LMFDB - Embedded newform 6039.2.a.e.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

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

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

chi := DirichletCharacter("6039.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$48.2216577807$
Analytic rank:	$1$
Dimension:	$12$
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} - x^{11} - 16 x^{10} + 13 x^{9} + 93 x^{8} - 59 x^{7} - 238 x^{6} + 108 x^{5} + 257 x^{4} + \cdots + 1 $ x^12 - x^11 - 16x^10 + 13x^9 + 93x^8 - 59x^7 - 238x^6 + 108x^5 + 257x^4 - 71x^3 - 93x^2 + 13x + 1
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			1.3
Root			$2.01091$ 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)

$f(q)$	$=$	$q-2.01091 q^{2} +2.04375 q^{4} -2.29360 q^{5} -4.38501 q^{7} -0.0879817 q^{8} +O(q^{10})$	\(q-2.01091 q^{2} +2.04375 q^{4} -2.29360 q^{5} -4.38501 q^{7} -0.0879817 q^{8} +4.61222 q^{10} +1.00000 q^{11} -1.69111 q^{13} +8.81784 q^{14} -3.91058 q^{16} +1.38158 q^{17} +0.110526 q^{19} -4.68755 q^{20} -2.01091 q^{22} +0.503803 q^{23} +0.260612 q^{25} +3.40067 q^{26} -8.96186 q^{28} +4.90215 q^{29} -3.94262 q^{31} +8.03978 q^{32} -2.77824 q^{34} +10.0575 q^{35} -3.22228 q^{37} -0.222258 q^{38} +0.201795 q^{40} -12.4079 q^{41} +1.40887 q^{43} +2.04375 q^{44} -1.01310 q^{46} +9.04166 q^{47} +12.2283 q^{49} -0.524067 q^{50} -3.45621 q^{52} +8.78013 q^{53} -2.29360 q^{55} +0.385800 q^{56} -9.85777 q^{58} +5.26459 q^{59} -1.00000 q^{61} +7.92825 q^{62} -8.34611 q^{64} +3.87874 q^{65} -8.70560 q^{67} +2.82362 q^{68} -20.2246 q^{70} -4.64701 q^{71} -1.09932 q^{73} +6.47971 q^{74} +0.225888 q^{76} -4.38501 q^{77} +14.4489 q^{79} +8.96932 q^{80} +24.9511 q^{82} +0.315020 q^{83} -3.16880 q^{85} -2.83311 q^{86} -0.0879817 q^{88} -2.05956 q^{89} +7.41553 q^{91} +1.02965 q^{92} -18.1819 q^{94} -0.253503 q^{95} +16.1329 q^{97} -24.5899 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - q^{2} + 9 q^{4} + 3 q^{5} - 9 q^{7} - 6 q^{8}+O(q^{10}) $ 12 * q - q^2 + 9 * q^4 + 3 * q^5 - 9 * q^7 - 6 * q^8	$ 12 q - q^{2} + 9 q^{4} + 3 q^{5} - 9 q^{7} - 6 q^{8} - 8 q^{10} + 12 q^{11} - q^{13} + 3 q^{14} + 3 q^{16} - 9 q^{17} - 20 q^{19} + 9 q^{20} - q^{22} + 9 q^{23} + 3 q^{25} + 18 q^{26} - 31 q^{28} - 18 q^{29} - 21 q^{31} - 18 q^{32} - 12 q^{34} + 4 q^{35} - 18 q^{37} + 2 q^{38} - 26 q^{40} - 15 q^{41} - 33 q^{43} + 9 q^{44} - 28 q^{46} + 20 q^{47} + 15 q^{49} + 2 q^{50} - 27 q^{52} + 3 q^{55} + 8 q^{56} - 11 q^{58} + 21 q^{59} - 12 q^{61} + 9 q^{62} - 12 q^{64} - 17 q^{65} - 34 q^{67} + 16 q^{68} - 36 q^{70} + 5 q^{71} - 2 q^{73} - 6 q^{74} - 27 q^{76} - 9 q^{77} - 31 q^{79} + 60 q^{80} - 12 q^{82} + 32 q^{83} - 40 q^{85} - 18 q^{86} - 6 q^{88} - 27 q^{89} - 45 q^{91} + 78 q^{92} - 13 q^{94} - 37 q^{95} - 19 q^{97} - 4 q^{98}+O(q^{100}) $ 12 * q - q^2 + 9 * q^4 + 3 * q^5 - 9 * q^7 - 6 * q^8 - 8 * q^10 + 12 * q^11 - q^13 + 3 * q^14 + 3 * q^16 - 9 * q^17 - 20 * q^19 + 9 * q^20 - q^22 + 9 * q^23 + 3 * q^25 + 18 * q^26 - 31 * q^28 - 18 * q^29 - 21 * q^31 - 18 * q^32 - 12 * q^34 + 4 * q^35 - 18 * q^37 + 2 * q^38 - 26 * q^40 - 15 * q^41 - 33 * q^43 + 9 * q^44 - 28 * q^46 + 20 * q^47 + 15 * q^49 + 2 * q^50 - 27 * q^52 + 3 * q^55 + 8 * q^56 - 11 * q^58 + 21 * q^59 - 12 * q^61 + 9 * q^62 - 12 * q^64 - 17 * q^65 - 34 * q^67 + 16 * q^68 - 36 * q^70 + 5 * q^71 - 2 * q^73 - 6 * q^74 - 27 * q^76 - 9 * q^77 - 31 * q^79 + 60 * q^80 - 12 * q^82 + 32 * q^83 - 40 * q^85 - 18 * q^86 - 6 * q^88 - 27 * q^89 - 45 * q^91 + 78 * q^92 - 13 * q^94 - 37 * q^95 - 19 * q^97 - 4 * 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 $

$2$	−2.01091	−1.42193	−0.710963	−	0.703229i	$-0.751741\pi$
$2$	−2.01091	−1.42193	−0.710963	+	0.703229i	$0.751741\pi$
$3$	0	0
$3$	0	0
$4$	2.04375	1.02188
$5$	−2.29360	−1.02573	−0.512865	−	0.858469i	$-0.671416\pi$
$5$	−2.29360	−1.02573	−0.512865	+	0.858469i	$0.671416\pi$
$6$	0	0
$7$	−4.38501	−1.65738	−0.828688	−	0.559711i	$-0.810912\pi$
$7$	−4.38501	−1.65738	−0.828688	+	0.559711i	$0.810912\pi$
$8$	−0.0879817	−0.0311062
$9$	0	0
$10$	4.61222	1.45851
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	−1.69111	−0.469030	−0.234515	−	0.972113i	$-0.575350\pi$
$13$	−1.69111	−0.469030	−0.234515	+	0.972113i	$0.575350\pi$
$14$	8.81784	2.35667
$15$	0	0
$16$	−3.91058	−0.977645
$17$	1.38158	0.335083	0.167542	−	0.985865i	$-0.446417\pi$
$17$	1.38158	0.335083	0.167542	+	0.985865i	$0.446417\pi$
$18$	0	0
$19$	0.110526	0.0253565	0.0126782	−	0.999920i	$-0.495964\pi$
$19$	0.110526	0.0253565	0.0126782	+	0.999920i	$0.495964\pi$
$20$	−4.68755	−1.04817
$21$	0	0
$22$	−2.01091	−0.428727
$23$	0.503803	0.105050	0.0525251	−	0.998620i	$-0.483273\pi$
$23$	0.503803	0.105050	0.0525251	+	0.998620i	$0.483273\pi$
$24$	0	0
$25$	0.260612	0.0521224
$26$	3.40067	0.666926
$27$	0	0
$28$	−8.96186	−1.69363
$29$	4.90215	0.910306	0.455153	−	0.890413i	$-0.349585\pi$
$29$	4.90215	0.910306	0.455153	+	0.890413i	$0.349585\pi$
$30$	0	0
$31$	−3.94262	−0.708116	−0.354058	−	0.935224i	$-0.615198\pi$
$31$	−3.94262	−0.708116	−0.354058	+	0.935224i	$0.615198\pi$
$32$	8.03978	1.42125
$33$	0	0
$34$	−2.77824	−0.476464
$35$	10.0575	1.70002
$36$	0	0
$37$	−3.22228	−0.529740	−0.264870	−	0.964284i	$-0.585329\pi$
$37$	−3.22228	−0.529740	−0.264870	+	0.964284i	$0.585329\pi$
$38$	−0.222258	−0.0360550
$39$	0	0
$40$	0.201795	0.0319066
$41$	−12.4079	−1.93778	−0.968891	−	0.247489i	$-0.920395\pi$
$41$	−12.4079	−1.93778	−0.968891	+	0.247489i	$0.920395\pi$
$42$	0	0
$43$	1.40887	0.214851	0.107425	−	0.994213i	$-0.465739\pi$
$43$	1.40887	0.214851	0.107425	+	0.994213i	$0.465739\pi$
$44$	2.04375	0.308107
$45$	0	0
$46$	−1.01310	−0.149374
$47$	9.04166	1.31886	0.659431	−	0.751765i	$-0.270797\pi$
$47$	9.04166	1.31886	0.659431	+	0.751765i	$0.270797\pi$
$48$	0	0
$49$	12.2283	1.74690
$50$	−0.524067	−0.0741142
$51$	0	0
$52$	−3.45621	−0.479290
$53$	8.78013	1.20604	0.603022	−	0.797725i	$-0.293963\pi$
$53$	8.78013	1.20604	0.603022	+	0.797725i	$0.293963\pi$
$54$	0	0
$55$	−2.29360	−0.309269
$56$	0.385800	0.0515547
$57$	0	0
$58$	−9.85777	−1.29439
$59$	5.26459	0.685392	0.342696	−	0.939446i	$-0.388660\pi$
$59$	5.26459	0.685392	0.342696	+	0.939446i	$0.388660\pi$
$60$	0	0
$61$	−1.00000	−0.128037
$61$	−1.00000	−0.128037
$62$	7.92825	1.00689
$63$	0	0
$64$	−8.34611	−1.04326
$65$	3.87874	0.481098
$66$	0	0
$67$	−8.70560	−1.06356	−0.531779	−	0.846883i	$-0.678476\pi$
$67$	−8.70560	−1.06356	−0.531779	+	0.846883i	$0.678476\pi$
$68$	2.82362	0.342414
$69$	0	0
$70$	−20.2246	−2.41731
$71$	−4.64701	−0.551499	−0.275749	−	0.961230i	$-0.588926\pi$
$71$	−4.64701	−0.551499	−0.275749	+	0.961230i	$0.588926\pi$
$72$	0	0
$73$	−1.09932	−0.128666	−0.0643331	−	0.997928i	$-0.520492\pi$
$73$	−1.09932	−0.128666	−0.0643331	+	0.997928i	$0.520492\pi$
$74$	6.47971	0.753251
$75$	0	0
$76$	0.225888	0.0259112
$77$	−4.38501	−0.499718
$78$	0	0
$79$	14.4489	1.62563	0.812814	−	0.582524i	$-0.197935\pi$
$79$	14.4489	1.62563	0.812814	+	0.582524i	$0.197935\pi$
$80$	8.96932	1.00280
$81$	0	0
$82$	24.9511	2.75538
$83$	0.315020	0.0345780	0.0172890	−	0.999851i	$-0.494496\pi$
$83$	0.315020	0.0345780	0.0172890	+	0.999851i	$0.494496\pi$
$84$	0	0
$85$	−3.16880	−0.343705
$86$	−2.83311	−0.305502
$87$	0	0
$88$	−0.0879817	−0.00937888
$89$	−2.05956	−0.218313	−0.109156	−	0.994025i	$-0.534815\pi$
$89$	−2.05956	−0.218313	−0.109156	+	0.994025i	$0.534815\pi$
$90$	0	0
$91$	7.41553	0.777359
$92$	1.02965	0.107348
$93$	0	0
$94$	−18.1819	−1.87532
$95$	−0.253503	−0.0260089
$96$	0	0
$97$	16.1329	1.63804	0.819022	−	0.573762i	$-0.194516\pi$
$97$	16.1329	1.63804	0.819022	+	0.573762i	$0.194516\pi$
$98$	−24.5899	−2.48396
$99$	0	0
$100$	0.532626	0.0532626
$101$	0.0715942	0.00712389	0.00356195	−	0.999994i	$-0.498866\pi$
$101$	0.0715942	0.00712389	0.00356195	+	0.999994i	$0.498866\pi$
$102$	0	0
$103$	−4.03904	−0.397978	−0.198989	−	0.980002i	$-0.563766\pi$
$103$	−4.03904	−0.397978	−0.198989	+	0.980002i	$0.563766\pi$
$104$	0.148787	0.0145897
$105$	0	0
$106$	−17.6560	−1.71491
$107$	18.0417	1.74416	0.872078	−	0.489367i	$-0.162772\pi$
$107$	18.0417	1.74416	0.872078	+	0.489367i	$0.162772\pi$
$108$	0	0
$109$	15.7912	1.51253	0.756263	−	0.654267i	$-0.227023\pi$
$109$	15.7912	1.51253	0.756263	+	0.654267i	$0.227023\pi$
$110$	4.61222	0.439758
$111$	0	0
$112$	17.1479	1.62033
$113$	5.62641	0.529288	0.264644	−	0.964346i	$-0.414745\pi$
$113$	5.62641	0.529288	0.264644	+	0.964346i	$0.414745\pi$
$114$	0	0
$115$	−1.15552	−0.107753
$116$	10.0188	0.930220
$117$	0	0
$118$	−10.5866	−0.974577
$119$	−6.05825	−0.555359
$120$	0	0
$121$	1.00000	0.0909091
$122$	2.01091	0.182059
$123$	0	0
$124$	−8.05774	−0.723606
$125$	10.8703	0.972267
$126$	0	0
$127$	−4.42664	−0.392801	−0.196400	−	0.980524i	$-0.562925\pi$
$127$	−4.42664	−0.392801	−0.196400	+	0.980524i	$0.562925\pi$
$128$	0.703685	0.0621976
$129$	0	0
$130$	−7.79978	−0.684086
$131$	−3.64937	−0.318847	−0.159423	−	0.987210i	$-0.550964\pi$
$131$	−3.64937	−0.318847	−0.159423	+	0.987210i	$0.550964\pi$
$132$	0	0
$133$	−0.484658	−0.0420252
$134$	17.5062	1.51230
$135$	0	0
$136$	−0.121554	−0.0104232
$137$	11.5989	0.990965	0.495483	−	0.868618i	$-0.334991\pi$
$137$	11.5989	0.990965	0.495483	+	0.868618i	$0.334991\pi$
$138$	0	0
$139$	−15.1002	−1.28078	−0.640390	−	0.768050i	$-0.721227\pi$
$139$	−15.1002	−1.28078	−0.640390	+	0.768050i	$0.721227\pi$
$140$	20.5550	1.73721
$141$	0	0
$142$	9.34471	0.784191
$143$	−1.69111	−0.141418
$144$	0	0
$145$	−11.2436	−0.933728
$146$	2.21064	0.182954
$147$	0	0
$148$	−6.58555	−0.541329
$149$	−13.2433	−1.08494	−0.542468	−	0.840076i	$-0.682510\pi$
$149$	−13.2433	−1.08494	−0.542468	+	0.840076i	$0.682510\pi$
$150$	0	0
$151$	16.6453	1.35457	0.677286	−	0.735720i	$-0.263156\pi$
$151$	16.6453	1.35457	0.677286	+	0.735720i	$0.263156\pi$
$152$	−0.00972429	−0.000788744 0
$153$	0	0
$154$	8.81784	0.710562
$155$	9.04280	0.726335
$156$	0	0
$157$	11.9301	0.952127	0.476063	−	0.879411i	$-0.342063\pi$
$157$	11.9301	0.952127	0.476063	+	0.879411i	$0.342063\pi$
$158$	−29.0554	−2.31152
$159$	0	0
$160$	−18.4401	−1.45782
$161$	−2.20918	−0.174108
$162$	0	0
$163$	−0.505535	−0.0395966	−0.0197983	−	0.999804i	$-0.506302\pi$
$163$	−0.505535	−0.0395966	−0.0197983	+	0.999804i	$0.506302\pi$
$164$	−25.3586	−1.98017
$165$	0	0
$166$	−0.633477	−0.0491673
$167$	7.02034	0.543250	0.271625	−	0.962403i	$-0.412439\pi$
$167$	7.02034	0.543250	0.271625	+	0.962403i	$0.412439\pi$
$168$	0	0
$169$	−10.1401	−0.780011
$170$	6.37218	0.488724
$171$	0	0
$172$	2.87938	0.219551
$173$	5.71737	0.434684	0.217342	−	0.976096i	$-0.430261\pi$
$173$	5.71737	0.434684	0.217342	+	0.976096i	$0.430261\pi$
$174$	0	0
$175$	−1.14278	−0.0863864
$176$	−3.91058	−0.294771
$177$	0	0
$178$	4.14158	0.310425
$179$	−5.77529	−0.431665	−0.215833	−	0.976430i	$-0.569247\pi$
$179$	−5.77529	−0.431665	−0.215833	+	0.976430i	$0.569247\pi$
$180$	0	0
$181$	8.98069	0.667530	0.333765	−	0.942656i	$-0.391681\pi$
$181$	8.98069	0.667530	0.333765	+	0.942656i	$0.391681\pi$
$182$	−14.9120	−1.10535
$183$	0	0
$184$	−0.0443255	−0.00326772
$185$	7.39063	0.543370
$186$	0	0
$187$	1.38158	0.101031
$188$	18.4789	1.34771
$189$	0	0
$190$	0.509772	0.0369827
$191$	2.79919	0.202542	0.101271	−	0.994859i	$-0.467709\pi$
$191$	2.79919	0.202542	0.101271	+	0.994859i	$0.467709\pi$
$192$	0	0
$193$	−10.5823	−0.761729	−0.380865	−	0.924631i	$-0.624374\pi$
$193$	−10.5823	−0.761729	−0.380865	+	0.924631i	$0.624374\pi$
$194$	−32.4417	−2.32918
$195$	0	0
$196$	24.9916	1.78511
$197$	−23.6369	−1.68406	−0.842031	−	0.539429i	$-0.818640\pi$
$197$	−23.6369	−1.68406	−0.842031	+	0.539429i	$0.818640\pi$
$198$	0	0
$199$	0.311922	0.0221116	0.0110558	−	0.999939i	$-0.496481\pi$
$199$	0.311922	0.0221116	0.0110558	+	0.999939i	$0.496481\pi$
$200$	−0.0229291	−0.00162133
$201$	0	0
$202$	−0.143969	−0.0101297
$203$	−21.4959	−1.50872
$204$	0	0
$205$	28.4587	1.98764
$206$	8.12214	0.565896
$207$	0	0
$208$	6.61323	0.458545
$209$	0.110526	0.00764526
$210$	0	0
$211$	−26.1628	−1.80112	−0.900562	−	0.434728i	$-0.856845\pi$
$211$	−26.1628	−1.80112	−0.900562	+	0.434728i	$0.856845\pi$
$212$	17.9444	1.23243
$213$	0	0
$214$	−36.2802	−2.48006
$215$	−3.23139	−0.220379
$216$	0	0
$217$	17.2884	1.17361
$218$	−31.7547	−2.15070
$219$	0	0
$220$	−4.68755	−0.316035
$221$	−2.33641	−0.157164
$222$	0	0
$223$	15.1197	1.01249	0.506244	−	0.862390i	$-0.331034\pi$
$223$	15.1197	1.01249	0.506244	+	0.862390i	$0.331034\pi$
$224$	−35.2545	−2.35554
$225$	0	0
$226$	−11.3142	−0.752609
$227$	−8.01271	−0.531822	−0.265911	−	0.963998i	$-0.585673\pi$
$227$	−8.01271	−0.531822	−0.265911	+	0.963998i	$0.585673\pi$
$228$	0	0
$229$	−18.0492	−1.19272	−0.596362	−	0.802716i	$-0.703388\pi$
$229$	−18.0492	−1.19272	−0.596362	+	0.802716i	$0.703388\pi$
$230$	2.32365	0.153217
$231$	0	0
$232$	−0.431299	−0.0283162
$233$	11.3922	0.746330	0.373165	−	0.927765i	$-0.378273\pi$
$233$	11.3922	0.746330	0.373165	+	0.927765i	$0.378273\pi$
$234$	0	0
$235$	−20.7380	−1.35280
$236$	10.7595	0.700386
$237$	0	0
$238$	12.1826	0.789680
$239$	−3.97892	−0.257375	−0.128688	−	0.991685i	$-0.541076\pi$
$239$	−3.97892	−0.257375	−0.128688	+	0.991685i	$0.541076\pi$
$240$	0	0
$241$	−5.89109	−0.379478	−0.189739	−	0.981835i	$-0.560764\pi$
$241$	−5.89109	−0.379478	−0.189739	+	0.981835i	$0.560764\pi$
$242$	−2.01091	−0.129266
$243$	0	0
$244$	−2.04375	−0.130838
$245$	−28.0468	−1.79184
$246$	0	0
$247$	−0.186912	−0.0118929
$248$	0.346878	0.0220268
$249$	0	0
$250$	−21.8591	−1.38249
$251$	21.6615	1.36726	0.683631	−	0.729828i	$-0.260400\pi$
$251$	21.6615	1.36726	0.683631	+	0.729828i	$0.260400\pi$
$252$	0	0
$253$	0.503803	0.0316738
$254$	8.90156	0.558534
$255$	0	0
$256$	15.2772	0.954823
$257$	8.24914	0.514567	0.257284	−	0.966336i	$-0.417173\pi$
$257$	8.24914	0.514567	0.257284	+	0.966336i	$0.417173\pi$
$258$	0	0
$259$	14.1297	0.877978
$260$	7.92717	0.491623
$261$	0	0
$262$	7.33855	0.453377
$263$	−26.5861	−1.63937	−0.819685	−	0.572815i	$-0.805851\pi$
$263$	−26.5861	−1.63937	−0.819685	+	0.572815i	$0.805851\pi$
$264$	0	0
$265$	−20.1381	−1.23708
$266$	0.974603	0.0597567
$267$	0	0
$268$	−17.7921	−1.08682
$269$	13.7490	0.838293	0.419147	−	0.907919i	$-0.362329\pi$
$269$	13.7490	0.838293	0.419147	+	0.907919i	$0.362329\pi$
$270$	0	0
$271$	−8.02467	−0.487464	−0.243732	−	0.969843i	$-0.578372\pi$
$271$	−8.02467	−0.487464	−0.243732	+	0.969843i	$0.578372\pi$
$272$	−5.40280	−0.327593
$273$	0	0
$274$	−23.3244	−1.40908
$275$	0.260612	0.0157155
$276$	0	0
$277$	15.6333	0.939314	0.469657	−	0.882849i	$-0.344378\pi$
$277$	15.6333	0.939314	0.469657	+	0.882849i	$0.344378\pi$
$278$	30.3651	1.82117
$279$	0	0
$280$	−0.884873	−0.0528812
$281$	−31.2843	−1.86626	−0.933132	−	0.359534i	$-0.882936\pi$
$281$	−31.2843	−1.86626	−0.933132	+	0.359534i	$0.882936\pi$
$282$	0	0
$283$	13.7147	0.815255	0.407628	−	0.913148i	$-0.366356\pi$
$283$	13.7147	0.815255	0.407628	+	0.913148i	$0.366356\pi$
$284$	−9.49734	−0.563563
$285$	0	0
$286$	3.40067	0.201086
$287$	54.4085	3.21163
$288$	0	0
$289$	−15.0912	−0.887719
$290$	22.6098	1.32769
$291$	0	0
$292$	−2.24675	−0.131481
$293$	−9.57188	−0.559195	−0.279598	−	0.960117i	$-0.590201\pi$
$293$	−9.57188	−0.559195	−0.279598	+	0.960117i	$0.590201\pi$
$294$	0	0
$295$	−12.0749	−0.703027
$296$	0.283502	0.0164782
$297$	0	0
$298$	26.6311	1.54270
$299$	−0.851987	−0.0492717
$300$	0	0
$301$	−6.17790	−0.356088
$302$	−33.4721	−1.92610
$303$	0	0
$304$	−0.432222	−0.0247896
$305$	2.29360	0.131331
$306$	0	0
$307$	−11.7940	−0.673120	−0.336560	−	0.941662i	$-0.609263\pi$
$307$	−11.7940	−0.673120	−0.336560	+	0.941662i	$0.609263\pi$
$308$	−8.96186	−0.510650
$309$	0	0
$310$	−18.1842	−1.03280
$311$	6.65342	0.377281	0.188640	−	0.982046i	$-0.439592\pi$
$311$	6.65342	0.377281	0.188640	+	0.982046i	$0.439592\pi$
$312$	0	0
$313$	−5.96584	−0.337209	−0.168605	−	0.985684i	$-0.553926\pi$
$313$	−5.96584	−0.337209	−0.168605	+	0.985684i	$0.553926\pi$
$314$	−23.9904	−1.35385
$315$	0	0
$316$	29.5300	1.66119
$317$	−16.5911	−0.931848	−0.465924	−	0.884825i	$-0.654278\pi$
$317$	−16.5911	−0.931848	−0.465924	+	0.884825i	$0.654278\pi$
$318$	0	0
$319$	4.90215	0.274468
$320$	19.1426	1.07011
$321$	0	0
$322$	4.44246	0.247568
$323$	0.152701	0.00849653
$324$	0	0
$325$	−0.440724	−0.0244469
$326$	1.01658	0.0563034
$327$	0	0
$328$	1.09166	0.0602771
$329$	−39.6477	−2.18585
$330$	0	0
$331$	−23.9520	−1.31652	−0.658259	−	0.752791i	$-0.728707\pi$
$331$	−23.9520	−1.31652	−0.658259	+	0.752791i	$0.728707\pi$
$332$	0.643823	0.0353344
$333$	0	0
$334$	−14.1173	−0.772462
$335$	19.9672	1.09092
$336$	0	0
$337$	−6.96307	−0.379302	−0.189651	−	0.981852i	$-0.560736\pi$
$337$	−6.96307	−0.379302	−0.189651	+	0.981852i	$0.560736\pi$
$338$	20.3909	1.10912
$339$	0	0
$340$	−6.47625	−0.351224
$341$	−3.94262	−0.213505
$342$	0	0
$343$	−22.9260	−1.23789
$344$	−0.123955	−0.00668319
$345$	0	0
$346$	−11.4971	−0.618088
$347$	−20.8345	−1.11845	−0.559226	−	0.829015i	$-0.688902\pi$
$347$	−20.8345	−1.11845	−0.559226	+	0.829015i	$0.688902\pi$
$348$	0	0
$349$	22.1809	1.18732	0.593659	−	0.804717i	$-0.297683\pi$
$349$	22.1809	1.18732	0.593659	+	0.804717i	$0.297683\pi$
$350$	2.29803	0.122835
$351$	0	0
$352$	8.03978	0.428522
$353$	−15.6354	−0.832190	−0.416095	−	0.909321i	$-0.636602\pi$
$353$	−15.6354	−0.832190	−0.416095	+	0.909321i	$0.636602\pi$
$354$	0	0
$355$	10.6584	0.565689
$356$	−4.20923	−0.223089
$357$	0	0
$358$	11.6136	0.613796
$359$	22.1078	1.16680	0.583402	−	0.812183i	$-0.301721\pi$
$359$	22.1078	1.16680	0.583402	+	0.812183i	$0.301721\pi$
$360$	0	0
$361$	−18.9878	−0.999357
$362$	−18.0593	−0.949178
$363$	0	0
$364$	15.1555	0.794364
$365$	2.52141	0.131977
$366$	0	0
$367$	9.46184	0.493904	0.246952	−	0.969028i	$-0.420571\pi$
$367$	9.46184	0.493904	0.246952	+	0.969028i	$0.420571\pi$
$368$	−1.97016	−0.102702
$369$	0	0
$370$	−14.8619	−0.772633
$371$	−38.5009	−1.99887
$372$	0	0
$373$	−4.34508	−0.224980	−0.112490	−	0.993653i	$-0.535883\pi$
$373$	−4.34508	−0.224980	−0.112490	+	0.993653i	$0.535883\pi$
$374$	−2.77824	−0.143659
$375$	0	0
$376$	−0.795501	−0.0410248
$377$	−8.29008	−0.426961
$378$	0	0
$379$	−4.12788	−0.212035	−0.106017	−	0.994364i	$-0.533810\pi$
$379$	−4.12788	−0.212035	−0.106017	+	0.994364i	$0.533810\pi$
$380$	−0.518098	−0.0265779
$381$	0	0
$382$	−5.62890	−0.288000
$383$	24.1984	1.23648	0.618241	−	0.785989i	$-0.287846\pi$
$383$	24.1984	1.23648	0.618241	+	0.785989i	$0.287846\pi$
$384$	0	0
$385$	10.0575	0.512576
$386$	21.2800	1.08312
$387$	0	0
$388$	32.9716	1.67388
$389$	−19.9003	−1.00899	−0.504493	−	0.863416i	$-0.668320\pi$
$389$	−19.9003	−1.00899	−0.504493	+	0.863416i	$0.668320\pi$
$390$	0	0
$391$	0.696046	0.0352006
$392$	−1.07586	−0.0543394
$393$	0	0
$394$	47.5317	2.39461
$395$	−33.1400	−1.66745
$396$	0	0
$397$	−14.7767	−0.741620	−0.370810	−	0.928709i	$-0.620920\pi$
$397$	−14.7767	−0.741620	−0.370810	+	0.928709i	$0.620920\pi$
$398$	−0.627247	−0.0314411
$399$	0	0
$400$	−1.01914	−0.0509572
$401$	32.5681	1.62637	0.813187	−	0.582002i	$-0.197731\pi$
$401$	32.5681	1.62637	0.813187	+	0.582002i	$0.197731\pi$
$402$	0	0
$403$	6.66741	0.332127
$404$	0.146321	0.00727973
$405$	0	0
$406$	43.2264	2.14529
$407$	−3.22228	−0.159723
$408$	0	0
$409$	12.4675	0.616476	0.308238	−	0.951309i	$-0.400261\pi$
$409$	12.4675	0.616476	0.308238	+	0.951309i	$0.400261\pi$
$410$	−57.2278	−2.82628
$411$	0	0
$412$	−8.25480	−0.406685
$413$	−23.0853	−1.13595
$414$	0	0
$415$	−0.722531	−0.0354677
$416$	−13.5962	−0.666607
$417$	0	0
$418$	−0.222258	−0.0108710
$419$	6.15342	0.300614	0.150307	−	0.988639i	$-0.451974\pi$
$419$	6.15342	0.300614	0.150307	+	0.988639i	$0.451974\pi$
$420$	0	0
$421$	2.12112	0.103377	0.0516886	−	0.998663i	$-0.483540\pi$
$421$	2.12112	0.103377	0.0516886	+	0.998663i	$0.483540\pi$
$422$	52.6111	2.56107
$423$	0	0
$424$	−0.772491	−0.0375155
$425$	0.360057	0.0174653
$426$	0	0
$427$	4.38501	0.212205
$428$	36.8727	1.78231
$429$	0	0
$430$	6.49802	0.313363
$431$	6.48815	0.312523	0.156262	−	0.987716i	$-0.450056\pi$
$431$	6.48815	0.312523	0.156262	+	0.987716i	$0.450056\pi$
$432$	0	0
$433$	27.8259	1.33723	0.668614	−	0.743610i	$-0.266888\pi$
$433$	27.8259	1.33723	0.668614	+	0.743610i	$0.266888\pi$
$434$	−34.7654	−1.66879
$435$	0	0
$436$	32.2734	1.54561
$437$	0.0556835	0.00266370
$438$	0	0
$439$	−18.7530	−0.895032	−0.447516	−	0.894276i	$-0.647691\pi$
$439$	−18.7530	−0.895032	−0.447516	+	0.894276i	$0.647691\pi$
$440$	0.201795	0.00962020
$441$	0	0
$442$	4.69831	0.223476
$443$	7.88037	0.374408	0.187204	−	0.982321i	$-0.440058\pi$
$443$	7.88037	0.374408	0.187204	+	0.982321i	$0.440058\pi$
$444$	0	0
$445$	4.72381	0.223930
$446$	−30.4043	−1.43968
$447$	0	0
$448$	36.5977	1.72908
$449$	−23.4421	−1.10630	−0.553150	−	0.833082i	$-0.686574\pi$
$449$	−23.4421	−1.10630	−0.553150	+	0.833082i	$0.686574\pi$
$450$	0	0
$451$	−12.4079	−0.584263
$452$	11.4990	0.540867
$453$	0	0
$454$	16.1128	0.756212
$455$	−17.0083	−0.797360
$456$	0	0
$457$	21.7612	1.01795	0.508974	−	0.860782i	$-0.330025\pi$
$457$	21.7612	1.01795	0.508974	+	0.860782i	$0.330025\pi$
$458$	36.2953	1.69597
$459$	0	0
$460$	−2.36160	−0.110110
$461$	−3.03355	−0.141287	−0.0706433	−	0.997502i	$-0.522505\pi$
$461$	−3.03355	−0.141287	−0.0706433	+	0.997502i	$0.522505\pi$
$462$	0	0
$463$	28.2066	1.31087	0.655435	−	0.755252i	$-0.272485\pi$
$463$	28.2066	1.31087	0.655435	+	0.755252i	$0.272485\pi$
$464$	−19.1703	−0.889957
$465$	0	0
$466$	−22.9087	−1.06123
$467$	37.5563	1.73790	0.868949	−	0.494901i	$-0.164796\pi$
$467$	37.5563	1.73790	0.868949	+	0.494901i	$0.164796\pi$
$468$	0	0
$469$	38.1741	1.76272
$470$	41.7021	1.92358
$471$	0	0
$472$	−0.463188	−0.0213200
$473$	1.40887	0.0647799
$474$	0	0
$475$	0.0288044	0.00132164
$476$	−12.3816	−0.567508
$477$	0	0
$478$	8.00125	0.365969
$479$	8.43821	0.385552	0.192776	−	0.981243i	$-0.438251\pi$
$479$	8.43821	0.385552	0.192776	+	0.981243i	$0.438251\pi$
$480$	0	0
$481$	5.44924	0.248464
$482$	11.8464	0.539590
$483$	0	0
$484$	2.04375	0.0928978
$485$	−37.0024	−1.68019
$486$	0	0
$487$	−12.6724	−0.574240	−0.287120	−	0.957895i	$-0.592698\pi$
$487$	−12.6724	−0.574240	−0.287120	+	0.957895i	$0.592698\pi$
$488$	0.0879817	0.00398275
$489$	0	0
$490$	56.3995	2.54787
$491$	−13.8805	−0.626420	−0.313210	−	0.949684i	$-0.601404\pi$
$491$	−13.8805	−0.626420	−0.313210	+	0.949684i	$0.601404\pi$
$492$	0	0
$493$	6.77273	0.305028
$494$	0.375863	0.0169109
$495$	0	0
$496$	15.4179	0.692286
$497$	20.3772	0.914041
$498$	0	0
$499$	−12.0067	−0.537496	−0.268748	−	0.963211i	$-0.586610\pi$
$499$	−12.0067	−0.537496	−0.268748	+	0.963211i	$0.586610\pi$
$500$	22.2161	0.993536
$501$	0	0
$502$	−43.5593	−1.94415
$503$	15.4221	0.687639	0.343819	−	0.939036i	$-0.388279\pi$
$503$	15.4221	0.687639	0.343819	+	0.939036i	$0.388279\pi$
$504$	0	0
$505$	−0.164209	−0.00730719
$506$	−1.01310	−0.0450379
$507$	0	0
$508$	−9.04695	−0.401394
$509$	−5.48166	−0.242970	−0.121485	−	0.992593i	$-0.538766\pi$
$509$	−5.48166	−0.242970	−0.121485	+	0.992593i	$0.538766\pi$
$510$	0	0
$511$	4.82054	0.213248
$512$	−32.1283	−1.41989
$513$	0	0
$514$	−16.5883	−0.731677
$515$	9.26395	0.408218
$516$	0	0
$517$	9.04166	0.397652
$518$	−28.4136	−1.24842
$519$	0	0
$520$	−0.341258	−0.0149651
$521$	14.2914	0.626116	0.313058	−	0.949734i	$-0.398647\pi$
$521$	14.2914	0.626116	0.313058	+	0.949734i	$0.398647\pi$
$522$	0	0
$523$	−34.2760	−1.49878	−0.749392	−	0.662127i	$-0.769654\pi$
$523$	−34.2760	−1.49878	−0.749392	+	0.662127i	$0.769654\pi$
$524$	−7.45841	−0.325822
$525$	0	0
$526$	53.4622	2.33106
$527$	−5.44706	−0.237278
$528$	0	0
$529$	−22.7462	−0.988964
$530$	40.4959	1.75903
$531$	0	0
$532$	−0.990521	−0.0429445
$533$	20.9831	0.908877
$534$	0	0
$535$	−41.3804	−1.78903
$536$	0.765933	0.0330833
$537$	0	0
$538$	−27.6480	−1.19199
$539$	12.2283	0.526709
$540$	0	0
$541$	−10.0880	−0.433718	−0.216859	−	0.976203i	$-0.569581\pi$
$541$	−10.0880	−0.433718	−0.216859	+	0.976203i	$0.569581\pi$
$542$	16.1369	0.693138
$543$	0	0
$544$	11.1076	0.476236
$545$	−36.2188	−1.55144
$546$	0	0
$547$	−13.4837	−0.576521	−0.288261	−	0.957552i	$-0.593077\pi$
$547$	−13.4837	−0.576521	−0.288261	+	0.957552i	$0.593077\pi$
$548$	23.7054	1.01264
$549$	0	0
$550$	−0.524067	−0.0223463
$551$	0.541816	0.0230821
$552$	0	0
$553$	−63.3585	−2.69428
$554$	−31.4371	−1.33564
$555$	0	0
$556$	−30.8610	−1.30880
$557$	−4.31291	−0.182744	−0.0913719	−	0.995817i	$-0.529125\pi$
$557$	−4.31291	−0.182744	−0.0913719	+	0.995817i	$0.529125\pi$
$558$	0	0
$559$	−2.38255	−0.100771
$560$	−39.3305	−1.66202
$561$	0	0
$562$	62.9098	2.65369
$563$	−19.9662	−0.841474	−0.420737	−	0.907183i	$-0.638228\pi$
$563$	−19.9662	−0.841474	−0.420737	+	0.907183i	$0.638228\pi$
$564$	0	0
$565$	−12.9048	−0.542907
$566$	−27.5790	−1.15923
$567$	0	0
$568$	0.408852	0.0171550
$569$	24.9722	1.04689	0.523445	−	0.852060i	$-0.324647\pi$
$569$	24.9722	1.04689	0.523445	+	0.852060i	$0.324647\pi$
$570$	0	0
$571$	−14.2393	−0.595895	−0.297947	−	0.954582i	$-0.596302\pi$
$571$	−14.2393	−0.595895	−0.297947	+	0.954582i	$0.596302\pi$
$572$	−3.45621	−0.144511
$573$	0	0
$574$	−109.411	−4.56671
$575$	0.131297	0.00547547
$576$	0	0
$577$	−30.9745	−1.28948	−0.644742	−	0.764400i	$-0.723035\pi$
$577$	−30.9745	−1.28948	−0.644742	+	0.764400i	$0.723035\pi$
$578$	30.3471	1.26227
$579$	0	0
$580$	−22.9791	−0.954155
$581$	−1.38137	−0.0573087
$582$	0	0
$583$	8.78013	0.363636
$584$	0.0967204	0.00400232
$585$	0	0
$586$	19.2482	0.795135
$587$	9.67605	0.399373	0.199687	−	0.979860i	$-0.436008\pi$
$587$	9.67605	0.399373	0.199687	+	0.979860i	$0.436008\pi$
$588$	0	0
$589$	−0.435763	−0.0179553
$590$	24.2815	0.999653
$591$	0	0
$592$	12.6010	0.517898
$593$	−21.0897	−0.866052	−0.433026	−	0.901381i	$-0.642554\pi$
$593$	−21.0897	−0.866052	−0.433026	+	0.901381i	$0.642554\pi$
$594$	0	0
$595$	13.8952	0.569649
$596$	−27.0661	−1.10867
$597$	0	0
$598$	1.71327	0.0700607
$599$	−45.4706	−1.85788	−0.928938	−	0.370234i	$-0.879277\pi$
$599$	−45.4706	−1.85788	−0.928938	+	0.370234i	$0.879277\pi$
$600$	0	0
$601$	−47.2990	−1.92937	−0.964684	−	0.263411i	$-0.915152\pi$
$601$	−47.2990	−1.92937	−0.964684	+	0.263411i	$0.915152\pi$
$602$	12.4232	0.506332
$603$	0	0
$604$	34.0188	1.38420
$605$	−2.29360	−0.0932482
$606$	0	0
$607$	14.6907	0.596279	0.298139	−	0.954522i	$-0.403634\pi$
$607$	14.6907	0.596279	0.298139	+	0.954522i	$0.403634\pi$
$608$	0.888607	0.0360378
$609$	0	0
$610$	−4.61222	−0.186743
$611$	−15.2904	−0.618585
$612$	0	0
$613$	−18.0321	−0.728308	−0.364154	−	0.931339i	$-0.618642\pi$
$613$	−18.0321	−0.728308	−0.364154	+	0.931339i	$0.618642\pi$
$614$	23.7167	0.957127
$615$	0	0
$616$	0.385800	0.0155443
$617$	−18.5435	−0.746534	−0.373267	−	0.927724i	$-0.621762\pi$
$617$	−18.5435	−0.746534	−0.373267	+	0.927724i	$0.621762\pi$
$618$	0	0
$619$	26.7079	1.07348	0.536741	−	0.843747i	$-0.319655\pi$
$619$	26.7079	1.07348	0.536741	+	0.843747i	$0.319655\pi$
$620$	18.4812	0.742225
$621$	0	0
$622$	−13.3794	−0.536465
$623$	9.03118	0.361826
$624$	0	0
$625$	−26.2351	−1.04941
$626$	11.9968	0.479487
$627$	0	0
$628$	24.3822	0.972956
$629$	−4.45185	−0.177507
$630$	0	0
$631$	−12.8556	−0.511775	−0.255887	−	0.966707i	$-0.582368\pi$
$631$	−12.8556	−0.511775	−0.255887	+	0.966707i	$0.582368\pi$
$632$	−1.27124	−0.0505671
$633$	0	0
$634$	33.3631	1.32502
$635$	10.1529	0.402908
$636$	0	0
$637$	−20.6794	−0.819346
$638$	−9.85777	−0.390273
$639$	0	0
$640$	−1.61397	−0.0637979
$641$	−3.35864	−0.132658	−0.0663292	−	0.997798i	$-0.521129\pi$
$641$	−3.35864	−0.132658	−0.0663292	+	0.997798i	$0.521129\pi$
$642$	0	0
$643$	24.3997	0.962232	0.481116	−	0.876657i	$-0.340232\pi$
$643$	24.3997	0.962232	0.481116	+	0.876657i	$0.340232\pi$
$644$	−4.51502	−0.177917
$645$	0	0
$646$	−0.307068	−0.0120814
$647$	−21.3923	−0.841018	−0.420509	−	0.907288i	$-0.638148\pi$
$647$	−21.3923	−0.841018	−0.420509	+	0.907288i	$0.638148\pi$
$648$	0	0
$649$	5.26459	0.206653
$650$	0.886255	0.0347618
$651$	0	0
$652$	−1.03319	−0.0404628
$653$	19.6838	0.770287	0.385143	−	0.922857i	$-0.374152\pi$
$653$	19.6838	0.770287	0.385143	+	0.922857i	$0.374152\pi$
$654$	0	0
$655$	8.37020	0.327051
$656$	48.5219	1.89446
$657$	0	0
$658$	79.7279	3.10812
$659$	−41.7847	−1.62770	−0.813851	−	0.581073i	$-0.802633\pi$
$659$	−41.7847	−1.62770	−0.813851	+	0.581073i	$0.802633\pi$
$660$	0	0
$661$	−49.1659	−1.91233	−0.956166	−	0.292827i	$-0.905404\pi$
$661$	−49.1659	−1.91233	−0.956166	+	0.292827i	$0.905404\pi$
$662$	48.1652	1.87199
$663$	0	0
$664$	−0.0277160	−0.00107559
$665$	1.11161	0.0431065
$666$	0	0
$667$	2.46972	0.0956279
$668$	14.3478	0.555135
$669$	0	0
$670$	−40.1522	−1.55121
$671$	−1.00000	−0.0386046
$672$	0	0
$673$	−26.0028	−1.00233	−0.501166	−	0.865351i	$-0.667096\pi$
$673$	−26.0028	−1.00233	−0.501166	+	0.865351i	$0.667096\pi$
$674$	14.0021	0.539340
$675$	0	0
$676$	−20.7239	−0.797075
$677$	39.4872	1.51762	0.758808	−	0.651314i	$-0.225782\pi$
$677$	39.4872	1.51762	0.758808	+	0.651314i	$0.225782\pi$
$678$	0	0
$679$	−70.7427	−2.71486
$680$	0.278797	0.0106914
$681$	0	0
$682$	7.92825	0.303588
$683$	30.4225	1.16408	0.582042	−	0.813159i	$-0.302254\pi$
$683$	30.4225	1.16408	0.582042	+	0.813159i	$0.302254\pi$
$684$	0	0
$685$	−26.6034	−1.01646
$686$	46.1021	1.76019
$687$	0	0
$688$	−5.50950	−0.210048
$689$	−14.8482	−0.565670
$690$	0	0
$691$	−15.2900	−0.581661	−0.290830	−	0.956775i	$-0.593931\pi$
$691$	−15.2900	−0.581661	−0.290830	+	0.956775i	$0.593931\pi$
$692$	11.6849	0.444193
$693$	0	0
$694$	41.8962	1.59036
$695$	34.6338	1.31373
$696$	0	0
$697$	−17.1425	−0.649318
$698$	−44.6038	−1.68828
$699$	0	0
$700$	−2.33557	−0.0882762
$701$	−46.4671	−1.75504	−0.877519	−	0.479542i	$-0.840803\pi$
$701$	−46.4671	−1.75504	−0.877519	+	0.479542i	$0.840803\pi$
$702$	0	0
$703$	−0.356147	−0.0134323
$704$	−8.34611	−0.314556
$705$	0	0
$706$	31.4414	1.18331
$707$	−0.313941	−0.0118070
$708$	0	0
$709$	−21.4793	−0.806671	−0.403335	−	0.915052i	$-0.632149\pi$
$709$	−21.4793	−0.806671	−0.403335	+	0.915052i	$0.632149\pi$
$710$	−21.4331	−0.804368
$711$	0	0
$712$	0.181204	0.00679089
$713$	−1.98630	−0.0743877
$714$	0	0
$715$	3.87874	0.145056
$716$	−11.8033	−0.441108
$717$	0	0
$718$	−44.4568	−1.65911
$719$	−26.8421	−1.00104	−0.500520	−	0.865725i	$-0.666858\pi$
$719$	−26.8421	−1.00104	−0.500520	+	0.865725i	$0.666858\pi$
$720$	0	0
$721$	17.7112	0.659600
$722$	38.1827	1.42101
$723$	0	0
$724$	18.3543	0.682133
$725$	1.27756	0.0474473
$726$	0	0
$727$	−3.22418	−0.119578	−0.0597892	−	0.998211i	$-0.519043\pi$
$727$	−3.22418	−0.119578	−0.0597892	+	0.998211i	$0.519043\pi$
$728$	−0.652431	−0.0241807
$729$	0	0
$730$	−5.07033	−0.187661
$731$	1.94647	0.0719929
$732$	0	0
$733$	−13.1514	−0.485758	−0.242879	−	0.970057i	$-0.578092\pi$
$733$	−13.1514	−0.485758	−0.242879	+	0.970057i	$0.578092\pi$
$734$	−19.0269	−0.702295
$735$	0	0
$736$	4.05047	0.149302
$737$	−8.70560	−0.320675
$738$	0	0
$739$	−11.4422	−0.420908	−0.210454	−	0.977604i	$-0.567494\pi$
$739$	−11.4422	−0.420908	−0.210454	+	0.977604i	$0.567494\pi$
$740$	15.1046	0.555257
$741$	0	0
$742$	77.4218	2.84224
$743$	5.62720	0.206442	0.103221	−	0.994658i	$-0.467085\pi$
$743$	5.62720	0.206442	0.103221	+	0.994658i	$0.467085\pi$
$744$	0	0
$745$	30.3749	1.11285
$746$	8.73755	0.319904
$747$	0	0
$748$	2.82362	0.103242
$749$	−79.1129	−2.89072
$750$	0	0
$751$	−30.1861	−1.10151	−0.550753	−	0.834668i	$-0.685659\pi$
$751$	−30.1861	−1.10151	−0.550753	+	0.834668i	$0.685659\pi$
$752$	−35.3581	−1.28938
$753$	0	0
$754$	16.6706	0.607107
$755$	−38.1776	−1.38943
$756$	0	0
$757$	−20.9910	−0.762931	−0.381466	−	0.924383i	$-0.624581\pi$
$757$	−20.9910	−0.762931	−0.381466	+	0.924383i	$0.624581\pi$
$758$	8.30078	0.301498
$759$	0	0
$760$	0.0223036	0.000809038 0
$761$	41.3479	1.49886	0.749429	−	0.662084i	$-0.230328\pi$
$761$	41.3479	1.49886	0.749429	+	0.662084i	$0.230328\pi$
$762$	0	0
$763$	−69.2447	−2.50683
$764$	5.72084	0.206973
$765$	0	0
$766$	−48.6608	−1.75819
$767$	−8.90301	−0.321469
$768$	0	0
$769$	−38.4423	−1.38627	−0.693133	−	0.720810i	$-0.743770\pi$
$769$	−38.4423	−1.38627	−0.693133	+	0.720810i	$0.743770\pi$
$770$	−20.2246	−0.728845
$771$	0	0
$772$	−21.6276	−0.778393
$773$	−13.3063	−0.478593	−0.239297	−	0.970946i	$-0.576917\pi$
$773$	−13.3063	−0.478593	−0.239297	+	0.970946i	$0.576917\pi$
$774$	0	0
$775$	−1.02749	−0.0369087
$776$	−1.41940	−0.0509534
$777$	0	0
$778$	40.0177	1.43470
$779$	−1.37139	−0.0491353
$780$	0	0
$781$	−4.64701	−0.166283
$782$	−1.39969	−0.0500527
$783$	0	0
$784$	−47.8197	−1.70784
$785$	−27.3629	−0.976625
$786$	0	0
$787$	−35.4872	−1.26498	−0.632491	−	0.774568i	$-0.717967\pi$
$787$	−35.4872	−1.26498	−0.632491	+	0.774568i	$0.717967\pi$
$788$	−48.3080	−1.72090
$789$	0	0
$790$	66.6415	2.37100
$791$	−24.6719	−0.877230
$792$	0	0
$793$	1.69111	0.0600531
$794$	29.7145	1.05453
$795$	0	0
$796$	0.637492	0.0225953
$797$	47.3262	1.67638	0.838190	−	0.545379i	$-0.183614\pi$
$797$	47.3262	1.67638	0.838190	+	0.545379i	$0.183614\pi$
$798$	0	0
$799$	12.4918	0.441928
$800$	2.09526	0.0740787
$801$	0	0
$802$	−65.4915	−2.31259
$803$	−1.09932	−0.0387943
$804$	0	0
$805$	5.06698	0.178588
$806$	−13.4075	−0.472261
$807$	0	0
$808$	−0.00629898	−0.000221597 0
$809$	−18.1512	−0.638161	−0.319080	−	0.947728i	$-0.603374\pi$
$809$	−18.1512	−0.638161	−0.319080	+	0.947728i	$0.603374\pi$
$810$	0	0
$811$	−28.2877	−0.993317	−0.496658	−	0.867946i	$-0.665440\pi$
$811$	−28.2877	−0.993317	−0.496658	+	0.867946i	$0.665440\pi$
$812$	−43.9324	−1.54172
$813$	0	0
$814$	6.47971	0.227114
$815$	1.15950	0.0406154
$816$	0	0
$817$	0.155717	0.00544785
$818$	−25.0709	−0.876584
$819$	0	0
$820$	58.1625	2.03112
$821$	−12.7438	−0.444761	−0.222380	−	0.974960i	$-0.571383\pi$
$821$	−12.7438	−0.444761	−0.222380	+	0.974960i	$0.571383\pi$
$822$	0	0
$823$	18.2656	0.636699	0.318350	−	0.947973i	$-0.396871\pi$
$823$	18.2656	0.636699	0.318350	+	0.947973i	$0.396871\pi$
$824$	0.355362	0.0123796
$825$	0	0
$826$	46.4224	1.61524
$827$	24.3560	0.846940	0.423470	−	0.905910i	$-0.360812\pi$
$827$	24.3560	0.846940	0.423470	+	0.905910i	$0.360812\pi$
$828$	0	0
$829$	48.5630	1.68666	0.843331	−	0.537394i	$-0.180591\pi$
$829$	48.5630	1.68666	0.843331	+	0.537394i	$0.180591\pi$
$830$	1.45294	0.0504324
$831$	0	0
$832$	14.1142	0.489321
$833$	16.8944	0.585356
$834$	0	0
$835$	−16.1019	−0.557228
$836$	0.225888	0.00781251
$837$	0	0
$838$	−12.3740	−0.427451
$839$	46.5295	1.60638	0.803189	−	0.595724i	$-0.203135\pi$
$839$	46.5295	1.60638	0.803189	+	0.595724i	$0.203135\pi$
$840$	0	0
$841$	−4.96894	−0.171343
$842$	−4.26539	−0.146995
$843$	0	0
$844$	−53.4704	−1.84053
$845$	23.2575	0.800081
$846$	0	0
$847$	−4.38501	−0.150671
$848$	−34.3354	−1.17908
$849$	0	0
$850$	−0.724042	−0.0248344
$851$	−1.62340	−0.0556493
$852$	0	0
$853$	−53.8273	−1.84301	−0.921506	−	0.388364i	$-0.873040\pi$
$853$	−53.8273	−1.84301	−0.921506	+	0.388364i	$0.873040\pi$
$854$	−8.81784	−0.301740
$855$	0	0
$856$	−1.58734	−0.0542541
$857$	11.7256	0.400538	0.200269	−	0.979741i	$-0.435818\pi$
$857$	11.7256	0.400538	0.200269	+	0.979741i	$0.435818\pi$
$858$	0	0
$859$	35.5984	1.21460	0.607300	−	0.794472i	$-0.292253\pi$
$859$	35.5984	1.21460	0.607300	+	0.794472i	$0.292253\pi$
$860$	−6.60415	−0.225200
$861$	0	0
$862$	−13.0471	−0.444385
$863$	25.4808	0.867376	0.433688	−	0.901063i	$-0.357212\pi$
$863$	25.4808	0.867376	0.433688	+	0.901063i	$0.357212\pi$
$864$	0	0
$865$	−13.1134	−0.445868
$866$	−55.9553	−1.90144
$867$	0	0
$868$	35.3332	1.19929
$869$	14.4489	0.490145
$870$	0	0
$871$	14.7221	0.498840
$872$	−1.38934	−0.0470490
$873$	0	0
$874$	−0.111974	−0.00378759
$875$	−47.6662	−1.61141
$876$	0	0
$877$	−25.4906	−0.860756	−0.430378	−	0.902649i	$-0.641620\pi$
$877$	−25.4906	−0.860756	−0.430378	+	0.902649i	$0.641620\pi$
$878$	37.7106	1.27267
$879$	0	0
$880$	8.96932	0.302356
$881$	57.5717	1.93964	0.969820	−	0.243821i	$-0.0784009\pi$
$881$	57.5717	1.93964	0.969820	+	0.243821i	$0.0784009\pi$
$882$	0	0
$883$	20.7139	0.697077	0.348538	−	0.937295i	$-0.386678\pi$
$883$	20.7139	0.697077	0.348538	+	0.937295i	$0.386678\pi$
$884$	−4.77505	−0.160602
$885$	0	0
$886$	−15.8467	−0.532380
$887$	48.1203	1.61572	0.807861	−	0.589373i	$-0.200625\pi$
$887$	48.1203	1.61572	0.807861	+	0.589373i	$0.200625\pi$
$888$	0	0
$889$	19.4108	0.651019
$890$	−9.49915	−0.318412
$891$	0	0
$892$	30.9008	1.03464
$893$	0.999340	0.0334416
$894$	0	0
$895$	13.2462	0.442772
$896$	−3.08566	−0.103085
$897$	0	0
$898$	47.1398	1.57308
$899$	−19.3273	−0.644602
$900$	0	0
$901$	12.1305	0.404125
$902$	24.9511	0.830779
$903$	0	0
$904$	−0.495021	−0.0164642
$905$	−20.5981	−0.684705
$906$	0	0
$907$	−27.0508	−0.898205	−0.449103	−	0.893480i	$-0.648256\pi$
$907$	−27.0508	−0.898205	−0.449103	+	0.893480i	$0.648256\pi$
$908$	−16.3760	−0.543456
$909$	0	0
$910$	34.2021	1.13379
$911$	−5.04031	−0.166993	−0.0834965	−	0.996508i	$-0.526609\pi$
$911$	−5.04031	−0.166993	−0.0834965	+	0.996508i	$0.526609\pi$
$912$	0	0
$913$	0.315020	0.0104256
$914$	−43.7598	−1.44745
$915$	0	0
$916$	−36.8881	−1.21882
$917$	16.0025	0.528449
$918$	0	0
$919$	11.2458	0.370965	0.185483	−	0.982648i	$-0.440615\pi$
$919$	11.2458	0.370965	0.185483	+	0.982648i	$0.440615\pi$
$920$	0.101665	0.00335180
$921$	0	0
$922$	6.10020	0.200899
$923$	7.85861	0.258669
$924$	0	0
$925$	−0.839765	−0.0276113
$926$	−56.7208	−1.86396
$927$	0	0
$928$	39.4122	1.29377
$929$	0.644532	0.0211464	0.0105732	−	0.999944i	$-0.496634\pi$
$929$	0.644532	0.0211464	0.0105732	+	0.999944i	$0.496634\pi$
$930$	0	0
$931$	1.35155	0.0442951
$932$	23.2829	0.762657
$933$	0	0
$934$	−75.5223	−2.47116
$935$	−3.16880	−0.103631
$936$	0	0
$937$	−42.3215	−1.38258	−0.691291	−	0.722576i	$-0.742958\pi$
$937$	−42.3215	−1.38258	−0.691291	+	0.722576i	$0.742958\pi$
$938$	−76.7646	−2.50645
$939$	0	0
$940$	−42.3833	−1.38239
$941$	17.3107	0.564314	0.282157	−	0.959368i	$-0.408950\pi$
$941$	17.3107	0.564314	0.282157	+	0.959368i	$0.408950\pi$
$942$	0	0
$943$	−6.25112	−0.203564
$944$	−20.5876	−0.670070
$945$	0	0
$946$	−2.83311	−0.0921123
$947$	36.6997	1.19258	0.596290	−	0.802769i	$-0.296641\pi$
$947$	36.6997	1.19258	0.596290	+	0.802769i	$0.296641\pi$
$948$	0	0
$949$	1.85908	0.0603482
$950$	−0.0579231	−0.00187927
$951$	0	0
$952$	0.533016	0.0172751
$953$	−3.27584	−0.106115	−0.0530575	−	0.998591i	$-0.516897\pi$
$953$	−3.27584	−0.106115	−0.0530575	+	0.998591i	$0.516897\pi$
$954$	0	0
$955$	−6.42022	−0.207753
$956$	−8.13194	−0.263006
$957$	0	0
$958$	−16.9685	−0.548226
$959$	−50.8615	−1.64240
$960$	0	0
$961$	−15.4557	−0.498572
$962$	−10.9579	−0.353297
$963$	0	0
$964$	−12.0399	−0.387780
$965$	24.2715	0.781329
$966$	0	0
$967$	12.4126	0.399163	0.199582	−	0.979881i	$-0.436042\pi$
$967$	12.4126	0.399163	0.199582	+	0.979881i	$0.436042\pi$
$968$	−0.0879817	−0.00282784
$969$	0	0
$970$	74.4084	2.38911
$971$	−6.15591	−0.197553	−0.0987763	−	0.995110i	$-0.531493\pi$
$971$	−6.15591	−0.197553	−0.0987763	+	0.995110i	$0.531493\pi$
$972$	0	0
$973$	66.2143	2.12273
$974$	25.4830	0.816527
$975$	0	0
$976$	3.91058	0.125175
$977$	−57.1570	−1.82861	−0.914307	−	0.405021i	$-0.867264\pi$
$977$	−57.1570	−1.82861	−0.914307	+	0.405021i	$0.867264\pi$
$978$	0	0
$979$	−2.05956	−0.0658238
$980$	−57.3207	−1.83104
$981$	0	0
$982$	27.9125	0.890723
$983$	−39.1647	−1.24916	−0.624580	−	0.780961i	$-0.714730\pi$
$983$	−39.1647	−1.24916	−0.624580	+	0.780961i	$0.714730\pi$
$984$	0	0
$985$	54.2137	1.72739
$986$	−13.6193	−0.433728
$987$	0	0
$988$	−0.382002	−0.0121531
$989$	0.709793	0.0225701
$990$	0	0
$991$	−13.8422	−0.439712	−0.219856	−	0.975532i	$-0.570559\pi$
$991$	−13.8422	−0.439712	−0.219856	+	0.975532i	$0.570559\pi$
$992$	−31.6978	−1.00641
$993$	0	0
$994$	−40.9766	−1.29970
$995$	−0.715426	−0.0226805
$996$	0	0
$997$	−16.8632	−0.534064	−0.267032	−	0.963688i	$-0.586043\pi$
$997$	−16.8632	−0.534064	−0.267032	+	0.963688i	$0.586043\pi$
$998$	24.1445	0.764279
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6039.2.a.e.1.3		12
3.2	odd	2		2013.2.a.d.1.10	✓	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2013.2.a.d.1.10	✓	12	3.2	odd	2
6039.2.a.e.1.3		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)

\(f(q)\)	\(=\)	\(q-2.01091 q^{2} +2.04375 q^{4} -2.29360 q^{5} -4.38501 q^{7} -0.0879817 q^{8} +O(q^{10})\)	\(q-2.01091 q^{2} +2.04375 q^{4} -2.29360 q^{5} -4.38501 q^{7} -0.0879817 q^{8} +4.61222 q^{10} +1.00000 q^{11} -1.69111 q^{13} +8.81784 q^{14} -3.91058 q^{16} +1.38158 q^{17} +0.110526 q^{19} -4.68755 q^{20} -2.01091 q^{22} +0.503803 q^{23} +0.260612 q^{25} +3.40067 q^{26} -8.96186 q^{28} +4.90215 q^{29} -3.94262 q^{31} +8.03978 q^{32} -2.77824 q^{34} +10.0575 q^{35} -3.22228 q^{37} -0.222258 q^{38} +0.201795 q^{40} -12.4079 q^{41} +1.40887 q^{43} +2.04375 q^{44} -1.01310 q^{46} +9.04166 q^{47} +12.2283 q^{49} -0.524067 q^{50} -3.45621 q^{52} +8.78013 q^{53} -2.29360 q^{55} +0.385800 q^{56} -9.85777 q^{58} +5.26459 q^{59} -1.00000 q^{61} +7.92825 q^{62} -8.34611 q^{64} +3.87874 q^{65} -8.70560 q^{67} +2.82362 q^{68} -20.2246 q^{70} -4.64701 q^{71} -1.09932 q^{73} +6.47971 q^{74} +0.225888 q^{76} -4.38501 q^{77} +14.4489 q^{79} +8.96932 q^{80} +24.9511 q^{82} +0.315020 q^{83} -3.16880 q^{85} -2.83311 q^{86} -0.0879817 q^{88} -2.05956 q^{89} +7.41553 q^{91} +1.02965 q^{92} -18.1819 q^{94} -0.253503 q^{95} +16.1329 q^{97} -24.5899 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - q^{2} + 9 q^{4} + 3 q^{5} - 9 q^{7} - 6 q^{8}+O(q^{10}) \) 12 * q - q^2 + 9 * q^4 + 3 * q^5 - 9 * q^7 - 6 * q^8	\( 12 q - q^{2} + 9 q^{4} + 3 q^{5} - 9 q^{7} - 6 q^{8} - 8 q^{10} + 12 q^{11} - q^{13} + 3 q^{14} + 3 q^{16} - 9 q^{17} - 20 q^{19} + 9 q^{20} - q^{22} + 9 q^{23} + 3 q^{25} + 18 q^{26} - 31 q^{28} - 18 q^{29} - 21 q^{31} - 18 q^{32} - 12 q^{34} + 4 q^{35} - 18 q^{37} + 2 q^{38} - 26 q^{40} - 15 q^{41} - 33 q^{43} + 9 q^{44} - 28 q^{46} + 20 q^{47} + 15 q^{49} + 2 q^{50} - 27 q^{52} + 3 q^{55} + 8 q^{56} - 11 q^{58} + 21 q^{59} - 12 q^{61} + 9 q^{62} - 12 q^{64} - 17 q^{65} - 34 q^{67} + 16 q^{68} - 36 q^{70} + 5 q^{71} - 2 q^{73} - 6 q^{74} - 27 q^{76} - 9 q^{77} - 31 q^{79} + 60 q^{80} - 12 q^{82} + 32 q^{83} - 40 q^{85} - 18 q^{86} - 6 q^{88} - 27 q^{89} - 45 q^{91} + 78 q^{92} - 13 q^{94} - 37 q^{95} - 19 q^{97} - 4 q^{98}+O(q^{100}) \) 12 * q - q^2 + 9 * q^4 + 3 * q^5 - 9 * q^7 - 6 * q^8 - 8 * q^10 + 12 * q^11 - q^13 + 3 * q^14 + 3 * q^16 - 9 * q^17 - 20 * q^19 + 9 * q^20 - q^22 + 9 * q^23 + 3 * q^25 + 18 * q^26 - 31 * q^28 - 18 * q^29 - 21 * q^31 - 18 * q^32 - 12 * q^34 + 4 * q^35 - 18 * q^37 + 2 * q^38 - 26 * q^40 - 15 * q^41 - 33 * q^43 + 9 * q^44 - 28 * q^46 + 20 * q^47 + 15 * q^49 + 2 * q^50 - 27 * q^52 + 3 * q^55 + 8 * q^56 - 11 * q^58 + 21 * q^59 - 12 * q^61 + 9 * q^62 - 12 * q^64 - 17 * q^65 - 34 * q^67 + 16 * q^68 - 36 * q^70 + 5 * q^71 - 2 * q^73 - 6 * q^74 - 27 * q^76 - 9 * q^77 - 31 * q^79 + 60 * q^80 - 12 * q^82 + 32 * q^83 - 40 * q^85 - 18 * q^86 - 6 * q^88 - 27 * q^89 - 45 * q^91 + 78 * q^92 - 13 * q^94 - 37 * q^95 - 19 * q^97 - 4 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more