LMFDB - Embedded newform 31.11.b.a.30.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [31,11,Mod(30,31)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(31, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1])) N = Newforms(chi, 11, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("31.30"); S:= CuspForms(chi, 11); N := Newforms(S);

Level:	$ N $	$=$	$ 31 $
Weight:	$ k $	$=$	$ 11 $
Character orbit:	$[\chi]$	$=$	31.b (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)

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

Self dual:	yes
Analytic conductor:	$19.6960748329$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.837.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 6x - 1 $ x^3 - 6*x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			30.3
Root			$-2.36147$ of defining polynomial
Character	$\chi$	$=$	31.30

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+57.4935 q^{2} +2281.50 q^{4} +1394.76 q^{5} -2750.44 q^{7} +72298.4 q^{8} +59049.0 q^{9} +80189.9 q^{10} -158132. q^{14} +1.82043e6 q^{16} +3.39493e6 q^{18} +2.05577e6 q^{19} +3.18216e6 q^{20} -7.82026e6 q^{25} -6.27513e6 q^{28} -2.86292e7 q^{31} +3.06292e7 q^{32} -3.83621e6 q^{35} +1.34721e8 q^{36} +1.18193e8 q^{38} +1.00839e8 q^{40} +6.60613e7 q^{41} +8.23595e7 q^{45} -4.58037e8 q^{47} -2.74910e8 q^{49} -4.49614e8 q^{50} -1.98852e8 q^{56} -7.66846e8 q^{59} -1.64599e9 q^{62} -1.62410e8 q^{63} -1.03136e8 q^{64} +9.85211e8 q^{67} -2.20557e8 q^{70} -1.29409e9 q^{71} +4.26915e9 q^{72} +4.69025e9 q^{76} +2.53907e9 q^{80} +3.48678e9 q^{81} +3.79810e9 q^{82} +4.73513e9 q^{90} -2.63342e10 q^{94} +2.86731e9 q^{95} -1.17935e10 q^{97} -1.58056e10 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3072 q^{4} + 40275 q^{8} + 177147 q^{9} - 136653 q^{10} - 1650069 q^{14} + 3145728 q^{16} + 17403147 q^{20} + 29296875 q^{25} + 76042875 q^{28} - 85887453 q^{31} + 41241600 q^{32} + 300399678 q^{35}+ \cdots - 51955098525 q^{98}+O(q^{100}) $ 3 * q + 3072 * q^4 + 40275 * q^8 + 177147 * q^9 - 136653 * q^10 - 1650069 * q^14 + 3145728 * q^16 + 17403147 * q^20 + 29296875 * q^25 + 76042875 * q^28 - 85887453 * q^31 + 41241600 * q^32 + 300399678 * q^35 + 181398528 * q^36 - 12716325 * q^38 - 139932672 * q^40 - 1374111450 * q^47 + 847425747 * q^49 - 1158243453 * q^50 - 1689670656 * q^56 - 2680533597 * q^64 + 2955633150 * q^67 - 8334387597 * q^70 + 2378198475 * q^72 + 14801849619 * q^76 - 1834566525 * q^80 + 10460353203 * q^81 + 15063733275 * q^82 - 8069222997 * q^90 + 45473345478 * q^95 - 51955098525 * q^98

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/31\mathbb{Z}\right)^\times$.

$n$	$3$
$\chi(n)$	$-1$

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 $

$2$	57.4935	1.79667	0.898336	−	0.439309i	$-0.144777\pi$
$2$	57.4935	1.79667	0.898336	+	0.439309i	$0.144777\pi$
$3$	0	0	1.00000			$0$
$3$	0	0	−1.00000			$\pi$
$4$	2281.50	2.22803
$5$	1394.76	0.446325	0.223162	−	0.974781i	$-0.428362\pi$
$5$	1394.76	0.446325	0.223162	+	0.974781i	$0.428362\pi$
$6$	0	0
$7$	−2750.44	−0.163648	−0.0818241	−	0.996647i	$-0.526075\pi$
$7$	−2750.44	−0.163648	−0.0818241	+	0.996647i	$0.526075\pi$
$8$	72298.4	2.20637
$9$	59049.0	1.00000
$10$	80189.9	0.801899
$11$	0	0	1.00000			$0$
$11$	0	0	−1.00000			$\pi$
$12$	0	0
$13$	0	0	1.00000			$0$
$13$	0	0	−1.00000			$\pi$
$14$	−158132.	−0.294022
$15$	0	0
$16$	1.82043e6	1.73609
$17$	0	0	1.00000			$0$
$17$	0	0	−1.00000			$\pi$
$18$	3.39493e6	1.79667
$19$	2.05577e6	0.830245	0.415123	−	0.909765i	$-0.363739\pi$
$19$	2.05577e6	0.830245	0.415123	+	0.909765i	$0.363739\pi$
$20$	3.18216e6	0.994426
$21$	0	0
$22$	0	0
$23$	0	0	1.00000			$0$
$23$	0	0	−1.00000			$\pi$
$24$	0	0
$25$	−7.82026e6	−0.800794
$26$	0	0
$27$	0	0
$28$	−6.27513e6	−0.364613
$29$	0	0	1.00000			$0$
$29$	0	0	−1.00000			$\pi$
$30$	0	0
$31$	−2.86292e7	−1.00000
$31$	−2.86292e7	−1.00000
$32$	3.06292e7	0.912821
$33$	0	0
$34$	0	0
$35$	−3.83621e6	−0.0730402
$36$	1.34721e8	2.22803
$37$	0	0	1.00000			$0$
$37$	0	0	−1.00000			$\pi$
$38$	1.18193e8	1.49168
$39$	0	0
$40$	1.00839e8	0.984758
$41$	6.60613e7	0.570201	0.285101	−	0.958498i	$-0.407973\pi$
$41$	6.60613e7	0.570201	0.285101	+	0.958498i	$0.407973\pi$
$42$	0	0
$43$	0	0	1.00000			$0$
$43$	0	0	−1.00000			$\pi$
$44$	0	0
$45$	8.23595e7	0.446325
$46$	0	0
$47$	−4.58037e8	−1.99715	−0.998577	−	0.0533350i	$-0.983015\pi$
$47$	−4.58037e8	−1.99715	−0.998577	+	0.0533350i	$0.983015\pi$
$48$	0	0
$49$	−2.74910e8	−0.973219
$50$	−4.49614e8	−1.43877
$51$	0	0
$52$	0	0
$53$	0	0	1.00000			$0$
$53$	0	0	−1.00000			$\pi$
$54$	0	0
$55$	0	0
$56$	−1.98852e8	−0.361069
$57$	0	0
$58$	0	0
$59$	−7.66846e8	−1.07263	−0.536313	−	0.844019i	$-0.680183\pi$
$59$	−7.66846e8	−1.07263	−0.536313	+	0.844019i	$0.680183\pi$
$60$	0	0
$61$	0	0	1.00000			$0$
$61$	0	0	−1.00000			$\pi$
$62$	−1.64599e9	−1.79667
$63$	−1.62410e8	−0.163648
$64$	−1.03136e8	−0.0960532
$65$	0	0
$66$	0	0
$67$	9.85211e8	0.729718	0.364859	−	0.931063i	$-0.381117\pi$
$67$	9.85211e8	0.729718	0.364859	+	0.931063i	$0.381117\pi$
$68$	0	0
$69$	0	0
$70$	−2.20557e8	−0.131229
$71$	−1.29409e9	−0.717253	−0.358627	−	0.933481i	$-0.616755\pi$
$71$	−1.29409e9	−0.717253	−0.358627	+	0.933481i	$0.616755\pi$
$72$	4.26915e9	2.20637
$73$	0	0	1.00000			$0$
$73$	0	0	−1.00000			$\pi$
$74$	0	0
$75$	0	0
$76$	4.69025e9	1.84981
$77$	0	0
$78$	0	0
$79$	0	0	1.00000			$0$
$79$	0	0	−1.00000			$\pi$
$80$	2.53907e9	0.774862
$81$	3.48678e9	1.00000
$82$	3.79810e9	1.02446
$83$	0	0	1.00000			$0$
$83$	0	0	−1.00000			$\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0	0
$88$	0	0
$89$	0	0	1.00000			$0$
$89$	0	0	−1.00000			$\pi$
$90$	4.73513e9	0.801899
$91$	0	0
$92$	0	0
$93$	0	0
$94$	−2.63342e10	−3.58823
$95$	2.86731e9	0.370559
$96$	0	0
$97$	−1.17935e10	−1.37336	−0.686679	−	0.726960i	$-0.740932\pi$
$97$	−1.17935e10	−1.37336	−0.686679	+	0.726960i	$0.740932\pi$
$98$	−1.58056e10	−1.74856
$99$	0	0
$100$	−1.78420e10	−1.78420
$101$	1.88643e10	1.79487	0.897436	−	0.441145i	$-0.145427\pi$
$101$	1.88643e10	1.79487	0.897436	+	0.441145i	$0.145427\pi$
$102$	0	0
$103$	9.82901e9	0.847859	0.423929	−	0.905695i	$-0.360650\pi$
$103$	9.82901e9	0.847859	0.423929	+	0.905695i	$0.360650\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	2.53783e10	1.80944	0.904719	−	0.426009i	$-0.140081\pi$
$107$	2.53783e10	1.80944	0.904719	+	0.426009i	$0.140081\pi$
$108$	0	0
$109$	3.95218e9	0.256865	0.128432	−	0.991718i	$-0.459005\pi$
$109$	3.95218e9	0.256865	0.128432	+	0.991718i	$0.459005\pi$
$110$	0	0
$111$	0	0
$112$	−5.00697e9	−0.284109
$113$	3.67783e10	1.99618	0.998091	−	0.0617661i	$-0.0196733\pi$
$113$	3.67783e10	1.99618	0.998091	+	0.0617661i	$0.0196733\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	−4.40887e10	−1.92716
$119$	0	0
$120$	0	0
$121$	2.59374e10	1.00000
$122$	0	0
$123$	0	0
$124$	−6.53175e10	−2.22803
$125$	−2.45282e10	−0.803739
$126$	−9.33755e9	−0.294022
$127$	0	0	1.00000			$0$
$127$	0	0	−1.00000			$\pi$
$128$	−3.72940e10	−1.08540
$129$	0	0
$130$	0	0
$131$	2.77522e10	0.719352	0.359676	−	0.933077i	$-0.382887\pi$
$131$	2.77522e10	0.719352	0.359676	+	0.933077i	$0.382887\pi$
$132$	0	0
$133$	−5.65426e9	−0.135868
$134$	5.66432e10	1.31106
$135$	0	0
$136$	0	0
$137$	0	0	1.00000			$0$
$137$	0	0	−1.00000			$\pi$
$138$	0	0
$139$	0	0	1.00000			$0$
$139$	0	0	−1.00000			$\pi$
$140$	−8.75233e9	−0.162736
$141$	0	0
$142$	−7.44017e10	−1.28867
$143$	0	0
$144$	1.07494e11	1.73609
$145$	0	0
$146$	0	0
$147$	0	0
$148$	0	0
$149$	6.93920e10	0.944882	0.472441	−	0.881362i	$-0.343373\pi$
$149$	6.93920e10	0.944882	0.472441	+	0.881362i	$0.343373\pi$
$150$	0	0
$151$	0	0	1.00000			$0$
$151$	0	0	−1.00000			$\pi$
$152$	1.48629e11	1.83183
$153$	0	0
$154$	0	0
$155$	−3.99309e10	−0.446325
$156$	0	0
$157$	−8.49695e10	−0.890769	−0.445384	−	0.895339i	$-0.646933\pi$
$157$	−8.49695e10	−0.890769	−0.445384	+	0.895339i	$0.646933\pi$
$158$	0	0
$159$	0	0
$160$	4.27205e10	0.407415
$161$	0	0
$162$	2.00468e11	1.79667
$163$	−2.26925e11	−1.97217	−0.986083	−	0.166254i	$-0.946833\pi$
$163$	−2.26925e11	−1.97217	−0.986083	+	0.166254i	$0.946833\pi$
$164$	1.50719e11	1.27043
$165$	0	0
$166$	0	0
$167$	0	0	1.00000			$0$
$167$	0	0	−1.00000			$\pi$
$168$	0	0
$169$	1.37858e11	1.00000
$170$	0	0
$171$	1.21391e11	0.830245
$172$	0	0
$173$	−2.42694e11	−1.56614	−0.783068	−	0.621937i	$-0.786346\pi$
$173$	−2.42694e11	−1.56614	−0.783068	+	0.621937i	$0.786346\pi$
$174$	0	0
$175$	2.15091e10	0.131049
$176$	0	0
$177$	0	0
$178$	0	0
$179$	0	0	1.00000			$0$
$179$	0	0	−1.00000			$\pi$
$180$	1.87903e11	0.994426
$181$	0	0	1.00000			$0$
$181$	0	0	−1.00000			$\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	0	0
$188$	−1.04501e12	−4.44972
$189$	0	0
$190$	1.64852e11	0.665773
$191$	−1.73249e11	−0.681558	−0.340779	−	0.940143i	$-0.610691\pi$
$191$	−1.73249e11	−0.681558	−0.340779	+	0.940143i	$0.610691\pi$
$192$	0	0
$193$	−3.64948e11	−1.36284	−0.681419	−	0.731894i	$-0.738637\pi$
$193$	−3.64948e11	−1.36284	−0.681419	+	0.731894i	$0.738637\pi$
$194$	−6.78050e11	−2.46748
$195$	0	0
$196$	−6.27209e11	−2.16836
$197$	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			$\pi$
$200$	−5.65392e11	−1.76685
$201$	0	0
$202$	1.08457e12	3.22480
$203$	0	0
$204$	0	0
$205$	9.21400e10	0.254495
$206$	5.65104e11	1.52332
$207$	0	0
$208$	0	0
$209$	0	0
$210$	0	0
$211$	7.77731e11	1.85959	0.929795	−	0.368079i	$-0.119984\pi$
$211$	7.77731e11	1.85959	0.929795	+	0.368079i	$0.119984\pi$
$212$	0	0
$213$	0	0
$214$	1.45909e12	3.25097
$215$	0	0
$216$	0	0
$217$	7.87426e10	0.163648
$218$	2.27225e11	0.461502
$219$	0	0
$220$	0	0
$221$	0	0
$222$	0	0
$223$	0	0	1.00000			$0$
$223$	0	0	−1.00000			$\pi$
$224$	−8.42436e10	−0.149382
$225$	−4.61778e11	−0.800794
$226$	2.11452e12	3.58648
$227$	−9.64307e11	−1.59987	−0.799937	−	0.600084i	$-0.795134\pi$
$227$	−9.64307e11	−1.59987	−0.799937	+	0.600084i	$0.795134\pi$
$228$	0	0
$229$	0	0	1.00000			$0$
$229$	0	0	−1.00000			$\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	1.32381e12	1.92772	0.963862	−	0.266401i	$-0.0858346\pi$
$233$	1.32381e12	1.92772	0.963862	+	0.266401i	$0.0858346\pi$
$234$	0	0
$235$	−6.38854e11	−0.891379
$236$	−1.74956e12	−2.38984
$237$	0	0
$238$	0	0
$239$	0	0	1.00000			$0$
$239$	0	0	−1.00000			$\pi$
$240$	0	0
$241$	0	0	1.00000			$0$
$241$	0	0	−1.00000			$\pi$
$242$	1.49123e12	1.79667
$243$	0	0
$244$	0	0
$245$	−3.83435e11	−0.434372
$246$	0	0
$247$	0	0
$248$	−2.06984e12	−2.20637
$249$	0	0
$250$	−1.41021e12	−1.44406
$251$	0	0	1.00000			$0$
$251$	0	0	−1.00000			$\pi$
$252$	−3.70540e11	−0.364613
$253$	0	0
$254$	0	0
$255$	0	0
$256$	−2.03855e12	−1.85405
$257$	1.60008e12	1.42717	0.713584	−	0.700570i	$-0.247071\pi$
$257$	1.60008e12	1.42717	0.713584	+	0.700570i	$0.247071\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	0	0
$262$	1.59557e12	1.29244
$263$	0	0	1.00000			$0$
$263$	0	0	−1.00000			$\pi$
$264$	0	0
$265$	0	0
$266$	−3.25083e11	−0.244111
$267$	0	0
$268$	2.24776e12	1.62584
$269$	0	0	1.00000			$0$
$269$	0	0	−1.00000			$\pi$
$270$	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
$276$	0	0
$277$	0	0	1.00000			$0$
$277$	0	0	−1.00000			$\pi$
$278$	0	0
$279$	−1.69052e12	−1.00000
$280$	−2.77352e11	−0.161154
$281$	−3.41157e12	−1.94726	−0.973628	−	0.228143i	$-0.926735\pi$
$281$	−3.41157e12	−1.94726	−0.973628	+	0.228143i	$0.926735\pi$
$282$	0	0
$283$	−1.34358e12	−0.740171	−0.370086	−	0.928998i	$-0.620672\pi$
$283$	−1.34358e12	−0.740171	−0.370086	+	0.928998i	$0.620672\pi$
$284$	−2.95247e12	−1.59806
$285$	0	0
$286$	0	0
$287$	−1.81698e11	−0.0933124
$288$	1.80862e12	0.912821
$289$	2.01599e12	1.00000
$290$	0	0
$291$	0	0
$292$	0	0
$293$	3.00951e12	1.39366	0.696831	−	0.717235i	$-0.254593\pi$
$293$	3.00951e12	1.39366	0.696831	+	0.717235i	$0.254593\pi$
$294$	0	0
$295$	−1.06957e12	−0.478739
$296$	0	0
$297$	0	0
$298$	3.98959e12	1.69764
$299$	0	0
$300$	0	0
$301$	0	0
$302$	0	0
$303$	0	0
$304$	3.74238e12	1.44138
$305$	0	0
$306$	0	0
$307$	3.90369e12	1.43147	0.715736	−	0.698371i	$-0.246091\pi$
$307$	3.90369e12	1.43147	0.715736	+	0.698371i	$0.246091\pi$
$308$	0	0
$309$	0	0
$310$	−2.29577e12	−0.801899
$311$	−5.72295e12	−1.96706	−0.983530	−	0.180745i	$-0.942149\pi$
$311$	−5.72295e12	−1.96706	−0.983530	+	0.180745i	$0.942149\pi$
$312$	0	0
$313$	0	0	1.00000			$0$
$313$	0	0	−1.00000			$\pi$
$314$	−4.88520e12	−1.60042
$315$	−2.26524e11	−0.0730402
$316$	0	0
$317$	−7.75645e11	−0.242307	−0.121154	−	0.992634i	$-0.538659\pi$
$317$	−7.75645e11	−0.242307	−0.121154	+	0.992634i	$0.538659\pi$
$318$	0	0
$319$	0	0
$320$	−1.43851e11	−0.0428709
$321$	0	0
$322$	0	0
$323$	0	0
$324$	7.95511e12	2.22803
$325$	0	0
$326$	−1.30467e13	−3.54334
$327$	0	0
$328$	4.77613e12	1.25808
$329$	1.25980e12	0.326831
$330$	0	0
$331$	0	0	1.00000			$0$
$331$	0	0	−1.00000			$\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	1.37414e12	0.325691
$336$	0	0
$337$	0	0	1.00000			$0$
$337$	0	0	−1.00000			$\pi$
$338$	7.92597e12	1.79667
$339$	0	0
$340$	0	0
$341$	0	0
$342$	6.97920e12	1.49168
$343$	1.53305e12	0.322914
$344$	0	0
$345$	0	0
$346$	−1.39534e13	−2.81383
$347$	0	0	1.00000			$0$
$347$	0	0	−1.00000			$\pi$
$348$	0	0
$349$	1.03005e13	1.98944	0.994720	−	0.102625i	$-0.0327240\pi$
$349$	1.03005e13	1.98944	0.994720	+	0.102625i	$0.0327240\pi$
$350$	1.23663e12	0.235451
$351$	0	0
$352$	0	0
$353$	0	0	1.00000			$0$
$353$	0	0	−1.00000			$\pi$
$354$	0	0
$355$	−1.80495e12	−0.320128
$356$	0	0
$357$	0	0
$358$	0	0
$359$	1.19216e13	1.99923	0.999615	−	0.0277619i	$-0.00883801\pi$
$359$	1.19216e13	1.99923	0.999615	+	0.0277619i	$0.00883801\pi$
$360$	5.95445e12	0.984758
$361$	−1.90488e12	−0.310693
$362$	0	0
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	0	0	1.00000			$0$
$367$	0	0	−1.00000			$\pi$
$368$	0	0
$369$	3.90086e12	0.570201
$370$	0	0
$371$	0	0
$372$	0	0
$373$	1.28027e13	1.77320	0.886599	−	0.462540i	$-0.153062\pi$
$373$	1.28027e13	1.77320	0.886599	+	0.462540i	$0.153062\pi$
$374$	0	0
$375$	0	0
$376$	−3.31153e13	−4.40646
$377$	0	0
$378$	0	0
$379$	−9.85992e12	−1.26089	−0.630445	−	0.776234i	$-0.717128\pi$
$379$	−9.85992e12	−1.26089	−0.630445	+	0.776234i	$0.717128\pi$
$380$	6.54179e12	0.825617
$381$	0	0
$382$	−9.96067e12	−1.22454
$383$	0	0	1.00000			$0$
$383$	0	0	−1.00000			$\pi$
$384$	0	0
$385$	0	0
$386$	−2.09821e13	−2.44857
$387$	0	0
$388$	−2.69069e13	−3.05989
$389$	0	0	1.00000			$0$
$389$	0	0	−1.00000			$\pi$
$390$	0	0
$391$	0	0
$392$	−1.98756e13	−2.14728
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−1.16072e10	−0.00117700	−0.000588498	−	1.00000i	$-0.500187\pi$
$397$	−1.16072e10	−0.00117700	−0.000588498		1.00000i	$0.500187\pi$
$398$	0	0
$399$	0	0
$400$	−1.42362e13	−1.39025
$401$	0	0	1.00000			$0$
$401$	0	0	−1.00000			$\pi$
$402$	0	0
$403$	0	0
$404$	4.30390e13	3.99903
$405$	4.86324e12	0.446325
$406$	0	0
$407$	0	0
$408$	0	0
$409$	0	0	1.00000			$0$
$409$	0	0	−1.00000			$\pi$
$410$	5.29745e12	0.457244
$411$	0	0
$412$	2.24249e13	1.88906
$413$	2.10916e12	0.175533
$414$	0	0
$415$	0	0
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−2.15757e13	−1.67069	−0.835343	−	0.549730i	$-0.814731\pi$
$419$	−2.15757e13	−1.67069	−0.835343	+	0.549730i	$0.814731\pi$
$420$	0	0
$421$	2.14464e13	1.62160	0.810802	−	0.585320i	$-0.199031\pi$
$421$	2.14464e13	1.62160	0.810802	+	0.585320i	$0.199031\pi$
$422$	4.47145e13	3.34107
$423$	−2.70466e13	−1.99715
$424$	0	0
$425$	0	0
$426$	0	0
$427$	0	0
$428$	5.79007e13	4.03148
$429$	0	0
$430$	0	0
$431$	−2.70811e13	−1.82087	−0.910437	−	0.413649i	$-0.864254\pi$
$431$	−2.70811e13	−1.82087	−0.910437	+	0.413649i	$0.864254\pi$
$432$	0	0
$433$	0	0	1.00000			$0$
$433$	0	0	−1.00000			$\pi$
$434$	4.52719e12	0.294022
$435$	0	0
$436$	9.01692e12	0.572303
$437$	0	0
$438$	0	0
$439$	5.67978e12	0.348344	0.174172	−	0.984715i	$-0.444275\pi$
$439$	5.67978e12	0.348344	0.174172	+	0.984715i	$0.444275\pi$
$440$	0	0
$441$	−1.62332e13	−0.973219
$442$	0	0
$443$	8.82190e12	0.517063	0.258531	−	0.966003i	$-0.416761\pi$
$443$	8.82190e12	0.517063	0.258531	+	0.966003i	$0.416761\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	2.83670e11	0.0157189
$449$	0	0	1.00000			$0$
$449$	0	0	−1.00000			$\pi$
$450$	−2.65493e13	−1.43877
$451$	0	0
$452$	8.39100e13	4.44756
$453$	0	0
$454$	−5.54414e13	−2.87445
$455$	0	0
$456$	0	0
$457$	0	0	1.00000			$0$
$457$	0	0	−1.00000			$\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	0	0	1.00000			$0$
$461$	0	0	−1.00000			$\pi$
$462$	0	0
$463$	0	0	1.00000			$0$
$463$	0	0	−1.00000			$\pi$
$464$	0	0
$465$	0	0
$466$	7.61103e13	3.46349
$467$	−3.56038e13	−1.60292	−0.801460	−	0.598048i	$-0.795943\pi$
$467$	−3.56038e13	−1.60292	−0.801460	+	0.598048i	$0.795943\pi$
$468$	0	0
$469$	−2.70976e12	−0.119417
$470$	−3.67300e13	−1.60152
$471$	0	0
$472$	−5.54417e13	−2.36661
$473$	0	0
$474$	0	0
$475$	−1.60766e13	−0.664856
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−2.70690e13	−1.07348	−0.536741	−	0.843747i	$-0.680345\pi$
$479$	−2.70690e13	−1.07348	−0.536741	+	0.843747i	$0.680345\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	0	0
$484$	5.91764e13	2.22803
$485$	−1.64492e13	−0.612964
$486$	0	0
$487$	0	0	1.00000			$0$
$487$	0	0	−1.00000			$\pi$
$488$	0	0
$489$	0	0
$490$	−2.20450e13	−0.780424
$491$	0	0	1.00000			$0$
$491$	0	0	−1.00000			$\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	0	0
$496$	−5.21173e13	−1.73609
$497$	3.55931e12	0.117377
$498$	0	0
$499$	0	0	1.00000			$0$
$499$	0	0	−1.00000			$\pi$
$500$	−5.59611e13	−1.79076
$501$	0	0
$502$	0	0
$503$	−3.76250e13	−1.16852	−0.584261	−	0.811566i	$-0.698615\pi$
$503$	−3.76250e13	−1.16852	−0.584261	+	0.811566i	$0.698615\pi$
$504$	−1.17420e13	−0.361069
$505$	2.63112e13	0.801096
$506$	0	0
$507$	0	0
$508$	0	0
$509$	0	0	1.00000			$0$
$509$	0	0	−1.00000			$\pi$
$510$	0	0
$511$	0	0
$512$	−7.90144e13	−2.24572
$513$	0	0
$514$	9.19940e13	2.56415
$515$	1.37092e13	0.378420
$516$	0	0
$517$	0	0
$518$	0	0
$519$	0	0
$520$	0	0
$521$	4.57048e13	1.19062	0.595310	−	0.803496i	$-0.297029\pi$
$521$	4.57048e13	1.19062	0.595310	+	0.803496i	$0.297029\pi$
$522$	0	0
$523$	0	0	1.00000			$0$
$523$	0	0	−1.00000			$\pi$
$524$	6.33168e13	1.60274
$525$	0	0
$526$	0	0
$527$	0	0
$528$	0	0
$529$	4.14265e13	1.00000
$530$	0	0
$531$	−4.52815e13	−1.07263
$532$	−1.29002e13	−0.302719
$533$	0	0
$534$	0	0
$535$	3.53968e13	0.807597
$536$	7.12291e13	1.61003
$537$	0	0
$538$	0	0
$539$	0	0
$540$	0	0
$541$	8.07859e13	1.74321	0.871604	−	0.490211i	$-0.163080\pi$
$541$	8.07859e13	1.74321	0.871604	+	0.490211i	$0.163080\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	5.51236e12	0.114645
$546$	0	0
$547$	9.41521e13	1.92262	0.961310	−	0.275470i	$-0.0888336\pi$
$547$	9.41521e13	1.92262	0.961310	+	0.275470i	$0.0888336\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	0	0
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	0	0
$557$	0	0	1.00000			$0$
$557$	0	0	−1.00000			$\pi$
$558$	−9.71941e13	−1.79667
$559$	0	0
$560$	−6.98354e12	−0.126805
$561$	0	0
$562$	−1.96143e14	−3.49858
$563$	−1.11470e14	−1.97068	−0.985338	−	0.170612i	$-0.945426\pi$
$563$	−1.11470e14	−1.97068	−0.985338	+	0.170612i	$0.945426\pi$
$564$	0	0
$565$	5.12971e13	0.890945
$566$	−7.72473e13	−1.32985
$567$	−9.59018e12	−0.163648
$568$	−9.35605e13	−1.58253
$569$	0	0	1.00000			$0$
$569$	0	0	−1.00000			$\pi$
$570$	0	0
$571$	0	0	1.00000			$0$
$571$	0	0	−1.00000			$\pi$
$572$	0	0
$573$	0	0
$574$	−1.04464e13	−0.167652
$575$	0	0
$576$	−6.09010e12	−0.0960532
$577$	9.79252e13	1.53114	0.765571	−	0.643352i	$-0.222457\pi$
$577$	9.79252e13	1.53114	0.765571	+	0.643352i	$0.222457\pi$
$578$	1.15907e14	1.79667
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	1.73027e14	2.50395
$587$	0	0	1.00000			$0$
$587$	0	0	−1.00000			$\pi$
$588$	0	0
$589$	−5.88549e13	−0.830245
$590$	−6.14933e13	−0.860137
$591$	0	0
$592$	0	0
$593$	−1.41598e13	−0.193101	−0.0965503	−	0.995328i	$-0.530781\pi$
$593$	−1.41598e13	−0.193101	−0.0965503	+	0.995328i	$0.530781\pi$
$594$	0	0
$595$	0	0
$596$	1.58318e14	2.10523
$597$	0	0
$598$	0	0
$599$	1.44859e14	1.87851	0.939253	−	0.343226i	$-0.111520\pi$
$599$	1.44859e14	1.87851	0.939253	+	0.343226i	$0.111520\pi$
$600$	0	0
$601$	0	0	1.00000			$0$
$601$	0	0	−1.00000			$\pi$
$602$	0	0
$603$	5.81757e13	0.729718
$604$	0	0
$605$	3.61766e13	0.446325
$606$	0	0
$607$	−1.07274e14	−1.30182	−0.650909	−	0.759156i	$-0.725612\pi$
$607$	−1.07274e14	−1.30182	−0.650909	+	0.759156i	$0.725612\pi$
$608$	6.29666e13	0.757866
$609$	0	0
$610$	0	0
$611$	0	0
$612$	0	0
$613$	0	0	1.00000			$0$
$613$	0	0	−1.00000			$\pi$
$614$	2.24437e14	2.57189
$615$	0	0
$616$	0	0
$617$	2.22457e13	0.248783	0.124392	−	0.992233i	$-0.460302\pi$
$617$	2.22457e13	0.248783	0.124392	+	0.992233i	$0.460302\pi$
$618$	0	0
$619$	0	0	1.00000			$0$
$619$	0	0	−1.00000			$\pi$
$620$	−9.11026e13	−0.994426
$621$	0	0
$622$	−3.29032e14	−3.53416
$623$	0	0
$624$	0	0
$625$	4.21587e13	0.442066
$626$	0	0
$627$	0	0
$628$	−1.93858e14	−1.98466
$629$	0	0
$630$	−1.30237e13	−0.131229
$631$	0	0	1.00000			$0$
$631$	0	0	−1.00000			$\pi$
$632$	0	0
$633$	0	0
$634$	−4.45945e13	−0.435347
$635$	0	0
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−7.64147e13	−0.717253
$640$	−5.20163e13	−0.484440
$641$	0	0	1.00000			$0$
$641$	0	0	−1.00000			$\pi$
$642$	0	0
$643$	0	0	1.00000			$0$
$643$	0	0	−1.00000			$\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	0	0	1.00000			$0$
$647$	0	0	−1.00000			$\pi$
$648$	2.52089e14	2.20637
$649$	0	0
$650$	0	0
$651$	0	0
$652$	−5.17729e14	−4.39405
$653$	−4.79879e13	−0.404172	−0.202086	−	0.979368i	$-0.564772\pi$
$653$	−4.79879e13	−0.404172	−0.202086	+	0.979368i	$0.564772\pi$
$654$	0	0
$655$	3.87078e13	0.321065
$656$	1.20260e14	0.989923
$657$	0	0
$658$	7.24304e13	0.587208
$659$	−2.45475e14	−1.97506	−0.987531	−	0.157426i	$-0.949680\pi$
$659$	−2.45475e14	−1.97506	−0.987531	+	0.157426i	$0.949680\pi$
$660$	0	0
$661$	−2.20343e14	−1.74619	−0.873097	−	0.487547i	$-0.837892\pi$
$661$	−2.20343e14	−1.74619	−0.873097	+	0.487547i	$0.837892\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−7.88637e12	−0.0606413
$666$	0	0
$667$	0	0
$668$	0	0
$669$	0	0
$670$	7.90040e13	0.585161
$671$	0	0
$672$	0	0
$673$	0	0	1.00000			$0$
$673$	0	0	−1.00000			$\pi$
$674$	0	0
$675$	0	0
$676$	3.14525e14	2.22803
$677$	0	0	1.00000			$0$
$677$	0	0	−1.00000			$\pi$
$678$	0	0
$679$	3.24373e13	0.224748
$680$	0	0
$681$	0	0
$682$	0	0
$683$	5.29627e12	0.0356341	0.0178171	−	0.999841i	$-0.494328\pi$
$683$	5.29627e12	0.0356341	0.0178171	+	0.999841i	$0.494328\pi$
$684$	2.76954e14	1.84981
$685$	0	0
$686$	8.81406e13	0.580170
$687$	0	0
$688$	0	0
$689$	0	0
$690$	0	0
$691$	2.92616e14	1.85741	0.928705	−	0.370819i	$-0.120923\pi$
$691$	2.92616e14	1.85741	0.928705	+	0.370819i	$0.120923\pi$
$692$	−5.53708e14	−3.48940
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	0	0
$698$	5.92212e14	3.57437
$699$	0	0
$700$	4.90731e13	0.291980
$701$	−3.06681e14	−1.81174	−0.905871	−	0.423553i	$-0.860783\pi$
$701$	−3.06681e14	−1.81174	−0.905871	+	0.423553i	$0.860783\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−5.18850e13	−0.293728
$708$	0	0
$709$	0	0	1.00000			$0$
$709$	0	0	−1.00000			$\pi$
$710$	−1.03773e14	−0.575165
$711$	0	0
$712$	0	0
$713$	0	0
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	6.85415e14	3.59196
$719$	0	0	1.00000			$0$
$719$	0	0	−1.00000			$\pi$
$720$	1.49929e14	0.774862
$721$	−2.70341e13	−0.138751
$722$	−1.09518e14	−0.558213
$723$	0	0
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−2.02104e14	−0.995181	−0.497590	−	0.867412i	$-0.665782\pi$
$727$	−2.02104e14	−0.995181	−0.497590	+	0.867412i	$0.665782\pi$
$728$	0	0
$729$	2.05891e14	1.00000
$730$	0	0
$731$	0	0
$732$	0	0
$733$	3.39728e14	1.60550	0.802752	−	0.596312i	$-0.203368\pi$
$733$	3.39728e14	1.60550	0.802752	+	0.596312i	$0.203368\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	0	0
$738$	2.24274e14	1.02446
$739$	0	0	1.00000			$0$
$739$	0	0	−1.00000			$\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	0	0	1.00000			$0$
$743$	0	0	−1.00000			$\pi$
$744$	0	0
$745$	9.67854e13	0.421724
$746$	7.36072e14	3.18585
$747$	0	0
$748$	0	0
$749$	−6.98014e13	−0.296111
$750$	0	0
$751$	4.83554e13	0.202416	0.101208	−	0.994865i	$-0.467729\pi$
$751$	4.83554e13	0.202416	0.101208	+	0.994865i	$0.467729\pi$
$752$	−8.33823e14	−3.46725
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	0	0	1.00000			$0$
$757$	0	0	−1.00000			$\pi$
$758$	−5.66881e14	−2.26541
$759$	0	0
$760$	2.07302e14	0.817591
$761$	0	0	1.00000			$0$
$761$	0	0	−1.00000			$\pi$
$762$	0	0
$763$	−1.08702e13	−0.0420354
$764$	−3.95267e14	−1.51853
$765$	0	0
$766$	0	0
$767$	0	0
$768$	0	0
$769$	−5.37825e14	−1.99990	−0.999952	−	0.00976081i	$-0.996893\pi$
$769$	−5.37825e14	−1.99990	−0.999952	+	0.00976081i	$0.996893\pi$
$770$	0	0
$771$	0	0
$772$	−8.32630e14	−3.03645
$773$	0	0	1.00000			$0$
$773$	0	0	−1.00000			$\pi$
$774$	0	0
$775$	2.23887e14	0.800794
$776$	−8.52651e14	−3.03014
$777$	0	0
$778$	0	0
$779$	1.35807e14	0.473407
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	−5.00454e14	−1.68960
$785$	−1.18513e14	−0.397572
$786$	0	0
$787$	0	0	1.00000			$0$
$787$	0	0	−1.00000			$\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−1.01156e14	−0.326672
$792$	0	0
$793$	0	0
$794$	−6.67339e11	−0.00211468
$795$	0	0
$796$	0	0
$797$	0	0	1.00000			$0$
$797$	0	0	−1.00000			$\pi$
$798$	0	0
$799$	0	0
$800$	−2.39528e14	−0.730982
$801$	0	0
$802$	0	0
$803$	0	0
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	1.36386e15	3.96015
$809$	0	0	1.00000			$0$
$809$	0	0	−1.00000			$\pi$
$810$	2.79605e14	0.801899
$811$	3.67948e14	1.04878	0.524388	−	0.851480i	$-0.324294\pi$
$811$	3.67948e14	1.04878	0.524388	+	0.851480i	$0.324294\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−3.16506e14	−0.880226
$816$	0	0
$817$	0	0
$818$	0	0
$819$	0	0
$820$	2.10218e14	0.567023
$821$	0	0	1.00000			$0$
$821$	0	0	−1.00000			$\pi$
$822$	0	0
$823$	0	0	1.00000			$0$
$823$	0	0	−1.00000			$\pi$
$824$	7.10621e14	1.87069
$825$	0	0
$826$	1.21263e14	0.315376
$827$	0	0	1.00000			$0$
$827$	0	0	−1.00000			$\pi$
$828$	0	0
$829$	0	0	1.00000			$0$
$829$	0	0	−1.00000			$\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	0	0
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	−1.24046e15	−3.00167
$839$	7.82938e14	1.88329	0.941646	−	0.336605i	$-0.109279\pi$
$839$	7.82938e14	1.88329	0.941646	+	0.336605i	$0.109279\pi$
$840$	0	0
$841$	4.20707e14	1.00000
$842$	1.23303e15	2.91349
$843$	0	0
$844$	1.77440e15	4.14322
$845$	1.92280e14	0.446325
$846$	−1.55501e15	−3.58823
$847$	−7.13392e13	−0.163648
$848$	0	0
$849$	0	0
$850$	0	0
$851$	0	0
$852$	0	0
$853$	−6.48316e14	−1.43563	−0.717814	−	0.696235i	$-0.754857\pi$
$853$	−6.48316e14	−1.43563	−0.717814	+	0.696235i	$0.754857\pi$
$854$	0	0
$855$	1.69312e14	0.370559
$856$	1.83481e15	3.99229
$857$	−6.57365e14	−1.42201	−0.711004	−	0.703187i	$-0.751759\pi$
$857$	−6.57365e14	−1.42201	−0.711004	+	0.703187i	$0.751759\pi$
$858$	0	0
$859$	0	0	1.00000			$0$
$859$	0	0	−1.00000			$\pi$
$860$	0	0
$861$	0	0
$862$	−1.55699e15	−3.27151
$863$	0	0	1.00000			$0$
$863$	0	0	−1.00000			$\pi$
$864$	0	0
$865$	−3.38502e14	−0.699005
$866$	0	0
$867$	0	0
$868$	1.79652e14	0.364613
$869$	0	0
$870$	0	0
$871$	0	0
$872$	2.85736e14	0.566739
$873$	−6.96394e14	−1.37336
$874$	0	0
$875$	6.74631e13	0.131530
$876$	0	0
$877$	−7.92753e14	−1.52806	−0.764029	−	0.645182i	$-0.776781\pi$
$877$	−7.92753e14	−1.52806	−0.764029	+	0.645182i	$0.776781\pi$
$878$	3.26550e14	0.625861
$879$	0	0
$880$	0	0
$881$	0	0	1.00000			$0$
$881$	0	0	−1.00000			$\pi$
$882$	−9.33303e14	−1.74856
$883$	0	0	1.00000			$0$
$883$	0	0	−1.00000			$\pi$
$884$	0	0
$885$	0	0
$886$	5.07202e14	0.928993
$887$	2.11270e14	0.384786	0.192393	−	0.981318i	$-0.438375\pi$
$887$	2.11270e14	0.384786	0.192393	+	0.981318i	$0.438375\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−9.41619e14	−1.65813
$894$	0	0
$895$	0	0
$896$	1.02575e14	0.177623
$897$	0	0
$898$	0	0
$899$	0	0
$900$	−1.05355e15	−1.78420
$901$	0	0
$902$	0	0
$903$	0	0
$904$	2.65901e15	4.40432
$905$	0	0
$906$	0	0
$907$	1.20333e15	1.96042	0.980212	−	0.197952i	$-0.0634289\pi$
$907$	1.20333e15	1.96042	0.980212	+	0.197952i	$0.0634289\pi$
$908$	−2.20007e15	−3.56457
$909$	1.11392e15	1.79487
$910$	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
$916$	0	0
$917$	−7.63307e13	−0.117721
$918$	0	0
$919$	7.85587e14	1.19844	0.599221	−	0.800584i	$-0.295477\pi$
$919$	7.85587e14	1.19844	0.599221	+	0.800584i	$0.295477\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	0	0
$924$	0	0
$925$	0	0
$926$	0	0
$927$	5.80393e14	0.847859
$928$	0	0
$929$	0	0	1.00000			$0$
$929$	0	0	−1.00000			$\pi$
$930$	0	0
$931$	−5.65152e14	−0.808011
$932$	3.02027e15	4.29503
$933$	0	0
$934$	−2.04699e15	−2.87992
$935$	0	0
$936$	0	0
$937$	−4.18129e14	−0.578912	−0.289456	−	0.957191i	$-0.593474\pi$
$937$	−4.18129e14	−0.578912	−0.289456	+	0.957191i	$0.593474\pi$
$938$	−1.55794e14	−0.214553
$939$	0	0
$940$	−1.45755e15	−1.98602
$941$	0	0	1.00000			$0$
$941$	0	0	−1.00000			$\pi$
$942$	0	0
$943$	0	0
$944$	−1.39599e15	−1.86218
$945$	0	0
$946$	0	0
$947$	0	0	1.00000			$0$
$947$	0	0	−1.00000			$\pi$
$948$	0	0
$949$	0	0
$950$	−9.24303e14	−1.19453
$951$	0	0
$952$	0	0
$953$	0	0	1.00000			$0$
$953$	0	0	−1.00000			$\pi$
$954$	0	0
$955$	−2.41641e14	−0.304196
$956$	0	0
$957$	0	0
$958$	−1.55629e15	−1.92870
$959$	0	0
$960$	0	0
$961$	8.19628e14	1.00000
$962$	0	0
$963$	1.49856e15	1.80944
$964$	0	0
$965$	−5.09016e14	−0.608268
$966$	0	0
$967$	0	0	1.00000			$0$
$967$	0	0	−1.00000			$\pi$
$968$	1.87523e15	2.20637
$969$	0	0
$970$	−9.45720e14	−1.10130
$971$	9.10755e13	0.105513	0.0527564	−	0.998607i	$-0.483199\pi$
$971$	9.10755e13	0.105513	0.0527564	+	0.998607i	$0.483199\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	1.72308e15	1.93567	0.967837	−	0.251578i	$-0.0809494\pi$
$977$	1.72308e15	1.93567	0.967837	+	0.251578i	$0.0809494\pi$
$978$	0	0
$979$	0	0
$980$	−8.74809e14	−0.967794
$981$	2.33372e14	0.256865
$982$	0	0
$983$	0	0	1.00000			$0$
$983$	0	0	−1.00000			$\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	0	0
$990$	0	0
$991$	0	0	1.00000			$0$
$991$	0	0	−1.00000			$\pi$
$992$	−8.76888e14	−0.912821
$993$	0	0
$994$	2.04637e14	0.210888
$995$	0	0
$996$	0	0
$997$	−3.09046e14	−0.313724	−0.156862	−	0.987621i	$-0.550138\pi$
$997$	−3.09046e14	−0.313724	−0.156862	+	0.987621i	$0.550138\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	31.11.b.a.30.3	✓	3
31.30	odd	2	CM	31.11.b.a.30.3	✓	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
31.11.b.a.30.3	✓	3	1.1	even	1	trivial
31.11.b.a.30.3	✓	3	31.30	odd	2	CM

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:	\( N \)	\(=\)	\( 31 \)
Weight:	\( k \)	\(=\)	\( 11 \)
Character orbit:	\([\chi]\)	\(=\)	31.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+57.4935 q^{2} +2281.50 q^{4} +1394.76 q^{5} -2750.44 q^{7} +72298.4 q^{8} +59049.0 q^{9} +80189.9 q^{10} -158132. q^{14} +1.82043e6 q^{16} +3.39493e6 q^{18} +2.05577e6 q^{19} +3.18216e6 q^{20} -7.82026e6 q^{25} -6.27513e6 q^{28} -2.86292e7 q^{31} +3.06292e7 q^{32} -3.83621e6 q^{35} +1.34721e8 q^{36} +1.18193e8 q^{38} +1.00839e8 q^{40} +6.60613e7 q^{41} +8.23595e7 q^{45} -4.58037e8 q^{47} -2.74910e8 q^{49} -4.49614e8 q^{50} -1.98852e8 q^{56} -7.66846e8 q^{59} -1.64599e9 q^{62} -1.62410e8 q^{63} -1.03136e8 q^{64} +9.85211e8 q^{67} -2.20557e8 q^{70} -1.29409e9 q^{71} +4.26915e9 q^{72} +4.69025e9 q^{76} +2.53907e9 q^{80} +3.48678e9 q^{81} +3.79810e9 q^{82} +4.73513e9 q^{90} -2.63342e10 q^{94} +2.86731e9 q^{95} -1.17935e10 q^{97} -1.58056e10 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3072 q^{4} + 40275 q^{8} + 177147 q^{9} - 136653 q^{10} - 1650069 q^{14} + 3145728 q^{16} + 17403147 q^{20} + 29296875 q^{25} + 76042875 q^{28} - 85887453 q^{31} + 41241600 q^{32} + 300399678 q^{35}+ \cdots - 51955098525 q^{98}+O(q^{100}) \) 3 * q + 3072 * q^4 + 40275 * q^8 + 177147 * q^9 - 136653 * q^10 - 1650069 * q^14 + 3145728 * q^16 + 17403147 * q^20 + 29296875 * q^25 + 76042875 * q^28 - 85887453 * q^31 + 41241600 * q^32 + 300399678 * q^35 + 181398528 * q^36 - 12716325 * q^38 - 139932672 * q^40 - 1374111450 * q^47 + 847425747 * q^49 - 1158243453 * q^50 - 1689670656 * q^56 - 2680533597 * q^64 + 2955633150 * q^67 - 8334387597 * q^70 + 2378198475 * q^72 + 14801849619 * q^76 - 1834566525 * q^80 + 10460353203 * q^81 + 15063733275 * q^82 - 8069222997 * q^90 + 45473345478 * q^95 - 51955098525 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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