LMFDB - Embedded newform 6040.2.a.n.1.6

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:	$1$
Dimension:	$13$
Coefficient field:	$\mathbb{Q}[x]/(x^{13} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{13} - 4 x^{12} - 14 x^{11} + 70 x^{10} + 41 x^{9} - 403 x^{8} + 109 x^{7} + 870 x^{6} - 444 x^{5} + \cdots + 11 $ x^13 - 4x^12 - 14x^11 + 70x^10 + 41x^9 - 403x^8 + 109x^7 + 870x^6 - 444x^5 - 708x^4 + 373x^3 + 108x^2 - 82x + 11
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			1.6
Root			$0.429184$ 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)

$f(q)$	$=$	$q-0.429184 q^{3} -1.00000 q^{5} -1.03124 q^{7} -2.81580 q^{9} +O(q^{10})$	$q-0.429184 q^{3} -1.00000 q^{5} -1.03124 q^{7} -2.81580 q^{9} +2.79610 q^{11} +4.59287 q^{13} +0.429184 q^{15} -2.11030 q^{17} -3.83172 q^{19} +0.442594 q^{21} -4.89655 q^{23} +1.00000 q^{25} +2.49605 q^{27} +0.574761 q^{29} +9.21400 q^{31} -1.20004 q^{33} +1.03124 q^{35} -4.56636 q^{37} -1.97119 q^{39} +9.36561 q^{41} +7.45905 q^{43} +2.81580 q^{45} -4.05026 q^{47} -5.93653 q^{49} +0.905709 q^{51} -5.14237 q^{53} -2.79610 q^{55} +1.64451 q^{57} -11.3129 q^{59} -0.885488 q^{61} +2.90378 q^{63} -4.59287 q^{65} +1.10450 q^{67} +2.10152 q^{69} +5.95731 q^{71} +3.87590 q^{73} -0.429184 q^{75} -2.88346 q^{77} +5.23466 q^{79} +7.37614 q^{81} -3.06196 q^{83} +2.11030 q^{85} -0.246679 q^{87} -4.05762 q^{89} -4.73637 q^{91} -3.95451 q^{93} +3.83172 q^{95} +7.56813 q^{97} -7.87325 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 13 q - 4 q^{3} - 13 q^{5} + 5 q^{9}+O(q^{10}) $ 13 * q - 4 * q^3 - 13 * q^5 + 5 * q^9	$ 13 q - 4 q^{3} - 13 q^{5} + 5 q^{9} - 14 q^{11} + 5 q^{13} + 4 q^{15} - 8 q^{17} + 16 q^{19} - 5 q^{21} - 4 q^{23} + 13 q^{25} + 2 q^{27} - 6 q^{29} + 11 q^{31} - 19 q^{33} + 6 q^{37} + 7 q^{39} - 18 q^{41} + 7 q^{43} - 5 q^{45} - 22 q^{47} - q^{49} + 12 q^{51} - 17 q^{53} + 14 q^{55} - 16 q^{57} - 6 q^{59} + 10 q^{61} - 5 q^{65} + 12 q^{67} + 13 q^{69} - 16 q^{71} - 24 q^{73} - 4 q^{75} - 11 q^{77} + 36 q^{79} - 19 q^{81} + q^{83} + 8 q^{85} - 8 q^{87} - 53 q^{89} + 23 q^{91} - 9 q^{93} - 16 q^{95} - 21 q^{97} - 9 q^{99}+O(q^{100}) $ 13 * q - 4 * q^3 - 13 * q^5 + 5 * q^9 - 14 * q^11 + 5 * q^13 + 4 * q^15 - 8 * q^17 + 16 * q^19 - 5 * q^21 - 4 * q^23 + 13 * q^25 + 2 * q^27 - 6 * q^29 + 11 * q^31 - 19 * q^33 + 6 * q^37 + 7 * q^39 - 18 * q^41 + 7 * q^43 - 5 * q^45 - 22 * q^47 - q^49 + 12 * q^51 - 17 * q^53 + 14 * q^55 - 16 * q^57 - 6 * q^59 + 10 * q^61 - 5 * q^65 + 12 * q^67 + 13 * q^69 - 16 * q^71 - 24 * q^73 - 4 * q^75 - 11 * q^77 + 36 * q^79 - 19 * q^81 + q^83 + 8 * q^85 - 8 * q^87 - 53 * q^89 + 23 * q^91 - 9 * q^93 - 16 * q^95 - 21 * q^97 - 9 * 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
$3$	−0.429184	−0.247790	−0.123895	−	0.992295i	$-0.539539\pi$
$3$	−0.429184	−0.247790	−0.123895	+	0.992295i	$0.539539\pi$
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−1.03124	−0.389774	−0.194887	−	0.980826i	$-0.562434\pi$
$7$	−1.03124	−0.389774	−0.194887	+	0.980826i	$0.562434\pi$
$8$	0	0
$9$	−2.81580	−0.938600
$10$	0	0
$11$	2.79610	0.843055	0.421528	−	0.906816i	$-0.361494\pi$
$11$	2.79610	0.843055	0.421528	+	0.906816i	$0.361494\pi$
$12$	0	0
$13$	4.59287	1.27383	0.636916	−	0.770933i	$-0.280210\pi$
$13$	4.59287	1.27383	0.636916	+	0.770933i	$0.280210\pi$
$14$	0	0
$15$	0.429184	0.110815
$16$	0	0
$17$	−2.11030	−0.511823	−0.255912	−	0.966700i	$-0.582376\pi$
$17$	−2.11030	−0.511823	−0.255912	+	0.966700i	$0.582376\pi$
$18$	0	0
$19$	−3.83172	−0.879057	−0.439528	−	0.898229i	$-0.644854\pi$
$19$	−3.83172	−0.879057	−0.439528	+	0.898229i	$0.644854\pi$
$20$	0	0
$21$	0.442594	0.0965819
$22$	0	0
$23$	−4.89655	−1.02100	−0.510501	−	0.859877i	$-0.670540\pi$
$23$	−4.89655	−1.02100	−0.510501	+	0.859877i	$0.670540\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	2.49605	0.480365
$28$	0	0
$29$	0.574761	0.106730	0.0533652	−	0.998575i	$-0.483005\pi$
$29$	0.574761	0.106730	0.0533652	+	0.998575i	$0.483005\pi$
$30$	0	0
$31$	9.21400	1.65488	0.827442	−	0.561552i	$-0.189795\pi$
$31$	9.21400	1.65488	0.827442	+	0.561552i	$0.189795\pi$
$32$	0	0
$33$	−1.20004	−0.208900
$34$	0	0
$35$	1.03124	0.174312
$36$	0	0
$37$	−4.56636	−0.750705	−0.375353	−	0.926882i	$-0.622478\pi$
$37$	−4.56636	−0.750705	−0.375353	+	0.926882i	$0.622478\pi$
$38$	0	0
$39$	−1.97119	−0.315642
$40$	0	0
$41$	9.36561	1.46266	0.731331	−	0.682023i	$-0.238900\pi$
$41$	9.36561	1.46266	0.731331	+	0.682023i	$0.238900\pi$
$42$	0	0
$43$	7.45905	1.13749	0.568747	−	0.822513i	$-0.307428\pi$
$43$	7.45905	1.13749	0.568747	+	0.822513i	$0.307428\pi$
$44$	0	0
$45$	2.81580	0.419755
$46$	0	0
$47$	−4.05026	−0.590791	−0.295395	−	0.955375i	$-0.595451\pi$
$47$	−4.05026	−0.590791	−0.295395	+	0.955375i	$0.595451\pi$
$48$	0	0
$49$	−5.93653	−0.848076
$50$	0	0
$51$	0.905709	0.126825
$52$	0	0
$53$	−5.14237	−0.706358	−0.353179	−	0.935556i	$-0.614899\pi$
$53$	−5.14237	−0.706358	−0.353179	+	0.935556i	$0.614899\pi$
$54$	0	0
$55$	−2.79610	−0.377026
$56$	0	0
$57$	1.64451	0.217821
$58$	0	0
$59$	−11.3129	−1.47281	−0.736407	−	0.676538i	$-0.763479\pi$
$59$	−11.3129	−1.47281	−0.736407	+	0.676538i	$0.763479\pi$
$60$	0	0
$61$	−0.885488	−0.113375	−0.0566875	−	0.998392i	$-0.518054\pi$
$61$	−0.885488	−0.113375	−0.0566875	+	0.998392i	$0.518054\pi$
$62$	0	0
$63$	2.90378	0.365842
$64$	0	0
$65$	−4.59287	−0.569675
$66$	0	0
$67$	1.10450	0.134936	0.0674679	−	0.997721i	$-0.478508\pi$
$67$	1.10450	0.134936	0.0674679	+	0.997721i	$0.478508\pi$
$68$	0	0
$69$	2.10152	0.252994
$70$	0	0
$71$	5.95731	0.707003	0.353501	−	0.935434i	$-0.384991\pi$
$71$	5.95731	0.707003	0.353501	+	0.935434i	$0.384991\pi$
$72$	0	0
$73$	3.87590	0.453639	0.226820	−	0.973937i	$-0.427167\pi$
$73$	3.87590	0.453639	0.226820	+	0.973937i	$0.427167\pi$
$74$	0	0
$75$	−0.429184	−0.0495579
$76$	0	0
$77$	−2.88346	−0.328601
$78$	0	0
$79$	5.23466	0.588945	0.294473	−	0.955660i	$-0.404856\pi$
$79$	5.23466	0.588945	0.294473	+	0.955660i	$0.404856\pi$
$80$	0	0
$81$	7.37614	0.819571
$82$	0	0
$83$	−3.06196	−0.336093	−0.168047	−	0.985779i	$-0.553746\pi$
$83$	−3.06196	−0.336093	−0.168047	+	0.985779i	$0.553746\pi$
$84$	0	0
$85$	2.11030	0.228894
$86$	0	0
$87$	−0.246679	−0.0264467
$88$	0	0
$89$	−4.05762	−0.430107	−0.215054	−	0.976602i	$-0.568993\pi$
$89$	−4.05762	−0.430107	−0.215054	+	0.976602i	$0.568993\pi$
$90$	0	0
$91$	−4.73637	−0.496506
$92$	0	0
$93$	−3.95451	−0.410063
$94$	0	0
$95$	3.83172	0.393126
$96$	0	0
$97$	7.56813	0.768428	0.384214	−	0.923244i	$-0.374473\pi$
$97$	7.56813	0.768428	0.384214	+	0.923244i	$0.374473\pi$
$98$	0	0
$99$	−7.87325	−0.791292
$100$	0	0
$101$	−13.6406	−1.35729	−0.678645	−	0.734466i	$-0.737432\pi$
$101$	−13.6406	−1.35729	−0.678645	+	0.734466i	$0.737432\pi$
$102$	0	0
$103$	−4.19356	−0.413204	−0.206602	−	0.978425i	$-0.566240\pi$
$103$	−4.19356	−0.413204	−0.206602	+	0.978425i	$0.566240\pi$
$104$	0	0
$105$	−0.442594	−0.0431928
$106$	0	0
$107$	1.84311	0.178180	0.0890899	−	0.996024i	$-0.471604\pi$
$107$	1.84311	0.178180	0.0890899	+	0.996024i	$0.471604\pi$
$108$	0	0
$109$	−0.586405	−0.0561674	−0.0280837	−	0.999606i	$-0.508940\pi$
$109$	−0.586405	−0.0561674	−0.0280837	+	0.999606i	$0.508940\pi$
$110$	0	0
$111$	1.95981	0.186017
$112$	0	0
$113$	−15.7693	−1.48345	−0.741725	−	0.670704i	$-0.765992\pi$
$113$	−15.7693	−1.48345	−0.741725	+	0.670704i	$0.765992\pi$
$114$	0	0
$115$	4.89655	0.456606
$116$	0	0
$117$	−12.9326	−1.19562
$118$	0	0
$119$	2.17624	0.199495
$120$	0	0
$121$	−3.18184	−0.289258
$122$	0	0
$123$	−4.01957	−0.362433
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−7.07800	−0.628071	−0.314036	−	0.949411i	$-0.601681\pi$
$127$	−7.07800	−0.628071	−0.314036	+	0.949411i	$0.601681\pi$
$128$	0	0
$129$	−3.20131	−0.281859
$130$	0	0
$131$	6.38010	0.557432	0.278716	−	0.960374i	$-0.410091\pi$
$131$	6.38010	0.557432	0.278716	+	0.960374i	$0.410091\pi$
$132$	0	0
$133$	3.95144	0.342633
$134$	0	0
$135$	−2.49605	−0.214826
$136$	0	0
$137$	11.3659	0.971052	0.485526	−	0.874222i	$-0.338628\pi$
$137$	11.3659	0.971052	0.485526	+	0.874222i	$0.338628\pi$
$138$	0	0
$139$	−11.5816	−0.982338	−0.491169	−	0.871064i	$-0.663430\pi$
$139$	−11.5816	−0.982338	−0.491169	+	0.871064i	$0.663430\pi$
$140$	0	0
$141$	1.73831	0.146392
$142$	0	0
$143$	12.8421	1.07391
$144$	0	0
$145$	−0.574761	−0.0477313
$146$	0	0
$147$	2.54787	0.210145
$148$	0	0
$149$	19.6750	1.61184	0.805918	−	0.592027i	$-0.201672\pi$
$149$	19.6750	1.61184	0.805918	+	0.592027i	$0.201672\pi$
$150$	0	0
$151$	1.00000	0.0813788
$151$	1.00000	0.0813788
$152$	0	0
$153$	5.94219	0.480398
$154$	0	0
$155$	−9.21400	−0.740086
$156$	0	0
$157$	−3.96139	−0.316153	−0.158077	−	0.987427i	$-0.550529\pi$
$157$	−3.96139	−0.316153	−0.158077	+	0.987427i	$0.550529\pi$
$158$	0	0
$159$	2.20702	0.175028
$160$	0	0
$161$	5.04954	0.397960
$162$	0	0
$163$	7.64465	0.598775	0.299388	−	0.954132i	$-0.403218\pi$
$163$	7.64465	0.598775	0.299388	+	0.954132i	$0.403218\pi$
$164$	0	0
$165$	1.20004	0.0934231
$166$	0	0
$167$	−5.21001	−0.403163	−0.201581	−	0.979472i	$-0.564608\pi$
$167$	−5.21001	−0.403163	−0.201581	+	0.979472i	$0.564608\pi$
$168$	0	0
$169$	8.09442	0.622648
$170$	0	0
$171$	10.7894	0.825083
$172$	0	0
$173$	−0.0242302	−0.00184219	−0.000921093	−	1.00000i	$-0.500293\pi$
$173$	−0.0242302	−0.00184219	−0.000921093		1.00000i	$0.500293\pi$
$174$	0	0
$175$	−1.03124	−0.0779548
$176$	0	0
$177$	4.85532	0.364948
$178$	0	0
$179$	−14.2659	−1.06628	−0.533142	−	0.846026i	$-0.678989\pi$
$179$	−14.2659	−1.06628	−0.533142	+	0.846026i	$0.678989\pi$
$180$	0	0
$181$	−14.5883	−1.08434	−0.542171	−	0.840268i	$-0.682397\pi$
$181$	−14.5883	−1.08434	−0.542171	+	0.840268i	$0.682397\pi$
$182$	0	0
$183$	0.380037	0.0280932
$184$	0	0
$185$	4.56636	0.335726
$186$	0	0
$187$	−5.90061	−0.431495
$188$	0	0
$189$	−2.57404	−0.187234
$190$	0	0
$191$	−7.55972	−0.547002	−0.273501	−	0.961872i	$-0.588182\pi$
$191$	−7.55972	−0.547002	−0.273501	+	0.961872i	$0.588182\pi$
$192$	0	0
$193$	−8.31247	−0.598344	−0.299172	−	0.954199i	$-0.596711\pi$
$193$	−8.31247	−0.598344	−0.299172	+	0.954199i	$0.596711\pi$
$194$	0	0
$195$	1.97119	0.141160
$196$	0	0
$197$	−16.9432	−1.20715	−0.603577	−	0.797305i	$-0.706258\pi$
$197$	−16.9432	−1.20715	−0.603577	+	0.797305i	$0.706258\pi$
$198$	0	0
$199$	−7.39238	−0.524032	−0.262016	−	0.965064i	$-0.584387\pi$
$199$	−7.39238	−0.524032	−0.262016	+	0.965064i	$0.584387\pi$
$200$	0	0
$201$	−0.474033	−0.0334357
$202$	0	0
$203$	−0.592719	−0.0416008
$204$	0	0
$205$	−9.36561	−0.654122
$206$	0	0
$207$	13.7877	0.958312
$208$	0	0
$209$	−10.7139	−0.741093
$210$	0	0
$211$	−19.8835	−1.36884	−0.684419	−	0.729089i	$-0.739944\pi$
$211$	−19.8835	−1.36884	−0.684419	+	0.729089i	$0.739944\pi$
$212$	0	0
$213$	−2.55678	−0.175188
$214$	0	0
$215$	−7.45905	−0.508703
$216$	0	0
$217$	−9.50189	−0.645030
$218$	0	0
$219$	−1.66347	−0.112407
$220$	0	0
$221$	−9.69234	−0.651977
$222$	0	0
$223$	1.15734	0.0775013	0.0387507	−	0.999249i	$-0.487662\pi$
$223$	1.15734	0.0775013	0.0387507	+	0.999249i	$0.487662\pi$
$224$	0	0
$225$	−2.81580	−0.187720
$226$	0	0
$227$	19.7196	1.30883	0.654416	−	0.756134i	$-0.272914\pi$
$227$	19.7196	1.30883	0.654416	+	0.756134i	$0.272914\pi$
$228$	0	0
$229$	−20.4133	−1.34895	−0.674474	−	0.738298i	$-0.735630\pi$
$229$	−20.4133	−1.34895	−0.674474	+	0.738298i	$0.735630\pi$
$230$	0	0
$231$	1.23754	0.0814239
$232$	0	0
$233$	−21.1845	−1.38785	−0.693923	−	0.720049i	$-0.744119\pi$
$233$	−21.1845	−1.38785	−0.693923	+	0.720049i	$0.744119\pi$
$234$	0	0
$235$	4.05026	0.264210
$236$	0	0
$237$	−2.24664	−0.145935
$238$	0	0
$239$	−3.69860	−0.239243	−0.119621	−	0.992820i	$-0.538168\pi$
$239$	−3.69860	−0.239243	−0.119621	+	0.992820i	$0.538168\pi$
$240$	0	0
$241$	−6.87556	−0.442894	−0.221447	−	0.975172i	$-0.571078\pi$
$241$	−6.87556	−0.442894	−0.221447	+	0.975172i	$0.571078\pi$
$242$	0	0
$243$	−10.6539	−0.683446
$244$	0	0
$245$	5.93653	0.379271
$246$	0	0
$247$	−17.5986	−1.11977
$248$	0	0
$249$	1.31414	0.0832805
$250$	0	0
$251$	−6.83979	−0.431724	−0.215862	−	0.976424i	$-0.569256\pi$
$251$	−6.83979	−0.431724	−0.215862	+	0.976424i	$0.569256\pi$
$252$	0	0
$253$	−13.6912	−0.860760
$254$	0	0
$255$	−0.905709	−0.0567177
$256$	0	0
$257$	−28.5620	−1.78165	−0.890825	−	0.454346i	$-0.849873\pi$
$257$	−28.5620	−1.78165	−0.890825	+	0.454346i	$0.849873\pi$
$258$	0	0
$259$	4.70904	0.292605
$260$	0	0
$261$	−1.61841	−0.100177
$262$	0	0
$263$	−15.3816	−0.948471	−0.474236	−	0.880398i	$-0.657275\pi$
$263$	−15.3816	−0.948471	−0.474236	+	0.880398i	$0.657275\pi$
$264$	0	0
$265$	5.14237	0.315893
$266$	0	0
$267$	1.74147	0.106576
$268$	0	0
$269$	−10.6434	−0.648940	−0.324470	−	0.945896i	$-0.605186\pi$
$269$	−10.6434	−0.648940	−0.324470	+	0.945896i	$0.605186\pi$
$270$	0	0
$271$	20.9931	1.27524	0.637620	−	0.770351i	$-0.279919\pi$
$271$	20.9931	1.27524	0.637620	+	0.770351i	$0.279919\pi$
$272$	0	0
$273$	2.03278	0.123029
$274$	0	0
$275$	2.79610	0.168611
$276$	0	0
$277$	12.4720	0.749368	0.374684	−	0.927153i	$-0.377751\pi$
$277$	12.4720	0.749368	0.374684	+	0.927153i	$0.377751\pi$
$278$	0	0
$279$	−25.9448	−1.55327
$280$	0	0
$281$	9.20543	0.549150	0.274575	−	0.961566i	$-0.411463\pi$
$281$	9.20543	0.549150	0.274575	+	0.961566i	$0.411463\pi$
$282$	0	0
$283$	18.1890	1.08123	0.540613	−	0.841271i	$-0.318192\pi$
$283$	18.1890	1.08123	0.540613	+	0.841271i	$0.318192\pi$
$284$	0	0
$285$	−1.64451	−0.0974126
$286$	0	0
$287$	−9.65823	−0.570107
$288$	0	0
$289$	−12.5466	−0.738037
$290$	0	0
$291$	−3.24812	−0.190408
$292$	0	0
$293$	−27.3687	−1.59890	−0.799449	−	0.600733i	$-0.794875\pi$
$293$	−27.3687	−1.59890	−0.799449	+	0.600733i	$0.794875\pi$
$294$	0	0
$295$	11.3129	0.658663
$296$	0	0
$297$	6.97920	0.404974
$298$	0	0
$299$	−22.4892	−1.30058
$300$	0	0
$301$	−7.69210	−0.443365
$302$	0	0
$303$	5.85433	0.336322
$304$	0	0
$305$	0.885488	0.0507029
$306$	0	0
$307$	4.72234	0.269518	0.134759	−	0.990878i	$-0.456974\pi$
$307$	4.72234	0.269518	0.134759	+	0.990878i	$0.456974\pi$
$308$	0	0
$309$	1.79981	0.102388
$310$	0	0
$311$	14.8011	0.839293	0.419647	−	0.907688i	$-0.362154\pi$
$311$	14.8011	0.839293	0.419647	+	0.907688i	$0.362154\pi$
$312$	0	0
$313$	−15.5477	−0.878805	−0.439403	−	0.898290i	$-0.644810\pi$
$313$	−15.5477	−0.878805	−0.439403	+	0.898290i	$0.644810\pi$
$314$	0	0
$315$	−2.90378	−0.163609
$316$	0	0
$317$	−26.0389	−1.46249	−0.731245	−	0.682115i	$-0.761060\pi$
$317$	−26.0389	−1.46249	−0.731245	+	0.682115i	$0.761060\pi$
$318$	0	0
$319$	1.60709	0.0899797
$320$	0	0
$321$	−0.791032	−0.0441511
$322$	0	0
$323$	8.08609	0.449922
$324$	0	0
$325$	4.59287	0.254766
$326$	0	0
$327$	0.251676	0.0139177
$328$	0	0
$329$	4.17681	0.230275
$330$	0	0
$331$	−5.98918	−0.329195	−0.164597	−	0.986361i	$-0.552633\pi$
$331$	−5.98918	−0.329195	−0.164597	+	0.986361i	$0.552633\pi$
$332$	0	0
$333$	12.8580	0.704612
$334$	0	0
$335$	−1.10450	−0.0603451
$336$	0	0
$337$	−10.0611	−0.548065	−0.274033	−	0.961720i	$-0.588358\pi$
$337$	−10.0611	−0.548065	−0.274033	+	0.961720i	$0.588358\pi$
$338$	0	0
$339$	6.76794	0.367584
$340$	0	0
$341$	25.7633	1.39516
$342$	0	0
$343$	13.3407	0.720332
$344$	0	0
$345$	−2.10152	−0.113142
$346$	0	0
$347$	12.7890	0.686548	0.343274	−	0.939235i	$-0.388464\pi$
$347$	12.7890	0.686548	0.343274	+	0.939235i	$0.388464\pi$
$348$	0	0
$349$	9.03064	0.483399	0.241700	−	0.970351i	$-0.422295\pi$
$349$	9.03064	0.483399	0.241700	+	0.970351i	$0.422295\pi$
$350$	0	0
$351$	11.4640	0.611905
$352$	0	0
$353$	−25.6431	−1.36484	−0.682422	−	0.730958i	$-0.739073\pi$
$353$	−25.6431	−1.36484	−0.682422	+	0.730958i	$0.739073\pi$
$354$	0	0
$355$	−5.95731	−0.316181
$356$	0	0
$357$	−0.934007	−0.0494329
$358$	0	0
$359$	15.9654	0.842624	0.421312	−	0.906916i	$-0.361570\pi$
$359$	15.9654	0.842624	0.421312	+	0.906916i	$0.361570\pi$
$360$	0	0
$361$	−4.31792	−0.227259
$362$	0	0
$363$	1.36559	0.0716751
$364$	0	0
$365$	−3.87590	−0.202874
$366$	0	0
$367$	−2.79887	−0.146100	−0.0730500	−	0.997328i	$-0.523273\pi$
$367$	−2.79887	−0.146100	−0.0730500	+	0.997328i	$0.523273\pi$
$368$	0	0
$369$	−26.3717	−1.37285
$370$	0	0
$371$	5.30304	0.275320
$372$	0	0
$373$	−14.5309	−0.752381	−0.376191	−	0.926542i	$-0.622766\pi$
$373$	−14.5309	−0.752381	−0.376191	+	0.926542i	$0.622766\pi$
$374$	0	0
$375$	0.429184	0.0221630
$376$	0	0
$377$	2.63980	0.135957
$378$	0	0
$379$	24.1407	1.24002	0.620012	−	0.784592i	$-0.287128\pi$
$379$	24.1407	1.24002	0.620012	+	0.784592i	$0.287128\pi$
$380$	0	0
$381$	3.03777	0.155630
$382$	0	0
$383$	−11.8810	−0.607089	−0.303544	−	0.952817i	$-0.598170\pi$
$383$	−11.8810	−0.607089	−0.303544	+	0.952817i	$0.598170\pi$
$384$	0	0
$385$	2.88346	0.146955
$386$	0	0
$387$	−21.0032	−1.06765
$388$	0	0
$389$	−0.836454	−0.0424099	−0.0212049	−	0.999775i	$-0.506750\pi$
$389$	−0.836454	−0.0424099	−0.0212049	+	0.999775i	$0.506750\pi$
$390$	0	0
$391$	10.3332	0.522572
$392$	0	0
$393$	−2.73824	−0.138126
$394$	0	0
$395$	−5.23466	−0.263384
$396$	0	0
$397$	−24.7669	−1.24302	−0.621508	−	0.783408i	$-0.713480\pi$
$397$	−24.7669	−1.24302	−0.621508	+	0.783408i	$0.713480\pi$
$398$	0	0
$399$	−1.69590	−0.0849010
$400$	0	0
$401$	−10.6066	−0.529670	−0.264835	−	0.964294i	$-0.585318\pi$
$401$	−10.6066	−0.529670	−0.264835	+	0.964294i	$0.585318\pi$
$402$	0	0
$403$	42.3187	2.10804
$404$	0	0
$405$	−7.37614	−0.366523
$406$	0	0
$407$	−12.7680	−0.632886
$408$	0	0
$409$	18.7737	0.928299	0.464149	−	0.885757i	$-0.346360\pi$
$409$	18.7737	0.928299	0.464149	+	0.885757i	$0.346360\pi$
$410$	0	0
$411$	−4.87806	−0.240617
$412$	0	0
$413$	11.6664	0.574065
$414$	0	0
$415$	3.06196	0.150306
$416$	0	0
$417$	4.97064	0.243413
$418$	0	0
$419$	21.3281	1.04195	0.520974	−	0.853573i	$-0.325569\pi$
$419$	21.3281	1.04195	0.520974	+	0.853573i	$0.325569\pi$
$420$	0	0
$421$	−9.51535	−0.463750	−0.231875	−	0.972746i	$-0.574486\pi$
$421$	−9.51535	−0.463750	−0.231875	+	0.972746i	$0.574486\pi$
$422$	0	0
$423$	11.4047	0.554516
$424$	0	0
$425$	−2.11030	−0.102365
$426$	0	0
$427$	0.913154	0.0441906
$428$	0	0
$429$	−5.51163	−0.266104
$430$	0	0
$431$	−11.6070	−0.559088	−0.279544	−	0.960133i	$-0.590183\pi$
$431$	−11.6070	−0.559088	−0.279544	+	0.960133i	$0.590183\pi$
$432$	0	0
$433$	−25.1647	−1.20934	−0.604670	−	0.796476i	$-0.706695\pi$
$433$	−25.1647	−1.20934	−0.604670	+	0.796476i	$0.706695\pi$
$434$	0	0
$435$	0.246679	0.0118273
$436$	0	0
$437$	18.7622	0.897518
$438$	0	0
$439$	0.617047	0.0294500	0.0147250	−	0.999892i	$-0.495313\pi$
$439$	0.617047	0.0294500	0.0147250	+	0.999892i	$0.495313\pi$
$440$	0	0
$441$	16.7161	0.796005
$442$	0	0
$443$	9.31727	0.442677	0.221339	−	0.975197i	$-0.428957\pi$
$443$	9.31727	0.442677	0.221339	+	0.975197i	$0.428957\pi$
$444$	0	0
$445$	4.05762	0.192350
$446$	0	0
$447$	−8.44419	−0.399396
$448$	0	0
$449$	11.9040	0.561787	0.280893	−	0.959739i	$-0.409369\pi$
$449$	11.9040	0.561787	0.280893	+	0.959739i	$0.409369\pi$
$450$	0	0
$451$	26.1871	1.23310
$452$	0	0
$453$	−0.429184	−0.0201648
$454$	0	0
$455$	4.73637	0.222044
$456$	0	0
$457$	−24.7123	−1.15599	−0.577996	−	0.816040i	$-0.696165\pi$
$457$	−24.7123	−1.15599	−0.577996	+	0.816040i	$0.696165\pi$
$458$	0	0
$459$	−5.26742	−0.245862
$460$	0	0
$461$	27.5606	1.28363	0.641814	−	0.766861i	$-0.278182\pi$
$461$	27.5606	1.28363	0.641814	+	0.766861i	$0.278182\pi$
$462$	0	0
$463$	8.07323	0.375195	0.187597	−	0.982246i	$-0.439930\pi$
$463$	8.07323	0.375195	0.187597	+	0.982246i	$0.439930\pi$
$464$	0	0
$465$	3.95451	0.183386
$466$	0	0
$467$	−28.2450	−1.30702	−0.653510	−	0.756918i	$-0.726704\pi$
$467$	−28.2450	−1.30702	−0.653510	+	0.756918i	$0.726704\pi$
$468$	0	0
$469$	−1.13901	−0.0525945
$470$	0	0
$471$	1.70017	0.0783395
$472$	0	0
$473$	20.8562	0.958970
$474$	0	0
$475$	−3.83172	−0.175811
$476$	0	0
$477$	14.4799	0.662988
$478$	0	0
$479$	41.9988	1.91898	0.959488	−	0.281750i	$-0.0909149\pi$
$479$	41.9988	1.91898	0.959488	+	0.281750i	$0.0909149\pi$
$480$	0	0
$481$	−20.9727	−0.956273
$482$	0	0
$483$	−2.16718	−0.0986103
$484$	0	0
$485$	−7.56813	−0.343651
$486$	0	0
$487$	28.2570	1.28045	0.640223	−	0.768189i	$-0.278842\pi$
$487$	28.2570	1.28045	0.640223	+	0.768189i	$0.278842\pi$
$488$	0	0
$489$	−3.28096	−0.148370
$490$	0	0
$491$	−27.2268	−1.22873	−0.614364	−	0.789023i	$-0.710587\pi$
$491$	−27.2268	−1.22873	−0.614364	+	0.789023i	$0.710587\pi$
$492$	0	0
$493$	−1.21292	−0.0546272
$494$	0	0
$495$	7.87325	0.353876
$496$	0	0
$497$	−6.14344	−0.275571
$498$	0	0
$499$	6.04093	0.270429	0.135215	−	0.990816i	$-0.456828\pi$
$499$	6.04093	0.270429	0.135215	+	0.990816i	$0.456828\pi$
$500$	0	0
$501$	2.23606	0.0998996
$502$	0	0
$503$	−30.8276	−1.37453	−0.687267	−	0.726405i	$-0.741190\pi$
$503$	−30.8276	−1.37453	−0.687267	+	0.726405i	$0.741190\pi$
$504$	0	0
$505$	13.6406	0.606998
$506$	0	0
$507$	−3.47400	−0.154286
$508$	0	0
$509$	13.0343	0.577736	0.288868	−	0.957369i	$-0.406721\pi$
$509$	13.0343	0.577736	0.288868	+	0.957369i	$0.406721\pi$
$510$	0	0
$511$	−3.99700	−0.176817
$512$	0	0
$513$	−9.56417	−0.422268
$514$	0	0
$515$	4.19356	0.184790
$516$	0	0
$517$	−11.3249	−0.498069
$518$	0	0
$519$	0.0103992	0.000456475 0
$520$	0	0
$521$	35.4615	1.55360	0.776799	−	0.629749i	$-0.216842\pi$
$521$	35.4615	1.55360	0.776799	+	0.629749i	$0.216842\pi$
$522$	0	0
$523$	20.5344	0.897906	0.448953	−	0.893555i	$-0.351797\pi$
$523$	20.5344	0.897906	0.448953	+	0.893555i	$0.351797\pi$
$524$	0	0
$525$	0.442594	0.0193164
$526$	0	0
$527$	−19.4443	−0.847008
$528$	0	0
$529$	0.976203	0.0424436
$530$	0	0
$531$	31.8549	1.38238
$532$	0	0
$533$	43.0150	1.86319
$534$	0	0
$535$	−1.84311	−0.0796844
$536$	0	0
$537$	6.12270	0.264214
$538$	0	0
$539$	−16.5991	−0.714975
$540$	0	0
$541$	14.1838	0.609811	0.304906	−	0.952383i	$-0.401375\pi$
$541$	14.1838	0.609811	0.304906	+	0.952383i	$0.401375\pi$
$542$	0	0
$543$	6.26108	0.268689
$544$	0	0
$545$	0.586405	0.0251188
$546$	0	0
$547$	−10.5763	−0.452209	−0.226105	−	0.974103i	$-0.572599\pi$
$547$	−10.5763	−0.452209	−0.226105	+	0.974103i	$0.572599\pi$
$548$	0	0
$549$	2.49336	0.106414
$550$	0	0
$551$	−2.20232	−0.0938222
$552$	0	0
$553$	−5.39822	−0.229556
$554$	0	0
$555$	−1.95981	−0.0831894
$556$	0	0
$557$	20.7471	0.879082	0.439541	−	0.898222i	$-0.355141\pi$
$557$	20.7471	0.879082	0.439541	+	0.898222i	$0.355141\pi$
$558$	0	0
$559$	34.2584	1.44898
$560$	0	0
$561$	2.53245	0.106920
$562$	0	0
$563$	−37.6011	−1.58470	−0.792350	−	0.610067i	$-0.791142\pi$
$563$	−37.6011	−1.58470	−0.792350	+	0.610067i	$0.791142\pi$
$564$	0	0
$565$	15.7693	0.663419
$566$	0	0
$567$	−7.60660	−0.319447
$568$	0	0
$569$	−4.20488	−0.176278	−0.0881388	−	0.996108i	$-0.528092\pi$
$569$	−4.20488	−0.176278	−0.0881388	+	0.996108i	$0.528092\pi$
$570$	0	0
$571$	−28.3600	−1.18683	−0.593414	−	0.804897i	$-0.702220\pi$
$571$	−28.3600	−1.18683	−0.593414	+	0.804897i	$0.702220\pi$
$572$	0	0
$573$	3.24451	0.135542
$574$	0	0
$575$	−4.89655	−0.204200
$576$	0	0
$577$	−42.6957	−1.77744	−0.888722	−	0.458447i	$-0.848406\pi$
$577$	−42.6957	−1.77744	−0.888722	+	0.458447i	$0.848406\pi$
$578$	0	0
$579$	3.56758	0.148264
$580$	0	0
$581$	3.15763	0.131000
$582$	0	0
$583$	−14.3786	−0.595499
$584$	0	0
$585$	12.9326	0.534697
$586$	0	0
$587$	−39.2766	−1.62112	−0.810560	−	0.585656i	$-0.800837\pi$
$587$	−39.2766	−1.62112	−0.810560	+	0.585656i	$0.800837\pi$
$588$	0	0
$589$	−35.3055	−1.45474
$590$	0	0
$591$	7.27176	0.299120
$592$	0	0
$593$	−41.5001	−1.70421	−0.852103	−	0.523375i	$-0.824673\pi$
$593$	−41.5001	−1.70421	−0.852103	+	0.523375i	$0.824673\pi$
$594$	0	0
$595$	−2.17624	−0.0892170
$596$	0	0
$597$	3.17269	0.129850
$598$	0	0
$599$	21.0974	0.862017	0.431008	−	0.902348i	$-0.358158\pi$
$599$	21.0974	0.862017	0.431008	+	0.902348i	$0.358158\pi$
$600$	0	0
$601$	−8.49597	−0.346558	−0.173279	−	0.984873i	$-0.555436\pi$
$601$	−8.49597	−0.346558	−0.173279	+	0.984873i	$0.555436\pi$
$602$	0	0
$603$	−3.11004	−0.126651
$604$	0	0
$605$	3.18184	0.129360
$606$	0	0
$607$	2.75901	0.111985	0.0559923	−	0.998431i	$-0.482168\pi$
$607$	2.75901	0.111985	0.0559923	+	0.998431i	$0.482168\pi$
$608$	0	0
$609$	0.254386	0.0103082
$610$	0	0
$611$	−18.6023	−0.752568
$612$	0	0
$613$	−5.78620	−0.233702	−0.116851	−	0.993149i	$-0.537280\pi$
$613$	−5.78620	−0.233702	−0.116851	+	0.993149i	$0.537280\pi$
$614$	0	0
$615$	4.01957	0.162085
$616$	0	0
$617$	−14.0370	−0.565110	−0.282555	−	0.959251i	$-0.591182\pi$
$617$	−14.0370	−0.565110	−0.282555	+	0.959251i	$0.591182\pi$
$618$	0	0
$619$	36.7675	1.47781	0.738906	−	0.673809i	$-0.235343\pi$
$619$	36.7675	1.47781	0.738906	+	0.673809i	$0.235343\pi$
$620$	0	0
$621$	−12.2220	−0.490453
$622$	0	0
$623$	4.18440	0.167645
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	4.59822	0.183635
$628$	0	0
$629$	9.63640	0.384229
$630$	0	0
$631$	−0.827650	−0.0329482	−0.0164741	−	0.999864i	$-0.505244\pi$
$631$	−0.827650	−0.0329482	−0.0164741	+	0.999864i	$0.505244\pi$
$632$	0	0
$633$	8.53370	0.339184
$634$	0	0
$635$	7.07800	0.280882
$636$	0	0
$637$	−27.2657	−1.08031
$638$	0	0
$639$	−16.7746	−0.663593
$640$	0	0
$641$	6.74693	0.266488	0.133244	−	0.991083i	$-0.457461\pi$
$641$	6.74693	0.266488	0.133244	+	0.991083i	$0.457461\pi$
$642$	0	0
$643$	−11.2998	−0.445619	−0.222810	−	0.974862i	$-0.571523\pi$
$643$	−11.2998	−0.445619	−0.222810	+	0.974862i	$0.571523\pi$
$644$	0	0
$645$	3.20131	0.126051
$646$	0	0
$647$	−23.7186	−0.932473	−0.466236	−	0.884660i	$-0.654390\pi$
$647$	−23.7186	−0.932473	−0.466236	+	0.884660i	$0.654390\pi$
$648$	0	0
$649$	−31.6320	−1.24166
$650$	0	0
$651$	4.07806	0.159832
$652$	0	0
$653$	16.0632	0.628601	0.314300	−	0.949324i	$-0.398230\pi$
$653$	16.0632	0.628601	0.314300	+	0.949324i	$0.398230\pi$
$654$	0	0
$655$	−6.38010	−0.249291
$656$	0	0
$657$	−10.9138	−0.425786
$658$	0	0
$659$	2.82555	0.110068	0.0550339	−	0.998484i	$-0.482473\pi$
$659$	2.82555	0.110068	0.0550339	+	0.998484i	$0.482473\pi$
$660$	0	0
$661$	−30.0281	−1.16796	−0.583980	−	0.811768i	$-0.698505\pi$
$661$	−30.0281	−1.16796	−0.583980	+	0.811768i	$0.698505\pi$
$662$	0	0
$663$	4.15980	0.161553
$664$	0	0
$665$	−3.95144	−0.153230
$666$	0	0
$667$	−2.81435	−0.108972
$668$	0	0
$669$	−0.496713	−0.0192040
$670$	0	0
$671$	−2.47591	−0.0955814
$672$	0	0
$673$	22.4486	0.865331	0.432665	−	0.901555i	$-0.357573\pi$
$673$	22.4486	0.865331	0.432665	+	0.901555i	$0.357573\pi$
$674$	0	0
$675$	2.49605	0.0960730
$676$	0	0
$677$	−2.84293	−0.109263	−0.0546313	−	0.998507i	$-0.517398\pi$
$677$	−2.84293	−0.109263	−0.0546313	+	0.998507i	$0.517398\pi$
$678$	0	0
$679$	−7.80460	−0.299513
$680$	0	0
$681$	−8.46332	−0.324315
$682$	0	0
$683$	−31.9604	−1.22293	−0.611466	−	0.791271i	$-0.709420\pi$
$683$	−31.9604	−1.22293	−0.611466	+	0.791271i	$0.709420\pi$
$684$	0	0
$685$	−11.3659	−0.434268
$686$	0	0
$687$	8.76107	0.334256
$688$	0	0
$689$	−23.6182	−0.899782
$690$	0	0
$691$	−0.522791	−0.0198879	−0.00994396	−	0.999951i	$-0.503165\pi$
$691$	−0.522791	−0.0198879	−0.00994396	+	0.999951i	$0.503165\pi$
$692$	0	0
$693$	8.11925	0.308425
$694$	0	0
$695$	11.5816	0.439315
$696$	0	0
$697$	−19.7643	−0.748625
$698$	0	0
$699$	9.09208	0.343894
$700$	0	0
$701$	27.6420	1.04402	0.522012	−	0.852938i	$-0.325182\pi$
$701$	27.6420	1.04402	0.522012	+	0.852938i	$0.325182\pi$
$702$	0	0
$703$	17.4970	0.659913
$704$	0	0
$705$	−1.73831	−0.0654685
$706$	0	0
$707$	14.0668	0.529036
$708$	0	0
$709$	22.2529	0.835726	0.417863	−	0.908510i	$-0.362779\pi$
$709$	22.2529	0.835726	0.417863	+	0.908510i	$0.362779\pi$
$710$	0	0
$711$	−14.7398	−0.552784
$712$	0	0
$713$	−45.1168	−1.68964
$714$	0	0
$715$	−12.8421	−0.480267
$716$	0	0
$717$	1.58738	0.0592819
$718$	0	0
$719$	−8.96506	−0.334340	−0.167170	−	0.985928i	$-0.553463\pi$
$719$	−8.96506	−0.334340	−0.167170	+	0.985928i	$0.553463\pi$
$720$	0	0
$721$	4.32459	0.161056
$722$	0	0
$723$	2.95088	0.109745
$724$	0	0
$725$	0.574761	0.0213461
$726$	0	0
$727$	−37.6947	−1.39802	−0.699009	−	0.715113i	$-0.746375\pi$
$727$	−37.6947	−1.39802	−0.699009	+	0.715113i	$0.746375\pi$
$728$	0	0
$729$	−17.5559	−0.650220
$730$	0	0
$731$	−15.7408	−0.582196
$732$	0	0
$733$	12.7544	0.471094	0.235547	−	0.971863i	$-0.424312\pi$
$733$	12.7544	0.471094	0.235547	+	0.971863i	$0.424312\pi$
$734$	0	0
$735$	−2.54787	−0.0939795
$736$	0	0
$737$	3.08828	0.113758
$738$	0	0
$739$	48.8373	1.79651	0.898255	−	0.439476i	$-0.144836\pi$
$739$	48.8373	1.79651	0.898255	+	0.439476i	$0.144836\pi$
$740$	0	0
$741$	7.55303	0.277468
$742$	0	0
$743$	44.8791	1.64645	0.823227	−	0.567712i	$-0.192171\pi$
$743$	44.8791	1.64645	0.823227	+	0.567712i	$0.192171\pi$
$744$	0	0
$745$	−19.6750	−0.720835
$746$	0	0
$747$	8.62186	0.315457
$748$	0	0
$749$	−1.90069	−0.0694498
$750$	0	0
$751$	−47.1854	−1.72182	−0.860910	−	0.508757i	$-0.830105\pi$
$751$	−47.1854	−1.72182	−0.860910	+	0.508757i	$0.830105\pi$
$752$	0	0
$753$	2.93553	0.106977
$754$	0	0
$755$	−1.00000	−0.0363937
$756$	0	0
$757$	38.0906	1.38443	0.692213	−	0.721694i	$-0.256636\pi$
$757$	38.0906	1.38443	0.692213	+	0.721694i	$0.256636\pi$
$758$	0	0
$759$	5.87606	0.213288
$760$	0	0
$761$	0.309052	0.0112031	0.00560157	−	0.999984i	$-0.498217\pi$
$761$	0.309052	0.0112031	0.00560157	+	0.999984i	$0.498217\pi$
$762$	0	0
$763$	0.604727	0.0218926
$764$	0	0
$765$	−5.94219	−0.214840
$766$	0	0
$767$	−51.9587	−1.87612
$768$	0	0
$769$	−24.9236	−0.898769	−0.449384	−	0.893339i	$-0.648357\pi$
$769$	−24.9236	−0.898769	−0.449384	+	0.893339i	$0.648357\pi$
$770$	0	0
$771$	12.2584	0.441475
$772$	0	0
$773$	13.2923	0.478092	0.239046	−	0.971008i	$-0.423165\pi$
$773$	13.2923	0.478092	0.239046	+	0.971008i	$0.423165\pi$
$774$	0	0
$775$	9.21400	0.330977
$776$	0	0
$777$	−2.02105	−0.0725046
$778$	0	0
$779$	−35.8864	−1.28576
$780$	0	0
$781$	16.6572	0.596042
$782$	0	0
$783$	1.43463	0.0512696
$784$	0	0
$785$	3.96139	0.141388
$786$	0	0
$787$	39.3875	1.40401	0.702007	−	0.712170i	$-0.252288\pi$
$787$	39.3875	1.40401	0.702007	+	0.712170i	$0.252288\pi$
$788$	0	0
$789$	6.60155	0.235021
$790$	0	0
$791$	16.2620	0.578210
$792$	0	0
$793$	−4.06693	−0.144421
$794$	0	0
$795$	−2.20702	−0.0782751
$796$	0	0
$797$	38.5446	1.36532	0.682661	−	0.730736i	$-0.260823\pi$
$797$	38.5446	1.36532	0.682661	+	0.730736i	$0.260823\pi$
$798$	0	0
$799$	8.54727	0.302381
$800$	0	0
$801$	11.4255	0.403699
$802$	0	0
$803$	10.8374	0.382443
$804$	0	0
$805$	−5.04954	−0.177973
$806$	0	0
$807$	4.56798	0.160801
$808$	0	0
$809$	−18.2803	−0.642702	−0.321351	−	0.946960i	$-0.604137\pi$
$809$	−18.2803	−0.642702	−0.321351	+	0.946960i	$0.604137\pi$
$810$	0	0
$811$	37.0137	1.29973	0.649864	−	0.760051i	$-0.274826\pi$
$811$	37.0137	1.29973	0.649864	+	0.760051i	$0.274826\pi$
$812$	0	0
$813$	−9.00991	−0.315991
$814$	0	0
$815$	−7.64465	−0.267780
$816$	0	0
$817$	−28.5810	−0.999922
$818$	0	0
$819$	13.3367	0.466021
$820$	0	0
$821$	20.7072	0.722687	0.361344	−	0.932433i	$-0.382318\pi$
$821$	20.7072	0.722687	0.361344	+	0.932433i	$0.382318\pi$
$822$	0	0
$823$	33.2718	1.15978	0.579891	−	0.814694i	$-0.303095\pi$
$823$	33.2718	1.15978	0.579891	+	0.814694i	$0.303095\pi$
$824$	0	0
$825$	−1.20004	−0.0417801
$826$	0	0
$827$	−32.7163	−1.13766	−0.568828	−	0.822456i	$-0.692603\pi$
$827$	−32.7163	−1.13766	−0.568828	+	0.822456i	$0.692603\pi$
$828$	0	0
$829$	−0.914309	−0.0317553	−0.0158776	−	0.999874i	$-0.505054\pi$
$829$	−0.914309	−0.0317553	−0.0158776	+	0.999874i	$0.505054\pi$
$830$	0	0
$831$	−5.35278	−0.185686
$832$	0	0
$833$	12.5279	0.434065
$834$	0	0
$835$	5.21001	0.180300
$836$	0	0
$837$	22.9986	0.794948
$838$	0	0
$839$	12.9768	0.448009	0.224005	−	0.974588i	$-0.428087\pi$
$839$	12.9768	0.448009	0.224005	+	0.974588i	$0.428087\pi$
$840$	0	0
$841$	−28.6696	−0.988609
$842$	0	0
$843$	−3.95083	−0.136074
$844$	0	0
$845$	−8.09442	−0.278457
$846$	0	0
$847$	3.28125	0.112745
$848$	0	0
$849$	−7.80645	−0.267917
$850$	0	0
$851$	22.3594	0.766471
$852$	0	0
$853$	7.74912	0.265325	0.132662	−	0.991161i	$-0.457647\pi$
$853$	7.74912	0.265325	0.132662	+	0.991161i	$0.457647\pi$
$854$	0	0
$855$	−10.7894	−0.368988
$856$	0	0
$857$	2.51285	0.0858372	0.0429186	−	0.999079i	$-0.486334\pi$
$857$	2.51285	0.0858372	0.0429186	+	0.999079i	$0.486334\pi$
$858$	0	0
$859$	28.9389	0.987384	0.493692	−	0.869637i	$-0.335647\pi$
$859$	28.9389	0.987384	0.493692	+	0.869637i	$0.335647\pi$
$860$	0	0
$861$	4.14516	0.141267
$862$	0	0
$863$	24.2527	0.825571	0.412785	−	0.910828i	$-0.364556\pi$
$863$	24.2527	0.825571	0.412785	+	0.910828i	$0.364556\pi$
$864$	0	0
$865$	0.0242302	0.000823851 0
$866$	0	0
$867$	5.38482	0.182878
$868$	0	0
$869$	14.6366	0.496514
$870$	0	0
$871$	5.07281	0.171886
$872$	0	0
$873$	−21.3104	−0.721246
$874$	0	0
$875$	1.03124	0.0348624
$876$	0	0
$877$	20.2712	0.684509	0.342254	−	0.939607i	$-0.388810\pi$
$877$	20.2712	0.684509	0.342254	+	0.939607i	$0.388810\pi$
$878$	0	0
$879$	11.7462	0.396191
$880$	0	0
$881$	−8.02092	−0.270232	−0.135116	−	0.990830i	$-0.543141\pi$
$881$	−8.02092	−0.270232	−0.135116	+	0.990830i	$0.543141\pi$
$882$	0	0
$883$	0.322644	0.0108578	0.00542891	−	0.999985i	$-0.498272\pi$
$883$	0.322644	0.0108578	0.00542891	+	0.999985i	$0.498272\pi$
$884$	0	0
$885$	−4.85532	−0.163210
$886$	0	0
$887$	−14.8786	−0.499575	−0.249787	−	0.968301i	$-0.580361\pi$
$887$	−14.8786	−0.499575	−0.249787	+	0.968301i	$0.580361\pi$
$888$	0	0
$889$	7.29915	0.244806
$890$	0	0
$891$	20.6244	0.690943
$892$	0	0
$893$	15.5195	0.519339
$894$	0	0
$895$	14.2659	0.476857
$896$	0	0
$897$	9.65201	0.322271
$898$	0	0
$899$	5.29585	0.176627
$900$	0	0
$901$	10.8519	0.361531
$902$	0	0
$903$	3.30133	0.109861
$904$	0	0
$905$	14.5883	0.484932
$906$	0	0
$907$	−12.6141	−0.418845	−0.209422	−	0.977825i	$-0.567158\pi$
$907$	−12.6141	−0.418845	−0.209422	+	0.977825i	$0.567158\pi$
$908$	0	0
$909$	38.4092	1.27395
$910$	0	0
$911$	2.24315	0.0743188	0.0371594	−	0.999309i	$-0.488169\pi$
$911$	2.24315	0.0743188	0.0371594	+	0.999309i	$0.488169\pi$
$912$	0	0
$913$	−8.56153	−0.283345
$914$	0	0
$915$	−0.380037	−0.0125636
$916$	0	0
$917$	−6.57945	−0.217273
$918$	0	0
$919$	25.8628	0.853136	0.426568	−	0.904455i	$-0.359722\pi$
$919$	25.8628	0.853136	0.426568	+	0.904455i	$0.359722\pi$
$920$	0	0
$921$	−2.02675	−0.0667838
$922$	0	0
$923$	27.3611	0.900602
$924$	0	0
$925$	−4.56636	−0.150141
$926$	0	0
$927$	11.8082	0.387833
$928$	0	0
$929$	−52.4950	−1.72230	−0.861152	−	0.508347i	$-0.830257\pi$
$929$	−52.4950	−1.72230	−0.861152	+	0.508347i	$0.830257\pi$
$930$	0	0
$931$	22.7471	0.745507
$932$	0	0
$933$	−6.35240	−0.207968
$934$	0	0
$935$	5.90061	0.192971
$936$	0	0
$937$	35.9522	1.17451	0.587253	−	0.809403i	$-0.300209\pi$
$937$	35.9522	1.17451	0.587253	+	0.809403i	$0.300209\pi$
$938$	0	0
$939$	6.67281	0.217759
$940$	0	0
$941$	−59.2819	−1.93254	−0.966268	−	0.257540i	$-0.917088\pi$
$941$	−59.2819	−1.93254	−0.966268	+	0.257540i	$0.917088\pi$
$942$	0	0
$943$	−45.8592	−1.49338
$944$	0	0
$945$	2.57404	0.0837335
$946$	0	0
$947$	6.45202	0.209663	0.104831	−	0.994490i	$-0.466570\pi$
$947$	6.45202	0.209663	0.104831	+	0.994490i	$0.466570\pi$
$948$	0	0
$949$	17.8015	0.577860
$950$	0	0
$951$	11.1755	0.362390
$952$	0	0
$953$	1.62747	0.0527189	0.0263594	−	0.999653i	$-0.491609\pi$
$953$	1.62747	0.0527189	0.0263594	+	0.999653i	$0.491609\pi$
$954$	0	0
$955$	7.55972	0.244627
$956$	0	0
$957$	−0.689737	−0.0222960
$958$	0	0
$959$	−11.7210	−0.378491
$960$	0	0
$961$	53.8978	1.73864
$962$	0	0
$963$	−5.18982	−0.167240
$964$	0	0
$965$	8.31247	0.267588
$966$	0	0
$967$	13.5981	0.437287	0.218643	−	0.975805i	$-0.429837\pi$
$967$	13.5981	0.437287	0.218643	+	0.975805i	$0.429837\pi$
$968$	0	0
$969$	−3.47042	−0.111486
$970$	0	0
$971$	24.7575	0.794507	0.397253	−	0.917709i	$-0.369963\pi$
$971$	24.7575	0.794507	0.397253	+	0.917709i	$0.369963\pi$
$972$	0	0
$973$	11.9435	0.382890
$974$	0	0
$975$	−1.97119	−0.0631285
$976$	0	0
$977$	−46.9659	−1.50257	−0.751285	−	0.659977i	$-0.770566\pi$
$977$	−46.9659	−1.50257	−0.751285	+	0.659977i	$0.770566\pi$
$978$	0	0
$979$	−11.3455	−0.362604
$980$	0	0
$981$	1.65120	0.0527187
$982$	0	0
$983$	4.96101	0.158232	0.0791158	−	0.996865i	$-0.474790\pi$
$983$	4.96101	0.158232	0.0791158	+	0.996865i	$0.474790\pi$
$984$	0	0
$985$	16.9432	0.539855
$986$	0	0
$987$	−1.79262	−0.0570597
$988$	0	0
$989$	−36.5236	−1.16138
$990$	0	0
$991$	−6.41269	−0.203706	−0.101853	−	0.994799i	$-0.532477\pi$
$991$	−6.41269	−0.203706	−0.101853	+	0.994799i	$0.532477\pi$
$992$	0	0
$993$	2.57046	0.0815711
$994$	0	0
$995$	7.39238	0.234354
$996$	0	0
$997$	15.7495	0.498792	0.249396	−	0.968402i	$-0.419768\pi$
$997$	15.7495	0.498792	0.249396	+	0.968402i	$0.419768\pi$
$998$	0	0
$999$	−11.3979	−0.360613

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6040.2.a.n.1.6	✓	13

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6040.2.a.n.1.6	✓	13	1.1	even	1	trivial

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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)

\(f(q)\)	\(=\)	\(q-0.429184 q^{3} -1.00000 q^{5} -1.03124 q^{7} -2.81580 q^{9} +O(q^{10})\)	\(q-0.429184 q^{3} -1.00000 q^{5} -1.03124 q^{7} -2.81580 q^{9} +2.79610 q^{11} +4.59287 q^{13} +0.429184 q^{15} -2.11030 q^{17} -3.83172 q^{19} +0.442594 q^{21} -4.89655 q^{23} +1.00000 q^{25} +2.49605 q^{27} +0.574761 q^{29} +9.21400 q^{31} -1.20004 q^{33} +1.03124 q^{35} -4.56636 q^{37} -1.97119 q^{39} +9.36561 q^{41} +7.45905 q^{43} +2.81580 q^{45} -4.05026 q^{47} -5.93653 q^{49} +0.905709 q^{51} -5.14237 q^{53} -2.79610 q^{55} +1.64451 q^{57} -11.3129 q^{59} -0.885488 q^{61} +2.90378 q^{63} -4.59287 q^{65} +1.10450 q^{67} +2.10152 q^{69} +5.95731 q^{71} +3.87590 q^{73} -0.429184 q^{75} -2.88346 q^{77} +5.23466 q^{79} +7.37614 q^{81} -3.06196 q^{83} +2.11030 q^{85} -0.246679 q^{87} -4.05762 q^{89} -4.73637 q^{91} -3.95451 q^{93} +3.83172 q^{95} +7.56813 q^{97} -7.87325 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 13 q - 4 q^{3} - 13 q^{5} + 5 q^{9}+O(q^{10}) \) 13 * q - 4 * q^3 - 13 * q^5 + 5 * q^9	\( 13 q - 4 q^{3} - 13 q^{5} + 5 q^{9} - 14 q^{11} + 5 q^{13} + 4 q^{15} - 8 q^{17} + 16 q^{19} - 5 q^{21} - 4 q^{23} + 13 q^{25} + 2 q^{27} - 6 q^{29} + 11 q^{31} - 19 q^{33} + 6 q^{37} + 7 q^{39} - 18 q^{41} + 7 q^{43} - 5 q^{45} - 22 q^{47} - q^{49} + 12 q^{51} - 17 q^{53} + 14 q^{55} - 16 q^{57} - 6 q^{59} + 10 q^{61} - 5 q^{65} + 12 q^{67} + 13 q^{69} - 16 q^{71} - 24 q^{73} - 4 q^{75} - 11 q^{77} + 36 q^{79} - 19 q^{81} + q^{83} + 8 q^{85} - 8 q^{87} - 53 q^{89} + 23 q^{91} - 9 q^{93} - 16 q^{95} - 21 q^{97} - 9 q^{99}+O(q^{100}) \) 13 * q - 4 * q^3 - 13 * q^5 + 5 * q^9 - 14 * q^11 + 5 * q^13 + 4 * q^15 - 8 * q^17 + 16 * q^19 - 5 * q^21 - 4 * q^23 + 13 * q^25 + 2 * q^27 - 6 * q^29 + 11 * q^31 - 19 * q^33 + 6 * q^37 + 7 * q^39 - 18 * q^41 + 7 * q^43 - 5 * q^45 - 22 * q^47 - q^49 + 12 * q^51 - 17 * q^53 + 14 * q^55 - 16 * q^57 - 6 * q^59 + 10 * q^61 - 5 * q^65 + 12 * q^67 + 13 * q^69 - 16 * q^71 - 24 * q^73 - 4 * q^75 - 11 * q^77 + 36 * q^79 - 19 * q^81 + q^83 + 8 * q^85 - 8 * q^87 - 53 * q^89 + 23 * q^91 - 9 * q^93 - 16 * q^95 - 21 * q^97 - 9 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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