LMFDB - Embedded newform 6039.2.a.j.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6039,2,Mod(1,6039)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6039, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("6039.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6039 = 3^{2} \cdot 11 \cdot 61 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6039.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:	$48.2216577807$
Analytic rank:	$0$
Dimension:	$14$
Coefficient field:	$\mathbb{Q}[x]/(x^{14} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{14} - x^{13} - 21 x^{12} + 20 x^{11} + 167 x^{10} - 148 x^{9} - 627 x^{8} + 497 x^{7} + 1123 x^{6} + \cdots - 1 $ x^14 - x^13 - 21x^12 + 20x^11 + 167x^10 - 148x^9 - 627x^8 + 497x^7 + 1123x^6 - 745x^5 - 802x^4 + 386x^3 + 74x^2 - 15x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 2013)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2.57087$ of defining polynomial
Character	$\chi$	$=$	6039.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.57087 q^{2} +4.60938 q^{4} -4.00721 q^{5} +2.04273 q^{7} -6.70837 q^{8} +O(q^{10})$	\(q-2.57087 q^{2} +4.60938 q^{4} -4.00721 q^{5} +2.04273 q^{7} -6.70837 q^{8} +10.3020 q^{10} +1.00000 q^{11} -4.75237 q^{13} -5.25159 q^{14} +8.02760 q^{16} +0.339679 q^{17} +4.54116 q^{19} -18.4707 q^{20} -2.57087 q^{22} +4.75749 q^{23} +11.0577 q^{25} +12.2177 q^{26} +9.41571 q^{28} +4.28494 q^{29} +1.81765 q^{31} -7.22118 q^{32} -0.873272 q^{34} -8.18565 q^{35} -2.11204 q^{37} -11.6747 q^{38} +26.8819 q^{40} +10.2585 q^{41} -9.89546 q^{43} +4.60938 q^{44} -12.2309 q^{46} +5.22451 q^{47} -2.82726 q^{49} -28.4280 q^{50} -21.9055 q^{52} -9.68455 q^{53} -4.00721 q^{55} -13.7034 q^{56} -11.0160 q^{58} +2.04679 q^{59} +1.00000 q^{61} -4.67295 q^{62} +2.50953 q^{64} +19.0437 q^{65} +5.30981 q^{67} +1.56571 q^{68} +21.0442 q^{70} -11.4906 q^{71} -4.51365 q^{73} +5.42978 q^{74} +20.9319 q^{76} +2.04273 q^{77} +5.49270 q^{79} -32.1683 q^{80} -26.3733 q^{82} -14.2207 q^{83} -1.36117 q^{85} +25.4399 q^{86} -6.70837 q^{88} +6.05470 q^{89} -9.70780 q^{91} +21.9291 q^{92} -13.4315 q^{94} -18.1974 q^{95} +13.6004 q^{97} +7.26851 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 14 q + q^{2} + 15 q^{4} - q^{5} + 9 q^{7}+O(q^{10}) $ 14 * q + q^2 + 15 * q^4 - q^5 + 9 * q^7	$ 14 q + q^{2} + 15 q^{4} - q^{5} + 9 q^{7} + 6 q^{10} + 14 q^{11} + q^{13} + 7 q^{14} + 17 q^{16} + 9 q^{17} + 22 q^{19} - 23 q^{20} + q^{22} - q^{23} + 25 q^{25} - 4 q^{26} + 37 q^{28} + 6 q^{29} + 9 q^{31} - 4 q^{32} + 8 q^{34} - 18 q^{35} + 18 q^{37} - 8 q^{38} + 16 q^{40} + 25 q^{41} + 25 q^{43} + 15 q^{44} + 20 q^{46} - 36 q^{47} + 25 q^{49} - 2 q^{50} - 13 q^{52} - q^{55} + 40 q^{56} + 33 q^{58} - 17 q^{59} + 14 q^{61} + 13 q^{62} - 6 q^{64} + 61 q^{65} + 22 q^{67} - 66 q^{68} + 44 q^{70} + 13 q^{71} + 20 q^{73} + 12 q^{74} + 49 q^{76} + 9 q^{77} + 31 q^{79} - 88 q^{80} + 2 q^{82} - 32 q^{83} + 2 q^{85} + 14 q^{86} + 21 q^{89} + 45 q^{91} + 14 q^{92} - 31 q^{94} - 23 q^{95} + 37 q^{97} + 38 q^{98}+O(q^{100}) $ 14 * q + q^2 + 15 * q^4 - q^5 + 9 * q^7 + 6 * q^10 + 14 * q^11 + q^13 + 7 * q^14 + 17 * q^16 + 9 * q^17 + 22 * q^19 - 23 * q^20 + q^22 - q^23 + 25 * q^25 - 4 * q^26 + 37 * q^28 + 6 * q^29 + 9 * q^31 - 4 * q^32 + 8 * q^34 - 18 * q^35 + 18 * q^37 - 8 * q^38 + 16 * q^40 + 25 * q^41 + 25 * q^43 + 15 * q^44 + 20 * q^46 - 36 * q^47 + 25 * q^49 - 2 * q^50 - 13 * q^52 - q^55 + 40 * q^56 + 33 * q^58 - 17 * q^59 + 14 * q^61 + 13 * q^62 - 6 * q^64 + 61 * q^65 + 22 * q^67 - 66 * q^68 + 44 * q^70 + 13 * q^71 + 20 * q^73 + 12 * q^74 + 49 * q^76 + 9 * q^77 + 31 * q^79 - 88 * q^80 + 2 * q^82 - 32 * q^83 + 2 * q^85 + 14 * q^86 + 21 * q^89 + 45 * q^91 + 14 * q^92 - 31 * q^94 - 23 * q^95 + 37 * q^97 + 38 * q^98

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 $

$2$	−2.57087	−1.81788	−0.908940	−	0.416927i	$-0.863107\pi$
$2$	−2.57087	−1.81788	−0.908940	+	0.416927i	$0.863107\pi$
$3$	0	0
$3$	0	0
$4$	4.60938	2.30469
$5$	−4.00721	−1.79208	−0.896040	−	0.443974i	$-0.853568\pi$
$5$	−4.00721	−1.79208	−0.896040	+	0.443974i	$0.853568\pi$
$6$	0	0
$7$	2.04273	0.772079	0.386040	−	0.922482i	$-0.373843\pi$
$7$	2.04273	0.772079	0.386040	+	0.922482i	$0.373843\pi$
$8$	−6.70837	−2.37177
$9$	0	0
$10$	10.3020	3.25779
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	−4.75237	−1.31807	−0.659035	−	0.752112i	$-0.729035\pi$
$13$	−4.75237	−1.31807	−0.659035	+	0.752112i	$0.729035\pi$
$14$	−5.25159	−1.40355
$15$	0	0
$16$	8.02760	2.00690
$17$	0.339679	0.0823844	0.0411922	−	0.999151i	$-0.486884\pi$
$17$	0.339679	0.0823844	0.0411922	+	0.999151i	$0.486884\pi$
$18$	0	0
$19$	4.54116	1.04181	0.520907	−	0.853613i	$-0.325594\pi$
$19$	4.54116	1.04181	0.520907	+	0.853613i	$0.325594\pi$
$20$	−18.4707	−4.13018
$21$	0	0
$22$	−2.57087	−0.548112
$23$	4.75749	0.992006	0.496003	−	0.868321i	$-0.334800\pi$
$23$	4.75749	0.992006	0.496003	+	0.868321i	$0.334800\pi$
$24$	0	0
$25$	11.0577	2.21155
$26$	12.2177	2.39609
$27$	0	0
$28$	9.41571	1.77940
$29$	4.28494	0.795693	0.397846	−	0.917452i	$-0.369758\pi$
$29$	4.28494	0.795693	0.397846	+	0.917452i	$0.369758\pi$
$30$	0	0
$31$	1.81765	0.326460	0.163230	−	0.986588i	$-0.447809\pi$
$31$	1.81765	0.326460	0.163230	+	0.986588i	$0.447809\pi$
$32$	−7.22118	−1.27654
$33$	0	0
$34$	−0.873272	−0.149765
$35$	−8.18565	−1.38363
$36$	0	0
$37$	−2.11204	−0.347217	−0.173609	−	0.984815i	$-0.555543\pi$
$37$	−2.11204	−0.347217	−0.173609	+	0.984815i	$0.555543\pi$
$38$	−11.6747	−1.89389
$39$	0	0
$40$	26.8819	4.25039
$41$	10.2585	1.60211	0.801056	−	0.598589i	$-0.204272\pi$
$41$	10.2585	1.60211	0.801056	+	0.598589i	$0.204272\pi$
$42$	0	0
$43$	−9.89546	−1.50904	−0.754522	−	0.656275i	$-0.772131\pi$
$43$	−9.89546	−1.50904	−0.754522	+	0.656275i	$0.772131\pi$
$44$	4.60938	0.694890
$45$	0	0
$46$	−12.2309	−1.80335
$47$	5.22451	0.762074	0.381037	−	0.924560i	$-0.375567\pi$
$47$	5.22451	0.762074	0.381037	+	0.924560i	$0.375567\pi$
$48$	0	0
$49$	−2.82726	−0.403894
$50$	−28.4280	−4.02033
$51$	0	0
$52$	−21.9055	−3.03774
$53$	−9.68455	−1.33028	−0.665138	−	0.746721i	$-0.731627\pi$
$53$	−9.68455	−1.33028	−0.665138	+	0.746721i	$0.731627\pi$
$54$	0	0
$55$	−4.00721	−0.540332
$56$	−13.7034	−1.83119
$57$	0	0
$58$	−11.0160	−1.44647
$59$	2.04679	0.266469	0.133235	−	0.991085i	$-0.457464\pi$
$59$	2.04679	0.266469	0.133235	+	0.991085i	$0.457464\pi$
$60$	0	0
$61$	1.00000	0.128037
$61$	1.00000	0.128037
$62$	−4.67295	−0.593465
$63$	0	0
$64$	2.50953	0.313691
$65$	19.0437	2.36209
$66$	0	0
$67$	5.30981	0.648697	0.324348	−	0.945938i	$-0.394855\pi$
$67$	5.30981	0.648697	0.324348	+	0.945938i	$0.394855\pi$
$68$	1.56571	0.189870
$69$	0	0
$70$	21.0442	2.51527
$71$	−11.4906	−1.36368	−0.681841	−	0.731500i	$-0.738820\pi$
$71$	−11.4906	−1.36368	−0.681841	+	0.731500i	$0.738820\pi$
$72$	0	0
$73$	−4.51365	−0.528283	−0.264142	−	0.964484i	$-0.585089\pi$
$73$	−4.51365	−0.528283	−0.264142	+	0.964484i	$0.585089\pi$
$74$	5.42978	0.631199
$75$	0	0
$76$	20.9319	2.40106
$77$	2.04273	0.232791
$78$	0	0
$79$	5.49270	0.617977	0.308989	−	0.951066i	$-0.400009\pi$
$79$	5.49270	0.617977	0.308989	+	0.951066i	$0.400009\pi$
$80$	−32.1683	−3.59652
$81$	0	0
$82$	−26.3733	−2.91245
$83$	−14.2207	−1.56092	−0.780460	−	0.625206i	$-0.785015\pi$
$83$	−14.2207	−1.56092	−0.780460	+	0.625206i	$0.785015\pi$
$84$	0	0
$85$	−1.36117	−0.147639
$86$	25.4399	2.74326
$87$	0	0
$88$	−6.70837	−0.715115
$89$	6.05470	0.641797	0.320899	−	0.947114i	$-0.396015\pi$
$89$	6.05470	0.641797	0.320899	+	0.947114i	$0.396015\pi$
$90$	0	0
$91$	−9.70780	−1.01765
$92$	21.9291	2.28626
$93$	0	0
$94$	−13.4315	−1.38536
$95$	−18.1974	−1.86701
$96$	0	0
$97$	13.6004	1.38092	0.690458	−	0.723373i	$-0.257409\pi$
$97$	13.6004	1.38092	0.690458	+	0.723373i	$0.257409\pi$
$98$	7.26851	0.734231
$99$	0	0
$100$	50.9693	5.09693
$101$	10.2656	1.02147	0.510735	−	0.859738i	$-0.329373\pi$
$101$	10.2656	1.02147	0.510735	+	0.859738i	$0.329373\pi$
$102$	0	0
$103$	10.8307	1.06718	0.533590	−	0.845743i	$-0.320843\pi$
$103$	10.8307	1.06718	0.533590	+	0.845743i	$0.320843\pi$
$104$	31.8807	3.12616
$105$	0	0
$106$	24.8977	2.41828
$107$	10.3745	1.00294	0.501471	−	0.865174i	$-0.332792\pi$
$107$	10.3745	1.00294	0.501471	+	0.865174i	$0.332792\pi$
$108$	0	0
$109$	−15.9251	−1.52534	−0.762672	−	0.646786i	$-0.776113\pi$
$109$	−15.9251	−1.52534	−0.762672	+	0.646786i	$0.776113\pi$
$110$	10.3020	0.982259
$111$	0	0
$112$	16.3982	1.54949
$113$	−4.50235	−0.423546	−0.211773	−	0.977319i	$-0.567924\pi$
$113$	−4.50235	−0.423546	−0.211773	+	0.977319i	$0.567924\pi$
$114$	0	0
$115$	−19.0643	−1.77775
$116$	19.7509	1.83382
$117$	0	0
$118$	−5.26203	−0.484409
$119$	0.693873	0.0636072
$120$	0	0
$121$	1.00000	0.0909091
$122$	−2.57087	−0.232756
$123$	0	0
$124$	8.37824	0.752388
$125$	−24.2746	−2.17119
$126$	0	0
$127$	−12.2578	−1.08770	−0.543852	−	0.839181i	$-0.683035\pi$
$127$	−12.2578	−1.08770	−0.543852	+	0.839181i	$0.683035\pi$
$128$	7.99069	0.706284
$129$	0	0
$130$	−48.9590	−4.29399
$131$	−14.1653	−1.23763	−0.618814	−	0.785538i	$-0.712386\pi$
$131$	−14.1653	−1.23763	−0.618814	+	0.785538i	$0.712386\pi$
$132$	0	0
$133$	9.27637	0.804363
$134$	−13.6508	−1.17925
$135$	0	0
$136$	−2.27870	−0.195397
$137$	−11.8311	−1.01080	−0.505401	−	0.862885i	$-0.668655\pi$
$137$	−11.8311	−1.01080	−0.505401	+	0.862885i	$0.668655\pi$
$138$	0	0
$139$	21.4652	1.82066	0.910328	−	0.413887i	$-0.135829\pi$
$139$	21.4652	1.82066	0.910328	+	0.413887i	$0.135829\pi$
$140$	−37.7307	−3.18883
$141$	0	0
$142$	29.5408	2.47901
$143$	−4.75237	−0.397413
$144$	0	0
$145$	−17.1706	−1.42594
$146$	11.6040	0.960356
$147$	0	0
$148$	−9.73519	−0.800227
$149$	−5.71869	−0.468493	−0.234247	−	0.972177i	$-0.575262\pi$
$149$	−5.71869	−0.468493	−0.234247	+	0.972177i	$0.575262\pi$
$150$	0	0
$151$	4.53058	0.368694	0.184347	−	0.982861i	$-0.440983\pi$
$151$	4.53058	0.368694	0.184347	+	0.982861i	$0.440983\pi$
$152$	−30.4638	−2.47094
$153$	0	0
$154$	−5.25159	−0.423185
$155$	−7.28371	−0.585042
$156$	0	0
$157$	−10.4733	−0.835859	−0.417930	−	0.908479i	$-0.637244\pi$
$157$	−10.4733	−0.835859	−0.417930	+	0.908479i	$0.637244\pi$
$158$	−14.1210	−1.12341
$159$	0	0
$160$	28.9368	2.28766
$161$	9.71827	0.765907
$162$	0	0
$163$	12.0156	0.941136	0.470568	−	0.882364i	$-0.344049\pi$
$163$	12.0156	0.941136	0.470568	+	0.882364i	$0.344049\pi$
$164$	47.2854	3.69237
$165$	0	0
$166$	36.5595	2.83756
$167$	−7.72247	−0.597582	−0.298791	−	0.954318i	$-0.596583\pi$
$167$	−7.72247	−0.597582	−0.298791	+	0.954318i	$0.596583\pi$
$168$	0	0
$169$	9.58501	0.737308
$170$	3.49938	0.268391
$171$	0	0
$172$	−45.6119	−3.47787
$173$	−14.5915	−1.10937	−0.554684	−	0.832061i	$-0.687161\pi$
$173$	−14.5915	−1.10937	−0.554684	+	0.832061i	$0.687161\pi$
$174$	0	0
$175$	22.5880	1.70749
$176$	8.02760	0.605103
$177$	0	0
$178$	−15.5659	−1.16671
$179$	13.1655	0.984037	0.492019	−	0.870585i	$-0.336259\pi$
$179$	13.1655	0.984037	0.492019	+	0.870585i	$0.336259\pi$
$180$	0	0
$181$	−7.26104	−0.539709	−0.269854	−	0.962901i	$-0.586976\pi$
$181$	−7.26104	−0.539709	−0.269854	+	0.962901i	$0.586976\pi$
$182$	24.9575	1.84997
$183$	0	0
$184$	−31.9150	−2.35281
$185$	8.46339	0.622241
$186$	0	0
$187$	0.339679	0.0248398
$188$	24.0818	1.75634
$189$	0	0
$190$	46.7832	3.39401
$191$	−3.04371	−0.220235	−0.110118	−	0.993919i	$-0.535123\pi$
$191$	−3.04371	−0.220235	−0.110118	+	0.993919i	$0.535123\pi$
$192$	0	0
$193$	6.85641	0.493535	0.246768	−	0.969075i	$-0.420632\pi$
$193$	6.85641	0.493535	0.246768	+	0.969075i	$0.420632\pi$
$194$	−34.9650	−2.51034
$195$	0	0
$196$	−13.0319	−0.930850
$197$	12.1375	0.864763	0.432381	−	0.901691i	$-0.357673\pi$
$197$	12.1375	0.864763	0.432381	+	0.901691i	$0.357673\pi$
$198$	0	0
$199$	−13.8050	−0.978613	−0.489306	−	0.872112i	$-0.662750\pi$
$199$	−13.8050	−0.978613	−0.489306	+	0.872112i	$0.662750\pi$
$200$	−74.1794	−5.24528
$201$	0	0
$202$	−26.3916	−1.85691
$203$	8.75297	0.614338
$204$	0	0
$205$	−41.1081	−2.87111
$206$	−27.8443	−1.94001
$207$	0	0
$208$	−38.1501	−2.64523
$209$	4.54116	0.314119
$210$	0	0
$211$	13.4936	0.928938	0.464469	−	0.885589i	$-0.346245\pi$
$211$	13.4936	0.928938	0.464469	+	0.885589i	$0.346245\pi$
$212$	−44.6398	−3.06587
$213$	0	0
$214$	−26.6715	−1.82323
$215$	39.6532	2.70432
$216$	0	0
$217$	3.71297	0.252053
$218$	40.9413	2.77289
$219$	0	0
$220$	−18.4707	−1.24530
$221$	−1.61428	−0.108588
$222$	0	0
$223$	−3.65818	−0.244970	−0.122485	−	0.992470i	$-0.539086\pi$
$223$	−3.65818	−0.244970	−0.122485	+	0.992470i	$0.539086\pi$
$224$	−14.7509	−0.985588
$225$	0	0
$226$	11.5750	0.769956
$227$	6.72165	0.446131	0.223066	−	0.974803i	$-0.428394\pi$
$227$	6.72165	0.446131	0.223066	+	0.974803i	$0.428394\pi$
$228$	0	0
$229$	−1.23766	−0.0817871	−0.0408935	−	0.999164i	$-0.513020\pi$
$229$	−1.23766	−0.0817871	−0.0408935	+	0.999164i	$0.513020\pi$
$230$	49.0118	3.23174
$231$	0	0
$232$	−28.7449	−1.88720
$233$	−12.9299	−0.847065	−0.423532	−	0.905881i	$-0.639210\pi$
$233$	−12.9299	−0.847065	−0.423532	+	0.905881i	$0.639210\pi$
$234$	0	0
$235$	−20.9357	−1.36570
$236$	9.43442	0.614128
$237$	0	0
$238$	−1.78386	−0.115630
$239$	18.5458	1.19963	0.599814	−	0.800140i	$-0.295241\pi$
$239$	18.5458	1.19963	0.599814	+	0.800140i	$0.295241\pi$
$240$	0	0
$241$	−27.7936	−1.79034	−0.895171	−	0.445723i	$-0.852947\pi$
$241$	−27.7936	−1.79034	−0.895171	+	0.445723i	$0.852947\pi$
$242$	−2.57087	−0.165262
$243$	0	0
$244$	4.60938	0.295085
$245$	11.3294	0.723810
$246$	0	0
$247$	−21.5813	−1.37318
$248$	−12.1935	−0.774287
$249$	0	0
$250$	62.4070	3.94696
$251$	4.56250	0.287982	0.143991	−	0.989579i	$-0.454006\pi$
$251$	4.56250	0.287982	0.143991	+	0.989579i	$0.454006\pi$
$252$	0	0
$253$	4.75749	0.299101
$254$	31.5132	1.97732
$255$	0	0
$256$	−25.5621	−1.59763
$257$	16.8127	1.04874	0.524372	−	0.851489i	$-0.324300\pi$
$257$	16.8127	1.04874	0.524372	+	0.851489i	$0.324300\pi$
$258$	0	0
$259$	−4.31433	−0.268079
$260$	87.7798	5.44387
$261$	0	0
$262$	36.4171	2.24986
$263$	8.19724	0.505463	0.252732	−	0.967536i	$-0.418671\pi$
$263$	8.19724	0.505463	0.252732	+	0.967536i	$0.418671\pi$
$264$	0	0
$265$	38.8081	2.38396
$266$	−23.8483	−1.46224
$267$	0	0
$268$	24.4749	1.49504
$269$	−12.7223	−0.775690	−0.387845	−	0.921725i	$-0.626780\pi$
$269$	−12.7223	−0.775690	−0.387845	+	0.921725i	$0.626780\pi$
$270$	0	0
$271$	28.9215	1.75686	0.878428	−	0.477875i	$-0.158593\pi$
$271$	28.9215	1.75686	0.878428	+	0.477875i	$0.158593\pi$
$272$	2.72681	0.165337
$273$	0	0
$274$	30.4163	1.83752
$275$	11.0577	0.666807
$276$	0	0
$277$	9.01259	0.541514	0.270757	−	0.962648i	$-0.412726\pi$
$277$	9.01259	0.541514	0.270757	+	0.962648i	$0.412726\pi$
$278$	−55.1843	−3.30974
$279$	0	0
$280$	54.9124	3.28164
$281$	20.0391	1.19543	0.597715	−	0.801708i	$-0.296075\pi$
$281$	20.0391	1.19543	0.597715	+	0.801708i	$0.296075\pi$
$282$	0	0
$283$	29.2872	1.74094	0.870471	−	0.492221i	$-0.163815\pi$
$283$	29.2872	1.74094	0.870471	+	0.492221i	$0.163815\pi$
$284$	−52.9645	−3.14286
$285$	0	0
$286$	12.2177	0.722449
$287$	20.9554	1.23696
$288$	0	0
$289$	−16.8846	−0.993213
$290$	44.1435	2.59220
$291$	0	0
$292$	−20.8051	−1.21753
$293$	−31.2913	−1.82806	−0.914029	−	0.405649i	$-0.867046\pi$
$293$	−31.2913	−1.82806	−0.914029	+	0.405649i	$0.867046\pi$
$294$	0	0
$295$	−8.20191	−0.477534
$296$	14.1683	0.823518
$297$	0	0
$298$	14.7020	0.851665
$299$	−22.6094	−1.30753
$300$	0	0
$301$	−20.2137	−1.16510
$302$	−11.6475	−0.670241
$303$	0	0
$304$	36.4547	2.09082
$305$	−4.00721	−0.229452
$306$	0	0
$307$	0.203717	0.0116268	0.00581338	−	0.999983i	$-0.498150\pi$
$307$	0.203717	0.0116268	0.00581338	+	0.999983i	$0.498150\pi$
$308$	9.41571	0.536510
$309$	0	0
$310$	18.7255	1.06354
$311$	15.4938	0.878573	0.439287	−	0.898347i	$-0.355231\pi$
$311$	15.4938	0.878573	0.439287	+	0.898347i	$0.355231\pi$
$312$	0	0
$313$	−27.2007	−1.53747	−0.768737	−	0.639565i	$-0.779114\pi$
$313$	−27.2007	−1.53747	−0.768737	+	0.639565i	$0.779114\pi$
$314$	26.9255	1.51949
$315$	0	0
$316$	25.3179	1.42425
$317$	11.9050	0.668653	0.334327	−	0.942457i	$-0.391491\pi$
$317$	11.9050	0.668653	0.334327	+	0.942457i	$0.391491\pi$
$318$	0	0
$319$	4.28494	0.239910
$320$	−10.0562	−0.562159
$321$	0	0
$322$	−24.9844	−1.39233
$323$	1.54254	0.0858292
$324$	0	0
$325$	−52.5505	−2.91497
$326$	−30.8906	−1.71087
$327$	0	0
$328$	−68.8180	−3.79984
$329$	10.6723	0.588381
$330$	0	0
$331$	−5.44265	−0.299155	−0.149577	−	0.988750i	$-0.547791\pi$
$331$	−5.44265	−0.299155	−0.149577	+	0.988750i	$0.547791\pi$
$332$	−65.5483	−3.59743
$333$	0	0
$334$	19.8535	1.08633
$335$	−21.2775	−1.16252
$336$	0	0
$337$	11.5595	0.629687	0.314843	−	0.949144i	$-0.398048\pi$
$337$	11.5595	0.629687	0.314843	+	0.949144i	$0.398048\pi$
$338$	−24.6418	−1.34034
$339$	0	0
$340$	−6.27413	−0.340263
$341$	1.81765	0.0984314
$342$	0	0
$343$	−20.0744	−1.08392
$344$	66.3824	3.57910
$345$	0	0
$346$	37.5128	2.01670
$347$	34.9338	1.87535	0.937673	−	0.347519i	$-0.112976\pi$
$347$	34.9338	1.87535	0.937673	+	0.347519i	$0.112976\pi$
$348$	0	0
$349$	−16.4637	−0.881281	−0.440641	−	0.897684i	$-0.645249\pi$
$349$	−16.4637	−0.881281	−0.440641	+	0.897684i	$0.645249\pi$
$350$	−58.0708	−3.10401
$351$	0	0
$352$	−7.22118	−0.384890
$353$	−36.1064	−1.92175	−0.960876	−	0.276980i	$-0.910667\pi$
$353$	−36.1064	−1.92175	−0.960876	+	0.276980i	$0.910667\pi$
$354$	0	0
$355$	46.0452	2.44383
$356$	27.9084	1.47914
$357$	0	0
$358$	−33.8469	−1.78886
$359$	16.1146	0.850498	0.425249	−	0.905076i	$-0.360187\pi$
$359$	16.1146	0.850498	0.425249	+	0.905076i	$0.360187\pi$
$360$	0	0
$361$	1.62217	0.0853773
$362$	18.6672	0.981126
$363$	0	0
$364$	−44.7469	−2.34538
$365$	18.0872	0.946725
$366$	0	0
$367$	35.8180	1.86969	0.934843	−	0.355061i	$-0.115540\pi$
$367$	35.8180	1.86969	0.934843	+	0.355061i	$0.115540\pi$
$368$	38.1913	1.99086
$369$	0	0
$370$	−21.7583	−1.13116
$371$	−19.7829	−1.02708
$372$	0	0
$373$	−18.3914	−0.952273	−0.476137	−	0.879371i	$-0.657963\pi$
$373$	−18.3914	−0.952273	−0.476137	+	0.879371i	$0.657963\pi$
$374$	−0.873272	−0.0451558
$375$	0	0
$376$	−35.0480	−1.80746
$377$	−20.3636	−1.04878
$378$	0	0
$379$	−5.27053	−0.270729	−0.135364	−	0.990796i	$-0.543221\pi$
$379$	−5.27053	−0.270729	−0.135364	+	0.990796i	$0.543221\pi$
$380$	−83.8787	−4.30289
$381$	0	0
$382$	7.82499	0.400361
$383$	−32.5167	−1.66153	−0.830763	−	0.556626i	$-0.812096\pi$
$383$	−32.5167	−1.66153	−0.830763	+	0.556626i	$0.812096\pi$
$384$	0	0
$385$	−8.18565	−0.417179
$386$	−17.6269	−0.897188
$387$	0	0
$388$	62.6895	3.18258
$389$	24.3758	1.23590	0.617950	−	0.786217i	$-0.287963\pi$
$389$	24.3758	1.23590	0.617950	+	0.786217i	$0.287963\pi$
$390$	0	0
$391$	1.61602	0.0817258
$392$	18.9663	0.957942
$393$	0	0
$394$	−31.2040	−1.57203
$395$	−22.0104	−1.10746
$396$	0	0
$397$	−18.0276	−0.904779	−0.452390	−	0.891820i	$-0.649428\pi$
$397$	−18.0276	−0.904779	−0.452390	+	0.891820i	$0.649428\pi$
$398$	35.4909	1.77900
$399$	0	0
$400$	88.7671	4.43836
$401$	29.1981	1.45808	0.729042	−	0.684469i	$-0.239966\pi$
$401$	29.1981	1.45808	0.729042	+	0.684469i	$0.239966\pi$
$402$	0	0
$403$	−8.63815	−0.430297
$404$	47.3182	2.35417
$405$	0	0
$406$	−22.5027	−1.11679
$407$	−2.11204	−0.104690
$408$	0	0
$409$	2.63476	0.130280	0.0651401	−	0.997876i	$-0.479251\pi$
$409$	2.63476	0.130280	0.0651401	+	0.997876i	$0.479251\pi$
$410$	105.684	5.21934
$411$	0	0
$412$	49.9227	2.45952
$413$	4.18103	0.205735
$414$	0	0
$415$	56.9852	2.79729
$416$	34.3177	1.68257
$417$	0	0
$418$	−11.6747	−0.571030
$419$	−32.0724	−1.56684	−0.783420	−	0.621492i	$-0.786527\pi$
$419$	−32.0724	−1.56684	−0.783420	+	0.621492i	$0.786527\pi$
$420$	0	0
$421$	6.02368	0.293576	0.146788	−	0.989168i	$-0.453106\pi$
$421$	6.02368	0.293576	0.146788	+	0.989168i	$0.453106\pi$
$422$	−34.6903	−1.68870
$423$	0	0
$424$	64.9676	3.15510
$425$	3.75609	0.182197
$426$	0	0
$427$	2.04273	0.0988546
$428$	47.8200	2.31147
$429$	0	0
$430$	−101.943	−4.91614
$431$	19.6288	0.945487	0.472744	−	0.881200i	$-0.343264\pi$
$431$	19.6288	0.945487	0.472744	+	0.881200i	$0.343264\pi$
$432$	0	0
$433$	27.8579	1.33876	0.669382	−	0.742919i	$-0.266559\pi$
$433$	27.8579	1.33876	0.669382	+	0.742919i	$0.266559\pi$
$434$	−9.54557	−0.458202
$435$	0	0
$436$	−73.4046	−3.51544
$437$	21.6046	1.03349
$438$	0	0
$439$	−25.4823	−1.21620	−0.608102	−	0.793859i	$-0.708069\pi$
$439$	−25.4823	−1.21620	−0.608102	+	0.793859i	$0.708069\pi$
$440$	26.8819	1.28154
$441$	0	0
$442$	4.15011	0.197401
$443$	−29.0256	−1.37905	−0.689524	−	0.724263i	$-0.742180\pi$
$443$	−29.0256	−1.37905	−0.689524	+	0.724263i	$0.742180\pi$
$444$	0	0
$445$	−24.2625	−1.15015
$446$	9.40471	0.445326
$447$	0	0
$448$	5.12629	0.242194
$449$	−23.3018	−1.09968	−0.549840	−	0.835270i	$-0.685311\pi$
$449$	−23.3018	−1.09968	−0.549840	+	0.835270i	$0.685311\pi$
$450$	0	0
$451$	10.2585	0.483055
$452$	−20.7530	−0.976141
$453$	0	0
$454$	−17.2805	−0.811013
$455$	38.9012	1.82372
$456$	0	0
$457$	38.8508	1.81737	0.908683	−	0.417487i	$-0.137089\pi$
$457$	38.8508	1.81737	0.908683	+	0.417487i	$0.137089\pi$
$458$	3.18187	0.148679
$459$	0	0
$460$	−87.8745	−4.09717
$461$	−2.42547	−0.112965	−0.0564827	−	0.998404i	$-0.517989\pi$
$461$	−2.42547	−0.112965	−0.0564827	+	0.998404i	$0.517989\pi$
$462$	0	0
$463$	37.3362	1.73516	0.867581	−	0.497295i	$-0.165673\pi$
$463$	37.3362	1.73516	0.867581	+	0.497295i	$0.165673\pi$
$464$	34.3978	1.59688
$465$	0	0
$466$	33.2411	1.53986
$467$	7.55898	0.349788	0.174894	−	0.984587i	$-0.444042\pi$
$467$	7.55898	0.349788	0.174894	+	0.984587i	$0.444042\pi$
$468$	0	0
$469$	10.8465	0.500845
$470$	53.8231	2.48267
$471$	0	0
$472$	−13.7306	−0.632003
$473$	−9.89546	−0.454994
$474$	0	0
$475$	50.2150	2.30402
$476$	3.19832	0.146595
$477$	0	0
$478$	−47.6788	−2.18078
$479$	35.7614	1.63398	0.816990	−	0.576651i	$-0.195641\pi$
$479$	35.7614	1.63398	0.816990	+	0.576651i	$0.195641\pi$
$480$	0	0
$481$	10.0372	0.457657
$482$	71.4537	3.25463
$483$	0	0
$484$	4.60938	0.209517
$485$	−54.4998	−2.47471
$486$	0	0
$487$	−6.39232	−0.289664	−0.144832	−	0.989456i	$-0.546264\pi$
$487$	−6.39232	−0.289664	−0.144832	+	0.989456i	$0.546264\pi$
$488$	−6.70837	−0.303674
$489$	0	0
$490$	−29.1265	−1.31580
$491$	−16.7893	−0.757691	−0.378846	−	0.925460i	$-0.623679\pi$
$491$	−16.7893	−0.757691	−0.378846	+	0.925460i	$0.623679\pi$
$492$	0	0
$493$	1.45550	0.0655526
$494$	55.4827	2.49628
$495$	0	0
$496$	14.5914	0.655172
$497$	−23.4722	−1.05287
$498$	0	0
$499$	2.39493	0.107212	0.0536060	−	0.998562i	$-0.482929\pi$
$499$	2.39493	0.107212	0.0536060	+	0.998562i	$0.482929\pi$
$500$	−111.891	−5.00392
$501$	0	0
$502$	−11.7296	−0.523517
$503$	−2.97796	−0.132781	−0.0663903	−	0.997794i	$-0.521148\pi$
$503$	−2.97796	−0.132781	−0.0663903	+	0.997794i	$0.521148\pi$
$504$	0	0
$505$	−41.1366	−1.83055
$506$	−12.2309	−0.543730
$507$	0	0
$508$	−56.5008	−2.50682
$509$	13.6035	0.602964	0.301482	−	0.953472i	$-0.402519\pi$
$509$	13.6035	0.602964	0.301482	+	0.953472i	$0.402519\pi$
$510$	0	0
$511$	−9.22017	−0.407876
$512$	49.7355	2.19802
$513$	0	0
$514$	−43.2232	−1.90649
$515$	−43.4009	−1.91247
$516$	0	0
$517$	5.22451	0.229774
$518$	11.0916	0.487336
$519$	0	0
$520$	−127.752	−5.60232
$521$	−4.86305	−0.213054	−0.106527	−	0.994310i	$-0.533973\pi$
$521$	−4.86305	−0.213054	−0.106527	+	0.994310i	$0.533973\pi$
$522$	0	0
$523$	4.06198	0.177618	0.0888091	−	0.996049i	$-0.471694\pi$
$523$	4.06198	0.177618	0.0888091	+	0.996049i	$0.471694\pi$
$524$	−65.2932	−2.85234
$525$	0	0
$526$	−21.0740	−0.918872
$527$	0.617419	0.0268952
$528$	0	0
$529$	−0.366250	−0.0159239
$530$	−99.7705	−4.33375
$531$	0	0
$532$	42.7583	1.85381
$533$	−48.7523	−2.11170
$534$	0	0
$535$	−41.5729	−1.79735
$536$	−35.6202	−1.53856
$537$	0	0
$538$	32.7073	1.41011
$539$	−2.82726	−0.121779
$540$	0	0
$541$	43.0563	1.85113	0.925566	−	0.378585i	$-0.123589\pi$
$541$	43.0563	1.85113	0.925566	+	0.378585i	$0.123589\pi$
$542$	−74.3534	−3.19375
$543$	0	0
$544$	−2.45289	−0.105167
$545$	63.8150	2.73354
$546$	0	0
$547$	8.25809	0.353090	0.176545	−	0.984293i	$-0.443508\pi$
$547$	8.25809	0.353090	0.176545	+	0.984293i	$0.443508\pi$
$548$	−54.5341	−2.32958
$549$	0	0
$550$	−28.4280	−1.21217
$551$	19.4586	0.828964
$552$	0	0
$553$	11.2201	0.477127
$554$	−23.1702	−0.984408
$555$	0	0
$556$	98.9413	4.19605
$557$	12.2925	0.520852	0.260426	−	0.965494i	$-0.416137\pi$
$557$	12.2925	0.520852	0.260426	+	0.965494i	$0.416137\pi$
$558$	0	0
$559$	47.0269	1.98902
$560$	−65.7111	−2.77680
$561$	0	0
$562$	−51.5179	−2.17315
$563$	20.3279	0.856719	0.428359	−	0.903609i	$-0.359092\pi$
$563$	20.3279	0.856719	0.428359	+	0.903609i	$0.359092\pi$
$564$	0	0
$565$	18.0419	0.759028
$566$	−75.2935	−3.16482
$567$	0	0
$568$	77.0831	3.23434
$569$	6.99933	0.293427	0.146714	−	0.989179i	$-0.453130\pi$
$569$	6.99933	0.293427	0.146714	+	0.989179i	$0.453130\pi$
$570$	0	0
$571$	−28.0828	−1.17523	−0.587615	−	0.809141i	$-0.699933\pi$
$571$	−28.0828	−1.17523	−0.587615	+	0.809141i	$0.699933\pi$
$572$	−21.9055	−0.915913
$573$	0	0
$574$	−53.8736	−2.24864
$575$	52.6071	2.19387
$576$	0	0
$577$	−20.0459	−0.834523	−0.417262	−	0.908786i	$-0.637010\pi$
$577$	−20.0459	−0.834523	−0.417262	+	0.908786i	$0.637010\pi$
$578$	43.4082	1.80554
$579$	0	0
$580$	−79.1460	−3.28636
$581$	−29.0489	−1.20515
$582$	0	0
$583$	−9.68455	−0.401093
$584$	30.2793	1.25296
$585$	0	0
$586$	80.4459	3.32319
$587$	16.2870	0.672235	0.336118	−	0.941820i	$-0.390886\pi$
$587$	16.2870	0.672235	0.336118	+	0.941820i	$0.390886\pi$
$588$	0	0
$589$	8.25425	0.340111
$590$	21.0861	0.868099
$591$	0	0
$592$	−16.9546	−0.696830
$593$	36.0721	1.48130	0.740652	−	0.671889i	$-0.234517\pi$
$593$	36.0721	1.48130	0.740652	+	0.671889i	$0.234517\pi$
$594$	0	0
$595$	−2.78050	−0.113989
$596$	−26.3596	−1.07973
$597$	0	0
$598$	58.1258	2.37694
$599$	12.5585	0.513125	0.256563	−	0.966528i	$-0.417410\pi$
$599$	12.5585	0.513125	0.256563	+	0.966528i	$0.417410\pi$
$600$	0	0
$601$	35.2090	1.43621	0.718103	−	0.695936i	$-0.245010\pi$
$601$	35.2090	1.43621	0.718103	+	0.695936i	$0.245010\pi$
$602$	51.9669	2.11801
$603$	0	0
$604$	20.8832	0.849724
$605$	−4.00721	−0.162916
$606$	0	0
$607$	1.60580	0.0651773	0.0325886	−	0.999469i	$-0.489625\pi$
$607$	1.60580	0.0651773	0.0325886	+	0.999469i	$0.489625\pi$
$608$	−32.7926	−1.32991
$609$	0	0
$610$	10.3020	0.417117
$611$	−24.8288	−1.00447
$612$	0	0
$613$	−2.16291	−0.0873591	−0.0436796	−	0.999046i	$-0.513908\pi$
$613$	−2.16291	−0.0873591	−0.0436796	+	0.999046i	$0.513908\pi$
$614$	−0.523731	−0.0211361
$615$	0	0
$616$	−13.7034	−0.552125
$617$	14.6864	0.591253	0.295627	−	0.955304i	$-0.404472\pi$
$617$	14.6864	0.591253	0.295627	+	0.955304i	$0.404472\pi$
$618$	0	0
$619$	47.1847	1.89651	0.948257	−	0.317505i	$-0.102845\pi$
$619$	47.1847	1.89651	0.948257	+	0.317505i	$0.102845\pi$
$620$	−33.5734	−1.34834
$621$	0	0
$622$	−39.8326	−1.59714
$623$	12.3681	0.495518
$624$	0	0
$625$	41.9849	1.67940
$626$	69.9294	2.79494
$627$	0	0
$628$	−48.2753	−1.92639
$629$	−0.717416	−0.0286053
$630$	0	0
$631$	44.0652	1.75421	0.877104	−	0.480300i	$-0.159472\pi$
$631$	44.0652	1.75421	0.877104	+	0.480300i	$0.159472\pi$
$632$	−36.8471	−1.46570
$633$	0	0
$634$	−30.6063	−1.21553
$635$	49.1196	1.94925
$636$	0	0
$637$	13.4362	0.532360
$638$	−11.0160	−0.436128
$639$	0	0
$640$	−32.0204	−1.26572
$641$	32.9295	1.30064	0.650319	−	0.759661i	$-0.274635\pi$
$641$	32.9295	1.30064	0.650319	+	0.759661i	$0.274635\pi$
$642$	0	0
$643$	34.3960	1.35645	0.678223	−	0.734856i	$-0.262750\pi$
$643$	34.3960	1.35645	0.678223	+	0.734856i	$0.262750\pi$
$644$	44.7952	1.76518
$645$	0	0
$646$	−3.96567	−0.156027
$647$	−40.6293	−1.59730	−0.798652	−	0.601793i	$-0.794453\pi$
$647$	−40.6293	−1.59730	−0.798652	+	0.601793i	$0.794453\pi$
$648$	0	0
$649$	2.04679	0.0803435
$650$	135.100	5.29908
$651$	0	0
$652$	55.3845	2.16903
$653$	0.882411	0.0345314	0.0172657	−	0.999851i	$-0.494504\pi$
$653$	0.882411	0.0345314	0.0172657	+	0.999851i	$0.494504\pi$
$654$	0	0
$655$	56.7633	2.21793
$656$	82.3514	3.21528
$657$	0	0
$658$	−27.4370	−1.06961
$659$	−24.8985	−0.969906	−0.484953	−	0.874540i	$-0.661163\pi$
$659$	−24.8985	−0.969906	−0.484953	+	0.874540i	$0.661163\pi$
$660$	0	0
$661$	40.3972	1.57127	0.785635	−	0.618690i	$-0.212336\pi$
$661$	40.3972	1.57127	0.785635	+	0.618690i	$0.212336\pi$
$662$	13.9923	0.543828
$663$	0	0
$664$	95.3974	3.70214
$665$	−37.1724	−1.44148
$666$	0	0
$667$	20.3856	0.789332
$668$	−35.5958	−1.37724
$669$	0	0
$670$	54.7018	2.11331
$671$	1.00000	0.0386046
$672$	0	0
$673$	−4.08661	−0.157527	−0.0787636	−	0.996893i	$-0.525097\pi$
$673$	−4.08661	−0.157527	−0.0787636	+	0.996893i	$0.525097\pi$
$674$	−29.7180	−1.14469
$675$	0	0
$676$	44.1809	1.69927
$677$	3.16188	0.121521	0.0607604	−	0.998152i	$-0.480647\pi$
$677$	3.16188	0.121521	0.0607604	+	0.998152i	$0.480647\pi$
$678$	0	0
$679$	27.7820	1.06618
$680$	9.13121	0.350166
$681$	0	0
$682$	−4.67295	−0.178936
$683$	2.84581	0.108892	0.0544459	−	0.998517i	$-0.482661\pi$
$683$	2.84581	0.108892	0.0544459	+	0.998517i	$0.482661\pi$
$684$	0	0
$685$	47.4098	1.81144
$686$	51.6088	1.97043
$687$	0	0
$688$	−79.4368	−3.02850
$689$	46.0246	1.75340
$690$	0	0
$691$	24.9625	0.949620	0.474810	−	0.880088i	$-0.342517\pi$
$691$	24.9625	0.949620	0.474810	+	0.880088i	$0.342517\pi$
$692$	−67.2576	−2.55675
$693$	0	0
$694$	−89.8104	−3.40915
$695$	−86.0157	−3.26276
$696$	0	0
$697$	3.48461	0.131989
$698$	42.3260	1.60206
$699$	0	0
$700$	104.116	3.93523
$701$	−3.42189	−0.129243	−0.0646216	−	0.997910i	$-0.520584\pi$
$701$	−3.42189	−0.129243	−0.0646216	+	0.997910i	$0.520584\pi$
$702$	0	0
$703$	−9.59112	−0.361736
$704$	2.50953	0.0945815
$705$	0	0
$706$	92.8250	3.49351
$707$	20.9699	0.788655
$708$	0	0
$709$	−52.0319	−1.95410	−0.977050	−	0.213008i	$-0.931674\pi$
$709$	−52.0319	−1.95410	−0.977050	+	0.213008i	$0.931674\pi$
$710$	−118.376	−4.44258
$711$	0	0
$712$	−40.6172	−1.52219
$713$	8.64747	0.323850
$714$	0	0
$715$	19.0437	0.712196
$716$	60.6849	2.26790
$717$	0	0
$718$	−41.4287	−1.54610
$719$	10.9759	0.409332	0.204666	−	0.978832i	$-0.434389\pi$
$719$	10.9759	0.409332	0.204666	+	0.978832i	$0.434389\pi$
$720$	0	0
$721$	22.1242	0.823947
$722$	−4.17038	−0.155206
$723$	0	0
$724$	−33.4689	−1.24386
$725$	47.3817	1.75971
$726$	0	0
$727$	32.1350	1.19182	0.595911	−	0.803051i	$-0.296791\pi$
$727$	32.1350	1.19182	0.595911	+	0.803051i	$0.296791\pi$
$728$	65.1235	2.41364
$729$	0	0
$730$	−46.4998	−1.72103
$731$	−3.36128	−0.124322
$732$	0	0
$733$	17.1024	0.631693	0.315846	−	0.948810i	$-0.397712\pi$
$733$	17.1024	0.631693	0.315846	+	0.948810i	$0.397712\pi$
$734$	−92.0835	−3.39887
$735$	0	0
$736$	−34.3547	−1.26633
$737$	5.30981	0.195589
$738$	0	0
$739$	43.8812	1.61420	0.807098	−	0.590417i	$-0.201037\pi$
$739$	43.8812	1.61420	0.807098	+	0.590417i	$0.201037\pi$
$740$	39.0109	1.43407
$741$	0	0
$742$	50.8593	1.86711
$743$	34.0979	1.25093	0.625465	−	0.780252i	$-0.284909\pi$
$743$	34.0979	1.25093	0.625465	+	0.780252i	$0.284909\pi$
$744$	0	0
$745$	22.9160	0.839577
$746$	47.2820	1.73112
$747$	0	0
$748$	1.56571	0.0572480
$749$	21.1923	0.774351
$750$	0	0
$751$	−0.462726	−0.0168851	−0.00844255	−	0.999964i	$-0.502687\pi$
$751$	−0.462726	−0.0168851	−0.00844255	+	0.999964i	$0.502687\pi$
$752$	41.9403	1.52941
$753$	0	0
$754$	52.3522	1.90655
$755$	−18.1550	−0.660728
$756$	0	0
$757$	−15.0488	−0.546957	−0.273479	−	0.961878i	$-0.588174\pi$
$757$	−15.0488	−0.546957	−0.273479	+	0.961878i	$0.588174\pi$
$758$	13.5498	0.492153
$759$	0	0
$760$	122.075	4.42812
$761$	3.40565	0.123455	0.0617274	−	0.998093i	$-0.480339\pi$
$761$	3.40565	0.123455	0.0617274	+	0.998093i	$0.480339\pi$
$762$	0	0
$763$	−32.5306	−1.17769
$764$	−14.0296	−0.507574
$765$	0	0
$766$	83.5963	3.02046
$767$	−9.72709	−0.351225
$768$	0	0
$769$	16.3158	0.588363	0.294182	−	0.955750i	$-0.404953\pi$
$769$	16.3158	0.588363	0.294182	+	0.955750i	$0.404953\pi$
$770$	21.0442	0.758382
$771$	0	0
$772$	31.6038	1.13744
$773$	−33.0741	−1.18959	−0.594797	−	0.803876i	$-0.702768\pi$
$773$	−33.0741	−1.18959	−0.594797	+	0.803876i	$0.702768\pi$
$774$	0	0
$775$	20.0991	0.721982
$776$	−91.2368	−3.27521
$777$	0	0
$778$	−62.6669	−2.24672
$779$	46.5857	1.66910
$780$	0	0
$781$	−11.4906	−0.411166
$782$	−4.15459	−0.148568
$783$	0	0
$784$	−22.6961	−0.810575
$785$	41.9687	1.49793
$786$	0	0
$787$	13.5591	0.483328	0.241664	−	0.970360i	$-0.422307\pi$
$787$	13.5591	0.483328	0.241664	+	0.970360i	$0.422307\pi$
$788$	55.9464	1.99301
$789$	0	0
$790$	56.5860	2.01324
$791$	−9.19709	−0.327011
$792$	0	0
$793$	−4.75237	−0.168762
$794$	46.3466	1.64478
$795$	0	0
$796$	−63.6326	−2.25540
$797$	−18.0188	−0.638257	−0.319129	−	0.947711i	$-0.603390\pi$
$797$	−18.0188	−0.638257	−0.319129	+	0.947711i	$0.603390\pi$
$798$	0	0
$799$	1.77466	0.0627829
$800$	−79.8500	−2.82312
$801$	0	0
$802$	−75.0646	−2.65062
$803$	−4.51365	−0.159283
$804$	0	0
$805$	−38.9432	−1.37257
$806$	22.2076	0.782228
$807$	0	0
$808$	−68.8657	−2.42269
$809$	−11.1179	−0.390883	−0.195441	−	0.980715i	$-0.562614\pi$
$809$	−11.1179	−0.390883	−0.195441	+	0.980715i	$0.562614\pi$
$810$	0	0
$811$	9.75699	0.342614	0.171307	−	0.985218i	$-0.445201\pi$
$811$	9.75699	0.342614	0.171307	+	0.985218i	$0.445201\pi$
$812$	40.3457	1.41586
$813$	0	0
$814$	5.42978	0.190314
$815$	−48.1491	−1.68659
$816$	0	0
$817$	−44.9369	−1.57214
$818$	−6.77362	−0.236834
$819$	0	0
$820$	−189.483	−6.61702
$821$	−17.6656	−0.616535	−0.308268	−	0.951300i	$-0.599749\pi$
$821$	−17.6656	−0.616535	−0.308268	+	0.951300i	$0.599749\pi$
$822$	0	0
$823$	45.2415	1.57702	0.788510	−	0.615022i	$-0.210853\pi$
$823$	45.2415	1.57702	0.788510	+	0.615022i	$0.210853\pi$
$824$	−72.6563	−2.53110
$825$	0	0
$826$	−10.7489	−0.374002
$827$	26.0944	0.907392	0.453696	−	0.891157i	$-0.350105\pi$
$827$	26.0944	0.907392	0.453696	+	0.891157i	$0.350105\pi$
$828$	0	0
$829$	−36.7769	−1.27732	−0.638658	−	0.769491i	$-0.720510\pi$
$829$	−36.7769	−1.27732	−0.638658	+	0.769491i	$0.720510\pi$
$830$	−146.501	−5.08514
$831$	0	0
$832$	−11.9262	−0.413467
$833$	−0.960361	−0.0332745
$834$	0	0
$835$	30.9456	1.07092
$836$	20.9319	0.723946
$837$	0	0
$838$	82.4541	2.84833
$839$	5.35112	0.184741	0.0923706	−	0.995725i	$-0.470556\pi$
$839$	5.35112	0.184741	0.0923706	+	0.995725i	$0.470556\pi$
$840$	0	0
$841$	−10.6393	−0.366873
$842$	−15.4861	−0.533686
$843$	0	0
$844$	62.1971	2.14091
$845$	−38.4092	−1.32131
$846$	0	0
$847$	2.04273	0.0701890
$848$	−77.7437	−2.66973
$849$	0	0
$850$	−9.65641	−0.331212
$851$	−10.0480	−0.344442
$852$	0	0
$853$	44.6105	1.52744	0.763718	−	0.645550i	$-0.223372\pi$
$853$	44.6105	1.52744	0.763718	+	0.645550i	$0.223372\pi$
$854$	−5.25159	−0.179706
$855$	0	0
$856$	−69.5961	−2.37875
$857$	7.03968	0.240471	0.120235	−	0.992745i	$-0.461635\pi$
$857$	7.03968	0.240471	0.120235	+	0.992745i	$0.461635\pi$
$858$	0	0
$859$	55.9231	1.90807	0.954036	−	0.299693i	$-0.0968841\pi$
$859$	55.9231	1.90807	0.954036	+	0.299693i	$0.0968841\pi$
$860$	182.776	6.23263
$861$	0	0
$862$	−50.4632	−1.71878
$863$	47.1465	1.60489	0.802443	−	0.596729i	$-0.203533\pi$
$863$	47.1465	1.60489	0.802443	+	0.596729i	$0.203533\pi$
$864$	0	0
$865$	58.4711	1.98808
$866$	−71.6190	−2.43371
$867$	0	0
$868$	17.1145	0.580903
$869$	5.49270	0.186327
$870$	0	0
$871$	−25.2342	−0.855028
$872$	106.831	3.61776
$873$	0	0
$874$	−55.5425	−1.87875
$875$	−49.5865	−1.67633
$876$	0	0
$877$	−15.8127	−0.533957	−0.266979	−	0.963702i	$-0.586025\pi$
$877$	−15.8127	−0.533957	−0.266979	+	0.963702i	$0.586025\pi$
$878$	65.5117	2.21091
$879$	0	0
$880$	−32.1683	−1.08439
$881$	−13.2185	−0.445343	−0.222672	−	0.974894i	$-0.571478\pi$
$881$	−13.2185	−0.445343	−0.222672	+	0.974894i	$0.571478\pi$
$882$	0	0
$883$	−31.4226	−1.05746	−0.528728	−	0.848791i	$-0.677331\pi$
$883$	−31.4226	−1.05746	−0.528728	+	0.848791i	$0.677331\pi$
$884$	−7.44083	−0.250262
$885$	0	0
$886$	74.6211	2.50694
$887$	26.4566	0.888326	0.444163	−	0.895946i	$-0.353501\pi$
$887$	26.4566	0.888326	0.444163	+	0.895946i	$0.353501\pi$
$888$	0	0
$889$	−25.0394	−0.839793
$890$	62.3757	2.09084
$891$	0	0
$892$	−16.8619	−0.564579
$893$	23.7254	0.793939
$894$	0	0
$895$	−52.7570	−1.76347
$896$	16.3228	0.545307
$897$	0	0
$898$	59.9059	1.99909
$899$	7.78852	0.259762
$900$	0	0
$901$	−3.28964	−0.109594
$902$	−26.3733	−0.878136
$903$	0	0
$904$	30.2035	1.00455
$905$	29.0965	0.967201
$906$	0	0
$907$	−34.1071	−1.13251	−0.566254	−	0.824231i	$-0.691608\pi$
$907$	−34.1071	−1.13251	−0.566254	+	0.824231i	$0.691608\pi$
$908$	30.9826	1.02819
$909$	0	0
$910$	−100.010	−3.31530
$911$	−9.45811	−0.313361	−0.156681	−	0.987649i	$-0.550079\pi$
$911$	−9.45811	−0.313361	−0.156681	+	0.987649i	$0.550079\pi$
$912$	0	0
$913$	−14.2207	−0.470635
$914$	−99.8805	−3.30375
$915$	0	0
$916$	−5.70485	−0.188494
$917$	−28.9359	−0.955546
$918$	0	0
$919$	21.9708	0.724750	0.362375	−	0.932032i	$-0.381966\pi$
$919$	21.9708	0.724750	0.362375	+	0.932032i	$0.381966\pi$
$920$	127.890	4.21642
$921$	0	0
$922$	6.23557	0.205357
$923$	54.6075	1.79743
$924$	0	0
$925$	−23.3544	−0.767887
$926$	−95.9867	−3.15432
$927$	0	0
$928$	−30.9423	−1.01573
$929$	0.381863	0.0125285	0.00626425	−	0.999980i	$-0.498006\pi$
$929$	0.381863	0.0125285	0.00626425	+	0.999980i	$0.498006\pi$
$930$	0	0
$931$	−12.8390	−0.420782
$932$	−59.5987	−1.95222
$933$	0	0
$934$	−19.4332	−0.635873
$935$	−1.36117	−0.0445149
$936$	0	0
$937$	−45.5441	−1.48786	−0.743931	−	0.668256i	$-0.767041\pi$
$937$	−45.5441	−1.48786	−0.743931	+	0.668256i	$0.767041\pi$
$938$	−27.8850	−0.910477
$939$	0	0
$940$	−96.5007	−3.14750
$941$	24.7833	0.807911	0.403956	−	0.914779i	$-0.367635\pi$
$941$	24.7833	0.807911	0.403956	+	0.914779i	$0.367635\pi$
$942$	0	0
$943$	48.8049	1.58931
$944$	16.4308	0.534777
$945$	0	0
$946$	25.4399	0.827124
$947$	−3.66692	−0.119159	−0.0595794	−	0.998224i	$-0.518976\pi$
$947$	−3.66692	−0.119159	−0.0595794	+	0.998224i	$0.518976\pi$
$948$	0	0
$949$	21.4505	0.696314
$950$	−129.096	−4.18844
$951$	0	0
$952$	−4.65476	−0.150862
$953$	55.9404	1.81209	0.906044	−	0.423184i	$-0.139088\pi$
$953$	55.9404	1.81209	0.906044	+	0.423184i	$0.139088\pi$
$954$	0	0
$955$	12.1968	0.394679
$956$	85.4845	2.76477
$957$	0	0
$958$	−91.9380	−2.97038
$959$	−24.1678	−0.780419
$960$	0	0
$961$	−27.6961	−0.893424
$962$	−25.8043	−0.831965
$963$	0	0
$964$	−128.111	−4.12618
$965$	−27.4751	−0.884454
$966$	0	0
$967$	40.3407	1.29727	0.648634	−	0.761100i	$-0.275341\pi$
$967$	40.3407	1.29727	0.648634	+	0.761100i	$0.275341\pi$
$968$	−6.70837	−0.215615
$969$	0	0
$970$	140.112	4.49873
$971$	−23.6883	−0.760193	−0.380096	−	0.924947i	$-0.624109\pi$
$971$	−23.6883	−0.760193	−0.380096	+	0.924947i	$0.624109\pi$
$972$	0	0
$973$	43.8476	1.40569
$974$	16.4338	0.526574
$975$	0	0
$976$	8.02760	0.256957
$977$	−10.2404	−0.327620	−0.163810	−	0.986492i	$-0.552378\pi$
$977$	−10.2404	−0.327620	−0.163810	+	0.986492i	$0.552378\pi$
$978$	0	0
$979$	6.05470	0.193509
$980$	52.2215	1.66816
$981$	0	0
$982$	43.1632	1.37739
$983$	26.3822	0.841461	0.420730	−	0.907186i	$-0.361774\pi$
$983$	26.3822	0.841461	0.420730	+	0.907186i	$0.361774\pi$
$984$	0	0
$985$	−48.6376	−1.54972
$986$	−3.74191	−0.119167
$987$	0	0
$988$	−99.4763	−3.16476
$989$	−47.0776	−1.49698
$990$	0	0
$991$	35.8606	1.13915	0.569575	−	0.821939i	$-0.307108\pi$
$991$	35.8606	1.13915	0.569575	+	0.821939i	$0.307108\pi$
$992$	−13.1256	−0.416738
$993$	0	0
$994$	60.3439	1.91399
$995$	55.3197	1.75375
$996$	0	0
$997$	7.80670	0.247241	0.123620	−	0.992330i	$-0.460550\pi$
$997$	7.80670	0.247241	0.123620	+	0.992330i	$0.460550\pi$
$998$	−6.15707	−0.194899
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6039.2.a.j.1.1		14
3.2	odd	2		2013.2.a.h.1.14	✓	14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2013.2.a.h.1.14	✓	14	3.2	odd	2
6039.2.a.j.1.1		14	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:	\( N \)	\(=\)	\( 6039 = 3^{2} \cdot 11 \cdot 61 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6039.a (trivial)

\(f(q)\)	\(=\)	\(q-2.57087 q^{2} +4.60938 q^{4} -4.00721 q^{5} +2.04273 q^{7} -6.70837 q^{8} +O(q^{10})\)	\(q-2.57087 q^{2} +4.60938 q^{4} -4.00721 q^{5} +2.04273 q^{7} -6.70837 q^{8} +10.3020 q^{10} +1.00000 q^{11} -4.75237 q^{13} -5.25159 q^{14} +8.02760 q^{16} +0.339679 q^{17} +4.54116 q^{19} -18.4707 q^{20} -2.57087 q^{22} +4.75749 q^{23} +11.0577 q^{25} +12.2177 q^{26} +9.41571 q^{28} +4.28494 q^{29} +1.81765 q^{31} -7.22118 q^{32} -0.873272 q^{34} -8.18565 q^{35} -2.11204 q^{37} -11.6747 q^{38} +26.8819 q^{40} +10.2585 q^{41} -9.89546 q^{43} +4.60938 q^{44} -12.2309 q^{46} +5.22451 q^{47} -2.82726 q^{49} -28.4280 q^{50} -21.9055 q^{52} -9.68455 q^{53} -4.00721 q^{55} -13.7034 q^{56} -11.0160 q^{58} +2.04679 q^{59} +1.00000 q^{61} -4.67295 q^{62} +2.50953 q^{64} +19.0437 q^{65} +5.30981 q^{67} +1.56571 q^{68} +21.0442 q^{70} -11.4906 q^{71} -4.51365 q^{73} +5.42978 q^{74} +20.9319 q^{76} +2.04273 q^{77} +5.49270 q^{79} -32.1683 q^{80} -26.3733 q^{82} -14.2207 q^{83} -1.36117 q^{85} +25.4399 q^{86} -6.70837 q^{88} +6.05470 q^{89} -9.70780 q^{91} +21.9291 q^{92} -13.4315 q^{94} -18.1974 q^{95} +13.6004 q^{97} +7.26851 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q + q^{2} + 15 q^{4} - q^{5} + 9 q^{7}+O(q^{10}) \) 14 * q + q^2 + 15 * q^4 - q^5 + 9 * q^7	\( 14 q + q^{2} + 15 q^{4} - q^{5} + 9 q^{7} + 6 q^{10} + 14 q^{11} + q^{13} + 7 q^{14} + 17 q^{16} + 9 q^{17} + 22 q^{19} - 23 q^{20} + q^{22} - q^{23} + 25 q^{25} - 4 q^{26} + 37 q^{28} + 6 q^{29} + 9 q^{31} - 4 q^{32} + 8 q^{34} - 18 q^{35} + 18 q^{37} - 8 q^{38} + 16 q^{40} + 25 q^{41} + 25 q^{43} + 15 q^{44} + 20 q^{46} - 36 q^{47} + 25 q^{49} - 2 q^{50} - 13 q^{52} - q^{55} + 40 q^{56} + 33 q^{58} - 17 q^{59} + 14 q^{61} + 13 q^{62} - 6 q^{64} + 61 q^{65} + 22 q^{67} - 66 q^{68} + 44 q^{70} + 13 q^{71} + 20 q^{73} + 12 q^{74} + 49 q^{76} + 9 q^{77} + 31 q^{79} - 88 q^{80} + 2 q^{82} - 32 q^{83} + 2 q^{85} + 14 q^{86} + 21 q^{89} + 45 q^{91} + 14 q^{92} - 31 q^{94} - 23 q^{95} + 37 q^{97} + 38 q^{98}+O(q^{100}) \) 14 * q + q^2 + 15 * q^4 - q^5 + 9 * q^7 + 6 * q^10 + 14 * q^11 + q^13 + 7 * q^14 + 17 * q^16 + 9 * q^17 + 22 * q^19 - 23 * q^20 + q^22 - q^23 + 25 * q^25 - 4 * q^26 + 37 * q^28 + 6 * q^29 + 9 * q^31 - 4 * q^32 + 8 * q^34 - 18 * q^35 + 18 * q^37 - 8 * q^38 + 16 * q^40 + 25 * q^41 + 25 * q^43 + 15 * q^44 + 20 * q^46 - 36 * q^47 + 25 * q^49 - 2 * q^50 - 13 * q^52 - q^55 + 40 * q^56 + 33 * q^58 - 17 * q^59 + 14 * q^61 + 13 * q^62 - 6 * q^64 + 61 * q^65 + 22 * q^67 - 66 * q^68 + 44 * q^70 + 13 * q^71 + 20 * q^73 + 12 * q^74 + 49 * q^76 + 9 * q^77 + 31 * q^79 - 88 * q^80 + 2 * q^82 - 32 * q^83 + 2 * q^85 + 14 * q^86 + 21 * q^89 + 45 * q^91 + 14 * q^92 - 31 * q^94 - 23 * q^95 + 37 * q^97 + 38 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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