LMFDB - Embedded newform 324.4.i.a.37.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [324,4,Mod(37,324)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(324, base_ring=CyclotomicField(18))

chi = DirichletCharacter(H, H._module([0, 14]))

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("324.37");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 324 = 2^{2} \cdot 3^{4} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	324.i (of order $9$, degree $6$, not minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$19.1166188419$
Analytic rank:	$0$
Dimension:	$54$
Relative dimension:	$9$ over $\Q(\zeta_{9})$
Twist minimal:	no (minimal twist has level 108)
Sato-Tate group:	$\mathrm{SU}(2)[C_{9}]$

Embedding invariants

Embedding label			37.1
Character	$\chi$	$=$	324.37
Dual form			324.4.i.a.289.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-19.7528 + 7.18944i) q^{5} +(0.723940 + 4.10567i) q^{7} +O(q^{10})$	$q+(-19.7528 + 7.18944i) q^{5} +(0.723940 + 4.10567i) q^{7} +(5.68204 + 2.06810i) q^{11} +(22.8525 + 19.1756i) q^{13} +(-48.7365 + 84.4141i) q^{17} +(-63.0708 - 109.242i) q^{19} +(26.2243 - 148.725i) q^{23} +(242.731 - 203.675i) q^{25} +(179.445 - 150.572i) q^{29} +(-2.62143 + 14.8669i) q^{31} +(-43.8173 - 75.8939i) q^{35} +(-0.488402 + 0.845938i) q^{37} +(-185.869 - 155.962i) q^{41} +(31.0450 + 11.2995i) q^{43} +(-66.6849 - 378.189i) q^{47} +(305.982 - 111.368i) q^{49} -245.113 q^{53} -127.105 q^{55} +(399.605 - 145.444i) q^{59} +(134.131 + 760.693i) q^{61} +(-589.264 - 214.475i) q^{65} +(-321.243 - 269.555i) q^{67} +(113.989 - 197.435i) q^{71} +(265.093 + 459.155i) q^{73} +(-4.37745 + 24.8258i) q^{77} +(848.570 - 712.034i) q^{79} +(-406.829 + 341.370i) q^{83} +(355.794 - 2017.81i) q^{85} +(-84.1224 - 145.704i) q^{89} +(-62.1846 + 107.707i) q^{91} +(2031.21 + 1704.39i) q^{95} +(-1375.96 - 500.807i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 54 q - 12 q^{5}+O(q^{10}) $ 54 * q - 12 * q^5	$ 54 q - 12 q^{5} + 87 q^{11} - 204 q^{17} - 96 q^{23} - 216 q^{25} - 318 q^{29} - 54 q^{31} - 6 q^{35} - 867 q^{41} - 513 q^{43} + 1548 q^{47} + 594 q^{49} + 1068 q^{53} + 1218 q^{59} - 54 q^{61} - 96 q^{65} - 2997 q^{67} + 120 q^{71} - 216 q^{73} - 3480 q^{77} + 2808 q^{79} - 4464 q^{83} + 2160 q^{85} - 4029 q^{89} + 270 q^{91} + 1650 q^{95} - 3483 q^{97}+O(q^{100}) $ 54 * q - 12 * q^5 + 87 * q^11 - 204 * q^17 - 96 * q^23 - 216 * q^25 - 318 * q^29 - 54 * q^31 - 6 * q^35 - 867 * q^41 - 513 * q^43 + 1548 * q^47 + 594 * q^49 + 1068 * q^53 + 1218 * q^59 - 54 * q^61 - 96 * q^65 - 2997 * q^67 + 120 * q^71 - 216 * q^73 - 3480 * q^77 + 2808 * q^79 - 4464 * q^83 + 2160 * q^85 - 4029 * q^89 + 270 * q^91 + 1650 * q^95 - 3483 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/324\mathbb{Z}\right)^\times$.

$n$	$163$	$245$
$\chi(n)$	$1$	$e\left(\frac{7}{9}\right)$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	−19.7528	+	7.18944i	−1.76675	+	0.643043i	−0.766747	+	0.641949i	$0.778126\pi$
$5$	−19.7528	+	7.18944i	−1.76675	+	0.643043i	−0.999999	+	0.00109400i	$0.999652\pi$
$6$		0			0
$7$	0.723940	+	4.10567i	0.0390891	+	0.221685i	0.998095	−	0.0617021i	$-0.0196529\pi$
$7$	0.723940	+	4.10567i	0.0390891	+	0.221685i	−0.959006	+	0.283387i	$0.908542\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	5.68204	+	2.06810i	0.155746	+	0.0566867i	0.418716	−	0.908117i	$-0.362480\pi$
$11$	5.68204	+	2.06810i	0.155746	+	0.0566867i	−0.262971	+	0.964804i	$0.584702\pi$
$12$		0			0
$13$	22.8525	+	19.1756i	0.487550	+	0.409103i	0.853147	−	0.521670i	$-0.174691\pi$
$13$	22.8525	+	19.1756i	0.487550	+	0.409103i	−0.365597	+	0.930773i	$0.619135\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−48.7365	+	84.4141i	−0.695314	+	1.20432i	0.274760	+	0.961513i	$0.411401\pi$
$17$	−48.7365	+	84.4141i	−0.695314	+	1.20432i	−0.970075	+	0.242807i	$0.921932\pi$
$18$		0			0
$19$	−63.0708	−	109.242i	−0.761549	−	1.31904i	−0.942052	−	0.335467i	$-0.891106\pi$
$19$	−63.0708	−	109.242i	−0.761549	−	1.31904i	0.180503	−	0.983574i	$-0.442227\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	26.2243	−	148.725i	0.237745	−	1.34832i	−0.599009	−	0.800743i	$-0.704439\pi$
$23$	26.2243	−	148.725i	0.237745	−	1.34832i	0.836754	−	0.547579i	$-0.184450\pi$
$24$		0			0
$25$	242.731	−	203.675i	1.94184	−	1.62940i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	179.445	−	150.572i	1.14904	−	0.964155i	0.149339	−	0.988786i	$-0.452285\pi$
$29$	179.445	−	150.572i	1.14904	−	0.964155i	0.999697	+	0.0246306i	$0.00784096\pi$
$30$		0			0
$31$	−2.62143	+	14.8669i	−0.0151879	+	0.0861346i	−0.991460	−	0.130415i	$-0.958369\pi$
$31$	−2.62143	+	14.8669i	−0.0151879	+	0.0861346i	0.976272	+	0.216549i	$0.0694802\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−43.8173	−	75.8939i	−0.211614	−	0.366526i
$36$		0			0
$37$	−0.488402	+	0.845938i	−0.00217008	+	0.00375868i	−0.867108	−	0.498119i	$-0.834024\pi$
$37$	−0.488402	+	0.845938i	−0.00217008	+	0.00375868i	0.864938	+	0.501878i	$0.167357\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−185.869	−	155.962i	−0.707995	−	0.594078i	0.216041	−	0.976384i	$-0.430686\pi$
$41$	−185.869	−	155.962i	−0.707995	−	0.594078i	−0.924036	+	0.382306i	$0.875130\pi$
$42$		0			0
$43$	31.0450	+	11.2995i	0.110101	+	0.0400733i	0.396483	−	0.918042i	$-0.370231\pi$
$43$	31.0450	+	11.2995i	0.110101	+	0.0400733i	−0.286382	+	0.958115i	$0.592453\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−66.6849	−	378.189i	−0.206957	−	1.17371i	−0.894330	−	0.447409i	$-0.852347\pi$
$47$	−66.6849	−	378.189i	−0.206957	−	1.17371i	0.687372	−	0.726305i	$-0.258764\pi$
$48$		0			0
$49$	305.982	−	111.368i	0.892076	−	0.324689i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−245.113			−0.635261			−0.317631	−	0.948215i	$-0.602887\pi$
$53$	−245.113			−0.635261			−0.317631	+	0.948215i	$0.602887\pi$
$54$		0			0
$55$	−127.105			−0.311615
$56$		0			0
$57$		0			0
$58$		0			0
$59$	399.605	−	145.444i	0.881765	−	0.320936i	0.138843	−	0.990314i	$-0.455662\pi$
$59$	399.605	−	145.444i	0.881765	−	0.320936i	0.742922	+	0.669378i	$0.233439\pi$
$60$		0			0
$61$	134.131	+	760.693i	0.281536	+	1.59667i	0.717403	+	0.696658i	$0.245331\pi$
$61$	134.131	+	760.693i	0.281536	+	1.59667i	−0.435867	+	0.900011i	$0.643558\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−589.264	−	214.475i	−1.12445	−	0.409266i
$66$		0			0
$67$	−321.243	−	269.555i	−0.585762	−	0.491512i	0.301072	−	0.953601i	$-0.402656\pi$
$67$	−321.243	−	269.555i	−0.585762	−	0.491512i	−0.886833	+	0.462089i	$0.847100\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	113.989	−	197.435i	0.190535	−	0.330016i	−0.754893	−	0.655848i	$-0.772311\pi$
$71$	113.989	−	197.435i	0.190535	−	0.330016i	0.945428	+	0.325832i	$0.105644\pi$
$72$		0			0
$73$	265.093	+	459.155i	0.425025	+	0.736165i	0.996423	−	0.0845083i	$-0.0269320\pi$
$73$	265.093	+	459.155i	0.425025	+	0.736165i	−0.571398	+	0.820673i	$0.693599\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−4.37745	+	24.8258i	−0.00647866	+	0.0367423i
$78$		0			0
$79$	848.570	−	712.034i	1.20850	−	1.01405i	0.209155	−	0.977883i	$-0.432929\pi$
$79$	848.570	−	712.034i	1.20850	−	1.01405i	0.999346	−	0.0361698i	$-0.0115157\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−406.829	+	341.370i	−0.538015	+	0.451449i	−0.870859	−	0.491534i	$-0.836436\pi$
$83$	−406.829	+	341.370i	−0.538015	+	0.451449i	0.332843	+	0.942982i	$0.391992\pi$
$84$		0			0
$85$	355.794	−	2017.81i	0.454015	−	2.57484i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−84.1224	−	145.704i	−0.100190	−	0.173535i	0.811573	−	0.584252i	$-0.198612\pi$
$89$	−84.1224	−	145.704i	−0.100190	−	0.173535i	−0.911763	+	0.410717i	$0.865279\pi$
$90$		0			0
$91$	−62.1846	+	107.707i	−0.0716343	+	0.124074i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	2031.21	+	1704.39i	2.19366	+	1.84070i
$96$		0			0
$97$	−1375.96	−	500.807i	−1.44028	−	0.524219i	−0.500421	−	0.865782i	$-0.666822\pi$
$97$	−1375.96	−	500.807i	−1.44028	−	0.524219i	−0.939859	+	0.341563i	$0.889044\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−180.078	−	1021.27i	−0.177410	−	1.00614i	−0.935325	−	0.353788i	$-0.884893\pi$
$101$	−180.078	−	1021.27i	−0.177410	−	1.00614i	0.757916	−	0.652353i	$-0.226218\pi$
$102$		0			0
$103$	159.205	−	57.9460i	0.152301	−	0.0554329i	−0.264745	−	0.964318i	$-0.585288\pi$
$103$	159.205	−	57.9460i	0.152301	−	0.0554329i	0.417046	+	0.908886i	$0.363066\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	125.729			0.113595			0.0567974	−	0.998386i	$-0.481911\pi$
$107$	125.729			0.113595			0.0567974	+	0.998386i	$0.481911\pi$
$108$		0			0
$109$	−722.542			−0.634927			−0.317463	−	0.948271i	$-0.602831\pi$
$109$	−722.542			−0.634927			−0.317463	+	0.948271i	$0.602831\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	372.502	−	135.580i	0.310107	−	0.112870i	−0.182279	−	0.983247i	$-0.558347\pi$
$113$	372.502	−	135.580i	0.310107	−	0.112870i	0.492385	+	0.870377i	$0.336125\pi$
$114$		0			0
$115$	551.248	+	3126.29i	0.446993	+	2.53502i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−381.859	−	138.985i	−0.294159	−	0.107065i
$120$		0			0
$121$	−991.597	−	832.048i	−0.745001	−	0.625130i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−2016.52	+	3492.72i	−1.44291	+	2.49919i
$126$		0			0
$127$	24.8188	+	42.9874i	0.0173410	+	0.0300356i	0.874566	−	0.484907i	$-0.161147\pi$
$127$	24.8188	+	42.9874i	0.0173410	+	0.0300356i	−0.857225	+	0.514943i	$0.827813\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−142.167	+	806.266i	−0.0948179	+	0.537739i	0.899985	+	0.435921i	$0.143577\pi$
$131$	−142.167	+	806.266i	−0.0948179	+	0.537739i	−0.994803	+	0.101818i	$0.967534\pi$
$132$		0			0
$133$	402.851	−	338.032i	0.262644	−	0.220384i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	446.492	−	374.651i	0.278441	−	0.233639i	−0.492863	−	0.870107i	$-0.664050\pi$
$137$	446.492	−	374.651i	0.278441	−	0.233639i	0.771303	+	0.636468i	$0.219605\pi$
$138$		0			0
$139$	−330.670	+	1875.32i	−0.201777	+	1.14433i	0.700654	+	0.713501i	$0.252892\pi$
$139$	−330.670	+	1875.32i	−0.201777	+	1.14433i	−0.902431	+	0.430834i	$0.858219\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	90.1923	+	156.218i	0.0527431	+	0.0913537i
$144$		0			0
$145$	−2462.01	+	4264.33i	−1.41006	+	2.44230i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−979.169	−	821.621i	−0.538367	−	0.451744i	0.332612	−	0.943064i	$-0.392070\pi$
$149$	−979.169	−	821.621i	−0.538367	−	0.451744i	−0.870979	+	0.491320i	$0.836514\pi$
$150$		0			0
$151$	2893.47	+	1053.14i	1.55939	+	0.567571i	0.970596	−	0.240713i	$-0.0773814\pi$
$151$	2893.47	+	1053.14i	1.55939	+	0.567571i	0.588792	+	0.808284i	$0.299604\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−55.1039	−	312.510i	−0.0285552	−	0.161944i
$156$		0			0
$157$	−909.442	+	331.010i	−0.462302	+	0.168264i	−0.562662	−	0.826687i	$-0.690223\pi$
$157$	−909.442	+	331.010i	−0.462302	+	0.168264i	0.100360	+	0.994951i	$0.468000\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	629.602			0.308196
$162$		0			0
$163$	−3732.36			−1.79350			−0.896752	−	0.442534i	$-0.854079\pi$
$163$	−3732.36			−1.79350			−0.896752	+	0.442534i	$0.854079\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	3395.70	−	1235.93i	1.57346	−	0.572691i	0.599687	−	0.800235i	$-0.295292\pi$
$167$	3395.70	−	1235.93i	1.57346	−	0.572691i	0.973768	+	0.227544i	$0.0730695\pi$
$168$		0			0
$169$	−226.968	−	1287.20i	−0.103308	−	0.585891i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	949.268	+	345.505i	0.417176	+	0.151840i	0.542076	−	0.840330i	$-0.317639\pi$
$173$	949.268	+	345.505i	0.417176	+	0.151840i	−0.124899	+	0.992169i	$0.539861\pi$
$174$		0			0
$175$	1011.95	+	849.123i	0.437119	+	0.366787i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−2154.60	+	3731.88i	−0.899678	+	1.55829i	−0.0717728	+	0.997421i	$0.522866\pi$
$179$	−2154.60	+	3731.88i	−0.899678	+	1.55829i	−0.827906	+	0.560868i	$0.810468\pi$
$180$		0			0
$181$	−1997.35	−	3459.52i	−0.820232	−	1.42068i	−0.905509	−	0.424327i	$-0.860511\pi$
$181$	−1997.35	−	3459.52i	−0.820232	−	1.42068i	0.0852767	−	0.996357i	$-0.472823\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	3.56551	−	20.2210i	0.00141698	−	0.00803609i
$186$		0			0
$187$	−451.500	+	378.853i	−0.176561	+	0.148152i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	3232.23	−	2712.16i	1.22448	−	1.02746i	0.225903	−	0.974150i	$-0.427467\pi$
$191$	3232.23	−	2712.16i	1.22448	−	1.02746i	0.998578	−	0.0533113i	$-0.0169776\pi$
$192$		0			0
$193$	399.764	−	2267.17i	0.149097	−	0.845569i	−0.814890	−	0.579616i	$-0.803202\pi$
$193$	399.764	−	2267.17i	0.149097	−	0.845569i	0.963986	−	0.265952i	$-0.0856864\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	709.523	+	1228.93i	0.256606	+	0.444455i	0.965331	−	0.261030i	$-0.0840623\pi$
$197$	709.523	+	1228.93i	0.256606	+	0.444455i	−0.708724	+	0.705486i	$0.750729\pi$
$198$		0			0
$199$	994.545	−	1722.60i	0.354278	−	0.613628i	−0.632716	−	0.774384i	$-0.718060\pi$
$199$	994.545	−	1722.60i	0.354278	−	0.613628i	0.986994	+	0.160756i	$0.0513932\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	748.106	+	627.735i	0.258654	+	0.217036i
$204$		0			0
$205$	4792.71	+	1744.40i	1.63287	+	0.594314i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−132.449	−	751.153i	−0.0438357	−	0.248605i
$210$		0			0
$211$	280.745	−	102.183i	0.0915985	−	0.0333391i	−0.295814	−	0.955245i	$-0.595591\pi$
$211$	280.745	−	102.183i	0.0915985	−	0.0333391i	0.387413	+	0.921906i	$0.373369\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−694.464			−0.220289
$216$		0			0
$217$	−62.9363			−0.0196884
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−2732.44	+	994.527i	−0.831692	+	0.302711i
$222$		0			0
$223$	−559.273	−	3171.80i	−0.167945	−	0.952463i	−0.945976	−	0.324237i	$-0.894893\pi$
$223$	−559.273	−	3171.80i	−0.167945	−	0.952463i	0.778031	−	0.628226i	$-0.216219\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−3153.27	−	1147.70i	−0.921983	−	0.335574i	−0.162956	−	0.986633i	$-0.552103\pi$
$227$	−3153.27	−	1147.70i	−0.921983	−	0.335574i	−0.759027	+	0.651059i	$0.774325\pi$
$228$		0			0
$229$	−3053.24	−	2561.97i	−0.881064	−	0.739301i	0.0853333	−	0.996352i	$-0.472804\pi$
$229$	−3053.24	−	2561.97i	−0.881064	−	0.739301i	−0.966398	+	0.257052i	$0.917249\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−180.603	+	312.813i	−0.0507797	+	0.0879531i	−0.890298	−	0.455378i	$-0.849504\pi$
$233$	−180.603	+	312.813i	−0.0507797	+	0.0879531i	0.839518	+	0.543331i	$0.182837\pi$
$234$		0			0
$235$	4036.18	+	6990.88i	1.12039	+	1.94057i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	66.7780	−	378.717i	0.0180733	−	0.102499i	−0.974437	−	0.224662i	$-0.927872\pi$
$239$	66.7780	−	378.717i	0.0180733	−	0.102499i	0.992510	+	0.122164i	$0.0389833\pi$
$240$		0			0
$241$	−1868.30	+	1567.69i	−0.499368	+	0.419019i	−0.857369	−	0.514702i	$-0.827903\pi$
$241$	−1868.30	+	1567.69i	−0.499368	+	0.419019i	0.358002	+	0.933721i	$0.383458\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−5243.34	+	4399.68i	−1.36728	+	1.14729i
$246$		0			0
$247$	653.445	−	3705.87i	0.168331	−	0.954651i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−3340.30	−	5785.57i	−0.839992	−	1.45491i	−0.889900	−	0.456155i	$-0.849226\pi$
$251$	−3340.30	−	5785.57i	−0.839992	−	1.45491i	0.0499080	−	0.998754i	$-0.484107\pi$
$252$		0			0
$253$	456.586	−	790.830i	0.113460	−	0.196518i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	3562.33	+	2989.15i	0.864638	+	0.725517i	0.962962	−	0.269637i	$-0.0869037\pi$
$257$	3562.33	+	2989.15i	0.864638	+	0.725517i	−0.0983240	+	0.995154i	$0.531348\pi$
$258$		0			0
$259$	−3.82671	−	1.39281i	−0.000918071	−	0.000334151i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−407.176	−	2309.21i	−0.0954659	−	0.541414i	−0.994604	−	0.103748i	$-0.966916\pi$
$263$	−407.176	−	2309.21i	−0.0954659	−	0.541414i	0.899138	−	0.437666i	$-0.144195\pi$
$264$		0			0
$265$	4841.67	−	1762.22i	1.12235	−	0.408500i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	5351.31			1.21292			0.606459	−	0.795115i	$-0.292589\pi$
$269$	5351.31			1.21292			0.606459	+	0.795115i	$0.292589\pi$
$270$		0			0
$271$	2895.83			0.649112			0.324556	−	0.945867i	$-0.394785\pi$
$271$	2895.83			0.649112			0.324556	+	0.945867i	$0.394785\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	1800.43	−	655.301i	0.394799	−	0.143695i
$276$		0			0
$277$	−461.311	−	2616.22i	−0.100063	−	0.567486i	−0.993078	−	0.117458i	$-0.962525\pi$
$277$	−461.311	−	2616.22i	−0.100063	−	0.567486i	0.893015	−	0.450027i	$-0.148586\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−3820.36	−	1390.50i	−0.811046	−	0.295196i	−0.0969900	−	0.995285i	$-0.530921\pi$
$281$	−3820.36	−	1390.50i	−0.811046	−	0.295196i	−0.714056	+	0.700089i	$0.753144\pi$
$282$		0			0
$283$	1665.70	+	1397.69i	0.349879	+	0.293583i	0.800741	−	0.599010i	$-0.204439\pi$
$283$	1665.70	+	1397.69i	0.349879	+	0.293583i	−0.450863	+	0.892593i	$0.648884\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	505.771	−	876.022i	0.104024	−	0.180174i
$288$		0			0
$289$	−2294.00	−	3973.32i	−0.466924	−	0.808736i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	705.493	−	4001.05i	0.140667	−	0.797760i	−0.830078	−	0.557647i	$-0.811704\pi$
$293$	705.493	−	4001.05i	0.140667	−	0.797760i	0.970745	−	0.240113i	$-0.0771846\pi$
$294$		0			0
$295$	−6847.67	+	5745.87i	−1.35148	+	1.13403i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	3451.18	−	2895.89i	0.667516	−	0.560112i
$300$		0			0
$301$	−23.9171	+	135.641i	−0.00457993	+	0.0259741i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−8118.42	−	14061.5i	−1.52413	−	2.63987i
$306$		0			0
$307$	3875.27	−	6712.17i	0.720435	−	1.24783i	−0.240391	−	0.970676i	$-0.577276\pi$
$307$	3875.27	−	6712.17i	0.720435	−	1.24783i	0.960826	−	0.277154i	$-0.0893912\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−4511.08	−	3785.24i	−0.822507	−	0.690166i	0.131050	−	0.991376i	$-0.458165\pi$
$311$	−4511.08	−	3785.24i	−0.822507	−	0.690166i	−0.953558	+	0.301210i	$0.902609\pi$
$312$		0			0
$313$	2898.37	+	1054.92i	0.523404	+	0.190504i	0.590191	−	0.807264i	$-0.299052\pi$
$313$	2898.37	+	1054.92i	0.523404	+	0.190504i	−0.0667865	+	0.997767i	$0.521275\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−73.4339	−	416.464i	−0.0130109	−	0.0737885i	0.977611	−	0.210421i	$-0.0674833\pi$
$317$	−73.4339	−	416.464i	−0.0130109	−	0.0737885i	−0.990622	+	0.136632i	$0.956372\pi$
$318$		0			0
$319$	1331.01	−	484.448i	0.233612	−	0.0850278i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	12295.4			2.11806
$324$		0			0
$325$	9452.60			1.61334
$326$		0			0
$327$		0			0
$328$		0			0
$329$	1504.44	−	547.573i	0.252105	−	0.0917588i
$330$		0			0
$331$	−1281.12	−	7265.62i	−0.212740	−	1.20651i	−0.884785	−	0.465999i	$-0.845695\pi$
$331$	−1281.12	−	7265.62i	−0.212740	−	1.20651i	0.672045	−	0.740510i	$-0.265416\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	8283.40	+	3014.91i	1.35096	+	0.491708i
$336$		0			0
$337$	−1222.42	−	1025.74i	−0.197596	−	0.165802i	0.538621	−	0.842548i	$-0.318945\pi$
$337$	−1222.42	−	1025.74i	−0.197596	−	0.165802i	−0.736217	+	0.676745i	$0.763390\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−45.6412	+	79.0529i	−0.00724813	+	0.0125541i
$342$		0			0
$343$	1393.74	+	2414.03i	0.219402	+	0.380015i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−1332.82	+	7558.82i	−0.206195	+	1.16939i	0.689352	+	0.724426i	$0.257895\pi$
$347$	−1332.82	+	7558.82i	−0.206195	+	1.16939i	−0.895548	+	0.444966i	$0.853216\pi$
$348$		0			0
$349$	3679.04	−	3087.08i	0.564282	−	0.473489i	−0.315461	−	0.948939i	$-0.602159\pi$
$349$	3679.04	−	3087.08i	0.564282	−	0.473489i	0.879743	+	0.475450i	$0.157715\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−3351.24	+	2812.02i	−0.505293	+	0.423991i	−0.859469	−	0.511187i	$-0.829206\pi$
$353$	−3351.24	+	2812.02i	−0.505293	+	0.423991i	0.354176	+	0.935179i	$0.384761\pi$
$354$		0			0
$355$	−832.159	+	4719.41i	−0.124412	+	0.705578i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	161.158	+	279.135i	0.0236925	+	0.0410366i	0.877629	−	0.479341i	$-0.159124\pi$
$359$	161.158	+	279.135i	0.0236925	+	0.0410366i	−0.853936	+	0.520378i	$0.825791\pi$
$360$		0			0
$361$	−4526.35	+	7839.86i	−0.659914	+	1.14300i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−8537.41	−	7163.74i	−1.22430	−	1.02731i
$366$		0			0
$367$	−5880.80	−	2140.43i	−0.836444	−	0.304441i	−0.111943	−	0.993715i	$-0.535708\pi$
$367$	−5880.80	−	2140.43i	−0.836444	−	0.304441i	−0.724501	+	0.689274i	$0.757930\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−177.447	−	1006.35i	−0.0248318	−	0.140828i
$372$		0			0
$373$	−10764.1	+	3917.82i	−1.49422	+	0.543852i	−0.954557	−	0.298028i	$-0.903671\pi$
$373$	−10764.1	+	3917.82i	−1.49422	+	0.543852i	−0.539664	+	0.841880i	$0.681449\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	6988.07			0.954652
$378$		0			0
$379$	−2145.56			−0.290791			−0.145396	−	0.989374i	$-0.546446\pi$
$379$	−2145.56			−0.290791			−0.145396	+	0.989374i	$0.546446\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−8574.14	+	3120.73i	−1.14391	+	0.416350i	−0.843324	−	0.537405i	$-0.819405\pi$
$383$	−8574.14	+	3120.73i	−1.14391	+	0.416350i	−0.300587	+	0.953754i	$0.597183\pi$
$384$		0			0
$385$	−92.0164	−	521.851i	−0.0121807	−	0.0690804i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	1226.29	+	446.334i	0.159834	+	0.0581749i	0.420698	−	0.907201i	$-0.361785\pi$
$389$	1226.29	+	446.334i	0.159834	+	0.0581749i	−0.260864	+	0.965376i	$0.584007\pi$
$390$		0			0
$391$	11276.4	+	9462.06i	1.45850	+	1.22383i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−11642.5	+	20165.4i	−1.48303	+	2.56869i
$396$		0			0
$397$	−5594.33	−	9689.67i	−0.707233	−	1.22496i	−0.965880	−	0.258991i	$-0.916610\pi$
$397$	−5594.33	−	9689.67i	−0.707233	−	1.22496i	0.258647	−	0.965972i	$-0.416723\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	1540.68	−	8737.63i	0.191865	−	1.08812i	−0.724948	−	0.688804i	$-0.758136\pi$
$401$	1540.68	−	8737.63i	0.191865	−	1.08812i	0.916813	−	0.399317i	$-0.130753\pi$
$402$		0			0
$403$	−344.987	+	289.479i	−0.0426428	+	0.0357815i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−4.52460	+	3.79659i	−0.000551047	+	0.000462384i
$408$		0			0
$409$	−2427.17	+	13765.2i	−0.293437	+	1.66417i	0.380049	+	0.924967i	$0.375907\pi$
$409$	−2427.17	+	13765.2i	−0.293437	+	1.66417i	−0.673486	+	0.739200i	$0.735204\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	886.437	+	1535.35i	0.105614	+	0.182929i
$414$		0			0
$415$	5581.76	−	9667.90i	0.660236	−	1.14356i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	2230.07	+	1871.25i	0.260014	+	0.218178i	0.763470	−	0.645843i	$-0.223494\pi$
$419$	2230.07	+	1871.25i	0.260014	+	0.218178i	−0.503456	+	0.864021i	$0.667938\pi$
$420$		0			0
$421$	1669.95	+	607.813i	0.193322	+	0.0703634i	0.436867	−	0.899526i	$-0.356088\pi$
$421$	1669.95	+	607.813i	0.193322	+	0.0703634i	−0.243545	+	0.969890i	$0.578310\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	5363.22	+	30416.3i	0.612127	+	3.47155i
$426$		0			0
$427$	−3026.05	+	1101.39i	−0.342953	+	0.124825i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−2943.86			−0.329004			−0.164502	−	0.986377i	$-0.552602\pi$
$431$	−2943.86			−0.329004			−0.164502	+	0.986377i	$0.552602\pi$
$432$		0			0
$433$	−5093.05			−0.565257			−0.282628	−	0.959229i	$-0.591206\pi$
$433$	−5093.05			−0.565257			−0.282628	+	0.959229i	$0.591206\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−17901.0	+	6515.44i	−1.95955	+	0.713217i
$438$		0			0
$439$	−114.587	−	649.855i	−0.0124577	−	0.0706512i	0.977945	−	0.208862i	$-0.0669760\pi$
$439$	−114.587	−	649.855i	−0.0124577	−	0.0706512i	−0.990403	+	0.138211i	$0.955865\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	14988.3	+	5455.28i	1.60748	+	0.585075i	0.980939	−	0.194314i	$-0.0622480\pi$
$443$	14988.3	+	5455.28i	1.60748	+	0.585075i	0.626541	+	0.779389i	$0.284470\pi$
$444$		0			0
$445$	2709.19	+	2273.28i	0.288602	+	0.242166i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	6982.02	−	12093.2i	0.733858	−	1.27108i	−0.221365	−	0.975191i	$-0.571051\pi$
$449$	6982.02	−	12093.2i	0.733858	−	1.27108i	0.955223	−	0.295888i	$-0.0956154\pi$
$450$		0			0
$451$	−733.568	−	1270.58i	−0.0765907	−	0.132659i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	453.970	−	2574.59i	0.0467746	−	0.265272i
$456$		0			0
$457$	9944.76	−	8344.65i	1.01794	−	0.854149i	0.0285686	−	0.999592i	$-0.490905\pi$
$457$	9944.76	−	8344.65i	1.01794	−	0.854149i	0.989367	+	0.145443i	$0.0464607\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	3090.41	−	2593.17i	0.312223	−	0.261986i	−0.473187	−	0.880962i	$-0.656896\pi$
$461$	3090.41	−	2593.17i	0.312223	−	0.261986i	0.785410	+	0.618976i	$0.212452\pi$
$462$		0			0
$463$	−1711.84	+	9708.30i	−0.171827	+	0.974477i	0.769917	+	0.638145i	$0.220298\pi$
$463$	−1711.84	+	9708.30i	−0.171827	+	0.974477i	−0.941743	+	0.336333i	$0.890813\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−3974.08	−	6883.30i	−0.393787	−	0.682058i	0.599159	−	0.800630i	$-0.295502\pi$
$467$	−3974.08	−	6883.30i	−0.393787	−	0.682058i	−0.992945	+	0.118572i	$0.962168\pi$
$468$		0			0
$469$	874.141	−	1514.06i	0.0860642	−	0.149068i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	153.031	+	128.408i	0.0148760	+	0.0124825i
$474$		0			0
$475$	−37559.0	−	13670.4i	−3.62806	−	1.32050i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−324.074	−	1837.91i	−0.0309130	−	0.175316i	0.965442	−	0.260617i	$-0.0839260\pi$
$479$	−324.074	−	1837.91i	−0.0309130	−	0.175316i	−0.996355	+	0.0853009i	$0.972815\pi$
$480$		0			0
$481$	−27.3826	+	9.96644i	−0.00259571	+	0.000944762i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	30779.5			2.88171
$486$		0			0
$487$	1560.17			0.145171			0.0725854	−	0.997362i	$-0.476875\pi$
$487$	1560.17			0.145171			0.0725854	+	0.997362i	$0.476875\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	3956.47	−	1440.04i	0.363652	−	0.132358i	−0.153731	−	0.988113i	$-0.549129\pi$
$491$	3956.47	−	1440.04i	0.363652	−	0.132358i	0.517382	+	0.855754i	$0.326907\pi$
$492$		0			0
$493$	3964.89	+	22486.0i	0.362210	+	2.05420i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	893.122	+	325.070i	0.0806076	+	0.0293388i
$498$		0			0
$499$	−8986.51	−	7540.58i	−0.806196	−	0.676478i	0.143501	−	0.989650i	$-0.454164\pi$
$499$	−8986.51	−	7540.58i	−0.806196	−	0.676478i	−0.949697	+	0.313172i	$0.898608\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−1066.88	+	1847.89i	−0.0945720	+	0.163803i	−0.909430	−	0.415857i	$-0.863482\pi$
$503$	−1066.88	+	1847.89i	−0.0945720	+	0.163803i	0.814858	+	0.579661i	$0.196815\pi$
$504$		0			0
$505$	10899.4	+	18878.3i	0.960430	+	1.66351i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−2281.35	+	12938.2i	−0.198662	+	1.12667i	0.708443	+	0.705768i	$0.249398\pi$
$509$	−2281.35	+	12938.2i	−0.198662	+	1.12667i	−0.907106	+	0.420903i	$0.861713\pi$
$510$		0			0
$511$	−1693.23	+	1420.79i	−0.146583	+	0.122998i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−2728.15	+	2289.19i	−0.233431	+	0.195872i
$516$		0			0
$517$	403.224	−	2286.80i	0.0343013	−	0.194532i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−305.345	−	528.873i	−0.0256764	−	0.0444728i	0.852902	−	0.522072i	$-0.174841\pi$
$521$	−305.345	−	528.873i	−0.0256764	−	0.0444728i	−0.878578	+	0.477599i	$0.841507\pi$
$522$		0			0
$523$	4132.80	−	7158.22i	0.345535	−	0.598484i	−0.639916	−	0.768445i	$-0.721031\pi$
$523$	4132.80	−	7158.22i	0.345535	−	0.598484i	0.985451	+	0.169961i	$0.0543641\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−1127.22	−	945.846i	−0.0931732	−	0.0781816i
$528$		0			0
$529$	−9998.29	−	3639.08i	−0.821755	−	0.299094i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−1256.91	−	7128.27i	−0.102144	−	0.579286i
$534$		0			0
$535$	−2483.50	+	903.919i	−0.200693	+	0.0730464i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	1968.92			0.157342
$540$		0			0
$541$	11983.2			0.952304			0.476152	−	0.879363i	$-0.342031\pi$
$541$	11983.2			0.952304			0.476152	+	0.879363i	$0.342031\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	14272.3	−	5194.68i	1.12175	−	0.408285i
$546$		0			0
$547$	2258.20	+	12806.9i	0.176515	+	1.00107i	0.936381	+	0.350986i	$0.114154\pi$
$547$	2258.20	+	12806.9i	0.176515	+	1.00107i	−0.759866	+	0.650080i	$0.774735\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−27766.5	−	10106.2i	−2.14681	−	0.781374i
$552$		0			0
$553$	3537.69	+	2968.48i	0.272040	+	0.228268i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−7371.70	+	12768.2i	−0.560770	+	0.971283i	0.436659	+	0.899627i	$0.356162\pi$
$557$	−7371.70	+	12768.2i	−0.560770	+	0.971283i	−0.997429	+	0.0716556i	$0.977172\pi$
$558$		0			0
$559$	492.784	+	853.527i	0.0372854	+	0.0645803i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−2093.29	+	11871.6i	−0.156699	+	0.888683i	0.800517	+	0.599310i	$0.204558\pi$
$563$	−2093.29	+	11871.6i	−0.156699	+	0.888683i	−0.957216	+	0.289374i	$0.906553\pi$
$564$		0			0
$565$	−6383.23	+	5356.17i	−0.475300	+	0.398824i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−14312.2	+	12009.4i	−1.05448	+	0.884813i	−0.993557	−	0.113330i	$-0.963848\pi$
$569$	−14312.2	+	12009.4i	−1.05448	+	0.884813i	−0.0609216	+	0.998143i	$0.519404\pi$
$570$		0			0
$571$	−2434.29	+	13805.6i	−0.178410	+	1.01181i	0.755724	+	0.654890i	$0.227285\pi$
$571$	−2434.29	+	13805.6i	−0.178410	+	1.01181i	−0.934134	+	0.356923i	$0.883826\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−23926.2	−	41441.4i	−1.73529	−	3.00561i
$576$		0			0
$577$	−8845.23	+	15320.4i	−0.638183	+	1.10537i	0.347648	+	0.937625i	$0.386980\pi$
$577$	−8845.23	+	15320.4i	−0.638183	+	1.10537i	−0.985831	+	0.167741i	$0.946353\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−1696.07	−	1423.17i	−0.121110	−	0.101623i
$582$		0			0
$583$	−1392.74	−	506.917i	−0.0989391	−	0.0360109i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	1937.34	+	10987.2i	0.136222	+	0.772555i	0.974001	+	0.226545i	$0.0727430\pi$
$587$	1937.34	+	10987.2i	0.136222	+	0.772555i	−0.837778	+	0.546010i	$0.816146\pi$
$588$		0			0
$589$	1789.42	−	651.296i	0.125181	−	0.0455623i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	10171.0			0.704336			0.352168	−	0.935937i	$-0.385445\pi$
$593$	10171.0			0.704336			0.352168	+	0.935937i	$0.385445\pi$
$594$		0			0
$595$	8542.02			0.588552
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−4550.73	+	1656.33i	−0.310414	+	0.112981i	−0.492529	−	0.870296i	$-0.663928\pi$
$599$	−4550.73	+	1656.33i	−0.310414	+	0.112981i	0.182116	+	0.983277i	$0.441705\pi$
$600$		0			0
$601$	−2049.47	−	11623.1i	−0.139101	−	0.788880i	−0.971916	−	0.235329i	$-0.924383\pi$
$601$	−2049.47	−	11623.1i	−0.139101	−	0.788880i	0.832815	−	0.553552i	$-0.186728\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	25568.8	+	9306.28i	1.71821	+	0.625379i
$606$		0			0
$607$	3586.38	+	3009.33i	0.239813	+	0.201227i	0.754771	−	0.655988i	$-0.227748\pi$
$607$	3586.38	+	3009.33i	0.239813	+	0.201227i	−0.514958	+	0.857216i	$0.672192\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	5728.07	−	9921.31i	0.379268	−	0.656912i
$612$		0			0
$613$	1434.25	+	2484.19i	0.0945005	+	0.163680i	0.909400	−	0.415923i	$-0.136541\pi$
$613$	1434.25	+	2484.19i	0.0945005	+	0.163680i	−0.814900	+	0.579602i	$0.803208\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	955.679	−	5419.93i	0.0623569	−	0.353643i	−0.937625	−	0.347648i	$-0.886980\pi$
$617$	955.679	−	5419.93i	0.0623569	−	0.353643i	0.999982	−	0.00599552i	$-0.00190845\pi$
$618$		0			0
$619$	7221.69	−	6059.72i	0.468925	−	0.393475i	−0.377477	−	0.926019i	$-0.623208\pi$
$619$	7221.69	−	6059.72i	0.468925	−	0.393475i	0.846402	+	0.532544i	$0.178764\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	537.314	−	450.860i	0.0345538	−	0.0289941i
$624$		0			0
$625$	7843.50	−	44482.7i	0.501984	−	2.84689i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−47.6061	−	82.4561i	−0.00301777	−	0.00522693i
$630$		0			0
$631$	12997.8	−	22512.8i	0.820022	−	1.42032i	−0.0856432	−	0.996326i	$-0.527295\pi$
$631$	12997.8	−	22512.8i	0.820022	−	1.42032i	0.905665	−	0.423994i	$-0.139372\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−799.297	−	670.690i	−0.0499514	−	0.0419142i
$636$		0			0
$637$	9128.02	+	3322.33i	0.567764	+	0.206649i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−1345.12	−	7628.57i	−0.0828847	−	0.470063i	−0.997793	−	0.0664010i	$-0.978848\pi$
$641$	−1345.12	−	7628.57i	−0.0828847	−	0.470063i	0.914908	−	0.403662i	$-0.132263\pi$
$642$		0			0
$643$	6591.22	−	2399.01i	0.404249	−	0.147135i	−0.131890	−	0.991264i	$-0.542105\pi$
$643$	6591.22	−	2399.01i	0.404249	−	0.147135i	0.536139	+	0.844130i	$0.319882\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	6294.68			0.382488			0.191244	−	0.981543i	$-0.438748\pi$
$647$	6294.68			0.382488			0.191244	+	0.981543i	$0.438748\pi$
$648$		0			0
$649$	2571.37			0.155524
$650$		0			0
$651$		0			0
$652$		0			0
$653$	4566.21	−	1661.97i	0.273644	−	0.0995984i	−0.201553	−	0.979478i	$-0.564599\pi$
$653$	4566.21	−	1661.97i	0.273644	−	0.0995984i	0.475197	+	0.879879i	$0.342377\pi$
$654$		0			0
$655$	−2988.41	−	16948.1i	−0.178270	−	1.01102i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−16715.1	−	6083.80i	−0.988054	−	0.359622i	−0.203088	−	0.979161i	$-0.565098\pi$
$659$	−16715.1	−	6083.80i	−0.988054	−	0.359622i	−0.784966	+	0.619538i	$0.787320\pi$
$660$		0			0
$661$	−6238.48	−	5234.71i	−0.367093	−	0.308028i	0.440517	−	0.897744i	$-0.354795\pi$
$661$	−6238.48	−	5234.71i	−0.367093	−	0.308028i	−0.807610	+	0.589716i	$0.799240\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−5527.19	+	9573.37i	−0.322308	+	0.558255i
$666$		0			0
$667$	−17688.1	−	30636.6i	−1.02681	−	1.77849i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−811.049	+	4599.69i	−0.0466620	+	0.264633i
$672$		0			0
$673$	730.692	−	613.124i	0.0418516	−	0.0351177i	−0.621622	−	0.783317i	$-0.713526\pi$
$673$	730.692	−	613.124i	0.0418516	−	0.0351177i	0.663474	+	0.748200i	$0.269082\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	17325.9	−	14538.2i	0.983589	−	0.825329i	−0.00103773	−	0.999999i	$-0.500330\pi$
$677$	17325.9	−	14538.2i	0.983589	−	0.825329i	0.984627	+	0.174670i	$0.0558859\pi$
$678$		0			0
$679$	1060.04	−	6011.78i	0.0599124	−	0.339780i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	12017.4	+	20814.8i	0.673257	+	1.16611i	0.976975	+	0.213353i	$0.0684385\pi$
$683$	12017.4	+	20814.8i	0.673257	+	1.16611i	−0.303718	+	0.952762i	$0.598228\pi$
$684$		0			0
$685$	−6125.94	+	10610.4i	−0.341694	+	0.591831i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−5601.45	−	4700.18i	−0.309722	−	0.259887i
$690$		0			0
$691$	24335.6	+	8857.43i	1.33975	+	0.487630i	0.909735	−	0.415190i	$-0.136285\pi$
$691$	24335.6	+	8857.43i	1.33975	+	0.487630i	0.430019	+	0.902820i	$0.358507\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−6950.84	−	39420.2i	−0.379368	−	2.15150i
$696$		0			0
$697$	22224.0	−	8088.87i	1.20774	−	0.439581i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	5879.83			0.316802			0.158401	−	0.987375i	$-0.449366\pi$
$701$	5879.83			0.316802			0.158401	+	0.987375i	$0.449366\pi$
$702$		0			0
$703$	123.216			0.00661048
$704$		0			0
$705$		0			0
$706$		0			0
$707$	4062.63	−	1478.68i	0.216112	−	0.0786583i
$708$		0			0
$709$	386.211	+	2190.31i	0.0204576	+	0.116021i	0.993327	−	0.115336i	$-0.0367945\pi$
$709$	386.211	+	2190.31i	0.0204576	+	0.116021i	−0.972869	+	0.231357i	$0.925683\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	2142.34	+	779.747i	0.112526	+	0.0409562i
$714$		0			0
$715$	−2904.67	−	2437.31i	−0.151928	−	0.127483i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−14710.5	+	25479.4i	−0.763018	+	1.32159i	0.178270	+	0.983982i	$0.442950\pi$
$719$	−14710.5	+	25479.4i	−0.763018	+	1.32159i	−0.941288	+	0.337605i	$0.890383\pi$
$720$		0			0
$721$	353.162	+	611.694i	0.0182419	+	0.0315960i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	12888.9	−	73096.8i	0.660253	−	3.74448i
$726$		0			0
$727$	25940.8	−	21766.9i	1.32337	−	1.11044i	0.337794	−	0.941220i	$-0.390319\pi$
$727$	25940.8	−	21766.9i	1.32337	−	1.11044i	0.985578	−	0.169220i	$-0.0541250\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−2466.86	+	2069.94i	−0.124816	+	0.104733i
$732$		0			0
$733$	−2518.64	+	14283.9i	−0.126914	+	0.719766i	0.853238	+	0.521521i	$0.174635\pi$
$733$	−2518.64	+	14283.9i	−0.126914	+	0.719766i	−0.980153	+	0.198245i	$0.936476\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−1267.85	−	2195.98i	−0.0633675	−	0.109756i
$738$		0			0
$739$	−5275.76	+	9137.89i	−0.262614	+	0.454862i	−0.966936	−	0.255020i	$-0.917918\pi$
$739$	−5275.76	+	9137.89i	−0.262614	+	0.454862i	0.704321	+	0.709881i	$0.251251\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	17080.5	+	14332.3i	0.843371	+	0.707672i	0.958319	−	0.285699i	$-0.0922259\pi$
$743$	17080.5	+	14332.3i	0.843371	+	0.707672i	−0.114949	+	0.993371i	$0.536670\pi$
$744$		0			0
$745$	25248.4	+	9189.65i	1.24165	+	0.451923i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	91.0200	+	516.200i	0.00444032	+	0.0251823i
$750$		0			0
$751$	−11137.3	+	4053.63i	−0.541151	+	0.196963i	−0.598110	−	0.801414i	$-0.704082\pi$
$751$	−11137.3	+	4053.63i	−0.541151	+	0.196963i	0.0569597	+	0.998376i	$0.481859\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−64725.8			−3.12002
$756$		0			0
$757$	−33298.7			−1.59876			−0.799379	−	0.600827i	$-0.794838\pi$
$757$	−33298.7			−1.59876			−0.799379	+	0.600827i	$0.794838\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−37817.9	+	13764.6i	−1.80144	+	0.655672i	−0.803247	+	0.595647i	$0.796896\pi$
$761$	−37817.9	+	13764.6i	−1.80144	+	0.655672i	−0.998197	+	0.0600251i	$0.980882\pi$
$762$		0			0
$763$	−523.077	−	2966.52i	−0.0248187	−	0.140754i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	11921.0	+	4338.88i	0.561201	+	0.204260i
$768$		0			0
$769$	−5901.10	−	4951.61i	−0.276722	−	0.232197i	0.493855	−	0.869544i	$-0.335587\pi$
$769$	−5901.10	−	4951.61i	−0.276722	−	0.232197i	−0.770577	+	0.637347i	$0.780032\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	2618.83	−	4535.94i	0.121853	−	0.211056i	−0.798645	−	0.601802i	$-0.794450\pi$
$773$	2618.83	−	4535.94i	0.121853	−	0.211056i	0.920499	+	0.390746i	$0.127783\pi$
$774$		0			0
$775$	2391.71	+	4142.57i	0.110855	+	0.192007i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−5314.72	+	30141.3i	−0.244441	+	1.38629i
$780$		0			0
$781$	1056.00	−	886.092i	0.0483826	−	0.0405978i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	15584.3	−	13076.8i	0.708569	−	0.594560i
$786$		0			0
$787$	−5676.14	+	32191.0i	−0.257093	+	1.45805i	0.533549	+	0.845769i	$0.320858\pi$
$787$	−5676.14	+	32191.0i	−0.257093	+	1.45805i	−0.790642	+	0.612279i	$0.790253\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	826.315	+	1431.22i	0.0371433	+	0.0643341i
$792$		0			0
$793$	−11521.5	+	19955.8i	−0.515940	+	0.893634i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−15837.0	−	13288.8i	−0.703857	−	0.590606i	0.219011	−	0.975722i	$-0.429717\pi$
$797$	−15837.0	−	13288.8i	−0.703857	−	0.590606i	−0.922868	+	0.385116i	$0.874161\pi$
$798$		0			0
$799$	35174.5	+	12802.5i	1.55743	+	0.566857i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	556.696	+	3157.18i	0.0244650	+	0.138748i
$804$		0			0
$805$	−12436.4	+	4526.49i	−0.544505	+	0.198184i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	2618.04			0.113777			0.0568884	−	0.998381i	$-0.481882\pi$
$809$	2618.04			0.113777			0.0568884	+	0.998381i	$0.481882\pi$
$810$		0			0
$811$	−28096.0			−1.21650			−0.608251	−	0.793744i	$-0.708129\pi$
$811$	−28096.0			−1.21650			−0.608251	+	0.793744i	$0.708129\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	73724.7	−	26833.6i	3.16867	−	1.15330i
$816$		0			0
$817$	−723.660	−	4104.08i	−0.0309886	−	0.175745i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−26718.4	−	9724.72i	−1.13579	−	0.413392i	−0.295396	−	0.955375i	$-0.595452\pi$
$821$	−26718.4	−	9724.72i	−1.13579	−	0.413392i	−0.840390	+	0.541983i	$0.817674\pi$
$822$		0			0
$823$	19859.2	+	16663.9i	0.841128	+	0.705791i	0.957817	−	0.287378i	$-0.0927836\pi$
$823$	19859.2	+	16663.9i	0.841128	+	0.705791i	−0.116689	+	0.993169i	$0.537228\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−5254.81	+	9101.60i	−0.220952	+	0.382701i	−0.955097	−	0.296292i	$-0.904250\pi$
$827$	−5254.81	+	9101.60i	−0.220952	+	0.382701i	0.734145	+	0.678993i	$0.237583\pi$
$828$		0			0
$829$	6930.26	+	12003.6i	0.290347	+	0.502896i	0.973892	−	0.227012i	$-0.0728958\pi$
$829$	6930.26	+	12003.6i	0.290347	+	0.502896i	−0.683544	+	0.729909i	$0.739562\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−5511.44	+	31256.9i	−0.229244	+	1.30011i
$834$		0			0
$835$	−58189.0	+	48826.3i	−2.41163	+	2.02360i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	28428.4	−	23854.3i	1.16980	−	0.981575i	0.169804	−	0.985478i	$-0.445687\pi$
$839$	28428.4	−	23854.3i	1.16980	−	0.981575i	0.999992	+	0.00390245i	$0.00124219\pi$
$840$		0			0
$841$	5293.37	−	30020.2i	0.217039	−	1.23089i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	13737.5	+	23794.1i	0.559273	+	0.968689i
$846$		0			0
$847$	2698.26	−	4673.52i	0.109461	−	0.189592i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	113.004	+	94.8219i	0.00455199	+	0.00381957i
$852$		0			0
$853$	20206.7	+	7354.62i	0.811093	+	0.295214i	0.714075	−	0.700069i	$-0.246847\pi$
$853$	20206.7	+	7354.62i	0.811093	+	0.295214i	0.0970180	+	0.995283i	$0.469070\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−4386.32	−	24876.0i	−0.174835	−	0.991540i	−0.938334	−	0.345730i	$-0.887631\pi$
$857$	−4386.32	−	24876.0i	−0.174835	−	0.991540i	0.763499	−	0.645809i	$-0.223480\pi$
$858$		0			0
$859$	12789.1	−	4654.85i	0.507984	−	0.184891i	−0.0752975	−	0.997161i	$-0.523991\pi$
$859$	12789.1	−	4654.85i	0.507984	−	0.184891i	0.583282	+	0.812270i	$0.301768\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	15562.5			0.613850			0.306925	−	0.951734i	$-0.400700\pi$
$863$	15562.5			0.613850			0.306925	+	0.951734i	$0.400700\pi$
$864$		0			0
$865$	−21234.7			−0.834684
$866$		0			0
$867$		0			0
$868$		0			0
$869$	6294.17	−	2290.89i	0.245702	−	0.0894282i
$870$		0			0
$871$	−2172.35	−	12320.0i	−0.0845089	−	0.479274i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−15799.8	−	5750.65i	−0.610435	−	0.222180i
$876$		0			0
$877$	−18024.2	−	15124.1i	−0.693995	−	0.582331i	0.226063	−	0.974113i	$-0.427414\pi$
$877$	−18024.2	−	15124.1i	−0.693995	−	0.582331i	−0.920058	+	0.391782i	$0.871859\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−11773.4	+	20392.2i	−0.450234	+	0.779829i	−0.998400	−	0.0565407i	$-0.981993\pi$
$881$	−11773.4	+	20392.2i	−0.450234	+	0.779829i	0.548166	+	0.836370i	$0.315326\pi$
$882$		0			0
$883$	3943.58	+	6830.49i	0.150297	+	0.260322i	0.931337	−	0.364159i	$-0.118644\pi$
$883$	3943.58	+	6830.49i	0.150297	+	0.260322i	−0.781040	+	0.624481i	$0.785310\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−1461.95	+	8291.13i	−0.0553410	+	0.313855i	−0.999895	−	0.0145021i	$-0.995384\pi$
$887$	−1461.95	+	8291.13i	−0.0553410	+	0.313855i	0.944554	+	0.328357i	$0.106495\pi$
$888$		0			0
$889$	−158.525	+	133.018i	−0.00598060	+	0.00501832i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−37108.2	+	31137.5i	−1.39057	+	1.16683i
$894$		0			0
$895$	15729.3	−	89205.5i	0.587457	−	3.33163i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	1768.13	+	3062.50i	0.0655957	+	0.113615i
$900$		0			0
$901$	11945.9	−	20691.0i	0.441706	−	0.765057i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	64325.4	+	53975.4i	2.36270	+	1.98254i
$906$		0			0
$907$	−4157.15	−	1513.08i	−0.152190	−	0.0553925i	0.264802	−	0.964303i	$-0.414693\pi$
$907$	−4157.15	−	1513.08i	−0.152190	−	0.0553925i	−0.416992	+	0.908910i	$0.636916\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	503.055	+	2852.97i	0.0182952	+	0.103757i	0.992588	−	0.121528i	$-0.0387795\pi$
$911$	503.055	+	2852.97i	0.0182952	+	0.103757i	−0.974293	+	0.225286i	$0.927668\pi$
$912$		0			0
$913$	−3017.61	+	1098.32i	−0.109385	+	0.0398128i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−3413.18			−0.122915
$918$		0			0
$919$	24120.5			0.865790			0.432895	−	0.901444i	$-0.357492\pi$
$919$	24120.5			0.865790			0.432895	+	0.901444i	$0.357492\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	6390.85	−	2326.08i	0.227906	−	0.0829511i
$924$		0			0
$925$	53.7463	+	304.810i	0.00191045	+	0.0108347i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−15087.5	−	5491.40i	−0.532836	−	0.193936i	0.0615682	−	0.998103i	$-0.480390\pi$
$929$	−15087.5	−	5491.40i	−0.532836	−	0.193936i	−0.594404	+	0.804166i	$0.702612\pi$
$930$		0			0
$931$	−31464.6	−	26401.9i	−1.10764	−	0.929419i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	6194.65	−	10729.4i	0.216670	−	0.375284i
$936$		0			0
$937$	16037.8	+	27778.3i	0.559160	+	0.968494i	0.997567	+	0.0697170i	$0.0222096\pi$
$937$	16037.8	+	27778.3i	0.559160	+	0.968494i	−0.438407	+	0.898777i	$0.644457\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	310.666	−	1761.87i	0.0107624	−	0.0610366i	−0.978954	−	0.204083i	$-0.934579\pi$
$941$	310.666	−	1761.87i	0.0107624	−	0.0610366i	0.989716	+	0.143046i	$0.0456898\pi$
$942$		0			0
$943$	−28069.8	+	23553.4i	−0.969331	+	0.813365i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−17838.2	+	14968.0i	−0.612104	+	0.513616i	−0.895310	−	0.445443i	$-0.853046\pi$
$947$	−17838.2	+	14968.0i	−0.612104	+	0.513616i	0.283207	+	0.959059i	$0.408602\pi$
$948$		0			0
$949$	−2746.50	+	15576.2i	−0.0939464	+	0.532797i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−378.222	−	655.100i	−0.0128561	−	0.0222673i	0.859526	−	0.511092i	$-0.170759\pi$
$953$	−378.222	−	655.100i	−0.0128561	−	0.0222673i	−0.872382	+	0.488825i	$0.837426\pi$
$954$		0			0
$955$	−44346.7	+	76810.8i	−1.50264	+	2.60266i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	1861.43	+	1561.92i	0.0626784	+	0.0525934i
$960$		0			0
$961$	27780.2	+	10111.2i	0.932504	+	0.339404i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	8403.25	+	47657.2i	0.280321	+	1.58978i
$966$		0			0
$967$	−18140.2	+	6602.48i	−0.603256	+	0.219567i	−0.625550	−	0.780184i	$-0.715125\pi$
$967$	−18140.2	+	6602.48i	−0.603256	+	0.219567i	0.0222937	+	0.999751i	$0.492903\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	11825.1			0.390818			0.195409	−	0.980722i	$-0.437397\pi$
$971$	11825.1			0.390818			0.195409	+	0.980722i	$0.437397\pi$
$972$		0			0
$973$	−7938.83			−0.261570
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−14594.2	+	5311.86i	−0.477902	+	0.173942i	−0.569729	−	0.821833i	$-0.692952\pi$
$977$	−14594.2	+	5311.86i	−0.477902	+	0.173942i	0.0918267	+	0.995775i	$0.470729\pi$
$978$		0			0
$979$	−176.657	−	1001.87i	−0.00576709	−	0.0327068i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	36816.8	+	13400.2i	1.19458	+	0.434792i	0.861330	−	0.508046i	$-0.169632\pi$
$983$	36816.8	+	13400.2i	1.19458	+	0.434792i	0.333251	+	0.942838i	$0.391854\pi$
$984$		0			0
$985$	−22850.4	−	19173.8i	−0.739162	−	0.620231i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	2494.65	−	4320.86i	0.0802076	−	0.138924i
$990$		0			0
$991$	−20152.2	−	34904.6i	−0.645969	−	1.11885i	−0.984077	−	0.177743i	$-0.943120\pi$
$991$	−20152.2	−	34904.6i	−0.645969	−	1.11885i	0.338108	−	0.941107i	$-0.390213\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−7260.52	+	41176.5i	−0.231331	+	1.31194i
$996$		0			0
$997$	−13367.5	+	11216.6i	−0.424626	+	0.356303i	−0.829919	−	0.557883i	$-0.811614\pi$
$997$	−13367.5	+	11216.6i	−0.424626	+	0.356303i	0.405294	+	0.914186i	$0.367169\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	324.4.i.a.37.1		54
3.2	odd	2		108.4.i.a.49.5	✓	54
27.11	odd	18		108.4.i.a.97.5	yes	54
27.16	even	9	inner	324.4.i.a.289.1		54

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
108.4.i.a.49.5	✓	54	3.2	odd	2
108.4.i.a.97.5	yes	54	27.11	odd	18
324.4.i.a.37.1		54	1.1	even	1	trivial
324.4.i.a.289.1		54	27.16	even	9	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 324 = 2^{2} \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	324.i (of order \(9\), degree \(6\), not minimal)

\(f(q)\)	\(=\)	\(q+(-19.7528 + 7.18944i) q^{5} +(0.723940 + 4.10567i) q^{7} +O(q^{10})\)	\(q+(-19.7528 + 7.18944i) q^{5} +(0.723940 + 4.10567i) q^{7} +(5.68204 + 2.06810i) q^{11} +(22.8525 + 19.1756i) q^{13} +(-48.7365 + 84.4141i) q^{17} +(-63.0708 - 109.242i) q^{19} +(26.2243 - 148.725i) q^{23} +(242.731 - 203.675i) q^{25} +(179.445 - 150.572i) q^{29} +(-2.62143 + 14.8669i) q^{31} +(-43.8173 - 75.8939i) q^{35} +(-0.488402 + 0.845938i) q^{37} +(-185.869 - 155.962i) q^{41} +(31.0450 + 11.2995i) q^{43} +(-66.6849 - 378.189i) q^{47} +(305.982 - 111.368i) q^{49} -245.113 q^{53} -127.105 q^{55} +(399.605 - 145.444i) q^{59} +(134.131 + 760.693i) q^{61} +(-589.264 - 214.475i) q^{65} +(-321.243 - 269.555i) q^{67} +(113.989 - 197.435i) q^{71} +(265.093 + 459.155i) q^{73} +(-4.37745 + 24.8258i) q^{77} +(848.570 - 712.034i) q^{79} +(-406.829 + 341.370i) q^{83} +(355.794 - 2017.81i) q^{85} +(-84.1224 - 145.704i) q^{89} +(-62.1846 + 107.707i) q^{91} +(2031.21 + 1704.39i) q^{95} +(-1375.96 - 500.807i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 54 q - 12 q^{5}+O(q^{10}) \) 54 * q - 12 * q^5	\( 54 q - 12 q^{5} + 87 q^{11} - 204 q^{17} - 96 q^{23} - 216 q^{25} - 318 q^{29} - 54 q^{31} - 6 q^{35} - 867 q^{41} - 513 q^{43} + 1548 q^{47} + 594 q^{49} + 1068 q^{53} + 1218 q^{59} - 54 q^{61} - 96 q^{65} - 2997 q^{67} + 120 q^{71} - 216 q^{73} - 3480 q^{77} + 2808 q^{79} - 4464 q^{83} + 2160 q^{85} - 4029 q^{89} + 270 q^{91} + 1650 q^{95} - 3483 q^{97}+O(q^{100}) \) 54 * q - 12 * q^5 + 87 * q^11 - 204 * q^17 - 96 * q^23 - 216 * q^25 - 318 * q^29 - 54 * q^31 - 6 * q^35 - 867 * q^41 - 513 * q^43 + 1548 * q^47 + 594 * q^49 + 1068 * q^53 + 1218 * q^59 - 54 * q^61 - 96 * q^65 - 2997 * q^67 + 120 * q^71 - 216 * q^73 - 3480 * q^77 + 2808 * q^79 - 4464 * q^83 + 2160 * q^85 - 4029 * q^89 + 270 * q^91 + 1650 * q^95 - 3483 * q^97

Introduction

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