LMFDB - Embedded newform 1620.3.l.f.1297.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1620,3,Mod(973,1620)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1620, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([0, 0, 3]))

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

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

chi := DirichletCharacter("1620.973");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1620 = 2^{2} \cdot 3^{4} \cdot 5 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1620.l (of order $4$, degree $2$, 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:	$24$
Relative dimension:	$12$ over $\Q(i)$
Twist minimal:	no (minimal twist has level 180)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			1297.3
Character	$\chi$	$=$	1620.1297
Dual form			1620.3.l.f.973.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-3.87041 - 3.16543i) q^{5} +(3.43863 + 3.43863i) q^{7} +O(q^{10})$	$q+(-3.87041 - 3.16543i) q^{5} +(3.43863 + 3.43863i) q^{7} +7.26535 q^{11} +(-3.06083 + 3.06083i) q^{13} +(-4.69512 - 4.69512i) q^{17} -10.1112i q^{19} +(-29.8592 + 29.8592i) q^{23} +(4.96012 + 24.5030i) q^{25} -33.7361i q^{29} +23.3466 q^{31} +(-2.42417 - 24.1937i) q^{35} +(40.1926 + 40.1926i) q^{37} -10.5684 q^{41} +(-52.7100 + 52.7100i) q^{43} +(-17.0667 - 17.0667i) q^{47} -25.3516i q^{49} +(44.6296 - 44.6296i) q^{53} +(-28.1199 - 22.9980i) q^{55} +13.1167i q^{59} -54.2365 q^{61} +(21.5355 - 2.15782i) q^{65} +(46.6532 + 46.6532i) q^{67} -15.1754 q^{71} +(31.1199 - 31.1199i) q^{73} +(24.9829 + 24.9829i) q^{77} +144.092i q^{79} +(-58.5951 + 58.5951i) q^{83} +(3.30997 + 33.0341i) q^{85} +124.203i q^{89} -21.0501 q^{91} +(-32.0063 + 39.1345i) q^{95} +(-20.8047 - 20.8047i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q+O(q^{10}) $ 24 * q	$ 24 q - 12 q^{11} + 18 q^{17} + 12 q^{23} + 6 q^{25} + 36 q^{35} - 42 q^{37} - 72 q^{41} + 78 q^{47} - 156 q^{53} + 66 q^{55} + 96 q^{61} + 132 q^{65} + 78 q^{67} - 156 q^{71} + 240 q^{77} - 132 q^{83} - 96 q^{85} + 84 q^{91} + 168 q^{95} + 90 q^{97}+O(q^{100}) $ 24 * q - 12 * q^11 + 18 * q^17 + 12 * q^23 + 6 * q^25 + 36 * q^35 - 42 * q^37 - 72 * q^41 + 78 * q^47 - 156 * q^53 + 66 * q^55 + 96 * q^61 + 132 * q^65 + 78 * q^67 - 156 * q^71 + 240 * q^77 - 132 * q^83 - 96 * q^85 + 84 * q^91 + 168 * q^95 + 90 * q^97

Display 100 coefficients 10 coefficients

Character values

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

$n$	$811$	$1297$	$1541$
$\chi(n)$	$1$	$e\left(\frac{1}{4}\right)$	$1$

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
$5$	−3.87041	−	3.16543i	−0.774082	−	0.633086i
$5$	−3.87041	−	3.16543i	−0.774082	−	0.633086i
$6$		0			0
$7$	3.43863	+	3.43863i	0.491233	+	0.491233i	0.908695	−	0.417462i	$-0.137080\pi$
$7$	3.43863	+	3.43863i	0.491233	+	0.491233i	−0.417462	+	0.908695i	$0.637080\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	7.26535			0.660487			0.330243	−	0.943896i	$-0.392869\pi$
$11$	7.26535			0.660487			0.330243	+	0.943896i	$0.392869\pi$
$12$		0			0
$13$	−3.06083	+	3.06083i	−0.235448	+	0.235448i	−0.814962	−	0.579514i	$-0.803242\pi$
$13$	−3.06083	+	3.06083i	−0.235448	+	0.235448i	0.579514	+	0.814962i	$0.303242\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−4.69512	−	4.69512i	−0.276184	−	0.276184i	0.555400	−	0.831584i	$-0.312565\pi$
$17$	−4.69512	−	4.69512i	−0.276184	−	0.276184i	−0.831584	+	0.555400i	$0.812565\pi$
$18$		0			0
$19$		−	10.1112i		−	0.532169i	−0.963950	−	0.266084i	$-0.914270\pi$
$19$		−	10.1112i		−	0.532169i	0.963950	−	0.266084i	$-0.0857300\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−29.8592	+	29.8592i	−1.29822	+	1.29822i	−0.368660	+	0.929564i	$0.620183\pi$
$23$	−29.8592	+	29.8592i	−1.29822	+	1.29822i	−0.929564	+	0.368660i	$0.879817\pi$
$24$		0			0
$25$	4.96012	+	24.5030i	0.198405	+	0.980120i
$26$		0			0
$27$		0			0
$28$		0			0
$29$		−	33.7361i		−	1.16332i	−0.813434	−	0.581658i	$-0.802404\pi$
$29$		−	33.7361i		−	1.16332i	0.813434	−	0.581658i	$-0.197596\pi$
$30$		0			0
$31$	23.3466			0.753116			0.376558	−	0.926393i	$-0.377108\pi$
$31$	23.3466			0.753116			0.376558	+	0.926393i	$0.377108\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−2.42417	−	24.1937i	−0.0692619	−	0.691247i
$36$		0			0
$37$	40.1926	+	40.1926i	1.08629	+	1.08629i	0.995907	+	0.0903805i	$0.0288083\pi$
$37$	40.1926	+	40.1926i	1.08629	+	1.08629i	0.0903805	+	0.995907i	$0.471192\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−10.5684			−0.257766			−0.128883	−	0.991660i	$-0.541139\pi$
$41$	−10.5684			−0.257766			−0.128883	+	0.991660i	$0.541139\pi$
$42$		0			0
$43$	−52.7100	+	52.7100i	−1.22581	+	1.22581i	−0.260280	+	0.965533i	$0.583815\pi$
$43$	−52.7100	+	52.7100i	−1.22581	+	1.22581i	−0.965533	+	0.260280i	$0.916185\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−17.0667	−	17.0667i	−0.363121	−	0.363121i	0.501840	−	0.864961i	$-0.332657\pi$
$47$	−17.0667	−	17.0667i	−0.363121	−	0.363121i	−0.864961	+	0.501840i	$0.832657\pi$
$48$		0			0
$49$		−	25.3516i		−	0.517380i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	44.6296	−	44.6296i	0.842069	−	0.842069i	−0.147059	−	0.989128i	$-0.546981\pi$
$53$	44.6296	−	44.6296i	0.842069	−	0.842069i	0.989128	+	0.147059i	$0.0469808\pi$
$54$		0			0
$55$	−28.1199	−	22.9980i	−0.511271	−	0.418145i
$56$		0			0
$57$		0			0
$58$		0			0
$59$			13.1167i			0.222318i	0.993803	+	0.111159i	$0.0354562\pi$
$59$			13.1167i			0.222318i	−0.993803	+	0.111159i	$0.964544\pi$
$60$		0			0
$61$	−54.2365			−0.889123			−0.444561	−	0.895748i	$-0.646640\pi$
$61$	−54.2365			−0.889123			−0.444561	+	0.895748i	$0.646640\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	21.5355	−	2.15782i	0.331315	−	0.0331973i
$66$		0			0
$67$	46.6532	+	46.6532i	0.696317	+	0.696317i	0.963614	−	0.267297i	$-0.0861306\pi$
$67$	46.6532	+	46.6532i	0.696317	+	0.696317i	−0.267297	+	0.963614i	$0.586131\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−15.1754			−0.213737			−0.106869	−	0.994273i	$-0.534082\pi$
$71$	−15.1754			−0.213737			−0.106869	+	0.994273i	$0.534082\pi$
$72$		0			0
$73$	31.1199	−	31.1199i	0.426300	−	0.426300i	−0.461066	−	0.887366i	$-0.652533\pi$
$73$	31.1199	−	31.1199i	0.426300	−	0.426300i	0.887366	+	0.461066i	$0.152533\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	24.9829	+	24.9829i	0.324453	+	0.324453i
$78$		0			0
$79$			144.092i			1.82395i	0.410249	+	0.911973i	$0.365442\pi$
$79$			144.092i			1.82395i	−0.410249	+	0.911973i	$0.634558\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−58.5951	+	58.5951i	−0.705965	+	0.705965i	−0.965684	−	0.259719i	$-0.916370\pi$
$83$	−58.5951	+	58.5951i	−0.705965	+	0.705965i	0.259719	+	0.965684i	$0.416370\pi$
$84$		0			0
$85$	3.30997	+	33.0341i	0.0389408	+	0.388637i
$86$		0			0
$87$		0			0
$88$		0			0
$89$			124.203i			1.39554i	0.716324	+	0.697768i	$0.245823\pi$
$89$			124.203i			1.39554i	−0.716324	+	0.697768i	$0.754177\pi$
$90$		0			0
$91$	−21.0501			−0.231320
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−32.0063	+	39.1345i	−0.336909	+	0.411942i
$96$		0			0
$97$	−20.8047	−	20.8047i	−0.214481	−	0.214481i	0.591687	−	0.806168i	$-0.298462\pi$
$97$	−20.8047	−	20.8047i	−0.214481	−	0.214481i	−0.806168	+	0.591687i	$0.798462\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	136.461			1.35110			0.675549	−	0.737315i	$-0.263907\pi$
$101$	136.461			1.35110			0.675549	+	0.737315i	$0.263907\pi$
$102$		0			0
$103$	−49.0738	+	49.0738i	−0.476445	+	0.476445i	−0.903993	−	0.427548i	$-0.859378\pi$
$103$	−49.0738	+	49.0738i	−0.476445	+	0.476445i	0.427548	+	0.903993i	$0.359378\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	28.8128	+	28.8128i	0.269279	+	0.269279i	0.828810	−	0.559531i	$-0.189019\pi$
$107$	28.8128	+	28.8128i	0.269279	+	0.269279i	−0.559531	+	0.828810i	$0.689019\pi$
$108$		0			0
$109$			178.917i			1.64144i	0.571332	+	0.820719i	$0.306427\pi$
$109$			178.917i			1.64144i	−0.571332	+	0.820719i	$0.693573\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−25.6615	+	25.6615i	−0.227093	+	0.227093i	−0.811477	−	0.584384i	$-0.801336\pi$
$113$	−25.6615	+	25.6615i	−0.227093	+	0.227093i	0.584384	+	0.811477i	$0.301336\pi$
$114$		0			0
$115$	210.084	−	21.0501i	1.82682	−	0.183044i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		−	32.2896i		−	0.271341i
$120$		0			0
$121$	−68.2147			−0.563758
$122$		0			0
$123$		0			0
$124$		0			0
$125$	58.3648	−	110.538i	0.466918	−	0.884300i
$126$		0			0
$127$	82.5876	+	82.5876i	0.650296	+	0.650296i	0.953064	−	0.302768i	$-0.0979108\pi$
$127$	82.5876	+	82.5876i	0.650296	+	0.650296i	−0.302768	+	0.953064i	$0.597911\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	146.464			1.11805			0.559024	−	0.829152i	$-0.311176\pi$
$131$	146.464			1.11805			0.559024	+	0.829152i	$0.311176\pi$
$132$		0			0
$133$	34.7687	−	34.7687i	0.261419	−	0.261419i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	116.396	+	116.396i	0.849606	+	0.849606i	0.990084	−	0.140478i	$-0.0448639\pi$
$137$	116.396	+	116.396i	0.849606	+	0.849606i	−0.140478	+	0.990084i	$0.544864\pi$
$138$		0			0
$139$			167.551i			1.20541i	0.797966	+	0.602703i	$0.205910\pi$
$139$			167.551i			1.20541i	−0.797966	+	0.602703i	$0.794090\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−22.2380	+	22.2380i	−0.155511	+	0.155511i
$144$		0			0
$145$	−106.789	+	130.573i	−0.736478	+	0.900501i
$146$		0			0
$147$		0			0
$148$		0			0
$149$		−	187.948i		−	1.26140i	−0.776028	−	0.630699i	$-0.782768\pi$
$149$		−	187.948i		−	1.26140i	0.776028	−	0.630699i	$-0.217232\pi$
$150$		0			0
$151$	77.7893			0.515161			0.257580	−	0.966257i	$-0.417075\pi$
$151$	77.7893			0.515161			0.257580	+	0.966257i	$0.417075\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−90.3608	−	73.9020i	−0.582973	−	0.476787i
$156$		0			0
$157$	97.9050	+	97.9050i	0.623599	+	0.623599i	0.946450	−	0.322851i	$-0.104641\pi$
$157$	97.9050	+	97.9050i	0.623599	+	0.623599i	−0.322851	+	0.946450i	$0.604641\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−205.349			−1.27546
$162$		0			0
$163$	−160.492	+	160.492i	−0.984616	+	0.984616i	−0.999883	−	0.0152671i	$-0.995140\pi$
$163$	−160.492	+	160.492i	−0.984616	+	0.984616i	0.0152671	+	0.999883i	$0.495140\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	36.7194	+	36.7194i	0.219877	+	0.219877i	0.808446	−	0.588570i	$-0.200309\pi$
$167$	36.7194	+	36.7194i	0.219877	+	0.219877i	−0.588570	+	0.808446i	$0.700309\pi$
$168$		0			0
$169$			150.263i			0.889128i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−118.749	+	118.749i	−0.686411	+	0.686411i	−0.961437	−	0.275026i	$-0.911314\pi$
$173$	−118.749	+	118.749i	−0.686411	+	0.686411i	0.275026	+	0.961437i	$0.411314\pi$
$174$		0			0
$175$	−67.2008	+	101.313i	−0.384004	+	0.578930i
$176$		0			0
$177$		0			0
$178$		0			0
$179$			293.293i			1.63851i	0.573429	+	0.819255i	$0.305613\pi$
$179$			293.293i			1.63851i	−0.573429	+	0.819255i	$0.694387\pi$
$180$		0			0
$181$	182.818			1.01005			0.505023	−	0.863106i	$-0.331484\pi$
$181$	182.818			1.01005			0.505023	+	0.863106i	$0.331484\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−28.3350	−	282.789i	−0.153162	−	1.52859i
$186$		0			0
$187$	−34.1117	−	34.1117i	−0.182416	−	0.182416i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−289.810			−1.51733			−0.758666	−	0.651480i	$-0.774149\pi$
$191$	−289.810			−1.51733			−0.758666	+	0.651480i	$0.774149\pi$
$192$		0			0
$193$	−196.785	+	196.785i	−1.01961	+	1.01961i	−0.0198065	+	0.999804i	$0.506305\pi$
$193$	−196.785	+	196.785i	−1.01961	+	1.01961i	−0.999804	+	0.0198065i	$0.993695\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−181.377	−	181.377i	−0.920694	−	0.920694i	0.0763840	−	0.997078i	$-0.475663\pi$
$197$	−181.377	−	181.377i	−0.920694	−	0.920694i	−0.997078	+	0.0763840i	$0.975663\pi$
$198$		0			0
$199$		−	160.532i		−	0.806694i	−0.915047	−	0.403347i	$-0.867847\pi$
$199$		−	160.532i		−	0.806694i	0.915047	−	0.403347i	$-0.132153\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	116.006	−	116.006i	0.571459	−	0.571459i
$204$		0			0
$205$	40.9040	+	33.4535i	0.199532	+	0.163188i
$206$		0			0
$207$		0			0
$208$		0			0
$209$		−	73.4615i		−	0.351490i
$210$		0			0
$211$	404.607			1.91757			0.958783	−	0.284138i	$-0.0917074\pi$
$211$	404.607			1.91757			0.958783	+	0.284138i	$0.0917074\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	370.859	−	37.1595i	1.72492	−	0.172835i
$216$		0			0
$217$	80.2803	+	80.2803i	0.369955	+	0.369955i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	28.7419			0.130054
$222$		0			0
$223$	207.081	−	207.081i	0.928614	−	0.928614i	−0.0690026	−	0.997616i	$-0.521982\pi$
$223$	207.081	−	207.081i	0.928614	−	0.928614i	0.997616	+	0.0690026i	$0.0219817\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	9.85057	+	9.85057i	0.0433946	+	0.0433946i	0.728471	−	0.685077i	$-0.240231\pi$
$227$	9.85057	+	9.85057i	0.0433946	+	0.0433946i	−0.685077	+	0.728471i	$0.740231\pi$
$228$		0			0
$229$		−	119.872i		−	0.523459i	−0.965141	−	0.261729i	$-0.915707\pi$
$229$		−	119.872i		−	0.523459i	0.965141	−	0.261729i	$-0.0842928\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−37.3376	+	37.3376i	−0.160247	+	0.160247i	−0.782676	−	0.622429i	$-0.786146\pi$
$233$	−37.3376	+	37.3376i	−0.160247	+	0.160247i	0.622429	+	0.782676i	$0.286146\pi$
$234$		0			0
$235$	12.0317	+	120.078i	0.0511986	+	0.510972i
$236$		0			0
$237$		0			0
$238$		0			0
$239$		−	176.707i		−	0.739359i	−0.929159	−	0.369679i	$-0.879468\pi$
$239$		−	176.707i		−	0.739359i	0.929159	−	0.369679i	$-0.120532\pi$
$240$		0			0
$241$	−292.782			−1.21486			−0.607431	−	0.794372i	$-0.707800\pi$
$241$	−292.782			−1.21486			−0.607431	+	0.794372i	$0.707800\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−80.2488	+	98.1212i	−0.327546	+	0.400495i
$246$		0			0
$247$	30.9487	+	30.9487i	0.125298	+	0.125298i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−279.420			−1.11323			−0.556614	−	0.830771i	$-0.687900\pi$
$251$	−279.420			−1.11323			−0.556614	+	0.830771i	$0.687900\pi$
$252$		0			0
$253$	−216.937	+	216.937i	−0.857460	+	0.857460i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	183.160	+	183.160i	0.712684	+	0.712684i	0.967096	−	0.254412i	$-0.0818820\pi$
$257$	183.160	+	183.160i	0.712684	+	0.712684i	−0.254412	+	0.967096i	$0.581882\pi$
$258$		0			0
$259$			276.415i			1.06724i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	125.277	−	125.277i	0.476340	−	0.476340i	−0.427619	−	0.903959i	$-0.640647\pi$
$263$	125.277	−	125.277i	0.476340	−	0.476340i	0.903959	+	0.427619i	$0.140647\pi$
$264$		0			0
$265$	−314.007	+	31.4630i	−1.18493	+	0.118728i
$266$		0			0
$267$		0			0
$268$		0			0
$269$		−	232.364i		−	0.863806i	−0.901920	−	0.431903i	$-0.857842\pi$
$269$		−	232.364i		−	0.863806i	0.901920	−	0.431903i	$-0.142158\pi$
$270$		0			0
$271$	2.09756			0.00774009			0.00387005	−	0.999993i	$-0.498768\pi$
$271$	2.09756			0.00774009			0.00387005	+	0.999993i	$0.498768\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	36.0370	+	178.023i	0.131044	+	0.647356i
$276$		0			0
$277$	−127.091	−	127.091i	−0.458812	−	0.458812i	0.439453	−	0.898265i	$-0.355172\pi$
$277$	−127.091	−	127.091i	−0.458812	−	0.458812i	−0.898265	+	0.439453i	$0.855172\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	47.6809			0.169683			0.0848414	−	0.996394i	$-0.472962\pi$
$281$	47.6809			0.169683			0.0848414	+	0.996394i	$0.472962\pi$
$282$		0			0
$283$	172.324	−	172.324i	0.608919	−	0.608919i	−0.333744	−	0.942664i	$-0.608312\pi$
$283$	172.324	−	172.324i	0.608919	−	0.608919i	0.942664	+	0.333744i	$0.108312\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−36.3408	−	36.3408i	−0.126623	−	0.126623i
$288$		0			0
$289$		−	244.912i		−	0.847445i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−397.647	+	397.647i	−1.35716	+	1.35716i	−0.479755	+	0.877402i	$0.659275\pi$
$293$	−397.647	+	397.647i	−1.35716	+	1.35716i	−0.877402	+	0.479755i	$0.840725\pi$
$294$		0			0
$295$	41.5201	−	50.7671i	0.140746	−	0.172092i
$296$		0			0
$297$		0			0
$298$		0			0
$299$		−	182.788i		−	0.611330i
$300$		0			0
$301$	−362.500			−1.20432
$302$		0			0
$303$		0			0
$304$		0			0
$305$	209.917	+	171.682i	0.688253	+	0.562891i
$306$		0			0
$307$	−25.0548	−	25.0548i	−0.0816117	−	0.0816117i	0.665123	−	0.746734i	$-0.268379\pi$
$307$	−25.0548	−	25.0548i	−0.0816117	−	0.0816117i	−0.746734	+	0.665123i	$0.768379\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−549.785			−1.76780			−0.883899	−	0.467678i	$-0.845091\pi$
$311$	−549.785			−1.76780			−0.883899	+	0.467678i	$0.845091\pi$
$312$		0			0
$313$	47.6266	−	47.6266i	0.152162	−	0.152162i	−0.626921	−	0.779083i	$-0.715685\pi$
$313$	47.6266	−	47.6266i	0.152162	−	0.152162i	0.779083	+	0.626921i	$0.215685\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	343.911	+	343.911i	1.08489	+	1.08489i	0.996045	+	0.0888492i	$0.0283189\pi$
$317$	343.911	+	343.911i	1.08489	+	1.08489i	0.0888492	+	0.996045i	$0.471681\pi$
$318$		0			0
$319$		−	245.105i		−	0.768354i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−47.4734	+	47.4734i	−0.146976	+	0.146976i
$324$		0			0
$325$	−90.1816	−	59.8174i	−0.277482	−	0.184054i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		−	117.372i		−	0.356754i
$330$		0			0
$331$	−29.2220			−0.0882840			−0.0441420	−	0.999025i	$-0.514055\pi$
$331$	−29.2220			−0.0882840			−0.0441420	+	0.999025i	$0.514055\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−32.8896	−	328.244i	−0.0981779	−	0.979834i
$336$		0			0
$337$	77.9144	+	77.9144i	0.231200	+	0.231200i	0.813193	−	0.581993i	$-0.197727\pi$
$337$	77.9144	+	77.9144i	0.231200	+	0.231200i	−0.581993	+	0.813193i	$0.697727\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	169.621			0.497423
$342$		0			0
$343$	255.668	−	255.668i	0.745387	−	0.745387i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	279.836	+	279.836i	0.806445	+	0.806445i	0.984094	−	0.177649i	$-0.0568491\pi$
$347$	279.836	+	279.836i	0.806445	+	0.806445i	−0.177649	+	0.984094i	$0.556849\pi$
$348$		0			0
$349$			660.500i			1.89255i	0.323361	+	0.946276i	$0.395187\pi$
$349$			660.500i			1.89255i	−0.323361	+	0.946276i	$0.604813\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	79.8004	−	79.8004i	0.226063	−	0.226063i	−0.584983	−	0.811046i	$-0.698899\pi$
$353$	79.8004	−	79.8004i	0.226063	−	0.226063i	0.811046	+	0.584983i	$0.198899\pi$
$354$		0			0
$355$	58.7348	+	48.0365i	0.165450	+	0.135314i
$356$		0			0
$357$		0			0
$358$		0			0
$359$		−	7.67587i		−	0.0213812i	−0.999943	−	0.0106906i	$-0.996597\pi$
$359$		−	7.67587i		−	0.0213812i	0.999943	−	0.0106906i	$-0.00340299\pi$
$360$		0			0
$361$	258.763			0.716796
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−218.955	+	21.9389i	−0.599875	+	0.0601066i
$366$		0			0
$367$	31.7561	+	31.7561i	0.0865288	+	0.0865288i	0.749046	−	0.662518i	$-0.230512\pi$
$367$	31.7561	+	31.7561i	0.0865288	+	0.0865288i	−0.662518	+	0.749046i	$0.730512\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	306.930			0.827304
$372$		0			0
$373$	209.534	−	209.534i	0.561754	−	0.561754i	−0.368052	−	0.929805i	$-0.619975\pi$
$373$	209.534	−	209.534i	0.561754	−	0.561754i	0.929805	+	0.368052i	$0.119975\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	103.261	+	103.261i	0.273901	+	0.273901i
$378$		0			0
$379$		−	509.263i		−	1.34370i	−0.740686	−	0.671851i	$-0.765500\pi$
$379$		−	509.263i		−	1.34370i	0.740686	−	0.671851i	$-0.234500\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	82.6383	−	82.6383i	0.215766	−	0.215766i	−0.590946	−	0.806711i	$-0.701245\pi$
$383$	82.6383	−	82.6383i	0.215766	−	0.215766i	0.806711	+	0.590946i	$0.201245\pi$
$384$		0			0
$385$	−17.6124	−	175.775i	−0.0457465	−	0.456559i
$386$		0			0
$387$		0			0
$388$		0			0
$389$		−	222.388i		−	0.571692i	−0.958276	−	0.285846i	$-0.907725\pi$
$389$		−	222.388i		−	0.571692i	0.958276	−	0.285846i	$-0.0922747\pi$
$390$		0			0
$391$	280.385			0.717097
$392$		0			0
$393$		0			0
$394$		0			0
$395$	456.112	−	557.694i	1.15471	−	1.41188i
$396$		0			0
$397$	−282.792	−	282.792i	−0.712323	−	0.712323i	0.254698	−	0.967021i	$-0.418024\pi$
$397$	−282.792	−	282.792i	−0.712323	−	0.712323i	−0.967021	+	0.254698i	$0.918024\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	104.462			0.260504			0.130252	−	0.991481i	$-0.458421\pi$
$401$	104.462			0.260504			0.130252	+	0.991481i	$0.458421\pi$
$402$		0			0
$403$	−71.4599	+	71.4599i	−0.177320	+	0.177320i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	292.014	+	292.014i	0.717478	+	0.717478i
$408$		0			0
$409$		−	198.833i		−	0.486144i	−0.970008	−	0.243072i	$-0.921845\pi$
$409$		−	198.833i		−	0.486144i	0.970008	−	0.243072i	$-0.0781551\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−45.1036	+	45.1036i	−0.109210	+	0.109210i
$414$		0			0
$415$	412.266	−	41.3084i	0.993412	−	0.0995383i
$416$		0			0
$417$		0			0
$418$		0			0
$419$			205.468i			0.490377i	0.969475	+	0.245189i	$0.0788499\pi$
$419$			205.468i			0.490377i	−0.969475	+	0.245189i	$0.921150\pi$
$420$		0			0
$421$	−628.299			−1.49240			−0.746199	−	0.665723i	$-0.768123\pi$
$421$	−628.299			−1.49240			−0.746199	+	0.665723i	$0.768123\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	91.7562	−	138.333i	0.215897	−	0.325490i
$426$		0			0
$427$	−186.499	−	186.499i	−0.436766	−	0.436766i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−325.388			−0.754961			−0.377481	−	0.926017i	$-0.623210\pi$
$431$	−325.388			−0.754961			−0.377481	+	0.926017i	$0.623210\pi$
$432$		0			0
$433$	−42.3633	+	42.3633i	−0.0978366	+	0.0978366i	−0.754331	−	0.656494i	$-0.772039\pi$
$433$	−42.3633	+	42.3633i	−0.0978366	+	0.0978366i	0.656494	+	0.754331i	$0.272039\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	301.912	+	301.912i	0.690875	+	0.690875i
$438$		0			0
$439$		−	239.832i		−	0.546315i	−0.961969	−	0.273158i	$-0.911932\pi$
$439$		−	239.832i		−	0.546315i	0.961969	−	0.273158i	$-0.0880681\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−345.233	+	345.233i	−0.779307	+	0.779307i	−0.979713	−	0.200406i	$-0.935774\pi$
$443$	−345.233	+	345.233i	−0.779307	+	0.779307i	0.200406	+	0.979713i	$0.435774\pi$
$444$		0			0
$445$	393.155	−	480.715i	0.883494	−	1.08026i
$446$		0			0
$447$		0			0
$448$		0			0
$449$			29.5708i			0.0658593i	0.999458	+	0.0329296i	$0.0104837\pi$
$449$			29.5708i			0.0658593i	−0.999458	+	0.0329296i	$0.989516\pi$
$450$		0			0
$451$	−76.7831			−0.170251
$452$		0			0
$453$		0			0
$454$		0			0
$455$	81.4726	+	66.6327i	0.179061	+	0.146445i
$456$		0			0
$457$	−351.775	−	351.775i	−0.769749	−	0.769749i	0.208313	−	0.978062i	$-0.433203\pi$
$457$	−351.775	−	351.775i	−0.769749	−	0.769749i	−0.978062	+	0.208313i	$0.933203\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−832.442			−1.80573			−0.902865	−	0.429923i	$-0.858541\pi$
$461$	−832.442			−1.80573			−0.902865	+	0.429923i	$0.858541\pi$
$462$		0			0
$463$	129.749	−	129.749i	0.280235	−	0.280235i	−0.552968	−	0.833203i	$-0.686505\pi$
$463$	129.749	−	129.749i	0.280235	−	0.280235i	0.833203	+	0.552968i	$0.186505\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	310.447	+	310.447i	0.664769	+	0.664769i	0.956500	−	0.291731i	$-0.0942313\pi$
$467$	310.447	+	310.447i	0.664769	+	0.664769i	−0.291731	+	0.956500i	$0.594231\pi$
$468$		0			0
$469$			320.846i			0.684108i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−382.956	+	382.956i	−0.809633	+	0.809633i
$474$		0			0
$475$	247.755	−	50.1529i	0.521590	−	0.105585i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		−	137.105i		−	0.286231i	−0.989706	−	0.143115i	$-0.954288\pi$
$479$		−	137.105i		−	0.286231i	0.989706	−	0.143115i	$-0.0457120\pi$
$480$		0			0
$481$	−246.046			−0.511529
$482$		0			0
$483$		0			0
$484$		0			0
$485$	14.6669	+	146.378i	0.0302410	+	0.301811i
$486$		0			0
$487$	187.010	+	187.010i	0.384003	+	0.384003i	0.872542	−	0.488539i	$-0.162470\pi$
$487$	187.010	+	187.010i	0.384003	+	0.384003i	−0.488539	+	0.872542i	$0.662470\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−97.2541			−0.198073			−0.0990367	−	0.995084i	$-0.531576\pi$
$491$	−97.2541			−0.198073			−0.0990367	+	0.995084i	$0.531576\pi$
$492$		0			0
$493$	−158.395	+	158.395i	−0.321289	+	0.321289i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−52.1824	−	52.1824i	−0.104995	−	0.104995i
$498$		0			0
$499$		−	175.306i		−	0.351314i	−0.984451	−	0.175657i	$-0.943795\pi$
$499$		−	175.306i		−	0.351314i	0.984451	−	0.175657i	$-0.0562049\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	252.151	−	252.151i	0.501293	−	0.501293i	−0.410546	−	0.911840i	$-0.634662\pi$
$503$	252.151	−	252.151i	0.501293	−	0.501293i	0.911840	+	0.410546i	$0.134662\pi$
$504$		0			0
$505$	−528.159	−	431.957i	−1.04586	−	0.855361i
$506$		0			0
$507$		0			0
$508$		0			0
$509$		−	185.839i		−	0.365106i	−0.983196	−	0.182553i	$-0.941564\pi$
$509$		−	185.839i		−	0.365106i	0.983196	−	0.182553i	$-0.0584362\pi$
$510$		0			0
$511$	214.020			0.418825
$512$		0			0
$513$		0			0
$514$		0			0
$515$	345.275	−	34.5960i	0.670437	−	0.0671768i
$516$		0			0
$517$	−123.996	−	123.996i	−0.239837	−	0.239837i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−877.064			−1.68343			−0.841713	−	0.539926i	$-0.818452\pi$
$521$	−877.064			−1.68343			−0.841713	+	0.539926i	$0.818452\pi$
$522$		0			0
$523$	526.019	−	526.019i	1.00577	−	1.00577i	0.00578841	−	0.999983i	$-0.498157\pi$
$523$	526.019	−	526.019i	1.00577	−	1.00577i	0.999983	−	0.00578841i	$-0.00184252\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−109.615	−	109.615i	−0.207998	−	0.207998i
$528$		0			0
$529$		−	1254.14i		−	2.37077i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	32.3481	−	32.3481i	0.0606905	−	0.0606905i
$534$		0			0
$535$	−20.3125	−	202.723i	−0.0379673	−	0.378921i
$536$		0			0
$537$		0			0
$538$		0			0
$539$		−	184.188i		−	0.341723i
$540$		0			0
$541$	−253.279			−0.468168			−0.234084	−	0.972216i	$-0.575209\pi$
$541$	−253.279			−0.468168			−0.234084	+	0.972216i	$0.575209\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	566.348	−	692.481i	1.03917	−	1.27061i
$546$		0			0
$547$	58.1608	+	58.1608i	0.106327	+	0.106327i	0.758269	−	0.651942i	$-0.226045\pi$
$547$	58.1608	+	58.1608i	0.106327	+	0.106327i	−0.651942	+	0.758269i	$0.726045\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−341.113			−0.619080
$552$		0			0
$553$	−495.479	+	495.479i	−0.895983	+	0.895983i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	595.191	+	595.191i	1.06857	+	1.06857i	0.997469	+	0.0710965i	$0.0226498\pi$
$557$	595.191	+	595.191i	1.06857	+	1.06857i	0.0710965	+	0.997469i	$0.477350\pi$
$558$		0			0
$559$		−	322.672i		−	0.577232i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	353.520	−	353.520i	0.627922	−	0.627922i	−0.319623	−	0.947545i	$-0.603556\pi$
$563$	353.520	−	353.520i	0.627922	−	0.627922i	0.947545	+	0.319623i	$0.103556\pi$
$564$		0			0
$565$	180.550	−	18.0909i	0.319558	−	0.0320192i
$566$		0			0
$567$		0			0
$568$		0			0
$569$		−	513.090i		−	0.901739i	−0.892590	−	0.450870i	$-0.851114\pi$
$569$		−	513.090i		−	0.901739i	0.892590	−	0.450870i	$-0.148886\pi$
$570$		0			0
$571$	135.817			0.237858			0.118929	−	0.992903i	$-0.462054\pi$
$571$	135.817			0.237858			0.118929	+	0.992903i	$0.462054\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−879.744	−	583.534i	−1.52999	−	1.01484i
$576$		0			0
$577$	−610.805	−	610.805i	−1.05859	−	1.05859i	−0.998173	−	0.0604145i	$-0.980758\pi$
$577$	−610.805	−	610.805i	−1.05859	−	1.05859i	−0.0604145	−	0.998173i	$-0.519242\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−402.974			−0.693587
$582$		0			0
$583$	324.250	−	324.250i	0.556175	−	0.556175i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	394.607	+	394.607i	0.672243	+	0.672243i	0.958233	−	0.285989i	$-0.0923222\pi$
$587$	394.607	+	394.607i	0.672243	+	0.672243i	−0.285989	+	0.958233i	$0.592322\pi$
$588$		0			0
$589$		−	236.062i		−	0.400785i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−343.990	+	343.990i	−0.580084	+	0.580084i	−0.934926	−	0.354842i	$-0.884535\pi$
$593$	−343.990	+	343.990i	−0.580084	+	0.580084i	0.354842	+	0.934926i	$0.384535\pi$
$594$		0			0
$595$	−102.210	+	124.974i	−0.171782	+	0.210040i
$596$		0			0
$597$		0			0
$598$		0			0
$599$		−	852.943i		−	1.42395i	−0.702207	−	0.711973i	$-0.747802\pi$
$599$		−	852.943i		−	1.42395i	0.702207	−	0.711973i	$-0.252198\pi$
$600$		0			0
$601$	708.388			1.17868			0.589341	−	0.807885i	$-0.299388\pi$
$601$	708.388			1.17868			0.589341	+	0.807885i	$0.299388\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	264.019	+	215.929i	0.436394	+	0.356907i
$606$		0			0
$607$	−345.248	−	345.248i	−0.568778	−	0.568778i	0.363008	−	0.931786i	$-0.381750\pi$
$607$	−345.248	−	345.248i	−0.568778	−	0.568778i	−0.931786	+	0.363008i	$0.881750\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	104.476			0.170993
$612$		0			0
$613$	−375.137	+	375.137i	−0.611969	+	0.611969i	−0.943459	−	0.331489i	$-0.892449\pi$
$613$	−375.137	+	375.137i	−0.611969	+	0.611969i	0.331489	+	0.943459i	$0.392449\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−147.878	−	147.878i	−0.239673	−	0.239673i	0.577042	−	0.816715i	$-0.304207\pi$
$617$	−147.878	−	147.878i	−0.239673	−	0.239673i	−0.816715	+	0.577042i	$0.804207\pi$
$618$		0			0
$619$		−	1013.22i		−	1.63686i	−0.574606	−	0.818430i	$-0.694845\pi$
$619$		−	1013.22i		−	1.63686i	0.574606	−	0.818430i	$-0.305155\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−427.087	+	427.087i	−0.685533	+	0.685533i
$624$		0			0
$625$	−575.794	+	243.076i	−0.921271	+	0.388921i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		−	377.419i		−	0.600030i
$630$		0			0
$631$	−656.253			−1.04002			−0.520010	−	0.854160i	$-0.674072\pi$
$631$	−656.253			−1.04002			−0.520010	+	0.854160i	$0.674072\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−58.2226	−	581.073i	−0.0916891	−	0.915075i
$636$		0			0
$637$	77.5970	+	77.5970i	0.121816	+	0.121816i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−64.9961			−0.101398			−0.0506990	−	0.998714i	$-0.516145\pi$
$641$	−64.9961			−0.101398			−0.0506990	+	0.998714i	$0.516145\pi$
$642$		0			0
$643$	−183.525	+	183.525i	−0.285420	+	0.285420i	−0.835266	−	0.549846i	$-0.814686\pi$
$643$	−183.525	+	183.525i	−0.285420	+	0.285420i	0.549846	+	0.835266i	$0.314686\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	153.861	+	153.861i	0.237807	+	0.237807i	0.815941	−	0.578135i	$-0.196219\pi$
$647$	153.861	+	153.861i	0.237807	+	0.237807i	−0.578135	+	0.815941i	$0.696219\pi$
$648$		0			0
$649$			95.2977i			0.146838i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	51.2563	−	51.2563i	0.0784935	−	0.0784935i	−0.666770	−	0.745264i	$-0.732324\pi$
$653$	51.2563	−	51.2563i	0.0784935	−	0.0784935i	0.745264	+	0.666770i	$0.232324\pi$
$654$		0			0
$655$	−566.876	−	463.622i	−0.865460	−	0.707820i
$656$		0			0
$657$		0			0
$658$		0			0
$659$			577.821i			0.876815i	0.898776	+	0.438407i	$0.144457\pi$
$659$			577.821i			0.876815i	−0.898776	+	0.438407i	$0.855543\pi$
$660$		0			0
$661$	−376.303			−0.569294			−0.284647	−	0.958632i	$-0.591876\pi$
$661$	−376.303			−0.569294			−0.284647	+	0.958632i	$0.591876\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−244.627	+	24.5113i	−0.367860	+	0.0368590i
$666$		0			0
$667$	1007.33	+	1007.33i	1.51024	+	1.51024i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−394.047			−0.587253
$672$		0			0
$673$	321.691	−	321.691i	0.477996	−	0.477996i	−0.426494	−	0.904490i	$-0.640252\pi$
$673$	321.691	−	321.691i	0.477996	−	0.477996i	0.904490	+	0.426494i	$0.140252\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−600.196	−	600.196i	−0.886552	−	0.886552i	0.107638	−	0.994190i	$-0.465671\pi$
$677$	−600.196	−	600.196i	−0.886552	−	0.886552i	−0.994190	+	0.107638i	$0.965671\pi$
$678$		0			0
$679$		−	143.079i		−	0.210721i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−731.092	+	731.092i	−1.07041	+	1.07041i	−0.0730866	+	0.997326i	$0.523285\pi$
$683$	−731.092	+	731.092i	−1.07041	+	1.07041i	−0.997326	+	0.0730866i	$0.976715\pi$
$684$		0			0
$685$	−82.0569	−	818.943i	−0.119791	−	1.19554i
$686$		0			0
$687$		0			0
$688$		0			0
$689$			273.207i			0.396527i
$690$		0			0
$691$	1104.64			1.59862			0.799308	−	0.600921i	$-0.205199\pi$
$691$	1104.64			1.59862			0.799308	+	0.600921i	$0.205199\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	530.372	−	648.493i	0.763125	−	0.933083i
$696$		0			0
$697$	49.6199	+	49.6199i	0.0711907	+	0.0711907i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	664.424			0.947823			0.473911	−	0.880573i	$-0.342842\pi$
$701$	664.424			0.947823			0.473911	+	0.880573i	$0.342842\pi$
$702$		0			0
$703$	406.396	−	406.396i	0.578089	−	0.578089i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	469.239	+	469.239i	0.663704	+	0.663704i
$708$		0			0
$709$		−	647.253i		−	0.912909i	−0.889747	−	0.456455i	$-0.849119\pi$
$709$		−	647.253i		−	0.912909i	0.889747	−	0.456455i	$-0.150881\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−697.110	+	697.110i	−0.977713	+	0.977713i
$714$		0			0
$715$	156.463	−	15.6773i	0.218829	−	0.0219264i
$716$		0			0
$717$		0			0
$718$		0			0
$719$		−	42.0429i		−	0.0584742i	−0.999573	−	0.0292371i	$-0.990692\pi$
$719$		−	42.0429i		−	0.0584742i	0.999573	−	0.0292371i	$-0.00930778\pi$
$720$		0			0
$721$	−337.493			−0.468091
$722$		0			0
$723$		0			0
$724$		0			0
$725$	826.637	−	167.335i	1.14019	−	0.230807i
$726$		0			0
$727$	−193.225	−	193.225i	−0.265784	−	0.265784i	0.561615	−	0.827399i	$-0.310180\pi$
$727$	−193.225	−	193.225i	−0.265784	−	0.265784i	−0.827399	+	0.561615i	$0.810180\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	494.960			0.677099
$732$		0			0
$733$	951.899	−	951.899i	1.29863	−	1.29863i	0.369340	−	0.929294i	$-0.379584\pi$
$733$	951.899	−	951.899i	1.29863	−	1.29863i	0.929294	−	0.369340i	$-0.120416\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	338.952	+	338.952i	0.459908	+	0.459908i
$738$		0			0
$739$			1019.58i			1.37968i	0.723963	+	0.689839i	$0.242319\pi$
$739$			1019.58i			1.37968i	−0.723963	+	0.689839i	$0.757681\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−418.295	+	418.295i	−0.562981	+	0.562981i	−0.930153	−	0.367172i	$-0.880326\pi$
$743$	−418.295	+	418.295i	−0.562981	+	0.562981i	0.367172	+	0.930153i	$0.380326\pi$
$744$		0			0
$745$	−594.937	+	727.436i	−0.798573	+	0.976424i
$746$		0			0
$747$		0			0
$748$		0			0
$749$			198.154i			0.264557i
$750$		0			0
$751$	7.35093			0.00978818			0.00489409	−	0.999988i	$-0.498442\pi$
$751$	7.35093			0.00978818			0.00489409	+	0.999988i	$0.498442\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−301.076	−	246.236i	−0.398777	−	0.326141i
$756$		0			0
$757$	−575.920	−	575.920i	−0.760793	−	0.760793i	0.215673	−	0.976466i	$-0.430806\pi$
$757$	−575.920	−	575.920i	−0.760793	−	0.760793i	−0.976466	+	0.215673i	$0.930806\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	1243.54			1.63408			0.817042	−	0.576578i	$-0.195612\pi$
$761$	1243.54			1.63408			0.817042	+	0.576578i	$0.195612\pi$
$762$		0			0
$763$	−615.229	+	615.229i	−0.806329	+	0.806329i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−40.1481	−	40.1481i	−0.0523443	−	0.0523443i
$768$		0			0
$769$			872.201i			1.13420i	0.823648	+	0.567101i	$0.191935\pi$
$769$			872.201i			1.13420i	−0.823648	+	0.567101i	$0.808065\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	502.677	−	502.677i	0.650294	−	0.650294i	−0.302770	−	0.953064i	$-0.597911\pi$
$773$	502.677	−	502.677i	0.650294	−	0.650294i	0.953064	+	0.302770i	$0.0979114\pi$
$774$		0			0
$775$	115.802	+	572.062i	0.149422	+	0.738144i
$776$		0			0
$777$		0			0
$778$		0			0
$779$			106.859i			0.137175i
$780$		0			0
$781$	−110.254			−0.141171
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−69.0210	−	688.843i	−0.0879249	−	0.877508i
$786$		0			0
$787$	249.776	+	249.776i	0.317377	+	0.317377i	0.847759	−	0.530382i	$-0.177951\pi$
$787$	249.776	+	249.776i	0.317377	+	0.317377i	−0.530382	+	0.847759i	$0.677951\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−176.481			−0.223111
$792$		0			0
$793$	166.009	−	166.009i	0.209342	−	0.209342i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	21.2255	+	21.2255i	0.0266317	+	0.0266317i	0.720297	−	0.693666i	$-0.244005\pi$
$797$	21.2255	+	21.2255i	0.0266317	+	0.0266317i	−0.693666	+	0.720297i	$0.744005\pi$
$798$		0			0
$799$			160.260i			0.200576i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	226.097	−	226.097i	0.281565	−	0.281565i
$804$		0			0
$805$	794.786	+	650.019i	0.987311	+	0.807476i
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	218.418i		−	0.269986i	−0.990847	−	0.134993i	$-0.956899\pi$
$809$		−	218.418i		−	0.269986i	0.990847	−	0.134993i	$-0.0431011\pi$
$810$		0			0
$811$	708.023			0.873024			0.436512	−	0.899698i	$-0.356214\pi$
$811$	708.023			0.873024			0.436512	+	0.899698i	$0.356214\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	1129.20	−	113.144i	1.38552	−	0.138827i
$816$		0			0
$817$	532.962	+	532.962i	0.652340	+	0.652340i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−1021.92			−1.24472			−0.622361	−	0.782731i	$-0.713826\pi$
$821$	−1021.92			−1.24472			−0.622361	+	0.782731i	$0.713826\pi$
$822$		0			0
$823$	196.702	−	196.702i	0.239007	−	0.239007i	−0.577432	−	0.816439i	$-0.695945\pi$
$823$	196.702	−	196.702i	0.239007	−	0.239007i	0.816439	+	0.577432i	$0.195945\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	870.595	+	870.595i	1.05271	+	1.05271i	0.998531	+	0.0541833i	$0.0172555\pi$
$827$	870.595	+	870.595i	1.05271	+	1.05271i	0.0541833	+	0.998531i	$0.482744\pi$
$828$		0			0
$829$			460.400i			0.555368i	0.960672	+	0.277684i	$0.0895668\pi$
$829$			460.400i			0.555368i	−0.960672	+	0.277684i	$0.910433\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−119.029	+	119.029i	−0.142892	+	0.142892i
$834$		0			0
$835$	−25.8864	−	258.352i	−0.0310017	−	0.309403i
$836$		0			0
$837$		0			0
$838$		0			0
$839$			1386.65i			1.65274i	0.563130	+	0.826368i	$0.309597\pi$
$839$			1386.65i			1.65274i	−0.563130	+	0.826368i	$0.690403\pi$
$840$		0			0
$841$	−297.127			−0.353302
$842$		0			0
$843$		0			0
$844$		0			0
$845$	475.646	−	581.578i	0.562894	−	0.688258i
$846$		0			0
$847$	−234.565	−	234.565i	−0.276936	−	0.276936i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−2400.24			−2.82049
$852$		0			0
$853$	−808.704	+	808.704i	−0.948070	+	0.948070i	−0.998717	−	0.0506467i	$-0.983872\pi$
$853$	−808.704	+	808.704i	−0.948070	+	0.948070i	0.0506467	+	0.998717i	$0.483872\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	945.140	+	945.140i	1.10285	+	1.10285i	0.994066	+	0.108781i	$0.0346948\pi$
$857$	945.140	+	945.140i	1.10285	+	1.10285i	0.108781	+	0.994066i	$0.465305\pi$
$858$		0			0
$859$			924.339i			1.07606i	0.842924	+	0.538032i	$0.180832\pi$
$859$			924.339i			1.07606i	−0.842924	+	0.538032i	$0.819168\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−788.779	+	788.779i	−0.913997	+	0.913997i	−0.996584	−	0.0825868i	$-0.973682\pi$
$863$	−788.779	+	788.779i	−0.913997	+	0.913997i	0.0825868	+	0.996584i	$0.473682\pi$
$864$		0			0
$865$	835.499	−	83.7157i	0.965895	−	0.0967812i
$866$		0			0
$867$		0			0
$868$		0			0
$869$			1046.88i			1.20469i
$870$		0			0
$871$	−285.595			−0.327893
$872$		0			0
$873$		0			0
$874$		0			0
$875$	580.793	−	179.403i	0.663763	−	0.205032i
$876$		0			0
$877$	−862.503	−	862.503i	−0.983470	−	0.983470i	0.0163953	−	0.999866i	$-0.494781\pi$
$877$	−862.503	−	862.503i	−0.983470	−	0.983470i	−0.999866	+	0.0163953i	$0.994781\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	356.941			0.405154			0.202577	−	0.979266i	$-0.435068\pi$
$881$	356.941			0.405154			0.202577	+	0.979266i	$0.435068\pi$
$882$		0			0
$883$	314.661	−	314.661i	0.356354	−	0.356354i	−0.506113	−	0.862467i	$-0.668918\pi$
$883$	314.661	−	314.661i	0.356354	−	0.356354i	0.862467	+	0.506113i	$0.168918\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	407.418	+	407.418i	0.459321	+	0.459321i	0.898432	−	0.439112i	$-0.144707\pi$
$887$	407.418	+	407.418i	0.459321	+	0.459321i	−0.439112	+	0.898432i	$0.644707\pi$
$888$		0			0
$889$			567.977i			0.638894i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−172.565	+	172.565i	−0.193242	+	0.193242i
$894$		0			0
$895$	928.399	−	1135.17i	1.03732	−	1.26834i
$896$		0			0
$897$		0			0
$898$		0			0
$899$		−	787.624i		−	0.876111i
$900$		0			0
$901$	−419.083			−0.465131
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−707.581	−	578.698i	−0.781858	−	0.639445i
$906$		0			0
$907$	361.599	+	361.599i	0.398676	+	0.398676i	0.877766	−	0.479090i	$-0.159033\pi$
$907$	361.599	+	361.599i	0.398676	+	0.398676i	−0.479090	+	0.877766i	$0.659033\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	59.3980			0.0652009			0.0326004	−	0.999468i	$-0.489621\pi$
$911$	59.3980			0.0652009			0.0326004	+	0.999468i	$0.489621\pi$
$912$		0			0
$913$	−425.714	+	425.714i	−0.466281	+	0.466281i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	503.636	+	503.636i	0.549222	+	0.549222i
$918$		0			0
$919$		−	272.183i		−	0.296173i	−0.988974	−	0.148087i	$-0.952689\pi$
$919$		−	272.183i		−	0.296173i	0.988974	−	0.148087i	$-0.0473115\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	46.4492	−	46.4492i	0.0503241	−	0.0503241i
$924$		0			0
$925$	−785.480	+	1184.20i	−0.849168	+	1.28022i
$926$		0			0
$927$		0			0
$928$		0			0
$929$		−	1690.97i		−	1.82020i	−0.414383	−	0.910102i	$-0.636003\pi$
$929$		−	1690.97i		−	1.82020i	0.414383	−	0.910102i	$-0.363997\pi$
$930$		0			0
$931$	−256.336			−0.275334
$932$		0			0
$933$		0			0
$934$		0			0
$935$	24.0481	+	240.005i	0.0257199	+	0.256689i
$936$		0			0
$937$	−658.662	−	658.662i	−0.702947	−	0.702947i	0.262095	−	0.965042i	$-0.415587\pi$
$937$	−658.662	−	658.662i	−0.702947	−	0.702947i	−0.965042	+	0.262095i	$0.915587\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	155.326			0.165065			0.0825326	−	0.996588i	$-0.473699\pi$
$941$	155.326			0.165065			0.0825326	+	0.996588i	$0.473699\pi$
$942$		0			0
$943$	315.563	−	315.563i	0.334638	−	0.334638i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−125.521	−	125.521i	−0.132546	−	0.132546i	0.637721	−	0.770267i	$-0.279877\pi$
$947$	−125.521	−	125.521i	−0.132546	−	0.132546i	−0.770267	+	0.637721i	$0.779877\pi$
$948$		0			0
$949$			190.505i			0.200743i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−1133.35	+	1133.35i	−1.18924	+	1.18924i	−0.211965	+	0.977277i	$0.567986\pi$
$953$	−1133.35	+	1133.35i	−1.18924	+	1.18924i	−0.977277	+	0.211965i	$0.932014\pi$
$954$		0			0
$955$	1121.68	+	917.374i	1.17454	+	0.960601i
$956$		0			0
$957$		0			0
$958$		0			0
$959$			800.486i			0.834709i
$960$		0			0
$961$	−415.937			−0.432817
$962$		0			0
$963$		0			0
$964$		0			0
$965$	1384.55	−	138.729i	1.43476	−	0.143761i
$966$		0			0
$967$	112.878	+	112.878i	0.116730	+	0.116730i	0.763059	−	0.646329i	$-0.223697\pi$
$967$	112.878	+	112.878i	0.116730	+	0.116730i	−0.646329	+	0.763059i	$0.723697\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	1054.48			1.08597			0.542986	−	0.839742i	$-0.317294\pi$
$971$	1054.48			1.08597			0.542986	+	0.839742i	$0.317294\pi$
$972$		0			0
$973$	−576.148	+	576.148i	−0.592135	+	0.592135i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−277.019	−	277.019i	−0.283540	−	0.283540i	0.550979	−	0.834519i	$-0.314254\pi$
$977$	−277.019	−	277.019i	−0.283540	−	0.283540i	−0.834519	+	0.550979i	$0.814254\pi$
$978$		0			0
$979$			902.376i			0.921733i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	824.165	−	824.165i	0.838418	−	0.838418i	−0.150233	−	0.988651i	$-0.548002\pi$
$983$	824.165	−	824.165i	0.838418	−	0.838418i	0.988651	+	0.150233i	$0.0480023\pi$
$984$		0			0
$985$	127.867	+	1276.14i	0.129814	+	1.29557i
$986$		0			0
$987$		0			0
$988$		0			0
$989$		−	3147.75i		−	3.18276i
$990$		0			0
$991$	130.497			0.131682			0.0658412	−	0.997830i	$-0.479027\pi$
$991$	130.497			0.131682			0.0658412	+	0.997830i	$0.479027\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−508.153	+	621.325i	−0.510706	+	0.624447i
$996$		0			0
$997$	1139.64	+	1139.64i	1.14306	+	1.14306i	0.987887	+	0.155178i	$0.0495951\pi$
$997$	1139.64	+	1139.64i	1.14306	+	1.14306i	0.155178	+	0.987887i	$0.450405\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1620.3.l.f.1297.3		24
3.2	odd	2		1620.3.l.g.1297.10		24
5.3	odd	4	inner	1620.3.l.f.973.3		24
9.2	odd	6		540.3.v.a.37.3		48
9.4	even	3		180.3.u.a.97.10	yes	48
9.5	odd	6		540.3.v.a.397.6		48
9.7	even	3		180.3.u.a.157.4	yes	48
15.8	even	4		1620.3.l.g.973.10		24
45.13	odd	12		180.3.u.a.133.4	yes	48
45.23	even	12		540.3.v.a.73.3		48
45.38	even	12		540.3.v.a.253.6		48
45.43	odd	12		180.3.u.a.13.10	✓	48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
180.3.u.a.13.10	✓	48	45.43	odd	12
180.3.u.a.97.10	yes	48	9.4	even	3
180.3.u.a.133.4	yes	48	45.13	odd	12
180.3.u.a.157.4	yes	48	9.7	even	3
540.3.v.a.37.3		48	9.2	odd	6
540.3.v.a.73.3		48	45.23	even	12
540.3.v.a.253.6		48	45.38	even	12
540.3.v.a.397.6		48	9.5	odd	6
1620.3.l.f.973.3		24	5.3	odd	4	inner
1620.3.l.f.1297.3		24	1.1	even	1	trivial
1620.3.l.g.973.10		24	15.8	even	4
1620.3.l.g.1297.10		24	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1620 = 2^{2} \cdot 3^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1620.l (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-3.87041 - 3.16543i) q^{5} +(3.43863 + 3.43863i) q^{7} +O(q^{10})\)	\(q+(-3.87041 - 3.16543i) q^{5} +(3.43863 + 3.43863i) q^{7} +7.26535 q^{11} +(-3.06083 + 3.06083i) q^{13} +(-4.69512 - 4.69512i) q^{17} -10.1112i q^{19} +(-29.8592 + 29.8592i) q^{23} +(4.96012 + 24.5030i) q^{25} -33.7361i q^{29} +23.3466 q^{31} +(-2.42417 - 24.1937i) q^{35} +(40.1926 + 40.1926i) q^{37} -10.5684 q^{41} +(-52.7100 + 52.7100i) q^{43} +(-17.0667 - 17.0667i) q^{47} -25.3516i q^{49} +(44.6296 - 44.6296i) q^{53} +(-28.1199 - 22.9980i) q^{55} +13.1167i q^{59} -54.2365 q^{61} +(21.5355 - 2.15782i) q^{65} +(46.6532 + 46.6532i) q^{67} -15.1754 q^{71} +(31.1199 - 31.1199i) q^{73} +(24.9829 + 24.9829i) q^{77} +144.092i q^{79} +(-58.5951 + 58.5951i) q^{83} +(3.30997 + 33.0341i) q^{85} +124.203i q^{89} -21.0501 q^{91} +(-32.0063 + 39.1345i) q^{95} +(-20.8047 - 20.8047i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q+O(q^{10}) \) 24 * q	\( 24 q - 12 q^{11} + 18 q^{17} + 12 q^{23} + 6 q^{25} + 36 q^{35} - 42 q^{37} - 72 q^{41} + 78 q^{47} - 156 q^{53} + 66 q^{55} + 96 q^{61} + 132 q^{65} + 78 q^{67} - 156 q^{71} + 240 q^{77} - 132 q^{83} - 96 q^{85} + 84 q^{91} + 168 q^{95} + 90 q^{97}+O(q^{100}) \) 24 * q - 12 * q^11 + 18 * q^17 + 12 * q^23 + 6 * q^25 + 36 * q^35 - 42 * q^37 - 72 * q^41 + 78 * q^47 - 156 * q^53 + 66 * q^55 + 96 * q^61 + 132 * q^65 + 78 * q^67 - 156 * q^71 + 240 * q^77 - 132 * q^83 - 96 * q^85 + 84 * q^91 + 168 * q^95 + 90 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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