LMFDB - Embedded newform 847.2.l.c.699.2

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			699.2
Root			$1.10362 - 0.884319i$ of defining polynomial
Character	$\chi$	$=$	847.699
Dual form			847.2.l.c.475.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(2.59471 - 0.843073i) q^{2} +(4.40373 - 3.19950i) q^{4} +(-1.55513 - 2.14046i) q^{7} +(5.52176 - 7.60006i) q^{8} +(0.927051 + 2.85317i) q^{9} +O(q^{10})$	$q+(2.59471 - 0.843073i) q^{2} +(4.40373 - 3.19950i) q^{4} +(-1.55513 - 2.14046i) q^{7} +(5.52176 - 7.60006i) q^{8} +(0.927051 + 2.85317i) q^{9} +(-5.83969 - 4.24278i) q^{14} +(4.55584 - 14.0214i) q^{16} +(4.81086 + 6.62158i) q^{18} +3.36187 q^{23} +(-4.04508 - 2.93893i) q^{25} +(-13.6968 - 4.45035i) q^{28} +(3.04028 + 4.18459i) q^{29} -21.4341i q^{32} +(13.2112 + 9.59849i) q^{36} +(-8.95957 + 6.50951i) q^{37} +13.0478i q^{43} +(8.72309 - 2.83430i) q^{46} +(-2.16312 + 6.65740i) q^{49} +(-12.9736 - 4.21537i) q^{50} +(-2.15535 - 6.63349i) q^{53} -24.8547 q^{56} +(11.4166 + 8.29464i) q^{58} +(4.66540 - 6.42137i) q^{63} +(-8.95886 - 27.5725i) q^{64} +13.8615 q^{67} +(0.0272696 - 0.0839271i) q^{71} +(26.8032 + 8.70889i) q^{72} +(-17.7595 + 24.4439i) q^{74} +(2.57067 - 0.835260i) q^{79} +(-7.28115 + 5.29007i) q^{81} +(11.0003 + 33.8554i) q^{86} +(14.8048 - 10.7563i) q^{92} +19.0977i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 10 q^{4} + 10 q^{8} - 6 q^{9}+O(q^{10}) $ 8 * q + 10 * q^4 + 10 * q^8 - 6 * q^9	$ 8 q + 10 q^{4} + 10 q^{8} - 6 q^{9} - 21 q^{14} + 8 q^{16} + 15 q^{18} + 16 q^{23} - 10 q^{25} - 35 q^{28} + 30 q^{36} - 18 q^{37} - 15 q^{46} + 14 q^{49} + 30 q^{53} - 42 q^{56} + 19 q^{58} - 34 q^{64} + 8 q^{67} - 48 q^{71} + 75 q^{72} + 40 q^{79} - 18 q^{81} + 23 q^{86} + 25 q^{92}+O(q^{100}) $ 8 * q + 10 * q^4 + 10 * q^8 - 6 * q^9 - 21 * q^14 + 8 * q^16 + 15 * q^18 + 16 * q^23 - 10 * q^25 - 35 * q^28 + 30 * q^36 - 18 * q^37 - 15 * q^46 + 14 * q^49 + 30 * q^53 - 42 * q^56 + 19 * q^58 - 34 * q^64 + 8 * q^67 - 48 * q^71 + 75 * q^72 + 40 * q^79 - 18 * q^81 + 23 * q^86 + 25 * q^92

Display 100 coefficients 10 coefficients

Character values

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

$n$	$122$	$365$
$\chi(n)$	$-1$	$e\left(\frac{9}{10}\right)$

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 $

$2$	2.59471	−	0.843073i	1.83474	−	0.596143i	0.835853	−	0.548953i	$-0.184973\pi$
$2$	2.59471	−	0.843073i	1.83474	−	0.596143i	0.998886	−	0.0471903i	$-0.0150267\pi$
$3$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$3$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$4$	4.40373	−	3.19950i	2.20187	−	1.59975i
$5$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$5$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$6$		0			0
$7$	−1.55513	−	2.14046i	−0.587785	−	0.809017i
$7$	−1.55513	−	2.14046i	−0.587785	−	0.809017i
$8$	5.52176	−	7.60006i	1.95224	−	2.68703i
$9$	0.927051	+	2.85317i	0.309017	+	0.951057i
$10$		0			0
$11$		0			0
$11$		0			0
$12$		0			0
$13$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$13$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$14$	−5.83969	−	4.24278i	−1.56072	−	1.13393i
$15$		0			0
$16$	4.55584	−	14.0214i	1.13896	−	3.50536i
$17$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$17$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$18$	4.81086	+	6.62158i	1.13393	+	1.56072i
$19$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$19$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$20$		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
$25$	−4.04508	−	2.93893i	−0.809017	−	0.587785i
$26$		0			0
$27$		0			0
$28$	−13.6968	−	4.45035i	−2.58845	−	0.841038i
$29$	3.04028	+	4.18459i	0.564567	+	0.777059i	0.991898	−	0.127036i	$-0.0405463\pi$
$29$	3.04028	+	4.18459i	0.564567	+	0.777059i	−0.427331	+	0.904095i	$0.640546\pi$
$30$		0			0
$31$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$31$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$32$		−	21.4341i		−	3.78905i
$33$		0			0
$34$		0			0
$35$		0			0
$36$	13.2112	+	9.59849i	2.20187	+	1.59975i
$37$	−8.95957	+	6.50951i	−1.47294	+	1.07016i	−0.493197	+	0.869918i	$0.664172\pi$
$37$	−8.95957	+	6.50951i	−1.47294	+	1.07016i	−0.979747	+	0.200239i	$0.935828\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$41$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$42$		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
$46$	8.72309	−	2.83430i	1.28615	−	0.417895i
$47$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$47$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$48$		0			0
$49$	−2.16312	+	6.65740i	−0.309017	+	0.951057i
$50$	−12.9736	−	4.21537i	−1.83474	−	0.596143i
$51$		0			0
$52$		0			0
$53$	−2.15535	−	6.63349i	−0.296060	−	0.911180i	−0.982863	−	0.184336i	$-0.940986\pi$
$53$	−2.15535	−	6.63349i	−0.296060	−	0.911180i	0.686803	−	0.726844i	$-0.259014\pi$
$54$		0			0
$55$		0			0
$56$	−24.8547			−3.32135
$57$		0			0
$58$	11.4166	+	8.29464i	1.49907	+	1.08914i
$59$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$59$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$60$		0			0
$61$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$61$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$62$		0			0
$63$	4.66540	−	6.42137i	0.587785	−	0.809017i
$64$	−8.95886	−	27.5725i	−1.11986	−	3.44657i
$65$		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
$71$	0.0272696	−	0.0839271i	0.00323630	−	0.00996031i	−0.949425	−	0.313993i	$-0.898333\pi$
$71$	0.0272696	−	0.0839271i	0.00323630	−	0.00996031i	0.952662	+	0.304033i	$0.0983332\pi$
$72$	26.8032	+	8.70889i	3.15879	+	1.02635i
$73$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$73$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$74$	−17.7595	+	24.4439i	−2.06450	+	2.84154i
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$	2.57067	−	0.835260i	0.289223	−	0.0939741i	−0.160813	−	0.986985i	$-0.551411\pi$
$79$	2.57067	−	0.835260i	0.289223	−	0.0939741i	0.450035	+	0.893011i	$0.351411\pi$
$80$		0			0
$81$	−7.28115	+	5.29007i	−0.809017	+	0.587785i
$82$		0			0
$83$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$83$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$84$		0			0
$85$		0			0
$86$	11.0003	+	33.8554i	1.18619	+	3.65072i
$87$		0			0
$88$		0			0
$89$		0			0		1.00000			$0$
$89$		0			0		−1.00000			$\pi$
$90$		0			0
$91$		0			0
$92$	14.8048	−	10.7563i	1.54350	−	1.12142i
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$97$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$98$			19.0977i			1.92916i
$99$		0			0
$100$	−27.2166			−2.72166
$101$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$101$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$102$		0			0
$103$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$103$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$104$		0			0
$105$		0			0
$106$	−11.1850	−	15.3949i	−1.08639	−	1.49528i
$107$	−4.39357	+	6.04723i	−0.424743	+	0.584608i	−0.966736	−	0.255774i	$-0.917670\pi$
$107$	−4.39357	+	6.04723i	−0.424743	+	0.584608i	0.541994	+	0.840382i	$0.317670\pi$
$108$		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
$112$	−37.0972	+	12.0536i	−3.50536	+	1.13896i
$113$	−16.7856	−	12.1954i	−1.57905	−	1.14725i	−0.917769	−	0.397114i	$-0.870012\pi$
$113$	−16.7856	−	12.1954i	−1.57905	−	1.14725i	−0.661285	−	0.750135i	$-0.729988\pi$
$114$		0			0
$115$		0			0
$116$	26.7772	+	8.70044i	2.48620	+	0.807815i
$117$		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
$126$	6.69169	−	20.5949i	0.596143	−	1.83474i
$127$	3.26991	+	1.06246i	0.290157	+	0.0942778i	0.450479	−	0.892787i	$-0.351253\pi$
$127$	3.26991	+	1.06246i	0.290157	+	0.0942778i	−0.160322	+	0.987065i	$0.551253\pi$
$128$	−21.2940	−	29.3087i	−1.88215	−	2.59055i
$129$		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
$134$	35.9666	−	11.6863i	3.10704	−	1.00954i
$135$		0			0
$136$		0			0
$137$	−1.34450	+	4.13795i	−0.114869	+	0.353529i	−0.991920	−	0.126868i	$-0.959507\pi$
$137$	−1.34450	+	4.13795i	−0.114869	+	0.353529i	0.877051	+	0.480397i	$0.159507\pi$
$138$		0			0
$139$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$139$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$140$		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
$148$	−18.6284	+	57.3322i	−1.53124	+	4.71268i
$149$	−10.0650	−	3.27033i	−0.824560	−	0.267916i	−0.133808	−	0.991007i	$-0.542720\pi$
$149$	−10.0650	−	3.27033i	−0.824560	−	0.267916i	−0.690752	+	0.723092i	$0.742720\pi$
$150$		0			0
$151$	10.8081	−	14.8760i	0.879547	−	1.21059i	−0.0969991	−	0.995284i	$-0.530924\pi$
$151$	10.8081	−	14.8760i	0.879547	−	1.21059i	0.976546	−	0.215308i	$-0.0690756\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$157$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$158$	5.96596	−	4.33452i	0.474626	−	0.344836i
$159$		0			0
$160$		0			0
$161$	−5.22816	−	7.19594i	−0.412036	−	0.567119i
$162$	−14.4326	+	19.8648i	−1.13393	+	1.56072i
$163$	−7.88338	−	24.2625i	−0.617474	−	1.90039i	−0.349225	−	0.937039i	$-0.613555\pi$
$163$	−7.88338	−	24.2625i	−0.617474	−	1.90039i	−0.268249	−	0.963350i	$-0.586445\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$167$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$168$		0			0
$169$	10.5172	−	7.64121i	0.809017	−	0.587785i
$170$		0			0
$171$		0			0
$172$	41.7465	+	57.4592i	3.18314	+	4.38122i
$173$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$173$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$174$		0			0
$175$			13.2288i			1.00000i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	15.1993	+	11.0430i	1.13605	+	0.825390i	0.986564	−	0.163374i	$-0.0522378\pi$
$179$	15.1993	+	11.0430i	1.13605	+	0.825390i	0.149487	+	0.988764i	$0.452238\pi$
$180$		0			0
$181$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$181$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$182$		0			0
$183$		0			0
$184$	18.5634	−	25.5504i	1.36852	−	1.88360i
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−22.3570	+	16.2433i	−1.61769	+	1.17532i	−0.796781	+	0.604268i	$0.793466\pi$
$191$	−22.3570	+	16.2433i	−1.61769	+	1.17532i	−0.820912	+	0.571055i	$0.806534\pi$
$192$		0			0
$193$	−10.0606	−	3.26889i	−0.724179	−	0.235300i	−0.0763450	−	0.997081i	$-0.524325\pi$
$193$	−10.0606	−	3.26889i	−0.724179	−	0.235300i	−0.647834	+	0.761781i	$0.724325\pi$
$194$		0			0
$195$		0			0
$196$	11.7745	+	36.2383i	0.841038	+	2.58845i
$197$		−	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			$\pi$
$200$	−44.6720	+	14.5148i	−3.15879	+	1.02635i
$201$		0			0
$202$		0			0
$203$	4.22890	−	13.0152i	0.296810	−	0.913488i
$204$		0			0
$205$		0			0
$206$		0			0
$207$	3.11662	+	9.59198i	0.216620	+	0.666689i
$208$		0			0
$209$		0			0
$210$		0			0
$211$	−13.6488	+	4.43475i	−0.939619	+	0.305301i	−0.738490	−	0.674264i	$-0.764461\pi$
$211$	−13.6488	+	4.43475i	−0.939619	+	0.305301i	−0.201129	+	0.979565i	$0.564461\pi$
$212$	−30.7154	−	22.3161i	−2.10954	−	1.53267i
$213$		0			0
$214$	−6.30180	+	19.3949i	−0.430782	+	1.32581i
$215$		0			0
$216$		0			0
$217$		0			0
$218$	1.70158	+	5.23692i	0.115245	+	0.354689i
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$223$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$224$	−45.8788	+	33.3329i	−3.06541	+	2.22715i
$225$	4.63525	−	14.2658i	0.309017	−	0.951057i
$226$	−53.8354	−	17.4922i	−3.58108	−	1.16356i
$227$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$227$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$228$		0			0
$229$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$229$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$230$		0			0
$231$		0			0
$232$	48.5909			3.19015
$233$	−20.1301	+	6.54066i	−1.31876	+	0.428493i	−0.882071	−	0.471117i	$-0.843851\pi$
$233$	−20.1301	+	6.54066i	−1.31876	+	0.428493i	−0.436694	+	0.899610i	$0.643851\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$	13.7499	−	18.9251i	0.889407	−	1.22416i	−0.0843185	−	0.996439i	$-0.526871\pi$
$239$	13.7499	−	18.9251i	0.889407	−	1.22416i	0.973726	−	0.227725i	$-0.0731287\pi$
$240$		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		−0.309017	−	0.951057i	$-0.600000\pi$
$251$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$252$		−	43.2049i		−	2.72166i
$253$		0			0
$254$	9.38020			0.588566
$255$		0			0
$256$	−33.0521	−	24.0138i	−2.06576	−	1.50086i
$257$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$257$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$258$		0			0
$259$	27.8667	+	9.05442i	1.73155	+	0.562615i
$260$		0			0
$261$	−9.12085	+	12.5538i	−0.564567	+	0.777059i
$262$		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
$268$	61.0422	−	44.3498i	3.72875	−	2.70909i
$269$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$269$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$270$		0			0
$271$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$271$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$272$		0			0
$273$		0			0
$274$			11.8703i			0.717112i
$275$		0			0
$276$		0			0
$277$	18.8382	−	6.12090i	1.13188	−	0.367769i	0.317588	−	0.948229i	$-0.397127\pi$
$277$	18.8382	−	6.12090i	1.13188	−	0.367769i	0.814289	+	0.580460i	$0.197127\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	29.7378	+	9.66238i	1.77401	+	0.576409i	0.998491	−	0.0549198i	$-0.0174903\pi$
$281$	29.7378	+	9.66238i	1.77401	+	0.576409i	0.775515	+	0.631329i	$0.217490\pi$
$282$		0			0
$283$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$283$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$284$	−0.148437	−	0.456841i	−0.00880809	−	0.0271085i
$285$		0			0
$286$		0			0
$287$		0			0
$288$	61.1552	−	19.8705i	3.60360	−	1.17088i
$289$	13.7533	+	9.99235i	0.809017	+	0.587785i
$290$		0			0
$291$		0			0
$292$		0			0
$293$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$293$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$294$		0			0
$295$		0			0
$296$			104.037i			6.04704i
$297$		0			0
$298$	−28.8730			−1.67257
$299$		0			0
$300$		0			0
$301$	27.9284	−	20.2911i	1.60976	−	1.16956i
$302$	15.5022	−	47.7110i	0.892053	−	2.74546i
$303$		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		−0.809017	−	0.587785i	$-0.800000\pi$
$311$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$312$		0			0
$313$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$313$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$314$		0			0
$315$		0			0
$316$	8.64811	−	11.9031i	0.486494	−	0.669602i
$317$	−6.35698	−	19.5648i	−0.357043	−	1.09887i	−0.954815	−	0.297200i	$-0.903947\pi$
$317$	−6.35698	−	19.5648i	−0.357043	−	1.09887i	0.597772	−	0.801666i	$-0.296053\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$	−19.6323	−	14.2637i	−1.09406	−	0.794884i
$323$		0			0
$324$	−15.1387	+	46.5921i	−0.841038	+	2.58845i
$325$		0			0
$326$	−40.9102	−	56.3081i	−2.26581	−	3.11862i
$327$		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
$333$	−26.8787	−	19.5285i	−1.47294	−	1.07016i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−20.4298	−	28.1192i	−1.11288	−	1.53175i	−0.817102	−	0.576493i	$-0.804421\pi$
$337$	−20.4298	−	28.1192i	−1.11288	−	1.53175i	−0.295779	−	0.955256i	$-0.595579\pi$
$338$	20.8471	−	28.6935i	1.13393	−	1.56072i
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	17.6138	−	5.72307i	0.951057	−	0.309017i
$344$	99.1643	+	72.0471i	5.34658	+	3.88452i
$345$		0			0
$346$		0			0
$347$	−7.26790	−	2.36148i	−0.390161	−	0.126771i	0.107366	−	0.994220i	$-0.465758\pi$
$347$	−7.26790	−	2.36148i	−0.390161	−	0.126771i	−0.497527	+	0.867448i	$0.665758\pi$
$348$		0			0
$349$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$349$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$350$	11.1528	+	34.3248i	0.596143	+	1.83474i
$351$		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
$358$	48.7479	+	15.8392i	2.57641	+	0.837126i
$359$	14.8499	+	20.4391i	0.783747	+	1.07873i	0.994859	+	0.101273i	$0.0322915\pi$
$359$	14.8499	+	20.4391i	0.783747	+	1.07873i	−0.211112	+	0.977462i	$0.567708\pi$
$360$		0			0
$361$	−5.87132	−	18.0701i	−0.309017	−	0.951057i
$362$		0			0
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$367$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$368$	15.3161	−	47.1382i	0.798409	−	2.45725i
$369$		0			0
$370$		0			0
$371$	−10.8468	+	14.9294i	−0.563140	+	0.775096i
$372$		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
$379$	3.72788	−	11.4732i	0.191488	−	0.589340i	−0.808511	−	0.588481i	$-0.799726\pi$
$379$	3.72788	−	11.4732i	0.191488	−	0.589340i	1.00000		0.000859657i	$-0.000273637\pi$
$380$		0			0
$381$		0			0
$382$	−44.3156	+	60.9952i	−2.26738	+	3.12079i
$383$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$383$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$384$		0			0
$385$		0			0
$386$	−28.8603			−1.46895
$387$	−37.2277	+	12.0960i	−1.89239	+	0.614875i
$388$		0			0
$389$	19.8388	−	14.4137i	1.00587	−	0.730805i	0.0425291	−	0.999095i	$-0.486458\pi$
$389$	19.8388	−	14.4137i	1.00587	−	0.730805i	0.963338	+	0.268290i	$0.0864585\pi$
$390$		0			0
$391$		0			0
$392$	38.6524	+	53.2004i	1.95224	+	2.68703i
$393$		0			0
$394$	−20.1025	−	61.8690i	−1.01275	−	3.11691i
$395$		0			0
$396$		0			0
$397$		0			0		1.00000			$0$
$397$		0			0		−1.00000			$\pi$
$398$		0			0
$399$		0			0
$400$	−59.6367	+	43.3286i	−2.98184	+	2.16643i
$401$	−12.3445	+	37.9925i	−0.616455	+	1.89725i	−0.240287	+	0.970702i	$0.577242\pi$
$401$	−12.3445	+	37.9925i	−0.616455	+	1.89725i	−0.376168	+	0.926552i	$0.622758\pi$
$402$		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		0.951057	−	0.309017i	$-0.100000\pi$
$409$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$	16.1735	+	22.2609i	0.794884	+	1.09406i
$415$		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
$421$	1.91967	+	1.39472i	0.0935588	+	0.0679745i	0.633581	−	0.773676i	$-0.281584\pi$
$421$	1.91967	+	1.39472i	0.0935588	+	0.0679745i	−0.540022	+	0.841651i	$0.681584\pi$
$422$	−31.6758	+	23.0138i	−1.54195	+	1.12029i
$423$		0			0
$424$	−62.3163	−	20.2478i	−3.02635	−	0.983319i
$425$		0			0
$426$		0			0
$427$		0			0
$428$			40.6876i			1.96671i
$429$		0			0
$430$		0			0
$431$	−38.2455	+	12.4267i	−1.84222	+	0.598574i	−0.844177	+	0.536065i	$0.819910\pi$
$431$	−38.2455	+	12.4267i	−1.84222	+	0.598574i	−0.998044	+	0.0625092i	$0.980090\pi$
$432$		0			0
$433$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$433$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$434$		0			0
$435$		0			0
$436$	6.45756	+	8.88806i	0.309261	+	0.425661i
$437$		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
$443$	33.4997	+	24.3390i	1.59162	+	1.15638i	0.901582	+	0.432608i	$0.142407\pi$
$443$	33.4997	+	24.3390i	1.59162	+	1.15638i	0.690039	+	0.723772i	$0.257593\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$	−45.0856	+	62.0551i	−2.13010	+	2.93183i
$449$	8.18900	+	25.2032i	0.386463	+	1.18941i	0.935413	+	0.353556i	$0.115028\pi$
$449$	8.18900	+	25.2032i	0.386463	+	1.18941i	−0.548950	+	0.835855i	$0.684972\pi$
$450$		−	40.9236i		−	1.92916i
$451$		0			0
$452$	−112.938			−5.31217
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−29.2170	−	9.49319i	−1.36672	−	0.444073i	−0.468436	−	0.883497i	$-0.655182\pi$
$457$	−29.2170	−	9.49319i	−1.36672	−	0.444073i	−0.898279	+	0.439425i	$0.855182\pi$
$458$		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	$0.279743\pi$
$464$	72.5250	−	23.5648i	3.36689	−	1.09397i
$465$		0			0
$466$	−46.7175	+	33.9423i	−2.16415	+	1.57234i
$467$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$467$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$468$		0			0
$469$	−21.5565	−	29.6699i	−0.995385	−	1.37003i
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$	16.9284	−	12.2992i	0.775096	−	0.563140i
$478$	19.7218	−	60.6974i	0.902053	−	2.77624i
$479$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$479$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$		0			0
$485$		0			0
$486$		0			0
$487$	22.4997	+	16.3470i	1.01956	+	0.740754i	0.966193	−	0.257821i	$-0.0830043\pi$
$487$	22.4997	+	16.3470i	1.01956	+	0.740754i	0.0533681	+	0.998575i	$0.483004\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−3.11027	−	4.28092i	−0.140364	−	0.193195i	0.733047	−	0.680178i	$-0.238097\pi$
$491$	−3.11027	−	4.28092i	−0.140364	−	0.193195i	−0.873412	+	0.486983i	$0.838097\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−0.222050	+	0.0721485i	−0.00996031	+	0.00323630i
$498$		0			0
$499$	−11.3570	+	8.25132i	−0.508407	+	0.369380i	−0.812219	−	0.583352i	$-0.801741\pi$
$499$	−11.3570	+	8.25132i	−0.508407	+	0.369380i	0.303812	+	0.952732i	$0.401741\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$503$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$504$	−23.0416	−	70.9146i	−1.02635	−	3.15879i
$505$		0			0
$506$		0			0
$507$		0			0
$508$	17.7991	−	5.78328i	0.789708	−	0.256592i
$509$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$509$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$510$		0			0
$511$		0			0
$512$	−37.0972	−	12.0536i	−1.63948	−	0.532700i
$513$		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		0.809017	−	0.587785i	$-0.200000\pi$
$521$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$522$	−13.0822	+	40.2630i	−0.572594	+	1.76226i
$523$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$523$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$524$		0			0
$525$		0			0
$526$	−24.2794	−	74.7242i	−1.05863	−	3.25813i
$527$		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
$536$	76.5399	−	105.348i	3.30602	−	4.55034i
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	43.4349	−	14.1129i	1.86741	−	0.606759i	0.874951	−	0.484211i	$-0.160893\pi$
$541$	43.4349	−	14.1129i	1.86741	−	0.606759i	0.992463	−	0.122548i	$-0.0391066\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	9.33080	−	12.8427i	0.398956	−	0.549116i	−0.561525	−	0.827460i	$-0.689785\pi$
$547$	9.33080	−	12.8427i	0.398956	−	0.549116i	0.960482	+	0.278343i	$0.0897853\pi$
$548$	7.31854	+	22.5241i	0.312633	+	0.962184i
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$	−5.78557	−	4.20346i	−0.246027	−	0.178749i
$554$	43.7193	−	31.7639i	1.85746	−	1.34952i
$555$		0			0
$556$		0			0
$557$	27.6370	+	38.0391i	1.17102	+	1.61177i	0.656634	+	0.754209i	$0.271980\pi$
$557$	27.6370	+	38.0391i	1.17102	+	1.61177i	0.514384	+	0.857560i	$0.328020\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$	85.3070			3.59846
$563$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$563$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	22.6463	+	7.35824i	0.951057	+	0.309017i
$568$	−0.487274	−	0.670676i	−0.0204456	−	0.0281409i
$569$	−24.8821	+	34.2473i	−1.04311	+	1.43572i	−0.148483	+	0.988915i	$0.547439\pi$
$569$	−24.8821	+	34.2473i	−1.04311	+	1.43572i	−0.894630	+	0.446808i	$0.852561\pi$
$570$		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
$575$	−13.5990	−	9.88028i	−0.567119	−	0.412036i
$576$	70.3638	−	51.1223i	2.93183	−	2.13010i
$577$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$577$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$578$	44.1101	+	14.3322i	1.83474	+	0.596143i
$579$		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		0.809017	−	0.587785i	$-0.200000\pi$
$587$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$	50.4543	+	155.282i	2.07366	+	6.38206i
$593$		0			0			−	1.00000i	$-0.5\pi$
$593$		0			0				1.00000i	$0.5\pi$
$594$		0			0
$595$		0			0
$596$	−54.7871	+	17.8014i	−2.24417	+	0.729174i
$597$		0			0
$598$		0			0
$599$	14.7279	−	45.3277i	0.601765	−	1.85204i	0.0841014	−	0.996457i	$-0.473198\pi$
$599$	14.7279	−	45.3277i	0.601765	−	1.85204i	0.517663	−	0.855584i	$-0.326802\pi$
$600$		0			0
$601$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$601$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$602$	55.3592	−	76.1953i	2.25627	−	3.10549i
$603$	12.8503	+	39.5492i	0.523305	+	1.61057i
$604$		−	100.090i		−	4.07262i
$605$		0			0
$606$		0			0
$607$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$607$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	15.4759	−	21.3007i	0.625065	−	0.860329i	−0.372644	−	0.927974i	$-0.621549\pi$
$613$	15.4759	−	21.3007i	0.625065	−	0.860329i	0.997709	+	0.0676456i	$0.0215487\pi$
$614$		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		−0.809017	−	0.587785i	$-0.800000\pi$
$619$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	7.72542	+	23.7764i	0.309017	+	0.951057i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$	33.1185	−	24.0620i	1.31843	−	0.957892i	0.318475	−	0.947931i	$-0.396829\pi$
$631$	33.1185	−	24.0620i	1.31843	−	0.957892i	0.999950	−	0.00996082i	$-0.00317068\pi$
$632$	7.84659	−	24.1493i	0.312121	−	0.960609i
$633$		0			0
$634$	−32.9891	−	45.4056i	−1.31016	−	1.80329i
$635$		0			0
$636$		0			0
$637$		0			0
$638$		0			0
$639$	0.264738			0.0104729
$640$		0			0
$641$	−27.7856	−	20.1874i	−1.09746	−	0.797354i	−0.116820	−	0.993153i	$-0.537270\pi$
$641$	−27.7856	−	20.1874i	−1.09746	−	0.797354i	−0.980644	+	0.195799i	$0.937270\pi$
$642$		0			0
$643$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$643$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$644$	−46.0468	−	14.9615i	−1.81450	−	0.589566i
$645$		0			0
$646$		0			0
$647$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$647$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$648$			84.5477i			3.32135i
$649$		0			0
$650$		0			0
$651$		0			0
$652$	−112.344	−	81.6228i	−4.39974	−	3.19660i
$653$	−37.7579	+	27.4328i	−1.47758	+	1.07353i	−0.499257	+	0.866454i	$0.666394\pi$
$653$	−37.7579	+	27.4328i	−1.47758	+	1.07353i	−0.978326	+	0.207072i	$0.933606\pi$
$654$		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			$\pi$
$662$	−83.6403	+	27.1764i	−3.25077	+	1.05624i
$663$		0			0
$664$		0			0
$665$		0			0
$666$	−86.2065	−	28.0102i	−3.34043	−	1.08537i
$667$	10.2210	+	14.0681i	0.395760	+	0.544717i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	39.5763	−	12.8591i	1.52556	−	0.495683i	0.578208	−	0.815890i	$-0.303752\pi$
$673$	39.5763	−	12.8591i	1.52556	−	0.495683i	0.947348	+	0.320207i	$0.103752\pi$
$674$	−76.7160	−	55.7374i	−2.95499	−	2.14692i
$675$		0			0
$676$	21.8670	−	67.2996i	0.841038	−	2.58845i
$677$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$677$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$678$		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
$686$	40.8778	−	29.6995i	1.56072	−	1.13393i
$687$		0			0
$688$	182.949	+	59.4439i	6.97488	+	2.26628i
$689$		0			0
$690$		0			0
$691$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$691$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$692$		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
$700$	42.3254	+	58.2559i	1.59975	+	2.20187i
$701$	−24.4716	+	33.6823i	−0.924280	+	1.27216i	0.0377695	+	0.999286i	$0.487975\pi$
$701$	−24.4716	+	33.6823i	−0.924280	+	1.27216i	−0.962049	+	0.272876i	$0.912025\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−14.9784	+	46.0987i	−0.562525	+	1.73127i	0.112667	+	0.993633i	$0.464061\pi$
$709$	−14.9784	+	46.0987i	−0.562525	+	1.73127i	−0.675192	+	0.737642i	$0.735939\pi$
$710$		0			0
$711$	4.76628	+	6.56022i	0.178749	+	0.246027i
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$	102.266			3.82185
$717$		0			0
$718$	55.7629	+	40.5141i	2.08105	+	1.51197i
$719$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$719$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$720$		0			0
$721$		0			0
$722$	−30.4688	−	41.9367i	−1.13393	−	1.56072i
$723$		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
$729$	−21.8435	−	15.8702i	−0.809017	−	0.587785i
$730$		0			0
$731$		0			0
$732$		0			0
$733$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$733$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$734$		0			0
$735$		0			0
$736$		−	72.0587i		−	2.65612i
$737$		0			0
$738$		0			0
$739$	42.3690	−	13.7665i	1.55857	−	0.506410i	0.602145	−	0.798387i	$-0.294313\pi$
$739$	42.3690	−	13.7665i	1.55857	−	0.506410i	0.956425	+	0.291977i	$0.0943129\pi$
$740$		0			0
$741$		0			0
$742$	−15.5579	+	47.8822i	−0.571148	+	1.75781i
$743$	−47.0663	−	15.2928i	−1.72669	−	0.561037i	−0.733729	−	0.679442i	$-0.762222\pi$
$743$	−47.0663	−	15.2928i	−1.72669	−	0.561037i	−0.992965	+	0.118405i	$0.962222\pi$
$744$		0			0
$745$		0			0
$746$	−26.7668	−	82.3796i	−0.980000	−	3.01613i
$747$		0			0
$748$		0			0
$749$	19.7764			0.722615
$750$		0			0
$751$	43.9977	+	31.9662i	1.60550	+	1.16646i	0.875772	+	0.482724i	$0.160353\pi$
$751$	43.9977	+	31.9662i	1.60550	+	1.16646i	0.729727	+	0.683739i	$0.239647\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	2.04627	+	6.29778i	0.0743731	+	0.228897i	0.981332	−	0.192323i	$-0.0616021\pi$
$757$	2.04627	+	6.29778i	0.0743731	+	0.228897i	−0.906959	+	0.421220i	$0.861602\pi$
$758$		−	32.9126i		−	1.19544i
$759$		0			0
$760$		0			0
$761$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$761$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$762$		0			0
$763$	4.32009	−	3.13873i	0.156398	−	0.113630i
$764$	−46.4837	+	143.062i	−1.68172	+	5.17580i
$765$		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
$772$	−54.7631	+	17.7936i	−1.97097	+	0.640406i
$773$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$773$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$774$	−86.3974	+	62.7714i	−3.10549	+	2.25627i
$775$		0			0
$776$		0			0
$777$		0			0
$778$	39.3242	−	54.1251i	1.40984	−	1.94048i
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	83.4914	+	60.6600i	2.98184	+	2.16643i
$785$		0			0
$786$		0			0
$787$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$787$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$788$	−76.2896	−	105.004i	−2.71770	−	3.74060i
$789$		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		0.309017	−	0.951057i	$-0.400000\pi$
$797$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$798$		0			0
$799$		0			0
$800$	−62.9933	+	86.7029i	−2.22715	+	3.06541i
$801$		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
$809$	−53.8138	−	17.4852i	−1.89199	−	0.614745i	−0.977832	−	0.209393i	$-0.932851\pi$
$809$	−53.8138	−	17.4852i	−1.89199	−	0.614745i	−0.914160	−	0.405353i	$-0.867149\pi$
$810$		0			0
$811$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$811$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$812$	−23.0192	−	70.8458i	−0.807815	−	2.48620i
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$	31.1027	+	42.8092i	1.08549	+	1.49405i	0.853329	+	0.521373i	$0.174580\pi$
$821$	31.1027	+	42.8092i	1.08549	+	1.49405i	0.232162	+	0.972677i	$0.425420\pi$
$822$		0			0
$823$	17.0519	+	52.4804i	0.594393	+	1.82935i	0.557725	+	0.830026i	$0.311674\pi$
$823$	17.0519	+	52.4804i	0.594393	+	1.82935i	0.0366680	+	0.999328i	$0.488326\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	35.2276	−	11.4461i	1.22498	−	0.398022i	0.376090	−	0.926583i	$-0.377268\pi$
$827$	35.2276	−	11.4461i	1.22498	−	0.398022i	0.848895	+	0.528562i	$0.177268\pi$
$828$	44.4143	+	32.2689i	1.54350	+	1.12142i
$829$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$829$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$830$		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		−0.809017	−	0.587785i	$-0.800000\pi$
$839$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$840$		0			0
$841$	0.694005	−	2.13593i	0.0239312	−	0.0736527i
$842$	6.15684	+	2.00048i	0.212179	+	0.0689410i
$843$		0			0
$844$	−45.9165	+	63.1986i	−1.58051	+	2.17539i
$845$		0			0
$846$		0			0
$847$		0			0
$848$	−102.830			−3.53121
$849$		0			0
$850$		0			0
$851$	−30.1209	+	21.8841i	−1.03253	+	0.750178i
$852$		0			0
$853$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$853$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$854$		0			0
$855$		0			0
$856$	21.6990	+	66.7828i	0.741658	+	2.28259i
$857$		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
$862$	−88.7595	+	64.4875i	−3.02316	+	2.19645i
$863$	2.47214	−	7.60845i	0.0841525	−	0.258995i	−0.900123	−	0.435636i	$-0.856523\pi$
$863$	2.47214	−	7.60845i	0.0841525	−	0.258995i	0.984275	+	0.176642i	$0.0565234\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$	15.3392	+	11.1446i	0.519452	+	0.377404i
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−33.7176	+	46.4083i	−1.13856	+	1.56710i	−0.367885	+	0.929871i	$0.619918\pi$
$877$	−33.7176	+	46.4083i	−1.13856	+	1.56710i	−0.770677	+	0.637226i	$0.780082\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0		1.00000			$0$
$881$		0			0		−1.00000			$\pi$
$882$	−54.4890	+	17.7045i	−1.83474	+	0.596143i
$883$	9.70820	+	7.05342i	0.326707	+	0.237367i	0.739032	−	0.673670i	$-0.235283\pi$
$883$	9.70820	+	7.05342i	0.326707	+	0.237367i	−0.412325	+	0.911037i	$0.635283\pi$
$884$		0			0
$885$		0			0
$886$	107.442	+	34.9100i	3.60958	+	1.17282i
$887$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$887$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$888$		0			0
$889$	−2.81100	−	8.65136i	−0.0942778	−	0.290157i
$890$		0			0
$891$		0			0
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$	−29.6190	+	91.1580i	−0.989502	+	3.04537i
$897$		0			0
$898$	42.4962	+	58.4910i	1.41812	+	1.95187i
$899$		0			0
$900$	−25.2311	−	77.6534i	−0.841038	−	2.58845i
$901$		0			0
$902$		0			0
$903$		0			0
$904$	−185.372	+	60.2310i	−6.16538	+	2.00325i
$905$		0			0
$906$		0			0
$907$	−4.17436	+	12.8473i	−0.138607	+	0.426589i	−0.996134	−	0.0878507i	$-0.972000\pi$
$907$	−4.17436	+	12.8473i	−0.138607	+	0.426589i	0.857526	+	0.514440i	$0.172000\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−4.94427	−	15.2169i	−0.163811	−	0.504159i	0.835136	−	0.550044i	$-0.185389\pi$
$911$	−4.94427	−	15.2169i	−0.163811	−	0.504159i	−0.998947	+	0.0458855i	$0.985389\pi$
$912$		0			0
$913$		0			0
$914$	−83.8133			−2.77230
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	32.5305	+	10.5698i	1.07308	+	0.348665i	0.791687	−	0.610927i	$-0.209203\pi$
$919$	32.5305	+	10.5698i	1.07308	+	0.348665i	0.281394	+	0.959592i	$0.409203\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$	55.3732			1.82066
$926$	71.2462	−	23.1493i	2.34130	−	0.760733i
$927$		0			0
$928$	89.6931	−	65.1658i	2.94432	−	2.13917i
$929$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$929$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$930$		0			0
$931$		0			0
$932$	−67.7206	+	93.2094i	−2.21826	+	3.05318i
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$937$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$938$	−80.9468	−	58.8113i	−2.64301	−	1.92026i
$939$		0			0
$940$		0			0
$941$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$941$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$942$		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
$953$	9.97347	+	13.7273i	0.323072	+	0.444671i	0.939402	−	0.342817i	$-0.111381\pi$
$953$	9.97347	+	13.7273i	0.323072	+	0.444671i	−0.616330	+	0.787488i	$0.711381\pi$
$954$	33.5551	−	46.1847i	1.08639	−	1.49528i
$955$		0			0
$956$		−	127.334i		−	4.11827i
$957$		0			0
$958$		0			0
$959$	10.9480	−	3.55722i	0.353529	−	0.114869i
$960$		0			0
$961$	−25.0795	+	18.2213i	−0.809017	+	0.587785i
$962$		0			0
$963$	−21.3268	−	6.92951i	−0.687248	−	0.223300i
$964$		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		−0.809017	−	0.587785i	$-0.800000\pi$
$971$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$972$		0			0
$973$		0			0
$974$	72.1621	+	23.4469i	2.31222	+	0.751287i
$975$		0			0
$976$		0			0
$977$	19.1890	+	59.0577i	0.613911	+	1.88942i	0.416632	+	0.909075i	$0.363210\pi$
$977$	19.1890	+	59.0577i	0.613911	+	1.88942i	0.197278	+	0.980348i	$0.436790\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	−5.75856	+	1.87107i	−0.183857	+	0.0597387i
$982$	−11.6794	−	8.48556i	−0.372704	−	0.270785i
$983$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$983$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$984$		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
$994$	−0.515330	+	0.374409i	−0.0163453	+	0.0118755i
$995$		0			0
$996$		0			0
$997$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$997$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$998$	−22.5116	+	30.9846i	−0.712592	+	0.980799i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	847.2.l.c.699.2		8
7.6	odd	2	CM	847.2.l.c.699.2		8
11.2	odd	10	inner	847.2.l.c.475.2		8
11.3	even	5		847.2.b.b.846.1		8
11.4	even	5		847.2.l.d.118.2		8
11.5	even	5		847.2.l.a.524.1		8
11.6	odd	10		847.2.l.d.524.2		8
11.7	odd	10		847.2.l.a.118.1		8
11.8	odd	10		847.2.b.b.846.8		8
11.9	even	5		77.2.l.a.13.1	yes	8
11.10	odd	2		77.2.l.a.6.1	✓	8
33.20	odd	10		693.2.bu.a.244.2		8
33.32	even	2		693.2.bu.a.622.2		8
77.6	even	10		847.2.l.d.524.2		8
77.9	even	15		539.2.s.a.178.2		16
77.10	even	6		539.2.s.a.215.2		16
77.13	even	10	inner	847.2.l.c.475.2		8
77.20	odd	10		77.2.l.a.13.1	yes	8
77.27	odd	10		847.2.l.a.524.1		8
77.31	odd	30		539.2.s.a.68.2		16
77.32	odd	6		539.2.s.a.215.2		16
77.41	even	10		847.2.b.b.846.8		8
77.48	odd	10		847.2.l.d.118.2		8
77.53	even	15		539.2.s.a.68.2		16
77.54	even	6		539.2.s.a.325.2		16
77.62	even	10		847.2.l.a.118.1		8
77.65	odd	6		539.2.s.a.325.2		16
77.69	odd	10		847.2.b.b.846.1		8
77.75	odd	30		539.2.s.a.178.2		16
77.76	even	2		77.2.l.a.6.1	✓	8
231.20	even	10		693.2.bu.a.244.2		8
231.230	odd	2		693.2.bu.a.622.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
77.2.l.a.6.1	✓	8	11.10	odd	2
77.2.l.a.6.1	✓	8	77.76	even	2
77.2.l.a.13.1	yes	8	11.9	even	5
77.2.l.a.13.1	yes	8	77.20	odd	10
539.2.s.a.68.2		16	77.31	odd	30
539.2.s.a.68.2		16	77.53	even	15
539.2.s.a.178.2		16	77.9	even	15
539.2.s.a.178.2		16	77.75	odd	30
539.2.s.a.215.2		16	77.10	even	6
539.2.s.a.215.2		16	77.32	odd	6
539.2.s.a.325.2		16	77.54	even	6
539.2.s.a.325.2		16	77.65	odd	6
693.2.bu.a.244.2		8	33.20	odd	10
693.2.bu.a.244.2		8	231.20	even	10
693.2.bu.a.622.2		8	33.32	even	2
693.2.bu.a.622.2		8	231.230	odd	2
847.2.b.b.846.1		8	11.3	even	5
847.2.b.b.846.1		8	77.69	odd	10
847.2.b.b.846.8		8	11.8	odd	10
847.2.b.b.846.8		8	77.41	even	10
847.2.l.a.118.1		8	11.7	odd	10
847.2.l.a.118.1		8	77.62	even	10
847.2.l.a.524.1		8	11.5	even	5
847.2.l.a.524.1		8	77.27	odd	10
847.2.l.c.475.2		8	11.2	odd	10	inner
847.2.l.c.475.2		8	77.13	even	10	inner
847.2.l.c.699.2		8	1.1	even	1	trivial
847.2.l.c.699.2		8	7.6	odd	2	CM
847.2.l.d.118.2		8	11.4	even	5
847.2.l.d.118.2		8	77.48	odd	10
847.2.l.d.524.2		8	11.6	odd	10
847.2.l.d.524.2		8	77.6	even	10

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)

\(f(q)\)	\(=\)	\(q+(2.59471 - 0.843073i) q^{2} +(4.40373 - 3.19950i) q^{4} +(-1.55513 - 2.14046i) q^{7} +(5.52176 - 7.60006i) q^{8} +(0.927051 + 2.85317i) q^{9} +O(q^{10})\)	\(q+(2.59471 - 0.843073i) q^{2} +(4.40373 - 3.19950i) q^{4} +(-1.55513 - 2.14046i) q^{7} +(5.52176 - 7.60006i) q^{8} +(0.927051 + 2.85317i) q^{9} +(-5.83969 - 4.24278i) q^{14} +(4.55584 - 14.0214i) q^{16} +(4.81086 + 6.62158i) q^{18} +3.36187 q^{23} +(-4.04508 - 2.93893i) q^{25} +(-13.6968 - 4.45035i) q^{28} +(3.04028 + 4.18459i) q^{29} -21.4341i q^{32} +(13.2112 + 9.59849i) q^{36} +(-8.95957 + 6.50951i) q^{37} +13.0478i q^{43} +(8.72309 - 2.83430i) q^{46} +(-2.16312 + 6.65740i) q^{49} +(-12.9736 - 4.21537i) q^{50} +(-2.15535 - 6.63349i) q^{53} -24.8547 q^{56} +(11.4166 + 8.29464i) q^{58} +(4.66540 - 6.42137i) q^{63} +(-8.95886 - 27.5725i) q^{64} +13.8615 q^{67} +(0.0272696 - 0.0839271i) q^{71} +(26.8032 + 8.70889i) q^{72} +(-17.7595 + 24.4439i) q^{74} +(2.57067 - 0.835260i) q^{79} +(-7.28115 + 5.29007i) q^{81} +(11.0003 + 33.8554i) q^{86} +(14.8048 - 10.7563i) q^{92} +19.0977i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 10 q^{4} + 10 q^{8} - 6 q^{9}+O(q^{10}) \) 8 * q + 10 * q^4 + 10 * q^8 - 6 * q^9	\( 8 q + 10 q^{4} + 10 q^{8} - 6 q^{9} - 21 q^{14} + 8 q^{16} + 15 q^{18} + 16 q^{23} - 10 q^{25} - 35 q^{28} + 30 q^{36} - 18 q^{37} - 15 q^{46} + 14 q^{49} + 30 q^{53} - 42 q^{56} + 19 q^{58} - 34 q^{64} + 8 q^{67} - 48 q^{71} + 75 q^{72} + 40 q^{79} - 18 q^{81} + 23 q^{86} + 25 q^{92}+O(q^{100}) \) 8 * q + 10 * q^4 + 10 * q^8 - 6 * q^9 - 21 * q^14 + 8 * q^16 + 15 * q^18 + 16 * q^23 - 10 * q^25 - 35 * q^28 + 30 * q^36 - 18 * q^37 - 15 * q^46 + 14 * q^49 + 30 * q^53 - 42 * q^56 + 19 * q^58 - 34 * q^64 + 8 * q^67 - 48 * q^71 + 75 * q^72 + 40 * q^79 - 18 * q^81 + 23 * q^86 + 25 * q^92

Introduction

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Database

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