LMFDB - Embedded newform 847.2.l.d.475.2

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			475.2
Root			$1.10362 - 0.884319i$ of defining polynomial
Character	$\chi$	$=$	847.475
Dual form			847.2.l.d.699.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.68208 + 0.546539i) q^{2} +(0.912638 + 0.663070i) q^{4} +(1.55513 - 2.14046i) q^{7} +(-0.906428 - 1.24759i) q^{8} +(0.927051 - 2.85317i) q^{9} +O(q^{10})$	$q+(1.68208 + 0.546539i) q^{2} +(0.912638 + 0.663070i) q^{4} +(1.55513 - 2.14046i) q^{7} +(-0.906428 - 1.24759i) q^{8} +(0.927051 - 2.85317i) q^{9} +(3.78570 - 2.75047i) q^{14} +(-1.54002 - 4.73968i) q^{16} +(3.11874 - 4.29258i) q^{18} +2.56038 q^{23} +(-4.04508 + 2.93893i) q^{25} +(2.83855 - 0.922300i) q^{28} +(4.34154 - 5.97561i) q^{29} -5.72996i q^{32} +(2.73791 - 1.98921i) q^{36} +(9.64279 + 7.00589i) q^{37} +2.77251i q^{43} +(4.30676 + 1.39935i) q^{46} +(-2.16312 - 6.65740i) q^{49} +(-8.41038 + 2.73270i) q^{50} +(-4.42225 + 13.6103i) q^{53} -4.08003 q^{56} +(10.5687 - 7.67861i) q^{58} +(-4.66540 - 6.42137i) q^{63} +(0.0516205 - 0.158872i) q^{64} +12.5669 q^{67} +(4.96113 + 15.2688i) q^{71} +(-4.39989 + 1.42961i) q^{72} +(12.3909 + 17.0546i) q^{74} +(-16.6863 - 5.42171i) q^{79} +(-7.28115 - 5.29007i) q^{81} +(-1.51528 + 4.66357i) q^{86} +(2.33670 + 1.69771i) q^{92} -12.3805i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $\left(\mathbb{Z}/847\mathbb{Z}\right)^\times$.

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

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 $

$2$	1.68208	+	0.546539i	1.18941	+	0.386462i	0.835853	−	0.548953i	$-0.184973\pi$
$2$	1.68208	+	0.546539i	1.18941	+	0.386462i	0.353553	+	0.935414i	$0.384973\pi$
$3$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$3$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$4$	0.912638	+	0.663070i	0.456319	+	0.331535i
$5$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$5$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$6$		0			0
$7$	1.55513	−	2.14046i	0.587785	−	0.809017i
$7$	1.55513	−	2.14046i	0.587785	−	0.809017i
$8$	−0.906428	−	1.24759i	−0.320471	−	0.441090i
$9$	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	$-0.900000\pi$
$13$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$14$	3.78570	−	2.75047i	1.01177	−	0.735094i
$15$		0			0
$16$	−1.54002	−	4.73968i	−0.385004	−	1.18492i
$17$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$17$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$18$	3.11874	−	4.29258i	0.735094	−	1.01177i
$19$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$19$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	2.56038			0.533877			0.266938	−	0.963714i	$-0.413988\pi$
$23$	2.56038			0.533877			0.266938	+	0.963714i	$0.413988\pi$
$24$		0			0
$25$	−4.04508	+	2.93893i	−0.809017	+	0.587785i
$26$		0			0
$27$		0			0
$28$	2.83855	−	0.922300i	0.536435	−	0.174298i
$29$	4.34154	−	5.97561i	0.806203	−	1.10964i	−0.185695	−	0.982607i	$-0.559454\pi$
$29$	4.34154	−	5.97561i	0.806203	−	1.10964i	0.991898	−	0.127036i	$-0.0405463\pi$
$30$		0			0
$31$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$31$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$32$		−	5.72996i		−	1.01292i
$33$		0			0
$34$		0			0
$35$		0			0
$36$	2.73791	−	1.98921i	0.456319	−	0.331535i
$37$	9.64279	+	7.00589i	1.58526	+	1.15176i	0.910330	+	0.413884i	$0.135828\pi$
$37$	9.64279	+	7.00589i	1.58526	+	1.15176i	0.674935	+	0.737878i	$0.264172\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$41$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$42$		0			0
$43$			2.77251i			0.422803i	0.977399	+	0.211402i	$0.0678028\pi$
$43$			2.77251i			0.422803i	−0.977399	+	0.211402i	$0.932197\pi$
$44$		0			0
$45$		0			0
$46$	4.30676	+	1.39935i	0.634996	+	0.206323i
$47$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$47$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$48$		0			0
$49$	−2.16312	−	6.65740i	−0.309017	−	0.951057i
$50$	−8.41038	+	2.73270i	−1.18941	+	0.386462i
$51$		0			0
$52$		0			0
$53$	−4.42225	+	13.6103i	−0.607443	+	1.86952i	−0.128407	+	0.991722i	$0.540986\pi$
$53$	−4.42225	+	13.6103i	−0.607443	+	1.86952i	−0.479036	+	0.877795i	$0.659014\pi$
$54$		0			0
$55$		0			0
$56$	−4.08003			−0.545217
$57$		0			0
$58$	10.5687	−	7.67861i	1.38774	−	1.00825i
$59$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$59$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$60$		0			0
$61$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$61$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$62$		0			0
$63$	−4.66540	−	6.42137i	−0.587785	−	0.809017i
$64$	0.0516205	−	0.158872i	0.00645256	−	0.0198590i
$65$		0			0
$66$		0			0
$67$	12.5669			1.53529			0.767644	−	0.640877i	$-0.221429\pi$
$67$	12.5669			1.53529			0.767644	+	0.640877i	$0.221429\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	4.96113	+	15.2688i	0.588777	+	1.81207i	0.583541	+	0.812084i	$0.301667\pi$
$71$	4.96113	+	15.2688i	0.588777	+	1.81207i	0.00523645	+	0.999986i	$0.498333\pi$
$72$	−4.39989	+	1.42961i	−0.518532	+	0.168481i
$73$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$73$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$74$	12.3909	+	17.0546i	1.44041	+	1.98256i
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$	−16.6863	−	5.42171i	−1.87736	−	0.609990i	−0.988372	−	0.152053i	$-0.951411\pi$
$79$	−16.6863	−	5.42171i	−1.87736	−	0.609990i	−0.888985	−	0.457937i	$-0.848589\pi$
$80$		0			0
$81$	−7.28115	−	5.29007i	−0.809017	−	0.587785i
$82$		0			0
$83$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$83$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$84$		0			0
$85$		0			0
$86$	−1.51528	+	4.66357i	−0.163397	+	0.502885i
$87$		0			0
$88$		0			0
$89$		0			0		1.00000			$0$
$89$		0			0		−1.00000			$\pi$
$90$		0			0
$91$		0			0
$92$	2.33670	+	1.69771i	0.243618	+	0.176999i
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$97$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$98$		−	12.3805i		−	1.25062i
$99$		0			0
$100$	−5.64041			−0.564041
$101$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$101$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$102$		0			0
$103$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$103$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$104$		0			0
$105$		0			0
$106$	−14.8771	+	20.4766i	−1.44499	+	1.98886i
$107$	−10.2192	−	14.0655i	−0.987929	−	1.35977i	−0.932447	−	0.361308i	$-0.882330\pi$
$107$	−10.2192	−	14.0655i	−0.987929	−	1.35977i	−0.0554821	−	0.998460i	$-0.517670\pi$
$108$		0			0
$109$		−	13.8487i		−	1.32646i	−0.748414	−	0.663232i	$-0.769184\pi$
$109$		−	13.8487i		−	1.32646i	0.748414	−	0.663232i	$-0.230816\pi$
$110$		0			0
$111$		0			0
$112$	−12.5400	−	4.07450i	−1.18492	−	0.385004i
$113$	−8.75602	+	6.36162i	−0.823697	+	0.598451i	−0.917769	−	0.397114i	$-0.870012\pi$
$113$	−8.75602	+	6.36162i	−0.823697	+	0.598451i	0.0940721	+	0.995565i	$0.470012\pi$
$114$		0			0
$115$		0			0
$116$	7.92450	−	2.57483i	0.735771	−	0.239067i
$117$		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
$126$	−4.33802	−	13.3511i	−0.386462	−	1.18941i
$127$	−9.80674	+	3.18640i	−0.870207	+	0.282747i	−0.709885	−	0.704317i	$-0.751253\pi$
$127$	−9.80674	+	3.18640i	−0.870207	+	0.282747i	−0.160322	+	0.987065i	$0.551253\pi$
$128$	−6.56232	+	9.03226i	−0.580032	+	0.798346i
$129$		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
$134$	21.1384	+	6.86829i	1.82608	+	0.593330i
$135$		0			0
$136$		0			0
$137$	5.26562	+	16.2059i	0.449872	+	1.38456i	0.877051	+	0.480397i	$0.159507\pi$
$137$	5.26562	+	16.2059i	0.449872	+	1.38456i	−0.427179	+	0.904167i	$0.640493\pi$
$138$		0			0
$139$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$139$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$140$		0			0
$141$		0			0
$142$			28.3947i			2.38283i
$143$		0			0
$144$	−14.9508			−1.24590
$145$		0			0
$146$		0			0
$147$		0			0
$148$	4.15497	+	12.7877i	0.341537	+	1.05114i
$149$	10.0650	−	3.27033i	0.824560	−	0.267916i	0.133808	−	0.991007i	$-0.457280\pi$
$149$	10.0650	−	3.27033i	0.824560	−	0.267916i	0.690752	+	0.723092i	$0.257280\pi$
$150$		0			0
$151$	14.3775	+	19.7890i	1.17003	+	1.61040i	0.663487	+	0.748187i	$0.269076\pi$
$151$	14.3775	+	19.7890i	1.17003	+	1.61040i	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		0.809017	−	0.587785i	$-0.200000\pi$
$157$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$158$	−25.1045	−	18.2395i	−1.99720	−	1.45105i
$159$		0			0
$160$		0			0
$161$	3.98174	−	5.48039i	0.313805	−	0.431915i
$162$	−9.35622	−	12.8777i	−0.735094	−	1.01177i
$163$	6.57523	−	20.2365i	0.515012	−	1.58504i	−0.268249	−	0.963350i	$-0.586445\pi$
$163$	6.57523	−	20.2365i	0.515012	−	1.58504i	0.783260	−	0.621694i	$-0.213555\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$167$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$168$		0			0
$169$	10.5172	+	7.64121i	0.809017	+	0.587785i
$170$		0			0
$171$		0			0
$172$	−1.83837	+	2.53030i	−0.140174	+	0.192933i
$173$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$173$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$174$		0			0
$175$			13.2288i			1.00000i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−21.3570	+	15.5167i	−1.59629	+	1.15978i	−0.702118	+	0.712060i	$0.747762\pi$
$179$	−21.3570	+	15.5167i	−1.59629	+	1.15978i	−0.894176	+	0.447715i	$0.852238\pi$
$180$		0			0
$181$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$181$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$182$		0			0
$183$		0			0
$184$	−2.32080	−	3.19431i	−0.171092	−	0.235488i
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−7.34523	−	5.33662i	−0.531482	−	0.386144i	0.289430	−	0.957199i	$-0.406534\pi$
$191$	−7.34523	−	5.33662i	−0.531482	−	0.386144i	−0.820912	+	0.571055i	$0.806534\pi$
$192$		0			0
$193$	26.3479	−	8.56094i	1.89656	−	0.616230i	0.924689	−	0.380724i	$-0.124325\pi$
$193$	26.3479	−	8.56094i	1.89656	−	0.616230i	0.971873	−	0.235507i	$-0.0756750\pi$
$194$		0			0
$195$		0			0
$196$	2.44018	−	7.51009i	0.174298	−	0.536435i
$197$			27.9978i			1.99476i	0.0723369	+	0.997380i	$0.476954\pi$
$197$			27.9978i			1.99476i	−0.0723369	+	0.997380i	$0.523046\pi$
$198$		0			0
$199$		0			0		1.00000			$0$
$199$		0			0		−1.00000			$\pi$
$200$	7.33315	+	2.38269i	0.518532	+	0.168481i
$201$		0			0
$202$		0			0
$203$	−6.03887	−	18.5857i	−0.423846	−	1.30446i
$204$		0			0
$205$		0			0
$206$		0			0
$207$	2.37360	−	7.30520i	0.164977	−	0.507747i
$208$		0			0
$209$		0			0
$210$		0			0
$211$	3.07843	+	1.00024i	0.211928	+	0.0688596i	0.413057	−	0.910705i	$-0.364461\pi$
$211$	3.07843	+	1.00024i	0.211928	+	0.0688596i	−0.201129	+	0.979565i	$0.564461\pi$
$212$	−13.0605	+	9.48900i	−0.896998	+	0.651707i
$213$		0			0
$214$	−9.50212	−	29.2445i	−0.649551	−	1.99911i
$215$		0			0
$216$		0			0
$217$		0			0
$218$	7.56885	−	23.2945i	0.512627	−	1.57771i
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$223$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$224$	−12.2647	−	8.91086i	−0.819473	−	0.595382i
$225$	4.63525	+	14.2658i	0.309017	+	0.951057i
$226$	−18.2052	+	5.91521i	−1.21099	+	0.393474i
$227$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$227$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$228$		0			0
$229$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$229$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$230$		0			0
$231$		0			0
$232$	−11.3904			−0.747817
$233$	20.1301	+	6.54066i	1.31876	+	0.428493i	0.882071	−	0.471117i	$-0.156149\pi$
$233$	20.1301	+	6.54066i	1.31876	+	0.428493i	0.436694	+	0.899610i	$0.356149\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$	7.05343	+	9.70822i	0.456249	+	0.627972i	0.973726	−	0.227725i	$-0.0731287\pi$
$239$	7.05343	+	9.70822i	0.456249	+	0.627972i	−0.517477	+	0.855697i	$0.673129\pi$
$240$		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	$-0.400000\pi$
$251$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$252$		−	8.95388i		−	0.564041i
$253$		0			0
$254$	−18.2372			−1.14430
$255$		0			0
$256$	−16.2451	+	11.8027i	−1.01532	+	0.737672i
$257$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$257$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$258$		0			0
$259$	29.9916	−	9.74487i	1.86359	−	0.605517i
$260$		0			0
$261$	−13.0246	−	17.9268i	−0.806203	−	1.10964i
$262$		0			0
$263$		−	23.0900i		−	1.42379i	−0.702284	−	0.711897i	$-0.747836\pi$
$263$		−	23.0900i		−	1.42379i	0.702284	−	0.711897i	$-0.252164\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$	11.4690	+	8.33272i	0.700581	+	0.509002i
$269$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$269$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$270$		0			0
$271$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$271$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$272$		0			0
$273$		0			0
$274$			30.1374i			1.82067i
$275$		0			0
$276$		0			0
$277$	0.285717	+	0.0928350i	0.0171670	+	0.00557791i	0.317588	−	0.948229i	$-0.397127\pi$
$277$	0.285717	+	0.0928350i	0.0171670	+	0.00557791i	−0.300421	+	0.953807i	$0.597127\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	1.75113	−	0.568976i	0.104464	−	0.0339423i	−0.256319	−	0.966592i	$-0.582510\pi$
$281$	1.75113	−	0.568976i	0.104464	−	0.0339423i	0.360782	+	0.932650i	$0.382510\pi$
$282$		0			0
$283$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$283$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$284$	−5.59656	+	17.2244i	−0.332095	+	1.02208i
$285$		0			0
$286$		0			0
$287$		0			0
$288$	−16.3486	−	5.31197i	−0.963348	−	0.313011i
$289$	13.7533	−	9.99235i	0.809017	−	0.587785i
$290$		0			0
$291$		0			0
$292$		0			0
$293$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$293$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$294$		0			0
$295$		0			0
$296$		−	18.3806i		−	1.06835i
$297$		0			0
$298$	18.7175			1.08428
$299$		0			0
$300$		0			0
$301$	5.93444	+	4.31162i	0.342055	+	0.248518i
$302$	13.3686	+	41.1444i	0.769279	+	2.36760i
$303$		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	$-0.200000\pi$
$311$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$312$		0			0
$313$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$313$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$314$		0			0
$315$		0			0
$316$	−11.6336	−	16.0123i	−0.654440	−	0.900760i
$317$	6.57775	−	20.2442i	0.369443	−	1.13703i	−0.577708	−	0.816243i	$-0.696053\pi$
$317$	6.57775	−	20.2442i	0.369443	−	1.13703i	0.947152	−	0.320786i	$-0.103947\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$	9.69283	−	7.04225i	0.540160	−	0.392449i
$323$		0			0
$324$	−3.13737	−	9.65583i	−0.174298	−	0.536435i
$325$		0			0
$326$	22.1201	−	30.4457i	1.22512	−	1.68623i
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	16.1571			0.888076			0.444038	−	0.896008i	$-0.353545\pi$
$331$	16.1571			0.888076			0.444038	+	0.896008i	$0.353545\pi$
$332$		0			0
$333$	28.9284	−	21.0177i	1.58526	−	1.15176i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−20.6150	+	28.3741i	−1.12297	+	1.54564i	−0.322195	+	0.946673i	$0.604421\pi$
$337$	−20.6150	+	28.3741i	−1.12297	+	1.54564i	−0.800776	+	0.598964i	$0.795579\pi$
$338$	13.5145	+	18.6012i	0.735094	+	1.01177i
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	−17.6138	−	5.72307i	−0.951057	−	0.309017i
$344$	3.45895	−	2.51308i	0.186494	−	0.135496i
$345$		0			0
$346$		0			0
$347$	−30.7358	+	9.98667i	−1.64999	+	0.536113i	−0.978737	−	0.205120i	$-0.934242\pi$
$347$	−30.7358	+	9.98667i	−1.64999	+	0.536113i	−0.671248	+	0.741233i	$0.734242\pi$
$348$		0			0
$349$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$349$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$350$	−7.23004	+	22.2518i	−0.386462	+	1.18941i
$351$		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
$358$	−44.4045	+	14.4279i	−2.34685	+	0.762539i
$359$	2.25574	−	3.10476i	0.119054	−	0.163863i	−0.745331	−	0.666695i	$-0.767708\pi$
$359$	2.25574	−	3.10476i	0.119054	−	0.163863i	0.864384	+	0.502832i	$0.167708\pi$
$360$		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	$-0.800000\pi$
$367$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$368$	−3.94303	−	12.1354i	−0.205545	−	0.632601i
$369$		0			0
$370$		0			0
$371$	22.2551	+	30.6315i	1.15542	+	1.59031i
$372$		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
$379$	−9.74004	−	29.9768i	−0.500312	−	1.53980i	−0.808511	−	0.588481i	$-0.799726\pi$
$379$	−9.74004	−	29.9768i	−0.500312	−	1.53980i	0.308199	−	0.951322i	$-0.400274\pi$
$380$		0			0
$381$		0			0
$382$	−9.43855	−	12.9911i	−0.482918	−	0.664680i
$383$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$383$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$384$		0			0
$385$		0			0
$386$	48.9980			2.49393
$387$	7.91043	+	2.57026i	0.402110	+	0.130653i
$388$		0			0
$389$	−1.35721	−	0.986074i	−0.0688135	−	0.0499959i	0.552847	−	0.833283i	$-0.313542\pi$
$389$	−1.35721	−	0.986074i	−0.0688135	−	0.0499959i	−0.621660	+	0.783287i	$0.713542\pi$
$390$		0			0
$391$		0			0
$392$	−6.34499	+	8.73313i	−0.320471	+	0.441090i
$393$		0			0
$394$	−15.3019	+	47.0944i	−0.770899	+	2.37258i
$395$		0			0
$396$		0			0
$397$		0			0		1.00000			$0$
$397$		0			0		−1.00000			$\pi$
$398$		0			0
$399$		0			0
$400$	20.1591	+	14.6464i	1.00795	+	0.732321i
$401$	9.46725	+	29.1372i	0.472772	+	1.45504i	0.848939	+	0.528490i	$0.177242\pi$
$401$	9.46725	+	29.1372i	0.472772	+	1.45504i	−0.376168	+	0.926552i	$0.622758\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$		−	34.5631i		−	1.71534i
$407$		0			0
$408$		0			0
$409$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$409$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$	7.98516	−	10.9906i	0.392449	−	0.540160i
$415$		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
$421$	17.9284	−	13.0257i	0.873775	−	0.634834i	−0.0578225	−	0.998327i	$-0.518416\pi$
$421$	17.9284	−	13.0257i	0.873775	−	0.634834i	0.931597	+	0.363492i	$0.118416\pi$
$422$	4.63149	+	3.36497i	0.225457	+	0.163804i
$423$		0			0
$424$	20.9885	−	6.81958i	1.01929	−	0.331188i
$425$		0			0
$426$		0			0
$427$		0			0
$428$		−	19.6128i		−	0.948021i
$429$		0			0
$430$		0			0
$431$	−36.7199	−	11.9310i	−1.76874	−	0.574698i	−0.770693	−	0.637207i	$-0.780090\pi$
$431$	−36.7199	−	11.9310i	−1.76874	−	0.574698i	−0.998044	+	0.0625092i	$0.980090\pi$
$432$		0			0
$433$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$433$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$434$		0			0
$435$		0			0
$436$	9.18265	−	12.6388i	0.439769	−	0.605291i
$437$		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
$443$	4.52364	−	3.28662i	0.214925	−	0.156152i	−0.475114	−	0.879924i	$-0.657593\pi$
$443$	4.52364	−	3.28662i	0.214925	−	0.156152i	0.690039	+	0.723772i	$0.257593\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$	−0.259781	−	0.357558i	−0.0122735	−	0.0168930i
$449$	−12.6321	+	38.8775i	−0.596143	+	1.83474i	−0.0471929	+	0.998886i	$0.515028\pi$
$449$	−12.6321	+	38.8775i	−0.596143	+	1.83474i	−0.548950	+	0.835855i	$0.684972\pi$
$450$			26.5296i			1.25062i
$451$		0			0
$452$	−12.2093			−0.574276
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−7.01402	+	2.27899i	−0.328102	+	0.106607i	−0.468436	−	0.883497i	$-0.655182\pi$
$457$	−7.01402	+	2.27899i	−0.328102	+	0.106607i	0.140334	+	0.990104i	$0.455182\pi$
$458$		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
$463$	−23.0299			−1.07029			−0.535145	−	0.844760i	$-0.679743\pi$
$463$	−23.0299			−1.07029			−0.535145	+	0.844760i	$0.679743\pi$
$464$	−35.0085	−	11.3750i	−1.62523	−	0.528069i
$465$		0			0
$466$	30.2856	+	22.0038i	1.40295	+	1.01930i
$467$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$467$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$468$		0			0
$469$	19.5432	−	26.8989i	0.902419	−	1.24207i
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$	34.7328	+	25.2349i	1.59031	+	1.15542i
$478$	6.55848	+	20.1849i	0.299978	+	0.923237i
$479$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$479$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$		0			0
$485$		0			0
$486$		0			0
$487$	33.3220	−	24.2099i	1.50996	−	1.09705i	0.543772	−	0.839233i	$-0.316996\pi$
$487$	33.3220	−	24.2099i	1.50996	−	1.09705i	0.966193	−	0.257821i	$-0.0830043\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	3.11027	−	4.28092i	0.140364	−	0.193195i	−0.733047	−	0.680178i	$-0.761903\pi$
$491$	3.11027	−	4.28092i	0.140364	−	0.193195i	0.873412	+	0.486983i	$0.161903\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$	40.3974	+	13.1259i	1.81207	+	0.588777i
$498$		0			0
$499$	−36.1436	−	26.2599i	−1.61801	−	1.17555i	−0.812219	−	0.583352i	$-0.801741\pi$
$499$	−36.1436	−	26.2599i	−1.61801	−	1.17555i	−0.805791	−	0.592200i	$-0.798259\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$503$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$504$	−3.78240	+	11.6410i	−0.168481	+	0.518532i
$505$		0			0
$506$		0			0
$507$		0			0
$508$	−11.0628	−	3.59452i	−0.490833	−	0.159481i
$509$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$509$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$510$		0			0
$511$		0			0
$512$	−12.5400	+	4.07450i	−0.554196	+	0.180069i
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$	55.7742			2.45058
$519$		0			0
$520$		0			0
$521$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$521$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$522$	−12.1106	−	37.2727i	−0.530069	−	1.63138i
$523$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$523$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$524$		0			0
$525$		0			0
$526$	12.6196	−	38.8392i	0.550242	−	1.69347i
$527$		0			0
$528$		0			0
$529$	−16.4444			−0.714976
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$	−11.3910	−	15.6783i	−0.492014	−	0.677200i
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	40.0841	+	13.0241i	1.72335	+	0.559950i	0.992463	−	0.122548i	$-0.0391066\pi$
$541$	40.0841	+	13.0241i	1.72335	+	0.559950i	0.730887	+	0.682498i	$0.239107\pi$
$542$		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	$-0.310215\pi$
$547$	−9.33080	−	12.8427i	−0.398956	−	0.549116i	−0.960482	+	0.278343i	$0.910215\pi$
$548$	−5.94005	+	18.2816i	−0.253747	+	0.780951i
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$	−37.5544	+	27.2849i	−1.59698	+	1.16027i
$554$	0.429859	+	0.312311i	0.0182630	+	0.0132688i
$555$		0			0
$556$		0			0
$557$	−10.8601	+	14.9476i	−0.460157	+	0.633352i	−0.974541	−	0.224208i	$-0.928020\pi$
$557$	−10.8601	+	14.9476i	−0.460157	+	0.633352i	0.514384	+	0.857560i	$0.328020\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$	3.25650			0.137367
$563$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$563$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	−22.6463	+	7.35824i	−0.951057	+	0.309017i
$568$	14.5523	−	20.0295i	0.610600	−	0.840419i
$569$	24.8821	+	34.2473i	1.04311	+	1.43572i	0.894630	+	0.446808i	$0.147439\pi$
$569$	24.8821	+	34.2473i	1.04311	+	1.43572i	0.148483	+	0.988915i	$0.452561\pi$
$570$		0			0
$571$		−	10.9123i		−	0.456664i	−0.973583	−	0.228332i	$-0.926673\pi$
$571$		−	10.9123i		−	0.456664i	0.973583	−	0.228332i	$-0.0733271\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−10.3570	+	7.52477i	−0.431915	+	0.313805i
$576$	−0.405433	−	0.294564i	−0.0168930	−	0.0122735i
$577$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$577$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$578$	28.5953	−	9.29117i	1.18941	−	0.386462i
$579$		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	$-0.800000\pi$
$587$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$	18.3557	−	56.4929i	0.754413	−	2.32184i
$593$		0			0			−	1.00000i	$-0.5\pi$
$593$		0			0				1.00000i	$0.5\pi$
$594$		0			0
$595$		0			0
$596$	11.3542	+	3.68920i	0.465086	+	0.151116i
$597$		0			0
$598$		0			0
$599$	−13.9417	−	42.9080i	−0.569641	−	1.75317i	−0.653742	−	0.756717i	$-0.726802\pi$
$599$	−13.9417	−	42.9080i	−0.569641	−	1.75317i	0.0841014	−	0.996457i	$-0.473198\pi$
$600$		0			0
$601$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$601$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$602$	7.62570	+	10.4959i	0.310800	+	0.427780i
$603$	11.6501	−	35.8554i	0.474430	−	1.46015i
$604$			27.5935i			1.12276i
$605$		0			0
$606$		0			0
$607$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$607$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	−28.2262	−	38.8501i	−1.14005	−	1.56914i	−0.767403	−	0.641165i	$-0.778451\pi$
$613$	−28.2262	−	38.8501i	−1.14005	−	1.56914i	−0.372644	−	0.927974i	$-0.621549\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−3.84770			−0.154902			−0.0774512	−	0.996996i	$-0.524678\pi$
$617$	−3.84770			−0.154902			−0.0774512	+	0.996996i	$0.524678\pi$
$618$		0			0
$619$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$619$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$620$		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
$631$	−40.6425	−	29.5285i	−1.61795	−	1.17551i	−0.814832	−	0.579698i	$-0.803171\pi$
$631$	−40.6425	−	29.5285i	−1.61795	−	1.17551i	−0.803122	−	0.595815i	$-0.796829\pi$
$632$	8.36086	+	25.7321i	0.332577	+	1.02357i
$633$		0			0
$634$	22.1285	−	30.4573i	0.878837	−	1.20961i
$635$		0			0
$636$		0			0
$637$		0			0
$638$		0			0
$639$	48.1636			1.90532
$640$		0			0
$641$	20.0424	−	14.5616i	0.791625	−	0.575150i	−0.116820	−	0.993153i	$-0.537270\pi$
$641$	20.0424	−	14.5616i	0.791625	−	0.575150i	0.908445	+	0.418004i	$0.137270\pi$
$642$		0			0
$643$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$643$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$644$	7.26777	−	2.36144i	0.286390	−	0.0930538i
$645$		0			0
$646$		0			0
$647$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$647$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$648$			13.8790i			0.545217i
$649$		0			0
$650$		0			0
$651$		0			0
$652$	19.4190	−	14.1087i	0.760507	−	0.552541i
$653$	20.6428	+	14.9979i	0.807815	+	0.586912i	0.913196	−	0.407520i	$-0.133606\pi$
$653$	20.6428	+	14.9979i	0.807815	+	0.586912i	−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			$\pi$
$662$	27.1775	+	8.83051i	1.05628	+	0.343207i
$663$		0			0
$664$		0			0
$665$		0			0
$666$	60.1467	−	19.5428i	2.33064	−	0.757269i
$667$	11.1160	−	15.2998i	0.430413	−	0.592412i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	15.8006	+	5.13394i	0.609070	+	0.197899i	0.597281	−	0.802032i	$-0.296248\pi$
$673$	15.8006	+	5.13394i	0.609070	+	0.197899i	0.0117883	+	0.999931i	$0.496248\pi$
$674$	−50.1836	+	36.4605i	−1.93300	+	1.40441i
$675$		0			0
$676$	4.53176	+	13.9473i	0.174298	+	0.536435i
$677$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$677$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−11.0364			−0.422295			−0.211147	−	0.977454i	$-0.567720\pi$
$683$	−11.0364			−0.422295			−0.211147	+	0.977454i	$0.567720\pi$
$684$		0			0
$685$		0			0
$686$	−26.4999	−	19.2533i	−1.01177	−	0.735094i
$687$		0			0
$688$	13.1408	−	4.26970i	0.500988	−	0.162781i
$689$		0			0
$690$		0			0
$691$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$691$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$692$		0			0
$693$		0			0
$694$	−57.1581			−2.16969
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	−8.77160	+	12.0731i	−0.331535	+	0.456319i
$701$	−8.49322	−	11.6899i	−0.320785	−	0.441522i	0.617922	−	0.786239i	$-0.287975\pi$
$701$	−8.49322	−	11.6899i	−0.320785	−	0.441522i	−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
$709$	−11.1113	−	34.1969i	−0.417292	−	1.28429i	−0.910185	−	0.414202i	$-0.864061\pi$
$709$	−11.1113	−	34.1969i	−0.417292	−	1.28429i	0.492893	−	0.870090i	$-0.335939\pi$
$710$		0			0
$711$	−30.9381	+	42.5827i	−1.16027	+	1.59698i
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$	−29.7799			−1.11293
$717$		0			0
$718$	5.49120	−	3.98959i	0.204930	−	0.148890i
$719$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$719$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$720$		0			0
$721$		0			0
$722$	−19.7520	+	27.1863i	−0.735094	+	1.01177i
$723$		0			0
$724$		0			0
$725$			36.9313i			1.37159i
$726$		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	$-0.300000\pi$
$733$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$734$		0			0
$735$		0			0
$736$		−	14.6709i		−	0.540777i
$737$		0			0
$738$		0			0
$739$	−41.2831	−	13.4137i	−1.51862	−	0.493430i	−0.573238	−	0.819389i	$-0.694313\pi$
$739$	−41.2831	−	13.4137i	−1.51862	−	0.493430i	−0.945384	+	0.325959i	$0.894313\pi$
$740$		0			0
$741$		0			0
$742$	20.6934	+	63.6877i	0.759678	+	2.33805i
$743$	−6.13907	+	1.99470i	−0.225221	+	0.0731786i	−0.419453	−	0.907777i	$-0.637778\pi$
$743$	−6.13907	+	1.99470i	−0.225221	+	0.0731786i	0.194233	+	0.980955i	$0.437778\pi$
$744$		0			0
$745$		0			0
$746$	17.3521	−	53.4042i	0.635305	−	1.95527i
$747$		0			0
$748$		0			0
$749$	−45.9989			−1.68076
$750$		0			0
$751$	−32.3570	+	23.5087i	−1.18072	+	0.857845i	−0.992253	−	0.124234i	$-0.960353\pi$
$751$	−32.3570	+	23.5087i	−1.18072	+	0.857845i	−0.188469	+	0.982079i	$0.560353\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−15.4223	+	47.4648i	−0.560531	+	1.72514i	0.120338	+	0.992733i	$0.461602\pi$
$757$	−15.4223	+	47.4648i	−0.560531	+	1.72514i	−0.680869	+	0.732405i	$0.738398\pi$
$758$		−	55.7465i		−	2.02480i
$759$		0			0
$760$		0			0
$761$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$761$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$762$		0			0
$763$	−29.6425	−	21.5366i	−1.07313	−	0.779676i
$764$	−3.16498	−	9.74080i	−0.114505	−	0.352410i
$765$		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
$772$	29.7226	+	9.65745i	1.06974	+	0.347579i
$773$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$773$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$774$	11.9012	+	8.64673i	0.427780	+	0.310800i
$775$		0			0
$776$		0			0
$777$		0			0
$778$	−1.74401	−	2.40042i	−0.0625258	−	0.0860593i
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	−28.2227	+	20.5050i	−1.00795	+	0.732321i
$785$		0			0
$786$		0			0
$787$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$787$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$788$	−18.5645	+	25.5519i	−0.661333	+	0.910247i
$789$		0			0
$790$		0			0
$791$			28.6351i			1.01815i
$792$		0			0
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$797$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$798$		0			0
$799$		0			0
$800$	16.8399	+	23.1782i	0.595382	+	0.819473i
$801$		0			0
$802$			54.1852i			1.91334i
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−46.8124	+	15.2103i	−1.64584	+	0.534765i	−0.977832	−	0.209393i	$-0.932851\pi$
$809$	−46.8124	+	15.2103i	−1.64584	+	0.534765i	−0.668004	+	0.744157i	$0.732851\pi$
$810$		0			0
$811$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$811$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$812$	6.81235	−	20.9663i	0.239067	−	0.735771i
$813$		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	−0.232162	+	0.972677i	$0.574580\pi$
$821$	−31.1027	+	42.8092i	−1.08549	+	1.49405i	−0.853329	+	0.521373i	$0.825420\pi$
$822$		0			0
$823$	0.650129	−	2.00089i	0.0226621	−	0.0697467i	−0.939086	−	0.343683i	$-0.888326\pi$
$823$	0.650129	−	2.00089i	0.0226621	−	0.0697467i	0.961748	+	0.273936i	$0.0883256\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−35.2276	−	11.4461i	−1.22498	−	0.398022i	−0.376090	−	0.926583i	$-0.622732\pi$
$827$	−35.2276	−	11.4461i	−1.22498	−	0.398022i	−0.848895	+	0.528562i	$0.822732\pi$
$828$	7.01011	−	5.09314i	0.243618	−	0.176999i
$829$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$829$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$830$		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	$-0.200000\pi$
$839$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$840$		0			0
$841$	−7.89750	−	24.3060i	−0.272328	−	0.838138i
$842$	37.2759	−	12.1117i	1.28461	−	0.417396i
$843$		0			0
$844$	2.14626	+	2.95408i	0.0738774	+	0.101684i
$845$		0			0
$846$		0			0
$847$		0			0
$848$	71.3188			2.44910
$849$		0			0
$850$		0			0
$851$	24.6892	+	17.9378i	0.846335	+	0.614899i
$852$		0			0
$853$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$853$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$854$		0			0
$855$		0			0
$856$	−8.28506	+	25.4988i	−0.283178	+	0.871531i
$857$		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
$862$	−55.2449	−	40.1378i	−1.88165	−	1.36710i
$863$	2.47214	+	7.60845i	0.0841525	+	0.258995i	0.984275	−	0.176642i	$-0.0565234\pi$
$863$	2.47214	+	7.60845i	0.0841525	+	0.258995i	−0.900123	+	0.435636i	$0.856523\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$	−17.2775	+	12.5528i	−0.585090	+	0.425093i
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	2.17702	+	2.99641i	0.0735128	+	0.101182i	0.844190	−	0.536044i	$-0.180082\pi$
$877$	2.17702	+	2.99641i	0.0735128	+	0.101182i	−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			$\pi$
$882$	−35.3236	−	11.4773i	−1.18941	−	0.386462i
$883$	9.70820	−	7.05342i	0.326707	−	0.237367i	−0.412325	−	0.911037i	$-0.635283\pi$
$883$	9.70820	−	7.05342i	0.326707	−	0.237367i	0.739032	+	0.673670i	$0.235283\pi$
$884$		0			0
$885$		0			0
$886$	9.40538	−	3.05599i	0.315980	−	0.102668i
$887$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$887$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$888$		0			0
$889$	−8.43043	+	25.9462i	−0.282747	+	0.870207i
$890$		0			0
$891$		0			0
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$	9.12788	+	28.0927i	0.304941	+	0.938512i
$897$		0			0
$898$	−42.4961	+	58.4909i	−1.41811	+	1.95187i
$899$		0			0
$900$	−5.22895	+	16.0931i	−0.174298	+	0.536435i
$901$		0			0
$902$		0			0
$903$		0			0
$904$	15.8734	+	5.15758i	0.527941	+	0.171539i
$905$		0			0
$906$		0			0
$907$	15.9611	+	49.1233i	0.529980	+	1.63111i	0.754252	+	0.656585i	$0.228000\pi$
$907$	15.9611	+	49.1233i	0.529980	+	1.63111i	−0.224271	+	0.974527i	$0.572000\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−4.94427	+	15.2169i	−0.163811	+	0.504159i	−0.998947	−	0.0458855i	$-0.985389\pi$
$911$	−4.94427	+	15.2169i	−0.163811	+	0.504159i	0.835136	+	0.550044i	$0.185389\pi$
$912$		0			0
$913$		0			0
$914$	−13.0437			−0.431446
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	−55.3326	+	17.9786i	−1.82525	+	0.593061i	−0.825671	+	0.564152i	$0.809203\pi$
$919$	−55.3326	+	17.9786i	−1.82525	+	0.593061i	−0.999582	+	0.0289084i	$0.990797\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$	−59.5957			−1.95949
$926$	−38.7380	−	12.5867i	−1.27301	−	0.413626i
$927$		0			0
$928$	−34.2400	−	24.8768i	−1.12398	−	0.816622i
$929$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$929$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$930$		0			0
$931$		0			0
$932$	14.0346	+	19.3169i	0.459717	+	0.632746i
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$937$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$938$	47.5744	−	34.5648i	1.55336	−	1.12858i
$939$		0			0
$940$		0			0
$941$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$941$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$942$		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
$953$	28.5785	−	39.3349i	0.925748	−	1.27418i	−0.0357473	−	0.999361i	$-0.511381\pi$
$953$	28.5785	−	39.3349i	0.925748	−	1.27418i	0.961495	−	0.274822i	$-0.0886189\pi$
$954$	44.6314	+	61.4298i	1.44499	+	1.98886i
$955$		0			0
$956$			13.5370i			0.437818i
$957$		0			0
$958$		0			0
$959$	42.8768	+	13.9315i	1.38456	+	0.449872i
$960$		0			0
$961$	−25.0795	−	18.2213i	−0.809017	−	0.587785i
$962$		0			0
$963$	−49.6051	+	16.1177i	−1.59850	+	0.519385i
$964$		0			0
$965$		0			0
$966$		0			0
$967$		−	52.7587i		−	1.69661i	−0.529511	−	0.848303i	$-0.677624\pi$
$967$		−	52.7587i		−	1.69661i	0.529511	−	0.848303i	$-0.322376\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$971$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$972$		0			0
$973$		0			0
$974$	69.2818	−	22.5110i	2.21993	−	0.721300i
$975$		0			0
$976$		0			0
$977$	−16.8337	+	51.8087i	−0.538557	+	1.65751i	0.197278	+	0.980348i	$0.436790\pi$
$977$	−16.8337	+	51.8087i	−0.538557	+	1.65751i	−0.735835	+	0.677161i	$0.763210\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	−39.5127	−	12.8384i	−1.26154	−	0.409900i
$982$	7.57139	−	5.50094i	0.241613	−	0.175542i
$983$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$983$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$			7.09868i			0.225725i
$990$		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
$994$	60.7776	+	44.1575i	1.92775	+	1.40059i
$995$		0			0
$996$		0			0
$997$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$997$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$998$	−46.4442	−	63.9250i	−1.47017	−	2.02351i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	847.2.l.d.475.2		8
7.6	odd	2	CM	847.2.l.d.475.2		8
11.2	odd	10		77.2.l.a.41.2	✓	8
11.3	even	5		77.2.l.a.62.2	yes	8
11.4	even	5		847.2.b.b.846.7		8
11.5	even	5		847.2.l.a.699.1		8
11.6	odd	10	inner	847.2.l.d.699.2		8
11.7	odd	10		847.2.b.b.846.2		8
11.8	odd	10		847.2.l.c.524.1		8
11.9	even	5		847.2.l.c.118.1		8
11.10	odd	2		847.2.l.a.475.1		8
33.2	even	10		693.2.bu.a.118.1		8
33.14	odd	10		693.2.bu.a.370.1		8
77.2	odd	30		539.2.s.a.129.2		16
77.3	odd	30		539.2.s.a.117.2		16
77.6	even	10	inner	847.2.l.d.699.2		8
77.13	even	10		77.2.l.a.41.2	✓	8
77.20	odd	10		847.2.l.c.118.1		8
77.24	even	30		539.2.s.a.19.1		16
77.25	even	15		539.2.s.a.117.2		16
77.27	odd	10		847.2.l.a.699.1		8
77.41	even	10		847.2.l.c.524.1		8
77.46	odd	30		539.2.s.a.19.1		16
77.47	odd	30		539.2.s.a.227.1		16
77.48	odd	10		847.2.b.b.846.7		8
77.58	even	15		539.2.s.a.227.1		16
77.62	even	10		847.2.b.b.846.2		8
77.68	even	30		539.2.s.a.129.2		16
77.69	odd	10		77.2.l.a.62.2	yes	8
77.76	even	2		847.2.l.a.475.1		8
231.146	even	10		693.2.bu.a.370.1		8
231.167	odd	10		693.2.bu.a.118.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
77.2.l.a.41.2	✓	8	11.2	odd	10
77.2.l.a.41.2	✓	8	77.13	even	10
77.2.l.a.62.2	yes	8	11.3	even	5
77.2.l.a.62.2	yes	8	77.69	odd	10
539.2.s.a.19.1		16	77.24	even	30
539.2.s.a.19.1		16	77.46	odd	30
539.2.s.a.117.2		16	77.3	odd	30
539.2.s.a.117.2		16	77.25	even	15
539.2.s.a.129.2		16	77.2	odd	30
539.2.s.a.129.2		16	77.68	even	30
539.2.s.a.227.1		16	77.47	odd	30
539.2.s.a.227.1		16	77.58	even	15
693.2.bu.a.118.1		8	33.2	even	10
693.2.bu.a.118.1		8	231.167	odd	10
693.2.bu.a.370.1		8	33.14	odd	10
693.2.bu.a.370.1		8	231.146	even	10
847.2.b.b.846.2		8	11.7	odd	10
847.2.b.b.846.2		8	77.62	even	10
847.2.b.b.846.7		8	11.4	even	5
847.2.b.b.846.7		8	77.48	odd	10
847.2.l.a.475.1		8	11.10	odd	2
847.2.l.a.475.1		8	77.76	even	2
847.2.l.a.699.1		8	11.5	even	5
847.2.l.a.699.1		8	77.27	odd	10
847.2.l.c.118.1		8	11.9	even	5
847.2.l.c.118.1		8	77.20	odd	10
847.2.l.c.524.1		8	11.8	odd	10
847.2.l.c.524.1		8	77.41	even	10
847.2.l.d.475.2		8	1.1	even	1	trivial
847.2.l.d.475.2		8	7.6	odd	2	CM
847.2.l.d.699.2		8	11.6	odd	10	inner
847.2.l.d.699.2		8	77.6	even	10	inner

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)

\(f(q)\)	\(=\)	\(q+(1.68208 + 0.546539i) q^{2} +(0.912638 + 0.663070i) q^{4} +(1.55513 - 2.14046i) q^{7} +(-0.906428 - 1.24759i) q^{8} +(0.927051 - 2.85317i) q^{9} +O(q^{10})\)	\(q+(1.68208 + 0.546539i) q^{2} +(0.912638 + 0.663070i) q^{4} +(1.55513 - 2.14046i) q^{7} +(-0.906428 - 1.24759i) q^{8} +(0.927051 - 2.85317i) q^{9} +(3.78570 - 2.75047i) q^{14} +(-1.54002 - 4.73968i) q^{16} +(3.11874 - 4.29258i) q^{18} +2.56038 q^{23} +(-4.04508 + 2.93893i) q^{25} +(2.83855 - 0.922300i) q^{28} +(4.34154 - 5.97561i) q^{29} -5.72996i q^{32} +(2.73791 - 1.98921i) q^{36} +(9.64279 + 7.00589i) q^{37} +2.77251i q^{43} +(4.30676 + 1.39935i) q^{46} +(-2.16312 - 6.65740i) q^{49} +(-8.41038 + 2.73270i) q^{50} +(-4.42225 + 13.6103i) q^{53} -4.08003 q^{56} +(10.5687 - 7.67861i) q^{58} +(-4.66540 - 6.42137i) q^{63} +(0.0516205 - 0.158872i) q^{64} +12.5669 q^{67} +(4.96113 + 15.2688i) q^{71} +(-4.39989 + 1.42961i) q^{72} +(12.3909 + 17.0546i) q^{74} +(-16.6863 - 5.42171i) q^{79} +(-7.28115 - 5.29007i) q^{81} +(-1.51528 + 4.66357i) q^{86} +(2.33670 + 1.69771i) q^{92} -12.3805i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 5 q^{2} - 5 q^{4} - 25 q^{8} - 6 q^{9}+O(q^{10}) \) 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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