LMFDB - Embedded newform 847.2.l.a.524.1

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			524.1
Root			$-0.373058 - 1.36412i$ of defining polynomial
Character	$\chi$	$=$	847.524
Dual form			847.2.l.a.118.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.60362 + 2.20719i) q^{2} +(-1.68208 - 5.17690i) q^{4} +(-2.51626 + 0.817582i) q^{7} +(8.93440 + 2.90296i) q^{8} +(-2.42705 - 1.76336i) q^{9} +O(q^{10})$	$q+(-1.60362 + 2.20719i) q^{2} +(-1.68208 - 5.17690i) q^{4} +(-2.51626 + 0.817582i) q^{7} +(8.93440 + 2.90296i) q^{8} +(-2.42705 - 1.76336i) q^{9} +(2.23056 - 6.86497i) q^{14} +(-11.9273 + 8.66572i) q^{16} +(7.78414 - 2.52922i) q^{18} +3.36187 q^{23} +(1.54508 - 4.75528i) q^{25} +(8.46507 + 11.6512i) q^{28} +(4.91928 - 1.59837i) q^{29} -21.4341i q^{32} +(-5.04623 + 15.5307i) q^{36} +(3.42225 + 10.5326i) q^{37} +13.0478i q^{43} +(-5.39116 + 7.42030i) q^{46} +(5.66312 - 4.11450i) q^{49} +(8.01811 + 11.0360i) q^{50} +(5.64279 + 4.09972i) q^{53} -24.8547 q^{56} +(-4.36075 + 13.4210i) q^{58} +(7.54878 + 2.45275i) q^{63} +(23.4546 + 17.0408i) q^{64} +13.8615 q^{67} +(-0.0713926 + 0.0518698i) q^{71} +(-16.5653 - 22.8002i) q^{72} +(-28.7355 - 9.33673i) q^{74} +(-1.58876 + 2.18674i) q^{79} +(2.78115 + 8.55951i) q^{81} +(-28.7991 - 20.9238i) q^{86} +(-5.65492 - 17.4040i) q^{92} +19.0977i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−1.60362	+	2.20719i	−1.13393	+	1.56072i	−0.353553	+	0.935414i	$0.615027\pi$
$2$	−1.60362	+	2.20719i	−1.13393	+	1.56072i	−0.780378	+	0.625308i	$0.784973\pi$
$3$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$3$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$4$	−1.68208	−	5.17690i	−0.841038	−	2.58845i
$5$		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
$8$	8.93440	+	2.90296i	3.15879	+	1.02635i
$9$	−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	$0.700000\pi$
$14$	2.23056	−	6.86497i	0.596143	−	1.83474i
$15$		0			0
$16$	−11.9273	+	8.66572i	−2.98184	+	2.16643i
$17$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$17$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$18$	7.78414	−	2.52922i	1.83474	−	0.596143i
$19$		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
$23$	3.36187			0.700998			0.350499	−	0.936563i	$-0.386012\pi$
$23$	3.36187			0.700998			0.350499	+	0.936563i	$0.386012\pi$
$24$		0			0
$25$	1.54508	−	4.75528i	0.309017	−	0.951057i
$26$		0			0
$27$		0			0
$28$	8.46507	+	11.6512i	1.59975	+	2.20187i
$29$	4.91928	−	1.59837i	0.913488	−	0.296810i	0.185695	−	0.982607i	$-0.440546\pi$
$29$	4.91928	−	1.59837i	0.913488	−	0.296810i	0.727793	+	0.685797i	$0.240546\pi$
$30$		0			0
$31$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$31$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$32$		−	21.4341i		−	3.78905i
$33$		0			0
$34$		0			0
$35$		0			0
$36$	−5.04623	+	15.5307i	−0.841038	+	2.58845i
$37$	3.42225	+	10.5326i	0.562615	+	1.73155i	0.674935	+	0.737878i	$0.264172\pi$
$37$	3.42225	+	10.5326i	0.562615	+	1.73155i	−0.112320	+	0.993672i	$0.535828\pi$
$38$		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
$43$			13.0478i			1.98978i	0.100978	+	0.994889i	$0.467803\pi$
$43$			13.0478i			1.98978i	−0.100978	+	0.994889i	$0.532197\pi$
$44$		0			0
$45$		0			0
$46$	−5.39116	+	7.42030i	−0.794884	+	1.09406i
$47$		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
$50$	8.01811	+	11.0360i	1.13393	+	1.56072i
$51$		0			0
$52$		0			0
$53$	5.64279	+	4.09972i	0.775096	+	0.563140i	0.903503	−	0.428581i	$-0.140986\pi$
$53$	5.64279	+	4.09972i	0.775096	+	0.563140i	−0.128407	+	0.991722i	$0.540986\pi$
$54$		0			0
$55$		0			0
$56$	−24.8547			−3.32135
$57$		0			0
$58$	−4.36075	+	13.4210i	−0.572594	+	1.76226i
$59$		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
$64$	23.4546	+	17.0408i	2.93183	+	2.13010i
$65$		0			0
$66$		0			0
$67$	13.8615			1.69345			0.846725	−	0.532031i	$-0.178571\pi$
$67$	13.8615			1.69345			0.846725	+	0.532031i	$0.178571\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−0.0713926	+	0.0518698i	−0.00847275	+	0.00615581i	−0.592014	−	0.805928i	$-0.701667\pi$
$71$	−0.0713926	+	0.0518698i	−0.00847275	+	0.00615581i	0.583541	+	0.812084i	$0.301667\pi$
$72$	−16.5653	−	22.8002i	−1.95224	−	2.68703i
$73$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$73$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$74$	−28.7355	−	9.33673i	−3.34043	−	1.08537i
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$	−1.58876	+	2.18674i	−0.178749	+	0.246027i	−0.888985	−	0.457937i	$-0.848589\pi$
$79$	−1.58876	+	2.18674i	−0.178749	+	0.246027i	0.710235	+	0.703964i	$0.248589\pi$
$80$		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
$86$	−28.7991	−	20.9238i	−3.10549	−	2.25627i
$87$		0			0
$88$		0			0
$89$		0			0		1.00000			$0$
$89$		0			0		−1.00000			$\pi$
$90$		0			0
$91$		0			0
$92$	−5.65492	−	17.4040i	−0.589566	−	1.81450i
$93$		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	$0.200000\pi$
$98$			19.0977i			1.92916i
$99$		0			0
$100$	−27.2166			−2.72166
$101$		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
$106$	−18.0978	+	5.88032i	−1.75781	+	0.571148i
$107$	−7.10895	−	2.30984i	−0.687248	−	0.223300i	−0.0554821	−	0.998460i	$-0.517670\pi$
$107$	−7.10895	−	2.30984i	−0.687248	−	0.223300i	−0.631766	+	0.775159i	$0.717670\pi$
$108$		0			0
$109$			2.01830i			0.193318i	0.995318	+	0.0966592i	$0.0308157\pi$
$109$			2.01830i			0.193318i	−0.995318	+	0.0966592i	$0.969184\pi$
$110$		0			0
$111$		0			0
$112$	22.9273	−	31.5568i	2.16643	−	2.98184i
$113$	6.41152	−	19.7326i	0.603145	−	1.85629i	0.0940721	−	0.995565i	$-0.470012\pi$
$113$	6.41152	−	19.7326i	0.603145	−	1.85629i	0.509073	−	0.860724i	$-0.329988\pi$
$114$		0			0
$115$		0			0
$116$	−16.5492	−	22.7780i	−1.53656	−	2.11489i
$117$		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
$126$	−17.5191	+	12.7283i	−1.56072	+	1.13393i
$127$	−2.02091	−	2.78155i	−0.179327	−	0.246822i	0.709885	−	0.704317i	$-0.248747\pi$
$127$	−2.02091	−	2.78155i	−0.179327	−	0.246822i	−0.889212	+	0.457495i	$0.848747\pi$
$128$	−34.4545	+	11.1949i	−3.04537	+	0.989502i
$129$		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
$134$	−22.2286	+	30.5950i	−1.92026	+	2.64301i
$135$		0			0
$136$		0			0
$137$	3.51995	−	2.55739i	0.300730	−	0.218493i	−0.427179	−	0.904167i	$-0.640493\pi$
$137$	3.51995	−	2.55739i	0.300730	−	0.218493i	0.727909	+	0.685674i	$0.240493\pi$
$138$		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
$142$		−	0.240757i		−	0.0202039i
$143$		0			0
$144$	44.2290			3.68575
$145$		0			0
$146$		0			0
$147$		0			0
$148$	48.7697	−	35.4333i	4.00885	−	2.91260i
$149$	6.22053	+	8.56183i	0.509606	+	0.701413i	0.983853	−	0.178979i	$-0.0572796\pi$
$149$	6.22053	+	8.56183i	0.509606	+	0.701413i	−0.474247	+	0.880392i	$0.657280\pi$
$150$		0			0
$151$	17.4878	+	5.68213i	1.42314	+	0.462405i	0.916597	−	0.399811i	$-0.130924\pi$
$151$	17.4878	+	5.68213i	1.42314	+	0.462405i	0.506540	+	0.862217i	$0.330924\pi$
$152$		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	$0.600000\pi$
$158$	−2.27879	−	7.01340i	−0.181291	−	0.557957i
$159$		0			0
$160$		0			0
$161$	−8.45933	+	2.74860i	−0.666689	+	0.216620i
$162$	−23.3524	−	7.58766i	−1.83474	−	0.596143i
$163$	20.6389	+	14.9951i	1.61657	+	1.17450i	0.833307	+	0.552811i	$0.186445\pi$
$163$	20.6389	+	14.9951i	1.61657	+	1.17450i	0.783260	+	0.621694i	$0.213555\pi$
$164$		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
$172$	67.5473	−	21.9475i	5.15043	−	1.67348i
$173$		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
$179$	−5.80563	+	17.8679i	−0.433933	+	1.33551i	0.460243	+	0.887793i	$0.347762\pi$
$179$	−5.80563	+	17.8679i	−0.433933	+	1.33551i	−0.894176	+	0.447715i	$0.852238\pi$
$180$		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
$184$	30.0363	+	9.75938i	2.21430	+	0.719471i
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$	8.53960	+	26.2822i	0.617904	+	1.90171i	0.328474	+	0.944513i	$0.393466\pi$
$191$	8.53960	+	26.2822i	0.617904	+	1.90171i	0.289430	+	0.957199i	$0.406534\pi$
$192$		0			0
$193$	6.21780	+	8.55807i	0.447567	+	0.616024i	0.971873	−	0.235507i	$-0.0756750\pi$
$193$	6.21780	+	8.55807i	0.447567	+	0.616024i	−0.524305	+	0.851530i	$0.675675\pi$
$194$		0			0
$195$		0			0
$196$	−30.8261	−	22.3965i	−2.20187	−	1.59975i
$197$		−	23.8442i		−	1.69883i	−0.527724	−	0.849416i	$-0.676954\pi$
$197$		−	23.8442i		−	1.69883i	0.527724	−	0.849416i	$-0.323046\pi$
$198$		0			0
$199$		0			0		1.00000			$0$
$199$		0			0		−1.00000			$\pi$
$200$	27.6088	−	38.0003i	1.95224	−	2.68703i
$201$		0			0
$202$		0			0
$203$	−11.0714	+	8.04384i	−0.777059	+	0.564567i
$204$		0			0
$205$		0			0
$206$		0			0
$207$	−8.15943	−	5.92817i	−0.567119	−	0.412036i
$208$		0			0
$209$		0			0
$210$		0			0
$211$	8.43540	−	11.6103i	0.580717	−	0.799288i	−0.413057	−	0.910705i	$-0.635539\pi$
$211$	8.43540	−	11.6103i	0.580717	−	0.799288i	0.993774	+	0.111417i	$0.0355390\pi$
$212$	11.7323	−	36.1082i	0.805774	−	2.47992i
$213$		0			0
$214$	16.4983	−	11.9867i	1.12780	−	0.819396i
$215$		0			0
$216$		0			0
$217$		0			0
$218$	−4.45479	−	3.23659i	−0.301716	−	0.219210i
$219$		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	$0.600000\pi$
$224$	17.5242	+	53.9338i	1.17088	+	3.60360i
$225$	−12.1353	+	8.81678i	−0.809017	+	0.587785i
$226$	33.2721	+	45.7951i	2.21323	+	3.04625i
$227$		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
$232$	48.5909			3.19015
$233$	12.4411	−	17.1237i	0.815042	−	1.12181i	−0.175484	−	0.984482i	$-0.556149\pi$
$233$	12.4411	−	17.1237i	0.815042	−	1.12181i	0.990526	−	0.137326i	$-0.0438509\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$	22.2478	+	7.22875i	1.43909	+	0.467589i	0.921614	−	0.388108i	$-0.126871\pi$
$239$	22.2478	+	7.22875i	1.43909	+	0.467589i	0.517477	+	0.855697i	$0.326871\pi$
$240$		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	$0.200000\pi$
$252$		−	43.2049i		−	2.72166i
$253$		0			0
$254$	9.38020			0.588566
$255$		0			0
$256$	12.6248	−	38.8551i	0.789049	−	2.42844i
$257$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$257$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$258$		0			0
$259$	−17.2225	−	23.7048i	−1.07016	−	1.47294i
$260$		0			0
$261$	−14.7579	−	4.79512i	−0.913488	−	0.296810i
$262$		0			0
$263$		−	28.7986i		−	1.77580i	−0.460036	−	0.887900i	$-0.652164\pi$
$263$		−	28.7986i		−	1.77580i	0.460036	−	0.887900i	$-0.347836\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$	−23.3161	−	71.7595i	−1.42426	−	4.38341i
$269$		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
$274$			11.8703i			0.717112i
$275$		0			0
$276$		0			0
$277$	−11.6426	+	16.0247i	−0.699538	+	0.962832i	0.300421	+	0.953807i	$0.402873\pi$
$277$	−11.6426	+	16.0247i	−0.699538	+	0.962832i	−0.999959	+	0.00902525i	$0.997127\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−18.3789	−	25.2964i	−1.09640	−	1.50906i	−0.840077	−	0.542467i	$-0.817490\pi$
$281$	−18.3789	−	25.2964i	−1.09640	−	1.50906i	−0.256319	−	0.966592i	$-0.582510\pi$
$282$		0			0
$283$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$283$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$284$	0.388612	+	0.282343i	0.0230599	+	0.0167540i
$285$		0			0
$286$		0			0
$287$		0			0
$288$	−37.7960	+	52.0217i	−2.22715	+	3.06541i
$289$	−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
$296$			104.037i			6.04704i
$297$		0			0
$298$	−28.8730			−1.67257
$299$		0			0
$300$		0			0
$301$	−10.6677	−	32.8318i	−0.614875	−	1.89239i
$302$	−40.5854	+	29.4870i	−2.33543	+	1.69679i
$303$		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
$316$	13.9929	+	4.54658i	0.787164	+	0.255765i
$317$	16.6428	+	12.0917i	0.934752	+	0.679137i	0.947152	−	0.320786i	$-0.103947\pi$
$317$	16.6428	+	12.0917i	0.934752	+	0.679137i	−0.0123997	+	0.999923i	$0.503947\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$	7.49886	−	23.0791i	0.417895	−	1.28615i
$323$		0			0
$324$	39.6336	−	28.7955i	2.20187	−	1.59975i
$325$		0			0
$326$	−66.1941	+	21.5078i	−3.66615	+	1.19121i
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−32.2349			−1.77179			−0.885895	−	0.463887i	$-0.846455\pi$
$331$	−32.2349			−1.77179			−0.885895	+	0.463887i	$0.846455\pi$
$332$		0			0
$333$	10.2668	−	31.5978i	0.562615	−	1.73155i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−33.0561	+	10.7406i	−1.80068	+	0.585076i	−0.999904	−	0.0138879i	$-0.995579\pi$
$337$	−33.0561	+	10.7406i	−1.80068	+	0.585076i	−0.800776	+	0.598964i	$0.795579\pi$
$338$	33.7313	+	10.9600i	1.83474	+	0.596143i
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	−10.8859	+	14.9832i	−0.587785	+	0.809017i
$344$	−37.8774	+	116.575i	−2.04221	+	6.28529i
$345$		0			0
$346$		0			0
$347$	4.49181	+	6.18245i	0.241133	+	0.331891i	0.912381	−	0.409342i	$-0.134242\pi$
$347$	4.49181	+	6.18245i	0.241133	+	0.331891i	−0.671248	+	0.741233i	$0.734242\pi$
$348$		0			0
$349$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$349$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$350$	−29.1984	−	21.2139i	−1.56072	−	1.13393i
$351$		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
$358$	−30.1279	−	41.4675i	−1.59231	−	2.19162i
$359$	24.0276	−	7.80704i	1.26813	−	0.412040i	0.403745	−	0.914872i	$-0.367708\pi$
$359$	24.0276	−	7.80704i	1.26813	−	0.412040i	0.864384	+	0.502832i	$0.167708\pi$
$360$		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	$0.400000\pi$
$368$	−40.0982	+	29.1330i	−2.09026	+	1.51866i
$369$		0			0
$370$		0			0
$371$	−17.5506	−	5.70253i	−0.911180	−	0.296060i
$372$		0			0
$373$		−	31.7490i		−	1.64390i	−0.569558	−	0.821951i	$-0.692886\pi$
$373$		−	31.7490i		−	1.64390i	0.569558	−	0.821951i	$-0.307114\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−9.75971	+	7.09085i	−0.501323	+	0.364232i	−0.809522	−	0.587090i	$-0.800274\pi$
$379$	−9.75971	+	7.09085i	−0.501323	+	0.364232i	0.308199	+	0.951322i	$0.400274\pi$
$380$		0			0
$381$		0			0
$382$	−71.7042	−	23.2981i	−3.66871	−	1.19203i
$383$		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
$386$	−28.8603			−1.46895
$387$	23.0080	−	31.6678i	1.16956	−	1.60976i
$388$		0			0
$389$	−7.57775	−	23.3219i	−0.384207	−	1.18247i	−0.937054	−	0.349185i	$-0.886458\pi$
$389$	−7.57775	−	23.3219i	−0.384207	−	1.18247i	0.552847	−	0.833283i	$-0.313542\pi$
$390$		0			0
$391$		0			0
$392$	62.5408	−	20.3207i	3.15879	−	1.02635i
$393$		0			0
$394$	52.6289	+	38.2371i	2.65141	+	1.92636i
$395$		0			0
$396$		0			0
$397$		0			0		1.00000			$0$
$397$		0			0		−1.00000			$\pi$
$398$		0			0
$399$		0			0
$400$	22.7792	+	70.1071i	1.13896	+	3.50536i
$401$	32.3183	−	23.4806i	1.61390	−	1.17257i	0.764961	−	0.644077i	$-0.222758\pi$
$401$	32.3183	−	23.4806i	1.61390	−	1.17257i	0.848939	−	0.528490i	$-0.177242\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$		−	37.3360i		−	1.85295i
$407$		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
$414$	26.1693	−	8.50291i	1.28615	−	0.417895i
$415$		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
$421$	−0.733247	+	2.25670i	−0.0357363	+	0.109985i	−0.967333	−	0.253507i	$-0.918416\pi$
$421$	−0.733247	+	2.25670i	−0.0357363	+	0.109985i	0.931597	+	0.363492i	$0.118416\pi$
$422$	12.0991	+	37.2371i	0.588974	+	1.81268i
$423$		0			0
$424$	38.5136	+	53.0094i	1.87038	+	2.57436i
$425$		0			0
$426$		0			0
$427$		0			0
$428$			40.6876i			1.96671i
$429$		0			0
$430$		0			0
$431$	23.6370	−	32.5336i	1.13856	−	1.56709i	0.367862	−	0.929880i	$-0.380090\pi$
$431$	23.6370	−	32.5336i	1.13856	−	1.56709i	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
$436$	10.4485	−	3.39494i	0.500395	−	0.162588i
$437$		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
$443$	−12.7958	+	39.3813i	−0.607945	+	1.87106i	−0.132831	+	0.991139i	$0.542407\pi$
$443$	−12.7958	+	39.3813i	−0.607945	+	1.87106i	−0.475114	+	0.879924i	$0.657593\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$	−72.9501	−	23.7029i	−3.44657	−	1.11986i
$449$	−21.4391	−	15.5764i	−1.01177	−	0.735096i	−0.0471929	−	0.998886i	$-0.515028\pi$
$449$	−21.4391	−	15.5764i	−1.01177	−	0.735096i	−0.964580	+	0.263790i	$0.915028\pi$
$450$		−	40.9236i		−	1.92916i
$451$		0			0
$452$	−112.938			−5.31217
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	18.0571	+	24.8535i	0.844677	+	1.16260i	0.985011	+	0.172493i	$0.0551823\pi$
$457$	18.0571	+	24.8535i	0.844677	+	1.16260i	−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
$463$	27.4582			1.27609			0.638046	−	0.769998i	$-0.279743\pi$
$463$	27.4582			1.27609			0.638046	+	0.769998i	$0.279743\pi$
$464$	−44.8229	+	61.6935i	−2.08085	+	2.86405i
$465$		0			0
$466$	17.8445	+	54.9197i	0.826631	+	2.54411i
$467$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$467$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$468$		0			0
$469$	−34.8791	+	11.3329i	−1.61057	+	0.523305i
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$	−6.46606	−	19.9005i	−0.296060	−	0.911180i
$478$	−51.6323	+	37.5131i	−2.36161	+	1.71581i
$479$		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
$487$	−8.59414	+	26.4500i	−0.389438	+	1.19857i	0.543772	+	0.839233i	$0.316996\pi$
$487$	−8.59414	+	26.4500i	−0.389438	+	1.19857i	−0.933210	+	0.359333i	$0.883004\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−5.03252	+	1.63516i	−0.227114	+	0.0737939i	−0.420363	−	0.907356i	$-0.638097\pi$
$491$	−5.03252	+	1.63516i	−0.227114	+	0.0737939i	0.193249	+	0.981150i	$0.438097\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$	0.137235	−	0.188887i	0.00615581	−	0.00847275i
$498$		0			0
$499$	4.33797	+	13.3509i	0.194194	+	0.597669i	0.999985	+	0.00546838i	$0.00174065\pi$
$499$	4.33797	+	13.3509i	0.194194	+	0.597669i	−0.805791	+	0.592200i	$0.798259\pi$
$500$		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	$0.100000\pi$
$504$	60.3236	+	43.8276i	2.68703	+	1.95224i
$505$		0			0
$506$		0			0
$507$		0			0
$508$	−11.0005	+	15.1408i	−0.488066	+	0.671766i
$509$		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
$512$	22.9273	+	31.5568i	1.01325	+	1.39463i
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$	79.9395			3.51234
$519$		0			0
$520$		0			0
$521$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$521$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$522$	34.2498	−	24.8839i	1.49907	−	1.08914i
$523$		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
$526$	63.5642	+	46.1821i	2.77153	+	2.01364i
$527$		0			0
$528$		0			0
$529$	−11.6978			−0.508602
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$	123.844	+	40.2394i	5.34925	+	1.73808i
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−26.8443	+	36.9480i	−1.15413	+	1.58852i	−0.423238	+	0.906019i	$0.639107\pi$
$541$	−26.8443	+	36.9480i	−1.15413	+	1.58852i	−0.730887	+	0.682498i	$0.760893\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	15.0976	+	4.90549i	0.645525	+	0.209744i	0.613440	−	0.789741i	$-0.289785\pi$
$547$	15.0976	+	4.90549i	0.645525	+	0.209744i	0.0320849	+	0.999485i	$0.489785\pi$
$548$	−19.1602	−	13.9207i	−0.818483	−	0.594662i
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$	2.20989	−	6.80135i	0.0939741	−	0.289223i
$554$	−16.6993	−	51.3951i	−0.709485	−	2.18357i
$555$		0			0
$556$		0			0
$557$	44.7177	−	14.5296i	1.89475	−	0.615641i	0.920207	−	0.391433i	$-0.128020\pi$
$557$	44.7177	−	14.5296i	1.89475	−	0.615641i	0.974541	−	0.224208i	$-0.0719796\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$	85.3070			3.59846
$563$		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
$568$	−0.788427	+	0.256175i	−0.0330816	+	0.0107489i
$569$	−40.2601	−	13.0813i	−1.68779	−	0.548397i	−0.701395	−	0.712773i	$-0.747439\pi$
$569$	−40.2601	−	13.0813i	−1.68779	−	0.548397i	−0.986398	+	0.164375i	$0.947439\pi$
$570$		0			0
$571$			36.1771i			1.51396i	0.653435	+	0.756982i	$0.273327\pi$
$571$			36.1771i			1.51396i	−0.653435	+	0.756982i	$0.726673\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	5.19437	−	15.9866i	0.216620	−	0.666689i
$576$	−26.8766	−	82.7176i	−1.11986	−	3.44657i
$577$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$577$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$578$	−27.2616	−	37.5223i	−1.13393	−	1.56072i
$579$		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
$592$	−132.091	−	95.9697i	−5.42891	−	3.94433i
$593$		0			0			−	1.00000i	$-0.5\pi$
$593$		0			0				1.00000i	$0.5\pi$
$594$		0			0
$595$		0			0
$596$	33.8603	−	46.6047i	1.38697	−	1.90900i
$597$		0			0
$598$		0			0
$599$	−38.5581	+	28.0141i	−1.57544	+	1.14462i	−0.653742	+	0.756717i	$0.726802\pi$
$599$	−38.5581	+	28.0141i	−1.57544	+	1.14462i	−0.921698	+	0.387907i	$0.873198\pi$
$600$		0			0
$601$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$601$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$602$	89.5730	+	29.1040i	3.65072	+	1.18619i
$603$	−33.6425	−	24.4427i	−1.37003	−	0.995385i
$604$		−	100.090i		−	4.07262i
$605$		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
$613$	25.0405	+	8.13616i	1.01138	+	0.328616i	0.767403	−	0.641165i	$-0.221549\pi$
$613$	25.0405	+	8.13616i	1.01138	+	0.328616i	0.243974	+	0.969782i	$0.421549\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−48.2946			−1.94427			−0.972133	−	0.234428i	$-0.924678\pi$
$617$	−48.2946			−1.94427			−0.972133	+	0.234428i	$0.924678\pi$
$618$		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
$631$	−12.6501	−	38.9331i	−0.503594	−	1.54990i	−0.803122	−	0.595815i	$-0.796829\pi$
$631$	−12.6501	−	38.9331i	−0.503594	−	1.54990i	0.299528	−	0.954087i	$-0.403171\pi$
$632$	−20.5426	+	14.9251i	−0.817143	+	0.593689i
$633$		0			0
$634$	−53.3774	+	17.3434i	−2.11989	+	0.688794i
$635$		0			0
$636$		0			0
$637$		0			0
$638$		0			0
$639$	0.264738			0.0104729
$640$		0			0
$641$	10.6131	−	32.6639i	0.419194	−	1.29015i	−0.489251	−	0.872143i	$-0.662730\pi$
$641$	10.6131	−	32.6639i	0.419194	−	1.29015i	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	$0.800000\pi$
$644$	28.4585	+	39.1697i	1.12142	+	1.54350i
$645$		0			0
$646$		0			0
$647$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$647$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$648$			84.5477i			3.32135i
$649$		0			0
$650$		0			0
$651$		0			0
$652$	42.9117	−	132.069i	1.68055	−	5.17220i
$653$	14.4223	+	44.3871i	0.564386	+	1.73700i	0.669768	+	0.742571i	$0.266394\pi$
$653$	14.4223	+	44.3871i	0.564386	+	1.73700i	−0.105382	+	0.994432i	$0.533606\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$			26.4575i			1.03064i	0.856998	+	0.515319i	$0.172327\pi$
$659$			26.4575i			1.03064i	−0.856998	+	0.515319i	$0.827673\pi$
$660$		0			0
$661$		0			0		1.00000			$0$
$661$		0			0		−1.00000			$\pi$
$662$	51.6925	−	71.1487i	2.00909	−	2.76527i
$663$		0			0
$664$		0			0
$665$		0			0
$666$	53.2786	+	73.3316i	2.06450	+	2.84154i
$667$	16.5380	−	5.37352i	0.640353	−	0.208063i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−24.4595	+	33.6656i	−0.942845	+	1.29771i	0.0117883	+	0.999931i	$0.496248\pi$
$673$	−24.4595	+	33.6656i	−0.942845	+	1.29771i	−0.954633	+	0.297784i	$0.903752\pi$
$674$	29.3029	−	90.1850i	1.12871	−	3.47380i
$675$		0			0
$676$	−57.2485	+	41.5935i	−2.20187	+	1.59975i
$677$		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
$683$	38.9586			1.49071			0.745355	−	0.666668i	$-0.232280\pi$
$683$	38.9586			1.49071			0.745355	+	0.666668i	$0.232280\pi$
$684$		0			0
$685$		0			0
$686$	−15.6139	−	48.0548i	−0.596143	−	1.83474i
$687$		0			0
$688$	−113.069	−	155.626i	−4.31071	−	5.93319i
$689$		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
$694$	−20.8490			−0.791418
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	68.4839	−	22.2518i	2.58845	−	0.841038i
$701$	−39.5959	−	12.8655i	−1.49552	−	0.485923i	−0.556810	−	0.830640i	$-0.687975\pi$
$701$	−39.5959	−	12.8655i	−1.49552	−	0.485923i	−0.938707	+	0.344717i	$0.887975\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	39.2139	−	28.4906i	1.47271	−	1.06999i	0.492893	−	0.870090i	$-0.335939\pi$
$709$	39.2139	−	28.4906i	1.47271	−	1.06999i	0.979817	−	0.199896i	$-0.0640606\pi$
$710$		0			0
$711$	7.71200	−	2.50578i	0.289223	−	0.0939741i
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$	102.266			3.82185
$717$		0			0
$718$	−21.2995	+	65.5532i	−0.794891	+	2.44642i
$719$		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
$722$	−49.2996	+	16.0184i	−1.83474	+	0.596143i
$723$		0			0
$724$		0			0
$725$		−	25.8622i		−	0.960498i
$726$		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
$736$		−	72.0587i		−	2.65612i
$737$		0			0
$738$		0			0
$739$	−26.1855	+	36.0413i	−0.963249	+	1.32580i	−0.0178655	+	0.999840i	$0.505687\pi$
$739$	−26.1855	+	36.0413i	−0.963249	+	1.32580i	−0.945384	+	0.325959i	$0.894313\pi$
$740$		0			0
$741$		0			0
$742$	40.7310	−	29.5928i	1.49528	−	1.08639i
$743$	29.0886	+	40.0370i	1.06716	+	1.46881i	0.872923	+	0.487858i	$0.162222\pi$
$743$	29.0886	+	40.0370i	1.06716	+	1.46881i	0.194233	+	0.980955i	$0.437778\pi$
$744$		0			0
$745$		0			0
$746$	70.0763	+	50.9134i	2.56567	+	1.86407i
$747$		0			0
$748$		0			0
$749$	19.7764			0.722615
$750$		0			0
$751$	−16.8056	+	51.7224i	−0.613246	+	1.88738i	−0.188469	+	0.982079i	$0.560353\pi$
$751$	−16.8056	+	51.7224i	−0.613246	+	1.88738i	−0.424777	+	0.905298i	$0.639647\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−5.35721	−	3.89224i	−0.194711	−	0.141466i	0.486158	−	0.873871i	$-0.338398\pi$
$757$	−5.35721	−	3.89224i	−0.194711	−	0.141466i	−0.680869	+	0.732405i	$0.738398\pi$
$758$		−	32.9126i		−	1.19544i
$759$		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
$763$	−1.65013	−	5.07857i	−0.0597387	−	0.183857i
$764$	121.696	−	88.4172i	4.40280	−	3.19882i
$765$		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
$772$	33.8454	−	46.5842i	1.21812	−	1.67660i
$773$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$773$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$774$	33.0009	+	101.566i	1.18619	+	3.65072i
$775$		0			0
$776$		0			0
$777$		0			0
$778$	63.6278	+	20.6739i	2.28117	+	0.741197i
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	−31.8909	+	98.1500i	−1.13896	+	3.50536i
$785$		0			0
$786$		0			0
$787$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$787$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$788$	−123.439	+	40.1078i	−4.39734	+	1.42878i
$789$		0			0
$790$		0			0
$791$			54.8943i			1.95182i
$792$		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
$800$	−101.925	−	33.1175i	−3.60360	−	1.17088i
$801$		0			0
$802$			108.987i			3.84846i
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	33.2587	+	45.7767i	1.16932	+	1.60942i	0.668004	+	0.744157i	$0.267149\pi$
$809$	33.2587	+	45.7767i	1.16932	+	1.60942i	0.501311	+	0.865267i	$0.332851\pi$
$810$		0			0
$811$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$811$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$812$	60.2650	+	43.7851i	2.11489	+	1.53656i
$813$		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	0.759548	−	0.650451i	$-0.225420\pi$
$821$	50.3252	−	16.3516i	1.75636	−	0.570676i	0.996813	+	0.0797750i	$0.0254202\pi$
$822$		0			0
$823$	−44.6425	−	32.4347i	−1.55614	−	1.13060i	−0.939086	−	0.343683i	$-0.888326\pi$
$823$	−44.6425	−	32.4347i	−1.55614	−	1.13060i	−0.617055	−	0.786920i	$-0.711674\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−21.7719	+	29.9664i	−0.757082	+	1.04203i	0.240369	+	0.970682i	$0.422732\pi$
$827$	−21.7719	+	29.9664i	−0.757082	+	1.04203i	−0.997451	+	0.0713526i	$0.977268\pi$
$828$	−16.9647	+	52.2121i	−0.589566	+	1.81450i
$829$		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
$841$	−1.81693	+	1.32008i	−0.0626528	+	0.0455199i
$842$	−3.80513	−	5.23732i	−0.131134	−	0.180490i
$843$		0			0
$844$	−74.2944	−	24.1397i	−2.55732	−	0.830924i
$845$		0			0
$846$		0			0
$847$		0			0
$848$	−102.830			−3.53121
$849$		0			0
$850$		0			0
$851$	11.5052	+	35.4092i	0.394392	+	1.21381i
$852$		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
$856$	−56.8088	−	41.2740i	−1.94169	−	1.41072i
$857$		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
$862$	33.9031	+	104.343i	1.15474	+	3.55394i
$863$	−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
$872$	−5.85906	+	18.0323i	−0.198413	+	0.610652i
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−54.5562	−	17.7264i	−1.84223	−	0.598578i	−0.998043	−	0.0625337i	$-0.980082\pi$
$877$	−54.5562	−	17.7264i	−1.84223	−	0.598578i	−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			$\pi$
$882$	33.6760	−	46.3511i	1.13393	−	1.56072i
$883$	−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
$886$	−66.4027	−	91.3955i	−2.23084	−	3.07049i
$887$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$887$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$888$		0			0
$889$	7.35928	+	5.34683i	0.246822	+	0.179327i
$890$		0			0
$891$		0			0
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$	77.5436	−	56.3387i	2.59055	−	1.88215i
$897$		0			0
$898$	68.7603	−	22.3416i	2.29456	−	0.745549i
$899$		0			0
$900$	66.0560	+	47.9925i	2.20187	+	1.59975i
$901$		0			0
$902$		0			0
$903$		0			0
$904$	114.566	−	157.687i	3.81041	−	5.24458i
$905$		0			0
$906$		0			0
$907$	10.9286	−	7.94010i	0.362878	−	0.263647i	−0.391373	−	0.920232i	$-0.628000\pi$
$907$	10.9286	−	7.94010i	0.362878	−	0.263647i	0.754252	+	0.656585i	$0.228000\pi$
$908$		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
$914$	−83.8133			−2.77230
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	−20.1049	−	27.6721i	−0.663201	−	0.912817i	0.336381	−	0.941726i	$-0.390797\pi$
$919$	−20.1049	−	27.6721i	−0.663201	−	0.912817i	−0.999582	+	0.0289084i	$0.990797\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$	55.3732			1.82066
$926$	−44.0326	+	60.6057i	−1.44700	+	1.99163i
$927$		0			0
$928$	−34.2597	−	105.441i	−1.12463	−	3.46126i
$929$		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
$932$	−109.574	−	35.6028i	−3.58922	−	1.16621i
$933$		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	$0.700000\pi$
$938$	30.9189	−	95.1586i	1.00954	−	3.10704i
$939$		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
$953$	16.1374	−	5.24336i	0.522742	−	0.169849i	−0.0357473	−	0.999361i	$-0.511381\pi$
$953$	16.1374	−	5.24336i	0.522742	−	0.169849i	0.558489	+	0.829512i	$0.311381\pi$
$954$	54.2933	+	17.6410i	1.75781	+	0.571148i
$955$		0			0
$956$		−	127.334i		−	4.11827i
$957$		0			0
$958$		0			0
$959$	−6.76623	+	9.31292i	−0.218493	+	0.300730i
$960$		0			0
$961$	9.57953	+	29.4828i	0.309017	+	0.951057i
$962$		0			0
$963$	13.1807	+	18.1417i	0.424743	+	0.584608i
$964$		0			0
$965$		0			0
$966$		0			0
$967$			62.0397i			1.99506i	0.0702371	+	0.997530i	$0.477624\pi$
$967$			62.0397i			1.99506i	−0.0702371	+	0.997530i	$0.522376\pi$
$968$		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
$974$	−44.5987	−	61.3848i	−1.42903	−	1.96690i
$975$		0			0
$976$		0			0
$977$	−50.2375	−	36.4997i	−1.60724	−	1.16773i	−0.871404	−	0.490567i	$-0.836790\pi$
$977$	−50.2375	−	36.4997i	−1.60724	−	1.16773i	−0.735835	−	0.677161i	$-0.763210\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	3.55899	−	4.89853i	0.113630	−	0.156398i
$982$	4.46113	−	13.7299i	0.142360	−	0.438140i
$983$		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
$989$			43.8651i			1.39483i
$990$		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
$994$	0.196839	+	0.605807i	0.00624334	+	0.0192150i
$995$		0			0
$996$		0			0
$997$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$997$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$998$	−36.4245	−	11.8350i	−1.15300	−	0.374632i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	847.2.l.a.524.1		8
7.6	odd	2	CM	847.2.l.a.524.1		8
11.2	odd	10		77.2.l.a.6.1	✓	8
11.3	even	5		847.2.l.d.118.2		8
11.4	even	5		77.2.l.a.13.1	yes	8
11.5	even	5		847.2.b.b.846.1		8
11.6	odd	10		847.2.b.b.846.8		8
11.7	odd	10		847.2.l.c.475.2		8
11.8	odd	10	inner	847.2.l.a.118.1		8
11.9	even	5		847.2.l.c.699.2		8
11.10	odd	2		847.2.l.d.524.2		8
33.2	even	10		693.2.bu.a.622.2		8
33.26	odd	10		693.2.bu.a.244.2		8
77.2	odd	30		539.2.s.a.325.2		16
77.4	even	15		539.2.s.a.68.2		16
77.6	even	10		847.2.b.b.846.8		8
77.13	even	10		77.2.l.a.6.1	✓	8
77.20	odd	10		847.2.l.c.699.2		8
77.24	even	30		539.2.s.a.215.2		16
77.26	odd	30		539.2.s.a.178.2		16
77.27	odd	10		847.2.b.b.846.1		8
77.37	even	15		539.2.s.a.178.2		16
77.41	even	10	inner	847.2.l.a.118.1		8
77.46	odd	30		539.2.s.a.215.2		16
77.48	odd	10		77.2.l.a.13.1	yes	8
77.59	odd	30		539.2.s.a.68.2		16
77.62	even	10		847.2.l.c.475.2		8
77.68	even	30		539.2.s.a.325.2		16
77.69	odd	10		847.2.l.d.118.2		8
77.76	even	2		847.2.l.d.524.2		8
231.125	even	10		693.2.bu.a.244.2		8
231.167	odd	10		693.2.bu.a.622.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
77.2.l.a.6.1	✓	8	11.2	odd	10
77.2.l.a.6.1	✓	8	77.13	even	10
77.2.l.a.13.1	yes	8	11.4	even	5
77.2.l.a.13.1	yes	8	77.48	odd	10
539.2.s.a.68.2		16	77.4	even	15
539.2.s.a.68.2		16	77.59	odd	30
539.2.s.a.178.2		16	77.26	odd	30
539.2.s.a.178.2		16	77.37	even	15
539.2.s.a.215.2		16	77.24	even	30
539.2.s.a.215.2		16	77.46	odd	30
539.2.s.a.325.2		16	77.2	odd	30
539.2.s.a.325.2		16	77.68	even	30
693.2.bu.a.244.2		8	33.26	odd	10
693.2.bu.a.244.2		8	231.125	even	10
693.2.bu.a.622.2		8	33.2	even	10
693.2.bu.a.622.2		8	231.167	odd	10
847.2.b.b.846.1		8	11.5	even	5
847.2.b.b.846.1		8	77.27	odd	10
847.2.b.b.846.8		8	11.6	odd	10
847.2.b.b.846.8		8	77.6	even	10
847.2.l.a.118.1		8	11.8	odd	10	inner
847.2.l.a.118.1		8	77.41	even	10	inner
847.2.l.a.524.1		8	1.1	even	1	trivial
847.2.l.a.524.1		8	7.6	odd	2	CM
847.2.l.c.475.2		8	11.7	odd	10
847.2.l.c.475.2		8	77.62	even	10
847.2.l.c.699.2		8	11.9	even	5
847.2.l.c.699.2		8	77.20	odd	10
847.2.l.d.118.2		8	11.3	even	5
847.2.l.d.118.2		8	77.69	odd	10
847.2.l.d.524.2		8	11.10	odd	2
847.2.l.d.524.2		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)

\(f(q)\)	\(=\)	\(q+(-1.60362 + 2.20719i) q^{2} +(-1.68208 - 5.17690i) q^{4} +(-2.51626 + 0.817582i) q^{7} +(8.93440 + 2.90296i) q^{8} +(-2.42705 - 1.76336i) q^{9} +O(q^{10})\)	\(q+(-1.60362 + 2.20719i) q^{2} +(-1.68208 - 5.17690i) q^{4} +(-2.51626 + 0.817582i) q^{7} +(8.93440 + 2.90296i) q^{8} +(-2.42705 - 1.76336i) q^{9} +(2.23056 - 6.86497i) q^{14} +(-11.9273 + 8.66572i) q^{16} +(7.78414 - 2.52922i) q^{18} +3.36187 q^{23} +(1.54508 - 4.75528i) q^{25} +(8.46507 + 11.6512i) q^{28} +(4.91928 - 1.59837i) q^{29} -21.4341i q^{32} +(-5.04623 + 15.5307i) q^{36} +(3.42225 + 10.5326i) q^{37} +13.0478i q^{43} +(-5.39116 + 7.42030i) q^{46} +(5.66312 - 4.11450i) q^{49} +(8.01811 + 11.0360i) q^{50} +(5.64279 + 4.09972i) q^{53} -24.8547 q^{56} +(-4.36075 + 13.4210i) q^{58} +(7.54878 + 2.45275i) q^{63} +(23.4546 + 17.0408i) q^{64} +13.8615 q^{67} +(-0.0713926 + 0.0518698i) q^{71} +(-16.5653 - 22.8002i) q^{72} +(-28.7355 - 9.33673i) q^{74} +(-1.58876 + 2.18674i) q^{79} +(2.78115 + 8.55951i) q^{81} +(-28.7991 - 20.9238i) q^{86} +(-5.65492 - 17.4040i) q^{92} +19.0977i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 5 q^{2} - 5 q^{4} + 25 q^{8} - 6 q^{9}+O(q^{10}) \) 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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