LMFDB - Embedded newform 121.4.c.a.81.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [121,4,Mod(3,121)] 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("121.3"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(121, base_ring=CyclotomicField(10)) chi = DirichletCharacter(H, H._module([8])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 121 = 11^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	121.c (of order $5$, degree $4$, not minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,-8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

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

Self dual:	no
Analytic conductor:	$7.13923111069$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{10})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} + x^{2} - x + 1 $ x^4 - x^3 + x^2 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{9}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{5}]$

Embedding invariants

Embedding label			81.1
Root			$0.809017 - 0.587785i$ of defining polynomial
Character	$\chi$	$=$	121.81
Dual form			121.4.c.a.3.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(2.47214 - 7.60845i) q^{3} +(-2.47214 - 7.60845i) q^{4} +(-14.5623 + 10.5801i) q^{5} +(-29.9336 - 21.7481i) q^{9} -64.0000 q^{12} +(44.4984 + 136.952i) q^{15} +(-51.7771 + 37.6183i) q^{16} +(116.498 + 84.6411i) q^{20} -108.000 q^{23} +(61.4944 - 189.260i) q^{25} +(-64.7214 + 47.0228i) q^{27} +(-275.066 - 199.847i) q^{31} +(-91.4690 + 281.513i) q^{36} +(-134.113 - 412.759i) q^{37} +666.000 q^{45} +(-11.1246 + 34.2380i) q^{47} +(158.217 + 486.941i) q^{48} +(277.493 - 201.610i) q^{49} +(597.055 + 433.786i) q^{53} +(-222.492 - 684.761i) q^{59} +(931.988 - 677.129i) q^{60} +(414.217 + 300.946i) q^{64} -416.000 q^{67} +(-266.991 + 821.713i) q^{69} +(-495.118 + 359.725i) q^{71} +(-1287.96 - 935.754i) q^{75} +(355.988 - 1095.62i) q^{80} +(-110.937 - 341.429i) q^{81} +1674.00 q^{89} +(266.991 + 821.713i) q^{92} +(-2200.53 + 1598.78i) q^{93} +(27.5066 + 19.9847i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 8 q^{3} + 8 q^{4} - 18 q^{5} - 37 q^{9} - 256 q^{12} - 144 q^{15} - 64 q^{16} + 144 q^{20} - 432 q^{23} - 199 q^{25} - 80 q^{27} - 340 q^{31} + 296 q^{36} + 434 q^{37} + 2664 q^{45} + 36 q^{47} - 512 q^{48}+ \cdots + 34 q^{97}+O(q^{100}) $ 4 * q - 8 * q^3 + 8 * q^4 - 18 * q^5 - 37 * q^9 - 256 * q^12 - 144 * q^15 - 64 * q^16 + 144 * q^20 - 432 * q^23 - 199 * q^25 - 80 * q^27 - 340 * q^31 + 296 * q^36 + 434 * q^37 + 2664 * q^45 + 36 * q^47 - 512 * q^48 + 343 * q^49 + 738 * q^53 + 720 * q^59 + 1152 * q^60 + 512 * q^64 - 1664 * q^67 + 864 * q^69 - 612 * q^71 - 1592 * q^75 - 1152 * q^80 + 359 * q^81 + 6696 * q^89 - 864 * q^92 - 2720 * q^93 + 34 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/121\mathbb{Z}\right)^\times$.

$n$	$2$
$\chi(n)$	$e\left(\frac{1}{5}\right)$

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 $

$2$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$2$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$3$	2.47214	−	7.60845i	0.475763	−	1.46425i	−0.369163	−	0.929364i	$-0.620356\pi$
$3$	2.47214	−	7.60845i	0.475763	−	1.46425i	0.844926	−	0.534883i	$-0.179644\pi$
$4$	−2.47214	−	7.60845i	−0.309017	−	0.951057i
$5$	−14.5623	+	10.5801i	−1.30249	+	0.946316i	−0.999977	−	0.00682845i	$-0.997826\pi$
$5$	−14.5623	+	10.5801i	−1.30249	+	0.946316i	−0.302516	+	0.953144i	$0.597826\pi$
$6$		0			0
$7$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$7$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$8$		0			0
$9$	−29.9336	−	21.7481i	−1.10865	−	0.805483i
$10$		0			0
$11$		0			0
$11$		0			0
$12$	−64.0000			−1.53960
$13$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$13$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$14$		0			0
$15$	44.4984	+	136.952i	0.765963	+	2.35739i
$16$	−51.7771	+	37.6183i	−0.809017	+	0.587785i
$17$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$17$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$18$		0			0
$19$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$19$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$20$	116.498	+	84.6411i	1.30249	+	0.946316i
$21$		0			0
$22$		0			0
$23$	−108.000			−0.979111			−0.489556	−	0.871972i	$-0.662841\pi$
$23$	−108.000			−0.979111			−0.489556	+	0.871972i	$0.662841\pi$
$24$		0			0
$25$	61.4944	−	189.260i	0.491955	−	1.51408i
$26$		0			0
$27$	−64.7214	+	47.0228i	−0.461320	+	0.335168i
$28$		0			0
$29$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$29$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$30$		0			0
$31$	−275.066	−	199.847i	−1.59365	−	1.15786i	−0.898479	−	0.439017i	$-0.855327\pi$
$31$	−275.066	−	199.847i	−1.59365	−	1.15786i	−0.695175	−	0.718840i	$-0.744673\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$	−91.4690	+	281.513i	−0.423468	+	1.30330i
$37$	−134.113	−	412.759i	−0.595895	−	1.83398i	−0.550221	−	0.835019i	$-0.685457\pi$
$37$	−134.113	−	412.759i	−0.595895	−	1.83398i	−0.0456733	−	0.998956i	$-0.514543\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$41$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$42$		0			0
$43$		0			0			−	1.00000i	$-0.5\pi$
$43$		0			0				1.00000i	$0.5\pi$
$44$		0			0
$45$	666.000			2.20625
$46$		0			0
$47$	−11.1246	+	34.2380i	−0.0345253	+	0.106258i	−0.966834	−	0.255405i	$-0.917791\pi$
$47$	−11.1246	+	34.2380i	−0.0345253	+	0.106258i	0.932309	+	0.361663i	$0.117791\pi$
$48$	158.217	+	486.941i	0.475763	+	1.46425i
$49$	277.493	−	201.610i	0.809017	−	0.587785i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	597.055	+	433.786i	1.54739	+	1.12425i	0.945479	+	0.325684i	$0.105595\pi$
$53$	597.055	+	433.786i	1.54739	+	1.12425i	0.601913	+	0.798562i	$0.294405\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−222.492	−	684.761i	−0.490950	−	1.51099i	−0.823175	−	0.567787i	$-0.807800\pi$
$59$	−222.492	−	684.761i	−0.490950	−	1.51099i	0.332226	−	0.943200i	$-0.392200\pi$
$60$	931.988	−	677.129i	2.00532	−	1.45695i
$61$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$61$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$62$		0			0
$63$		0			0
$64$	414.217	+	300.946i	0.809017	+	0.587785i
$65$		0			0
$66$		0			0
$67$	−416.000			−0.758545			−0.379272	−	0.925285i	$-0.623826\pi$
$67$	−416.000			−0.758545			−0.379272	+	0.925285i	$0.623826\pi$
$68$		0			0
$69$	−266.991	+	821.713i	−0.465825	+	1.43366i
$70$		0			0
$71$	−495.118	+	359.725i	−0.827602	+	0.601288i	−0.918880	−	0.394537i	$-0.870905\pi$
$71$	−495.118	+	359.725i	−0.827602	+	0.601288i	0.0912779	+	0.995825i	$0.470905\pi$
$72$		0			0
$73$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$73$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$74$		0			0
$75$	−1287.96	−	935.754i	−1.98294	−	1.44069i
$76$		0			0
$77$		0			0
$78$		0			0
$79$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$79$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$80$	355.988	−	1095.62i	0.497508	−	1.53117i
$81$	−110.937	−	341.429i	−0.152177	−	0.468353i
$82$		0			0
$83$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$83$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$	1674.00			1.99375			0.996874	−	0.0790026i	$-0.0251735\pi$
$89$	1674.00			1.99375			0.996874	+	0.0790026i	$0.0251735\pi$
$90$		0			0
$91$		0			0
$92$	266.991	+	821.713i	0.302562	+	0.931190i
$93$	−2200.53	+	1598.78i	−2.45359	+	1.78264i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	27.5066	+	19.9847i	0.0287925	+	0.0209190i	0.602088	−	0.798429i	$-0.294335\pi$
$97$	27.5066	+	19.9847i	0.0287925	+	0.0209190i	−0.573296	+	0.819348i	$0.694335\pi$
$98$		0			0
$99$		0			0
$100$	−1592.00			−1.59200
$101$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$101$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$102$		0			0
$103$	362.168	+	1114.64i	0.346461	+	1.06630i	0.960797	+	0.277252i	$0.0894238\pi$
$103$	362.168	+	1114.64i	0.346461	+	1.06630i	−0.614336	+	0.789044i	$0.710576\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$107$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$108$	517.771	+	376.183i	0.461320	+	0.335168i
$109$		0			0			−	1.00000i	$-0.5\pi$
$109$		0			0				1.00000i	$0.5\pi$
$110$		0			0
$111$	−3472.00			−2.96890
$112$		0			0
$113$	661.914	−	2037.16i	0.551041	−	1.69593i	−0.155134	−	0.987893i	$-0.549581\pi$
$113$	661.914	−	2037.16i	0.551041	−	1.69593i	0.706175	−	0.708037i	$-0.250419\pi$
$114$		0			0
$115$	1572.73	−	1142.65i	1.27528	−	0.926549i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$		0			0
$122$		0			0
$123$		0			0
$124$	−840.526	+	2586.87i	−0.608722	+	1.87345i
$125$	411.611	+	1266.81i	0.294525	+	0.906454i
$126$		0			0
$127$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$127$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$128$		0			0
$129$		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
$135$	444.984	−	1369.52i	0.283690	−	0.873108i
$136$		0			0
$137$	−975.674	+	708.869i	−0.608449	+	0.442064i	−0.848868	−	0.528605i	$-0.822715\pi$
$137$	−975.674	+	708.869i	−0.608449	+	0.442064i	0.240419	+	0.970669i	$0.422715\pi$
$138$		0			0
$139$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$139$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$140$		0			0
$141$	232.997	+	169.282i	0.139162	+	0.101107i
$142$		0			0
$143$		0			0
$144$	2368.00			1.37037
$145$		0			0
$146$		0			0
$147$	−847.943	−	2609.70i	−0.475763	−	1.46425i
$148$	−2808.91	+	2040.79i	−1.56007	+	1.13346i
$149$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$149$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$150$		0			0
$151$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$151$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	6120.00			3.17142
$156$		0			0
$157$	−412.229	+	1268.71i	−0.209551	+	0.644930i	0.789945	+	0.613177i	$0.210109\pi$
$157$	−412.229	+	1268.71i	−0.209551	+	0.644930i	−0.999496	+	0.0317528i	$0.989891\pi$
$158$		0			0
$159$	4776.44	−	3470.28i	2.38237	−	1.73089i
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−3016.02	−	2191.26i	−1.44928	−	1.05296i	−0.985999	−	0.166751i	$-0.946672\pi$
$163$	−3016.02	−	2191.26i	−1.44928	−	1.05296i	−0.463281	−	0.886212i	$-0.653328\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$167$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$168$		0			0
$169$	−678.910	−	2089.47i	−0.309017	−	0.951057i
$170$		0			0
$171$		0			0
$172$		0			0
$173$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$173$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	−5760.00			−2.44603
$178$		0			0
$179$	−622.978	+	1917.33i	−0.260132	+	0.800603i	0.732643	+	0.680613i	$0.238286\pi$
$179$	−622.978	+	1917.33i	−0.260132	+	0.800603i	−0.992775	+	0.119990i	$0.961714\pi$
$180$	−1646.44	−	5067.23i	−0.681770	−	2.09827i
$181$	1658.48	−	1204.96i	0.681073	−	0.494828i	−0.192641	−	0.981269i	$-0.561705\pi$
$181$	1658.48	−	1204.96i	0.681073	−	0.494828i	0.873713	+	0.486441i	$0.161705\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	6320.04	+	4591.78i	2.51167	+	1.82483i
$186$		0			0
$187$		0			0
$188$	288.000			0.111726
$189$		0			0
$190$		0			0
$191$	1613.07	+	4964.52i	0.611087	+	1.88073i	0.447743	+	0.894162i	$0.352228\pi$
$191$	1613.07	+	4964.52i	0.611087	+	1.88073i	0.163343	+	0.986569i	$0.447772\pi$
$192$	3313.73	−	2407.57i	1.24556	−	0.904955i
$193$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$193$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$194$		0			0
$195$		0			0
$196$	−2219.94	−	1612.88i	−0.809017	−	0.587785i
$197$		0			0			−	1.00000i	$-0.5\pi$
$197$		0			0				1.00000i	$0.5\pi$
$198$		0			0
$199$	3940.00			1.40351			0.701757	−	0.712417i	$-0.252399\pi$
$199$	3940.00			1.40351			0.701757	+	0.712417i	$0.252399\pi$
$200$		0			0
$201$	−1028.41	+	3165.12i	−0.360887	+	1.11070i
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$	3232.83	+	2348.79i	1.08549	+	0.788658i
$208$		0			0
$209$		0			0
$210$		0			0
$211$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$211$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$212$	1824.44	−	5615.04i	0.591051	−	1.81907i
$213$	1512.95	+	4656.37i	0.486692	+	1.49788i
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$	206.423	−	635.306i	0.0619871	−	0.190777i	−0.915267	−	0.402847i	$-0.868021\pi$
$223$	206.423	−	635.306i	0.0619871	−	0.190777i	0.977254	+	0.212070i	$0.0680206\pi$
$224$		0			0
$225$	−5956.79	+	4327.86i	−1.76498	+	1.28233i
$226$		0			0
$227$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$227$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$228$		0			0
$229$	−2677.85	−	1945.57i	−0.772738	−	0.561427i	0.130053	−	0.991507i	$-0.458485\pi$
$229$	−2677.85	−	1945.57i	−0.772738	−	0.561427i	−0.902791	+	0.430080i	$0.858485\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$233$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$234$		0			0
$235$	−200.243	−	616.285i	−0.0555847	−	0.171072i
$236$	−4659.94	+	3385.64i	−1.28532	+	0.933841i
$237$		0			0
$238$		0			0
$239$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$239$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$240$	−7455.90	−	5417.03i	−2.00532	−	1.45695i
$241$		0			0			−	1.00000i	$-0.5\pi$
$241$		0			0				1.00000i	$0.5\pi$
$242$		0			0
$243$	−5032.00			−1.32841
$244$		0			0
$245$	−1907.87	+	5871.82i	−0.497508	+	1.53117i
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−524.243	−	380.885i	−0.131832	−	0.0957818i	0.519915	−	0.854218i	$-0.325964\pi$
$251$	−524.243	−	380.885i	−0.131832	−	0.0957818i	−0.651747	+	0.758436i	$0.725964\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$	1265.73	−	3895.53i	0.309017	−	0.951057i
$257$	−2486.35	−	7652.20i	−0.603480	−	1.85732i	−0.506921	−	0.861992i	$-0.669216\pi$
$257$	−2486.35	−	7652.20i	−0.603480	−	1.85732i	−0.0965586	−	0.995327i	$-0.530784\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0			−	1.00000i	$-0.5\pi$
$263$		0			0				1.00000i	$0.5\pi$
$264$		0			0
$265$	−13284.0			−3.07936
$266$		0			0
$267$	4138.36	−	12736.5i	0.948551	−	2.91934i
$268$	1028.41	+	3165.12i	0.234403	+	0.721419i
$269$	−2257.16	+	1639.92i	−0.511603	+	0.371702i	−0.813431	−	0.581661i	$-0.802403\pi$
$269$	−2257.16	+	1639.92i	−0.511603	+	0.371702i	0.301828	+	0.953362i	$0.402403\pi$
$270$		0			0
$271$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$271$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$	6912.00			1.50744
$277$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$277$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$278$		0			0
$279$	3887.43	+	11964.3i	0.834174	+	2.56732i
$280$		0			0
$281$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$281$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$282$		0			0
$283$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$283$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$284$	3960.95	+	2877.80i	0.827602	+	0.601288i
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−1518.20	+	4672.54i	−0.309017	+	0.951057i
$290$		0			0
$291$	220.053	−	159.878i	0.0443289	−	0.0322068i
$292$		0			0
$293$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$293$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$294$		0			0
$295$	10484.9	+	7617.70i	2.06933	+	1.50346i
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$	−3935.64	+	12112.7i	−0.757414	+	2.33108i
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$		0			0			−	1.00000i	$-0.5\pi$
$307$		0			0				1.00000i	$0.5\pi$
$308$		0			0
$309$	9376.00			1.72616
$310$		0			0
$311$	2925.77	−	9004.60i	0.533458	−	1.64181i	−0.213500	−	0.976943i	$-0.568486\pi$
$311$	2925.77	−	9004.60i	0.533458	−	1.64181i	0.746958	−	0.664871i	$-0.231514\pi$
$312$		0			0
$313$	8884.62	−	6455.06i	1.60444	−	1.16569i	0.726177	−	0.687508i	$-0.241295\pi$
$313$	8884.62	−	6455.06i	1.60444	−	1.16569i	0.878260	−	0.478183i	$-0.158705\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−4849.25	−	3523.18i	−0.859183	−	0.624233i	0.0684798	−	0.997653i	$-0.478185\pi$
$317$	−4849.25	−	3523.18i	−0.859183	−	0.624233i	−0.927662	+	0.373420i	$0.878185\pi$
$318$		0			0
$319$		0			0
$320$	−9216.00			−1.60997
$321$		0			0
$322$		0			0
$323$		0			0
$324$	−2323.50	+	1688.12i	−0.398405	+	0.289458i
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	8120.00			1.34839			0.674193	−	0.738555i	$-0.264492\pi$
$331$	8120.00			1.34839			0.674193	+	0.738555i	$0.264492\pi$
$332$		0			0
$333$	−4962.19	+	15272.1i	−0.816596	+	2.51323i
$334$		0			0
$335$	6057.92	−	4401.34i	0.987998	−	0.717823i
$336$		0			0
$337$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$337$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$338$		0			0
$339$	−13863.3	−	10072.3i	−2.22110	−	1.61372i
$340$		0			0
$341$		0			0
$342$		0			0
$343$		0			0
$344$		0			0
$345$	−4805.83	−	14790.8i	−0.749963	−	2.30815i
$346$		0			0
$347$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$347$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$348$		0			0
$349$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$349$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	8802.00			1.32715			0.663574	−	0.748111i	$-0.269039\pi$
$353$	8802.00			1.32715			0.663574	+	0.748111i	$0.269039\pi$
$354$		0			0
$355$	3404.13	−	10476.8i	0.508937	−	1.56635i
$356$	−4138.36	−	12736.5i	−0.616102	−	1.89617i
$357$		0			0
$358$		0			0
$359$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$359$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$360$		0			0
$361$	5549.05	+	4031.62i	0.809017	+	0.587785i
$362$		0			0
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$	−3064.21	−	9430.68i	−0.435833	−	1.34136i	−0.892231	−	0.451579i	$-0.850861\pi$
$367$	−3064.21	−	9430.68i	−0.435833	−	1.34136i	0.456398	−	0.889776i	$-0.349139\pi$
$368$	5591.93	−	4062.77i	0.792118	−	0.575507i
$369$		0			0
$370$		0			0
$371$		0			0
$372$	17604.2	+	12790.2i	2.45359	+	1.78264i
$373$		0			0			−	1.00000i	$-0.5\pi$
$373$		0			0				1.00000i	$0.5\pi$
$374$		0			0
$375$	10656.0			1.46740
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−10355.4	+	7523.65i	−1.40349	+	1.01969i	−0.409259	+	0.912418i	$0.634213\pi$
$379$	−10355.4	+	7523.65i	−1.40349	+	1.01969i	−0.994229	+	0.107276i	$0.965787\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−11737.2	−	8527.59i	−1.56591	−	1.13770i	−0.930944	−	0.365161i	$-0.881014\pi$
$383$	−11737.2	−	8527.59i	−1.56591	−	1.13770i	−0.634967	−	0.772540i	$-0.718986\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$	84.0526	−	258.687i	0.0109977	−	0.0338476i
$389$	4366.41	+	13438.4i	0.569115	+	1.75156i	0.655396	+	0.755285i	$0.272502\pi$
$389$	4366.41	+	13438.4i	0.569115	+	1.75156i	−0.0862809	+	0.996271i	$0.527498\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−2374.00			−0.300120			−0.150060	−	0.988677i	$-0.547947\pi$
$397$	−2374.00			−0.300120			−0.150060	+	0.988677i	$0.547947\pi$
$398$		0			0
$399$		0			0
$400$	3935.64	+	12112.7i	0.491955	+	1.51408i
$401$	7353.96	−	5342.97i	0.915809	−	0.665374i	−0.0266682	−	0.999644i	$-0.508490\pi$
$401$	7353.96	−	5342.97i	0.915809	−	0.665374i	0.942477	+	0.334270i	$0.108490\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$	5227.87	+	3798.27i	0.641419	+	0.466018i
$406$		0			0
$407$		0			0
$408$		0			0
$409$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$409$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$410$		0			0
$411$	2981.40	+	9175.79i	0.357814	+	1.10124i
$412$	7585.34	−	5511.07i	0.907046	−	0.659008i
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−16344.0			−1.90562			−0.952812	−	0.303560i	$-0.901825\pi$
$419$	−16344.0			−1.90562			−0.952812	+	0.303560i	$0.901825\pi$
$420$		0			0
$421$	−3593.87	+	11060.8i	−0.416044	+	1.28045i	0.495270	+	0.868739i	$0.335069\pi$
$421$	−3593.87	+	11060.8i	−0.416044	+	1.28045i	−0.911314	+	0.411712i	$0.864931\pi$
$422$		0			0
$423$	1077.61	−	782.930i	0.123866	−	0.0899938i
$424$		0			0
$425$		0			0
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$431$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$432$	1582.17	−	4869.41i	0.176208	−	0.542314i
$433$	−4104.36	−	12631.9i	−0.455527	−	1.40197i	−0.870516	−	0.492141i	$-0.836214\pi$
$433$	−4104.36	−	12631.9i	−0.455527	−	1.40197i	0.414989	−	0.909827i	$-0.363786\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$		0			0			−	1.00000i	$-0.5\pi$
$439$		0			0				1.00000i	$0.5\pi$
$440$		0			0
$441$	−12691.0			−1.37037
$442$		0			0
$443$	−5762.55	+	17735.3i	−0.618029	+	1.90210i	−0.305293	+	0.952258i	$0.598755\pi$
$443$	−5762.55	+	17735.3i	−0.618029	+	1.90210i	−0.312736	+	0.949840i	$0.601245\pi$
$444$	8583.26	+	26416.5i	0.917440	+	2.82359i
$445$	−24377.3	+	17711.1i	−2.59684	+	1.88672i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−5489.99	−	3988.71i	−0.577035	−	0.419240i	0.260619	−	0.965442i	$-0.416073\pi$
$449$	−5489.99	−	3988.71i	−0.577035	−	0.419240i	−0.837654	+	0.546201i	$0.816073\pi$
$450$		0			0
$451$		0			0
$452$	−17136.0			−1.78321
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$457$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$458$		0			0
$459$		0			0
$460$	−12581.8	−	9141.24i	−1.27528	−	0.926549i
$461$		0			0			−	1.00000i	$-0.5\pi$
$461$		0			0				1.00000i	$0.5\pi$
$462$		0			0
$463$	13268.0			1.33178			0.665892	−	0.746048i	$-0.268051\pi$
$463$	13268.0			1.33178			0.665892	+	0.746048i	$0.268051\pi$
$464$		0			0
$465$	15129.5	−	46563.7i	1.50884	−	4.64374i
$466$		0			0
$467$	−3378.45	+	2454.59i	−0.334767	+	0.243223i	−0.742451	−	0.669901i	$-0.766337\pi$
$467$	−3378.45	+	2454.59i	−0.334767	+	0.243223i	0.407684	+	0.913123i	$0.366337\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$	8633.83	+	6272.84i	0.844641	+	0.613668i
$472$		0			0
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$	−8438.02	−	25969.5i	−0.809959	−	2.49280i
$478$		0			0
$479$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$479$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−612.000			−0.0572979
$486$		0			0
$487$	5155.64	−	15867.4i	0.479721	−	1.47643i	−0.359761	−	0.933044i	$-0.617142\pi$
$487$	5155.64	−	15867.4i	0.479721	−	1.47643i	0.839483	−	0.543386i	$-0.182858\pi$
$488$		0			0
$489$	−24128.1	+	17530.1i	−2.23131	+	1.62114i
$490$		0			0
$491$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$491$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$	21760.0			1.96986
$497$		0			0
$498$		0			0
$499$	1273.15	+	3918.35i	0.114216	+	0.351522i	0.991783	−	0.127933i	$-0.0408343\pi$
$499$	1273.15	+	3918.35i	0.114216	+	0.351522i	−0.877566	+	0.479455i	$0.840834\pi$
$500$	8620.89	−	6263.44i	0.771075	−	0.560219i
$501$		0			0
$502$		0			0
$503$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$503$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−17576.0			−1.53960
$508$		0			0
$509$	−6925.07	+	21313.2i	−0.603042	+	1.85597i	−0.0933141	+	0.995637i	$0.529746\pi$
$509$	−6925.07	+	21313.2i	−0.603042	+	1.85597i	−0.509728	+	0.860336i	$0.670254\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−17067.0	−	12399.9i	−1.46032	−	1.06098i
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$	6201.97	+	19087.7i	0.521523	+	1.60508i	0.771091	+	0.636725i	$0.219711\pi$
$521$	6201.97	+	19087.7i	0.521523	+	1.60508i	−0.249569	+	0.968357i	$0.580289\pi$
$522$		0			0
$523$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$523$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−503.000			−0.0413413
$530$		0			0
$531$	−8232.21	+	25336.1i	−0.672783	+	2.07061i
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$	13047.8	+	9479.80i	1.04852	+	0.761794i
$538$		0			0
$539$		0			0
$540$	−11520.0			−0.918040
$541$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$541$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$542$		0			0
$543$	−5067.88	−	15597.3i	−0.400522	−	1.23268i
$544$		0			0
$545$		0			0
$546$		0			0
$547$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$547$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$548$	7805.40	+	5670.95i	0.608449	+	0.442064i
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$	50560.3	−	36734.2i	3.86697	−	2.80952i
$556$		0			0
$557$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$557$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$563$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$564$	711.975	−	2191.23i	0.0531553	−	0.163595i
$565$	11914.5	+	36668.9i	0.887159	+	2.73040i
$566$		0			0
$567$		0			0
$568$		0			0
$569$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$569$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$570$		0			0
$571$		0			0			−	1.00000i	$-0.5\pi$
$571$		0			0				1.00000i	$0.5\pi$
$572$		0			0
$573$	41760.0			3.04459
$574$		0			0
$575$	−6641.39	+	20440.1i	−0.481679	+	1.48245i
$576$	−5854.02	−	18016.8i	−0.423468	−	1.30330i
$577$	18175.4	−	13205.2i	1.31135	−	0.952754i	0.311356	−	0.950293i	$-0.399217\pi$
$577$	18175.4	−	13205.2i	1.31135	−	0.952754i	0.999997	−	0.00246060i	$-0.000783233\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−8054.22	−	24788.3i	−0.566326	−	1.74297i	−0.663979	−	0.747751i	$-0.731134\pi$
$587$	−8054.22	−	24788.3i	−0.566326	−	1.74297i	0.0976537	−	0.995220i	$-0.468866\pi$
$588$	−17759.5	+	12903.1i	−1.24556	+	0.904955i
$589$		0			0
$590$		0			0
$591$		0			0
$592$	22471.3	+	16326.3i	1.56007	+	1.13346i
$593$		0			0			−	1.00000i	$-0.5\pi$
$593$		0			0				1.00000i	$0.5\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	9740.22	−	29977.3i	0.667740	−	2.05509i
$598$		0			0
$599$	−14591.4	+	10601.3i	−0.995308	+	0.723134i	−0.961077	−	0.276280i	$-0.910898\pi$
$599$	−14591.4	+	10601.3i	−0.995308	+	0.723134i	−0.0342312	+	0.999414i	$0.510898\pi$
$600$		0			0
$601$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$601$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$602$		0			0
$603$	12452.4	+	9047.19i	0.840963	+	0.610995i
$604$		0			0
$605$		0			0
$606$		0			0
$607$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$607$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$613$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	3654.00			0.238419			0.119209	−	0.992869i	$-0.461964\pi$
$617$	3654.00			0.238419			0.119209	+	0.992869i	$0.461964\pi$
$618$		0			0
$619$	573.536	−	1765.16i	0.0372413	−	0.114617i	−0.930708	−	0.365764i	$-0.880808\pi$
$619$	573.536	−	1765.16i	0.0372413	−	0.114617i	0.967949	+	0.251147i	$0.0808078\pi$
$620$	−15129.5	−	46563.7i	−0.980023	−	3.01620i
$621$	6989.91	−	5078.46i	0.451683	−	0.328167i
$622$		0			0
$623$		0			0
$624$		0			0
$625$	727.306	+	528.419i	0.0465476	+	0.0338188i
$626$		0			0
$627$		0			0
$628$	10672.0			0.678120
$629$		0			0
$630$		0			0
$631$	−3988.79	−	12276.2i	−0.251650	−	0.774500i	−0.994471	−	0.105010i	$-0.966513\pi$
$631$	−3988.79	−	12276.2i	−0.251650	−	0.774500i	0.742821	−	0.669490i	$-0.233487\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$	−38211.5	−	27762.3i	−2.38237	−	1.73089i
$637$		0			0
$638$		0			0
$639$	22644.0			1.40185
$640$		0			0
$641$	−1418.39	+	4365.35i	−0.0873993	+	0.268987i	−0.985198	−	0.171418i	$-0.945165\pi$
$641$	−1418.39	+	4365.35i	−0.0873993	+	0.268987i	0.897799	+	0.440405i	$0.145165\pi$
$642$		0			0
$643$	8226.08	−	5976.60i	0.504518	−	0.366554i	−0.306222	−	0.951960i	$-0.599065\pi$
$643$	8226.08	−	5976.60i	0.504518	−	0.366554i	0.810740	+	0.585406i	$0.199065\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−26474.3	−	19234.7i	−1.60867	−	1.16877i	−0.867473	−	0.497483i	$-0.834258\pi$
$647$	−26474.3	−	19234.7i	−1.60867	−	1.16877i	−0.741199	−	0.671285i	$-0.765742\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		0			0
$652$	−9216.12	+	28364.3i	−0.553576	+	1.70373i
$653$	10117.8	+	31139.5i	0.606342	+	1.86613i	0.487287	+	0.873242i	$0.337986\pi$
$653$	10117.8	+	31139.5i	0.606342	+	1.86613i	0.119055	+	0.992888i	$0.462014\pi$
$654$		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
$661$	−23582.0			−1.38765			−0.693823	−	0.720146i	$-0.744075\pi$
$661$	−23582.0			−1.38765			−0.693823	+	0.720146i	$0.744075\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$	−4323.39	−	3141.12i	−0.249853	−	0.181529i
$670$		0			0
$671$		0			0
$672$		0			0
$673$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$673$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$674$		0			0
$675$	4919.55	+	15140.8i	0.280524	+	0.863364i
$676$	−14219.3	+	10330.9i	−0.809017	+	0.587785i
$677$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$677$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−35352.0			−1.98054			−0.990268	−	0.139170i	$-0.955556\pi$
$683$	−35352.0			−1.98054			−0.990268	+	0.139170i	$0.955556\pi$
$684$		0			0
$685$	6708.14	−	20645.5i	0.374168	−	1.15157i
$686$		0			0
$687$	−21422.8	+	15564.6i	−1.18971	+	0.864374i
$688$		0			0
$689$		0			0
$690$		0			0
$691$	24535.9	+	17826.4i	1.35078	+	0.981398i	0.998972	+	0.0453259i	$0.0144326\pi$
$691$	24535.9	+	17826.4i	1.35078	+	0.981398i	0.351807	+	0.936073i	$0.385567\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$		0			0
$701$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$701$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$	−5184.00			−0.276937
$706$		0			0
$707$		0			0
$708$	14239.5	+	43824.7i	0.755866	+	2.32632i
$709$	27145.8	−	19722.5i	1.43791	−	1.04470i	0.449439	−	0.893311i	$-0.351624\pi$
$709$	27145.8	−	19722.5i	1.43791	−	1.04470i	0.988473	−	0.151394i	$-0.0483763\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	29707.1	+	21583.5i	1.56036	+	1.13367i
$714$		0			0
$715$		0			0
$716$	16128.0			0.841804
$717$		0			0
$718$		0			0
$719$	6997.38	+	21535.7i	0.362946	+	1.11703i	0.951257	+	0.308399i	$0.0997931\pi$
$719$	6997.38	+	21535.7i	0.362946	+	1.11703i	−0.588311	+	0.808635i	$0.700207\pi$
$720$	−34483.5	+	25053.8i	−1.78490	+	1.29680i
$721$		0			0
$722$		0			0
$723$		0			0
$724$	−13267.9	−	9639.68i	−0.681073	−	0.494828i
$725$		0			0
$726$		0			0
$727$	−33284.0			−1.69799			−0.848993	−	0.528405i	$-0.822790\pi$
$727$	−33284.0			−1.69799			−0.848993	+	0.528405i	$0.822790\pi$
$728$		0			0
$729$	−9444.49	+	29067.1i	−0.479830	+	1.47676i
$730$		0			0
$731$		0			0
$732$		0			0
$733$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$733$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$734$		0			0
$735$	39959.0	+	29031.9i	2.00532	+	1.45695i
$736$		0			0
$737$		0			0
$738$		0			0
$739$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$739$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$740$	19312.3	−	59437.2i	0.959372	−	2.95264i
$741$		0			0
$742$		0			0
$743$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$743$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	12253.1	−	37711.3i	0.595371	−	1.83236i	0.0424993	−	0.999096i	$-0.486468\pi$
$751$	12253.1	−	37711.3i	0.595371	−	1.83236i	0.552872	−	0.833267i	$-0.313532\pi$
$752$	−711.975	−	2191.23i	−0.0345253	−	0.106258i
$753$	−4193.94	+	3047.08i	−0.202969	+	0.147466i
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−25424.2	−	18471.7i	−1.22068	−	0.886878i	−0.224526	−	0.974468i	$-0.572084\pi$
$757$	−25424.2	−	18471.7i	−1.22068	−	0.886878i	−0.996157	+	0.0875898i	$0.972084\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$761$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$762$		0			0
$763$		0			0
$764$	33784.5	−	24545.9i	1.59985	−	1.16236i
$765$		0			0
$766$		0			0
$767$		0			0
$768$	−26509.9	−	19260.5i	−1.24556	−	0.904955i
$769$		0			0			−	1.00000i	$-0.5\pi$
$769$		0			0				1.00000i	$0.5\pi$
$770$		0			0
$771$	−64368.0			−3.00669
$772$		0			0
$773$	9962.09	−	30660.2i	0.463534	−	1.42661i	−0.397284	−	0.917696i	$-0.630047\pi$
$773$	9962.09	−	30660.2i	0.463534	−	1.42661i	0.860817	−	0.508914i	$-0.169953\pi$
$774$		0			0
$775$	−54738.1	+	39769.6i	−2.53710	+	1.84331i
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	−6783.54	+	20877.6i	−0.309017	+	0.951057i
$785$	−7420.12	−	22836.8i	−0.337370	−	1.03832i
$786$		0			0
$787$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$787$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$		0			0
$795$	−32839.9	+	101071.i	−1.46504	+	4.50894i
$796$	−9740.22	−	29977.3i	−0.433710	−	1.33482i
$797$	−5781.24	+	4200.31i	−0.256941	+	0.186678i	−0.708797	−	0.705412i	$-0.750762\pi$
$797$	−5781.24	+	4200.31i	−0.256941	+	0.186678i	0.451856	+	0.892091i	$0.350762\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$	−50108.9	−	36406.2i	−2.21038	−	1.60593i
$802$		0			0
$803$		0			0
$804$	26624.0			1.16786
$805$		0			0
$806$		0			0
$807$	6897.26	+	21227.6i	0.300861	+	0.925956i
$808$		0			0
$809$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$809$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$810$		0			0
$811$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$811$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	67104.0			2.88411
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$821$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$822$		0			0
$823$	−2695.64	−	1958.50i	−0.114173	−	0.0829514i	0.529234	−	0.848476i	$-0.322479\pi$
$823$	−2695.64	−	1958.50i	−0.114173	−	0.0829514i	−0.643407	+	0.765525i	$0.722479\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$827$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$828$	9878.66	−	30403.4i	0.414622	−	1.27608i
$829$	−14750.6	−	45397.7i	−0.617986	−	1.90196i	−0.320886	−	0.947118i	$-0.603981\pi$
$829$	−14750.6	−	45397.7i	−0.617986	−	1.90196i	−0.297099	−	0.954847i	$-0.596019\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$	27200.0			1.12326
$838$		0			0
$839$	6841.64	−	21056.4i	0.281525	−	0.866445i	−0.705894	−	0.708318i	$-0.749454\pi$
$839$	6841.64	−	21056.4i	0.281525	−	0.866445i	0.987419	−	0.158127i	$-0.0505456\pi$
$840$		0			0
$841$	19731.1	−	14335.5i	0.809017	−	0.587785i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	31993.4	+	23244.6i	1.30249	+	0.946316i
$846$		0			0
$847$		0			0
$848$	−47232.0			−1.91268
$849$		0			0
$850$		0			0
$851$	14484.2	+	44577.9i	0.583447	+	1.79567i
$852$	31687.6	−	23022.4i	1.27418	−	0.925744i
$853$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$853$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0			−	1.00000i	$-0.5\pi$
$857$		0			0				1.00000i	$0.5\pi$
$858$		0			0
$859$	50096.0			1.98982			0.994909	−	0.100779i	$-0.0321334\pi$
$859$	50096.0			1.98982			0.994909	+	0.100779i	$0.0321334\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−37658.1	+	27360.2i	−1.48540	+	1.07920i	−0.509629	+	0.860394i	$0.670217\pi$
$863$	−37658.1	+	27360.2i	−1.48540	+	1.07920i	−0.975768	+	0.218810i	$0.929783\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	31797.6	+	23102.3i	1.24556	+	0.904955i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$	−388.743	−	1196.43i	−0.0150710	−	0.0463837i
$874$		0			0
$875$		0			0
$876$		0			0
$877$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$877$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	34542.0			1.32094			0.660471	−	0.750852i	$-0.270357\pi$
$881$	34542.0			1.32094			0.660471	+	0.750852i	$0.270357\pi$
$882$		0			0
$883$	−8427.51	+	25937.2i	−0.321187	+	0.988513i	0.651945	+	0.758266i	$0.273953\pi$
$883$	−8427.51	+	25937.2i	−0.321187	+	0.988513i	−0.973132	+	0.230247i	$0.926047\pi$
$884$		0			0
$885$	83878.9	−	60941.6i	3.18594	−	2.31472i
$886$		0			0
$887$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$887$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$	−5344.00			−0.200595
$893$		0			0
$894$		0			0
$895$	−11213.6	−	34511.9i	−0.418804	−	1.28895i
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$	47654.3	+	34622.9i	1.76498	+	1.28233i
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−11402.7	+	35094.0i	−0.418828	+	1.28902i
$906$		0			0
$907$	17196.5	−	12494.0i	0.629547	−	0.457393i	−0.226696	−	0.973966i	$-0.572792\pi$
$907$	17196.5	−	12494.0i	0.629547	−	0.457393i	0.856243	+	0.516573i	$0.172792\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	42085.1	+	30576.6i	1.53056	+	1.11202i	0.955926	+	0.293607i	$0.0948555\pi$
$911$	42085.1	+	30576.6i	1.53056	+	1.11202i	0.574634	+	0.818411i	$0.305144\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$	−8182.77	+	25184.0i	−0.295160	+	0.908408i
$917$		0			0
$918$		0			0
$919$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$919$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$	−86366.0			−3.06994
$926$		0			0
$927$	13400.2	−	41241.6i	0.474780	−	1.46122i
$928$		0			0
$929$	45798.5	−	33274.5i	1.61744	−	1.17514i	0.792745	−	0.609553i	$-0.208651\pi$
$929$	45798.5	−	33274.5i	1.61744	−	1.17514i	0.824691	−	0.565584i	$-0.191349\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$	−61278.2	−	44521.2i	−2.15022	−	1.56223i
$934$		0			0
$935$		0			0
$936$		0			0
$937$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$937$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$938$		0			0
$939$	−27149.0	−	83556.0i	−0.943529	−	2.90388i
$940$	−4193.94	+	3047.08i	−0.145523	+	0.105728i
$941$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$941$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$942$		0			0
$943$		0			0
$944$	37279.5	+	27085.1i	1.28532	+	0.933841i
$945$		0			0
$946$		0			0
$947$	23256.0			0.798013			0.399007	−	0.916948i	$-0.369355\pi$
$947$	23256.0			0.798013			0.399007	+	0.916948i	$0.369355\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$	−38794.0	+	28185.5i	−1.32280	+	0.961069i
$952$		0			0
$953$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$953$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$954$		0			0
$955$	−76015.2	−	55228.3i	−2.57570	−	1.87136i
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$	−22783.2	+	70119.5i	−0.765963	+	2.35739i
$961$	26516.4	+	81609.2i	0.890082	+	2.73939i
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$		0			0			−	1.00000i	$-0.5\pi$
$967$		0			0				1.00000i	$0.5\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	12348.3	−	38004.2i	0.408112	−	1.25604i	−0.510157	−	0.860081i	$-0.670413\pi$
$971$	12348.3	−	38004.2i	0.408112	−	1.25604i	0.918269	−	0.395957i	$-0.129587\pi$
$972$	12439.8	+	38285.7i	0.410500	+	1.26339i
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−48099.3	−	34946.2i	−1.57506	−	1.14435i	−0.922098	−	0.386956i	$-0.873527\pi$
$977$	−48099.3	−	34946.2i	−1.57506	−	1.14435i	−0.652961	−	0.757392i	$-0.726473\pi$
$978$		0			0
$979$		0			0
$980$	49392.0			1.60997
$981$		0			0
$982$		0			0
$983$	−5484.43	−	16879.4i	−0.177951	−	0.547678i	0.821805	−	0.569770i	$-0.192968\pi$
$983$	−5484.43	−	16879.4i	−0.177951	−	0.547678i	−0.999756	+	0.0220913i	$0.992968\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$	51460.0			1.64953			0.824763	−	0.565478i	$-0.191308\pi$
$991$	51460.0			1.64953			0.824763	+	0.565478i	$0.191308\pi$
$992$		0			0
$993$	20073.7	−	61780.6i	0.641512	−	1.97437i
$994$		0			0
$995$	−57375.5	+	41685.7i	−1.82807	+	1.32817i
$996$		0			0
$997$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$997$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$998$		0			0
$999$	28089.1	+	20407.9i	0.889588	+	0.646324i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	121.4.c.a.81.1		4
11.2	odd	10	inner	121.4.c.a.27.1		4
11.3	even	5	inner	121.4.c.a.3.1		4
11.4	even	5	inner	121.4.c.a.9.1		4
11.5	even	5		121.4.a.a.1.1	✓	1
11.6	odd	10		121.4.a.a.1.1	✓	1
11.7	odd	10	inner	121.4.c.a.9.1		4
11.8	odd	10	inner	121.4.c.a.3.1		4
11.9	even	5	inner	121.4.c.a.27.1		4
11.10	odd	2	CM	121.4.c.a.81.1		4
33.5	odd	10		1089.4.a.f.1.1		1
33.17	even	10		1089.4.a.f.1.1		1
44.27	odd	10		1936.4.a.a.1.1		1
44.39	even	10		1936.4.a.a.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
121.4.a.a.1.1	✓	1	11.5	even	5
121.4.a.a.1.1	✓	1	11.6	odd	10
121.4.c.a.3.1		4	11.3	even	5	inner
121.4.c.a.3.1		4	11.8	odd	10	inner
121.4.c.a.9.1		4	11.4	even	5	inner
121.4.c.a.9.1		4	11.7	odd	10	inner
121.4.c.a.27.1		4	11.2	odd	10	inner
121.4.c.a.27.1		4	11.9	even	5	inner
121.4.c.a.81.1		4	1.1	even	1	trivial
121.4.c.a.81.1		4	11.10	odd	2	CM
1089.4.a.f.1.1		1	33.5	odd	10
1089.4.a.f.1.1		1	33.17	even	10
1936.4.a.a.1.1		1	44.27	odd	10
1936.4.a.a.1.1		1	44.39	even	10

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:	\( N \)	\(=\)	\( 121 = 11^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	121.c (of order \(5\), degree \(4\), not minimal)

\(f(q)\)	\(=\)	\(q+(2.47214 - 7.60845i) q^{3} +(-2.47214 - 7.60845i) q^{4} +(-14.5623 + 10.5801i) q^{5} +(-29.9336 - 21.7481i) q^{9} -64.0000 q^{12} +(44.4984 + 136.952i) q^{15} +(-51.7771 + 37.6183i) q^{16} +(116.498 + 84.6411i) q^{20} -108.000 q^{23} +(61.4944 - 189.260i) q^{25} +(-64.7214 + 47.0228i) q^{27} +(-275.066 - 199.847i) q^{31} +(-91.4690 + 281.513i) q^{36} +(-134.113 - 412.759i) q^{37} +666.000 q^{45} +(-11.1246 + 34.2380i) q^{47} +(158.217 + 486.941i) q^{48} +(277.493 - 201.610i) q^{49} +(597.055 + 433.786i) q^{53} +(-222.492 - 684.761i) q^{59} +(931.988 - 677.129i) q^{60} +(414.217 + 300.946i) q^{64} -416.000 q^{67} +(-266.991 + 821.713i) q^{69} +(-495.118 + 359.725i) q^{71} +(-1287.96 - 935.754i) q^{75} +(355.988 - 1095.62i) q^{80} +(-110.937 - 341.429i) q^{81} +1674.00 q^{89} +(266.991 + 821.713i) q^{92} +(-2200.53 + 1598.78i) q^{93} +(27.5066 + 19.9847i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 8 q^{3} + 8 q^{4} - 18 q^{5} - 37 q^{9} - 256 q^{12} - 144 q^{15} - 64 q^{16} + 144 q^{20} - 432 q^{23} - 199 q^{25} - 80 q^{27} - 340 q^{31} + 296 q^{36} + 434 q^{37} + 2664 q^{45} + 36 q^{47} - 512 q^{48}+ \cdots + 34 q^{97}+O(q^{100}) \) 4 * q - 8 * q^3 + 8 * q^4 - 18 * q^5 - 37 * q^9 - 256 * q^12 - 144 * q^15 - 64 * q^16 + 144 * q^20 - 432 * q^23 - 199 * q^25 - 80 * q^27 - 340 * q^31 + 296 * q^36 + 434 * q^37 + 2664 * q^45 + 36 * q^47 - 512 * q^48 + 343 * q^49 + 738 * q^53 + 720 * q^59 + 1152 * q^60 + 512 * q^64 - 1664 * q^67 + 864 * q^69 - 612 * q^71 - 1592 * q^75 - 1152 * q^80 + 359 * q^81 + 6696 * q^89 - 864 * q^92 - 2720 * q^93 + 34 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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