LMFDB - Embedded newform 4014.2.a.v.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4014,2,Mod(1,4014)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4014, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("4014.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4014 = 2 \cdot 3^{2} \cdot 223 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4014.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:	$32.0519513713$
Analytic rank:	$0$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - 18x^{5} - 8x^{4} + 51x^{3} + 47x^{2} - 2x - 8 $ x^7 - 18x^5 - 8x^4 + 51x^3 + 47x^2 - 2*x - 8
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1338)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$-1.31546$ of defining polynomial
Character	$\chi$	$=$	4014.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} +1.60399 q^{5} -4.86544 q^{7} -1.00000 q^{8} +O(q^{10})$	\(q-1.00000 q^{2} +1.00000 q^{4} +1.60399 q^{5} -4.86544 q^{7} -1.00000 q^{8} -1.60399 q^{10} -1.44000 q^{11} -0.471330 q^{13} +4.86544 q^{14} +1.00000 q^{16} -5.16677 q^{17} +1.41454 q^{19} +1.60399 q^{20} +1.44000 q^{22} -6.51679 q^{23} -2.42720 q^{25} +0.471330 q^{26} -4.86544 q^{28} +1.09134 q^{29} +8.90666 q^{31} -1.00000 q^{32} +5.16677 q^{34} -7.80414 q^{35} +9.93358 q^{37} -1.41454 q^{38} -1.60399 q^{40} +4.74735 q^{41} -8.99506 q^{43} -1.44000 q^{44} +6.51679 q^{46} -10.5698 q^{47} +16.6725 q^{49} +2.42720 q^{50} -0.471330 q^{52} -6.81188 q^{53} -2.30975 q^{55} +4.86544 q^{56} -1.09134 q^{58} +3.13173 q^{59} +5.09134 q^{61} -8.90666 q^{62} +1.00000 q^{64} -0.756010 q^{65} -1.33867 q^{67} -5.16677 q^{68} +7.80414 q^{70} +16.5276 q^{71} +13.2485 q^{73} -9.93358 q^{74} +1.41454 q^{76} +7.00623 q^{77} -14.0259 q^{79} +1.60399 q^{80} -4.74735 q^{82} +5.99096 q^{83} -8.28747 q^{85} +8.99506 q^{86} +1.44000 q^{88} +1.98251 q^{89} +2.29323 q^{91} -6.51679 q^{92} +10.5698 q^{94} +2.26892 q^{95} +15.5440 q^{97} -16.6725 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q - 7 q^{2} + 7 q^{4} - 6 q^{5} + 3 q^{7} - 7 q^{8}+O(q^{10}) $ 7 * q - 7 * q^2 + 7 * q^4 - 6 * q^5 + 3 * q^7 - 7 * q^8	$ 7 q - 7 q^{2} + 7 q^{4} - 6 q^{5} + 3 q^{7} - 7 q^{8} + 6 q^{10} + q^{11} + 8 q^{13} - 3 q^{14} + 7 q^{16} - 16 q^{17} + 2 q^{19} - 6 q^{20} - q^{22} - 8 q^{23} + 19 q^{25} - 8 q^{26} + 3 q^{28} - 4 q^{29} + 11 q^{31} - 7 q^{32} + 16 q^{34} + 17 q^{37} - 2 q^{38} + 6 q^{40} - 18 q^{41} - q^{43} + q^{44} + 8 q^{46} - 5 q^{47} + 24 q^{49} - 19 q^{50} + 8 q^{52} - 6 q^{53} + 21 q^{55} - 3 q^{56} + 4 q^{58} + 7 q^{59} + 24 q^{61} - 11 q^{62} + 7 q^{64} - 11 q^{65} - 4 q^{67} - 16 q^{68} + 7 q^{71} + 28 q^{73} - 17 q^{74} + 2 q^{76} + 2 q^{77} - 6 q^{80} + 18 q^{82} + 2 q^{83} + 4 q^{85} + q^{86} - q^{88} - 5 q^{89} + 2 q^{91} - 8 q^{92} + 5 q^{94} + 14 q^{95} + 23 q^{97} - 24 q^{98}+O(q^{100}) $ 7 * q - 7 * q^2 + 7 * q^4 - 6 * q^5 + 3 * q^7 - 7 * q^8 + 6 * q^10 + q^11 + 8 * q^13 - 3 * q^14 + 7 * q^16 - 16 * q^17 + 2 * q^19 - 6 * q^20 - q^22 - 8 * q^23 + 19 * q^25 - 8 * q^26 + 3 * q^28 - 4 * q^29 + 11 * q^31 - 7 * q^32 + 16 * q^34 + 17 * q^37 - 2 * q^38 + 6 * q^40 - 18 * q^41 - q^43 + q^44 + 8 * q^46 - 5 * q^47 + 24 * q^49 - 19 * q^50 + 8 * q^52 - 6 * q^53 + 21 * q^55 - 3 * q^56 + 4 * q^58 + 7 * q^59 + 24 * q^61 - 11 * q^62 + 7 * q^64 - 11 * q^65 - 4 * q^67 - 16 * q^68 + 7 * q^71 + 28 * q^73 - 17 * q^74 + 2 * q^76 + 2 * q^77 - 6 * q^80 + 18 * q^82 + 2 * q^83 + 4 * q^85 + q^86 - q^88 - 5 * q^89 + 2 * q^91 - 8 * q^92 + 5 * q^94 + 14 * q^95 + 23 * q^97 - 24 * 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 $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	1.60399	0.717328	0.358664	−	0.933467i	$-0.383232\pi$
$5$	1.60399	0.717328	0.358664	+	0.933467i	$0.383232\pi$
$6$	0	0
$7$	−4.86544	−1.83896	−0.919482	−	0.393132i	$-0.871392\pi$
$7$	−4.86544	−1.83896	−0.919482	+	0.393132i	$0.871392\pi$
$8$	−1.00000	−0.353553
$9$	0	0
$10$	−1.60399	−0.507227
$11$	−1.44000	−0.434176	−0.217088	−	0.976152i	$-0.569656\pi$
$11$	−1.44000	−0.434176	−0.217088	+	0.976152i	$0.569656\pi$
$12$	0	0
$13$	−0.471330	−0.130723	−0.0653617	−	0.997862i	$-0.520820\pi$
$13$	−0.471330	−0.130723	−0.0653617	+	0.997862i	$0.520820\pi$
$14$	4.86544	1.30034
$15$	0	0
$16$	1.00000	0.250000
$17$	−5.16677	−1.25313	−0.626563	−	0.779371i	$-0.715539\pi$
$17$	−5.16677	−1.25313	−0.626563	+	0.779371i	$0.715539\pi$
$18$	0	0
$19$	1.41454	0.324518	0.162259	−	0.986748i	$-0.448122\pi$
$19$	1.41454	0.324518	0.162259	+	0.986748i	$0.448122\pi$
$20$	1.60399	0.358664
$21$	0	0
$22$	1.44000	0.307009
$23$	−6.51679	−1.35884	−0.679422	−	0.733748i	$-0.737769\pi$
$23$	−6.51679	−1.35884	−0.679422	+	0.733748i	$0.737769\pi$
$24$	0	0
$25$	−2.42720	−0.485441
$26$	0.471330	0.0924353
$27$	0	0
$28$	−4.86544	−0.919482
$29$	1.09134	0.202657	0.101329	−	0.994853i	$-0.467691\pi$
$29$	1.09134	0.202657	0.101329	+	0.994853i	$0.467691\pi$
$30$	0	0
$31$	8.90666	1.59968	0.799841	−	0.600211i	$-0.204917\pi$
$31$	8.90666	1.59968	0.799841	+	0.600211i	$0.204917\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	5.16677	0.886094
$35$	−7.80414	−1.31914
$36$	0	0
$37$	9.93358	1.63307	0.816535	−	0.577296i	$-0.195892\pi$
$37$	9.93358	1.63307	0.816535	+	0.577296i	$0.195892\pi$
$38$	−1.41454	−0.229469
$39$	0	0
$40$	−1.60399	−0.253614
$41$	4.74735	0.741412	0.370706	−	0.928750i	$-0.379116\pi$
$41$	4.74735	0.741412	0.370706	+	0.928750i	$0.379116\pi$
$42$	0	0
$43$	−8.99506	−1.37173	−0.685867	−	0.727727i	$-0.740577\pi$
$43$	−8.99506	−1.37173	−0.685867	+	0.727727i	$0.740577\pi$
$44$	−1.44000	−0.217088
$45$	0	0
$46$	6.51679	0.960848
$47$	−10.5698	−1.54177	−0.770884	−	0.636975i	$-0.780185\pi$
$47$	−10.5698	−1.54177	−0.770884	+	0.636975i	$0.780185\pi$
$48$	0	0
$49$	16.6725	2.38179
$50$	2.42720	0.343258
$51$	0	0
$52$	−0.471330	−0.0653617
$53$	−6.81188	−0.935684	−0.467842	−	0.883812i	$-0.654968\pi$
$53$	−6.81188	−0.935684	−0.467842	+	0.883812i	$0.654968\pi$
$54$	0	0
$55$	−2.30975	−0.311446
$56$	4.86544	0.650172
$57$	0	0
$58$	−1.09134	−0.143300
$59$	3.13173	0.407716	0.203858	−	0.979000i	$-0.434652\pi$
$59$	3.13173	0.407716	0.203858	+	0.979000i	$0.434652\pi$
$60$	0	0
$61$	5.09134	0.651880	0.325940	−	0.945390i	$-0.394319\pi$
$61$	5.09134	0.651880	0.325940	+	0.945390i	$0.394319\pi$
$62$	−8.90666	−1.13115
$63$	0	0
$64$	1.00000	0.125000
$65$	−0.756010	−0.0937715
$66$	0	0
$67$	−1.33867	−0.163544	−0.0817720	−	0.996651i	$-0.526058\pi$
$67$	−1.33867	−0.163544	−0.0817720	+	0.996651i	$0.526058\pi$
$68$	−5.16677	−0.626563
$69$	0	0
$70$	7.80414	0.932773
$71$	16.5276	1.96146	0.980730	−	0.195368i	$-0.0625902\pi$
$71$	16.5276	1.96146	0.980730	+	0.195368i	$0.0625902\pi$
$72$	0	0
$73$	13.2485	1.55063	0.775313	−	0.631578i	$-0.217592\pi$
$73$	13.2485	1.55063	0.775313	+	0.631578i	$0.217592\pi$
$74$	−9.93358	−1.15475
$75$	0	0
$76$	1.41454	0.162259
$77$	7.00623	0.798434
$78$	0	0
$79$	−14.0259	−1.57803	−0.789017	−	0.614372i	$-0.789410\pi$
$79$	−14.0259	−1.57803	−0.789017	+	0.614372i	$0.789410\pi$
$80$	1.60399	0.179332
$81$	0	0
$82$	−4.74735	−0.524257
$83$	5.99096	0.657593	0.328797	−	0.944401i	$-0.393357\pi$
$83$	5.99096	0.657593	0.328797	+	0.944401i	$0.393357\pi$
$84$	0	0
$85$	−8.28747	−0.898903
$86$	8.99506	0.969962
$87$	0	0
$88$	1.44000	0.153504
$89$	1.98251	0.210146	0.105073	−	0.994465i	$-0.466492\pi$
$89$	1.98251	0.210146	0.105073	+	0.994465i	$0.466492\pi$
$90$	0	0
$91$	2.29323	0.240395
$92$	−6.51679	−0.679422
$93$	0	0
$94$	10.5698	1.09020
$95$	2.26892	0.232786
$96$	0	0
$97$	15.5440	1.57826	0.789128	−	0.614229i	$-0.210533\pi$
$97$	15.5440	1.57826	0.789128	+	0.614229i	$0.210533\pi$
$98$	−16.6725	−1.68418
$99$	0	0
$100$	−2.42720	−0.242720
$101$	−7.82134	−0.778253	−0.389126	−	0.921184i	$-0.627223\pi$
$101$	−7.82134	−0.778253	−0.389126	+	0.921184i	$0.627223\pi$
$102$	0	0
$103$	8.70624	0.857851	0.428926	−	0.903340i	$-0.358892\pi$
$103$	8.70624	0.857851	0.428926	+	0.903340i	$0.358892\pi$
$104$	0.471330	0.0462177
$105$	0	0
$106$	6.81188	0.661628
$107$	0.0236553	0.00228685	0.00114342	−	0.999999i	$-0.499636\pi$
$107$	0.0236553	0.00228685	0.00114342	+	0.999999i	$0.499636\pi$
$108$	0	0
$109$	8.56901	0.820762	0.410381	−	0.911914i	$-0.365396\pi$
$109$	8.56901	0.820762	0.410381	+	0.911914i	$0.365396\pi$
$110$	2.30975	0.220226
$111$	0	0
$112$	−4.86544	−0.459741
$113$	−1.70777	−0.160653	−0.0803267	−	0.996769i	$-0.525596\pi$
$113$	−1.70777	−0.160653	−0.0803267	+	0.996769i	$0.525596\pi$
$114$	0	0
$115$	−10.4529	−0.974737
$116$	1.09134	0.101329
$117$	0	0
$118$	−3.13173	−0.288299
$119$	25.1386	2.30445
$120$	0	0
$121$	−8.92640	−0.811491
$122$	−5.09134	−0.460949
$123$	0	0
$124$	8.90666	0.799841
$125$	−11.9132	−1.06555
$126$	0	0
$127$	3.99904	0.354857	0.177429	−	0.984134i	$-0.443222\pi$
$127$	3.99904	0.354857	0.177429	+	0.984134i	$0.443222\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	0.756010	0.0663064
$131$	20.1830	1.76340	0.881700	−	0.471810i	$-0.156399\pi$
$131$	20.1830	1.76340	0.881700	+	0.471810i	$0.156399\pi$
$132$	0	0
$133$	−6.88237	−0.596777
$134$	1.33867	0.115643
$135$	0	0
$136$	5.16677	0.443047
$137$	5.33292	0.455622	0.227811	−	0.973705i	$-0.426843\pi$
$137$	5.33292	0.455622	0.227811	+	0.973705i	$0.426843\pi$
$138$	0	0
$139$	18.9900	1.61071	0.805353	−	0.592795i	$-0.201976\pi$
$139$	18.9900	1.61071	0.805353	+	0.592795i	$0.201976\pi$
$140$	−7.80414	−0.659570
$141$	0	0
$142$	−16.5276	−1.38696
$143$	0.678714	0.0567569
$144$	0	0
$145$	1.75051	0.145372
$146$	−13.2485	−1.09646
$147$	0	0
$148$	9.93358	0.816535
$149$	10.2096	0.836400	0.418200	−	0.908355i	$-0.362661\pi$
$149$	10.2096	0.836400	0.418200	+	0.908355i	$0.362661\pi$
$150$	0	0
$151$	0.0580448	0.00472362	0.00236181	−	0.999997i	$-0.499248\pi$
$151$	0.0580448	0.00472362	0.00236181	+	0.999997i	$0.499248\pi$
$152$	−1.41454	−0.114735
$153$	0	0
$154$	−7.00623	−0.564578
$155$	14.2862	1.14750
$156$	0	0
$157$	−5.77444	−0.460850	−0.230425	−	0.973090i	$-0.574012\pi$
$157$	−5.77444	−0.460850	−0.230425	+	0.973090i	$0.574012\pi$
$158$	14.0259	1.11584
$159$	0	0
$160$	−1.60399	−0.126807
$161$	31.7070	2.49887
$162$	0	0
$163$	−1.20564	−0.0944329	−0.0472165	−	0.998885i	$-0.515035\pi$
$163$	−1.20564	−0.0944329	−0.0472165	+	0.998885i	$0.515035\pi$
$164$	4.74735	0.370706
$165$	0	0
$166$	−5.99096	−0.464989
$167$	−15.7942	−1.22220	−0.611098	−	0.791555i	$-0.709272\pi$
$167$	−15.7942	−1.22220	−0.611098	+	0.791555i	$0.709272\pi$
$168$	0	0
$169$	−12.7778	−0.982911
$170$	8.28747	0.635620
$171$	0	0
$172$	−8.99506	−0.685867
$173$	−6.15714	−0.468119	−0.234059	−	0.972222i	$-0.575201\pi$
$173$	−6.15714	−0.468119	−0.234059	+	0.972222i	$0.575201\pi$
$174$	0	0
$175$	11.8094	0.892708
$176$	−1.44000	−0.108544
$177$	0	0
$178$	−1.98251	−0.148596
$179$	−19.9049	−1.48776	−0.743881	−	0.668313i	$-0.767017\pi$
$179$	−19.9049	−1.48776	−0.743881	+	0.668313i	$0.767017\pi$
$180$	0	0
$181$	23.0165	1.71080	0.855400	−	0.517967i	$-0.173311\pi$
$181$	23.0165	1.71080	0.855400	+	0.517967i	$0.173311\pi$
$182$	−2.29323	−0.169985
$183$	0	0
$184$	6.51679	0.480424
$185$	15.9334	1.17145
$186$	0	0
$187$	7.44015	0.544077
$188$	−10.5698	−0.770884
$189$	0	0
$190$	−2.26892	−0.164605
$191$	−12.6440	−0.914887	−0.457444	−	0.889239i	$-0.651235\pi$
$191$	−12.6440	−0.914887	−0.457444	+	0.889239i	$0.651235\pi$
$192$	0	0
$193$	7.48011	0.538430	0.269215	−	0.963080i	$-0.413236\pi$
$193$	7.48011	0.538430	0.269215	+	0.963080i	$0.413236\pi$
$194$	−15.5440	−1.11600
$195$	0	0
$196$	16.6725	1.19089
$197$	4.02066	0.286460	0.143230	−	0.989689i	$-0.454251\pi$
$197$	4.02066	0.286460	0.143230	+	0.989689i	$0.454251\pi$
$198$	0	0
$199$	5.12390	0.363224	0.181612	−	0.983370i	$-0.441869\pi$
$199$	5.12390	0.363224	0.181612	+	0.983370i	$0.441869\pi$
$200$	2.42720	0.171629
$201$	0	0
$202$	7.82134	0.550308
$203$	−5.30987	−0.372680
$204$	0	0
$205$	7.61472	0.531835
$206$	−8.70624	−0.606592
$207$	0	0
$208$	−0.471330	−0.0326808
$209$	−2.03694	−0.140898
$210$	0	0
$211$	−12.8974	−0.887897	−0.443949	−	0.896052i	$-0.646423\pi$
$211$	−12.8974	−0.887897	−0.443949	+	0.896052i	$0.646423\pi$
$212$	−6.81188	−0.467842
$213$	0	0
$214$	−0.0236553	−0.00161704
$215$	−14.4280	−0.983982
$216$	0	0
$217$	−43.3348	−2.94176
$218$	−8.56901	−0.580367
$219$	0	0
$220$	−2.30975	−0.155723
$221$	2.43525	0.163813
$222$	0	0
$223$	−1.00000	−0.0669650
$223$	−1.00000	−0.0669650
$224$	4.86544	0.325086
$225$	0	0
$226$	1.70777	0.113599
$227$	23.7498	1.57633	0.788165	−	0.615463i	$-0.211031\pi$
$227$	23.7498	1.57633	0.788165	+	0.615463i	$0.211031\pi$
$228$	0	0
$229$	22.4453	1.48322	0.741612	−	0.670829i	$-0.234061\pi$
$229$	22.4453	1.48322	0.741612	+	0.670829i	$0.234061\pi$
$230$	10.4529	0.689243
$231$	0	0
$232$	−1.09134	−0.0716502
$233$	−9.22110	−0.604094	−0.302047	−	0.953293i	$-0.597670\pi$
$233$	−9.22110	−0.604094	−0.302047	+	0.953293i	$0.597670\pi$
$234$	0	0
$235$	−16.9539	−1.10595
$236$	3.13173	0.203858
$237$	0	0
$238$	−25.1386	−1.62950
$239$	−3.77220	−0.244003	−0.122002	−	0.992530i	$-0.538931\pi$
$239$	−3.77220	−0.244003	−0.122002	+	0.992530i	$0.538931\pi$
$240$	0	0
$241$	−23.5440	−1.51660	−0.758302	−	0.651903i	$-0.773971\pi$
$241$	−23.5440	−1.51660	−0.758302	+	0.651903i	$0.773971\pi$
$242$	8.92640	0.573811
$243$	0	0
$244$	5.09134	0.325940
$245$	26.7426	1.70852
$246$	0	0
$247$	−0.666716	−0.0424221
$248$	−8.90666	−0.565573
$249$	0	0
$250$	11.9132	0.753456
$251$	9.77531	0.617012	0.308506	−	0.951222i	$-0.400171\pi$
$251$	9.77531	0.617012	0.308506	+	0.951222i	$0.400171\pi$
$252$	0	0
$253$	9.38416	0.589977
$254$	−3.99904	−0.250922
$255$	0	0
$256$	1.00000	0.0625000
$257$	−24.5706	−1.53267	−0.766336	−	0.642440i	$-0.777922\pi$
$257$	−24.5706	−1.53267	−0.766336	+	0.642440i	$0.777922\pi$
$258$	0	0
$259$	−48.3312	−3.00316
$260$	−0.756010	−0.0468857
$261$	0	0
$262$	−20.1830	−1.24691
$263$	7.34923	0.453173	0.226586	−	0.973991i	$-0.427243\pi$
$263$	7.34923	0.453173	0.226586	+	0.973991i	$0.427243\pi$
$264$	0	0
$265$	−10.9262	−0.671192
$266$	6.88237	0.421985
$267$	0	0
$268$	−1.33867	−0.0817720
$269$	26.9039	1.64036	0.820179	−	0.572107i	$-0.193874\pi$
$269$	26.9039	1.64036	0.820179	+	0.572107i	$0.193874\pi$
$270$	0	0
$271$	−26.8892	−1.63340	−0.816701	−	0.577060i	$-0.804200\pi$
$271$	−26.8892	−1.63340	−0.816701	+	0.577060i	$0.804200\pi$
$272$	−5.16677	−0.313282
$273$	0	0
$274$	−5.33292	−0.322173
$275$	3.49517	0.210767
$276$	0	0
$277$	23.9665	1.44001	0.720004	−	0.693970i	$-0.244140\pi$
$277$	23.9665	1.44001	0.720004	+	0.693970i	$0.244140\pi$
$278$	−18.9900	−1.13894
$279$	0	0
$280$	7.80414	0.466387
$281$	−8.91839	−0.532026	−0.266013	−	0.963969i	$-0.585706\pi$
$281$	−8.91839	−0.532026	−0.266013	+	0.963969i	$0.585706\pi$
$282$	0	0
$283$	8.76145	0.520814	0.260407	−	0.965499i	$-0.416143\pi$
$283$	8.76145	0.520814	0.260407	+	0.965499i	$0.416143\pi$
$284$	16.5276	0.980730
$285$	0	0
$286$	−0.678714	−0.0401332
$287$	−23.0980	−1.36343
$288$	0	0
$289$	9.69554	0.570326
$290$	−1.75051	−0.102793
$291$	0	0
$292$	13.2485	0.775313
$293$	14.9430	0.872978	0.436489	−	0.899710i	$-0.356222\pi$
$293$	14.9430	0.872978	0.436489	+	0.899710i	$0.356222\pi$
$294$	0	0
$295$	5.02327	0.292466
$296$	−9.93358	−0.577377
$297$	0	0
$298$	−10.2096	−0.591424
$299$	3.07155	0.177633
$300$	0	0
$301$	43.7649	2.52257
$302$	−0.0580448	−0.00334011
$303$	0	0
$304$	1.41454	0.0811296
$305$	8.16648	0.467612
$306$	0	0
$307$	−15.0546	−0.859209	−0.429604	−	0.903017i	$-0.641347\pi$
$307$	−15.0546	−0.859209	−0.429604	+	0.903017i	$0.641347\pi$
$308$	7.00623	0.399217
$309$	0	0
$310$	−14.2862	−0.811403
$311$	−17.5808	−0.996913	−0.498457	−	0.866915i	$-0.666100\pi$
$311$	−17.5808	−0.996913	−0.498457	+	0.866915i	$0.666100\pi$
$312$	0	0
$313$	13.3753	0.756019	0.378009	−	0.925802i	$-0.376609\pi$
$313$	13.3753	0.756019	0.378009	+	0.925802i	$0.376609\pi$
$314$	5.77444	0.325870
$315$	0	0
$316$	−14.0259	−0.789017
$317$	5.12331	0.287754	0.143877	−	0.989596i	$-0.454043\pi$
$317$	5.12331	0.287754	0.143877	+	0.989596i	$0.454043\pi$
$318$	0	0
$319$	−1.57153	−0.0879890
$320$	1.60399	0.0896660
$321$	0	0
$322$	−31.7070	−1.76696
$323$	−7.30862	−0.406662
$324$	0	0
$325$	1.14401	0.0634584
$326$	1.20564	0.0667741
$327$	0	0
$328$	−4.74735	−0.262129
$329$	51.4269	2.83526
$330$	0	0
$331$	−14.5745	−0.801088	−0.400544	−	0.916277i	$-0.631179\pi$
$331$	−14.5745	−0.801088	−0.400544	+	0.916277i	$0.631179\pi$
$332$	5.99096	0.328797
$333$	0	0
$334$	15.7942	0.864222
$335$	−2.14721	−0.117315
$336$	0	0
$337$	1.63369	0.0889925	0.0444963	−	0.999010i	$-0.485832\pi$
$337$	1.63369	0.0889925	0.0444963	+	0.999010i	$0.485832\pi$
$338$	12.7778	0.695023
$339$	0	0
$340$	−8.28747	−0.449451
$341$	−12.8256	−0.694544
$342$	0	0
$343$	−47.0611	−2.54106
$344$	8.99506	0.484981
$345$	0	0
$346$	6.15714	0.331010
$347$	21.0414	1.12956	0.564781	−	0.825241i	$-0.308961\pi$
$347$	21.0414	1.12956	0.564781	+	0.825241i	$0.308961\pi$
$348$	0	0
$349$	−4.82308	−0.258174	−0.129087	−	0.991633i	$-0.541205\pi$
$349$	−4.82308	−0.258174	−0.129087	+	0.991633i	$0.541205\pi$
$350$	−11.8094	−0.631240
$351$	0	0
$352$	1.44000	0.0767522
$353$	19.6185	1.04419	0.522093	−	0.852888i	$-0.325151\pi$
$353$	19.6185	1.04419	0.522093	+	0.852888i	$0.325151\pi$
$354$	0	0
$355$	26.5101	1.40701
$356$	1.98251	0.105073
$357$	0	0
$358$	19.9049	1.05201
$359$	2.71271	0.143171	0.0715856	−	0.997434i	$-0.477194\pi$
$359$	2.71271	0.143171	0.0715856	+	0.997434i	$0.477194\pi$
$360$	0	0
$361$	−16.9991	−0.894688
$362$	−23.0165	−1.20972
$363$	0	0
$364$	2.29323	0.120198
$365$	21.2506	1.11231
$366$	0	0
$367$	27.3420	1.42724	0.713622	−	0.700531i	$-0.247054\pi$
$367$	27.3420	1.42724	0.713622	+	0.700531i	$0.247054\pi$
$368$	−6.51679	−0.339711
$369$	0	0
$370$	−15.9334	−0.828338
$371$	33.1428	1.72069
$372$	0	0
$373$	0.178537	0.00924431	0.00462215	−	0.999989i	$-0.498529\pi$
$373$	0.178537	0.00924431	0.00462215	+	0.999989i	$0.498529\pi$
$374$	−7.44015	−0.384721
$375$	0	0
$376$	10.5698	0.545098
$377$	−0.514383	−0.0264920
$378$	0	0
$379$	25.1988	1.29437	0.647186	−	0.762332i	$-0.275945\pi$
$379$	25.1988	1.29437	0.647186	+	0.762332i	$0.275945\pi$
$380$	2.26892	0.116393
$381$	0	0
$382$	12.6440	0.646923
$383$	24.7131	1.26278	0.631390	−	0.775465i	$-0.282485\pi$
$383$	24.7131	1.26278	0.631390	+	0.775465i	$0.282485\pi$
$384$	0	0
$385$	11.2379	0.572739
$386$	−7.48011	−0.380728
$387$	0	0
$388$	15.5440	0.789128
$389$	5.90447	0.299369	0.149684	−	0.988734i	$-0.452174\pi$
$389$	5.90447	0.299369	0.149684	+	0.988734i	$0.452174\pi$
$390$	0	0
$391$	33.6708	1.70280
$392$	−16.6725	−0.842090
$393$	0	0
$394$	−4.02066	−0.202558
$395$	−22.4974	−1.13197
$396$	0	0
$397$	32.0893	1.61052	0.805259	−	0.592923i	$-0.202026\pi$
$397$	32.0893	1.61052	0.805259	+	0.592923i	$0.202026\pi$
$398$	−5.12390	−0.256838
$399$	0	0
$400$	−2.42720	−0.121360
$401$	−15.3722	−0.767652	−0.383826	−	0.923405i	$-0.625394\pi$
$401$	−15.3722	−0.767652	−0.383826	+	0.923405i	$0.625394\pi$
$402$	0	0
$403$	−4.19797	−0.209116
$404$	−7.82134	−0.389126
$405$	0	0
$406$	5.30987	0.263524
$407$	−14.3043	−0.709040
$408$	0	0
$409$	4.14741	0.205076	0.102538	−	0.994729i	$-0.467304\pi$
$409$	4.14741	0.205076	0.102538	+	0.994729i	$0.467304\pi$
$410$	−7.61472	−0.376064
$411$	0	0
$412$	8.70624	0.428926
$413$	−15.2372	−0.749776
$414$	0	0
$415$	9.60946	0.471710
$416$	0.471330	0.0231088
$417$	0	0
$418$	2.03694	0.0996299
$419$	26.2690	1.28333	0.641663	−	0.766987i	$-0.278245\pi$
$419$	26.2690	1.28333	0.641663	+	0.766987i	$0.278245\pi$
$420$	0	0
$421$	−9.17005	−0.446921	−0.223460	−	0.974713i	$-0.571735\pi$
$421$	−9.17005	−0.446921	−0.223460	+	0.974713i	$0.571735\pi$
$422$	12.8974	0.627838
$423$	0	0
$424$	6.81188	0.330814
$425$	12.5408	0.608319
$426$	0	0
$427$	−24.7716	−1.19878
$428$	0.0236553	0.00114342
$429$	0	0
$430$	14.4280	0.695781
$431$	−19.2622	−0.927827	−0.463913	−	0.885881i	$-0.653555\pi$
$431$	−19.2622	−0.927827	−0.463913	+	0.885881i	$0.653555\pi$
$432$	0	0
$433$	30.6252	1.47175	0.735877	−	0.677115i	$-0.236770\pi$
$433$	30.6252	1.47175	0.735877	+	0.677115i	$0.236770\pi$
$434$	43.3348	2.08014
$435$	0	0
$436$	8.56901	0.410381
$437$	−9.21827	−0.440970
$438$	0	0
$439$	−17.2313	−0.822407	−0.411204	−	0.911544i	$-0.634891\pi$
$439$	−17.2313	−0.822407	−0.411204	+	0.911544i	$0.634891\pi$
$440$	2.30975	0.110113
$441$	0	0
$442$	−2.43525	−0.115833
$443$	15.6230	0.742273	0.371136	−	0.928578i	$-0.378968\pi$
$443$	15.6230	0.742273	0.371136	+	0.928578i	$0.378968\pi$
$444$	0	0
$445$	3.17994	0.150744
$446$	1.00000	0.0473514
$447$	0	0
$448$	−4.86544	−0.229871
$449$	17.3813	0.820276	0.410138	−	0.912024i	$-0.365481\pi$
$449$	17.3813	0.820276	0.410138	+	0.912024i	$0.365481\pi$
$450$	0	0
$451$	−6.83618	−0.321903
$452$	−1.70777	−0.0803267
$453$	0	0
$454$	−23.7498	−1.11463
$455$	3.67832	0.172442
$456$	0	0
$457$	36.3057	1.69831	0.849156	−	0.528143i	$-0.177111\pi$
$457$	36.3057	1.69831	0.849156	+	0.528143i	$0.177111\pi$
$458$	−22.4453	−1.04880
$459$	0	0
$460$	−10.4529	−0.487368
$461$	−2.29878	−0.107065	−0.0535323	−	0.998566i	$-0.517048\pi$
$461$	−2.29878	−0.107065	−0.0535323	+	0.998566i	$0.517048\pi$
$462$	0	0
$463$	14.7034	0.683327	0.341663	−	0.939822i	$-0.389010\pi$
$463$	14.7034	0.683327	0.341663	+	0.939822i	$0.389010\pi$
$464$	1.09134	0.0506644
$465$	0	0
$466$	9.22110	0.427159
$467$	−18.6263	−0.861921	−0.430961	−	0.902371i	$-0.641825\pi$
$467$	−18.6263	−0.861921	−0.430961	+	0.902371i	$0.641825\pi$
$468$	0	0
$469$	6.51320	0.300752
$470$	16.9539	0.782027
$471$	0	0
$472$	−3.13173	−0.144149
$473$	12.9529	0.595573
$474$	0	0
$475$	−3.43338	−0.157534
$476$	25.1386	1.15223
$477$	0	0
$478$	3.77220	0.172536
$479$	−14.3139	−0.654017	−0.327008	−	0.945021i	$-0.606041\pi$
$479$	−14.3139	−0.654017	−0.327008	+	0.945021i	$0.606041\pi$
$480$	0	0
$481$	−4.68199	−0.213480
$482$	23.5440	1.07240
$483$	0	0
$484$	−8.92640	−0.405746
$485$	24.9325	1.13213
$486$	0	0
$487$	16.2734	0.737418	0.368709	−	0.929545i	$-0.379800\pi$
$487$	16.2734	0.737418	0.368709	+	0.929545i	$0.379800\pi$
$488$	−5.09134	−0.230474
$489$	0	0
$490$	−26.7426	−1.20811
$491$	−21.7849	−0.983138	−0.491569	−	0.870839i	$-0.663576\pi$
$491$	−21.7849	−0.983138	−0.491569	+	0.870839i	$0.663576\pi$
$492$	0	0
$493$	−5.63872	−0.253955
$494$	0.666716	0.0299970
$495$	0	0
$496$	8.90666	0.399921
$497$	−80.4139	−3.60705
$498$	0	0
$499$	20.1791	0.903341	0.451671	−	0.892185i	$-0.350828\pi$
$499$	20.1791	0.903341	0.451671	+	0.892185i	$0.350828\pi$
$500$	−11.9132	−0.532774
$501$	0	0
$502$	−9.77531	−0.436293
$503$	−12.7329	−0.567733	−0.283866	−	0.958864i	$-0.591617\pi$
$503$	−12.7329	−0.567733	−0.283866	+	0.958864i	$0.591617\pi$
$504$	0	0
$505$	−12.5454	−0.558262
$506$	−9.38416	−0.417177
$507$	0	0
$508$	3.99904	0.177429
$509$	20.0279	0.887720	0.443860	−	0.896096i	$-0.353609\pi$
$509$	20.0279	0.887720	0.443860	+	0.896096i	$0.353609\pi$
$510$	0	0
$511$	−64.4600	−2.85154
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	24.5706	1.08376
$515$	13.9648	0.615361
$516$	0	0
$517$	15.2205	0.669399
$518$	48.3312	2.12355
$519$	0	0
$520$	0.756010	0.0331532
$521$	10.4503	0.457834	0.228917	−	0.973446i	$-0.426482\pi$
$521$	10.4503	0.457834	0.228917	+	0.973446i	$0.426482\pi$
$522$	0	0
$523$	−32.9255	−1.43973	−0.719866	−	0.694113i	$-0.755797\pi$
$523$	−32.9255	−1.43973	−0.719866	+	0.694113i	$0.755797\pi$
$524$	20.1830	0.881700
$525$	0	0
$526$	−7.34923	−0.320442
$527$	−46.0187	−2.00460
$528$	0	0
$529$	19.4685	0.846457
$530$	10.9262	0.474604
$531$	0	0
$532$	−6.88237	−0.298389
$533$	−2.23757	−0.0969198
$534$	0	0
$535$	0.0379430	0.00164042
$536$	1.33867	0.0578215
$537$	0	0
$538$	−26.9039	−1.15991
$539$	−24.0084	−1.03412
$540$	0	0
$541$	−9.84978	−0.423475	−0.211737	−	0.977327i	$-0.567912\pi$
$541$	−9.84978	−0.423475	−0.211737	+	0.977327i	$0.567912\pi$
$542$	26.8892	1.15499
$543$	0	0
$544$	5.16677	0.221524
$545$	13.7446	0.588756
$546$	0	0
$547$	40.3280	1.72430	0.862151	−	0.506651i	$-0.169117\pi$
$547$	40.3280	1.72430	0.862151	+	0.506651i	$0.169117\pi$
$548$	5.33292	0.227811
$549$	0	0
$550$	−3.49517	−0.149035
$551$	1.54375	0.0657660
$552$	0	0
$553$	68.2421	2.90195
$554$	−23.9665	−1.01824
$555$	0	0
$556$	18.9900	0.805353
$557$	33.2417	1.40850	0.704249	−	0.709953i	$-0.251284\pi$
$557$	33.2417	1.40850	0.704249	+	0.709953i	$0.251284\pi$
$558$	0	0
$559$	4.23964	0.179317
$560$	−7.80414	−0.329785
$561$	0	0
$562$	8.91839	0.376199
$563$	31.0168	1.30720	0.653602	−	0.756839i	$-0.273257\pi$
$563$	31.0168	1.30720	0.653602	+	0.756839i	$0.273257\pi$
$564$	0	0
$565$	−2.73925	−0.115241
$566$	−8.76145	−0.368271
$567$	0	0
$568$	−16.5276	−0.693481
$569$	42.3943	1.77726	0.888630	−	0.458625i	$-0.151658\pi$
$569$	42.3943	1.77726	0.888630	+	0.458625i	$0.151658\pi$
$570$	0	0
$571$	−31.4331	−1.31543	−0.657717	−	0.753265i	$-0.728478\pi$
$571$	−31.4331	−1.31543	−0.657717	+	0.753265i	$0.728478\pi$
$572$	0.678714	0.0283785
$573$	0	0
$574$	23.0980	0.964090
$575$	15.8176	0.659638
$576$	0	0
$577$	−3.26694	−0.136004	−0.0680022	−	0.997685i	$-0.521662\pi$
$577$	−3.26694	−0.136004	−0.0680022	+	0.997685i	$0.521662\pi$
$578$	−9.69554	−0.403281
$579$	0	0
$580$	1.75051	0.0726859
$581$	−29.1487	−1.20929
$582$	0	0
$583$	9.80910	0.406251
$584$	−13.2485	−0.548229
$585$	0	0
$586$	−14.9430	−0.617289
$587$	−2.88389	−0.119031	−0.0595155	−	0.998227i	$-0.518956\pi$
$587$	−2.88389	−0.119031	−0.0595155	+	0.998227i	$0.518956\pi$
$588$	0	0
$589$	12.5988	0.519126
$590$	−5.02327	−0.206805
$591$	0	0
$592$	9.93358	0.408268
$593$	−18.3304	−0.752739	−0.376370	−	0.926470i	$-0.622828\pi$
$593$	−18.3304	−0.752739	−0.376370	+	0.926470i	$0.622828\pi$
$594$	0	0
$595$	40.3222	1.65305
$596$	10.2096	0.418200
$597$	0	0
$598$	−3.07155	−0.125605
$599$	−16.9619	−0.693046	−0.346523	−	0.938041i	$-0.612638\pi$
$599$	−16.9619	−0.693046	−0.346523	+	0.938041i	$0.612638\pi$
$600$	0	0
$601$	15.4554	0.630440	0.315220	−	0.949019i	$-0.397922\pi$
$601$	15.4554	0.630440	0.315220	+	0.949019i	$0.397922\pi$
$602$	−43.7649	−1.78373
$603$	0	0
$604$	0.0580448	0.00236181
$605$	−14.3179	−0.582105
$606$	0	0
$607$	8.86403	0.359780	0.179890	−	0.983687i	$-0.442426\pi$
$607$	8.86403	0.359780	0.179890	+	0.983687i	$0.442426\pi$
$608$	−1.41454	−0.0573673
$609$	0	0
$610$	−8.16648	−0.330651
$611$	4.98188	0.201545
$612$	0	0
$613$	−16.8152	−0.679159	−0.339580	−	0.940577i	$-0.610285\pi$
$613$	−16.8152	−0.679159	−0.339580	+	0.940577i	$0.610285\pi$
$614$	15.0546	0.607552
$615$	0	0
$616$	−7.00623	−0.282289
$617$	−22.9365	−0.923389	−0.461694	−	0.887039i	$-0.652758\pi$
$617$	−22.9365	−0.923389	−0.461694	+	0.887039i	$0.652758\pi$
$618$	0	0
$619$	41.0803	1.65115	0.825577	−	0.564289i	$-0.190850\pi$
$619$	41.0803	1.65115	0.825577	+	0.564289i	$0.190850\pi$
$620$	14.2862	0.573748
$621$	0	0
$622$	17.5808	0.704924
$623$	−9.64581	−0.386451
$624$	0	0
$625$	−6.97266	−0.278907
$626$	−13.3753	−0.534586
$627$	0	0
$628$	−5.77444	−0.230425
$629$	−51.3245	−2.04644
$630$	0	0
$631$	−17.2338	−0.686068	−0.343034	−	0.939323i	$-0.611455\pi$
$631$	−17.2338	−0.686068	−0.343034	+	0.939323i	$0.611455\pi$
$632$	14.0259	0.557919
$633$	0	0
$634$	−5.12331	−0.203473
$635$	6.41443	0.254549
$636$	0	0
$637$	−7.85825	−0.311355
$638$	1.57153	0.0622176
$639$	0	0
$640$	−1.60399	−0.0634034
$641$	−14.8661	−0.587176	−0.293588	−	0.955932i	$-0.594849\pi$
$641$	−14.8661	−0.587176	−0.293588	+	0.955932i	$0.594849\pi$
$642$	0	0
$643$	−14.0895	−0.555637	−0.277819	−	0.960634i	$-0.589611\pi$
$643$	−14.0895	−0.555637	−0.277819	+	0.960634i	$0.589611\pi$
$644$	31.7070	1.24943
$645$	0	0
$646$	7.30862	0.287554
$647$	43.2485	1.70027	0.850137	−	0.526562i	$-0.176519\pi$
$647$	43.2485	1.70027	0.850137	+	0.526562i	$0.176519\pi$
$648$	0	0
$649$	−4.50968	−0.177021
$650$	−1.14401	−0.0448719
$651$	0	0
$652$	−1.20564	−0.0472165
$653$	−28.5181	−1.11600	−0.557999	−	0.829842i	$-0.688431\pi$
$653$	−28.5181	−1.11600	−0.557999	+	0.829842i	$0.688431\pi$
$654$	0	0
$655$	32.3735	1.26494
$656$	4.74735	0.185353
$657$	0	0
$658$	−51.4269	−2.00483
$659$	−24.1337	−0.940114	−0.470057	−	0.882636i	$-0.655767\pi$
$659$	−24.1337	−0.940114	−0.470057	+	0.882636i	$0.655767\pi$
$660$	0	0
$661$	−3.80662	−0.148060	−0.0740301	−	0.997256i	$-0.523586\pi$
$661$	−3.80662	−0.148060	−0.0740301	+	0.997256i	$0.523586\pi$
$662$	14.5745	0.566455
$663$	0	0
$664$	−5.99096	−0.232494
$665$	−11.0393	−0.428085
$666$	0	0
$667$	−7.11205	−0.275380
$668$	−15.7942	−0.611098
$669$	0	0
$670$	2.14721	0.0829540
$671$	−7.33153	−0.283030
$672$	0	0
$673$	−39.5792	−1.52567	−0.762833	−	0.646595i	$-0.776192\pi$
$673$	−39.5792	−1.52567	−0.762833	+	0.646595i	$0.776192\pi$
$674$	−1.63369	−0.0629272
$675$	0	0
$676$	−12.7778	−0.491456
$677$	−20.4562	−0.786195	−0.393097	−	0.919497i	$-0.628596\pi$
$677$	−20.4562	−0.786195	−0.393097	+	0.919497i	$0.628596\pi$
$678$	0	0
$679$	−75.6285	−2.90236
$680$	8.28747	0.317810
$681$	0	0
$682$	12.8256	0.491117
$683$	28.7459	1.09993	0.549966	−	0.835187i	$-0.314641\pi$
$683$	28.7459	1.09993	0.549966	+	0.835187i	$0.314641\pi$
$684$	0	0
$685$	8.55396	0.326830
$686$	47.0611	1.79680
$687$	0	0
$688$	−8.99506	−0.342933
$689$	3.21064	0.122316
$690$	0	0
$691$	−6.17738	−0.234999	−0.117499	−	0.993073i	$-0.537488\pi$
$691$	−6.17738	−0.234999	−0.117499	+	0.993073i	$0.537488\pi$
$692$	−6.15714	−0.234059
$693$	0	0
$694$	−21.0414	−0.798721
$695$	30.4598	1.15540
$696$	0	0
$697$	−24.5285	−0.929083
$698$	4.82308	0.182556
$699$	0	0
$700$	11.8094	0.446354
$701$	−28.9279	−1.09259	−0.546296	−	0.837592i	$-0.683963\pi$
$701$	−28.9279	−1.09259	−0.546296	+	0.837592i	$0.683963\pi$
$702$	0	0
$703$	14.0515	0.529961
$704$	−1.44000	−0.0542720
$705$	0	0
$706$	−19.6185	−0.738351
$707$	38.0543	1.43118
$708$	0	0
$709$	1.52014	0.0570901	0.0285451	−	0.999593i	$-0.490913\pi$
$709$	1.52014	0.0570901	0.0285451	+	0.999593i	$0.490913\pi$
$710$	−26.5101	−0.994906
$711$	0	0
$712$	−1.98251	−0.0742979
$713$	−58.0428	−2.17372
$714$	0	0
$715$	1.08865	0.0407133
$716$	−19.9049	−0.743881
$717$	0	0
$718$	−2.71271	−0.101237
$719$	−33.0945	−1.23422	−0.617109	−	0.786878i	$-0.711696\pi$
$719$	−33.0945	−1.23422	−0.617109	+	0.786878i	$0.711696\pi$
$720$	0	0
$721$	−42.3597	−1.57756
$722$	16.9991	0.632640
$723$	0	0
$724$	23.0165	0.855400
$725$	−2.64891	−0.0983782
$726$	0	0
$727$	8.63118	0.320113	0.160056	−	0.987108i	$-0.448832\pi$
$727$	8.63118	0.320113	0.160056	+	0.987108i	$0.448832\pi$
$728$	−2.29323	−0.0849926
$729$	0	0
$730$	−21.2506	−0.786520
$731$	46.4754	1.71896
$732$	0	0
$733$	−36.1566	−1.33547	−0.667737	−	0.744397i	$-0.732737\pi$
$733$	−36.1566	−1.33547	−0.667737	+	0.744397i	$0.732737\pi$
$734$	−27.3420	−1.00921
$735$	0	0
$736$	6.51679	0.240212
$737$	1.92768	0.0710069
$738$	0	0
$739$	−24.8358	−0.913601	−0.456800	−	0.889569i	$-0.651005\pi$
$739$	−24.8358	−0.913601	−0.456800	+	0.889569i	$0.651005\pi$
$740$	15.9334	0.585723
$741$	0	0
$742$	−33.1428	−1.21671
$743$	−24.1245	−0.885042	−0.442521	−	0.896758i	$-0.645916\pi$
$743$	−24.1245	−0.885042	−0.442521	+	0.896758i	$0.645916\pi$
$744$	0	0
$745$	16.3761	0.599973
$746$	−0.178537	−0.00653671
$747$	0	0
$748$	7.44015	0.272039
$749$	−0.115094	−0.00420543
$750$	0	0
$751$	9.26005	0.337904	0.168952	−	0.985624i	$-0.445962\pi$
$751$	9.26005	0.337904	0.168952	+	0.985624i	$0.445962\pi$
$752$	−10.5698	−0.385442
$753$	0	0
$754$	0.514383	0.0187327
$755$	0.0931036	0.00338839
$756$	0	0
$757$	20.9782	0.762464	0.381232	−	0.924479i	$-0.375500\pi$
$757$	20.9782	0.762464	0.381232	+	0.924479i	$0.375500\pi$
$758$	−25.1988	−0.915260
$759$	0	0
$760$	−2.26892	−0.0823023
$761$	−28.8802	−1.04691	−0.523454	−	0.852054i	$-0.675357\pi$
$761$	−28.8802	−1.04691	−0.523454	+	0.852054i	$0.675357\pi$
$762$	0	0
$763$	−41.6920	−1.50935
$764$	−12.6440	−0.457444
$765$	0	0
$766$	−24.7131	−0.892921
$767$	−1.47608	−0.0532980
$768$	0	0
$769$	−20.9370	−0.755006	−0.377503	−	0.926008i	$-0.623217\pi$
$769$	−20.9370	−0.755006	−0.377503	+	0.926008i	$0.623217\pi$
$770$	−11.2379	−0.404988
$771$	0	0
$772$	7.48011	0.269215
$773$	27.7474	0.998005	0.499002	−	0.866601i	$-0.333700\pi$
$773$	27.7474	0.998005	0.499002	+	0.866601i	$0.333700\pi$
$774$	0	0
$775$	−21.6183	−0.776551
$776$	−15.5440	−0.557998
$777$	0	0
$778$	−5.90447	−0.211686
$779$	6.71533	0.240602
$780$	0	0
$781$	−23.7997	−0.851619
$782$	−33.6708	−1.20406
$783$	0	0
$784$	16.6725	0.595447
$785$	−9.26217	−0.330581
$786$	0	0
$787$	18.6784	0.665813	0.332906	−	0.942960i	$-0.391971\pi$
$787$	18.6784	0.665813	0.332906	+	0.942960i	$0.391971\pi$
$788$	4.02066	0.143230
$789$	0	0
$790$	22.4974	0.800422
$791$	8.30905	0.295436
$792$	0	0
$793$	−2.39970	−0.0852159
$794$	−32.0893	−1.13881
$795$	0	0
$796$	5.12390	0.181612
$797$	16.7230	0.592360	0.296180	−	0.955132i	$-0.404287\pi$
$797$	16.7230	0.592360	0.296180	+	0.955132i	$0.404287\pi$
$798$	0	0
$799$	54.6119	1.93203
$800$	2.42720	0.0858146
$801$	0	0
$802$	15.3722	0.542812
$803$	−19.0779	−0.673244
$804$	0	0
$805$	50.8579	1.79251
$806$	4.19797	0.147867
$807$	0	0
$808$	7.82134	0.275154
$809$	−5.88388	−0.206866	−0.103433	−	0.994636i	$-0.532983\pi$
$809$	−5.88388	−0.206866	−0.103433	+	0.994636i	$0.532983\pi$
$810$	0	0
$811$	−5.43403	−0.190815	−0.0954073	−	0.995438i	$-0.530415\pi$
$811$	−5.43403	−0.190815	−0.0954073	+	0.995438i	$0.530415\pi$
$812$	−5.30987	−0.186340
$813$	0	0
$814$	14.3043	0.501367
$815$	−1.93384	−0.0677394
$816$	0	0
$817$	−12.7239	−0.445152
$818$	−4.14741	−0.145011
$819$	0	0
$820$	7.61472	0.265918
$821$	40.0297	1.39705	0.698523	−	0.715588i	$-0.253841\pi$
$821$	40.0297	1.39705	0.698523	+	0.715588i	$0.253841\pi$
$822$	0	0
$823$	−30.3053	−1.05638	−0.528188	−	0.849128i	$-0.677128\pi$
$823$	−30.3053	−1.05638	−0.528188	+	0.849128i	$0.677128\pi$
$824$	−8.70624	−0.303296
$825$	0	0
$826$	15.2372	0.530172
$827$	16.1666	0.562167	0.281083	−	0.959683i	$-0.409306\pi$
$827$	16.1666	0.562167	0.281083	+	0.959683i	$0.409306\pi$
$828$	0	0
$829$	51.4058	1.78540	0.892699	−	0.450653i	$-0.148809\pi$
$829$	51.4058	1.78540	0.892699	+	0.450653i	$0.148809\pi$
$830$	−9.60946	−0.333549
$831$	0	0
$832$	−0.471330	−0.0163404
$833$	−86.1432	−2.98468
$834$	0	0
$835$	−25.3339	−0.876715
$836$	−2.03694	−0.0704490
$837$	0	0
$838$	−26.2690	−0.907449
$839$	−47.1568	−1.62803	−0.814016	−	0.580842i	$-0.802723\pi$
$839$	−47.1568	−1.62803	−0.814016	+	0.580842i	$0.802723\pi$
$840$	0	0
$841$	−27.8090	−0.958930
$842$	9.17005	0.316021
$843$	0	0
$844$	−12.8974	−0.443949
$845$	−20.4956	−0.705070
$846$	0	0
$847$	43.4309	1.49230
$848$	−6.81188	−0.233921
$849$	0	0
$850$	−12.5408	−0.430146
$851$	−64.7350	−2.21909
$852$	0	0
$853$	−11.3507	−0.388640	−0.194320	−	0.980938i	$-0.562250\pi$
$853$	−11.3507	−0.388640	−0.194320	+	0.980938i	$0.562250\pi$
$854$	24.7716	0.847668
$855$	0	0
$856$	−0.0236553	−0.000808522 0
$857$	12.1842	0.416206	0.208103	−	0.978107i	$-0.433271\pi$
$857$	12.1842	0.416206	0.208103	+	0.978107i	$0.433271\pi$
$858$	0	0
$859$	25.8712	0.882713	0.441357	−	0.897332i	$-0.354497\pi$
$859$	25.8712	0.882713	0.441357	+	0.897332i	$0.354497\pi$
$860$	−14.4280	−0.491991
$861$	0	0
$862$	19.2622	0.656073
$863$	36.2938	1.23545	0.617727	−	0.786392i	$-0.288054\pi$
$863$	36.2938	1.23545	0.617727	+	0.786392i	$0.288054\pi$
$864$	0	0
$865$	−9.87601	−0.335794
$866$	−30.6252	−1.04069
$867$	0	0
$868$	−43.3348	−1.47088
$869$	20.1972	0.685144
$870$	0	0
$871$	0.630953	0.0213790
$872$	−8.56901	−0.290183
$873$	0	0
$874$	9.21827	0.311813
$875$	57.9629	1.95950
$876$	0	0
$877$	−49.8608	−1.68368	−0.841839	−	0.539729i	$-0.818527\pi$
$877$	−49.8608	−1.68368	−0.841839	+	0.539729i	$0.818527\pi$
$878$	17.2313	0.581530
$879$	0	0
$880$	−2.30975	−0.0778616
$881$	−12.6707	−0.426887	−0.213443	−	0.976955i	$-0.568468\pi$
$881$	−12.6707	−0.426887	−0.213443	+	0.976955i	$0.568468\pi$
$882$	0	0
$883$	−1.98777	−0.0668937	−0.0334468	−	0.999440i	$-0.510648\pi$
$883$	−1.98777	−0.0668937	−0.0334468	+	0.999440i	$0.510648\pi$
$884$	2.43525	0.0819064
$885$	0	0
$886$	−15.6230	−0.524866
$887$	−38.7243	−1.30023	−0.650117	−	0.759834i	$-0.725280\pi$
$887$	−38.7243	−1.30023	−0.650117	+	0.759834i	$0.725280\pi$
$888$	0	0
$889$	−19.4571	−0.652570
$890$	−3.17994	−0.106592
$891$	0	0
$892$	−1.00000	−0.0334825
$893$	−14.9515	−0.500332
$894$	0	0
$895$	−31.9273	−1.06721
$896$	4.86544	0.162543
$897$	0	0
$898$	−17.3813	−0.580023
$899$	9.72022	0.324188
$900$	0	0
$901$	35.1954	1.17253
$902$	6.83618	0.227620
$903$	0	0
$904$	1.70777	0.0567996
$905$	36.9183	1.22721
$906$	0	0
$907$	−45.2865	−1.50371	−0.751856	−	0.659327i	$-0.770841\pi$
$907$	−45.2865	−1.50371	−0.751856	+	0.659327i	$0.770841\pi$
$908$	23.7498	0.788165
$909$	0	0
$910$	−3.67832	−0.121935
$911$	−48.1113	−1.59400	−0.797000	−	0.603979i	$-0.793581\pi$
$911$	−48.1113	−1.59400	−0.797000	+	0.603979i	$0.793581\pi$
$912$	0	0
$913$	−8.62697	−0.285511
$914$	−36.3057	−1.20089
$915$	0	0
$916$	22.4453	0.741612
$917$	−98.1994	−3.24283
$918$	0	0
$919$	21.8411	0.720473	0.360236	−	0.932861i	$-0.382696\pi$
$919$	21.8411	0.720473	0.360236	+	0.932861i	$0.382696\pi$
$920$	10.4529	0.344621
$921$	0	0
$922$	2.29878	0.0757061
$923$	−7.78993	−0.256409
$924$	0	0
$925$	−24.1108	−0.792759
$926$	−14.7034	−0.483185
$927$	0	0
$928$	−1.09134	−0.0358251
$929$	54.8410	1.79928	0.899638	−	0.436637i	$-0.143830\pi$
$929$	54.8410	1.79928	0.899638	+	0.436637i	$0.143830\pi$
$930$	0	0
$931$	23.5840	0.772934
$932$	−9.22110	−0.302047
$933$	0	0
$934$	18.6263	0.609470
$935$	11.9339	0.390282
$936$	0	0
$937$	49.5647	1.61921	0.809603	−	0.586977i	$-0.199682\pi$
$937$	49.5647	1.61921	0.809603	+	0.586977i	$0.199682\pi$
$938$	−6.51320	−0.212663
$939$	0	0
$940$	−16.9539	−0.552977
$941$	−45.7119	−1.49016	−0.745082	−	0.666973i	$-0.767590\pi$
$941$	−45.7119	−1.49016	−0.745082	+	0.666973i	$0.767590\pi$
$942$	0	0
$943$	−30.9375	−1.00746
$944$	3.13173	0.101929
$945$	0	0
$946$	−12.9529	−0.421134
$947$	17.6559	0.573738	0.286869	−	0.957970i	$-0.407385\pi$
$947$	17.6559	0.573738	0.286869	+	0.957970i	$0.407385\pi$
$948$	0	0
$949$	−6.24443	−0.202703
$950$	3.43338	0.111394
$951$	0	0
$952$	−25.1386	−0.814748
$953$	33.0512	1.07063	0.535316	−	0.844652i	$-0.320193\pi$
$953$	33.0512	1.07063	0.535316	+	0.844652i	$0.320193\pi$
$954$	0	0
$955$	−20.2809	−0.656274
$956$	−3.77220	−0.122002
$957$	0	0
$958$	14.3139	0.462460
$959$	−25.9470	−0.837872
$960$	0	0
$961$	48.3285	1.55898
$962$	4.68199	0.150953
$963$	0	0
$964$	−23.5440	−0.758302
$965$	11.9981	0.386231
$966$	0	0
$967$	1.07857	0.0346845	0.0173423	−	0.999850i	$-0.494480\pi$
$967$	1.07857	0.0346845	0.0173423	+	0.999850i	$0.494480\pi$
$968$	8.92640	0.286905
$969$	0	0
$970$	−24.9325	−0.800535
$971$	22.9356	0.736039	0.368020	−	0.929818i	$-0.380036\pi$
$971$	22.9356	0.736039	0.368020	+	0.929818i	$0.380036\pi$
$972$	0	0
$973$	−92.3945	−2.96203
$974$	−16.2734	−0.521434
$975$	0	0
$976$	5.09134	0.162970
$977$	−7.11650	−0.227677	−0.113839	−	0.993499i	$-0.536315\pi$
$977$	−7.11650	−0.227677	−0.113839	+	0.993499i	$0.536315\pi$
$978$	0	0
$979$	−2.85482	−0.0912404
$980$	26.7426	0.854262
$981$	0	0
$982$	21.7849	0.695183
$983$	51.7171	1.64952	0.824759	−	0.565485i	$-0.191311\pi$
$983$	51.7171	1.64952	0.824759	+	0.565485i	$0.191311\pi$
$984$	0	0
$985$	6.44911	0.205486
$986$	5.63872	0.179574
$987$	0	0
$988$	−0.666716	−0.0212111
$989$	58.6189	1.86397
$990$	0	0
$991$	−31.3919	−0.997198	−0.498599	−	0.866833i	$-0.666152\pi$
$991$	−31.3919	−0.997198	−0.498599	+	0.866833i	$0.666152\pi$
$992$	−8.90666	−0.282787
$993$	0	0
$994$	80.4139	2.55057
$995$	8.21871	0.260551
$996$	0	0
$997$	1.71435	0.0542940	0.0271470	−	0.999631i	$-0.491358\pi$
$997$	1.71435	0.0542940	0.0271470	+	0.999631i	$0.491358\pi$
$998$	−20.1791	−0.638759
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4014.2.a.v.1.6		7
3.2	odd	2		1338.2.a.j.1.2	✓	7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1338.2.a.j.1.2	✓	7	3.2	odd	2
4014.2.a.v.1.6		7	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:	\( N \)	\(=\)	\( 4014 = 2 \cdot 3^{2} \cdot 223 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4014.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} +1.60399 q^{5} -4.86544 q^{7} -1.00000 q^{8} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{4} +1.60399 q^{5} -4.86544 q^{7} -1.00000 q^{8} -1.60399 q^{10} -1.44000 q^{11} -0.471330 q^{13} +4.86544 q^{14} +1.00000 q^{16} -5.16677 q^{17} +1.41454 q^{19} +1.60399 q^{20} +1.44000 q^{22} -6.51679 q^{23} -2.42720 q^{25} +0.471330 q^{26} -4.86544 q^{28} +1.09134 q^{29} +8.90666 q^{31} -1.00000 q^{32} +5.16677 q^{34} -7.80414 q^{35} +9.93358 q^{37} -1.41454 q^{38} -1.60399 q^{40} +4.74735 q^{41} -8.99506 q^{43} -1.44000 q^{44} +6.51679 q^{46} -10.5698 q^{47} +16.6725 q^{49} +2.42720 q^{50} -0.471330 q^{52} -6.81188 q^{53} -2.30975 q^{55} +4.86544 q^{56} -1.09134 q^{58} +3.13173 q^{59} +5.09134 q^{61} -8.90666 q^{62} +1.00000 q^{64} -0.756010 q^{65} -1.33867 q^{67} -5.16677 q^{68} +7.80414 q^{70} +16.5276 q^{71} +13.2485 q^{73} -9.93358 q^{74} +1.41454 q^{76} +7.00623 q^{77} -14.0259 q^{79} +1.60399 q^{80} -4.74735 q^{82} +5.99096 q^{83} -8.28747 q^{85} +8.99506 q^{86} +1.44000 q^{88} +1.98251 q^{89} +2.29323 q^{91} -6.51679 q^{92} +10.5698 q^{94} +2.26892 q^{95} +15.5440 q^{97} -16.6725 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q - 7 q^{2} + 7 q^{4} - 6 q^{5} + 3 q^{7} - 7 q^{8}+O(q^{10}) \) 7 * q - 7 * q^2 + 7 * q^4 - 6 * q^5 + 3 * q^7 - 7 * q^8	\( 7 q - 7 q^{2} + 7 q^{4} - 6 q^{5} + 3 q^{7} - 7 q^{8} + 6 q^{10} + q^{11} + 8 q^{13} - 3 q^{14} + 7 q^{16} - 16 q^{17} + 2 q^{19} - 6 q^{20} - q^{22} - 8 q^{23} + 19 q^{25} - 8 q^{26} + 3 q^{28} - 4 q^{29} + 11 q^{31} - 7 q^{32} + 16 q^{34} + 17 q^{37} - 2 q^{38} + 6 q^{40} - 18 q^{41} - q^{43} + q^{44} + 8 q^{46} - 5 q^{47} + 24 q^{49} - 19 q^{50} + 8 q^{52} - 6 q^{53} + 21 q^{55} - 3 q^{56} + 4 q^{58} + 7 q^{59} + 24 q^{61} - 11 q^{62} + 7 q^{64} - 11 q^{65} - 4 q^{67} - 16 q^{68} + 7 q^{71} + 28 q^{73} - 17 q^{74} + 2 q^{76} + 2 q^{77} - 6 q^{80} + 18 q^{82} + 2 q^{83} + 4 q^{85} + q^{86} - q^{88} - 5 q^{89} + 2 q^{91} - 8 q^{92} + 5 q^{94} + 14 q^{95} + 23 q^{97} - 24 q^{98}+O(q^{100}) \) 7 * q - 7 * q^2 + 7 * q^4 - 6 * q^5 + 3 * q^7 - 7 * q^8 + 6 * q^10 + q^11 + 8 * q^13 - 3 * q^14 + 7 * q^16 - 16 * q^17 + 2 * q^19 - 6 * q^20 - q^22 - 8 * q^23 + 19 * q^25 - 8 * q^26 + 3 * q^28 - 4 * q^29 + 11 * q^31 - 7 * q^32 + 16 * q^34 + 17 * q^37 - 2 * q^38 + 6 * q^40 - 18 * q^41 - q^43 + q^44 + 8 * q^46 - 5 * q^47 + 24 * q^49 - 19 * q^50 + 8 * q^52 - 6 * q^53 + 21 * q^55 - 3 * q^56 + 4 * q^58 + 7 * q^59 + 24 * q^61 - 11 * q^62 + 7 * q^64 - 11 * q^65 - 4 * q^67 - 16 * q^68 + 7 * q^71 + 28 * q^73 - 17 * q^74 + 2 * q^76 + 2 * q^77 - 6 * q^80 + 18 * q^82 + 2 * q^83 + 4 * q^85 + q^86 - q^88 - 5 * q^89 + 2 * q^91 - 8 * q^92 + 5 * q^94 + 14 * q^95 + 23 * q^97 - 24 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more