LMFDB - Embedded newform 6039.2.a.g.1.9

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:	$1$
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} - 4 x^{12} - 11 x^{11} + 55 x^{10} + 32 x^{9} - 266 x^{8} + 13 x^{7} + 534 x^{6} - 141 x^{5} + \cdots - 11 $ x^13 - 4x^12 - 11x^11 + 55x^10 + 32x^9 - 266x^8 + 13x^7 + 534x^6 - 141x^5 - 404x^4 + 98x^3 + 118x^2 - 16x - 11
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			1.9
Root			$-0.534142$ 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)

$f(q)$	$=$	$q+0.534142 q^{2} -1.71469 q^{4} -3.62612 q^{5} +4.31243 q^{7} -1.98417 q^{8} +O(q^{10})$	\(q+0.534142 q^{2} -1.71469 q^{4} -3.62612 q^{5} +4.31243 q^{7} -1.98417 q^{8} -1.93686 q^{10} +1.00000 q^{11} -1.21915 q^{13} +2.30345 q^{14} +2.36956 q^{16} -0.492960 q^{17} +0.909227 q^{19} +6.21767 q^{20} +0.534142 q^{22} -9.05847 q^{23} +8.14872 q^{25} -0.651199 q^{26} -7.39450 q^{28} +3.36455 q^{29} -1.80170 q^{31} +5.23402 q^{32} -0.263311 q^{34} -15.6374 q^{35} -2.52082 q^{37} +0.485656 q^{38} +7.19484 q^{40} +3.50527 q^{41} +1.17586 q^{43} -1.71469 q^{44} -4.83851 q^{46} +6.47062 q^{47} +11.5971 q^{49} +4.35257 q^{50} +2.09047 q^{52} -7.40180 q^{53} -3.62612 q^{55} -8.55661 q^{56} +1.79715 q^{58} +12.4433 q^{59} -1.00000 q^{61} -0.962363 q^{62} -1.94340 q^{64} +4.42078 q^{65} -1.39825 q^{67} +0.845275 q^{68} -8.35258 q^{70} +12.1125 q^{71} -9.92547 q^{73} -1.34647 q^{74} -1.55905 q^{76} +4.31243 q^{77} +4.62242 q^{79} -8.59228 q^{80} +1.87231 q^{82} +2.80042 q^{83} +1.78753 q^{85} +0.628078 q^{86} -1.98417 q^{88} -12.0828 q^{89} -5.25750 q^{91} +15.5325 q^{92} +3.45623 q^{94} -3.29696 q^{95} +2.52978 q^{97} +6.19449 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 13 q - 4 q^{2} + 12 q^{4} - 7 q^{5} + 5 q^{7} - 15 q^{8}+O(q^{10}) $ 13 * q - 4 * q^2 + 12 * q^4 - 7 * q^5 + 5 * q^7 - 15 * q^8	$ 13 q - 4 q^{2} + 12 q^{4} - 7 q^{5} + 5 q^{7} - 15 q^{8} + 8 q^{10} + 13 q^{11} - 9 q^{13} - 19 q^{14} + 18 q^{16} - 7 q^{17} + 2 q^{19} - 15 q^{20} - 4 q^{22} - 23 q^{23} + 10 q^{25} - 8 q^{26} + 9 q^{28} - 16 q^{29} + 9 q^{31} - 29 q^{32} + 2 q^{34} - 16 q^{35} + 14 q^{37} - 8 q^{38} + 16 q^{40} - 19 q^{41} + 7 q^{43} + 12 q^{44} + 4 q^{46} - 26 q^{47} + 8 q^{49} + 15 q^{50} - 17 q^{52} - 18 q^{53} - 7 q^{55} - 44 q^{56} - q^{58} - 31 q^{59} - 13 q^{61} + 5 q^{62} - 17 q^{64} - 31 q^{65} + 14 q^{67} + 32 q^{68} - 20 q^{70} - 37 q^{71} - 16 q^{73} + 6 q^{74} - 7 q^{76} + 5 q^{77} - 17 q^{79} + 2 q^{80} - 2 q^{82} - 30 q^{83} - 16 q^{85} + 22 q^{86} - 15 q^{88} - 35 q^{89} - q^{91} - 24 q^{92} - 11 q^{94} - 13 q^{95} - q^{97} + q^{98}+O(q^{100}) $ 13 * q - 4 * q^2 + 12 * q^4 - 7 * q^5 + 5 * q^7 - 15 * q^8 + 8 * q^10 + 13 * q^11 - 9 * q^13 - 19 * q^14 + 18 * q^16 - 7 * q^17 + 2 * q^19 - 15 * q^20 - 4 * q^22 - 23 * q^23 + 10 * q^25 - 8 * q^26 + 9 * q^28 - 16 * q^29 + 9 * q^31 - 29 * q^32 + 2 * q^34 - 16 * q^35 + 14 * q^37 - 8 * q^38 + 16 * q^40 - 19 * q^41 + 7 * q^43 + 12 * q^44 + 4 * q^46 - 26 * q^47 + 8 * q^49 + 15 * q^50 - 17 * q^52 - 18 * q^53 - 7 * q^55 - 44 * q^56 - q^58 - 31 * q^59 - 13 * q^61 + 5 * q^62 - 17 * q^64 - 31 * q^65 + 14 * q^67 + 32 * q^68 - 20 * q^70 - 37 * q^71 - 16 * q^73 + 6 * q^74 - 7 * q^76 + 5 * q^77 - 17 * q^79 + 2 * q^80 - 2 * q^82 - 30 * q^83 - 16 * q^85 + 22 * q^86 - 15 * q^88 - 35 * q^89 - q^91 - 24 * q^92 - 11 * q^94 - 13 * q^95 - q^97 + 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 $

$2$	0.534142	0.377695	0.188848	−	0.982006i	$-0.439525\pi$
$2$	0.534142	0.377695	0.188848	+	0.982006i	$0.439525\pi$
$3$	0	0
$3$	0	0
$4$	−1.71469	−0.857346
$5$	−3.62612	−1.62165	−0.810824	−	0.585290i	$-0.800981\pi$
$5$	−3.62612	−1.62165	−0.810824	+	0.585290i	$0.800981\pi$
$6$	0	0
$7$	4.31243	1.62995	0.814973	−	0.579498i	$-0.196751\pi$
$7$	4.31243	1.62995	0.814973	+	0.579498i	$0.196751\pi$
$8$	−1.98417	−0.701511
$9$	0	0
$10$	−1.93686	−0.612489
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	−1.21915	−0.338131	−0.169066	−	0.985605i	$-0.554075\pi$
$13$	−1.21915	−0.338131	−0.169066	+	0.985605i	$0.554075\pi$
$14$	2.30345	0.615623
$15$	0	0
$16$	2.36956	0.592389
$17$	−0.492960	−0.119560	−0.0597802	−	0.998212i	$-0.519040\pi$
$17$	−0.492960	−0.119560	−0.0597802	+	0.998212i	$0.519040\pi$
$18$	0	0
$19$	0.909227	0.208591	0.104296	−	0.994546i	$-0.466741\pi$
$19$	0.909227	0.208591	0.104296	+	0.994546i	$0.466741\pi$
$20$	6.21767	1.39031
$21$	0	0
$22$	0.534142	0.113879
$23$	−9.05847	−1.88882	−0.944411	−	0.328768i	$-0.893367\pi$
$23$	−9.05847	−1.88882	−0.944411	+	0.328768i	$0.893367\pi$
$24$	0	0
$25$	8.14872	1.62974
$26$	−0.651199	−0.127711
$27$	0	0
$28$	−7.39450	−1.39743
$29$	3.36455	0.624781	0.312391	−	0.949954i	$-0.398870\pi$
$29$	3.36455	0.624781	0.312391	+	0.949954i	$0.398870\pi$
$30$	0	0
$31$	−1.80170	−0.323595	−0.161797	−	0.986824i	$-0.551729\pi$
$31$	−1.80170	−0.323595	−0.161797	+	0.986824i	$0.551729\pi$
$32$	5.23402	0.925253
$33$	0	0
$34$	−0.263311	−0.0451574
$35$	−15.6374	−2.64320
$36$	0	0
$37$	−2.52082	−0.414420	−0.207210	−	0.978296i	$-0.566438\pi$
$37$	−2.52082	−0.414420	−0.207210	+	0.978296i	$0.566438\pi$
$38$	0.485656	0.0787839
$39$	0	0
$40$	7.19484	1.13760
$41$	3.50527	0.547432	0.273716	−	0.961811i	$-0.411747\pi$
$41$	3.50527	0.547432	0.273716	+	0.961811i	$0.411747\pi$
$42$	0	0
$43$	1.17586	0.179318	0.0896588	−	0.995973i	$-0.471422\pi$
$43$	1.17586	0.179318	0.0896588	+	0.995973i	$0.471422\pi$
$44$	−1.71469	−0.258500
$45$	0	0
$46$	−4.83851	−0.713399
$47$	6.47062	0.943837	0.471919	−	0.881642i	$-0.343562\pi$
$47$	6.47062	0.943837	0.471919	+	0.881642i	$0.343562\pi$
$48$	0	0
$49$	11.5971	1.65673
$50$	4.35257	0.615546
$51$	0	0
$52$	2.09047	0.289896
$53$	−7.40180	−1.01672	−0.508358	−	0.861146i	$-0.669747\pi$
$53$	−7.40180	−1.01672	−0.508358	+	0.861146i	$0.669747\pi$
$54$	0	0
$55$	−3.62612	−0.488945
$56$	−8.55661	−1.14343
$57$	0	0
$58$	1.79715	0.235977
$59$	12.4433	1.61997	0.809987	−	0.586448i	$-0.199474\pi$
$59$	12.4433	1.61997	0.809987	+	0.586448i	$0.199474\pi$
$60$	0	0
$61$	−1.00000	−0.128037
$61$	−1.00000	−0.128037
$62$	−0.962363	−0.122220
$63$	0	0
$64$	−1.94340	−0.242925
$65$	4.42078	0.548330
$66$	0	0
$67$	−1.39825	−0.170824	−0.0854118	−	0.996346i	$-0.527221\pi$
$67$	−1.39825	−0.170824	−0.0854118	+	0.996346i	$0.527221\pi$
$68$	0.845275	0.102505
$69$	0	0
$70$	−8.35258	−0.998324
$71$	12.1125	1.43749	0.718747	−	0.695272i	$-0.244716\pi$
$71$	12.1125	1.43749	0.718747	+	0.695272i	$0.244716\pi$
$72$	0	0
$73$	−9.92547	−1.16169	−0.580844	−	0.814015i	$-0.697277\pi$
$73$	−9.92547	−1.16169	−0.580844	+	0.814015i	$0.697277\pi$
$74$	−1.34647	−0.156524
$75$	0	0
$76$	−1.55905	−0.178835
$77$	4.31243	0.491447
$78$	0	0
$79$	4.62242	0.520063	0.260031	−	0.965600i	$-0.416267\pi$
$79$	4.62242	0.520063	0.260031	+	0.965600i	$0.416267\pi$
$80$	−8.59228	−0.960647
$81$	0	0
$82$	1.87231	0.206762
$83$	2.80042	0.307386	0.153693	−	0.988119i	$-0.450883\pi$
$83$	2.80042	0.307386	0.153693	+	0.988119i	$0.450883\pi$
$84$	0	0
$85$	1.78753	0.193885
$86$	0.628078	0.0677274
$87$	0	0
$88$	−1.98417	−0.211513
$89$	−12.0828	−1.28077	−0.640387	−	0.768053i	$-0.721226\pi$
$89$	−12.0828	−1.28077	−0.640387	+	0.768053i	$0.721226\pi$
$90$	0	0
$91$	−5.25750	−0.551136
$92$	15.5325	1.61937
$93$	0	0
$94$	3.45623	0.356483
$95$	−3.29696	−0.338261
$96$	0	0
$97$	2.52978	0.256861	0.128430	−	0.991719i	$-0.459006\pi$
$97$	2.52978	0.256861	0.128430	+	0.991719i	$0.459006\pi$
$98$	6.19449	0.625738
$99$	0	0
$100$	−13.9725	−1.39725
$101$	3.20857	0.319264	0.159632	−	0.987177i	$-0.448969\pi$
$101$	3.20857	0.319264	0.159632	+	0.987177i	$0.448969\pi$
$102$	0	0
$103$	−11.5401	−1.13708	−0.568542	−	0.822654i	$-0.692492\pi$
$103$	−11.5401	−1.13708	−0.568542	+	0.822654i	$0.692492\pi$
$104$	2.41900	0.237203
$105$	0	0
$106$	−3.95361	−0.384009
$107$	10.8292	1.04690	0.523450	−	0.852056i	$-0.324645\pi$
$107$	10.8292	1.04690	0.523450	+	0.852056i	$0.324645\pi$
$108$	0	0
$109$	6.23060	0.596783	0.298391	−	0.954444i	$-0.403550\pi$
$109$	6.23060	0.596783	0.298391	+	0.954444i	$0.403550\pi$
$110$	−1.93686	−0.184672
$111$	0	0
$112$	10.2186	0.965562
$113$	−2.26940	−0.213487	−0.106744	−	0.994287i	$-0.534042\pi$
$113$	−2.26940	−0.213487	−0.106744	+	0.994287i	$0.534042\pi$
$114$	0	0
$115$	32.8471	3.06300
$116$	−5.76917	−0.535654
$117$	0	0
$118$	6.64646	0.611856
$119$	−2.12586	−0.194877
$120$	0	0
$121$	1.00000	0.0909091
$122$	−0.534142	−0.0483589
$123$	0	0
$124$	3.08936	0.277433
$125$	−11.4176	−1.02122
$126$	0	0
$127$	3.48898	0.309597	0.154798	−	0.987946i	$-0.450527\pi$
$127$	3.48898	0.309597	0.154798	+	0.987946i	$0.450527\pi$
$128$	−11.5061	−1.01701
$129$	0	0
$130$	2.36132	0.207102
$131$	−21.6610	−1.89253	−0.946266	−	0.323390i	$-0.895177\pi$
$131$	−21.6610	−1.89253	−0.946266	+	0.323390i	$0.895177\pi$
$132$	0	0
$133$	3.92098	0.339992
$134$	−0.746864	−0.0645192
$135$	0	0
$136$	0.978118	0.0838730
$137$	−21.3976	−1.82812	−0.914062	−	0.405575i	$-0.867071\pi$
$137$	−21.3976	−1.82812	−0.914062	+	0.405575i	$0.867071\pi$
$138$	0	0
$139$	−6.56067	−0.556469	−0.278235	−	0.960513i	$-0.589749\pi$
$139$	−6.56067	−0.556469	−0.278235	+	0.960513i	$0.589749\pi$
$140$	26.8133	2.26614
$141$	0	0
$142$	6.46981	0.542934
$143$	−1.21915	−0.101950
$144$	0	0
$145$	−12.2003	−1.01318
$146$	−5.30161	−0.438764
$147$	0	0
$148$	4.32243	0.355301
$149$	−15.9721	−1.30849	−0.654244	−	0.756283i	$-0.727013\pi$
$149$	−15.9721	−1.30849	−0.654244	+	0.756283i	$0.727013\pi$
$150$	0	0
$151$	−17.1351	−1.39444	−0.697218	−	0.716860i	$-0.745579\pi$
$151$	−17.1351	−1.39444	−0.697218	+	0.716860i	$0.745579\pi$
$152$	−1.80406	−0.146329
$153$	0	0
$154$	2.30345	0.185617
$155$	6.53317	0.524757
$156$	0	0
$157$	1.67936	0.134027	0.0670137	−	0.997752i	$-0.478653\pi$
$157$	1.67936	0.134027	0.0670137	+	0.997752i	$0.478653\pi$
$158$	2.46903	0.196425
$159$	0	0
$160$	−18.9792	−1.50044
$161$	−39.0640	−3.07868
$162$	0	0
$163$	5.03166	0.394110	0.197055	−	0.980392i	$-0.436862\pi$
$163$	5.03166	0.394110	0.197055	+	0.980392i	$0.436862\pi$
$164$	−6.01047	−0.469339
$165$	0	0
$166$	1.49582	0.116098
$167$	−16.5787	−1.28290	−0.641450	−	0.767165i	$-0.721667\pi$
$167$	−16.5787	−1.28290	−0.641450	+	0.767165i	$0.721667\pi$
$168$	0	0
$169$	−11.5137	−0.885667
$170$	0.954795	0.0732294
$171$	0	0
$172$	−2.01625	−0.153737
$173$	10.7498	0.817289	0.408644	−	0.912694i	$-0.366002\pi$
$173$	10.7498	0.817289	0.408644	+	0.912694i	$0.366002\pi$
$174$	0	0
$175$	35.1408	2.65640
$176$	2.36956	0.178612
$177$	0	0
$178$	−6.45392	−0.483742
$179$	−15.8180	−1.18229	−0.591145	−	0.806565i	$-0.701324\pi$
$179$	−15.8180	−1.18229	−0.591145	+	0.806565i	$0.701324\pi$
$180$	0	0
$181$	−15.4710	−1.14995	−0.574975	−	0.818171i	$-0.694988\pi$
$181$	−15.4710	−1.14995	−0.574975	+	0.818171i	$0.694988\pi$
$182$	−2.80825	−0.208162
$183$	0	0
$184$	17.9736	1.32503
$185$	9.14078	0.672043
$186$	0	0
$187$	−0.492960	−0.0360488
$188$	−11.0951	−0.809195
$189$	0	0
$190$	−1.76105	−0.127760
$191$	−18.2591	−1.32119	−0.660593	−	0.750745i	$-0.729695\pi$
$191$	−18.2591	−1.32119	−0.660593	+	0.750745i	$0.729695\pi$
$192$	0	0
$193$	−14.9462	−1.07585	−0.537926	−	0.842992i	$-0.680792\pi$
$193$	−14.9462	−1.07585	−0.537926	+	0.842992i	$0.680792\pi$
$194$	1.35126	0.0970150
$195$	0	0
$196$	−19.8854	−1.42039
$197$	5.53663	0.394469	0.197234	−	0.980356i	$-0.436804\pi$
$197$	5.53663	0.394469	0.197234	+	0.980356i	$0.436804\pi$
$198$	0	0
$199$	1.42160	0.100774	0.0503872	−	0.998730i	$-0.483954\pi$
$199$	1.42160	0.100774	0.0503872	+	0.998730i	$0.483954\pi$
$200$	−16.1685	−1.14328
$201$	0	0
$202$	1.71383	0.120585
$203$	14.5094	1.01836
$204$	0	0
$205$	−12.7105	−0.887742
$206$	−6.16407	−0.429471
$207$	0	0
$208$	−2.88884	−0.200305
$209$	0.909227	0.0628926
$210$	0	0
$211$	−18.4855	−1.27259	−0.636296	−	0.771445i	$-0.719534\pi$
$211$	−18.4855	−1.27259	−0.636296	+	0.771445i	$0.719534\pi$
$212$	12.6918	0.871677
$213$	0	0
$214$	5.78434	0.395409
$215$	−4.26382	−0.290790
$216$	0	0
$217$	−7.76971	−0.527442
$218$	3.32802	0.225402
$219$	0	0
$220$	6.21767	0.419196
$221$	0.600993	0.0404271
$222$	0	0
$223$	11.2160	0.751077	0.375538	−	0.926807i	$-0.377458\pi$
$223$	11.2160	0.751077	0.375538	+	0.926807i	$0.377458\pi$
$224$	22.5714	1.50811
$225$	0	0
$226$	−1.21218	−0.0806331
$227$	−22.4464	−1.48982	−0.744911	−	0.667163i	$-0.767508\pi$
$227$	−22.4464	−1.48982	−0.744911	+	0.667163i	$0.767508\pi$
$228$	0	0
$229$	21.4818	1.41955	0.709777	−	0.704426i	$-0.248795\pi$
$229$	21.4818	1.41955	0.709777	+	0.704426i	$0.248795\pi$
$230$	17.5450	1.15688
$231$	0	0
$232$	−6.67585	−0.438291
$233$	4.14079	0.271272	0.135636	−	0.990759i	$-0.456692\pi$
$233$	4.14079	0.271272	0.135636	+	0.990759i	$0.456692\pi$
$234$	0	0
$235$	−23.4632	−1.53057
$236$	−21.3363	−1.38888
$237$	0	0
$238$	−1.13551	−0.0736042
$239$	10.5358	0.681507	0.340753	−	0.940153i	$-0.389318\pi$
$239$	10.5358	0.681507	0.340753	+	0.940153i	$0.389318\pi$
$240$	0	0
$241$	−11.8063	−0.760513	−0.380257	−	0.924881i	$-0.624164\pi$
$241$	−11.8063	−0.760513	−0.380257	+	0.924881i	$0.624164\pi$
$242$	0.534142	0.0343359
$243$	0	0
$244$	1.71469	0.109772
$245$	−42.0524	−2.68663
$246$	0	0
$247$	−1.10848	−0.0705312
$248$	3.57488	0.227005
$249$	0	0
$250$	−6.09862	−0.385711
$251$	−13.7725	−0.869314	−0.434657	−	0.900596i	$-0.643130\pi$
$251$	−13.7725	−0.869314	−0.434657	+	0.900596i	$0.643130\pi$
$252$	0	0
$253$	−9.05847	−0.569501
$254$	1.86361	0.116933
$255$	0	0
$256$	−2.25909	−0.141193
$257$	29.9006	1.86515	0.932574	−	0.360980i	$-0.117558\pi$
$257$	29.9006	1.86515	0.932574	+	0.360980i	$0.117558\pi$
$258$	0	0
$259$	−10.8709	−0.675482
$260$	−7.58028	−0.470109
$261$	0	0
$262$	−11.5701	−0.714800
$263$	6.38562	0.393754	0.196877	−	0.980428i	$-0.436920\pi$
$263$	6.38562	0.393754	0.196877	+	0.980428i	$0.436920\pi$
$264$	0	0
$265$	26.8398	1.64875
$266$	2.09436	0.128413
$267$	0	0
$268$	2.39757	0.146455
$269$	−26.0111	−1.58592	−0.792962	−	0.609272i	$-0.791462\pi$
$269$	−26.0111	−1.58592	−0.792962	+	0.609272i	$0.791462\pi$
$270$	0	0
$271$	−0.817061	−0.0496329	−0.0248165	−	0.999692i	$-0.507900\pi$
$271$	−0.817061	−0.0496329	−0.0248165	+	0.999692i	$0.507900\pi$
$272$	−1.16810	−0.0708263
$273$	0	0
$274$	−11.4294	−0.690474
$275$	8.14872	0.491386
$276$	0	0
$277$	4.93273	0.296379	0.148190	−	0.988959i	$-0.452655\pi$
$277$	4.93273	0.296379	0.148190	+	0.988959i	$0.452655\pi$
$278$	−3.50433	−0.210176
$279$	0	0
$280$	31.0273	1.85423
$281$	21.8269	1.30209	0.651043	−	0.759041i	$-0.274332\pi$
$281$	21.8269	1.30209	0.651043	+	0.759041i	$0.274332\pi$
$282$	0	0
$283$	−5.17215	−0.307452	−0.153726	−	0.988113i	$-0.549127\pi$
$283$	−5.17215	−0.307452	−0.153726	+	0.988113i	$0.549127\pi$
$284$	−20.7693	−1.23243
$285$	0	0
$286$	−0.651199	−0.0385062
$287$	15.1163	0.892285
$288$	0	0
$289$	−16.7570	−0.985705
$290$	−6.51666	−0.382672
$291$	0	0
$292$	17.0191	0.995969
$293$	9.66881	0.564858	0.282429	−	0.959288i	$-0.408860\pi$
$293$	9.66881	0.564858	0.282429	+	0.959288i	$0.408860\pi$
$294$	0	0
$295$	−45.1207	−2.62703
$296$	5.00174	0.290720
$297$	0	0
$298$	−8.53138	−0.494210
$299$	11.0436	0.638670
$300$	0	0
$301$	5.07084	0.292278
$302$	−9.15258	−0.526672
$303$	0	0
$304$	2.15447	0.123567
$305$	3.62612	0.207631
$306$	0	0
$307$	19.6167	1.11958	0.559791	−	0.828634i	$-0.310881\pi$
$307$	19.6167	1.11958	0.559791	+	0.828634i	$0.310881\pi$
$308$	−7.39450	−0.421341
$309$	0	0
$310$	3.48964	0.198198
$311$	−14.3311	−0.812641	−0.406321	−	0.913731i	$-0.633188\pi$
$311$	−14.3311	−0.812641	−0.406321	+	0.913731i	$0.633188\pi$
$312$	0	0
$313$	−32.1674	−1.81821	−0.909103	−	0.416570i	$-0.863232\pi$
$313$	−32.1674	−1.81821	−0.909103	+	0.416570i	$0.863232\pi$
$314$	0.897016	0.0506215
$315$	0	0
$316$	−7.92603	−0.445874
$317$	25.8443	1.45156	0.725781	−	0.687926i	$-0.241479\pi$
$317$	25.8443	1.45156	0.725781	+	0.687926i	$0.241479\pi$
$318$	0	0
$319$	3.36455	0.188379
$320$	7.04700	0.393939
$321$	0	0
$322$	−20.8657	−1.16280
$323$	−0.448213	−0.0249392
$324$	0	0
$325$	−9.93451	−0.551067
$326$	2.68762	0.148854
$327$	0	0
$328$	−6.95507	−0.384029
$329$	27.9041	1.53840
$330$	0	0
$331$	6.95974	0.382542	0.191271	−	0.981537i	$-0.438739\pi$
$331$	6.95974	0.382542	0.191271	+	0.981537i	$0.438739\pi$
$332$	−4.80186	−0.263536
$333$	0	0
$334$	−8.85538	−0.484545
$335$	5.07022	0.277016
$336$	0	0
$337$	−27.1628	−1.47965	−0.739825	−	0.672799i	$-0.765092\pi$
$337$	−27.1628	−1.47965	−0.739825	+	0.672799i	$0.765092\pi$
$338$	−6.14993	−0.334512
$339$	0	0
$340$	−3.06507	−0.166227
$341$	−1.80170	−0.0975675
$342$	0	0
$343$	19.8246	1.07043
$344$	−2.33312	−0.125793
$345$	0	0
$346$	5.74189	0.308686
$347$	24.5634	1.31863	0.659315	−	0.751867i	$-0.270846\pi$
$347$	24.5634	1.31863	0.659315	+	0.751867i	$0.270846\pi$
$348$	0	0
$349$	19.9223	1.06642	0.533209	−	0.845984i	$-0.320986\pi$
$349$	19.9223	1.06642	0.533209	+	0.845984i	$0.320986\pi$
$350$	18.7702	1.00331
$351$	0	0
$352$	5.23402	0.278974
$353$	−1.49470	−0.0795546	−0.0397773	−	0.999209i	$-0.512665\pi$
$353$	−1.49470	−0.0795546	−0.0397773	+	0.999209i	$0.512665\pi$
$354$	0	0
$355$	−43.9215	−2.33111
$356$	20.7183	1.09807
$357$	0	0
$358$	−8.44903	−0.446545
$359$	−13.4291	−0.708762	−0.354381	−	0.935101i	$-0.615308\pi$
$359$	−13.4291	−0.708762	−0.354381	+	0.935101i	$0.615308\pi$
$360$	0	0
$361$	−18.1733	−0.956490
$362$	−8.26371	−0.434331
$363$	0	0
$364$	9.01500	0.472515
$365$	35.9909	1.88385
$366$	0	0
$367$	−17.3863	−0.907557	−0.453778	−	0.891115i	$-0.649924\pi$
$367$	−17.3863	−0.907557	−0.453778	+	0.891115i	$0.649924\pi$
$368$	−21.4645	−1.11892
$369$	0	0
$370$	4.88247	0.253828
$371$	−31.9198	−1.65719
$372$	0	0
$373$	−10.2747	−0.532002	−0.266001	−	0.963973i	$-0.585702\pi$
$373$	−10.2747	−0.532002	−0.266001	+	0.963973i	$0.585702\pi$
$374$	−0.263311	−0.0136155
$375$	0	0
$376$	−12.8388	−0.662112
$377$	−4.10189	−0.211258
$378$	0	0
$379$	−14.6983	−0.755003	−0.377501	−	0.926009i	$-0.623217\pi$
$379$	−14.6983	−0.755003	−0.377501	+	0.926009i	$0.623217\pi$
$380$	5.65328	0.290007
$381$	0	0
$382$	−9.75297	−0.499005
$383$	−28.1057	−1.43613	−0.718067	−	0.695973i	$-0.754973\pi$
$383$	−28.1057	−1.43613	−0.718067	+	0.695973i	$0.754973\pi$
$384$	0	0
$385$	−15.6374	−0.796955
$386$	−7.98340	−0.406344
$387$	0	0
$388$	−4.33780	−0.220218
$389$	−26.0871	−1.32267	−0.661334	−	0.750091i	$-0.730009\pi$
$389$	−26.0871	−1.32267	−0.661334	+	0.750091i	$0.730009\pi$
$390$	0	0
$391$	4.46547	0.225828
$392$	−23.0106	−1.16221
$393$	0	0
$394$	2.95735	0.148989
$395$	−16.7614	−0.843359
$396$	0	0
$397$	28.9690	1.45391	0.726957	−	0.686683i	$-0.240934\pi$
$397$	28.9690	1.45391	0.726957	+	0.686683i	$0.240934\pi$
$398$	0.759335	0.0380620
$399$	0	0
$400$	19.3088	0.965442
$401$	−15.6855	−0.783297	−0.391648	−	0.920115i	$-0.628095\pi$
$401$	−15.6855	−0.783297	−0.391648	+	0.920115i	$0.628095\pi$
$402$	0	0
$403$	2.19654	0.109418
$404$	−5.50171	−0.273720
$405$	0	0
$406$	7.75008	0.384630
$407$	−2.52082	−0.124952
$408$	0	0
$409$	−27.7465	−1.37197	−0.685987	−	0.727614i	$-0.740629\pi$
$409$	−27.7465	−1.37197	−0.685987	+	0.727614i	$0.740629\pi$
$410$	−6.78922	−0.335296
$411$	0	0
$412$	19.7878	0.974875
$413$	53.6607	2.64047
$414$	0	0
$415$	−10.1546	−0.498472
$416$	−6.38106	−0.312857
$417$	0	0
$418$	0.485656	0.0237542
$419$	−29.3305	−1.43289	−0.716445	−	0.697643i	$-0.754232\pi$
$419$	−29.3305	−1.43289	−0.716445	+	0.697643i	$0.754232\pi$
$420$	0	0
$421$	20.6444	1.00615	0.503073	−	0.864244i	$-0.332203\pi$
$421$	20.6444	1.00615	0.503073	+	0.864244i	$0.332203\pi$
$422$	−9.87387	−0.480652
$423$	0	0
$424$	14.6864	0.713237
$425$	−4.01699	−0.194853
$426$	0	0
$427$	−4.31243	−0.208693
$428$	−18.5688	−0.897556
$429$	0	0
$430$	−2.27748	−0.109830
$431$	29.1418	1.40371	0.701857	−	0.712318i	$-0.252355\pi$
$431$	29.1418	1.40371	0.701857	+	0.712318i	$0.252355\pi$
$432$	0	0
$433$	18.2972	0.879307	0.439654	−	0.898167i	$-0.355101\pi$
$433$	18.2972	0.879307	0.439654	+	0.898167i	$0.355101\pi$
$434$	−4.15013	−0.199213
$435$	0	0
$436$	−10.6836	−0.511650
$437$	−8.23621	−0.393991
$438$	0	0
$439$	21.3407	1.01854	0.509268	−	0.860608i	$-0.329916\pi$
$439$	21.3407	1.01854	0.509268	+	0.860608i	$0.329916\pi$
$440$	7.19484	0.343001
$441$	0	0
$442$	0.321015	0.0152691
$443$	−20.8404	−0.990159	−0.495080	−	0.868848i	$-0.664861\pi$
$443$	−20.8404	−0.990159	−0.495080	+	0.868848i	$0.664861\pi$
$444$	0	0
$445$	43.8136	2.07696
$446$	5.99092	0.283678
$447$	0	0
$448$	−8.38079	−0.395955
$449$	13.0657	0.616607	0.308304	−	0.951288i	$-0.400239\pi$
$449$	13.0657	0.616607	0.308304	+	0.951288i	$0.400239\pi$
$450$	0	0
$451$	3.50527	0.165057
$452$	3.89132	0.183032
$453$	0	0
$454$	−11.9896	−0.562699
$455$	19.0643	0.893749
$456$	0	0
$457$	−31.7457	−1.48500	−0.742500	−	0.669846i	$-0.766360\pi$
$457$	−31.7457	−1.48500	−0.742500	+	0.669846i	$0.766360\pi$
$458$	11.4743	0.536159
$459$	0	0
$460$	−56.3226	−2.62606
$461$	21.5585	1.00408	0.502041	−	0.864844i	$-0.332583\pi$
$461$	21.5585	1.00408	0.502041	+	0.864844i	$0.332583\pi$
$462$	0	0
$463$	−20.3433	−0.945434	−0.472717	−	0.881214i	$-0.656727\pi$
$463$	−20.3433	−0.945434	−0.472717	+	0.881214i	$0.656727\pi$
$464$	7.97249	0.370114
$465$	0	0
$466$	2.21177	0.102458
$467$	1.24480	0.0576027	0.0288013	−	0.999585i	$-0.490831\pi$
$467$	1.24480	0.0576027	0.0288013	+	0.999585i	$0.490831\pi$
$468$	0	0
$469$	−6.02986	−0.278433
$470$	−12.5327	−0.578090
$471$	0	0
$472$	−24.6896	−1.13643
$473$	1.17586	0.0540663
$474$	0	0
$475$	7.40904	0.339950
$476$	3.64519	0.167077
$477$	0	0
$478$	5.62763	0.257402
$479$	−14.9723	−0.684102	−0.342051	−	0.939681i	$-0.611122\pi$
$479$	−14.9723	−0.684102	−0.342051	+	0.939681i	$0.611122\pi$
$480$	0	0
$481$	3.07326	0.140128
$482$	−6.30626	−0.287242
$483$	0	0
$484$	−1.71469	−0.0779406
$485$	−9.17329	−0.416538
$486$	0	0
$487$	−19.3857	−0.878450	−0.439225	−	0.898377i	$-0.644747\pi$
$487$	−19.3857	−0.878450	−0.439225	+	0.898377i	$0.644747\pi$
$488$	1.98417	0.0898193
$489$	0	0
$490$	−22.4619	−1.01473
$491$	12.6872	0.572565	0.286282	−	0.958145i	$-0.407580\pi$
$491$	12.6872	0.572565	0.286282	+	0.958145i	$0.407580\pi$
$492$	0	0
$493$	−1.65859	−0.0746991
$494$	−0.592088	−0.0266393
$495$	0	0
$496$	−4.26923	−0.191694
$497$	52.2345	2.34304
$498$	0	0
$499$	4.49897	0.201402	0.100701	−	0.994917i	$-0.467892\pi$
$499$	4.49897	0.201402	0.100701	+	0.994917i	$0.467892\pi$
$500$	19.5777	0.875541
$501$	0	0
$502$	−7.35648	−0.328336
$503$	35.1228	1.56605	0.783024	−	0.621991i	$-0.213676\pi$
$503$	35.1228	1.56605	0.783024	+	0.621991i	$0.213676\pi$
$504$	0	0
$505$	−11.6346	−0.517734
$506$	−4.83851	−0.215098
$507$	0	0
$508$	−5.98253	−0.265432
$509$	0.331366	0.0146875	0.00734377	−	0.999973i	$-0.497662\pi$
$509$	0.331366	0.0146875	0.00734377	+	0.999973i	$0.497662\pi$
$510$	0	0
$511$	−42.8029	−1.89349
$512$	21.8055	0.963677
$513$	0	0
$514$	15.9712	0.704457
$515$	41.8459	1.84395
$516$	0	0
$517$	6.47062	0.284578
$518$	−5.80658	−0.255127
$519$	0	0
$520$	−8.77159	−0.384660
$521$	−11.5442	−0.505762	−0.252881	−	0.967497i	$-0.581378\pi$
$521$	−11.5442	−0.505762	−0.252881	+	0.967497i	$0.581378\pi$
$522$	0	0
$523$	−16.4599	−0.719741	−0.359871	−	0.933002i	$-0.617179\pi$
$523$	−16.4599	−0.719741	−0.359871	+	0.933002i	$0.617179\pi$
$524$	37.1420	1.62255
$525$	0	0
$526$	3.41083	0.148719
$527$	0.888167	0.0386892
$528$	0	0
$529$	59.0559	2.56765
$530$	14.3362	0.622727
$531$	0	0
$532$	−6.72328	−0.291491
$533$	−4.27345	−0.185104
$534$	0	0
$535$	−39.2680	−1.69770
$536$	2.77437	0.119835
$537$	0	0
$538$	−13.8936	−0.598996
$539$	11.5971	0.499522
$540$	0	0
$541$	−19.3268	−0.830924	−0.415462	−	0.909611i	$-0.636380\pi$
$541$	−19.3268	−0.830924	−0.415462	+	0.909611i	$0.636380\pi$
$542$	−0.436426	−0.0187461
$543$	0	0
$544$	−2.58017	−0.110624
$545$	−22.5929	−0.967772
$546$	0	0
$547$	9.18213	0.392599	0.196300	−	0.980544i	$-0.437107\pi$
$547$	9.18213	0.392599	0.196300	+	0.980544i	$0.437107\pi$
$548$	36.6904	1.56734
$549$	0	0
$550$	4.35257	0.185594
$551$	3.05914	0.130324
$552$	0	0
$553$	19.9339	0.847675
$554$	2.63478	0.111941
$555$	0	0
$556$	11.2495	0.477087
$557$	27.0866	1.14770	0.573848	−	0.818962i	$-0.305450\pi$
$557$	27.0866	1.14770	0.573848	+	0.818962i	$0.305450\pi$
$558$	0	0
$559$	−1.43356	−0.0606329
$560$	−37.0537	−1.56580
$561$	0	0
$562$	11.6587	0.491792
$563$	15.3958	0.648854	0.324427	−	0.945911i	$-0.394829\pi$
$563$	15.3958	0.648854	0.324427	+	0.945911i	$0.394829\pi$
$564$	0	0
$565$	8.22910	0.346201
$566$	−2.76266	−0.116123
$567$	0	0
$568$	−24.0334	−1.00842
$569$	−8.58984	−0.360105	−0.180052	−	0.983657i	$-0.557627\pi$
$569$	−8.58984	−0.360105	−0.180052	+	0.983657i	$0.557627\pi$
$570$	0	0
$571$	30.5350	1.27785	0.638924	−	0.769270i	$-0.279380\pi$
$571$	30.5350	1.27785	0.638924	+	0.769270i	$0.279380\pi$
$572$	2.09047	0.0874068
$573$	0	0
$574$	8.07423	0.337012
$575$	−73.8149	−3.07829
$576$	0	0
$577$	−30.0446	−1.25077	−0.625387	−	0.780315i	$-0.715059\pi$
$577$	−30.0446	−1.25077	−0.625387	+	0.780315i	$0.715059\pi$
$578$	−8.95061	−0.372296
$579$	0	0
$580$	20.9197	0.868642
$581$	12.0766	0.501023
$582$	0	0
$583$	−7.40180	−0.306551
$584$	19.6938	0.814937
$585$	0	0
$586$	5.16452	0.213344
$587$	−29.0280	−1.19811	−0.599057	−	0.800706i	$-0.704458\pi$
$587$	−29.0280	−1.19811	−0.599057	+	0.800706i	$0.704458\pi$
$588$	0	0
$589$	−1.63816	−0.0674990
$590$	−24.1008	−0.992216
$591$	0	0
$592$	−5.97322	−0.245498
$593$	40.3838	1.65836	0.829182	−	0.558979i	$-0.188807\pi$
$593$	40.3838	1.65836	0.829182	+	0.558979i	$0.188807\pi$
$594$	0	0
$595$	7.70861	0.316022
$596$	27.3873	1.12183
$597$	0	0
$598$	5.89887	0.241223
$599$	−34.3217	−1.40235	−0.701173	−	0.712992i	$-0.747340\pi$
$599$	−34.3217	−1.40235	−0.701173	+	0.712992i	$0.747340\pi$
$600$	0	0
$601$	−27.5446	−1.12357	−0.561785	−	0.827284i	$-0.689885\pi$
$601$	−27.5446	−1.12357	−0.561785	+	0.827284i	$0.689885\pi$
$602$	2.70855	0.110392
$603$	0	0
$604$	29.3814	1.19551
$605$	−3.62612	−0.147423
$606$	0	0
$607$	−7.48437	−0.303781	−0.151891	−	0.988397i	$-0.548536\pi$
$607$	−7.48437	−0.303781	−0.151891	+	0.988397i	$0.548536\pi$
$608$	4.75892	0.193000
$609$	0	0
$610$	1.93686	0.0784212
$611$	−7.88866	−0.319141
$612$	0	0
$613$	42.2994	1.70846	0.854229	−	0.519897i	$-0.174030\pi$
$613$	42.2994	1.70846	0.854229	+	0.519897i	$0.174030\pi$
$614$	10.4781	0.422861
$615$	0	0
$616$	−8.55661	−0.344756
$617$	8.67487	0.349237	0.174619	−	0.984636i	$-0.444131\pi$
$617$	8.67487	0.349237	0.174619	+	0.984636i	$0.444131\pi$
$618$	0	0
$619$	18.7219	0.752496	0.376248	−	0.926519i	$-0.377214\pi$
$619$	18.7219	0.752496	0.376248	+	0.926519i	$0.377214\pi$
$620$	−11.2024	−0.449899
$621$	0	0
$622$	−7.65483	−0.306931
$623$	−52.1063	−2.08759
$624$	0	0
$625$	0.658006	0.0263202
$626$	−17.1819	−0.686728
$627$	0	0
$628$	−2.87958	−0.114908
$629$	1.24266	0.0495482
$630$	0	0
$631$	−2.06390	−0.0821626	−0.0410813	−	0.999156i	$-0.513080\pi$
$631$	−2.06390	−0.0821626	−0.0410813	+	0.999156i	$0.513080\pi$
$632$	−9.17168	−0.364830
$633$	0	0
$634$	13.8045	0.548248
$635$	−12.6514	−0.502057
$636$	0	0
$637$	−14.1386	−0.560191
$638$	1.79715	0.0711497
$639$	0	0
$640$	41.7224	1.64922
$641$	−1.83897	−0.0726350	−0.0363175	−	0.999340i	$-0.511563\pi$
$641$	−1.83897	−0.0726350	−0.0363175	+	0.999340i	$0.511563\pi$
$642$	0	0
$643$	−20.5058	−0.808670	−0.404335	−	0.914611i	$-0.632497\pi$
$643$	−20.5058	−0.808670	−0.404335	+	0.914611i	$0.632497\pi$
$644$	66.9828	2.63949
$645$	0	0
$646$	−0.239409	−0.00941943
$647$	5.70352	0.224228	0.112114	−	0.993695i	$-0.464238\pi$
$647$	5.70352	0.224228	0.112114	+	0.993695i	$0.464238\pi$
$648$	0	0
$649$	12.4433	0.488440
$650$	−5.30644	−0.208136
$651$	0	0
$652$	−8.62775	−0.337889
$653$	45.7322	1.78964	0.894819	−	0.446429i	$-0.147304\pi$
$653$	45.7322	1.78964	0.894819	+	0.446429i	$0.147304\pi$
$654$	0	0
$655$	78.5453	3.06902
$656$	8.30594	0.324293
$657$	0	0
$658$	14.9048	0.581048
$659$	2.76046	0.107532	0.0537661	−	0.998554i	$-0.482877\pi$
$659$	2.76046	0.107532	0.0537661	+	0.998554i	$0.482877\pi$
$660$	0	0
$661$	−39.3593	−1.53090	−0.765449	−	0.643497i	$-0.777483\pi$
$661$	−39.3593	−1.53090	−0.765449	+	0.643497i	$0.777483\pi$
$662$	3.71749	0.144484
$663$	0	0
$664$	−5.55651	−0.215635
$665$	−14.2179	−0.551348
$666$	0	0
$667$	−30.4777	−1.18010
$668$	28.4274	1.09989
$669$	0	0
$670$	2.70822	0.104628
$671$	−1.00000	−0.0386046
$672$	0	0
$673$	37.2480	1.43580	0.717902	−	0.696144i	$-0.245103\pi$
$673$	37.2480	1.43580	0.717902	+	0.696144i	$0.245103\pi$
$674$	−14.5088	−0.558857
$675$	0	0
$676$	19.7424	0.759323
$677$	2.89189	0.111144	0.0555721	−	0.998455i	$-0.482302\pi$
$677$	2.89189	0.111144	0.0555721	+	0.998455i	$0.482302\pi$
$678$	0	0
$679$	10.9095	0.418669
$680$	−3.54677	−0.136012
$681$	0	0
$682$	−0.962363	−0.0368508
$683$	−51.1425	−1.95691	−0.978457	−	0.206449i	$-0.933809\pi$
$683$	−51.1425	−1.95691	−0.978457	+	0.206449i	$0.933809\pi$
$684$	0	0
$685$	77.5903	2.96457
$686$	10.5892	0.404296
$687$	0	0
$688$	2.78628	0.106226
$689$	9.02390	0.343783
$690$	0	0
$691$	−5.49790	−0.209150	−0.104575	−	0.994517i	$-0.533348\pi$
$691$	−5.49790	−0.209150	−0.104575	+	0.994517i	$0.533348\pi$
$692$	−18.4325	−0.700699
$693$	0	0
$694$	13.1203	0.498041
$695$	23.7898	0.902397
$696$	0	0
$697$	−1.72796	−0.0654512
$698$	10.6413	0.402781
$699$	0	0
$700$	−60.2557	−2.27745
$701$	−2.46512	−0.0931063	−0.0465531	−	0.998916i	$-0.514824\pi$
$701$	−2.46512	−0.0931063	−0.0465531	+	0.998916i	$0.514824\pi$
$702$	0	0
$703$	−2.29200	−0.0864443
$704$	−1.94340	−0.0732447
$705$	0	0
$706$	−0.798379	−0.0300474
$707$	13.8367	0.520384
$708$	0	0
$709$	−9.77187	−0.366990	−0.183495	−	0.983021i	$-0.558741\pi$
$709$	−9.77187	−0.366990	−0.183495	+	0.983021i	$0.558741\pi$
$710$	−23.4603	−0.880449
$711$	0	0
$712$	23.9743	0.898477
$713$	16.3206	0.611213
$714$	0	0
$715$	4.42078	0.165328
$716$	27.1229	1.01363
$717$	0	0
$718$	−7.17305	−0.267696
$719$	5.87214	0.218994	0.109497	−	0.993987i	$-0.465076\pi$
$719$	5.87214	0.218994	0.109497	+	0.993987i	$0.465076\pi$
$720$	0	0
$721$	−49.7661	−1.85339
$722$	−9.70712	−0.361262
$723$	0	0
$724$	26.5280	0.985906
$725$	27.4168	1.01823
$726$	0	0
$727$	−24.9562	−0.925576	−0.462788	−	0.886469i	$-0.653151\pi$
$727$	−24.9562	−0.925576	−0.462788	+	0.886469i	$0.653151\pi$
$728$	10.4318	0.386628
$729$	0	0
$730$	19.2242	0.711521
$731$	−0.579655	−0.0214393
$732$	0	0
$733$	−3.63477	−0.134253	−0.0671266	−	0.997744i	$-0.521383\pi$
$733$	−3.63477	−0.134253	−0.0671266	+	0.997744i	$0.521383\pi$
$734$	−9.28674	−0.342780
$735$	0	0
$736$	−47.4122	−1.74764
$737$	−1.39825	−0.0515052
$738$	0	0
$739$	−54.0129	−1.98690	−0.993448	−	0.114282i	$-0.963543\pi$
$739$	−54.0129	−1.98690	−0.993448	+	0.114282i	$0.963543\pi$
$740$	−15.6736	−0.576174
$741$	0	0
$742$	−17.0497	−0.625913
$743$	20.2219	0.741870	0.370935	−	0.928659i	$-0.379037\pi$
$743$	20.2219	0.741870	0.370935	+	0.928659i	$0.379037\pi$
$744$	0	0
$745$	57.9168	2.12191
$746$	−5.48813	−0.200935
$747$	0	0
$748$	0.845275	0.0309063
$749$	46.7003	1.70639
$750$	0	0
$751$	−6.50255	−0.237281	−0.118641	−	0.992937i	$-0.537854\pi$
$751$	−6.50255	−0.237281	−0.118641	+	0.992937i	$0.537854\pi$
$752$	15.3325	0.559119
$753$	0	0
$754$	−2.19099	−0.0797912
$755$	62.1339	2.26128
$756$	0	0
$757$	−25.3404	−0.921013	−0.460507	−	0.887656i	$-0.652332\pi$
$757$	−25.3404	−0.921013	−0.460507	+	0.887656i	$0.652332\pi$
$758$	−7.85100	−0.285161
$759$	0	0
$760$	6.54174	0.237294
$761$	−38.1316	−1.38227	−0.691135	−	0.722725i	$-0.742889\pi$
$761$	−38.1316	−1.38227	−0.691135	+	0.722725i	$0.742889\pi$
$762$	0	0
$763$	26.8690	0.972724
$764$	31.3088	1.13271
$765$	0	0
$766$	−15.0124	−0.542421
$767$	−15.1702	−0.547764
$768$	0	0
$769$	−25.0980	−0.905058	−0.452529	−	0.891750i	$-0.649478\pi$
$769$	−25.0980	−0.905058	−0.452529	+	0.891750i	$0.649478\pi$
$770$	−8.35258	−0.301006
$771$	0	0
$772$	25.6282	0.922378
$773$	−14.8328	−0.533500	−0.266750	−	0.963766i	$-0.585950\pi$
$773$	−14.8328	−0.533500	−0.266750	+	0.963766i	$0.585950\pi$
$774$	0	0
$775$	−14.6815	−0.527377
$776$	−5.01953	−0.180190
$777$	0	0
$778$	−13.9342	−0.499566
$779$	3.18709	0.114189
$780$	0	0
$781$	12.1125	0.433421
$782$	2.38519	0.0852943
$783$	0	0
$784$	27.4799	0.981427
$785$	−6.08955	−0.217345
$786$	0	0
$787$	20.9211	0.745758	0.372879	−	0.927880i	$-0.378371\pi$
$787$	20.9211	0.745758	0.372879	+	0.927880i	$0.378371\pi$
$788$	−9.49363	−0.338196
$789$	0	0
$790$	−8.95298	−0.318533
$791$	−9.78663	−0.347973
$792$	0	0
$793$	1.21915	0.0432933
$794$	15.4736	0.549137
$795$	0	0
$796$	−2.43760	−0.0863986
$797$	50.6551	1.79430	0.897148	−	0.441730i	$-0.145635\pi$
$797$	50.6551	1.79430	0.897148	+	0.441730i	$0.145635\pi$
$798$	0	0
$799$	−3.18976	−0.112846
$800$	42.6506	1.50793
$801$	0	0
$802$	−8.37828	−0.295847
$803$	−9.92547	−0.350262
$804$	0	0
$805$	141.651	4.99253
$806$	1.17327	0.0413265
$807$	0	0
$808$	−6.36635	−0.223967
$809$	17.7140	0.622792	0.311396	−	0.950280i	$-0.399203\pi$
$809$	17.7140	0.622792	0.311396	+	0.950280i	$0.399203\pi$
$810$	0	0
$811$	28.1935	0.990009	0.495004	−	0.868890i	$-0.335166\pi$
$811$	28.1935	0.990009	0.495004	+	0.868890i	$0.335166\pi$
$812$	−24.8792	−0.873087
$813$	0	0
$814$	−1.34647	−0.0471939
$815$	−18.2454	−0.639108
$816$	0	0
$817$	1.06913	0.0374041
$818$	−14.8205	−0.518188
$819$	0	0
$820$	21.7946	0.761102
$821$	43.3355	1.51242	0.756210	−	0.654329i	$-0.227049\pi$
$821$	43.3355	1.51242	0.756210	+	0.654329i	$0.227049\pi$
$822$	0	0
$823$	39.0773	1.36215	0.681075	−	0.732214i	$-0.261513\pi$
$823$	39.0773	1.36215	0.681075	+	0.732214i	$0.261513\pi$
$824$	22.8976	0.797677
$825$	0	0
$826$	28.6624	0.997293
$827$	−2.15211	−0.0748363	−0.0374181	−	0.999300i	$-0.511913\pi$
$827$	−2.15211	−0.0748363	−0.0374181	+	0.999300i	$0.511913\pi$
$828$	0	0
$829$	33.0029	1.14624	0.573119	−	0.819472i	$-0.305733\pi$
$829$	33.0029	1.14624	0.573119	+	0.819472i	$0.305733\pi$
$830$	−5.42402	−0.188270
$831$	0	0
$832$	2.36930	0.0821406
$833$	−5.71690	−0.198079
$834$	0	0
$835$	60.1163	2.08041
$836$	−1.55905	−0.0539207
$837$	0	0
$838$	−15.6667	−0.541196
$839$	−17.0342	−0.588086	−0.294043	−	0.955792i	$-0.595001\pi$
$839$	−17.0342	−0.588086	−0.294043	+	0.955792i	$0.595001\pi$
$840$	0	0
$841$	−17.6798	−0.609648
$842$	11.0270	0.380016
$843$	0	0
$844$	31.6969	1.09105
$845$	41.7499	1.43624
$846$	0	0
$847$	4.31243	0.148177
$848$	−17.5390	−0.602291
$849$	0	0
$850$	−2.14564	−0.0735950
$851$	22.8348	0.782765
$852$	0	0
$853$	46.1109	1.57881	0.789404	−	0.613873i	$-0.210390\pi$
$853$	46.1109	1.57881	0.789404	+	0.613873i	$0.210390\pi$
$854$	−2.30345	−0.0788225
$855$	0	0
$856$	−21.4870	−0.734412
$857$	19.3678	0.661591	0.330796	−	0.943702i	$-0.392683\pi$
$857$	19.3678	0.661591	0.330796	+	0.943702i	$0.392683\pi$
$858$	0	0
$859$	26.8064	0.914622	0.457311	−	0.889307i	$-0.348813\pi$
$859$	26.8064	0.914622	0.457311	+	0.889307i	$0.348813\pi$
$860$	7.31114	0.249308
$861$	0	0
$862$	15.5659	0.530176
$863$	−19.7597	−0.672627	−0.336313	−	0.941750i	$-0.609180\pi$
$863$	−19.7597	−0.672627	−0.336313	+	0.941750i	$0.609180\pi$
$864$	0	0
$865$	−38.9799	−1.32535
$866$	9.77330	0.332110
$867$	0	0
$868$	13.3227	0.452201
$869$	4.62242	0.156805
$870$	0	0
$871$	1.70468	0.0577608
$872$	−12.3626	−0.418650
$873$	0	0
$874$	−4.39930	−0.148809
$875$	−49.2377	−1.66454
$876$	0	0
$877$	55.2507	1.86568	0.932842	−	0.360285i	$-0.117321\pi$
$877$	55.2507	1.86568	0.932842	+	0.360285i	$0.117321\pi$
$878$	11.3990	0.384696
$879$	0	0
$880$	−8.59228	−0.289646
$881$	7.93825	0.267446	0.133723	−	0.991019i	$-0.457307\pi$
$881$	7.93825	0.267446	0.133723	+	0.991019i	$0.457307\pi$
$882$	0	0
$883$	−28.2610	−0.951058	−0.475529	−	0.879700i	$-0.657743\pi$
$883$	−28.2610	−0.951058	−0.475529	+	0.879700i	$0.657743\pi$
$884$	−1.03052	−0.0346601
$885$	0	0
$886$	−11.1318	−0.373979
$887$	−42.0820	−1.41297	−0.706487	−	0.707726i	$-0.749721\pi$
$887$	−42.0820	−1.41297	−0.706487	+	0.707726i	$0.749721\pi$
$888$	0	0
$889$	15.0460	0.504626
$890$	23.4027	0.784460
$891$	0	0
$892$	−19.2319	−0.643933
$893$	5.88327	0.196876
$894$	0	0
$895$	57.3578	1.91726
$896$	−49.6193	−1.65766
$897$	0	0
$898$	6.97892	0.232890
$899$	−6.06191	−0.202176
$900$	0	0
$901$	3.64879	0.121559
$902$	1.87231	0.0623412
$903$	0	0
$904$	4.50288	0.149764
$905$	56.0997	1.86482
$906$	0	0
$907$	−21.4152	−0.711079	−0.355540	−	0.934661i	$-0.615703\pi$
$907$	−21.4152	−0.711079	−0.355540	+	0.934661i	$0.615703\pi$
$908$	38.4887	1.27729
$909$	0	0
$910$	10.1830	0.337565
$911$	−6.72715	−0.222880	−0.111440	−	0.993771i	$-0.535546\pi$
$911$	−6.72715	−0.222880	−0.111440	+	0.993771i	$0.535546\pi$
$912$	0	0
$913$	2.80042	0.0926803
$914$	−16.9567	−0.560878
$915$	0	0
$916$	−36.8346	−1.21705
$917$	−93.4117	−3.08473
$918$	0	0
$919$	−44.7382	−1.47578	−0.737888	−	0.674923i	$-0.764177\pi$
$919$	−44.7382	−1.47578	−0.737888	+	0.674923i	$0.764177\pi$
$920$	−65.1742	−2.14873
$921$	0	0
$922$	11.5153	0.379237
$923$	−14.7670	−0.486062
$924$	0	0
$925$	−20.5414	−0.675398
$926$	−10.8662	−0.357086
$927$	0	0
$928$	17.6101	0.578081
$929$	38.6489	1.26803	0.634014	−	0.773322i	$-0.281406\pi$
$929$	38.6489	1.26803	0.634014	+	0.773322i	$0.281406\pi$
$930$	0	0
$931$	10.5444	0.345578
$932$	−7.10018	−0.232574
$933$	0	0
$934$	0.664902	0.0217563
$935$	1.78753	0.0584585
$936$	0	0
$937$	−39.2625	−1.28265	−0.641325	−	0.767269i	$-0.721615\pi$
$937$	−39.2625	−1.28265	−0.641325	+	0.767269i	$0.721615\pi$
$938$	−3.22080	−0.105163
$939$	0	0
$940$	40.2322	1.31223
$941$	−19.1212	−0.623332	−0.311666	−	0.950192i	$-0.600887\pi$
$941$	−19.1212	−0.623332	−0.311666	+	0.950192i	$0.600887\pi$
$942$	0	0
$943$	−31.7524	−1.03400
$944$	29.4850	0.959654
$945$	0	0
$946$	0.628078	0.0204206
$947$	−19.6358	−0.638078	−0.319039	−	0.947742i	$-0.603360\pi$
$947$	−19.6358	−0.638078	−0.319039	+	0.947742i	$0.603360\pi$
$948$	0	0
$949$	12.1006	0.392803
$950$	3.95748	0.128397
$951$	0	0
$952$	4.21807	0.136708
$953$	25.5350	0.827160	0.413580	−	0.910468i	$-0.364278\pi$
$953$	25.5350	0.827160	0.413580	+	0.910468i	$0.364278\pi$
$954$	0	0
$955$	66.2098	2.14250
$956$	−18.0657	−0.584287
$957$	0	0
$958$	−7.99733	−0.258382
$959$	−92.2759	−2.97974
$960$	0	0
$961$	−27.7539	−0.895286
$962$	1.64155	0.0529258
$963$	0	0
$964$	20.2442	0.652023
$965$	54.1967	1.74465
$966$	0	0
$967$	55.0924	1.77165	0.885826	−	0.464018i	$-0.153593\pi$
$967$	55.0924	1.77165	0.885826	+	0.464018i	$0.153593\pi$
$968$	−1.98417	−0.0637737
$969$	0	0
$970$	−4.89984	−0.157324
$971$	7.43173	0.238496	0.119248	−	0.992865i	$-0.461952\pi$
$971$	7.43173	0.238496	0.119248	+	0.992865i	$0.461952\pi$
$972$	0	0
$973$	−28.2925	−0.907015
$974$	−10.3547	−0.331786
$975$	0	0
$976$	−2.36956	−0.0758476
$977$	−22.8141	−0.729889	−0.364945	−	0.931029i	$-0.618912\pi$
$977$	−22.8141	−0.729889	−0.364945	+	0.931029i	$0.618912\pi$
$978$	0	0
$979$	−12.0828	−0.386168
$980$	72.1069	2.30337
$981$	0	0
$982$	6.77676	0.216255
$983$	−28.4396	−0.907081	−0.453541	−	0.891236i	$-0.649839\pi$
$983$	−28.4396	−0.907081	−0.453541	+	0.891236i	$0.649839\pi$
$984$	0	0
$985$	−20.0765	−0.639690
$986$	−0.885922	−0.0282135
$987$	0	0
$988$	1.90071	0.0604697
$989$	−10.6515	−0.338699
$990$	0	0
$991$	−48.9700	−1.55558	−0.777791	−	0.628523i	$-0.783660\pi$
$991$	−48.9700	−1.55558	−0.777791	+	0.628523i	$0.783660\pi$
$992$	−9.43014	−0.299407
$993$	0	0
$994$	27.9006	0.884954
$995$	−5.15488	−0.163421
$996$	0	0
$997$	10.7047	0.339022	0.169511	−	0.985528i	$-0.445781\pi$
$997$	10.7047	0.339022	0.169511	+	0.985528i	$0.445781\pi$
$998$	2.40309	0.0760684
$999$	0	0

Twists

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

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

\(f(q)\)	\(=\)	\(q+0.534142 q^{2} -1.71469 q^{4} -3.62612 q^{5} +4.31243 q^{7} -1.98417 q^{8} +O(q^{10})\)	\(q+0.534142 q^{2} -1.71469 q^{4} -3.62612 q^{5} +4.31243 q^{7} -1.98417 q^{8} -1.93686 q^{10} +1.00000 q^{11} -1.21915 q^{13} +2.30345 q^{14} +2.36956 q^{16} -0.492960 q^{17} +0.909227 q^{19} +6.21767 q^{20} +0.534142 q^{22} -9.05847 q^{23} +8.14872 q^{25} -0.651199 q^{26} -7.39450 q^{28} +3.36455 q^{29} -1.80170 q^{31} +5.23402 q^{32} -0.263311 q^{34} -15.6374 q^{35} -2.52082 q^{37} +0.485656 q^{38} +7.19484 q^{40} +3.50527 q^{41} +1.17586 q^{43} -1.71469 q^{44} -4.83851 q^{46} +6.47062 q^{47} +11.5971 q^{49} +4.35257 q^{50} +2.09047 q^{52} -7.40180 q^{53} -3.62612 q^{55} -8.55661 q^{56} +1.79715 q^{58} +12.4433 q^{59} -1.00000 q^{61} -0.962363 q^{62} -1.94340 q^{64} +4.42078 q^{65} -1.39825 q^{67} +0.845275 q^{68} -8.35258 q^{70} +12.1125 q^{71} -9.92547 q^{73} -1.34647 q^{74} -1.55905 q^{76} +4.31243 q^{77} +4.62242 q^{79} -8.59228 q^{80} +1.87231 q^{82} +2.80042 q^{83} +1.78753 q^{85} +0.628078 q^{86} -1.98417 q^{88} -12.0828 q^{89} -5.25750 q^{91} +15.5325 q^{92} +3.45623 q^{94} -3.29696 q^{95} +2.52978 q^{97} +6.19449 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 13 q - 4 q^{2} + 12 q^{4} - 7 q^{5} + 5 q^{7} - 15 q^{8}+O(q^{10}) \) 13 * q - 4 * q^2 + 12 * q^4 - 7 * q^5 + 5 * q^7 - 15 * q^8	\( 13 q - 4 q^{2} + 12 q^{4} - 7 q^{5} + 5 q^{7} - 15 q^{8} + 8 q^{10} + 13 q^{11} - 9 q^{13} - 19 q^{14} + 18 q^{16} - 7 q^{17} + 2 q^{19} - 15 q^{20} - 4 q^{22} - 23 q^{23} + 10 q^{25} - 8 q^{26} + 9 q^{28} - 16 q^{29} + 9 q^{31} - 29 q^{32} + 2 q^{34} - 16 q^{35} + 14 q^{37} - 8 q^{38} + 16 q^{40} - 19 q^{41} + 7 q^{43} + 12 q^{44} + 4 q^{46} - 26 q^{47} + 8 q^{49} + 15 q^{50} - 17 q^{52} - 18 q^{53} - 7 q^{55} - 44 q^{56} - q^{58} - 31 q^{59} - 13 q^{61} + 5 q^{62} - 17 q^{64} - 31 q^{65} + 14 q^{67} + 32 q^{68} - 20 q^{70} - 37 q^{71} - 16 q^{73} + 6 q^{74} - 7 q^{76} + 5 q^{77} - 17 q^{79} + 2 q^{80} - 2 q^{82} - 30 q^{83} - 16 q^{85} + 22 q^{86} - 15 q^{88} - 35 q^{89} - q^{91} - 24 q^{92} - 11 q^{94} - 13 q^{95} - q^{97} + q^{98}+O(q^{100}) \) 13 * q - 4 * q^2 + 12 * q^4 - 7 * q^5 + 5 * q^7 - 15 * q^8 + 8 * q^10 + 13 * q^11 - 9 * q^13 - 19 * q^14 + 18 * q^16 - 7 * q^17 + 2 * q^19 - 15 * q^20 - 4 * q^22 - 23 * q^23 + 10 * q^25 - 8 * q^26 + 9 * q^28 - 16 * q^29 + 9 * q^31 - 29 * q^32 + 2 * q^34 - 16 * q^35 + 14 * q^37 - 8 * q^38 + 16 * q^40 - 19 * q^41 + 7 * q^43 + 12 * q^44 + 4 * q^46 - 26 * q^47 + 8 * q^49 + 15 * q^50 - 17 * q^52 - 18 * q^53 - 7 * q^55 - 44 * q^56 - q^58 - 31 * q^59 - 13 * q^61 + 5 * q^62 - 17 * q^64 - 31 * q^65 + 14 * q^67 + 32 * q^68 - 20 * q^70 - 37 * q^71 - 16 * q^73 + 6 * q^74 - 7 * q^76 + 5 * q^77 - 17 * q^79 + 2 * q^80 - 2 * q^82 - 30 * q^83 - 16 * q^85 + 22 * q^86 - 15 * q^88 - 35 * q^89 - q^91 - 24 * q^92 - 11 * q^94 - 13 * q^95 - q^97 + q^98

Introduction

L-functions

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