LMFDB - Embedded newform 6024.2.a.r.1.15

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6024,2,Mod(1,6024)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6024, 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("6024.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6024 = 2^{3} \cdot 3 \cdot 251 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6024.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:	$48.1018821776$
Analytic rank:	$0$
Dimension:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - 9 x^{19} - 31 x^{18} + 471 x^{17} - 82 x^{16} - 9476 x^{15} + 12881 x^{14} + 94079 x^{13} + \cdots + 24832 $ x^20 - 9x^19 - 31x^18 + 471x^17 - 82x^16 - 9476x^15 + 12881x^14 + 94079x^13 - 181272x^12 - 504914x^11 + 1108012x^10 + 1591114x^9 - 3399127x^8 - 3224089x^7 + 5089722x^6 + 4287384x^5 - 2840216x^4 - 2820864x^3 - 208576x^2 + 204992*x + 24832
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{5} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.15
Root			$2.60908$ of defining polynomial
Character	$\chi$	$=$	6024.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} +2.60908 q^{5} +3.89020 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +2.60908 q^{5} +3.89020 q^{7} +1.00000 q^{9} +4.77326 q^{11} -3.03256 q^{13} +2.60908 q^{15} -0.454622 q^{17} -7.72714 q^{19} +3.89020 q^{21} +2.99400 q^{23} +1.80730 q^{25} +1.00000 q^{27} -7.59405 q^{29} +0.00502606 q^{31} +4.77326 q^{33} +10.1498 q^{35} +8.22864 q^{37} -3.03256 q^{39} +4.08397 q^{41} -0.188947 q^{43} +2.60908 q^{45} +5.48986 q^{47} +8.13367 q^{49} -0.454622 q^{51} +2.37697 q^{53} +12.4538 q^{55} -7.72714 q^{57} +8.71709 q^{59} -7.54578 q^{61} +3.89020 q^{63} -7.91220 q^{65} -3.67342 q^{67} +2.99400 q^{69} +9.29571 q^{71} +11.7271 q^{73} +1.80730 q^{75} +18.5689 q^{77} +4.89448 q^{79} +1.00000 q^{81} +10.9485 q^{83} -1.18614 q^{85} -7.59405 q^{87} +8.59172 q^{89} -11.7973 q^{91} +0.00502606 q^{93} -20.1607 q^{95} -3.38072 q^{97} +4.77326 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q + 20 q^{3} + 9 q^{5} + 9 q^{7} + 20 q^{9}+O(q^{10}) $ 20 * q + 20 * q^3 + 9 * q^5 + 9 * q^7 + 20 * q^9	$ 20 q + 20 q^{3} + 9 q^{5} + 9 q^{7} + 20 q^{9} + 4 q^{11} + 21 q^{13} + 9 q^{15} + 10 q^{17} + 8 q^{19} + 9 q^{21} + 9 q^{23} + 43 q^{25} + 20 q^{27} + 18 q^{29} + 27 q^{31} + 4 q^{33} - 7 q^{35} + 33 q^{37} + 21 q^{39} + 14 q^{41} - 6 q^{43} + 9 q^{45} + 21 q^{47} + 47 q^{49} + 10 q^{51} + 23 q^{53} + 24 q^{55} + 8 q^{57} + 10 q^{59} + 28 q^{61} + 9 q^{63} + 2 q^{65} + 15 q^{67} + 9 q^{69} + 33 q^{71} + 50 q^{73} + 43 q^{75} + 20 q^{77} + 17 q^{79} + 20 q^{81} - 19 q^{83} + 41 q^{85} + 18 q^{87} + 21 q^{89} + 30 q^{91} + 27 q^{93} + 27 q^{95} + 47 q^{97} + 4 q^{99}+O(q^{100}) $ 20 * q + 20 * q^3 + 9 * q^5 + 9 * q^7 + 20 * q^9 + 4 * q^11 + 21 * q^13 + 9 * q^15 + 10 * q^17 + 8 * q^19 + 9 * q^21 + 9 * q^23 + 43 * q^25 + 20 * q^27 + 18 * q^29 + 27 * q^31 + 4 * q^33 - 7 * q^35 + 33 * q^37 + 21 * q^39 + 14 * q^41 - 6 * q^43 + 9 * q^45 + 21 * q^47 + 47 * q^49 + 10 * q^51 + 23 * q^53 + 24 * q^55 + 8 * q^57 + 10 * q^59 + 28 * q^61 + 9 * q^63 + 2 * q^65 + 15 * q^67 + 9 * q^69 + 33 * q^71 + 50 * q^73 + 43 * q^75 + 20 * q^77 + 17 * q^79 + 20 * q^81 - 19 * q^83 + 41 * q^85 + 18 * q^87 + 21 * q^89 + 30 * q^91 + 27 * q^93 + 27 * q^95 + 47 * q^97 + 4 * q^99

Display 100 coefficients 10 coefficients

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
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	2.60908	1.16682	0.583408	−	0.812179i	$-0.301719\pi$
$5$	2.60908	1.16682	0.583408	+	0.812179i	$0.301719\pi$
$6$	0	0
$7$	3.89020	1.47036	0.735179	−	0.677873i	$-0.237098\pi$
$7$	3.89020	1.47036	0.735179	+	0.677873i	$0.237098\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	4.77326	1.43919	0.719595	−	0.694394i	$-0.244327\pi$
$11$	4.77326	1.43919	0.719595	+	0.694394i	$0.244327\pi$
$12$	0	0
$13$	−3.03256	−0.841081	−0.420541	−	0.907274i	$-0.638160\pi$
$13$	−3.03256	−0.841081	−0.420541	+	0.907274i	$0.638160\pi$
$14$	0	0
$15$	2.60908	0.673661
$16$	0	0
$17$	−0.454622	−0.110262	−0.0551310	−	0.998479i	$-0.517558\pi$
$17$	−0.454622	−0.110262	−0.0551310	+	0.998479i	$0.517558\pi$
$18$	0	0
$19$	−7.72714	−1.77273	−0.886364	−	0.462989i	$-0.846777\pi$
$19$	−7.72714	−1.77273	−0.886364	+	0.462989i	$0.846777\pi$
$20$	0	0
$21$	3.89020	0.848912
$22$	0	0
$23$	2.99400	0.624292	0.312146	−	0.950034i	$-0.398952\pi$
$23$	2.99400	0.624292	0.312146	+	0.950034i	$0.398952\pi$
$24$	0	0
$25$	1.80730	0.361459
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−7.59405	−1.41018	−0.705090	−	0.709118i	$-0.749094\pi$
$29$	−7.59405	−1.41018	−0.705090	+	0.709118i	$0.749094\pi$
$30$	0	0
$31$	0.00502606	0.000902707 0	0.000451353	−	1.00000i	$-0.499856\pi$
$31$	0.00502606	0.000902707 0	0.000451353		1.00000i	$0.499856\pi$
$32$	0	0
$33$	4.77326	0.830917
$34$	0	0
$35$	10.1498	1.71564
$36$	0	0
$37$	8.22864	1.35278	0.676390	−	0.736543i	$-0.263543\pi$
$37$	8.22864	1.35278	0.676390	+	0.736543i	$0.263543\pi$
$38$	0	0
$39$	−3.03256	−0.485599
$40$	0	0
$41$	4.08397	0.637808	0.318904	−	0.947787i	$-0.396685\pi$
$41$	4.08397	0.637808	0.318904	+	0.947787i	$0.396685\pi$
$42$	0	0
$43$	−0.188947	−0.0288142	−0.0144071	−	0.999896i	$-0.504586\pi$
$43$	−0.188947	−0.0288142	−0.0144071	+	0.999896i	$0.504586\pi$
$44$	0	0
$45$	2.60908	0.388939
$46$	0	0
$47$	5.48986	0.800778	0.400389	−	0.916345i	$-0.368875\pi$
$47$	5.48986	0.800778	0.400389	+	0.916345i	$0.368875\pi$
$48$	0	0
$49$	8.13367	1.16195
$50$	0	0
$51$	−0.454622	−0.0636598
$52$	0	0
$53$	2.37697	0.326502	0.163251	−	0.986585i	$-0.447802\pi$
$53$	2.37697	0.326502	0.163251	+	0.986585i	$0.447802\pi$
$54$	0	0
$55$	12.4538	1.67927
$56$	0	0
$57$	−7.72714	−1.02348
$58$	0	0
$59$	8.71709	1.13487	0.567434	−	0.823419i	$-0.307936\pi$
$59$	8.71709	1.13487	0.567434	+	0.823419i	$0.307936\pi$
$60$	0	0
$61$	−7.54578	−0.966139	−0.483069	−	0.875582i	$-0.660478\pi$
$61$	−7.54578	−0.966139	−0.483069	+	0.875582i	$0.660478\pi$
$62$	0	0
$63$	3.89020	0.490119
$64$	0	0
$65$	−7.91220	−0.981387
$66$	0	0
$67$	−3.67342	−0.448780	−0.224390	−	0.974499i	$-0.572039\pi$
$67$	−3.67342	−0.448780	−0.224390	+	0.974499i	$0.572039\pi$
$68$	0	0
$69$	2.99400	0.360435
$70$	0	0
$71$	9.29571	1.10320	0.551599	−	0.834109i	$-0.314018\pi$
$71$	9.29571	1.10320	0.551599	+	0.834109i	$0.314018\pi$
$72$	0	0
$73$	11.7271	1.37255	0.686277	−	0.727340i	$-0.259244\pi$
$73$	11.7271	1.37255	0.686277	+	0.727340i	$0.259244\pi$
$74$	0	0
$75$	1.80730	0.208689
$76$	0	0
$77$	18.5689	2.11613
$78$	0	0
$79$	4.89448	0.550672	0.275336	−	0.961348i	$-0.411211\pi$
$79$	4.89448	0.550672	0.275336	+	0.961348i	$0.411211\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	10.9485	1.20175	0.600876	−	0.799342i	$-0.294818\pi$
$83$	10.9485	1.20175	0.600876	+	0.799342i	$0.294818\pi$
$84$	0	0
$85$	−1.18614	−0.128655
$86$	0	0
$87$	−7.59405	−0.814168
$88$	0	0
$89$	8.59172	0.910721	0.455360	−	0.890307i	$-0.349510\pi$
$89$	8.59172	0.910721	0.455360	+	0.890307i	$0.349510\pi$
$90$	0	0
$91$	−11.7973	−1.23669
$92$	0	0
$93$	0.00502606	0.000521178 0
$94$	0	0
$95$	−20.1607	−2.06845
$96$	0	0
$97$	−3.38072	−0.343260	−0.171630	−	0.985161i	$-0.554903\pi$
$97$	−3.38072	−0.343260	−0.171630	+	0.985161i	$0.554903\pi$
$98$	0	0
$99$	4.77326	0.479730
$100$	0	0
$101$	−9.02834	−0.898353	−0.449177	−	0.893443i	$-0.648283\pi$
$101$	−9.02834	−0.898353	−0.449177	+	0.893443i	$0.648283\pi$
$102$	0	0
$103$	−0.251833	−0.0248138	−0.0124069	−	0.999923i	$-0.503949\pi$
$103$	−0.251833	−0.0248138	−0.0124069	+	0.999923i	$0.503949\pi$
$104$	0	0
$105$	10.1498	0.990524
$106$	0	0
$107$	5.53647	0.535231	0.267616	−	0.963526i	$-0.413764\pi$
$107$	5.53647	0.535231	0.267616	+	0.963526i	$0.413764\pi$
$108$	0	0
$109$	−0.302664	−0.0289900	−0.0144950	−	0.999895i	$-0.504614\pi$
$109$	−0.302664	−0.0289900	−0.0144950	+	0.999895i	$0.504614\pi$
$110$	0	0
$111$	8.22864	0.781028
$112$	0	0
$113$	−1.94620	−0.183083	−0.0915415	−	0.995801i	$-0.529179\pi$
$113$	−1.94620	−0.183083	−0.0915415	+	0.995801i	$0.529179\pi$
$114$	0	0
$115$	7.81159	0.728434
$116$	0	0
$117$	−3.03256	−0.280360
$118$	0	0
$119$	−1.76857	−0.162125
$120$	0	0
$121$	11.7840	1.07127
$122$	0	0
$123$	4.08397	0.368239
$124$	0	0
$125$	−8.33002	−0.745060
$126$	0	0
$127$	14.6262	1.29786	0.648932	−	0.760846i	$-0.275216\pi$
$127$	14.6262	1.29786	0.648932	+	0.760846i	$0.275216\pi$
$128$	0	0
$129$	−0.188947	−0.0166359
$130$	0	0
$131$	−0.621210	−0.0542754	−0.0271377	−	0.999632i	$-0.508639\pi$
$131$	−0.621210	−0.0542754	−0.0271377	+	0.999632i	$0.508639\pi$
$132$	0	0
$133$	−30.0601	−2.60654
$134$	0	0
$135$	2.60908	0.224554
$136$	0	0
$137$	−3.65703	−0.312441	−0.156220	−	0.987722i	$-0.549931\pi$
$137$	−3.65703	−0.312441	−0.156220	+	0.987722i	$0.549931\pi$
$138$	0	0
$139$	−9.48887	−0.804835	−0.402418	−	0.915456i	$-0.631830\pi$
$139$	−9.48887	−0.804835	−0.402418	+	0.915456i	$0.631830\pi$
$140$	0	0
$141$	5.48986	0.462329
$142$	0	0
$143$	−14.4752	−1.21048
$144$	0	0
$145$	−19.8135	−1.64542
$146$	0	0
$147$	8.13367	0.670854
$148$	0	0
$149$	−12.9718	−1.06269	−0.531345	−	0.847155i	$-0.678313\pi$
$149$	−12.9718	−1.06269	−0.531345	+	0.847155i	$0.678313\pi$
$150$	0	0
$151$	1.15492	0.0939862	0.0469931	−	0.998895i	$-0.485036\pi$
$151$	1.15492	0.0939862	0.0469931	+	0.998895i	$0.485036\pi$
$152$	0	0
$153$	−0.454622	−0.0367540
$154$	0	0
$155$	0.0131134	0.00105329
$156$	0	0
$157$	12.3875	0.988634	0.494317	−	0.869282i	$-0.335418\pi$
$157$	12.3875	0.988634	0.494317	+	0.869282i	$0.335418\pi$
$158$	0	0
$159$	2.37697	0.188506
$160$	0	0
$161$	11.6473	0.917933
$162$	0	0
$163$	−21.1628	−1.65760	−0.828800	−	0.559546i	$-0.810976\pi$
$163$	−21.1628	−1.65760	−0.828800	+	0.559546i	$0.810976\pi$
$164$	0	0
$165$	12.4538	0.969527
$166$	0	0
$167$	−10.6419	−0.823495	−0.411747	−	0.911298i	$-0.635081\pi$
$167$	−10.6419	−0.823495	−0.411747	+	0.911298i	$0.635081\pi$
$168$	0	0
$169$	−3.80357	−0.292582
$170$	0	0
$171$	−7.72714	−0.590909
$172$	0	0
$173$	13.1309	0.998324	0.499162	−	0.866509i	$-0.333641\pi$
$173$	13.1309	0.998324	0.499162	+	0.866509i	$0.333641\pi$
$174$	0	0
$175$	7.03075	0.531474
$176$	0	0
$177$	8.71709	0.655216
$178$	0	0
$179$	−11.8989	−0.889366	−0.444683	−	0.895688i	$-0.646684\pi$
$179$	−11.8989	−0.889366	−0.444683	+	0.895688i	$0.646684\pi$
$180$	0	0
$181$	−7.26972	−0.540354	−0.270177	−	0.962811i	$-0.587082\pi$
$181$	−7.26972	−0.540354	−0.270177	+	0.962811i	$0.587082\pi$
$182$	0	0
$183$	−7.54578	−0.557800
$184$	0	0
$185$	21.4692	1.57845
$186$	0	0
$187$	−2.17003	−0.158688
$188$	0	0
$189$	3.89020	0.282971
$190$	0	0
$191$	−15.9031	−1.15071	−0.575353	−	0.817905i	$-0.695135\pi$
$191$	−15.9031	−1.15071	−0.575353	+	0.817905i	$0.695135\pi$
$192$	0	0
$193$	20.1946	1.45364	0.726818	−	0.686830i	$-0.240998\pi$
$193$	20.1946	1.45364	0.726818	+	0.686830i	$0.240998\pi$
$194$	0	0
$195$	−7.91220	−0.566604
$196$	0	0
$197$	8.38272	0.597244	0.298622	−	0.954371i	$-0.403473\pi$
$197$	8.38272	0.597244	0.298622	+	0.954371i	$0.403473\pi$
$198$	0	0
$199$	−9.93511	−0.704281	−0.352141	−	0.935947i	$-0.614546\pi$
$199$	−9.93511	−0.704281	−0.352141	+	0.935947i	$0.614546\pi$
$200$	0	0
$201$	−3.67342	−0.259103
$202$	0	0
$203$	−29.5424	−2.07347
$204$	0	0
$205$	10.6554	0.744205
$206$	0	0
$207$	2.99400	0.208097
$208$	0	0
$209$	−36.8836	−2.55129
$210$	0	0
$211$	−9.43905	−0.649811	−0.324906	−	0.945746i	$-0.605333\pi$
$211$	−9.43905	−0.649811	−0.324906	+	0.945746i	$0.605333\pi$
$212$	0	0
$213$	9.29571	0.636932
$214$	0	0
$215$	−0.492979	−0.0336209
$216$	0	0
$217$	0.0195524	0.00132730
$218$	0	0
$219$	11.7271	0.792445
$220$	0	0
$221$	1.37867	0.0927393
$222$	0	0
$223$	13.6643	0.915031	0.457516	−	0.889202i	$-0.348739\pi$
$223$	13.6643	0.915031	0.457516	+	0.889202i	$0.348739\pi$
$224$	0	0
$225$	1.80730	0.120486
$226$	0	0
$227$	−25.6140	−1.70006	−0.850031	−	0.526732i	$-0.823417\pi$
$227$	−25.6140	−1.70006	−0.850031	+	0.526732i	$0.823417\pi$
$228$	0	0
$229$	10.1840	0.672980	0.336490	−	0.941687i	$-0.390760\pi$
$229$	10.1840	0.672980	0.336490	+	0.941687i	$0.390760\pi$
$230$	0	0
$231$	18.5689	1.22175
$232$	0	0
$233$	−0.898428	−0.0588580	−0.0294290	−	0.999567i	$-0.509369\pi$
$233$	−0.898428	−0.0588580	−0.0294290	+	0.999567i	$0.509369\pi$
$234$	0	0
$235$	14.3235	0.934360
$236$	0	0
$237$	4.89448	0.317931
$238$	0	0
$239$	−16.6531	−1.07720	−0.538601	−	0.842561i	$-0.681047\pi$
$239$	−16.6531	−1.07720	−0.538601	+	0.842561i	$0.681047\pi$
$240$	0	0
$241$	12.9837	0.836354	0.418177	−	0.908366i	$-0.362669\pi$
$241$	12.9837	0.836354	0.418177	+	0.908366i	$0.362669\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	21.2214	1.35579
$246$	0	0
$247$	23.4330	1.49101
$248$	0	0
$249$	10.9485	0.693832
$250$	0	0
$251$	−1.00000	−0.0631194
$251$	−1.00000	−0.0631194
$252$	0	0
$253$	14.2911	0.898476
$254$	0	0
$255$	−1.18614	−0.0742792
$256$	0	0
$257$	14.1092	0.880108	0.440054	−	0.897971i	$-0.354959\pi$
$257$	14.1092	0.880108	0.440054	+	0.897971i	$0.354959\pi$
$258$	0	0
$259$	32.0111	1.98907
$260$	0	0
$261$	−7.59405	−0.470060
$262$	0	0
$263$	−7.53490	−0.464622	−0.232311	−	0.972642i	$-0.574629\pi$
$263$	−7.53490	−0.464622	−0.232311	+	0.972642i	$0.574629\pi$
$264$	0	0
$265$	6.20171	0.380968
$266$	0	0
$267$	8.59172	0.525805
$268$	0	0
$269$	−5.34690	−0.326006	−0.163003	−	0.986626i	$-0.552118\pi$
$269$	−5.34690	−0.326006	−0.163003	+	0.986626i	$0.552118\pi$
$270$	0	0
$271$	−25.9694	−1.57753	−0.788764	−	0.614697i	$-0.789278\pi$
$271$	−25.9694	−1.57753	−0.788764	+	0.614697i	$0.789278\pi$
$272$	0	0
$273$	−11.7973	−0.714004
$274$	0	0
$275$	8.62668	0.520209
$276$	0	0
$277$	−7.87959	−0.473439	−0.236719	−	0.971578i	$-0.576072\pi$
$277$	−7.87959	−0.473439	−0.236719	+	0.971578i	$0.576072\pi$
$278$	0	0
$279$	0.00502606	0.000300902 0
$280$	0	0
$281$	5.28721	0.315408	0.157704	−	0.987486i	$-0.449591\pi$
$281$	5.28721	0.315408	0.157704	+	0.987486i	$0.449591\pi$
$282$	0	0
$283$	4.23515	0.251753	0.125877	−	0.992046i	$-0.459826\pi$
$283$	4.23515	0.251753	0.125877	+	0.992046i	$0.459826\pi$
$284$	0	0
$285$	−20.1607	−1.19422
$286$	0	0
$287$	15.8875	0.937807
$288$	0	0
$289$	−16.7933	−0.987842
$290$	0	0
$291$	−3.38072	−0.198181
$292$	0	0
$293$	−29.9559	−1.75004	−0.875020	−	0.484087i	$-0.839152\pi$
$293$	−29.9559	−1.75004	−0.875020	+	0.484087i	$0.839152\pi$
$294$	0	0
$295$	22.7436	1.32418
$296$	0	0
$297$	4.77326	0.276972
$298$	0	0
$299$	−9.07950	−0.525081
$300$	0	0
$301$	−0.735044	−0.0423672
$302$	0	0
$303$	−9.02834	−0.518665
$304$	0	0
$305$	−19.6876	−1.12731
$306$	0	0
$307$	−20.0757	−1.14578	−0.572891	−	0.819631i	$-0.694178\pi$
$307$	−20.0757	−1.14578	−0.572891	+	0.819631i	$0.694178\pi$
$308$	0	0
$309$	−0.251833	−0.0143263
$310$	0	0
$311$	−10.2285	−0.580003	−0.290002	−	0.957026i	$-0.593656\pi$
$311$	−10.2285	−0.580003	−0.290002	+	0.957026i	$0.593656\pi$
$312$	0	0
$313$	−17.0910	−0.966040	−0.483020	−	0.875609i	$-0.660460\pi$
$313$	−17.0910	−0.966040	−0.483020	+	0.875609i	$0.660460\pi$
$314$	0	0
$315$	10.1498	0.571879
$316$	0	0
$317$	35.4743	1.99243	0.996217	−	0.0869058i	$-0.0276979\pi$
$317$	35.4743	1.99243	0.996217	+	0.0869058i	$0.0276979\pi$
$318$	0	0
$319$	−36.2484	−2.02952
$320$	0	0
$321$	5.53647	0.309016
$322$	0	0
$323$	3.51293	0.195464
$324$	0	0
$325$	−5.48074	−0.304017
$326$	0	0
$327$	−0.302664	−0.0167374
$328$	0	0
$329$	21.3566	1.17743
$330$	0	0
$331$	9.95302	0.547067	0.273534	−	0.961862i	$-0.411808\pi$
$331$	9.95302	0.547067	0.273534	+	0.961862i	$0.411808\pi$
$332$	0	0
$333$	8.22864	0.450927
$334$	0	0
$335$	−9.58426	−0.523644
$336$	0	0
$337$	−11.5707	−0.630295	−0.315147	−	0.949043i	$-0.602054\pi$
$337$	−11.5707	−0.630295	−0.315147	+	0.949043i	$0.602054\pi$
$338$	0	0
$339$	−1.94620	−0.105703
$340$	0	0
$341$	0.0239907	0.00129917
$342$	0	0
$343$	4.41021	0.238129
$344$	0	0
$345$	7.81159	0.420562
$346$	0	0
$347$	−19.4596	−1.04465	−0.522323	−	0.852748i	$-0.674934\pi$
$347$	−19.4596	−1.04465	−0.522323	+	0.852748i	$0.674934\pi$
$348$	0	0
$349$	8.63455	0.462197	0.231099	−	0.972930i	$-0.425768\pi$
$349$	8.63455	0.462197	0.231099	+	0.972930i	$0.425768\pi$
$350$	0	0
$351$	−3.03256	−0.161866
$352$	0	0
$353$	22.4219	1.19340	0.596700	−	0.802464i	$-0.296478\pi$
$353$	22.4219	1.19340	0.596700	+	0.802464i	$0.296478\pi$
$354$	0	0
$355$	24.2533	1.28723
$356$	0	0
$357$	−1.76857	−0.0936026
$358$	0	0
$359$	−15.6318	−0.825013	−0.412506	−	0.910955i	$-0.635347\pi$
$359$	−15.6318	−0.825013	−0.412506	+	0.910955i	$0.635347\pi$
$360$	0	0
$361$	40.7087	2.14256
$362$	0	0
$363$	11.7840	0.618498
$364$	0	0
$365$	30.5970	1.60152
$366$	0	0
$367$	16.1970	0.845478	0.422739	−	0.906252i	$-0.361069\pi$
$367$	16.1970	0.845478	0.422739	+	0.906252i	$0.361069\pi$
$368$	0	0
$369$	4.08397	0.212603
$370$	0	0
$371$	9.24690	0.480075
$372$	0	0
$373$	−12.1656	−0.629909	−0.314954	−	0.949107i	$-0.601989\pi$
$373$	−12.1656	−0.629909	−0.314954	+	0.949107i	$0.601989\pi$
$374$	0	0
$375$	−8.33002	−0.430160
$376$	0	0
$377$	23.0294	1.18608
$378$	0	0
$379$	4.44413	0.228280	0.114140	−	0.993465i	$-0.463589\pi$
$379$	4.44413	0.228280	0.114140	+	0.993465i	$0.463589\pi$
$380$	0	0
$381$	14.6262	0.749322
$382$	0	0
$383$	−27.0969	−1.38459	−0.692293	−	0.721616i	$-0.743400\pi$
$383$	−27.0969	−1.38459	−0.692293	+	0.721616i	$0.743400\pi$
$384$	0	0
$385$	48.4478	2.46913
$386$	0	0
$387$	−0.188947	−0.00960474
$388$	0	0
$389$	−33.0589	−1.67615	−0.838077	−	0.545552i	$-0.816320\pi$
$389$	−33.0589	−1.67615	−0.838077	+	0.545552i	$0.816320\pi$
$390$	0	0
$391$	−1.36114	−0.0688357
$392$	0	0
$393$	−0.621210	−0.0313359
$394$	0	0
$395$	12.7701	0.642533
$396$	0	0
$397$	15.7480	0.790368	0.395184	−	0.918602i	$-0.370681\pi$
$397$	15.7480	0.790368	0.395184	+	0.918602i	$0.370681\pi$
$398$	0	0
$399$	−30.0601	−1.50489
$400$	0	0
$401$	−3.98082	−0.198793	−0.0993964	−	0.995048i	$-0.531691\pi$
$401$	−3.98082	−0.198793	−0.0993964	+	0.995048i	$0.531691\pi$
$402$	0	0
$403$	−0.0152418	−0.000759250 0
$404$	0	0
$405$	2.60908	0.129646
$406$	0	0
$407$	39.2774	1.94691
$408$	0	0
$409$	−4.63482	−0.229177	−0.114588	−	0.993413i	$-0.536555\pi$
$409$	−4.63482	−0.229177	−0.114588	+	0.993413i	$0.536555\pi$
$410$	0	0
$411$	−3.65703	−0.180388
$412$	0	0
$413$	33.9112	1.66866
$414$	0	0
$415$	28.5655	1.40222
$416$	0	0
$417$	−9.48887	−0.464672
$418$	0	0
$419$	31.3591	1.53199	0.765997	−	0.642844i	$-0.222246\pi$
$419$	31.3591	1.53199	0.765997	+	0.642844i	$0.222246\pi$
$420$	0	0
$421$	2.88257	0.140488	0.0702439	−	0.997530i	$-0.477622\pi$
$421$	2.88257	0.140488	0.0702439	+	0.997530i	$0.477622\pi$
$422$	0	0
$423$	5.48986	0.266926
$424$	0	0
$425$	−0.821636	−0.0398552
$426$	0	0
$427$	−29.3546	−1.42057
$428$	0	0
$429$	−14.4752	−0.698869
$430$	0	0
$431$	6.96178	0.335337	0.167668	−	0.985843i	$-0.446376\pi$
$431$	6.96178	0.335337	0.167668	+	0.985843i	$0.446376\pi$
$432$	0	0
$433$	−16.1338	−0.775342	−0.387671	−	0.921798i	$-0.626720\pi$
$433$	−16.1338	−0.775342	−0.387671	+	0.921798i	$0.626720\pi$
$434$	0	0
$435$	−19.8135	−0.949984
$436$	0	0
$437$	−23.1351	−1.10670
$438$	0	0
$439$	10.1497	0.484417	0.242208	−	0.970224i	$-0.422128\pi$
$439$	10.1497	0.484417	0.242208	+	0.970224i	$0.422128\pi$
$440$	0	0
$441$	8.13367	0.387318
$442$	0	0
$443$	−30.0008	−1.42538	−0.712691	−	0.701478i	$-0.752524\pi$
$443$	−30.0008	−1.42538	−0.712691	+	0.701478i	$0.752524\pi$
$444$	0	0
$445$	22.4165	1.06264
$446$	0	0
$447$	−12.9718	−0.613544
$448$	0	0
$449$	−29.5390	−1.39403	−0.697016	−	0.717055i	$-0.745490\pi$
$449$	−29.5390	−1.39403	−0.697016	+	0.717055i	$0.745490\pi$
$450$	0	0
$451$	19.4938	0.917928
$452$	0	0
$453$	1.15492	0.0542630
$454$	0	0
$455$	−30.7800	−1.44299
$456$	0	0
$457$	32.7276	1.53093	0.765466	−	0.643477i	$-0.222509\pi$
$457$	32.7276	1.53093	0.765466	+	0.643477i	$0.222509\pi$
$458$	0	0
$459$	−0.454622	−0.0212199
$460$	0	0
$461$	16.1835	0.753743	0.376871	−	0.926266i	$-0.377000\pi$
$461$	16.1835	0.753743	0.376871	+	0.926266i	$0.377000\pi$
$462$	0	0
$463$	−25.9582	−1.20638	−0.603190	−	0.797598i	$-0.706104\pi$
$463$	−25.9582	−1.20638	−0.603190	+	0.797598i	$0.706104\pi$
$464$	0	0
$465$	0.0131134	0.000608119 0
$466$	0	0
$467$	22.4440	1.03859	0.519293	−	0.854596i	$-0.326195\pi$
$467$	22.4440	1.03859	0.519293	+	0.854596i	$0.326195\pi$
$468$	0	0
$469$	−14.2904	−0.659868
$470$	0	0
$471$	12.3875	0.570788
$472$	0	0
$473$	−0.901895	−0.0414692
$474$	0	0
$475$	−13.9652	−0.640769
$476$	0	0
$477$	2.37697	0.108834
$478$	0	0
$479$	−43.4414	−1.98489	−0.992444	−	0.122702i	$-0.960844\pi$
$479$	−43.4414	−1.98489	−0.992444	+	0.122702i	$0.960844\pi$
$480$	0	0
$481$	−24.9539	−1.13780
$482$	0	0
$483$	11.6473	0.529969
$484$	0	0
$485$	−8.82056	−0.400521
$486$	0	0
$487$	10.3545	0.469207	0.234604	−	0.972091i	$-0.424621\pi$
$487$	10.3545	0.469207	0.234604	+	0.972091i	$0.424621\pi$
$488$	0	0
$489$	−21.1628	−0.957015
$490$	0	0
$491$	7.98224	0.360233	0.180117	−	0.983645i	$-0.442352\pi$
$491$	7.98224	0.360233	0.180117	+	0.983645i	$0.442352\pi$
$492$	0	0
$493$	3.45242	0.155489
$494$	0	0
$495$	12.4538	0.559757
$496$	0	0
$497$	36.1622	1.62210
$498$	0	0
$499$	−15.1414	−0.677824	−0.338912	−	0.940818i	$-0.610059\pi$
$499$	−15.1414	−0.677824	−0.338912	+	0.940818i	$0.610059\pi$
$500$	0	0
$501$	−10.6419	−0.475445
$502$	0	0
$503$	−26.3623	−1.17544	−0.587718	−	0.809066i	$-0.699974\pi$
$503$	−26.3623	−1.17544	−0.587718	+	0.809066i	$0.699974\pi$
$504$	0	0
$505$	−23.5557	−1.04821
$506$	0	0
$507$	−3.80357	−0.168922
$508$	0	0
$509$	−13.7572	−0.609779	−0.304889	−	0.952388i	$-0.598619\pi$
$509$	−13.7572	−0.609779	−0.304889	+	0.952388i	$0.598619\pi$
$510$	0	0
$511$	45.6208	2.01815
$512$	0	0
$513$	−7.72714	−0.341162
$514$	0	0
$515$	−0.657052	−0.0289532
$516$	0	0
$517$	26.2045	1.15247
$518$	0	0
$519$	13.1309	0.576383
$520$	0	0
$521$	20.2303	0.886305	0.443153	−	0.896446i	$-0.353860\pi$
$521$	20.2303	0.886305	0.443153	+	0.896446i	$0.353860\pi$
$522$	0	0
$523$	−39.4495	−1.72501	−0.862504	−	0.506050i	$-0.831105\pi$
$523$	−39.4495	−1.72501	−0.862504	+	0.506050i	$0.831105\pi$
$524$	0	0
$525$	7.03075	0.306847
$526$	0	0
$527$	−0.00228496	−9.95342e−5 0
$528$	0	0
$529$	−14.0360	−0.610259
$530$	0	0
$531$	8.71709	0.378289
$532$	0	0
$533$	−12.3849	−0.536449
$534$	0	0
$535$	14.4451	0.624516
$536$	0	0
$537$	−11.8989	−0.513476
$538$	0	0
$539$	38.8241	1.67227
$540$	0	0
$541$	30.4973	1.31118	0.655591	−	0.755116i	$-0.272419\pi$
$541$	30.4973	1.31118	0.655591	+	0.755116i	$0.272419\pi$
$542$	0	0
$543$	−7.26972	−0.311974
$544$	0	0
$545$	−0.789676	−0.0338260
$546$	0	0
$547$	13.4636	0.575664	0.287832	−	0.957681i	$-0.407066\pi$
$547$	13.4636	0.575664	0.287832	+	0.957681i	$0.407066\pi$
$548$	0	0
$549$	−7.54578	−0.322046
$550$	0	0
$551$	58.6803	2.49987
$552$	0	0
$553$	19.0405	0.809685
$554$	0	0
$555$	21.4692	0.911316
$556$	0	0
$557$	−31.0997	−1.31774	−0.658868	−	0.752259i	$-0.728964\pi$
$557$	−31.0997	−1.31774	−0.658868	+	0.752259i	$0.728964\pi$
$558$	0	0
$559$	0.572995	0.0242351
$560$	0	0
$561$	−2.17003	−0.0916185
$562$	0	0
$563$	3.18045	0.134040	0.0670200	−	0.997752i	$-0.478651\pi$
$563$	3.18045	0.134040	0.0670200	+	0.997752i	$0.478651\pi$
$564$	0	0
$565$	−5.07779	−0.213624
$566$	0	0
$567$	3.89020	0.163373
$568$	0	0
$569$	−18.1816	−0.762214	−0.381107	−	0.924531i	$-0.624457\pi$
$569$	−18.1816	−0.762214	−0.381107	+	0.924531i	$0.624457\pi$
$570$	0	0
$571$	−1.06971	−0.0447661	−0.0223830	−	0.999749i	$-0.507125\pi$
$571$	−1.06971	−0.0447661	−0.0223830	+	0.999749i	$0.507125\pi$
$572$	0	0
$573$	−15.9031	−0.664361
$574$	0	0
$575$	5.41105	0.225656
$576$	0	0
$577$	7.72746	0.321698	0.160849	−	0.986979i	$-0.448577\pi$
$577$	7.72746	0.321698	0.160849	+	0.986979i	$0.448577\pi$
$578$	0	0
$579$	20.1946	0.839257
$580$	0	0
$581$	42.5918	1.76701
$582$	0	0
$583$	11.3459	0.469899
$584$	0	0
$585$	−7.91220	−0.327129
$586$	0	0
$587$	7.61825	0.314439	0.157219	−	0.987564i	$-0.449747\pi$
$587$	7.61825	0.314439	0.157219	+	0.987564i	$0.449747\pi$
$588$	0	0
$589$	−0.0388371	−0.00160025
$590$	0	0
$591$	8.38272	0.344819
$592$	0	0
$593$	−43.9586	−1.80516	−0.902581	−	0.430520i	$-0.858330\pi$
$593$	−43.9586	−1.80516	−0.902581	+	0.430520i	$0.858330\pi$
$594$	0	0
$595$	−4.61434	−0.189169
$596$	0	0
$597$	−9.93511	−0.406617
$598$	0	0
$599$	1.56917	0.0641147	0.0320574	−	0.999486i	$-0.489794\pi$
$599$	1.56917	0.0641147	0.0320574	+	0.999486i	$0.489794\pi$
$600$	0	0
$601$	−4.80718	−0.196089	−0.0980445	−	0.995182i	$-0.531259\pi$
$601$	−4.80718	−0.196089	−0.0980445	+	0.995182i	$0.531259\pi$
$602$	0	0
$603$	−3.67342	−0.149593
$604$	0	0
$605$	30.7453	1.24998
$606$	0	0
$607$	23.7501	0.963985	0.481992	−	0.876175i	$-0.339913\pi$
$607$	23.7501	0.963985	0.481992	+	0.876175i	$0.339913\pi$
$608$	0	0
$609$	−29.5424	−1.19712
$610$	0	0
$611$	−16.6483	−0.673519
$612$	0	0
$613$	−25.0125	−1.01024	−0.505122	−	0.863048i	$-0.668552\pi$
$613$	−25.0125	−1.01024	−0.505122	+	0.863048i	$0.668552\pi$
$614$	0	0
$615$	10.6554	0.429667
$616$	0	0
$617$	−6.58409	−0.265066	−0.132533	−	0.991179i	$-0.542311\pi$
$617$	−6.58409	−0.265066	−0.132533	+	0.991179i	$0.542311\pi$
$618$	0	0
$619$	−31.5144	−1.26667	−0.633335	−	0.773878i	$-0.718315\pi$
$619$	−31.5144	−1.26667	−0.633335	+	0.773878i	$0.718315\pi$
$620$	0	0
$621$	2.99400	0.120145
$622$	0	0
$623$	33.4235	1.33909
$624$	0	0
$625$	−30.7702	−1.23081
$626$	0	0
$627$	−36.8836	−1.47299
$628$	0	0
$629$	−3.74092	−0.149160
$630$	0	0
$631$	−40.5888	−1.61581	−0.807907	−	0.589310i	$-0.799400\pi$
$631$	−40.5888	−1.61581	−0.807907	+	0.589310i	$0.799400\pi$
$632$	0	0
$633$	−9.43905	−0.375169
$634$	0	0
$635$	38.1609	1.51437
$636$	0	0
$637$	−24.6659	−0.977297
$638$	0	0
$639$	9.29571	0.367733
$640$	0	0
$641$	−6.45178	−0.254830	−0.127415	−	0.991850i	$-0.540668\pi$
$641$	−6.45178	−0.254830	−0.127415	+	0.991850i	$0.540668\pi$
$642$	0	0
$643$	23.0168	0.907695	0.453848	−	0.891079i	$-0.350051\pi$
$643$	23.0168	0.907695	0.453848	+	0.891079i	$0.350051\pi$
$644$	0	0
$645$	−0.492979	−0.0194110
$646$	0	0
$647$	17.9720	0.706552	0.353276	−	0.935519i	$-0.385068\pi$
$647$	17.9720	0.706552	0.353276	+	0.935519i	$0.385068\pi$
$648$	0	0
$649$	41.6089	1.63329
$650$	0	0
$651$	0.0195524	0.000766318 0
$652$	0	0
$653$	−3.47016	−0.135798	−0.0678989	−	0.997692i	$-0.521630\pi$
$653$	−3.47016	−0.135798	−0.0678989	+	0.997692i	$0.521630\pi$
$654$	0	0
$655$	−1.62079	−0.0633293
$656$	0	0
$657$	11.7271	0.457518
$658$	0	0
$659$	−24.7096	−0.962551	−0.481275	−	0.876569i	$-0.659826\pi$
$659$	−24.7096	−0.962551	−0.481275	+	0.876569i	$0.659826\pi$
$660$	0	0
$661$	−27.1725	−1.05689	−0.528443	−	0.848969i	$-0.677224\pi$
$661$	−27.1725	−1.05689	−0.528443	+	0.848969i	$0.677224\pi$
$662$	0	0
$663$	1.37867	0.0535430
$664$	0	0
$665$	−78.4293	−3.04136
$666$	0	0
$667$	−22.7366	−0.880365
$668$	0	0
$669$	13.6643	0.528294
$670$	0	0
$671$	−36.0180	−1.39046
$672$	0	0
$673$	48.2941	1.86160	0.930801	−	0.365526i	$-0.119111\pi$
$673$	48.2941	1.86160	0.930801	+	0.365526i	$0.119111\pi$
$674$	0	0
$675$	1.80730	0.0695628
$676$	0	0
$677$	40.9527	1.57394	0.786969	−	0.616992i	$-0.211649\pi$
$677$	40.9527	1.57394	0.786969	+	0.616992i	$0.211649\pi$
$678$	0	0
$679$	−13.1517	−0.504715
$680$	0	0
$681$	−25.6140	−0.981531
$682$	0	0
$683$	9.24074	0.353587	0.176794	−	0.984248i	$-0.443428\pi$
$683$	9.24074	0.353587	0.176794	+	0.984248i	$0.443428\pi$
$684$	0	0
$685$	−9.54148	−0.364561
$686$	0	0
$687$	10.1840	0.388545
$688$	0	0
$689$	−7.20831	−0.274615
$690$	0	0
$691$	7.38316	0.280869	0.140434	−	0.990090i	$-0.455150\pi$
$691$	7.38316	0.280869	0.140434	+	0.990090i	$0.455150\pi$
$692$	0	0
$693$	18.5689	0.705375
$694$	0	0
$695$	−24.7572	−0.939095
$696$	0	0
$697$	−1.85666	−0.0703260
$698$	0	0
$699$	−0.898428	−0.0339817
$700$	0	0
$701$	−8.00568	−0.302371	−0.151185	−	0.988505i	$-0.548309\pi$
$701$	−8.00568	−0.302371	−0.151185	+	0.988505i	$0.548309\pi$
$702$	0	0
$703$	−63.5839	−2.39811
$704$	0	0
$705$	14.3235	0.539453
$706$	0	0
$707$	−35.1221	−1.32090
$708$	0	0
$709$	44.9851	1.68945	0.844725	−	0.535201i	$-0.179764\pi$
$709$	44.9851	1.68945	0.844725	+	0.535201i	$0.179764\pi$
$710$	0	0
$711$	4.89448	0.183557
$712$	0	0
$713$	0.0150480	0.000563553 0
$714$	0	0
$715$	−37.7669	−1.41240
$716$	0	0
$717$	−16.6531	−0.621923
$718$	0	0
$719$	39.7217	1.48137	0.740685	−	0.671853i	$-0.234501\pi$
$719$	39.7217	1.48137	0.740685	+	0.671853i	$0.234501\pi$
$720$	0	0
$721$	−0.979681	−0.0364852
$722$	0	0
$723$	12.9837	0.482869
$724$	0	0
$725$	−13.7247	−0.509723
$726$	0	0
$727$	11.8162	0.438239	0.219119	−	0.975698i	$-0.429682\pi$
$727$	11.8162	0.438239	0.219119	+	0.975698i	$0.429682\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	0.0858996	0.00317711
$732$	0	0
$733$	14.9323	0.551537	0.275768	−	0.961224i	$-0.411068\pi$
$733$	14.9323	0.551537	0.275768	+	0.961224i	$0.411068\pi$
$734$	0	0
$735$	21.2214	0.782763
$736$	0	0
$737$	−17.5342	−0.645880
$738$	0	0
$739$	−8.70218	−0.320115	−0.160057	−	0.987108i	$-0.551168\pi$
$739$	−8.70218	−0.320115	−0.160057	+	0.987108i	$0.551168\pi$
$740$	0	0
$741$	23.4330	0.860834
$742$	0	0
$743$	−16.3179	−0.598645	−0.299323	−	0.954152i	$-0.596761\pi$
$743$	−16.3179	−0.598645	−0.299323	+	0.954152i	$0.596761\pi$
$744$	0	0
$745$	−33.8444	−1.23996
$746$	0	0
$747$	10.9485	0.400584
$748$	0	0
$749$	21.5380	0.786982
$750$	0	0
$751$	49.4911	1.80596	0.902978	−	0.429688i	$-0.141376\pi$
$751$	49.4911	1.80596	0.902978	+	0.429688i	$0.141376\pi$
$752$	0	0
$753$	−1.00000	−0.0364420
$754$	0	0
$755$	3.01328	0.109665
$756$	0	0
$757$	−20.8490	−0.757771	−0.378885	−	0.925444i	$-0.623693\pi$
$757$	−20.8490	−0.757771	−0.378885	+	0.925444i	$0.623693\pi$
$758$	0	0
$759$	14.2911	0.518735
$760$	0	0
$761$	−4.51584	−0.163699	−0.0818495	−	0.996645i	$-0.526083\pi$
$761$	−4.51584	−0.163699	−0.0818495	+	0.996645i	$0.526083\pi$
$762$	0	0
$763$	−1.17743	−0.0426257
$764$	0	0
$765$	−1.18614	−0.0428851
$766$	0	0
$767$	−26.4351	−0.954517
$768$	0	0
$769$	−13.5263	−0.487773	−0.243886	−	0.969804i	$-0.578422\pi$
$769$	−13.5263	−0.487773	−0.243886	+	0.969804i	$0.578422\pi$
$770$	0	0
$771$	14.1092	0.508130
$772$	0	0
$773$	44.3081	1.59365	0.796827	−	0.604208i	$-0.206510\pi$
$773$	44.3081	1.59365	0.796827	+	0.604208i	$0.206510\pi$
$774$	0	0
$775$	0.00908358	0.000326292 0
$776$	0	0
$777$	32.0111	1.14839
$778$	0	0
$779$	−31.5574	−1.13066
$780$	0	0
$781$	44.3708	1.58771
$782$	0	0
$783$	−7.59405	−0.271389
$784$	0	0
$785$	32.3201	1.15355
$786$	0	0
$787$	−42.7984	−1.52560	−0.762799	−	0.646635i	$-0.776176\pi$
$787$	−42.7984	−1.52560	−0.762799	+	0.646635i	$0.776176\pi$
$788$	0	0
$789$	−7.53490	−0.268250
$790$	0	0
$791$	−7.57111	−0.269198
$792$	0	0
$793$	22.8831	0.812601
$794$	0	0
$795$	6.20171	0.219952
$796$	0	0
$797$	3.71728	0.131673	0.0658365	−	0.997830i	$-0.479028\pi$
$797$	3.71728	0.131673	0.0658365	+	0.997830i	$0.479028\pi$
$798$	0	0
$799$	−2.49581	−0.0882953
$800$	0	0
$801$	8.59172	0.303574
$802$	0	0
$803$	55.9765	1.97537
$804$	0	0
$805$	30.3887	1.07106
$806$	0	0
$807$	−5.34690	−0.188220
$808$	0	0
$809$	−4.68712	−0.164790	−0.0823951	−	0.996600i	$-0.526257\pi$
$809$	−4.68712	−0.164790	−0.0823951	+	0.996600i	$0.526257\pi$
$810$	0	0
$811$	28.9158	1.01537	0.507686	−	0.861542i	$-0.330501\pi$
$811$	28.9158	1.01537	0.507686	+	0.861542i	$0.330501\pi$
$812$	0	0
$813$	−25.9694	−0.910786
$814$	0	0
$815$	−55.2155	−1.93411
$816$	0	0
$817$	1.46002	0.0510798
$818$	0	0
$819$	−11.7973	−0.412230
$820$	0	0
$821$	16.6455	0.580932	0.290466	−	0.956885i	$-0.406190\pi$
$821$	16.6455	0.580932	0.290466	+	0.956885i	$0.406190\pi$
$822$	0	0
$823$	−29.8576	−1.04077	−0.520385	−	0.853932i	$-0.674211\pi$
$823$	−29.8576	−1.04077	−0.520385	+	0.853932i	$0.674211\pi$
$824$	0	0
$825$	8.62668	0.300343
$826$	0	0
$827$	−32.5057	−1.13033	−0.565167	−	0.824976i	$-0.691188\pi$
$827$	−32.5057	−1.13033	−0.565167	+	0.824976i	$0.691188\pi$
$828$	0	0
$829$	15.5527	0.540168	0.270084	−	0.962837i	$-0.412949\pi$
$829$	15.5527	0.540168	0.270084	+	0.962837i	$0.412949\pi$
$830$	0	0
$831$	−7.87959	−0.273340
$832$	0	0
$833$	−3.69774	−0.128119
$834$	0	0
$835$	−27.7656	−0.960867
$836$	0	0
$837$	0.00502606	0.000173726 0
$838$	0	0
$839$	37.5992	1.29807	0.649034	−	0.760759i	$-0.275173\pi$
$839$	37.5992	1.29807	0.649034	+	0.760759i	$0.275173\pi$
$840$	0	0
$841$	28.6697	0.988609
$842$	0	0
$843$	5.28721	0.182101
$844$	0	0
$845$	−9.92381	−0.341389
$846$	0	0
$847$	45.8420	1.57515
$848$	0	0
$849$	4.23515	0.145350
$850$	0	0
$851$	24.6366	0.844531
$852$	0	0
$853$	42.2153	1.44543	0.722713	−	0.691149i	$-0.242895\pi$
$853$	42.2153	1.44543	0.722713	+	0.691149i	$0.242895\pi$
$854$	0	0
$855$	−20.1607	−0.689482
$856$	0	0
$857$	−5.75358	−0.196538	−0.0982692	−	0.995160i	$-0.531331\pi$
$857$	−5.75358	−0.196538	−0.0982692	+	0.995160i	$0.531331\pi$
$858$	0	0
$859$	31.0306	1.05875	0.529375	−	0.848388i	$-0.322426\pi$
$859$	31.0306	1.05875	0.529375	+	0.848388i	$0.322426\pi$
$860$	0	0
$861$	15.8875	0.541443
$862$	0	0
$863$	−29.5641	−1.00637	−0.503187	−	0.864177i	$-0.667839\pi$
$863$	−29.5641	−1.00637	−0.503187	+	0.864177i	$0.667839\pi$
$864$	0	0
$865$	34.2596	1.16486
$866$	0	0
$867$	−16.7933	−0.570331
$868$	0	0
$869$	23.3626	0.792522
$870$	0	0
$871$	11.1399	0.377461
$872$	0	0
$873$	−3.38072	−0.114420
$874$	0	0
$875$	−32.4055	−1.09550
$876$	0	0
$877$	44.3683	1.49821	0.749106	−	0.662450i	$-0.230483\pi$
$877$	44.3683	1.49821	0.749106	+	0.662450i	$0.230483\pi$
$878$	0	0
$879$	−29.9559	−1.01039
$880$	0	0
$881$	−29.8916	−1.00707	−0.503537	−	0.863974i	$-0.667968\pi$
$881$	−29.8916	−1.00707	−0.503537	+	0.863974i	$0.667968\pi$
$882$	0	0
$883$	−36.1849	−1.21772	−0.608860	−	0.793278i	$-0.708373\pi$
$883$	−36.1849	−1.21772	−0.608860	+	0.793278i	$0.708373\pi$
$884$	0	0
$885$	22.7436	0.764517
$886$	0	0
$887$	15.2285	0.511322	0.255661	−	0.966766i	$-0.417707\pi$
$887$	15.2285	0.511322	0.255661	+	0.966766i	$0.417707\pi$
$888$	0	0
$889$	56.8988	1.90833
$890$	0	0
$891$	4.77326	0.159910
$892$	0	0
$893$	−42.4209	−1.41956
$894$	0	0
$895$	−31.0452	−1.03773
$896$	0	0
$897$	−9.07950	−0.303156
$898$	0	0
$899$	−0.0381682	−0.00127298
$900$	0	0
$901$	−1.08062	−0.0360007
$902$	0	0
$903$	−0.735044	−0.0244607
$904$	0	0
$905$	−18.9673	−0.630494
$906$	0	0
$907$	−15.0107	−0.498423	−0.249211	−	0.968449i	$-0.580171\pi$
$907$	−15.0107	−0.498423	−0.249211	+	0.968449i	$0.580171\pi$
$908$	0	0
$909$	−9.02834	−0.299451
$910$	0	0
$911$	−3.80434	−0.126043	−0.0630216	−	0.998012i	$-0.520074\pi$
$911$	−3.80434	−0.126043	−0.0630216	+	0.998012i	$0.520074\pi$
$912$	0	0
$913$	52.2599	1.72955
$914$	0	0
$915$	−19.6876	−0.650850
$916$	0	0
$917$	−2.41663	−0.0798042
$918$	0	0
$919$	−6.72016	−0.221678	−0.110839	−	0.993838i	$-0.535354\pi$
$919$	−6.72016	−0.221678	−0.110839	+	0.993838i	$0.535354\pi$
$920$	0	0
$921$	−20.0757	−0.661518
$922$	0	0
$923$	−28.1898	−0.927879
$924$	0	0
$925$	14.8716	0.488975
$926$	0	0
$927$	−0.251833	−0.00827128
$928$	0	0
$929$	−42.8036	−1.40434	−0.702170	−	0.712010i	$-0.747785\pi$
$929$	−42.8036	−1.40434	−0.702170	+	0.712010i	$0.747785\pi$
$930$	0	0
$931$	−62.8500	−2.05983
$932$	0	0
$933$	−10.2285	−0.334865
$934$	0	0
$935$	−5.66177	−0.185160
$936$	0	0
$937$	−24.9060	−0.813643	−0.406821	−	0.913508i	$-0.633363\pi$
$937$	−24.9060	−0.813643	−0.406821	+	0.913508i	$0.633363\pi$
$938$	0	0
$939$	−17.0910	−0.557743
$940$	0	0
$941$	−54.2478	−1.76843	−0.884214	−	0.467082i	$-0.845305\pi$
$941$	−54.2478	−1.76843	−0.884214	+	0.467082i	$0.845305\pi$
$942$	0	0
$943$	12.2274	0.398179
$944$	0	0
$945$	10.1498	0.330175
$946$	0	0
$947$	27.6162	0.897404	0.448702	−	0.893681i	$-0.351886\pi$
$947$	27.6162	0.897404	0.448702	+	0.893681i	$0.351886\pi$
$948$	0	0
$949$	−35.5632	−1.15443
$950$	0	0
$951$	35.4743	1.15033
$952$	0	0
$953$	−11.6992	−0.378973	−0.189487	−	0.981883i	$-0.560682\pi$
$953$	−11.6992	−0.378973	−0.189487	+	0.981883i	$0.560682\pi$
$954$	0	0
$955$	−41.4924	−1.34266
$956$	0	0
$957$	−36.2484	−1.17174
$958$	0	0
$959$	−14.2266	−0.459400
$960$	0	0
$961$	−31.0000	−0.999999
$962$	0	0
$963$	5.53647	0.178410
$964$	0	0
$965$	52.6892	1.69612
$966$	0	0
$967$	14.6113	0.469867	0.234934	−	0.972011i	$-0.424513\pi$
$967$	14.6113	0.469867	0.234934	+	0.972011i	$0.424513\pi$
$968$	0	0
$969$	3.51293	0.112851
$970$	0	0
$971$	22.4913	0.721781	0.360891	−	0.932608i	$-0.382473\pi$
$971$	22.4913	0.721781	0.360891	+	0.932608i	$0.382473\pi$
$972$	0	0
$973$	−36.9136	−1.18340
$974$	0	0
$975$	−5.48074	−0.175524
$976$	0	0
$977$	−36.6910	−1.17385	−0.586924	−	0.809642i	$-0.699661\pi$
$977$	−36.6910	−1.17385	−0.586924	+	0.809642i	$0.699661\pi$
$978$	0	0
$979$	41.0105	1.31070
$980$	0	0
$981$	−0.302664	−0.00966333
$982$	0	0
$983$	−21.3562	−0.681159	−0.340579	−	0.940216i	$-0.610623\pi$
$983$	−21.3562	−0.681159	−0.340579	+	0.940216i	$0.610623\pi$
$984$	0	0
$985$	21.8712	0.696874
$986$	0	0
$987$	21.3566	0.679790
$988$	0	0
$989$	−0.565709	−0.0179885
$990$	0	0
$991$	−43.5711	−1.38408	−0.692042	−	0.721858i	$-0.743289\pi$
$991$	−43.5711	−1.38408	−0.692042	+	0.721858i	$0.743289\pi$
$992$	0	0
$993$	9.95302	0.315849
$994$	0	0
$995$	−25.9215	−0.821766
$996$	0	0
$997$	−47.3461	−1.49947	−0.749733	−	0.661740i	$-0.769818\pi$
$997$	−47.3461	−1.49947	−0.749733	+	0.661740i	$0.769818\pi$
$998$	0	0
$999$	8.22864	0.260343

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6024.2.a.r.1.15	✓	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6024.2.a.r.1.15	✓	20	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 6024 = 2^{3} \cdot 3 \cdot 251 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6024.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +2.60908 q^{5} +3.89020 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +2.60908 q^{5} +3.89020 q^{7} +1.00000 q^{9} +4.77326 q^{11} -3.03256 q^{13} +2.60908 q^{15} -0.454622 q^{17} -7.72714 q^{19} +3.89020 q^{21} +2.99400 q^{23} +1.80730 q^{25} +1.00000 q^{27} -7.59405 q^{29} +0.00502606 q^{31} +4.77326 q^{33} +10.1498 q^{35} +8.22864 q^{37} -3.03256 q^{39} +4.08397 q^{41} -0.188947 q^{43} +2.60908 q^{45} +5.48986 q^{47} +8.13367 q^{49} -0.454622 q^{51} +2.37697 q^{53} +12.4538 q^{55} -7.72714 q^{57} +8.71709 q^{59} -7.54578 q^{61} +3.89020 q^{63} -7.91220 q^{65} -3.67342 q^{67} +2.99400 q^{69} +9.29571 q^{71} +11.7271 q^{73} +1.80730 q^{75} +18.5689 q^{77} +4.89448 q^{79} +1.00000 q^{81} +10.9485 q^{83} -1.18614 q^{85} -7.59405 q^{87} +8.59172 q^{89} -11.7973 q^{91} +0.00502606 q^{93} -20.1607 q^{95} -3.38072 q^{97} +4.77326 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 20 q^{3} + 9 q^{5} + 9 q^{7} + 20 q^{9}+O(q^{10}) \) 20 * q + 20 * q^3 + 9 * q^5 + 9 * q^7 + 20 * q^9	\( 20 q + 20 q^{3} + 9 q^{5} + 9 q^{7} + 20 q^{9} + 4 q^{11} + 21 q^{13} + 9 q^{15} + 10 q^{17} + 8 q^{19} + 9 q^{21} + 9 q^{23} + 43 q^{25} + 20 q^{27} + 18 q^{29} + 27 q^{31} + 4 q^{33} - 7 q^{35} + 33 q^{37} + 21 q^{39} + 14 q^{41} - 6 q^{43} + 9 q^{45} + 21 q^{47} + 47 q^{49} + 10 q^{51} + 23 q^{53} + 24 q^{55} + 8 q^{57} + 10 q^{59} + 28 q^{61} + 9 q^{63} + 2 q^{65} + 15 q^{67} + 9 q^{69} + 33 q^{71} + 50 q^{73} + 43 q^{75} + 20 q^{77} + 17 q^{79} + 20 q^{81} - 19 q^{83} + 41 q^{85} + 18 q^{87} + 21 q^{89} + 30 q^{91} + 27 q^{93} + 27 q^{95} + 47 q^{97} + 4 q^{99}+O(q^{100}) \) 20 * q + 20 * q^3 + 9 * q^5 + 9 * q^7 + 20 * q^9 + 4 * q^11 + 21 * q^13 + 9 * q^15 + 10 * q^17 + 8 * q^19 + 9 * q^21 + 9 * q^23 + 43 * q^25 + 20 * q^27 + 18 * q^29 + 27 * q^31 + 4 * q^33 - 7 * q^35 + 33 * q^37 + 21 * q^39 + 14 * q^41 - 6 * q^43 + 9 * q^45 + 21 * q^47 + 47 * q^49 + 10 * q^51 + 23 * q^53 + 24 * q^55 + 8 * q^57 + 10 * q^59 + 28 * q^61 + 9 * q^63 + 2 * q^65 + 15 * q^67 + 9 * q^69 + 33 * q^71 + 50 * q^73 + 43 * q^75 + 20 * q^77 + 17 * q^79 + 20 * q^81 - 19 * q^83 + 41 * q^85 + 18 * q^87 + 21 * q^89 + 30 * q^91 + 27 * q^93 + 27 * q^95 + 47 * q^97 + 4 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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