LMFDB - Embedded newform 847.2.b.b.846.4

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			846.4
Root			$-1.41264 - 0.0667372i$ of defining polynomial
Character	$\chi$	$=$	847.846
Dual form			847.2.b.b.846.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.133474i q^{2} +1.98218 q^{4} +2.64575i q^{7} -0.531520i q^{8} +3.00000 q^{9} +O(q^{10})$	$q-0.133474i q^{2} +1.98218 q^{4} +2.64575i q^{7} -0.531520i q^{8} +3.00000 q^{9} +0.353140 q^{14} +3.89342 q^{16} -0.400423i q^{18} -7.50465 q^{23} +5.00000 q^{25} +5.24437i q^{28} +9.73740i q^{29} -1.58271i q^{32} +5.94655 q^{36} +8.21093 q^{37} -11.3343i q^{43} +1.00168i q^{46} -7.00000 q^{49} -0.667372i q^{50} -1.86964 q^{53} +1.40627 q^{56} +1.29969 q^{58} +7.93725i q^{63} +7.57560 q^{64} -6.09473 q^{67} +9.83400 q^{71} -1.59456i q^{72} -1.09595i q^{74} -8.14046i q^{79} +9.00000 q^{81} -1.51284 q^{86} -14.8756 q^{92} +0.934321i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	− 0.133474i	− 0.0943807i	−0.998886	−	0.0471903i	$-0.984973\pi$
$2$	− 0.133474i	− 0.0943807i	0.998886	−	0.0471903i	$-0.0150267\pi$
$3$	0	0	1.00000			$0$
$3$	0	0	−1.00000			$\pi$
$4$	1.98218	0.991092
$5$	0	0	1.00000			$0$
$5$	0	0	−1.00000			$\pi$
$6$	0	0
$7$	2.64575i	1.00000i
$7$	2.64575i	1.00000i
$8$	− 0.531520i	− 0.187921i
$9$	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	$0.5\pi$
$14$	0.353140	0.0943807
$15$	0	0
$16$	3.89342	0.973356
$17$	0	0		−	1.00000i	$-0.5\pi$
$17$	0	0			1.00000i	$0.5\pi$
$18$	− 0.400423i	− 0.0943807i
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−7.50465	−1.56483	−0.782414	−	0.622758i	$-0.786012\pi$
$23$	−7.50465	−1.56483	−0.782414	+	0.622758i	$0.786012\pi$
$24$	0	0
$25$	5.00000	1.00000
$26$	0	0
$27$	0	0
$28$	5.24437i	0.991092i
$29$	9.73740i	1.80819i	0.427331	+	0.904095i	$0.359454\pi$
$29$	9.73740i	1.80819i	−0.427331	+	0.904095i	$0.640546\pi$
$30$	0	0
$31$	0	0	1.00000			$0$
$31$	0	0	−1.00000			$\pi$
$32$	− 1.58271i	− 0.279787i
$33$	0	0
$34$	0	0
$35$	0	0
$36$	5.94655	0.991092
$37$	8.21093	1.34987	0.674935	−	0.737878i	$-0.264172\pi$
$37$	8.21093	1.34987	0.674935	+	0.737878i	$0.264172\pi$
$38$	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
$43$	− 11.3343i	− 1.72847i	−0.503088	−	0.864235i	$-0.667803\pi$
$43$	− 11.3343i	− 1.72847i	0.503088	−	0.864235i	$-0.332197\pi$
$44$	0	0
$45$	0	0
$46$	1.00168i	0.147690i
$47$	0	0	1.00000			$0$
$47$	0	0	−1.00000			$\pi$
$48$	0	0
$49$	−7.00000	−1.00000
$50$	− 0.667372i	− 0.0943807i
$51$	0	0
$52$	0	0
$53$	−1.86964	−0.256814	−0.128407	−	0.991722i	$-0.540986\pi$
$53$	−1.86964	−0.256814	−0.128407	+	0.991722i	$0.540986\pi$
$54$	0	0
$55$	0	0
$56$	1.40627	0.187921
$57$	0	0
$58$	1.29969	0.170658
$59$	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
$64$	7.57560	0.946950
$65$	0	0
$66$	0	0
$67$	−6.09473	−0.744590	−0.372295	−	0.928114i	$-0.621429\pi$
$67$	−6.09473	−0.744590	−0.372295	+	0.928114i	$0.621429\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	9.83400	1.16708	0.583541	−	0.812084i	$-0.301667\pi$
$71$	9.83400	1.16708	0.583541	+	0.812084i	$0.301667\pi$
$72$	− 1.59456i	− 0.187921i
$73$	0	0		−	1.00000i	$-0.5\pi$
$73$	0	0			1.00000i	$0.5\pi$
$74$	− 1.09595i	− 0.127402i
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	− 8.14046i	− 0.915874i	−0.888985	−	0.457937i	$-0.848589\pi$
$79$	− 8.14046i	− 0.915874i	0.888985	−	0.457937i	$-0.151411\pi$
$80$	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
$86$	−1.51284	−0.163134
$87$	0	0
$88$	0	0
$89$	0	0	1.00000			$0$
$89$	0	0	−1.00000			$\pi$
$90$	0	0
$91$	0	0
$92$	−14.8756	−1.55089
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	0	0	1.00000			$0$
$97$	0	0	−1.00000			$\pi$
$98$	0.934321i	0.0943807i
$99$	0	0
$100$	9.91092	0.991092
$101$	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
$106$	0.249548i	0.0242383i
$107$	− 20.6563i	− 1.99692i	−0.0554821	−	0.998460i	$-0.517670\pi$
$107$	− 20.6563i	− 1.99692i	0.0554821	−	0.998460i	$-0.482330\pi$
$108$	0	0
$109$	20.3893i	1.95295i	0.215642	+	0.976473i	$0.430816\pi$
$109$	20.3893i	1.95295i	−0.215642	+	0.976473i	$0.569184\pi$
$110$	0	0
$111$	0	0
$112$	10.3010i	0.973356i
$113$	−14.0591	−1.32257	−0.661285	−	0.750135i	$-0.729988\pi$
$113$	−14.0591	−1.32257	−0.661285	+	0.750135i	$0.729988\pi$
$114$	0	0
$115$	0	0
$116$	19.3013i	1.79208i
$117$	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
$126$	1.05942	0.0943807
$127$	− 20.1224i	− 1.78557i	−0.450479	−	0.892787i	$-0.648747\pi$
$127$	− 20.1224i	− 1.78557i	0.450479	−	0.892787i	$-0.351253\pi$
$128$	− 4.17657i	− 0.369160i
$129$	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
$134$	0.813491i	0.0702749i
$135$	0	0
$136$	0	0
$137$	−23.2202	−1.98384	−0.991920	−	0.126868i	$-0.959507\pi$
$137$	−23.2202	−1.98384	−0.991920	+	0.126868i	$0.959507\pi$
$138$	0	0
$139$	0	0		−	1.00000i	$-0.5\pi$
$139$	0	0			1.00000i	$0.5\pi$
$140$	0	0
$141$	0	0
$142$	− 1.31259i	− 0.110150i
$143$	0	0
$144$	11.6803	0.973356
$145$	0	0
$146$	0	0
$147$	0	0
$148$	16.2756	1.33785
$149$	− 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
$151$	21.1902i	1.72443i	0.506540	+	0.862217i	$0.330924\pi$
$151$	21.1902i	1.72443i	−0.506540	+	0.862217i	$0.669076\pi$
$152$	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			$\pi$
$158$	−1.08654	−0.0864408
$159$	0	0
$160$	0	0
$161$	− 19.8554i	− 1.56483i
$162$	− 1.20127i	− 0.0943807i
$163$	−8.91721	−0.698450	−0.349225	−	0.937039i	$-0.613555\pi$
$163$	−8.91721	−0.698450	−0.349225	+	0.937039i	$0.613555\pi$
$164$	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
$172$	− 22.4668i	− 1.71307i
$173$	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
$179$	−23.9265	−1.78835	−0.894176	−	0.447715i	$-0.852238\pi$
$179$	−23.9265	−1.78835	−0.894176	+	0.447715i	$0.852238\pi$
$180$	0	0
$181$	0	0	1.00000			$0$
$181$	0	0	−1.00000			$\pi$
$182$	0	0
$183$	0	0
$184$	3.98887i	0.294063i
$185$	0	0
$186$	0	0
$187$	0	0
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−22.0235	−1.59356	−0.796781	−	0.604268i	$-0.793466\pi$
$191$	−22.0235	−1.59356	−0.796781	+	0.604268i	$0.793466\pi$
$192$	0	0
$193$	− 6.54353i	− 0.471013i	−0.971873	−	0.235507i	$-0.924325\pi$
$193$	− 6.54353i	− 0.471013i	0.971873	−	0.235507i	$-0.0756750\pi$
$194$	0	0
$195$	0	0
$196$	−13.8753	−0.991092
$197$	− 21.4571i	− 1.52876i	−0.644767	−	0.764379i	$-0.723046\pi$
$197$	− 21.4571i	− 1.52876i	0.644767	−	0.764379i	$-0.276954\pi$
$198$	0	0
$199$	0	0	1.00000			$0$
$199$	0	0	−1.00000			$\pi$
$200$	− 2.65760i	− 0.187921i
$201$	0	0
$202$	0	0
$203$	−25.7627	−1.80819
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−22.5140	−1.56483
$208$	0	0
$209$	0	0
$210$	0	0
$211$	− 19.5885i	− 1.34853i	−0.738490	−	0.674264i	$-0.764461\pi$
$211$	− 19.5885i	− 1.34853i	0.738490	−	0.674264i	$-0.235539\pi$
$212$	−3.70596	−0.254527
$213$	0	0
$214$	−2.75709	−0.188471
$215$	0	0
$216$	0	0
$217$	0	0
$218$	2.72146	0.184320
$219$	0	0
$220$	0	0
$221$	0	0
$222$	0	0
$223$	0	0	1.00000			$0$
$223$	0	0	−1.00000			$\pi$
$224$	4.18746	0.279787
$225$	15.0000	1.00000
$226$	1.87653i	0.124825i
$227$	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
$232$	5.17562	0.339796
$233$	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
$239$	− 30.8091i	− 1.99288i	−0.0843185	−	0.996439i	$-0.526871\pi$
$239$	− 30.8091i	− 1.99288i	0.0843185	−	0.996439i	$-0.473129\pi$
$240$	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			$\pi$
$252$	15.7331i	0.991092i
$253$	0	0
$254$	−2.68582	−0.168524
$255$	0	0
$256$	14.5937	0.912108
$257$	0	0	1.00000			$0$
$257$	0	0	−1.00000			$\pi$
$258$	0	0
$259$	21.7241i	1.34987i
$260$	0	0
$261$	29.2122i	1.80819i
$262$	0	0
$263$	14.5282i	0.895848i	0.894072	+	0.447924i	$0.147836\pi$
$263$	14.5282i	0.895848i	−0.894072	+	0.447924i	$0.852164\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	−12.0809	−0.737958
$269$	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
$274$	3.09931i	0.187236i
$275$	0	0
$276$	0	0
$277$	19.3216i	1.16092i	0.814289	+	0.580460i	$0.197127\pi$
$277$	19.3216i	1.16092i	−0.814289	+	0.580460i	$0.802873\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	− 32.4061i	− 1.93318i	−0.256319	−	0.966592i	$-0.582510\pi$
$281$	− 32.4061i	− 1.93318i	0.256319	−	0.966592i	$-0.417490\pi$
$282$	0	0
$283$	0	0		−	1.00000i	$-0.5\pi$
$283$	0	0			1.00000i	$0.5\pi$
$284$	19.4928	1.15669
$285$	0	0
$286$	0	0
$287$	0	0
$288$	− 4.74814i	− 0.279787i
$289$	−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
$296$	− 4.36427i	− 0.253668i
$297$	0	0
$298$	−1.41256	−0.0818274
$299$	0	0
$300$	0	0
$301$	29.9878	1.72847
$302$	2.82835	0.162753
$303$	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
$316$	− 16.1359i	− 0.907715i
$317$	33.7271	1.89430	0.947152	−	0.320786i	$-0.103947\pi$
$317$	33.7271	1.89430	0.947152	+	0.320786i	$0.103947\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	0	0
$322$	−2.65019	−0.147690
$323$	0	0
$324$	17.8397	0.991092
$325$	0	0
$326$	1.19022i	0.0659202i
$327$	0	0
$328$	0	0
$329$	0	0
$330$	0	0
$331$	6.09209	0.334852	0.167426	−	0.985885i	$-0.446455\pi$
$331$	6.09209	0.334852	0.167426	+	0.985885i	$0.446455\pi$
$332$	0	0
$333$	24.6328	1.34987
$334$	0	0
$335$	0	0
$336$	0	0
$337$	21.9910i	1.19793i	0.800776	+	0.598964i	$0.204421\pi$
$337$	21.9910i	1.19793i	−0.800776	+	0.598964i	$0.795579\pi$
$338$	1.73517i	0.0943807i
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	− 18.5203i	− 1.00000i
$344$	−6.02442	−0.324815
$345$	0	0
$346$	0	0
$347$	27.6153i	1.48247i	0.671248	+	0.741233i	$0.265758\pi$
$347$	27.6153i	1.48247i	−0.671248	+	0.741233i	$0.734242\pi$
$348$	0	0
$349$	0	0		−	1.00000i	$-0.5\pi$
$349$	0	0			1.00000i	$0.5\pi$
$350$	1.76570	0.0943807
$351$	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
$358$	3.19358i	0.168786i
$359$	19.0546i	1.00566i	0.864384	+	0.502832i	$0.167708\pi$
$359$	19.0546i	1.00566i	−0.864384	+	0.502832i	$0.832292\pi$
$360$	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			$\pi$
$368$	−29.2188	−1.52314
$369$	0	0
$370$	0	0
$371$	− 4.94659i	− 0.256814i
$372$	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
$379$	38.9358	2.00000	1.00000		0.000859657i	$-0.000273637\pi$
$379$	38.9358	2.00000	1.00000		0.000859657i	$0.000273637\pi$
$380$	0	0
$381$	0	0
$382$	2.93957i	0.150401i
$383$	0	0	1.00000			$0$
$383$	0	0	−1.00000			$\pi$
$384$	0	0
$385$	0	0
$386$	−0.873393	−0.0444546
$387$	− 34.0030i	− 1.72847i
$388$	0	0
$389$	21.8077	1.10569	0.552847	−	0.833283i	$-0.313542\pi$
$389$	21.8077	1.10569	0.552847	+	0.833283i	$0.313542\pi$
$390$	0	0
$391$	0	0
$392$	3.72064i	0.187921i
$393$	0	0
$394$	−2.86398	−0.144285
$395$	0	0
$396$	0	0
$397$	0	0	1.00000			$0$
$397$	0	0	−1.00000			$\pi$
$398$	0	0
$399$	0	0
$400$	19.4671	0.973356
$401$	−9.62349	−0.480574	−0.240287	−	0.970702i	$-0.577242\pi$
$401$	−9.62349	−0.480574	−0.240287	+	0.970702i	$0.577242\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	0	0
$406$	3.43867i	0.170658i
$407$	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
$414$	3.00504i	0.147690i
$415$	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
$421$	38.2296	1.86319	0.931597	−	0.363492i	$-0.118416\pi$
$421$	38.2296	1.86319	0.931597	+	0.363492i	$0.118416\pi$
$422$	−2.61456	−0.127275
$423$	0	0
$424$	0.993748i	0.0482607i
$425$	0	0
$426$	0	0
$427$	0	0
$428$	− 40.9446i	− 1.97913i
$429$	0	0
$430$	0	0
$431$	22.2580i	1.07213i	0.844177	+	0.536065i	$0.180090\pi$
$431$	22.2580i	1.07213i	−0.844177	+	0.536065i	$0.819910\pi$
$432$	0	0
$433$	0	0	1.00000			$0$
$433$	0	0	−1.00000			$\pi$
$434$	0	0
$435$	0	0
$436$	40.4154i	1.93555i
$437$	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
$443$	37.9522	1.80316	0.901582	−	0.432608i	$-0.142407\pi$
$443$	37.9522	1.80316	0.901582	+	0.432608i	$0.142407\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	20.0431i	0.946950i
$449$	39.6421	1.87083	0.935413	−	0.353556i	$-0.115028\pi$
$449$	39.6421	1.87083	0.935413	+	0.353556i	$0.115028\pi$
$450$	− 2.00212i	− 0.0943807i
$451$	0	0
$452$	−27.8677	−1.31079
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	18.7877i	0.878849i	0.898279	+	0.439425i	$0.144818\pi$
$457$	18.7877i	0.878849i	−0.898279	+	0.439425i	$0.855182\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	0	0		−	1.00000i	$-0.5\pi$
$461$	0	0			1.00000i	$0.5\pi$
$462$	0	0
$463$	−41.6915	−1.93757	−0.968784	−	0.247907i	$-0.920257\pi$
$463$	−41.6915	−1.93757	−0.968784	+	0.247907i	$0.920257\pi$
$464$	37.9118i	1.76001i
$465$	0	0
$466$	2.82512	0.130871
$467$	0	0	1.00000			$0$
$467$	0	0	−1.00000			$\pi$
$468$	0	0
$469$	− 16.1251i	− 0.744590i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	0	0
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−5.60891	−0.256814
$478$	−4.11223	−0.188089
$479$	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
$487$	2.35546	0.106736	0.0533681	−	0.998575i	$-0.483004\pi$
$487$	2.35546	0.106736	0.0533681	+	0.998575i	$0.483004\pi$
$488$	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
$497$	26.0183i	1.16708i
$498$	0	0
$499$	13.5733	0.607623	0.303812	−	0.952732i	$-0.401741\pi$
$499$	13.5733	0.607623	0.303812	+	0.952732i	$0.401741\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	0	0		−	1.00000i	$-0.5\pi$
$503$	0	0			1.00000i	$0.5\pi$
$504$	4.21881	0.187921
$505$	0	0
$506$	0	0
$507$	0	0
$508$	− 39.8863i	− 1.76967i
$509$	0	0	1.00000			$0$
$509$	0	0	−1.00000			$\pi$
$510$	0	0
$511$	0	0
$512$	− 10.3010i	− 0.455246i
$513$	0	0
$514$	0	0
$515$	0	0
$516$	0	0
$517$	0	0
$518$	2.89961	0.127402
$519$	0	0
$520$	0	0
$521$	0	0	1.00000			$0$
$521$	0	0	−1.00000			$\pi$
$522$	3.89908	0.170658
$523$	0	0		−	1.00000i	$-0.5\pi$
$523$	0	0			1.00000i	$0.5\pi$
$524$	0	0
$525$	0	0
$526$	1.93914	0.0845507
$527$	0	0
$528$	0	0
$529$	33.3198	1.44869
$530$	0	0
$531$	0	0
$532$	0	0
$533$	0	0
$534$	0	0
$535$	0	0
$536$	3.23947i	0.139924i
$537$	0	0
$538$	0	0
$539$	0	0
$540$	0	0
$541$	− 22.5249i	− 0.968423i	−0.874951	−	0.484211i	$-0.839107\pi$
$541$	− 22.5249i	− 0.968423i	0.874951	−	0.484211i	$-0.160893\pi$
$542$	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	$0.889785\pi$
$548$	−46.0268	−1.96617
$549$	0	0
$550$	0	0
$551$	0	0
$552$	0	0
$553$	21.5376	0.915874
$554$	2.57893	0.109568
$555$	0	0
$556$	0	0
$557$	35.6000i	1.50842i	0.656634	+	0.754209i	$0.271980\pi$
$557$	35.6000i	1.50842i	−0.656634	+	0.754209i	$0.728020\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0	0
$562$	−4.32538	−0.182455
$563$	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
$568$	− 5.22697i	− 0.219319i
$569$	− 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
$571$	− 18.5207i	− 0.775067i	−0.921856	−	0.387534i	$-0.873327\pi$
$571$	− 18.5207i	− 0.775067i	0.921856	−	0.387534i	$-0.126673\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−37.5233	−1.56483
$576$	22.7268	0.946950
$577$	0	0	1.00000			$0$
$577$	0	0	−1.00000			$\pi$
$578$	2.26906i	0.0943807i
$579$	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
$592$	31.9687	1.31390
$593$	0	0		−	1.00000i	$-0.5\pi$
$593$	0	0			1.00000i	$0.5\pi$
$594$	0	0
$595$	0	0
$596$	− 20.9775i	− 0.859271i
$597$	0	0
$598$	0	0
$599$	25.3391	1.03533	0.517663	−	0.855584i	$-0.326802\pi$
$599$	25.3391	1.03533	0.517663	+	0.855584i	$0.326802\pi$
$600$	0	0
$601$	0	0		−	1.00000i	$-0.5\pi$
$601$	0	0			1.00000i	$0.5\pi$
$602$	− 4.00261i	− 0.163134i
$603$	−18.2842	−0.744590
$604$	42.0029i	1.70907i
$605$	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
$613$	3.34965i	0.135291i	0.997709	+	0.0676456i	$0.0215487\pi$
$613$	3.34965i	0.135291i	−0.997709	+	0.0676456i	$0.978451\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	45.9166	1.84853	0.924266	−	0.381749i	$-0.124678\pi$
$617$	45.9166	1.84853	0.924266	+	0.381749i	$0.124678\pi$
$618$	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
$631$	−40.3484	−1.60624	−0.803122	−	0.595815i	$-0.796829\pi$
$631$	−40.3484	−1.60624	−0.803122	+	0.595815i	$0.796829\pi$
$632$	−4.32682	−0.172112
$633$	0	0
$634$	− 4.50171i	− 0.178786i
$635$	0	0
$636$	0	0
$637$	0	0
$638$	0	0
$639$	29.5020	1.16708
$640$	0	0
$641$	−49.6559	−1.96129	−0.980644	−	0.195799i	$-0.937270\pi$
$641$	−49.6559	−1.96129	−0.980644	+	0.195799i	$0.937270\pi$
$642$	0	0
$643$	0	0	1.00000			$0$
$643$	0	0	−1.00000			$\pi$
$644$	− 39.3572i	− 1.55089i
$645$	0	0
$646$	0	0
$647$	0	0	1.00000			$0$
$647$	0	0	−1.00000			$\pi$
$648$	− 4.78368i	− 0.187921i
$649$	0	0
$650$	0	0
$651$	0	0
$652$	−17.6756	−0.692229
$653$	−5.38581	−0.210763	−0.105382	−	0.994432i	$-0.533606\pi$
$653$	−5.38581	−0.210763	−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			$\pi$
$662$	− 0.813139i	− 0.0316035i
$663$	0	0
$664$	0	0
$665$	0	0
$666$	− 3.28785i	− 0.127402i
$667$	− 73.0758i	− 2.82951i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	0	0
$672$	0	0
$673$	51.8809i	1.99986i	0.0117883	+	0.999931i	$0.496248\pi$
$673$	51.8809i	1.99986i	−0.0117883	+	0.999931i	$0.503752\pi$
$674$	2.93524	0.113061
$675$	0	0
$676$	−25.7684	−0.991092
$677$	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
$683$	−21.1014	−0.807423	−0.403711	−	0.914886i	$-0.632280\pi$
$683$	−21.1014	−0.807423	−0.403711	+	0.914886i	$0.632280\pi$
$684$	0	0
$685$	0	0
$686$	−2.47198	−0.0943807
$687$	0	0
$688$	− 44.1294i	− 1.68242i
$689$	0	0
$690$	0	0
$691$	0	0	1.00000			$0$
$691$	0	0	−1.00000			$\pi$
$692$	0	0
$693$	0	0
$694$	3.68593	0.139916
$695$	0	0
$696$	0	0
$697$	0	0
$698$	0	0
$699$	0	0
$700$	26.2218i	0.991092i
$701$	18.2538i	0.689435i	0.938707	+	0.344717i	$0.112025\pi$
$701$	18.2538i	0.689435i	−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
$709$	26.2486	0.985786	0.492893	−	0.870090i	$-0.335939\pi$
$709$	26.2486	0.985786	0.492893	+	0.870090i	$0.335939\pi$
$710$	0	0
$711$	− 24.4214i	− 0.915874i
$712$	0	0
$713$	0	0
$714$	0	0
$715$	0	0
$716$	−47.4268	−1.77242
$717$	0	0
$718$	2.54330	0.0949152
$719$	0	0	1.00000			$0$
$719$	0	0	−1.00000			$\pi$
$720$	0	0
$721$	0	0
$722$	2.53601i	0.0943807i
$723$	0	0
$724$	0	0
$725$	48.6870i	1.80819i
$726$	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
$736$	11.8777i	0.437818i
$737$	0	0
$738$	0	0
$739$	17.7221i	0.651917i	0.945384	+	0.325959i	$0.105687\pi$
$739$	17.7221i	0.651917i	−0.945384	+	0.325959i	$0.894313\pi$
$740$	0	0
$741$	0	0
$742$	−0.660243	−0.0242383
$743$	53.4778i	1.96191i	0.194233	+	0.980955i	$0.437778\pi$
$743$	53.4778i	1.96191i	−0.194233	+	0.980955i	$0.562222\pi$
$744$	0	0
$745$	0	0
$746$	−4.23768	−0.155153
$747$	0	0
$748$	0	0
$749$	54.6514	1.99692
$750$	0	0
$751$	−10.3298	−0.376939	−0.188469	−	0.982079i	$-0.560353\pi$
$751$	−10.3298	−0.376939	−0.188469	+	0.982079i	$0.560353\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−37.4664	−1.36174	−0.680869	−	0.732405i	$-0.738398\pi$
$757$	−37.4664	−1.36174	−0.680869	+	0.732405i	$0.738398\pi$
$758$	− 5.19694i	− 0.188761i
$759$	0	0
$760$	0	0
$761$	0	0		−	1.00000i	$-0.5\pi$
$761$	0	0			1.00000i	$0.5\pi$
$762$	0	0
$763$	−53.9451	−1.95295
$764$	−43.6546	−1.57937
$765$	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
$772$	− 12.9705i	− 0.466818i
$773$	0	0	1.00000			$0$
$773$	0	0	−1.00000			$\pi$
$774$	−4.53853	−0.163134
$775$	0	0
$776$	0	0
$777$	0	0
$778$	− 2.91077i	− 0.104356i
$779$	0	0
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	−27.2540	−0.973356
$785$	0	0
$786$	0	0
$787$	0	0		−	1.00000i	$-0.5\pi$
$787$	0	0			1.00000i	$0.5\pi$
$788$	− 42.5320i	− 1.51514i
$789$	0	0
$790$	0	0
$791$	− 37.1969i	− 1.32257i
$792$	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
$800$	− 7.91356i	− 0.279787i
$801$	0	0
$802$	1.28449i	0.0453569i
$803$	0	0
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	− 23.0588i	− 0.810705i	−0.914160	−	0.405353i	$-0.867149\pi$
$809$	− 23.0588i	− 0.810705i	0.914160	−	0.405353i	$-0.132851\pi$
$810$	0	0
$811$	0	0		−	1.00000i	$-0.5\pi$
$811$	0	0			1.00000i	$0.5\pi$
$812$	−51.0665	−1.79208
$813$	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
$823$	−53.8809	−1.87817	−0.939086	−	0.343683i	$-0.888326\pi$
$823$	−53.8809	−1.87817	−0.939086	+	0.343683i	$0.888326\pi$
$824$	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	$-0.222732\pi$
$828$	−44.6268	−1.55089
$829$	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
$841$	−65.8170	−2.26955
$842$	− 5.10267i	− 0.175849i
$843$	0	0
$844$	− 38.8280i	− 1.33652i
$845$	0	0
$846$	0	0
$847$	0	0
$848$	−7.27929	−0.249972
$849$	0	0
$850$	0	0
$851$	−61.6202	−2.11231
$852$	0	0
$853$	0	0		−	1.00000i	$-0.5\pi$
$853$	0	0			1.00000i	$0.5\pi$
$854$	0	0
$855$	0	0
$856$	−10.9792	−0.375262
$857$	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
$862$	2.97087	0.101188
$863$	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
$872$	10.8373	0.366999
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	55.0748i	1.85974i	0.367885	+	0.929871i	$0.380082\pi$
$877$	55.0748i	1.85974i	−0.367885	+	0.929871i	$0.619918\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	0	0	1.00000			$0$
$881$	0	0	−1.00000			$\pi$
$882$	2.80296i	0.0943807i
$883$	−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
$886$	− 5.06565i	− 0.170184i
$887$	0	0		−	1.00000i	$-0.5\pi$
$887$	0	0			1.00000i	$0.5\pi$
$888$	0	0
$889$	53.2389	1.78557
$890$	0	0
$891$	0	0
$892$	0	0
$893$	0	0
$894$	0	0
$895$	0	0
$896$	11.0502	0.369160
$897$	0	0
$898$	− 5.29121i	− 0.176570i
$899$	0	0
$900$	29.7328	0.991092
$901$	0	0
$902$	0	0
$903$	0	0
$904$	7.47269i	0.248538i
$905$	0	0
$906$	0	0
$907$	45.4308	1.50850	0.754252	−	0.656585i	$-0.228000\pi$
$907$	45.4308	1.50850	0.754252	+	0.656585i	$0.228000\pi$
$908$	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
$914$	2.50767	0.0829464
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	1.75272i	0.0578168i	0.999582	+	0.0289084i	$0.00920311\pi$
$919$	1.75272i	0.0578168i	−0.999582	+	0.0289084i	$0.990797\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	0	0
$924$	0	0
$925$	41.0547	1.34987
$926$	5.56475i	0.182869i
$927$	0	0
$928$	15.4115	0.505907
$929$	0	0	1.00000			$0$
$929$	0	0	−1.00000			$\pi$
$930$	0	0
$931$	0	0
$932$	41.9549i	1.37428i
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	0	0		−	1.00000i	$-0.5\pi$
$937$	0	0			1.00000i	$0.5\pi$
$938$	−2.15229	−0.0702749
$939$	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
$953$	− 61.7019i	− 1.99872i	−0.0357473	−	0.999361i	$-0.511381\pi$
$953$	− 61.7019i	− 1.99872i	0.0357473	−	0.999361i	$-0.488619\pi$
$954$	0.748645i	0.0242383i
$955$	0	0
$956$	− 61.0694i	− 1.97513i
$957$	0	0
$958$	0	0
$959$	− 61.4350i	− 1.98384i
$960$	0	0
$961$	31.0000	1.00000
$962$	0	0
$963$	− 61.9689i	− 1.99692i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	23.3258i	0.750107i	0.927003	+	0.375053i	$0.122376\pi$
$967$	23.3258i	0.750107i	−0.927003	+	0.375053i	$0.877624\pi$
$968$	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
$974$	− 0.314394i	− 0.0100738i
$975$	0	0
$976$	0	0
$977$	26.0454	0.833265	0.416632	−	0.909075i	$-0.363210\pi$
$977$	26.0454	0.833265	0.416632	+	0.909075i	$0.363210\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	61.1680i	1.95295i
$982$	0.706280	0.0225383
$983$	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
$989$	85.0603i	2.70476i
$990$	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
$994$	3.47278	0.110150
$995$	0	0
$996$	0	0
$997$	0	0		−	1.00000i	$-0.5\pi$
$997$	0	0			1.00000i	$0.5\pi$
$998$	− 1.81168i	− 0.0573479i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	847.2.b.b.846.4		8
7.6	odd	2	CM	847.2.b.b.846.4		8
11.2	odd	10		77.2.l.a.62.1	yes	8
11.3	even	5		847.2.l.a.475.2		8
11.4	even	5		847.2.l.d.699.1		8
11.5	even	5		77.2.l.a.41.1	✓	8
11.6	odd	10		847.2.l.c.118.2		8
11.7	odd	10		847.2.l.a.699.2		8
11.8	odd	10		847.2.l.d.475.1		8
11.9	even	5		847.2.l.c.524.2		8
11.10	odd	2	inner	847.2.b.b.846.5		8
33.2	even	10		693.2.bu.a.370.2		8
33.5	odd	10		693.2.bu.a.118.2		8
77.2	odd	30		539.2.s.a.227.2		16
77.5	odd	30		539.2.s.a.129.1		16
77.6	even	10		847.2.l.c.118.2		8
77.13	even	10		77.2.l.a.62.1	yes	8
77.16	even	15		539.2.s.a.129.1		16
77.20	odd	10		847.2.l.c.524.2		8
77.24	even	30		539.2.s.a.117.1		16
77.27	odd	10		77.2.l.a.41.1	✓	8
77.38	odd	30		539.2.s.a.19.2		16
77.41	even	10		847.2.l.d.475.1		8
77.46	odd	30		539.2.s.a.117.1		16
77.48	odd	10		847.2.l.d.699.1		8
77.60	even	15		539.2.s.a.19.2		16
77.62	even	10		847.2.l.a.699.2		8
77.68	even	30		539.2.s.a.227.2		16
77.69	odd	10		847.2.l.a.475.2		8
77.76	even	2	inner	847.2.b.b.846.5		8
231.104	even	10		693.2.bu.a.118.2		8
231.167	odd	10		693.2.bu.a.370.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
77.2.l.a.41.1	✓	8	11.5	even	5
77.2.l.a.41.1	✓	8	77.27	odd	10
77.2.l.a.62.1	yes	8	11.2	odd	10
77.2.l.a.62.1	yes	8	77.13	even	10
539.2.s.a.19.2		16	77.38	odd	30
539.2.s.a.19.2		16	77.60	even	15
539.2.s.a.117.1		16	77.24	even	30
539.2.s.a.117.1		16	77.46	odd	30
539.2.s.a.129.1		16	77.5	odd	30
539.2.s.a.129.1		16	77.16	even	15
539.2.s.a.227.2		16	77.2	odd	30
539.2.s.a.227.2		16	77.68	even	30
693.2.bu.a.118.2		8	33.5	odd	10
693.2.bu.a.118.2		8	231.104	even	10
693.2.bu.a.370.2		8	33.2	even	10
693.2.bu.a.370.2		8	231.167	odd	10
847.2.b.b.846.4		8	1.1	even	1	trivial
847.2.b.b.846.4		8	7.6	odd	2	CM
847.2.b.b.846.5		8	11.10	odd	2	inner
847.2.b.b.846.5		8	77.76	even	2	inner
847.2.l.a.475.2		8	11.3	even	5
847.2.l.a.475.2		8	77.69	odd	10
847.2.l.a.699.2		8	11.7	odd	10
847.2.l.a.699.2		8	77.62	even	10
847.2.l.c.118.2		8	11.6	odd	10
847.2.l.c.118.2		8	77.6	even	10
847.2.l.c.524.2		8	11.9	even	5
847.2.l.c.524.2		8	77.20	odd	10
847.2.l.d.475.1		8	11.8	odd	10
847.2.l.d.475.1		8	77.41	even	10
847.2.l.d.699.1		8	11.4	even	5
847.2.l.d.699.1		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)

\(f(q)\)	\(=\)	\(q-0.133474i q^{2} +1.98218 q^{4} +2.64575i q^{7} -0.531520i q^{8} +3.00000 q^{9} +O(q^{10})\)	\(q-0.133474i q^{2} +1.98218 q^{4} +2.64575i q^{7} -0.531520i q^{8} +3.00000 q^{9} +0.353140 q^{14} +3.89342 q^{16} -0.400423i q^{18} -7.50465 q^{23} +5.00000 q^{25} +5.24437i q^{28} +9.73740i q^{29} -1.58271i q^{32} +5.94655 q^{36} +8.21093 q^{37} -11.3343i q^{43} +1.00168i q^{46} -7.00000 q^{49} -0.667372i q^{50} -1.86964 q^{53} +1.40627 q^{56} +1.29969 q^{58} +7.93725i q^{63} +7.57560 q^{64} -6.09473 q^{67} +9.83400 q^{71} -1.59456i q^{72} -1.09595i q^{74} -8.14046i q^{79} +9.00000 q^{81} -1.51284 q^{86} -14.8756 q^{92} +0.934321i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 10 q^{4} + 24 q^{9}+O(q^{10}) \) 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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