LMFDB - Embedded newform 6040.2.a.o.1.13

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6040,2,Mod(1,6040)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6040, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("6040.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6040 = 2^{3} \cdot 5 \cdot 151 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6040.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:	$48.2296428209$
Analytic rank:	$0$
Dimension:	$15$
Coefficient field:	$\mathbb{Q}[x]/(x^{15} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{15} - 5 x^{14} - 19 x^{13} + 119 x^{12} + 106 x^{11} - 1063 x^{10} - 48 x^{9} + 4510 x^{8} + \cdots + 272 $ x^15 - 5x^14 - 19x^13 + 119x^12 + 106x^11 - 1063x^10 - 48x^9 + 4510x^8 - 1130x^7 - 9512x^6 + 2839x^5 + 9383x^4 - 1781x^3 - 3692x^2 - 92x + 272
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.13
Root			$2.79329$ of defining polynomial
Character	$\chi$	$=$	6040.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.79329 q^{3} +1.00000 q^{5} +4.28460 q^{7} +4.80245 q^{9} +O(q^{10})$	$q+2.79329 q^{3} +1.00000 q^{5} +4.28460 q^{7} +4.80245 q^{9} -3.82563 q^{11} -1.37044 q^{13} +2.79329 q^{15} +1.08602 q^{17} +1.88873 q^{19} +11.9681 q^{21} +0.393873 q^{23} +1.00000 q^{25} +5.03477 q^{27} +4.59286 q^{29} +2.75319 q^{31} -10.6861 q^{33} +4.28460 q^{35} -6.07448 q^{37} -3.82804 q^{39} +1.05063 q^{41} +7.29222 q^{43} +4.80245 q^{45} -1.97922 q^{47} +11.3578 q^{49} +3.03355 q^{51} +3.67975 q^{53} -3.82563 q^{55} +5.27577 q^{57} +8.88518 q^{59} -5.24945 q^{61} +20.5766 q^{63} -1.37044 q^{65} -5.54012 q^{67} +1.10020 q^{69} +2.99609 q^{71} -1.44371 q^{73} +2.79329 q^{75} -16.3913 q^{77} -6.78720 q^{79} -0.343794 q^{81} -2.39181 q^{83} +1.08602 q^{85} +12.8292 q^{87} +14.5194 q^{89} -5.87179 q^{91} +7.69046 q^{93} +1.88873 q^{95} +1.16658 q^{97} -18.3724 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 15 q + 5 q^{3} + 15 q^{5} + 7 q^{7} + 18 q^{9}+O(q^{10}) $ 15 * q + 5 * q^3 + 15 * q^5 + 7 * q^7 + 18 * q^9	$ 15 q + 5 q^{3} + 15 q^{5} + 7 q^{7} + 18 q^{9} + 7 q^{11} + 2 q^{13} + 5 q^{15} - 3 q^{17} + 8 q^{19} + 7 q^{21} + 15 q^{23} + 15 q^{25} + 23 q^{27} + 5 q^{29} + 27 q^{31} - 5 q^{33} + 7 q^{35} - 4 q^{37} + 11 q^{39} + 20 q^{41} + 25 q^{43} + 18 q^{45} + 35 q^{47} - 14 q^{49} + 25 q^{51} - 2 q^{53} + 7 q^{55} - 24 q^{57} + 39 q^{59} + 23 q^{61} + 39 q^{63} + 2 q^{65} + 32 q^{67} + 13 q^{69} + 30 q^{71} + 7 q^{73} + 5 q^{75} - 4 q^{77} + 38 q^{79} + 11 q^{81} + 29 q^{83} - 3 q^{85} + 4 q^{87} + 19 q^{89} + 16 q^{91} + 8 q^{93} + 8 q^{95} - 8 q^{97} + 16 q^{99}+O(q^{100}) $ 15 * q + 5 * q^3 + 15 * q^5 + 7 * q^7 + 18 * q^9 + 7 * q^11 + 2 * q^13 + 5 * q^15 - 3 * q^17 + 8 * q^19 + 7 * q^21 + 15 * q^23 + 15 * q^25 + 23 * q^27 + 5 * q^29 + 27 * q^31 - 5 * q^33 + 7 * q^35 - 4 * q^37 + 11 * q^39 + 20 * q^41 + 25 * q^43 + 18 * q^45 + 35 * q^47 - 14 * q^49 + 25 * q^51 - 2 * q^53 + 7 * q^55 - 24 * q^57 + 39 * q^59 + 23 * q^61 + 39 * q^63 + 2 * q^65 + 32 * q^67 + 13 * q^69 + 30 * q^71 + 7 * q^73 + 5 * q^75 - 4 * q^77 + 38 * q^79 + 11 * q^81 + 29 * q^83 - 3 * q^85 + 4 * q^87 + 19 * q^89 + 16 * q^91 + 8 * q^93 + 8 * q^95 - 8 * q^97 + 16 * q^99

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	0
$2$	0	0
$3$	2.79329	1.61271	0.806353	−	0.591435i	$-0.201438\pi$
$3$	2.79329	1.61271	0.806353	+	0.591435i	$0.201438\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	4.28460	1.61943	0.809713	−	0.586827i	$-0.199623\pi$
$7$	4.28460	1.61943	0.809713	+	0.586827i	$0.199623\pi$
$8$	0	0
$9$	4.80245	1.60082
$10$	0	0
$11$	−3.82563	−1.15347	−0.576735	−	0.816931i	$-0.695673\pi$
$11$	−3.82563	−1.15347	−0.576735	+	0.816931i	$0.695673\pi$
$12$	0	0
$13$	−1.37044	−0.380092	−0.190046	−	0.981775i	$-0.560864\pi$
$13$	−1.37044	−0.380092	−0.190046	+	0.981775i	$0.560864\pi$
$14$	0	0
$15$	2.79329	0.721224
$16$	0	0
$17$	1.08602	0.263397	0.131699	−	0.991290i	$-0.457957\pi$
$17$	1.08602	0.263397	0.131699	+	0.991290i	$0.457957\pi$
$18$	0	0
$19$	1.88873	0.433305	0.216652	−	0.976249i	$-0.430486\pi$
$19$	1.88873	0.433305	0.216652	+	0.976249i	$0.430486\pi$
$20$	0	0
$21$	11.9681	2.61166
$22$	0	0
$23$	0.393873	0.0821283	0.0410641	−	0.999157i	$-0.486925\pi$
$23$	0.393873	0.0821283	0.0410641	+	0.999157i	$0.486925\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	5.03477	0.968943
$28$	0	0
$29$	4.59286	0.852872	0.426436	−	0.904518i	$-0.359769\pi$
$29$	4.59286	0.852872	0.426436	+	0.904518i	$0.359769\pi$
$30$	0	0
$31$	2.75319	0.494488	0.247244	−	0.968953i	$-0.420475\pi$
$31$	2.75319	0.494488	0.247244	+	0.968953i	$0.420475\pi$
$32$	0	0
$33$	−10.6861	−1.86021
$34$	0	0
$35$	4.28460	0.724229
$36$	0	0
$37$	−6.07448	−0.998638	−0.499319	−	0.866418i	$-0.666416\pi$
$37$	−6.07448	−0.998638	−0.499319	+	0.866418i	$0.666416\pi$
$38$	0	0
$39$	−3.82804	−0.612976
$40$	0	0
$41$	1.05063	0.164081	0.0820403	−	0.996629i	$-0.473856\pi$
$41$	1.05063	0.164081	0.0820403	+	0.996629i	$0.473856\pi$
$42$	0	0
$43$	7.29222	1.11205	0.556027	−	0.831165i	$-0.312325\pi$
$43$	7.29222	1.11205	0.556027	+	0.831165i	$0.312325\pi$
$44$	0	0
$45$	4.80245	0.715908
$46$	0	0
$47$	−1.97922	−0.288698	−0.144349	−	0.989527i	$-0.546109\pi$
$47$	−1.97922	−0.288698	−0.144349	+	0.989527i	$0.546109\pi$
$48$	0	0
$49$	11.3578	1.62254
$50$	0	0
$51$	3.03355	0.424782
$52$	0	0
$53$	3.67975	0.505452	0.252726	−	0.967538i	$-0.418673\pi$
$53$	3.67975	0.505452	0.252726	+	0.967538i	$0.418673\pi$
$54$	0	0
$55$	−3.82563	−0.515847
$56$	0	0
$57$	5.27577	0.698793
$58$	0	0
$59$	8.88518	1.15675	0.578376	−	0.815770i	$-0.303687\pi$
$59$	8.88518	1.15675	0.578376	+	0.815770i	$0.303687\pi$
$60$	0	0
$61$	−5.24945	−0.672124	−0.336062	−	0.941840i	$-0.609095\pi$
$61$	−5.24945	−0.672124	−0.336062	+	0.941840i	$0.609095\pi$
$62$	0	0
$63$	20.5766	2.59241
$64$	0	0
$65$	−1.37044	−0.169982
$66$	0	0
$67$	−5.54012	−0.676833	−0.338416	−	0.940996i	$-0.609891\pi$
$67$	−5.54012	−0.676833	−0.338416	+	0.940996i	$0.609891\pi$
$68$	0	0
$69$	1.10020	0.132449
$70$	0	0
$71$	2.99609	0.355570	0.177785	−	0.984069i	$-0.443107\pi$
$71$	2.99609	0.355570	0.177785	+	0.984069i	$0.443107\pi$
$72$	0	0
$73$	−1.44371	−0.168974	−0.0844868	−	0.996425i	$-0.526925\pi$
$73$	−1.44371	−0.168974	−0.0844868	+	0.996425i	$0.526925\pi$
$74$	0	0
$75$	2.79329	0.322541
$76$	0	0
$77$	−16.3913	−1.86796
$78$	0	0
$79$	−6.78720	−0.763619	−0.381810	−	0.924241i	$-0.624699\pi$
$79$	−6.78720	−0.763619	−0.381810	+	0.924241i	$0.624699\pi$
$80$	0	0
$81$	−0.343794	−0.0381993
$82$	0	0
$83$	−2.39181	−0.262535	−0.131268	−	0.991347i	$-0.541905\pi$
$83$	−2.39181	−0.262535	−0.131268	+	0.991347i	$0.541905\pi$
$84$	0	0
$85$	1.08602	0.117795
$86$	0	0
$87$	12.8292	1.37543
$88$	0	0
$89$	14.5194	1.53905	0.769527	−	0.638614i	$-0.220492\pi$
$89$	14.5194	1.53905	0.769527	+	0.638614i	$0.220492\pi$
$90$	0	0
$91$	−5.87179	−0.615531
$92$	0	0
$93$	7.69046	0.797464
$94$	0	0
$95$	1.88873	0.193780
$96$	0	0
$97$	1.16658	0.118449	0.0592243	−	0.998245i	$-0.481137\pi$
$97$	1.16658	0.118449	0.0592243	+	0.998245i	$0.481137\pi$
$98$	0	0
$99$	−18.3724	−1.84649
$100$	0	0
$101$	12.3543	1.22930	0.614649	−	0.788801i	$-0.289298\pi$
$101$	12.3543	1.22930	0.614649	+	0.788801i	$0.289298\pi$
$102$	0	0
$103$	3.63180	0.357852	0.178926	−	0.983863i	$-0.442738\pi$
$103$	3.63180	0.357852	0.178926	+	0.983863i	$0.442738\pi$
$104$	0	0
$105$	11.9681	1.16797
$106$	0	0
$107$	7.39447	0.714850	0.357425	−	0.933942i	$-0.383655\pi$
$107$	7.39447	0.714850	0.357425	+	0.933942i	$0.383655\pi$
$108$	0	0
$109$	8.29703	0.794711	0.397356	−	0.917665i	$-0.369928\pi$
$109$	8.29703	0.794711	0.397356	+	0.917665i	$0.369928\pi$
$110$	0	0
$111$	−16.9678	−1.61051
$112$	0	0
$113$	−2.66630	−0.250825	−0.125412	−	0.992105i	$-0.540025\pi$
$113$	−2.66630	−0.250825	−0.125412	+	0.992105i	$0.540025\pi$
$114$	0	0
$115$	0.393873	0.0367289
$116$	0	0
$117$	−6.58148	−0.608458
$118$	0	0
$119$	4.65314	0.426552
$120$	0	0
$121$	3.63541	0.330492
$122$	0	0
$123$	2.93471	0.264614
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−12.1721	−1.08010	−0.540048	−	0.841634i	$-0.681594\pi$
$127$	−12.1721	−1.08010	−0.540048	+	0.841634i	$0.681594\pi$
$128$	0	0
$129$	20.3693	1.79341
$130$	0	0
$131$	−3.44241	−0.300765	−0.150382	−	0.988628i	$-0.548050\pi$
$131$	−3.44241	−0.300765	−0.150382	+	0.988628i	$0.548050\pi$
$132$	0	0
$133$	8.09245	0.701704
$134$	0	0
$135$	5.03477	0.433324
$136$	0	0
$137$	−7.53354	−0.643634	−0.321817	−	0.946802i	$-0.604294\pi$
$137$	−7.53354	−0.643634	−0.321817	+	0.946802i	$0.604294\pi$
$138$	0	0
$139$	−1.67890	−0.142403	−0.0712013	−	0.997462i	$-0.522683\pi$
$139$	−1.67890	−0.142403	−0.0712013	+	0.997462i	$0.522683\pi$
$140$	0	0
$141$	−5.52852	−0.465585
$142$	0	0
$143$	5.24280	0.438425
$144$	0	0
$145$	4.59286	0.381416
$146$	0	0
$147$	31.7255	2.61668
$148$	0	0
$149$	−9.98124	−0.817695	−0.408847	−	0.912603i	$-0.634069\pi$
$149$	−9.98124	−0.817695	−0.408847	+	0.912603i	$0.634069\pi$
$150$	0	0
$151$	−1.00000	−0.0813788
$151$	−1.00000	−0.0813788
$152$	0	0
$153$	5.21554	0.421651
$154$	0	0
$155$	2.75319	0.221142
$156$	0	0
$157$	14.0298	1.11970	0.559849	−	0.828595i	$-0.310859\pi$
$157$	14.0298	1.11970	0.559849	+	0.828595i	$0.310859\pi$
$158$	0	0
$159$	10.2786	0.815145
$160$	0	0
$161$	1.68759	0.133001
$162$	0	0
$163$	−4.11737	−0.322497	−0.161249	−	0.986914i	$-0.551552\pi$
$163$	−4.11737	−0.322497	−0.161249	+	0.986914i	$0.551552\pi$
$164$	0	0
$165$	−10.6861	−0.831910
$166$	0	0
$167$	5.52019	0.427165	0.213582	−	0.976925i	$-0.431487\pi$
$167$	5.52019	0.427165	0.213582	+	0.976925i	$0.431487\pi$
$168$	0	0
$169$	−11.1219	−0.855530
$170$	0	0
$171$	9.07054	0.693642
$172$	0	0
$173$	14.0517	1.06833	0.534165	−	0.845380i	$-0.320626\pi$
$173$	14.0517	1.06833	0.534165	+	0.845380i	$0.320626\pi$
$174$	0	0
$175$	4.28460	0.323885
$176$	0	0
$177$	24.8189	1.86550
$178$	0	0
$179$	−3.99318	−0.298464	−0.149232	−	0.988802i	$-0.547680\pi$
$179$	−3.99318	−0.298464	−0.149232	+	0.988802i	$0.547680\pi$
$180$	0	0
$181$	4.50458	0.334822	0.167411	−	0.985887i	$-0.446459\pi$
$181$	4.50458	0.334822	0.167411	+	0.985887i	$0.446459\pi$
$182$	0	0
$183$	−14.6632	−1.08394
$184$	0	0
$185$	−6.07448	−0.446604
$186$	0	0
$187$	−4.15469	−0.303821
$188$	0	0
$189$	21.5720	1.56913
$190$	0	0
$191$	12.6793	0.917439	0.458719	−	0.888581i	$-0.348308\pi$
$191$	12.6793	0.917439	0.458719	+	0.888581i	$0.348308\pi$
$192$	0	0
$193$	−13.5798	−0.977497	−0.488749	−	0.872425i	$-0.662546\pi$
$193$	−13.5798	−0.977497	−0.488749	+	0.872425i	$0.662546\pi$
$194$	0	0
$195$	−3.82804	−0.274131
$196$	0	0
$197$	−7.55994	−0.538624	−0.269312	−	0.963053i	$-0.586796\pi$
$197$	−7.55994	−0.538624	−0.269312	+	0.963053i	$0.586796\pi$
$198$	0	0
$199$	−21.6294	−1.53326	−0.766632	−	0.642086i	$-0.778069\pi$
$199$	−21.6294	−1.53326	−0.766632	+	0.642086i	$0.778069\pi$
$200$	0	0
$201$	−15.4751	−1.09153
$202$	0	0
$203$	19.6785	1.38116
$204$	0	0
$205$	1.05063	0.0733791
$206$	0	0
$207$	1.89156	0.131472
$208$	0	0
$209$	−7.22558	−0.499804
$210$	0	0
$211$	−21.1979	−1.45932	−0.729661	−	0.683809i	$-0.760322\pi$
$211$	−21.1979	−1.45932	−0.729661	+	0.683809i	$0.760322\pi$
$212$	0	0
$213$	8.36894	0.573430
$214$	0	0
$215$	7.29222	0.497325
$216$	0	0
$217$	11.7963	0.800787
$218$	0	0
$219$	−4.03270	−0.272505
$220$	0	0
$221$	−1.48832	−0.100115
$222$	0	0
$223$	−19.9841	−1.33824	−0.669118	−	0.743156i	$-0.733328\pi$
$223$	−19.9841	−1.33824	−0.669118	+	0.743156i	$0.733328\pi$
$224$	0	0
$225$	4.80245	0.320164
$226$	0	0
$227$	15.1783	1.00742	0.503708	−	0.863874i	$-0.331969\pi$
$227$	15.1783	1.00742	0.503708	+	0.863874i	$0.331969\pi$
$228$	0	0
$229$	0.360349	0.0238125	0.0119063	−	0.999929i	$-0.496210\pi$
$229$	0.360349	0.0238125	0.0119063	+	0.999929i	$0.496210\pi$
$230$	0	0
$231$	−45.7855	−3.01246
$232$	0	0
$233$	−7.42083	−0.486155	−0.243077	−	0.970007i	$-0.578157\pi$
$233$	−7.42083	−0.486155	−0.243077	+	0.970007i	$0.578157\pi$
$234$	0	0
$235$	−1.97922	−0.129110
$236$	0	0
$237$	−18.9586	−1.23149
$238$	0	0
$239$	−18.6679	−1.20752	−0.603762	−	0.797165i	$-0.706332\pi$
$239$	−18.6679	−1.20752	−0.603762	+	0.797165i	$0.706332\pi$
$240$	0	0
$241$	−23.6535	−1.52366	−0.761828	−	0.647779i	$-0.775698\pi$
$241$	−23.6535	−1.52366	−0.761828	+	0.647779i	$0.775698\pi$
$242$	0	0
$243$	−16.0646	−1.03055
$244$	0	0
$245$	11.3578	0.725621
$246$	0	0
$247$	−2.58839	−0.164696
$248$	0	0
$249$	−6.68102	−0.423392
$250$	0	0
$251$	−4.99172	−0.315074	−0.157537	−	0.987513i	$-0.550355\pi$
$251$	−4.99172	−0.315074	−0.157537	+	0.987513i	$0.550355\pi$
$252$	0	0
$253$	−1.50681	−0.0947325
$254$	0	0
$255$	3.03355	0.189968
$256$	0	0
$257$	5.32855	0.332386	0.166193	−	0.986093i	$-0.446853\pi$
$257$	5.32855	0.332386	0.166193	+	0.986093i	$0.446853\pi$
$258$	0	0
$259$	−26.0267	−1.61722
$260$	0	0
$261$	22.0570	1.36529
$262$	0	0
$263$	−18.7198	−1.15431	−0.577155	−	0.816635i	$-0.695837\pi$
$263$	−18.7198	−1.15431	−0.577155	+	0.816635i	$0.695837\pi$
$264$	0	0
$265$	3.67975	0.226045
$266$	0	0
$267$	40.5569	2.48204
$268$	0	0
$269$	−6.81770	−0.415683	−0.207841	−	0.978163i	$-0.566644\pi$
$269$	−6.81770	−0.415683	−0.207841	+	0.978163i	$0.566644\pi$
$270$	0	0
$271$	−28.8082	−1.74997	−0.874986	−	0.484148i	$-0.839130\pi$
$271$	−28.8082	−1.74997	−0.874986	+	0.484148i	$0.839130\pi$
$272$	0	0
$273$	−16.4016	−0.992669
$274$	0	0
$275$	−3.82563	−0.230694
$276$	0	0
$277$	−15.1593	−0.910833	−0.455417	−	0.890278i	$-0.650510\pi$
$277$	−15.1593	−0.910833	−0.455417	+	0.890278i	$0.650510\pi$
$278$	0	0
$279$	13.2221	0.791586
$280$	0	0
$281$	5.58698	0.333291	0.166646	−	0.986017i	$-0.446706\pi$
$281$	5.58698	0.333291	0.166646	+	0.986017i	$0.446706\pi$
$282$	0	0
$283$	−2.39978	−0.142652	−0.0713261	−	0.997453i	$-0.522723\pi$
$283$	−2.39978	−0.142652	−0.0713261	+	0.997453i	$0.522723\pi$
$284$	0	0
$285$	5.27577	0.312510
$286$	0	0
$287$	4.50152	0.265716
$288$	0	0
$289$	−15.8206	−0.930622
$290$	0	0
$291$	3.25861	0.191023
$292$	0	0
$293$	−4.94939	−0.289147	−0.144573	−	0.989494i	$-0.546181\pi$
$293$	−4.94939	−0.289147	−0.144573	+	0.989494i	$0.546181\pi$
$294$	0	0
$295$	8.88518	0.517315
$296$	0	0
$297$	−19.2612	−1.11765
$298$	0	0
$299$	−0.539780	−0.0312163
$300$	0	0
$301$	31.2442	1.80089
$302$	0	0
$303$	34.5091	1.98250
$304$	0	0
$305$	−5.24945	−0.300583
$306$	0	0
$307$	12.0297	0.686573	0.343286	−	0.939231i	$-0.388460\pi$
$307$	12.0297	0.686573	0.343286	+	0.939231i	$0.388460\pi$
$308$	0	0
$309$	10.1447	0.577110
$310$	0	0
$311$	−5.92870	−0.336186	−0.168093	−	0.985771i	$-0.553761\pi$
$311$	−5.92870	−0.336186	−0.168093	+	0.985771i	$0.553761\pi$
$312$	0	0
$313$	−9.51341	−0.537730	−0.268865	−	0.963178i	$-0.586648\pi$
$313$	−9.51341	−0.537730	−0.268865	+	0.963178i	$0.586648\pi$
$314$	0	0
$315$	20.5766	1.15936
$316$	0	0
$317$	2.23378	0.125462	0.0627308	−	0.998030i	$-0.480019\pi$
$317$	2.23378	0.125462	0.0627308	+	0.998030i	$0.480019\pi$
$318$	0	0
$319$	−17.5705	−0.983762
$320$	0	0
$321$	20.6549	1.15284
$322$	0	0
$323$	2.05119	0.114131
$324$	0	0
$325$	−1.37044	−0.0760184
$326$	0	0
$327$	23.1760	1.28163
$328$	0	0
$329$	−8.48014	−0.467525
$330$	0	0
$331$	27.7364	1.52453	0.762266	−	0.647264i	$-0.224087\pi$
$331$	27.7364	1.52453	0.762266	+	0.647264i	$0.224087\pi$
$332$	0	0
$333$	−29.1724	−1.59864
$334$	0	0
$335$	−5.54012	−0.302689
$336$	0	0
$337$	−18.1109	−0.986562	−0.493281	−	0.869870i	$-0.664203\pi$
$337$	−18.1109	−0.986562	−0.493281	+	0.869870i	$0.664203\pi$
$338$	0	0
$339$	−7.44775	−0.404506
$340$	0	0
$341$	−10.5327	−0.570377
$342$	0	0
$343$	18.6713	1.00815
$344$	0	0
$345$	1.10020	0.0592329
$346$	0	0
$347$	23.0603	1.23794	0.618970	−	0.785414i	$-0.287550\pi$
$347$	23.0603	1.23794	0.618970	+	0.785414i	$0.287550\pi$
$348$	0	0
$349$	27.6973	1.48260	0.741302	−	0.671171i	$-0.234209\pi$
$349$	27.6973	1.48260	0.741302	+	0.671171i	$0.234209\pi$
$350$	0	0
$351$	−6.89986	−0.368287
$352$	0	0
$353$	−4.18928	−0.222973	−0.111486	−	0.993766i	$-0.535561\pi$
$353$	−4.18928	−0.222973	−0.111486	+	0.993766i	$0.535561\pi$
$354$	0	0
$355$	2.99609	0.159016
$356$	0	0
$357$	12.9976	0.687903
$358$	0	0
$359$	9.48831	0.500774	0.250387	−	0.968146i	$-0.419442\pi$
$359$	9.48831	0.500774	0.250387	+	0.968146i	$0.419442\pi$
$360$	0	0
$361$	−15.4327	−0.812247
$362$	0	0
$363$	10.1547	0.532986
$364$	0	0
$365$	−1.44371	−0.0755673
$366$	0	0
$367$	2.97290	0.155184	0.0775921	−	0.996985i	$-0.475277\pi$
$367$	2.97290	0.155184	0.0775921	+	0.996985i	$0.475277\pi$
$368$	0	0
$369$	5.04560	0.262663
$370$	0	0
$371$	15.7662	0.818542
$372$	0	0
$373$	11.8018	0.611072	0.305536	−	0.952181i	$-0.401164\pi$
$373$	11.8018	0.611072	0.305536	+	0.952181i	$0.401164\pi$
$374$	0	0
$375$	2.79329	0.144245
$376$	0	0
$377$	−6.29424	−0.324170
$378$	0	0
$379$	21.6094	1.11000	0.554999	−	0.831851i	$-0.312719\pi$
$379$	21.6094	1.11000	0.554999	+	0.831851i	$0.312719\pi$
$380$	0	0
$381$	−34.0001	−1.74188
$382$	0	0
$383$	−2.42878	−0.124105	−0.0620524	−	0.998073i	$-0.519765\pi$
$383$	−2.42878	−0.124105	−0.0620524	+	0.998073i	$0.519765\pi$
$384$	0	0
$385$	−16.3913	−0.835376
$386$	0	0
$387$	35.0206	1.78019
$388$	0	0
$389$	−20.3063	−1.02957	−0.514786	−	0.857319i	$-0.672129\pi$
$389$	−20.3063	−1.02957	−0.514786	+	0.857319i	$0.672129\pi$
$390$	0	0
$391$	0.427753	0.0216324
$392$	0	0
$393$	−9.61564	−0.485045
$394$	0	0
$395$	−6.78720	−0.341501
$396$	0	0
$397$	31.8304	1.59752	0.798761	−	0.601648i	$-0.205489\pi$
$397$	31.8304	1.59752	0.798761	+	0.601648i	$0.205489\pi$
$398$	0	0
$399$	22.6045	1.13164
$400$	0	0
$401$	−7.11145	−0.355129	−0.177565	−	0.984109i	$-0.556822\pi$
$401$	−7.11145	−0.355129	−0.177565	+	0.984109i	$0.556822\pi$
$402$	0	0
$403$	−3.77309	−0.187951
$404$	0	0
$405$	−0.343794	−0.0170833
$406$	0	0
$407$	23.2387	1.15190
$408$	0	0
$409$	−0.549251	−0.0271587	−0.0135794	−	0.999908i	$-0.504323\pi$
$409$	−0.549251	−0.0271587	−0.0135794	+	0.999908i	$0.504323\pi$
$410$	0	0
$411$	−21.0433	−1.03799
$412$	0	0
$413$	38.0694	1.87327
$414$	0	0
$415$	−2.39181	−0.117409
$416$	0	0
$417$	−4.68965	−0.229653
$418$	0	0
$419$	−21.5140	−1.05103	−0.525513	−	0.850786i	$-0.676126\pi$
$419$	−21.5140	−1.05103	−0.525513	+	0.850786i	$0.676126\pi$
$420$	0	0
$421$	9.07932	0.442499	0.221250	−	0.975217i	$-0.428986\pi$
$421$	9.07932	0.442499	0.221250	+	0.975217i	$0.428986\pi$
$422$	0	0
$423$	−9.50509	−0.462153
$424$	0	0
$425$	1.08602	0.0526795
$426$	0	0
$427$	−22.4918	−1.08845
$428$	0	0
$429$	14.6446	0.707050
$430$	0	0
$431$	33.8556	1.63077	0.815384	−	0.578920i	$-0.196526\pi$
$431$	33.8556	1.63077	0.815384	+	0.578920i	$0.196526\pi$
$432$	0	0
$433$	−19.1767	−0.921573	−0.460787	−	0.887511i	$-0.652433\pi$
$433$	−19.1767	−0.921573	−0.460787	+	0.887511i	$0.652433\pi$
$434$	0	0
$435$	12.8292	0.615111
$436$	0	0
$437$	0.743921	0.0355866
$438$	0	0
$439$	−18.5588	−0.885764	−0.442882	−	0.896580i	$-0.646044\pi$
$439$	−18.5588	−0.885764	−0.442882	+	0.896580i	$0.646044\pi$
$440$	0	0
$441$	54.5451	2.59739
$442$	0	0
$443$	23.1586	1.10030	0.550149	−	0.835067i	$-0.314571\pi$
$443$	23.1586	1.10030	0.550149	+	0.835067i	$0.314571\pi$
$444$	0	0
$445$	14.5194	0.688286
$446$	0	0
$447$	−27.8805	−1.31870
$448$	0	0
$449$	−13.8047	−0.651483	−0.325741	−	0.945459i	$-0.605614\pi$
$449$	−13.8047	−0.651483	−0.325741	+	0.945459i	$0.605614\pi$
$450$	0	0
$451$	−4.01931	−0.189262
$452$	0	0
$453$	−2.79329	−0.131240
$454$	0	0
$455$	−5.87179	−0.275274
$456$	0	0
$457$	−24.5918	−1.15036	−0.575179	−	0.818028i	$-0.695068\pi$
$457$	−24.5918	−1.15036	−0.575179	+	0.818028i	$0.695068\pi$
$458$	0	0
$459$	5.46784	0.255217
$460$	0	0
$461$	22.9952	1.07099	0.535497	−	0.844537i	$-0.320124\pi$
$461$	22.9952	1.07099	0.535497	+	0.844537i	$0.320124\pi$
$462$	0	0
$463$	34.9723	1.62530	0.812651	−	0.582750i	$-0.198023\pi$
$463$	34.9723	1.62530	0.812651	+	0.582750i	$0.198023\pi$
$464$	0	0
$465$	7.69046	0.356637
$466$	0	0
$467$	−21.9556	−1.01599	−0.507993	−	0.861361i	$-0.669612\pi$
$467$	−21.9556	−1.01599	−0.507993	+	0.861361i	$0.669612\pi$
$468$	0	0
$469$	−23.7372	−1.09608
$470$	0	0
$471$	39.1892	1.80574
$472$	0	0
$473$	−27.8973	−1.28272
$474$	0	0
$475$	1.88873	0.0866609
$476$	0	0
$477$	17.6718	0.809137
$478$	0	0
$479$	7.03665	0.321513	0.160756	−	0.986994i	$-0.448607\pi$
$479$	7.03665	0.321513	0.160756	+	0.986994i	$0.448607\pi$
$480$	0	0
$481$	8.32471	0.379574
$482$	0	0
$483$	4.71392	0.214491
$484$	0	0
$485$	1.16658	0.0529719
$486$	0	0
$487$	−27.2465	−1.23466	−0.617329	−	0.786705i	$-0.711785\pi$
$487$	−27.2465	−1.23466	−0.617329	+	0.786705i	$0.711785\pi$
$488$	0	0
$489$	−11.5010	−0.520093
$490$	0	0
$491$	38.7294	1.74783	0.873916	−	0.486077i	$-0.161572\pi$
$491$	38.7294	1.74783	0.873916	+	0.486077i	$0.161572\pi$
$492$	0	0
$493$	4.98791	0.224644
$494$	0	0
$495$	−18.3724	−0.825778
$496$	0	0
$497$	12.8370	0.575819
$498$	0	0
$499$	−37.0427	−1.65826	−0.829131	−	0.559055i	$-0.811164\pi$
$499$	−37.0427	−1.65826	−0.829131	+	0.559055i	$0.811164\pi$
$500$	0	0
$501$	15.4195	0.688891
$502$	0	0
$503$	18.5535	0.827258	0.413629	−	0.910445i	$-0.364261\pi$
$503$	18.5535	0.827258	0.413629	+	0.910445i	$0.364261\pi$
$504$	0	0
$505$	12.3543	0.549759
$506$	0	0
$507$	−31.0666	−1.37972
$508$	0	0
$509$	−8.48624	−0.376146	−0.188073	−	0.982155i	$-0.560224\pi$
$509$	−8.48624	−0.376146	−0.188073	+	0.982155i	$0.560224\pi$
$510$	0	0
$511$	−6.18572	−0.273640
$512$	0	0
$513$	9.50933	0.419847
$514$	0	0
$515$	3.63180	0.160036
$516$	0	0
$517$	7.57174	0.333005
$518$	0	0
$519$	39.2504	1.72290
$520$	0	0
$521$	−12.1693	−0.533146	−0.266573	−	0.963815i	$-0.585891\pi$
$521$	−12.1693	−0.533146	−0.266573	+	0.963815i	$0.585891\pi$
$522$	0	0
$523$	−30.9551	−1.35357	−0.676787	−	0.736179i	$-0.736628\pi$
$523$	−30.9551	−1.35357	−0.676787	+	0.736179i	$0.736628\pi$
$524$	0	0
$525$	11.9681	0.522331
$526$	0	0
$527$	2.99001	0.130247
$528$	0	0
$529$	−22.8449	−0.993255
$530$	0	0
$531$	42.6707	1.85175
$532$	0	0
$533$	−1.43983	−0.0623658
$534$	0	0
$535$	7.39447	0.319691
$536$	0	0
$537$	−11.1541	−0.481335
$538$	0	0
$539$	−43.4506	−1.87155
$540$	0	0
$541$	0.376046	0.0161675	0.00808373	−	0.999967i	$-0.497427\pi$
$541$	0.376046	0.0161675	0.00808373	+	0.999967i	$0.497427\pi$
$542$	0	0
$543$	12.5826	0.539970
$544$	0	0
$545$	8.29703	0.355406
$546$	0	0
$547$	−32.1788	−1.37587	−0.687933	−	0.725774i	$-0.741482\pi$
$547$	−32.1788	−1.37587	−0.687933	+	0.725774i	$0.741482\pi$
$548$	0	0
$549$	−25.2103	−1.07595
$550$	0	0
$551$	8.67467	0.369553
$552$	0	0
$553$	−29.0804	−1.23662
$554$	0	0
$555$	−16.9678	−0.720241
$556$	0	0
$557$	−5.63729	−0.238859	−0.119430	−	0.992843i	$-0.538107\pi$
$557$	−5.63729	−0.238859	−0.119430	+	0.992843i	$0.538107\pi$
$558$	0	0
$559$	−9.99356	−0.422683
$560$	0	0
$561$	−11.6052	−0.489974
$562$	0	0
$563$	33.0141	1.39138	0.695690	−	0.718342i	$-0.255099\pi$
$563$	33.0141	1.39138	0.695690	+	0.718342i	$0.255099\pi$
$564$	0	0
$565$	−2.66630	−0.112172
$566$	0	0
$567$	−1.47302	−0.0618609
$568$	0	0
$569$	−5.26665	−0.220790	−0.110395	−	0.993888i	$-0.535212\pi$
$569$	−5.26665	−0.220790	−0.110395	+	0.993888i	$0.535212\pi$
$570$	0	0
$571$	−3.11004	−0.130151	−0.0650756	−	0.997880i	$-0.520729\pi$
$571$	−3.11004	−0.130151	−0.0650756	+	0.997880i	$0.520729\pi$
$572$	0	0
$573$	35.4168	1.47956
$574$	0	0
$575$	0.393873	0.0164257
$576$	0	0
$577$	37.5752	1.56428	0.782138	−	0.623106i	$-0.214129\pi$
$577$	37.5752	1.56428	0.782138	+	0.623106i	$0.214129\pi$
$578$	0	0
$579$	−37.9324	−1.57642
$580$	0	0
$581$	−10.2479	−0.425157
$582$	0	0
$583$	−14.0773	−0.583024
$584$	0	0
$585$	−6.58148	−0.272111
$586$	0	0
$587$	−28.8132	−1.18925	−0.594624	−	0.804004i	$-0.702699\pi$
$587$	−28.8132	−1.18925	−0.594624	+	0.804004i	$0.702699\pi$
$588$	0	0
$589$	5.20004	0.214264
$590$	0	0
$591$	−21.1171	−0.868641
$592$	0	0
$593$	−0.698635	−0.0286895	−0.0143448	−	0.999897i	$-0.504566\pi$
$593$	−0.698635	−0.0286895	−0.0143448	+	0.999897i	$0.504566\pi$
$594$	0	0
$595$	4.65314	0.190760
$596$	0	0
$597$	−60.4170	−2.47270
$598$	0	0
$599$	−27.3220	−1.11634	−0.558172	−	0.829725i	$-0.688497\pi$
$599$	−27.3220	−1.11634	−0.558172	+	0.829725i	$0.688497\pi$
$600$	0	0
$601$	−27.7380	−1.13146	−0.565728	−	0.824592i	$-0.691405\pi$
$601$	−27.7380	−1.13146	−0.565728	+	0.824592i	$0.691405\pi$
$602$	0	0
$603$	−26.6062	−1.08349
$604$	0	0
$605$	3.63541	0.147800
$606$	0	0
$607$	45.2664	1.83731	0.918654	−	0.395063i	$-0.129277\pi$
$607$	45.2664	1.83731	0.918654	+	0.395063i	$0.129277\pi$
$608$	0	0
$609$	54.9678	2.22741
$610$	0	0
$611$	2.71240	0.109732
$612$	0	0
$613$	13.7748	0.556359	0.278180	−	0.960529i	$-0.410269\pi$
$613$	13.7748	0.556359	0.278180	+	0.960529i	$0.410269\pi$
$614$	0	0
$615$	2.93471	0.118339
$616$	0	0
$617$	−15.6805	−0.631272	−0.315636	−	0.948880i	$-0.602218\pi$
$617$	−15.6805	−0.631272	−0.315636	+	0.948880i	$0.602218\pi$
$618$	0	0
$619$	10.9402	0.439726	0.219863	−	0.975531i	$-0.429439\pi$
$619$	10.9402	0.439726	0.219863	+	0.975531i	$0.429439\pi$
$620$	0	0
$621$	1.98306	0.0795776
$622$	0	0
$623$	62.2098	2.49238
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−20.1831	−0.806036
$628$	0	0
$629$	−6.59698	−0.263039
$630$	0	0
$631$	44.4939	1.77128	0.885638	−	0.464376i	$-0.153721\pi$
$631$	44.4939	1.77128	0.885638	+	0.464376i	$0.153721\pi$
$632$	0	0
$633$	−59.2117	−2.35346
$634$	0	0
$635$	−12.1721	−0.483034
$636$	0	0
$637$	−15.5651	−0.616714
$638$	0	0
$639$	14.3886	0.569203
$640$	0	0
$641$	−1.81871	−0.0718346	−0.0359173	−	0.999355i	$-0.511435\pi$
$641$	−1.81871	−0.0718346	−0.0359173	+	0.999355i	$0.511435\pi$
$642$	0	0
$643$	−11.0013	−0.433848	−0.216924	−	0.976188i	$-0.569602\pi$
$643$	−11.0013	−0.433848	−0.216924	+	0.976188i	$0.569602\pi$
$644$	0	0
$645$	20.3693	0.802039
$646$	0	0
$647$	48.2576	1.89720	0.948601	−	0.316474i	$-0.102499\pi$
$647$	48.2576	1.89720	0.948601	+	0.316474i	$0.102499\pi$
$648$	0	0
$649$	−33.9914	−1.33428
$650$	0	0
$651$	32.9505	1.29143
$652$	0	0
$653$	4.25101	0.166355	0.0831774	−	0.996535i	$-0.473493\pi$
$653$	4.25101	0.166355	0.0831774	+	0.996535i	$0.473493\pi$
$654$	0	0
$655$	−3.44241	−0.134506
$656$	0	0
$657$	−6.93336	−0.270496
$658$	0	0
$659$	−18.3500	−0.714815	−0.357407	−	0.933949i	$-0.616339\pi$
$659$	−18.3500	−0.714815	−0.357407	+	0.933949i	$0.616339\pi$
$660$	0	0
$661$	−0.304952	−0.0118613	−0.00593064	−	0.999982i	$-0.501888\pi$
$661$	−0.304952	−0.0118613	−0.00593064	+	0.999982i	$0.501888\pi$
$662$	0	0
$663$	−4.15731	−0.161456
$664$	0	0
$665$	8.09245	0.313812
$666$	0	0
$667$	1.80900	0.0700449
$668$	0	0
$669$	−55.8214	−2.15818
$670$	0	0
$671$	20.0824	0.775274
$672$	0	0
$673$	−36.3615	−1.40163	−0.700816	−	0.713342i	$-0.747181\pi$
$673$	−36.3615	−1.40163	−0.700816	+	0.713342i	$0.747181\pi$
$674$	0	0
$675$	5.03477	0.193789
$676$	0	0
$677$	1.87705	0.0721407	0.0360704	−	0.999349i	$-0.488516\pi$
$677$	1.87705	0.0721407	0.0360704	+	0.999349i	$0.488516\pi$
$678$	0	0
$679$	4.99834	0.191819
$680$	0	0
$681$	42.3972	1.62467
$682$	0	0
$683$	−9.03613	−0.345758	−0.172879	−	0.984943i	$-0.555307\pi$
$683$	−9.03613	−0.345758	−0.172879	+	0.984943i	$0.555307\pi$
$684$	0	0
$685$	−7.53354	−0.287842
$686$	0	0
$687$	1.00656	0.0384026
$688$	0	0
$689$	−5.04288	−0.192118
$690$	0	0
$691$	49.9605	1.90059	0.950293	−	0.311356i	$-0.100783\pi$
$691$	49.9605	1.90059	0.950293	+	0.311356i	$0.100783\pi$
$692$	0	0
$693$	−78.7183	−2.99026
$694$	0	0
$695$	−1.67890	−0.0636844
$696$	0	0
$697$	1.14100	0.0432184
$698$	0	0
$699$	−20.7285	−0.784025
$700$	0	0
$701$	3.38965	0.128025	0.0640126	−	0.997949i	$-0.479610\pi$
$701$	3.38965	0.128025	0.0640126	+	0.997949i	$0.479610\pi$
$702$	0	0
$703$	−11.4731	−0.432714
$704$	0	0
$705$	−5.52852	−0.208216
$706$	0	0
$707$	52.9332	1.99076
$708$	0	0
$709$	−3.26379	−0.122574	−0.0612871	−	0.998120i	$-0.519521\pi$
$709$	−3.26379	−0.122574	−0.0612871	+	0.998120i	$0.519521\pi$
$710$	0	0
$711$	−32.5952	−1.22242
$712$	0	0
$713$	1.08441	0.0406115
$714$	0	0
$715$	5.24280	0.196069
$716$	0	0
$717$	−52.1447	−1.94738
$718$	0	0
$719$	11.7467	0.438076	0.219038	−	0.975716i	$-0.429708\pi$
$719$	11.7467	0.438076	0.219038	+	0.975716i	$0.429708\pi$
$720$	0	0
$721$	15.5608	0.579515
$722$	0	0
$723$	−66.0711	−2.45721
$724$	0	0
$725$	4.59286	0.170574
$726$	0	0
$727$	46.2021	1.71354	0.856770	−	0.515699i	$-0.172468\pi$
$727$	46.2021	1.71354	0.856770	+	0.515699i	$0.172468\pi$
$728$	0	0
$729$	−43.8418	−1.62377
$730$	0	0
$731$	7.91946	0.292912
$732$	0	0
$733$	−17.8735	−0.660173	−0.330087	−	0.943951i	$-0.607078\pi$
$733$	−17.8735	−0.660173	−0.330087	+	0.943951i	$0.607078\pi$
$734$	0	0
$735$	31.7255	1.17021
$736$	0	0
$737$	21.1944	0.780706
$738$	0	0
$739$	−10.0028	−0.367960	−0.183980	−	0.982930i	$-0.558898\pi$
$739$	−10.0028	−0.367960	−0.183980	+	0.982930i	$0.558898\pi$
$740$	0	0
$741$	−7.23013	−0.265605
$742$	0	0
$743$	39.4709	1.44805	0.724024	−	0.689775i	$-0.242291\pi$
$743$	39.4709	1.44805	0.724024	+	0.689775i	$0.242291\pi$
$744$	0	0
$745$	−9.98124	−0.365684
$746$	0	0
$747$	−11.4866	−0.420272
$748$	0	0
$749$	31.6823	1.15765
$750$	0	0
$751$	12.8824	0.470087	0.235043	−	0.971985i	$-0.424477\pi$
$751$	12.8824	0.470087	0.235043	+	0.971985i	$0.424477\pi$
$752$	0	0
$753$	−13.9433	−0.508122
$754$	0	0
$755$	−1.00000	−0.0363937
$756$	0	0
$757$	30.1419	1.09553	0.547764	−	0.836633i	$-0.315479\pi$
$757$	30.1419	1.09553	0.547764	+	0.836633i	$0.315479\pi$
$758$	0	0
$759$	−4.20896	−0.152776
$760$	0	0
$761$	−14.7219	−0.533668	−0.266834	−	0.963742i	$-0.585978\pi$
$761$	−14.7219	−0.533668	−0.266834	+	0.963742i	$0.585978\pi$
$762$	0	0
$763$	35.5494	1.28698
$764$	0	0
$765$	5.21554	0.188568
$766$	0	0
$767$	−12.1766	−0.439672
$768$	0	0
$769$	3.93316	0.141833	0.0709167	−	0.997482i	$-0.477408\pi$
$769$	3.93316	0.141833	0.0709167	+	0.997482i	$0.477408\pi$
$770$	0	0
$771$	14.8842	0.536041
$772$	0	0
$773$	−30.5494	−1.09879	−0.549393	−	0.835564i	$-0.685141\pi$
$773$	−30.5494	−1.09879	−0.549393	+	0.835564i	$0.685141\pi$
$774$	0	0
$775$	2.75319	0.0988976
$776$	0	0
$777$	−72.7000	−2.60810
$778$	0	0
$779$	1.98436	0.0710969
$780$	0	0
$781$	−11.4619	−0.410139
$782$	0	0
$783$	23.1240	0.826384
$784$	0	0
$785$	14.0298	0.500744
$786$	0	0
$787$	−21.2146	−0.756220	−0.378110	−	0.925761i	$-0.623426\pi$
$787$	−21.2146	−0.756220	−0.378110	+	0.925761i	$0.623426\pi$
$788$	0	0
$789$	−52.2897	−1.86156
$790$	0	0
$791$	−11.4240	−0.406192
$792$	0	0
$793$	7.19407	0.255469
$794$	0	0
$795$	10.2786	0.364544
$796$	0	0
$797$	−47.7176	−1.69025	−0.845123	−	0.534572i	$-0.820473\pi$
$797$	−47.7176	−1.69025	−0.845123	+	0.534572i	$0.820473\pi$
$798$	0	0
$799$	−2.14946	−0.0760424
$800$	0	0
$801$	69.7288	2.46375
$802$	0	0
$803$	5.52310	0.194906
$804$	0	0
$805$	1.68759	0.0594797
$806$	0	0
$807$	−19.0438	−0.670374
$808$	0	0
$809$	−24.3666	−0.856686	−0.428343	−	0.903616i	$-0.640902\pi$
$809$	−24.3666	−0.856686	−0.428343	+	0.903616i	$0.640902\pi$
$810$	0	0
$811$	22.8372	0.801922	0.400961	−	0.916095i	$-0.368676\pi$
$811$	22.8372	0.801922	0.400961	+	0.916095i	$0.368676\pi$
$812$	0	0
$813$	−80.4695	−2.82219
$814$	0	0
$815$	−4.11737	−0.144225
$816$	0	0
$817$	13.7730	0.481858
$818$	0	0
$819$	−28.1990	−0.985353
$820$	0	0
$821$	−16.8237	−0.587151	−0.293576	−	0.955936i	$-0.594845\pi$
$821$	−16.8237	−0.587151	−0.293576	+	0.955936i	$0.594845\pi$
$822$	0	0
$823$	−20.9510	−0.730307	−0.365154	−	0.930947i	$-0.618984\pi$
$823$	−20.9510	−0.730307	−0.365154	+	0.930947i	$0.618984\pi$
$824$	0	0
$825$	−10.6861	−0.372041
$826$	0	0
$827$	−23.2061	−0.806955	−0.403478	−	0.914990i	$-0.632199\pi$
$827$	−23.2061	−0.806955	−0.403478	+	0.914990i	$0.632199\pi$
$828$	0	0
$829$	1.13628	0.0394645	0.0197323	−	0.999805i	$-0.493719\pi$
$829$	1.13628	0.0394645	0.0197323	+	0.999805i	$0.493719\pi$
$830$	0	0
$831$	−42.3443	−1.46891
$832$	0	0
$833$	12.3347	0.427372
$834$	0	0
$835$	5.52019	0.191034
$836$	0	0
$837$	13.8617	0.479131
$838$	0	0
$839$	−46.3691	−1.60084	−0.800420	−	0.599440i	$-0.795390\pi$
$839$	−46.3691	−1.60084	−0.800420	+	0.599440i	$0.795390\pi$
$840$	0	0
$841$	−7.90568	−0.272610
$842$	0	0
$843$	15.6060	0.537501
$844$	0	0
$845$	−11.1219	−0.382605
$846$	0	0
$847$	15.5763	0.535207
$848$	0	0
$849$	−6.70328	−0.230056
$850$	0	0
$851$	−2.39257	−0.0820164
$852$	0	0
$853$	−24.7957	−0.848990	−0.424495	−	0.905430i	$-0.639548\pi$
$853$	−24.7957	−0.848990	−0.424495	+	0.905430i	$0.639548\pi$
$854$	0	0
$855$	9.07054	0.310206
$856$	0	0
$857$	6.22220	0.212546	0.106273	−	0.994337i	$-0.466108\pi$
$857$	6.22220	0.212546	0.106273	+	0.994337i	$0.466108\pi$
$858$	0	0
$859$	0.577367	0.0196995	0.00984975	−	0.999951i	$-0.496865\pi$
$859$	0.577367	0.0196995	0.00984975	+	0.999951i	$0.496865\pi$
$860$	0	0
$861$	12.5740	0.428522
$862$	0	0
$863$	−13.4425	−0.457590	−0.228795	−	0.973475i	$-0.573478\pi$
$863$	−13.4425	−0.457590	−0.228795	+	0.973475i	$0.573478\pi$
$864$	0	0
$865$	14.0517	0.477772
$866$	0	0
$867$	−44.1914	−1.50082
$868$	0	0
$869$	25.9653	0.880812
$870$	0	0
$871$	7.59240	0.257259
$872$	0	0
$873$	5.60247	0.189615
$874$	0	0
$875$	4.28460	0.144846
$876$	0	0
$877$	3.92597	0.132571	0.0662853	−	0.997801i	$-0.478885\pi$
$877$	3.92597	0.132571	0.0662853	+	0.997801i	$0.478885\pi$
$878$	0	0
$879$	−13.8251	−0.466308
$880$	0	0
$881$	−5.95386	−0.200591	−0.100295	−	0.994958i	$-0.531979\pi$
$881$	−5.95386	−0.200591	−0.100295	+	0.994958i	$0.531979\pi$
$882$	0	0
$883$	−3.30158	−0.111107	−0.0555535	−	0.998456i	$-0.517692\pi$
$883$	−3.30158	−0.111107	−0.0555535	+	0.998456i	$0.517692\pi$
$884$	0	0
$885$	24.8189	0.834277
$886$	0	0
$887$	32.0294	1.07544	0.537721	−	0.843123i	$-0.319285\pi$
$887$	32.0294	1.07544	0.537721	+	0.843123i	$0.319285\pi$
$888$	0	0
$889$	−52.1524	−1.74913
$890$	0	0
$891$	1.31523	0.0440617
$892$	0	0
$893$	−3.73821	−0.125094
$894$	0	0
$895$	−3.99318	−0.133477
$896$	0	0
$897$	−1.50776	−0.0503427
$898$	0	0
$899$	12.6450	0.421735
$900$	0	0
$901$	3.99626	0.133135
$902$	0	0
$903$	87.2741	2.90430
$904$	0	0
$905$	4.50458	0.149737
$906$	0	0
$907$	25.0625	0.832187	0.416094	−	0.909322i	$-0.363399\pi$
$907$	25.0625	0.832187	0.416094	+	0.909322i	$0.363399\pi$
$908$	0	0
$909$	59.3309	1.96788
$910$	0	0
$911$	26.1424	0.866137	0.433068	−	0.901361i	$-0.357431\pi$
$911$	26.1424	0.866137	0.433068	+	0.901361i	$0.357431\pi$
$912$	0	0
$913$	9.15018	0.302827
$914$	0	0
$915$	−14.6632	−0.484752
$916$	0	0
$917$	−14.7493	−0.487066
$918$	0	0
$919$	34.3311	1.13248	0.566240	−	0.824241i	$-0.308398\pi$
$919$	34.3311	1.13248	0.566240	+	0.824241i	$0.308398\pi$
$920$	0	0
$921$	33.6025	1.10724
$922$	0	0
$923$	−4.10596	−0.135149
$924$	0	0
$925$	−6.07448	−0.199728
$926$	0	0
$927$	17.4416	0.572856
$928$	0	0
$929$	12.4668	0.409022	0.204511	−	0.978864i	$-0.434440\pi$
$929$	12.4668	0.409022	0.204511	+	0.978864i	$0.434440\pi$
$930$	0	0
$931$	21.4518	0.703053
$932$	0	0
$933$	−16.5606	−0.542169
$934$	0	0
$935$	−4.15469	−0.135873
$936$	0	0
$937$	−12.1367	−0.396490	−0.198245	−	0.980153i	$-0.563524\pi$
$937$	−12.1367	−0.396490	−0.198245	+	0.980153i	$0.563524\pi$
$938$	0	0
$939$	−26.5737	−0.867199
$940$	0	0
$941$	−11.4344	−0.372750	−0.186375	−	0.982479i	$-0.559674\pi$
$941$	−11.4344	−0.372750	−0.186375	+	0.982479i	$0.559674\pi$
$942$	0	0
$943$	0.413815	0.0134757
$944$	0	0
$945$	21.5720	0.701736
$946$	0	0
$947$	16.3805	0.532295	0.266148	−	0.963932i	$-0.414249\pi$
$947$	16.3805	0.532295	0.266148	+	0.963932i	$0.414249\pi$
$948$	0	0
$949$	1.97852	0.0642256
$950$	0	0
$951$	6.23959	0.202332
$952$	0	0
$953$	−32.4075	−1.04978	−0.524892	−	0.851169i	$-0.675894\pi$
$953$	−32.4075	−1.04978	−0.524892	+	0.851169i	$0.675894\pi$
$954$	0	0
$955$	12.6793	0.410291
$956$	0	0
$957$	−49.0796	−1.58652
$958$	0	0
$959$	−32.2782	−1.04232
$960$	0	0
$961$	−23.4199	−0.755481
$962$	0	0
$963$	35.5116	1.14435
$964$	0	0
$965$	−13.5798	−0.437150
$966$	0	0
$967$	−8.78343	−0.282456	−0.141228	−	0.989977i	$-0.545105\pi$
$967$	−8.78343	−0.282456	−0.141228	+	0.989977i	$0.545105\pi$
$968$	0	0
$969$	5.72957	0.184060
$970$	0	0
$971$	41.1815	1.32158	0.660788	−	0.750572i	$-0.270222\pi$
$971$	41.1815	1.32158	0.660788	+	0.750572i	$0.270222\pi$
$972$	0	0
$973$	−7.19341	−0.230610
$974$	0	0
$975$	−3.82804	−0.122595
$976$	0	0
$977$	40.7652	1.30419	0.652097	−	0.758136i	$-0.273890\pi$
$977$	40.7652	1.30419	0.652097	+	0.758136i	$0.273890\pi$
$978$	0	0
$979$	−55.5458	−1.77525
$980$	0	0
$981$	39.8461	1.27219
$982$	0	0
$983$	50.8638	1.62230	0.811152	−	0.584835i	$-0.198841\pi$
$983$	50.8638	1.62230	0.811152	+	0.584835i	$0.198841\pi$
$984$	0	0
$985$	−7.55994	−0.240880
$986$	0	0
$987$	−23.6875	−0.753980
$988$	0	0
$989$	2.87221	0.0913310
$990$	0	0
$991$	4.27128	0.135682	0.0678409	−	0.997696i	$-0.478389\pi$
$991$	4.27128	0.135682	0.0678409	+	0.997696i	$0.478389\pi$
$992$	0	0
$993$	77.4758	2.45862
$994$	0	0
$995$	−21.6294	−0.685697
$996$	0	0
$997$	37.6303	1.19176	0.595881	−	0.803073i	$-0.296803\pi$
$997$	37.6303	1.19176	0.595881	+	0.803073i	$0.296803\pi$
$998$	0	0
$999$	−30.5836	−0.967623

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6040.2.a.o.1.13	✓	15

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6040.2.a.o.1.13	✓	15	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:	\( N \)	\(=\)	\( 6040 = 2^{3} \cdot 5 \cdot 151 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6040.a (trivial)

\(f(q)\)	\(=\)	\(q+2.79329 q^{3} +1.00000 q^{5} +4.28460 q^{7} +4.80245 q^{9} +O(q^{10})\)	\(q+2.79329 q^{3} +1.00000 q^{5} +4.28460 q^{7} +4.80245 q^{9} -3.82563 q^{11} -1.37044 q^{13} +2.79329 q^{15} +1.08602 q^{17} +1.88873 q^{19} +11.9681 q^{21} +0.393873 q^{23} +1.00000 q^{25} +5.03477 q^{27} +4.59286 q^{29} +2.75319 q^{31} -10.6861 q^{33} +4.28460 q^{35} -6.07448 q^{37} -3.82804 q^{39} +1.05063 q^{41} +7.29222 q^{43} +4.80245 q^{45} -1.97922 q^{47} +11.3578 q^{49} +3.03355 q^{51} +3.67975 q^{53} -3.82563 q^{55} +5.27577 q^{57} +8.88518 q^{59} -5.24945 q^{61} +20.5766 q^{63} -1.37044 q^{65} -5.54012 q^{67} +1.10020 q^{69} +2.99609 q^{71} -1.44371 q^{73} +2.79329 q^{75} -16.3913 q^{77} -6.78720 q^{79} -0.343794 q^{81} -2.39181 q^{83} +1.08602 q^{85} +12.8292 q^{87} +14.5194 q^{89} -5.87179 q^{91} +7.69046 q^{93} +1.88873 q^{95} +1.16658 q^{97} -18.3724 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 15 q + 5 q^{3} + 15 q^{5} + 7 q^{7} + 18 q^{9}+O(q^{10}) \) 15 * q + 5 * q^3 + 15 * q^5 + 7 * q^7 + 18 * q^9	\( 15 q + 5 q^{3} + 15 q^{5} + 7 q^{7} + 18 q^{9} + 7 q^{11} + 2 q^{13} + 5 q^{15} - 3 q^{17} + 8 q^{19} + 7 q^{21} + 15 q^{23} + 15 q^{25} + 23 q^{27} + 5 q^{29} + 27 q^{31} - 5 q^{33} + 7 q^{35} - 4 q^{37} + 11 q^{39} + 20 q^{41} + 25 q^{43} + 18 q^{45} + 35 q^{47} - 14 q^{49} + 25 q^{51} - 2 q^{53} + 7 q^{55} - 24 q^{57} + 39 q^{59} + 23 q^{61} + 39 q^{63} + 2 q^{65} + 32 q^{67} + 13 q^{69} + 30 q^{71} + 7 q^{73} + 5 q^{75} - 4 q^{77} + 38 q^{79} + 11 q^{81} + 29 q^{83} - 3 q^{85} + 4 q^{87} + 19 q^{89} + 16 q^{91} + 8 q^{93} + 8 q^{95} - 8 q^{97} + 16 q^{99}+O(q^{100}) \) 15 * q + 5 * q^3 + 15 * q^5 + 7 * q^7 + 18 * q^9 + 7 * q^11 + 2 * q^13 + 5 * q^15 - 3 * q^17 + 8 * q^19 + 7 * q^21 + 15 * q^23 + 15 * q^25 + 23 * q^27 + 5 * q^29 + 27 * q^31 - 5 * q^33 + 7 * q^35 - 4 * q^37 + 11 * q^39 + 20 * q^41 + 25 * q^43 + 18 * q^45 + 35 * q^47 - 14 * q^49 + 25 * q^51 - 2 * q^53 + 7 * q^55 - 24 * q^57 + 39 * q^59 + 23 * q^61 + 39 * q^63 + 2 * q^65 + 32 * q^67 + 13 * q^69 + 30 * q^71 + 7 * q^73 + 5 * q^75 - 4 * q^77 + 38 * q^79 + 11 * q^81 + 29 * q^83 - 3 * q^85 + 4 * q^87 + 19 * q^89 + 16 * q^91 + 8 * q^93 + 8 * q^95 - 8 * q^97 + 16 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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