LMFDB - Embedded newform 588.8.i.c.361.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [588,8,Mod(361,588)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(588, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 4]))

N = Newforms(chi, 8, names="a")

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

chi := DirichletCharacter("588.361");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 588 = 2^{2} \cdot 3 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	588.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:	$183.682394985$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 1 $ x^2 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 84)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			361.1
Root			$0.500000 + 0.866025i$ of defining polynomial
Character	$\chi$	$=$	588.361
Dual form			588.8.i.c.373.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-13.5000 + 23.3827i) q^{3} +(50.0000 + 86.6025i) q^{5} +(-364.500 - 631.333i) q^{9} +(-1387.00 + 2402.35i) q^{11} +3294.00 q^{13} -2700.00 q^{15} +(2950.00 - 5109.55i) q^{17} +(3322.00 + 5753.87i) q^{19} +(-991.000 - 1716.46i) q^{23} +(34062.5 - 58998.0i) q^{25} +19683.0 q^{27} -208106. q^{29} +(-58896.0 + 102011. i) q^{31} +(-37449.0 - 64863.6i) q^{33} +(167843. + 290713. i) q^{37} +(-44469.0 + 77022.6i) q^{39} +265488. q^{41} -93292.0 q^{43} +(36450.0 - 63133.3i) q^{45} +(-328758. - 569426. i) q^{47} +(79650.0 + 137958. i) q^{51} +(304359. - 527165. i) q^{53} -277400. q^{55} -179388. q^{57} +(-268060. + 464294. i) q^{59} +(-898545. - 1.55633e6i) q^{61} +(164700. + 285269. i) q^{65} +(-1.06159e6 + 1.83872e6i) q^{67} +53514.0 q^{69} -1.19121e6 q^{71} +(528215. - 914895. i) q^{73} +(919688. + 1.59295e6i) q^{75} +(-499242. - 864713. i) q^{79} +(-265720. + 460241. i) q^{81} -3.89800e6 q^{83} +590000. q^{85} +(2.80943e6 - 4.86608e6i) q^{87} +(-2.31118e6 - 4.00307e6i) q^{89} +(-1.59019e6 - 2.75429e6i) q^{93} +(-332200. + 575387. i) q^{95} -1.52877e7 q^{97} +2.02225e6 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 27 q^{3} + 100 q^{5} - 729 q^{9} - 2774 q^{11} + 6588 q^{13} - 5400 q^{15} + 5900 q^{17} + 6644 q^{19} - 1982 q^{23} + 68125 q^{25} + 39366 q^{27} - 416212 q^{29} - 117792 q^{31} - 74898 q^{33} + 335686 q^{37}+ \cdots + 4044492 q^{99}+O(q^{100}) $ 2 * q - 27 * q^3 + 100 * q^5 - 729 * q^9 - 2774 * q^11 + 6588 * q^13 - 5400 * q^15 + 5900 * q^17 + 6644 * q^19 - 1982 * q^23 + 68125 * q^25 + 39366 * q^27 - 416212 * q^29 - 117792 * q^31 - 74898 * q^33 + 335686 * q^37 - 88938 * q^39 + 530976 * q^41 - 186584 * q^43 + 72900 * q^45 - 657516 * q^47 + 159300 * q^51 + 608718 * q^53 - 554800 * q^55 - 358776 * q^57 - 536120 * q^59 - 1797090 * q^61 + 329400 * q^65 - 2123176 * q^67 + 107028 * q^69 - 2382428 * q^71 + 1056430 * q^73 + 1839375 * q^75 - 998484 * q^79 - 531441 * q^81 - 7796008 * q^83 + 1180000 * q^85 + 5618862 * q^87 - 4622352 * q^89 - 3180384 * q^93 - 664400 * q^95 - 30575420 * q^97 + 4044492 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/588\mathbb{Z}\right)^\times$.

$n$	$197$	$295$	$493$
$\chi(n)$	$1$	$1$	$e\left(\frac{2}{3}\right)$

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
$3$	−13.5000	+	23.3827i	−0.288675	+	0.500000i
$3$	−13.5000	+	23.3827i	−0.288675	+	0.500000i
$4$		0			0
$5$	50.0000	+	86.6025i	0.178885	+	0.309839i	0.941499	−	0.337016i	$-0.109418\pi$
$5$	50.0000	+	86.6025i	0.178885	+	0.309839i	−0.762614	+	0.646854i	$0.776084\pi$
$6$		0			0
$7$		0			0
$7$		0			0
$8$		0			0
$9$	−364.500	−	631.333i	−0.166667	−	0.288675i
$10$		0			0
$11$	−1387.00	+	2402.35i	−0.314197	+	0.544205i	−0.979266	−	0.202576i	$-0.935069\pi$
$11$	−1387.00	+	2402.35i	−0.314197	+	0.544205i	0.665069	+	0.746782i	$0.268402\pi$
$12$		0			0
$13$	3294.00			0.415836			0.207918	−	0.978146i	$-0.433331\pi$
$13$	3294.00			0.415836			0.207918	+	0.978146i	$0.433331\pi$
$14$		0			0
$15$	−2700.00			−0.206559
$16$		0			0
$17$	2950.00	−	5109.55i	0.145630	−	0.252239i	−0.783978	−	0.620789i	$-0.786812\pi$
$17$	2950.00	−	5109.55i	0.145630	−	0.252239i	0.929608	+	0.368550i	$0.120146\pi$
$18$		0			0
$19$	3322.00	+	5753.87i	0.111112	+	0.192452i	0.916219	−	0.400678i	$-0.131225\pi$
$19$	3322.00	+	5753.87i	0.111112	+	0.192452i	−0.805107	+	0.593130i	$0.797892\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−991.000	−	1716.46i	−0.0169835	−	0.0294162i	0.857409	−	0.514636i	$-0.172073\pi$
$23$	−991.000	−	1716.46i	−0.0169835	−	0.0294162i	−0.874392	+	0.485220i	$0.838740\pi$
$24$		0			0
$25$	34062.5	−	58998.0i	0.436000	−	0.755174i
$26$		0			0
$27$	19683.0			0.192450
$28$		0			0
$29$	−208106.			−1.58450			−0.792249	−	0.610198i	$-0.791090\pi$
$29$	−208106.			−1.58450			−0.792249	+	0.610198i	$0.791090\pi$
$30$		0			0
$31$	−58896.0	+	102011.i	−0.355075	+	0.615008i	−0.987131	−	0.159915i	$-0.948878\pi$
$31$	−58896.0	+	102011.i	−0.355075	+	0.615008i	0.632056	+	0.774923i	$0.282211\pi$
$32$		0			0
$33$	−37449.0	−	64863.6i	−0.181402	−	0.314197i
$34$		0			0
$35$		0			0
$36$		0			0
$37$	167843.	+	290713.i	0.544750	+	0.943535i	0.998623	+	0.0524680i	$0.0167088\pi$
$37$	167843.	+	290713.i	0.544750	+	0.943535i	−0.453873	+	0.891067i	$0.649958\pi$
$38$		0			0
$39$	−44469.0	+	77022.6i	−0.120041	+	0.207918i
$40$		0			0
$41$	265488.			0.601591			0.300796	−	0.953689i	$-0.402748\pi$
$41$	265488.			0.601591			0.300796	+	0.953689i	$0.402748\pi$
$42$		0			0
$43$	−93292.0			−0.178939			−0.0894695	−	0.995990i	$-0.528517\pi$
$43$	−93292.0			−0.178939			−0.0894695	+	0.995990i	$0.528517\pi$
$44$		0			0
$45$	36450.0	−	63133.3i	0.0596285	−	0.103280i
$46$		0			0
$47$	−328758.	−	569426.i	−0.461885	−	0.800008i	0.537170	−	0.843474i	$-0.319493\pi$
$47$	−328758.	−	569426.i	−0.461885	−	0.800008i	−0.999055	+	0.0434658i	$0.986160\pi$
$48$		0			0
$49$		0			0
$50$		0			0
$51$	79650.0	+	137958.i	0.0840795	+	0.145630i
$52$		0			0
$53$	304359.	−	527165.i	0.280815	−	0.486386i	−0.690771	−	0.723074i	$-0.742729\pi$
$53$	304359.	−	527165.i	0.280815	−	0.486386i	0.971586	+	0.236688i	$0.0760619\pi$
$54$		0			0
$55$	−277400.			−0.224821
$56$		0			0
$57$	−179388.			−0.128301
$58$		0			0
$59$	−268060.	+	464294.i	−0.169922	+	0.294314i	−0.938392	−	0.345572i	$-0.887685\pi$
$59$	−268060.	+	464294.i	−0.169922	+	0.294314i	0.768470	+	0.639886i	$0.221018\pi$
$60$		0			0
$61$	−898545.	−	1.55633e6i	−0.506857	−	0.877902i	−0.999969	−	0.00793591i	$-0.997474\pi$
$61$	−898545.	−	1.55633e6i	−0.506857	−	0.877902i	0.493112	−	0.869966i	$-0.335859\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	164700.	+	285269.i	0.0743870	+	0.128842i
$66$		0			0
$67$	−1.06159e6	+	1.83872e6i	−0.431215	+	0.746887i	−0.996978	−	0.0776811i	$-0.975248\pi$
$67$	−1.06159e6	+	1.83872e6i	−0.431215	+	0.746887i	0.565763	+	0.824568i	$0.308582\pi$
$68$		0			0
$69$	53514.0			0.0196108
$70$		0			0
$71$	−1.19121e6			−0.394990			−0.197495	−	0.980304i	$-0.563281\pi$
$71$	−1.19121e6			−0.394990			−0.197495	+	0.980304i	$0.563281\pi$
$72$		0			0
$73$	528215.	−	914895.i	0.158921	−	0.275259i	−0.775559	−	0.631275i	$-0.782532\pi$
$73$	528215.	−	914895.i	0.158921	−	0.275259i	0.934480	+	0.356016i	$0.115865\pi$
$74$		0			0
$75$	919688.	+	1.59295e6i	0.251725	+	0.436000i
$76$		0			0
$77$		0			0
$78$		0			0
$79$	−499242.	−	864713.i	−0.113924	−	0.197323i	0.803425	−	0.595406i	$-0.203009\pi$
$79$	−499242.	−	864713.i	−0.113924	−	0.197323i	−0.917349	+	0.398083i	$0.869675\pi$
$80$		0			0
$81$	−265720.	+	460241.i	−0.0555556	+	0.0962250i
$82$		0			0
$83$	−3.89800e6			−0.748288			−0.374144	−	0.927371i	$-0.622063\pi$
$83$	−3.89800e6			−0.748288			−0.374144	+	0.927371i	$0.622063\pi$
$84$		0			0
$85$	590000.			0.104204
$86$		0			0
$87$	2.80943e6	−	4.86608e6i	0.457405	−	0.792249i
$88$		0			0
$89$	−2.31118e6	−	4.00307e6i	−0.347511	−	0.601906i	0.638296	−	0.769791i	$-0.279640\pi$
$89$	−2.31118e6	−	4.00307e6i	−0.347511	−	0.601906i	−0.985807	+	0.167885i	$0.946306\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$	−1.59019e6	−	2.75429e6i	−0.205003	−	0.355075i
$94$		0			0
$95$	−332200.	+	575387.i	−0.0397527	+	0.0688538i
$96$		0			0
$97$	−1.52877e7			−1.70075			−0.850377	−	0.526174i	$-0.823626\pi$
$97$	−1.52877e7			−1.70075			−0.850377	+	0.526174i	$0.823626\pi$
$98$		0			0
$99$	2.02225e6			0.209465
$100$		0			0
$101$	1.11934e6	−	1.93876e6i	0.108103	−	0.187240i	−0.806899	−	0.590690i	$-0.798856\pi$
$101$	1.11934e6	−	1.93876e6i	0.108103	−	0.187240i	0.915002	+	0.403450i	$0.132189\pi$
$102$		0			0
$103$	6.22511e6	+	1.07822e7i	0.561328	+	0.972249i	0.997381	+	0.0723279i	$0.0230428\pi$
$103$	6.22511e6	+	1.07822e7i	0.561328	+	0.972249i	−0.436053	+	0.899921i	$0.643624\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	3.79430e6	+	6.57192e6i	0.299425	+	0.518620i	0.976005	−	0.217750i	$-0.0698717\pi$
$107$	3.79430e6	+	6.57192e6i	0.299425	+	0.518620i	−0.676579	+	0.736370i	$0.736538\pi$
$108$		0			0
$109$	−2.58548e6	+	4.47819e6i	−0.191227	+	0.331215i	−0.945657	−	0.325166i	$-0.894580\pi$
$109$	−2.58548e6	+	4.47819e6i	−0.191227	+	0.331215i	0.754430	+	0.656380i	$0.227913\pi$
$110$		0			0
$111$	−9.06352e6			−0.629023
$112$		0			0
$113$	−9.63868e6			−0.628410			−0.314205	−	0.949355i	$-0.601738\pi$
$113$	−9.63868e6			−0.628410			−0.314205	+	0.949355i	$0.601738\pi$
$114$		0			0
$115$	99100.0	−	171646.i	0.00607619	−	0.0105243i
$116$		0			0
$117$	−1.20066e6	−	2.07961e6i	−0.0693060	−	0.120041i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	5.89605e6	+	1.02123e7i	0.302560	+	0.524050i
$122$		0			0
$123$	−3.58409e6	+	6.20782e6i	−0.173664	+	0.300796i
$124$		0			0
$125$	1.46250e7			0.669747
$126$		0			0
$127$	8.08309e6			0.350158			0.175079	−	0.984554i	$-0.443982\pi$
$127$	8.08309e6			0.350158			0.175079	+	0.984554i	$0.443982\pi$
$128$		0			0
$129$	1.25944e6	−	2.18142e6i	0.0516552	−	0.0894695i
$130$		0			0
$131$	−9.87927e6	−	1.71114e7i	−0.383951	−	0.665022i	0.607672	−	0.794188i	$-0.292103\pi$
$131$	−9.87927e6	−	1.71114e7i	−0.383951	−	0.665022i	−0.991623	+	0.129166i	$0.958770\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$	984150.	+	1.70460e6i	0.0344265	+	0.0596285i
$136$		0			0
$137$	2.40448e7	−	4.16468e7i	0.798913	−	1.38376i	−0.121412	−	0.992602i	$-0.538742\pi$
$137$	2.40448e7	−	4.16468e7i	0.798913	−	1.38376i	0.920325	−	0.391155i	$-0.127924\pi$
$138$		0			0
$139$	−1.37173e7			−0.433229			−0.216615	−	0.976257i	$-0.569501\pi$
$139$	−1.37173e7			−0.433229			−0.216615	+	0.976257i	$0.569501\pi$
$140$		0			0
$141$	1.77529e7			0.533339
$142$		0			0
$143$	−4.56878e6	+	7.91336e6i	−0.130654	+	0.226300i
$144$		0			0
$145$	−1.04053e7	−	1.80225e7i	−0.283444	−	0.490939i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	3.53552e7	+	6.12370e7i	0.875591	+	1.51657i	0.856132	+	0.516758i	$0.172861\pi$
$149$	3.53552e7	+	6.12370e7i	0.875591	+	1.51657i	0.0194598	+	0.999811i	$0.493805\pi$
$150$		0			0
$151$	1.28485e7	−	2.22542e7i	0.303691	−	0.526009i	−0.673278	−	0.739390i	$-0.735114\pi$
$151$	1.28485e7	−	2.22542e7i	0.303691	−	0.526009i	0.976969	+	0.213381i	$0.0684475\pi$
$152$		0			0
$153$	−4.30110e6			−0.0970867
$154$		0			0
$155$	−1.17792e7			−0.254071
$156$		0			0
$157$	1.04590e7	−	1.81155e7i	0.215695	−	0.373595i	−0.737792	−	0.675028i	$-0.764132\pi$
$157$	1.04590e7	−	1.81155e7i	0.215695	−	0.373595i	0.953487	+	0.301433i	$0.0974649\pi$
$158$		0			0
$159$	8.21769e6	+	1.42335e7i	0.162129	+	0.280815i
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−2.72080e7	−	4.71256e7i	−0.492084	−	0.852315i	0.507874	−	0.861431i	$-0.330432\pi$
$163$	−2.72080e7	−	4.71256e7i	−0.492084	−	0.852315i	−0.999958	+	0.00911613i	$0.997098\pi$
$164$		0			0
$165$	3.74490e6	−	6.48636e6i	0.0649003	−	0.112411i
$166$		0			0
$167$	−6.41888e6			−0.106648			−0.0533238	−	0.998577i	$-0.516982\pi$
$167$	−6.41888e6			−0.106648			−0.0533238	+	0.998577i	$0.516982\pi$
$168$		0			0
$169$	−5.18981e7			−0.827081
$170$		0			0
$171$	2.42174e6	−	4.19457e6i	0.0370374	−	0.0641507i
$172$		0			0
$173$	−2.73266e7	−	4.73311e7i	−0.401259	−	0.695000i	0.592619	−	0.805483i	$-0.298094\pi$
$173$	−2.73266e7	−	4.73311e7i	−0.401259	−	0.695000i	−0.993878	+	0.110482i	$0.964760\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	−7.23762e6	−	1.25359e7i	−0.0981046	−	0.169922i
$178$		0			0
$179$	1.88329e7	−	3.26196e7i	0.245433	−	0.425102i	−0.716821	−	0.697258i	$-0.754403\pi$
$179$	1.88329e7	−	3.26196e7i	0.245433	−	0.425102i	0.962253	+	0.272156i	$0.0877367\pi$
$180$		0			0
$181$	−1.76788e7			−0.221604			−0.110802	−	0.993842i	$-0.535342\pi$
$181$	−1.76788e7			−0.221604			−0.110802	+	0.993842i	$0.535342\pi$
$182$		0			0
$183$	4.85214e7			0.585268
$184$		0			0
$185$	−1.67843e7	+	2.90713e7i	−0.194896	+	0.337569i
$186$		0			0
$187$	8.18330e6	+	1.41739e7i	0.0915130	+	0.158505i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−2.04588e7	−	3.54357e7i	−0.212454	−	0.367980i	0.740028	−	0.672576i	$-0.234812\pi$
$191$	−2.04588e7	−	3.54357e7i	−0.212454	−	0.367980i	−0.952482	+	0.304595i	$0.901479\pi$
$192$		0			0
$193$	8.19134e7	−	1.41878e8i	0.820171	−	1.42058i	−0.0853830	−	0.996348i	$-0.527211\pi$
$193$	8.19134e7	−	1.41878e8i	0.820171	−	1.42058i	0.905554	−	0.424230i	$-0.139455\pi$
$194$		0			0
$195$	−8.89380e6			−0.0858947
$196$		0			0
$197$	8.02076e7			0.747453			0.373726	−	0.927539i	$-0.378080\pi$
$197$	8.02076e7			0.747453			0.373726	+	0.927539i	$0.378080\pi$
$198$		0			0
$199$	4.91610e7	−	8.51493e7i	0.442216	−	0.765941i	−0.555637	−	0.831425i	$-0.687526\pi$
$199$	4.91610e7	−	8.51493e7i	0.442216	−	0.765941i	0.997854	+	0.0654838i	$0.0208591\pi$
$200$		0			0
$201$	−2.86629e7	−	4.96456e7i	−0.248962	−	0.431215i
$202$		0			0
$203$		0			0
$204$		0			0
$205$	1.32744e7	+	2.29919e7i	0.107616	+	0.186396i
$206$		0			0
$207$	−722439.	+	1.25130e6i	−0.00566115	+	0.00980541i
$208$		0			0
$209$	−1.84305e7			−0.139645
$210$		0			0
$211$	1.36321e8			0.999021			0.499510	−	0.866308i	$-0.333513\pi$
$211$	1.36321e8			0.999021			0.499510	+	0.866308i	$0.333513\pi$
$212$		0			0
$213$	1.60814e7	−	2.78538e7i	0.114024	−	0.197495i
$214$		0			0
$215$	−4.66460e6	−	8.07932e6i	−0.0320096	−	0.0554422i
$216$		0			0
$217$		0			0
$218$		0			0
$219$	1.42618e7	+	2.47022e7i	0.0917530	+	0.158921i
$220$		0			0
$221$	9.71730e6	−	1.68309e7i	0.0605582	−	0.104890i
$222$		0			0
$223$	−1.26358e8			−0.763019			−0.381510	−	0.924365i	$-0.624596\pi$
$223$	−1.26358e8			−0.763019			−0.381510	+	0.924365i	$0.624596\pi$
$224$		0			0
$225$	−4.96631e7			−0.290667
$226$		0			0
$227$	1.17004e8	−	2.02656e8i	0.663909	−	1.14992i	−0.315671	−	0.948869i	$-0.602230\pi$
$227$	1.17004e8	−	2.02656e8i	0.663909	−	1.14992i	0.979580	−	0.201055i	$-0.0644370\pi$
$228$		0			0
$229$	3.41803e7	+	5.92020e7i	0.188084	+	0.325771i	0.944611	−	0.328191i	$-0.106439\pi$
$229$	3.41803e7	+	5.92020e7i	0.188084	+	0.325771i	−0.756527	+	0.653962i	$0.773106\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−9.01923e7	−	1.56218e8i	−0.467115	−	0.809067i	0.532179	−	0.846632i	$-0.321373\pi$
$233$	−9.01923e7	−	1.56218e8i	−0.467115	−	0.809067i	−0.999294	+	0.0375648i	$0.988040\pi$
$234$		0			0
$235$	3.28758e7	−	5.69426e7i	0.165249	−	0.286220i
$236$		0			0
$237$	2.69591e7			0.131548
$238$		0			0
$239$	2.61131e8			1.23727			0.618637	−	0.785677i	$-0.287685\pi$
$239$	2.61131e8			1.23727			0.618637	+	0.785677i	$0.287685\pi$
$240$		0			0
$241$	9.88781e6	−	1.71262e7i	0.0455031	−	0.0788136i	−0.842377	−	0.538889i	$-0.818844\pi$
$241$	9.88781e6	−	1.71262e7i	0.0455031	−	0.0788136i	0.887880	+	0.460075i	$0.152178\pi$
$242$		0			0
$243$	−7.17445e6	−	1.24265e7i	−0.0320750	−	0.0555556i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	1.09427e7	+	1.89533e7i	0.0462045	+	0.0800285i
$248$		0			0
$249$	5.26231e7	−	9.11458e7i	0.216012	−	0.374144i
$250$		0			0
$251$	−1.53770e8			−0.613779			−0.306890	−	0.951745i	$-0.599288\pi$
$251$	−1.53770e8			−0.613779			−0.306890	+	0.951745i	$0.599288\pi$
$252$		0			0
$253$	5.49807e6			0.0213446
$254$		0			0
$255$	−7.96500e6	+	1.37958e7i	−0.0300812	+	0.0521022i
$256$		0			0
$257$	−7.16802e7	−	1.24154e8i	−0.263411	−	0.456241i	0.703735	−	0.710462i	$-0.251514\pi$
$257$	−7.16802e7	−	1.24154e8i	−0.263411	−	0.456241i	−0.967146	+	0.254222i	$0.918181\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	7.58546e7	+	1.31384e8i	0.264083	+	0.457405i
$262$		0			0
$263$	2.53650e8	−	4.39336e8i	0.859786	−	1.48919i	−0.0123459	−	0.999924i	$-0.503930\pi$
$263$	2.53650e8	−	4.39336e8i	0.859786	−	1.48919i	0.872132	−	0.489270i	$-0.162737\pi$
$264$		0			0
$265$	6.08718e7			0.200935
$266$		0			0
$267$	1.24804e8			0.401271
$268$		0			0
$269$	2.63943e8	−	4.57163e8i	0.826755	−	1.43198i	−0.0738149	−	0.997272i	$-0.523517\pi$
$269$	2.63943e8	−	4.57163e8i	0.826755	−	1.43198i	0.900570	−	0.434710i	$-0.143149\pi$
$270$		0			0
$271$	−2.52918e8	−	4.38067e8i	−0.771947	−	1.33705i	−0.936495	−	0.350682i	$-0.885950\pi$
$271$	−2.52918e8	−	4.38067e8i	−0.771947	−	1.33705i	0.164548	−	0.986369i	$-0.447384\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	9.44894e7	+	1.63660e8i	0.273980	+	0.474547i
$276$		0			0
$277$	2.94293e8	−	5.09730e8i	0.831956	−	1.44099i	−0.0645284	−	0.997916i	$-0.520554\pi$
$277$	2.94293e8	−	5.09730e8i	0.831956	−	1.44099i	0.896485	−	0.443075i	$-0.146112\pi$
$278$		0			0
$279$	8.58704e7			0.236717
$280$		0			0
$281$	8.65142e7			0.232603			0.116301	−	0.993214i	$-0.462896\pi$
$281$	8.65142e7			0.232603			0.116301	+	0.993214i	$0.462896\pi$
$282$		0			0
$283$	−7.86504e7	+	1.36227e8i	−0.206276	+	0.357280i	−0.950538	−	0.310607i	$-0.899468\pi$
$283$	−7.86504e7	+	1.36227e8i	−0.206276	+	0.357280i	0.744263	+	0.667887i	$0.232801\pi$
$284$		0			0
$285$	−8.96940e6	−	1.55355e7i	−0.0229513	−	0.0397527i
$286$		0			0
$287$		0			0
$288$		0			0
$289$	1.87764e8	+	3.25217e8i	0.457584	+	0.792558i
$290$		0			0
$291$	2.06384e8	−	3.57468e8i	0.490965	−	0.850377i
$292$		0			0
$293$	−6.80964e8			−1.58157			−0.790783	−	0.612097i	$-0.790326\pi$
$293$	−6.80964e8			−1.58157			−0.790783	+	0.612097i	$0.790326\pi$
$294$		0			0
$295$	−5.36120e7			−0.121586
$296$		0			0
$297$	−2.73003e7	+	4.72855e7i	−0.0604672	+	0.104732i
$298$		0			0
$299$	−3.26435e6	−	5.65403e6i	−0.00706233	−	0.0122323i
$300$		0			0
$301$		0			0
$302$		0			0
$303$	3.02223e7	+	5.23465e7i	0.0624134	+	0.108103i
$304$		0			0
$305$	8.98545e7	−	1.55633e8i	0.181339	−	0.314088i
$306$		0			0
$307$	−2.81734e8			−0.555718			−0.277859	−	0.960622i	$-0.589625\pi$
$307$	−2.81734e8			−0.555718			−0.277859	+	0.960622i	$0.589625\pi$
$308$		0			0
$309$	−3.36156e8			−0.648166
$310$		0			0
$311$	−2.09860e8	+	3.63487e8i	−0.395610	+	0.685217i	−0.993179	−	0.116601i	$-0.962800\pi$
$311$	−2.09860e8	+	3.63487e8i	−0.395610	+	0.685217i	0.597569	+	0.801818i	$0.296134\pi$
$312$		0			0
$313$	−1.14342e8	−	1.98046e8i	−0.210766	−	0.365057i	0.741188	−	0.671297i	$-0.234262\pi$
$313$	−1.14342e8	−	1.98046e8i	−0.210766	−	0.365057i	−0.951954	+	0.306240i	$0.900929\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−1.33001e8	−	2.30365e8i	−0.234503	−	0.406171i	0.724625	−	0.689143i	$-0.242013\pi$
$317$	−1.33001e8	−	2.30365e8i	−0.234503	−	0.406171i	−0.959128	+	0.282972i	$0.908680\pi$
$318$		0			0
$319$	2.88643e8	−	4.99944e8i	0.497844	−	0.862292i
$320$		0			0
$321$	−2.04892e8			−0.345747
$322$		0			0
$323$	3.91996e7			0.0647251
$324$		0			0
$325$	1.12202e8	−	1.94339e8i	0.181304	−	0.314028i
$326$		0			0
$327$	−6.98080e7	−	1.20911e8i	−0.110405	−	0.191227i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	3.58054e8	+	6.20167e8i	0.542688	+	0.939963i	0.998749	+	0.0500141i	$0.0159266\pi$
$331$	3.58054e8	+	6.20167e8i	0.542688	+	0.939963i	−0.456061	+	0.889949i	$0.650740\pi$
$332$		0			0
$333$	1.22358e8	−	2.11929e8i	0.181583	−	0.314512i
$334$		0			0
$335$	−2.12318e8			−0.308553
$336$		0			0
$337$	5.70266e8			0.811657			0.405829	−	0.913949i	$-0.366983\pi$
$337$	5.70266e8			0.811657			0.405829	+	0.913949i	$0.366983\pi$
$338$		0			0
$339$	1.30122e8	−	2.25378e8i	0.181406	−	0.314205i
$340$		0			0
$341$	−1.63378e8	−	2.82978e8i	−0.223127	−	0.386467i
$342$		0			0
$343$		0			0
$344$		0			0
$345$	2.67570e6	+	4.63445e6i	0.00350809	+	0.00607619i
$346$		0			0
$347$	8.23203e7	−	1.42583e8i	0.105768	−	0.183195i	−0.808284	−	0.588793i	$-0.799603\pi$
$347$	8.23203e7	−	1.42583e8i	0.105768	−	0.183195i	0.914052	+	0.405598i	$0.132937\pi$
$348$		0			0
$349$	−1.31564e9			−1.65671			−0.828357	−	0.560200i	$-0.810724\pi$
$349$	−1.31564e9			−1.65671			−0.828357	+	0.560200i	$0.810724\pi$
$350$		0			0
$351$	6.48358e7			0.0800276
$352$		0			0
$353$	7.80588e8	−	1.35202e9i	0.944518	−	1.63595i	0.187805	−	0.982206i	$-0.439863\pi$
$353$	7.80588e8	−	1.35202e9i	0.944518	−	1.63595i	0.756713	−	0.653747i	$-0.226804\pi$
$354$		0			0
$355$	−5.95607e7	−	1.03162e8i	−0.0706579	−	0.122383i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	4.30358e8	+	7.45402e8i	0.490907	+	0.850276i	0.999945	−	0.0104681i	$-0.00333215\pi$
$359$	4.30358e8	+	7.45402e8i	0.490907	+	0.850276i	−0.509038	+	0.860744i	$0.669999\pi$
$360$		0			0
$361$	4.24865e8	−	7.35887e8i	0.475308	−	0.823258i
$362$		0			0
$363$	−3.18387e8			−0.349367
$364$		0			0
$365$	1.05643e8			0.113714
$366$		0			0
$367$	4.56066e8	−	7.89930e8i	0.481612	−	0.834176i	−0.518166	−	0.855280i	$-0.673385\pi$
$367$	4.56066e8	−	7.89930e8i	0.481612	−	0.834176i	0.999777	+	0.0211045i	$0.00671828\pi$
$368$		0			0
$369$	−9.67704e7	−	1.67611e8i	−0.100265	−	0.173664i
$370$		0			0
$371$		0			0
$372$		0			0
$373$	−1.18002e8	−	2.04386e8i	−0.117736	−	0.203925i	0.801134	−	0.598485i	$-0.204230\pi$
$373$	−1.18002e8	−	2.04386e8i	−0.117736	−	0.203925i	−0.918870	+	0.394560i	$0.870897\pi$
$374$		0			0
$375$	−1.97438e8	+	3.41972e8i	−0.193339	+	0.334874i
$376$		0			0
$377$	−6.85501e8			−0.658891
$378$		0			0
$379$	8.29313e8			0.782495			0.391247	−	0.920286i	$-0.372044\pi$
$379$	8.29313e8			0.782495			0.391247	+	0.920286i	$0.372044\pi$
$380$		0			0
$381$	−1.09122e8	+	1.89004e8i	−0.101082	+	0.175079i
$382$		0			0
$383$	−6.08316e8	−	1.05363e9i	−0.553265	−	0.958284i	−0.998036	−	0.0626393i	$-0.980048\pi$
$383$	−6.08316e8	−	1.05363e9i	−0.553265	−	0.958284i	0.444771	−	0.895644i	$-0.353285\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	3.40049e7	+	5.88983e7i	0.0298232	+	0.0516552i
$388$		0			0
$389$	5.91064e7	−	1.02375e8i	0.0509109	−	0.0881803i	−0.839447	−	0.543442i	$-0.817121\pi$
$389$	5.91064e7	−	1.02375e8i	0.0509109	−	0.0881803i	0.890358	+	0.455261i	$0.150454\pi$
$390$		0			0
$391$	−1.16938e7			−0.00989321
$392$		0			0
$393$	5.33481e8			0.443348
$394$		0			0
$395$	4.99242e7	−	8.64713e7i	0.0407588	−	0.0705963i
$396$		0			0
$397$	1.81283e7	+	3.13991e7i	0.0145408	+	0.0251855i	0.873204	−	0.487354i	$-0.162038\pi$
$397$	1.81283e7	+	3.13991e7i	0.0145408	+	0.0251855i	−0.858663	+	0.512540i	$0.828705\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−2.53425e8	−	4.38945e8i	−0.196265	−	0.339942i	0.751049	−	0.660246i	$-0.229548\pi$
$401$	−2.53425e8	−	4.38945e8i	−0.196265	−	0.339942i	−0.947315	+	0.320305i	$0.896215\pi$
$402$		0			0
$403$	−1.94003e8	+	3.36024e8i	−0.147653	+	0.255742i
$404$		0			0
$405$	−5.31441e7			−0.0397523
$406$		0			0
$407$	−9.31193e8			−0.684635
$408$		0			0
$409$	8.40605e8	−	1.45597e9i	0.607519	−	1.05225i	−0.384128	−	0.923280i	$-0.625498\pi$
$409$	8.40605e8	−	1.45597e9i	0.607519	−	1.05225i	0.991648	−	0.128975i	$-0.0411687\pi$
$410$		0			0
$411$	6.49210e8	+	1.12446e9i	0.461252	+	0.798913i
$412$		0			0
$413$		0			0
$414$		0			0
$415$	−1.94900e8	−	3.37577e8i	−0.133858	−	0.231849i
$416$		0			0
$417$	1.85184e8	−	3.20748e8i	0.125062	−	0.216615i
$418$		0			0
$419$	−4.94962e8			−0.328718			−0.164359	−	0.986401i	$-0.552556\pi$
$419$	−4.94962e8			−0.328718			−0.164359	+	0.986401i	$0.552556\pi$
$420$		0			0
$421$	−6.57487e7			−0.0429437			−0.0214719	−	0.999769i	$-0.506835\pi$
$421$	−6.57487e7			−0.0429437			−0.0214719	+	0.999769i	$0.506835\pi$
$422$		0			0
$423$	−2.39665e8	+	4.15111e8i	−0.153962	+	0.266669i
$424$		0			0
$425$	−2.00969e8	−	3.48088e8i	−0.126989	−	0.219952i
$426$		0			0
$427$		0			0
$428$		0			0
$429$	−1.23357e8	−	2.13661e8i	−0.0754333	−	0.130654i
$430$		0			0
$431$	6.67089e8	−	1.15543e9i	0.401341	−	0.695143i	−0.592547	−	0.805536i	$-0.701878\pi$
$431$	6.67089e8	−	1.15543e9i	0.401341	−	0.695143i	0.993888	+	0.110393i	$0.0352109\pi$
$432$		0			0
$433$	2.47903e7			0.0146749			0.00733745	−	0.999973i	$-0.497664\pi$
$433$	2.47903e7			0.0146749			0.00733745	+	0.999973i	$0.497664\pi$
$434$		0			0
$435$	5.61886e8			0.327292
$436$		0			0
$437$	6.58420e6	−	1.14042e7i	0.00377414	−	0.00653701i
$438$		0			0
$439$	−3.44664e8	−	5.96975e8i	−0.194433	−	0.336768i	0.752282	−	0.658842i	$-0.228953\pi$
$439$	−3.44664e8	−	5.96975e8i	−0.194433	−	0.336768i	−0.946714	+	0.322074i	$0.895620\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	2.87308e8	+	4.97633e8i	0.157013	+	0.271954i	0.933790	−	0.357821i	$-0.116480\pi$
$443$	2.87308e8	+	4.97633e8i	0.157013	+	0.271954i	−0.776777	+	0.629775i	$0.783147\pi$
$444$		0			0
$445$	2.31118e8	−	4.00307e8i	0.124329	−	0.215344i
$446$		0			0
$447$	−1.90918e9			−1.01105
$448$		0			0
$449$	9.56884e7			0.0498881			0.0249440	−	0.999689i	$-0.492059\pi$
$449$	9.56884e7			0.0498881			0.0249440	+	0.999689i	$0.492059\pi$
$450$		0			0
$451$	−3.68232e8	+	6.37796e8i	−0.189018	+	0.327389i
$452$		0			0
$453$	3.46909e8	+	6.00864e8i	0.175336	+	0.303691i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	2.36545e7	+	4.09707e7i	0.0115933	+	0.0200802i	0.871764	−	0.489926i	$-0.162976\pi$
$457$	2.36545e7	+	4.09707e7i	0.0115933	+	0.0200802i	−0.860171	+	0.510006i	$0.829643\pi$
$458$		0			0
$459$	5.80648e7	−	1.00571e8i	0.0280265	−	0.0485433i
$460$		0			0
$461$	2.13567e9			1.01527			0.507634	−	0.861573i	$-0.330520\pi$
$461$	2.13567e9			1.01527			0.507634	+	0.861573i	$0.330520\pi$
$462$		0			0
$463$	2.92675e8			0.137042			0.0685208	−	0.997650i	$-0.478172\pi$
$463$	2.92675e8			0.137042			0.0685208	+	0.997650i	$0.478172\pi$
$464$		0			0
$465$	1.59019e8	−	2.75429e8i	0.0733439	−	0.127035i
$466$		0			0
$467$	−1.76917e9	−	3.06429e9i	−0.803824	−	1.39226i	−0.917082	−	0.398698i	$-0.869462\pi$
$467$	−1.76917e9	−	3.06429e9i	−0.803824	−	1.39226i	0.113259	−	0.993566i	$-0.463871\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$	2.82392e8	+	4.89118e8i	0.124532	+	0.215695i
$472$		0			0
$473$	1.29396e8	−	2.24120e8i	0.0562221	−	0.0973795i
$474$		0			0
$475$	4.52622e8			0.193780
$476$		0			0
$477$	−4.43755e8			−0.187210
$478$		0			0
$479$	−2.07849e9	+	3.60005e9i	−0.864120	+	1.49670i	0.00379880	+	0.999993i	$0.498791\pi$
$479$	−2.07849e9	+	3.60005e9i	−0.864120	+	1.49670i	−0.867919	+	0.496707i	$0.834543\pi$
$480$		0			0
$481$	5.52875e8	+	9.57607e8i	0.226527	+	0.392355i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−7.64386e8	−	1.32395e9i	−0.304240	−	0.526959i
$486$		0			0
$487$	−6.20460e7	+	1.07467e8i	−0.0243423	+	0.0421622i	−0.877940	−	0.478771i	$-0.841083\pi$
$487$	−6.20460e7	+	1.07467e8i	−0.0243423	+	0.0421622i	0.853598	+	0.520933i	$0.174416\pi$
$488$		0			0
$489$	1.46923e9			0.568210
$490$		0			0
$491$	−3.29218e9			−1.25516			−0.627578	−	0.778554i	$-0.715954\pi$
$491$	−3.29218e9			−1.25516			−0.627578	+	0.778554i	$0.715954\pi$
$492$		0			0
$493$	−6.13913e8	+	1.06333e9i	−0.230750	+	0.399671i
$494$		0			0
$495$	1.01112e8	+	1.75132e8i	0.0374702	+	0.0649003i
$496$		0			0
$497$		0			0
$498$		0			0
$499$	1.80219e8	+	3.12149e8i	0.0649306	+	0.112463i	0.896663	−	0.442713i	$-0.145984\pi$
$499$	1.80219e8	+	3.12149e8i	0.0649306	+	0.112463i	−0.831733	+	0.555176i	$0.812651\pi$
$500$		0			0
$501$	8.66548e7	−	1.50091e8i	0.0307865	−	0.0533238i
$502$		0			0
$503$	4.29760e9			1.50570			0.752849	−	0.658193i	$-0.228679\pi$
$503$	4.29760e9			1.50570			0.752849	+	0.658193i	$0.228679\pi$
$504$		0			0
$505$	2.23869e8			0.0773524
$506$		0			0
$507$	7.00624e8	−	1.21352e9i	0.238758	−	0.413540i
$508$		0			0
$509$	−2.71294e8	−	4.69895e8i	−0.0911860	−	0.157939i	0.816824	−	0.576886i	$-0.195732\pi$
$509$	−2.71294e8	−	4.69895e8i	−0.0911860	−	0.157939i	−0.908010	+	0.418948i	$0.862399\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$	6.53869e7	+	1.13253e8i	0.0213836	+	0.0370374i
$514$		0			0
$515$	−6.22511e8	+	1.07822e9i	−0.200827	+	0.347842i
$516$		0			0
$517$	1.82395e9			0.580492
$518$		0			0
$519$	1.47564e9			0.463334
$520$		0			0
$521$	−1.18378e8	+	2.05036e8i	−0.0366722	+	0.0635182i	−0.883779	−	0.467905i	$-0.845009\pi$
$521$	−1.18378e8	+	2.05036e8i	−0.0366722	+	0.0635182i	0.847107	+	0.531423i	$0.178342\pi$
$522$		0			0
$523$	3.24162e9	+	5.61465e9i	0.990846	+	1.71619i	0.612337	+	0.790597i	$0.290230\pi$
$523$	3.24162e9	+	5.61465e9i	0.990846	+	1.71619i	0.378509	+	0.925598i	$0.376437\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	3.47486e8	+	6.01864e8i	0.103419	+	0.179127i
$528$		0			0
$529$	1.70045e9	−	2.94526e9i	0.499423	−	0.865026i
$530$		0			0
$531$	3.90831e8			0.113281
$532$		0			0
$533$	8.74517e8			0.250163
$534$		0			0
$535$	−3.79430e8	+	6.57192e8i	−0.107126	+	0.185547i
$536$		0			0
$537$	5.08489e8	+	8.80729e8i	0.141701	+	0.245433i
$538$		0			0
$539$		0			0
$540$		0			0
$541$	7.50494e7	+	1.29989e8i	0.0203778	+	0.0352954i	0.876034	−	0.482248i	$-0.160180\pi$
$541$	7.50494e7	+	1.29989e8i	0.0203778	+	0.0352954i	−0.855657	+	0.517544i	$0.826846\pi$
$542$		0			0
$543$	2.38664e8	−	4.13378e8i	0.0639716	−	0.110802i
$544$		0			0
$545$	−5.17097e8			−0.136831
$546$		0			0
$547$	3.32631e9			0.868974			0.434487	−	0.900678i	$-0.356930\pi$
$547$	3.32631e9			0.868974			0.434487	+	0.900678i	$0.356930\pi$
$548$		0			0
$549$	−6.55039e8	+	1.13456e9i	−0.168952	+	0.292634i
$550$		0			0
$551$	−6.91328e8	−	1.19742e9i	−0.176057	−	0.304940i
$552$		0			0
$553$		0			0
$554$		0			0
$555$	−4.53176e8	−	7.84924e8i	−0.112523	−	0.194896i
$556$		0			0
$557$	−2.36256e9	+	4.09208e9i	−0.579282	+	1.00335i	0.416280	+	0.909236i	$0.363334\pi$
$557$	−2.36256e9	+	4.09208e9i	−0.579282	+	1.00335i	−0.995562	+	0.0941092i	$0.970000\pi$
$558$		0			0
$559$	−3.07304e8			−0.0744092
$560$		0			0
$561$	−4.41898e8			−0.105670
$562$		0			0
$563$	3.55522e9	−	6.15782e9i	0.839628	−	1.45428i	−0.0505779	−	0.998720i	$-0.516106\pi$
$563$	3.55522e9	−	6.15782e9i	0.839628	−	1.45428i	0.890206	−	0.455558i	$-0.150560\pi$
$564$		0			0
$565$	−4.81934e8	−	8.34734e8i	−0.112413	−	0.194706i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−3.27657e9	−	5.67519e9i	−0.745636	−	1.29148i	−0.949897	−	0.312564i	$-0.898812\pi$
$569$	−3.27657e9	−	5.67519e9i	−0.745636	−	1.29148i	0.204260	−	0.978917i	$-0.434521\pi$
$570$		0			0
$571$	−4.14639e9	+	7.18175e9i	−0.932059	+	1.61437i	−0.152263	+	0.988340i	$0.548656\pi$
$571$	−4.14639e9	+	7.18175e9i	−0.932059	+	1.61437i	−0.779796	+	0.626034i	$0.784677\pi$
$572$		0			0
$573$	1.10478e9			0.245320
$574$		0			0
$575$	−1.35024e8			−0.0296192
$576$		0			0
$577$	−2.94000e9	+	5.09223e9i	−0.637135	+	1.10355i	0.348923	+	0.937151i	$0.386547\pi$
$577$	−2.94000e9	+	5.09223e9i	−0.637135	+	1.10355i	−0.986058	+	0.166400i	$0.946786\pi$
$578$		0			0
$579$	2.21166e9	+	3.83071e9i	0.473526	+	0.820171i
$580$		0			0
$581$		0			0
$582$		0			0
$583$	8.44292e8	+	1.46236e9i	0.176463	+	0.305642i
$584$		0			0
$585$	1.20066e8	−	2.07961e8i	0.0247957	−	0.0429473i
$586$		0			0
$587$	1.43194e9			0.292207			0.146103	−	0.989269i	$-0.453327\pi$
$587$	1.43194e9			0.292207			0.146103	+	0.989269i	$0.453327\pi$
$588$		0			0
$589$	−7.82610e8			−0.157813
$590$		0			0
$591$	−1.08280e9	+	1.87547e9i	−0.215771	+	0.373726i
$592$		0			0
$593$	1.74069e9	+	3.01497e9i	0.342792	+	0.593733i	0.984950	−	0.172839i	$-0.0552941\pi$
$593$	1.74069e9	+	3.01497e9i	0.342792	+	0.593733i	−0.642158	+	0.766572i	$0.721961\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	1.32735e9	+	2.29903e9i	0.255314	+	0.442216i
$598$		0			0
$599$	−3.96799e9	+	6.87275e9i	−0.754356	+	1.30658i	0.191338	+	0.981524i	$0.438717\pi$
$599$	−3.96799e9	+	6.87275e9i	−0.754356	+	1.30658i	−0.945694	+	0.325058i	$0.894616\pi$
$600$		0			0
$601$	4.05169e9			0.761335			0.380668	−	0.924712i	$-0.375694\pi$
$601$	4.05169e9			0.761335			0.380668	+	0.924712i	$0.375694\pi$
$602$		0			0
$603$	1.54780e9			0.287477
$604$		0			0
$605$	−5.89605e8	+	1.02123e9i	−0.108247	+	0.187490i
$606$		0			0
$607$	2.59580e9	+	4.49605e9i	0.471097	+	0.815964i	0.999453	−	0.0330588i	$-0.0105249\pi$
$607$	2.59580e9	+	4.49605e9i	0.471097	+	0.815964i	−0.528356	+	0.849023i	$0.677192\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−1.08293e9	−	1.87569e9i	−0.192068	−	0.332672i
$612$		0			0
$613$	−1.84903e9	+	3.20262e9i	−0.324215	+	0.561557i	−0.981353	−	0.192213i	$-0.938434\pi$
$613$	−1.84903e9	+	3.20262e9i	−0.324215	+	0.561557i	0.657138	+	0.753770i	$0.271767\pi$
$614$		0			0
$615$	−7.16818e8			−0.124264
$616$		0			0
$617$	−6.19879e9			−1.06245			−0.531226	−	0.847230i	$-0.678268\pi$
$617$	−6.19879e9			−1.06245			−0.531226	+	0.847230i	$0.678268\pi$
$618$		0			0
$619$	−1.42569e9	+	2.46936e9i	−0.241605	+	0.418473i	−0.961172	−	0.275951i	$-0.911007\pi$
$619$	−1.42569e9	+	2.46936e9i	−0.241605	+	0.418473i	0.719566	+	0.694424i	$0.244341\pi$
$620$		0			0
$621$	−1.95059e7	−	3.37851e7i	−0.00326847	−	0.00566115i
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−1.92988e9	−	3.34266e9i	−0.316192	−	0.547661i
$626$		0			0
$627$	2.48811e8	−	4.30954e8i	0.0403119	−	0.0698223i
$628$		0			0
$629$	1.98055e9			0.317328
$630$		0			0
$631$	−7.56414e9			−1.19855			−0.599276	−	0.800543i	$-0.704545\pi$
$631$	−7.56414e9			−1.19855			−0.599276	+	0.800543i	$0.704545\pi$
$632$		0			0
$633$	−1.84033e9	+	3.18755e9i	−0.288392	+	0.499510i
$634$		0			0
$635$	4.04155e8	+	7.00016e8i	0.0626382	+	0.108493i
$636$		0			0
$637$		0			0
$638$		0			0
$639$	4.34198e8	+	7.52052e8i	0.0658316	+	0.114024i
$640$		0			0
$641$	−5.60093e9	+	9.70109e9i	−0.839957	+	1.45485i	0.0499734	+	0.998751i	$0.484086\pi$
$641$	−5.60093e9	+	9.70109e9i	−0.839957	+	1.45485i	−0.889930	+	0.456097i	$0.849247\pi$
$642$		0			0
$643$	−7.87742e9			−1.16855			−0.584273	−	0.811557i	$-0.698620\pi$
$643$	−7.87742e9			−1.16855			−0.584273	+	0.811557i	$0.698620\pi$
$644$		0			0
$645$	2.51888e8			0.0369615
$646$		0			0
$647$	6.24724e8	−	1.08205e9i	0.0906825	−	0.157067i	−0.817116	−	0.576473i	$-0.804428\pi$
$647$	6.24724e8	−	1.08205e9i	0.0906825	−	0.157067i	0.907798	+	0.419407i	$0.137762\pi$
$648$		0			0
$649$	−7.43598e8	−	1.28795e9i	−0.106778	−	0.184945i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−3.76348e9	−	6.51854e9i	−0.528924	−	0.916124i	−0.999431	−	0.0337275i	$-0.989262\pi$
$653$	−3.76348e9	−	6.51854e9i	−0.528924	−	0.916124i	0.470507	−	0.882396i	$-0.344071\pi$
$654$		0			0
$655$	9.87927e8	−	1.71114e9i	0.137366	−	0.237926i
$656$		0			0
$657$	−7.70137e8			−0.105947
$658$		0			0
$659$	2.29417e9			0.312268			0.156134	−	0.987736i	$-0.450097\pi$
$659$	2.29417e9			0.312268			0.156134	+	0.987736i	$0.450097\pi$
$660$		0			0
$661$	−2.54192e9	+	4.40274e9i	−0.342339	+	0.592949i	−0.984867	−	0.173314i	$-0.944553\pi$
$661$	−2.54192e9	+	4.40274e9i	−0.342339	+	0.592949i	0.642527	+	0.766263i	$0.277886\pi$
$662$		0			0
$663$	2.62367e8	+	4.54433e8i	0.0349633	+	0.0605582i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	2.06233e8	+	3.57206e8i	0.0269103	+	0.0466099i
$668$		0			0
$669$	1.70583e9	−	2.95459e9i	0.220265	−	0.381510i
$670$		0			0
$671$	4.98513e9			0.637012
$672$		0			0
$673$	−5.62649e9			−0.711516			−0.355758	−	0.934578i	$-0.615777\pi$
$673$	−5.62649e9			−0.711516			−0.355758	+	0.934578i	$0.615777\pi$
$674$		0			0
$675$	6.70452e8	−	1.16126e9i	0.0839082	−	0.145333i
$676$		0			0
$677$	−5.13727e9	−	8.89801e9i	−0.636314	−	1.10213i	−0.986235	−	0.165349i	$-0.947125\pi$
$677$	−5.13727e9	−	8.89801e9i	−0.636314	−	1.10213i	0.349921	−	0.936779i	$-0.386209\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$	3.15909e9	+	5.47171e9i	0.383308	+	0.663909i
$682$		0			0
$683$	−7.76228e9	+	1.34447e10i	−0.932217	+	1.61465i	−0.152693	+	0.988274i	$0.548795\pi$
$683$	−7.76228e9	+	1.34447e10i	−0.932217	+	1.61465i	−0.779524	+	0.626373i	$0.784539\pi$
$684$		0			0
$685$	4.80896e9			0.571655
$686$		0			0
$687$	−1.84574e9			−0.217181
$688$		0			0
$689$	1.00256e9	−	1.73648e9i	0.116773	−	0.202257i
$690$		0			0
$691$	4.11725e9	+	7.13128e9i	0.474716	+	0.822232i	0.999581	−	0.0289536i	$-0.00921751\pi$
$691$	4.11725e9	+	7.13128e9i	0.474716	+	0.822232i	−0.524865	+	0.851186i	$0.675884\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−6.85866e8	−	1.18796e9i	−0.0774984	−	0.134231i
$696$		0			0
$697$	7.83190e8	−	1.35652e9i	0.0876097	−	0.151744i
$698$		0			0
$699$	4.87039e9			0.539378
$700$		0			0
$701$	−4.88820e9			−0.535964			−0.267982	−	0.963424i	$-0.586357\pi$
$701$	−4.88820e9			−0.535964			−0.267982	+	0.963424i	$0.586357\pi$
$702$		0			0
$703$	−1.11515e9	+	1.93149e9i	−0.121057	+	0.209677i
$704$		0			0
$705$	8.87647e8	+	1.53745e9i	0.0954065	+	0.165249i
$706$		0			0
$707$		0			0
$708$		0			0
$709$	3.97553e9	+	6.88582e9i	0.418922	+	0.725594i	0.995831	−	0.0912141i	$-0.0290748\pi$
$709$	3.97553e9	+	6.88582e9i	0.418922	+	0.725594i	−0.576909	+	0.816808i	$0.695741\pi$
$710$		0			0
$711$	−3.63947e8	+	6.30375e8i	−0.0379748	+	0.0657742i
$712$		0			0
$713$	2.33464e8			0.0241216
$714$		0			0
$715$	−9.13756e8			−0.0934887
$716$		0			0
$717$	−3.52527e9	+	6.10594e9i	−0.357170	+	0.618637i
$718$		0			0
$719$	−6.50173e9	−	1.12613e10i	−0.652346	−	1.12990i	−0.982552	−	0.185987i	$-0.940452\pi$
$719$	−6.50173e9	−	1.12613e10i	−0.652346	−	1.12990i	0.330206	−	0.943909i	$-0.392882\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$	2.66971e8	+	4.62407e8i	0.0262712	+	0.0455031i
$724$		0			0
$725$	−7.08861e9	+	1.22778e10i	−0.690841	+	1.19657i
$726$		0			0
$727$	1.73805e10			1.67761			0.838805	−	0.544431i	$-0.183255\pi$
$727$	1.73805e10			1.67761			0.838805	+	0.544431i	$0.183255\pi$
$728$		0			0
$729$	3.87420e8			0.0370370
$730$		0			0
$731$	−2.75211e8	+	4.76680e8i	−0.0260589	+	0.0451353i
$732$		0			0
$733$	4.73557e9	+	8.20225e9i	0.444128	+	0.769253i	0.997991	−	0.0633553i	$-0.0201801\pi$
$733$	4.73557e9	+	8.20225e9i	0.444128	+	0.769253i	−0.553863	+	0.832608i	$0.686847\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−2.94485e9	−	5.10062e9i	−0.270973	−	0.469339i
$738$		0			0
$739$	−2.76457e9	+	4.78837e9i	−0.251983	+	0.436448i	−0.964072	−	0.265642i	$-0.914416\pi$
$739$	−2.76457e9	+	4.78837e9i	−0.251983	+	0.436448i	0.712088	+	0.702090i	$0.247749\pi$
$740$		0			0
$741$	−5.90904e8			−0.0533523
$742$		0			0
$743$	−1.53701e10			−1.37472			−0.687362	−	0.726315i	$-0.741231\pi$
$743$	−1.53701e10			−1.37472			−0.687362	+	0.726315i	$0.741231\pi$
$744$		0			0
$745$	−3.53552e9	+	6.12370e9i	−0.313261	+	0.542584i
$746$		0			0
$747$	1.42082e9	+	2.46094e9i	0.124715	+	0.216012i
$748$		0			0
$749$		0			0
$750$		0			0
$751$	4.25975e9	+	7.37811e9i	0.366982	+	0.635631i	0.989092	−	0.147299i	$-0.0470578\pi$
$751$	4.25975e9	+	7.37811e9i	0.366982	+	0.635631i	−0.622110	+	0.782930i	$0.713724\pi$
$752$		0			0
$753$	2.07589e9	−	3.59555e9i	0.177183	−	0.306890i
$754$		0			0
$755$	2.56970e9			0.217304
$756$		0			0
$757$	−4.72648e8			−0.0396006			−0.0198003	−	0.999804i	$-0.506303\pi$
$757$	−4.72648e8			−0.0396006			−0.0198003	+	0.999804i	$0.506303\pi$
$758$		0			0
$759$	−7.42239e7	+	1.28560e8i	−0.00616166	+	0.0106723i
$760$		0			0
$761$	−7.17326e9	−	1.24245e10i	−0.590025	−	1.02195i	−0.994228	−	0.107284i	$-0.965785\pi$
$761$	−7.17326e9	−	1.24245e10i	−0.590025	−	1.02195i	0.404203	−	0.914669i	$-0.367549\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$	−2.15055e8	−	3.72486e8i	−0.0173674	−	0.0300812i
$766$		0			0
$767$	−8.82990e8	+	1.52938e9i	−0.0706597	+	0.122386i
$768$		0			0
$769$	−1.39480e10			−1.10603			−0.553017	−	0.833170i	$-0.686524\pi$
$769$	−1.39480e10			−1.10603			−0.553017	+	0.833170i	$0.686524\pi$
$770$		0			0
$771$	3.87073e9			0.304160
$772$		0			0
$773$	−2.16169e9	+	3.74415e9i	−0.168331	+	0.291558i	−0.937833	−	0.347086i	$-0.887171\pi$
$773$	−2.16169e9	+	3.74415e9i	−0.168331	+	0.291558i	0.769502	+	0.638644i	$0.220504\pi$
$774$		0			0
$775$	4.01229e9	+	6.94949e9i	0.309625	+	0.536287i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	8.81951e8	+	1.52758e9i	0.0668442	+	0.115777i
$780$		0			0
$781$	1.65221e9	−	2.86172e9i	0.124105	−	0.214955i
$782$		0			0
$783$	−4.09615e9			−0.304937

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:	\( N \)	\(=\)	\( 588 = 2^{2} \cdot 3 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	588.i (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-13.5000 + 23.3827i) q^{3} +(50.0000 + 86.6025i) q^{5} +(-364.500 - 631.333i) q^{9} +(-1387.00 + 2402.35i) q^{11} +3294.00 q^{13} -2700.00 q^{15} +(2950.00 - 5109.55i) q^{17} +(3322.00 + 5753.87i) q^{19} +(-991.000 - 1716.46i) q^{23} +(34062.5 - 58998.0i) q^{25} +19683.0 q^{27} -208106. q^{29} +(-58896.0 + 102011. i) q^{31} +(-37449.0 - 64863.6i) q^{33} +(167843. + 290713. i) q^{37} +(-44469.0 + 77022.6i) q^{39} +265488. q^{41} -93292.0 q^{43} +(36450.0 - 63133.3i) q^{45} +(-328758. - 569426. i) q^{47} +(79650.0 + 137958. i) q^{51} +(304359. - 527165. i) q^{53} -277400. q^{55} -179388. q^{57} +(-268060. + 464294. i) q^{59} +(-898545. - 1.55633e6i) q^{61} +(164700. + 285269. i) q^{65} +(-1.06159e6 + 1.83872e6i) q^{67} +53514.0 q^{69} -1.19121e6 q^{71} +(528215. - 914895. i) q^{73} +(919688. + 1.59295e6i) q^{75} +(-499242. - 864713. i) q^{79} +(-265720. + 460241. i) q^{81} -3.89800e6 q^{83} +590000. q^{85} +(2.80943e6 - 4.86608e6i) q^{87} +(-2.31118e6 - 4.00307e6i) q^{89} +(-1.59019e6 - 2.75429e6i) q^{93} +(-332200. + 575387. i) q^{95} -1.52877e7 q^{97} +2.02225e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 27 q^{3} + 100 q^{5} - 729 q^{9} - 2774 q^{11} + 6588 q^{13} - 5400 q^{15} + 5900 q^{17} + 6644 q^{19} - 1982 q^{23} + 68125 q^{25} + 39366 q^{27} - 416212 q^{29} - 117792 q^{31} - 74898 q^{33} + 335686 q^{37}+ \cdots + 4044492 q^{99}+O(q^{100}) \) 2 * q - 27 * q^3 + 100 * q^5 - 729 * q^9 - 2774 * q^11 + 6588 * q^13 - 5400 * q^15 + 5900 * q^17 + 6644 * q^19 - 1982 * q^23 + 68125 * q^25 + 39366 * q^27 - 416212 * q^29 - 117792 * q^31 - 74898 * q^33 + 335686 * q^37 - 88938 * q^39 + 530976 * q^41 - 186584 * q^43 + 72900 * q^45 - 657516 * q^47 + 159300 * q^51 + 608718 * q^53 - 554800 * q^55 - 358776 * q^57 - 536120 * q^59 - 1797090 * q^61 + 329400 * q^65 - 2123176 * q^67 + 107028 * q^69 - 2382428 * q^71 + 1056430 * q^73 + 1839375 * q^75 - 998484 * q^79 - 531441 * q^81 - 7796008 * q^83 + 1180000 * q^85 + 5618862 * q^87 - 4622352 * q^89 - 3180384 * q^93 - 664400 * q^95 - 30575420 * q^97 + 4044492 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more