LMFDB - Embedded newform 100.7.b.d.51.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [100,7,Mod(51,100)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(100, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0])) N = Newforms(chi, 7, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("100.51"); S:= CuspForms(chi, 7); N := Newforms(S);

Level:	$ N $	$=$	$ 100 = 2^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 7 $
Character orbit:	$[\chi]$	$=$	100.b (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [2,0] 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:	no
Analytic conductor:	$23.0054083620$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 20)
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			51.1
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	100.51
Dual form			100.7.b.d.51.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-8.00000i q^{2} +44.0000i q^{3} -64.0000 q^{4} +352.000 q^{6} +524.000i q^{7} +512.000i q^{8} -1207.00 q^{9} -2816.00i q^{12} +4192.00 q^{14} +4096.00 q^{16} +9656.00i q^{18} -23056.0 q^{21} -15356.0i q^{23} -22528.0 q^{24} -21032.0i q^{27} -33536.0i q^{28} -44858.0 q^{29} -32768.0i q^{32} +77248.0 q^{36} -74338.0 q^{41} +184448. i q^{42} +17404.0i q^{43} -122848. q^{46} +26444.0i q^{47} +180224. i q^{48} -156927. q^{49} -168256. q^{54} -268288. q^{56} +358864. i q^{58} +452342. q^{61} -632468. i q^{63} -262144. q^{64} -1276.00i q^{67} +675664. q^{69} -617984. i q^{72} +45505.0 q^{81} +594704. i q^{82} -1.13172e6i q^{83} +1.47558e6 q^{84} +139232. q^{86} -1.97375e6i q^{87} -511058. q^{89} +982784. i q^{92} +211552. q^{94} +1.44179e6 q^{96} +1.25542e6i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 128 q^{4} + 704 q^{6} - 2414 q^{9} + 8384 q^{14} + 8192 q^{16} - 46112 q^{21} - 45056 q^{24} - 89716 q^{29} + 154496 q^{36} - 148676 q^{41} - 245696 q^{46} - 313854 q^{49} - 336512 q^{54} - 536576 q^{56}+ \cdots + 2883584 q^{96}+O(q^{100}) $ 2 * q - 128 * q^4 + 704 * q^6 - 2414 * q^9 + 8384 * q^14 + 8192 * q^16 - 46112 * q^21 - 45056 * q^24 - 89716 * q^29 + 154496 * q^36 - 148676 * q^41 - 245696 * q^46 - 313854 * q^49 - 336512 * q^54 - 536576 * q^56 + 904684 * q^61 - 524288 * q^64 + 1351328 * q^69 + 91010 * q^81 + 2951168 * q^84 + 278464 * q^86 - 1022116 * q^89 + 423104 * q^94 + 2883584 * q^96

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/100\mathbb{Z}\right)^\times$.

$n$	$51$	$77$
$\chi(n)$	$-1$	$1$

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 $

$2$	− 8.00000i	− 1.00000i
$2$	− 8.00000i	− 1.00000i
$3$	44.0000i	1.62963i	0.579721	+	0.814815i	$0.303161\pi$
$3$	44.0000i	1.62963i	−0.579721	+	0.814815i	$0.696839\pi$
$4$	−64.0000	−1.00000
$5$	0	0
$5$	0	0
$6$	352.000	1.62963
$7$	524.000i	1.52770i	0.645396	+	0.763848i	$0.276692\pi$
$7$	524.000i	1.52770i	−0.645396	+	0.763848i	$0.723308\pi$
$8$	512.000i	1.00000i
$9$	−1207.00	−1.65569
$10$	0	0
$11$	0	0	1.00000			$0$
$11$	0	0	−1.00000			$\pi$
$12$	− 2816.00i	− 1.62963i
$13$	0	0		−	1.00000i	$-0.5\pi$
$13$	0	0			1.00000i	$0.5\pi$
$14$	4192.00	1.52770
$15$	0	0
$16$	4096.00	1.00000
$17$	0	0		−	1.00000i	$-0.5\pi$
$17$	0	0			1.00000i	$0.5\pi$
$18$	9656.00i	1.65569i
$19$	0	0	1.00000			$0$
$19$	0	0	−1.00000			$\pi$
$20$	0	0
$21$	−23056.0	−2.48958
$22$	0	0
$23$	− 15356.0i	− 1.26210i	−0.775741	−	0.631051i	$-0.782624\pi$
$23$	− 15356.0i	− 1.26210i	0.775741	−	0.631051i	$-0.217376\pi$
$24$	−22528.0	−1.62963
$25$	0	0
$26$	0	0
$27$	− 21032.0i	− 1.06854i
$28$	− 33536.0i	− 1.52770i
$29$	−44858.0	−1.83927	−0.919636	−	0.392772i	$-0.871516\pi$
$29$	−44858.0	−1.83927	−0.919636	+	0.392772i	$0.871516\pi$
$30$	0	0
$31$	0	0	1.00000			$0$
$31$	0	0	−1.00000			$\pi$
$32$	− 32768.0i	− 1.00000i
$33$	0	0
$34$	0	0
$35$	0	0
$36$	77248.0	1.65569
$37$	0	0		−	1.00000i	$-0.5\pi$
$37$	0	0			1.00000i	$0.5\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−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.i	2.48958i
$43$	17404.0i	0.218899i	0.993992	+	0.109449i	$0.0349088\pi$
$43$	17404.0i	0.218899i	−0.993992	+	0.109449i	$0.965091\pi$
$44$	0	0
$45$	0	0
$46$	−122848.	−1.26210
$47$	26444.0i	0.254703i	0.991858	+	0.127351i	$0.0406476\pi$
$47$	26444.0i	0.254703i	−0.991858	+	0.127351i	$0.959352\pi$
$48$	180224.i	1.62963i
$49$	−156927.	−1.33386
$50$	0	0
$51$	0	0
$52$	0	0
$53$	0	0		−	1.00000i	$-0.5\pi$
$53$	0	0			1.00000i	$0.5\pi$
$54$	−168256.	−1.06854
$55$	0	0
$56$	−268288.	−1.52770
$57$	0	0
$58$	358864.i	1.83927i
$59$	0	0	1.00000			$0$
$59$	0	0	−1.00000			$\pi$
$60$	0	0
$61$	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.i	− 2.52940i
$64$	−262144.	−1.00000
$65$	0	0
$66$	0	0
$67$	− 1276.00i	− 0.00424254i	−0.999998	−	0.00212127i	$-0.999325\pi$
$67$	− 1276.00i	− 0.00424254i	0.999998	−	0.00212127i	$-0.000675222\pi$
$68$	0	0
$69$	675664.	2.05676
$70$	0	0
$71$	0	0	1.00000			$0$
$71$	0	0	−1.00000			$\pi$
$72$	− 617984.i	− 1.65569i
$73$	0	0		−	1.00000i	$-0.5\pi$
$73$	0	0			1.00000i	$0.5\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	0	0	1.00000			$0$
$79$	0	0	−1.00000			$\pi$
$80$	0	0
$81$	45505.0	0.0856257
$82$	594704.i	1.07860i
$83$	− 1.13172e6i	− 1.97926i	−0.143635	−	0.989631i	$-0.545879\pi$
$83$	− 1.13172e6i	− 1.97926i	0.143635	−	0.989631i	$-0.454121\pi$
$84$	1.47558e6	2.48958
$85$	0	0
$86$	139232.	0.218899
$87$	− 1.97375e6i	− 2.99733i
$88$	0	0
$89$	−511058.	−0.724937	−0.362468	−	0.931996i	$-0.618066\pi$
$89$	−511058.	−0.724937	−0.362468	+	0.931996i	$0.618066\pi$
$90$	0	0
$91$	0	0
$92$	982784.i	1.26210i
$93$	0	0
$94$	211552.	0.254703
$95$	0	0
$96$	1.44179e6	1.62963
$97$	0	0		−	1.00000i	$-0.5\pi$
$97$	0	0			1.00000i	$0.5\pi$
$98$	1.25542e6i	1.33386i
$99$	0	0
$100$	0	0
$101$	−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
$103$	1.31520e6i	1.20360i	0.798648	+	0.601799i	$0.205549\pi$
$103$	1.31520e6i	1.20360i	−0.798648	+	0.601799i	$0.794451\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	2.35240e6i	1.92026i	0.279550	+	0.960131i	$0.409815\pi$
$107$	2.35240e6i	1.92026i	−0.279550	+	0.960131i	$0.590185\pi$
$108$	1.34605e6i	1.06854i
$109$	1.29956e6	1.00350	0.501750	−	0.865013i	$-0.332690\pi$
$109$	1.29956e6	1.00350	0.501750	+	0.865013i	$0.332690\pi$
$110$	0	0
$111$	0	0
$112$	2.14630e6i	1.52770i
$113$	0	0		−	1.00000i	$-0.5\pi$
$113$	0	0			1.00000i	$0.5\pi$
$114$	0	0
$115$	0	0
$116$	2.87091e6	1.83927
$117$	0	0
$118$	0	0
$119$	0	0
$120$	0	0
$121$	1.77156e6	1.00000
$122$	− 3.61874e6i	− 1.99286i
$123$	− 3.27087e6i	− 1.75771i
$124$	0	0
$125$	0	0
$126$	−5.05974e6	−2.52940
$127$	1.72492e6i	0.842091i	0.907040	+	0.421045i	$0.138337\pi$
$127$	1.72492e6i	0.842091i	−0.907040	+	0.421045i	$0.861663\pi$
$128$	2.09715e6i	1.00000i
$129$	−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
$135$	0	0
$136$	0	0
$137$	0	0		−	1.00000i	$-0.5\pi$
$137$	0	0			1.00000i	$0.5\pi$
$138$	− 5.40531e6i	− 2.05676i
$139$	0	0	1.00000			$0$
$139$	0	0	−1.00000			$\pi$
$140$	0	0
$141$	−1.16354e6	−0.415071
$142$	0	0
$143$	0	0
$144$	−4.94387e6	−1.65569
$145$	0	0
$146$	0	0
$147$	− 6.90479e6i	− 2.17369i
$148$	0	0
$149$	−2.96932e6	−0.897631	−0.448816	−	0.893624i	$-0.648154\pi$
$149$	−2.96932e6	−0.897631	−0.448816	+	0.893624i	$0.648154\pi$
$150$	0	0
$151$	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		−	1.00000i	$-0.5\pi$
$157$	0	0			1.00000i	$0.5\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	8.04654e6	1.92811
$162$	− 364040.i	− 0.0856257i
$163$	8.66100e6i	1.99989i	0.0106368	+	0.999943i	$0.496614\pi$
$163$	8.66100e6i	1.99989i	−0.0106368	+	0.999943i	$0.503386\pi$
$164$	4.75763e6	1.07860
$165$	0	0
$166$	−9.05373e6	−1.97926
$167$	5.88796e6i	1.26420i	0.774887	+	0.632100i	$0.217807\pi$
$167$	5.88796e6i	1.26420i	−0.774887	+	0.632100i	$0.782193\pi$
$168$	− 1.18047e7i	− 2.48958i
$169$	−4.82681e6	−1.00000
$170$	0	0
$171$	0	0
$172$	− 1.11386e6i	− 0.218899i
$173$	0	0		−	1.00000i	$-0.5\pi$
$173$	0	0			1.00000i	$0.5\pi$
$174$	−1.57900e7	−2.99733
$175$	0	0
$176$	0	0
$177$	0	0
$178$	4.08846e6i	0.724937i
$179$	0	0	1.00000			$0$
$179$	0	0	−1.00000			$\pi$
$180$	0	0
$181$	−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
$183$	1.99030e7i	3.24763i
$184$	7.86227e6	1.26210
$185$	0	0
$186$	0	0
$187$	0	0
$188$	− 1.69242e6i	− 0.254703i
$189$	1.10208e7	1.63240
$190$	0	0
$191$	0	0	1.00000			$0$
$191$	0	0	−1.00000			$\pi$
$192$	− 1.15343e7i	− 1.62963i
$193$	0	0		−	1.00000i	$-0.5\pi$
$193$	0	0			1.00000i	$0.5\pi$
$194$	0	0
$195$	0	0
$196$	1.00433e7	1.33386
$197$	0	0		−	1.00000i	$-0.5\pi$
$197$	0	0			1.00000i	$0.5\pi$
$198$	0	0
$199$	0	0	1.00000			$0$
$199$	0	0	−1.00000			$\pi$
$200$	0	0
$201$	56144.0	0.00691377
$202$	1.53417e7i	1.86132i
$203$	− 2.35056e7i	− 2.80985i
$204$	0	0
$205$	0	0
$206$	1.05216e7	1.20360
$207$	1.85347e7i	2.08965i
$208$	0	0
$209$	0	0
$210$	0	0
$211$	0	0	1.00000			$0$
$211$	0	0	−1.00000			$\pi$
$212$	0	0
$213$	0	0
$214$	1.88192e7	1.92026
$215$	0	0
$216$	1.07684e7	1.06854
$217$	0	0
$218$	− 1.03965e7i	− 1.00350i
$219$	0	0
$220$	0	0
$221$	0	0
$222$	0	0
$223$	1.78363e7i	1.60839i	0.594367	+	0.804194i	$0.297403\pi$
$223$	1.78363e7i	1.60839i	−0.594367	+	0.804194i	$0.702597\pi$
$224$	1.71704e7	1.52770
$225$	0	0
$226$	0	0
$227$	− 9.91496e6i	− 0.847643i	−0.905746	−	0.423822i	$-0.860688\pi$
$227$	− 9.91496e6i	− 0.847643i	0.905746	−	0.423822i	$-0.139312\pi$
$228$	0	0
$229$	−2.32339e7	−1.93471	−0.967354	−	0.253427i	$-0.918442\pi$
$229$	−2.32339e7	−1.93471	−0.967354	+	0.253427i	$0.918442\pi$
$230$	0	0
$231$	0	0
$232$	− 2.29673e7i	− 1.83927i
$233$	0	0		−	1.00000i	$-0.5\pi$
$233$	0	0			1.00000i	$0.5\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0	0
$239$	0	0	1.00000			$0$
$239$	0	0	−1.00000			$\pi$
$240$	0	0
$241$	−2.50778e7	−1.79159	−0.895794	−	0.444470i	$-0.853392\pi$
$241$	−2.50778e7	−1.79159	−0.895794	+	0.444470i	$0.853392\pi$
$242$	− 1.41725e7i	− 1.00000i
$243$	− 1.33301e7i	− 0.928998i
$244$	−2.89499e7	−1.99286
$245$	0	0
$246$	−2.61670e7	−1.75771
$247$	0	0
$248$	0	0
$249$	4.97955e7	3.22546
$250$	0	0
$251$	0	0	1.00000			$0$
$251$	0	0	−1.00000			$\pi$
$252$	4.04780e7i	2.52940i
$253$	0	0
$254$	1.37994e7	0.842091
$255$	0	0
$256$	1.67772e7	1.00000
$257$	0	0		−	1.00000i	$-0.5\pi$
$257$	0	0			1.00000i	$0.5\pi$
$258$	6.12621e6i	0.356724i
$259$	0	0
$260$	0	0
$261$	5.41436e7	3.04527
$262$	0	0
$263$	3.60257e7i	1.98036i	0.139785	+	0.990182i	$0.455359\pi$
$263$	3.60257e7i	1.98036i	−0.139785	+	0.990182i	$0.544641\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	− 2.24866e7i	− 1.18138i
$268$	81664.0i	0.00424254i
$269$	8.19428e6	0.420973	0.210486	−	0.977597i	$-0.432495\pi$
$269$	8.19428e6	0.420973	0.210486	+	0.977597i	$0.432495\pi$
$270$	0	0
$271$	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		−	1.00000i	$-0.5\pi$
$277$	0	0			1.00000i	$0.5\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	2.59825e7	1.17101	0.585506	−	0.810668i	$-0.300896\pi$
$281$	2.59825e7	1.17101	0.585506	+	0.810668i	$0.300896\pi$
$282$	9.30829e6i	0.415071i
$283$	− 4.43699e7i	− 1.95762i	−0.204765	−	0.978811i	$-0.565643\pi$
$283$	− 4.43699e7i	− 1.95762i	0.204765	−	0.978811i	$-0.434357\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 3.89531e7i	− 1.64777i
$288$	3.95510e7i	1.65569i
$289$	−2.41376e7	−1.00000
$290$	0	0
$291$	0	0
$292$	0	0
$293$	0	0		−	1.00000i	$-0.5\pi$
$293$	0	0			1.00000i	$0.5\pi$
$294$	−5.52383e7	−2.17369
$295$	0	0
$296$	0	0
$297$	0	0
$298$	2.37545e7i	0.897631i
$299$	0	0
$300$	0	0
$301$	−9.11970e6	−0.334411
$302$	0	0
$303$	− 8.43796e7i	− 3.03326i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	4.31915e7i	1.49274i	0.665533	+	0.746369i	$0.268204\pi$
$307$	4.31915e7i	1.49274i	−0.665533	+	0.746369i	$0.731796\pi$
$308$	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		−	1.00000i	$-0.5\pi$
$313$	0	0			1.00000i	$0.5\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	0	0		−	1.00000i	$-0.5\pi$
$317$	0	0			1.00000i	$0.5\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	−1.03506e8	−3.12932
$322$	− 6.43724e7i	− 1.92811i
$323$	0	0
$324$	−2.91232e6	−0.0856257
$325$	0	0
$326$	6.92880e7	1.99989
$327$	5.71807e7i	1.63533i
$328$	− 3.80611e7i	− 1.07860i
$329$	−1.38567e7	−0.389109
$330$	0	0
$331$	0	0	1.00000			$0$
$331$	0	0	−1.00000			$\pi$
$332$	7.24298e7i	1.97926i
$333$	0	0
$334$	4.71037e7	1.26420
$335$	0	0
$336$	−9.44374e7	−2.48958
$337$	0	0		−	1.00000i	$-0.5\pi$
$337$	0	0			1.00000i	$0.5\pi$
$338$	3.86145e7i	1.00000i
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	− 2.05817e7i	− 0.510033i
$344$	−8.91085e6	−0.218899
$345$	0	0
$346$	0	0
$347$	− 4.03414e7i	− 0.965524i	−0.875752	−	0.482762i	$-0.839634\pi$
$347$	− 4.03414e7i	− 0.965524i	0.875752	−	0.482762i	$-0.160366\pi$
$348$	1.26320e8i	2.99733i
$349$	−8.02822e6	−0.188861	−0.0944306	−	0.995531i	$-0.530103\pi$
$349$	−8.02822e6	−0.188861	−0.0944306	+	0.995531i	$0.530103\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	0	0		−	1.00000i	$-0.5\pi$
$353$	0	0			1.00000i	$0.5\pi$
$354$	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			$\pi$
$360$	0	0
$361$	4.70459e7	1.00000
$362$	8.55792e7i	1.80402i
$363$	7.79487e7i	1.62963i
$364$	0	0
$365$	0	0
$366$	1.59224e8	3.24763
$367$	− 8.69589e7i	− 1.75920i	−0.475711	−	0.879601i	$-0.657809\pi$
$367$	− 8.69589e7i	− 1.75920i	0.475711	−	0.879601i	$-0.342191\pi$
$368$	− 6.28982e7i	− 1.26210i
$369$	8.97260e7	1.78583
$370$	0	0
$371$	0	0
$372$	0	0
$373$	0	0		−	1.00000i	$-0.5\pi$
$373$	0	0			1.00000i	$0.5\pi$
$374$	0	0
$375$	0	0
$376$	−1.35393e7	−0.254703
$377$	0	0
$378$	− 8.81661e7i	− 1.63240i
$379$	0	0	1.00000			$0$
$379$	0	0	−1.00000			$\pi$
$380$	0	0
$381$	−7.58967e7	−1.37230
$382$	0	0
$383$	1.92778e7i	0.343131i	0.985173	+	0.171566i	$0.0548826\pi$
$383$	1.92778e7i	0.343131i	−0.985173	+	0.171566i	$0.945117\pi$
$384$	−9.22747e7	−1.62963
$385$	0	0
$386$	0	0
$387$	− 2.10066e7i	− 0.362429i
$388$	0	0
$389$	−8.34581e7	−1.41782	−0.708908	−	0.705301i	$-0.750812\pi$
$389$	−8.34581e7	−1.41782	−0.708908	+	0.705301i	$0.750812\pi$
$390$	0	0
$391$	0	0
$392$	− 8.03466e7i	− 1.33386i
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	0	0		−	1.00000i	$-0.5\pi$
$397$	0	0			1.00000i	$0.5\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	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.i	− 0.00691377i
$403$	0	0
$404$	1.22734e8	1.86132
$405$	0	0
$406$	−1.88045e8	−2.80985
$407$	0	0
$408$	0	0
$409$	1.13372e8	1.65704	0.828522	−	0.559956i	$-0.189182\pi$
$409$	1.13372e8	1.65704	0.828522	+	0.559956i	$0.189182\pi$
$410$	0	0
$411$	0	0
$412$	− 8.41731e7i	− 1.20360i
$413$	0	0
$414$	1.48278e8	2.08965
$415$	0	0
$416$	0	0
$417$	0	0
$418$	0	0
$419$	0	0	1.00000			$0$
$419$	0	0	−1.00000			$\pi$
$420$	0	0
$421$	−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
$423$	− 3.19179e7i	− 0.421709i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	2.37027e8i	3.04449i
$428$	− 1.50554e8i	− 1.92026i
$429$	0	0
$430$	0	0
$431$	0	0	1.00000			$0$
$431$	0	0	−1.00000			$\pi$
$432$	− 8.61471e7i	− 1.06854i
$433$	0	0		−	1.00000i	$-0.5\pi$
$433$	0	0			1.00000i	$0.5\pi$
$434$	0	0
$435$	0	0
$436$	−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
$443$	3.57157e7i	0.410817i	0.978676	+	0.205408i	$0.0658523\pi$
$443$	3.57157e7i	0.410817i	−0.978676	+	0.205408i	$0.934148\pi$
$444$	0	0
$445$	0	0
$446$	1.42691e8	1.60839
$447$	− 1.30650e8i	− 1.46281i
$448$	− 1.37363e8i	− 1.52770i
$449$	1.77667e8	1.96276	0.981380	−	0.192076i	$-0.0615219\pi$
$449$	1.77667e8	1.96276	0.981380	+	0.192076i	$0.0615219\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	0	0
$454$	−7.93196e7	−0.847643
$455$	0	0
$456$	0	0
$457$	0	0		−	1.00000i	$-0.5\pi$
$457$	0	0			1.00000i	$0.5\pi$
$458$	1.85871e8i	1.93471i
$459$	0	0
$460$	0	0
$461$	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
$463$	4.53900e7i	0.457317i	0.973507	+	0.228658i	$0.0734339\pi$
$463$	4.53900e7i	0.457317i	−0.973507	+	0.228658i	$0.926566\pi$
$464$	−1.83738e8	−1.83927
$465$	0	0
$466$	0	0
$467$	7.92225e7i	0.777854i	0.921269	+	0.388927i	$0.127154\pi$
$467$	7.92225e7i	0.777854i	−0.921269	+	0.388927i	$0.872846\pi$
$468$	0	0
$469$	668624.	0.00648132
$470$	0	0
$471$	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			$\pi$
$480$	0	0
$481$	0	0
$482$	2.00622e8i	1.79159i
$483$	3.54048e8i	3.14210i
$484$	−1.13380e8	−1.00000
$485$	0	0
$486$	−1.06641e8	−0.928998
$487$	2.30989e8i	1.99989i	0.0106586	+	0.999943i	$0.496607\pi$
$487$	2.30989e8i	1.99989i	−0.0106586	+	0.999943i	$0.503393\pi$
$488$	2.31599e8i	1.99286i
$489$	−3.81084e8	−3.25907
$490$	0	0
$491$	0	0	1.00000			$0$
$491$	0	0	−1.00000			$\pi$
$492$	2.09336e8i	1.75771i
$493$	0	0
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	− 3.98364e8i	− 3.22546i
$499$	0	0	1.00000			$0$
$499$	0	0	−1.00000			$\pi$
$500$	0	0
$501$	−2.59070e8	−2.06018
$502$	0	0
$503$	7.33066e7i	0.576022i	0.957627	+	0.288011i	$0.0929939\pi$
$503$	7.33066e7i	0.576022i	−0.957627	+	0.288011i	$0.907006\pi$
$504$	3.23824e8	2.52940
$505$	0	0
$506$	0	0
$507$	− 2.12380e8i	− 1.62963i
$508$	− 1.10395e8i	− 0.842091i
$509$	1.83714e8	1.39312	0.696559	−	0.717500i	$-0.254714\pi$
$509$	1.83714e8	1.39312	0.696559	+	0.717500i	$0.254714\pi$
$510$	0	0
$511$	0	0
$512$	− 1.34218e8i	− 1.00000i
$513$	0	0
$514$	0	0
$515$	0	0
$516$	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	$0.769000\pi$
$522$	− 4.33149e8i	− 3.04527i
$523$	1.30165e8i	0.909893i	0.890519	+	0.454946i	$0.150342\pi$
$523$	1.30165e8i	0.909893i	−0.890519	+	0.454946i	$0.849658\pi$
$524$	0	0
$525$	0	0
$526$	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
$535$	0	0
$536$	653312.	0.00424254
$537$	0	0
$538$	− 6.55543e7i	− 0.420973i
$539$	0	0
$540$	0	0
$541$	−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
$543$	− 4.70686e8i	− 2.93989i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 3.00733e8i	− 1.83747i	−0.394880	−	0.918733i	$-0.629214\pi$
$547$	− 3.00733e8i	− 1.83747i	0.394880	−	0.918733i	$-0.370786\pi$
$548$	0	0
$549$	−5.45977e8	−3.29957
$550$	0	0
$551$	0	0
$552$	3.45940e8i	2.05676i
$553$	0	0
$554$	0	0
$555$	0	0
$556$	0	0
$557$	0	0		−	1.00000i	$-0.5\pi$
$557$	0	0			1.00000i	$0.5\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0	0
$562$	− 2.07860e8i	− 1.17101i
$563$	− 3.51215e8i	− 1.96810i	−0.177881	−	0.984052i	$-0.556924\pi$
$563$	− 3.51215e8i	− 1.96810i	0.177881	−	0.984052i	$-0.443076\pi$
$564$	7.44663e7	0.415071
$565$	0	0
$566$	−3.54959e8	−1.95762
$567$	2.38446e7i	0.130810i
$568$	0	0
$569$	1.49518e8	0.811630	0.405815	−	0.913955i	$-0.366988\pi$
$569$	1.49518e8	0.811630	0.405815	+	0.913955i	$0.366988\pi$
$570$	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
$575$	0	0
$576$	3.16408e8	1.65569
$577$	0	0		−	1.00000i	$-0.5\pi$
$577$	0	0			1.00000i	$0.5\pi$
$578$	1.93101e8i	1.00000i
$579$	0	0
$580$	0	0
$581$	5.93019e8	3.02371
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	0	0
$587$	1.33498e8i	0.660026i	0.943976	+	0.330013i	$0.107053\pi$
$587$	1.33498e8i	0.660026i	−0.943976	+	0.330013i	$0.892947\pi$
$588$	4.41906e8i	2.17369i
$589$	0	0
$590$	0	0
$591$	0	0
$592$	0	0
$593$	0	0		−	1.00000i	$-0.5\pi$
$593$	0	0			1.00000i	$0.5\pi$
$594$	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			$\pi$
$600$	0	0
$601$	3.79911e8	1.75008	0.875042	−	0.484047i	$-0.160834\pi$
$601$	3.79911e8	1.75008	0.875042	+	0.484047i	$0.160834\pi$
$602$	7.29576e7i	0.334411i
$603$	1.54013e6i	0.00702435i
$604$	0	0
$605$	0	0
$606$	−6.75037e8	−3.03326
$607$	1.69713e8i	0.758839i	0.925225	+	0.379419i	$0.123876\pi$
$607$	1.69713e8i	0.758839i	−0.925225	+	0.379419i	$0.876124\pi$
$608$	0	0
$609$	1.03425e9	4.57901
$610$	0	0
$611$	0	0
$612$	0	0
$613$	0	0		−	1.00000i	$-0.5\pi$
$613$	0	0			1.00000i	$0.5\pi$
$614$	3.45532e8	1.49274
$615$	0	0
$616$	0	0
$617$	0	0		−	1.00000i	$-0.5\pi$
$617$	0	0			1.00000i	$0.5\pi$
$618$	4.62952e8i	1.96142i
$619$	0	0	1.00000			$0$
$619$	0	0	−1.00000			$\pi$
$620$	0	0
$621$	−3.22967e8	−1.34860
$622$	0	0
$623$	− 2.67794e8i	− 1.10748i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	0	0
$628$	0	0
$629$	0	0
$630$	0	0
$631$	0	0	1.00000			$0$
$631$	0	0	−1.00000			$\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	0	0
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−7.10123e7	−0.269625	−0.134812	−	0.990871i	$-0.543043\pi$
$641$	−7.10123e7	−0.269625	−0.134812	+	0.990871i	$0.543043\pi$
$642$	8.28046e8i	3.12932i
$643$	4.35161e8i	1.63688i	0.574592	+	0.818440i	$0.305161\pi$
$643$	4.35161e8i	1.63688i	−0.574592	+	0.818440i	$0.694839\pi$
$644$	−5.14979e8	−1.92811
$645$	0	0
$646$	0	0
$647$	− 3.26848e8i	− 1.20679i	−0.797441	−	0.603396i	$-0.793814\pi$
$647$	− 3.26848e8i	− 1.20679i	0.797441	−	0.603396i	$-0.206186\pi$
$648$	2.32986e7i	0.0856257i
$649$	0	0
$650$	0	0
$651$	0	0
$652$	− 5.54304e8i	− 1.99989i
$653$	0	0		−	1.00000i	$-0.5\pi$
$653$	0	0			1.00000i	$0.5\pi$
$654$	4.57446e8	1.63533
$655$	0	0
$656$	−3.04488e8	−1.07860
$657$	0	0
$658$	1.10853e8i	0.389109i
$659$	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
$667$	6.88839e8i	2.32135i
$668$	− 3.76830e8i	− 1.26420i
$669$	−7.84798e8	−2.62108
$670$	0	0
$671$	0	0
$672$	7.55499e8i	2.48958i
$673$	0	0		−	1.00000i	$-0.5\pi$
$673$	0	0			1.00000i	$0.5\pi$
$674$	0	0
$675$	0	0
$676$	3.08916e8	1.00000
$677$	0	0		−	1.00000i	$-0.5\pi$
$677$	0	0			1.00000i	$0.5\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	4.36258e8	1.38134
$682$	0	0
$683$	− 6.06224e8i	− 1.90270i	−0.308107	−	0.951352i	$-0.599696\pi$
$683$	− 6.06224e8i	− 1.90270i	0.308107	−	0.951352i	$-0.400304\pi$
$684$	0	0
$685$	0	0
$686$	−1.64653e8	−0.510033
$687$	− 1.02229e9i	− 3.15286i
$688$	7.12868e7i	0.218899i
$689$	0	0
$690$	0	0
$691$	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
$698$	6.42257e7i	0.188861i
$699$	0	0
$700$	0	0
$701$	−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
$705$	0	0
$706$	0	0
$707$	− 1.00488e9i	− 2.84353i
$708$	0	0
$709$	7.12546e8	1.99928	0.999641	−	0.0268005i	$-0.00853187\pi$
$709$	7.12546e8	1.99928	0.999641	+	0.0268005i	$0.00853187\pi$
$710$	0	0
$711$	0	0
$712$	− 2.61662e8i	− 0.724937i
$713$	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			$\pi$
$720$	0	0
$721$	−6.89167e8	−1.83873
$722$	− 3.76367e8i	− 1.00000i
$723$	− 1.10342e9i	− 2.91962i
$724$	6.84633e8	1.80402
$725$	0	0
$726$	6.23589e8	1.62963
$727$	6.84267e8i	1.78083i	0.455150	+	0.890415i	$0.349586\pi$
$727$	6.84267e8i	1.78083i	−0.455150	+	0.890415i	$0.650414\pi$
$728$	0	0
$729$	6.19698e8	1.59955
$730$	0	0
$731$	0	0
$732$	− 1.27380e9i	− 3.24763i
$733$	0	0		−	1.00000i	$-0.5\pi$
$733$	0	0			1.00000i	$0.5\pi$
$734$	−6.95671e8	−1.75920
$735$	0	0
$736$	−5.03185e8	−1.26210
$737$	0	0
$738$	− 7.17808e8i	− 1.78583i
$739$	0	0	1.00000			$0$
$739$	0	0	−1.00000			$\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	8.19353e8i	1.99758i	0.0491695	+	0.998790i	$0.484343\pi$
$743$	8.19353e8i	1.99758i	−0.0491695	+	0.998790i	$0.515657\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	1.36598e9i	3.27705i
$748$	0	0
$749$	−1.23266e9	−2.93358
$750$	0	0
$751$	0	0	1.00000			$0$
$751$	0	0	−1.00000			$\pi$
$752$	1.08315e8i	0.254703i
$753$	0	0
$754$	0	0
$755$	0	0
$756$	−7.05329e8	−1.63240
$757$	0	0		−	1.00000i	$-0.5\pi$
$757$	0	0			1.00000i	$0.5\pi$
$758$	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	$0.652482\pi$
$762$	6.07173e8i	1.37230i
$763$	6.80970e8i	1.53304i
$764$	0	0
$765$	0	0
$766$	1.54222e8	0.343131
$767$	0	0
$768$	7.38198e8i	1.62963i
$769$	3.60743e7	0.0793266	0.0396633	−	0.999213i	$-0.487371\pi$
$769$	3.60743e7	0.0793266	0.0396633	+	0.999213i	$0.487371\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	0	0		−	1.00000i	$-0.5\pi$
$773$	0	0			1.00000i	$0.5\pi$
$774$	−1.68053e8	−0.362429
$775$	0	0
$776$	0	0
$777$	0	0
$778$	6.67665e8i	1.41782i
$779$	0	0
$780$	0	0
$781$	0	0
$782$	0	0
$783$	9.43453e8i	1.96533i
$784$	−6.42773e8	−1.33386
$785$	0	0
$786$	0	0
$787$	− 2.13980e8i	− 0.438983i	−0.975614	−	0.219492i	$-0.929560\pi$
$787$	− 2.13980e8i	− 0.438983i	0.975614	−	0.219492i	$-0.0704399\pi$
$788$	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		−	1.00000i	$-0.5\pi$
$797$	0	0			1.00000i	$0.5\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	6.16847e8	1.20027
$802$	− 9.70986e8i	− 1.88230i
$803$	0	0
$804$	−3.59322e6	−0.00691377
$805$	0	0
$806$	0	0
$807$	3.60548e8i	0.686030i
$808$	− 9.81872e8i	− 1.86132i
$809$	3.61673e8	0.683079	0.341540	−	0.939867i	$-0.389052\pi$
$809$	3.61673e8	0.683079	0.341540	+	0.939867i	$0.389052\pi$
$810$	0	0
$811$	0	0	1.00000			$0$
$811$	0	0	−1.00000			$\pi$
$812$	1.50436e9i	2.80985i
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	0	0
$818$	− 9.06972e8i	− 1.65704i
$819$	0	0
$820$	0	0
$821$	−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
$823$	− 1.16107e8i	− 0.208285i	−0.994562	−	0.104142i	$-0.966790\pi$
$823$	− 1.16107e8i	− 0.208285i	0.994562	−	0.104142i	$-0.0332098\pi$
$824$	−6.73384e8	−1.20360
$825$	0	0
$826$	0	0
$827$	− 1.01116e9i	− 1.78773i	−0.448337	−	0.893865i	$-0.647983\pi$
$827$	− 1.01116e9i	− 1.78773i	0.448337	−	0.893865i	$-0.352017\pi$
$828$	− 1.18622e9i	− 2.08965i
$829$	−1.81441e8	−0.318472	−0.159236	−	0.987241i	$-0.550903\pi$
$829$	−1.81441e8	−0.318472	−0.159236	+	0.987241i	$0.550903\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	0	0
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	0	0
$839$	0	0	1.00000			$0$
$839$	0	0	−1.00000			$\pi$
$840$	0	0
$841$	1.41742e9	2.38292
$842$	4.57844e8i	0.766975i
$843$	1.14323e9i	1.90832i
$844$	0	0
$845$	0	0
$846$	−2.55343e8	−0.421709
$847$	9.28298e8i	1.52770i
$848$	0	0
$849$	1.95227e9	3.19020
$850$	0	0
$851$	0	0
$852$	0	0
$853$	0	0		−	1.00000i	$-0.5\pi$
$853$	0	0			1.00000i	$0.5\pi$
$854$	1.89622e9	3.04449
$855$	0	0
$856$	−1.20443e9	−1.92026
$857$	0	0		−	1.00000i	$-0.5\pi$
$857$	0	0			1.00000i	$0.5\pi$
$858$	0	0
$859$	0	0	1.00000			$0$
$859$	0	0	−1.00000			$\pi$
$860$	0	0
$861$	1.71394e9	2.68525
$862$	0	0
$863$	7.23421e8i	1.12553i	0.826615	+	0.562767i	$0.190263\pi$
$863$	7.23421e8i	1.12553i	−0.826615	+	0.562767i	$0.809737\pi$
$864$	−6.89177e8	−1.06854
$865$	0	0
$866$	0	0
$867$	− 1.06205e9i	− 1.62963i
$868$	0	0
$869$	0	0
$870$	0	0
$871$	0	0
$872$	6.65376e8i	1.00350i
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	0	0		−	1.00000i	$-0.5\pi$
$877$	0	0			1.00000i	$0.5\pi$
$878$	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	$0.611252\pi$
$882$	− 1.51529e9i	− 2.20846i
$883$	− 9.07097e8i	− 1.31756i	−0.752334	−	0.658782i	$-0.771072\pi$
$883$	− 9.07097e8i	− 1.31756i	0.752334	−	0.658782i	$-0.228928\pi$
$884$	0	0
$885$	0	0
$886$	2.85726e8	0.410817
$887$	1.37949e9i	1.97674i	0.152074	+	0.988369i	$0.451405\pi$
$887$	1.37949e9i	1.97674i	−0.152074	+	0.988369i	$0.548595\pi$
$888$	0	0
$889$	−9.03860e8	−1.28646
$890$	0	0
$891$	0	0
$892$	− 1.14152e9i	− 1.60839i
$893$	0	0
$894$	−1.04520e9	−1.46281
$895$	0	0
$896$	−1.09891e9	−1.52770
$897$	0	0
$898$	− 1.42133e9i	− 1.96276i
$899$	0	0
$900$	0	0
$901$	0	0
$902$	0	0
$903$	− 4.01267e8i	− 0.544966i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	1.36077e9i	1.82374i	0.410474	+	0.911872i	$0.365363\pi$
$907$	1.36077e9i	1.82374i	−0.410474	+	0.911872i	$0.634637\pi$
$908$	6.34557e8i	0.847643i
$909$	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
$915$	0	0
$916$	1.48697e9	1.93471
$917$	0	0
$918$	0	0
$919$	0	0	1.00000			$0$
$919$	0	0	−1.00000			$\pi$
$920$	0	0
$921$	−1.90043e9	−2.43261
$922$	− 4.80957e8i	− 0.613640i
$923$	0	0
$924$	0	0
$925$	0	0
$926$	3.63120e8	0.457317
$927$	− 1.58745e9i	− 1.99279i
$928$	1.46991e9i	1.83927i
$929$	−1.27758e9	−1.59346	−0.796731	−	0.604334i	$-0.793439\pi$
$929$	−1.27758e9	−1.59346	−0.796731	+	0.604334i	$0.793439\pi$
$930$	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		−	1.00000i	$-0.5\pi$
$937$	0	0			1.00000i	$0.5\pi$
$938$	− 5.34899e6i	− 0.00648132i
$939$	0	0
$940$	0	0
$941$	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
$943$	1.14153e9i	1.36130i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 1.01744e9i	− 1.19800i	−0.800749	−	0.599001i	$-0.795565\pi$
$947$	− 1.01744e9i	− 1.19800i	0.800749	−	0.599001i	$-0.204435\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	0	0
$952$	0	0
$953$	0	0		−	1.00000i	$-0.5\pi$
$953$	0	0			1.00000i	$0.5\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	8.87504e8	1.00000
$962$	0	0
$963$	− 2.83935e9i	− 3.17936i
$964$	1.60498e9	1.79159
$965$	0	0
$966$	2.83238e9	3.14210
$967$	6.69958e8i	0.740915i	0.928849	+	0.370458i	$0.120799\pi$
$967$	6.69958e8i	0.740915i	−0.928849	+	0.370458i	$0.879201\pi$
$968$	9.07039e8i	1.00000i
$969$	0	0
$970$	0	0
$971$	0	0	1.00000			$0$
$971$	0	0	−1.00000			$\pi$
$972$	8.53127e8i	0.928998i
$973$	0	0
$974$	1.84792e9	1.99989
$975$	0	0
$976$	1.85279e9	1.99286
$977$	0	0		−	1.00000i	$-0.5\pi$
$977$	0	0			1.00000i	$0.5\pi$
$978$	3.04867e9i	3.25907i
$979$	0	0
$980$	0	0
$981$	−1.56857e9	−1.66149
$982$	0	0
$983$	8.00372e8i	0.842619i	0.906917	+	0.421310i	$0.138429\pi$
$983$	8.00372e8i	0.842619i	−0.906917	+	0.421310i	$0.861571\pi$
$984$	1.67469e9	1.75771
$985$	0	0
$986$	0	0
$987$	− 6.09693e8i	− 0.634103i
$988$	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		−	1.00000i	$-0.5\pi$
$997$	0	0			1.00000i	$0.5\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	100.7.b.d.51.1		2
4.3	odd	2	inner	100.7.b.d.51.2		2
5.2	odd	4		20.7.d.b.19.1	yes	1
5.3	odd	4		20.7.d.a.19.1	✓	1
5.4	even	2	inner	100.7.b.d.51.2		2
15.2	even	4		180.7.f.a.19.1		1
15.8	even	4		180.7.f.b.19.1		1
20.3	even	4		20.7.d.b.19.1	yes	1
20.7	even	4		20.7.d.a.19.1	✓	1
20.19	odd	2	CM	100.7.b.d.51.1		2
40.3	even	4		320.7.h.a.319.1		1
40.13	odd	4		320.7.h.b.319.1		1
40.27	even	4		320.7.h.b.319.1		1
40.37	odd	4		320.7.h.a.319.1		1
60.23	odd	4		180.7.f.a.19.1		1
60.47	odd	4		180.7.f.b.19.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
20.7.d.a.19.1	✓	1	5.3	odd	4
20.7.d.a.19.1	✓	1	20.7	even	4
20.7.d.b.19.1	yes	1	5.2	odd	4
20.7.d.b.19.1	yes	1	20.3	even	4
100.7.b.d.51.1		2	1.1	even	1	trivial
100.7.b.d.51.1		2	20.19	odd	2	CM
100.7.b.d.51.2		2	4.3	odd	2	inner
100.7.b.d.51.2		2	5.4	even	2	inner
180.7.f.a.19.1		1	15.2	even	4
180.7.f.a.19.1		1	60.23	odd	4
180.7.f.b.19.1		1	15.8	even	4
180.7.f.b.19.1		1	60.47	odd	4
320.7.h.a.319.1		1	40.3	even	4
320.7.h.a.319.1		1	40.37	odd	4
320.7.h.b.319.1		1	40.13	odd	4
320.7.h.b.319.1		1	40.27	even	4

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:	\( N \)	\(=\)	\( 100 = 2^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	100.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-8.00000i q^{2} +44.0000i q^{3} -64.0000 q^{4} +352.000 q^{6} +524.000i q^{7} +512.000i q^{8} -1207.00 q^{9} -2816.00i q^{12} +4192.00 q^{14} +4096.00 q^{16} +9656.00i q^{18} -23056.0 q^{21} -15356.0i q^{23} -22528.0 q^{24} -21032.0i q^{27} -33536.0i q^{28} -44858.0 q^{29} -32768.0i q^{32} +77248.0 q^{36} -74338.0 q^{41} +184448. i q^{42} +17404.0i q^{43} -122848. q^{46} +26444.0i q^{47} +180224. i q^{48} -156927. q^{49} -168256. q^{54} -268288. q^{56} +358864. i q^{58} +452342. q^{61} -632468. i q^{63} -262144. q^{64} -1276.00i q^{67} +675664. q^{69} -617984. i q^{72} +45505.0 q^{81} +594704. i q^{82} -1.13172e6i q^{83} +1.47558e6 q^{84} +139232. q^{86} -1.97375e6i q^{87} -511058. q^{89} +982784. i q^{92} +211552. q^{94} +1.44179e6 q^{96} +1.25542e6i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 128 q^{4} + 704 q^{6} - 2414 q^{9} + 8384 q^{14} + 8192 q^{16} - 46112 q^{21} - 45056 q^{24} - 89716 q^{29} + 154496 q^{36} - 148676 q^{41} - 245696 q^{46} - 313854 q^{49} - 336512 q^{54} - 536576 q^{56}+ \cdots + 2883584 q^{96}+O(q^{100}) \) 2 * q - 128 * q^4 + 704 * q^6 - 2414 * q^9 + 8384 * q^14 + 8192 * q^16 - 46112 * q^21 - 45056 * q^24 - 89716 * q^29 + 154496 * q^36 - 148676 * q^41 - 245696 * q^46 - 313854 * q^49 - 336512 * q^54 - 536576 * q^56 + 904684 * q^61 - 524288 * q^64 + 1351328 * q^69 + 91010 * q^81 + 2951168 * q^84 + 278464 * q^86 - 1022116 * q^89 + 423104 * q^94 + 2883584 * q^96

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