LMFDB - Embedded newform 20.7.d.a.19.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [20,7,Mod(19,20)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(20, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1])) N = Newforms(chi, 7, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("20.19"); S:= CuspForms(chi, 7); N := Newforms(S);

Level:	$ N $	$=$	$ 20 = 2^{2} \cdot 5 $
Weight:	$ k $	$=$	$ 7 $
Character orbit:	$[\chi]$	$=$	20.d (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [1,-8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$4.60108167240$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			19.1
Character	$\chi$	$=$	20.19

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$

$=$

$q-8.00000 q^{2} -44.0000 q^{3} +64.0000 q^{4} -125.000 q^{5} +352.000 q^{6} +524.000 q^{7} -512.000 q^{8} +1207.00 q^{9} +1000.00 q^{10} -2816.00 q^{12} -4192.00 q^{14} +5500.00 q^{15} +4096.00 q^{16} -9656.00 q^{18} -8000.00 q^{20} -23056.0 q^{21} +15356.0 q^{23} +22528.0 q^{24} +15625.0 q^{25} -21032.0 q^{27} +33536.0 q^{28} +44858.0 q^{29} -44000.0 q^{30} -32768.0 q^{32} -65500.0 q^{35} +77248.0 q^{36} +64000.0 q^{40} -74338.0 q^{41} +184448. q^{42} -17404.0 q^{43} -150875. q^{45} -122848. q^{46} +26444.0 q^{47} -180224. q^{48} +156927. q^{49} -125000. q^{50} +168256. q^{54} -268288. q^{56} -358864. q^{58} +352000. q^{60} +452342. q^{61} +632468. q^{63} +262144. q^{64} -1276.00 q^{67} -675664. q^{69} +524000. q^{70} -617984. q^{72} -687500. q^{75} -512000. q^{80} +45505.0 q^{81} +594704. q^{82} +1.13172e6 q^{83} -1.47558e6 q^{84} +139232. q^{86} -1.97375e6 q^{87} +511058. q^{89} +1.20700e6 q^{90} +982784. q^{92} -211552. q^{94} +1.44179e6 q^{96} -1.25542e6 q^{98} +O(q^{100})$

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/20\mathbb{Z}\right)^\times$.

$n$	$11$	$17$
$\chi(n)$	$-1$	$-1$

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 $

$2$	−8.00000	−1.00000
$2$	−8.00000	−1.00000
$3$	−44.0000	−1.62963	−0.814815	−	0.579721i	$-0.803161\pi$
$3$	−44.0000	−1.62963	−0.814815	+	0.579721i	$0.803161\pi$
$4$	64.0000	1.00000
$5$	−125.000	−1.00000
$5$	−125.000	−1.00000
$6$	352.000	1.62963
$7$	524.000	1.52770	0.763848	−	0.645396i	$-0.223308\pi$
$7$	524.000	1.52770	0.763848	+	0.645396i	$0.223308\pi$
$8$	−512.000	−1.00000
$9$	1207.00	1.65569
$10$	1000.00	1.00000
$11$	0	0	1.00000			$0$
$11$	0	0	−1.00000			$\pi$
$12$	−2816.00	−1.62963
$13$	0	0	1.00000			$0$
$13$	0	0	−1.00000			$\pi$
$14$	−4192.00	−1.52770
$15$	5500.00	1.62963
$16$	4096.00	1.00000
$17$	0	0	1.00000			$0$
$17$	0	0	−1.00000			$\pi$
$18$	−9656.00	−1.65569
$19$	0	0	1.00000			$0$
$19$	0	0	−1.00000			$\pi$
$20$	−8000.00	−1.00000
$21$	−23056.0	−2.48958
$22$	0	0
$23$	15356.0	1.26210	0.631051	−	0.775741i	$-0.282624\pi$
$23$	15356.0	1.26210	0.631051	+	0.775741i	$0.282624\pi$
$24$	22528.0	1.62963
$25$	15625.0	1.00000
$26$	0	0
$27$	−21032.0	−1.06854
$28$	33536.0	1.52770
$29$	44858.0	1.83927	0.919636	−	0.392772i	$-0.128484\pi$
$29$	44858.0	1.83927	0.919636	+	0.392772i	$0.128484\pi$
$30$	−44000.0	−1.62963
$31$	0	0	1.00000			$0$
$31$	0	0	−1.00000			$\pi$
$32$	−32768.0	−1.00000
$33$	0	0
$34$	0	0
$35$	−65500.0	−1.52770
$36$	77248.0	1.65569
$37$	0	0	1.00000			$0$
$37$	0	0	−1.00000			$\pi$
$38$	0	0
$39$	0	0
$40$	64000.0	1.00000
$41$	−74338.0	−1.07860	−0.539299	−	0.842115i	$-0.681311\pi$
$41$	−74338.0	−1.07860	−0.539299	+	0.842115i	$0.681311\pi$
$42$	184448.	2.48958
$43$	−17404.0	−0.218899	−0.109449	−	0.993992i	$-0.534909\pi$
$43$	−17404.0	−0.218899	−0.109449	+	0.993992i	$0.534909\pi$
$44$	0	0
$45$	−150875.	−1.65569
$46$	−122848.	−1.26210
$47$	26444.0	0.254703	0.127351	−	0.991858i	$-0.459352\pi$
$47$	26444.0	0.254703	0.127351	+	0.991858i	$0.459352\pi$
$48$	−180224.	−1.62963
$49$	156927.	1.33386
$50$	−125000.	−1.00000
$51$	0	0
$52$	0	0
$53$	0	0	1.00000			$0$
$53$	0	0	−1.00000			$\pi$
$54$	168256.	1.06854
$55$	0	0
$56$	−268288.	−1.52770
$57$	0	0
$58$	−358864.	−1.83927
$59$	0	0	1.00000			$0$
$59$	0	0	−1.00000			$\pi$
$60$	352000.	1.62963
$61$	452342.	1.99286	0.996431	−	0.0844063i	$-0.0268994\pi$
$61$	452342.	1.99286	0.996431	+	0.0844063i	$0.0268994\pi$
$62$	0	0
$63$	632468.	2.52940
$64$	262144.	1.00000
$65$	0	0
$66$	0	0
$67$	−1276.00	−0.00424254	−0.00212127	−	0.999998i	$-0.500675\pi$
$67$	−1276.00	−0.00424254	−0.00212127	+	0.999998i	$0.500675\pi$
$68$	0	0
$69$	−675664.	−2.05676
$70$	524000.	1.52770
$71$	0	0	1.00000			$0$
$71$	0	0	−1.00000			$\pi$
$72$	−617984.	−1.65569
$73$	0	0	1.00000			$0$
$73$	0	0	−1.00000			$\pi$
$74$	0	0
$75$	−687500.	−1.62963
$76$	0	0
$77$	0	0
$78$	0	0
$79$	0	0	1.00000			$0$
$79$	0	0	−1.00000			$\pi$
$80$	−512000.	−1.00000
$81$	45505.0	0.0856257
$82$	594704.	1.07860
$83$	1.13172e6	1.97926	0.989631	−	0.143635i	$-0.0458791\pi$
$83$	1.13172e6	1.97926	0.989631	+	0.143635i	$0.0458791\pi$
$84$	−1.47558e6	−2.48958
$85$	0	0
$86$	139232.	0.218899
$87$	−1.97375e6	−2.99733
$88$	0	0
$89$	511058.	0.724937	0.362468	−	0.931996i	$-0.381934\pi$
$89$	511058.	0.724937	0.362468	+	0.931996i	$0.381934\pi$
$90$	1.20700e6	1.65569
$91$	0	0
$92$	982784.	1.26210
$93$	0	0
$94$	−211552.	−0.254703
$95$	0	0
$96$	1.44179e6	1.62963
$97$	0	0	1.00000			$0$
$97$	0	0	−1.00000			$\pi$
$98$	−1.25542e6	−1.33386
$99$	0	0
$100$	1.00000e6	1.00000
$101$	−1.91772e6	−1.86132	−0.930659	−	0.365887i	$-0.880766\pi$
$101$	−1.91772e6	−1.86132	−0.930659	+	0.365887i	$0.880766\pi$
$102$	0	0
$103$	−1.31520e6	−1.20360	−0.601799	−	0.798648i	$-0.705549\pi$
$103$	−1.31520e6	−1.20360	−0.601799	+	0.798648i	$0.705549\pi$
$104$	0	0
$105$	2.88200e6	2.48958
$106$	0	0
$107$	2.35240e6	1.92026	0.960131	−	0.279550i	$-0.0901852\pi$
$107$	2.35240e6	1.92026	0.960131	+	0.279550i	$0.0901852\pi$
$108$	−1.34605e6	−1.06854
$109$	−1.29956e6	−1.00350	−0.501750	−	0.865013i	$-0.667310\pi$
$109$	−1.29956e6	−1.00350	−0.501750	+	0.865013i	$0.667310\pi$
$110$	0	0
$111$	0	0
$112$	2.14630e6	1.52770
$113$	0	0	1.00000			$0$
$113$	0	0	−1.00000			$\pi$
$114$	0	0
$115$	−1.91950e6	−1.26210
$116$	2.87091e6	1.83927
$117$	0	0
$118$	0	0
$119$	0	0
$120$	−2.81600e6	−1.62963
$121$	1.77156e6	1.00000
$122$	−3.61874e6	−1.99286
$123$	3.27087e6	1.75771
$124$	0	0
$125$	−1.95312e6	−1.00000
$126$	−5.05974e6	−2.52940
$127$	1.72492e6	0.842091	0.421045	−	0.907040i	$-0.361663\pi$
$127$	1.72492e6	0.842091	0.421045	+	0.907040i	$0.361663\pi$
$128$	−2.09715e6	−1.00000
$129$	765776.	0.356724
$130$	0	0
$131$	0	0	1.00000			$0$
$131$	0	0	−1.00000			$\pi$
$132$	0	0
$133$	0	0
$134$	10208.0	0.00424254
$135$	2.62900e6	1.06854
$136$	0	0
$137$	0	0	1.00000			$0$
$137$	0	0	−1.00000			$\pi$
$138$	5.40531e6	2.05676
$139$	0	0	1.00000			$0$
$139$	0	0	−1.00000			$\pi$
$140$	−4.19200e6	−1.52770
$141$	−1.16354e6	−0.415071
$142$	0	0
$143$	0	0
$144$	4.94387e6	1.65569
$145$	−5.60725e6	−1.83927
$146$	0	0
$147$	−6.90479e6	−2.17369
$148$	0	0
$149$	2.96932e6	0.897631	0.448816	−	0.893624i	$-0.351846\pi$
$149$	2.96932e6	0.897631	0.448816	+	0.893624i	$0.351846\pi$
$150$	5.50000e6	1.62963
$151$	0	0	1.00000			$0$
$151$	0	0	−1.00000			$\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	0	0	1.00000			$0$
$157$	0	0	−1.00000			$\pi$
$158$	0	0
$159$	0	0
$160$	4.09600e6	1.00000
$161$	8.04654e6	1.92811
$162$	−364040.	−0.0856257
$163$	−8.66100e6	−1.99989	−0.999943	−	0.0106368i	$-0.996614\pi$
$163$	−8.66100e6	−1.99989	−0.999943	+	0.0106368i	$0.996614\pi$
$164$	−4.75763e6	−1.07860
$165$	0	0
$166$	−9.05373e6	−1.97926
$167$	5.88796e6	1.26420	0.632100	−	0.774887i	$-0.282193\pi$
$167$	5.88796e6	1.26420	0.632100	+	0.774887i	$0.282193\pi$
$168$	1.18047e7	2.48958
$169$	4.82681e6	1.00000
$170$	0	0
$171$	0	0
$172$	−1.11386e6	−0.218899
$173$	0	0	1.00000			$0$
$173$	0	0	−1.00000			$\pi$
$174$	1.57900e7	2.99733
$175$	8.18750e6	1.52770
$176$	0	0
$177$	0	0
$178$	−4.08846e6	−0.724937
$179$	0	0	1.00000			$0$
$179$	0	0	−1.00000			$\pi$
$180$	−9.65600e6	−1.65569
$181$	−1.06974e7	−1.80402	−0.902012	−	0.431710i	$-0.857910\pi$
$181$	−1.06974e7	−1.80402	−0.902012	+	0.431710i	$0.857910\pi$
$182$	0	0
$183$	−1.99030e7	−3.24763
$184$	−7.86227e6	−1.26210
$185$	0	0
$186$	0	0
$187$	0	0
$188$	1.69242e6	0.254703
$189$	−1.10208e7	−1.63240
$190$	0	0
$191$	0	0	1.00000			$0$
$191$	0	0	−1.00000			$\pi$
$192$	−1.15343e7	−1.62963
$193$	0	0	1.00000			$0$
$193$	0	0	−1.00000			$\pi$
$194$	0	0
$195$	0	0
$196$	1.00433e7	1.33386
$197$	0	0	1.00000			$0$
$197$	0	0	−1.00000			$\pi$
$198$	0	0
$199$	0	0	1.00000			$0$
$199$	0	0	−1.00000			$\pi$
$200$	−8.00000e6	−1.00000
$201$	56144.0	0.00691377
$202$	1.53417e7	1.86132
$203$	2.35056e7	2.80985
$204$	0	0
$205$	9.29225e6	1.07860
$206$	1.05216e7	1.20360
$207$	1.85347e7	2.08965
$208$	0	0
$209$	0	0
$210$	−2.30560e7	−2.48958
$211$	0	0	1.00000			$0$
$211$	0	0	−1.00000			$\pi$
$212$	0	0
$213$	0	0
$214$	−1.88192e7	−1.92026
$215$	2.17550e6	0.218899
$216$	1.07684e7	1.06854
$217$	0	0
$218$	1.03965e7	1.00350
$219$	0	0
$220$	0	0
$221$	0	0
$222$	0	0
$223$	−1.78363e7	−1.60839	−0.804194	−	0.594367i	$-0.797403\pi$
$223$	−1.78363e7	−1.60839	−0.804194	+	0.594367i	$0.797403\pi$
$224$	−1.71704e7	−1.52770
$225$	1.88594e7	1.65569
$226$	0	0
$227$	−9.91496e6	−0.847643	−0.423822	−	0.905746i	$-0.639312\pi$
$227$	−9.91496e6	−0.847643	−0.423822	+	0.905746i	$0.639312\pi$
$228$	0	0
$229$	2.32339e7	1.93471	0.967354	−	0.253427i	$-0.0815578\pi$
$229$	2.32339e7	1.93471	0.967354	+	0.253427i	$0.0815578\pi$
$230$	1.53560e7	1.26210
$231$	0	0
$232$	−2.29673e7	−1.83927
$233$	0	0	1.00000			$0$
$233$	0	0	−1.00000			$\pi$
$234$	0	0
$235$	−3.30550e6	−0.254703
$236$	0	0
$237$	0	0
$238$	0	0
$239$	0	0	1.00000			$0$
$239$	0	0	−1.00000			$\pi$
$240$	2.25280e7	1.62963
$241$	−2.50778e7	−1.79159	−0.895794	−	0.444470i	$-0.853392\pi$
$241$	−2.50778e7	−1.79159	−0.895794	+	0.444470i	$0.853392\pi$
$242$	−1.41725e7	−1.00000
$243$	1.33301e7	0.928998
$244$	2.89499e7	1.99286
$245$	−1.96159e7	−1.33386
$246$	−2.61670e7	−1.75771
$247$	0	0
$248$	0	0
$249$	−4.97955e7	−3.22546
$250$	1.56250e7	1.00000
$251$	0	0	1.00000			$0$
$251$	0	0	−1.00000			$\pi$
$252$	4.04780e7	2.52940
$253$	0	0
$254$	−1.37994e7	−0.842091
$255$	0	0
$256$	1.67772e7	1.00000
$257$	0	0	1.00000			$0$
$257$	0	0	−1.00000			$\pi$
$258$	−6.12621e6	−0.356724
$259$	0	0
$260$	0	0
$261$	5.41436e7	3.04527
$262$	0	0
$263$	−3.60257e7	−1.98036	−0.990182	−	0.139785i	$-0.955359\pi$
$263$	−3.60257e7	−1.98036	−0.990182	+	0.139785i	$0.955359\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−2.24866e7	−1.18138
$268$	−81664.0	−0.00424254
$269$	−8.19428e6	−0.420973	−0.210486	−	0.977597i	$-0.567505\pi$
$269$	−8.19428e6	−0.420973	−0.210486	+	0.977597i	$0.567505\pi$
$270$	−2.10320e7	−1.06854
$271$	0	0	1.00000			$0$
$271$	0	0	−1.00000			$\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	−4.32425e7	−2.05676
$277$	0	0	1.00000			$0$
$277$	0	0	−1.00000			$\pi$
$278$	0	0
$279$	0	0
$280$	3.35360e7	1.52770
$281$	2.59825e7	1.17101	0.585506	−	0.810668i	$-0.300896\pi$
$281$	2.59825e7	1.17101	0.585506	+	0.810668i	$0.300896\pi$
$282$	9.30829e6	0.415071
$283$	4.43699e7	1.95762	0.978811	−	0.204765i	$-0.0656430\pi$
$283$	4.43699e7	1.95762	0.978811	+	0.204765i	$0.0656430\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−3.89531e7	−1.64777
$288$	−3.95510e7	−1.65569
$289$	2.41376e7	1.00000
$290$	4.48580e7	1.83927
$291$	0	0
$292$	0	0
$293$	0	0	1.00000			$0$
$293$	0	0	−1.00000			$\pi$
$294$	5.52383e7	2.17369
$295$	0	0
$296$	0	0
$297$	0	0
$298$	−2.37545e7	−0.897631
$299$	0	0
$300$	−4.40000e7	−1.62963
$301$	−9.11970e6	−0.334411
$302$	0	0
$303$	8.43796e7	3.03326
$304$	0	0
$305$	−5.65428e7	−1.99286
$306$	0	0
$307$	4.31915e7	1.49274	0.746369	−	0.665533i	$-0.231796\pi$
$307$	4.31915e7	1.49274	0.746369	+	0.665533i	$0.231796\pi$
$308$	0	0
$309$	5.78690e7	1.96142
$310$	0	0
$311$	0	0	1.00000			$0$
$311$	0	0	−1.00000			$\pi$
$312$	0	0
$313$	0	0	1.00000			$0$
$313$	0	0	−1.00000			$\pi$
$314$	0	0
$315$	−7.90585e7	−2.52940
$316$	0	0
$317$	0	0	1.00000			$0$
$317$	0	0	−1.00000			$\pi$
$318$	0	0
$319$	0	0
$320$	−3.27680e7	−1.00000
$321$	−1.03506e8	−3.12932
$322$	−6.43724e7	−1.92811
$323$	0	0
$324$	2.91232e6	0.0856257
$325$	0	0
$326$	6.92880e7	1.99989
$327$	5.71807e7	1.63533
$328$	3.80611e7	1.07860
$329$	1.38567e7	0.389109
$330$	0	0
$331$	0	0	1.00000			$0$
$331$	0	0	−1.00000			$\pi$
$332$	7.24298e7	1.97926
$333$	0	0
$334$	−4.71037e7	−1.26420
$335$	159500.	0.00424254
$336$	−9.44374e7	−2.48958
$337$	0	0	1.00000			$0$
$337$	0	0	−1.00000			$\pi$
$338$	−3.86145e7	−1.00000
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	2.05817e7	0.510033
$344$	8.91085e6	0.218899
$345$	8.44580e7	2.05676
$346$	0	0
$347$	−4.03414e7	−0.965524	−0.482762	−	0.875752i	$-0.660366\pi$
$347$	−4.03414e7	−0.965524	−0.482762	+	0.875752i	$0.660366\pi$
$348$	−1.26320e8	−2.99733
$349$	8.02822e6	0.188861	0.0944306	−	0.995531i	$-0.469897\pi$
$349$	8.02822e6	0.188861	0.0944306	+	0.995531i	$0.469897\pi$
$350$	−6.55000e7	−1.52770
$351$	0	0
$352$	0	0
$353$	0	0	1.00000			$0$
$353$	0	0	−1.00000			$\pi$
$354$	0	0
$355$	0	0
$356$	3.27077e7	0.724937
$357$	0	0
$358$	0	0
$359$	0	0	1.00000			$0$
$359$	0	0	−1.00000			$\pi$
$360$	7.72480e7	1.65569
$361$	4.70459e7	1.00000
$362$	8.55792e7	1.80402
$363$	−7.79487e7	−1.62963
$364$	0	0
$365$	0	0
$366$	1.59224e8	3.24763
$367$	−8.69589e7	−1.75920	−0.879601	−	0.475711i	$-0.842191\pi$
$367$	−8.69589e7	−1.75920	−0.879601	+	0.475711i	$0.842191\pi$
$368$	6.28982e7	1.26210
$369$	−8.97260e7	−1.78583
$370$	0	0
$371$	0	0
$372$	0	0
$373$	0	0	1.00000			$0$
$373$	0	0	−1.00000			$\pi$
$374$	0	0
$375$	8.59375e7	1.62963
$376$	−1.35393e7	−0.254703
$377$	0	0
$378$	8.81661e7	1.63240
$379$	0	0	1.00000			$0$
$379$	0	0	−1.00000			$\pi$
$380$	0	0
$381$	−7.58967e7	−1.37230
$382$	0	0
$383$	−1.92778e7	−0.343131	−0.171566	−	0.985173i	$-0.554883\pi$
$383$	−1.92778e7	−0.343131	−0.171566	+	0.985173i	$0.554883\pi$
$384$	9.22747e7	1.62963
$385$	0	0
$386$	0	0
$387$	−2.10066e7	−0.362429
$388$	0	0
$389$	8.34581e7	1.41782	0.708908	−	0.705301i	$-0.249188\pi$
$389$	8.34581e7	1.41782	0.708908	+	0.705301i	$0.249188\pi$
$390$	0	0
$391$	0	0
$392$	−8.03466e7	−1.33386
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	0	0	1.00000			$0$
$397$	0	0	−1.00000			$\pi$
$398$	0	0
$399$	0	0
$400$	6.40000e7	1.00000
$401$	1.21373e8	1.88230	0.941152	−	0.337982i	$-0.109744\pi$
$401$	1.21373e8	1.88230	0.941152	+	0.337982i	$0.109744\pi$
$402$	−449152.	−0.00691377
$403$	0	0
$404$	−1.22734e8	−1.86132
$405$	−5.68812e6	−0.0856257
$406$	−1.88045e8	−2.80985
$407$	0	0
$408$	0	0
$409$	−1.13372e8	−1.65704	−0.828522	−	0.559956i	$-0.810818\pi$
$409$	−1.13372e8	−1.65704	−0.828522	+	0.559956i	$0.810818\pi$
$410$	−7.43380e7	−1.07860
$411$	0	0
$412$	−8.41731e7	−1.20360
$413$	0	0
$414$	−1.48278e8	−2.08965
$415$	−1.41464e8	−1.97926
$416$	0	0
$417$	0	0
$418$	0	0
$419$	0	0	1.00000			$0$
$419$	0	0	−1.00000			$\pi$
$420$	1.84448e8	2.48958
$421$	−5.72305e7	−0.766975	−0.383487	−	0.923546i	$-0.625277\pi$
$421$	−5.72305e7	−0.766975	−0.383487	+	0.923546i	$0.625277\pi$
$422$	0	0
$423$	3.19179e7	0.421709
$424$	0	0
$425$	0	0
$426$	0	0
$427$	2.37027e8	3.04449
$428$	1.50554e8	1.92026
$429$	0	0
$430$	−1.74040e7	−0.218899
$431$	0	0	1.00000			$0$
$431$	0	0	−1.00000			$\pi$
$432$	−8.61471e7	−1.06854
$433$	0	0	1.00000			$0$
$433$	0	0	−1.00000			$\pi$
$434$	0	0
$435$	2.46719e8	2.99733
$436$	−8.31720e7	−1.00350
$437$	0	0
$438$	0	0
$439$	0	0	1.00000			$0$
$439$	0	0	−1.00000			$\pi$
$440$	0	0
$441$	1.89411e8	2.20846
$442$	0	0
$443$	−3.57157e7	−0.410817	−0.205408	−	0.978676i	$-0.565852\pi$
$443$	−3.57157e7	−0.410817	−0.205408	+	0.978676i	$0.565852\pi$
$444$	0	0
$445$	−6.38822e7	−0.724937
$446$	1.42691e8	1.60839
$447$	−1.30650e8	−1.46281
$448$	1.37363e8	1.52770
$449$	−1.77667e8	−1.96276	−0.981380	−	0.192076i	$-0.938478\pi$
$449$	−1.77667e8	−1.96276	−0.981380	+	0.192076i	$0.938478\pi$
$450$	−1.50875e8	−1.65569
$451$	0	0
$452$	0	0
$453$	0	0
$454$	7.93196e7	0.847643
$455$	0	0
$456$	0	0
$457$	0	0	1.00000			$0$
$457$	0	0	−1.00000			$\pi$
$458$	−1.85871e8	−1.93471
$459$	0	0
$460$	−1.22848e8	−1.26210
$461$	6.01196e7	0.613640	0.306820	−	0.951768i	$-0.400735\pi$
$461$	6.01196e7	0.613640	0.306820	+	0.951768i	$0.400735\pi$
$462$	0	0
$463$	−4.53900e7	−0.457317	−0.228658	−	0.973507i	$-0.573434\pi$
$463$	−4.53900e7	−0.457317	−0.228658	+	0.973507i	$0.573434\pi$
$464$	1.83738e8	1.83927
$465$	0	0
$466$	0	0
$467$	7.92225e7	0.777854	0.388927	−	0.921269i	$-0.372846\pi$
$467$	7.92225e7	0.777854	0.388927	+	0.921269i	$0.372846\pi$
$468$	0	0
$469$	−668624.	−0.00648132
$470$	2.64440e7	0.254703
$471$	0	0
$472$	0	0
$473$	0	0
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	0	0
$479$	0	0	1.00000			$0$
$479$	0	0	−1.00000			$\pi$
$480$	−1.80224e8	−1.62963
$481$	0	0
$482$	2.00622e8	1.79159
$483$	−3.54048e8	−3.14210
$484$	1.13380e8	1.00000
$485$	0	0
$486$	−1.06641e8	−0.928998
$487$	2.30989e8	1.99989	0.999943	−	0.0106586i	$-0.00339280\pi$
$487$	2.30989e8	1.99989	0.999943	+	0.0106586i	$0.00339280\pi$
$488$	−2.31599e8	−1.99286
$489$	3.81084e8	3.25907
$490$	1.56927e8	1.33386
$491$	0	0	1.00000			$0$
$491$	0	0	−1.00000			$\pi$
$492$	2.09336e8	1.75771
$493$	0	0
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	3.98364e8	3.22546
$499$	0	0	1.00000			$0$
$499$	0	0	−1.00000			$\pi$
$500$	−1.25000e8	−1.00000
$501$	−2.59070e8	−2.06018
$502$	0	0
$503$	−7.33066e7	−0.576022	−0.288011	−	0.957627i	$-0.592994\pi$
$503$	−7.33066e7	−0.576022	−0.288011	+	0.957627i	$0.592994\pi$
$504$	−3.23824e8	−2.52940
$505$	2.39715e8	1.86132
$506$	0	0
$507$	−2.12380e8	−1.62963
$508$	1.10395e8	0.842091
$509$	−1.83714e8	−1.39312	−0.696559	−	0.717500i	$-0.745286\pi$
$509$	−1.83714e8	−1.39312	−0.696559	+	0.717500i	$0.745286\pi$
$510$	0	0
$511$	0	0
$512$	−1.34218e8	−1.00000
$513$	0	0
$514$	0	0
$515$	1.64400e8	1.20360
$516$	4.90097e7	0.356724
$517$	0	0
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−2.11574e8	−1.49606	−0.748031	−	0.663664i	$-0.769000\pi$
$521$	−2.11574e8	−1.49606	−0.748031	+	0.663664i	$0.769000\pi$
$522$	−4.33149e8	−3.04527
$523$	−1.30165e8	−0.909893	−0.454946	−	0.890519i	$-0.650342\pi$
$523$	−1.30165e8	−0.909893	−0.454946	+	0.890519i	$0.650342\pi$
$524$	0	0
$525$	−3.60250e8	−2.48958
$526$	2.88205e8	1.98036
$527$	0	0
$528$	0	0
$529$	8.77708e7	0.592902
$530$	0	0
$531$	0	0
$532$	0	0
$533$	0	0
$534$	1.79892e8	1.18138
$535$	−2.94050e8	−1.92026
$536$	653312.	0.00424254
$537$	0	0
$538$	6.55543e7	0.420973
$539$	0	0
$540$	1.68256e8	1.06854
$541$	−2.70414e8	−1.70780	−0.853900	−	0.520438i	$-0.825769\pi$
$541$	−2.70414e8	−1.70780	−0.853900	+	0.520438i	$0.825769\pi$
$542$	0	0
$543$	4.70686e8	2.93989
$544$	0	0
$545$	1.62445e8	1.00350
$546$	0	0
$547$	−3.00733e8	−1.83747	−0.918733	−	0.394880i	$-0.870786\pi$
$547$	−3.00733e8	−1.83747	−0.918733	+	0.394880i	$0.870786\pi$
$548$	0	0
$549$	5.45977e8	3.29957
$550$	0	0
$551$	0	0
$552$	3.45940e8	2.05676
$553$	0	0
$554$	0	0
$555$	0	0
$556$	0	0
$557$	0	0	1.00000			$0$
$557$	0	0	−1.00000			$\pi$
$558$	0	0
$559$	0	0
$560$	−2.68288e8	−1.52770
$561$	0	0
$562$	−2.07860e8	−1.17101
$563$	3.51215e8	1.96810	0.984052	−	0.177881i	$-0.0569241\pi$
$563$	3.51215e8	1.96810	0.984052	+	0.177881i	$0.0569241\pi$
$564$	−7.44663e7	−0.415071
$565$	0	0
$566$	−3.54959e8	−1.95762
$567$	2.38446e7	0.130810
$568$	0	0
$569$	−1.49518e8	−0.811630	−0.405815	−	0.913955i	$-0.633012\pi$
$569$	−1.49518e8	−0.811630	−0.405815	+	0.913955i	$0.633012\pi$
$570$	0	0
$571$	0	0	1.00000			$0$
$571$	0	0	−1.00000			$\pi$
$572$	0	0
$573$	0	0
$574$	3.11625e8	1.64777
$575$	2.39938e8	1.26210
$576$	3.16408e8	1.65569
$577$	0	0	1.00000			$0$
$577$	0	0	−1.00000			$\pi$
$578$	−1.93101e8	−1.00000
$579$	0	0
$580$	−3.58864e8	−1.83927
$581$	5.93019e8	3.02371
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	0	0
$587$	1.33498e8	0.660026	0.330013	−	0.943976i	$-0.392947\pi$
$587$	1.33498e8	0.660026	0.330013	+	0.943976i	$0.392947\pi$
$588$	−4.41906e8	−2.17369
$589$	0	0
$590$	0	0
$591$	0	0
$592$	0	0
$593$	0	0	1.00000			$0$
$593$	0	0	−1.00000			$\pi$
$594$	0	0
$595$	0	0
$596$	1.90036e8	0.897631
$597$	0	0
$598$	0	0
$599$	0	0	1.00000			$0$
$599$	0	0	−1.00000			$\pi$
$600$	3.52000e8	1.62963
$601$	3.79911e8	1.75008	0.875042	−	0.484047i	$-0.160834\pi$
$601$	3.79911e8	1.75008	0.875042	+	0.484047i	$0.160834\pi$
$602$	7.29576e7	0.334411
$603$	−1.54013e6	−0.00702435
$604$	0	0
$605$	−2.21445e8	−1.00000
$606$	−6.75037e8	−3.03326
$607$	1.69713e8	0.758839	0.379419	−	0.925225i	$-0.376124\pi$
$607$	1.69713e8	0.758839	0.379419	+	0.925225i	$0.376124\pi$
$608$	0	0
$609$	−1.03425e9	−4.57901
$610$	4.52342e8	1.99286
$611$	0	0
$612$	0	0
$613$	0	0	1.00000			$0$
$613$	0	0	−1.00000			$\pi$
$614$	−3.45532e8	−1.49274
$615$	−4.08859e8	−1.75771
$616$	0	0
$617$	0	0	1.00000			$0$
$617$	0	0	−1.00000			$\pi$
$618$	−4.62952e8	−1.96142
$619$	0	0	1.00000			$0$
$619$	0	0	−1.00000			$\pi$
$620$	0	0
$621$	−3.22967e8	−1.34860
$622$	0	0
$623$	2.67794e8	1.10748
$624$	0	0
$625$	2.44141e8	1.00000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	0	0
$630$	6.32468e8	2.52940
$631$	0	0	1.00000			$0$
$631$	0	0	−1.00000			$\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−2.15616e8	−0.842091
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	2.62144e8	1.00000
$641$	−7.10123e7	−0.269625	−0.134812	−	0.990871i	$-0.543043\pi$
$641$	−7.10123e7	−0.269625	−0.134812	+	0.990871i	$0.543043\pi$
$642$	8.28046e8	3.12932
$643$	−4.35161e8	−1.63688	−0.818440	−	0.574592i	$-0.805161\pi$
$643$	−4.35161e8	−1.63688	−0.818440	+	0.574592i	$0.805161\pi$
$644$	5.14979e8	1.92811
$645$	−9.57220e7	−0.356724
$646$	0	0
$647$	−3.26848e8	−1.20679	−0.603396	−	0.797441i	$-0.706186\pi$
$647$	−3.26848e8	−1.20679	−0.603396	+	0.797441i	$0.706186\pi$
$648$	−2.32986e7	−0.0856257
$649$	0	0
$650$	0	0
$651$	0	0
$652$	−5.54304e8	−1.99989
$653$	0	0	1.00000			$0$
$653$	0	0	−1.00000			$\pi$
$654$	−4.57446e8	−1.63533
$655$	0	0
$656$	−3.04488e8	−1.07860
$657$	0	0
$658$	−1.10853e8	−0.389109
$659$	0	0	1.00000			$0$
$659$	0	0	−1.00000			$\pi$
$660$	0	0
$661$	3.64144e8	1.26086	0.630432	−	0.776244i	$-0.282878\pi$
$661$	3.64144e8	1.26086	0.630432	+	0.776244i	$0.282878\pi$
$662$	0	0
$663$	0	0
$664$	−5.79439e8	−1.97926
$665$	0	0
$666$	0	0
$667$	6.88839e8	2.32135
$668$	3.76830e8	1.26420
$669$	7.84798e8	2.62108
$670$	−1.27600e6	−0.00424254
$671$	0	0
$672$	7.55499e8	2.48958
$673$	0	0	1.00000			$0$
$673$	0	0	−1.00000			$\pi$
$674$	0	0
$675$	−3.28625e8	−1.06854
$676$	3.08916e8	1.00000
$677$	0	0	1.00000			$0$
$677$	0	0	−1.00000			$\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	4.36258e8	1.38134
$682$	0	0
$683$	6.06224e8	1.90270	0.951352	−	0.308107i	$-0.0996956\pi$
$683$	6.06224e8	1.90270	0.951352	+	0.308107i	$0.0996956\pi$
$684$	0	0
$685$	0	0
$686$	−1.64653e8	−0.510033
$687$	−1.02229e9	−3.15286
$688$	−7.12868e7	−0.218899
$689$	0	0
$690$	−6.75664e8	−2.05676
$691$	0	0	1.00000			$0$
$691$	0	0	−1.00000			$\pi$
$692$	0	0
$693$	0	0
$694$	3.22731e8	0.965524
$695$	0	0
$696$	1.01056e9	2.99733
$697$	0	0
$698$	−6.42257e7	−0.188861
$699$	0	0
$700$	5.24000e8	1.52770
$701$	−5.95860e8	−1.72978	−0.864889	−	0.501963i	$-0.832611\pi$
$701$	−5.95860e8	−1.72978	−0.864889	+	0.501963i	$0.832611\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	1.45442e8	0.415071
$706$	0	0
$707$	−1.00488e9	−2.84353
$708$	0	0
$709$	−7.12546e8	−1.99928	−0.999641	−	0.0268005i	$-0.991468\pi$
$709$	−7.12546e8	−1.99928	−0.999641	+	0.0268005i	$0.991468\pi$
$710$	0	0
$711$	0	0
$712$	−2.61662e8	−0.724937
$713$	0	0
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	0	0	1.00000			$0$
$719$	0	0	−1.00000			$\pi$
$720$	−6.17984e8	−1.65569
$721$	−6.89167e8	−1.83873
$722$	−3.76367e8	−1.00000
$723$	1.10342e9	2.91962
$724$	−6.84633e8	−1.80402
$725$	7.00906e8	1.83927
$726$	6.23589e8	1.62963
$727$	6.84267e8	1.78083	0.890415	−	0.455150i	$-0.150414\pi$
$727$	6.84267e8	1.78083	0.890415	+	0.455150i	$0.150414\pi$
$728$	0	0
$729$	−6.19698e8	−1.59955
$730$	0	0
$731$	0	0
$732$	−1.27380e9	−3.24763
$733$	0	0	1.00000			$0$
$733$	0	0	−1.00000			$\pi$
$734$	6.95671e8	1.75920
$735$	8.63098e8	2.17369
$736$	−5.03185e8	−1.26210
$737$	0	0
$738$	7.17808e8	1.78583
$739$	0	0	1.00000			$0$
$739$	0	0	−1.00000			$\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−8.19353e8	−1.99758	−0.998790	−	0.0491695i	$-0.984343\pi$
$743$	−8.19353e8	−1.99758	−0.998790	+	0.0491695i	$0.984343\pi$
$744$	0	0
$745$	−3.71165e8	−0.897631
$746$	0	0
$747$	1.36598e9	3.27705
$748$	0	0
$749$	1.23266e9	2.93358
$750$	−6.87500e8	−1.62963
$751$	0	0	1.00000			$0$
$751$	0	0	−1.00000			$\pi$
$752$	1.08315e8	0.254703
$753$	0	0
$754$	0	0
$755$	0	0
$756$	−7.05329e8	−1.63240
$757$	0	0	1.00000			$0$
$757$	0	0	−1.00000			$\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−4.06269e8	−0.921850	−0.460925	−	0.887439i	$-0.652482\pi$
$761$	−4.06269e8	−0.921850	−0.460925	+	0.887439i	$0.652482\pi$
$762$	6.07173e8	1.37230
$763$	−6.80970e8	−1.53304
$764$	0	0
$765$	0	0
$766$	1.54222e8	0.343131
$767$	0	0
$768$	−7.38198e8	−1.62963
$769$	−3.60743e7	−0.0793266	−0.0396633	−	0.999213i	$-0.512629\pi$
$769$	−3.60743e7	−0.0793266	−0.0396633	+	0.999213i	$0.512629\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	0	0	1.00000			$0$
$773$	0	0	−1.00000			$\pi$
$774$	1.68053e8	0.362429
$775$	0	0
$776$	0	0
$777$	0	0
$778$	−6.67665e8	−1.41782
$779$	0	0
$780$	0	0
$781$	0	0
$782$	0	0
$783$	−9.43453e8	−1.96533
$784$	6.42773e8	1.33386
$785$	0	0
$786$	0	0
$787$	−2.13980e8	−0.438983	−0.219492	−	0.975614i	$-0.570440\pi$
$787$	−2.13980e8	−0.438983	−0.219492	+	0.975614i	$0.570440\pi$
$788$	0	0
$789$	1.58513e9	3.22726
$790$	0	0
$791$	0	0
$792$	0	0
$793$	0	0
$794$	0	0
$795$	0	0
$796$	0	0
$797$	0	0	1.00000			$0$
$797$	0	0	−1.00000			$\pi$
$798$	0	0
$799$	0	0
$800$	−5.12000e8	−1.00000
$801$	6.16847e8	1.20027
$802$	−9.70986e8	−1.88230
$803$	0	0
$804$	3.59322e6	0.00691377
$805$	−1.00582e9	−1.92811
$806$	0	0
$807$	3.60548e8	0.686030
$808$	9.81872e8	1.86132
$809$	−3.61673e8	−0.683079	−0.341540	−	0.939867i	$-0.610948\pi$
$809$	−3.61673e8	−0.683079	−0.341540	+	0.939867i	$0.610948\pi$
$810$	4.55050e7	0.0856257
$811$	0	0	1.00000			$0$
$811$	0	0	−1.00000			$\pi$
$812$	1.50436e9	2.80985
$813$	0	0
$814$	0	0
$815$	1.08263e9	1.99989
$816$	0	0
$817$	0	0
$818$	9.06972e8	1.65704
$819$	0	0
$820$	5.94704e8	1.07860
$821$	−1.04853e9	−1.89474	−0.947372	−	0.320135i	$-0.896272\pi$
$821$	−1.04853e9	−1.89474	−0.947372	+	0.320135i	$0.896272\pi$
$822$	0	0
$823$	1.16107e8	0.208285	0.104142	−	0.994562i	$-0.466790\pi$
$823$	1.16107e8	0.208285	0.104142	+	0.994562i	$0.466790\pi$
$824$	6.73384e8	1.20360
$825$	0	0
$826$	0	0
$827$	−1.01116e9	−1.78773	−0.893865	−	0.448337i	$-0.852017\pi$
$827$	−1.01116e9	−1.78773	−0.893865	+	0.448337i	$0.852017\pi$
$828$	1.18622e9	2.08965
$829$	1.81441e8	0.318472	0.159236	−	0.987241i	$-0.449097\pi$
$829$	1.81441e8	0.318472	0.159236	+	0.987241i	$0.449097\pi$
$830$	1.13172e9	1.97926
$831$	0	0
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−7.35996e8	−1.26420
$836$	0	0
$837$	0	0
$838$	0	0
$839$	0	0	1.00000			$0$
$839$	0	0	−1.00000			$\pi$
$840$	−1.47558e9	−2.48958
$841$	1.41742e9	2.38292
$842$	4.57844e8	0.766975
$843$	−1.14323e9	−1.90832
$844$	0	0
$845$	−6.03351e8	−1.00000
$846$	−2.55343e8	−0.421709
$847$	9.28298e8	1.52770
$848$	0	0
$849$	−1.95227e9	−3.19020
$850$	0	0
$851$	0	0
$852$	0	0
$853$	0	0	1.00000			$0$
$853$	0	0	−1.00000			$\pi$
$854$	−1.89622e9	−3.04449
$855$	0	0
$856$	−1.20443e9	−1.92026
$857$	0	0	1.00000			$0$
$857$	0	0	−1.00000			$\pi$
$858$	0	0
$859$	0	0	1.00000			$0$
$859$	0	0	−1.00000			$\pi$
$860$	1.39232e8	0.218899
$861$	1.71394e9	2.68525
$862$	0	0
$863$	−7.23421e8	−1.12553	−0.562767	−	0.826615i	$-0.690263\pi$
$863$	−7.23421e8	−1.12553	−0.562767	+	0.826615i	$0.690263\pi$
$864$	6.89177e8	1.06854
$865$	0	0
$866$	0	0
$867$	−1.06205e9	−1.62963
$868$	0	0
$869$	0	0
$870$	−1.97375e9	−2.99733
$871$	0	0
$872$	6.65376e8	1.00350
$873$	0	0
$874$	0	0
$875$	−1.02344e9	−1.52770
$876$	0	0
$877$	0	0	1.00000			$0$
$877$	0	0	−1.00000			$\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−4.68316e8	−0.684874	−0.342437	−	0.939541i	$-0.611252\pi$
$881$	−4.68316e8	−0.684874	−0.342437	+	0.939541i	$0.611252\pi$
$882$	−1.51529e9	−2.20846
$883$	9.07097e8	1.31756	0.658782	−	0.752334i	$-0.271072\pi$
$883$	9.07097e8	1.31756	0.658782	+	0.752334i	$0.271072\pi$
$884$	0	0
$885$	0	0
$886$	2.85726e8	0.410817
$887$	1.37949e9	1.97674	0.988369	−	0.152074i	$-0.0485953\pi$
$887$	1.37949e9	1.97674	0.988369	+	0.152074i	$0.0485953\pi$
$888$	0	0
$889$	9.03860e8	1.28646
$890$	5.11058e8	0.724937
$891$	0	0
$892$	−1.14152e9	−1.60839
$893$	0	0
$894$	1.04520e9	1.46281
$895$	0	0
$896$	−1.09891e9	−1.52770
$897$	0	0
$898$	1.42133e9	1.96276
$899$	0	0
$900$	1.20700e9	1.65569
$901$	0	0
$902$	0	0
$903$	4.01267e8	0.544966
$904$	0	0
$905$	1.33717e9	1.80402
$906$	0	0
$907$	1.36077e9	1.82374	0.911872	−	0.410474i	$-0.134637\pi$
$907$	1.36077e9	1.82374	0.911872	+	0.410474i	$0.134637\pi$
$908$	−6.34557e8	−0.847643
$909$	−2.31469e9	−3.08177
$910$	0	0
$911$	0	0	1.00000			$0$
$911$	0	0	−1.00000			$\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	2.48788e9	3.24763
$916$	1.48697e9	1.93471
$917$	0	0
$918$	0	0
$919$	0	0	1.00000			$0$
$919$	0	0	−1.00000			$\pi$
$920$	9.82784e8	1.26210
$921$	−1.90043e9	−2.43261
$922$	−4.80957e8	−0.613640
$923$	0	0
$924$	0	0
$925$	0	0
$926$	3.63120e8	0.457317
$927$	−1.58745e9	−1.99279
$928$	−1.46991e9	−1.83927
$929$	1.27758e9	1.59346	0.796731	−	0.604334i	$-0.206561\pi$
$929$	1.27758e9	1.59346	0.796731	+	0.604334i	$0.206561\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	0	0
$934$	−6.33780e8	−0.777854
$935$	0	0
$936$	0	0
$937$	0	0	1.00000			$0$
$937$	0	0	−1.00000			$\pi$
$938$	5.34899e6	0.00648132
$939$	0	0
$940$	−2.11552e8	−0.254703
$941$	3.11817e8	0.374224	0.187112	−	0.982339i	$-0.440087\pi$
$941$	3.11817e8	0.374224	0.187112	+	0.982339i	$0.440087\pi$
$942$	0	0
$943$	−1.14153e9	−1.36130
$944$	0	0
$945$	1.37760e9	1.63240
$946$	0	0
$947$	−1.01744e9	−1.19800	−0.599001	−	0.800749i	$-0.704435\pi$
$947$	−1.01744e9	−1.19800	−0.599001	+	0.800749i	$0.704435\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	0	0
$952$	0	0
$953$	0	0	1.00000			$0$
$953$	0	0	−1.00000			$\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	1.44179e9	1.62963
$961$	8.87504e8	1.00000
$962$	0	0
$963$	2.83935e9	3.17936
$964$	−1.60498e9	−1.79159
$965$	0	0
$966$	2.83238e9	3.14210
$967$	6.69958e8	0.740915	0.370458	−	0.928849i	$-0.379201\pi$
$967$	6.69958e8	0.740915	0.370458	+	0.928849i	$0.379201\pi$
$968$	−9.07039e8	−1.00000
$969$	0	0
$970$	0	0
$971$	0	0	1.00000			$0$
$971$	0	0	−1.00000			$\pi$
$972$	8.53127e8	0.928998
$973$	0	0
$974$	−1.84792e9	−1.99989
$975$	0	0
$976$	1.85279e9	1.99286
$977$	0	0	1.00000			$0$
$977$	0	0	−1.00000			$\pi$
$978$	−3.04867e9	−3.25907
$979$	0	0
$980$	−1.25542e9	−1.33386
$981$	−1.56857e9	−1.66149
$982$	0	0
$983$	−8.00372e8	−0.842619	−0.421310	−	0.906917i	$-0.638429\pi$
$983$	−8.00372e8	−0.842619	−0.421310	+	0.906917i	$0.638429\pi$
$984$	−1.67469e9	−1.75771
$985$	0	0
$986$	0	0
$987$	−6.09693e8	−0.634103
$988$	0	0
$989$	−2.67256e8	−0.276273
$990$	0	0
$991$	0	0	1.00000			$0$
$991$	0	0	−1.00000			$\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	0	0
$996$	−3.18691e9	−3.22546
$997$	0	0	1.00000			$0$
$997$	0	0	−1.00000			$\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	20.7.d.a.19.1	✓	1
3.2	odd	2		180.7.f.b.19.1		1
4.3	odd	2		20.7.d.b.19.1	yes	1
5.2	odd	4		100.7.b.d.51.1		2
5.3	odd	4		100.7.b.d.51.2		2
5.4	even	2		20.7.d.b.19.1	yes	1
8.3	odd	2		320.7.h.a.319.1		1
8.5	even	2		320.7.h.b.319.1		1
12.11	even	2		180.7.f.a.19.1		1
15.14	odd	2		180.7.f.a.19.1		1
20.3	even	4		100.7.b.d.51.1		2
20.7	even	4		100.7.b.d.51.2		2
20.19	odd	2	CM	20.7.d.a.19.1	✓	1
40.19	odd	2		320.7.h.b.319.1		1
40.29	even	2		320.7.h.a.319.1		1
60.59	even	2		180.7.f.b.19.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
20.7.d.a.19.1	✓	1	1.1	even	1	trivial
20.7.d.a.19.1	✓	1	20.19	odd	2	CM
20.7.d.b.19.1	yes	1	4.3	odd	2
20.7.d.b.19.1	yes	1	5.4	even	2
100.7.b.d.51.1		2	5.2	odd	4
100.7.b.d.51.1		2	20.3	even	4
100.7.b.d.51.2		2	5.3	odd	4
100.7.b.d.51.2		2	20.7	even	4
180.7.f.a.19.1		1	12.11	even	2
180.7.f.a.19.1		1	15.14	odd	2
180.7.f.b.19.1		1	3.2	odd	2
180.7.f.b.19.1		1	60.59	even	2
320.7.h.a.319.1		1	8.3	odd	2
320.7.h.a.319.1		1	40.29	even	2
320.7.h.b.319.1		1	8.5	even	2
320.7.h.b.319.1		1	40.19	odd	2

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 20 = 2^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	20.d (of order \(2\), degree \(1\), minimal)

\(n\)	\(11\)	\(17\)
\(\chi(n)\)	\(-1\)	\(-1\)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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