LMFDB - Embedded newform 135.4.b.a.109.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [135,4,Mod(109,135)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("135.109"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(135, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 135 = 3^{3} \cdot 5 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	135.b (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [4] 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:	no
Analytic conductor:	$7.96525785077$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{5})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 3x^{2} + 1 $ x^4 + 3*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 3^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			109.3
Root			$-1.61803i$ of defining polynomial
Character	$\chi$	$=$	135.109
Dual form			135.4.b.a.109.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+3.38197i q^{2} -3.43769 q^{4} +11.1803i q^{5} +15.4296i q^{8} -37.8115 q^{10} -79.6838 q^{16} +87.3181i q^{17} -102.125 q^{19} -38.4346i q^{20} -121.807i q^{23} -125.000 q^{25} +337.371 q^{31} -146.051i q^{32} -295.307 q^{34} -345.382i q^{38} -172.508 q^{40} +411.945 q^{46} +545.601i q^{47} +343.000 q^{49} -422.746i q^{50} +706.813i q^{53} +943.735 q^{61} +1140.98i q^{62} -143.530 q^{64} -300.173i q^{68} +351.073 q^{76} +1339.35 q^{79} -890.892i q^{80} -1346.04i q^{83} -976.246 q^{85} +418.734i q^{92} -1845.20 q^{94} -1141.79i q^{95} +1160.01i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 54 q^{4} + 50 q^{10} + 446 q^{16} - 328 q^{19} - 500 q^{25} + 464 q^{31} - 14 q^{34} - 2300 q^{40} + 3298 q^{46} + 1372 q^{49} + 716 q^{61} - 6692 q^{64} + 3618 q^{76} + 608 q^{79} - 3100 q^{85} + 2440 q^{94}+O(q^{100}) $ 4 * q - 54 * q^4 + 50 * q^10 + 446 * q^16 - 328 * q^19 - 500 * q^25 + 464 * q^31 - 14 * q^34 - 2300 * q^40 + 3298 * q^46 + 1372 * q^49 + 716 * q^61 - 6692 * q^64 + 3618 * q^76 + 608 * q^79 - 3100 * q^85 + 2440 * q^94

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/135\mathbb{Z}\right)^\times$.

$n$	$56$	$82$
$\chi(n)$	$1$	$-1$

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 $

$2$	3.38197i	1.19571i	0.801606	+	0.597853i	$0.203979\pi$
$2$	3.38197i	1.19571i	−0.801606	+	0.597853i	$0.796021\pi$
$3$	0	0
$3$	0	0
$4$	−3.43769	−0.429712
$5$	11.1803i	1.00000i
$5$	11.1803i	1.00000i
$6$	0	0
$7$	0	0	1.00000			$0$
$7$	0	0	−1.00000			$\pi$
$8$	15.4296i	0.681897i
$9$	0	0
$10$	−37.8115	−1.19571
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	0	0
$13$	0	0	1.00000			$0$
$13$	0	0	−1.00000			$\pi$
$14$	0	0
$15$	0	0
$16$	−79.6838	−1.24506
$17$	87.3181i	1.24575i	0.782321	+	0.622875i	$0.214036\pi$
$17$	87.3181i	1.24575i	−0.782321	+	0.622875i	$0.785964\pi$
$18$	0	0
$19$	−102.125	−1.23310	−0.616552	−	0.787314i	$-0.711471\pi$
$19$	−102.125	−1.23310	−0.616552	+	0.787314i	$0.711471\pi$
$20$	− 38.4346i	− 0.429712i
$21$	0	0
$22$	0	0
$23$	− 121.807i	− 1.10428i	−0.833752	−	0.552139i	$-0.813812\pi$
$23$	− 121.807i	− 1.10428i	0.833752	−	0.552139i	$-0.186188\pi$
$24$	0	0
$25$	−125.000	−1.00000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	0	0		−	1.00000i	$-0.5\pi$
$29$	0	0			1.00000i	$0.5\pi$
$30$	0	0
$31$	337.371	1.95463	0.977316	−	0.211788i	$-0.0679286\pi$
$31$	337.371	1.95463	0.977316	+	0.211788i	$0.0679286\pi$
$32$	− 146.051i	− 0.806828i
$33$	0	0
$34$	−295.307	−1.48955
$35$	0	0
$36$	0	0
$37$	0	0	1.00000			$0$
$37$	0	0	−1.00000			$\pi$
$38$	− 345.382i	− 1.47443i
$39$	0	0
$40$	−172.508	−0.681897
$41$	0	0		−	1.00000i	$-0.5\pi$
$41$	0	0			1.00000i	$0.5\pi$
$42$	0	0
$43$	0	0	1.00000			$0$
$43$	0	0	−1.00000			$\pi$
$44$	0	0
$45$	0	0
$46$	411.945	1.32039
$47$	545.601i	1.69328i	0.532168	+	0.846639i	$0.321377\pi$
$47$	545.601i	1.69328i	−0.532168	+	0.846639i	$0.678623\pi$
$48$	0	0
$49$	343.000	1.00000
$50$	− 422.746i	− 1.19571i
$51$	0	0
$52$	0	0
$53$	706.813i	1.83185i	0.401344	+	0.915927i	$0.368543\pi$
$53$	706.813i	1.83185i	−0.401344	+	0.915927i	$0.631457\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	0	0
$61$	943.735	1.98087	0.990434	−	0.137989i	$-0.0440639\pi$
$61$	943.735	1.98087	0.990434	+	0.137989i	$0.0440639\pi$
$62$	1140.98i	2.33716i
$63$	0	0
$64$	−143.530	−0.280331
$65$	0	0
$66$	0	0
$67$	0	0	1.00000			$0$
$67$	0	0	−1.00000			$\pi$
$68$	− 300.173i	− 0.535313i
$69$	0	0
$70$	0	0
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	0	0
$73$	0	0	1.00000			$0$
$73$	0	0	−1.00000			$\pi$
$74$	0	0
$75$	0	0
$76$	351.073	0.529880
$77$	0	0
$78$	0	0
$79$	1339.35	1.90745	0.953727	−	0.300674i	$-0.0972115\pi$
$79$	1339.35	1.90745	0.953727	+	0.300674i	$0.0972115\pi$
$80$	− 890.892i	− 1.24506i
$81$	0	0
$82$	0	0
$83$	− 1346.04i	− 1.78009i	−0.455873	−	0.890045i	$-0.650673\pi$
$83$	− 1346.04i	− 1.78009i	0.455873	−	0.890045i	$-0.349327\pi$
$84$	0	0
$85$	−976.246	−1.24575
$86$	0	0
$87$	0	0
$88$	0	0
$89$	0	0		−	1.00000i	$-0.5\pi$
$89$	0	0			1.00000i	$0.5\pi$
$90$	0	0
$91$	0	0
$92$	418.734i	0.474522i
$93$	0	0
$94$	−1845.20	−2.02466
$95$	− 1141.79i	− 1.23310i
$96$	0	0
$97$	0	0	1.00000			$0$
$97$	0	0	−1.00000			$\pi$
$98$	1160.01i	1.19571i
$99$	0	0
$100$	429.712	0.429712
$101$	0	0		−	1.00000i	$-0.5\pi$
$101$	0	0			1.00000i	$0.5\pi$
$102$	0	0
$103$	0	0	1.00000			$0$
$103$	0	0	−1.00000			$\pi$
$104$	0	0
$105$	0	0
$106$	−2390.42	−2.19036
$107$	− 17.8885i	− 0.0161622i	−0.999967	−	0.00808108i	$-0.997428\pi$
$107$	− 17.8885i	− 0.0161622i	0.999967	−	0.00808108i	$-0.00257232\pi$
$108$	0	0
$109$	−250.227	−0.219885	−0.109942	−	0.993938i	$-0.535067\pi$
$109$	−250.227	−0.219885	−0.109942	+	0.993938i	$0.535067\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 1426.61i	− 1.18765i	−0.804595	−	0.593824i	$-0.797617\pi$
$113$	− 1426.61i	− 1.18765i	0.804595	−	0.593824i	$-0.202383\pi$
$114$	0	0
$115$	1361.84	1.10428
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−1331.00	−1.00000
$122$	3191.68i	2.36853i
$123$	0	0
$124$	−1159.78	−0.839928
$125$	− 1397.54i	− 1.00000i
$126$	0	0
$127$	0	0	1.00000			$0$
$127$	0	0	−1.00000			$\pi$
$128$	− 1653.82i	− 1.14202i
$129$	0	0
$130$	0	0
$131$	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
$136$	−1347.28	−0.849473
$137$	3173.05i	1.97877i	0.145306	+	0.989387i	$0.453583\pi$
$137$	3173.05i	1.97877i	−0.145306	+	0.989387i	$0.546417\pi$
$138$	0	0
$139$	1604.00	0.978773	0.489387	−	0.872067i	$-0.337221\pi$
$139$	1604.00	0.978773	0.489387	+	0.872067i	$0.337221\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	0	0
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0	0
$148$	0	0
$149$	0	0		−	1.00000i	$-0.5\pi$
$149$	0	0			1.00000i	$0.5\pi$
$150$	0	0
$151$	−3112.00	−1.67716	−0.838579	−	0.544779i	$-0.816613\pi$
$151$	−3112.00	−1.67716	−0.838579	+	0.544779i	$0.816613\pi$
$152$	− 1575.74i	− 0.840850i
$153$	0	0
$154$	0	0
$155$	3771.92i	1.95463i
$156$	0	0
$157$	0	0	1.00000			$0$
$157$	0	0	−1.00000			$\pi$
$158$	4529.64i	2.28075i
$159$	0	0
$160$	1632.90	0.806828
$161$	0	0
$162$	0	0
$163$	0	0	1.00000			$0$
$163$	0	0	−1.00000			$\pi$
$164$	0	0
$165$	0	0
$166$	4552.27	2.12846
$167$	− 55.7692i	− 0.0258416i	−0.999917	−	0.0129208i	$-0.995887\pi$
$167$	− 55.7692i	− 0.0258416i	0.999917	−	0.0129208i	$-0.00411294\pi$
$168$	0	0
$169$	2197.00	1.00000
$170$	− 3301.63i	− 1.48955i
$171$	0	0
$172$	0	0
$173$	− 4136.43i	− 1.81784i	−0.416968	−	0.908921i	$-0.636907\pi$
$173$	− 4136.43i	− 1.81784i	0.416968	−	0.908921i	$-0.363093\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	0	0		−	1.00000i	$-0.5\pi$
$179$	0	0			1.00000i	$0.5\pi$
$180$	0	0
$181$	−4714.17	−1.93592	−0.967960	−	0.251103i	$-0.919207\pi$
$181$	−4714.17	−1.93592	−0.967960	+	0.251103i	$0.919207\pi$
$182$	0	0
$183$	0	0
$184$	1879.42	0.753004
$185$	0	0
$186$	0	0
$187$	0	0
$188$	− 1875.61i	− 0.727621i
$189$	0	0
$190$	3861.49	1.47443
$191$	0	0		−	1.00000i	$-0.5\pi$
$191$	0	0			1.00000i	$0.5\pi$
$192$	0	0
$193$	0	0	1.00000			$0$
$193$	0	0	−1.00000			$\pi$
$194$	0	0
$195$	0	0
$196$	−1179.13	−0.429712
$197$	− 2971.21i	− 1.07457i	−0.843402	−	0.537283i	$-0.819451\pi$
$197$	− 2971.21i	− 1.07457i	0.843402	−	0.537283i	$-0.180549\pi$
$198$	0	0
$199$	5456.00	1.94355	0.971773	−	0.235919i	$-0.0758099\pi$
$199$	5456.00	1.94355	0.971773	+	0.235919i	$0.0758099\pi$
$200$	− 1928.70i	− 0.681897i
$201$	0	0
$202$	0	0
$203$	0	0
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	0	0
$209$	0	0
$210$	0	0
$211$	5957.80	1.94385	0.971924	−	0.235295i	$-0.0756056\pi$
$211$	5957.80	1.94385	0.971924	+	0.235295i	$0.0756056\pi$
$212$	− 2429.81i	− 0.787169i
$213$	0	0
$214$	60.4984	0.0193252
$215$	0	0
$216$	0	0
$217$	0	0
$218$	− 846.261i	− 0.262918i
$219$	0	0
$220$	0	0
$221$	0	0
$222$	0	0
$223$	0	0	1.00000			$0$
$223$	0	0	−1.00000			$\pi$
$224$	0	0
$225$	0	0
$226$	4824.75	1.42008
$227$	1275.88i	0.373054i	0.982450	+	0.186527i	$0.0597232\pi$
$227$	1275.88i	0.373054i	−0.982450	+	0.186527i	$0.940277\pi$
$228$	0	0
$229$	−5854.13	−1.68931	−0.844655	−	0.535311i	$-0.820195\pi$
$229$	−5854.13	−1.68931	−0.844655	+	0.535311i	$0.820195\pi$
$230$	4605.69i	1.32039i
$231$	0	0
$232$	0	0
$233$	4449.78i	1.25114i	0.780170	+	0.625568i	$0.215133\pi$
$233$	4449.78i	1.25114i	−0.780170	+	0.625568i	$0.784867\pi$
$234$	0	0
$235$	−6100.00	−1.69328
$236$	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
$241$	−5640.38	−1.50759	−0.753794	−	0.657111i	$-0.771778\pi$
$241$	−5640.38	−1.50759	−0.753794	+	0.657111i	$0.771778\pi$
$242$	− 4501.40i	− 1.19571i
$243$	0	0
$244$	−3244.27	−0.851202
$245$	3834.86i	1.00000i
$246$	0	0
$247$	0	0
$248$	5205.48i	1.33286i
$249$	0	0
$250$	4726.44	1.19571
$251$	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
$256$	4444.94	1.08519
$257$	1066.63i	0.258890i	0.991587	+	0.129445i	$0.0413196\pi$
$257$	1066.63i	0.258890i	−0.991587	+	0.129445i	$0.958680\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	0	0
$262$	0	0
$263$	6341.49i	1.48682i	0.668837	+	0.743409i	$0.266792\pi$
$263$	6341.49i	1.48682i	−0.668837	+	0.743409i	$0.733208\pi$
$264$	0	0
$265$	−7902.41	−1.83185
$266$	0	0
$267$	0	0
$268$	0	0
$269$	0	0		−	1.00000i	$-0.5\pi$
$269$	0	0			1.00000i	$0.5\pi$
$270$	0	0
$271$	1675.28	0.375520	0.187760	−	0.982215i	$-0.439877\pi$
$271$	1675.28	0.375520	0.187760	+	0.982215i	$0.439877\pi$
$272$	− 6957.84i	− 1.55103i
$273$	0	0
$274$	−10731.1	−2.36603
$275$	0	0
$276$	0	0
$277$	0	0	1.00000			$0$
$277$	0	0	−1.00000			$\pi$
$278$	5424.67i	1.17032i
$279$	0	0
$280$	0	0
$281$	0	0		−	1.00000i	$-0.5\pi$
$281$	0	0			1.00000i	$0.5\pi$
$282$	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
$289$	−2711.45	−0.551893
$290$	0	0
$291$	0	0
$292$	0	0
$293$	− 9824.90i	− 1.95897i	−0.201529	−	0.979483i	$-0.564591\pi$
$293$	− 9824.90i	− 1.95897i	0.201529	−	0.979483i	$-0.435409\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	0	0
$300$	0	0
$301$	0	0
$302$	− 10524.7i	− 2.00539i
$303$	0	0
$304$	8137.68	1.53529
$305$	10551.3i	1.98087i
$306$	0	0
$307$	0	0	1.00000			$0$
$307$	0	0	−1.00000			$\pi$
$308$	0	0
$309$	0	0
$310$	−12756.5	−2.33716
$311$	0	0		−	1.00000i	$-0.5\pi$
$311$	0	0			1.00000i	$0.5\pi$
$312$	0	0
$313$	0	0	1.00000			$0$
$313$	0	0	−1.00000			$\pi$
$314$	0	0
$315$	0	0
$316$	−4604.28	−0.819656
$317$	− 650.333i	− 0.115225i	−0.998339	−	0.0576125i	$-0.981651\pi$
$317$	− 650.333i	− 0.115225i	0.998339	−	0.0576125i	$-0.0183488\pi$
$318$	0	0
$319$	0	0
$320$	− 1604.71i	− 0.280331i
$321$	0	0
$322$	0	0
$323$	− 8917.33i	− 1.53614i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0	0
$328$	0	0
$329$	0	0
$330$	0	0
$331$	5852.00	0.971767	0.485884	−	0.874023i	$-0.338498\pi$
$331$	5852.00	0.971767	0.485884	+	0.874023i	$0.338498\pi$
$332$	4627.29i	0.764925i
$333$	0	0
$334$	188.610	0.0308990
$335$	0	0
$336$	0	0
$337$	0	0	1.00000			$0$
$337$	0	0	−1.00000			$\pi$
$338$	7430.18i	1.19571i
$339$	0	0
$340$	3356.04	0.535313
$341$	0	0
$342$	0	0
$343$	0	0
$344$	0	0
$345$	0	0
$346$	13989.3	2.17360
$347$	− 8550.72i	− 1.32284i	−0.750014	−	0.661422i	$-0.769953\pi$
$347$	− 8550.72i	− 1.32284i	0.750014	−	0.661422i	$-0.230047\pi$
$348$	0	0
$349$	−8974.04	−1.37642	−0.688208	−	0.725513i	$-0.741602\pi$
$349$	−8974.04	−1.37642	−0.688208	+	0.725513i	$0.741602\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 2473.09i	− 0.372888i	−0.982466	−	0.186444i	$-0.940304\pi$
$353$	− 2473.09i	− 0.372888i	0.982466	−	0.186444i	$-0.0596962\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	0	0		−	1.00000i	$-0.5\pi$
$359$	0	0			1.00000i	$0.5\pi$
$360$	0	0
$361$	3570.44	0.520548
$362$	− 15943.2i	− 2.31479i
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	0	0	1.00000			$0$
$367$	0	0	−1.00000			$\pi$
$368$	9706.01i	1.37489i
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	0	0	1.00000			$0$
$373$	0	0	−1.00000			$\pi$
$374$	0	0
$375$	0	0
$376$	−8418.38	−1.15464
$377$	0	0
$378$	0	0
$379$	9933.39	1.34629	0.673145	−	0.739510i	$-0.264943\pi$
$379$	9933.39	1.34629	0.673145	+	0.739510i	$0.264943\pi$
$380$	3925.12i	0.529880i
$381$	0	0
$382$	0	0
$383$	− 14779.7i	− 1.97182i	−0.167269	−	0.985911i	$-0.553495\pi$
$383$	− 14779.7i	− 1.97182i	0.167269	−	0.985911i	$-0.446505\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	0	0		−	1.00000i	$-0.5\pi$
$389$	0	0			1.00000i	$0.5\pi$
$390$	0	0
$391$	10635.9	1.37566
$392$	5292.34i	0.681897i
$393$	0	0
$394$	10048.5	1.28487
$395$	14974.4i	1.90745i
$396$	0	0
$397$	0	0	1.00000			$0$
$397$	0	0	−1.00000			$\pi$
$398$	18452.0i	2.32391i
$399$	0	0
$400$	9960.48	1.24506
$401$	0	0		−	1.00000i	$-0.5\pi$
$401$	0	0			1.00000i	$0.5\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0	0
$408$	0	0
$409$	14327.4	1.73213	0.866067	−	0.499929i	$-0.166640\pi$
$409$	14327.4	1.73213	0.866067	+	0.499929i	$0.166640\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	0	0
$414$	0	0
$415$	15049.2	1.78009
$416$	0	0
$417$	0	0
$418$	0	0
$419$	0	0		−	1.00000i	$-0.5\pi$
$419$	0	0			1.00000i	$0.5\pi$
$420$	0	0
$421$	−17164.0	−1.98699	−0.993493	−	0.113890i	$-0.963669\pi$
$421$	−17164.0	−1.98699	−0.993493	+	0.113890i	$0.963669\pi$
$422$	20149.1i	2.32427i
$423$	0	0
$424$	−10905.8	−1.24914
$425$	− 10914.8i	− 1.24575i
$426$	0	0
$427$	0	0
$428$	61.4953i	0.00694507i
$429$	0	0
$430$	0	0
$431$	0	0		−	1.00000i	$-0.5\pi$
$431$	0	0			1.00000i	$0.5\pi$
$432$	0	0
$433$	0	0	1.00000			$0$
$433$	0	0	−1.00000			$\pi$
$434$	0	0
$435$	0	0
$436$	860.206	0.0944871
$437$	12439.4i	1.36169i
$438$	0	0
$439$	−2349.99	−0.255488	−0.127744	−	0.991807i	$-0.540774\pi$
$439$	−2349.99	−0.255488	−0.127744	+	0.991807i	$0.540774\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	4965.77i	0.532575i	0.963894	+	0.266288i	$0.0857971\pi$
$443$	4965.77i	0.532575i	−0.963894	+	0.266288i	$0.914203\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	0	0
$449$	0	0		−	1.00000i	$-0.5\pi$
$449$	0	0			1.00000i	$0.5\pi$
$450$	0	0
$451$	0	0
$452$	4904.25i	0.510347i
$453$	0	0
$454$	−4314.99	−0.446063
$455$	0	0
$456$	0	0
$457$	0	0	1.00000			$0$
$457$	0	0	−1.00000			$\pi$
$458$	− 19798.5i	− 2.01992i
$459$	0	0
$460$	−4681.58	−0.474522
$461$	0	0		−	1.00000i	$-0.5\pi$
$461$	0	0			1.00000i	$0.5\pi$
$462$	0	0
$463$	0	0	1.00000			$0$
$463$	0	0	−1.00000			$\pi$
$464$	0	0
$465$	0	0
$466$	−15049.0	−1.49599
$467$	− 19237.5i	− 1.90622i	−0.302620	−	0.953111i	$-0.597861\pi$
$467$	− 19237.5i	− 1.90622i	0.302620	−	0.953111i	$-0.402139\pi$
$468$	0	0
$469$	0	0
$470$	− 20630.0i	− 2.02466i
$471$	0	0
$472$	0	0
$473$	0	0
$474$	0	0
$475$	12765.6	1.23310
$476$	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
$481$	0	0
$482$	− 19075.6i	− 1.80263i
$483$	0	0
$484$	4575.57	0.429712
$485$	0	0
$486$	0	0
$487$	0	0	1.00000			$0$
$487$	0	0	−1.00000			$\pi$
$488$	14561.4i	1.35075i
$489$	0	0
$490$	−12969.4	−1.19571
$491$	0	0		−	1.00000i	$-0.5\pi$
$491$	0	0			1.00000i	$0.5\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	0	0
$496$	−26883.0	−2.43363
$497$	0	0
$498$	0	0
$499$	−18.3109	−0.00164271	−0.000821353	−	1.00000i	$-0.500261\pi$
$499$	−18.3109	−0.00164271	−0.000821353		1.00000i	$0.500261\pi$
$500$	4804.32i	0.429712i
$501$	0	0
$502$	0	0
$503$	− 15470.1i	− 1.37133i	−0.727919	−	0.685663i	$-0.759512\pi$
$503$	− 15470.1i	− 1.37133i	0.727919	−	0.685663i	$-0.240488\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	0	0
$508$	0	0
$509$	0	0		−	1.00000i	$-0.5\pi$
$509$	0	0			1.00000i	$0.5\pi$
$510$	0	0
$511$	0	0
$512$	1802.04i	0.155547i
$513$	0	0
$514$	−3607.32	−0.309556
$515$	0	0
$516$	0	0
$517$	0	0
$518$	0	0
$519$	0	0
$520$	0	0
$521$	0	0		−	1.00000i	$-0.5\pi$
$521$	0	0			1.00000i	$0.5\pi$
$522$	0	0
$523$	0	0	1.00000			$0$
$523$	0	0	−1.00000			$\pi$
$524$	0	0
$525$	0	0
$526$	−21446.7	−1.77780
$527$	29458.6i	2.43498i
$528$	0	0
$529$	−2669.82	−0.219432
$530$	− 26725.7i	− 2.19036i
$531$	0	0
$532$	0	0
$533$	0	0
$534$	0	0
$535$	200.000	0.0161622
$536$	0	0
$537$	0	0
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−3238.00	−0.257324	−0.128662	−	0.991688i	$-0.541068\pi$
$541$	−3238.00	−0.257324	−0.128662	+	0.991688i	$0.541068\pi$
$542$	5665.73i	0.449011i
$543$	0	0
$544$	12752.9	1.00511
$545$	− 2797.63i	− 0.219885i
$546$	0	0
$547$	0	0	1.00000			$0$
$547$	0	0	−1.00000			$\pi$
$548$	− 10908.0i	− 0.850302i
$549$	0	0
$550$	0	0
$551$	0	0
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	−5514.06	−0.420590
$557$	26184.4i	1.99186i	0.0901226	+	0.995931i	$0.471274\pi$
$557$	26184.4i	1.99186i	−0.0901226	+	0.995931i	$0.528726\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0	0
$562$	0	0
$563$	14632.8i	1.09538i	0.836681	+	0.547691i	$0.184493\pi$
$563$	14632.8i	1.09538i	−0.836681	+	0.547691i	$0.815507\pi$
$564$	0	0
$565$	15950.0	1.18765
$566$	0	0
$567$	0	0
$568$	0	0
$569$	0	0		−	1.00000i	$-0.5\pi$
$569$	0	0			1.00000i	$0.5\pi$
$570$	0	0
$571$	−20150.2	−1.47681	−0.738407	−	0.674355i	$-0.764422\pi$
$571$	−20150.2	−1.47681	−0.738407	+	0.674355i	$0.764422\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	15225.8i	1.10428i
$576$	0	0
$577$	0	0	1.00000			$0$
$577$	0	0	−1.00000			$\pi$
$578$	− 9170.04i	− 0.659902i
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	33227.5	2.34235
$587$	23301.7i	1.63844i	0.573480	+	0.819220i	$0.305593\pi$
$587$	23301.7i	1.63844i	−0.573480	+	0.819220i	$0.694407\pi$
$588$	0	0
$589$	−34453.9	−2.41027
$590$	0	0
$591$	0	0
$592$	0	0
$593$	− 20132.8i	− 1.39419i	−0.716980	−	0.697094i	$-0.754476\pi$
$593$	− 20132.8i	− 1.39419i	0.716980	−	0.697094i	$-0.245524\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	0	0		−	1.00000i	$-0.5\pi$
$599$	0	0			1.00000i	$0.5\pi$
$600$	0	0
$601$	29441.1	1.99821	0.999107	−	0.0422630i	$-0.0134567\pi$
$601$	29441.1	1.99821	0.999107	+	0.0422630i	$0.0134567\pi$
$602$	0	0
$603$	0	0
$604$	10698.1	0.720695
$605$	− 14881.0i	− 1.00000i
$606$	0	0
$607$	0	0	1.00000			$0$
$607$	0	0	−1.00000			$\pi$
$608$	14915.4i	0.994903i
$609$	0	0
$610$	−35684.1	−2.36853
$611$	0	0
$612$	0	0
$613$	0	0	1.00000			$0$
$613$	0	0	−1.00000			$\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	24813.5i	1.61905i	0.587083	+	0.809527i	$0.300276\pi$
$617$	24813.5i	1.61905i	−0.587083	+	0.809527i	$0.699724\pi$
$618$	0	0
$619$	3476.00	0.225706	0.112853	−	0.993612i	$-0.464001\pi$
$619$	3476.00	0.225706	0.112853	+	0.993612i	$0.464001\pi$
$620$	− 12966.7i	− 0.839928i
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	15625.0	1.00000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	0	0
$630$	0	0
$631$	722.379	0.0455744	0.0227872	−	0.999740i	$-0.492746\pi$
$631$	722.379	0.0455744	0.0227872	+	0.999740i	$0.492746\pi$
$632$	20665.6i	1.30069i
$633$	0	0
$634$	2199.41	0.137775
$635$	0	0
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	18490.3	1.14202
$641$	0	0		−	1.00000i	$-0.5\pi$
$641$	0	0			1.00000i	$0.5\pi$
$642$	0	0
$643$	0	0	1.00000			$0$
$643$	0	0	−1.00000			$\pi$
$644$	0	0
$645$	0	0
$646$	30158.1	1.83677
$647$	− 29732.3i	− 1.80664i	−0.428967	−	0.903320i	$-0.641122\pi$
$647$	− 29732.3i	− 1.80664i	0.428967	−	0.903320i	$-0.358878\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 9864.90i	− 0.591184i	−0.955314	−	0.295592i	$-0.904483\pi$
$653$	− 9864.90i	− 0.591184i	0.955314	−	0.295592i	$-0.0955169\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0	0
$658$	0	0
$659$	0	0		−	1.00000i	$-0.5\pi$
$659$	0	0			1.00000i	$0.5\pi$
$660$	0	0
$661$	32978.0	1.94054	0.970269	−	0.242029i	$-0.0778130\pi$
$661$	32978.0	1.94054	0.970269	+	0.242029i	$0.0778130\pi$
$662$	19791.3i	1.16195i
$663$	0	0
$664$	20768.9	1.21384
$665$	0	0
$666$	0	0
$667$	0	0
$668$	191.718i	0.0111045i
$669$	0	0
$670$	0	0
$671$	0	0
$672$	0	0
$673$	0	0	1.00000			$0$
$673$	0	0	−1.00000			$\pi$
$674$	0	0
$675$	0	0
$676$	−7552.61	−0.429712
$677$	− 32901.5i	− 1.86781i	−0.357521	−	0.933905i	$-0.616378\pi$
$677$	− 32901.5i	− 1.86781i	0.357521	−	0.933905i	$-0.383622\pi$
$678$	0	0
$679$	0	0
$680$	− 15063.1i	− 0.849473i
$681$	0	0
$682$	0	0
$683$	5944.71i	0.333042i	0.986038	+	0.166521i	$0.0532534\pi$
$683$	5944.71i	0.333042i	−0.986038	+	0.166521i	$0.946747\pi$
$684$	0	0
$685$	−35475.8	−1.97877
$686$	0	0
$687$	0	0
$688$	0	0
$689$	0	0
$690$	0	0
$691$	29909.6	1.64662	0.823311	−	0.567591i	$-0.192124\pi$
$691$	29909.6	1.64662	0.823311	+	0.567591i	$0.192124\pi$
$692$	14219.8i	0.781148i
$693$	0	0
$694$	28918.3	1.58173
$695$	17933.3i	0.978773i
$696$	0	0
$697$	0	0
$698$	− 30349.9i	− 1.64579i
$699$	0	0
$700$	0	0
$701$	0	0		−	1.00000i	$-0.5\pi$
$701$	0	0			1.00000i	$0.5\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	0	0
$706$	8363.91	0.445864
$707$	0	0
$708$	0	0
$709$	−37726.0	−1.99835	−0.999175	−	0.0406201i	$-0.987067\pi$
$709$	−37726.0	−1.99835	−0.999175	+	0.0406201i	$0.987067\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	− 41093.9i	− 2.15846i
$714$	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
$722$	12075.1i	0.622422i
$723$	0	0
$724$	16205.9	0.831888
$725$	0	0
$726$	0	0
$727$	0	0	1.00000			$0$
$727$	0	0	−1.00000			$\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	0	0
$732$	0	0
$733$	0	0	1.00000			$0$
$733$	0	0	−1.00000			$\pi$
$734$	0	0
$735$	0	0
$736$	−17790.0	−0.890963
$737$	0	0
$738$	0	0
$739$	−38352.2	−1.90908	−0.954539	−	0.298085i	$-0.903652\pi$
$739$	−38352.2	−1.90908	−0.954539	+	0.298085i	$0.903652\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	− 10241.2i	− 0.505670i	−0.967509	−	0.252835i	$-0.918637\pi$
$743$	− 10241.2i	− 0.505670i	0.967509	−	0.252835i	$-0.0813630\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−23503.8	−1.14203	−0.571015	−	0.820940i	$-0.693450\pi$
$751$	−23503.8	−1.14203	−0.571015	+	0.820940i	$0.693450\pi$
$752$	− 43475.5i	− 2.10823i
$753$	0	0
$754$	0	0
$755$	− 34793.2i	− 1.67716i
$756$	0	0
$757$	0	0	1.00000			$0$
$757$	0	0	−1.00000			$\pi$
$758$	33594.4i	1.60977i
$759$	0	0
$760$	17617.3	0.840850
$761$	0	0		−	1.00000i	$-0.5\pi$
$761$	0	0			1.00000i	$0.5\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	49984.5	2.35772
$767$	0	0
$768$	0	0
$769$	31750.8	1.48890	0.744449	−	0.667679i	$-0.232712\pi$
$769$	31750.8	1.48890	0.744449	+	0.667679i	$0.232712\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	41284.7i	1.92097i	0.278335	+	0.960484i	$0.410217\pi$
$773$	41284.7i	1.92097i	−0.278335	+	0.960484i	$0.589783\pi$
$774$	0	0
$775$	−42171.3	−1.95463
$776$	0	0
$777$	0	0
$778$	0	0
$779$	0	0
$780$	0	0
$781$	0	0
$782$	35970.3i	1.64488i
$783$	0	0
$784$	−27331.5	−1.24506
$785$	0	0
$786$	0	0
$787$	0	0	1.00000			$0$
$787$	0	0	−1.00000			$\pi$
$788$	10214.1i	0.461754i
$789$	0	0
$790$	−50643.0	−2.28075
$791$	0	0
$792$	0	0
$793$	0	0
$794$	0	0
$795$	0	0
$796$	−18756.1	−0.835164
$797$	39665.3i	1.76288i	0.472294	+	0.881441i	$0.343426\pi$
$797$	39665.3i	1.76288i	−0.472294	+	0.881441i	$0.656574\pi$
$798$	0	0
$799$	−47640.8	−2.10940
$800$	18256.4i	0.806828i
$801$	0	0
$802$	0	0
$803$	0	0
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	0	0		−	1.00000i	$-0.5\pi$
$809$	0	0			1.00000i	$0.5\pi$
$810$	0	0
$811$	−14092.0	−0.610157	−0.305078	−	0.952327i	$-0.598683\pi$
$811$	−14092.0	−0.610157	−0.305078	+	0.952327i	$0.598683\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	0	0
$818$	48454.7i	2.07112i
$819$	0	0
$820$	0	0
$821$	0	0		−	1.00000i	$-0.5\pi$
$821$	0	0			1.00000i	$0.5\pi$
$822$	0	0
$823$	0	0	1.00000			$0$
$823$	0	0	−1.00000			$\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 39151.1i	− 1.64621i	−0.567887	−	0.823107i	$-0.692239\pi$
$827$	− 39151.1i	− 1.64621i	0.567887	−	0.823107i	$-0.307761\pi$
$828$	0	0
$829$	45254.0	1.89594	0.947971	−	0.318356i	$-0.103131\pi$
$829$	45254.0	1.89594	0.947971	+	0.318356i	$0.103131\pi$
$830$	50896.0i	2.12846i
$831$	0	0
$832$	0	0
$833$	29950.1i	1.24575i
$834$	0	0
$835$	623.519	0.0258416
$836$	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
$841$	−24389.0	−1.00000
$842$	− 58048.0i	− 2.37585i
$843$	0	0
$844$	−20481.1	−0.835294
$845$	24563.2i	1.00000i
$846$	0	0
$847$	0	0
$848$	− 56321.6i	− 2.28077i
$849$	0	0
$850$	36913.4	1.48955
$851$	0	0
$852$	0	0
$853$	0	0	1.00000			$0$
$853$	0	0	−1.00000			$\pi$
$854$	0	0
$855$	0	0
$856$	276.012	0.0110209
$857$	− 8378.73i	− 0.333969i	−0.985959	−	0.166985i	$-0.946597\pi$
$857$	− 8378.73i	− 0.333969i	0.985959	−	0.166985i	$-0.0534031\pi$
$858$	0	0
$859$	−22021.7	−0.874703	−0.437352	−	0.899291i	$-0.644083\pi$
$859$	−22021.7	−0.874703	−0.437352	+	0.899291i	$0.644083\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	50691.4i	1.99949i	0.0226629	+	0.999743i	$0.492786\pi$
$863$	50691.4i	1.99949i	−0.0226629	+	0.999743i	$0.507214\pi$
$864$	0	0
$865$	46246.7	1.81784
$866$	0	0
$867$	0	0
$868$	0	0
$869$	0	0
$870$	0	0
$871$	0	0
$872$	− 3860.90i	− 0.149939i
$873$	0	0
$874$	−42069.8	−1.62818
$875$	0	0
$876$	0	0
$877$	0	0	1.00000			$0$
$877$	0	0	−1.00000			$\pi$
$878$	− 7947.60i	− 0.305488i
$879$	0	0
$880$	0	0
$881$	0	0		−	1.00000i	$-0.5\pi$
$881$	0	0			1.00000i	$0.5\pi$
$882$	0	0
$883$	0	0	1.00000			$0$
$883$	0	0	−1.00000			$\pi$
$884$	0	0
$885$	0	0
$886$	−16794.1	−0.636803
$887$	− 52699.6i	− 1.99490i	−0.0713489	−	0.997451i	$-0.522730\pi$
$887$	− 52699.6i	− 1.99490i	0.0713489	−	0.997451i	$-0.477270\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 55719.2i	− 2.08799i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	0	0
$900$	0	0
$901$	−61717.6	−2.28203
$902$	0	0
$903$	0	0
$904$	22012.0	0.809854
$905$	− 52706.0i	− 1.93592i
$906$	0	0
$907$	0	0	1.00000			$0$
$907$	0	0	−1.00000			$\pi$
$908$	− 4386.09i	− 0.160306i
$909$	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
$914$	0	0
$915$	0	0
$916$	20124.7	0.725917
$917$	0	0
$918$	0	0
$919$	21224.0	0.761823	0.380911	−	0.924612i	$-0.375610\pi$
$919$	21224.0	0.761823	0.380911	+	0.924612i	$0.375610\pi$
$920$	21012.6i	0.753004i
$921$	0	0
$922$	0	0
$923$	0	0
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0	0
$928$	0	0
$929$	0	0		−	1.00000i	$-0.5\pi$
$929$	0	0			1.00000i	$0.5\pi$
$930$	0	0
$931$	−35028.7	−1.23310
$932$	− 15297.0i	− 0.537627i
$933$	0	0
$934$	65060.6	2.27928
$935$	0	0
$936$	0	0
$937$	0	0	1.00000			$0$
$937$	0	0	−1.00000			$\pi$
$938$	0	0
$939$	0	0
$940$	20969.9	0.727621
$941$	0	0		−	1.00000i	$-0.5\pi$
$941$	0	0			1.00000i	$0.5\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 9428.49i	− 0.323532i	−0.986829	−	0.161766i	$-0.948281\pi$
$947$	− 9428.49i	− 0.323532i	0.986829	−	0.161766i	$-0.0517190\pi$
$948$	0	0
$949$	0	0
$950$	43172.7i	1.47443i
$951$	0	0
$952$	0	0
$953$	30289.8i	1.02957i	0.857319	+	0.514786i	$0.172129\pi$
$953$	30289.8i	1.02957i	−0.857319	+	0.514786i	$0.827871\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	84028.0	2.82058
$962$	0	0
$963$	0	0
$964$	19389.9	0.647828
$965$	0	0
$966$	0	0
$967$	0	0	1.00000			$0$
$967$	0	0	−1.00000			$\pi$
$968$	− 20536.7i	− 0.681897i
$969$	0	0
$970$	0	0
$971$	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
$976$	−75200.4	−2.46630
$977$	− 13018.4i	− 0.426300i	−0.977019	−	0.213150i	$-0.931628\pi$
$977$	− 13018.4i	− 0.426300i	0.977019	−	0.213150i	$-0.0683723\pi$
$978$	0	0
$979$	0	0
$980$	− 13183.1i	− 0.429712i
$981$	0	0
$982$	0	0
$983$	21586.1i	0.700397i	0.936676	+	0.350198i	$0.113886\pi$
$983$	21586.1i	0.700397i	−0.936676	+	0.350198i	$0.886114\pi$
$984$	0	0
$985$	33219.1	1.07457
$986$	0	0
$987$	0	0
$988$	0	0
$989$	0	0
$990$	0	0
$991$	32733.6	1.04926	0.524630	−	0.851330i	$-0.324204\pi$
$991$	32733.6	1.04926	0.524630	+	0.851330i	$0.324204\pi$
$992$	− 49273.5i	− 1.57705i
$993$	0	0
$994$	0	0
$995$	60999.9i	1.94355i
$996$	0	0
$997$	0	0	1.00000			$0$
$997$	0	0	−1.00000			$\pi$
$998$	− 61.9270i	− 0.00196419i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	135.4.b.a.109.3	yes	4
3.2	odd	2	inner	135.4.b.a.109.2	✓	4
5.2	odd	4		675.4.a.k.1.2		2
5.3	odd	4		675.4.a.o.1.1		2
5.4	even	2	inner	135.4.b.a.109.2	✓	4
15.2	even	4		675.4.a.o.1.1		2
15.8	even	4		675.4.a.k.1.2		2
15.14	odd	2	CM	135.4.b.a.109.3	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
135.4.b.a.109.2	✓	4	3.2	odd	2	inner
135.4.b.a.109.2	✓	4	5.4	even	2	inner
135.4.b.a.109.3	yes	4	1.1	even	1	trivial
135.4.b.a.109.3	yes	4	15.14	odd	2	CM
675.4.a.k.1.2		2	5.2	odd	4
675.4.a.k.1.2		2	15.8	even	4
675.4.a.o.1.1		2	5.3	odd	4
675.4.a.o.1.1		2	15.2	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:	\( N \)	\(=\)	\( 135 = 3^{3} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	135.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+3.38197i q^{2} -3.43769 q^{4} +11.1803i q^{5} +15.4296i q^{8} -37.8115 q^{10} -79.6838 q^{16} +87.3181i q^{17} -102.125 q^{19} -38.4346i q^{20} -121.807i q^{23} -125.000 q^{25} +337.371 q^{31} -146.051i q^{32} -295.307 q^{34} -345.382i q^{38} -172.508 q^{40} +411.945 q^{46} +545.601i q^{47} +343.000 q^{49} -422.746i q^{50} +706.813i q^{53} +943.735 q^{61} +1140.98i q^{62} -143.530 q^{64} -300.173i q^{68} +351.073 q^{76} +1339.35 q^{79} -890.892i q^{80} -1346.04i q^{83} -976.246 q^{85} +418.734i q^{92} -1845.20 q^{94} -1141.79i q^{95} +1160.01i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 54 q^{4} + 50 q^{10} + 446 q^{16} - 328 q^{19} - 500 q^{25} + 464 q^{31} - 14 q^{34} - 2300 q^{40} + 3298 q^{46} + 1372 q^{49} + 716 q^{61} - 6692 q^{64} + 3618 q^{76} + 608 q^{79} - 3100 q^{85} + 2440 q^{94}+O(q^{100}) \) 4 * q - 54 * q^4 + 50 * q^10 + 446 * q^16 - 328 * q^19 - 500 * q^25 + 464 * q^31 - 14 * q^34 - 2300 * q^40 + 3298 * q^46 + 1372 * q^49 + 716 * q^61 - 6692 * q^64 + 3618 * q^76 + 608 * q^79 - 3100 * q^85 + 2440 * q^94

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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