LMFDB - Embedded newform 6040.2.a.n.1.5

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			1.5
Root			$1.74594$ 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)

$f(q)$	$=$	$q-1.74594 q^{3} -1.00000 q^{5} +3.52041 q^{7} +0.0483066 q^{9} +O(q^{10})$	$q-1.74594 q^{3} -1.00000 q^{5} +3.52041 q^{7} +0.0483066 q^{9} -2.84362 q^{11} +4.69769 q^{13} +1.74594 q^{15} -3.79757 q^{17} -4.08433 q^{19} -6.14643 q^{21} -0.0192373 q^{23} +1.00000 q^{25} +5.15348 q^{27} -0.0360313 q^{29} +8.93493 q^{31} +4.96479 q^{33} -3.52041 q^{35} +4.06547 q^{37} -8.20188 q^{39} -11.7715 q^{41} -2.51333 q^{43} -0.0483066 q^{45} -5.06076 q^{47} +5.39332 q^{49} +6.63033 q^{51} +4.12555 q^{53} +2.84362 q^{55} +7.13100 q^{57} -2.35163 q^{59} -7.08485 q^{61} +0.170059 q^{63} -4.69769 q^{65} +10.0679 q^{67} +0.0335873 q^{69} -14.6452 q^{71} +7.51052 q^{73} -1.74594 q^{75} -10.0107 q^{77} -11.4673 q^{79} -9.14259 q^{81} +8.82146 q^{83} +3.79757 q^{85} +0.0629085 q^{87} +3.62332 q^{89} +16.5378 q^{91} -15.5999 q^{93} +4.08433 q^{95} -15.9239 q^{97} -0.137366 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−1.74594	−1.00802	−0.504009	−	0.863698i	$-0.668142\pi$
$3$	−1.74594	−1.00802	−0.504009	+	0.863698i	$0.668142\pi$
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	3.52041	1.33059	0.665296	−	0.746580i	$-0.268305\pi$
$7$	3.52041	1.33059	0.665296	+	0.746580i	$0.268305\pi$
$8$	0	0
$9$	0.0483066	0.0161022
$10$	0	0
$11$	−2.84362	−0.857384	−0.428692	−	0.903451i	$-0.641025\pi$
$11$	−2.84362	−0.857384	−0.428692	+	0.903451i	$0.641025\pi$
$12$	0	0
$13$	4.69769	1.30290	0.651452	−	0.758690i	$-0.274160\pi$
$13$	4.69769	1.30290	0.651452	+	0.758690i	$0.274160\pi$
$14$	0	0
$15$	1.74594	0.450800
$16$	0	0
$17$	−3.79757	−0.921046	−0.460523	−	0.887648i	$-0.652338\pi$
$17$	−3.79757	−0.921046	−0.460523	+	0.887648i	$0.652338\pi$
$18$	0	0
$19$	−4.08433	−0.937010	−0.468505	−	0.883461i	$-0.655207\pi$
$19$	−4.08433	−0.937010	−0.468505	+	0.883461i	$0.655207\pi$
$20$	0	0
$21$	−6.14643	−1.34126
$22$	0	0
$23$	−0.0192373	−0.00401126	−0.00200563	−	0.999998i	$-0.500638\pi$
$23$	−0.0192373	−0.00401126	−0.00200563	+	0.999998i	$0.500638\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	5.15348	0.991788
$28$	0	0
$29$	−0.0360313	−0.00669085	−0.00334542	−	0.999994i	$-0.501065\pi$
$29$	−0.0360313	−0.00669085	−0.00334542	+	0.999994i	$0.501065\pi$
$30$	0	0
$31$	8.93493	1.60476	0.802380	−	0.596813i	$-0.203567\pi$
$31$	8.93493	1.60476	0.802380	+	0.596813i	$0.203567\pi$
$32$	0	0
$33$	4.96479	0.864259
$34$	0	0
$35$	−3.52041	−0.595059
$36$	0	0
$37$	4.06547	0.668358	0.334179	−	0.942510i	$-0.391541\pi$
$37$	4.06547	0.668358	0.334179	+	0.942510i	$0.391541\pi$
$38$	0	0
$39$	−8.20188	−1.31335
$40$	0	0
$41$	−11.7715	−1.83840	−0.919202	−	0.393786i	$-0.871165\pi$
$41$	−11.7715	−1.83840	−0.919202	+	0.393786i	$0.871165\pi$
$42$	0	0
$43$	−2.51333	−0.383279	−0.191639	−	0.981465i	$-0.561380\pi$
$43$	−2.51333	−0.383279	−0.191639	+	0.981465i	$0.561380\pi$
$44$	0	0
$45$	−0.0483066	−0.00720112
$46$	0	0
$47$	−5.06076	−0.738187	−0.369094	−	0.929392i	$-0.620332\pi$
$47$	−5.06076	−0.738187	−0.369094	+	0.929392i	$0.620332\pi$
$48$	0	0
$49$	5.39332	0.770474
$50$	0	0
$51$	6.63033	0.928432
$52$	0	0
$53$	4.12555	0.566688	0.283344	−	0.959018i	$-0.408556\pi$
$53$	4.12555	0.566688	0.283344	+	0.959018i	$0.408556\pi$
$54$	0	0
$55$	2.84362	0.383434
$56$	0	0
$57$	7.13100	0.944524
$58$	0	0
$59$	−2.35163	−0.306157	−0.153078	−	0.988214i	$-0.548919\pi$
$59$	−2.35163	−0.306157	−0.153078	+	0.988214i	$0.548919\pi$
$60$	0	0
$61$	−7.08485	−0.907122	−0.453561	−	0.891225i	$-0.649846\pi$
$61$	−7.08485	−0.907122	−0.453561	+	0.891225i	$0.649846\pi$
$62$	0	0
$63$	0.170059	0.0214254
$64$	0	0
$65$	−4.69769	−0.582676
$66$	0	0
$67$	10.0679	1.22999	0.614994	−	0.788532i	$-0.289158\pi$
$67$	10.0679	1.22999	0.614994	+	0.788532i	$0.289158\pi$
$68$	0	0
$69$	0.0335873	0.00404343
$70$	0	0
$71$	−14.6452	−1.73807	−0.869034	−	0.494753i	$-0.835259\pi$
$71$	−14.6452	−1.73807	−0.869034	+	0.494753i	$0.835259\pi$
$72$	0	0
$73$	7.51052	0.879040	0.439520	−	0.898233i	$-0.355149\pi$
$73$	7.51052	0.879040	0.439520	+	0.898233i	$0.355149\pi$
$74$	0	0
$75$	−1.74594	−0.201604
$76$	0	0
$77$	−10.0107	−1.14083
$78$	0	0
$79$	−11.4673	−1.29017	−0.645086	−	0.764110i	$-0.723178\pi$
$79$	−11.4673	−1.29017	−0.645086	+	0.764110i	$0.723178\pi$
$80$	0	0
$81$	−9.14259	−1.01584
$82$	0	0
$83$	8.82146	0.968281	0.484140	−	0.874990i	$-0.339132\pi$
$83$	8.82146	0.968281	0.484140	+	0.874990i	$0.339132\pi$
$84$	0	0
$85$	3.79757	0.411904
$86$	0	0
$87$	0.0629085	0.00674450
$88$	0	0
$89$	3.62332	0.384071	0.192035	−	0.981388i	$-0.438491\pi$
$89$	3.62332	0.384071	0.192035	+	0.981388i	$0.438491\pi$
$90$	0	0
$91$	16.5378	1.73363
$92$	0	0
$93$	−15.5999	−1.61763
$94$	0	0
$95$	4.08433	0.419044
$96$	0	0
$97$	−15.9239	−1.61683	−0.808415	−	0.588613i	$-0.799674\pi$
$97$	−15.9239	−1.61683	−0.808415	+	0.588613i	$0.799674\pi$
$98$	0	0
$99$	−0.137366	−0.0138058
$100$	0	0
$101$	7.77310	0.773453	0.386726	−	0.922195i	$-0.373606\pi$
$101$	7.77310	0.773453	0.386726	+	0.922195i	$0.373606\pi$
$102$	0	0
$103$	−1.44274	−0.142158	−0.0710788	−	0.997471i	$-0.522644\pi$
$103$	−1.44274	−0.142158	−0.0710788	+	0.997471i	$0.522644\pi$
$104$	0	0
$105$	6.14643	0.599830
$106$	0	0
$107$	5.29980	0.512351	0.256176	−	0.966630i	$-0.417537\pi$
$107$	5.29980	0.512351	0.256176	+	0.966630i	$0.417537\pi$
$108$	0	0
$109$	13.2727	1.27130	0.635649	−	0.771978i	$-0.280733\pi$
$109$	13.2727	1.27130	0.635649	+	0.771978i	$0.280733\pi$
$110$	0	0
$111$	−7.09806	−0.673718
$112$	0	0
$113$	−3.55823	−0.334730	−0.167365	−	0.985895i	$-0.553526\pi$
$113$	−3.55823	−0.334730	−0.167365	+	0.985895i	$0.553526\pi$
$114$	0	0
$115$	0.0192373	0.00179389
$116$	0	0
$117$	0.226929	0.0209796
$118$	0	0
$119$	−13.3690	−1.22554
$120$	0	0
$121$	−2.91382	−0.264893
$122$	0	0
$123$	20.5524	1.85315
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	0.966461	0.0857596	0.0428798	−	0.999080i	$-0.486347\pi$
$127$	0.966461	0.0857596	0.0428798	+	0.999080i	$0.486347\pi$
$128$	0	0
$129$	4.38812	0.386352
$130$	0	0
$131$	18.7016	1.63397	0.816983	−	0.576662i	$-0.195645\pi$
$131$	18.7016	1.63397	0.816983	+	0.576662i	$0.195645\pi$
$132$	0	0
$133$	−14.3785	−1.24678
$134$	0	0
$135$	−5.15348	−0.443541
$136$	0	0
$137$	−9.59357	−0.819634	−0.409817	−	0.912168i	$-0.634407\pi$
$137$	−9.59357	−0.819634	−0.409817	+	0.912168i	$0.634407\pi$
$138$	0	0
$139$	21.3666	1.81229	0.906147	−	0.422963i	$-0.139010\pi$
$139$	21.3666	1.81229	0.906147	+	0.422963i	$0.139010\pi$
$140$	0	0
$141$	8.83578	0.744107
$142$	0	0
$143$	−13.3584	−1.11709
$144$	0	0
$145$	0.0360313	0.00299224
$146$	0	0
$147$	−9.41641	−0.776653
$148$	0	0
$149$	−15.3055	−1.25388	−0.626939	−	0.779068i	$-0.715692\pi$
$149$	−15.3055	−1.25388	−0.626939	+	0.779068i	$0.715692\pi$
$150$	0	0
$151$	1.00000	0.0813788
$151$	1.00000	0.0813788
$152$	0	0
$153$	−0.183448	−0.0148309
$154$	0	0
$155$	−8.93493	−0.717671
$156$	0	0
$157$	3.82297	0.305106	0.152553	−	0.988295i	$-0.451250\pi$
$157$	3.82297	0.305106	0.152553	+	0.988295i	$0.451250\pi$
$158$	0	0
$159$	−7.20296	−0.571232
$160$	0	0
$161$	−0.0677234	−0.00533736
$162$	0	0
$163$	16.6167	1.30152	0.650761	−	0.759283i	$-0.274450\pi$
$163$	16.6167	1.30152	0.650761	+	0.759283i	$0.274450\pi$
$164$	0	0
$165$	−4.96479	−0.386508
$166$	0	0
$167$	−7.03095	−0.544071	−0.272036	−	0.962287i	$-0.587697\pi$
$167$	−7.03095	−0.544071	−0.272036	+	0.962287i	$0.587697\pi$
$168$	0	0
$169$	9.06825	0.697558
$170$	0	0
$171$	−0.197300	−0.0150879
$172$	0	0
$173$	−5.39551	−0.410213	−0.205106	−	0.978740i	$-0.565754\pi$
$173$	−5.39551	−0.410213	−0.205106	+	0.978740i	$0.565754\pi$
$174$	0	0
$175$	3.52041	0.266118
$176$	0	0
$177$	4.10581	0.308612
$178$	0	0
$179$	−4.23831	−0.316786	−0.158393	−	0.987376i	$-0.550631\pi$
$179$	−4.23831	−0.316786	−0.158393	+	0.987376i	$0.550631\pi$
$180$	0	0
$181$	−7.04069	−0.523330	−0.261665	−	0.965159i	$-0.584272\pi$
$181$	−7.04069	−0.523330	−0.261665	+	0.965159i	$0.584272\pi$
$182$	0	0
$183$	12.3697	0.914396
$184$	0	0
$185$	−4.06547	−0.298899
$186$	0	0
$187$	10.7989	0.789690
$188$	0	0
$189$	18.1424	1.31966
$190$	0	0
$191$	3.88900	0.281398	0.140699	−	0.990052i	$-0.455065\pi$
$191$	3.88900	0.281398	0.140699	+	0.990052i	$0.455065\pi$
$192$	0	0
$193$	−6.51215	−0.468754	−0.234377	−	0.972146i	$-0.575305\pi$
$193$	−6.51215	−0.468754	−0.234377	+	0.972146i	$0.575305\pi$
$194$	0	0
$195$	8.20188	0.587349
$196$	0	0
$197$	−11.7988	−0.840626	−0.420313	−	0.907379i	$-0.638080\pi$
$197$	−11.7988	−0.840626	−0.420313	+	0.907379i	$0.638080\pi$
$198$	0	0
$199$	6.69256	0.474423	0.237212	−	0.971458i	$-0.423767\pi$
$199$	6.69256	0.474423	0.237212	+	0.971458i	$0.423767\pi$
$200$	0	0
$201$	−17.5779	−1.23985
$202$	0	0
$203$	−0.126845	−0.00890279
$204$	0	0
$205$	11.7715	0.822159
$206$	0	0
$207$	−0.000929290 0	−6.45901e−5 0
$208$	0	0
$209$	11.6143	0.803378
$210$	0	0
$211$	−10.7269	−0.738471	−0.369236	−	0.929336i	$-0.620381\pi$
$211$	−10.7269	−0.738471	−0.369236	+	0.929336i	$0.620381\pi$
$212$	0	0
$213$	25.5697	1.75200
$214$	0	0
$215$	2.51333	0.171408
$216$	0	0
$217$	31.4547	2.13528
$218$	0	0
$219$	−13.1129	−0.886089
$220$	0	0
$221$	−17.8398	−1.20003
$222$	0	0
$223$	−15.2777	−1.02307	−0.511534	−	0.859263i	$-0.670922\pi$
$223$	−15.2777	−1.02307	−0.511534	+	0.859263i	$0.670922\pi$
$224$	0	0
$225$	0.0483066	0.00322044
$226$	0	0
$227$	−28.1257	−1.86677	−0.933383	−	0.358881i	$-0.883158\pi$
$227$	−28.1257	−1.86677	−0.933383	+	0.358881i	$0.883158\pi$
$228$	0	0
$229$	−18.8231	−1.24386	−0.621932	−	0.783071i	$-0.713652\pi$
$229$	−18.8231	−1.24386	−0.621932	+	0.783071i	$0.713652\pi$
$230$	0	0
$231$	17.4781	1.14998
$232$	0	0
$233$	8.53850	0.559376	0.279688	−	0.960091i	$-0.409769\pi$
$233$	8.53850	0.559376	0.279688	+	0.960091i	$0.409769\pi$
$234$	0	0
$235$	5.06076	0.330127
$236$	0	0
$237$	20.0212	1.30052
$238$	0	0
$239$	19.1947	1.24160	0.620799	−	0.783969i	$-0.286808\pi$
$239$	19.1947	1.24160	0.620799	+	0.783969i	$0.286808\pi$
$240$	0	0
$241$	−26.0690	−1.67925	−0.839627	−	0.543163i	$-0.817227\pi$
$241$	−26.0690	−1.67925	−0.839627	+	0.543163i	$0.817227\pi$
$242$	0	0
$243$	0.501968	0.0322013
$244$	0	0
$245$	−5.39332	−0.344567
$246$	0	0
$247$	−19.1869	−1.22083
$248$	0	0
$249$	−15.4017	−0.976045
$250$	0	0
$251$	−0.993417	−0.0627039	−0.0313520	−	0.999508i	$-0.509981\pi$
$251$	−0.993417	−0.0627039	−0.0313520	+	0.999508i	$0.509981\pi$
$252$	0	0
$253$	0.0547037	0.00343919
$254$	0	0
$255$	−6.63033	−0.415207
$256$	0	0
$257$	10.6674	0.665411	0.332706	−	0.943031i	$-0.392038\pi$
$257$	10.6674	0.665411	0.332706	+	0.943031i	$0.392038\pi$
$258$	0	0
$259$	14.3121	0.889312
$260$	0	0
$261$	−0.00174055	−0.000107737 0
$262$	0	0
$263$	17.9283	1.10551	0.552754	−	0.833344i	$-0.313577\pi$
$263$	17.9283	1.10551	0.552754	+	0.833344i	$0.313577\pi$
$264$	0	0
$265$	−4.12555	−0.253430
$266$	0	0
$267$	−6.32609	−0.387151
$268$	0	0
$269$	−20.8374	−1.27048	−0.635238	−	0.772316i	$-0.719098\pi$
$269$	−20.8374	−1.27048	−0.635238	+	0.772316i	$0.719098\pi$
$270$	0	0
$271$	−18.0651	−1.09738	−0.548689	−	0.836027i	$-0.684873\pi$
$271$	−18.0651	−1.09738	−0.548689	+	0.836027i	$0.684873\pi$
$272$	0	0
$273$	−28.8740	−1.74753
$274$	0	0
$275$	−2.84362	−0.171477
$276$	0	0
$277$	−30.3190	−1.82169	−0.910845	−	0.412749i	$-0.864569\pi$
$277$	−30.3190	−1.82169	−0.910845	+	0.412749i	$0.864569\pi$
$278$	0	0
$279$	0.431616	0.0258402
$280$	0	0
$281$	7.73848	0.461639	0.230820	−	0.972997i	$-0.425859\pi$
$281$	7.73848	0.461639	0.230820	+	0.972997i	$0.425859\pi$
$282$	0	0
$283$	−8.87697	−0.527681	−0.263840	−	0.964566i	$-0.584989\pi$
$283$	−8.87697	−0.527681	−0.263840	+	0.964566i	$0.584989\pi$
$284$	0	0
$285$	−7.13100	−0.422404
$286$	0	0
$287$	−41.4407	−2.44617
$288$	0	0
$289$	−2.57846	−0.151674
$290$	0	0
$291$	27.8022	1.62980
$292$	0	0
$293$	5.52861	0.322985	0.161492	−	0.986874i	$-0.448369\pi$
$293$	5.52861	0.322985	0.161492	+	0.986874i	$0.448369\pi$
$294$	0	0
$295$	2.35163	0.136917
$296$	0	0
$297$	−14.6545	−0.850343
$298$	0	0
$299$	−0.0903710	−0.00522629
$300$	0	0
$301$	−8.84796	−0.509988
$302$	0	0
$303$	−13.5714	−0.779655
$304$	0	0
$305$	7.08485	0.405677
$306$	0	0
$307$	−16.1953	−0.924315	−0.462158	−	0.886798i	$-0.652925\pi$
$307$	−16.1953	−0.924315	−0.462158	+	0.886798i	$0.652925\pi$
$308$	0	0
$309$	2.51894	0.143297
$310$	0	0
$311$	−23.9869	−1.36017	−0.680086	−	0.733132i	$-0.738058\pi$
$311$	−23.9869	−1.36017	−0.680086	+	0.733132i	$0.738058\pi$
$312$	0	0
$313$	−30.7967	−1.74073	−0.870366	−	0.492405i	$-0.836118\pi$
$313$	−30.7967	−1.74073	−0.870366	+	0.492405i	$0.836118\pi$
$314$	0	0
$315$	−0.170059	−0.00958175
$316$	0	0
$317$	−9.41410	−0.528749	−0.264374	−	0.964420i	$-0.585165\pi$
$317$	−9.41410	−0.528749	−0.264374	+	0.964420i	$0.585165\pi$
$318$	0	0
$319$	0.102459	0.00573663
$320$	0	0
$321$	−9.25314	−0.516460
$322$	0	0
$323$	15.5105	0.863030
$324$	0	0
$325$	4.69769	0.260581
$326$	0	0
$327$	−23.1734	−1.28149
$328$	0	0
$329$	−17.8160	−0.982226
$330$	0	0
$331$	2.26810	0.124666	0.0623331	−	0.998055i	$-0.480146\pi$
$331$	2.26810	0.124666	0.0623331	+	0.998055i	$0.480146\pi$
$332$	0	0
$333$	0.196389	0.0107620
$334$	0	0
$335$	−10.0679	−0.550067
$336$	0	0
$337$	−31.1636	−1.69759	−0.848794	−	0.528723i	$-0.822671\pi$
$337$	−31.1636	−1.69759	−0.848794	+	0.528723i	$0.822671\pi$
$338$	0	0
$339$	6.21246	0.337414
$340$	0	0
$341$	−25.4076	−1.37590
$342$	0	0
$343$	−5.65618	−0.305405
$344$	0	0
$345$	−0.0335873	−0.00180828
$346$	0	0
$347$	−32.7218	−1.75660	−0.878299	−	0.478112i	$-0.841321\pi$
$347$	−32.7218	−1.75660	−0.878299	+	0.478112i	$0.841321\pi$
$348$	0	0
$349$	−13.2035	−0.706768	−0.353384	−	0.935478i	$-0.614969\pi$
$349$	−13.2035	−0.706768	−0.353384	+	0.935478i	$0.614969\pi$
$350$	0	0
$351$	24.2094	1.29220
$352$	0	0
$353$	−26.1083	−1.38961	−0.694803	−	0.719200i	$-0.744509\pi$
$353$	−26.1083	−1.38961	−0.694803	+	0.719200i	$0.744509\pi$
$354$	0	0
$355$	14.6452	0.777287
$356$	0	0
$357$	23.3415	1.23536
$358$	0	0
$359$	−25.1002	−1.32474	−0.662368	−	0.749179i	$-0.730448\pi$
$359$	−25.1002	−1.32474	−0.662368	+	0.749179i	$0.730448\pi$
$360$	0	0
$361$	−2.31822	−0.122011
$362$	0	0
$363$	5.08736	0.267017
$364$	0	0
$365$	−7.51052	−0.393119
$366$	0	0
$367$	−14.2531	−0.744008	−0.372004	−	0.928231i	$-0.621329\pi$
$367$	−14.2531	−0.744008	−0.372004	+	0.928231i	$0.621329\pi$
$368$	0	0
$369$	−0.568642	−0.0296023
$370$	0	0
$371$	14.5236	0.754030
$372$	0	0
$373$	9.01055	0.466549	0.233274	−	0.972411i	$-0.425056\pi$
$373$	9.01055	0.466549	0.233274	+	0.972411i	$0.425056\pi$
$374$	0	0
$375$	1.74594	0.0901600
$376$	0	0
$377$	−0.169264	−0.00871753
$378$	0	0
$379$	4.31076	0.221429	0.110714	−	0.993852i	$-0.464686\pi$
$379$	4.31076	0.221429	0.110714	+	0.993852i	$0.464686\pi$
$380$	0	0
$381$	−1.68738	−0.0864473
$382$	0	0
$383$	29.1776	1.49090	0.745452	−	0.666559i	$-0.232234\pi$
$383$	29.1776	1.49090	0.745452	+	0.666559i	$0.232234\pi$
$384$	0	0
$385$	10.0107	0.510194
$386$	0	0
$387$	−0.121410	−0.00617163
$388$	0	0
$389$	−1.01832	−0.0516307	−0.0258153	−	0.999667i	$-0.508218\pi$
$389$	−1.01832	−0.0516307	−0.0258153	+	0.999667i	$0.508218\pi$
$390$	0	0
$391$	0.0730552	0.00369456
$392$	0	0
$393$	−32.6518	−1.64707
$394$	0	0
$395$	11.4673	0.576982
$396$	0	0
$397$	2.05148	0.102961	0.0514805	−	0.998674i	$-0.483606\pi$
$397$	2.05148	0.102961	0.0514805	+	0.998674i	$0.483606\pi$
$398$	0	0
$399$	25.1041	1.25678
$400$	0	0
$401$	26.8834	1.34249	0.671247	−	0.741233i	$-0.265759\pi$
$401$	26.8834	1.34249	0.671247	+	0.741233i	$0.265759\pi$
$402$	0	0
$403$	41.9735	2.09085
$404$	0	0
$405$	9.14259	0.454299
$406$	0	0
$407$	−11.5606	−0.573040
$408$	0	0
$409$	19.6098	0.969643	0.484821	−	0.874613i	$-0.338885\pi$
$409$	19.6098	0.969643	0.484821	+	0.874613i	$0.338885\pi$
$410$	0	0
$411$	16.7498	0.826207
$412$	0	0
$413$	−8.27873	−0.407370
$414$	0	0
$415$	−8.82146	−0.433028
$416$	0	0
$417$	−37.3049	−1.82683
$418$	0	0
$419$	−6.28627	−0.307104	−0.153552	−	0.988141i	$-0.549071\pi$
$419$	−6.28627	−0.307104	−0.153552	+	0.988141i	$0.549071\pi$
$420$	0	0
$421$	−31.9793	−1.55857	−0.779287	−	0.626667i	$-0.784419\pi$
$421$	−31.9793	−1.55857	−0.779287	+	0.626667i	$0.784419\pi$
$422$	0	0
$423$	−0.244468	−0.0118864
$424$	0	0
$425$	−3.79757	−0.184209
$426$	0	0
$427$	−24.9416	−1.20701
$428$	0	0
$429$	23.3230	1.12605
$430$	0	0
$431$	41.1161	1.98049	0.990247	−	0.139323i	$-0.0444925\pi$
$431$	41.1161	1.98049	0.990247	+	0.139323i	$0.0444925\pi$
$432$	0	0
$433$	39.1136	1.87968	0.939840	−	0.341615i	$-0.110974\pi$
$433$	39.1136	1.87968	0.939840	+	0.341615i	$0.110974\pi$
$434$	0	0
$435$	−0.0629085	−0.00301623
$436$	0	0
$437$	0.0785718	0.00375860
$438$	0	0
$439$	32.7551	1.56332	0.781659	−	0.623706i	$-0.214374\pi$
$439$	32.7551	1.56332	0.781659	+	0.623706i	$0.214374\pi$
$440$	0	0
$441$	0.260533	0.0124063
$442$	0	0
$443$	21.6787	1.02999	0.514994	−	0.857194i	$-0.327794\pi$
$443$	21.6787	1.02999	0.514994	+	0.857194i	$0.327794\pi$
$444$	0	0
$445$	−3.62332	−0.171762
$446$	0	0
$447$	26.7225	1.26393
$448$	0	0
$449$	−33.4382	−1.57805	−0.789024	−	0.614363i	$-0.789413\pi$
$449$	−33.4382	−1.57805	−0.789024	+	0.614363i	$0.789413\pi$
$450$	0	0
$451$	33.4738	1.57622
$452$	0	0
$453$	−1.74594	−0.0820314
$454$	0	0
$455$	−16.5378	−0.775304
$456$	0	0
$457$	−22.9020	−1.07131	−0.535654	−	0.844437i	$-0.679935\pi$
$457$	−22.9020	−1.07131	−0.535654	+	0.844437i	$0.679935\pi$
$458$	0	0
$459$	−19.5707	−0.913482
$460$	0	0
$461$	−17.3106	−0.806234	−0.403117	−	0.915148i	$-0.632073\pi$
$461$	−17.3106	−0.806234	−0.403117	+	0.915148i	$0.632073\pi$
$462$	0	0
$463$	−22.8893	−1.06376	−0.531878	−	0.846821i	$-0.678513\pi$
$463$	−22.8893	−1.06376	−0.531878	+	0.846821i	$0.678513\pi$
$464$	0	0
$465$	15.5999	0.723426
$466$	0	0
$467$	−20.6435	−0.955268	−0.477634	−	0.878559i	$-0.658506\pi$
$467$	−20.6435	−0.955268	−0.477634	+	0.878559i	$0.658506\pi$
$468$	0	0
$469$	35.4431	1.63661
$470$	0	0
$471$	−6.67468	−0.307553
$472$	0	0
$473$	7.14695	0.328617
$474$	0	0
$475$	−4.08433	−0.187402
$476$	0	0
$477$	0.199291	0.00912492
$478$	0	0
$479$	13.6053	0.621643	0.310822	−	0.950468i	$-0.399396\pi$
$479$	13.6053	0.621643	0.310822	+	0.950468i	$0.399396\pi$
$480$	0	0
$481$	19.0983	0.870806
$482$	0	0
$483$	0.118241	0.00538016
$484$	0	0
$485$	15.9239	0.723068
$486$	0	0
$487$	33.8599	1.53434	0.767169	−	0.641445i	$-0.221665\pi$
$487$	33.8599	1.53434	0.767169	+	0.641445i	$0.221665\pi$
$488$	0	0
$489$	−29.0118	−1.31196
$490$	0	0
$491$	−23.0655	−1.04093	−0.520465	−	0.853883i	$-0.674241\pi$
$491$	−23.0655	−1.04093	−0.520465	+	0.853883i	$0.674241\pi$
$492$	0	0
$493$	0.136832	0.00616258
$494$	0	0
$495$	0.137366	0.00617412
$496$	0	0
$497$	−51.5572	−2.31266
$498$	0	0
$499$	21.6051	0.967176	0.483588	−	0.875296i	$-0.339333\pi$
$499$	21.6051	0.967176	0.483588	+	0.875296i	$0.339333\pi$
$500$	0	0
$501$	12.2756	0.548434
$502$	0	0
$503$	−3.32746	−0.148364	−0.0741820	−	0.997245i	$-0.523635\pi$
$503$	−3.32746	−0.148364	−0.0741820	+	0.997245i	$0.523635\pi$
$504$	0	0
$505$	−7.77310	−0.345899
$506$	0	0
$507$	−15.8326	−0.703151
$508$	0	0
$509$	−27.6906	−1.22736	−0.613681	−	0.789554i	$-0.710312\pi$
$509$	−27.6906	−1.22736	−0.613681	+	0.789554i	$0.710312\pi$
$510$	0	0
$511$	26.4402	1.16964
$512$	0	0
$513$	−21.0485	−0.929315
$514$	0	0
$515$	1.44274	0.0635748
$516$	0	0
$517$	14.3909	0.632910
$518$	0	0
$519$	9.42023	0.413502
$520$	0	0
$521$	28.4500	1.24642	0.623208	−	0.782057i	$-0.285829\pi$
$521$	28.4500	1.24642	0.623208	+	0.782057i	$0.285829\pi$
$522$	0	0
$523$	−5.02752	−0.219838	−0.109919	−	0.993941i	$-0.535059\pi$
$523$	−5.02752	−0.219838	−0.109919	+	0.993941i	$0.535059\pi$
$524$	0	0
$525$	−6.14643	−0.268252
$526$	0	0
$527$	−33.9310	−1.47806
$528$	0	0
$529$	−22.9996	−0.999984
$530$	0	0
$531$	−0.113599	−0.00492979
$532$	0	0
$533$	−55.2990	−2.39526
$534$	0	0
$535$	−5.29980	−0.229131
$536$	0	0
$537$	7.39983	0.319326
$538$	0	0
$539$	−15.3366	−0.660592
$540$	0	0
$541$	−15.5143	−0.667013	−0.333507	−	0.942748i	$-0.608232\pi$
$541$	−15.5143	−0.667013	−0.333507	+	0.942748i	$0.608232\pi$
$542$	0	0
$543$	12.2926	0.527527
$544$	0	0
$545$	−13.2727	−0.568542
$546$	0	0
$547$	23.2028	0.992082	0.496041	−	0.868299i	$-0.334787\pi$
$547$	23.2028	0.992082	0.496041	+	0.868299i	$0.334787\pi$
$548$	0	0
$549$	−0.342245	−0.0146066
$550$	0	0
$551$	0.147164	0.00626940
$552$	0	0
$553$	−40.3696	−1.71669
$554$	0	0
$555$	7.09806	0.301296
$556$	0	0
$557$	−37.5439	−1.59079	−0.795393	−	0.606094i	$-0.792736\pi$
$557$	−37.5439	−1.59079	−0.795393	+	0.606094i	$0.792736\pi$
$558$	0	0
$559$	−11.8068	−0.499375
$560$	0	0
$561$	−18.8541	−0.796023
$562$	0	0
$563$	43.3465	1.82684	0.913419	−	0.407021i	$-0.133432\pi$
$563$	43.3465	1.82684	0.913419	+	0.407021i	$0.133432\pi$
$564$	0	0
$565$	3.55823	0.149696
$566$	0	0
$567$	−32.1857	−1.35167
$568$	0	0
$569$	−1.26500	−0.0530314	−0.0265157	−	0.999648i	$-0.508441\pi$
$569$	−1.26500	−0.0530314	−0.0265157	+	0.999648i	$0.508441\pi$
$570$	0	0
$571$	28.6649	1.19959	0.599795	−	0.800154i	$-0.295249\pi$
$571$	28.6649	1.19959	0.599795	+	0.800154i	$0.295249\pi$
$572$	0	0
$573$	−6.78996	−0.283655
$574$	0	0
$575$	−0.0192373	−0.000802253 0
$576$	0	0
$577$	11.5968	0.482782	0.241391	−	0.970428i	$-0.422396\pi$
$577$	11.5968	0.482782	0.241391	+	0.970428i	$0.422396\pi$
$578$	0	0
$579$	11.3698	0.472513
$580$	0	0
$581$	31.0552	1.28839
$582$	0	0
$583$	−11.7315	−0.485869
$584$	0	0
$585$	−0.226929	−0.00938236
$586$	0	0
$587$	−21.8656	−0.902489	−0.451245	−	0.892400i	$-0.649020\pi$
$587$	−21.8656	−0.902489	−0.451245	+	0.892400i	$0.649020\pi$
$588$	0	0
$589$	−36.4932	−1.50368
$590$	0	0
$591$	20.5999	0.847367
$592$	0	0
$593$	−3.60432	−0.148012	−0.0740059	−	0.997258i	$-0.523578\pi$
$593$	−3.60432	−0.148012	−0.0740059	+	0.997258i	$0.523578\pi$
$594$	0	0
$595$	13.3690	0.548077
$596$	0	0
$597$	−11.6848	−0.478228
$598$	0	0
$599$	6.66340	0.272259	0.136130	−	0.990691i	$-0.456534\pi$
$599$	6.66340	0.272259	0.136130	+	0.990691i	$0.456534\pi$
$600$	0	0
$601$	−26.2130	−1.06925	−0.534626	−	0.845089i	$-0.679547\pi$
$601$	−26.2130	−1.06925	−0.534626	+	0.845089i	$0.679547\pi$
$602$	0	0
$603$	0.486345	0.0198055
$604$	0	0
$605$	2.91382	0.118464
$606$	0	0
$607$	−31.2361	−1.26783	−0.633917	−	0.773401i	$-0.718554\pi$
$607$	−31.2361	−1.26783	−0.633917	+	0.773401i	$0.718554\pi$
$608$	0	0
$609$	0.221464	0.00897418
$610$	0	0
$611$	−23.7738	−0.961787
$612$	0	0
$613$	45.0010	1.81757	0.908787	−	0.417260i	$-0.137010\pi$
$613$	45.0010	1.81757	0.908787	+	0.417260i	$0.137010\pi$
$614$	0	0
$615$	−20.5524	−0.828752
$616$	0	0
$617$	−34.6011	−1.39299	−0.696493	−	0.717563i	$-0.745257\pi$
$617$	−34.6011	−1.39299	−0.696493	+	0.717563i	$0.745257\pi$
$618$	0	0
$619$	−36.8891	−1.48270	−0.741349	−	0.671120i	$-0.765813\pi$
$619$	−36.8891	−1.48270	−0.741349	+	0.671120i	$0.765813\pi$
$620$	0	0
$621$	−0.0991393	−0.00397832
$622$	0	0
$623$	12.7556	0.511041
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−20.2779	−0.809820
$628$	0	0
$629$	−15.4389	−0.615589
$630$	0	0
$631$	−28.2873	−1.12610	−0.563050	−	0.826423i	$-0.690372\pi$
$631$	−28.2873	−1.12610	−0.563050	+	0.826423i	$0.690372\pi$
$632$	0	0
$633$	18.7286	0.744393
$634$	0	0
$635$	−0.966461	−0.0383528
$636$	0	0
$637$	25.3361	1.00385
$638$	0	0
$639$	−0.707460	−0.0279867
$640$	0	0
$641$	34.1944	1.35060	0.675300	−	0.737543i	$-0.264014\pi$
$641$	34.1944	1.35060	0.675300	+	0.737543i	$0.264014\pi$
$642$	0	0
$643$	4.57808	0.180542	0.0902710	−	0.995917i	$-0.471227\pi$
$643$	4.57808	0.180542	0.0902710	+	0.995917i	$0.471227\pi$
$644$	0	0
$645$	−4.38812	−0.172782
$646$	0	0
$647$	−10.7716	−0.423476	−0.211738	−	0.977326i	$-0.567912\pi$
$647$	−10.7716	−0.423476	−0.211738	+	0.977326i	$0.567912\pi$
$648$	0	0
$649$	6.68716	0.262494
$650$	0	0
$651$	−54.9180	−2.15240
$652$	0	0
$653$	−5.27098	−0.206269	−0.103135	−	0.994667i	$-0.532887\pi$
$653$	−5.27098	−0.206269	−0.103135	+	0.994667i	$0.532887\pi$
$654$	0	0
$655$	−18.7016	−0.730731
$656$	0	0
$657$	0.362808	0.0141545
$658$	0	0
$659$	15.3833	0.599250	0.299625	−	0.954057i	$-0.403139\pi$
$659$	15.3833	0.599250	0.299625	+	0.954057i	$0.403139\pi$
$660$	0	0
$661$	−19.5753	−0.761391	−0.380695	−	0.924701i	$-0.624315\pi$
$661$	−19.5753	−0.761391	−0.380695	+	0.924701i	$0.624315\pi$
$662$	0	0
$663$	31.1472	1.20966
$664$	0	0
$665$	14.3785	0.557576
$666$	0	0
$667$	0.000693147 0	2.68388e−5 0
$668$	0	0
$669$	26.6739	1.03127
$670$	0	0
$671$	20.1466	0.777751
$672$	0	0
$673$	−34.3971	−1.32591	−0.662955	−	0.748660i	$-0.730698\pi$
$673$	−34.3971	−1.32591	−0.662955	+	0.748660i	$0.730698\pi$
$674$	0	0
$675$	5.15348	0.198358
$676$	0	0
$677$	33.4462	1.28544	0.642721	−	0.766100i	$-0.277806\pi$
$677$	33.4462	1.28544	0.642721	+	0.766100i	$0.277806\pi$
$678$	0	0
$679$	−56.0588	−2.15134
$680$	0	0
$681$	49.1057	1.88174
$682$	0	0
$683$	−34.3135	−1.31297	−0.656486	−	0.754339i	$-0.727958\pi$
$683$	−34.3135	−1.31297	−0.656486	+	0.754339i	$0.727958\pi$
$684$	0	0
$685$	9.59357	0.366552
$686$	0	0
$687$	32.8640	1.25384
$688$	0	0
$689$	19.3805	0.738340
$690$	0	0
$691$	5.54373	0.210894	0.105447	−	0.994425i	$-0.466373\pi$
$691$	5.54373	0.210894	0.105447	+	0.994425i	$0.466373\pi$
$692$	0	0
$693$	−0.483584	−0.0183698
$694$	0	0
$695$	−21.3666	−0.810483
$696$	0	0
$697$	44.7032	1.69326
$698$	0	0
$699$	−14.9077	−0.563861
$700$	0	0
$701$	9.75405	0.368405	0.184203	−	0.982888i	$-0.441030\pi$
$701$	9.75405	0.368405	0.184203	+	0.982888i	$0.441030\pi$
$702$	0	0
$703$	−16.6047	−0.626259
$704$	0	0
$705$	−8.83578	−0.332775
$706$	0	0
$707$	27.3646	1.02915
$708$	0	0
$709$	14.1763	0.532401	0.266201	−	0.963918i	$-0.414232\pi$
$709$	14.1763	0.532401	0.266201	+	0.963918i	$0.414232\pi$
$710$	0	0
$711$	−0.553946	−0.0207746
$712$	0	0
$713$	−0.171884	−0.00643712
$714$	0	0
$715$	13.3584	0.499577
$716$	0	0
$717$	−33.5127	−1.25156
$718$	0	0
$719$	25.1246	0.936991	0.468495	−	0.883466i	$-0.344796\pi$
$719$	25.1246	0.936991	0.468495	+	0.883466i	$0.344796\pi$
$720$	0	0
$721$	−5.07905	−0.189154
$722$	0	0
$723$	45.5150	1.69272
$724$	0	0
$725$	−0.0360313	−0.00133817
$726$	0	0
$727$	30.8107	1.14271	0.571354	−	0.820704i	$-0.306418\pi$
$727$	30.8107	1.14271	0.571354	+	0.820704i	$0.306418\pi$
$728$	0	0
$729$	26.5514	0.983383
$730$	0	0
$731$	9.54454	0.353018
$732$	0	0
$733$	−16.4594	−0.607941	−0.303970	−	0.952681i	$-0.598312\pi$
$733$	−16.4594	−0.607941	−0.303970	+	0.952681i	$0.598312\pi$
$734$	0	0
$735$	9.41641	0.347330
$736$	0	0
$737$	−28.6293	−1.05457
$738$	0	0
$739$	36.5486	1.34446	0.672231	−	0.740342i	$-0.265336\pi$
$739$	36.5486	1.34446	0.672231	+	0.740342i	$0.265336\pi$
$740$	0	0
$741$	33.4992	1.23062
$742$	0	0
$743$	−47.6224	−1.74709	−0.873547	−	0.486739i	$-0.838186\pi$
$743$	−47.6224	−1.74709	−0.873547	+	0.486739i	$0.838186\pi$
$744$	0	0
$745$	15.3055	0.560751
$746$	0	0
$747$	0.426134	0.0155914
$748$	0	0
$749$	18.6575	0.681731
$750$	0	0
$751$	12.1948	0.444993	0.222496	−	0.974934i	$-0.428579\pi$
$751$	12.1948	0.444993	0.222496	+	0.974934i	$0.428579\pi$
$752$	0	0
$753$	1.73445	0.0632067
$754$	0	0
$755$	−1.00000	−0.0363937
$756$	0	0
$757$	−26.1225	−0.949439	−0.474719	−	0.880137i	$-0.657450\pi$
$757$	−26.1225	−0.949439	−0.474719	+	0.880137i	$0.657450\pi$
$758$	0	0
$759$	−0.0955094	−0.00346677
$760$	0	0
$761$	26.8672	0.973937	0.486968	−	0.873420i	$-0.338103\pi$
$761$	26.8672	0.973937	0.486968	+	0.873420i	$0.338103\pi$
$762$	0	0
$763$	46.7256	1.69158
$764$	0	0
$765$	0.183448	0.00663256
$766$	0	0
$767$	−11.0472	−0.398893
$768$	0	0
$769$	14.2401	0.513509	0.256755	−	0.966477i	$-0.417347\pi$
$769$	14.2401	0.513509	0.256755	+	0.966477i	$0.417347\pi$
$770$	0	0
$771$	−18.6246	−0.670747
$772$	0	0
$773$	43.9862	1.58208	0.791038	−	0.611768i	$-0.209541\pi$
$773$	43.9862	1.58208	0.791038	+	0.611768i	$0.209541\pi$
$774$	0	0
$775$	8.93493	0.320952
$776$	0	0
$777$	−24.9881	−0.896443
$778$	0	0
$779$	48.0789	1.72260
$780$	0	0
$781$	41.6454	1.49019
$782$	0	0
$783$	−0.185687	−0.00663590
$784$	0	0
$785$	−3.82297	−0.136448
$786$	0	0
$787$	20.3060	0.723832	0.361916	−	0.932211i	$-0.382123\pi$
$787$	20.3060	0.723832	0.361916	+	0.932211i	$0.382123\pi$
$788$	0	0
$789$	−31.3018	−1.11437
$790$	0	0
$791$	−12.5265	−0.445389
$792$	0	0
$793$	−33.2824	−1.18189
$794$	0	0
$795$	7.20296	0.255463
$796$	0	0
$797$	−27.3928	−0.970304	−0.485152	−	0.874430i	$-0.661236\pi$
$797$	−27.3928	−0.970304	−0.485152	+	0.874430i	$0.661236\pi$
$798$	0	0
$799$	19.2186	0.679905
$800$	0	0
$801$	0.175030	0.00618438
$802$	0	0
$803$	−21.3571	−0.753675
$804$	0	0
$805$	0.0677234	0.00238694
$806$	0	0
$807$	36.3808	1.28066
$808$	0	0
$809$	−15.8660	−0.557821	−0.278910	−	0.960317i	$-0.589973\pi$
$809$	−15.8660	−0.557821	−0.278910	+	0.960317i	$0.589973\pi$
$810$	0	0
$811$	1.13439	0.0398337	0.0199169	−	0.999802i	$-0.493660\pi$
$811$	1.13439	0.0398337	0.0199169	+	0.999802i	$0.493660\pi$
$812$	0	0
$813$	31.5406	1.10618
$814$	0	0
$815$	−16.6167	−0.582058
$816$	0	0
$817$	10.2653	0.359136
$818$	0	0
$819$	0.798884	0.0279153
$820$	0	0
$821$	−8.72590	−0.304536	−0.152268	−	0.988339i	$-0.548658\pi$
$821$	−8.72590	−0.304536	−0.152268	+	0.988339i	$0.548658\pi$
$822$	0	0
$823$	−3.06388	−0.106800	−0.0534001	−	0.998573i	$-0.517006\pi$
$823$	−3.06388	−0.106800	−0.0534001	+	0.998573i	$0.517006\pi$
$824$	0	0
$825$	4.96479	0.172852
$826$	0	0
$827$	2.24954	0.0782243	0.0391121	−	0.999235i	$-0.487547\pi$
$827$	2.24954	0.0782243	0.0391121	+	0.999235i	$0.487547\pi$
$828$	0	0
$829$	26.0604	0.905116	0.452558	−	0.891735i	$-0.350512\pi$
$829$	26.0604	0.905116	0.452558	+	0.891735i	$0.350512\pi$
$830$	0	0
$831$	52.9351	1.83630
$832$	0	0
$833$	−20.4815	−0.709642
$834$	0	0
$835$	7.03095	0.243316
$836$	0	0
$837$	46.0460	1.59158
$838$	0	0
$839$	−3.49504	−0.120662	−0.0603310	−	0.998178i	$-0.519216\pi$
$839$	−3.49504	−0.120662	−0.0603310	+	0.998178i	$0.519216\pi$
$840$	0	0
$841$	−28.9987	−0.999955
$842$	0	0
$843$	−13.5109	−0.465341
$844$	0	0
$845$	−9.06825	−0.311957
$846$	0	0
$847$	−10.2579	−0.352464
$848$	0	0
$849$	15.4987	0.531912
$850$	0	0
$851$	−0.0782088	−0.00268096
$852$	0	0
$853$	37.4887	1.28359	0.641795	−	0.766876i	$-0.278190\pi$
$853$	37.4887	1.28359	0.641795	+	0.766876i	$0.278190\pi$
$854$	0	0
$855$	0.197300	0.00674752
$856$	0	0
$857$	31.3075	1.06944	0.534722	−	0.845028i	$-0.320416\pi$
$857$	31.3075	1.06944	0.534722	+	0.845028i	$0.320416\pi$
$858$	0	0
$859$	−20.8508	−0.711419	−0.355710	−	0.934597i	$-0.615761\pi$
$859$	−20.8508	−0.711419	−0.355710	+	0.934597i	$0.615761\pi$
$860$	0	0
$861$	72.3529	2.46578
$862$	0	0
$863$	24.4356	0.831798	0.415899	−	0.909411i	$-0.363467\pi$
$863$	24.4356	0.831798	0.415899	+	0.909411i	$0.363467\pi$
$864$	0	0
$865$	5.39551	0.183453
$866$	0	0
$867$	4.50183	0.152890
$868$	0	0
$869$	32.6086	1.10617
$870$	0	0
$871$	47.2958	1.60256
$872$	0	0
$873$	−0.769230	−0.0260345
$874$	0	0
$875$	−3.52041	−0.119012
$876$	0	0
$877$	−35.9860	−1.21516	−0.607581	−	0.794258i	$-0.707860\pi$
$877$	−35.9860	−1.21516	−0.607581	+	0.794258i	$0.707860\pi$
$878$	0	0
$879$	−9.65262	−0.325575
$880$	0	0
$881$	1.52243	0.0512919	0.0256459	−	0.999671i	$-0.491836\pi$
$881$	1.52243	0.0512919	0.0256459	+	0.999671i	$0.491836\pi$
$882$	0	0
$883$	−51.2317	−1.72409	−0.862043	−	0.506836i	$-0.830815\pi$
$883$	−51.2317	−1.72409	−0.862043	+	0.506836i	$0.830815\pi$
$884$	0	0
$885$	−4.10581	−0.138015
$886$	0	0
$887$	12.7146	0.426913	0.213457	−	0.976953i	$-0.431528\pi$
$887$	12.7146	0.426913	0.213457	+	0.976953i	$0.431528\pi$
$888$	0	0
$889$	3.40234	0.114111
$890$	0	0
$891$	25.9980	0.870967
$892$	0	0
$893$	20.6698	0.691689
$894$	0	0
$895$	4.23831	0.141671
$896$	0	0
$897$	0.157782	0.00526820
$898$	0	0
$899$	−0.321937	−0.0107372
$900$	0	0
$901$	−15.6671	−0.521946
$902$	0	0
$903$	15.4480	0.514077
$904$	0	0
$905$	7.04069	0.234041
$906$	0	0
$907$	−34.3131	−1.13935	−0.569675	−	0.821870i	$-0.692931\pi$
$907$	−34.3131	−1.13935	−0.569675	+	0.821870i	$0.692931\pi$
$908$	0	0
$909$	0.375492	0.0124543
$910$	0	0
$911$	33.0276	1.09425	0.547126	−	0.837050i	$-0.315722\pi$
$911$	33.0276	1.09425	0.547126	+	0.837050i	$0.315722\pi$
$912$	0	0
$913$	−25.0849	−0.830188
$914$	0	0
$915$	−12.3697	−0.408930
$916$	0	0
$917$	65.8373	2.17414
$918$	0	0
$919$	−47.4633	−1.56567	−0.782835	−	0.622230i	$-0.786227\pi$
$919$	−47.4633	−1.56567	−0.782835	+	0.622230i	$0.786227\pi$
$920$	0	0
$921$	28.2760	0.931728
$922$	0	0
$923$	−68.7986	−2.26453
$924$	0	0
$925$	4.06547	0.133672
$926$	0	0
$927$	−0.0696939	−0.00228905
$928$	0	0
$929$	21.4205	0.702783	0.351391	−	0.936229i	$-0.385709\pi$
$929$	21.4205	0.702783	0.351391	+	0.936229i	$0.385709\pi$
$930$	0	0
$931$	−22.0281	−0.721942
$932$	0	0
$933$	41.8797	1.37108
$934$	0	0
$935$	−10.7989	−0.353160
$936$	0	0
$937$	−1.54831	−0.0505809	−0.0252905	−	0.999680i	$-0.508051\pi$
$937$	−1.54831	−0.0505809	−0.0252905	+	0.999680i	$0.508051\pi$
$938$	0	0
$939$	53.7692	1.75469
$940$	0	0
$941$	13.1603	0.429014	0.214507	−	0.976722i	$-0.431185\pi$
$941$	13.1603	0.429014	0.214507	+	0.976722i	$0.431185\pi$
$942$	0	0
$943$	0.226453	0.00737433
$944$	0	0
$945$	−18.1424	−0.590172
$946$	0	0
$947$	16.1880	0.526038	0.263019	−	0.964791i	$-0.415282\pi$
$947$	16.1880	0.526038	0.263019	+	0.964791i	$0.415282\pi$
$948$	0	0
$949$	35.2821	1.14530
$950$	0	0
$951$	16.4365	0.532989
$952$	0	0
$953$	−46.5834	−1.50898	−0.754492	−	0.656309i	$-0.772117\pi$
$953$	−46.5834	−1.50898	−0.754492	+	0.656309i	$0.772117\pi$
$954$	0	0
$955$	−3.88900	−0.125845
$956$	0	0
$957$	−0.178888	−0.00578263
$958$	0	0
$959$	−33.7734	−1.09060
$960$	0	0
$961$	48.8330	1.57526
$962$	0	0
$963$	0.256015	0.00824998
$964$	0	0
$965$	6.51215	0.209633
$966$	0	0
$967$	−37.8017	−1.21562	−0.607810	−	0.794083i	$-0.707952\pi$
$967$	−37.8017	−1.21562	−0.607810	+	0.794083i	$0.707952\pi$
$968$	0	0
$969$	−27.0805	−0.869950
$970$	0	0
$971$	20.9751	0.673123	0.336561	−	0.941662i	$-0.390736\pi$
$971$	20.9751	0.673123	0.336561	+	0.941662i	$0.390736\pi$
$972$	0	0
$973$	75.2194	2.41142
$974$	0	0
$975$	−8.20188	−0.262670
$976$	0	0
$977$	−23.5851	−0.754554	−0.377277	−	0.926101i	$-0.623139\pi$
$977$	−23.5851	−0.754554	−0.377277	+	0.926101i	$0.623139\pi$
$978$	0	0
$979$	−10.3033	−0.329296
$980$	0	0
$981$	0.641161	0.0204707
$982$	0	0
$983$	−44.0736	−1.40573	−0.702864	−	0.711324i	$-0.748096\pi$
$983$	−44.0736	−1.40573	−0.702864	+	0.711324i	$0.748096\pi$
$984$	0	0
$985$	11.7988	0.375939
$986$	0	0
$987$	31.1056	0.990102
$988$	0	0
$989$	0.0483498	0.00153743
$990$	0	0
$991$	30.3709	0.964762	0.482381	−	0.875962i	$-0.339772\pi$
$991$	30.3709	0.964762	0.482381	+	0.875962i	$0.339772\pi$
$992$	0	0
$993$	−3.95997	−0.125666
$994$	0	0
$995$	−6.69256	−0.212169
$996$	0	0
$997$	38.7489	1.22719	0.613594	−	0.789621i	$-0.289723\pi$
$997$	38.7489	1.22719	0.613594	+	0.789621i	$0.289723\pi$
$998$	0	0
$999$	20.9513	0.662870

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6040.2.a.n.1.5	✓	13

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6040.2.a.n.1.5	✓	13	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)

\(f(q)\)	\(=\)	\(q-1.74594 q^{3} -1.00000 q^{5} +3.52041 q^{7} +0.0483066 q^{9} +O(q^{10})\)	\(q-1.74594 q^{3} -1.00000 q^{5} +3.52041 q^{7} +0.0483066 q^{9} -2.84362 q^{11} +4.69769 q^{13} +1.74594 q^{15} -3.79757 q^{17} -4.08433 q^{19} -6.14643 q^{21} -0.0192373 q^{23} +1.00000 q^{25} +5.15348 q^{27} -0.0360313 q^{29} +8.93493 q^{31} +4.96479 q^{33} -3.52041 q^{35} +4.06547 q^{37} -8.20188 q^{39} -11.7715 q^{41} -2.51333 q^{43} -0.0483066 q^{45} -5.06076 q^{47} +5.39332 q^{49} +6.63033 q^{51} +4.12555 q^{53} +2.84362 q^{55} +7.13100 q^{57} -2.35163 q^{59} -7.08485 q^{61} +0.170059 q^{63} -4.69769 q^{65} +10.0679 q^{67} +0.0335873 q^{69} -14.6452 q^{71} +7.51052 q^{73} -1.74594 q^{75} -10.0107 q^{77} -11.4673 q^{79} -9.14259 q^{81} +8.82146 q^{83} +3.79757 q^{85} +0.0629085 q^{87} +3.62332 q^{89} +16.5378 q^{91} -15.5999 q^{93} +4.08433 q^{95} -15.9239 q^{97} -0.137366 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 13 q - 4 q^{3} - 13 q^{5} + 5 q^{9}+O(q^{10}) \) 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

Related objects

Downloads

Learn more