LMFDB - Embedded newform 783.2.a.b.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [783,2,Mod(1,783)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(783, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("783.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 783 = 3^{3} \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	783.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,-1,0,5] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(4)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$6.25228647827$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{17}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 4 $ x^2 - x - 4
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.56155$ of defining polynomial
Character	$\chi$	$=$	783.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.56155 q^{2} +0.438447 q^{4} -2.00000 q^{5} +0.561553 q^{7} -2.43845 q^{8} -3.12311 q^{10} -5.12311 q^{11} -2.12311 q^{13} +0.876894 q^{14} -4.68466 q^{16} -0.438447 q^{17} +3.00000 q^{19} -0.876894 q^{20} -8.00000 q^{22} -1.56155 q^{23} -1.00000 q^{25} -3.31534 q^{26} +0.246211 q^{28} -1.00000 q^{29} -9.56155 q^{31} -2.43845 q^{32} -0.684658 q^{34} -1.12311 q^{35} +9.68466 q^{37} +4.68466 q^{38} +4.87689 q^{40} -2.68466 q^{41} +7.80776 q^{43} -2.24621 q^{44} -2.43845 q^{46} +8.00000 q^{47} -6.68466 q^{49} -1.56155 q^{50} -0.930870 q^{52} -10.0000 q^{53} +10.2462 q^{55} -1.36932 q^{56} -1.56155 q^{58} -4.43845 q^{59} -1.43845 q^{61} -14.9309 q^{62} +5.56155 q^{64} +4.24621 q^{65} +6.56155 q^{67} -0.192236 q^{68} -1.75379 q^{70} -3.56155 q^{71} -0.315342 q^{73} +15.1231 q^{74} +1.31534 q^{76} -2.87689 q^{77} +1.43845 q^{79} +9.36932 q^{80} -4.19224 q^{82} -11.5616 q^{83} +0.876894 q^{85} +12.1922 q^{86} +12.4924 q^{88} -2.00000 q^{89} -1.19224 q^{91} -0.684658 q^{92} +12.4924 q^{94} -6.00000 q^{95} +3.68466 q^{97} -10.4384 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - q^{2} + 5 q^{4} - 4 q^{5} - 3 q^{7} - 9 q^{8} + 2 q^{10} - 2 q^{11} + 4 q^{13} + 10 q^{14} + 3 q^{16} - 5 q^{17} + 6 q^{19} - 10 q^{20} - 16 q^{22} + q^{23} - 2 q^{25} - 19 q^{26} - 16 q^{28} - 2 q^{29}+ \cdots - 25 q^{98}+O(q^{100}) $ 2 * q - q^2 + 5 * q^4 - 4 * q^5 - 3 * q^7 - 9 * q^8 + 2 * q^10 - 2 * q^11 + 4 * q^13 + 10 * q^14 + 3 * q^16 - 5 * q^17 + 6 * q^19 - 10 * q^20 - 16 * q^22 + q^23 - 2 * q^25 - 19 * q^26 - 16 * q^28 - 2 * q^29 - 15 * q^31 - 9 * q^32 + 11 * q^34 + 6 * q^35 + 7 * q^37 - 3 * q^38 + 18 * q^40 + 7 * q^41 - 5 * q^43 + 12 * q^44 - 9 * q^46 + 16 * q^47 - q^49 + q^50 + 27 * q^52 - 20 * q^53 + 4 * q^55 + 22 * q^56 + q^58 - 13 * q^59 - 7 * q^61 - q^62 + 7 * q^64 - 8 * q^65 + 9 * q^67 - 21 * q^68 - 20 * q^70 - 3 * q^71 - 13 * q^73 + 22 * q^74 + 15 * q^76 - 14 * q^77 + 7 * q^79 - 6 * q^80 - 29 * q^82 - 19 * q^83 + 10 * q^85 + 45 * q^86 - 8 * q^88 - 4 * q^89 - 23 * q^91 + 11 * q^92 - 8 * q^94 - 12 * q^95 - 5 * q^97 - 25 * q^98

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 $

$2$	1.56155	1.10418	0.552092	−	0.833783i	$-0.313830\pi$
$2$	1.56155	1.10418	0.552092	+	0.833783i	$0.313830\pi$
$3$	0	0
$3$	0	0
$4$	0.438447	0.219224
$5$	−2.00000	−0.894427	−0.447214	−	0.894427i	$-0.647584\pi$
$5$	−2.00000	−0.894427	−0.447214	+	0.894427i	$0.647584\pi$
$6$	0	0
$7$	0.561553	0.212247	0.106124	−	0.994353i	$-0.466156\pi$
$7$	0.561553	0.212247	0.106124	+	0.994353i	$0.466156\pi$
$8$	−2.43845	−0.862121
$9$	0	0
$10$	−3.12311	−0.987613
$11$	−5.12311	−1.54467	−0.772337	−	0.635213i	$-0.780912\pi$
$11$	−5.12311	−1.54467	−0.772337	+	0.635213i	$0.780912\pi$
$12$	0	0
$13$	−2.12311	−0.588844	−0.294422	−	0.955676i	$-0.595127\pi$
$13$	−2.12311	−0.588844	−0.294422	+	0.955676i	$0.595127\pi$
$14$	0.876894	0.234360
$15$	0	0
$16$	−4.68466	−1.17116
$17$	−0.438447	−0.106339	−0.0531695	−	0.998586i	$-0.516932\pi$
$17$	−0.438447	−0.106339	−0.0531695	+	0.998586i	$0.516932\pi$
$18$	0	0
$19$	3.00000	0.688247	0.344124	−	0.938924i	$-0.388176\pi$
$19$	3.00000	0.688247	0.344124	+	0.938924i	$0.388176\pi$
$20$	−0.876894	−0.196080
$21$	0	0
$22$	−8.00000	−1.70561
$23$	−1.56155	−0.325606	−0.162803	−	0.986659i	$-0.552054\pi$
$23$	−1.56155	−0.325606	−0.162803	+	0.986659i	$0.552054\pi$
$24$	0	0
$25$	−1.00000	−0.200000
$26$	−3.31534	−0.650192
$27$	0	0
$28$	0.246211	0.0465296
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	0	0
$31$	−9.56155	−1.71731	−0.858653	−	0.512558i	$-0.828698\pi$
$31$	−9.56155	−1.71731	−0.858653	+	0.512558i	$0.828698\pi$
$32$	−2.43845	−0.431061
$33$	0	0
$34$	−0.684658	−0.117418
$35$	−1.12311	−0.189839
$36$	0	0
$37$	9.68466	1.59215	0.796074	−	0.605199i	$-0.206907\pi$
$37$	9.68466	1.59215	0.796074	+	0.605199i	$0.206907\pi$
$38$	4.68466	0.759952
$39$	0	0
$40$	4.87689	0.771105
$41$	−2.68466	−0.419273	−0.209637	−	0.977779i	$-0.567228\pi$
$41$	−2.68466	−0.419273	−0.209637	+	0.977779i	$0.567228\pi$
$42$	0	0
$43$	7.80776	1.19067	0.595336	−	0.803477i	$-0.297019\pi$
$43$	7.80776	1.19067	0.595336	+	0.803477i	$0.297019\pi$
$44$	−2.24621	−0.338629
$45$	0	0
$46$	−2.43845	−0.359529
$47$	8.00000	1.16692	0.583460	−	0.812142i	$-0.301699\pi$
$47$	8.00000	1.16692	0.583460	+	0.812142i	$0.301699\pi$
$48$	0	0
$49$	−6.68466	−0.954951
$50$	−1.56155	−0.220837
$51$	0	0
$52$	−0.930870	−0.129088
$53$	−10.0000	−1.37361	−0.686803	−	0.726844i	$-0.740986\pi$
$53$	−10.0000	−1.37361	−0.686803	+	0.726844i	$0.740986\pi$
$54$	0	0
$55$	10.2462	1.38160
$56$	−1.36932	−0.182983
$57$	0	0
$58$	−1.56155	−0.205042
$59$	−4.43845	−0.577837	−0.288918	−	0.957354i	$-0.593296\pi$
$59$	−4.43845	−0.577837	−0.288918	+	0.957354i	$0.593296\pi$
$60$	0	0
$61$	−1.43845	−0.184174	−0.0920871	−	0.995751i	$-0.529354\pi$
$61$	−1.43845	−0.184174	−0.0920871	+	0.995751i	$0.529354\pi$
$62$	−14.9309	−1.89622
$63$	0	0
$64$	5.56155	0.695194
$65$	4.24621	0.526678
$66$	0	0
$67$	6.56155	0.801621	0.400811	−	0.916161i	$-0.368729\pi$
$67$	6.56155	0.801621	0.400811	+	0.916161i	$0.368729\pi$
$68$	−0.192236	−0.0233120
$69$	0	0
$70$	−1.75379	−0.209618
$71$	−3.56155	−0.422679	−0.211339	−	0.977413i	$-0.567782\pi$
$71$	−3.56155	−0.422679	−0.211339	+	0.977413i	$0.567782\pi$
$72$	0	0
$73$	−0.315342	−0.0369079	−0.0184540	−	0.999830i	$-0.505874\pi$
$73$	−0.315342	−0.0369079	−0.0184540	+	0.999830i	$0.505874\pi$
$74$	15.1231	1.75803
$75$	0	0
$76$	1.31534	0.150880
$77$	−2.87689	−0.327853
$78$	0	0
$79$	1.43845	0.161838	0.0809190	−	0.996721i	$-0.474214\pi$
$79$	1.43845	0.161838	0.0809190	+	0.996721i	$0.474214\pi$
$80$	9.36932	1.04752
$81$	0	0
$82$	−4.19224	−0.462955
$83$	−11.5616	−1.26905	−0.634523	−	0.772904i	$-0.718803\pi$
$83$	−11.5616	−1.26905	−0.634523	+	0.772904i	$0.718803\pi$
$84$	0	0
$85$	0.876894	0.0951125
$86$	12.1922	1.31472
$87$	0	0
$88$	12.4924	1.33170
$89$	−2.00000	−0.212000	−0.106000	−	0.994366i	$-0.533804\pi$
$89$	−2.00000	−0.212000	−0.106000	+	0.994366i	$0.533804\pi$
$90$	0	0
$91$	−1.19224	−0.124980
$92$	−0.684658	−0.0713806
$93$	0	0
$94$	12.4924	1.28849
$95$	−6.00000	−0.615587
$96$	0	0
$97$	3.68466	0.374120	0.187060	−	0.982348i	$-0.440104\pi$
$97$	3.68466	0.374120	0.187060	+	0.982348i	$0.440104\pi$
$98$	−10.4384	−1.05444
$99$	0	0
$100$	−0.438447	−0.0438447
$101$	7.80776	0.776902	0.388451	−	0.921469i	$-0.373010\pi$
$101$	7.80776	0.776902	0.388451	+	0.921469i	$0.373010\pi$
$102$	0	0
$103$	4.56155	0.449463	0.224732	−	0.974421i	$-0.427849\pi$
$103$	4.56155	0.449463	0.224732	+	0.974421i	$0.427849\pi$
$104$	5.17708	0.507655
$105$	0	0
$106$	−15.6155	−1.51671
$107$	1.12311	0.108575	0.0542874	−	0.998525i	$-0.482711\pi$
$107$	1.12311	0.108575	0.0542874	+	0.998525i	$0.482711\pi$
$108$	0	0
$109$	19.5616	1.87366	0.936828	−	0.349789i	$-0.113747\pi$
$109$	19.5616	1.87366	0.936828	+	0.349789i	$0.113747\pi$
$110$	16.0000	1.52554
$111$	0	0
$112$	−2.63068	−0.248576
$113$	1.80776	0.170060	0.0850301	−	0.996378i	$-0.472901\pi$
$113$	1.80776	0.170060	0.0850301	+	0.996378i	$0.472901\pi$
$114$	0	0
$115$	3.12311	0.291231
$116$	−0.438447	−0.0407088
$117$	0	0
$118$	−6.93087	−0.638038
$119$	−0.246211	−0.0225701
$120$	0	0
$121$	15.2462	1.38602
$122$	−2.24621	−0.203362
$123$	0	0
$124$	−4.19224	−0.376474
$125$	12.0000	1.07331
$126$	0	0
$127$	−20.4924	−1.81841	−0.909204	−	0.416350i	$-0.863309\pi$
$127$	−20.4924	−1.81841	−0.909204	+	0.416350i	$0.863309\pi$
$128$	13.5616	1.19868
$129$	0	0
$130$	6.63068	0.581549
$131$	−1.75379	−0.153229	−0.0766146	−	0.997061i	$-0.524411\pi$
$131$	−1.75379	−0.153229	−0.0766146	+	0.997061i	$0.524411\pi$
$132$	0	0
$133$	1.68466	0.146078
$134$	10.2462	0.885138
$135$	0	0
$136$	1.06913	0.0916772
$137$	−16.6847	−1.42547	−0.712733	−	0.701435i	$-0.752543\pi$
$137$	−16.6847	−1.42547	−0.712733	+	0.701435i	$0.752543\pi$
$138$	0	0
$139$	−5.68466	−0.482166	−0.241083	−	0.970504i	$-0.577503\pi$
$139$	−5.68466	−0.482166	−0.241083	+	0.970504i	$0.577503\pi$
$140$	−0.492423	−0.0416173
$141$	0	0
$142$	−5.56155	−0.466715
$143$	10.8769	0.909572
$144$	0	0
$145$	2.00000	0.166091
$146$	−0.492423	−0.0407532
$147$	0	0
$148$	4.24621	0.349036
$149$	−18.2462	−1.49479	−0.747394	−	0.664381i	$-0.768695\pi$
$149$	−18.2462	−1.49479	−0.747394	+	0.664381i	$0.768695\pi$
$150$	0	0
$151$	−12.5616	−1.02224	−0.511122	−	0.859508i	$-0.670770\pi$
$151$	−12.5616	−1.02224	−0.511122	+	0.859508i	$0.670770\pi$
$152$	−7.31534	−0.593353
$153$	0	0
$154$	−4.49242	−0.362010
$155$	19.1231	1.53600
$156$	0	0
$157$	−18.0000	−1.43656	−0.718278	−	0.695756i	$-0.755069\pi$
$157$	−18.0000	−1.43656	−0.718278	+	0.695756i	$0.755069\pi$
$158$	2.24621	0.178699
$159$	0	0
$160$	4.87689	0.385552
$161$	−0.876894	−0.0691090
$162$	0	0
$163$	−2.12311	−0.166294	−0.0831472	−	0.996537i	$-0.526497\pi$
$163$	−2.12311	−0.166294	−0.0831472	+	0.996537i	$0.526497\pi$
$164$	−1.17708	−0.0919146
$165$	0	0
$166$	−18.0540	−1.40126
$167$	21.5616	1.66848	0.834242	−	0.551399i	$-0.185906\pi$
$167$	21.5616	1.66848	0.834242	+	0.551399i	$0.185906\pi$
$168$	0	0
$169$	−8.49242	−0.653263
$170$	1.36932	0.105022
$171$	0	0
$172$	3.42329	0.261024
$173$	14.2462	1.08312	0.541560	−	0.840662i	$-0.317834\pi$
$173$	14.2462	1.08312	0.541560	+	0.840662i	$0.317834\pi$
$174$	0	0
$175$	−0.561553	−0.0424494
$176$	24.0000	1.80907
$177$	0	0
$178$	−3.12311	−0.234087
$179$	−0.684658	−0.0511738	−0.0255869	−	0.999673i	$-0.508145\pi$
$179$	−0.684658	−0.0511738	−0.0255869	+	0.999673i	$0.508145\pi$
$180$	0	0
$181$	4.12311	0.306468	0.153234	−	0.988190i	$-0.451031\pi$
$181$	4.12311	0.306468	0.153234	+	0.988190i	$0.451031\pi$
$182$	−1.86174	−0.138001
$183$	0	0
$184$	3.80776	0.280712
$185$	−19.3693	−1.42406
$186$	0	0
$187$	2.24621	0.164259
$188$	3.50758	0.255816
$189$	0	0
$190$	−9.36932	−0.679722
$191$	15.6155	1.12990	0.564950	−	0.825125i	$-0.308895\pi$
$191$	15.6155	1.12990	0.564950	+	0.825125i	$0.308895\pi$
$192$	0	0
$193$	0.807764	0.0581441	0.0290721	−	0.999577i	$-0.490745\pi$
$193$	0.807764	0.0581441	0.0290721	+	0.999577i	$0.490745\pi$
$194$	5.75379	0.413098
$195$	0	0
$196$	−2.93087	−0.209348
$197$	−23.3693	−1.66499	−0.832497	−	0.554029i	$-0.813090\pi$
$197$	−23.3693	−1.66499	−0.832497	+	0.554029i	$0.813090\pi$
$198$	0	0
$199$	−17.6847	−1.25363	−0.626816	−	0.779167i	$-0.715642\pi$
$199$	−17.6847	−1.25363	−0.626816	+	0.779167i	$0.715642\pi$
$200$	2.43845	0.172424
$201$	0	0
$202$	12.1922	0.857843
$203$	−0.561553	−0.0394133
$204$	0	0
$205$	5.36932	0.375009
$206$	7.12311	0.496290
$207$	0	0
$208$	9.94602	0.689633
$209$	−15.3693	−1.06312
$210$	0	0
$211$	−13.9309	−0.959041	−0.479520	−	0.877531i	$-0.659189\pi$
$211$	−13.9309	−0.959041	−0.479520	+	0.877531i	$0.659189\pi$
$212$	−4.38447	−0.301127
$213$	0	0
$214$	1.75379	0.119887
$215$	−15.6155	−1.06497
$216$	0	0
$217$	−5.36932	−0.364493
$218$	30.5464	2.06886
$219$	0	0
$220$	4.49242	0.302879
$221$	0.930870	0.0626171
$222$	0	0
$223$	19.1231	1.28058	0.640289	−	0.768134i	$-0.278815\pi$
$223$	19.1231	1.28058	0.640289	+	0.768134i	$0.278815\pi$
$224$	−1.36932	−0.0914913
$225$	0	0
$226$	2.82292	0.187778
$227$	6.93087	0.460018	0.230009	−	0.973189i	$-0.426124\pi$
$227$	6.93087	0.460018	0.230009	+	0.973189i	$0.426124\pi$
$228$	0	0
$229$	−6.00000	−0.396491	−0.198246	−	0.980152i	$-0.563524\pi$
$229$	−6.00000	−0.396491	−0.198246	+	0.980152i	$0.563524\pi$
$230$	4.87689	0.321573
$231$	0	0
$232$	2.43845	0.160092
$233$	−14.4924	−0.949430	−0.474715	−	0.880140i	$-0.657449\pi$
$233$	−14.4924	−0.949430	−0.474715	+	0.880140i	$0.657449\pi$
$234$	0	0
$235$	−16.0000	−1.04372
$236$	−1.94602	−0.126675
$237$	0	0
$238$	−0.384472	−0.0249216
$239$	5.56155	0.359747	0.179873	−	0.983690i	$-0.442431\pi$
$239$	5.56155	0.359747	0.179873	+	0.983690i	$0.442431\pi$
$240$	0	0
$241$	−18.8078	−1.21151	−0.605757	−	0.795649i	$-0.707130\pi$
$241$	−18.8078	−1.21151	−0.605757	+	0.795649i	$0.707130\pi$
$242$	23.8078	1.53042
$243$	0	0
$244$	−0.630683	−0.0403753
$245$	13.3693	0.854134
$246$	0	0
$247$	−6.36932	−0.405270
$248$	23.3153	1.48053
$249$	0	0
$250$	18.7386	1.18514
$251$	19.6155	1.23812	0.619061	−	0.785343i	$-0.287514\pi$
$251$	19.6155	1.23812	0.619061	+	0.785343i	$0.287514\pi$
$252$	0	0
$253$	8.00000	0.502956
$254$	−32.0000	−2.00786
$255$	0	0
$256$	10.0540	0.628373
$257$	−10.4924	−0.654499	−0.327250	−	0.944938i	$-0.606122\pi$
$257$	−10.4924	−0.654499	−0.327250	+	0.944938i	$0.606122\pi$
$258$	0	0
$259$	5.43845	0.337929
$260$	1.86174	0.115460
$261$	0	0
$262$	−2.73863	−0.169193
$263$	−10.0000	−0.616626	−0.308313	−	0.951285i	$-0.599764\pi$
$263$	−10.0000	−0.616626	−0.308313	+	0.951285i	$0.599764\pi$
$264$	0	0
$265$	20.0000	1.22859
$266$	2.63068	0.161298
$267$	0	0
$268$	2.87689	0.175734
$269$	22.6847	1.38311	0.691554	−	0.722325i	$-0.256926\pi$
$269$	22.6847	1.38311	0.691554	+	0.722325i	$0.256926\pi$
$270$	0	0
$271$	−19.2462	−1.16912	−0.584562	−	0.811349i	$-0.698734\pi$
$271$	−19.2462	−1.16912	−0.584562	+	0.811349i	$0.698734\pi$
$272$	2.05398	0.124541
$273$	0	0
$274$	−26.0540	−1.57398
$275$	5.12311	0.308935
$276$	0	0
$277$	−15.1771	−0.911902	−0.455951	−	0.890005i	$-0.650701\pi$
$277$	−15.1771	−0.911902	−0.455951	+	0.890005i	$0.650701\pi$
$278$	−8.87689	−0.532401
$279$	0	0
$280$	2.73863	0.163665
$281$	−24.7386	−1.47578	−0.737892	−	0.674919i	$-0.764178\pi$
$281$	−24.7386	−1.47578	−0.737892	+	0.674919i	$0.764178\pi$
$282$	0	0
$283$	16.8769	1.00323	0.501614	−	0.865092i	$-0.332740\pi$
$283$	16.8769	1.00323	0.501614	+	0.865092i	$0.332740\pi$
$284$	−1.56155	−0.0926611
$285$	0	0
$286$	16.9848	1.00433
$287$	−1.50758	−0.0889895
$288$	0	0
$289$	−16.8078	−0.988692
$290$	3.12311	0.183395
$291$	0	0
$292$	−0.138261	−0.00809109
$293$	−16.2462	−0.949114	−0.474557	−	0.880225i	$-0.657392\pi$
$293$	−16.2462	−0.949114	−0.474557	+	0.880225i	$0.657392\pi$
$294$	0	0
$295$	8.87689	0.516833
$296$	−23.6155	−1.37262
$297$	0	0
$298$	−28.4924	−1.65052
$299$	3.31534	0.191731
$300$	0	0
$301$	4.38447	0.252717
$302$	−19.6155	−1.12875
$303$	0	0
$304$	−14.0540	−0.806051
$305$	2.87689	0.164730
$306$	0	0
$307$	−8.49242	−0.484688	−0.242344	−	0.970190i	$-0.577916\pi$
$307$	−8.49242	−0.484688	−0.242344	+	0.970190i	$0.577916\pi$
$308$	−1.26137	−0.0718730
$309$	0	0
$310$	29.8617	1.69603
$311$	−10.2462	−0.581009	−0.290505	−	0.956874i	$-0.593823\pi$
$311$	−10.2462	−0.581009	−0.290505	+	0.956874i	$0.593823\pi$
$312$	0	0
$313$	32.8617	1.85746	0.928728	−	0.370763i	$-0.120904\pi$
$313$	32.8617	1.85746	0.928728	+	0.370763i	$0.120904\pi$
$314$	−28.1080	−1.58622
$315$	0	0
$316$	0.630683	0.0354787
$317$	−22.6847	−1.27410	−0.637049	−	0.770824i	$-0.719845\pi$
$317$	−22.6847	−1.27410	−0.637049	+	0.770824i	$0.719845\pi$
$318$	0	0
$319$	5.12311	0.286839
$320$	−11.1231	−0.621801
$321$	0	0
$322$	−1.36932	−0.0763090
$323$	−1.31534	−0.0731876
$324$	0	0
$325$	2.12311	0.117769
$326$	−3.31534	−0.183620
$327$	0	0
$328$	6.54640	0.361464
$329$	4.49242	0.247675
$330$	0	0
$331$	4.80776	0.264259	0.132129	−	0.991232i	$-0.457819\pi$
$331$	4.80776	0.264259	0.132129	+	0.991232i	$0.457819\pi$
$332$	−5.06913	−0.278205
$333$	0	0
$334$	33.6695	1.84231
$335$	−13.1231	−0.716992
$336$	0	0
$337$	−31.9309	−1.73939	−0.869693	−	0.493594i	$-0.835683\pi$
$337$	−31.9309	−1.73939	−0.869693	+	0.493594i	$0.835683\pi$
$338$	−13.2614	−0.721323
$339$	0	0
$340$	0.384472	0.0208509
$341$	48.9848	2.65268
$342$	0	0
$343$	−7.68466	−0.414933
$344$	−19.0388	−1.02650
$345$	0	0
$346$	22.2462	1.19596
$347$	26.0540	1.39865	0.699325	−	0.714804i	$-0.253484\pi$
$347$	26.0540	1.39865	0.699325	+	0.714804i	$0.253484\pi$
$348$	0	0
$349$	14.8078	0.792641	0.396321	−	0.918112i	$-0.370287\pi$
$349$	14.8078	0.792641	0.396321	+	0.918112i	$0.370287\pi$
$350$	−0.876894	−0.0468720
$351$	0	0
$352$	12.4924	0.665848
$353$	14.7386	0.784458	0.392229	−	0.919868i	$-0.371704\pi$
$353$	14.7386	0.784458	0.392229	+	0.919868i	$0.371704\pi$
$354$	0	0
$355$	7.12311	0.378055
$356$	−0.876894	−0.0464753
$357$	0	0
$358$	−1.06913	−0.0565053
$359$	7.75379	0.409229	0.204615	−	0.978843i	$-0.434406\pi$
$359$	7.75379	0.409229	0.204615	+	0.978843i	$0.434406\pi$
$360$	0	0
$361$	−10.0000	−0.526316
$362$	6.43845	0.338397
$363$	0	0
$364$	−0.522732	−0.0273986
$365$	0.630683	0.0330115
$366$	0	0
$367$	−7.68466	−0.401136	−0.200568	−	0.979680i	$-0.564279\pi$
$367$	−7.68466	−0.401136	−0.200568	+	0.979680i	$0.564279\pi$
$368$	7.31534	0.381339
$369$	0	0
$370$	−30.2462	−1.57243
$371$	−5.61553	−0.291544
$372$	0	0
$373$	12.5616	0.650413	0.325206	−	0.945643i	$-0.394566\pi$
$373$	12.5616	0.650413	0.325206	+	0.945643i	$0.394566\pi$
$374$	3.50758	0.181373
$375$	0	0
$376$	−19.5076	−1.00603
$377$	2.12311	0.109346
$378$	0	0
$379$	−0.315342	−0.0161980	−0.00809900	−	0.999967i	$-0.502578\pi$
$379$	−0.315342	−0.0161980	−0.00809900	+	0.999967i	$0.502578\pi$
$380$	−2.63068	−0.134951
$381$	0	0
$382$	24.3845	1.24762
$383$	−30.7386	−1.57067	−0.785335	−	0.619071i	$-0.787509\pi$
$383$	−30.7386	−1.57067	−0.785335	+	0.619071i	$0.787509\pi$
$384$	0	0
$385$	5.75379	0.293240
$386$	1.26137	0.0642019
$387$	0	0
$388$	1.61553	0.0820160
$389$	−15.3153	−0.776519	−0.388259	−	0.921550i	$-0.626924\pi$
$389$	−15.3153	−0.776519	−0.388259	+	0.921550i	$0.626924\pi$
$390$	0	0
$391$	0.684658	0.0346247
$392$	16.3002	0.823284
$393$	0	0
$394$	−36.4924	−1.83846
$395$	−2.87689	−0.144752
$396$	0	0
$397$	21.8078	1.09450	0.547250	−	0.836969i	$-0.315675\pi$
$397$	21.8078	1.09450	0.547250	+	0.836969i	$0.315675\pi$
$398$	−27.6155	−1.38424
$399$	0	0
$400$	4.68466	0.234233
$401$	−11.7538	−0.586956	−0.293478	−	0.955966i	$-0.594813\pi$
$401$	−11.7538	−0.586956	−0.293478	+	0.955966i	$0.594813\pi$
$402$	0	0
$403$	20.3002	1.01122
$404$	3.42329	0.170315
$405$	0	0
$406$	−0.876894	−0.0435195
$407$	−49.6155	−2.45935
$408$	0	0
$409$	−12.3153	−0.608954	−0.304477	−	0.952520i	$-0.598482\pi$
$409$	−12.3153	−0.608954	−0.304477	+	0.952520i	$0.598482\pi$
$410$	8.38447	0.414080
$411$	0	0
$412$	2.00000	0.0985329
$413$	−2.49242	−0.122644
$414$	0	0
$415$	23.1231	1.13507
$416$	5.17708	0.253827
$417$	0	0
$418$	−24.0000	−1.17388
$419$	28.6847	1.40134	0.700669	−	0.713487i	$-0.252885\pi$
$419$	28.6847	1.40134	0.700669	+	0.713487i	$0.252885\pi$
$420$	0	0
$421$	21.9309	1.06885	0.534423	−	0.845217i	$-0.320529\pi$
$421$	21.9309	1.06885	0.534423	+	0.845217i	$0.320529\pi$
$422$	−21.7538	−1.05896
$423$	0	0
$424$	24.3845	1.18421
$425$	0.438447	0.0212678
$426$	0	0
$427$	−0.807764	−0.0390904
$428$	0.492423	0.0238021
$429$	0	0
$430$	−24.3845	−1.17592
$431$	33.8078	1.62846	0.814231	−	0.580541i	$-0.197159\pi$
$431$	33.8078	1.62846	0.814231	+	0.580541i	$0.197159\pi$
$432$	0	0
$433$	4.24621	0.204060	0.102030	−	0.994781i	$-0.467466\pi$
$433$	4.24621	0.204060	0.102030	+	0.994781i	$0.467466\pi$
$434$	−8.38447	−0.402468
$435$	0	0
$436$	8.57671	0.410750
$437$	−4.68466	−0.224098
$438$	0	0
$439$	−19.1231	−0.912696	−0.456348	−	0.889801i	$-0.650843\pi$
$439$	−19.1231	−0.912696	−0.456348	+	0.889801i	$0.650843\pi$
$440$	−24.9848	−1.19111
$441$	0	0
$442$	1.45360	0.0691408
$443$	−29.6155	−1.40708	−0.703538	−	0.710658i	$-0.748398\pi$
$443$	−29.6155	−1.40708	−0.703538	+	0.710658i	$0.748398\pi$
$444$	0	0
$445$	4.00000	0.189618
$446$	29.8617	1.41399
$447$	0	0
$448$	3.12311	0.147553
$449$	−21.3693	−1.00848	−0.504240	−	0.863563i	$-0.668227\pi$
$449$	−21.3693	−1.00848	−0.504240	+	0.863563i	$0.668227\pi$
$450$	0	0
$451$	13.7538	0.647641
$452$	0.792609	0.0372812
$453$	0	0
$454$	10.8229	0.507945
$455$	2.38447	0.111786
$456$	0	0
$457$	−22.6847	−1.06114	−0.530572	−	0.847640i	$-0.678023\pi$
$457$	−22.6847	−1.06114	−0.530572	+	0.847640i	$0.678023\pi$
$458$	−9.36932	−0.437799
$459$	0	0
$460$	1.36932	0.0638447
$461$	17.3153	0.806456	0.403228	−	0.915100i	$-0.367888\pi$
$461$	17.3153	0.806456	0.403228	+	0.915100i	$0.367888\pi$
$462$	0	0
$463$	−31.0540	−1.44320	−0.721600	−	0.692310i	$-0.756593\pi$
$463$	−31.0540	−1.44320	−0.721600	+	0.692310i	$0.756593\pi$
$464$	4.68466	0.217480
$465$	0	0
$466$	−22.6307	−1.04835
$467$	14.2462	0.659236	0.329618	−	0.944114i	$-0.393080\pi$
$467$	14.2462	0.659236	0.329618	+	0.944114i	$0.393080\pi$
$468$	0	0
$469$	3.68466	0.170142
$470$	−24.9848	−1.15246
$471$	0	0
$472$	10.8229	0.498165
$473$	−40.0000	−1.83920
$474$	0	0
$475$	−3.00000	−0.137649
$476$	−0.107951	−0.00494791
$477$	0	0
$478$	8.68466	0.397227
$479$	25.1231	1.14790	0.573952	−	0.818889i	$-0.305410\pi$
$479$	25.1231	1.14790	0.573952	+	0.818889i	$0.305410\pi$
$480$	0	0
$481$	−20.5616	−0.937526
$482$	−29.3693	−1.33774
$483$	0	0
$484$	6.68466	0.303848
$485$	−7.36932	−0.334623
$486$	0	0
$487$	33.0540	1.49782	0.748909	−	0.662673i	$-0.230578\pi$
$487$	33.0540	1.49782	0.748909	+	0.662673i	$0.230578\pi$
$488$	3.50758	0.158781
$489$	0	0
$490$	20.8769	0.943122
$491$	26.7386	1.20670	0.603349	−	0.797477i	$-0.293833\pi$
$491$	26.7386	1.20670	0.603349	+	0.797477i	$0.293833\pi$
$492$	0	0
$493$	0.438447	0.0197467
$494$	−9.94602	−0.447493
$495$	0	0
$496$	44.7926	2.01125
$497$	−2.00000	−0.0897123
$498$	0	0
$499$	38.7386	1.73418	0.867090	−	0.498152i	$-0.165988\pi$
$499$	38.7386	1.73418	0.867090	+	0.498152i	$0.165988\pi$
$500$	5.26137	0.235295
$501$	0	0
$502$	30.6307	1.36711
$503$	38.7386	1.72727	0.863635	−	0.504117i	$-0.168182\pi$
$503$	38.7386	1.72727	0.863635	+	0.504117i	$0.168182\pi$
$504$	0	0
$505$	−15.6155	−0.694882
$506$	12.4924	0.555356
$507$	0	0
$508$	−8.98485	−0.398638
$509$	−20.0000	−0.886484	−0.443242	−	0.896402i	$-0.646172\pi$
$509$	−20.0000	−0.886484	−0.443242	+	0.896402i	$0.646172\pi$
$510$	0	0
$511$	−0.177081	−0.00783360
$512$	−11.4233	−0.504843
$513$	0	0
$514$	−16.3845	−0.722688
$515$	−9.12311	−0.402012
$516$	0	0
$517$	−40.9848	−1.80251
$518$	8.49242	0.373136
$519$	0	0
$520$	−10.3542	−0.454060
$521$	−1.12311	−0.0492042	−0.0246021	−	0.999697i	$-0.507832\pi$
$521$	−1.12311	−0.0492042	−0.0246021	+	0.999697i	$0.507832\pi$
$522$	0	0
$523$	13.1922	0.576856	0.288428	−	0.957502i	$-0.406867\pi$
$523$	13.1922	0.576856	0.288428	+	0.957502i	$0.406867\pi$
$524$	−0.768944	−0.0335915
$525$	0	0
$526$	−15.6155	−0.680869
$527$	4.19224	0.182617
$528$	0	0
$529$	−20.5616	−0.893981
$530$	31.2311	1.35659
$531$	0	0
$532$	0.738634	0.0320238
$533$	5.69981	0.246886
$534$	0	0
$535$	−2.24621	−0.0971122
$536$	−16.0000	−0.691095
$537$	0	0
$538$	35.4233	1.52721
$539$	34.2462	1.47509
$540$	0	0
$541$	−24.8078	−1.06657	−0.533285	−	0.845936i	$-0.679042\pi$
$541$	−24.8078	−1.06657	−0.533285	+	0.845936i	$0.679042\pi$
$542$	−30.0540	−1.29093
$543$	0	0
$544$	1.06913	0.0458386
$545$	−39.1231	−1.67585
$546$	0	0
$547$	−23.9309	−1.02321	−0.511605	−	0.859221i	$-0.670949\pi$
$547$	−23.9309	−1.02321	−0.511605	+	0.859221i	$0.670949\pi$
$548$	−7.31534	−0.312496
$549$	0	0
$550$	8.00000	0.341121
$551$	−3.00000	−0.127804
$552$	0	0
$553$	0.807764	0.0343496
$554$	−23.6998	−1.00691
$555$	0	0
$556$	−2.49242	−0.105702
$557$	4.49242	0.190350	0.0951750	−	0.995461i	$-0.469659\pi$
$557$	4.49242	0.190350	0.0951750	+	0.995461i	$0.469659\pi$
$558$	0	0
$559$	−16.5767	−0.701120
$560$	5.26137	0.222333
$561$	0	0
$562$	−38.6307	−1.62954
$563$	−28.7386	−1.21119	−0.605595	−	0.795773i	$-0.707065\pi$
$563$	−28.7386	−1.21119	−0.605595	+	0.795773i	$0.707065\pi$
$564$	0	0
$565$	−3.61553	−0.152106
$566$	26.3542	1.10775
$567$	0	0
$568$	8.68466	0.364400
$569$	32.3002	1.35409	0.677047	−	0.735940i	$-0.263259\pi$
$569$	32.3002	1.35409	0.677047	+	0.735940i	$0.263259\pi$
$570$	0	0
$571$	−13.4384	−0.562382	−0.281191	−	0.959652i	$-0.590729\pi$
$571$	−13.4384	−0.562382	−0.281191	+	0.959652i	$0.590729\pi$
$572$	4.76894	0.199400
$573$	0	0
$574$	−2.35416	−0.0982608
$575$	1.56155	0.0651213
$576$	0	0
$577$	−16.1771	−0.673461	−0.336730	−	0.941601i	$-0.609321\pi$
$577$	−16.1771	−0.673461	−0.336730	+	0.941601i	$0.609321\pi$
$578$	−26.2462	−1.09170
$579$	0	0
$580$	0.876894	0.0364111
$581$	−6.49242	−0.269351
$582$	0	0
$583$	51.2311	2.12177
$584$	0.768944	0.0318191
$585$	0	0
$586$	−25.3693	−1.04800
$587$	32.4924	1.34111	0.670553	−	0.741862i	$-0.266057\pi$
$587$	32.4924	1.34111	0.670553	+	0.741862i	$0.266057\pi$
$588$	0	0
$589$	−28.6847	−1.18193
$590$	13.8617	0.570679
$591$	0	0
$592$	−45.3693	−1.86467
$593$	2.00000	0.0821302	0.0410651	−	0.999156i	$-0.486925\pi$
$593$	2.00000	0.0821302	0.0410651	+	0.999156i	$0.486925\pi$
$594$	0	0
$595$	0.492423	0.0201874
$596$	−8.00000	−0.327693
$597$	0	0
$598$	5.17708	0.211707
$599$	27.1231	1.10822	0.554110	−	0.832443i	$-0.313059\pi$
$599$	27.1231	1.10822	0.554110	+	0.832443i	$0.313059\pi$
$600$	0	0
$601$	−9.50758	−0.387822	−0.193911	−	0.981019i	$-0.562117\pi$
$601$	−9.50758	−0.387822	−0.193911	+	0.981019i	$0.562117\pi$
$602$	6.84658	0.279046
$603$	0	0
$604$	−5.50758	−0.224100
$605$	−30.4924	−1.23969
$606$	0	0
$607$	−7.00000	−0.284121	−0.142061	−	0.989858i	$-0.545373\pi$
$607$	−7.00000	−0.284121	−0.142061	+	0.989858i	$0.545373\pi$
$608$	−7.31534	−0.296676
$609$	0	0
$610$	4.49242	0.181893
$611$	−16.9848	−0.687133
$612$	0	0
$613$	−21.0000	−0.848182	−0.424091	−	0.905620i	$-0.639406\pi$
$613$	−21.0000	−0.848182	−0.424091	+	0.905620i	$0.639406\pi$
$614$	−13.2614	−0.535185
$615$	0	0
$616$	7.01515	0.282649
$617$	−1.50758	−0.0606928	−0.0303464	−	0.999539i	$-0.509661\pi$
$617$	−1.50758	−0.0606928	−0.0303464	+	0.999539i	$0.509661\pi$
$618$	0	0
$619$	−43.3542	−1.74255	−0.871275	−	0.490794i	$-0.836707\pi$
$619$	−43.3542	−1.74255	−0.871275	+	0.490794i	$0.836707\pi$
$620$	8.38447	0.336728
$621$	0	0
$622$	−16.0000	−0.641542
$623$	−1.12311	−0.0449963
$624$	0	0
$625$	−19.0000	−0.760000
$626$	51.3153	2.05097
$627$	0	0
$628$	−7.89205	−0.314927
$629$	−4.24621	−0.169308
$630$	0	0
$631$	−21.4384	−0.853451	−0.426726	−	0.904381i	$-0.640333\pi$
$631$	−21.4384	−0.853451	−0.426726	+	0.904381i	$0.640333\pi$
$632$	−3.50758	−0.139524
$633$	0	0
$634$	−35.4233	−1.40684
$635$	40.9848	1.62643
$636$	0	0
$637$	14.1922	0.562317
$638$	8.00000	0.316723
$639$	0	0
$640$	−27.1231	−1.07213
$641$	24.3002	0.959800	0.479900	−	0.877323i	$-0.340673\pi$
$641$	24.3002	0.959800	0.479900	+	0.877323i	$0.340673\pi$
$642$	0	0
$643$	33.3693	1.31596	0.657979	−	0.753037i	$-0.271412\pi$
$643$	33.3693	1.31596	0.657979	+	0.753037i	$0.271412\pi$
$644$	−0.384472	−0.0151503
$645$	0	0
$646$	−2.05398	−0.0808126
$647$	39.8617	1.56713	0.783563	−	0.621312i	$-0.213400\pi$
$647$	39.8617	1.56713	0.783563	+	0.621312i	$0.213400\pi$
$648$	0	0
$649$	22.7386	0.892569
$650$	3.31534	0.130038
$651$	0	0
$652$	−0.930870	−0.0364557
$653$	22.9309	0.897354	0.448677	−	0.893694i	$-0.351895\pi$
$653$	22.9309	0.897354	0.448677	+	0.893694i	$0.351895\pi$
$654$	0	0
$655$	3.50758	0.137052
$656$	12.5767	0.491038
$657$	0	0
$658$	7.01515	0.273479
$659$	−9.61553	−0.374568	−0.187284	−	0.982306i	$-0.559968\pi$
$659$	−9.61553	−0.374568	−0.187284	+	0.982306i	$0.559968\pi$
$660$	0	0
$661$	−11.9309	−0.464057	−0.232029	−	0.972709i	$-0.574536\pi$
$661$	−11.9309	−0.464057	−0.232029	+	0.972709i	$0.574536\pi$
$662$	7.50758	0.291790
$663$	0	0
$664$	28.1922	1.09407
$665$	−3.36932	−0.130657
$666$	0	0
$667$	1.56155	0.0604636
$668$	9.45360	0.365771
$669$	0	0
$670$	−20.4924	−0.791691
$671$	7.36932	0.284489
$672$	0	0
$673$	−28.6155	−1.10305	−0.551524	−	0.834159i	$-0.685953\pi$
$673$	−28.6155	−1.10305	−0.551524	+	0.834159i	$0.685953\pi$
$674$	−49.8617	−1.92060
$675$	0	0
$676$	−3.72348	−0.143211
$677$	−16.0540	−0.617004	−0.308502	−	0.951224i	$-0.599828\pi$
$677$	−16.0540	−0.617004	−0.308502	+	0.951224i	$0.599828\pi$
$678$	0	0
$679$	2.06913	0.0794059
$680$	−2.13826	−0.0819986
$681$	0	0
$682$	76.4924	2.92905
$683$	26.7386	1.02313	0.511563	−	0.859246i	$-0.329067\pi$
$683$	26.7386	1.02313	0.511563	+	0.859246i	$0.329067\pi$
$684$	0	0
$685$	33.3693	1.27498
$686$	−12.0000	−0.458162
$687$	0	0
$688$	−36.5767	−1.39447
$689$	21.2311	0.808839
$690$	0	0
$691$	−28.1080	−1.06928	−0.534638	−	0.845081i	$-0.679552\pi$
$691$	−28.1080	−1.06928	−0.534638	+	0.845081i	$0.679552\pi$
$692$	6.24621	0.237445
$693$	0	0
$694$	40.6847	1.54437
$695$	11.3693	0.431263
$696$	0	0
$697$	1.17708	0.0445851
$698$	23.1231	0.875222
$699$	0	0
$700$	−0.246211	−0.00930591
$701$	11.6155	0.438712	0.219356	−	0.975645i	$-0.429604\pi$
$701$	11.6155	0.438712	0.219356	+	0.975645i	$0.429604\pi$
$702$	0	0
$703$	29.0540	1.09579
$704$	−28.4924	−1.07385
$705$	0	0
$706$	23.0152	0.866187
$707$	4.38447	0.164895
$708$	0	0
$709$	−22.1231	−0.830851	−0.415425	−	0.909627i	$-0.636367\pi$
$709$	−22.1231	−0.830851	−0.415425	+	0.909627i	$0.636367\pi$
$710$	11.1231	0.417443
$711$	0	0
$712$	4.87689	0.182769
$713$	14.9309	0.559165
$714$	0	0
$715$	−21.7538	−0.813546
$716$	−0.300187	−0.0112185
$717$	0	0
$718$	12.1080	0.451865
$719$	−19.8617	−0.740718	−0.370359	−	0.928889i	$-0.620765\pi$
$719$	−19.8617	−0.740718	−0.370359	+	0.928889i	$0.620765\pi$
$720$	0	0
$721$	2.56155	0.0953972
$722$	−15.6155	−0.581150
$723$	0	0
$724$	1.80776	0.0671850
$725$	1.00000	0.0371391
$726$	0	0
$727$	34.4384	1.27725	0.638626	−	0.769518i	$-0.279503\pi$
$727$	34.4384	1.27725	0.638626	+	0.769518i	$0.279503\pi$
$728$	2.90720	0.107748
$729$	0	0
$730$	0.984845	0.0364507
$731$	−3.42329	−0.126615
$732$	0	0
$733$	9.61553	0.355158	0.177579	−	0.984107i	$-0.443174\pi$
$733$	9.61553	0.355158	0.177579	+	0.984107i	$0.443174\pi$
$734$	−12.0000	−0.442928
$735$	0	0
$736$	3.80776	0.140356
$737$	−33.6155	−1.23824
$738$	0	0
$739$	23.8078	0.875783	0.437891	−	0.899028i	$-0.355725\pi$
$739$	23.8078	0.875783	0.437891	+	0.899028i	$0.355725\pi$
$740$	−8.49242	−0.312188
$741$	0	0
$742$	−8.76894	−0.321918
$743$	13.1231	0.481440	0.240720	−	0.970595i	$-0.422616\pi$
$743$	13.1231	0.481440	0.240720	+	0.970595i	$0.422616\pi$
$744$	0	0
$745$	36.4924	1.33698
$746$	19.6155	0.718176
$747$	0	0
$748$	0.984845	0.0360095
$749$	0.630683	0.0230447
$750$	0	0
$751$	24.8617	0.907218	0.453609	−	0.891201i	$-0.350136\pi$
$751$	24.8617	0.907218	0.453609	+	0.891201i	$0.350136\pi$
$752$	−37.4773	−1.36666
$753$	0	0
$754$	3.31534	0.120738
$755$	25.1231	0.914323
$756$	0	0
$757$	14.3153	0.520300	0.260150	−	0.965568i	$-0.416228\pi$
$757$	14.3153	0.520300	0.260150	+	0.965568i	$0.416228\pi$
$758$	−0.492423	−0.0178856
$759$	0	0
$760$	14.6307	0.530711
$761$	22.4924	0.815350	0.407675	−	0.913127i	$-0.366340\pi$
$761$	22.4924	0.815350	0.407675	+	0.913127i	$0.366340\pi$
$762$	0	0
$763$	10.9848	0.397678
$764$	6.84658	0.247701
$765$	0	0
$766$	−48.0000	−1.73431
$767$	9.42329	0.340255
$768$	0	0
$769$	44.8078	1.61581	0.807905	−	0.589313i	$-0.200602\pi$
$769$	44.8078	1.61581	0.807905	+	0.589313i	$0.200602\pi$
$770$	8.98485	0.323791
$771$	0	0
$772$	0.354162	0.0127466
$773$	36.0540	1.29677	0.648386	−	0.761312i	$-0.275444\pi$
$773$	36.0540	1.29677	0.648386	+	0.761312i	$0.275444\pi$
$774$	0	0
$775$	9.56155	0.343461
$776$	−8.98485	−0.322537
$777$	0	0
$778$	−23.9157	−0.857420
$779$	−8.05398	−0.288564
$780$	0	0
$781$	18.2462	0.652901
$782$	1.06913	0.0382320
$783$	0	0
$784$	31.3153	1.11841
$785$	36.0000	1.28490
$786$	0	0
$787$	44.8078	1.59722	0.798612	−	0.601846i	$-0.205568\pi$
$787$	44.8078	1.59722	0.798612	+	0.601846i	$0.205568\pi$
$788$	−10.2462	−0.365006
$789$	0	0
$790$	−4.49242	−0.159833
$791$	1.01515	0.0360948
$792$	0	0
$793$	3.05398	0.108450
$794$	34.0540	1.20853
$795$	0	0
$796$	−7.75379	−0.274826
$797$	−9.80776	−0.347409	−0.173704	−	0.984798i	$-0.555574\pi$
$797$	−9.80776	−0.347409	−0.173704	+	0.984798i	$0.555574\pi$
$798$	0	0
$799$	−3.50758	−0.124089
$800$	2.43845	0.0862121
$801$	0	0
$802$	−18.3542	−0.648108
$803$	1.61553	0.0570107
$804$	0	0
$805$	1.75379	0.0618129
$806$	31.6998	1.11658
$807$	0	0
$808$	−19.0388	−0.669783
$809$	48.2462	1.69625	0.848123	−	0.529799i	$-0.177733\pi$
$809$	48.2462	1.69625	0.848123	+	0.529799i	$0.177733\pi$
$810$	0	0
$811$	43.2311	1.51805	0.759024	−	0.651063i	$-0.225677\pi$
$811$	43.2311	1.51805	0.759024	+	0.651063i	$0.225677\pi$
$812$	−0.246211	−0.00864032
$813$	0	0
$814$	−77.4773	−2.71558
$815$	4.24621	0.148738
$816$	0	0
$817$	23.4233	0.819477
$818$	−19.2311	−0.672398
$819$	0	0
$820$	2.35416	0.0822109
$821$	5.36932	0.187390	0.0936952	−	0.995601i	$-0.470132\pi$
$821$	5.36932	0.187390	0.0936952	+	0.995601i	$0.470132\pi$
$822$	0	0
$823$	−9.49242	−0.330885	−0.165443	−	0.986219i	$-0.552905\pi$
$823$	−9.49242	−0.330885	−0.165443	+	0.986219i	$0.552905\pi$
$824$	−11.1231	−0.387492
$825$	0	0
$826$	−3.89205	−0.135422
$827$	−43.6155	−1.51666	−0.758330	−	0.651871i	$-0.773985\pi$
$827$	−43.6155	−1.51666	−0.758330	+	0.651871i	$0.773985\pi$
$828$	0	0
$829$	17.9309	0.622765	0.311382	−	0.950285i	$-0.399208\pi$
$829$	17.9309	0.622765	0.311382	+	0.950285i	$0.399208\pi$
$830$	36.1080	1.25333
$831$	0	0
$832$	−11.8078	−0.409361
$833$	2.93087	0.101549
$834$	0	0
$835$	−43.1231	−1.49234
$836$	−6.73863	−0.233061
$837$	0	0
$838$	44.7926	1.54734
$839$	26.2462	0.906120	0.453060	−	0.891480i	$-0.350332\pi$
$839$	26.2462	0.906120	0.453060	+	0.891480i	$0.350332\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	34.2462	1.18020
$843$	0	0
$844$	−6.10795	−0.210244
$845$	16.9848	0.584296
$846$	0	0
$847$	8.56155	0.294178
$848$	46.8466	1.60872
$849$	0	0
$850$	0.684658	0.0234836
$851$	−15.1231	−0.518413
$852$	0	0
$853$	11.0540	0.378481	0.189240	−	0.981931i	$-0.439397\pi$
$853$	11.0540	0.378481	0.189240	+	0.981931i	$0.439397\pi$
$854$	−1.26137	−0.0431631
$855$	0	0
$856$	−2.73863	−0.0936046
$857$	−37.1231	−1.26810	−0.634051	−	0.773292i	$-0.718609\pi$
$857$	−37.1231	−1.26810	−0.634051	+	0.773292i	$0.718609\pi$
$858$	0	0
$859$	33.7926	1.15299	0.576494	−	0.817101i	$-0.304420\pi$
$859$	33.7926	1.15299	0.576494	+	0.817101i	$0.304420\pi$
$860$	−6.84658	−0.233467
$861$	0	0
$862$	52.7926	1.79812
$863$	50.5464	1.72062	0.860310	−	0.509772i	$-0.170270\pi$
$863$	50.5464	1.72062	0.860310	+	0.509772i	$0.170270\pi$
$864$	0	0
$865$	−28.4924	−0.968771
$866$	6.63068	0.225320
$867$	0	0
$868$	−2.35416	−0.0799055
$869$	−7.36932	−0.249987
$870$	0	0
$871$	−13.9309	−0.472030
$872$	−47.6998	−1.61532
$873$	0	0
$874$	−7.31534	−0.247445
$875$	6.73863	0.227807
$876$	0	0
$877$	54.2311	1.83125	0.915626	−	0.402030i	$-0.131695\pi$
$877$	54.2311	1.83125	0.915626	+	0.402030i	$0.131695\pi$
$878$	−29.8617	−1.00778
$879$	0	0
$880$	−48.0000	−1.61808
$881$	−52.0540	−1.75374	−0.876871	−	0.480725i	$-0.840374\pi$
$881$	−52.0540	−1.75374	−0.876871	+	0.480725i	$0.840374\pi$
$882$	0	0
$883$	−41.6847	−1.40280	−0.701400	−	0.712768i	$-0.747441\pi$
$883$	−41.6847	−1.40280	−0.701400	+	0.712768i	$0.747441\pi$
$884$	0.408137	0.0137271
$885$	0	0
$886$	−46.2462	−1.55367
$887$	7.50758	0.252080	0.126040	−	0.992025i	$-0.459773\pi$
$887$	7.50758	0.252080	0.126040	+	0.992025i	$0.459773\pi$
$888$	0	0
$889$	−11.5076	−0.385952
$890$	6.24621	0.209373
$891$	0	0
$892$	8.38447	0.280733
$893$	24.0000	0.803129
$894$	0	0
$895$	1.36932	0.0457712
$896$	7.61553	0.254417
$897$	0	0
$898$	−33.3693	−1.11355
$899$	9.56155	0.318896
$900$	0	0
$901$	4.38447	0.146068
$902$	21.4773	0.715115
$903$	0	0
$904$	−4.40814	−0.146612
$905$	−8.24621	−0.274113
$906$	0	0
$907$	−35.2462	−1.17033	−0.585166	−	0.810914i	$-0.698971\pi$
$907$	−35.2462	−1.17033	−0.585166	+	0.810914i	$0.698971\pi$
$908$	3.03882	0.100847
$909$	0	0
$910$	3.72348	0.123432
$911$	−29.7538	−0.985787	−0.492894	−	0.870090i	$-0.664061\pi$
$911$	−29.7538	−0.985787	−0.492894	+	0.870090i	$0.664061\pi$
$912$	0	0
$913$	59.2311	1.96026
$914$	−35.4233	−1.17170
$915$	0	0
$916$	−2.63068	−0.0869202
$917$	−0.984845	−0.0325224
$918$	0	0
$919$	59.6155	1.96653	0.983267	−	0.182168i	$-0.0583115\pi$
$919$	59.6155	1.96653	0.983267	+	0.182168i	$0.0583115\pi$
$920$	−7.61553	−0.251077
$921$	0	0
$922$	27.0388	0.890476
$923$	7.56155	0.248892
$924$	0	0
$925$	−9.68466	−0.318430
$926$	−48.4924	−1.59356
$927$	0	0
$928$	2.43845	0.0800460
$929$	−48.3542	−1.58645	−0.793224	−	0.608930i	$-0.791599\pi$
$929$	−48.3542	−1.58645	−0.793224	+	0.608930i	$0.791599\pi$
$930$	0	0
$931$	−20.0540	−0.657242
$932$	−6.35416	−0.208137
$933$	0	0
$934$	22.2462	0.727918
$935$	−4.49242	−0.146918
$936$	0	0
$937$	−41.9848	−1.37159	−0.685793	−	0.727797i	$-0.740544\pi$
$937$	−41.9848	−1.37159	−0.685793	+	0.727797i	$0.740544\pi$
$938$	5.75379	0.187868
$939$	0	0
$940$	−7.01515	−0.228809
$941$	34.2462	1.11639	0.558197	−	0.829708i	$-0.311493\pi$
$941$	34.2462	1.11639	0.558197	+	0.829708i	$0.311493\pi$
$942$	0	0
$943$	4.19224	0.136518
$944$	20.7926	0.676742
$945$	0	0
$946$	−62.4621	−2.03082
$947$	−7.12311	−0.231470	−0.115735	−	0.993280i	$-0.536922\pi$
$947$	−7.12311	−0.231470	−0.115735	+	0.993280i	$0.536922\pi$
$948$	0	0
$949$	0.669503	0.0217330
$950$	−4.68466	−0.151990
$951$	0	0
$952$	0.600373	0.0194582
$953$	−18.4924	−0.599028	−0.299514	−	0.954092i	$-0.596825\pi$
$953$	−18.4924	−0.599028	−0.299514	+	0.954092i	$0.596825\pi$
$954$	0	0
$955$	−31.2311	−1.01061
$956$	2.43845	0.0788650
$957$	0	0
$958$	39.2311	1.26750
$959$	−9.36932	−0.302551
$960$	0	0
$961$	60.4233	1.94914
$962$	−32.1080	−1.03520
$963$	0	0
$964$	−8.24621	−0.265593
$965$	−1.61553	−0.0520057
$966$	0	0
$967$	17.7386	0.570436	0.285218	−	0.958463i	$-0.407934\pi$
$967$	17.7386	0.570436	0.285218	+	0.958463i	$0.407934\pi$
$968$	−37.1771	−1.19492
$969$	0	0
$970$	−11.5076	−0.369486
$971$	−61.2311	−1.96500	−0.982499	−	0.186268i	$-0.940361\pi$
$971$	−61.2311	−1.96500	−0.982499	+	0.186268i	$0.940361\pi$
$972$	0	0
$973$	−3.19224	−0.102338
$974$	51.6155	1.65387
$975$	0	0
$976$	6.73863	0.215698
$977$	1.86174	0.0595623	0.0297812	−	0.999556i	$-0.490519\pi$
$977$	1.86174	0.0595623	0.0297812	+	0.999556i	$0.490519\pi$
$978$	0	0
$979$	10.2462	0.327470
$980$	5.86174	0.187246
$981$	0	0
$982$	41.7538	1.33242
$983$	−30.4924	−0.972557	−0.486279	−	0.873804i	$-0.661646\pi$
$983$	−30.4924	−0.972557	−0.486279	+	0.873804i	$0.661646\pi$
$984$	0	0
$985$	46.7386	1.48922
$986$	0.684658	0.0218040
$987$	0	0
$988$	−2.79261	−0.0888447
$989$	−12.1922	−0.387691
$990$	0	0
$991$	23.1922	0.736726	0.368363	−	0.929682i	$-0.379918\pi$
$991$	23.1922	0.736726	0.368363	+	0.929682i	$0.379918\pi$
$992$	23.3153	0.740263
$993$	0	0
$994$	−3.12311	−0.0990589
$995$	35.3693	1.12128
$996$	0	0
$997$	−48.3542	−1.53139	−0.765696	−	0.643203i	$-0.777605\pi$
$997$	−48.3542	−1.53139	−0.765696	+	0.643203i	$0.777605\pi$
$998$	60.4924	1.91485
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	783.2.a.b.1.2	✓	2
3.2	odd	2		783.2.a.d.1.1	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
783.2.a.b.1.2	✓	2	1.1	even	1	trivial
783.2.a.d.1.1	yes	2	3.2	odd	2

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 783 = 3^{3} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	783.a (trivial)

\(f(q)\)	\(=\)	\(q+1.56155 q^{2} +0.438447 q^{4} -2.00000 q^{5} +0.561553 q^{7} -2.43845 q^{8} -3.12311 q^{10} -5.12311 q^{11} -2.12311 q^{13} +0.876894 q^{14} -4.68466 q^{16} -0.438447 q^{17} +3.00000 q^{19} -0.876894 q^{20} -8.00000 q^{22} -1.56155 q^{23} -1.00000 q^{25} -3.31534 q^{26} +0.246211 q^{28} -1.00000 q^{29} -9.56155 q^{31} -2.43845 q^{32} -0.684658 q^{34} -1.12311 q^{35} +9.68466 q^{37} +4.68466 q^{38} +4.87689 q^{40} -2.68466 q^{41} +7.80776 q^{43} -2.24621 q^{44} -2.43845 q^{46} +8.00000 q^{47} -6.68466 q^{49} -1.56155 q^{50} -0.930870 q^{52} -10.0000 q^{53} +10.2462 q^{55} -1.36932 q^{56} -1.56155 q^{58} -4.43845 q^{59} -1.43845 q^{61} -14.9309 q^{62} +5.56155 q^{64} +4.24621 q^{65} +6.56155 q^{67} -0.192236 q^{68} -1.75379 q^{70} -3.56155 q^{71} -0.315342 q^{73} +15.1231 q^{74} +1.31534 q^{76} -2.87689 q^{77} +1.43845 q^{79} +9.36932 q^{80} -4.19224 q^{82} -11.5616 q^{83} +0.876894 q^{85} +12.1922 q^{86} +12.4924 q^{88} -2.00000 q^{89} -1.19224 q^{91} -0.684658 q^{92} +12.4924 q^{94} -6.00000 q^{95} +3.68466 q^{97} -10.4384 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{2} + 5 q^{4} - 4 q^{5} - 3 q^{7} - 9 q^{8} + 2 q^{10} - 2 q^{11} + 4 q^{13} + 10 q^{14} + 3 q^{16} - 5 q^{17} + 6 q^{19} - 10 q^{20} - 16 q^{22} + q^{23} - 2 q^{25} - 19 q^{26} - 16 q^{28} - 2 q^{29}+ \cdots - 25 q^{98}+O(q^{100}) \) 2 * q - q^2 + 5 * q^4 - 4 * q^5 - 3 * q^7 - 9 * q^8 + 2 * q^10 - 2 * q^11 + 4 * q^13 + 10 * q^14 + 3 * q^16 - 5 * q^17 + 6 * q^19 - 10 * q^20 - 16 * q^22 + q^23 - 2 * q^25 - 19 * q^26 - 16 * q^28 - 2 * q^29 - 15 * q^31 - 9 * q^32 + 11 * q^34 + 6 * q^35 + 7 * q^37 - 3 * q^38 + 18 * q^40 + 7 * q^41 - 5 * q^43 + 12 * q^44 - 9 * q^46 + 16 * q^47 - q^49 + q^50 + 27 * q^52 - 20 * q^53 + 4 * q^55 + 22 * q^56 + q^58 - 13 * q^59 - 7 * q^61 - q^62 + 7 * q^64 - 8 * q^65 + 9 * q^67 - 21 * q^68 - 20 * q^70 - 3 * q^71 - 13 * q^73 + 22 * q^74 + 15 * q^76 - 14 * q^77 + 7 * q^79 - 6 * q^80 - 29 * q^82 - 19 * q^83 + 10 * q^85 + 45 * q^86 - 8 * q^88 - 4 * q^89 - 23 * q^91 + 11 * q^92 - 8 * q^94 - 12 * q^95 - 5 * q^97 - 25 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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