LMFDB - Embedded newform 6039.2.a.i.1.5

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:	$13$
Coefficient field:	$\mathbb{Q}[x]/(x^{13} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{13} - 2 x^{12} - 19 x^{11} + 35 x^{10} + 136 x^{9} - 220 x^{8} - 469 x^{7} + 610 x^{6} + 841 x^{5} + \cdots - 47 $ x^13 - 2x^12 - 19x^11 + 35x^10 + 136x^9 - 220x^8 - 469x^7 + 610x^6 + 841x^5 - 760x^4 - 742x^3 + 366x^2 + 236x - 47
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 2013)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$1.50067$ 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)

$f(q)$	$=$	$q-1.50067 q^{2} +0.252024 q^{4} +0.569898 q^{5} -3.97554 q^{7} +2.62314 q^{8} +O(q^{10})$	\(q-1.50067 q^{2} +0.252024 q^{4} +0.569898 q^{5} -3.97554 q^{7} +2.62314 q^{8} -0.855232 q^{10} -1.00000 q^{11} -3.48724 q^{13} +5.96599 q^{14} -4.44053 q^{16} -7.75664 q^{17} +1.56002 q^{19} +0.143628 q^{20} +1.50067 q^{22} -4.91362 q^{23} -4.67522 q^{25} +5.23321 q^{26} -1.00193 q^{28} -6.29679 q^{29} +2.87665 q^{31} +1.41751 q^{32} +11.6402 q^{34} -2.26565 q^{35} -9.34414 q^{37} -2.34109 q^{38} +1.49492 q^{40} -1.58419 q^{41} +5.31329 q^{43} -0.252024 q^{44} +7.37374 q^{46} -3.46571 q^{47} +8.80489 q^{49} +7.01598 q^{50} -0.878869 q^{52} -10.8254 q^{53} -0.569898 q^{55} -10.4284 q^{56} +9.44943 q^{58} -0.418492 q^{59} -1.00000 q^{61} -4.31692 q^{62} +6.75385 q^{64} -1.98737 q^{65} +9.84375 q^{67} -1.95486 q^{68} +3.40001 q^{70} -9.93167 q^{71} +7.11859 q^{73} +14.0225 q^{74} +0.393164 q^{76} +3.97554 q^{77} -13.1270 q^{79} -2.53065 q^{80} +2.37735 q^{82} -4.04331 q^{83} -4.42050 q^{85} -7.97351 q^{86} -2.62314 q^{88} +13.4410 q^{89} +13.8636 q^{91} -1.23835 q^{92} +5.20090 q^{94} +0.889055 q^{95} -1.77923 q^{97} -13.2133 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 13 q - 2 q^{2} + 16 q^{4} - 3 q^{5} + 11 q^{7} - 9 q^{8}+O(q^{10}) $ 13 * q - 2 * q^2 + 16 * q^4 - 3 * q^5 + 11 * q^7 - 9 * q^8	$ 13 q - 2 q^{2} + 16 q^{4} - 3 q^{5} + 11 q^{7} - 9 q^{8} + 6 q^{10} - 13 q^{11} + 13 q^{13} - q^{14} + 18 q^{16} - 17 q^{17} + 14 q^{19} + 7 q^{20} + 2 q^{22} - 7 q^{23} + 18 q^{25} + 10 q^{26} + 19 q^{28} + 6 q^{29} + 27 q^{31} - 5 q^{32} + 6 q^{34} - 14 q^{35} + 10 q^{37} - 2 q^{38} + 8 q^{40} - 3 q^{41} + 29 q^{43} - 16 q^{44} - 24 q^{46} - 8 q^{47} + 8 q^{49} + 27 q^{50} + 37 q^{52} + 24 q^{53} + 3 q^{55} - 24 q^{56} - 5 q^{58} - 13 q^{59} - 13 q^{61} - 39 q^{62} + 47 q^{64} + 11 q^{65} + 44 q^{67} + 8 q^{68} - 12 q^{70} - 3 q^{71} + 48 q^{73} + 22 q^{74} + 47 q^{76} - 11 q^{77} - 17 q^{79} + 26 q^{80} + 56 q^{82} - 50 q^{83} + 8 q^{85} - 18 q^{86} + 9 q^{88} + 15 q^{89} + 47 q^{91} - 14 q^{92} + 45 q^{94} + q^{95} + 27 q^{97} - 47 q^{98}+O(q^{100}) $ 13 * q - 2 * q^2 + 16 * q^4 - 3 * q^5 + 11 * q^7 - 9 * q^8 + 6 * q^10 - 13 * q^11 + 13 * q^13 - q^14 + 18 * q^16 - 17 * q^17 + 14 * q^19 + 7 * q^20 + 2 * q^22 - 7 * q^23 + 18 * q^25 + 10 * q^26 + 19 * q^28 + 6 * q^29 + 27 * q^31 - 5 * q^32 + 6 * q^34 - 14 * q^35 + 10 * q^37 - 2 * q^38 + 8 * q^40 - 3 * q^41 + 29 * q^43 - 16 * q^44 - 24 * q^46 - 8 * q^47 + 8 * q^49 + 27 * q^50 + 37 * q^52 + 24 * q^53 + 3 * q^55 - 24 * q^56 - 5 * q^58 - 13 * q^59 - 13 * q^61 - 39 * q^62 + 47 * q^64 + 11 * q^65 + 44 * q^67 + 8 * q^68 - 12 * q^70 - 3 * q^71 + 48 * q^73 + 22 * q^74 + 47 * q^76 - 11 * q^77 - 17 * q^79 + 26 * q^80 + 56 * q^82 - 50 * q^83 + 8 * q^85 - 18 * q^86 + 9 * q^88 + 15 * q^89 + 47 * q^91 - 14 * q^92 + 45 * q^94 + q^95 + 27 * q^97 - 47 * 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 $

$2$	−1.50067	−1.06114	−0.530569	−	0.847642i	$-0.678022\pi$
$2$	−1.50067	−1.06114	−0.530569	+	0.847642i	$0.678022\pi$
$3$	0	0
$3$	0	0
$4$	0.252024	0.126012
$5$	0.569898	0.254866	0.127433	−	0.991847i	$-0.459326\pi$
$5$	0.569898	0.254866	0.127433	+	0.991847i	$0.459326\pi$
$6$	0	0
$7$	−3.97554	−1.50261	−0.751306	−	0.659954i	$-0.770576\pi$
$7$	−3.97554	−1.50261	−0.751306	+	0.659954i	$0.770576\pi$
$8$	2.62314	0.927421
$9$	0	0
$10$	−0.855232	−0.270448
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	−3.48724	−0.967186	−0.483593	−	0.875293i	$-0.660669\pi$
$13$	−3.48724	−0.967186	−0.483593	+	0.875293i	$0.660669\pi$
$14$	5.96599	1.59448
$15$	0	0
$16$	−4.44053	−1.11013
$17$	−7.75664	−1.88126	−0.940631	−	0.339430i	$-0.889766\pi$
$17$	−7.75664	−1.88126	−0.940631	+	0.339430i	$0.889766\pi$
$18$	0	0
$19$	1.56002	0.357894	0.178947	−	0.983859i	$-0.442731\pi$
$19$	1.56002	0.357894	0.178947	+	0.983859i	$0.442731\pi$
$20$	0.143628	0.0321162
$21$	0	0
$22$	1.50067	0.319945
$23$	−4.91362	−1.02456	−0.512280	−	0.858819i	$-0.671199\pi$
$23$	−4.91362	−1.02456	−0.512280	+	0.858819i	$0.671199\pi$
$24$	0	0
$25$	−4.67522	−0.935043
$26$	5.23321	1.02632
$27$	0	0
$28$	−1.00193	−0.189347
$29$	−6.29679	−1.16928	−0.584642	−	0.811291i	$-0.698765\pi$
$29$	−6.29679	−1.16928	−0.584642	+	0.811291i	$0.698765\pi$
$30$	0	0
$31$	2.87665	0.516662	0.258331	−	0.966056i	$-0.416828\pi$
$31$	2.87665	0.516662	0.258331	+	0.966056i	$0.416828\pi$
$32$	1.41751	0.250583
$33$	0	0
$34$	11.6402	1.99628
$35$	−2.26565	−0.382965
$36$	0	0
$37$	−9.34414	−1.53617	−0.768083	−	0.640350i	$-0.778789\pi$
$37$	−9.34414	−1.53617	−0.768083	+	0.640350i	$0.778789\pi$
$38$	−2.34109	−0.379775
$39$	0	0
$40$	1.49492	0.236368
$41$	−1.58419	−0.247409	−0.123704	−	0.992319i	$-0.539477\pi$
$41$	−1.58419	−0.247409	−0.123704	+	0.992319i	$0.539477\pi$
$42$	0	0
$43$	5.31329	0.810269	0.405134	−	0.914257i	$-0.367225\pi$
$43$	5.31329	0.810269	0.405134	+	0.914257i	$0.367225\pi$
$44$	−0.252024	−0.0379941
$45$	0	0
$46$	7.37374	1.08720
$47$	−3.46571	−0.505526	−0.252763	−	0.967528i	$-0.581339\pi$
$47$	−3.46571	−0.505526	−0.252763	+	0.967528i	$0.581339\pi$
$48$	0	0
$49$	8.80489	1.25784
$50$	7.01598	0.992209
$51$	0	0
$52$	−0.878869	−0.121877
$53$	−10.8254	−1.48699	−0.743493	−	0.668744i	$-0.766832\pi$
$53$	−10.8254	−1.48699	−0.743493	+	0.668744i	$0.766832\pi$
$54$	0	0
$55$	−0.569898	−0.0768451
$56$	−10.4284	−1.39355
$57$	0	0
$58$	9.44943	1.24077
$59$	−0.418492	−0.0544831	−0.0272415	−	0.999629i	$-0.508672\pi$
$59$	−0.418492	−0.0544831	−0.0272415	+	0.999629i	$0.508672\pi$
$60$	0	0
$61$	−1.00000	−0.128037
$61$	−1.00000	−0.128037
$62$	−4.31692	−0.548249
$63$	0	0
$64$	6.75385	0.844231
$65$	−1.98737	−0.246503
$66$	0	0
$67$	9.84375	1.20260	0.601302	−	0.799022i	$-0.294649\pi$
$67$	9.84375	1.20260	0.601302	+	0.799022i	$0.294649\pi$
$68$	−1.95486	−0.237062
$69$	0	0
$70$	3.40001	0.406378
$71$	−9.93167	−1.17867	−0.589336	−	0.807888i	$-0.700611\pi$
$71$	−9.93167	−1.17867	−0.589336	+	0.807888i	$0.700611\pi$
$72$	0	0
$73$	7.11859	0.833168	0.416584	−	0.909097i	$-0.363227\pi$
$73$	7.11859	0.833168	0.416584	+	0.909097i	$0.363227\pi$
$74$	14.0225	1.63008
$75$	0	0
$76$	0.393164	0.0450990
$77$	3.97554	0.453054
$78$	0	0
$79$	−13.1270	−1.47690	−0.738450	−	0.674308i	$-0.764442\pi$
$79$	−13.1270	−1.47690	−0.738450	+	0.674308i	$0.764442\pi$
$80$	−2.53065	−0.282935
$81$	0	0
$82$	2.37735	0.262535
$83$	−4.04331	−0.443811	−0.221906	−	0.975068i	$-0.571228\pi$
$83$	−4.04331	−0.443811	−0.221906	+	0.975068i	$0.571228\pi$
$84$	0	0
$85$	−4.42050	−0.479470
$86$	−7.97351	−0.859806
$87$	0	0
$88$	−2.62314	−0.279628
$89$	13.4410	1.42474	0.712371	−	0.701803i	$-0.247621\pi$
$89$	13.4410	1.42474	0.712371	+	0.701803i	$0.247621\pi$
$90$	0	0
$91$	13.8636	1.45331
$92$	−1.23835	−0.129107
$93$	0	0
$94$	5.20090	0.536432
$95$	0.889055	0.0912151
$96$	0	0
$97$	−1.77923	−0.180653	−0.0903267	−	0.995912i	$-0.528791\pi$
$97$	−1.77923	−0.180653	−0.0903267	+	0.995912i	$0.528791\pi$
$98$	−13.2133	−1.33474
$99$	0	0
$100$	−1.17827	−0.117827
$101$	18.9035	1.88096	0.940482	−	0.339844i	$-0.110374\pi$
$101$	18.9035	1.88096	0.940482	+	0.339844i	$0.110374\pi$
$102$	0	0
$103$	2.41802	0.238254	0.119127	−	0.992879i	$-0.461990\pi$
$103$	2.41802	0.238254	0.119127	+	0.992879i	$0.461990\pi$
$104$	−9.14753	−0.896989
$105$	0	0
$106$	16.2454	1.57790
$107$	−10.7538	−1.03961	−0.519806	−	0.854285i	$-0.673996\pi$
$107$	−10.7538	−1.03961	−0.519806	+	0.854285i	$0.673996\pi$
$108$	0	0
$109$	−14.6227	−1.40060	−0.700298	−	0.713851i	$-0.746950\pi$
$109$	−14.6227	−1.40060	−0.700298	+	0.713851i	$0.746950\pi$
$110$	0.855232	0.0815432
$111$	0	0
$112$	17.6535	1.66810
$113$	8.88757	0.836072	0.418036	−	0.908430i	$-0.362719\pi$
$113$	8.88757	0.836072	0.418036	+	0.908430i	$0.362719\pi$
$114$	0	0
$115$	−2.80026	−0.261126
$116$	−1.58694	−0.147344
$117$	0	0
$118$	0.628021	0.0578140
$119$	30.8368	2.82681
$120$	0	0
$121$	1.00000	0.0909091
$122$	1.50067	0.135865
$123$	0	0
$124$	0.724986	0.0651057
$125$	−5.51389	−0.493177
$126$	0	0
$127$	−16.2612	−1.44295	−0.721473	−	0.692443i	$-0.756535\pi$
$127$	−16.2612	−1.44295	−0.721473	+	0.692443i	$0.756535\pi$
$128$	−12.9703	−1.14643
$129$	0	0
$130$	2.98240	0.261574
$131$	−16.7549	−1.46389	−0.731943	−	0.681366i	$-0.761386\pi$
$131$	−16.7549	−1.46389	−0.731943	+	0.681366i	$0.761386\pi$
$132$	0	0
$133$	−6.20193	−0.537776
$134$	−14.7723	−1.27613
$135$	0	0
$136$	−20.3468	−1.74472
$137$	−21.7052	−1.85440	−0.927198	−	0.374571i	$-0.877790\pi$
$137$	−21.7052	−1.85440	−0.927198	+	0.374571i	$0.877790\pi$
$138$	0	0
$139$	9.86673	0.836885	0.418442	−	0.908243i	$-0.362576\pi$
$139$	9.86673	0.836885	0.418442	+	0.908243i	$0.362576\pi$
$140$	−0.570999	−0.0482582
$141$	0	0
$142$	14.9042	1.25073
$143$	3.48724	0.291618
$144$	0	0
$145$	−3.58853	−0.298011
$146$	−10.6827	−0.884106
$147$	0	0
$148$	−2.35495	−0.193576
$149$	−16.9431	−1.38803	−0.694017	−	0.719959i	$-0.744161\pi$
$149$	−16.9431	−1.38803	−0.694017	+	0.719959i	$0.744161\pi$
$150$	0	0
$151$	16.5384	1.34587	0.672937	−	0.739700i	$-0.265032\pi$
$151$	16.5384	1.34587	0.672937	+	0.739700i	$0.265032\pi$
$152$	4.09217	0.331918
$153$	0	0
$154$	−5.96599	−0.480753
$155$	1.63940	0.131680
$156$	0	0
$157$	17.8806	1.42703	0.713515	−	0.700640i	$-0.247102\pi$
$157$	17.8806	1.42703	0.713515	+	0.700640i	$0.247102\pi$
$158$	19.6993	1.56719
$159$	0	0
$160$	0.807836	0.0638650
$161$	19.5343	1.53952
$162$	0	0
$163$	−0.855017	−0.0669701	−0.0334851	−	0.999439i	$-0.510661\pi$
$163$	−0.855017	−0.0669701	−0.0334851	+	0.999439i	$0.510661\pi$
$164$	−0.399254	−0.0311765
$165$	0	0
$166$	6.06770	0.470945
$167$	−13.4101	−1.03770	−0.518852	−	0.854864i	$-0.673640\pi$
$167$	−13.4101	−1.03770	−0.518852	+	0.854864i	$0.673640\pi$
$168$	0	0
$169$	−0.839160	−0.0645508
$170$	6.63373	0.508784
$171$	0	0
$172$	1.33908	0.102104
$173$	9.03817	0.687160	0.343580	−	0.939123i	$-0.388360\pi$
$173$	9.03817	0.687160	0.343580	+	0.939123i	$0.388360\pi$
$174$	0	0
$175$	18.5865	1.40501
$176$	4.44053	0.334718
$177$	0	0
$178$	−20.1706	−1.51185
$179$	26.1042	1.95112	0.975559	−	0.219736i	$-0.0705196\pi$
$179$	26.1042	1.95112	0.975559	+	0.219736i	$0.0705196\pi$
$180$	0	0
$181$	3.16435	0.235204	0.117602	−	0.993061i	$-0.462479\pi$
$181$	3.16435	0.235204	0.117602	+	0.993061i	$0.462479\pi$
$182$	−20.8048	−1.54216
$183$	0	0
$184$	−12.8891	−0.950198
$185$	−5.32521	−0.391517
$186$	0	0
$187$	7.75664	0.567222
$188$	−0.873443	−0.0637024
$189$	0	0
$190$	−1.33418	−0.0967917
$191$	0.339788	0.0245862	0.0122931	−	0.999924i	$-0.496087\pi$
$191$	0.339788	0.0245862	0.0122931	+	0.999924i	$0.496087\pi$
$192$	0	0
$193$	−10.9988	−0.791714	−0.395857	−	0.918312i	$-0.629552\pi$
$193$	−10.9988	−0.791714	−0.395857	+	0.918312i	$0.629552\pi$
$194$	2.67005	0.191698
$195$	0	0
$196$	2.21905	0.158503
$197$	−13.6647	−0.973571	−0.486786	−	0.873522i	$-0.661831\pi$
$197$	−13.6647	−0.973571	−0.486786	+	0.873522i	$0.661831\pi$
$198$	0	0
$199$	−5.05352	−0.358235	−0.179117	−	0.983828i	$-0.557324\pi$
$199$	−5.05352	−0.358235	−0.179117	+	0.983828i	$0.557324\pi$
$200$	−12.2638	−0.867179
$201$	0	0
$202$	−28.3679	−1.99596
$203$	25.0331	1.75698
$204$	0	0
$205$	−0.902827	−0.0630561
$206$	−3.62865	−0.252820
$207$	0	0
$208$	15.4852	1.07371
$209$	−1.56002	−0.107909
$210$	0	0
$211$	0.418379	0.0288024	0.0144012	−	0.999896i	$-0.495416\pi$
$211$	0.418379	0.0288024	0.0144012	+	0.999896i	$0.495416\pi$
$212$	−2.72827	−0.187378
$213$	0	0
$214$	16.1380	1.10317
$215$	3.02803	0.206510
$216$	0	0
$217$	−11.4362	−0.776342
$218$	21.9438	1.48622
$219$	0	0
$220$	−0.143628	−0.00968341
$221$	27.0493	1.81953
$222$	0	0
$223$	8.46953	0.567162	0.283581	−	0.958948i	$-0.408478\pi$
$223$	8.46953	0.567162	0.283581	+	0.958948i	$0.408478\pi$
$224$	−5.63536	−0.376528
$225$	0	0
$226$	−13.3373	−0.887187
$227$	1.28958	0.0855927	0.0427963	−	0.999084i	$-0.486373\pi$
$227$	1.28958	0.0855927	0.0427963	+	0.999084i	$0.486373\pi$
$228$	0	0
$229$	18.0318	1.19157	0.595787	−	0.803143i	$-0.296840\pi$
$229$	18.0318	1.19157	0.595787	+	0.803143i	$0.296840\pi$
$230$	4.20228	0.277090
$231$	0	0
$232$	−16.5174	−1.08442
$233$	−15.1799	−0.994470	−0.497235	−	0.867616i	$-0.665651\pi$
$233$	−15.1799	−0.994470	−0.497235	+	0.867616i	$0.665651\pi$
$234$	0	0
$235$	−1.97510	−0.128841
$236$	−0.105470	−0.00686553
$237$	0	0
$238$	−46.2760	−2.99963
$239$	−5.61796	−0.363396	−0.181698	−	0.983354i	$-0.558159\pi$
$239$	−5.61796	−0.363396	−0.181698	+	0.983354i	$0.558159\pi$
$240$	0	0
$241$	−9.85967	−0.635117	−0.317559	−	0.948239i	$-0.602863\pi$
$241$	−9.85967	−0.635117	−0.317559	+	0.948239i	$0.602863\pi$
$242$	−1.50067	−0.0964670
$243$	0	0
$244$	−0.252024	−0.0161342
$245$	5.01789	0.320581
$246$	0	0
$247$	−5.44018	−0.346150
$248$	7.54587	0.479163
$249$	0	0
$250$	8.27455	0.523329
$251$	−31.0856	−1.96210	−0.981052	−	0.193743i	$-0.937937\pi$
$251$	−31.0856	−1.96210	−0.981052	+	0.193743i	$0.937937\pi$
$252$	0	0
$253$	4.91362	0.308916
$254$	24.4027	1.53116
$255$	0	0
$256$	5.95657	0.372286
$257$	10.0493	0.626855	0.313428	−	0.949612i	$-0.398523\pi$
$257$	10.0493	0.626855	0.313428	+	0.949612i	$0.398523\pi$
$258$	0	0
$259$	37.1480	2.30826
$260$	−0.500866	−0.0310624
$261$	0	0
$262$	25.1437	1.55338
$263$	−19.8284	−1.22267	−0.611337	−	0.791370i	$-0.709368\pi$
$263$	−19.8284	−1.22267	−0.611337	+	0.791370i	$0.709368\pi$
$264$	0	0
$265$	−6.16938	−0.378982
$266$	9.30708	0.570654
$267$	0	0
$268$	2.48086	0.151543
$269$	11.7548	0.716702	0.358351	−	0.933587i	$-0.383339\pi$
$269$	11.7548	0.716702	0.358351	+	0.933587i	$0.383339\pi$
$270$	0	0
$271$	−19.0082	−1.15467	−0.577334	−	0.816508i	$-0.695907\pi$
$271$	−19.0082	−1.15467	−0.577334	+	0.816508i	$0.695907\pi$
$272$	34.4436	2.08845
$273$	0	0
$274$	32.5724	1.96777
$275$	4.67522	0.281926
$276$	0	0
$277$	−27.7437	−1.66695	−0.833477	−	0.552553i	$-0.813654\pi$
$277$	−27.7437	−1.66695	−0.833477	+	0.552553i	$0.813654\pi$
$278$	−14.8067	−0.888050
$279$	0	0
$280$	−5.94313	−0.355170
$281$	26.9692	1.60885	0.804424	−	0.594056i	$-0.202474\pi$
$281$	26.9692	1.60885	0.804424	+	0.594056i	$0.202474\pi$
$282$	0	0
$283$	30.2378	1.79745	0.898725	−	0.438513i	$-0.144495\pi$
$283$	30.2378	1.79745	0.898725	+	0.438513i	$0.144495\pi$
$284$	−2.50302	−0.148527
$285$	0	0
$286$	−5.23321	−0.309446
$287$	6.29800	0.371759
$288$	0	0
$289$	43.1655	2.53915
$290$	5.38521	0.316230
$291$	0	0
$292$	1.79406	0.104989
$293$	−6.99534	−0.408672	−0.204336	−	0.978901i	$-0.565504\pi$
$293$	−6.99534	−0.408672	−0.204336	+	0.978901i	$0.565504\pi$
$294$	0	0
$295$	−0.238498	−0.0138859
$296$	−24.5110	−1.42467
$297$	0	0
$298$	25.4261	1.47289
$299$	17.1350	0.990940
$300$	0	0
$301$	−21.1232	−1.21752
$302$	−24.8187	−1.42816
$303$	0	0
$304$	−6.92734	−0.397310
$305$	−0.569898	−0.0326323
$306$	0	0
$307$	7.80207	0.445288	0.222644	−	0.974900i	$-0.428531\pi$
$307$	7.80207	0.445288	0.222644	+	0.974900i	$0.428531\pi$
$308$	1.00193	0.0570904
$309$	0	0
$310$	−2.46020	−0.139730
$311$	1.36849	0.0775998	0.0387999	−	0.999247i	$-0.487647\pi$
$311$	1.36849	0.0775998	0.0387999	+	0.999247i	$0.487647\pi$
$312$	0	0
$313$	−5.14485	−0.290804	−0.145402	−	0.989373i	$-0.546448\pi$
$313$	−5.14485	−0.290804	−0.145402	+	0.989373i	$0.546448\pi$
$314$	−26.8330	−1.51427
$315$	0	0
$316$	−3.30832	−0.186107
$317$	−30.3916	−1.70696	−0.853482	−	0.521122i	$-0.825514\pi$
$317$	−30.3916	−1.70696	−0.853482	+	0.521122i	$0.825514\pi$
$318$	0	0
$319$	6.29679	0.352552
$320$	3.84900	0.215166
$321$	0	0
$322$	−29.3146	−1.63364
$323$	−12.1006	−0.673293
$324$	0	0
$325$	16.3036	0.904361
$326$	1.28310	0.0710645
$327$	0	0
$328$	−4.15555	−0.229452
$329$	13.7781	0.759609
$330$	0	0
$331$	17.3933	0.956024	0.478012	−	0.878353i	$-0.341357\pi$
$331$	17.3933	0.956024	0.478012	+	0.878353i	$0.341357\pi$
$332$	−1.01901	−0.0559256
$333$	0	0
$334$	20.1242	1.10115
$335$	5.60993	0.306503
$336$	0	0
$337$	3.38221	0.184241	0.0921203	−	0.995748i	$-0.470636\pi$
$337$	3.38221	0.184241	0.0921203	+	0.995748i	$0.470636\pi$
$338$	1.25931	0.0684972
$339$	0	0
$340$	−1.11407	−0.0604191
$341$	−2.87665	−0.155779
$342$	0	0
$343$	−7.17541	−0.387436
$344$	13.9375	0.751460
$345$	0	0
$346$	−13.5634	−0.729171
$347$	−22.0213	−1.18217	−0.591084	−	0.806610i	$-0.701300\pi$
$347$	−22.0213	−1.18217	−0.591084	+	0.806610i	$0.701300\pi$
$348$	0	0
$349$	−6.15318	−0.329372	−0.164686	−	0.986346i	$-0.552661\pi$
$349$	−6.15318	−0.329372	−0.164686	+	0.986346i	$0.552661\pi$
$350$	−27.8923	−1.49090
$351$	0	0
$352$	−1.41751	−0.0755535
$353$	2.81549	0.149853	0.0749267	−	0.997189i	$-0.476128\pi$
$353$	2.81549	0.149853	0.0749267	+	0.997189i	$0.476128\pi$
$354$	0	0
$355$	−5.66004	−0.300404
$356$	3.38746	0.179535
$357$	0	0
$358$	−39.1739	−2.07040
$359$	19.3450	1.02099	0.510494	−	0.859881i	$-0.329463\pi$
$359$	19.3450	1.02099	0.510494	+	0.859881i	$0.329463\pi$
$360$	0	0
$361$	−16.5663	−0.871912
$362$	−4.74865	−0.249584
$363$	0	0
$364$	3.49398	0.183134
$365$	4.05687	0.212346
$366$	0	0
$367$	9.48578	0.495154	0.247577	−	0.968868i	$-0.420366\pi$
$367$	9.48578	0.495154	0.247577	+	0.968868i	$0.420366\pi$
$368$	21.8191	1.13740
$369$	0	0
$370$	7.99140	0.415453
$371$	43.0368	2.23436
$372$	0	0
$373$	−28.1396	−1.45701	−0.728507	−	0.685038i	$-0.759785\pi$
$373$	−28.1396	−1.45701	−0.728507	+	0.685038i	$0.759785\pi$
$374$	−11.6402	−0.601900
$375$	0	0
$376$	−9.09105	−0.468835
$377$	21.9584	1.13092
$378$	0	0
$379$	−28.6718	−1.47277	−0.736386	−	0.676561i	$-0.763469\pi$
$379$	−28.6718	−1.47277	−0.736386	+	0.676561i	$0.763469\pi$
$380$	0.224063	0.0114942
$381$	0	0
$382$	−0.509911	−0.0260893
$383$	35.9467	1.83679	0.918395	−	0.395666i	$-0.129486\pi$
$383$	35.9467	1.83679	0.918395	+	0.395666i	$0.129486\pi$
$384$	0	0
$385$	2.26565	0.115468
$386$	16.5057	0.840118
$387$	0	0
$388$	−0.448409	−0.0227645
$389$	36.5774	1.85455	0.927274	−	0.374383i	$-0.122145\pi$
$389$	36.5774	1.85455	0.927274	+	0.374383i	$0.122145\pi$
$390$	0	0
$391$	38.1132	1.92747
$392$	23.0965	1.16655
$393$	0	0
$394$	20.5063	1.03309
$395$	−7.48104	−0.376412
$396$	0	0
$397$	17.4290	0.874738	0.437369	−	0.899282i	$-0.355910\pi$
$397$	17.4290	0.874738	0.437369	+	0.899282i	$0.355910\pi$
$398$	7.58370	0.380136
$399$	0	0
$400$	20.7604	1.03802
$401$	28.3816	1.41731	0.708655	−	0.705556i	$-0.249303\pi$
$401$	28.3816	1.41731	0.708655	+	0.705556i	$0.249303\pi$
$402$	0	0
$403$	−10.0316	−0.499708
$404$	4.76413	0.237024
$405$	0	0
$406$	−37.5665	−1.86440
$407$	9.34414	0.463172
$408$	0	0
$409$	27.6614	1.36777	0.683883	−	0.729592i	$-0.260290\pi$
$409$	27.6614	1.36777	0.683883	+	0.729592i	$0.260290\pi$
$410$	1.35485	0.0669112
$411$	0	0
$412$	0.609399	0.0300229
$413$	1.66373	0.0818669
$414$	0	0
$415$	−2.30428	−0.113113
$416$	−4.94319	−0.242360
$417$	0	0
$418$	2.34109	0.114506
$419$	1.14196	0.0557883	0.0278942	−	0.999611i	$-0.491120\pi$
$419$	1.14196	0.0557883	0.0278942	+	0.999611i	$0.491120\pi$
$420$	0	0
$421$	5.24502	0.255627	0.127813	−	0.991798i	$-0.459204\pi$
$421$	5.24502	0.255627	0.127813	+	0.991798i	$0.459204\pi$
$422$	−0.627850	−0.0305633
$423$	0	0
$424$	−28.3966	−1.37906
$425$	36.2640	1.75906
$426$	0	0
$427$	3.97554	0.192390
$428$	−2.71022	−0.131004
$429$	0	0
$430$	−4.54409	−0.219136
$431$	−39.7537	−1.91487	−0.957434	−	0.288651i	$-0.906793\pi$
$431$	−39.7537	−1.91487	−0.957434	+	0.288651i	$0.906793\pi$
$432$	0	0
$433$	−22.4352	−1.07817	−0.539084	−	0.842252i	$-0.681229\pi$
$433$	−22.4352	−1.07817	−0.539084	+	0.842252i	$0.681229\pi$
$434$	17.1621	0.823806
$435$	0	0
$436$	−3.68526	−0.176492
$437$	−7.66536	−0.366684
$438$	0	0
$439$	−5.46960	−0.261050	−0.130525	−	0.991445i	$-0.541666\pi$
$439$	−5.46960	−0.261050	−0.130525	+	0.991445i	$0.541666\pi$
$440$	−1.49492	−0.0712677
$441$	0	0
$442$	−40.5922	−1.93077
$443$	4.21494	0.200258	0.100129	−	0.994974i	$-0.468074\pi$
$443$	4.21494	0.200258	0.100129	+	0.994974i	$0.468074\pi$
$444$	0	0
$445$	7.66000	0.363119
$446$	−12.7100	−0.601837
$447$	0	0
$448$	−26.8502	−1.26855
$449$	−23.8577	−1.12592	−0.562958	−	0.826485i	$-0.690337\pi$
$449$	−23.8577	−1.12592	−0.562958	+	0.826485i	$0.690337\pi$
$450$	0	0
$451$	1.58419	0.0745966
$452$	2.23988	0.105355
$453$	0	0
$454$	−1.93525	−0.0908256
$455$	7.90087	0.370398
$456$	0	0
$457$	35.7911	1.67424	0.837119	−	0.547021i	$-0.184238\pi$
$457$	35.7911	1.67424	0.837119	+	0.547021i	$0.184238\pi$
$458$	−27.0598	−1.26442
$459$	0	0
$460$	−0.705734	−0.0329050
$461$	−33.0708	−1.54026	−0.770131	−	0.637886i	$-0.779809\pi$
$461$	−33.0708	−1.54026	−0.770131	+	0.637886i	$0.779809\pi$
$462$	0	0
$463$	−30.1684	−1.40205	−0.701023	−	0.713139i	$-0.747273\pi$
$463$	−30.1684	−1.40205	−0.701023	+	0.713139i	$0.747273\pi$
$464$	27.9611	1.29806
$465$	0	0
$466$	22.7801	1.05527
$467$	−13.3041	−0.615642	−0.307821	−	0.951444i	$-0.599600\pi$
$467$	−13.3041	−0.615642	−0.307821	+	0.951444i	$0.599600\pi$
$468$	0	0
$469$	−39.1342	−1.80705
$470$	2.96399	0.136718
$471$	0	0
$472$	−1.09777	−0.0505287
$473$	−5.31329	−0.244305
$474$	0	0
$475$	−7.29345	−0.334646
$476$	7.77163	0.356212
$477$	0	0
$478$	8.43073	0.385613
$479$	−36.2381	−1.65576	−0.827881	−	0.560904i	$-0.810454\pi$
$479$	−36.2381	−1.65576	−0.827881	+	0.560904i	$0.810454\pi$
$480$	0	0
$481$	32.5852	1.48576
$482$	14.7962	0.673947
$483$	0	0
$484$	0.252024	0.0114557
$485$	−1.01398	−0.0460425
$486$	0	0
$487$	4.94148	0.223920	0.111960	−	0.993713i	$-0.464287\pi$
$487$	4.94148	0.223920	0.111960	+	0.993713i	$0.464287\pi$
$488$	−2.62314	−0.118744
$489$	0	0
$490$	−7.53022	−0.340181
$491$	14.0150	0.632489	0.316245	−	0.948678i	$-0.397578\pi$
$491$	14.0150	0.632489	0.316245	+	0.948678i	$0.397578\pi$
$492$	0	0
$493$	48.8419	2.19973
$494$	8.16394	0.367313
$495$	0	0
$496$	−12.7739	−0.573563
$497$	39.4837	1.77109
$498$	0	0
$499$	15.0317	0.672913	0.336456	−	0.941699i	$-0.390772\pi$
$499$	15.0317	0.672913	0.336456	+	0.941699i	$0.390772\pi$
$500$	−1.38963	−0.0621463
$501$	0	0
$502$	46.6493	2.08206
$503$	4.69369	0.209281	0.104641	−	0.994510i	$-0.466631\pi$
$503$	4.69369	0.209281	0.104641	+	0.994510i	$0.466631\pi$
$504$	0	0
$505$	10.7730	0.479394
$506$	−7.37374	−0.327803
$507$	0	0
$508$	−4.09821	−0.181829
$509$	−2.29316	−0.101643	−0.0508214	−	0.998708i	$-0.516184\pi$
$509$	−2.29316	−0.101643	−0.0508214	+	0.998708i	$0.516184\pi$
$510$	0	0
$511$	−28.3002	−1.25193
$512$	17.0018	0.751381
$513$	0	0
$514$	−15.0807	−0.665179
$515$	1.37802	0.0607229
$516$	0	0
$517$	3.46571	0.152422
$518$	−55.7470	−2.44938
$519$	0	0
$520$	−5.21316	−0.228612
$521$	−6.53621	−0.286356	−0.143178	−	0.989697i	$-0.545732\pi$
$521$	−6.53621	−0.286356	−0.143178	+	0.989697i	$0.545732\pi$
$522$	0	0
$523$	28.4759	1.24516	0.622582	−	0.782554i	$-0.286084\pi$
$523$	28.4759	1.24516	0.622582	+	0.782554i	$0.286084\pi$
$524$	−4.22265	−0.184467
$525$	0	0
$526$	29.7560	1.29742
$527$	−22.3132	−0.971977
$528$	0	0
$529$	1.14362	0.0497224
$530$	9.25824	0.402152
$531$	0	0
$532$	−1.56304	−0.0677663
$533$	5.52445	0.239290
$534$	0	0
$535$	−6.12858	−0.264962
$536$	25.8215	1.11532
$537$	0	0
$538$	−17.6401	−0.760519
$539$	−8.80489	−0.379253
$540$	0	0
$541$	−22.4670	−0.965931	−0.482965	−	0.875640i	$-0.660440\pi$
$541$	−22.4670	−0.965931	−0.482965	+	0.875640i	$0.660440\pi$
$542$	28.5252	1.22526
$543$	0	0
$544$	−10.9951	−0.471412
$545$	−8.33342	−0.356965
$546$	0	0
$547$	−21.9796	−0.939779	−0.469889	−	0.882725i	$-0.655706\pi$
$547$	−21.9796	−0.939779	−0.469889	+	0.882725i	$0.655706\pi$
$548$	−5.47023	−0.233677
$549$	0	0
$550$	−7.01598	−0.299162
$551$	−9.82314	−0.418480
$552$	0	0
$553$	52.1868	2.21921
$554$	41.6342	1.76887
$555$	0	0
$556$	2.48665	0.105458
$557$	0.280608	0.0118897	0.00594487	−	0.999982i	$-0.498108\pi$
$557$	0.280608	0.0118897	0.00594487	+	0.999982i	$0.498108\pi$
$558$	0	0
$559$	−18.5287	−0.783681
$560$	10.0607	0.425142
$561$	0	0
$562$	−40.4720	−1.70721
$563$	−5.01496	−0.211355	−0.105678	−	0.994400i	$-0.533701\pi$
$563$	−5.01496	−0.211355	−0.105678	+	0.994400i	$0.533701\pi$
$564$	0	0
$565$	5.06501	0.213087
$566$	−45.3771	−1.90734
$567$	0	0
$568$	−26.0522	−1.09313
$569$	0.592920	0.0248565	0.0124282	−	0.999923i	$-0.496044\pi$
$569$	0.592920	0.0248565	0.0124282	+	0.999923i	$0.496044\pi$
$570$	0	0
$571$	0.318940	0.0133472	0.00667360	−	0.999978i	$-0.497876\pi$
$571$	0.318940	0.0133472	0.00667360	+	0.999978i	$0.497876\pi$
$572$	0.878869	0.0367474
$573$	0	0
$574$	−9.45125	−0.394488
$575$	22.9722	0.958008
$576$	0	0
$577$	36.7863	1.53143	0.765716	−	0.643179i	$-0.222385\pi$
$577$	36.7863	1.53143	0.765716	+	0.643179i	$0.222385\pi$
$578$	−64.7774	−2.69439
$579$	0	0
$580$	−0.904396	−0.0375530
$581$	16.0743	0.666876
$582$	0	0
$583$	10.8254	0.448343
$584$	18.6731	0.772698
$585$	0	0
$586$	10.4977	0.433657
$587$	−35.9317	−1.48306	−0.741530	−	0.670920i	$-0.765899\pi$
$587$	−35.9317	−1.48306	−0.741530	+	0.670920i	$0.765899\pi$
$588$	0	0
$589$	4.48765	0.184910
$590$	0.357908	0.0147348
$591$	0	0
$592$	41.4929	1.70535
$593$	−5.63179	−0.231270	−0.115635	−	0.993292i	$-0.536890\pi$
$593$	−5.63179	−0.231270	−0.115635	+	0.993292i	$0.536890\pi$
$594$	0	0
$595$	17.5739	0.720458
$596$	−4.27008	−0.174909
$597$	0	0
$598$	−25.7140	−1.05152
$599$	25.5479	1.04386	0.521929	−	0.852989i	$-0.325213\pi$
$599$	25.5479	1.04386	0.521929	+	0.852989i	$0.325213\pi$
$600$	0	0
$601$	39.1627	1.59748	0.798741	−	0.601675i	$-0.205500\pi$
$601$	39.1627	1.59748	0.798741	+	0.601675i	$0.205500\pi$
$602$	31.6990	1.29195
$603$	0	0
$604$	4.16807	0.169597
$605$	0.569898	0.0231697
$606$	0	0
$607$	8.19398	0.332583	0.166292	−	0.986077i	$-0.446821\pi$
$607$	8.19398	0.332583	0.166292	+	0.986077i	$0.446821\pi$
$608$	2.21135	0.0896820
$609$	0	0
$610$	0.855232	0.0346273
$611$	12.0858	0.488938
$612$	0	0
$613$	29.6138	1.19609	0.598046	−	0.801462i	$-0.295944\pi$
$613$	29.6138	1.19609	0.598046	+	0.801462i	$0.295944\pi$
$614$	−11.7084	−0.472511
$615$	0	0
$616$	10.4284	0.420172
$617$	−24.1413	−0.971894	−0.485947	−	0.873988i	$-0.661525\pi$
$617$	−24.1413	−0.971894	−0.485947	+	0.873988i	$0.661525\pi$
$618$	0	0
$619$	−15.2803	−0.614168	−0.307084	−	0.951682i	$-0.599353\pi$
$619$	−15.2803	−0.614168	−0.307084	+	0.951682i	$0.599353\pi$
$620$	0.413168	0.0165932
$621$	0	0
$622$	−2.05365	−0.0823440
$623$	−53.4352	−2.14084
$624$	0	0
$625$	20.2337	0.809349
$626$	7.72075	0.308583
$627$	0	0
$628$	4.50635	0.179823
$629$	72.4791	2.88993
$630$	0	0
$631$	−25.2273	−1.00428	−0.502142	−	0.864785i	$-0.667454\pi$
$631$	−25.2273	−1.00428	−0.502142	+	0.864785i	$0.667454\pi$
$632$	−34.4339	−1.36971
$633$	0	0
$634$	45.6080	1.81132
$635$	−9.26722	−0.367758
$636$	0	0
$637$	−30.7048	−1.21657
$638$	−9.44943	−0.374106
$639$	0	0
$640$	−7.39177	−0.292186
$641$	11.1758	0.441417	0.220709	−	0.975340i	$-0.429163\pi$
$641$	11.1758	0.441417	0.220709	+	0.975340i	$0.429163\pi$
$642$	0	0
$643$	33.0083	1.30172	0.650861	−	0.759197i	$-0.274408\pi$
$643$	33.0083	1.30172	0.650861	+	0.759197i	$0.274408\pi$
$644$	4.92311	0.193998
$645$	0	0
$646$	18.1590	0.714456
$647$	10.2919	0.404616	0.202308	−	0.979322i	$-0.435156\pi$
$647$	10.2919	0.404616	0.202308	+	0.979322i	$0.435156\pi$
$648$	0	0
$649$	0.418492	0.0164273
$650$	−24.4664	−0.959651
$651$	0	0
$652$	−0.215485	−0.00843905
$653$	8.56890	0.335327	0.167664	−	0.985844i	$-0.446378\pi$
$653$	8.56890	0.335327	0.167664	+	0.985844i	$0.446378\pi$
$654$	0	0
$655$	−9.54861	−0.373095
$656$	7.03464	0.274657
$657$	0	0
$658$	−20.6764	−0.806049
$659$	−0.185337	−0.00721972	−0.00360986	−	0.999993i	$-0.501149\pi$
$659$	−0.185337	−0.00721972	−0.00360986	+	0.999993i	$0.501149\pi$
$660$	0	0
$661$	6.58896	0.256281	0.128141	−	0.991756i	$-0.459099\pi$
$661$	6.58896	0.256281	0.128141	+	0.991756i	$0.459099\pi$
$662$	−26.1017	−1.01447
$663$	0	0
$664$	−10.6062	−0.411600
$665$	−3.53447	−0.137061
$666$	0	0
$667$	30.9400	1.19800
$668$	−3.37967	−0.130763
$669$	0	0
$670$	−8.41868	−0.325242
$671$	1.00000	0.0386046
$672$	0	0
$673$	32.8732	1.26717	0.633584	−	0.773674i	$-0.281583\pi$
$673$	32.8732	1.26717	0.633584	+	0.773674i	$0.281583\pi$
$674$	−5.07559	−0.195505
$675$	0	0
$676$	−0.211489	−0.00813418
$677$	49.0282	1.88431	0.942154	−	0.335181i	$-0.108797\pi$
$677$	49.0282	1.88431	0.942154	+	0.335181i	$0.108797\pi$
$678$	0	0
$679$	7.07339	0.271452
$680$	−11.5956	−0.444671
$681$	0	0
$682$	4.31692	0.165303
$683$	24.9821	0.955912	0.477956	−	0.878384i	$-0.341378\pi$
$683$	24.9821	0.955912	0.477956	+	0.878384i	$0.341378\pi$
$684$	0	0
$685$	−12.3697	−0.472623
$686$	10.7680	0.411122
$687$	0	0
$688$	−23.5938	−0.899506
$689$	37.7508	1.43819
$690$	0	0
$691$	8.91705	0.339221	0.169610	−	0.985511i	$-0.445749\pi$
$691$	8.91705	0.339221	0.169610	+	0.985511i	$0.445749\pi$
$692$	2.27784	0.0865905
$693$	0	0
$694$	33.0469	1.25444
$695$	5.62303	0.213294
$696$	0	0
$697$	12.2880	0.465441
$698$	9.23392	0.349509
$699$	0	0
$700$	4.68425	0.177048
$701$	19.3770	0.731861	0.365930	−	0.930642i	$-0.380751\pi$
$701$	19.3770	0.731861	0.365930	+	0.930642i	$0.380751\pi$
$702$	0	0
$703$	−14.5771	−0.549785
$704$	−6.75385	−0.254545
$705$	0	0
$706$	−4.22513	−0.159015
$707$	−75.1514	−2.82636
$708$	0	0
$709$	−31.7098	−1.19089	−0.595444	−	0.803397i	$-0.703024\pi$
$709$	−31.7098	−1.19089	−0.595444	+	0.803397i	$0.703024\pi$
$710$	8.49388	0.318770
$711$	0	0
$712$	35.2577	1.32134
$713$	−14.1348	−0.529351
$714$	0	0
$715$	1.98737	0.0743235
$716$	6.57889	0.245865
$717$	0	0
$718$	−29.0305	−1.08341
$719$	29.9033	1.11521	0.557603	−	0.830108i	$-0.311721\pi$
$719$	29.9033	1.11521	0.557603	+	0.830108i	$0.311721\pi$
$720$	0	0
$721$	−9.61291	−0.358003
$722$	24.8607	0.925218
$723$	0	0
$724$	0.797492	0.0296386
$725$	29.4388	1.09333
$726$	0	0
$727$	−11.3206	−0.419859	−0.209930	−	0.977717i	$-0.567323\pi$
$727$	−11.3206	−0.419859	−0.209930	+	0.977717i	$0.567323\pi$
$728$	36.3663	1.34783
$729$	0	0
$730$	−6.08805	−0.225329
$731$	−41.2133	−1.52433
$732$	0	0
$733$	46.5548	1.71954	0.859771	−	0.510680i	$-0.170606\pi$
$733$	46.5548	1.71954	0.859771	+	0.510680i	$0.170606\pi$
$734$	−14.2351	−0.525426
$735$	0	0
$736$	−6.96509	−0.256737
$737$	−9.84375	−0.362599
$738$	0	0
$739$	3.40428	0.125229	0.0626143	−	0.998038i	$-0.480056\pi$
$739$	3.40428	0.125229	0.0626143	+	0.998038i	$0.480056\pi$
$740$	−1.34208	−0.0493359
$741$	0	0
$742$	−64.5843	−2.37096
$743$	−3.46368	−0.127070	−0.0635350	−	0.997980i	$-0.520237\pi$
$743$	−3.46368	−0.127070	−0.0635350	+	0.997980i	$0.520237\pi$
$744$	0	0
$745$	−9.65585	−0.353763
$746$	42.2284	1.54609
$747$	0	0
$748$	1.95486	0.0714769
$749$	42.7522	1.56213
$750$	0	0
$751$	−1.89930	−0.0693065	−0.0346532	−	0.999399i	$-0.511033\pi$
$751$	−1.89930	−0.0693065	−0.0346532	+	0.999399i	$0.511033\pi$
$752$	15.3896	0.561201
$753$	0	0
$754$	−32.9524	−1.20006
$755$	9.42519	0.343018
$756$	0	0
$757$	8.67210	0.315193	0.157596	−	0.987504i	$-0.449626\pi$
$757$	8.67210	0.315193	0.157596	+	0.987504i	$0.449626\pi$
$758$	43.0271	1.56281
$759$	0	0
$760$	2.33212	0.0845948
$761$	−37.5110	−1.35977	−0.679887	−	0.733317i	$-0.737971\pi$
$761$	−37.5110	−1.35977	−0.679887	+	0.733317i	$0.737971\pi$
$762$	0	0
$763$	58.1329	2.10455
$764$	0.0856348	0.00309816
$765$	0	0
$766$	−53.9443	−1.94909
$767$	1.45938	0.0526953
$768$	0	0
$769$	−30.1552	−1.08743	−0.543713	−	0.839271i	$-0.682982\pi$
$769$	−30.1552	−1.08743	−0.543713	+	0.839271i	$0.682982\pi$
$770$	−3.40001	−0.122528
$771$	0	0
$772$	−2.77198	−0.0997656
$773$	23.2997	0.838031	0.419015	−	0.907979i	$-0.362375\pi$
$773$	23.2997	0.838031	0.419015	+	0.907979i	$0.362375\pi$
$774$	0	0
$775$	−13.4490	−0.483101
$776$	−4.66717	−0.167542
$777$	0	0
$778$	−54.8908	−1.96793
$779$	−2.47137	−0.0885461
$780$	0	0
$781$	9.93167	0.355383
$782$	−57.1955	−2.04531
$783$	0	0
$784$	−39.0984	−1.39637
$785$	10.1901	0.363702
$786$	0	0
$787$	51.3754	1.83134	0.915668	−	0.401936i	$-0.131663\pi$
$787$	51.3754	1.83134	0.915668	+	0.401936i	$0.131663\pi$
$788$	−3.44384	−0.122682
$789$	0	0
$790$	11.2266	0.399425
$791$	−35.3329	−1.25629
$792$	0	0
$793$	3.48724	0.123836
$794$	−26.1553	−0.928217
$795$	0	0
$796$	−1.27361	−0.0451419
$797$	−17.7177	−0.627593	−0.313796	−	0.949490i	$-0.601601\pi$
$797$	−17.7177	−0.627593	−0.313796	+	0.949490i	$0.601601\pi$
$798$	0	0
$799$	26.8823	0.951027
$800$	−6.62716	−0.234305
$801$	0	0
$802$	−42.5915	−1.50396
$803$	−7.11859	−0.251210
$804$	0	0
$805$	11.1325	0.392370
$806$	15.0541	0.530259
$807$	0	0
$808$	49.5864	1.74445
$809$	−16.0592	−0.564612	−0.282306	−	0.959324i	$-0.591099\pi$
$809$	−16.0592	−0.564612	−0.282306	+	0.959324i	$0.591099\pi$
$810$	0	0
$811$	−12.0185	−0.422028	−0.211014	−	0.977483i	$-0.567677\pi$
$811$	−12.0185	−0.422028	−0.211014	+	0.977483i	$0.567677\pi$
$812$	6.30895	0.221401
$813$	0	0
$814$	−14.0225	−0.491489
$815$	−0.487273	−0.0170684
$816$	0	0
$817$	8.28885	0.289990
$818$	−41.5107	−1.45139
$819$	0	0
$820$	−0.227534	−0.00794584
$821$	12.6542	0.441636	0.220818	−	0.975315i	$-0.429127\pi$
$821$	12.6542	0.441636	0.220818	+	0.975315i	$0.429127\pi$
$822$	0	0
$823$	−0.0537888	−0.00187496	−0.000937480	−	1.00000i	$-0.500298\pi$
$823$	−0.0537888	−0.00187496	−0.000937480		1.00000i	$0.500298\pi$
$824$	6.34280	0.220962
$825$	0	0
$826$	−2.49672	−0.0868720
$827$	31.9496	1.11100	0.555498	−	0.831518i	$-0.312528\pi$
$827$	31.9496	1.11100	0.555498	+	0.831518i	$0.312528\pi$
$828$	0	0
$829$	35.7815	1.24274	0.621372	−	0.783516i	$-0.286576\pi$
$829$	35.7815	1.24274	0.621372	+	0.783516i	$0.286576\pi$
$830$	3.45797	0.120028
$831$	0	0
$832$	−23.5523	−0.816528
$833$	−68.2964	−2.36633
$834$	0	0
$835$	−7.64238	−0.264476
$836$	−0.393164	−0.0135979
$837$	0	0
$838$	−1.71371	−0.0591990
$839$	−6.85181	−0.236551	−0.118275	−	0.992981i	$-0.537737\pi$
$839$	−6.85181	−0.236551	−0.118275	+	0.992981i	$0.537737\pi$
$840$	0	0
$841$	10.6495	0.367224
$842$	−7.87107	−0.271255
$843$	0	0
$844$	0.105442	0.00362945
$845$	−0.478236	−0.0164518
$846$	0	0
$847$	−3.97554	−0.136601
$848$	48.0706	1.65075
$849$	0	0
$850$	−54.4204	−1.86661
$851$	45.9135	1.57389
$852$	0	0
$853$	−42.9971	−1.47219	−0.736096	−	0.676877i	$-0.763333\pi$
$853$	−42.9971	−1.47219	−0.736096	+	0.676877i	$0.763333\pi$
$854$	−5.96599	−0.204152
$855$	0	0
$856$	−28.2088	−0.964157
$857$	3.32058	0.113429	0.0567145	−	0.998390i	$-0.481938\pi$
$857$	3.32058	0.113429	0.0567145	+	0.998390i	$0.481938\pi$
$858$	0	0
$859$	46.4438	1.58464	0.792321	−	0.610105i	$-0.208873\pi$
$859$	46.4438	1.58464	0.792321	+	0.610105i	$0.208873\pi$
$860$	0.763138	0.0260228
$861$	0	0
$862$	59.6574	2.03194
$863$	−11.1536	−0.379672	−0.189836	−	0.981816i	$-0.560796\pi$
$863$	−11.1536	−0.379672	−0.189836	+	0.981816i	$0.560796\pi$
$864$	0	0
$865$	5.15084	0.175134
$866$	33.6679	1.14408
$867$	0	0
$868$	−2.88221	−0.0978286
$869$	13.1270	0.445302
$870$	0	0
$871$	−34.3275	−1.16314
$872$	−38.3573	−1.29894
$873$	0	0
$874$	11.5032	0.389102
$875$	21.9207	0.741054
$876$	0	0
$877$	−29.0494	−0.980929	−0.490464	−	0.871461i	$-0.663173\pi$
$877$	−29.0494	−0.980929	−0.490464	+	0.871461i	$0.663173\pi$
$878$	8.20810	0.277010
$879$	0	0
$880$	2.53065	0.0853082
$881$	21.4381	0.722269	0.361134	−	0.932514i	$-0.382390\pi$
$881$	21.4381	0.722269	0.361134	+	0.932514i	$0.382390\pi$
$882$	0	0
$883$	−35.2149	−1.18508	−0.592538	−	0.805542i	$-0.701874\pi$
$883$	−35.2149	−1.18508	−0.592538	+	0.805542i	$0.701874\pi$
$884$	6.81708	0.229283
$885$	0	0
$886$	−6.32525	−0.212501
$887$	−41.4951	−1.39327	−0.696635	−	0.717426i	$-0.745320\pi$
$887$	−41.4951	−1.39327	−0.696635	+	0.717426i	$0.745320\pi$
$888$	0	0
$889$	64.6469	2.16819
$890$	−11.4952	−0.385319
$891$	0	0
$892$	2.13453	0.0714693
$893$	−5.40659	−0.180925
$894$	0	0
$895$	14.8767	0.497274
$896$	51.5641	1.72263
$897$	0	0
$898$	35.8027	1.19475
$899$	−18.1137	−0.604124
$900$	0	0
$901$	83.9689	2.79741
$902$	−2.37735	−0.0791572
$903$	0	0
$904$	23.3134	0.775391
$905$	1.80336	0.0599456
$906$	0	0
$907$	−21.7745	−0.723009	−0.361505	−	0.932370i	$-0.617737\pi$
$907$	−21.7745	−0.723009	−0.361505	+	0.932370i	$0.617737\pi$
$908$	0.325007	0.0107857
$909$	0	0
$910$	−11.8566	−0.393044
$911$	−26.9675	−0.893472	−0.446736	−	0.894666i	$-0.647414\pi$
$911$	−26.9675	−0.893472	−0.446736	+	0.894666i	$0.647414\pi$
$912$	0	0
$913$	4.04331	0.133814
$914$	−53.7108	−1.77660
$915$	0	0
$916$	4.54445	0.150153
$917$	66.6099	2.19965
$918$	0	0
$919$	25.4015	0.837917	0.418958	−	0.908005i	$-0.362395\pi$
$919$	25.4015	0.837917	0.418958	+	0.908005i	$0.362395\pi$
$920$	−7.34548	−0.242173
$921$	0	0
$922$	49.6285	1.63443
$923$	34.6341	1.14000
$924$	0	0
$925$	43.6859	1.43638
$926$	45.2730	1.48776
$927$	0	0
$928$	−8.92575	−0.293002
$929$	29.0759	0.953949	0.476975	−	0.878917i	$-0.341733\pi$
$929$	29.0759	0.953949	0.476975	+	0.878917i	$0.341733\pi$
$930$	0	0
$931$	13.7358	0.450174
$932$	−3.82571	−0.125315
$933$	0	0
$934$	19.9652	0.653280
$935$	4.42050	0.144566
$936$	0	0
$937$	−21.2285	−0.693506	−0.346753	−	0.937957i	$-0.612716\pi$
$937$	−21.2285	−0.693506	−0.346753	+	0.937957i	$0.612716\pi$
$938$	58.7277	1.91753
$939$	0	0
$940$	−0.497774	−0.0162356
$941$	−53.6582	−1.74921	−0.874604	−	0.484839i	$-0.838878\pi$
$941$	−53.6582	−1.74921	−0.874604	+	0.484839i	$0.838878\pi$
$942$	0	0
$943$	7.78410	0.253485
$944$	1.85833	0.0604835
$945$	0	0
$946$	7.97351	0.259241
$947$	−42.9691	−1.39631	−0.698154	−	0.715947i	$-0.745995\pi$
$947$	−42.9691	−1.39631	−0.698154	+	0.715947i	$0.745995\pi$
$948$	0	0
$949$	−24.8242	−0.805829
$950$	10.9451	0.355106
$951$	0	0
$952$	80.8894	2.62164
$953$	−20.7043	−0.670678	−0.335339	−	0.942098i	$-0.608851\pi$
$953$	−20.7043	−0.670678	−0.335339	+	0.942098i	$0.608851\pi$
$954$	0	0
$955$	0.193644	0.00626619
$956$	−1.41586	−0.0457923
$957$	0	0
$958$	54.3816	1.75699
$959$	86.2896	2.78644
$960$	0	0
$961$	−22.7249	−0.733060
$962$	−48.8998	−1.57659
$963$	0	0
$964$	−2.48488	−0.0800325
$965$	−6.26822	−0.201781
$966$	0	0
$967$	−28.9634	−0.931401	−0.465701	−	0.884942i	$-0.654198\pi$
$967$	−28.9634	−0.931401	−0.465701	+	0.884942i	$0.654198\pi$
$968$	2.62314	0.0843110
$969$	0	0
$970$	1.52165	0.0488574
$971$	−41.1027	−1.31905	−0.659524	−	0.751684i	$-0.729242\pi$
$971$	−41.1027	−1.31905	−0.659524	+	0.751684i	$0.729242\pi$
$972$	0	0
$973$	−39.2255	−1.25751
$974$	−7.41556	−0.237610
$975$	0	0
$976$	4.44053	0.142138
$977$	−0.544347	−0.0174152	−0.00870761	−	0.999962i	$-0.502772\pi$
$977$	−0.544347	−0.0174152	−0.00870761	+	0.999962i	$0.502772\pi$
$978$	0	0
$979$	−13.4410	−0.429576
$980$	1.26463	0.0403971
$981$	0	0
$982$	−21.0320	−0.671158
$983$	−19.5398	−0.623224	−0.311612	−	0.950209i	$-0.600869\pi$
$983$	−19.5398	−0.623224	−0.311612	+	0.950209i	$0.600869\pi$
$984$	0	0
$985$	−7.78750	−0.248130
$986$	−73.2958	−2.33422
$987$	0	0
$988$	−1.37106	−0.0436191
$989$	−26.1074	−0.830169
$990$	0	0
$991$	−45.5206	−1.44601	−0.723004	−	0.690843i	$-0.757239\pi$
$991$	−45.5206	−1.44601	−0.723004	+	0.690843i	$0.757239\pi$
$992$	4.07768	0.129466
$993$	0	0
$994$	−59.2522	−1.87937
$995$	−2.87999	−0.0913020
$996$	0	0
$997$	8.42755	0.266903	0.133452	−	0.991055i	$-0.457394\pi$
$997$	8.42755	0.266903	0.133452	+	0.991055i	$0.457394\pi$
$998$	−22.5577	−0.714052
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6039.2.a.i.1.5		13
3.2	odd	2		2013.2.a.e.1.9	✓	13

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2013.2.a.e.1.9	✓	13	3.2	odd	2
6039.2.a.i.1.5		13	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)

\(f(q)\)	\(=\)	\(q-1.50067 q^{2} +0.252024 q^{4} +0.569898 q^{5} -3.97554 q^{7} +2.62314 q^{8} +O(q^{10})\)	\(q-1.50067 q^{2} +0.252024 q^{4} +0.569898 q^{5} -3.97554 q^{7} +2.62314 q^{8} -0.855232 q^{10} -1.00000 q^{11} -3.48724 q^{13} +5.96599 q^{14} -4.44053 q^{16} -7.75664 q^{17} +1.56002 q^{19} +0.143628 q^{20} +1.50067 q^{22} -4.91362 q^{23} -4.67522 q^{25} +5.23321 q^{26} -1.00193 q^{28} -6.29679 q^{29} +2.87665 q^{31} +1.41751 q^{32} +11.6402 q^{34} -2.26565 q^{35} -9.34414 q^{37} -2.34109 q^{38} +1.49492 q^{40} -1.58419 q^{41} +5.31329 q^{43} -0.252024 q^{44} +7.37374 q^{46} -3.46571 q^{47} +8.80489 q^{49} +7.01598 q^{50} -0.878869 q^{52} -10.8254 q^{53} -0.569898 q^{55} -10.4284 q^{56} +9.44943 q^{58} -0.418492 q^{59} -1.00000 q^{61} -4.31692 q^{62} +6.75385 q^{64} -1.98737 q^{65} +9.84375 q^{67} -1.95486 q^{68} +3.40001 q^{70} -9.93167 q^{71} +7.11859 q^{73} +14.0225 q^{74} +0.393164 q^{76} +3.97554 q^{77} -13.1270 q^{79} -2.53065 q^{80} +2.37735 q^{82} -4.04331 q^{83} -4.42050 q^{85} -7.97351 q^{86} -2.62314 q^{88} +13.4410 q^{89} +13.8636 q^{91} -1.23835 q^{92} +5.20090 q^{94} +0.889055 q^{95} -1.77923 q^{97} -13.2133 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 13 q - 2 q^{2} + 16 q^{4} - 3 q^{5} + 11 q^{7} - 9 q^{8}+O(q^{10}) \) 13 * q - 2 * q^2 + 16 * q^4 - 3 * q^5 + 11 * q^7 - 9 * q^8	\( 13 q - 2 q^{2} + 16 q^{4} - 3 q^{5} + 11 q^{7} - 9 q^{8} + 6 q^{10} - 13 q^{11} + 13 q^{13} - q^{14} + 18 q^{16} - 17 q^{17} + 14 q^{19} + 7 q^{20} + 2 q^{22} - 7 q^{23} + 18 q^{25} + 10 q^{26} + 19 q^{28} + 6 q^{29} + 27 q^{31} - 5 q^{32} + 6 q^{34} - 14 q^{35} + 10 q^{37} - 2 q^{38} + 8 q^{40} - 3 q^{41} + 29 q^{43} - 16 q^{44} - 24 q^{46} - 8 q^{47} + 8 q^{49} + 27 q^{50} + 37 q^{52} + 24 q^{53} + 3 q^{55} - 24 q^{56} - 5 q^{58} - 13 q^{59} - 13 q^{61} - 39 q^{62} + 47 q^{64} + 11 q^{65} + 44 q^{67} + 8 q^{68} - 12 q^{70} - 3 q^{71} + 48 q^{73} + 22 q^{74} + 47 q^{76} - 11 q^{77} - 17 q^{79} + 26 q^{80} + 56 q^{82} - 50 q^{83} + 8 q^{85} - 18 q^{86} + 9 q^{88} + 15 q^{89} + 47 q^{91} - 14 q^{92} + 45 q^{94} + q^{95} + 27 q^{97} - 47 q^{98}+O(q^{100}) \) 13 * q - 2 * q^2 + 16 * q^4 - 3 * q^5 + 11 * q^7 - 9 * q^8 + 6 * q^10 - 13 * q^11 + 13 * q^13 - q^14 + 18 * q^16 - 17 * q^17 + 14 * q^19 + 7 * q^20 + 2 * q^22 - 7 * q^23 + 18 * q^25 + 10 * q^26 + 19 * q^28 + 6 * q^29 + 27 * q^31 - 5 * q^32 + 6 * q^34 - 14 * q^35 + 10 * q^37 - 2 * q^38 + 8 * q^40 - 3 * q^41 + 29 * q^43 - 16 * q^44 - 24 * q^46 - 8 * q^47 + 8 * q^49 + 27 * q^50 + 37 * q^52 + 24 * q^53 + 3 * q^55 - 24 * q^56 - 5 * q^58 - 13 * q^59 - 13 * q^61 - 39 * q^62 + 47 * q^64 + 11 * q^65 + 44 * q^67 + 8 * q^68 - 12 * q^70 - 3 * q^71 + 48 * q^73 + 22 * q^74 + 47 * q^76 - 11 * q^77 - 17 * q^79 + 26 * q^80 + 56 * q^82 - 50 * q^83 + 8 * q^85 - 18 * q^86 + 9 * q^88 + 15 * q^89 + 47 * q^91 - 14 * q^92 + 45 * q^94 + q^95 + 27 * q^97 - 47 * q^98

Introduction

L-functions

Modular forms

Varieties

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Database

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