−	1274.65i		−	1.57559i	0.615940	−	0.787793i	$-0.288776\pi$
$810$		0			0
$811$	866.001			1.06782			0.533909	−	0.845542i	$-0.320722\pi$
$811$	866.001			1.06782			0.533909	+	0.845542i	$0.320722\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	157.160	+	90.7364i	0.192835	+	0.111333i
$816$		0			0
$817$	−423.236	−	733.067i	−0.518037	−	0.897266i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	246.588	−	142.368i	0.300351	−	0.173408i	−0.342249	−	0.939609i	$-0.611189\pi$
$821$	246.588	−	142.368i	0.300351	−	0.173408i	0.642601	+	0.766201i	$0.277855\pi$
$822$		0			0
$823$	−302.851	+	524.553i	−0.367984	+	0.637367i	−0.989250	−	0.146233i	$-0.953285\pi$
$823$	−302.851	+	524.553i	−0.367984	+	0.637367i	0.621266	+	0.783600i	$0.286619\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	1545.19i		−	1.86843i	−0.356708	−	0.934216i	$-0.616101\pi$
$827$		−	1545.19i		−	1.86843i	0.356708	−	0.934216i	$-0.383899\pi$
$828$		0			0
$829$	−361.126			−0.435617			−0.217808	−	0.975992i	$-0.569891\pi$
$829$	−361.126			−0.435617			−0.217808	+	0.975992i	$0.569891\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	859.361	+	496.153i	1.03165	+	0.595621i
$834$		0			0
$835$	320.582	+	555.264i	0.383930	+	0.664987i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	507.689	−	293.115i	0.605113	−	0.349362i	−0.165938	−	0.986136i	$-0.553065\pi$
$839$	507.689	−	293.115i	0.605113	−	0.349362i	0.771050	+	0.636774i	$0.219732\pi$
$840$		0			0
$841$	445.719	−	772.007i	0.529986	−	0.917963i
$842$		0			0
$843$		0			0
$844$		0			0
$845$			798.814i			0.945342i
$846$		0			0
$847$	−251.658			−0.297117
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−1208.14	−	697.520i	−1.41967	−	0.819648i
$852$		0			0
$853$	663.111	+	1148.54i	0.777387	+	1.34647i	0.933443	+	0.358726i	$0.116789\pi$
$853$	663.111	+	1148.54i	0.777387	+	1.34647i	−0.156056	+	0.987748i	$0.549878\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	257.372	−	148.594i	0.300317	−	0.173388i	−0.342268	−	0.939602i	$-0.611195\pi$
$857$	257.372	−	148.594i	0.300317	−	0.173388i	0.642585	+	0.766214i	$0.277862\pi$
$858$		0			0
$859$	421.190	−	729.522i	0.490325	−	0.849269i	−0.509613	−	0.860404i	$-0.670211\pi$
$859$	421.190	−	729.522i	0.490325	−	0.849269i	0.999938	+	0.0111355i	$0.00354460\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			1603.54i			1.85810i	0.369957	+	0.929049i	$0.379373\pi$
$863$			1603.54i			1.85810i	−0.369957	+	0.929049i	$0.620627\pi$
$864$		0			0
$865$	−363.335			−0.420040
$866$		0			0
$867$		0			0
$868$		0			0
$869$	0.0864541	+	0.0499143i	9.94868e−5	+	5.74387e-5i
$870$		0			0
$871$	−939.095	−	1626.56i	−1.07818	−	1.86746i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−18.8567	+	10.8869i	−0.0215505	+	0.0124422i
$876$		0			0
$877$	159.307	−	275.928i	0.181650	−	0.314627i	−0.760793	−	0.648995i	$-0.775190\pi$
$877$	159.307	−	275.928i	0.181650	−	0.314627i	0.942442	+	0.334368i	$0.108523\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	646.569i		−	0.733903i	−0.930240	−	0.366952i	$-0.880401\pi$
$881$		−	646.569i		−	0.733903i	0.930240	−	0.366952i	$-0.119599\pi$
$882$		0			0
$883$	−359.976			−0.407674			−0.203837	−	0.979005i	$-0.565341\pi$
$883$	−359.976			−0.407674			−0.203837	+	0.979005i	$0.565341\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	114.926	+	66.3526i	0.129567	+	0.0748056i	0.563383	−	0.826196i	$-0.309500\pi$
$887$	114.926	+	66.3526i	0.129567	+	0.0748056i	−0.433815	+	0.901002i	$0.642833\pi$
$888$		0			0
$889$	−92.8110	−	160.753i	−0.104399	−	0.180825i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	1037.44	−	598.964i	1.16174	−	0.670732i
$894$		0			0
$895$	−145.725	+	252.402i	−0.162821	+	0.282014i
$896$		0			0
$897$		0			0
$898$		0			0
$899$			2164.06i			2.40718i
$900$		0			0
$901$	477.056			0.529473
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−362.838	−	209.484i	−0.400925	−	0.231474i
$906$		0			0
$907$	188.629	+	326.715i	0.207970	+	0.360215i	0.951075	−	0.308960i	$-0.0999809\pi$
$907$	188.629	+	326.715i	0.207970	+	0.360215i	−0.743105	+	0.669175i	$0.766648\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−71.2370	+	41.1287i	−0.0781965	+	0.0451468i	−0.538588	−	0.842569i	$-0.681042\pi$
$911$	−71.2370	+	41.1287i	−0.0781965	+	0.0451468i	0.460392	+	0.887716i	$0.347709\pi$
$912$		0			0
$913$	507.422	−	878.880i	0.555774	−	0.962629i
$914$		0			0
$915$		0			0
$916$		0			0
$917$			251.699i			0.274481i
$918$		0			0
$919$	722.940			0.786659			0.393330	−	0.919398i	$-0.371323\pi$
$919$	722.940			0.786659			0.393330	+	0.919398i	$0.371323\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	1234.71	+	712.858i	1.33771	+	0.772327i
$924$		0			0
$925$	−124.602	−	215.816i	−0.134705	−	0.233315i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	452.602	−	261.310i	0.487193	−	0.281281i	−0.236216	−	0.971700i	$-0.575907\pi$
$929$	452.602	−	261.310i	0.487193	−	0.281281i	0.723409	+	0.690420i	$0.242574\pi$
$930$		0			0
$931$	380.337	−	658.763i	0.408525	−	0.707587i
$932$		0			0
$933$		0			0
$934$		0			0
$935$			776.397i			0.830371i
$936$		0			0
$937$	413.215			0.440998			0.220499	−	0.975387i	$-0.429231\pi$
$937$	413.215			0.440998			0.220499	+	0.975387i	$0.429231\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	230.665	+	133.175i	0.245128	+	0.141525i	0.617531	−	0.786546i	$-0.288133\pi$
$941$	230.665	+	133.175i	0.245128	+	0.141525i	−0.372404	+	0.928071i	$0.621466\pi$
$942$		0			0
$943$	−841.213	−	1457.02i	−0.892061	−	1.54509i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	544.962	−	314.634i	0.575462	−	0.332243i	−0.183866	−	0.982951i	$-0.558861\pi$
$947$	544.962	−	314.634i	0.575462	−	0.332243i	0.759328	+	0.650708i	$0.225528\pi$
$948$		0			0
$949$	538.191	−	932.173i	0.567113	−	0.982269i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	138.019i		−	0.144826i	−0.997375	−	0.0724130i	$-0.976930\pi$
$953$		−	138.019i		−	0.144826i	0.997375	−	0.0724130i	$-0.0230700\pi$
$954$		0			0
$955$	−607.104			−0.635711
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−89.1585	−	51.4757i	−0.0929703	−	0.0536764i
$960$		0			0
$961$	−871.108	−	1508.80i	−0.906460	−	1.57003i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−56.1091	+	32.3946i	−0.0581442	+	0.0335696i
$966$		0			0
$967$	−127.299	+	220.488i	−0.131643	+	0.228012i	−0.924310	−	0.381643i	$-0.875358\pi$
$967$	−127.299	+	220.488i	−0.131643	+	0.228012i	0.792667	+	0.609655i	$0.208692\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		−	1430.25i		−	1.47296i	−0.676457	−	0.736482i	$-0.736486\pi$
$971$		−	1430.25i		−	1.47296i	0.676457	−	0.736482i	$-0.263514\pi$
$972$		0			0
$973$	−106.910			−0.109877
$974$		0			0
$975$		0			0
$976$		0			0
$977$	336.866	+	194.490i	0.344796	+	0.199068i	0.662391	−	0.749158i	$-0.269542\pi$
$977$	336.866	+	194.490i	0.344796	+	0.199068i	−0.317595	+	0.948227i	$0.602875\pi$
$978$		0			0
$979$	−403.685	−	699.204i	−0.412345	−	0.714202i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	136.571	−	78.8493i	0.138933	−	0.0802130i	−0.428922	−	0.903341i	$-0.641107\pi$
$983$	136.571	−	78.8493i	0.138933	−	0.0802130i	0.567855	+	0.823128i	$0.307773\pi$
$984$		0			0
$985$	33.1607	−	57.4360i	0.0336657	−	0.0583107i
$986$		0			0
$987$		0			0
$988$		0			0
$989$		−	1408.07i		−	1.42373i
$990$		0			0
$991$	730.868			0.737506			0.368753	−	0.929527i	$-0.379785\pi$
$991$	730.868			0.737506			0.368753	+	0.929527i	$0.379785\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−524.742	−	302.960i	−0.527379	−	0.304482i
$996$		0			0
$997$	−791.004	−	1370.06i	−0.793384	−	1.37418i	−0.923860	−	0.382730i	$-0.874984\pi$
$997$	−791.004	−	1370.06i	−0.793384	−	1.37418i	0.130476	−	0.991451i	$-0.458349\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1620.3.o.g.701.3		32
3.2	odd	2	inner	1620.3.o.g.701.15		32
9.2	odd	6	inner	1620.3.o.g.1241.3		32
9.4	even	3		1620.3.g.c.161.12	yes	16
9.5	odd	6		1620.3.g.c.161.4	✓	16
9.7	even	3	inner	1620.3.o.g.1241.15		32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1620.3.g.c.161.4	✓	16	9.5	odd	6
1620.3.g.c.161.12	yes	16	9.4	even	3
1620.3.o.g.701.3		32	1.1	even	1	trivial
1620.3.o.g.701.15		32	3.2	odd	2	inner
1620.3.o.g.1241.3		32	9.2	odd	6	inner
1620.3.o.g.1241.15		32	9.7	even	3	inner

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:	\( N \)	\(=\)	\( 1620 = 2^{2} \cdot 3^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1620.o (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-1.93649 - 1.11803i) q^{5} +(-0.973755 - 1.68659i) q^{7} +O(q^{10})\)	\(q+(-1.93649 - 1.11803i) q^{5} +(-0.973755 - 1.68659i) q^{7} +(-13.6991 + 7.90918i) q^{11} +(11.4700 - 19.8666i) q^{13} +21.9502i q^{17} +16.8264 q^{19} +(24.2401 + 13.9950i) q^{23} +(2.50000 + 4.33013i) q^{25} +(36.0462 - 20.8113i) q^{29} +(-25.9962 + 45.0268i) q^{31} +4.35476i q^{35} -49.8407 q^{37} +(-52.0551 - 30.0541i) q^{41} +(-25.1531 - 43.5665i) q^{43} +(61.6553 - 35.5967i) q^{47} +(22.6036 - 39.1506i) q^{49} -21.7336i q^{53} +35.3709 q^{55} +(-34.4517 - 19.8907i) q^{59} +(-53.7423 - 93.0844i) q^{61} +(-44.4230 + 25.6476i) q^{65} +(40.9371 - 70.9052i) q^{67} +62.1499i q^{71} +46.9217 q^{73} +(26.6791 + 15.4032i) q^{77} +(-0.00315546 - 0.00546542i) q^{79} +(-55.5608 + 32.0780i) q^{83} +(24.5410 - 42.5063i) q^{85} +51.0401i q^{89} -44.6757 q^{91} +(-32.5842 - 18.8125i) q^{95} +(-22.3365 - 38.6879i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - 8 q^{7}+O(q^{10}) \) 32 * q - 8 * q^7	\( 32 q - 8 q^{7} + 40 q^{13} + 112 q^{19} + 80 q^{25} + 64 q^{31} - 176 q^{37} - 128 q^{43} - 216 q^{49} - 8 q^{61} + 40 q^{67} + 112 q^{73} + 136 q^{79} - 784 q^{91} - 128 q^{97}+O(q^{100}) \) 32 * q - 8 * q^7 + 40 * q^13 + 112 * q^19 + 80 * q^25 + 64 * q^31 - 176 * q^37 - 128 * q^43 - 216 * q^49 - 8 * q^61 + 40 * q^67 + 112 * q^73 + 136 * q^79 - 784 * q^91 - 128 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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