LMFDB - Embedded newform 6039.2.a.i.1.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

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

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

chi := DirichletCharacter("6039.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$48.2216577807$
Analytic rank:	$0$
Dimension:	$13$
Coefficient field:	$\mathbb{Q}[x]/(x^{13} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{13} - 2 x^{12} - 19 x^{11} + 35 x^{10} + 136 x^{9} - 220 x^{8} - 469 x^{7} + 610 x^{6} + 841 x^{5} + \cdots - 47 $ x^13 - 2x^12 - 19x^11 + 35x^10 + 136x^9 - 220x^8 - 469x^7 + 610x^6 + 841x^5 - 760x^4 - 742x^3 + 366x^2 + 236x - 47
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 2013)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$0.171582$ 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)

$f(q)$	$=$	$q-0.171582 q^{2} -1.97056 q^{4} +0.133072 q^{5} +0.615329 q^{7} +0.681275 q^{8} +O(q^{10})$	\(q-0.171582 q^{2} -1.97056 q^{4} +0.133072 q^{5} +0.615329 q^{7} +0.681275 q^{8} -0.0228327 q^{10} -1.00000 q^{11} -4.70459 q^{13} -0.105579 q^{14} +3.82423 q^{16} -2.34281 q^{17} -7.85906 q^{19} -0.262226 q^{20} +0.171582 q^{22} -1.86581 q^{23} -4.98229 q^{25} +0.807221 q^{26} -1.21254 q^{28} -4.53951 q^{29} +3.06900 q^{31} -2.01872 q^{32} +0.401984 q^{34} +0.0818830 q^{35} +9.24562 q^{37} +1.34847 q^{38} +0.0906587 q^{40} -1.52969 q^{41} +1.17504 q^{43} +1.97056 q^{44} +0.320139 q^{46} +7.78342 q^{47} -6.62137 q^{49} +0.854870 q^{50} +9.27067 q^{52} +11.1113 q^{53} -0.133072 q^{55} +0.419208 q^{56} +0.778896 q^{58} -11.0751 q^{59} -1.00000 q^{61} -0.526584 q^{62} -7.30207 q^{64} -0.626049 q^{65} +11.2239 q^{67} +4.61665 q^{68} -0.0140496 q^{70} +6.59487 q^{71} +4.36946 q^{73} -1.58638 q^{74} +15.4867 q^{76} -0.615329 q^{77} +11.7882 q^{79} +0.508897 q^{80} +0.262467 q^{82} -10.8873 q^{83} -0.311763 q^{85} -0.201616 q^{86} -0.681275 q^{88} -15.1533 q^{89} -2.89487 q^{91} +3.67669 q^{92} -1.33549 q^{94} -1.04582 q^{95} +15.1739 q^{97} +1.13611 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 13 q - 2 q^{2} + 16 q^{4} - 3 q^{5} + 11 q^{7} - 9 q^{8}+O(q^{10}) $ 13 * q - 2 * q^2 + 16 * q^4 - 3 * q^5 + 11 * q^7 - 9 * q^8	$ 13 q - 2 q^{2} + 16 q^{4} - 3 q^{5} + 11 q^{7} - 9 q^{8} + 6 q^{10} - 13 q^{11} + 13 q^{13} - q^{14} + 18 q^{16} - 17 q^{17} + 14 q^{19} + 7 q^{20} + 2 q^{22} - 7 q^{23} + 18 q^{25} + 10 q^{26} + 19 q^{28} + 6 q^{29} + 27 q^{31} - 5 q^{32} + 6 q^{34} - 14 q^{35} + 10 q^{37} - 2 q^{38} + 8 q^{40} - 3 q^{41} + 29 q^{43} - 16 q^{44} - 24 q^{46} - 8 q^{47} + 8 q^{49} + 27 q^{50} + 37 q^{52} + 24 q^{53} + 3 q^{55} - 24 q^{56} - 5 q^{58} - 13 q^{59} - 13 q^{61} - 39 q^{62} + 47 q^{64} + 11 q^{65} + 44 q^{67} + 8 q^{68} - 12 q^{70} - 3 q^{71} + 48 q^{73} + 22 q^{74} + 47 q^{76} - 11 q^{77} - 17 q^{79} + 26 q^{80} + 56 q^{82} - 50 q^{83} + 8 q^{85} - 18 q^{86} + 9 q^{88} + 15 q^{89} + 47 q^{91} - 14 q^{92} + 45 q^{94} + q^{95} + 27 q^{97} - 47 q^{98}+O(q^{100}) $ 13 * q - 2 * q^2 + 16 * q^4 - 3 * q^5 + 11 * q^7 - 9 * q^8 + 6 * q^10 - 13 * q^11 + 13 * q^13 - q^14 + 18 * q^16 - 17 * q^17 + 14 * q^19 + 7 * q^20 + 2 * q^22 - 7 * q^23 + 18 * q^25 + 10 * q^26 + 19 * q^28 + 6 * q^29 + 27 * q^31 - 5 * q^32 + 6 * q^34 - 14 * q^35 + 10 * q^37 - 2 * q^38 + 8 * q^40 - 3 * q^41 + 29 * q^43 - 16 * q^44 - 24 * q^46 - 8 * q^47 + 8 * q^49 + 27 * q^50 + 37 * q^52 + 24 * q^53 + 3 * q^55 - 24 * q^56 - 5 * q^58 - 13 * q^59 - 13 * q^61 - 39 * q^62 + 47 * q^64 + 11 * q^65 + 44 * q^67 + 8 * q^68 - 12 * q^70 - 3 * q^71 + 48 * q^73 + 22 * q^74 + 47 * q^76 - 11 * q^77 - 17 * q^79 + 26 * q^80 + 56 * q^82 - 50 * q^83 + 8 * q^85 - 18 * q^86 + 9 * q^88 + 15 * q^89 + 47 * q^91 - 14 * q^92 + 45 * q^94 + q^95 + 27 * q^97 - 47 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−0.171582	−0.121327	−0.0606633	−	0.998158i	$-0.519322\pi$
$2$	−0.171582	−0.121327	−0.0606633	+	0.998158i	$0.519322\pi$
$3$	0	0
$3$	0	0
$4$	−1.97056	−0.985280
$5$	0.133072	0.0595116	0.0297558	−	0.999557i	$-0.490527\pi$
$5$	0.133072	0.0595116	0.0297558	+	0.999557i	$0.490527\pi$
$6$	0	0
$7$	0.615329	0.232572	0.116286	−	0.993216i	$-0.462901\pi$
$7$	0.615329	0.232572	0.116286	+	0.993216i	$0.462901\pi$
$8$	0.681275	0.240867
$9$	0	0
$10$	−0.0228327	−0.00722034
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	−4.70459	−1.30482	−0.652409	−	0.757867i	$-0.726241\pi$
$13$	−4.70459	−1.30482	−0.652409	+	0.757867i	$0.726241\pi$
$14$	−0.105579	−0.0282172
$15$	0	0
$16$	3.82423	0.956056
$17$	−2.34281	−0.568215	−0.284108	−	0.958792i	$-0.591697\pi$
$17$	−2.34281	−0.568215	−0.284108	+	0.958792i	$0.591697\pi$
$18$	0	0
$19$	−7.85906	−1.80299	−0.901496	−	0.432788i	$-0.857530\pi$
$19$	−7.85906	−1.80299	−0.901496	+	0.432788i	$0.857530\pi$
$20$	−0.262226	−0.0586356
$21$	0	0
$22$	0.171582	0.0365813
$23$	−1.86581	−0.389048	−0.194524	−	0.980898i	$-0.562316\pi$
$23$	−1.86581	−0.389048	−0.194524	+	0.980898i	$0.562316\pi$
$24$	0	0
$25$	−4.98229	−0.996458
$26$	0.807221	0.158309
$27$	0	0
$28$	−1.21254	−0.229149
$29$	−4.53951	−0.842965	−0.421482	−	0.906837i	$-0.638490\pi$
$29$	−4.53951	−0.842965	−0.421482	+	0.906837i	$0.638490\pi$
$30$	0	0
$31$	3.06900	0.551208	0.275604	−	0.961271i	$-0.411122\pi$
$31$	3.06900	0.551208	0.275604	+	0.961271i	$0.411122\pi$
$32$	−2.01872	−0.356862
$33$	0	0
$34$	0.401984	0.0689396
$35$	0.0818830	0.0138408
$36$	0	0
$37$	9.24562	1.51997	0.759986	−	0.649940i	$-0.225206\pi$
$37$	9.24562	1.51997	0.759986	+	0.649940i	$0.225206\pi$
$38$	1.34847	0.218751
$39$	0	0
$40$	0.0906587	0.0143344
$41$	−1.52969	−0.238897	−0.119449	−	0.992840i	$-0.538113\pi$
$41$	−1.52969	−0.238897	−0.119449	+	0.992840i	$0.538113\pi$
$42$	0	0
$43$	1.17504	0.179193	0.0895963	−	0.995978i	$-0.471442\pi$
$43$	1.17504	0.179193	0.0895963	+	0.995978i	$0.471442\pi$
$44$	1.97056	0.297073
$45$	0	0
$46$	0.320139	0.0472019
$47$	7.78342	1.13533	0.567664	−	0.823260i	$-0.307847\pi$
$47$	7.78342	1.13533	0.567664	+	0.823260i	$0.307847\pi$
$48$	0	0
$49$	−6.62137	−0.945910
$50$	0.854870	0.120897
$51$	0	0
$52$	9.27067	1.28561
$53$	11.1113	1.52626	0.763130	−	0.646246i	$-0.223662\pi$
$53$	11.1113	1.52626	0.763130	+	0.646246i	$0.223662\pi$
$54$	0	0
$55$	−0.133072	−0.0179434
$56$	0.419208	0.0560191
$57$	0	0
$58$	0.778896	0.102274
$59$	−11.0751	−1.44185	−0.720925	−	0.693013i	$-0.756283\pi$
$59$	−11.0751	−1.44185	−0.720925	+	0.693013i	$0.756283\pi$
$60$	0	0
$61$	−1.00000	−0.128037
$61$	−1.00000	−0.128037
$62$	−0.526584	−0.0668762
$63$	0	0
$64$	−7.30207	−0.912759
$65$	−0.626049	−0.0776518
$66$	0	0
$67$	11.2239	1.37121	0.685607	−	0.727972i	$-0.259537\pi$
$67$	11.2239	1.37121	0.685607	+	0.727972i	$0.259537\pi$
$68$	4.61665	0.559851
$69$	0	0
$70$	−0.0140496	−0.00167925
$71$	6.59487	0.782668	0.391334	−	0.920249i	$-0.372014\pi$
$71$	6.59487	0.782668	0.391334	+	0.920249i	$0.372014\pi$
$72$	0	0
$73$	4.36946	0.511406	0.255703	−	0.966755i	$-0.417693\pi$
$73$	4.36946	0.511406	0.255703	+	0.966755i	$0.417693\pi$
$74$	−1.58638	−0.184413
$75$	0	0
$76$	15.4867	1.77645
$77$	−0.615329	−0.0701232
$78$	0	0
$79$	11.7882	1.32628	0.663141	−	0.748495i	$-0.269223\pi$
$79$	11.7882	1.32628	0.663141	+	0.748495i	$0.269223\pi$
$80$	0.508897	0.0568964
$81$	0	0
$82$	0.262467	0.0289846
$83$	−10.8873	−1.19503	−0.597517	−	0.801856i	$-0.703846\pi$
$83$	−10.8873	−1.19503	−0.597517	+	0.801856i	$0.703846\pi$
$84$	0	0
$85$	−0.311763	−0.0338154
$86$	−0.201616	−0.0217408
$87$	0	0
$88$	−0.681275	−0.0726242
$89$	−15.1533	−1.60625	−0.803123	−	0.595813i	$-0.796830\pi$
$89$	−15.1533	−1.60625	−0.803123	+	0.595813i	$0.796830\pi$
$90$	0	0
$91$	−2.89487	−0.303465
$92$	3.67669	0.383321
$93$	0	0
$94$	−1.33549	−0.137745
$95$	−1.04582	−0.107299
$96$	0	0
$97$	15.1739	1.54068	0.770340	−	0.637633i	$-0.220086\pi$
$97$	15.1739	1.54068	0.770340	+	0.637633i	$0.220086\pi$
$98$	1.13611	0.114764
$99$	0	0
$100$	9.81790	0.981790
$101$	−14.9729	−1.48986	−0.744931	−	0.667141i	$-0.767518\pi$
$101$	−14.9729	−1.48986	−0.744931	+	0.667141i	$0.767518\pi$
$102$	0	0
$103$	2.65906	0.262005	0.131003	−	0.991382i	$-0.458180\pi$
$103$	2.65906	0.262005	0.131003	+	0.991382i	$0.458180\pi$
$104$	−3.20512	−0.314288
$105$	0	0
$106$	−1.90650	−0.185176
$107$	−9.97635	−0.964450	−0.482225	−	0.876047i	$-0.660171\pi$
$107$	−9.97635	−0.964450	−0.482225	+	0.876047i	$0.660171\pi$
$108$	0	0
$109$	−8.83359	−0.846104	−0.423052	−	0.906105i	$-0.639041\pi$
$109$	−8.83359	−0.846104	−0.423052	+	0.906105i	$0.639041\pi$
$110$	0.0228327	0.00217701
$111$	0	0
$112$	2.35316	0.222352
$113$	4.24584	0.399415	0.199707	−	0.979856i	$-0.436001\pi$
$113$	4.24584	0.399415	0.199707	+	0.979856i	$0.436001\pi$
$114$	0	0
$115$	−0.248287	−0.0231529
$116$	8.94537	0.830556
$117$	0	0
$118$	1.90028	0.174935
$119$	−1.44160	−0.132151
$120$	0	0
$121$	1.00000	0.0909091
$122$	0.171582	0.0155343
$123$	0	0
$124$	−6.04764	−0.543094
$125$	−1.32836	−0.118812
$126$	0	0
$127$	19.3085	1.71335	0.856675	−	0.515856i	$-0.172526\pi$
$127$	19.3085	1.71335	0.856675	+	0.515856i	$0.172526\pi$
$128$	5.29034	0.467604
$129$	0	0
$130$	0.107419	0.00942123
$131$	−16.1882	−1.41437	−0.707187	−	0.707027i	$-0.750036\pi$
$131$	−16.1882	−1.41437	−0.707187	+	0.707027i	$0.750036\pi$
$132$	0	0
$133$	−4.83590	−0.419326
$134$	−1.92581	−0.166365
$135$	0	0
$136$	−1.59610	−0.136864
$137$	16.9876	1.45135	0.725674	−	0.688038i	$-0.241528\pi$
$137$	16.9876	1.45135	0.725674	+	0.688038i	$0.241528\pi$
$138$	0	0
$139$	0.981625	0.0832604	0.0416302	−	0.999133i	$-0.486745\pi$
$139$	0.981625	0.0832604	0.0416302	+	0.999133i	$0.486745\pi$
$140$	−0.161355	−0.0136370
$141$	0	0
$142$	−1.13156	−0.0949584
$143$	4.70459	0.393417
$144$	0	0
$145$	−0.604081	−0.0501662
$146$	−0.749719	−0.0620472
$147$	0	0
$148$	−18.2191	−1.49760
$149$	−19.9181	−1.63175	−0.815876	−	0.578227i	$-0.803745\pi$
$149$	−19.9181	−1.63175	−0.815876	+	0.578227i	$0.803745\pi$
$150$	0	0
$151$	−9.27179	−0.754528	−0.377264	−	0.926106i	$-0.623135\pi$
$151$	−9.27179	−0.754528	−0.377264	+	0.926106i	$0.623135\pi$
$152$	−5.35418	−0.434282
$153$	0	0
$154$	0.105579	0.00850781
$155$	0.408397	0.0328033
$156$	0	0
$157$	9.62702	0.768320	0.384160	−	0.923267i	$-0.374491\pi$
$157$	9.62702	0.768320	0.384160	+	0.923267i	$0.374491\pi$
$158$	−2.02265	−0.160913
$159$	0	0
$160$	−0.268635	−0.0212374
$161$	−1.14809	−0.0904818
$162$	0	0
$163$	17.0593	1.33619	0.668093	−	0.744077i	$-0.267111\pi$
$163$	17.0593	1.33619	0.668093	+	0.744077i	$0.267111\pi$
$164$	3.01434	0.235381
$165$	0	0
$166$	1.86806	0.144989
$167$	−8.92176	−0.690387	−0.345193	−	0.938532i	$-0.612187\pi$
$167$	−8.92176	−0.690387	−0.345193	+	0.938532i	$0.612187\pi$
$168$	0	0
$169$	9.13314	0.702549
$170$	0.0534928	0.00410271
$171$	0	0
$172$	−2.31549	−0.176555
$173$	−3.85738	−0.293271	−0.146636	−	0.989191i	$-0.546844\pi$
$173$	−3.85738	−0.293271	−0.146636	+	0.989191i	$0.546844\pi$
$174$	0	0
$175$	−3.06575	−0.231749
$176$	−3.82423	−0.288262
$177$	0	0
$178$	2.60003	0.194880
$179$	10.4383	0.780194	0.390097	−	0.920774i	$-0.372442\pi$
$179$	10.4383	0.780194	0.390097	+	0.920774i	$0.372442\pi$
$180$	0	0
$181$	13.9705	1.03842	0.519208	−	0.854648i	$-0.326227\pi$
$181$	13.9705	1.03842	0.519208	+	0.854648i	$0.326227\pi$
$182$	0.496706	0.0368183
$183$	0	0
$184$	−1.27113	−0.0937089
$185$	1.23033	0.0904559
$186$	0	0
$187$	2.34281	0.171323
$188$	−15.3377	−1.11862
$189$	0	0
$190$	0.179444	0.0130182
$191$	24.4448	1.76876	0.884382	−	0.466763i	$-0.154580\pi$
$191$	24.4448	1.76876	0.884382	+	0.466763i	$0.154580\pi$
$192$	0	0
$193$	−6.16239	−0.443578	−0.221789	−	0.975095i	$-0.571190\pi$
$193$	−6.16239	−0.443578	−0.221789	+	0.975095i	$0.571190\pi$
$194$	−2.60357	−0.186926
$195$	0	0
$196$	13.0478	0.931986
$197$	16.1438	1.15020	0.575100	−	0.818083i	$-0.304963\pi$
$197$	16.1438	1.15020	0.575100	+	0.818083i	$0.304963\pi$
$198$	0	0
$199$	−11.7030	−0.829600	−0.414800	−	0.909913i	$-0.636149\pi$
$199$	−11.7030	−0.829600	−0.414800	+	0.909913i	$0.636149\pi$
$200$	−3.39431	−0.240014
$201$	0	0
$202$	2.56908	0.180760
$203$	−2.79329	−0.196050
$204$	0	0
$205$	−0.203559	−0.0142172
$206$	−0.456247	−0.0317882
$207$	0	0
$208$	−17.9914	−1.24748
$209$	7.85906	0.543622
$210$	0	0
$211$	−13.0881	−0.901019	−0.450509	−	0.892772i	$-0.648758\pi$
$211$	−13.0881	−0.901019	−0.450509	+	0.892772i	$0.648758\pi$
$212$	−21.8955	−1.50379
$213$	0	0
$214$	1.71176	0.117013
$215$	0.156365	0.0106640
$216$	0	0
$217$	1.88844	0.128196
$218$	1.51568	0.102655
$219$	0	0
$220$	0.262226	0.0176793
$221$	11.0220	0.741417
$222$	0	0
$223$	2.06775	0.138467	0.0692334	−	0.997600i	$-0.477945\pi$
$223$	2.06775	0.138467	0.0692334	+	0.997600i	$0.477945\pi$
$224$	−1.24218	−0.0829963
$225$	0	0
$226$	−0.728508	−0.0484596
$227$	6.15200	0.408322	0.204161	−	0.978937i	$-0.434553\pi$
$227$	6.15200	0.408322	0.204161	+	0.978937i	$0.434553\pi$
$228$	0	0
$229$	19.5963	1.29496	0.647479	−	0.762083i	$-0.275823\pi$
$229$	19.5963	1.29496	0.647479	+	0.762083i	$0.275823\pi$
$230$	0.0426015	0.00280906
$231$	0	0
$232$	−3.09265	−0.203043
$233$	20.5752	1.34792	0.673962	−	0.738766i	$-0.264591\pi$
$233$	20.5752	1.34792	0.673962	+	0.738766i	$0.264591\pi$
$234$	0	0
$235$	1.03575	0.0675652
$236$	21.8241	1.42063
$237$	0	0
$238$	0.247352	0.0160335
$239$	23.1378	1.49666	0.748330	−	0.663327i	$-0.230856\pi$
$239$	23.1378	1.49666	0.748330	+	0.663327i	$0.230856\pi$
$240$	0	0
$241$	0.0983366	0.00633442	0.00316721	−	0.999995i	$-0.498992\pi$
$241$	0.0983366	0.00633442	0.00316721	+	0.999995i	$0.498992\pi$
$242$	−0.171582	−0.0110297
$243$	0	0
$244$	1.97056	0.126152
$245$	−0.881119	−0.0562926
$246$	0	0
$247$	36.9736	2.35257
$248$	2.09083	0.132768
$249$	0	0
$250$	0.227923	0.0144151
$251$	3.43655	0.216913	0.108457	−	0.994101i	$-0.465409\pi$
$251$	3.43655	0.216913	0.108457	+	0.994101i	$0.465409\pi$
$252$	0	0
$253$	1.86581	0.117302
$254$	−3.31298	−0.207875
$255$	0	0
$256$	13.6964	0.856027
$257$	12.4152	0.774439	0.387220	−	0.921988i	$-0.373436\pi$
$257$	12.4152	0.774439	0.387220	+	0.921988i	$0.373436\pi$
$258$	0	0
$259$	5.68910	0.353503
$260$	1.23367	0.0765087
$261$	0	0
$262$	2.77761	0.171601
$263$	−4.10959	−0.253408	−0.126704	−	0.991941i	$-0.540440\pi$
$263$	−4.10959	−0.253408	−0.126704	+	0.991941i	$0.540440\pi$
$264$	0	0
$265$	1.47861	0.0908301
$266$	0.829753	0.0508754
$267$	0	0
$268$	−22.1173	−1.35103
$269$	−17.7222	−1.08054	−0.540270	−	0.841492i	$-0.681678\pi$
$269$	−17.7222	−1.08054	−0.540270	+	0.841492i	$0.681678\pi$
$270$	0	0
$271$	6.97502	0.423702	0.211851	−	0.977302i	$-0.432051\pi$
$271$	6.97502	0.423702	0.211851	+	0.977302i	$0.432051\pi$
$272$	−8.95944	−0.543246
$273$	0	0
$274$	−2.91476	−0.176087
$275$	4.98229	0.300444
$276$	0	0
$277$	−2.67748	−0.160874	−0.0804371	−	0.996760i	$-0.525632\pi$
$277$	−2.67748	−0.160874	−0.0804371	+	0.996760i	$0.525632\pi$
$278$	−0.168429	−0.0101017
$279$	0	0
$280$	0.0557849	0.00333379
$281$	16.1747	0.964905	0.482452	−	0.875922i	$-0.339746\pi$
$281$	16.1747	0.964905	0.482452	+	0.875922i	$0.339746\pi$
$282$	0	0
$283$	14.1430	0.840715	0.420357	−	0.907359i	$-0.361905\pi$
$283$	14.1430	0.840715	0.420357	+	0.907359i	$0.361905\pi$
$284$	−12.9956	−0.771147
$285$	0	0
$286$	−0.807221	−0.0477320
$287$	−0.941262	−0.0555609
$288$	0	0
$289$	−11.5112	−0.677131
$290$	0.103649	0.00608649
$291$	0	0
$292$	−8.61028	−0.503878
$293$	8.25000	0.481970	0.240985	−	0.970529i	$-0.422529\pi$
$293$	8.25000	0.481970	0.240985	+	0.970529i	$0.422529\pi$
$294$	0	0
$295$	−1.47378	−0.0858068
$296$	6.29882	0.366111
$297$	0	0
$298$	3.41758	0.197975
$299$	8.77786	0.507637
$300$	0	0
$301$	0.723038	0.0416752
$302$	1.59087	0.0915443
$303$	0	0
$304$	−30.0548	−1.72376
$305$	−0.133072	−0.00761968
$306$	0	0
$307$	−5.17139	−0.295147	−0.147573	−	0.989051i	$-0.547146\pi$
$307$	−5.17139	−0.295147	−0.147573	+	0.989051i	$0.547146\pi$
$308$	1.21254	0.0690910
$309$	0	0
$310$	−0.0700735	−0.00397991
$311$	−5.56505	−0.315565	−0.157783	−	0.987474i	$-0.550435\pi$
$311$	−5.56505	−0.315565	−0.157783	+	0.987474i	$0.550435\pi$
$312$	0	0
$313$	−26.1339	−1.47718	−0.738589	−	0.674156i	$-0.764507\pi$
$313$	−26.1339	−1.47718	−0.738589	+	0.674156i	$0.764507\pi$
$314$	−1.65182	−0.0932176
$315$	0	0
$316$	−23.2294	−1.30676
$317$	9.26111	0.520156	0.260078	−	0.965588i	$-0.416252\pi$
$317$	9.26111	0.520156	0.260078	+	0.965588i	$0.416252\pi$
$318$	0	0
$319$	4.53951	0.254163
$320$	−0.971702	−0.0543198
$321$	0	0
$322$	0.196991	0.0109779
$323$	18.4123	1.02449
$324$	0	0
$325$	23.4396	1.30020
$326$	−2.92706	−0.162115
$327$	0	0
$328$	−1.04214	−0.0575425
$329$	4.78936	0.264046
$330$	0	0
$331$	−2.06884	−0.113714	−0.0568569	−	0.998382i	$-0.518108\pi$
$331$	−2.06884	−0.113714	−0.0568569	+	0.998382i	$0.518108\pi$
$332$	21.4540	1.17744
$333$	0	0
$334$	1.53081	0.0837622
$335$	1.49358	0.0816031
$336$	0	0
$337$	−0.238290	−0.0129805	−0.00649024	−	0.999979i	$-0.502066\pi$
$337$	−0.238290	−0.0129805	−0.00649024	+	0.999979i	$0.502066\pi$
$338$	−1.56708	−0.0852379
$339$	0	0
$340$	0.614347	0.0333176
$341$	−3.06900	−0.166195
$342$	0	0
$343$	−8.38162	−0.452565
$344$	0.800529	0.0431616
$345$	0	0
$346$	0.661856	0.0355816
$347$	−27.9023	−1.49787	−0.748937	−	0.662641i	$-0.769436\pi$
$347$	−27.9023	−1.49787	−0.748937	+	0.662641i	$0.769436\pi$
$348$	0	0
$349$	30.6969	1.64317	0.821585	−	0.570086i	$-0.193090\pi$
$349$	30.6969	1.64317	0.821585	+	0.570086i	$0.193090\pi$
$350$	0.526026	0.0281173
$351$	0	0
$352$	2.01872	0.107598
$353$	−32.2120	−1.71447	−0.857236	−	0.514924i	$-0.827820\pi$
$353$	−32.2120	−1.71447	−0.857236	+	0.514924i	$0.827820\pi$
$354$	0	0
$355$	0.877593	0.0465778
$356$	29.8605	1.58260
$357$	0	0
$358$	−1.79102	−0.0946582
$359$	25.4721	1.34437	0.672184	−	0.740384i	$-0.265356\pi$
$359$	25.4721	1.34437	0.672184	+	0.740384i	$0.265356\pi$
$360$	0	0
$361$	42.7648	2.25078
$362$	−2.39708	−0.125987
$363$	0	0
$364$	5.70451	0.298998
$365$	0.581452	0.0304346
$366$	0	0
$367$	35.8891	1.87340	0.936698	−	0.350138i	$-0.113865\pi$
$367$	35.8891	1.87340	0.936698	+	0.350138i	$0.113865\pi$
$368$	−7.13527	−0.371952
$369$	0	0
$370$	−0.211103	−0.0109747
$371$	6.83713	0.354966
$372$	0	0
$373$	26.2640	1.35990	0.679950	−	0.733259i	$-0.262002\pi$
$373$	26.2640	1.35990	0.679950	+	0.733259i	$0.262002\pi$
$374$	−0.401984	−0.0207861
$375$	0	0
$376$	5.30265	0.273463
$377$	21.3565	1.09992
$378$	0	0
$379$	−22.5811	−1.15991	−0.579957	−	0.814647i	$-0.696931\pi$
$379$	−22.5811	−1.15991	−0.579957	+	0.814647i	$0.696931\pi$
$380$	2.06085	0.105719
$381$	0	0
$382$	−4.19428	−0.214598
$383$	−1.52359	−0.0778520	−0.0389260	−	0.999242i	$-0.512394\pi$
$383$	−1.52359	−0.0778520	−0.0389260	+	0.999242i	$0.512394\pi$
$384$	0	0
$385$	−0.0818830	−0.00417315
$386$	1.05735	0.0538179
$387$	0	0
$388$	−29.9012	−1.51800
$389$	21.4415	1.08713	0.543563	−	0.839369i	$-0.317075\pi$
$389$	21.4415	1.08713	0.543563	+	0.839369i	$0.317075\pi$
$390$	0	0
$391$	4.37124	0.221063
$392$	−4.51098	−0.227839
$393$	0	0
$394$	−2.76999	−0.139550
$395$	1.56869	0.0789291
$396$	0	0
$397$	−7.89768	−0.396373	−0.198187	−	0.980164i	$-0.563505\pi$
$397$	−7.89768	−0.396373	−0.198187	+	0.980164i	$0.563505\pi$
$398$	2.00801	0.100653
$399$	0	0
$400$	−19.0534	−0.952670
$401$	−11.0900	−0.553807	−0.276903	−	0.960898i	$-0.589308\pi$
$401$	−11.0900	−0.553807	−0.276903	+	0.960898i	$0.589308\pi$
$402$	0	0
$403$	−14.4384	−0.719226
$404$	29.5051	1.46793
$405$	0	0
$406$	0.479277	0.0237861
$407$	−9.24562	−0.458289
$408$	0	0
$409$	1.82984	0.0904797	0.0452399	−	0.998976i	$-0.485595\pi$
$409$	1.82984	0.0904797	0.0452399	+	0.998976i	$0.485595\pi$
$410$	0.0349270	0.00172492
$411$	0	0
$412$	−5.23985	−0.258149
$413$	−6.81481	−0.335335
$414$	0	0
$415$	−1.44879	−0.0711183
$416$	9.49723	0.465640
$417$	0	0
$418$	−1.34847	−0.0659558
$419$	−29.0901	−1.42115	−0.710573	−	0.703623i	$-0.751564\pi$
$419$	−29.0901	−1.42115	−0.710573	+	0.703623i	$0.751564\pi$
$420$	0	0
$421$	−0.0560143	−0.00272997	−0.00136499	−	0.999999i	$-0.500434\pi$
$421$	−0.0560143	−0.00272997	−0.00136499	+	0.999999i	$0.500434\pi$
$422$	2.24567	0.109318
$423$	0	0
$424$	7.56988	0.367626
$425$	11.6726	0.566203
$426$	0	0
$427$	−0.615329	−0.0297779
$428$	19.6590	0.950253
$429$	0	0
$430$	−0.0268294	−0.00129383
$431$	2.07838	0.100112	0.0500561	−	0.998746i	$-0.484060\pi$
$431$	2.07838	0.100112	0.0500561	+	0.998746i	$0.484060\pi$
$432$	0	0
$433$	−5.73899	−0.275798	−0.137899	−	0.990446i	$-0.544035\pi$
$433$	−5.73899	−0.275798	−0.137899	+	0.990446i	$0.544035\pi$
$434$	−0.324022	−0.0155536
$435$	0	0
$436$	17.4071	0.833650
$437$	14.6635	0.701450
$438$	0	0
$439$	32.6721	1.55935	0.779677	−	0.626182i	$-0.215383\pi$
$439$	32.6721	1.55935	0.779677	+	0.626182i	$0.215383\pi$
$440$	−0.0906587	−0.00432198
$441$	0	0
$442$	−1.89117	−0.0899537
$443$	25.8218	1.22683	0.613415	−	0.789761i	$-0.289795\pi$
$443$	25.8218	1.22683	0.613415	+	0.789761i	$0.289795\pi$
$444$	0	0
$445$	−2.01648	−0.0955903
$446$	−0.354788	−0.0167997
$447$	0	0
$448$	−4.49318	−0.212283
$449$	18.6591	0.880579	0.440290	−	0.897856i	$-0.354876\pi$
$449$	18.6591	0.880579	0.440290	+	0.897856i	$0.354876\pi$
$450$	0	0
$451$	1.52969	0.0720303
$452$	−8.36667	−0.393535
$453$	0	0
$454$	−1.05557	−0.0495404
$455$	−0.385226	−0.0180597
$456$	0	0
$457$	−4.37660	−0.204729	−0.102364	−	0.994747i	$-0.532641\pi$
$457$	−4.37660	−0.204729	−0.102364	+	0.994747i	$0.532641\pi$
$458$	−3.36236	−0.157113
$459$	0	0
$460$	0.489264	0.0228120
$461$	36.4443	1.69738	0.848689	−	0.528891i	$-0.177392\pi$
$461$	36.4443	1.69738	0.848689	+	0.528891i	$0.177392\pi$
$462$	0	0
$463$	14.5793	0.677557	0.338779	−	0.940866i	$-0.389986\pi$
$463$	14.5793	0.677557	0.338779	+	0.940866i	$0.389986\pi$
$464$	−17.3601	−0.805922
$465$	0	0
$466$	−3.53032	−0.163539
$467$	−26.5701	−1.22952	−0.614760	−	0.788714i	$-0.710747\pi$
$467$	−26.5701	−1.22952	−0.614760	+	0.788714i	$0.710747\pi$
$468$	0	0
$469$	6.90637	0.318906
$470$	−0.177717	−0.00819745
$471$	0	0
$472$	−7.54517	−0.347295
$473$	−1.17504	−0.0540286
$474$	0	0
$475$	39.1561	1.79661
$476$	2.84076	0.130206
$477$	0	0
$478$	−3.97002	−0.181585
$479$	−4.66442	−0.213123	−0.106561	−	0.994306i	$-0.533984\pi$
$479$	−4.66442	−0.213123	−0.106561	+	0.994306i	$0.533984\pi$
$480$	0	0
$481$	−43.4968	−1.98329
$482$	−0.0168728	−0.000768533 0
$483$	0	0
$484$	−1.97056	−0.0895709
$485$	2.01923	0.0916884
$486$	0	0
$487$	16.2253	0.735240	0.367620	−	0.929976i	$-0.380173\pi$
$487$	16.2253	0.735240	0.367620	+	0.929976i	$0.380173\pi$
$488$	−0.681275	−0.0308399
$489$	0	0
$490$	0.151184	0.00682979
$491$	−37.8484	−1.70808	−0.854038	−	0.520210i	$-0.825854\pi$
$491$	−37.8484	−1.70808	−0.854038	+	0.520210i	$0.825854\pi$
$492$	0	0
$493$	10.6352	0.478986
$494$	−6.34400	−0.285430
$495$	0	0
$496$	11.7365	0.526986
$497$	4.05802	0.182027
$498$	0	0
$499$	−12.9470	−0.579586	−0.289793	−	0.957089i	$-0.593587\pi$
$499$	−12.9470	−0.579586	−0.289793	+	0.957089i	$0.593587\pi$
$500$	2.61762	0.117063
$501$	0	0
$502$	−0.589650	−0.0263174
$503$	22.8525	1.01894	0.509471	−	0.860488i	$-0.329841\pi$
$503$	22.8525	1.01894	0.509471	+	0.860488i	$0.329841\pi$
$504$	0	0
$505$	−1.99248	−0.0886641
$506$	−0.320139	−0.0142319
$507$	0	0
$508$	−38.0485	−1.68813
$509$	−1.26053	−0.0558720	−0.0279360	−	0.999610i	$-0.508893\pi$
$509$	−1.26053	−0.0558720	−0.0279360	+	0.999610i	$0.508893\pi$
$510$	0	0
$511$	2.68865	0.118939
$512$	−12.9307	−0.571463
$513$	0	0
$514$	−2.13022	−0.0939601
$515$	0.353847	0.0155924
$516$	0	0
$517$	−7.78342	−0.342314
$518$	−0.976146	−0.0428894
$519$	0	0
$520$	−0.426512	−0.0187038
$521$	−33.2078	−1.45486	−0.727430	−	0.686182i	$-0.759285\pi$
$521$	−33.2078	−1.45486	−0.727430	+	0.686182i	$0.759285\pi$
$522$	0	0
$523$	22.8888	1.00086	0.500428	−	0.865778i	$-0.333176\pi$
$523$	22.8888	1.00086	0.500428	+	0.865778i	$0.333176\pi$
$524$	31.8999	1.39355
$525$	0	0
$526$	0.705131	0.0307452
$527$	−7.19008	−0.313205
$528$	0	0
$529$	−19.5188	−0.848642
$530$	−0.253702	−0.0110201
$531$	0	0
$532$	9.52944	0.413154
$533$	7.19656	0.311717
$534$	0	0
$535$	−1.32757	−0.0573959
$536$	7.64654	0.330280
$537$	0	0
$538$	3.04080	0.131098
$539$	6.62137	0.285203
$540$	0	0
$541$	−7.47250	−0.321268	−0.160634	−	0.987014i	$-0.551354\pi$
$541$	−7.47250	−0.321268	−0.160634	+	0.987014i	$0.551354\pi$
$542$	−1.19679	−0.0514064
$543$	0	0
$544$	4.72948	0.202775
$545$	−1.17550	−0.0503530
$546$	0	0
$547$	42.4596	1.81544	0.907721	−	0.419574i	$-0.137820\pi$
$547$	42.4596	1.81544	0.907721	+	0.419574i	$0.137820\pi$
$548$	−33.4751	−1.42998
$549$	0	0
$550$	−0.854870	−0.0364518
$551$	35.6762	1.51986
$552$	0	0
$553$	7.25365	0.308457
$554$	0.459407	0.0195183
$555$	0	0
$556$	−1.93435	−0.0820348
$557$	−33.5690	−1.42236	−0.711182	−	0.703008i	$-0.751840\pi$
$557$	−33.5690	−1.42236	−0.711182	+	0.703008i	$0.751840\pi$
$558$	0	0
$559$	−5.52810	−0.233814
$560$	0.313139	0.0132325
$561$	0	0
$562$	−2.77529	−0.117069
$563$	6.90978	0.291212	0.145606	−	0.989343i	$-0.453487\pi$
$563$	6.90978	0.291212	0.145606	+	0.989343i	$0.453487\pi$
$564$	0	0
$565$	0.565002	0.0237698
$566$	−2.42668	−0.102001
$567$	0	0
$568$	4.49293	0.188519
$569$	43.2342	1.81247	0.906236	−	0.422772i	$-0.138943\pi$
$569$	43.2342	1.81247	0.906236	+	0.422772i	$0.138943\pi$
$570$	0	0
$571$	36.9917	1.54805	0.774027	−	0.633153i	$-0.218240\pi$
$571$	36.9917	1.54805	0.774027	+	0.633153i	$0.218240\pi$
$572$	−9.27067	−0.387626
$573$	0	0
$574$	0.161503	0.00674102
$575$	9.29600	0.387670
$576$	0	0
$577$	−43.0272	−1.79125	−0.895623	−	0.444815i	$-0.853270\pi$
$577$	−43.0272	−1.79125	−0.895623	+	0.444815i	$0.853270\pi$
$578$	1.97512	0.0821540
$579$	0	0
$580$	1.19038	0.0494277
$581$	−6.69925	−0.277932
$582$	0	0
$583$	−11.1113	−0.460184
$584$	2.97680	0.123181
$585$	0	0
$586$	−1.41555	−0.0584758
$587$	−2.31390	−0.0955048	−0.0477524	−	0.998859i	$-0.515206\pi$
$587$	−2.31390	−0.0955048	−0.0477524	+	0.998859i	$0.515206\pi$
$588$	0	0
$589$	−24.1194	−0.993823
$590$	0.252874	0.0104107
$591$	0	0
$592$	35.3573	1.45318
$593$	−10.2698	−0.421731	−0.210865	−	0.977515i	$-0.567628\pi$
$593$	−10.2698	−0.421731	−0.210865	+	0.977515i	$0.567628\pi$
$594$	0	0
$595$	−0.191837	−0.00786453
$596$	39.2498	1.60773
$597$	0	0
$598$	−1.50612	−0.0615898
$599$	−21.7905	−0.890335	−0.445167	−	0.895447i	$-0.646856\pi$
$599$	−21.7905	−0.890335	−0.445167	+	0.895447i	$0.646856\pi$
$600$	0	0
$601$	−20.7713	−0.847279	−0.423640	−	0.905831i	$-0.639248\pi$
$601$	−20.7713	−0.847279	−0.423640	+	0.905831i	$0.639248\pi$
$602$	−0.124060	−0.00505632
$603$	0	0
$604$	18.2706	0.743421
$605$	0.133072	0.00541015
$606$	0	0
$607$	1.14578	0.0465057	0.0232528	−	0.999730i	$-0.492598\pi$
$607$	1.14578	0.0465057	0.0232528	+	0.999730i	$0.492598\pi$
$608$	15.8652	0.643420
$609$	0	0
$610$	0.0228327	0.000924470 0
$611$	−36.6178	−1.48140
$612$	0	0
$613$	13.2402	0.534766	0.267383	−	0.963590i	$-0.413841\pi$
$613$	13.2402	0.534766	0.267383	+	0.963590i	$0.413841\pi$
$614$	0.887316	0.0358092
$615$	0	0
$616$	−0.419208	−0.0168904
$617$	19.1708	0.771787	0.385894	−	0.922543i	$-0.373893\pi$
$617$	19.1708	0.771787	0.385894	+	0.922543i	$0.373893\pi$
$618$	0	0
$619$	3.63574	0.146133	0.0730663	−	0.997327i	$-0.476722\pi$
$619$	3.63574	0.146133	0.0730663	+	0.997327i	$0.476722\pi$
$620$	−0.804771	−0.0323204
$621$	0	0
$622$	0.954861	0.0382864
$623$	−9.32426	−0.373569
$624$	0	0
$625$	24.7347	0.989388
$626$	4.48411	0.179221
$627$	0	0
$628$	−18.9706	−0.757010
$629$	−21.6608	−0.863671
$630$	0	0
$631$	34.0488	1.35546	0.677732	−	0.735309i	$-0.262963\pi$
$631$	34.0488	1.35546	0.677732	+	0.735309i	$0.262963\pi$
$632$	8.03105	0.319458
$633$	0	0
$634$	−1.58904	−0.0631087
$635$	2.56942	0.101964
$636$	0	0
$637$	31.1508	1.23424
$638$	−0.778896	−0.0308368
$639$	0	0
$640$	0.703996	0.0278279
$641$	−1.72492	−0.0681302	−0.0340651	−	0.999420i	$-0.510845\pi$
$641$	−1.72492	−0.0681302	−0.0340651	+	0.999420i	$0.510845\pi$
$642$	0	0
$643$	−37.6517	−1.48484	−0.742420	−	0.669935i	$-0.766322\pi$
$643$	−37.6517	−1.48484	−0.742420	+	0.669935i	$0.766322\pi$
$644$	2.26237	0.0891499
$645$	0	0
$646$	−3.15921	−0.124298
$647$	−46.6191	−1.83279	−0.916393	−	0.400279i	$-0.868913\pi$
$647$	−46.6191	−1.83279	−0.916393	+	0.400279i	$0.868913\pi$
$648$	0	0
$649$	11.0751	0.434734
$650$	−4.02181	−0.157748
$651$	0	0
$652$	−33.6164	−1.31652
$653$	32.6575	1.27799	0.638993	−	0.769212i	$-0.279351\pi$
$653$	32.6575	1.27799	0.638993	+	0.769212i	$0.279351\pi$
$654$	0	0
$655$	−2.15420	−0.0841716
$656$	−5.84988	−0.228399
$657$	0	0
$658$	−0.821767	−0.0320358
$659$	−36.3322	−1.41530	−0.707651	−	0.706562i	$-0.750245\pi$
$659$	−36.3322	−1.41530	−0.707651	+	0.706562i	$0.750245\pi$
$660$	0	0
$661$	20.6126	0.801736	0.400868	−	0.916136i	$-0.368709\pi$
$661$	20.6126	0.801736	0.400868	+	0.916136i	$0.368709\pi$
$662$	0.354975	0.0137965
$663$	0	0
$664$	−7.41723	−0.287844
$665$	−0.643523	−0.0249548
$666$	0	0
$667$	8.46985	0.327954
$668$	17.5809	0.680224
$669$	0	0
$670$	−0.256271	−0.00990063
$671$	1.00000	0.0386046
$672$	0	0
$673$	−5.41924	−0.208896	−0.104448	−	0.994530i	$-0.533308\pi$
$673$	−5.41924	−0.208896	−0.104448	+	0.994530i	$0.533308\pi$
$674$	0.0408862	0.00157488
$675$	0	0
$676$	−17.9974	−0.692208
$677$	15.1357	0.581713	0.290857	−	0.956767i	$-0.406060\pi$
$677$	15.1357	0.581713	0.290857	+	0.956767i	$0.406060\pi$
$678$	0	0
$679$	9.33697	0.358320
$680$	−0.212396	−0.00814502
$681$	0	0
$682$	0.526584	0.0201639
$683$	−5.25700	−0.201154	−0.100577	−	0.994929i	$-0.532069\pi$
$683$	−5.25700	−0.201154	−0.100577	+	0.994929i	$0.532069\pi$
$684$	0	0
$685$	2.26057	0.0863721
$686$	1.43813	0.0549082
$687$	0	0
$688$	4.49363	0.171318
$689$	−52.2742	−1.99149
$690$	0	0
$691$	−37.8441	−1.43966	−0.719828	−	0.694152i	$-0.755780\pi$
$691$	−37.8441	−1.43966	−0.719828	+	0.694152i	$0.755780\pi$
$692$	7.60120	0.288954
$693$	0	0
$694$	4.78753	0.181732
$695$	0.130627	0.00495496
$696$	0	0
$697$	3.58377	0.135745
$698$	−5.26703	−0.199360
$699$	0	0
$700$	6.04124	0.228337
$701$	26.2157	0.990154	0.495077	−	0.868849i	$-0.335140\pi$
$701$	26.2157	0.990154	0.495077	+	0.868849i	$0.335140\pi$
$702$	0	0
$703$	−72.6619	−2.74049
$704$	7.30207	0.275207
$705$	0	0
$706$	5.52699	0.208011
$707$	−9.21328	−0.346501
$708$	0	0
$709$	−22.3870	−0.840762	−0.420381	−	0.907348i	$-0.638104\pi$
$709$	−22.3870	−0.840762	−0.420381	+	0.907348i	$0.638104\pi$
$710$	−0.150579	−0.00565113
$711$	0	0
$712$	−10.3236	−0.386892
$713$	−5.72616	−0.214446
$714$	0	0
$715$	0.626049	0.0234129
$716$	−20.5693	−0.768709
$717$	0	0
$718$	−4.37055	−0.163108
$719$	21.7396	0.810751	0.405376	−	0.914150i	$-0.367141\pi$
$719$	21.7396	0.810751	0.405376	+	0.914150i	$0.367141\pi$
$720$	0	0
$721$	1.63620	0.0609352
$722$	−7.33765	−0.273079
$723$	0	0
$724$	−27.5296	−1.02313
$725$	22.6171	0.839979
$726$	0	0
$727$	−24.4460	−0.906653	−0.453327	−	0.891344i	$-0.649763\pi$
$727$	−24.4460	−0.906653	−0.453327	+	0.891344i	$0.649763\pi$
$728$	−1.97220	−0.0730947
$729$	0	0
$730$	−0.0997666	−0.00369253
$731$	−2.75291	−0.101820
$732$	0	0
$733$	13.6034	0.502452	0.251226	−	0.967928i	$-0.419166\pi$
$733$	13.6034	0.502452	0.251226	+	0.967928i	$0.419166\pi$
$734$	−6.15792	−0.227293
$735$	0	0
$736$	3.76654	0.138837
$737$	−11.2239	−0.413436
$738$	0	0
$739$	1.68561	0.0620061	0.0310031	−	0.999519i	$-0.490130\pi$
$739$	1.68561	0.0620061	0.0310031	+	0.999519i	$0.490130\pi$
$740$	−2.42445	−0.0891244
$741$	0	0
$742$	−1.17313	−0.0430668
$743$	−45.8835	−1.68330	−0.841652	−	0.540020i	$-0.818417\pi$
$743$	−45.8835	−1.68330	−0.841652	+	0.540020i	$0.818417\pi$
$744$	0	0
$745$	−2.65054	−0.0971082
$746$	−4.50643	−0.164992
$747$	0	0
$748$	−4.61665	−0.168801
$749$	−6.13873	−0.224304
$750$	0	0
$751$	−47.2695	−1.72489	−0.862445	−	0.506151i	$-0.831068\pi$
$751$	−47.2695	−1.72489	−0.862445	+	0.506151i	$0.831068\pi$
$752$	29.7655	1.08544
$753$	0	0
$754$	−3.66438	−0.133449
$755$	−1.23382	−0.0449032
$756$	0	0
$757$	−33.4045	−1.21411	−0.607054	−	0.794661i	$-0.707649\pi$
$757$	−33.4045	−1.21411	−0.607054	+	0.794661i	$0.707649\pi$
$758$	3.87451	0.140728
$759$	0	0
$760$	−0.712492	−0.0258448
$761$	−19.2821	−0.698974	−0.349487	−	0.936941i	$-0.613644\pi$
$761$	−19.2821	−0.698974	−0.349487	+	0.936941i	$0.613644\pi$
$762$	0	0
$763$	−5.43556	−0.196781
$764$	−48.1700	−1.74273
$765$	0	0
$766$	0.261421	0.00944551
$767$	52.1036	1.88135
$768$	0	0
$769$	−27.7997	−1.00248	−0.501242	−	0.865307i	$-0.667123\pi$
$769$	−27.7997	−1.00248	−0.501242	+	0.865307i	$0.667123\pi$
$770$	0.0140496	0.000506314 0
$771$	0	0
$772$	12.1434	0.437049
$773$	−15.4428	−0.555438	−0.277719	−	0.960662i	$-0.589578\pi$
$773$	−15.4428	−0.555438	−0.277719	+	0.960662i	$0.589578\pi$
$774$	0	0
$775$	−15.2906	−0.549256
$776$	10.3376	0.371100
$777$	0	0
$778$	−3.67896	−0.131897
$779$	12.0219	0.430730
$780$	0	0
$781$	−6.59487	−0.235983
$782$	−0.750024	−0.0268208
$783$	0	0
$784$	−25.3216	−0.904343
$785$	1.28109	0.0457239
$786$	0	0
$787$	−39.8594	−1.42083	−0.710417	−	0.703781i	$-0.751494\pi$
$787$	−39.8594	−1.42083	−0.710417	+	0.703781i	$0.751494\pi$
$788$	−31.8124	−1.13327
$789$	0	0
$790$	−0.269158	−0.00957620
$791$	2.61259	0.0928929
$792$	0	0
$793$	4.70459	0.167065
$794$	1.35510	0.0480906
$795$	0	0
$796$	23.0614	0.817388
$797$	17.9931	0.637348	0.318674	−	0.947864i	$-0.396762\pi$
$797$	17.9931	0.637348	0.318674	+	0.947864i	$0.396762\pi$
$798$	0	0
$799$	−18.2351	−0.645111
$800$	10.0578	0.355598
$801$	0	0
$802$	1.90284	0.0671915
$803$	−4.36946	−0.154195
$804$	0	0
$805$	−0.152778	−0.00538472
$806$	2.47736	0.0872612
$807$	0	0
$808$	−10.2007	−0.358859
$809$	36.0517	1.26751	0.633755	−	0.773534i	$-0.281513\pi$
$809$	36.0517	1.26751	0.633755	+	0.773534i	$0.281513\pi$
$810$	0	0
$811$	17.6858	0.621032	0.310516	−	0.950568i	$-0.399498\pi$
$811$	17.6858	0.621032	0.310516	+	0.950568i	$0.399498\pi$
$812$	5.50434	0.193165
$813$	0	0
$814$	1.58638	0.0556026
$815$	2.27011	0.0795186
$816$	0	0
$817$	−9.23474	−0.323082
$818$	−0.313967	−0.0109776
$819$	0	0
$820$	0.401125	0.0140079
$821$	14.5517	0.507858	0.253929	−	0.967223i	$-0.418277\pi$
$821$	14.5517	0.507858	0.253929	+	0.967223i	$0.418277\pi$
$822$	0	0
$823$	48.1384	1.67800	0.838999	−	0.544133i	$-0.183141\pi$
$823$	48.1384	1.67800	0.838999	+	0.544133i	$0.183141\pi$
$824$	1.81156	0.0631085
$825$	0	0
$826$	1.16930	0.0406850
$827$	−27.5769	−0.958943	−0.479472	−	0.877557i	$-0.659172\pi$
$827$	−27.5769	−0.958943	−0.479472	+	0.877557i	$0.659172\pi$
$828$	0	0
$829$	−10.6222	−0.368926	−0.184463	−	0.982839i	$-0.559055\pi$
$829$	−10.6222	−0.368926	−0.184463	+	0.982839i	$0.559055\pi$
$830$	0.248586	0.00862855
$831$	0	0
$832$	34.3532	1.19098
$833$	15.5126	0.537481
$834$	0	0
$835$	−1.18724	−0.0410860
$836$	−15.4867	−0.535620
$837$	0	0
$838$	4.99134	0.172423
$839$	48.1933	1.66382	0.831909	−	0.554912i	$-0.187248\pi$
$839$	48.1933	1.66382	0.831909	+	0.554912i	$0.187248\pi$
$840$	0	0
$841$	−8.39289	−0.289410
$842$	0.00961104	0.000331218 0
$843$	0	0
$844$	25.7908	0.887756
$845$	1.21536	0.0418098
$846$	0	0
$847$	0.615329	0.0211430
$848$	42.4922	1.45919
$849$	0	0
$850$	−2.00280	−0.0686955
$851$	−17.2506	−0.591342
$852$	0	0
$853$	46.5258	1.59301	0.796507	−	0.604629i	$-0.206679\pi$
$853$	46.5258	1.59301	0.796507	+	0.604629i	$0.206679\pi$
$854$	0.105579	0.00361285
$855$	0	0
$856$	−6.79664	−0.232304
$857$	10.3918	0.354979	0.177489	−	0.984123i	$-0.443202\pi$
$857$	10.3918	0.354979	0.177489	+	0.984123i	$0.443202\pi$
$858$	0	0
$859$	57.1315	1.94930	0.974651	−	0.223732i	$-0.0718241\pi$
$859$	57.1315	1.94930	0.974651	+	0.223732i	$0.0718241\pi$
$860$	−0.308127	−0.0105071
$861$	0	0
$862$	−0.356613	−0.0121463
$863$	−21.5752	−0.734429	−0.367214	−	0.930136i	$-0.619688\pi$
$863$	−21.5752	−0.734429	−0.367214	+	0.930136i	$0.619688\pi$
$864$	0	0
$865$	−0.513309	−0.0174530
$866$	0.984706	0.0334617
$867$	0	0
$868$	−3.72129	−0.126309
$869$	−11.7882	−0.399889
$870$	0	0
$871$	−52.8036	−1.78918
$872$	−6.01811	−0.203799
$873$	0	0
$874$	−2.51599	−0.0851045
$875$	−0.817380	−0.0276325
$876$	0	0
$877$	−19.6961	−0.665092	−0.332546	−	0.943087i	$-0.607908\pi$
$877$	−19.6961	−0.665092	−0.332546	+	0.943087i	$0.607908\pi$
$878$	−5.60594	−0.189191
$879$	0	0
$880$	−0.508897	−0.0171549
$881$	13.2882	0.447691	0.223845	−	0.974625i	$-0.428139\pi$
$881$	13.2882	0.447691	0.223845	+	0.974625i	$0.428139\pi$
$882$	0	0
$883$	12.8872	0.433688	0.216844	−	0.976206i	$-0.430424\pi$
$883$	12.8872	0.433688	0.216844	+	0.976206i	$0.430424\pi$
$884$	−21.7194	−0.730504
$885$	0	0
$886$	−4.43055	−0.148847
$887$	−1.36425	−0.0458071	−0.0229036	−	0.999738i	$-0.507291\pi$
$887$	−1.36425	−0.0458071	−0.0229036	+	0.999738i	$0.507291\pi$
$888$	0	0
$889$	11.8811	0.398478
$890$	0.345991	0.0115976
$891$	0	0
$892$	−4.07463	−0.136429
$893$	−61.1703	−2.04699
$894$	0	0
$895$	1.38904	0.0464306
$896$	3.25530	0.108752
$897$	0	0
$898$	−3.20157	−0.106838
$899$	−13.9317	−0.464649
$900$	0	0
$901$	−26.0318	−0.867244
$902$	−0.262467	−0.00873919
$903$	0	0
$904$	2.89258	0.0962059
$905$	1.85908	0.0617978
$906$	0	0
$907$	52.3304	1.73760	0.868802	−	0.495160i	$-0.164890\pi$
$907$	52.3304	1.73760	0.868802	+	0.495160i	$0.164890\pi$
$908$	−12.1229	−0.402312
$909$	0	0
$910$	0.0660977	0.00219112
$911$	−4.50180	−0.149151	−0.0745756	−	0.997215i	$-0.523760\pi$
$911$	−4.50180	−0.149151	−0.0745756	+	0.997215i	$0.523760\pi$
$912$	0	0
$913$	10.8873	0.360316
$914$	0.750944	0.0248390
$915$	0	0
$916$	−38.6156	−1.27590
$917$	−9.96110	−0.328944
$918$	0	0
$919$	25.3477	0.836143	0.418072	−	0.908414i	$-0.362706\pi$
$919$	25.3477	0.836143	0.418072	+	0.908414i	$0.362706\pi$
$920$	−0.169152	−0.00557677
$921$	0	0
$922$	−6.25317	−0.205937
$923$	−31.0262	−1.02124
$924$	0	0
$925$	−46.0644	−1.51459
$926$	−2.50154	−0.0822057
$927$	0	0
$928$	9.16398	0.300822
$929$	−20.9203	−0.686372	−0.343186	−	0.939267i	$-0.611506\pi$
$929$	−20.9203	−0.686372	−0.343186	+	0.939267i	$0.611506\pi$
$930$	0	0
$931$	52.0377	1.70547
$932$	−40.5446	−1.32808
$933$	0	0
$934$	4.55895	0.149173
$935$	0.311763	0.0101957
$936$	0	0
$937$	33.7225	1.10167	0.550833	−	0.834615i	$-0.314310\pi$
$937$	33.7225	1.10167	0.550833	+	0.834615i	$0.314310\pi$
$938$	−1.18501	−0.0386918
$939$	0	0
$940$	−2.04102	−0.0665706
$941$	45.8406	1.49436	0.747180	−	0.664621i	$-0.231407\pi$
$941$	45.8406	1.49436	0.747180	+	0.664621i	$0.231407\pi$
$942$	0	0
$943$	2.85411	0.0929425
$944$	−42.3535	−1.37849
$945$	0	0
$946$	0.201616	0.00655510
$947$	31.2864	1.01667	0.508336	−	0.861159i	$-0.330261\pi$
$947$	31.2864	1.01667	0.508336	+	0.861159i	$0.330261\pi$
$948$	0	0
$949$	−20.5565	−0.667292
$950$	−6.71847	−0.217976
$951$	0	0
$952$	−0.982127	−0.0318309
$953$	−58.6406	−1.89955	−0.949777	−	0.312926i	$-0.898691\pi$
$953$	−58.6406	−1.89955	−0.949777	+	0.312926i	$0.898691\pi$
$954$	0	0
$955$	3.25292	0.105262
$956$	−45.5944	−1.47463
$957$	0	0
$958$	0.800329	0.0258574
$959$	10.4530	0.337544
$960$	0	0
$961$	−21.5813	−0.696170
$962$	7.46326	0.240625
$963$	0	0
$964$	−0.193778	−0.00624117
$965$	−0.820041	−0.0263981
$966$	0	0
$967$	−35.3660	−1.13729	−0.568647	−	0.822582i	$-0.692533\pi$
$967$	−35.3660	−1.13729	−0.568647	+	0.822582i	$0.692533\pi$
$968$	0.681275	0.0218970
$969$	0	0
$970$	−0.346462	−0.0111242
$971$	−21.4189	−0.687366	−0.343683	−	0.939086i	$-0.611675\pi$
$971$	−21.4189	−0.687366	−0.343683	+	0.939086i	$0.611675\pi$
$972$	0	0
$973$	0.604022	0.0193641
$974$	−2.78397	−0.0892042
$975$	0	0
$976$	−3.82423	−0.122410
$977$	50.4860	1.61519	0.807595	−	0.589737i	$-0.200769\pi$
$977$	50.4860	1.61519	0.807595	+	0.589737i	$0.200769\pi$
$978$	0	0
$979$	15.1533	0.484301
$980$	1.73630	0.0554640
$981$	0	0
$982$	6.49410	0.207235
$983$	4.20465	0.134107	0.0670537	−	0.997749i	$-0.478640\pi$
$983$	4.20465	0.134107	0.0670537	+	0.997749i	$0.478640\pi$
$984$	0	0
$985$	2.14829	0.0684503
$986$	−1.82481	−0.0581137
$987$	0	0
$988$	−72.8587	−2.31794
$989$	−2.19241	−0.0697145
$990$	0	0
$991$	14.4063	0.457631	0.228816	−	0.973470i	$-0.426515\pi$
$991$	14.4063	0.457631	0.228816	+	0.973470i	$0.426515\pi$
$992$	−6.19544	−0.196705
$993$	0	0
$994$	−0.696282	−0.0220847
$995$	−1.55734	−0.0493708
$996$	0	0
$997$	25.4235	0.805169	0.402585	−	0.915383i	$-0.368112\pi$
$997$	25.4235	0.805169	0.402585	+	0.915383i	$0.368112\pi$
$998$	2.22147	0.0703193
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6039.2.a.i.1.7		13
3.2	odd	2		2013.2.a.e.1.7	✓	13

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2013.2.a.e.1.7	✓	13	3.2	odd	2
6039.2.a.i.1.7		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)

\(f(q)\)	\(=\)	\(q-0.171582 q^{2} -1.97056 q^{4} +0.133072 q^{5} +0.615329 q^{7} +0.681275 q^{8} +O(q^{10})\)	\(q-0.171582 q^{2} -1.97056 q^{4} +0.133072 q^{5} +0.615329 q^{7} +0.681275 q^{8} -0.0228327 q^{10} -1.00000 q^{11} -4.70459 q^{13} -0.105579 q^{14} +3.82423 q^{16} -2.34281 q^{17} -7.85906 q^{19} -0.262226 q^{20} +0.171582 q^{22} -1.86581 q^{23} -4.98229 q^{25} +0.807221 q^{26} -1.21254 q^{28} -4.53951 q^{29} +3.06900 q^{31} -2.01872 q^{32} +0.401984 q^{34} +0.0818830 q^{35} +9.24562 q^{37} +1.34847 q^{38} +0.0906587 q^{40} -1.52969 q^{41} +1.17504 q^{43} +1.97056 q^{44} +0.320139 q^{46} +7.78342 q^{47} -6.62137 q^{49} +0.854870 q^{50} +9.27067 q^{52} +11.1113 q^{53} -0.133072 q^{55} +0.419208 q^{56} +0.778896 q^{58} -11.0751 q^{59} -1.00000 q^{61} -0.526584 q^{62} -7.30207 q^{64} -0.626049 q^{65} +11.2239 q^{67} +4.61665 q^{68} -0.0140496 q^{70} +6.59487 q^{71} +4.36946 q^{73} -1.58638 q^{74} +15.4867 q^{76} -0.615329 q^{77} +11.7882 q^{79} +0.508897 q^{80} +0.262467 q^{82} -10.8873 q^{83} -0.311763 q^{85} -0.201616 q^{86} -0.681275 q^{88} -15.1533 q^{89} -2.89487 q^{91} +3.67669 q^{92} -1.33549 q^{94} -1.04582 q^{95} +15.1739 q^{97} +1.13611 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 13 q - 2 q^{2} + 16 q^{4} - 3 q^{5} + 11 q^{7} - 9 q^{8}+O(q^{10}) \) 13 * q - 2 * q^2 + 16 * q^4 - 3 * q^5 + 11 * q^7 - 9 * q^8	\( 13 q - 2 q^{2} + 16 q^{4} - 3 q^{5} + 11 q^{7} - 9 q^{8} + 6 q^{10} - 13 q^{11} + 13 q^{13} - q^{14} + 18 q^{16} - 17 q^{17} + 14 q^{19} + 7 q^{20} + 2 q^{22} - 7 q^{23} + 18 q^{25} + 10 q^{26} + 19 q^{28} + 6 q^{29} + 27 q^{31} - 5 q^{32} + 6 q^{34} - 14 q^{35} + 10 q^{37} - 2 q^{38} + 8 q^{40} - 3 q^{41} + 29 q^{43} - 16 q^{44} - 24 q^{46} - 8 q^{47} + 8 q^{49} + 27 q^{50} + 37 q^{52} + 24 q^{53} + 3 q^{55} - 24 q^{56} - 5 q^{58} - 13 q^{59} - 13 q^{61} - 39 q^{62} + 47 q^{64} + 11 q^{65} + 44 q^{67} + 8 q^{68} - 12 q^{70} - 3 q^{71} + 48 q^{73} + 22 q^{74} + 47 q^{76} - 11 q^{77} - 17 q^{79} + 26 q^{80} + 56 q^{82} - 50 q^{83} + 8 q^{85} - 18 q^{86} + 9 q^{88} + 15 q^{89} + 47 q^{91} - 14 q^{92} + 45 q^{94} + q^{95} + 27 q^{97} - 47 q^{98}+O(q^{100}) \) 13 * q - 2 * q^2 + 16 * q^4 - 3 * q^5 + 11 * q^7 - 9 * q^8 + 6 * q^10 - 13 * q^11 + 13 * q^13 - q^14 + 18 * q^16 - 17 * q^17 + 14 * q^19 + 7 * q^20 + 2 * q^22 - 7 * q^23 + 18 * q^25 + 10 * q^26 + 19 * q^28 + 6 * q^29 + 27 * q^31 - 5 * q^32 + 6 * q^34 - 14 * q^35 + 10 * q^37 - 2 * q^38 + 8 * q^40 - 3 * q^41 + 29 * q^43 - 16 * q^44 - 24 * q^46 - 8 * q^47 + 8 * q^49 + 27 * q^50 + 37 * q^52 + 24 * q^53 + 3 * q^55 - 24 * q^56 - 5 * q^58 - 13 * q^59 - 13 * q^61 - 39 * q^62 + 47 * q^64 + 11 * q^65 + 44 * q^67 + 8 * q^68 - 12 * q^70 - 3 * q^71 + 48 * q^73 + 22 * q^74 + 47 * q^76 - 11 * q^77 - 17 * q^79 + 26 * q^80 + 56 * q^82 - 50 * q^83 + 8 * q^85 - 18 * q^86 + 9 * q^88 + 15 * q^89 + 47 * q^91 - 14 * q^92 + 45 * q^94 + q^95 + 27 * q^97 - 47 * q^98

Introduction

L-functions

Modular forms

Varieties

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Database

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