LMFDB - Embedded newform 135.4.b.c.109.2

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 = [12] 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:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} + \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} + 255x^{8} + 1289x^{4} + 1600 $ x^12 + 255x^8 + 1289x^4 + 1600
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{4}\cdot 3^{18}\cdot 5 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			109.2
Root			$0.930240 + 0.930240i$ of defining polynomial
Character	$\chi$	$=$	135.109
Dual form			135.4.b.c.109.12

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-4.92742i q^{2} -16.2795 q^{4} +(7.57192 - 8.22594i) q^{5} -34.7816i q^{7} +40.7965i q^{8} +(-40.5327 - 37.3100i) q^{10} +12.4586 q^{11} +37.1137i q^{13} -171.384 q^{14} +70.7858 q^{16} +102.680i q^{17} -63.7858 q^{19} +(-123.267 + 133.914i) q^{20} -61.3890i q^{22} -100.592i q^{23} +(-10.3321 - 124.572i) q^{25} +182.875 q^{26} +566.227i q^{28} +138.980 q^{29} +24.8911 q^{31} -22.4195i q^{32} +505.947 q^{34} +(-286.111 - 263.363i) q^{35} -186.668i q^{37} +314.300i q^{38} +(335.590 + 308.908i) q^{40} +255.169 q^{41} -108.471i q^{43} -202.820 q^{44} -495.659 q^{46} -223.781i q^{47} -866.760 q^{49} +(-613.820 + 50.9107i) q^{50} -604.192i q^{52} +40.5151i q^{53} +(94.3358 - 102.484i) q^{55} +1418.97 q^{56} -684.812i q^{58} +131.550 q^{59} -336.171 q^{61} -122.649i q^{62} +455.817 q^{64} +(305.295 + 281.022i) q^{65} -17.8284i q^{67} -1671.58i q^{68} +(-1297.70 + 1409.79i) q^{70} +992.473 q^{71} -769.455i q^{73} -919.792 q^{74} +1038.40 q^{76} -433.331i q^{77} +252.252 q^{79} +(535.985 - 582.280i) q^{80} -1257.33i q^{82} -234.606i q^{83} +(844.639 + 777.484i) q^{85} -534.484 q^{86} +508.269i q^{88} +405.641 q^{89} +1290.87 q^{91} +1637.58i q^{92} -1102.67 q^{94} +(-482.981 + 524.698i) q^{95} +1067.80i q^{97} +4270.89i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 60 q^{4} - 36 q^{10} + 84 q^{16} - 348 q^{25} + 252 q^{31} + 1068 q^{34} + 1320 q^{40} - 1668 q^{46} - 2868 q^{49} + 684 q^{55} + 792 q^{61} + 2268 q^{64} - 5652 q^{70} + 1824 q^{76} + 2196 q^{79}+ \cdots - 5148 q^{94}+O(q^{100}) $ 12 * q - 60 * q^4 - 36 * q^10 + 84 * q^16 - 348 * q^25 + 252 * q^31 + 1068 * q^34 + 1320 * q^40 - 1668 * q^46 - 2868 * q^49 + 684 * q^55 + 792 * q^61 + 2268 * q^64 - 5652 * q^70 + 1824 * q^76 + 2196 * q^79 + 3816 * q^85 - 3384 * q^91 - 5148 * 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 $

$2$		−	4.92742i		−	1.74211i	−0.491188	−	0.871053i	$-0.663437\pi$
$2$		−	4.92742i		−	1.74211i	0.491188	−	0.871053i	$-0.336563\pi$
$3$		0			0
$3$		0			0
$4$	−16.2795			−2.03494
$5$	7.57192	−	8.22594i	0.677253	−	0.735750i
$5$	7.57192	−	8.22594i	0.677253	−	0.735750i
$6$		0			0
$7$		−	34.7816i		−	1.87803i	−0.343876	−	0.939015i	$-0.611740\pi$
$7$		−	34.7816i		−	1.87803i	0.343876	−	0.939015i	$-0.388260\pi$
$8$			40.7965i			1.80297i
$9$		0			0
$10$	−40.5327	−	37.3100i	−1.28176	−	1.17985i
$11$	12.4586			0.341493			0.170746	−	0.985315i	$-0.445382\pi$
$11$	12.4586			0.341493			0.170746	+	0.985315i	$0.445382\pi$
$12$		0			0
$13$			37.1137i			0.791807i	0.918292	+	0.395904i	$0.129569\pi$
$13$			37.1137i			0.791807i	−0.918292	+	0.395904i	$0.870431\pi$
$14$	−171.384			−3.27173
$15$		0			0
$16$	70.7858			1.10603
$17$			102.680i			1.46491i	0.680813	+	0.732457i	$0.261627\pi$
$17$			102.680i			1.46491i	−0.680813	+	0.732457i	$0.738373\pi$
$18$		0			0
$19$	−63.7858			−0.770183			−0.385092	−	0.922878i	$-0.625830\pi$
$19$	−63.7858			−0.770183			−0.385092	+	0.922878i	$0.625830\pi$
$20$	−123.267	+	133.914i	−1.37817	+	1.49720i
$21$		0			0
$22$		−	61.3890i		−	0.594917i
$23$		−	100.592i		−	0.911951i	−0.889992	−	0.455975i	$-0.849291\pi$
$23$		−	100.592i		−	0.911951i	0.889992	−	0.455975i	$-0.150709\pi$
$24$		0			0
$25$	−10.3321	−	124.572i	−0.0826569	−	0.996578i
$26$	182.875			1.37941
$27$		0			0
$28$			566.227i			3.82167i
$29$	138.980			0.889927			0.444964	−	0.895549i	$-0.353217\pi$
$29$	138.980			0.889927			0.444964	+	0.895549i	$0.353217\pi$
$30$		0			0
$31$	24.8911			0.144212			0.0721060	−	0.997397i	$-0.477028\pi$
$31$	24.8911			0.144212			0.0721060	+	0.997397i	$0.477028\pi$
$32$		−	22.4195i		−	0.123851i
$33$		0			0
$34$	505.947			2.55204
$35$	−286.111	−	263.363i	−1.38176	−	1.27190i
$36$		0			0
$37$		−	186.668i		−	0.829406i	−0.909957	−	0.414703i	$-0.863886\pi$
$37$		−	186.668i		−	0.829406i	0.909957	−	0.414703i	$-0.136114\pi$
$38$			314.300i			1.34174i
$39$		0			0
$40$	335.590	+	308.908i	1.32654	+	1.22107i
$41$	255.169			0.971970			0.485985	−	0.873967i	$-0.338461\pi$
$41$	255.169			0.971970			0.485985	+	0.873967i	$0.338461\pi$
$42$		0			0
$43$		−	108.471i		−	0.384691i	−0.981327	−	0.192345i	$-0.938391\pi$
$43$		−	108.471i		−	0.384691i	0.981327	−	0.192345i	$-0.0616094\pi$
$44$	−202.820			−0.694916
$45$		0			0
$46$	−495.659			−1.58872
$47$		−	223.781i		−	0.694508i	−0.937771	−	0.347254i	$-0.887114\pi$
$47$		−	223.781i		−	0.694508i	0.937771	−	0.347254i	$-0.112886\pi$
$48$		0			0
$49$	−866.760			−2.52700
$50$	−613.820	+	50.9107i	−1.73615	+	0.143997i
$51$		0			0
$52$		−	604.192i		−	1.61128i
$53$			40.5151i			0.105003i	0.998621	+	0.0525017i	$0.0167195\pi$
$53$			40.5151i			0.105003i	−0.998621	+	0.0525017i	$0.983280\pi$
$54$		0			0
$55$	94.3358	−	102.484i	0.231277	−	0.251253i
$56$	1418.97			3.38603
$57$		0			0
$58$		−	684.812i		−	1.55035i
$59$	131.550			0.290278			0.145139	−	0.989411i	$-0.453637\pi$
$59$	131.550			0.290278			0.145139	+	0.989411i	$0.453637\pi$
$60$		0			0
$61$	−336.171			−0.705610			−0.352805	−	0.935697i	$-0.614772\pi$
$61$	−336.171			−0.705610			−0.352805	+	0.935697i	$0.614772\pi$
$62$		−	122.649i		−	0.251233i
$63$		0			0
$64$	455.817			0.890267
$65$	305.295	+	281.022i	0.582572	+	0.536254i
$66$		0			0
$67$		−	17.8284i		−	0.0325087i	−0.999868	−	0.0162543i	$-0.994826\pi$
$67$		−	17.8284i		−	0.0325087i	0.999868	−	0.0162543i	$-0.00517414\pi$
$68$		−	1671.58i		−	2.98101i
$69$		0			0
$70$	−1297.70	+	1409.79i	−2.21579	+	2.40718i
$71$	992.473			1.65894			0.829471	−	0.558549i	$-0.188642\pi$
$71$	992.473			1.65894			0.829471	+	0.558549i	$0.188642\pi$
$72$		0			0
$73$		−	769.455i		−	1.23367i	−0.787092	−	0.616835i	$-0.788415\pi$
$73$		−	769.455i		−	1.23367i	0.787092	−	0.616835i	$-0.211585\pi$
$74$	−919.792			−1.44491
$75$		0			0
$76$	1038.40			1.56727
$77$		−	433.331i		−	0.641334i
$78$		0			0
$79$	252.252			0.359248			0.179624	−	0.983735i	$-0.442512\pi$
$79$	252.252			0.359248			0.179624	+	0.983735i	$0.442512\pi$
$80$	535.985	−	582.280i	0.749061	−	0.813761i
$81$		0			0
$82$		−	1257.33i		−	1.69328i
$83$		−	234.606i		−	0.310258i	−0.987894	−	0.155129i	$-0.950421\pi$
$83$		−	234.606i		−	0.310258i	0.987894	−	0.155129i	$-0.0495793\pi$
$84$		0			0
$85$	844.639	+	777.484i	1.07781	+	0.992117i
$86$	−534.484			−0.670173
$87$		0			0
$88$			508.269i			0.615701i
$89$	405.641			0.483121			0.241561	−	0.970386i	$-0.422341\pi$
$89$	405.641			0.483121			0.241561	+	0.970386i	$0.422341\pi$
$90$		0			0
$91$	1290.87			1.48704
$92$			1637.58i			1.85576i
$93$		0			0
$94$	−1102.67			−1.20991
$95$	−482.981	+	524.698i	−0.521609	+	0.566662i
$96$		0			0
$97$			1067.80i			1.11772i	0.829262	+	0.558860i	$0.188761\pi$
$97$			1067.80i			1.11772i	−0.829262	+	0.558860i	$0.811239\pi$
$98$			4270.89i			4.40230i
$99$		0			0
$100$	168.201	+	2027.97i	0.168201	+	2.02797i
$101$	−1460.41			−1.43877			−0.719386	−	0.694611i	$-0.755577\pi$
$101$	−1460.41			−1.43877			−0.719386	+	0.694611i	$0.755577\pi$
$102$		0			0
$103$			1503.55i			1.43834i	0.694835	+	0.719170i	$0.255478\pi$
$103$			1503.55i			1.43834i	−0.694835	+	0.719170i	$0.744522\pi$
$104$	−1514.11			−1.42760
$105$		0			0
$106$	199.635			0.182927
$107$		−	1862.54i		−	1.68279i	−0.540418	−	0.841396i	$-0.681734\pi$
$107$		−	1862.54i		−	1.68279i	0.540418	−	0.841396i	$-0.318266\pi$
$108$		0			0
$109$	1489.05			1.30848			0.654242	−	0.756286i	$-0.272988\pi$
$109$	1489.05			1.30848			0.654242	+	0.756286i	$0.272988\pi$
$110$	−504.982	−	464.832i	−0.437710	−	0.402909i
$111$		0			0
$112$		−	2462.05i		−	2.07716i
$113$		−	936.618i		−	0.779731i	−0.920872	−	0.389865i	$-0.872522\pi$
$113$		−	936.618i		−	0.779731i	0.920872	−	0.389865i	$-0.127478\pi$
$114$		0			0
$115$	−827.463	−	761.674i	−0.670968	−	0.617621i
$116$	−2262.52			−1.81094
$117$		0			0
$118$		−	648.205i		−	0.505696i
$119$	3571.37			2.75115
$120$		0			0
$121$	−1175.78			−0.883383
$122$			1656.45i			1.22925i
$123$		0			0
$124$	−405.214			−0.293462
$125$	−1102.96	−	858.260i	−0.789212	−	0.614121i
$126$		0			0
$127$		−	285.249i		−	0.199305i	−0.995022	−	0.0996525i	$-0.968227\pi$
$127$		−	285.249i		−	0.199305i	0.995022	−	0.0996525i	$-0.0317731\pi$
$128$		−	2425.36i		−	1.67479i
$129$		0			0
$130$	1384.71	−	1504.32i	0.934211	−	1.01490i
$131$	−697.411			−0.465138			−0.232569	−	0.972580i	$-0.574713\pi$
$131$	−697.411			−0.465138			−0.232569	+	0.972580i	$0.574713\pi$
$132$		0			0
$133$			2218.57i			1.44643i
$134$	−87.8479			−0.0566336
$135$		0			0
$136$	−4188.99			−2.64120
$137$			1557.87i			0.971514i	0.874094	+	0.485757i	$0.161456\pi$
$137$			1557.87i			0.971514i	−0.874094	+	0.485757i	$0.838544\pi$
$138$		0			0
$139$	315.325			0.192414			0.0962068	−	0.995361i	$-0.469329\pi$
$139$	315.325			0.192414			0.0962068	+	0.995361i	$0.469329\pi$
$140$	4657.75	+	4287.42i	2.81180	+	2.58824i
$141$		0			0
$142$		−	4890.34i		−	2.89006i
$143$			462.386i			0.270396i
$144$		0			0
$145$	1052.34	−	1143.24i	0.602706	−	0.654764i
$146$	−3791.43			−2.14919
$147$		0			0
$148$			3038.86i			1.68779i
$149$	1983.67			1.09066			0.545331	−	0.838221i	$-0.316404\pi$
$149$	1983.67			1.09066			0.545331	+	0.838221i	$0.316404\pi$
$150$		0			0
$151$	1057.25			0.569787			0.284894	−	0.958559i	$-0.408042\pi$
$151$	1057.25			0.569787			0.284894	+	0.958559i	$0.408042\pi$
$152$		−	2602.24i		−	1.38862i
$153$		0			0
$154$	−2135.21			−1.11727
$155$	188.473	−	204.753i	0.0976680	−	0.106104i
$156$		0			0
$157$		−	2171.68i		−	1.10394i	−0.833862	−	0.551972i	$-0.813875\pi$
$157$		−	2171.68i		−	1.10394i	0.833862	−	0.551972i	$-0.186125\pi$
$158$		−	1242.95i		−	0.625848i
$159$		0			0
$160$	−184.421	−	169.758i	−0.0911235	−	0.0838786i
$161$	−3498.75			−1.71267
$162$		0			0
$163$			1219.04i			0.585781i	0.956146	+	0.292890i	$0.0946171\pi$
$163$			1219.04i			0.585781i	−0.956146	+	0.292890i	$0.905383\pi$
$164$	−4154.03			−1.97790
$165$		0			0
$166$	−1156.01			−0.540502
$167$			3577.08i			1.65750i	0.559619	+	0.828750i	$0.310948\pi$
$167$			3577.08i			1.65750i	−0.559619	+	0.828750i	$0.689052\pi$
$168$		0			0
$169$	819.572			0.373041
$170$	3830.99	−	4161.89i	1.72837	−	1.87766i
$171$		0			0
$172$			1765.86i			0.782822i
$173$			2372.95i			1.04284i	0.853299	+	0.521422i	$0.174598\pi$
$173$			2372.95i			1.04284i	−0.853299	+	0.521422i	$0.825402\pi$
$174$		0			0
$175$	−4332.82	+	359.367i	−1.87160	+	0.155232i
$176$	881.895			0.377701
$177$		0			0
$178$		−	1998.76i		−	0.841649i
$179$	774.792			0.323524			0.161762	−	0.986830i	$-0.448282\pi$
$179$	774.792			0.323524			0.161762	+	0.986830i	$0.448282\pi$
$180$		0			0
$181$	−1250.05			−0.513344			−0.256672	−	0.966499i	$-0.582626\pi$
$181$	−1250.05			−0.513344			−0.256672	+	0.966499i	$0.582626\pi$
$182$		−	6360.68i		−	2.59058i
$183$		0			0
$184$	4103.80			1.64422
$185$	−1535.52	−	1413.43i	−0.610235	−	0.561717i
$186$		0			0
$187$			1279.25i			0.500258i
$188$			3643.05i			1.41328i
$189$		0			0
$190$	2585.41	+	2379.85i	0.987186	+	0.908698i
$191$	2141.03			0.811097			0.405548	−	0.914074i	$-0.367081\pi$
$191$	2141.03			0.811097			0.405548	+	0.914074i	$0.367081\pi$
$192$		0			0
$193$			1829.86i			0.682466i	0.939979	+	0.341233i	$0.110844\pi$
$193$			1829.86i			0.682466i	−0.939979	+	0.341233i	$0.889156\pi$
$194$	5261.51			1.94719
$195$		0			0
$196$	14110.4			5.14228
$197$			1142.23i			0.413099i	0.978436	+	0.206550i	$0.0662235\pi$
$197$			1142.23i			0.413099i	−0.978436	+	0.206550i	$0.933777\pi$
$198$		0			0
$199$	−284.078			−0.101195			−0.0505973	−	0.998719i	$-0.516113\pi$
$199$	−284.078			−0.101195			−0.0505973	+	0.998719i	$0.516113\pi$
$200$	5082.12	−	421.514i	1.79680	−	0.149028i
$201$		0			0
$202$			7196.04i			2.50649i
$203$		−	4833.94i		−	1.67131i
$204$		0			0
$205$	1932.12	−	2099.01i	0.658270	−	0.715127i
$206$	7408.61			2.50574
$207$		0			0
$208$			2627.13i			0.875762i
$209$	−794.685			−0.263012
$210$		0			0
$211$	813.075			0.265281			0.132641	−	0.991164i	$-0.457654\pi$
$211$	813.075			0.265281			0.132641	+	0.991164i	$0.457654\pi$
$212$		−	659.566i		−	0.213675i
$213$		0			0
$214$	−9177.54			−2.93160
$215$	−892.278	−	821.336i	−0.283036	−	0.260533i
$216$		0			0
$217$		−	865.752i		−	0.270834i
$218$		−	7337.16i		−	2.27952i
$219$		0			0
$220$	−1535.74	+	1668.39i	−0.470634	+	0.511285i
$221$	−3810.83			−1.15993
$222$		0			0
$223$		−	4936.43i		−	1.48237i	−0.671303	−	0.741183i	$-0.734265\pi$
$223$		−	4936.43i		−	1.48237i	0.671303	−	0.741183i	$-0.265735\pi$
$224$	−779.785			−0.232596
$225$		0			0
$226$	−4615.11			−1.35837
$227$		−	3589.88i		−	1.04964i	−0.851213	−	0.524820i	$-0.824133\pi$
$227$		−	3589.88i		−	1.04964i	0.851213	−	0.524820i	$-0.175867\pi$
$228$		0			0
$229$	3869.82			1.11670			0.558352	−	0.829604i	$-0.311434\pi$
$229$	3869.82			1.11670			0.558352	+	0.829604i	$0.311434\pi$
$230$	−3753.09	+	4077.26i	−1.07596	+	1.16890i
$231$		0			0
$232$			5669.89i			1.60451i
$233$			2983.29i			0.838805i	0.907800	+	0.419403i	$0.137760\pi$
$233$			2983.29i			0.838805i	−0.907800	+	0.419403i	$0.862240\pi$
$234$		0			0
$235$	−1840.81	−	1694.46i	−0.510985	−	0.470358i
$236$	−2141.57			−0.590698
$237$		0			0
$238$		−	17597.7i		−	4.79280i
$239$	2627.57			0.711145			0.355572	−	0.934649i	$-0.384286\pi$
$239$	2627.57			0.711145			0.355572	+	0.934649i	$0.384286\pi$
$240$		0			0
$241$	−4236.43			−1.13233			−0.566167	−	0.824291i	$-0.691574\pi$
$241$	−4236.43			−1.13233			−0.566167	+	0.824291i	$0.691574\pi$
$242$			5793.58i			1.53895i
$243$		0			0
$244$	5472.68			1.43587
$245$	−6563.04	+	7129.92i	−1.71142	+	1.85924i
$246$		0			0
$247$		−	2367.33i		−	0.609836i
$248$			1015.47i			0.260010i
$249$		0			0
$250$	−4229.01	+	5434.74i	−1.06986	+	1.37489i
$251$	−6298.35			−1.58386			−0.791929	−	0.610614i	$-0.790923\pi$
$251$	−6298.35			−1.58386			−0.791929	+	0.610614i	$0.790923\pi$
$252$		0			0
$253$		−	1253.24i		−	0.311425i
$254$	−1405.54			−0.347211
$255$		0			0
$256$	−8304.22			−2.02740
$257$			3034.13i			0.736434i	0.929740	+	0.368217i	$0.120032\pi$
$257$			3034.13i			0.736434i	−0.929740	+	0.368217i	$0.879968\pi$
$258$		0			0
$259$	−6492.61			−1.55765
$260$	−4970.05	−	4574.89i	−1.18550	−	1.09124i
$261$		0			0
$262$			3436.44i			0.810320i
$263$			5212.09i			1.22202i	0.791623	+	0.611010i	$0.209237\pi$
$263$			5212.09i			1.22202i	−0.791623	+	0.611010i	$0.790763\pi$
$264$		0			0
$265$	333.275	+	306.777i	0.0772563	+	0.0711139i
$266$	10931.9			2.51983
$267$		0			0
$268$			290.237i			0.0661531i
$269$	1851.11			0.419571			0.209785	−	0.977747i	$-0.432724\pi$
$269$	1851.11			0.419571			0.209785	+	0.977747i	$0.432724\pi$
$270$		0			0
$271$	922.751			0.206838			0.103419	−	0.994638i	$-0.467022\pi$
$271$	922.751			0.206838			0.103419	+	0.994638i	$0.467022\pi$
$272$			7268.29i			1.62024i
$273$		0			0
$274$	7676.26			1.69248
$275$	−128.724	−	1552.00i	−0.0282267	−	0.340324i
$276$		0			0
$277$		−	1932.26i		−	0.419126i	−0.977795	−	0.209563i	$-0.932796\pi$
$277$		−	1932.26i		−	0.419126i	0.977795	−	0.209563i	$-0.0672042\pi$
$278$		−	1553.74i		−	0.335205i
$279$		0			0
$280$	10744.3	−	11672.4i	2.29320	−	2.49127i
$281$	7020.02			1.49032			0.745159	−	0.666887i	$-0.232374\pi$
$281$	7020.02			1.49032			0.745159	+	0.666887i	$0.232374\pi$
$282$		0			0
$283$			1497.15i			0.314475i	0.987561	+	0.157237i	$0.0502588\pi$
$283$			1497.15i			0.314475i	−0.987561	+	0.157237i	$0.949741\pi$
$284$	−16157.0			−3.37584
$285$		0			0
$286$	2278.37			0.471059
$287$		−	8875.21i		−	1.82539i
$288$		0			0
$289$	−5630.17			−1.14597
$290$	−5633.22	−	5185.34i	−1.14067	−	1.04998i
$291$		0			0
$292$			12526.3i			2.51044i
$293$			7003.84i			1.39648i	0.715863	+	0.698240i	$0.246033\pi$
$293$			7003.84i			1.39648i	−0.715863	+	0.698240i	$0.753967\pi$
$294$		0			0
$295$	996.089	−	1082.13i	0.196592	−	0.213572i
$296$	7615.40			1.49539
$297$		0			0
$298$		−	9774.38i		−	1.90005i
$299$	3733.34			0.722089
$300$		0			0
$301$	−3772.80			−0.722461
$302$		−	5209.52i		−	0.992630i
$303$		0			0
$304$	−4515.14			−0.851845
$305$	−2545.46	+	2765.32i	−0.477877	+	0.519153i
$306$		0			0
$307$		−	1297.09i		−	0.241137i	−0.992705	−	0.120568i	$-0.961528\pi$
$307$		−	1297.09i		−	0.241137i	0.992705	−	0.120568i	$-0.0384717\pi$
$308$			7054.41i			1.30507i
$309$		0			0
$310$	−1008.90	−	928.687i	−0.184844	−	0.170148i
$311$	3073.25			0.560348			0.280174	−	0.959949i	$-0.409608\pi$
$311$	3073.25			0.560348			0.280174	+	0.959949i	$0.409608\pi$
$312$		0			0
$313$			8290.40i			1.49713i	0.663062	+	0.748564i	$0.269256\pi$
$313$			8290.40i			1.49713i	−0.663062	+	0.748564i	$0.730744\pi$
$314$	−10700.8			−1.92319
$315$		0			0
$316$	−4106.53			−0.731046
$317$			2882.26i			0.510675i	0.966852	+	0.255338i	$0.0821866\pi$
$317$			2882.26i			0.510675i	−0.966852	+	0.255338i	$0.917813\pi$
$318$		0			0
$319$	1731.50			0.303904
$320$	3451.41	−	3749.52i	0.602936	−	0.655014i
$321$		0			0
$322$			17239.8i			2.98366i
$323$		−	6549.53i		−	1.12825i
$324$		0			0
$325$	4623.34	−	383.463i	0.789098	−	0.0654483i
$326$	6006.71			1.02049
$327$		0			0
$328$			10410.0i			1.75243i
$329$	−7783.48			−1.30431
$330$		0			0
$331$	9505.77			1.57850			0.789251	−	0.614071i	$-0.210469\pi$
$331$	9505.77			1.57850			0.789251	+	0.614071i	$0.210469\pi$
$332$			3819.27i			0.631355i
$333$		0			0
$334$	17625.8			2.88754
$335$	−146.655	−	134.995i	−0.0239183	−	0.0220166i
$336$		0			0
$337$		−	5929.45i		−	0.958451i	−0.877692	−	0.479225i	$-0.840918\pi$
$337$		−	5929.45i		−	0.958451i	0.877692	−	0.479225i	$-0.159082\pi$
$338$		−	4038.38i		−	0.649878i
$339$		0			0
$340$	−13750.3	−	12657.0i	−2.19328	−	2.01890i
$341$	310.109			0.0492473
$342$		0			0
$343$			18217.2i			2.86775i
$344$	4425.25			0.693586
$345$		0			0
$346$	11692.5			1.81675
$347$			4199.08i			0.649621i	0.945779	+	0.324811i	$0.105301\pi$
$347$			4199.08i			0.649621i	−0.945779	+	0.324811i	$0.894699\pi$
$348$		0			0
$349$	4490.59			0.688756			0.344378	−	0.938831i	$-0.388090\pi$
$349$	4490.59			0.688756			0.344378	+	0.938831i	$0.388090\pi$
$350$	1770.75	+	21349.7i	0.270431	+	3.26053i
$351$		0			0
$352$		−	279.316i		−	0.0422943i
$353$		−	3992.72i		−	0.602014i	−0.953622	−	0.301007i	$-0.902677\pi$
$353$		−	3992.72i		−	0.602014i	0.953622	−	0.301007i	$-0.0973227\pi$
$354$		0			0
$355$	7514.93	−	8164.02i	1.12352	−	1.22057i
$356$	−6603.62			−0.983121
$357$		0			0
$358$		−	3817.73i		−	0.563613i
$359$	1630.18			0.239659			0.119830	−	0.992794i	$-0.461765\pi$
$359$	1630.18			0.239659			0.119830	+	0.992794i	$0.461765\pi$
$360$		0			0
$361$	−2790.37			−0.406818
$362$			6159.51i			0.894301i
$363$		0			0
$364$	−21014.8			−3.02603
$365$	−6329.49	−	5826.25i	−0.907673	−	0.835507i
$366$		0			0
$367$			1402.36i			0.199462i	0.995014	+	0.0997312i	$0.0317983\pi$
$367$			1402.36i			0.199462i	−0.995014	+	0.0997312i	$0.968202\pi$
$368$		−	7120.48i		−	1.00864i
$369$		0			0
$370$	−6964.59	+	7566.15i	−0.978572	+	1.06310i
$371$	1409.18			0.197200
$372$		0			0
$373$			4988.43i			0.692469i	0.938148	+	0.346235i	$0.112540\pi$
$373$			4988.43i			0.692469i	−0.938148	+	0.346235i	$0.887460\pi$
$374$	6303.41			0.871502
$375$		0			0
$376$	9129.51			1.25218
$377$			5158.05i			0.704651i
$378$		0			0
$379$	−9475.33			−1.28421			−0.642104	−	0.766617i	$-0.721938\pi$
$379$	−9475.33			−1.28421			−0.642104	+	0.766617i	$0.721938\pi$
$380$	7862.69	−	8541.82i	1.06144	−	1.15312i
$381$		0			0
$382$		−	10549.8i		−	1.41302i
$383$		−	7172.68i		−	0.956937i	−0.878105	−	0.478469i	$-0.841192\pi$
$383$		−	7172.68i		−	0.956937i	0.878105	−	0.478469i	$-0.158808\pi$
$384$		0			0
$385$	−3564.56	−	3281.15i	−0.471861	−	0.434345i
$386$	9016.47			1.18893
$387$		0			0
$388$		−	17383.3i		−	2.27449i
$389$	64.9700			0.00846815			0.00423407	−	0.999991i	$-0.498652\pi$
$389$	64.9700			0.00846815			0.00423407	+	0.999991i	$0.498652\pi$
$390$		0			0
$391$	10328.8			1.33593
$392$		−	35360.8i		−	4.55610i
$393$		0			0
$394$	5628.25			0.719663
$395$	1910.03	−	2075.01i	0.243302	−	0.264317i
$396$		0			0
$397$			9638.18i			1.21845i	0.792996	+	0.609227i	$0.208520\pi$
$397$			9638.18i			1.21845i	−0.792996	+	0.609227i	$0.791480\pi$
$398$			1399.77i			0.176292i
$399$		0			0
$400$	−731.367	−	8817.95i	−0.0914209	−	1.10224i
$401$	−7130.33			−0.887959			−0.443979	−	0.896037i	$-0.646434\pi$
$401$	−7130.33			−0.887959			−0.443979	+	0.896037i	$0.646434\pi$
$402$		0			0
$403$			923.801i			0.114188i
$404$	23774.7			2.92781
$405$		0			0
$406$	−23818.9			−2.91160
$407$		−	2325.63i		−	0.283236i
$408$		0			0
$409$	950.932			0.114965			0.0574824	−	0.998347i	$-0.481693\pi$
$409$	950.932			0.114965			0.0574824	+	0.998347i	$0.481693\pi$
$410$	−10342.7	−	9520.38i	−1.24583	−	1.14678i
$411$		0			0
$412$		−	24477.0i		−	2.92693i
$413$		−	4575.54i		−	0.545151i
$414$		0			0
$415$	−1929.86	−	1776.42i	−0.228272	−	0.210123i
$416$	832.069			0.0980662
$417$		0			0
$418$			3915.75i			0.458195i
$419$	16381.7			1.91002			0.955008	−	0.296579i	$-0.0958457\pi$
$419$	16381.7			1.91002			0.955008	+	0.296579i	$0.0958457\pi$
$420$		0			0
$421$	−408.506			−0.0472907			−0.0236453	−	0.999720i	$-0.507527\pi$
$421$	−408.506			−0.0472907			−0.0236453	+	0.999720i	$0.507527\pi$
$422$		−	4006.36i		−	0.462148i
$423$		0			0
$424$	−1652.88			−0.189318
$425$	12791.1	−	1060.90i	1.45990	−	0.121085i
$426$		0			0
$427$			11692.6i			1.32516i
$428$			30321.3i			3.42438i
$429$		0			0
$430$	−4047.07	+	4396.63i	−0.453876	+	0.493080i
$431$	8202.47			0.916703			0.458352	−	0.888771i	$-0.348440\pi$
$431$	8202.47			0.916703			0.458352	+	0.888771i	$0.348440\pi$
$432$		0			0
$433$			1435.71i			0.159343i	0.996821	+	0.0796716i	$0.0253872\pi$
$433$			1435.71i			0.159343i	−0.996821	+	0.0796716i	$0.974613\pi$
$434$	−4265.93			−0.471823
$435$		0			0
$436$	−24240.9			−2.66268
$437$			6416.34i			0.702369i
$438$		0			0
$439$	13629.3			1.48176			0.740880	−	0.671637i	$-0.234409\pi$
$439$	13629.3			1.48176			0.740880	+	0.671637i	$0.234409\pi$
$440$	4180.99	+	3848.57i	0.453002	+	0.416985i
$441$		0			0
$442$			18777.6i			2.02072i
$443$			8832.42i			0.947271i	0.880721	+	0.473636i	$0.157059\pi$
$443$			8832.42i			0.947271i	−0.880721	+	0.473636i	$0.842941\pi$
$444$		0			0
$445$	3071.48	−	3336.77i	0.327195	−	0.355457i
$446$	−24323.9			−2.58244
$447$		0			0
$448$		−	15854.0i		−	1.67195i
$449$	−11128.7			−1.16970			−0.584848	−	0.811143i	$-0.698846\pi$
$449$	−11128.7			−1.16970			−0.584848	+	0.811143i	$0.698846\pi$
$450$		0			0
$451$	3179.06			0.331921
$452$			15247.7i			1.58670i
$453$		0			0
$454$	−17688.8			−1.82859
$455$	9774.40	−	10618.7i	1.00710	−	1.09409i
$456$		0			0
$457$		−	16984.0i		−	1.73846i	−0.494405	−	0.869231i	$-0.664614\pi$
$457$		−	16984.0i		−	1.73846i	0.494405	−	0.869231i	$-0.335386\pi$
$458$		−	19068.3i		−	1.94542i
$459$		0			0
$460$	13470.7	+	12399.7i	1.36538	+	1.25682i
$461$	−15698.3			−1.58600			−0.792999	−	0.609223i	$-0.791481\pi$
$461$	−15698.3			−1.58600			−0.792999	+	0.609223i	$0.791481\pi$
$462$		0			0
$463$		−	7996.32i		−	0.802636i	−0.915939	−	0.401318i	$-0.868552\pi$
$463$		−	7996.32i		−	0.802636i	0.915939	−	0.401318i	$-0.131448\pi$
$464$	9837.80			0.984285
$465$		0			0
$466$	14699.9			1.46129
$467$			6347.66i			0.628982i	0.949260	+	0.314491i	$0.101834\pi$
$467$			6347.66i			0.628982i	−0.949260	+	0.314491i	$0.898166\pi$
$468$		0			0
$469$	−620.099			−0.0610523
$470$	−8349.30	+	9070.46i	−0.819413	+	0.890190i
$471$		0			0
$472$			5366.80i			0.523363i
$473$		−	1351.40i		−	0.131369i
$474$		0			0
$475$	659.042	+	7945.95i	0.0636609	+	0.767547i
$476$	−58140.1			−5.59842
$477$		0			0
$478$		−	12947.2i		−	1.23889i
$479$	−5794.01			−0.552683			−0.276341	−	0.961060i	$-0.589122\pi$
$479$	−5794.01			−0.552683			−0.276341	+	0.961060i	$0.589122\pi$
$480$		0			0
$481$	6927.94			0.656729
$482$			20874.7i			1.97265i
$483$		0			0
$484$	19141.1			1.79763
$485$	8783.68	+	8085.31i	0.822363	+	0.756980i
$486$		0			0
$487$		−	10363.7i		−	0.964323i	−0.876082	−	0.482162i	$-0.839852\pi$
$487$		−	10363.7i		−	0.964323i	0.876082	−	0.482162i	$-0.160148\pi$
$488$		−	13714.6i		−	1.27219i
$489$		0			0
$490$	35132.1	+	32338.9i	3.23899	+	2.98147i
$491$	21420.5			1.96882			0.984412	−	0.175878i	$-0.0562766\pi$
$491$	21420.5			1.96882			0.984412	+	0.175878i	$0.0562766\pi$
$492$		0			0
$493$			14270.4i			1.30367i
$494$	−11664.8			−1.06240
$495$		0			0
$496$	1761.94			0.159503
$497$		−	34519.8i		−	3.11554i
$498$		0			0
$499$	−1661.78			−0.149081			−0.0745404	−	0.997218i	$-0.523749\pi$
$499$	−1661.78			−0.149081			−0.0745404	+	0.997218i	$0.523749\pi$
$500$	17955.6	+	13972.0i	1.60600	+	1.24970i
$501$		0			0
$502$			31034.6i			2.75925i
$503$		−	13256.5i		−	1.17510i	−0.809187	−	0.587551i	$-0.800092\pi$
$503$		−	13256.5i		−	1.17510i	0.809187	−	0.587551i	$-0.199908\pi$
$504$		0			0
$505$	−11058.1	+	12013.2i	−0.974412	+	1.05858i
$506$	−6175.23			−0.542535
$507$		0			0
$508$			4643.70i			0.405573i
$509$	−13818.0			−1.20329			−0.601643	−	0.798765i	$-0.705487\pi$
$509$	−13818.0			−1.20329			−0.601643	+	0.798765i	$0.705487\pi$
$510$		0			0
$511$	−26762.9			−2.31687
$512$			21515.6i			1.85715i
$513$		0			0
$514$	14950.4			1.28295
$515$	12368.1	+	11384.7i	1.05826	+	0.974120i
$516$		0			0
$517$		−	2788.01i		−	0.237170i
$518$			31991.8i			2.71359i
$519$		0			0
$520$	−11464.7	+	12455.0i	−0.966849	+	1.05036i
$521$	11817.3			0.993718			0.496859	−	0.867831i	$-0.334487\pi$
$521$	11817.3			0.993718			0.496859	+	0.867831i	$0.334487\pi$
$522$		0			0
$523$		−	3358.57i		−	0.280803i	−0.990095	−	0.140401i	$-0.955161\pi$
$523$		−	3358.57i		−	0.280803i	0.990095	−	0.140401i	$-0.0448393\pi$
$524$	11353.5			0.946526
$525$		0			0
$526$	25682.2			2.12889
$527$			2555.81i			0.211258i
$528$		0			0
$529$	2048.27			0.168346
$530$	1511.62	−	1642.19i	0.123888	−	0.134589i
$531$		0			0
$532$		−	36117.3i		−	2.94339i
$533$			9470.29i			0.769613i
$534$		0			0
$535$	−15321.2	−	14103.0i	−1.23812	−	1.13968i
$536$	727.335			0.0586121
$537$		0			0
$538$		−	9121.23i		−	0.730937i
$539$	−10798.7			−0.862952
$540$		0			0
$541$	−1111.42			−0.0883243			−0.0441622	−	0.999024i	$-0.514062\pi$
$541$	−1111.42			−0.0883243			−0.0441622	+	0.999024i	$0.514062\pi$
$542$		−	4546.78i		−	0.360334i
$543$		0			0
$544$	2302.03			0.181431
$545$	11274.9	−	12248.8i	0.886174	−	0.962717i
$546$		0			0
$547$			4216.48i			0.329586i	0.986328	+	0.164793i	$0.0526956\pi$
$547$			4216.48i			0.329586i	−0.986328	+	0.164793i	$0.947304\pi$
$548$		−	25361.2i		−	1.97697i
$549$		0			0
$550$	−7647.36	+	634.277i	−0.592881	+	0.0491740i
$551$	−8864.94			−0.685407
$552$		0			0
$553$		−	8773.73i		−	0.674678i
$554$	−9521.04			−0.730163
$555$		0			0
$556$	−5133.33			−0.391549
$557$			1403.06i			0.106732i	0.998575	+	0.0533658i	$0.0169949\pi$
$557$			1403.06i			0.106732i	−0.998575	+	0.0533658i	$0.983005\pi$
$558$		0			0
$559$	4025.77			0.304601
$560$	−20252.6	−	18642.4i	−1.52827	−	1.40676i
$561$		0			0
$562$		−	34590.6i		−	2.59629i
$563$			5088.40i			0.380907i	0.981696	+	0.190453i	$0.0609958\pi$
$563$			5088.40i			0.380907i	−0.981696	+	0.190453i	$0.939004\pi$
$564$		0			0
$565$	−7704.56	−	7091.99i	−0.573687	−	0.528075i
$566$	7377.10			0.547849
$567$		0			0
$568$			40489.5i			2.99102i
$569$	8310.91			0.612323			0.306161	−	0.951980i	$-0.400955\pi$
$569$	8310.91			0.612323			0.306161	+	0.951980i	$0.400955\pi$
$570$		0			0
$571$	−24264.2			−1.77833			−0.889166	−	0.457585i	$-0.848714\pi$
$571$	−24264.2			−1.77833			−0.889166	+	0.457585i	$0.848714\pi$
$572$		−	7527.41i		−	0.550239i
$573$		0			0
$574$	−43731.9			−3.18002
$575$	−12531.0	+	1039.33i	−0.908830	+	0.0753790i
$576$		0			0
$577$		−	2740.03i		−	0.197693i	−0.995103	−	0.0988464i	$-0.968485\pi$
$577$		−	2740.03i		−	0.197693i	0.995103	−	0.0988464i	$-0.0315153\pi$
$578$			27742.2i			1.99641i
$579$		0			0
$580$	−17131.6	+	18611.3i	−1.22647	+	1.33240i
$581$	−8159.99			−0.582674
$582$		0			0
$583$			504.763i			0.0358579i
$584$	31391.1			2.22427
$585$		0			0
$586$	34510.9			2.43282
$587$		−	23433.9i		−	1.64773i	−0.566784	−	0.823867i	$-0.691812\pi$
$587$		−	23433.9i		−	1.64773i	0.566784	−	0.823867i	$-0.308188\pi$
$588$		0			0
$589$	−1587.70			−0.111070
$590$	−5332.09	−	4908.15i	−0.372066	−	0.342484i
$591$		0			0
$592$		−	13213.4i		−	0.917347i
$593$			18399.2i			1.27414i	0.770807	+	0.637068i	$0.219853\pi$
$593$			18399.2i			1.27414i	−0.770807	+	0.637068i	$0.780147\pi$
$594$		0			0
$595$	27042.1	−	29377.9i	1.86323	−	2.02416i
$596$	−32293.1			−2.21943
$597$		0			0
$598$		−	18395.7i		−	1.25796i
$599$	−20464.1			−1.39589			−0.697946	−	0.716151i	$-0.745902\pi$
$599$	−20464.1			−1.39589			−0.697946	+	0.716151i	$0.745902\pi$
$600$		0			0
$601$	20627.9			1.40005			0.700023	−	0.714120i	$-0.253173\pi$
$601$	20627.9			1.40005			0.700023	+	0.714120i	$0.253173\pi$
$602$			18590.2i			1.25860i
$603$		0			0
$604$	−17211.5			−1.15948
$605$	−8902.93	+	9671.91i	−0.598274	+	0.649949i
$606$		0			0
$607$			4733.11i			0.316492i	0.987400	+	0.158246i	$0.0505840\pi$
$607$			4733.11i			0.316492i	−0.987400	+	0.158246i	$0.949416\pi$
$608$			1430.04i			0.0953881i
$609$		0			0
$610$	13625.9	+	12542.5i	0.904420	+	0.832512i
$611$	8305.36			0.549917
$612$		0			0
$613$		−	13741.6i		−	0.905414i	−0.891659	−	0.452707i	$-0.850458\pi$
$613$		−	13741.6i		−	0.905414i	0.891659	−	0.452707i	$-0.149542\pi$
$614$	−6391.32			−0.420086
$615$		0			0
$616$	17678.4			1.15631
$617$			18762.1i			1.22420i	0.790780	+	0.612100i	$0.209675\pi$
$617$			18762.1i			1.22420i	−0.790780	+	0.612100i	$0.790325\pi$
$618$		0			0
$619$	−8550.03			−0.555177			−0.277589	−	0.960700i	$-0.589535\pi$
$619$	−8550.03			−0.555177			−0.277589	+	0.960700i	$0.589535\pi$
$620$	−3068.25	+	3333.27i	−0.198748	+	0.215915i
$621$		0			0
$622$		−	15143.2i		−	0.976186i
$623$		−	14108.8i		−	0.907317i
$624$		0			0
$625$	−15411.5	+	2574.19i	−0.986336	+	0.164748i
$626$	40850.3			2.60816
$627$		0			0
$628$			35353.9i			2.24646i
$629$	19167.0			1.21501
$630$		0			0
$631$	−11772.7			−0.742730			−0.371365	−	0.928487i	$-0.621110\pi$
$631$	−11772.7			−0.742730			−0.371365	+	0.928487i	$0.621110\pi$
$632$			10291.0i			0.647713i
$633$		0			0
$634$	14202.1			0.889651
$635$	−2346.44	−	2159.88i	−0.146639	−	0.134980i
$636$		0			0
$637$		−	32168.7i		−	2.00090i
$638$		−	8531.82i		−	0.529433i
$639$		0			0
$640$	−19950.8	−	18364.6i	−1.23223	−	1.13426i
$641$	13219.7			0.814585			0.407292	−	0.913298i	$-0.366473\pi$
$641$	13219.7			0.814585			0.407292	+	0.913298i	$0.366473\pi$
$642$		0			0
$643$			29569.6i			1.81355i	0.421617	+	0.906774i	$0.361463\pi$
$643$			29569.6i			1.81355i	−0.421617	+	0.906774i	$0.638537\pi$
$644$	56957.8			3.48518
$645$		0			0
$646$	−32272.3			−1.96554
$647$		−	16887.3i		−	1.02613i	−0.858349	−	0.513066i	$-0.828510\pi$
$647$		−	16887.3i		−	1.02613i	0.858349	−	0.513066i	$-0.171490\pi$
$648$		0			0
$649$	1638.94			0.0991279
$650$	−1889.48	−	22781.1i	−0.114018	−	1.37469i
$651$		0			0
$652$		−	19845.3i		−	1.19203i
$653$		−	20847.4i		−	1.24934i	−0.780888	−	0.624671i	$-0.785233\pi$
$653$		−	20847.4i		−	1.24934i	0.780888	−	0.624671i	$-0.214767\pi$
$654$		0			0
$655$	−5280.74	+	5736.86i	−0.315016	+	0.342225i
$656$	18062.4			1.07503
$657$		0			0
$658$			38352.5i			2.27224i
$659$	−21697.2			−1.28255			−0.641277	−	0.767309i	$-0.721595\pi$
$659$	−21697.2			−1.28255			−0.641277	+	0.767309i	$0.721595\pi$
$660$		0			0
$661$	−2987.43			−0.175790			−0.0878952	−	0.996130i	$-0.528014\pi$
$661$	−2987.43			−0.175790			−0.0878952	+	0.996130i	$0.528014\pi$
$662$		−	46838.9i		−	2.74992i
$663$		0			0
$664$	9571.13			0.559385
$665$	18249.9	+	16798.9i	1.06421	+	0.979597i
$666$		0			0
$667$		−	13980.2i		−	0.811570i
$668$		−	58233.0i		−	3.37291i
$669$		0			0
$670$	−665.177	+	722.631i	−0.0383553	+	0.0416682i
$671$	−4188.23			−0.240961
$672$		0			0
$673$			23532.4i			1.34785i	0.738798	+	0.673927i	$0.235394\pi$
$673$			23532.4i			1.34785i	−0.738798	+	0.673927i	$0.764606\pi$
$674$	−29216.9			−1.66972
$675$		0			0
$676$	−13342.2			−0.759116
$677$			18598.3i			1.05582i	0.849300	+	0.527911i	$0.177025\pi$
$677$			18598.3i			1.05582i	−0.849300	+	0.527911i	$0.822975\pi$
$678$		0			0
$679$	37139.9			2.09911
$680$	−31718.7	+	34458.3i	−1.78876	+	1.94326i
$681$		0			0
$682$		−	1528.04i		−	0.0857941i
$683$			474.740i			0.0265965i	0.999912	+	0.0132983i	$0.00423309\pi$
$683$			474.740i			0.0265965i	−0.999912	+	0.0132983i	$0.995767\pi$
$684$		0			0
$685$	12814.9	+	11796.0i	0.714792	+	0.657961i
$686$	89764.0			4.99592
$687$		0			0
$688$		−	7678.23i		−	0.425479i
$689$	−1503.67			−0.0831425
$690$		0			0
$691$	1586.50			0.0873418			0.0436709	−	0.999046i	$-0.486095\pi$
$691$	1586.50			0.0873418			0.0436709	+	0.999046i	$0.486095\pi$
$692$		−	38630.4i		−	2.12212i
$693$		0			0
$694$	20690.6			1.13171
$695$	2387.61	−	2593.84i	0.130313	−	0.141568i
$696$		0			0
$697$			26200.8i			1.42385i
$698$		−	22127.0i		−	1.19989i
$699$		0			0
$700$	70536.2	−	5850.32i	3.80859	−	0.315887i
$701$	12342.4			0.664999			0.332499	−	0.943103i	$-0.392108\pi$
$701$	12342.4			0.664999			0.332499	+	0.943103i	$0.392108\pi$
$702$		0			0
$703$			11906.8i			0.638794i
$704$	5678.85			0.304020
$705$		0			0
$706$	−19673.8			−1.04877
$707$			50795.3i			2.70206i
$708$		0			0
$709$	−27714.9			−1.46806			−0.734031	−	0.679116i	$-0.762363\pi$
$709$	−27714.9			−1.46806			−0.734031	+	0.679116i	$0.762363\pi$
$710$	−40227.6	−	37029.2i	−2.12636	−	1.95730i
$711$		0			0
$712$			16548.7i			0.871053i
$713$		−	2503.84i		−	0.131514i
$714$		0			0
$715$	3803.56	+	3501.15i	0.198944	+	0.183127i
$716$	−12613.2			−0.658350
$717$		0			0
$718$		−	8032.60i		−	0.417512i
$719$	−29126.8			−1.51077			−0.755387	−	0.655279i	$-0.772551\pi$
$719$	−29126.8			−1.51077			−0.755387	+	0.655279i	$0.772551\pi$
$720$		0			0
$721$	52295.8			2.70124
$722$			13749.3i			0.708721i
$723$		0			0
$724$	20350.1			1.04462
$725$	−1435.95	−	17313.0i	−0.0735586	−	0.886882i
$726$		0			0
$727$			32102.8i			1.63772i	0.573990	+	0.818862i	$0.305395\pi$
$727$			32102.8i			1.63772i	−0.573990	+	0.818862i	$0.694605\pi$
$728$			52663.2i			2.68108i
$729$		0			0
$730$	−28708.4	+	31188.1i	−1.45554	+	1.58126i
$731$	11137.8			0.563539
$732$		0			0
$733$			18658.3i			0.940190i	0.882616	+	0.470095i	$0.155780\pi$
$733$			18658.3i			0.940190i	−0.882616	+	0.470095i	$0.844220\pi$
$734$	6910.03			0.347485
$735$		0			0
$736$	−2255.22			−0.112946
$737$		−	222.117i		−	0.0111015i
$738$		0			0
$739$	−19105.8			−0.951040			−0.475520	−	0.879705i	$-0.657740\pi$
$739$	−19105.8			−0.951040			−0.475520	+	0.879705i	$0.657740\pi$
$740$	24997.5	+	23010.0i	1.24179	+	1.14306i
$741$		0			0
$742$		−	6943.63i		−	0.343543i
$743$			34177.1i			1.68753i	0.536712	+	0.843765i	$0.319666\pi$
$743$			34177.1i			1.68753i	−0.536712	+	0.843765i	$0.680334\pi$
$744$		0			0
$745$	15020.2	−	16317.5i	0.738654	−	0.802455i
$746$	24580.1			1.20636
$747$		0			0
$748$		−	20825.6i		−	1.01799i
$749$	−64782.2			−3.16034
$750$		0			0
$751$	−25906.0			−1.25875			−0.629376	−	0.777101i	$-0.716690\pi$
$751$	−25906.0			−1.25875			−0.629376	+	0.777101i	$0.716690\pi$
$752$		−	15840.6i		−	0.768146i
$753$		0			0
$754$	25415.9			1.22758
$755$	8005.42	−	8696.88i	0.385890	−	0.419221i
$756$		0			0
$757$		−	12734.0i		−	0.611391i	−0.952129	−	0.305696i	$-0.901111\pi$
$757$		−	12734.0i		−	0.611391i	0.952129	−	0.305696i	$-0.0988890\pi$
$758$			46689.0i			2.23723i
$759$		0			0
$760$	−21405.9	−	19704.0i	−1.02167	−	0.940444i
$761$	26680.6			1.27092			0.635461	−	0.772133i	$-0.280810\pi$
$761$	26680.6			1.27092			0.635461	+	0.772133i	$0.280810\pi$
$762$		0			0
$763$		−	51791.4i		−	2.45737i
$764$	−34854.9			−1.65053
$765$		0			0
$766$	−35342.8			−1.66709
$767$			4882.33i			0.229844i
$768$		0			0
$769$	15123.8			0.709203			0.354602	−	0.935017i	$-0.384616\pi$
$769$	15123.8			0.709203			0.354602	+	0.935017i	$0.384616\pi$
$770$	−16167.6	+	17564.1i	−0.756676	+	0.822033i
$771$		0			0
$772$		−	29789.1i		−	1.38877i
$773$			32394.6i			1.50731i	0.657269	+	0.753656i	$0.271712\pi$
$773$			32394.6i			1.50731i	−0.657269	+	0.753656i	$0.728288\pi$
$774$		0			0
$775$	−257.177	−	3100.74i	−0.0119201	−	0.143718i
$776$	−43562.6			−2.01522
$777$		0			0
$778$		−	320.135i		−	0.0147524i
$779$	−16276.2			−0.748595
$780$		0			0
$781$	12364.9			0.566517
$782$		−	50894.2i		−	2.32733i
$783$		0			0
$784$	−61354.4			−2.79493
$785$	−17864.1	−	16443.8i	−0.812228	−	0.747650i
$786$		0			0
$787$			13328.5i			0.603699i	0.953356	+	0.301849i	$0.0976040\pi$
$787$			13328.5i			0.603699i	−0.953356	+	0.301849i	$0.902396\pi$
$788$		−	18594.9i		−	0.840630i
$789$		0			0
$790$	−10224.4	−	9411.53i	−0.460468	−	0.423857i
$791$	−32577.1			−1.46436
$792$		0			0
$793$		−	12476.5i		−	0.558707i
$794$	47491.4			2.12268
$795$		0			0
$796$	4624.64			0.205925
$797$			3904.61i			0.173536i	0.996229	+	0.0867681i	$0.0276539\pi$
$797$			3904.61i			0.173536i	−0.996229	+	0.0867681i	$0.972346\pi$
$798$		0			0
$799$	22977.9			1.01739
$800$	−2792.84	+	231.640i	−0.123427	+	0.0102371i
$801$		0			0
$802$			35134.1i			1.54692i
$803$		−	9586.36i		−	0.421289i
$804$		0			0
$805$	−26492.2	+	28780.5i	−1.15991	+	1.26010i
$806$	4551.96			0.198928
$807$		0			0
$808$		−	59579.6i		−	2.59406i
$809$	27021.7			1.17433			0.587165	−	0.809467i	$-0.300244\pi$
$809$	27021.7			1.17433			0.587165	+	0.809467i	$0.300244\pi$
$810$		0			0
$811$	13873.1			0.600679			0.300340	−	0.953832i	$-0.402900\pi$
$811$	13873.1			0.600679			0.300340	+	0.953832i	$0.402900\pi$
$812$			78694.0i			3.40101i
$813$		0			0
$814$	−11459.3			−0.493427
$815$	10027.7	+	9230.44i	0.430988	+	0.396722i
$816$		0			0
$817$			6918.93i			0.296282i
$818$		−	4685.65i		−	0.200281i
$819$		0			0
$820$	−31454.0	+	34170.8i	−1.33954	+	1.45524i
$821$	−11284.2			−0.479684			−0.239842	−	0.970812i	$-0.577096\pi$
$821$	−11284.2			−0.479684			−0.239842	+	0.970812i	$0.577096\pi$
$822$		0			0
$823$		−	30236.7i		−	1.28066i	−0.768098	−	0.640332i	$-0.778797\pi$
$823$		−	30236.7i		−	1.28066i	0.768098	−	0.640332i	$-0.221203\pi$
$824$	−61339.5			−2.59328
$825$		0			0
$826$	−22545.6			−0.949712
$827$			2610.71i			0.109774i	0.998493	+	0.0548871i	$0.0174799\pi$
$827$			2610.71i			0.109774i	−0.998493	+	0.0548871i	$0.982520\pi$
$828$		0			0
$829$	−43070.0			−1.80444			−0.902222	−	0.431272i	$-0.858065\pi$
$829$	−43070.0			−1.80444			−0.902222	+	0.431272i	$0.858065\pi$
$830$	−8753.18	+	9509.23i	−0.366057	+	0.397675i
$831$		0			0
$832$			16917.0i			0.704920i
$833$		−	88998.9i		−	3.70183i
$834$		0			0
$835$	29424.8	+	27085.3i	1.21951	+	1.12255i
$836$	12937.1			0.535212
$837$		0			0
$838$		−	80719.4i		−	3.32745i
$839$	19767.0			0.813388			0.406694	−	0.913564i	$-0.366681\pi$
$839$	19767.0			0.813388			0.406694	+	0.913564i	$0.366681\pi$
$840$		0			0
$841$	−5073.64			−0.208030
$842$			2012.88i			0.0823854i
$843$		0			0
$844$	−13236.4			−0.539831
$845$	6205.73	−	6741.75i	0.252643	−	0.274465i
$846$		0			0
$847$			40895.6i			1.65902i
$848$			2867.90i			0.116137i
$849$		0			0
$850$	−5227.50	−	63027.0i	−0.210943	−	2.54330i
$851$	−18777.3			−0.756377
$852$		0			0
$853$			28437.0i			1.14146i	0.821139	+	0.570729i	$0.193339\pi$
$853$			28437.0i			1.14146i	−0.821139	+	0.570729i	$0.806661\pi$
$854$	57614.1			2.30857
$855$		0			0
$856$	75985.3			3.03402
$857$		−	45202.3i		−	1.80173i	−0.434101	−	0.900864i	$-0.642934\pi$
$857$		−	45202.3i		−	1.80173i	0.434101	−	0.900864i	$-0.357066\pi$
$858$		0			0
$859$	−39304.1			−1.56116			−0.780582	−	0.625054i	$-0.785077\pi$
$859$	−39304.1			−1.56116			−0.780582	+	0.625054i	$0.785077\pi$
$860$	14525.8	+	13370.9i	0.575961	+	0.530168i
$861$		0			0
$862$		−	40417.0i		−	1.59699i
$863$			41102.8i			1.62127i	0.585552	+	0.810635i	$0.300878\pi$
$863$			41102.8i			1.62127i	−0.585552	+	0.810635i	$0.699122\pi$
$864$		0			0
$865$	19519.7	+	17967.8i	0.767273	+	0.706270i
$866$	7074.33			0.277593
$867$		0			0
$868$			14094.0i			0.551131i
$869$	3142.72			0.122680
$870$		0			0
$871$	661.677			0.0257406
$872$			60747.9i			2.35916i
$873$		0			0
$874$	31616.0			1.22360
$875$	−29851.7	+	38362.6i	−1.15334	+	1.48216i
$876$		0			0
$877$		−	48649.4i		−	1.87317i	−0.350436	−	0.936587i	$-0.613967\pi$
$877$		−	48649.4i		−	1.87317i	0.350436	−	0.936587i	$-0.386033\pi$
$878$		−	67157.5i		−	2.58139i
$879$		0			0
$880$	6677.64	−	7254.42i	0.255799	−	0.277894i
$881$	1470.17			0.0562217			0.0281108	−	0.999605i	$-0.491051\pi$
$881$	1470.17			0.0562217			0.0281108	+	0.999605i	$0.491051\pi$
$882$		0			0
$883$			103.543i			0.00394622i	0.999998	+	0.00197311i	$0.000628061\pi$
$883$			103.543i			0.00394622i	−0.999998	+	0.00197311i	$0.999372\pi$
$884$	62038.4			2.36038
$885$		0			0
$886$	43521.1			1.65025
$887$			21162.1i			0.801074i	0.916281	+	0.400537i	$0.131176\pi$
$887$			21162.1i			0.801074i	−0.916281	+	0.400537i	$0.868824\pi$
$888$		0			0
$889$	−9921.41			−0.374301
$890$	−16441.7	−	15134.5i	−0.619244	−	0.570009i
$891$		0			0
$892$			80362.5i			3.01652i
$893$			14274.1i			0.534898i
$894$		0			0
$895$	5866.67	−	6373.39i	0.219107	−	0.238033i
$896$	−84357.8			−3.14531
$897$		0			0
$898$			54835.6i			2.03774i
$899$	3459.36			0.128338
$900$		0			0
$901$	−4160.09			−0.153821
$902$		−	15664.6i		−	0.578242i
$903$		0			0
$904$	38210.8			1.40583
$905$	−9465.26	+	10282.8i	−0.347664	+	0.377693i
$906$		0			0
$907$			31376.9i			1.14868i	0.818617	+	0.574340i	$0.194741\pi$
$907$			31376.9i			1.14868i	−0.818617	+	0.574340i	$0.805259\pi$
$908$			58441.3i			2.13595i
$909$		0			0
$910$	−52322.6	−	48162.6i	−1.90602	−	1.75448i
$911$	9271.17			0.337176			0.168588	−	0.985687i	$-0.446079\pi$
$911$	9271.17			0.337176			0.168588	+	0.985687i	$0.446079\pi$
$912$		0			0
$913$		−	2922.88i		−	0.105951i
$914$	−83687.3			−3.02859
$915$		0			0
$916$	−62998.8			−2.27242
$917$			24257.1i			0.873544i
$918$		0			0
$919$	39558.3			1.41992			0.709962	−	0.704240i	$-0.248712\pi$
$919$	39558.3			1.41992			0.709962	+	0.704240i	$0.248712\pi$
$920$	31073.6	−	33757.6i	1.11355	−	1.20973i
$921$		0			0
$922$			77352.4i			2.76298i
$923$			36834.4i			1.31356i
$924$		0			0
$925$	−23253.6	+	1928.67i	−0.826568	+	0.0685561i
$926$	−39401.2			−1.39828
$927$		0			0
$928$		−	3115.85i		−	0.110219i
$929$	37330.3			1.31837			0.659186	−	0.751980i	$-0.270901\pi$
$929$	37330.3			1.31837			0.659186	+	0.751980i	$0.270901\pi$
$930$		0			0
$931$	55287.0			1.94625
$932$		−	48566.4i		−	1.70692i
$933$		0			0
$934$	31277.6			1.09575
$935$	10523.0	+	9686.39i	0.368065	+	0.338801i
$936$		0			0
$937$		−	32301.0i		−	1.12618i	−0.826397	−	0.563088i	$-0.809613\pi$
$937$		−	32301.0i		−	1.12618i	0.826397	−	0.563088i	$-0.190387\pi$
$938$			3055.49i			0.106360i
$939$		0			0
$940$	29967.5	+	27584.9i	1.03982	+	0.957148i
$941$	−11473.3			−0.397468			−0.198734	−	0.980053i	$-0.563683\pi$
$941$	−11473.3			−0.397468			−0.198734	+	0.980053i	$0.563683\pi$
$942$		0			0
$943$		−	25668.0i		−	0.886389i
$944$	9311.91			0.321056
$945$		0			0
$946$	−6658.94			−0.228859
$947$		−	34000.9i		−	1.16672i	−0.812215	−	0.583359i	$-0.801738\pi$
$947$		−	34000.9i		−	1.16672i	0.812215	−	0.583359i	$-0.198262\pi$
$948$		0			0
$949$	28557.3			0.976829
$950$	39153.0	−	3247.38i	1.33715	−	0.110904i
$951$		0			0
$952$			145700.i			4.96024i
$953$			2018.64i			0.0686149i	0.999411	+	0.0343075i	$0.0109225\pi$
$953$			2018.64i			0.0686149i	−0.999411	+	0.0343075i	$0.989077\pi$
$954$		0			0
$955$	16211.7	−	17612.0i	0.549318	−	0.596765i
$956$	−42775.6			−1.44713
$957$		0			0
$958$			28549.5i			0.962833i
$959$	54185.1			1.82453
$960$		0			0
$961$	−29171.4			−0.979203
$962$		−	34136.9i		−	1.14409i
$963$		0			0
$964$	68966.9			2.30423
$965$	15052.3	+	13855.5i	0.502124	+	0.462202i
$966$		0			0
$967$		−	18525.1i		−	0.616056i	−0.951377	−	0.308028i	$-0.900331\pi$
$967$		−	18525.1i		−	0.616056i	0.951377	−	0.308028i	$-0.0996690\pi$
$968$		−	47967.8i		−	1.59271i
$969$		0			0
$970$	39839.8	−	43280.9i	1.31874	−	1.43264i
$971$	−48076.1			−1.58891			−0.794457	−	0.607320i	$-0.792245\pi$
$971$	−48076.1			−1.58891			−0.794457	+	0.607320i	$0.792245\pi$
$972$		0			0
$973$		−	10967.5i		−	0.361359i
$974$	−51066.5			−1.67995
$975$		0			0
$976$	−23796.1			−0.780425
$977$		−	47135.6i		−	1.54350i	−0.635925	−	0.771751i	$-0.719381\pi$
$977$		−	47135.6i		−	1.54350i	0.635925	−	0.771751i	$-0.280619\pi$
$978$		0			0
$979$	5053.73			0.164982
$980$	106843.	−	116071.i	3.48262	−	3.78343i
$981$		0			0
$982$		−	105548.i		−	3.42990i
$983$			20180.8i			0.654799i	0.944886	+	0.327399i	$0.106172\pi$
$983$			20180.8i			0.654799i	−0.944886	+	0.327399i	$0.893828\pi$
$984$		0			0
$985$	9395.91	+	8648.87i	0.303938	+	0.279773i
$986$	70316.4			2.27113
$987$		0			0
$988$			38538.9i			1.24098i
$989$	−10911.3			−0.350819
$990$		0			0
$991$	−21110.9			−0.676701			−0.338351	−	0.941020i	$-0.609869\pi$
$991$	−21110.9			−0.676701			−0.338351	+	0.941020i	$0.609869\pi$
$992$		−	558.045i		−	0.0178608i
$993$		0			0
$994$	−170094.			−5.42761
$995$	−2151.01	+	2336.81i	−0.0685344	+	0.0744540i
$996$		0			0
$997$			11304.0i			0.359079i	0.983751	+	0.179540i	$0.0574608\pi$
$997$			11304.0i			0.359079i	−0.983751	+	0.179540i	$0.942539\pi$
$998$			8188.28i			0.259715i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	135.4.b.c.109.2	yes	12
3.2	odd	2	inner	135.4.b.c.109.11	yes	12
5.2	odd	4		675.4.a.bc.1.6		6
5.3	odd	4		675.4.a.bb.1.1		6
5.4	even	2	inner	135.4.b.c.109.12	yes	12
15.2	even	4		675.4.a.bc.1.1		6
15.8	even	4		675.4.a.bb.1.6		6
15.14	odd	2	inner	135.4.b.c.109.1	✓	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
135.4.b.c.109.1	✓	12	15.14	odd	2	inner
135.4.b.c.109.2	yes	12	1.1	even	1	trivial
135.4.b.c.109.11	yes	12	3.2	odd	2	inner
135.4.b.c.109.12	yes	12	5.4	even	2	inner
675.4.a.bb.1.1		6	5.3	odd	4
675.4.a.bb.1.6		6	15.8	even	4
675.4.a.bc.1.1		6	15.2	even	4
675.4.a.bc.1.6		6	5.2	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 $\F_{q}$
Belyi maps

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

\(f(q)\)	\(=\)	\(q-4.92742i q^{2} -16.2795 q^{4} +(7.57192 - 8.22594i) q^{5} -34.7816i q^{7} +40.7965i q^{8} +(-40.5327 - 37.3100i) q^{10} +12.4586 q^{11} +37.1137i q^{13} -171.384 q^{14} +70.7858 q^{16} +102.680i q^{17} -63.7858 q^{19} +(-123.267 + 133.914i) q^{20} -61.3890i q^{22} -100.592i q^{23} +(-10.3321 - 124.572i) q^{25} +182.875 q^{26} +566.227i q^{28} +138.980 q^{29} +24.8911 q^{31} -22.4195i q^{32} +505.947 q^{34} +(-286.111 - 263.363i) q^{35} -186.668i q^{37} +314.300i q^{38} +(335.590 + 308.908i) q^{40} +255.169 q^{41} -108.471i q^{43} -202.820 q^{44} -495.659 q^{46} -223.781i q^{47} -866.760 q^{49} +(-613.820 + 50.9107i) q^{50} -604.192i q^{52} +40.5151i q^{53} +(94.3358 - 102.484i) q^{55} +1418.97 q^{56} -684.812i q^{58} +131.550 q^{59} -336.171 q^{61} -122.649i q^{62} +455.817 q^{64} +(305.295 + 281.022i) q^{65} -17.8284i q^{67} -1671.58i q^{68} +(-1297.70 + 1409.79i) q^{70} +992.473 q^{71} -769.455i q^{73} -919.792 q^{74} +1038.40 q^{76} -433.331i q^{77} +252.252 q^{79} +(535.985 - 582.280i) q^{80} -1257.33i q^{82} -234.606i q^{83} +(844.639 + 777.484i) q^{85} -534.484 q^{86} +508.269i q^{88} +405.641 q^{89} +1290.87 q^{91} +1637.58i q^{92} -1102.67 q^{94} +(-482.981 + 524.698i) q^{95} +1067.80i q^{97} +4270.89i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 60 q^{4} - 36 q^{10} + 84 q^{16} - 348 q^{25} + 252 q^{31} + 1068 q^{34} + 1320 q^{40} - 1668 q^{46} - 2868 q^{49} + 684 q^{55} + 792 q^{61} + 2268 q^{64} - 5652 q^{70} + 1824 q^{76} + 2196 q^{79}+ \cdots - 5148 q^{94}+O(q^{100}) \) 12 * q - 60 * q^4 - 36 * q^10 + 84 * q^16 - 348 * q^25 + 252 * q^31 + 1068 * q^34 + 1320 * q^40 - 1668 * q^46 - 2868 * q^49 + 684 * q^55 + 792 * q^61 + 2268 * q^64 - 5652 * q^70 + 1824 * q^76 + 2196 * q^79 + 3816 * q^85 - 3384 * q^91 - 5148 * q^94

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