LMFDB - Embedded newform 847.2.b.b.846.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [847,2,Mod(846,847)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(847, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("847.846");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 847 = 7 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	847.b (of order $2$, degree $1$, 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:	$8$
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:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2^{4}\cdot 11^{2} $
Twist minimal:	no (minimal twist has level 77)
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			846.1
Root			$-0.373058 - 1.36412i$ of defining polynomial
Character	$\chi$	$=$	847.846
Dual form			847.2.b.b.846.8

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.72824i q^{2} -5.44331 q^{4} +2.64575i q^{7} +9.39419i q^{8} +3.00000 q^{9} +O(q^{10})$	$q-2.72824i q^{2} -5.44331 q^{4} +2.64575i q^{7} +9.39419i q^{8} +3.00000 q^{9} +7.21825 q^{14} +14.7430 q^{16} -8.18473i q^{18} +3.36187 q^{23} +5.00000 q^{25} -14.4016i q^{28} -5.17244i q^{29} -21.4341i q^{32} -16.3299 q^{36} +11.0746 q^{37} +13.0478i q^{43} -9.17199i q^{46} -7.00000 q^{49} -13.6412i q^{50} -6.97487 q^{53} -24.8547 q^{56} -14.1117 q^{58} +7.93725i q^{63} -28.9915 q^{64} +13.8615 q^{67} +0.0882461 q^{71} +28.1826i q^{72} -30.2143i q^{74} -2.70296i q^{79} +9.00000 q^{81} +35.5977 q^{86} -18.2997 q^{92} +19.0977i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 10 q^{4} + 24 q^{9}+O(q^{10}) $ 8 * q - 10 * q^4 + 24 * q^9	$ 8 q - 10 q^{4} + 24 q^{9} + 14 q^{14} + 18 q^{16} + 16 q^{23} + 40 q^{25} - 30 q^{36} + 12 q^{37} - 56 q^{49} - 20 q^{53} - 42 q^{56} - 56 q^{58} - 54 q^{64} + 8 q^{67} + 32 q^{71} + 72 q^{81} + 28 q^{86} - 80 q^{92}+O(q^{100}) $ 8 * q - 10 * q^4 + 24 * q^9 + 14 * q^14 + 18 * q^16 + 16 * q^23 + 40 * q^25 - 30 * q^36 + 12 * q^37 - 56 * q^49 - 20 * q^53 - 42 * q^56 - 56 * q^58 - 54 * q^64 + 8 * q^67 + 32 * q^71 + 72 * q^81 + 28 * q^86 - 80 * q^92

Display 100 coefficients 10 coefficients

Character values

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

$n$	$122$	$365$
$\chi(n)$	$-1$	$-1$

Coefficient data

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

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

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

$2$	− 2.72824i	− 1.92916i	−0.263792	−	0.964580i	$-0.584973\pi$
$2$	− 2.72824i	− 1.92916i	0.263792	−	0.964580i	$-0.415027\pi$
$3$	0	0	1.00000			$0$
$3$	0	0	−1.00000			$\pi$
$4$	−5.44331	−2.72166
$5$	0	0	1.00000			$0$
$5$	0	0	−1.00000			$\pi$
$6$	0	0
$7$	2.64575i	1.00000i
$7$	2.64575i	1.00000i
$8$	9.39419i	3.32135i
$9$	3.00000	1.00000
$10$	0	0
$11$	0	0
$11$	0	0
$12$	0	0
$13$	0	0		−	1.00000i	$-0.5\pi$
$13$	0	0			1.00000i	$0.5\pi$
$14$	7.21825	1.92916
$15$	0	0
$16$	14.7430	3.68575
$17$	0	0		−	1.00000i	$-0.5\pi$
$17$	0	0			1.00000i	$0.5\pi$
$18$	− 8.18473i	− 1.92916i
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	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
$25$	5.00000	1.00000
$26$	0	0
$27$	0	0
$28$	− 14.4016i	− 2.72166i
$29$	− 5.17244i	− 0.960498i	−0.877132	−	0.480249i	$-0.840546\pi$
$29$	− 5.17244i	− 0.960498i	0.877132	−	0.480249i	$-0.159454\pi$
$30$	0	0
$31$	0	0	1.00000			$0$
$31$	0	0	−1.00000			$\pi$
$32$	− 21.4341i	− 3.78905i
$33$	0	0
$34$	0	0
$35$	0	0
$36$	−16.3299	−2.72166
$37$	11.0746	1.82066	0.910330	−	0.413884i	$-0.135828\pi$
$37$	11.0746	1.82066	0.910330	+	0.413884i	$0.135828\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	0	0		−	1.00000i	$-0.5\pi$
$41$	0	0			1.00000i	$0.5\pi$
$42$	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
$46$	− 9.17199i	− 1.35234i
$47$	0	0	1.00000			$0$
$47$	0	0	−1.00000			$\pi$
$48$	0	0
$49$	−7.00000	−1.00000
$50$	− 13.6412i	− 1.92916i
$51$	0	0
$52$	0	0
$53$	−6.97487	−0.958072	−0.479036	−	0.877795i	$-0.659014\pi$
$53$	−6.97487	−0.958072	−0.479036	+	0.877795i	$0.659014\pi$
$54$	0	0
$55$	0	0
$56$	−24.8547	−3.32135
$57$	0	0
$58$	−14.1117	−1.85295
$59$	0	0	1.00000			$0$
$59$	0	0	−1.00000			$\pi$
$60$	0	0
$61$	0	0		−	1.00000i	$-0.5\pi$
$61$	0	0			1.00000i	$0.5\pi$
$62$	0	0
$63$	7.93725i	1.00000i
$64$	−28.9915	−3.62394
$65$	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
$71$	0.0882461	0.0104729	0.00523645	−	0.999986i	$-0.498333\pi$
$71$	0.0882461	0.0104729	0.00523645	+	0.999986i	$0.498333\pi$
$72$	28.1826i	3.32135i
$73$	0	0		−	1.00000i	$-0.5\pi$
$73$	0	0			1.00000i	$0.5\pi$
$74$	− 30.2143i	− 3.51234i
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	− 2.70296i	− 0.304107i	−0.988372	−	0.152053i	$-0.951411\pi$
$79$	− 2.70296i	− 0.304107i	0.988372	−	0.152053i	$-0.0485886\pi$
$80$	0	0
$81$	9.00000	1.00000
$82$	0	0
$83$	0	0		−	1.00000i	$-0.5\pi$
$83$	0	0			1.00000i	$0.5\pi$
$84$	0	0
$85$	0	0
$86$	35.5977	3.83860
$87$	0	0
$88$	0	0
$89$	0	0	1.00000			$0$
$89$	0	0	−1.00000			$\pi$
$90$	0	0
$91$	0	0
$92$	−18.2997	−1.90787
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	0	0	1.00000			$0$
$97$	0	0	−1.00000			$\pi$
$98$	19.0977i	1.92916i
$99$	0	0
$100$	−27.2166	−2.72166
$101$	0	0		−	1.00000i	$-0.5\pi$
$101$	0	0			1.00000i	$0.5\pi$
$102$	0	0
$103$	0	0	1.00000			$0$
$103$	0	0	−1.00000			$\pi$
$104$	0	0
$105$	0	0
$106$	19.0291i	1.84827i
$107$	− 7.47479i	− 0.722615i	−0.932447	−	0.361308i	$-0.882330\pi$
$107$	− 7.47479i	− 0.722615i	0.932447	−	0.361308i	$-0.117670\pi$
$108$	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
$112$	39.0063i	3.68575i
$113$	20.7481	1.95182	0.975909	−	0.218179i	$-0.0700116\pi$
$113$	20.7481	1.95182	0.975909	+	0.218179i	$0.0700116\pi$
$114$	0	0
$115$	0	0
$116$	28.1552i	2.61414i
$117$	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
$126$	21.6548	1.92916
$127$	3.43818i	0.305089i	0.988297	+	0.152545i	$0.0487468\pi$
$127$	3.43818i	0.305089i	−0.988297	+	0.152545i	$0.951253\pi$
$128$	36.2276i	3.20210i
$129$	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
$134$	− 37.8175i	− 3.26693i
$135$	0	0
$136$	0	0
$137$	−4.35090	−0.371722	−0.185861	−	0.982576i	$-0.559507\pi$
$137$	−4.35090	−0.371722	−0.185861	+	0.982576i	$0.559507\pi$
$138$	0	0
$139$	0	0		−	1.00000i	$-0.5\pi$
$139$	0	0			1.00000i	$0.5\pi$
$140$	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
$148$	−60.2827	−4.95521
$149$	− 10.5830i	− 0.866994i	−0.901155	−	0.433497i	$-0.857280\pi$
$149$	− 10.5830i	− 0.866994i	0.901155	−	0.433497i	$-0.142720\pi$
$150$	0	0
$151$	18.3878i	1.49637i	0.663487	+	0.748187i	$0.269076\pi$
$151$	18.3878i	1.49637i	−0.663487	+	0.748187i	$0.730924\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	0	0	1.00000			$0$
$157$	0	0	−1.00000			$\pi$
$158$	−7.37433	−0.586670
$159$	0	0
$160$	0	0
$161$	8.89467i	0.700998i
$162$	− 24.5542i	− 1.92916i
$163$	−25.5111	−1.99819	−0.999093	−	0.0425718i	$-0.986445\pi$
$163$	−25.5111	−1.99819	−0.999093	+	0.0425718i	$0.986445\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	0	0		−	1.00000i	$-0.5\pi$
$167$	0	0			1.00000i	$0.5\pi$
$168$	0	0
$169$	−13.0000	−1.00000
$170$	0	0
$171$	0	0
$172$	− 71.0235i	− 5.41549i
$173$	0	0		−	1.00000i	$-0.5\pi$
$173$	0	0			1.00000i	$0.5\pi$
$174$	0	0
$175$	13.2288i	1.00000i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−18.7874	−1.40424	−0.702118	−	0.712060i	$-0.747762\pi$
$179$	−18.7874	−1.40424	−0.702118	+	0.712060i	$0.747762\pi$
$180$	0	0
$181$	0	0	1.00000			$0$
$181$	0	0	−1.00000			$\pi$
$182$	0	0
$183$	0	0
$184$	31.5820i	2.32826i
$185$	0	0
$186$	0	0
$187$	0	0
$188$	0	0
$189$	0	0
$190$	0	0
$191$	27.6347	1.99958	0.999789	−	0.0205267i	$-0.00653431\pi$
$191$	27.6347	1.99958	0.999789	+	0.0205267i	$0.00653431\pi$
$192$	0	0
$193$	− 10.5784i	− 0.761447i	−0.924689	−	0.380724i	$-0.875675\pi$
$193$	− 10.5784i	− 0.761447i	0.924689	−	0.380724i	$-0.124325\pi$
$194$	0	0
$195$	0	0
$196$	38.1032	2.72166
$197$	− 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			$\pi$
$200$	46.9709i	3.32135i
$201$	0	0
$202$	0	0
$203$	13.6850	0.960498
$204$	0	0
$205$	0	0
$206$	0	0
$207$	10.0856	0.700998
$208$	0	0
$209$	0	0
$210$	0	0
$211$	14.3512i	0.987974i	0.869469	+	0.493987i	$0.164461\pi$
$211$	14.3512i	0.987974i	−0.869469	+	0.493987i	$0.835539\pi$
$212$	37.9664	2.60754
$213$	0	0
$214$	−20.3930	−1.39404
$215$	0	0
$216$	0	0
$217$	0	0
$218$	5.50642	0.372942
$219$	0	0
$220$	0	0
$221$	0	0
$222$	0	0
$223$	0	0	1.00000			$0$
$223$	0	0	−1.00000			$\pi$
$224$	56.7094	3.78905
$225$	15.0000	1.00000
$226$	− 56.6059i	− 3.76537i
$227$	0	0		−	1.00000i	$-0.5\pi$
$227$	0	0			1.00000i	$0.5\pi$
$228$	0	0
$229$	0	0	1.00000			$0$
$229$	0	0	−1.00000			$\pi$
$230$	0	0
$231$	0	0
$232$	48.5909	3.19015
$233$	21.1660i	1.38663i	0.720634	+	0.693316i	$0.243851\pi$
$233$	21.1660i	1.38663i	−0.720634	+	0.693316i	$0.756149\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0	0
$239$	23.3927i	1.51315i	0.653907	+	0.756575i	$0.273129\pi$
$239$	23.3927i	1.51315i	−0.653907	+	0.756575i	$0.726871\pi$
$240$	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	1.00000			$0$
$251$	0	0	−1.00000			$\pi$
$252$	− 43.2049i	− 2.72166i
$253$	0	0
$254$	9.38020	0.588566
$255$	0	0
$256$	40.8547	2.55342
$257$	0	0	1.00000			$0$
$257$	0	0	−1.00000			$\pi$
$258$	0	0
$259$	29.3007i	1.82066i
$260$	0	0
$261$	− 15.5173i	− 0.960498i
$262$	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
$268$	−75.4524	−4.60899
$269$	0	0	1.00000			$0$
$269$	0	0	−1.00000			$\pi$
$270$	0	0
$271$	0	0		−	1.00000i	$-0.5\pi$
$271$	0	0			1.00000i	$0.5\pi$
$272$	0	0
$273$	0	0
$274$	11.8703i	0.717112i
$275$	0	0
$276$	0	0
$277$	− 19.8076i	− 1.19013i	−0.803679	−	0.595063i	$-0.797127\pi$
$277$	− 19.8076i	− 1.19013i	0.803679	−	0.595063i	$-0.202873\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	31.2681i	1.86530i	0.360782	+	0.932650i	$0.382510\pi$
$281$	31.2681i	1.86530i	−0.360782	+	0.932650i	$0.617490\pi$
$282$	0	0
$283$	0	0		−	1.00000i	$-0.5\pi$
$283$	0	0			1.00000i	$0.5\pi$
$284$	−0.480351	−0.0285036
$285$	0	0
$286$	0	0
$287$	0	0
$288$	− 64.3024i	− 3.78905i
$289$	−17.0000	−1.00000
$290$	0	0
$291$	0	0
$292$	0	0
$293$	0	0		−	1.00000i	$-0.5\pi$
$293$	0	0			1.00000i	$0.5\pi$
$294$	0	0
$295$	0	0
$296$	104.037i	6.04704i
$297$	0	0
$298$	−28.8730	−1.67257
$299$	0	0
$300$	0	0
$301$	−34.5213	−1.98978
$302$	50.1663	2.88675
$303$	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	1.00000			$0$
$311$	0	0	−1.00000			$\pi$
$312$	0	0
$313$	0	0	1.00000			$0$
$313$	0	0	−1.00000			$\pi$
$314$	0	0
$315$	0	0
$316$	14.7130i	0.827674i
$317$	−20.5716	−1.15542	−0.577708	−	0.816243i	$-0.696053\pi$
$317$	−20.5716	−1.15542	−0.577708	+	0.816243i	$0.696053\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	0	0
$322$	24.2668	1.35234
$323$	0	0
$324$	−48.9898	−2.72166
$325$	0	0
$326$	69.6006i	3.85482i
$327$	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
$333$	33.2239	1.82066
$334$	0	0
$335$	0	0
$336$	0	0
$337$	34.7572i	1.89335i	0.322195	+	0.946673i	$0.395579\pi$
$337$	34.7572i	1.89335i	−0.322195	+	0.946673i	$0.604421\pi$
$338$	35.4672i	1.92916i
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	− 18.5203i	− 1.00000i
$344$	−122.574	−6.60874
$345$	0	0
$346$	0	0
$347$	− 7.64192i	− 0.410240i	−0.978737	−	0.205120i	$-0.934242\pi$
$347$	− 7.64192i	− 0.410240i	0.978737	−	0.205120i	$-0.0657585\pi$
$348$	0	0
$349$	0	0		−	1.00000i	$-0.5\pi$
$349$	0	0			1.00000i	$0.5\pi$
$350$	36.0913	1.92916
$351$	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
$358$	51.2566i	2.70900i
$359$	− 25.2641i	− 1.33339i	−0.745331	−	0.666695i	$-0.767708\pi$
$359$	− 25.2641i	− 1.33339i	0.745331	−	0.666695i	$-0.232292\pi$
$360$	0	0
$361$	−19.0000	−1.00000
$362$	0	0
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	0	0	1.00000			$0$
$367$	0	0	−1.00000			$\pi$
$368$	49.5640	2.58370
$369$	0	0
$370$	0	0
$371$	− 18.4538i	− 0.958072i
$372$	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
$379$	12.0637	0.619669	0.309834	−	0.950791i	$-0.399726\pi$
$379$	12.0637	0.619669	0.309834	+	0.950791i	$0.399726\pi$
$380$	0	0
$381$	0	0
$382$	− 75.3942i	− 3.85751i
$383$	0	0	1.00000			$0$
$383$	0	0	−1.00000			$\pi$
$384$	0	0
$385$	0	0
$386$	−28.8603	−1.46895
$387$	39.1435i	1.98978i
$388$	0	0
$389$	−24.5221	−1.24332	−0.621660	−	0.783287i	$-0.713542\pi$
$389$	−24.5221	−1.24332	−0.621660	+	0.783287i	$0.713542\pi$
$390$	0	0
$391$	0	0
$392$	− 65.7593i	− 3.32135i
$393$	0	0
$394$	−65.0529	−3.27732
$395$	0	0
$396$	0	0
$397$	0	0	1.00000			$0$
$397$	0	0	−1.00000			$\pi$
$398$	0	0
$399$	0	0
$400$	73.7150	3.68575
$401$	−39.9476	−1.99489	−0.997445	−	0.0714367i	$-0.977242\pi$
$401$	−39.9476	−1.99489	−0.997445	+	0.0714367i	$0.977242\pi$
$402$	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		−	1.00000i	$-0.5\pi$
$409$	0	0			1.00000i	$0.5\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	0	0
$414$	− 27.5160i	− 1.35234i
$415$	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
$421$	−2.37284	−0.115645	−0.0578225	−	0.998327i	$-0.518416\pi$
$421$	−2.37284	−0.115645	−0.0578225	+	0.998327i	$0.518416\pi$
$422$	39.1534	1.90596
$423$	0	0
$424$	− 65.5232i	− 3.18209i
$425$	0	0
$426$	0	0
$427$	0	0
$428$	40.6876i	1.96671i
$429$	0	0
$430$	0	0
$431$	40.2137i	1.93703i	0.248963	+	0.968513i	$0.419910\pi$
$431$	40.2137i	1.93703i	−0.248963	+	0.968513i	$0.580090\pi$
$432$	0	0
$433$	0	0	1.00000			$0$
$433$	0	0	−1.00000			$\pi$
$434$	0	0
$435$	0	0
$436$	− 10.9863i	− 0.526146i
$437$	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
$443$	−41.4080	−1.96735	−0.983676	−	0.179949i	$-0.942407\pi$
$443$	−41.4080	−1.96735	−0.983676	+	0.179949i	$0.942407\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	− 76.7043i	− 3.62394i
$449$	26.5002	1.25062	0.625310	−	0.780376i	$-0.284972\pi$
$449$	26.5002	1.25062	0.625310	+	0.780376i	$0.284972\pi$
$450$	− 40.9236i	− 1.92916i
$451$	0	0
$452$	−112.938	−5.31217
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 30.7206i	− 1.43705i	−0.695501	−	0.718525i	$-0.744818\pi$
$457$	− 30.7206i	− 1.43705i	0.695501	−	0.718525i	$-0.255182\pi$
$458$	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	$0.279743\pi$
$464$	− 76.2573i	− 3.54016i
$465$	0	0
$466$	57.7460	2.67503
$467$	0	0	1.00000			$0$
$467$	0	0	−1.00000			$\pi$
$468$	0	0
$469$	36.6740i	1.69345i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	0	0
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−20.9246	−0.958072
$478$	63.8210	2.91911
$479$	0	0		−	1.00000i	$-0.5\pi$
$479$	0	0			1.00000i	$0.5\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−27.8112	−1.26025	−0.630123	−	0.776495i	$-0.716996\pi$
$487$	−27.8112	−1.26025	−0.630123	+	0.776495i	$0.716996\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	5.29150i	0.238802i	0.992846	+	0.119401i	$0.0380974\pi$
$491$	5.29150i	0.238802i	−0.992846	+	0.119401i	$0.961903\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0.233477i	0.0104729i
$498$	0	0
$499$	14.0380	0.628426	0.314213	−	0.949352i	$-0.398259\pi$
$499$	14.0380	0.628426	0.314213	+	0.949352i	$0.398259\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	0	0		−	1.00000i	$-0.5\pi$
$503$	0	0			1.00000i	$0.5\pi$
$504$	−74.5640	−3.32135
$505$	0	0
$506$	0	0
$507$	0	0
$508$	− 18.7151i	− 0.830348i
$509$	0	0	1.00000			$0$
$509$	0	0	−1.00000			$\pi$
$510$	0	0
$511$	0	0
$512$	− 39.0063i	− 1.72385i
$513$	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	1.00000			$0$
$521$	0	0	−1.00000			$\pi$
$522$	−42.3350	−1.85295
$523$	0	0		−	1.00000i	$-0.5\pi$
$523$	0	0			1.00000i	$0.5\pi$
$524$	0	0
$525$	0	0
$526$	−78.5697	−3.42580
$527$	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
$536$	130.217i	5.62453i
$537$	0	0
$538$	0	0
$539$	0	0
$540$	0	0
$541$	− 45.6702i	− 1.96351i	−0.190138	−	0.981757i	$-0.560893\pi$
$541$	− 45.6702i	− 1.96351i	0.190138	−	0.981757i	$-0.439107\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	15.8745i	0.678745i	0.940652	+	0.339372i	$0.110215\pi$
$547$	15.8745i	0.678745i	−0.940652	+	0.339372i	$0.889785\pi$
$548$	23.6833	1.01170
$549$	0	0
$550$	0	0
$551$	0	0
$552$	0	0
$553$	7.15136	0.304107
$554$	−54.0401	−2.29594
$555$	0	0
$556$	0	0
$557$	− 47.0189i	− 1.99226i	−0.0879152	−	0.996128i	$-0.528020\pi$
$557$	− 47.0189i	− 1.99226i	0.0879152	−	0.996128i	$-0.471980\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0	0
$562$	85.3070	3.59846
$563$	0	0		−	1.00000i	$-0.5\pi$
$563$	0	0			1.00000i	$0.5\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	23.8118i	1.00000i
$568$	0.829001i	0.0347841i
$569$	− 42.3320i	− 1.77465i	−0.461144	−	0.887325i	$-0.652561\pi$
$569$	− 42.3320i	− 1.77465i	0.461144	−	0.887325i	$-0.347439\pi$
$570$	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
$575$	16.8093	0.700998
$576$	−86.9745	−3.62394
$577$	0	0	1.00000			$0$
$577$	0	0	−1.00000			$\pi$
$578$	46.3801i	1.92916i
$579$	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	1.00000			$0$
$587$	0	0	−1.00000			$\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	0	0
$592$	163.273	6.71050
$593$	0	0		−	1.00000i	$-0.5\pi$
$593$	0	0			1.00000i	$0.5\pi$
$594$	0	0
$595$	0	0
$596$	57.6066i	2.35966i
$597$	0	0
$598$	0	0
$599$	47.6604	1.94735	0.973676	−	0.227937i	$-0.0731980\pi$
$599$	47.6604	1.94735	0.973676	+	0.227937i	$0.0731980\pi$
$600$	0	0
$601$	0	0		−	1.00000i	$-0.5\pi$
$601$	0	0			1.00000i	$0.5\pi$
$602$	94.1826i	3.83860i
$603$	41.5845	1.69345
$604$	− 100.090i	− 4.07262i
$605$	0	0
$606$	0	0
$607$	0	0		−	1.00000i	$-0.5\pi$
$607$	0	0			1.00000i	$0.5\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	0	0
$612$	0	0
$613$	26.3292i	1.06342i	0.846925	+	0.531712i	$0.178451\pi$
$613$	26.3292i	1.06342i	−0.846925	+	0.531712i	$0.821549\pi$
$614$	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	1.00000			$0$
$619$	0	0	−1.00000			$\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	25.0000	1.00000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	0	0
$630$	0	0
$631$	−40.9367	−1.62966	−0.814832	−	0.579698i	$-0.803171\pi$
$631$	−40.9367	−1.62966	−0.814832	+	0.579698i	$0.803171\pi$
$632$	25.3921	1.01004
$633$	0	0
$634$	56.1244i	2.22898i
$635$	0	0
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0.264738	0.0104729
$640$	0	0
$641$	34.3449	1.35654	0.678270	−	0.734813i	$-0.262730\pi$
$641$	34.3449	1.35654	0.678270	+	0.734813i	$0.262730\pi$
$642$	0	0
$643$	0	0	1.00000			$0$
$643$	0	0	−1.00000			$\pi$
$644$	− 48.4164i	− 1.90787i
$645$	0	0
$646$	0	0
$647$	0	0	1.00000			$0$
$647$	0	0	−1.00000			$\pi$
$648$	84.5477i	3.32135i
$649$	0	0
$650$	0	0
$651$	0	0
$652$	138.865	5.43838
$653$	46.6714	1.82639	0.913196	−	0.407520i	$-0.133606\pi$
$653$	46.6714	1.82639	0.913196	+	0.407520i	$0.133606\pi$
$654$	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			$\pi$
$662$	87.9446i	3.41806i
$663$	0	0
$664$	0	0
$665$	0	0
$666$	− 90.6429i	− 3.51234i
$667$	− 17.3891i	− 0.673307i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	0	0
$672$	0	0
$673$	− 41.6130i	− 1.60406i	−0.597281	−	0.802032i	$-0.703752\pi$
$673$	− 41.6130i	− 1.60406i	0.597281	−	0.802032i	$-0.296248\pi$
$674$	94.8261	3.65257
$675$	0	0
$676$	70.7630	2.72166
$677$	0	0		−	1.00000i	$-0.5\pi$
$677$	0	0			1.00000i	$0.5\pi$
$678$	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
$686$	−50.5278	−1.92916
$687$	0	0
$688$	192.364i	7.33382i
$689$	0	0
$690$	0	0
$691$	0	0	1.00000			$0$
$691$	0	0	−1.00000			$\pi$
$692$	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
$700$	− 72.0082i	− 2.72166i
$701$	− 41.6336i	− 1.57248i	−0.617922	−	0.786239i	$-0.712025\pi$
$701$	− 41.6336i	− 1.57248i	0.617922	−	0.786239i	$-0.287975\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−48.4711	−1.82037	−0.910185	−	0.414202i	$-0.864061\pi$
$709$	−48.4711	−1.82037	−0.910185	+	0.414202i	$0.864061\pi$
$710$	0	0
$711$	− 8.10888i	− 0.304107i
$712$	0	0
$713$	0	0
$714$	0	0
$715$	0	0
$716$	102.266	3.82185
$717$	0	0
$718$	−68.9267	−2.57232
$719$	0	0	1.00000			$0$
$719$	0	0	−1.00000			$\pi$
$720$	0	0
$721$	0	0
$722$	51.8366i	1.92916i
$723$	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
$729$	27.0000	1.00000
$730$	0	0
$731$	0	0
$732$	0	0
$733$	0	0		−	1.00000i	$-0.5\pi$
$733$	0	0			1.00000i	$0.5\pi$
$734$	0	0
$735$	0	0
$736$	− 72.0587i	− 2.65612i
$737$	0	0
$738$	0	0
$739$	− 44.5494i	− 1.63878i	−0.573238	−	0.819389i	$-0.694313\pi$
$739$	− 44.5494i	− 1.63878i	0.573238	−	0.819389i	$-0.305687\pi$
$740$	0	0
$741$	0	0
$742$	−50.3463	−1.84827
$743$	− 49.4884i	− 1.81555i	−0.419453	−	0.907777i	$-0.637778\pi$
$743$	− 49.4884i	− 1.81555i	0.419453	−	0.907777i	$-0.362222\pi$
$744$	0	0
$745$	0	0
$746$	−86.6190	−3.17135
$747$	0	0
$748$	0	0
$749$	19.7764	0.722615
$750$	0	0
$751$	−54.3842	−1.98451	−0.992253	−	0.124234i	$-0.960353\pi$
$751$	−54.3842	−1.98451	−0.992253	+	0.124234i	$0.960353\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	6.62188	0.240676	0.120338	−	0.992733i	$-0.461602\pi$
$757$	6.62188	0.240676	0.120338	+	0.992733i	$0.461602\pi$
$758$	− 32.9126i	− 1.19544i
$759$	0	0
$760$	0	0
$761$	0	0		−	1.00000i	$-0.5\pi$
$761$	0	0			1.00000i	$0.5\pi$
$762$	0	0
$763$	−5.33993	−0.193318
$764$	−150.424	−5.44216
$765$	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
$772$	57.5813i	2.07240i
$773$	0	0	1.00000			$0$
$773$	0	0	−1.00000			$\pi$
$774$	106.793	3.83860
$775$	0	0
$776$	0	0
$777$	0	0
$778$	66.9023i	2.39856i
$779$	0	0
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	−103.201	−3.68575
$785$	0	0
$786$	0	0
$787$	0	0		−	1.00000i	$-0.5\pi$
$787$	0	0			1.00000i	$0.5\pi$
$788$	129.792i	4.62364i
$789$	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	1.00000			$0$
$797$	0	0	−1.00000			$\pi$
$798$	0	0
$799$	0	0
$800$	− 107.171i	− 3.78905i
$801$	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
$809$	− 56.5832i	− 1.98936i	−0.103022	−	0.994679i	$-0.532851\pi$
$809$	− 56.5832i	− 1.98936i	0.103022	−	0.994679i	$-0.467149\pi$
$810$	0	0
$811$	0	0		−	1.00000i	$-0.5\pi$
$811$	0	0			1.00000i	$0.5\pi$
$812$	−74.4917	−2.61414
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	0	0
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 52.9150i	− 1.84675i	−0.383903	−	0.923374i	$-0.625420\pi$
$821$	− 52.9150i	− 1.84675i	0.383903	−	0.923374i	$-0.374580\pi$
$822$	0	0
$823$	55.1812	1.92350	0.961748	−	0.273936i	$-0.0883256\pi$
$823$	55.1812	1.92350	0.961748	+	0.273936i	$0.0883256\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 37.0405i	− 1.28803i	−0.765015	−	0.644013i	$-0.777268\pi$
$827$	− 37.0405i	− 1.28803i	0.765015	−	0.644013i	$-0.222732\pi$
$828$	−54.8991	−1.90787
$829$	0	0	1.00000			$0$
$829$	0	0	−1.00000			$\pi$
$830$	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	1.00000			$0$
$839$	0	0	−1.00000			$\pi$
$840$	0	0
$841$	2.24585	0.0774431
$842$	6.47368i	0.223098i
$843$	0	0
$844$	− 78.1178i	− 2.68893i
$845$	0	0
$846$	0	0
$847$	0	0
$848$	−102.830	−3.53121
$849$	0	0
$850$	0	0
$851$	37.2315	1.27628
$852$	0	0
$853$	0	0		−	1.00000i	$-0.5\pi$
$853$	0	0			1.00000i	$0.5\pi$
$854$	0	0
$855$	0	0
$856$	70.2196	2.40006
$857$	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
$862$	109.713	3.73683
$863$	8.00000	0.272323	0.136162	−	0.990687i	$-0.456523\pi$
$863$	8.00000	0.272323	0.136162	+	0.990687i	$0.456523\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	0	0
$868$	0	0
$869$	0	0
$870$	0	0
$871$	0	0
$872$	−18.9603	−0.642077
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 57.3638i	− 1.93704i	−0.248939	−	0.968519i	$-0.580082\pi$
$877$	− 57.3638i	− 1.93704i	0.248939	−	0.968519i	$-0.419918\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	0	0	1.00000			$0$
$881$	0	0	−1.00000			$\pi$
$882$	57.2931i	1.92916i
$883$	−12.0000	−0.403832	−0.201916	−	0.979403i	$-0.564717\pi$
$883$	−12.0000	−0.403832	−0.201916	+	0.979403i	$0.564717\pi$
$884$	0	0
$885$	0	0
$886$	112.971i	3.79533i
$887$	0	0		−	1.00000i	$-0.5\pi$
$887$	0	0			1.00000i	$0.5\pi$
$888$	0	0
$889$	−9.09658	−0.305089
$890$	0	0
$891$	0	0
$892$	0	0
$893$	0	0
$894$	0	0
$895$	0	0
$896$	−95.8492	−3.20210
$897$	0	0
$898$	− 72.2989i	− 2.41265i
$899$	0	0
$900$	−81.6497	−2.72166
$901$	0	0
$902$	0	0
$903$	0	0
$904$	194.912i	6.48266i
$905$	0	0
$906$	0	0
$907$	−13.5085	−0.448542	−0.224271	−	0.974527i	$-0.572000\pi$
$907$	−13.5085	−0.448542	−0.224271	+	0.974527i	$0.572000\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−16.0000	−0.530104	−0.265052	−	0.964234i	$-0.585389\pi$
$911$	−16.0000	−0.530104	−0.265052	+	0.964234i	$0.585389\pi$
$912$	0	0
$913$	0	0
$914$	−83.8133	−2.77230
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	34.2046i	1.12830i	0.825671	+	0.564152i	$0.190797\pi$
$919$	34.2046i	1.12830i	−0.825671	+	0.564152i	$0.809203\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	0	0
$924$	0	0
$925$	55.3732	1.82066
$926$	− 74.9127i	− 2.46179i
$927$	0	0
$928$	−110.867	−3.63938
$929$	0	0	1.00000			$0$
$929$	0	0	−1.00000			$\pi$
$930$	0	0
$931$	0	0
$932$	− 115.213i	− 3.77393i
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	0	0		−	1.00000i	$-0.5\pi$
$937$	0	0			1.00000i	$0.5\pi$
$938$	100.056	3.26693
$939$	0	0
$940$	0	0
$941$	0	0		−	1.00000i	$-0.5\pi$
$941$	0	0			1.00000i	$0.5\pi$
$942$	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
$953$	− 16.9679i	− 0.549644i	−0.961495	−	0.274822i	$-0.911381\pi$
$953$	− 16.9679i	− 0.549644i	0.961495	−	0.274822i	$-0.0886189\pi$
$954$	57.0874i	1.84827i
$955$	0	0
$956$	− 127.334i	− 4.11827i
$957$	0	0
$958$	0	0
$959$	− 11.5114i	− 0.371722i
$960$	0	0
$961$	31.0000	1.00000
$962$	0	0
$963$	− 22.4244i	− 0.722615i
$964$	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	1.00000			$0$
$971$	0	0	−1.00000			$\pi$
$972$	0	0
$973$	0	0
$974$	75.8758i	2.43122i
$975$	0	0
$976$	0	0
$977$	62.0969	1.98666	0.993328	−	0.115321i	$-0.0367898\pi$
$977$	62.0969	1.98666	0.993328	+	0.115321i	$0.0367898\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	6.05491i	0.193318i
$982$	14.4365	0.460687
$983$	0	0	1.00000			$0$
$983$	0	0	−1.00000			$\pi$
$984$	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
$994$	0.636983	0.0202039
$995$	0	0
$996$	0	0
$997$	0	0		−	1.00000i	$-0.5\pi$
$997$	0	0			1.00000i	$0.5\pi$
$998$	− 38.2990i	− 1.21233i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	847.2.b.b.846.1		8
7.6	odd	2	CM	847.2.b.b.846.1		8
11.2	odd	10		847.2.l.d.524.2		8
11.3	even	5		77.2.l.a.13.1	yes	8
11.4	even	5		847.2.l.c.699.2		8
11.5	even	5		847.2.l.d.118.2		8
11.6	odd	10		847.2.l.a.118.1		8
11.7	odd	10		77.2.l.a.6.1	✓	8
11.8	odd	10		847.2.l.c.475.2		8
11.9	even	5		847.2.l.a.524.1		8
11.10	odd	2	inner	847.2.b.b.846.8		8
33.14	odd	10		693.2.bu.a.244.2		8
33.29	even	10		693.2.bu.a.622.2		8
77.3	odd	30		539.2.s.a.68.2		16
77.6	even	10		847.2.l.a.118.1		8
77.13	even	10		847.2.l.d.524.2		8
77.18	odd	30		539.2.s.a.215.2		16
77.20	odd	10		847.2.l.a.524.1		8
77.25	even	15		539.2.s.a.68.2		16
77.27	odd	10		847.2.l.d.118.2		8
77.40	even	30		539.2.s.a.325.2		16
77.41	even	10		847.2.l.c.475.2		8
77.47	odd	30		539.2.s.a.178.2		16
77.48	odd	10		847.2.l.c.699.2		8
77.51	odd	30		539.2.s.a.325.2		16
77.58	even	15		539.2.s.a.178.2		16
77.62	even	10		77.2.l.a.6.1	✓	8
77.69	odd	10		77.2.l.a.13.1	yes	8
77.73	even	30		539.2.s.a.215.2		16
77.76	even	2	inner	847.2.b.b.846.8		8
231.62	odd	10		693.2.bu.a.622.2		8
231.146	even	10		693.2.bu.a.244.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
77.2.l.a.6.1	✓	8	11.7	odd	10
77.2.l.a.6.1	✓	8	77.62	even	10
77.2.l.a.13.1	yes	8	11.3	even	5
77.2.l.a.13.1	yes	8	77.69	odd	10
539.2.s.a.68.2		16	77.3	odd	30
539.2.s.a.68.2		16	77.25	even	15
539.2.s.a.178.2		16	77.47	odd	30
539.2.s.a.178.2		16	77.58	even	15
539.2.s.a.215.2		16	77.18	odd	30
539.2.s.a.215.2		16	77.73	even	30
539.2.s.a.325.2		16	77.40	even	30
539.2.s.a.325.2		16	77.51	odd	30
693.2.bu.a.244.2		8	33.14	odd	10
693.2.bu.a.244.2		8	231.146	even	10
693.2.bu.a.622.2		8	33.29	even	10
693.2.bu.a.622.2		8	231.62	odd	10
847.2.b.b.846.1		8	1.1	even	1	trivial
847.2.b.b.846.1		8	7.6	odd	2	CM
847.2.b.b.846.8		8	11.10	odd	2	inner
847.2.b.b.846.8		8	77.76	even	2	inner
847.2.l.a.118.1		8	11.6	odd	10
847.2.l.a.118.1		8	77.6	even	10
847.2.l.a.524.1		8	11.9	even	5
847.2.l.a.524.1		8	77.20	odd	10
847.2.l.c.475.2		8	11.8	odd	10
847.2.l.c.475.2		8	77.41	even	10
847.2.l.c.699.2		8	11.4	even	5
847.2.l.c.699.2		8	77.48	odd	10
847.2.l.d.118.2		8	11.5	even	5
847.2.l.d.118.2		8	77.27	odd	10
847.2.l.d.524.2		8	11.2	odd	10
847.2.l.d.524.2		8	77.13	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.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-2.72824i q^{2} -5.44331 q^{4} +2.64575i q^{7} +9.39419i q^{8} +3.00000 q^{9} +O(q^{10})\)	\(q-2.72824i q^{2} -5.44331 q^{4} +2.64575i q^{7} +9.39419i q^{8} +3.00000 q^{9} +7.21825 q^{14} +14.7430 q^{16} -8.18473i q^{18} +3.36187 q^{23} +5.00000 q^{25} -14.4016i q^{28} -5.17244i q^{29} -21.4341i q^{32} -16.3299 q^{36} +11.0746 q^{37} +13.0478i q^{43} -9.17199i q^{46} -7.00000 q^{49} -13.6412i q^{50} -6.97487 q^{53} -24.8547 q^{56} -14.1117 q^{58} +7.93725i q^{63} -28.9915 q^{64} +13.8615 q^{67} +0.0882461 q^{71} +28.1826i q^{72} -30.2143i q^{74} -2.70296i q^{79} +9.00000 q^{81} +35.5977 q^{86} -18.2997 q^{92} +19.0977i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 10 q^{4} + 24 q^{9}+O(q^{10}) \) 8 * q - 10 * q^4 + 24 * q^9	\( 8 q - 10 q^{4} + 24 q^{9} + 14 q^{14} + 18 q^{16} + 16 q^{23} + 40 q^{25} - 30 q^{36} + 12 q^{37} - 56 q^{49} - 20 q^{53} - 42 q^{56} - 56 q^{58} - 54 q^{64} + 8 q^{67} + 32 q^{71} + 72 q^{81} + 28 q^{86} - 80 q^{92}+O(q^{100}) \) 8 * q - 10 * q^4 + 24 * q^9 + 14 * q^14 + 18 * q^16 + 16 * q^23 + 40 * q^25 - 30 * q^36 + 12 * q^37 - 56 * q^49 - 20 * q^53 - 42 * q^56 - 56 * q^58 - 54 * q^64 + 8 * q^67 + 32 * q^71 + 72 * q^81 + 28 * q^86 - 80 * q^92

Introduction

L-functions

Modular forms

Varieties

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Database

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