LMFDB - Embedded newform 135.4.b.c.109.6

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			109.6
Root			$-2.81133 + 2.81133i$ of defining polynomial
Character	$\chi$	$=$	135.109
Dual form			135.4.b.c.109.8

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.25118i q^{2} +6.43456 q^{4} +(9.04484 - 6.57198i) q^{5} +5.67029i q^{7} -18.0602i q^{8} +(-8.22271 - 11.3167i) q^{10} +21.2537 q^{11} +40.9479i q^{13} +7.09453 q^{14} +28.8800 q^{16} -59.1853i q^{17} -21.8800 q^{19} +(58.1995 - 42.2878i) q^{20} -26.5922i q^{22} -98.7941i q^{23} +(38.6182 - 118.885i) q^{25} +51.2331 q^{26} +36.4858i q^{28} +159.643 q^{29} -69.4873 q^{31} -180.615i q^{32} -74.0512 q^{34} +(37.2650 + 51.2868i) q^{35} +235.618i q^{37} +27.3757i q^{38} +(-118.691 - 163.351i) q^{40} -491.487 q^{41} -95.3080i q^{43} +136.758 q^{44} -123.609 q^{46} +548.796i q^{47} +310.848 q^{49} +(-148.746 - 48.3182i) q^{50} +263.482i q^{52} +509.829i q^{53} +(192.236 - 139.679i) q^{55} +102.406 q^{56} -199.742i q^{58} -741.098 q^{59} +387.157 q^{61} +86.9409i q^{62} +5.05795 q^{64} +(269.109 + 370.367i) q^{65} +1060.45i q^{67} -380.831i q^{68} +(64.1689 - 46.6251i) q^{70} -508.998 q^{71} -914.906i q^{73} +294.800 q^{74} -140.788 q^{76} +120.515i q^{77} -925.717 q^{79} +(261.215 - 189.799i) q^{80} +614.937i q^{82} +708.964i q^{83} +(-388.964 - 535.321i) q^{85} -119.247 q^{86} -383.846i q^{88} -273.516 q^{89} -232.186 q^{91} -635.696i q^{92} +686.641 q^{94} +(-197.901 + 143.795i) q^{95} +1450.22i q^{97} -388.926i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$		−	1.25118i		−	0.442358i	−0.975233	−	0.221179i	$-0.929010\pi$
$2$		−	1.25118i		−	0.442358i	0.975233	−	0.221179i	$-0.0709905\pi$
$3$		0			0
$3$		0			0
$4$	6.43456			0.804320
$5$	9.04484	−	6.57198i	0.808995	−	0.587816i
$5$	9.04484	−	6.57198i	0.808995	−	0.587816i
$6$		0			0
$7$			5.67029i			0.306167i	0.988213	+	0.153083i	$0.0489203\pi$
$7$			5.67029i			0.306167i	−0.988213	+	0.153083i	$0.951080\pi$
$8$		−	18.0602i		−	0.798155i
$9$		0			0
$10$	−8.22271	−	11.3167i	−0.260025	−	0.357865i
$11$	21.2537			0.582567			0.291283	−	0.956637i	$-0.405918\pi$
$11$	21.2537			0.582567			0.291283	+	0.956637i	$0.405918\pi$
$12$		0			0
$13$			40.9479i			0.873608i	0.899557	+	0.436804i	$0.143890\pi$
$13$			40.9479i			0.873608i	−0.899557	+	0.436804i	$0.856110\pi$
$14$	7.09453			0.135435
$15$		0			0
$16$	28.8800			0.451250
$17$		−	59.1853i		−	0.844384i	−0.906506	−	0.422192i	$-0.861261\pi$
$17$		−	59.1853i		−	0.844384i	0.906506	−	0.422192i	$-0.138739\pi$
$18$		0			0
$19$	−21.8800			−0.264190			−0.132095	−	0.991237i	$-0.542170\pi$
$19$	−21.8800			−0.264190			−0.132095	+	0.991237i	$0.542170\pi$
$20$	58.1995	−	42.2878i	0.650690	−	0.472792i
$21$		0			0
$22$		−	26.5922i		−	0.257703i
$23$		−	98.7941i		−	0.895652i	−0.894121	−	0.447826i	$-0.852198\pi$
$23$		−	98.7941i		−	0.895652i	0.894121	−	0.447826i	$-0.147802\pi$
$24$		0			0
$25$	38.6182	−	118.885i	0.308946	−	0.951080i
$26$	51.2331			0.386447
$27$		0			0
$28$			36.4858i			0.246256i
$29$	159.643			1.02224			0.511120	−	0.859509i	$-0.329231\pi$
$29$	159.643			1.02224			0.511120	+	0.859509i	$0.329231\pi$
$30$		0			0
$31$	−69.4873			−0.402590			−0.201295	−	0.979531i	$-0.564515\pi$
$31$	−69.4873			−0.402590			−0.201295	+	0.979531i	$0.564515\pi$
$32$		−	180.615i		−	0.997769i
$33$		0			0
$34$	−74.0512			−0.373520
$35$	37.2650	+	51.2868i	0.179970	+	0.247687i
$36$		0			0
$37$			235.618i			1.04690i	0.852056	+	0.523451i	$0.175356\pi$
$37$			235.618i			1.04690i	−0.852056	+	0.523451i	$0.824644\pi$
$38$			27.3757i			0.116866i
$39$		0			0
$40$	−118.691	−	163.351i	−0.469168	−	0.645703i
$41$	−491.487			−1.87213			−0.936066	−	0.351825i	$-0.885561\pi$
$41$	−491.487			−1.87213			−0.936066	+	0.351825i	$0.885561\pi$
$42$		0			0
$43$		−	95.3080i		−	0.338008i	−0.985615	−	0.169004i	$-0.945945\pi$
$43$		−	95.3080i		−	0.338008i	0.985615	−	0.169004i	$-0.0540550\pi$
$44$	136.758			0.468570
$45$		0			0
$46$	−123.609			−0.396199
$47$			548.796i			1.70319i	0.524197	+	0.851597i	$0.324366\pi$
$47$			548.796i			1.70319i	−0.524197	+	0.851597i	$0.675634\pi$
$48$		0			0
$49$	310.848			0.906262
$50$	−148.746	−	48.3182i	−0.420718	−	0.136664i
$51$		0			0
$52$			263.482i			0.702660i
$53$			509.829i			1.32133i	0.750682	+	0.660664i	$0.229725\pi$
$53$			509.829i			1.32133i	−0.750682	+	0.660664i	$0.770275\pi$
$54$		0			0
$55$	192.236	−	139.679i	0.471294	−	0.342442i
$56$	102.406			0.244368
$57$		0			0
$58$		−	199.742i		−	0.452196i
$59$	−741.098			−1.63530			−0.817650	−	0.575715i	$-0.804724\pi$
$59$	−741.098			−1.63530			−0.817650	+	0.575715i	$0.804724\pi$
$60$		0			0
$61$	387.157			0.812629			0.406314	−	0.913733i	$-0.366814\pi$
$61$	387.157			0.812629			0.406314	+	0.913733i	$0.366814\pi$
$62$			86.9409i			0.178089i
$63$		0			0
$64$	5.05795			0.00987881
$65$	269.109	+	370.367i	0.513521	+	0.706745i
$66$		0			0
$67$			1060.45i			1.93364i	0.255451	+	0.966822i	$0.417776\pi$
$67$			1060.45i			1.93364i	−0.255451	+	0.966822i	$0.582224\pi$
$68$		−	380.831i		−	0.679155i
$69$		0			0
$70$	64.1689	−	46.6251i	0.109566	−	0.0796109i
$71$	−508.998			−0.850802			−0.425401	−	0.905005i	$-0.639867\pi$
$71$	−508.998			−0.850802			−0.425401	+	0.905005i	$0.639867\pi$
$72$		0			0
$73$		−	914.906i		−	1.46687i	−0.679758	−	0.733436i	$-0.737915\pi$
$73$		−	914.906i		−	1.46687i	0.679758	−	0.733436i	$-0.262085\pi$
$74$	294.800			0.463105
$75$		0			0
$76$	−140.788			−0.212493
$77$			120.515i			0.178363i
$78$		0			0
$79$	−925.717			−1.31837			−0.659185	−	0.751981i	$-0.729099\pi$
$79$	−925.717			−1.31837			−0.659185	+	0.751981i	$0.729099\pi$
$80$	261.215	−	189.799i	0.365059	−	0.265252i
$81$		0			0
$82$			614.937i			0.828152i
$83$			708.964i			0.937577i	0.883310	+	0.468788i	$0.155309\pi$
$83$			708.964i			0.937577i	−0.883310	+	0.468788i	$0.844691\pi$
$84$		0			0
$85$	−388.964	−	535.321i	−0.496342	−	0.683103i
$86$	−119.247			−0.149520
$87$		0			0
$88$		−	383.846i		−	0.464979i
$89$	−273.516			−0.325760			−0.162880	−	0.986646i	$-0.552078\pi$
$89$	−273.516			−0.325760			−0.162880	+	0.986646i	$0.552078\pi$
$90$		0			0
$91$	−232.186			−0.267470
$92$		−	635.696i		−	0.720390i
$93$		0			0
$94$	686.641			0.753421
$95$	−197.901	+	143.795i	−0.213728	+	0.155295i
$96$		0			0
$97$			1450.22i			1.51802i	0.651081	+	0.759008i	$0.274316\pi$
$97$			1450.22i			1.51802i	−0.651081	+	0.759008i	$0.725684\pi$
$98$		−	388.926i		−	0.400892i
$99$		0			0
$100$	248.491	−	764.972i	0.248491	−	0.764972i
$101$	676.246			0.666228			0.333114	−	0.942887i	$-0.391901\pi$
$101$	676.246			0.666228			0.333114	+	0.942887i	$0.391901\pi$
$102$		0			0
$103$		−	1069.47i		−	1.02308i	−0.859258	−	0.511542i	$-0.829074\pi$
$103$		−	1069.47i		−	1.02308i	0.859258	−	0.511542i	$-0.170926\pi$
$104$	739.527			0.697275
$105$		0			0
$106$	637.886			0.584499
$107$		−	209.566i		−	0.189341i	−0.995509	−	0.0946707i	$-0.969820\pi$
$107$		−	209.566i		−	0.189341i	0.995509	−	0.0946707i	$-0.0301798\pi$
$108$		0			0
$109$	455.127			0.399938			0.199969	−	0.979802i	$-0.435916\pi$
$109$	455.127			0.399938			0.199969	+	0.979802i	$0.435916\pi$
$110$	−174.763	−	240.522i	−0.151482	−	0.208480i
$111$		0			0
$112$			163.758i			0.138158i
$113$		−	1130.83i		−	0.941413i	−0.882290	−	0.470706i	$-0.843999\pi$
$113$		−	1130.83i		−	0.941413i	0.882290	−	0.470706i	$-0.156001\pi$
$114$		0			0
$115$	−649.273	−	893.577i	−0.526478	−	0.724578i
$116$	1027.23			0.822208
$117$		0			0
$118$			927.245i			0.723388i
$119$	335.597			0.258522
$120$		0			0
$121$	−879.280			−0.660616
$122$		−	484.402i		−	0.359473i
$123$		0			0
$124$	−447.120			−0.323811
$125$	−432.014	−	1329.09i	−0.309124	−	0.951022i
$126$		0			0
$127$			1277.62i			0.892682i	0.894863	+	0.446341i	$0.147273\pi$
$127$			1277.62i			0.892682i	−0.894863	+	0.446341i	$0.852727\pi$
$128$		−	1451.25i		−	1.00214i
$129$		0			0
$130$	463.395	−	336.703i	0.312634	−	0.227160i
$131$	1358.29			0.905909			0.452954	−	0.891534i	$-0.350370\pi$
$131$	1358.29			0.905909			0.452954	+	0.891534i	$0.350370\pi$
$132$		0			0
$133$		−	124.066i		−	0.0808862i
$134$	1326.81			0.855363
$135$		0			0
$136$	−1068.90			−0.673949
$137$		−	956.545i		−	0.596519i	−0.954485	−	0.298260i	$-0.903594\pi$
$137$		−	956.545i		−	0.596519i	0.954485	−	0.298260i	$-0.0964061\pi$
$138$		0			0
$139$	−27.3274			−0.0166754			−0.00833769	−	0.999965i	$-0.502654\pi$
$139$	−27.3274			−0.0166754			−0.00833769	+	0.999965i	$0.502654\pi$
$140$	239.784	+	330.008i	0.144753	+	0.199220i
$141$		0			0
$142$			636.847i			0.376359i
$143$			870.295i			0.508935i
$144$		0			0
$145$	1443.95	−	1049.17i	0.826987	−	0.600889i
$146$	−1144.71			−0.648882
$147$		0			0
$148$			1516.10i			0.842044i
$149$	216.410			0.118987			0.0594933	−	0.998229i	$-0.481052\pi$
$149$	216.410			0.118987			0.0594933	+	0.998229i	$0.481052\pi$
$150$		0			0
$151$	2675.41			1.44187			0.720933	−	0.693004i	$-0.243713\pi$
$151$	2675.41			1.44187			0.720933	+	0.693004i	$0.243713\pi$
$152$			395.156i			0.210865i
$153$		0			0
$154$	150.785			0.0789001
$155$	−628.501	+	456.669i	−0.325693	+	0.236649i
$156$		0			0
$157$			956.987i			0.486471i	0.969967	+	0.243235i	$0.0782087\pi$
$157$			956.987i			0.486471i	−0.969967	+	0.243235i	$0.921791\pi$
$158$			1158.24i			0.583192i
$159$		0			0
$160$	−1187.00	−	1633.64i	−0.586504	−	0.807190i
$161$	560.191			0.274219
$162$		0			0
$163$		−	478.098i		−	0.229739i	−0.993381	−	0.114870i	$-0.963355\pi$
$163$		−	478.098i		−	0.229739i	0.993381	−	0.114870i	$-0.0366450\pi$
$164$	−3162.50			−1.50579
$165$		0			0
$166$	887.039			0.414744
$167$			2666.34i			1.23549i	0.786377	+	0.617746i	$0.211954\pi$
$167$			2666.34i			1.23549i	−0.786377	+	0.617746i	$0.788046\pi$
$168$		0			0
$169$	520.269			0.236809
$170$	−669.781	+	486.663i	−0.302176	+	0.219561i
$171$		0			0
$172$		−	613.265i		−	0.271866i
$173$		−	492.589i		−	0.216479i	−0.994125	−	0.108239i	$-0.965479\pi$
$173$		−	492.589i		−	0.216479i	0.994125	−	0.108239i	$-0.0345213\pi$
$174$		0			0
$175$	674.112	+	218.976i	0.291189	+	0.0945888i
$176$	613.807			0.262883
$177$		0			0
$178$			342.217i			0.144103i
$179$	−3890.48			−1.62451			−0.812257	−	0.583299i	$-0.801762\pi$
$179$	−3890.48			−1.62451			−0.812257	+	0.583299i	$0.801762\pi$
$180$		0			0
$181$	−4371.58			−1.79523			−0.897616	−	0.440779i	$-0.854702\pi$
$181$	−4371.58			−1.79523			−0.897616	+	0.440779i	$0.854702\pi$
$182$			290.506i			0.118317i
$183$		0			0
$184$	−1784.24			−0.714869
$185$	1548.48	+	2131.13i	0.615386	+	0.846939i
$186$		0			0
$187$		−	1257.91i		−	0.491910i
$188$			3531.26i			1.36991i
$189$		0			0
$190$	179.913	+	247.609i	0.0686960	+	0.0945444i
$191$	−328.351			−0.124391			−0.0621954	−	0.998064i	$-0.519810\pi$
$191$	−328.351			−0.124391			−0.0621954	+	0.998064i	$0.519810\pi$
$192$		0			0
$193$		−	944.655i		−	0.352320i	−0.984362	−	0.176160i	$-0.943632\pi$
$193$		−	944.655i		−	0.352320i	0.984362	−	0.176160i	$-0.0563676\pi$
$194$	1814.48			0.671506
$195$		0			0
$196$	2000.17			0.728924
$197$		−	2757.66i		−	0.997337i	−0.866793	−	0.498668i	$-0.833823\pi$
$197$		−	2757.66i		−	0.997337i	0.866793	−	0.498668i	$-0.166177\pi$
$198$		0			0
$199$	1529.88			0.544978			0.272489	−	0.962159i	$-0.412153\pi$
$199$	1529.88			0.544978			0.272489	+	0.962159i	$0.412153\pi$
$200$	−2147.08	−	697.452i	−0.759109	−	0.246586i
$201$		0			0
$202$		−	846.103i		−	0.294711i
$203$			905.222i			0.312976i
$204$		0			0
$205$	−4445.42	+	3230.04i	−1.51454	+	1.10047i
$206$	−1338.09			−0.452569
$207$		0			0
$208$			1182.57i			0.394215i
$209$	−465.031			−0.153908
$210$		0			0
$211$	−150.761			−0.0491886			−0.0245943	−	0.999698i	$-0.507829\pi$
$211$	−150.761			−0.0491886			−0.0245943	+	0.999698i	$0.507829\pi$
$212$			3280.52i			1.06277i
$213$		0			0
$214$	−262.204			−0.0837567
$215$	−626.362	−	862.045i	−0.198686	−	0.273447i
$216$		0			0
$217$		−	394.013i		−	0.123260i
$218$		−	569.444i		−	0.176916i
$219$		0			0
$220$	1236.96	−	898.772i	0.379071	−	0.275433i
$221$	2423.51			0.737661
$222$		0			0
$223$		−	2573.87i		−	0.772910i	−0.922308	−	0.386455i	$-0.873699\pi$
$223$		−	2573.87i		−	0.772910i	0.922308	−	0.386455i	$-0.126301\pi$
$224$	1024.14			0.305483
$225$		0			0
$226$	−1414.87			−0.416441
$227$			2491.62i			0.728523i	0.931297	+	0.364261i	$0.118679\pi$
$227$			2491.62i			0.728523i	−0.931297	+	0.364261i	$0.881321\pi$
$228$		0			0
$229$	−1400.56			−0.404154			−0.202077	−	0.979370i	$-0.564769\pi$
$229$	−1400.56			−0.404154			−0.202077	+	0.979370i	$0.564769\pi$
$230$	−1118.02	+	812.355i	−0.320523	+	0.232892i
$231$		0			0
$232$		−	2883.18i		−	0.815906i
$233$		−	3258.86i		−	0.916289i	−0.888878	−	0.458144i	$-0.848514\pi$
$233$		−	3258.86i		−	0.916289i	0.888878	−	0.458144i	$-0.151486\pi$
$234$		0			0
$235$	3606.68	+	4963.77i	1.00116	+	1.37788i
$236$	−4768.64			−1.31530
$237$		0			0
$238$		−	419.892i		−	0.114359i
$239$	1660.80			0.449491			0.224746	−	0.974417i	$-0.427845\pi$
$239$	1660.80			0.449491			0.224746	+	0.974417i	$0.427845\pi$
$240$		0			0
$241$	5966.20			1.59467			0.797337	−	0.603534i	$-0.206241\pi$
$241$	5966.20			1.59467			0.797337	+	0.603534i	$0.206241\pi$
$242$			1100.13i			0.292229i
$243$		0			0
$244$	2491.18			0.653613
$245$	2811.57	−	2042.89i	0.733161	−	0.532715i
$246$		0			0
$247$		−	895.939i		−	0.230799i
$248$			1254.95i			0.321329i
$249$		0			0
$250$	−1662.93	+	540.526i	−0.420692	+	0.136743i
$251$	−3051.45			−0.767354			−0.383677	−	0.923467i	$-0.625342\pi$
$251$	−3051.45			−0.767354			−0.383677	+	0.923467i	$0.625342\pi$
$252$		0			0
$253$		−	2099.74i		−	0.521777i
$254$	1598.53			0.394885
$255$		0			0
$256$	−1775.31			−0.433425
$257$		−	5559.16i		−	1.34930i	−0.738136	−	0.674652i	$-0.764294\pi$
$257$		−	5559.16i		−	1.34930i	0.738136	−	0.674652i	$-0.235706\pi$
$258$		0			0
$259$	−1336.02			−0.320527
$260$	1731.60	+	2383.15i	0.413035	+	0.568449i
$261$		0			0
$262$		−	1699.46i		−	0.400736i
$263$		−	956.467i		−	0.224252i	−0.993694	−	0.112126i	$-0.964234\pi$
$263$		−	956.467i		−	0.224252i	0.993694	−	0.112126i	$-0.0357660\pi$
$264$		0			0
$265$	3350.58	+	4611.32i	0.776697	+	1.06895i
$266$	−155.228			−0.0357806
$267$		0			0
$268$			6823.50i			1.55527i
$269$	4740.41			1.07445			0.537226	−	0.843438i	$-0.319472\pi$
$269$	4740.41			1.07445			0.537226	+	0.843438i	$0.319472\pi$
$270$		0			0
$271$	−3814.78			−0.855097			−0.427548	−	0.903992i	$-0.640623\pi$
$271$	−3814.78			−0.855097			−0.427548	+	0.903992i	$0.640623\pi$
$272$		−	1709.27i		−	0.381028i
$273$		0			0
$274$	−1196.81			−0.263875
$275$	820.780	−	2526.75i	0.179981	−	0.554068i
$276$		0			0
$277$		−	1579.65i		−	0.342642i	−0.985215	−	0.171321i	$-0.945196\pi$
$277$		−	1579.65i		−	0.342642i	0.985215	−	0.171321i	$-0.0548035\pi$
$278$			34.1914i			0.00737649i
$279$		0			0
$280$	926.249	−	673.013i	0.197693	−	0.143644i
$281$	7876.16			1.67207			0.836036	−	0.548674i	$-0.184867\pi$
$281$	7876.16			1.67207			0.836036	+	0.548674i	$0.184867\pi$
$282$		0			0
$283$			905.509i			0.190201i	0.995468	+	0.0951006i	$0.0303173\pi$
$283$			905.509i			0.190201i	−0.995468	+	0.0951006i	$0.969683\pi$
$284$	−3275.18			−0.684317
$285$		0			0
$286$	1088.89			0.225131
$287$		−	2786.87i		−	0.573184i
$288$		0			0
$289$	1410.11			0.287015
$290$	−1312.70	−	1806.63i	−0.265808	−	0.365824i
$291$		0			0
$292$		−	5887.02i		−	1.17983i
$293$		−	7691.27i		−	1.53355i	−0.641919	−	0.766773i	$-0.721861\pi$
$293$		−	7691.27i		−	1.53355i	0.641919	−	0.766773i	$-0.278139\pi$
$294$		0			0
$295$	−6703.11	+	4870.48i	−1.32295	+	0.961255i
$296$	4255.31			0.835590
$297$		0			0
$298$		−	270.767i		−	0.0526346i
$299$	4045.41			0.782449
$300$		0			0
$301$	540.424			0.103487
$302$		−	3347.41i		−	0.637821i
$303$		0			0
$304$	−631.893			−0.119216
$305$	3501.77	−	2544.39i	0.657412	−	0.477676i
$306$		0			0
$307$		−	8464.71i		−	1.57364i	−0.617185	−	0.786818i	$-0.711727\pi$
$307$		−	8464.71i		−	1.57364i	0.617185	−	0.786818i	$-0.288273\pi$
$308$			775.458i			0.143460i
$309$		0			0
$310$	571.374	+	786.366i	0.104683	+	0.144073i
$311$	−8388.01			−1.52939			−0.764695	−	0.644392i	$-0.777111\pi$
$311$	−8388.01			−1.52939			−0.764695	+	0.644392i	$0.777111\pi$
$312$		0			0
$313$		−	2384.69i		−	0.430640i	−0.976544	−	0.215320i	$-0.930920\pi$
$313$		−	2384.69i		−	0.430640i	0.976544	−	0.215320i	$-0.0690795\pi$
$314$	1197.36			0.215194
$315$		0			0
$316$	−5956.58			−1.06039
$317$		−	3211.58i		−	0.569023i	−0.958673	−	0.284511i	$-0.908169\pi$
$317$		−	3211.58i		−	0.569023i	0.958673	−	0.284511i	$-0.0918313\pi$
$318$		0			0
$319$	3393.01			0.595523
$320$	45.7484	−	33.2408i	0.00799191	−	0.00580692i
$321$		0			0
$322$		−	700.898i		−	0.121303i
$323$			1294.97i			0.223078i
$324$		0			0
$325$	4868.09	+	1581.33i	0.830871	+	0.269897i
$326$	−598.185			−0.101627
$327$		0			0
$328$			8876.35i			1.49425i
$329$	−3111.83			−0.521461
$330$		0			0
$331$	7193.90			1.19460			0.597300	−	0.802018i	$-0.296240\pi$
$331$	7193.90			1.19460			0.597300	+	0.802018i	$0.296240\pi$
$332$			4561.87i			0.754111i
$333$		0			0
$334$	3336.06			0.546530
$335$	6969.23	+	9591.57i	1.13663	+	1.56431i
$336$		0			0
$337$		−	6913.84i		−	1.11757i	−0.829313	−	0.558785i	$-0.811268\pi$
$337$		−	6913.84i		−	1.11757i	0.829313	−	0.558785i	$-0.188732\pi$
$338$		−	650.948i		−	0.104754i
$339$		0			0
$340$	−2502.81	−	3444.55i	−0.399218	−	0.549433i
$341$	−1476.86			−0.234536
$342$		0			0
$343$			3707.50i			0.583634i
$344$	−1721.28			−0.269783
$345$		0			0
$346$	−616.316			−0.0957611
$347$			4009.84i			0.620344i	0.950681	+	0.310172i	$0.100387\pi$
$347$			4009.84i			0.620344i	−0.950681	+	0.310172i	$0.899613\pi$
$348$		0			0
$349$	−6884.76			−1.05597			−0.527984	−	0.849254i	$-0.677052\pi$
$349$	−6884.76			−1.05597			−0.527984	+	0.849254i	$0.677052\pi$
$350$	273.978	−	843.433i	0.0418421	−	0.128810i
$351$		0			0
$352$		−	3838.75i		−	0.581267i
$353$			7783.83i			1.17363i	0.809721	+	0.586815i	$0.199618\pi$
$353$			7783.83i			1.17363i	−0.809721	+	0.586815i	$0.800382\pi$
$354$		0			0
$355$	−4603.81	+	3345.12i	−0.688295	+	0.500115i
$356$	−1759.96			−0.262016
$357$		0			0
$358$			4867.68i			0.718617i
$359$	−8188.19			−1.20378			−0.601889	−	0.798580i	$-0.705585\pi$
$359$	−8188.19			−1.20378			−0.601889	+	0.798580i	$0.705585\pi$
$360$		0			0
$361$	−6380.27			−0.930204
$362$			5469.62i			0.794135i
$363$		0			0
$364$	−1494.02			−0.215131
$365$	−6012.74	−	8275.18i	−0.862250	−	1.18669i
$366$		0			0
$367$		−	3688.79i		−	0.524668i	−0.964977	−	0.262334i	$-0.915508\pi$
$367$		−	3688.79i		−	0.524668i	0.964977	−	0.262334i	$-0.0844922\pi$
$368$		−	2853.17i		−	0.404162i
$369$		0			0
$370$	2666.42	−	1937.42i	0.374650	−	0.272221i
$371$	−2890.87			−0.404546
$372$		0			0
$373$		−	4127.16i		−	0.572912i	−0.958093	−	0.286456i	$-0.907523\pi$
$373$		−	4127.16i		−	0.572912i	0.958093	−	0.286456i	$-0.0924773\pi$
$374$	−1573.86			−0.217600
$375$		0			0
$376$	9911.36			1.35941
$377$			6537.05i			0.893038i
$378$		0			0
$379$	−4145.52			−0.561850			−0.280925	−	0.959730i	$-0.590641\pi$
$379$	−4145.52			−0.561850			−0.280925	+	0.959730i	$0.590641\pi$
$380$	−1273.40	+	925.255i	−0.171906	+	0.124907i
$381$		0			0
$382$			410.825i			0.0550253i
$383$			9983.95i			1.33200i	0.745952	+	0.666000i	$0.231995\pi$
$383$			9983.95i			1.33200i	−0.745952	+	0.666000i	$0.768005\pi$
$384$		0			0
$385$	792.020	+	1090.04i	0.104844	+	0.144294i
$386$	−1181.93			−0.155852
$387$		0			0
$388$			9331.53i			1.22097i
$389$	7868.51			1.02558			0.512788	−	0.858515i	$-0.328613\pi$
$389$	7868.51			1.02558			0.512788	+	0.858515i	$0.328613\pi$
$390$		0			0
$391$	−5847.15			−0.756274
$392$		−	5613.97i		−	0.723337i
$393$		0			0
$394$	−3450.32			−0.441180
$395$	−8372.96	+	6083.79i	−1.06656	+	0.774959i
$396$		0			0
$397$			8137.21i			1.02870i	0.857580	+	0.514351i	$0.171967\pi$
$397$			8137.21i			1.02870i	−0.857580	+	0.514351i	$0.828033\pi$
$398$		−	1914.16i		−	0.241075i
$399$		0			0
$400$	1115.29	−	3433.39i	0.139412	−	0.429174i
$401$	−4204.86			−0.523643			−0.261822	−	0.965116i	$-0.584323\pi$
$401$	−4204.86			−0.523643			−0.261822	+	0.965116i	$0.584323\pi$
$402$		0			0
$403$		−	2845.36i		−	0.351706i
$404$	4351.34			0.535860
$405$		0			0
$406$	1132.59			0.138447
$407$			5007.76i			0.609891i
$408$		0			0
$409$	8318.77			1.00571			0.502857	−	0.864370i	$-0.332282\pi$
$409$	8318.77			1.00571			0.502857	+	0.864370i	$0.332282\pi$
$410$	4041.35	+	5562.01i	0.486801	+	0.669971i
$411$		0			0
$412$		−	6881.54i		−	0.822886i
$413$		−	4202.24i		−	0.500675i
$414$		0			0
$415$	4659.29	+	6412.46i	0.551122	+	0.758495i
$416$	7395.82			0.871659
$417$		0			0
$418$			581.836i			0.0680826i
$419$	−6938.90			−0.809040			−0.404520	−	0.914529i	$-0.632561\pi$
$419$	−6938.90			−0.809040			−0.404520	+	0.914529i	$0.632561\pi$
$420$		0			0
$421$	9856.88			1.14108			0.570541	−	0.821269i	$-0.306734\pi$
$421$	9856.88			1.14108			0.570541	+	0.821269i	$0.306734\pi$
$422$			188.628i			0.0217590i
$423$		0			0
$424$	9207.60			1.05462
$425$	−7036.24	−	2285.63i	−0.803077	−	0.260869i
$426$		0			0
$427$			2195.29i			0.248800i
$428$		−	1348.47i		−	0.152291i
$429$		0			0
$430$	−1078.57	+	783.690i	−0.120961	+	0.0878904i
$431$	3013.40			0.336775			0.168388	−	0.985721i	$-0.446144\pi$
$431$	3013.40			0.336775			0.168388	+	0.985721i	$0.446144\pi$
$432$		0			0
$433$		−	3902.19i		−	0.433088i	−0.976273	−	0.216544i	$-0.930521\pi$
$433$		−	3902.19i		−	0.433088i	0.976273	−	0.216544i	$-0.0694786\pi$
$434$	−492.980			−0.0545249
$435$		0			0
$436$	2928.54			0.321678
$437$			2161.61i			0.236622i
$438$		0			0
$439$	−3169.12			−0.344542			−0.172271	−	0.985050i	$-0.555110\pi$
$439$	−3169.12			−0.344542			−0.172271	+	0.985050i	$0.555110\pi$
$440$	−2522.63	−	3471.82i	−0.273322	−	0.376165i
$441$		0			0
$442$		−	3032.24i		−	0.326310i
$443$			10732.6i			1.15107i	0.817778	+	0.575534i	$0.195206\pi$
$443$			10732.6i			1.15107i	−0.817778	+	0.575534i	$0.804794\pi$
$444$		0			0
$445$	−2473.91	+	1797.54i	−0.263539	+	0.191487i
$446$	−3220.36			−0.341903
$447$		0			0
$448$			28.6800i			0.00302456i
$449$	294.508			0.0309547			0.0154774	−	0.999880i	$-0.495073\pi$
$449$	294.508			0.0309547			0.0154774	+	0.999880i	$0.495073\pi$
$450$		0			0
$451$	−10445.9			−1.09064
$452$		−	7276.40i		−	0.757197i
$453$		0			0
$454$	3117.46			0.322268
$455$	−2100.09	+	1525.92i	−0.216382	+	0.157223i
$456$		0			0
$457$			5419.66i			0.554751i	0.960762	+	0.277376i	$0.0894646\pi$
$457$			5419.66i			0.554751i	−0.960762	+	0.277376i	$0.910535\pi$
$458$			1752.34i			0.178781i
$459$		0			0
$460$	−4177.78	−	5749.77i	−0.423457	−	0.582792i
$461$	1002.66			0.101299			0.0506493	−	0.998717i	$-0.483871\pi$
$461$	1002.66			0.101299			0.0506493	+	0.998717i	$0.483871\pi$
$462$		0			0
$463$		−	13615.3i		−	1.36665i	−0.730116	−	0.683323i	$-0.760534\pi$
$463$		−	13615.3i		−	1.36665i	0.730116	−	0.683323i	$-0.239466\pi$
$464$	4610.49			0.461285
$465$		0			0
$466$	−4077.42			−0.405328
$467$			1335.52i			0.132336i	0.997809	+	0.0661678i	$0.0210773\pi$
$467$			1335.52i			0.132336i	−0.997809	+	0.0661678i	$0.978923\pi$
$468$		0			0
$469$	−6013.04			−0.592017
$470$	6210.56	−	4512.59i	0.609514	−	0.442873i
$471$		0			0
$472$			13384.4i			1.30522i
$473$		−	2025.65i		−	0.196912i
$474$		0			0
$475$	−844.965	+	2601.20i	−0.0816203	+	0.251266i
$476$	2159.42			0.207935
$477$		0			0
$478$		−	2077.96i		−	0.198836i
$479$	−13852.1			−1.32134			−0.660668	−	0.750678i	$-0.729727\pi$
$479$	−13852.1			−1.32134			−0.660668	+	0.750678i	$0.729727\pi$
$480$		0			0
$481$	−9648.07			−0.914583
$482$		−	7464.77i		−	0.705417i
$483$		0			0
$484$	−5657.77			−0.531346
$485$	9530.82	+	13117.0i	0.892314	+	1.22807i
$486$		0			0
$487$			6113.26i			0.568826i	0.958702	+	0.284413i	$0.0917987\pi$
$487$			6113.26i			0.568826i	−0.958702	+	0.284413i	$0.908201\pi$
$488$		−	6992.12i		−	0.648603i
$489$		0			0
$490$	−2556.01	−	3517.77i	−0.235651	−	0.324320i
$491$	16356.5			1.50338			0.751691	−	0.659515i	$-0.229238\pi$
$491$	16356.5			1.50338			0.751691	+	0.659515i	$0.229238\pi$
$492$		0			0
$493$		−	9448.51i		−	0.863164i
$494$	−1120.98			−0.102096
$495$		0			0
$496$	−2006.79			−0.181669
$497$		−	2886.17i		−	0.260487i
$498$		0			0
$499$	9262.82			0.830983			0.415492	−	0.909597i	$-0.363610\pi$
$499$	9262.82			0.830983			0.415492	+	0.909597i	$0.363610\pi$
$500$	−2779.82	−	8552.13i	−0.248635	−	0.764925i
$501$		0			0
$502$			3817.90i			0.339445i
$503$		−	12498.5i		−	1.10792i	−0.832544	−	0.553959i	$-0.813116\pi$
$503$		−	12498.5i		−	1.10792i	0.832544	−	0.553959i	$-0.186884\pi$
$504$		0			0
$505$	6116.54	−	4444.27i	0.538975	−	0.391619i
$506$	−2627.15			−0.230812
$507$		0			0
$508$			8220.93i			0.718002i
$509$	7370.25			0.641809			0.320904	−	0.947112i	$-0.396013\pi$
$509$	7370.25			0.641809			0.320904	+	0.947112i	$0.396013\pi$
$510$		0			0
$511$	5187.78			0.449107
$512$		−	9388.79i		−	0.810410i
$513$		0			0
$514$	−6955.49			−0.596875
$515$	−7028.51	−	9673.15i	−0.601385	−	0.827670i
$516$		0			0
$517$			11664.0i			0.992224i
$518$			1671.60i			0.141787i
$519$		0			0
$520$	6688.90	−	4860.15i	0.564092	−	0.409869i
$521$	5426.50			0.456313			0.228157	−	0.973624i	$-0.426730\pi$
$521$	5426.50			0.456313			0.228157	+	0.973624i	$0.426730\pi$
$522$		0			0
$523$		−	19681.7i		−	1.64555i	−0.568371	−	0.822773i	$-0.692426\pi$
$523$		−	19681.7i		−	1.64555i	0.568371	−	0.822773i	$-0.307574\pi$
$524$	8739.97			0.728640
$525$		0			0
$526$	−1196.71			−0.0991996
$527$			4112.62i			0.339941i
$528$		0			0
$529$	2406.73			0.197808
$530$	5769.57	−	4192.17i	0.472857	−	0.343578i
$531$		0			0
$532$		−	798.308i		−	0.0650583i
$533$		−	20125.4i		−	1.63551i
$534$		0			0
$535$	−1377.27	−	1895.49i	−0.111298	−	0.153176i
$536$	19151.9			1.54335
$537$		0			0
$538$		−	5931.09i		−	0.475293i
$539$	6606.67			0.527958
$540$		0			0
$541$	16726.1			1.32923			0.664613	−	0.747188i	$-0.268596\pi$
$541$	16726.1			1.32923			0.664613	+	0.747188i	$0.268596\pi$
$542$			4772.96i			0.378259i
$543$		0			0
$544$	−10689.8			−0.842500
$545$	4116.55	−	2991.08i	0.323548	−	0.235090i
$546$		0			0
$547$			19292.6i			1.50803i	0.656859	+	0.754014i	$0.271885\pi$
$547$			19292.6i			1.50803i	−0.656859	+	0.754014i	$0.728115\pi$
$548$		−	6154.94i		−	0.479792i
$549$		0			0
$550$	−3161.41	−	1026.94i	−0.245096	−	0.0796162i
$551$	−3492.99			−0.270066
$552$		0			0
$553$		−	5249.08i		−	0.403641i
$554$	−1976.42			−0.151570
$555$		0			0
$556$	−175.840			−0.0134123
$557$		−	7161.62i		−	0.544789i	−0.962186	−	0.272395i	$-0.912184\pi$
$557$		−	7161.62i		−	0.544789i	0.962186	−	0.272395i	$-0.0878157\pi$
$558$		0			0
$559$	3902.66			0.295286
$560$	1076.21	+	1481.16i	0.0812112	+	0.111769i
$561$		0			0
$562$		−	9854.47i		−	0.739655i
$563$			17096.8i			1.27983i	0.768446	+	0.639914i	$0.221030\pi$
$563$			17096.8i			1.27983i	−0.768446	+	0.639914i	$0.778970\pi$
$564$		0			0
$565$	−7431.80	−	10228.2i	−0.553377	−	0.761598i
$566$	1132.95			0.0841369
$567$		0			0
$568$			9192.60i			0.679072i
$569$	−11508.9			−0.847942			−0.423971	−	0.905676i	$-0.639364\pi$
$569$	−11508.9			−0.847942			−0.423971	+	0.905676i	$0.639364\pi$
$570$		0			0
$571$	10695.3			0.783863			0.391931	−	0.919994i	$-0.371807\pi$
$571$	10695.3			0.783863			0.391931	+	0.919994i	$0.371807\pi$
$572$			5599.96i			0.409347i
$573$		0			0
$574$	−3486.87			−0.253553
$575$	−11745.1	−	3815.25i	−0.851836	−	0.276708i
$576$		0			0
$577$			23047.9i			1.66291i	0.555595	+	0.831453i	$0.312491\pi$
$577$			23047.9i			1.66291i	−0.555595	+	0.831453i	$0.687509\pi$
$578$		−	1764.29i		−	0.126963i
$579$		0			0
$580$	9291.15	−	6750.95i	0.665162	−	0.483307i
$581$	−4020.03			−0.287055
$582$		0			0
$583$			10835.8i			0.769762i
$584$	−16523.4			−1.17079
$585$		0			0
$586$	−9623.14			−0.678376
$587$		−	12449.5i		−	0.875378i	−0.899126	−	0.437689i	$-0.855797\pi$
$587$		−	12449.5i		−	0.875378i	0.899126	−	0.437689i	$-0.144203\pi$
$588$		0			0
$589$	1520.38			0.106360
$590$	6093.83	+	8386.78i	0.425219	+	0.585217i
$591$		0			0
$592$			6804.65i			0.472414i
$593$			18331.7i			1.26946i	0.772732	+	0.634732i	$0.218890\pi$
$593$			18331.7i			1.26946i	−0.772732	+	0.634732i	$0.781110\pi$
$594$		0			0
$595$	3035.42	−	2205.54i	0.209143	−	0.151963i
$596$	1392.50			0.0957032
$597$		0			0
$598$		−	5061.53i		−	0.346122i
$599$	23905.9			1.63067			0.815333	−	0.578992i	$-0.196554\pi$
$599$	23905.9			1.63067			0.815333	+	0.578992i	$0.196554\pi$
$600$		0			0
$601$	−16524.6			−1.12155			−0.560777	−	0.827967i	$-0.689498\pi$
$601$	−16524.6			−1.12155			−0.560777	+	0.827967i	$0.689498\pi$
$602$		−	676.165i		−	0.0457782i
$603$		0			0
$604$	17215.1			1.15972
$605$	−7952.94	+	5778.61i	−0.534435	+	0.388320i
$606$		0			0
$607$			2610.92i			0.174587i	0.996183	+	0.0872933i	$0.0278217\pi$
$607$			2610.92i			0.174587i	−0.996183	+	0.0872933i	$0.972178\pi$
$608$			3951.86i			0.263600i
$609$		0			0
$610$	−3183.48	−	4381.33i	−0.211304	−	0.290811i
$611$	−22472.0			−1.48792
$612$		0			0
$613$		−	5036.27i		−	0.331832i	−0.986140	−	0.165916i	$-0.946942\pi$
$613$		−	5036.27i		−	0.331832i	0.986140	−	0.165916i	$-0.0530581\pi$
$614$	−10590.8			−0.696110
$615$		0			0
$616$	2176.52			0.142361
$617$			15805.7i			1.03130i	0.856800	+	0.515650i	$0.172449\pi$
$617$			15805.7i			1.03130i	−0.856800	+	0.515650i	$0.827551\pi$
$618$		0			0
$619$	5929.52			0.385020			0.192510	−	0.981295i	$-0.438337\pi$
$619$	5929.52			0.385020			0.192510	+	0.981295i	$0.438337\pi$
$620$	−4044.13	+	2938.46i	−0.261961	+	0.190341i
$621$		0			0
$622$			10494.9i			0.676538i
$623$		−	1550.92i		−	0.0997370i
$624$		0			0
$625$	−12642.3	−	9182.25i	−0.809105	−	0.587664i
$626$	−2983.66			−0.190497
$627$		0			0
$628$			6157.79i			0.391278i
$629$	13945.1			0.883988
$630$		0			0
$631$	12427.8			0.784060			0.392030	−	0.919952i	$-0.371773\pi$
$631$	12427.8			0.784060			0.392030	+	0.919952i	$0.371773\pi$
$632$			16718.6i			1.05226i
$633$		0			0
$634$	−4018.25			−0.251712
$635$	8396.51	+	11555.9i	0.524733	+	0.722175i
$636$		0			0
$637$			12728.6i			0.791718i
$638$		−	4245.25i		−	0.263434i
$639$		0			0
$640$	−9537.59	−	13126.3i	−0.589073	−	0.810725i
$641$	−28418.8			−1.75113			−0.875566	−	0.483098i	$-0.839511\pi$
$641$	−28418.8			−1.75113			−0.875566	+	0.483098i	$0.839511\pi$
$642$		0			0
$643$			13882.4i			0.851431i	0.904857	+	0.425715i	$0.139978\pi$
$643$			13882.4i			0.851431i	−0.904857	+	0.425715i	$0.860022\pi$
$644$	3604.58			0.220559
$645$		0			0
$646$	1620.24			0.0986802
$647$			13037.5i			0.792207i	0.918206	+	0.396104i	$0.129638\pi$
$647$			13037.5i			0.792207i	−0.918206	+	0.396104i	$0.870362\pi$
$648$		0			0
$649$	−15751.1			−0.952672
$650$	1978.53	−	6090.84i	0.119391	−	0.367542i
$651$		0			0
$652$		−	3076.35i		−	0.184784i
$653$			4574.74i			0.274155i	0.990560	+	0.137078i	$0.0437710\pi$
$653$			4574.74i			0.274155i	−0.990560	+	0.137078i	$0.956229\pi$
$654$		0			0
$655$	12285.5	−	8926.63i	0.732876	−	0.532507i
$656$	−14194.1			−0.844798
$657$		0			0
$658$			3893.45i			0.230672i
$659$	7182.71			0.424581			0.212290	−	0.977207i	$-0.431908\pi$
$659$	7182.71			0.424581			0.212290	+	0.977207i	$0.431908\pi$
$660$		0			0
$661$	1611.10			0.0948026			0.0474013	−	0.998876i	$-0.484906\pi$
$661$	1611.10			0.0948026			0.0474013	+	0.998876i	$0.484906\pi$
$662$		−	9000.84i		−	0.528441i
$663$		0			0
$664$	12804.0			0.748332
$665$	−815.357	−	1122.15i	−0.0475462	−	0.0654365i
$666$		0			0
$667$		−	15771.8i		−	0.915571i
$668$			17156.7i			0.993731i
$669$		0			0
$670$	12000.7	−	8719.74i	0.691984	−	0.502796i
$671$	8228.52			0.473410
$672$		0			0
$673$		−	17721.1i		−	1.01501i	−0.861650	−	0.507503i	$-0.830569\pi$
$673$		−	17721.1i		−	1.01501i	0.861650	−	0.507503i	$-0.169431\pi$
$674$	−8650.44			−0.494365
$675$		0			0
$676$	3347.70			0.190470
$677$		−	12774.3i		−	0.725196i	−0.931946	−	0.362598i	$-0.881890\pi$
$677$		−	12774.3i		−	0.725196i	0.931946	−	0.362598i	$-0.118110\pi$
$678$		0			0
$679$	−8223.17			−0.464766
$680$	−9668.00	+	7024.76i	−0.545222	+	0.396158i
$681$		0			0
$682$			1847.82i			0.103749i
$683$		−	7065.27i		−	0.395820i	−0.980220	−	0.197910i	$-0.936585\pi$
$683$		−	7065.27i		−	0.395820i	0.980220	−	0.197910i	$-0.0634153\pi$
$684$		0			0
$685$	−6286.39	−	8651.79i	−0.350643	−	0.482581i
$686$	4638.74			0.258175
$687$		0			0
$688$		−	2752.49i		−	0.152526i
$689$	−20876.4			−1.15432
$690$		0			0
$691$	−26270.4			−1.44627			−0.723136	−	0.690706i	$-0.757300\pi$
$691$	−26270.4			−1.44627			−0.723136	+	0.690706i	$0.757300\pi$
$692$		−	3169.59i		−	0.174118i
$693$		0			0
$694$	5017.01			0.274414
$695$	−247.172	+	179.595i	−0.0134903	+	0.00980205i
$696$		0			0
$697$			29088.8i			1.58080i
$698$			8614.05i			0.467116i
$699$		0			0
$700$	4337.61	+	1409.01i	0.234209	+	0.0760797i
$701$	32274.9			1.73895			0.869476	−	0.493975i	$-0.164456\pi$
$701$	32274.9			1.73895			0.869476	+	0.493975i	$0.164456\pi$
$702$		0			0
$703$		−	5155.32i		−	0.276581i
$704$	107.500			0.00575507
$705$		0			0
$706$	9738.94			0.519164
$707$			3834.51i			0.203977i
$708$		0			0
$709$	13697.1			0.725539			0.362770	−	0.931879i	$-0.381831\pi$
$709$	13697.1			0.725539			0.362770	+	0.931879i	$0.381831\pi$
$710$	4185.34	+	5760.18i	0.221230	+	0.304473i
$711$		0			0
$712$			4939.76i			0.260007i
$713$			6864.94i			0.360580i
$714$		0			0
$715$	5719.56	+	7871.68i	0.299160	+	0.411726i
$716$	−25033.5			−1.30663
$717$		0			0
$718$			10244.9i			0.532501i
$719$	−6826.36			−0.354076			−0.177038	−	0.984204i	$-0.556651\pi$
$719$	−6826.36			−0.354076			−0.177038	+	0.984204i	$0.556651\pi$
$720$		0			0
$721$	6064.18			0.313234
$722$			7982.84i			0.411483i
$723$		0			0
$724$	−28129.2			−1.44394
$725$	6165.13	−	18979.2i	0.315817	−	0.972232i
$726$		0			0
$727$		−	22413.2i		−	1.14341i	−0.820458	−	0.571706i	$-0.806282\pi$
$727$		−	22413.2i		−	1.14341i	0.820458	−	0.571706i	$-0.193718\pi$
$728$			4193.33i			0.213482i
$729$		0			0
$730$	−10353.7	+	7523.01i	−0.524943	+	0.381423i
$731$	−5640.83			−0.285408
$732$		0			0
$733$		−	167.208i		−	0.00842562i	−0.999991	−	0.00421281i	$-0.998659\pi$
$733$		−	167.208i		−	0.00842562i	0.999991	−	0.00421281i	$-0.00134098\pi$
$734$	−4615.33			−0.232091
$735$		0			0
$736$	−17843.7			−0.893653
$737$			22538.4i			1.12648i
$738$		0			0
$739$	−27695.8			−1.37863			−0.689313	−	0.724463i	$-0.742088\pi$
$739$	−27695.8			−1.37863			−0.689313	+	0.724463i	$0.742088\pi$
$740$	9963.77	+	13712.9i	0.494967	+	0.681209i
$741$		0			0
$742$			3616.99i			0.178954i
$743$		−	35379.4i		−	1.74689i	−0.486919	−	0.873447i	$-0.661879\pi$
$743$		−	35379.4i		−	1.74689i	0.486919	−	0.873447i	$-0.338121\pi$
$744$		0			0
$745$	1957.39	−	1422.24i	0.0962595	−	0.0699421i
$746$	−5163.81			−0.253432
$747$		0			0
$748$		−	8094.07i		−	0.395653i
$749$	1188.30			0.0579700
$750$		0			0
$751$	−5408.34			−0.262787			−0.131394	−	0.991330i	$-0.541945\pi$
$751$	−5408.34			−0.262787			−0.131394	+	0.991330i	$0.541945\pi$
$752$			15849.2i			0.768566i
$753$		0			0
$754$	8179.00			0.395042
$755$	24198.7	−	17582.7i	1.16646	−	0.847552i
$756$		0			0
$757$		−	701.285i		−	0.0336706i	−0.999858	−	0.0168353i	$-0.994641\pi$
$757$		−	701.285i		−	0.0336706i	0.999858	−	0.0168353i	$-0.00535909\pi$
$758$			5186.78i			0.248539i
$759$		0			0
$760$	2596.96	+	3574.12i	0.123949	+	0.170588i
$761$	−23271.0			−1.10850			−0.554252	−	0.832349i	$-0.686996\pi$
$761$	−23271.0			−1.10850			−0.554252	+	0.832349i	$0.686996\pi$
$762$		0			0
$763$			2580.70i			0.122448i
$764$	−2112.79			−0.100050
$765$		0			0
$766$	12491.7			0.589221
$767$		−	30346.4i		−	1.42861i
$768$		0			0
$769$	10456.3			0.490332			0.245166	−	0.969481i	$-0.421158\pi$
$769$	10456.3			0.490332			0.245166	+	0.969481i	$0.421158\pi$
$770$	1363.83	−	990.956i	0.0638298	−	0.0463787i
$771$		0			0
$772$		−	6078.44i		−	0.283378i
$773$			18324.8i			0.852650i	0.904570	+	0.426325i	$0.140192\pi$
$773$			18324.8i			0.852650i	−0.904570	+	0.426325i	$0.859808\pi$
$774$		0			0
$775$	−2683.47	+	8261.00i	−0.124378	+	0.382895i
$776$	26191.3			1.21161
$777$		0			0
$778$		−	9844.89i		−	0.453672i
$779$	10753.7			0.494598
$780$		0			0
$781$	−10818.1			−0.495649
$782$			7315.82i			0.334544i
$783$		0			0
$784$	8977.28			0.408950
$785$	6289.30	+	8655.79i	0.285955	+	0.393552i
$786$		0			0
$787$		−	14839.7i		−	0.672147i	−0.941836	−	0.336073i	$-0.890901\pi$
$787$		−	14839.7i		−	0.672147i	0.941836	−	0.336073i	$-0.109099\pi$
$788$		−	17744.3i		−	0.802178i
$789$		0			0
$790$	7611.90	+	10476.1i	0.342809	+	0.471799i
$791$	6412.14			0.288229
$792$		0			0
$793$			15853.3i			0.709919i
$794$	10181.1			0.455055
$795$		0			0
$796$	9844.13			0.438337
$797$		−	38086.1i		−	1.69269i	−0.532631	−	0.846347i	$-0.678797\pi$
$797$		−	38086.1i		−	1.69269i	0.532631	−	0.846347i	$-0.321203\pi$
$798$		0			0
$799$	32480.6			1.43815
$800$	−21472.5	−	6975.04i	−0.948957	−	0.308256i
$801$		0			0
$802$			5261.03i			0.231638i
$803$		−	19445.2i		−	0.854551i
$804$		0			0
$805$	5066.83	−	3681.56i	0.221842	−	0.161190i
$806$	−3560.05			−0.155580
$807$		0			0
$808$		−	12213.1i		−	0.531753i
$809$	37170.3			1.61537			0.807686	−	0.589612i	$-0.200719\pi$
$809$	37170.3			1.61537			0.807686	+	0.589612i	$0.200719\pi$
$810$		0			0
$811$	37402.9			1.61947			0.809736	−	0.586794i	$-0.199610\pi$
$811$	37402.9			1.61947			0.809736	+	0.586794i	$0.199610\pi$
$812$			5824.70i			0.251733i
$813$		0			0
$814$	6265.60			0.269790
$815$	−3142.05	−	4324.32i	−0.135044	−	0.185858i
$816$		0			0
$817$			2085.34i			0.0892983i
$818$		−	10408.3i		−	0.444885i
$819$		0			0
$820$	−28604.3	+	20783.9i	−1.21818	+	0.885128i
$821$	15002.4			0.637743			0.318872	−	0.947798i	$-0.396696\pi$
$821$	15002.4			0.637743			0.318872	+	0.947798i	$0.396696\pi$
$822$		0			0
$823$		−	11648.5i		−	0.493366i	−0.969096	−	0.246683i	$-0.920659\pi$
$823$		−	11648.5i		−	0.493366i	0.969096	−	0.246683i	$-0.0793406\pi$
$824$	−19314.8			−0.816579
$825$		0			0
$826$	−5257.74			−0.221477
$827$		−	46483.9i		−	1.95454i	−0.212003	−	0.977269i	$-0.567999\pi$
$827$		−	46483.9i		−	1.95454i	0.212003	−	0.977269i	$-0.432001\pi$
$828$		0			0
$829$	−1565.64			−0.0655933			−0.0327967	−	0.999462i	$-0.510441\pi$
$829$	−1565.64			−0.0655933			−0.0327967	+	0.999462i	$0.510441\pi$
$830$	8023.12	−	5829.60i	0.335526	−	0.243793i
$831$		0			0
$832$			207.113i			0.00863021i
$833$		−	18397.6i		−	0.765233i
$834$		0			0
$835$	17523.1	+	24116.6i	0.726242	+	0.999507i
$836$	−2992.27			−0.123791
$837$		0			0
$838$			8681.79i			0.357885i
$839$	32926.2			1.35487			0.677436	−	0.735582i	$-0.263091\pi$
$839$	32926.2			1.35487			0.677436	+	0.735582i	$0.263091\pi$
$840$		0			0
$841$	1096.90			0.0449754
$842$		−	12332.7i		−	0.504766i
$843$		0			0
$844$	−970.078			−0.0395633
$845$	4705.75	−	3419.19i	0.191577	−	0.139200i
$846$		0			0
$847$		−	4985.77i		−	0.202259i
$848$			14723.8i			0.596248i
$849$		0			0
$850$	−2859.72	+	8803.58i	−0.115397	+	0.355247i
$851$	23277.7			0.937660
$852$		0			0
$853$		−	14963.0i		−	0.600613i	−0.953843	−	0.300306i	$-0.902911\pi$
$853$		−	14963.0i		−	0.600613i	0.953843	−	0.300306i	$-0.0970889\pi$
$854$	2746.70			0.110059
$855$		0			0
$856$	−3784.81			−0.151124
$857$			21868.7i			0.871668i	0.900027	+	0.435834i	$0.143546\pi$
$857$			21868.7i			0.871668i	−0.900027	+	0.435834i	$0.856454\pi$
$858$		0			0
$859$	−13542.5			−0.537909			−0.268954	−	0.963153i	$-0.586678\pi$
$859$	−13542.5			−0.537909			−0.268954	+	0.963153i	$0.586678\pi$
$860$	−4030.36	−	5546.88i	−0.159807	−	0.219938i
$861$		0			0
$862$		−	3770.29i		−	0.148975i
$863$			2080.25i			0.0820539i	0.999158	+	0.0410269i	$0.0130629\pi$
$863$			2080.25i			0.0820539i	−0.999158	+	0.0410269i	$0.986937\pi$
$864$		0			0
$865$	−3237.28	−	4455.39i	−0.127250	−	0.175130i
$866$	−4882.33			−0.191580
$867$		0			0
$868$		−	2535.30i		−	0.0991401i
$869$	−19674.9			−0.768039
$870$		0			0
$871$	−43423.1			−1.68925
$872$		−	8219.67i		−	0.319212i
$873$		0			0
$874$	2704.56			0.104672
$875$	7536.34	−	2449.64i	0.291171	−	0.0946435i
$876$		0			0
$877$		−	28076.9i		−	1.08106i	−0.841325	−	0.540530i	$-0.818224\pi$
$877$		−	28076.9i		−	1.08106i	0.841325	−	0.540530i	$-0.181776\pi$
$878$			3965.13i			0.152411i
$879$		0			0
$880$	5551.78	−	4033.92i	0.212671	−	0.154527i
$881$	−2105.32			−0.0805110			−0.0402555	−	0.999189i	$-0.512817\pi$
$881$	−2105.32			−0.0805110			−0.0402555	+	0.999189i	$0.512817\pi$
$882$		0			0
$883$			23829.7i			0.908190i	0.890953	+	0.454095i	$0.150037\pi$
$883$			23829.7i			0.908190i	−0.890953	+	0.454095i	$0.849963\pi$
$884$	15594.2			0.593315
$885$		0			0
$886$	13428.4			0.509184
$887$			12634.7i			0.478277i	0.970985	+	0.239139i	$0.0768651\pi$
$887$			12634.7i			0.478277i	−0.970985	+	0.239139i	$0.923135\pi$
$888$		0			0
$889$	−7244.48			−0.273310
$890$	2249.05	+	3095.30i	0.0847058	+	0.116578i
$891$		0			0
$892$		−	16561.7i		−	0.621667i
$893$		−	12007.6i		−	0.449967i
$894$		0			0
$895$	−35188.8	+	25568.2i	−1.31422	+	0.954915i
$896$	8229.01			0.306821
$897$		0			0
$898$		−	368.481i		−	0.0136931i
$899$	−11093.2			−0.411544
$900$		0			0
$901$	30174.3			1.11571
$902$			13069.7i			0.482454i
$903$		0			0
$904$	−20423.0			−0.751393
$905$	−39540.2	+	28729.9i	−1.45233	+	1.05527i
$906$		0			0
$907$		−	31942.7i		−	1.16939i	−0.811252	−	0.584697i	$-0.801213\pi$
$907$		−	31942.7i		−	1.16939i	0.811252	−	0.584697i	$-0.198787\pi$
$908$			16032.5i			0.585965i
$909$		0			0
$910$	1909.20	+	2627.58i	0.0695488	+	0.0957181i
$911$	22605.6			0.822128			0.411064	−	0.911607i	$-0.365157\pi$
$911$	22605.6			0.822128			0.411064	+	0.911607i	$0.365157\pi$
$912$		0			0
$913$			15068.1i			0.546201i
$914$	6780.96			0.245398
$915$		0			0
$916$	−9011.95			−0.325069
$917$			7701.87i			0.277359i
$918$		0			0
$919$	−45168.0			−1.62128			−0.810638	−	0.585547i	$-0.800880\pi$
$919$	−45168.0			−1.62128			−0.810638	+	0.585547i	$0.800880\pi$
$920$	−16138.2	+	11726.0i	−0.578325	+	0.420211i
$921$		0			0
$922$		−	1254.51i		−	0.0448102i
$923$		−	20842.4i		−	0.743268i
$924$		0			0
$925$	28011.5	+	9099.15i	0.995688	+	0.323436i
$926$	−17035.2			−0.604547
$927$		0			0
$928$		−	28834.0i		−	1.01996i
$929$	−32891.6			−1.16161			−0.580806	−	0.814042i	$-0.697262\pi$
$929$	−32891.6			−1.16161			−0.580806	+	0.814042i	$0.697262\pi$
$930$		0			0
$931$	−6801.34			−0.239425
$932$		−	20969.3i		−	0.736989i
$933$		0			0
$934$	1670.98			0.0585397
$935$	−8266.93	−	11377.6i	−0.289153	−	0.397953i
$936$		0			0
$937$		−	34702.8i		−	1.20992i	−0.796257	−	0.604958i	$-0.793190\pi$
$937$		−	34702.8i		−	1.20992i	0.796257	−	0.604958i	$-0.206810\pi$
$938$			7523.37i			0.261884i
$939$		0			0
$940$	23207.4	+	31939.7i	0.805256	+	1.10825i
$941$	−27172.9			−0.941351			−0.470675	−	0.882306i	$-0.655990\pi$
$941$	−27172.9			−0.941351			−0.470675	+	0.882306i	$0.655990\pi$
$942$		0			0
$943$			48556.0i			1.67678i
$944$	−21402.9			−0.737929
$945$		0			0
$946$	−2534.44			−0.0871056
$947$			39846.1i			1.36729i	0.729815	+	0.683645i	$0.239606\pi$
$947$			39846.1i			1.36729i	−0.729815	+	0.683645i	$0.760394\pi$
$948$		0			0
$949$	37463.5			1.28147
$950$	3254.56	+	1057.20i	0.111149	+	0.0361054i
$951$		0			0
$952$		−	6060.95i		−	0.206341i
$953$			45983.5i			1.56301i	0.623896	+	0.781507i	$0.285549\pi$
$953$			45983.5i			1.56301i	−0.623896	+	0.781507i	$0.714451\pi$
$954$		0			0
$955$	−2969.88	+	2157.92i	−0.100632	+	0.0731189i
$956$	10686.5			0.361534
$957$		0			0
$958$			17331.5i			0.584503i
$959$	5423.88			0.182634
$960$		0			0
$961$	−24962.5			−0.837921
$962$			12071.4i			0.404573i
$963$		0			0
$964$	38389.8			1.28263
$965$	−6208.25	−	8544.25i	−0.207099	−	0.285025i
$966$		0			0
$967$		−	52188.1i		−	1.73553i	−0.496976	−	0.867765i	$-0.665556\pi$
$967$		−	52188.1i		−	1.73553i	0.496976	−	0.867765i	$-0.334444\pi$
$968$			15880.0i			0.527274i
$969$		0			0
$970$	16411.7	−	11924.7i	0.543245	−	0.394722i
$971$	2777.37			0.0917921			0.0458961	−	0.998946i	$-0.485386\pi$
$971$	2777.37			0.0917921			0.0458961	+	0.998946i	$0.485386\pi$
$972$		0			0
$973$		−	154.954i		−	0.00510545i
$974$	7648.77			0.251625
$975$		0			0
$976$	11181.1			0.366698
$977$			33242.3i			1.08855i	0.838907	+	0.544275i	$0.183195\pi$
$977$			33242.3i			1.08855i	−0.838907	+	0.544275i	$0.816805\pi$
$978$		0			0
$979$	−5813.24			−0.189777
$980$	18091.2	−	13145.1i	0.589696	−	0.428473i
$981$		0			0
$982$		−	20464.9i		−	0.665033i
$983$			18130.3i			0.588268i	0.955764	+	0.294134i	$0.0950313\pi$
$983$			18130.3i			0.588268i	−0.955764	+	0.294134i	$0.904969\pi$
$984$		0			0
$985$	−18123.3	−	24942.6i	−0.586250	−	0.806841i
$986$	−11821.8			−0.381827
$987$		0			0
$988$		−	5764.97i		−	0.185636i
$989$	−9415.87			−0.302737
$990$		0			0
$991$	30597.2			0.980778			0.490389	−	0.871504i	$-0.336855\pi$
$991$	30597.2			0.980778			0.490389	+	0.871504i	$0.336855\pi$
$992$			12550.5i			0.401692i
$993$		0			0
$994$	−3611.10			−0.115229
$995$	13837.6	−	10054.4i	0.440884	−	0.320347i
$996$		0			0
$997$			23839.4i			0.757273i	0.925545	+	0.378637i	$0.123607\pi$
$997$			23839.4i			0.757273i	−0.925545	+	0.378637i	$0.876393\pi$
$998$		−	11589.4i		−	0.367592i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	135.4.b.c.109.6	yes	12
3.2	odd	2	inner	135.4.b.c.109.7	yes	12
5.2	odd	4		675.4.a.bb.1.4		6
5.3	odd	4		675.4.a.bc.1.3		6
5.4	even	2	inner	135.4.b.c.109.8	yes	12
15.2	even	4		675.4.a.bb.1.3		6
15.8	even	4		675.4.a.bc.1.4		6
15.14	odd	2	inner	135.4.b.c.109.5	✓	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
135.4.b.c.109.5	✓	12	15.14	odd	2	inner
135.4.b.c.109.6	yes	12	1.1	even	1	trivial
135.4.b.c.109.7	yes	12	3.2	odd	2	inner
135.4.b.c.109.8	yes	12	5.4	even	2	inner
675.4.a.bb.1.3		6	15.2	even	4
675.4.a.bb.1.4		6	5.2	odd	4
675.4.a.bc.1.3		6	5.3	odd	4
675.4.a.bc.1.4		6	15.8	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

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

\(f(q)\)	\(=\)	\(q-1.25118i q^{2} +6.43456 q^{4} +(9.04484 - 6.57198i) q^{5} +5.67029i q^{7} -18.0602i q^{8} +(-8.22271 - 11.3167i) q^{10} +21.2537 q^{11} +40.9479i q^{13} +7.09453 q^{14} +28.8800 q^{16} -59.1853i q^{17} -21.8800 q^{19} +(58.1995 - 42.2878i) q^{20} -26.5922i q^{22} -98.7941i q^{23} +(38.6182 - 118.885i) q^{25} +51.2331 q^{26} +36.4858i q^{28} +159.643 q^{29} -69.4873 q^{31} -180.615i q^{32} -74.0512 q^{34} +(37.2650 + 51.2868i) q^{35} +235.618i q^{37} +27.3757i q^{38} +(-118.691 - 163.351i) q^{40} -491.487 q^{41} -95.3080i q^{43} +136.758 q^{44} -123.609 q^{46} +548.796i q^{47} +310.848 q^{49} +(-148.746 - 48.3182i) q^{50} +263.482i q^{52} +509.829i q^{53} +(192.236 - 139.679i) q^{55} +102.406 q^{56} -199.742i q^{58} -741.098 q^{59} +387.157 q^{61} +86.9409i q^{62} +5.05795 q^{64} +(269.109 + 370.367i) q^{65} +1060.45i q^{67} -380.831i q^{68} +(64.1689 - 46.6251i) q^{70} -508.998 q^{71} -914.906i q^{73} +294.800 q^{74} -140.788 q^{76} +120.515i q^{77} -925.717 q^{79} +(261.215 - 189.799i) q^{80} +614.937i q^{82} +708.964i q^{83} +(-388.964 - 535.321i) q^{85} -119.247 q^{86} -383.846i q^{88} -273.516 q^{89} -232.186 q^{91} -635.696i q^{92} +686.641 q^{94} +(-197.901 + 143.795i) q^{95} +1450.22i q^{97} -388.926i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 60 q^{4} - 36 q^{10} + 84 q^{16} - 348 q^{25} + 252 q^{31} + 1068 q^{34} + 1320 q^{40} - 1668 q^{46} - 2868 q^{49} + 684 q^{55} + 792 q^{61} + 2268 q^{64} - 5652 q^{70} + 1824 q^{76} + 2196 q^{79}+ \cdots - 5148 q^{94}+O(q^{100}) \) 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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