LMFDB - Embedded newform 1620.3.o.g.701.7

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			701.7
Character	$\chi$	$=$	1620.701
Dual form			1620.3.o.g.1241.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.93649 - 1.11803i) q^{5} +(3.48846 + 6.04219i) q^{7} +O(q^{10})$	$q+(-1.93649 - 1.11803i) q^{5} +(3.48846 + 6.04219i) q^{7} +(11.6406 - 6.72068i) q^{11} +(9.06955 - 15.7089i) q^{13} +19.1923i q^{17} -0.280881 q^{19} +(10.3350 + 5.96693i) q^{23} +(2.50000 + 4.33013i) q^{25} +(-41.3003 + 23.8447i) q^{29} +(9.80054 - 16.9750i) q^{31} -15.6009i q^{35} +55.9566 q^{37} +(-24.7372 - 14.2820i) q^{41} +(-34.7700 - 60.2233i) q^{43} +(-13.6845 + 7.90074i) q^{47} +(0.161293 - 0.279367i) q^{49} -57.3307i q^{53} -30.0558 q^{55} +(34.7812 + 20.0809i) q^{59} +(54.2890 + 94.0314i) q^{61} +(-35.1262 + 20.2801i) q^{65} +(-5.09891 + 8.83158i) q^{67} -89.0429i q^{71} +90.9902 q^{73} +(81.2153 + 46.8896i) q^{77} +(67.2671 + 116.510i) q^{79} +(-24.2534 + 14.0027i) q^{83} +(21.4576 - 37.1657i) q^{85} +60.1404i q^{89} +126.555 q^{91} +(0.543924 + 0.314035i) q^{95} +(2.59486 + 4.49443i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$7$	3.48846	+	6.04219i	0.498351	+	0.863170i	0.999998	−	0.00190255i	$-0.000605601\pi$
$7$	3.48846	+	6.04219i	0.498351	+	0.863170i	−0.501647	+	0.865073i	$0.667272\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	11.6406	−	6.72068i	1.05823	−	0.610971i	0.133290	−	0.991077i	$-0.457446\pi$
$11$	11.6406	−	6.72068i	1.05823	−	0.610971i	0.924943	+	0.380106i	$0.124113\pi$
$12$		0			0
$13$	9.06955	−	15.7089i	0.697658	−	1.20838i	−0.271619	−	0.962405i	$-0.587559\pi$
$13$	9.06955	−	15.7089i	0.697658	−	1.20838i	0.969276	−	0.245974i	$-0.0791078\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$			19.1923i			1.12896i	0.825448	+	0.564478i	$0.190923\pi$
$17$			19.1923i			1.12896i	−0.825448	+	0.564478i	$0.809077\pi$
$18$		0			0
$19$	−0.280881			−0.0147832			−0.00739161	−	0.999973i	$-0.502353\pi$
$19$	−0.280881			−0.0147832			−0.00739161	+	0.999973i	$0.502353\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	10.3350	+	5.96693i	0.449349	+	0.259432i	0.707555	−	0.706658i	$-0.249798\pi$
$23$	10.3350	+	5.96693i	0.449349	+	0.259432i	−0.258206	+	0.966090i	$0.583131\pi$
$24$		0			0
$25$	2.50000	+	4.33013i	0.100000	+	0.173205i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−41.3003	+	23.8447i	−1.42415	+	0.822232i	−0.996650	−	0.0817847i	$-0.973938\pi$
$29$	−41.3003	+	23.8447i	−1.42415	+	0.822232i	−0.427497	+	0.904017i	$0.640605\pi$
$30$		0			0
$31$	9.80054	−	16.9750i	0.316146	−	0.547582i	−0.663534	−	0.748146i	$-0.730944\pi$
$31$	9.80054	−	16.9750i	0.316146	−	0.547582i	0.979681	+	0.200564i	$0.0642776\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		−	15.6009i		−	0.445739i
$36$		0			0
$37$	55.9566			1.51234			0.756171	−	0.654374i	$-0.227068\pi$
$37$	55.9566			1.51234			0.756171	+	0.654374i	$0.227068\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−24.7372	−	14.2820i	−0.603345	−	0.348341i	0.167011	−	0.985955i	$-0.446588\pi$
$41$	−24.7372	−	14.2820i	−0.603345	−	0.348341i	−0.770356	+	0.637613i	$0.779922\pi$
$42$		0			0
$43$	−34.7700	−	60.2233i	−0.808604	−	1.40054i	−0.913831	−	0.406095i	$-0.866890\pi$
$43$	−34.7700	−	60.2233i	−0.808604	−	1.40054i	0.105227	−	0.994448i	$-0.466443\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−13.6845	+	7.90074i	−0.291159	+	0.168101i	−0.638464	−	0.769651i	$-0.720430\pi$
$47$	−13.6845	+	7.90074i	−0.291159	+	0.168101i	0.347305	+	0.937752i	$0.387097\pi$
$48$		0			0
$49$	0.161293	−	0.279367i	0.00329169	−	0.00570137i
$50$		0			0
$51$		0			0
$52$		0			0
$53$		−	57.3307i		−	1.08171i	−0.841116	−	0.540856i	$-0.818101\pi$
$53$		−	57.3307i		−	1.08171i	0.841116	−	0.540856i	$-0.181899\pi$
$54$		0			0
$55$	−30.0558			−0.546469
$56$		0			0
$57$		0			0
$58$		0			0
$59$	34.7812	+	20.0809i	0.589512	+	0.340355i	0.764904	−	0.644144i	$-0.222786\pi$
$59$	34.7812	+	20.0809i	0.589512	+	0.340355i	−0.175393	+	0.984499i	$0.556119\pi$
$60$		0			0
$61$	54.2890	+	94.0314i	0.889984	+	1.54150i	0.839892	+	0.542753i	$0.182618\pi$
$61$	54.2890	+	94.0314i	0.889984	+	1.54150i	0.0500920	+	0.998745i	$0.484049\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−35.1262	+	20.2801i	−0.540403	+	0.312002i
$66$		0			0
$67$	−5.09891	+	8.83158i	−0.0761032	+	0.131815i	−0.901565	−	0.432643i	$-0.857581\pi$
$67$	−5.09891	+	8.83158i	−0.0761032	+	0.131815i	0.825462	+	0.564457i	$0.190915\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	89.0429i		−	1.25412i	−0.778969	−	0.627062i	$-0.784257\pi$
$71$		−	89.0429i		−	1.25412i	0.778969	−	0.627062i	$-0.215743\pi$
$72$		0			0
$73$	90.9902			1.24644			0.623221	−	0.782046i	$-0.285824\pi$
$73$	90.9902			1.24644			0.623221	+	0.782046i	$0.285824\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	81.2153	+	46.8896i	1.05474	+	0.608956i
$78$		0			0
$79$	67.2671	+	116.510i	0.851483	+	1.47481i	0.879870	+	0.475215i	$0.157630\pi$
$79$	67.2671	+	116.510i	0.851483	+	1.47481i	−0.0283871	+	0.999597i	$0.509037\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−24.2534	+	14.0027i	−0.292210	+	0.168707i	−0.638938	−	0.769258i	$-0.720626\pi$
$83$	−24.2534	+	14.0027i	−0.292210	+	0.168707i	0.346728	+	0.937966i	$0.387293\pi$
$84$		0			0
$85$	21.4576	−	37.1657i	0.252442	−	0.437243i
$86$		0			0
$87$		0			0
$88$		0			0
$89$			60.1404i			0.675735i	0.941194	+	0.337867i	$0.109706\pi$
$89$			60.1404i			0.675735i	−0.941194	+	0.337867i	$0.890294\pi$
$90$		0			0
$91$	126.555			1.39072
$92$		0			0
$93$		0			0
$94$		0			0
$95$	0.543924	+	0.314035i	0.00572551	+	0.00330563i
$96$		0			0
$97$	2.59486	+	4.49443i	0.0267511	+	0.0463343i	0.879091	−	0.476654i	$-0.158151\pi$
$97$	2.59486	+	4.49443i	0.0267511	+	0.0463343i	−0.852340	+	0.522988i	$0.824817\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	85.9474	−	49.6217i	0.850964	−	0.491304i	−0.0100120	−	0.999950i	$-0.503187\pi$
$101$	85.9474	−	49.6217i	0.850964	−	0.491304i	0.860976	+	0.508646i	$0.169854\pi$
$102$		0			0
$103$	−16.7365	+	28.9885i	−0.162490	+	0.281442i	−0.935761	−	0.352634i	$-0.885286\pi$
$103$	−16.7365	+	28.9885i	−0.162490	+	0.281442i	0.773271	+	0.634076i	$0.218619\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	49.4591i		−	0.462235i	−0.972926	−	0.231117i	$-0.925762\pi$
$107$		−	49.4591i		−	0.462235i	0.972926	−	0.231117i	$-0.0742381\pi$
$108$		0			0
$109$	96.2524			0.883049			0.441525	−	0.897249i	$-0.354438\pi$
$109$	96.2524			0.883049			0.441525	+	0.897249i	$0.354438\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	99.6712	+	57.5452i	0.882046	+	0.509250i	0.871333	−	0.490693i	$-0.163256\pi$
$113$	99.6712	+	57.5452i	0.882046	+	0.509250i	0.0107136	+	0.999943i	$0.496590\pi$
$114$		0			0
$115$	−13.3425	−	23.1098i	−0.116021	−	0.200955i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−115.963	+	66.9515i	−0.974482	+	0.562617i
$120$		0			0
$121$	29.8351	−	51.6759i	0.246571	−	0.427073i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		−	11.1803i		−	0.0894427i
$126$		0			0
$127$	54.9123			0.432380			0.216190	−	0.976351i	$-0.430637\pi$
$127$	54.9123			0.432380			0.216190	+	0.976351i	$0.430637\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	217.340	+	125.482i	1.65909	+	0.957874i	0.973137	+	0.230225i	$0.0739463\pi$
$131$	217.340	+	125.482i	1.65909	+	0.957874i	0.685950	+	0.727649i	$0.259387\pi$
$132$		0			0
$133$	−0.979842	−	1.69714i	−0.00736724	−	0.0127604i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−27.7304	+	16.0101i	−0.202411	+	0.116862i	−0.597780	−	0.801660i	$-0.703950\pi$
$137$	−27.7304	+	16.0101i	−0.202411	+	0.116862i	0.395368	+	0.918523i	$0.370617\pi$
$138$		0			0
$139$	135.382	−	234.489i	0.973974	−	1.68697i	0.290694	−	0.956816i	$-0.406114\pi$
$139$	135.382	−	234.489i	0.973974	−	1.68697i	0.683280	−	0.730156i	$-0.260553\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		−	243.814i		−	1.70499i
$144$		0			0
$145$	106.637			0.735427
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−133.348	−	76.9883i	−0.894951	−	0.516700i	−0.0193920	−	0.999812i	$-0.506173\pi$
$149$	−133.348	−	76.9883i	−0.894951	−	0.516700i	−0.875559	+	0.483112i	$0.839506\pi$
$150$		0			0
$151$	−19.8176	−	34.3251i	−0.131243	−	0.227319i	0.792913	−	0.609335i	$-0.208563\pi$
$151$	−19.8176	−	34.3251i	−0.131243	−	0.227319i	−0.924156	+	0.382016i	$0.875230\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−37.9573	+	21.9147i	−0.244886	+	0.141385i
$156$		0			0
$157$	107.303	−	185.855i	0.683462	−	1.18379i	−0.290456	−	0.956888i	$-0.593807\pi$
$157$	107.303	−	185.855i	0.683462	−	1.18379i	0.973918	−	0.226902i	$-0.0728597\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$			83.2617i			0.517153i
$162$		0			0
$163$	243.002			1.49081			0.745405	−	0.666612i	$-0.232256\pi$
$163$	243.002			1.49081			0.745405	+	0.666612i	$0.232256\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−28.1595	−	16.2579i	−0.168620	−	0.0973527i	0.413315	−	0.910588i	$-0.364371\pi$
$167$	−28.1595	−	16.2579i	−0.168620	−	0.0973527i	−0.581935	+	0.813235i	$0.697704\pi$
$168$		0			0
$169$	−80.0136	−	138.588i	−0.473453	−	0.820045i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−14.6784	+	8.47456i	−0.0848461	+	0.0489859i	−0.541823	−	0.840493i	$-0.682266\pi$
$173$	−14.6784	+	8.47456i	−0.0848461	+	0.0489859i	0.456977	+	0.889479i	$0.348932\pi$
$174$		0			0
$175$	−17.4423	+	30.2110i	−0.0996703	+	0.172634i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		−	120.995i		−	0.675951i	−0.941155	−	0.337975i	$-0.890258\pi$
$179$		−	120.995i		−	0.675951i	0.941155	−	0.337975i	$-0.109742\pi$
$180$		0			0
$181$	77.2509			0.426801			0.213400	−	0.976965i	$-0.431546\pi$
$181$	77.2509			0.426801			0.213400	+	0.976965i	$0.431546\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−108.360	−	62.5614i	−0.585727	−	0.338170i
$186$		0			0
$187$	128.985	+	223.409i	0.689760	+	1.19470i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−276.298	+	159.521i	−1.44659	+	0.835187i	−0.998276	−	0.0586914i	$-0.981307\pi$
$191$	−276.298	+	159.521i	−1.44659	+	0.835187i	−0.448310	+	0.893878i	$0.647974\pi$
$192$		0			0
$193$	−24.5723	+	42.5606i	−0.127318	+	0.220521i	−0.922637	−	0.385670i	$-0.873970\pi$
$193$	−24.5723	+	42.5606i	−0.127318	+	0.220521i	0.795319	+	0.606191i	$0.207303\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	149.741i		−	0.760109i	−0.924964	−	0.380054i	$-0.875905\pi$
$197$		−	149.741i		−	0.760109i	0.924964	−	0.380054i	$-0.124095\pi$
$198$		0			0
$199$	42.4521			0.213327			0.106664	−	0.994295i	$-0.465983\pi$
$199$	42.4521			0.213327			0.106664	+	0.994295i	$0.465983\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−288.149	−	166.363i	−1.41945	−	0.819521i
$204$		0			0
$205$	31.9355	+	55.3140i	0.155783	+	0.269824i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−3.26961	+	1.88771i	−0.0156441	+	0.00903211i
$210$		0			0
$211$	14.7416	−	25.5332i	0.0698654	−	0.121010i	−0.828977	−	0.559283i	$-0.811076\pi$
$211$	14.7416	−	25.5332i	0.0698654	−	0.121010i	0.898842	+	0.438273i	$0.144410\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$			155.496i			0.723237i
$216$		0			0
$217$	136.755			0.630208
$218$		0			0
$219$		0			0
$220$		0			0
$221$	301.490	+	174.065i	1.36421	+	0.787626i
$222$		0			0
$223$	128.130	+	221.927i	0.574572	+	0.995188i	0.996088	+	0.0883672i	$0.0281649\pi$
$223$	128.130	+	221.927i	0.574572	+	0.995188i	−0.421516	+	0.906821i	$0.638502\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	155.095	−	89.5440i	0.683237	−	0.394467i	−0.117837	−	0.993033i	$-0.537596\pi$
$227$	155.095	−	89.5440i	0.683237	−	0.394467i	0.801073	+	0.598566i	$0.204263\pi$
$228$		0			0
$229$	56.9440	−	98.6300i	0.248664	−	0.430699i	−0.714491	−	0.699644i	$-0.753342\pi$
$229$	56.9440	−	98.6300i	0.248664	−	0.430699i	0.963155	+	0.268946i	$0.0866752\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			112.366i			0.482258i	0.970493	+	0.241129i	$0.0775178\pi$
$233$			112.366i			0.482258i	−0.970493	+	0.241129i	$0.922482\pi$
$234$		0			0
$235$	35.3332			0.150354
$236$		0			0
$237$		0			0
$238$		0			0
$239$	381.823	+	220.446i	1.59759	+	0.922366i	0.991951	+	0.126623i	$0.0404139\pi$
$239$	381.823	+	220.446i	1.59759	+	0.922366i	0.605634	+	0.795743i	$0.292919\pi$
$240$		0			0
$241$	−34.5110	−	59.7748i	−0.143199	−	0.248028i	0.785501	−	0.618861i	$-0.212406\pi$
$241$	−34.5110	−	59.7748i	−0.143199	−	0.248028i	−0.928700	+	0.370833i	$0.879072\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−0.624684	+	0.360662i	−0.00254973	+	0.00147209i
$246$		0			0
$247$	−2.54747	+	4.41234i	−0.0103136	+	0.0178637i
$248$		0			0
$249$		0			0
$250$		0			0
$251$			255.113i			1.01638i	0.861244	+	0.508192i	$0.169686\pi$
$251$			255.113i			1.01638i	−0.861244	+	0.508192i	$0.830314\pi$
$252$		0			0
$253$	160.407			0.634021
$254$		0			0
$255$		0			0
$256$		0			0
$257$	235.657	+	136.057i	0.916953	+	0.529403i	0.882662	−	0.470009i	$-0.155749\pi$
$257$	235.657	+	136.057i	0.916953	+	0.529403i	0.0342911	+	0.999412i	$0.489083\pi$
$258$		0			0
$259$	195.203	+	338.101i	0.753678	+	1.30541i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−352.403	+	203.460i	−1.33994	+	0.773613i	−0.986798	−	0.161958i	$-0.948219\pi$
$263$	−352.403	+	203.460i	−1.33994	+	0.773613i	−0.353139	+	0.935571i	$0.614886\pi$
$264$		0			0
$265$	−64.0977	+	111.020i	−0.241878	+	0.418945i
$266$		0			0
$267$		0			0
$268$		0			0
$269$		−	147.882i		−	0.549749i	−0.961480	−	0.274874i	$-0.911364\pi$
$269$		−	147.882i		−	0.549749i	0.961480	−	0.274874i	$-0.0886362\pi$
$270$		0			0
$271$	−315.408			−1.16387			−0.581934	−	0.813236i	$-0.697704\pi$
$271$	−315.408			−1.16387			−0.581934	+	0.813236i	$0.697704\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	58.2028	+	33.6034i	0.211647	+	0.122194i
$276$		0			0
$277$	−43.7452	−	75.7688i	−0.157925	−	0.273534i	0.776195	−	0.630492i	$-0.217147\pi$
$277$	−43.7452	−	75.7688i	−0.157925	−	0.273534i	−0.934120	+	0.356959i	$0.883814\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	206.929	−	119.470i	0.736402	−	0.425162i	−0.0843578	−	0.996436i	$-0.526884\pi$
$281$	206.929	−	119.470i	0.736402	−	0.425162i	0.820760	+	0.571274i	$0.193551\pi$
$282$		0			0
$283$	−148.098	+	256.514i	−0.523316	+	0.906410i	0.476316	+	0.879274i	$0.341972\pi$
$283$	−148.098	+	256.514i	−0.523316	+	0.906410i	−0.999632	+	0.0271359i	$0.991361\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		−	199.289i		−	0.694386i
$288$		0			0
$289$	−79.3431			−0.274544
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−467.873	−	270.126i	−1.59683	−	0.921933i	−0.992092	−	0.125512i	$-0.959943\pi$
$293$	−467.873	−	270.126i	−1.59683	−	0.921933i	−0.604743	−	0.796421i	$-0.706724\pi$
$294$		0			0
$295$	−44.9023	−	77.7731i	−0.152211	−	0.263638i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	187.468	−	108.235i	0.626984	−	0.361989i
$300$		0			0
$301$	242.587	−	420.173i	0.805938	−	1.39593i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		−	242.788i		−	0.796026i
$306$		0			0
$307$	−595.754			−1.94057			−0.970284	−	0.241969i	$-0.922207\pi$
$307$	−595.754			−1.94057			−0.970284	+	0.241969i	$0.922207\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	157.392	+	90.8702i	0.506083	+	0.292187i	0.731222	−	0.682139i	$-0.238950\pi$
$311$	157.392	+	90.8702i	0.506083	+	0.292187i	−0.225139	+	0.974327i	$0.572284\pi$
$312$		0			0
$313$	−143.232	−	248.085i	−0.457611	−	0.792605i	0.541223	−	0.840879i	$-0.317961\pi$
$313$	−143.232	−	248.085i	−0.457611	−	0.792605i	−0.998834	+	0.0482738i	$0.984628\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−398.060	+	229.820i	−1.25571	+	0.724984i	−0.972237	−	0.233997i	$-0.924820\pi$
$317$	−398.060	+	229.820i	−1.25571	+	0.724984i	−0.283472	+	0.958981i	$0.591486\pi$
$318$		0			0
$319$	−320.506	+	555.132i	−1.00472	+	1.74023i
$320$		0			0
$321$		0			0
$322$		0			0
$323$		−	5.39074i		−	0.0166896i
$324$		0			0
$325$	90.6955			0.279063
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−95.4755	−	55.1228i	−0.290199	−	0.167547i
$330$		0			0
$331$	62.1644	+	107.672i	0.187808	+	0.325293i	0.944519	−	0.328457i	$-0.106528\pi$
$331$	62.1644	+	107.672i	0.187808	+	0.325293i	−0.756711	+	0.653749i	$0.773195\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	19.7480	−	11.4015i	0.0589493	−	0.0340344i
$336$		0			0
$337$	198.729	−	344.209i	0.589701	−	1.02139i	−0.404571	−	0.914507i	$-0.632579\pi$
$337$	198.729	−	344.209i	0.589701	−	1.02139i	0.994271	−	0.106885i	$-0.0340877\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		−	263.465i		−	0.772625i
$342$		0			0
$343$	344.120			1.00326
$344$		0			0
$345$		0			0
$346$		0			0
$347$	89.4758	+	51.6589i	0.257855	+	0.148873i	0.623356	−	0.781938i	$-0.285769\pi$
$347$	89.4758	+	51.6589i	0.257855	+	0.148873i	−0.365501	+	0.930811i	$0.619102\pi$
$348$		0			0
$349$	−163.565	−	283.303i	−0.468668	−	0.811757i	0.530690	−	0.847566i	$-0.321933\pi$
$349$	−163.565	−	283.303i	−0.468668	−	0.811757i	−0.999359	+	0.0358084i	$0.988599\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	46.9233	−	27.0912i	0.132927	−	0.0767456i	−0.432062	−	0.901844i	$-0.642214\pi$
$353$	46.9233	−	27.0912i	0.132927	−	0.0767456i	0.564989	+	0.825098i	$0.308880\pi$
$354$		0			0
$355$	−99.5529	+	172.431i	−0.280431	+	0.485720i
$356$		0			0
$357$		0			0
$358$		0			0
$359$		−	18.9630i		−	0.0528216i	−0.999651	−	0.0264108i	$-0.991592\pi$
$359$		−	18.9630i		−	0.0528216i	0.999651	−	0.0264108i	$-0.00840780\pi$
$360$		0			0
$361$	−360.921			−0.999781
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−176.202	−	101.730i	−0.482745	−	0.278713i
$366$		0			0
$367$	−151.647	−	262.661i	−0.413208	−	0.715697i	0.582031	−	0.813167i	$-0.302258\pi$
$367$	−151.647	−	262.661i	−0.413208	−	0.715697i	−0.995238	+	0.0974699i	$0.968925\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	346.403	−	199.996i	0.933700	−	0.539072i
$372$		0			0
$373$	145.899	−	252.704i	0.391150	−	0.677491i	−0.601452	−	0.798909i	$-0.705411\pi$
$373$	145.899	−	252.704i	0.391150	−	0.677491i	0.992601	+	0.121418i	$0.0387441\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$			865.044i			2.29455i
$378$		0			0
$379$	147.620			0.389497			0.194749	−	0.980853i	$-0.437611\pi$
$379$	147.620			0.389497			0.194749	+	0.980853i	$0.437611\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−400.127	−	231.014i	−1.04472	−	0.603169i	−0.123553	−	0.992338i	$-0.539429\pi$
$383$	−400.127	−	231.014i	−1.04472	−	0.603169i	−0.921166	+	0.389169i	$0.872762\pi$
$384$		0			0
$385$	−104.848	−	181.603i	−0.272334	−	0.471696i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	384.072	−	221.744i	0.987332	−	0.570036i	0.0828562	−	0.996562i	$-0.473596\pi$
$389$	384.072	−	221.744i	0.987332	−	0.570036i	0.904476	+	0.426525i	$0.140262\pi$
$390$		0			0
$391$	−114.519	+	198.353i	−0.292887	+	0.507296i
$392$		0			0
$393$		0			0
$394$		0			0
$395$		−	300.828i		−	0.761589i
$396$		0			0
$397$	256.214			0.645377			0.322688	−	0.946505i	$-0.395413\pi$
$397$	256.214			0.645377			0.322688	+	0.946505i	$0.395413\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−154.004	−	88.9142i	−0.384050	−	0.221731i	0.295529	−	0.955334i	$-0.404504\pi$
$401$	−154.004	−	88.9142i	−0.384050	−	0.221731i	−0.679579	+	0.733603i	$0.737837\pi$
$402$		0			0
$403$	−177.773	−	307.912i	−0.441124	−	0.764050i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	651.367	−	376.067i	1.60041	−	0.923997i
$408$		0			0
$409$	−170.500	+	295.314i	−0.416869	+	0.722039i	−0.995623	−	0.0934637i	$-0.970206\pi$
$409$	−170.500	+	295.314i	−0.416869	+	0.722039i	0.578753	+	0.815503i	$0.303539\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$			280.206i			0.678465i
$414$		0			0
$415$	62.6220			0.150896
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−92.1537	−	53.2050i	−0.219937	−	0.126981i	0.385984	−	0.922505i	$-0.373862\pi$
$419$	−92.1537	−	53.2050i	−0.219937	−	0.126981i	−0.605921	+	0.795525i	$0.707195\pi$
$420$		0			0
$421$	350.631	+	607.311i	0.832853	+	1.44254i	0.895766	+	0.444525i	$0.146628\pi$
$421$	350.631	+	607.311i	0.832853	+	1.44254i	−0.0629131	+	0.998019i	$0.520039\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−83.1050	+	47.9807i	−0.195541	+	0.112896i
$426$		0			0
$427$	−378.770	+	656.049i	−0.887050	+	1.53642i
$428$		0			0
$429$		0			0
$430$		0			0
$431$			546.935i			1.26899i	0.772926	+	0.634496i	$0.218792\pi$
$431$			546.935i			1.26899i	−0.772926	+	0.634496i	$0.781208\pi$
$432$		0			0
$433$	−585.896			−1.35311			−0.676555	−	0.736392i	$-0.736528\pi$
$433$	−585.896			−1.35311			−0.676555	+	0.736392i	$0.736528\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−2.90292	−	1.67600i	−0.00664283	−	0.00383524i
$438$		0			0
$439$	−221.778	−	384.131i	−0.505189	−	0.875013i	−0.999982	−	0.00600204i	$-0.998089\pi$
$439$	−221.778	−	384.131i	−0.505189	−	0.875013i	0.494793	−	0.869011i	$-0.335244\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−715.933	+	413.344i	−1.61610	+	0.933057i	−0.628186	+	0.778063i	$0.716202\pi$
$443$	−715.933	+	413.344i	−1.61610	+	0.933057i	−0.987915	+	0.154993i	$0.950464\pi$
$444$		0			0
$445$	67.2390	−	116.461i	0.151099	−	0.261711i
$446$		0			0
$447$		0			0
$448$		0			0
$449$			341.820i			0.761292i	0.924721	+	0.380646i	$0.124298\pi$
$449$			341.820i			0.761292i	−0.924721	+	0.380646i	$0.875702\pi$
$450$		0			0
$451$	−383.939			−0.851306
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−245.073	−	141.493i	−0.538622	−	0.310973i
$456$		0			0
$457$	75.9740	+	131.591i	0.166245	+	0.287945i	0.937097	−	0.349070i	$-0.113502\pi$
$457$	75.9740	+	131.591i	0.166245	+	0.287945i	−0.770852	+	0.637015i	$0.780169\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	622.153	−	359.200i	1.34957	−	0.779177i	0.361384	−	0.932417i	$-0.382304\pi$
$461$	622.153	−	359.200i	1.34957	−	0.779177i	0.988189	+	0.153240i	$0.0489709\pi$
$462$		0			0
$463$	−262.971	+	455.479i	−0.567972	+	0.983756i	0.428795	+	0.903402i	$0.358938\pi$
$463$	−262.971	+	455.479i	−0.567972	+	0.983756i	−0.996766	+	0.0803538i	$0.974395\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$			395.715i			0.847356i	0.905813	+	0.423678i	$0.139261\pi$
$467$			395.715i			0.847356i	−0.905813	+	0.423678i	$0.860739\pi$
$468$		0			0
$469$	−71.1494			−0.151705
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−809.484	−	467.356i	−1.71138	−	0.988067i
$474$		0			0
$475$	−0.702203	−	1.21625i	−0.00147832	−	0.00256053i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	143.982	−	83.1283i	0.300590	−	0.173545i	−0.342118	−	0.939657i	$-0.611144\pi$
$479$	143.982	−	83.1283i	0.300590	−	0.173545i	0.642708	+	0.766111i	$0.277811\pi$
$480$		0			0
$481$	507.502	−	879.019i	1.05510	−	1.82748i
$482$		0			0
$483$		0			0
$484$		0			0
$485$		−	11.6046i		−	0.0239269i
$486$		0			0
$487$	−319.963			−0.657008			−0.328504	−	0.944503i	$-0.606544\pi$
$487$	−319.963			−0.657008			−0.328504	+	0.944503i	$0.606544\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	491.823	+	283.954i	1.00168	+	0.578318i	0.908744	−	0.417355i	$-0.137043\pi$
$491$	491.823	+	283.954i	1.00168	+	0.578318i	0.0929321	+	0.995672i	$0.470376\pi$
$492$		0			0
$493$	−457.634	−	792.646i	−0.928264	−	1.60780i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	538.014	−	310.622i	1.08252	−	0.624995i
$498$		0			0
$499$	249.737	−	432.557i	0.500475	−	0.866848i	−0.499525	−	0.866299i	$-0.666492\pi$
$499$	249.737	−	432.557i	0.500475	−	0.866848i	1.00000		0.000548216i	$-0.000174503\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	858.979i		−	1.70771i	−0.520509	−	0.853856i	$-0.674258\pi$
$503$		−	858.979i		−	1.70771i	0.520509	−	0.853856i	$-0.325742\pi$
$504$		0			0
$505$	−221.915			−0.439436
$506$		0			0
$507$		0			0
$508$		0			0
$509$	524.997	+	303.107i	1.03143	+	0.595496i	0.917394	−	0.397981i	$-0.130289\pi$
$509$	524.997	+	303.107i	1.03143	+	0.595496i	0.114035	+	0.993477i	$0.463622\pi$
$510$		0			0
$511$	317.416	+	549.780i	0.621166	+	1.07589i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	64.8202	−	37.4240i	0.125865	−	0.0726679i
$516$		0			0
$517$	−106.197	+	183.938i	−0.205409	+	0.355779i
$518$		0			0
$519$		0			0
$520$		0			0
$521$			413.782i			0.794208i	0.917774	+	0.397104i	$0.129985\pi$
$521$			413.782i			0.794208i	−0.917774	+	0.397104i	$0.870015\pi$
$522$		0			0
$523$	−739.740			−1.41442			−0.707208	−	0.707005i	$-0.750046\pi$
$523$	−739.740			−1.41442			−0.707208	+	0.707005i	$0.750046\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	325.789	+	188.095i	0.618196	+	0.356916i
$528$		0			0
$529$	−193.291	−	334.790i	−0.365390	−	0.632874i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−448.710	+	259.063i	−0.841857	+	0.486046i
$534$		0			0
$535$	−55.2970	+	95.7771i	−0.103359	+	0.179023i
$536$		0			0
$537$		0			0
$538$		0			0
$539$		−	4.33599i		−	0.00804451i
$540$		0			0
$541$	−939.815			−1.73718			−0.868591	−	0.495530i	$-0.834974\pi$
$541$	−939.815			−1.73718			−0.868591	+	0.495530i	$0.834974\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−186.392	−	107.613i	−0.342003	−	0.197456i
$546$		0			0
$547$	358.500	+	620.939i	0.655392	+	1.13517i	0.981795	+	0.189942i	$0.0608300\pi$
$547$	358.500	+	620.939i	0.655392	+	1.13517i	−0.326403	+	0.945231i	$0.605837\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	11.6005	−	6.69753i	0.0210535	−	0.0121552i
$552$		0			0
$553$	−469.318	+	812.882i	−0.848675	+	1.46995i
$554$		0			0
$555$		0			0
$556$		0			0
$557$			507.982i			0.911996i	0.889981	+	0.455998i	$0.150718\pi$
$557$			507.982i			0.911996i	−0.889981	+	0.455998i	$0.849282\pi$
$558$		0			0
$559$	−1261.39			−2.25652
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−41.2767	−	23.8311i	−0.0733156	−	0.0423288i	0.462894	−	0.886414i	$-0.346811\pi$
$563$	−41.2767	−	23.8311i	−0.0733156	−	0.0423288i	−0.536210	+	0.844085i	$0.680144\pi$
$564$		0			0
$565$	−128.675	−	222.872i	−0.227743	−	0.394463i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	611.620	−	353.119i	1.07490	−	0.620596i	0.145387	−	0.989375i	$-0.453557\pi$
$569$	611.620	−	353.119i	1.07490	−	0.620596i	0.929517	+	0.368779i	$0.120224\pi$
$570$		0			0
$571$	−525.508	+	910.206i	−0.920329	+	1.59406i	−0.121422	+	0.992601i	$0.538745\pi$
$571$	−525.508	+	910.206i	−0.920329	+	1.59406i	−0.798907	+	0.601455i	$0.794588\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$			59.6693i			0.103773i
$576$		0			0
$577$	−568.036			−0.984464			−0.492232	−	0.870464i	$-0.663819\pi$
$577$	−568.036			−0.984464			−0.492232	+	0.870464i	$0.663819\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−169.214	−	97.6958i	−0.291246	−	0.168151i
$582$		0			0
$583$	−385.301	−	667.361i	−0.660894	−	1.14470i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−782.794	+	451.947i	−1.33355	+	0.769926i	−0.985842	−	0.167677i	$-0.946374\pi$
$587$	−782.794	+	451.947i	−1.33355	+	0.769926i	−0.347709	+	0.937603i	$0.613040\pi$
$588$		0			0
$589$	−2.75279	+	4.76797i	−0.00467366	+	0.00809502i
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	741.637i		−	1.25065i	−0.780363	−	0.625326i	$-0.784966\pi$
$593$		−	741.637i		−	1.25065i	0.780363	−	0.625326i	$-0.215034\pi$
$594$		0			0
$595$	299.416			0.503220
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−502.797	−	290.290i	−0.839393	−	0.484624i	0.0176647	−	0.999844i	$-0.494377\pi$
$599$	−502.797	−	290.290i	−0.839393	−	0.484624i	−0.857058	+	0.515220i	$0.827710\pi$
$600$		0			0
$601$	−144.236	−	249.825i	−0.239994	−	0.415682i	0.720718	−	0.693228i	$-0.243812\pi$
$601$	−144.236	−	249.825i	−0.239994	−	0.415682i	−0.960712	+	0.277546i	$0.910479\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−115.551	+	66.7133i	−0.190993	+	0.110270i
$606$		0			0
$607$	178.840	−	309.760i	0.294629	−	0.510313i	−0.680269	−	0.732962i	$-0.738137\pi$
$607$	178.840	−	309.760i	0.294629	−	0.510313i	0.974899	+	0.222649i	$0.0714705\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$			286.625i			0.469107i
$612$		0			0
$613$	1108.76			1.80875			0.904375	−	0.426738i	$-0.140337\pi$
$613$	1108.76			1.80875			0.904375	+	0.426738i	$0.140337\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−435.556	−	251.468i	−0.705925	−	0.407566i	0.103625	−	0.994616i	$-0.466956\pi$
$617$	−435.556	−	251.468i	−0.705925	−	0.407566i	−0.809550	+	0.587050i	$0.800289\pi$
$618$		0			0
$619$	319.916	+	554.111i	0.516827	+	0.895171i	0.999809	+	0.0195407i	$0.00622038\pi$
$619$	319.916	+	554.111i	0.516827	+	0.895171i	−0.482982	+	0.875630i	$0.660446\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−363.380	+	209.797i	−0.583274	+	0.336753i
$624$		0			0
$625$	−12.5000	+	21.6506i	−0.0200000	+	0.0346410i
$626$		0			0
$627$		0			0
$628$		0			0
$629$			1073.93i			1.70737i
$630$		0			0
$631$	−134.214			−0.212700			−0.106350	−	0.994329i	$-0.533916\pi$
$631$	−134.214			−0.212700			−0.106350	+	0.994329i	$0.533916\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−106.337	−	61.3938i	−0.167460	−	0.0966831i
$636$		0			0
$637$	−2.92571	−	5.06747i	−0.00459295	−	0.00795522i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	510.974	−	295.011i	0.797152	−	0.460236i	−0.0453223	−	0.998972i	$-0.514431\pi$
$641$	510.974	−	295.011i	0.797152	−	0.460236i	0.842474	+	0.538736i	$0.181098\pi$
$642$		0			0
$643$	−416.792	+	721.904i	−0.648199	+	1.12271i	0.335354	+	0.942092i	$0.391144\pi$
$643$	−416.792	+	721.904i	−0.648199	+	1.12271i	−0.983553	+	0.180621i	$0.942189\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	816.003i		−	1.26121i	−0.776104	−	0.630605i	$-0.782807\pi$
$647$		−	816.003i		−	1.26121i	0.776104	−	0.630605i	$-0.217193\pi$
$648$		0			0
$649$	539.830			0.831788
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−1112.91	−	642.537i	−1.70430	−	0.983977i	−0.941298	−	0.337578i	$-0.890392\pi$
$653$	−1112.91	−	642.537i	−1.70430	−	0.983977i	−0.763000	−	0.646399i	$-0.776274\pi$
$654$		0			0
$655$	−280.585	−	485.988i	−0.428374	−	0.741966i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−962.550	+	555.729i	−1.46062	+	0.843291i	−0.999040	−	0.0438064i	$-0.986052\pi$
$659$	−962.550	+	555.729i	−1.46062	+	0.843291i	−0.461583	+	0.887097i	$0.652718\pi$
$660$		0			0
$661$	−361.008	+	625.284i	−0.546154	+	0.945967i	0.452379	+	0.891826i	$0.350575\pi$
$661$	−361.008	+	625.284i	−0.546154	+	0.945967i	−0.998533	+	0.0541413i	$0.982758\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$			4.38199i			0.00658946i
$666$		0			0
$667$	−569.120			−0.853253
$668$		0			0
$669$		0			0
$670$		0			0
$671$	1263.91	+	729.718i	1.88362	+	1.08751i
$672$		0			0
$673$	−95.0443	−	164.622i	−0.141225	−	0.244609i	0.786733	−	0.617293i	$-0.211771\pi$
$673$	−95.0443	−	164.622i	−0.141225	−	0.244609i	−0.927958	+	0.372684i	$0.878437\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	966.444	−	557.977i	1.42754	−	0.824190i	0.430614	−	0.902536i	$-0.358297\pi$
$677$	966.444	−	557.977i	1.42754	−	0.824190i	0.996926	+	0.0783459i	$0.0249639\pi$
$678$		0			0
$679$	−18.1041	+	31.3572i	−0.0266629	+	0.0461815i
$680$		0			0
$681$		0			0
$682$		0			0
$683$			717.452i			1.05044i	0.850966	+	0.525221i	$0.176017\pi$
$683$			717.452i			1.05044i	−0.850966	+	0.525221i	$0.823983\pi$
$684$		0			0
$685$	71.5995			0.104525
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−900.603	−	519.964i	−1.30712	−	0.754664i
$690$		0			0
$691$	214.212	+	371.026i	0.310003	+	0.536941i	0.978363	−	0.206897i	$-0.0663366\pi$
$691$	214.212	+	371.026i	0.310003	+	0.536941i	−0.668360	+	0.743838i	$0.733003\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−524.334	+	302.724i	−0.754437	+	0.435574i
$696$		0			0
$697$	274.104	−	474.762i	0.393263	−	0.681151i
$698$		0			0
$699$		0			0
$700$		0			0
$701$			816.554i			1.16484i	0.812887	+	0.582421i	$0.197894\pi$
$701$			816.554i			1.16484i	−0.812887	+	0.582421i	$0.802106\pi$
$702$		0			0
$703$	−15.7172			−0.0223573
$704$		0			0
$705$		0			0
$706$		0			0
$707$	599.648	+	346.207i	0.848158	+	0.489684i
$708$		0			0
$709$	371.473	+	643.411i	0.523940	+	0.907490i	0.999612	+	0.0278674i	$0.00887163\pi$
$709$	371.473	+	643.411i	0.523940	+	0.907490i	−0.475672	+	0.879623i	$0.657795\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	202.578	−	116.958i	0.284120	−	0.164037i
$714$		0			0
$715$	−272.593	+	472.144i	−0.381248	+	0.660342i
$716$		0			0
$717$		0			0
$718$		0			0
$719$		−	448.063i		−	0.623175i	−0.950217	−	0.311587i	$-0.899139\pi$
$719$		−	448.063i		−	0.623175i	0.950217	−	0.311587i	$-0.100861\pi$
$720$		0			0
$721$	−233.539			−0.323909
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−206.501	−	119.224i	−0.284829	−	0.164446i
$726$		0			0
$727$	−309.249	−	535.636i	−0.425377	−	0.736775i	0.571078	−	0.820896i	$-0.306525\pi$
$727$	−309.249	−	535.636i	−0.425377	−	0.736775i	−0.996456	+	0.0841204i	$0.973192\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	1155.82	−	667.314i	1.58115	−	0.912879i
$732$		0			0
$733$	−43.0113	+	74.4978i	−0.0586785	+	0.101634i	−0.893872	−	0.448321i	$-0.852022\pi$
$733$	−43.0113	+	74.4978i	−0.0586785	+	0.101634i	0.835194	+	0.549956i	$0.185355\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			137.073i			0.185987i
$738$		0			0
$739$	−236.149			−0.319552			−0.159776	−	0.987153i	$-0.551077\pi$
$739$	−236.149			−0.319552			−0.159776	+	0.987153i	$0.551077\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−893.700	−	515.978i	−1.20283	−	0.694452i	−0.241643	−	0.970365i	$-0.577686\pi$
$743$	−893.700	−	515.978i	−1.20283	−	0.694452i	−0.961183	+	0.275913i	$0.911020\pi$
$744$		0			0
$745$	172.151	+	298.174i	0.231075	+	0.400234i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	298.841	−	172.536i	0.398987	−	0.230355i
$750$		0			0
$751$	710.255	−	1230.20i	0.945746	−	1.63808i	0.191494	−	0.981494i	$-0.438667\pi$
$751$	710.255	−	1230.20i	0.945746	−	1.63808i	0.754251	−	0.656586i	$-0.228000\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$			88.6272i			0.117387i
$756$		0			0
$757$	−601.752			−0.794916			−0.397458	−	0.917620i	$-0.630108\pi$
$757$	−601.752			−0.794916			−0.397458	+	0.917620i	$0.630108\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	1204.13	+	695.206i	1.58230	+	0.913542i	0.994523	+	0.104521i	$0.0333310\pi$
$761$	1204.13	+	695.206i	1.58230	+	0.913542i	0.587779	+	0.809021i	$0.300002\pi$
$762$		0			0
$763$	335.772	+	581.575i	0.440069	+	0.762221i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	630.900	−	364.250i	0.822555	−	0.474903i
$768$		0			0
$769$	388.062	−	672.143i	0.504632	−	0.874049i	−0.495353	−	0.868692i	$-0.664961\pi$
$769$	388.062	−	672.143i	0.504632	−	0.874049i	0.999986	−	0.00535708i	$-0.00170522\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		−	781.817i		−	1.01141i	−0.862708	−	0.505703i	$-0.831233\pi$
$773$		−	781.817i		−	1.01141i	0.862708	−	0.505703i	$-0.168767\pi$
$774$		0			0
$775$	98.0054			0.126459
$776$		0			0
$777$		0			0
$778$		0			0
$779$	6.94820	+	4.01154i	0.00891938	+	0.00514961i
$780$		0			0
$781$	−598.429	−	1036.51i	−0.766234	−	1.32716i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−415.585	+	239.938i	−0.529407	+	0.305653i
$786$		0			0
$787$	276.669	−	479.205i	0.351549	−	0.608901i	−0.634972	−	0.772535i	$-0.718988\pi$
$787$	276.669	−	479.205i	0.351549	−	0.608901i	0.986521	+	0.163634i	$0.0523218\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$			802.977i			1.01514i
$792$		0			0
$793$	1969.51			2.48362
$794$		0			0
$795$		0			0
$796$		0			0
$797$	113.965	+	65.7976i	0.142992	+	0.0825566i	0.569789	−	0.821791i	$-0.307025\pi$
$797$	113.965	+	65.7976i	0.142992	+	0.0825566i	−0.426797	+	0.904347i	$0.640358\pi$
$798$		0			0
$799$	−151.633	−	262.636i	−0.189779	−	0.328706i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	1059.18	−	611.516i	1.31903	−	0.761540i
$804$		0			0
$805$	93.0894	−	161.235i	0.115639	−	0.200293i
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	758.331i		−	0.937368i	−0.883366	−	0.468684i	$-0.844728\pi$
$809$		−	758.331i		−	0.937368i	0.883366	−	0.468684i	$-0.155272\pi$
$810$		0			0
$811$	−157.400			−0.194081			−0.0970406	−	0.995280i	$-0.530938\pi$
$811$	−157.400			−0.194081			−0.0970406	+	0.995280i	$0.530938\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−470.571	−	271.684i	−0.577388	−	0.333355i
$816$		0			0
$817$	9.76622	+	16.9156i	0.0119538	+	0.0207045i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−422.503	+	243.932i	−0.514620	+	0.297116i	−0.734731	−	0.678359i	$-0.762691\pi$
$821$	−422.503	+	243.932i	−0.514620	+	0.297116i	0.220111	+	0.975475i	$0.429358\pi$
$822$		0			0
$823$	−124.503	+	215.645i	−0.151279	+	0.262023i	−0.931698	−	0.363234i	$-0.881673\pi$
$823$	−124.503	+	215.645i	−0.151279	+	0.262023i	0.780419	+	0.625257i	$0.215006\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	1404.93i		−	1.69883i	−0.527724	−	0.849416i	$-0.676954\pi$
$827$		−	1404.93i		−	1.69883i	0.527724	−	0.849416i	$-0.323046\pi$
$828$		0			0
$829$	−554.816			−0.669260			−0.334630	−	0.942350i	$-0.608611\pi$
$829$	−554.816			−0.669260			−0.334630	+	0.942350i	$0.608611\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	5.36169	+	3.09557i	0.00643660	+	0.00371618i
$834$		0			0
$835$	36.3538	+	62.9666i	0.0435374	+	0.0754091i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−628.881	+	363.085i	−0.749561	+	0.432759i	−0.825535	−	0.564351i	$-0.809127\pi$
$839$	−628.881	+	363.085i	−0.749561	+	0.432759i	0.0759744	+	0.997110i	$0.475793\pi$
$840$		0			0
$841$	716.642	−	1241.26i	0.852131	−	1.47593i
$842$		0			0
$843$		0			0
$844$		0			0
$845$			357.832i			0.423469i
$846$		0			0
$847$	416.314			0.491516
$848$		0			0
$849$		0			0
$850$		0			0
$851$	578.314	+	333.890i	0.679570	+	0.392350i
$852$		0			0
$853$	−729.317	−	1263.21i	−0.855002	−	1.48091i	−0.876643	−	0.481142i	$-0.840222\pi$
$853$	−729.317	−	1263.21i	−0.855002	−	1.48091i	0.0216407	−	0.999766i	$-0.493111\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−511.856	+	295.520i	−0.597264	+	0.344831i	−0.767965	−	0.640492i	$-0.778730\pi$
$857$	−511.856	+	295.520i	−0.597264	+	0.344831i	0.170700	+	0.985323i	$0.445397\pi$
$858$		0			0
$859$	−589.337	+	1020.76i	−0.686074	+	1.18831i	0.287025	+	0.957923i	$0.407334\pi$
$859$	−589.337	+	1020.76i	−0.686074	+	1.18831i	−0.973098	+	0.230391i	$0.925999\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	610.718i		−	0.707669i	−0.935308	−	0.353834i	$-0.884878\pi$
$863$		−	610.718i		−	0.707669i	0.935308	−	0.353834i	$-0.115122\pi$
$864$		0			0
$865$	37.8994			0.0438143
$866$		0			0
$867$		0			0
$868$		0			0
$869$	1566.05	+	904.162i	1.80213	+	1.04046i
$870$		0			0
$871$	92.4897	+	160.197i	0.106188	+	0.183923i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	67.5537	−	39.0022i	0.0772043	−	0.0445739i
$876$		0			0
$877$	162.043	−	280.666i	0.184769	−	0.320030i	−0.758729	−	0.651406i	$-0.774180\pi$
$877$	162.043	−	280.666i	0.184769	−	0.320030i	0.943499	+	0.331376i	$0.107513\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	246.392i		−	0.279673i	−0.990175	−	0.139837i	$-0.955342\pi$
$881$		−	246.392i		−	0.279673i	0.990175	−	0.139837i	$-0.0446577\pi$
$882$		0			0
$883$	267.884			0.303379			0.151690	−	0.988428i	$-0.451529\pi$
$883$	267.884			0.303379			0.151690	+	0.988428i	$0.451529\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	962.933	+	555.950i	1.08561	+	0.626775i	0.932403	−	0.361419i	$-0.117708\pi$
$887$	962.933	+	555.950i	1.08561	+	0.626775i	0.153203	+	0.988195i	$0.451041\pi$
$888$		0			0
$889$	191.559	+	331.790i	0.215477	+	0.373217i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	3.84371	−	2.21917i	0.00430427	−	0.00248507i
$894$		0			0
$895$	−135.277	+	234.306i	−0.151147	+	0.261795i
$896$		0			0
$897$		0			0
$898$		0			0
$899$			934.765i			1.03978i
$900$		0			0
$901$	1100.31			1.22121
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−149.596	−	86.3692i	−0.165299	−	0.0954356i
$906$		0			0
$907$	398.494	+	690.211i	0.439354	+	0.760983i	0.997640	−	0.0686657i	$-0.0218742\pi$
$907$	398.494	+	690.211i	0.439354	+	0.760983i	−0.558286	+	0.829648i	$0.688541\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−1342.32	+	774.989i	−1.47346	+	0.850701i	−0.999554	−	0.0298758i	$-0.990489\pi$
$911$	−1342.32	+	774.989i	−1.47346	+	0.850701i	−0.473904	+	0.880577i	$0.657155\pi$
$912$		0			0
$913$	−188.216	+	325.999i	−0.206151	+	0.357063i
$914$		0			0
$915$		0			0
$916$		0			0
$917$			1750.95i			1.90943i
$918$		0			0
$919$	−761.899			−0.829052			−0.414526	−	0.910037i	$-0.636053\pi$
$919$	−761.899			−0.829052			−0.414526	+	0.910037i	$0.636053\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−1398.77	−	807.579i	−1.51546	−	0.874950i
$924$		0			0
$925$	139.892	+	242.299i	0.151234	+	0.261945i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−526.186	+	303.793i	−0.566400	+	0.327011i	−0.755710	−	0.654906i	$-0.772708\pi$
$929$	−526.186	+	303.793i	−0.566400	+	0.327011i	0.189310	+	0.981917i	$0.439375\pi$
$930$		0			0
$931$	−0.0453041	+	0.0784690i	−4.86617e−5	+	8.42846e-5i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		−	576.839i		−	0.616940i
$936$		0			0
$937$	173.756			0.185438			0.0927192	−	0.995692i	$-0.470444\pi$
$937$	173.756			0.185438			0.0927192	+	0.995692i	$0.470444\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	991.293	+	572.323i	1.05345	+	0.608208i	0.923612	−	0.383328i	$-0.125222\pi$
$941$	991.293	+	572.323i	1.05345	+	0.608208i	0.129834	+	0.991536i	$0.458555\pi$
$942$		0			0
$943$	−170.440	−	295.210i	−0.180742	−	0.313054i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	874.931	−	505.142i	0.923897	−	0.533412i	0.0390210	−	0.999238i	$-0.487576\pi$
$947$	874.931	−	505.142i	0.923897	−	0.533412i	0.884876	+	0.465826i	$0.154243\pi$
$948$		0			0
$949$	825.241	−	1429.36i	0.869590	−	1.50617i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	377.477i		−	0.396093i	−0.980193	−	0.198047i	$-0.936540\pi$
$953$		−	377.477i		−	0.396093i	0.980193	−	0.198047i	$-0.0634598\pi$
$954$		0			0
$955$	713.398			0.747014
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−193.472	−	111.701i	−0.201744	−	0.116477i
$960$		0			0
$961$	288.399	+	499.521i	0.300103	+	0.519793i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	95.1683	−	54.9454i	0.0986200	−	0.0569383i
$966$		0			0
$967$	−370.091	+	641.016i	−0.382721	+	0.662891i	−0.991450	−	0.130486i	$-0.958346\pi$
$967$	−370.091	+	641.016i	−0.382721	+	0.662891i	0.608730	+	0.793378i	$0.291679\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		−	1451.96i		−	1.49532i	−0.664082	−	0.747660i	$-0.731177\pi$
$971$		−	1451.96i		−	1.49532i	0.664082	−	0.747660i	$-0.268823\pi$
$972$		0			0
$973$	1889.10			1.94153
$974$		0			0
$975$		0			0
$976$		0			0
$977$	4.54636	+	2.62484i	0.00465339	+	0.00268664i	0.502325	−	0.864679i	$-0.332478\pi$
$977$	4.54636	+	2.62484i	0.00465339	+	0.00268664i	−0.497672	+	0.867366i	$0.665811\pi$
$978$		0			0
$979$	404.184	+	700.068i	0.412854	+	0.715084i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−1092.21	+	630.590i	−1.11110	+	0.641495i	−0.939115	−	0.343604i	$-0.888352\pi$
$983$	−1092.21	+	630.590i	−1.11110	+	0.641495i	−0.171987	+	0.985099i	$0.555019\pi$
$984$		0			0
$985$	−167.416	+	289.973i	−0.169965	+	0.294389i
$986$		0			0
$987$		0			0
$988$		0			0
$989$		−	829.880i		−	0.839111i
$990$		0			0
$991$	−1028.16			−1.03749			−0.518746	−	0.854928i	$-0.673601\pi$
$991$	−1028.16			−1.03749			−0.518746	+	0.854928i	$0.673601\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−82.2081	−	47.4629i	−0.0826212	−	0.0477014i
$996$		0			0
$997$	421.466	+	730.001i	0.422735	+	0.732198i	0.996206	−	0.0870278i	$-0.0277369\pi$
$997$	421.466	+	730.001i	0.422735	+	0.732198i	−0.573471	+	0.819226i	$0.694404\pi$
$998$		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.701.7		32
3.2	odd	2	inner	1620.3.o.g.701.9		32
9.2	odd	6	inner	1620.3.o.g.1241.6		32
9.4	even	3		1620.3.g.c.161.11	yes	16
9.5	odd	6		1620.3.g.c.161.3	✓	16
9.7	even	3	inner	1620.3.o.g.1241.9		32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1620.3.g.c.161.3	✓	16	9.5	odd	6
1620.3.g.c.161.11	yes	16	9.4	even	3
1620.3.o.g.701.7		32	1.1	even	1	trivial
1620.3.o.g.701.9		32	3.2	odd	2	inner
1620.3.o.g.1241.6		32	9.2	odd	6	inner
1620.3.o.g.1241.9		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)

\(f(q)\)	\(=\)	\(q+(-1.93649 - 1.11803i) q^{5} +(3.48846 + 6.04219i) q^{7} +O(q^{10})\)	\(q+(-1.93649 - 1.11803i) q^{5} +(3.48846 + 6.04219i) q^{7} +(11.6406 - 6.72068i) q^{11} +(9.06955 - 15.7089i) q^{13} +19.1923i q^{17} -0.280881 q^{19} +(10.3350 + 5.96693i) q^{23} +(2.50000 + 4.33013i) q^{25} +(-41.3003 + 23.8447i) q^{29} +(9.80054 - 16.9750i) q^{31} -15.6009i q^{35} +55.9566 q^{37} +(-24.7372 - 14.2820i) q^{41} +(-34.7700 - 60.2233i) q^{43} +(-13.6845 + 7.90074i) q^{47} +(0.161293 - 0.279367i) q^{49} -57.3307i q^{53} -30.0558 q^{55} +(34.7812 + 20.0809i) q^{59} +(54.2890 + 94.0314i) q^{61} +(-35.1262 + 20.2801i) q^{65} +(-5.09891 + 8.83158i) q^{67} -89.0429i q^{71} +90.9902 q^{73} +(81.2153 + 46.8896i) q^{77} +(67.2671 + 116.510i) q^{79} +(-24.2534 + 14.0027i) q^{83} +(21.4576 - 37.1657i) q^{85} +60.1404i q^{89} +126.555 q^{91} +(0.543924 + 0.314035i) q^{95} +(2.59486 + 4.49443i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - 8 q^{7}+O(q^{10}) \) 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

Related objects

Downloads

Learn more