LMFDB - Embedded newform 1296.3.q.j.593.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1296,3,Mod(593,1296)] 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("1296.593"); S:= CuspForms(chi, 3); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1296, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 3, names="a")

Level:	$ N $	$=$	$ 1296 = 2^{4} \cdot 3^{4} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1296.q (of order $6$, degree $2$, not minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,0,0,0,0,10,0,0,0,0,0,20] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

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

Self dual:	no
Analytic conductor:	$35.3134422611$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{6})$
Coefficient field:	$\Q(\zeta_{12})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{2} + 1 $ x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 3^{2} $
Twist minimal:	no (minimal twist has level 27)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			593.2
Root			$0.866025 - 0.500000i$ of defining polynomial
Character	$\chi$	$=$	1296.593
Dual form			1296.3.q.j.1025.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(2.59808 - 1.50000i) q^{5} +(2.50000 - 4.33013i) q^{7} +(12.9904 + 7.50000i) q^{11} +(5.00000 + 8.66025i) q^{13} -18.0000i q^{17} +16.0000 q^{19} +(-10.3923 + 6.00000i) q^{23} +(-8.00000 + 13.8564i) q^{25} +(25.9808 + 15.0000i) q^{29} +(-0.500000 - 0.866025i) q^{31} -15.0000i q^{35} +20.0000 q^{37} +(-51.9615 + 30.0000i) q^{41} +(25.0000 - 43.3013i) q^{43} +(5.19615 + 3.00000i) q^{47} +(12.0000 + 20.7846i) q^{49} +27.0000i q^{53} +45.0000 q^{55} +(-25.9808 + 15.0000i) q^{59} +(38.0000 - 65.8179i) q^{61} +(25.9808 + 15.0000i) q^{65} +(-5.00000 - 8.66025i) q^{67} -90.0000i q^{71} +65.0000 q^{73} +(64.9519 - 37.5000i) q^{77} +(7.00000 - 12.1244i) q^{79} +(-2.59808 - 1.50000i) q^{83} +(-27.0000 - 46.7654i) q^{85} -90.0000i q^{89} +50.0000 q^{91} +(41.5692 - 24.0000i) q^{95} +(42.5000 - 73.6122i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 10 q^{7} + 20 q^{13} + 64 q^{19} - 32 q^{25} - 2 q^{31} + 80 q^{37} + 100 q^{43} + 48 q^{49} + 180 q^{55} + 152 q^{61} - 20 q^{67} + 260 q^{73} + 28 q^{79} - 108 q^{85} + 200 q^{91} + 170 q^{97}+O(q^{100}) $ 4 * q + 10 * q^7 + 20 * q^13 + 64 * q^19 - 32 * q^25 - 2 * q^31 + 80 * q^37 + 100 * q^43 + 48 * q^49 + 180 * q^55 + 152 * q^61 - 20 * q^67 + 260 * q^73 + 28 * q^79 - 108 * q^85 + 200 * q^91 + 170 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1296\mathbb{Z}\right)^\times$.

$n$	$325$	$1135$	$1217$
$\chi(n)$	$1$	$1$	$e\left(\frac{1}{6}\right)$

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
$5$	2.59808	−	1.50000i	0.519615	−	0.300000i	−0.217162	−	0.976136i	$-0.569680\pi$
$5$	2.59808	−	1.50000i	0.519615	−	0.300000i	0.736777	+	0.676136i	$0.236347\pi$
$6$		0			0
$7$	2.50000	−	4.33013i	0.357143	−	0.618590i	−0.630339	−	0.776320i	$-0.717084\pi$
$7$	2.50000	−	4.33013i	0.357143	−	0.618590i	0.987482	+	0.157730i	$0.0504176\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	12.9904	+	7.50000i	1.18094	+	0.681818i	0.956233	−	0.292607i	$-0.0945229\pi$
$11$	12.9904	+	7.50000i	1.18094	+	0.681818i	0.224711	+	0.974425i	$0.427856\pi$
$12$		0			0
$13$	5.00000	+	8.66025i	0.384615	+	0.666173i	0.991716	−	0.128452i	$-0.0410008\pi$
$13$	5.00000	+	8.66025i	0.384615	+	0.666173i	−0.607100	+	0.794625i	$0.707667\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		−	18.0000i		−	1.05882i	−0.848365	−	0.529412i	$-0.822413\pi$
$17$		−	18.0000i		−	1.05882i	0.848365	−	0.529412i	$-0.177587\pi$
$18$		0			0
$19$	16.0000			0.842105			0.421053	−	0.907036i	$-0.361661\pi$
$19$	16.0000			0.842105			0.421053	+	0.907036i	$0.361661\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−10.3923	+	6.00000i	−0.451839	+	0.260870i	−0.708607	−	0.705604i	$-0.750676\pi$
$23$	−10.3923	+	6.00000i	−0.451839	+	0.260870i	0.256767	+	0.966473i	$0.417343\pi$
$24$		0			0
$25$	−8.00000	+	13.8564i	−0.320000	+	0.554256i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	25.9808	+	15.0000i	0.895888	+	0.517241i	0.875864	−	0.482558i	$-0.160292\pi$
$29$	25.9808	+	15.0000i	0.895888	+	0.517241i	0.0200244	+	0.999799i	$0.493626\pi$
$30$		0			0
$31$	−0.500000	−	0.866025i	−0.0161290	−	0.0279363i	0.857848	−	0.513903i	$-0.171801\pi$
$31$	−0.500000	−	0.866025i	−0.0161290	−	0.0279363i	−0.873977	+	0.485967i	$0.838468\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		−	15.0000i		−	0.428571i
$36$		0			0
$37$	20.0000			0.540541			0.270270	−	0.962784i	$-0.412887\pi$
$37$	20.0000			0.540541			0.270270	+	0.962784i	$0.412887\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−51.9615	+	30.0000i	−1.26735	+	0.731707i	−0.974487	−	0.224446i	$-0.927943\pi$
$41$	−51.9615	+	30.0000i	−1.26735	+	0.731707i	−0.292868	+	0.956153i	$0.594610\pi$
$42$		0			0
$43$	25.0000	−	43.3013i	0.581395	−	1.00701i	−0.413919	−	0.910314i	$-0.635840\pi$
$43$	25.0000	−	43.3013i	0.581395	−	1.00701i	0.995314	−	0.0966925i	$-0.0308264\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	5.19615	+	3.00000i	0.110556	+	0.0638298i	0.554259	−	0.832345i	$-0.313002\pi$
$47$	5.19615	+	3.00000i	0.110556	+	0.0638298i	−0.443702	+	0.896174i	$0.646335\pi$
$48$		0			0
$49$	12.0000	+	20.7846i	0.244898	+	0.424176i
$50$		0			0
$51$		0			0
$52$		0			0
$53$			27.0000i			0.509434i	0.967016	+	0.254717i	$0.0819823\pi$
$53$			27.0000i			0.509434i	−0.967016	+	0.254717i	$0.918018\pi$
$54$		0			0
$55$	45.0000			0.818182
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−25.9808	+	15.0000i	−0.440352	+	0.254237i	−0.703747	−	0.710451i	$-0.748491\pi$
$59$	−25.9808	+	15.0000i	−0.440352	+	0.254237i	0.263395	+	0.964688i	$0.415158\pi$
$60$		0			0
$61$	38.0000	−	65.8179i	0.622951	−	1.07898i	−0.365982	−	0.930622i	$-0.619267\pi$
$61$	38.0000	−	65.8179i	0.622951	−	1.07898i	0.988933	−	0.148361i	$-0.0473997\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	25.9808	+	15.0000i	0.399704	+	0.230769i
$66$		0			0
$67$	−5.00000	−	8.66025i	−0.0746269	−	0.129258i	0.826297	−	0.563235i	$-0.190443\pi$
$67$	−5.00000	−	8.66025i	−0.0746269	−	0.129258i	−0.900924	+	0.433977i	$0.857110\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	90.0000i		−	1.26761i	−0.773495	−	0.633803i	$-0.781493\pi$
$71$		−	90.0000i		−	1.26761i	0.773495	−	0.633803i	$-0.218507\pi$
$72$		0			0
$73$	65.0000			0.890411			0.445205	−	0.895428i	$-0.353131\pi$
$73$	65.0000			0.890411			0.445205	+	0.895428i	$0.353131\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	64.9519	−	37.5000i	0.843531	−	0.487013i
$78$		0			0
$79$	7.00000	−	12.1244i	0.0886076	−	0.153473i	−0.818315	−	0.574770i	$-0.805092\pi$
$79$	7.00000	−	12.1244i	0.0886076	−	0.153473i	0.906923	+	0.421297i	$0.138425\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−2.59808	−	1.50000i	−0.0313021	−	0.0180723i	0.484267	−	0.874920i	$-0.339086\pi$
$83$	−2.59808	−	1.50000i	−0.0313021	−	0.0180723i	−0.515569	+	0.856848i	$0.672420\pi$
$84$		0			0
$85$	−27.0000	−	46.7654i	−0.317647	−	0.550181i
$86$		0			0
$87$		0			0
$88$		0			0
$89$		−	90.0000i		−	1.01124i	−0.862757	−	0.505618i	$-0.831265\pi$
$89$		−	90.0000i		−	1.01124i	0.862757	−	0.505618i	$-0.168735\pi$
$90$		0			0
$91$	50.0000			0.549451
$92$		0			0
$93$		0			0
$94$		0			0
$95$	41.5692	−	24.0000i	0.437571	−	0.252632i
$96$		0			0
$97$	42.5000	−	73.6122i	0.438144	−	0.758888i	−0.559402	−	0.828896i	$-0.688969\pi$
$97$	42.5000	−	73.6122i	0.438144	−	0.758888i	0.997546	+	0.0700082i	$0.0223025\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−168.875	−	97.5000i	−1.67203	−	0.965347i	−0.966500	−	0.256665i	$-0.917376\pi$
$101$	−168.875	−	97.5000i	−1.67203	−	0.965347i	−0.705529	−	0.708681i	$-0.749291\pi$
$102$		0			0
$103$	85.0000	+	147.224i	0.825243	+	1.42936i	0.901734	+	0.432292i	$0.142295\pi$
$103$	85.0000	+	147.224i	0.825243	+	1.42936i	−0.0764909	+	0.997070i	$0.524372\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			189.000i			1.76636i	0.469039	+	0.883178i	$0.344600\pi$
$107$			189.000i			1.76636i	−0.469039	+	0.883178i	$0.655400\pi$
$108$		0			0
$109$	164.000			1.50459			0.752294	−	0.658828i	$-0.228948\pi$
$109$	164.000			1.50459			0.752294	+	0.658828i	$0.228948\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−20.7846	+	12.0000i	−0.183935	+	0.106195i	−0.589140	−	0.808031i	$-0.700533\pi$
$113$	−20.7846	+	12.0000i	−0.183935	+	0.106195i	0.405205	+	0.914226i	$0.367200\pi$
$114$		0			0
$115$	−18.0000	+	31.1769i	−0.156522	+	0.271104i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−77.9423	−	45.0000i	−0.654977	−	0.378151i
$120$		0			0
$121$	52.0000	+	90.0666i	0.429752	+	0.744352i
$122$		0			0
$123$		0			0
$124$		0			0
$125$			123.000i			0.984000i
$126$		0			0
$127$	205.000			1.61417			0.807087	−	0.590433i	$-0.201043\pi$
$127$	205.000			1.61417			0.807087	+	0.590433i	$0.201043\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	12.9904	−	7.50000i	0.0991632	−	0.0572519i	−0.449598	−	0.893231i	$-0.648433\pi$
$131$	12.9904	−	7.50000i	0.0991632	−	0.0572519i	0.548761	+	0.835979i	$0.315100\pi$
$132$		0			0
$133$	40.0000	−	69.2820i	0.300752	−	0.520918i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	119.512	+	69.0000i	0.872347	+	0.503650i	0.868127	−	0.496341i	$-0.165324\pi$
$137$	119.512	+	69.0000i	0.872347	+	0.503650i	0.00421937	+	0.999991i	$0.498657\pi$
$138$		0			0
$139$	−14.0000	−	24.2487i	−0.100719	−	0.174451i	0.811262	−	0.584683i	$-0.198781\pi$
$139$	−14.0000	−	24.2487i	−0.100719	−	0.174451i	−0.911981	+	0.410232i	$0.865448\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$			150.000i			1.04895i
$144$		0			0
$145$	90.0000			0.620690
$146$		0			0
$147$		0			0
$148$		0			0
$149$	64.9519	−	37.5000i	0.435919	−	0.251678i	−0.265946	−	0.963988i	$-0.585684\pi$
$149$	64.9519	−	37.5000i	0.435919	−	0.251678i	0.701865	+	0.712310i	$0.252351\pi$
$150$		0			0
$151$	38.5000	−	66.6840i	0.254967	−	0.441616i	−0.709920	−	0.704283i	$-0.751269\pi$
$151$	38.5000	−	66.6840i	0.254967	−	0.441616i	0.964887	+	0.262667i	$0.0846021\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−2.59808	−	1.50000i	−0.0167618	−	0.00967742i
$156$		0			0
$157$	50.0000	+	86.6025i	0.318471	+	0.551609i	0.980169	−	0.198162i	$-0.0634972\pi$
$157$	50.0000	+	86.6025i	0.318471	+	0.551609i	−0.661698	+	0.749771i	$0.730164\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$			60.0000i			0.372671i
$162$		0			0
$163$	−110.000			−0.674847			−0.337423	−	0.941353i	$-0.609555\pi$
$163$	−110.000			−0.674847			−0.337423	+	0.941353i	$0.609555\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	67.5500	−	39.0000i	0.404491	−	0.233533i	−0.283929	−	0.958845i	$-0.591638\pi$
$167$	67.5500	−	39.0000i	0.404491	−	0.233533i	0.688420	+	0.725312i	$0.258305\pi$
$168$		0			0
$169$	34.5000	−	59.7558i	0.204142	−	0.353584i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−153.286	−	88.5000i	−0.886049	−	0.511561i	−0.0134010	−	0.999910i	$-0.504266\pi$
$173$	−153.286	−	88.5000i	−0.886049	−	0.511561i	−0.872648	+	0.488350i	$0.837599\pi$
$174$		0			0
$175$	40.0000	+	69.2820i	0.228571	+	0.395897i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		−	225.000i		−	1.25698i	−0.777816	−	0.628492i	$-0.783673\pi$
$179$		−	225.000i		−	1.25698i	0.777816	−	0.628492i	$-0.216327\pi$
$180$		0			0
$181$	−16.0000			−0.0883978			−0.0441989	−	0.999023i	$-0.514074\pi$
$181$	−16.0000			−0.0883978			−0.0441989	+	0.999023i	$0.514074\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	51.9615	−	30.0000i	0.280873	−	0.162162i
$186$		0			0
$187$	135.000	−	233.827i	0.721925	−	1.25041i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−25.9808	−	15.0000i	−0.136025	−	0.0785340i	0.430443	−	0.902618i	$-0.358357\pi$
$191$	−25.9808	−	15.0000i	−0.136025	−	0.0785340i	−0.566468	+	0.824084i	$0.691691\pi$
$192$		0			0
$193$	−107.500	−	186.195i	−0.556995	−	0.964743i	−0.997745	−	0.0671137i	$-0.978621\pi$
$193$	−107.500	−	186.195i	−0.556995	−	0.964743i	0.440751	−	0.897630i	$-0.354712\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			207.000i			1.05076i	0.850867	+	0.525381i	$0.176077\pi$
$197$			207.000i			1.05076i	−0.850867	+	0.525381i	$0.823923\pi$
$198$		0			0
$199$	223.000			1.12060			0.560302	−	0.828289i	$-0.310685\pi$
$199$	223.000			1.12060			0.560302	+	0.828289i	$0.310685\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	129.904	−	75.0000i	0.639920	−	0.369458i
$204$		0			0
$205$	−90.0000	+	155.885i	−0.439024	+	0.760413i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	207.846	+	120.000i	0.994479	+	0.574163i
$210$		0			0
$211$	−158.000	−	273.664i	−0.748815	−	1.29699i	−0.948391	−	0.317104i	$-0.897290\pi$
$211$	−158.000	−	273.664i	−0.748815	−	1.29699i	0.199576	−	0.979882i	$-0.436044\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		−	150.000i		−	0.697674i
$216$		0			0
$217$	−5.00000			−0.0230415
$218$		0			0
$219$		0			0
$220$		0			0
$221$	155.885	−	90.0000i	0.705360	−	0.407240i
$222$		0			0
$223$	−65.0000	+	112.583i	−0.291480	+	0.504858i	−0.974160	−	0.225860i	$-0.927481\pi$
$223$	−65.0000	+	112.583i	−0.291480	+	0.504858i	0.682680	+	0.730717i	$0.260814\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	36.3731	+	21.0000i	0.160234	+	0.0925110i	0.577973	−	0.816056i	$-0.303844\pi$
$227$	36.3731	+	21.0000i	0.160234	+	0.0925110i	−0.417739	+	0.908567i	$0.637177\pi$
$228$		0			0
$229$	113.000	+	195.722i	0.493450	+	0.854680i	0.999972	−	0.00754710i	$-0.00240234\pi$
$229$	113.000	+	195.722i	0.493450	+	0.854680i	−0.506522	+	0.862227i	$0.669069\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	234.000i		−	1.00429i	−0.864783	−	0.502146i	$-0.832544\pi$
$233$		−	234.000i		−	1.00429i	0.864783	−	0.502146i	$-0.167456\pi$
$234$		0			0
$235$	18.0000			0.0765957
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−103.923	+	60.0000i	−0.434824	+	0.251046i	−0.701400	−	0.712768i	$-0.747441\pi$
$239$	−103.923	+	60.0000i	−0.434824	+	0.251046i	0.266575	+	0.963814i	$0.414108\pi$
$240$		0			0
$241$	−7.00000	+	12.1244i	−0.0290456	+	0.0503085i	−0.880183	−	0.474635i	$-0.842580\pi$
$241$	−7.00000	+	12.1244i	−0.0290456	+	0.0503085i	0.851137	+	0.524943i	$0.175913\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	62.3538	+	36.0000i	0.254505	+	0.146939i
$246$		0			0
$247$	80.0000	+	138.564i	0.323887	+	0.560988i
$248$		0			0
$249$		0			0
$250$		0			0
$251$			90.0000i			0.358566i	0.983798	+	0.179283i	$0.0573777\pi$
$251$			90.0000i			0.358566i	−0.983798	+	0.179283i	$0.942622\pi$
$252$		0			0
$253$	−180.000			−0.711462
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−379.319	+	219.000i	−1.47595	+	0.852140i	−0.999632	−	0.0271330i	$-0.991362\pi$
$257$	−379.319	+	219.000i	−1.47595	+	0.852140i	−0.476318	+	0.879273i	$0.658029\pi$
$258$		0			0
$259$	50.0000	−	86.6025i	0.193050	−	0.334373i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	239.023	+	138.000i	0.908833	+	0.524715i	0.880055	−	0.474871i	$-0.157505\pi$
$263$	239.023	+	138.000i	0.908833	+	0.524715i	0.0287773	+	0.999586i	$0.490839\pi$
$264$		0			0
$265$	40.5000	+	70.1481i	0.152830	+	0.264710i
$266$		0			0
$267$		0			0
$268$		0			0
$269$		−	270.000i		−	1.00372i	−0.864950	−	0.501859i	$-0.832650\pi$
$269$		−	270.000i		−	1.00372i	0.864950	−	0.501859i	$-0.167350\pi$
$270$		0			0
$271$	−299.000			−1.10332			−0.551661	−	0.834069i	$-0.686006\pi$
$271$	−299.000			−1.10332			−0.551661	+	0.834069i	$0.686006\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−207.846	+	120.000i	−0.755804	+	0.436364i
$276$		0			0
$277$	−70.0000	+	121.244i	−0.252708	+	0.437702i	−0.964270	−	0.264920i	$-0.914654\pi$
$277$	−70.0000	+	121.244i	−0.252708	+	0.437702i	0.711563	+	0.702623i	$0.247988\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−129.904	−	75.0000i	−0.462291	−	0.266904i	0.250716	−	0.968061i	$-0.419334\pi$
$281$	−129.904	−	75.0000i	−0.462291	−	0.266904i	−0.713007	+	0.701157i	$0.752667\pi$
$282$		0			0
$283$	−140.000	−	242.487i	−0.494700	−	0.856845i	0.505282	−	0.862954i	$-0.331389\pi$
$283$	−140.000	−	242.487i	−0.494700	−	0.856845i	−0.999981	+	0.00610955i	$0.998055\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$			300.000i			1.04530i
$288$		0			0
$289$	−35.0000			−0.121107
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−223.435	+	129.000i	−0.762575	+	0.440273i	−0.830220	−	0.557436i	$-0.811785\pi$
$293$	−223.435	+	129.000i	−0.762575	+	0.440273i	0.0676443	+	0.997709i	$0.478452\pi$
$294$		0			0
$295$	−45.0000	+	77.9423i	−0.152542	+	0.264211i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−103.923	−	60.0000i	−0.347569	−	0.200669i
$300$		0			0
$301$	−125.000	−	216.506i	−0.415282	−	0.719290i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		−	228.000i		−	0.747541i
$306$		0			0
$307$	−290.000			−0.944625			−0.472313	−	0.881431i	$-0.656581\pi$
$307$	−290.000			−0.944625			−0.472313	+	0.881431i	$0.656581\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−415.692	+	240.000i	−1.33663	+	0.771704i	−0.986306	−	0.164924i	$-0.947262\pi$
$311$	−415.692	+	240.000i	−1.33663	+	0.771704i	−0.350325	+	0.936628i	$0.613929\pi$
$312$		0			0
$313$	−92.5000	+	160.215i	−0.295527	+	0.511868i	−0.975107	−	0.221733i	$-0.928829\pi$
$313$	−92.5000	+	160.215i	−0.295527	+	0.511868i	0.679580	+	0.733601i	$0.262162\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	158.483	+	91.5000i	0.499945	+	0.288644i	0.728691	−	0.684843i	$-0.240129\pi$
$317$	158.483	+	91.5000i	0.499945	+	0.288644i	−0.228746	+	0.973486i	$0.573462\pi$
$318$		0			0
$319$	225.000	+	389.711i	0.705329	+	1.22167i
$320$		0			0
$321$		0			0
$322$		0			0
$323$		−	288.000i		−	0.891641i
$324$		0			0
$325$	−160.000			−0.492308
$326$		0			0
$327$		0			0
$328$		0			0
$329$	25.9808	−	15.0000i	0.0789689	−	0.0455927i
$330$		0			0
$331$	−119.000	+	206.114i	−0.359517	+	0.622701i	−0.987880	−	0.155219i	$-0.950392\pi$
$331$	−119.000	+	206.114i	−0.359517	+	0.622701i	0.628363	+	0.777920i	$0.283725\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−25.9808	−	15.0000i	−0.0775545	−	0.0447761i
$336$		0			0
$337$	5.00000	+	8.66025i	0.0148368	+	0.0256981i	0.873348	−	0.487096i	$-0.161944\pi$
$337$	5.00000	+	8.66025i	0.0148368	+	0.0256981i	−0.858512	+	0.512794i	$0.828610\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		−	15.0000i		−	0.0439883i
$342$		0			0
$343$	365.000			1.06414
$344$		0			0
$345$		0			0
$346$		0			0
$347$	59.7558	−	34.5000i	0.172207	−	0.0994236i	−0.411419	−	0.911446i	$-0.634967\pi$
$347$	59.7558	−	34.5000i	0.172207	−	0.0994236i	0.583626	+	0.812023i	$0.301633\pi$
$348$		0			0
$349$	128.000	−	221.703i	0.366762	−	0.635251i	−0.622295	−	0.782783i	$-0.713800\pi$
$349$	128.000	−	221.703i	0.366762	−	0.635251i	0.989057	+	0.147532i	$0.0471329\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−394.908	−	228.000i	−1.11872	−	0.645892i	−0.177645	−	0.984095i	$-0.556848\pi$
$353$	−394.908	−	228.000i	−1.11872	−	0.645892i	−0.941073	+	0.338202i	$0.890181\pi$
$354$		0			0
$355$	−135.000	−	233.827i	−0.380282	−	0.658667i
$356$		0			0
$357$		0			0
$358$		0			0
$359$		−	450.000i		−	1.25348i	−0.779228	−	0.626741i	$-0.784388\pi$
$359$		−	450.000i		−	1.25348i	0.779228	−	0.626741i	$-0.215612\pi$
$360$		0			0
$361$	−105.000			−0.290859
$362$		0			0
$363$		0			0
$364$		0			0
$365$	168.875	−	97.5000i	0.462671	−	0.267123i
$366$		0			0
$367$	−312.500	+	541.266i	−0.851499	+	1.47484i	0.0283570	+	0.999598i	$0.490972\pi$
$367$	−312.500	+	541.266i	−0.851499	+	1.47484i	−0.879856	+	0.475241i	$0.842361\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	116.913	+	67.5000i	0.315131	+	0.181941i
$372$		0			0
$373$	−85.0000	−	147.224i	−0.227882	−	0.394703i	0.729298	−	0.684196i	$-0.239847\pi$
$373$	−85.0000	−	147.224i	−0.227882	−	0.394703i	−0.957180	+	0.289493i	$0.906513\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$			300.000i			0.795756i
$378$		0			0
$379$	−704.000			−1.85752			−0.928760	−	0.370682i	$-0.879124\pi$
$379$	−704.000			−1.85752			−0.928760	+	0.370682i	$0.879124\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	535.204	−	309.000i	1.39740	−	0.806789i	0.403279	−	0.915077i	$-0.367870\pi$
$383$	535.204	−	309.000i	1.39740	−	0.806789i	0.994120	+	0.108288i	$0.0345370\pi$
$384$		0			0
$385$	112.500	−	194.856i	0.292208	−	0.506119i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	454.663	+	262.500i	1.16880	+	0.674807i	0.953397	−	0.301718i	$-0.0975601\pi$
$389$	454.663	+	262.500i	1.16880	+	0.674807i	0.215403	+	0.976525i	$0.430893\pi$
$390$		0			0
$391$	108.000	+	187.061i	0.276215	+	0.478418i
$392$		0			0
$393$		0			0
$394$		0			0
$395$		−	42.0000i		−	0.106329i
$396$		0			0
$397$	−70.0000			−0.176322			−0.0881612	−	0.996106i	$-0.528099\pi$
$397$	−70.0000			−0.176322			−0.0881612	+	0.996106i	$0.528099\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	103.923	−	60.0000i	0.259160	−	0.149626i	−0.364791	−	0.931089i	$-0.618860\pi$
$401$	103.923	−	60.0000i	0.259160	−	0.149626i	0.623951	+	0.781463i	$0.285526\pi$
$402$		0			0
$403$	5.00000	−	8.66025i	0.0124069	−	0.0214895i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	259.808	+	150.000i	0.638348	+	0.368550i
$408$		0			0
$409$	−134.500	−	232.961i	−0.328851	−	0.569586i	0.653433	−	0.756984i	$-0.273328\pi$
$409$	−134.500	−	232.961i	−0.328851	−	0.569586i	−0.982284	+	0.187398i	$0.939995\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$			150.000i			0.363196i
$414$		0			0
$415$	−9.00000			−0.0216867
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−181.865	+	105.000i	−0.434046	+	0.250597i	−0.701069	−	0.713094i	$-0.747293\pi$
$419$	−181.865	+	105.000i	−0.434046	+	0.250597i	0.267023	+	0.963690i	$0.413960\pi$
$420$		0			0
$421$	−322.000	+	557.720i	−0.764846	+	1.32475i	0.175483	+	0.984483i	$0.443851\pi$
$421$	−322.000	+	557.720i	−0.764846	+	1.32475i	−0.940328	+	0.340269i	$0.889482\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	249.415	+	144.000i	0.586860	+	0.338824i
$426$		0			0
$427$	−190.000	−	329.090i	−0.444965	−	0.770702i
$428$		0			0
$429$		0			0
$430$		0			0
$431$			270.000i			0.626450i	0.949679	+	0.313225i	$0.101409\pi$
$431$			270.000i			0.626450i	−0.949679	+	0.313225i	$0.898591\pi$
$432$		0			0
$433$	−565.000			−1.30485			−0.652425	−	0.757853i	$-0.726248\pi$
$433$	−565.000			−1.30485			−0.652425	+	0.757853i	$0.726248\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−166.277	+	96.0000i	−0.380496	+	0.219680i
$438$		0			0
$439$	−105.500	+	182.731i	−0.240319	+	0.416245i	−0.960805	−	0.277225i	$-0.910585\pi$
$439$	−105.500	+	182.731i	−0.240319	+	0.416245i	0.720486	+	0.693469i	$0.243919\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−431.281	−	249.000i	−0.973545	−	0.562077i	−0.0732302	−	0.997315i	$-0.523331\pi$
$443$	−431.281	−	249.000i	−0.973545	−	0.562077i	−0.900315	+	0.435238i	$0.856664\pi$
$444$		0			0
$445$	−135.000	−	233.827i	−0.303371	−	0.525454i
$446$		0			0
$447$		0			0
$448$		0			0
$449$			360.000i			0.801782i	0.916126	+	0.400891i	$0.131299\pi$
$449$			360.000i			0.801782i	−0.916126	+	0.400891i	$0.868701\pi$
$450$		0			0
$451$	−900.000			−1.99557
$452$		0			0
$453$		0			0
$454$		0			0
$455$	129.904	−	75.0000i	0.285503	−	0.164835i
$456$		0			0
$457$	−182.500	+	316.099i	−0.399344	+	0.691683i	−0.993645	−	0.112559i	$-0.964095\pi$
$457$	−182.500	+	316.099i	−0.399344	+	0.691683i	0.594301	+	0.804242i	$0.297429\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−90.9327	−	52.5000i	−0.197251	−	0.113883i	0.398122	−	0.917333i	$-0.369662\pi$
$461$	−90.9327	−	52.5000i	−0.197251	−	0.113883i	−0.595373	+	0.803450i	$0.702996\pi$
$462$		0			0
$463$	107.500	+	186.195i	0.232181	+	0.402150i	0.958450	−	0.285261i	$-0.0920804\pi$
$463$	107.500	+	186.195i	0.232181	+	0.402150i	−0.726268	+	0.687411i	$0.758747\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$			63.0000i			0.134904i	0.997723	+	0.0674518i	$0.0214869\pi$
$467$			63.0000i			0.134904i	−0.997723	+	0.0674518i	$0.978513\pi$
$468$		0			0
$469$	−50.0000			−0.106610
$470$		0			0
$471$		0			0
$472$		0			0
$473$	649.519	−	375.000i	1.37319	−	0.792812i
$474$		0			0
$475$	−128.000	+	221.703i	−0.269474	+	0.466742i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−649.519	−	375.000i	−1.35599	−	0.782881i	−0.366909	−	0.930257i	$-0.619584\pi$
$479$	−649.519	−	375.000i	−1.35599	−	0.782881i	−0.989081	+	0.147376i	$0.952917\pi$
$480$		0			0
$481$	100.000	+	173.205i	0.207900	+	0.360094i
$482$		0			0
$483$		0			0
$484$		0			0
$485$		−	255.000i		−	0.525773i
$486$		0			0
$487$	−110.000			−0.225873			−0.112936	−	0.993602i	$-0.536026\pi$
$487$	−110.000			−0.225873			−0.112936	+	0.993602i	$0.536026\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	558.586	−	322.500i	1.13765	−	0.656823i	0.191803	−	0.981433i	$-0.438567\pi$
$491$	558.586	−	322.500i	1.13765	−	0.656823i	0.945848	+	0.324611i	$0.105233\pi$
$492$		0			0
$493$	270.000	−	467.654i	0.547667	−	0.948588i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−389.711	−	225.000i	−0.784128	−	0.452716i
$498$		0			0
$499$	−383.000	−	663.375i	−0.767535	−	1.32941i	−0.938896	−	0.344201i	$-0.888150\pi$
$499$	−383.000	−	663.375i	−0.767535	−	1.32941i	0.171361	−	0.985208i	$-0.445184\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$			828.000i			1.64612i	0.567952	+	0.823062i	$0.307736\pi$
$503$			828.000i			1.64612i	−0.567952	+	0.823062i	$0.692264\pi$
$504$		0			0
$505$	−585.000			−1.15842
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−480.644	+	277.500i	−0.944291	+	0.545187i	−0.891303	−	0.453409i	$-0.850208\pi$
$509$	−480.644	+	277.500i	−0.944291	+	0.545187i	−0.0529881	+	0.998595i	$0.516875\pi$
$510$		0			0
$511$	162.500	−	281.458i	0.318004	−	0.550799i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	441.673	+	255.000i	0.857617	+	0.495146i
$516$		0			0
$517$	45.0000	+	77.9423i	0.0870406	+	0.150759i
$518$		0			0
$519$		0			0
$520$		0			0
$521$			450.000i			0.863724i	0.901940	+	0.431862i	$0.142143\pi$
$521$			450.000i			0.863724i	−0.901940	+	0.431862i	$0.857857\pi$
$522$		0			0
$523$	250.000			0.478011			0.239006	−	0.971018i	$-0.423179\pi$
$523$	250.000			0.478011			0.239006	+	0.971018i	$0.423179\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−15.5885	+	9.00000i	−0.0295796	+	0.0170778i
$528$		0			0
$529$	−192.500	+	333.420i	−0.363894	+	0.630283i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−519.615	−	300.000i	−0.974888	−	0.562852i
$534$		0			0
$535$	283.500	+	491.036i	0.529907	+	0.917825i
$536$		0			0
$537$		0			0
$538$		0			0
$539$			360.000i			0.667904i
$540$		0			0
$541$	−268.000			−0.495379			−0.247689	−	0.968839i	$-0.579671\pi$
$541$	−268.000			−0.495379			−0.247689	+	0.968839i	$0.579671\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	426.084	−	246.000i	0.781806	−	0.451376i
$546$		0			0
$547$	205.000	−	355.070i	0.374771	−	0.649123i	−0.615521	−	0.788120i	$-0.711055\pi$
$547$	205.000	−	355.070i	0.374771	−	0.649123i	0.990293	+	0.138997i	$0.0443878\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	415.692	+	240.000i	0.754432	+	0.435572i
$552$		0			0
$553$	−35.0000	−	60.6218i	−0.0632911	−	0.109623i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		−	639.000i		−	1.14722i	−0.819130	−	0.573609i	$-0.805543\pi$
$557$		−	639.000i		−	1.14722i	0.819130	−	0.573609i	$-0.194457\pi$
$558$		0			0
$559$	500.000			0.894454
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−174.071	+	100.500i	−0.309185	+	0.178508i	−0.646562	−	0.762862i	$-0.723794\pi$
$563$	−174.071	+	100.500i	−0.309185	+	0.178508i	0.337377	+	0.941370i	$0.390460\pi$
$564$		0			0
$565$	−36.0000	+	62.3538i	−0.0637168	+	0.110361i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−207.846	−	120.000i	−0.365283	−	0.210896i	0.306113	−	0.951995i	$-0.400972\pi$
$569$	−207.846	−	120.000i	−0.365283	−	0.210896i	−0.671396	+	0.741099i	$0.734305\pi$
$570$		0			0
$571$	−473.000	−	819.260i	−0.828371	−	1.43478i	−0.899315	−	0.437301i	$-0.855934\pi$
$571$	−473.000	−	819.260i	−0.828371	−	1.43478i	0.0709440	−	0.997480i	$-0.477399\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		−	192.000i		−	0.333913i
$576$		0			0
$577$	830.000			1.43847			0.719237	−	0.694764i	$-0.244491\pi$
$577$	830.000			1.43847			0.719237	+	0.694764i	$0.244491\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−12.9904	+	7.50000i	−0.0223587	+	0.0129088i
$582$		0			0
$583$	−202.500	+	350.740i	−0.347341	+	0.601613i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−392.310	−	226.500i	−0.668330	−	0.385860i	0.127114	−	0.991888i	$-0.459429\pi$
$587$	−392.310	−	226.500i	−0.668330	−	0.385860i	−0.795443	+	0.606028i	$0.792762\pi$
$588$		0			0
$589$	−8.00000	−	13.8564i	−0.0135823	−	0.0235253i
$590$		0			0
$591$		0			0
$592$		0			0
$593$			702.000i			1.18381i	0.806007	+	0.591906i	$0.201624\pi$
$593$			702.000i			1.18381i	−0.806007	+	0.591906i	$0.798376\pi$
$594$		0			0
$595$	−270.000			−0.453782
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−961.288	+	555.000i	−1.60482	+	0.926544i	−0.614318	+	0.789059i	$0.710569\pi$
$599$	−961.288	+	555.000i	−1.60482	+	0.926544i	−0.990504	+	0.137486i	$0.956098\pi$
$600$		0			0
$601$	−434.500	+	752.576i	−0.722962	+	1.25221i	0.236846	+	0.971547i	$0.423886\pi$
$601$	−434.500	+	752.576i	−0.722962	+	1.25221i	−0.959807	+	0.280659i	$0.909447\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	270.200	+	156.000i	0.446611	+	0.257851i
$606$		0			0
$607$	265.000	+	458.993i	0.436573	+	0.756167i	0.997423	−	0.0717508i	$-0.0228586\pi$
$607$	265.000	+	458.993i	0.436573	+	0.756167i	−0.560849	+	0.827918i	$0.689525\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$			60.0000i			0.0981997i
$612$		0			0
$613$	−70.0000			−0.114192			−0.0570962	−	0.998369i	$-0.518184\pi$
$613$	−70.0000			−0.114192			−0.0570962	+	0.998369i	$0.518184\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	478.046	−	276.000i	0.774791	−	0.447326i	−0.0597901	−	0.998211i	$-0.519043\pi$
$617$	478.046	−	276.000i	0.774791	−	0.447326i	0.834581	+	0.550885i	$0.185710\pi$
$618$		0			0
$619$	331.000	−	573.309i	0.534733	−	0.926185i	−0.464443	−	0.885603i	$-0.653745\pi$
$619$	331.000	−	573.309i	0.534733	−	0.926185i	0.999176	−	0.0405823i	$-0.0129213\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−389.711	−	225.000i	−0.625540	−	0.361156i
$624$		0			0
$625$	−15.5000	−	26.8468i	−0.0248000	−	0.0429549i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		−	360.000i		−	0.572337i
$630$		0			0
$631$	331.000			0.524564			0.262282	−	0.964991i	$-0.415525\pi$
$631$	331.000			0.524564			0.262282	+	0.964991i	$0.415525\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	532.606	−	307.500i	0.838749	−	0.484252i
$636$		0			0
$637$	−120.000	+	207.846i	−0.188383	+	0.326289i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−51.9615	−	30.0000i	−0.0810632	−	0.0468019i	0.458920	−	0.888477i	$-0.348236\pi$
$641$	−51.9615	−	30.0000i	−0.0810632	−	0.0468019i	−0.539984	+	0.841675i	$0.681570\pi$
$642$		0			0
$643$	220.000	+	381.051i	0.342146	+	0.592615i	0.984831	−	0.173516i	$-0.0555129\pi$
$643$	220.000	+	381.051i	0.342146	+	0.592615i	−0.642685	+	0.766131i	$0.722180\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	972.000i		−	1.50232i	−0.660121	−	0.751159i	$-0.729495\pi$
$647$		−	972.000i		−	1.50232i	0.660121	−	0.751159i	$-0.270505\pi$
$648$		0			0
$649$	−450.000			−0.693374
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−418.290	+	241.500i	−0.640567	+	0.369832i	−0.784833	−	0.619707i	$-0.787251\pi$
$653$	−418.290	+	241.500i	−0.640567	+	0.369832i	0.144266	+	0.989539i	$0.453918\pi$
$654$		0			0
$655$	22.5000	−	38.9711i	0.0343511	−	0.0594979i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	714.471	+	412.500i	1.08417	+	0.625948i	0.932019	−	0.362408i	$-0.118045\pi$
$659$	714.471	+	412.500i	1.08417	+	0.625948i	0.152155	+	0.988357i	$0.451379\pi$
$660$		0			0
$661$	464.000	+	803.672i	0.701967	+	1.21584i	0.967775	+	0.251816i	$0.0810278\pi$
$661$	464.000	+	803.672i	0.701967	+	1.21584i	−0.265808	+	0.964026i	$0.585639\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		−	240.000i		−	0.360902i
$666$		0			0
$667$	−360.000			−0.539730
$668$		0			0
$669$		0			0
$670$		0			0
$671$	987.269	−	570.000i	1.47134	−	0.849478i
$672$		0			0
$673$	492.500	−	853.035i	0.731798	−	1.26751i	−0.224316	−	0.974516i	$-0.572015\pi$
$673$	492.500	−	853.035i	0.731798	−	1.26751i	0.956114	−	0.292995i	$-0.0946518\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	306.573	+	177.000i	0.452840	+	0.261448i	0.709029	−	0.705179i	$-0.249133\pi$
$677$	306.573	+	177.000i	0.452840	+	0.261448i	−0.256189	+	0.966627i	$0.582467\pi$
$678$		0			0
$679$	−212.500	−	368.061i	−0.312960	−	0.542063i
$680$		0			0
$681$		0			0
$682$		0			0
$683$			198.000i			0.289898i	0.989439	+	0.144949i	$0.0463017\pi$
$683$			198.000i			0.289898i	−0.989439	+	0.144949i	$0.953698\pi$
$684$		0			0
$685$	414.000			0.604380
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−233.827	+	135.000i	−0.339371	+	0.195936i
$690$		0			0
$691$	−218.000	+	377.587i	−0.315485	+	0.546436i	−0.979540	−	0.201247i	$-0.935500\pi$
$691$	−218.000	+	377.587i	−0.315485	+	0.546436i	0.664056	+	0.747683i	$0.268834\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−72.7461	−	42.0000i	−0.104671	−	0.0604317i
$696$		0			0
$697$	540.000	+	935.307i	0.774749	+	1.34190i
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	135.000i		−	0.192582i	−0.995353	−	0.0962910i	$-0.969302\pi$
$701$		−	135.000i		−	0.192582i	0.995353	−	0.0962910i	$-0.0306979\pi$
$702$		0			0
$703$	320.000			0.455192
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−844.375	+	487.500i	−1.19431	+	0.689533i
$708$		0			0
$709$	−16.0000	+	27.7128i	−0.0225670	+	0.0390872i	−0.877088	−	0.480329i	$-0.840517\pi$
$709$	−16.0000	+	27.7128i	−0.0225670	+	0.0390872i	0.854521	+	0.519416i	$0.173851\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	10.3923	+	6.00000i	0.0145755	+	0.00841515i
$714$		0			0
$715$	225.000	+	389.711i	0.314685	+	0.545051i
$716$		0			0
$717$		0			0
$718$		0			0
$719$		−	900.000i		−	1.25174i	−0.779928	−	0.625869i	$-0.784744\pi$
$719$		−	900.000i		−	1.25174i	0.779928	−	0.625869i	$-0.215256\pi$
$720$		0			0
$721$	850.000			1.17892
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−415.692	+	240.000i	−0.573369	+	0.331034i
$726$		0			0
$727$	−87.5000	+	151.554i	−0.120358	+	0.208466i	−0.919909	−	0.392133i	$-0.871737\pi$
$727$	−87.5000	+	151.554i	−0.120358	+	0.208466i	0.799551	+	0.600598i	$0.205071\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−779.423	−	450.000i	−1.06624	−	0.615595i
$732$		0			0
$733$	−580.000	−	1004.59i	−0.791269	−	1.37052i	−0.925182	−	0.379525i	$-0.876088\pi$
$733$	−580.000	−	1004.59i	−0.791269	−	1.37052i	0.133913	−	0.990993i	$-0.457246\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		−	150.000i		−	0.203528i
$738$		0			0
$739$	1006.00			1.36130			0.680650	−	0.732609i	$-0.261698\pi$
$739$	1006.00			1.36130			0.680650	+	0.732609i	$0.261698\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	98.7269	−	57.0000i	0.132876	−	0.0767160i	−0.432088	−	0.901831i	$-0.642223\pi$
$743$	98.7269	−	57.0000i	0.132876	−	0.0767160i	0.564965	+	0.825115i	$0.308890\pi$
$744$		0			0
$745$	112.500	−	194.856i	0.151007	−	0.261551i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	818.394	+	472.500i	1.09265	+	0.630841i
$750$		0			0
$751$	179.500	+	310.903i	0.239015	+	0.413986i	0.960432	−	0.278515i	$-0.0898423\pi$
$751$	179.500	+	310.903i	0.239015	+	0.413986i	−0.721417	+	0.692501i	$0.756509\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		−	231.000i		−	0.305960i
$756$		0			0
$757$	−430.000			−0.568032			−0.284016	−	0.958820i	$-0.591667\pi$
$757$	−430.000			−0.568032			−0.284016	+	0.958820i	$0.591667\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−1143.15	+	660.000i	−1.50217	+	0.867280i	−0.502176	+	0.864765i	$0.667467\pi$
$761$	−1143.15	+	660.000i	−1.50217	+	0.867280i	−0.999997	+	0.00251446i	$0.999200\pi$
$762$		0			0
$763$	410.000	−	710.141i	0.537353	−	0.930722i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−259.808	−	150.000i	−0.338732	−	0.195567i
$768$		0			0
$769$	−629.500	−	1090.33i	−0.818596	−	1.41785i	−0.906717	−	0.421739i	$-0.861420\pi$
$769$	−629.500	−	1090.33i	−0.818596	−	1.41785i	0.0881215	−	0.996110i	$-0.471914\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$			522.000i			0.675291i	0.941273	+	0.337646i	$0.109631\pi$
$773$			522.000i			0.675291i	−0.941273	+	0.337646i	$0.890369\pi$
$774$		0			0
$775$	16.0000			0.0206452
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−831.384	+	480.000i	−1.06725	+	0.616175i
$780$		0			0
$781$	675.000	−	1169.13i	0.864277	−	1.49697i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	259.808	+	150.000i	0.330965	+	0.191083i
$786$		0			0
$787$	−230.000	−	398.372i	−0.292249	−	0.506190i	0.682092	−	0.731266i	$-0.261070\pi$
$787$	−230.000	−	398.372i	−0.292249	−	0.506190i	−0.974341	+	0.225076i	$0.927737\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$			120.000i			0.151707i
$792$		0			0
$793$	760.000			0.958386
$794$		0			0
$795$		0			0
$796$		0			0
$797$	205.248	−	118.500i	0.257526	−	0.148683i	−0.365680	−	0.930741i	$-0.619163\pi$
$797$	205.248	−	118.500i	0.257526	−	0.148683i	0.623205	+	0.782058i	$0.285830\pi$
$798$		0			0
$799$	54.0000	−	93.5307i	0.0675845	−	0.117060i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	844.375	+	487.500i	1.05153	+	0.607098i
$804$		0			0
$805$	90.0000	+	155.885i	0.111801	+	0.193645i
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	810.000i		−	1.00124i	−0.865668	−	0.500618i	$-0.833106\pi$
$809$		−	810.000i		−	1.00124i	0.865668	−	0.500618i	$-0.166894\pi$
$810$		0			0
$811$	−272.000			−0.335388			−0.167694	−	0.985839i	$-0.553632\pi$
$811$	−272.000			−0.335388			−0.167694	+	0.985839i	$0.553632\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−285.788	+	165.000i	−0.350661	+	0.202454i
$816$		0			0
$817$	400.000	−	692.820i	0.489596	−	0.848005i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	337.750	+	195.000i	0.411388	+	0.237515i	0.691386	−	0.722485i	$-0.257000\pi$
$821$	337.750	+	195.000i	0.411388	+	0.237515i	−0.279998	+	0.960001i	$0.590334\pi$
$822$		0			0
$823$	602.500	+	1043.56i	0.732078	+	1.26800i	0.955994	+	0.293387i	$0.0947826\pi$
$823$	602.500	+	1043.56i	0.732078	+	1.26800i	−0.223916	+	0.974608i	$0.571884\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			18.0000i			0.0217654i	0.999941	+	0.0108827i	$0.00346414\pi$
$827$			18.0000i			0.0217654i	−0.999941	+	0.0108827i	$0.996536\pi$
$828$		0			0
$829$	1442.00			1.73945			0.869723	−	0.493541i	$-0.164298\pi$
$829$	1442.00			1.73945			0.869723	+	0.493541i	$0.164298\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	374.123	−	216.000i	0.449127	−	0.259304i
$834$		0			0
$835$	117.000	−	202.650i	0.140120	−	0.242695i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	1376.98	+	795.000i	1.64122	+	0.947557i	0.980402	+	0.197007i	$0.0631222\pi$
$839$	1376.98	+	795.000i	1.64122	+	0.947557i	0.660814	+	0.750550i	$0.270211\pi$
$840$		0			0
$841$	29.5000	+	51.0955i	0.0350773	+	0.0607556i
$842$		0			0
$843$		0			0
$844$		0			0
$845$		−	207.000i		−	0.244970i
$846$		0			0
$847$	520.000			0.613932
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−207.846	+	120.000i	−0.244237	+	0.141011i
$852$		0			0
$853$	−295.000	+	510.955i	−0.345838	+	0.599009i	−0.985506	−	0.169642i	$-0.945739\pi$
$853$	−295.000	+	510.955i	−0.345838	+	0.599009i	0.639667	+	0.768652i	$0.279072\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−1127.57	−	651.000i	−1.31571	−	0.759627i	−0.332676	−	0.943041i	$-0.607952\pi$
$857$	−1127.57	−	651.000i	−1.31571	−	0.759627i	−0.983036	+	0.183415i	$0.941285\pi$
$858$		0			0
$859$	−158.000	−	273.664i	−0.183935	−	0.318584i	0.759282	−	0.650761i	$-0.225550\pi$
$859$	−158.000	−	273.664i	−0.183935	−	0.318584i	−0.943217	+	0.332177i	$0.892217\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			1188.00i			1.37659i	0.725429	+	0.688297i	$0.241641\pi$
$863$			1188.00i			1.37659i	−0.725429	+	0.688297i	$0.758359\pi$
$864$		0			0
$865$	−531.000			−0.613873
$866$		0			0
$867$		0			0
$868$		0			0
$869$	181.865	−	105.000i	0.209281	−	0.120829i
$870$		0			0
$871$	50.0000	−	86.6025i	0.0574053	−	0.0994289i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	532.606	+	307.500i	0.608692	+	0.351429i
$876$		0			0
$877$	275.000	+	476.314i	0.313569	+	0.543117i	0.979132	−	0.203224i	$-0.0651420\pi$
$877$	275.000	+	476.314i	0.313569	+	0.543117i	−0.665563	+	0.746341i	$0.731809\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$			90.0000i			0.102157i	0.998695	+	0.0510783i	$0.0162658\pi$
$881$			90.0000i			0.102157i	−0.998695	+	0.0510783i	$0.983734\pi$
$882$		0			0
$883$	880.000			0.996602			0.498301	−	0.867004i	$-0.333957\pi$
$883$	880.000			0.996602			0.498301	+	0.867004i	$0.333957\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−244.219	+	141.000i	−0.275332	+	0.158963i	−0.631308	−	0.775532i	$-0.717482\pi$
$887$	−244.219	+	141.000i	−0.275332	+	0.158963i	0.355976	+	0.934495i	$0.384148\pi$
$888$		0			0
$889$	512.500	−	887.676i	0.576490	−	0.998511i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	83.1384	+	48.0000i	0.0931002	+	0.0537514i
$894$		0			0
$895$	−337.500	−	584.567i	−0.377095	−	0.653148i
$896$		0			0
$897$		0			0
$898$		0			0
$899$		−	30.0000i		−	0.0333704i
$900$		0			0
$901$	486.000			0.539401
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−41.5692	+	24.0000i	−0.0459328	+	0.0265193i
$906$		0			0
$907$	−650.000	+	1125.83i	−0.716648	+	1.24127i	0.245672	+	0.969353i	$0.420991\pi$
$907$	−650.000	+	1125.83i	−0.716648	+	1.24127i	−0.962320	+	0.271918i	$0.912342\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−181.865	−	105.000i	−0.199633	−	0.115258i	0.396851	−	0.917883i	$-0.370103\pi$
$911$	−181.865	−	105.000i	−0.199633	−	0.115258i	−0.596484	+	0.802625i	$0.703436\pi$
$912$		0			0
$913$	−22.5000	−	38.9711i	−0.0246440	−	0.0426847i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		−	75.0000i		−	0.0817884i
$918$		0			0
$919$	−137.000			−0.149075			−0.0745375	−	0.997218i	$-0.523748\pi$
$919$	−137.000			−0.149075			−0.0745375	+	0.997218i	$0.523748\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	779.423	−	450.000i	0.844445	−	0.487541i
$924$		0			0
$925$	−160.000	+	277.128i	−0.172973	+	0.299598i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	571.577	+	330.000i	0.615260	+	0.355221i	0.775021	−	0.631935i	$-0.217739\pi$
$929$	571.577	+	330.000i	0.615260	+	0.355221i	−0.159761	+	0.987156i	$0.551072\pi$
$930$		0			0
$931$	192.000	+	332.554i	0.206230	+	0.357201i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		−	810.000i		−	0.866310i
$936$		0			0
$937$	605.000			0.645678			0.322839	−	0.946454i	$-0.395363\pi$
$937$	605.000			0.645678			0.322839	+	0.946454i	$0.395363\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	1389.97	−	802.500i	1.47712	−	0.852816i	0.477455	−	0.878656i	$-0.341559\pi$
$941$	1389.97	−	802.500i	1.47712	−	0.852816i	0.999666	+	0.0258401i	$0.00822609\pi$
$942$		0			0
$943$	360.000	−	623.538i	0.381760	−	0.661228i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−470.252	−	271.500i	−0.496570	−	0.286695i	0.230726	−	0.973019i	$-0.425890\pi$
$947$	−470.252	−	271.500i	−0.496570	−	0.286695i	−0.727296	+	0.686324i	$0.759223\pi$
$948$		0			0
$949$	325.000	+	562.917i	0.342466	+	0.593168i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	144.000i		−	0.151102i	−0.997142	−	0.0755509i	$-0.975928\pi$
$953$		−	144.000i		−	0.151102i	0.997142	−	0.0755509i	$-0.0240715\pi$
$954$		0			0
$955$	−90.0000			−0.0942408
$956$		0			0
$957$		0			0
$958$		0			0
$959$	597.558	−	345.000i	0.623105	−	0.359750i
$960$		0			0
$961$	480.000	−	831.384i	0.499480	−	0.865124i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−558.586	−	322.500i	−0.578846	−	0.334197i
$966$		0			0
$967$	422.500	+	731.791i	0.436918	+	0.756765i	0.997450	−	0.0713682i	$-0.0227365\pi$
$967$	422.500	+	731.791i	0.436918	+	0.756765i	−0.560532	+	0.828133i	$0.689403\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$			405.000i			0.417096i	0.978012	+	0.208548i	$0.0668737\pi$
$971$			405.000i			0.417096i	−0.978012	+	0.208548i	$0.933126\pi$
$972$		0			0
$973$	−140.000			−0.143885
$974$		0			0
$975$		0			0
$976$		0			0
$977$	213.042	−	123.000i	0.218058	−	0.125896i	−0.386993	−	0.922083i	$-0.626486\pi$
$977$	213.042	−	123.000i	0.218058	−	0.125896i	0.605051	+	0.796187i	$0.293153\pi$
$978$		0			0
$979$	675.000	−	1169.13i	0.689479	−	1.19421i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−898.934	−	519.000i	−0.914481	−	0.527976i	−0.0326105	−	0.999468i	$-0.510382\pi$
$983$	−898.934	−	519.000i	−0.914481	−	0.527976i	−0.881870	+	0.471493i	$0.843715\pi$
$984$		0			0
$985$	310.500	+	537.802i	0.315228	+	0.545992i
$986$		0			0
$987$		0			0
$988$		0			0
$989$			600.000i			0.606673i
$990$		0			0
$991$	1501.00			1.51463			0.757316	−	0.653049i	$-0.226510\pi$
$991$	1501.00			1.51463			0.757316	+	0.653049i	$0.226510\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	579.371	−	334.500i	0.582282	−	0.336181i
$996$		0			0
$997$	−385.000	+	666.840i	−0.386158	+	0.668846i	−0.991929	−	0.126793i	$-0.959532\pi$
$997$	−385.000	+	666.840i	−0.386158	+	0.668846i	0.605771	+	0.795639i	$0.292865\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1296.3.q.j.593.2		4
3.2	odd	2	inner	1296.3.q.j.593.1		4
4.3	odd	2		81.3.d.b.26.2		4
9.2	odd	6		432.3.e.c.161.1		2
9.4	even	3	inner	1296.3.q.j.1025.1		4
9.5	odd	6	inner	1296.3.q.j.1025.2		4
9.7	even	3		432.3.e.c.161.2		2
12.11	even	2		81.3.d.b.26.1		4
36.7	odd	6		27.3.b.b.26.1	✓	2
36.11	even	6		27.3.b.b.26.2	yes	2
36.23	even	6		81.3.d.b.53.2		4
36.31	odd	6		81.3.d.b.53.1		4
72.11	even	6		1728.3.e.m.1025.2		2
72.29	odd	6		1728.3.e.g.1025.2		2
72.43	odd	6		1728.3.e.m.1025.1		2
72.61	even	6		1728.3.e.g.1025.1		2
180.7	even	12		675.3.d.d.674.2		2
180.43	even	12		675.3.d.a.674.1		2
180.47	odd	12		675.3.d.a.674.2		2
180.79	odd	6		675.3.c.h.26.2		2
180.83	odd	12		675.3.d.d.674.1		2
180.119	even	6		675.3.c.h.26.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
27.3.b.b.26.1	✓	2	36.7	odd	6
27.3.b.b.26.2	yes	2	36.11	even	6
81.3.d.b.26.1		4	12.11	even	2
81.3.d.b.26.2		4	4.3	odd	2
81.3.d.b.53.1		4	36.31	odd	6
81.3.d.b.53.2		4	36.23	even	6
432.3.e.c.161.1		2	9.2	odd	6
432.3.e.c.161.2		2	9.7	even	3
675.3.c.h.26.1		2	180.119	even	6
675.3.c.h.26.2		2	180.79	odd	6
675.3.d.a.674.1		2	180.43	even	12
675.3.d.a.674.2		2	180.47	odd	12
675.3.d.d.674.1		2	180.83	odd	12
675.3.d.d.674.2		2	180.7	even	12
1296.3.q.j.593.1		4	3.2	odd	2	inner
1296.3.q.j.593.2		4	1.1	even	1	trivial
1296.3.q.j.1025.1		4	9.4	even	3	inner
1296.3.q.j.1025.2		4	9.5	odd	6	inner
1728.3.e.g.1025.1		2	72.61	even	6
1728.3.e.g.1025.2		2	72.29	odd	6
1728.3.e.m.1025.1		2	72.43	odd	6
1728.3.e.m.1025.2		2	72.11	even	6

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:	\( N \)	\(=\)	\( 1296 = 2^{4} \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1296.q (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(2.59808 - 1.50000i) q^{5} +(2.50000 - 4.33013i) q^{7} +(12.9904 + 7.50000i) q^{11} +(5.00000 + 8.66025i) q^{13} -18.0000i q^{17} +16.0000 q^{19} +(-10.3923 + 6.00000i) q^{23} +(-8.00000 + 13.8564i) q^{25} +(25.9808 + 15.0000i) q^{29} +(-0.500000 - 0.866025i) q^{31} -15.0000i q^{35} +20.0000 q^{37} +(-51.9615 + 30.0000i) q^{41} +(25.0000 - 43.3013i) q^{43} +(5.19615 + 3.00000i) q^{47} +(12.0000 + 20.7846i) q^{49} +27.0000i q^{53} +45.0000 q^{55} +(-25.9808 + 15.0000i) q^{59} +(38.0000 - 65.8179i) q^{61} +(25.9808 + 15.0000i) q^{65} +(-5.00000 - 8.66025i) q^{67} -90.0000i q^{71} +65.0000 q^{73} +(64.9519 - 37.5000i) q^{77} +(7.00000 - 12.1244i) q^{79} +(-2.59808 - 1.50000i) q^{83} +(-27.0000 - 46.7654i) q^{85} -90.0000i q^{89} +50.0000 q^{91} +(41.5692 - 24.0000i) q^{95} +(42.5000 - 73.6122i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 10 q^{7} + 20 q^{13} + 64 q^{19} - 32 q^{25} - 2 q^{31} + 80 q^{37} + 100 q^{43} + 48 q^{49} + 180 q^{55} + 152 q^{61} - 20 q^{67} + 260 q^{73} + 28 q^{79} - 108 q^{85} + 200 q^{91} + 170 q^{97}+O(q^{100}) \) 4 * q + 10 * q^7 + 20 * q^13 + 64 * q^19 - 32 * q^25 - 2 * q^31 + 80 * q^37 + 100 * q^43 + 48 * q^49 + 180 * q^55 + 152 * q^61 - 20 * q^67 + 260 * q^73 + 28 * q^79 - 108 * q^85 + 200 * q^91 + 170 * q^97

Introduction

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