LMFDB - Embedded newform 608.6.b.b.303.19

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			303.19
Character	$\chi$	$=$	608.303
Dual form			608.6.b.b.303.78

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-19.2136i q^{3} -61.7747i q^{5} +34.9784i q^{7} -126.164 q^{9} +O(q^{10})$	$q-19.2136i q^{3} -61.7747i q^{5} +34.9784i q^{7} -126.164 q^{9} +733.529 q^{11} -336.837 q^{13} -1186.92 q^{15} +1573.27 q^{17} +(-1233.74 + 976.721i) q^{19} +672.062 q^{21} -3337.62i q^{23} -691.109 q^{25} -2244.84i q^{27} +7405.66 q^{29} +3413.12 q^{31} -14093.8i q^{33} +2160.78 q^{35} +1830.03 q^{37} +6471.87i q^{39} +17828.4i q^{41} +5440.96 q^{43} +7793.74i q^{45} -16954.5i q^{47} +15583.5 q^{49} -30228.2i q^{51} +21506.1 q^{53} -45313.5i q^{55} +(18766.4 + 23704.6i) q^{57} +19540.8i q^{59} -34163.7i q^{61} -4413.01i q^{63} +20808.0i q^{65} -54614.1i q^{67} -64127.8 q^{69} +31860.4 q^{71} +42303.5 q^{73} +13278.7i q^{75} +25657.7i q^{77} +12971.4 q^{79} -73789.5 q^{81} -23806.4 q^{83} -97187.9i q^{85} -142290. i q^{87} +29525.6i q^{89} -11782.0i q^{91} -65578.5i q^{93} +(60336.6 + 76213.9i) q^{95} +59242.8i q^{97} -92544.9 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$		−	19.2136i		−	1.23256i	−0.787529	−	0.616278i	$-0.788640\pi$
$3$		−	19.2136i		−	1.23256i	0.787529	−	0.616278i	$-0.211360\pi$
$4$		0			0
$5$		−	61.7747i		−	1.10506i	−0.833493	−	0.552529i	$-0.813663\pi$
$5$		−	61.7747i		−	1.10506i	0.833493	−	0.552529i	$-0.186337\pi$
$6$		0			0
$7$			34.9784i			0.269808i	0.990859	+	0.134904i	$0.0430726\pi$
$7$			34.9784i			0.269808i	−0.990859	+	0.134904i	$0.956927\pi$
$8$		0			0
$9$	−126.164			−0.519193
$10$		0			0
$11$	733.529			1.82783			0.913914	−	0.405907i	$-0.133044\pi$
$11$	733.529			1.82783			0.913914	+	0.405907i	$0.133044\pi$
$12$		0			0
$13$	−336.837			−0.552792			−0.276396	−	0.961044i	$-0.589140\pi$
$13$	−336.837			−0.552792			−0.276396	+	0.961044i	$0.589140\pi$
$14$		0			0
$15$	−1186.92			−1.36205
$16$		0			0
$17$	1573.27			1.32032			0.660161	−	0.751124i	$-0.270488\pi$
$17$	1573.27			1.32032			0.660161	+	0.751124i	$0.270488\pi$
$18$		0			0
$19$	−1233.74	+	976.721i	−0.784043	+	0.620707i
$19$	−1233.74	+	976.721i	−0.784043	+	0.620707i
$20$		0			0
$21$	672.062			0.332553
$22$		0			0
$23$		−	3337.62i		−	1.31558i	−0.753202	−	0.657789i	$-0.771492\pi$
$23$		−	3337.62i		−	1.31558i	0.753202	−	0.657789i	$-0.228508\pi$
$24$		0			0
$25$	−691.109			−0.221155
$26$		0			0
$27$		−	2244.84i		−	0.592621i
$28$		0			0
$29$	7405.66			1.63519			0.817596	−	0.575793i	$-0.195306\pi$
$29$	7405.66			1.63519			0.817596	+	0.575793i	$0.195306\pi$
$30$		0			0
$31$	3413.12			0.637892			0.318946	−	0.947773i	$-0.396671\pi$
$31$	3413.12			0.637892			0.318946	+	0.947773i	$0.396671\pi$
$32$		0			0
$33$		−	14093.8i		−	2.25290i
$34$		0			0
$35$	2160.78			0.298154
$36$		0			0
$37$	1830.03			0.219763			0.109881	−	0.993945i	$-0.464953\pi$
$37$	1830.03			0.219763			0.109881	+	0.993945i	$0.464953\pi$
$38$		0			0
$39$			6471.87i			0.681347i
$40$		0			0
$41$			17828.4i			1.65635i	0.560468	+	0.828176i	$0.310621\pi$
$41$			17828.4i			1.65635i	−0.560468	+	0.828176i	$0.689379\pi$
$42$		0			0
$43$	5440.96			0.448750			0.224375	−	0.974503i	$-0.427966\pi$
$43$	5440.96			0.448750			0.224375	+	0.974503i	$0.427966\pi$
$44$		0			0
$45$			7793.74i			0.573739i
$46$		0			0
$47$		−	16954.5i		−	1.11954i	−0.828648	−	0.559771i	$-0.810889\pi$
$47$		−	16954.5i		−	1.11954i	0.828648	−	0.559771i	$-0.189111\pi$
$48$		0			0
$49$	15583.5			0.927204
$50$		0			0
$51$		−	30228.2i		−	1.62737i
$52$		0			0
$53$	21506.1			1.05165			0.525826	−	0.850592i	$-0.323756\pi$
$53$	21506.1			1.05165			0.525826	+	0.850592i	$0.323756\pi$
$54$		0			0
$55$		−	45313.5i		−	2.01986i
$56$		0			0
$57$	18766.4	+	23704.6i	0.765056	+	0.966376i
$58$		0			0
$59$			19540.8i			0.730823i	0.930846	+	0.365411i	$0.119072\pi$
$59$			19540.8i			0.730823i	−0.930846	+	0.365411i	$0.880928\pi$
$60$		0			0
$61$		−	34163.7i		−	1.17555i	−0.809026	−	0.587773i	$-0.800005\pi$
$61$		−	34163.7i		−	1.17555i	0.809026	−	0.587773i	$-0.199995\pi$
$62$		0			0
$63$		−	4413.01i		−	0.140082i
$64$		0			0
$65$			20808.0i			0.610868i
$66$		0			0
$67$		−	54614.1i		−	1.48634i	−0.669104	−	0.743169i	$-0.733322\pi$
$67$		−	54614.1i		−	1.48634i	0.669104	−	0.743169i	$-0.266678\pi$
$68$		0			0
$69$	−64127.8			−1.62152
$70$		0			0
$71$	31860.4			0.750077			0.375038	−	0.927009i	$-0.377629\pi$
$71$	31860.4			0.750077			0.375038	+	0.927009i	$0.377629\pi$
$72$		0			0
$73$	42303.5			0.929114			0.464557	−	0.885543i	$-0.346214\pi$
$73$	42303.5			0.929114			0.464557	+	0.885543i	$0.346214\pi$
$74$		0			0
$75$			13278.7i			0.272586i
$76$		0			0
$77$			25657.7i			0.493163i
$78$		0			0
$79$	12971.4			0.233840			0.116920	−	0.993141i	$-0.462698\pi$
$79$	12971.4			0.233840			0.116920	+	0.993141i	$0.462698\pi$
$80$		0			0
$81$	−73789.5			−1.24963
$82$		0			0
$83$	−23806.4			−0.379313			−0.189657	−	0.981851i	$-0.560737\pi$
$83$	−23806.4			−0.379313			−0.189657	+	0.981851i	$0.560737\pi$
$84$		0			0
$85$		−	97187.9i		−	1.45903i
$86$		0			0
$87$		−	142290.i		−	2.01546i
$88$		0			0
$89$			29525.6i			0.395116i	0.980291	+	0.197558i	$0.0633010\pi$
$89$			29525.6i			0.395116i	−0.980291	+	0.197558i	$0.936699\pi$
$90$		0			0
$91$		−	11782.0i		−	0.149148i
$92$		0			0
$93$		−	65578.5i		−	0.786238i
$94$		0			0
$95$	60336.6	+	76213.9i	0.685918	+	0.866413i
$96$		0			0
$97$			59242.8i			0.639302i	0.947535	+	0.319651i	$0.103566\pi$
$97$			59242.8i			0.639302i	−0.947535	+	0.319651i	$0.896434\pi$
$98$		0			0
$99$	−92544.9			−0.948996
$100$		0			0
$101$			145430.i			1.41857i	0.704920	+	0.709287i	$0.250983\pi$
$101$			145430.i			1.41857i	−0.704920	+	0.709287i	$0.749017\pi$
$102$		0			0
$103$	−122437.			−1.13715			−0.568577	−	0.822630i	$-0.692506\pi$
$103$	−122437.			−1.13715			−0.568577	+	0.822630i	$0.692506\pi$
$104$		0			0
$105$		−	41516.4i		−	0.367491i
$106$		0			0
$107$			106826.i			0.902020i	0.892519	+	0.451010i	$0.148936\pi$
$107$			106826.i			0.902020i	−0.892519	+	0.451010i	$0.851064\pi$
$108$		0			0
$109$	−215623.			−1.73832			−0.869159	−	0.494533i	$-0.835339\pi$
$109$	−215623.			−1.73832			−0.869159	+	0.494533i	$0.835339\pi$
$110$		0			0
$111$		−	35161.5i		−	0.270870i
$112$		0			0
$113$			51915.9i			0.382476i	0.981544	+	0.191238i	$0.0612503\pi$
$113$			51915.9i			0.382476i	−0.981544	+	0.191238i	$0.938750\pi$
$114$		0			0
$115$	−206180.			−1.45379
$116$		0			0
$117$	42496.7			0.287006
$118$		0			0
$119$			55030.3i			0.356233i
$120$		0			0
$121$	377013.			2.34096
$122$		0			0
$123$	342548.			2.04155
$124$		0			0
$125$		−	150353.i		−	0.860670i
$126$		0			0
$127$	51306.2			0.282267			0.141134	−	0.989991i	$-0.454925\pi$
$127$	51306.2			0.282267			0.141134	+	0.989991i	$0.454925\pi$
$128$		0			0
$129$		−	104541.i		−	0.553110i
$130$		0			0
$131$	−210419.			−1.07129			−0.535644	−	0.844444i	$-0.679931\pi$
$131$	−210419.			−1.07129			−0.535644	+	0.844444i	$0.679931\pi$
$132$		0			0
$133$	−34164.1	−	43154.2i	−0.167472	−	0.211541i
$134$		0			0
$135$	−138675.			−0.654881
$136$		0			0
$137$	135419.			0.616420			0.308210	−	0.951318i	$-0.400270\pi$
$137$	135419.			0.616420			0.308210	+	0.951318i	$0.400270\pi$
$138$		0			0
$139$	−272611.			−1.19676			−0.598378	−	0.801214i	$-0.704188\pi$
$139$	−272611.			−1.19676			−0.598378	+	0.801214i	$0.704188\pi$
$140$		0			0
$141$	−325757.			−1.37990
$142$		0			0
$143$	−247080.			−1.01041
$144$		0			0
$145$		−	457482.i		−	1.80698i
$146$		0			0
$147$		−	299416.i		−	1.14283i
$148$		0			0
$149$			402331.i			1.48463i	0.670053	+	0.742313i	$0.266271\pi$
$149$			402331.i			1.48463i	−0.670053	+	0.742313i	$0.733729\pi$
$150$		0			0
$151$	−511632.			−1.82606			−0.913031	−	0.407889i	$-0.866265\pi$
$151$	−511632.			−1.82606			−0.913031	+	0.407889i	$0.866265\pi$
$152$		0			0
$153$	−198489.			−0.685502
$154$		0			0
$155$		−	210844.i		−	0.704908i
$156$		0			0
$157$		−	233033.i		−	0.754517i	−0.926108	−	0.377259i	$-0.876867\pi$
$157$		−	233033.i		−	0.754517i	0.926108	−	0.377259i	$-0.123133\pi$
$158$		0			0
$159$		−	413211.i		−	1.29622i
$160$		0			0
$161$	116744.			0.354954
$162$		0			0
$163$	148258.			0.437069			0.218534	−	0.975829i	$-0.429872\pi$
$163$	148258.			0.437069			0.218534	+	0.975829i	$0.429872\pi$
$164$		0			0
$165$	−870637.			−2.48959
$166$		0			0
$167$	256424.			0.711487			0.355744	−	0.934584i	$-0.384228\pi$
$167$	256424.			0.711487			0.355744	+	0.934584i	$0.384228\pi$
$168$		0			0
$169$	−257834.			−0.694421
$170$		0			0
$171$	155654.	−	123227.i	0.407070	−	0.322267i
$172$		0			0
$173$	58026.6			0.147405			0.0737024	−	0.997280i	$-0.476518\pi$
$173$	58026.6			0.147405			0.0737024	+	0.997280i	$0.476518\pi$
$174$		0			0
$175$		−	24173.9i		−	0.0596693i
$176$		0			0
$177$	375450.			0.900780
$178$		0			0
$179$		−	169553.i		−	0.395524i	−0.980250	−	0.197762i	$-0.936633\pi$
$179$		−	169553.i		−	0.395524i	0.980250	−	0.197762i	$-0.0633674\pi$
$180$		0			0
$181$	−120996.			−0.274520			−0.137260	−	0.990535i	$-0.543830\pi$
$181$	−120996.			−0.274520			−0.137260	+	0.990535i	$0.543830\pi$
$182$		0			0
$183$	−656409.			−1.44893
$184$		0			0
$185$		−	113049.i		−	0.242851i
$186$		0			0
$187$	1.15404e6			2.41332
$188$		0			0
$189$	78521.1			0.159894
$190$		0			0
$191$		−	556615.i		−	1.10401i	−0.833842	−	0.552003i	$-0.813864\pi$
$191$		−	556615.i		−	1.10401i	0.833842	−	0.552003i	$-0.186136\pi$
$192$		0			0
$193$		−	236209.i		−	0.456461i	−0.973607	−	0.228231i	$-0.926706\pi$
$193$		−	236209.i		−	0.456461i	0.973607	−	0.228231i	$-0.0732940\pi$
$194$		0			0
$195$	399798.			0.752928
$196$		0			0
$197$		−	114983.i		−	0.211091i	−0.994414	−	0.105545i	$-0.966341\pi$
$197$		−	114983.i		−	0.211091i	0.994414	−	0.105545i	$-0.0336589\pi$
$198$		0			0
$199$			855759.i			1.53186i	0.642924	+	0.765930i	$0.277721\pi$
$199$			855759.i			1.53186i	−0.642924	+	0.765930i	$0.722279\pi$
$200$		0			0
$201$	−1.04933e6			−1.83199
$202$		0			0
$203$			259038.i			0.441188i
$204$		0			0
$205$	1.10134e6			1.83037
$206$		0			0
$207$			421087.i			0.683040i
$208$		0			0
$209$	−904984.	+	716453.i	−1.43310	+	1.13455i
$210$		0			0
$211$			173469.i			0.268235i	0.990965	+	0.134117i	$0.0428200\pi$
$211$			173469.i			0.268235i	−0.990965	+	0.134117i	$0.957180\pi$
$212$		0			0
$213$		−	612155.i		−	0.924512i
$214$		0			0
$215$		−	336114.i		−	0.495895i
$216$		0			0
$217$			119385.i			0.172108i
$218$		0			0
$219$		−	812804.i		−	1.14518i
$220$		0			0
$221$	−529934.			−0.729863
$222$		0			0
$223$	−1.37021e6			−1.84513			−0.922563	−	0.385846i	$-0.873910\pi$
$223$	−1.37021e6			−1.84513			−0.922563	+	0.385846i	$0.873910\pi$
$224$		0			0
$225$	87193.0			0.114822
$226$		0			0
$227$			826653.i			1.06478i	0.846500	+	0.532388i	$0.178705\pi$
$227$			826653.i			1.06478i	−0.846500	+	0.532388i	$0.821295\pi$
$228$		0			0
$229$		−	569289.i		−	0.717371i	−0.933458	−	0.358686i	$-0.883225\pi$
$229$		−	569289.i		−	0.717371i	0.933458	−	0.358686i	$-0.116775\pi$
$230$		0			0
$231$	492977.			0.607850
$232$		0			0
$233$	700677.			0.845528			0.422764	−	0.906240i	$-0.361060\pi$
$233$	700677.			0.845528			0.422764	+	0.906240i	$0.361060\pi$
$234$		0			0
$235$	−1.04736e6			−1.23716
$236$		0			0
$237$		−	249227.i		−	0.288220i
$238$		0			0
$239$		−	231957.i		−	0.262672i	−0.991338	−	0.131336i	$-0.958073\pi$
$239$		−	231957.i		−	0.262672i	0.991338	−	0.131336i	$-0.0419266\pi$
$240$		0			0
$241$		−	22964.3i		−	0.0254690i	−0.999919	−	0.0127345i	$-0.995946\pi$
$241$		−	22964.3i		−	0.0254690i	0.999919	−	0.0127345i	$-0.00405362\pi$
$242$		0			0
$243$			872268.i			0.947620i
$244$		0			0
$245$		−	962666.i		−	1.02461i
$246$		0			0
$247$	415570.	−	328996.i	0.433412	−	0.343122i
$248$		0			0
$249$			457407.i			0.467524i
$250$		0			0
$251$	674545.			0.675813			0.337907	−	0.941180i	$-0.390281\pi$
$251$	674545.			0.675813			0.337907	+	0.941180i	$0.390281\pi$
$252$		0			0
$253$		−	2.44824e6i		−	2.40465i
$254$		0			0
$255$	−1.86733e6			−1.79834
$256$		0			0
$257$		−	449738.i		−	0.424743i	−0.977189	−	0.212372i	$-0.931881\pi$
$257$		−	449738.i		−	0.424743i	0.977189	−	0.212372i	$-0.0681187\pi$
$258$		0			0
$259$			64011.5i			0.0592937i
$260$		0			0
$261$	−934327.			−0.848981
$262$		0			0
$263$		−	2.05651e6i		−	1.83333i	−0.399652	−	0.916667i	$-0.630869\pi$
$263$		−	2.05651e6i		−	1.83333i	0.399652	−	0.916667i	$-0.369131\pi$
$264$		0			0
$265$		−	1.32853e6i		−	1.16214i
$266$		0			0
$267$	567295.			0.487002
$268$		0			0
$269$	−180961.			−0.152477			−0.0762384	−	0.997090i	$-0.524291\pi$
$269$	−180961.			−0.152477			−0.0762384	+	0.997090i	$0.524291\pi$
$270$		0			0
$271$		−	1.31587e6i		−	1.08840i	−0.838956	−	0.544200i	$-0.816833\pi$
$271$		−	1.31587e6i		−	1.08840i	0.838956	−	0.544200i	$-0.183167\pi$
$272$		0			0
$273$	−226376.			−0.183833
$274$		0			0
$275$	−506948.			−0.404233
$276$		0			0
$277$			1.81937e6i			1.42469i	0.701828	+	0.712347i	$0.252368\pi$
$277$			1.81937e6i			1.42469i	−0.701828	+	0.712347i	$0.747632\pi$
$278$		0			0
$279$	−430613.			−0.331189
$280$		0			0
$281$		−	33089.6i		−	0.0249991i	−0.999922	−	0.0124996i	$-0.996021\pi$
$281$		−	33089.6i		−	0.0249991i	0.999922	−	0.0124996i	$-0.00397884\pi$
$282$		0			0
$283$	−182534.			−0.135481			−0.0677403	−	0.997703i	$-0.521579\pi$
$283$	−182534.			−0.135481			−0.0677403	+	0.997703i	$0.521579\pi$
$284$		0			0
$285$	1.46435e6	−	1.15929e6i	1.06790	−	0.845432i
$286$		0			0
$287$	−623609.			−0.446897
$288$		0			0
$289$	1.05531e6			0.743249
$290$		0			0
$291$	1.13827e6			0.787975
$292$		0			0
$293$	−1.77453e6			−1.20757			−0.603787	−	0.797146i	$-0.706342\pi$
$293$	−1.77453e6			−1.20757			−0.603787	+	0.797146i	$0.706342\pi$
$294$		0			0
$295$	1.20713e6			0.807602
$296$		0			0
$297$		−	1.64666e6i		−	1.08321i
$298$		0			0
$299$			1.12423e6i			0.727241i
$300$		0			0
$301$			190316.i			0.121076i
$302$		0			0
$303$	2.79425e6			1.74847
$304$		0			0
$305$	−2.11045e6			−1.29905
$306$		0			0
$307$			1.86468e6i			1.12917i	0.825375	+	0.564584i	$0.190963\pi$
$307$			1.86468e6i			1.12917i	−0.825375	+	0.564584i	$0.809037\pi$
$308$		0			0
$309$			2.35246e6i			1.40161i
$310$		0			0
$311$		−	757288.i		−	0.443976i	−0.975049	−	0.221988i	$-0.928745\pi$
$311$		−	757288.i		−	0.443976i	0.975049	−	0.221988i	$-0.0712547\pi$
$312$		0			0
$313$	2.30626e6			1.33060			0.665300	−	0.746576i	$-0.268304\pi$
$313$	2.30626e6			1.33060			0.665300	+	0.746576i	$0.268304\pi$
$314$		0			0
$315$	−272612.			−0.154799
$316$		0			0
$317$	924112.			0.516507			0.258254	−	0.966077i	$-0.416853\pi$
$317$	924112.			0.516507			0.258254	+	0.966077i	$0.416853\pi$
$318$		0			0
$319$	5.43226e6			2.98885
$320$		0			0
$321$	2.05251e6			1.11179
$322$		0			0
$323$	−1.94100e6	+	1.53664e6i	−1.03519	+	0.819533i
$324$		0			0
$325$	232791.			0.122253
$326$		0			0
$327$			4.14291e6i			2.14257i
$328$		0			0
$329$	593041.			0.302061
$330$		0			0
$331$		−	3.02450e6i		−	1.51734i	−0.651474	−	0.758671i	$-0.725849\pi$
$331$		−	3.02450e6i		−	1.51734i	0.651474	−	0.758671i	$-0.274151\pi$
$332$		0			0
$333$	−230884.			−0.114099
$334$		0			0
$335$	−3.37376e6			−1.64249
$336$		0			0
$337$		−	559982.i		−	0.268596i	−0.990941	−	0.134298i	$-0.957122\pi$
$337$		−	559982.i		−	0.268596i	0.990941	−	0.134298i	$-0.0428779\pi$
$338$		0			0
$339$	997494.			0.471423
$340$		0			0
$341$	2.50362e6			1.16596
$342$		0			0
$343$			1.13297e6i			0.519975i
$344$		0			0
$345$			3.96147e6i			1.79188i
$346$		0			0
$347$	−793967.			−0.353980			−0.176990	−	0.984213i	$-0.556636\pi$
$347$	−793967.			−0.353980			−0.176990	+	0.984213i	$0.556636\pi$
$348$		0			0
$349$			2.30316e6i			1.01219i	0.862478	+	0.506094i	$0.168911\pi$
$349$			2.30316e6i			1.01219i	−0.862478	+	0.506094i	$0.831089\pi$
$350$		0			0
$351$			756147.i			0.327596i
$352$		0			0
$353$	−1.22225e6			−0.522063			−0.261031	−	0.965330i	$-0.584063\pi$
$353$	−1.22225e6			−0.522063			−0.261031	+	0.965330i	$0.584063\pi$
$354$		0			0
$355$		−	1.96817e6i		−	0.828879i
$356$		0			0
$357$	1.05733e6			0.439077
$358$		0			0
$359$			986167.i			0.403845i	0.979402	+	0.201922i	$0.0647188\pi$
$359$			986167.i			0.403845i	−0.979402	+	0.201922i	$0.935281\pi$
$360$		0			0
$361$	568130.	−	2.41004e6i	0.229446	−	0.973321i
$362$		0			0
$363$		−	7.24380e6i		−	2.88536i
$364$		0			0
$365$		−	2.61328e6i		−	1.02673i
$366$		0			0
$367$			2.93192e6i			1.13629i	0.822930	+	0.568143i	$0.192338\pi$
$367$			2.93192e6i			1.13629i	−0.822930	+	0.568143i	$0.807662\pi$
$368$		0			0
$369$		−	2.24930e6i		−	0.859967i
$370$		0			0
$371$			752249.i			0.283744i
$372$		0			0
$373$	−2.77043e6			−1.03104			−0.515519	−	0.856878i	$-0.672401\pi$
$373$	−2.77043e6			−1.03104			−0.515519	+	0.856878i	$0.672401\pi$
$374$		0			0
$375$	−2.88882e6			−1.06082
$376$		0			0
$377$	−2.49450e6			−0.903921
$378$		0			0
$379$			2.50076e6i			0.894282i	0.894463	+	0.447141i	$0.147558\pi$
$379$			2.50076e6i			0.894282i	−0.894463	+	0.447141i	$0.852442\pi$
$380$		0			0
$381$		−	985779.i		−	0.347910i
$382$		0			0
$383$	−426706.			−0.148639			−0.0743193	−	0.997234i	$-0.523678\pi$
$383$	−426706.			−0.148639			−0.0743193	+	0.997234i	$0.523678\pi$
$384$		0			0
$385$	1.58499e6			0.544974
$386$		0			0
$387$	−686454.			−0.232988
$388$		0			0
$389$		−	643478.i		−	0.215605i	−0.994172	−	0.107803i	$-0.965619\pi$
$389$		−	643478.i		−	0.215605i	0.994172	−	0.107803i	$-0.0343815\pi$
$390$		0			0
$391$		−	5.25096e6i		−	1.73699i
$392$		0			0
$393$			4.04291e6i			1.32042i
$394$		0			0
$395$		−	801302.i		−	0.258407i
$396$		0			0
$397$			2.74515e6i			0.874159i	0.899423	+	0.437079i	$0.143987\pi$
$397$			2.74515e6i			0.874159i	−0.899423	+	0.437079i	$0.856013\pi$
$398$		0			0
$399$	−829150.	+	656417.i	−0.260736	+	0.206418i
$400$		0			0
$401$			2.83865e6i			0.881557i	0.897616	+	0.440779i	$0.145298\pi$
$401$			2.83865e6i			0.881557i	−0.897616	+	0.440779i	$0.854702\pi$
$402$		0			0
$403$	−1.14967e6			−0.352622
$404$		0			0
$405$			4.55832e6i			1.38092i
$406$		0			0
$407$	1.34238e6			0.401688
$408$		0			0
$409$			4.03884e6i			1.19385i	0.802299	+	0.596923i	$0.203610\pi$
$409$			4.03884e6i			1.19385i	−0.802299	+	0.596923i	$0.796390\pi$
$410$		0			0
$411$		−	2.60188e6i		−	0.759772i
$412$		0			0
$413$	−683505.			−0.197182
$414$		0			0
$415$			1.47063e6i			0.419163i
$416$		0			0
$417$			5.23784e6i			1.47507i
$418$		0			0
$419$	2.72843e6			0.759237			0.379619	−	0.925143i	$-0.376055\pi$
$419$	2.72843e6			0.759237			0.379619	+	0.925143i	$0.376055\pi$
$420$		0			0
$421$	−4.53833e6			−1.24793			−0.623966	−	0.781452i	$-0.714480\pi$
$421$	−4.53833e6			−1.24793			−0.623966	+	0.781452i	$0.714480\pi$
$422$		0			0
$423$			2.13905e6i			0.581258i
$424$		0			0
$425$	−1.08730e6			−0.291995
$426$		0			0
$427$	1.19499e6			0.317172
$428$		0			0
$429$			4.74730e6i			1.24539i
$430$		0			0
$431$	373044.			0.0967314			0.0483657	−	0.998830i	$-0.484599\pi$
$431$	373044.			0.0967314			0.0483657	+	0.998830i	$0.484599\pi$
$432$		0			0
$433$		−	3.20117e6i		−	0.820520i	−0.911969	−	0.410260i	$-0.865438\pi$
$433$		−	3.20117e6i		−	0.820520i	0.911969	−	0.410260i	$-0.134562\pi$
$434$		0			0
$435$	−8.78989e6			−2.22721
$436$		0			0
$437$	3.25992e6	+	4.11775e6i	0.816589	+	1.03147i
$438$		0			0
$439$	4.20752e6			1.04199			0.520996	−	0.853559i	$-0.325561\pi$
$439$	4.20752e6			1.04199			0.520996	+	0.853559i	$0.325561\pi$
$440$		0			0
$441$	−1.96608e6			−0.481398
$442$		0			0
$443$	−1.87922e6			−0.454954			−0.227477	−	0.973783i	$-0.573048\pi$
$443$	−1.87922e6			−0.454954			−0.227477	+	0.973783i	$0.573048\pi$
$444$		0			0
$445$	1.82394e6			0.436626
$446$		0			0
$447$	7.73023e6			1.82989
$448$		0			0
$449$		−	1.26905e6i		−	0.297073i	−0.988907	−	0.148537i	$-0.952544\pi$
$449$		−	1.26905e6i		−	0.297073i	0.988907	−	0.148537i	$-0.0474562\pi$
$450$		0			0
$451$			1.30776e7i			3.02753i
$452$		0			0
$453$			9.83032e6i			2.25072i
$454$		0			0
$455$	−727830.			−0.164817
$456$		0			0
$457$	−2.17594e6			−0.487368			−0.243684	−	0.969855i	$-0.578356\pi$
$457$	−2.17594e6			−0.487368			−0.243684	+	0.969855i	$0.578356\pi$
$458$		0			0
$459$		−	3.53174e6i		−	0.782450i
$460$		0			0
$461$			1.93199e6i			0.423402i	0.977335	+	0.211701i	$0.0679003\pi$
$461$			1.93199e6i			0.423402i	−0.977335	+	0.211701i	$0.932100\pi$
$462$		0			0
$463$		−	1.85633e6i		−	0.402441i	−0.979546	−	0.201221i	$-0.935509\pi$
$463$		−	1.85633e6i		−	0.402441i	0.979546	−	0.201221i	$-0.0644908\pi$
$464$		0			0
$465$	−4.05109e6			−0.868839
$466$		0			0
$467$	184963.			0.0392457			0.0196228	−	0.999807i	$-0.493753\pi$
$467$	184963.			0.0392457			0.0196228	+	0.999807i	$0.493753\pi$
$468$		0			0
$469$	1.91031e6			0.401026
$470$		0			0
$471$	−4.47742e6			−0.929985
$472$		0			0
$473$	3.99110e6			0.820238
$474$		0			0
$475$	852648.	−	675020.i	0.173395	−	0.137272i
$476$		0			0
$477$	−2.71330e6			−0.546011
$478$		0			0
$479$		−	3.04829e6i		−	0.607041i	−0.952825	−	0.303520i	$-0.901838\pi$
$479$		−	3.04829e6i		−	0.607041i	0.952825	−	0.303520i	$-0.0981620\pi$
$480$		0			0
$481$	−616422.			−0.121483
$482$		0			0
$483$		−	2.24309e6i		−	0.437500i
$484$		0			0
$485$	3.65970e6			0.706466
$486$		0			0
$487$	3.44956e6			0.659085			0.329542	−	0.944141i	$-0.393106\pi$
$487$	3.44956e6			0.659085			0.329542	+	0.944141i	$0.393106\pi$
$488$		0			0
$489$		−	2.84858e6i		−	0.538712i
$490$		0			0
$491$	2.45605e6			0.459762			0.229881	−	0.973219i	$-0.426166\pi$
$491$	2.45605e6			0.459762			0.229881	+	0.973219i	$0.426166\pi$
$492$		0			0
$493$	1.16511e7			2.15898
$494$		0			0
$495$			5.71693e6i			1.04870i
$496$		0			0
$497$			1.11443e6i			0.202377i
$498$		0			0
$499$	−1.05393e7			−1.89478			−0.947392	−	0.320077i	$-0.896291\pi$
$499$	−1.05393e7			−1.89478			−0.947392	+	0.320077i	$0.896291\pi$
$500$		0			0
$501$		−	4.92683e6i		−	0.876948i
$502$		0			0
$503$			3.72259e6i			0.656033i	0.944672	+	0.328016i	$0.106380\pi$
$503$			3.72259e6i			0.656033i	−0.944672	+	0.328016i	$0.893620\pi$
$504$		0			0
$505$	8.98391e6			1.56761
$506$		0			0
$507$			4.95392e6i			0.855913i
$508$		0			0
$509$	7.95464e6			1.36090			0.680450	−	0.732794i	$-0.261784\pi$
$509$	7.95464e6			1.36090			0.680450	+	0.732794i	$0.261784\pi$
$510$		0			0
$511$			1.47971e6i			0.250682i
$512$		0			0
$513$	2.19259e6	+	2.76956e6i	0.367844	+	0.464640i
$514$		0			0
$515$			7.56350e6i			1.25662i
$516$		0			0
$517$		−	1.24366e7i		−	2.04633i
$518$		0			0
$519$		−	1.11490e6i		−	0.181685i
$520$		0			0
$521$		−	2.33025e6i		−	0.376105i	−0.982159	−	0.188052i	$-0.939783\pi$
$521$		−	2.33025e6i		−	0.376105i	0.982159	−	0.188052i	$-0.0602175\pi$
$522$		0			0
$523$		−	2.58540e6i		−	0.413308i	−0.978414	−	0.206654i	$-0.933742\pi$
$523$		−	2.58540e6i		−	0.413308i	0.978414	−	0.206654i	$-0.0662575\pi$
$524$		0			0
$525$	−464468.			−0.0735457
$526$		0			0
$527$	5.36974e6			0.842223
$528$		0			0
$529$	−4.70334e6			−0.730748
$530$		0			0
$531$		−	2.46534e6i		−	0.379438i
$532$		0			0
$533$		−	6.00527e6i		−	0.915618i
$534$		0			0
$535$	6.59912e6			0.996785
$536$		0			0
$537$	−3.25773e6			−0.487506
$538$		0			0
$539$	1.14310e7			1.69477
$540$		0			0
$541$			6.61022e6i			0.971009i	0.874234	+	0.485504i	$0.161364\pi$
$541$			6.61022e6i			0.971009i	−0.874234	+	0.485504i	$0.838636\pi$
$542$		0			0
$543$			2.32477e6i			0.338361i
$544$		0			0
$545$			1.33200e7i			1.92094i
$546$		0			0
$547$		−	1.87191e6i		−	0.267495i	−0.991015	−	0.133748i	$-0.957299\pi$
$547$		−	1.87191e6i		−	0.267495i	0.991015	−	0.133748i	$-0.0427012\pi$
$548$		0			0
$549$			4.31023e6i			0.610336i
$550$		0			0
$551$	−9.13666e6	+	7.23326e6i	−1.28206	+	1.01497i
$552$		0			0
$553$			453718.i			0.0630918i
$554$		0			0
$555$	−2.17209e6			−0.299327
$556$		0			0
$557$		−	1.06790e7i		−	1.45845i	−0.684272	−	0.729227i	$-0.739880\pi$
$557$		−	1.06790e7i		−	1.45845i	0.684272	−	0.729227i	$-0.260120\pi$
$558$		0			0
$559$	−1.83272e6			−0.248065
$560$		0			0
$561$		−	2.21732e7i		−	2.97455i
$562$		0			0
$563$		−	3.49950e6i		−	0.465302i	−0.972560	−	0.232651i	$-0.925260\pi$
$563$		−	3.49950e6i		−	0.465302i	0.972560	−	0.232651i	$-0.0747401\pi$
$564$		0			0
$565$	3.20709e6			0.422659
$566$		0			0
$567$		−	2.58104e6i		−	0.337160i
$568$		0			0
$569$		−	2.32301e6i		−	0.300795i	−0.988626	−	0.150398i	$-0.951945\pi$
$569$		−	2.32301e6i		−	0.300795i	0.988626	−	0.150398i	$-0.0480554\pi$
$570$		0			0
$571$	6.49877e6			0.834143			0.417072	−	0.908874i	$-0.363056\pi$
$571$	6.49877e6			0.834143			0.417072	+	0.908874i	$0.363056\pi$
$572$		0			0
$573$	−1.06946e7			−1.36075
$574$		0			0
$575$			2.30666e6i			0.290946i
$576$		0			0
$577$	9.15580e6			1.14487			0.572436	−	0.819950i	$-0.305999\pi$
$577$	9.15580e6			1.14487			0.572436	+	0.819950i	$0.305999\pi$
$578$		0			0
$579$	−4.53844e6			−0.562614
$580$		0			0
$581$		−	832708.i		−	0.102342i
$582$		0			0
$583$	1.57753e7			1.92224
$584$		0			0
$585$		−	2.62522e6i		−	0.317158i
$586$		0			0
$587$	1.48513e7			1.77897			0.889485	−	0.456964i	$-0.151063\pi$
$587$	1.48513e7			1.77897			0.889485	+	0.456964i	$0.151063\pi$
$588$		0			0
$589$	−4.21090e6	+	3.33367e6i	−0.500135	+	0.395944i
$590$		0			0
$591$	−2.20925e6			−0.260181
$592$		0			0
$593$	−4.31062e6			−0.503388			−0.251694	−	0.967807i	$-0.580988\pi$
$593$	−4.31062e6			−0.503388			−0.251694	+	0.967807i	$0.580988\pi$
$594$		0			0
$595$	3.39948e6			0.393659
$596$		0			0
$597$	1.64423e7			1.88810
$598$		0			0
$599$	8.48126e6			0.965813			0.482907	−	0.875672i	$-0.339581\pi$
$599$	8.48126e6			0.965813			0.482907	+	0.875672i	$0.339581\pi$
$600$		0			0
$601$		−	8.66899e6i		−	0.978999i	−0.872004	−	0.489499i	$-0.837180\pi$
$601$		−	8.66899e6i		−	0.978999i	0.872004	−	0.489499i	$-0.162820\pi$
$602$		0			0
$603$			6.89033e6i			0.771697i
$604$		0			0
$605$		−	2.32899e7i		−	2.58689i
$606$		0			0
$607$	−4.26228e6			−0.469538			−0.234769	−	0.972051i	$-0.575433\pi$
$607$	−4.26228e6			−0.469538			−0.234769	+	0.972051i	$0.575433\pi$
$608$		0			0
$609$	4.97706e6			0.543788
$610$		0			0
$611$			5.71090e6i			0.618873i
$612$		0			0
$613$		−	27655.7i		−	0.00297258i	−0.999999	−	0.00148629i	$-0.999527\pi$
$613$		−	27655.7i		−	0.00297258i	0.999999	−	0.00148629i	$-0.000473101\pi$
$614$		0			0
$615$		−	2.11608e7i		−	2.25603i
$616$		0			0
$617$	−1.57537e7			−1.66598			−0.832989	−	0.553289i	$-0.813373\pi$
$617$	−1.57537e7			−1.66598			−0.832989	+	0.553289i	$0.813373\pi$
$618$		0			0
$619$	5.01194e6			0.525750			0.262875	−	0.964830i	$-0.415329\pi$
$619$	5.01194e6			0.525750			0.262875	+	0.964830i	$0.415329\pi$
$620$		0			0
$621$	−7.49243e6			−0.779639
$622$		0			0
$623$	−1.03276e6			−0.106605
$624$		0			0
$625$	−1.14477e7			−1.17225
$626$		0			0
$627$	1.37657e7	+	1.73880e7i	1.39839	+	1.76637i
$628$		0			0
$629$	2.87912e6			0.290157
$630$		0			0
$631$		−	3.66483e6i		−	0.366421i	−0.983074	−	0.183211i	$-0.941351\pi$
$631$		−	3.66483e6i		−	0.366421i	0.983074	−	0.183211i	$-0.0586490\pi$
$632$		0			0
$633$	3.33297e6			0.330615
$634$		0			0
$635$		−	3.16942e6i		−	0.311922i
$636$		0			0
$637$	−5.24911e6			−0.512551
$638$		0			0
$639$	−4.01964e6			−0.389435
$640$		0			0
$641$		−	1.10672e7i		−	1.06388i	−0.846783	−	0.531938i	$-0.821464\pi$
$641$		−	1.10672e7i		−	1.06388i	0.846783	−	0.531938i	$-0.178536\pi$
$642$		0			0
$643$	−1.78003e7			−1.69786			−0.848928	−	0.528508i	$-0.822752\pi$
$643$	−1.78003e7			−1.69786			−0.848928	+	0.528508i	$0.822752\pi$
$644$		0			0
$645$	−6.45797e6			−0.611219
$646$		0			0
$647$			7.37630e6i			0.692752i	0.938096	+	0.346376i	$0.112588\pi$
$647$			7.37630e6i			0.692752i	−0.938096	+	0.346376i	$0.887412\pi$
$648$		0			0
$649$			1.43337e7i			1.33582i
$650$		0			0
$651$	2.29383e6			0.212133
$652$		0			0
$653$			9.21106e6i			0.845331i	0.906286	+	0.422665i	$0.138905\pi$
$653$			9.21106e6i			0.845331i	−0.906286	+	0.422665i	$0.861095\pi$
$654$		0			0
$655$			1.29985e7i			1.18384i
$656$		0			0
$657$	−5.33718e6			−0.482390
$658$		0			0
$659$		−	1.93618e7i		−	1.73673i	−0.495923	−	0.868366i	$-0.665170\pi$
$659$		−	1.93618e7i		−	1.73673i	0.495923	−	0.868366i	$-0.334830\pi$
$660$		0			0
$661$	−1.05673e7			−0.940724			−0.470362	−	0.882474i	$-0.655877\pi$
$661$	−1.05673e7			−0.940724			−0.470362	+	0.882474i	$0.655877\pi$
$662$		0			0
$663$			1.01820e7i			0.899597i
$664$		0			0
$665$	−2.66584e6	+	2.11048e6i	−0.233765	+	0.185066i
$666$		0			0
$667$		−	2.47172e7i		−	2.15122i
$668$		0			0
$669$			2.63268e7i			2.27422i
$670$		0			0
$671$		−	2.50600e7i		−	2.14870i
$672$		0			0
$673$			5.46352e6i			0.464981i	0.972599	+	0.232490i	$0.0746875\pi$
$673$			5.46352e6i			0.464981i	−0.972599	+	0.232490i	$0.925313\pi$
$674$		0			0
$675$			1.55143e6i			0.131061i
$676$		0			0
$677$	2.12861e7			1.78494			0.892472	−	0.451102i	$-0.148969\pi$
$677$	2.12861e7			1.78494			0.892472	+	0.451102i	$0.148969\pi$
$678$		0			0
$679$	−2.07222e6			−0.172489
$680$		0			0
$681$	1.58830e7			1.31240
$682$		0			0
$683$			2.10602e7i			1.72747i	0.503949	+	0.863734i	$0.331880\pi$
$683$			2.10602e7i			1.72747i	−0.503949	+	0.863734i	$0.668120\pi$
$684$		0			0
$685$		−	8.36543e6i		−	0.681180i
$686$		0			0
$687$	−1.09381e7			−0.884200
$688$		0			0
$689$	−7.24406e6			−0.581345
$690$		0			0
$691$	−1.80165e7			−1.43541			−0.717704	−	0.696348i	$-0.754807\pi$
$691$	−1.80165e7			−1.43541			−0.717704	+	0.696348i	$0.754807\pi$
$692$		0			0
$693$		−	3.23707e6i		−	0.256047i
$694$		0			0
$695$			1.68404e7i			1.32249i
$696$		0			0
$697$			2.80488e7i			2.18692i
$698$		0			0
$699$		−	1.34626e7i		−	1.04216i
$700$		0			0
$701$			6.85939e6i			0.527218i	0.964630	+	0.263609i	$0.0849129\pi$
$701$			6.85939e6i			0.527218i	−0.964630	+	0.263609i	$0.915087\pi$
$702$		0			0
$703$	−2.25778e6	+	1.78743e6i	−0.172303	+	0.136408i
$704$		0			0
$705$			2.01236e7i			1.52487i
$706$		0			0
$707$	−5.08692e6			−0.382742
$708$		0			0
$709$		−	1.41272e7i		−	1.05545i	−0.849414	−	0.527727i	$-0.823044\pi$
$709$		−	1.41272e7i		−	1.05545i	0.849414	−	0.527727i	$-0.176956\pi$
$710$		0			0
$711$	−1.63652e6			−0.121408
$712$		0			0
$713$		−	1.13917e7i		−	0.839197i
$714$		0			0
$715$			1.52633e7i			1.11656i
$716$		0			0
$717$	−4.45674e6			−0.323757
$718$		0			0
$719$			3.92042e6i			0.282820i	0.989951	+	0.141410i	$0.0451636\pi$
$719$			3.92042e6i			0.282820i	−0.989951	+	0.141410i	$0.954836\pi$
$720$		0			0
$721$		−	4.28265e6i		−	0.306813i
$722$		0			0
$723$	−441228.			−0.0313919
$724$		0			0
$725$	−5.11811e6			−0.361630
$726$		0			0
$727$		−	1.34940e7i		−	0.946903i	−0.880820	−	0.473452i	$-0.843008\pi$
$727$		−	1.34940e7i		−	0.946903i	0.880820	−	0.473452i	$-0.156992\pi$
$728$		0			0
$729$	−1.17141e6			−0.0816377
$730$		0			0
$731$	8.56008e6			0.592495
$732$		0			0
$733$		−	1.16355e6i		−	0.0799881i	−0.999200	−	0.0399940i	$-0.987266\pi$
$733$		−	1.16355e6i		−	0.0799881i	0.999200	−	0.0399940i	$-0.0127339\pi$
$734$		0			0
$735$	−1.84963e7			−1.26289
$736$		0			0
$737$		−	4.00610e7i		−	2.71677i
$738$		0			0
$739$	9.93879e6			0.669457			0.334728	−	0.942315i	$-0.391355\pi$
$739$	9.93879e6			0.669457			0.334728	+	0.942315i	$0.391355\pi$
$740$		0			0
$741$	−6.32121e6	−	7.98460e6i	−0.422917	−	0.534205i
$742$		0			0
$743$	−1.79938e7			−1.19578			−0.597889	−	0.801579i	$-0.703993\pi$
$743$	−1.79938e7			−1.19578			−0.597889	+	0.801579i	$0.703993\pi$
$744$		0			0
$745$	2.48538e7			1.64060
$746$		0			0
$747$	3.00351e6			0.196937
$748$		0			0
$749$	−3.73659e6			−0.243372
$750$		0			0
$751$	−1.17254e7			−0.758629			−0.379314	−	0.925268i	$-0.623840\pi$
$751$	−1.17254e7			−0.758629			−0.379314	+	0.925268i	$0.623840\pi$
$752$		0			0
$753$		−	1.29605e7i		−	0.832977i
$754$		0			0
$755$			3.16059e7i			2.01791i
$756$		0			0
$757$			2.38116e6i			0.151025i	0.997145	+	0.0755124i	$0.0240593\pi$
$757$			2.38116e6i			0.151025i	−0.997145	+	0.0755124i	$0.975941\pi$
$758$		0			0
$759$	−4.70396e7			−2.96387
$760$		0			0
$761$	1.10542e7			0.691935			0.345968	−	0.938246i	$-0.387551\pi$
$761$	1.10542e7			0.691935			0.345968	+	0.938246i	$0.387551\pi$
$762$		0			0
$763$		−	7.54215e6i		−	0.469012i
$764$		0			0
$765$			1.22616e7i			0.757520i
$766$		0			0
$767$		−	6.58207e6i		−	0.403993i
$768$		0			0
$769$	−2.91057e7			−1.77485			−0.887426	−	0.460950i	$-0.847509\pi$
$769$	−2.91057e7			−1.77485			−0.887426	+	0.460950i	$0.847509\pi$
$770$		0			0
$771$	−8.64110e6			−0.523520
$772$		0			0
$773$	−1.61628e7			−0.972900			−0.486450	−	0.873709i	$-0.661708\pi$
$773$	−1.61628e7			−0.972900			−0.486450	+	0.873709i	$0.661708\pi$
$774$		0			0
$775$	−2.35884e6			−0.141073
$776$		0			0
$777$	1.22989e6			0.0730828
$778$		0			0
$779$	−1.74134e7	−	2.19956e7i	−1.02811	−	1.29865i
$780$		0			0
$781$	2.33705e7			1.37101
$782$		0			0
$783$		−	1.66246e7i		−	0.969049i
$784$		0			0
$785$	−1.43956e7			−0.833786
$786$		0			0
$787$		−	2.95097e7i		−	1.69835i	−0.528111	−	0.849175i	$-0.677099\pi$
$787$		−	2.95097e7i		−	1.69835i	0.528111	−	0.849175i	$-0.322901\pi$
$788$		0			0
$789$	−3.95131e7			−2.25969
$790$		0			0
$791$	−1.81594e6			−0.103195
$792$		0			0
$793$			1.15076e7i			0.649833i
$794$		0			0
$795$	−2.55259e7			−1.43240
$796$		0			0
$797$	−2.87000e7			−1.60043			−0.800214	−	0.599714i	$-0.795281\pi$
$797$	−2.87000e7			−1.60043			−0.800214	+	0.599714i	$0.795281\pi$
$798$		0			0
$799$		−	2.66739e7i		−	1.47815i
$800$		0			0
$801$		−	3.72507e6i		−	0.205142i
$802$		0			0
$803$	3.10308e7			1.69826
$804$		0			0
$805$		−	7.21185e6i		−	0.392245i
$806$		0			0
$807$			3.47691e6i			0.187936i
$808$		0			0
$809$	−1.23204e6			−0.0661843			−0.0330922	−	0.999452i	$-0.510535\pi$
$809$	−1.23204e6			−0.0661843			−0.0330922	+	0.999452i	$0.510535\pi$
$810$		0			0
$811$			2.32829e7i			1.24304i	0.783399	+	0.621520i	$0.213484\pi$
$811$			2.32829e7i			1.24304i	−0.783399	+	0.621520i	$0.786516\pi$
$812$		0			0
$813$	−2.52826e7			−1.34151
$814$		0			0
$815$		−	9.15860e6i		−	0.482987i
$816$		0			0
$817$	−6.71273e6	+	5.31430e6i	−0.351839	+	0.278542i
$818$		0			0
$819$			1.48647e6i			0.0774365i
$820$		0			0
$821$		−	2.15225e7i		−	1.11439i	−0.830383	−	0.557193i	$-0.811878\pi$
$821$		−	2.15225e7i		−	1.11439i	0.830383	−	0.557193i	$-0.188122\pi$
$822$		0			0
$823$		−	4.80998e6i		−	0.247539i	−0.992311	−	0.123769i	$-0.960502\pi$
$823$		−	4.80998e6i		−	0.247539i	0.992311	−	0.123769i	$-0.0394983\pi$
$824$		0			0
$825$			9.74032e6i			0.498240i
$826$		0			0
$827$		−	3.14275e7i		−	1.59788i	−0.601407	−	0.798942i	$-0.705393\pi$
$827$		−	3.14275e7i		−	1.59788i	0.601407	−	0.798942i	$-0.294607\pi$
$828$		0			0
$829$	5.53293e6			0.279620			0.139810	−	0.990178i	$-0.455351\pi$
$829$	5.53293e6			0.279620			0.139810	+	0.990178i	$0.455351\pi$
$830$		0			0
$831$	3.49567e7			1.75601
$832$		0			0
$833$	2.45170e7			1.22421
$834$		0			0
$835$		−	1.58405e7i		−	0.786235i
$836$		0			0
$837$		−	7.66192e6i		−	0.378028i
$838$		0			0
$839$	3.21431e7			1.57646			0.788229	−	0.615382i	$-0.210998\pi$
$839$	3.21431e7			1.57646			0.788229	+	0.615382i	$0.210998\pi$
$840$		0			0
$841$	3.43326e7			1.67385
$842$		0			0
$843$	−635771.			−0.0308128
$844$		0			0
$845$			1.59276e7i			0.767376i
$846$		0			0
$847$			1.31873e7i			0.631609i
$848$		0			0
$849$			3.50714e6i			0.166987i
$850$		0			0
$851$		−	6.10794e6i		−	0.289115i
$852$		0			0
$853$			2.12570e7i			1.00030i	0.865940	+	0.500148i	$0.166721\pi$
$853$			2.12570e7i			1.00030i	−0.865940	+	0.500148i	$0.833279\pi$
$854$		0			0
$855$	−7.61231e6	−	9.61545e6i	−0.356124	−	0.449836i
$856$		0			0
$857$			1.05471e7i			0.490547i	0.969454	+	0.245273i	$0.0788777\pi$
$857$			1.05471e7i			0.490547i	−0.969454	+	0.245273i	$0.921122\pi$
$858$		0			0
$859$	2.22465e7			1.02867			0.514337	−	0.857588i	$-0.328038\pi$
$859$	2.22465e7			1.02867			0.514337	+	0.857588i	$0.328038\pi$
$860$		0			0
$861$			1.19818e7i			0.550825i
$862$		0			0
$863$	1.93334e7			0.883651			0.441826	−	0.897101i	$-0.354331\pi$
$863$	1.93334e7			0.883651			0.441826	+	0.897101i	$0.354331\pi$
$864$		0			0
$865$		−	3.58457e6i		−	0.162891i
$866$		0			0
$867$		−	2.02763e7i		−	0.916095i
$868$		0			0
$869$	9.51488e6			0.427419
$870$		0			0
$871$			1.83960e7i			0.821635i
$872$		0			0
$873$		−	7.47430e6i		−	0.331921i
$874$		0			0
$875$	5.25910e6			0.232216
$876$		0			0
$877$	−2.38641e7			−1.04772			−0.523862	−	0.851803i	$-0.675509\pi$
$877$	−2.38641e7			−1.04772			−0.523862	+	0.851803i	$0.675509\pi$
$878$		0			0
$879$			3.40951e7i			1.48840i
$880$		0			0
$881$	−443019.			−0.0192302			−0.00961508	−	0.999954i	$-0.503061\pi$
$881$	−443019.			−0.0192302			−0.00961508	+	0.999954i	$0.503061\pi$
$882$		0			0
$883$	8.72910e6			0.376763			0.188381	−	0.982096i	$-0.439676\pi$
$883$	8.72910e6			0.376763			0.188381	+	0.982096i	$0.439676\pi$
$884$		0			0
$885$		−	2.31933e7i		−	0.995414i
$886$		0			0
$887$	1.97159e6			0.0841411			0.0420706	−	0.999115i	$-0.486605\pi$
$887$	1.97159e6			0.0841411			0.0420706	+	0.999115i	$0.486605\pi$
$888$		0			0
$889$			1.79461e6i			0.0761579i
$890$		0			0
$891$	−5.41267e7			−2.28411
$892$		0			0
$893$	1.65598e7	+	2.09174e7i	0.694907	+	0.877768i
$894$		0			0
$895$	−1.04741e7			−0.437077
$896$		0			0
$897$	2.16006e7			0.896365
$898$		0			0
$899$	2.52764e7			1.04308
$900$		0			0
$901$	3.38348e7			1.38852
$902$		0			0
$903$	3.65667e6			0.149233
$904$		0			0
$905$			7.47447e6i			0.303360i
$906$		0			0
$907$			1.34438e7i			0.542630i	0.962491	+	0.271315i	$0.0874586\pi$
$907$			1.34438e7i			0.542630i	−0.962491	+	0.271315i	$0.912541\pi$
$908$		0			0
$909$		−	1.83481e7i		−	0.736514i
$910$		0			0
$911$	−2.80168e7			−1.11847			−0.559233	−	0.829010i	$-0.688904\pi$
$911$	−2.80168e7			−1.11847			−0.559233	+	0.829010i	$0.688904\pi$
$912$		0			0
$913$	−1.74626e7			−0.693319
$914$		0			0
$915$			4.05494e7i			1.60115i
$916$		0			0
$917$		−	7.36011e6i		−	0.289042i
$918$		0			0
$919$		−	8.53567e6i		−	0.333387i	−0.986009	−	0.166694i	$-0.946691\pi$
$919$		−	8.53567e6i		−	0.333387i	0.986009	−	0.166694i	$-0.0533091\pi$
$920$		0			0
$921$	3.58273e7			1.39176
$922$		0			0
$923$	−1.07318e7			−0.414636
$924$		0			0
$925$	−1.26475e6			−0.0486016
$926$		0			0
$927$	1.54471e7			0.590403
$928$		0			0
$929$	2.12034e7			0.806057			0.403029	−	0.915187i	$-0.367958\pi$
$929$	2.12034e7			0.806057			0.403029	+	0.915187i	$0.367958\pi$
$930$		0			0
$931$	−1.92260e7	+	1.52207e7i	−0.726967	+	0.575522i
$932$		0			0
$933$	−1.45503e7			−0.547226
$934$		0			0
$935$		−	7.12901e7i		−	2.66686i
$936$		0			0
$937$	972475.			0.0361851			0.0180925	−	0.999836i	$-0.494241\pi$
$937$	972475.			0.0361851			0.0180925	+	0.999836i	$0.494241\pi$
$938$		0			0
$939$		−	4.43117e7i		−	1.64004i
$940$		0			0
$941$	−2.03628e7			−0.749657			−0.374829	−	0.927094i	$-0.622298\pi$
$941$	−2.03628e7			−0.749657			−0.374829	+	0.927094i	$0.622298\pi$
$942$		0			0
$943$	5.95044e7			2.17906
$944$		0			0
$945$		−	4.85061e6i		−	0.176692i
$946$		0			0
$947$	1.47459e7			0.534314			0.267157	−	0.963653i	$-0.413916\pi$
$947$	1.47459e7			0.534314			0.267157	+	0.963653i	$0.413916\pi$
$948$		0			0
$949$	−1.42494e7			−0.513607
$950$		0			0
$951$		−	1.77555e7i		−	0.636624i
$952$		0			0
$953$		−	2.16052e7i		−	0.770595i	−0.922792	−	0.385297i	$-0.874099\pi$
$953$		−	2.16052e7i		−	0.770595i	0.922792	−	0.385297i	$-0.125901\pi$
$954$		0			0
$955$	−3.43847e7			−1.21999
$956$		0			0
$957$		−	1.04374e8i		−	3.68392i
$958$		0			0
$959$			4.73672e6i			0.166315i
$960$		0			0
$961$	−1.69798e7			−0.593094
$962$		0			0
$963$		−	1.34776e7i		−	0.468323i
$964$		0			0
$965$	−1.45918e7			−0.504417
$966$		0			0
$967$		−	1.09463e7i		−	0.376443i	−0.982127	−	0.188221i	$-0.939728\pi$
$967$		−	1.09463e7i		−	0.376443i	0.982127	−	0.188221i	$-0.0602723\pi$
$968$		0			0
$969$	2.95245e7	+	3.72937e7i	1.01012	+	1.27593i
$970$		0			0
$971$		−	3.74957e7i		−	1.27624i	−0.769935	−	0.638122i	$-0.779711\pi$
$971$		−	3.74957e7i		−	1.27624i	0.769935	−	0.638122i	$-0.220289\pi$
$972$		0			0
$973$		−	9.53548e6i		−	0.322894i
$974$		0			0
$975$		−	4.47276e6i		−	0.150683i
$976$		0			0
$977$		−	2.79099e7i		−	0.935453i	−0.883873	−	0.467727i	$-0.845073\pi$
$977$		−	2.79099e7i		−	0.935453i	0.883873	−	0.467727i	$-0.154927\pi$
$978$		0			0
$979$			2.16579e7i			0.722204i
$980$		0			0
$981$	2.72039e7			0.902523
$982$		0			0
$983$	−4.28228e7			−1.41349			−0.706743	−	0.707470i	$-0.749836\pi$
$983$	−4.28228e7			−1.41349			−0.706743	+	0.707470i	$0.749836\pi$
$984$		0			0
$985$	−7.10306e6			−0.233268
$986$		0			0
$987$		−	1.13945e7i		−	0.372307i
$988$		0			0
$989$		−	1.81598e7i		−	0.590366i
$990$		0			0
$991$	3.33542e6			0.107886			0.0539431	−	0.998544i	$-0.482821\pi$
$991$	3.33542e6			0.107886			0.0539431	+	0.998544i	$0.482821\pi$
$992$		0			0
$993$	−5.81117e7			−1.87021
$994$		0			0
$995$	5.28642e7			1.69279
$996$		0			0
$997$			4.26994e7i			1.36045i	0.733001	+	0.680227i	$0.238119\pi$
$997$			4.26994e7i			1.36045i	−0.733001	+	0.680227i	$0.761881\pi$
$998$		0			0
$999$		−	4.10813e6i		−	0.130236i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	608.6.b.b.303.19		96
4.3	odd	2		152.6.b.b.75.10	yes	96
8.3	odd	2	inner	608.6.b.b.303.20		96
8.5	even	2		152.6.b.b.75.88	yes	96
19.18	odd	2	inner	608.6.b.b.303.77		96
76.75	even	2		152.6.b.b.75.87	yes	96
152.37	odd	2		152.6.b.b.75.9	✓	96
152.75	even	2	inner	608.6.b.b.303.78		96

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
152.6.b.b.75.9	✓	96	152.37	odd	2
152.6.b.b.75.10	yes	96	4.3	odd	2
152.6.b.b.75.87	yes	96	76.75	even	2
152.6.b.b.75.88	yes	96	8.5	even	2
608.6.b.b.303.19		96	1.1	even	1	trivial
608.6.b.b.303.20		96	8.3	odd	2	inner
608.6.b.b.303.77		96	19.18	odd	2	inner
608.6.b.b.303.78		96	152.75	even	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)

\(f(q)\)	\(=\)	\(q-19.2136i q^{3} -61.7747i q^{5} +34.9784i q^{7} -126.164 q^{9} +O(q^{10})\)	\(q-19.2136i q^{3} -61.7747i q^{5} +34.9784i q^{7} -126.164 q^{9} +733.529 q^{11} -336.837 q^{13} -1186.92 q^{15} +1573.27 q^{17} +(-1233.74 + 976.721i) q^{19} +672.062 q^{21} -3337.62i q^{23} -691.109 q^{25} -2244.84i q^{27} +7405.66 q^{29} +3413.12 q^{31} -14093.8i q^{33} +2160.78 q^{35} +1830.03 q^{37} +6471.87i q^{39} +17828.4i q^{41} +5440.96 q^{43} +7793.74i q^{45} -16954.5i q^{47} +15583.5 q^{49} -30228.2i q^{51} +21506.1 q^{53} -45313.5i q^{55} +(18766.4 + 23704.6i) q^{57} +19540.8i q^{59} -34163.7i q^{61} -4413.01i q^{63} +20808.0i q^{65} -54614.1i q^{67} -64127.8 q^{69} +31860.4 q^{71} +42303.5 q^{73} +13278.7i q^{75} +25657.7i q^{77} +12971.4 q^{79} -73789.5 q^{81} -23806.4 q^{83} -97187.9i q^{85} -142290. i q^{87} +29525.6i q^{89} -11782.0i q^{91} -65578.5i q^{93} +(60336.6 + 76213.9i) q^{95} +59242.8i q^{97} -92544.9 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 96 q - 6168 q^{9}+O(q^{10}) \) 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