LMFDB - Embedded newform 3015.2.a.l.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3015,2,Mod(1,3015)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3015, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("3015.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3015 = 3^{2} \cdot 5 \cdot 67 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3015.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:	$24.0748962094$
Analytic rank:	$1$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - 3x^{6} - 6x^{5} + 15x^{4} + 14x^{3} - 15x^{2} - 6x + 4 $ x^7 - 3x^6 - 6x^5 + 15x^4 + 14x^3 - 15x^2 - 6x + 4
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 1005)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$2.42013$ of defining polynomial
Character	$\chi$	$=$	3015.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.42013 q^{2} +0.0167760 q^{4} -1.00000 q^{5} +3.66225 q^{7} -2.81644 q^{8} +O(q^{10})$	\(q+1.42013 q^{2} +0.0167760 q^{4} -1.00000 q^{5} +3.66225 q^{7} -2.81644 q^{8} -1.42013 q^{10} -3.26380 q^{11} -2.79966 q^{13} +5.20088 q^{14} -4.03327 q^{16} -2.12064 q^{17} +6.67086 q^{19} -0.0167760 q^{20} -4.63502 q^{22} +1.64333 q^{23} +1.00000 q^{25} -3.97589 q^{26} +0.0614380 q^{28} -0.0189191 q^{29} -10.5995 q^{31} -0.0948971 q^{32} -3.01159 q^{34} -3.66225 q^{35} -1.71472 q^{37} +9.47350 q^{38} +2.81644 q^{40} -1.81274 q^{41} -9.11183 q^{43} -0.0547536 q^{44} +2.33375 q^{46} -3.76947 q^{47} +6.41206 q^{49} +1.42013 q^{50} -0.0469673 q^{52} -11.3124 q^{53} +3.26380 q^{55} -10.3145 q^{56} -0.0268676 q^{58} -6.08164 q^{59} -7.18949 q^{61} -15.0527 q^{62} +7.93177 q^{64} +2.79966 q^{65} +1.00000 q^{67} -0.0355759 q^{68} -5.20088 q^{70} +8.11351 q^{71} +5.96747 q^{73} -2.43513 q^{74} +0.111911 q^{76} -11.9528 q^{77} -6.08780 q^{79} +4.03327 q^{80} -2.57432 q^{82} +1.95253 q^{83} +2.12064 q^{85} -12.9400 q^{86} +9.19229 q^{88} -16.2631 q^{89} -10.2531 q^{91} +0.0275686 q^{92} -5.35315 q^{94} -6.67086 q^{95} +4.43089 q^{97} +9.10598 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q - 4 q^{2} + 8 q^{4} - 7 q^{5} + 3 q^{7} - 9 q^{8}+O(q^{10}) $ 7 * q - 4 * q^2 + 8 * q^4 - 7 * q^5 + 3 * q^7 - 9 * q^8	$ 7 q - 4 q^{2} + 8 q^{4} - 7 q^{5} + 3 q^{7} - 9 q^{8} + 4 q^{10} - 5 q^{11} - q^{13} + 4 q^{14} + 6 q^{16} - 11 q^{17} + 8 q^{19} - 8 q^{20} - 3 q^{22} - 11 q^{23} + 7 q^{25} + 5 q^{26} - 17 q^{28} - 3 q^{31} - 22 q^{32} + 4 q^{34} - 3 q^{35} - 5 q^{37} + 9 q^{40} - q^{41} - 3 q^{43} + 9 q^{44} - 12 q^{46} - 10 q^{47} - 4 q^{49} - 4 q^{50} - 6 q^{52} - 3 q^{53} + 5 q^{55} + 12 q^{56} - 24 q^{58} + 4 q^{59} - 9 q^{61} - 20 q^{62} - 3 q^{64} + q^{65} + 7 q^{67} + 3 q^{68} - 4 q^{70} - 6 q^{71} - 9 q^{73} - 2 q^{74} + 2 q^{76} - 17 q^{77} - 11 q^{79} - 6 q^{80} - 16 q^{82} - 30 q^{83} + 11 q^{85} + 11 q^{86} - 25 q^{88} - 13 q^{89} - 5 q^{91} - 10 q^{92} - 25 q^{94} - 8 q^{95} - 7 q^{97} + 10 q^{98}+O(q^{100}) $ 7 * q - 4 * q^2 + 8 * q^4 - 7 * q^5 + 3 * q^7 - 9 * q^8 + 4 * q^10 - 5 * q^11 - q^13 + 4 * q^14 + 6 * q^16 - 11 * q^17 + 8 * q^19 - 8 * q^20 - 3 * q^22 - 11 * q^23 + 7 * q^25 + 5 * q^26 - 17 * q^28 - 3 * q^31 - 22 * q^32 + 4 * q^34 - 3 * q^35 - 5 * q^37 + 9 * q^40 - q^41 - 3 * q^43 + 9 * q^44 - 12 * q^46 - 10 * q^47 - 4 * q^49 - 4 * q^50 - 6 * q^52 - 3 * q^53 + 5 * q^55 + 12 * q^56 - 24 * q^58 + 4 * q^59 - 9 * q^61 - 20 * q^62 - 3 * q^64 + q^65 + 7 * q^67 + 3 * q^68 - 4 * q^70 - 6 * q^71 - 9 * q^73 - 2 * q^74 + 2 * q^76 - 17 * q^77 - 11 * q^79 - 6 * q^80 - 16 * q^82 - 30 * q^83 + 11 * q^85 + 11 * q^86 - 25 * q^88 - 13 * q^89 - 5 * q^91 - 10 * q^92 - 25 * q^94 - 8 * q^95 - 7 * q^97 + 10 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	1.42013	1.00419	0.502093	−	0.864814i	$-0.332564\pi$
$2$	1.42013	1.00419	0.502093	+	0.864814i	$0.332564\pi$
$3$	0	0
$3$	0	0
$4$	0.0167760	0.00838802
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	3.66225	1.38420	0.692100	−	0.721802i	$-0.256686\pi$
$7$	3.66225	1.38420	0.692100	+	0.721802i	$0.256686\pi$
$8$	−2.81644	−0.995762
$9$	0	0
$10$	−1.42013	−0.449085
$11$	−3.26380	−0.984072	−0.492036	−	0.870575i	$-0.663747\pi$
$11$	−3.26380	−0.984072	−0.492036	+	0.870575i	$0.663747\pi$
$12$	0	0
$13$	−2.79966	−0.776487	−0.388244	−	0.921557i	$-0.626918\pi$
$13$	−2.79966	−0.776487	−0.388244	+	0.921557i	$0.626918\pi$
$14$	5.20088	1.38999
$15$	0	0
$16$	−4.03327	−1.00832
$17$	−2.12064	−0.514331	−0.257165	−	0.966367i	$-0.582789\pi$
$17$	−2.12064	−0.514331	−0.257165	+	0.966367i	$0.582789\pi$
$18$	0	0
$19$	6.67086	1.53040	0.765200	−	0.643793i	$-0.222640\pi$
$19$	6.67086	1.53040	0.765200	+	0.643793i	$0.222640\pi$
$20$	−0.0167760	−0.00375124
$21$	0	0
$22$	−4.63502	−0.988190
$23$	1.64333	0.342658	0.171329	−	0.985214i	$-0.445194\pi$
$23$	1.64333	0.342658	0.171329	+	0.985214i	$0.445194\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	−3.97589	−0.779737
$27$	0	0
$28$	0.0614380	0.0116107
$29$	−0.0189191	−0.00351319	−0.00175660	−	0.999998i	$-0.500559\pi$
$29$	−0.0189191	−0.00351319	−0.00175660	+	0.999998i	$0.500559\pi$
$30$	0	0
$31$	−10.5995	−1.90373	−0.951866	−	0.306516i	$-0.900837\pi$
$31$	−10.5995	−1.90373	−0.951866	+	0.306516i	$0.900837\pi$
$32$	−0.0948971	−0.0167756
$33$	0	0
$34$	−3.01159	−0.516483
$35$	−3.66225	−0.619033
$36$	0	0
$37$	−1.71472	−0.281898	−0.140949	−	0.990017i	$-0.545015\pi$
$37$	−1.71472	−0.281898	−0.140949	+	0.990017i	$0.545015\pi$
$38$	9.47350	1.53681
$39$	0	0
$40$	2.81644	0.445318
$41$	−1.81274	−0.283102	−0.141551	−	0.989931i	$-0.545209\pi$
$41$	−1.81274	−0.283102	−0.141551	+	0.989931i	$0.545209\pi$
$42$	0	0
$43$	−9.11183	−1.38954	−0.694770	−	0.719232i	$-0.744494\pi$
$43$	−9.11183	−1.38954	−0.694770	+	0.719232i	$0.744494\pi$
$44$	−0.0547536	−0.00825441
$45$	0	0
$46$	2.33375	0.344092
$47$	−3.76947	−0.549834	−0.274917	−	0.961468i	$-0.588650\pi$
$47$	−3.76947	−0.549834	−0.274917	+	0.961468i	$0.588650\pi$
$48$	0	0
$49$	6.41206	0.916009
$50$	1.42013	0.200837
$51$	0	0
$52$	−0.0469673	−0.00651319
$53$	−11.3124	−1.55387	−0.776936	−	0.629579i	$-0.783227\pi$
$53$	−11.3124	−1.55387	−0.776936	+	0.629579i	$0.783227\pi$
$54$	0	0
$55$	3.26380	0.440090
$56$	−10.3145	−1.37833
$57$	0	0
$58$	−0.0268676	−0.00352789
$59$	−6.08164	−0.791762	−0.395881	−	0.918302i	$-0.629561\pi$
$59$	−6.08164	−0.791762	−0.395881	+	0.918302i	$0.629561\pi$
$60$	0	0
$61$	−7.18949	−0.920519	−0.460260	−	0.887784i	$-0.652244\pi$
$61$	−7.18949	−0.920519	−0.460260	+	0.887784i	$0.652244\pi$
$62$	−15.0527	−1.91170
$63$	0	0
$64$	7.93177	0.991472
$65$	2.79966	0.347256
$66$	0	0
$67$	1.00000	0.122169
$67$	1.00000	0.122169
$68$	−0.0355759	−0.00431422
$69$	0	0
$70$	−5.20088	−0.621624
$71$	8.11351	0.962896	0.481448	−	0.876475i	$-0.340111\pi$
$71$	8.11351	0.962896	0.481448	+	0.876475i	$0.340111\pi$
$72$	0	0
$73$	5.96747	0.698440	0.349220	−	0.937041i	$-0.386447\pi$
$73$	5.96747	0.698440	0.349220	+	0.937041i	$0.386447\pi$
$74$	−2.43513	−0.283078
$75$	0	0
$76$	0.111911	0.0128370
$77$	−11.9528	−1.36215
$78$	0	0
$79$	−6.08780	−0.684932	−0.342466	−	0.939530i	$-0.611262\pi$
$79$	−6.08780	−0.684932	−0.342466	+	0.939530i	$0.611262\pi$
$80$	4.03327	0.450933
$81$	0	0
$82$	−2.57432	−0.284287
$83$	1.95253	0.214318	0.107159	−	0.994242i	$-0.465825\pi$
$83$	1.95253	0.214318	0.107159	+	0.994242i	$0.465825\pi$
$84$	0	0
$85$	2.12064	0.230016
$86$	−12.9400	−1.39536
$87$	0	0
$88$	9.19229	0.979902
$89$	−16.2631	−1.72388	−0.861942	−	0.507007i	$-0.830752\pi$
$89$	−16.2631	−1.72388	−0.861942	+	0.507007i	$0.830752\pi$
$90$	0	0
$91$	−10.2531	−1.07481
$92$	0.0275686	0.00287422
$93$	0	0
$94$	−5.35315	−0.552135
$95$	−6.67086	−0.684416
$96$	0	0
$97$	4.43089	0.449888	0.224944	−	0.974372i	$-0.427780\pi$
$97$	4.43089	0.449888	0.224944	+	0.974372i	$0.427780\pi$
$98$	9.10598	0.919843
$99$	0	0
$100$	0.0167760	0.00167760
$101$	15.6077	1.55302	0.776510	−	0.630105i	$-0.216988\pi$
$101$	15.6077	1.55302	0.776510	+	0.630105i	$0.216988\pi$
$102$	0	0
$103$	−1.97784	−0.194882	−0.0974411	−	0.995241i	$-0.531066\pi$
$103$	−1.97784	−0.194882	−0.0974411	+	0.995241i	$0.531066\pi$
$104$	7.88509	0.773197
$105$	0	0
$106$	−16.0651	−1.56038
$107$	−14.1608	−1.36898	−0.684488	−	0.729024i	$-0.739974\pi$
$107$	−14.1608	−1.36898	−0.684488	+	0.729024i	$0.739974\pi$
$108$	0	0
$109$	7.47058	0.715552	0.357776	−	0.933807i	$-0.383535\pi$
$109$	7.47058	0.715552	0.357776	+	0.933807i	$0.383535\pi$
$110$	4.63502	0.441932
$111$	0	0
$112$	−14.7708	−1.39571
$113$	−9.85628	−0.927200	−0.463600	−	0.886044i	$-0.653443\pi$
$113$	−9.85628	−0.927200	−0.463600	+	0.886044i	$0.653443\pi$
$114$	0	0
$115$	−1.64333	−0.153241
$116$	−0.000317388 0	−2.94687e−5 0
$117$	0	0
$118$	−8.63674	−0.795076
$119$	−7.76631	−0.711937
$120$	0	0
$121$	−0.347627	−0.0316025
$122$	−10.2100	−0.924372
$123$	0	0
$124$	−0.177818	−0.0159685
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	2.58185	0.229102	0.114551	−	0.993417i	$-0.463457\pi$
$127$	2.58185	0.229102	0.114551	+	0.993417i	$0.463457\pi$
$128$	11.4540	1.01240
$129$	0	0
$130$	3.97589	0.348709
$131$	6.97007	0.608978	0.304489	−	0.952516i	$-0.401514\pi$
$131$	6.97007	0.608978	0.304489	+	0.952516i	$0.401514\pi$
$132$	0	0
$133$	24.4303	2.11838
$134$	1.42013	0.122681
$135$	0	0
$136$	5.97266	0.512151
$137$	−11.0644	−0.945295	−0.472648	−	0.881251i	$-0.656702\pi$
$137$	−11.0644	−0.945295	−0.472648	+	0.881251i	$0.656702\pi$
$138$	0	0
$139$	−3.51297	−0.297966	−0.148983	−	0.988840i	$-0.547600\pi$
$139$	−3.51297	−0.297966	−0.148983	+	0.988840i	$0.547600\pi$
$140$	−0.0614380	−0.00519246
$141$	0	0
$142$	11.5223	0.966926
$143$	9.13754	0.764119
$144$	0	0
$145$	0.0189191	0.00157115
$146$	8.47460	0.701363
$147$	0	0
$148$	−0.0287662	−0.00236457
$149$	7.82118	0.640736	0.320368	−	0.947293i	$-0.396193\pi$
$149$	7.82118	0.640736	0.320368	+	0.947293i	$0.396193\pi$
$150$	0	0
$151$	−8.04607	−0.654780	−0.327390	−	0.944889i	$-0.606169\pi$
$151$	−8.04607	−0.654780	−0.327390	+	0.944889i	$0.606169\pi$
$152$	−18.7881	−1.52391
$153$	0	0
$154$	−16.9746	−1.36785
$155$	10.5995	0.851375
$156$	0	0
$157$	−2.78927	−0.222608	−0.111304	−	0.993786i	$-0.535503\pi$
$157$	−2.78927	−0.222608	−0.111304	+	0.993786i	$0.535503\pi$
$158$	−8.64549	−0.687798
$159$	0	0
$160$	0.0948971	0.00750227
$161$	6.01828	0.474307
$162$	0	0
$163$	21.6024	1.69203	0.846016	−	0.533158i	$-0.178995\pi$
$163$	21.6024	1.69203	0.846016	+	0.533158i	$0.178995\pi$
$164$	−0.0304105	−0.00237466
$165$	0	0
$166$	2.77285	0.215215
$167$	−18.3413	−1.41930	−0.709648	−	0.704557i	$-0.751146\pi$
$167$	−18.3413	−1.41930	−0.709648	+	0.704557i	$0.751146\pi$
$168$	0	0
$169$	−5.16188	−0.397068
$170$	3.01159	0.230978
$171$	0	0
$172$	−0.152860	−0.0116555
$173$	21.7221	1.65150	0.825751	−	0.564035i	$-0.190752\pi$
$173$	21.7221	1.65150	0.825751	+	0.564035i	$0.190752\pi$
$174$	0	0
$175$	3.66225	0.276840
$176$	13.1638	0.992257
$177$	0	0
$178$	−23.0957	−1.73110
$179$	16.1880	1.20995	0.604973	−	0.796246i	$-0.293184\pi$
$179$	16.1880	1.20995	0.604973	+	0.796246i	$0.293184\pi$
$180$	0	0
$181$	−0.670680	−0.0498513	−0.0249256	−	0.999689i	$-0.507935\pi$
$181$	−0.670680	−0.0498513	−0.0249256	+	0.999689i	$0.507935\pi$
$182$	−14.5607	−1.07931
$183$	0	0
$184$	−4.62834	−0.341206
$185$	1.71472	0.126069
$186$	0	0
$187$	6.92134	0.506138
$188$	−0.0632368	−0.00461202
$189$	0	0
$190$	−9.47350	−0.687280
$191$	−10.6974	−0.774036	−0.387018	−	0.922072i	$-0.626495\pi$
$191$	−10.6974	−0.774036	−0.387018	+	0.922072i	$0.626495\pi$
$192$	0	0
$193$	−7.09859	−0.510968	−0.255484	−	0.966813i	$-0.582235\pi$
$193$	−7.09859	−0.510968	−0.255484	+	0.966813i	$0.582235\pi$
$194$	6.29244	0.451771
$195$	0	0
$196$	0.107569	0.00768350
$197$	13.9254	0.992147	0.496074	−	0.868281i	$-0.334775\pi$
$197$	13.9254	0.992147	0.496074	+	0.868281i	$0.334775\pi$
$198$	0	0
$199$	−23.9299	−1.69635	−0.848173	−	0.529719i	$-0.822297\pi$
$199$	−23.9299	−1.69635	−0.848173	+	0.529719i	$0.822297\pi$
$200$	−2.81644	−0.199152
$201$	0	0
$202$	22.1649	1.55952
$203$	−0.0692865	−0.00486296
$204$	0	0
$205$	1.81274	0.126607
$206$	−2.80879	−0.195698
$207$	0	0
$208$	11.2918	0.782946
$209$	−21.7723	−1.50602
$210$	0	0
$211$	−18.8988	−1.30105	−0.650524	−	0.759485i	$-0.725451\pi$
$211$	−18.8988	−1.30105	−0.650524	+	0.759485i	$0.725451\pi$
$212$	−0.189777	−0.0130339
$213$	0	0
$214$	−20.1102	−1.37471
$215$	9.11183	0.621422
$216$	0	0
$217$	−38.8181	−2.63514
$218$	10.6092	0.718547
$219$	0	0
$220$	0.0547536	0.00369149
$221$	5.93708	0.399371
$222$	0	0
$223$	3.78003	0.253129	0.126565	−	0.991958i	$-0.459605\pi$
$223$	3.78003	0.253129	0.126565	+	0.991958i	$0.459605\pi$
$224$	−0.347537	−0.0232208
$225$	0	0
$226$	−13.9972	−0.931081
$227$	−4.02424	−0.267098	−0.133549	−	0.991042i	$-0.542637\pi$
$227$	−4.02424	−0.267098	−0.133549	+	0.991042i	$0.542637\pi$
$228$	0	0
$229$	−15.3718	−1.01580	−0.507899	−	0.861416i	$-0.669578\pi$
$229$	−15.3718	−1.01580	−0.507899	+	0.861416i	$0.669578\pi$
$230$	−2.33375	−0.153883
$231$	0	0
$232$	0.0532845	0.00349830
$233$	16.8263	1.10233	0.551164	−	0.834397i	$-0.314184\pi$
$233$	16.8263	1.10233	0.551164	+	0.834397i	$0.314184\pi$
$234$	0	0
$235$	3.76947	0.245893
$236$	−0.102026	−0.00664132
$237$	0	0
$238$	−11.0292	−0.714916
$239$	−18.1984	−1.17715	−0.588577	−	0.808441i	$-0.700312\pi$
$239$	−18.1984	−1.17715	−0.588577	+	0.808441i	$0.700312\pi$
$240$	0	0
$241$	9.42628	0.607200	0.303600	−	0.952800i	$-0.401811\pi$
$241$	9.42628	0.607200	0.303600	+	0.952800i	$0.401811\pi$
$242$	−0.493677	−0.0317347
$243$	0	0
$244$	−0.120611	−0.00772133
$245$	−6.41206	−0.409652
$246$	0	0
$247$	−18.6762	−1.18834
$248$	29.8529	1.89566
$249$	0	0
$250$	−1.42013	−0.0898171
$251$	−5.65618	−0.357015	−0.178507	−	0.983939i	$-0.557127\pi$
$251$	−5.65618	−0.357015	−0.178507	+	0.983939i	$0.557127\pi$
$252$	0	0
$253$	−5.36349	−0.337200
$254$	3.66657	0.230061
$255$	0	0
$256$	0.402597	0.0251623
$257$	19.5519	1.21961	0.609806	−	0.792551i	$-0.291247\pi$
$257$	19.5519	1.21961	0.609806	+	0.792551i	$0.291247\pi$
$258$	0	0
$259$	−6.27973	−0.390203
$260$	0.0469673	0.00291279
$261$	0	0
$262$	9.89842	0.611526
$263$	2.84131	0.175203	0.0876013	−	0.996156i	$-0.472080\pi$
$263$	2.84131	0.175203	0.0876013	+	0.996156i	$0.472080\pi$
$264$	0	0
$265$	11.3124	0.694913
$266$	34.6943	2.12725
$267$	0	0
$268$	0.0167760	0.00102476
$269$	18.4970	1.12778	0.563890	−	0.825850i	$-0.309304\pi$
$269$	18.4970	1.12778	0.563890	+	0.825850i	$0.309304\pi$
$270$	0	0
$271$	5.41652	0.329030	0.164515	−	0.986375i	$-0.447394\pi$
$271$	5.41652	0.329030	0.164515	+	0.986375i	$0.447394\pi$
$272$	8.55311	0.518609
$273$	0	0
$274$	−15.7129	−0.949252
$275$	−3.26380	−0.196814
$276$	0	0
$277$	31.1357	1.87076	0.935381	−	0.353643i	$-0.115057\pi$
$277$	31.1357	1.87076	0.935381	+	0.353643i	$0.115057\pi$
$278$	−4.98888	−0.299213
$279$	0	0
$280$	10.3145	0.616410
$281$	−0.812247	−0.0484546	−0.0242273	−	0.999706i	$-0.507713\pi$
$281$	−0.812247	−0.0484546	−0.0242273	+	0.999706i	$0.507713\pi$
$282$	0	0
$283$	−7.04330	−0.418681	−0.209340	−	0.977843i	$-0.567132\pi$
$283$	−7.04330	−0.418681	−0.209340	+	0.977843i	$0.567132\pi$
$284$	0.136113	0.00807679
$285$	0	0
$286$	12.9765	0.767317
$287$	−6.63869	−0.391869
$288$	0	0
$289$	−12.5029	−0.735464
$290$	0.0268676	0.00157772
$291$	0	0
$292$	0.100110	0.00585852
$293$	−13.0361	−0.761578	−0.380789	−	0.924662i	$-0.624348\pi$
$293$	−13.0361	−0.761578	−0.380789	+	0.924662i	$0.624348\pi$
$294$	0	0
$295$	6.08164	0.354087
$296$	4.82941	0.280704
$297$	0	0
$298$	11.1071	0.643418
$299$	−4.60077	−0.266069
$300$	0	0
$301$	−33.3698	−1.92340
$302$	−11.4265	−0.657520
$303$	0	0
$304$	−26.9054	−1.54313
$305$	7.18949	0.411669
$306$	0	0
$307$	18.9833	1.08343	0.541716	−	0.840562i	$-0.317775\pi$
$307$	18.9833	1.08343	0.541716	+	0.840562i	$0.317775\pi$
$308$	−0.200521	−0.0114258
$309$	0	0
$310$	15.0527	0.854938
$311$	1.11836	0.0634166	0.0317083	−	0.999497i	$-0.489905\pi$
$311$	1.11836	0.0634166	0.0317083	+	0.999497i	$0.489905\pi$
$312$	0	0
$313$	−26.5721	−1.50194	−0.750972	−	0.660334i	$-0.770415\pi$
$313$	−26.5721	−1.50194	−0.750972	+	0.660334i	$0.770415\pi$
$314$	−3.96113	−0.223540
$315$	0	0
$316$	−0.102129	−0.00574522
$317$	10.2185	0.573930	0.286965	−	0.957941i	$-0.407354\pi$
$317$	10.2185	0.573930	0.286965	+	0.957941i	$0.407354\pi$
$318$	0	0
$319$	0.0617481	0.00345723
$320$	−7.93177	−0.443400
$321$	0	0
$322$	8.54676	0.476292
$323$	−14.1465	−0.787132
$324$	0	0
$325$	−2.79966	−0.155297
$326$	30.6783	1.69911
$327$	0	0
$328$	5.10546	0.281902
$329$	−13.8047	−0.761080
$330$	0	0
$331$	−3.73565	−0.205330	−0.102665	−	0.994716i	$-0.532737\pi$
$331$	−3.73565	−0.205330	−0.102665	+	0.994716i	$0.532737\pi$
$332$	0.0327557	0.00179771
$333$	0	0
$334$	−26.0471	−1.42524
$335$	−1.00000	−0.0546358
$336$	0	0
$337$	−34.0269	−1.85356	−0.926782	−	0.375599i	$-0.877437\pi$
$337$	−34.0269	−1.85356	−0.926782	+	0.375599i	$0.877437\pi$
$338$	−7.33055	−0.398729
$339$	0	0
$340$	0.0355759	0.00192938
$341$	34.5947	1.87341
$342$	0	0
$343$	−2.15317	−0.116260
$344$	25.6629	1.38365
$345$	0	0
$346$	30.8483	1.65841
$347$	29.2097	1.56806	0.784031	−	0.620722i	$-0.213160\pi$
$347$	29.2097	1.56806	0.784031	+	0.620722i	$0.213160\pi$
$348$	0	0
$349$	21.3114	1.14077	0.570386	−	0.821377i	$-0.306794\pi$
$349$	21.3114	1.14077	0.570386	+	0.821377i	$0.306794\pi$
$350$	5.20088	0.277999
$351$	0	0
$352$	0.309725	0.0165084
$353$	6.32580	0.336688	0.168344	−	0.985728i	$-0.446158\pi$
$353$	6.32580	0.336688	0.168344	+	0.985728i	$0.446158\pi$
$354$	0	0
$355$	−8.11351	−0.430620
$356$	−0.272830	−0.0144600
$357$	0	0
$358$	22.9891	1.21501
$359$	14.7835	0.780245	0.390123	−	0.920763i	$-0.372433\pi$
$359$	14.7835	0.780245	0.390123	+	0.920763i	$0.372433\pi$
$360$	0	0
$361$	25.5004	1.34212
$362$	−0.952454	−0.0500599
$363$	0	0
$364$	−0.172006	−0.00901555
$365$	−5.96747	−0.312352
$366$	0	0
$367$	22.8315	1.19180	0.595898	−	0.803060i	$-0.296796\pi$
$367$	22.8315	1.19180	0.595898	+	0.803060i	$0.296796\pi$
$368$	−6.62799	−0.345508
$369$	0	0
$370$	2.43513	0.126596
$371$	−41.4287	−2.15087
$372$	0	0
$373$	29.0985	1.50666	0.753332	−	0.657640i	$-0.228445\pi$
$373$	29.0985	1.50666	0.753332	+	0.657640i	$0.228445\pi$
$374$	9.82922	0.508257
$375$	0	0
$376$	10.6165	0.547504
$377$	0.0529672	0.00272795
$378$	0	0
$379$	16.6978	0.857708	0.428854	−	0.903374i	$-0.358917\pi$
$379$	16.6978	0.857708	0.428854	+	0.903374i	$0.358917\pi$
$380$	−0.111911	−0.00574089
$381$	0	0
$382$	−15.1917	−0.777275
$383$	−24.5499	−1.25444	−0.627220	−	0.778842i	$-0.715807\pi$
$383$	−24.5499	−1.25444	−0.627220	+	0.778842i	$0.715807\pi$
$384$	0	0
$385$	11.9528	0.609173
$386$	−10.0809	−0.513106
$387$	0	0
$388$	0.0743327	0.00377367
$389$	35.1081	1.78005	0.890025	−	0.455912i	$-0.150687\pi$
$389$	35.1081	1.78005	0.890025	+	0.455912i	$0.150687\pi$
$390$	0	0
$391$	−3.48491	−0.176239
$392$	−18.0592	−0.912127
$393$	0	0
$394$	19.7760	0.996299
$395$	6.08780	0.306311
$396$	0	0
$397$	−17.8265	−0.894686	−0.447343	−	0.894363i	$-0.647630\pi$
$397$	−17.8265	−0.894686	−0.447343	+	0.894363i	$0.647630\pi$
$398$	−33.9836	−1.70345
$399$	0	0
$400$	−4.03327	−0.201664
$401$	−3.97752	−0.198628	−0.0993140	−	0.995056i	$-0.531665\pi$
$401$	−3.97752	−0.198628	−0.0993140	+	0.995056i	$0.531665\pi$
$402$	0	0
$403$	29.6751	1.47822
$404$	0.261835	0.0130268
$405$	0	0
$406$	−0.0983960	−0.00488331
$407$	5.59650	0.277408
$408$	0	0
$409$	35.9212	1.77619	0.888094	−	0.459661i	$-0.152029\pi$
$409$	35.9212	1.77619	0.888094	+	0.459661i	$0.152029\pi$
$410$	2.57432	0.127137
$411$	0	0
$412$	−0.0331803	−0.00163468
$413$	−22.2725	−1.09596
$414$	0	0
$415$	−1.95253	−0.0958460
$416$	0.265680	0.0130260
$417$	0	0
$418$	−30.9196	−1.51233
$419$	13.8740	0.677789	0.338895	−	0.940824i	$-0.389947\pi$
$419$	13.8740	0.677789	0.338895	+	0.940824i	$0.389947\pi$
$420$	0	0
$421$	40.5473	1.97616	0.988078	−	0.153952i	$-0.0492000\pi$
$421$	40.5473	1.97616	0.988078	+	0.153952i	$0.0492000\pi$
$422$	−26.8388	−1.30649
$423$	0	0
$424$	31.8606	1.54729
$425$	−2.12064	−0.102866
$426$	0	0
$427$	−26.3297	−1.27418
$428$	−0.237562	−0.0114830
$429$	0	0
$430$	12.9400	0.624022
$431$	9.79550	0.471833	0.235916	−	0.971773i	$-0.424191\pi$
$431$	9.79550	0.471833	0.235916	+	0.971773i	$0.424191\pi$
$432$	0	0
$433$	−19.1370	−0.919667	−0.459833	−	0.888005i	$-0.652091\pi$
$433$	−19.1370	−0.919667	−0.459833	+	0.888005i	$0.652091\pi$
$434$	−55.1268	−2.64617
$435$	0	0
$436$	0.125327	0.00600206
$437$	10.9624	0.524404
$438$	0	0
$439$	8.03843	0.383653	0.191827	−	0.981429i	$-0.438559\pi$
$439$	8.03843	0.383653	0.191827	+	0.981429i	$0.438559\pi$
$440$	−9.19229	−0.438225
$441$	0	0
$442$	8.43144	0.401043
$443$	13.0740	0.621162	0.310581	−	0.950547i	$-0.399476\pi$
$443$	13.0740	0.621162	0.310581	+	0.950547i	$0.399476\pi$
$444$	0	0
$445$	16.2631	0.770944
$446$	5.36814	0.254189
$447$	0	0
$448$	29.0481	1.37240
$449$	25.1929	1.18892	0.594462	−	0.804124i	$-0.297365\pi$
$449$	25.1929	1.18892	0.594462	+	0.804124i	$0.297365\pi$
$450$	0	0
$451$	5.91640	0.278592
$452$	−0.165349	−0.00777737
$453$	0	0
$454$	−5.71495	−0.268216
$455$	10.2531	0.480671
$456$	0	0
$457$	4.80210	0.224633	0.112316	−	0.993672i	$-0.464173\pi$
$457$	4.80210	0.224633	0.112316	+	0.993672i	$0.464173\pi$
$458$	−21.8300	−1.02005
$459$	0	0
$460$	−0.0275686	−0.00128539
$461$	6.82376	0.317814	0.158907	−	0.987294i	$-0.449203\pi$
$461$	6.82376	0.317814	0.158907	+	0.987294i	$0.449203\pi$
$462$	0	0
$463$	−6.34009	−0.294649	−0.147324	−	0.989088i	$-0.547066\pi$
$463$	−6.34009	−0.294649	−0.147324	+	0.989088i	$0.547066\pi$
$464$	0.0763059	0.00354241
$465$	0	0
$466$	23.8956	1.10694
$467$	13.1557	0.608774	0.304387	−	0.952549i	$-0.401548\pi$
$467$	13.1557	0.608774	0.304387	+	0.952549i	$0.401548\pi$
$468$	0	0
$469$	3.66225	0.169107
$470$	5.35315	0.246922
$471$	0	0
$472$	17.1286	0.788407
$473$	29.7392	1.36741
$474$	0	0
$475$	6.67086	0.306080
$476$	−0.130288	−0.00597174
$477$	0	0
$478$	−25.8441	−1.18208
$479$	34.9990	1.59915	0.799574	−	0.600568i	$-0.205059\pi$
$479$	34.9990	1.59915	0.799574	+	0.600568i	$0.205059\pi$
$480$	0	0
$481$	4.80064	0.218890
$482$	13.3866	0.609741
$483$	0	0
$484$	−0.00583181	−0.000265082 0
$485$	−4.43089	−0.201196
$486$	0	0
$487$	−35.5178	−1.60946	−0.804732	−	0.593639i	$-0.797691\pi$
$487$	−35.5178	−1.60946	−0.804732	+	0.593639i	$0.797691\pi$
$488$	20.2488	0.916618
$489$	0	0
$490$	−9.10598	−0.411366
$491$	−25.2985	−1.14171	−0.570853	−	0.821052i	$-0.693387\pi$
$491$	−25.2985	−1.14171	−0.570853	+	0.821052i	$0.693387\pi$
$492$	0	0
$493$	0.0401206	0.00180694
$494$	−26.5226	−1.19331
$495$	0	0
$496$	42.7508	1.91957
$497$	29.7137	1.33284
$498$	0	0
$499$	−6.05417	−0.271022	−0.135511	−	0.990776i	$-0.543268\pi$
$499$	−6.05417	−0.271022	−0.135511	+	0.990776i	$0.543268\pi$
$500$	−0.0167760	−0.000750247 0
$501$	0	0
$502$	−8.03252	−0.358509
$503$	−17.6401	−0.786533	−0.393266	−	0.919425i	$-0.628655\pi$
$503$	−17.6401	−0.786533	−0.393266	+	0.919425i	$0.628655\pi$
$504$	0	0
$505$	−15.6077	−0.694532
$506$	−7.61687	−0.338611
$507$	0	0
$508$	0.0433132	0.00192171
$509$	−38.6106	−1.71138	−0.855692	−	0.517486i	$-0.826868\pi$
$509$	−38.6106	−1.71138	−0.855692	+	0.517486i	$0.826868\pi$
$510$	0	0
$511$	21.8544	0.966780
$512$	−22.3362	−0.987129
$513$	0	0
$514$	27.7662	1.22472
$515$	1.97784	0.0871540
$516$	0	0
$517$	12.3028	0.541076
$518$	−8.91805	−0.391837
$519$	0	0
$520$	−7.88509	−0.345784
$521$	−0.934016	−0.0409200	−0.0204600	−	0.999791i	$-0.506513\pi$
$521$	−0.934016	−0.0409200	−0.0204600	+	0.999791i	$0.506513\pi$
$522$	0	0
$523$	44.0708	1.92708	0.963542	−	0.267558i	$-0.0862168\pi$
$523$	44.0708	1.92708	0.963542	+	0.267558i	$0.0862168\pi$
$524$	0.116930	0.00510812
$525$	0	0
$526$	4.03503	0.175936
$527$	22.4778	0.979148
$528$	0	0
$529$	−20.2995	−0.882586
$530$	16.0651	0.697821
$531$	0	0
$532$	0.409844	0.0177690
$533$	5.07505	0.219825
$534$	0	0
$535$	14.1608	0.612225
$536$	−2.81644	−0.121652
$537$	0	0
$538$	26.2681	1.13250
$539$	−20.9277	−0.901419
$540$	0	0
$541$	−23.5988	−1.01459	−0.507295	−	0.861772i	$-0.669355\pi$
$541$	−23.5988	−1.01459	−0.507295	+	0.861772i	$0.669355\pi$
$542$	7.69218	0.330407
$543$	0	0
$544$	0.201243	0.00862820
$545$	−7.47058	−0.320005
$546$	0	0
$547$	25.0627	1.07160	0.535801	−	0.844344i	$-0.320010\pi$
$547$	25.0627	1.07160	0.535801	+	0.844344i	$0.320010\pi$
$548$	−0.185617	−0.00792915
$549$	0	0
$550$	−4.63502	−0.197638
$551$	−0.126207	−0.00537659
$552$	0	0
$553$	−22.2951	−0.948082
$554$	44.2168	1.87859
$555$	0	0
$556$	−0.0589337	−0.00249934
$557$	−34.0542	−1.44292	−0.721461	−	0.692456i	$-0.756529\pi$
$557$	−34.0542	−1.44292	−0.721461	+	0.692456i	$0.756529\pi$
$558$	0	0
$559$	25.5101	1.07896
$560$	14.7708	0.624182
$561$	0	0
$562$	−1.15350	−0.0486574
$563$	12.2856	0.517777	0.258889	−	0.965907i	$-0.416644\pi$
$563$	12.2856	0.517777	0.258889	+	0.965907i	$0.416644\pi$
$564$	0	0
$565$	9.85628	0.414657
$566$	−10.0024	−0.420433
$567$	0	0
$568$	−22.8512	−0.958816
$569$	−28.8290	−1.20858	−0.604288	−	0.796766i	$-0.706542\pi$
$569$	−28.8290	−1.20858	−0.604288	+	0.796766i	$0.706542\pi$
$570$	0	0
$571$	−2.57238	−0.107651	−0.0538253	−	0.998550i	$-0.517141\pi$
$571$	−2.57238	−0.107651	−0.0538253	+	0.998550i	$0.517141\pi$
$572$	0.153292	0.00640945
$573$	0	0
$574$	−9.42782	−0.393509
$575$	1.64333	0.0685316
$576$	0	0
$577$	−22.9339	−0.954752	−0.477376	−	0.878699i	$-0.658412\pi$
$577$	−22.9339	−0.954752	−0.477376	+	0.878699i	$0.658412\pi$
$578$	−17.7558	−0.738542
$579$	0	0
$580$	0.000317388 0	1.31788e−5 0
$581$	7.15066	0.296659
$582$	0	0
$583$	36.9213	1.52912
$584$	−16.8070	−0.695480
$585$	0	0
$586$	−18.5130	−0.764765
$587$	−28.1157	−1.16046	−0.580229	−	0.814453i	$-0.697037\pi$
$587$	−28.1157	−1.16046	−0.580229	+	0.814453i	$0.697037\pi$
$588$	0	0
$589$	−70.7079	−2.91347
$590$	8.63674	0.355569
$591$	0	0
$592$	6.91593	0.284243
$593$	17.2239	0.707301	0.353650	−	0.935378i	$-0.384940\pi$
$593$	17.2239	0.707301	0.353650	+	0.935378i	$0.384940\pi$
$594$	0	0
$595$	7.76631	0.318388
$596$	0.131208	0.00537451
$597$	0	0
$598$	−6.53370	−0.267183
$599$	27.8806	1.13917	0.569585	−	0.821933i	$-0.307104\pi$
$599$	27.8806	1.13917	0.569585	+	0.821933i	$0.307104\pi$
$600$	0	0
$601$	−8.48766	−0.346219	−0.173109	−	0.984903i	$-0.555381\pi$
$601$	−8.48766	−0.346219	−0.173109	+	0.984903i	$0.555381\pi$
$602$	−47.3895	−1.93145
$603$	0	0
$604$	−0.134981	−0.00549230
$605$	0.347627	0.0141331
$606$	0	0
$607$	44.8744	1.82140	0.910698	−	0.413074i	$-0.135545\pi$
$607$	44.8744	1.82140	0.910698	+	0.413074i	$0.135545\pi$
$608$	−0.633045	−0.0256734
$609$	0	0
$610$	10.2100	0.413392
$611$	10.5533	0.426939
$612$	0	0
$613$	−10.6398	−0.429738	−0.214869	−	0.976643i	$-0.568932\pi$
$613$	−10.6398	−0.429738	−0.214869	+	0.976643i	$0.568932\pi$
$614$	26.9587	1.08797
$615$	0	0
$616$	33.6645	1.35638
$617$	−36.3155	−1.46201	−0.731003	−	0.682374i	$-0.760948\pi$
$617$	−36.3155	−1.46201	−0.731003	+	0.682374i	$0.760948\pi$
$618$	0	0
$619$	−26.4333	−1.06244	−0.531221	−	0.847233i	$-0.678267\pi$
$619$	−26.4333	−1.06244	−0.531221	+	0.847233i	$0.678267\pi$
$620$	0.177818	0.00714134
$621$	0	0
$622$	1.58823	0.0636821
$623$	−59.5595	−2.38620
$624$	0	0
$625$	1.00000	0.0400000
$626$	−37.7359	−1.50823
$627$	0	0
$628$	−0.0467929	−0.00186724
$629$	3.63630	0.144989
$630$	0	0
$631$	25.0074	0.995529	0.497765	−	0.867312i	$-0.334154\pi$
$631$	25.0074	0.995529	0.497765	+	0.867312i	$0.334154\pi$
$632$	17.1459	0.682029
$633$	0	0
$634$	14.5117	0.576332
$635$	−2.58185	−0.102458
$636$	0	0
$637$	−17.9516	−0.711269
$638$	0.0876905	0.00347170
$639$	0	0
$640$	−11.4540	−0.452758
$641$	42.6060	1.68283	0.841417	−	0.540386i	$-0.181722\pi$
$641$	42.6060	1.68283	0.841417	+	0.540386i	$0.181722\pi$
$642$	0	0
$643$	−20.6664	−0.815003	−0.407502	−	0.913205i	$-0.633600\pi$
$643$	−20.6664	−0.815003	−0.407502	+	0.913205i	$0.633600\pi$
$644$	0.100963	0.00397850
$645$	0	0
$646$	−20.0899	−0.790426
$647$	−39.4826	−1.55222	−0.776111	−	0.630596i	$-0.782810\pi$
$647$	−39.4826	−1.55222	−0.776111	+	0.630596i	$0.782810\pi$
$648$	0	0
$649$	19.8492	0.779151
$650$	−3.97589	−0.155947
$651$	0	0
$652$	0.362403	0.0141928
$653$	9.79998	0.383503	0.191751	−	0.981444i	$-0.438583\pi$
$653$	9.79998	0.383503	0.191751	+	0.981444i	$0.438583\pi$
$654$	0	0
$655$	−6.97007	−0.272343
$656$	7.31125	0.285456
$657$	0	0
$658$	−19.6046	−0.764266
$659$	−2.40593	−0.0937218	−0.0468609	−	0.998901i	$-0.514922\pi$
$659$	−2.40593	−0.0937218	−0.0468609	+	0.998901i	$0.514922\pi$
$660$	0	0
$661$	18.3264	0.712813	0.356407	−	0.934331i	$-0.384002\pi$
$661$	18.3264	0.712813	0.356407	+	0.934331i	$0.384002\pi$
$662$	−5.30511	−0.206189
$663$	0	0
$664$	−5.49919	−0.213410
$665$	−24.4303	−0.947368
$666$	0	0
$667$	−0.0310903	−0.00120382
$668$	−0.307695	−0.0119051
$669$	0	0
$670$	−1.42013	−0.0548645
$671$	23.4650	0.905857
$672$	0	0
$673$	−18.7300	−0.721988	−0.360994	−	0.932568i	$-0.617563\pi$
$673$	−18.7300	−0.721988	−0.360994	+	0.932568i	$0.617563\pi$
$674$	−48.3227	−1.86132
$675$	0	0
$676$	−0.0865959	−0.00333061
$677$	−38.8504	−1.49314	−0.746570	−	0.665307i	$-0.768301\pi$
$677$	−38.8504	−1.49314	−0.746570	+	0.665307i	$0.768301\pi$
$678$	0	0
$679$	16.2270	0.622735
$680$	−5.97266	−0.229041
$681$	0	0
$682$	49.1291	1.88125
$683$	−21.5365	−0.824070	−0.412035	−	0.911168i	$-0.635182\pi$
$683$	−21.5365	−0.824070	−0.412035	+	0.911168i	$0.635182\pi$
$684$	0	0
$685$	11.0644	0.422749
$686$	−3.05778	−0.116747
$687$	0	0
$688$	36.7505	1.40110
$689$	31.6708	1.20656
$690$	0	0
$691$	−35.4334	−1.34795	−0.673974	−	0.738755i	$-0.735414\pi$
$691$	−35.4334	−1.34795	−0.673974	+	0.738755i	$0.735414\pi$
$692$	0.364411	0.0138528
$693$	0	0
$694$	41.4817	1.57462
$695$	3.51297	0.133254
$696$	0	0
$697$	3.84416	0.145608
$698$	30.2650	1.14555
$699$	0	0
$700$	0.0614380	0.00232214
$701$	−48.8588	−1.84537	−0.922686	−	0.385551i	$-0.874011\pi$
$701$	−48.8588	−1.84537	−0.922686	+	0.385551i	$0.874011\pi$
$702$	0	0
$703$	−11.4387	−0.431417
$704$	−25.8877	−0.975680
$705$	0	0
$706$	8.98348	0.338098
$707$	57.1591	2.14969
$708$	0	0
$709$	−24.4737	−0.919128	−0.459564	−	0.888145i	$-0.651994\pi$
$709$	−24.4737	−0.919128	−0.459564	+	0.888145i	$0.651994\pi$
$710$	−11.5223	−0.432423
$711$	0	0
$712$	45.8040	1.71658
$713$	−17.4185	−0.652329
$714$	0	0
$715$	−9.13754	−0.341725
$716$	0.271570	0.0101490
$717$	0	0
$718$	20.9946	0.783511
$719$	6.98045	0.260327	0.130163	−	0.991493i	$-0.458450\pi$
$719$	6.98045	0.260327	0.130163	+	0.991493i	$0.458450\pi$
$720$	0	0
$721$	−7.24334	−0.269756
$722$	36.2139	1.34774
$723$	0	0
$724$	−0.0112514	−0.000418153 0
$725$	−0.0189191	−0.000702638 0
$726$	0	0
$727$	−23.9628	−0.888731	−0.444366	−	0.895846i	$-0.646571\pi$
$727$	−23.9628	−0.888731	−0.444366	+	0.895846i	$0.646571\pi$
$728$	28.8772	1.07026
$729$	0	0
$730$	−8.47460	−0.313659
$731$	19.3229	0.714684
$732$	0	0
$733$	−23.1099	−0.853582	−0.426791	−	0.904350i	$-0.640356\pi$
$733$	−23.1099	−0.853582	−0.426791	+	0.904350i	$0.640356\pi$
$734$	32.4238	1.19678
$735$	0	0
$736$	−0.155947	−0.00574829
$737$	−3.26380	−0.120224
$738$	0	0
$739$	33.9813	1.25002	0.625011	−	0.780616i	$-0.285095\pi$
$739$	33.9813	1.25002	0.625011	+	0.780616i	$0.285095\pi$
$740$	0.0287662	0.00105747
$741$	0	0
$742$	−58.8342	−2.15987
$743$	48.0961	1.76447	0.882237	−	0.470805i	$-0.156036\pi$
$743$	48.0961	1.76447	0.882237	+	0.470805i	$0.156036\pi$
$744$	0	0
$745$	−7.82118	−0.286546
$746$	41.3238	1.51297
$747$	0	0
$748$	0.116113	0.00424550
$749$	−51.8604	−1.89494
$750$	0	0
$751$	−54.3642	−1.98378	−0.991888	−	0.127113i	$-0.959429\pi$
$751$	−54.3642	−1.98378	−0.991888	+	0.127113i	$0.959429\pi$
$752$	15.2033	0.554408
$753$	0	0
$754$	0.0752204	0.00273936
$755$	8.04607	0.292826
$756$	0	0
$757$	−33.9263	−1.23307	−0.616537	−	0.787326i	$-0.711465\pi$
$757$	−33.9263	−1.23307	−0.616537	+	0.787326i	$0.711465\pi$
$758$	23.7131	0.861298
$759$	0	0
$760$	18.7881	0.681515
$761$	35.4554	1.28526	0.642628	−	0.766178i	$-0.277844\pi$
$761$	35.4554	1.28526	0.642628	+	0.766178i	$0.277844\pi$
$762$	0	0
$763$	27.3591	0.990467
$764$	−0.179460	−0.00649262
$765$	0	0
$766$	−34.8641	−1.25969
$767$	17.0266	0.614793
$768$	0	0
$769$	−31.2069	−1.12535	−0.562675	−	0.826678i	$-0.690228\pi$
$769$	−31.2069	−1.12535	−0.562675	+	0.826678i	$0.690228\pi$
$770$	16.9746	0.611723
$771$	0	0
$772$	−0.119086	−0.00428601
$773$	−27.8700	−1.00241	−0.501207	−	0.865328i	$-0.667110\pi$
$773$	−27.8700	−1.00241	−0.501207	+	0.865328i	$0.667110\pi$
$774$	0	0
$775$	−10.5995	−0.380746
$776$	−12.4793	−0.447982
$777$	0	0
$778$	49.8581	1.78750
$779$	−12.0925	−0.433259
$780$	0	0
$781$	−26.4808	−0.947559
$782$	−4.94903	−0.176977
$783$	0	0
$784$	−25.8616	−0.923628
$785$	2.78927	0.0995533
$786$	0	0
$787$	−30.5100	−1.08756	−0.543782	−	0.839227i	$-0.683008\pi$
$787$	−30.5100	−1.08756	−0.543782	+	0.839227i	$0.683008\pi$
$788$	0.233614	0.00832215
$789$	0	0
$790$	8.64549	0.307593
$791$	−36.0961	−1.28343
$792$	0	0
$793$	20.1282	0.714772
$794$	−25.3160	−0.898430
$795$	0	0
$796$	−0.401449	−0.0142290
$797$	−5.47315	−0.193869	−0.0969345	−	0.995291i	$-0.530904\pi$
$797$	−5.47315	−0.193869	−0.0969345	+	0.995291i	$0.530904\pi$
$798$	0	0
$799$	7.99370	0.282797
$800$	−0.0948971	−0.00335512
$801$	0	0
$802$	−5.64861	−0.199459
$803$	−19.4766	−0.687315
$804$	0	0
$805$	−6.01828	−0.212117
$806$	42.1426	1.48441
$807$	0	0
$808$	−43.9580	−1.54644
$809$	2.36549	0.0831661	0.0415830	−	0.999135i	$-0.486760\pi$
$809$	2.36549	0.0831661	0.0415830	+	0.999135i	$0.486760\pi$
$810$	0	0
$811$	−21.2573	−0.746443	−0.373222	−	0.927742i	$-0.621747\pi$
$811$	−21.2573	−0.746443	−0.373222	+	0.927742i	$0.621747\pi$
$812$	−0.00116235	−4.07906e−5 0
$813$	0	0
$814$	7.94777	0.278569
$815$	−21.6024	−0.756700
$816$	0	0
$817$	−60.7837	−2.12655
$818$	51.0129	1.78362
$819$	0	0
$820$	0.0304105	0.00106198
$821$	27.6538	0.965124	0.482562	−	0.875862i	$-0.339706\pi$
$821$	27.6538	0.965124	0.482562	+	0.875862i	$0.339706\pi$
$822$	0	0
$823$	−1.58378	−0.0552070	−0.0276035	−	0.999619i	$-0.508788\pi$
$823$	−1.58378	−0.0552070	−0.0276035	+	0.999619i	$0.508788\pi$
$824$	5.57047	0.194056
$825$	0	0
$826$	−31.6299	−1.10054
$827$	−13.1210	−0.456261	−0.228130	−	0.973631i	$-0.573261\pi$
$827$	−13.1210	−0.456261	−0.228130	+	0.973631i	$0.573261\pi$
$828$	0	0
$829$	5.81240	0.201873	0.100937	−	0.994893i	$-0.467816\pi$
$829$	5.81240	0.201873	0.100937	+	0.994893i	$0.467816\pi$
$830$	−2.77285	−0.0962472
$831$	0	0
$832$	−22.2063	−0.769865
$833$	−13.5977	−0.471132
$834$	0	0
$835$	18.3413	0.634728
$836$	−0.365253	−0.0126326
$837$	0	0
$838$	19.7029	0.680626
$839$	−15.3259	−0.529108	−0.264554	−	0.964371i	$-0.585225\pi$
$839$	−15.3259	−0.529108	−0.264554	+	0.964371i	$0.585225\pi$
$840$	0	0
$841$	−28.9996	−0.999988
$842$	57.5826	1.98443
$843$	0	0
$844$	−0.317047	−0.0109132
$845$	5.16188	0.177574
$846$	0	0
$847$	−1.27310	−0.0437441
$848$	45.6258	1.56680
$849$	0	0
$850$	−3.01159	−0.103297
$851$	−2.81785	−0.0965946
$852$	0	0
$853$	5.98528	0.204932	0.102466	−	0.994737i	$-0.467327\pi$
$853$	5.98528	0.204932	0.102466	+	0.994737i	$0.467327\pi$
$854$	−37.3916	−1.27952
$855$	0	0
$856$	39.8831	1.36318
$857$	31.0336	1.06009	0.530044	−	0.847970i	$-0.322175\pi$
$857$	31.0336	1.06009	0.530044	+	0.847970i	$0.322175\pi$
$858$	0	0
$859$	26.3730	0.899834	0.449917	−	0.893070i	$-0.351454\pi$
$859$	26.3730	0.899834	0.449917	+	0.893070i	$0.351454\pi$
$860$	0.152860	0.00521250
$861$	0	0
$862$	13.9109	0.473807
$863$	20.2885	0.690630	0.345315	−	0.938487i	$-0.387772\pi$
$863$	20.2885	0.690630	0.345315	+	0.938487i	$0.387772\pi$
$864$	0	0
$865$	−21.7221	−0.738574
$866$	−27.1771	−0.923516
$867$	0	0
$868$	−0.651214	−0.0221036
$869$	19.8694	0.674022
$870$	0	0
$871$	−2.79966	−0.0948630
$872$	−21.0405	−0.712520
$873$	0	0
$874$	15.5681	0.526598
$875$	−3.66225	−0.123807
$876$	0	0
$877$	−45.1636	−1.52507	−0.762533	−	0.646949i	$-0.776045\pi$
$877$	−45.1636	−1.52507	−0.762533	+	0.646949i	$0.776045\pi$
$878$	11.4156	0.385259
$879$	0	0
$880$	−13.1638	−0.443751
$881$	12.0663	0.406524	0.203262	−	0.979124i	$-0.434846\pi$
$881$	12.0663	0.406524	0.203262	+	0.979124i	$0.434846\pi$
$882$	0	0
$883$	−7.10948	−0.239253	−0.119627	−	0.992819i	$-0.538170\pi$
$883$	−7.10948	−0.239253	−0.119627	+	0.992819i	$0.538170\pi$
$884$	0.0996007	0.00334993
$885$	0	0
$886$	18.5667	0.623762
$887$	14.8013	0.496980	0.248490	−	0.968634i	$-0.420066\pi$
$887$	14.8013	0.496980	0.248490	+	0.968634i	$0.420066\pi$
$888$	0	0
$889$	9.45538	0.317123
$890$	23.0957	0.774171
$891$	0	0
$892$	0.0634139	0.00212325
$893$	−25.1456	−0.841466
$894$	0	0
$895$	−16.1880	−0.541104
$896$	41.9473	1.40136
$897$	0	0
$898$	35.7772	1.19390
$899$	0.200534	0.00668817
$900$	0	0
$901$	23.9894	0.799204
$902$	8.40207	0.279758
$903$	0	0
$904$	27.7596	0.923271
$905$	0.670680	0.0222942
$906$	0	0
$907$	37.2992	1.23850	0.619250	−	0.785194i	$-0.287437\pi$
$907$	37.2992	1.23850	0.619250	+	0.785194i	$0.287437\pi$
$908$	−0.0675108	−0.00224042
$909$	0	0
$910$	14.5607	0.482683
$911$	−22.6810	−0.751455	−0.375728	−	0.926730i	$-0.622607\pi$
$911$	−22.6810	−0.751455	−0.375728	+	0.926730i	$0.622607\pi$
$912$	0	0
$913$	−6.37267	−0.210905
$914$	6.81962	0.225573
$915$	0	0
$916$	−0.257878	−0.00852054
$917$	25.5261	0.842947
$918$	0	0
$919$	−18.0709	−0.596103	−0.298052	−	0.954550i	$-0.596337\pi$
$919$	−18.0709	−0.596103	−0.298052	+	0.954550i	$0.596337\pi$
$920$	4.62834	0.152592
$921$	0	0
$922$	9.69064	0.319144
$923$	−22.7151	−0.747677
$924$	0	0
$925$	−1.71472	−0.0563796
$926$	−9.00376	−0.295882
$927$	0	0
$928$	0.00179537	5.89359e−5 0
$929$	−7.79262	−0.255667	−0.127834	−	0.991796i	$-0.540802\pi$
$929$	−7.79262	−0.255667	−0.127834	+	0.991796i	$0.540802\pi$
$930$	0	0
$931$	42.7740	1.40186
$932$	0.282279	0.00924635
$933$	0	0
$934$	18.6828	0.611321
$935$	−6.92134	−0.226352
$936$	0	0
$937$	−10.5694	−0.345289	−0.172644	−	0.984984i	$-0.555231\pi$
$937$	−10.5694	−0.345289	−0.172644	+	0.984984i	$0.555231\pi$
$938$	5.20088	0.169815
$939$	0	0
$940$	0.0632368	0.00206256
$941$	28.2843	0.922040	0.461020	−	0.887390i	$-0.347484\pi$
$941$	28.2843	0.922040	0.461020	+	0.887390i	$0.347484\pi$
$942$	0	0
$943$	−2.97892	−0.0970070
$944$	24.5289	0.798348
$945$	0	0
$946$	42.2335	1.37313
$947$	−36.7152	−1.19308	−0.596541	−	0.802582i	$-0.703459\pi$
$947$	−36.7152	−1.19308	−0.596541	+	0.802582i	$0.703459\pi$
$948$	0	0
$949$	−16.7069	−0.542329
$950$	9.47350	0.307361
$951$	0	0
$952$	21.8734	0.708919
$953$	−1.66770	−0.0540220	−0.0270110	−	0.999635i	$-0.508599\pi$
$953$	−1.66770	−0.0540220	−0.0270110	+	0.999635i	$0.508599\pi$
$954$	0	0
$955$	10.6974	0.346159
$956$	−0.305296	−0.00987398
$957$	0	0
$958$	49.7033	1.60584
$959$	−40.5206	−1.30848
$960$	0	0
$961$	81.3500	2.62419
$962$	6.81754	0.219806
$963$	0	0
$964$	0.158136	0.00509320
$965$	7.09859	0.228512
$966$	0	0
$967$	−4.98539	−0.160319	−0.0801597	−	0.996782i	$-0.525543\pi$
$967$	−4.98539	−0.160319	−0.0801597	+	0.996782i	$0.525543\pi$
$968$	0.979071	0.0314685
$969$	0	0
$970$	−6.29244	−0.202038
$971$	40.0306	1.28464	0.642321	−	0.766436i	$-0.277972\pi$
$971$	40.0306	1.28464	0.642321	+	0.766436i	$0.277972\pi$
$972$	0	0
$973$	−12.8654	−0.412445
$974$	−50.4399	−1.61620
$975$	0	0
$976$	28.9971	0.928176
$977$	−58.9565	−1.88619	−0.943094	−	0.332527i	$-0.892099\pi$
$977$	−58.9565	−1.88619	−0.943094	+	0.332527i	$0.892099\pi$
$978$	0	0
$979$	53.0794	1.69643
$980$	−0.107569	−0.00343617
$981$	0	0
$982$	−35.9272	−1.14648
$983$	23.1568	0.738587	0.369293	−	0.929313i	$-0.379600\pi$
$983$	23.1568	0.738587	0.369293	+	0.929313i	$0.379600\pi$
$984$	0	0
$985$	−13.9254	−0.443702
$986$	0.0569766	0.00181450
$987$	0	0
$988$	−0.313312	−0.00996778
$989$	−14.9737	−0.476137
$990$	0	0
$991$	−23.9504	−0.760810	−0.380405	−	0.924820i	$-0.624215\pi$
$991$	−23.9504	−0.760810	−0.380405	+	0.924820i	$0.624215\pi$
$992$	1.00586	0.0319362
$993$	0	0
$994$	42.1974	1.33842
$995$	23.9299	0.758629
$996$	0	0
$997$	−46.4764	−1.47192	−0.735961	−	0.677024i	$-0.763269\pi$
$997$	−46.4764	−1.47192	−0.735961	+	0.677024i	$0.763269\pi$
$998$	−8.59773	−0.272156
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3015.2.a.l.1.6		7
3.2	odd	2		1005.2.a.i.1.2	✓	7
15.14	odd	2		5025.2.a.bb.1.6		7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1005.2.a.i.1.2	✓	7	3.2	odd	2
3015.2.a.l.1.6		7	1.1	even	1	trivial
5025.2.a.bb.1.6		7	15.14	odd	2

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:	\( N \)	\(=\)	\( 3015 = 3^{2} \cdot 5 \cdot 67 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3015.a (trivial)

\(f(q)\)	\(=\)	\(q+1.42013 q^{2} +0.0167760 q^{4} -1.00000 q^{5} +3.66225 q^{7} -2.81644 q^{8} +O(q^{10})\)	\(q+1.42013 q^{2} +0.0167760 q^{4} -1.00000 q^{5} +3.66225 q^{7} -2.81644 q^{8} -1.42013 q^{10} -3.26380 q^{11} -2.79966 q^{13} +5.20088 q^{14} -4.03327 q^{16} -2.12064 q^{17} +6.67086 q^{19} -0.0167760 q^{20} -4.63502 q^{22} +1.64333 q^{23} +1.00000 q^{25} -3.97589 q^{26} +0.0614380 q^{28} -0.0189191 q^{29} -10.5995 q^{31} -0.0948971 q^{32} -3.01159 q^{34} -3.66225 q^{35} -1.71472 q^{37} +9.47350 q^{38} +2.81644 q^{40} -1.81274 q^{41} -9.11183 q^{43} -0.0547536 q^{44} +2.33375 q^{46} -3.76947 q^{47} +6.41206 q^{49} +1.42013 q^{50} -0.0469673 q^{52} -11.3124 q^{53} +3.26380 q^{55} -10.3145 q^{56} -0.0268676 q^{58} -6.08164 q^{59} -7.18949 q^{61} -15.0527 q^{62} +7.93177 q^{64} +2.79966 q^{65} +1.00000 q^{67} -0.0355759 q^{68} -5.20088 q^{70} +8.11351 q^{71} +5.96747 q^{73} -2.43513 q^{74} +0.111911 q^{76} -11.9528 q^{77} -6.08780 q^{79} +4.03327 q^{80} -2.57432 q^{82} +1.95253 q^{83} +2.12064 q^{85} -12.9400 q^{86} +9.19229 q^{88} -16.2631 q^{89} -10.2531 q^{91} +0.0275686 q^{92} -5.35315 q^{94} -6.67086 q^{95} +4.43089 q^{97} +9.10598 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q - 4 q^{2} + 8 q^{4} - 7 q^{5} + 3 q^{7} - 9 q^{8}+O(q^{10}) \) 7 * q - 4 * q^2 + 8 * q^4 - 7 * q^5 + 3 * q^7 - 9 * q^8	\( 7 q - 4 q^{2} + 8 q^{4} - 7 q^{5} + 3 q^{7} - 9 q^{8} + 4 q^{10} - 5 q^{11} - q^{13} + 4 q^{14} + 6 q^{16} - 11 q^{17} + 8 q^{19} - 8 q^{20} - 3 q^{22} - 11 q^{23} + 7 q^{25} + 5 q^{26} - 17 q^{28} - 3 q^{31} - 22 q^{32} + 4 q^{34} - 3 q^{35} - 5 q^{37} + 9 q^{40} - q^{41} - 3 q^{43} + 9 q^{44} - 12 q^{46} - 10 q^{47} - 4 q^{49} - 4 q^{50} - 6 q^{52} - 3 q^{53} + 5 q^{55} + 12 q^{56} - 24 q^{58} + 4 q^{59} - 9 q^{61} - 20 q^{62} - 3 q^{64} + q^{65} + 7 q^{67} + 3 q^{68} - 4 q^{70} - 6 q^{71} - 9 q^{73} - 2 q^{74} + 2 q^{76} - 17 q^{77} - 11 q^{79} - 6 q^{80} - 16 q^{82} - 30 q^{83} + 11 q^{85} + 11 q^{86} - 25 q^{88} - 13 q^{89} - 5 q^{91} - 10 q^{92} - 25 q^{94} - 8 q^{95} - 7 q^{97} + 10 q^{98}+O(q^{100}) \) 7 * q - 4 * q^2 + 8 * q^4 - 7 * q^5 + 3 * q^7 - 9 * q^8 + 4 * q^10 - 5 * q^11 - q^13 + 4 * q^14 + 6 * q^16 - 11 * q^17 + 8 * q^19 - 8 * q^20 - 3 * q^22 - 11 * q^23 + 7 * q^25 + 5 * q^26 - 17 * q^28 - 3 * q^31 - 22 * q^32 + 4 * q^34 - 3 * q^35 - 5 * q^37 + 9 * q^40 - q^41 - 3 * q^43 + 9 * q^44 - 12 * q^46 - 10 * q^47 - 4 * q^49 - 4 * q^50 - 6 * q^52 - 3 * q^53 + 5 * q^55 + 12 * q^56 - 24 * q^58 + 4 * q^59 - 9 * q^61 - 20 * q^62 - 3 * q^64 + q^65 + 7 * q^67 + 3 * q^68 - 4 * q^70 - 6 * q^71 - 9 * q^73 - 2 * q^74 + 2 * q^76 - 17 * q^77 - 11 * q^79 - 6 * q^80 - 16 * q^82 - 30 * q^83 + 11 * q^85 + 11 * q^86 - 25 * q^88 - 13 * q^89 - 5 * q^91 - 10 * q^92 - 25 * q^94 - 8 * q^95 - 7 * q^97 + 10 * q^98

Introduction

L-functions

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