LMFDB - Embedded newform 6039.2.a.e.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

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

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

chi := DirichletCharacter("6039.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$48.2216577807$
Analytic rank:	$1$
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			1.5
Root			$0.852887$ 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)

$f(q)$	$=$	$q-0.852887 q^{2} -1.27258 q^{4} -2.61325 q^{5} -1.65804 q^{7} +2.79114 q^{8} +O(q^{10})$	\(q-0.852887 q^{2} -1.27258 q^{4} -2.61325 q^{5} -1.65804 q^{7} +2.79114 q^{8} +2.22881 q^{10} +1.00000 q^{11} +2.92626 q^{13} +1.41412 q^{14} +0.164638 q^{16} -1.17789 q^{17} -6.88660 q^{19} +3.32559 q^{20} -0.852887 q^{22} +1.74930 q^{23} +1.82910 q^{25} -2.49577 q^{26} +2.11000 q^{28} -3.91346 q^{29} -5.07153 q^{31} -5.72271 q^{32} +1.00461 q^{34} +4.33289 q^{35} +6.35732 q^{37} +5.87349 q^{38} -7.29397 q^{40} +11.5987 q^{41} +1.20545 q^{43} -1.27258 q^{44} -1.49195 q^{46} +11.3152 q^{47} -4.25089 q^{49} -1.56001 q^{50} -3.72392 q^{52} -3.35005 q^{53} -2.61325 q^{55} -4.62784 q^{56} +3.33774 q^{58} +5.28722 q^{59} -1.00000 q^{61} +4.32544 q^{62} +4.55154 q^{64} -7.64707 q^{65} +4.96873 q^{67} +1.49896 q^{68} -3.69546 q^{70} +8.80321 q^{71} -2.70379 q^{73} -5.42208 q^{74} +8.76378 q^{76} -1.65804 q^{77} -11.4997 q^{79} -0.430242 q^{80} -9.89238 q^{82} -6.15717 q^{83} +3.07813 q^{85} -1.02811 q^{86} +2.79114 q^{88} +14.3727 q^{89} -4.85187 q^{91} -2.22613 q^{92} -9.65056 q^{94} +17.9964 q^{95} -10.1570 q^{97} +3.62553 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−0.852887	−0.603082	−0.301541	−	0.953453i	$-0.597501\pi$
$2$	−0.852887	−0.603082	−0.301541	+	0.953453i	$0.597501\pi$
$3$	0	0
$3$	0	0
$4$	−1.27258	−0.636292
$5$	−2.61325	−1.16868	−0.584341	−	0.811508i	$-0.698647\pi$
$5$	−2.61325	−1.16868	−0.584341	+	0.811508i	$0.698647\pi$
$6$	0	0
$7$	−1.65804	−0.626681	−0.313341	−	0.949641i	$-0.601448\pi$
$7$	−1.65804	−0.626681	−0.313341	+	0.949641i	$0.601448\pi$
$8$	2.79114	0.986818
$9$	0	0
$10$	2.22881	0.704812
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	2.92626	0.811600	0.405800	−	0.913962i	$-0.366993\pi$
$13$	2.92626	0.811600	0.405800	+	0.913962i	$0.366993\pi$
$14$	1.41412	0.377940
$15$	0	0
$16$	0.164638	0.0411596
$17$	−1.17789	−0.285680	−0.142840	−	0.989746i	$-0.545623\pi$
$17$	−1.17789	−0.285680	−0.142840	+	0.989746i	$0.545623\pi$
$18$	0	0
$19$	−6.88660	−1.57989	−0.789947	−	0.613175i	$-0.789892\pi$
$19$	−6.88660	−1.57989	−0.789947	+	0.613175i	$0.789892\pi$
$20$	3.32559	0.743624
$21$	0	0
$22$	−0.852887	−0.181836
$23$	1.74930	0.364754	0.182377	−	0.983229i	$-0.441621\pi$
$23$	1.74930	0.364754	0.182377	+	0.983229i	$0.441621\pi$
$24$	0	0
$25$	1.82910	0.365819
$26$	−2.49577	−0.489461
$27$	0	0
$28$	2.11000	0.398752
$29$	−3.91346	−0.726712	−0.363356	−	0.931650i	$-0.618369\pi$
$29$	−3.91346	−0.726712	−0.363356	+	0.931650i	$0.618369\pi$
$30$	0	0
$31$	−5.07153	−0.910874	−0.455437	−	0.890268i	$-0.650517\pi$
$31$	−5.07153	−0.910874	−0.455437	+	0.890268i	$0.650517\pi$
$32$	−5.72271	−1.01164
$33$	0	0
$34$	1.00461	0.172289
$35$	4.33289	0.732392
$36$	0	0
$37$	6.35732	1.04514	0.522569	−	0.852597i	$-0.324974\pi$
$37$	6.35732	1.04514	0.522569	+	0.852597i	$0.324974\pi$
$38$	5.87349	0.952806
$39$	0	0
$40$	−7.29397	−1.15328
$41$	11.5987	1.81141	0.905707	−	0.423905i	$-0.139341\pi$
$41$	11.5987	1.81141	0.905707	+	0.423905i	$0.139341\pi$
$42$	0	0
$43$	1.20545	0.183829	0.0919145	−	0.995767i	$-0.470701\pi$
$43$	1.20545	0.183829	0.0919145	+	0.995767i	$0.470701\pi$
$44$	−1.27258	−0.191849
$45$	0	0
$46$	−1.49195	−0.219976
$47$	11.3152	1.65049	0.825244	−	0.564777i	$-0.191038\pi$
$47$	11.3152	1.65049	0.825244	+	0.564777i	$0.191038\pi$
$48$	0	0
$49$	−4.25089	−0.607270
$50$	−1.56001	−0.220619
$51$	0	0
$52$	−3.72392	−0.516414
$53$	−3.35005	−0.460164	−0.230082	−	0.973171i	$-0.573899\pi$
$53$	−3.35005	−0.460164	−0.230082	+	0.973171i	$0.573899\pi$
$54$	0	0
$55$	−2.61325	−0.352371
$56$	−4.62784	−0.618421
$57$	0	0
$58$	3.33774	0.438267
$59$	5.28722	0.688337	0.344169	−	0.938908i	$-0.388161\pi$
$59$	5.28722	0.688337	0.344169	+	0.938908i	$0.388161\pi$
$60$	0	0
$61$	−1.00000	−0.128037
$61$	−1.00000	−0.128037
$62$	4.32544	0.549332
$63$	0	0
$64$	4.55154	0.568943
$65$	−7.64707	−0.948503
$66$	0	0
$67$	4.96873	0.607027	0.303514	−	0.952827i	$-0.401840\pi$
$67$	4.96873	0.607027	0.303514	+	0.952827i	$0.401840\pi$
$68$	1.49896	0.181776
$69$	0	0
$70$	−3.69546	−0.441692
$71$	8.80321	1.04475	0.522375	−	0.852716i	$-0.325046\pi$
$71$	8.80321	1.04475	0.522375	+	0.852716i	$0.325046\pi$
$72$	0	0
$73$	−2.70379	−0.316454	−0.158227	−	0.987403i	$-0.550578\pi$
$73$	−2.70379	−0.316454	−0.158227	+	0.987403i	$0.550578\pi$
$74$	−5.42208	−0.630304
$75$	0	0
$76$	8.76378	1.00527
$77$	−1.65804	−0.188952
$78$	0	0
$79$	−11.4997	−1.29382	−0.646911	−	0.762566i	$-0.723939\pi$
$79$	−11.4997	−1.29382	−0.646911	+	0.762566i	$0.723939\pi$
$80$	−0.430242	−0.0481025
$81$	0	0
$82$	−9.89238	−1.09243
$83$	−6.15717	−0.675837	−0.337919	−	0.941175i	$-0.609723\pi$
$83$	−6.15717	−0.675837	−0.337919	+	0.941175i	$0.609723\pi$
$84$	0	0
$85$	3.07813	0.333870
$86$	−1.02811	−0.110864
$87$	0	0
$88$	2.79114	0.297537
$89$	14.3727	1.52350	0.761751	−	0.647870i	$-0.224340\pi$
$89$	14.3727	1.52350	0.761751	+	0.647870i	$0.224340\pi$
$90$	0	0
$91$	−4.85187	−0.508615
$92$	−2.22613	−0.232090
$93$	0	0
$94$	−9.65056	−0.995379
$95$	17.9964	1.84640
$96$	0	0
$97$	−10.1570	−1.03129	−0.515643	−	0.856804i	$-0.672447\pi$
$97$	−10.1570	−1.03129	−0.515643	+	0.856804i	$0.672447\pi$
$98$	3.62553	0.366234
$99$	0	0
$100$	−2.32768	−0.232768
$101$	9.03436	0.898953	0.449476	−	0.893292i	$-0.351611\pi$
$101$	9.03436	0.898953	0.449476	+	0.893292i	$0.351611\pi$
$102$	0	0
$103$	9.59612	0.945533	0.472767	−	0.881188i	$-0.343255\pi$
$103$	9.59612	0.945533	0.472767	+	0.881188i	$0.343255\pi$
$104$	8.16763	0.800902
$105$	0	0
$106$	2.85721	0.277517
$107$	4.13595	0.399837	0.199918	−	0.979813i	$-0.435932\pi$
$107$	4.13595	0.399837	0.199918	+	0.979813i	$0.435932\pi$
$108$	0	0
$109$	11.2269	1.07534	0.537672	−	0.843154i	$-0.319304\pi$
$109$	11.2269	1.07534	0.537672	+	0.843154i	$0.319304\pi$
$110$	2.22881	0.212509
$111$	0	0
$112$	−0.272978	−0.0257940
$113$	2.17483	0.204591	0.102295	−	0.994754i	$-0.467381\pi$
$113$	2.17483	0.204591	0.102295	+	0.994754i	$0.467381\pi$
$114$	0	0
$115$	−4.57136	−0.426281
$116$	4.98021	0.462401
$117$	0	0
$118$	−4.50940	−0.415124
$119$	1.95299	0.179030
$120$	0	0
$121$	1.00000	0.0909091
$122$	0.852887	0.0772167
$123$	0	0
$124$	6.45395	0.579582
$125$	8.28637	0.741156
$126$	0	0
$127$	−20.7653	−1.84262	−0.921310	−	0.388830i	$-0.872879\pi$
$127$	−20.7653	−1.84262	−0.921310	+	0.388830i	$0.872879\pi$
$128$	7.56346	0.668522
$129$	0	0
$130$	6.52209	0.572025
$131$	1.05438	0.0921219	0.0460610	−	0.998939i	$-0.485333\pi$
$131$	1.05438	0.0921219	0.0460610	+	0.998939i	$0.485333\pi$
$132$	0	0
$133$	11.4183	0.990091
$134$	−4.23777	−0.366087
$135$	0	0
$136$	−3.28766	−0.281914
$137$	−6.28167	−0.536679	−0.268340	−	0.963324i	$-0.586475\pi$
$137$	−6.28167	−0.536679	−0.268340	+	0.963324i	$0.586475\pi$
$138$	0	0
$139$	8.00703	0.679147	0.339574	−	0.940579i	$-0.389717\pi$
$139$	8.00703	0.679147	0.339574	+	0.940579i	$0.389717\pi$
$140$	−5.51397	−0.466015
$141$	0	0
$142$	−7.50814	−0.630069
$143$	2.92626	0.244707
$144$	0	0
$145$	10.2269	0.849296
$146$	2.30602	0.190848
$147$	0	0
$148$	−8.09023	−0.665013
$149$	−5.73276	−0.469646	−0.234823	−	0.972038i	$-0.575451\pi$
$149$	−5.73276	−0.469646	−0.234823	+	0.972038i	$0.575451\pi$
$150$	0	0
$151$	12.1978	0.992645	0.496323	−	0.868138i	$-0.334683\pi$
$151$	12.1978	0.992645	0.496323	+	0.868138i	$0.334683\pi$
$152$	−19.2215	−1.55907
$153$	0	0
$154$	1.41412	0.113953
$155$	13.2532	1.06452
$156$	0	0
$157$	−11.1536	−0.890154	−0.445077	−	0.895492i	$-0.646824\pi$
$157$	−11.1536	−0.890154	−0.445077	+	0.895492i	$0.646824\pi$
$158$	9.80798	0.780281
$159$	0	0
$160$	14.9549	1.18229
$161$	−2.90041	−0.228584
$162$	0	0
$163$	7.92753	0.620932	0.310466	−	0.950584i	$-0.399515\pi$
$163$	7.92753	0.620932	0.310466	+	0.950584i	$0.399515\pi$
$164$	−14.7603	−1.15259
$165$	0	0
$166$	5.25137	0.407585
$167$	3.31990	0.256902	0.128451	−	0.991716i	$-0.459000\pi$
$167$	3.31990	0.256902	0.128451	+	0.991716i	$0.459000\pi$
$168$	0	0
$169$	−4.43698	−0.341306
$170$	−2.62529	−0.201351
$171$	0	0
$172$	−1.53403	−0.116969
$173$	−7.09656	−0.539541	−0.269771	−	0.962925i	$-0.586948\pi$
$173$	−7.09656	−0.539541	−0.269771	+	0.962925i	$0.586948\pi$
$174$	0	0
$175$	−3.03272	−0.229252
$176$	0.164638	0.0124101
$177$	0	0
$178$	−12.2583	−0.918797
$179$	−2.31718	−0.173194	−0.0865969	−	0.996243i	$-0.527599\pi$
$179$	−2.31718	−0.173194	−0.0865969	+	0.996243i	$0.527599\pi$
$180$	0	0
$181$	−16.4141	−1.22005	−0.610027	−	0.792381i	$-0.708841\pi$
$181$	−16.4141	−1.22005	−0.610027	+	0.792381i	$0.708841\pi$
$182$	4.13810	0.306736
$183$	0	0
$184$	4.88254	0.359946
$185$	−16.6133	−1.22143
$186$	0	0
$187$	−1.17789	−0.0861358
$188$	−14.3995	−1.05019
$189$	0	0
$190$	−15.3489	−1.11353
$191$	2.28030	0.164997	0.0824983	−	0.996591i	$-0.473710\pi$
$191$	2.28030	0.164997	0.0824983	+	0.996591i	$0.473710\pi$
$192$	0	0
$193$	−5.86985	−0.422521	−0.211260	−	0.977430i	$-0.567757\pi$
$193$	−5.86985	−0.422521	−0.211260	+	0.977430i	$0.567757\pi$
$194$	8.66275	0.621950
$195$	0	0
$196$	5.40962	0.386401
$197$	−1.15359	−0.0821896	−0.0410948	−	0.999155i	$-0.513085\pi$
$197$	−1.15359	−0.0821896	−0.0410948	+	0.999155i	$0.513085\pi$
$198$	0	0
$199$	8.61477	0.610685	0.305342	−	0.952243i	$-0.401229\pi$
$199$	8.61477	0.610685	0.305342	+	0.952243i	$0.401229\pi$
$200$	5.10527	0.360997
$201$	0	0
$202$	−7.70529	−0.542142
$203$	6.48869	0.455417
$204$	0	0
$205$	−30.3104	−2.11697
$206$	−8.18440	−0.570234
$207$	0	0
$208$	0.481775	0.0334051
$209$	−6.88660	−0.476356
$210$	0	0
$211$	19.4862	1.34148	0.670742	−	0.741691i	$-0.265976\pi$
$211$	19.4862	1.34148	0.670742	+	0.741691i	$0.265976\pi$
$212$	4.26321	0.292799
$213$	0	0
$214$	−3.52749	−0.241134
$215$	−3.15014	−0.214838
$216$	0	0
$217$	8.40882	0.570828
$218$	−9.57529	−0.648521
$219$	0	0
$220$	3.32559	0.224211
$221$	−3.44682	−0.231858
$222$	0	0
$223$	−14.2284	−0.952803	−0.476402	−	0.879228i	$-0.658059\pi$
$223$	−14.2284	−0.952803	−0.476402	+	0.879228i	$0.658059\pi$
$224$	9.48849	0.633977
$225$	0	0
$226$	−1.85488	−0.123385
$227$	−21.5420	−1.42979	−0.714897	−	0.699230i	$-0.753527\pi$
$227$	−21.5420	−1.42979	−0.714897	+	0.699230i	$0.753527\pi$
$228$	0	0
$229$	3.19112	0.210875	0.105437	−	0.994426i	$-0.466376\pi$
$229$	3.19112	0.210875	0.105437	+	0.994426i	$0.466376\pi$
$230$	3.89885	0.257083
$231$	0	0
$232$	−10.9230	−0.717133
$233$	1.49468	0.0979198	0.0489599	−	0.998801i	$-0.484409\pi$
$233$	1.49468	0.0979198	0.0489599	+	0.998801i	$0.484409\pi$
$234$	0	0
$235$	−29.5694	−1.92890
$236$	−6.72843	−0.437984
$237$	0	0
$238$	−1.66568	−0.107970
$239$	0.810667	0.0524377	0.0262188	−	0.999656i	$-0.491653\pi$
$239$	0.810667	0.0524377	0.0262188	+	0.999656i	$0.491653\pi$
$240$	0	0
$241$	10.8379	0.698133	0.349066	−	0.937098i	$-0.386499\pi$
$241$	10.8379	0.698133	0.349066	+	0.937098i	$0.386499\pi$
$242$	−0.852887	−0.0548256
$243$	0	0
$244$	1.27258	0.0814688
$245$	11.1087	0.709706
$246$	0	0
$247$	−20.1520	−1.28224
$248$	−14.1554	−0.898867
$249$	0	0
$250$	−7.06734	−0.446978
$251$	0.139416	0.00879988	0.00439994	−	0.999990i	$-0.498599\pi$
$251$	0.139416	0.00879988	0.00439994	+	0.999990i	$0.498599\pi$
$252$	0	0
$253$	1.74930	0.109977
$254$	17.7104	1.11125
$255$	0	0
$256$	−15.5539	−0.972116
$257$	−13.3160	−0.830631	−0.415316	−	0.909677i	$-0.636329\pi$
$257$	−13.3160	−0.830631	−0.415316	+	0.909677i	$0.636329\pi$
$258$	0	0
$259$	−10.5407	−0.654968
$260$	9.73154	0.603525
$261$	0	0
$262$	−0.899270	−0.0555571
$263$	−17.7610	−1.09519	−0.547596	−	0.836743i	$-0.684457\pi$
$263$	−17.7610	−1.09519	−0.547596	+	0.836743i	$0.684457\pi$
$264$	0	0
$265$	8.75452	0.537786
$266$	−9.73850	−0.597106
$267$	0	0
$268$	−6.32313	−0.386247
$269$	1.12034	0.0683082	0.0341541	−	0.999417i	$-0.489126\pi$
$269$	1.12034	0.0683082	0.0341541	+	0.999417i	$0.489126\pi$
$270$	0	0
$271$	−0.810668	−0.0492446	−0.0246223	−	0.999697i	$-0.507838\pi$
$271$	−0.810668	−0.0492446	−0.0246223	+	0.999697i	$0.507838\pi$
$272$	−0.193926	−0.0117585
$273$	0	0
$274$	5.35755	0.323662
$275$	1.82910	0.110299
$276$	0	0
$277$	−27.9816	−1.68125	−0.840627	−	0.541614i	$-0.817813\pi$
$277$	−27.9816	−1.68125	−0.840627	+	0.541614i	$0.817813\pi$
$278$	−6.82909	−0.409582
$279$	0	0
$280$	12.0937	0.722738
$281$	−14.2163	−0.848075	−0.424037	−	0.905645i	$-0.639387\pi$
$281$	−14.2163	−0.848075	−0.424037	+	0.905645i	$0.639387\pi$
$282$	0	0
$283$	−7.69514	−0.457429	−0.228714	−	0.973494i	$-0.573452\pi$
$283$	−7.69514	−0.457429	−0.228714	+	0.973494i	$0.573452\pi$
$284$	−11.2028	−0.664766
$285$	0	0
$286$	−2.49577	−0.147578
$287$	−19.2312	−1.13518
$288$	0	0
$289$	−15.6126	−0.918387
$290$	−8.72237	−0.512195
$291$	0	0
$292$	3.44080	0.201357
$293$	−29.6378	−1.73146	−0.865730	−	0.500511i	$-0.833146\pi$
$293$	−29.6378	−1.73146	−0.865730	+	0.500511i	$0.833146\pi$
$294$	0	0
$295$	−13.8168	−0.804448
$296$	17.7442	1.03136
$297$	0	0
$298$	4.88939	0.283235
$299$	5.11891	0.296034
$300$	0	0
$301$	−1.99868	−0.115202
$302$	−10.4034	−0.598647
$303$	0	0
$304$	−1.13380	−0.0650278
$305$	2.61325	0.149634
$306$	0	0
$307$	27.3638	1.56174	0.780869	−	0.624695i	$-0.214777\pi$
$307$	27.3638	1.56174	0.780869	+	0.624695i	$0.214777\pi$
$308$	2.11000	0.120228
$309$	0	0
$310$	−11.3035	−0.641995
$311$	−19.4732	−1.10423	−0.552113	−	0.833770i	$-0.686178\pi$
$311$	−19.4732	−1.10423	−0.552113	+	0.833770i	$0.686178\pi$
$312$	0	0
$313$	−12.5841	−0.711295	−0.355647	−	0.934620i	$-0.615740\pi$
$313$	−12.5841	−0.711295	−0.355647	+	0.934620i	$0.615740\pi$
$314$	9.51276	0.536836
$315$	0	0
$316$	14.6344	0.823248
$317$	25.2783	1.41977	0.709886	−	0.704317i	$-0.248746\pi$
$317$	25.2783	1.41977	0.709886	+	0.704317i	$0.248746\pi$
$318$	0	0
$319$	−3.91346	−0.219112
$320$	−11.8943	−0.664914
$321$	0	0
$322$	2.47372	0.137855
$323$	8.11166	0.451345
$324$	0	0
$325$	5.35242	0.296899
$326$	−6.76129	−0.374473
$327$	0	0
$328$	32.3737	1.78754
$329$	−18.7610	−1.03433
$330$	0	0
$331$	−26.0660	−1.43272	−0.716360	−	0.697731i	$-0.754193\pi$
$331$	−26.0660	−1.43272	−0.716360	+	0.697731i	$0.754193\pi$
$332$	7.83551	0.430030
$333$	0	0
$334$	−2.83150	−0.154933
$335$	−12.9846	−0.709422
$336$	0	0
$337$	14.9043	0.811888	0.405944	−	0.913898i	$-0.366943\pi$
$337$	14.9043	0.811888	0.405944	+	0.913898i	$0.366943\pi$
$338$	3.78424	0.205835
$339$	0	0
$340$	−3.91717	−0.212439
$341$	−5.07153	−0.274639
$342$	0	0
$343$	18.6545	1.00725
$344$	3.36458	0.181406
$345$	0	0
$346$	6.05256	0.325388
$347$	−11.8261	−0.634859	−0.317430	−	0.948282i	$-0.602820\pi$
$347$	−11.8261	−0.634859	−0.317430	+	0.948282i	$0.602820\pi$
$348$	0	0
$349$	7.57138	0.405287	0.202643	−	0.979253i	$-0.435047\pi$
$349$	7.57138	0.405287	0.202643	+	0.979253i	$0.435047\pi$
$350$	2.58657	0.138258
$351$	0	0
$352$	−5.72271	−0.305021
$353$	2.91563	0.155184	0.0775918	−	0.996985i	$-0.475277\pi$
$353$	2.91563	0.155184	0.0775918	+	0.996985i	$0.475277\pi$
$354$	0	0
$355$	−23.0050	−1.22098
$356$	−18.2905	−0.969392
$357$	0	0
$358$	1.97629	0.104450
$359$	−23.2034	−1.22463	−0.612315	−	0.790614i	$-0.709761\pi$
$359$	−23.2034	−1.22463	−0.612315	+	0.790614i	$0.709761\pi$
$360$	0	0
$361$	28.4253	1.49607
$362$	13.9994	0.735792
$363$	0	0
$364$	6.17442	0.323627
$365$	7.06568	0.369835
$366$	0	0
$367$	−11.8922	−0.620766	−0.310383	−	0.950612i	$-0.600457\pi$
$367$	−11.8922	−0.620766	−0.310383	+	0.950612i	$0.600457\pi$
$368$	0.288001	0.0150131
$369$	0	0
$370$	14.1693	0.736625
$371$	5.55452	0.288376
$372$	0	0
$373$	−19.1837	−0.993295	−0.496647	−	0.867952i	$-0.665436\pi$
$373$	−19.1837	−0.993295	−0.496647	+	0.867952i	$0.665436\pi$
$374$	1.00461	0.0519470
$375$	0	0
$376$	31.5823	1.62873
$377$	−11.4518	−0.589799
$378$	0	0
$379$	30.2989	1.55635	0.778174	−	0.628049i	$-0.216146\pi$
$379$	30.2989	1.55635	0.778174	+	0.628049i	$0.216146\pi$
$380$	−22.9020	−1.17485
$381$	0	0
$382$	−1.94484	−0.0995065
$383$	11.4317	0.584133	0.292066	−	0.956398i	$-0.405657\pi$
$383$	11.4317	0.584133	0.292066	+	0.956398i	$0.405657\pi$
$384$	0	0
$385$	4.33289	0.220824
$386$	5.00632	0.254815
$387$	0	0
$388$	12.9256	0.656198
$389$	−34.6014	−1.75436	−0.877181	−	0.480160i	$-0.840579\pi$
$389$	−34.6014	−1.75436	−0.877181	+	0.480160i	$0.840579\pi$
$390$	0	0
$391$	−2.06048	−0.104203
$392$	−11.8649	−0.599265
$393$	0	0
$394$	0.983878	0.0495671
$395$	30.0517	1.51207
$396$	0	0
$397$	−28.9693	−1.45393	−0.726964	−	0.686675i	$-0.759069\pi$
$397$	−28.9693	−1.45393	−0.726964	+	0.686675i	$0.759069\pi$
$398$	−7.34742	−0.368293
$399$	0	0
$400$	0.301140	0.0150570
$401$	−33.2606	−1.66095	−0.830477	−	0.557053i	$-0.811932\pi$
$401$	−33.2606	−1.66095	−0.830477	+	0.557053i	$0.811932\pi$
$402$	0	0
$403$	−14.8406	−0.739265
$404$	−11.4970	−0.571996
$405$	0	0
$406$	−5.53412	−0.274654
$407$	6.35732	0.315121
$408$	0	0
$409$	−37.9905	−1.87851	−0.939254	−	0.343223i	$-0.888481\pi$
$409$	−37.9905	−1.87851	−0.939254	+	0.343223i	$0.888481\pi$
$410$	25.8513	1.27671
$411$	0	0
$412$	−12.2119	−0.601635
$413$	−8.76644	−0.431368
$414$	0	0
$415$	16.0902	0.789839
$416$	−16.7462	−0.821048
$417$	0	0
$418$	5.87349	0.287282
$419$	−25.3617	−1.23900	−0.619501	−	0.784996i	$-0.712665\pi$
$419$	−25.3617	−1.23900	−0.619501	+	0.784996i	$0.712665\pi$
$420$	0	0
$421$	27.5715	1.34375	0.671876	−	0.740664i	$-0.265489\pi$
$421$	27.5715	1.34375	0.671876	+	0.740664i	$0.265489\pi$
$422$	−16.6195	−0.809024
$423$	0	0
$424$	−9.35046	−0.454098
$425$	−2.15447	−0.104507
$426$	0	0
$427$	1.65804	0.0802383
$428$	−5.26334	−0.254413
$429$	0	0
$430$	2.68671	0.129565
$431$	18.0518	0.869526	0.434763	−	0.900545i	$-0.356832\pi$
$431$	18.0518	0.869526	0.434763	+	0.900545i	$0.356832\pi$
$432$	0	0
$433$	2.22871	0.107105	0.0535526	−	0.998565i	$-0.482946\pi$
$433$	2.22871	0.107105	0.0535526	+	0.998565i	$0.482946\pi$
$434$	−7.17177	−0.344256
$435$	0	0
$436$	−14.2872	−0.684233
$437$	−12.0467	−0.576272
$438$	0	0
$439$	3.17517	0.151542	0.0757712	−	0.997125i	$-0.475858\pi$
$439$	3.17517	0.151542	0.0757712	+	0.997125i	$0.475858\pi$
$440$	−7.29397	−0.347726
$441$	0	0
$442$	2.93974	0.139829
$443$	−36.3852	−1.72872	−0.864358	−	0.502878i	$-0.832275\pi$
$443$	−36.3852	−1.72872	−0.864358	+	0.502878i	$0.832275\pi$
$444$	0	0
$445$	−37.5595	−1.78049
$446$	12.1352	0.574619
$447$	0	0
$448$	−7.54666	−0.356546
$449$	4.46618	0.210772	0.105386	−	0.994431i	$-0.466392\pi$
$449$	4.46618	0.210772	0.105386	+	0.994431i	$0.466392\pi$
$450$	0	0
$451$	11.5987	0.546162
$452$	−2.76766	−0.130180
$453$	0	0
$454$	18.3729	0.862283
$455$	12.6792	0.594409
$456$	0	0
$457$	6.82174	0.319108	0.159554	−	0.987189i	$-0.448994\pi$
$457$	6.82174	0.319108	0.159554	+	0.987189i	$0.448994\pi$
$458$	−2.72166	−0.127175
$459$	0	0
$460$	5.81744	0.271239
$461$	−8.33514	−0.388206	−0.194103	−	0.980981i	$-0.562180\pi$
$461$	−8.33514	−0.388206	−0.194103	+	0.980981i	$0.562180\pi$
$462$	0	0
$463$	38.4228	1.78566	0.892830	−	0.450394i	$-0.148716\pi$
$463$	38.4228	1.78566	0.892830	+	0.450394i	$0.148716\pi$
$464$	−0.644306	−0.0299112
$465$	0	0
$466$	−1.27479	−0.0590537
$467$	−3.83114	−0.177284	−0.0886420	−	0.996064i	$-0.528253\pi$
$467$	−3.83114	−0.177284	−0.0886420	+	0.996064i	$0.528253\pi$
$468$	0	0
$469$	−8.23837	−0.380413
$470$	25.2194	1.16328
$471$	0	0
$472$	14.7574	0.679264
$473$	1.20545	0.0554265
$474$	0	0
$475$	−12.5963	−0.577956
$476$	−2.48535	−0.113916
$477$	0	0
$478$	−0.691407	−0.0316242
$479$	−41.5821	−1.89993	−0.949967	−	0.312350i	$-0.898884\pi$
$479$	−41.5821	−1.89993	−0.949967	+	0.312350i	$0.898884\pi$
$480$	0	0
$481$	18.6032	0.848233
$482$	−9.24353	−0.421031
$483$	0	0
$484$	−1.27258	−0.0578447
$485$	26.5428	1.20525
$486$	0	0
$487$	−28.1377	−1.27504	−0.637520	−	0.770434i	$-0.720040\pi$
$487$	−28.1377	−1.27504	−0.637520	+	0.770434i	$0.720040\pi$
$488$	−2.79114	−0.126349
$489$	0	0
$490$	−9.47443	−0.428011
$491$	40.4392	1.82499	0.912497	−	0.409083i	$-0.134151\pi$
$491$	40.4392	1.82499	0.912497	+	0.409083i	$0.134151\pi$
$492$	0	0
$493$	4.60963	0.207607
$494$	17.1874	0.773297
$495$	0	0
$496$	−0.834969	−0.0374912
$497$	−14.5961	−0.654725
$498$	0	0
$499$	12.7398	0.570310	0.285155	−	0.958481i	$-0.407955\pi$
$499$	12.7398	0.570310	0.285155	+	0.958481i	$0.407955\pi$
$500$	−10.5451	−0.471592
$501$	0	0
$502$	−0.118906	−0.00530705
$503$	28.0766	1.25187	0.625937	−	0.779874i	$-0.284717\pi$
$503$	28.0766	1.25187	0.625937	+	0.779874i	$0.284717\pi$
$504$	0	0
$505$	−23.6091	−1.05059
$506$	−1.49195	−0.0663254
$507$	0	0
$508$	26.4255	1.17244
$509$	−15.1082	−0.669661	−0.334830	−	0.942278i	$-0.608679\pi$
$509$	−15.1082	−0.669661	−0.334830	+	0.942278i	$0.608679\pi$
$510$	0	0
$511$	4.48300	0.198316
$512$	−1.86124	−0.0822558
$513$	0	0
$514$	11.3571	0.500939
$515$	−25.0771	−1.10503
$516$	0	0
$517$	11.3152	0.497641
$518$	8.99004	0.395000
$519$	0	0
$520$	−21.3441	−0.936000
$521$	13.3197	0.583547	0.291774	−	0.956487i	$-0.405755\pi$
$521$	13.3197	0.583547	0.291774	+	0.956487i	$0.405755\pi$
$522$	0	0
$523$	28.2491	1.23525	0.617624	−	0.786474i	$-0.288096\pi$
$523$	28.2491	1.23525	0.617624	+	0.786474i	$0.288096\pi$
$524$	−1.34179	−0.0586164
$525$	0	0
$526$	15.1482	0.660491
$527$	5.97371	0.260219
$528$	0	0
$529$	−19.9400	−0.866955
$530$	−7.46661	−0.324329
$531$	0	0
$532$	−14.5307	−0.629987
$533$	33.9409	1.47014
$534$	0	0
$535$	−10.8083	−0.467283
$536$	13.8684	0.599026
$537$	0	0
$538$	−0.955521	−0.0411954
$539$	−4.25089	−0.183099
$540$	0	0
$541$	−22.4690	−0.966019	−0.483009	−	0.875615i	$-0.660456\pi$
$541$	−22.4690	−0.966019	−0.483009	+	0.875615i	$0.660456\pi$
$542$	0.691408	0.0296985
$543$	0	0
$544$	6.74072	0.289006
$545$	−29.3388	−1.25674
$546$	0	0
$547$	2.45383	0.104918	0.0524592	−	0.998623i	$-0.483294\pi$
$547$	2.45383	0.104918	0.0524592	+	0.998623i	$0.483294\pi$
$548$	7.99395	0.341485
$549$	0	0
$550$	−1.56001	−0.0665192
$551$	26.9505	1.14813
$552$	0	0
$553$	19.0671	0.810814
$554$	23.8652	1.01393
$555$	0	0
$556$	−10.1896	−0.432136
$557$	39.1872	1.66042	0.830208	−	0.557453i	$-0.188221\pi$
$557$	39.1872	1.66042	0.830208	+	0.557453i	$0.188221\pi$
$558$	0	0
$559$	3.52746	0.149196
$560$	0.713360	0.0301450
$561$	0	0
$562$	12.1249	0.511459
$563$	24.5231	1.03353	0.516763	−	0.856128i	$-0.327137\pi$
$563$	24.5231	1.03353	0.516763	+	0.856128i	$0.327137\pi$
$564$	0	0
$565$	−5.68339	−0.239102
$566$	6.56308	0.275867
$567$	0	0
$568$	24.5710	1.03098
$569$	25.1073	1.05255	0.526276	−	0.850314i	$-0.323588\pi$
$569$	25.1073	1.05255	0.526276	+	0.850314i	$0.323588\pi$
$570$	0	0
$571$	5.61101	0.234813	0.117407	−	0.993084i	$-0.462542\pi$
$571$	5.61101	0.234813	0.117407	+	0.993084i	$0.462542\pi$
$572$	−3.72392	−0.155705
$573$	0	0
$574$	16.4020	0.684606
$575$	3.19963	0.133434
$576$	0	0
$577$	−15.9216	−0.662826	−0.331413	−	0.943486i	$-0.607525\pi$
$577$	−15.9216	−0.662826	−0.331413	+	0.943486i	$0.607525\pi$
$578$	13.3158	0.553863
$579$	0	0
$580$	−13.0146	−0.540400
$581$	10.2089	0.423535
$582$	0	0
$583$	−3.35005	−0.138745
$584$	−7.54666	−0.312283
$585$	0	0
$586$	25.2777	1.04421
$587$	18.8277	0.777102	0.388551	−	0.921427i	$-0.372976\pi$
$587$	18.8277	0.777102	0.388551	+	0.921427i	$0.372976\pi$
$588$	0	0
$589$	34.9256	1.43909
$590$	11.7842	0.485148
$591$	0	0
$592$	1.04666	0.0430174
$593$	−4.73556	−0.194466	−0.0972332	−	0.995262i	$-0.530999\pi$
$593$	−4.73556	−0.194466	−0.0972332	+	0.995262i	$0.530999\pi$
$594$	0	0
$595$	−5.10366	−0.209230
$596$	7.29541	0.298832
$597$	0	0
$598$	−4.36585	−0.178533
$599$	41.5097	1.69604	0.848019	−	0.529965i	$-0.177795\pi$
$599$	41.5097	1.69604	0.848019	+	0.529965i	$0.177795\pi$
$600$	0	0
$601$	2.65743	0.108399	0.0541994	−	0.998530i	$-0.482739\pi$
$601$	2.65743	0.108399	0.0541994	+	0.998530i	$0.482739\pi$
$602$	1.70465	0.0694764
$603$	0	0
$604$	−15.5228	−0.631612
$605$	−2.61325	−0.106244
$606$	0	0
$607$	−38.9990	−1.58292	−0.791461	−	0.611220i	$-0.790679\pi$
$607$	−38.9990	−1.58292	−0.791461	+	0.611220i	$0.790679\pi$
$608$	39.4100	1.59829
$609$	0	0
$610$	−2.22881	−0.0902419
$611$	33.1112	1.33954
$612$	0	0
$613$	−4.35101	−0.175736	−0.0878679	−	0.996132i	$-0.528005\pi$
$613$	−4.35101	−0.175736	−0.0878679	+	0.996132i	$0.528005\pi$
$614$	−23.3383	−0.941856
$615$	0	0
$616$	−4.62784	−0.186461
$617$	−0.975178	−0.0392592	−0.0196296	−	0.999807i	$-0.506249\pi$
$617$	−0.975178	−0.0392592	−0.0196296	+	0.999807i	$0.506249\pi$
$618$	0	0
$619$	−14.7988	−0.594813	−0.297407	−	0.954751i	$-0.596122\pi$
$619$	−14.7988	−0.594813	−0.297407	+	0.954751i	$0.596122\pi$
$620$	−16.8658	−0.677347
$621$	0	0
$622$	16.6085	0.665938
$623$	−23.8305	−0.954751
$624$	0	0
$625$	−30.7999	−1.23200
$626$	10.7328	0.428969
$627$	0	0
$628$	14.1939	0.566398
$629$	−7.48823	−0.298575
$630$	0	0
$631$	−11.7278	−0.466878	−0.233439	−	0.972371i	$-0.574998\pi$
$631$	−11.7278	−0.466878	−0.233439	+	0.972371i	$0.574998\pi$
$632$	−32.0974	−1.27677
$633$	0	0
$634$	−21.5595	−0.856239
$635$	54.2649	2.15344
$636$	0	0
$637$	−12.4392	−0.492860
$638$	3.33774	0.132142
$639$	0	0
$640$	−19.7652	−0.781290
$641$	−12.2366	−0.483317	−0.241658	−	0.970361i	$-0.577691\pi$
$641$	−12.2366	−0.483317	−0.241658	+	0.970361i	$0.577691\pi$
$642$	0	0
$643$	−28.0664	−1.10683	−0.553415	−	0.832906i	$-0.686676\pi$
$643$	−28.0664	−1.10683	−0.553415	+	0.832906i	$0.686676\pi$
$644$	3.69102	0.145446
$645$	0	0
$646$	−6.91833	−0.272198
$647$	39.8291	1.56584	0.782922	−	0.622119i	$-0.213728\pi$
$647$	39.8291	1.56584	0.782922	+	0.622119i	$0.213728\pi$
$648$	0	0
$649$	5.28722	0.207542
$650$	−4.56501	−0.179054
$651$	0	0
$652$	−10.0885	−0.395094
$653$	32.9858	1.29084	0.645418	−	0.763830i	$-0.276683\pi$
$653$	32.9858	1.29084	0.645418	+	0.763830i	$0.276683\pi$
$654$	0	0
$655$	−2.75537	−0.107661
$656$	1.90959	0.0745570
$657$	0	0
$658$	16.0010	0.623786
$659$	−43.4532	−1.69270	−0.846349	−	0.532629i	$-0.821204\pi$
$659$	−43.4532	−1.69270	−0.846349	+	0.532629i	$0.821204\pi$
$660$	0	0
$661$	36.0735	1.40310	0.701548	−	0.712623i	$-0.252493\pi$
$661$	36.0735	1.40310	0.701548	+	0.712623i	$0.252493\pi$
$662$	22.2314	0.864047
$663$	0	0
$664$	−17.1855	−0.666928
$665$	−29.8389	−1.15710
$666$	0	0
$667$	−6.84581	−0.265071
$668$	−4.22485	−0.163465
$669$	0	0
$670$	11.0744	0.427840
$671$	−1.00000	−0.0386046
$672$	0	0
$673$	9.71795	0.374600	0.187300	−	0.982303i	$-0.440026\pi$
$673$	9.71795	0.374600	0.187300	+	0.982303i	$0.440026\pi$
$674$	−12.7117	−0.489635
$675$	0	0
$676$	5.64643	0.217170
$677$	−24.9453	−0.958725	−0.479362	−	0.877617i	$-0.659132\pi$
$677$	−24.9453	−0.958725	−0.479362	+	0.877617i	$0.659132\pi$
$678$	0	0
$679$	16.8407	0.646287
$680$	8.59149	0.329469
$681$	0	0
$682$	4.32544	0.165630
$683$	−21.3978	−0.818764	−0.409382	−	0.912363i	$-0.634256\pi$
$683$	−21.3978	−0.818764	−0.409382	+	0.912363i	$0.634256\pi$
$684$	0	0
$685$	16.4156	0.627208
$686$	−15.9101	−0.607452
$687$	0	0
$688$	0.198463	0.00756632
$689$	−9.80312	−0.373469
$690$	0	0
$691$	9.21723	0.350640	0.175320	−	0.984512i	$-0.443904\pi$
$691$	9.21723	0.350640	0.175320	+	0.984512i	$0.443904\pi$
$692$	9.03097	0.343306
$693$	0	0
$694$	10.0863	0.382872
$695$	−20.9244	−0.793708
$696$	0	0
$697$	−13.6620	−0.517485
$698$	−6.45753	−0.244421
$699$	0	0
$700$	3.85939	0.145871
$701$	12.3879	0.467883	0.233942	−	0.972251i	$-0.424838\pi$
$701$	12.3879	0.467883	0.233942	+	0.972251i	$0.424838\pi$
$702$	0	0
$703$	−43.7804	−1.65121
$704$	4.55154	0.171543
$705$	0	0
$706$	−2.48671	−0.0935884
$707$	−14.9794	−0.563357
$708$	0	0
$709$	−11.6785	−0.438594	−0.219297	−	0.975658i	$-0.570376\pi$
$709$	−11.6785	−0.438594	−0.219297	+	0.975658i	$0.570376\pi$
$710$	19.6207	0.736351
$711$	0	0
$712$	40.1162	1.50342
$713$	−8.87162	−0.332245
$714$	0	0
$715$	−7.64707	−0.285984
$716$	2.94880	0.110202
$717$	0	0
$718$	19.7899	0.738552
$719$	−7.26001	−0.270753	−0.135376	−	0.990794i	$-0.543224\pi$
$719$	−7.26001	−0.270753	−0.135376	+	0.990794i	$0.543224\pi$
$720$	0	0
$721$	−15.9108	−0.592548
$722$	−24.2436	−0.902252
$723$	0	0
$724$	20.8884	0.776310
$725$	−7.15811	−0.265845
$726$	0	0
$727$	−14.8511	−0.550796	−0.275398	−	0.961330i	$-0.588810\pi$
$727$	−14.8511	−0.550796	−0.275398	+	0.961330i	$0.588810\pi$
$728$	−13.5423	−0.501910
$729$	0	0
$730$	−6.02623	−0.223041
$731$	−1.41988	−0.0525163
$732$	0	0
$733$	29.6409	1.09481	0.547405	−	0.836868i	$-0.315616\pi$
$733$	29.6409	1.09481	0.547405	+	0.836868i	$0.315616\pi$
$734$	10.1427	0.374373
$735$	0	0
$736$	−10.0107	−0.369000
$737$	4.96873	0.183026
$738$	0	0
$739$	−47.1383	−1.73401	−0.867005	−	0.498300i	$-0.833958\pi$
$739$	−47.1383	−1.73401	−0.867005	+	0.498300i	$0.833958\pi$
$740$	21.1418	0.777189
$741$	0	0
$742$	−4.73738	−0.173915
$743$	−28.2612	−1.03680	−0.518402	−	0.855137i	$-0.673473\pi$
$743$	−28.2612	−1.03680	−0.518402	+	0.855137i	$0.673473\pi$
$744$	0	0
$745$	14.9811	0.548867
$746$	16.3615	0.599038
$747$	0	0
$748$	1.49896	0.0548075
$749$	−6.85758	−0.250570
$750$	0	0
$751$	−2.38559	−0.0870513	−0.0435256	−	0.999052i	$-0.513859\pi$
$751$	−2.38559	−0.0870513	−0.0435256	+	0.999052i	$0.513859\pi$
$752$	1.86291	0.0679334
$753$	0	0
$754$	9.76712	0.355697
$755$	−31.8760	−1.16009
$756$	0	0
$757$	2.00925	0.0730275	0.0365138	−	0.999333i	$-0.488375\pi$
$757$	2.00925	0.0730275	0.0365138	+	0.999333i	$0.488375\pi$
$758$	−25.8415	−0.938606
$759$	0	0
$760$	50.2307	1.82206
$761$	23.0243	0.834629	0.417314	−	0.908762i	$-0.362971\pi$
$761$	23.0243	0.834629	0.417314	+	0.908762i	$0.362971\pi$
$762$	0	0
$763$	−18.6147	−0.673898
$764$	−2.90187	−0.104986
$765$	0	0
$766$	−9.74995	−0.352280
$767$	15.4718	0.558654
$768$	0	0
$769$	−2.79586	−0.100821	−0.0504106	−	0.998729i	$-0.516053\pi$
$769$	−2.79586	−0.100821	−0.0504106	+	0.998729i	$0.516053\pi$
$770$	−3.69546	−0.133175
$771$	0	0
$772$	7.46988	0.268847
$773$	32.9938	1.18671	0.593353	−	0.804943i	$-0.297804\pi$
$773$	32.9938	1.18671	0.593353	+	0.804943i	$0.297804\pi$
$774$	0	0
$775$	−9.27633	−0.333216
$776$	−28.3496	−1.01769
$777$	0	0
$778$	29.5111	1.05802
$779$	−79.8757	−2.86184
$780$	0	0
$781$	8.80321	0.315004
$782$	1.75736	0.0628429
$783$	0	0
$784$	−0.699860	−0.0249950
$785$	29.1472	1.04031
$786$	0	0
$787$	−49.1211	−1.75098	−0.875489	−	0.483238i	$-0.839461\pi$
$787$	−49.1211	−1.75098	−0.875489	+	0.483238i	$0.839461\pi$
$788$	1.46803	0.0522966
$789$	0	0
$790$	−25.6307	−0.911901
$791$	−3.60596	−0.128213
$792$	0	0
$793$	−2.92626	−0.103915
$794$	24.7076	0.876838
$795$	0	0
$796$	−10.9630	−0.388574
$797$	16.3614	0.579552	0.289776	−	0.957094i	$-0.406419\pi$
$797$	16.3614	0.579552	0.289776	+	0.957094i	$0.406419\pi$
$798$	0	0
$799$	−13.3280	−0.471512
$800$	−10.4674	−0.370078
$801$	0	0
$802$	28.3675	1.00169
$803$	−2.70379	−0.0954146
$804$	0	0
$805$	7.57951	0.267143
$806$	12.6574	0.445838
$807$	0	0
$808$	25.2162	0.887103
$809$	−25.1180	−0.883101	−0.441550	−	0.897236i	$-0.645571\pi$
$809$	−25.1180	−0.883101	−0.441550	+	0.897236i	$0.645571\pi$
$810$	0	0
$811$	−5.80475	−0.203832	−0.101916	−	0.994793i	$-0.532497\pi$
$811$	−5.80475	−0.203832	−0.101916	+	0.994793i	$0.532497\pi$
$812$	−8.25741	−0.289778
$813$	0	0
$814$	−5.42208	−0.190044
$815$	−20.7167	−0.725673
$816$	0	0
$817$	−8.30143	−0.290430
$818$	32.4016	1.13289
$819$	0	0
$820$	38.5725	1.34701
$821$	−18.3415	−0.640122	−0.320061	−	0.947397i	$-0.603703\pi$
$821$	−18.3415	−0.640122	−0.320061	+	0.947397i	$0.603703\pi$
$822$	0	0
$823$	−3.59592	−0.125346	−0.0626730	−	0.998034i	$-0.519963\pi$
$823$	−3.59592	−0.125346	−0.0626730	+	0.998034i	$0.519963\pi$
$824$	26.7841	0.933070
$825$	0	0
$826$	7.47678	0.260150
$827$	4.32831	0.150510	0.0752551	−	0.997164i	$-0.476023\pi$
$827$	4.32831	0.150510	0.0752551	+	0.997164i	$0.476023\pi$
$828$	0	0
$829$	−8.50149	−0.295269	−0.147634	−	0.989042i	$-0.547166\pi$
$829$	−8.50149	−0.295269	−0.147634	+	0.989042i	$0.547166\pi$
$830$	−13.7232	−0.476338
$831$	0	0
$832$	13.3190	0.461754
$833$	5.00708	0.173485
$834$	0	0
$835$	−8.67575	−0.300237
$836$	8.76378	0.303102
$837$	0	0
$838$	21.6307	0.747219
$839$	−9.96602	−0.344065	−0.172033	−	0.985091i	$-0.555033\pi$
$839$	−9.96602	−0.344065	−0.172033	+	0.985091i	$0.555033\pi$
$840$	0	0
$841$	−13.6848	−0.471889
$842$	−23.5154	−0.810393
$843$	0	0
$844$	−24.7978	−0.853575
$845$	11.5949	0.398878
$846$	0	0
$847$	−1.65804	−0.0569710
$848$	−0.551546	−0.0189402
$849$	0	0
$850$	1.83752	0.0630265
$851$	11.1208	0.381218
$852$	0	0
$853$	25.6693	0.878899	0.439450	−	0.898267i	$-0.355173\pi$
$853$	25.6693	0.878899	0.439450	+	0.898267i	$0.355173\pi$
$854$	−1.41412	−0.0483903
$855$	0	0
$856$	11.5440	0.394566
$857$	−8.69052	−0.296863	−0.148431	−	0.988923i	$-0.547422\pi$
$857$	−8.69052	−0.296863	−0.148431	+	0.988923i	$0.547422\pi$
$858$	0	0
$859$	−50.5865	−1.72599	−0.862994	−	0.505213i	$-0.831414\pi$
$859$	−50.5865	−1.72599	−0.862994	+	0.505213i	$0.831414\pi$
$860$	4.00882	0.136700
$861$	0	0
$862$	−15.3962	−0.524396
$863$	−44.4249	−1.51224	−0.756121	−	0.654432i	$-0.772908\pi$
$863$	−44.4249	−1.51224	−0.756121	+	0.654432i	$0.772908\pi$
$864$	0	0
$865$	18.5451	0.630553
$866$	−1.90084	−0.0645932
$867$	0	0
$868$	−10.7009	−0.363213
$869$	−11.4997	−0.390102
$870$	0	0
$871$	14.5398	0.492663
$872$	31.3360	1.06117
$873$	0	0
$874$	10.2745	0.347540
$875$	−13.7392	−0.464469
$876$	0	0
$877$	−16.1552	−0.545522	−0.272761	−	0.962082i	$-0.587937\pi$
$877$	−16.1552	−0.545522	−0.272761	+	0.962082i	$0.587937\pi$
$878$	−2.70806	−0.0913926
$879$	0	0
$880$	−0.430242	−0.0145035
$881$	13.6638	0.460344	0.230172	−	0.973150i	$-0.426071\pi$
$881$	13.6638	0.460344	0.230172	+	0.973150i	$0.426071\pi$
$882$	0	0
$883$	40.3550	1.35805	0.679027	−	0.734113i	$-0.262402\pi$
$883$	40.3550	1.35805	0.679027	+	0.734113i	$0.262402\pi$
$884$	4.38636	0.147529
$885$	0	0
$886$	31.0325	1.04256
$887$	48.0709	1.61406	0.807031	−	0.590509i	$-0.201073\pi$
$887$	48.0709	1.61406	0.807031	+	0.590509i	$0.201073\pi$
$888$	0	0
$889$	34.4297	1.15474
$890$	32.0340	1.07378
$891$	0	0
$892$	18.1068	0.606261
$893$	−77.9231	−2.60760
$894$	0	0
$895$	6.05537	0.202409
$896$	−12.5405	−0.418950
$897$	0	0
$898$	−3.80915	−0.127113
$899$	19.8473	0.661943
$900$	0	0
$901$	3.94598	0.131460
$902$	−9.89238	−0.329380
$903$	0	0
$904$	6.07027	0.201894
$905$	42.8943	1.42586
$906$	0	0
$907$	−6.89910	−0.229081	−0.114540	−	0.993419i	$-0.536540\pi$
$907$	−6.89910	−0.229081	−0.114540	+	0.993419i	$0.536540\pi$
$908$	27.4140	0.909767
$909$	0	0
$910$	−10.8139	−0.358477
$911$	−30.7794	−1.01977	−0.509884	−	0.860243i	$-0.670312\pi$
$911$	−30.7794	−1.01977	−0.509884	+	0.860243i	$0.670312\pi$
$912$	0	0
$913$	−6.15717	−0.203773
$914$	−5.81818	−0.192448
$915$	0	0
$916$	−4.06096	−0.134178
$917$	−1.74821	−0.0577311
$918$	0	0
$919$	17.9352	0.591627	0.295813	−	0.955246i	$-0.404409\pi$
$919$	17.9352	0.591627	0.295813	+	0.955246i	$0.404409\pi$
$920$	−12.7593	−0.420662
$921$	0	0
$922$	7.10893	0.234120
$923$	25.7605	0.847918
$924$	0	0
$925$	11.6282	0.382332
$926$	−32.7703	−1.07690
$927$	0	0
$928$	22.3956	0.735172
$929$	−43.3770	−1.42315	−0.711576	−	0.702609i	$-0.752018\pi$
$929$	−43.3770	−1.42315	−0.711576	+	0.702609i	$0.752018\pi$
$930$	0	0
$931$	29.2742	0.959423
$932$	−1.90211	−0.0623056
$933$	0	0
$934$	3.26753	0.106917
$935$	3.07813	0.100665
$936$	0	0
$937$	24.9911	0.816424	0.408212	−	0.912887i	$-0.366152\pi$
$937$	24.9911	0.816424	0.408212	+	0.912887i	$0.366152\pi$
$938$	7.02640	0.229420
$939$	0	0
$940$	37.6296	1.22734
$941$	−17.6309	−0.574751	−0.287375	−	0.957818i	$-0.592783\pi$
$941$	−17.6309	−0.574751	−0.287375	+	0.957818i	$0.592783\pi$
$942$	0	0
$943$	20.2896	0.660720
$944$	0.870479	0.0283317
$945$	0	0
$946$	−1.02811	−0.0334267
$947$	36.4841	1.18558	0.592788	−	0.805359i	$-0.298027\pi$
$947$	36.4841	1.18558	0.592788	+	0.805359i	$0.298027\pi$
$948$	0	0
$949$	−7.91199	−0.256834
$950$	10.7432	0.348555
$951$	0	0
$952$	5.45108	0.176671
$953$	−4.60213	−0.149077	−0.0745387	−	0.997218i	$-0.523748\pi$
$953$	−4.60213	−0.149077	−0.0745387	+	0.997218i	$0.523748\pi$
$954$	0	0
$955$	−5.95900	−0.192829
$956$	−1.03164	−0.0333657
$957$	0	0
$958$	35.4648	1.14582
$959$	10.4153	0.336327
$960$	0	0
$961$	−5.27956	−0.170308
$962$	−15.8664	−0.511554
$963$	0	0
$964$	−13.7922	−0.444216
$965$	15.3394	0.493793
$966$	0	0
$967$	−44.1589	−1.42005	−0.710027	−	0.704174i	$-0.751317\pi$
$967$	−44.1589	−1.42005	−0.710027	+	0.704174i	$0.751317\pi$
$968$	2.79114	0.0897108
$969$	0	0
$970$	−22.6380	−0.726862
$971$	−20.1147	−0.645512	−0.322756	−	0.946482i	$-0.604609\pi$
$971$	−20.1147	−0.645512	−0.322756	+	0.946482i	$0.604609\pi$
$972$	0	0
$973$	−13.2760	−0.425609
$974$	23.9983	0.768954
$975$	0	0
$976$	−0.164638	−0.00526995
$977$	39.5575	1.26556	0.632778	−	0.774333i	$-0.281915\pi$
$977$	39.5575	1.26556	0.632778	+	0.774333i	$0.281915\pi$
$978$	0	0
$979$	14.3727	0.459353
$980$	−14.1367	−0.451581
$981$	0	0
$982$	−34.4900	−1.10062
$983$	10.4833	0.334364	0.167182	−	0.985926i	$-0.446533\pi$
$983$	10.4833	0.334364	0.167182	+	0.985926i	$0.446533\pi$
$984$	0	0
$985$	3.01461	0.0960536
$986$	−3.93149	−0.125204
$987$	0	0
$988$	25.6451	0.815881
$989$	2.10868	0.0670523
$990$	0	0
$991$	−39.4578	−1.25342	−0.626710	−	0.779253i	$-0.715599\pi$
$991$	−39.4578	−1.25342	−0.626710	+	0.779253i	$0.715599\pi$
$992$	29.0229	0.921478
$993$	0	0
$994$	12.4488	0.394853
$995$	−22.5126	−0.713697
$996$	0	0
$997$	−14.8061	−0.468915	−0.234458	−	0.972126i	$-0.575331\pi$
$997$	−14.8061	−0.468915	−0.234458	+	0.972126i	$0.575331\pi$
$998$	−10.8656	−0.343944
$999$	0	0

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2013.2.a.d.1.8	✓	12	3.2	odd	2
6039.2.a.e.1.5		12	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-0.852887 q^{2} -1.27258 q^{4} -2.61325 q^{5} -1.65804 q^{7} +2.79114 q^{8} +O(q^{10})\)	\(q-0.852887 q^{2} -1.27258 q^{4} -2.61325 q^{5} -1.65804 q^{7} +2.79114 q^{8} +2.22881 q^{10} +1.00000 q^{11} +2.92626 q^{13} +1.41412 q^{14} +0.164638 q^{16} -1.17789 q^{17} -6.88660 q^{19} +3.32559 q^{20} -0.852887 q^{22} +1.74930 q^{23} +1.82910 q^{25} -2.49577 q^{26} +2.11000 q^{28} -3.91346 q^{29} -5.07153 q^{31} -5.72271 q^{32} +1.00461 q^{34} +4.33289 q^{35} +6.35732 q^{37} +5.87349 q^{38} -7.29397 q^{40} +11.5987 q^{41} +1.20545 q^{43} -1.27258 q^{44} -1.49195 q^{46} +11.3152 q^{47} -4.25089 q^{49} -1.56001 q^{50} -3.72392 q^{52} -3.35005 q^{53} -2.61325 q^{55} -4.62784 q^{56} +3.33774 q^{58} +5.28722 q^{59} -1.00000 q^{61} +4.32544 q^{62} +4.55154 q^{64} -7.64707 q^{65} +4.96873 q^{67} +1.49896 q^{68} -3.69546 q^{70} +8.80321 q^{71} -2.70379 q^{73} -5.42208 q^{74} +8.76378 q^{76} -1.65804 q^{77} -11.4997 q^{79} -0.430242 q^{80} -9.89238 q^{82} -6.15717 q^{83} +3.07813 q^{85} -1.02811 q^{86} +2.79114 q^{88} +14.3727 q^{89} -4.85187 q^{91} -2.22613 q^{92} -9.65056 q^{94} +17.9964 q^{95} -10.1570 q^{97} +3.62553 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - q^{2} + 9 q^{4} + 3 q^{5} - 9 q^{7} - 6 q^{8}+O(q^{10}) \) 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

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Database

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