LMFDB - Embedded newform 608.6.b.b.303.28

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			303.28
Character	$\chi$	$=$	608.303
Dual form			608.6.b.b.303.69

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-14.2121i q^{3} +9.24558i q^{5} +124.323i q^{7} +41.0170 q^{9} +O(q^{10})$	$q-14.2121i q^{3} +9.24558i q^{5} +124.323i q^{7} +41.0170 q^{9} -365.060 q^{11} -366.232 q^{13} +131.399 q^{15} +1513.21 q^{17} +(794.804 + 1358.08i) q^{19} +1766.89 q^{21} -2261.20i q^{23} +3039.52 q^{25} -4036.47i q^{27} -2188.28 q^{29} -7701.88 q^{31} +5188.26i q^{33} -1149.44 q^{35} -2009.12 q^{37} +5204.91i q^{39} +10836.3i q^{41} -6494.00 q^{43} +379.226i q^{45} +11509.4i q^{47} +1350.71 q^{49} -21505.8i q^{51} -31071.7 q^{53} -3375.19i q^{55} +(19301.2 - 11295.8i) q^{57} -34078.2i q^{59} -54487.2i q^{61} +5099.37i q^{63} -3386.03i q^{65} +40891.1i q^{67} -32136.3 q^{69} +3893.87 q^{71} -79369.5 q^{73} -43197.9i q^{75} -45385.5i q^{77} -18972.8 q^{79} -47399.5 q^{81} +77330.3 q^{83} +13990.5i q^{85} +31100.0i q^{87} +90614.0i q^{89} -45531.2i q^{91} +109460. i q^{93} +(-12556.2 + 7348.42i) q^{95} -41798.6i q^{97} -14973.7 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$		−	14.2121i		−	0.911705i	−0.890055	−	0.455852i	$-0.849334\pi$
$3$		−	14.2121i		−	0.911705i	0.890055	−	0.455852i	$-0.150666\pi$
$4$		0			0
$5$			9.24558i			0.165390i	0.996575	+	0.0826949i	$0.0263527\pi$
$5$			9.24558i			0.165390i	−0.996575	+	0.0826949i	$0.973647\pi$
$6$		0			0
$7$			124.323i			0.958975i	0.877549	+	0.479488i	$0.159178\pi$
$7$			124.323i			0.958975i	−0.877549	+	0.479488i	$0.840822\pi$
$8$		0			0
$9$	41.0170			0.168794
$10$		0			0
$11$	−365.060			−0.909667			−0.454834	−	0.890576i	$-0.650301\pi$
$11$	−365.060			−0.909667			−0.454834	+	0.890576i	$0.650301\pi$
$12$		0			0
$13$	−366.232			−0.601032			−0.300516	−	0.953777i	$-0.597159\pi$
$13$	−366.232			−0.601032			−0.300516	+	0.953777i	$0.597159\pi$
$14$		0			0
$15$	131.399			0.150787
$16$		0			0
$17$	1513.21			1.26992			0.634960	−	0.772545i	$-0.281017\pi$
$17$	1513.21			1.26992			0.634960	+	0.772545i	$0.281017\pi$
$18$		0			0
$19$	794.804	+	1358.08i	0.505098	+	0.863062i
$19$	794.804	+	1358.08i	0.505098	+	0.863062i
$20$		0			0
$21$	1766.89			0.874303
$22$		0			0
$23$		−	2261.20i		−	0.891290i	−0.895210	−	0.445645i	$-0.852974\pi$
$23$		−	2261.20i		−	0.891290i	0.895210	−	0.445645i	$-0.147026\pi$
$24$		0			0
$25$	3039.52			0.972646
$26$		0			0
$27$		−	4036.47i		−	1.06560i
$28$		0			0
$29$	−2188.28			−0.483178			−0.241589	−	0.970379i	$-0.577669\pi$
$29$	−2188.28			−0.483178			−0.241589	+	0.970379i	$0.577669\pi$
$30$		0			0
$31$	−7701.88			−1.43944			−0.719719	−	0.694266i	$-0.755729\pi$
$31$	−7701.88			−1.43944			−0.719719	+	0.694266i	$0.755729\pi$
$32$		0			0
$33$			5188.26i			0.829348i
$34$		0			0
$35$	−1149.44			−0.158605
$36$		0			0
$37$	−2009.12			−0.241269			−0.120635	−	0.992697i	$-0.538493\pi$
$37$	−2009.12			−0.241269			−0.120635	+	0.992697i	$0.538493\pi$
$38$		0			0
$39$			5204.91i			0.547964i
$40$		0			0
$41$			10836.3i			1.00675i	0.864068	+	0.503375i	$0.167909\pi$
$41$			10836.3i			1.00675i	−0.864068	+	0.503375i	$0.832091\pi$
$42$		0			0
$43$	−6494.00			−0.535600			−0.267800	−	0.963474i	$-0.586297\pi$
$43$	−6494.00			−0.535600			−0.267800	+	0.963474i	$0.586297\pi$
$44$		0			0
$45$			379.226i			0.0279169i
$46$		0			0
$47$			11509.4i			0.759993i	0.924988	+	0.379997i	$0.124075\pi$
$47$			11509.4i			0.759993i	−0.924988	+	0.379997i	$0.875925\pi$
$48$		0			0
$49$	1350.71			0.0803660
$50$		0			0
$51$		−	21505.8i		−	1.15779i
$52$		0			0
$53$	−31071.7			−1.51941			−0.759705	−	0.650268i	$-0.774657\pi$
$53$	−31071.7			−1.51941			−0.759705	+	0.650268i	$0.774657\pi$
$54$		0			0
$55$		−	3375.19i		−	0.150450i
$56$		0			0
$57$	19301.2	−	11295.8i	0.786858	−	0.460501i
$58$		0			0
$59$		−	34078.2i		−	1.27452i	−0.770648	−	0.637261i	$-0.780068\pi$
$59$		−	34078.2i		−	1.27452i	0.770648	−	0.637261i	$-0.219932\pi$
$60$		0			0
$61$		−	54487.2i		−	1.87487i	−0.348165	−	0.937433i	$-0.613195\pi$
$61$		−	54487.2i		−	1.87487i	0.348165	−	0.937433i	$-0.386805\pi$
$62$		0			0
$63$			5099.37i			0.161870i
$64$		0			0
$65$		−	3386.03i		−	0.0994047i
$66$		0			0
$67$			40891.1i			1.11286i	0.830894	+	0.556431i	$0.187830\pi$
$67$			40891.1i			1.11286i	−0.830894	+	0.556431i	$0.812170\pi$
$68$		0			0
$69$	−32136.3			−0.812593
$70$		0			0
$71$	3893.87			0.0916719			0.0458359	−	0.998949i	$-0.485405\pi$
$71$	3893.87			0.0916719			0.0458359	+	0.998949i	$0.485405\pi$
$72$		0			0
$73$	−79369.5			−1.74320			−0.871598	−	0.490221i	$-0.836916\pi$
$73$	−79369.5			−1.74320			−0.871598	+	0.490221i	$0.836916\pi$
$74$		0			0
$75$		−	43197.9i		−	0.886766i
$76$		0			0
$77$		−	45385.5i		−	0.872349i
$78$		0			0
$79$	−18972.8			−0.342030			−0.171015	−	0.985268i	$-0.554705\pi$
$79$	−18972.8			−0.342030			−0.171015	+	0.985268i	$0.554705\pi$
$80$		0			0
$81$	−47399.5			−0.802714
$82$		0			0
$83$	77330.3			1.23212			0.616062	−	0.787698i	$-0.288727\pi$
$83$	77330.3			1.23212			0.616062	+	0.787698i	$0.288727\pi$
$84$		0			0
$85$			13990.5i			0.210032i
$86$		0			0
$87$			31100.0i			0.440516i
$88$		0			0
$89$			90614.0i			1.21261i	0.795233	+	0.606304i	$0.207348\pi$
$89$			90614.0i			1.21261i	−0.795233	+	0.606304i	$0.792652\pi$
$90$		0			0
$91$		−	45531.2i		−	0.576375i
$92$		0			0
$93$			109460.i			1.31234i
$94$		0			0
$95$	−12556.2	+	7348.42i	−0.142742	+	0.0835382i
$96$		0			0
$97$		−	41798.6i		−	0.451058i	−0.974236	−	0.225529i	$-0.927589\pi$
$97$		−	41798.6i		−	0.451058i	0.974236	−	0.225529i	$-0.0724110\pi$
$98$		0			0
$99$	−14973.7			−0.153547
$100$		0			0
$101$			132278.i			1.29028i	0.764063	+	0.645141i	$0.223202\pi$
$101$			132278.i			1.29028i	−0.764063	+	0.645141i	$0.776798\pi$
$102$		0			0
$103$	−94980.3			−0.882146			−0.441073	−	0.897471i	$-0.645402\pi$
$103$	−94980.3			−0.882146			−0.441073	+	0.897471i	$0.645402\pi$
$104$		0			0
$105$			16335.9i			0.144601i
$106$		0			0
$107$		−	26503.0i		−	0.223787i	−0.993720	−	0.111894i	$-0.964308\pi$
$107$		−	26503.0i		−	0.223787i	0.993720	−	0.111894i	$-0.0356916\pi$
$108$		0			0
$109$	−189293.			−1.52605			−0.763026	−	0.646368i	$-0.776287\pi$
$109$	−189293.			−1.52605			−0.763026	+	0.646368i	$0.776287\pi$
$110$		0			0
$111$			28553.8i			0.219966i
$112$		0			0
$113$			2785.22i			0.0205194i	0.999947	+	0.0102597i	$0.00326582\pi$
$113$			2785.22i			0.0205194i	−0.999947	+	0.0102597i	$0.996734\pi$
$114$		0			0
$115$	20906.1			0.147410
$116$		0			0
$117$	−15021.7			−0.101451
$118$		0			0
$119$			188127.i			1.21782i
$120$		0			0
$121$	−27782.1			−0.172505
$122$		0			0
$123$	154006.			0.917859
$124$		0			0
$125$			56994.5i			0.326256i
$126$		0			0
$127$	256840.			1.41304			0.706518	−	0.707695i	$-0.250265\pi$
$127$	256840.			1.41304			0.706518	+	0.707695i	$0.250265\pi$
$128$		0			0
$129$			92293.2i			0.488310i
$130$		0			0
$131$	−243483.			−1.23963			−0.619814	−	0.784749i	$-0.712792\pi$
$131$	−243483.			−1.23963			−0.619814	+	0.784749i	$0.712792\pi$
$132$		0			0
$133$	−168841.	+	98812.7i	−0.827655	+	0.484377i
$134$		0			0
$135$	37319.5			0.176239
$136$		0			0
$137$	−179062.			−0.815082			−0.407541	−	0.913187i	$-0.633614\pi$
$137$	−179062.			−0.815082			−0.407541	+	0.913187i	$0.633614\pi$
$138$		0			0
$139$	199170.			0.874351			0.437176	−	0.899376i	$-0.355979\pi$
$139$	199170.			0.874351			0.437176	+	0.899376i	$0.355979\pi$
$140$		0			0
$141$	163573.			0.692890
$142$		0			0
$143$	133697.			0.546740
$144$		0			0
$145$		−	20231.9i		−	0.0799128i
$146$		0			0
$147$		−	19196.4i		−	0.0732701i
$148$		0			0
$149$			54501.3i			0.201114i	0.994931	+	0.100557i	$0.0320624\pi$
$149$			54501.3i			0.201114i	−0.994931	+	0.100557i	$0.967938\pi$
$150$		0			0
$151$	121160.			0.432432			0.216216	−	0.976346i	$-0.430628\pi$
$151$	121160.			0.432432			0.216216	+	0.976346i	$0.430628\pi$
$152$		0			0
$153$	62067.2			0.214355
$154$		0			0
$155$		−	71208.3i		−	0.238068i
$156$		0			0
$157$			471348.i			1.52613i	0.646320	+	0.763067i	$0.276307\pi$
$157$			471348.i			1.52613i	−0.646320	+	0.763067i	$0.723693\pi$
$158$		0			0
$159$			441593.i			1.38525i
$160$		0			0
$161$	281120.			0.854725
$162$		0			0
$163$	−502273.			−1.48071			−0.740356	−	0.672215i	$-0.765343\pi$
$163$	−502273.			−1.48071			−0.740356	+	0.672215i	$0.765343\pi$
$164$		0			0
$165$	−47968.5			−0.137166
$166$		0			0
$167$	−373307.			−1.03580			−0.517898	−	0.855442i	$-0.673285\pi$
$167$	−373307.			−1.03580			−0.517898	+	0.855442i	$0.673285\pi$
$168$		0			0
$169$	−237167.			−0.638760
$170$		0			0
$171$	32600.5	+	55704.4i	0.0852577	+	0.145680i
$172$		0			0
$173$	−365898.			−0.929491			−0.464745	−	0.885444i	$-0.653854\pi$
$173$	−365898.			−0.929491			−0.464745	+	0.885444i	$0.653854\pi$
$174$		0			0
$175$			377883.i			0.932744i
$176$		0			0
$177$	−484322.			−1.16199
$178$		0			0
$179$		−	156783.i		−	0.365735i	−0.983138	−	0.182867i	$-0.941462\pi$
$179$		−	156783.i		−	0.365735i	0.983138	−	0.182867i	$-0.0585379\pi$
$180$		0			0
$181$	391539.			0.888339			0.444169	−	0.895943i	$-0.353499\pi$
$181$	391539.			0.888339			0.444169	+	0.895943i	$0.353499\pi$
$182$		0			0
$183$	−774377.			−1.70932
$184$		0			0
$185$		−	18575.5i		−	0.0399035i
$186$		0			0
$187$	−552412.			−1.15520
$188$		0			0
$189$	501827.			1.02188
$190$		0			0
$191$			559199.i			1.10913i	0.832140	+	0.554566i	$0.187116\pi$
$191$			559199.i			1.10913i	−0.832140	+	0.554566i	$0.812884\pi$
$192$		0			0
$193$		−	36191.3i		−	0.0699377i	−0.999388	−	0.0349688i	$-0.988867\pi$
$193$		−	36191.3i		−	0.0699377i	0.999388	−	0.0349688i	$-0.0111332\pi$
$194$		0			0
$195$	−48122.4			−0.0906277
$196$		0			0
$197$			616594.i			1.13197i	0.824417	+	0.565983i	$0.191503\pi$
$197$			616594.i			1.13197i	−0.824417	+	0.565983i	$0.808497\pi$
$198$		0			0
$199$			155213.i			0.277840i	0.990304	+	0.138920i	$0.0443631\pi$
$199$			155213.i			0.277840i	−0.990304	+	0.138920i	$0.955637\pi$
$200$		0			0
$201$	581147.			1.01460
$202$		0			0
$203$		−	272054.i		−	0.463356i
$204$		0			0
$205$	−100188.			−0.166506
$206$		0			0
$207$		−	92747.5i		−	0.150445i
$208$		0			0
$209$	−290151.	−	495781.i	−0.459472	−	0.785099i
$210$		0			0
$211$			339665.i			0.525225i	0.964901	+	0.262612i	$0.0845840\pi$
$211$			339665.i			0.525225i	−0.964901	+	0.262612i	$0.915416\pi$
$212$		0			0
$213$		−	55340.0i		−	0.0835777i
$214$		0			0
$215$		−	60040.7i		−	0.0885829i
$216$		0			0
$217$		−	957524.i		−	1.38038i
$218$		0			0
$219$			1.12800e6i			1.58928i
$220$		0			0
$221$	−554185.			−0.763263
$222$		0			0
$223$	−1.08038e6			−1.45483			−0.727415	−	0.686198i	$-0.759279\pi$
$223$	−1.08038e6			−1.45483			−0.727415	+	0.686198i	$0.759279\pi$
$224$		0			0
$225$	124672.			0.164177
$226$		0			0
$227$			105829.i			0.136314i	0.997675	+	0.0681569i	$0.0217119\pi$
$227$			105829.i			0.136314i	−0.997675	+	0.0681569i	$0.978288\pi$
$228$		0			0
$229$			285269.i			0.359473i	0.983715	+	0.179736i	$0.0575245\pi$
$229$			285269.i			0.359473i	−0.983715	+	0.179736i	$0.942475\pi$
$230$		0			0
$231$	−645022.			−0.795325
$232$		0			0
$233$	981311.			1.18418			0.592089	−	0.805873i	$-0.298303\pi$
$233$	981311.			1.18418			0.592089	+	0.805873i	$0.298303\pi$
$234$		0			0
$235$	−106411.			−0.125695
$236$		0			0
$237$			269643.i			0.311831i
$238$		0			0
$239$			1.21758e6i			1.37881i	0.724377	+	0.689404i	$0.242127\pi$
$239$			1.21758e6i			1.37881i	−0.724377	+	0.689404i	$0.757873\pi$
$240$		0			0
$241$			696863.i			0.772867i	0.922317	+	0.386434i	$0.126293\pi$
$241$			696863.i			0.772867i	−0.922317	+	0.386434i	$0.873707\pi$
$242$		0			0
$243$		−	307217.i		−	0.333757i
$244$		0			0
$245$			12488.1i			0.0132917i
$246$		0			0
$247$	−291083.	−	497373.i	−0.303580	−	0.518728i
$248$		0			0
$249$		−	1.09902e6i		−	1.12333i
$250$		0			0
$251$	−567572.			−0.568639			−0.284319	−	0.958730i	$-0.591768\pi$
$251$	−567572.			−0.568639			−0.284319	+	0.958730i	$0.591768\pi$
$252$		0			0
$253$			825473.i			0.810777i
$254$		0			0
$255$	198834.			0.191487
$256$		0			0
$257$			172685.i			0.163088i	0.996670	+	0.0815441i	$0.0259851\pi$
$257$			172685.i			0.163088i	−0.996670	+	0.0815441i	$0.974015\pi$
$258$		0			0
$259$		−	249781.i		−	0.231371i
$260$		0			0
$261$	−89756.6			−0.0815577
$262$		0			0
$263$			678009.i			0.604430i	0.953240	+	0.302215i	$0.0977261\pi$
$263$			678009.i			0.604430i	−0.953240	+	0.302215i	$0.902274\pi$
$264$		0			0
$265$		−	287276.i		−	0.251295i
$266$		0			0
$267$	1.28781e6			1.10554
$268$		0			0
$269$	−1.02017e6			−0.859591			−0.429796	−	0.902926i	$-0.641414\pi$
$269$	−1.02017e6			−0.859591			−0.429796	+	0.902926i	$0.641414\pi$
$270$		0			0
$271$		−	2.30195e6i		−	1.90403i	−0.306055	−	0.952014i	$-0.599009\pi$
$271$		−	2.30195e6i		−	1.90403i	0.306055	−	0.952014i	$-0.400991\pi$
$272$		0			0
$273$	−647092.			−0.525484
$274$		0			0
$275$	−1.10961e6			−0.884785
$276$		0			0
$277$		−	608040.i		−	0.476138i	−0.971248	−	0.238069i	$-0.923486\pi$
$277$		−	608040.i		−	0.476138i	0.971248	−	0.238069i	$-0.0765145\pi$
$278$		0			0
$279$	−315908.			−0.242969
$280$		0			0
$281$		−	49508.2i		−	0.0374034i	−0.999825	−	0.0187017i	$-0.994047\pi$
$281$		−	49508.2i		−	0.0374034i	0.999825	−	0.0187017i	$-0.00595328\pi$
$282$		0			0
$283$	300557.			0.223080			0.111540	−	0.993760i	$-0.464422\pi$
$283$	300557.			0.223080			0.111540	+	0.993760i	$0.464422\pi$
$284$		0			0
$285$	104436.	+	178450.i	0.0761621	+	0.130138i
$286$		0			0
$287$	−1.34721e6			−0.965449
$288$		0			0
$289$	869939.			0.612695
$290$		0			0
$291$	−594044.			−0.411232
$292$		0			0
$293$	−1.81073e6			−1.23221			−0.616104	−	0.787664i	$-0.711290\pi$
$293$	−1.81073e6			−1.23221			−0.616104	+	0.787664i	$0.711290\pi$
$294$		0			0
$295$	315073.			0.210793
$296$		0			0
$297$			1.47355e6i			0.969337i
$298$		0			0
$299$			828123.i			0.535694i
$300$		0			0
$301$		−	807355.i		−	0.513628i
$302$		0			0
$303$	1.87995e6			1.17636
$304$		0			0
$305$	503766.			0.310084
$306$		0			0
$307$		−	2.34882e6i		−	1.42234i	−0.703018	−	0.711172i	$-0.748165\pi$
$307$		−	2.34882e6i		−	1.42234i	0.703018	−	0.711172i	$-0.251835\pi$
$308$		0			0
$309$			1.34987e6i			0.804256i
$310$		0			0
$311$		−	3.13408e6i		−	1.83743i	−0.394926	−	0.918713i	$-0.629230\pi$
$311$		−	3.13408e6i		−	1.83743i	0.394926	−	0.918713i	$-0.370770\pi$
$312$		0			0
$313$	−2.28638e6			−1.31913			−0.659565	−	0.751648i	$-0.729259\pi$
$313$	−2.28638e6			−1.31913			−0.659565	+	0.751648i	$0.729259\pi$
$314$		0			0
$315$	−47146.6			−0.0267716
$316$		0			0
$317$	96347.3			0.0538507			0.0269254	−	0.999637i	$-0.491428\pi$
$317$	96347.3			0.0538507			0.0269254	+	0.999637i	$0.491428\pi$
$318$		0			0
$319$	798853.			0.439532
$320$		0			0
$321$	−376662.			−0.204028
$322$		0			0
$323$	1.20270e6	+	2.05506e6i	0.641434	+	1.09602i
$324$		0			0
$325$	−1.11317e6			−0.584592
$326$		0			0
$327$			2.69025e6i			1.39131i
$328$		0			0
$329$	−1.43089e6			−0.728815
$330$		0			0
$331$		−	601438.i		−	0.301731i	−0.988554	−	0.150866i	$-0.951794\pi$
$331$		−	601438.i		−	0.301731i	0.988554	−	0.150866i	$-0.0482061\pi$
$332$		0			0
$333$	−82408.1			−0.0407249
$334$		0			0
$335$	−378062.			−0.184056
$336$		0			0
$337$		−	3.82139e6i		−	1.83294i	−0.400109	−	0.916468i	$-0.631028\pi$
$337$		−	3.82139e6i		−	1.83294i	0.400109	−	0.916468i	$-0.368972\pi$
$338$		0			0
$339$	39583.8			0.0187076
$340$		0			0
$341$	2.81165e6			1.30941
$342$		0			0
$343$			2.25743e6i			1.03604i
$344$		0			0
$345$		−	297119.i		−	0.134395i
$346$		0			0
$347$	1.55563e6			0.693556			0.346778	−	0.937947i	$-0.387276\pi$
$347$	1.55563e6			0.693556			0.346778	+	0.937947i	$0.387276\pi$
$348$		0			0
$349$			1.01085e6i			0.444246i	0.975019	+	0.222123i	$0.0712987\pi$
$349$			1.01085e6i			0.444246i	−0.975019	+	0.222123i	$0.928701\pi$
$350$		0			0
$351$			1.47828e6i			0.640457i
$352$		0			0
$353$	1.24723e6			0.532731			0.266366	−	0.963872i	$-0.414177\pi$
$353$	1.24723e6			0.532731			0.266366	+	0.963872i	$0.414177\pi$
$354$		0			0
$355$			36001.1i			0.0151616i
$356$		0			0
$357$	2.67367e6			1.11029
$358$		0			0
$359$		−	1.07100e6i		−	0.438584i	−0.975659	−	0.219292i	$-0.929625\pi$
$359$		−	1.07100e6i		−	0.438584i	0.975659	−	0.219292i	$-0.0703747\pi$
$360$		0			0
$361$	−1.21267e6	+	2.15882e6i	−0.489751	+	0.871862i
$362$		0			0
$363$			394842.i			0.157274i
$364$		0			0
$365$		−	733816.i		−	0.288307i
$366$		0			0
$367$		−	2.89160e6i		−	1.12066i	−0.828270	−	0.560329i	$-0.810675\pi$
$367$		−	2.89160e6i		−	1.12066i	0.828270	−	0.560329i	$-0.189325\pi$
$368$		0			0
$369$			444473.i			0.169934i
$370$		0			0
$371$		−	3.86294e6i		−	1.45708i
$372$		0			0
$373$	−1.47047e6			−0.547247			−0.273624	−	0.961837i	$-0.588222\pi$
$373$	−1.47047e6			−0.547247			−0.273624	+	0.961837i	$0.588222\pi$
$374$		0			0
$375$	810010.			0.297449
$376$		0			0
$377$	801417.			0.290406
$378$		0			0
$379$			1.89549e6i			0.677833i	0.940816	+	0.338917i	$0.110060\pi$
$379$			1.89549e6i			0.677833i	−0.940816	+	0.338917i	$0.889940\pi$
$380$		0			0
$381$		−	3.65023e6i		−	1.28827i
$382$		0			0
$383$	3.50444e6			1.22073			0.610367	−	0.792119i	$-0.291022\pi$
$383$	3.50444e6			1.22073			0.610367	+	0.792119i	$0.291022\pi$
$384$		0			0
$385$	419615.			0.144278
$386$		0			0
$387$	−266364.			−0.0904063
$388$		0			0
$389$		−	4.31747e6i		−	1.44662i	−0.690523	−	0.723311i	$-0.742619\pi$
$389$		−	4.31747e6i		−	1.44662i	0.690523	−	0.723311i	$-0.257381\pi$
$390$		0			0
$391$		−	3.42166e6i		−	1.13187i
$392$		0			0
$393$			3.46041e6i			1.13017i
$394$		0			0
$395$		−	175415.i		−	0.0565684i
$396$		0			0
$397$			391751.i			0.124748i	0.998053	+	0.0623740i	$0.0198671\pi$
$397$			391751.i			0.124748i	−0.998053	+	0.0623740i	$0.980133\pi$
$398$		0			0
$399$	1.40433e6	+	2.39958e6i	0.441609	+	0.754577i
$400$		0			0
$401$			5.15441e6i			1.60073i	0.599514	+	0.800364i	$0.295361\pi$
$401$			5.15441e6i			1.60073i	−0.599514	+	0.800364i	$0.704639\pi$
$402$		0			0
$403$	2.82068e6			0.865148
$404$		0			0
$405$		−	438235.i		−	0.132761i
$406$		0			0
$407$	733450.			0.219475
$408$		0			0
$409$			4.89993e6i			1.44838i	0.689602	+	0.724189i	$0.257786\pi$
$409$			4.89993e6i			1.44838i	−0.689602	+	0.724189i	$0.742214\pi$
$410$		0			0
$411$			2.54484e6i			0.743114i
$412$		0			0
$413$	4.23672e6			1.22223
$414$		0			0
$415$			714963.i			0.203781i
$416$		0			0
$417$		−	2.83061e6i		−	0.797150i
$418$		0			0
$419$	−4.28386e6			−1.19207			−0.596033	−	0.802960i	$-0.703257\pi$
$419$	−4.28386e6			−1.19207			−0.596033	+	0.802960i	$0.703257\pi$
$420$		0			0
$421$	2.52522e6			0.694375			0.347188	−	0.937796i	$-0.387137\pi$
$421$	2.52522e6			0.694375			0.347188	+	0.937796i	$0.387137\pi$
$422$		0			0
$423$			472083.i			0.128282i
$424$		0			0
$425$	4.59942e6			1.23518
$426$		0			0
$427$	6.77404e6			1.79795
$428$		0			0
$429$		−	1.90011e6i		−	0.498465i
$430$		0			0
$431$	−1.42046e6			−0.368328			−0.184164	−	0.982896i	$-0.558958\pi$
$431$	−1.42046e6			−0.368328			−0.184164	+	0.982896i	$0.558958\pi$
$432$		0			0
$433$			178506.i			0.0457543i	0.999738	+	0.0228772i	$0.00728266\pi$
$433$			178506.i			0.0457543i	−0.999738	+	0.0228772i	$0.992717\pi$
$434$		0			0
$435$	−287537.			−0.0728569
$436$		0			0
$437$	3.07089e6	−	1.79721e6i	0.769238	−	0.450189i
$438$		0			0
$439$	6.49147e6			1.60761			0.803807	−	0.594891i	$-0.202805\pi$
$439$	6.49147e6			1.60761			0.803807	+	0.594891i	$0.202805\pi$
$440$		0			0
$441$	55402.1			0.0135653
$442$		0			0
$443$	1.04834e6			0.253800			0.126900	−	0.991915i	$-0.459497\pi$
$443$	1.04834e6			0.253800			0.126900	+	0.991915i	$0.459497\pi$
$444$		0			0
$445$	−837779.			−0.200553
$446$		0			0
$447$	774577.			0.183356
$448$		0			0
$449$		−	1.14659e6i		−	0.268407i	−0.990954	−	0.134203i	$-0.957152\pi$
$449$		−	1.14659e6i		−	0.268407i	0.990954	−	0.134203i	$-0.0428476\pi$
$450$		0			0
$451$		−	3.95590e6i		−	0.915808i
$452$		0			0
$453$		−	1.72194e6i		−	0.394251i
$454$		0			0
$455$	420962.			0.0953266
$456$		0			0
$457$	−3.64223e6			−0.815786			−0.407893	−	0.913030i	$-0.633736\pi$
$457$	−3.64223e6			−0.815786			−0.407893	+	0.913030i	$0.633736\pi$
$458$		0			0
$459$		−	6.10802e6i		−	1.35322i
$460$		0			0
$461$		−	2.78881e6i		−	0.611177i	−0.952164	−	0.305589i	$-0.901147\pi$
$461$		−	2.78881e6i		−	0.611177i	0.952164	−	0.305589i	$-0.0988533\pi$
$462$		0			0
$463$			1.15763e6i			0.250968i	0.992096	+	0.125484i	$0.0400483\pi$
$463$			1.15763e6i			0.250968i	−0.992096	+	0.125484i	$0.959952\pi$
$464$		0			0
$465$	−1.01202e6			−0.217048
$466$		0			0
$467$	−5.45656e6			−1.15778			−0.578891	−	0.815405i	$-0.696514\pi$
$467$	−5.45656e6			−1.15778			−0.578891	+	0.815405i	$0.696514\pi$
$468$		0			0
$469$	−5.08371e6			−1.06721
$470$		0			0
$471$	6.69883e6			1.39138
$472$		0			0
$473$	2.37070e6			0.487218
$474$		0			0
$475$	2.41582e6	+	4.12792e6i	0.491282	+	0.839454i
$476$		0			0
$477$	−1.27447e6			−0.256468
$478$		0			0
$479$		−	4.48505e6i		−	0.893157i	−0.894744	−	0.446579i	$-0.852642\pi$
$479$		−	4.48505e6i		−	0.893157i	0.894744	−	0.446579i	$-0.147358\pi$
$480$		0			0
$481$	735805.			0.145011
$482$		0			0
$483$		−	3.99529e6i		−	0.779257i
$484$		0			0
$485$	386452.			0.0746004
$486$		0			0
$487$	−3.10015e6			−0.592326			−0.296163	−	0.955137i	$-0.595707\pi$
$487$	−3.10015e6			−0.592326			−0.296163	+	0.955137i	$0.595707\pi$
$488$		0			0
$489$			7.13834e6i			1.34997i
$490$		0			0
$491$	−1.43332e6			−0.268312			−0.134156	−	0.990960i	$-0.542832\pi$
$491$	−1.43332e6			−0.268312			−0.134156	+	0.990960i	$0.542832\pi$
$492$		0			0
$493$	−3.31132e6			−0.613597
$494$		0			0
$495$		−	138440.i		−	0.0253951i
$496$		0			0
$497$			484099.i			0.0879111i
$498$		0			0
$499$	−3.39259e6			−0.609930			−0.304965	−	0.952364i	$-0.598645\pi$
$499$	−3.39259e6			−0.609930			−0.304965	+	0.952364i	$0.598645\pi$
$500$		0			0
$501$			5.30546e6i			0.944341i
$502$		0			0
$503$		−	5.68066e6i		−	1.00110i	−0.865707	−	0.500552i	$-0.833131\pi$
$503$		−	5.68066e6i		−	1.00110i	0.865707	−	0.500552i	$-0.166869\pi$
$504$		0			0
$505$	−1.22299e6			−0.213400
$506$		0			0
$507$			3.37064e6i			0.582361i
$508$		0			0
$509$	3.22322e6			0.551436			0.275718	−	0.961239i	$-0.411084\pi$
$509$	3.22322e6			0.551436			0.275718	+	0.961239i	$0.411084\pi$
$510$		0			0
$511$		−	9.86747e6i		−	1.67168i
$512$		0			0
$513$	5.48186e6	−	3.20820e6i	0.919675	−	0.538230i
$514$		0			0
$515$		−	878147.i		−	0.145898i
$516$		0			0
$517$		−	4.20164e6i		−	0.691341i
$518$		0			0
$519$			5.20017e6i			0.847421i
$520$		0			0
$521$		−	1.11994e7i		−	1.80759i	−0.427962	−	0.903797i	$-0.640768\pi$
$521$		−	1.11994e7i		−	1.80759i	0.427962	−	0.903797i	$-0.359232\pi$
$522$		0			0
$523$			2.39158e6i			0.382323i	0.981559	+	0.191161i	$0.0612254\pi$
$523$			2.39158e6i			0.382323i	−0.981559	+	0.191161i	$0.938775\pi$
$524$		0			0
$525$	5.37050e6			0.850387
$526$		0			0
$527$	−1.16545e7			−1.82797
$528$		0			0
$529$	1.32333e6			0.205603
$530$		0			0
$531$		−	1.39779e6i		−	0.215132i
$532$		0			0
$533$		−	3.96860e6i		−	0.605089i
$534$		0			0
$535$	245035.			0.0370121
$536$		0			0
$537$	−2.22821e6			−0.333442
$538$		0			0
$539$	−493091.			−0.0731063
$540$		0			0
$541$		−	5.07981e6i		−	0.746198i	−0.927792	−	0.373099i	$-0.878295\pi$
$541$		−	5.07981e6i		−	0.746198i	0.927792	−	0.373099i	$-0.121705\pi$
$542$		0			0
$543$		−	5.56458e6i		−	0.809903i
$544$		0			0
$545$		−	1.75013e6i		−	0.252394i
$546$		0			0
$547$			4.98971e6i			0.713028i	0.934290	+	0.356514i	$0.116035\pi$
$547$			4.98971e6i			0.713028i	−0.934290	+	0.356514i	$0.883965\pi$
$548$		0			0
$549$		−	2.23490e6i		−	0.316467i
$550$		0			0
$551$	−1.73925e6	−	2.97186e6i	−0.244053	−	0.417013i
$552$		0			0
$553$		−	2.35877e6i		−	0.327999i
$554$		0			0
$555$	−263996.			−0.0363802
$556$		0			0
$557$			4.63835e6i			0.633469i	0.948514	+	0.316735i	$0.102586\pi$
$557$			4.63835e6i			0.633469i	−0.948514	+	0.316735i	$0.897414\pi$
$558$		0			0
$559$	2.37831e6			0.321913
$560$		0			0
$561$			7.85091e6i			1.05321i
$562$		0			0
$563$			1.49382e7i			1.98622i	0.117175	+	0.993111i	$0.462616\pi$
$563$			1.49382e7i			1.98622i	−0.117175	+	0.993111i	$0.537384\pi$
$564$		0			0
$565$	−25751.0			−0.00339370
$566$		0			0
$567$		−	5.89286e6i		−	0.769783i
$568$		0			0
$569$		−	1.32859e7i		−	1.72033i	−0.510017	−	0.860164i	$-0.670361\pi$
$569$		−	1.32859e7i		−	1.72033i	0.510017	−	0.860164i	$-0.329639\pi$
$570$		0			0
$571$	9.39535e6			1.20593			0.602966	−	0.797767i	$-0.293986\pi$
$571$	9.39535e6			1.20593			0.602966	+	0.797767i	$0.293986\pi$
$572$		0			0
$573$	7.94738e6			1.01120
$574$		0			0
$575$		−	6.87295e6i		−	0.866909i
$576$		0			0
$577$	−837200.			−0.104686			−0.0523431	−	0.998629i	$-0.516669\pi$
$577$	−837200.			−0.104686			−0.0523431	+	0.998629i	$0.516669\pi$
$578$		0			0
$579$	−514354.			−0.0637625
$580$		0			0
$581$			9.61395e6i			1.18158i
$582$		0			0
$583$	1.13430e7			1.38216
$584$		0			0
$585$		−	138885.i		−	0.0167789i
$586$		0			0
$587$	3.77948e6			0.452727			0.226364	−	0.974043i	$-0.427316\pi$
$587$	3.77948e6			0.452727			0.226364	+	0.974043i	$0.427316\pi$
$588$		0			0
$589$	−6.12149e6	−	1.04598e7i	−0.727057	−	1.24232i
$590$		0			0
$591$	8.76308e6			1.03202
$592$		0			0
$593$	−5.96912e6			−0.697065			−0.348532	−	0.937297i	$-0.613320\pi$
$593$	−5.96912e6			−0.697065			−0.348532	+	0.937297i	$0.613320\pi$
$594$		0			0
$595$	−1.73934e6			−0.201415
$596$		0			0
$597$	2.20589e6			0.253308
$598$		0			0
$599$	−1.72710e7			−1.96675			−0.983376	−	0.181581i	$-0.941879\pi$
$599$	−1.72710e7			−1.96675			−0.983376	+	0.181581i	$0.941879\pi$
$600$		0			0
$601$		−	3.08241e6i		−	0.348100i	−0.984737	−	0.174050i	$-0.944315\pi$
$601$		−	3.08241e6i		−	0.348100i	0.984737	−	0.174050i	$-0.0556854\pi$
$602$		0			0
$603$			1.67723e6i			0.187845i
$604$		0			0
$605$		−	256862.i		−	0.0285306i
$606$		0			0
$607$	−1.07721e7			−1.18667			−0.593334	−	0.804957i	$-0.702189\pi$
$607$	−1.07721e7			−1.18667			−0.593334	+	0.804957i	$0.702189\pi$
$608$		0			0
$609$	−3.86645e6			−0.422444
$610$		0			0
$611$		−	4.21513e6i		−	0.456781i
$612$		0			0
$613$			3.17483e6i			0.341248i	0.985336	+	0.170624i	$0.0545783\pi$
$613$			3.17483e6i			0.341248i	−0.985336	+	0.170624i	$0.945422\pi$
$614$		0			0
$615$			1.42388e6i			0.151805i
$616$		0			0
$617$	3.57680e6			0.378253			0.189126	−	0.981953i	$-0.439434\pi$
$617$	3.57680e6			0.378253			0.189126	+	0.981953i	$0.439434\pi$
$618$		0			0
$619$	1.81270e7			1.90151			0.950757	−	0.309937i	$-0.100308\pi$
$619$	1.81270e7			1.90151			0.950757	+	0.309937i	$0.100308\pi$
$620$		0			0
$621$	−9.12726e6			−0.949754
$622$		0			0
$623$	−1.12654e7			−1.16286
$624$		0			0
$625$	8.97155e6			0.918687
$626$		0			0
$627$	−7.04608e6	+	4.12365e6i	−0.715779	+	0.418902i
$628$		0			0
$629$	−3.04022e6			−0.306392
$630$		0			0
$631$			5.80712e6i			0.580614i	0.956934	+	0.290307i	$0.0937574\pi$
$631$			5.80712e6i			0.580614i	−0.956934	+	0.290307i	$0.906243\pi$
$632$		0			0
$633$	4.82735e6			0.478850
$634$		0			0
$635$			2.37463e6i			0.233702i
$636$		0			0
$637$	−494674.			−0.0483026
$638$		0			0
$639$	159715.			0.0154737
$640$		0			0
$641$		−	1.04768e7i		−	1.00712i	−0.863960	−	0.503560i	$-0.832023\pi$
$641$		−	1.04768e7i		−	1.00712i	0.863960	−	0.503560i	$-0.167977\pi$
$642$		0			0
$643$	3.09642e6			0.295347			0.147674	−	0.989036i	$-0.452821\pi$
$643$	3.09642e6			0.295347			0.147674	+	0.989036i	$0.452821\pi$
$644$		0			0
$645$	−853303.			−0.0807615
$646$		0			0
$647$			6.31465e6i			0.593046i	0.955026	+	0.296523i	$0.0958271\pi$
$647$			6.31465e6i			0.593046i	−0.955026	+	0.296523i	$0.904173\pi$
$648$		0			0
$649$			1.24406e7i			1.15939i
$650$		0			0
$651$	−1.36084e7			−1.25850
$652$		0			0
$653$			9.24440e6i			0.848390i	0.905571	+	0.424195i	$0.139443\pi$
$653$			9.24440e6i			0.848390i	−0.905571	+	0.424195i	$0.860557\pi$
$654$		0			0
$655$		−	2.25115e6i		−	0.205022i
$656$		0			0
$657$	−3.25550e6			−0.294241
$658$		0			0
$659$		−	2.03128e6i		−	0.182203i	−0.995842	−	0.0911015i	$-0.970961\pi$
$659$		−	2.03128e6i		−	0.182203i	0.995842	−	0.0911015i	$-0.0290388\pi$
$660$		0			0
$661$	7.84907e6			0.698738			0.349369	−	0.936985i	$-0.386396\pi$
$661$	7.84907e6			0.698738			0.349369	+	0.936985i	$0.386396\pi$
$662$		0			0
$663$			7.87612e6i			0.695870i
$664$		0			0
$665$	−913580.	−	1.56103e6i	−0.0801110	−	0.136886i
$666$		0			0
$667$			4.94813e6i			0.430652i
$668$		0			0
$669$			1.53544e7i			1.32638i
$670$		0			0
$671$			1.98911e7i			1.70550i
$672$		0			0
$673$		−	1.23851e7i		−	1.05405i	−0.849850	−	0.527025i	$-0.823307\pi$
$673$		−	1.23851e7i		−	1.05405i	0.849850	−	0.527025i	$-0.176693\pi$
$674$		0			0
$675$		−	1.22689e7i		−	1.03645i
$676$		0			0
$677$	−4.80364e6			−0.402808			−0.201404	−	0.979508i	$-0.564550\pi$
$677$	−4.80364e6			−0.402808			−0.201404	+	0.979508i	$0.564550\pi$
$678$		0			0
$679$	5.19654e6			0.432553
$680$		0			0
$681$	1.50405e6			0.124278
$682$		0			0
$683$			1.17375e7i			0.962769i	0.876510	+	0.481384i	$0.159866\pi$
$683$			1.17375e7i			0.962769i	−0.876510	+	0.481384i	$0.840134\pi$
$684$		0			0
$685$		−	1.65553e6i		−	0.134806i
$686$		0			0
$687$	4.05427e6			0.327733
$688$		0			0
$689$	1.13794e7			0.913215
$690$		0			0
$691$	−1.58528e7			−1.26302			−0.631512	−	0.775366i	$-0.717565\pi$
$691$	−1.58528e7			−1.26302			−0.631512	+	0.775366i	$0.717565\pi$
$692$		0			0
$693$		−	1.86158e6i		−	0.147247i
$694$		0			0
$695$			1.84144e6i			0.144609i
$696$		0			0
$697$			1.63976e7i			1.27849i
$698$		0			0
$699$		−	1.39465e7i		−	1.07962i
$700$		0			0
$701$			6.17197e6i			0.474382i	0.971463	+	0.237191i	$0.0762268\pi$
$701$			6.17197e6i			0.474382i	−0.971463	+	0.237191i	$0.923773\pi$
$702$		0			0
$703$	−1.59686e6	−	2.72855e6i	−0.121865	−	0.208230i
$704$		0			0
$705$			1.51233e6i			0.114597i
$706$		0			0
$707$	−1.64453e7			−1.23735
$708$		0			0
$709$		−	1.38638e7i		−	1.03577i	−0.855449	−	0.517887i	$-0.826719\pi$
$709$		−	1.38638e7i		−	1.03577i	0.855449	−	0.517887i	$-0.173281\pi$
$710$		0			0
$711$	−778209.			−0.0577327
$712$		0			0
$713$			1.74155e7i			1.28296i
$714$		0			0
$715$			1.23610e6i			0.0904252i
$716$		0			0
$717$	1.73044e7			1.25707
$718$		0			0
$719$			1.32215e7i			0.953804i	0.878956	+	0.476902i	$0.158240\pi$
$719$			1.32215e7i			0.953804i	−0.878956	+	0.476902i	$0.841760\pi$
$720$		0			0
$721$		−	1.18083e7i		−	0.845956i
$722$		0			0
$723$	9.90387e6			0.704627
$724$		0			0
$725$	−6.65131e6			−0.469962
$726$		0			0
$727$		−	1.08625e7i		−	0.762242i	−0.924525	−	0.381121i	$-0.875538\pi$
$727$		−	1.08625e7i		−	0.762242i	0.924525	−	0.381121i	$-0.124462\pi$
$728$		0			0
$729$	−1.58843e7			−1.10700
$730$		0			0
$731$	−9.82676e6			−0.680169
$732$		0			0
$733$			5.85557e6i			0.402540i	0.979536	+	0.201270i	$0.0645069\pi$
$733$			5.85557e6i			0.402540i	−0.979536	+	0.201270i	$0.935493\pi$
$734$		0			0
$735$	177482.			0.0121181
$736$		0			0
$737$		−	1.49277e7i		−	1.01233i
$738$		0			0
$739$	−1.67168e7			−1.12601			−0.563005	−	0.826453i	$-0.690355\pi$
$739$	−1.67168e7			−1.12601			−0.563005	+	0.826453i	$0.690355\pi$
$740$		0			0
$741$	−7.06870e6	+	4.13689e6i	−0.472927	+	0.276776i
$742$		0			0
$743$	5.07962e6			0.337566			0.168783	−	0.985653i	$-0.446016\pi$
$743$	5.07962e6			0.337566			0.168783	+	0.985653i	$0.446016\pi$
$744$		0			0
$745$	−503896.			−0.0332621
$746$		0			0
$747$	3.17185e6			0.207975
$748$		0			0
$749$	3.29494e6			0.214606
$750$		0			0
$751$	6.62925e6			0.428908			0.214454	−	0.976734i	$-0.431203\pi$
$751$	6.62925e6			0.428908			0.214454	+	0.976734i	$0.431203\pi$
$752$		0			0
$753$			8.06637e6i			0.518431i
$754$		0			0
$755$			1.12020e6i			0.0715199i
$756$		0			0
$757$			1.78443e7i			1.13177i	0.824483	+	0.565887i	$0.191466\pi$
$757$			1.78443e7i			1.13177i	−0.824483	+	0.565887i	$0.808534\pi$
$758$		0			0
$759$	1.17317e7			0.739190
$760$		0			0
$761$	−5.27818e6			−0.330387			−0.165193	−	0.986261i	$-0.552825\pi$
$761$	−5.27818e6			−0.330387			−0.165193	+	0.986261i	$0.552825\pi$
$762$		0			0
$763$		−	2.35336e7i		−	1.46345i
$764$		0			0
$765$			573847.i			0.0354522i
$766$		0			0
$767$			1.24805e7i			0.766028i
$768$		0			0
$769$	1.13899e7			0.694549			0.347275	−	0.937763i	$-0.387107\pi$
$769$	1.13899e7			0.694549			0.347275	+	0.937763i	$0.387107\pi$
$770$		0			0
$771$	2.45422e6			0.148688
$772$		0			0
$773$	2.00043e6			0.120414			0.0602068	−	0.998186i	$-0.480824\pi$
$773$	2.00043e6			0.120414			0.0602068	+	0.998186i	$0.480824\pi$
$774$		0			0
$775$	−2.34100e7			−1.40006
$776$		0			0
$777$	−3.54990e6			−0.210942
$778$		0			0
$779$	−1.47166e7	+	8.61274e6i	−0.868888	+	0.508508i
$780$		0			0
$781$	−1.42150e6			−0.0833909
$782$		0			0
$783$			8.83292e6i			0.514873i
$784$		0			0
$785$	−4.35788e6			−0.252407
$786$		0			0
$787$		−	6.23690e6i		−	0.358948i	−0.983763	−	0.179474i	$-0.942560\pi$
$787$		−	6.23690e6i		−	0.358948i	0.983763	−	0.179474i	$-0.0574397\pi$
$788$		0			0
$789$	9.63591e6			0.551062
$790$		0			0
$791$	−346268.			−0.0196776
$792$		0			0
$793$			1.99550e7i			1.12686i
$794$		0			0
$795$	−4.08278e6			−0.229107
$796$		0			0
$797$	−9.80871e6			−0.546973			−0.273487	−	0.961876i	$-0.588177\pi$
$797$	−9.80871e6			−0.546973			−0.273487	+	0.961876i	$0.588177\pi$
$798$		0			0
$799$			1.74162e7i			0.965130i
$800$		0			0
$801$			3.71671e6i			0.204681i
$802$		0			0
$803$	2.89746e7			1.58573
$804$		0			0
$805$			2.59911e6i			0.141363i
$806$		0			0
$807$			1.44987e7i			0.783693i
$808$		0			0
$809$	−1.57251e7			−0.844738			−0.422369	−	0.906424i	$-0.638801\pi$
$809$	−1.57251e7			−0.844738			−0.422369	+	0.906424i	$0.638801\pi$
$810$		0			0
$811$		−	6.29338e6i		−	0.335994i	−0.985788	−	0.167997i	$-0.946270\pi$
$811$		−	6.29338e6i		−	0.335994i	0.985788	−	0.167997i	$-0.0537299\pi$
$812$		0			0
$813$	−3.27155e7			−1.73591
$814$		0			0
$815$		−	4.64380e6i		−	0.244895i
$816$		0			0
$817$	−5.16145e6	−	8.81938e6i	−0.270531	−	0.462256i
$818$		0			0
$819$		−	1.86755e6i		−	0.0972888i
$820$		0			0
$821$			2.12117e7i			1.09829i	0.835726	+	0.549146i	$0.185047\pi$
$821$			2.12117e7i			1.09829i	−0.835726	+	0.549146i	$0.814953\pi$
$822$		0			0
$823$			1.52175e7i			0.783146i	0.920147	+	0.391573i	$0.128069\pi$
$823$			1.52175e7i			0.783146i	−0.920147	+	0.391573i	$0.871931\pi$
$824$		0			0
$825$			1.57698e7i			0.806662i
$826$		0			0
$827$		−	4.76448e6i		−	0.242243i	−0.992638	−	0.121122i	$-0.961351\pi$
$827$		−	4.76448e6i		−	0.242243i	0.992638	−	0.121122i	$-0.0386491\pi$
$828$		0			0
$829$	2.38161e7			1.20361			0.601804	−	0.798644i	$-0.294449\pi$
$829$	2.38161e7			1.20361			0.601804	+	0.798644i	$0.294449\pi$
$830$		0			0
$831$	−8.64151e6			−0.434097
$832$		0			0
$833$	2.04391e6			0.102058
$834$		0			0
$835$		−	3.45144e6i		−	0.171310i
$836$		0			0
$837$			3.10884e7i			1.53386i
$838$		0			0
$839$	−1.66134e7			−0.814805			−0.407402	−	0.913249i	$-0.633565\pi$
$839$	−1.66134e7			−0.814805			−0.407402	+	0.913249i	$0.633565\pi$
$840$		0			0
$841$	−1.57226e7			−0.766539
$842$		0			0
$843$	−703614.			−0.0341009
$844$		0			0
$845$		−	2.19275e6i		−	0.105644i
$846$		0			0
$847$		−	3.45397e6i		−	0.165428i
$848$		0			0
$849$		−	4.27153e6i		−	0.203383i
$850$		0			0
$851$			4.54302e6i			0.215041i
$852$		0			0
$853$		−	3.11169e7i		−	1.46428i	−0.681155	−	0.732139i	$-0.738522\pi$
$853$		−	3.11169e7i		−	1.46428i	0.681155	−	0.732139i	$-0.261478\pi$
$854$		0			0
$855$	−515019.	+	301410.i	−0.0240940	+	0.0141008i
$856$		0			0
$857$		−	2.33524e7i		−	1.08612i	−0.839692	−	0.543062i	$-0.817265\pi$
$857$		−	2.33524e7i		−	1.08612i	0.839692	−	0.543062i	$-0.182735\pi$
$858$		0			0
$859$	8.80191e6			0.407000			0.203500	−	0.979075i	$-0.434768\pi$
$859$	8.80191e6			0.407000			0.203500	+	0.979075i	$0.434768\pi$
$860$		0			0
$861$			1.91466e7i			0.880204i
$862$		0			0
$863$	4.16374e7			1.90308			0.951539	−	0.307530i	$-0.0995023\pi$
$863$	4.16374e7			1.90308			0.951539	+	0.307530i	$0.0995023\pi$
$864$		0			0
$865$		−	3.38294e6i		−	0.153728i
$866$		0			0
$867$		−	1.23636e7i		−	0.558597i
$868$		0			0
$869$	6.92623e6			0.311134
$870$		0			0
$871$		−	1.49756e7i		−	0.668866i
$872$		0			0
$873$		−	1.71445e6i		−	0.0761359i
$874$		0			0
$875$	−7.08575e6			−0.312871
$876$		0			0
$877$	−2.62664e7			−1.15319			−0.576597	−	0.817029i	$-0.695620\pi$
$877$	−2.62664e7			−1.15319			−0.576597	+	0.817029i	$0.695620\pi$
$878$		0			0
$879$			2.57342e7i			1.12341i
$880$		0			0
$881$	3.51136e6			0.152418			0.0762088	−	0.997092i	$-0.475718\pi$
$881$	3.51136e6			0.152418			0.0762088	+	0.997092i	$0.475718\pi$
$882$		0			0
$883$	3.99212e7			1.72307			0.861533	−	0.507702i	$-0.169505\pi$
$883$	3.99212e7			1.72307			0.861533	+	0.507702i	$0.169505\pi$
$884$		0			0
$885$		−	4.47784e6i		−	0.192181i
$886$		0			0
$887$	−1.51954e7			−0.648492			−0.324246	−	0.945973i	$-0.605111\pi$
$887$	−1.51954e7			−0.648492			−0.324246	+	0.945973i	$0.605111\pi$
$888$		0			0
$889$			3.19312e7i			1.35507i
$890$		0			0
$891$	1.73037e7			0.730203
$892$		0			0
$893$	−1.56308e7	+	9.14775e6i	−0.655921	+	0.383871i
$894$		0			0
$895$	1.44955e6			0.0604888
$896$		0			0
$897$	1.17693e7			0.488395
$898$		0			0
$899$	1.68539e7			0.695505
$900$		0			0
$901$	−4.70179e7			−1.92953
$902$		0			0
$903$	−1.14742e7			−0.468277
$904$		0			0
$905$			3.62000e6i			0.146922i
$906$		0			0
$907$			1.67286e6i			0.0675215i	0.999430	+	0.0337607i	$0.0107484\pi$
$907$			1.67286e6i			0.0675215i	−0.999430	+	0.0337607i	$0.989252\pi$
$908$		0			0
$909$			5.42565e6i			0.217792i
$910$		0			0
$911$	4.59483e6			0.183431			0.0917156	−	0.995785i	$-0.470765\pi$
$911$	4.59483e6			0.183431			0.0917156	+	0.995785i	$0.470765\pi$
$912$		0			0
$913$	−2.82302e7			−1.12082
$914$		0			0
$915$		−	7.15956e6i		−	0.282705i
$916$		0			0
$917$		−	3.02707e7i		−	1.18877i
$918$		0			0
$919$			3.90980e7i			1.52709i	0.645752	+	0.763547i	$0.276544\pi$
$919$			3.90980e7i			1.52709i	−0.645752	+	0.763547i	$0.723456\pi$
$920$		0			0
$921$	−3.33817e7			−1.29676
$922$		0			0
$923$	−1.42606e6			−0.0550977
$924$		0			0
$925$	−6.10676e6			−0.234670
$926$		0			0
$927$	−3.89580e6			−0.148901
$928$		0			0
$929$	1.09135e7			0.414881			0.207440	−	0.978248i	$-0.433487\pi$
$929$	1.09135e7			0.414881			0.207440	+	0.978248i	$0.433487\pi$
$930$		0			0
$931$	1.07355e6	+	1.83438e6i	0.0405927	+	0.0693608i
$932$		0			0
$933$	−4.45418e7			−1.67519
$934$		0			0
$935$		−	5.10736e6i		−	0.191059i
$936$		0			0
$937$	−2.77150e7			−1.03126			−0.515628	−	0.856813i	$-0.672441\pi$
$937$	−2.77150e7			−1.03126			−0.515628	+	0.856813i	$0.672441\pi$
$938$		0			0
$939$			3.24942e7i			1.20266i
$940$		0			0
$941$	−5.89140e6			−0.216893			−0.108446	−	0.994102i	$-0.534588\pi$
$941$	−5.89140e6			−0.216893			−0.108446	+	0.994102i	$0.534588\pi$
$942$		0			0
$943$	2.45030e7			0.897306
$944$		0			0
$945$			4.63968e6i			0.169009i
$946$		0			0
$947$	1.18149e6			0.0428111			0.0214055	−	0.999771i	$-0.493186\pi$
$947$	1.18149e6			0.0428111			0.0214055	+	0.999771i	$0.493186\pi$
$948$		0			0
$949$	2.90676e7			1.04772
$950$		0			0
$951$		−	1.36929e6i		−	0.0490960i
$952$		0			0
$953$			1.68084e7i			0.599508i	0.954017	+	0.299754i	$0.0969046\pi$
$953$			1.68084e7i			0.599508i	−0.954017	+	0.299754i	$0.903095\pi$
$954$		0			0
$955$	−5.17012e6			−0.183439
$956$		0			0
$957$		−	1.13534e7i		−	0.400723i
$958$		0			0
$959$		−	2.22616e7i		−	0.781644i
$960$		0			0
$961$	3.06898e7			1.07198
$962$		0			0
$963$		−	1.08707e6i		−	0.0377740i
$964$		0			0
$965$	334610.			0.0115670
$966$		0			0
$967$		−	4.47806e7i		−	1.54001i	−0.638037	−	0.770006i	$-0.720253\pi$
$967$		−	4.47806e7i		−	1.54001i	0.638037	−	0.770006i	$-0.279747\pi$
$968$		0			0
$969$	2.92066e7	−	1.70929e7i	0.999246	−	0.584799i
$970$		0			0
$971$			4.12614e7i			1.40441i	0.711972	+	0.702207i	$0.247802\pi$
$971$			4.12614e7i			1.40441i	−0.711972	+	0.702207i	$0.752198\pi$
$972$		0			0
$973$			2.47614e7i			0.838481i
$974$		0			0
$975$			1.58204e7i			0.532975i
$976$		0			0
$977$		−	5.25883e7i		−	1.76260i	−0.472560	−	0.881299i	$-0.656670\pi$
$977$		−	5.25883e7i		−	1.76260i	0.472560	−	0.881299i	$-0.343330\pi$
$978$		0			0
$979$		−	3.30796e7i		−	1.10307i
$980$		0			0
$981$	−7.76425e6			−0.257589
$982$		0			0
$983$	−2.31566e7			−0.764347			−0.382173	−	0.924091i	$-0.624824\pi$
$983$	−2.31566e7			−0.764347			−0.382173	+	0.924091i	$0.624824\pi$
$984$		0			0
$985$	−5.70076e6			−0.187216
$986$		0			0
$987$			2.03360e7i			0.664464i
$988$		0			0
$989$			1.46842e7i			0.477375i
$990$		0			0
$991$	−2.99104e7			−0.967472			−0.483736	−	0.875214i	$-0.660721\pi$
$991$	−2.99104e7			−0.967472			−0.483736	+	0.875214i	$0.660721\pi$
$992$		0			0
$993$	−8.54768e6			−0.275090
$994$		0			0
$995$	−1.43503e6			−0.0459519
$996$		0			0
$997$		−	5.73919e7i		−	1.82858i	−0.405065	−	0.914288i	$-0.632751\pi$
$997$		−	5.73919e7i		−	1.82858i	0.405065	−	0.914288i	$-0.367249\pi$
$998$		0			0
$999$			8.10976e6i			0.257095i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	608.6.b.b.303.28		96
4.3	odd	2		152.6.b.b.75.85	yes	96
8.3	odd	2	inner	608.6.b.b.303.27		96
8.5	even	2		152.6.b.b.75.11	✓	96
19.18	odd	2	inner	608.6.b.b.303.70		96
76.75	even	2		152.6.b.b.75.12	yes	96
152.37	odd	2		152.6.b.b.75.86	yes	96
152.75	even	2	inner	608.6.b.b.303.69		96

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
152.6.b.b.75.11	✓	96	8.5	even	2
152.6.b.b.75.12	yes	96	76.75	even	2
152.6.b.b.75.85	yes	96	4.3	odd	2
152.6.b.b.75.86	yes	96	152.37	odd	2
608.6.b.b.303.27		96	8.3	odd	2	inner
608.6.b.b.303.28		96	1.1	even	1	trivial
608.6.b.b.303.69		96	152.75	even	2	inner
608.6.b.b.303.70		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)

\(f(q)\)	\(=\)	\(q-14.2121i q^{3} +9.24558i q^{5} +124.323i q^{7} +41.0170 q^{9} +O(q^{10})\)	\(q-14.2121i q^{3} +9.24558i q^{5} +124.323i q^{7} +41.0170 q^{9} -365.060 q^{11} -366.232 q^{13} +131.399 q^{15} +1513.21 q^{17} +(794.804 + 1358.08i) q^{19} +1766.89 q^{21} -2261.20i q^{23} +3039.52 q^{25} -4036.47i q^{27} -2188.28 q^{29} -7701.88 q^{31} +5188.26i q^{33} -1149.44 q^{35} -2009.12 q^{37} +5204.91i q^{39} +10836.3i q^{41} -6494.00 q^{43} +379.226i q^{45} +11509.4i q^{47} +1350.71 q^{49} -21505.8i q^{51} -31071.7 q^{53} -3375.19i q^{55} +(19301.2 - 11295.8i) q^{57} -34078.2i q^{59} -54487.2i q^{61} +5099.37i q^{63} -3386.03i q^{65} +40891.1i q^{67} -32136.3 q^{69} +3893.87 q^{71} -79369.5 q^{73} -43197.9i q^{75} -45385.5i q^{77} -18972.8 q^{79} -47399.5 q^{81} +77330.3 q^{83} +13990.5i q^{85} +31100.0i q^{87} +90614.0i q^{89} -45531.2i q^{91} +109460. i q^{93} +(-12556.2 + 7348.42i) q^{95} -41798.6i q^{97} -14973.7 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 96 q - 6168 q^{9}+O(q^{10}) \) 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

Related objects

Downloads

Learn more