LMFDB - Embedded newform 847.2.l.a.524.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [847,2,Mod(118,847)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(847, base_ring=CyclotomicField(10))

chi = DirichletCharacter(H, H._module([5, 3]))

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

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

chi := DirichletCharacter("847.118");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 847 = 7 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	847.l (of order $10$, degree $4$, not minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$6.76332905120$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$2$ over $\Q(\zeta_{10})$
Coefficient field:	8.0.37515625.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{7} - x^{6} + 3x^{5} - x^{4} + 6x^{3} - 4x^{2} - 8x + 16 $ x^8 - x^7 - x^6 + 3x^5 - x^4 + 6x^3 - 4x^2 - 8x + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 77)
Sato-Tate group:	$\mathrm{U}(1)[D_{10}]$

Embedding invariants

Embedding label			524.2
Root			$1.18208 + 0.776336i$ of defining polynomial
Character	$\chi$	$=$	847.524
Dual form			847.2.l.a.118.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.912638 - 1.25614i) q^{2} +(-0.126942 - 0.390686i) q^{4} +(2.51626 - 0.817582i) q^{7} +(2.34675 + 0.762506i) q^{8} +(-2.42705 - 1.76336i) q^{9} +O(q^{10})$	$q+(0.912638 - 1.25614i) q^{2} +(-0.126942 - 0.390686i) q^{4} +(2.51626 - 0.817582i) q^{7} +(2.34675 + 0.762506i) q^{8} +(-2.42705 - 1.76336i) q^{9} +(1.26944 - 3.90693i) q^{14} +(3.76422 - 2.73487i) q^{16} +(-4.43004 + 1.43941i) q^{18} +9.58240 q^{23} +(1.54508 - 4.75528i) q^{25} +(-0.638836 - 0.879283i) q^{28} +(-1.30125 + 0.422802i) q^{29} -2.28929i q^{32} +(-0.380825 + 1.17206i) q^{36} +(-0.422251 - 1.29955i) q^{37} +9.77751i q^{43} +(8.74527 - 12.0368i) q^{46} +(5.66312 - 4.11450i) q^{49} +(-4.56319 - 6.28069i) q^{50} +(-10.6428 - 7.73244i) q^{53} +6.52844 q^{56} +(-0.656473 + 2.02042i) q^{58} +(-7.54878 - 2.45275i) q^{63} +(4.65278 + 3.38044i) q^{64} -16.3336 q^{67} +(8.07139 - 5.86421i) q^{71} +(-4.35111 - 5.98880i) q^{72} +(-2.01778 - 0.655617i) q^{74} +(-7.35551 + 10.1240i) q^{79} +(2.78115 + 8.55951i) q^{81} +(12.2819 + 8.92333i) q^{86} +(-1.21641 - 3.74371i) q^{92} -10.8687i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 5 q^{2} - 5 q^{4} + 25 q^{8} - 6 q^{9}+O(q^{10}) $ 8 * q - 5 * q^2 - 5 * q^4 + 25 * q^8 - 6 * q^9	$ 8 q - 5 q^{2} - 5 q^{4} + 25 q^{8} - 6 q^{9} + 14 q^{14} - 17 q^{16} + 16 q^{23} - 10 q^{25} + 10 q^{29} - 15 q^{36} + 12 q^{37} + 14 q^{49} + 25 q^{50} - 20 q^{53} - 42 q^{56} + 9 q^{58} + 61 q^{64} + 8 q^{67} + 32 q^{71} - 30 q^{72} - 85 q^{74} - 18 q^{81} - 37 q^{86} + 15 q^{92}+O(q^{100}) $ 8 * q - 5 * q^2 - 5 * q^4 + 25 * q^8 - 6 * q^9 + 14 * q^14 - 17 * q^16 + 16 * q^23 - 10 * q^25 + 10 * q^29 - 15 * q^36 + 12 * q^37 + 14 * q^49 + 25 * q^50 - 20 * q^53 - 42 * q^56 + 9 * q^58 + 61 * q^64 + 8 * q^67 + 32 * q^71 - 30 * q^72 - 85 * q^74 - 18 * q^81 - 37 * q^86 + 15 * q^92

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/847\mathbb{Z}\right)^\times$.

$n$	$122$	$365$
$\chi(n)$	$-1$	$e\left(\frac{7}{10}\right)$

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 $

$2$	0.912638	−	1.25614i	0.645333	−	0.888224i	−0.353553	−	0.935414i	$-0.615027\pi$
$2$	0.912638	−	1.25614i	0.645333	−	0.888224i	0.998886	+	0.0471903i	$0.0150267\pi$
$3$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$3$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$4$	−0.126942	−	0.390686i	−0.0634708	−	0.195343i
$5$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$5$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$6$		0			0
$7$	2.51626	−	0.817582i	0.951057	−	0.309017i
$7$	2.51626	−	0.817582i	0.951057	−	0.309017i
$8$	2.34675	+	0.762506i	0.829702	+	0.269586i
$9$	−2.42705	−	1.76336i	−0.809017	−	0.587785i
$10$		0			0
$11$		0			0
$11$		0			0
$12$		0			0
$13$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$13$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$14$	1.26944	−	3.90693i	0.339271	−	1.04417i
$15$		0			0
$16$	3.76422	−	2.73487i	0.941055	−	0.683717i
$17$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$17$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$18$	−4.43004	+	1.43941i	−1.04417	+	0.339271i
$19$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$19$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	9.58240			1.99807			0.999035	−	0.0439305i	$-0.0139880\pi$
$23$	9.58240			1.99807			0.999035	+	0.0439305i	$0.0139880\pi$
$24$		0			0
$25$	1.54508	−	4.75528i	0.309017	−	0.951057i
$26$		0			0
$27$		0			0
$28$	−0.638836	−	0.879283i	−0.120729	−	0.166169i
$29$	−1.30125	+	0.422802i	−0.241636	+	0.0785123i	−0.427331	−	0.904095i	$-0.640546\pi$
$29$	−1.30125	+	0.422802i	−0.241636	+	0.0785123i	0.185695	+	0.982607i	$0.440546\pi$
$30$		0			0
$31$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$31$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$32$		−	2.28929i		−	0.404693i
$33$		0			0
$34$		0			0
$35$		0			0
$36$	−0.380825	+	1.17206i	−0.0634708	+	0.195343i
$37$	−0.422251	−	1.29955i	−0.0694176	−	0.213645i	0.910330	−	0.413884i	$-0.135828\pi$
$37$	−0.422251	−	1.29955i	−0.0694176	−	0.213645i	−0.979747	+	0.200239i	$0.935828\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$41$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$42$		0			0
$43$			9.77751i			1.49106i	0.666474	+	0.745528i	$0.267803\pi$
$43$			9.77751i			1.49106i	−0.666474	+	0.745528i	$0.732197\pi$
$44$		0			0
$45$		0			0
$46$	8.74527	−	12.0368i	1.28942	−	1.77473i
$47$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$47$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$48$		0			0
$49$	5.66312	−	4.11450i	0.809017	−	0.587785i
$50$	−4.56319	−	6.28069i	−0.645333	−	0.888224i
$51$		0			0
$52$		0			0
$53$	−10.6428	−	7.73244i	−1.46190	−	1.06213i	−0.982863	−	0.184336i	$-0.940986\pi$
$53$	−10.6428	−	7.73244i	−1.46190	−	1.06213i	−0.479036	−	0.877795i	$-0.659014\pi$
$54$		0			0
$55$		0			0
$56$	6.52844			0.872400
$57$		0			0
$58$	−0.656473	+	2.02042i	−0.0861991	+	0.265294i
$59$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$59$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$60$		0			0
$61$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$61$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$62$		0			0
$63$	−7.54878	−	2.45275i	−0.951057	−	0.309017i
$64$	4.65278	+	3.38044i	0.581598	+	0.422555i
$65$		0			0
$66$		0			0
$67$	−16.3336			−1.99547			−0.997735	−	0.0672706i	$-0.978571\pi$
$67$	−16.3336			−1.99547			−0.997735	+	0.0672706i	$0.978571\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	8.07139	−	5.86421i	0.957898	−	0.695954i	0.00523645	−	0.999986i	$-0.498333\pi$
$71$	8.07139	−	5.86421i	0.957898	−	0.695954i	0.952662	+	0.304033i	$0.0983332\pi$
$72$	−4.35111	−	5.98880i	−0.512784	−	0.705786i
$73$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$73$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$74$	−2.01778	−	0.655617i	−0.234562	−	0.0762140i
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$	−7.35551	+	10.1240i	−0.827560	+	1.13904i	0.160813	+	0.986985i	$0.448589\pi$
$79$	−7.35551	+	10.1240i	−0.827560	+	1.13904i	−0.988372	+	0.152053i	$0.951411\pi$
$80$		0			0
$81$	2.78115	+	8.55951i	0.309017	+	0.951057i
$82$		0			0
$83$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$83$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$84$		0			0
$85$		0			0
$86$	12.2819	+	8.92333i	1.32439	+	0.962228i
$87$		0			0
$88$		0			0
$89$		0			0		1.00000			$0$
$89$		0			0		−1.00000			$\pi$
$90$		0			0
$91$		0			0
$92$	−1.21641	−	3.74371i	−0.126819	−	0.390309i
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$97$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$98$		−	10.8687i		−	1.09791i
$99$		0			0
$100$	−2.05396			−0.205396
$101$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$101$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$102$		0			0
$103$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$103$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$104$		0			0
$105$		0			0
$106$	−19.4260	+	6.31190i	−1.88682	+	0.613066i
$107$	−15.2517	−	4.95559i	−1.47444	−	0.479075i	−0.541994	−	0.840382i	$-0.682330\pi$
$107$	−15.2517	−	4.95559i	−1.47444	−	0.479075i	−0.932447	+	0.361308i	$0.882330\pi$
$108$		0			0
$109$			19.1420i			1.83347i	0.399498	+	0.916734i	$0.369184\pi$
$109$			19.1420i			1.83347i	−0.399498	+	0.916734i	$0.630816\pi$
$110$		0			0
$111$		0			0
$112$	7.23578	−	9.95919i	0.683717	−	0.941055i
$113$	−6.02955	+	18.5571i	−0.567212	+	1.74570i	0.0940721	+	0.995565i	$0.470012\pi$
$113$	−6.02955	+	18.5571i	−0.567212	+	1.74570i	−0.661285	+	0.750135i	$0.729988\pi$
$114$		0			0
$115$		0			0
$116$	0.330366	+	0.454710i	0.0306737	+	0.0422187i
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$		0			0
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$	−9.97029	+	7.24384i	−0.888224	+	0.645333i
$127$	13.0766	+	17.9985i	1.16036	+	1.59710i	0.709885	+	0.704317i	$0.248747\pi$
$127$	13.0766	+	17.9985i	1.16036	+	1.59710i	0.450479	+	0.892787i	$0.351253\pi$
$128$	12.8471	−	4.17427i	1.13553	−	0.368957i
$129$		0			0
$130$		0			0
$131$		0			0			−	1.00000i	$-0.5\pi$
$131$		0			0				1.00000i	$0.5\pi$
$132$		0			0
$133$		0			0
$134$	−14.9067	+	20.5173i	−1.28774	+	1.77242i
$135$		0			0
$136$		0			0
$137$	−16.6101	+	12.0680i	−1.41910	+	1.03104i	−0.427179	+	0.904167i	$0.640493\pi$
$137$	−16.6101	+	12.0680i	−1.41910	+	1.03104i	−0.991920	+	0.126868i	$0.959507\pi$
$138$		0			0
$139$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$139$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$140$		0			0
$141$		0			0
$142$		−	15.4907i		−	1.29995i
$143$		0			0
$144$	−13.9585			−1.16321
$145$		0			0
$146$		0			0
$147$		0			0
$148$	−0.454117	+	0.329935i	−0.0373282	+	0.0271205i
$149$	−6.22053	−	8.56183i	−0.509606	−	0.701413i	0.474247	−	0.880392i	$-0.342720\pi$
$149$	−6.22053	−	8.56183i	−0.509606	−	0.701413i	−0.983853	+	0.178979i	$0.942720\pi$
$150$		0			0
$151$	9.34502	+	3.03638i	0.760487	+	0.247097i	0.663487	−	0.748187i	$-0.269076\pi$
$151$	9.34502	+	3.03638i	0.760487	+	0.247097i	0.0969991	+	0.995284i	$0.469076\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$157$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$158$	6.00422	+	18.4791i	0.477670	+	1.47012i
$159$		0			0
$160$		0			0
$161$	24.1118	−	7.83440i	1.90028	−	0.617437i
$162$	13.2901	+	4.31822i	1.04417	+	0.339271i
$163$	5.54139	+	4.02606i	0.434035	+	0.315345i	0.783260	−	0.621694i	$-0.213555\pi$
$163$	5.54139	+	4.02606i	0.434035	+	0.315345i	−0.349225	+	0.937039i	$0.613555\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$167$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$168$		0			0
$169$	−4.01722	−	12.3637i	−0.309017	−	0.951057i
$170$		0			0
$171$		0			0
$172$	3.81994	−	1.24117i	0.291268	−	0.0946386i
$173$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$173$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$174$		0			0
$175$		−	13.2288i		−	1.00000i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	3.80563	−	11.7125i	0.284446	−	0.875435i	−0.702118	−	0.712060i	$-0.747762\pi$
$179$	3.80563	−	11.7125i	0.284446	−	0.875435i	0.986564	−	0.163374i	$-0.0522378\pi$
$180$		0			0
$181$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$181$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$182$		0			0
$183$		0			0
$184$	22.4875	+	7.30664i	1.65780	+	0.538652i
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−7.01174	−	21.5799i	−0.507351	−	1.56147i	−0.796781	−	0.604268i	$-0.793466\pi$
$191$	−7.01174	−	21.5799i	−0.507351	−	1.56147i	0.289430	−	0.957199i	$-0.406534\pi$
$192$		0			0
$193$	13.9068	+	19.1411i	1.00103	+	1.37780i	0.924689	+	0.380724i	$0.124325\pi$
$193$	13.9068	+	19.1411i	1.00103	+	1.37780i	0.0763450	+	0.997081i	$0.475675\pi$
$194$		0			0
$195$		0			0
$196$	−2.32636	−	1.69020i	−0.166169	−	0.120729i
$197$		−	6.72059i		−	0.478822i	−0.970918	−	0.239411i	$-0.923046\pi$
$197$		−	6.72059i		−	0.478822i	0.970918	−	0.239411i	$-0.0769543\pi$
$198$		0			0
$199$		0			0		1.00000			$0$
$199$		0			0		−1.00000			$\pi$
$200$	7.25186	−	9.98133i	0.512784	−	0.705786i
$201$		0			0
$202$		0			0
$203$	−2.92861	+	2.12776i	−0.205548	+	0.149339i
$204$		0			0
$205$		0			0
$206$		0			0
$207$	−23.2570	−	16.8972i	−1.61647	−	1.17444i
$208$		0			0
$209$		0			0
$210$		0			0
$211$	−16.7272	+	23.0230i	−1.15155	+	1.58497i	−0.413057	+	0.910705i	$0.635539\pi$
$211$	−16.7272	+	23.0230i	−1.15155	+	1.58497i	−0.738490	+	0.674264i	$0.764461\pi$
$212$	−1.66994	+	5.13956i	−0.114692	+	0.352986i
$213$		0			0
$214$	−20.1442	+	14.6356i	−1.37703	+	1.00047i
$215$		0			0
$216$		0			0
$217$		0			0
$218$	24.0450	+	17.4697i	1.62853	+	1.18320i
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$223$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$224$	−1.87168	−	5.76044i	−0.125057	−	0.384886i
$225$	−12.1353	+	8.81678i	−0.809017	+	0.587785i
$226$	17.8074	+	24.5098i	1.18453	+	1.63037i
$227$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$227$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$228$		0			0
$229$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$229$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$230$		0			0
$231$		0			0
$232$	−3.37610			−0.221652
$233$	−12.4411	+	17.1237i	−0.815042	+	1.12181i	0.175484	+	0.984482i	$0.443851\pi$
$233$	−12.4411	+	17.1237i	−0.815042	+	1.12181i	−0.990526	+	0.137326i	$0.956149\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$	6.69647	+	2.17581i	0.433159	+	0.140742i	0.517477	−	0.855697i	$-0.326871\pi$
$239$	6.69647	+	2.17581i	0.433159	+	0.140742i	−0.0843185	+	0.996439i	$0.526871\pi$
$240$		0			0
$241$		0			0			−	1.00000i	$-0.5\pi$
$241$		0			0				1.00000i	$0.5\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$251$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$252$			3.26056i			0.205396i
$253$		0			0
$254$	34.5428			2.16741
$255$		0			0
$256$	2.92687	−	9.00799i	0.182929	−	0.562999i
$257$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$257$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$258$		0			0
$259$	−2.12499	−	2.92479i	−0.132040	−	0.181738i
$260$		0			0
$261$	3.90375	+	1.26841i	0.241636	+	0.0785123i
$262$		0			0
$263$		−	32.0690i		−	1.97746i	−0.149718	−	0.988729i	$-0.547836\pi$
$263$		−	32.0690i		−	1.97746i	0.149718	−	0.988729i	$-0.452164\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$	2.07342	+	6.38132i	0.126654	+	0.389801i
$269$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$269$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$270$		0			0
$271$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$271$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$272$		0			0
$273$		0			0
$274$			31.8783i			1.92584i
$275$		0			0
$276$		0			0
$277$	18.5525	−	25.5353i	1.11471	−	1.53427i	0.300421	−	0.953807i	$-0.402873\pi$
$277$	18.5525	−	25.5353i	1.11471	−	1.53427i	0.814289	−	0.580460i	$-0.197127\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−10.6899	−	14.7134i	−0.637708	−	0.877730i	0.360782	−	0.932650i	$-0.382510\pi$
$281$	−10.6899	−	14.7134i	−0.637708	−	0.877730i	−0.998491	+	0.0549198i	$0.982510\pi$
$282$		0			0
$283$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$283$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$284$	−3.31566	−	2.40897i	−0.196748	−	0.142946i
$285$		0			0
$286$		0			0
$287$		0			0
$288$	−4.03683	+	5.55622i	−0.237872	+	0.327403i
$289$	−5.25329	+	16.1680i	−0.309017	+	0.951057i
$290$		0			0
$291$		0			0
$292$		0			0
$293$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$293$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$294$		0			0
$295$		0			0
$296$		−	3.37170i		−	0.195976i
$297$		0			0
$298$	−16.4319			−0.951877
$299$		0			0
$300$		0			0
$301$	7.99392	+	24.6028i	0.460762	+	1.41808i
$302$	12.3427	−	8.96752i	0.710244	−	0.516023i
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$		0			0			−	1.00000i	$-0.5\pi$
$307$		0			0				1.00000i	$0.5\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$311$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$312$		0			0
$313$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$313$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$314$		0			0
$315$		0			0
$316$	4.88903	+	1.58854i	0.275029	+	0.0893624i
$317$	0.357215	+	0.259532i	0.0200632	+	0.0145768i	0.597772	−	0.801666i	$-0.296053\pi$
$317$	0.357215	+	0.259532i	0.0200632	+	0.0145768i	−0.577708	+	0.816243i	$0.696053\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$	12.1643	−	37.4377i	0.677888	−	2.08632i
$323$		0			0
$324$	2.99104	−	2.17312i	0.166169	−	0.120729i
$325$		0			0
$326$	10.1146	−	3.28642i	0.560194	−	0.182018i
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−26.0143			−1.42988			−0.714939	−	0.699187i	$-0.753545\pi$
$331$	−26.0143			−1.42988			−0.714939	+	0.699187i	$0.753545\pi$
$332$		0			0
$333$	−1.26675	+	3.89866i	−0.0694176	+	0.213645i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−0.484939	+	0.157566i	−0.0264163	+	0.00858319i	−0.322195	−	0.946673i	$-0.604421\pi$
$337$	−0.484939	+	0.157566i	−0.0264163	+	0.00858319i	0.295779	+	0.955256i	$0.404421\pi$
$338$	−19.1968	−	6.23743i	−1.04417	−	0.339271i
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	10.8859	−	14.9832i	0.587785	−	0.809017i
$344$	−7.45541	+	22.9454i	−0.401969	+	1.23713i
$345$		0			0
$346$		0			0
$347$	−8.96395	−	12.3378i	−0.481210	−	0.662329i	0.497527	−	0.867448i	$-0.334242\pi$
$347$	−8.96395	−	12.3378i	−0.481210	−	0.662329i	−0.978737	+	0.205120i	$0.934242\pi$
$348$		0			0
$349$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$349$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$350$	−16.6171	−	12.0731i	−0.888224	−	0.645333i
$351$		0			0
$352$		0			0
$353$		0			0		1.00000			$0$
$353$		0			0		−1.00000			$\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$	−11.2394	−	15.4697i	−0.594020	−	0.817598i
$359$	−32.9719	+	10.7132i	−1.74019	+	0.565422i	−0.994859	−	0.101273i	$-0.967708\pi$
$359$	−32.9719	+	10.7132i	−1.74019	+	0.565422i	−0.745331	+	0.666695i	$0.767708\pi$
$360$		0			0
$361$	15.3713	+	11.1679i	0.809017	+	0.587785i
$362$		0			0
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$367$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$368$	36.0703	−	26.2066i	1.88029	−	1.36611i
$369$		0			0
$370$		0			0
$371$	−33.1019	−	10.7555i	−1.71857	−	0.558396i
$372$		0			0
$373$			31.7490i			1.64390i	0.569558	+	0.821951i	$0.307114\pi$
$373$			31.7490i			1.64390i	−0.569558	+	0.821951i	$0.692886\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	25.4679	−	18.5035i	1.30820	−	0.950462i	0.308199	−	0.951322i	$-0.400274\pi$
$379$	25.4679	−	18.5035i	1.30820	−	0.950462i	1.00000		0.000859657i	$0.000273637\pi$
$380$		0			0
$381$		0			0
$382$	−33.5065	−	10.8869i	−1.71434	−	0.557024i
$383$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$383$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$384$		0			0
$385$		0			0
$386$	36.7357			1.86980
$387$	17.2412	−	23.7305i	0.876421	−	1.20629i
$388$		0			0
$389$	−11.4223	−	35.1541i	−0.579131	−	1.78238i	−0.621660	−	0.783287i	$-0.713542\pi$
$389$	−11.4223	−	35.1541i	−0.579131	−	1.78238i	0.0425291	−	0.999095i	$-0.486458\pi$
$390$		0			0
$391$		0			0
$392$	16.4273	−	5.33754i	0.829702	−	0.269586i
$393$		0			0
$394$	−8.44199	−	6.13346i	−0.425301	−	0.308999i
$395$		0			0
$396$		0			0
$397$		0			0		1.00000			$0$
$397$		0			0		−1.00000			$\pi$
$398$		0			0
$399$		0			0
$400$	−7.18902	−	22.1255i	−0.359451	−	1.10628i
$401$	12.1883	−	8.85528i	0.608652	−	0.442212i	−0.240287	−	0.970702i	$-0.577242\pi$
$401$	12.1883	−	8.85528i	0.608652	−	0.442212i	0.848939	+	0.528490i	$0.177242\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$			5.62061i			0.278946i
$407$		0			0
$408$		0			0
$409$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$409$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$	−42.4504	+	13.7930i	−2.08632	+	0.677888i
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$		0			0		1.00000			$0$
$419$		0			0		−1.00000			$\pi$
$420$		0			0
$421$	−12.2668	+	37.7532i	−0.597845	+	1.83998i	−0.0578225	+	0.998327i	$0.518416\pi$
$421$	−12.2668	+	37.7532i	−0.597845	+	1.83998i	−0.540022	+	0.841651i	$0.681584\pi$
$422$	13.6542	+	42.0233i	0.664677	+	2.04566i
$423$		0			0
$424$	−19.0799	−	26.2613i	−0.926604	−	1.27536i
$425$		0			0
$426$		0			0
$427$		0			0
$428$			6.58771i			0.318429i
$429$		0			0
$430$		0			0
$431$	−1.52557	+	2.09976i	−0.0734840	+	0.101142i	−0.844177	−	0.536065i	$-0.819910\pi$
$431$	−1.52557	+	2.09976i	−0.0734840	+	0.101142i	0.770693	+	0.637207i	$0.219910\pi$
$432$		0			0
$433$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$433$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$434$		0			0
$435$		0			0
$436$	7.47851	−	2.42991i	0.358155	−	0.116372i
$437$		0			0
$438$		0			0
$439$		0			0			−	1.00000i	$-0.5\pi$
$439$		0			0				1.00000i	$0.5\pi$
$440$		0			0
$441$	−21.0000			−1.00000
$442$		0			0
$443$	8.97611	−	27.6256i	0.426468	−	1.31253i	−0.475114	−	0.879924i	$-0.657593\pi$
$443$	8.97611	−	27.6256i	0.426468	−	1.31253i	0.901582	−	0.432608i	$-0.142407\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$	14.4714	+	4.70204i	0.683709	+	0.222151i
$449$	18.8211	+	13.6743i	0.888221	+	0.645330i	0.935413	−	0.353556i	$-0.115028\pi$
$449$	18.8211	+	13.6743i	0.888221	+	0.645330i	−0.0471929	+	0.998886i	$0.515028\pi$
$450$			23.2901i			1.09791i
$451$		0			0
$452$	8.01539			0.377012
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−22.2030	−	30.5598i	−1.03861	−	1.42953i	−0.898279	−	0.439425i	$-0.855182\pi$
$457$	−22.2030	−	30.5598i	−1.03861	−	1.42953i	−0.140334	−	0.990104i	$-0.544818\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0			−	1.00000i	$-0.5\pi$
$461$		0			0				1.00000i	$0.5\pi$
$462$		0			0
$463$	−2.73687			−0.127193			−0.0635967	−	0.997976i	$-0.520257\pi$
$463$	−2.73687			−0.127193			−0.0635967	+	0.997976i	$0.520257\pi$
$464$	−3.74189	+	5.15027i	−0.173713	+	0.239095i
$465$		0			0
$466$	10.1555	+	31.2554i	0.470444	+	1.44788i
$467$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$467$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$468$		0			0
$469$	−41.0996	+	13.3541i	−1.89780	+	0.616634i
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$	12.1955	+	37.5340i	0.558396	+	1.71857i
$478$	8.84458	−	6.42596i	0.404541	−	0.293917i
$479$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$479$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$		0			0
$485$		0			0
$486$		0			0
$487$	13.1777	−	40.5569i	0.597140	−	1.83781i	0.0533681	−	0.998575i	$-0.483004\pi$
$487$	13.1777	−	40.5569i	0.597140	−	1.83781i	0.543772	−	0.839233i	$-0.316996\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	5.03252	−	1.63516i	0.227114	−	0.0737939i	−0.193249	−	0.981150i	$-0.561903\pi$
$491$	5.03252	−	1.63516i	0.227114	−	0.0737939i	0.420363	+	0.907356i	$0.361903\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$	15.5152	−	21.3549i	0.695954	−	0.957898i
$498$		0			0
$499$	−11.2134	−	34.5112i	−0.501979	−	1.54493i	−0.805791	−	0.592200i	$-0.798259\pi$
$499$	−11.2134	−	34.5112i	−0.501979	−	1.54493i	0.303812	−	0.952732i	$-0.401741\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$503$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$504$	−15.8449	−	11.5120i	−0.705786	−	0.512784i
$505$		0			0
$506$		0			0
$507$		0			0
$508$	5.37178	−	7.39362i	0.238334	−	0.328039i
$509$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$509$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$510$		0			0
$511$		0			0
$512$	7.23578	+	9.95919i	0.319779	+	0.440138i
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$	−5.61328			−0.246634
$519$		0			0
$520$		0			0
$521$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$521$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$522$	5.15600	−	3.74606i	0.225672	−	0.163960i
$523$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$523$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$524$		0			0
$525$		0			0
$526$	−40.2831	−	29.2674i	−1.75643	−	1.27612i
$527$		0			0
$528$		0			0
$529$	68.8225			2.99228
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$	−38.3309	−	12.4545i	−1.65564	−	0.537951i
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	3.35084	−	4.61204i	0.144064	−	0.198287i	−0.730887	−	0.682498i	$-0.760893\pi$
$541$	3.35084	−	4.61204i	0.144064	−	0.198287i	0.874951	+	0.484211i	$0.160893\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−15.0976	−	4.90549i	−0.645525	−	0.209744i	−0.0320849	−	0.999485i	$-0.510215\pi$
$547$	−15.0976	−	4.90549i	−0.645525	−	0.209744i	−0.613440	+	0.789741i	$0.710215\pi$
$548$	6.82330	+	4.95742i	0.291477	+	0.211771i
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$	−10.2312	+	31.4883i	−0.435074	+	1.33902i
$554$	−15.1442	−	46.6089i	−0.643414	−	1.98022i
$555$		0			0
$556$		0			0
$557$	38.4971	−	12.5085i	1.63118	−	0.530001i	0.656634	−	0.754209i	$-0.271980\pi$
$557$	38.4971	−	12.5085i	1.63118	−	0.530001i	0.974541	+	0.224208i	$0.0719796\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$	−28.2382			−1.19116
$563$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$563$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	13.9962	+	19.2641i	0.587785	+	0.809017i
$568$	23.4130	−	7.60736i	0.982389	−	0.319198i
$569$	40.2601	+	13.0813i	1.68779	+	0.548397i	0.986398	−	0.164375i	$-0.0525608\pi$
$569$	40.2601	+	13.0813i	1.68779	+	0.548397i	0.701395	+	0.712773i	$0.252561\pi$
$570$		0			0
$571$		−	40.8794i		−	1.71075i	−0.518010	−	0.855374i	$-0.673327\pi$
$571$		−	40.8794i		−	1.71075i	0.518010	−	0.855374i	$-0.326673\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	14.8056	−	45.5670i	0.617437	−	1.90028i
$576$	−5.33161	−	16.4090i	−0.222151	−	0.683709i
$577$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$577$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$578$	15.5148	+	21.3544i	0.645333	+	0.888224i
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$587$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$	−5.14355	−	3.73701i	−0.211399	−	0.153590i
$593$		0			0			−	1.00000i	$-0.5\pi$
$593$		0			0				1.00000i	$0.5\pi$
$594$		0			0
$595$		0			0
$596$	−2.55535	+	3.51713i	−0.104671	+	0.144067i
$597$		0			0
$598$		0			0
$599$	−3.33046	+	2.41972i	−0.136079	+	0.0988671i	−0.653742	−	0.756717i	$-0.726802\pi$
$599$	−3.33046	+	2.41972i	−0.136079	+	0.0988671i	0.517663	+	0.855584i	$0.326802\pi$
$600$		0			0
$601$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$601$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$602$	38.2000	+	12.4119i	1.55692	+	0.505873i
$603$	39.6425	+	28.8020i	1.61437	+	1.17291i
$604$		−	4.03641i		−	0.164239i
$605$		0			0
$606$		0			0
$607$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$607$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	43.7021	+	14.1997i	1.76511	+	0.573520i	0.997709	−	0.0676456i	$-0.0215487\pi$
$613$	43.7021	+	14.1997i	1.76511	+	0.573520i	0.767403	+	0.641165i	$0.221549\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	32.2257			1.29736			0.648679	−	0.761062i	$-0.275322\pi$
$617$	32.2257			1.29736			0.648679	+	0.761062i	$0.275322\pi$
$618$		0			0
$619$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$619$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−20.2254	−	14.6946i	−0.809017	−	0.587785i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$	4.65013	+	14.3116i	0.185119	+	0.569737i	0.999950	−	0.00996082i	$-0.00317068\pi$
$631$	4.65013	+	14.3116i	0.185119	+	0.569737i	−0.814832	+	0.579698i	$0.803171\pi$
$632$	−24.9812	+	18.1499i	−0.993697	+	0.721963i
$633$		0			0
$634$	0.652016	−	0.211853i	0.0258948	−	0.00841374i
$635$		0			0
$636$		0			0
$637$		0			0
$638$		0			0
$639$	−29.9304			−1.18403
$640$		0			0
$641$	−1.82793	+	5.62578i	−0.0721987	+	0.222205i	−0.980644	−	0.195799i	$-0.937270\pi$
$641$	−1.82793	+	5.62578i	−0.0721987	+	0.222205i	0.908445	+	0.418004i	$0.137270\pi$
$642$		0			0
$643$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$643$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$644$	−6.12159	−	8.42564i	−0.241224	−	0.332017i
$645$		0			0
$646$		0			0
$647$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$647$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$648$			22.2077i			0.872400i
$649$		0			0
$650$		0			0
$651$		0			0
$652$	0.869492	−	2.67602i	0.0340519	−	0.104801i
$653$	10.5777	+	32.5550i	0.413939	+	1.27397i	0.913196	+	0.407520i	$0.133606\pi$
$653$	10.5777	+	32.5550i	0.413939	+	1.27397i	−0.499257	+	0.866454i	$0.666394\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$		−	26.4575i		−	1.03064i	−0.856998	−	0.515319i	$-0.827673\pi$
$659$		−	26.4575i		−	1.03064i	0.856998	−	0.515319i	$-0.172327\pi$
$660$		0			0
$661$		0			0		1.00000			$0$
$661$		0			0		−1.00000			$\pi$
$662$	−23.7417	+	32.6776i	−0.922747	+	1.27005i
$663$		0			0
$664$		0			0
$665$		0			0
$666$	3.74117	+	5.14929i	0.144968	+	0.199531i
$667$	−12.4691	+	4.05146i	−0.482806	+	0.156873i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−9.08151	+	12.4996i	−0.350066	+	0.481825i	−0.947348	−	0.320207i	$-0.896248\pi$
$673$	−9.08151	+	12.4996i	−0.350066	+	0.481825i	0.597281	+	0.802032i	$0.296248\pi$
$674$	−0.244649	+	0.752952i	−0.00942352	+	0.0290026i
$675$		0			0
$676$	−4.32039	+	3.13895i	−0.166169	+	0.120729i
$677$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$677$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	45.1792			1.72873			0.864366	−	0.502863i	$-0.167720\pi$
$683$	45.1792			1.72873			0.864366	+	0.502863i	$0.167720\pi$
$684$		0			0
$685$		0			0
$686$	−8.88606	−	27.3485i	−0.339271	−	1.04417i
$687$		0			0
$688$	26.7402	+	36.8047i	1.01946	+	1.40317i
$689$		0			0
$690$		0			0
$691$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$691$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$692$		0			0
$693$		0			0
$694$	−23.6788			−0.898836
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	−5.16829	+	1.67928i	−0.195343	+	0.0634708i
$701$	41.8320	+	13.5920i	1.57997	+	0.513364i	0.962049	−	0.272876i	$-0.0879747\pi$
$701$	41.8320	+	13.5920i	1.57997	+	0.513364i	0.617922	+	0.786239i	$0.287975\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−42.2139	+	30.6702i	−1.58538	+	1.15184i	−0.675192	+	0.737642i	$0.735939\pi$
$709$	−42.2139	+	30.6702i	−1.58538	+	1.15184i	−0.910185	+	0.414202i	$0.864061\pi$
$710$		0			0
$711$	35.7044	−	11.6011i	1.33902	−	0.435074i
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$	−5.05901			−0.189064
$717$		0			0
$718$	−16.6341	+	51.1945i	−0.620780	+	1.91056i
$719$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$719$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$720$		0			0
$721$		0			0
$722$	28.0569	−	9.11624i	1.04417	−	0.339271i
$723$		0			0
$724$		0			0
$725$			6.84108i			0.254071i
$726$		0			0
$727$		0			0		1.00000			$0$
$727$		0			0		−1.00000			$\pi$
$728$		0			0
$729$	8.34346	−	25.6785i	0.309017	−	0.951057i
$730$		0			0
$731$		0			0
$732$		0			0
$733$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$733$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$734$		0			0
$735$		0			0
$736$		−	21.9369i		−	0.808604i
$737$		0			0
$738$		0			0
$739$	−31.9523	+	43.9785i	−1.17538	+	1.61778i	−0.573238	+	0.819389i	$0.694313\pi$
$739$	−31.9523	+	43.9785i	−1.17538	+	1.61778i	−0.602145	+	0.798387i	$0.705687\pi$
$740$		0			0
$741$		0			0
$742$	−43.7204	+	31.7647i	−1.60503	+	1.16612i
$743$	15.6328	+	21.5167i	0.573512	+	0.789371i	0.992965	−	0.118405i	$-0.0377783\pi$
$743$	15.6328	+	21.5167i	0.573512	+	0.789371i	−0.419453	+	0.907777i	$0.637778\pi$
$744$		0			0
$745$		0			0
$746$	39.8812	+	28.9754i	1.46015	+	1.06086i
$747$		0			0
$748$		0			0
$749$	−42.4289			−1.55032
$750$		0			0
$751$	−7.19437	+	22.1420i	−0.262526	+	0.807973i	0.729727	+	0.683739i	$0.239647\pi$
$751$	−7.19437	+	22.1420i	−0.262526	+	0.807973i	−0.992253	+	0.124234i	$0.960353\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−21.6428	−	15.7244i	−0.786620	−	0.571513i	0.120338	−	0.992733i	$-0.461602\pi$
$757$	−21.6428	−	15.7244i	−0.786620	−	0.571513i	−0.906959	+	0.421220i	$0.861602\pi$
$758$		−	48.8782i		−	1.77534i
$759$		0			0
$760$		0			0
$761$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$761$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$762$		0			0
$763$	15.6501	+	48.1661i	0.566573	+	1.74373i
$764$	−7.54089	+	5.47878i	−0.272820	+	0.198215i
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$		0			0			−	1.00000i	$-0.5\pi$
$769$		0			0				1.00000i	$0.5\pi$
$770$		0			0
$771$		0			0
$772$	5.71280	−	7.86300i	0.205608	−	0.282996i
$773$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$773$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$774$	−14.0738	−	43.3148i	−0.505873	−	1.55692i
$775$		0			0
$776$		0			0
$777$		0			0
$778$	−54.5828	−	17.7350i	−1.95689	−	0.635831i
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	10.0646	−	30.9758i	0.359451	−	1.10628i
$785$		0			0
$786$		0			0
$787$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$787$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$788$	−2.62564	+	0.853123i	−0.0935346	+	0.0303912i
$789$		0			0
$790$		0			0
$791$			51.6240i			1.83554i
$792$		0			0
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$797$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$798$		0			0
$799$		0			0
$800$	−10.8862	−	3.53714i	−0.384886	−	0.125057i
$801$		0			0
$802$		−	23.3918i		−	0.825993i
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−7.00140	−	9.63659i	−0.246156	−	0.338805i	0.668004	−	0.744157i	$-0.267149\pi$
$809$	−7.00140	−	9.63659i	−0.246156	−	0.338805i	−0.914160	+	0.405353i	$0.867149\pi$
$810$		0			0
$811$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$811$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$812$	1.20305	+	0.874066i	0.0422187	+	0.0306737i
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−50.3252	+	16.3516i	−1.75636	+	0.570676i	−0.996813	−	0.0797750i	$-0.974580\pi$
$821$	−50.3252	+	16.3516i	−1.75636	+	0.570676i	−0.759548	+	0.650451i	$0.774580\pi$
$822$		0			0
$823$	28.6425	+	20.8100i	0.998416	+	0.725392i	0.961748	−	0.273936i	$-0.0883256\pi$
$823$	28.6425	+	20.8100i	0.998416	+	0.725392i	0.0366680	+	0.999328i	$0.488326\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	21.7719	−	29.9664i	0.757082	−	1.04203i	−0.240369	−	0.970682i	$-0.577268\pi$
$827$	21.7719	−	29.9664i	0.757082	−	1.04203i	0.997451	−	0.0713526i	$-0.0227315\pi$
$828$	−3.64922	+	11.2311i	−0.126819	+	0.390309i
$829$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$829$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$839$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$840$		0			0
$841$	−21.9470	+	15.9454i	−0.756793	+	0.549842i
$842$	36.2281	+	49.8637i	1.24850	+	1.71842i
$843$		0			0
$844$	11.1182	+	3.61251i	0.382703	+	0.124348i
$845$		0			0
$846$		0			0
$847$		0			0
$848$	−61.2090			−2.10193
$849$		0			0
$850$		0			0
$851$	−4.04618	−	12.4529i	−0.138701	−	0.426878i
$852$		0			0
$853$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$853$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$854$		0			0
$855$		0			0
$856$	−32.0133	−	23.2591i	−1.09419	−	0.794978i
$857$		0			0			−	1.00000i	$-0.5\pi$
$857$		0			0				1.00000i	$0.5\pi$
$858$		0			0
$859$		0			0		1.00000			$0$
$859$		0			0		−1.00000			$\pi$
$860$		0			0
$861$		0			0
$862$	1.24530	+	3.83265i	0.0424152	+	0.130540i
$863$	−6.47214	+	4.70228i	−0.220314	+	0.160068i	−0.692468	−	0.721449i	$-0.743477\pi$
$863$	−6.47214	+	4.70228i	−0.220314	+	0.160068i	0.472154	+	0.881516i	$0.343477\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$	−14.5959	+	44.9214i	−0.494278	+	1.52123i
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−35.8946	−	11.6629i	−1.21208	−	0.393827i	−0.367885	−	0.929871i	$-0.619918\pi$
$877$	−35.8946	−	11.6629i	−1.21208	−	0.393827i	−0.844190	+	0.536044i	$0.819918\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0		1.00000			$0$
$881$		0			0		−1.00000			$\pi$
$882$	−19.1654	+	26.3789i	−0.645333	+	0.888224i
$883$	−3.70820	+	11.4127i	−0.124791	+	0.384067i	−0.993863	−	0.110619i	$-0.964717\pi$
$883$	−3.70820	+	11.4127i	−0.124791	+	0.384067i	0.869072	+	0.494686i	$0.164717\pi$
$884$		0			0
$885$		0			0
$886$	−26.5097	−	36.4874i	−0.890609	−	1.22582i
$887$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$887$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$888$		0			0
$889$	47.6194	+	34.5975i	1.59710	+	1.16036i
$890$		0			0
$891$		0			0
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$	28.9138	−	21.0071i	0.965942	−	0.701798i
$897$		0			0
$898$	34.3536	−	11.1622i	1.14640	−	0.372486i
$899$		0			0
$900$	4.98507	+	3.62186i	0.166169	+	0.120729i
$901$		0			0
$902$		0			0
$903$		0			0
$904$	−28.2997	+	38.9512i	−0.941234	+	1.29550i
$905$		0			0
$906$		0			0
$907$	19.0714	−	13.8562i	0.633255	−	0.460087i	−0.224271	−	0.974527i	$-0.572000\pi$
$907$	19.0714	−	13.8562i	0.633255	−	0.460087i	0.857526	+	0.514440i	$0.172000\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	12.9443	+	9.40456i	0.428863	+	0.311587i	0.781194	−	0.624288i	$-0.214611\pi$
$911$	12.9443	+	9.40456i	0.428863	+	0.311587i	−0.352331	+	0.935875i	$0.614611\pi$
$912$		0			0
$913$		0			0
$914$	−58.6507			−1.93999
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	−33.5607	−	46.1923i	−1.10707	−	1.52374i	−0.825671	−	0.564152i	$-0.809203\pi$
$919$	−33.5607	−	46.1923i	−1.10707	−	1.52374i	−0.281394	−	0.959592i	$-0.590797\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$	−6.83216			−0.224640
$926$	−2.49778	+	3.43789i	−0.0820820	+	0.112976i
$927$		0			0
$928$	0.967915	+	2.97894i	0.0317734	+	0.0977884i
$929$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$929$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$930$		0			0
$931$		0			0
$932$	8.26927	+	2.68685i	0.270869	+	0.0880107i
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$937$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$938$	−20.7345	+	63.8143i	−0.677006	+	2.08361i
$939$		0			0
$940$		0			0
$941$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$941$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−20.0000			−0.649913			−0.324956	−	0.945729i	$-0.605350\pi$
$947$	−20.0000			−0.649913			−0.324956	+	0.945729i	$0.605350\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$	48.7086	−	15.8264i	1.57782	−	0.512666i	0.616330	−	0.787488i	$-0.288619\pi$
$953$	48.7086	−	15.8264i	1.57782	−	0.512666i	0.961495	+	0.274822i	$0.0886189\pi$
$954$	58.2781	+	18.9357i	1.88682	+	0.613066i
$955$		0			0
$956$		−	2.89242i		−	0.0935476i
$957$		0			0
$958$		0			0
$959$	−31.9288	+	43.9462i	−1.03104	+	1.41910i
$960$		0			0
$961$	9.57953	+	29.4828i	0.309017	+	0.951057i
$962$		0			0
$963$	28.2783	+	38.9217i	0.911254	+	1.25423i
$964$		0			0
$965$		0			0
$966$		0			0
$967$		−	15.0168i		−	0.482909i	−0.970412	−	0.241454i	$-0.922376\pi$
$967$		−	15.0168i		−	0.482909i	0.970412	−	0.241454i	$-0.0776244\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$971$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$972$		0			0
$973$		0			0
$974$	−38.9186	−	53.5668i	−1.24703	−	1.71639i
$975$		0			0
$976$		0			0
$977$	−9.97732	−	7.24895i	−0.319203	−	0.231914i	0.416632	−	0.909075i	$-0.363210\pi$
$977$	−9.97732	−	7.24895i	−0.319203	−	0.231914i	−0.735835	+	0.677161i	$0.763210\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	33.7541	−	46.4585i	1.07769	−	1.48331i
$982$	2.53887	−	7.81385i	0.0810187	−	0.249350i
$983$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$983$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$			93.6921i			2.97923i
$990$		0			0
$991$	24.0000			0.762385			0.381193	−	0.924496i	$-0.375513\pi$
$991$	24.0000			0.762385			0.381193	+	0.924496i	$0.375513\pi$
$992$		0			0
$993$		0			0
$994$	−12.6649	−	38.9786i	−0.401707	−	1.23633i
$995$		0			0
$996$		0			0
$997$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$997$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$998$	−53.5846	−	17.4107i	−1.69619	−	0.551125i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	847.2.l.a.524.2		8
7.6	odd	2	CM	847.2.l.a.524.2		8
11.2	odd	10		77.2.l.a.6.2	✓	8
11.3	even	5		847.2.l.d.118.1		8
11.4	even	5		77.2.l.a.13.2	yes	8
11.5	even	5		847.2.b.b.846.6		8
11.6	odd	10		847.2.b.b.846.3		8
11.7	odd	10		847.2.l.c.475.1		8
11.8	odd	10	inner	847.2.l.a.118.2		8
11.9	even	5		847.2.l.c.699.1		8
11.10	odd	2		847.2.l.d.524.1		8
33.2	even	10		693.2.bu.a.622.1		8
33.26	odd	10		693.2.bu.a.244.1		8
77.2	odd	30		539.2.s.a.325.1		16
77.4	even	15		539.2.s.a.68.1		16
77.6	even	10		847.2.b.b.846.3		8
77.13	even	10		77.2.l.a.6.2	✓	8
77.20	odd	10		847.2.l.c.699.1		8
77.24	even	30		539.2.s.a.215.1		16
77.26	odd	30		539.2.s.a.178.1		16
77.27	odd	10		847.2.b.b.846.6		8
77.37	even	15		539.2.s.a.178.1		16
77.41	even	10	inner	847.2.l.a.118.2		8
77.46	odd	30		539.2.s.a.215.1		16
77.48	odd	10		77.2.l.a.13.2	yes	8
77.59	odd	30		539.2.s.a.68.1		16
77.62	even	10		847.2.l.c.475.1		8
77.68	even	30		539.2.s.a.325.1		16
77.69	odd	10		847.2.l.d.118.1		8
77.76	even	2		847.2.l.d.524.1		8
231.125	even	10		693.2.bu.a.244.1		8
231.167	odd	10		693.2.bu.a.622.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
77.2.l.a.6.2	✓	8	11.2	odd	10
77.2.l.a.6.2	✓	8	77.13	even	10
77.2.l.a.13.2	yes	8	11.4	even	5
77.2.l.a.13.2	yes	8	77.48	odd	10
539.2.s.a.68.1		16	77.4	even	15
539.2.s.a.68.1		16	77.59	odd	30
539.2.s.a.178.1		16	77.26	odd	30
539.2.s.a.178.1		16	77.37	even	15
539.2.s.a.215.1		16	77.24	even	30
539.2.s.a.215.1		16	77.46	odd	30
539.2.s.a.325.1		16	77.2	odd	30
539.2.s.a.325.1		16	77.68	even	30
693.2.bu.a.244.1		8	33.26	odd	10
693.2.bu.a.244.1		8	231.125	even	10
693.2.bu.a.622.1		8	33.2	even	10
693.2.bu.a.622.1		8	231.167	odd	10
847.2.b.b.846.3		8	11.6	odd	10
847.2.b.b.846.3		8	77.6	even	10
847.2.b.b.846.6		8	11.5	even	5
847.2.b.b.846.6		8	77.27	odd	10
847.2.l.a.118.2		8	11.8	odd	10	inner
847.2.l.a.118.2		8	77.41	even	10	inner
847.2.l.a.524.2		8	1.1	even	1	trivial
847.2.l.a.524.2		8	7.6	odd	2	CM
847.2.l.c.475.1		8	11.7	odd	10
847.2.l.c.475.1		8	77.62	even	10
847.2.l.c.699.1		8	11.9	even	5
847.2.l.c.699.1		8	77.20	odd	10
847.2.l.d.118.1		8	11.3	even	5
847.2.l.d.118.1		8	77.69	odd	10
847.2.l.d.524.1		8	11.10	odd	2
847.2.l.d.524.1		8	77.76	even	2

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:	\( N \)	\(=\)	\( 847 = 7 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	847.l (of order \(10\), degree \(4\), not minimal)

\(f(q)\)	\(=\)	\(q+(0.912638 - 1.25614i) q^{2} +(-0.126942 - 0.390686i) q^{4} +(2.51626 - 0.817582i) q^{7} +(2.34675 + 0.762506i) q^{8} +(-2.42705 - 1.76336i) q^{9} +O(q^{10})\)	\(q+(0.912638 - 1.25614i) q^{2} +(-0.126942 - 0.390686i) q^{4} +(2.51626 - 0.817582i) q^{7} +(2.34675 + 0.762506i) q^{8} +(-2.42705 - 1.76336i) q^{9} +(1.26944 - 3.90693i) q^{14} +(3.76422 - 2.73487i) q^{16} +(-4.43004 + 1.43941i) q^{18} +9.58240 q^{23} +(1.54508 - 4.75528i) q^{25} +(-0.638836 - 0.879283i) q^{28} +(-1.30125 + 0.422802i) q^{29} -2.28929i q^{32} +(-0.380825 + 1.17206i) q^{36} +(-0.422251 - 1.29955i) q^{37} +9.77751i q^{43} +(8.74527 - 12.0368i) q^{46} +(5.66312 - 4.11450i) q^{49} +(-4.56319 - 6.28069i) q^{50} +(-10.6428 - 7.73244i) q^{53} +6.52844 q^{56} +(-0.656473 + 2.02042i) q^{58} +(-7.54878 - 2.45275i) q^{63} +(4.65278 + 3.38044i) q^{64} -16.3336 q^{67} +(8.07139 - 5.86421i) q^{71} +(-4.35111 - 5.98880i) q^{72} +(-2.01778 - 0.655617i) q^{74} +(-7.35551 + 10.1240i) q^{79} +(2.78115 + 8.55951i) q^{81} +(12.2819 + 8.92333i) q^{86} +(-1.21641 - 3.74371i) q^{92} -10.8687i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 5 q^{2} - 5 q^{4} + 25 q^{8} - 6 q^{9}+O(q^{10}) \) 8 * q - 5 * q^2 - 5 * q^4 + 25 * q^8 - 6 * q^9	\( 8 q - 5 q^{2} - 5 q^{4} + 25 q^{8} - 6 q^{9} + 14 q^{14} - 17 q^{16} + 16 q^{23} - 10 q^{25} + 10 q^{29} - 15 q^{36} + 12 q^{37} + 14 q^{49} + 25 q^{50} - 20 q^{53} - 42 q^{56} + 9 q^{58} + 61 q^{64} + 8 q^{67} + 32 q^{71} - 30 q^{72} - 85 q^{74} - 18 q^{81} - 37 q^{86} + 15 q^{92}+O(q^{100}) \) 8 * q - 5 * q^2 - 5 * q^4 + 25 * q^8 - 6 * q^9 + 14 * q^14 - 17 * q^16 + 16 * q^23 - 10 * q^25 + 10 * q^29 - 15 * q^36 + 12 * q^37 + 14 * q^49 + 25 * q^50 - 20 * q^53 - 42 * q^56 + 9 * q^58 + 61 * q^64 + 8 * q^67 + 32 * q^71 - 30 * q^72 - 85 * q^74 - 18 * q^81 - 37 * q^86 + 15 * q^92

Introduction

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