LMFDB - Embedded newform 4014.2.a.w.1.3

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} - x^{6} - 14x^{5} + 12x^{4} + 50x^{3} - 36x^{2} - 38x + 18 $ x^7 - x^6 - 14x^5 + 12x^4 + 50x^3 - 36x^2 - 38*x + 18
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 446)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-2.01250$ 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)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} -1.52490 q^{5} +0.908231 q^{7} -1.00000 q^{8} +O(q^{10})$	\(q-1.00000 q^{2} +1.00000 q^{4} -1.52490 q^{5} +0.908231 q^{7} -1.00000 q^{8} +1.52490 q^{10} -5.03528 q^{11} -5.51425 q^{13} -0.908231 q^{14} +1.00000 q^{16} -5.38428 q^{17} +4.02500 q^{19} -1.52490 q^{20} +5.03528 q^{22} -0.290419 q^{23} -2.67470 q^{25} +5.51425 q^{26} +0.908231 q^{28} -8.72171 q^{29} +10.2440 q^{31} -1.00000 q^{32} +5.38428 q^{34} -1.38496 q^{35} -6.80493 q^{37} -4.02500 q^{38} +1.52490 q^{40} +6.15244 q^{41} -7.03714 q^{43} -5.03528 q^{44} +0.290419 q^{46} +10.5033 q^{47} -6.17512 q^{49} +2.67470 q^{50} -5.51425 q^{52} -4.35234 q^{53} +7.67827 q^{55} -0.908231 q^{56} +8.72171 q^{58} +10.0783 q^{59} -1.04184 q^{61} -10.2440 q^{62} +1.00000 q^{64} +8.40865 q^{65} +15.3492 q^{67} -5.38428 q^{68} +1.38496 q^{70} +2.68271 q^{71} -10.2734 q^{73} +6.80493 q^{74} +4.02500 q^{76} -4.57320 q^{77} +10.5693 q^{79} -1.52490 q^{80} -6.15244 q^{82} +11.5427 q^{83} +8.21046 q^{85} +7.03714 q^{86} +5.03528 q^{88} +13.7906 q^{89} -5.00821 q^{91} -0.290419 q^{92} -10.5033 q^{94} -6.13771 q^{95} +2.46450 q^{97} +6.17512 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q - 7 q^{2} + 7 q^{4} - 2 q^{5} + 6 q^{7} - 7 q^{8}+O(q^{10}) $ 7 * q - 7 * q^2 + 7 * q^4 - 2 * q^5 + 6 * q^7 - 7 * q^8	$ 7 q - 7 q^{2} + 7 q^{4} - 2 q^{5} + 6 q^{7} - 7 q^{8} + 2 q^{10} - 9 q^{11} - 2 q^{13} - 6 q^{14} + 7 q^{16} + 7 q^{17} - 2 q^{19} - 2 q^{20} + 9 q^{22} - 15 q^{23} + 13 q^{25} + 2 q^{26} + 6 q^{28} - 9 q^{29} - 2 q^{31} - 7 q^{32} - 7 q^{34} + 4 q^{35} + 5 q^{37} + 2 q^{38} + 2 q^{40} + 33 q^{41} + 20 q^{43} - 9 q^{44} + 15 q^{46} + 2 q^{47} + 3 q^{49} - 13 q^{50} - 2 q^{52} + 13 q^{53} - 18 q^{55} - 6 q^{56} + 9 q^{58} - 9 q^{59} + 8 q^{61} + 2 q^{62} + 7 q^{64} + 44 q^{65} + 29 q^{67} + 7 q^{68} - 4 q^{70} - 37 q^{73} - 5 q^{74} - 2 q^{76} + 18 q^{77} + 32 q^{79} - 2 q^{80} - 33 q^{82} + 6 q^{83} - 4 q^{85} - 20 q^{86} + 9 q^{88} + 17 q^{89} - 4 q^{91} - 15 q^{92} - 2 q^{94} + 12 q^{95} + 12 q^{97} - 3 q^{98}+O(q^{100}) $ 7 * q - 7 * q^2 + 7 * q^4 - 2 * q^5 + 6 * q^7 - 7 * q^8 + 2 * q^10 - 9 * q^11 - 2 * q^13 - 6 * q^14 + 7 * q^16 + 7 * q^17 - 2 * q^19 - 2 * q^20 + 9 * q^22 - 15 * q^23 + 13 * q^25 + 2 * q^26 + 6 * q^28 - 9 * q^29 - 2 * q^31 - 7 * q^32 - 7 * q^34 + 4 * q^35 + 5 * q^37 + 2 * q^38 + 2 * q^40 + 33 * q^41 + 20 * q^43 - 9 * q^44 + 15 * q^46 + 2 * q^47 + 3 * q^49 - 13 * q^50 - 2 * q^52 + 13 * q^53 - 18 * q^55 - 6 * q^56 + 9 * q^58 - 9 * q^59 + 8 * q^61 + 2 * q^62 + 7 * q^64 + 44 * q^65 + 29 * q^67 + 7 * q^68 - 4 * q^70 - 37 * q^73 - 5 * q^74 - 2 * q^76 + 18 * q^77 + 32 * q^79 - 2 * q^80 - 33 * q^82 + 6 * q^83 - 4 * q^85 - 20 * q^86 + 9 * q^88 + 17 * q^89 - 4 * q^91 - 15 * q^92 - 2 * q^94 + 12 * q^95 + 12 * q^97 - 3 * 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
$5$	−1.52490	−0.681954	−0.340977	−	0.940072i	$-0.610758\pi$
$5$	−1.52490	−0.681954	−0.340977	+	0.940072i	$0.610758\pi$
$6$	0	0
$7$	0.908231	0.343279	0.171639	−	0.985160i	$-0.445094\pi$
$7$	0.908231	0.343279	0.171639	+	0.985160i	$0.445094\pi$
$8$	−1.00000	−0.353553
$9$	0	0
$10$	1.52490	0.482214
$11$	−5.03528	−1.51819	−0.759097	−	0.650977i	$-0.774359\pi$
$11$	−5.03528	−1.51819	−0.759097	+	0.650977i	$0.774359\pi$
$12$	0	0
$13$	−5.51425	−1.52938	−0.764689	−	0.644400i	$-0.777107\pi$
$13$	−5.51425	−1.52938	−0.764689	+	0.644400i	$0.777107\pi$
$14$	−0.908231	−0.242735
$15$	0	0
$16$	1.00000	0.250000
$17$	−5.38428	−1.30588	−0.652939	−	0.757410i	$-0.726464\pi$
$17$	−5.38428	−1.30588	−0.652939	+	0.757410i	$0.726464\pi$
$18$	0	0
$19$	4.02500	0.923399	0.461699	−	0.887036i	$-0.347240\pi$
$19$	4.02500	0.923399	0.461699	+	0.887036i	$0.347240\pi$
$20$	−1.52490	−0.340977
$21$	0	0
$22$	5.03528	1.07353
$23$	−0.290419	−0.0605566	−0.0302783	−	0.999542i	$-0.509639\pi$
$23$	−0.290419	−0.0605566	−0.0302783	+	0.999542i	$0.509639\pi$
$24$	0	0
$25$	−2.67470	−0.534939
$26$	5.51425	1.08143
$27$	0	0
$28$	0.908231	0.171639
$29$	−8.72171	−1.61958	−0.809791	−	0.586719i	$-0.800419\pi$
$29$	−8.72171	−1.61958	−0.809791	+	0.586719i	$0.800419\pi$
$30$	0	0
$31$	10.2440	1.83987	0.919936	−	0.392069i	$-0.128241\pi$
$31$	10.2440	1.83987	0.919936	+	0.392069i	$0.128241\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	5.38428	0.923396
$35$	−1.38496	−0.234100
$36$	0	0
$37$	−6.80493	−1.11872	−0.559362	−	0.828924i	$-0.688954\pi$
$37$	−6.80493	−1.11872	−0.559362	+	0.828924i	$0.688954\pi$
$38$	−4.02500	−0.652942
$39$	0	0
$40$	1.52490	0.241107
$41$	6.15244	0.960850	0.480425	−	0.877036i	$-0.340482\pi$
$41$	6.15244	0.960850	0.480425	+	0.877036i	$0.340482\pi$
$42$	0	0
$43$	−7.03714	−1.07315	−0.536577	−	0.843852i	$-0.680283\pi$
$43$	−7.03714	−1.07315	−0.536577	+	0.843852i	$0.680283\pi$
$44$	−5.03528	−0.759097
$45$	0	0
$46$	0.290419	0.0428200
$47$	10.5033	1.53206	0.766031	−	0.642803i	$-0.222229\pi$
$47$	10.5033	1.53206	0.766031	+	0.642803i	$0.222229\pi$
$48$	0	0
$49$	−6.17512	−0.882160
$50$	2.67470	0.378259
$51$	0	0
$52$	−5.51425	−0.764689
$53$	−4.35234	−0.597840	−0.298920	−	0.954278i	$-0.596626\pi$
$53$	−4.35234	−0.597840	−0.298920	+	0.954278i	$0.596626\pi$
$54$	0	0
$55$	7.67827	1.03534
$56$	−0.908231	−0.121367
$57$	0	0
$58$	8.72171	1.14522
$59$	10.0783	1.31209	0.656045	−	0.754722i	$-0.272228\pi$
$59$	10.0783	1.31209	0.656045	+	0.754722i	$0.272228\pi$
$60$	0	0
$61$	−1.04184	−0.133394	−0.0666968	−	0.997773i	$-0.521246\pi$
$61$	−1.04184	−0.133394	−0.0666968	+	0.997773i	$0.521246\pi$
$62$	−10.2440	−1.30099
$63$	0	0
$64$	1.00000	0.125000
$65$	8.40865	1.04296
$66$	0	0
$67$	15.3492	1.87520	0.937599	−	0.347718i	$-0.113043\pi$
$67$	15.3492	1.87520	0.937599	+	0.347718i	$0.113043\pi$
$68$	−5.38428	−0.652939
$69$	0	0
$70$	1.38496	0.165534
$71$	2.68271	0.318380	0.159190	−	0.987248i	$-0.449112\pi$
$71$	2.68271	0.318380	0.159190	+	0.987248i	$0.449112\pi$
$72$	0	0
$73$	−10.2734	−1.20242	−0.601208	−	0.799093i	$-0.705314\pi$
$73$	−10.2734	−1.20242	−0.601208	+	0.799093i	$0.705314\pi$
$74$	6.80493	0.791057
$75$	0	0
$76$	4.02500	0.461699
$77$	−4.57320	−0.521164
$78$	0	0
$79$	10.5693	1.18914	0.594568	−	0.804045i	$-0.297323\pi$
$79$	10.5693	1.18914	0.594568	+	0.804045i	$0.297323\pi$
$80$	−1.52490	−0.170488
$81$	0	0
$82$	−6.15244	−0.679423
$83$	11.5427	1.26697	0.633485	−	0.773755i	$-0.281624\pi$
$83$	11.5427	1.26697	0.633485	+	0.773755i	$0.281624\pi$
$84$	0	0
$85$	8.21046	0.890549
$86$	7.03714	0.758834
$87$	0	0
$88$	5.03528	0.536763
$89$	13.7906	1.46180	0.730899	−	0.682486i	$-0.239101\pi$
$89$	13.7906	1.46180	0.730899	+	0.682486i	$0.239101\pi$
$90$	0	0
$91$	−5.00821	−0.525003
$92$	−0.290419	−0.0302783
$93$	0	0
$94$	−10.5033	−1.08333
$95$	−6.13771	−0.629715
$96$	0	0
$97$	2.46450	0.250232	0.125116	−	0.992142i	$-0.460070\pi$
$97$	2.46450	0.250232	0.125116	+	0.992142i	$0.460070\pi$
$98$	6.17512	0.623781
$99$	0	0
$100$	−2.67470	−0.267470
$101$	14.6472	1.45745	0.728726	−	0.684805i	$-0.240113\pi$
$101$	14.6472	1.45745	0.728726	+	0.684805i	$0.240113\pi$
$102$	0	0
$103$	0.320702	0.0315997	0.0157998	−	0.999875i	$-0.494971\pi$
$103$	0.320702	0.0315997	0.0157998	+	0.999875i	$0.494971\pi$
$104$	5.51425	0.540717
$105$	0	0
$106$	4.35234	0.422737
$107$	−13.9377	−1.34741	−0.673705	−	0.739000i	$-0.735298\pi$
$107$	−13.9377	−1.34741	−0.673705	+	0.739000i	$0.735298\pi$
$108$	0	0
$109$	−0.707255	−0.0677427	−0.0338714	−	0.999426i	$-0.510784\pi$
$109$	−0.707255	−0.0677427	−0.0338714	+	0.999426i	$0.510784\pi$
$110$	−7.67827	−0.732095
$111$	0	0
$112$	0.908231	0.0858197
$113$	−10.8700	−1.02256	−0.511280	−	0.859414i	$-0.670829\pi$
$113$	−10.8700	−1.02256	−0.511280	+	0.859414i	$0.670829\pi$
$114$	0	0
$115$	0.442859	0.0412968
$116$	−8.72171	−0.809791
$117$	0	0
$118$	−10.0783	−0.927787
$119$	−4.89017	−0.448281
$120$	0	0
$121$	14.3541	1.30491
$122$	1.04184	0.0943236
$123$	0	0
$124$	10.2440	0.919936
$125$	11.7031	1.04676
$126$	0	0
$127$	−9.08270	−0.805959	−0.402979	−	0.915209i	$-0.632025\pi$
$127$	−9.08270	−0.805959	−0.402979	+	0.915209i	$0.632025\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	−8.40865	−0.737488
$131$	−13.8183	−1.20731	−0.603657	−	0.797244i	$-0.706290\pi$
$131$	−13.8183	−1.20731	−0.603657	+	0.797244i	$0.706290\pi$
$132$	0	0
$133$	3.65563	0.316983
$134$	−15.3492	−1.32597
$135$	0	0
$136$	5.38428	0.461698
$137$	13.2046	1.12815	0.564073	−	0.825725i	$-0.309234\pi$
$137$	13.2046	1.12815	0.564073	+	0.825725i	$0.309234\pi$
$138$	0	0
$139$	−7.84073	−0.665042	−0.332521	−	0.943096i	$-0.607899\pi$
$139$	−7.84073	−0.665042	−0.332521	+	0.943096i	$0.607899\pi$
$140$	−1.38496	−0.117050
$141$	0	0
$142$	−2.68271	−0.225128
$143$	27.7658	2.32189
$144$	0	0
$145$	13.2997	1.10448
$146$	10.2734	0.850236
$147$	0	0
$148$	−6.80493	−0.559362
$149$	−18.2537	−1.49540	−0.747701	−	0.664036i	$-0.768842\pi$
$149$	−18.2537	−1.49540	−0.747701	+	0.664036i	$0.768842\pi$
$150$	0	0
$151$	−7.78861	−0.633828	−0.316914	−	0.948454i	$-0.602647\pi$
$151$	−7.78861	−0.633828	−0.316914	+	0.948454i	$0.602647\pi$
$152$	−4.02500	−0.326471
$153$	0	0
$154$	4.57320	0.368519
$155$	−15.6210	−1.25471
$156$	0	0
$157$	9.14015	0.729464	0.364732	−	0.931113i	$-0.381161\pi$
$157$	9.14015	0.729464	0.364732	+	0.931113i	$0.381161\pi$
$158$	−10.5693	−0.840846
$159$	0	0
$160$	1.52490	0.120554
$161$	−0.263768	−0.0207878
$162$	0	0
$163$	−20.5330	−1.60827	−0.804134	−	0.594449i	$-0.797370\pi$
$163$	−20.5330	−1.60827	−0.804134	+	0.594449i	$0.797370\pi$
$164$	6.15244	0.480425
$165$	0	0
$166$	−11.5427	−0.895884
$167$	2.99767	0.231967	0.115983	−	0.993251i	$-0.462998\pi$
$167$	2.99767	0.231967	0.115983	+	0.993251i	$0.462998\pi$
$168$	0	0
$169$	17.4070	1.33900
$170$	−8.21046	−0.629713
$171$	0	0
$172$	−7.03714	−0.536577
$173$	−8.96255	−0.681410	−0.340705	−	0.940170i	$-0.610666\pi$
$173$	−8.96255	−0.681410	−0.340705	+	0.940170i	$0.610666\pi$
$174$	0	0
$175$	−2.42924	−0.183633
$176$	−5.03528	−0.379549
$177$	0	0
$178$	−13.7906	−1.03365
$179$	−0.418318	−0.0312665	−0.0156333	−	0.999878i	$-0.504976\pi$
$179$	−0.418318	−0.0312665	−0.0156333	+	0.999878i	$0.504976\pi$
$180$	0	0
$181$	3.84683	0.285932	0.142966	−	0.989728i	$-0.454336\pi$
$181$	3.84683	0.285932	0.142966	+	0.989728i	$0.454336\pi$
$182$	5.00821	0.371233
$183$	0	0
$184$	0.290419	0.0214100
$185$	10.3768	0.762918
$186$	0	0
$187$	27.1113	1.98258
$188$	10.5033	0.766031
$189$	0	0
$190$	6.13771	0.445276
$191$	12.5869	0.910757	0.455379	−	0.890298i	$-0.349504\pi$
$191$	12.5869	0.910757	0.455379	+	0.890298i	$0.349504\pi$
$192$	0	0
$193$	2.54364	0.183096	0.0915478	−	0.995801i	$-0.470819\pi$
$193$	2.54364	0.183096	0.0915478	+	0.995801i	$0.470819\pi$
$194$	−2.46450	−0.176941
$195$	0	0
$196$	−6.17512	−0.441080
$197$	−14.2791	−1.01734	−0.508672	−	0.860960i	$-0.669863\pi$
$197$	−14.2791	−1.01734	−0.508672	+	0.860960i	$0.669863\pi$
$198$	0	0
$199$	4.63234	0.328378	0.164189	−	0.986429i	$-0.447499\pi$
$199$	4.63234	0.328378	0.164189	+	0.986429i	$0.447499\pi$
$200$	2.67470	0.189130
$201$	0	0
$202$	−14.6472	−1.03057
$203$	−7.92133	−0.555968
$204$	0	0
$205$	−9.38182	−0.655255
$206$	−0.320702	−0.0223443
$207$	0	0
$208$	−5.51425	−0.382344
$209$	−20.2670	−1.40190
$210$	0	0
$211$	18.4454	1.26983	0.634916	−	0.772581i	$-0.281035\pi$
$211$	18.4454	1.26983	0.634916	+	0.772581i	$0.281035\pi$
$212$	−4.35234	−0.298920
$213$	0	0
$214$	13.9377	0.952763
$215$	10.7309	0.731841
$216$	0	0
$217$	9.30389	0.631589
$218$	0.707255	0.0479013
$219$	0	0
$220$	7.67827	0.517669
$221$	29.6902	1.99718
$222$	0	0
$223$	−1.00000	−0.0669650
$223$	−1.00000	−0.0669650
$224$	−0.908231	−0.0606837
$225$	0	0
$226$	10.8700	0.723059
$227$	24.1870	1.60535	0.802673	−	0.596420i	$-0.203411\pi$
$227$	24.1870	1.60535	0.802673	+	0.596420i	$0.203411\pi$
$228$	0	0
$229$	−4.01632	−0.265406	−0.132703	−	0.991156i	$-0.542366\pi$
$229$	−4.01632	−0.265406	−0.132703	+	0.991156i	$0.542366\pi$
$230$	−0.442859	−0.0292013
$231$	0	0
$232$	8.72171	0.572609
$233$	26.4607	1.73350	0.866751	−	0.498742i	$-0.166204\pi$
$233$	26.4607	1.73350	0.866751	+	0.498742i	$0.166204\pi$
$234$	0	0
$235$	−16.0164	−1.04480
$236$	10.0783	0.656045
$237$	0	0
$238$	4.89017	0.316982
$239$	4.41114	0.285333	0.142667	−	0.989771i	$-0.454432\pi$
$239$	4.41114	0.285333	0.142667	+	0.989771i	$0.454432\pi$
$240$	0	0
$241$	−13.9584	−0.899140	−0.449570	−	0.893245i	$-0.648423\pi$
$241$	−13.9584	−0.899140	−0.449570	+	0.893245i	$0.648423\pi$
$242$	−14.3541	−0.922713
$243$	0	0
$244$	−1.04184	−0.0666968
$245$	9.41640	0.601592
$246$	0	0
$247$	−22.1949	−1.41223
$248$	−10.2440	−0.650493
$249$	0	0
$250$	−11.7031	−0.740169
$251$	7.55760	0.477032	0.238516	−	0.971139i	$-0.423339\pi$
$251$	7.55760	0.477032	0.238516	+	0.971139i	$0.423339\pi$
$252$	0	0
$253$	1.46234	0.0919367
$254$	9.08270	0.569899
$255$	0	0
$256$	1.00000	0.0625000
$257$	−14.1502	−0.882664	−0.441332	−	0.897344i	$-0.645494\pi$
$257$	−14.1502	−0.882664	−0.441332	+	0.897344i	$0.645494\pi$
$258$	0	0
$259$	−6.18045	−0.384034
$260$	8.40865	0.521482
$261$	0	0
$262$	13.8183	0.853700
$263$	−6.88918	−0.424805	−0.212403	−	0.977182i	$-0.568129\pi$
$263$	−6.88918	−0.424805	−0.212403	+	0.977182i	$0.568129\pi$
$264$	0	0
$265$	6.63687	0.407700
$266$	−3.65563	−0.224141
$267$	0	0
$268$	15.3492	0.937599
$269$	−12.5832	−0.767208	−0.383604	−	0.923498i	$-0.625317\pi$
$269$	−12.5832	−0.767208	−0.383604	+	0.923498i	$0.625317\pi$
$270$	0	0
$271$	22.8747	1.38954	0.694769	−	0.719232i	$-0.255506\pi$
$271$	22.8747	1.38954	0.694769	+	0.719232i	$0.255506\pi$
$272$	−5.38428	−0.326470
$273$	0	0
$274$	−13.2046	−0.797719
$275$	13.4678	0.812141
$276$	0	0
$277$	9.30561	0.559120	0.279560	−	0.960128i	$-0.409811\pi$
$277$	9.30561	0.559120	0.279560	+	0.960128i	$0.409811\pi$
$278$	7.84073	0.470256
$279$	0	0
$280$	1.38496	0.0827670
$281$	−0.0277078	−0.00165291	−0.000826453	−	1.00000i	$-0.500263\pi$
$281$	−0.0277078	−0.00165291	−0.000826453		1.00000i	$0.500263\pi$
$282$	0	0
$283$	20.3178	1.20777	0.603884	−	0.797072i	$-0.293619\pi$
$283$	20.3178	1.20777	0.603884	+	0.797072i	$0.293619\pi$
$284$	2.68271	0.159190
$285$	0	0
$286$	−27.7658	−1.64183
$287$	5.58784	0.329840
$288$	0	0
$289$	11.9904	0.705319
$290$	−13.2997	−0.780985
$291$	0	0
$292$	−10.2734	−0.601208
$293$	−0.635336	−0.0371167	−0.0185584	−	0.999828i	$-0.505908\pi$
$293$	−0.635336	−0.0371167	−0.0185584	+	0.999828i	$0.505908\pi$
$294$	0	0
$295$	−15.3684	−0.894784
$296$	6.80493	0.395528
$297$	0	0
$298$	18.2537	1.05741
$299$	1.60145	0.0926140
$300$	0	0
$301$	−6.39134	−0.368391
$302$	7.78861	0.448184
$303$	0	0
$304$	4.02500	0.230850
$305$	1.58869	0.0909683
$306$	0	0
$307$	−13.1386	−0.749860	−0.374930	−	0.927053i	$-0.622333\pi$
$307$	−13.1386	−0.749860	−0.374930	+	0.927053i	$0.622333\pi$
$308$	−4.57320	−0.260582
$309$	0	0
$310$	15.6210	0.887212
$311$	13.8305	0.784253	0.392127	−	0.919911i	$-0.371739\pi$
$311$	13.8305	0.784253	0.392127	+	0.919911i	$0.371739\pi$
$312$	0	0
$313$	24.5707	1.38882	0.694409	−	0.719581i	$-0.255666\pi$
$313$	24.5707	1.38882	0.694409	+	0.719581i	$0.255666\pi$
$314$	−9.14015	−0.515809
$315$	0	0
$316$	10.5693	0.594568
$317$	21.9969	1.23547	0.617734	−	0.786387i	$-0.288051\pi$
$317$	21.9969	1.23547	0.617734	+	0.786387i	$0.288051\pi$
$318$	0	0
$319$	43.9163	2.45884
$320$	−1.52490	−0.0852442
$321$	0	0
$322$	0.263768	0.0146992
$323$	−21.6717	−1.20585
$324$	0	0
$325$	14.7489	0.818124
$326$	20.5330	1.13722
$327$	0	0
$328$	−6.15244	−0.339712
$329$	9.53941	0.525925
$330$	0	0
$331$	19.2357	1.05729	0.528645	−	0.848843i	$-0.322700\pi$
$331$	19.2357	1.05729	0.528645	+	0.848843i	$0.322700\pi$
$332$	11.5427	0.633485
$333$	0	0
$334$	−2.99767	−0.164025
$335$	−23.4059	−1.27880
$336$	0	0
$337$	−18.5164	−1.00865	−0.504326	−	0.863513i	$-0.668259\pi$
$337$	−18.5164	−1.00865	−0.504326	+	0.863513i	$0.668259\pi$
$338$	−17.4070	−0.946813
$339$	0	0
$340$	8.21046	0.445274
$341$	−51.5813	−2.79328
$342$	0	0
$343$	−11.9660	−0.646106
$344$	7.03714	0.379417
$345$	0	0
$346$	8.96255	0.481829
$347$	−12.3571	−0.663363	−0.331681	−	0.943391i	$-0.607616\pi$
$347$	−12.3571	−0.663363	−0.331681	+	0.943391i	$0.607616\pi$
$348$	0	0
$349$	−1.72893	−0.0925476	−0.0462738	−	0.998929i	$-0.514735\pi$
$349$	−1.72893	−0.0925476	−0.0462738	+	0.998929i	$0.514735\pi$
$350$	2.42924	0.129848
$351$	0	0
$352$	5.03528	0.268381
$353$	31.0336	1.65175	0.825876	−	0.563852i	$-0.190681\pi$
$353$	31.0336	1.65175	0.825876	+	0.563852i	$0.190681\pi$
$354$	0	0
$355$	−4.09086	−0.217120
$356$	13.7906	0.730899
$357$	0	0
$358$	0.418318	0.0221088
$359$	−8.32739	−0.439503	−0.219751	−	0.975556i	$-0.570525\pi$
$359$	−8.32739	−0.439503	−0.219751	+	0.975556i	$0.570525\pi$
$360$	0	0
$361$	−2.79935	−0.147334
$362$	−3.84683	−0.202185
$363$	0	0
$364$	−5.00821	−0.262502
$365$	15.6659	0.819992
$366$	0	0
$367$	−13.8316	−0.722004	−0.361002	−	0.932565i	$-0.617565\pi$
$367$	−13.8316	−0.722004	−0.361002	+	0.932565i	$0.617565\pi$
$368$	−0.290419	−0.0151392
$369$	0	0
$370$	−10.3768	−0.539464
$371$	−3.95293	−0.205226
$372$	0	0
$373$	12.7740	0.661411	0.330705	−	0.943734i	$-0.392713\pi$
$373$	12.7740	0.661411	0.330705	+	0.943734i	$0.392713\pi$
$374$	−27.1113	−1.40189
$375$	0	0
$376$	−10.5033	−0.541666
$377$	48.0937	2.47695
$378$	0	0
$379$	1.95926	0.100640	0.0503202	−	0.998733i	$-0.483976\pi$
$379$	1.95926	0.100640	0.0503202	+	0.998733i	$0.483976\pi$
$380$	−6.13771	−0.314858
$381$	0	0
$382$	−12.5869	−0.644003
$383$	19.1951	0.980824	0.490412	−	0.871491i	$-0.336846\pi$
$383$	19.1951	0.980824	0.490412	+	0.871491i	$0.336846\pi$
$384$	0	0
$385$	6.97365	0.355410
$386$	−2.54364	−0.129468
$387$	0	0
$388$	2.46450	0.125116
$389$	−8.18936	−0.415217	−0.207609	−	0.978212i	$-0.566568\pi$
$389$	−8.18936	−0.415217	−0.207609	+	0.978212i	$0.566568\pi$
$390$	0	0
$391$	1.56370	0.0790796
$392$	6.17512	0.311890
$393$	0	0
$394$	14.2791	0.719371
$395$	−16.1170	−0.810936
$396$	0	0
$397$	11.7832	0.591381	0.295690	−	0.955284i	$-0.404450\pi$
$397$	11.7832	0.591381	0.295690	+	0.955284i	$0.404450\pi$
$398$	−4.63234	−0.232198
$399$	0	0
$400$	−2.67470	−0.133735
$401$	12.9957	0.648976	0.324488	−	0.945890i	$-0.394808\pi$
$401$	12.9957	0.648976	0.324488	+	0.945890i	$0.394808\pi$
$402$	0	0
$403$	−56.4878	−2.81386
$404$	14.6472	0.728726
$405$	0	0
$406$	7.92133	0.393129
$407$	34.2647	1.69844
$408$	0	0
$409$	−30.1795	−1.49228	−0.746140	−	0.665789i	$-0.768095\pi$
$409$	−30.1795	−1.49228	−0.746140	+	0.665789i	$0.768095\pi$
$410$	9.38182	0.463335
$411$	0	0
$412$	0.320702	0.0157998
$413$	9.15347	0.450413
$414$	0	0
$415$	−17.6013	−0.864016
$416$	5.51425	0.270358
$417$	0	0
$418$	20.2670	0.991292
$419$	−6.69506	−0.327075	−0.163538	−	0.986537i	$-0.552291\pi$
$419$	−6.69506	−0.327075	−0.163538	+	0.986537i	$0.552291\pi$
$420$	0	0
$421$	11.6958	0.570017	0.285009	−	0.958525i	$-0.408004\pi$
$421$	11.6958	0.570017	0.285009	+	0.958525i	$0.408004\pi$
$422$	−18.4454	−0.897906
$423$	0	0
$424$	4.35234	0.211368
$425$	14.4013	0.698565
$426$	0	0
$427$	−0.946229	−0.0457913
$428$	−13.9377	−0.673705
$429$	0	0
$430$	−10.7309	−0.517490
$431$	−33.6223	−1.61953	−0.809764	−	0.586755i	$-0.800405\pi$
$431$	−33.6223	−1.61953	−0.809764	+	0.586755i	$0.800405\pi$
$432$	0	0
$433$	−2.58333	−0.124147	−0.0620735	−	0.998072i	$-0.519771\pi$
$433$	−2.58333	−0.124147	−0.0620735	+	0.998072i	$0.519771\pi$
$434$	−9.30389	−0.446601
$435$	0	0
$436$	−0.707255	−0.0338714
$437$	−1.16894	−0.0559179
$438$	0	0
$439$	31.5324	1.50496	0.752479	−	0.658616i	$-0.228858\pi$
$439$	31.5324	1.50496	0.752479	+	0.658616i	$0.228858\pi$
$440$	−7.67827	−0.366047
$441$	0	0
$442$	−29.6902	−1.41222
$443$	−9.94030	−0.472278	−0.236139	−	0.971719i	$-0.575882\pi$
$443$	−9.94030	−0.472278	−0.236139	+	0.971719i	$0.575882\pi$
$444$	0	0
$445$	−21.0292	−0.996878
$446$	1.00000	0.0473514
$447$	0	0
$448$	0.908231	0.0429099
$449$	−8.86967	−0.418586	−0.209293	−	0.977853i	$-0.567116\pi$
$449$	−8.86967	−0.418586	−0.209293	+	0.977853i	$0.567116\pi$
$450$	0	0
$451$	−30.9793	−1.45876
$452$	−10.8700	−0.511280
$453$	0	0
$454$	−24.1870	−1.13515
$455$	7.63700	0.358028
$456$	0	0
$457$	−14.3932	−0.673287	−0.336643	−	0.941632i	$-0.609292\pi$
$457$	−14.3932	−0.673287	−0.336643	+	0.941632i	$0.609292\pi$
$458$	4.01632	0.187670
$459$	0	0
$460$	0.442859	0.0206484
$461$	−8.86502	−0.412885	−0.206443	−	0.978459i	$-0.566189\pi$
$461$	−8.86502	−0.412885	−0.206443	+	0.978459i	$0.566189\pi$
$462$	0	0
$463$	27.4206	1.27434	0.637172	−	0.770722i	$-0.280104\pi$
$463$	27.4206	1.27434	0.637172	+	0.770722i	$0.280104\pi$
$464$	−8.72171	−0.404895
$465$	0	0
$466$	−26.4607	−1.22577
$467$	3.28901	0.152197	0.0760987	−	0.997100i	$-0.475754\pi$
$467$	3.28901	0.152197	0.0760987	+	0.997100i	$0.475754\pi$
$468$	0	0
$469$	13.9406	0.643716
$470$	16.0164	0.738782
$471$	0	0
$472$	−10.0783	−0.463894
$473$	35.4340	1.62926
$474$	0	0
$475$	−10.7657	−0.493962
$476$	−4.89017	−0.224140
$477$	0	0
$478$	−4.41114	−0.201761
$479$	22.5687	1.03119	0.515595	−	0.856832i	$-0.327571\pi$
$479$	22.5687	1.03119	0.515595	+	0.856832i	$0.327571\pi$
$480$	0	0
$481$	37.5241	1.71095
$482$	13.9584	0.635788
$483$	0	0
$484$	14.3541	0.652457
$485$	−3.75811	−0.170647
$486$	0	0
$487$	−35.2461	−1.59715	−0.798576	−	0.601894i	$-0.794413\pi$
$487$	−35.2461	−1.59715	−0.798576	+	0.601894i	$0.794413\pi$
$488$	1.04184	0.0471618
$489$	0	0
$490$	−9.41640	−0.425390
$491$	42.7367	1.92868	0.964340	−	0.264667i	$-0.0852620\pi$
$491$	42.7367	1.92868	0.964340	+	0.264667i	$0.0852620\pi$
$492$	0	0
$493$	46.9601	2.11498
$494$	22.1949	0.998594
$495$	0	0
$496$	10.2440	0.459968
$497$	2.43652	0.109293
$498$	0	0
$499$	10.9919	0.492067	0.246034	−	0.969261i	$-0.420873\pi$
$499$	10.9919	0.492067	0.246034	+	0.969261i	$0.420873\pi$
$500$	11.7031	0.523379
$501$	0	0
$502$	−7.55760	−0.337312
$503$	14.2266	0.634333	0.317166	−	0.948370i	$-0.397269\pi$
$503$	14.2266	0.634333	0.317166	+	0.948370i	$0.397269\pi$
$504$	0	0
$505$	−22.3355	−0.993915
$506$	−1.46234	−0.0650091
$507$	0	0
$508$	−9.08270	−0.402979
$509$	−15.3802	−0.681716	−0.340858	−	0.940115i	$-0.610718\pi$
$509$	−15.3802	−0.681716	−0.340858	+	0.940115i	$0.610718\pi$
$510$	0	0
$511$	−9.33066	−0.412764
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	14.1502	0.624137
$515$	−0.489036	−0.0215495
$516$	0	0
$517$	−52.8870	−2.32597
$518$	6.18045	0.271553
$519$	0	0
$520$	−8.40865	−0.368744
$521$	9.90684	0.434027	0.217013	−	0.976169i	$-0.430368\pi$
$521$	9.90684	0.434027	0.217013	+	0.976169i	$0.430368\pi$
$522$	0	0
$523$	−26.2497	−1.14782	−0.573910	−	0.818919i	$-0.694574\pi$
$523$	−26.2497	−1.14782	−0.573910	+	0.818919i	$0.694574\pi$
$524$	−13.8183	−0.603657
$525$	0	0
$526$	6.88918	0.300383
$527$	−55.1564	−2.40265
$528$	0	0
$529$	−22.9157	−0.996333
$530$	−6.63687	−0.288287
$531$	0	0
$532$	3.65563	0.158492
$533$	−33.9261	−1.46950
$534$	0	0
$535$	21.2536	0.918872
$536$	−15.3492	−0.662983
$537$	0	0
$538$	12.5832	0.542498
$539$	31.0934	1.33929
$540$	0	0
$541$	−28.3950	−1.22080	−0.610398	−	0.792094i	$-0.708991\pi$
$541$	−28.3950	−1.22080	−0.610398	+	0.792094i	$0.708991\pi$
$542$	−22.8747	−0.982552
$543$	0	0
$544$	5.38428	0.230849
$545$	1.07849	0.0461974
$546$	0	0
$547$	21.1196	0.903009	0.451504	−	0.892269i	$-0.350887\pi$
$547$	21.1196	0.903009	0.451504	+	0.892269i	$0.350887\pi$
$548$	13.2046	0.564073
$549$	0	0
$550$	−13.4678	−0.574271
$551$	−35.1049	−1.49552
$552$	0	0
$553$	9.59934	0.408206
$554$	−9.30561	−0.395357
$555$	0	0
$556$	−7.84073	−0.332521
$557$	30.5016	1.29239	0.646197	−	0.763170i	$-0.276358\pi$
$557$	30.5016	1.29239	0.646197	+	0.763170i	$0.276358\pi$
$558$	0	0
$559$	38.8045	1.64126
$560$	−1.38496	−0.0585251
$561$	0	0
$562$	0.0277078	0.00116878
$563$	−15.5201	−0.654094	−0.327047	−	0.945008i	$-0.606053\pi$
$563$	−15.5201	−0.654094	−0.327047	+	0.945008i	$0.606053\pi$
$564$	0	0
$565$	16.5756	0.697339
$566$	−20.3178	−0.854021
$567$	0	0
$568$	−2.68271	−0.112564
$569$	16.3185	0.684106	0.342053	−	0.939681i	$-0.388878\pi$
$569$	16.3185	0.684106	0.342053	+	0.939681i	$0.388878\pi$
$570$	0	0
$571$	5.45287	0.228195	0.114098	−	0.993470i	$-0.463602\pi$
$571$	5.45287	0.228195	0.114098	+	0.993470i	$0.463602\pi$
$572$	27.7658	1.16095
$573$	0	0
$574$	−5.58784	−0.233232
$575$	0.776783	0.0323941
$576$	0	0
$577$	33.9033	1.41141	0.705707	−	0.708503i	$-0.250629\pi$
$577$	33.9033	1.41141	0.705707	+	0.708503i	$0.250629\pi$
$578$	−11.9904	−0.498736
$579$	0	0
$580$	13.2997	0.552240
$581$	10.4834	0.434925
$582$	0	0
$583$	21.9153	0.907638
$584$	10.2734	0.425118
$585$	0	0
$586$	0.635336	0.0262455
$587$	−9.34004	−0.385505	−0.192752	−	0.981247i	$-0.561741\pi$
$587$	−9.34004	−0.385505	−0.192752	+	0.981247i	$0.561741\pi$
$588$	0	0
$589$	41.2320	1.69894
$590$	15.3684	0.632708
$591$	0	0
$592$	−6.80493	−0.279681
$593$	−28.4943	−1.17012	−0.585061	−	0.810989i	$-0.698929\pi$
$593$	−28.4943	−1.17012	−0.585061	+	0.810989i	$0.698929\pi$
$594$	0	0
$595$	7.45699	0.305707
$596$	−18.2537	−0.747701
$597$	0	0
$598$	−1.60145	−0.0654880
$599$	3.57992	0.146271	0.0731357	−	0.997322i	$-0.476699\pi$
$599$	3.57992	0.146271	0.0731357	+	0.997322i	$0.476699\pi$
$600$	0	0
$601$	−16.3665	−0.667603	−0.333802	−	0.942643i	$-0.608332\pi$
$601$	−16.3665	−0.667603	−0.333802	+	0.942643i	$0.608332\pi$
$602$	6.39134	0.260492
$603$	0	0
$604$	−7.78861	−0.316914
$605$	−21.8884	−0.889891
$606$	0	0
$607$	12.6254	0.512448	0.256224	−	0.966617i	$-0.417521\pi$
$607$	12.6254	0.512448	0.256224	+	0.966617i	$0.417521\pi$
$608$	−4.02500	−0.163235
$609$	0	0
$610$	−1.58869	−0.0643243
$611$	−57.9178	−2.34310
$612$	0	0
$613$	4.72129	0.190691	0.0953456	−	0.995444i	$-0.469604\pi$
$613$	4.72129	0.190691	0.0953456	+	0.995444i	$0.469604\pi$
$614$	13.1386	0.530231
$615$	0	0
$616$	4.57320	0.184259
$617$	29.6369	1.19314	0.596569	−	0.802562i	$-0.296530\pi$
$617$	29.6369	1.19314	0.596569	+	0.802562i	$0.296530\pi$
$618$	0	0
$619$	25.0913	1.00850	0.504252	−	0.863556i	$-0.331768\pi$
$619$	25.0913	1.00850	0.504252	+	0.863556i	$0.331768\pi$
$620$	−15.6210	−0.627354
$621$	0	0
$622$	−13.8305	−0.554551
$623$	12.5250	0.501804
$624$	0	0
$625$	−4.47253	−0.178901
$626$	−24.5707	−0.982042
$627$	0	0
$628$	9.14015	0.364732
$629$	36.6396	1.46092
$630$	0	0
$631$	3.69600	0.147135	0.0735676	−	0.997290i	$-0.476562\pi$
$631$	3.69600	0.147135	0.0735676	+	0.997290i	$0.476562\pi$
$632$	−10.5693	−0.420423
$633$	0	0
$634$	−21.9969	−0.873608
$635$	13.8502	0.549627
$636$	0	0
$637$	34.0511	1.34916
$638$	−43.9163	−1.73866
$639$	0	0
$640$	1.52490	0.0602768
$641$	−21.2458	−0.839160	−0.419580	−	0.907718i	$-0.637823\pi$
$641$	−21.2458	−0.839160	−0.419580	+	0.907718i	$0.637823\pi$
$642$	0	0
$643$	5.22312	0.205980	0.102990	−	0.994682i	$-0.467159\pi$
$643$	5.22312	0.205980	0.102990	+	0.994682i	$0.467159\pi$
$644$	−0.263768	−0.0103939
$645$	0	0
$646$	21.6717	0.852663
$647$	−2.07581	−0.0816084	−0.0408042	−	0.999167i	$-0.512992\pi$
$647$	−2.07581	−0.0816084	−0.0408042	+	0.999167i	$0.512992\pi$
$648$	0	0
$649$	−50.7473	−1.99201
$650$	−14.7489	−0.578501
$651$	0	0
$652$	−20.5330	−0.804134
$653$	−9.83876	−0.385020	−0.192510	−	0.981295i	$-0.561663\pi$
$653$	−9.83876	−0.385020	−0.192510	+	0.981295i	$0.561663\pi$
$654$	0	0
$655$	21.0715	0.823332
$656$	6.15244	0.240212
$657$	0	0
$658$	−9.53941	−0.371885
$659$	16.2116	0.631515	0.315758	−	0.948840i	$-0.397741\pi$
$659$	16.2116	0.631515	0.315758	+	0.948840i	$0.397741\pi$
$660$	0	0
$661$	41.9954	1.63343	0.816715	−	0.577041i	$-0.195793\pi$
$661$	41.9954	1.63343	0.816715	+	0.577041i	$0.195793\pi$
$662$	−19.2357	−0.747617
$663$	0	0
$664$	−11.5427	−0.447942
$665$	−5.57445	−0.216168
$666$	0	0
$667$	2.53296	0.0980764
$668$	2.99767	0.115983
$669$	0	0
$670$	23.4059	0.904247
$671$	5.24595	0.202518
$672$	0	0
$673$	34.5836	1.33310	0.666550	−	0.745460i	$-0.267770\pi$
$673$	34.5836	1.33310	0.666550	+	0.745460i	$0.267770\pi$
$674$	18.5164	0.713225
$675$	0	0
$676$	17.4070	0.669498
$677$	−23.8982	−0.918481	−0.459241	−	0.888312i	$-0.651878\pi$
$677$	−23.8982	−0.918481	−0.459241	+	0.888312i	$0.651878\pi$
$678$	0	0
$679$	2.23834	0.0858996
$680$	−8.21046	−0.314857
$681$	0	0
$682$	51.5813	1.97515
$683$	−5.70608	−0.218337	−0.109169	−	0.994023i	$-0.534819\pi$
$683$	−5.70608	−0.218337	−0.109169	+	0.994023i	$0.534819\pi$
$684$	0	0
$685$	−20.1356	−0.769343
$686$	11.9660	0.456866
$687$	0	0
$688$	−7.03714	−0.268288
$689$	23.9999	0.914324
$690$	0	0
$691$	35.9937	1.36926	0.684632	−	0.728889i	$-0.259963\pi$
$691$	35.9937	1.36926	0.684632	+	0.728889i	$0.259963\pi$
$692$	−8.96255	−0.340705
$693$	0	0
$694$	12.3571	0.469068
$695$	11.9563	0.453528
$696$	0	0
$697$	−33.1264	−1.25475
$698$	1.72893	0.0654410
$699$	0	0
$700$	−2.42924	−0.0918167
$701$	0.382733	0.0144556	0.00722780	−	0.999974i	$-0.497699\pi$
$701$	0.382733	0.0144556	0.00722780	+	0.999974i	$0.497699\pi$
$702$	0	0
$703$	−27.3899	−1.03303
$704$	−5.03528	−0.189774
$705$	0	0
$706$	−31.0336	−1.16797
$707$	13.3030	0.500313
$708$	0	0
$709$	−25.6028	−0.961533	−0.480766	−	0.876849i	$-0.659641\pi$
$709$	−25.6028	−0.961533	−0.480766	+	0.876849i	$0.659641\pi$
$710$	4.09086	0.153527
$711$	0	0
$712$	−13.7906	−0.516823
$713$	−2.97505	−0.111416
$714$	0	0
$715$	−42.3399	−1.58342
$716$	−0.418318	−0.0156333
$717$	0	0
$718$	8.32739	0.310776
$719$	14.7259	0.549183	0.274592	−	0.961561i	$-0.411457\pi$
$719$	14.7259	0.549183	0.274592	+	0.961561i	$0.411457\pi$
$720$	0	0
$721$	0.291271	0.0108475
$722$	2.79935	0.104181
$723$	0	0
$724$	3.84683	0.142966
$725$	23.3279	0.866377
$726$	0	0
$727$	−6.91099	−0.256314	−0.128157	−	0.991754i	$-0.540906\pi$
$727$	−6.91099	−0.256314	−0.128157	+	0.991754i	$0.540906\pi$
$728$	5.00821	0.185617
$729$	0	0
$730$	−15.6659	−0.579822
$731$	37.8899	1.40141
$732$	0	0
$733$	−16.5110	−0.609847	−0.304923	−	0.952377i	$-0.598631\pi$
$733$	−16.5110	−0.609847	−0.304923	+	0.952377i	$0.598631\pi$
$734$	13.8316	0.510534
$735$	0	0
$736$	0.290419	0.0107050
$737$	−77.2873	−2.84691
$738$	0	0
$739$	18.8814	0.694565	0.347283	−	0.937761i	$-0.387104\pi$
$739$	18.8814	0.694565	0.347283	+	0.937761i	$0.387104\pi$
$740$	10.3768	0.381459
$741$	0	0
$742$	3.95293	0.145117
$743$	−31.2072	−1.14488	−0.572440	−	0.819947i	$-0.694003\pi$
$743$	−31.2072	−1.14488	−0.572440	+	0.819947i	$0.694003\pi$
$744$	0	0
$745$	27.8350	1.01979
$746$	−12.7740	−0.467688
$747$	0	0
$748$	27.1113	0.991289
$749$	−12.6587	−0.462538
$750$	0	0
$751$	−38.5905	−1.40819	−0.704093	−	0.710108i	$-0.748646\pi$
$751$	−38.5905	−1.40819	−0.704093	+	0.710108i	$0.748646\pi$
$752$	10.5033	0.383016
$753$	0	0
$754$	−48.0937	−1.75147
$755$	11.8768	0.432242
$756$	0	0
$757$	−22.2037	−0.807009	−0.403504	−	0.914978i	$-0.632208\pi$
$757$	−22.2037	−0.807009	−0.403504	+	0.914978i	$0.632208\pi$
$758$	−1.95926	−0.0711635
$759$	0	0
$760$	6.13771	0.222638
$761$	27.3025	0.989713	0.494857	−	0.868975i	$-0.335221\pi$
$761$	27.3025	0.989713	0.494857	+	0.868975i	$0.335221\pi$
$762$	0	0
$763$	−0.642351	−0.0232547
$764$	12.5869	0.455379
$765$	0	0
$766$	−19.1951	−0.693548
$767$	−55.5745	−2.00668
$768$	0	0
$769$	−24.7062	−0.890927	−0.445464	−	0.895300i	$-0.646961\pi$
$769$	−24.7062	−0.890927	−0.445464	+	0.895300i	$0.646961\pi$
$770$	−6.97365	−0.251313
$771$	0	0
$772$	2.54364	0.0915478
$773$	36.8687	1.32607	0.663037	−	0.748587i	$-0.269267\pi$
$773$	36.8687	1.32607	0.663037	+	0.748587i	$0.269267\pi$
$774$	0	0
$775$	−27.3995	−0.984219
$776$	−2.46450	−0.0884705
$777$	0	0
$778$	8.18936	0.293603
$779$	24.7636	0.887248
$780$	0	0
$781$	−13.5082	−0.483362
$782$	−1.56370	−0.0559177
$783$	0	0
$784$	−6.17512	−0.220540
$785$	−13.9378	−0.497461
$786$	0	0
$787$	−19.7605	−0.704387	−0.352193	−	0.935927i	$-0.614564\pi$
$787$	−19.7605	−0.704387	−0.352193	+	0.935927i	$0.614564\pi$
$788$	−14.2791	−0.508672
$789$	0	0
$790$	16.1170	0.573418
$791$	−9.87244	−0.351023
$792$	0	0
$793$	5.74496	0.204009
$794$	−11.7832	−0.418169
$795$	0	0
$796$	4.63234	0.164189
$797$	−31.6917	−1.12258	−0.561289	−	0.827620i	$-0.689694\pi$
$797$	−31.6917	−1.12258	−0.561289	+	0.827620i	$0.689694\pi$
$798$	0	0
$799$	−56.5526	−2.00069
$800$	2.67470	0.0945648
$801$	0	0
$802$	−12.9957	−0.458895
$803$	51.7297	1.82550
$804$	0	0
$805$	0.402218	0.0141763
$806$	56.4878	1.98970
$807$	0	0
$808$	−14.6472	−0.515287
$809$	−26.4988	−0.931647	−0.465823	−	0.884878i	$-0.654242\pi$
$809$	−26.4988	−0.931647	−0.465823	+	0.884878i	$0.654242\pi$
$810$	0	0
$811$	36.9381	1.29707	0.648536	−	0.761184i	$-0.275382\pi$
$811$	36.9381	1.29707	0.648536	+	0.761184i	$0.275382\pi$
$812$	−7.92133	−0.277984
$813$	0	0
$814$	−34.2647	−1.20098
$815$	31.3106	1.09676
$816$	0	0
$817$	−28.3245	−0.990949
$818$	30.1795	1.05520
$819$	0	0
$820$	−9.38182	−0.327628
$821$	−29.6401	−1.03445	−0.517223	−	0.855851i	$-0.673034\pi$
$821$	−29.6401	−1.03445	−0.517223	+	0.855851i	$0.673034\pi$
$822$	0	0
$823$	6.64120	0.231498	0.115749	−	0.993279i	$-0.463073\pi$
$823$	6.64120	0.231498	0.115749	+	0.993279i	$0.463073\pi$
$824$	−0.320702	−0.0111722
$825$	0	0
$826$	−9.15347	−0.318490
$827$	−20.4247	−0.710235	−0.355118	−	0.934822i	$-0.615559\pi$
$827$	−20.4247	−0.710235	−0.355118	+	0.934822i	$0.615559\pi$
$828$	0	0
$829$	−31.5687	−1.09643	−0.548214	−	0.836338i	$-0.684692\pi$
$829$	−31.5687	−1.09643	−0.548214	+	0.836338i	$0.684692\pi$
$830$	17.6013	0.610951
$831$	0	0
$832$	−5.51425	−0.191172
$833$	33.2485	1.15199
$834$	0	0
$835$	−4.57114	−0.158191
$836$	−20.2670	−0.700950
$837$	0	0
$838$	6.69506	0.231277
$839$	−8.97841	−0.309969	−0.154984	−	0.987917i	$-0.549533\pi$
$839$	−8.97841	−0.309969	−0.154984	+	0.987917i	$0.549533\pi$
$840$	0	0
$841$	47.0683	1.62305
$842$	−11.6958	−0.403063
$843$	0	0
$844$	18.4454	0.634916
$845$	−26.5438	−0.913134
$846$	0	0
$847$	13.0368	0.447949
$848$	−4.35234	−0.149460
$849$	0	0
$850$	−14.4013	−0.493960
$851$	1.97628	0.0677461
$852$	0	0
$853$	13.1879	0.451544	0.225772	−	0.974180i	$-0.427510\pi$
$853$	13.1879	0.451544	0.225772	+	0.974180i	$0.427510\pi$
$854$	0.946229	0.0323793
$855$	0	0
$856$	13.9377	0.476382
$857$	−29.7611	−1.01662	−0.508310	−	0.861174i	$-0.669730\pi$
$857$	−29.7611	−1.01662	−0.508310	+	0.861174i	$0.669730\pi$
$858$	0	0
$859$	36.6554	1.25067	0.625333	−	0.780358i	$-0.284963\pi$
$859$	36.6554	1.25067	0.625333	+	0.780358i	$0.284963\pi$
$860$	10.7309	0.365920
$861$	0	0
$862$	33.6223	1.14518
$863$	−4.53228	−0.154280	−0.0771402	−	0.997020i	$-0.524579\pi$
$863$	−4.53228	−0.154280	−0.0771402	+	0.997020i	$0.524579\pi$
$864$	0	0
$865$	13.6669	0.464690
$866$	2.58333	0.0877852
$867$	0	0
$868$	9.30389	0.315795
$869$	−53.2193	−1.80534
$870$	0	0
$871$	−84.6391	−2.86789
$872$	0.707255	0.0239507
$873$	0	0
$874$	1.16894	0.0395400
$875$	10.6291	0.359330
$876$	0	0
$877$	33.4562	1.12974	0.564868	−	0.825182i	$-0.308927\pi$
$877$	33.4562	1.12974	0.564868	+	0.825182i	$0.308927\pi$
$878$	−31.5324	−1.06417
$879$	0	0
$880$	7.67827	0.258835
$881$	−52.4396	−1.76673	−0.883367	−	0.468682i	$-0.844729\pi$
$881$	−52.4396	−1.76673	−0.883367	+	0.468682i	$0.844729\pi$
$882$	0	0
$883$	21.7867	0.733181	0.366591	−	0.930382i	$-0.380525\pi$
$883$	21.7867	0.733181	0.366591	+	0.930382i	$0.380525\pi$
$884$	29.6902	0.998591
$885$	0	0
$886$	9.94030	0.333951
$887$	4.88790	0.164120	0.0820598	−	0.996627i	$-0.473850\pi$
$887$	4.88790	0.164120	0.0820598	+	0.996627i	$0.473850\pi$
$888$	0	0
$889$	−8.24918	−0.276669
$890$	21.0292	0.704899
$891$	0	0
$892$	−1.00000	−0.0334825
$893$	42.2758	1.41470
$894$	0	0
$895$	0.637891	0.0213223
$896$	−0.908231	−0.0303419
$897$	0	0
$898$	8.86967	0.295985
$899$	−89.3450	−2.97982
$900$	0	0
$901$	23.4342	0.780707
$902$	30.9793	1.03150
$903$	0	0
$904$	10.8700	0.361530
$905$	−5.86601	−0.194993
$906$	0	0
$907$	−53.5480	−1.77803	−0.889016	−	0.457877i	$-0.848610\pi$
$907$	−53.5480	−1.77803	−0.889016	+	0.457877i	$0.848610\pi$
$908$	24.1870	0.802673
$909$	0	0
$910$	−7.63700	−0.253164
$911$	−1.21606	−0.0402900	−0.0201450	−	0.999797i	$-0.506413\pi$
$911$	−1.21606	−0.0402900	−0.0201450	+	0.999797i	$0.506413\pi$
$912$	0	0
$913$	−58.1205	−1.92351
$914$	14.3932	0.476086
$915$	0	0
$916$	−4.01632	−0.132703
$917$	−12.5502	−0.414446
$918$	0	0
$919$	−0.768926	−0.0253645	−0.0126823	−	0.999920i	$-0.504037\pi$
$919$	−0.768926	−0.0253645	−0.0126823	+	0.999920i	$0.504037\pi$
$920$	−0.442859	−0.0146006
$921$	0	0
$922$	8.86502	0.291954
$923$	−14.7932	−0.486923
$924$	0	0
$925$	18.2011	0.598449
$926$	−27.4206	−0.901097
$927$	0	0
$928$	8.72171	0.286304
$929$	9.48407	0.311162	0.155581	−	0.987823i	$-0.450275\pi$
$929$	9.48407	0.311162	0.155581	+	0.987823i	$0.450275\pi$
$930$	0	0
$931$	−24.8549	−0.814585
$932$	26.4607	0.866751
$933$	0	0
$934$	−3.28901	−0.107620
$935$	−41.3419	−1.35203
$936$	0	0
$937$	8.32884	0.272091	0.136046	−	0.990703i	$-0.456561\pi$
$937$	8.32884	0.272091	0.136046	+	0.990703i	$0.456561\pi$
$938$	−13.9406	−0.455176
$939$	0	0
$940$	−16.0164	−0.522398
$941$	15.5635	0.507356	0.253678	−	0.967289i	$-0.418360\pi$
$941$	15.5635	0.507356	0.253678	+	0.967289i	$0.418360\pi$
$942$	0	0
$943$	−1.78679	−0.0581858
$944$	10.0783	0.328022
$945$	0	0
$946$	−35.4340	−1.15206
$947$	28.0142	0.910340	0.455170	−	0.890405i	$-0.349578\pi$
$947$	28.0142	0.910340	0.455170	+	0.890405i	$0.349578\pi$
$948$	0	0
$949$	56.6503	1.83895
$950$	10.7657	0.349284
$951$	0	0
$952$	4.89017	0.158491
$953$	21.0679	0.682458	0.341229	−	0.939980i	$-0.389157\pi$
$953$	21.0679	0.682458	0.341229	+	0.939980i	$0.389157\pi$
$954$	0	0
$955$	−19.1937	−0.621095
$956$	4.41114	0.142667
$957$	0	0
$958$	−22.5687	−0.729162
$959$	11.9928	0.387269
$960$	0	0
$961$	73.9389	2.38513
$962$	−37.5241	−1.20982
$963$	0	0
$964$	−13.9584	−0.449570
$965$	−3.87879	−0.124863
$966$	0	0
$967$	40.1650	1.29162	0.645810	−	0.763498i	$-0.276520\pi$
$967$	40.1650	1.29162	0.645810	+	0.763498i	$0.276520\pi$
$968$	−14.3541	−0.461357
$969$	0	0
$970$	3.75811	0.120666
$971$	−22.7819	−0.731107	−0.365553	−	0.930790i	$-0.619120\pi$
$971$	−22.7819	−0.731107	−0.365553	+	0.930790i	$0.619120\pi$
$972$	0	0
$973$	−7.12119	−0.228295
$974$	35.2461	1.12936
$975$	0	0
$976$	−1.04184	−0.0333484
$977$	−20.8218	−0.666148	−0.333074	−	0.942901i	$-0.608086\pi$
$977$	−20.8218	−0.666148	−0.333074	+	0.942901i	$0.608086\pi$
$978$	0	0
$979$	−69.4394	−2.21929
$980$	9.41640	0.300796
$981$	0	0
$982$	−42.7367	−1.36378
$983$	−27.2774	−0.870014	−0.435007	−	0.900427i	$-0.643254\pi$
$983$	−27.2774	−0.870014	−0.435007	+	0.900427i	$0.643254\pi$
$984$	0	0
$985$	21.7741	0.693782
$986$	−46.9601	−1.49551
$987$	0	0
$988$	−22.1949	−0.706113
$989$	2.04372	0.0649866
$990$	0	0
$991$	−14.1352	−0.449020	−0.224510	−	0.974472i	$-0.572078\pi$
$991$	−14.1352	−0.449020	−0.224510	+	0.974472i	$0.572078\pi$
$992$	−10.2440	−0.325246
$993$	0	0
$994$	−2.43652	−0.0772818
$995$	−7.06383	−0.223938
$996$	0	0
$997$	−43.9727	−1.39263	−0.696314	−	0.717737i	$-0.745178\pi$
$997$	−43.9727	−1.39263	−0.696314	+	0.717737i	$0.745178\pi$
$998$	−10.9919	−0.347944
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4014.2.a.w.1.3		7
3.2	odd	2		446.2.a.e.1.2	✓	7
12.11	even	2		3568.2.a.l.1.6		7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
446.2.a.e.1.2	✓	7	3.2	odd	2
3568.2.a.l.1.6		7	12.11	even	2
4014.2.a.w.1.3		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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} -1.52490 q^{5} +0.908231 q^{7} -1.00000 q^{8} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{4} -1.52490 q^{5} +0.908231 q^{7} -1.00000 q^{8} +1.52490 q^{10} -5.03528 q^{11} -5.51425 q^{13} -0.908231 q^{14} +1.00000 q^{16} -5.38428 q^{17} +4.02500 q^{19} -1.52490 q^{20} +5.03528 q^{22} -0.290419 q^{23} -2.67470 q^{25} +5.51425 q^{26} +0.908231 q^{28} -8.72171 q^{29} +10.2440 q^{31} -1.00000 q^{32} +5.38428 q^{34} -1.38496 q^{35} -6.80493 q^{37} -4.02500 q^{38} +1.52490 q^{40} +6.15244 q^{41} -7.03714 q^{43} -5.03528 q^{44} +0.290419 q^{46} +10.5033 q^{47} -6.17512 q^{49} +2.67470 q^{50} -5.51425 q^{52} -4.35234 q^{53} +7.67827 q^{55} -0.908231 q^{56} +8.72171 q^{58} +10.0783 q^{59} -1.04184 q^{61} -10.2440 q^{62} +1.00000 q^{64} +8.40865 q^{65} +15.3492 q^{67} -5.38428 q^{68} +1.38496 q^{70} +2.68271 q^{71} -10.2734 q^{73} +6.80493 q^{74} +4.02500 q^{76} -4.57320 q^{77} +10.5693 q^{79} -1.52490 q^{80} -6.15244 q^{82} +11.5427 q^{83} +8.21046 q^{85} +7.03714 q^{86} +5.03528 q^{88} +13.7906 q^{89} -5.00821 q^{91} -0.290419 q^{92} -10.5033 q^{94} -6.13771 q^{95} +2.46450 q^{97} +6.17512 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q - 7 q^{2} + 7 q^{4} - 2 q^{5} + 6 q^{7} - 7 q^{8}+O(q^{10}) \) 7 * q - 7 * q^2 + 7 * q^4 - 2 * q^5 + 6 * q^7 - 7 * q^8	\( 7 q - 7 q^{2} + 7 q^{4} - 2 q^{5} + 6 q^{7} - 7 q^{8} + 2 q^{10} - 9 q^{11} - 2 q^{13} - 6 q^{14} + 7 q^{16} + 7 q^{17} - 2 q^{19} - 2 q^{20} + 9 q^{22} - 15 q^{23} + 13 q^{25} + 2 q^{26} + 6 q^{28} - 9 q^{29} - 2 q^{31} - 7 q^{32} - 7 q^{34} + 4 q^{35} + 5 q^{37} + 2 q^{38} + 2 q^{40} + 33 q^{41} + 20 q^{43} - 9 q^{44} + 15 q^{46} + 2 q^{47} + 3 q^{49} - 13 q^{50} - 2 q^{52} + 13 q^{53} - 18 q^{55} - 6 q^{56} + 9 q^{58} - 9 q^{59} + 8 q^{61} + 2 q^{62} + 7 q^{64} + 44 q^{65} + 29 q^{67} + 7 q^{68} - 4 q^{70} - 37 q^{73} - 5 q^{74} - 2 q^{76} + 18 q^{77} + 32 q^{79} - 2 q^{80} - 33 q^{82} + 6 q^{83} - 4 q^{85} - 20 q^{86} + 9 q^{88} + 17 q^{89} - 4 q^{91} - 15 q^{92} - 2 q^{94} + 12 q^{95} + 12 q^{97} - 3 q^{98}+O(q^{100}) \) 7 * q - 7 * q^2 + 7 * q^4 - 2 * q^5 + 6 * q^7 - 7 * q^8 + 2 * q^10 - 9 * q^11 - 2 * q^13 - 6 * q^14 + 7 * q^16 + 7 * q^17 - 2 * q^19 - 2 * q^20 + 9 * q^22 - 15 * q^23 + 13 * q^25 + 2 * q^26 + 6 * q^28 - 9 * q^29 - 2 * q^31 - 7 * q^32 - 7 * q^34 + 4 * q^35 + 5 * q^37 + 2 * q^38 + 2 * q^40 + 33 * q^41 + 20 * q^43 - 9 * q^44 + 15 * q^46 + 2 * q^47 + 3 * q^49 - 13 * q^50 - 2 * q^52 + 13 * q^53 - 18 * q^55 - 6 * q^56 + 9 * q^58 - 9 * q^59 + 8 * q^61 + 2 * q^62 + 7 * q^64 + 44 * q^65 + 29 * q^67 + 7 * q^68 - 4 * q^70 - 37 * q^73 - 5 * q^74 - 2 * q^76 + 18 * q^77 + 32 * q^79 - 2 * q^80 - 33 * q^82 + 6 * q^83 - 4 * q^85 - 20 * q^86 + 9 * q^88 + 17 * q^89 - 4 * q^91 - 15 * q^92 - 2 * q^94 + 12 * q^95 + 12 * q^97 - 3 * q^98

Introduction

L-functions

Modular forms

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Database

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