LMFDB - Embedded newform 847.2.b.b.846.2

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			846.2
Root			$1.10362 - 0.884319i$ of defining polynomial
Character	$\chi$	$=$	847.846
Dual form			847.2.b.b.846.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.76864i q^{2} -1.12808 q^{4} -2.64575i q^{7} -1.54211i q^{8} +3.00000 q^{9} +O(q^{10})$	$q-1.76864i q^{2} -1.12808 q^{4} -2.64575i q^{7} -1.54211i q^{8} +3.00000 q^{9} -4.67938 q^{14} -4.98359 q^{16} -5.30592i q^{18} +2.56038 q^{23} +5.00000 q^{25} +2.98463i q^{28} -7.38626i q^{29} +5.72996i q^{32} -3.38425 q^{36} -11.9191 q^{37} -2.77251i q^{43} -4.52839i q^{46} -7.00000 q^{49} -8.84319i q^{50} -14.3107 q^{53} -4.08003 q^{56} -13.0636 q^{58} -7.93725i q^{63} +0.167048 q^{64} +12.5669 q^{67} +16.0545 q^{71} -4.62632i q^{72} +21.0806i q^{74} +17.5450i q^{79} +9.00000 q^{81} -4.90356 q^{86} -2.88832 q^{92} +12.3805i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	− 1.76864i	− 1.25062i	−0.780378	−	0.625308i	$-0.784973\pi$
$2$	− 1.76864i	− 1.25062i	0.780378	−	0.625308i	$-0.215027\pi$
$3$	0	0	1.00000			$0$
$3$	0	0	−1.00000			$\pi$
$4$	−1.12808	−0.564041
$5$	0	0	1.00000			$0$
$5$	0	0	−1.00000			$\pi$
$6$	0	0
$7$	− 2.64575i	− 1.00000i
$7$	− 2.64575i	− 1.00000i
$8$	− 1.54211i	− 0.545217i
$9$	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	$0.5\pi$
$14$	−4.67938	−1.25062
$15$	0	0
$16$	−4.98359	−1.24590
$17$	0	0		−	1.00000i	$-0.5\pi$
$17$	0	0			1.00000i	$0.5\pi$
$18$	− 5.30592i	− 1.25062i
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	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
$25$	5.00000	1.00000
$26$	0	0
$27$	0	0
$28$	2.98463i	0.564041i
$29$	− 7.38626i	− 1.37159i	−0.727793	−	0.685797i	$-0.759454\pi$
$29$	− 7.38626i	− 1.37159i	0.727793	−	0.685797i	$-0.240546\pi$
$30$	0	0
$31$	0	0	1.00000			$0$
$31$	0	0	−1.00000			$\pi$
$32$	5.72996i	1.01292i
$33$	0	0
$34$	0	0
$35$	0	0
$36$	−3.38425	−0.564041
$37$	−11.9191	−1.95949	−0.979747	−	0.200239i	$-0.935828\pi$
$37$	−11.9191	−1.95949	−0.979747	+	0.200239i	$0.935828\pi$
$38$	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
$43$	− 2.77251i	− 0.422803i	−0.977399	−	0.211402i	$-0.932197\pi$
$43$	− 2.77251i	− 0.422803i	0.977399	−	0.211402i	$-0.0678028\pi$
$44$	0	0
$45$	0	0
$46$	− 4.52839i	− 0.667675i
$47$	0	0	1.00000			$0$
$47$	0	0	−1.00000			$\pi$
$48$	0	0
$49$	−7.00000	−1.00000
$50$	− 8.84319i	− 1.25062i
$51$	0	0
$52$	0	0
$53$	−14.3107	−1.96573	−0.982863	−	0.184336i	$-0.940986\pi$
$53$	−14.3107	−1.96573	−0.982863	+	0.184336i	$0.940986\pi$
$54$	0	0
$55$	0	0
$56$	−4.08003	−0.545217
$57$	0	0
$58$	−13.0636	−1.71534
$59$	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
$64$	0.167048	0.0208809
$65$	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
$71$	16.0545	1.90532	0.952662	−	0.304033i	$-0.0983332\pi$
$71$	16.0545	1.90532	0.952662	+	0.304033i	$0.0983332\pi$
$72$	− 4.62632i	− 0.545217i
$73$	0	0		−	1.00000i	$-0.5\pi$
$73$	0	0			1.00000i	$0.5\pi$
$74$	21.0806i	2.45058i
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	17.5450i	1.97397i	0.160813	+	0.986985i	$0.448589\pi$
$79$	17.5450i	1.97397i	−0.160813	+	0.986985i	$0.551411\pi$
$80$	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
$86$	−4.90356	−0.528765
$87$	0	0
$88$	0	0
$89$	0	0	1.00000			$0$
$89$	0	0	−1.00000			$\pi$
$90$	0	0
$91$	0	0
$92$	−2.88832	−0.301128
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	0	0	1.00000			$0$
$97$	0	0	−1.00000			$\pi$
$98$	12.3805i	1.25062i
$99$	0	0
$100$	−5.64041	−0.564041
$101$	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
$106$	25.3105i	2.45837i
$107$	− 17.3860i	− 1.68076i	−0.541994	−	0.840382i	$-0.682330\pi$
$107$	− 17.3860i	− 1.68076i	0.541994	−	0.840382i	$-0.317670\pi$
$108$	0	0
$109$	13.8487i	1.32646i	0.748414	+	0.663232i	$0.230816\pi$
$109$	13.8487i	1.32646i	−0.748414	+	0.663232i	$0.769184\pi$
$110$	0	0
$111$	0	0
$112$	13.1854i	1.24590i
$113$	10.8230	1.01815	0.509073	−	0.860724i	$-0.329988\pi$
$113$	10.8230	1.01815	0.509073	+	0.860724i	$0.329988\pi$
$114$	0	0
$115$	0	0
$116$	8.33231i	0.773636i
$117$	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
$126$	−14.0381	−1.25062
$127$	− 10.3114i	− 0.914990i	−0.889212	−	0.457495i	$-0.848747\pi$
$127$	− 10.3114i	− 0.914990i	0.889212	−	0.457495i	$-0.151253\pi$
$128$	11.1645i	0.986810i
$129$	0	0
$130$	0	0
$131$	0	0		−	1.00000i	$-0.5\pi$
$131$	0	0			1.00000i	$0.5\pi$
$132$	0	0
$133$	0	0
$134$	− 22.2263i	− 1.92006i
$135$	0	0
$136$	0	0
$137$	17.0399	1.45582	0.727909	−	0.685674i	$-0.240493\pi$
$137$	17.0399	1.45582	0.727909	+	0.685674i	$0.240493\pi$
$138$	0	0
$139$	0	0		−	1.00000i	$-0.5\pi$
$139$	0	0			1.00000i	$0.5\pi$
$140$	0	0
$141$	0	0
$142$	− 28.3947i	− 2.38283i
$143$	0	0
$144$	−14.9508	−1.24590
$145$	0	0
$146$	0	0
$147$	0	0
$148$	13.4458	1.10524
$149$	10.5830i	0.866994i	0.901155	+	0.433497i	$0.142720\pi$
$149$	10.5830i	0.866994i	−0.901155	+	0.433497i	$0.857280\pi$
$150$	0	0
$151$	24.4605i	1.99057i	0.0969991	+	0.995284i	$0.469076\pi$
$151$	24.4605i	1.99057i	−0.0969991	+	0.995284i	$0.530924\pi$
$152$	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			$\pi$
$158$	31.0308	2.46868
$159$	0	0
$160$	0	0
$161$	− 6.77413i	− 0.533877i
$162$	− 15.9177i	− 1.25062i
$163$	21.2779	1.66661	0.833307	−	0.552811i	$-0.186445\pi$
$163$	21.2779	1.66661	0.833307	+	0.552811i	$0.186445\pi$
$164$	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
$172$	3.12762i	0.238479i
$173$	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
$179$	26.3987	1.97313	0.986564	−	0.163374i	$-0.0522378\pi$
$179$	26.3987	1.97313	0.986564	+	0.163374i	$0.0522378\pi$
$180$	0	0
$181$	0	0	1.00000			$0$
$181$	0	0	−1.00000			$\pi$
$182$	0	0
$183$	0	0
$184$	− 3.94838i	− 0.291079i
$185$	0	0
$186$	0	0
$187$	0	0
$188$	0	0
$189$	0	0
$190$	0	0
$191$	9.07920	0.656948	0.328474	−	0.944513i	$-0.393466\pi$
$191$	9.07920	0.656948	0.328474	+	0.944513i	$0.393466\pi$
$192$	0	0
$193$	27.7038i	1.99416i	0.0763450	+	0.997081i	$0.475675\pi$
$193$	27.7038i	1.99416i	−0.0763450	+	0.997081i	$0.524325\pi$
$194$	0	0
$195$	0	0
$196$	7.89658	0.564041
$197$	− 27.9978i	− 1.99476i	−0.0723369	−	0.997380i	$-0.523046\pi$
$197$	− 27.9978i	− 1.99476i	0.0723369	−	0.997380i	$-0.476954\pi$
$198$	0	0
$199$	0	0	1.00000			$0$
$199$	0	0	−1.00000			$\pi$
$200$	− 7.71053i	− 0.545217i
$201$	0	0
$202$	0	0
$203$	−19.5422	−1.37159
$204$	0	0
$205$	0	0
$206$	0	0
$207$	7.68115	0.533877
$208$	0	0
$209$	0	0
$210$	0	0
$211$	− 3.23686i	− 0.222834i	−0.993774	−	0.111417i	$-0.964461\pi$
$211$	− 3.23686i	− 0.222834i	0.993774	−	0.111417i	$-0.0355390\pi$
$212$	16.1437	1.10875
$213$	0	0
$214$	−30.7495	−2.10199
$215$	0	0
$216$	0	0
$217$	0	0
$218$	24.4933	1.65890
$219$	0	0
$220$	0	0
$221$	0	0
$222$	0	0
$223$	0	0	1.00000			$0$
$223$	0	0	−1.00000			$\pi$
$224$	15.1601	1.01292
$225$	15.0000	1.00000
$226$	− 19.1420i	− 1.27331i
$227$	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
$232$	−11.3904	−0.747817
$233$	− 21.1660i	− 1.38663i	−0.720634	−	0.693316i	$-0.756149\pi$
$233$	− 21.1660i	− 1.38663i	0.720634	−	0.693316i	$-0.243851\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0	0
$239$	12.0000i	0.776216i	0.921614	+	0.388108i	$0.126871\pi$
$239$	12.0000i	0.776216i	−0.921614	+	0.388108i	$0.873129\pi$
$240$	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			$\pi$
$252$	8.95388i	0.564041i
$253$	0	0
$254$	−18.2372	−1.14430
$255$	0	0
$256$	20.0800	1.25500
$257$	0	0	1.00000			$0$
$257$	0	0	−1.00000			$\pi$
$258$	0	0
$259$	31.5351i	1.95949i
$260$	0	0
$261$	− 22.1588i	− 1.37159i
$262$	0	0
$263$	23.0900i	1.42379i	0.702284	+	0.711897i	$0.252164\pi$
$263$	23.0900i	1.42379i	−0.702284	+	0.711897i	$0.747836\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	−14.1765	−0.865966
$269$	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
$274$	− 30.1374i	− 1.82067i
$275$	0	0
$276$	0	0
$277$	− 0.300420i	− 0.0180505i	−0.999959	−	0.00902525i	$-0.997127\pi$
$277$	− 0.300420i	− 0.0180505i	0.999959	−	0.00902525i	$-0.00287287\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	1.84125i	0.109840i	0.998491	+	0.0549198i	$0.0174903\pi$
$281$	1.84125i	0.109840i	−0.998491	+	0.0549198i	$0.982510\pi$
$282$	0	0
$283$	0	0		−	1.00000i	$-0.5\pi$
$283$	0	0			1.00000i	$0.5\pi$
$284$	−18.1108	−1.07468
$285$	0	0
$286$	0	0
$287$	0	0
$288$	17.1899i	1.01292i
$289$	−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
$296$	18.3806i	1.06835i
$297$	0	0
$298$	18.7175	1.08428
$299$	0	0
$300$	0	0
$301$	−7.33537	−0.422803
$302$	43.2618	2.48944
$303$	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
$316$	− 19.7922i	− 1.11340i
$317$	21.2860	1.19554	0.597772	−	0.801666i	$-0.296053\pi$
$317$	21.2860	1.19554	0.597772	+	0.801666i	$0.296053\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	0	0
$322$	−11.9810	−0.667675
$323$	0	0
$324$	−10.1527	−0.564041
$325$	0	0
$326$	− 37.6329i	− 2.08429i
$327$	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
$333$	−35.7574	−1.95949
$334$	0	0
$335$	0	0
$336$	0	0
$337$	35.0724i	1.91051i	0.295779	+	0.955256i	$0.404421\pi$
$337$	35.0724i	1.91051i	−0.295779	+	0.955256i	$0.595579\pi$
$338$	22.9923i	1.25062i
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	18.5203i	1.00000i
$344$	−4.27550	−0.230520
$345$	0	0
$346$	0	0
$347$	− 32.3175i	− 1.73490i	−0.497527	−	0.867448i	$-0.665758\pi$
$347$	− 32.3175i	− 1.73490i	0.497527	−	0.867448i	$-0.334242\pi$
$348$	0	0
$349$	0	0		−	1.00000i	$-0.5\pi$
$349$	0	0			1.00000i	$0.5\pi$
$350$	−23.3969	−1.25062
$351$	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
$358$	− 46.6897i	− 2.46763i
$359$	− 3.83770i	− 0.202546i	−0.994859	−	0.101273i	$-0.967708\pi$
$359$	− 3.83770i	− 0.202546i	0.994859	−	0.101273i	$-0.0322915\pi$
$360$	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			$\pi$
$368$	−12.7599	−0.665156
$369$	0	0
$370$	0	0
$371$	37.8626i	1.96573i
$372$	0	0
$373$	31.7490i	1.64390i	0.569558	+	0.821951i	$0.307114\pi$
$373$	31.7490i	1.64390i	−0.569558	+	0.821951i	$0.692886\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	0	0
$378$	0	0
$379$	−31.5194	−1.61904	−0.809522	−	0.587090i	$-0.800274\pi$
$379$	−31.5194	−1.61904	−0.809522	+	0.587090i	$0.800274\pi$
$380$	0	0
$381$	0	0
$382$	− 16.0578i	− 0.821590i
$383$	0	0	1.00000			$0$
$383$	0	0	−1.00000			$\pi$
$384$	0	0
$385$	0	0
$386$	48.9980	2.49393
$387$	− 8.31752i	− 0.422803i
$388$	0	0
$389$	1.67761	0.0850582	0.0425291	−	0.999095i	$-0.486458\pi$
$389$	1.67761	0.0850582	0.0425291	+	0.999095i	$0.486458\pi$
$390$	0	0
$391$	0	0
$392$	10.7947i	0.545217i
$393$	0	0
$394$	−49.5180	−2.49468
$395$	0	0
$396$	0	0
$397$	0	0	1.00000			$0$
$397$	0	0	−1.00000			$\pi$
$398$	0	0
$399$	0	0
$400$	−24.9180	−1.24590
$401$	30.6367	1.52992	0.764961	−	0.644077i	$-0.222758\pi$
$401$	30.6367	1.52992	0.764961	+	0.644077i	$0.222758\pi$
$402$	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		−	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
$414$	− 13.5852i	− 0.667675i
$415$	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
$421$	−22.1607	−1.08004	−0.540022	−	0.841651i	$-0.681584\pi$
$421$	−22.1607	−1.08004	−0.540022	+	0.841651i	$0.681584\pi$
$422$	−5.72483	−0.278680
$423$	0	0
$424$	22.0686i	1.07175i
$425$	0	0
$426$	0	0
$427$	0	0
$428$	19.6128i	0.948021i
$429$	0	0
$430$	0	0
$431$	38.6096i	1.85976i	0.367862	+	0.929880i	$0.380090\pi$
$431$	38.6096i	1.85976i	−0.367862	+	0.929880i	$0.619910\pi$
$432$	0	0
$433$	0	0	1.00000			$0$
$433$	0	0	−1.00000			$\pi$
$434$	0	0
$435$	0	0
$436$	− 15.6225i	− 0.748180i
$437$	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
$443$	−5.59153	−0.265662	−0.132831	−	0.991139i	$-0.542407\pi$
$443$	−5.59153	−0.265662	−0.132831	+	0.991139i	$0.542407\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	− 0.441966i	− 0.0208809i
$449$	−40.8782	−1.92916	−0.964580	−	0.263790i	$-0.915028\pi$
$449$	−40.8782	−1.92916	−0.964580	+	0.263790i	$0.915028\pi$
$450$	− 26.5296i	− 1.25062i
$451$	0	0
$452$	−12.2093	−0.574276
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 7.37497i	− 0.344987i	−0.985011	−	0.172493i	$-0.944818\pi$
$457$	− 7.37497i	− 0.344987i	0.985011	−	0.172493i	$-0.0551823\pi$
$458$	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	$0.679743\pi$
$464$	36.8101i	1.70887i
$465$	0	0
$466$	−37.4350	−1.73414
$467$	0	0	1.00000			$0$
$467$	0	0	−1.00000			$\pi$
$468$	0	0
$469$	− 33.2488i	− 1.53529i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	0	0
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−42.9321	−1.96573
$478$	21.2237	0.970749
$479$	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
$487$	−41.1883	−1.86642	−0.933210	−	0.359333i	$-0.883004\pi$
$487$	−41.1883	−1.86642	−0.933210	+	0.359333i	$0.883004\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	− 5.29150i	− 0.238802i	−0.992846	−	0.119401i	$-0.961903\pi$
$491$	− 5.29150i	− 0.238802i	0.992846	−	0.119401i	$-0.0380974\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 42.4763i	− 1.90532i
$498$	0	0
$499$	44.6759	1.99997	0.999985	−	0.00546838i	$-0.00174065\pi$
$499$	44.6759	1.99997	0.999985	+	0.00546838i	$0.00174065\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	0	0		−	1.00000i	$-0.5\pi$
$503$	0	0			1.00000i	$0.5\pi$
$504$	−12.2401	−0.545217
$505$	0	0
$506$	0	0
$507$	0	0
$508$	11.6321i	0.516092i
$509$	0	0	1.00000			$0$
$509$	0	0	−1.00000			$\pi$
$510$	0	0
$511$	0	0
$512$	− 13.1854i	− 0.582716i
$513$	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	1.00000			$0$
$521$	0	0	−1.00000			$\pi$
$522$	−39.1909	−1.71534
$523$	0	0		−	1.00000i	$-0.5\pi$
$523$	0	0			1.00000i	$0.5\pi$
$524$	0	0
$525$	0	0
$526$	40.8379	1.78062
$527$	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
$536$	− 19.3795i	− 0.837065i
$537$	0	0
$538$	0	0
$539$	0	0
$540$	0	0
$541$	− 42.1469i	− 1.81204i	−0.423238	−	0.906019i	$-0.639107\pi$
$541$	− 42.1469i	− 1.81204i	0.423238	−	0.906019i	$-0.360893\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 15.8745i	− 0.678745i	−0.940652	−	0.339372i	$-0.889785\pi$
$547$	− 15.8745i	− 0.678745i	0.940652	−	0.339372i	$-0.110215\pi$
$548$	−19.2224	−0.821141
$549$	0	0
$550$	0	0
$551$	0	0
$552$	0	0
$553$	46.4198	1.97397
$554$	−0.531335	−0.0225743
$555$	0	0
$556$	0	0
$557$	18.4763i	0.782866i	0.920207	+	0.391433i	$0.128020\pi$
$557$	18.4763i	0.782866i	−0.920207	+	0.391433i	$0.871980\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0	0
$562$	3.25650	0.137367
$563$	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
$568$	− 24.7578i	− 1.03881i
$569$	42.3320i	1.77465i	0.461144	+	0.887325i	$0.347439\pi$
$569$	42.3320i	1.77465i	−0.461144	+	0.887325i	$0.652561\pi$
$570$	0	0
$571$	10.9123i	0.456664i	0.973583	+	0.228332i	$0.0733271\pi$
$571$	10.9123i	0.456664i	−0.973583	+	0.228332i	$0.926673\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	12.8019	0.533877
$576$	0.501143	0.0208809
$577$	0	0	1.00000			$0$
$577$	0	0	−1.00000			$\pi$
$578$	30.0669i	1.25062i
$579$	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
$592$	59.4002	2.44133
$593$	0	0		−	1.00000i	$-0.5\pi$
$593$	0	0			1.00000i	$0.5\pi$
$594$	0	0
$595$	0	0
$596$	− 11.9385i	− 0.489020i
$597$	0	0
$598$	0	0
$599$	−45.1162	−1.84340	−0.921698	−	0.387907i	$-0.873198\pi$
$599$	−45.1162	−1.84340	−0.921698	+	0.387907i	$0.873198\pi$
$600$	0	0
$601$	0	0		−	1.00000i	$-0.5\pi$
$601$	0	0			1.00000i	$0.5\pi$
$602$	12.9736i	0.528765i
$603$	37.7006	1.53529
$604$	− 27.5935i	− 1.12276i
$605$	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
$613$	− 48.0213i	− 1.93956i	−0.243974	−	0.969782i	$-0.578451\pi$
$613$	− 48.0213i	− 1.93956i	0.243974	−	0.969782i	$-0.421549\pi$
$614$	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	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
$631$	50.2369	1.99990	0.999950	−	0.00996082i	$-0.00317068\pi$
$631$	50.2369	1.99990	0.999950	+	0.00996082i	$0.00317068\pi$
$632$	27.0563	1.07624
$633$	0	0
$634$	− 37.6473i	− 1.49517i
$635$	0	0
$636$	0	0
$637$	0	0
$638$	0	0
$639$	48.1636	1.90532
$640$	0	0
$641$	−24.7737	−0.978503	−0.489251	−	0.872143i	$-0.662730\pi$
$641$	−24.7737	−0.978503	−0.489251	+	0.872143i	$0.662730\pi$
$642$	0	0
$643$	0	0	1.00000			$0$
$643$	0	0	−1.00000			$\pi$
$644$	7.64178i	0.301128i
$645$	0	0
$646$	0	0
$647$	0	0	1.00000			$0$
$647$	0	0	−1.00000			$\pi$
$648$	− 13.8790i	− 0.545217i
$649$	0	0
$650$	0	0
$651$	0	0
$652$	−24.0032	−0.940039
$653$	−25.5159	−0.998514	−0.499257	−	0.866454i	$-0.666394\pi$
$653$	−25.5159	−0.998514	−0.499257	+	0.866454i	$0.666394\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0	0
$658$	0	0
$659$	− 26.4575i	− 1.03064i	−0.856998	−	0.515319i	$-0.827673\pi$
$659$	− 26.4575i	− 1.03064i	0.856998	−	0.515319i	$-0.172327\pi$
$660$	0	0
$661$	0	0	1.00000			$0$
$661$	0	0	−1.00000			$\pi$
$662$	− 28.5761i	− 1.11064i
$663$	0	0
$664$	0	0
$665$	0	0
$666$	63.2419i	2.45058i
$667$	− 18.9116i	− 0.732262i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	0	0
$672$	0	0
$673$	− 16.6138i	− 0.640414i	−0.947348	−	0.320207i	$-0.896248\pi$
$673$	− 16.6138i	− 0.640414i	0.947348	−	0.320207i	$-0.103752\pi$
$674$	62.0303	2.38932
$675$	0	0
$676$	14.6651	0.564041
$677$	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
$683$	−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
$686$	32.7556	1.25062
$687$	0	0
$688$	13.8171i	0.526770i
$689$	0	0
$690$	0	0
$691$	0	0	1.00000			$0$
$691$	0	0	−1.00000			$\pi$
$692$	0	0
$693$	0	0
$694$	−57.1581	−2.16969
$695$	0	0
$696$	0	0
$697$	0	0
$698$	0	0
$699$	0	0
$700$	14.9231i	0.564041i
$701$	− 14.4495i	− 0.545751i	−0.962049	−	0.272876i	$-0.912025\pi$
$701$	− 14.4495i	− 0.545751i	0.962049	−	0.272876i	$-0.0879747\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−35.9568	−1.35038	−0.675192	−	0.737642i	$-0.735939\pi$
$709$	−35.9568	−1.35038	−0.675192	+	0.737642i	$0.735939\pi$
$710$	0	0
$711$	52.6351i	1.97397i
$712$	0	0
$713$	0	0
$714$	0	0
$715$	0	0
$716$	−29.7799	−1.11293
$717$	0	0
$718$	−6.78750	−0.253307
$719$	0	0	1.00000			$0$
$719$	0	0	−1.00000			$\pi$
$720$	0	0
$721$	0	0
$722$	33.6041i	1.25062i
$723$	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
$729$	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
$736$	14.6709i	0.540777i
$737$	0	0
$738$	0	0
$739$	43.4076i	1.59677i	0.602145	+	0.798387i	$0.294313\pi$
$739$	43.4076i	1.59677i	−0.602145	+	0.798387i	$0.705687\pi$
$740$	0	0
$741$	0	0
$742$	66.9652	2.45837
$743$	− 6.45500i	− 0.236811i	−0.992965	−	0.118405i	$-0.962222\pi$
$743$	− 6.45500i	− 0.236811i	0.992965	−	0.118405i	$-0.0377783\pi$
$744$	0	0
$745$	0	0
$746$	56.1525	2.05589
$747$	0	0
$748$	0	0
$749$	−45.9989	−1.68076
$750$	0	0
$751$	39.9954	1.45945	0.729727	−	0.683739i	$-0.239647\pi$
$751$	39.9954	1.45945	0.729727	+	0.683739i	$0.239647\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−49.9075	−1.81392	−0.906959	−	0.421220i	$-0.861602\pi$
$757$	−49.9075	−1.81392	−0.906959	+	0.421220i	$0.861602\pi$
$758$	55.7465i	2.02480i
$759$	0	0
$760$	0	0
$761$	0	0		−	1.00000i	$-0.5\pi$
$761$	0	0			1.00000i	$0.5\pi$
$762$	0	0
$763$	36.6402	1.32646
$764$	−10.2421	−0.370546
$765$	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
$772$	− 31.2522i	− 1.12479i
$773$	0	0	1.00000			$0$
$773$	0	0	−1.00000			$\pi$
$774$	−14.7107	−0.528765
$775$	0	0
$776$	0	0
$777$	0	0
$778$	− 2.96709i	− 0.106375i
$779$	0	0
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	34.8852	1.24590
$785$	0	0
$786$	0	0
$787$	0	0		−	1.00000i	$-0.5\pi$
$787$	0	0			1.00000i	$0.5\pi$
$788$	31.5838i	1.12513i
$789$	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	1.00000			$0$
$797$	0	0	−1.00000			$\pi$
$798$	0	0
$799$	0	0
$800$	28.6498i	1.01292i
$801$	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
$809$	− 49.2215i	− 1.73053i	−0.501311	−	0.865267i	$-0.667149\pi$
$809$	− 49.2215i	− 1.73053i	0.501311	−	0.865267i	$-0.332851\pi$
$810$	0	0
$811$	0	0		−	1.00000i	$-0.5\pi$
$811$	0	0			1.00000i	$0.5\pi$
$812$	22.0452	0.773636
$813$	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	$0.374580\pi$
$821$	52.9150i	1.84675i	−0.383903	+	0.923374i	$0.625420\pi$
$822$	0	0
$823$	2.10386	0.0733360	0.0366680	−	0.999328i	$-0.488326\pi$
$823$	2.10386	0.0733360	0.0366680	+	0.999328i	$0.488326\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	37.0405i	1.28803i	0.765015	+	0.644013i	$0.222732\pi$
$827$	37.0405i	1.28803i	−0.765015	+	0.644013i	$0.777268\pi$
$828$	−8.66497	−0.301128
$829$	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
$841$	−25.5568	−0.881271
$842$	39.1942i	1.35072i
$843$	0	0
$844$	3.65144i	0.125688i
$845$	0	0
$846$	0	0
$847$	0	0
$848$	71.3188	2.44910
$849$	0	0
$850$	0	0
$851$	−30.5175	−1.04613
$852$	0	0
$853$	0	0		−	1.00000i	$-0.5\pi$
$853$	0	0			1.00000i	$0.5\pi$
$854$	0	0
$855$	0	0
$856$	−26.8110	−0.916382
$857$	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
$862$	68.2865	2.32585
$863$	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
$872$	21.3562	0.723211
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	3.70377i	0.125067i	0.998043	+	0.0625337i	$0.0199181\pi$
$877$	3.70377i	0.125067i	−0.998043	+	0.0625337i	$0.980082\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	0	0	1.00000			$0$
$881$	0	0	−1.00000			$\pi$
$882$	37.1414i	1.25062i
$883$	−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
$886$	9.88940i	0.332241i
$887$	0	0		−	1.00000i	$-0.5\pi$
$887$	0	0			1.00000i	$0.5\pi$
$888$	0	0
$889$	−27.2814	−0.914990
$890$	0	0
$891$	0	0
$892$	0	0
$893$	0	0
$894$	0	0
$895$	0	0
$896$	29.5384	0.986810
$897$	0	0
$898$	72.2987i	2.41264i
$899$	0	0
$900$	−16.9212	−0.564041
$901$	0	0
$902$	0	0
$903$	0	0
$904$	− 16.6903i	− 0.555110i
$905$	0	0
$906$	0	0
$907$	51.6513	1.71505	0.857526	−	0.514440i	$-0.172000\pi$
$907$	51.6513	1.71505	0.857526	+	0.514440i	$0.172000\pi$
$908$	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
$914$	−13.0437	−0.431446
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	− 58.1801i	− 1.91918i	−0.281394	−	0.959592i	$-0.590797\pi$
$919$	− 58.1801i	− 1.91918i	0.281394	−	0.959592i	$-0.409203\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	0	0
$924$	0	0
$925$	−59.5957	−1.95949
$926$	40.7315i	1.33852i
$927$	0	0
$928$	42.3230	1.38932
$929$	0	0	1.00000			$0$
$929$	0	0	−1.00000			$\pi$
$930$	0	0
$931$	0	0
$932$	23.8770i	0.782117i
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	0	0		−	1.00000i	$-0.5\pi$
$937$	0	0			1.00000i	$0.5\pi$
$938$	−58.8051	−1.92006
$939$	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
$953$	− 48.6206i	− 1.57498i	−0.616330	−	0.787488i	$-0.711381\pi$
$953$	− 48.6206i	− 1.57498i	0.616330	−	0.787488i	$-0.288619\pi$
$954$	75.9314i	2.45837i
$955$	0	0
$956$	− 13.5370i	− 0.437818i
$957$	0	0
$958$	0	0
$959$	− 45.0833i	− 1.45582i
$960$	0	0
$961$	31.0000	1.00000
$962$	0	0
$963$	− 52.1579i	− 1.68076i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	52.7587i	1.69661i	0.529511	+	0.848303i	$0.322376\pi$
$967$	52.7587i	1.69661i	−0.529511	+	0.848303i	$0.677624\pi$
$968$	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
$974$	72.8472i	2.33417i
$975$	0	0
$976$	0	0
$977$	−54.4749	−1.74281	−0.871404	−	0.490567i	$-0.836790\pi$
$977$	−54.4749	−1.74281	−0.871404	+	0.490567i	$0.836790\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	41.5461i	1.32646i
$982$	−9.35876	−0.298650
$983$	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
$989$	− 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
$994$	−75.1253	−2.38283
$995$	0	0
$996$	0	0
$997$	0	0		−	1.00000i	$-0.5\pi$
$997$	0	0			1.00000i	$0.5\pi$
$998$	− 79.0156i	− 2.50120i
$999$	0	0

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
77.2.l.a.41.2	✓	8	11.5	even	5
77.2.l.a.41.2	✓	8	77.27	odd	10
77.2.l.a.62.2	yes	8	11.2	odd	10
77.2.l.a.62.2	yes	8	77.13	even	10
539.2.s.a.19.1		16	77.38	odd	30
539.2.s.a.19.1		16	77.60	even	15
539.2.s.a.117.2		16	77.24	even	30
539.2.s.a.117.2		16	77.46	odd	30
539.2.s.a.129.2		16	77.5	odd	30
539.2.s.a.129.2		16	77.16	even	15
539.2.s.a.227.1		16	77.2	odd	30
539.2.s.a.227.1		16	77.68	even	30
693.2.bu.a.118.1		8	33.5	odd	10
693.2.bu.a.118.1		8	231.104	even	10
693.2.bu.a.370.1		8	33.2	even	10
693.2.bu.a.370.1		8	231.167	odd	10
847.2.b.b.846.2		8	1.1	even	1	trivial
847.2.b.b.846.2		8	7.6	odd	2	CM
847.2.b.b.846.7		8	11.10	odd	2	inner
847.2.b.b.846.7		8	77.76	even	2	inner
847.2.l.a.475.1		8	11.3	even	5
847.2.l.a.475.1		8	77.69	odd	10
847.2.l.a.699.1		8	11.7	odd	10
847.2.l.a.699.1		8	77.62	even	10
847.2.l.c.118.1		8	11.6	odd	10
847.2.l.c.118.1		8	77.6	even	10
847.2.l.c.524.1		8	11.9	even	5
847.2.l.c.524.1		8	77.20	odd	10
847.2.l.d.475.2		8	11.8	odd	10
847.2.l.d.475.2		8	77.41	even	10
847.2.l.d.699.2		8	11.4	even	5
847.2.l.d.699.2		8	77.48	odd	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)

\(f(q)\)	\(=\)	\(q-1.76864i q^{2} -1.12808 q^{4} -2.64575i q^{7} -1.54211i q^{8} +3.00000 q^{9} +O(q^{10})\)	\(q-1.76864i q^{2} -1.12808 q^{4} -2.64575i q^{7} -1.54211i q^{8} +3.00000 q^{9} -4.67938 q^{14} -4.98359 q^{16} -5.30592i q^{18} +2.56038 q^{23} +5.00000 q^{25} +2.98463i q^{28} -7.38626i q^{29} +5.72996i q^{32} -3.38425 q^{36} -11.9191 q^{37} -2.77251i q^{43} -4.52839i q^{46} -7.00000 q^{49} -8.84319i q^{50} -14.3107 q^{53} -4.08003 q^{56} -13.0636 q^{58} -7.93725i q^{63} +0.167048 q^{64} +12.5669 q^{67} +16.0545 q^{71} -4.62632i q^{72} +21.0806i q^{74} +17.5450i q^{79} +9.00000 q^{81} -4.90356 q^{86} -2.88832 q^{92} +12.3805i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 10 q^{4} + 24 q^{9}+O(q^{10}) \) 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

Fields

Representations

Groups

Database

Properties

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