LMFDB - Embedded newform 4284.2.z.b.3277.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4284,2,Mod(3277,4284)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4284, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([0, 0, 0, 1]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("4284.3277");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4284 = 2^{2} \cdot 3^{2} \cdot 7 \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4284.z (of order $4$, degree $2$, 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:	$34.2079122259$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$8$ over $\Q(i)$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} + 18x^{14} + 127x^{12} + 444x^{10} + 801x^{8} + 708x^{6} + 263x^{4} + 30x^{2} + 1 $ x^16 + 18x^14 + 127x^12 + 444x^10 + 801x^8 + 708x^6 + 263x^4 + 30*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 476)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			3277.4
Root			$-2.05350i$ of defining polynomial
Character	$\chi$	$=$	4284.3277
Dual form			4284.2.z.b.4033.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.25265 - 1.25265i) q^{5} +(0.707107 - 0.707107i) q^{7} +O(q^{10})$	$q+(-1.25265 - 1.25265i) q^{5} +(0.707107 - 0.707107i) q^{7} +(2.12117 - 2.12117i) q^{11} -2.59917 q^{13} +(-1.08609 + 3.97749i) q^{17} -4.57015i q^{19} +(-3.09262 + 3.09262i) q^{23} -1.86172i q^{25} +(-1.70313 - 1.70313i) q^{29} +(1.86978 + 1.86978i) q^{31} -1.77152 q^{35} +(0.372597 + 0.372597i) q^{37} +(8.52683 - 8.52683i) q^{41} -2.63796i q^{43} -9.67463 q^{47} -1.00000i q^{49} +3.88549i q^{53} -5.31420 q^{55} +2.63190i q^{59} +(-6.61920 + 6.61920i) q^{61} +(3.25586 + 3.25586i) q^{65} -14.9050 q^{67} +(-10.1101 - 10.1101i) q^{71} +(2.22856 + 2.22856i) q^{73} -2.99979i q^{77} +(12.2409 - 12.2409i) q^{79} +12.6749i q^{83} +(6.34291 - 3.62191i) q^{85} -2.05547 q^{89} +(-1.83789 + 1.83789i) q^{91} +(-5.72482 + 5.72482i) q^{95} +(5.08951 + 5.08951i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 4 q^{5}+O(q^{10}) $ 16 * q - 4 * q^5	$ 16 q - 4 q^{5} + 12 q^{11} - 8 q^{17} + 12 q^{29} - 12 q^{31} + 8 q^{35} + 28 q^{41} - 40 q^{47} + 8 q^{55} + 12 q^{61} + 8 q^{65} - 40 q^{67} - 8 q^{71} + 4 q^{73} + 4 q^{79} - 44 q^{85} + 16 q^{89} + 4 q^{91} + 20 q^{95} + 4 q^{97}+O(q^{100}) $ 16 * q - 4 * q^5 + 12 * q^11 - 8 * q^17 + 12 * q^29 - 12 * q^31 + 8 * q^35 + 28 * q^41 - 40 * q^47 + 8 * q^55 + 12 * q^61 + 8 * q^65 - 40 * q^67 - 8 * q^71 + 4 * q^73 + 4 * q^79 - 44 * q^85 + 16 * q^89 + 4 * q^91 + 20 * q^95 + 4 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4284\mathbb{Z}\right)^\times$.

$n$	$1261$	$1837$	$2143$	$3809$
$\chi(n)$	$e\left(\frac{1}{4}\right)$	$1$	$1$	$1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	−1.25265	−	1.25265i	−0.560204	−	0.560204i	0.369161	−	0.929365i	$-0.379645\pi$
$5$	−1.25265	−	1.25265i	−0.560204	−	0.560204i	−0.929365	+	0.369161i	$0.879645\pi$
$6$		0			0
$7$	0.707107	−	0.707107i	0.267261	−	0.267261i
$7$	0.707107	−	0.707107i	0.267261	−	0.267261i
$8$		0			0
$9$		0			0
$10$		0			0
$11$	2.12117	−	2.12117i	0.639558	−	0.639558i	−0.310888	−	0.950446i	$-0.600626\pi$
$11$	2.12117	−	2.12117i	0.639558	−	0.639558i	0.950446	+	0.310888i	$0.100626\pi$
$12$		0			0
$13$	−2.59917			−0.720879			−0.360440	−	0.932783i	$-0.617373\pi$
$13$	−2.59917			−0.720879			−0.360440	+	0.932783i	$0.617373\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−1.08609	+	3.97749i	−0.263417	+	0.964682i
$17$	−1.08609	+	3.97749i	−0.263417	+	0.964682i
$18$		0			0
$19$		−	4.57015i		−	1.04846i	−0.851575	−	0.524232i	$-0.824352\pi$
$19$		−	4.57015i		−	1.04846i	0.851575	−	0.524232i	$-0.175648\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−3.09262	+	3.09262i	−0.644856	+	0.644856i	−0.951745	−	0.306889i	$-0.900712\pi$
$23$	−3.09262	+	3.09262i	−0.644856	+	0.644856i	0.306889	+	0.951745i	$0.400712\pi$
$24$		0			0
$25$		−	1.86172i		−	0.372343i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−1.70313	−	1.70313i	−0.316264	−	0.316264i	0.531066	−	0.847330i	$-0.321792\pi$
$29$	−1.70313	−	1.70313i	−0.316264	−	0.316264i	−0.847330	+	0.531066i	$0.821792\pi$
$30$		0			0
$31$	1.86978	+	1.86978i	0.335823	+	0.335823i	0.854793	−	0.518970i	$-0.173684\pi$
$31$	1.86978	+	1.86978i	0.335823	+	0.335823i	−0.518970	+	0.854793i	$0.673684\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−1.77152			−0.299442
$36$		0			0
$37$	0.372597	+	0.372597i	0.0612546	+	0.0612546i	0.737070	−	0.675816i	$-0.236209\pi$
$37$	0.372597	+	0.372597i	0.0612546	+	0.0612546i	−0.675816	+	0.737070i	$0.736209\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	8.52683	−	8.52683i	1.33167	−	1.33167i	0.427789	−	0.903879i	$-0.359293\pi$
$41$	8.52683	−	8.52683i	1.33167	−	1.33167i	0.903879	−	0.427789i	$-0.140707\pi$
$42$		0			0
$43$		−	2.63796i		−	0.402285i	−0.979562	−	0.201143i	$-0.935535\pi$
$43$		−	2.63796i		−	0.402285i	0.979562	−	0.201143i	$-0.0644655\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−9.67463			−1.41119			−0.705595	−	0.708616i	$-0.749320\pi$
$47$	−9.67463			−1.41119			−0.705595	+	0.708616i	$0.749320\pi$
$48$		0			0
$49$		−	1.00000i		−	0.142857i
$50$		0			0
$51$		0			0
$52$		0			0
$53$			3.88549i			0.533713i	0.963736	+	0.266857i	$0.0859850\pi$
$53$			3.88549i			0.533713i	−0.963736	+	0.266857i	$0.914015\pi$
$54$		0			0
$55$	−5.31420			−0.716566
$56$		0			0
$57$		0			0
$58$		0			0
$59$			2.63190i			0.342645i	0.985215	+	0.171322i	$0.0548040\pi$
$59$			2.63190i			0.342645i	−0.985215	+	0.171322i	$0.945196\pi$
$60$		0			0
$61$	−6.61920	+	6.61920i	−0.847501	+	0.847501i	−0.989821	−	0.142319i	$-0.954544\pi$
$61$	−6.61920	+	6.61920i	−0.847501	+	0.847501i	0.142319	+	0.989821i	$0.454544\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	3.25586	+	3.25586i	0.403839	+	0.403839i
$66$		0			0
$67$	−14.9050			−1.82093			−0.910467	−	0.413581i	$-0.864278\pi$
$67$	−14.9050			−1.82093			−0.910467	+	0.413581i	$0.864278\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−10.1101	−	10.1101i	−1.19985	−	1.19985i	−0.974211	−	0.225638i	$-0.927553\pi$
$71$	−10.1101	−	10.1101i	−1.19985	−	1.19985i	−0.225638	−	0.974211i	$-0.572447\pi$
$72$		0			0
$73$	2.22856	+	2.22856i	0.260834	+	0.260834i	0.825393	−	0.564559i	$-0.190954\pi$
$73$	2.22856	+	2.22856i	0.260834	+	0.260834i	−0.564559	+	0.825393i	$0.690954\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		−	2.99979i		−	0.341858i
$78$		0			0
$79$	12.2409	−	12.2409i	1.37721	−	1.37721i	0.527896	−	0.849309i	$-0.322981\pi$
$79$	12.2409	−	12.2409i	1.37721	−	1.37721i	0.849309	−	0.527896i	$-0.177019\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$			12.6749i			1.39125i	0.718406	+	0.695624i	$0.244872\pi$
$83$			12.6749i			1.39125i	−0.718406	+	0.695624i	$0.755128\pi$
$84$		0			0
$85$	6.34291	−	3.62191i	0.687986	−	0.392852i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−2.05547			−0.217879			−0.108940	−	0.994048i	$-0.534746\pi$
$89$	−2.05547			−0.217879			−0.108940	+	0.994048i	$0.534746\pi$
$90$		0			0
$91$	−1.83789	+	1.83789i	−0.192663	+	0.192663i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−5.72482	+	5.72482i	−0.587354	+	0.587354i
$96$		0			0
$97$	5.08951	+	5.08951i	0.516761	+	0.516761i	0.916590	−	0.399829i	$-0.130930\pi$
$97$	5.08951	+	5.08951i	0.516761	+	0.516761i	−0.399829	+	0.916590i	$0.630930\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	1.15980			0.115404			0.0577021	−	0.998334i	$-0.481623\pi$
$101$	1.15980			0.115404			0.0577021	+	0.998334i	$0.481623\pi$
$102$		0			0
$103$	−8.73357			−0.860545			−0.430272	−	0.902699i	$-0.641582\pi$
$103$	−8.73357			−0.860545			−0.430272	+	0.902699i	$0.641582\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	9.34809	+	9.34809i	0.903714	+	0.903714i	0.995755	−	0.0920415i	$-0.0293392\pi$
$107$	9.34809	+	9.34809i	0.903714	+	0.903714i	−0.0920415	+	0.995755i	$0.529339\pi$
$108$		0			0
$109$	−3.28189	+	3.28189i	−0.314348	+	0.314348i	−0.846591	−	0.532244i	$-0.821349\pi$
$109$	−3.28189	+	3.28189i	−0.314348	+	0.314348i	0.532244	+	0.846591i	$0.321349\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−0.283749	+	0.283749i	−0.0266928	+	0.0266928i	−0.720327	−	0.693634i	$-0.756008\pi$
$113$	−0.283749	+	0.283749i	−0.0266928	+	0.0266928i	0.693634	+	0.720327i	$0.256008\pi$
$114$		0			0
$115$	7.74797			0.722502
$116$		0			0
$117$		0			0
$118$		0			0
$119$	2.04452	+	3.58049i	0.187421	+	0.328223i
$120$		0			0
$121$			2.00124i			0.181931i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−8.59536	+	8.59536i	−0.768792	+	0.768792i
$126$		0			0
$127$			0.350103i			0.0310666i	0.999879	+	0.0155333i	$0.00494461\pi$
$127$			0.350103i			0.0310666i	−0.999879	+	0.0155333i	$0.995055\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	5.57703	+	5.57703i	0.487267	+	0.487267i	0.907443	−	0.420176i	$-0.138031\pi$
$131$	5.57703	+	5.57703i	0.487267	+	0.487267i	−0.420176	+	0.907443i	$0.638031\pi$
$132$		0			0
$133$	−3.23159	−	3.23159i	−0.280214	−	0.280214i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−9.64144			−0.823724			−0.411862	−	0.911246i	$-0.635121\pi$
$137$	−9.64144			−0.823724			−0.411862	+	0.911246i	$0.635121\pi$
$138$		0			0
$139$	−9.12776	−	9.12776i	−0.774206	−	0.774206i	0.204633	−	0.978839i	$-0.434400\pi$
$139$	−9.12776	−	9.12776i	−0.774206	−	0.774206i	−0.978839	+	0.204633i	$0.934400\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−5.51329	+	5.51329i	−0.461044	+	0.461044i
$144$		0			0
$145$			4.26688i			0.354345i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	3.94430			0.323129			0.161565	−	0.986862i	$-0.448346\pi$
$149$	3.94430			0.323129			0.161565	+	0.986862i	$0.448346\pi$
$150$		0			0
$151$		−	23.8865i		−	1.94385i	−0.235281	−	0.971927i	$-0.575601\pi$
$151$		−	23.8865i		−	1.94385i	0.235281	−	0.971927i	$-0.424399\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		−	4.68438i		−	0.376258i
$156$		0			0
$157$	−18.4697			−1.47404			−0.737020	−	0.675871i	$-0.763768\pi$
$157$	−18.4697			−1.47404			−0.737020	+	0.675871i	$0.763768\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$			4.37363i			0.344690i
$162$		0			0
$163$	−13.0105	+	13.0105i	−1.01906	+	1.01906i	−0.0192472	+	0.999815i	$0.506127\pi$
$163$	−13.0105	+	13.0105i	−1.01906	+	1.01906i	−0.999815	+	0.0192472i	$0.993873\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−16.8488	−	16.8488i	−1.30380	−	1.30380i	−0.925810	−	0.377988i	$-0.876616\pi$
$167$	−16.8488	−	16.8488i	−1.30380	−	1.30380i	−0.377988	−	0.925810i	$-0.623384\pi$
$168$		0			0
$169$	−6.24433			−0.480333
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−4.43253	−	4.43253i	−0.336999	−	0.336999i	0.518238	−	0.855237i	$-0.326588\pi$
$173$	−4.43253	−	4.43253i	−0.336999	−	0.336999i	−0.855237	+	0.518238i	$0.826588\pi$
$174$		0			0
$175$	−1.31643	−	1.31643i	−0.0995129	−	0.0995129i
$176$		0			0
$177$		0			0
$178$		0			0
$179$			12.8732i			0.962190i	0.876668	+	0.481095i	$0.159761\pi$
$179$			12.8732i			0.962190i	−0.876668	+	0.481095i	$0.840239\pi$
$180$		0			0
$181$	−15.1228	+	15.1228i	−1.12407	+	1.12407i	−0.132948	+	0.991123i	$0.542444\pi$
$181$	−15.1228	+	15.1228i	−1.12407	+	1.12407i	−0.991123	+	0.132948i	$0.957556\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		−	0.933470i		−	0.0686301i
$186$		0			0
$187$	6.13315	+	10.7407i	0.448500	+	0.785441i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	17.5198			1.26769			0.633846	−	0.773460i	$-0.281475\pi$
$191$	17.5198			1.26769			0.633846	+	0.773460i	$0.281475\pi$
$192$		0			0
$193$	3.64600	−	3.64600i	0.262444	−	0.262444i	−0.563602	−	0.826046i	$-0.690585\pi$
$193$	3.64600	−	3.64600i	0.262444	−	0.262444i	0.826046	+	0.563602i	$0.190585\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	6.48158	−	6.48158i	0.461794	−	0.461794i	−0.437449	−	0.899243i	$-0.644118\pi$
$197$	6.48158	−	6.48158i	0.461794	−	0.461794i	0.899243	+	0.437449i	$0.144118\pi$
$198$		0			0
$199$	8.23149	+	8.23149i	0.583515	+	0.583515i	0.935867	−	0.352353i	$-0.114618\pi$
$199$	8.23149	+	8.23149i	0.583515	+	0.583515i	−0.352353	+	0.935867i	$0.614618\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−2.40860			−0.169050
$204$		0			0
$205$	−21.3623			−1.49201
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−9.69409	−	9.69409i	−0.670554	−	0.670554i
$210$		0			0
$211$	2.52188	−	2.52188i	0.173613	−	0.173613i	−0.614952	−	0.788565i	$-0.710824\pi$
$211$	2.52188	−	2.52188i	0.173613	−	0.173613i	0.788565	+	0.614952i	$0.210824\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−3.30445	+	3.30445i	−0.225362	+	0.225362i
$216$		0			0
$217$	2.64427			0.179505
$218$		0			0
$219$		0			0
$220$		0			0
$221$	2.82294	−	10.3381i	0.189892	−	0.695419i
$222$		0			0
$223$			21.4094i			1.43368i	0.697238	+	0.716840i	$0.254412\pi$
$223$			21.4094i			1.43368i	−0.697238	+	0.716840i	$0.745588\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−14.9150	+	14.9150i	−0.989946	+	0.989946i	−0.999950	−	0.0100041i	$-0.996816\pi$
$227$	−14.9150	+	14.9150i	−0.989946	+	0.989946i	0.0100041	+	0.999950i	$0.496816\pi$
$228$		0			0
$229$			10.2882i			0.679866i	0.940450	+	0.339933i	$0.110404\pi$
$229$			10.2882i			0.679866i	−0.940450	+	0.339933i	$0.889596\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−8.07234	−	8.07234i	−0.528837	−	0.528837i	0.391389	−	0.920225i	$-0.371995\pi$
$233$	−8.07234	−	8.07234i	−0.528837	−	0.528837i	−0.920225	+	0.391389i	$0.871995\pi$
$234$		0			0
$235$	12.1190	+	12.1190i	0.790554	+	0.790554i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	0.00572125			0.000370077			0.000185039	−	1.00000i	$-0.499941\pi$
$239$	0.00572125			0.000370077			0.000185039		1.00000i	$0.499941\pi$
$240$		0			0
$241$	−18.1682	−	18.1682i	−1.17031	−	1.17031i	−0.982135	−	0.188180i	$-0.939741\pi$
$241$	−18.1682	−	18.1682i	−1.17031	−	1.17031i	−0.188180	−	0.982135i	$-0.560259\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−1.25265	+	1.25265i	−0.0800291	+	0.0800291i
$246$		0			0
$247$			11.8786i			0.755816i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−8.56127			−0.540383			−0.270191	−	0.962807i	$-0.587087\pi$
$251$	−8.56127			−0.540383			−0.270191	+	0.962807i	$0.587087\pi$
$252$		0			0
$253$			13.1200i			0.824846i
$254$		0			0
$255$		0			0
$256$		0			0
$257$		−	15.3786i		−	0.959291i	−0.877462	−	0.479646i	$-0.840765\pi$
$257$		−	15.3786i		−	0.959291i	0.877462	−	0.479646i	$-0.159235\pi$
$258$		0			0
$259$	0.526932			0.0327419
$260$		0			0
$261$		0			0
$262$		0			0
$263$		−	1.18787i		−	0.0732471i	−0.999329	−	0.0366236i	$-0.988340\pi$
$263$		−	1.18787i		−	0.0732471i	0.999329	−	0.0366236i	$-0.0116603\pi$
$264$		0			0
$265$	4.86717	−	4.86717i	0.298988	−	0.298988i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	3.16855	+	3.16855i	0.193190	+	0.193190i	0.797073	−	0.603883i	$-0.206381\pi$
$269$	3.16855	+	3.16855i	0.193190	+	0.193190i	−0.603883	+	0.797073i	$0.706381\pi$
$270$		0			0
$271$	−15.5763			−0.946192			−0.473096	−	0.881011i	$-0.656864\pi$
$271$	−15.5763			−0.946192			−0.473096	+	0.881011i	$0.656864\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−3.94903	−	3.94903i	−0.238135	−	0.238135i
$276$		0			0
$277$	−12.8334	−	12.8334i	−0.771082	−	0.771082i	0.207214	−	0.978296i	$-0.433560\pi$
$277$	−12.8334	−	12.8334i	−0.771082	−	0.771082i	−0.978296	+	0.207214i	$0.933560\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	27.0114i		−	1.61137i	−0.592347	−	0.805683i	$-0.701798\pi$
$281$		−	27.0114i		−	1.61137i	0.592347	−	0.805683i	$-0.298202\pi$
$282$		0			0
$283$	−4.54071	+	4.54071i	−0.269917	+	0.269917i	−0.829067	−	0.559150i	$-0.811128\pi$
$283$	−4.54071	+	4.54071i	−0.269917	+	0.269917i	0.559150	+	0.829067i	$0.311128\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		−	12.0588i		−	0.711806i
$288$		0			0
$289$	−14.6408	−	8.63985i	−0.861223	−	0.508227i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−12.4808			−0.729134			−0.364567	−	0.931177i	$-0.618783\pi$
$293$	−12.4808			−0.729134			−0.364567	+	0.931177i	$0.618783\pi$
$294$		0			0
$295$	3.29686	−	3.29686i	0.191951	−	0.191951i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	8.03824	−	8.03824i	0.464863	−	0.464863i
$300$		0			0
$301$	−1.86532	−	1.86532i	−0.107515	−	0.107515i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	16.5831			0.949547
$306$		0			0
$307$	29.3966			1.67775			0.838877	−	0.544321i	$-0.183213\pi$
$307$	29.3966			1.67775			0.838877	+	0.544321i	$0.183213\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−11.2210	−	11.2210i	−0.636286	−	0.636286i	0.313352	−	0.949637i	$-0.398548\pi$
$311$	−11.2210	−	11.2210i	−0.636286	−	0.636286i	−0.949637	+	0.313352i	$0.898548\pi$
$312$		0			0
$313$	−23.2386	+	23.2386i	−1.31352	+	1.31352i	−0.394722	+	0.918800i	$0.629159\pi$
$313$	−23.2386	+	23.2386i	−1.31352	+	1.31352i	−0.918800	+	0.394722i	$0.870841\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−18.7827	+	18.7827i	−1.05494	+	1.05494i	−0.0565423	+	0.998400i	$0.518008\pi$
$317$	−18.7827	+	18.7827i	−1.05494	+	1.05494i	−0.998400	+	0.0565423i	$0.981992\pi$
$318$		0			0
$319$	−7.22529			−0.404539
$320$		0			0
$321$		0			0
$322$		0			0
$323$	18.1777	+	4.96362i	1.01144	+	0.276183i
$324$		0			0
$325$			4.83891i			0.268414i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−6.84099	+	6.84099i	−0.377156	+	0.377156i
$330$		0			0
$331$		−	17.9872i		−	0.988666i	−0.869273	−	0.494333i	$-0.835412\pi$
$331$		−	17.9872i		−	0.988666i	0.869273	−	0.494333i	$-0.164588\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	18.6708	+	18.6708i	1.02009	+	1.02009i
$336$		0			0
$337$	−19.8367	−	19.8367i	−1.08058	−	1.08058i	−0.996456	−	0.0841196i	$-0.973192\pi$
$337$	−19.8367	−	19.8367i	−1.08058	−	1.08058i	−0.0841196	−	0.996456i	$-0.526808\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	7.93227			0.429556
$342$		0			0
$343$	−0.707107	−	0.707107i	−0.0381802	−	0.0381802i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	13.5196	−	13.5196i	0.725773	−	0.725773i	−0.244002	−	0.969775i	$-0.578460\pi$
$347$	13.5196	−	13.5196i	0.725773	−	0.725773i	0.969775	+	0.244002i	$0.0784604\pi$
$348$		0			0
$349$			12.4741i			0.667725i	0.942622	+	0.333863i	$0.108352\pi$
$349$			12.4741i			0.667725i	−0.942622	+	0.333863i	$0.891648\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	32.0136			1.70391			0.851956	−	0.523613i	$-0.175416\pi$
$353$	32.0136			1.70391			0.851956	+	0.523613i	$0.175416\pi$
$354$		0			0
$355$			25.3289i			1.34432i
$356$		0			0
$357$		0			0
$358$		0			0
$359$			11.5673i			0.610501i	0.952272	+	0.305251i	$0.0987402\pi$
$359$			11.5673i			0.610501i	−0.952272	+	0.305251i	$0.901260\pi$
$360$		0			0
$361$	−1.88628			−0.0992781
$362$		0			0
$363$		0			0
$364$		0			0
$365$		−	5.58324i		−	0.292240i
$366$		0			0
$367$	−3.99418	+	3.99418i	−0.208495	+	0.208495i	−0.803627	−	0.595133i	$-0.797099\pi$
$367$	−3.99418	+	3.99418i	−0.208495	+	0.208495i	0.595133	+	0.803627i	$0.297099\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	2.74746	+	2.74746i	0.142641	+	0.142641i
$372$		0			0
$373$	−18.0723			−0.935750			−0.467875	−	0.883795i	$-0.654980\pi$
$373$	−18.0723			−0.935750			−0.467875	+	0.883795i	$0.654980\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	4.42673	+	4.42673i	0.227988	+	0.227988i
$378$		0			0
$379$	−3.02455	−	3.02455i	−0.155361	−	0.155361i	0.625147	−	0.780507i	$-0.285039\pi$
$379$	−3.02455	−	3.02455i	−0.155361	−	0.155361i	−0.780507	+	0.625147i	$0.785039\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		−	2.28017i		−	0.116511i	−0.998302	−	0.0582557i	$-0.981446\pi$
$383$		−	2.28017i		−	0.116511i	0.998302	−	0.0582557i	$-0.0185539\pi$
$384$		0			0
$385$	−3.75770	+	3.75770i	−0.191510	+	0.191510i
$386$		0			0
$387$		0			0
$388$		0			0
$389$			3.16537i			0.160491i	0.996775	+	0.0802453i	$0.0255704\pi$
$389$			3.16537i			0.160491i	−0.996775	+	0.0802453i	$0.974430\pi$
$390$		0			0
$391$	−8.94198	−	15.6597i	−0.452215	−	0.791947i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−30.6671			−1.54303
$396$		0			0
$397$	−1.03989	+	1.03989i	−0.0521904	+	0.0521904i	−0.732720	−	0.680530i	$-0.761750\pi$
$397$	−1.03989	+	1.03989i	−0.0521904	+	0.0521904i	0.680530	+	0.732720i	$0.261750\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−23.2831	+	23.2831i	−1.16270	+	1.16270i	−0.178821	+	0.983882i	$0.557228\pi$
$401$	−23.2831	+	23.2831i	−1.16270	+	1.16270i	−0.983882	+	0.178821i	$0.942772\pi$
$402$		0			0
$403$	−4.85987	−	4.85987i	−0.242088	−	0.242088i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	1.58069			0.0783517
$408$		0			0
$409$	11.6136			0.574255			0.287128	−	0.957892i	$-0.407300\pi$
$409$	11.6136			0.574255			0.287128	+	0.957892i	$0.407300\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	1.86104	+	1.86104i	0.0915756	+	0.0915756i
$414$		0			0
$415$	15.8772	−	15.8772i	0.779382	−	0.779382i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	24.0737	−	24.0737i	1.17608	−	1.17608i	0.195342	−	0.980735i	$-0.437418\pi$
$419$	24.0737	−	24.0737i	1.17608	−	1.17608i	0.980735	−	0.195342i	$-0.0625818\pi$
$420$		0			0
$421$	16.4720			0.802797			0.401399	−	0.915903i	$-0.368524\pi$
$421$	16.4720			0.802797			0.401399	+	0.915903i	$0.368524\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	7.40495	+	2.02200i	0.359193	+	0.0980814i
$426$		0			0
$427$			9.36096i			0.453009i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	2.52263	−	2.52263i	0.121511	−	0.121511i	−0.643736	−	0.765247i	$-0.722617\pi$
$431$	2.52263	−	2.52263i	0.121511	−	0.121511i	0.765247	+	0.643736i	$0.222617\pi$
$432$		0			0
$433$		−	13.7012i		−	0.658436i	−0.944254	−	0.329218i	$-0.893215\pi$
$433$		−	13.7012i		−	0.658436i	0.944254	−	0.329218i	$-0.106785\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	14.1338	+	14.1338i	0.676109	+	0.676109i
$438$		0			0
$439$	25.5344	+	25.5344i	1.21869	+	1.21869i	0.968090	+	0.250601i	$0.0806283\pi$
$439$	25.5344	+	25.5344i	1.21869	+	1.21869i	0.250601	+	0.968090i	$0.419372\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−6.17342			−0.293308			−0.146654	−	0.989188i	$-0.546850\pi$
$443$	−6.17342			−0.293308			−0.146654	+	0.989188i	$0.546850\pi$
$444$		0			0
$445$	2.57479	+	2.57479i	0.122057	+	0.122057i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	14.2385	−	14.2385i	0.671957	−	0.671957i	−0.286210	−	0.958167i	$-0.592396\pi$
$449$	14.2385	−	14.2385i	0.671957	−	0.671957i	0.958167	+	0.286210i	$0.0923955\pi$
$450$		0			0
$451$		−	36.1738i		−	1.70336i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	4.60448			0.215861
$456$		0			0
$457$			13.0583i			0.610842i	0.952217	+	0.305421i	$0.0987973\pi$
$457$			13.0583i			0.610842i	−0.952217	+	0.305421i	$0.901203\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		−	15.0878i		−	0.702710i	−0.936242	−	0.351355i	$-0.885721\pi$
$461$		−	15.0878i		−	0.702710i	0.936242	−	0.351355i	$-0.114279\pi$
$462$		0			0
$463$	30.0923			1.39851			0.699255	−	0.714873i	$-0.253515\pi$
$463$	30.0923			1.39851			0.699255	+	0.714873i	$0.253515\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		−	13.0220i		−	0.602587i	−0.953531	−	0.301294i	$-0.902582\pi$
$467$		−	13.0220i		−	0.602587i	0.953531	−	0.301294i	$-0.0974185\pi$
$468$		0			0
$469$	−10.5394	+	10.5394i	−0.486665	+	0.486665i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−5.59558	−	5.59558i	−0.257285	−	0.257285i
$474$		0			0
$475$	−8.50833			−0.390389
$476$		0			0
$477$		0			0
$478$		0			0
$479$	26.5488	+	26.5488i	1.21304	+	1.21304i	0.970020	+	0.243024i	$0.0781395\pi$
$479$	26.5488	+	26.5488i	1.21304	+	1.21304i	0.243024	+	0.970020i	$0.421861\pi$
$480$		0			0
$481$	−0.968442	−	0.968442i	−0.0441571	−	0.0441571i
$482$		0			0
$483$		0			0
$484$		0			0
$485$		−	12.7508i		−	0.578983i
$486$		0			0
$487$	25.7940	−	25.7940i	1.16884	−	1.16884i	0.186354	−	0.982483i	$-0.440333\pi$
$487$	25.7940	−	25.7940i	1.16884	−	1.16884i	0.982483	−	0.186354i	$-0.0596670\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		−	21.2512i		−	0.959053i	−0.877527	−	0.479526i	$-0.840808\pi$
$491$		−	21.2512i		−	0.959053i	0.877527	−	0.479526i	$-0.159192\pi$
$492$		0			0
$493$	8.62396	−	4.92443i	0.388404	−	0.221785i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−14.2979			−0.641347
$498$		0			0
$499$	−16.3182	+	16.3182i	−0.730503	+	0.730503i	−0.970719	−	0.240216i	$-0.922782\pi$
$499$	−16.3182	+	16.3182i	−0.730503	+	0.730503i	0.240216	+	0.970719i	$0.422782\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	0.460191	−	0.460191i	0.0205189	−	0.0205189i	−0.696773	−	0.717292i	$-0.745381\pi$
$503$	0.460191	−	0.460191i	0.0205189	−	0.0205189i	0.717292	+	0.696773i	$0.245381\pi$
$504$		0			0
$505$	−1.45282	−	1.45282i	−0.0646498	−	0.0646498i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−25.6793			−1.13822			−0.569108	−	0.822263i	$-0.692711\pi$
$509$	−25.6793			−1.13822			−0.569108	+	0.822263i	$0.692711\pi$
$510$		0			0
$511$	3.15166			0.139421
$512$		0			0
$513$		0			0
$514$		0			0
$515$	10.9401	+	10.9401i	0.482080	+	0.482080i
$516$		0			0
$517$	−20.5216	+	20.5216i	−0.902538	+	0.902538i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	6.03446	−	6.03446i	0.264374	−	0.264374i	−0.562454	−	0.826829i	$-0.690143\pi$
$521$	6.03446	−	6.03446i	0.264374	−	0.264374i	0.826829	+	0.562454i	$0.190143\pi$
$522$		0			0
$523$	18.7723			0.820856			0.410428	−	0.911893i	$-0.365379\pi$
$523$	18.7723			0.820856			0.410428	+	0.911893i	$0.365379\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−9.46779	+	5.40627i	−0.412424	+	0.235501i
$528$		0			0
$529$			3.87138i			0.168321i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−22.1627	+	22.1627i	−0.959971	+	0.959971i
$534$		0			0
$535$		−	23.4198i		−	1.01253i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−2.12117	−	2.12117i	−0.0913655	−	0.0913655i
$540$		0			0
$541$	9.76691	+	9.76691i	0.419912	+	0.419912i	0.885173	−	0.465261i	$-0.154040\pi$
$541$	9.76691	+	9.76691i	0.419912	+	0.419912i	−0.465261	+	0.885173i	$0.654040\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	8.22213			0.352197
$546$		0			0
$547$	−7.12577	−	7.12577i	−0.304676	−	0.304676i	0.538164	−	0.842840i	$-0.319118\pi$
$547$	−7.12577	−	7.12577i	−0.304676	−	0.304676i	−0.842840	+	0.538164i	$0.819118\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−7.78358	+	7.78358i	−0.331592	+	0.331592i
$552$		0			0
$553$		−	17.3112i		−	0.736147i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−36.0805			−1.52878			−0.764389	−	0.644756i	$-0.776959\pi$
$557$	−36.0805			−1.52878			−0.764389	+	0.644756i	$0.776959\pi$
$558$		0			0
$559$			6.85650i			0.289999i
$560$		0			0
$561$		0			0
$562$		0			0
$563$		−	21.6232i		−	0.911309i	−0.890157	−	0.455655i	$-0.849405\pi$
$563$		−	21.6232i		−	0.911309i	0.890157	−	0.455655i	$-0.150595\pi$
$564$		0			0
$565$	0.710878			0.0299069
$566$		0			0
$567$		0			0
$568$		0			0
$569$		−	8.92834i		−	0.374296i	−0.982332	−	0.187148i	$-0.940076\pi$
$569$		−	8.92834i		−	0.374296i	0.982332	−	0.187148i	$-0.0599243\pi$
$570$		0			0
$571$	26.7616	−	26.7616i	1.11994	−	1.11994i	0.128188	−	0.991750i	$-0.459084\pi$
$571$	26.7616	−	26.7616i	1.11994	−	1.11994i	0.991750	−	0.128188i	$-0.0409160\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	5.75758	+	5.75758i	0.240108	+	0.240108i
$576$		0			0
$577$	32.3336			1.34607			0.673033	−	0.739612i	$-0.264991\pi$
$577$	32.3336			1.34607			0.673033	+	0.739612i	$0.264991\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	8.96249	+	8.96249i	0.371827	+	0.371827i
$582$		0			0
$583$	8.24180	+	8.24180i	0.341341	+	0.341341i
$584$		0			0
$585$		0			0
$586$		0			0
$587$		−	18.0489i		−	0.744957i	−0.928041	−	0.372478i	$-0.878508\pi$
$587$		−	18.0489i		−	0.744957i	0.928041	−	0.372478i	$-0.121492\pi$
$588$		0			0
$589$	8.54519	−	8.54519i	0.352098	−	0.352098i
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	8.95732i		−	0.367833i	−0.982942	−	0.183917i	$-0.941122\pi$
$593$		−	8.95732i		−	0.367833i	0.982942	−	0.183917i	$-0.0588776\pi$
$594$		0			0
$595$	1.92404	−	7.04620i	0.0788779	−	0.288866i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	1.29732			0.0530070			0.0265035	−	0.999649i	$-0.491563\pi$
$599$	1.29732			0.0530070			0.0265035	+	0.999649i	$0.491563\pi$
$600$		0			0
$601$	−22.2262	+	22.2262i	−0.906626	+	0.906626i	−0.995998	−	0.0893720i	$-0.971514\pi$
$601$	−22.2262	+	22.2262i	−0.906626	+	0.906626i	0.0893720	+	0.995998i	$0.471514\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	2.50686	−	2.50686i	0.101918	−	0.101918i
$606$		0			0
$607$	0.747075	+	0.747075i	0.0303228	+	0.0303228i	0.722106	−	0.691783i	$-0.243174\pi$
$607$	0.747075	+	0.747075i	0.0303228	+	0.0303228i	−0.691783	+	0.722106i	$0.743174\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	25.1460			1.01730
$612$		0			0
$613$	−25.7088			−1.03837			−0.519183	−	0.854663i	$-0.673764\pi$
$613$	−25.7088			−1.03837			−0.519183	+	0.854663i	$0.673764\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	5.37161	+	5.37161i	0.216253	+	0.216253i	0.806917	−	0.590664i	$-0.201134\pi$
$617$	5.37161	+	5.37161i	0.216253	+	0.216253i	−0.590664	+	0.806917i	$0.701134\pi$
$618$		0			0
$619$	−12.4781	+	12.4781i	−0.501538	+	0.501538i	−0.911916	−	0.410377i	$-0.865397\pi$
$619$	−12.4781	+	12.4781i	−0.501538	+	0.501538i	0.410377	+	0.911916i	$0.365397\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−1.45344	+	1.45344i	−0.0582307	+	0.0582307i
$624$		0			0
$625$	12.2254			0.489017
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−1.88668	+	1.07732i	−0.0752267	+	0.0429557i
$630$		0			0
$631$			20.2378i			0.805653i	0.915276	+	0.402827i	$0.131972\pi$
$631$			20.2378i			0.805653i	−0.915276	+	0.402827i	$0.868028\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	0.438558	−	0.438558i	0.0174036	−	0.0174036i
$636$		0			0
$637$			2.59917i			0.102983i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	26.3727	+	26.3727i	1.04166	+	1.04166i	0.999094	+	0.0425664i	$0.0135534\pi$
$641$	26.3727	+	26.3727i	1.04166	+	1.04166i	0.0425664	+	0.999094i	$0.486447\pi$
$642$		0			0
$643$	−16.7438	−	16.7438i	−0.660310	−	0.660310i	0.295143	−	0.955453i	$-0.404633\pi$
$643$	−16.7438	−	16.7438i	−0.660310	−	0.660310i	−0.955453	+	0.295143i	$0.904633\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	28.2713			1.11146			0.555730	−	0.831363i	$-0.312439\pi$
$647$	28.2713			1.11146			0.555730	+	0.831363i	$0.312439\pi$
$648$		0			0
$649$	5.58273	+	5.58273i	0.219141	+	0.219141i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−9.67767	+	9.67767i	−0.378716	+	0.378716i	−0.870639	−	0.491923i	$-0.836294\pi$
$653$	−9.67767	+	9.67767i	−0.378716	+	0.378716i	0.491923	+	0.870639i	$0.336294\pi$
$654$		0			0
$655$		−	13.9722i		−	0.545938i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	22.7157			0.884878			0.442439	−	0.896799i	$-0.354113\pi$
$659$	22.7157			0.884878			0.442439	+	0.896799i	$0.354113\pi$
$660$		0			0
$661$		−	47.6821i		−	1.85462i	−0.374294	−	0.927310i	$-0.622115\pi$
$661$		−	47.6821i		−	1.85462i	0.374294	−	0.927310i	$-0.377885\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$			8.09612i			0.313954i
$666$		0			0
$667$	10.5343			0.407890
$668$		0			0
$669$		0			0
$670$		0			0
$671$			28.0809i			1.08405i
$672$		0			0
$673$	33.7768	−	33.7768i	1.30200	−	1.30200i	0.374961	−	0.927041i	$-0.377656\pi$
$673$	33.7768	−	33.7768i	1.30200	−	1.30200i	0.927041	−	0.374961i	$-0.122344\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−16.8811	−	16.8811i	−0.648793	−	0.648793i	0.303908	−	0.952701i	$-0.401708\pi$
$677$	−16.8811	−	16.8811i	−0.648793	−	0.648793i	−0.952701	+	0.303908i	$0.901708\pi$
$678$		0			0
$679$	7.19765			0.276220
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−14.7364	−	14.7364i	−0.563871	−	0.563871i	0.366534	−	0.930405i	$-0.380544\pi$
$683$	−14.7364	−	14.7364i	−0.563871	−	0.563871i	−0.930405	+	0.366534i	$0.880544\pi$
$684$		0			0
$685$	12.0774	+	12.0774i	0.461453	+	0.461453i
$686$		0			0
$687$		0			0
$688$		0			0
$689$		−	10.0990i		−	0.384743i
$690$		0			0
$691$	7.85086	−	7.85086i	0.298661	−	0.298661i	−0.541828	−	0.840489i	$-0.682268\pi$
$691$	7.85086	−	7.85086i	0.298661	−	0.298661i	0.840489	+	0.541828i	$0.182268\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$			22.8678i			0.867427i
$696$		0			0
$697$	24.6544	+	43.1763i	0.933852	+	1.63542i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−19.3141			−0.729482			−0.364741	−	0.931109i	$-0.618842\pi$
$701$	−19.3141			−0.729482			−0.364741	+	0.931109i	$0.618842\pi$
$702$		0			0
$703$	1.70282	−	1.70282i	0.0642233	−	0.0642233i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	0.820100	−	0.820100i	0.0308431	−	0.0308431i
$708$		0			0
$709$	4.84764	+	4.84764i	0.182057	+	0.182057i	0.792252	−	0.610195i	$-0.208909\pi$
$709$	4.84764	+	4.84764i	0.182057	+	0.182057i	−0.610195	+	0.792252i	$0.708909\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−11.5651			−0.433115
$714$		0			0
$715$	13.8125			0.516557
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−11.4094	−	11.4094i	−0.425499	−	0.425499i	0.461593	−	0.887092i	$-0.347278\pi$
$719$	−11.4094	−	11.4094i	−0.425499	−	0.425499i	−0.887092	+	0.461593i	$0.847278\pi$
$720$		0			0
$721$	−6.17557	+	6.17557i	−0.229990	+	0.229990i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−3.17075	+	3.17075i	−0.117759	+	0.117759i
$726$		0			0
$727$	−10.8301			−0.401665			−0.200832	−	0.979626i	$-0.564365\pi$
$727$	−10.8301			−0.401665			−0.200832	+	0.979626i	$0.564365\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	10.4925	+	2.86508i	0.388077	+	0.105969i
$732$		0			0
$733$			27.4997i			1.01573i	0.861438	+	0.507863i	$0.169564\pi$
$733$			27.4997i			1.01573i	−0.861438	+	0.507863i	$0.830436\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−31.6161	+	31.6161i	−1.16459	+	1.16459i
$738$		0			0
$739$		−	6.42719i		−	0.236428i	−0.992988	−	0.118214i	$-0.962283\pi$
$739$		−	6.42719i		−	0.236428i	0.992988	−	0.118214i	$-0.0377169\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−19.6633	−	19.6633i	−0.721377	−	0.721377i	0.247509	−	0.968886i	$-0.420388\pi$
$743$	−19.6633	−	19.6633i	−0.721377	−	0.721377i	−0.968886	+	0.247509i	$0.920388\pi$
$744$		0			0
$745$	−4.94084	−	4.94084i	−0.181018	−	0.181018i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	13.2202			0.483055
$750$		0			0
$751$	−12.6745	−	12.6745i	−0.462498	−	0.462498i	0.436975	−	0.899473i	$-0.356050\pi$
$751$	−12.6745	−	12.6745i	−0.462498	−	0.462498i	−0.899473	+	0.436975i	$0.856050\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−29.9215	+	29.9215i	−1.08895	+	1.08895i
$756$		0			0
$757$		−	9.96117i		−	0.362045i	−0.983479	−	0.181023i	$-0.942059\pi$
$757$		−	9.96117i		−	0.362045i	0.983479	−	0.181023i	$-0.0579407\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	0.279750			0.0101409			0.00507046	−	0.999987i	$-0.498386\pi$
$761$	0.279750			0.0101409			0.00507046	+	0.999987i	$0.498386\pi$
$762$		0			0
$763$			4.64129i			0.168026i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		−	6.84076i		−	0.247005i
$768$		0			0
$769$	18.8011			0.677986			0.338993	−	0.940789i	$-0.389914\pi$
$769$	18.8011			0.677986			0.338993	+	0.940789i	$0.389914\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$			14.9445i			0.537518i	0.963207	+	0.268759i	$0.0866135\pi$
$773$			14.9445i			0.537518i	−0.963207	+	0.268759i	$0.913387\pi$
$774$		0			0
$775$	3.48100	−	3.48100i	0.125041	−	0.125041i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−38.9689	−	38.9689i	−1.39621	−	1.39621i
$780$		0			0
$781$	−42.8906			−1.53475
$782$		0			0
$783$		0			0
$784$		0			0
$785$	23.1361	+	23.1361i	0.825763	+	0.825763i
$786$		0			0
$787$	37.3686	+	37.3686i	1.33205	+	1.33205i	0.903536	+	0.428511i	$0.140962\pi$
$787$	37.3686	+	37.3686i	1.33205	+	1.33205i	0.428511	+	0.903536i	$0.359038\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$			0.401281i			0.0142679i
$792$		0			0
$793$	17.2044	−	17.2044i	0.610946	−	0.610946i
$794$		0			0
$795$		0			0
$796$		0			0
$797$		−	21.3791i		−	0.757286i	−0.925543	−	0.378643i	$-0.876391\pi$
$797$		−	21.3791i		−	0.757286i	0.925543	−	0.378643i	$-0.123609\pi$
$798$		0			0
$799$	10.5076	−	38.4807i	0.371731	−	1.36135i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	9.45434			0.333636
$804$		0			0
$805$	5.47864	−	5.47864i	0.193097	−	0.193097i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−4.85381	+	4.85381i	−0.170651	+	0.170651i	−0.787265	−	0.616614i	$-0.788504\pi$
$809$	−4.85381	+	4.85381i	−0.170651	+	0.170651i	0.616614	+	0.787265i	$0.288504\pi$
$810$		0			0
$811$	22.1746	+	22.1746i	0.778655	+	0.778655i	0.979602	−	0.200947i	$-0.0644020\pi$
$811$	22.1746	+	22.1746i	0.778655	+	0.778655i	−0.200947	+	0.979602i	$0.564402\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	32.5953			1.14176
$816$		0			0
$817$	−12.0559			−0.421782
$818$		0			0
$819$		0			0
$820$		0			0
$821$	18.3362	+	18.3362i	0.639936	+	0.639936i	0.950540	−	0.310603i	$-0.100531\pi$
$821$	18.3362	+	18.3362i	0.639936	+	0.639936i	−0.310603	+	0.950540i	$0.600531\pi$
$822$		0			0
$823$	26.9188	−	26.9188i	0.938330	−	0.938330i	−0.0598760	−	0.998206i	$-0.519071\pi$
$823$	26.9188	−	26.9188i	0.938330	−	0.938330i	0.998206	+	0.0598760i	$0.0190705\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	18.6471	−	18.6471i	0.648424	−	0.648424i	−0.304188	−	0.952612i	$-0.598385\pi$
$827$	18.6471	−	18.6471i	0.648424	−	0.648424i	0.952612	+	0.304188i	$0.0983850\pi$
$828$		0			0
$829$	−11.7846			−0.409298			−0.204649	−	0.978835i	$-0.565605\pi$
$829$	−11.7846			−0.409298			−0.204649	+	0.978835i	$0.565605\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	3.97749	+	1.08609i	0.137812	+	0.0376310i
$834$		0			0
$835$			42.2114i			1.46079i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−30.1798	+	30.1798i	−1.04192	+	1.04192i	−0.0428405	+	0.999082i	$0.513641\pi$
$839$	−30.1798	+	30.1798i	−1.04192	+	1.04192i	−0.999082	+	0.0428405i	$0.986359\pi$
$840$		0			0
$841$		−	23.1987i		−	0.799954i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	7.82199	+	7.82199i	0.269085	+	0.269085i
$846$		0			0
$847$	1.41509	+	1.41509i	0.0486230	+	0.0486230i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−2.30460			−0.0790008
$852$		0			0
$853$	−7.02358	−	7.02358i	−0.240483	−	0.240483i	0.576567	−	0.817050i	$-0.304392\pi$
$853$	−7.02358	−	7.02358i	−0.240483	−	0.240483i	−0.817050	+	0.576567i	$0.804392\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−20.6172	+	20.6172i	−0.704271	+	0.704271i	−0.965324	−	0.261053i	$-0.915930\pi$
$857$	−20.6172	+	20.6172i	−0.704271	+	0.704271i	0.261053	+	0.965324i	$0.415930\pi$
$858$		0			0
$859$			25.5644i			0.872246i	0.899887	+	0.436123i	$0.143649\pi$
$859$			25.5644i			0.872246i	−0.899887	+	0.436123i	$0.856351\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	4.40501			0.149948			0.0749741	−	0.997185i	$-0.476113\pi$
$863$	4.40501			0.149948			0.0749741	+	0.997185i	$0.476113\pi$
$864$		0			0
$865$			11.1048i			0.377576i
$866$		0			0
$867$		0			0
$868$		0			0
$869$		−	51.9300i		−	1.76161i
$870$		0			0
$871$	38.7406			1.31267
$872$		0			0
$873$		0			0
$874$		0			0
$875$			12.1557i			0.410937i
$876$		0			0
$877$	18.0662	−	18.0662i	0.610052	−	0.610052i	−0.332908	−	0.942959i	$-0.608030\pi$
$877$	18.0662	−	18.0662i	0.610052	−	0.610052i	0.942959	+	0.332908i	$0.108030\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	27.7536	+	27.7536i	0.935041	+	0.935041i	0.998015	−	0.0629740i	$-0.0200585\pi$
$881$	27.7536	+	27.7536i	0.935041	+	0.935041i	−0.0629740	+	0.998015i	$0.520059\pi$
$882$		0			0
$883$	33.1225			1.11466			0.557330	−	0.830291i	$-0.311826\pi$
$883$	33.1225			1.11466			0.557330	+	0.830291i	$0.311826\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	19.1858	+	19.1858i	0.644195	+	0.644195i	0.951584	−	0.307389i	$-0.0994552\pi$
$887$	19.1858	+	19.1858i	0.644195	+	0.644195i	−0.307389	+	0.951584i	$0.599455\pi$
$888$		0			0
$889$	0.247560	+	0.247560i	0.00830291	+	0.00830291i
$890$		0			0
$891$		0			0
$892$		0			0
$893$			44.2145i			1.47958i
$894$		0			0
$895$	16.1257	−	16.1257i	0.539023	−	0.539023i
$896$		0			0
$897$		0			0
$898$		0			0
$899$		−	6.36898i		−	0.212417i
$900$		0			0
$901$	−15.4545	−	4.22001i	−0.514864	−	0.140589i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	37.8873			1.25942
$906$		0			0
$907$	39.7896	−	39.7896i	1.32119	−	1.32119i	0.408380	−	0.912812i	$-0.366094\pi$
$907$	39.7896	−	39.7896i	1.32119	−	1.32119i	0.912812	−	0.408380i	$-0.133906\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−32.0658	+	32.0658i	−1.06239	+	1.06239i	−0.0644668	+	0.997920i	$0.520535\pi$
$911$	−32.0658	+	32.0658i	−1.06239	+	1.06239i	−0.997920	+	0.0644668i	$0.979465\pi$
$912$		0			0
$913$	26.8856	+	26.8856i	0.889784	+	0.889784i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	7.88711			0.260455
$918$		0			0
$919$	−30.5827			−1.00883			−0.504415	−	0.863461i	$-0.668292\pi$
$919$	−30.5827			−1.00883			−0.504415	+	0.863461i	$0.668292\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	26.2779	+	26.2779i	0.864946	+	0.864946i
$924$		0			0
$925$	0.693670	−	0.693670i	0.0228077	−	0.0228077i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−19.4123	+	19.4123i	−0.636896	+	0.636896i	−0.949789	−	0.312892i	$-0.898702\pi$
$929$	−19.4123	+	19.4123i	−0.636896	+	0.636896i	0.312892	+	0.949789i	$0.398702\pi$
$930$		0			0
$931$	−4.57015			−0.149781
$932$		0			0
$933$		0			0
$934$		0			0
$935$	5.77172	−	21.1371i	0.188755	−	0.691258i
$936$		0			0
$937$		−	18.3154i		−	0.598337i	−0.954200	−	0.299169i	$-0.903291\pi$
$937$		−	18.3154i		−	0.598337i	0.954200	−	0.299169i	$-0.0967093\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	21.8095	−	21.8095i	0.710969	−	0.710969i	−0.255769	−	0.966738i	$-0.582329\pi$
$941$	21.8095	−	21.8095i	0.710969	−	0.710969i	0.966738	+	0.255769i	$0.0823287\pi$
$942$		0			0
$943$			52.7405i			1.71747i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	11.1276	+	11.1276i	0.361599	+	0.361599i	0.864402	−	0.502802i	$-0.167698\pi$
$947$	11.1276	+	11.1276i	0.361599	+	0.361599i	−0.502802	+	0.864402i	$0.667698\pi$
$948$		0			0
$949$	−5.79241	−	5.79241i	−0.188029	−	0.188029i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	51.5074			1.66849			0.834244	−	0.551395i	$-0.185904\pi$
$953$	51.5074			1.66849			0.834244	+	0.551395i	$0.185904\pi$
$954$		0			0
$955$	−21.9463	−	21.9463i	−0.710165	−	0.710165i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−6.81753	+	6.81753i	−0.220149	+	0.220149i
$960$		0			0
$961$		−	24.0078i		−	0.774446i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−9.13434			−0.294045
$966$		0			0
$967$		−	21.6915i		−	0.697552i	−0.937206	−	0.348776i	$-0.886597\pi$
$967$		−	21.6915i		−	0.697552i	0.937206	−	0.348776i	$-0.113403\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$			14.7980i			0.474889i	0.971401	+	0.237444i	$0.0763098\pi$
$971$			14.7980i			0.474889i	−0.971401	+	0.237444i	$0.923690\pi$
$972$		0			0
$973$	−12.9086			−0.413831
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	23.5724i		−	0.754147i	−0.926183	−	0.377074i	$-0.876930\pi$
$977$		−	23.5724i		−	0.754147i	0.926183	−	0.377074i	$-0.123070\pi$
$978$		0			0
$979$	−4.36001	+	4.36001i	−0.139347	+	0.139347i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−14.1621	−	14.1621i	−0.451702	−	0.451702i	0.444217	−	0.895919i	$-0.353482\pi$
$983$	−14.1621	−	14.1621i	−0.451702	−	0.451702i	−0.895919	+	0.444217i	$0.853482\pi$
$984$		0			0
$985$	−16.2384			−0.517397
$986$		0			0
$987$		0			0
$988$		0			0
$989$	8.15822	+	8.15822i	0.259416	+	0.259416i
$990$		0			0
$991$	10.9668	+	10.9668i	0.348372	+	0.348372i	0.859503	−	0.511131i	$-0.170773\pi$
$991$	10.9668	+	10.9668i	0.348372	+	0.348372i	−0.511131	+	0.859503i	$0.670773\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		−	20.6224i		−	0.653774i
$996$		0			0
$997$	19.4332	−	19.4332i	0.615457	−	0.615457i	−0.328906	−	0.944363i	$-0.606680\pi$
$997$	19.4332	−	19.4332i	0.615457	−	0.615457i	0.944363	+	0.328906i	$0.106680\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4284.2.z.b.3277.4		16
3.2	odd	2		476.2.l.a.421.8	yes	16
17.4	even	4	inner	4284.2.z.b.4033.4		16
51.2	odd	8		8092.2.a.t.1.8		8
51.32	odd	8		8092.2.a.s.1.1		8
51.38	odd	4		476.2.l.a.225.8	✓	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
476.2.l.a.225.8	✓	16	51.38	odd	4
476.2.l.a.421.8	yes	16	3.2	odd	2
4284.2.z.b.3277.4		16	1.1	even	1	trivial
4284.2.z.b.4033.4		16	17.4	even	4	inner
8092.2.a.s.1.1		8	51.32	odd	8
8092.2.a.t.1.8		8	51.2	odd	8

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:	\( N \)	\(=\)	\( 4284 = 2^{2} \cdot 3^{2} \cdot 7 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4284.z (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.25265 - 1.25265i) q^{5} +(0.707107 - 0.707107i) q^{7} +O(q^{10})\)	\(q+(-1.25265 - 1.25265i) q^{5} +(0.707107 - 0.707107i) q^{7} +(2.12117 - 2.12117i) q^{11} -2.59917 q^{13} +(-1.08609 + 3.97749i) q^{17} -4.57015i q^{19} +(-3.09262 + 3.09262i) q^{23} -1.86172i q^{25} +(-1.70313 - 1.70313i) q^{29} +(1.86978 + 1.86978i) q^{31} -1.77152 q^{35} +(0.372597 + 0.372597i) q^{37} +(8.52683 - 8.52683i) q^{41} -2.63796i q^{43} -9.67463 q^{47} -1.00000i q^{49} +3.88549i q^{53} -5.31420 q^{55} +2.63190i q^{59} +(-6.61920 + 6.61920i) q^{61} +(3.25586 + 3.25586i) q^{65} -14.9050 q^{67} +(-10.1101 - 10.1101i) q^{71} +(2.22856 + 2.22856i) q^{73} -2.99979i q^{77} +(12.2409 - 12.2409i) q^{79} +12.6749i q^{83} +(6.34291 - 3.62191i) q^{85} -2.05547 q^{89} +(-1.83789 + 1.83789i) q^{91} +(-5.72482 + 5.72482i) q^{95} +(5.08951 + 5.08951i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 4 q^{5}+O(q^{10}) \) 16 * q - 4 * q^5	\( 16 q - 4 q^{5} + 12 q^{11} - 8 q^{17} + 12 q^{29} - 12 q^{31} + 8 q^{35} + 28 q^{41} - 40 q^{47} + 8 q^{55} + 12 q^{61} + 8 q^{65} - 40 q^{67} - 8 q^{71} + 4 q^{73} + 4 q^{79} - 44 q^{85} + 16 q^{89} + 4 q^{91} + 20 q^{95} + 4 q^{97}+O(q^{100}) \) 16 * q - 4 * q^5 + 12 * q^11 - 8 * q^17 + 12 * q^29 - 12 * q^31 + 8 * q^35 + 28 * q^41 - 40 * q^47 + 8 * q^55 + 12 * q^61 + 8 * q^65 - 40 * q^67 - 8 * q^71 + 4 * q^73 + 4 * q^79 - 44 * q^85 + 16 * q^89 + 4 * q^91 + 20 * q^95 + 4 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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