LMFDB - Embedded newform 600.3.c.d.449.7

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [600,3,Mod(449,600)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("600.449"); S:= CuspForms(chi, 3); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(600, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1, 1])) N = Newforms(chi, 3, names="a")

Level:	$ N $	$=$	$ 600 = 2^{3} \cdot 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	600.c (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [16,0,0,0,0,0,0,0,-40,0,0,0,0,0,0,0,0,0,16] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(19)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$16.3488158616$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} + \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} + 138x^{12} + 3393x^{8} + 15208x^{4} + 1296 $ x^16 + 138x^12 + 3393x^8 + 15208*x^4 + 1296
Coefficient ring:	$\Z[a_1, \ldots, a_{37}]$
Coefficient ring index:	$ 2^{26}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 120)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			449.7
Root			$0.383916 + 0.383916i$ of defining polynomial
Character	$\chi$	$=$	600.449
Dual form			600.3.c.d.449.8

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.323191 - 2.98254i) q^{3} +4.72640i q^{7} +(-8.79110 + 1.92786i) q^{9} +4.76442i q^{11} +1.06692i q^{13} +26.7847 q^{17} +8.12938 q^{19} +(14.0967 - 1.52753i) q^{21} +40.0468 q^{23} +(8.59112 + 25.5967i) q^{27} +20.8744i q^{29} -33.7860 q^{31} +(14.2101 - 1.53982i) q^{33} -60.4351i q^{37} +(3.18213 - 0.344819i) q^{39} -59.2611i q^{41} +56.4424i q^{43} -9.68942 q^{47} +26.6611 q^{49} +(-8.65657 - 79.8864i) q^{51} +93.1378 q^{53} +(-2.62734 - 24.2462i) q^{57} +17.4907i q^{59} +57.7400 q^{61} +(-9.11184 - 41.5503i) q^{63} +101.531i q^{67} +(-12.9428 - 119.441i) q^{69} -90.1745i q^{71} -40.0700i q^{73} -22.5186 q^{77} -65.3727 q^{79} +(73.5667 - 33.8960i) q^{81} +117.888 q^{83} +(62.2587 - 6.74641i) q^{87} +119.679i q^{89} -5.04269 q^{91} +(10.9193 + 100.768i) q^{93} -15.2522i q^{97} +(-9.18513 - 41.8845i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 40 q^{9} + 16 q^{19} + 56 q^{21} + 240 q^{31} + 144 q^{39} - 128 q^{49} + 128 q^{51} + 16 q^{61} - 200 q^{69} - 176 q^{79} + 448 q^{81} + 1120 q^{91} - 64 q^{99}+O(q^{100}) $ 16 * q - 40 * q^9 + 16 * q^19 + 56 * q^21 + 240 * q^31 + 144 * q^39 - 128 * q^49 + 128 * q^51 + 16 * q^61 - 200 * q^69 - 176 * q^79 + 448 * q^81 + 1120 * q^91 - 64 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/600\mathbb{Z}\right)^\times$.

$n$	$151$	$301$	$401$	$577$
$\chi(n)$	$1$	$1$	$-1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$	−0.323191	−	2.98254i	−0.107730	−	0.994180i
$3$	−0.323191	−	2.98254i	−0.107730	−	0.994180i
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$			4.72640i			0.675201i	0.941290	+	0.337600i	$0.109615\pi$
$7$			4.72640i			0.675201i	−0.941290	+	0.337600i	$0.890385\pi$
$8$		0			0
$9$	−8.79110	+	1.92786i	−0.976788	+	0.214207i
$10$		0			0
$11$			4.76442i			0.433129i	0.976268	+	0.216565i	$0.0694852\pi$
$11$			4.76442i			0.433129i	−0.976268	+	0.216565i	$0.930515\pi$
$12$		0			0
$13$			1.06692i			0.0820707i	0.999158	+	0.0410354i	$0.0130656\pi$
$13$			1.06692i			0.0820707i	−0.999158	+	0.0410354i	$0.986934\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	26.7847			1.57557			0.787785	−	0.615950i	$-0.211228\pi$
$17$	26.7847			1.57557			0.787785	+	0.615950i	$0.211228\pi$
$18$		0			0
$19$	8.12938			0.427862			0.213931	−	0.976849i	$-0.431373\pi$
$19$	8.12938			0.427862			0.213931	+	0.976849i	$0.431373\pi$
$20$		0			0
$21$	14.0967	−	1.52753i	0.671271	−	0.0727395i
$22$		0			0
$23$	40.0468			1.74117			0.870583	−	0.492021i	$-0.163742\pi$
$23$	40.0468			1.74117			0.870583	+	0.492021i	$0.163742\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$	8.59112	+	25.5967i	0.318190	+	0.948027i
$28$		0			0
$29$			20.8744i			0.719807i	0.932990	+	0.359903i	$0.117190\pi$
$29$			20.8744i			0.719807i	−0.932990	+	0.359903i	$0.882810\pi$
$30$		0			0
$31$	−33.7860			−1.08987			−0.544936	−	0.838478i	$-0.683446\pi$
$31$	−33.7860			−1.08987			−0.544936	+	0.838478i	$0.683446\pi$
$32$		0			0
$33$	14.2101	−	1.53982i	0.430608	−	0.0466611i
$34$		0			0
$35$		0			0
$36$		0			0
$37$		−	60.4351i		−	1.63338i	−0.577075	−	0.816691i	$-0.695806\pi$
$37$		−	60.4351i		−	1.63338i	0.577075	−	0.816691i	$-0.304194\pi$
$38$		0			0
$39$	3.18213	−	0.344819i	0.0815931	−	0.00884150i
$40$		0			0
$41$		−	59.2611i		−	1.44539i	−0.691166	−	0.722696i	$-0.742903\pi$
$41$		−	59.2611i		−	1.44539i	0.691166	−	0.722696i	$-0.257097\pi$
$42$		0			0
$43$			56.4424i			1.31261i	0.754494	+	0.656307i	$0.227883\pi$
$43$			56.4424i			1.31261i	−0.754494	+	0.656307i	$0.772117\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−9.68942			−0.206158			−0.103079	−	0.994673i	$-0.532869\pi$
$47$	−9.68942			−0.206158			−0.103079	+	0.994673i	$0.532869\pi$
$48$		0			0
$49$	26.6611			0.544104
$50$		0			0
$51$	−8.65657	−	79.8864i	−0.169737	−	1.56640i
$52$		0			0
$53$	93.1378			1.75732			0.878659	−	0.477451i	$-0.158439\pi$
$53$	93.1378			1.75732			0.878659	+	0.477451i	$0.158439\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	−2.62734	−	24.2462i	−0.0460937	−	0.425372i
$58$		0			0
$59$			17.4907i			0.296453i	0.988953	+	0.148227i	$0.0473565\pi$
$59$			17.4907i			0.296453i	−0.988953	+	0.148227i	$0.952643\pi$
$60$		0			0
$61$	57.7400			0.946558			0.473279	−	0.880913i	$-0.343070\pi$
$61$	57.7400			0.946558			0.473279	+	0.880913i	$0.343070\pi$
$62$		0			0
$63$	−9.11184	−	41.5503i	−0.144632	−	0.659528i
$64$		0			0
$65$		0			0
$66$		0			0
$67$			101.531i			1.51539i	0.652609	+	0.757695i	$0.273675\pi$
$67$			101.531i			1.51539i	−0.652609	+	0.757695i	$0.726325\pi$
$68$		0			0
$69$	−12.9428	−	119.441i	−0.187576	−	1.73103i
$70$		0			0
$71$		−	90.1745i		−	1.27006i	−0.772486	−	0.635032i	$-0.780987\pi$
$71$		−	90.1745i		−	1.27006i	0.772486	−	0.635032i	$-0.219013\pi$
$72$		0			0
$73$		−	40.0700i		−	0.548904i	−0.961601	−	0.274452i	$-0.911504\pi$
$73$		−	40.0700i		−	0.548904i	0.961601	−	0.274452i	$-0.0884965\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−22.5186			−0.292449
$78$		0			0
$79$	−65.3727			−0.827502			−0.413751	−	0.910390i	$-0.635782\pi$
$79$	−65.3727			−0.827502			−0.413751	+	0.910390i	$0.635782\pi$
$80$		0			0
$81$	73.5667	−	33.8960i	0.908231	−	0.418469i
$82$		0			0
$83$	117.888			1.42033			0.710166	−	0.704034i	$-0.248620\pi$
$83$	117.888			1.42033			0.710166	+	0.704034i	$0.248620\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	62.2587	−	6.74641i	0.715618	−	0.0775450i
$88$		0			0
$89$			119.679i			1.34471i	0.740228	+	0.672356i	$0.234718\pi$
$89$			119.679i			1.34471i	−0.740228	+	0.672356i	$0.765282\pi$
$90$		0			0
$91$	−5.04269			−0.0554142
$92$		0			0
$93$	10.9193	+	100.768i	0.117412	+	1.08353i
$94$		0			0
$95$		0			0
$96$		0			0
$97$		−	15.2522i		−	0.157239i	−0.996905	−	0.0786196i	$-0.974949\pi$
$97$		−	15.2522i		−	0.157239i	0.996905	−	0.0786196i	$-0.0250513\pi$
$98$		0			0
$99$	−9.18513	−	41.8845i	−0.0927791	−	0.423075i
$100$		0			0
$101$			72.0047i			0.712918i	0.934311	+	0.356459i	$0.116016\pi$
$101$			72.0047i			0.712918i	−0.934311	+	0.356459i	$0.883984\pi$
$102$		0			0
$103$		−	110.950i		−	1.07718i	−0.842567	−	0.538591i	$-0.818957\pi$
$103$		−	110.950i		−	1.07718i	0.842567	−	0.538591i	$-0.181043\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−30.6020			−0.286000			−0.143000	−	0.989723i	$-0.545675\pi$
$107$	−30.6020			−0.286000			−0.143000	+	0.989723i	$0.545675\pi$
$108$		0			0
$109$	17.3694			0.159353			0.0796763	−	0.996821i	$-0.474611\pi$
$109$	17.3694			0.159353			0.0796763	+	0.996821i	$0.474611\pi$
$110$		0			0
$111$	−180.250	+	19.5321i	−1.62388	+	0.175965i
$112$		0			0
$113$	−4.71526			−0.0417280			−0.0208640	−	0.999782i	$-0.506642\pi$
$113$	−4.71526			−0.0417280			−0.0208640	+	0.999782i	$0.506642\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	−2.05687	−	9.37939i	−0.0175801	−	0.0801657i
$118$		0			0
$119$			126.595i			1.06383i
$120$		0			0
$121$	98.3003			0.812399
$122$		0			0
$123$	−176.749	+	19.1526i	−1.43698	+	0.155712i
$124$		0			0
$125$		0			0
$126$		0			0
$127$		−	1.39528i		−	0.0109865i	−0.999985	−	0.00549325i	$-0.998251\pi$
$127$		−	1.39528i		−	0.0109865i	0.999985	−	0.00549325i	$-0.00174856\pi$
$128$		0			0
$129$	168.342	−	18.2417i	1.30498	−	0.141408i
$130$		0			0
$131$			226.220i			1.72687i	0.504460	+	0.863435i	$0.331692\pi$
$131$			226.220i			1.72687i	−0.504460	+	0.863435i	$0.668308\pi$
$132$		0			0
$133$			38.4227i			0.288893i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	76.2589			0.556634			0.278317	−	0.960489i	$-0.410223\pi$
$137$	76.2589			0.556634			0.278317	+	0.960489i	$0.410223\pi$
$138$		0			0
$139$	−127.660			−0.918417			−0.459208	−	0.888329i	$-0.651867\pi$
$139$	−127.660			−0.918417			−0.459208	+	0.888329i	$0.651867\pi$
$140$		0			0
$141$	3.13153	+	28.8991i	0.0222094	+	0.204958i
$142$		0			0
$143$	−5.08325			−0.0355472
$144$		0			0
$145$		0			0
$146$		0			0
$147$	−8.61662	−	79.5178i	−0.0586165	−	0.540938i
$148$		0			0
$149$			63.3790i			0.425362i	0.977122	+	0.212681i	$0.0682195\pi$
$149$			63.3790i			0.425362i	−0.977122	+	0.212681i	$0.931780\pi$
$150$		0			0
$151$	−115.233			−0.763134			−0.381567	−	0.924341i	$-0.624615\pi$
$151$	−115.233			−0.763134			−0.381567	+	0.924341i	$0.624615\pi$
$152$		0			0
$153$	−235.467	+	51.6371i	−1.53900	+	0.337498i
$154$		0			0
$155$		0			0
$156$		0			0
$157$		−	18.3695i		−	0.117003i	−0.998287	−	0.0585017i	$-0.981368\pi$
$157$		−	18.3695i		−	0.117003i	0.998287	−	0.0585017i	$-0.0186323\pi$
$158$		0			0
$159$	−30.1013	−	277.787i	−0.189316	−	1.74709i
$160$		0			0
$161$			189.278i			1.17564i
$162$		0			0
$163$			163.693i			1.00425i	0.864794	+	0.502126i	$0.167449\pi$
$163$			163.693i			1.00425i	−0.864794	+	0.502126i	$0.832551\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	48.7025			0.291632			0.145816	−	0.989312i	$-0.453419\pi$
$167$	48.7025			0.291632			0.145816	+	0.989312i	$0.453419\pi$
$168$		0			0
$169$	167.862			0.993264
$170$		0			0
$171$	−71.4662	+	15.6723i	−0.417931	+	0.0916509i
$172$		0			0
$173$	−140.785			−0.813787			−0.406894	−	0.913476i	$-0.633388\pi$
$173$	−140.785			−0.813787			−0.406894	+	0.913476i	$0.633388\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	52.1669	−	5.65285i	0.294728	−	0.0319370i
$178$		0			0
$179$			11.1350i			0.0622065i	0.999516	+	0.0311033i	$0.00990207\pi$
$179$			11.1350i			0.0622065i	−0.999516	+	0.0311033i	$0.990098\pi$
$180$		0			0
$181$	−150.235			−0.830028			−0.415014	−	0.909815i	$-0.636223\pi$
$181$	−150.235			−0.830028			−0.415014	+	0.909815i	$0.636223\pi$
$182$		0			0
$183$	−18.6610	−	172.212i	−0.101973	−	0.941049i
$184$		0			0
$185$		0			0
$186$		0			0
$187$			127.614i			0.682425i
$188$		0			0
$189$	−120.981	+	40.6051i	−0.640108	+	0.214842i
$190$		0			0
$191$			168.060i			0.879895i	0.898023	+	0.439948i	$0.145003\pi$
$191$			168.060i			0.879895i	−0.898023	+	0.439948i	$0.854997\pi$
$192$		0			0
$193$		−	312.926i		−	1.62138i	−0.585479	−	0.810688i	$-0.699093\pi$
$193$		−	312.926i		−	1.62138i	0.585479	−	0.810688i	$-0.300907\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−166.150			−0.843400			−0.421700	−	0.906735i	$-0.638566\pi$
$197$	−166.150			−0.843400			−0.421700	+	0.906735i	$0.638566\pi$
$198$		0			0
$199$	105.535			0.530327			0.265163	−	0.964204i	$-0.414574\pi$
$199$	105.535			0.530327			0.265163	+	0.964204i	$0.414574\pi$
$200$		0			0
$201$	302.821	−	32.8139i	1.50657	−	0.163253i
$202$		0			0
$203$	−98.6608			−0.486014
$204$		0			0
$205$		0			0
$206$		0			0
$207$	−352.056	+	77.2046i	−1.70075	+	0.372969i
$208$		0			0
$209$			38.7318i			0.185320i
$210$		0			0
$211$	283.373			1.34300			0.671500	−	0.741004i	$-0.265650\pi$
$211$	283.373			1.34300			0.671500	+	0.741004i	$0.265650\pi$
$212$		0			0
$213$	−268.949	+	29.1436i	−1.26267	+	0.136824i
$214$		0			0
$215$		0			0
$216$		0			0
$217$		−	159.686i		−	0.735882i
$218$		0			0
$219$	−119.510	+	12.9503i	−0.545710	+	0.0591336i
$220$		0			0
$221$			28.5771i			0.129308i
$222$		0			0
$223$			100.108i			0.448915i	0.974484	+	0.224458i	$0.0720610\pi$
$223$			100.108i			0.448915i	−0.974484	+	0.224458i	$0.927939\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−239.794			−1.05636			−0.528181	−	0.849132i	$-0.677126\pi$
$227$	−239.794			−1.05636			−0.528181	+	0.849132i	$0.677126\pi$
$228$		0			0
$229$	−393.036			−1.71632			−0.858158	−	0.513386i	$-0.828391\pi$
$229$	−393.036			−1.71632			−0.858158	+	0.513386i	$0.828391\pi$
$230$		0			0
$231$	7.27779	+	67.1626i	0.0315056	+	0.290747i
$232$		0			0
$233$	278.691			1.19610			0.598050	−	0.801459i	$-0.295942\pi$
$233$	278.691			1.19610			0.598050	+	0.801459i	$0.295942\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	21.1278	+	194.977i	0.0891470	+	0.822686i
$238$		0			0
$239$			196.594i			0.822570i	0.911507	+	0.411285i	$0.134920\pi$
$239$			196.594i			0.822570i	−0.911507	+	0.411285i	$0.865080\pi$
$240$		0			0
$241$	−231.153			−0.959139			−0.479570	−	0.877504i	$-0.659207\pi$
$241$	−231.153			−0.959139			−0.479570	+	0.877504i	$0.659207\pi$
$242$		0			0
$243$	−124.872	−	208.461i	−0.513877	−	0.857864i
$244$		0			0
$245$		0			0
$246$		0			0
$247$			8.67340i			0.0351150i
$248$		0			0
$249$	−38.1002	−	351.604i	−0.153013	−	1.41207i
$250$		0			0
$251$		−	243.442i		−	0.969887i	−0.874545	−	0.484944i	$-0.838840\pi$
$251$		−	243.442i		−	0.969887i	0.874545	−	0.484944i	$-0.161160\pi$
$252$		0			0
$253$			190.800i			0.754150i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−358.482			−1.39487			−0.697437	−	0.716646i	$-0.745676\pi$
$257$	−358.482			−1.39487			−0.697437	+	0.716646i	$0.745676\pi$
$258$		0			0
$259$	285.641			1.10286
$260$		0			0
$261$	−40.2429	−	183.509i	−0.154187	−	0.703099i
$262$		0			0
$263$	−330.913			−1.25823			−0.629113	−	0.777314i	$-0.716582\pi$
$263$	−330.913			−1.25823			−0.629113	+	0.777314i	$0.716582\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	356.949	−	38.6793i	1.33689	−	0.144866i
$268$		0			0
$269$		−	97.0354i		−	0.360727i	−0.983600	−	0.180363i	$-0.942273\pi$
$269$		−	97.0354i		−	0.360727i	0.983600	−	0.180363i	$-0.0577273\pi$
$270$		0			0
$271$	67.1851			0.247915			0.123958	−	0.992288i	$-0.460441\pi$
$271$	67.1851			0.247915			0.123958	+	0.992288i	$0.460441\pi$
$272$		0			0
$273$	1.62975	+	15.0400i	0.00596979	+	0.0550917i
$274$		0			0
$275$		0			0
$276$		0			0
$277$		−	361.801i		−	1.30614i	−0.757298	−	0.653070i	$-0.773481\pi$
$277$		−	361.801i		−	1.30614i	0.757298	−	0.653070i	$-0.226519\pi$
$278$		0			0
$279$	297.016	−	65.1347i	1.06457	−	0.233458i
$280$		0			0
$281$			288.193i			1.02560i	0.858509	+	0.512798i	$0.171391\pi$
$281$			288.193i			1.02560i	−0.858509	+	0.512798i	$0.828609\pi$
$282$		0			0
$283$			272.474i			0.962805i	0.876500	+	0.481402i	$0.159872\pi$
$283$			272.474i			0.962805i	−0.876500	+	0.481402i	$0.840128\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	280.092			0.975930
$288$		0			0
$289$	428.420			1.48242
$290$		0			0
$291$	−45.4903	+	4.92937i	−0.156324	+	0.0169394i
$292$		0			0
$293$	70.5674			0.240845			0.120422	−	0.992723i	$-0.461575\pi$
$293$	70.5674			0.240845			0.120422	+	0.992723i	$0.461575\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	−121.954	+	40.9317i	−0.410618	+	0.137817i
$298$		0			0
$299$			42.7267i			0.142899i
$300$		0			0
$301$	−266.770			−0.886278
$302$		0			0
$303$	214.757	−	23.2713i	0.708769	−	0.0768028i
$304$		0			0
$305$		0			0
$306$		0			0
$307$		−	124.274i		−	0.404801i	−0.979303	−	0.202401i	$-0.935126\pi$
$307$		−	124.274i		−	0.404801i	0.979303	−	0.202401i	$-0.0648744\pi$
$308$		0			0
$309$	−330.912	+	35.8580i	−1.07091	+	0.116045i
$310$		0			0
$311$		−	229.006i		−	0.736353i	−0.929756	−	0.368176i	$-0.879982\pi$
$311$		−	229.006i		−	0.736353i	0.929756	−	0.368176i	$-0.120018\pi$
$312$		0			0
$313$		−	465.490i		−	1.48719i	−0.668631	−	0.743594i	$-0.733120\pi$
$313$		−	465.490i		−	1.48719i	0.668631	−	0.743594i	$-0.266880\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	405.458			1.27905			0.639524	−	0.768771i	$-0.279131\pi$
$317$	405.458			1.27905			0.639524	+	0.768771i	$0.279131\pi$
$318$		0			0
$319$	−99.4544			−0.311769
$320$		0			0
$321$	9.89027	+	91.2716i	0.0308108	+	0.284335i
$322$		0			0
$323$	217.743			0.674127
$324$		0			0
$325$		0			0
$326$		0			0
$327$	−5.61364	−	51.8051i	−0.0171671	−	0.158425i
$328$		0			0
$329$		−	45.7961i		−	0.139198i
$330$		0			0
$331$	45.9271			0.138753			0.0693763	−	0.997591i	$-0.477899\pi$
$331$	45.9271			0.138753			0.0693763	+	0.997591i	$0.477899\pi$
$332$		0			0
$333$	116.510	+	531.291i	0.349881	+	1.59547i
$334$		0			0
$335$		0			0
$336$		0			0
$337$			516.924i			1.53390i	0.641707	+	0.766950i	$0.278226\pi$
$337$			516.924i			1.53390i	−0.641707	+	0.766950i	$0.721774\pi$
$338$		0			0
$339$	1.52393	+	14.0635i	0.00449537	+	0.0414852i
$340$		0			0
$341$		−	160.971i		−	0.472055i
$342$		0			0
$343$			357.605i			1.04258i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−5.39280			−0.0155412			−0.00777061	−	0.999970i	$-0.502473\pi$
$347$	−5.39280			−0.0155412			−0.00777061	+	0.999970i	$0.502473\pi$
$348$		0			0
$349$	−284.894			−0.816315			−0.408157	−	0.912912i	$-0.633829\pi$
$349$	−284.894			−0.816315			−0.408157	+	0.912912i	$0.633829\pi$
$350$		0			0
$351$	−27.3097	+	9.16603i	−0.0778053	+	0.0261141i
$352$		0			0
$353$	73.2882			0.207615			0.103808	−	0.994597i	$-0.466897\pi$
$353$	73.2882			0.207615			0.103808	+	0.994597i	$0.466897\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$	377.576	−	40.9144i	1.05763	−	0.114606i
$358$		0			0
$359$			361.674i			1.00745i	0.863865	+	0.503724i	$0.168037\pi$
$359$			361.674i			1.00745i	−0.863865	+	0.503724i	$0.831963\pi$
$360$		0			0
$361$	−294.913			−0.816934
$362$		0			0
$363$	−31.7697	−	293.185i	−0.0875200	−	0.807671i
$364$		0			0
$365$		0			0
$366$		0			0
$367$		−	131.432i		−	0.358124i	−0.983838	−	0.179062i	$-0.942694\pi$
$367$		−	131.432i		−	0.358124i	0.983838	−	0.179062i	$-0.0573063\pi$
$368$		0			0
$369$	114.247	+	520.970i	0.309612	+	1.41184i
$370$		0			0
$371$			440.207i			1.18654i
$372$		0			0
$373$		−	211.216i		−	0.566262i	−0.959081	−	0.283131i	$-0.908627\pi$
$373$		−	211.216i		−	0.566262i	0.959081	−	0.283131i	$-0.0913731\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−22.2713			−0.0590751
$378$		0			0
$379$	−47.4989			−0.125327			−0.0626635	−	0.998035i	$-0.519959\pi$
$379$	−47.4989			−0.125327			−0.0626635	+	0.998035i	$0.519959\pi$
$380$		0			0
$381$	−4.16149	+	0.450943i	−0.0109226	+	0.00118358i
$382$		0			0
$383$	517.991			1.35246			0.676228	−	0.736692i	$-0.263613\pi$
$383$	517.991			1.35246			0.676228	+	0.736692i	$0.263613\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−108.813	−	496.191i	−0.281171	−	1.28215i
$388$		0			0
$389$			253.951i			0.652830i	0.945227	+	0.326415i	$0.105841\pi$
$389$			253.951i			0.652830i	−0.945227	+	0.326415i	$0.894159\pi$
$390$		0			0
$391$	1072.64			2.74333
$392$		0			0
$393$	674.710	−	73.1122i	1.71682	−	0.186036i
$394$		0			0
$395$		0			0
$396$		0			0
$397$			527.905i			1.32974i	0.746961	+	0.664868i	$0.231512\pi$
$397$			527.905i			1.32974i	−0.746961	+	0.664868i	$0.768488\pi$
$398$		0			0
$399$	114.597	−	12.4179i	0.287212	−	0.0311225i
$400$		0			0
$401$		−	664.097i		−	1.65610i	−0.560653	−	0.828051i	$-0.689450\pi$
$401$		−	664.097i		−	1.65610i	0.560653	−	0.828051i	$-0.310550\pi$
$402$		0			0
$403$		−	36.0470i		−	0.0894466i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	287.938			0.707465
$408$		0			0
$409$	−79.2471			−0.193758			−0.0968791	−	0.995296i	$-0.530886\pi$
$409$	−79.2471			−0.193758			−0.0968791	+	0.995296i	$0.530886\pi$
$410$		0			0
$411$	−24.6462	−	227.445i	−0.0599664	−	0.553395i
$412$		0			0
$413$	−82.6683			−0.200165
$414$		0			0
$415$		0			0
$416$		0			0
$417$	41.2585	+	380.751i	0.0989413	+	0.913072i
$418$		0			0
$419$		−	666.530i		−	1.59076i	−0.606109	−	0.795381i	$-0.707271\pi$
$419$		−	666.530i		−	1.59076i	0.606109	−	0.795381i	$-0.292729\pi$
$420$		0			0
$421$	−306.220			−0.727364			−0.363682	−	0.931523i	$-0.618480\pi$
$421$	−306.220			−0.727364			−0.363682	+	0.931523i	$0.618480\pi$
$422$		0			0
$423$	85.1806	−	18.6798i	0.201373	−	0.0441604i
$424$		0			0
$425$		0			0
$426$		0			0
$427$			272.903i			0.639116i
$428$		0			0
$429$	1.64286	+	15.1610i	0.00382951	+	0.0353403i
$430$		0			0
$431$		−	254.551i		−	0.590605i	−0.955404	−	0.295303i	$-0.904579\pi$
$431$		−	254.551i		−	0.590605i	0.955404	−	0.295303i	$-0.0954205\pi$
$432$		0			0
$433$		−	442.391i		−	1.02169i	−0.859673	−	0.510845i	$-0.829333\pi$
$433$		−	442.391i		−	1.02169i	0.859673	−	0.510845i	$-0.170667\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	325.556			0.744979
$438$		0			0
$439$	−561.568			−1.27920			−0.639599	−	0.768709i	$-0.720899\pi$
$439$	−561.568			−1.27920			−0.639599	+	0.768709i	$0.720899\pi$
$440$		0			0
$441$	−234.380	+	51.3988i	−0.531475	+	0.116551i
$442$		0			0
$443$	564.841			1.27504			0.637518	−	0.770436i	$-0.279961\pi$
$443$	564.841			1.27504			0.637518	+	0.770436i	$0.279961\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	189.030	−	20.4835i	0.422887	−	0.0458244i
$448$		0			0
$449$		−	512.215i		−	1.14079i	−0.821370	−	0.570396i	$-0.806790\pi$
$449$		−	512.215i		−	1.14079i	0.821370	−	0.570396i	$-0.193210\pi$
$450$		0			0
$451$	282.345			0.626041
$452$		0			0
$453$	37.2423	+	343.688i	0.0822126	+	0.758692i
$454$		0			0
$455$		0			0
$456$		0			0
$457$			52.5358i			0.114958i	0.998347	+	0.0574790i	$0.0183062\pi$
$457$			52.5358i			0.114958i	−0.998347	+	0.0574790i	$0.981694\pi$
$458$		0			0
$459$	230.110	+	685.601i	0.501330	+	1.49368i
$460$		0			0
$461$		−	625.737i		−	1.35735i	−0.734441	−	0.678673i	$-0.762555\pi$
$461$		−	625.737i		−	1.35735i	0.734441	−	0.678673i	$-0.237445\pi$
$462$		0			0
$463$			49.8782i			0.107728i	0.998548	+	0.0538641i	$0.0171538\pi$
$463$			49.8782i			0.107728i	−0.998548	+	0.0538641i	$0.982846\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	64.6312			0.138397			0.0691983	−	0.997603i	$-0.477956\pi$
$467$	64.6312			0.138397			0.0691983	+	0.997603i	$0.477956\pi$
$468$		0			0
$469$	−479.877			−1.02319
$470$		0			0
$471$	−54.7879	+	5.93686i	−0.116322	+	0.0126048i
$472$		0			0
$473$	−268.915			−0.568532
$474$		0			0
$475$		0			0
$476$		0			0
$477$	−818.783	+	179.557i	−1.71653	+	0.376429i
$478$		0			0
$479$		−	872.673i		−	1.82186i	−0.412556	−	0.910932i	$-0.635364\pi$
$479$		−	872.673i		−	1.82186i	0.412556	−	0.910932i	$-0.364636\pi$
$480$		0			0
$481$	64.4794			0.134053
$482$		0			0
$483$	564.528	−	61.1727i	1.16879	−	0.126652i
$484$		0			0
$485$		0			0
$486$		0			0
$487$		−	883.613i		−	1.81440i	−0.420698	−	0.907201i	$-0.638215\pi$
$487$		−	883.613i		−	1.81440i	0.420698	−	0.907201i	$-0.361785\pi$
$488$		0			0
$489$	488.221	−	52.9041i	0.998408	−	0.108188i
$490$		0			0
$491$		−	596.247i		−	1.21435i	−0.794567	−	0.607177i	$-0.792302\pi$
$491$		−	596.247i		−	1.21435i	0.794567	−	0.607177i	$-0.207698\pi$
$492$		0			0
$493$			559.114i			1.13411i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	426.201			0.857548
$498$		0			0
$499$	560.109			1.12246			0.561231	−	0.827659i	$-0.310328\pi$
$499$	560.109			1.12246			0.561231	+	0.827659i	$0.310328\pi$
$500$		0			0
$501$	−15.7402	−	145.257i	−0.0314175	−	0.289934i
$502$		0			0
$503$	−505.038			−1.00405			−0.502026	−	0.864853i	$-0.667412\pi$
$503$	−505.038			−1.00405			−0.502026	+	0.864853i	$0.667412\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−54.2513	−	500.654i	−0.107005	−	0.987484i
$508$		0			0
$509$			13.3027i			0.0261350i	0.999915	+	0.0130675i	$0.00415963\pi$
$509$			13.3027i			0.0261350i	−0.999915	+	0.0130675i	$0.995840\pi$
$510$		0			0
$511$	189.387			0.370620
$512$		0			0
$513$	69.8405	+	208.086i	0.136141	+	0.405625i
$514$		0			0
$515$		0			0
$516$		0			0
$517$		−	46.1645i		−	0.0892930i
$518$		0			0
$519$	45.5005	+	419.898i	0.0876695	+	0.809051i
$520$		0			0
$521$		−	267.898i		−	0.514200i	−0.966385	−	0.257100i	$-0.917233\pi$
$521$		−	267.898i		−	0.514200i	0.966385	−	0.257100i	$-0.0827670\pi$
$522$		0			0
$523$		−	176.493i		−	0.337463i	−0.985662	−	0.168731i	$-0.946033\pi$
$523$		−	176.493i		−	0.337463i	0.985662	−	0.168731i	$-0.0539671\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−904.949			−1.71717
$528$		0			0
$529$	1074.75			2.03166
$530$		0			0
$531$	−33.7197	−	153.763i	−0.0635022	−	0.289572i
$532$		0			0
$533$	63.2268			0.118624
$534$		0			0
$535$		0			0
$536$		0			0
$537$	33.2105	−	3.59872i	0.0618445	−	0.00670152i
$538$		0			0
$539$			127.025i			0.235667i
$540$		0			0
$541$	−790.757			−1.46166			−0.730829	−	0.682561i	$-0.760866\pi$
$541$	−790.757			−1.46166			−0.730829	+	0.682561i	$0.760866\pi$
$542$		0			0
$543$	48.5546	+	448.082i	0.0894191	+	0.825198i
$544$		0			0
$545$		0			0
$546$		0			0
$547$		−	44.1668i		−	0.0807436i	−0.999185	−	0.0403718i	$-0.987146\pi$
$547$		−	44.1668i		−	0.0807436i	0.999185	−	0.0403718i	$-0.0128542\pi$
$548$		0			0
$549$	−507.598	+	111.315i	−0.924587	+	0.202759i
$550$		0			0
$551$			169.696i			0.307978i
$552$		0			0
$553$		−	308.978i		−	0.558730i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−753.685			−1.35312			−0.676558	−	0.736390i	$-0.736529\pi$
$557$	−753.685			−1.35312			−0.676558	+	0.736390i	$0.736529\pi$
$558$		0			0
$559$	−60.2195			−0.107727
$560$		0			0
$561$	380.613	−	41.2435i	0.678454	−	0.0735179i
$562$		0			0
$563$	−609.590			−1.08275			−0.541376	−	0.840780i	$-0.682097\pi$
$563$	−609.590			−1.08275			−0.541376	+	0.840780i	$0.682097\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	160.206	+	347.706i	0.282550	+	0.613238i
$568$		0			0
$569$			528.082i			0.928088i	0.885812	+	0.464044i	$0.153602\pi$
$569$			528.082i			0.928088i	−0.885812	+	0.464044i	$0.846398\pi$
$570$		0			0
$571$	−708.097			−1.24010			−0.620050	−	0.784562i	$-0.712888\pi$
$571$	−708.097			−1.24010			−0.620050	+	0.784562i	$0.712888\pi$
$572$		0			0
$573$	501.246	−	54.3154i	0.874774	−	0.0947913i
$574$		0			0
$575$		0			0
$576$		0			0
$577$		−	55.0939i		−	0.0954834i	−0.998860	−	0.0477417i	$-0.984798\pi$
$577$		−	55.0939i		−	0.0954834i	0.998860	−	0.0477417i	$-0.0152024\pi$
$578$		0			0
$579$	−933.313	+	101.135i	−1.61194	+	0.174671i
$580$		0			0
$581$			557.184i			0.959009i
$582$		0			0
$583$			443.748i			0.761145i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	596.377			1.01598			0.507988	−	0.861364i	$-0.330390\pi$
$587$	596.377			1.01598			0.507988	+	0.861364i	$0.330390\pi$
$588$		0			0
$589$	−274.660			−0.466315
$590$		0			0
$591$	53.6981	+	495.549i	0.0908597	+	0.838492i
$592$		0			0
$593$	−269.915			−0.455169			−0.227584	−	0.973758i	$-0.573083\pi$
$593$	−269.915			−0.455169			−0.227584	+	0.973758i	$0.573083\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−34.1079	−	314.762i	−0.0571322	−	0.527240i
$598$		0			0
$599$		−	622.634i		−	1.03946i	−0.854332	−	0.519728i	$-0.826033\pi$
$599$		−	622.634i		−	1.03946i	0.854332	−	0.519728i	$-0.173967\pi$
$600$		0			0
$601$	865.760			1.44053			0.720266	−	0.693698i	$-0.244020\pi$
$601$	865.760			1.44053			0.720266	+	0.693698i	$0.244020\pi$
$602$		0			0
$603$	−195.738	−	892.570i	−0.324606	−	1.48022i
$604$		0			0
$605$		0			0
$606$		0			0
$607$			390.780i			0.643790i	0.946775	+	0.321895i	$0.104320\pi$
$607$			390.780i			0.643790i	−0.946775	+	0.321895i	$0.895680\pi$
$608$		0			0
$609$	31.8863	+	294.260i	0.0523584	+	0.483185i
$610$		0			0
$611$		−	10.3378i		−	0.0169195i
$612$		0			0
$613$			398.441i			0.649985i	0.945717	+	0.324993i	$0.105362\pi$
$613$			398.441i			0.649985i	−0.945717	+	0.324993i	$0.894638\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	178.765			0.289732			0.144866	−	0.989451i	$-0.453725\pi$
$617$	178.765			0.289732			0.144866	+	0.989451i	$0.453725\pi$
$618$		0			0
$619$	224.867			0.363274			0.181637	−	0.983366i	$-0.441860\pi$
$619$	224.867			0.363274			0.181637	+	0.983366i	$0.441860\pi$
$620$		0			0
$621$	344.047	+	1025.07i	0.554021	+	1.65067i
$622$		0			0
$623$	−565.653			−0.907950
$624$		0			0
$625$		0			0
$626$		0			0
$627$	115.519	−	12.5178i	0.184241	−	0.0199645i
$628$		0			0
$629$		−	1618.74i		−	2.57351i
$630$		0			0
$631$	−9.36019			−0.0148339			−0.00741695	−	0.999972i	$-0.502361\pi$
$631$	−9.36019			−0.0148339			−0.00741695	+	0.999972i	$0.502361\pi$
$632$		0			0
$633$	−91.5836	−	845.172i	−0.144682	−	1.33518i
$634$		0			0
$635$		0			0
$636$		0			0
$637$			28.4453i			0.0446550i
$638$		0			0
$639$	173.844	+	792.733i	0.272056	+	1.24058i
$640$		0			0
$641$			785.281i			1.22509i	0.790437	+	0.612543i	$0.209854\pi$
$641$			785.281i			1.22509i	−0.790437	+	0.612543i	$0.790146\pi$
$642$		0			0
$643$		−	131.320i		−	0.204230i	−0.994773	−	0.102115i	$-0.967439\pi$
$643$		−	131.320i		−	0.204230i	0.994773	−	0.102115i	$-0.0325610\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	155.225			0.239915			0.119958	−	0.992779i	$-0.461724\pi$
$647$	155.225			0.239915			0.119958	+	0.992779i	$0.461724\pi$
$648$		0			0
$649$	−83.3333			−0.128403
$650$		0			0
$651$	−476.271	+	51.6092i	−0.731599	+	0.0792768i
$652$		0			0
$653$	−668.464			−1.02368			−0.511841	−	0.859080i	$-0.671036\pi$
$653$	−668.464			−1.02368			−0.511841	+	0.859080i	$0.671036\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	77.2493	+	352.259i	0.117579	+	0.536163i
$658$		0			0
$659$			579.510i			0.879377i	0.898150	+	0.439689i	$0.144911\pi$
$659$			579.510i			0.879377i	−0.898150	+	0.439689i	$0.855089\pi$
$660$		0			0
$661$	−307.070			−0.464553			−0.232277	−	0.972650i	$-0.574617\pi$
$661$	−307.070			−0.464553			−0.232277	+	0.972650i	$0.574617\pi$
$662$		0			0
$663$	85.2324	−	9.23586i	0.128556	−	0.0139304i
$664$		0			0
$665$		0			0
$666$		0			0
$667$			835.953i			1.25330i
$668$		0			0
$669$	298.576	−	32.3540i	0.446302	−	0.0483617i
$670$		0			0
$671$			275.098i			0.409982i
$672$		0			0
$673$		−	7.02018i		−	0.0104312i	−0.999986	−	0.00521559i	$-0.998340\pi$
$673$		−	7.02018i		−	0.0104312i	0.999986	−	0.00521559i	$-0.00166018\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−1.21742			−0.00179825			−0.000899127	−	1.00000i	$-0.500286\pi$
$677$	−1.21742			−0.00179825			−0.000899127		1.00000i	$0.500286\pi$
$678$		0			0
$679$	72.0881			0.106168
$680$		0			0
$681$	77.4993	+	715.196i	0.113802	+	1.05021i
$682$		0			0
$683$	−1261.23			−1.84661			−0.923305	−	0.384067i	$-0.874523\pi$
$683$	−1261.23			−1.84661			−0.923305	+	0.384067i	$0.874523\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	127.026	+	1172.25i	0.184899	+	1.70633i
$688$		0			0
$689$			99.3706i			0.144224i
$690$		0			0
$691$	−158.177			−0.228910			−0.114455	−	0.993428i	$-0.536512\pi$
$691$	−158.177			−0.228910			−0.114455	+	0.993428i	$0.536512\pi$
$692$		0			0
$693$	197.963	−	43.4126i	0.285661	−	0.0626445i
$694$		0			0
$695$		0			0
$696$		0			0
$697$		−	1587.29i		−	2.27732i
$698$		0			0
$699$	−90.0705	−	831.208i	−0.128856	−	1.18914i
$700$		0			0
$701$			331.746i			0.473247i	0.971601	+	0.236623i	$0.0760408\pi$
$701$			331.746i			0.473247i	−0.971601	+	0.236623i	$0.923959\pi$
$702$		0			0
$703$		−	491.300i		−	0.698863i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−340.323			−0.481363
$708$		0			0
$709$	−854.366			−1.20503			−0.602515	−	0.798108i	$-0.705835\pi$
$709$	−854.366			−1.20503			−0.602515	+	0.798108i	$0.705835\pi$
$710$		0			0
$711$	574.697	−	126.029i	0.808295	−	0.177256i
$712$		0			0
$713$	−1353.02			−1.89765
$714$		0			0
$715$		0			0
$716$		0			0
$717$	586.350	−	63.5374i	0.817782	−	0.0886156i
$718$		0			0
$719$			218.882i			0.304426i	0.988348	+	0.152213i	$0.0486400\pi$
$719$			218.882i			0.304426i	−0.988348	+	0.152213i	$0.951360\pi$
$720$		0			0
$721$	524.394			0.727314
$722$		0			0
$723$	74.7064	+	689.422i	0.103328	+	0.953557i
$724$		0			0
$725$		0			0
$726$		0			0
$727$			12.2420i			0.0168391i	0.999965	+	0.00841954i	$0.00268006\pi$
$727$			12.2420i			0.0168391i	−0.999965	+	0.00841954i	$0.997320\pi$
$728$		0			0
$729$	−581.385	+	439.809i	−0.797511	+	0.603305i
$730$		0			0
$731$			1511.79i			2.06812i
$732$		0			0
$733$		−	86.8331i		−	0.118463i	−0.998244	−	0.0592313i	$-0.981135\pi$
$733$		−	86.8331i		−	0.118463i	0.998244	−	0.0592313i	$-0.0188650\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−483.737			−0.656360
$738$		0			0
$739$	187.725			0.254026			0.127013	−	0.991901i	$-0.459461\pi$
$739$	187.725			0.254026			0.127013	+	0.991901i	$0.459461\pi$
$740$		0			0
$741$	25.8688	−	2.80316i	0.0349106	−	0.00378294i
$742$		0			0
$743$	178.264			0.239925			0.119963	−	0.992778i	$-0.461723\pi$
$743$	178.264			0.239925			0.119963	+	0.992778i	$0.461723\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	−1036.36	+	227.271i	−1.38736	+	0.304244i
$748$		0			0
$749$		−	144.637i		−	0.193107i
$750$		0			0
$751$	−1328.32			−1.76874			−0.884370	−	0.466786i	$-0.845412\pi$
$751$	−1328.32			−1.76874			−0.884370	+	0.466786i	$0.845412\pi$
$752$		0			0
$753$	−726.075	+	78.6781i	−0.964243	+	0.104486i
$754$		0			0
$755$		0			0
$756$		0			0
$757$		−	545.957i		−	0.721211i	−0.932718	−	0.360606i	$-0.882570\pi$
$757$		−	545.957i		−	0.721211i	0.932718	−	0.360606i	$-0.117430\pi$
$758$		0			0
$759$	569.068	−	61.6648i	0.749761	−	0.0812448i
$760$		0			0
$761$		−	828.655i		−	1.08890i	−0.838793	−	0.544451i	$-0.816738\pi$
$761$		−	828.655i		−	1.08890i	0.838793	−	0.544451i	$-0.183262\pi$
$762$		0			0
$763$			82.0950i			0.107595i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−18.6612			−0.0243301
$768$		0			0
$769$	303.757			0.395002			0.197501	−	0.980303i	$-0.436717\pi$
$769$	303.757			0.395002			0.197501	+	0.980303i	$0.436717\pi$
$770$		0			0
$771$	115.858	+	1069.19i	0.150270	+	1.38676i
$772$		0			0
$773$	−59.2137			−0.0766025			−0.0383013	−	0.999266i	$-0.512195\pi$
$773$	−59.2137			−0.0766025			−0.0383013	+	0.999266i	$0.512195\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	−92.3165	−	851.936i	−0.118811	−	1.09644i
$778$		0			0
$779$		−	481.756i		−	0.618429i
$780$		0			0
$781$	429.629			0.550102
$782$		0			0
$783$	−534.316	+	179.334i	−0.682396	+	0.229035i
$784$		0			0
$785$		0			0
$786$		0			0
$787$			214.281i			0.272276i	0.990690	+	0.136138i	$0.0434691\pi$
$787$			214.281i			0.272276i	−0.990690	+	0.136138i	$0.956531\pi$
$788$		0			0
$789$	106.948	+	986.963i	0.135549	+	1.25090i
$790$		0			0
$791$		−	22.2862i		−	0.0281748i
$792$		0			0
$793$			61.6040i			0.0776847i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	124.487			0.156195			0.0780974	−	0.996946i	$-0.475115\pi$
$797$	124.487			0.156195			0.0780974	+	0.996946i	$0.475115\pi$
$798$		0			0
$799$	−259.528			−0.324816
$800$		0			0
$801$	−230.725	−	1052.11i	−0.288046	−	1.31350i
$802$		0			0
$803$	190.910			0.237746
$804$		0			0
$805$		0			0
$806$		0			0
$807$	−289.412	+	31.3610i	−0.358627	+	0.0388612i
$808$		0			0
$809$			44.3423i			0.0548113i	0.999624	+	0.0274056i	$0.00872458\pi$
$809$			44.3423i			0.0548113i	−0.999624	+	0.0274056i	$0.991275\pi$
$810$		0			0
$811$	−686.962			−0.847056			−0.423528	−	0.905883i	$-0.639208\pi$
$811$	−686.962			−0.847056			−0.423528	+	0.905883i	$0.639208\pi$
$812$		0			0
$813$	−21.7136	−	200.382i	−0.0267080	−	0.246473i
$814$		0			0
$815$		0			0
$816$		0			0
$817$			458.842i			0.561618i
$818$		0			0
$819$	44.3308	−	9.72160i	0.0541280	−	0.0118701i
$820$		0			0
$821$			865.772i			1.05453i	0.849700	+	0.527267i	$0.176783\pi$
$821$			865.772i			1.05453i	−0.849700	+	0.527267i	$0.823217\pi$
$822$		0			0
$823$			240.142i			0.291789i	0.989300	+	0.145895i	$0.0466060\pi$
$823$			240.142i			0.291789i	−0.989300	+	0.145895i	$0.953394\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−311.704			−0.376910			−0.188455	−	0.982082i	$-0.560348\pi$
$827$	−311.704			−0.376910			−0.188455	+	0.982082i	$0.560348\pi$
$828$		0			0
$829$	1125.26			1.35737			0.678684	−	0.734430i	$-0.262551\pi$
$829$	1125.26			1.35737			0.678684	+	0.734430i	$0.262551\pi$
$830$		0			0
$831$	−1079.08	+	116.931i	−1.29854	+	0.140711i
$832$		0			0
$833$	714.110			0.857274
$834$		0			0
$835$		0			0
$836$		0			0
$837$	−290.260	−	864.812i	−0.346786	−	1.03323i
$838$		0			0
$839$		−	40.3794i		−	0.0481280i	−0.999710	−	0.0240640i	$-0.992339\pi$
$839$		−	40.3794i		−	0.0481280i	0.999710	−	0.0240640i	$-0.00766055\pi$
$840$		0			0
$841$	405.259			0.481878
$842$		0			0
$843$	859.546	−	93.1412i	1.01963	−	0.110488i
$844$		0			0
$845$		0			0
$846$		0			0
$847$			464.607i			0.548532i
$848$		0			0
$849$	812.664	−	88.0610i	0.957201	−	0.103723i
$850$		0			0
$851$		−	2420.24i		−	2.84399i
$852$		0			0
$853$			290.487i			0.340547i	0.985397	+	0.170274i	$0.0544652\pi$
$853$			290.487i			0.340547i	−0.985397	+	0.170274i	$0.945535\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	1111.79			1.29731			0.648655	−	0.761083i	$-0.275332\pi$
$857$	1111.79			1.29731			0.648655	+	0.761083i	$0.275332\pi$
$858$		0			0
$859$	159.069			0.185179			0.0925894	−	0.995704i	$-0.470486\pi$
$859$	159.069			0.185179			0.0925894	+	0.995704i	$0.470486\pi$
$860$		0			0
$861$	−90.5231	−	835.385i	−0.105137	−	0.970250i
$862$		0			0
$863$	−75.4630			−0.0874426			−0.0437213	−	0.999044i	$-0.513921\pi$
$863$	−75.4630			−0.0874426			−0.0437213	+	0.999044i	$0.513921\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	−138.461	−	1277.78i	−0.159702	−	1.47379i
$868$		0			0
$869$		−	311.463i		−	0.358415i
$870$		0			0
$871$	−108.326			−0.124369
$872$		0			0
$873$	29.4041	+	134.084i	0.0336817	+	0.153589i
$874$		0			0
$875$		0			0
$876$		0			0
$877$		−	511.799i		−	0.583579i	−0.956483	−	0.291790i	$-0.905749\pi$
$877$		−	511.799i		−	0.583579i	0.956483	−	0.291790i	$-0.0942507\pi$
$878$		0			0
$879$	−22.8067	−	210.470i	−0.0259462	−	0.239443i
$880$		0			0
$881$			855.549i			0.971111i	0.874206	+	0.485556i	$0.161383\pi$
$881$			855.549i			0.971111i	−0.874206	+	0.485556i	$0.838617\pi$
$882$		0			0
$883$		−	11.2299i		−	0.0127179i	−0.999980	−	0.00635894i	$-0.997976\pi$
$883$		−	11.2299i		−	0.0127179i	0.999980	−	0.00635894i	$-0.00202413\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−733.251			−0.826664			−0.413332	−	0.910580i	$-0.635635\pi$
$887$	−733.251			−0.826664			−0.413332	+	0.910580i	$0.635635\pi$
$888$		0			0
$889$	6.59468			0.00741808
$890$		0			0
$891$	161.495	+	350.503i	0.181251	+	0.393381i
$892$		0			0
$893$	−78.7690			−0.0882072
$894$		0			0
$895$		0			0
$896$		0			0
$897$	127.434	−	13.8089i	0.142067	−	0.0153945i
$898$		0			0
$899$		−	705.263i		−	0.784497i
$900$		0			0
$901$	2494.67			2.76878
$902$		0			0
$903$	86.2175	+	795.652i	0.0954790	+	0.881120i
$904$		0			0
$905$		0			0
$906$		0			0
$907$			1141.39i			1.25843i	0.777232	+	0.629214i	$0.216623\pi$
$907$			1141.39i			1.25843i	−0.777232	+	0.629214i	$0.783377\pi$
$908$		0			0
$909$	−138.815	−	633.000i	−0.152712	−	0.696370i
$910$		0			0
$911$		−	157.145i		−	0.172497i	−0.996274	−	0.0862487i	$-0.972512\pi$
$911$		−	157.145i		−	0.172497i	0.996274	−	0.0862487i	$-0.0274880\pi$
$912$		0			0
$913$			561.666i			0.615187i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−1069.21			−1.16598
$918$		0			0
$919$	−1542.94			−1.67893			−0.839466	−	0.543412i	$-0.817132\pi$
$919$	−1542.94			−1.67893			−0.839466	+	0.543412i	$0.817132\pi$
$920$		0			0
$921$	−370.652	+	40.1642i	−0.402446	+	0.0436094i
$922$		0			0
$923$	96.2090			0.104235
$924$		0			0
$925$		0			0
$926$		0			0
$927$	213.896	+	975.371i	0.230740	+	1.05218i
$928$		0			0
$929$		−	165.617i		−	0.178274i	−0.996019	−	0.0891372i	$-0.971589\pi$
$929$		−	165.617i		−	0.178274i	0.996019	−	0.0891372i	$-0.0284110\pi$
$930$		0			0
$931$	216.738			0.232802
$932$		0			0
$933$	−683.019	+	74.0125i	−0.732067	+	0.0793275i
$934$		0			0
$935$		0			0
$936$		0			0
$937$			1418.05i			1.51339i	0.653768	+	0.756695i	$0.273187\pi$
$937$			1418.05i			1.51339i	−0.653768	+	0.756695i	$0.726813\pi$
$938$		0			0
$939$	−1388.34	+	150.442i	−1.47853	+	0.160215i
$940$		0			0
$941$		−	1274.33i		−	1.35423i	−0.735878	−	0.677114i	$-0.763231\pi$
$941$		−	1274.33i		−	1.35423i	0.735878	−	0.677114i	$-0.236769\pi$
$942$		0			0
$943$		−	2373.22i		−	2.51667i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	1406.16			1.48486			0.742428	−	0.669926i	$-0.233674\pi$
$947$	1406.16			1.48486			0.742428	+	0.669926i	$0.233674\pi$
$948$		0			0
$949$	42.7515			0.0450490
$950$		0			0
$951$	−131.040	−	1209.30i	−0.137792	−	1.27160i
$952$		0			0
$953$	1415.36			1.48516			0.742579	−	0.669759i	$-0.233602\pi$
$953$	1415.36			1.48516			0.742579	+	0.669759i	$0.233602\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$	32.1427	+	296.627i	0.0335870	+	0.309955i
$958$		0			0
$959$			360.430i			0.375840i
$960$		0			0
$961$	180.496			0.187821
$962$		0			0
$963$	269.025	−	58.9963i	0.279361	−	0.0612630i
$964$		0			0
$965$		0			0
$966$		0			0
$967$		−	1662.07i		−	1.71879i	−0.511315	−	0.859394i	$-0.670841\pi$
$967$		−	1662.07i		−	1.71879i	0.511315	−	0.859394i	$-0.329159\pi$
$968$		0			0
$969$	−70.3725	−	649.427i	−0.0726239	−	0.670204i
$970$		0			0
$971$		−	868.901i		−	0.894852i	−0.894321	−	0.447426i	$-0.852341\pi$
$971$		−	868.901i		−	0.894852i	0.894321	−	0.447426i	$-0.147659\pi$
$972$		0			0
$973$		−	603.373i		−	0.620116i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−1336.81			−1.36828			−0.684140	−	0.729350i	$-0.739822\pi$
$977$	−1336.81			−1.36828			−0.684140	+	0.729350i	$0.739822\pi$
$978$		0			0
$979$	−570.203			−0.582434
$980$		0			0
$981$	−152.696	+	33.4858i	−0.155654	+	0.0341344i
$982$		0			0
$983$	1193.86			1.21451			0.607254	−	0.794508i	$-0.292271\pi$
$983$	1193.86			1.21451			0.607254	+	0.794508i	$0.292271\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	−136.589	+	14.8009i	−0.138388	+	0.0149958i
$988$		0			0
$989$			2260.34i			2.28548i
$990$		0			0
$991$	−460.690			−0.464874			−0.232437	−	0.972611i	$-0.574670\pi$
$991$	−460.690			−0.464874			−0.232437	+	0.972611i	$0.574670\pi$
$992$		0			0
$993$	−14.8432	−	136.980i	−0.0149479	−	0.137945i
$994$		0			0
$995$		0			0
$996$		0			0
$997$			673.924i			0.675952i	0.941155	+	0.337976i	$0.109742\pi$
$997$			673.924i			0.675952i	−0.941155	+	0.337976i	$0.890258\pi$
$998$		0			0
$999$	1546.94	−	519.205i	1.54849	−	0.519725i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	600.3.c.d.449.7		16
3.2	odd	2	inner	600.3.c.d.449.9		16
4.3	odd	2		1200.3.c.m.449.10		16
5.2	odd	4		600.3.l.f.401.2		8
5.3	odd	4		120.3.l.a.41.7	✓	8
5.4	even	2	inner	600.3.c.d.449.10		16
12.11	even	2		1200.3.c.m.449.8		16
15.2	even	4		600.3.l.f.401.1		8
15.8	even	4		120.3.l.a.41.8	yes	8
15.14	odd	2	inner	600.3.c.d.449.8		16
20.3	even	4		240.3.l.d.161.2		8
20.7	even	4		1200.3.l.x.401.7		8
20.19	odd	2		1200.3.c.m.449.7		16
40.3	even	4		960.3.l.g.641.7		8
40.13	odd	4		960.3.l.h.641.2		8
60.23	odd	4		240.3.l.d.161.1		8
60.47	odd	4		1200.3.l.x.401.8		8
60.59	even	2		1200.3.c.m.449.9		16
120.53	even	4		960.3.l.h.641.1		8
120.83	odd	4		960.3.l.g.641.8		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
120.3.l.a.41.7	✓	8	5.3	odd	4
120.3.l.a.41.8	yes	8	15.8	even	4
240.3.l.d.161.1		8	60.23	odd	4
240.3.l.d.161.2		8	20.3	even	4
600.3.c.d.449.7		16	1.1	even	1	trivial
600.3.c.d.449.8		16	15.14	odd	2	inner
600.3.c.d.449.9		16	3.2	odd	2	inner
600.3.c.d.449.10		16	5.4	even	2	inner
600.3.l.f.401.1		8	15.2	even	4
600.3.l.f.401.2		8	5.2	odd	4
960.3.l.g.641.7		8	40.3	even	4
960.3.l.g.641.8		8	120.83	odd	4
960.3.l.h.641.1		8	120.53	even	4
960.3.l.h.641.2		8	40.13	odd	4
1200.3.c.m.449.7		16	20.19	odd	2
1200.3.c.m.449.8		16	12.11	even	2
1200.3.c.m.449.9		16	60.59	even	2
1200.3.c.m.449.10		16	4.3	odd	2
1200.3.l.x.401.7		8	20.7	even	4
1200.3.l.x.401.8		8	60.47	odd	4

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 600 = 2^{3} \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	600.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+(-0.323191 - 2.98254i) q^{3} +4.72640i q^{7} +(-8.79110 + 1.92786i) q^{9} +4.76442i q^{11} +1.06692i q^{13} +26.7847 q^{17} +8.12938 q^{19} +(14.0967 - 1.52753i) q^{21} +40.0468 q^{23} +(8.59112 + 25.5967i) q^{27} +20.8744i q^{29} -33.7860 q^{31} +(14.2101 - 1.53982i) q^{33} -60.4351i q^{37} +(3.18213 - 0.344819i) q^{39} -59.2611i q^{41} +56.4424i q^{43} -9.68942 q^{47} +26.6611 q^{49} +(-8.65657 - 79.8864i) q^{51} +93.1378 q^{53} +(-2.62734 - 24.2462i) q^{57} +17.4907i q^{59} +57.7400 q^{61} +(-9.11184 - 41.5503i) q^{63} +101.531i q^{67} +(-12.9428 - 119.441i) q^{69} -90.1745i q^{71} -40.0700i q^{73} -22.5186 q^{77} -65.3727 q^{79} +(73.5667 - 33.8960i) q^{81} +117.888 q^{83} +(62.2587 - 6.74641i) q^{87} +119.679i q^{89} -5.04269 q^{91} +(10.9193 + 100.768i) q^{93} -15.2522i q^{97} +(-9.18513 - 41.8845i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 40 q^{9} + 16 q^{19} + 56 q^{21} + 240 q^{31} + 144 q^{39} - 128 q^{49} + 128 q^{51} + 16 q^{61} - 200 q^{69} - 176 q^{79} + 448 q^{81} + 1120 q^{91} - 64 q^{99}+O(q^{100}) \) 16 * q - 40 * q^9 + 16 * q^19 + 56 * q^21 + 240 * q^31 + 144 * q^39 - 128 * q^49 + 128 * q^51 + 16 * q^61 - 200 * q^69 - 176 * q^79 + 448 * q^81 + 1120 * q^91 - 64 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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