LMFDB - Embedded newform 1620.4.i.u.1081.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1620,4,Mod(541,1620)]

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, 4, 0]))

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("1620.541");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1620 = 2^{2} \cdot 3^{4} \cdot 5 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1620.i (of order $3$, 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:	$95.5830942093$
Analytic rank:	$0$
Dimension:	$6$
Relative dimension:	$3$ over $\Q(\zeta_{3})$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{5} + 91x^{4} - 294x^{3} + 8292x^{2} - 17280x + 36864 $ x^6 - x^5 + 91x^4 - 294x^3 + 8292x^2 - 17280x + 36864
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2}\cdot 3^{3} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			1081.3
Root			$-4.98253 - 8.62999i$ of defining polynomial
Character	$\chi$	$=$	1620.1081
Dual form			1620.4.i.u.541.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(2.50000 - 4.33013i) q^{5} +(9.81683 + 17.0033i) q^{7} +O(q^{10})$	$q+(2.50000 - 4.33013i) q^{5} +(9.81683 + 17.0033i) q^{7} +(-14.5783 - 25.2504i) q^{11} +(6.81683 - 11.8071i) q^{13} -39.0460 q^{17} +43.7443 q^{19} +(-79.6072 + 137.884i) q^{23} +(-12.5000 - 21.6506i) q^{25} +(12.5725 + 21.7762i) q^{29} +(58.3686 - 101.097i) q^{31} +98.1683 q^{35} +329.383 q^{37} +(-90.4010 + 156.579i) q^{41} +(106.836 + 185.046i) q^{43} +(89.7062 + 155.376i) q^{47} +(-21.2404 + 36.7895i) q^{49} +60.8635 q^{53} -145.783 q^{55} +(191.289 - 331.322i) q^{59} +(-216.841 - 375.579i) q^{61} +(-34.0842 - 59.0355i) q^{65} +(-62.5534 + 108.346i) q^{67} -30.0873 q^{71} +676.299 q^{73} +(286.226 - 495.758i) q^{77} +(416.924 + 722.134i) q^{79} +(176.729 + 306.103i) q^{83} +(-97.6149 + 169.074i) q^{85} -948.224 q^{89} +267.679 q^{91} +(109.361 - 189.419i) q^{95} +(142.919 + 247.543i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 15 q^{5} - 15 q^{7}+O(q^{10}) $ 6 * q + 15 * q^5 - 15 * q^7	$ 6 q + 15 q^{5} - 15 q^{7} + 24 q^{11} - 33 q^{13} - 84 q^{17} + 42 q^{19} - 33 q^{23} - 75 q^{25} + 222 q^{29} - 132 q^{31} - 150 q^{35} + 348 q^{37} - 99 q^{41} + 120 q^{43} + 537 q^{47} - 492 q^{49} - 534 q^{53} + 240 q^{55} - 225 q^{59} + 480 q^{61} + 165 q^{65} - 12 q^{67} - 1140 q^{71} + 2124 q^{73} + 312 q^{77} + 1026 q^{79} + 702 q^{83} - 210 q^{85} - 2280 q^{89} + 3222 q^{91} + 105 q^{95} + 2556 q^{97}+O(q^{100}) $ 6 * q + 15 * q^5 - 15 * q^7 + 24 * q^11 - 33 * q^13 - 84 * q^17 + 42 * q^19 - 33 * q^23 - 75 * q^25 + 222 * q^29 - 132 * q^31 - 150 * q^35 + 348 * q^37 - 99 * q^41 + 120 * q^43 + 537 * q^47 - 492 * q^49 - 534 * q^53 + 240 * q^55 - 225 * q^59 + 480 * q^61 + 165 * q^65 - 12 * q^67 - 1140 * q^71 + 2124 * q^73 + 312 * q^77 + 1026 * q^79 + 702 * q^83 - 210 * q^85 - 2280 * q^89 + 3222 * q^91 + 105 * q^95 + 2556 * q^97

Display 100 coefficients 10 coefficients

Character values

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

$n$	$811$	$1297$	$1541$
$\chi(n)$	$1$	$1$	$e\left(\frac{1}{3}\right)$

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
$5$	2.50000	−	4.33013i	0.223607	−	0.387298i
$5$	2.50000	−	4.33013i	0.223607	−	0.387298i
$6$		0			0
$7$	9.81683	+	17.0033i	0.530059	+	0.918089i	0.999385	+	0.0350644i	$0.0111636\pi$
$7$	9.81683	+	17.0033i	0.530059	+	0.918089i	−0.469326	+	0.883025i	$0.655503\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−14.5783	−	25.2504i	−0.399594	−	0.692117i	0.594082	−	0.804404i	$-0.297515\pi$
$11$	−14.5783	−	25.2504i	−0.399594	−	0.692117i	−0.993676	+	0.112288i	$0.964182\pi$
$12$		0			0
$13$	6.81683	−	11.8071i	0.145435	−	0.251900i	−0.784100	−	0.620634i	$-0.786875\pi$
$13$	6.81683	−	11.8071i	0.145435	−	0.251900i	0.929535	+	0.368734i	$0.120209\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−39.0460			−0.557061			−0.278531	−	0.960427i	$-0.589847\pi$
$17$	−39.0460			−0.557061			−0.278531	+	0.960427i	$0.589847\pi$
$18$		0			0
$19$	43.7443			0.528192			0.264096	−	0.964496i	$-0.414926\pi$
$19$	43.7443			0.528192			0.264096	+	0.964496i	$0.414926\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−79.6072	+	137.884i	−0.721706	+	1.25003i	0.238610	+	0.971116i	$0.423308\pi$
$23$	−79.6072	+	137.884i	−0.721706	+	1.25003i	−0.960316	+	0.278916i	$0.910025\pi$
$24$		0			0
$25$	−12.5000	−	21.6506i	−0.100000	−	0.173205i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	12.5725	+	21.7762i	0.0805052	+	0.139439i	0.903467	−	0.428658i	$-0.141013\pi$
$29$	12.5725	+	21.7762i	0.0805052	+	0.139439i	−0.822962	+	0.568097i	$0.807680\pi$
$30$		0			0
$31$	58.3686	−	101.097i	0.338172	−	0.585730i	−0.645917	−	0.763407i	$-0.723525\pi$
$31$	58.3686	−	101.097i	0.338172	−	0.585730i	0.984089	+	0.177677i	$0.0568583\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	98.1683			0.474099
$36$		0			0
$37$	329.383			1.46352			0.731759	−	0.681563i	$-0.238700\pi$
$37$	329.383			1.46352			0.731759	+	0.681563i	$0.238700\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−90.4010	+	156.579i	−0.344348	+	0.596428i	−0.985235	−	0.171207i	$-0.945233\pi$
$41$	−90.4010	+	156.579i	−0.344348	+	0.596428i	0.640887	+	0.767635i	$0.278567\pi$
$42$		0			0
$43$	106.836	+	185.046i	0.378893	+	0.656261i	0.990901	−	0.134590i	$-0.0429716\pi$
$43$	106.836	+	185.046i	0.378893	+	0.656261i	−0.612009	+	0.790851i	$0.709638\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	89.7062	+	155.376i	0.278404	+	0.482210i	0.970988	−	0.239127i	$-0.0768613\pi$
$47$	89.7062	+	155.376i	0.278404	+	0.482210i	−0.692584	+	0.721337i	$0.743528\pi$
$48$		0			0
$49$	−21.2404	+	36.7895i	−0.0619255	+	0.107258i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	60.8635			0.157740			0.0788702	−	0.996885i	$-0.474869\pi$
$53$	60.8635			0.157740			0.0788702	+	0.996885i	$0.474869\pi$
$54$		0			0
$55$	−145.783			−0.357407
$56$		0			0
$57$		0			0
$58$		0			0
$59$	191.289	−	331.322i	0.422097	−	0.731093i	−0.574047	−	0.818822i	$-0.694628\pi$
$59$	191.289	−	331.322i	0.422097	−	0.731093i	0.996144	+	0.0877287i	$0.0279609\pi$
$60$		0			0
$61$	−216.841	−	375.579i	−0.455142	−	0.788328i	0.543555	−	0.839374i	$-0.317078\pi$
$61$	−216.841	−	375.579i	−0.455142	−	0.788328i	−0.998696	+	0.0510454i	$0.983745\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−34.0842	−	59.0355i	−0.0650403	−	0.112653i
$66$		0			0
$67$	−62.5534	+	108.346i	−0.114061	+	0.197560i	−0.917404	−	0.397957i	$-0.869719\pi$
$67$	−62.5534	+	108.346i	−0.114061	+	0.197560i	0.803343	+	0.595517i	$0.203053\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−30.0873			−0.0502917			−0.0251458	−	0.999684i	$-0.508005\pi$
$71$	−30.0873			−0.0502917			−0.0251458	+	0.999684i	$0.508005\pi$
$72$		0			0
$73$	676.299			1.08431			0.542156	−	0.840278i	$-0.317608\pi$
$73$	676.299			1.08431			0.542156	+	0.840278i	$0.317608\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	286.226	−	495.758i	0.423617	−	0.733725i
$78$		0			0
$79$	416.924	+	722.134i	0.593768	+	1.02844i	0.993719	+	0.111900i	$0.0356936\pi$
$79$	416.924	+	722.134i	0.593768	+	1.02844i	−0.399952	+	0.916536i	$0.630973\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	176.729	+	306.103i	0.233717	+	0.404810i	0.958899	−	0.283748i	$-0.0915778\pi$
$83$	176.729	+	306.103i	0.233717	+	0.404810i	−0.725182	+	0.688557i	$0.758244\pi$
$84$		0			0
$85$	−97.6149	+	169.074i	−0.124563	+	0.215749i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−948.224			−1.12934			−0.564671	−	0.825316i	$-0.690997\pi$
$89$	−948.224			−1.12934			−0.564671	+	0.825316i	$0.690997\pi$
$90$		0			0
$91$	267.679			0.308356
$92$		0			0
$93$		0			0
$94$		0			0
$95$	109.361	−	189.419i	0.118107	−	0.204568i
$96$		0			0
$97$	142.919	+	247.543i	0.149600	+	0.259115i	0.931080	−	0.364816i	$-0.118868\pi$
$97$	142.919	+	247.543i	0.149600	+	0.259115i	−0.781480	+	0.623931i	$0.785535\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	737.354	+	1277.13i	0.726430	+	1.25821i	0.958383	+	0.285487i	$0.0921554\pi$
$101$	737.354	+	1277.13i	0.726430	+	1.25821i	−0.231952	+	0.972727i	$0.574511\pi$
$102$		0			0
$103$	−877.649	+	1520.13i	−0.839586	+	1.45421i	0.0506558	+	0.998716i	$0.483869\pi$
$103$	−877.649	+	1520.13i	−0.839586	+	1.45421i	−0.890241	+	0.455489i	$0.849464\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−657.811			−0.594327			−0.297164	−	0.954827i	$-0.596041\pi$
$107$	−657.811			−0.594327			−0.297164	+	0.954827i	$0.596041\pi$
$108$		0			0
$109$	−1042.07			−0.915706			−0.457853	−	0.889028i	$-0.651381\pi$
$109$	−1042.07			−0.915706			−0.457853	+	0.889028i	$0.651381\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−281.763	+	488.028i	−0.234567	+	0.406281i	−0.959147	−	0.282909i	$-0.908700\pi$
$113$	−281.763	+	488.028i	−0.234567	+	0.406281i	0.724580	+	0.689191i	$0.242034\pi$
$114$		0			0
$115$	398.036	+	689.418i	0.322757	+	0.559031i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−383.308	−	663.909i	−0.295275	−	0.511432i
$120$		0			0
$121$	240.445	−	416.463i	0.180650	−	0.312895i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−125.000			−0.0894427
$126$		0			0
$127$	202.776			0.141681			0.0708406	−	0.997488i	$-0.477432\pi$
$127$	202.776			0.141681			0.0708406	+	0.997488i	$0.477432\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	555.945	−	962.926i	0.370788	−	0.642223i	−0.618899	−	0.785470i	$-0.712421\pi$
$131$	555.945	−	962.926i	0.370788	−	0.642223i	0.989687	+	0.143247i	$0.0457545\pi$
$132$		0			0
$133$	429.431	+	743.796i	0.279973	+	0.484927i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	272.703	+	472.335i	0.170063	+	0.294557i	0.938442	−	0.345438i	$-0.112270\pi$
$137$	272.703	+	472.335i	0.170063	+	0.294557i	−0.768379	+	0.639995i	$0.778936\pi$
$138$		0			0
$139$	678.287	−	1174.83i	0.413896	−	0.716889i	−0.581416	−	0.813606i	$-0.697501\pi$
$139$	678.287	−	1174.83i	0.413896	−	0.716889i	0.995312	+	0.0967178i	$0.0308344\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−397.512			−0.232459
$144$		0			0
$145$	125.725			0.0720061
$146$		0			0
$147$		0			0
$148$		0			0
$149$	1736.30	−	3007.36i	0.954653	−	1.65351i	0.219494	−	0.975614i	$-0.429559\pi$
$149$	1736.30	−	3007.36i	0.954653	−	1.65351i	0.735159	−	0.677894i	$-0.237107\pi$
$150$		0			0
$151$	1701.70	+	2947.43i	0.917101	+	1.58847i	0.803796	+	0.594905i	$0.202810\pi$
$151$	1701.70	+	2947.43i	0.917101	+	1.58847i	0.113305	+	0.993560i	$0.463856\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−291.843	−	505.487i	−0.151235	−	0.261947i
$156$		0			0
$157$	181.097	−	313.669i	0.0920579	−	0.159449i	−0.816319	−	0.577601i	$-0.803989\pi$
$157$	181.097	−	313.669i	0.0920579	−	0.159449i	0.908377	+	0.418152i	$0.137322\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−3125.96			−1.53019
$162$		0			0
$163$	1830.32			0.879520			0.439760	−	0.898115i	$-0.355063\pi$
$163$	1830.32			0.879520			0.439760	+	0.898115i	$0.355063\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−417.759	+	723.580i	−0.193576	+	0.335283i	−0.946433	−	0.322901i	$-0.895342\pi$
$167$	−417.759	+	723.580i	−0.193576	+	0.335283i	0.752857	+	0.658184i	$0.228675\pi$
$168$		0			0
$169$	1005.56	+	1741.68i	0.457698	+	0.792755i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	859.902	+	1489.39i	0.377903	+	0.654547i	0.990757	−	0.135649i	$-0.0433120\pi$
$173$	859.902	+	1489.39i	0.377903	+	0.654547i	−0.612854	+	0.790196i	$0.709979\pi$
$174$		0			0
$175$	245.421	−	425.081i	0.106012	−	0.183618i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	2291.42			0.956809			0.478404	−	0.878140i	$-0.341215\pi$
$179$	2291.42			0.956809			0.478404	+	0.878140i	$0.341215\pi$
$180$		0			0
$181$	4228.30			1.73639			0.868197	−	0.496221i	$-0.165279\pi$
$181$	4228.30			1.73639			0.868197	+	0.496221i	$0.165279\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	823.457	−	1426.27i	0.327253	−	0.566818i
$186$		0			0
$187$	569.225	+	985.926i	0.222598	+	0.385551i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	1023.22	+	1772.27i	0.387632	+	0.671398i	0.992131	−	0.125208i	$-0.0399598\pi$
$191$	1023.22	+	1772.27i	0.387632	+	0.671398i	−0.604499	+	0.796606i	$0.706627\pi$
$192$		0			0
$193$	−548.462	+	949.964i	−0.204555	+	0.354300i	−0.949991	−	0.312277i	$-0.898908\pi$
$193$	−548.462	+	949.964i	−0.204555	+	0.354300i	0.745436	+	0.666578i	$0.232241\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−1593.40			−0.576269			−0.288135	−	0.957590i	$-0.593035\pi$
$197$	−1593.40			−0.576269			−0.288135	+	0.957590i	$0.593035\pi$
$198$		0			0
$199$	4936.25			1.75840			0.879201	−	0.476452i	$-0.158077\pi$
$199$	4936.25			1.75840			0.879201	+	0.476452i	$0.158077\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−246.844	+	427.546i	−0.0853451	+	0.147822i
$204$		0			0
$205$	452.005	+	782.896i	0.153997	+	0.266731i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−637.719	−	1104.56i	−0.211062	−	0.365570i
$210$		0			0
$211$	−664.709	+	1151.31i	−0.216874	+	0.375637i	−0.953851	−	0.300281i	$-0.902919\pi$
$211$	−664.709	+	1151.31i	−0.216874	+	0.375637i	0.736977	+	0.675918i	$0.236253\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	1068.36			0.338892
$216$		0			0
$217$	2291.98			0.717004
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−266.170	+	461.020i	−0.0810159	+	0.140324i
$222$		0			0
$223$	512.410	+	887.520i	0.153872	+	0.266515i	0.932648	−	0.360788i	$-0.117492\pi$
$223$	512.410	+	887.520i	0.153872	+	0.266515i	−0.778776	+	0.627303i	$0.784159\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	266.844	+	462.188i	0.0780224	+	0.135139i	0.902397	−	0.430906i	$-0.141806\pi$
$227$	266.844	+	462.188i	0.0780224	+	0.135139i	−0.824374	+	0.566045i	$0.808473\pi$
$228$		0			0
$229$	25.5468	−	44.2484i	0.00737198	−	0.0127686i	−0.862316	−	0.506371i	$-0.830987\pi$
$229$	25.5468	−	44.2484i	0.00737198	−	0.0127686i	0.869688	+	0.493602i	$0.164320\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−509.227			−0.143178			−0.0715892	−	0.997434i	$-0.522807\pi$
$233$	−509.227			−0.143178			−0.0715892	+	0.997434i	$0.522807\pi$
$234$		0			0
$235$	897.062			0.249012
$236$		0			0
$237$		0			0
$238$		0			0
$239$	1392.38	−	2411.67i	0.376843	−	0.652711i	−0.613758	−	0.789494i	$-0.710343\pi$
$239$	1392.38	−	2411.67i	0.376843	−	0.652711i	0.990601	+	0.136783i	$0.0436763\pi$
$240$		0			0
$241$	41.5097	+	71.8969i	0.0110949	+	0.0192170i	0.871520	−	0.490361i	$-0.163135\pi$
$241$	41.5097	+	71.8969i	0.0110949	+	0.0192170i	−0.860425	+	0.509578i	$0.829802\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	106.202	+	183.948i	0.0276939	+	0.0479673i
$246$		0			0
$247$	298.198	−	516.494i	0.0768173	−	0.133052i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−1286.35			−0.323480			−0.161740	−	0.986833i	$-0.551711\pi$
$251$	−1286.35			−0.323480			−0.161740	+	0.986833i	$0.551711\pi$
$252$		0			0
$253$	4642.16			1.15356
$254$		0			0
$255$		0			0
$256$		0			0
$257$	772.063	−	1337.25i	0.187393	−	0.324574i	−0.756987	−	0.653429i	$-0.773330\pi$
$257$	772.063	−	1337.25i	0.187393	−	0.324574i	0.944380	+	0.328856i	$0.106663\pi$
$258$		0			0
$259$	3233.49	+	5600.58i	0.775751	+	1.34364i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	4161.79	+	7208.43i	0.975767	+	1.69008i	0.677380	+	0.735634i	$0.263116\pi$
$263$	4161.79	+	7208.43i	0.975767	+	1.69008i	0.298388	+	0.954445i	$0.403551\pi$
$264$		0			0
$265$	152.159	−	263.547i	0.0352718	−	0.0610926i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−3203.80			−0.726168			−0.363084	−	0.931756i	$-0.618276\pi$
$269$	−3203.80			−0.726168			−0.363084	+	0.931756i	$0.618276\pi$
$270$		0			0
$271$	−1419.39			−0.318161			−0.159081	−	0.987266i	$-0.550853\pi$
$271$	−1419.39			−0.318161			−0.159081	+	0.987266i	$0.550853\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−364.458	+	631.260i	−0.0799187	+	0.138423i
$276$		0			0
$277$	4438.45	+	7687.62i	0.962745	+	1.66752i	0.715555	+	0.698557i	$0.246174\pi$
$277$	4438.45	+	7687.62i	0.962745	+	1.66752i	0.247191	+	0.968967i	$0.420493\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−1303.27	−	2257.33i	−0.276679	−	0.479221i	0.693879	−	0.720092i	$-0.255900\pi$
$281$	−1303.27	−	2257.33i	−0.276679	−	0.479221i	−0.970557	+	0.240871i	$0.922567\pi$
$282$		0			0
$283$	−703.389	+	1218.31i	−0.147746	+	0.255904i	−0.930394	−	0.366561i	$-0.880535\pi$
$283$	−703.389	+	1218.31i	−0.147746	+	0.255904i	0.782648	+	0.622465i	$0.213869\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−3549.81			−0.730099
$288$		0			0
$289$	−3388.41			−0.689683
$290$		0			0
$291$		0			0
$292$		0			0
$293$	3585.00	−	6209.41i	0.714806	−	1.23808i	−0.248229	−	0.968701i	$-0.579848\pi$
$293$	3585.00	−	6209.41i	0.714806	−	1.23808i	0.963034	−	0.269378i	$-0.0868183\pi$
$294$		0			0
$295$	−956.445	−	1656.61i	−0.188767	−	0.326955i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	1085.34	+	1879.86i	0.209922	+	0.363596i
$300$		0			0
$301$	−2097.59	+	3633.13i	−0.401671	+	0.695715i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−2168.41			−0.407091
$306$		0			0
$307$	3695.22			0.686963			0.343481	−	0.939160i	$-0.388394\pi$
$307$	3695.22			0.686963			0.343481	+	0.939160i	$0.388394\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−3042.83	+	5270.34i	−0.554801	+	0.960944i	0.443118	+	0.896463i	$0.353872\pi$
$311$	−3042.83	+	5270.34i	−0.554801	+	0.960944i	−0.997919	+	0.0644805i	$0.979461\pi$
$312$		0			0
$313$	−887.284	−	1536.82i	−0.160231	−	0.277528i	0.774721	−	0.632304i	$-0.217891\pi$
$313$	−887.284	−	1536.82i	−0.160231	−	0.277528i	−0.934951	+	0.354776i	$0.884557\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	4176.76	+	7234.37i	0.740033	+	1.28177i	0.952480	+	0.304602i	$0.0985234\pi$
$317$	4176.76	+	7234.37i	0.740033	+	1.28177i	−0.212447	+	0.977173i	$0.568143\pi$
$318$		0			0
$319$	366.572	−	634.921i	0.0643388	−	0.111438i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−1708.04			−0.294235
$324$		0			0
$325$	−340.842			−0.0581738
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−1761.26	+	3050.59i	−0.295141	+	0.511199i
$330$		0			0
$331$	1892.90	+	3278.60i	0.314330	+	0.544436i	0.979295	−	0.202439i	$-0.0648867\pi$
$331$	1892.90	+	3278.60i	0.314330	+	0.544436i	−0.664965	+	0.746875i	$0.731553\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	312.767	+	541.728i	0.0510098	+	0.0883516i
$336$		0			0
$337$	−3731.54	+	6463.22i	−0.603176	+	1.04473i	0.389161	+	0.921170i	$0.372765\pi$
$337$	−3731.54	+	6463.22i	−0.603176	+	1.04473i	−0.992337	+	0.123561i	$0.960569\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−3403.67			−0.540525
$342$		0			0
$343$	5900.29			0.928822
$344$		0			0
$345$		0			0
$346$		0			0
$347$	1710.53	−	2962.73i	0.264629	−	0.458350i	−0.702838	−	0.711350i	$-0.748084\pi$
$347$	1710.53	−	2962.73i	0.264629	−	0.458350i	0.967466	+	0.253000i	$0.0814173\pi$
$348$		0			0
$349$	1300.41	+	2252.37i	0.199453	+	0.345463i	0.948351	−	0.317222i	$-0.102750\pi$
$349$	1300.41	+	2252.37i	0.199453	+	0.345463i	−0.748898	+	0.662685i	$0.769417\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−1108.89	−	1920.65i	−0.167196	−	0.289592i	0.770237	−	0.637758i	$-0.220138\pi$
$353$	−1108.89	−	1920.65i	−0.167196	−	0.289592i	−0.937433	+	0.348166i	$0.886805\pi$
$354$		0			0
$355$	−75.2183	+	130.282i	−0.0112456	+	0.0194779i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−10423.6			−1.53241			−0.766207	−	0.642594i	$-0.777858\pi$
$359$	−10423.6			−1.53241			−0.766207	+	0.642594i	$0.777858\pi$
$360$		0			0
$361$	−4945.43			−0.721014
$362$		0			0
$363$		0			0
$364$		0			0
$365$	1690.75	−	2928.46i	0.242460	−	0.419953i
$366$		0			0
$367$	−15.9681	−	27.6576i	−0.00227120	−	0.00393383i	0.864888	−	0.501966i	$-0.167390\pi$
$367$	−15.9681	−	27.6576i	−0.00227120	−	0.00393383i	−0.867159	+	0.498032i	$0.834056\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	597.487	+	1034.88i	0.0836118	+	0.144820i
$372$		0			0
$373$	−116.674	+	202.086i	−0.0161961	+	0.0280526i	−0.874010	−	0.485908i	$-0.838489\pi$
$373$	−116.674	+	202.086i	−0.0161961	+	0.0280526i	0.857814	+	0.513961i	$0.171822\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	342.818			0.0468330
$378$		0			0
$379$	5558.90			0.753408			0.376704	−	0.926334i	$-0.377057\pi$
$379$	5558.90			0.753408			0.376704	+	0.926334i	$0.377057\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	4082.12	−	7070.44i	0.544613	−	0.943297i	−0.454019	−	0.890992i	$-0.650010\pi$
$383$	4082.12	−	7070.44i	0.544613	−	0.943297i	0.998631	−	0.0523045i	$-0.0166566\pi$
$384$		0			0
$385$	−1431.13	−	2478.79i	−0.189447	−	0.328132i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	5098.55	+	8830.95i	0.664542	+	1.15102i	0.979409	+	0.201884i	$0.0647066\pi$
$389$	5098.55	+	8830.95i	0.664542	+	1.15102i	−0.314868	+	0.949136i	$0.601960\pi$
$390$		0			0
$391$	3108.34	−	5383.80i	0.402034	−	0.696344i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	4169.24			0.531082
$396$		0			0
$397$	−2601.86			−0.328926			−0.164463	−	0.986383i	$-0.552589\pi$
$397$	−2601.86			−0.328926			−0.164463	+	0.986383i	$0.552589\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	2592.64	−	4490.58i	0.322868	−	0.559224i	−0.658211	−	0.752834i	$-0.728686\pi$
$401$	2592.64	−	4490.58i	0.322868	−	0.559224i	0.981079	+	0.193610i	$0.0620197\pi$
$402$		0			0
$403$	−795.779	−	1378.33i	−0.0983637	−	0.170371i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−4801.85	−	8317.04i	−0.584813	−	1.01293i
$408$		0			0
$409$	−3591.73	+	6221.06i	−0.434229	+	0.752107i	−0.997232	−	0.0743476i	$-0.976313\pi$
$409$	−3591.73	+	6221.06i	−0.434229	+	0.752107i	0.563003	+	0.826455i	$0.309646\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	7511.41			0.894945
$414$		0			0
$415$	1767.29			0.209043
$416$		0			0
$417$		0			0
$418$		0			0
$419$	341.614	−	591.693i	0.0398304	−	0.0689883i	−0.845423	−	0.534097i	$-0.820652\pi$
$419$	341.614	−	591.693i	0.0398304	−	0.0689883i	0.885253	+	0.465109i	$0.153985\pi$
$420$		0			0
$421$	5372.54	+	9305.51i	0.621951	+	1.07725i	0.989122	+	0.147098i	$0.0469932\pi$
$421$	5372.54	+	9305.51i	0.621951	+	1.07725i	−0.367171	+	0.930154i	$0.619674\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	488.075	+	845.370i	0.0557061	+	0.0964858i
$426$		0			0
$427$	4257.38	−	7374.00i	0.482504	−	0.835721i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	1055.60			0.117974			0.0589869	−	0.998259i	$-0.481213\pi$
$431$	1055.60			0.117974			0.0589869	+	0.998259i	$0.481213\pi$
$432$		0			0
$433$	−4536.95			−0.503538			−0.251769	−	0.967787i	$-0.581012\pi$
$433$	−4536.95			−0.503538			−0.251769	+	0.967787i	$0.581012\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−3482.36	+	6031.63i	−0.381199	+	0.660256i
$438$		0			0
$439$	−3783.93	−	6553.95i	−0.411383	−	0.712536i	0.583659	−	0.811999i	$-0.301621\pi$
$439$	−3783.93	−	6553.95i	−0.411383	−	0.712536i	−0.995041	+	0.0994636i	$0.968287\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−5172.57	−	8959.15i	−0.554754	−	0.960863i	−0.997923	−	0.0644244i	$-0.979479\pi$
$443$	−5172.57	−	8959.15i	−0.554754	−	0.960863i	0.443168	−	0.896439i	$-0.353854\pi$
$444$		0			0
$445$	−2370.56	+	4105.93i	−0.252529	+	0.437393i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	11122.4			1.16904			0.584520	−	0.811379i	$-0.301283\pi$
$449$	11122.4			1.16904			0.584520	+	0.811379i	$0.301283\pi$
$450$		0			0
$451$	5271.58			0.550397
$452$		0			0
$453$		0			0
$454$		0			0
$455$	669.197	−	1159.08i	0.0689504	−	0.119426i
$456$		0			0
$457$	869.224	+	1505.54i	0.0889728	+	0.154105i	0.907077	−	0.420964i	$-0.138308\pi$
$457$	869.224	+	1505.54i	0.0889728	+	0.154105i	−0.818104	+	0.575070i	$0.804975\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−3786.38	−	6558.20i	−0.382536	−	0.662572i	0.608888	−	0.793256i	$-0.291616\pi$
$461$	−3786.38	−	6558.20i	−0.382536	−	0.662572i	−0.991424	+	0.130684i	$0.958283\pi$
$462$		0			0
$463$	6358.96	−	11014.0i	0.638285	−	1.10554i	−0.347525	−	0.937671i	$-0.612978\pi$
$463$	6358.96	−	11014.0i	0.638285	−	1.10554i	0.985809	−	0.167870i	$-0.0536890\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	4641.96			0.459967			0.229983	−	0.973195i	$-0.426133\pi$
$467$	4641.96			0.459967			0.229983	+	0.973195i	$0.426133\pi$
$468$		0			0
$469$	−2456.31			−0.241837
$470$		0			0
$471$		0			0
$472$		0			0
$473$	3114.99	−	5395.32i	0.302806	−	0.524476i
$474$		0			0
$475$	−546.804	−	947.093i	−0.0528192	−	0.0914855i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−2817.98	−	4880.89i	−0.268804	−	0.465581i	0.699750	−	0.714388i	$-0.253295\pi$
$479$	−2817.98	−	4880.89i	−0.268804	−	0.465581i	−0.968553	+	0.248807i	$0.919962\pi$
$480$		0			0
$481$	2245.35	−	3889.05i	0.212846	−	0.368660i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	1429.19			0.133807
$486$		0			0
$487$	−1557.90			−0.144960			−0.0724799	−	0.997370i	$-0.523091\pi$
$487$	−1557.90			−0.144960			−0.0724799	+	0.997370i	$0.523091\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−3374.51	+	5844.83i	−0.310162	+	0.537217i	−0.978397	−	0.206733i	$-0.933717\pi$
$491$	−3374.51	+	5844.83i	−0.310162	+	0.537217i	0.668235	+	0.743950i	$0.267050\pi$
$492$		0			0
$493$	−490.905	−	850.272i	−0.0448463	−	0.0776761i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−295.362	−	511.582i	−0.0266576	−	0.0461722i
$498$		0			0
$499$	−1114.74	+	1930.79i	−0.100005	+	0.173214i	−0.911687	−	0.410886i	$-0.865219\pi$
$499$	−1114.74	+	1930.79i	−0.100005	+	0.173214i	0.811681	+	0.584101i	$0.198553\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	13685.7			1.21315			0.606577	−	0.795024i	$-0.292542\pi$
$503$	13685.7			1.21315			0.606577	+	0.795024i	$0.292542\pi$
$504$		0			0
$505$	7373.54			0.649739
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−6813.72	+	11801.7i	−0.593346	+	1.02770i	0.400432	+	0.916326i	$0.368860\pi$
$509$	−6813.72	+	11801.7i	−0.593346	+	1.02770i	−0.993778	+	0.111379i	$0.964473\pi$
$510$		0			0
$511$	6639.12	+	11499.3i	0.574750	+	0.995496i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	4388.24	+	7600.66i	0.375474	+	0.650340i
$516$		0			0
$517$	2615.53	−	4530.23i	0.222497	−	0.385376i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−21425.1			−1.80163			−0.900817	−	0.434200i	$-0.857031\pi$
$521$	−21425.1			−1.80163			−0.900817	+	0.434200i	$0.857031\pi$
$522$		0			0
$523$	8954.62			0.748677			0.374338	−	0.927292i	$-0.377870\pi$
$523$	8954.62			0.748677			0.374338	+	0.927292i	$0.377870\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−2279.06	+	3947.45i	−0.188382	+	0.326288i
$528$		0			0
$529$	−6591.10	−	11416.1i	−0.541719	−	0.938285i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	1232.50	+	2134.75i	0.100160	+	0.173483i
$534$		0			0
$535$	−1644.53	+	2848.41i	−0.132896	+	0.230182i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	1238.60			0.0989801
$540$		0			0
$541$	−10982.6			−0.872788			−0.436394	−	0.899756i	$-0.643745\pi$
$541$	−10982.6			−0.872788			−0.436394	+	0.899756i	$0.643745\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−2605.17	+	4512.28i	−0.204758	+	0.354651i
$546$		0			0
$547$	−7606.61	−	13175.0i	−0.594580	−	1.02984i	−0.993606	−	0.112903i	$-0.963985\pi$
$547$	−7606.61	−	13175.0i	−0.594580	−	1.02984i	0.399026	−	0.916940i	$-0.369348\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	549.975	+	952.585i	0.0425222	+	0.0736506i
$552$		0			0
$553$	−8185.75	+	14178.1i	−0.629464	+	1.09026i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−12732.7			−0.968588			−0.484294	−	0.874905i	$-0.660923\pi$
$557$	−12732.7			−0.968588			−0.484294	+	0.874905i	$0.660923\pi$
$558$		0			0
$559$	2913.14			0.220416
$560$		0			0
$561$		0			0
$562$		0			0
$563$	5444.61	−	9430.34i	0.407572	−	0.705935i	−0.587045	−	0.809554i	$-0.699709\pi$
$563$	5444.61	−	9430.34i	0.407572	−	0.705935i	0.994617	+	0.103619i	$0.0330422\pi$
$564$		0			0
$565$	1408.81	+	2440.14i	0.104901	+	0.181694i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−784.009	−	1357.94i	−0.0577634	−	0.100049i	0.835698	−	0.549190i	$-0.185064\pi$
$569$	−784.009	−	1357.94i	−0.0577634	−	0.100049i	−0.893461	+	0.449141i	$0.851730\pi$
$570$		0			0
$571$	12209.1	−	21146.8i	0.894809	−	1.54986i	0.0607693	−	0.998152i	$-0.480645\pi$
$571$	12209.1	−	21146.8i	0.894809	−	1.54986i	0.834040	−	0.551704i	$-0.186022\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	3980.36			0.288682
$576$		0			0
$577$	−21305.8			−1.53721			−0.768606	−	0.639722i	$-0.779049\pi$
$577$	−21305.8			−1.53721			−0.768606	+	0.639722i	$0.779049\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−3469.83	+	6009.93i	−0.247768	+	0.429146i
$582$		0			0
$583$	−887.288	−	1536.83i	−0.0630321	−	0.109175i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−12731.5	−	22051.5i	−0.895202	−	1.55054i	−0.833554	−	0.552437i	$-0.813698\pi$
$587$	−12731.5	−	22051.5i	−0.895202	−	1.55054i	−0.0616476	−	0.998098i	$-0.519635\pi$
$588$		0			0
$589$	2553.30	−	4422.44i	0.178619	−	0.309378i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	26914.5			1.86382			0.931911	−	0.362686i	$-0.118140\pi$
$593$	26914.5			1.86382			0.931911	+	0.362686i	$0.118140\pi$
$594$		0			0
$595$	−3833.08			−0.264102
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−10476.2	+	18145.4i	−0.714604	+	1.23773i	0.248509	+	0.968630i	$0.420060\pi$
$599$	−10476.2	+	18145.4i	−0.714604	+	1.23773i	−0.963112	+	0.269100i	$0.913274\pi$
$600$		0			0
$601$	−12895.9	−	22336.3i	−0.875265	−	1.51600i	−0.856480	−	0.516180i	$-0.827354\pi$
$601$	−12895.9	−	22336.3i	−0.875265	−	1.51600i	−0.0187845	−	0.999824i	$-0.505980\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−1202.22	−	2082.31i	−0.0807890	−	0.139931i
$606$		0			0
$607$	3268.48	−	5661.18i	0.218556	−	0.378551i	−0.735810	−	0.677188i	$-0.763199\pi$
$607$	3268.48	−	5661.18i	0.218556	−	0.378551i	0.954367	+	0.298637i	$0.0965319\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	2446.05			0.161958
$612$		0			0
$613$	−22102.9			−1.45633			−0.728163	−	0.685404i	$-0.759626\pi$
$613$	−22102.9			−1.45633			−0.728163	+	0.685404i	$0.759626\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−1393.97	+	2414.43i	−0.0909549	+	0.157539i	−0.907913	−	0.419158i	$-0.862325\pi$
$617$	−1393.97	+	2414.43i	−0.0909549	+	0.157539i	0.816958	+	0.576697i	$0.195659\pi$
$618$		0			0
$619$	−3901.30	−	6757.24i	−0.253322	−	0.438767i	0.711116	−	0.703074i	$-0.248190\pi$
$619$	−3901.30	−	6757.24i	−0.253322	−	0.438767i	−0.964438	+	0.264308i	$0.914857\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−9308.55	−	16122.9i	−0.598618	−	1.03684i
$624$		0			0
$625$	−312.500	+	541.266i	−0.0200000	+	0.0346410i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−12861.1			−0.815269
$630$		0			0
$631$	−18267.8			−1.15251			−0.576253	−	0.817272i	$-0.695486\pi$
$631$	−18267.8			−1.15251			−0.576253	+	0.817272i	$0.695486\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	506.941	−	878.048i	0.0316809	−	0.0548729i
$636$		0			0
$637$	289.585	+	501.576i	0.0180122	+	0.0311981i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−1804.48	−	3125.46i	−0.111190	−	0.192587i	0.805060	−	0.593193i	$-0.202133\pi$
$641$	−1804.48	−	3125.46i	−0.111190	−	0.192587i	−0.916250	+	0.400606i	$0.868800\pi$
$642$		0			0
$643$	−9925.67	+	17191.8i	−0.608756	+	1.05440i	0.382690	+	0.923877i	$0.374998\pi$
$643$	−9925.67	+	17191.8i	−0.608756	+	1.05440i	−0.991446	+	0.130520i	$0.958335\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−9352.11			−0.568268			−0.284134	−	0.958785i	$-0.591706\pi$
$647$	−9352.11			−0.568268			−0.284134	+	0.958785i	$0.591706\pi$
$648$		0			0
$649$	−11154.7			−0.674669
$650$		0			0
$651$		0			0
$652$		0			0
$653$	13108.3	−	22704.2i	0.785553	−	1.36062i	−0.143115	−	0.989706i	$-0.545712\pi$
$653$	13108.3	−	22704.2i	0.785553	−	1.36062i	0.928668	−	0.370912i	$-0.120955\pi$
$654$		0			0
$655$	−2779.73	−	4814.63i	−0.165821	−	0.287211i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	7691.89	+	13322.8i	0.454679	+	0.787528i	0.998670	−	0.0515638i	$-0.0164205\pi$
$659$	7691.89	+	13322.8i	0.454679	+	0.787528i	−0.543990	+	0.839091i	$0.683087\pi$
$660$		0			0
$661$	−6855.48	+	11874.0i	−0.403400	+	0.698709i	−0.994134	−	0.108157i	$-0.965505\pi$
$661$	−6855.48	+	11874.0i	−0.403400	+	0.698709i	0.590734	+	0.806866i	$0.298838\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	4294.31			0.250415
$666$		0			0
$667$	−4003.44			−0.232404
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−6322.35	+	10950.6i	−0.363743	+	0.630022i
$672$		0			0
$673$	−13898.4	−	24072.7i	−0.796054	−	1.37881i	−0.922168	−	0.386790i	$-0.873584\pi$
$673$	−13898.4	−	24072.7i	−0.796054	−	1.37881i	0.126114	−	0.992016i	$-0.459749\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	14296.7	+	24762.6i	0.811621	+	1.40577i	0.911729	+	0.410792i	$0.134748\pi$
$677$	14296.7	+	24762.6i	0.811621	+	1.40577i	−0.100108	+	0.994977i	$0.531919\pi$
$678$		0			0
$679$	−2806.02	+	4860.18i	−0.158594	+	0.274693i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	33510.5			1.87737			0.938686	−	0.344774i	$-0.112044\pi$
$683$	33510.5			1.87737			0.938686	+	0.344774i	$0.112044\pi$
$684$		0			0
$685$	2727.03			0.152109
$686$		0			0
$687$		0			0
$688$		0			0
$689$	414.896	−	718.622i	0.0229409	−	0.0397348i
$690$		0			0
$691$	−4784.99	−	8287.84i	−0.263429	−	0.456273i	0.703722	−	0.710476i	$-0.251520\pi$
$691$	−4784.99	−	8287.84i	−0.263429	−	0.456273i	−0.967151	+	0.254203i	$0.918187\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−3391.43	−	5874.13i	−0.185100	−	0.320602i
$696$		0			0
$697$	3529.79	−	6113.78i	0.191823	−	0.332247i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−6945.59			−0.374224			−0.187112	−	0.982339i	$-0.559913\pi$
$701$	−6945.59			−0.374224			−0.187112	+	0.982339i	$0.559913\pi$
$702$		0			0
$703$	14408.6			0.773018
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−14477.0	+	25074.8i	−0.770102	+	1.33386i
$708$		0			0
$709$	−2969.13	−	5142.68i	−0.157275	−	0.272408i	0.776610	−	0.629982i	$-0.216938\pi$
$709$	−2969.13	−	5142.68i	−0.157275	−	0.272408i	−0.933885	+	0.357573i	$0.883604\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	9293.12	+	16096.2i	0.488121	+	0.845450i
$714$		0			0
$715$	−993.780	+	1721.28i	−0.0519794	+	0.0900310i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	7669.30			0.397798			0.198899	−	0.980020i	$-0.436263\pi$
$719$	7669.30			0.397798			0.198899	+	0.980020i	$0.436263\pi$
$720$		0			0
$721$	−34462.9			−1.78012
$722$		0			0
$723$		0			0
$724$		0			0
$725$	314.312	−	544.405i	0.0161010	−	0.0278878i
$726$		0			0
$727$	−3929.73	−	6806.50i	−0.200476	−	0.347234i	0.748206	−	0.663466i	$-0.230915\pi$
$727$	−3929.73	−	6806.50i	−0.200476	−	0.347234i	−0.948682	+	0.316232i	$0.897582\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−4171.53	−	7225.30i	−0.211066	−	0.365578i
$732$		0			0
$733$	−15104.4	+	26161.6i	−0.761111	+	1.31828i	0.181167	+	0.983452i	$0.442013\pi$
$733$	−15104.4	+	26161.6i	−0.761111	+	1.31828i	−0.942278	+	0.334831i	$0.891321\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	3647.70			0.182313
$738$		0			0
$739$	6648.76			0.330959			0.165480	−	0.986213i	$-0.447083\pi$
$739$	6648.76			0.330959			0.165480	+	0.986213i	$0.447083\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−3823.34	+	6622.22i	−0.188782	+	0.326979i	−0.944844	−	0.327520i	$-0.893787\pi$
$743$	−3823.34	+	6622.22i	−0.188782	+	0.326979i	0.756063	+	0.654499i	$0.227121\pi$
$744$		0			0
$745$	−8681.51	−	15036.8i	−0.426934	−	0.739471i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−6457.62	−	11184.9i	−0.315029	−	0.545646i
$750$		0			0
$751$	15813.4	−	27389.7i	0.768363	−	1.33084i	−0.170088	−	0.985429i	$-0.554405\pi$
$751$	15813.4	−	27389.7i	0.768363	−	1.33084i	0.938450	−	0.345414i	$-0.112262\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	17017.0			0.820280
$756$		0			0
$757$	−6859.10			−0.329324			−0.164662	−	0.986350i	$-0.552653\pi$
$757$	−6859.10			−0.329324			−0.164662	+	0.986350i	$0.552653\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−8989.30	+	15569.9i	−0.428202	+	0.741668i	−0.996714	−	0.0810073i	$-0.974186\pi$
$761$	−8989.30	+	15569.9i	−0.428202	+	0.741668i	0.568511	+	0.822676i	$0.307520\pi$
$762$		0			0
$763$	−10229.8	−	17718.5i	−0.485378	−	0.840700i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−2607.97	−	4517.14i	−0.122775	−	0.212652i
$768$		0			0
$769$	7584.52	−	13136.8i	0.355663	−	0.616027i	−0.631568	−	0.775320i	$-0.717588\pi$
$769$	7584.52	−	13136.8i	0.355663	−	0.616027i	0.987231	+	0.159294i	$0.0509217\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	2921.73			0.135947			0.0679736	−	0.997687i	$-0.478347\pi$
$773$	2921.73			0.135947			0.0679736	+	0.997687i	$0.478347\pi$
$774$		0			0
$775$	−2918.43			−0.135269
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−3954.53	+	6849.45i	−0.181882	+	0.315028i
$780$		0			0
$781$	438.623	+	759.717i	0.0200962	+	0.0348077i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−905.483	−	1568.34i	−0.0411695	−	0.0713077i
$786$		0			0
$787$	−8027.38	+	13903.8i	−0.363590	+	0.629756i	−0.988549	−	0.150901i	$-0.951782\pi$
$787$	−8027.38	+	13903.8i	−0.363590	+	0.629756i	0.624959	+	0.780658i	$0.285116\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−11064.1			−0.497337
$792$		0			0
$793$	−5912.67			−0.264773
$794$		0			0
$795$		0			0
$796$		0			0
$797$	8717.49	−	15099.1i	0.387440	−	0.671065i	−0.604665	−	0.796480i	$-0.706693\pi$
$797$	8717.49	−	15099.1i	0.387440	−	0.671065i	0.992104	+	0.125415i	$0.0400262\pi$
$798$		0			0
$799$	−3502.66	−	6066.79i	−0.155088	−	0.268620i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−9859.31	−	17076.8i	−0.433285	−	0.750471i
$804$		0			0
$805$	−7814.90	+	13535.8i	−0.342160	+	0.592639i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	22185.0			0.964132			0.482066	−	0.876135i	$-0.339887\pi$
$809$	22185.0			0.964132			0.482066	+	0.876135i	$0.339887\pi$
$810$		0			0
$811$	−18738.5			−0.811341			−0.405671	−	0.914019i	$-0.632962\pi$
$811$	−18738.5			−0.811341			−0.405671	+	0.914019i	$0.632962\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	4575.80	−	7925.52i	0.196667	−	0.340637i
$816$		0			0
$817$	4673.48	+	8094.71i	0.200128	+	0.346632i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−8204.34	−	14210.3i	−0.348762	−	0.604073i	0.637268	−	0.770642i	$-0.280064\pi$
$821$	−8204.34	−	14210.3i	−0.348762	−	0.604073i	−0.986030	+	0.166569i	$0.946731\pi$
$822$		0			0
$823$	3908.12	−	6769.06i	0.165527	−	0.286701i	−0.771315	−	0.636453i	$-0.780401\pi$
$823$	3908.12	−	6769.06i	0.165527	−	0.286701i	0.936842	+	0.349752i	$0.113734\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−12444.2			−0.523248			−0.261624	−	0.965170i	$-0.584258\pi$
$827$	−12444.2			−0.523248			−0.261624	+	0.965170i	$0.584258\pi$
$828$		0			0
$829$	−21850.1			−0.915422			−0.457711	−	0.889101i	$-0.651331\pi$
$829$	−21850.1			−0.915422			−0.457711	+	0.889101i	$0.651331\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	829.353	−	1436.48i	0.0344963	−	0.0597493i
$834$		0			0
$835$	2088.79	+	3617.90i	0.0865697	+	0.149943i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−19445.7	−	33680.9i	−0.800166	−	1.38593i	−0.919506	−	0.393075i	$-0.871411\pi$
$839$	−19445.7	−	33680.9i	−0.800166	−	1.38593i	0.119340	−	0.992853i	$-0.461922\pi$
$840$		0			0
$841$	11878.4	−	20573.9i	0.487038	−	0.843574i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	10055.6			0.409377
$846$		0			0
$847$	9441.63			0.383020
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−26221.2	+	45416.5i	−1.05623	+	1.82944i
$852$		0			0
$853$	−14665.7	−	25401.8i	−0.588680	−	1.01962i	−0.994406	−	0.105629i	$-0.966314\pi$
$853$	−14665.7	−	25401.8i	−0.588680	−	1.01962i	0.405725	−	0.913995i	$-0.367019\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−7009.11	−	12140.1i	−0.279377	−	0.483896i	0.691853	−	0.722039i	$-0.256795\pi$
$857$	−7009.11	−	12140.1i	−0.279377	−	0.483896i	−0.971230	+	0.238143i	$0.923461\pi$
$858$		0			0
$859$	3908.31	−	6769.40i	0.155239	−	0.268881i	−0.777907	−	0.628379i	$-0.783719\pi$
$859$	3908.31	−	6769.40i	0.155239	−	0.268881i	0.933146	+	0.359498i	$0.117052\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	39764.9			1.56850			0.784248	−	0.620448i	$-0.213049\pi$
$863$	39764.9			1.56850			0.784248	+	0.620448i	$0.213049\pi$
$864$		0			0
$865$	8599.02			0.338006
$866$		0			0
$867$		0			0
$868$		0			0
$869$	12156.1	−	21055.0i	0.474532	−	0.821913i
$870$		0			0
$871$	852.832	+	1477.15i	0.0331769	+	0.0574642i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−1227.10	−	2125.41i	−0.0474099	−	0.0821164i
$876$		0			0
$877$	18359.0	−	31798.7i	0.706887	−	1.22436i	−0.259119	−	0.965845i	$-0.583432\pi$
$877$	18359.0	−	31798.7i	0.706887	−	1.22436i	0.966006	−	0.258519i	$-0.0832344\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−49326.5			−1.88633			−0.943163	−	0.332331i	$-0.892165\pi$
$881$	−49326.5			−1.88633			−0.943163	+	0.332331i	$0.892165\pi$
$882$		0			0
$883$	32093.1			1.22312			0.611562	−	0.791197i	$-0.290542\pi$
$883$	32093.1			1.22312			0.611562	+	0.791197i	$0.290542\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−10575.4	+	18317.2i	−0.400324	+	0.693382i	−0.993765	−	0.111496i	$-0.964436\pi$
$887$	−10575.4	+	18317.2i	−0.400324	+	0.693382i	0.593441	+	0.804878i	$0.297769\pi$
$888$		0			0
$889$	1990.62	+	3447.86i	0.0750994	+	0.130076i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	3924.14	+	6796.81i	0.147051	+	0.254699i
$894$		0			0
$895$	5728.55	−	9922.14i	0.213949	−	0.370570i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	2935.36			0.108898
$900$		0			0
$901$	−2376.47			−0.0878711
$902$		0			0
$903$		0			0
$904$		0			0
$905$	10570.8	−	18309.1i	0.388269	−	0.672502i
$906$		0			0
$907$	8791.29	+	15227.0i	0.321841	+	0.557445i	0.980868	−	0.194674i	$-0.0623649\pi$
$907$	8791.29	+	15227.0i	0.321841	+	0.557445i	−0.659027	+	0.752119i	$0.729032\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−6849.45	−	11863.6i	−0.249102	−	0.431458i	0.714175	−	0.699968i	$-0.246802\pi$
$911$	−6849.45	−	11863.6i	−0.249102	−	0.431458i	−0.963277	+	0.268510i	$0.913469\pi$
$912$		0			0
$913$	5152.82	−	8924.95i	0.186784	−	0.323519i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	21830.5			0.786157
$918$		0			0
$919$	−14632.7			−0.525231			−0.262616	−	0.964901i	$-0.584585\pi$
$919$	−14632.7			−0.525231			−0.262616	+	0.964901i	$0.584585\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−205.100	+	355.244i	−0.00731415	+	0.0126685i
$924$		0			0
$925$	−4117.28	−	7131.34i	−0.146352	−	0.253489i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	7531.24	+	13044.5i	0.265976	+	0.460685i	0.967819	−	0.251647i	$-0.0809723\pi$
$929$	7531.24	+	13044.5i	0.265976	+	0.460685i	−0.701843	+	0.712332i	$0.747639\pi$
$930$		0			0
$931$	−929.149	+	1609.33i	−0.0327085	+	0.0566528i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	5692.25			0.199098
$936$		0			0
$937$	−52963.9			−1.84659			−0.923296	−	0.384090i	$-0.874515\pi$
$937$	−52963.9			−1.84659			−0.923296	+	0.384090i	$0.874515\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−12431.0	+	21531.1i	−0.430647	+	0.745903i	−0.996929	−	0.0783086i	$-0.975048\pi$
$941$	−12431.0	+	21531.1i	−0.430647	+	0.745903i	0.566282	+	0.824212i	$0.308381\pi$
$942$		0			0
$943$	−14393.1	−	24929.6i	−0.497036	−	0.860891i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−14898.2	−	25804.4i	−0.511221	−	0.885461i	−0.999915	−	0.0130059i	$-0.995860\pi$
$947$	−14898.2	−	25804.4i	−0.511221	−	0.885461i	0.488694	−	0.872455i	$-0.337473\pi$
$948$		0			0
$949$	4610.22	−	7985.13i	0.157697	−	0.273138i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	10042.2			0.341341			0.170670	−	0.985328i	$-0.445407\pi$
$953$	10042.2			0.341341			0.170670	+	0.985328i	$0.445407\pi$
$954$		0			0
$955$	10232.2			0.346708
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−5354.16	+	9273.68i	−0.180287	+	0.312266i
$960$		0			0
$961$	8081.70	+	13997.9i	0.271280	+	0.469871i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	2742.31	+	4749.82i	0.0914799	+	0.158448i
$966$		0			0
$967$	12537.7	−	21716.0i	0.416945	−	0.722170i	−0.578686	−	0.815551i	$-0.696434\pi$
$967$	12537.7	−	21716.0i	0.416945	−	0.722170i	0.995630	+	0.0933811i	$0.0297675\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	6632.06			0.219189			0.109595	−	0.993976i	$-0.465045\pi$
$971$	6632.06			0.219189			0.109595	+	0.993976i	$0.465045\pi$
$972$		0			0
$973$	26634.5			0.877557
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−20484.0	+	35479.3i	−0.670768	+	1.16180i	0.306919	+	0.951736i	$0.400702\pi$
$977$	−20484.0	+	35479.3i	−0.670768	+	1.16180i	−0.977687	+	0.210069i	$0.932631\pi$
$978$		0			0
$979$	13823.5	+	23943.0i	0.451278	+	0.781637i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	28721.3	+	49746.8i	0.931910	+	1.61412i	0.780054	+	0.625713i	$0.215192\pi$
$983$	28721.3	+	49746.8i	0.931910	+	1.61412i	0.151856	+	0.988403i	$0.451475\pi$
$984$		0			0
$985$	−3983.50	+	6899.63i	−0.128858	+	0.223188i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−34019.7			−1.09380
$990$		0			0
$991$	34805.7			1.11568			0.557840	−	0.829948i	$-0.311630\pi$
$991$	34805.7			1.11568			0.557840	+	0.829948i	$0.311630\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	12340.6	−	21374.6i	0.393190	−	0.681026i
$996$		0			0
$997$	−28494.8	−	49354.4i	−0.905155	−	1.56777i	−0.820710	−	0.571346i	$-0.806422\pi$
$997$	−28494.8	−	49354.4i	−0.905155	−	1.56777i	−0.0844449	−	0.996428i	$-0.526912\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1620.4.i.u.1081.3		6
3.2	odd	2		1620.4.i.s.1081.3		6
9.2	odd	6		1620.4.i.s.541.3		6
9.4	even	3		1620.4.a.d.1.1	✓	3
9.5	odd	6		1620.4.a.f.1.1	yes	3
9.7	even	3	inner	1620.4.i.u.541.3		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1620.4.a.d.1.1	✓	3	9.4	even	3
1620.4.a.f.1.1	yes	3	9.5	odd	6
1620.4.i.s.541.3		6	9.2	odd	6
1620.4.i.s.1081.3		6	3.2	odd	2
1620.4.i.u.541.3		6	9.7	even	3	inner
1620.4.i.u.1081.3		6	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+(2.50000 - 4.33013i) q^{5} +(9.81683 + 17.0033i) q^{7} +O(q^{10})\)	\(q+(2.50000 - 4.33013i) q^{5} +(9.81683 + 17.0033i) q^{7} +(-14.5783 - 25.2504i) q^{11} +(6.81683 - 11.8071i) q^{13} -39.0460 q^{17} +43.7443 q^{19} +(-79.6072 + 137.884i) q^{23} +(-12.5000 - 21.6506i) q^{25} +(12.5725 + 21.7762i) q^{29} +(58.3686 - 101.097i) q^{31} +98.1683 q^{35} +329.383 q^{37} +(-90.4010 + 156.579i) q^{41} +(106.836 + 185.046i) q^{43} +(89.7062 + 155.376i) q^{47} +(-21.2404 + 36.7895i) q^{49} +60.8635 q^{53} -145.783 q^{55} +(191.289 - 331.322i) q^{59} +(-216.841 - 375.579i) q^{61} +(-34.0842 - 59.0355i) q^{65} +(-62.5534 + 108.346i) q^{67} -30.0873 q^{71} +676.299 q^{73} +(286.226 - 495.758i) q^{77} +(416.924 + 722.134i) q^{79} +(176.729 + 306.103i) q^{83} +(-97.6149 + 169.074i) q^{85} -948.224 q^{89} +267.679 q^{91} +(109.361 - 189.419i) q^{95} +(142.919 + 247.543i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 15 q^{5} - 15 q^{7}+O(q^{10}) \) 6 * q + 15 * q^5 - 15 * q^7	\( 6 q + 15 q^{5} - 15 q^{7} + 24 q^{11} - 33 q^{13} - 84 q^{17} + 42 q^{19} - 33 q^{23} - 75 q^{25} + 222 q^{29} - 132 q^{31} - 150 q^{35} + 348 q^{37} - 99 q^{41} + 120 q^{43} + 537 q^{47} - 492 q^{49} - 534 q^{53} + 240 q^{55} - 225 q^{59} + 480 q^{61} + 165 q^{65} - 12 q^{67} - 1140 q^{71} + 2124 q^{73} + 312 q^{77} + 1026 q^{79} + 702 q^{83} - 210 q^{85} - 2280 q^{89} + 3222 q^{91} + 105 q^{95} + 2556 q^{97}+O(q^{100}) \) 6 * q + 15 * q^5 - 15 * q^7 + 24 * q^11 - 33 * q^13 - 84 * q^17 + 42 * q^19 - 33 * q^23 - 75 * q^25 + 222 * q^29 - 132 * q^31 - 150 * q^35 + 348 * q^37 - 99 * q^41 + 120 * q^43 + 537 * q^47 - 492 * q^49 - 534 * q^53 + 240 * q^55 - 225 * q^59 + 480 * q^61 + 165 * q^65 - 12 * q^67 - 1140 * q^71 + 2124 * q^73 + 312 * q^77 + 1026 * q^79 + 702 * q^83 - 210 * q^85 - 2280 * q^89 + 3222 * q^91 + 105 * q^95 + 2556 * q^97

Introduction

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