LMFDB - Embedded newform 608.6.b.b.303.52

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			303.52
Character	$\chi$	$=$	608.303
Dual form			608.6.b.b.303.45

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.79162i q^{3} +26.9679i q^{5} +21.5599i q^{7} +239.790 q^{9} +O(q^{10})$	$q+1.79162i q^{3} +26.9679i q^{5} +21.5599i q^{7} +239.790 q^{9} +90.3444 q^{11} -419.826 q^{13} -48.3163 q^{15} -215.062 q^{17} +(-1355.34 - 799.470i) q^{19} -38.6273 q^{21} +3077.62i q^{23} +2397.73 q^{25} +864.978i q^{27} -6155.92 q^{29} -3059.38 q^{31} +161.863i q^{33} -581.426 q^{35} +9295.40 q^{37} -752.170i q^{39} +17772.2i q^{41} -7152.73 q^{43} +6466.64i q^{45} -23161.1i q^{47} +16342.2 q^{49} -385.311i q^{51} +11807.4 q^{53} +2436.40i q^{55} +(1432.35 - 2428.26i) q^{57} -42428.8i q^{59} +45648.8i q^{61} +5169.86i q^{63} -11321.8i q^{65} +33058.2i q^{67} -5513.94 q^{69} -29886.8 q^{71} -39654.0 q^{73} +4295.83i q^{75} +1947.82i q^{77} -62295.5 q^{79} +56719.3 q^{81} -107005. q^{83} -5799.78i q^{85} -11029.1i q^{87} -92408.3i q^{89} -9051.43i q^{91} -5481.25i q^{93} +(21560.0 - 36550.7i) q^{95} -76257.9i q^{97} +21663.7 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$			1.79162i			0.114933i	0.998347	+	0.0574664i	$0.0183022\pi$
$3$			1.79162i			0.114933i	−0.998347	+	0.0574664i	$0.981698\pi$
$4$		0			0
$5$			26.9679i			0.482417i	0.970473	+	0.241208i	$0.0775437\pi$
$5$			26.9679i			0.482417i	−0.970473	+	0.241208i	$0.922456\pi$
$6$		0			0
$7$			21.5599i			0.166304i	0.996537	+	0.0831520i	$0.0264987\pi$
$7$			21.5599i			0.166304i	−0.996537	+	0.0831520i	$0.973501\pi$
$8$		0			0
$9$	239.790			0.986790
$10$		0			0
$11$	90.3444			0.225123			0.112561	−	0.993645i	$-0.464094\pi$
$11$	90.3444			0.225123			0.112561	+	0.993645i	$0.464094\pi$
$12$		0			0
$13$	−419.826			−0.688987			−0.344493	−	0.938789i	$-0.611949\pi$
$13$	−419.826			−0.688987			−0.344493	+	0.938789i	$0.611949\pi$
$14$		0			0
$15$	−48.3163			−0.0554454
$16$		0			0
$17$	−215.062			−0.180485			−0.0902427	−	0.995920i	$-0.528764\pi$
$17$	−215.062			−0.180485			−0.0902427	+	0.995920i	$0.528764\pi$
$18$		0			0
$19$	−1355.34	−	799.470i	−0.861320	−	0.508064i
$19$	−1355.34	−	799.470i	−0.861320	−	0.508064i
$20$		0			0
$21$	−38.6273			−0.0191138
$22$		0			0
$23$			3077.62i			1.21310i	0.795046	+	0.606549i	$0.207446\pi$
$23$			3077.62i			1.21310i	−0.795046	+	0.606549i	$0.792554\pi$
$24$		0			0
$25$	2397.73			0.767274
$26$		0			0
$27$			864.978i			0.228347i
$28$		0			0
$29$	−6155.92			−1.35925			−0.679623	−	0.733561i	$-0.737857\pi$
$29$	−6155.92			−1.35925			−0.679623	+	0.733561i	$0.737857\pi$
$30$		0			0
$31$	−3059.38			−0.571779			−0.285890	−	0.958263i	$-0.592289\pi$
$31$	−3059.38			−0.571779			−0.285890	+	0.958263i	$0.592289\pi$
$32$		0			0
$33$			161.863i			0.0258740i
$34$		0			0
$35$	−581.426			−0.0802278
$36$		0			0
$37$	9295.40			1.11626			0.558128	−	0.829755i	$-0.311520\pi$
$37$	9295.40			1.11626			0.558128	+	0.829755i	$0.311520\pi$
$38$		0			0
$39$		−	752.170i		−	0.0791871i
$40$		0			0
$41$			17772.2i			1.65113i	0.564307	+	0.825565i	$0.309143\pi$
$41$			17772.2i			1.65113i	−0.564307	+	0.825565i	$0.690857\pi$
$42$		0			0
$43$	−7152.73			−0.589930			−0.294965	−	0.955508i	$-0.595308\pi$
$43$	−7152.73			−0.589930			−0.294965	+	0.955508i	$0.595308\pi$
$44$		0			0
$45$			6466.64i			0.476044i
$46$		0			0
$47$		−	23161.1i		−	1.52938i	−0.644400	−	0.764689i	$-0.722893\pi$
$47$		−	23161.1i		−	1.52938i	0.644400	−	0.764689i	$-0.277107\pi$
$48$		0			0
$49$	16342.2			0.972343
$50$		0			0
$51$		−	385.311i		−	0.0207437i
$52$		0			0
$53$	11807.4			0.577383			0.288691	−	0.957422i	$-0.406780\pi$
$53$	11807.4			0.577383			0.288691	+	0.957422i	$0.406780\pi$
$54$		0			0
$55$			2436.40i			0.108603i
$56$		0			0
$57$	1432.35	−	2428.26i	0.0583931	−	0.0989938i
$58$		0			0
$59$		−	42428.8i		−	1.58683i	−0.608680	−	0.793416i	$-0.708301\pi$
$59$		−	42428.8i		−	1.58683i	0.608680	−	0.793416i	$-0.291699\pi$
$60$		0			0
$61$			45648.8i			1.57074i	0.619025	+	0.785372i	$0.287528\pi$
$61$			45648.8i			1.57074i	−0.619025	+	0.785372i	$0.712472\pi$
$62$		0			0
$63$			5169.86i			0.164107i
$64$		0			0
$65$		−	11321.8i		−	0.332379i
$66$		0			0
$67$			33058.2i			0.899690i	0.893107	+	0.449845i	$0.148521\pi$
$67$			33058.2i			0.899690i	−0.893107	+	0.449845i	$0.851479\pi$
$68$		0			0
$69$	−5513.94			−0.139425
$70$		0			0
$71$	−29886.8			−0.703612			−0.351806	−	0.936073i	$-0.614432\pi$
$71$	−29886.8			−0.703612			−0.351806	+	0.936073i	$0.614432\pi$
$72$		0			0
$73$	−39654.0			−0.870924			−0.435462	−	0.900207i	$-0.643415\pi$
$73$	−39654.0			−0.870924			−0.435462	+	0.900207i	$0.643415\pi$
$74$		0			0
$75$			4295.83i			0.0881849i
$76$		0			0
$77$			1947.82i			0.0374388i
$78$		0			0
$79$	−62295.5			−1.12302			−0.561512	−	0.827469i	$-0.689780\pi$
$79$	−62295.5			−1.12302			−0.561512	+	0.827469i	$0.689780\pi$
$80$		0			0
$81$	56719.3			0.960546
$82$		0			0
$83$	−107005.			−1.70494			−0.852468	−	0.522780i	$-0.824895\pi$
$83$	−107005.			−1.70494			−0.852468	+	0.522780i	$0.824895\pi$
$84$		0			0
$85$		−	5799.78i		−	0.0870692i
$86$		0			0
$87$		−	11029.1i		−	0.156222i
$88$		0			0
$89$		−	92408.3i		−	1.23662i	−0.785935	−	0.618310i	$-0.787818\pi$
$89$		−	92408.3i		−	1.23662i	0.785935	−	0.618310i	$-0.212182\pi$
$90$		0			0
$91$		−	9051.43i		−	0.114581i
$92$		0			0
$93$		−	5481.25i		−	0.0657162i
$94$		0			0
$95$	21560.0	−	36550.7i	0.245098	−	0.415515i
$96$		0			0
$97$		−	76257.9i		−	0.822916i	−0.911429	−	0.411458i	$-0.865020\pi$
$97$		−	76257.9i		−	0.822916i	0.911429	−	0.411458i	$-0.134980\pi$
$98$		0			0
$99$	21663.7			0.222149
$100$		0			0
$101$			144178.i			1.40636i	0.711012	+	0.703180i	$0.248237\pi$
$101$			144178.i			1.40636i	−0.711012	+	0.703180i	$0.751763\pi$
$102$		0			0
$103$	−44423.8			−0.412594			−0.206297	−	0.978489i	$-0.566141\pi$
$103$	−44423.8			−0.412594			−0.206297	+	0.978489i	$0.566141\pi$
$104$		0			0
$105$		−	1041.70i		−	0.00922079i
$106$		0			0
$107$			64037.4i			0.540722i	0.962759	+	0.270361i	$0.0871431\pi$
$107$			64037.4i			0.540722i	−0.962759	+	0.270361i	$0.912857\pi$
$108$		0			0
$109$	−53855.6			−0.434175			−0.217087	−	0.976152i	$-0.569656\pi$
$109$	−53855.6			−0.434175			−0.217087	+	0.976152i	$0.569656\pi$
$110$		0			0
$111$			16653.9i			0.128294i
$112$		0			0
$113$		−	63828.3i		−	0.470238i	−0.971967	−	0.235119i	$-0.924452\pi$
$113$		−	63828.3i		−	0.470238i	0.971967	−	0.235119i	$-0.0755479\pi$
$114$		0			0
$115$	−82997.0			−0.585218
$116$		0			0
$117$	−100670.			−0.679886
$118$		0			0
$119$		−	4636.73i		−	0.0300154i
$120$		0			0
$121$	−152889.			−0.949320
$122$		0			0
$123$	−31841.1			−0.189769
$124$		0			0
$125$			148937.i			0.852562i
$126$		0			0
$127$	−245386.			−1.35002			−0.675010	−	0.737808i	$-0.735861\pi$
$127$	−245386.			−1.35002			−0.675010	+	0.737808i	$0.735861\pi$
$128$		0			0
$129$		−	12815.0i		−	0.0678023i
$130$		0			0
$131$	184118.			0.937387			0.468693	−	0.883361i	$-0.344725\pi$
$131$	184118.			0.937387			0.468693	+	0.883361i	$0.344725\pi$
$132$		0			0
$133$	17236.5	−	29221.1i	0.0844930	−	0.143241i
$134$		0			0
$135$	−23326.6			−0.110158
$136$		0			0
$137$	164046.			0.746732			0.373366	−	0.927684i	$-0.378204\pi$
$137$	164046.			0.746732			0.373366	+	0.927684i	$0.378204\pi$
$138$		0			0
$139$	87851.9			0.385668			0.192834	−	0.981231i	$-0.438232\pi$
$139$	87851.9			0.385668			0.192834	+	0.981231i	$0.438232\pi$
$140$		0			0
$141$	41496.0			0.175776
$142$		0			0
$143$	−37928.9			−0.155107
$144$		0			0
$145$		−	166012.i		−	0.655723i
$146$		0			0
$147$			29279.0i			0.111754i
$148$		0			0
$149$			12580.6i			0.0464232i	0.999731	+	0.0232116i	$0.00738915\pi$
$149$			12580.6i			0.0464232i	−0.999731	+	0.0232116i	$0.992611\pi$
$150$		0			0
$151$	6755.65			0.0241115			0.0120558	−	0.999927i	$-0.496162\pi$
$151$	6755.65			0.0241115			0.0120558	+	0.999927i	$0.496162\pi$
$152$		0			0
$153$	−51569.8			−0.178101
$154$		0			0
$155$		−	82504.9i		−	0.275836i
$156$		0			0
$157$			298532.i			0.966588i	0.875458	+	0.483294i	$0.160560\pi$
$157$			298532.i			0.966588i	−0.875458	+	0.483294i	$0.839440\pi$
$158$		0			0
$159$			21154.4i			0.0663601i
$160$		0			0
$161$	−66353.3			−0.201743
$162$		0			0
$163$	−225736.			−0.665476			−0.332738	−	0.943019i	$-0.607972\pi$
$163$	−225736.			−0.665476			−0.332738	+	0.943019i	$0.607972\pi$
$164$		0			0
$165$	−4365.11			−0.0124820
$166$		0			0
$167$	−581701.			−1.61402			−0.807009	−	0.590539i	$-0.798915\pi$
$167$	−581701.			−1.61402			−0.807009	+	0.590539i	$0.798915\pi$
$168$		0			0
$169$	−195039.			−0.525297
$170$		0			0
$171$	−324997.	−	191705.i	−0.849942	−	0.501352i
$172$		0			0
$173$	−626577.			−1.59169			−0.795846	−	0.605499i	$-0.792974\pi$
$173$	−626577.			−1.59169			−0.795846	+	0.605499i	$0.792974\pi$
$174$		0			0
$175$			51695.0i			0.127601i
$176$		0			0
$177$	76016.4			0.182379
$178$		0			0
$179$		−	446020.i		−	1.04045i	−0.854029	−	0.520226i	$-0.825848\pi$
$179$		−	446020.i		−	1.04045i	0.854029	−	0.520226i	$-0.174152\pi$
$180$		0			0
$181$	−645175.			−1.46380			−0.731899	−	0.681413i	$-0.761366\pi$
$181$	−645175.			−1.46380			−0.731899	+	0.681413i	$0.761366\pi$
$182$		0			0
$183$	−81785.5			−0.180530
$184$		0			0
$185$			250677.i			0.538500i
$186$		0			0
$187$	−19429.7			−0.0406314
$188$		0			0
$189$	−18648.9			−0.0379750
$190$		0			0
$191$			497589.i			0.986932i	0.869765	+	0.493466i	$0.164270\pi$
$191$			497589.i			0.986932i	−0.869765	+	0.493466i	$0.835730\pi$
$192$		0			0
$193$		−	41507.7i		−	0.0802112i	−0.999195	−	0.0401056i	$-0.987231\pi$
$193$		−	41507.7i		−	0.0802112i	0.999195	−	0.0401056i	$-0.0127694\pi$
$194$		0			0
$195$	20284.5			0.0382012
$196$		0			0
$197$		−	140183.i		−	0.257354i	−0.991687	−	0.128677i	$-0.958927\pi$
$197$		−	140183.i		−	0.257354i	0.991687	−	0.128677i	$-0.0410730\pi$
$198$		0			0
$199$			745266.i			1.33407i	0.745026	+	0.667035i	$0.232437\pi$
$199$			745266.i			1.33407i	−0.745026	+	0.667035i	$0.767563\pi$
$200$		0			0
$201$	−59227.9			−0.103404
$202$		0			0
$203$		−	132721.i		−	0.226048i
$204$		0			0
$205$	−479279.			−0.796533
$206$		0			0
$207$			737983.i			1.19707i
$208$		0			0
$209$	−122447.	−	72227.7i	−0.193903	−	0.114377i
$210$		0			0
$211$		−	759403.i		−	1.17426i	−0.809491	−	0.587132i	$-0.800257\pi$
$211$		−	759403.i		−	1.17426i	0.809491	−	0.587132i	$-0.199743\pi$
$212$		0			0
$213$		−	53545.9i		−	0.0808680i
$214$		0			0
$215$		−	192894.i		−	0.284592i
$216$		0			0
$217$		−	65960.0i		−	0.0950892i
$218$		0			0
$219$		−	71045.1i		−	0.100098i
$220$		0			0
$221$	90288.8			0.124352
$222$		0			0
$223$	−354973.			−0.478006			−0.239003	−	0.971019i	$-0.576821\pi$
$223$	−354973.			−0.478006			−0.239003	+	0.971019i	$0.576821\pi$
$224$		0			0
$225$	574952.			0.757139
$226$		0			0
$227$		−	521607.i		−	0.671860i	−0.941887	−	0.335930i	$-0.890949\pi$
$227$		−	521607.i		−	0.671860i	0.941887	−	0.335930i	$-0.109051\pi$
$228$		0			0
$229$		−	921989.i		−	1.16182i	−0.813969	−	0.580908i	$-0.802698\pi$
$229$		−	921989.i		−	1.16182i	0.813969	−	0.580908i	$-0.197302\pi$
$230$		0			0
$231$	−3489.76			−0.00430295
$232$		0			0
$233$	−1.42390e6			−1.71826			−0.859132	−	0.511754i	$-0.828996\pi$
$233$	−1.42390e6			−1.71826			−0.859132	+	0.511754i	$0.828996\pi$
$234$		0			0
$235$	624607.			0.737797
$236$		0			0
$237$		−	111610.i		−	0.129072i
$238$		0			0
$239$		−	316371.i		−	0.358263i	−0.983825	−	0.179132i	$-0.942671\pi$
$239$		−	316371.i		−	0.358263i	0.983825	−	0.179132i	$-0.0573288\pi$
$240$		0			0
$241$		−	485506.i		−	0.538458i	−0.963076	−	0.269229i	$-0.913231\pi$
$241$		−	485506.i		−	0.538458i	0.963076	−	0.269229i	$-0.0867689\pi$
$242$		0			0
$243$			311809.i			0.338745i
$244$		0			0
$245$			440714.i			0.469074i
$246$		0			0
$247$	569007.	+	335638.i	0.593438	+	0.350049i
$248$		0			0
$249$		−	191712.i		−	0.195953i
$250$		0			0
$251$	−788575.			−0.790058			−0.395029	−	0.918669i	$-0.629265\pi$
$251$	−788575.			−0.790058			−0.395029	+	0.918669i	$0.629265\pi$
$252$		0			0
$253$			278046.i			0.273096i
$254$		0			0
$255$	10391.0			0.0100071
$256$		0			0
$257$			743793.i			0.702456i	0.936290	+	0.351228i	$0.114236\pi$
$257$			743793.i			0.702456i	−0.936290	+	0.351228i	$0.885764\pi$
$258$		0			0
$259$			200408.i			0.185638i
$260$		0			0
$261$	−1.47613e6			−1.34129
$262$		0			0
$263$			1.46320e6i			1.30441i	0.758043	+	0.652205i	$0.226156\pi$
$263$			1.46320e6i			1.30441i	−0.758043	+	0.652205i	$0.773844\pi$
$264$		0			0
$265$			318420.i			0.278539i
$266$		0			0
$267$	165561.			0.142128
$268$		0			0
$269$	397571.			0.334992			0.167496	−	0.985873i	$-0.446432\pi$
$269$	397571.			0.334992			0.167496	+	0.985873i	$0.446432\pi$
$270$		0			0
$271$		−	1.02186e6i		−	0.845217i	−0.906312	−	0.422608i	$-0.861115\pi$
$271$		−	1.02186e6i		−	0.845217i	0.906312	−	0.422608i	$-0.138885\pi$
$272$		0			0
$273$	16216.7			0.0131691
$274$		0			0
$275$	216622.			0.172731
$276$		0			0
$277$			482020.i			0.377455i	0.982029	+	0.188728i	$0.0604363\pi$
$277$			482020.i			0.377455i	−0.982029	+	0.188728i	$0.939564\pi$
$278$		0			0
$279$	−733608.			−0.564227
$280$		0			0
$281$		−	1.37395e6i		−	1.03802i	−0.854768	−	0.519010i	$-0.826301\pi$
$281$		−	1.37395e6i		−	1.03802i	0.854768	−	0.519010i	$-0.173699\pi$
$282$		0			0
$283$	539885.			0.400715			0.200357	−	0.979723i	$-0.435790\pi$
$283$	539885.			0.400715			0.200357	+	0.979723i	$0.435790\pi$
$284$		0			0
$285$	65485.1	+	38627.5i	0.0477562	+	0.0281698i
$286$		0			0
$287$	−383167.			−0.274589
$288$		0			0
$289$	−1.37361e6			−0.967425
$290$		0			0
$291$	136626.			0.0945800
$292$		0			0
$293$	−522263.			−0.355402			−0.177701	−	0.984085i	$-0.556866\pi$
$293$	−522263.			−0.355402			−0.177701	+	0.984085i	$0.556866\pi$
$294$		0			0
$295$	1.14422e6			0.765514
$296$		0			0
$297$			78146.0i			0.0514062i
$298$		0			0
$299$		−	1.29207e6i		−	0.835808i
$300$		0			0
$301$		−	154212.i		−	0.0981077i
$302$		0			0
$303$	−258313.			−0.161637
$304$		0			0
$305$	−1.23105e6			−0.757752
$306$		0			0
$307$			1.90603e6i			1.15421i	0.816670	+	0.577105i	$0.195818\pi$
$307$			1.90603e6i			1.15421i	−0.816670	+	0.577105i	$0.804182\pi$
$308$		0			0
$309$		−	79590.7i		−	0.0474205i
$310$		0			0
$311$			37265.3i			0.0218476i	0.999940	+	0.0109238i	$0.00347723\pi$
$311$			37265.3i			0.0218476i	−0.999940	+	0.0109238i	$0.996523\pi$
$312$		0			0
$313$	1.34915e6			0.778396			0.389198	−	0.921154i	$-0.372752\pi$
$313$	1.34915e6			0.778396			0.389198	+	0.921154i	$0.372752\pi$
$314$		0			0
$315$	−139420.			−0.0791680
$316$		0			0
$317$	695473.			0.388716			0.194358	−	0.980931i	$-0.437738\pi$
$317$	695473.			0.388716			0.194358	+	0.980931i	$0.437738\pi$
$318$		0			0
$319$	−556153.			−0.305998
$320$		0			0
$321$	−114731.			−0.0621467
$322$		0			0
$323$	291483.	+	171936.i	0.155456	+	0.0916981i
$324$		0			0
$325$	−1.00663e6			−0.528642
$326$		0			0
$327$		−	96488.9i		−	0.0499009i
$328$		0			0
$329$	499352.			0.254342
$330$		0			0
$331$			2.30186e6i			1.15481i	0.816459	+	0.577404i	$0.195934\pi$
$331$			2.30186e6i			1.15481i	−0.816459	+	0.577404i	$0.804066\pi$
$332$		0			0
$333$	2.22894e6			1.10151
$334$		0			0
$335$	−891512.			−0.434025
$336$		0			0
$337$		−	3.72058e6i		−	1.78458i	−0.451464	−	0.892290i	$-0.649098\pi$
$337$		−	3.72058e6i		−	1.78458i	0.451464	−	0.892290i	$-0.350902\pi$
$338$		0			0
$339$	114356.			0.0540457
$340$		0			0
$341$	−276398.			−0.128721
$342$		0			0
$343$			714694.i			0.328008i
$344$		0			0
$345$		−	148699.i		−	0.0672607i
$346$		0			0
$347$	102106.			0.0455225			0.0227613	−	0.999741i	$-0.492754\pi$
$347$	102106.			0.0455225			0.0227613	+	0.999741i	$0.492754\pi$
$348$		0			0
$349$			2.09248e6i			0.919597i	0.888023	+	0.459799i	$0.152078\pi$
$349$			2.09248e6i			0.919597i	−0.888023	+	0.459799i	$0.847922\pi$
$350$		0			0
$351$		−	363140.i		−	0.157328i
$352$		0			0
$353$	1.29274e6			0.552173			0.276086	−	0.961133i	$-0.410962\pi$
$353$	1.29274e6			0.552173			0.276086	+	0.961133i	$0.410962\pi$
$354$		0			0
$355$		−	805984.i		−	0.339434i
$356$		0			0
$357$	8307.28			0.00344976
$358$		0			0
$359$		−	4.19203e6i		−	1.71667i	−0.513086	−	0.858337i	$-0.671498\pi$
$359$		−	4.19203e6i		−	1.71667i	0.513086	−	0.858337i	$-0.328502\pi$
$360$		0			0
$361$	1.19779e6	+	2.16711e6i	0.483743	+	0.875210i
$362$		0			0
$363$		−	273919.i		−	0.109108i
$364$		0			0
$365$		−	1.06939e6i		−	0.420148i
$366$		0			0
$367$			1.08261e6i			0.419571i	0.977747	+	0.209785i	$0.0672765\pi$
$367$			1.08261e6i			0.419571i	−0.977747	+	0.209785i	$0.932723\pi$
$368$		0			0
$369$			4.26160e6i			1.62932i
$370$		0			0
$371$			254566.i			0.0960210i
$372$		0			0
$373$	2.51466e6			0.935851			0.467925	−	0.883768i	$-0.345002\pi$
$373$	2.51466e6			0.935851			0.467925	+	0.883768i	$0.345002\pi$
$374$		0			0
$375$	−266838.			−0.0979873
$376$		0			0
$377$	2.58442e6			0.936503
$378$		0			0
$379$			2.81311e6i			1.00598i	0.864292	+	0.502990i	$0.167767\pi$
$379$			2.81311e6i			1.00598i	−0.864292	+	0.502990i	$0.832233\pi$
$380$		0			0
$381$		−	439639.i		−	0.155161i
$382$		0			0
$383$	4.38136e6			1.52620			0.763101	−	0.646279i	$-0.223676\pi$
$383$	4.38136e6			1.52620			0.763101	+	0.646279i	$0.223676\pi$
$384$		0			0
$385$	−52528.6			−0.0180611
$386$		0			0
$387$	−1.71515e6			−0.582137
$388$		0			0
$389$		−	1.39789e6i		−	0.468381i	−0.972191	−	0.234191i	$-0.924756\pi$
$389$		−	1.39789e6i		−	0.468381i	0.972191	−	0.234191i	$-0.0752440\pi$
$390$		0			0
$391$		−	661881.i		−	0.218946i
$392$		0			0
$393$			329871.i			0.107736i
$394$		0			0
$395$		−	1.67998e6i		−	0.541765i
$396$		0			0
$397$			139153.i			0.0443115i	0.999755	+	0.0221557i	$0.00705297\pi$
$397$			139153.i			0.0443115i	−0.999755	+	0.0221557i	$0.992947\pi$
$398$		0			0
$399$	52353.1	+	30881.4i	0.0164631	+	0.00971101i
$400$		0			0
$401$			2.89612e6i			0.899404i	0.893179	+	0.449702i	$0.148470\pi$
$401$			2.89612e6i			0.899404i	−0.893179	+	0.449702i	$0.851530\pi$
$402$		0			0
$403$	1.28441e6			0.393949
$404$		0			0
$405$			1.52960e6i			0.463383i
$406$		0			0
$407$	839787.			0.251295
$408$		0			0
$409$		−	4.18231e6i		−	1.23626i	−0.786078	−	0.618128i	$-0.787891\pi$
$409$		−	4.18231e6i		−	1.23626i	0.786078	−	0.618128i	$-0.212109\pi$
$410$		0			0
$411$			293909.i			0.0858239i
$412$		0			0
$413$	914762.			0.263896
$414$		0			0
$415$		−	2.88569e6i		−	0.822489i
$416$		0			0
$417$			157397.i			0.0443259i
$418$		0			0
$419$	−5.33921e6			−1.48574			−0.742869	−	0.669437i	$-0.766535\pi$
$419$	−5.33921e6			−1.48574			−0.742869	+	0.669437i	$0.766535\pi$
$420$		0			0
$421$	5.84059e6			1.60602			0.803011	−	0.595964i	$-0.203230\pi$
$421$	5.84059e6			1.60602			0.803011	+	0.595964i	$0.203230\pi$
$422$		0			0
$423$		−	5.55381e6i		−	1.50918i
$424$		0			0
$425$	−515662.			−0.138482
$426$		0			0
$427$	−984186.			−0.261221
$428$		0			0
$429$		−	67954.4i		−	0.0178268i
$430$		0			0
$431$	3.18075e6			0.824777			0.412388	−	0.911008i	$-0.364695\pi$
$431$	3.18075e6			0.824777			0.412388	+	0.911008i	$0.364695\pi$
$432$		0			0
$433$			2.87889e6i			0.737913i	0.929447	+	0.368956i	$0.120285\pi$
$433$			2.87889e6i			0.737913i	−0.929447	+	0.368956i	$0.879715\pi$
$434$		0			0
$435$	297432.			0.0753640
$436$		0			0
$437$	2.46047e6	−	4.17122e6i	0.616330	−	1.04486i
$438$		0			0
$439$	−4.54057e6			−1.12447			−0.562236	−	0.826977i	$-0.690059\pi$
$439$	−4.54057e6			−1.12447			−0.562236	+	0.826977i	$0.690059\pi$
$440$		0			0
$441$	3.91869e6			0.959499
$442$		0			0
$443$	3.58650e6			0.868284			0.434142	−	0.900845i	$-0.357052\pi$
$443$	3.58650e6			0.868284			0.434142	+	0.900845i	$0.357052\pi$
$444$		0			0
$445$	2.49206e6			0.596566
$446$		0			0
$447$	−22539.7			−0.00533555
$448$		0			0
$449$			2.78600e6i			0.652178i	0.945339	+	0.326089i	$0.105731\pi$
$449$			2.78600e6i			0.652178i	−0.945339	+	0.326089i	$0.894269\pi$
$450$		0			0
$451$			1.60562e6i			0.371707i
$452$		0			0
$453$			12103.6i			0.00277120i
$454$		0			0
$455$	244098.			0.0552759
$456$		0			0
$457$	6.01508e6			1.34726			0.673629	−	0.739069i	$-0.264734\pi$
$457$	6.01508e6			1.34726			0.673629	+	0.739069i	$0.264734\pi$
$458$		0			0
$459$		−	186024.i		−	0.0412134i
$460$		0			0
$461$			5.49794e6i			1.20489i	0.798160	+	0.602445i	$0.205807\pi$
$461$			5.49794e6i			1.20489i	−0.798160	+	0.602445i	$0.794193\pi$
$462$		0			0
$463$		−	6.80325e6i		−	1.47490i	−0.675400	−	0.737452i	$-0.736029\pi$
$463$		−	6.80325e6i		−	1.47490i	0.675400	−	0.737452i	$-0.263971\pi$
$464$		0			0
$465$	147818.			0.0317026
$466$		0			0
$467$	3.92402e6			0.832606			0.416303	−	0.909226i	$-0.363326\pi$
$467$	3.92402e6			0.832606			0.416303	+	0.909226i	$0.363326\pi$
$468$		0			0
$469$	−712734.			−0.149622
$470$		0			0
$471$	−534857.			−0.111093
$472$		0			0
$473$	−646209.			−0.132807
$474$		0			0
$475$	−3.24974e6	−	1.91691e6i	−0.660868	−	0.389824i
$476$		0			0
$477$	2.83129e6			0.569756
$478$		0			0
$479$			4.25311e6i			0.846969i	0.905903	+	0.423485i	$0.139193\pi$
$479$			4.25311e6i			0.846969i	−0.905903	+	0.423485i	$0.860807\pi$
$480$		0			0
$481$	−3.90245e6			−0.769086
$482$		0			0
$483$		−	118880.i		−	0.0231868i
$484$		0			0
$485$	2.05652e6			0.396988
$486$		0			0
$487$	1.02183e7			1.95235			0.976174	−	0.216988i	$-0.0696232\pi$
$487$	1.02183e7			1.95235			0.976174	+	0.216988i	$0.0696232\pi$
$488$		0			0
$489$		−	404434.i		−	0.0764849i
$490$		0			0
$491$	−3.32334e6			−0.622116			−0.311058	−	0.950391i	$-0.600683\pi$
$491$	−3.32334e6			−0.622116			−0.311058	+	0.950391i	$0.600683\pi$
$492$		0			0
$493$	1.32391e6			0.245324
$494$		0			0
$495$			584225.i			0.107168i
$496$		0			0
$497$		−	644357.i		−	0.117013i
$498$		0			0
$499$	5.52536e6			0.993366			0.496683	−	0.867932i	$-0.334551\pi$
$499$	5.52536e6			0.993366			0.496683	+	0.867932i	$0.334551\pi$
$500$		0			0
$501$		−	1.04219e6i		−	0.185504i
$502$		0			0
$503$			2.18871e6i			0.385716i	0.981227	+	0.192858i	$0.0617756\pi$
$503$			2.18871e6i			0.385716i	−0.981227	+	0.192858i	$0.938224\pi$
$504$		0			0
$505$	−3.88819e6			−0.678451
$506$		0			0
$507$		−	349437.i		−	0.0603738i
$508$		0			0
$509$	−6.27300e6			−1.07320			−0.536600	−	0.843836i	$-0.680292\pi$
$509$	−6.27300e6			−1.07320			−0.536600	+	0.843836i	$0.680292\pi$
$510$		0			0
$511$		−	854939.i		−	0.144838i
$512$		0			0
$513$	691524.	−	1.17234e6i	0.116015	−	0.196680i
$514$		0			0
$515$		−	1.19802e6i		−	0.199042i
$516$		0			0
$517$		−	2.09248e6i		−	0.344298i
$518$		0			0
$519$		−	1.12259e6i		−	0.182938i
$520$		0			0
$521$			1.45769e6i			0.235273i	0.993057	+	0.117636i	$0.0375317\pi$
$521$			1.45769e6i			0.235273i	−0.993057	+	0.117636i	$0.962468\pi$
$522$		0			0
$523$			1.24010e7i			1.98245i	0.132178	+	0.991226i	$0.457803\pi$
$523$			1.24010e7i			1.98245i	−0.132178	+	0.991226i	$0.542197\pi$
$524$		0			0
$525$	−92617.9			−0.0146655
$526$		0			0
$527$	657957.			0.103198
$528$		0			0
$529$	−3.03541e6			−0.471605
$530$		0			0
$531$		−	1.01740e7i		−	1.56587i
$532$		0			0
$533$		−	7.46123e6i		−	1.13761i
$534$		0			0
$535$	−1.72695e6			−0.260853
$536$		0			0
$537$	799100.			0.119582
$538$		0			0
$539$	1.47642e6			0.218897
$540$		0			0
$541$		−	6.25172e6i		−	0.918346i	−0.888347	−	0.459173i	$-0.848146\pi$
$541$		−	6.25172e6i		−	0.918346i	0.888347	−	0.459173i	$-0.151854\pi$
$542$		0			0
$543$		−	1.15591e6i		−	0.168238i
$544$		0			0
$545$		−	1.45237e6i		−	0.209453i
$546$		0			0
$547$			8.68527e6i			1.24112i	0.784158	+	0.620561i	$0.213095\pi$
$547$			8.68527e6i			1.24112i	−0.784158	+	0.620561i	$0.786905\pi$
$548$		0			0
$549$			1.09461e7i			1.54999i
$550$		0			0
$551$	8.34337e6	+	4.92147e6i	1.17075	+	0.690584i
$552$		0			0
$553$		−	1.34309e6i		−	0.186763i
$554$		0			0
$555$	−449120.			−0.0618913
$556$		0			0
$557$		−	1.26351e7i		−	1.72560i	−0.505544	−	0.862801i	$-0.668708\pi$
$557$		−	1.26351e7i		−	1.72560i	0.505544	−	0.862801i	$-0.331292\pi$
$558$		0			0
$559$	3.00290e6			0.406454
$560$		0			0
$561$		−	34810.7i		−	0.00466988i
$562$		0			0
$563$			2.22227e6i			0.295478i	0.989026	+	0.147739i	$0.0471996\pi$
$563$			2.22227e6i			0.295478i	−0.989026	+	0.147739i	$0.952800\pi$
$564$		0			0
$565$	1.72132e6			0.226850
$566$		0			0
$567$			1.22286e6i			0.159743i
$568$		0			0
$569$			8.80479e6i			1.14009i	0.821614	+	0.570044i	$0.193074\pi$
$569$			8.80479e6i			1.14009i	−0.821614	+	0.570044i	$0.806926\pi$
$570$		0			0
$571$	1.03222e7			1.32490			0.662450	−	0.749106i	$-0.269517\pi$
$571$	1.03222e7			1.32490			0.662450	+	0.749106i	$0.269517\pi$
$572$		0			0
$573$	−891491.			−0.113431
$574$		0			0
$575$			7.37931e6i			0.930778i
$576$		0			0
$577$	−2.38619e6			−0.298378			−0.149189	−	0.988809i	$-0.547666\pi$
$577$	−2.38619e6			−0.298378			−0.149189	+	0.988809i	$0.547666\pi$
$578$		0			0
$579$	74366.1			0.00921889
$580$		0			0
$581$		−	2.30702e6i		−	0.283537i
$582$		0			0
$583$	1.06673e6			0.129982
$584$		0			0
$585$		−	2.71486e6i		−	0.327988i
$586$		0			0
$587$	4.97228e6			0.595608			0.297804	−	0.954627i	$-0.403746\pi$
$587$	4.97228e6			0.595608			0.297804	+	0.954627i	$0.403746\pi$
$588$		0			0
$589$	4.14649e6	+	2.44588e6i	0.492485	+	0.290500i
$590$		0			0
$591$	251156.			0.0295784
$592$		0			0
$593$	550883.			0.0643313			0.0321657	−	0.999483i	$-0.489760\pi$
$593$	550883.			0.0643313			0.0321657	+	0.999483i	$0.489760\pi$
$594$		0			0
$595$	125043.			0.0144799
$596$		0			0
$597$	−1.33524e6			−0.153328
$598$		0			0
$599$	−4.35875e6			−0.496357			−0.248179	−	0.968714i	$-0.579832\pi$
$599$	−4.35875e6			−0.496357			−0.248179	+	0.968714i	$0.579832\pi$
$600$		0			0
$601$			387678.i			0.0437809i	0.999760	+	0.0218905i	$0.00696851\pi$
$601$			387678.i			0.0437809i	−0.999760	+	0.0218905i	$0.993031\pi$
$602$		0			0
$603$			7.92704e6i			0.887805i
$604$		0			0
$605$		−	4.12309e6i		−	0.457968i
$606$		0			0
$607$	3.69831e6			0.407410			0.203705	−	0.979032i	$-0.434702\pi$
$607$	3.69831e6			0.407410			0.203705	+	0.979032i	$0.434702\pi$
$608$		0			0
$609$	237787.			0.0259803
$610$		0			0
$611$			9.72364e6i			1.05372i
$612$		0			0
$613$		−	2.46872e6i		−	0.265351i	−0.991160	−	0.132675i	$-0.957643\pi$
$613$		−	2.46872e6i		−	0.265351i	0.991160	−	0.132675i	$-0.0423568\pi$
$614$		0			0
$615$		−	858687.i		−	0.0915476i
$616$		0			0
$617$	−6.46330e6			−0.683504			−0.341752	−	0.939790i	$-0.611020\pi$
$617$	−6.46330e6			−0.683504			−0.341752	+	0.939790i	$0.611020\pi$
$618$		0			0
$619$	−1.08579e6			−0.113899			−0.0569494	−	0.998377i	$-0.518137\pi$
$619$	−1.08579e6			−0.113899			−0.0569494	+	0.998377i	$0.518137\pi$
$620$		0			0
$621$	−2.66207e6			−0.277007
$622$		0			0
$623$	1.99232e6			0.205655
$624$		0			0
$625$	3.47641e6			0.355984
$626$		0			0
$627$	129405.	−	219380.i	0.0131456	−	0.0222858i
$628$		0			0
$629$	−1.99909e6			−0.201468
$630$		0			0
$631$		−	4.86063e6i		−	0.485981i	−0.970029	−	0.242991i	$-0.921872\pi$
$631$		−	4.86063e6i		−	0.485981i	0.970029	−	0.242991i	$-0.0781284\pi$
$632$		0			0
$633$	1.36056e6			0.134961
$634$		0			0
$635$		−	6.61754e6i		−	0.651272i
$636$		0			0
$637$	−6.86087e6			−0.669932
$638$		0			0
$639$	−7.16655e6			−0.694318
$640$		0			0
$641$		−	7.52432e6i		−	0.723306i	−0.932313	−	0.361653i	$-0.882212\pi$
$641$		−	7.52432e6i		−	0.723306i	0.932313	−	0.361653i	$-0.117788\pi$
$642$		0			0
$643$	−3.17853e6			−0.303178			−0.151589	−	0.988444i	$-0.548439\pi$
$643$	−3.17853e6			−0.303178			−0.151589	+	0.988444i	$0.548439\pi$
$644$		0			0
$645$	345594.			0.0327089
$646$		0			0
$647$			1.27831e7i			1.20054i	0.799798	+	0.600269i	$0.204940\pi$
$647$			1.27831e7i			1.20054i	−0.799798	+	0.600269i	$0.795060\pi$
$648$		0			0
$649$		−	3.83321e6i		−	0.357232i
$650$		0			0
$651$	118175.			0.0109289
$652$		0			0
$653$			1.25243e7i			1.14940i	0.818365	+	0.574699i	$0.194881\pi$
$653$			1.25243e7i			1.14940i	−0.818365	+	0.574699i	$0.805119\pi$
$654$		0			0
$655$			4.96529e6i			0.452211i
$656$		0			0
$657$	−9.50865e6			−0.859420
$658$		0			0
$659$		−	1.95227e7i		−	1.75116i	−0.483075	−	0.875579i	$-0.660480\pi$
$659$		−	1.95227e7i		−	1.75116i	0.483075	−	0.875579i	$-0.339520\pi$
$660$		0			0
$661$	1.17749e6			0.104822			0.0524110	−	0.998626i	$-0.483309\pi$
$661$	1.17749e6			0.104822			0.0524110	+	0.998626i	$0.483309\pi$
$662$		0			0
$663$			161764.i			0.0142921i
$664$		0			0
$665$	788031.	+	464833.i	0.0691017	+	0.0407608i
$666$		0			0
$667$		−	1.89456e7i		−	1.64890i
$668$		0			0
$669$		−	635978.i		−	0.0549385i
$670$		0			0
$671$			4.12412e6i			0.353610i
$672$		0			0
$673$			1.92406e7i			1.63750i	0.574151	+	0.818749i	$0.305332\pi$
$673$			1.92406e7i			1.63750i	−0.574151	+	0.818749i	$0.694668\pi$
$674$		0			0
$675$			2.07399e6i			0.175205i
$676$		0			0
$677$	−1.22877e7			−1.03039			−0.515194	−	0.857074i	$-0.672280\pi$
$677$	−1.22877e7			−1.03039			−0.515194	+	0.857074i	$0.672280\pi$
$678$		0			0
$679$	1.64412e6			0.136854
$680$		0			0
$681$	934523.			0.0772187
$682$		0			0
$683$		−	2.78444e6i		−	0.228395i	−0.993458	−	0.114197i	$-0.963570\pi$
$683$		−	2.78444e6i		−	0.228395i	0.993458	−	0.114197i	$-0.0364296\pi$
$684$		0			0
$685$			4.42398e6i			0.360236i
$686$		0			0
$687$	1.65186e6			0.133531
$688$		0			0
$689$	−4.95704e6			−0.397809
$690$		0			0
$691$	5.06792e6			0.403771			0.201885	−	0.979409i	$-0.435293\pi$
$691$	5.06792e6			0.403771			0.201885	+	0.979409i	$0.435293\pi$
$692$		0			0
$693$			467068.i			0.0369443i
$694$		0			0
$695$			2.36918e6i			0.186053i
$696$		0			0
$697$		−	3.82213e6i		−	0.298005i
$698$		0			0
$699$		−	2.55109e6i		−	0.197485i
$700$		0			0
$701$		−	1.84189e7i		−	1.41569i	−0.706366	−	0.707847i	$-0.749667\pi$
$701$		−	1.84189e7i		−	1.41569i	0.706366	−	0.707847i	$-0.250333\pi$
$702$		0			0
$703$	−1.25984e7	−	7.43139e6i	−0.961453	−	0.567129i
$704$		0			0
$705$			1.11906e6i			0.0847970i
$706$		0			0
$707$	−3.10848e6			−0.233883
$708$		0			0
$709$			1.41054e7i			1.05383i	0.849918	+	0.526915i	$0.176651\pi$
$709$			1.41054e7i			1.05383i	−0.849918	+	0.526915i	$0.823349\pi$
$710$		0			0
$711$	−1.49378e7			−1.10819
$712$		0			0
$713$		−	9.41560e6i		−	0.693624i
$714$		0			0
$715$		−	1.02286e6i		−	0.0748261i
$716$		0			0
$717$	566819.			0.0411762
$718$		0			0
$719$		−	3.59772e6i		−	0.259540i	−0.991544	−	0.129770i	$-0.958576\pi$
$719$		−	3.59772e6i		−	0.259540i	0.991544	−	0.129770i	$-0.0414240\pi$
$720$		0			0
$721$		−	957774.i		−	0.0686159i
$722$		0			0
$723$	869844.			0.0618865
$724$		0			0
$725$	−1.47603e7			−1.04291
$726$		0			0
$727$		−	4.42128e6i		−	0.310250i	−0.987895	−	0.155125i	$-0.950422\pi$
$727$		−	4.42128e6i		−	0.310250i	0.987895	−	0.155125i	$-0.0495781\pi$
$728$		0			0
$729$	1.32241e7			0.921613
$730$		0			0
$731$	1.53828e6			0.106474
$732$		0			0
$733$			5.89336e6i			0.405138i	0.979268	+	0.202569i	$0.0649291\pi$
$733$			5.89336e6i			0.405138i	−0.979268	+	0.202569i	$0.935071\pi$
$734$		0			0
$735$	−789594.			−0.0539120
$736$		0			0
$737$			2.98663e6i			0.202541i
$738$		0			0
$739$	−2.27527e7			−1.53258			−0.766288	−	0.642497i	$-0.777898\pi$
$739$	−2.27527e7			−1.53258			−0.766288	+	0.642497i	$0.777898\pi$
$740$		0			0
$741$	−601337.	+	1.01945e6i	−0.0402321	+	0.0682054i
$742$		0			0
$743$	−341178.			−0.0226730			−0.0113365	−	0.999936i	$-0.503609\pi$
$743$	−341178.			−0.0226730			−0.0113365	+	0.999936i	$0.503609\pi$
$744$		0			0
$745$	−339272.			−0.0223953
$746$		0			0
$747$	−2.56587e7			−1.68241
$748$		0			0
$749$	−1.38064e6			−0.0899242
$750$		0			0
$751$	−6.71971e6			−0.434761			−0.217380	−	0.976087i	$-0.569751\pi$
$751$	−6.71971e6			−0.434761			−0.217380	+	0.976087i	$0.569751\pi$
$752$		0			0
$753$		−	1.41283e6i		−	0.0908035i
$754$		0			0
$755$			182186.i			0.0116318i
$756$		0			0
$757$			8.93104e6i			0.566451i	0.959053	+	0.283226i	$0.0914045\pi$
$757$			8.93104e6i			0.566451i	−0.959053	+	0.283226i	$0.908595\pi$
$758$		0			0
$759$	−498154.			−0.0313877
$760$		0			0
$761$	−1.57336e7			−0.984839			−0.492419	−	0.870358i	$-0.663887\pi$
$761$	−1.57336e7			−0.984839			−0.492419	+	0.870358i	$0.663887\pi$
$762$		0			0
$763$		−	1.16112e6i		−	0.0722050i
$764$		0			0
$765$		−	1.39073e6i		−	0.0859190i
$766$		0			0
$767$			1.78127e7i			1.09331i
$768$		0			0
$769$	5.08185e6			0.309889			0.154944	−	0.987923i	$-0.450480\pi$
$769$	5.08185e6			0.309889			0.154944	+	0.987923i	$0.450480\pi$
$770$		0			0
$771$	−1.33260e6			−0.0807352
$772$		0			0
$773$	1.30205e7			0.783755			0.391877	−	0.920017i	$-0.371826\pi$
$773$	1.30205e7			0.783755			0.391877	+	0.920017i	$0.371826\pi$
$774$		0			0
$775$	−7.33556e6			−0.438712
$776$		0			0
$777$	−359056.			−0.0213358
$778$		0			0
$779$	1.42083e7	−	2.40874e7i	0.838879	−	1.42215i
$780$		0			0
$781$	−2.70010e6			−0.158399
$782$		0			0
$783$		−	5.32474e6i		−	0.310380i
$784$		0			0
$785$	−8.05078e6			−0.466298
$786$		0			0
$787$		−	5.56173e6i		−	0.320091i	−0.987110	−	0.160045i	$-0.948836\pi$
$787$		−	5.56173e6i		−	0.320091i	0.987110	−	0.160045i	$-0.0511641\pi$
$788$		0			0
$789$	−2.62150e6			−0.149919
$790$		0			0
$791$	1.37614e6			0.0782024
$792$		0			0
$793$		−	1.91646e7i		−	1.08222i
$794$		0			0
$795$	−570489.			−0.0320132
$796$		0			0
$797$	−2.16448e7			−1.20700			−0.603502	−	0.797361i	$-0.706228\pi$
$797$	−2.16448e7			−1.20700			−0.603502	+	0.797361i	$0.706228\pi$
$798$		0			0
$799$			4.98109e6i			0.276031i
$800$		0			0
$801$		−	2.21586e7i		−	1.22028i
$802$		0			0
$803$	−3.58252e6			−0.196065
$804$		0			0
$805$		−	1.78941e6i		−	0.0973241i
$806$		0			0
$807$			712298.i			0.0385015i
$808$		0			0
$809$	2.25215e7			1.20983			0.604917	−	0.796289i	$-0.293206\pi$
$809$	2.25215e7			1.20983			0.604917	+	0.796289i	$0.293206\pi$
$810$		0			0
$811$			8.93759e6i			0.477165i	0.971122	+	0.238582i	$0.0766827\pi$
$811$			8.93759e6i			0.477165i	−0.971122	+	0.238582i	$0.923317\pi$
$812$		0			0
$813$	1.83079e6			0.0971430
$814$		0			0
$815$		−	6.08763e6i		−	0.321036i
$816$		0			0
$817$	9.69438e6	+	5.71839e6i	0.508118	+	0.299722i
$818$		0			0
$819$		−	2.17044e6i		−	0.113068i
$820$		0			0
$821$			3.29211e7i			1.70457i	0.523074	+	0.852287i	$0.324785\pi$
$821$			3.29211e7i			1.70457i	−0.523074	+	0.852287i	$0.675215\pi$
$822$		0			0
$823$			6.73875e6i			0.346801i	0.984851	+	0.173400i	$0.0554754\pi$
$823$			6.73875e6i			0.346801i	−0.984851	+	0.173400i	$0.944525\pi$
$824$		0			0
$825$			388105.i			0.0198524i
$826$		0			0
$827$			2.63122e7i			1.33781i	0.743349	+	0.668904i	$0.233236\pi$
$827$			2.63122e7i			1.33781i	−0.743349	+	0.668904i	$0.766764\pi$
$828$		0			0
$829$	5.20905e6			0.263252			0.131626	−	0.991299i	$-0.457980\pi$
$829$	5.20905e6			0.263252			0.131626	+	0.991299i	$0.457980\pi$
$830$		0			0
$831$	−863598.			−0.0433819
$832$		0			0
$833$	−3.51459e6			−0.175494
$834$		0			0
$835$		−	1.56873e7i		−	0.778629i
$836$		0			0
$837$		−	2.64629e6i		−	0.130564i
$838$		0			0
$839$	−3.75422e7			−1.84126			−0.920630	−	0.390437i	$-0.872324\pi$
$839$	−3.75422e7			−1.84126			−0.920630	+	0.390437i	$0.872324\pi$
$840$		0			0
$841$	1.73842e7			0.847551
$842$		0			0
$843$	2.46161e6			0.119302
$844$		0			0
$845$		−	5.25980e6i		−	0.253412i
$846$		0			0
$847$		−	3.29628e6i		−	0.157876i
$848$		0			0
$849$			967271.i			0.0460552i
$850$		0			0
$851$			2.86077e7i			1.35413i
$852$		0			0
$853$			4.33902e6i			0.204183i	0.994775	+	0.102091i	$0.0325534\pi$
$853$			4.33902e6i			0.204183i	−0.994775	+	0.102091i	$0.967447\pi$
$854$		0			0
$855$	5.16988e6	−	8.76449e6i	0.241861	−	0.410026i
$856$		0			0
$857$			3.19143e7i			1.48434i	0.670213	+	0.742169i	$0.266203\pi$
$857$			3.19143e7i			1.48434i	−0.670213	+	0.742169i	$0.733797\pi$
$858$		0			0
$859$	−2.97907e7			−1.37752			−0.688761	−	0.724989i	$-0.741845\pi$
$859$	−2.97907e7			−1.37752			−0.688761	+	0.724989i	$0.741845\pi$
$860$		0			0
$861$		−	686492.i		−	0.0315593i
$862$		0			0
$863$	1.38627e6			0.0633609			0.0316804	−	0.999498i	$-0.489914\pi$
$863$	1.38627e6			0.0633609			0.0316804	+	0.999498i	$0.489914\pi$
$864$		0			0
$865$		−	1.68975e7i		−	0.767859i
$866$		0			0
$867$		−	2.46098e6i		−	0.111189i
$868$		0			0
$869$	−5.62805e6			−0.252818
$870$		0			0
$871$		−	1.38787e7i		−	0.619875i
$872$		0			0
$873$		−	1.82859e7i		−	0.812046i
$874$		0			0
$875$	−3.21106e6			−0.141784
$876$		0			0
$877$	1.65815e6			0.0727987			0.0363994	−	0.999337i	$-0.488411\pi$
$877$	1.65815e6			0.0727987			0.0363994	+	0.999337i	$0.488411\pi$
$878$		0			0
$879$		−	935698.i		−	0.0408473i
$880$		0			0
$881$	−2.95505e7			−1.28270			−0.641349	−	0.767249i	$-0.721625\pi$
$881$	−2.95505e7			−1.28270			−0.641349	+	0.767249i	$0.721625\pi$
$882$		0			0
$883$	1.95346e7			0.843146			0.421573	−	0.906794i	$-0.361478\pi$
$883$	1.95346e7			0.843146			0.421573	+	0.906794i	$0.361478\pi$
$884$		0			0
$885$			2.05000e6i			0.0879825i
$886$		0			0
$887$	−1.11186e7			−0.474506			−0.237253	−	0.971448i	$-0.576247\pi$
$887$	−1.11186e7			−0.474506			−0.237253	+	0.971448i	$0.576247\pi$
$888$		0			0
$889$		−	5.29051e6i		−	0.224514i
$890$		0			0
$891$	5.12427e6			0.216241
$892$		0			0
$893$	−1.85166e7	+	3.13912e7i	−0.777021	+	1.31728i
$894$		0			0
$895$	1.20282e7			0.501931
$896$		0			0
$897$	2.31489e6			0.0960617
$898$		0			0
$899$	1.88333e7			0.777189
$900$		0			0
$901$	−2.53932e6			−0.104209
$902$		0			0
$903$	276291.			0.0112758
$904$		0			0
$905$		−	1.73990e7i		−	0.706160i
$906$		0			0
$907$			2.35927e7i			0.952270i	0.879372	+	0.476135i	$0.157963\pi$
$907$			2.35927e7i			0.952270i	−0.879372	+	0.476135i	$0.842037\pi$
$908$		0			0
$909$			3.45725e7i			1.38778i
$910$		0			0
$911$	−3.38240e7			−1.35030			−0.675148	−	0.737682i	$-0.735920\pi$
$911$	−3.38240e7			−1.35030			−0.675148	+	0.737682i	$0.735920\pi$
$912$		0			0
$913$	−9.66729e6			−0.383820
$914$		0			0
$915$		−	2.20558e6i		−	0.0870905i
$916$		0			0
$917$			3.96958e6i			0.155891i
$918$		0			0
$919$		−	5.37265e6i		−	0.209845i	−0.994480	−	0.104923i	$-0.966540\pi$
$919$		−	5.37265e6i		−	0.209845i	0.994480	−	0.104923i	$-0.0334595\pi$
$920$		0			0
$921$	−3.41490e6			−0.132656
$922$		0			0
$923$	1.25472e7			0.484779
$924$		0			0
$925$	2.22879e7			0.856474
$926$		0			0
$927$	−1.06524e7			−0.407143
$928$		0			0
$929$	−1.92859e7			−0.733164			−0.366582	−	0.930386i	$-0.619472\pi$
$929$	−1.92859e7			−0.733164			−0.366582	+	0.930386i	$0.619472\pi$
$930$		0			0
$931$	−2.21492e7	−	1.30651e7i	−0.837498	−	0.494012i
$932$		0			0
$933$	−66765.5			−0.00251101
$934$		0			0
$935$		−	523978.i		−	0.0196013i
$936$		0			0
$937$	1.41288e7			0.525721			0.262861	−	0.964834i	$-0.415334\pi$
$937$	1.41288e7			0.525721			0.262861	+	0.964834i	$0.415334\pi$
$938$		0			0
$939$			2.41717e6i			0.0894631i
$940$		0			0
$941$	−2.19479e7			−0.808012			−0.404006	−	0.914756i	$-0.632383\pi$
$941$	−2.19479e7			−0.808012			−0.404006	+	0.914756i	$0.632383\pi$
$942$		0			0
$943$	−5.46961e7			−2.00298
$944$		0			0
$945$		−	502921.i		−	0.0183198i
$946$		0			0
$947$	8.42285e6			0.305200			0.152600	−	0.988288i	$-0.451235\pi$
$947$	8.42285e6			0.305200			0.152600	+	0.988288i	$0.451235\pi$
$948$		0			0
$949$	1.66478e7			0.600055
$950$		0			0
$951$			1.24603e6i			0.0446762i
$952$		0			0
$953$		−	4.19088e7i		−	1.49477i	−0.664394	−	0.747383i	$-0.731310\pi$
$953$		−	4.19088e7i		−	1.49477i	0.664394	−	0.747383i	$-0.268690\pi$
$954$		0			0
$955$	−1.34189e7			−0.476112
$956$		0			0
$957$		−	996418.i		−	0.0351691i
$958$		0			0
$959$			3.53682e6i			0.124184i
$960$		0			0
$961$	−1.92694e7			−0.673068
$962$		0			0
$963$			1.53555e7i			0.533580i
$964$		0			0
$965$	1.11937e6			0.0386952
$966$		0			0
$967$			1.10867e7i			0.381272i	0.981661	+	0.190636i	$0.0610550\pi$
$967$			1.10867e7i			0.381272i	−0.981661	+	0.190636i	$0.938945\pi$
$968$		0			0
$969$	−308044.	+	522227.i	−0.0105391	+	0.0178669i
$970$		0			0
$971$		−	2.03318e7i		−	0.692035i	−0.938228	−	0.346018i	$-0.887534\pi$
$971$		−	2.03318e7i		−	0.692035i	0.938228	−	0.346018i	$-0.112466\pi$
$972$		0			0
$973$			1.89408e6i			0.0641382i
$974$		0			0
$975$		−	1.80350e6i		−	0.0607582i
$976$		0			0
$977$			1.13639e7i			0.380884i	0.981698	+	0.190442i	$0.0609921\pi$
$977$			1.13639e7i			0.380884i	−0.981698	+	0.190442i	$0.939008\pi$
$978$		0			0
$979$		−	8.34858e6i		−	0.278391i
$980$		0			0
$981$	−1.29140e7			−0.428439
$982$		0			0
$983$	3.96913e7			1.31012			0.655060	−	0.755577i	$-0.272643\pi$
$983$	3.96913e7			1.31012			0.655060	+	0.755577i	$0.272643\pi$
$984$		0			0
$985$	3.78045e6			0.124152
$986$		0			0
$987$			894651.i			0.0292322i
$988$		0			0
$989$		−	2.20134e7i		−	0.715643i
$990$		0			0
$991$	5.69524e7			1.84216			0.921081	−	0.389372i	$-0.127308\pi$
$991$	5.69524e7			1.84216			0.921081	+	0.389372i	$0.127308\pi$
$992$		0			0
$993$	−4.12407e6			−0.132725
$994$		0			0
$995$	−2.00983e7			−0.643578
$996$		0			0
$997$		−	1.02156e7i		−	0.325483i	−0.986669	−	0.162741i	$-0.947966\pi$
$997$		−	1.02156e7i		−	0.325483i	0.986669	−	0.162741i	$-0.0520336\pi$
$998$		0			0
$999$			8.04031e6i			0.254894i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	608.6.b.b.303.52		96
4.3	odd	2		152.6.b.b.75.16	yes	96
8.3	odd	2	inner	608.6.b.b.303.51		96
8.5	even	2		152.6.b.b.75.82	yes	96
19.18	odd	2	inner	608.6.b.b.303.46		96
76.75	even	2		152.6.b.b.75.81	yes	96
152.37	odd	2		152.6.b.b.75.15	✓	96
152.75	even	2	inner	608.6.b.b.303.45		96

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
152.6.b.b.75.15	✓	96	152.37	odd	2
152.6.b.b.75.16	yes	96	4.3	odd	2
152.6.b.b.75.81	yes	96	76.75	even	2
152.6.b.b.75.82	yes	96	8.5	even	2
608.6.b.b.303.45		96	152.75	even	2	inner
608.6.b.b.303.46		96	19.18	odd	2	inner
608.6.b.b.303.51		96	8.3	odd	2	inner
608.6.b.b.303.52		96	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+1.79162i q^{3} +26.9679i q^{5} +21.5599i q^{7} +239.790 q^{9} +O(q^{10})\)	\(q+1.79162i q^{3} +26.9679i q^{5} +21.5599i q^{7} +239.790 q^{9} +90.3444 q^{11} -419.826 q^{13} -48.3163 q^{15} -215.062 q^{17} +(-1355.34 - 799.470i) q^{19} -38.6273 q^{21} +3077.62i q^{23} +2397.73 q^{25} +864.978i q^{27} -6155.92 q^{29} -3059.38 q^{31} +161.863i q^{33} -581.426 q^{35} +9295.40 q^{37} -752.170i q^{39} +17772.2i q^{41} -7152.73 q^{43} +6466.64i q^{45} -23161.1i q^{47} +16342.2 q^{49} -385.311i q^{51} +11807.4 q^{53} +2436.40i q^{55} +(1432.35 - 2428.26i) q^{57} -42428.8i q^{59} +45648.8i q^{61} +5169.86i q^{63} -11321.8i q^{65} +33058.2i q^{67} -5513.94 q^{69} -29886.8 q^{71} -39654.0 q^{73} +4295.83i q^{75} +1947.82i q^{77} -62295.5 q^{79} +56719.3 q^{81} -107005. q^{83} -5799.78i q^{85} -11029.1i q^{87} -92408.3i q^{89} -9051.43i q^{91} -5481.25i q^{93} +(21560.0 - 36550.7i) q^{95} -76257.9i q^{97} +21663.7 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 96 q - 6168 q^{9}+O(q^{10}) \) 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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