LMFDB - Embedded newform 6039.2.a.f.1.11

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:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 5x^{11} - 5x^{10} + 48x^{9} - 173x^{7} + 29x^{6} + 281x^{5} - 41x^{4} - 201x^{3} + 8x^{2} + 49x + 8 $ x^12 - 5x^11 - 5x^10 + 48x^9 - 173x^7 + 29x^6 + 281x^5 - 41x^4 - 201x^3 + 8x^2 + 49x + 8
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			1.11
Root			$-1.62825$ of defining polynomial
Character	$\chi$	$=$	6039.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.62825 q^{2} +4.90768 q^{4} -2.26825 q^{5} -0.677820 q^{7} +7.64209 q^{8} +O(q^{10})$	\(q+2.62825 q^{2} +4.90768 q^{4} -2.26825 q^{5} -0.677820 q^{7} +7.64209 q^{8} -5.96152 q^{10} -1.00000 q^{11} -6.23689 q^{13} -1.78148 q^{14} +10.2699 q^{16} +4.02340 q^{17} +5.38012 q^{19} -11.1318 q^{20} -2.62825 q^{22} +7.11412 q^{23} +0.144966 q^{25} -16.3921 q^{26} -3.32652 q^{28} +9.43948 q^{29} +8.36505 q^{31} +11.7077 q^{32} +10.5745 q^{34} +1.53747 q^{35} +1.53516 q^{37} +14.1403 q^{38} -17.3342 q^{40} -2.74126 q^{41} -8.78677 q^{43} -4.90768 q^{44} +18.6977 q^{46} +4.57528 q^{47} -6.54056 q^{49} +0.381006 q^{50} -30.6086 q^{52} +8.79479 q^{53} +2.26825 q^{55} -5.17996 q^{56} +24.8093 q^{58} +13.7649 q^{59} -1.00000 q^{61} +21.9854 q^{62} +10.2309 q^{64} +14.1468 q^{65} -0.817631 q^{67} +19.7455 q^{68} +4.04084 q^{70} +8.00895 q^{71} -5.45065 q^{73} +4.03477 q^{74} +26.4039 q^{76} +0.677820 q^{77} +15.6070 q^{79} -23.2948 q^{80} -7.20471 q^{82} -8.66843 q^{83} -9.12608 q^{85} -23.0938 q^{86} -7.64209 q^{88} -3.13352 q^{89} +4.22749 q^{91} +34.9138 q^{92} +12.0250 q^{94} -12.2035 q^{95} +11.6733 q^{97} -17.1902 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 7 q^{2} + 13 q^{4} + 7 q^{5} - 15 q^{7} + 18 q^{8}+O(q^{10}) $ 12 * q + 7 * q^2 + 13 * q^4 + 7 * q^5 - 15 * q^7 + 18 * q^8	$ 12 q + 7 q^{2} + 13 q^{4} + 7 q^{5} - 15 q^{7} + 18 q^{8} - 6 q^{10} - 12 q^{11} - 11 q^{13} - 3 q^{14} + 19 q^{16} + 33 q^{17} - 24 q^{19} + 11 q^{20} - 7 q^{22} + 9 q^{23} + 11 q^{25} + 16 q^{26} - 41 q^{28} + 16 q^{29} + q^{31} + 28 q^{32} + 32 q^{34} + 22 q^{35} - 6 q^{37} - 12 q^{38} + 26 q^{40} + 21 q^{41} - 39 q^{43} - 13 q^{44} + 18 q^{47} + 31 q^{49} + 44 q^{50} + 3 q^{52} + 14 q^{53} - 7 q^{55} - 16 q^{56} + 33 q^{58} + 23 q^{59} - 12 q^{61} + 25 q^{62} + 12 q^{64} + 29 q^{65} + 96 q^{68} + 44 q^{70} + 19 q^{71} - 42 q^{73} - 38 q^{74} + 11 q^{76} + 15 q^{77} - 11 q^{79} + 44 q^{80} - 14 q^{82} + 56 q^{83} + 16 q^{85} + 18 q^{86} - 18 q^{88} + 55 q^{89} + 11 q^{91} + 4 q^{92} - 5 q^{94} - 15 q^{95} - 7 q^{97} - 6 q^{98}+O(q^{100}) $ 12 * q + 7 * q^2 + 13 * q^4 + 7 * q^5 - 15 * q^7 + 18 * q^8 - 6 * q^10 - 12 * q^11 - 11 * q^13 - 3 * q^14 + 19 * q^16 + 33 * q^17 - 24 * q^19 + 11 * q^20 - 7 * q^22 + 9 * q^23 + 11 * q^25 + 16 * q^26 - 41 * q^28 + 16 * q^29 + q^31 + 28 * q^32 + 32 * q^34 + 22 * q^35 - 6 * q^37 - 12 * q^38 + 26 * q^40 + 21 * q^41 - 39 * q^43 - 13 * q^44 + 18 * q^47 + 31 * q^49 + 44 * q^50 + 3 * q^52 + 14 * q^53 - 7 * q^55 - 16 * q^56 + 33 * q^58 + 23 * q^59 - 12 * q^61 + 25 * q^62 + 12 * q^64 + 29 * q^65 + 96 * q^68 + 44 * q^70 + 19 * q^71 - 42 * q^73 - 38 * q^74 + 11 * q^76 + 15 * q^77 - 11 * q^79 + 44 * q^80 - 14 * q^82 + 56 * q^83 + 16 * q^85 + 18 * q^86 - 18 * q^88 + 55 * q^89 + 11 * q^91 + 4 * q^92 - 5 * q^94 - 15 * q^95 - 7 * q^97 - 6 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	2.62825	1.85845	0.929225	−	0.369514i	$-0.120476\pi$
$2$	2.62825	1.85845	0.929225	+	0.369514i	$0.120476\pi$
$3$	0	0
$3$	0	0
$4$	4.90768	2.45384
$5$	−2.26825	−1.01439	−0.507197	−	0.861830i	$-0.669318\pi$
$5$	−2.26825	−1.01439	−0.507197	+	0.861830i	$0.669318\pi$
$6$	0	0
$7$	−0.677820	−0.256192	−0.128096	−	0.991762i	$-0.540887\pi$
$7$	−0.677820	−0.256192	−0.128096	+	0.991762i	$0.540887\pi$
$8$	7.64209	2.70189
$9$	0	0
$10$	−5.96152	−1.88520
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	−6.23689	−1.72980	−0.864901	−	0.501942i	$-0.832619\pi$
$13$	−6.23689	−1.72980	−0.864901	+	0.501942i	$0.832619\pi$
$14$	−1.78148	−0.476120
$15$	0	0
$16$	10.2699	2.56748
$17$	4.02340	0.975817	0.487909	−	0.872895i	$-0.337760\pi$
$17$	4.02340	0.975817	0.487909	+	0.872895i	$0.337760\pi$
$18$	0	0
$19$	5.38012	1.23428	0.617142	−	0.786851i	$-0.288290\pi$
$19$	5.38012	1.23428	0.617142	+	0.786851i	$0.288290\pi$
$20$	−11.1318	−2.48916
$21$	0	0
$22$	−2.62825	−0.560344
$23$	7.11412	1.48340	0.741699	−	0.670733i	$-0.234020\pi$
$23$	7.11412	1.48340	0.741699	+	0.670733i	$0.234020\pi$
$24$	0	0
$25$	0.144966	0.0289932
$26$	−16.3921	−3.21475
$27$	0	0
$28$	−3.32652	−0.628653
$29$	9.43948	1.75287	0.876434	−	0.481522i	$-0.159916\pi$
$29$	9.43948	1.75287	0.876434	+	0.481522i	$0.159916\pi$
$30$	0	0
$31$	8.36505	1.50241	0.751204	−	0.660071i	$-0.229474\pi$
$31$	8.36505	1.50241	0.751204	+	0.660071i	$0.229474\pi$
$32$	11.7077	2.06965
$33$	0	0
$34$	10.5745	1.81351
$35$	1.53747	0.259879
$36$	0	0
$37$	1.53516	0.252378	0.126189	−	0.992006i	$-0.459725\pi$
$37$	1.53516	0.252378	0.126189	+	0.992006i	$0.459725\pi$
$38$	14.1403	2.29386
$39$	0	0
$40$	−17.3342	−2.74077
$41$	−2.74126	−0.428113	−0.214057	−	0.976821i	$-0.568668\pi$
$41$	−2.74126	−0.428113	−0.214057	+	0.976821i	$0.568668\pi$
$42$	0	0
$43$	−8.78677	−1.33997	−0.669985	−	0.742375i	$-0.733699\pi$
$43$	−8.78677	−1.33997	−0.669985	+	0.742375i	$0.733699\pi$
$44$	−4.90768	−0.739860
$45$	0	0
$46$	18.6977	2.75682
$47$	4.57528	0.667373	0.333687	−	0.942684i	$-0.391707\pi$
$47$	4.57528	0.667373	0.333687	+	0.942684i	$0.391707\pi$
$48$	0	0
$49$	−6.54056	−0.934366
$50$	0.381006	0.0538824
$51$	0	0
$52$	−30.6086	−4.24465
$53$	8.79479	1.20806	0.604028	−	0.796963i	$-0.293561\pi$
$53$	8.79479	1.20806	0.604028	+	0.796963i	$0.293561\pi$
$54$	0	0
$55$	2.26825	0.305851
$56$	−5.17996	−0.692201
$57$	0	0
$58$	24.8093	3.25762
$59$	13.7649	1.79204	0.896018	−	0.444019i	$-0.146448\pi$
$59$	13.7649	1.79204	0.896018	+	0.444019i	$0.146448\pi$
$60$	0	0
$61$	−1.00000	−0.128037
$61$	−1.00000	−0.128037
$62$	21.9854	2.79215
$63$	0	0
$64$	10.2309	1.27887
$65$	14.1468	1.75470
$66$	0	0
$67$	−0.817631	−0.0998896	−0.0499448	−	0.998752i	$-0.515905\pi$
$67$	−0.817631	−0.0998896	−0.0499448	+	0.998752i	$0.515905\pi$
$68$	19.7455	2.39450
$69$	0	0
$70$	4.04084	0.482973
$71$	8.00895	0.950487	0.475244	−	0.879854i	$-0.342360\pi$
$71$	8.00895	0.950487	0.475244	+	0.879854i	$0.342360\pi$
$72$	0	0
$73$	−5.45065	−0.637950	−0.318975	−	0.947763i	$-0.603339\pi$
$73$	−5.45065	−0.637950	−0.318975	+	0.947763i	$0.603339\pi$
$74$	4.03477	0.469032
$75$	0	0
$76$	26.4039	3.02874
$77$	0.677820	0.0772448
$78$	0	0
$79$	15.6070	1.75592	0.877962	−	0.478730i	$-0.158903\pi$
$79$	15.6070	1.75592	0.877962	+	0.478730i	$0.158903\pi$
$80$	−23.2948	−2.60444
$81$	0	0
$82$	−7.20471	−0.795627
$83$	−8.66843	−0.951484	−0.475742	−	0.879585i	$-0.657820\pi$
$83$	−8.66843	−0.951484	−0.475742	+	0.879585i	$0.657820\pi$
$84$	0	0
$85$	−9.12608	−0.989862
$86$	−23.0938	−2.49027
$87$	0	0
$88$	−7.64209	−0.814649
$89$	−3.13352	−0.332152	−0.166076	−	0.986113i	$-0.553110\pi$
$89$	−3.13352	−0.332152	−0.166076	+	0.986113i	$0.553110\pi$
$90$	0	0
$91$	4.22749	0.443161
$92$	34.9138	3.64002
$93$	0	0
$94$	12.0250	1.24028
$95$	−12.2035	−1.25205
$96$	0	0
$97$	11.6733	1.18525	0.592623	−	0.805480i	$-0.298092\pi$
$97$	11.6733	1.18525	0.592623	+	0.805480i	$0.298092\pi$
$98$	−17.1902	−1.73647
$99$	0	0
$100$	0.711446	0.0711446
$101$	−15.9134	−1.58344	−0.791721	−	0.610883i	$-0.790815\pi$
$101$	−15.9134	−1.58344	−0.791721	+	0.610883i	$0.790815\pi$
$102$	0	0
$103$	−1.24885	−0.123053	−0.0615266	−	0.998105i	$-0.519597\pi$
$103$	−1.24885	−0.123053	−0.0615266	+	0.998105i	$0.519597\pi$
$104$	−47.6629	−4.67373
$105$	0	0
$106$	23.1149	2.24511
$107$	−0.638486	−0.0617248	−0.0308624	−	0.999524i	$-0.509825\pi$
$107$	−0.638486	−0.0617248	−0.0308624	+	0.999524i	$0.509825\pi$
$108$	0	0
$109$	−8.47237	−0.811505	−0.405753	−	0.913983i	$-0.632991\pi$
$109$	−8.47237	−0.811505	−0.405753	+	0.913983i	$0.632991\pi$
$110$	5.96152	0.568409
$111$	0	0
$112$	−6.96116	−0.657768
$113$	9.38250	0.882631	0.441315	−	0.897352i	$-0.354512\pi$
$113$	9.38250	0.882631	0.441315	+	0.897352i	$0.354512\pi$
$114$	0	0
$115$	−16.1366	−1.50475
$116$	46.3259	4.30125
$117$	0	0
$118$	36.1775	3.33041
$119$	−2.72714	−0.249996
$120$	0	0
$121$	1.00000	0.0909091
$122$	−2.62825	−0.237950
$123$	0	0
$124$	41.0529	3.68666
$125$	11.0124	0.984983
$126$	0	0
$127$	14.6159	1.29695	0.648475	−	0.761236i	$-0.275407\pi$
$127$	14.6159	1.29695	0.648475	+	0.761236i	$0.275407\pi$
$128$	3.47393	0.307055
$129$	0	0
$130$	37.1814	3.26102
$131$	11.3821	0.994458	0.497229	−	0.867619i	$-0.334351\pi$
$131$	11.3821	0.994458	0.497229	+	0.867619i	$0.334351\pi$
$132$	0	0
$133$	−3.64675	−0.316214
$134$	−2.14894	−0.185640
$135$	0	0
$136$	30.7471	2.63655
$137$	−0.921880	−0.0787616	−0.0393808	−	0.999224i	$-0.512539\pi$
$137$	−0.921880	−0.0787616	−0.0393808	+	0.999224i	$0.512539\pi$
$138$	0	0
$139$	0.260606	0.0221043	0.0110521	−	0.999939i	$-0.496482\pi$
$139$	0.260606	0.0221043	0.0110521	+	0.999939i	$0.496482\pi$
$140$	7.54539	0.637702
$141$	0	0
$142$	21.0495	1.76643
$143$	6.23689	0.521555
$144$	0	0
$145$	−21.4111	−1.77810
$146$	−14.3256	−1.18560
$147$	0	0
$148$	7.53405	0.619295
$149$	−0.816498	−0.0668901	−0.0334451	−	0.999441i	$-0.510648\pi$
$149$	−0.816498	−0.0668901	−0.0334451	+	0.999441i	$0.510648\pi$
$150$	0	0
$151$	−6.49046	−0.528186	−0.264093	−	0.964497i	$-0.585073\pi$
$151$	−6.49046	−0.528186	−0.264093	+	0.964497i	$0.585073\pi$
$152$	41.1154	3.33490
$153$	0	0
$154$	1.78148	0.143556
$155$	−18.9740	−1.52403
$156$	0	0
$157$	5.04806	0.402879	0.201440	−	0.979501i	$-0.435438\pi$
$157$	5.04806	0.402879	0.201440	+	0.979501i	$0.435438\pi$
$158$	41.0190	3.26330
$159$	0	0
$160$	−26.5561	−2.09944
$161$	−4.82210	−0.380034
$162$	0	0
$163$	−1.07237	−0.0839947	−0.0419973	−	0.999118i	$-0.513372\pi$
$163$	−1.07237	−0.0839947	−0.0419973	+	0.999118i	$0.513372\pi$
$164$	−13.4532	−1.05052
$165$	0	0
$166$	−22.7828	−1.76829
$167$	−21.1967	−1.64025	−0.820123	−	0.572187i	$-0.806095\pi$
$167$	−21.1967	−1.64025	−0.820123	+	0.572187i	$0.806095\pi$
$168$	0	0
$169$	25.8988	1.99222
$170$	−23.9856	−1.83961
$171$	0	0
$172$	−43.1226	−3.28807
$173$	−20.9012	−1.58909	−0.794544	−	0.607207i	$-0.792290\pi$
$173$	−20.9012	−1.58909	−0.794544	+	0.607207i	$0.792290\pi$
$174$	0	0
$175$	−0.0982608	−0.00742782
$176$	−10.2699	−0.774125
$177$	0	0
$178$	−8.23565	−0.617288
$179$	21.3155	1.59320	0.796598	−	0.604509i	$-0.206631\pi$
$179$	21.3155	1.59320	0.796598	+	0.604509i	$0.206631\pi$
$180$	0	0
$181$	−21.2619	−1.58039	−0.790194	−	0.612857i	$-0.790020\pi$
$181$	−21.2619	−1.58039	−0.790194	+	0.612857i	$0.790020\pi$
$182$	11.1109	0.823593
$183$	0	0
$184$	54.3668	4.00797
$185$	−3.48212	−0.256011
$186$	0	0
$187$	−4.02340	−0.294220
$188$	22.4540	1.63763
$189$	0	0
$190$	−32.0737	−2.32687
$191$	−13.1923	−0.954565	−0.477282	−	0.878750i	$-0.658378\pi$
$191$	−13.1923	−0.954565	−0.477282	+	0.878750i	$0.658378\pi$
$192$	0	0
$193$	−20.9460	−1.50772	−0.753861	−	0.657033i	$-0.771811\pi$
$193$	−20.9460	−1.50772	−0.753861	+	0.657033i	$0.771811\pi$
$194$	30.6804	2.20272
$195$	0	0
$196$	−32.0989	−2.29278
$197$	−0.600918	−0.0428136	−0.0214068	−	0.999771i	$-0.506815\pi$
$197$	−0.600918	−0.0428136	−0.0214068	+	0.999771i	$0.506815\pi$
$198$	0	0
$199$	5.66754	0.401762	0.200881	−	0.979616i	$-0.435620\pi$
$199$	5.66754	0.401762	0.200881	+	0.979616i	$0.435620\pi$
$200$	1.10784	0.0783363
$201$	0	0
$202$	−41.8243	−2.94275
$203$	−6.39827	−0.449070
$204$	0	0
$205$	6.21787	0.434275
$206$	−3.28229	−0.228688
$207$	0	0
$208$	−64.0524	−4.44124
$209$	−5.38012	−0.372151
$210$	0	0
$211$	−9.53062	−0.656115	−0.328057	−	0.944658i	$-0.606394\pi$
$211$	−9.53062	−0.656115	−0.328057	+	0.944658i	$0.606394\pi$
$212$	43.1620	2.96438
$213$	0	0
$214$	−1.67810	−0.114712
$215$	19.9306	1.35926
$216$	0	0
$217$	−5.67000	−0.384904
$218$	−22.2675	−1.50814
$219$	0	0
$220$	11.1318	0.750509
$221$	−25.0935	−1.68797
$222$	0	0
$223$	22.7244	1.52174	0.760868	−	0.648906i	$-0.224773\pi$
$223$	22.7244	1.52174	0.760868	+	0.648906i	$0.224773\pi$
$224$	−7.93573	−0.530228
$225$	0	0
$226$	24.6595	1.64033
$227$	17.5715	1.16626	0.583132	−	0.812378i	$-0.301827\pi$
$227$	17.5715	1.16626	0.583132	+	0.812378i	$0.301827\pi$
$228$	0	0
$229$	0.281355	0.0185925	0.00929623	−	0.999957i	$-0.497041\pi$
$229$	0.281355	0.0185925	0.00929623	+	0.999957i	$0.497041\pi$
$230$	−42.4110	−2.79650
$231$	0	0
$232$	72.1373	4.73605
$233$	14.4135	0.944261	0.472130	−	0.881529i	$-0.343485\pi$
$233$	14.4135	0.944261	0.472130	+	0.881529i	$0.343485\pi$
$234$	0	0
$235$	−10.3779	−0.676979
$236$	67.5536	4.39736
$237$	0	0
$238$	−7.16759	−0.464606
$239$	2.81273	0.181940	0.0909702	−	0.995854i	$-0.471003\pi$
$239$	2.81273	0.181940	0.0909702	+	0.995854i	$0.471003\pi$
$240$	0	0
$241$	−18.8316	−1.21305	−0.606525	−	0.795064i	$-0.707437\pi$
$241$	−18.8316	−1.21305	−0.606525	+	0.795064i	$0.707437\pi$
$242$	2.62825	0.168950
$243$	0	0
$244$	−4.90768	−0.314182
$245$	14.8356	0.947814
$246$	0	0
$247$	−33.5552	−2.13507
$248$	63.9264	4.05933
$249$	0	0
$250$	28.9434	1.83054
$251$	0.844957	0.0533332	0.0266666	−	0.999644i	$-0.491511\pi$
$251$	0.844957	0.0533332	0.0266666	+	0.999644i	$0.491511\pi$
$252$	0	0
$253$	−7.11412	−0.447261
$254$	38.4141	2.41032
$255$	0	0
$256$	−11.3315	−0.708220
$257$	−31.4775	−1.96351	−0.981757	−	0.190141i	$-0.939106\pi$
$257$	−31.4775	−1.96351	−0.981757	+	0.190141i	$0.939106\pi$
$258$	0	0
$259$	−1.04056	−0.0646572
$260$	69.4281	4.30575
$261$	0	0
$262$	29.9149	1.84815
$263$	−17.3883	−1.07221	−0.536103	−	0.844153i	$-0.680104\pi$
$263$	−17.3883	−1.07221	−0.536103	+	0.844153i	$0.680104\pi$
$264$	0	0
$265$	−19.9488	−1.22544
$266$	−9.58457	−0.587668
$267$	0	0
$268$	−4.01267	−0.245113
$269$	−18.0331	−1.09950	−0.549748	−	0.835330i	$-0.685276\pi$
$269$	−18.0331	−1.09950	−0.549748	+	0.835330i	$0.685276\pi$
$270$	0	0
$271$	−29.0923	−1.76723	−0.883616	−	0.468213i	$-0.844898\pi$
$271$	−29.0923	−1.76723	−0.883616	+	0.468213i	$0.844898\pi$
$272$	41.3200	2.50539
$273$	0	0
$274$	−2.42293	−0.146374
$275$	−0.144966	−0.00874178
$276$	0	0
$277$	7.16090	0.430257	0.215128	−	0.976586i	$-0.430983\pi$
$277$	7.16090	0.430257	0.215128	+	0.976586i	$0.430983\pi$
$278$	0.684936	0.0410797
$279$	0	0
$280$	11.7495	0.702164
$281$	27.9912	1.66982	0.834908	−	0.550390i	$-0.185521\pi$
$281$	27.9912	1.66982	0.834908	+	0.550390i	$0.185521\pi$
$282$	0	0
$283$	−18.9093	−1.12404	−0.562021	−	0.827123i	$-0.689976\pi$
$283$	−18.9093	−1.12404	−0.562021	+	0.827123i	$0.689976\pi$
$284$	39.3053	2.33234
$285$	0	0
$286$	16.3921	0.969284
$287$	1.85808	0.109679
$288$	0	0
$289$	−0.812277	−0.0477810
$290$	−56.2737	−3.30450
$291$	0	0
$292$	−26.7500	−1.56543
$293$	−27.0653	−1.58117	−0.790585	−	0.612353i	$-0.790223\pi$
$293$	−27.0653	−1.58117	−0.790585	+	0.612353i	$0.790223\pi$
$294$	0	0
$295$	−31.2222	−1.81783
$296$	11.7318	0.681897
$297$	0	0
$298$	−2.14596	−0.124312
$299$	−44.3700	−2.56598
$300$	0	0
$301$	5.95585	0.343289
$302$	−17.0585	−0.981608
$303$	0	0
$304$	55.2535	3.16900
$305$	2.26825	0.129880
$306$	0	0
$307$	7.55343	0.431097	0.215549	−	0.976493i	$-0.430846\pi$
$307$	7.55343	0.431097	0.215549	+	0.976493i	$0.430846\pi$
$308$	3.32652	0.189546
$309$	0	0
$310$	−49.8684	−2.83234
$311$	30.7650	1.74452	0.872262	−	0.489038i	$-0.162652\pi$
$311$	30.7650	1.74452	0.872262	+	0.489038i	$0.162652\pi$
$312$	0	0
$313$	−5.65307	−0.319531	−0.159765	−	0.987155i	$-0.551074\pi$
$313$	−5.65307	−0.319531	−0.159765	+	0.987155i	$0.551074\pi$
$314$	13.2675	0.748731
$315$	0	0
$316$	76.5941	4.30875
$317$	−9.21715	−0.517687	−0.258843	−	0.965919i	$-0.583341\pi$
$317$	−9.21715	−0.517687	−0.258843	+	0.965919i	$0.583341\pi$
$318$	0	0
$319$	−9.43948	−0.528509
$320$	−23.2063	−1.29727
$321$	0	0
$322$	−12.6737	−0.706275
$323$	21.6464	1.20444
$324$	0	0
$325$	−0.904137	−0.0501525
$326$	−2.81846	−0.156100
$327$	0	0
$328$	−20.9490	−1.15671
$329$	−3.10122	−0.170976
$330$	0	0
$331$	15.0509	0.827272	0.413636	−	0.910442i	$-0.364259\pi$
$331$	15.0509	0.827272	0.413636	+	0.910442i	$0.364259\pi$
$332$	−42.5419	−2.33479
$333$	0	0
$334$	−55.7100	−3.04832
$335$	1.85459	0.101327
$336$	0	0
$337$	−6.36834	−0.346906	−0.173453	−	0.984842i	$-0.555492\pi$
$337$	−6.36834	−0.346906	−0.173453	+	0.984842i	$0.555492\pi$
$338$	68.0684	3.70243
$339$	0	0
$340$	−44.7878	−2.42896
$341$	−8.36505	−0.452993
$342$	0	0
$343$	9.17806	0.495569
$344$	−67.1492	−3.62044
$345$	0	0
$346$	−54.9334	−2.95324
$347$	23.1377	1.24209	0.621047	−	0.783773i	$-0.286707\pi$
$347$	23.1377	1.24209	0.621047	+	0.783773i	$0.286707\pi$
$348$	0	0
$349$	2.71080	0.145106	0.0725530	−	0.997365i	$-0.476885\pi$
$349$	2.71080	0.145106	0.0725530	+	0.997365i	$0.476885\pi$
$350$	−0.258254	−0.0138042
$351$	0	0
$352$	−11.7077	−0.624024
$353$	6.59067	0.350786	0.175393	−	0.984498i	$-0.443880\pi$
$353$	6.59067	0.350786	0.175393	+	0.984498i	$0.443880\pi$
$354$	0	0
$355$	−18.1663	−0.964167
$356$	−15.3783	−0.815047
$357$	0	0
$358$	56.0224	2.96088
$359$	25.4541	1.34342	0.671708	−	0.740816i	$-0.265561\pi$
$359$	25.4541	1.34342	0.671708	+	0.740816i	$0.265561\pi$
$360$	0	0
$361$	9.94573	0.523459
$362$	−55.8816	−2.93707
$363$	0	0
$364$	20.7471	1.08745
$365$	12.3634	0.647132
$366$	0	0
$367$	5.22788	0.272893	0.136447	−	0.990647i	$-0.456432\pi$
$367$	5.22788	0.272893	0.136447	+	0.990647i	$0.456432\pi$
$368$	73.0616	3.80860
$369$	0	0
$370$	−9.15187	−0.475783
$371$	−5.96128	−0.309494
$372$	0	0
$373$	9.64167	0.499227	0.249613	−	0.968346i	$-0.419696\pi$
$373$	9.64167	0.499227	0.249613	+	0.968346i	$0.419696\pi$
$374$	−10.5745	−0.546793
$375$	0	0
$376$	34.9647	1.80317
$377$	−58.8730	−3.03211
$378$	0	0
$379$	−28.6776	−1.47307	−0.736535	−	0.676400i	$-0.763539\pi$
$379$	−28.6776	−1.47307	−0.736535	+	0.676400i	$0.763539\pi$
$380$	−59.8907	−3.07233
$381$	0	0
$382$	−34.6727	−1.77401
$383$	−19.1657	−0.979324	−0.489662	−	0.871912i	$-0.662880\pi$
$383$	−19.1657	−0.979324	−0.489662	+	0.871912i	$0.662880\pi$
$384$	0	0
$385$	−1.53747	−0.0783565
$386$	−55.0511	−2.80203
$387$	0	0
$388$	57.2889	2.90840
$389$	−3.48289	−0.176589	−0.0882947	−	0.996094i	$-0.528142\pi$
$389$	−3.48289	−0.176589	−0.0882947	+	0.996094i	$0.528142\pi$
$390$	0	0
$391$	28.6229	1.44752
$392$	−49.9835	−2.52455
$393$	0	0
$394$	−1.57936	−0.0795670
$395$	−35.4006	−1.78120
$396$	0	0
$397$	−9.88415	−0.496071	−0.248036	−	0.968751i	$-0.579785\pi$
$397$	−9.88415	−0.496071	−0.248036	+	0.968751i	$0.579785\pi$
$398$	14.8957	0.746654
$399$	0	0
$400$	1.48879	0.0744395
$401$	−8.47759	−0.423350	−0.211675	−	0.977340i	$-0.567892\pi$
$401$	−8.47759	−0.423350	−0.211675	+	0.977340i	$0.567892\pi$
$402$	0	0
$403$	−52.1719	−2.59887
$404$	−78.0978	−3.88551
$405$	0	0
$406$	−16.8162	−0.834575
$407$	−1.53516	−0.0760949
$408$	0	0
$409$	35.5810	1.75937	0.879684	−	0.475558i	$-0.157754\pi$
$409$	35.5810	1.75937	0.879684	+	0.475558i	$0.157754\pi$
$410$	16.3421	0.807078
$411$	0	0
$412$	−6.12897	−0.301952
$413$	−9.33011	−0.459105
$414$	0	0
$415$	19.6622	0.965179
$416$	−73.0198	−3.58009
$417$	0	0
$418$	−14.1403	−0.691624
$419$	7.46345	0.364614	0.182307	−	0.983242i	$-0.441644\pi$
$419$	7.46345	0.364614	0.182307	+	0.983242i	$0.441644\pi$
$420$	0	0
$421$	3.24263	0.158036	0.0790180	−	0.996873i	$-0.474822\pi$
$421$	3.24263	0.158036	0.0790180	+	0.996873i	$0.474822\pi$
$422$	−25.0488	−1.21936
$423$	0	0
$424$	67.2105	3.26403
$425$	0.583256	0.0282921
$426$	0	0
$427$	0.677820	0.0328020
$428$	−3.13348	−0.151463
$429$	0	0
$430$	52.3825	2.52611
$431$	41.3187	1.99025	0.995125	−	0.0986235i	$-0.0314440\pi$
$431$	41.3187	1.99025	0.995125	+	0.0986235i	$0.0314440\pi$
$432$	0	0
$433$	26.3674	1.26714	0.633568	−	0.773687i	$-0.281590\pi$
$433$	26.3674	1.26714	0.633568	+	0.773687i	$0.281590\pi$
$434$	−14.9021	−0.715326
$435$	0	0
$436$	−41.5796	−1.99130
$437$	38.2749	1.83094
$438$	0	0
$439$	34.8899	1.66521	0.832603	−	0.553871i	$-0.186850\pi$
$439$	34.8899	1.66521	0.832603	+	0.553871i	$0.186850\pi$
$440$	17.3342	0.826374
$441$	0	0
$442$	−65.9518	−3.13701
$443$	22.0070	1.04558	0.522791	−	0.852461i	$-0.324891\pi$
$443$	22.0070	1.04558	0.522791	+	0.852461i	$0.324891\pi$
$444$	0	0
$445$	7.10760	0.336933
$446$	59.7252	2.82807
$447$	0	0
$448$	−6.93472	−0.327635
$449$	−7.28821	−0.343952	−0.171976	−	0.985101i	$-0.555015\pi$
$449$	−7.28821	−0.343952	−0.171976	+	0.985101i	$0.555015\pi$
$450$	0	0
$451$	2.74126	0.129081
$452$	46.0462	2.16583
$453$	0	0
$454$	46.1823	2.16744
$455$	−9.58901	−0.449540
$456$	0	0
$457$	14.4748	0.677104	0.338552	−	0.940948i	$-0.390063\pi$
$457$	14.4748	0.677104	0.338552	+	0.940948i	$0.390063\pi$
$458$	0.739470	0.0345532
$459$	0	0
$460$	−79.1933	−3.69241
$461$	−25.6071	−1.19264	−0.596321	−	0.802746i	$-0.703372\pi$
$461$	−25.6071	−1.19264	−0.596321	+	0.802746i	$0.703372\pi$
$462$	0	0
$463$	−12.4274	−0.577549	−0.288775	−	0.957397i	$-0.593248\pi$
$463$	−12.4274	−0.577549	−0.288775	+	0.957397i	$0.593248\pi$
$464$	96.9428	4.50046
$465$	0	0
$466$	37.8823	1.75486
$467$	1.07287	0.0496466	0.0248233	−	0.999692i	$-0.492098\pi$
$467$	1.07287	0.0496466	0.0248233	+	0.999692i	$0.492098\pi$
$468$	0	0
$469$	0.554207	0.0255909
$470$	−27.2756	−1.25813
$471$	0	0
$472$	105.192	4.84187
$473$	8.78677	0.404016
$474$	0	0
$475$	0.779935	0.0357859
$476$	−13.3839	−0.613451
$477$	0	0
$478$	7.39255	0.338127
$479$	−16.5398	−0.755721	−0.377861	−	0.925862i	$-0.623340\pi$
$479$	−16.5398	−0.755721	−0.377861	+	0.925862i	$0.623340\pi$
$480$	0	0
$481$	−9.57460	−0.436564
$482$	−49.4941	−2.25439
$483$	0	0
$484$	4.90768	0.223076
$485$	−26.4780	−1.20231
$486$	0	0
$487$	22.8545	1.03564	0.517818	−	0.855491i	$-0.326745\pi$
$487$	22.8545	1.03564	0.517818	+	0.855491i	$0.326745\pi$
$488$	−7.64209	−0.345941
$489$	0	0
$490$	38.9917	1.76147
$491$	11.4464	0.516570	0.258285	−	0.966069i	$-0.416843\pi$
$491$	11.4464	0.516570	0.258285	+	0.966069i	$0.416843\pi$
$492$	0	0
$493$	37.9788	1.71048
$494$	−88.1914	−3.96792
$495$	0	0
$496$	85.9085	3.85740
$497$	−5.42862	−0.243507
$498$	0	0
$499$	−26.9452	−1.20623	−0.603117	−	0.797652i	$-0.706075\pi$
$499$	−26.9452	−1.20623	−0.603117	+	0.797652i	$0.706075\pi$
$500$	54.0455	2.41699
$501$	0	0
$502$	2.22075	0.0991171
$503$	−5.28203	−0.235514	−0.117757	−	0.993042i	$-0.537570\pi$
$503$	−5.28203	−0.235514	−0.117757	+	0.993042i	$0.537570\pi$
$504$	0	0
$505$	36.0956	1.60623
$506$	−18.6977	−0.831213
$507$	0	0
$508$	71.7300	3.18250
$509$	2.80042	0.124126	0.0620632	−	0.998072i	$-0.480232\pi$
$509$	2.80042	0.124126	0.0620632	+	0.998072i	$0.480232\pi$
$510$	0	0
$511$	3.69456	0.163438
$512$	−36.7299	−1.62325
$513$	0	0
$514$	−82.7307	−3.64909
$515$	2.83271	0.124824
$516$	0	0
$517$	−4.57528	−0.201221
$518$	−2.73485	−0.120162
$519$	0	0
$520$	108.111	4.74100
$521$	19.6351	0.860230	0.430115	−	0.902774i	$-0.358473\pi$
$521$	19.6351	0.860230	0.430115	+	0.902774i	$0.358473\pi$
$522$	0	0
$523$	−8.93069	−0.390512	−0.195256	−	0.980752i	$-0.562554\pi$
$523$	−8.93069	−0.390512	−0.195256	+	0.980752i	$0.562554\pi$
$524$	55.8596	2.44024
$525$	0	0
$526$	−45.7006	−1.99264
$527$	33.6559	1.46607
$528$	0	0
$529$	27.6108	1.20047
$530$	−52.4303	−2.27743
$531$	0	0
$532$	−17.8971	−0.775937
$533$	17.0969	0.740551
$534$	0	0
$535$	1.44825	0.0626132
$536$	−6.24841	−0.269890
$537$	0	0
$538$	−47.3954	−2.04336
$539$	6.54056	0.281722
$540$	0	0
$541$	−11.8835	−0.510912	−0.255456	−	0.966821i	$-0.582226\pi$
$541$	−11.8835	−0.510912	−0.255456	+	0.966821i	$0.582226\pi$
$542$	−76.4617	−3.28431
$543$	0	0
$544$	47.1048	2.01960
$545$	19.2175	0.823186
$546$	0	0
$547$	−22.0270	−0.941807	−0.470904	−	0.882185i	$-0.656072\pi$
$547$	−22.0270	−0.941807	−0.470904	+	0.882185i	$0.656072\pi$
$548$	−4.52429	−0.193268
$549$	0	0
$550$	−0.381006	−0.0162462
$551$	50.7856	2.16354
$552$	0	0
$553$	−10.5787	−0.449853
$554$	18.8206	0.799611
$555$	0	0
$556$	1.27897	0.0542404
$557$	−4.77487	−0.202318	−0.101159	−	0.994870i	$-0.532255\pi$
$557$	−4.77487	−0.202318	−0.101159	+	0.994870i	$0.532255\pi$
$558$	0	0
$559$	54.8021	2.31788
$560$	15.7897	0.667235
$561$	0	0
$562$	73.5678	3.10327
$563$	−30.9141	−1.30287	−0.651436	−	0.758703i	$-0.725833\pi$
$563$	−30.9141	−1.30287	−0.651436	+	0.758703i	$0.725833\pi$
$564$	0	0
$565$	−21.2819	−0.895335
$566$	−49.6983	−2.08898
$567$	0	0
$568$	61.2051	2.56811
$569$	9.27498	0.388827	0.194414	−	0.980920i	$-0.437720\pi$
$569$	9.27498	0.388827	0.194414	+	0.980920i	$0.437720\pi$
$570$	0	0
$571$	36.5002	1.52749	0.763743	−	0.645521i	$-0.223360\pi$
$571$	36.5002	1.52749	0.763743	+	0.645521i	$0.223360\pi$
$572$	30.6086	1.27981
$573$	0	0
$574$	4.88349	0.203833
$575$	1.03131	0.0430084
$576$	0	0
$577$	3.79582	0.158022	0.0790110	−	0.996874i	$-0.474824\pi$
$577$	3.79582	0.158022	0.0790110	+	0.996874i	$0.474824\pi$
$578$	−2.13486	−0.0887986
$579$	0	0
$580$	−105.079	−4.36316
$581$	5.87564	0.243762
$582$	0	0
$583$	−8.79479	−0.364243
$584$	−41.6543	−1.72367
$585$	0	0
$586$	−71.1342	−2.93853
$587$	2.71327	0.111989	0.0559943	−	0.998431i	$-0.482167\pi$
$587$	2.71327	0.111989	0.0559943	+	0.998431i	$0.482167\pi$
$588$	0	0
$589$	45.0050	1.85440
$590$	−82.0597	−3.37834
$591$	0	0
$592$	15.7659	0.647977
$593$	14.5360	0.596923	0.298462	−	0.954422i	$-0.403527\pi$
$593$	14.5360	0.596923	0.298462	+	0.954422i	$0.403527\pi$
$594$	0	0
$595$	6.18584	0.253595
$596$	−4.00711	−0.164137
$597$	0	0
$598$	−116.615	−4.76875
$599$	−2.72166	−0.111204	−0.0556020	−	0.998453i	$-0.517708\pi$
$599$	−2.72166	−0.111204	−0.0556020	+	0.998453i	$0.517708\pi$
$600$	0	0
$601$	−44.6005	−1.81929	−0.909647	−	0.415383i	$-0.863647\pi$
$601$	−44.6005	−1.81929	−0.909647	+	0.415383i	$0.863647\pi$
$602$	15.6534	0.637986
$603$	0	0
$604$	−31.8531	−1.29608
$605$	−2.26825	−0.0922175
$606$	0	0
$607$	−44.7326	−1.81564	−0.907820	−	0.419360i	$-0.862255\pi$
$607$	−44.7326	−1.81564	−0.907820	+	0.419360i	$0.862255\pi$
$608$	62.9890	2.55454
$609$	0	0
$610$	5.96152	0.241375
$611$	−28.5355	−1.15442
$612$	0	0
$613$	14.2699	0.576358	0.288179	−	0.957577i	$-0.406950\pi$
$613$	14.2699	0.576358	0.288179	+	0.957577i	$0.406950\pi$
$614$	19.8523	0.801173
$615$	0	0
$616$	5.17996	0.208706
$617$	26.9261	1.08400	0.542001	−	0.840378i	$-0.317667\pi$
$617$	26.9261	1.08400	0.542001	+	0.840378i	$0.317667\pi$
$618$	0	0
$619$	33.2293	1.33560	0.667800	−	0.744341i	$-0.267236\pi$
$619$	33.2293	1.33560	0.667800	+	0.744341i	$0.267236\pi$
$620$	−93.1184	−3.73973
$621$	0	0
$622$	80.8581	3.24211
$623$	2.12396	0.0850947
$624$	0	0
$625$	−25.7038	−1.02815
$626$	−14.8577	−0.593832
$627$	0	0
$628$	24.7743	0.988600
$629$	6.17654	0.246275
$630$	0	0
$631$	−30.5417	−1.21584	−0.607922	−	0.793997i	$-0.707997\pi$
$631$	−30.5417	−1.21584	−0.607922	+	0.793997i	$0.707997\pi$
$632$	119.270	4.74431
$633$	0	0
$634$	−24.2249	−0.962095
$635$	−33.1525	−1.31562
$636$	0	0
$637$	40.7928	1.61627
$638$	−24.8093	−0.982209
$639$	0	0
$640$	−7.87974	−0.311474
$641$	−15.0097	−0.592849	−0.296424	−	0.955056i	$-0.595794\pi$
$641$	−15.0097	−0.592849	−0.296424	+	0.955056i	$0.595794\pi$
$642$	0	0
$643$	−6.49653	−0.256198	−0.128099	−	0.991761i	$-0.540888\pi$
$643$	−6.49653	−0.256198	−0.128099	+	0.991761i	$0.540888\pi$
$644$	−23.6653	−0.932543
$645$	0	0
$646$	56.8920	2.23839
$647$	18.1420	0.713235	0.356618	−	0.934250i	$-0.383930\pi$
$647$	18.1420	0.713235	0.356618	+	0.934250i	$0.383930\pi$
$648$	0	0
$649$	−13.7649	−0.540319
$650$	−2.37629	−0.0932059
$651$	0	0
$652$	−5.26285	−0.206109
$653$	−42.4886	−1.66271	−0.831354	−	0.555743i	$-0.812434\pi$
$653$	−42.4886	−1.66271	−0.831354	+	0.555743i	$0.812434\pi$
$654$	0	0
$655$	−25.8174	−1.00877
$656$	−28.1526	−1.09917
$657$	0	0
$658$	−8.15076	−0.317750
$659$	34.9887	1.36297	0.681484	−	0.731833i	$-0.261335\pi$
$659$	34.9887	1.36297	0.681484	+	0.731833i	$0.261335\pi$
$660$	0	0
$661$	−18.1116	−0.704460	−0.352230	−	0.935913i	$-0.614577\pi$
$661$	−18.1116	−0.704460	−0.352230	+	0.935913i	$0.614577\pi$
$662$	39.5574	1.53744
$663$	0	0
$664$	−66.2449	−2.57080
$665$	8.27176	0.320765
$666$	0	0
$667$	67.1536	2.60020
$668$	−104.026	−4.02490
$669$	0	0
$670$	4.87433	0.188312
$671$	1.00000	0.0386046
$672$	0	0
$673$	−48.0194	−1.85101	−0.925506	−	0.378734i	$-0.876360\pi$
$673$	−48.0194	−1.85101	−0.925506	+	0.378734i	$0.876360\pi$
$674$	−16.7376	−0.644707
$675$	0	0
$676$	127.103	4.88857
$677$	−33.3800	−1.28290	−0.641450	−	0.767165i	$-0.721667\pi$
$677$	−33.3800	−1.28290	−0.641450	+	0.767165i	$0.721667\pi$
$678$	0	0
$679$	−7.91241	−0.303650
$680$	−69.7423	−2.67449
$681$	0	0
$682$	−21.9854	−0.841865
$683$	19.0568	0.729187	0.364594	−	0.931167i	$-0.381208\pi$
$683$	19.0568	0.729187	0.364594	+	0.931167i	$0.381208\pi$
$684$	0	0
$685$	2.09106	0.0798952
$686$	24.1222	0.920990
$687$	0	0
$688$	−90.2395	−3.44035
$689$	−54.8521	−2.08970
$690$	0	0
$691$	−26.0922	−0.992596	−0.496298	−	0.868152i	$-0.665308\pi$
$691$	−26.0922	−0.992596	−0.496298	+	0.868152i	$0.665308\pi$
$692$	−102.576	−3.89936
$693$	0	0
$694$	60.8115	2.30837
$695$	−0.591120	−0.0224224
$696$	0	0
$697$	−11.0292	−0.417760
$698$	7.12465	0.269672
$699$	0	0
$700$	−0.482232	−0.0182267
$701$	12.5379	0.473550	0.236775	−	0.971565i	$-0.423910\pi$
$701$	12.5379	0.473550	0.236775	+	0.971565i	$0.423910\pi$
$702$	0	0
$703$	8.25933	0.311507
$704$	−10.2309	−0.385592
$705$	0	0
$706$	17.3219	0.651918
$707$	10.7864	0.405665
$708$	0	0
$709$	19.5002	0.732347	0.366173	−	0.930547i	$-0.380668\pi$
$709$	19.5002	0.732347	0.366173	+	0.930547i	$0.380668\pi$
$710$	−47.7455	−1.79186
$711$	0	0
$712$	−23.9466	−0.897437
$713$	59.5100	2.22867
$714$	0	0
$715$	−14.1468	−0.529062
$716$	104.610	3.90945
$717$	0	0
$718$	66.8996	2.49667
$719$	−9.69442	−0.361541	−0.180770	−	0.983525i	$-0.557859\pi$
$719$	−9.69442	−0.361541	−0.180770	+	0.983525i	$0.557859\pi$
$720$	0	0
$721$	0.846497	0.0315252
$722$	26.1398	0.972823
$723$	0	0
$724$	−104.347	−3.87802
$725$	1.36840	0.0508212
$726$	0	0
$727$	31.2453	1.15882	0.579411	−	0.815035i	$-0.303282\pi$
$727$	31.2453	1.15882	0.579411	+	0.815035i	$0.303282\pi$
$728$	32.3068	1.19737
$729$	0	0
$730$	32.4942	1.20266
$731$	−35.3527	−1.30757
$732$	0	0
$733$	−37.5135	−1.38559	−0.692796	−	0.721133i	$-0.743621\pi$
$733$	−37.5135	−1.38559	−0.692796	+	0.721133i	$0.743621\pi$
$734$	13.7402	0.507158
$735$	0	0
$736$	83.2902	3.07012
$737$	0.817631	0.0301178
$738$	0	0
$739$	25.3336	0.931911	0.465956	−	0.884808i	$-0.345711\pi$
$739$	25.3336	0.931911	0.465956	+	0.884808i	$0.345711\pi$
$740$	−17.0891	−0.628209
$741$	0	0
$742$	−15.6677	−0.575180
$743$	−18.8748	−0.692447	−0.346224	−	0.938152i	$-0.612536\pi$
$743$	−18.8748	−0.692447	−0.346224	+	0.938152i	$0.612536\pi$
$744$	0	0
$745$	1.85202	0.0678529
$746$	25.3407	0.927788
$747$	0	0
$748$	−19.7455	−0.721968
$749$	0.432779	0.0158134
$750$	0	0
$751$	−33.5250	−1.22334	−0.611672	−	0.791112i	$-0.709503\pi$
$751$	−33.5250	−1.22334	−0.611672	+	0.791112i	$0.709503\pi$
$752$	46.9878	1.71347
$753$	0	0
$754$	−154.733	−5.63503
$755$	14.7220	0.535789
$756$	0	0
$757$	−11.9328	−0.433705	−0.216853	−	0.976204i	$-0.569579\pi$
$757$	−11.9328	−0.433705	−0.216853	+	0.976204i	$0.569579\pi$
$758$	−75.3718	−2.73763
$759$	0	0
$760$	−93.2600	−3.38290
$761$	−35.8602	−1.29993	−0.649966	−	0.759963i	$-0.725217\pi$
$761$	−35.8602	−1.29993	−0.649966	+	0.759963i	$0.725217\pi$
$762$	0	0
$763$	5.74274	0.207901
$764$	−64.7438	−2.34235
$765$	0	0
$766$	−50.3723	−1.82002
$767$	−85.8501	−3.09987
$768$	0	0
$769$	6.21041	0.223953	0.111977	−	0.993711i	$-0.464282\pi$
$769$	6.21041	0.223953	0.111977	+	0.993711i	$0.464282\pi$
$770$	−4.04084	−0.145622
$771$	0	0
$772$	−102.796	−3.69971
$773$	3.59019	0.129130	0.0645651	−	0.997913i	$-0.479434\pi$
$773$	3.59019	0.129130	0.0645651	+	0.997913i	$0.479434\pi$
$774$	0	0
$775$	1.21265	0.0435596
$776$	89.2085	3.20240
$777$	0	0
$778$	−9.15388	−0.328183
$779$	−14.7483	−0.528413
$780$	0	0
$781$	−8.00895	−0.286583
$782$	75.2281	2.69015
$783$	0	0
$784$	−67.1711	−2.39897
$785$	−11.4503	−0.408678
$786$	0	0
$787$	−7.25410	−0.258581	−0.129290	−	0.991607i	$-0.541270\pi$
$787$	−7.25410	−0.258581	−0.129290	+	0.991607i	$0.541270\pi$
$788$	−2.94911	−0.105058
$789$	0	0
$790$	−93.0415	−3.31027
$791$	−6.35964	−0.226123
$792$	0	0
$793$	6.23689	0.221478
$794$	−25.9780	−0.921924
$795$	0	0
$796$	27.8145	0.985858
$797$	18.9025	0.669559	0.334780	−	0.942296i	$-0.391338\pi$
$797$	18.9025	0.669559	0.334780	+	0.942296i	$0.391338\pi$
$798$	0	0
$799$	18.4082	0.651234
$800$	1.69722	0.0600059
$801$	0	0
$802$	−22.2812	−0.786776
$803$	5.45065	0.192349
$804$	0	0
$805$	10.9377	0.385504
$806$	−137.121	−4.82987
$807$	0	0
$808$	−121.612	−4.27828
$809$	−18.1394	−0.637748	−0.318874	−	0.947797i	$-0.603305\pi$
$809$	−18.1394	−0.637748	−0.318874	+	0.947797i	$0.603305\pi$
$810$	0	0
$811$	23.7262	0.833140	0.416570	−	0.909104i	$-0.363232\pi$
$811$	23.7262	0.833140	0.416570	+	0.909104i	$0.363232\pi$
$812$	−31.4006	−1.10195
$813$	0	0
$814$	−4.03477	−0.141419
$815$	2.43241	0.0852036
$816$	0	0
$817$	−47.2739	−1.65390
$818$	93.5157	3.26970
$819$	0	0
$820$	30.5153	1.06564
$821$	3.45440	0.120559	0.0602797	−	0.998182i	$-0.480801\pi$
$821$	3.45440	0.120559	0.0602797	+	0.998182i	$0.480801\pi$
$822$	0	0
$823$	−36.5852	−1.27528	−0.637640	−	0.770334i	$-0.720089\pi$
$823$	−36.5852	−1.27528	−0.637640	+	0.770334i	$0.720089\pi$
$824$	−9.54384	−0.332476
$825$	0	0
$826$	−24.5218	−0.853224
$827$	22.9390	0.797666	0.398833	−	0.917023i	$-0.369415\pi$
$827$	22.9390	0.797666	0.398833	+	0.917023i	$0.369415\pi$
$828$	0	0
$829$	5.11745	0.177736	0.0888682	−	0.996043i	$-0.471675\pi$
$829$	5.11745	0.177736	0.0888682	+	0.996043i	$0.471675\pi$
$830$	51.6771	1.79374
$831$	0	0
$832$	−63.8091	−2.21218
$833$	−26.3153	−0.911770
$834$	0	0
$835$	48.0794	1.66385
$836$	−26.4039	−0.913198
$837$	0	0
$838$	19.6158	0.677616
$839$	10.9795	0.379054	0.189527	−	0.981875i	$-0.439305\pi$
$839$	10.9795	0.379054	0.189527	+	0.981875i	$0.439305\pi$
$840$	0	0
$841$	60.1038	2.07254
$842$	8.52242	0.293702
$843$	0	0
$844$	−46.7732	−1.61000
$845$	−58.7450	−2.02089
$846$	0	0
$847$	−0.677820	−0.0232902
$848$	90.3218	3.10166
$849$	0	0
$850$	1.53294	0.0525794
$851$	10.9213	0.374377
$852$	0	0
$853$	−21.2622	−0.728004	−0.364002	−	0.931398i	$-0.618590\pi$
$853$	−21.2622	−0.728004	−0.364002	+	0.931398i	$0.618590\pi$
$854$	1.78148	0.0609609
$855$	0	0
$856$	−4.87936	−0.166773
$857$	−27.2670	−0.931422	−0.465711	−	0.884937i	$-0.654201\pi$
$857$	−27.2670	−0.931422	−0.465711	+	0.884937i	$0.654201\pi$
$858$	0	0
$859$	18.3514	0.626142	0.313071	−	0.949730i	$-0.398642\pi$
$859$	18.3514	0.626142	0.313071	+	0.949730i	$0.398642\pi$
$860$	97.8129	3.33539
$861$	0	0
$862$	108.596	3.69878
$863$	−20.0649	−0.683017	−0.341509	−	0.939879i	$-0.610938\pi$
$863$	−20.0649	−0.683017	−0.341509	+	0.939879i	$0.610938\pi$
$864$	0	0
$865$	47.4091	1.61196
$866$	69.3000	2.35491
$867$	0	0
$868$	−27.8265	−0.944493
$869$	−15.6070	−0.529431
$870$	0	0
$871$	5.09948	0.172789
$872$	−64.7466	−2.19260
$873$	0	0
$874$	100.596	3.40270
$875$	−7.46445	−0.252345
$876$	0	0
$877$	17.6278	0.595250	0.297625	−	0.954683i	$-0.403806\pi$
$877$	17.6278	0.595250	0.297625	+	0.954683i	$0.403806\pi$
$878$	91.6993	3.09470
$879$	0	0
$880$	23.2948	0.785267
$881$	6.68212	0.225126	0.112563	−	0.993645i	$-0.464094\pi$
$881$	6.68212	0.225126	0.112563	+	0.993645i	$0.464094\pi$
$882$	0	0
$883$	3.45390	0.116233	0.0581165	−	0.998310i	$-0.481491\pi$
$883$	3.45390	0.116233	0.0581165	+	0.998310i	$0.481491\pi$
$884$	−123.151	−4.14201
$885$	0	0
$886$	57.8397	1.94316
$887$	−10.1154	−0.339641	−0.169821	−	0.985475i	$-0.554319\pi$
$887$	−10.1154	−0.339641	−0.169821	+	0.985475i	$0.554319\pi$
$888$	0	0
$889$	−9.90693	−0.332268
$890$	18.6805	0.626173
$891$	0	0
$892$	111.524	3.73409
$893$	24.6156	0.823729
$894$	0	0
$895$	−48.3490	−1.61613
$896$	−2.35470	−0.0786649
$897$	0	0
$898$	−19.1552	−0.639218
$899$	78.9617	2.63352
$900$	0	0
$901$	35.3849	1.17884
$902$	7.20471	0.239891
$903$	0	0
$904$	71.7018	2.38477
$905$	48.2274	1.60313
$906$	0	0
$907$	−23.5840	−0.783095	−0.391548	−	0.920158i	$-0.628060\pi$
$907$	−23.5840	−0.783095	−0.391548	+	0.920158i	$0.628060\pi$
$908$	86.2354	2.86182
$909$	0	0
$910$	−25.2023	−0.835447
$911$	−46.5930	−1.54370	−0.771848	−	0.635808i	$-0.780667\pi$
$911$	−46.5930	−1.54370	−0.771848	+	0.635808i	$0.780667\pi$
$912$	0	0
$913$	8.66843	0.286883
$914$	38.0434	1.25836
$915$	0	0
$916$	1.38080	0.0456229
$917$	−7.71501	−0.254772
$918$	0	0
$919$	−29.3294	−0.967486	−0.483743	−	0.875210i	$-0.660723\pi$
$919$	−29.3294	−0.967486	−0.483743	+	0.875210i	$0.660723\pi$
$920$	−123.317	−4.06566
$921$	0	0
$922$	−67.3018	−2.21647
$923$	−49.9509	−1.64415
$924$	0	0
$925$	0.222545	0.00731725
$926$	−32.6622	−1.07335
$927$	0	0
$928$	110.515	3.62783
$929$	−14.9563	−0.490701	−0.245350	−	0.969434i	$-0.578903\pi$
$929$	−14.9563	−0.490701	−0.245350	+	0.969434i	$0.578903\pi$
$930$	0	0
$931$	−35.1890	−1.15327
$932$	70.7369	2.31706
$933$	0	0
$934$	2.81977	0.0922657
$935$	9.12608	0.298455
$936$	0	0
$937$	−11.4240	−0.373205	−0.186602	−	0.982436i	$-0.559748\pi$
$937$	−11.4240	−0.373205	−0.186602	+	0.982436i	$0.559748\pi$
$938$	1.45659	0.0475594
$939$	0	0
$940$	−50.9313	−1.66120
$941$	8.45015	0.275467	0.137734	−	0.990469i	$-0.456018\pi$
$941$	8.45015	0.275467	0.137734	+	0.990469i	$0.456018\pi$
$942$	0	0
$943$	−19.5017	−0.635062
$944$	141.364	4.60102
$945$	0	0
$946$	23.0938	0.750844
$947$	−37.0384	−1.20359	−0.601793	−	0.798652i	$-0.705547\pi$
$947$	−37.0384	−1.20359	−0.601793	+	0.798652i	$0.705547\pi$
$948$	0	0
$949$	33.9951	1.10353
$950$	2.04986	0.0665062
$951$	0	0
$952$	−20.8410	−0.675462
$953$	−22.1687	−0.718115	−0.359058	−	0.933315i	$-0.616902\pi$
$953$	−22.1687	−0.718115	−0.359058	+	0.933315i	$0.616902\pi$
$954$	0	0
$955$	29.9236	0.968304
$956$	13.8040	0.446452
$957$	0	0
$958$	−43.4706	−1.40447
$959$	0.624869	0.0201781
$960$	0	0
$961$	38.9740	1.25723
$962$	−25.1644	−0.811333
$963$	0	0
$964$	−92.4194	−2.97663
$965$	47.5107	1.52942
$966$	0	0
$967$	−10.1901	−0.327691	−0.163845	−	0.986486i	$-0.552390\pi$
$967$	−10.1901	−0.327691	−0.163845	+	0.986486i	$0.552390\pi$
$968$	7.64209	0.245626
$969$	0	0
$970$	−69.5908	−2.23442
$971$	−32.9601	−1.05774	−0.528870	−	0.848703i	$-0.677384\pi$
$971$	−32.9601	−1.05774	−0.528870	+	0.848703i	$0.677384\pi$
$972$	0	0
$973$	−0.176644	−0.00566294
$974$	60.0672	1.92468
$975$	0	0
$976$	−10.2699	−0.328732
$977$	−21.0735	−0.674200	−0.337100	−	0.941469i	$-0.609446\pi$
$977$	−21.0735	−0.674200	−0.337100	+	0.941469i	$0.609446\pi$
$978$	0	0
$979$	3.13352	0.100148
$980$	72.8085	2.32578
$981$	0	0
$982$	30.0840	0.960019
$983$	23.3726	0.745470	0.372735	−	0.927938i	$-0.378420\pi$
$983$	23.3726	0.745470	0.372735	+	0.927938i	$0.378420\pi$
$984$	0	0
$985$	1.36303	0.0434299
$986$	99.8176	3.17884
$987$	0	0
$988$	−164.678	−5.23911
$989$	−62.5101	−1.98771
$990$	0	0
$991$	35.7069	1.13427	0.567134	−	0.823626i	$-0.308052\pi$
$991$	35.7069	1.13427	0.567134	+	0.823626i	$0.308052\pi$
$992$	97.9357	3.10946
$993$	0	0
$994$	−14.2678	−0.452546
$995$	−12.8554	−0.407544
$996$	0	0
$997$	−27.8299	−0.881382	−0.440691	−	0.897659i	$-0.645267\pi$
$997$	−27.8299	−0.881382	−0.440691	+	0.897659i	$0.645267\pi$
$998$	−70.8187	−2.24173
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6039.2.a.f.1.11		12
3.2	odd	2		2013.2.a.c.1.2	✓	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2013.2.a.c.1.2	✓	12	3.2	odd	2
6039.2.a.f.1.11		12	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 6039 = 3^{2} \cdot 11 \cdot 61 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6039.a (trivial)

\(f(q)\)	\(=\)	\(q+2.62825 q^{2} +4.90768 q^{4} -2.26825 q^{5} -0.677820 q^{7} +7.64209 q^{8} +O(q^{10})\)	\(q+2.62825 q^{2} +4.90768 q^{4} -2.26825 q^{5} -0.677820 q^{7} +7.64209 q^{8} -5.96152 q^{10} -1.00000 q^{11} -6.23689 q^{13} -1.78148 q^{14} +10.2699 q^{16} +4.02340 q^{17} +5.38012 q^{19} -11.1318 q^{20} -2.62825 q^{22} +7.11412 q^{23} +0.144966 q^{25} -16.3921 q^{26} -3.32652 q^{28} +9.43948 q^{29} +8.36505 q^{31} +11.7077 q^{32} +10.5745 q^{34} +1.53747 q^{35} +1.53516 q^{37} +14.1403 q^{38} -17.3342 q^{40} -2.74126 q^{41} -8.78677 q^{43} -4.90768 q^{44} +18.6977 q^{46} +4.57528 q^{47} -6.54056 q^{49} +0.381006 q^{50} -30.6086 q^{52} +8.79479 q^{53} +2.26825 q^{55} -5.17996 q^{56} +24.8093 q^{58} +13.7649 q^{59} -1.00000 q^{61} +21.9854 q^{62} +10.2309 q^{64} +14.1468 q^{65} -0.817631 q^{67} +19.7455 q^{68} +4.04084 q^{70} +8.00895 q^{71} -5.45065 q^{73} +4.03477 q^{74} +26.4039 q^{76} +0.677820 q^{77} +15.6070 q^{79} -23.2948 q^{80} -7.20471 q^{82} -8.66843 q^{83} -9.12608 q^{85} -23.0938 q^{86} -7.64209 q^{88} -3.13352 q^{89} +4.22749 q^{91} +34.9138 q^{92} +12.0250 q^{94} -12.2035 q^{95} +11.6733 q^{97} -17.1902 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 7 q^{2} + 13 q^{4} + 7 q^{5} - 15 q^{7} + 18 q^{8}+O(q^{10}) \) 12 * q + 7 * q^2 + 13 * q^4 + 7 * q^5 - 15 * q^7 + 18 * q^8	\( 12 q + 7 q^{2} + 13 q^{4} + 7 q^{5} - 15 q^{7} + 18 q^{8} - 6 q^{10} - 12 q^{11} - 11 q^{13} - 3 q^{14} + 19 q^{16} + 33 q^{17} - 24 q^{19} + 11 q^{20} - 7 q^{22} + 9 q^{23} + 11 q^{25} + 16 q^{26} - 41 q^{28} + 16 q^{29} + q^{31} + 28 q^{32} + 32 q^{34} + 22 q^{35} - 6 q^{37} - 12 q^{38} + 26 q^{40} + 21 q^{41} - 39 q^{43} - 13 q^{44} + 18 q^{47} + 31 q^{49} + 44 q^{50} + 3 q^{52} + 14 q^{53} - 7 q^{55} - 16 q^{56} + 33 q^{58} + 23 q^{59} - 12 q^{61} + 25 q^{62} + 12 q^{64} + 29 q^{65} + 96 q^{68} + 44 q^{70} + 19 q^{71} - 42 q^{73} - 38 q^{74} + 11 q^{76} + 15 q^{77} - 11 q^{79} + 44 q^{80} - 14 q^{82} + 56 q^{83} + 16 q^{85} + 18 q^{86} - 18 q^{88} + 55 q^{89} + 11 q^{91} + 4 q^{92} - 5 q^{94} - 15 q^{95} - 7 q^{97} - 6 q^{98}+O(q^{100}) \) 12 * q + 7 * q^2 + 13 * q^4 + 7 * q^5 - 15 * q^7 + 18 * q^8 - 6 * q^10 - 12 * q^11 - 11 * q^13 - 3 * q^14 + 19 * q^16 + 33 * q^17 - 24 * q^19 + 11 * q^20 - 7 * q^22 + 9 * q^23 + 11 * q^25 + 16 * q^26 - 41 * q^28 + 16 * q^29 + q^31 + 28 * q^32 + 32 * q^34 + 22 * q^35 - 6 * q^37 - 12 * q^38 + 26 * q^40 + 21 * q^41 - 39 * q^43 - 13 * q^44 + 18 * q^47 + 31 * q^49 + 44 * q^50 + 3 * q^52 + 14 * q^53 - 7 * q^55 - 16 * q^56 + 33 * q^58 + 23 * q^59 - 12 * q^61 + 25 * q^62 + 12 * q^64 + 29 * q^65 + 96 * q^68 + 44 * q^70 + 19 * q^71 - 42 * q^73 - 38 * q^74 + 11 * q^76 + 15 * q^77 - 11 * q^79 + 44 * q^80 - 14 * q^82 + 56 * q^83 + 16 * q^85 + 18 * q^86 - 18 * q^88 + 55 * q^89 + 11 * q^91 + 4 * q^92 - 5 * q^94 - 15 * q^95 - 7 * q^97 - 6 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more