LMFDB - Embedded newform 608.6.b.b.303.12

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			303.12
Character	$\chi$	$=$	608.303
Dual form			608.6.b.b.303.85

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-23.2069i q^{3} +79.1121i q^{5} +208.766i q^{7} -295.559 q^{9} +O(q^{10})$	$q-23.2069i q^{3} +79.1121i q^{5} +208.766i q^{7} -295.559 q^{9} -42.6699 q^{11} -1094.06 q^{13} +1835.95 q^{15} +923.831 q^{17} +(-887.137 + 1299.65i) q^{19} +4844.81 q^{21} +4470.12i q^{23} -3133.73 q^{25} +1219.73i q^{27} -702.397 q^{29} +8677.02 q^{31} +990.235i q^{33} -16515.9 q^{35} +1236.77 q^{37} +25389.7i q^{39} -3337.88i q^{41} -15464.5 q^{43} -23382.3i q^{45} -14789.1i q^{47} -26776.3 q^{49} -21439.2i q^{51} +1362.26 q^{53} -3375.71i q^{55} +(30160.8 + 20587.7i) q^{57} +16205.1i q^{59} -46237.6i q^{61} -61702.7i q^{63} -86553.4i q^{65} -20860.6i q^{67} +103738. q^{69} -11340.6 q^{71} +5870.39 q^{73} +72724.1i q^{75} -8908.02i q^{77} -43058.9 q^{79} -43514.7 q^{81} +65188.1 q^{83} +73086.2i q^{85} +16300.4i q^{87} -95213.6i q^{89} -228403. i q^{91} -201367. i q^{93} +(-102818. - 70183.3i) q^{95} -25719.9i q^{97} +12611.5 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$		−	23.2069i		−	1.48872i	−0.667778	−	0.744361i	$-0.732754\pi$
$3$		−	23.2069i		−	1.48872i	0.667778	−	0.744361i	$-0.267246\pi$
$4$		0			0
$5$			79.1121i			1.41520i	0.706613	+	0.707601i	$0.250222\pi$
$5$			79.1121i			1.41520i	−0.706613	+	0.707601i	$0.749778\pi$
$6$		0			0
$7$			208.766i			1.61033i	0.593051	+	0.805165i	$0.297923\pi$
$7$			208.766i			1.61033i	−0.593051	+	0.805165i	$0.702077\pi$
$8$		0			0
$9$	−295.559			−1.21629
$10$		0			0
$11$	−42.6699			−0.106326			−0.0531630	−	0.998586i	$-0.516930\pi$
$11$	−42.6699			−0.106326			−0.0531630	+	0.998586i	$0.516930\pi$
$12$		0			0
$13$	−1094.06			−1.79549			−0.897745	−	0.440516i	$-0.854795\pi$
$13$	−1094.06			−1.79549			−0.897745	+	0.440516i	$0.854795\pi$
$14$		0			0
$15$	1835.95			2.10684
$16$		0			0
$17$	923.831			0.775300			0.387650	−	0.921807i	$-0.373287\pi$
$17$	923.831			0.775300			0.387650	+	0.921807i	$0.373287\pi$
$18$		0			0
$19$	−887.137	+	1299.65i	−0.563776	+	0.825928i
$19$	−887.137	+	1299.65i	−0.563776	+	0.825928i
$20$		0			0
$21$	4844.81			2.39733
$22$		0			0
$23$			4470.12i			1.76197i	0.473140	+	0.880987i	$0.343121\pi$
$23$			4470.12i			1.76197i	−0.473140	+	0.880987i	$0.656879\pi$
$24$		0			0
$25$	−3133.73			−1.00279
$26$		0			0
$27$			1219.73i			0.322000i
$28$		0			0
$29$	−702.397			−0.155091			−0.0775457	−	0.996989i	$-0.524708\pi$
$29$	−702.397			−0.155091			−0.0775457	+	0.996989i	$0.524708\pi$
$30$		0			0
$31$	8677.02			1.62168			0.810842	−	0.585264i	$-0.199009\pi$
$31$	8677.02			1.62168			0.810842	+	0.585264i	$0.199009\pi$
$32$		0			0
$33$			990.235i			0.158290i
$34$		0			0
$35$	−16515.9			−2.27894
$36$		0			0
$37$	1236.77			0.148520			0.0742599	−	0.997239i	$-0.476341\pi$
$37$	1236.77			0.148520			0.0742599	+	0.997239i	$0.476341\pi$
$38$		0			0
$39$			25389.7i			2.67298i
$40$		0			0
$41$		−	3337.88i		−	0.310106i	−0.987906	−	0.155053i	$-0.950445\pi$
$41$		−	3337.88i		−	0.310106i	0.987906	−	0.155053i	$-0.0495549\pi$
$42$		0			0
$43$	−15464.5			−1.27545			−0.637726	−	0.770263i	$-0.720125\pi$
$43$	−15464.5			−1.27545			−0.637726	+	0.770263i	$0.720125\pi$
$44$		0			0
$45$		−	23382.3i		−	1.72130i
$46$		0			0
$47$		−	14789.1i		−	0.976553i	−0.872689	−	0.488277i	$-0.837626\pi$
$47$		−	14789.1i		−	0.976553i	0.872689	−	0.488277i	$-0.162374\pi$
$48$		0			0
$49$	−26776.3			−1.59316
$50$		0			0
$51$		−	21439.2i		−	1.15421i
$52$		0			0
$53$	1362.26			0.0666147			0.0333074	−	0.999445i	$-0.489396\pi$
$53$	1362.26			0.0666147			0.0333074	+	0.999445i	$0.489396\pi$
$54$		0			0
$55$		−	3375.71i		−	0.150473i
$56$		0			0
$57$	30160.8	+	20587.7i	1.22958	+	0.839305i
$58$		0			0
$59$			16205.1i			0.606067i	0.952980	+	0.303034i	$0.0979995\pi$
$59$			16205.1i			0.606067i	−0.952980	+	0.303034i	$0.902001\pi$
$60$		0			0
$61$		−	46237.6i		−	1.59100i	−0.605952	−	0.795501i	$-0.707208\pi$
$61$		−	46237.6i		−	1.59100i	0.605952	−	0.795501i	$-0.292792\pi$
$62$		0			0
$63$		−	61702.7i		−	1.95863i
$64$		0			0
$65$		−	86553.4i		−	2.54098i
$66$		0			0
$67$		−	20860.6i		−	0.567727i	−0.958865	−	0.283864i	$-0.908384\pi$
$67$		−	20860.6i		−	0.567727i	0.958865	−	0.283864i	$-0.0916163\pi$
$68$		0			0
$69$	103738.			2.62309
$70$		0			0
$71$	−11340.6			−0.266988			−0.133494	−	0.991050i	$-0.542620\pi$
$71$	−11340.6			−0.266988			−0.133494	+	0.991050i	$0.542620\pi$
$72$		0			0
$73$	5870.39			0.128932			0.0644658	−	0.997920i	$-0.479466\pi$
$73$	5870.39			0.128932			0.0644658	+	0.997920i	$0.479466\pi$
$74$		0			0
$75$			72724.1i			1.49288i
$76$		0			0
$77$		−	8908.02i		−	0.171220i
$78$		0			0
$79$	−43058.9			−0.776239			−0.388120	−	0.921609i	$-0.626875\pi$
$79$	−43058.9			−0.776239			−0.388120	+	0.921609i	$0.626875\pi$
$80$		0			0
$81$	−43514.7			−0.736925
$82$		0			0
$83$	65188.1			1.03866			0.519330	−	0.854574i	$-0.326182\pi$
$83$	65188.1			1.03866			0.519330	+	0.854574i	$0.326182\pi$
$84$		0			0
$85$			73086.2i			1.09721i
$86$		0			0
$87$			16300.4i			0.230888i
$88$		0			0
$89$		−	95213.6i		−	1.27416i	−0.770798	−	0.637080i	$-0.780142\pi$
$89$		−	95213.6i		−	1.27416i	0.770798	−	0.637080i	$-0.219858\pi$
$90$		0			0
$91$		−	228403.i		−	2.89133i
$92$		0			0
$93$		−	201367.i		−	2.41424i
$94$		0			0
$95$	−102818.	−	70183.3i	−1.16885	−	0.797856i
$96$		0			0
$97$		−	25719.9i		−	0.277549i	−0.990324	−	0.138774i	$-0.955684\pi$
$97$		−	25719.9i		−	0.277549i	0.990324	−	0.138774i	$-0.0443163\pi$
$98$		0			0
$99$	12611.5			0.129324
$100$		0			0
$101$			42811.5i			0.417597i	0.977959	+	0.208798i	$0.0669552\pi$
$101$			42811.5i			0.417597i	−0.977959	+	0.208798i	$0.933045\pi$
$102$		0			0
$103$	−101300.			−0.940842			−0.470421	−	0.882442i	$-0.655898\pi$
$103$	−101300.			−0.940842			−0.470421	+	0.882442i	$0.655898\pi$
$104$		0			0
$105$			383283.i			3.39271i
$106$		0			0
$107$			80169.9i			0.676943i	0.940977	+	0.338471i	$0.109910\pi$
$107$			80169.9i			0.676943i	−0.940977	+	0.338471i	$0.890090\pi$
$108$		0			0
$109$	−110280.			−0.889060			−0.444530	−	0.895764i	$-0.646629\pi$
$109$	−110280.			−0.889060			−0.444530	+	0.895764i	$0.646629\pi$
$110$		0			0
$111$		−	28701.6i		−	0.221105i
$112$		0			0
$113$		−	38800.4i		−	0.285851i	−0.989733	−	0.142926i	$-0.954349\pi$
$113$		−	38800.4i		−	0.285851i	0.989733	−	0.142926i	$-0.0456510\pi$
$114$		0			0
$115$	−353641.			−2.49355
$116$		0			0
$117$	323359.			2.18384
$118$		0			0
$119$			192864.i			1.24849i
$120$		0			0
$121$	−159230.			−0.988695
$122$		0			0
$123$	−77461.7			−0.461662
$124$		0			0
$125$		−	690.805i		−	0.00395440i
$126$		0			0
$127$	−20057.1			−0.110347			−0.0551733	−	0.998477i	$-0.517571\pi$
$127$	−20057.1			−0.110347			−0.0551733	+	0.998477i	$0.517571\pi$
$128$		0			0
$129$			358882.i			1.89879i
$130$		0			0
$131$	285953.			1.45585			0.727925	−	0.685657i	$-0.240485\pi$
$131$	285953.			1.45585			0.727925	+	0.685657i	$0.240485\pi$
$132$		0			0
$133$	−271323.	−	185204.i	−1.33002	−	0.907865i
$134$		0			0
$135$	−96495.7			−0.455694
$136$		0			0
$137$	39689.2			0.180664			0.0903318	−	0.995912i	$-0.471207\pi$
$137$	39689.2			0.180664			0.0903318	+	0.995912i	$0.471207\pi$
$138$		0			0
$139$	−116907.			−0.513222			−0.256611	−	0.966515i	$-0.582606\pi$
$139$	−116907.			−0.513222			−0.256611	+	0.966515i	$0.582606\pi$
$140$		0			0
$141$	−343208.			−1.45382
$142$		0			0
$143$	46683.4			0.190907
$144$		0			0
$145$		−	55568.2i		−	0.219486i
$146$		0			0
$147$			621394.i			2.37177i
$148$		0			0
$149$		−	329263.i		−	1.21500i	−0.794319	−	0.607501i	$-0.792172\pi$
$149$		−	329263.i		−	1.21500i	0.794319	−	0.607501i	$-0.207828\pi$
$150$		0			0
$151$	−412897.			−1.47367			−0.736834	−	0.676073i	$-0.763680\pi$
$151$	−412897.			−1.47367			−0.736834	+	0.676073i	$0.763680\pi$
$152$		0			0
$153$	−273047.			−0.942992
$154$		0			0
$155$			686458.i			2.29501i
$156$		0			0
$157$			340578.i			1.10273i	0.834266	+	0.551363i	$0.185892\pi$
$157$			340578.i			1.10273i	−0.834266	+	0.551363i	$0.814108\pi$
$158$		0			0
$159$		−	31613.8i		−	0.0991708i
$160$		0			0
$161$	−933210.			−2.83736
$162$		0			0
$163$	239840.			0.707053			0.353527	−	0.935424i	$-0.384982\pi$
$163$	239840.			0.707053			0.353527	+	0.935424i	$0.384982\pi$
$164$		0			0
$165$	−78339.6			−0.224012
$166$		0			0
$167$	518481.			1.43860			0.719302	−	0.694698i	$-0.244462\pi$
$167$	518481.			1.43860			0.719302	+	0.694698i	$0.244462\pi$
$168$		0			0
$169$	825674.			2.22378
$170$		0			0
$171$	262201.	−	384123.i	0.685716	−	1.00457i
$172$		0			0
$173$	−176280.			−0.447803			−0.223901	−	0.974612i	$-0.571879\pi$
$173$	−176280.			−0.447803			−0.223901	+	0.974612i	$0.571879\pi$
$174$		0			0
$175$		−	654217.i		−	1.61483i
$176$		0			0
$177$	376069.			0.902266
$178$		0			0
$179$			195531.i			0.456125i	0.973646	+	0.228063i	$0.0732391\pi$
$179$			195531.i			0.456125i	−0.973646	+	0.228063i	$0.926761\pi$
$180$		0			0
$181$	−307063.			−0.696677			−0.348338	−	0.937369i	$-0.613254\pi$
$181$	−307063.			−0.696677			−0.348338	+	0.937369i	$0.613254\pi$
$182$		0			0
$183$	−1.07303e6			−2.36856
$184$		0			0
$185$			97843.5i			0.210185i
$186$		0			0
$187$	−39419.7			−0.0824346
$188$		0			0
$189$	−254639.			−0.518525
$190$		0			0
$191$			794882.i			1.57659i	0.615297	+	0.788295i	$0.289036\pi$
$191$			794882.i			1.57659i	−0.615297	+	0.788295i	$0.710964\pi$
$192$		0			0
$193$		−	424903.i		−	0.821101i	−0.911838	−	0.410551i	$-0.865337\pi$
$193$		−	424903.i		−	0.821101i	0.911838	−	0.410551i	$-0.134663\pi$
$194$		0			0
$195$	−2.00863e6			−3.78281
$196$		0			0
$197$		−	138949.i		−	0.255087i	−0.991833	−	0.127544i	$-0.959291\pi$
$197$		−	138949.i		−	0.255087i	0.991833	−	0.127544i	$-0.0407092\pi$
$198$		0			0
$199$			549694.i			0.983985i	0.870600	+	0.491992i	$0.163731\pi$
$199$			549694.i			0.983985i	−0.870600	+	0.491992i	$0.836269\pi$
$200$		0			0
$201$	−484109.			−0.845188
$202$		0			0
$203$		−	146637.i		−	0.249748i
$204$		0			0
$205$	264067.			0.438863
$206$		0			0
$207$		−	1.32119e6i		−	2.14308i
$208$		0			0
$209$	37854.0	−	55455.9i	0.0599441	−	0.0878177i
$210$		0			0
$211$		−	1.19193e6i		−	1.84308i	−0.388279	−	0.921542i	$-0.626930\pi$
$211$		−	1.19193e6i		−	1.84308i	0.388279	−	0.921542i	$-0.373070\pi$
$212$		0			0
$213$			263181.i			0.397470i
$214$		0			0
$215$		−	1.22343e6i		−	1.80502i
$216$		0			0
$217$			1.81147e6i			2.61145i
$218$		0			0
$219$		−	136233.i		−	0.191943i
$220$		0			0
$221$	−1.01073e6			−1.39204
$222$		0			0
$223$	1.31561e6			1.77159			0.885796	−	0.464074i	$-0.153613\pi$
$223$	1.31561e6			1.77159			0.885796	+	0.464074i	$0.153613\pi$
$224$		0			0
$225$	926203.			1.21969
$226$		0			0
$227$			869105.i			1.11946i	0.828676	+	0.559729i	$0.189095\pi$
$227$			869105.i			1.11946i	−0.828676	+	0.559729i	$0.810905\pi$
$228$		0			0
$229$		−	432757.i		−	0.545325i	−0.962110	−	0.272663i	$-0.912096\pi$
$229$		−	432757.i		−	0.545325i	0.962110	−	0.272663i	$-0.0879043\pi$
$230$		0			0
$231$	−206727.			−0.254899
$232$		0			0
$233$	−552562.			−0.666793			−0.333396	−	0.942787i	$-0.608195\pi$
$233$	−552562.			−0.666793			−0.333396	+	0.942787i	$0.608195\pi$
$234$		0			0
$235$	1.16999e6			1.38202
$236$		0			0
$237$			999263.i			1.15560i
$238$		0			0
$239$		−	1.17262e6i		−	1.32790i	−0.747779	−	0.663948i	$-0.768880\pi$
$239$		−	1.17262e6i		−	1.32790i	0.747779	−	0.663948i	$-0.231120\pi$
$240$		0			0
$241$			157697.i			0.174896i	0.996169	+	0.0874479i	$0.0278711\pi$
$241$			157697.i			0.174896i	−0.996169	+	0.0874479i	$0.972129\pi$
$242$		0			0
$243$			1.30623e6i			1.41908i
$244$		0			0
$245$		−	2.11833e6i		−	2.25464i
$246$		0			0
$247$	970581.	−	1.42189e6i	1.01225	−	1.48294i
$248$		0			0
$249$		−	1.51281e6i		−	1.54628i
$250$		0			0
$251$	−749852.			−0.751261			−0.375631	−	0.926769i	$-0.622574\pi$
$251$	−749852.			−0.751261			−0.375631	+	0.926769i	$0.622574\pi$
$252$		0			0
$253$		−	190740.i		−	0.187344i
$254$		0			0
$255$	1.69610e6			1.63343
$256$		0			0
$257$		−	422720.i		−	0.399227i	−0.979875	−	0.199613i	$-0.936031\pi$
$257$		−	422720.i		−	0.399227i	0.979875	−	0.199613i	$-0.0639686\pi$
$258$		0			0
$259$			258196.i			0.239166i
$260$		0			0
$261$	207600.			0.188637
$262$		0			0
$263$			4895.42i			0.00436416i	0.999998	+	0.00218208i	$0.000694578\pi$
$263$			4895.42i			0.00436416i	−0.999998	+	0.00218208i	$0.999305\pi$
$264$		0			0
$265$			107771.i			0.0942732i
$266$		0			0
$267$	−2.20961e6			−1.89687
$268$		0			0
$269$	−401410.			−0.338226			−0.169113	−	0.985597i	$-0.554090\pi$
$269$	−401410.			−0.338226			−0.169113	+	0.985597i	$0.554090\pi$
$270$		0			0
$271$			1.40287e6i			1.16036i	0.814487	+	0.580181i	$0.197018\pi$
$271$			1.40287e6i			1.16036i	−0.814487	+	0.580181i	$0.802982\pi$
$272$		0			0
$273$	−5.30051e6			−4.30439
$274$		0			0
$275$	133716.			0.106623
$276$		0			0
$277$		−	2.01766e6i		−	1.57997i	−0.613128	−	0.789984i	$-0.710089\pi$
$277$		−	2.01766e6i		−	1.57997i	0.613128	−	0.789984i	$-0.289911\pi$
$278$		0			0
$279$	−2.56457e6			−1.97244
$280$		0			0
$281$		−	2.11022e6i		−	1.59427i	−0.603800	−	0.797136i	$-0.706347\pi$
$281$		−	2.11022e6i		−	1.59427i	0.603800	−	0.797136i	$-0.293653\pi$
$282$		0			0
$283$	1.13097e6			0.839428			0.419714	−	0.907656i	$-0.362130\pi$
$283$	1.13097e6			0.839428			0.419714	+	0.907656i	$0.362130\pi$
$284$		0			0
$285$	−1.62873e6	+	2.38609e6i	−1.18779	+	1.74010i
$286$		0			0
$287$	696835.			0.499373
$288$		0			0
$289$	−566394.			−0.398909
$290$		0			0
$291$	−596878.			−0.413193
$292$		0			0
$293$	885946.			0.602890			0.301445	−	0.953483i	$-0.402531\pi$
$293$	885946.			0.602890			0.301445	+	0.953483i	$0.402531\pi$
$294$		0			0
$295$	−1.28202e6			−0.857707
$296$		0			0
$297$		−	52045.9i		−	0.0342369i
$298$		0			0
$299$		−	4.89058e6i		−	3.16361i
$300$		0			0
$301$		−	3.22846e6i		−	2.05390i
$302$		0			0
$303$	993521.			0.621685
$304$		0			0
$305$	3.65796e6			2.25159
$306$		0			0
$307$		−	977353.i		−	0.591841i	−0.955213	−	0.295921i	$-0.904374\pi$
$307$		−	977353.i		−	0.591841i	0.955213	−	0.295921i	$-0.0956264\pi$
$308$		0			0
$309$			2.35086e6i			1.40065i
$310$		0			0
$311$			228260.i			0.133822i	0.997759	+	0.0669112i	$0.0213144\pi$
$311$			228260.i			0.133822i	−0.997759	+	0.0669112i	$0.978686\pi$
$312$		0			0
$313$	−2.39695e6			−1.38292			−0.691461	−	0.722414i	$-0.743033\pi$
$313$	−2.39695e6			−1.38292			−0.691461	+	0.722414i	$0.743033\pi$
$314$		0			0
$315$	4.88143e6			2.77186
$316$		0			0
$317$	499487.			0.279174			0.139587	−	0.990210i	$-0.455422\pi$
$317$	499487.			0.279174			0.139587	+	0.990210i	$0.455422\pi$
$318$		0			0
$319$	29971.2			0.0164903
$320$		0			0
$321$	1.86049e6			1.00778
$322$		0			0
$323$	−819564.	+	1.20066e6i	−0.437096	+	0.640342i
$324$		0			0
$325$	3.42849e6			1.80051
$326$		0			0
$327$			2.55926e6i			1.32356i
$328$		0			0
$329$	3.08745e6			1.57257
$330$		0			0
$331$			1.95157e6i			0.979072i	0.871983	+	0.489536i	$0.162834\pi$
$331$			1.95157e6i			0.979072i	−0.871983	+	0.489536i	$0.837166\pi$
$332$		0			0
$333$	−365538.			−0.180644
$334$		0			0
$335$	1.65033e6			0.803448
$336$		0			0
$337$		−	51256.6i		−	0.0245853i	−0.999924	−	0.0122926i	$-0.996087\pi$
$337$		−	51256.6i		−	0.0245853i	0.999924	−	0.0122926i	$-0.00391297\pi$
$338$		0			0
$339$	−900436.			−0.425553
$340$		0			0
$341$	−370248.			−0.172427
$342$		0			0
$343$		−	2.08125e6i		−	0.955186i
$344$		0			0
$345$			8.20690e6i			3.71220i
$346$		0			0
$347$	−1.11059e6			−0.495141			−0.247571	−	0.968870i	$-0.579632\pi$
$347$	−1.11059e6			−0.495141			−0.247571	+	0.968870i	$0.579632\pi$
$348$		0			0
$349$			3.56207e6i			1.56545i	0.622369	+	0.782724i	$0.286170\pi$
$349$			3.56207e6i			1.56545i	−0.622369	+	0.782724i	$0.713830\pi$
$350$		0			0
$351$		−	1.33446e6i		−	0.578147i
$352$		0			0
$353$	−62268.2			−0.0265968			−0.0132984	−	0.999912i	$-0.504233\pi$
$353$	−62268.2			−0.0265968			−0.0132984	+	0.999912i	$0.504233\pi$
$354$		0			0
$355$		−	897181.i		−	0.377841i
$356$		0			0
$357$	4.47578e6			1.85865
$358$		0			0
$359$			1.92470e6i			0.788182i	0.919071	+	0.394091i	$0.128940\pi$
$359$			1.92470e6i			0.788182i	−0.919071	+	0.394091i	$0.871060\pi$
$360$		0			0
$361$	−902076.	−	2.30593e6i	−0.364313	−	0.931276i
$362$		0			0
$363$			3.69524e6i			1.47189i
$364$		0			0
$365$			464419.i			0.182464i
$366$		0			0
$367$		−	3.02729e6i		−	1.17324i	−0.809861	−	0.586622i	$-0.800457\pi$
$367$		−	3.02729e6i		−	1.17324i	0.809861	−	0.586622i	$-0.199543\pi$
$368$		0			0
$369$			986540.i			0.377180i
$370$		0			0
$371$			284394.i			0.107272i
$372$		0			0
$373$	−1.80577e6			−0.672033			−0.336016	−	0.941856i	$-0.609080\pi$
$373$	−1.80577e6			−0.672033			−0.336016	+	0.941856i	$0.609080\pi$
$374$		0			0
$375$	−16031.4			−0.00588700
$376$		0			0
$377$	768465.			0.278465
$378$		0			0
$379$			1.34229e6i			0.480009i	0.970772	+	0.240005i	$0.0771489\pi$
$379$			1.34229e6i			0.480009i	−0.970772	+	0.240005i	$0.922851\pi$
$380$		0			0
$381$			465463.i			0.164275i
$382$		0			0
$383$	−4.74504e6			−1.65289			−0.826443	−	0.563021i	$-0.809639\pi$
$383$	−4.74504e6			−1.65289			−0.826443	+	0.563021i	$0.809639\pi$
$384$		0			0
$385$	704733.			0.242311
$386$		0			0
$387$	4.57067e6			1.55132
$388$		0			0
$389$			1.21902e6i			0.408446i	0.978924	+	0.204223i	$0.0654668\pi$
$389$			1.21902e6i			0.408446i	−0.978924	+	0.204223i	$0.934533\pi$
$390$		0			0
$391$			4.12963e6i			1.36606i
$392$		0			0
$393$		−	6.63608e6i		−	2.16736i
$394$		0			0
$395$		−	3.40648e6i		−	1.09853i
$396$		0			0
$397$			4.64281e6i			1.47844i	0.673463	+	0.739221i	$0.264806\pi$
$397$			4.64281e6i			1.47844i	−0.673463	+	0.739221i	$0.735194\pi$
$398$		0			0
$399$	−4.29801e6	+	6.29655e6i	−1.35156	+	1.98002i
$400$		0			0
$401$			1.92383e6i			0.597455i	0.954338	+	0.298727i	$0.0965621\pi$
$401$			1.92383e6i			0.597455i	−0.954338	+	0.298727i	$0.903438\pi$
$402$		0			0
$403$	−9.49318e6			−2.91172
$404$		0			0
$405$		−	3.44254e6i		−	1.04290i
$406$		0			0
$407$	−52772.8			−0.0157915
$408$		0			0
$409$		−	946349.i		−	0.279733i	−0.990170	−	0.139866i	$-0.955333\pi$
$409$		−	946349.i		−	0.279733i	0.990170	−	0.139866i	$-0.0446673\pi$
$410$		0			0
$411$		−	921062.i		−	0.268958i
$412$		0			0
$413$	−3.38307e6			−0.975969
$414$		0			0
$415$			5.15717e6i			1.46991i
$416$		0			0
$417$			2.71305e6i			0.764044i
$418$		0			0
$419$	−389444.			−0.108370			−0.0541851	−	0.998531i	$-0.517256\pi$
$419$	−389444.			−0.108370			−0.0541851	+	0.998531i	$0.517256\pi$
$420$		0			0
$421$	2.63190e6			0.723710			0.361855	−	0.932234i	$-0.382144\pi$
$421$	2.63190e6			0.723710			0.361855	+	0.932234i	$0.382144\pi$
$422$		0			0
$423$			4.37104e6i			1.18777i
$424$		0			0
$425$	−2.89504e6			−0.777467
$426$		0			0
$427$	9.65284e6			2.56204
$428$		0			0
$429$		−	1.08338e6i		−	0.284208i
$430$		0			0
$431$	−6.50192e6			−1.68596			−0.842982	−	0.537942i	$-0.819202\pi$
$431$	−6.50192e6			−1.68596			−0.842982	+	0.537942i	$0.819202\pi$
$432$		0			0
$433$		−	464494.i		−	0.119059i	−0.998227	−	0.0595293i	$-0.981040\pi$
$433$		−	464494.i		−	0.119059i	0.998227	−	0.0595293i	$-0.0189600\pi$
$434$		0			0
$435$	−1.28956e6			−0.326753
$436$		0			0
$437$	−5.80959e6	−	3.96561e6i	−1.45526	−	0.993359i
$438$		0			0
$439$	334337.			0.0827987			0.0413994	−	0.999143i	$-0.486818\pi$
$439$	334337.			0.0827987			0.0413994	+	0.999143i	$0.486818\pi$
$440$		0			0
$441$	7.91397e6			1.93775
$442$		0			0
$443$	5.54066e6			1.34138			0.670690	−	0.741737i	$-0.265998\pi$
$443$	5.54066e6			1.34138			0.670690	+	0.741737i	$0.265998\pi$
$444$		0			0
$445$	7.53255e6			1.80319
$446$		0			0
$447$	−7.64116e6			−1.80880
$448$		0			0
$449$		−	3.60099e6i		−	0.842959i	−0.906838	−	0.421480i	$-0.861511\pi$
$449$		−	3.60099e6i		−	0.842959i	0.906838	−	0.421480i	$-0.138489\pi$
$450$		0			0
$451$			142427.i			0.0329724i
$452$		0			0
$453$			9.58206e6i			2.19388i
$454$		0			0
$455$	1.80694e7			4.09181
$456$		0			0
$457$	1.27762e6			0.286162			0.143081	−	0.989711i	$-0.454299\pi$
$457$	1.27762e6			0.286162			0.143081	+	0.989711i	$0.454299\pi$
$458$		0			0
$459$			1.12683e6i			0.249646i
$460$		0			0
$461$		−	5.31168e6i		−	1.16407i	−0.813164	−	0.582035i	$-0.802257\pi$
$461$		−	5.31168e6i		−	1.16407i	0.813164	−	0.582035i	$-0.197743\pi$
$462$		0			0
$463$		−	1.78693e6i		−	0.387395i	−0.981061	−	0.193698i	$-0.937952\pi$
$463$		−	1.78693e6i		−	0.387395i	0.981061	−	0.193698i	$-0.0620481\pi$
$464$		0			0
$465$	1.59305e7			3.41663
$466$		0			0
$467$	−4.14061e6			−0.878562			−0.439281	−	0.898350i	$-0.644767\pi$
$467$	−4.14061e6			−0.878562			−0.439281	+	0.898350i	$0.644767\pi$
$468$		0			0
$469$	4.35498e6			0.914228
$470$		0			0
$471$	7.90375e6			1.64165
$472$		0			0
$473$	659868.			0.135614
$474$		0			0
$475$	2.78005e6	−	4.07275e6i	0.565351	−	0.828236i
$476$		0			0
$477$	−402628.			−0.0810230
$478$		0			0
$479$			1.36027e6i			0.270885i	0.990785	+	0.135443i	$0.0432457\pi$
$479$			1.36027e6i			0.270885i	−0.990785	+	0.135443i	$0.956754\pi$
$480$		0			0
$481$	−1.35310e6			−0.266666
$482$		0			0
$483$			2.16569e7i			4.22404i
$484$		0			0
$485$	2.03475e6			0.392787
$486$		0			0
$487$	−2.68364e6			−0.512746			−0.256373	−	0.966578i	$-0.582527\pi$
$487$	−2.68364e6			−0.512746			−0.256373	+	0.966578i	$0.582527\pi$
$488$		0			0
$489$		−	5.56593e6i		−	1.05261i
$490$		0			0
$491$	4.85247e6			0.908363			0.454181	−	0.890909i	$-0.349932\pi$
$491$	4.85247e6			0.908363			0.454181	+	0.890909i	$0.349932\pi$
$492$		0			0
$493$	−648896.			−0.120242
$494$		0			0
$495$			997721.i			0.183019i
$496$		0			0
$497$		−	2.36754e6i		−	0.429938i
$498$		0			0
$499$	6.98823e6			1.25637			0.628183	−	0.778066i	$-0.283799\pi$
$499$	6.98823e6			1.25637			0.628183	+	0.778066i	$0.283799\pi$
$500$		0			0
$501$		−	1.20323e7i		−	2.14168i
$502$		0			0
$503$		−	6.81985e6i		−	1.20186i	−0.799301	−	0.600931i	$-0.794796\pi$
$503$		−	6.81985e6i		−	1.20186i	0.799301	−	0.600931i	$-0.205204\pi$
$504$		0			0
$505$	−3.38691e6			−0.590983
$506$		0			0
$507$		−	1.91613e7i		−	3.31059i
$508$		0			0
$509$	2.77815e6			0.475292			0.237646	−	0.971352i	$-0.423624\pi$
$509$	2.77815e6			0.475292			0.237646	+	0.971352i	$0.423624\pi$
$510$		0			0
$511$			1.22554e6i			0.207622i
$512$		0			0
$513$	−1.58522e6	−	1.08207e6i	−0.265948	−	0.181536i
$514$		0			0
$515$		−	8.01407e6i		−	1.33148i
$516$		0			0
$517$			631048.i			0.103833i
$518$		0			0
$519$			4.09090e6i			0.666654i
$520$		0			0
$521$			4.52893e6i			0.730973i	0.930817	+	0.365487i	$0.119097\pi$
$521$			4.52893e6i			0.730973i	−0.930817	+	0.365487i	$0.880903\pi$
$522$		0			0
$523$		−	1.02176e7i		−	1.63342i	−0.577051	−	0.816708i	$-0.695797\pi$
$523$		−	1.02176e7i		−	1.63342i	0.577051	−	0.816708i	$-0.304203\pi$
$524$		0			0
$525$	−1.51823e7			−2.40403
$526$		0			0
$527$	8.01610e6			1.25729
$528$		0			0
$529$	−1.35456e7			−2.10456
$530$		0			0
$531$		−	4.78956e6i		−	0.737155i
$532$		0			0
$533$			3.65184e6i			0.556793i
$534$		0			0
$535$	−6.34241e6			−0.958010
$536$		0			0
$537$	4.53767e6			0.679043
$538$		0			0
$539$	1.14254e6			0.169395
$540$		0			0
$541$			632975.i			0.0929808i	0.998919	+	0.0464904i	$0.0148037\pi$
$541$			632975.i			0.0929808i	−0.998919	+	0.0464904i	$0.985196\pi$
$542$		0			0
$543$			7.12598e6i			1.03716i
$544$		0			0
$545$		−	8.72450e6i		−	1.25820i
$546$		0			0
$547$			2.99879e6i			0.428527i	0.976776	+	0.214264i	$0.0687351\pi$
$547$			2.99879e6i			0.428527i	−0.976776	+	0.214264i	$0.931265\pi$
$548$		0			0
$549$			1.36659e7i			1.93512i
$550$		0			0
$551$	623122.	−	912870.i	0.0874368	−	0.128094i
$552$		0			0
$553$		−	8.98924e6i		−	1.25000i
$554$		0			0
$555$	2.27064e6			0.312908
$556$		0			0
$557$			1.94328e6i			0.265398i	0.991156	+	0.132699i	$0.0423644\pi$
$557$			1.94328e6i			0.265398i	−0.991156	+	0.132699i	$0.957636\pi$
$558$		0			0
$559$	1.69191e7			2.29006
$560$		0			0
$561$			914809.i			0.122722i
$562$		0			0
$563$		−	6.21315e6i		−	0.826116i	−0.910705	−	0.413058i	$-0.864461\pi$
$563$		−	6.21315e6i		−	0.826116i	0.910705	−	0.413058i	$-0.135539\pi$
$564$		0			0
$565$	3.06958e6			0.404537
$566$		0			0
$567$		−	9.08439e6i		−	1.18669i
$568$		0			0
$569$		−	1.29234e6i		−	0.167338i	−0.996494	−	0.0836692i	$-0.973336\pi$
$569$		−	1.29234e6i		−	0.167338i	0.996494	−	0.0836692i	$-0.0266639\pi$
$570$		0			0
$571$	−4.41951e6			−0.567262			−0.283631	−	0.958934i	$-0.591539\pi$
$571$	−4.41951e6			−0.567262			−0.283631	+	0.958934i	$0.591539\pi$
$572$		0			0
$573$	1.84467e7			2.34711
$574$		0			0
$575$		−	1.40082e7i		−	1.76690i
$576$		0			0
$577$	−51903.2			−0.00649015			−0.00324507	−	0.999995i	$-0.501033\pi$
$577$	−51903.2			−0.00649015			−0.00324507	+	0.999995i	$0.501033\pi$
$578$		0			0
$579$	−9.86067e6			−1.22239
$580$		0			0
$581$			1.36091e7i			1.67258i
$582$		0			0
$583$	−58127.5			−0.00708288
$584$		0			0
$585$			2.55817e7i			3.09057i
$586$		0			0
$587$	−8.38391e6			−1.00427			−0.502136	−	0.864789i	$-0.667452\pi$
$587$	−8.38391e6			−1.00427			−0.502136	+	0.864789i	$0.667452\pi$
$588$		0			0
$589$	−7.69770e6	+	1.12771e7i	−0.914267	+	1.33939i
$590$		0			0
$591$	−3.22456e6			−0.379754
$592$		0			0
$593$	−1.56701e6			−0.182994			−0.0914968	−	0.995805i	$-0.529165\pi$
$593$	−1.56701e6			−0.182994			−0.0914968	+	0.995805i	$0.529165\pi$
$594$		0			0
$595$	−1.52579e7			−1.76686
$596$		0			0
$597$	1.27567e7			1.46488
$598$		0			0
$599$	−3.73596e6			−0.425436			−0.212718	−	0.977114i	$-0.568232\pi$
$599$	−3.73596e6			−0.425436			−0.212718	+	0.977114i	$0.568232\pi$
$600$		0			0
$601$			4.62374e6i			0.522164i	0.965317	+	0.261082i	$0.0840793\pi$
$601$			4.62374e6i			0.522164i	−0.965317	+	0.261082i	$0.915921\pi$
$602$		0			0
$603$			6.16554e6i			0.690522i
$604$		0			0
$605$		−	1.25970e7i		−	1.39920i
$606$		0			0
$607$	−5.87391e6			−0.647076			−0.323538	−	0.946215i	$-0.604872\pi$
$607$	−5.87391e6			−0.647076			−0.323538	+	0.946215i	$0.604872\pi$
$608$		0			0
$609$	−3.40298e6			−0.371806
$610$		0			0
$611$			1.61801e7i			1.75339i
$612$		0			0
$613$			4.52559e6i			0.486434i	0.969972	+	0.243217i	$0.0782028\pi$
$613$			4.52559e6i			0.486434i	−0.969972	+	0.243217i	$0.921797\pi$
$614$		0			0
$615$		−	6.12816e6i		−	0.653345i
$616$		0			0
$617$	−243553.			−0.0257561			−0.0128781	−	0.999917i	$-0.504099\pi$
$617$	−243553.			−0.0257561			−0.0128781	+	0.999917i	$0.504099\pi$
$618$		0			0
$619$	−585320.			−0.0613997			−0.0306999	−	0.999529i	$-0.509774\pi$
$619$	−585320.			−0.0613997			−0.0306999	+	0.999529i	$0.509774\pi$
$620$		0			0
$621$	−5.45235e6			−0.567355
$622$		0			0
$623$	1.98774e7			2.05182
$624$		0			0
$625$	−9.73826e6			−0.997198
$626$		0			0
$627$	−1.28696e6	−	878474.i	−0.130736	−	0.0892401i
$628$		0			0
$629$	1.14257e6			0.115147
$630$		0			0
$631$		−	1.82484e7i		−	1.82453i	−0.409597	−	0.912267i	$-0.634331\pi$
$631$		−	1.82484e7i		−	1.82453i	0.409597	−	0.912267i	$-0.365669\pi$
$632$		0			0
$633$	−2.76610e7			−2.74384
$634$		0			0
$635$		−	1.58676e6i		−	0.156163i
$636$		0			0
$637$	2.92948e7			2.86050
$638$		0			0
$639$	3.35183e6			0.324735
$640$		0			0
$641$		−	6.79087e6i		−	0.652800i	−0.945232	−	0.326400i	$-0.894164\pi$
$641$		−	6.79087e6i		−	0.652800i	0.945232	−	0.326400i	$-0.105836\pi$
$642$		0			0
$643$	2.68509e6			0.256113			0.128056	−	0.991767i	$-0.459126\pi$
$643$	2.68509e6			0.256113			0.128056	+	0.991767i	$0.459126\pi$
$644$		0			0
$645$	−2.83920e7			−2.68718
$646$		0			0
$647$		−	1.66246e7i		−	1.56132i	−0.624958	−	0.780658i	$-0.714884\pi$
$647$		−	1.66246e7i		−	1.56132i	0.624958	−	0.780658i	$-0.285116\pi$
$648$		0			0
$649$		−	691469.i		−	0.0644408i
$650$		0			0
$651$	4.20385e7			3.88772
$652$		0			0
$653$			8.57166e6i			0.786650i	0.919399	+	0.393325i	$0.128675\pi$
$653$			8.57166e6i			0.786650i	−0.919399	+	0.393325i	$0.871325\pi$
$654$		0			0
$655$			2.26224e7i			2.06032i
$656$		0			0
$657$	−1.73505e6			−0.156819
$658$		0			0
$659$			5.17045e6i			0.463783i	0.972742	+	0.231891i	$0.0744914\pi$
$659$			5.17045e6i			0.463783i	−0.972742	+	0.231891i	$0.925509\pi$
$660$		0			0
$661$	−9.72243e6			−0.865508			−0.432754	−	0.901512i	$-0.642458\pi$
$661$	−9.72243e6			−0.865508			−0.432754	+	0.901512i	$0.642458\pi$
$662$		0			0
$663$			2.34558e7i			2.07237i
$664$		0			0
$665$	1.46519e7	−	2.14649e7i	1.28481	−	1.88224i
$666$		0			0
$667$		−	3.13980e6i		−	0.273267i
$668$		0			0
$669$		−	3.05311e7i		−	2.63741i
$670$		0			0
$671$			1.97295e6i			0.169165i
$672$		0			0
$673$			1.37663e7i			1.17160i	0.810456	+	0.585800i	$0.199219\pi$
$673$			1.37663e7i			1.17160i	−0.810456	+	0.585800i	$0.800781\pi$
$674$		0			0
$675$		−	3.82231e6i		−	0.322899i
$676$		0			0
$677$	−1.01474e7			−0.850911			−0.425456	−	0.904979i	$-0.639886\pi$
$677$	−1.01474e7			−0.850911			−0.425456	+	0.904979i	$0.639886\pi$
$678$		0			0
$679$	5.36943e6			0.446945
$680$		0			0
$681$	2.01692e7			1.66656
$682$		0			0
$683$			1.93224e7i			1.58492i	0.609921	+	0.792462i	$0.291201\pi$
$683$			1.93224e7i			1.58492i	−0.609921	+	0.792462i	$0.708799\pi$
$684$		0			0
$685$			3.13990e6i			0.255675i
$686$		0			0
$687$	−1.00429e7			−0.811838
$688$		0			0
$689$	−1.49039e6			−0.119606
$690$		0			0
$691$	−1.94690e7			−1.55113			−0.775566	−	0.631267i	$-0.782535\pi$
$691$	−1.94690e7			−1.55113			−0.775566	+	0.631267i	$0.782535\pi$
$692$		0			0
$693$			2.63285e6i			0.208254i
$694$		0			0
$695$		−	9.24879e6i		−	0.726312i
$696$		0			0
$697$		−	3.08363e6i		−	0.240426i
$698$		0			0
$699$			1.28232e7i			0.992669i
$700$		0			0
$701$		−	8.89100e6i		−	0.683369i	−0.939815	−	0.341685i	$-0.889003\pi$
$701$		−	8.89100e6i		−	0.683369i	0.939815	−	0.341685i	$-0.110997\pi$
$702$		0			0
$703$	−1.09718e6	+	1.60737e6i	−0.0837319	+	0.122667i
$704$		0			0
$705$		−	2.71519e7i		−	2.05744i
$706$		0			0
$707$	−8.93759e6			−0.672468
$708$		0			0
$709$			7.84967e6i			0.586457i	0.956042	+	0.293228i	$0.0947296\pi$
$709$			7.84967e6i			0.586457i	−0.956042	+	0.293228i	$0.905270\pi$
$710$		0			0
$711$	1.27265e7			0.944134
$712$		0			0
$713$			3.87873e7i			2.85737i
$714$		0			0
$715$			3.69323e6i			0.270172i
$716$		0			0
$717$	−2.72129e7			−1.97687
$718$		0			0
$719$		−	2.51684e6i		−	0.181566i	−0.995871	−	0.0907828i	$-0.971063\pi$
$719$		−	2.51684e6i		−	0.181566i	0.995871	−	0.0907828i	$-0.0289369\pi$
$720$		0			0
$721$		−	2.11480e7i		−	1.51507i
$722$		0			0
$723$	3.65964e6			0.260371
$724$		0			0
$725$	2.20112e6			0.155525
$726$		0			0
$727$			1.06500e7i			0.747334i	0.927563	+	0.373667i	$0.121900\pi$
$727$			1.06500e7i			0.747334i	−0.927563	+	0.373667i	$0.878100\pi$
$728$		0			0
$729$	1.97396e7			1.37568
$730$		0			0
$731$	−1.42866e7			−0.988859
$732$		0			0
$733$			4.79856e6i			0.329876i	0.986304	+	0.164938i	$0.0527424\pi$
$733$			4.79856e6i			0.329876i	−0.986304	+	0.164938i	$0.947258\pi$
$734$		0			0
$735$	−4.91598e7			−3.35654
$736$		0			0
$737$			890119.i			0.0603642i
$738$		0			0
$739$	2.48325e7			1.67267			0.836333	−	0.548221i	$-0.184695\pi$
$739$	2.48325e7			1.67267			0.836333	+	0.548221i	$0.184695\pi$
$740$		0			0
$741$	−3.29977e7	−	2.25241e7i	−2.20769	−	1.50696i
$742$		0			0
$743$	−1.87670e7			−1.24716			−0.623581	−	0.781759i	$-0.714323\pi$
$743$	−1.87670e7			−1.24716			−0.623581	+	0.781759i	$0.714323\pi$
$744$		0			0
$745$	2.60487e7			1.71947
$746$		0			0
$747$	−1.92669e7			−1.26331
$748$		0			0
$749$	−1.67368e7			−1.09010
$750$		0			0
$751$	1.75017e7			1.13235			0.566174	−	0.824286i	$-0.308423\pi$
$751$	1.75017e7			1.13235			0.566174	+	0.824286i	$0.308423\pi$
$752$		0			0
$753$			1.74017e7i			1.11842i
$754$		0			0
$755$		−	3.26652e7i		−	2.08554i
$756$		0			0
$757$		−	626986.i		−	0.0397665i	−0.999802	−	0.0198833i	$-0.993671\pi$
$757$		−	626986.i		−	0.0397665i	0.999802	−	0.0198833i	$-0.00632946\pi$
$758$		0			0
$759$	−4.42647e6			−0.278903
$760$		0			0
$761$	−2.45039e7			−1.53382			−0.766909	−	0.641756i	$-0.778206\pi$
$761$	−2.45039e7			−1.53382			−0.766909	+	0.641756i	$0.778206\pi$
$762$		0			0
$763$		−	2.30227e7i		−	1.43168i
$764$		0			0
$765$		−	2.16013e7i		−	1.33452i
$766$		0			0
$767$		−	1.77293e7i		−	1.08819i
$768$		0			0
$769$	2.49376e7			1.52069			0.760343	−	0.649522i	$-0.225031\pi$
$769$	2.49376e7			1.52069			0.760343	+	0.649522i	$0.225031\pi$
$770$		0			0
$771$	−9.81001e6			−0.594338
$772$		0			0
$773$	−1.57842e7			−0.950112			−0.475056	−	0.879955i	$-0.657572\pi$
$773$	−1.57842e7			−0.950112			−0.475056	+	0.879955i	$0.657572\pi$
$774$		0			0
$775$	−2.71915e7			−1.62622
$776$		0			0
$777$	5.99191e6			0.356052
$778$		0			0
$779$	4.33807e6	+	2.96115e6i	0.256125	+	0.174830i
$780$		0			0
$781$	483903.			0.0283878
$782$		0			0
$783$		−	856737.i		−	0.0499394i
$784$		0			0
$785$	−2.69439e7			−1.56058
$786$		0			0
$787$		−	1.74748e6i		−	0.100572i	−0.998735	−	0.0502859i	$-0.983987\pi$
$787$		−	1.74748e6i		−	0.100572i	0.998735	−	0.0502859i	$-0.0160132\pi$
$788$		0			0
$789$	113607.			0.00649702
$790$		0			0
$791$	8.10021e6			0.460315
$792$		0			0
$793$			5.05867e7i			2.85663i
$794$		0			0
$795$	2.50103e6			0.140347
$796$		0			0
$797$	1.36499e7			0.761172			0.380586	−	0.924745i	$-0.375722\pi$
$797$	1.36499e7			0.761172			0.380586	+	0.924745i	$0.375722\pi$
$798$		0			0
$799$		−	1.36626e7i		−	0.757122i
$800$		0			0
$801$			2.81413e7i			1.54975i
$802$		0			0
$803$	−250489.			−0.0137088
$804$		0			0
$805$		−	7.38282e7i		−	4.01544i
$806$		0			0
$807$			9.31546e6i			0.503525i
$808$		0			0
$809$	−859304.			−0.0461611			−0.0230805	−	0.999734i	$-0.507347\pi$
$809$	−859304.			−0.0461611			−0.0230805	+	0.999734i	$0.507347\pi$
$810$		0			0
$811$		−	8.18973e6i		−	0.437238i	−0.975810	−	0.218619i	$-0.929845\pi$
$811$		−	8.18973e6i		−	0.437238i	0.975810	−	0.218619i	$-0.0701551\pi$
$812$		0			0
$813$	3.25562e7			1.72746
$814$		0			0
$815$			1.89742e7i			1.00062i
$816$		0			0
$817$	1.37191e7	−	2.00984e7i	0.719070	−	1.05343i
$818$		0			0
$819$			6.75065e7i			3.51670i
$820$		0			0
$821$		−	7.24829e6i		−	0.375299i	−0.982236	−	0.187649i	$-0.939913\pi$
$821$		−	7.24829e6i		−	0.375299i	0.982236	−	0.187649i	$-0.0600869\pi$
$822$		0			0
$823$		−	1.09949e7i		−	0.565837i	−0.959144	−	0.282918i	$-0.908697\pi$
$823$		−	1.09949e7i		−	0.565837i	0.959144	−	0.282918i	$-0.0913025\pi$
$824$		0			0
$825$		−	3.10313e6i		−	0.158732i
$826$		0			0
$827$			1.46356e7i			0.744129i	0.928207	+	0.372064i	$0.121350\pi$
$827$			1.46356e7i			0.744129i	−0.928207	+	0.372064i	$0.878650\pi$
$828$		0			0
$829$	3.19158e7			1.61294			0.806472	−	0.591272i	$-0.201374\pi$
$829$	3.19158e7			1.61294			0.806472	+	0.591272i	$0.201374\pi$
$830$		0			0
$831$	−4.68235e7			−2.35213
$832$		0			0
$833$	−2.47367e7			−1.23518
$834$		0			0
$835$			4.10181e7i			2.03591i
$836$		0			0
$837$			1.05836e7i			0.522182i
$838$		0			0
$839$	−1.74497e7			−0.855821			−0.427910	−	0.903821i	$-0.640750\pi$
$839$	−1.74497e7			−0.855821			−0.427910	+	0.903821i	$0.640750\pi$
$840$		0			0
$841$	−2.00178e7			−0.975947
$842$		0			0
$843$	−4.89717e7			−2.37343
$844$		0			0
$845$			6.53209e7i			3.14710i
$846$		0			0
$847$		−	3.32419e7i		−	1.59212i
$848$		0			0
$849$		−	2.62462e7i		−	1.24967i
$850$		0			0
$851$			5.52851e6i			0.261688i
$852$		0			0
$853$		−	4.10028e6i		−	0.192948i	−0.995335	−	0.0964741i	$-0.969244\pi$
$853$		−	4.10028e6i		−	0.192948i	0.995335	−	0.0964741i	$-0.0307565\pi$
$854$		0			0
$855$	3.03888e7	+	2.07433e7i	1.42167	+	0.970427i
$856$		0			0
$857$			4.18282e7i			1.94543i	0.231993	+	0.972717i	$0.425475\pi$
$857$			4.18282e7i			1.94543i	−0.231993	+	0.972717i	$0.574525\pi$
$858$		0			0
$859$	2.70400e7			1.25033			0.625163	−	0.780494i	$-0.285032\pi$
$859$	2.70400e7			1.25033			0.625163	+	0.780494i	$0.285032\pi$
$860$		0			0
$861$		−	1.61714e7i		−	0.743428i
$862$		0			0
$863$	2.71246e7			1.23976			0.619879	−	0.784697i	$-0.287182\pi$
$863$	2.71246e7			1.23976			0.619879	+	0.784697i	$0.287182\pi$
$864$		0			0
$865$		−	1.39459e7i		−	0.633731i
$866$		0			0
$867$			1.31442e7i			0.593865i
$868$		0			0
$869$	1.83732e6			0.0825345
$870$		0			0
$871$			2.28227e7i			1.01935i
$872$		0			0
$873$			7.60174e6i			0.337580i
$874$		0			0
$875$	144217.			0.00636789
$876$		0			0
$877$	2.87259e7			1.26117			0.630586	−	0.776120i	$-0.282815\pi$
$877$	2.87259e7			1.26117			0.630586	+	0.776120i	$0.282815\pi$
$878$		0			0
$879$		−	2.05600e7i		−	0.897536i
$880$		0			0
$881$	−2.88595e7			−1.25271			−0.626354	−	0.779539i	$-0.715453\pi$
$881$	−2.88595e7			−1.25271			−0.626354	+	0.779539i	$0.715453\pi$
$882$		0			0
$883$	3.58193e6			0.154602			0.0773011	−	0.997008i	$-0.475370\pi$
$883$	3.58193e6			0.154602			0.0773011	+	0.997008i	$0.475370\pi$
$884$		0			0
$885$			2.97516e7i			1.27689i
$886$		0			0
$887$	−4.73743e6			−0.202178			−0.101089	−	0.994877i	$-0.532233\pi$
$887$	−4.73743e6			−0.202178			−0.101089	+	0.994877i	$0.532233\pi$
$888$		0			0
$889$		−	4.18724e6i		−	0.177694i
$890$		0			0
$891$	1.85677e6			0.0783543
$892$		0			0
$893$	1.92206e7	+	1.31199e7i	0.806563	+	0.550557i
$894$		0			0
$895$	−1.54689e7			−0.645509
$896$		0			0
$897$	−1.13495e8			−4.70973
$898$		0			0
$899$	−6.09472e6			−0.251509
$900$		0			0
$901$	1.25850e6			0.0516464
$902$		0			0
$903$	−7.49225e7			−3.05769
$904$		0			0
$905$		−	2.42924e7i		−	0.985938i
$906$		0			0
$907$		−	2.82690e7i		−	1.14102i	−0.821292	−	0.570509i	$-0.806746\pi$
$907$		−	2.82690e7i		−	1.14102i	0.821292	−	0.570509i	$-0.193254\pi$
$908$		0			0
$909$		−	1.26533e7i		−	0.507920i
$910$		0			0
$911$	3.93598e7			1.57129			0.785647	−	0.618676i	$-0.212330\pi$
$911$	3.93598e7			1.57129			0.785647	+	0.618676i	$0.212330\pi$
$912$		0			0
$913$	−2.78157e6			−0.110437
$914$		0			0
$915$		−	8.48897e7i		−	3.35199i
$916$		0			0
$917$			5.96973e7i			2.34440i
$918$		0			0
$919$			2.04620e7i			0.799207i	0.916688	+	0.399604i	$0.130852\pi$
$919$			2.04620e7i			0.799207i	−0.916688	+	0.399604i	$0.869148\pi$
$920$		0			0
$921$	−2.26813e7			−0.881087
$922$		0			0
$923$	1.24073e7			0.479373
$924$		0			0
$925$	−3.87570e6			−0.148935
$926$		0			0
$927$	2.99402e7			1.14434
$928$		0			0
$929$	2.10640e7			0.800758			0.400379	−	0.916350i	$-0.368879\pi$
$929$	2.10640e7			0.800758			0.400379	+	0.916350i	$0.368879\pi$
$930$		0			0
$931$	2.37542e7	−	3.47998e7i	0.898186	−	1.31584i
$932$		0			0
$933$	5.29720e6			0.199224
$934$		0			0
$935$		−	3.11858e6i		−	0.116662i
$936$		0			0
$937$	1.13985e7			0.424129			0.212064	−	0.977256i	$-0.431981\pi$
$937$	1.13985e7			0.424129			0.212064	+	0.977256i	$0.431981\pi$
$938$		0			0
$939$			5.56257e7i			2.05879i
$940$		0			0
$941$	−1.09524e7			−0.403212			−0.201606	−	0.979467i	$-0.564616\pi$
$941$	−1.09524e7			−0.403212			−0.201606	+	0.979467i	$0.564616\pi$
$942$		0			0
$943$	1.49207e7			0.546400
$944$		0			0
$945$		−	2.01450e7i		−	0.733818i
$946$		0			0
$947$	−4.41831e7			−1.60096			−0.800482	−	0.599357i	$-0.795423\pi$
$947$	−4.41831e7			−1.60096			−0.800482	+	0.599357i	$0.795423\pi$
$948$		0			0
$949$	−6.42256e6			−0.231495
$950$		0			0
$951$		−	1.15915e7i		−	0.415613i
$952$		0			0
$953$		−	1.94042e7i		−	0.692091i	−0.938218	−	0.346045i	$-0.887524\pi$
$953$		−	1.94042e7i		−	0.692091i	0.938218	−	0.346045i	$-0.112476\pi$
$954$		0			0
$955$	−6.28848e7			−2.23119
$956$		0			0
$957$		−	695538.i		−	0.0245494i
$958$		0			0
$959$			8.28575e6i			0.290928i
$960$		0			0
$961$	4.66616e7			1.62986
$962$		0			0
$963$		−	2.36949e7i		−	0.823360i
$964$		0			0
$965$	3.36150e7			1.16202
$966$		0			0
$967$		−	1.89264e6i		−	0.0650883i	−0.999470	−	0.0325441i	$-0.989639\pi$
$967$		−	1.89264e6i		−	0.0650883i	0.999470	−	0.0325441i	$-0.0103609\pi$
$968$		0			0
$969$	2.78635e7	+	1.90195e7i	0.953291	+	0.650714i
$970$		0			0
$971$			1.79499e7i			0.610960i	0.952198	+	0.305480i	$0.0988170\pi$
$971$			1.79499e7i			0.610960i	−0.952198	+	0.305480i	$0.901183\pi$
$972$		0			0
$973$		−	2.44063e7i		−	0.826456i
$974$		0			0
$975$		−	7.95646e7i		−	2.68045i
$976$		0			0
$977$			6.01313e6i			0.201541i	0.994910	+	0.100771i	$0.0321308\pi$
$977$			6.01313e6i			0.201541i	−0.994910	+	0.100771i	$0.967869\pi$
$978$		0			0
$979$			4.06275e6i			0.135476i
$980$		0			0
$981$	3.25943e7			1.08136
$982$		0			0
$983$	9.57310e6			0.315987			0.157993	−	0.987440i	$-0.449498\pi$
$983$	9.57310e6			0.315987			0.157993	+	0.987440i	$0.449498\pi$
$984$		0			0
$985$	1.09925e7			0.360999
$986$		0			0
$987$		−	7.16502e7i		−	2.34112i
$988$		0			0
$989$		−	6.91281e7i		−	2.24732i
$990$		0			0
$991$	−1.21505e7			−0.393015			−0.196508	−	0.980502i	$-0.562960\pi$
$991$	−1.21505e7			−0.393015			−0.196508	+	0.980502i	$0.562960\pi$
$992$		0			0
$993$	4.52899e7			1.45757
$994$		0			0
$995$	−4.34875e7			−1.39254
$996$		0			0
$997$		−	2.06081e7i		−	0.656600i	−0.944573	−	0.328300i	$-0.893524\pi$
$997$		−	2.06081e7i		−	0.656600i	0.944573	−	0.328300i	$-0.106476\pi$
$998$		0			0
$999$			1.50853e6i			0.0478233i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	608.6.b.b.303.12		96
4.3	odd	2		152.6.b.b.75.25	✓	96
8.3	odd	2	inner	608.6.b.b.303.11		96
8.5	even	2		152.6.b.b.75.71	yes	96
19.18	odd	2	inner	608.6.b.b.303.86		96
76.75	even	2		152.6.b.b.75.72	yes	96
152.37	odd	2		152.6.b.b.75.26	yes	96
152.75	even	2	inner	608.6.b.b.303.85		96

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
152.6.b.b.75.25	✓	96	4.3	odd	2
152.6.b.b.75.26	yes	96	152.37	odd	2
152.6.b.b.75.71	yes	96	8.5	even	2
152.6.b.b.75.72	yes	96	76.75	even	2
608.6.b.b.303.11		96	8.3	odd	2	inner
608.6.b.b.303.12		96	1.1	even	1	trivial
608.6.b.b.303.85		96	152.75	even	2	inner
608.6.b.b.303.86		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)

\(f(q)\)	\(=\)	\(q-23.2069i q^{3} +79.1121i q^{5} +208.766i q^{7} -295.559 q^{9} +O(q^{10})\)	\(q-23.2069i q^{3} +79.1121i q^{5} +208.766i q^{7} -295.559 q^{9} -42.6699 q^{11} -1094.06 q^{13} +1835.95 q^{15} +923.831 q^{17} +(-887.137 + 1299.65i) q^{19} +4844.81 q^{21} +4470.12i q^{23} -3133.73 q^{25} +1219.73i q^{27} -702.397 q^{29} +8677.02 q^{31} +990.235i q^{33} -16515.9 q^{35} +1236.77 q^{37} +25389.7i q^{39} -3337.88i q^{41} -15464.5 q^{43} -23382.3i q^{45} -14789.1i q^{47} -26776.3 q^{49} -21439.2i q^{51} +1362.26 q^{53} -3375.71i q^{55} +(30160.8 + 20587.7i) q^{57} +16205.1i q^{59} -46237.6i q^{61} -61702.7i q^{63} -86553.4i q^{65} -20860.6i q^{67} +103738. q^{69} -11340.6 q^{71} +5870.39 q^{73} +72724.1i q^{75} -8908.02i q^{77} -43058.9 q^{79} -43514.7 q^{81} +65188.1 q^{83} +73086.2i q^{85} +16300.4i q^{87} -95213.6i q^{89} -228403. i q^{91} -201367. i q^{93} +(-102818. - 70183.3i) q^{95} -25719.9i q^{97} +12611.5 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 96 q - 6168 q^{9}+O(q^{10}) \) 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

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