LMFDB - Embedded newform 6039.2.a.f.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6039,2,Mod(1,6039)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6039, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("6039.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$48.2216577807$
Analytic rank:	$0$
Dimension:	$12$
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			1.5
Root			$1.10918$ 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)

$f(q)$	$=$	$q-0.109182 q^{2} -1.98808 q^{4} -0.0420221 q^{5} -3.35497 q^{7} +0.435426 q^{8} +O(q^{10})$	\(q-0.109182 q^{2} -1.98808 q^{4} -0.0420221 q^{5} -3.35497 q^{7} +0.435426 q^{8} +0.00458805 q^{10} -1.00000 q^{11} +3.70170 q^{13} +0.366302 q^{14} +3.92862 q^{16} -3.42925 q^{17} -2.07415 q^{19} +0.0835432 q^{20} +0.109182 q^{22} +3.46809 q^{23} -4.99823 q^{25} -0.404159 q^{26} +6.66995 q^{28} +3.55498 q^{29} -6.55374 q^{31} -1.29979 q^{32} +0.374412 q^{34} +0.140983 q^{35} +4.86670 q^{37} +0.226459 q^{38} -0.0182975 q^{40} -8.79360 q^{41} -12.4442 q^{43} +1.98808 q^{44} -0.378653 q^{46} +4.63928 q^{47} +4.25583 q^{49} +0.545716 q^{50} -7.35928 q^{52} -2.75640 q^{53} +0.0420221 q^{55} -1.46084 q^{56} -0.388139 q^{58} +15.2446 q^{59} -1.00000 q^{61} +0.715549 q^{62} -7.71532 q^{64} -0.155553 q^{65} -15.9254 q^{67} +6.81761 q^{68} -0.0153928 q^{70} -7.79929 q^{71} -10.5801 q^{73} -0.531355 q^{74} +4.12357 q^{76} +3.35497 q^{77} -2.91415 q^{79} -0.165089 q^{80} +0.960102 q^{82} +11.8437 q^{83} +0.144104 q^{85} +1.35868 q^{86} -0.435426 q^{88} +5.47005 q^{89} -12.4191 q^{91} -6.89484 q^{92} -0.506525 q^{94} +0.0871601 q^{95} +11.5423 q^{97} -0.464660 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−0.109182	−0.0772032	−0.0386016	−	0.999255i	$-0.512290\pi$
$2$	−0.109182	−0.0772032	−0.0386016	+	0.999255i	$0.512290\pi$
$3$	0	0
$3$	0	0
$4$	−1.98808	−0.994040
$5$	−0.0420221	−0.0187928	−0.00939642	−	0.999956i	$-0.502991\pi$
$5$	−0.0420221	−0.0187928	−0.00939642	+	0.999956i	$0.502991\pi$
$6$	0	0
$7$	−3.35497	−1.26806	−0.634030	−	0.773308i	$-0.718600\pi$
$7$	−3.35497	−1.26806	−0.634030	+	0.773308i	$0.718600\pi$
$8$	0.435426	0.153946
$9$	0	0
$10$	0.00458805	0.00145087
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	3.70170	1.02667	0.513334	−	0.858189i	$-0.328410\pi$
$13$	3.70170	1.02667	0.513334	+	0.858189i	$0.328410\pi$
$14$	0.366302	0.0978983
$15$	0	0
$16$	3.92862	0.982155
$17$	−3.42925	−0.831715	−0.415857	−	0.909430i	$-0.636518\pi$
$17$	−3.42925	−0.831715	−0.415857	+	0.909430i	$0.636518\pi$
$18$	0	0
$19$	−2.07415	−0.475843	−0.237921	−	0.971284i	$-0.576466\pi$
$19$	−2.07415	−0.475843	−0.237921	+	0.971284i	$0.576466\pi$
$20$	0.0835432	0.0186808
$21$	0	0
$22$	0.109182	0.0232776
$23$	3.46809	0.723147	0.361573	−	0.932344i	$-0.382240\pi$
$23$	3.46809	0.723147	0.361573	+	0.932344i	$0.382240\pi$
$24$	0	0
$25$	−4.99823	−0.999647
$26$	−0.404159	−0.0792621
$27$	0	0
$28$	6.66995	1.26050
$29$	3.55498	0.660143	0.330071	−	0.943956i	$-0.392927\pi$
$29$	3.55498	0.660143	0.330071	+	0.943956i	$0.392927\pi$
$30$	0	0
$31$	−6.55374	−1.17709	−0.588543	−	0.808466i	$-0.700298\pi$
$31$	−6.55374	−1.17709	−0.588543	+	0.808466i	$0.700298\pi$
$32$	−1.29979	−0.229772
$33$	0	0
$34$	0.374412	0.0642110
$35$	0.140983	0.0238304
$36$	0	0
$37$	4.86670	0.800081	0.400040	−	0.916498i	$-0.368996\pi$
$37$	4.86670	0.800081	0.400040	+	0.916498i	$0.368996\pi$
$38$	0.226459	0.0367366
$39$	0	0
$40$	−0.0182975	−0.00289309
$41$	−8.79360	−1.37333	−0.686665	−	0.726974i	$-0.740926\pi$
$41$	−8.79360	−1.37333	−0.686665	+	0.726974i	$0.740926\pi$
$42$	0	0
$43$	−12.4442	−1.89772	−0.948862	−	0.315691i	$-0.897764\pi$
$43$	−12.4442	−1.89772	−0.948862	+	0.315691i	$0.897764\pi$
$44$	1.98808	0.299714
$45$	0	0
$46$	−0.378653	−0.0558293
$47$	4.63928	0.676708	0.338354	−	0.941019i	$-0.390130\pi$
$47$	4.63928	0.676708	0.338354	+	0.941019i	$0.390130\pi$
$48$	0	0
$49$	4.25583	0.607976
$50$	0.545716	0.0771760
$51$	0	0
$52$	−7.35928	−1.02055
$53$	−2.75640	−0.378620	−0.189310	−	0.981917i	$-0.560625\pi$
$53$	−2.75640	−0.378620	−0.189310	+	0.981917i	$0.560625\pi$
$54$	0	0
$55$	0.0420221	0.00566625
$56$	−1.46084	−0.195213
$57$	0	0
$58$	−0.388139	−0.0509651
$59$	15.2446	1.98468	0.992342	−	0.123523i	$-0.0394194\pi$
$59$	15.2446	1.98468	0.992342	+	0.123523i	$0.0394194\pi$
$60$	0	0
$61$	−1.00000	−0.128037
$61$	−1.00000	−0.128037
$62$	0.715549	0.0908748
$63$	0	0
$64$	−7.71532	−0.964415
$65$	−0.155553	−0.0192940
$66$	0	0
$67$	−15.9254	−1.94559	−0.972797	−	0.231657i	$-0.925585\pi$
$67$	−15.9254	−1.94559	−0.972797	+	0.231657i	$0.925585\pi$
$68$	6.81761	0.826757
$69$	0	0
$70$	−0.0153928	−0.00183979
$71$	−7.79929	−0.925605	−0.462802	−	0.886461i	$-0.653156\pi$
$71$	−7.79929	−0.925605	−0.462802	+	0.886461i	$0.653156\pi$
$72$	0	0
$73$	−10.5801	−1.23831	−0.619153	−	0.785270i	$-0.712524\pi$
$73$	−10.5801	−1.23831	−0.619153	+	0.785270i	$0.712524\pi$
$74$	−0.531355	−0.0617688
$75$	0	0
$76$	4.12357	0.473006
$77$	3.35497	0.382334
$78$	0	0
$79$	−2.91415	−0.327868	−0.163934	−	0.986471i	$-0.552418\pi$
$79$	−2.91415	−0.327868	−0.163934	+	0.986471i	$0.552418\pi$
$80$	−0.165089	−0.0184575
$81$	0	0
$82$	0.960102	0.106025
$83$	11.8437	1.30002	0.650009	−	0.759927i	$-0.274765\pi$
$83$	11.8437	1.30002	0.650009	+	0.759927i	$0.274765\pi$
$84$	0	0
$85$	0.144104	0.0156303
$86$	1.35868	0.146510
$87$	0	0
$88$	−0.435426	−0.0464166
$89$	5.47005	0.579824	0.289912	−	0.957053i	$-0.406374\pi$
$89$	5.47005	0.579824	0.289912	+	0.957053i	$0.406374\pi$
$90$	0	0
$91$	−12.4191	−1.30188
$92$	−6.89484	−0.718837
$93$	0	0
$94$	−0.506525	−0.0522441
$95$	0.0871601	0.00894243
$96$	0	0
$97$	11.5423	1.17194	0.585972	−	0.810331i	$-0.300713\pi$
$97$	11.5423	1.17194	0.585972	+	0.810331i	$0.300713\pi$
$98$	−0.464660	−0.0469377
$99$	0	0
$100$	9.93689	0.993689
$101$	17.8919	1.78031	0.890154	−	0.455659i	$-0.150597\pi$
$101$	17.8919	1.78031	0.890154	+	0.455659i	$0.150597\pi$
$102$	0	0
$103$	12.7688	1.25815	0.629075	−	0.777345i	$-0.283434\pi$
$103$	12.7688	1.25815	0.629075	+	0.777345i	$0.283434\pi$
$104$	1.61182	0.158052
$105$	0	0
$106$	0.300949	0.0292307
$107$	−11.5184	−1.11352	−0.556762	−	0.830672i	$-0.687956\pi$
$107$	−11.5184	−1.11352	−0.556762	+	0.830672i	$0.687956\pi$
$108$	0	0
$109$	−11.9346	−1.14313	−0.571565	−	0.820556i	$-0.693664\pi$
$109$	−11.9346	−1.14313	−0.571565	+	0.820556i	$0.693664\pi$
$110$	−0.00458805	−0.000437453 0
$111$	0	0
$112$	−13.1804	−1.24543
$113$	0.101743	0.00957116	0.00478558	−	0.999989i	$-0.498477\pi$
$113$	0.101743	0.00957116	0.00478558	+	0.999989i	$0.498477\pi$
$114$	0	0
$115$	−0.145736	−0.0135900
$116$	−7.06758	−0.656208
$117$	0	0
$118$	−1.66444	−0.153224
$119$	11.5050	1.05466
$120$	0	0
$121$	1.00000	0.0909091
$122$	0.109182	0.00988486
$123$	0	0
$124$	13.0293	1.17007
$125$	0.420146	0.0375790
$126$	0	0
$127$	6.00768	0.533095	0.266548	−	0.963822i	$-0.414117\pi$
$127$	6.00768	0.533095	0.266548	+	0.963822i	$0.414117\pi$
$128$	3.44194	0.304228
$129$	0	0
$130$	0.0169836	0.00148956
$131$	−8.35659	−0.730119	−0.365059	−	0.930984i	$-0.618951\pi$
$131$	−8.35659	−0.730119	−0.365059	+	0.930984i	$0.618951\pi$
$132$	0	0
$133$	6.95871	0.603397
$134$	1.73876	0.150206
$135$	0	0
$136$	−1.49318	−0.128039
$137$	−12.4806	−1.06629	−0.533144	−	0.846025i	$-0.678990\pi$
$137$	−12.4806	−1.06629	−0.533144	+	0.846025i	$0.678990\pi$
$138$	0	0
$139$	11.9702	1.01530	0.507651	−	0.861563i	$-0.330514\pi$
$139$	11.9702	1.01530	0.507651	+	0.861563i	$0.330514\pi$
$140$	−0.280285	−0.0236884
$141$	0	0
$142$	0.851540	0.0714597
$143$	−3.70170	−0.309552
$144$	0	0
$145$	−0.149387	−0.0124060
$146$	1.15515	0.0956012
$147$	0	0
$148$	−9.67539	−0.795312
$149$	2.26209	0.185317	0.0926587	−	0.995698i	$-0.470463\pi$
$149$	2.26209	0.185317	0.0926587	+	0.995698i	$0.470463\pi$
$150$	0	0
$151$	−3.10607	−0.252768	−0.126384	−	0.991981i	$-0.540337\pi$
$151$	−3.10607	−0.252768	−0.126384	+	0.991981i	$0.540337\pi$
$152$	−0.903138	−0.0732542
$153$	0	0
$154$	−0.366302	−0.0295175
$155$	0.275402	0.0221208
$156$	0	0
$157$	18.7608	1.49727	0.748637	−	0.662980i	$-0.230709\pi$
$157$	18.7608	1.49727	0.748637	+	0.662980i	$0.230709\pi$
$158$	0.318173	0.0253125
$159$	0	0
$160$	0.0546197	0.00431806
$161$	−11.6353	−0.916993
$162$	0	0
$163$	−8.35781	−0.654634	−0.327317	−	0.944915i	$-0.606145\pi$
$163$	−8.35781	−0.654634	−0.327317	+	0.944915i	$0.606145\pi$
$164$	17.4824	1.36514
$165$	0	0
$166$	−1.29312	−0.100366
$167$	17.6636	1.36685	0.683425	−	0.730021i	$-0.260490\pi$
$167$	17.6636	1.36685	0.683425	+	0.730021i	$0.260490\pi$
$168$	0	0
$169$	0.702619	0.0540476
$170$	−0.0157335	−0.00120671
$171$	0	0
$172$	24.7401	1.88641
$173$	11.3812	0.865293	0.432647	−	0.901564i	$-0.357580\pi$
$173$	11.3812	0.865293	0.432647	+	0.901564i	$0.357580\pi$
$174$	0	0
$175$	16.7689	1.26761
$176$	−3.92862	−0.296131
$177$	0	0
$178$	−0.597231	−0.0447643
$179$	0.885576	0.0661911	0.0330955	−	0.999452i	$-0.489463\pi$
$179$	0.885576	0.0661911	0.0330955	+	0.999452i	$0.489463\pi$
$180$	0	0
$181$	16.8668	1.25370	0.626849	−	0.779141i	$-0.284344\pi$
$181$	16.8668	1.25370	0.626849	+	0.779141i	$0.284344\pi$
$182$	1.35594	0.100509
$183$	0	0
$184$	1.51010	0.111326
$185$	−0.204509	−0.0150358
$186$	0	0
$187$	3.42925	0.250771
$188$	−9.22325	−0.672675
$189$	0	0
$190$	−0.00951630	−0.000690385 0
$191$	11.6939	0.846144	0.423072	−	0.906096i	$-0.360952\pi$
$191$	11.6939	0.846144	0.423072	+	0.906096i	$0.360952\pi$
$192$	0	0
$193$	−16.0405	−1.15462	−0.577310	−	0.816525i	$-0.695898\pi$
$193$	−16.0405	−1.15462	−0.577310	+	0.816525i	$0.695898\pi$
$194$	−1.26021	−0.0904779
$195$	0	0
$196$	−8.46093	−0.604352
$197$	15.3309	1.09228	0.546139	−	0.837695i	$-0.316097\pi$
$197$	15.3309	1.09228	0.546139	+	0.837695i	$0.316097\pi$
$198$	0	0
$199$	6.84770	0.485421	0.242710	−	0.970099i	$-0.421964\pi$
$199$	6.84770	0.485421	0.242710	+	0.970099i	$0.421964\pi$
$200$	−2.17636	−0.153892
$201$	0	0
$202$	−1.95347	−0.137446
$203$	−11.9268	−0.837100
$204$	0	0
$205$	0.369525	0.0258088
$206$	−1.39412	−0.0971333
$207$	0	0
$208$	14.5426	1.00835
$209$	2.07415	0.143472
$210$	0	0
$211$	−15.4727	−1.06518	−0.532591	−	0.846373i	$-0.678781\pi$
$211$	−15.4727	−1.06518	−0.532591	+	0.846373i	$0.678781\pi$
$212$	5.47994	0.376364
$213$	0	0
$214$	1.25760	0.0859676
$215$	0.522931	0.0356636
$216$	0	0
$217$	21.9876	1.49262
$218$	1.30305	0.0882534
$219$	0	0
$220$	−0.0835432	−0.00563248
$221$	−12.6941	−0.853895
$222$	0	0
$223$	2.33178	0.156147	0.0780737	−	0.996948i	$-0.475123\pi$
$223$	2.33178	0.156147	0.0780737	+	0.996948i	$0.475123\pi$
$224$	4.36074	0.291364
$225$	0	0
$226$	−0.0111085	−0.000738924 0
$227$	−13.5845	−0.901633	−0.450816	−	0.892617i	$-0.648867\pi$
$227$	−13.5845	−0.901633	−0.450816	+	0.892617i	$0.648867\pi$
$228$	0	0
$229$	27.2625	1.80156	0.900779	−	0.434277i	$-0.142996\pi$
$229$	27.2625	1.80156	0.900779	+	0.434277i	$0.142996\pi$
$230$	0.0159118	0.00104919
$231$	0	0
$232$	1.54793	0.101627
$233$	14.2384	0.932790	0.466395	−	0.884577i	$-0.345553\pi$
$233$	14.2384	0.932790	0.466395	+	0.884577i	$0.345553\pi$
$234$	0	0
$235$	−0.194952	−0.0127173
$236$	−30.3076	−1.97285
$237$	0	0
$238$	−1.25614	−0.0814235
$239$	7.12128	0.460638	0.230319	−	0.973115i	$-0.426023\pi$
$239$	7.12128	0.460638	0.230319	+	0.973115i	$0.426023\pi$
$240$	0	0
$241$	24.0001	1.54598	0.772992	−	0.634416i	$-0.218759\pi$
$241$	24.0001	1.54598	0.772992	+	0.634416i	$0.218759\pi$
$242$	−0.109182	−0.00701848
$243$	0	0
$244$	1.98808	0.127274
$245$	−0.178839	−0.0114256
$246$	0	0
$247$	−7.67789	−0.488532
$248$	−2.85367	−0.181208
$249$	0	0
$250$	−0.0458724	−0.00290122
$251$	30.6044	1.93173	0.965867	−	0.259038i	$-0.0834054\pi$
$251$	30.6044	1.93173	0.965867	+	0.259038i	$0.0834054\pi$
$252$	0	0
$253$	−3.46809	−0.218037
$254$	−0.655930	−0.0411567
$255$	0	0
$256$	15.0548	0.940928
$257$	1.36326	0.0850377	0.0425189	−	0.999096i	$-0.486462\pi$
$257$	1.36326	0.0850377	0.0425189	+	0.999096i	$0.486462\pi$
$258$	0	0
$259$	−16.3276	−1.01455
$260$	0.309252	0.0191790
$261$	0	0
$262$	0.912388	0.0563675
$263$	−18.1634	−1.12000	−0.560001	−	0.828492i	$-0.689199\pi$
$263$	−18.1634	−1.12000	−0.560001	+	0.828492i	$0.689199\pi$
$264$	0	0
$265$	0.115830	0.00711535
$266$	−0.759765	−0.0465842
$267$	0	0
$268$	31.6609	1.93400
$269$	−14.1686	−0.863874	−0.431937	−	0.901904i	$-0.642170\pi$
$269$	−14.1686	−0.863874	−0.431937	+	0.901904i	$0.642170\pi$
$270$	0	0
$271$	−15.5255	−0.943108	−0.471554	−	0.881837i	$-0.656307\pi$
$271$	−15.5255	−0.943108	−0.471554	+	0.881837i	$0.656307\pi$
$272$	−13.4722	−0.816872
$273$	0	0
$274$	1.36265	0.0823208
$275$	4.99823	0.301405
$276$	0	0
$277$	7.94106	0.477132	0.238566	−	0.971126i	$-0.423323\pi$
$277$	7.94106	0.477132	0.238566	+	0.971126i	$0.423323\pi$
$278$	−1.30693	−0.0783847
$279$	0	0
$280$	0.0613876	0.00366861
$281$	6.34184	0.378322	0.189161	−	0.981946i	$-0.439423\pi$
$281$	6.34184	0.378322	0.189161	+	0.981946i	$0.439423\pi$
$282$	0	0
$283$	−1.01873	−0.0605569	−0.0302785	−	0.999542i	$-0.509639\pi$
$283$	−1.01873	−0.0605569	−0.0302785	+	0.999542i	$0.509639\pi$
$284$	15.5056	0.920088
$285$	0	0
$286$	0.404159	0.0238984
$287$	29.5023	1.74146
$288$	0	0
$289$	−5.24027	−0.308251
$290$	0.0163104	0.000957780 0
$291$	0	0
$292$	21.0341	1.23093
$293$	17.4375	1.01871	0.509354	−	0.860557i	$-0.329885\pi$
$293$	17.4375	1.01871	0.509354	+	0.860557i	$0.329885\pi$
$294$	0	0
$295$	−0.640611	−0.0372978
$296$	2.11909	0.123169
$297$	0	0
$298$	−0.246979	−0.0143071
$299$	12.8378	0.742432
$300$	0	0
$301$	41.7500	2.40643
$302$	0.339126	0.0195145
$303$	0	0
$304$	−8.14854	−0.467351
$305$	0.0420221	0.00240618
$306$	0	0
$307$	17.5386	1.00098	0.500489	−	0.865743i	$-0.333153\pi$
$307$	17.5386	1.00098	0.500489	+	0.865743i	$0.333153\pi$
$308$	−6.66995	−0.380056
$309$	0	0
$310$	−0.0300688	−0.00170780
$311$	−33.0617	−1.87476	−0.937378	−	0.348314i	$-0.886754\pi$
$311$	−33.0617	−1.87476	−0.937378	+	0.348314i	$0.886754\pi$
$312$	0	0
$313$	−10.4956	−0.593247	−0.296623	−	0.954995i	$-0.595861\pi$
$313$	−10.4956	−0.593247	−0.296623	+	0.954995i	$0.595861\pi$
$314$	−2.04834	−0.115594
$315$	0	0
$316$	5.79357	0.325914
$317$	4.95231	0.278150	0.139075	−	0.990282i	$-0.455587\pi$
$317$	4.95231	0.278150	0.139075	+	0.990282i	$0.455587\pi$
$318$	0	0
$319$	−3.55498	−0.199040
$320$	0.324214	0.0181241
$321$	0	0
$322$	1.27037	0.0707949
$323$	7.11277	0.395765
$324$	0	0
$325$	−18.5020	−1.02631
$326$	0.912521	0.0505399
$327$	0	0
$328$	−3.82896	−0.211419
$329$	−15.5646	−0.858107
$330$	0	0
$331$	27.9856	1.53823	0.769115	−	0.639110i	$-0.220697\pi$
$331$	27.9856	1.53823	0.769115	+	0.639110i	$0.220697\pi$
$332$	−23.5463	−1.29227
$333$	0	0
$334$	−1.92854	−0.105525
$335$	0.669217	0.0365633
$336$	0	0
$337$	12.3582	0.673195	0.336598	−	0.941649i	$-0.390724\pi$
$337$	12.3582	0.673195	0.336598	+	0.941649i	$0.390724\pi$
$338$	−0.0767132	−0.00417265
$339$	0	0
$340$	−0.286490	−0.0155371
$341$	6.55374	0.354905
$342$	0	0
$343$	9.20661	0.497110
$344$	−5.41853	−0.292148
$345$	0	0
$346$	−1.24262	−0.0668034
$347$	18.9130	1.01530	0.507651	−	0.861563i	$-0.330514\pi$
$347$	18.9130	1.01530	0.507651	+	0.861563i	$0.330514\pi$
$348$	0	0
$349$	−5.90314	−0.315988	−0.157994	−	0.987440i	$-0.550503\pi$
$349$	−5.90314	−0.315988	−0.157994	+	0.987440i	$0.550503\pi$
$350$	−1.83086	−0.0978637
$351$	0	0
$352$	1.29979	0.0692788
$353$	37.1836	1.97908	0.989542	−	0.144244i	$-0.0460749\pi$
$353$	37.1836	1.97908	0.989542	+	0.144244i	$0.0460749\pi$
$354$	0	0
$355$	0.327742	0.0173947
$356$	−10.8749	−0.576369
$357$	0	0
$358$	−0.0966889	−0.00511016
$359$	31.5382	1.66452	0.832262	−	0.554383i	$-0.187046\pi$
$359$	31.5382	1.66452	0.832262	+	0.554383i	$0.187046\pi$
$360$	0	0
$361$	−14.6979	−0.773574
$362$	−1.84155	−0.0967895
$363$	0	0
$364$	24.6902	1.29412
$365$	0.444597	0.0232713
$366$	0	0
$367$	−9.21065	−0.480792	−0.240396	−	0.970675i	$-0.577277\pi$
$367$	−9.21065	−0.480792	−0.240396	+	0.970675i	$0.577277\pi$
$368$	13.6248	0.710242
$369$	0	0
$370$	0.0223287	0.00116081
$371$	9.24764	0.480113
$372$	0	0
$373$	17.4502	0.903538	0.451769	−	0.892135i	$-0.350793\pi$
$373$	17.4502	0.903538	0.451769	+	0.892135i	$0.350793\pi$
$374$	−0.374412	−0.0193604
$375$	0	0
$376$	2.02006	0.104177
$377$	13.1595	0.677747
$378$	0	0
$379$	7.08636	0.364002	0.182001	−	0.983298i	$-0.441743\pi$
$379$	7.08636	0.364002	0.182001	+	0.983298i	$0.441743\pi$
$380$	−0.173281	−0.00888913
$381$	0	0
$382$	−1.27677	−0.0653250
$383$	7.41767	0.379025	0.189513	−	0.981878i	$-0.439309\pi$
$383$	7.41767	0.379025	0.189513	+	0.981878i	$0.439309\pi$
$384$	0	0
$385$	−0.140983	−0.00718515
$386$	1.75133	0.0891404
$387$	0	0
$388$	−22.9470	−1.16496
$389$	−8.40106	−0.425950	−0.212975	−	0.977058i	$-0.568315\pi$
$389$	−8.40106	−0.425950	−0.212975	+	0.977058i	$0.568315\pi$
$390$	0	0
$391$	−11.8929	−0.601452
$392$	1.85310	0.0935956
$393$	0	0
$394$	−1.67385	−0.0843274
$395$	0.122459	0.00616157
$396$	0	0
$397$	7.17104	0.359904	0.179952	−	0.983675i	$-0.442406\pi$
$397$	7.17104	0.359904	0.179952	+	0.983675i	$0.442406\pi$
$398$	−0.747644	−0.0374760
$399$	0	0
$400$	−19.6362	−0.981808
$401$	−32.7515	−1.63553	−0.817766	−	0.575550i	$-0.804788\pi$
$401$	−32.7515	−1.63553	−0.817766	+	0.575550i	$0.804788\pi$
$402$	0	0
$403$	−24.2600	−1.20848
$404$	−35.5705	−1.76970
$405$	0	0
$406$	1.30219	0.0646268
$407$	−4.86670	−0.241233
$408$	0	0
$409$	−37.7539	−1.86681	−0.933405	−	0.358824i	$-0.883178\pi$
$409$	−37.7539	−1.86681	−0.933405	+	0.358824i	$0.883178\pi$
$410$	−0.0403455	−0.00199252
$411$	0	0
$412$	−25.3854	−1.25065
$413$	−51.1453	−2.51670
$414$	0	0
$415$	−0.497698	−0.0244310
$416$	−4.81142	−0.235899
$417$	0	0
$418$	−0.226459	−0.0110765
$419$	−12.7212	−0.621470	−0.310735	−	0.950497i	$-0.600575\pi$
$419$	−12.7212	−0.621470	−0.310735	+	0.950497i	$0.600575\pi$
$420$	0	0
$421$	32.2847	1.57346	0.786729	−	0.617298i	$-0.211773\pi$
$421$	32.2847	1.57346	0.786729	+	0.617298i	$0.211773\pi$
$422$	1.68933	0.0822354
$423$	0	0
$424$	−1.20021	−0.0582872
$425$	17.1402	0.831421
$426$	0	0
$427$	3.35497	0.162358
$428$	22.8994	1.10689
$429$	0	0
$430$	−0.0570946	−0.00275335
$431$	−27.9610	−1.34683	−0.673416	−	0.739263i	$-0.735174\pi$
$431$	−27.9610	−1.34683	−0.673416	+	0.739263i	$0.735174\pi$
$432$	0	0
$433$	−27.3383	−1.31379	−0.656897	−	0.753980i	$-0.728131\pi$
$433$	−27.3383	−1.31379	−0.656897	+	0.753980i	$0.728131\pi$
$434$	−2.40065	−0.115235
$435$	0	0
$436$	23.7270	1.13632
$437$	−7.19334	−0.344104
$438$	0	0
$439$	3.57514	0.170632	0.0853161	−	0.996354i	$-0.472810\pi$
$439$	3.57514	0.170632	0.0853161	+	0.996354i	$0.472810\pi$
$440$	0.0182975	0.000872299 0
$441$	0	0
$442$	1.38596	0.0659234
$443$	4.56896	0.217078	0.108539	−	0.994092i	$-0.465383\pi$
$443$	4.56896	0.217078	0.108539	+	0.994092i	$0.465383\pi$
$444$	0	0
$445$	−0.229863	−0.0108965
$446$	−0.254588	−0.0120551
$447$	0	0
$448$	25.8847	1.22294
$449$	12.2367	0.577487	0.288744	−	0.957406i	$-0.406762\pi$
$449$	12.2367	0.577487	0.288744	+	0.957406i	$0.406762\pi$
$450$	0	0
$451$	8.79360	0.414075
$452$	−0.202273	−0.00951411
$453$	0	0
$454$	1.48318	0.0696089
$455$	0.521877	0.0244660
$456$	0	0
$457$	−19.4598	−0.910290	−0.455145	−	0.890417i	$-0.650412\pi$
$457$	−19.4598	−0.910290	−0.455145	+	0.890417i	$0.650412\pi$
$458$	−2.97657	−0.139086
$459$	0	0
$460$	0.289735	0.0135090
$461$	2.49974	0.116425	0.0582123	−	0.998304i	$-0.481460\pi$
$461$	2.49974	0.116425	0.0582123	+	0.998304i	$0.481460\pi$
$462$	0	0
$463$	10.8790	0.505592	0.252796	−	0.967520i	$-0.418650\pi$
$463$	10.8790	0.505592	0.252796	+	0.967520i	$0.418650\pi$
$464$	13.9661	0.648362
$465$	0	0
$466$	−1.55458	−0.0720144
$467$	−37.7497	−1.74685	−0.873424	−	0.486961i	$-0.838105\pi$
$467$	−37.7497	−1.74685	−0.873424	+	0.486961i	$0.838105\pi$
$468$	0	0
$469$	53.4292	2.46713
$470$	0.0212852	0.000981814 0
$471$	0	0
$472$	6.63791	0.305535
$473$	12.4442	0.572185
$474$	0	0
$475$	10.3671	0.475675
$476$	−22.8729	−1.04838
$477$	0	0
$478$	−0.777515	−0.0355627
$479$	−14.2594	−0.651526	−0.325763	−	0.945451i	$-0.605621\pi$
$479$	−14.2594	−0.651526	−0.325763	+	0.945451i	$0.605621\pi$
$480$	0	0
$481$	18.0151	0.821417
$482$	−2.62038	−0.119355
$483$	0	0
$484$	−1.98808	−0.0903672
$485$	−0.485032	−0.0220242
$486$	0	0
$487$	16.9175	0.766604	0.383302	−	0.923623i	$-0.374787\pi$
$487$	16.9175	0.766604	0.383302	+	0.923623i	$0.374787\pi$
$488$	−0.435426	−0.0197108
$489$	0	0
$490$	0.0195260	0.000882093 0
$491$	−31.6525	−1.42846	−0.714229	−	0.699912i	$-0.753222\pi$
$491$	−31.6525	−1.42846	−0.714229	+	0.699912i	$0.753222\pi$
$492$	0	0
$493$	−12.1909	−0.549050
$494$	0.838286	0.0377163
$495$	0	0
$496$	−25.7471	−1.15608
$497$	26.1664	1.17372
$498$	0	0
$499$	−8.13145	−0.364014	−0.182007	−	0.983297i	$-0.558259\pi$
$499$	−8.13145	−0.364014	−0.182007	+	0.983297i	$0.558259\pi$
$500$	−0.835285	−0.0373551
$501$	0	0
$502$	−3.34145	−0.149136
$503$	−22.8054	−1.01684	−0.508422	−	0.861108i	$-0.669771\pi$
$503$	−22.8054	−1.01684	−0.508422	+	0.861108i	$0.669771\pi$
$504$	0	0
$505$	−0.751854	−0.0334571
$506$	0.378653	0.0168332
$507$	0	0
$508$	−11.9437	−0.529918
$509$	32.6985	1.44933	0.724667	−	0.689099i	$-0.241994\pi$
$509$	32.6985	1.44933	0.724667	+	0.689099i	$0.241994\pi$
$510$	0	0
$511$	35.4959	1.57025
$512$	−8.52760	−0.376870
$513$	0	0
$514$	−0.148843	−0.00656519
$515$	−0.536573	−0.0236442
$516$	0	0
$517$	−4.63928	−0.204035
$518$	1.78268	0.0783266
$519$	0	0
$520$	−0.0677319	−0.00297024
$521$	18.9847	0.831734	0.415867	−	0.909425i	$-0.363478\pi$
$521$	18.9847	0.831734	0.415867	+	0.909425i	$0.363478\pi$
$522$	0	0
$523$	−8.68179	−0.379628	−0.189814	−	0.981820i	$-0.560789\pi$
$523$	−8.68179	−0.379628	−0.189814	+	0.981820i	$0.560789\pi$
$524$	16.6136	0.725767
$525$	0	0
$526$	1.98311	0.0864678
$527$	22.4744	0.978999
$528$	0	0
$529$	−10.9723	−0.477059
$530$	−0.0126465	−0.000549328 0
$531$	0	0
$532$	−13.8345	−0.599800
$533$	−32.5513	−1.40995
$534$	0	0
$535$	0.484026	0.0209263
$536$	−6.93432	−0.299517
$537$	0	0
$538$	1.54695	0.0666938
$539$	−4.25583	−0.183312
$540$	0	0
$541$	21.3293	0.917019	0.458509	−	0.888690i	$-0.348384\pi$
$541$	21.3293	0.917019	0.458509	+	0.888690i	$0.348384\pi$
$542$	1.69510	0.0728110
$543$	0	0
$544$	4.45729	0.191105
$545$	0.501518	0.0214827
$546$	0	0
$547$	7.72549	0.330318	0.165159	−	0.986267i	$-0.447186\pi$
$547$	7.72549	0.330318	0.165159	+	0.986267i	$0.447186\pi$
$548$	24.8124	1.05993
$549$	0	0
$550$	−0.545716	−0.0232694
$551$	−7.37355	−0.314124
$552$	0	0
$553$	9.77690	0.415756
$554$	−0.867020	−0.0368361
$555$	0	0
$556$	−23.7978	−1.00925
$557$	27.8758	1.18114	0.590568	−	0.806988i	$-0.298904\pi$
$557$	27.8758	1.18114	0.590568	+	0.806988i	$0.298904\pi$
$558$	0	0
$559$	−46.0648	−1.94833
$560$	0.553868	0.0234052
$561$	0	0
$562$	−0.692413	−0.0292077
$563$	13.8817	0.585045	0.292522	−	0.956259i	$-0.405505\pi$
$563$	13.8817	0.585045	0.292522	+	0.956259i	$0.405505\pi$
$564$	0	0
$565$	−0.00427544	−0.000179869 0
$566$	0.111226	0.00467519
$567$	0	0
$568$	−3.39601	−0.142493
$569$	40.3020	1.68955	0.844773	−	0.535125i	$-0.179736\pi$
$569$	40.3020	1.68955	0.844773	+	0.535125i	$0.179736\pi$
$570$	0	0
$571$	−43.4795	−1.81956	−0.909780	−	0.415090i	$-0.863750\pi$
$571$	−43.4795	−1.81956	−0.909780	+	0.415090i	$0.863750\pi$
$572$	7.35928	0.307707
$573$	0	0
$574$	−3.22111	−0.134447
$575$	−17.3343	−0.722891
$576$	0	0
$577$	15.5635	0.647919	0.323959	−	0.946071i	$-0.394986\pi$
$577$	15.5635	0.647919	0.323959	+	0.946071i	$0.394986\pi$
$578$	0.572142	0.0237980
$579$	0	0
$580$	0.296994	0.0123320
$581$	−39.7353	−1.64850
$582$	0	0
$583$	2.75640	0.114158
$584$	−4.60685	−0.190633
$585$	0	0
$586$	−1.90385	−0.0786475
$587$	26.9142	1.11087	0.555434	−	0.831560i	$-0.312552\pi$
$587$	26.9142	1.11087	0.555434	+	0.831560i	$0.312552\pi$
$588$	0	0
$589$	13.5934	0.560107
$590$	0.0699431	0.00287951
$591$	0	0
$592$	19.1194	0.785803
$593$	−38.4546	−1.57914	−0.789571	−	0.613660i	$-0.789697\pi$
$593$	−38.4546	−1.57914	−0.789571	+	0.613660i	$0.789697\pi$
$594$	0	0
$595$	−0.483465	−0.0198201
$596$	−4.49721	−0.184213
$597$	0	0
$598$	−1.40166	−0.0573181
$599$	−17.3140	−0.707431	−0.353716	−	0.935353i	$-0.615082\pi$
$599$	−17.3140	−0.707431	−0.353716	+	0.935353i	$0.615082\pi$
$600$	0	0
$601$	3.79656	0.154865	0.0774325	−	0.996998i	$-0.475328\pi$
$601$	3.79656	0.154865	0.0774325	+	0.996998i	$0.475328\pi$
$602$	−4.55834	−0.185784
$603$	0	0
$604$	6.17511	0.251262
$605$	−0.0420221	−0.00170844
$606$	0	0
$607$	21.4577	0.870942	0.435471	−	0.900203i	$-0.356582\pi$
$607$	21.4577	0.870942	0.435471	+	0.900203i	$0.356582\pi$
$608$	2.69595	0.109335
$609$	0	0
$610$	−0.00458805	−0.000185765 0
$611$	17.1732	0.694755
$612$	0	0
$613$	5.06400	0.204533	0.102266	−	0.994757i	$-0.467391\pi$
$613$	5.06400	0.204533	0.102266	+	0.994757i	$0.467391\pi$
$614$	−1.91489	−0.0772788
$615$	0	0
$616$	1.46084	0.0588590
$617$	7.41175	0.298386	0.149193	−	0.988808i	$-0.452332\pi$
$617$	7.41175	0.298386	0.149193	+	0.988808i	$0.452332\pi$
$618$	0	0
$619$	38.7188	1.55624	0.778121	−	0.628115i	$-0.216173\pi$
$619$	38.7188	1.55624	0.778121	+	0.628115i	$0.216173\pi$
$620$	−0.547520	−0.0219889
$621$	0	0
$622$	3.60973	0.144737
$623$	−18.3519	−0.735252
$624$	0	0
$625$	24.9735	0.998941
$626$	1.14593	0.0458006
$627$	0	0
$628$	−37.2979	−1.48835
$629$	−16.6891	−0.665439
$630$	0	0
$631$	−5.14182	−0.204693	−0.102346	−	0.994749i	$-0.532635\pi$
$631$	−5.14182	−0.204693	−0.102346	+	0.994749i	$0.532635\pi$
$632$	−1.26890	−0.0504740
$633$	0	0
$634$	−0.540703	−0.0214740
$635$	−0.252455	−0.0100184
$636$	0	0
$637$	15.7538	0.624190
$638$	0.388139	0.0153666
$639$	0	0
$640$	−0.144638	−0.00571730
$641$	39.5092	1.56052	0.780260	−	0.625455i	$-0.215087\pi$
$641$	39.5092	1.56052	0.780260	+	0.625455i	$0.215087\pi$
$642$	0	0
$643$	−11.4245	−0.450539	−0.225269	−	0.974297i	$-0.572326\pi$
$643$	−11.4245	−0.450539	−0.225269	+	0.974297i	$0.572326\pi$
$644$	23.1320	0.911528
$645$	0	0
$646$	−0.776586	−0.0305543
$647$	−27.2758	−1.07232	−0.536161	−	0.844116i	$-0.680126\pi$
$647$	−27.2758	−1.07232	−0.536161	+	0.844116i	$0.680126\pi$
$648$	0	0
$649$	−15.2446	−0.598405
$650$	2.02008	0.0792341
$651$	0	0
$652$	16.6160	0.650733
$653$	11.6626	0.456394	0.228197	−	0.973615i	$-0.426717\pi$
$653$	11.6626	0.456394	0.228197	+	0.973615i	$0.426717\pi$
$654$	0	0
$655$	0.351161	0.0137210
$656$	−34.5467	−1.34882
$657$	0	0
$658$	1.69938	0.0662486
$659$	27.3258	1.06446	0.532230	−	0.846600i	$-0.321354\pi$
$659$	27.3258	1.06446	0.532230	+	0.846600i	$0.321354\pi$
$660$	0	0
$661$	9.50496	0.369700	0.184850	−	0.982767i	$-0.440820\pi$
$661$	9.50496	0.369700	0.184850	+	0.982767i	$0.440820\pi$
$662$	−3.05552	−0.118756
$663$	0	0
$664$	5.15706	0.200133
$665$	−0.292420	−0.0113395
$666$	0	0
$667$	12.3290	0.477380
$668$	−35.1166	−1.35870
$669$	0	0
$670$	−0.0730664	−0.00282280
$671$	1.00000	0.0386046
$672$	0	0
$673$	13.6481	0.526096	0.263048	−	0.964783i	$-0.415272\pi$
$673$	13.6481	0.526096	0.263048	+	0.964783i	$0.415272\pi$
$674$	−1.34929	−0.0519728
$675$	0	0
$676$	−1.39686	−0.0537255
$677$	45.3132	1.74153	0.870764	−	0.491700i	$-0.163624\pi$
$677$	45.3132	1.74153	0.870764	+	0.491700i	$0.163624\pi$
$678$	0	0
$679$	−38.7241	−1.48610
$680$	0.0627466	0.00240622
$681$	0	0
$682$	−0.715549	−0.0273998
$683$	10.5598	0.404061	0.202031	−	0.979379i	$-0.435246\pi$
$683$	10.5598	0.404061	0.202031	+	0.979379i	$0.435246\pi$
$684$	0	0
$685$	0.524460	0.0200386
$686$	−1.00519	−0.0383785
$687$	0	0
$688$	−48.8885	−1.86386
$689$	−10.2034	−0.388717
$690$	0	0
$691$	16.0500	0.610572	0.305286	−	0.952261i	$-0.401248\pi$
$691$	16.0500	0.610572	0.305286	+	0.952261i	$0.401248\pi$
$692$	−22.6266	−0.860136
$693$	0	0
$694$	−2.06495	−0.0783846
$695$	−0.503014	−0.0190804
$696$	0	0
$697$	30.1554	1.14222
$698$	0.644516	0.0243953
$699$	0	0
$700$	−33.3380	−1.26006
$701$	44.3858	1.67643	0.838214	−	0.545342i	$-0.183600\pi$
$701$	44.3858	1.67643	0.838214	+	0.545342i	$0.183600\pi$
$702$	0	0
$703$	−10.0943	−0.380712
$704$	7.71532	0.290782
$705$	0	0
$706$	−4.05978	−0.152792
$707$	−60.0267	−2.25754
$708$	0	0
$709$	40.1351	1.50731	0.753653	−	0.657272i	$-0.228290\pi$
$709$	40.1351	1.50731	0.753653	+	0.657272i	$0.228290\pi$
$710$	−0.0357835	−0.00134293
$711$	0	0
$712$	2.38180	0.0892618
$713$	−22.7289	−0.851206
$714$	0	0
$715$	0.155553	0.00581736
$716$	−1.76060	−0.0657966
$717$	0	0
$718$	−3.44340	−0.128507
$719$	−27.0726	−1.00964	−0.504819	−	0.863225i	$-0.668441\pi$
$719$	−27.0726	−1.00964	−0.504819	+	0.863225i	$0.668441\pi$
$720$	0	0
$721$	−42.8391	−1.59541
$722$	1.60474	0.0597224
$723$	0	0
$724$	−33.5325	−1.24623
$725$	−17.7686	−0.659909
$726$	0	0
$727$	44.2715	1.64194	0.820970	−	0.570971i	$-0.193433\pi$
$727$	44.2715	1.64194	0.820970	+	0.570971i	$0.193433\pi$
$728$	−5.40760	−0.200419
$729$	0	0
$730$	−0.0485420	−0.00179662
$731$	42.6743	1.57836
$732$	0	0
$733$	−10.8756	−0.401700	−0.200850	−	0.979622i	$-0.564370\pi$
$733$	−10.8756	−0.401700	−0.200850	+	0.979622i	$0.564370\pi$
$734$	1.00564	0.0371187
$735$	0	0
$736$	−4.50777	−0.166159
$737$	15.9254	0.586619
$738$	0	0
$739$	11.6062	0.426941	0.213471	−	0.976949i	$-0.431523\pi$
$739$	11.6062	0.426941	0.213471	+	0.976949i	$0.431523\pi$
$740$	0.406580	0.0149462
$741$	0	0
$742$	−1.00967	−0.0370663
$743$	−32.9097	−1.20734	−0.603670	−	0.797234i	$-0.706295\pi$
$743$	−32.9097	−1.20734	−0.603670	+	0.797234i	$0.706295\pi$
$744$	0	0
$745$	−0.0950575	−0.00348264
$746$	−1.90525	−0.0697561
$747$	0	0
$748$	−6.81761	−0.249277
$749$	38.6438	1.41201
$750$	0	0
$751$	−52.6588	−1.92155	−0.960774	−	0.277333i	$-0.910550\pi$
$751$	−52.6588	−1.92155	−0.960774	+	0.277333i	$0.910550\pi$
$752$	18.2260	0.664632
$753$	0	0
$754$	−1.43678	−0.0523243
$755$	0.130523	0.00475023
$756$	0	0
$757$	−15.5537	−0.565307	−0.282654	−	0.959222i	$-0.591215\pi$
$757$	−15.5537	−0.565307	−0.282654	+	0.959222i	$0.591215\pi$
$758$	−0.773702	−0.0281021
$759$	0	0
$760$	0.0379517	0.00137665
$761$	−49.8476	−1.80697	−0.903486	−	0.428617i	$-0.859001\pi$
$761$	−49.8476	−1.80697	−0.903486	+	0.428617i	$0.859001\pi$
$762$	0	0
$763$	40.0404	1.44956
$764$	−23.2485	−0.841100
$765$	0	0
$766$	−0.809875	−0.0292620
$767$	56.4312	2.03761
$768$	0	0
$769$	−12.4450	−0.448779	−0.224389	−	0.974500i	$-0.572039\pi$
$769$	−12.4450	−0.448779	−0.224389	+	0.974500i	$0.572039\pi$
$770$	0.0153928	0.000554717 0
$771$	0	0
$772$	31.8898	1.14774
$773$	−27.0846	−0.974167	−0.487084	−	0.873355i	$-0.661939\pi$
$773$	−27.0846	−0.974167	−0.487084	+	0.873355i	$0.661939\pi$
$774$	0	0
$775$	32.7571	1.17667
$776$	5.02582	0.180417
$777$	0	0
$778$	0.917243	0.0328847
$779$	18.2392	0.653489
$780$	0	0
$781$	7.79929	0.279080
$782$	1.29849	0.0464340
$783$	0	0
$784$	16.7195	0.597126
$785$	−0.788367	−0.0281380
$786$	0	0
$787$	−8.45600	−0.301424	−0.150712	−	0.988578i	$-0.548157\pi$
$787$	−8.45600	−0.301424	−0.150712	+	0.988578i	$0.548157\pi$
$788$	−30.4789	−1.08577
$789$	0	0
$790$	−0.0133703	−0.000475693 0
$791$	−0.341344	−0.0121368
$792$	0	0
$793$	−3.70170	−0.131451
$794$	−0.782947	−0.0277857
$795$	0	0
$796$	−13.6138	−0.482527
$797$	38.1087	1.34988	0.674940	−	0.737873i	$-0.264170\pi$
$797$	38.1087	1.34988	0.674940	+	0.737873i	$0.264170\pi$
$798$	0	0
$799$	−15.9092	−0.562828
$800$	6.49663	0.229691
$801$	0	0
$802$	3.57587	0.126268
$803$	10.5801	0.373363
$804$	0	0
$805$	0.488941	0.0172329
$806$	2.64875	0.0932983
$807$	0	0
$808$	7.79059	0.274072
$809$	−25.7263	−0.904490	−0.452245	−	0.891894i	$-0.649377\pi$
$809$	−25.7263	−0.904490	−0.452245	+	0.891894i	$0.649377\pi$
$810$	0	0
$811$	40.9430	1.43770	0.718852	−	0.695164i	$-0.244668\pi$
$811$	40.9430	1.43770	0.718852	+	0.695164i	$0.244668\pi$
$812$	23.7115	0.832111
$813$	0	0
$814$	0.531355	0.0186240
$815$	0.351213	0.0123024
$816$	0	0
$817$	25.8111	0.903018
$818$	4.12204	0.144124
$819$	0	0
$820$	−0.734646	−0.0256549
$821$	−12.4881	−0.435838	−0.217919	−	0.975967i	$-0.569927\pi$
$821$	−12.4881	−0.435838	−0.217919	+	0.975967i	$0.569927\pi$
$822$	0	0
$823$	19.3863	0.675763	0.337882	−	0.941189i	$-0.390290\pi$
$823$	19.3863	0.675763	0.337882	+	0.941189i	$0.390290\pi$
$824$	5.55988	0.193688
$825$	0	0
$826$	5.58414	0.194297
$827$	−41.3003	−1.43615	−0.718076	−	0.695965i	$-0.754977\pi$
$827$	−41.3003	−1.43615	−0.718076	+	0.695965i	$0.754977\pi$
$828$	0	0
$829$	32.9112	1.14305	0.571526	−	0.820584i	$-0.306352\pi$
$829$	32.9112	1.14305	0.571526	+	0.820584i	$0.306352\pi$
$830$	0.0543396	0.00188615
$831$	0	0
$832$	−28.5598	−0.990135
$833$	−14.5943	−0.505662
$834$	0	0
$835$	−0.742260	−0.0256870
$836$	−4.12357	−0.142617
$837$	0	0
$838$	1.38892	0.0479795
$839$	26.8799	0.927997	0.463998	−	0.885836i	$-0.346414\pi$
$839$	26.8799	0.927997	0.463998	+	0.885836i	$0.346414\pi$
$840$	0	0
$841$	−16.3621	−0.564212
$842$	−3.52490	−0.121476
$843$	0	0
$844$	30.7609	1.05883
$845$	−0.0295255	−0.00101571
$846$	0	0
$847$	−3.35497	−0.115278
$848$	−10.8288	−0.371864
$849$	0	0
$850$	−1.87140	−0.0641884
$851$	16.8782	0.578576
$852$	0	0
$853$	−34.6492	−1.18637	−0.593183	−	0.805068i	$-0.702129\pi$
$853$	−34.6492	−1.18637	−0.593183	+	0.805068i	$0.702129\pi$
$854$	−0.366302	−0.0125346
$855$	0	0
$856$	−5.01540	−0.171423
$857$	21.4820	0.733810	0.366905	−	0.930258i	$-0.380417\pi$
$857$	21.4820	0.733810	0.366905	+	0.930258i	$0.380417\pi$
$858$	0	0
$859$	26.9346	0.918996	0.459498	−	0.888179i	$-0.348029\pi$
$859$	26.9346	0.918996	0.459498	+	0.888179i	$0.348029\pi$
$860$	−1.03963	−0.0354511
$861$	0	0
$862$	3.05283	0.103980
$863$	−52.9952	−1.80398	−0.901988	−	0.431760i	$-0.857893\pi$
$863$	−52.9952	−1.80398	−0.901988	+	0.431760i	$0.857893\pi$
$864$	0	0
$865$	−0.478260	−0.0162613
$866$	2.98484	0.101429
$867$	0	0
$868$	−43.7131	−1.48372
$869$	2.91415	0.0988559
$870$	0	0
$871$	−58.9511	−1.99748
$872$	−5.19665	−0.175981
$873$	0	0
$874$	0.785382	0.0265659
$875$	−1.40958	−0.0476525
$876$	0	0
$877$	13.9878	0.472336	0.236168	−	0.971712i	$-0.424108\pi$
$877$	13.9878	0.472336	0.236168	+	0.971712i	$0.424108\pi$
$878$	−0.390340	−0.0131734
$879$	0	0
$880$	0.165089	0.00556514
$881$	−32.6000	−1.09832	−0.549161	−	0.835716i	$-0.685053\pi$
$881$	−32.6000	−1.09832	−0.549161	+	0.835716i	$0.685053\pi$
$882$	0	0
$883$	−17.0645	−0.574266	−0.287133	−	0.957891i	$-0.592702\pi$
$883$	−17.0645	−0.574266	−0.287133	+	0.957891i	$0.592702\pi$
$884$	25.2368	0.848805
$885$	0	0
$886$	−0.498848	−0.0167591
$887$	−18.2692	−0.613420	−0.306710	−	0.951803i	$-0.599228\pi$
$887$	−18.2692	−0.613420	−0.306710	+	0.951803i	$0.599228\pi$
$888$	0	0
$889$	−20.1556	−0.675997
$890$	0.0250969	0.000841249 0
$891$	0	0
$892$	−4.63576	−0.155217
$893$	−9.62256	−0.322007
$894$	0	0
$895$	−0.0372137	−0.00124392
$896$	−11.5476	−0.385779
$897$	0	0
$898$	−1.33603	−0.0445839
$899$	−23.2984	−0.777044
$900$	0	0
$901$	9.45237	0.314904
$902$	−0.960102	−0.0319679
$903$	0	0
$904$	0.0443014	0.00147344
$905$	−0.708777	−0.0235605
$906$	0	0
$907$	−47.6812	−1.58323	−0.791615	−	0.611020i	$-0.790759\pi$
$907$	−47.6812	−1.58323	−0.791615	+	0.611020i	$0.790759\pi$
$908$	27.0070	0.896258
$909$	0	0
$910$	−0.0569795	−0.00188885
$911$	45.7429	1.51553	0.757765	−	0.652527i	$-0.226291\pi$
$911$	45.7429	1.51553	0.757765	+	0.652527i	$0.226291\pi$
$912$	0	0
$913$	−11.8437	−0.391970
$914$	2.12465	0.0702773
$915$	0	0
$916$	−54.2001	−1.79082
$917$	28.0361	0.925835
$918$	0	0
$919$	30.1048	0.993066	0.496533	−	0.868018i	$-0.334606\pi$
$919$	30.1048	0.993066	0.496533	+	0.868018i	$0.334606\pi$
$920$	−0.0634574	−0.00209213
$921$	0	0
$922$	−0.272927	−0.00898836
$923$	−28.8707	−0.950289
$924$	0	0
$925$	−24.3249	−0.799798
$926$	−1.18779	−0.0390333
$927$	0	0
$928$	−4.62071	−0.151682
$929$	−19.3823	−0.635914	−0.317957	−	0.948105i	$-0.602997\pi$
$929$	−19.3823	−0.635914	−0.317957	+	0.948105i	$0.602997\pi$
$930$	0	0
$931$	−8.82723	−0.289301
$932$	−28.3071	−0.927231
$933$	0	0
$934$	4.12158	0.134862
$935$	−0.144104	−0.00471271
$936$	0	0
$937$	17.8066	0.581718	0.290859	−	0.956766i	$-0.406059\pi$
$937$	17.8066	0.581718	0.290859	+	0.956766i	$0.406059\pi$
$938$	−5.83350	−0.190470
$939$	0	0
$940$	0.387580	0.0126415
$941$	21.0920	0.687580	0.343790	−	0.939046i	$-0.388289\pi$
$941$	21.0920	0.687580	0.343790	+	0.939046i	$0.388289\pi$
$942$	0	0
$943$	−30.4970	−0.993119
$944$	59.8904	1.94927
$945$	0	0
$946$	−1.35868	−0.0441745
$947$	12.8885	0.418821	0.209410	−	0.977828i	$-0.432846\pi$
$947$	12.8885	0.418821	0.209410	+	0.977828i	$0.432846\pi$
$948$	0	0
$949$	−39.1644	−1.27133
$950$	−1.13190	−0.0367236
$951$	0	0
$952$	5.00959	0.162362
$953$	−15.9044	−0.515195	−0.257597	−	0.966252i	$-0.582931\pi$
$953$	−15.9044	−0.515195	−0.257597	+	0.966252i	$0.582931\pi$
$954$	0	0
$955$	−0.491403	−0.0159014
$956$	−14.1577	−0.457892
$957$	0	0
$958$	1.55686	0.0502999
$959$	41.8720	1.35212
$960$	0	0
$961$	11.9515	0.385531
$962$	−1.96692	−0.0634161
$963$	0	0
$964$	−47.7141	−1.53677
$965$	0.674055	0.0216986
$966$	0	0
$967$	−30.3212	−0.975062	−0.487531	−	0.873106i	$-0.662102\pi$
$967$	−30.3212	−0.975062	−0.487531	+	0.873106i	$0.662102\pi$
$968$	0.435426	0.0139951
$969$	0	0
$970$	0.0529567	0.00170034
$971$	2.21981	0.0712370	0.0356185	−	0.999365i	$-0.488660\pi$
$971$	2.21981	0.0712370	0.0356185	+	0.999365i	$0.488660\pi$
$972$	0	0
$973$	−40.1598	−1.28746
$974$	−1.84708	−0.0591843
$975$	0	0
$976$	−3.92862	−0.125752
$977$	7.89192	0.252485	0.126242	−	0.991999i	$-0.459708\pi$
$977$	7.89192	0.252485	0.126242	+	0.991999i	$0.459708\pi$
$978$	0	0
$979$	−5.47005	−0.174824
$980$	0.355546	0.0113575
$981$	0	0
$982$	3.45588	0.110282
$983$	16.9485	0.540573	0.270287	−	0.962780i	$-0.412882\pi$
$983$	16.9485	0.540573	0.270287	+	0.962780i	$0.412882\pi$
$984$	0	0
$985$	−0.644234	−0.0205270
$986$	1.33102	0.0423884
$987$	0	0
$988$	15.2643	0.485621
$989$	−43.1576	−1.37233
$990$	0	0
$991$	−25.0193	−0.794763	−0.397381	−	0.917654i	$-0.630081\pi$
$991$	−25.0193	−0.794763	−0.397381	+	0.917654i	$0.630081\pi$
$992$	8.51845	0.270461
$993$	0	0
$994$	−2.85689	−0.0906152
$995$	−0.287754	−0.00912243
$996$	0	0
$997$	−14.0140	−0.443826	−0.221913	−	0.975066i	$-0.571230\pi$
$997$	−14.0140	−0.443826	−0.221913	+	0.975066i	$0.571230\pi$
$998$	0.887807	0.0281030
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6039.2.a.f.1.5		12
3.2	odd	2		2013.2.a.c.1.8	✓	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2013.2.a.c.1.8	✓	12	3.2	odd	2
6039.2.a.f.1.5		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)

\(f(q)\)	\(=\)	\(q-0.109182 q^{2} -1.98808 q^{4} -0.0420221 q^{5} -3.35497 q^{7} +0.435426 q^{8} +O(q^{10})\)	\(q-0.109182 q^{2} -1.98808 q^{4} -0.0420221 q^{5} -3.35497 q^{7} +0.435426 q^{8} +0.00458805 q^{10} -1.00000 q^{11} +3.70170 q^{13} +0.366302 q^{14} +3.92862 q^{16} -3.42925 q^{17} -2.07415 q^{19} +0.0835432 q^{20} +0.109182 q^{22} +3.46809 q^{23} -4.99823 q^{25} -0.404159 q^{26} +6.66995 q^{28} +3.55498 q^{29} -6.55374 q^{31} -1.29979 q^{32} +0.374412 q^{34} +0.140983 q^{35} +4.86670 q^{37} +0.226459 q^{38} -0.0182975 q^{40} -8.79360 q^{41} -12.4442 q^{43} +1.98808 q^{44} -0.378653 q^{46} +4.63928 q^{47} +4.25583 q^{49} +0.545716 q^{50} -7.35928 q^{52} -2.75640 q^{53} +0.0420221 q^{55} -1.46084 q^{56} -0.388139 q^{58} +15.2446 q^{59} -1.00000 q^{61} +0.715549 q^{62} -7.71532 q^{64} -0.155553 q^{65} -15.9254 q^{67} +6.81761 q^{68} -0.0153928 q^{70} -7.79929 q^{71} -10.5801 q^{73} -0.531355 q^{74} +4.12357 q^{76} +3.35497 q^{77} -2.91415 q^{79} -0.165089 q^{80} +0.960102 q^{82} +11.8437 q^{83} +0.144104 q^{85} +1.35868 q^{86} -0.435426 q^{88} +5.47005 q^{89} -12.4191 q^{91} -6.89484 q^{92} -0.506525 q^{94} +0.0871601 q^{95} +11.5423 q^{97} -0.464660 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 7 q^{2} + 13 q^{4} + 7 q^{5} - 15 q^{7} + 18 q^{8}+O(q^{10}) \) 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

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