LMFDB - Embedded newform 6039.2.a.f.1.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

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

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

chi := DirichletCharacter("6039.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$48.2216577807$
Analytic rank:	$0$
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			1.9
Root			$-1.03538$ of defining polynomial
Character	$\chi$	$=$	6039.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.03538 q^{2} +2.14278 q^{4} -3.64320 q^{5} -4.68983 q^{7} +0.290605 q^{8} +O(q^{10})$	\(q+2.03538 q^{2} +2.14278 q^{4} -3.64320 q^{5} -4.68983 q^{7} +0.290605 q^{8} -7.41529 q^{10} -1.00000 q^{11} +2.53815 q^{13} -9.54560 q^{14} -3.69406 q^{16} +7.26013 q^{17} -7.00236 q^{19} -7.80656 q^{20} -2.03538 q^{22} -2.30296 q^{23} +8.27289 q^{25} +5.16610 q^{26} -10.0493 q^{28} -4.04230 q^{29} -7.56494 q^{31} -8.10003 q^{32} +14.7771 q^{34} +17.0860 q^{35} -4.60383 q^{37} -14.2525 q^{38} -1.05873 q^{40} +3.50819 q^{41} -4.39873 q^{43} -2.14278 q^{44} -4.68740 q^{46} +12.2779 q^{47} +14.9945 q^{49} +16.8385 q^{50} +5.43869 q^{52} +5.26804 q^{53} +3.64320 q^{55} -1.36289 q^{56} -8.22761 q^{58} +1.23759 q^{59} -1.00000 q^{61} -15.3975 q^{62} -9.09853 q^{64} -9.24698 q^{65} +6.04060 q^{67} +15.5568 q^{68} +34.7765 q^{70} +12.6630 q^{71} +2.01935 q^{73} -9.37055 q^{74} -15.0045 q^{76} +4.68983 q^{77} +1.56355 q^{79} +13.4582 q^{80} +7.14050 q^{82} +13.7810 q^{83} -26.4501 q^{85} -8.95310 q^{86} -0.290605 q^{88} +2.49179 q^{89} -11.9035 q^{91} -4.93473 q^{92} +24.9902 q^{94} +25.5110 q^{95} +10.3814 q^{97} +30.5196 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 7 q^{2} + 13 q^{4} + 7 q^{5} - 15 q^{7} + 18 q^{8}+O(q^{10}) $ 12 * q + 7 * q^2 + 13 * q^4 + 7 * q^5 - 15 * q^7 + 18 * q^8	$ 12 q + 7 q^{2} + 13 q^{4} + 7 q^{5} - 15 q^{7} + 18 q^{8} - 6 q^{10} - 12 q^{11} - 11 q^{13} - 3 q^{14} + 19 q^{16} + 33 q^{17} - 24 q^{19} + 11 q^{20} - 7 q^{22} + 9 q^{23} + 11 q^{25} + 16 q^{26} - 41 q^{28} + 16 q^{29} + q^{31} + 28 q^{32} + 32 q^{34} + 22 q^{35} - 6 q^{37} - 12 q^{38} + 26 q^{40} + 21 q^{41} - 39 q^{43} - 13 q^{44} + 18 q^{47} + 31 q^{49} + 44 q^{50} + 3 q^{52} + 14 q^{53} - 7 q^{55} - 16 q^{56} + 33 q^{58} + 23 q^{59} - 12 q^{61} + 25 q^{62} + 12 q^{64} + 29 q^{65} + 96 q^{68} + 44 q^{70} + 19 q^{71} - 42 q^{73} - 38 q^{74} + 11 q^{76} + 15 q^{77} - 11 q^{79} + 44 q^{80} - 14 q^{82} + 56 q^{83} + 16 q^{85} + 18 q^{86} - 18 q^{88} + 55 q^{89} + 11 q^{91} + 4 q^{92} - 5 q^{94} - 15 q^{95} - 7 q^{97} - 6 q^{98}+O(q^{100}) $ 12 * q + 7 * q^2 + 13 * q^4 + 7 * q^5 - 15 * q^7 + 18 * q^8 - 6 * q^10 - 12 * q^11 - 11 * q^13 - 3 * q^14 + 19 * q^16 + 33 * q^17 - 24 * q^19 + 11 * q^20 - 7 * q^22 + 9 * q^23 + 11 * q^25 + 16 * q^26 - 41 * q^28 + 16 * q^29 + q^31 + 28 * q^32 + 32 * q^34 + 22 * q^35 - 6 * q^37 - 12 * q^38 + 26 * q^40 + 21 * q^41 - 39 * q^43 - 13 * q^44 + 18 * q^47 + 31 * q^49 + 44 * q^50 + 3 * q^52 + 14 * q^53 - 7 * q^55 - 16 * q^56 + 33 * q^58 + 23 * q^59 - 12 * q^61 + 25 * q^62 + 12 * q^64 + 29 * q^65 + 96 * q^68 + 44 * q^70 + 19 * q^71 - 42 * q^73 - 38 * q^74 + 11 * q^76 + 15 * q^77 - 11 * q^79 + 44 * q^80 - 14 * q^82 + 56 * q^83 + 16 * q^85 + 18 * q^86 - 18 * q^88 + 55 * q^89 + 11 * q^91 + 4 * q^92 - 5 * q^94 - 15 * q^95 - 7 * q^97 - 6 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	2.03538	1.43923	0.719616	−	0.694372i	$-0.244318\pi$
$2$	2.03538	1.43923	0.719616	+	0.694372i	$0.244318\pi$
$3$	0	0
$3$	0	0
$4$	2.14278	1.07139
$5$	−3.64320	−1.62929	−0.814644	−	0.579962i	$-0.803067\pi$
$5$	−3.64320	−1.62929	−0.814644	+	0.579962i	$0.803067\pi$
$6$	0	0
$7$	−4.68983	−1.77259	−0.886295	−	0.463121i	$-0.846730\pi$
$7$	−4.68983	−1.77259	−0.886295	+	0.463121i	$0.846730\pi$
$8$	0.290605	0.102744
$9$	0	0
$10$	−7.41529	−2.34492
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	2.53815	0.703956	0.351978	−	0.936008i	$-0.385509\pi$
$13$	2.53815	0.703956	0.351978	+	0.936008i	$0.385509\pi$
$14$	−9.54560	−2.55117
$15$	0	0
$16$	−3.69406	−0.923515
$17$	7.26013	1.76084	0.880420	−	0.474195i	$-0.157261\pi$
$17$	7.26013	1.76084	0.880420	+	0.474195i	$0.157261\pi$
$18$	0	0
$19$	−7.00236	−1.60645	−0.803226	−	0.595675i	$-0.796885\pi$
$19$	−7.00236	−1.60645	−0.803226	+	0.595675i	$0.796885\pi$
$20$	−7.80656	−1.74560
$21$	0	0
$22$	−2.03538	−0.433945
$23$	−2.30296	−0.480200	−0.240100	−	0.970748i	$-0.577180\pi$
$23$	−2.30296	−0.480200	−0.240100	+	0.970748i	$0.577180\pi$
$24$	0	0
$25$	8.27289	1.65458
$26$	5.16610	1.01316
$27$	0	0
$28$	−10.0493	−1.89913
$29$	−4.04230	−0.750635	−0.375318	−	0.926896i	$-0.622466\pi$
$29$	−4.04230	−0.750635	−0.375318	+	0.926896i	$0.622466\pi$
$30$	0	0
$31$	−7.56494	−1.35870	−0.679352	−	0.733813i	$-0.737739\pi$
$31$	−7.56494	−1.35870	−0.679352	+	0.733813i	$0.737739\pi$
$32$	−8.10003	−1.43190
$33$	0	0
$34$	14.7771	2.53426
$35$	17.0860	2.88806
$36$	0	0
$37$	−4.60383	−0.756865	−0.378433	−	0.925629i	$-0.623537\pi$
$37$	−4.60383	−0.756865	−0.378433	+	0.925629i	$0.623537\pi$
$38$	−14.2525	−2.31206
$39$	0	0
$40$	−1.05873	−0.167400
$41$	3.50819	0.547887	0.273943	−	0.961746i	$-0.411672\pi$
$41$	3.50819	0.547887	0.273943	+	0.961746i	$0.411672\pi$
$42$	0	0
$43$	−4.39873	−0.670801	−0.335400	−	0.942076i	$-0.608872\pi$
$43$	−4.39873	−0.670801	−0.335400	+	0.942076i	$0.608872\pi$
$44$	−2.14278	−0.323036
$45$	0	0
$46$	−4.68740	−0.691119
$47$	12.2779	1.79092	0.895458	−	0.445147i	$-0.146849\pi$
$47$	12.2779	1.79092	0.895458	+	0.445147i	$0.146849\pi$
$48$	0	0
$49$	14.9945	2.14208
$50$	16.8385	2.38132
$51$	0	0
$52$	5.43869	0.754210
$53$	5.26804	0.723621	0.361811	−	0.932252i	$-0.382159\pi$
$53$	5.26804	0.723621	0.361811	+	0.932252i	$0.382159\pi$
$54$	0	0
$55$	3.64320	0.491249
$56$	−1.36289	−0.182123
$57$	0	0
$58$	−8.22761	−1.08034
$59$	1.23759	0.161120	0.0805600	−	0.996750i	$-0.474329\pi$
$59$	1.23759	0.161120	0.0805600	+	0.996750i	$0.474329\pi$
$60$	0	0
$61$	−1.00000	−0.128037
$61$	−1.00000	−0.128037
$62$	−15.3975	−1.95549
$63$	0	0
$64$	−9.09853	−1.13732
$65$	−9.24698	−1.14695
$66$	0	0
$67$	6.04060	0.737977	0.368988	−	0.929434i	$-0.379704\pi$
$67$	6.04060	0.737977	0.368988	+	0.929434i	$0.379704\pi$
$68$	15.5568	1.88654
$69$	0	0
$70$	34.7765	4.15659
$71$	12.6630	1.50282	0.751412	−	0.659834i	$-0.229373\pi$
$71$	12.6630	1.50282	0.751412	+	0.659834i	$0.229373\pi$
$72$	0	0
$73$	2.01935	0.236347	0.118174	−	0.992993i	$-0.462296\pi$
$73$	2.01935	0.236347	0.118174	+	0.992993i	$0.462296\pi$
$74$	−9.37055	−1.08930
$75$	0	0
$76$	−15.0045	−1.72113
$77$	4.68983	0.534456
$78$	0	0
$79$	1.56355	0.175913	0.0879563	−	0.996124i	$-0.471966\pi$
$79$	1.56355	0.175913	0.0879563	+	0.996124i	$0.471966\pi$
$80$	13.4582	1.50467
$81$	0	0
$82$	7.14050	0.788536
$83$	13.7810	1.51267	0.756333	−	0.654187i	$-0.226989\pi$
$83$	13.7810	1.51267	0.756333	+	0.654187i	$0.226989\pi$
$84$	0	0
$85$	−26.4501	−2.86891
$86$	−8.95310	−0.965438
$87$	0	0
$88$	−0.290605	−0.0309786
$89$	2.49179	0.264129	0.132065	−	0.991241i	$-0.457839\pi$
$89$	2.49179	0.264129	0.132065	+	0.991241i	$0.457839\pi$
$90$	0	0
$91$	−11.9035	−1.24783
$92$	−4.93473	−0.514481
$93$	0	0
$94$	24.9902	2.57754
$95$	25.5110	2.61737
$96$	0	0
$97$	10.3814	1.05407	0.527035	−	0.849844i	$-0.323304\pi$
$97$	10.3814	1.05407	0.527035	+	0.849844i	$0.323304\pi$
$98$	30.5196	3.08294
$99$	0	0
$100$	17.7269	1.77269
$101$	−17.6969	−1.76091	−0.880456	−	0.474129i	$-0.842763\pi$
$101$	−17.6969	−1.76091	−0.880456	+	0.474129i	$0.842763\pi$
$102$	0	0
$103$	14.1535	1.39459	0.697294	−	0.716785i	$-0.254387\pi$
$103$	14.1535	1.39459	0.697294	+	0.716785i	$0.254387\pi$
$104$	0.737598	0.0723275
$105$	0	0
$106$	10.7225	1.04146
$107$	−8.84032	−0.854626	−0.427313	−	0.904104i	$-0.640540\pi$
$107$	−8.84032	−0.854626	−0.427313	+	0.904104i	$0.640540\pi$
$108$	0	0
$109$	6.08893	0.583214	0.291607	−	0.956538i	$-0.405810\pi$
$109$	6.08893	0.583214	0.291607	+	0.956538i	$0.405810\pi$
$110$	7.41529	0.707021
$111$	0	0
$112$	17.3245	1.63701
$113$	5.34589	0.502899	0.251449	−	0.967870i	$-0.419093\pi$
$113$	5.34589	0.502899	0.251449	+	0.967870i	$0.419093\pi$
$114$	0	0
$115$	8.39013	0.782384
$116$	−8.66174	−0.804222
$117$	0	0
$118$	2.51896	0.231889
$119$	−34.0488	−3.12125
$120$	0	0
$121$	1.00000	0.0909091
$122$	−2.03538	−0.184275
$123$	0	0
$124$	−16.2100	−1.45570
$125$	−11.9238	−1.06649
$126$	0	0
$127$	−11.5221	−1.02242	−0.511211	−	0.859455i	$-0.670803\pi$
$127$	−11.5221	−1.02242	−0.511211	+	0.859455i	$0.670803\pi$
$128$	−2.31891	−0.204965
$129$	0	0
$130$	−18.8211	−1.65072
$131$	−21.2523	−1.85682	−0.928409	−	0.371560i	$-0.878823\pi$
$131$	−21.2523	−1.85682	−0.928409	+	0.371560i	$0.878823\pi$
$132$	0	0
$133$	32.8399	2.84758
$134$	12.2949	1.06212
$135$	0	0
$136$	2.10983	0.180916
$137$	14.7406	1.25937	0.629686	−	0.776850i	$-0.283184\pi$
$137$	14.7406	1.25937	0.629686	+	0.776850i	$0.283184\pi$
$138$	0	0
$139$	−15.4607	−1.31136	−0.655681	−	0.755038i	$-0.727618\pi$
$139$	−15.4607	−1.31136	−0.655681	+	0.755038i	$0.727618\pi$
$140$	36.6115	3.09423
$141$	0	0
$142$	25.7741	2.16291
$143$	−2.53815	−0.212251
$144$	0	0
$145$	14.7269	1.22300
$146$	4.11015	0.340159
$147$	0	0
$148$	−9.86498	−0.810897
$149$	−6.25623	−0.512530	−0.256265	−	0.966607i	$-0.582492\pi$
$149$	−6.25623	−0.512530	−0.256265	+	0.966607i	$0.582492\pi$
$150$	0	0
$151$	12.3903	1.00831	0.504155	−	0.863613i	$-0.331804\pi$
$151$	12.3903	1.00831	0.504155	+	0.863613i	$0.331804\pi$
$152$	−2.03492	−0.165054
$153$	0	0
$154$	9.54560	0.769206
$155$	27.5606	2.21372
$156$	0	0
$157$	−6.06191	−0.483793	−0.241897	−	0.970302i	$-0.577769\pi$
$157$	−6.06191	−0.483793	−0.241897	+	0.970302i	$0.577769\pi$
$158$	3.18241	0.253179
$159$	0	0
$160$	29.5100	2.33297
$161$	10.8005	0.851198
$162$	0	0
$163$	−0.681180	−0.0533541	−0.0266771	−	0.999644i	$-0.508493\pi$
$163$	−0.681180	−0.0533541	−0.0266771	+	0.999644i	$0.508493\pi$
$164$	7.51726	0.587000
$165$	0	0
$166$	28.0497	2.17708
$167$	−9.15595	−0.708508	−0.354254	−	0.935149i	$-0.615265\pi$
$167$	−9.15595	−0.708508	−0.354254	+	0.935149i	$0.615265\pi$
$168$	0	0
$169$	−6.55779	−0.504446
$170$	−53.8360	−4.12903
$171$	0	0
$172$	−9.42550	−0.718688
$173$	24.7562	1.88218	0.941090	−	0.338157i	$-0.109804\pi$
$173$	24.7562	1.88218	0.941090	+	0.338157i	$0.109804\pi$
$174$	0	0
$175$	−38.7985	−2.93289
$176$	3.69406	0.278450
$177$	0	0
$178$	5.07174	0.380143
$179$	−0.378311	−0.0282763	−0.0141381	−	0.999900i	$-0.504500\pi$
$179$	−0.378311	−0.0282763	−0.0141381	+	0.999900i	$0.504500\pi$
$180$	0	0
$181$	−18.2472	−1.35631	−0.678153	−	0.734921i	$-0.737219\pi$
$181$	−18.2472	−1.35631	−0.678153	+	0.734921i	$0.737219\pi$
$182$	−24.2282	−1.79591
$183$	0	0
$184$	−0.669250	−0.0493378
$185$	16.7727	1.23315
$186$	0	0
$187$	−7.26013	−0.530913
$188$	26.3088	1.91877
$189$	0	0
$190$	51.9245	3.76700
$191$	−0.0423953	−0.00306762	−0.00153381	−	0.999999i	$-0.500488\pi$
$191$	−0.0423953	−0.00306762	−0.00153381	+	0.999999i	$0.500488\pi$
$192$	0	0
$193$	8.68788	0.625367	0.312684	−	0.949857i	$-0.398772\pi$
$193$	8.68788	0.625367	0.312684	+	0.949857i	$0.398772\pi$
$194$	21.1301	1.51705
$195$	0	0
$196$	32.1299	2.29500
$197$	15.2402	1.08582	0.542908	−	0.839792i	$-0.317323\pi$
$197$	15.2402	1.08582	0.542908	+	0.839792i	$0.317323\pi$
$198$	0	0
$199$	11.7486	0.832835	0.416417	−	0.909174i	$-0.363286\pi$
$199$	11.7486	0.832835	0.416417	+	0.909174i	$0.363286\pi$
$200$	2.40414	0.169998
$201$	0	0
$202$	−36.0200	−2.53436
$203$	18.9577	1.33057
$204$	0	0
$205$	−12.7810	−0.892665
$206$	28.8078	2.00713
$207$	0	0
$208$	−9.37608	−0.650115
$209$	7.00236	0.484363
$210$	0	0
$211$	−7.49488	−0.515969	−0.257984	−	0.966149i	$-0.583058\pi$
$211$	−7.49488	−0.515969	−0.257984	+	0.966149i	$0.583058\pi$
$212$	11.2882	0.775279
$213$	0	0
$214$	−17.9934	−1.23001
$215$	16.0255	1.09293
$216$	0	0
$217$	35.4783	2.40843
$218$	12.3933	0.839380
$219$	0	0
$220$	7.80656	0.526318
$221$	18.4273	1.23955
$222$	0	0
$223$	−19.6088	−1.31310	−0.656550	−	0.754282i	$-0.727985\pi$
$223$	−19.6088	−1.31310	−0.656550	+	0.754282i	$0.727985\pi$
$224$	37.9878	2.53817
$225$	0	0
$226$	10.8809	0.723788
$227$	10.2134	0.677889	0.338944	−	0.940806i	$-0.389930\pi$
$227$	10.2134	0.677889	0.338944	+	0.940806i	$0.389930\pi$
$228$	0	0
$229$	−16.8612	−1.11422	−0.557109	−	0.830439i	$-0.688090\pi$
$229$	−16.8612	−1.11422	−0.557109	+	0.830439i	$0.688090\pi$
$230$	17.0771	1.12603
$231$	0	0
$232$	−1.17471	−0.0771235
$233$	−22.2881	−1.46014	−0.730072	−	0.683370i	$-0.760514\pi$
$233$	−22.2881	−1.46014	−0.730072	+	0.683370i	$0.760514\pi$
$234$	0	0
$235$	−44.7308	−2.91792
$236$	2.65187	0.172622
$237$	0	0
$238$	−69.3023	−4.49220
$239$	−19.9170	−1.28833	−0.644163	−	0.764888i	$-0.722794\pi$
$239$	−19.9170	−1.28833	−0.644163	+	0.764888i	$0.722794\pi$
$240$	0	0
$241$	10.8476	0.698758	0.349379	−	0.936981i	$-0.386393\pi$
$241$	10.8476	0.698758	0.349379	+	0.936981i	$0.386393\pi$
$242$	2.03538	0.130839
$243$	0	0
$244$	−2.14278	−0.137177
$245$	−54.6280	−3.49006
$246$	0	0
$247$	−17.7730	−1.13087
$248$	−2.19841	−0.139599
$249$	0	0
$250$	−24.2694	−1.53493
$251$	5.74852	0.362843	0.181422	−	0.983405i	$-0.441930\pi$
$251$	5.74852	0.362843	0.181422	+	0.983405i	$0.441930\pi$
$252$	0	0
$253$	2.30296	0.144786
$254$	−23.4519	−1.47150
$255$	0	0
$256$	13.4772	0.842324
$257$	24.2003	1.50957	0.754786	−	0.655971i	$-0.227741\pi$
$257$	24.2003	1.50957	0.754786	+	0.655971i	$0.227741\pi$
$258$	0	0
$259$	21.5912	1.34161
$260$	−19.8142	−1.22883
$261$	0	0
$262$	−43.2564	−2.67239
$263$	17.0507	1.05139	0.525697	−	0.850672i	$-0.323805\pi$
$263$	17.0507	1.05139	0.525697	+	0.850672i	$0.323805\pi$
$264$	0	0
$265$	−19.1925	−1.17899
$266$	66.8417	4.09833
$267$	0	0
$268$	12.9437	0.790660
$269$	−15.6684	−0.955319	−0.477659	−	0.878545i	$-0.658515\pi$
$269$	−15.6684	−0.955319	−0.477659	+	0.878545i	$0.658515\pi$
$270$	0	0
$271$	8.11046	0.492675	0.246338	−	0.969184i	$-0.420773\pi$
$271$	8.11046	0.492675	0.246338	+	0.969184i	$0.420773\pi$
$272$	−26.8194	−1.62616
$273$	0	0
$274$	30.0027	1.81253
$275$	−8.27289	−0.498874
$276$	0	0
$277$	−0.347564	−0.0208831	−0.0104416	−	0.999945i	$-0.503324\pi$
$277$	−0.347564	−0.0208831	−0.0104416	+	0.999945i	$0.503324\pi$
$278$	−31.4685	−1.88735
$279$	0	0
$280$	4.96527	0.296731
$281$	31.5166	1.88012	0.940062	−	0.341003i	$-0.110767\pi$
$281$	31.5166	1.88012	0.940062	+	0.341003i	$0.110767\pi$
$282$	0	0
$283$	−23.2867	−1.38425	−0.692127	−	0.721776i	$-0.743326\pi$
$283$	−23.2867	−1.38425	−0.692127	+	0.721776i	$0.743326\pi$
$284$	27.1340	1.61011
$285$	0	0
$286$	−5.16610	−0.305478
$287$	−16.4528	−0.971179
$288$	0	0
$289$	35.7094	2.10056
$290$	29.9748	1.76018
$291$	0	0
$292$	4.32702	0.253220
$293$	13.8274	0.807803	0.403901	−	0.914803i	$-0.367654\pi$
$293$	13.8274	0.807803	0.403901	+	0.914803i	$0.367654\pi$
$294$	0	0
$295$	−4.50877	−0.262511
$296$	−1.33789	−0.0777636
$297$	0	0
$298$	−12.7338	−0.737650
$299$	−5.84526	−0.338040
$300$	0	0
$301$	20.6293	1.18905
$302$	25.2190	1.45119
$303$	0	0
$304$	25.8671	1.48358
$305$	3.64320	0.208609
$306$	0	0
$307$	−0.622350	−0.0355194	−0.0177597	−	0.999842i	$-0.505653\pi$
$307$	−0.622350	−0.0355194	−0.0177597	+	0.999842i	$0.505653\pi$
$308$	10.0493	0.572610
$309$	0	0
$310$	56.0963	3.18606
$311$	−32.0025	−1.81469	−0.907347	−	0.420383i	$-0.861896\pi$
$311$	−32.0025	−1.81469	−0.907347	+	0.420383i	$0.861896\pi$
$312$	0	0
$313$	9.75051	0.551132	0.275566	−	0.961282i	$-0.411135\pi$
$313$	9.75051	0.551132	0.275566	+	0.961282i	$0.411135\pi$
$314$	−12.3383	−0.696291
$315$	0	0
$316$	3.35033	0.188471
$317$	9.83781	0.552547	0.276273	−	0.961079i	$-0.410901\pi$
$317$	9.83781	0.552547	0.276273	+	0.961079i	$0.410901\pi$
$318$	0	0
$319$	4.04230	0.226325
$320$	33.1477	1.85302
$321$	0	0
$322$	21.9831	1.22507
$323$	−50.8380	−2.82870
$324$	0	0
$325$	20.9978	1.16475
$326$	−1.38646	−0.0767889
$327$	0	0
$328$	1.01950	0.0562922
$329$	−57.5813	−3.17456
$330$	0	0
$331$	10.4811	0.576096	0.288048	−	0.957616i	$-0.406994\pi$
$331$	10.4811	0.576096	0.288048	+	0.957616i	$0.406994\pi$
$332$	29.5297	1.62065
$333$	0	0
$334$	−18.6358	−1.01971
$335$	−22.0071	−1.20238
$336$	0	0
$337$	−7.34494	−0.400105	−0.200052	−	0.979785i	$-0.564111\pi$
$337$	−7.34494	−0.400105	−0.200052	+	0.979785i	$0.564111\pi$
$338$	−13.3476	−0.726014
$339$	0	0
$340$	−56.6766	−3.07372
$341$	7.56494	0.409665
$342$	0	0
$343$	−37.4930	−2.02443
$344$	−1.27829	−0.0689209
$345$	0	0
$346$	50.3883	2.70889
$347$	−14.3873	−0.772353	−0.386176	−	0.922425i	$-0.626204\pi$
$347$	−14.3873	−0.772353	−0.386176	+	0.922425i	$0.626204\pi$
$348$	0	0
$349$	−0.901220	−0.0482412	−0.0241206	−	0.999709i	$-0.507679\pi$
$349$	−0.901220	−0.0482412	−0.0241206	+	0.999709i	$0.507679\pi$
$350$	−78.9696	−4.22110
$351$	0	0
$352$	8.10003	0.431733
$353$	−14.6186	−0.778068	−0.389034	−	0.921223i	$-0.627191\pi$
$353$	−14.6186	−0.778068	−0.389034	+	0.921223i	$0.627191\pi$
$354$	0	0
$355$	−46.1339	−2.44853
$356$	5.33935	0.282985
$357$	0	0
$358$	−0.770007	−0.0406961
$359$	30.7091	1.62076	0.810382	−	0.585902i	$-0.199260\pi$
$359$	30.7091	1.62076	0.810382	+	0.585902i	$0.199260\pi$
$360$	0	0
$361$	30.0330	1.58068
$362$	−37.1401	−1.95204
$363$	0	0
$364$	−25.5065	−1.33691
$365$	−7.35690	−0.385078
$366$	0	0
$367$	−31.7475	−1.65720	−0.828602	−	0.559838i	$-0.810863\pi$
$367$	−31.7475	−1.65720	−0.828602	+	0.559838i	$0.810863\pi$
$368$	8.50727	0.443472
$369$	0	0
$370$	34.1388	1.77479
$371$	−24.7062	−1.28268
$372$	0	0
$373$	11.9713	0.619852	0.309926	−	0.950761i	$-0.399696\pi$
$373$	11.9713	0.619852	0.309926	+	0.950761i	$0.399696\pi$
$374$	−14.7771	−0.764107
$375$	0	0
$376$	3.56801	0.184006
$377$	−10.2600	−0.528415
$378$	0	0
$379$	7.27754	0.373822	0.186911	−	0.982377i	$-0.440152\pi$
$379$	7.27754	0.373822	0.186911	+	0.982377i	$0.440152\pi$
$380$	54.6643	2.80422
$381$	0	0
$382$	−0.0862907	−0.00441502
$383$	32.6838	1.67006	0.835032	−	0.550201i	$-0.185449\pi$
$383$	32.6838	1.67006	0.835032	+	0.550201i	$0.185449\pi$
$384$	0	0
$385$	−17.0860	−0.870783
$386$	17.6831	0.900048
$387$	0	0
$388$	22.2450	1.12932
$389$	−19.8941	−1.00867	−0.504335	−	0.863508i	$-0.668262\pi$
$389$	−19.8941	−1.00867	−0.504335	+	0.863508i	$0.668262\pi$
$390$	0	0
$391$	−16.7198	−0.845555
$392$	4.35748	0.220086
$393$	0	0
$394$	31.0195	1.56274
$395$	−5.69630	−0.286612
$396$	0	0
$397$	23.8399	1.19649	0.598244	−	0.801314i	$-0.295865\pi$
$397$	23.8399	1.19649	0.598244	+	0.801314i	$0.295865\pi$
$398$	23.9128	1.19864
$399$	0	0
$400$	−30.5606	−1.52803
$401$	8.95725	0.447304	0.223652	−	0.974669i	$-0.428202\pi$
$401$	8.95725	0.447304	0.223652	+	0.974669i	$0.428202\pi$
$402$	0	0
$403$	−19.2010	−0.956468
$404$	−37.9206	−1.88662
$405$	0	0
$406$	38.5861	1.91500
$407$	4.60383	0.228203
$408$	0	0
$409$	−2.16459	−0.107032	−0.0535160	−	0.998567i	$-0.517043\pi$
$409$	−2.16459	−0.107032	−0.0535160	+	0.998567i	$0.517043\pi$
$410$	−26.0142	−1.28475
$411$	0	0
$412$	30.3278	1.49414
$413$	−5.80408	−0.285600
$414$	0	0
$415$	−50.2071	−2.46457
$416$	−20.5591	−1.00799
$417$	0	0
$418$	14.2525	0.697111
$419$	14.3813	0.702571	0.351286	−	0.936268i	$-0.385745\pi$
$419$	14.3813	0.702571	0.351286	+	0.936268i	$0.385745\pi$
$420$	0	0
$421$	−17.9478	−0.874723	−0.437362	−	0.899286i	$-0.644087\pi$
$421$	−17.9478	−0.874723	−0.437362	+	0.899286i	$0.644087\pi$
$422$	−15.2549	−0.742599
$423$	0	0
$424$	1.53092	0.0743479
$425$	60.0622	2.91344
$426$	0	0
$427$	4.68983	0.226957
$428$	−18.9428	−0.915637
$429$	0	0
$430$	32.6179	1.57298
$431$	−3.90648	−0.188168	−0.0940841	−	0.995564i	$-0.529992\pi$
$431$	−3.90648	−0.188168	−0.0940841	+	0.995564i	$0.529992\pi$
$432$	0	0
$433$	11.9670	0.575098	0.287549	−	0.957766i	$-0.407160\pi$
$433$	11.9670	0.575098	0.287549	+	0.957766i	$0.407160\pi$
$434$	72.2119	3.46628
$435$	0	0
$436$	13.0472	0.624849
$437$	16.1261	0.771418
$438$	0	0
$439$	−4.57537	−0.218370	−0.109185	−	0.994021i	$-0.534824\pi$
$439$	−4.57537	−0.218370	−0.109185	+	0.994021i	$0.534824\pi$
$440$	1.05873	0.0504730
$441$	0	0
$442$	37.5066	1.78401
$443$	−8.48126	−0.402957	−0.201478	−	0.979493i	$-0.564575\pi$
$443$	−8.48126	−0.402957	−0.201478	+	0.979493i	$0.564575\pi$
$444$	0	0
$445$	−9.07808	−0.430342
$446$	−39.9113	−1.88986
$447$	0	0
$448$	42.6706	2.01600
$449$	−30.7884	−1.45299	−0.726496	−	0.687170i	$-0.758853\pi$
$449$	−30.7884	−1.45299	−0.726496	+	0.687170i	$0.758853\pi$
$450$	0	0
$451$	−3.50819	−0.165194
$452$	11.4550	0.538800
$453$	0	0
$454$	20.7882	0.975639
$455$	43.3668	2.03307
$456$	0	0
$457$	8.09871	0.378842	0.189421	−	0.981896i	$-0.439339\pi$
$457$	8.09871	0.378842	0.189421	+	0.981896i	$0.439339\pi$
$458$	−34.3189	−1.60362
$459$	0	0
$460$	17.9782	0.838237
$461$	−11.9563	−0.556859	−0.278430	−	0.960457i	$-0.589814\pi$
$461$	−11.9563	−0.556859	−0.278430	+	0.960457i	$0.589814\pi$
$462$	0	0
$463$	−21.2091	−0.985671	−0.492836	−	0.870122i	$-0.664040\pi$
$463$	−21.2091	−0.985671	−0.492836	+	0.870122i	$0.664040\pi$
$464$	14.9325	0.693223
$465$	0	0
$466$	−45.3648	−2.10149
$467$	16.4835	0.762766	0.381383	−	0.924417i	$-0.375448\pi$
$467$	16.4835	0.762766	0.381383	+	0.924417i	$0.375448\pi$
$468$	0	0
$469$	−28.3294	−1.30813
$470$	−91.0442	−4.19956
$471$	0	0
$472$	0.359648	0.0165542
$473$	4.39873	0.202254
$474$	0	0
$475$	−57.9297	−2.65800
$476$	−72.9589	−3.34407
$477$	0	0
$478$	−40.5388	−1.85420
$479$	5.89367	0.269289	0.134644	−	0.990894i	$-0.457011\pi$
$479$	5.89367	0.269289	0.134644	+	0.990894i	$0.457011\pi$
$480$	0	0
$481$	−11.6852	−0.532800
$482$	22.0791	1.00568
$483$	0	0
$484$	2.14278	0.0973989
$485$	−37.8214	−1.71738
$486$	0	0
$487$	4.70750	0.213317	0.106659	−	0.994296i	$-0.465985\pi$
$487$	4.70750	0.213317	0.106659	+	0.994296i	$0.465985\pi$
$488$	−0.290605	−0.0131551
$489$	0	0
$490$	−111.189	−5.02300
$491$	−2.88342	−0.130127	−0.0650635	−	0.997881i	$-0.520725\pi$
$491$	−2.88342	−0.130127	−0.0650635	+	0.997881i	$0.520725\pi$
$492$	0	0
$493$	−29.3476	−1.32175
$494$	−36.1749	−1.62759
$495$	0	0
$496$	27.9454	1.25478
$497$	−59.3874	−2.66389
$498$	0	0
$499$	33.5303	1.50102	0.750510	−	0.660859i	$-0.229808\pi$
$499$	33.5303	1.50102	0.750510	+	0.660859i	$0.229808\pi$
$500$	−25.5500	−1.14263
$501$	0	0
$502$	11.7004	0.522216
$503$	7.09540	0.316368	0.158184	−	0.987410i	$-0.449436\pi$
$503$	7.09540	0.316368	0.158184	+	0.987410i	$0.449436\pi$
$504$	0	0
$505$	64.4734	2.86903
$506$	4.68740	0.208380
$507$	0	0
$508$	−24.6893	−1.09541
$509$	−25.5826	−1.13393	−0.566965	−	0.823742i	$-0.691883\pi$
$509$	−25.5826	−1.13393	−0.566965	+	0.823742i	$0.691883\pi$
$510$	0	0
$511$	−9.47042	−0.418947
$512$	32.0690	1.41727
$513$	0	0
$514$	49.2568	2.17262
$515$	−51.5641	−2.27218
$516$	0	0
$517$	−12.2779	−0.539981
$518$	43.9463	1.93089
$519$	0	0
$520$	−2.68722	−0.117842
$521$	26.2764	1.15119	0.575596	−	0.817734i	$-0.304770\pi$
$521$	26.2764	1.15119	0.575596	+	0.817734i	$0.304770\pi$
$522$	0	0
$523$	30.3176	1.32570	0.662848	−	0.748754i	$-0.269348\pi$
$523$	30.3176	1.32570	0.662848	+	0.748754i	$0.269348\pi$
$524$	−45.5388	−1.98937
$525$	0	0
$526$	34.7047	1.51320
$527$	−54.9225	−2.39246
$528$	0	0
$529$	−17.6964	−0.769408
$530$	−39.0641	−1.69684
$531$	0	0
$532$	70.3685	3.05086
$533$	8.90431	0.385688
$534$	0	0
$535$	32.2070	1.39243
$536$	1.75543	0.0758229
$537$	0	0
$538$	−31.8911	−1.37492
$539$	−14.9945	−0.645860
$540$	0	0
$541$	20.4919	0.881017	0.440508	−	0.897748i	$-0.354798\pi$
$541$	20.4919	0.881017	0.440508	+	0.897748i	$0.354798\pi$
$542$	16.5079	0.709074
$543$	0	0
$544$	−58.8073	−2.52134
$545$	−22.1832	−0.950223
$546$	0	0
$547$	28.8237	1.23241	0.616206	−	0.787585i	$-0.288669\pi$
$547$	28.8237	1.23241	0.616206	+	0.787585i	$0.288669\pi$
$548$	31.5857	1.34928
$549$	0	0
$550$	−16.8385	−0.717995
$551$	28.3056	1.20586
$552$	0	0
$553$	−7.33277	−0.311821
$554$	−0.707426	−0.0300556
$555$	0	0
$556$	−33.1289	−1.40498
$557$	3.71481	0.157402	0.0787008	−	0.996898i	$-0.474923\pi$
$557$	3.71481	0.157402	0.0787008	+	0.996898i	$0.474923\pi$
$558$	0	0
$559$	−11.1646	−0.472214
$560$	−63.1167	−2.66717
$561$	0	0
$562$	64.1483	2.70593
$563$	−34.2645	−1.44408	−0.722038	−	0.691854i	$-0.756794\pi$
$563$	−34.2645	−1.44408	−0.722038	+	0.691854i	$0.756794\pi$
$564$	0	0
$565$	−19.4761	−0.819367
$566$	−47.3974	−1.99226
$567$	0	0
$568$	3.67993	0.154406
$569$	29.8315	1.25060	0.625300	−	0.780384i	$-0.284976\pi$
$569$	29.8315	1.25060	0.625300	+	0.780384i	$0.284976\pi$
$570$	0	0
$571$	7.62436	0.319069	0.159535	−	0.987192i	$-0.449001\pi$
$571$	7.62436	0.319069	0.159535	+	0.987192i	$0.449001\pi$
$572$	−5.43869	−0.227403
$573$	0	0
$574$	−33.4878	−1.39775
$575$	−19.0521	−0.794528
$576$	0	0
$577$	13.8227	0.575446	0.287723	−	0.957714i	$-0.407102\pi$
$577$	13.8227	0.575446	0.287723	+	0.957714i	$0.407102\pi$
$578$	72.6823	3.02319
$579$	0	0
$580$	31.5564	1.31031
$581$	−64.6308	−2.68134
$582$	0	0
$583$	−5.26804	−0.218180
$584$	0.586833	0.0242833
$585$	0	0
$586$	28.1439	1.16262
$587$	−30.3944	−1.25451	−0.627257	−	0.778813i	$-0.715822\pi$
$587$	−30.3944	−1.25451	−0.627257	+	0.778813i	$0.715822\pi$
$588$	0	0
$589$	52.9724	2.18269
$590$	−9.17707	−0.377814
$591$	0	0
$592$	17.0068	0.698977
$593$	19.1615	0.786867	0.393433	−	0.919353i	$-0.371287\pi$
$593$	19.1615	0.786867	0.393433	+	0.919353i	$0.371287\pi$
$594$	0	0
$595$	124.046	5.08541
$596$	−13.4057	−0.549119
$597$	0	0
$598$	−11.8973	−0.486518
$599$	32.1514	1.31367	0.656836	−	0.754034i	$-0.271894\pi$
$599$	32.1514	1.31367	0.656836	+	0.754034i	$0.271894\pi$
$600$	0	0
$601$	−36.8302	−1.50233	−0.751167	−	0.660112i	$-0.770509\pi$
$601$	−36.8302	−1.50233	−0.751167	+	0.660112i	$0.770509\pi$
$602$	41.9886	1.71133
$603$	0	0
$604$	26.5497	1.08029
$605$	−3.64320	−0.148117
$606$	0	0
$607$	−13.7738	−0.559060	−0.279530	−	0.960137i	$-0.590179\pi$
$607$	−13.7738	−0.559060	−0.279530	+	0.960137i	$0.590179\pi$
$608$	56.7193	2.30027
$609$	0	0
$610$	7.41529	0.300237
$611$	31.1631	1.26073
$612$	0	0
$613$	45.0115	1.81800	0.908999	−	0.416799i	$-0.136848\pi$
$613$	45.0115	1.81800	0.908999	+	0.416799i	$0.136848\pi$
$614$	−1.26672	−0.0511206
$615$	0	0
$616$	1.36289	0.0549123
$617$	−5.26769	−0.212069	−0.106035	−	0.994362i	$-0.533815\pi$
$617$	−5.26769	−0.212069	−0.106035	+	0.994362i	$0.533815\pi$
$618$	0	0
$619$	0.225956	0.00908194	0.00454097	−	0.999990i	$-0.498555\pi$
$619$	0.225956	0.00908194	0.00454097	+	0.999990i	$0.498555\pi$
$620$	59.0562	2.37175
$621$	0	0
$622$	−65.1372	−2.61176
$623$	−11.6861	−0.468193
$624$	0	0
$625$	2.07621	0.0830483
$626$	19.8460	0.793206
$627$	0	0
$628$	−12.9893	−0.518330
$629$	−33.4244	−1.33272
$630$	0	0
$631$	13.9516	0.555403	0.277701	−	0.960667i	$-0.410427\pi$
$631$	13.9516	0.555403	0.277701	+	0.960667i	$0.410427\pi$
$632$	0.454373	0.0180740
$633$	0	0
$634$	20.0237	0.795243
$635$	41.9773	1.66582
$636$	0	0
$637$	38.0584	1.50793
$638$	8.22761	0.325734
$639$	0	0
$640$	8.44826	0.333947
$641$	8.58714	0.339172	0.169586	−	0.985515i	$-0.445757\pi$
$641$	8.58714	0.339172	0.169586	+	0.985515i	$0.445757\pi$
$642$	0	0
$643$	5.01284	0.197687	0.0988435	−	0.995103i	$-0.468486\pi$
$643$	5.01284	0.197687	0.0988435	+	0.995103i	$0.468486\pi$
$644$	23.1430	0.911964
$645$	0	0
$646$	−103.475	−4.07116
$647$	17.2188	0.676942	0.338471	−	0.940977i	$-0.390090\pi$
$647$	17.2188	0.676942	0.338471	+	0.940977i	$0.390090\pi$
$648$	0	0
$649$	−1.23759	−0.0485795
$650$	42.7386	1.67635
$651$	0	0
$652$	−1.45962	−0.0571630
$653$	−40.0846	−1.56863	−0.784316	−	0.620361i	$-0.786986\pi$
$653$	−40.0846	−1.56863	−0.784316	+	0.620361i	$0.786986\pi$
$654$	0	0
$655$	77.4261	3.02529
$656$	−12.9595	−0.505982
$657$	0	0
$658$	−117.200	−4.56893
$659$	−44.6994	−1.74124	−0.870622	−	0.491953i	$-0.836283\pi$
$659$	−44.6994	−1.74124	−0.870622	+	0.491953i	$0.836283\pi$
$660$	0	0
$661$	18.9406	0.736703	0.368351	−	0.929687i	$-0.379922\pi$
$661$	18.9406	0.736703	0.368351	+	0.929687i	$0.379922\pi$
$662$	21.3331	0.829135
$663$	0	0
$664$	4.00484	0.155418
$665$	−119.642	−4.63953
$666$	0	0
$667$	9.30924	0.360455
$668$	−19.6191	−0.759088
$669$	0	0
$670$	−44.7928	−1.73050
$671$	1.00000	0.0386046
$672$	0	0
$673$	5.92807	0.228510	0.114255	−	0.993451i	$-0.463552\pi$
$673$	5.92807	0.228510	0.114255	+	0.993451i	$0.463552\pi$
$674$	−14.9498	−0.575843
$675$	0	0
$676$	−14.0519	−0.540457
$677$	−1.69010	−0.0649558	−0.0324779	−	0.999472i	$-0.510340\pi$
$677$	−1.69010	−0.0649558	−0.0324779	+	0.999472i	$0.510340\pi$
$678$	0	0
$679$	−48.6870	−1.86843
$680$	−7.68651	−0.294764
$681$	0	0
$682$	15.3975	0.589602
$683$	41.8817	1.60256	0.801279	−	0.598291i	$-0.204153\pi$
$683$	41.8817	1.60256	0.801279	+	0.598291i	$0.204153\pi$
$684$	0	0
$685$	−53.7028	−2.05188
$686$	−76.3126	−2.91363
$687$	0	0
$688$	16.2492	0.619495
$689$	13.3711	0.509398
$690$	0	0
$691$	8.60093	0.327195	0.163597	−	0.986527i	$-0.447690\pi$
$691$	8.60093	0.327195	0.163597	+	0.986527i	$0.447690\pi$
$692$	53.0470	2.01654
$693$	0	0
$694$	−29.2837	−1.11159
$695$	56.3265	2.13658
$696$	0	0
$697$	25.4699	0.964741
$698$	−1.83433	−0.0694303
$699$	0	0
$700$	−83.1364	−3.14226
$701$	−28.0591	−1.05978	−0.529888	−	0.848068i	$-0.677766\pi$
$701$	−28.0591	−1.05978	−0.529888	+	0.848068i	$0.677766\pi$
$702$	0	0
$703$	32.2377	1.21587
$704$	9.09853	0.342914
$705$	0	0
$706$	−29.7544	−1.11982
$707$	82.9957	3.12137
$708$	0	0
$709$	42.8777	1.61031	0.805154	−	0.593066i	$-0.202083\pi$
$709$	42.8777	1.61031	0.805154	+	0.593066i	$0.202083\pi$
$710$	−93.9000	−3.52400
$711$	0	0
$712$	0.724125	0.0271377
$713$	17.4218	0.652450
$714$	0	0
$715$	9.24698	0.345818
$716$	−0.810636	−0.0302949
$717$	0	0
$718$	62.5047	2.33265
$719$	−39.3106	−1.46604	−0.733019	−	0.680208i	$-0.761890\pi$
$719$	−39.3106	−1.46604	−0.733019	+	0.680208i	$0.761890\pi$
$720$	0	0
$721$	−66.3776	−2.47203
$722$	61.1286	2.27497
$723$	0	0
$724$	−39.0997	−1.45313
$725$	−33.4414	−1.24198
$726$	0	0
$727$	−20.5914	−0.763693	−0.381846	−	0.924226i	$-0.624712\pi$
$727$	−20.5914	−0.763693	−0.381846	+	0.924226i	$0.624712\pi$
$728$	−3.45921	−0.128207
$729$	0	0
$730$	−14.9741	−0.554216
$731$	−31.9354	−1.18117
$732$	0	0
$733$	−33.3897	−1.23328	−0.616639	−	0.787246i	$-0.711506\pi$
$733$	−33.3897	−1.23328	−0.616639	+	0.787246i	$0.711506\pi$
$734$	−64.6182	−2.38510
$735$	0	0
$736$	18.6540	0.687597
$737$	−6.04060	−0.222508
$738$	0	0
$739$	6.75881	0.248627	0.124313	−	0.992243i	$-0.460327\pi$
$739$	6.75881	0.248627	0.124313	+	0.992243i	$0.460327\pi$
$740$	35.9401	1.32118
$741$	0	0
$742$	−50.2866	−1.84608
$743$	24.0448	0.882119	0.441060	−	0.897478i	$-0.354603\pi$
$743$	24.0448	0.882119	0.441060	+	0.897478i	$0.354603\pi$
$744$	0	0
$745$	22.7927	0.835059
$746$	24.3662	0.892111
$747$	0	0
$748$	−15.5568	−0.568814
$749$	41.4596	1.51490
$750$	0	0
$751$	11.0501	0.403224	0.201612	−	0.979465i	$-0.435382\pi$
$751$	11.0501	0.403224	0.201612	+	0.979465i	$0.435382\pi$
$752$	−45.3553	−1.65394
$753$	0	0
$754$	−20.8829	−0.760511
$755$	−45.1403	−1.64283
$756$	0	0
$757$	−6.93705	−0.252131	−0.126066	−	0.992022i	$-0.540235\pi$
$757$	−6.93705	−0.252131	−0.126066	+	0.992022i	$0.540235\pi$
$758$	14.8126	0.538017
$759$	0	0
$760$	7.41361	0.268920
$761$	31.8639	1.15506	0.577532	−	0.816368i	$-0.304016\pi$
$761$	31.8639	1.15506	0.577532	+	0.816368i	$0.304016\pi$
$762$	0	0
$763$	−28.5561	−1.03380
$764$	−0.0908438	−0.00328661
$765$	0	0
$766$	66.5240	2.40361
$767$	3.14118	0.113421
$768$	0	0
$769$	−10.0504	−0.362427	−0.181214	−	0.983444i	$-0.558003\pi$
$769$	−10.0504	−0.362427	−0.181214	+	0.983444i	$0.558003\pi$
$770$	−34.7765	−1.25326
$771$	0	0
$772$	18.6162	0.670011
$773$	−13.4269	−0.482931	−0.241466	−	0.970409i	$-0.577628\pi$
$773$	−13.4269	−0.482931	−0.241466	+	0.970409i	$0.577628\pi$
$774$	0	0
$775$	−62.5839	−2.24808
$776$	3.01688	0.108300
$777$	0	0
$778$	−40.4920	−1.45171
$779$	−24.5656	−0.880153
$780$	0	0
$781$	−12.6630	−0.453118
$782$	−34.0311	−1.21695
$783$	0	0
$784$	−55.3907	−1.97824
$785$	22.0847	0.788238
$786$	0	0
$787$	−31.4820	−1.12221	−0.561107	−	0.827743i	$-0.689624\pi$
$787$	−31.4820	−1.12221	−0.561107	+	0.827743i	$0.689624\pi$
$788$	32.6562	1.16333
$789$	0	0
$790$	−11.5941	−0.412501
$791$	−25.0713	−0.891434
$792$	0	0
$793$	−2.53815	−0.0901324
$794$	48.5232	1.72202
$795$	0	0
$796$	25.1746	0.892289
$797$	15.0762	0.534025	0.267012	−	0.963693i	$-0.413964\pi$
$797$	15.0762	0.534025	0.267012	+	0.963693i	$0.413964\pi$
$798$	0	0
$799$	89.1391	3.15351
$800$	−67.0106	−2.36918
$801$	0	0
$802$	18.2314	0.643774
$803$	−2.01935	−0.0712614
$804$	0	0
$805$	−39.3483	−1.38685
$806$	−39.0813	−1.37658
$807$	0	0
$808$	−5.14281	−0.180923
$809$	49.6145	1.74435	0.872177	−	0.489190i	$-0.162708\pi$
$809$	49.6145	1.74435	0.872177	+	0.489190i	$0.162708\pi$
$810$	0	0
$811$	45.0796	1.58296	0.791480	−	0.611195i	$-0.209311\pi$
$811$	45.0796	1.58296	0.791480	+	0.611195i	$0.209311\pi$
$812$	40.6221	1.42556
$813$	0	0
$814$	9.37055	0.328438
$815$	2.48167	0.0869292
$816$	0	0
$817$	30.8015	1.07761
$818$	−4.40576	−0.154044
$819$	0	0
$820$	−27.3869	−0.956391
$821$	33.9818	1.18597	0.592987	−	0.805212i	$-0.297949\pi$
$821$	33.9818	1.18597	0.592987	+	0.805212i	$0.297949\pi$
$822$	0	0
$823$	13.7180	0.478179	0.239090	−	0.970998i	$-0.423151\pi$
$823$	13.7180	0.478179	0.239090	+	0.970998i	$0.423151\pi$
$824$	4.11308	0.143286
$825$	0	0
$826$	−11.8135	−0.411044
$827$	47.0665	1.63666	0.818332	−	0.574746i	$-0.194899\pi$
$827$	47.0665	1.63666	0.818332	+	0.574746i	$0.194899\pi$
$828$	0	0
$829$	−23.2955	−0.809086	−0.404543	−	0.914519i	$-0.632569\pi$
$829$	−23.2955	−0.809086	−0.404543	+	0.914519i	$0.632569\pi$
$830$	−102.191	−3.54708
$831$	0	0
$832$	−23.0934	−0.800621
$833$	108.862	3.77185
$834$	0	0
$835$	33.3569	1.15436
$836$	15.0045	0.518941
$837$	0	0
$838$	29.2714	1.01116
$839$	−56.1331	−1.93793	−0.968965	−	0.247199i	$-0.920490\pi$
$839$	−56.1331	−1.93793	−0.968965	+	0.247199i	$0.920490\pi$
$840$	0	0
$841$	−12.6598	−0.436546
$842$	−36.5307	−1.25893
$843$	0	0
$844$	−16.0598	−0.552803
$845$	23.8913	0.821887
$846$	0	0
$847$	−4.68983	−0.161145
$848$	−19.4605	−0.668275
$849$	0	0
$850$	122.249	4.19312
$851$	10.6024	0.363447
$852$	0	0
$853$	18.0999	0.619729	0.309864	−	0.950781i	$-0.399716\pi$
$853$	18.0999	0.619729	0.309864	+	0.950781i	$0.399716\pi$
$854$	9.54560	0.326644
$855$	0	0
$856$	−2.56904	−0.0878079
$857$	−7.44500	−0.254316	−0.127158	−	0.991882i	$-0.540586\pi$
$857$	−7.44500	−0.254316	−0.127158	+	0.991882i	$0.540586\pi$
$858$	0	0
$859$	54.7631	1.86849	0.934247	−	0.356626i	$-0.116073\pi$
$859$	54.7631	1.86849	0.934247	+	0.356626i	$0.116073\pi$
$860$	34.3390	1.17095
$861$	0	0
$862$	−7.95117	−0.270818
$863$	−18.6756	−0.635724	−0.317862	−	0.948137i	$-0.602965\pi$
$863$	−18.6756	−0.635724	−0.317862	+	0.948137i	$0.602965\pi$
$864$	0	0
$865$	−90.1917	−3.06661
$866$	24.3574	0.827700
$867$	0	0
$868$	76.0221	2.58036
$869$	−1.56355	−0.0530396
$870$	0	0
$871$	15.3320	0.519503
$872$	1.76947	0.0599219
$873$	0	0
$874$	32.8228	1.11025
$875$	55.9205	1.89046
$876$	0	0
$877$	36.5218	1.23325	0.616626	−	0.787256i	$-0.288499\pi$
$877$	36.5218	1.23325	0.616626	+	0.787256i	$0.288499\pi$
$878$	−9.31262	−0.314286
$879$	0	0
$880$	−13.4582	−0.453676
$881$	38.1820	1.28638	0.643192	−	0.765705i	$-0.277610\pi$
$881$	38.1820	1.28638	0.643192	+	0.765705i	$0.277610\pi$
$882$	0	0
$883$	13.5685	0.456618	0.228309	−	0.973589i	$-0.426680\pi$
$883$	13.5685	0.456618	0.228309	+	0.973589i	$0.426680\pi$
$884$	39.4856	1.32804
$885$	0	0
$886$	−17.2626	−0.579948
$887$	−16.2525	−0.545707	−0.272853	−	0.962056i	$-0.587967\pi$
$887$	−16.2525	−0.545707	−0.272853	+	0.962056i	$0.587967\pi$
$888$	0	0
$889$	54.0367	1.81233
$890$	−18.4773	−0.619362
$891$	0	0
$892$	−42.0172	−1.40684
$893$	−85.9742	−2.87702
$894$	0	0
$895$	1.37826	0.0460702
$896$	10.8753	0.363319
$897$	0	0
$898$	−62.6661	−2.09119
$899$	30.5797	1.01989
$900$	0	0
$901$	38.2467	1.27418
$902$	−7.14050	−0.237753
$903$	0	0
$904$	1.55354	0.0516700
$905$	66.4783	2.20981
$906$	0	0
$907$	41.4074	1.37491	0.687455	−	0.726227i	$-0.258728\pi$
$907$	41.4074	1.37491	0.687455	+	0.726227i	$0.258728\pi$
$908$	21.8851	0.726282
$909$	0	0
$910$	88.2680	2.92606
$911$	8.80278	0.291649	0.145824	−	0.989310i	$-0.453417\pi$
$911$	8.80278	0.291649	0.145824	+	0.989310i	$0.453417\pi$
$912$	0	0
$913$	−13.7810	−0.456086
$914$	16.4840	0.545241
$915$	0	0
$916$	−36.1297	−1.19376
$917$	99.6695	3.29138
$918$	0	0
$919$	−27.7544	−0.915534	−0.457767	−	0.889072i	$-0.651351\pi$
$919$	−27.7544	−0.915534	−0.457767	+	0.889072i	$0.651351\pi$
$920$	2.43821	0.0803855
$921$	0	0
$922$	−24.3356	−0.801450
$923$	32.1406	1.05792
$924$	0	0
$925$	−38.0870	−1.25229
$926$	−43.1686	−1.41861
$927$	0	0
$928$	32.7427	1.07483
$929$	16.2422	0.532889	0.266444	−	0.963850i	$-0.414151\pi$
$929$	16.2422	0.532889	0.266444	+	0.963850i	$0.414151\pi$
$930$	0	0
$931$	−104.997	−3.44114
$932$	−47.7585	−1.56438
$933$	0	0
$934$	33.5502	1.09780
$935$	26.4501	0.865010
$936$	0	0
$937$	27.3746	0.894289	0.447144	−	0.894462i	$-0.352441\pi$
$937$	27.3746	0.894289	0.447144	+	0.894462i	$0.352441\pi$
$938$	−57.6611	−1.88270
$939$	0	0
$940$	−95.8481	−3.12622
$941$	−32.4459	−1.05771	−0.528854	−	0.848713i	$-0.677378\pi$
$941$	−32.4459	−1.05771	−0.528854	+	0.848713i	$0.677378\pi$
$942$	0	0
$943$	−8.07921	−0.263095
$944$	−4.57172	−0.148797
$945$	0	0
$946$	8.95310	0.291090
$947$	45.6348	1.48293	0.741466	−	0.670990i	$-0.234131\pi$
$947$	45.6348	1.48293	0.741466	+	0.670990i	$0.234131\pi$
$948$	0	0
$949$	5.12542	0.166378
$950$	−117.909	−3.82547
$951$	0	0
$952$	−9.89473	−0.320690
$953$	11.1355	0.360715	0.180358	−	0.983601i	$-0.442274\pi$
$953$	11.1355	0.360715	0.180358	+	0.983601i	$0.442274\pi$
$954$	0	0
$955$	0.154455	0.00499803
$956$	−42.6777	−1.38030
$957$	0	0
$958$	11.9959	0.387569
$959$	−69.1308	−2.23235
$960$	0	0
$961$	26.2284	0.846077
$962$	−23.7839	−0.766823
$963$	0	0
$964$	23.2441	0.748641
$965$	−31.6517	−1.01890
$966$	0	0
$967$	33.3918	1.07381	0.536904	−	0.843643i	$-0.319594\pi$
$967$	33.3918	1.07381	0.536904	+	0.843643i	$0.319594\pi$
$968$	0.290605	0.00934039
$969$	0	0
$970$	−76.9810	−2.47171
$971$	56.3438	1.80816	0.904080	−	0.427363i	$-0.140557\pi$
$971$	56.3438	1.80816	0.904080	+	0.427363i	$0.140557\pi$
$972$	0	0
$973$	72.5082	2.32451
$974$	9.58156	0.307013
$975$	0	0
$976$	3.69406	0.118244
$977$	−47.5197	−1.52029	−0.760145	−	0.649753i	$-0.774872\pi$
$977$	−47.5197	−1.52029	−0.760145	+	0.649753i	$0.774872\pi$
$978$	0	0
$979$	−2.49179	−0.0796379
$980$	−117.056	−3.73921
$981$	0	0
$982$	−5.86887	−0.187283
$983$	20.7775	0.662700	0.331350	−	0.943508i	$-0.392496\pi$
$983$	20.7775	0.662700	0.331350	+	0.943508i	$0.392496\pi$
$984$	0	0
$985$	−55.5229	−1.76911
$986$	−59.7335	−1.90230
$987$	0	0
$988$	−38.0836	−1.21160
$989$	10.1301	0.322119
$990$	0	0
$991$	24.1060	0.765753	0.382877	−	0.923799i	$-0.374933\pi$
$991$	24.1060	0.765753	0.382877	+	0.923799i	$0.374933\pi$
$992$	61.2763	1.94552
$993$	0	0
$994$	−120.876	−3.83396
$995$	−42.8024	−1.35693
$996$	0	0
$997$	−13.9288	−0.441130	−0.220565	−	0.975372i	$-0.570790\pi$
$997$	−13.9288	−0.441130	−0.220565	+	0.975372i	$0.570790\pi$
$998$	68.2468	2.16032
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6039.2.a.f.1.9		12
3.2	odd	2		2013.2.a.c.1.4	✓	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2013.2.a.c.1.4	✓	12	3.2	odd	2
6039.2.a.f.1.9		12	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+2.03538 q^{2} +2.14278 q^{4} -3.64320 q^{5} -4.68983 q^{7} +0.290605 q^{8} +O(q^{10})\)	\(q+2.03538 q^{2} +2.14278 q^{4} -3.64320 q^{5} -4.68983 q^{7} +0.290605 q^{8} -7.41529 q^{10} -1.00000 q^{11} +2.53815 q^{13} -9.54560 q^{14} -3.69406 q^{16} +7.26013 q^{17} -7.00236 q^{19} -7.80656 q^{20} -2.03538 q^{22} -2.30296 q^{23} +8.27289 q^{25} +5.16610 q^{26} -10.0493 q^{28} -4.04230 q^{29} -7.56494 q^{31} -8.10003 q^{32} +14.7771 q^{34} +17.0860 q^{35} -4.60383 q^{37} -14.2525 q^{38} -1.05873 q^{40} +3.50819 q^{41} -4.39873 q^{43} -2.14278 q^{44} -4.68740 q^{46} +12.2779 q^{47} +14.9945 q^{49} +16.8385 q^{50} +5.43869 q^{52} +5.26804 q^{53} +3.64320 q^{55} -1.36289 q^{56} -8.22761 q^{58} +1.23759 q^{59} -1.00000 q^{61} -15.3975 q^{62} -9.09853 q^{64} -9.24698 q^{65} +6.04060 q^{67} +15.5568 q^{68} +34.7765 q^{70} +12.6630 q^{71} +2.01935 q^{73} -9.37055 q^{74} -15.0045 q^{76} +4.68983 q^{77} +1.56355 q^{79} +13.4582 q^{80} +7.14050 q^{82} +13.7810 q^{83} -26.4501 q^{85} -8.95310 q^{86} -0.290605 q^{88} +2.49179 q^{89} -11.9035 q^{91} -4.93473 q^{92} +24.9902 q^{94} +25.5110 q^{95} +10.3814 q^{97} +30.5196 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 7 q^{2} + 13 q^{4} + 7 q^{5} - 15 q^{7} + 18 q^{8}+O(q^{10}) \) 12 * q + 7 * q^2 + 13 * q^4 + 7 * q^5 - 15 * q^7 + 18 * q^8	\( 12 q + 7 q^{2} + 13 q^{4} + 7 q^{5} - 15 q^{7} + 18 q^{8} - 6 q^{10} - 12 q^{11} - 11 q^{13} - 3 q^{14} + 19 q^{16} + 33 q^{17} - 24 q^{19} + 11 q^{20} - 7 q^{22} + 9 q^{23} + 11 q^{25} + 16 q^{26} - 41 q^{28} + 16 q^{29} + q^{31} + 28 q^{32} + 32 q^{34} + 22 q^{35} - 6 q^{37} - 12 q^{38} + 26 q^{40} + 21 q^{41} - 39 q^{43} - 13 q^{44} + 18 q^{47} + 31 q^{49} + 44 q^{50} + 3 q^{52} + 14 q^{53} - 7 q^{55} - 16 q^{56} + 33 q^{58} + 23 q^{59} - 12 q^{61} + 25 q^{62} + 12 q^{64} + 29 q^{65} + 96 q^{68} + 44 q^{70} + 19 q^{71} - 42 q^{73} - 38 q^{74} + 11 q^{76} + 15 q^{77} - 11 q^{79} + 44 q^{80} - 14 q^{82} + 56 q^{83} + 16 q^{85} + 18 q^{86} - 18 q^{88} + 55 q^{89} + 11 q^{91} + 4 q^{92} - 5 q^{94} - 15 q^{95} - 7 q^{97} - 6 q^{98}+O(q^{100}) \) 12 * q + 7 * q^2 + 13 * q^4 + 7 * q^5 - 15 * q^7 + 18 * q^8 - 6 * q^10 - 12 * q^11 - 11 * q^13 - 3 * q^14 + 19 * q^16 + 33 * q^17 - 24 * q^19 + 11 * q^20 - 7 * q^22 + 9 * q^23 + 11 * q^25 + 16 * q^26 - 41 * q^28 + 16 * q^29 + q^31 + 28 * q^32 + 32 * q^34 + 22 * q^35 - 6 * q^37 - 12 * q^38 + 26 * q^40 + 21 * q^41 - 39 * q^43 - 13 * q^44 + 18 * q^47 + 31 * q^49 + 44 * q^50 + 3 * q^52 + 14 * q^53 - 7 * q^55 - 16 * q^56 + 33 * q^58 + 23 * q^59 - 12 * q^61 + 25 * q^62 + 12 * q^64 + 29 * q^65 + 96 * q^68 + 44 * q^70 + 19 * q^71 - 42 * q^73 - 38 * q^74 + 11 * q^76 + 15 * q^77 - 11 * q^79 + 44 * q^80 - 14 * q^82 + 56 * q^83 + 16 * q^85 + 18 * q^86 - 18 * q^88 + 55 * q^89 + 11 * q^91 + 4 * q^92 - 5 * q^94 - 15 * q^95 - 7 * q^97 - 6 * q^98

Introduction

L-functions

Modular forms

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Database

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