LMFDB - Embedded newform 6039.2.a.p.1.4

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:	$25$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
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)

$f(q)$	$=$	$q-1.92069 q^{2} +1.68905 q^{4} -2.68560 q^{5} -3.03402 q^{7} +0.597238 q^{8} +O(q^{10})$	\(q-1.92069 q^{2} +1.68905 q^{4} -2.68560 q^{5} -3.03402 q^{7} +0.597238 q^{8} +5.15821 q^{10} -1.00000 q^{11} -1.87365 q^{13} +5.82741 q^{14} -4.52521 q^{16} +5.85766 q^{17} +2.10407 q^{19} -4.53612 q^{20} +1.92069 q^{22} -5.06036 q^{23} +2.21246 q^{25} +3.59870 q^{26} -5.12461 q^{28} -0.755584 q^{29} -7.72934 q^{31} +7.49705 q^{32} -11.2508 q^{34} +8.14817 q^{35} -8.48252 q^{37} -4.04126 q^{38} -1.60394 q^{40} +7.29535 q^{41} +2.41940 q^{43} -1.68905 q^{44} +9.71939 q^{46} +11.9984 q^{47} +2.20527 q^{49} -4.24946 q^{50} -3.16469 q^{52} -14.2389 q^{53} +2.68560 q^{55} -1.81203 q^{56} +1.45124 q^{58} -10.2343 q^{59} +1.00000 q^{61} +14.8457 q^{62} -5.34909 q^{64} +5.03188 q^{65} -7.27474 q^{67} +9.89389 q^{68} -15.6501 q^{70} +9.06523 q^{71} +0.597050 q^{73} +16.2923 q^{74} +3.55387 q^{76} +3.03402 q^{77} -13.7941 q^{79} +12.1529 q^{80} -14.0121 q^{82} -10.5157 q^{83} -15.7314 q^{85} -4.64691 q^{86} -0.597238 q^{88} -4.45740 q^{89} +5.68469 q^{91} -8.54721 q^{92} -23.0452 q^{94} -5.65068 q^{95} -0.962072 q^{97} -4.23565 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 25 q + 5 q^{2} + 25 q^{4} + 12 q^{5} - 4 q^{7} + 15 q^{8}+O(q^{10}) $ 25 * q + 5 * q^2 + 25 * q^4 + 12 * q^5 - 4 * q^7 + 15 * q^8	$ 25 q + 5 q^{2} + 25 q^{4} + 12 q^{5} - 4 q^{7} + 15 q^{8} - 12 q^{10} - 25 q^{11} - 4 q^{13} + 14 q^{14} + 21 q^{16} + 16 q^{17} - 18 q^{19} + 28 q^{20} - 5 q^{22} + 8 q^{23} + 29 q^{25} + 16 q^{26} + 18 q^{28} + 28 q^{29} - 8 q^{31} + 35 q^{32} + 6 q^{34} + 22 q^{35} + 4 q^{37} - 4 q^{38} - 12 q^{40} + 58 q^{41} - 26 q^{43} - 25 q^{44} + 8 q^{46} + 20 q^{47} + 23 q^{49} + 27 q^{50} - 2 q^{52} + 36 q^{53} - 12 q^{55} + 70 q^{56} + 12 q^{58} + 18 q^{59} + 25 q^{61} + 42 q^{62} + 35 q^{64} + 76 q^{65} - 8 q^{67} + 28 q^{68} + 76 q^{70} + 24 q^{71} + 2 q^{73} + 40 q^{74} - 64 q^{76} + 4 q^{77} - 22 q^{79} + 36 q^{80} + 30 q^{82} + 14 q^{83} + 70 q^{86} - 15 q^{88} + 82 q^{89} - 6 q^{91} + 48 q^{92} - 16 q^{94} + 34 q^{95} + 16 q^{97} + 35 q^{98}+O(q^{100}) $ 25 * q + 5 * q^2 + 25 * q^4 + 12 * q^5 - 4 * q^7 + 15 * q^8 - 12 * q^10 - 25 * q^11 - 4 * q^13 + 14 * q^14 + 21 * q^16 + 16 * q^17 - 18 * q^19 + 28 * q^20 - 5 * q^22 + 8 * q^23 + 29 * q^25 + 16 * q^26 + 18 * q^28 + 28 * q^29 - 8 * q^31 + 35 * q^32 + 6 * q^34 + 22 * q^35 + 4 * q^37 - 4 * q^38 - 12 * q^40 + 58 * q^41 - 26 * q^43 - 25 * q^44 + 8 * q^46 + 20 * q^47 + 23 * q^49 + 27 * q^50 - 2 * q^52 + 36 * q^53 - 12 * q^55 + 70 * q^56 + 12 * q^58 + 18 * q^59 + 25 * q^61 + 42 * q^62 + 35 * q^64 + 76 * q^65 - 8 * q^67 + 28 * q^68 + 76 * q^70 + 24 * q^71 + 2 * q^73 + 40 * q^74 - 64 * q^76 + 4 * q^77 - 22 * q^79 + 36 * q^80 + 30 * q^82 + 14 * q^83 + 70 * q^86 - 15 * q^88 + 82 * q^89 - 6 * q^91 + 48 * q^92 - 16 * q^94 + 34 * q^95 + 16 * q^97 + 35 * 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 $

$2$	−1.92069	−1.35813	−0.679066	−	0.734077i	$-0.737615\pi$
$2$	−1.92069	−1.35813	−0.679066	+	0.734077i	$0.737615\pi$
$3$	0	0
$3$	0	0
$4$	1.68905	0.844525
$5$	−2.68560	−1.20104	−0.600519	−	0.799610i	$-0.705039\pi$
$5$	−2.68560	−1.20104	−0.600519	+	0.799610i	$0.705039\pi$
$6$	0	0
$7$	−3.03402	−1.14675	−0.573376	−	0.819293i	$-0.694366\pi$
$7$	−3.03402	−1.14675	−0.573376	+	0.819293i	$0.694366\pi$
$8$	0.597238	0.211155
$9$	0	0
$10$	5.15821	1.63117
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	−1.87365	−0.519657	−0.259829	−	0.965655i	$-0.583666\pi$
$13$	−1.87365	−0.519657	−0.259829	+	0.965655i	$0.583666\pi$
$14$	5.82741	1.55744
$15$	0	0
$16$	−4.52521	−1.13130
$17$	5.85766	1.42069	0.710346	−	0.703852i	$-0.248538\pi$
$17$	5.85766	1.42069	0.710346	+	0.703852i	$0.248538\pi$
$18$	0	0
$19$	2.10407	0.482706	0.241353	−	0.970437i	$-0.422409\pi$
$19$	2.10407	0.482706	0.241353	+	0.970437i	$0.422409\pi$
$20$	−4.53612	−1.01431
$21$	0	0
$22$	1.92069	0.409493
$23$	−5.06036	−1.05516	−0.527579	−	0.849506i	$-0.676900\pi$
$23$	−5.06036	−1.05516	−0.527579	+	0.849506i	$0.676900\pi$
$24$	0	0
$25$	2.21246	0.442493
$26$	3.59870	0.705764
$27$	0	0
$28$	−5.12461	−0.968461
$29$	−0.755584	−0.140308	−0.0701542	−	0.997536i	$-0.522349\pi$
$29$	−0.755584	−0.140308	−0.0701542	+	0.997536i	$0.522349\pi$
$30$	0	0
$31$	−7.72934	−1.38823	−0.694115	−	0.719864i	$-0.744204\pi$
$31$	−7.72934	−1.38823	−0.694115	+	0.719864i	$0.744204\pi$
$32$	7.49705	1.32530
$33$	0	0
$34$	−11.2508	−1.92949
$35$	8.14817	1.37729
$36$	0	0
$37$	−8.48252	−1.39452	−0.697259	−	0.716819i	$-0.745597\pi$
$37$	−8.48252	−1.39452	−0.697259	+	0.716819i	$0.745597\pi$
$38$	−4.04126	−0.655579
$39$	0	0
$40$	−1.60394	−0.253606
$41$	7.29535	1.13934	0.569671	−	0.821873i	$-0.307071\pi$
$41$	7.29535	1.13934	0.569671	+	0.821873i	$0.307071\pi$
$42$	0	0
$43$	2.41940	0.368954	0.184477	−	0.982837i	$-0.440941\pi$
$43$	2.41940	0.368954	0.184477	+	0.982837i	$0.440941\pi$
$44$	−1.68905	−0.254634
$45$	0	0
$46$	9.71939	1.43305
$47$	11.9984	1.75015	0.875075	−	0.483988i	$-0.160812\pi$
$47$	11.9984	1.75015	0.875075	+	0.483988i	$0.160812\pi$
$48$	0	0
$49$	2.20527	0.315039
$50$	−4.24946	−0.600964
$51$	0	0
$52$	−3.16469	−0.438864
$53$	−14.2389	−1.95587	−0.977934	−	0.208916i	$-0.933006\pi$
$53$	−14.2389	−1.95587	−0.977934	+	0.208916i	$0.933006\pi$
$54$	0	0
$55$	2.68560	0.362127
$56$	−1.81203	−0.242143
$57$	0	0
$58$	1.45124	0.190558
$59$	−10.2343	−1.33239	−0.666194	−	0.745779i	$-0.732078\pi$
$59$	−10.2343	−1.33239	−0.666194	+	0.745779i	$0.732078\pi$
$60$	0	0
$61$	1.00000	0.128037
$61$	1.00000	0.128037
$62$	14.8457	1.88540
$63$	0	0
$64$	−5.34909	−0.668636
$65$	5.03188	0.624128
$66$	0	0
$67$	−7.27474	−0.888751	−0.444376	−	0.895841i	$-0.646574\pi$
$67$	−7.27474	−0.888751	−0.444376	+	0.895841i	$0.646574\pi$
$68$	9.89389	1.19981
$69$	0	0
$70$	−15.6501	−1.87055
$71$	9.06523	1.07584	0.537922	−	0.842994i	$-0.319209\pi$
$71$	9.06523	1.07584	0.537922	+	0.842994i	$0.319209\pi$
$72$	0	0
$73$	0.597050	0.0698795	0.0349397	−	0.999389i	$-0.488876\pi$
$73$	0.597050	0.0698795	0.0349397	+	0.999389i	$0.488876\pi$
$74$	16.2923	1.89394
$75$	0	0
$76$	3.55387	0.407657
$77$	3.03402	0.345759
$78$	0	0
$79$	−13.7941	−1.55195	−0.775976	−	0.630762i	$-0.782742\pi$
$79$	−13.7941	−1.55195	−0.775976	+	0.630762i	$0.782742\pi$
$80$	12.1529	1.35874
$81$	0	0
$82$	−14.0121	−1.54738
$83$	−10.5157	−1.15425	−0.577123	−	0.816657i	$-0.695825\pi$
$83$	−10.5157	−1.15425	−0.577123	+	0.816657i	$0.695825\pi$
$84$	0	0
$85$	−15.7314	−1.70631
$86$	−4.64691	−0.501089
$87$	0	0
$88$	−0.597238	−0.0636658
$89$	−4.45740	−0.472483	−0.236241	−	0.971694i	$-0.575916\pi$
$89$	−4.45740	−0.472483	−0.236241	+	0.971694i	$0.575916\pi$
$90$	0	0
$91$	5.68469	0.595918
$92$	−8.54721	−0.891108
$93$	0	0
$94$	−23.0452	−2.37694
$95$	−5.65068	−0.579748
$96$	0	0
$97$	−0.962072	−0.0976836	−0.0488418	−	0.998807i	$-0.515553\pi$
$97$	−0.962072	−0.0976836	−0.0488418	+	0.998807i	$0.515553\pi$
$98$	−4.23565	−0.427865
$99$	0	0
$100$	3.73696	0.373696
$101$	4.30914	0.428775	0.214388	−	0.976749i	$-0.431224\pi$
$101$	4.30914	0.428775	0.214388	+	0.976749i	$0.431224\pi$
$102$	0	0
$103$	−6.82509	−0.672496	−0.336248	−	0.941773i	$-0.609158\pi$
$103$	−6.82509	−0.672496	−0.336248	+	0.941773i	$0.609158\pi$
$104$	−1.11902	−0.109728
$105$	0	0
$106$	27.3486	2.65633
$107$	8.27683	0.800151	0.400076	−	0.916482i	$-0.368984\pi$
$107$	8.27683	0.800151	0.400076	+	0.916482i	$0.368984\pi$
$108$	0	0
$109$	−8.76305	−0.839348	−0.419674	−	0.907675i	$-0.637856\pi$
$109$	−8.76305	−0.839348	−0.419674	+	0.907675i	$0.637856\pi$
$110$	−5.15821	−0.491816
$111$	0	0
$112$	13.7296	1.29732
$113$	−20.9404	−1.96991	−0.984955	−	0.172810i	$-0.944715\pi$
$113$	−20.9404	−1.96991	−0.984955	+	0.172810i	$0.944715\pi$
$114$	0	0
$115$	13.5901	1.26729
$116$	−1.27622	−0.118494
$117$	0	0
$118$	19.6569	1.80956
$119$	−17.7723	−1.62918
$120$	0	0
$121$	1.00000	0.0909091
$122$	−1.92069	−0.173891
$123$	0	0
$124$	−13.0552	−1.17240
$125$	7.48621	0.669587
$126$	0	0
$127$	−4.50514	−0.399767	−0.199883	−	0.979820i	$-0.564056\pi$
$127$	−4.50514	−0.399767	−0.199883	+	0.979820i	$0.564056\pi$
$128$	−4.72016	−0.417207
$129$	0	0
$130$	−9.66469	−0.847649
$131$	−7.64421	−0.667878	−0.333939	−	0.942595i	$-0.608378\pi$
$131$	−7.64421	−0.667878	−0.333939	+	0.942595i	$0.608378\pi$
$132$	0	0
$133$	−6.38377	−0.553543
$134$	13.9725	1.20704
$135$	0	0
$136$	3.49842	0.299987
$137$	16.9930	1.45181	0.725907	−	0.687793i	$-0.241420\pi$
$137$	16.9930	1.45181	0.725907	+	0.687793i	$0.241420\pi$
$138$	0	0
$139$	−14.0834	−1.19454	−0.597268	−	0.802042i	$-0.703747\pi$
$139$	−14.0834	−1.19454	−0.597268	+	0.802042i	$0.703747\pi$
$140$	13.7627	1.16316
$141$	0	0
$142$	−17.4115	−1.46114
$143$	1.87365	0.156683
$144$	0	0
$145$	2.02920	0.168516
$146$	−1.14675	−0.0949056
$147$	0	0
$148$	−14.3274	−1.17771
$149$	2.88300	0.236185	0.118092	−	0.993003i	$-0.462322\pi$
$149$	2.88300	0.236185	0.118092	+	0.993003i	$0.462322\pi$
$150$	0	0
$151$	−2.57560	−0.209599	−0.104800	−	0.994493i	$-0.533420\pi$
$151$	−2.57560	−0.209599	−0.104800	+	0.994493i	$0.533420\pi$
$152$	1.25663	0.101926
$153$	0	0
$154$	−5.82741	−0.469586
$155$	20.7579	1.66732
$156$	0	0
$157$	−0.562797	−0.0449161	−0.0224580	−	0.999748i	$-0.507149\pi$
$157$	−0.562797	−0.0449161	−0.0224580	+	0.999748i	$0.507149\pi$
$158$	26.4941	2.10776
$159$	0	0
$160$	−20.1341	−1.59174
$161$	15.3532	1.21000
$162$	0	0
$163$	−0.331152	−0.0259378	−0.0129689	−	0.999916i	$-0.504128\pi$
$163$	−0.331152	−0.0259378	−0.0129689	+	0.999916i	$0.504128\pi$
$164$	12.3222	0.962203
$165$	0	0
$166$	20.1974	1.56762
$167$	−15.8427	−1.22594	−0.612971	−	0.790106i	$-0.710026\pi$
$167$	−15.8427	−1.22594	−0.612971	+	0.790106i	$0.710026\pi$
$168$	0	0
$169$	−9.48943	−0.729956
$170$	30.2151	2.31739
$171$	0	0
$172$	4.08648	0.311591
$173$	−0.564460	−0.0429151	−0.0214576	−	0.999770i	$-0.506831\pi$
$173$	−0.564460	−0.0429151	−0.0214576	+	0.999770i	$0.506831\pi$
$174$	0	0
$175$	−6.71266	−0.507429
$176$	4.52521	0.341101
$177$	0	0
$178$	8.56128	0.641695
$179$	19.8246	1.48176	0.740882	−	0.671636i	$-0.234408\pi$
$179$	19.8246	1.48176	0.740882	+	0.671636i	$0.234408\pi$
$180$	0	0
$181$	−4.31610	−0.320813	−0.160407	−	0.987051i	$-0.551281\pi$
$181$	−4.31610	−0.320813	−0.160407	+	0.987051i	$0.551281\pi$
$182$	−10.9185	−0.809336
$183$	0	0
$184$	−3.02224	−0.222803
$185$	22.7807	1.67487
$186$	0	0
$187$	−5.85766	−0.428355
$188$	20.2659	1.47805
$189$	0	0
$190$	10.8532	0.787375
$191$	−23.3370	−1.68861	−0.844303	−	0.535867i	$-0.819985\pi$
$191$	−23.3370	−1.68861	−0.844303	+	0.535867i	$0.819985\pi$
$192$	0	0
$193$	−20.4439	−1.47158	−0.735792	−	0.677208i	$-0.763190\pi$
$193$	−20.4439	−1.47158	−0.735792	+	0.677208i	$0.763190\pi$
$194$	1.84784	0.132667
$195$	0	0
$196$	3.72482	0.266058
$197$	−23.3930	−1.66668	−0.833341	−	0.552759i	$-0.813575\pi$
$197$	−23.3930	−1.66668	−0.833341	+	0.552759i	$0.813575\pi$
$198$	0	0
$199$	21.7253	1.54006	0.770032	−	0.638006i	$-0.220240\pi$
$199$	21.7253	1.54006	0.770032	+	0.638006i	$0.220240\pi$
$200$	1.32137	0.0934348
$201$	0	0
$202$	−8.27652	−0.582334
$203$	2.29246	0.160899
$204$	0	0
$205$	−19.5924	−1.36839
$206$	13.1089	0.913339
$207$	0	0
$208$	8.47866	0.587889
$209$	−2.10407	−0.145541
$210$	0	0
$211$	1.53961	0.105991	0.0529956	−	0.998595i	$-0.483123\pi$
$211$	1.53961	0.105991	0.0529956	+	0.998595i	$0.483123\pi$
$212$	−24.0503	−1.65178
$213$	0	0
$214$	−15.8972	−1.08671
$215$	−6.49754	−0.443128
$216$	0	0
$217$	23.4510	1.59196
$218$	16.8311	1.13995
$219$	0	0
$220$	4.53612	0.305825
$221$	−10.9752	−0.738273
$222$	0	0
$223$	26.6963	1.78772	0.893858	−	0.448350i	$-0.147988\pi$
$223$	26.6963	1.78772	0.893858	+	0.448350i	$0.147988\pi$
$224$	−22.7462	−1.51979
$225$	0	0
$226$	40.2201	2.67540
$227$	0.182850	0.0121362	0.00606809	−	0.999982i	$-0.498068\pi$
$227$	0.182850	0.0121362	0.00606809	+	0.999982i	$0.498068\pi$
$228$	0	0
$229$	20.7203	1.36923	0.684617	−	0.728903i	$-0.259970\pi$
$229$	20.7203	1.36923	0.684617	+	0.728903i	$0.259970\pi$
$230$	−26.1024	−1.72114
$231$	0	0
$232$	−0.451263	−0.0296269
$233$	7.38599	0.483872	0.241936	−	0.970292i	$-0.422218\pi$
$233$	7.38599	0.483872	0.241936	+	0.970292i	$0.422218\pi$
$234$	0	0
$235$	−32.2230	−2.10200
$236$	−17.2862	−1.12524
$237$	0	0
$238$	34.1350	2.21264
$239$	27.7784	1.79683	0.898417	−	0.439143i	$-0.144718\pi$
$239$	27.7784	1.79683	0.898417	+	0.439143i	$0.144718\pi$
$240$	0	0
$241$	7.28366	0.469182	0.234591	−	0.972094i	$-0.424625\pi$
$241$	7.28366	0.469182	0.234591	+	0.972094i	$0.424625\pi$
$242$	−1.92069	−0.123467
$243$	0	0
$244$	1.68905	0.108130
$245$	−5.92249	−0.378374
$246$	0	0
$247$	−3.94228	−0.250842
$248$	−4.61625	−0.293132
$249$	0	0
$250$	−14.3787	−0.909389
$251$	−10.2814	−0.648955	−0.324478	−	0.945893i	$-0.605189\pi$
$251$	−10.2814	−0.648955	−0.324478	+	0.945893i	$0.605189\pi$
$252$	0	0
$253$	5.06036	0.318142
$254$	8.65298	0.542936
$255$	0	0
$256$	19.7641	1.23526
$257$	25.1638	1.56967	0.784837	−	0.619702i	$-0.212747\pi$
$257$	25.1638	1.56967	0.784837	+	0.619702i	$0.212747\pi$
$258$	0	0
$259$	25.7361	1.59917
$260$	8.49910	0.527092
$261$	0	0
$262$	14.6822	0.907067
$263$	−13.1562	−0.811247	−0.405623	−	0.914040i	$-0.632946\pi$
$263$	−13.1562	−0.811247	−0.405623	+	0.914040i	$0.632946\pi$
$264$	0	0
$265$	38.2401	2.34907
$266$	12.2613	0.751786
$267$	0	0
$268$	−12.2874	−0.750573
$269$	8.43413	0.514238	0.257119	−	0.966380i	$-0.417227\pi$
$269$	8.43413	0.514238	0.257119	+	0.966380i	$0.417227\pi$
$270$	0	0
$271$	−11.7857	−0.715930	−0.357965	−	0.933735i	$-0.616529\pi$
$271$	−11.7857	−0.715930	−0.357965	+	0.933735i	$0.616529\pi$
$272$	−26.5072	−1.60723
$273$	0	0
$274$	−32.6384	−1.97176
$275$	−2.21246	−0.133417
$276$	0	0
$277$	19.7247	1.18514	0.592572	−	0.805517i	$-0.298112\pi$
$277$	19.7247	1.18514	0.592572	+	0.805517i	$0.298112\pi$
$278$	27.0498	1.62234
$279$	0	0
$280$	4.86640	0.290823
$281$	6.19288	0.369436	0.184718	−	0.982792i	$-0.440863\pi$
$281$	6.19288	0.369436	0.184718	+	0.982792i	$0.440863\pi$
$282$	0	0
$283$	−3.99596	−0.237535	−0.118768	−	0.992922i	$-0.537894\pi$
$283$	−3.99596	−0.237535	−0.118768	+	0.992922i	$0.537894\pi$
$284$	15.3116	0.908578
$285$	0	0
$286$	−3.59870	−0.212796
$287$	−22.1342	−1.30654
$288$	0	0
$289$	17.3122	1.01837
$290$	−3.89746	−0.228867
$291$	0	0
$292$	1.00845	0.0590150
$293$	−0.952142	−0.0556247	−0.0278124	−	0.999613i	$-0.508854\pi$
$293$	−0.952142	−0.0556247	−0.0278124	+	0.999613i	$0.508854\pi$
$294$	0	0
$295$	27.4852	1.60025
$296$	−5.06608	−0.294460
$297$	0	0
$298$	−5.53735	−0.320770
$299$	9.48135	0.548321
$300$	0	0
$301$	−7.34049	−0.423099
$302$	4.94692	0.284663
$303$	0	0
$304$	−9.52134	−0.546086
$305$	−2.68560	−0.153777
$306$	0	0
$307$	−11.3966	−0.650440	−0.325220	−	0.945638i	$-0.605438\pi$
$307$	−11.3966	−0.650440	−0.325220	+	0.945638i	$0.605438\pi$
$308$	5.12461	0.292002
$309$	0	0
$310$	−39.8696	−2.26444
$311$	−32.4418	−1.83961	−0.919803	−	0.392381i	$-0.871651\pi$
$311$	−32.4418	−1.83961	−0.919803	+	0.392381i	$0.871651\pi$
$312$	0	0
$313$	22.2290	1.25646	0.628230	−	0.778028i	$-0.283780\pi$
$313$	22.2290	1.25646	0.628230	+	0.778028i	$0.283780\pi$
$314$	1.08096	0.0610020
$315$	0	0
$316$	−23.2989	−1.31066
$317$	−14.4511	−0.811654	−0.405827	−	0.913950i	$-0.633016\pi$
$317$	−14.4511	−0.811654	−0.405827	+	0.913950i	$0.633016\pi$
$318$	0	0
$319$	0.755584	0.0423046
$320$	14.3655	0.803058
$321$	0	0
$322$	−29.4888	−1.64335
$323$	12.3249	0.685776
$324$	0	0
$325$	−4.14539	−0.229945
$326$	0.636040	0.0352270
$327$	0	0
$328$	4.35706	0.240578
$329$	−36.4034	−2.00699
$330$	0	0
$331$	−13.3440	−0.733452	−0.366726	−	0.930329i	$-0.619521\pi$
$331$	−13.3440	−0.733452	−0.366726	+	0.930329i	$0.619521\pi$
$332$	−17.7615	−0.974790
$333$	0	0
$334$	30.4288	1.66499
$335$	19.5371	1.06742
$336$	0	0
$337$	7.01778	0.382283	0.191141	−	0.981563i	$-0.438781\pi$
$337$	7.01778	0.382283	0.191141	+	0.981563i	$0.438781\pi$
$338$	18.2263	0.991378
$339$	0	0
$340$	−26.5711	−1.44102
$341$	7.72934	0.418567
$342$	0	0
$343$	14.5473	0.785480
$344$	1.44495	0.0779067
$345$	0	0
$346$	1.08415	0.0582844
$347$	15.9871	0.858235	0.429117	−	0.903249i	$-0.358825\pi$
$347$	15.9871	0.858235	0.429117	+	0.903249i	$0.358825\pi$
$348$	0	0
$349$	−12.0688	−0.646026	−0.323013	−	0.946394i	$-0.604696\pi$
$349$	−12.0688	−0.646026	−0.323013	+	0.946394i	$0.604696\pi$
$350$	12.8929	0.689157
$351$	0	0
$352$	−7.49705	−0.399594
$353$	10.2129	0.543576	0.271788	−	0.962357i	$-0.412385\pi$
$353$	10.2129	0.543576	0.271788	+	0.962357i	$0.412385\pi$
$354$	0	0
$355$	−24.3456	−1.29213
$356$	−7.52877	−0.399024
$357$	0	0
$358$	−38.0770	−2.01243
$359$	5.92377	0.312645	0.156322	−	0.987706i	$-0.450036\pi$
$359$	5.92377	0.312645	0.156322	+	0.987706i	$0.450036\pi$
$360$	0	0
$361$	−14.5729	−0.766995
$362$	8.28989	0.435707
$363$	0	0
$364$	9.60173	0.503268
$365$	−1.60344	−0.0839279
$366$	0	0
$367$	−9.15928	−0.478111	−0.239055	−	0.971006i	$-0.576838\pi$
$367$	−9.15928	−0.478111	−0.239055	+	0.971006i	$0.576838\pi$
$368$	22.8992	1.19370
$369$	0	0
$370$	−43.7547	−2.27470
$371$	43.2012	2.24289
$372$	0	0
$373$	2.98476	0.154545	0.0772725	−	0.997010i	$-0.475379\pi$
$373$	2.98476	0.154545	0.0772725	+	0.997010i	$0.475379\pi$
$374$	11.2508	0.581763
$375$	0	0
$376$	7.16591	0.369554
$377$	1.41570	0.0729123
$378$	0	0
$379$	22.6319	1.16252	0.581260	−	0.813718i	$-0.302560\pi$
$379$	22.6319	1.16252	0.581260	+	0.813718i	$0.302560\pi$
$380$	−9.54429	−0.489612
$381$	0	0
$382$	44.8231	2.29335
$383$	−12.0801	−0.617266	−0.308633	−	0.951181i	$-0.599871\pi$
$383$	−12.0801	−0.617266	−0.308633	+	0.951181i	$0.599871\pi$
$384$	0	0
$385$	−8.14817	−0.415269
$386$	39.2664	1.99861
$387$	0	0
$388$	−1.62499	−0.0824963
$389$	9.74067	0.493872	0.246936	−	0.969032i	$-0.420576\pi$
$389$	9.74067	0.493872	0.246936	+	0.969032i	$0.420576\pi$
$390$	0	0
$391$	−29.6419	−1.49906
$392$	1.31707	0.0665222
$393$	0	0
$394$	44.9307	2.26358
$395$	37.0454	1.86395
$396$	0	0
$397$	−16.9031	−0.848345	−0.424172	−	0.905582i	$-0.639435\pi$
$397$	−16.9031	−0.848345	−0.424172	+	0.905582i	$0.639435\pi$
$398$	−41.7275	−2.09161
$399$	0	0
$400$	−10.0119	−0.500593
$401$	−2.33002	−0.116356	−0.0581778	−	0.998306i	$-0.518529\pi$
$401$	−2.33002	−0.116356	−0.0581778	+	0.998306i	$0.518529\pi$
$402$	0	0
$403$	14.4821	0.721404
$404$	7.27835	0.362111
$405$	0	0
$406$	−4.40310	−0.218522
$407$	8.48252	0.420463
$408$	0	0
$409$	28.7795	1.42305	0.711527	−	0.702658i	$-0.248004\pi$
$409$	28.7795	1.42305	0.711527	+	0.702658i	$0.248004\pi$
$410$	37.6310	1.85846
$411$	0	0
$412$	−11.5279	−0.567940
$413$	31.0510	1.52792
$414$	0	0
$415$	28.2409	1.38629
$416$	−14.0469	−0.688704
$417$	0	0
$418$	4.04126	0.197664
$419$	−5.14746	−0.251470	−0.125735	−	0.992064i	$-0.540129\pi$
$419$	−5.14746	−0.251470	−0.125735	+	0.992064i	$0.540129\pi$
$420$	0	0
$421$	−4.69336	−0.228740	−0.114370	−	0.993438i	$-0.536485\pi$
$421$	−4.69336	−0.228740	−0.114370	+	0.993438i	$0.536485\pi$
$422$	−2.95711	−0.143950
$423$	0	0
$424$	−8.50403	−0.412992
$425$	12.9599	0.628646
$426$	0	0
$427$	−3.03402	−0.146826
$428$	13.9800	0.675748
$429$	0	0
$430$	12.4798	0.601827
$431$	19.2769	0.928537	0.464269	−	0.885694i	$-0.346317\pi$
$431$	19.2769	0.928537	0.464269	+	0.885694i	$0.346317\pi$
$432$	0	0
$433$	1.53389	0.0737141	0.0368571	−	0.999321i	$-0.488265\pi$
$433$	1.53389	0.0737141	0.0368571	+	0.999321i	$0.488265\pi$
$434$	−45.0420	−2.16209
$435$	0	0
$436$	−14.8012	−0.708851
$437$	−10.6473	−0.509331
$438$	0	0
$439$	17.6572	0.842730	0.421365	−	0.906891i	$-0.361551\pi$
$439$	17.6572	0.842730	0.421365	+	0.906891i	$0.361551\pi$
$440$	1.60394	0.0764650
$441$	0	0
$442$	21.0800	1.00267
$443$	−36.3978	−1.72931	−0.864656	−	0.502364i	$-0.832464\pi$
$443$	−36.3978	−1.72931	−0.864656	+	0.502364i	$0.832464\pi$
$444$	0	0
$445$	11.9708	0.567470
$446$	−51.2753	−2.42796
$447$	0	0
$448$	16.2292	0.766760
$449$	41.2603	1.94719	0.973596	−	0.228277i	$-0.0733091\pi$
$449$	41.2603	1.94719	0.973596	+	0.228277i	$0.0733091\pi$
$450$	0	0
$451$	−7.29535	−0.343525
$452$	−35.3694	−1.66364
$453$	0	0
$454$	−0.351198	−0.0164826
$455$	−15.2668	−0.715720
$456$	0	0
$457$	−28.5702	−1.33646	−0.668228	−	0.743956i	$-0.732947\pi$
$457$	−28.5702	−1.33646	−0.668228	+	0.743956i	$0.732947\pi$
$458$	−39.7972	−1.85960
$459$	0	0
$460$	22.9544	1.07025
$461$	36.2422	1.68797	0.843983	−	0.536370i	$-0.180205\pi$
$461$	36.2422	1.68797	0.843983	+	0.536370i	$0.180205\pi$
$462$	0	0
$463$	28.6891	1.33330	0.666648	−	0.745373i	$-0.267729\pi$
$463$	28.6891	1.33330	0.666648	+	0.745373i	$0.267729\pi$
$464$	3.41918	0.158731
$465$	0	0
$466$	−14.1862	−0.657163
$467$	−10.6618	−0.493368	−0.246684	−	0.969096i	$-0.579341\pi$
$467$	−10.6618	−0.493368	−0.246684	+	0.969096i	$0.579341\pi$
$468$	0	0
$469$	22.0717	1.01918
$470$	61.8904	2.85479
$471$	0	0
$472$	−6.11229	−0.281341
$473$	−2.41940	−0.111244
$474$	0	0
$475$	4.65517	0.213594
$476$	−30.0183	−1.37588
$477$	0	0
$478$	−53.3537	−2.44034
$479$	−16.6293	−0.759813	−0.379907	−	0.925025i	$-0.624044\pi$
$479$	−16.6293	−0.759813	−0.379907	+	0.925025i	$0.624044\pi$
$480$	0	0
$481$	15.8933	0.724672
$482$	−13.9897	−0.637211
$483$	0	0
$484$	1.68905	0.0767750
$485$	2.58374	0.117322
$486$	0	0
$487$	−10.4927	−0.475472	−0.237736	−	0.971330i	$-0.576405\pi$
$487$	−10.4927	−0.475472	−0.237736	+	0.971330i	$0.576405\pi$
$488$	0.597238	0.0270357
$489$	0	0
$490$	11.3753	0.513882
$491$	−38.2621	−1.72675	−0.863373	−	0.504567i	$-0.831652\pi$
$491$	−38.2621	−1.72675	−0.863373	+	0.504567i	$0.831652\pi$
$492$	0	0
$493$	−4.42596	−0.199335
$494$	7.57191	0.340676
$495$	0	0
$496$	34.9769	1.57051
$497$	−27.5041	−1.23373
$498$	0	0
$499$	−6.61364	−0.296067	−0.148034	−	0.988982i	$-0.547294\pi$
$499$	−6.61364	−0.296067	−0.148034	+	0.988982i	$0.547294\pi$
$500$	12.6446	0.565483
$501$	0	0
$502$	19.7474	0.881368
$503$	28.3381	1.26353	0.631767	−	0.775158i	$-0.282330\pi$
$503$	28.3381	1.26353	0.631767	+	0.775158i	$0.282330\pi$
$504$	0	0
$505$	−11.5726	−0.514975
$506$	−9.71939	−0.432080
$507$	0	0
$508$	−7.60941	−0.337613
$509$	13.2744	0.588377	0.294188	−	0.955747i	$-0.404951\pi$
$509$	13.2744	0.588377	0.294188	+	0.955747i	$0.404951\pi$
$510$	0	0
$511$	−1.81146	−0.0801344
$512$	−28.5205	−1.26044
$513$	0	0
$514$	−48.3318	−2.13183
$515$	18.3295	0.807693
$516$	0	0
$517$	−11.9984	−0.527690
$518$	−49.4312	−2.17188
$519$	0	0
$520$	3.00523	0.131788
$521$	32.1863	1.41011	0.705053	−	0.709155i	$-0.250923\pi$
$521$	32.1863	1.41011	0.705053	+	0.709155i	$0.250923\pi$
$522$	0	0
$523$	−17.8761	−0.781665	−0.390833	−	0.920462i	$-0.627813\pi$
$523$	−17.8761	−0.781665	−0.390833	+	0.920462i	$0.627813\pi$
$524$	−12.9115	−0.564040
$525$	0	0
$526$	25.2690	1.10178
$527$	−45.2759	−1.97225
$528$	0	0
$529$	2.60728	0.113360
$530$	−73.4474	−3.19035
$531$	0	0
$532$	−10.7825	−0.467481
$533$	−13.6689	−0.592068
$534$	0	0
$535$	−22.2283	−0.961012
$536$	−4.34475	−0.187665
$537$	0	0
$538$	−16.1994	−0.698404
$539$	−2.20527	−0.0949878
$540$	0	0
$541$	6.53672	0.281035	0.140518	−	0.990078i	$-0.455123\pi$
$541$	6.53672	0.281035	0.140518	+	0.990078i	$0.455123\pi$
$542$	22.6367	0.972328
$543$	0	0
$544$	43.9152	1.88285
$545$	23.5341	1.00809
$546$	0	0
$547$	−27.4500	−1.17368	−0.586838	−	0.809704i	$-0.699628\pi$
$547$	−27.4500	−1.17368	−0.586838	+	0.809704i	$0.699628\pi$
$548$	28.7021	1.22609
$549$	0	0
$550$	4.24946	0.181198
$551$	−1.58980	−0.0677277
$552$	0	0
$553$	41.8514	1.77970
$554$	−37.8851	−1.60958
$555$	0	0
$556$	−23.7875	−1.00882
$557$	−10.9835	−0.465387	−0.232693	−	0.972550i	$-0.574754\pi$
$557$	−10.9835	−0.465387	−0.232693	+	0.972550i	$0.574754\pi$
$558$	0	0
$559$	−4.53310	−0.191730
$560$	−36.8722	−1.55813
$561$	0	0
$562$	−11.8946	−0.501743
$563$	31.2089	1.31530	0.657648	−	0.753325i	$-0.271551\pi$
$563$	31.2089	1.31530	0.657648	+	0.753325i	$0.271551\pi$
$564$	0	0
$565$	56.2377	2.36594
$566$	7.67501	0.322605
$567$	0	0
$568$	5.41410	0.227170
$569$	−7.00936	−0.293848	−0.146924	−	0.989148i	$-0.546937\pi$
$569$	−7.00936	−0.293848	−0.146924	+	0.989148i	$0.546937\pi$
$570$	0	0
$571$	−4.18952	−0.175326	−0.0876630	−	0.996150i	$-0.527940\pi$
$571$	−4.18952	−0.175326	−0.0876630	+	0.996150i	$0.527940\pi$
$572$	3.16469	0.132322
$573$	0	0
$574$	42.5130	1.77446
$575$	−11.1959	−0.466900
$576$	0	0
$577$	−12.6938	−0.528451	−0.264226	−	0.964461i	$-0.585116\pi$
$577$	−12.6938	−0.528451	−0.264226	+	0.964461i	$0.585116\pi$
$578$	−33.2514	−1.38308
$579$	0	0
$580$	3.42742	0.142316
$581$	31.9048	1.32363
$582$	0	0
$583$	14.2389	0.589716
$584$	0.356581	0.0147554
$585$	0	0
$586$	1.82877	0.0755458
$587$	16.5586	0.683446	0.341723	−	0.939801i	$-0.388990\pi$
$587$	16.5586	0.683446	0.341723	+	0.939801i	$0.388990\pi$
$588$	0	0
$589$	−16.2630	−0.670107
$590$	−52.7905	−2.17335
$591$	0	0
$592$	38.3852	1.57762
$593$	37.4310	1.53711	0.768553	−	0.639786i	$-0.220977\pi$
$593$	37.4310	1.53711	0.768553	+	0.639786i	$0.220977\pi$
$594$	0	0
$595$	47.7293	1.95671
$596$	4.86954	0.199464
$597$	0	0
$598$	−18.2107	−0.744693
$599$	4.94247	0.201944	0.100972	−	0.994889i	$-0.467805\pi$
$599$	4.94247	0.201944	0.100972	+	0.994889i	$0.467805\pi$
$600$	0	0
$601$	24.9371	1.01721	0.508603	−	0.861001i	$-0.330162\pi$
$601$	24.9371	1.01721	0.508603	+	0.861001i	$0.330162\pi$
$602$	14.0988	0.574625
$603$	0	0
$604$	−4.35031	−0.177012
$605$	−2.68560	−0.109185
$606$	0	0
$607$	1.85692	0.0753700	0.0376850	−	0.999290i	$-0.488002\pi$
$607$	1.85692	0.0753700	0.0376850	+	0.999290i	$0.488002\pi$
$608$	15.7743	0.639732
$609$	0	0
$610$	5.15821	0.208850
$611$	−22.4808	−0.909478
$612$	0	0
$613$	−33.8677	−1.36790	−0.683951	−	0.729528i	$-0.739740\pi$
$613$	−33.8677	−1.36790	−0.683951	+	0.729528i	$0.739740\pi$
$614$	21.8894	0.883384
$615$	0	0
$616$	1.81203	0.0730088
$617$	−11.3565	−0.457194	−0.228597	−	0.973521i	$-0.573414\pi$
$617$	−11.3565	−0.457194	−0.228597	+	0.973521i	$0.573414\pi$
$618$	0	0
$619$	26.5587	1.06748	0.533742	−	0.845647i	$-0.320785\pi$
$619$	26.5587	1.06748	0.533742	+	0.845647i	$0.320785\pi$
$620$	35.0612	1.40809
$621$	0	0
$622$	62.3106	2.49843
$623$	13.5238	0.541821
$624$	0	0
$625$	−31.1673	−1.24669
$626$	−42.6951	−1.70644
$627$	0	0
$628$	−0.950593	−0.0379328
$629$	−49.6878	−1.98118
$630$	0	0
$631$	15.3818	0.612341	0.306171	−	0.951977i	$-0.400952\pi$
$631$	15.3818	0.612341	0.306171	+	0.951977i	$0.400952\pi$
$632$	−8.23833	−0.327703
$633$	0	0
$634$	27.7561	1.10233
$635$	12.0990	0.480135
$636$	0	0
$637$	−4.13191	−0.163712
$638$	−1.45124	−0.0574553
$639$	0	0
$640$	12.6765	0.501081
$641$	14.3153	0.565422	0.282711	−	0.959205i	$-0.408766\pi$
$641$	14.3153	0.565422	0.282711	+	0.959205i	$0.408766\pi$
$642$	0	0
$643$	42.5606	1.67843	0.839213	−	0.543803i	$-0.183016\pi$
$643$	42.5606	1.67843	0.839213	+	0.543803i	$0.183016\pi$
$644$	25.9324	1.02188
$645$	0	0
$646$	−23.6723	−0.931375
$647$	−21.7415	−0.854746	−0.427373	−	0.904075i	$-0.640561\pi$
$647$	−21.7415	−0.854746	−0.427373	+	0.904075i	$0.640561\pi$
$648$	0	0
$649$	10.2343	0.401730
$650$	7.96200	0.312295
$651$	0	0
$652$	−0.559332	−0.0219051
$653$	−16.1812	−0.633218	−0.316609	−	0.948556i	$-0.602544\pi$
$653$	−16.1812	−0.633218	−0.316609	+	0.948556i	$0.602544\pi$
$654$	0	0
$655$	20.5293	0.802147
$656$	−33.0130	−1.28894
$657$	0	0
$658$	69.9197	2.72575
$659$	31.7172	1.23553	0.617763	−	0.786365i	$-0.288039\pi$
$659$	31.7172	1.23553	0.617763	+	0.786365i	$0.288039\pi$
$660$	0	0
$661$	49.8447	1.93873	0.969367	−	0.245618i	$-0.0789908\pi$
$661$	49.8447	1.93873	0.969367	+	0.245618i	$0.0789908\pi$
$662$	25.6297	0.996126
$663$	0	0
$664$	−6.28036	−0.243725
$665$	17.1443	0.664827
$666$	0	0
$667$	3.82353	0.148048
$668$	−26.7590	−1.03534
$669$	0	0
$670$	−37.5247	−1.44970
$671$	−1.00000	−0.0386046
$672$	0	0
$673$	21.1094	0.813708	0.406854	−	0.913493i	$-0.366626\pi$
$673$	21.1094	0.813708	0.406854	+	0.913493i	$0.366626\pi$
$674$	−13.4790	−0.519191
$675$	0	0
$676$	−16.0281	−0.616467
$677$	−13.7572	−0.528733	−0.264367	−	0.964422i	$-0.585163\pi$
$677$	−13.7572	−0.528733	−0.264367	+	0.964422i	$0.585163\pi$
$678$	0	0
$679$	2.91895	0.112019
$680$	−9.39537	−0.360296
$681$	0	0
$682$	−14.8457	−0.568470
$683$	−15.2682	−0.584221	−0.292111	−	0.956385i	$-0.594358\pi$
$683$	−15.2682	−0.584221	−0.292111	+	0.956385i	$0.594358\pi$
$684$	0	0
$685$	−45.6366	−1.74368
$686$	−27.9408	−1.06679
$687$	0	0
$688$	−10.9483	−0.417399
$689$	26.6788	1.01638
$690$	0	0
$691$	−43.1847	−1.64282	−0.821412	−	0.570335i	$-0.806813\pi$
$691$	−43.1847	−1.64282	−0.821412	+	0.570335i	$0.806813\pi$
$692$	−0.953402	−0.0362429
$693$	0	0
$694$	−30.7064	−1.16560
$695$	37.8224	1.43468
$696$	0	0
$697$	42.7337	1.61866
$698$	23.1804	0.877390
$699$	0	0
$700$	−11.3380	−0.428537
$701$	20.5633	0.776665	0.388333	−	0.921519i	$-0.373051\pi$
$701$	20.5633	0.776665	0.388333	+	0.921519i	$0.373051\pi$
$702$	0	0
$703$	−17.8478	−0.673142
$704$	5.34909	0.201601
$705$	0	0
$706$	−19.6158	−0.738249
$707$	−13.0740	−0.491699
$708$	0	0
$709$	5.99904	0.225299	0.112649	−	0.993635i	$-0.464066\pi$
$709$	5.99904	0.225299	0.112649	+	0.993635i	$0.464066\pi$
$710$	46.7604	1.75489
$711$	0	0
$712$	−2.66213	−0.0997674
$713$	39.1133	1.46480
$714$	0	0
$715$	−5.03188	−0.188182
$716$	33.4848	1.25139
$717$	0	0
$718$	−11.3777	−0.424613
$719$	−24.7195	−0.921880	−0.460940	−	0.887431i	$-0.652488\pi$
$719$	−24.7195	−0.921880	−0.460940	+	0.887431i	$0.652488\pi$
$720$	0	0
$721$	20.7074	0.771186
$722$	27.9900	1.04168
$723$	0	0
$724$	−7.29011	−0.270935
$725$	−1.67170	−0.0620855
$726$	0	0
$727$	35.6585	1.32250	0.661250	−	0.750166i	$-0.270026\pi$
$727$	35.6585	1.32250	0.661250	+	0.750166i	$0.270026\pi$
$728$	3.39511	0.125831
$729$	0	0
$730$	3.07971	0.113985
$731$	14.1720	0.524171
$732$	0	0
$733$	−32.8848	−1.21463	−0.607314	−	0.794462i	$-0.707753\pi$
$733$	−32.8848	−1.21463	−0.607314	+	0.794462i	$0.707753\pi$
$734$	17.5921	0.649338
$735$	0	0
$736$	−37.9378	−1.39841
$737$	7.27474	0.267969
$738$	0	0
$739$	33.2300	1.22238	0.611192	−	0.791482i	$-0.290690\pi$
$739$	33.2300	1.22238	0.611192	+	0.791482i	$0.290690\pi$
$740$	38.4777	1.41447
$741$	0	0
$742$	−82.9761	−3.04615
$743$	9.11787	0.334502	0.167251	−	0.985914i	$-0.446511\pi$
$743$	9.11787	0.334502	0.167251	+	0.985914i	$0.446511\pi$
$744$	0	0
$745$	−7.74260	−0.283667
$746$	−5.73280	−0.209893
$747$	0	0
$748$	−9.89389	−0.361756
$749$	−25.1121	−0.917575
$750$	0	0
$751$	16.3363	0.596121	0.298060	−	0.954547i	$-0.403660\pi$
$751$	16.3363	0.596121	0.298060	+	0.954547i	$0.403660\pi$
$752$	−54.2954	−1.97995
$753$	0	0
$754$	−2.71912	−0.0990246
$755$	6.91703	0.251736
$756$	0	0
$757$	−35.9268	−1.30578	−0.652890	−	0.757453i	$-0.726444\pi$
$757$	−35.9268	−1.30578	−0.652890	+	0.757453i	$0.726444\pi$
$758$	−43.4688	−1.57886
$759$	0	0
$760$	−3.37480	−0.122417
$761$	44.8610	1.62621	0.813106	−	0.582116i	$-0.197775\pi$
$761$	44.8610	1.62621	0.813106	+	0.582116i	$0.197775\pi$
$762$	0	0
$763$	26.5873	0.962524
$764$	−39.4174	−1.42607
$765$	0	0
$766$	23.2022	0.838329
$767$	19.1754	0.692385
$768$	0	0
$769$	41.9041	1.51110	0.755551	−	0.655090i	$-0.227369\pi$
$769$	41.9041	1.51110	0.755551	+	0.655090i	$0.227369\pi$
$770$	15.6501	0.563991
$771$	0	0
$772$	−34.5308	−1.24279
$773$	3.23022	0.116183	0.0580914	−	0.998311i	$-0.481499\pi$
$773$	3.23022	0.116183	0.0580914	+	0.998311i	$0.481499\pi$
$774$	0	0
$775$	−17.1009	−0.614282
$776$	−0.574586	−0.0206264
$777$	0	0
$778$	−18.7088	−0.670743
$779$	15.3499	0.549967
$780$	0	0
$781$	−9.06523	−0.324379
$782$	56.9329	2.03592
$783$	0	0
$784$	−9.97932	−0.356404
$785$	1.51145	0.0539459
$786$	0	0
$787$	52.3947	1.86767	0.933835	−	0.357704i	$-0.116440\pi$
$787$	52.3947	1.86767	0.933835	+	0.357704i	$0.116440\pi$
$788$	−39.5120	−1.40755
$789$	0	0
$790$	−71.1527	−2.53150
$791$	63.5337	2.25900
$792$	0	0
$793$	−1.87365	−0.0665353
$794$	32.4657	1.15216
$795$	0	0
$796$	36.6951	1.30062
$797$	56.0821	1.98653	0.993265	−	0.115863i	$-0.0369632\pi$
$797$	56.0821	1.98653	0.993265	+	0.115863i	$0.0369632\pi$
$798$	0	0
$799$	70.2827	2.48642
$800$	16.5870	0.586437
$801$	0	0
$802$	4.47524	0.158026
$803$	−0.597050	−0.0210694
$804$	0	0
$805$	−41.2327	−1.45326
$806$	−27.8156	−0.979762
$807$	0	0
$808$	2.57358	0.0905382
$809$	11.6450	0.409418	0.204709	−	0.978823i	$-0.434375\pi$
$809$	11.6450	0.409418	0.204709	+	0.978823i	$0.434375\pi$
$810$	0	0
$811$	−27.1242	−0.952460	−0.476230	−	0.879321i	$-0.657997\pi$
$811$	−27.1242	−0.952460	−0.476230	+	0.879321i	$0.657997\pi$
$812$	3.87208	0.135883
$813$	0	0
$814$	−16.2923	−0.571045
$815$	0.889342	0.0311523
$816$	0	0
$817$	5.09057	0.178096
$818$	−55.2765	−1.93270
$819$	0	0
$820$	−33.0926	−1.15564
$821$	−26.8199	−0.936021	−0.468010	−	0.883723i	$-0.655029\pi$
$821$	−26.8199	−0.936021	−0.468010	+	0.883723i	$0.655029\pi$
$822$	0	0
$823$	33.1761	1.15645	0.578223	−	0.815879i	$-0.303746\pi$
$823$	33.1761	1.15645	0.578223	+	0.815879i	$0.303746\pi$
$824$	−4.07620	−0.142001
$825$	0	0
$826$	−59.6393	−2.07512
$827$	−23.1367	−0.804540	−0.402270	−	0.915521i	$-0.631779\pi$
$827$	−23.1367	−0.804540	−0.402270	+	0.915521i	$0.631779\pi$
$828$	0	0
$829$	13.7877	0.478867	0.239434	−	0.970913i	$-0.423038\pi$
$829$	13.7877	0.478867	0.239434	+	0.970913i	$0.423038\pi$
$830$	−54.2421	−1.88277
$831$	0	0
$832$	10.0223	0.347462
$833$	12.9178	0.447574
$834$	0	0
$835$	42.5471	1.47240
$836$	−3.55387	−0.122913
$837$	0	0
$838$	9.88668	0.341530
$839$	4.88835	0.168765	0.0843824	−	0.996433i	$-0.473108\pi$
$839$	4.88835	0.168765	0.0843824	+	0.996433i	$0.473108\pi$
$840$	0	0
$841$	−28.4291	−0.980314
$842$	9.01449	0.310660
$843$	0	0
$844$	2.60048	0.0895122
$845$	25.4849	0.876706
$846$	0	0
$847$	−3.03402	−0.104250
$848$	64.4341	2.21268
$849$	0	0
$850$	−24.8919	−0.853785
$851$	42.9247	1.47144
$852$	0	0
$853$	−3.90084	−0.133562	−0.0667811	−	0.997768i	$-0.521273\pi$
$853$	−3.90084	−0.133562	−0.0667811	+	0.997768i	$0.521273\pi$
$854$	5.82741	0.199410
$855$	0	0
$856$	4.94324	0.168956
$857$	−26.5893	−0.908273	−0.454136	−	0.890932i	$-0.650052\pi$
$857$	−26.5893	−0.908273	−0.454136	+	0.890932i	$0.650052\pi$
$858$	0	0
$859$	−32.9465	−1.12412	−0.562060	−	0.827096i	$-0.689991\pi$
$859$	−32.9465	−1.12412	−0.562060	+	0.827096i	$0.689991\pi$
$860$	−10.9747	−0.374233
$861$	0	0
$862$	−37.0250	−1.26108
$863$	−30.0640	−1.02339	−0.511696	−	0.859167i	$-0.670982\pi$
$863$	−30.0640	−1.02339	−0.511696	+	0.859167i	$0.670982\pi$
$864$	0	0
$865$	1.51592	0.0515427
$866$	−2.94613	−0.100114
$867$	0	0
$868$	39.6099	1.34445
$869$	13.7941	0.467931
$870$	0	0
$871$	13.6303	0.461846
$872$	−5.23363	−0.177233
$873$	0	0
$874$	20.4502	0.691739
$875$	−22.7133	−0.767850
$876$	0	0
$877$	−43.5385	−1.47019	−0.735096	−	0.677963i	$-0.762863\pi$
$877$	−43.5385	−1.47019	−0.735096	+	0.677963i	$0.762863\pi$
$878$	−33.9139	−1.14454
$879$	0	0
$880$	−12.1529	−0.409675
$881$	−30.7647	−1.03649	−0.518245	−	0.855232i	$-0.673415\pi$
$881$	−30.7647	−1.03649	−0.518245	+	0.855232i	$0.673415\pi$
$882$	0	0
$883$	49.2278	1.65665	0.828324	−	0.560249i	$-0.189294\pi$
$883$	49.2278	1.65665	0.828324	+	0.560249i	$0.189294\pi$
$884$	−18.5377	−0.623490
$885$	0	0
$886$	69.9089	2.34864
$887$	51.0375	1.71367	0.856836	−	0.515590i	$-0.172427\pi$
$887$	51.0375	1.71367	0.856836	+	0.515590i	$0.172427\pi$
$888$	0	0
$889$	13.6687	0.458433
$890$	−22.9922	−0.770700
$891$	0	0
$892$	45.0914	1.50977
$893$	25.2455	0.844807
$894$	0	0
$895$	−53.2411	−1.77965
$896$	14.3210	0.478432
$897$	0	0
$898$	−79.2482	−2.64455
$899$	5.84017	0.194780
$900$	0	0
$901$	−83.4069	−2.77869
$902$	14.0121	0.466552
$903$	0	0
$904$	−12.5064	−0.415957
$905$	11.5913	0.385309
$906$	0	0
$907$	55.9556	1.85798	0.928988	−	0.370109i	$-0.120680\pi$
$907$	55.9556	1.85798	0.928988	+	0.370109i	$0.120680\pi$
$908$	0.308843	0.0102493
$909$	0	0
$910$	29.3228	0.972043
$911$	19.7794	0.655322	0.327661	−	0.944795i	$-0.393740\pi$
$911$	19.7794	0.655322	0.327661	+	0.944795i	$0.393740\pi$
$912$	0	0
$913$	10.5157	0.348018
$914$	54.8745	1.81509
$915$	0	0
$916$	34.9976	1.15635
$917$	23.1927	0.765890
$918$	0	0
$919$	11.5928	0.382411	0.191206	−	0.981550i	$-0.438760\pi$
$919$	11.5928	0.382411	0.191206	+	0.981550i	$0.438760\pi$
$920$	8.11654	0.267594
$921$	0	0
$922$	−69.6100	−2.29248
$923$	−16.9851	−0.559071
$924$	0	0
$925$	−18.7673	−0.617065
$926$	−55.1029	−1.81079
$927$	0	0
$928$	−5.66465	−0.185951
$929$	−4.84739	−0.159038	−0.0795189	−	0.996833i	$-0.525338\pi$
$929$	−4.84739	−0.159038	−0.0795189	+	0.996833i	$0.525338\pi$
$930$	0	0
$931$	4.64004	0.152071
$932$	12.4753	0.408642
$933$	0	0
$934$	20.4780	0.670059
$935$	15.7314	0.514471
$936$	0	0
$937$	42.2635	1.38069	0.690344	−	0.723481i	$-0.257459\pi$
$937$	42.2635	1.38069	0.690344	+	0.723481i	$0.257459\pi$
$938$	−42.3929	−1.38418
$939$	0	0
$940$	−54.4263	−1.77519
$941$	36.1693	1.17908	0.589542	−	0.807737i	$-0.299308\pi$
$941$	36.1693	1.17908	0.589542	+	0.807737i	$0.299308\pi$
$942$	0	0
$943$	−36.9171	−1.20219
$944$	46.3122	1.50733
$945$	0	0
$946$	4.64691	0.151084
$947$	−8.11503	−0.263703	−0.131852	−	0.991269i	$-0.542092\pi$
$947$	−8.11503	−0.263703	−0.131852	+	0.991269i	$0.542092\pi$
$948$	0	0
$949$	−1.11866	−0.0363134
$950$	−8.94114	−0.290089
$951$	0	0
$952$	−10.6143	−0.344010
$953$	−15.4660	−0.500994	−0.250497	−	0.968117i	$-0.580594\pi$
$953$	−15.4660	−0.500994	−0.250497	+	0.968117i	$0.580594\pi$
$954$	0	0
$955$	62.6739	2.02808
$956$	46.9191	1.51747
$957$	0	0
$958$	31.9398	1.03193
$959$	−51.5572	−1.66487
$960$	0	0
$961$	28.7427	0.927183
$962$	−30.5261	−0.984200
$963$	0	0
$964$	12.3025	0.396236
$965$	54.9042	1.76743
$966$	0	0
$967$	−27.5152	−0.884830	−0.442415	−	0.896811i	$-0.645878\pi$
$967$	−27.5152	−0.884830	−0.442415	+	0.896811i	$0.645878\pi$
$968$	0.597238	0.0191960
$969$	0	0
$970$	−4.96257	−0.159339
$971$	−27.3359	−0.877251	−0.438626	−	0.898670i	$-0.644535\pi$
$971$	−27.3359	−0.877251	−0.438626	+	0.898670i	$0.644535\pi$
$972$	0	0
$973$	42.7292	1.36984
$974$	20.1533	0.645754
$975$	0	0
$976$	−4.52521	−0.144848
$977$	50.8680	1.62741	0.813706	−	0.581277i	$-0.197447\pi$
$977$	50.8680	1.62741	0.813706	+	0.581277i	$0.197447\pi$
$978$	0	0
$979$	4.45740	0.142459
$980$	−10.0034	−0.319546
$981$	0	0
$982$	73.4897	2.34515
$983$	−61.6949	−1.96776	−0.983882	−	0.178821i	$-0.942772\pi$
$983$	−61.6949	−1.96776	−0.983882	+	0.178821i	$0.942772\pi$
$984$	0	0
$985$	62.8243	2.00175
$986$	8.50089	0.270724
$987$	0	0
$988$	−6.65872	−0.211842
$989$	−12.2430	−0.389306
$990$	0	0
$991$	−7.12464	−0.226322	−0.113161	−	0.993577i	$-0.536098\pi$
$991$	−7.12464	−0.226322	−0.113161	+	0.993577i	$0.536098\pi$
$992$	−57.9472	−1.83983
$993$	0	0
$994$	52.8268	1.67556
$995$	−58.3454	−1.84967
$996$	0	0
$997$	−40.8695	−1.29435	−0.647175	−	0.762342i	$-0.724050\pi$
$997$	−40.8695	−1.29435	−0.647175	+	0.762342i	$0.724050\pi$
$998$	12.7028	0.402099
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6039.2.a.p.1.4	yes	25
3.2	odd	2		6039.2.a.m.1.22	✓	25

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6039.2.a.m.1.22	✓	25	3.2	odd	2
6039.2.a.p.1.4	yes	25	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)

\(f(q)\)	\(=\)	\(q-1.92069 q^{2} +1.68905 q^{4} -2.68560 q^{5} -3.03402 q^{7} +0.597238 q^{8} +O(q^{10})\)	\(q-1.92069 q^{2} +1.68905 q^{4} -2.68560 q^{5} -3.03402 q^{7} +0.597238 q^{8} +5.15821 q^{10} -1.00000 q^{11} -1.87365 q^{13} +5.82741 q^{14} -4.52521 q^{16} +5.85766 q^{17} +2.10407 q^{19} -4.53612 q^{20} +1.92069 q^{22} -5.06036 q^{23} +2.21246 q^{25} +3.59870 q^{26} -5.12461 q^{28} -0.755584 q^{29} -7.72934 q^{31} +7.49705 q^{32} -11.2508 q^{34} +8.14817 q^{35} -8.48252 q^{37} -4.04126 q^{38} -1.60394 q^{40} +7.29535 q^{41} +2.41940 q^{43} -1.68905 q^{44} +9.71939 q^{46} +11.9984 q^{47} +2.20527 q^{49} -4.24946 q^{50} -3.16469 q^{52} -14.2389 q^{53} +2.68560 q^{55} -1.81203 q^{56} +1.45124 q^{58} -10.2343 q^{59} +1.00000 q^{61} +14.8457 q^{62} -5.34909 q^{64} +5.03188 q^{65} -7.27474 q^{67} +9.89389 q^{68} -15.6501 q^{70} +9.06523 q^{71} +0.597050 q^{73} +16.2923 q^{74} +3.55387 q^{76} +3.03402 q^{77} -13.7941 q^{79} +12.1529 q^{80} -14.0121 q^{82} -10.5157 q^{83} -15.7314 q^{85} -4.64691 q^{86} -0.597238 q^{88} -4.45740 q^{89} +5.68469 q^{91} -8.54721 q^{92} -23.0452 q^{94} -5.65068 q^{95} -0.962072 q^{97} -4.23565 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 25 q + 5 q^{2} + 25 q^{4} + 12 q^{5} - 4 q^{7} + 15 q^{8}+O(q^{10}) \) 25 * q + 5 * q^2 + 25 * q^4 + 12 * q^5 - 4 * q^7 + 15 * q^8	\( 25 q + 5 q^{2} + 25 q^{4} + 12 q^{5} - 4 q^{7} + 15 q^{8} - 12 q^{10} - 25 q^{11} - 4 q^{13} + 14 q^{14} + 21 q^{16} + 16 q^{17} - 18 q^{19} + 28 q^{20} - 5 q^{22} + 8 q^{23} + 29 q^{25} + 16 q^{26} + 18 q^{28} + 28 q^{29} - 8 q^{31} + 35 q^{32} + 6 q^{34} + 22 q^{35} + 4 q^{37} - 4 q^{38} - 12 q^{40} + 58 q^{41} - 26 q^{43} - 25 q^{44} + 8 q^{46} + 20 q^{47} + 23 q^{49} + 27 q^{50} - 2 q^{52} + 36 q^{53} - 12 q^{55} + 70 q^{56} + 12 q^{58} + 18 q^{59} + 25 q^{61} + 42 q^{62} + 35 q^{64} + 76 q^{65} - 8 q^{67} + 28 q^{68} + 76 q^{70} + 24 q^{71} + 2 q^{73} + 40 q^{74} - 64 q^{76} + 4 q^{77} - 22 q^{79} + 36 q^{80} + 30 q^{82} + 14 q^{83} + 70 q^{86} - 15 q^{88} + 82 q^{89} - 6 q^{91} + 48 q^{92} - 16 q^{94} + 34 q^{95} + 16 q^{97} + 35 q^{98}+O(q^{100}) \) 25 * q + 5 * q^2 + 25 * q^4 + 12 * q^5 - 4 * q^7 + 15 * q^8 - 12 * q^10 - 25 * q^11 - 4 * q^13 + 14 * q^14 + 21 * q^16 + 16 * q^17 - 18 * q^19 + 28 * q^20 - 5 * q^22 + 8 * q^23 + 29 * q^25 + 16 * q^26 + 18 * q^28 + 28 * q^29 - 8 * q^31 + 35 * q^32 + 6 * q^34 + 22 * q^35 + 4 * q^37 - 4 * q^38 - 12 * q^40 + 58 * q^41 - 26 * q^43 - 25 * q^44 + 8 * q^46 + 20 * q^47 + 23 * q^49 + 27 * q^50 - 2 * q^52 + 36 * q^53 - 12 * q^55 + 70 * q^56 + 12 * q^58 + 18 * q^59 + 25 * q^61 + 42 * q^62 + 35 * q^64 + 76 * q^65 - 8 * q^67 + 28 * q^68 + 76 * q^70 + 24 * q^71 + 2 * q^73 + 40 * q^74 - 64 * q^76 + 4 * q^77 - 22 * q^79 + 36 * q^80 + 30 * q^82 + 14 * q^83 + 70 * q^86 - 15 * q^88 + 82 * q^89 - 6 * q^91 + 48 * q^92 - 16 * q^94 + 34 * q^95 + 16 * q^97 + 35 * q^98

Introduction

L-functions

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