LMFDB - Embedded newform 608.6.b.b.303.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [608,6,Mod(303,608)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(608, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 1, 1]))

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("608.303");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 608 = 2^{5} \cdot 19 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	608.b (of order $2$, degree $1$, 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:	$97.5133624463$
Analytic rank:	$0$
Dimension:	$96$
Twist minimal:	no (minimal twist has level 152)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			303.4
Character	$\chi$	$=$	608.303
Dual form			608.6.b.b.303.93

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-27.1983i q^{3} +36.9812i q^{5} -150.607i q^{7} -496.748 q^{9} +O(q^{10})$	$q-27.1983i q^{3} +36.9812i q^{5} -150.607i q^{7} -496.748 q^{9} +603.531 q^{11} +566.677 q^{13} +1005.83 q^{15} -1989.92 q^{17} +(-1308.75 - 873.651i) q^{19} -4096.26 q^{21} -1465.34i q^{23} +1757.39 q^{25} +6901.50i q^{27} +4744.43 q^{29} -1904.96 q^{31} -16415.0i q^{33} +5569.64 q^{35} -9126.91 q^{37} -15412.6i q^{39} +2640.62i q^{41} -10081.0 q^{43} -18370.3i q^{45} -20444.0i q^{47} -5875.55 q^{49} +54122.5i q^{51} -30065.0 q^{53} +22319.3i q^{55} +(-23761.8 + 35595.8i) q^{57} -15138.2i q^{59} -38264.3i q^{61} +74813.8i q^{63} +20956.4i q^{65} +45900.3i q^{67} -39854.9 q^{69} -59869.1 q^{71} +21567.5 q^{73} -47798.0i q^{75} -90896.1i q^{77} -72787.8 q^{79} +66999.5 q^{81} -14628.7 q^{83} -73589.7i q^{85} -129041. i q^{87} -18633.0i q^{89} -85345.6i q^{91} +51811.7i q^{93} +(32308.7 - 48399.3i) q^{95} +20961.5i q^{97} -299802. q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 96 q - 6168 q^{9}+O(q^{10}) $ 96 * q - 6168 * q^9	$ 96 q - 6168 q^{9} - 944 q^{11} - 3832 q^{17} - 5240 q^{19} - 62504 q^{25} - 7720 q^{35} - 45096 q^{43} - 210840 q^{49} - 36336 q^{57} - 4336 q^{73} - 20624 q^{81} - 52152 q^{83} + 752768 q^{99}+O(q^{100}) $ 96 * q - 6168 * q^9 - 944 * q^11 - 3832 * q^17 - 5240 * q^19 - 62504 * q^25 - 7720 * q^35 - 45096 * q^43 - 210840 * q^49 - 36336 * q^57 - 4336 * q^73 - 20624 * q^81 - 52152 * q^83 + 752768 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/608\mathbb{Z}\right)^\times$.

$n$	$97$	$191$	$229$
$\chi(n)$	$-1$	$-1$	$-1$

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
$3$		−	27.1983i		−	1.74477i	−0.488818	−	0.872386i	$-0.662572\pi$
$3$		−	27.1983i		−	1.74477i	0.488818	−	0.872386i	$-0.337428\pi$
$4$		0			0
$5$			36.9812i			0.661540i	0.943711	+	0.330770i	$0.107308\pi$
$5$			36.9812i			0.661540i	−0.943711	+	0.330770i	$0.892692\pi$
$6$		0			0
$7$		−	150.607i		−	1.16172i	−0.814004	−	0.580859i	$-0.802717\pi$
$7$		−	150.607i		−	1.16172i	0.814004	−	0.580859i	$-0.197283\pi$
$8$		0			0
$9$	−496.748			−2.04423
$10$		0			0
$11$	603.531			1.50390			0.751948	−	0.659223i	$-0.229115\pi$
$11$	603.531			1.50390			0.751948	+	0.659223i	$0.229115\pi$
$12$		0			0
$13$	566.677			0.929987			0.464993	−	0.885314i	$-0.346057\pi$
$13$	566.677			0.929987			0.464993	+	0.885314i	$0.346057\pi$
$14$		0			0
$15$	1005.83			1.15424
$16$		0			0
$17$	−1989.92			−1.66999			−0.834995	−	0.550258i	$-0.814529\pi$
$17$	−1989.92			−1.66999			−0.834995	+	0.550258i	$0.814529\pi$
$18$		0			0
$19$	−1308.75	−	873.651i	−0.831713	−	0.555206i
$19$	−1308.75	−	873.651i	−0.831713	−	0.555206i
$20$		0			0
$21$	−4096.26			−2.02693
$22$		0			0
$23$		−	1465.34i		−	0.577591i	−0.957391	−	0.288795i	$-0.906745\pi$
$23$		−	1465.34i		−	0.577591i	0.957391	−	0.288795i	$-0.0932547\pi$
$24$		0			0
$25$	1757.39			0.562365
$26$		0			0
$27$			6901.50i			1.82194i
$28$		0			0
$29$	4744.43			1.04759			0.523793	−	0.851846i	$-0.324517\pi$
$29$	4744.43			1.04759			0.523793	+	0.851846i	$0.324517\pi$
$30$		0			0
$31$	−1904.96			−0.356026			−0.178013	−	0.984028i	$-0.556967\pi$
$31$	−1904.96			−0.356026			−0.178013	+	0.984028i	$0.556967\pi$
$32$		0			0
$33$		−	16415.0i		−	2.62395i
$34$		0			0
$35$	5569.64			0.768523
$36$		0			0
$37$	−9126.91			−1.09602			−0.548011	−	0.836471i	$-0.684615\pi$
$37$	−9126.91			−1.09602			−0.548011	+	0.836471i	$0.684615\pi$
$38$		0			0
$39$		−	15412.6i		−	1.62261i
$40$		0			0
$41$			2640.62i			0.245327i	0.992448	+	0.122664i	$0.0391436\pi$
$41$			2640.62i			0.245327i	−0.992448	+	0.122664i	$0.960856\pi$
$42$		0			0
$43$	−10081.0			−0.831443			−0.415721	−	0.909492i	$-0.636471\pi$
$43$	−10081.0			−0.831443			−0.415721	+	0.909492i	$0.636471\pi$
$44$		0			0
$45$		−	18370.3i		−	1.35234i
$46$		0			0
$47$		−	20444.0i		−	1.34996i	−0.737836	−	0.674980i	$-0.764152\pi$
$47$		−	20444.0i		−	1.34996i	0.737836	−	0.674980i	$-0.235848\pi$
$48$		0			0
$49$	−5875.55			−0.349589
$50$		0			0
$51$			54122.5i			2.91375i
$52$		0			0
$53$	−30065.0			−1.47018			−0.735092	−	0.677967i	$-0.762861\pi$
$53$	−30065.0			−1.47018			−0.735092	+	0.677967i	$0.762861\pi$
$54$		0			0
$55$			22319.3i			0.994887i
$56$		0			0
$57$	−23761.8	+	35595.8i	−0.968707	+	1.45115i
$58$		0			0
$59$		−	15138.2i		−	0.566167i	−0.959095	−	0.283084i	$-0.908643\pi$
$59$		−	15138.2i		−	0.566167i	0.959095	−	0.283084i	$-0.0913574\pi$
$60$		0			0
$61$		−	38264.3i		−	1.31665i	−0.752735	−	0.658323i	$-0.771266\pi$
$61$		−	38264.3i		−	1.31665i	0.752735	−	0.658323i	$-0.228734\pi$
$62$		0			0
$63$			74813.8i			2.37482i
$64$		0			0
$65$			20956.4i			0.615224i
$66$		0			0
$67$			45900.3i			1.24919i	0.780949	+	0.624595i	$0.214736\pi$
$67$			45900.3i			1.24919i	−0.780949	+	0.624595i	$0.785264\pi$
$68$		0			0
$69$	−39854.9			−1.00776
$70$		0			0
$71$	−59869.1			−1.40947			−0.704737	−	0.709468i	$-0.748935\pi$
$71$	−59869.1			−1.40947			−0.704737	+	0.709468i	$0.748935\pi$
$72$		0			0
$73$	21567.5			0.473688			0.236844	−	0.971548i	$-0.423887\pi$
$73$	21567.5			0.473688			0.236844	+	0.971548i	$0.423887\pi$
$74$		0			0
$75$		−	47798.0i		−	0.981198i
$76$		0			0
$77$		−	90896.1i		−	1.74710i
$78$		0			0
$79$	−72787.8			−1.31217			−0.656087	−	0.754686i	$-0.727789\pi$
$79$	−72787.8			−1.31217			−0.656087	+	0.754686i	$0.727789\pi$
$80$		0			0
$81$	66999.5			1.13464
$82$		0			0
$83$	−14628.7			−0.233083			−0.116541	−	0.993186i	$-0.537181\pi$
$83$	−14628.7			−0.233083			−0.116541	+	0.993186i	$0.537181\pi$
$84$		0			0
$85$		−	73589.7i		−	1.10476i
$86$		0			0
$87$		−	129041.i		−	1.82780i
$88$		0			0
$89$		−	18633.0i		−	0.249349i	−0.992198	−	0.124675i	$-0.960211\pi$
$89$		−	18633.0i		−	0.249349i	0.992198	−	0.124675i	$-0.0397887\pi$
$90$		0			0
$91$		−	85345.6i		−	1.08038i
$92$		0			0
$93$			51811.7i			0.621184i
$94$		0			0
$95$	32308.7	−	48399.3i	0.367291	−	0.550212i
$96$		0			0
$97$			20961.5i			0.226200i	0.993584	+	0.113100i	$0.0360780\pi$
$97$			20961.5i			0.226200i	−0.993584	+	0.113100i	$0.963922\pi$
$98$		0			0
$99$	−299802.			−3.07431
$100$		0			0
$101$			83763.7i			0.817057i	0.912745	+	0.408529i	$0.133958\pi$
$101$			83763.7i			0.817057i	−0.912745	+	0.408529i	$0.866042\pi$
$102$		0			0
$103$	171162.			1.58970			0.794848	−	0.606809i	$-0.207551\pi$
$103$	171162.			1.58970			0.794848	+	0.606809i	$0.207551\pi$
$104$		0			0
$105$		−	151485.i		−	1.34090i
$106$		0			0
$107$			106661.i			0.900628i	0.892870	+	0.450314i	$0.148688\pi$
$107$			106661.i			0.900628i	−0.892870	+	0.450314i	$0.851312\pi$
$108$		0			0
$109$	74162.6			0.597886			0.298943	−	0.954271i	$-0.403366\pi$
$109$	74162.6			0.597886			0.298943	+	0.954271i	$0.403366\pi$
$110$		0			0
$111$			248236.i			1.91231i
$112$		0			0
$113$		−	57356.7i		−	0.422560i	−0.977426	−	0.211280i	$-0.932237\pi$
$113$		−	57356.7i		−	0.422560i	0.977426	−	0.211280i	$-0.0677632\pi$
$114$		0			0
$115$	54190.2			0.382099
$116$		0			0
$117$	−281495.			−1.90111
$118$		0			0
$119$			299697.i			1.94006i
$120$		0			0
$121$	203198.			1.26170
$122$		0			0
$123$	71820.3			0.428040
$124$		0			0
$125$			180557.i			1.03357i
$126$		0			0
$127$	−263384.			−1.44904			−0.724518	−	0.689256i	$-0.757938\pi$
$127$	−263384.			−1.44904			−0.724518	+	0.689256i	$0.757938\pi$
$128$		0			0
$129$			274186.i			1.45068i
$130$		0			0
$131$	−30274.3			−0.154133			−0.0770665	−	0.997026i	$-0.524555\pi$
$131$	−30274.3			−0.154133			−0.0770665	+	0.997026i	$0.524555\pi$
$132$		0			0
$133$	−131578.	+	197108.i	−0.644993	+	0.966216i
$134$		0			0
$135$	−255226.			−1.20529
$136$		0			0
$137$	−249038.			−1.13361			−0.566807	−	0.823851i	$-0.691821\pi$
$137$	−249038.			−1.13361			−0.566807	+	0.823851i	$0.691821\pi$
$138$		0			0
$139$	423617.			1.85967			0.929837	−	0.367972i	$-0.119948\pi$
$139$	423617.			1.85967			0.929837	+	0.367972i	$0.119948\pi$
$140$		0			0
$141$	−556042.			−2.35537
$142$		0			0
$143$	342007.			1.39860
$144$		0			0
$145$			175455.i			0.693020i
$146$		0			0
$147$			159805.i			0.609953i
$148$		0			0
$149$		−	14258.2i		−	0.0526136i	−0.999654	−	0.0263068i	$-0.991625\pi$
$149$		−	14258.2i		−	0.0526136i	0.999654	−	0.0263068i	$-0.00837469\pi$
$150$		0			0
$151$	257437.			0.918814			0.459407	−	0.888226i	$-0.348062\pi$
$151$	257437.			0.918814			0.459407	+	0.888226i	$0.348062\pi$
$152$		0			0
$153$	988489.			3.41384
$154$		0			0
$155$		−	70447.7i		−	0.235525i
$156$		0			0
$157$			588545.i			1.90560i	0.303605	+	0.952798i	$0.401810\pi$
$157$			588545.i			1.90560i	−0.303605	+	0.952798i	$0.598190\pi$
$158$		0			0
$159$			817717.i			2.56514i
$160$		0			0
$161$	−220692.			−0.670998
$162$		0			0
$163$	410287.			1.20953			0.604767	−	0.796402i	$-0.293266\pi$
$163$	410287.			1.20953			0.604767	+	0.796402i	$0.293266\pi$
$164$		0			0
$165$	607047.			1.73585
$166$		0			0
$167$	130499.			0.362089			0.181045	−	0.983475i	$-0.442052\pi$
$167$	130499.			0.362089			0.181045	+	0.983475i	$0.442052\pi$
$168$		0			0
$169$	−50170.7			−0.135124
$170$		0			0
$171$	650120.	+	433984.i	1.70021	+	1.13497i
$172$		0			0
$173$	60471.8			0.153616			0.0768082	−	0.997046i	$-0.475527\pi$
$173$	60471.8			0.153616			0.0768082	+	0.997046i	$0.475527\pi$
$174$		0			0
$175$		−	264676.i		−	0.653309i
$176$		0			0
$177$	−411734.			−0.987832
$178$		0			0
$179$			202401.i			0.472150i	0.971735	+	0.236075i	$0.0758611\pi$
$179$			202401.i			0.472150i	−0.971735	+	0.236075i	$0.924139\pi$
$180$		0			0
$181$	−260264.			−0.590498			−0.295249	−	0.955420i	$-0.595403\pi$
$181$	−260264.			−0.590498			−0.295249	+	0.955420i	$0.595403\pi$
$182$		0			0
$183$	−1.04072e6			−2.29725
$184$		0			0
$185$		−	337524.i		−	0.725063i
$186$		0			0
$187$	−1.20098e6			−2.51149
$188$		0			0
$189$	1.03942e6			2.11658
$190$		0			0
$191$		−	33320.5i		−	0.0660888i	−0.999454	−	0.0330444i	$-0.989480\pi$
$191$		−	33320.5i		−	0.0660888i	0.999454	−	0.0330444i	$-0.0105203\pi$
$192$		0			0
$193$			689608.i			1.33263i	0.745671	+	0.666314i	$0.232129\pi$
$193$			689608.i			1.33263i	−0.745671	+	0.666314i	$0.767871\pi$
$194$		0			0
$195$	569978.			1.07342
$196$		0			0
$197$		−	636722.i		−	1.16892i	−0.811423	−	0.584460i	$-0.801306\pi$
$197$		−	636722.i		−	1.16892i	0.811423	−	0.584460i	$-0.198694\pi$
$198$		0			0
$199$		−	379147.i		−	0.678695i	−0.940661	−	0.339348i	$-0.889794\pi$
$199$		−	379147.i		−	0.678695i	0.940661	−	0.339348i	$-0.110206\pi$
$200$		0			0
$201$	1.24841e6			2.17955
$202$		0			0
$203$		−	714546.i		−	1.21700i
$204$		0			0
$205$	−97653.3			−0.162294
$206$		0			0
$207$			727906.i			1.18073i
$208$		0			0
$209$	−789872.	−	527275.i	−1.25081	−	0.834971i
$210$		0			0
$211$			1.10273e6i			1.70515i	0.522608	+	0.852573i	$0.324959\pi$
$211$			1.10273e6i			1.70515i	−0.522608	+	0.852573i	$0.675041\pi$
$212$		0			0
$213$			1.62834e6i			2.45921i
$214$		0			0
$215$		−	372808.i		−	0.550033i
$216$		0			0
$217$			286901.i			0.413602i
$218$		0			0
$219$		−	586599.i		−	0.826478i
$220$		0			0
$221$	−1.12764e6			−1.55307
$222$		0			0
$223$	−12624.3			−0.0169998			−0.00849991	−	0.999964i	$-0.502706\pi$
$223$	−12624.3			−0.0169998			−0.00849991	+	0.999964i	$0.502706\pi$
$224$		0			0
$225$	−872979.			−1.14960
$226$		0			0
$227$		−	581264.i		−	0.748701i	−0.927287	−	0.374351i	$-0.877866\pi$
$227$		−	581264.i		−	0.748701i	0.927287	−	0.374351i	$-0.122134\pi$
$228$		0			0
$229$		−	1.01092e6i		−	1.27388i	−0.770915	−	0.636938i	$-0.780201\pi$
$229$		−	1.01092e6i		−	1.27388i	0.770915	−	0.636938i	$-0.219799\pi$
$230$		0			0
$231$	−2.47222e6			−3.04830
$232$		0			0
$233$	−722717.			−0.872124			−0.436062	−	0.899917i	$-0.643627\pi$
$233$	−722717.			−0.872124			−0.436062	+	0.899917i	$0.643627\pi$
$234$		0			0
$235$	756044.			0.893053
$236$		0			0
$237$			1.97971e6i			2.28944i
$238$		0			0
$239$		−	586425.i		−	0.664076i	−0.943266	−	0.332038i	$-0.892264\pi$
$239$		−	586425.i		−	0.664076i	0.943266	−	0.332038i	$-0.107736\pi$
$240$		0			0
$241$			636052.i			0.705424i	0.935732	+	0.352712i	$0.114740\pi$
$241$			636052.i			0.705424i	−0.935732	+	0.352712i	$0.885260\pi$
$242$		0			0
$243$		−	145207.i		−	0.157751i
$244$		0			0
$245$		−	217285.i		−	0.231267i
$246$		0			0
$247$	−741639.	−	495077.i	−0.773482	−	0.516334i
$248$		0			0
$249$			397875.i			0.406676i
$250$		0			0
$251$	−395054.			−0.395797			−0.197898	−	0.980223i	$-0.563412\pi$
$251$	−395054.			−0.395797			−0.197898	+	0.980223i	$0.563412\pi$
$252$		0			0
$253$		−	884380.i		−	0.868636i
$254$		0			0
$255$	−2.00152e6			−1.92756
$256$		0			0
$257$			1.32766e6i			1.25388i	0.779069	+	0.626939i	$0.215692\pi$
$257$			1.32766e6i			1.25388i	−0.779069	+	0.626939i	$0.784308\pi$
$258$		0			0
$259$			1.37458e6i			1.27327i
$260$		0			0
$261$	−2.35679e6			−2.14150
$262$		0			0
$263$			689459.i			0.614638i	0.951607	+	0.307319i	$0.0994318\pi$
$263$			689459.i			0.614638i	−0.951607	+	0.307319i	$0.900568\pi$
$264$		0			0
$265$		−	1.11184e6i		−	0.972586i
$266$		0			0
$267$	−506787.			−0.435058
$268$		0			0
$269$	−977952.			−0.824018			−0.412009	−	0.911180i	$-0.635173\pi$
$269$	−977952.			−0.824018			−0.412009	+	0.911180i	$0.635173\pi$
$270$		0			0
$271$		−	1.49359e6i		−	1.23540i	−0.786413	−	0.617701i	$-0.788064\pi$
$271$		−	1.49359e6i		−	1.23540i	0.786413	−	0.617701i	$-0.211936\pi$
$272$		0			0
$273$	−2.32126e6			−1.88502
$274$		0			0
$275$	1.06064e6			0.845737
$276$		0			0
$277$			1.32182e6i			1.03507i	0.855661	+	0.517537i	$0.173151\pi$
$277$			1.32182e6i			1.03507i	−0.855661	+	0.517537i	$0.826849\pi$
$278$		0			0
$279$	946284.			0.727798
$280$		0			0
$281$		−	2.25632e6i		−	1.70465i	−0.523016	−	0.852323i	$-0.675193\pi$
$281$		−	2.25632e6i		−	1.70465i	0.523016	−	0.852323i	$-0.324807\pi$
$282$		0			0
$283$	−849766.			−0.630715			−0.315358	−	0.948973i	$-0.602124\pi$
$283$	−849766.			−0.630715			−0.315358	+	0.948973i	$0.602124\pi$
$284$		0			0
$285$	−1.31638e6	−	878741.i	−0.959994	−	0.640839i
$286$		0			0
$287$	397696.			0.285001
$288$		0			0
$289$	2.53993e6			1.78886
$290$		0			0
$291$	570116.			0.394667
$292$		0			0
$293$	−1.23388e6			−0.839659			−0.419830	−	0.907603i	$-0.637910\pi$
$293$	−1.23388e6			−0.839659			−0.419830	+	0.907603i	$0.637910\pi$
$294$		0			0
$295$	559830.			0.374542
$296$		0			0
$297$			4.16527e6i			2.74001i
$298$		0			0
$299$		−	830376.i		−	0.537152i
$300$		0			0
$301$			1.51827e6i			0.965902i
$302$		0			0
$303$	2.27823e6			1.42558
$304$		0			0
$305$	1.41506e6			0.871015
$306$		0			0
$307$			19289.4i			0.0116808i	0.999983	+	0.00584041i	$0.00185907\pi$
$307$			19289.4i			0.0116808i	−0.999983	+	0.00584041i	$0.998141\pi$
$308$		0			0
$309$		−	4.65531e6i		−	2.77366i
$310$		0			0
$311$		−	696484.i		−	0.408329i	−0.978937	−	0.204165i	$-0.934552\pi$
$311$		−	696484.i		−	0.408329i	0.978937	−	0.204165i	$-0.0654478\pi$
$312$		0			0
$313$	−1.56308e6			−0.901819			−0.450909	−	0.892570i	$-0.648900\pi$
$313$	−1.56308e6			−0.901819			−0.450909	+	0.892570i	$0.648900\pi$
$314$		0			0
$315$	−2.76671e6			−1.57104
$316$		0			0
$317$	1.55098e6			0.866880			0.433440	−	0.901183i	$-0.357300\pi$
$317$	1.55098e6			0.866880			0.433440	+	0.901183i	$0.357300\pi$
$318$		0			0
$319$	2.86341e6			1.57546
$320$		0			0
$321$	2.90099e6			1.57139
$322$		0			0
$323$	2.60431e6	+	1.73850e6i	1.38895	+	0.927188i
$324$		0			0
$325$	995871.			0.522992
$326$		0			0
$327$		−	2.01710e6i		−	1.04317i
$328$		0			0
$329$	−3.07901e6			−1.56827
$330$		0			0
$331$			515679.i			0.258708i	0.991598	+	0.129354i	$0.0412903\pi$
$331$			515679.i			0.258708i	−0.991598	+	0.129354i	$0.958710\pi$
$332$		0			0
$333$	4.53377e6			2.24052
$334$		0			0
$335$	−1.69745e6			−0.826389
$336$		0			0
$337$		−	180334.i		−	0.0864972i	−0.999064	−	0.0432486i	$-0.986229\pi$
$337$		−	180334.i		−	0.0864972i	0.999064	−	0.0432486i	$-0.0137708\pi$
$338$		0			0
$339$	−1.56001e6			−0.737271
$340$		0			0
$341$	−1.14970e6			−0.535425
$342$		0			0
$343$		−	1.64636e6i		−	0.755594i
$344$		0			0
$345$		−	1.47388e6i		−	0.666676i
$346$		0			0
$347$	−2.02177e6			−0.901379			−0.450690	−	0.892681i	$-0.648822\pi$
$347$	−2.02177e6			−0.901379			−0.450690	+	0.892681i	$0.648822\pi$
$348$		0			0
$349$		−	3.23137e6i		−	1.42011i	−0.704144	−	0.710057i	$-0.748669\pi$
$349$		−	3.23137e6i		−	1.42011i	0.704144	−	0.710057i	$-0.251331\pi$
$350$		0			0
$351$			3.91092e6i			1.69438i
$352$		0			0
$353$	368290.			0.157309			0.0786545	−	0.996902i	$-0.474938\pi$
$353$	368290.			0.157309			0.0786545	+	0.996902i	$0.474938\pi$
$354$		0			0
$355$		−	2.21403e6i		−	0.932424i
$356$		0			0
$357$	8.15124e6			3.38496
$358$		0			0
$359$			61762.2i			0.0252922i	0.999920	+	0.0126461i	$0.00402549\pi$
$359$			61762.2i			0.0252922i	−0.999920	+	0.0126461i	$0.995975\pi$
$360$		0			0
$361$	949567.	+	2.28679e6i	0.383493	+	0.923544i
$362$		0			0
$363$		−	5.52665e6i		−	2.20138i
$364$		0			0
$365$			797592.i			0.313364i
$366$		0			0
$367$			529501.i			0.205211i	0.994722	+	0.102606i	$0.0327180\pi$
$367$			529501.i			0.205211i	−0.994722	+	0.102606i	$0.967282\pi$
$368$		0			0
$369$		−	1.31172e6i		−	0.501505i
$370$		0			0
$371$			4.52801e6i			1.70794i
$372$		0			0
$373$	−4.19693e6			−1.56192			−0.780961	−	0.624580i	$-0.785270\pi$
$373$	−4.19693e6			−1.56192			−0.780961	+	0.624580i	$0.785270\pi$
$374$		0			0
$375$	4.91084e6			1.80334
$376$		0			0
$377$	2.68856e6			0.974241
$378$		0			0
$379$		−	870364.i		−	0.311245i	−0.987817	−	0.155623i	$-0.950262\pi$
$379$		−	870364.i		−	0.311245i	0.987817	−	0.155623i	$-0.0497384\pi$
$380$		0			0
$381$			7.16358e6i			2.52824i
$382$		0			0
$383$	−3.88095e6			−1.35189			−0.675945	−	0.736952i	$-0.736264\pi$
$383$	−3.88095e6			−1.35189			−0.675945	+	0.736952i	$0.736264\pi$
$384$		0			0
$385$	3.36145e6			1.15578
$386$		0			0
$387$	5.00771e6			1.69966
$388$		0			0
$389$			55797.8i			0.0186958i	0.999956	+	0.00934788i	$0.00297557\pi$
$389$			55797.8i			0.0186958i	−0.999956	+	0.00934788i	$0.997024\pi$
$390$		0			0
$391$			2.91592e6i			0.964570i
$392$		0			0
$393$			823409.i			0.268927i
$394$		0			0
$395$		−	2.69178e6i		−	0.868055i
$396$		0			0
$397$		−	2.63936e6i		−	0.840470i	−0.907415	−	0.420235i	$-0.861948\pi$
$397$		−	2.63936e6i		−	0.840470i	0.907415	−	0.420235i	$-0.138052\pi$
$398$		0			0
$399$	5.36099e6	+	3.57870e6i	1.68583	+	1.12536i
$400$		0			0
$401$		−	4.73912e6i		−	1.47176i	−0.677112	−	0.735880i	$-0.736769\pi$
$401$		−	4.73912e6i		−	1.47176i	0.677112	−	0.735880i	$-0.263231\pi$
$402$		0			0
$403$	−1.07950e6			−0.331099
$404$		0			0
$405$			2.47772e6i			0.750611i
$406$		0			0
$407$	−5.50837e6			−1.64830
$408$		0			0
$409$			1.03043e6i			0.304585i	0.988335	+	0.152293i	$0.0486656\pi$
$409$			1.03043e6i			0.304585i	−0.988335	+	0.152293i	$0.951334\pi$
$410$		0			0
$411$			6.77342e6i			1.97790i
$412$		0			0
$413$	−2.27992e6			−0.657727
$414$		0			0
$415$		−	540987.i		−	0.154194i
$416$		0			0
$417$		−	1.15217e7i		−	3.24471i
$418$		0			0
$419$	−792428.			−0.220508			−0.110254	−	0.993903i	$-0.535166\pi$
$419$	−792428.			−0.220508			−0.110254	+	0.993903i	$0.535166\pi$
$420$		0			0
$421$	−408226.			−0.112252			−0.0561262	−	0.998424i	$-0.517875\pi$
$421$	−408226.			−0.112252			−0.0561262	+	0.998424i	$0.517875\pi$
$422$		0			0
$423$			1.01555e7i			2.75963i
$424$		0			0
$425$	−3.49707e6			−0.939143
$426$		0			0
$427$	−5.76288e6			−1.52957
$428$		0			0
$429$		−	9.30200e6i		−	2.44024i
$430$		0			0
$431$	−1.00984e6			−0.261854			−0.130927	−	0.991392i	$-0.541795\pi$
$431$	−1.00984e6			−0.261854			−0.130927	+	0.991392i	$0.541795\pi$
$432$		0			0
$433$			1.59950e6i			0.409982i	0.978764	+	0.204991i	$0.0657165\pi$
$433$			1.59950e6i			0.409982i	−0.978764	+	0.204991i	$0.934283\pi$
$434$		0			0
$435$	4.77208e6			1.20916
$436$		0			0
$437$	−1.28020e6	+	1.91777e6i	−0.320682	+	0.480390i
$438$		0			0
$439$	6.61321e6			1.63776			0.818882	−	0.573962i	$-0.194594\pi$
$439$	6.61321e6			1.63776			0.818882	+	0.573962i	$0.194594\pi$
$440$		0			0
$441$	2.91866e6			0.714640
$442$		0			0
$443$	−2.90633e6			−0.703617			−0.351808	−	0.936072i	$-0.614433\pi$
$443$	−2.90633e6			−0.703617			−0.351808	+	0.936072i	$0.614433\pi$
$444$		0			0
$445$	689072.			0.164955
$446$		0			0
$447$	−387798.			−0.0917988
$448$		0			0
$449$		−	1.22462e6i		−	0.286671i	−0.989674	−	0.143336i	$-0.954217\pi$
$449$		−	1.22462e6i		−	0.286671i	0.989674	−	0.143336i	$-0.0457828\pi$
$450$		0			0
$451$			1.59369e6i			0.368947i
$452$		0			0
$453$		−	7.00184e6i		−	1.60312i
$454$		0			0
$455$	3.15618e6			0.714717
$456$		0			0
$457$	−3.30328e6			−0.739869			−0.369935	−	0.929058i	$-0.620620\pi$
$457$	−3.30328e6			−0.739869			−0.369935	+	0.929058i	$0.620620\pi$
$458$		0			0
$459$		−	1.37334e7i		−	3.04262i
$460$		0			0
$461$			7.00106e6i			1.53430i	0.641465	+	0.767152i	$0.278327\pi$
$461$			7.00106e6i			1.53430i	−0.641465	+	0.767152i	$0.721673\pi$
$462$		0			0
$463$		−	3.15394e6i		−	0.683755i	−0.939745	−	0.341878i	$-0.888937\pi$
$463$		−	3.15394e6i		−	0.683755i	0.939745	−	0.341878i	$-0.111063\pi$
$464$		0			0
$465$	−1.91606e6			−0.410938
$466$		0			0
$467$	−2.07779e6			−0.440869			−0.220435	−	0.975402i	$-0.570748\pi$
$467$	−2.07779e6			−0.440869			−0.220435	+	0.975402i	$0.570748\pi$
$468$		0			0
$469$	6.91291e6			1.45121
$470$		0			0
$471$	1.60074e7			3.32483
$472$		0			0
$473$	−6.08419e6			−1.25040
$474$		0			0
$475$	−2.29999e6	−	1.53534e6i	−0.467726	−	0.312228i
$476$		0			0
$477$	1.49347e7			3.00539
$478$		0			0
$479$		−	8.24274e6i		−	1.64147i	−0.571310	−	0.820734i	$-0.693565\pi$
$479$		−	8.24274e6i		−	1.64147i	0.571310	−	0.820734i	$-0.306435\pi$
$480$		0			0
$481$	−5.17201e6			−1.01929
$482$		0			0
$483$			6.00243e6i			1.17074i
$484$		0			0
$485$	−775180.			−0.149640
$486$		0			0
$487$	2.01001e6			0.384039			0.192020	−	0.981391i	$-0.438496\pi$
$487$	2.01001e6			0.384039			0.192020	+	0.981391i	$0.438496\pi$
$488$		0			0
$489$		−	1.11591e7i		−	2.11036i
$490$		0			0
$491$	6.35888e6			1.19036			0.595178	−	0.803594i	$-0.297082\pi$
$491$	6.35888e6			1.19036			0.595178	+	0.803594i	$0.297082\pi$
$492$		0			0
$493$	−9.44105e6			−1.74946
$494$		0			0
$495$		−	1.10871e7i		−	2.03378i
$496$		0			0
$497$			9.01673e6i			1.63741i
$498$		0			0
$499$	1.82256e6			0.327665			0.163833	−	0.986488i	$-0.447614\pi$
$499$	1.82256e6			0.327665			0.163833	+	0.986488i	$0.447614\pi$
$500$		0			0
$501$		−	3.54935e6i		−	0.631763i
$502$		0			0
$503$		−	5.97846e6i		−	1.05358i	−0.849994	−	0.526792i	$-0.823395\pi$
$503$		−	5.97846e6i		−	1.05358i	0.849994	−	0.526792i	$-0.176605\pi$
$504$		0			0
$505$	−3.09769e6			−0.540516
$506$		0			0
$507$			1.36456e6i			0.235761i
$508$		0			0
$509$	4.68133e6			0.800894			0.400447	−	0.916320i	$-0.368855\pi$
$509$	4.68133e6			0.800894			0.400447	+	0.916320i	$0.368855\pi$
$510$		0			0
$511$		−	3.24822e6i		−	0.550292i
$512$		0			0
$513$	6.02950e6	−	9.03236e6i	1.01155	−	1.51533i
$514$		0			0
$515$			6.32977e6i			1.05165i
$516$		0			0
$517$		−	1.23386e7i		−	2.03020i
$518$		0			0
$519$		−	1.64473e6i		−	0.268026i
$520$		0			0
$521$		−	2.59329e6i		−	0.418559i	−0.977856	−	0.209280i	$-0.932888\pi$
$521$		−	2.59329e6i		−	0.418559i	0.977856	−	0.209280i	$-0.0671119\pi$
$522$		0			0
$523$		−	1.21768e7i		−	1.94661i	−0.229511	−	0.973306i	$-0.573713\pi$
$523$		−	1.21768e7i		−	1.94661i	0.229511	−	0.973306i	$-0.426287\pi$
$524$		0			0
$525$	−7.19873e6			−1.13988
$526$		0			0
$527$	3.79072e6			0.594559
$528$		0			0
$529$	4.28911e6			0.666389
$530$		0			0
$531$			7.51987e6i			1.15737i
$532$		0			0
$533$			1.49638e6i			0.228151i
$534$		0			0
$535$	−3.94445e6			−0.595802
$536$		0			0
$537$	5.50496e6			0.823794
$538$		0			0
$539$	−3.54607e6			−0.525746
$540$		0			0
$541$		−	2.89110e6i		−	0.424688i	−0.977195	−	0.212344i	$-0.931890\pi$
$541$		−	2.89110e6i		−	0.424688i	0.977195	−	0.212344i	$-0.0681097\pi$
$542$		0			0
$543$			7.07875e6i			1.03028i
$544$		0			0
$545$			2.74262e6i			0.395526i
$546$		0			0
$547$		−	5.14595e6i		−	0.735355i	−0.929953	−	0.367678i	$-0.880153\pi$
$547$		−	5.14595e6i		−	0.735355i	0.929953	−	0.367678i	$-0.119847\pi$
$548$		0			0
$549$			1.90077e7i			2.69153i
$550$		0			0
$551$	−6.20929e6	−	4.14498e6i	−0.871290	−	0.581625i
$552$		0			0
$553$			1.09624e7i			1.52438i
$554$		0			0
$555$	−9.18009e6			−1.26507
$556$		0			0
$557$		−	1.15665e7i		−	1.57967i	−0.613323	−	0.789833i	$-0.710167\pi$
$557$		−	1.15665e7i		−	1.57967i	0.613323	−	0.789833i	$-0.289833\pi$
$558$		0			0
$559$	−5.71266e6			−0.773231
$560$		0			0
$561$			3.26646e7i			4.38197i
$562$		0			0
$563$			1.26700e7i			1.68463i	0.538986	+	0.842315i	$0.318807\pi$
$563$			1.26700e7i			1.68463i	−0.538986	+	0.842315i	$0.681193\pi$
$564$		0			0
$565$	2.12112e6			0.279540
$566$		0			0
$567$		−	1.00906e7i		−	1.31813i
$568$		0			0
$569$			6.08549e6i			0.787980i	0.919115	+	0.393990i	$0.128906\pi$
$569$			6.08549e6i			0.787980i	−0.919115	+	0.393990i	$0.871094\pi$
$570$		0			0
$571$	−9.25086e6			−1.18739			−0.593693	−	0.804692i	$-0.702331\pi$
$571$	−9.25086e6			−1.18739			−0.593693	+	0.804692i	$0.702331\pi$
$572$		0			0
$573$	−906260.			−0.115310
$574$		0			0
$575$		−	2.57518e6i		−	0.324817i
$576$		0			0
$577$	−3.88697e6			−0.486040			−0.243020	−	0.970021i	$-0.578138\pi$
$577$	−3.88697e6			−0.486040			−0.243020	+	0.970021i	$0.578138\pi$
$578$		0			0
$579$	1.87562e7			2.32513
$580$		0			0
$581$			2.20319e6i			0.270776i
$582$		0			0
$583$	−1.81452e7			−2.21100
$584$		0			0
$585$		−	1.04100e7i		−	1.25766i
$586$		0			0
$587$	−2.42989e6			−0.291066			−0.145533	−	0.989353i	$-0.546490\pi$
$587$	−2.42989e6			−0.291066			−0.145533	+	0.989353i	$0.546490\pi$
$588$		0			0
$589$	2.49312e6	+	1.66427e6i	0.296111	+	0.197668i
$590$		0			0
$591$	−1.73178e7			−2.03950
$592$		0			0
$593$	−1.21632e7			−1.42040			−0.710199	−	0.704001i	$-0.751395\pi$
$593$	−1.21632e7			−1.42040			−0.710199	+	0.704001i	$0.751395\pi$
$594$		0			0
$595$	−1.10831e7			−1.28343
$596$		0			0
$597$	−1.03121e7			−1.18417
$598$		0			0
$599$	−7.66883e6			−0.873297			−0.436649	−	0.899632i	$-0.643835\pi$
$599$	−7.66883e6			−0.873297			−0.436649	+	0.899632i	$0.643835\pi$
$600$		0			0
$601$		−	925541.i		−	0.104522i	−0.998633	−	0.0522612i	$-0.983357\pi$
$601$		−	925541.i		−	0.104522i	0.998633	−	0.0522612i	$-0.0166429\pi$
$602$		0			0
$603$		−	2.28008e7i		−	2.55363i
$604$		0			0
$605$			7.51452e6i			0.834666i
$606$		0			0
$607$	3.05032e6			0.336027			0.168013	−	0.985785i	$-0.446265\pi$
$607$	3.05032e6			0.336027			0.168013	+	0.985785i	$0.446265\pi$
$608$		0			0
$609$	−1.94344e7			−2.12339
$610$		0			0
$611$		−	1.15851e7i		−	1.25545i
$612$		0			0
$613$		−	9.58896e6i		−	1.03067i	−0.856988	−	0.515336i	$-0.827667\pi$
$613$		−	9.58896e6i		−	1.03067i	0.856988	−	0.515336i	$-0.172333\pi$
$614$		0			0
$615$			2.65600e6i			0.283166i
$616$		0			0
$617$	1.60113e7			1.69323			0.846613	−	0.532209i	$-0.178638\pi$
$617$	1.60113e7			1.69323			0.846613	+	0.532209i	$0.178638\pi$
$618$		0			0
$619$	1.43260e7			1.50279			0.751394	−	0.659854i	$-0.229382\pi$
$619$	1.43260e7			1.50279			0.751394	+	0.659854i	$0.229382\pi$
$620$		0			0
$621$	1.01131e7			1.05234
$622$		0			0
$623$	−2.80627e6			−0.289674
$624$		0			0
$625$	−1.18537e6			−0.121382
$626$		0			0
$627$	−1.43410e7	+	2.14832e7i	−1.45683	+	2.18238i
$628$		0			0
$629$	1.81618e7			1.83035
$630$		0			0
$631$			1.53824e6i			0.153798i	0.997039	+	0.0768990i	$0.0245019\pi$
$631$			1.53824e6i			0.153798i	−0.997039	+	0.0768990i	$0.975498\pi$
$632$		0			0
$633$	2.99923e7			2.97509
$634$		0			0
$635$		−	9.74025e6i		−	0.958596i
$636$		0			0
$637$	−3.32953e6			−0.325113
$638$		0			0
$639$	2.97398e7			2.88129
$640$		0			0
$641$			4.45367e6i			0.428128i	0.976820	+	0.214064i	$0.0686701\pi$
$641$			4.45367e6i			0.428128i	−0.976820	+	0.214064i	$0.931330\pi$
$642$		0			0
$643$	9.68595e6			0.923878			0.461939	−	0.886912i	$-0.347154\pi$
$643$	9.68595e6			0.923878			0.461939	+	0.886912i	$0.347154\pi$
$644$		0			0
$645$	−1.01397e7			−0.959682
$646$		0			0
$647$		−	1.23914e7i		−	1.16375i	−0.813277	−	0.581877i	$-0.802319\pi$
$647$		−	1.23914e7i		−	1.16375i	0.813277	−	0.581877i	$-0.197681\pi$
$648$		0			0
$649$		−	9.13637e6i		−	0.851456i
$650$		0			0
$651$	7.80321e6			0.721640
$652$		0			0
$653$		−	1.22140e7i		−	1.12093i	−0.828180	−	0.560463i	$-0.810623\pi$
$653$		−	1.22140e7i		−	1.12093i	0.828180	−	0.560463i	$-0.189377\pi$
$654$		0			0
$655$		−	1.11958e6i		−	0.101965i
$656$		0			0
$657$	−1.07136e7			−0.968327
$658$		0			0
$659$		−	1.71867e7i		−	1.54162i	−0.637064	−	0.770811i	$-0.719851\pi$
$659$		−	1.71867e7i		−	1.54162i	0.637064	−	0.770811i	$-0.280149\pi$
$660$		0			0
$661$	−1.24496e7			−1.10828			−0.554141	−	0.832423i	$-0.686953\pi$
$661$	−1.24496e7			−1.10828			−0.554141	+	0.832423i	$0.686953\pi$
$662$		0			0
$663$			3.06699e7i			2.70975i
$664$		0			0
$665$	−7.28928e6	−	4.86592e6i	−0.639191	−	0.426689i
$666$		0			0
$667$		−	6.95223e6i		−	0.605076i
$668$		0			0
$669$			343359.i			0.0296608i
$670$		0			0
$671$		−	2.30937e7i		−	1.98010i
$672$		0			0
$673$			3.43928e6i			0.292705i	0.989232	+	0.146353i	$0.0467534\pi$
$673$			3.43928e6i			0.292705i	−0.989232	+	0.146353i	$0.953247\pi$
$674$		0			0
$675$			1.21286e7i			1.02459i
$676$		0			0
$677$	9.71121e6			0.814332			0.407166	−	0.913354i	$-0.366517\pi$
$677$	9.71121e6			0.814332			0.407166	+	0.913354i	$0.366517\pi$
$678$		0			0
$679$	3.15695e6			0.262780
$680$		0			0
$681$	−1.58094e7			−1.30631
$682$		0			0
$683$			5.58761e6i			0.458325i	0.973388	+	0.229163i	$0.0735988\pi$
$683$			5.58761e6i			0.458325i	−0.973388	+	0.229163i	$0.926401\pi$
$684$		0			0
$685$		−	9.20975e6i		−	0.749931i
$686$		0			0
$687$	−2.74952e7			−2.22262
$688$		0			0
$689$	−1.70371e7			−1.36725
$690$		0			0
$691$	−1.96566e7			−1.56608			−0.783039	−	0.621973i	$-0.786331\pi$
$691$	−1.96566e7			−1.56608			−0.783039	+	0.621973i	$0.786331\pi$
$692$		0			0
$693$			4.51524e7i			3.57148i
$694$		0			0
$695$			1.56659e7i			1.23025i
$696$		0			0
$697$		−	5.25462e6i		−	0.409694i
$698$		0			0
$699$			1.96567e7i			1.52166i
$700$		0			0
$701$		−	7.29308e6i		−	0.560552i	−0.959919	−	0.280276i	$-0.909574\pi$
$701$		−	7.29308e6i		−	0.560552i	0.959919	−	0.280276i	$-0.0904260\pi$
$702$		0			0
$703$	1.19449e7	+	7.97373e6i	0.911576	+	0.608518i
$704$		0			0
$705$		−	2.05631e7i		−	1.55817i
$706$		0			0
$707$	1.26154e7			0.949190
$708$		0			0
$709$		−	8.20141e6i		−	0.612735i	−0.951913	−	0.306368i	$-0.900886\pi$
$709$		−	8.20141e6i		−	0.612735i	0.951913	−	0.306368i	$-0.0991136\pi$
$710$		0			0
$711$	3.61572e7			2.68238
$712$		0			0
$713$			2.79142e6i			0.205637i
$714$		0			0
$715$			1.26478e7i			0.925232i
$716$		0			0
$717$	−1.59498e7			−1.15866
$718$		0			0
$719$			8.13778e6i			0.587062i	0.955950	+	0.293531i	$0.0948303\pi$
$719$			8.13778e6i			0.587062i	−0.955950	+	0.293531i	$0.905170\pi$
$720$		0			0
$721$		−	2.57782e7i		−	1.84678i
$722$		0			0
$723$	1.72995e7			1.23080
$724$		0			0
$725$	8.33782e6			0.589125
$726$		0			0
$727$		−	3.38258e6i		−	0.237363i	−0.992932	−	0.118681i	$-0.962133\pi$
$727$		−	3.38258e6i		−	0.237363i	0.992932	−	0.118681i	$-0.0378667\pi$
$728$		0			0
$729$	1.23315e7			0.859403
$730$		0			0
$731$	2.00604e7			1.38850
$732$		0			0
$733$			2.60792e7i			1.79281i	0.443233	+	0.896407i	$0.353832\pi$
$733$			2.60792e7i			1.79281i	−0.443233	+	0.896407i	$0.646168\pi$
$734$		0			0
$735$	−5.90978e6			−0.403509
$736$		0			0
$737$			2.77022e7i			1.87865i
$738$		0			0
$739$	−9.29414e6			−0.626035			−0.313017	−	0.949747i	$-0.601340\pi$
$739$	−9.29414e6			−0.626035			−0.313017	+	0.949747i	$0.601340\pi$
$740$		0			0
$741$	−1.34653e7	+	2.01713e7i	−0.900885	+	1.34955i
$742$		0			0
$743$	−7.55420e6			−0.502015			−0.251007	−	0.967985i	$-0.580762\pi$
$743$	−7.55420e6			−0.502015			−0.251007	+	0.967985i	$0.580762\pi$
$744$		0			0
$745$	527285.			0.0348060
$746$		0			0
$747$	7.26676e6			0.476474
$748$		0			0
$749$	1.60639e7			1.04628
$750$		0			0
$751$	−4.11726e6			−0.266384			−0.133192	−	0.991090i	$-0.542523\pi$
$751$	−4.11726e6			−0.266384			−0.133192	+	0.991090i	$0.542523\pi$
$752$		0			0
$753$			1.07448e7i			0.690575i
$754$		0			0
$755$			9.52032e6i			0.607833i
$756$		0			0
$757$		−	2.99636e6i		−	0.190044i	−0.995475	−	0.0950219i	$-0.969708\pi$
$757$		−	2.99636e6i		−	0.190044i	0.995475	−	0.0950219i	$-0.0302921\pi$
$758$		0			0
$759$	−2.40536e7			−1.51557
$760$		0			0
$761$	1.82731e7			1.14380			0.571900	−	0.820323i	$-0.306206\pi$
$761$	1.82731e7			1.14380			0.571900	+	0.820323i	$0.306206\pi$
$762$		0			0
$763$		−	1.11694e7i		−	0.694575i
$764$		0			0
$765$			3.65555e7i			2.25839i
$766$		0			0
$767$		−	8.57847e6i		−	0.526528i
$768$		0			0
$769$	2.24831e7			1.37101			0.685505	−	0.728068i	$-0.259581\pi$
$769$	2.24831e7			1.37101			0.685505	+	0.728068i	$0.259581\pi$
$770$		0			0
$771$	3.61102e7			2.18773
$772$		0			0
$773$	2.42067e6			0.145709			0.0728545	−	0.997343i	$-0.476789\pi$
$773$	2.42067e6			0.145709			0.0728545	+	0.997343i	$0.476789\pi$
$774$		0			0
$775$	−3.34775e6			−0.200216
$776$		0			0
$777$	3.73862e7			2.22156
$778$		0			0
$779$	2.30698e6	−	3.45591e6i	0.136207	−	0.204042i
$780$		0			0
$781$	−3.61329e7			−2.11970
$782$		0			0
$783$			3.27437e7i			1.90864i
$784$		0			0
$785$	−2.17651e7			−1.26063
$786$		0			0
$787$			1.59992e7i			0.920794i	0.887713	+	0.460397i	$0.152293\pi$
$787$			1.59992e7i			0.920794i	−0.887713	+	0.460397i	$0.847707\pi$
$788$		0			0
$789$	1.87521e7			1.07240
$790$		0			0
$791$	−8.63834e6			−0.490896
$792$		0			0
$793$		−	2.16835e7i		−	1.22446i
$794$		0			0
$795$	−3.02402e7			−1.69694
$796$		0			0
$797$	−2.14331e7			−1.19519			−0.597597	−	0.801797i	$-0.703878\pi$
$797$	−2.14331e7			−1.19519			−0.597597	+	0.801797i	$0.703878\pi$
$798$		0			0
$799$			4.06819e7i			2.25442i
$800$		0			0
$801$			9.25591e6i			0.509727i
$802$		0			0
$803$	1.30166e7			0.712377
$804$		0			0
$805$		−	8.16144e6i		−	0.443892i
$806$		0			0
$807$			2.65986e7i			1.43772i
$808$		0			0
$809$	−2.78269e6			−0.149484			−0.0747418	−	0.997203i	$-0.523813\pi$
$809$	−2.78269e6			−0.149484			−0.0747418	+	0.997203i	$0.523813\pi$
$810$		0			0
$811$			1.93966e7i			1.03555i	0.855516	+	0.517777i	$0.173240\pi$
$811$			1.93966e7i			1.03555i	−0.855516	+	0.517777i	$0.826760\pi$
$812$		0			0
$813$	−4.06231e7			−2.15549
$814$		0			0
$815$			1.51729e7i			0.800156i
$816$		0			0
$817$	1.31935e7	+	8.80727e6i	0.691522	+	0.461622i
$818$		0			0
$819$			4.23952e7i			2.20855i
$820$		0			0
$821$		−	9.94941e6i		−	0.515157i	−0.966257	−	0.257579i	$-0.917075\pi$
$821$		−	9.94941e6i		−	0.515157i	0.966257	−	0.257579i	$-0.0829246\pi$
$822$		0			0
$823$		−	3.58026e7i		−	1.84253i	−0.388936	−	0.921265i	$-0.627157\pi$
$823$		−	3.58026e7i		−	1.84253i	0.388936	−	0.921265i	$-0.372843\pi$
$824$		0			0
$825$		−	2.88476e7i		−	1.47562i
$826$		0			0
$827$		−	1.95579e7i		−	0.994394i	−0.867638	−	0.497197i	$-0.834363\pi$
$827$		−	1.95579e7i		−	0.994394i	0.867638	−	0.497197i	$-0.165637\pi$
$828$		0			0
$829$	2.71049e7			1.36982			0.684908	−	0.728630i	$-0.259843\pi$
$829$	2.71049e7			1.36982			0.684908	+	0.728630i	$0.259843\pi$
$830$		0			0
$831$	3.59512e7			1.80597
$832$		0			0
$833$	1.16919e7			0.583810
$834$		0			0
$835$			4.82601e6i			0.239537i
$836$		0			0
$837$		−	1.31471e7i		−	0.648658i
$838$		0			0
$839$	−1.30660e7			−0.640820			−0.320410	−	0.947279i	$-0.603821\pi$
$839$	−1.30660e7			−0.640820			−0.320410	+	0.947279i	$0.603821\pi$
$840$		0			0
$841$	1.99851e6			0.0974353
$842$		0			0
$843$	−6.13679e7			−2.97422
$844$		0			0
$845$		−	1.85538e6i		−	0.0893902i
$846$		0			0
$847$		−	3.06031e7i		−	1.46574i
$848$		0			0
$849$			2.31122e7i			1.10045i
$850$		0			0
$851$			1.33741e7i			0.633052i
$852$		0			0
$853$		−	8.29122e6i		−	0.390163i	−0.980787	−	0.195081i	$-0.937503\pi$
$853$		−	8.29122e6i		−	0.390163i	0.980787	−	0.195081i	$-0.0624971\pi$
$854$		0			0
$855$	−1.60493e7	+	2.40422e7i	−0.750827	+	1.12476i
$856$		0			0
$857$			3.08173e7i			1.43332i	0.697424	+	0.716659i	$0.254329\pi$
$857$			3.08173e7i			1.43332i	−0.697424	+	0.716659i	$0.745671\pi$
$858$		0			0
$859$	−2.55864e7			−1.18311			−0.591557	−	0.806263i	$-0.701487\pi$
$859$	−2.55864e7			−1.18311			−0.591557	+	0.806263i	$0.701487\pi$
$860$		0			0
$861$		−	1.08167e7i		−	0.497262i
$862$		0			0
$863$	−1.18111e7			−0.539836			−0.269918	−	0.962883i	$-0.586997\pi$
$863$	−1.18111e7			−0.539836			−0.269918	+	0.962883i	$0.586997\pi$
$864$		0			0
$865$			2.23632e6i			0.101623i
$866$		0			0
$867$		−	6.90818e7i		−	3.12116i
$868$		0			0
$869$	−4.39297e7			−1.97337
$870$		0			0
$871$			2.60106e7i			1.16173i
$872$		0			0
$873$		−	1.04126e7i		−	0.462404i
$874$		0			0
$875$	2.71932e7			1.20071
$876$		0			0
$877$	2.57059e7			1.12858			0.564292	−	0.825575i	$-0.309149\pi$
$877$	2.57059e7			1.12858			0.564292	+	0.825575i	$0.309149\pi$
$878$		0			0
$879$			3.35594e7i			1.46501i
$880$		0			0
$881$	8.50822e6			0.369317			0.184658	−	0.982803i	$-0.440882\pi$
$881$	8.50822e6			0.369317			0.184658	+	0.982803i	$0.440882\pi$
$882$		0			0
$883$	−1.68717e7			−0.728212			−0.364106	−	0.931358i	$-0.618625\pi$
$883$	−1.68717e7			−0.728212			−0.364106	+	0.931358i	$0.618625\pi$
$884$		0			0
$885$		−	1.52264e7i		−	0.653491i
$886$		0			0
$887$	2.00106e6			0.0853986			0.0426993	−	0.999088i	$-0.486404\pi$
$887$	2.00106e6			0.0853986			0.0426993	+	0.999088i	$0.486404\pi$
$888$		0			0
$889$			3.96675e7i			1.68337i
$890$		0			0
$891$	4.04362e7			1.70638
$892$		0			0
$893$	−1.78609e7	+	2.67561e7i	−0.749506	+	1.12278i
$894$		0			0
$895$	−7.48504e6			−0.312346
$896$		0			0
$897$	−2.25848e7			−0.937207
$898$		0			0
$899$	−9.03795e6			−0.372967
$900$		0			0
$901$	5.98270e7			2.45519
$902$		0			0
$903$	4.12944e7			1.68528
$904$		0			0
$905$		−	9.62490e6i		−	0.390638i
$906$		0			0
$907$		−	1.33783e7i		−	0.539985i	−0.962862	−	0.269993i	$-0.912979\pi$
$907$		−	1.33783e7i		−	0.539985i	0.962862	−	0.269993i	$-0.0870213\pi$
$908$		0			0
$909$		−	4.16094e7i		−	1.67025i
$910$		0			0
$911$	1.78700e6			0.0713393			0.0356696	−	0.999364i	$-0.488644\pi$
$911$	1.78700e6			0.0713393			0.0356696	+	0.999364i	$0.488644\pi$
$912$		0			0
$913$	−8.82886e6			−0.350532
$914$		0			0
$915$		−	3.84873e7i		−	1.51972i
$916$		0			0
$917$			4.55953e6i			0.179059i
$918$		0			0
$919$		−	3.22979e7i		−	1.26149i	−0.775989	−	0.630747i	$-0.782749\pi$
$919$		−	3.22979e7i		−	1.26149i	0.775989	−	0.630747i	$-0.217251\pi$
$920$		0			0
$921$	524639.			0.0203804
$922$		0			0
$923$	−3.39264e7			−1.31079
$924$		0			0
$925$	−1.60395e7			−0.616364
$926$		0			0
$927$	−8.50242e7			−3.24970
$928$		0			0
$929$	2.14880e7			0.816875			0.408438	−	0.912786i	$-0.366074\pi$
$929$	2.14880e7			0.816875			0.408438	+	0.912786i	$0.366074\pi$
$930$		0			0
$931$	7.68964e6	+	5.13318e6i	0.290758	+	0.194094i
$932$		0			0
$933$	−1.89432e7			−0.712441
$934$		0			0
$935$		−	4.44137e7i		−	1.66145i
$936$		0			0
$937$	−3.69853e7			−1.37619			−0.688097	−	0.725619i	$-0.741554\pi$
$937$	−3.69853e7			−1.37619			−0.688097	+	0.725619i	$0.741554\pi$
$938$		0			0
$939$			4.25130e7i			1.57347i
$940$		0			0
$941$	−1.46476e7			−0.539252			−0.269626	−	0.962965i	$-0.586900\pi$
$941$	−1.46476e7			−0.539252			−0.269626	+	0.962965i	$0.586900\pi$
$942$		0			0
$943$	3.86941e6			0.141699
$944$		0			0
$945$			3.84389e7i			1.40020i
$946$		0			0
$947$	−1.57533e7			−0.570816			−0.285408	−	0.958406i	$-0.592129\pi$
$947$	−1.57533e7			−0.570816			−0.285408	+	0.958406i	$0.592129\pi$
$948$		0			0
$949$	1.22218e7			0.440524
$950$		0			0
$951$		−	4.21841e7i		−	1.51251i
$952$		0			0
$953$			5.45212e7i			1.94461i	0.233709	+	0.972307i	$0.424914\pi$
$953$			5.45212e7i			1.94461i	−0.233709	+	0.972307i	$0.575086\pi$
$954$		0			0
$955$	1.23223e6			0.0437204
$956$		0			0
$957$		−	7.78799e7i		−	2.74882i
$958$		0			0
$959$			3.75070e7i			1.31694i
$960$		0			0
$961$	−2.50003e7			−0.873246
$962$		0			0
$963$		−	5.29835e7i		−	1.84109i
$964$		0			0
$965$	−2.55025e7			−0.881587
$966$		0			0
$967$			5.94249e6i			0.204363i	0.994766	+	0.102181i	$0.0325822\pi$
$967$			5.94249e6i			0.204363i	−0.994766	+	0.102181i	$0.967418\pi$
$968$		0			0
$969$	4.72842e7	−	7.08329e7i	1.61773	−	2.42340i
$970$		0			0
$971$		−	758200.i		−	0.0258069i	−0.999917	−	0.0129034i	$-0.995893\pi$
$971$		−	758200.i		−	0.0258069i	0.999917	−	0.0129034i	$-0.00410741\pi$
$972$		0			0
$973$		−	6.37998e7i		−	2.16042i
$974$		0			0
$975$		−	2.70860e7i		−	0.912501i
$976$		0			0
$977$		−	1.14601e7i		−	0.384107i	−0.981384	−	0.192054i	$-0.938485\pi$
$977$		−	1.14601e7i		−	0.384107i	0.981384	−	0.192054i	$-0.0615148\pi$
$978$		0			0
$979$		−	1.12456e7i		−	0.374995i
$980$		0			0
$981$	−3.68401e7			−1.22222
$982$		0			0
$983$	−4.07508e7			−1.34509			−0.672546	−	0.740055i	$-0.734799\pi$
$983$	−4.07508e7			−1.34509			−0.672546	+	0.740055i	$0.734799\pi$
$984$		0			0
$985$	2.35468e7			0.773287
$986$		0			0
$987$			8.37440e7i			2.73628i
$988$		0			0
$989$			1.47721e7i			0.480234i
$990$		0			0
$991$	5.70617e7			1.84570			0.922849	−	0.385161i	$-0.125854\pi$
$991$	5.70617e7			1.84570			0.922849	+	0.385161i	$0.125854\pi$
$992$		0			0
$993$	1.40256e7			0.451386
$994$		0			0
$995$	1.40213e7			0.448984
$996$		0			0
$997$			3.72819e7i			1.18785i	0.804522	+	0.593923i	$0.202422\pi$
$997$			3.72819e7i			1.18785i	−0.804522	+	0.593923i	$0.797578\pi$
$998$		0			0
$999$		−	6.29894e7i		−	1.99689i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	608.6.b.b.303.4		96
4.3	odd	2		152.6.b.b.75.13	✓	96
8.3	odd	2	inner	608.6.b.b.303.3		96
8.5	even	2		152.6.b.b.75.83	yes	96
19.18	odd	2	inner	608.6.b.b.303.94		96
76.75	even	2		152.6.b.b.75.84	yes	96
152.37	odd	2		152.6.b.b.75.14	yes	96
152.75	even	2	inner	608.6.b.b.303.93		96

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
152.6.b.b.75.13	✓	96	4.3	odd	2
152.6.b.b.75.14	yes	96	152.37	odd	2
152.6.b.b.75.83	yes	96	8.5	even	2
152.6.b.b.75.84	yes	96	76.75	even	2
608.6.b.b.303.3		96	8.3	odd	2	inner
608.6.b.b.303.4		96	1.1	even	1	trivial
608.6.b.b.303.93		96	152.75	even	2	inner
608.6.b.b.303.94		96	19.18	odd	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 608 = 2^{5} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	608.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-27.1983i q^{3} +36.9812i q^{5} -150.607i q^{7} -496.748 q^{9} +O(q^{10})\)	\(q-27.1983i q^{3} +36.9812i q^{5} -150.607i q^{7} -496.748 q^{9} +603.531 q^{11} +566.677 q^{13} +1005.83 q^{15} -1989.92 q^{17} +(-1308.75 - 873.651i) q^{19} -4096.26 q^{21} -1465.34i q^{23} +1757.39 q^{25} +6901.50i q^{27} +4744.43 q^{29} -1904.96 q^{31} -16415.0i q^{33} +5569.64 q^{35} -9126.91 q^{37} -15412.6i q^{39} +2640.62i q^{41} -10081.0 q^{43} -18370.3i q^{45} -20444.0i q^{47} -5875.55 q^{49} +54122.5i q^{51} -30065.0 q^{53} +22319.3i q^{55} +(-23761.8 + 35595.8i) q^{57} -15138.2i q^{59} -38264.3i q^{61} +74813.8i q^{63} +20956.4i q^{65} +45900.3i q^{67} -39854.9 q^{69} -59869.1 q^{71} +21567.5 q^{73} -47798.0i q^{75} -90896.1i q^{77} -72787.8 q^{79} +66999.5 q^{81} -14628.7 q^{83} -73589.7i q^{85} -129041. i q^{87} -18633.0i q^{89} -85345.6i q^{91} +51811.7i q^{93} +(32308.7 - 48399.3i) q^{95} +20961.5i q^{97} -299802. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 96 q - 6168 q^{9}+O(q^{10}) \) 96 * q - 6168 * q^9	\( 96 q - 6168 q^{9} - 944 q^{11} - 3832 q^{17} - 5240 q^{19} - 62504 q^{25} - 7720 q^{35} - 45096 q^{43} - 210840 q^{49} - 36336 q^{57} - 4336 q^{73} - 20624 q^{81} - 52152 q^{83} + 752768 q^{99}+O(q^{100}) \) 96 * q - 6168 * q^9 - 944 * q^11 - 3832 * q^17 - 5240 * q^19 - 62504 * q^25 - 7720 * q^35 - 45096 * q^43 - 210840 * q^49 - 36336 * q^57 - 4336 * q^73 - 20624 * q^81 - 52152 * q^83 + 752768 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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