LMFDB - Embedded newform 36.2.b.a.35.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [36,2,Mod(35,36)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(36, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 1]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("36.35");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 36 = 2^{2} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	36.b (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$0.287461447277$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{-2}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 2 $ x^2 + 2
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			35.2
Root			$1.41421i$ of defining polynomial
Character	$\chi$	$=$	36.35
Dual form			36.2.b.a.35.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.41421i q^{2} -2.00000 q^{4} -1.41421i q^{5} -2.82843i q^{8} +O(q^{10})$	$q+1.41421i q^{2} -2.00000 q^{4} -1.41421i q^{5} -2.82843i q^{8} +2.00000 q^{10} -4.00000 q^{13} +4.00000 q^{16} +7.07107i q^{17} +2.82843i q^{20} +3.00000 q^{25} -5.65685i q^{26} -9.89949i q^{29} +5.65685i q^{32} -10.0000 q^{34} +2.00000 q^{37} -4.00000 q^{40} -1.41421i q^{41} +7.00000 q^{49} +4.24264i q^{50} +8.00000 q^{52} +7.07107i q^{53} +14.0000 q^{58} -10.0000 q^{61} -8.00000 q^{64} +5.65685i q^{65} -14.1421i q^{68} -16.0000 q^{73} +2.82843i q^{74} -5.65685i q^{80} +2.00000 q^{82} +10.0000 q^{85} -18.3848i q^{89} +8.00000 q^{97} +9.89949i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 4 q^{4}+O(q^{10}) $ 2 * q - 4 * q^4	$ 2 q - 4 q^{4} + 4 q^{10} - 8 q^{13} + 8 q^{16} + 6 q^{25} - 20 q^{34} + 4 q^{37} - 8 q^{40} + 14 q^{49} + 16 q^{52} + 28 q^{58} - 20 q^{61} - 16 q^{64} - 32 q^{73} + 4 q^{82} + 20 q^{85} + 16 q^{97}+O(q^{100}) $ 2 * q - 4 * q^4 + 4 * q^10 - 8 * q^13 + 8 * q^16 + 6 * q^25 - 20 * q^34 + 4 * q^37 - 8 * q^40 + 14 * q^49 + 16 * q^52 + 28 * q^58 - 20 * q^61 - 16 * q^64 - 32 * q^73 + 4 * q^82 + 20 * q^85 + 16 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/36\mathbb{Z}\right)^\times$.

$n$	$19$	$29$
$\chi(n)$	$-1$	$-1$

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 $

$2$	1.41421i	1.00000i
$2$	1.41421i	1.00000i
$3$	0	0
$3$	0	0
$4$	−2.00000	−1.00000
$5$	− 1.41421i	− 0.632456i	−0.948683	−	0.316228i	$-0.897584\pi$
$5$	− 1.41421i	− 0.632456i	0.948683	−	0.316228i	$-0.102416\pi$
$6$	0	0
$7$	0	0	1.00000			$0$
$7$	0	0	−1.00000			$\pi$
$8$	− 2.82843i	− 1.00000i
$9$	0	0
$10$	2.00000	0.632456
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	0	0
$13$	−4.00000	−1.10940	−0.554700	−	0.832050i	$-0.687167\pi$
$13$	−4.00000	−1.10940	−0.554700	+	0.832050i	$0.687167\pi$
$14$	0	0
$15$	0	0
$16$	4.00000	1.00000
$17$	7.07107i	1.71499i	0.514496	+	0.857493i	$0.327979\pi$
$17$	7.07107i	1.71499i	−0.514496	+	0.857493i	$0.672021\pi$
$18$	0	0
$19$	0	0	1.00000			$0$
$19$	0	0	−1.00000			$\pi$
$20$	2.82843i	0.632456i
$21$	0	0
$22$	0	0
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	0	0
$25$	3.00000	0.600000
$26$	− 5.65685i	− 1.10940i
$27$	0	0
$28$	0	0
$29$	− 9.89949i	− 1.83829i	−0.393919	−	0.919145i	$-0.628881\pi$
$29$	− 9.89949i	− 1.83829i	0.393919	−	0.919145i	$-0.371119\pi$
$30$	0	0
$31$	0	0	1.00000			$0$
$31$	0	0	−1.00000			$\pi$
$32$	5.65685i	1.00000i
$33$	0	0
$34$	−10.0000	−1.71499
$35$	0	0
$36$	0	0
$37$	2.00000	0.328798	0.164399	−	0.986394i	$-0.447432\pi$
$37$	2.00000	0.328798	0.164399	+	0.986394i	$0.447432\pi$
$38$	0	0
$39$	0	0
$40$	−4.00000	−0.632456
$41$	− 1.41421i	− 0.220863i	−0.993884	−	0.110432i	$-0.964777\pi$
$41$	− 1.41421i	− 0.220863i	0.993884	−	0.110432i	$-0.0352233\pi$
$42$	0	0
$43$	0	0	1.00000			$0$
$43$	0	0	−1.00000			$\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	0	0
$49$	7.00000	1.00000
$50$	4.24264i	0.600000i
$51$	0	0
$52$	8.00000	1.10940
$53$	7.07107i	0.971286i	0.874157	+	0.485643i	$0.161414\pi$
$53$	7.07107i	0.971286i	−0.874157	+	0.485643i	$0.838586\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	14.0000	1.83829
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	0	0
$61$	−10.0000	−1.28037	−0.640184	−	0.768221i	$-0.721142\pi$
$61$	−10.0000	−1.28037	−0.640184	+	0.768221i	$0.721142\pi$
$62$	0	0
$63$	0	0
$64$	−8.00000	−1.00000
$65$	5.65685i	0.701646i
$66$	0	0
$67$	0	0	1.00000			$0$
$67$	0	0	−1.00000			$\pi$
$68$	− 14.1421i	− 1.71499i
$69$	0	0
$70$	0	0
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	0	0
$73$	−16.0000	−1.87266	−0.936329	−	0.351123i	$-0.885800\pi$
$73$	−16.0000	−1.87266	−0.936329	+	0.351123i	$0.885800\pi$
$74$	2.82843i	0.328798i
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	0	0	1.00000			$0$
$79$	0	0	−1.00000			$\pi$
$80$	− 5.65685i	− 0.632456i
$81$	0	0
$82$	2.00000	0.220863
$83$	0	0		−	1.00000i	$-0.5\pi$
$83$	0	0			1.00000i	$0.5\pi$
$84$	0	0
$85$	10.0000	1.08465
$86$	0	0
$87$	0	0
$88$	0	0
$89$	− 18.3848i	− 1.94878i	−0.224860	−	0.974391i	$-0.572192\pi$
$89$	− 18.3848i	− 1.94878i	0.224860	−	0.974391i	$-0.427808\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	8.00000	0.812277	0.406138	−	0.913812i	$-0.366875\pi$
$97$	8.00000	0.812277	0.406138	+	0.913812i	$0.366875\pi$
$98$	9.89949i	1.00000i
$99$	0	0
$100$	−6.00000	−0.600000
$101$	15.5563i	1.54791i	0.633238	+	0.773957i	$0.281726\pi$
$101$	15.5563i	1.54791i	−0.633238	+	0.773957i	$0.718274\pi$
$102$	0	0
$103$	0	0	1.00000			$0$
$103$	0	0	−1.00000			$\pi$
$104$	11.3137i	1.10940i
$105$	0	0
$106$	−10.0000	−0.971286
$107$	0	0		−	1.00000i	$-0.5\pi$
$107$	0	0			1.00000i	$0.5\pi$
$108$	0	0
$109$	20.0000	1.91565	0.957826	−	0.287348i	$-0.0927736\pi$
$109$	20.0000	1.91565	0.957826	+	0.287348i	$0.0927736\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 1.41421i	− 0.133038i	−0.997785	−	0.0665190i	$-0.978811\pi$
$113$	− 1.41421i	− 0.133038i	0.997785	−	0.0665190i	$-0.0211893\pi$
$114$	0	0
$115$	0	0
$116$	19.7990i	1.83829i
$117$	0	0
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−11.0000	−1.00000
$122$	− 14.1421i	− 1.28037i
$123$	0	0
$124$	0	0
$125$	− 11.3137i	− 1.01193i
$126$	0	0
$127$	0	0	1.00000			$0$
$127$	0	0	−1.00000			$\pi$
$128$	− 11.3137i	− 1.00000i
$129$	0	0
$130$	−8.00000	−0.701646
$131$	0	0		−	1.00000i	$-0.5\pi$
$131$	0	0			1.00000i	$0.5\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	20.0000	1.71499
$137$	− 9.89949i	− 0.845771i	−0.906183	−	0.422885i	$-0.861017\pi$
$137$	− 9.89949i	− 0.845771i	0.906183	−	0.422885i	$-0.138983\pi$
$138$	0	0
$139$	0	0	1.00000			$0$
$139$	0	0	−1.00000			$\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	0	0
$144$	0	0
$145$	−14.0000	−1.16264
$146$	− 22.6274i	− 1.87266i
$147$	0	0
$148$	−4.00000	−0.328798
$149$	24.0416i	1.96957i	0.173785	+	0.984784i	$0.444400\pi$
$149$	24.0416i	1.96957i	−0.173785	+	0.984784i	$0.555600\pi$
$150$	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
$157$	−22.0000	−1.75579	−0.877896	−	0.478852i	$-0.841053\pi$
$157$	−22.0000	−1.75579	−0.877896	+	0.478852i	$0.841053\pi$
$158$	0	0
$159$	0	0
$160$	8.00000	0.632456
$161$	0	0
$162$	0	0
$163$	0	0	1.00000			$0$
$163$	0	0	−1.00000			$\pi$
$164$	2.82843i	0.220863i
$165$	0	0
$166$	0	0
$167$	0	0		−	1.00000i	$-0.5\pi$
$167$	0	0			1.00000i	$0.5\pi$
$168$	0	0
$169$	3.00000	0.230769
$170$	14.1421i	1.08465i
$171$	0	0
$172$	0	0
$173$	15.5563i	1.18273i	0.806405	+	0.591364i	$0.201410\pi$
$173$	15.5563i	1.18273i	−0.806405	+	0.591364i	$0.798590\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	26.0000	1.94878
$179$	0	0		−	1.00000i	$-0.5\pi$
$179$	0	0			1.00000i	$0.5\pi$
$180$	0	0
$181$	20.0000	1.48659	0.743294	−	0.668965i	$-0.233262\pi$
$181$	20.0000	1.48659	0.743294	+	0.668965i	$0.233262\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	− 2.82843i	− 0.207950i
$186$	0	0
$187$	0	0
$188$	0	0
$189$	0	0
$190$	0	0
$191$	0	0		−	1.00000i	$-0.5\pi$
$191$	0	0			1.00000i	$0.5\pi$
$192$	0	0
$193$	14.0000	1.00774	0.503871	−	0.863779i	$-0.331909\pi$
$193$	14.0000	1.00774	0.503871	+	0.863779i	$0.331909\pi$
$194$	11.3137i	0.812277i
$195$	0	0
$196$	−14.0000	−1.00000
$197$	− 18.3848i	− 1.30986i	−0.755689	−	0.654931i	$-0.772698\pi$
$197$	− 18.3848i	− 1.30986i	0.755689	−	0.654931i	$-0.227302\pi$
$198$	0	0
$199$	0	0	1.00000			$0$
$199$	0	0	−1.00000			$\pi$
$200$	− 8.48528i	− 0.600000i
$201$	0	0
$202$	−22.0000	−1.54791
$203$	0	0
$204$	0	0
$205$	−2.00000	−0.139686
$206$	0	0
$207$	0	0
$208$	−16.0000	−1.10940
$209$	0	0
$210$	0	0
$211$	0	0	1.00000			$0$
$211$	0	0	−1.00000			$\pi$
$212$	− 14.1421i	− 0.971286i
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	0	0
$218$	28.2843i	1.91565i
$219$	0	0
$220$	0	0
$221$	− 28.2843i	− 1.90261i
$222$	0	0
$223$	0	0	1.00000			$0$
$223$	0	0	−1.00000			$\pi$
$224$	0	0
$225$	0	0
$226$	2.00000	0.133038
$227$	0	0		−	1.00000i	$-0.5\pi$
$227$	0	0			1.00000i	$0.5\pi$
$228$	0	0
$229$	−4.00000	−0.264327	−0.132164	−	0.991228i	$-0.542192\pi$
$229$	−4.00000	−0.264327	−0.132164	+	0.991228i	$0.542192\pi$
$230$	0	0
$231$	0	0
$232$	−28.0000	−1.83829
$233$	7.07107i	0.463241i	0.972806	+	0.231621i	$0.0744028\pi$
$233$	7.07107i	0.463241i	−0.972806	+	0.231621i	$0.925597\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0	0
$239$	0	0		−	1.00000i	$-0.5\pi$
$239$	0	0			1.00000i	$0.5\pi$
$240$	0	0
$241$	8.00000	0.515325	0.257663	−	0.966235i	$-0.417048\pi$
$241$	8.00000	0.515325	0.257663	+	0.966235i	$0.417048\pi$
$242$	− 15.5563i	− 1.00000i
$243$	0	0
$244$	20.0000	1.28037
$245$	− 9.89949i	− 0.632456i
$246$	0	0
$247$	0	0
$248$	0	0
$249$	0	0
$250$	16.0000	1.01193
$251$	0	0		−	1.00000i	$-0.5\pi$
$251$	0	0			1.00000i	$0.5\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	0	0
$256$	16.0000	1.00000
$257$	24.0416i	1.49968i	0.661622	+	0.749838i	$0.269869\pi$
$257$	24.0416i	1.49968i	−0.661622	+	0.749838i	$0.730131\pi$
$258$	0	0
$259$	0	0
$260$	− 11.3137i	− 0.701646i
$261$	0	0
$262$	0	0
$263$	0	0		−	1.00000i	$-0.5\pi$
$263$	0	0			1.00000i	$0.5\pi$
$264$	0	0
$265$	10.0000	0.614295
$266$	0	0
$267$	0	0
$268$	0	0
$269$	32.5269i	1.98320i	0.129339	+	0.991600i	$0.458714\pi$
$269$	32.5269i	1.98320i	−0.129339	+	0.991600i	$0.541286\pi$
$270$	0	0
$271$	0	0	1.00000			$0$
$271$	0	0	−1.00000			$\pi$
$272$	28.2843i	1.71499i
$273$	0	0
$274$	14.0000	0.845771
$275$	0	0
$276$	0	0
$277$	−28.0000	−1.68236	−0.841178	−	0.540758i	$-0.818138\pi$
$277$	−28.0000	−1.68236	−0.841178	+	0.540758i	$0.818138\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	15.5563i	0.928014i	0.885832	+	0.464007i	$0.153589\pi$
$281$	15.5563i	0.928014i	−0.885832	+	0.464007i	$0.846411\pi$
$282$	0	0
$283$	0	0	1.00000			$0$
$283$	0	0	−1.00000			$\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−33.0000	−1.94118
$290$	− 19.7990i	− 1.16264i
$291$	0	0
$292$	32.0000	1.87266
$293$	− 26.8701i	− 1.56977i	−0.619644	−	0.784883i	$-0.712723\pi$
$293$	− 26.8701i	− 1.56977i	0.619644	−	0.784883i	$-0.287277\pi$
$294$	0	0
$295$	0	0
$296$	− 5.65685i	− 0.328798i
$297$	0	0
$298$	−34.0000	−1.96957
$299$	0	0
$300$	0	0
$301$	0	0
$302$	0	0
$303$	0	0
$304$	0	0
$305$	14.1421i	0.809776i
$306$	0	0
$307$	0	0	1.00000			$0$
$307$	0	0	−1.00000			$\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	0	0		−	1.00000i	$-0.5\pi$
$311$	0	0			1.00000i	$0.5\pi$
$312$	0	0
$313$	26.0000	1.46961	0.734803	−	0.678280i	$-0.237274\pi$
$313$	26.0000	1.46961	0.734803	+	0.678280i	$0.237274\pi$
$314$	− 31.1127i	− 1.75579i
$315$	0	0
$316$	0	0
$317$	− 35.3553i	− 1.98575i	−0.119145	−	0.992877i	$-0.538015\pi$
$317$	− 35.3553i	− 1.98575i	0.119145	−	0.992877i	$-0.461985\pi$
$318$	0	0
$319$	0	0
$320$	11.3137i	0.632456i
$321$	0	0
$322$	0	0
$323$	0	0
$324$	0	0
$325$	−12.0000	−0.665640
$326$	0	0
$327$	0	0
$328$	−4.00000	−0.220863
$329$	0	0
$330$	0	0
$331$	0	0	1.00000			$0$
$331$	0	0	−1.00000			$\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	32.0000	1.74315	0.871576	−	0.490261i	$-0.163099\pi$
$337$	32.0000	1.74315	0.871576	+	0.490261i	$0.163099\pi$
$338$	4.24264i	0.230769i
$339$	0	0
$340$	−20.0000	−1.08465
$341$	0	0
$342$	0	0
$343$	0	0
$344$	0	0
$345$	0	0
$346$	−22.0000	−1.18273
$347$	0	0		−	1.00000i	$-0.5\pi$
$347$	0	0			1.00000i	$0.5\pi$
$348$	0	0
$349$	−10.0000	−0.535288	−0.267644	−	0.963518i	$-0.586245\pi$
$349$	−10.0000	−0.535288	−0.267644	+	0.963518i	$0.586245\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 35.3553i	− 1.88177i	−0.338719	−	0.940887i	$-0.609994\pi$
$353$	− 35.3553i	− 1.88177i	0.338719	−	0.940887i	$-0.390006\pi$
$354$	0	0
$355$	0	0
$356$	36.7696i	1.94878i
$357$	0	0
$358$	0	0
$359$	0	0		−	1.00000i	$-0.5\pi$
$359$	0	0			1.00000i	$0.5\pi$
$360$	0	0
$361$	19.0000	1.00000
$362$	28.2843i	1.48659i
$363$	0	0
$364$	0	0
$365$	22.6274i	1.18437i
$366$	0	0
$367$	0	0	1.00000			$0$
$367$	0	0	−1.00000			$\pi$
$368$	0	0
$369$	0	0
$370$	4.00000	0.207950
$371$	0	0
$372$	0	0
$373$	14.0000	0.724893	0.362446	−	0.932005i	$-0.381942\pi$
$373$	14.0000	0.724893	0.362446	+	0.932005i	$0.381942\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	39.5980i	2.03940i
$378$	0	0
$379$	0	0	1.00000			$0$
$379$	0	0	−1.00000			$\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	0	0		−	1.00000i	$-0.5\pi$
$383$	0	0			1.00000i	$0.5\pi$
$384$	0	0
$385$	0	0
$386$	19.7990i	1.00774i
$387$	0	0
$388$	−16.0000	−0.812277
$389$	− 9.89949i	− 0.501924i	−0.967997	−	0.250962i	$-0.919253\pi$
$389$	− 9.89949i	− 0.501924i	0.967997	−	0.250962i	$-0.0807470\pi$
$390$	0	0
$391$	0	0
$392$	− 19.7990i	− 1.00000i
$393$	0	0
$394$	26.0000	1.30986
$395$	0	0
$396$	0	0
$397$	38.0000	1.90717	0.953583	−	0.301131i	$-0.0973643\pi$
$397$	38.0000	1.90717	0.953583	+	0.301131i	$0.0973643\pi$
$398$	0	0
$399$	0	0
$400$	12.0000	0.600000
$401$	− 26.8701i	− 1.34183i	−0.741536	−	0.670913i	$-0.765902\pi$
$401$	− 26.8701i	− 1.34183i	0.741536	−	0.670913i	$-0.234098\pi$
$402$	0	0
$403$	0	0
$404$	− 31.1127i	− 1.54791i
$405$	0	0
$406$	0	0
$407$	0	0
$408$	0	0
$409$	−40.0000	−1.97787	−0.988936	−	0.148340i	$-0.952607\pi$
$409$	−40.0000	−1.97787	−0.988936	+	0.148340i	$0.952607\pi$
$410$	− 2.82843i	− 0.139686i
$411$	0	0
$412$	0	0
$413$	0	0
$414$	0	0
$415$	0	0
$416$	− 22.6274i	− 1.10940i
$417$	0	0
$418$	0	0
$419$	0	0		−	1.00000i	$-0.5\pi$
$419$	0	0			1.00000i	$0.5\pi$
$420$	0	0
$421$	−28.0000	−1.36464	−0.682318	−	0.731055i	$-0.739028\pi$
$421$	−28.0000	−1.36464	−0.682318	+	0.731055i	$0.739028\pi$
$422$	0	0
$423$	0	0
$424$	20.0000	0.971286
$425$	21.2132i	1.02899i
$426$	0	0
$427$	0	0
$428$	0	0
$429$	0	0
$430$	0	0
$431$	0	0		−	1.00000i	$-0.5\pi$
$431$	0	0			1.00000i	$0.5\pi$
$432$	0	0
$433$	−34.0000	−1.63394	−0.816968	−	0.576683i	$-0.804347\pi$
$433$	−34.0000	−1.63394	−0.816968	+	0.576683i	$0.804347\pi$
$434$	0	0
$435$	0	0
$436$	−40.0000	−1.91565
$437$	0	0
$438$	0	0
$439$	0	0	1.00000			$0$
$439$	0	0	−1.00000			$\pi$
$440$	0	0
$441$	0	0
$442$	40.0000	1.90261
$443$	0	0		−	1.00000i	$-0.5\pi$
$443$	0	0			1.00000i	$0.5\pi$
$444$	0	0
$445$	−26.0000	−1.23252
$446$	0	0
$447$	0	0
$448$	0	0
$449$	− 18.3848i	− 0.867631i	−0.901002	−	0.433816i	$-0.857167\pi$
$449$	− 18.3848i	− 0.867631i	0.901002	−	0.433816i	$-0.142833\pi$
$450$	0	0
$451$	0	0
$452$	2.82843i	0.133038i
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	8.00000	0.374224	0.187112	−	0.982339i	$-0.440087\pi$
$457$	8.00000	0.374224	0.187112	+	0.982339i	$0.440087\pi$
$458$	− 5.65685i	− 0.264327i
$459$	0	0
$460$	0	0
$461$	41.0122i	1.91013i	0.296399	+	0.955064i	$0.404214\pi$
$461$	41.0122i	1.91013i	−0.296399	+	0.955064i	$0.595786\pi$
$462$	0	0
$463$	0	0	1.00000			$0$
$463$	0	0	−1.00000			$\pi$
$464$	− 39.5980i	− 1.83829i
$465$	0	0
$466$	−10.0000	−0.463241
$467$	0	0		−	1.00000i	$-0.5\pi$
$467$	0	0			1.00000i	$0.5\pi$
$468$	0	0
$469$	0	0
$470$	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		−	1.00000i	$-0.5\pi$
$479$	0	0			1.00000i	$0.5\pi$
$480$	0	0
$481$	−8.00000	−0.364769
$482$	11.3137i	0.515325i
$483$	0	0
$484$	22.0000	1.00000
$485$	− 11.3137i	− 0.513729i
$486$	0	0
$487$	0	0	1.00000			$0$
$487$	0	0	−1.00000			$\pi$
$488$	28.2843i	1.28037i
$489$	0	0
$490$	14.0000	0.632456
$491$	0	0		−	1.00000i	$-0.5\pi$
$491$	0	0			1.00000i	$0.5\pi$
$492$	0	0
$493$	70.0000	3.15264
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	0	0	1.00000			$0$
$499$	0	0	−1.00000			$\pi$
$500$	22.6274i	1.01193i
$501$	0	0
$502$	0	0
$503$	0	0		−	1.00000i	$-0.5\pi$
$503$	0	0			1.00000i	$0.5\pi$
$504$	0	0
$505$	22.0000	0.978987
$506$	0	0
$507$	0	0
$508$	0	0
$509$	24.0416i	1.06563i	0.846233	+	0.532813i	$0.178865\pi$
$509$	24.0416i	1.06563i	−0.846233	+	0.532813i	$0.821135\pi$
$510$	0	0
$511$	0	0
$512$	22.6274i	1.00000i
$513$	0	0
$514$	−34.0000	−1.49968
$515$	0	0
$516$	0	0
$517$	0	0
$518$	0	0
$519$	0	0
$520$	16.0000	0.701646
$521$	− 43.8406i	− 1.92069i	−0.278810	−	0.960346i	$-0.589940\pi$
$521$	− 43.8406i	− 1.92069i	0.278810	−	0.960346i	$-0.410060\pi$
$522$	0	0
$523$	0	0	1.00000			$0$
$523$	0	0	−1.00000			$\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	0	0
$528$	0	0
$529$	−23.0000	−1.00000
$530$	14.1421i	0.614295i
$531$	0	0
$532$	0	0
$533$	5.65685i	0.245026i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	−46.0000	−1.98320
$539$	0	0
$540$	0	0
$541$	20.0000	0.859867	0.429934	−	0.902861i	$-0.358537\pi$
$541$	20.0000	0.859867	0.429934	+	0.902861i	$0.358537\pi$
$542$	0	0
$543$	0	0
$544$	−40.0000	−1.71499
$545$	− 28.2843i	− 1.21157i
$546$	0	0
$547$	0	0	1.00000			$0$
$547$	0	0	−1.00000			$\pi$
$548$	19.7990i	0.845771i
$549$	0	0
$550$	0	0
$551$	0	0
$552$	0	0
$553$	0	0
$554$	− 39.5980i	− 1.68236i
$555$	0	0
$556$	0	0
$557$	7.07107i	0.299611i	0.988716	+	0.149805i	$0.0478647\pi$
$557$	7.07107i	0.299611i	−0.988716	+	0.149805i	$0.952135\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0	0
$562$	−22.0000	−0.928014
$563$	0	0		−	1.00000i	$-0.5\pi$
$563$	0	0			1.00000i	$0.5\pi$
$564$	0	0
$565$	−2.00000	−0.0841406
$566$	0	0
$567$	0	0
$568$	0	0
$569$	− 9.89949i	− 0.415008i	−0.978234	−	0.207504i	$-0.933466\pi$
$569$	− 9.89949i	− 0.415008i	0.978234	−	0.207504i	$-0.0665341\pi$
$570$	0	0
$571$	0	0	1.00000			$0$
$571$	0	0	−1.00000			$\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	2.00000	0.0832611	0.0416305	−	0.999133i	$-0.486745\pi$
$577$	2.00000	0.0832611	0.0416305	+	0.999133i	$0.486745\pi$
$578$	− 46.6690i	− 1.94118i
$579$	0	0
$580$	28.0000	1.16264
$581$	0	0
$582$	0	0
$583$	0	0
$584$	45.2548i	1.87266i
$585$	0	0
$586$	38.0000	1.56977
$587$	0	0		−	1.00000i	$-0.5\pi$
$587$	0	0			1.00000i	$0.5\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	0	0
$592$	8.00000	0.328798
$593$	− 43.8406i	− 1.80032i	−0.435561	−	0.900159i	$-0.643450\pi$
$593$	− 43.8406i	− 1.80032i	0.435561	−	0.900159i	$-0.356550\pi$
$594$	0	0
$595$	0	0
$596$	− 48.0833i	− 1.96957i
$597$	0	0
$598$	0	0
$599$	0	0		−	1.00000i	$-0.5\pi$
$599$	0	0			1.00000i	$0.5\pi$
$600$	0	0
$601$	−10.0000	−0.407909	−0.203954	−	0.978980i	$-0.565379\pi$
$601$	−10.0000	−0.407909	−0.203954	+	0.978980i	$0.565379\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	15.5563i	0.632456i
$606$	0	0
$607$	0	0	1.00000			$0$
$607$	0	0	−1.00000			$\pi$
$608$	0	0
$609$	0	0
$610$	−20.0000	−0.809776
$611$	0	0
$612$	0	0
$613$	−34.0000	−1.37325	−0.686624	−	0.727013i	$-0.740908\pi$
$613$	−34.0000	−1.37325	−0.686624	+	0.727013i	$0.740908\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	49.4975i	1.99269i	0.0854011	+	0.996347i	$0.472783\pi$
$617$	49.4975i	1.99269i	−0.0854011	+	0.996347i	$0.527217\pi$
$618$	0	0
$619$	0	0	1.00000			$0$
$619$	0	0	−1.00000			$\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−1.00000	−0.0400000
$626$	36.7696i	1.46961i
$627$	0	0
$628$	44.0000	1.75579
$629$	14.1421i	0.563884i
$630$	0	0
$631$	0	0	1.00000			$0$
$631$	0	0	−1.00000			$\pi$
$632$	0	0
$633$	0	0
$634$	50.0000	1.98575
$635$	0	0
$636$	0	0
$637$	−28.0000	−1.10940
$638$	0	0
$639$	0	0
$640$	−16.0000	−0.632456
$641$	41.0122i	1.61988i	0.586510	+	0.809942i	$0.300502\pi$
$641$	41.0122i	1.61988i	−0.586510	+	0.809942i	$0.699498\pi$
$642$	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		−	1.00000i	$-0.5\pi$
$647$	0	0			1.00000i	$0.5\pi$
$648$	0	0
$649$	0	0
$650$	− 16.9706i	− 0.665640i
$651$	0	0
$652$	0	0
$653$	49.4975i	1.93699i	0.249041	+	0.968493i	$0.419885\pi$
$653$	49.4975i	1.93699i	−0.249041	+	0.968493i	$0.580115\pi$
$654$	0	0
$655$	0	0
$656$	− 5.65685i	− 0.220863i
$657$	0	0
$658$	0	0
$659$	0	0		−	1.00000i	$-0.5\pi$
$659$	0	0			1.00000i	$0.5\pi$
$660$	0	0
$661$	50.0000	1.94477	0.972387	−	0.233373i	$-0.0749763\pi$
$661$	50.0000	1.94477	0.972387	+	0.233373i	$0.0749763\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	0	0
$668$	0	0
$669$	0	0
$670$	0	0
$671$	0	0
$672$	0	0
$673$	−46.0000	−1.77317	−0.886585	−	0.462566i	$-0.846929\pi$
$673$	−46.0000	−1.77317	−0.886585	+	0.462566i	$0.846929\pi$
$674$	45.2548i	1.74315i
$675$	0	0
$676$	−6.00000	−0.230769
$677$	− 35.3553i	− 1.35882i	−0.733761	−	0.679408i	$-0.762237\pi$
$677$	− 35.3553i	− 1.35882i	0.733761	−	0.679408i	$-0.237763\pi$
$678$	0	0
$679$	0	0
$680$	− 28.2843i	− 1.08465i
$681$	0	0
$682$	0	0
$683$	0	0		−	1.00000i	$-0.5\pi$
$683$	0	0			1.00000i	$0.5\pi$
$684$	0	0
$685$	−14.0000	−0.534913
$686$	0	0
$687$	0	0
$688$	0	0
$689$	− 28.2843i	− 1.07754i
$690$	0	0
$691$	0	0	1.00000			$0$
$691$	0	0	−1.00000			$\pi$
$692$	− 31.1127i	− 1.18273i
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	10.0000	0.378777
$698$	− 14.1421i	− 0.535288i
$699$	0	0
$700$	0	0
$701$	− 43.8406i	− 1.65584i	−0.560848	−	0.827919i	$-0.689525\pi$
$701$	− 43.8406i	− 1.65584i	0.560848	−	0.827919i	$-0.310475\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	0	0
$706$	50.0000	1.88177
$707$	0	0
$708$	0	0
$709$	44.0000	1.65245	0.826227	−	0.563337i	$-0.190483\pi$
$709$	44.0000	1.65245	0.826227	+	0.563337i	$0.190483\pi$
$710$	0	0
$711$	0	0
$712$	−52.0000	−1.94878
$713$	0	0
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	0	0		−	1.00000i	$-0.5\pi$
$719$	0	0			1.00000i	$0.5\pi$
$720$	0	0
$721$	0	0
$722$	26.8701i	1.00000i
$723$	0	0
$724$	−40.0000	−1.48659
$725$	− 29.6985i	− 1.10297i
$726$	0	0
$727$	0	0	1.00000			$0$
$727$	0	0	−1.00000			$\pi$
$728$	0	0
$729$	0	0
$730$	−32.0000	−1.18437
$731$	0	0
$732$	0	0
$733$	−4.00000	−0.147743	−0.0738717	−	0.997268i	$-0.523536\pi$
$733$	−4.00000	−0.147743	−0.0738717	+	0.997268i	$0.523536\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	0	0
$738$	0	0
$739$	0	0	1.00000			$0$
$739$	0	0	−1.00000			$\pi$
$740$	5.65685i	0.207950i
$741$	0	0
$742$	0	0
$743$	0	0		−	1.00000i	$-0.5\pi$
$743$	0	0			1.00000i	$0.5\pi$
$744$	0	0
$745$	34.0000	1.24566
$746$	19.7990i	0.724893i
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	0	0	1.00000			$0$
$751$	0	0	−1.00000			$\pi$
$752$	0	0
$753$	0	0
$754$	−56.0000	−2.03940
$755$	0	0
$756$	0	0
$757$	−52.0000	−1.88997	−0.944986	−	0.327111i	$-0.893925\pi$
$757$	−52.0000	−1.88997	−0.944986	+	0.327111i	$0.893925\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	− 1.41421i	− 0.0512652i	−0.999671	−	0.0256326i	$-0.991840\pi$
$761$	− 1.41421i	− 0.0512652i	0.999671	−	0.0256326i	$-0.00816000\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	0	0
$769$	50.0000	1.80305	0.901523	−	0.432731i	$-0.142450\pi$
$769$	50.0000	1.80305	0.901523	+	0.432731i	$0.142450\pi$
$770$	0	0
$771$	0	0
$772$	−28.0000	−1.00774
$773$	7.07107i	0.254329i	0.991882	+	0.127164i	$0.0405876\pi$
$773$	7.07107i	0.254329i	−0.991882	+	0.127164i	$0.959412\pi$
$774$	0	0
$775$	0	0
$776$	− 22.6274i	− 0.812277i
$777$	0	0
$778$	14.0000	0.501924
$779$	0	0
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	28.0000	1.00000
$785$	31.1127i	1.11046i
$786$	0	0
$787$	0	0	1.00000			$0$
$787$	0	0	−1.00000			$\pi$
$788$	36.7696i	1.30986i
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	40.0000	1.42044
$794$	53.7401i	1.90717i
$795$	0	0
$796$	0	0
$797$	− 52.3259i	− 1.85348i	−0.375705	−	0.926739i	$-0.622599\pi$
$797$	− 52.3259i	− 1.85348i	0.375705	−	0.926739i	$-0.377401\pi$
$798$	0	0
$799$	0	0
$800$	16.9706i	0.600000i
$801$	0	0
$802$	38.0000	1.34183
$803$	0	0
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	44.0000	1.54791
$809$	32.5269i	1.14359i	0.820398	+	0.571793i	$0.193752\pi$
$809$	32.5269i	1.14359i	−0.820398	+	0.571793i	$0.806248\pi$
$810$	0	0
$811$	0	0	1.00000			$0$
$811$	0	0	−1.00000			$\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	0	0
$818$	− 56.5685i	− 1.97787i
$819$	0	0
$820$	4.00000	0.139686
$821$	15.5563i	0.542920i	0.962450	+	0.271460i	$0.0875065\pi$
$821$	15.5563i	0.542920i	−0.962450	+	0.271460i	$0.912493\pi$
$822$	0	0
$823$	0	0	1.00000			$0$
$823$	0	0	−1.00000			$\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	0	0		−	1.00000i	$-0.5\pi$
$827$	0	0			1.00000i	$0.5\pi$
$828$	0	0
$829$	20.0000	0.694629	0.347314	−	0.937749i	$-0.387094\pi$
$829$	20.0000	0.694629	0.347314	+	0.937749i	$0.387094\pi$
$830$	0	0
$831$	0	0
$832$	32.0000	1.10940
$833$	49.4975i	1.71499i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	0	0
$839$	0	0		−	1.00000i	$-0.5\pi$
$839$	0	0			1.00000i	$0.5\pi$
$840$	0	0
$841$	−69.0000	−2.37931
$842$	− 39.5980i	− 1.36464i
$843$	0	0
$844$	0	0
$845$	− 4.24264i	− 0.145951i
$846$	0	0
$847$	0	0
$848$	28.2843i	0.971286i
$849$	0	0
$850$	−30.0000	−1.02899
$851$	0	0
$852$	0	0
$853$	−46.0000	−1.57501	−0.787505	−	0.616308i	$-0.788628\pi$
$853$	−46.0000	−1.57501	−0.787505	+	0.616308i	$0.788628\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 35.3553i	− 1.20772i	−0.797092	−	0.603858i	$-0.793630\pi$
$857$	− 35.3553i	− 1.20772i	0.797092	−	0.603858i	$-0.206370\pi$
$858$	0	0
$859$	0	0	1.00000			$0$
$859$	0	0	−1.00000			$\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	0	0		−	1.00000i	$-0.5\pi$
$863$	0	0			1.00000i	$0.5\pi$
$864$	0	0
$865$	22.0000	0.748022
$866$	− 48.0833i	− 1.63394i
$867$	0	0
$868$	0	0
$869$	0	0
$870$	0	0
$871$	0	0
$872$	− 56.5685i	− 1.91565i
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−58.0000	−1.95852	−0.979260	−	0.202606i	$-0.935059\pi$
$877$	−58.0000	−1.95852	−0.979260	+	0.202606i	$0.935059\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	57.9828i	1.95349i	0.214407	+	0.976744i	$0.431218\pi$
$881$	57.9828i	1.95349i	−0.214407	+	0.976744i	$0.568782\pi$
$882$	0	0
$883$	0	0	1.00000			$0$
$883$	0	0	−1.00000			$\pi$
$884$	56.5685i	1.90261i
$885$	0	0
$886$	0	0
$887$	0	0		−	1.00000i	$-0.5\pi$
$887$	0	0			1.00000i	$0.5\pi$
$888$	0	0
$889$	0	0
$890$	− 36.7696i	− 1.23252i
$891$	0	0
$892$	0	0
$893$	0	0
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	26.0000	0.867631
$899$	0	0
$900$	0	0
$901$	−50.0000	−1.66574
$902$	0	0
$903$	0	0
$904$	−4.00000	−0.133038
$905$	− 28.2843i	− 0.940201i
$906$	0	0
$907$	0	0	1.00000			$0$
$907$	0	0	−1.00000			$\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	0	0		−	1.00000i	$-0.5\pi$
$911$	0	0			1.00000i	$0.5\pi$
$912$	0	0
$913$	0	0
$914$	11.3137i	0.374224i
$915$	0	0
$916$	8.00000	0.264327
$917$	0	0
$918$	0	0
$919$	0	0	1.00000			$0$
$919$	0	0	−1.00000			$\pi$
$920$	0	0
$921$	0	0
$922$	−58.0000	−1.91013
$923$	0	0
$924$	0	0
$925$	6.00000	0.197279
$926$	0	0
$927$	0	0
$928$	56.0000	1.83829
$929$	− 60.8112i	− 1.99515i	−0.0695983	−	0.997575i	$-0.522172\pi$
$929$	− 60.8112i	− 1.99515i	0.0695983	−	0.997575i	$-0.477828\pi$
$930$	0	0
$931$	0	0
$932$	− 14.1421i	− 0.463241i
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	38.0000	1.24141	0.620703	−	0.784046i	$-0.286847\pi$
$937$	38.0000	1.24141	0.620703	+	0.784046i	$0.286847\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	− 26.8701i	− 0.875939i	−0.898990	−	0.437969i	$-0.855698\pi$
$941$	− 26.8701i	− 0.875939i	0.898990	−	0.437969i	$-0.144302\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	0	0
$946$	0	0
$947$	0	0		−	1.00000i	$-0.5\pi$
$947$	0	0			1.00000i	$0.5\pi$
$948$	0	0
$949$	64.0000	2.07753
$950$	0	0
$951$	0	0
$952$	0	0
$953$	57.9828i	1.87825i	0.343582	+	0.939123i	$0.388360\pi$
$953$	57.9828i	1.87825i	−0.343582	+	0.939123i	$0.611640\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	31.0000	1.00000
$962$	− 11.3137i	− 0.364769i
$963$	0	0
$964$	−16.0000	−0.515325
$965$	− 19.7990i	− 0.637352i
$966$	0	0
$967$	0	0	1.00000			$0$
$967$	0	0	−1.00000			$\pi$
$968$	31.1127i	1.00000i
$969$	0	0
$970$	16.0000	0.513729
$971$	0	0		−	1.00000i	$-0.5\pi$
$971$	0	0			1.00000i	$0.5\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0	0
$976$	−40.0000	−1.28037
$977$	49.4975i	1.58356i	0.610803	+	0.791782i	$0.290847\pi$
$977$	49.4975i	1.58356i	−0.610803	+	0.791782i	$0.709153\pi$
$978$	0	0
$979$	0	0
$980$	19.7990i	0.632456i
$981$	0	0
$982$	0	0
$983$	0	0		−	1.00000i	$-0.5\pi$
$983$	0	0			1.00000i	$0.5\pi$
$984$	0	0
$985$	−26.0000	−0.828429
$986$	98.9949i	3.15264i
$987$	0	0
$988$	0	0
$989$	0	0
$990$	0	0
$991$	0	0	1.00000			$0$
$991$	0	0	−1.00000			$\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	62.0000	1.96356	0.981780	−	0.190022i	$-0.0608559\pi$
$997$	62.0000	1.96356	0.981780	+	0.190022i	$0.0608559\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	36.2.b.a.35.2	yes	2
3.2	odd	2	inner	36.2.b.a.35.1	✓	2
4.3	odd	2	CM	36.2.b.a.35.2	yes	2
5.2	odd	4		900.2.h.a.899.2		4
5.3	odd	4		900.2.h.a.899.3		4
5.4	even	2		900.2.e.b.251.1		2
7.6	odd	2		1764.2.e.b.1079.2		2
8.3	odd	2		576.2.c.b.575.2		2
8.5	even	2		576.2.c.b.575.2		2
9.2	odd	6		324.2.h.c.215.1		4
9.4	even	3		324.2.h.c.107.1		4
9.5	odd	6		324.2.h.c.107.2		4
9.7	even	3		324.2.h.c.215.2		4
12.11	even	2	inner	36.2.b.a.35.1	✓	2
15.2	even	4		900.2.h.a.899.4		4
15.8	even	4		900.2.h.a.899.1		4
15.14	odd	2		900.2.e.b.251.2		2
16.3	odd	4		2304.2.f.d.1151.1		4
16.5	even	4		2304.2.f.d.1151.4		4
16.11	odd	4		2304.2.f.d.1151.4		4
16.13	even	4		2304.2.f.d.1151.1		4
20.3	even	4		900.2.h.a.899.3		4
20.7	even	4		900.2.h.a.899.2		4
20.19	odd	2		900.2.e.b.251.1		2
21.20	even	2		1764.2.e.b.1079.1		2
24.5	odd	2		576.2.c.b.575.1		2
24.11	even	2		576.2.c.b.575.1		2
28.27	even	2		1764.2.e.b.1079.2		2
36.7	odd	6		324.2.h.c.215.2		4
36.11	even	6		324.2.h.c.215.1		4
36.23	even	6		324.2.h.c.107.2		4
36.31	odd	6		324.2.h.c.107.1		4
48.5	odd	4		2304.2.f.d.1151.2		4
48.11	even	4		2304.2.f.d.1151.2		4
48.29	odd	4		2304.2.f.d.1151.3		4
48.35	even	4		2304.2.f.d.1151.3		4
60.23	odd	4		900.2.h.a.899.1		4
60.47	odd	4		900.2.h.a.899.4		4
60.59	even	2		900.2.e.b.251.2		2
84.83	odd	2		1764.2.e.b.1079.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
36.2.b.a.35.1	✓	2	3.2	odd	2	inner
36.2.b.a.35.1	✓	2	12.11	even	2	inner
36.2.b.a.35.2	yes	2	1.1	even	1	trivial
36.2.b.a.35.2	yes	2	4.3	odd	2	CM
324.2.h.c.107.1		4	9.4	even	3
324.2.h.c.107.1		4	36.31	odd	6
324.2.h.c.107.2		4	9.5	odd	6
324.2.h.c.107.2		4	36.23	even	6
324.2.h.c.215.1		4	9.2	odd	6
324.2.h.c.215.1		4	36.11	even	6
324.2.h.c.215.2		4	9.7	even	3
324.2.h.c.215.2		4	36.7	odd	6
576.2.c.b.575.1		2	24.5	odd	2
576.2.c.b.575.1		2	24.11	even	2
576.2.c.b.575.2		2	8.3	odd	2
576.2.c.b.575.2		2	8.5	even	2
900.2.e.b.251.1		2	5.4	even	2
900.2.e.b.251.1		2	20.19	odd	2
900.2.e.b.251.2		2	15.14	odd	2
900.2.e.b.251.2		2	60.59	even	2
900.2.h.a.899.1		4	15.8	even	4
900.2.h.a.899.1		4	60.23	odd	4
900.2.h.a.899.2		4	5.2	odd	4
900.2.h.a.899.2		4	20.7	even	4
900.2.h.a.899.3		4	5.3	odd	4
900.2.h.a.899.3		4	20.3	even	4
900.2.h.a.899.4		4	15.2	even	4
900.2.h.a.899.4		4	60.47	odd	4
1764.2.e.b.1079.1		2	21.20	even	2
1764.2.e.b.1079.1		2	84.83	odd	2
1764.2.e.b.1079.2		2	7.6	odd	2
1764.2.e.b.1079.2		2	28.27	even	2
2304.2.f.d.1151.1		4	16.3	odd	4
2304.2.f.d.1151.1		4	16.13	even	4
2304.2.f.d.1151.2		4	48.5	odd	4
2304.2.f.d.1151.2		4	48.11	even	4
2304.2.f.d.1151.3		4	48.29	odd	4
2304.2.f.d.1151.3		4	48.35	even	4
2304.2.f.d.1151.4		4	16.5	even	4
2304.2.f.d.1151.4		4	16.11	odd	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 36 = 2^{2} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	36.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.41421i q^{2} -2.00000 q^{4} -1.41421i q^{5} -2.82843i q^{8} +O(q^{10})\)	\(q+1.41421i q^{2} -2.00000 q^{4} -1.41421i q^{5} -2.82843i q^{8} +2.00000 q^{10} -4.00000 q^{13} +4.00000 q^{16} +7.07107i q^{17} +2.82843i q^{20} +3.00000 q^{25} -5.65685i q^{26} -9.89949i q^{29} +5.65685i q^{32} -10.0000 q^{34} +2.00000 q^{37} -4.00000 q^{40} -1.41421i q^{41} +7.00000 q^{49} +4.24264i q^{50} +8.00000 q^{52} +7.07107i q^{53} +14.0000 q^{58} -10.0000 q^{61} -8.00000 q^{64} +5.65685i q^{65} -14.1421i q^{68} -16.0000 q^{73} +2.82843i q^{74} -5.65685i q^{80} +2.00000 q^{82} +10.0000 q^{85} -18.3848i q^{89} +8.00000 q^{97} +9.89949i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 4 q^{4}+O(q^{10}) \) 2 * q - 4 * q^4	\( 2 q - 4 q^{4} + 4 q^{10} - 8 q^{13} + 8 q^{16} + 6 q^{25} - 20 q^{34} + 4 q^{37} - 8 q^{40} + 14 q^{49} + 16 q^{52} + 28 q^{58} - 20 q^{61} - 16 q^{64} - 32 q^{73} + 4 q^{82} + 20 q^{85} + 16 q^{97}+O(q^{100}) \) 2 * q - 4 * q^4 + 4 * q^10 - 8 * q^13 + 8 * q^16 + 6 * q^25 - 20 * q^34 + 4 * q^37 - 8 * q^40 + 14 * q^49 + 16 * q^52 + 28 * q^58 - 20 * q^61 - 16 * q^64 - 32 * q^73 + 4 * q^82 + 20 * q^85 + 16 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more