LMFDB - Embedded newform 4840.2.a.ba.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4840,2,Mod(1,4840)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4840, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))

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

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

chi := DirichletCharacter("4840.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4840 = 2^{3} \cdot 5 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4840.a (trivial)

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:	yes
Analytic conductor:	$38.6475945783$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	6.6.25903625.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 3x^{5} - 7x^{4} + 17x^{3} + 16x^{2} - 20x - 5 $ x^6 - 3x^5 - 7x^4 + 17x^3 + 16x^2 - 20*x - 5
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 440)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$3.29288$ of defining polynomial
Character	$\chi$	$=$	4840.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.29288 q^{3} -1.00000 q^{5} -3.51508 q^{7} +7.84307 q^{9} +4.27118 q^{13} +3.29288 q^{15} -1.58998 q^{17} -5.55094 q^{19} +11.5747 q^{21} -6.39042 q^{23} +1.00000 q^{25} -15.9477 q^{27} +0.381220 q^{29} -0.760465 q^{31} +3.51508 q^{35} -1.45191 q^{37} -14.0645 q^{39} +6.12208 q^{41} +7.19067 q^{43} -7.84307 q^{45} +4.32164 q^{47} +5.35577 q^{49} +5.23561 q^{51} -6.94904 q^{53} +18.2786 q^{57} +12.7110 q^{59} +8.89599 q^{61} -27.5690 q^{63} -4.27118 q^{65} +12.5135 q^{67} +21.0429 q^{69} +7.35890 q^{71} -7.03210 q^{73} -3.29288 q^{75} +6.99381 q^{79} +28.9846 q^{81} +10.7509 q^{83} +1.58998 q^{85} -1.25531 q^{87} +0.451594 q^{89} -15.0135 q^{91} +2.50412 q^{93} +5.55094 q^{95} -2.77956 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 3 q^{3} - 6 q^{5} - 7 q^{7} + 5 q^{9} + q^{13} + 3 q^{15} + 6 q^{17} - 7 q^{19} + 14 q^{21} - 9 q^{23} + 6 q^{25} - 21 q^{27} + 10 q^{29} + q^{31} + 7 q^{35} + 3 q^{37} - q^{39} - 6 q^{41} + 18 q^{43}+ \cdots - 33 q^{97}+O(q^{100}) $ 6 * q - 3 * q^3 - 6 * q^5 - 7 * q^7 + 5 * q^9 + q^13 + 3 * q^15 + 6 * q^17 - 7 * q^19 + 14 * q^21 - 9 * q^23 + 6 * q^25 - 21 * q^27 + 10 * q^29 + q^31 + 7 * q^35 + 3 * q^37 - q^39 - 6 * q^41 + 18 * q^43 - 5 * q^45 - 3 * q^47 + 17 * q^49 + 15 * q^51 - 23 * q^53 + 9 * q^57 - 2 * q^59 + 6 * q^61 - 49 * q^63 - q^65 - 22 * q^67 + 2 * q^69 - 13 * q^71 + 10 * q^73 - 3 * q^75 - 22 * q^79 + 10 * q^81 + 10 * q^83 - 6 * q^85 - 3 * q^87 - 25 * q^89 + 12 * q^91 + 19 * q^93 + 7 * q^95 - 33 * q^97

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	0
$2$	0	0
$3$	−3.29288	−1.90115	−0.950573	−	0.310501i	$-0.899503\pi$
$3$	−3.29288	−1.90115	−0.950573	+	0.310501i	$0.899503\pi$
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−3.51508	−1.32857	−0.664287	−	0.747477i	$-0.731265\pi$
$7$	−3.51508	−1.32857	−0.664287	+	0.747477i	$0.731265\pi$
$8$	0	0
$9$	7.84307	2.61436
$10$	0	0
$11$	0	0
$11$	0	0
$12$	0	0
$13$	4.27118	1.18461	0.592306	−	0.805713i	$-0.298218\pi$
$13$	4.27118	1.18461	0.592306	+	0.805713i	$0.298218\pi$
$14$	0	0
$15$	3.29288	0.850218
$16$	0	0
$17$	−1.58998	−0.385626	−0.192813	−	0.981236i	$-0.561761\pi$
$17$	−1.58998	−0.385626	−0.192813	+	0.981236i	$0.561761\pi$
$18$	0	0
$19$	−5.55094	−1.27347	−0.636736	−	0.771082i	$-0.719716\pi$
$19$	−5.55094	−1.27347	−0.636736	+	0.771082i	$0.719716\pi$
$20$	0	0
$21$	11.5747	2.52581
$22$	0	0
$23$	−6.39042	−1.33249	−0.666247	−	0.745731i	$-0.732100\pi$
$23$	−6.39042	−1.33249	−0.666247	+	0.745731i	$0.732100\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−15.9477	−3.06913
$28$	0	0
$29$	0.381220	0.0707908	0.0353954	−	0.999373i	$-0.488731\pi$
$29$	0.381220	0.0707908	0.0353954	+	0.999373i	$0.488731\pi$
$30$	0	0
$31$	−0.760465	−0.136584	−0.0682918	−	0.997665i	$-0.521755\pi$
$31$	−0.760465	−0.136584	−0.0682918	+	0.997665i	$0.521755\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	3.51508	0.594156
$36$	0	0
$37$	−1.45191	−0.238692	−0.119346	−	0.992853i	$-0.538080\pi$
$37$	−1.45191	−0.238692	−0.119346	+	0.992853i	$0.538080\pi$
$38$	0	0
$39$	−14.0645	−2.25212
$40$	0	0
$41$	6.12208	0.956108	0.478054	−	0.878330i	$-0.341342\pi$
$41$	6.12208	0.956108	0.478054	+	0.878330i	$0.341342\pi$
$42$	0	0
$43$	7.19067	1.09657	0.548284	−	0.836293i	$-0.315281\pi$
$43$	7.19067	1.09657	0.548284	+	0.836293i	$0.315281\pi$
$44$	0	0
$45$	−7.84307	−1.16918
$46$	0	0
$47$	4.32164	0.630376	0.315188	−	0.949029i	$-0.397932\pi$
$47$	4.32164	0.630376	0.315188	+	0.949029i	$0.397932\pi$
$48$	0	0
$49$	5.35577	0.765110
$50$	0	0
$51$	5.23561	0.733132
$52$	0	0
$53$	−6.94904	−0.954523	−0.477262	−	0.878761i	$-0.658371\pi$
$53$	−6.94904	−0.954523	−0.477262	+	0.878761i	$0.658371\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	18.2786	2.42106
$58$	0	0
$59$	12.7110	1.65484	0.827419	−	0.561586i	$-0.189808\pi$
$59$	12.7110	1.65484	0.827419	+	0.561586i	$0.189808\pi$
$60$	0	0
$61$	8.89599	1.13901	0.569507	−	0.821986i	$-0.307134\pi$
$61$	8.89599	1.13901	0.569507	+	0.821986i	$0.307134\pi$
$62$	0	0
$63$	−27.5690	−3.47337
$64$	0	0
$65$	−4.27118	−0.529775
$66$	0	0
$67$	12.5135	1.52877	0.764383	−	0.644763i	$-0.223044\pi$
$67$	12.5135	1.52877	0.764383	+	0.644763i	$0.223044\pi$
$68$	0	0
$69$	21.0429	2.53327
$70$	0	0
$71$	7.35890	0.873340	0.436670	−	0.899622i	$-0.356158\pi$
$71$	7.35890	0.873340	0.436670	+	0.899622i	$0.356158\pi$
$72$	0	0
$73$	−7.03210	−0.823045	−0.411523	−	0.911399i	$-0.635003\pi$
$73$	−7.03210	−0.823045	−0.411523	+	0.911399i	$0.635003\pi$
$74$	0	0
$75$	−3.29288	−0.380229
$76$	0	0
$77$	0	0
$78$	0	0
$79$	6.99381	0.786865	0.393432	−	0.919354i	$-0.371288\pi$
$79$	6.99381	0.786865	0.393432	+	0.919354i	$0.371288\pi$
$80$	0	0
$81$	28.9846	3.22051
$82$	0	0
$83$	10.7509	1.18007	0.590034	−	0.807379i	$-0.299114\pi$
$83$	10.7509	1.18007	0.590034	+	0.807379i	$0.299114\pi$
$84$	0	0
$85$	1.58998	0.172457
$86$	0	0
$87$	−1.25531	−0.134584
$88$	0	0
$89$	0.451594	0.0478689	0.0239344	−	0.999714i	$-0.492381\pi$
$89$	0.451594	0.0478689	0.0239344	+	0.999714i	$0.492381\pi$
$90$	0	0
$91$	−15.0135	−1.57385
$92$	0	0
$93$	2.50412	0.259665
$94$	0	0
$95$	5.55094	0.569514
$96$	0	0
$97$	−2.77956	−0.282222	−0.141111	−	0.989994i	$-0.545067\pi$
$97$	−2.77956	−0.282222	−0.141111	+	0.989994i	$0.545067\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−5.33731	−0.531082	−0.265541	−	0.964100i	$-0.585551\pi$
$101$	−5.33731	−0.531082	−0.265541	+	0.964100i	$0.585551\pi$
$102$	0	0
$103$	17.3817	1.71267	0.856334	−	0.516422i	$-0.172737\pi$
$103$	17.3817	1.71267	0.856334	+	0.516422i	$0.172737\pi$
$104$	0	0
$105$	−11.5747	−1.12958
$106$	0	0
$107$	−3.49783	−0.338148	−0.169074	−	0.985603i	$-0.554078\pi$
$107$	−3.49783	−0.338148	−0.169074	+	0.985603i	$0.554078\pi$
$108$	0	0
$109$	−4.91209	−0.470493	−0.235247	−	0.971936i	$-0.575590\pi$
$109$	−4.91209	−0.470493	−0.235247	+	0.971936i	$0.575590\pi$
$110$	0	0
$111$	4.78096	0.453789
$112$	0	0
$113$	−17.1718	−1.61539	−0.807694	−	0.589602i	$-0.799285\pi$
$113$	−17.1718	−1.61539	−0.807694	+	0.589602i	$0.799285\pi$
$114$	0	0
$115$	6.39042	0.595910
$116$	0	0
$117$	33.4992	3.09700
$118$	0	0
$119$	5.58889	0.512333
$120$	0	0
$121$	0	0
$122$	0	0
$123$	−20.1593	−1.81770
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−15.4883	−1.37437	−0.687183	−	0.726484i	$-0.741153\pi$
$127$	−15.4883	−1.37437	−0.687183	+	0.726484i	$0.741153\pi$
$128$	0	0
$129$	−23.6780	−2.08473
$130$	0	0
$131$	3.64734	0.318670	0.159335	−	0.987225i	$-0.449065\pi$
$131$	3.64734	0.318670	0.159335	+	0.987225i	$0.449065\pi$
$132$	0	0
$133$	19.5120	1.69190
$134$	0	0
$135$	15.9477	1.37256
$136$	0	0
$137$	−16.2579	−1.38901	−0.694503	−	0.719490i	$-0.744376\pi$
$137$	−16.2579	−1.38901	−0.694503	+	0.719490i	$0.744376\pi$
$138$	0	0
$139$	7.49232	0.635490	0.317745	−	0.948176i	$-0.397074\pi$
$139$	7.49232	0.635490	0.317745	+	0.948176i	$0.397074\pi$
$140$	0	0
$141$	−14.2306	−1.19844
$142$	0	0
$143$	0	0
$144$	0	0
$145$	−0.381220	−0.0316586
$146$	0	0
$147$	−17.6359	−1.45459
$148$	0	0
$149$	11.7025	0.958708	0.479354	−	0.877622i	$-0.340871\pi$
$149$	11.7025	0.958708	0.479354	+	0.877622i	$0.340871\pi$
$150$	0	0
$151$	−12.6007	−1.02543	−0.512715	−	0.858559i	$-0.671360\pi$
$151$	−12.6007	−1.02543	−0.512715	+	0.858559i	$0.671360\pi$
$152$	0	0
$153$	−12.4703	−1.00816
$154$	0	0
$155$	0.760465	0.0610820
$156$	0	0
$157$	−6.47653	−0.516883	−0.258441	−	0.966027i	$-0.583209\pi$
$157$	−6.47653	−0.516883	−0.258441	+	0.966027i	$0.583209\pi$
$158$	0	0
$159$	22.8824	1.81469
$160$	0	0
$161$	22.4628	1.77032
$162$	0	0
$163$	−21.7737	−1.70545	−0.852724	−	0.522361i	$-0.825051\pi$
$163$	−21.7737	−1.70545	−0.852724	+	0.522361i	$0.825051\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−19.0881	−1.47708	−0.738541	−	0.674208i	$-0.764485\pi$
$167$	−19.0881	−1.47708	−0.738541	+	0.674208i	$0.764485\pi$
$168$	0	0
$169$	5.24299	0.403307
$170$	0	0
$171$	−43.5364	−3.32931
$172$	0	0
$173$	1.44630	0.109960	0.0549800	−	0.998487i	$-0.482491\pi$
$173$	1.44630	0.109960	0.0549800	+	0.998487i	$0.482491\pi$
$174$	0	0
$175$	−3.51508	−0.265715
$176$	0	0
$177$	−41.8560	−3.14609
$178$	0	0
$179$	−9.02979	−0.674918	−0.337459	−	0.941340i	$-0.609567\pi$
$179$	−9.02979	−0.674918	−0.337459	+	0.941340i	$0.609567\pi$
$180$	0	0
$181$	−1.14725	−0.0852744	−0.0426372	−	0.999091i	$-0.513576\pi$
$181$	−1.14725	−0.0852744	−0.0426372	+	0.999091i	$0.513576\pi$
$182$	0	0
$183$	−29.2934	−2.16543
$184$	0	0
$185$	1.45191	0.106746
$186$	0	0
$187$	0	0
$188$	0	0
$189$	56.0573	4.07757
$190$	0	0
$191$	−14.9868	−1.08440	−0.542202	−	0.840248i	$-0.682409\pi$
$191$	−14.9868	−1.08440	−0.542202	+	0.840248i	$0.682409\pi$
$192$	0	0
$193$	15.1821	1.09283	0.546414	−	0.837515i	$-0.315993\pi$
$193$	15.1821	1.09283	0.546414	+	0.837515i	$0.315993\pi$
$194$	0	0
$195$	14.0645	1.00718
$196$	0	0
$197$	2.95776	0.210731	0.105366	−	0.994434i	$-0.466399\pi$
$197$	2.95776	0.210731	0.105366	+	0.994434i	$0.466399\pi$
$198$	0	0
$199$	−5.07974	−0.360093	−0.180046	−	0.983658i	$-0.557625\pi$
$199$	−5.07974	−0.360093	−0.180046	+	0.983658i	$0.557625\pi$
$200$	0	0
$201$	−41.2054	−2.90641
$202$	0	0
$203$	−1.34002	−0.0940509
$204$	0	0
$205$	−6.12208	−0.427585
$206$	0	0
$207$	−50.1205	−3.48362
$208$	0	0
$209$	0	0
$210$	0	0
$211$	8.50814	0.585724	0.292862	−	0.956155i	$-0.405392\pi$
$211$	8.50814	0.585724	0.292862	+	0.956155i	$0.405392\pi$
$212$	0	0
$213$	−24.2320	−1.66035
$214$	0	0
$215$	−7.19067	−0.490400
$216$	0	0
$217$	2.67309	0.181461
$218$	0	0
$219$	23.1559	1.56473
$220$	0	0
$221$	−6.79108	−0.456817
$222$	0	0
$223$	15.0443	1.00744	0.503719	−	0.863867i	$-0.331965\pi$
$223$	15.0443	1.00744	0.503719	+	0.863867i	$0.331965\pi$
$224$	0	0
$225$	7.84307	0.522871
$226$	0	0
$227$	15.7035	1.04228	0.521138	−	0.853473i	$-0.325508\pi$
$227$	15.7035	1.04228	0.521138	+	0.853473i	$0.325508\pi$
$228$	0	0
$229$	21.4760	1.41918	0.709588	−	0.704617i	$-0.248881\pi$
$229$	21.4760	1.41918	0.709588	+	0.704617i	$0.248881\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	7.07615	0.463574	0.231787	−	0.972767i	$-0.425543\pi$
$233$	7.07615	0.463574	0.231787	+	0.972767i	$0.425543\pi$
$234$	0	0
$235$	−4.32164	−0.281913
$236$	0	0
$237$	−23.0298	−1.49595
$238$	0	0
$239$	−12.9846	−0.839905	−0.419952	−	0.907546i	$-0.637953\pi$
$239$	−12.9846	−0.839905	−0.419952	+	0.907546i	$0.637953\pi$
$240$	0	0
$241$	−4.91442	−0.316565	−0.158283	−	0.987394i	$-0.550596\pi$
$241$	−4.91442	−0.316565	−0.158283	+	0.987394i	$0.550596\pi$
$242$	0	0
$243$	−47.5998	−3.05353
$244$	0	0
$245$	−5.35577	−0.342167
$246$	0	0
$247$	−23.7091	−1.50857
$248$	0	0
$249$	−35.4015	−2.24348
$250$	0	0
$251$	4.68561	0.295753	0.147877	−	0.989006i	$-0.452756\pi$
$251$	4.68561	0.295753	0.147877	+	0.989006i	$0.452756\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	−5.23561	−0.327866
$256$	0	0
$257$	1.92457	0.120051	0.0600255	−	0.998197i	$-0.480882\pi$
$257$	1.92457	0.120051	0.0600255	+	0.998197i	$0.480882\pi$
$258$	0	0
$259$	5.10357	0.317120
$260$	0	0
$261$	2.98994	0.185072
$262$	0	0
$263$	16.6865	1.02894	0.514468	−	0.857510i	$-0.327990\pi$
$263$	16.6865	1.02894	0.514468	+	0.857510i	$0.327990\pi$
$264$	0	0
$265$	6.94904	0.426876
$266$	0	0
$267$	−1.48705	−0.0910058
$268$	0	0
$269$	−4.58774	−0.279719	−0.139860	−	0.990171i	$-0.544665\pi$
$269$	−4.58774	−0.279719	−0.139860	+	0.990171i	$0.544665\pi$
$270$	0	0
$271$	−20.2427	−1.22966	−0.614829	−	0.788660i	$-0.710775\pi$
$271$	−20.2427	−1.22966	−0.614829	+	0.788660i	$0.710775\pi$
$272$	0	0
$273$	49.4378	2.99211
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−4.79906	−0.288348	−0.144174	−	0.989552i	$-0.546052\pi$
$277$	−4.79906	−0.288348	−0.144174	+	0.989552i	$0.546052\pi$
$278$	0	0
$279$	−5.96438	−0.357078
$280$	0	0
$281$	21.5980	1.28843	0.644214	−	0.764845i	$-0.277185\pi$
$281$	21.5980	1.28843	0.644214	+	0.764845i	$0.277185\pi$
$282$	0	0
$283$	12.2540	0.728422	0.364211	−	0.931316i	$-0.381339\pi$
$283$	12.2540	0.728422	0.364211	+	0.931316i	$0.381339\pi$
$284$	0	0
$285$	−18.2786	−1.08273
$286$	0	0
$287$	−21.5196	−1.27026
$288$	0	0
$289$	−14.4720	−0.851293
$290$	0	0
$291$	9.15277	0.536545
$292$	0	0
$293$	−14.3193	−0.836544	−0.418272	−	0.908322i	$-0.637364\pi$
$293$	−14.3193	−0.836544	−0.418272	+	0.908322i	$0.637364\pi$
$294$	0	0
$295$	−12.7110	−0.740066
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−27.2946	−1.57849
$300$	0	0
$301$	−25.2758	−1.45687
$302$	0	0
$303$	17.5751	1.00967
$304$	0	0
$305$	−8.89599	−0.509383
$306$	0	0
$307$	−0.568487	−0.0324453	−0.0162226	−	0.999868i	$-0.505164\pi$
$307$	−0.568487	−0.0324453	−0.0162226	+	0.999868i	$0.505164\pi$
$308$	0	0
$309$	−57.2358	−3.25603
$310$	0	0
$311$	10.2047	0.578654	0.289327	−	0.957230i	$-0.406568\pi$
$311$	10.2047	0.578654	0.289327	+	0.957230i	$0.406568\pi$
$312$	0	0
$313$	−9.33625	−0.527716	−0.263858	−	0.964561i	$-0.584995\pi$
$313$	−9.33625	−0.527716	−0.263858	+	0.964561i	$0.584995\pi$
$314$	0	0
$315$	27.5690	1.55334
$316$	0	0
$317$	−28.2884	−1.58884	−0.794418	−	0.607372i	$-0.792224\pi$
$317$	−28.2884	−1.58884	−0.794418	+	0.607372i	$0.792224\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	11.5179	0.642869
$322$	0	0
$323$	8.82586	0.491084
$324$	0	0
$325$	4.27118	0.236922
$326$	0	0
$327$	16.1749	0.894476
$328$	0	0
$329$	−15.1909	−0.837501
$330$	0	0
$331$	−5.31846	−0.292329	−0.146164	−	0.989260i	$-0.546693\pi$
$331$	−5.31846	−0.292329	−0.146164	+	0.989260i	$0.546693\pi$
$332$	0	0
$333$	−11.3874	−0.624026
$334$	0	0
$335$	−12.5135	−0.683685
$336$	0	0
$337$	17.6340	0.960584	0.480292	−	0.877109i	$-0.340531\pi$
$337$	17.6340	0.960584	0.480292	+	0.877109i	$0.340531\pi$
$338$	0	0
$339$	56.5448	3.07109
$340$	0	0
$341$	0	0
$342$	0	0
$343$	5.77961	0.312069
$344$	0	0
$345$	−21.0429	−1.13291
$346$	0	0
$347$	11.8289	0.635006	0.317503	−	0.948257i	$-0.397156\pi$
$347$	11.8289	0.635006	0.317503	+	0.948257i	$0.397156\pi$
$348$	0	0
$349$	9.35132	0.500565	0.250282	−	0.968173i	$-0.419477\pi$
$349$	9.35132	0.500565	0.250282	+	0.968173i	$0.419477\pi$
$350$	0	0
$351$	−68.1154	−3.63573
$352$	0	0
$353$	−17.7336	−0.943863	−0.471932	−	0.881635i	$-0.656443\pi$
$353$	−17.7336	−0.943863	−0.471932	+	0.881635i	$0.656443\pi$
$354$	0	0
$355$	−7.35890	−0.390570
$356$	0	0
$357$	−18.4036	−0.974020
$358$	0	0
$359$	18.8840	0.996658	0.498329	−	0.866988i	$-0.333947\pi$
$359$	18.8840	0.996658	0.498329	+	0.866988i	$0.333947\pi$
$360$	0	0
$361$	11.8129	0.621731
$362$	0	0
$363$	0	0
$364$	0	0
$365$	7.03210	0.368077
$366$	0	0
$367$	−26.3936	−1.37773	−0.688867	−	0.724887i	$-0.741892\pi$
$367$	−26.3936	−1.37773	−0.688867	+	0.724887i	$0.741892\pi$
$368$	0	0
$369$	48.0159	2.49961
$370$	0	0
$371$	24.4264	1.26816
$372$	0	0
$373$	19.2368	0.996043	0.498022	−	0.867165i	$-0.334060\pi$
$373$	19.2368	0.996043	0.498022	+	0.867165i	$0.334060\pi$
$374$	0	0
$375$	3.29288	0.170044
$376$	0	0
$377$	1.62826	0.0838597
$378$	0	0
$379$	−34.4665	−1.77042	−0.885212	−	0.465189i	$-0.845986\pi$
$379$	−34.4665	−1.77042	−0.885212	+	0.465189i	$0.845986\pi$
$380$	0	0
$381$	51.0012	2.61287
$382$	0	0
$383$	−22.3171	−1.14035	−0.570176	−	0.821523i	$-0.693125\pi$
$383$	−22.3171	−1.14035	−0.570176	+	0.821523i	$0.693125\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	56.3970	2.86682
$388$	0	0
$389$	−16.1067	−0.816644	−0.408322	−	0.912838i	$-0.633886\pi$
$389$	−16.1067	−0.816644	−0.408322	+	0.912838i	$0.633886\pi$
$390$	0	0
$391$	10.1606	0.513845
$392$	0	0
$393$	−12.0103	−0.605838
$394$	0	0
$395$	−6.99381	−0.351897
$396$	0	0
$397$	16.8311	0.844731	0.422365	−	0.906426i	$-0.361200\pi$
$397$	16.8311	0.844731	0.422365	+	0.906426i	$0.361200\pi$
$398$	0	0
$399$	−64.2506	−3.21655
$400$	0	0
$401$	5.26721	0.263032	0.131516	−	0.991314i	$-0.458016\pi$
$401$	5.26721	0.263032	0.131516	+	0.991314i	$0.458016\pi$
$402$	0	0
$403$	−3.24808	−0.161799
$404$	0	0
$405$	−28.9846	−1.44025
$406$	0	0
$407$	0	0
$408$	0	0
$409$	−23.3376	−1.15397	−0.576985	−	0.816755i	$-0.695771\pi$
$409$	−23.3376	−1.15397	−0.576985	+	0.816755i	$0.695771\pi$
$410$	0	0
$411$	53.5353	2.64070
$412$	0	0
$413$	−44.6803	−2.19857
$414$	0	0
$415$	−10.7509	−0.527742
$416$	0	0
$417$	−24.6713	−1.20816
$418$	0	0
$419$	9.21755	0.450307	0.225153	−	0.974323i	$-0.427712\pi$
$419$	9.21755	0.450307	0.225153	+	0.974323i	$0.427712\pi$
$420$	0	0
$421$	20.9901	1.02299	0.511496	−	0.859285i	$-0.329091\pi$
$421$	20.9901	1.02299	0.511496	+	0.859285i	$0.329091\pi$
$422$	0	0
$423$	33.8949	1.64803
$424$	0	0
$425$	−1.58998	−0.0771252
$426$	0	0
$427$	−31.2701	−1.51327
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−36.8179	−1.77346	−0.886729	−	0.462290i	$-0.847028\pi$
$431$	−36.8179	−1.77346	−0.886729	+	0.462290i	$0.847028\pi$
$432$	0	0
$433$	11.6339	0.559087	0.279544	−	0.960133i	$-0.409817\pi$
$433$	11.6339	0.559087	0.279544	+	0.960133i	$0.409817\pi$
$434$	0	0
$435$	1.25531	0.0601877
$436$	0	0
$437$	35.4728	1.69689
$438$	0	0
$439$	12.4765	0.595471	0.297735	−	0.954648i	$-0.403769\pi$
$439$	12.4765	0.595471	0.297735	+	0.954648i	$0.403769\pi$
$440$	0	0
$441$	42.0057	2.00027
$442$	0	0
$443$	8.22715	0.390884	0.195442	−	0.980715i	$-0.437386\pi$
$443$	8.22715	0.390884	0.195442	+	0.980715i	$0.437386\pi$
$444$	0	0
$445$	−0.451594	−0.0214076
$446$	0	0
$447$	−38.5350	−1.82264
$448$	0	0
$449$	−34.2395	−1.61586	−0.807930	−	0.589278i	$-0.799412\pi$
$449$	−34.2395	−1.61586	−0.807930	+	0.589278i	$0.799412\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	41.4926	1.94949
$454$	0	0
$455$	15.0135	0.703845
$456$	0	0
$457$	−24.4317	−1.14287	−0.571434	−	0.820648i	$-0.693613\pi$
$457$	−24.4317	−1.14287	−0.571434	+	0.820648i	$0.693613\pi$
$458$	0	0
$459$	25.3564	1.18354
$460$	0	0
$461$	15.4406	0.719141	0.359571	−	0.933118i	$-0.382923\pi$
$461$	15.4406	0.719141	0.359571	+	0.933118i	$0.382923\pi$
$462$	0	0
$463$	−25.6507	−1.19209	−0.596046	−	0.802951i	$-0.703262\pi$
$463$	−25.6507	−1.19209	−0.596046	+	0.802951i	$0.703262\pi$
$464$	0	0
$465$	−2.50412	−0.116126
$466$	0	0
$467$	−7.80052	−0.360965	−0.180482	−	0.983578i	$-0.557766\pi$
$467$	−7.80052	−0.360965	−0.180482	+	0.983578i	$0.557766\pi$
$468$	0	0
$469$	−43.9859	−2.03108
$470$	0	0
$471$	21.3264	0.982670
$472$	0	0
$473$	0	0
$474$	0	0
$475$	−5.55094	−0.254694
$476$	0	0
$477$	−54.5018	−2.49547
$478$	0	0
$479$	38.9353	1.77900	0.889501	−	0.456934i	$-0.151052\pi$
$479$	38.9353	1.77900	0.889501	+	0.456934i	$0.151052\pi$
$480$	0	0
$481$	−6.20136	−0.282758
$482$	0	0
$483$	−73.9674	−3.36563
$484$	0	0
$485$	2.77956	0.126213
$486$	0	0
$487$	−17.8252	−0.807737	−0.403869	−	0.914817i	$-0.632335\pi$
$487$	−17.8252	−0.807737	−0.403869	+	0.914817i	$0.632335\pi$
$488$	0	0
$489$	71.6983	3.24231
$490$	0	0
$491$	−4.37606	−0.197489	−0.0987443	−	0.995113i	$-0.531483\pi$
$491$	−4.37606	−0.197489	−0.0987443	+	0.995113i	$0.531483\pi$
$492$	0	0
$493$	−0.606131	−0.0272988
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−25.8671	−1.16030
$498$	0	0
$499$	40.6175	1.81829	0.909145	−	0.416479i	$-0.136736\pi$
$499$	40.6175	1.81829	0.909145	+	0.416479i	$0.136736\pi$
$500$	0	0
$501$	62.8549	2.80815
$502$	0	0
$503$	−6.32076	−0.281829	−0.140914	−	0.990022i	$-0.545004\pi$
$503$	−6.32076	−0.281829	−0.140914	+	0.990022i	$0.545004\pi$
$504$	0	0
$505$	5.33731	0.237507
$506$	0	0
$507$	−17.2645	−0.766745
$508$	0	0
$509$	−12.9331	−0.573250	−0.286625	−	0.958043i	$-0.592533\pi$
$509$	−12.9331	−0.573250	−0.286625	+	0.958043i	$0.592533\pi$
$510$	0	0
$511$	24.7184	1.09348
$512$	0	0
$513$	88.5245	3.90845
$514$	0	0
$515$	−17.3817	−0.765928
$516$	0	0
$517$	0	0
$518$	0	0
$519$	−4.76249	−0.209050
$520$	0	0
$521$	−16.4286	−0.719748	−0.359874	−	0.933001i	$-0.617180\pi$
$521$	−16.4286	−0.719748	−0.359874	+	0.933001i	$0.617180\pi$
$522$	0	0
$523$	21.3601	0.934012	0.467006	−	0.884254i	$-0.345333\pi$
$523$	21.3601	0.934012	0.467006	+	0.884254i	$0.345333\pi$
$524$	0	0
$525$	11.5747	0.505163
$526$	0	0
$527$	1.20912	0.0526702
$528$	0	0
$529$	17.8375	0.775542
$530$	0	0
$531$	99.6937	4.32634
$532$	0	0
$533$	26.1485	1.13262
$534$	0	0
$535$	3.49783	0.151224
$536$	0	0
$537$	29.7340	1.28312
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−24.1003	−1.03615	−0.518076	−	0.855334i	$-0.673352\pi$
$541$	−24.1003	−1.03615	−0.518076	+	0.855334i	$0.673352\pi$
$542$	0	0
$543$	3.77776	0.162119
$544$	0	0
$545$	4.91209	0.210411
$546$	0	0
$547$	−22.1889	−0.948727	−0.474364	−	0.880329i	$-0.657322\pi$
$547$	−22.1889	−0.948727	−0.474364	+	0.880329i	$0.657322\pi$
$548$	0	0
$549$	69.7719	2.97779
$550$	0	0
$551$	−2.11613	−0.0901501
$552$	0	0
$553$	−24.5838	−1.04541
$554$	0	0
$555$	−4.78096	−0.202940
$556$	0	0
$557$	−35.3751	−1.49889	−0.749445	−	0.662067i	$-0.769679\pi$
$557$	−35.3751	−1.49889	−0.749445	+	0.662067i	$0.769679\pi$
$558$	0	0
$559$	30.7127	1.29901
$560$	0	0
$561$	0	0
$562$	0	0
$563$	6.79389	0.286328	0.143164	−	0.989699i	$-0.454272\pi$
$563$	6.79389	0.286328	0.143164	+	0.989699i	$0.454272\pi$
$564$	0	0
$565$	17.1718	0.722424
$566$	0	0
$567$	−101.883	−4.27868
$568$	0	0
$569$	−36.5320	−1.53150	−0.765750	−	0.643139i	$-0.777632\pi$
$569$	−36.5320	−1.53150	−0.765750	+	0.643139i	$0.777632\pi$
$570$	0	0
$571$	−30.4606	−1.27474	−0.637369	−	0.770559i	$-0.719977\pi$
$571$	−30.4606	−1.27474	−0.637369	+	0.770559i	$0.719977\pi$
$572$	0	0
$573$	49.3496	2.06161
$574$	0	0
$575$	−6.39042	−0.266499
$576$	0	0
$577$	10.5075	0.437434	0.218717	−	0.975788i	$-0.429813\pi$
$577$	10.5075	0.437434	0.218717	+	0.975788i	$0.429813\pi$
$578$	0	0
$579$	−49.9927	−2.07763
$580$	0	0
$581$	−37.7903	−1.56781
$582$	0	0
$583$	0	0
$584$	0	0
$585$	−33.4992	−1.38502
$586$	0	0
$587$	−21.9284	−0.905083	−0.452542	−	0.891743i	$-0.649483\pi$
$587$	−21.9284	−0.905083	−0.452542	+	0.891743i	$0.649483\pi$
$588$	0	0
$589$	4.22129	0.173935
$590$	0	0
$591$	−9.73955	−0.400631
$592$	0	0
$593$	21.6482	0.888987	0.444493	−	0.895782i	$-0.353384\pi$
$593$	21.6482	0.888987	0.444493	+	0.895782i	$0.353384\pi$
$594$	0	0
$595$	−5.58889	−0.229122
$596$	0	0
$597$	16.7270	0.684589
$598$	0	0
$599$	−43.7105	−1.78596	−0.892982	−	0.450092i	$-0.851391\pi$
$599$	−43.7105	−1.78596	−0.892982	+	0.450092i	$0.851391\pi$
$600$	0	0
$601$	−4.09877	−0.167192	−0.0835962	−	0.996500i	$-0.526641\pi$
$601$	−4.09877	−0.167192	−0.0835962	+	0.996500i	$0.526641\pi$
$602$	0	0
$603$	98.1442	3.99674
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−42.1631	−1.71135	−0.855674	−	0.517515i	$-0.826857\pi$
$607$	−42.1631	−1.71135	−0.855674	+	0.517515i	$0.826857\pi$
$608$	0	0
$609$	4.41252	0.178804
$610$	0	0
$611$	18.4585	0.746751
$612$	0	0
$613$	10.6572	0.430441	0.215220	−	0.976565i	$-0.430953\pi$
$613$	10.6572	0.430441	0.215220	+	0.976565i	$0.430953\pi$
$614$	0	0
$615$	20.1593	0.812901
$616$	0	0
$617$	6.71898	0.270496	0.135248	−	0.990812i	$-0.456817\pi$
$617$	6.71898	0.270496	0.135248	+	0.990812i	$0.456817\pi$
$618$	0	0
$619$	−34.3204	−1.37945	−0.689727	−	0.724069i	$-0.742270\pi$
$619$	−34.3204	−1.37945	−0.689727	+	0.724069i	$0.742270\pi$
$620$	0	0
$621$	101.912	4.08960
$622$	0	0
$623$	−1.58739	−0.0635974
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	2.30850	0.0920459
$630$	0	0
$631$	11.0326	0.439201	0.219601	−	0.975590i	$-0.429525\pi$
$631$	11.0326	0.439201	0.219601	+	0.975590i	$0.429525\pi$
$632$	0	0
$633$	−28.0163	−1.11355
$634$	0	0
$635$	15.4883	0.614636
$636$	0	0
$637$	22.8755	0.906358
$638$	0	0
$639$	57.7163	2.28322
$640$	0	0
$641$	−25.2836	−0.998644	−0.499322	−	0.866417i	$-0.666417\pi$
$641$	−25.2836	−0.998644	−0.499322	+	0.866417i	$0.666417\pi$
$642$	0	0
$643$	−5.33932	−0.210562	−0.105281	−	0.994442i	$-0.533574\pi$
$643$	−5.33932	−0.210562	−0.105281	+	0.994442i	$0.533574\pi$
$644$	0	0
$645$	23.6780	0.932322
$646$	0	0
$647$	3.45155	0.135694	0.0678471	−	0.997696i	$-0.478387\pi$
$647$	3.45155	0.135694	0.0678471	+	0.997696i	$0.478387\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	−8.80218	−0.344985
$652$	0	0
$653$	11.3245	0.443161	0.221580	−	0.975142i	$-0.428878\pi$
$653$	11.3245	0.443161	0.221580	+	0.975142i	$0.428878\pi$
$654$	0	0
$655$	−3.64734	−0.142513
$656$	0	0
$657$	−55.1533	−2.15174
$658$	0	0
$659$	−12.0511	−0.469444	−0.234722	−	0.972063i	$-0.575418\pi$
$659$	−12.0511	−0.469444	−0.234722	+	0.972063i	$0.575418\pi$
$660$	0	0
$661$	−11.9301	−0.464027	−0.232013	−	0.972713i	$-0.574531\pi$
$661$	−11.9301	−0.464027	−0.232013	+	0.972713i	$0.574531\pi$
$662$	0	0
$663$	22.3622	0.868477
$664$	0	0
$665$	−19.5120	−0.756642
$666$	0	0
$667$	−2.43616	−0.0943284
$668$	0	0
$669$	−49.5390	−1.91529
$670$	0	0
$671$	0	0
$672$	0	0
$673$	33.3541	1.28571	0.642854	−	0.765989i	$-0.277750\pi$
$673$	33.3541	1.28571	0.642854	+	0.765989i	$0.277750\pi$
$674$	0	0
$675$	−15.9477	−0.613826
$676$	0	0
$677$	−29.9924	−1.15270	−0.576351	−	0.817202i	$-0.695524\pi$
$677$	−29.9924	−1.15270	−0.576351	+	0.817202i	$0.695524\pi$
$678$	0	0
$679$	9.77038	0.374953
$680$	0	0
$681$	−51.7096	−1.98152
$682$	0	0
$683$	13.1505	0.503190	0.251595	−	0.967833i	$-0.419045\pi$
$683$	13.1505	0.503190	0.251595	+	0.967833i	$0.419045\pi$
$684$	0	0
$685$	16.2579	0.621182
$686$	0	0
$687$	−70.7180	−2.69806
$688$	0	0
$689$	−29.6806	−1.13074
$690$	0	0
$691$	−26.1763	−0.995792	−0.497896	−	0.867237i	$-0.665894\pi$
$691$	−26.1763	−0.995792	−0.497896	+	0.867237i	$0.665894\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−7.49232	−0.284200
$696$	0	0
$697$	−9.73397	−0.368700
$698$	0	0
$699$	−23.3009	−0.881322
$700$	0	0
$701$	−1.41481	−0.0534368	−0.0267184	−	0.999643i	$-0.508506\pi$
$701$	−1.41481	−0.0534368	−0.0267184	+	0.999643i	$0.508506\pi$
$702$	0	0
$703$	8.05944	0.303968
$704$	0	0
$705$	14.2306	0.535957
$706$	0	0
$707$	18.7611	0.705582
$708$	0	0
$709$	3.68174	0.138271	0.0691353	−	0.997607i	$-0.477976\pi$
$709$	3.68174	0.138271	0.0691353	+	0.997607i	$0.477976\pi$
$710$	0	0
$711$	54.8529	2.05715
$712$	0	0
$713$	4.85969	0.181997
$714$	0	0
$715$	0	0
$716$	0	0
$717$	42.7568	1.59678
$718$	0	0
$719$	−7.95387	−0.296629	−0.148315	−	0.988940i	$-0.547385\pi$
$719$	−7.95387	−0.296629	−0.148315	+	0.988940i	$0.547385\pi$
$720$	0	0
$721$	−61.0980	−2.27541
$722$	0	0
$723$	16.1826	0.601837
$724$	0	0
$725$	0.381220	0.0141582
$726$	0	0
$727$	−35.6495	−1.32217	−0.661083	−	0.750313i	$-0.729903\pi$
$727$	−35.6495	−1.32217	−0.661083	+	0.750313i	$0.729903\pi$
$728$	0	0
$729$	69.7867	2.58469
$730$	0	0
$731$	−11.4330	−0.422865
$732$	0	0
$733$	−9.54263	−0.352465	−0.176233	−	0.984349i	$-0.556391\pi$
$733$	−9.54263	−0.352465	−0.176233	+	0.984349i	$0.556391\pi$
$734$	0	0
$735$	17.6359	0.650510
$736$	0	0
$737$	0	0
$738$	0	0
$739$	−34.9555	−1.28586	−0.642929	−	0.765926i	$-0.722281\pi$
$739$	−34.9555	−1.28586	−0.642929	+	0.765926i	$0.722281\pi$
$740$	0	0
$741$	78.0711	2.86801
$742$	0	0
$743$	−33.7663	−1.23876	−0.619382	−	0.785089i	$-0.712617\pi$
$743$	−33.7663	−1.23876	−0.619382	+	0.785089i	$0.712617\pi$
$744$	0	0
$745$	−11.7025	−0.428747
$746$	0	0
$747$	84.3203	3.08512
$748$	0	0
$749$	12.2951	0.449255
$750$	0	0
$751$	−13.2772	−0.484491	−0.242246	−	0.970215i	$-0.577884\pi$
$751$	−13.2772	−0.484491	−0.242246	+	0.970215i	$0.577884\pi$
$752$	0	0
$753$	−15.4292	−0.562270
$754$	0	0
$755$	12.6007	0.458586
$756$	0	0
$757$	16.3056	0.592636	0.296318	−	0.955089i	$-0.404241\pi$
$757$	16.3056	0.592636	0.296318	+	0.955089i	$0.404241\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	21.2704	0.771053	0.385526	−	0.922697i	$-0.374020\pi$
$761$	21.2704	0.771053	0.385526	+	0.922697i	$0.374020\pi$
$762$	0	0
$763$	17.2664	0.625085
$764$	0	0
$765$	12.4703	0.450865
$766$	0	0
$767$	54.2912	1.96034
$768$	0	0
$769$	5.81494	0.209692	0.104846	−	0.994488i	$-0.466565\pi$
$769$	5.81494	0.209692	0.104846	+	0.994488i	$0.466565\pi$
$770$	0	0
$771$	−6.33737	−0.228235
$772$	0	0
$773$	18.6709	0.671544	0.335772	−	0.941943i	$-0.391003\pi$
$773$	18.6709	0.671544	0.335772	+	0.941943i	$0.391003\pi$
$774$	0	0
$775$	−0.760465	−0.0273167
$776$	0	0
$777$	−16.8054	−0.602892
$778$	0	0
$779$	−33.9833	−1.21758
$780$	0	0
$781$	0	0
$782$	0	0
$783$	−6.07957	−0.217266
$784$	0	0
$785$	6.47653	0.231157
$786$	0	0
$787$	46.3213	1.65118	0.825588	−	0.564274i	$-0.190844\pi$
$787$	46.3213	1.65118	0.825588	+	0.564274i	$0.190844\pi$
$788$	0	0
$789$	−54.9468	−1.95616
$790$	0	0
$791$	60.3602	2.14616
$792$	0	0
$793$	37.9964	1.34929
$794$	0	0
$795$	−22.8824	−0.811553
$796$	0	0
$797$	−6.11777	−0.216702	−0.108351	−	0.994113i	$-0.534557\pi$
$797$	−6.11777	−0.216702	−0.108351	+	0.994113i	$0.534557\pi$
$798$	0	0
$799$	−6.87131	−0.243089
$800$	0	0
$801$	3.54189	0.125146
$802$	0	0
$803$	0	0
$804$	0	0
$805$	−22.4628	−0.791710
$806$	0	0
$807$	15.1069	0.531788
$808$	0	0
$809$	−14.2124	−0.499682	−0.249841	−	0.968287i	$-0.580378\pi$
$809$	−14.2124	−0.499682	−0.249841	+	0.968287i	$0.580378\pi$
$810$	0	0
$811$	−30.6405	−1.07593	−0.537967	−	0.842966i	$-0.680807\pi$
$811$	−30.6405	−1.07593	−0.537967	+	0.842966i	$0.680807\pi$
$812$	0	0
$813$	66.6570	2.33776
$814$	0	0
$815$	21.7737	0.762700
$816$	0	0
$817$	−39.9150	−1.39645
$818$	0	0
$819$	−117.752	−4.11460
$820$	0	0
$821$	3.53169	0.123257	0.0616285	−	0.998099i	$-0.480371\pi$
$821$	3.53169	0.123257	0.0616285	+	0.998099i	$0.480371\pi$
$822$	0	0
$823$	29.7928	1.03851	0.519256	−	0.854619i	$-0.326209\pi$
$823$	29.7928	1.03851	0.519256	+	0.854619i	$0.326209\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	50.4631	1.75477	0.877386	−	0.479785i	$-0.159285\pi$
$827$	50.4631	1.75477	0.877386	+	0.479785i	$0.159285\pi$
$828$	0	0
$829$	−19.2980	−0.670246	−0.335123	−	0.942174i	$-0.608778\pi$
$829$	−19.2980	−0.670246	−0.335123	+	0.942174i	$0.608778\pi$
$830$	0	0
$831$	15.8027	0.548191
$832$	0	0
$833$	−8.51555	−0.295046
$834$	0	0
$835$	19.0881	0.660572
$836$	0	0
$837$	12.1276	0.419193
$838$	0	0
$839$	−16.8989	−0.583414	−0.291707	−	0.956508i	$-0.594223\pi$
$839$	−16.8989	−0.583414	−0.291707	+	0.956508i	$0.594223\pi$
$840$	0	0
$841$	−28.8547	−0.994989
$842$	0	0
$843$	−71.1197	−2.44949
$844$	0	0
$845$	−5.24299	−0.180364
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−40.3508	−1.38484
$850$	0	0
$851$	9.27830	0.318056
$852$	0	0
$853$	47.5816	1.62916	0.814582	−	0.580048i	$-0.196966\pi$
$853$	47.5816	1.62916	0.814582	+	0.580048i	$0.196966\pi$
$854$	0	0
$855$	43.5364	1.48891
$856$	0	0
$857$	−21.8041	−0.744813	−0.372406	−	0.928070i	$-0.621467\pi$
$857$	−21.8041	−0.744813	−0.372406	+	0.928070i	$0.621467\pi$
$858$	0	0
$859$	−23.3331	−0.796116	−0.398058	−	0.917360i	$-0.630316\pi$
$859$	−23.3331	−0.796116	−0.398058	+	0.917360i	$0.630316\pi$
$860$	0	0
$861$	70.8615	2.41495
$862$	0	0
$863$	−21.9086	−0.745777	−0.372889	−	0.927876i	$-0.621633\pi$
$863$	−21.9086	−0.745777	−0.372889	+	0.927876i	$0.621633\pi$
$864$	0	0
$865$	−1.44630	−0.0491756
$866$	0	0
$867$	47.6545	1.61843
$868$	0	0
$869$	0	0
$870$	0	0
$871$	53.4474	1.81099
$872$	0	0
$873$	−21.8003	−0.737829
$874$	0	0
$875$	3.51508	0.118831
$876$	0	0
$877$	−52.7358	−1.78076	−0.890381	−	0.455216i	$-0.849562\pi$
$877$	−52.7358	−1.78076	−0.890381	+	0.455216i	$0.849562\pi$
$878$	0	0
$879$	47.1518	1.59039
$880$	0	0
$881$	34.3244	1.15642	0.578210	−	0.815888i	$-0.303752\pi$
$881$	34.3244	1.15642	0.578210	+	0.815888i	$0.303752\pi$
$882$	0	0
$883$	49.7184	1.67316	0.836579	−	0.547846i	$-0.184552\pi$
$883$	49.7184	1.67316	0.836579	+	0.547846i	$0.184552\pi$
$884$	0	0
$885$	41.8560	1.40697
$886$	0	0
$887$	11.9815	0.402300	0.201150	−	0.979560i	$-0.435532\pi$
$887$	11.9815	0.402300	0.201150	+	0.979560i	$0.435532\pi$
$888$	0	0
$889$	54.4427	1.82595
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−23.9891	−0.802766
$894$	0	0
$895$	9.02979	0.301833
$896$	0	0
$897$	89.8780	3.00094
$898$	0	0
$899$	−0.289905	−0.00966886
$900$	0	0
$901$	11.0488	0.368089
$902$	0	0
$903$	83.2301	2.76972
$904$	0	0
$905$	1.14725	0.0381359
$906$	0	0
$907$	47.2674	1.56949	0.784744	−	0.619820i	$-0.212794\pi$
$907$	47.2674	1.56949	0.784744	+	0.619820i	$0.212794\pi$
$908$	0	0
$909$	−41.8609	−1.38844
$910$	0	0
$911$	−8.85300	−0.293313	−0.146656	−	0.989187i	$-0.546851\pi$
$911$	−8.85300	−0.293313	−0.146656	+	0.989187i	$0.546851\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	29.2934	0.968411
$916$	0	0
$917$	−12.8207	−0.423376
$918$	0	0
$919$	16.5141	0.544750	0.272375	−	0.962191i	$-0.412191\pi$
$919$	16.5141	0.544750	0.272375	+	0.962191i	$0.412191\pi$
$920$	0	0
$921$	1.87196	0.0616832
$922$	0	0
$923$	31.4312	1.03457
$924$	0	0
$925$	−1.45191	−0.0477384
$926$	0	0
$927$	136.326	4.47753
$928$	0	0
$929$	−11.2898	−0.370408	−0.185204	−	0.982700i	$-0.559295\pi$
$929$	−11.2898	−0.370408	−0.185204	+	0.982700i	$0.559295\pi$
$930$	0	0
$931$	−29.7295	−0.974346
$932$	0	0
$933$	−33.6028	−1.10011
$934$	0	0
$935$	0	0
$936$	0	0
$937$	21.5727	0.704748	0.352374	−	0.935859i	$-0.385374\pi$
$937$	21.5727	0.704748	0.352374	+	0.935859i	$0.385374\pi$
$938$	0	0
$939$	30.7432	1.00327
$940$	0	0
$941$	−50.3062	−1.63994	−0.819968	−	0.572409i	$-0.806009\pi$
$941$	−50.3062	−1.63994	−0.819968	+	0.572409i	$0.806009\pi$
$942$	0	0
$943$	−39.1227	−1.27401
$944$	0	0
$945$	−56.0573	−1.82354
$946$	0	0
$947$	−41.7335	−1.35615	−0.678077	−	0.734990i	$-0.737187\pi$
$947$	−41.7335	−1.35615	−0.678077	+	0.734990i	$0.737187\pi$
$948$	0	0
$949$	−30.0354	−0.974990
$950$	0	0
$951$	93.1504	3.02061
$952$	0	0
$953$	35.2481	1.14180	0.570899	−	0.821021i	$-0.306595\pi$
$953$	35.2481	1.14180	0.570899	+	0.821021i	$0.306595\pi$
$954$	0	0
$955$	14.9868	0.484960
$956$	0	0
$957$	0	0
$958$	0	0
$959$	57.1477	1.84540
$960$	0	0
$961$	−30.4217	−0.981345
$962$	0	0
$963$	−27.4337	−0.884039
$964$	0	0
$965$	−15.1821	−0.488728
$966$	0	0
$967$	21.6156	0.695111	0.347556	−	0.937659i	$-0.387012\pi$
$967$	21.6156	0.695111	0.347556	+	0.937659i	$0.387012\pi$
$968$	0	0
$969$	−29.0625	−0.933623
$970$	0	0
$971$	−11.8335	−0.379755	−0.189878	−	0.981808i	$-0.560809\pi$
$971$	−11.8335	−0.379755	−0.189878	+	0.981808i	$0.560809\pi$
$972$	0	0
$973$	−26.3361	−0.844296
$974$	0	0
$975$	−14.0645	−0.450424
$976$	0	0
$977$	−51.6998	−1.65402	−0.827011	−	0.562185i	$-0.809961\pi$
$977$	−51.6998	−1.65402	−0.827011	+	0.562185i	$0.809961\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	−38.5259	−1.23004
$982$	0	0
$983$	−51.1243	−1.63061	−0.815306	−	0.579030i	$-0.803431\pi$
$983$	−51.1243	−1.63061	−0.815306	+	0.579030i	$0.803431\pi$
$984$	0	0
$985$	−2.95776	−0.0942420
$986$	0	0
$987$	50.0218	1.59221
$988$	0	0
$989$	−45.9514	−1.46117
$990$	0	0
$991$	3.16402	0.100508	0.0502542	−	0.998736i	$-0.483997\pi$
$991$	3.16402	0.100508	0.0502542	+	0.998736i	$0.483997\pi$
$992$	0	0
$993$	17.5130	0.555760
$994$	0	0
$995$	5.07974	0.161038
$996$	0	0
$997$	5.68777	0.180133	0.0900667	−	0.995936i	$-0.471292\pi$
$997$	5.68777	0.180133	0.0900667	+	0.995936i	$0.471292\pi$
$998$	0	0
$999$	23.1545	0.732577

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4840.2.a.ba.1.1		6
4.3	odd	2		9680.2.a.dd.1.6		6
11.2	odd	10		440.2.y.c.81.1	✓	12
11.6	odd	10		440.2.y.c.201.1	yes	12
11.10	odd	2		4840.2.a.bb.1.1		6
44.35	even	10		880.2.bo.i.81.3		12
44.39	even	10		880.2.bo.i.641.3		12
44.43	even	2		9680.2.a.dc.1.6		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
440.2.y.c.81.1	✓	12	11.2	odd	10
440.2.y.c.201.1	yes	12	11.6	odd	10
880.2.bo.i.81.3		12	44.35	even	10
880.2.bo.i.641.3		12	44.39	even	10
4840.2.a.ba.1.1		6	1.1	even	1	trivial
4840.2.a.bb.1.1		6	11.10	odd	2
9680.2.a.dc.1.6		6	44.43	even	2
9680.2.a.dd.1.6		6	4.3	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 4840 = 2^{3} \cdot 5 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4840.a (trivial)

\(f(q)\)	\(=\)	\(q-3.29288 q^{3} -1.00000 q^{5} -3.51508 q^{7} +7.84307 q^{9} +4.27118 q^{13} +3.29288 q^{15} -1.58998 q^{17} -5.55094 q^{19} +11.5747 q^{21} -6.39042 q^{23} +1.00000 q^{25} -15.9477 q^{27} +0.381220 q^{29} -0.760465 q^{31} +3.51508 q^{35} -1.45191 q^{37} -14.0645 q^{39} +6.12208 q^{41} +7.19067 q^{43} -7.84307 q^{45} +4.32164 q^{47} +5.35577 q^{49} +5.23561 q^{51} -6.94904 q^{53} +18.2786 q^{57} +12.7110 q^{59} +8.89599 q^{61} -27.5690 q^{63} -4.27118 q^{65} +12.5135 q^{67} +21.0429 q^{69} +7.35890 q^{71} -7.03210 q^{73} -3.29288 q^{75} +6.99381 q^{79} +28.9846 q^{81} +10.7509 q^{83} +1.58998 q^{85} -1.25531 q^{87} +0.451594 q^{89} -15.0135 q^{91} +2.50412 q^{93} +5.55094 q^{95} -2.77956 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 3 q^{3} - 6 q^{5} - 7 q^{7} + 5 q^{9} + q^{13} + 3 q^{15} + 6 q^{17} - 7 q^{19} + 14 q^{21} - 9 q^{23} + 6 q^{25} - 21 q^{27} + 10 q^{29} + q^{31} + 7 q^{35} + 3 q^{37} - q^{39} - 6 q^{41} + 18 q^{43}+ \cdots - 33 q^{97}+O(q^{100}) \) 6 * q - 3 * q^3 - 6 * q^5 - 7 * q^7 + 5 * q^9 + q^13 + 3 * q^15 + 6 * q^17 - 7 * q^19 + 14 * q^21 - 9 * q^23 + 6 * q^25 - 21 * q^27 + 10 * q^29 + q^31 + 7 * q^35 + 3 * q^37 - q^39 - 6 * q^41 + 18 * q^43 - 5 * q^45 - 3 * q^47 + 17 * q^49 + 15 * q^51 - 23 * q^53 + 9 * q^57 - 2 * q^59 + 6 * q^61 - 49 * q^63 - q^65 - 22 * q^67 + 2 * q^69 - 13 * q^71 + 10 * q^73 - 3 * q^75 - 22 * q^79 + 10 * q^81 + 10 * q^83 - 6 * q^85 - 3 * q^87 - 25 * q^89 + 12 * q^91 + 19 * q^93 + 7 * q^95 - 33 * q^97

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