LMFDB - Embedded newform 6003.2.a.s.1.13

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6003,2,Mod(1,6003)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6003, 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("6003.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6003 = 3^{2} \cdot 23 \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6003.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:	$47.9341963334$
Analytic rank:	$0$
Dimension:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - 2 x^{19} - 33 x^{18} + 64 x^{17} + 453 x^{16} - 846 x^{15} - 3353 x^{14} + 5985 x^{13} + \cdots - 256 $ x^20 - 2x^19 - 33x^18 + 64x^17 + 453x^16 - 846x^15 - 3353x^14 + 5985x^13 + 14484x^12 - 24566x^11 - 36791x^10 + 59410x^9 + 52109x^8 - 82362x^7 - 34967x^6 + 60661x^5 + 5201x^4 - 19624x^3 + 2768x^2 + 1408*x - 256
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{7} $
Twist minimal:	no (minimal twist has level 2001)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.13
Root			$-0.852674$ of defining polynomial
Character	$\chi$	$=$	6003.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.852674 q^{2} -1.27295 q^{4} +0.278169 q^{5} +2.70488 q^{7} -2.79076 q^{8} +O(q^{10})$	\(q+0.852674 q^{2} -1.27295 q^{4} +0.278169 q^{5} +2.70488 q^{7} -2.79076 q^{8} +0.237188 q^{10} -5.07059 q^{11} -0.519860 q^{13} +2.30638 q^{14} +0.166287 q^{16} -2.19587 q^{17} -0.104552 q^{19} -0.354095 q^{20} -4.32356 q^{22} -1.00000 q^{23} -4.92262 q^{25} -0.443271 q^{26} -3.44317 q^{28} -1.00000 q^{29} +7.35747 q^{31} +5.72330 q^{32} -1.87236 q^{34} +0.752416 q^{35} -0.259721 q^{37} -0.0891492 q^{38} -0.776303 q^{40} -0.509113 q^{41} +4.63055 q^{43} +6.45459 q^{44} -0.852674 q^{46} +12.4082 q^{47} +0.316385 q^{49} -4.19739 q^{50} +0.661754 q^{52} -1.04291 q^{53} -1.41048 q^{55} -7.54867 q^{56} -0.852674 q^{58} -0.323984 q^{59} +8.63178 q^{61} +6.27352 q^{62} +4.54754 q^{64} -0.144609 q^{65} -3.31184 q^{67} +2.79522 q^{68} +0.641565 q^{70} +14.1260 q^{71} -0.479599 q^{73} -0.221458 q^{74} +0.133090 q^{76} -13.7153 q^{77} +9.38238 q^{79} +0.0462559 q^{80} -0.434108 q^{82} +6.67576 q^{83} -0.610823 q^{85} +3.94835 q^{86} +14.1508 q^{88} -5.52034 q^{89} -1.40616 q^{91} +1.27295 q^{92} +10.5801 q^{94} -0.0290833 q^{95} +0.587030 q^{97} +0.269774 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q - 2 q^{2} + 30 q^{4} + q^{5} + 9 q^{7} - 6 q^{8}+O(q^{10}) $ 20 * q - 2 * q^2 + 30 * q^4 + q^5 + 9 * q^7 - 6 * q^8	$ 20 q - 2 q^{2} + 30 q^{4} + q^{5} + 9 q^{7} - 6 q^{8} + 7 q^{10} + 21 q^{13} + q^{14} + 58 q^{16} + 4 q^{17} + 7 q^{19} + 20 q^{20} + 7 q^{22} - 20 q^{23} + 47 q^{25} - 8 q^{26} + 11 q^{28} - 20 q^{29} + 28 q^{31} - 14 q^{32} + 16 q^{34} - 9 q^{35} + 14 q^{37} + 20 q^{38} + 34 q^{40} - 7 q^{41} + 3 q^{43} + q^{44} + 2 q^{46} - 3 q^{47} + 35 q^{49} + 24 q^{50} + 73 q^{52} + 19 q^{53} + 29 q^{55} + 30 q^{56} + 2 q^{58} - 20 q^{59} + 15 q^{61} - 12 q^{62} + 82 q^{64} + 28 q^{65} + 20 q^{67} + 23 q^{68} - 24 q^{70} - 63 q^{71} + 19 q^{73} - 16 q^{74} - 44 q^{76} + 7 q^{77} + 32 q^{79} + 56 q^{80} - 20 q^{82} + 21 q^{83} + 4 q^{85} + 6 q^{86} + 55 q^{88} + 13 q^{89} + 70 q^{91} - 30 q^{92} - 12 q^{94} - 9 q^{95} - 9 q^{97} - 31 q^{98}+O(q^{100}) $ 20 * q - 2 * q^2 + 30 * q^4 + q^5 + 9 * q^7 - 6 * q^8 + 7 * q^10 + 21 * q^13 + q^14 + 58 * q^16 + 4 * q^17 + 7 * q^19 + 20 * q^20 + 7 * q^22 - 20 * q^23 + 47 * q^25 - 8 * q^26 + 11 * q^28 - 20 * q^29 + 28 * q^31 - 14 * q^32 + 16 * q^34 - 9 * q^35 + 14 * q^37 + 20 * q^38 + 34 * q^40 - 7 * q^41 + 3 * q^43 + q^44 + 2 * q^46 - 3 * q^47 + 35 * q^49 + 24 * q^50 + 73 * q^52 + 19 * q^53 + 29 * q^55 + 30 * q^56 + 2 * q^58 - 20 * q^59 + 15 * q^61 - 12 * q^62 + 82 * q^64 + 28 * q^65 + 20 * q^67 + 23 * q^68 - 24 * q^70 - 63 * q^71 + 19 * q^73 - 16 * q^74 - 44 * q^76 + 7 * q^77 + 32 * q^79 + 56 * q^80 - 20 * q^82 + 21 * q^83 + 4 * q^85 + 6 * q^86 + 55 * q^88 + 13 * q^89 + 70 * q^91 - 30 * q^92 - 12 * q^94 - 9 * q^95 - 9 * q^97 - 31 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0.852674	0.602932	0.301466	−	0.953477i	$-0.402524\pi$
$2$	0.852674	0.602932	0.301466	+	0.953477i	$0.402524\pi$
$3$	0	0
$3$	0	0
$4$	−1.27295	−0.636473
$5$	0.278169	0.124401	0.0622006	−	0.998064i	$-0.480188\pi$
$5$	0.278169	0.124401	0.0622006	+	0.998064i	$0.480188\pi$
$6$	0	0
$7$	2.70488	1.02235	0.511175	−	0.859477i	$-0.329211\pi$
$7$	2.70488	1.02235	0.511175	+	0.859477i	$0.329211\pi$
$8$	−2.79076	−0.986682
$9$	0	0
$10$	0.237188	0.0750054
$11$	−5.07059	−1.52884	−0.764420	−	0.644718i	$-0.776975\pi$
$11$	−5.07059	−1.52884	−0.764420	+	0.644718i	$0.776975\pi$
$12$	0	0
$13$	−0.519860	−0.144183	−0.0720916	−	0.997398i	$-0.522967\pi$
$13$	−0.519860	−0.144183	−0.0720916	+	0.997398i	$0.522967\pi$
$14$	2.30638	0.616407
$15$	0	0
$16$	0.166287	0.0415717
$17$	−2.19587	−0.532576	−0.266288	−	0.963894i	$-0.585797\pi$
$17$	−2.19587	−0.532576	−0.266288	+	0.963894i	$0.585797\pi$
$18$	0	0
$19$	−0.104552	−0.0239860	−0.0119930	−	0.999928i	$-0.503818\pi$
$19$	−0.104552	−0.0239860	−0.0119930	+	0.999928i	$0.503818\pi$
$20$	−0.354095	−0.0791780
$21$	0	0
$22$	−4.32356	−0.921786
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	−4.92262	−0.984524
$26$	−0.443271	−0.0869326
$27$	0	0
$28$	−3.44317	−0.650698
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	0	0
$31$	7.35747	1.32144	0.660720	−	0.750632i	$-0.270251\pi$
$31$	7.35747	1.32144	0.660720	+	0.750632i	$0.270251\pi$
$32$	5.72330	1.01175
$33$	0	0
$34$	−1.87236	−0.321107
$35$	0.752416	0.127181
$36$	0	0
$37$	−0.259721	−0.0426979	−0.0213490	−	0.999772i	$-0.506796\pi$
$37$	−0.259721	−0.0426979	−0.0213490	+	0.999772i	$0.506796\pi$
$38$	−0.0891492	−0.0144619
$39$	0	0
$40$	−0.776303	−0.122744
$41$	−0.509113	−0.0795101	−0.0397551	−	0.999209i	$-0.512658\pi$
$41$	−0.509113	−0.0795101	−0.0397551	+	0.999209i	$0.512658\pi$
$42$	0	0
$43$	4.63055	0.706152	0.353076	−	0.935595i	$-0.385136\pi$
$43$	4.63055	0.706152	0.353076	+	0.935595i	$0.385136\pi$
$44$	6.45459	0.973066
$45$	0	0
$46$	−0.852674	−0.125720
$47$	12.4082	1.80992	0.904959	−	0.425499i	$-0.139902\pi$
$47$	12.4082	1.80992	0.904959	+	0.425499i	$0.139902\pi$
$48$	0	0
$49$	0.316385	0.0451979
$50$	−4.19739	−0.593601
$51$	0	0
$52$	0.661754	0.0917687
$53$	−1.04291	−0.143255	−0.0716274	−	0.997431i	$-0.522819\pi$
$53$	−1.04291	−0.143255	−0.0716274	+	0.997431i	$0.522819\pi$
$54$	0	0
$55$	−1.41048	−0.190189
$56$	−7.54867	−1.00873
$57$	0	0
$58$	−0.852674	−0.111962
$59$	−0.323984	−0.0421792	−0.0210896	−	0.999778i	$-0.506714\pi$
$59$	−0.323984	−0.0421792	−0.0210896	+	0.999778i	$0.506714\pi$
$60$	0	0
$61$	8.63178	1.10519	0.552593	−	0.833451i	$-0.313638\pi$
$61$	8.63178	1.10519	0.552593	+	0.833451i	$0.313638\pi$
$62$	6.27352	0.796738
$63$	0	0
$64$	4.54754	0.568442
$65$	−0.144609	−0.0179366
$66$	0	0
$67$	−3.31184	−0.404605	−0.202303	−	0.979323i	$-0.564842\pi$
$67$	−3.31184	−0.404605	−0.202303	+	0.979323i	$0.564842\pi$
$68$	2.79522	0.338970
$69$	0	0
$70$	0.641565	0.0766817
$71$	14.1260	1.67645	0.838226	−	0.545323i	$-0.183593\pi$
$71$	14.1260	1.67645	0.838226	+	0.545323i	$0.183593\pi$
$72$	0	0
$73$	−0.479599	−0.0561328	−0.0280664	−	0.999606i	$-0.508935\pi$
$73$	−0.479599	−0.0561328	−0.0280664	+	0.999606i	$0.508935\pi$
$74$	−0.221458	−0.0257439
$75$	0	0
$76$	0.133090	0.0152664
$77$	−13.7153	−1.56301
$78$	0	0
$79$	9.38238	1.05560	0.527800	−	0.849369i	$-0.323017\pi$
$79$	9.38238	1.05560	0.527800	+	0.849369i	$0.323017\pi$
$80$	0.0462559	0.00517156
$81$	0	0
$82$	−0.434108	−0.0479392
$83$	6.67576	0.732759	0.366380	−	0.930465i	$-0.380597\pi$
$83$	6.67576	0.732759	0.366380	+	0.930465i	$0.380597\pi$
$84$	0	0
$85$	−0.610823	−0.0662531
$86$	3.94835	0.425761
$87$	0	0
$88$	14.1508	1.50848
$89$	−5.52034	−0.585155	−0.292577	−	0.956242i	$-0.594513\pi$
$89$	−5.52034	−0.585155	−0.292577	+	0.956242i	$0.594513\pi$
$90$	0	0
$91$	−1.40616	−0.147406
$92$	1.27295	0.132714
$93$	0	0
$94$	10.5801	1.09126
$95$	−0.0290833	−0.00298388
$96$	0	0
$97$	0.587030	0.0596038	0.0298019	−	0.999556i	$-0.490512\pi$
$97$	0.587030	0.0596038	0.0298019	+	0.999556i	$0.490512\pi$
$98$	0.269774	0.0272513
$99$	0	0
$100$	6.26624	0.626624
$101$	−12.3741	−1.23127	−0.615634	−	0.788033i	$-0.711100\pi$
$101$	−12.3741	−1.23127	−0.615634	+	0.788033i	$0.711100\pi$
$102$	0	0
$103$	13.6687	1.34681	0.673407	−	0.739272i	$-0.264830\pi$
$103$	13.6687	1.34681	0.673407	+	0.739272i	$0.264830\pi$
$104$	1.45080	0.142263
$105$	0	0
$106$	−0.889263	−0.0863728
$107$	4.99088	0.482487	0.241243	−	0.970465i	$-0.422445\pi$
$107$	4.99088	0.482487	0.241243	+	0.970465i	$0.422445\pi$
$108$	0	0
$109$	8.14915	0.780547	0.390274	−	0.920699i	$-0.372380\pi$
$109$	8.14915	0.780547	0.390274	+	0.920699i	$0.372380\pi$
$110$	−1.20268	−0.114671
$111$	0	0
$112$	0.449786	0.0425008
$113$	−14.9087	−1.40250	−0.701248	−	0.712917i	$-0.747373\pi$
$113$	−14.9087	−1.40250	−0.701248	+	0.712917i	$0.747373\pi$
$114$	0	0
$115$	−0.278169	−0.0259394
$116$	1.27295	0.118190
$117$	0	0
$118$	−0.276253	−0.0254312
$119$	−5.93956	−0.544479
$120$	0	0
$121$	14.7109	1.33735
$122$	7.36010	0.666352
$123$	0	0
$124$	−9.36566	−0.841062
$125$	−2.76017	−0.246877
$126$	0	0
$127$	−9.41334	−0.835299	−0.417649	−	0.908608i	$-0.637146\pi$
$127$	−9.41334	−0.835299	−0.417649	+	0.908608i	$0.637146\pi$
$128$	−7.56904	−0.669015
$129$	0	0
$130$	−0.123304	−0.0108145
$131$	−2.66337	−0.232700	−0.116350	−	0.993208i	$-0.537119\pi$
$131$	−2.66337	−0.232700	−0.116350	+	0.993208i	$0.537119\pi$
$132$	0	0
$133$	−0.282802	−0.0245220
$134$	−2.82392	−0.243949
$135$	0	0
$136$	6.12813	0.525483
$137$	11.4148	0.975233	0.487617	−	0.873058i	$-0.337866\pi$
$137$	11.4148	0.975233	0.487617	+	0.873058i	$0.337866\pi$
$138$	0	0
$139$	12.6211	1.07051	0.535254	−	0.844691i	$-0.320216\pi$
$139$	12.6211	1.07051	0.535254	+	0.844691i	$0.320216\pi$
$140$	−0.957785	−0.0809476
$141$	0	0
$142$	12.0449	1.01079
$143$	2.63600	0.220433
$144$	0	0
$145$	−0.278169	−0.0231007
$146$	−0.408941	−0.0338442
$147$	0	0
$148$	0.330611	0.0271761
$149$	−10.6883	−0.875616	−0.437808	−	0.899068i	$-0.644245\pi$
$149$	−10.6883	−0.875616	−0.437808	+	0.899068i	$0.644245\pi$
$150$	0	0
$151$	−7.96037	−0.647806	−0.323903	−	0.946090i	$-0.604995\pi$
$151$	−7.96037	−0.647806	−0.323903	+	0.946090i	$0.604995\pi$
$152$	0.291780	0.0236665
$153$	0	0
$154$	−11.6947	−0.942387
$155$	2.04662	0.164389
$156$	0	0
$157$	6.59115	0.526031	0.263015	−	0.964792i	$-0.415283\pi$
$157$	6.59115	0.526031	0.263015	+	0.964792i	$0.415283\pi$
$158$	8.00011	0.636455
$159$	0	0
$160$	1.59205	0.125862
$161$	−2.70488	−0.213175
$162$	0	0
$163$	21.2877	1.66738	0.833691	−	0.552232i	$-0.186224\pi$
$163$	21.2877	1.66738	0.833691	+	0.552232i	$0.186224\pi$
$164$	0.648074	0.0506061
$165$	0	0
$166$	5.69225	0.441804
$167$	−18.6280	−1.44148	−0.720739	−	0.693207i	$-0.756197\pi$
$167$	−18.6280	−1.44148	−0.720739	+	0.693207i	$0.756197\pi$
$168$	0	0
$169$	−12.7297	−0.979211
$170$	−0.520833	−0.0399461
$171$	0	0
$172$	−5.89444	−0.449447
$173$	10.1837	0.774251	0.387125	−	0.922027i	$-0.373468\pi$
$173$	10.1837	0.774251	0.387125	+	0.922027i	$0.373468\pi$
$174$	0	0
$175$	−13.3151	−1.00653
$176$	−0.843171	−0.0635564
$177$	0	0
$178$	−4.70705	−0.352808
$179$	5.18551	0.387583	0.193792	−	0.981043i	$-0.437921\pi$
$179$	5.18551	0.387583	0.193792	+	0.981043i	$0.437921\pi$
$180$	0	0
$181$	11.3310	0.842224	0.421112	−	0.907009i	$-0.361640\pi$
$181$	11.3310	0.842224	0.421112	+	0.907009i	$0.361640\pi$
$182$	−1.19900	−0.0888755
$183$	0	0
$184$	2.79076	0.205737
$185$	−0.0722466	−0.00531167
$186$	0	0
$187$	11.1343	0.814224
$188$	−15.7949	−1.15196
$189$	0	0
$190$	−0.0247986	−0.00179908
$191$	15.0350	1.08789	0.543947	−	0.839119i	$-0.316929\pi$
$191$	15.0350	1.08789	0.543947	+	0.839119i	$0.316929\pi$
$192$	0	0
$193$	11.4680	0.825487	0.412743	−	0.910847i	$-0.364571\pi$
$193$	11.4680	0.825487	0.412743	+	0.910847i	$0.364571\pi$
$194$	0.500545	0.0359370
$195$	0	0
$196$	−0.402742	−0.0287673
$197$	25.0347	1.78365	0.891826	−	0.452379i	$-0.149425\pi$
$197$	25.0347	1.78365	0.891826	+	0.452379i	$0.149425\pi$
$198$	0	0
$199$	−5.75281	−0.407806	−0.203903	−	0.978991i	$-0.565363\pi$
$199$	−5.75281	−0.407806	−0.203903	+	0.978991i	$0.565363\pi$
$200$	13.7378	0.971412
$201$	0	0
$202$	−10.5511	−0.742370
$203$	−2.70488	−0.189845
$204$	0	0
$205$	−0.141620	−0.00989115
$206$	11.6549	0.812037
$207$	0	0
$208$	−0.0864458	−0.00599393
$209$	0.530143	0.0366707
$210$	0	0
$211$	3.52515	0.242682	0.121341	−	0.992611i	$-0.461281\pi$
$211$	3.52515	0.242682	0.121341	+	0.992611i	$0.461281\pi$
$212$	1.32757	0.0911778
$213$	0	0
$214$	4.25560	0.290907
$215$	1.28808	0.0878461
$216$	0	0
$217$	19.9011	1.35097
$218$	6.94857	0.470617
$219$	0	0
$220$	1.79547	0.121051
$221$	1.14154	0.0767885
$222$	0	0
$223$	−22.4725	−1.50487	−0.752436	−	0.658665i	$-0.771121\pi$
$223$	−22.4725	−1.50487	−0.752436	+	0.658665i	$0.771121\pi$
$224$	15.4809	1.03436
$225$	0	0
$226$	−12.7123	−0.845609
$227$	13.5150	0.897023	0.448511	−	0.893777i	$-0.351954\pi$
$227$	13.5150	0.897023	0.448511	+	0.893777i	$0.351954\pi$
$228$	0	0
$229$	−6.11059	−0.403799	−0.201900	−	0.979406i	$-0.564711\pi$
$229$	−6.11059	−0.403799	−0.201900	+	0.979406i	$0.564711\pi$
$230$	−0.237188	−0.0156397
$231$	0	0
$232$	2.79076	0.183222
$233$	17.3863	1.13902	0.569509	−	0.821985i	$-0.307133\pi$
$233$	17.3863	1.13902	0.569509	+	0.821985i	$0.307133\pi$
$234$	0	0
$235$	3.45157	0.225156
$236$	0.412415	0.0268459
$237$	0	0
$238$	−5.06451	−0.328283
$239$	−1.65727	−0.107200	−0.0535998	−	0.998562i	$-0.517070\pi$
$239$	−1.65727	−0.107200	−0.0535998	+	0.998562i	$0.517070\pi$
$240$	0	0
$241$	19.8147	1.27637	0.638187	−	0.769881i	$-0.279685\pi$
$241$	19.8147	1.27637	0.638187	+	0.769881i	$0.279685\pi$
$242$	12.5436	0.806332
$243$	0	0
$244$	−10.9878	−0.703422
$245$	0.0880088	0.00562267
$246$	0	0
$247$	0.0543526	0.00345837
$248$	−20.5329	−1.30384
$249$	0	0
$250$	−2.35353	−0.148850
$251$	9.99777	0.631054	0.315527	−	0.948917i	$-0.397819\pi$
$251$	9.99777	0.631054	0.315527	+	0.948917i	$0.397819\pi$
$252$	0	0
$253$	5.07059	0.318785
$254$	−8.02651	−0.503628
$255$	0	0
$256$	−15.5490	−0.971813
$257$	2.47265	0.154239	0.0771197	−	0.997022i	$-0.475428\pi$
$257$	2.47265	0.154239	0.0771197	+	0.997022i	$0.475428\pi$
$258$	0	0
$259$	−0.702516	−0.0436522
$260$	0.184080	0.0114161
$261$	0	0
$262$	−2.27099	−0.140302
$263$	−23.9486	−1.47673	−0.738367	−	0.674399i	$-0.764403\pi$
$263$	−23.9486	−1.47673	−0.738367	+	0.674399i	$0.764403\pi$
$264$	0	0
$265$	−0.290106	−0.0178211
$266$	−0.241138	−0.0147851
$267$	0	0
$268$	4.21579	0.257521
$269$	17.5956	1.07282	0.536412	−	0.843956i	$-0.319779\pi$
$269$	17.5956	1.07282	0.536412	+	0.843956i	$0.319779\pi$
$270$	0	0
$271$	12.5307	0.761183	0.380591	−	0.924743i	$-0.375720\pi$
$271$	12.5307	0.761183	0.380591	+	0.924743i	$0.375720\pi$
$272$	−0.365143	−0.0221401
$273$	0	0
$274$	9.73312	0.587999
$275$	24.9606	1.50518
$276$	0	0
$277$	7.17339	0.431007	0.215504	−	0.976503i	$-0.430861\pi$
$277$	7.17339	0.431007	0.215504	+	0.976503i	$0.430861\pi$
$278$	10.7617	0.645443
$279$	0	0
$280$	−2.09981	−0.125488
$281$	3.53636	0.210961	0.105481	−	0.994421i	$-0.466362\pi$
$281$	3.53636	0.210961	0.105481	+	0.994421i	$0.466362\pi$
$282$	0	0
$283$	−2.19773	−0.130642	−0.0653208	−	0.997864i	$-0.520807\pi$
$283$	−2.19773	−0.130642	−0.0653208	+	0.997864i	$0.520807\pi$
$284$	−17.9817	−1.06702
$285$	0	0
$286$	2.24765	0.132906
$287$	−1.37709	−0.0812871
$288$	0	0
$289$	−12.1782	−0.716363
$290$	−0.237188	−0.0139282
$291$	0	0
$292$	0.610504	0.0357270
$293$	3.48709	0.203718	0.101859	−	0.994799i	$-0.467521\pi$
$293$	3.48709	0.203718	0.101859	+	0.994799i	$0.467521\pi$
$294$	0	0
$295$	−0.0901226	−0.00524714
$296$	0.724819	0.0421293
$297$	0	0
$298$	−9.11360	−0.527937
$299$	0.519860	0.0300643
$300$	0	0
$301$	12.5251	0.721934
$302$	−6.78761	−0.390583
$303$	0	0
$304$	−0.0173857	−0.000997137 0
$305$	2.40110	0.137487
$306$	0	0
$307$	−10.3415	−0.590222	−0.295111	−	0.955463i	$-0.595357\pi$
$307$	−10.3415	−0.590222	−0.295111	+	0.955463i	$0.595357\pi$
$308$	17.4589	0.994813
$309$	0	0
$310$	1.74510	0.0991152
$311$	3.44449	0.195319	0.0976595	−	0.995220i	$-0.468864\pi$
$311$	3.44449	0.195319	0.0976595	+	0.995220i	$0.468864\pi$
$312$	0	0
$313$	9.39424	0.530994	0.265497	−	0.964112i	$-0.414464\pi$
$313$	9.39424	0.530994	0.265497	+	0.964112i	$0.414464\pi$
$314$	5.62010	0.317161
$315$	0	0
$316$	−11.9433	−0.671861
$317$	−20.4150	−1.14662	−0.573310	−	0.819338i	$-0.694341\pi$
$317$	−20.4150	−1.14662	−0.573310	+	0.819338i	$0.694341\pi$
$318$	0	0
$319$	5.07059	0.283898
$320$	1.26499	0.0707149
$321$	0	0
$322$	−2.30638	−0.128530
$323$	0.229583	0.0127744
$324$	0	0
$325$	2.55907	0.141952
$326$	18.1515	1.00532
$327$	0	0
$328$	1.42081	0.0784512
$329$	33.5626	1.85037
$330$	0	0
$331$	−1.63941	−0.0901101	−0.0450551	−	0.998985i	$-0.514346\pi$
$331$	−1.63941	−0.0901101	−0.0450551	+	0.998985i	$0.514346\pi$
$332$	−8.49788	−0.466382
$333$	0	0
$334$	−15.8836	−0.869113
$335$	−0.921252	−0.0503334
$336$	0	0
$337$	25.4684	1.38735	0.693677	−	0.720286i	$-0.255990\pi$
$337$	25.4684	1.38735	0.693677	+	0.720286i	$0.255990\pi$
$338$	−10.8543	−0.590397
$339$	0	0
$340$	0.777545	0.0421683
$341$	−37.3067	−2.02027
$342$	0	0
$343$	−18.0784	−0.976141
$344$	−12.9227	−0.696747
$345$	0	0
$346$	8.68336	0.466820
$347$	30.8489	1.65606	0.828029	−	0.560686i	$-0.189462\pi$
$347$	30.8489	1.65606	0.828029	+	0.560686i	$0.189462\pi$
$348$	0	0
$349$	7.05975	0.377900	0.188950	−	0.981987i	$-0.439492\pi$
$349$	7.05975	0.377900	0.188950	+	0.981987i	$0.439492\pi$
$350$	−11.3535	−0.606867
$351$	0	0
$352$	−29.0205	−1.54680
$353$	−16.3887	−0.872282	−0.436141	−	0.899878i	$-0.643655\pi$
$353$	−16.3887	−0.872282	−0.436141	+	0.899878i	$0.643655\pi$
$354$	0	0
$355$	3.92943	0.208553
$356$	7.02710	0.372435
$357$	0	0
$358$	4.42155	0.233686
$359$	−24.2910	−1.28203	−0.641014	−	0.767529i	$-0.721486\pi$
$359$	−24.2910	−1.28203	−0.641014	+	0.767529i	$0.721486\pi$
$360$	0	0
$361$	−18.9891	−0.999425
$362$	9.66162	0.507803
$363$	0	0
$364$	1.78997	0.0938197
$365$	−0.133410	−0.00698298
$366$	0	0
$367$	20.9932	1.09583	0.547917	−	0.836532i	$-0.315421\pi$
$367$	20.9932	1.09583	0.547917	+	0.836532i	$0.315421\pi$
$368$	−0.166287	−0.00866829
$369$	0	0
$370$	−0.0616028	−0.00320258
$371$	−2.82095	−0.146456
$372$	0	0
$373$	8.86668	0.459099	0.229550	−	0.973297i	$-0.426275\pi$
$373$	8.86668	0.459099	0.229550	+	0.973297i	$0.426275\pi$
$374$	9.49396	0.490921
$375$	0	0
$376$	−34.6282	−1.78581
$377$	0.519860	0.0267741
$378$	0	0
$379$	−18.0116	−0.925195	−0.462597	−	0.886568i	$-0.653082\pi$
$379$	−18.0116	−0.925195	−0.462597	+	0.886568i	$0.653082\pi$
$380$	0.0370215	0.00189916
$381$	0	0
$382$	12.8200	0.655926
$383$	−23.2088	−1.18591	−0.592957	−	0.805234i	$-0.702039\pi$
$383$	−23.2088	−1.18591	−0.592957	+	0.805234i	$0.702039\pi$
$384$	0	0
$385$	−3.81519	−0.194440
$386$	9.77849	0.497712
$387$	0	0
$388$	−0.747258	−0.0379363
$389$	12.7471	0.646306	0.323153	−	0.946347i	$-0.395257\pi$
$389$	12.7471	0.646306	0.323153	+	0.946347i	$0.395257\pi$
$390$	0	0
$391$	2.19587	0.111050
$392$	−0.882955	−0.0445960
$393$	0	0
$394$	21.3465	1.07542
$395$	2.60989	0.131318
$396$	0	0
$397$	0.267573	0.0134291	0.00671456	−	0.999977i	$-0.497863\pi$
$397$	0.267573	0.0134291	0.00671456	+	0.999977i	$0.497863\pi$
$398$	−4.90528	−0.245879
$399$	0	0
$400$	−0.818566	−0.0409283
$401$	25.0956	1.25321	0.626607	−	0.779335i	$-0.284443\pi$
$401$	25.0956	1.25321	0.626607	+	0.779335i	$0.284443\pi$
$402$	0	0
$403$	−3.82485	−0.190529
$404$	15.7515	0.783669
$405$	0	0
$406$	−2.30638	−0.114464
$407$	1.31694	0.0652783
$408$	0	0
$409$	−24.7033	−1.22150	−0.610750	−	0.791823i	$-0.709132\pi$
$409$	−24.7033	−1.22150	−0.610750	+	0.791823i	$0.709132\pi$
$410$	−0.120755	−0.00596369
$411$	0	0
$412$	−17.3995	−0.857211
$413$	−0.876339	−0.0431218
$414$	0	0
$415$	1.85699	0.0911561
$416$	−2.97532	−0.145877
$417$	0	0
$418$	0.452039	0.0221099
$419$	14.0991	0.688786	0.344393	−	0.938826i	$-0.388085\pi$
$419$	14.0991	0.688786	0.344393	+	0.938826i	$0.388085\pi$
$420$	0	0
$421$	18.9132	0.921772	0.460886	−	0.887459i	$-0.347532\pi$
$421$	18.9132	0.921772	0.460886	+	0.887459i	$0.347532\pi$
$422$	3.00581	0.146320
$423$	0	0
$424$	2.91051	0.141347
$425$	10.8094	0.524334
$426$	0	0
$427$	23.3480	1.12989
$428$	−6.35313	−0.307090
$429$	0	0
$430$	1.09831	0.0529652
$431$	7.03206	0.338722	0.169361	−	0.985554i	$-0.445830\pi$
$431$	7.03206	0.338722	0.169361	+	0.985554i	$0.445830\pi$
$432$	0	0
$433$	−20.4589	−0.983192	−0.491596	−	0.870823i	$-0.663586\pi$
$433$	−20.4589	−0.983192	−0.491596	+	0.870823i	$0.663586\pi$
$434$	16.9691	0.814545
$435$	0	0
$436$	−10.3734	−0.496798
$437$	0.104552	0.00500142
$438$	0	0
$439$	−28.6535	−1.36756	−0.683778	−	0.729690i	$-0.739664\pi$
$439$	−28.6535	−1.36756	−0.683778	+	0.729690i	$0.739664\pi$
$440$	3.93632	0.187656
$441$	0	0
$442$	0.973364	0.0462982
$443$	37.4338	1.77854	0.889268	−	0.457387i	$-0.151214\pi$
$443$	37.4338	1.77854	0.889268	+	0.457387i	$0.151214\pi$
$444$	0	0
$445$	−1.53559	−0.0727939
$446$	−19.1617	−0.907335
$447$	0	0
$448$	12.3006	0.581147
$449$	−4.16541	−0.196578	−0.0982889	−	0.995158i	$-0.531337\pi$
$449$	−4.16541	−0.196578	−0.0982889	+	0.995158i	$0.531337\pi$
$450$	0	0
$451$	2.58150	0.121558
$452$	18.9780	0.892651
$453$	0	0
$454$	11.5239	0.540843
$455$	−0.391151	−0.0183374
$456$	0	0
$457$	15.6569	0.732399	0.366200	−	0.930536i	$-0.380659\pi$
$457$	15.6569	0.732399	0.366200	+	0.930536i	$0.380659\pi$
$458$	−5.21035	−0.243463
$459$	0	0
$460$	0.354095	0.0165098
$461$	22.6379	1.05435	0.527177	−	0.849756i	$-0.323251\pi$
$461$	22.6379	1.05435	0.527177	+	0.849756i	$0.323251\pi$
$462$	0	0
$463$	−4.04786	−0.188120	−0.0940599	−	0.995567i	$-0.529985\pi$
$463$	−4.04786	−0.188120	−0.0940599	+	0.995567i	$0.529985\pi$
$464$	−0.166287	−0.00771966
$465$	0	0
$466$	14.8249	0.686750
$467$	2.50969	0.116135	0.0580673	−	0.998313i	$-0.481506\pi$
$467$	2.50969	0.116135	0.0580673	+	0.998313i	$0.481506\pi$
$468$	0	0
$469$	−8.95813	−0.413648
$470$	2.94307	0.135754
$471$	0	0
$472$	0.904162	0.0416174
$473$	−23.4796	−1.07959
$474$	0	0
$475$	0.514672	0.0236148
$476$	7.56074	0.346546
$477$	0	0
$478$	−1.41311	−0.0646341
$479$	−15.3017	−0.699154	−0.349577	−	0.936908i	$-0.613675\pi$
$479$	−15.3017	−0.699154	−0.349577	+	0.936908i	$0.613675\pi$
$480$	0	0
$481$	0.135019	0.00615632
$482$	16.8954	0.769566
$483$	0	0
$484$	−18.7262	−0.851189
$485$	0.163294	0.00741479
$486$	0	0
$487$	−12.4632	−0.564760	−0.282380	−	0.959303i	$-0.591124\pi$
$487$	−12.4632	−0.564760	−0.282380	+	0.959303i	$0.591124\pi$
$488$	−24.0892	−1.09047
$489$	0	0
$490$	0.0750428	0.00339009
$491$	−6.93040	−0.312764	−0.156382	−	0.987697i	$-0.549983\pi$
$491$	−6.93040	−0.312764	−0.156382	+	0.987697i	$0.549983\pi$
$492$	0	0
$493$	2.19587	0.0988969
$494$	0.0463451	0.00208516
$495$	0	0
$496$	1.22345	0.0549345
$497$	38.2093	1.71392
$498$	0	0
$499$	27.5543	1.23350	0.616749	−	0.787160i	$-0.288449\pi$
$499$	27.5543	1.23350	0.616749	+	0.787160i	$0.288449\pi$
$500$	3.51355	0.157131
$501$	0	0
$502$	8.52484	0.380482
$503$	7.21927	0.321891	0.160946	−	0.986963i	$-0.448546\pi$
$503$	7.21927	0.321891	0.160946	+	0.986963i	$0.448546\pi$
$504$	0	0
$505$	−3.44209	−0.153171
$506$	4.32356	0.192206
$507$	0	0
$508$	11.9827	0.531645
$509$	30.1844	1.33790	0.668950	−	0.743308i	$-0.266744\pi$
$509$	30.1844	1.33790	0.668950	+	0.743308i	$0.266744\pi$
$510$	0	0
$511$	−1.29726	−0.0573873
$512$	1.87984	0.0830780
$513$	0	0
$514$	2.10836	0.0929958
$515$	3.80221	0.167545
$516$	0	0
$517$	−62.9167	−2.76707
$518$	−0.599017	−0.0263193
$519$	0	0
$520$	0.403569	0.0176977
$521$	10.4106	0.456096	0.228048	−	0.973650i	$-0.426766\pi$
$521$	10.4106	0.456096	0.228048	+	0.973650i	$0.426766\pi$
$522$	0	0
$523$	−30.7711	−1.34552	−0.672762	−	0.739859i	$-0.734892\pi$
$523$	−30.7711	−1.34552	−0.672762	+	0.739859i	$0.734892\pi$
$524$	3.39033	0.148107
$525$	0	0
$526$	−20.4203	−0.890369
$527$	−16.1560	−0.703767
$528$	0	0
$529$	1.00000	0.0434783
$530$	−0.247366	−0.0107449
$531$	0	0
$532$	0.359992	0.0156076
$533$	0.264667	0.0114640
$534$	0	0
$535$	1.38831	0.0600219
$536$	9.24254	0.399217
$537$	0	0
$538$	15.0033	0.646840
$539$	−1.60426	−0.0691004
$540$	0	0
$541$	24.0739	1.03502	0.517509	−	0.855678i	$-0.326860\pi$
$541$	24.0739	1.03502	0.517509	+	0.855678i	$0.326860\pi$
$542$	10.6846	0.458941
$543$	0	0
$544$	−12.5676	−0.538832
$545$	2.26685	0.0971010
$546$	0	0
$547$	−2.60380	−0.111330	−0.0556652	−	0.998449i	$-0.517728\pi$
$547$	−2.60380	−0.111330	−0.0556652	+	0.998449i	$0.517728\pi$
$548$	−14.5304	−0.620710
$549$	0	0
$550$	21.2833	0.907521
$551$	0.104552	0.00445408
$552$	0	0
$553$	25.3782	1.07919
$554$	6.11656	0.259868
$555$	0	0
$556$	−16.0660	−0.681349
$557$	−16.5729	−0.702215	−0.351108	−	0.936335i	$-0.614195\pi$
$557$	−16.5729	−0.702215	−0.351108	+	0.936335i	$0.614195\pi$
$558$	0	0
$559$	−2.40724	−0.101815
$560$	0.125117	0.00528714
$561$	0	0
$562$	3.01536	0.127195
$563$	4.71445	0.198690	0.0993452	−	0.995053i	$-0.468325\pi$
$563$	4.71445	0.198690	0.0993452	+	0.995053i	$0.468325\pi$
$564$	0	0
$565$	−4.14716	−0.174472
$566$	−1.87395	−0.0787680
$567$	0	0
$568$	−39.4223	−1.65412
$569$	−2.24239	−0.0940058	−0.0470029	−	0.998895i	$-0.514967\pi$
$569$	−2.24239	−0.0940058	−0.0470029	+	0.998895i	$0.514967\pi$
$570$	0	0
$571$	38.4824	1.61044	0.805220	−	0.592977i	$-0.202047\pi$
$571$	38.4824	1.61044	0.805220	+	0.592977i	$0.202047\pi$
$572$	−3.35548	−0.140300
$573$	0	0
$574$	−1.17421	−0.0490106
$575$	4.92262	0.205288
$576$	0	0
$577$	−31.8247	−1.32488	−0.662439	−	0.749116i	$-0.730479\pi$
$577$	−31.8247	−1.32488	−0.662439	+	0.749116i	$0.730479\pi$
$578$	−10.3840	−0.431918
$579$	0	0
$580$	0.354095	0.0147030
$581$	18.0571	0.749136
$582$	0	0
$583$	5.28817	0.219014
$584$	1.33844	0.0553852
$585$	0	0
$586$	2.97335	0.122828
$587$	35.1398	1.45038	0.725188	−	0.688551i	$-0.241753\pi$
$587$	35.1398	1.45038	0.725188	+	0.688551i	$0.241753\pi$
$588$	0	0
$589$	−0.769241	−0.0316960
$590$	−0.0768452	−0.00316367
$591$	0	0
$592$	−0.0431882	−0.00177502
$593$	−13.1988	−0.542009	−0.271004	−	0.962578i	$-0.587356\pi$
$593$	−13.1988	−0.542009	−0.271004	+	0.962578i	$0.587356\pi$
$594$	0	0
$595$	−1.65220	−0.0677338
$596$	13.6056	0.557306
$597$	0	0
$598$	0.443271	0.0181267
$599$	42.6629	1.74316	0.871579	−	0.490254i	$-0.163096\pi$
$599$	42.6629	1.74316	0.871579	+	0.490254i	$0.163096\pi$
$600$	0	0
$601$	−4.17667	−0.170370	−0.0851850	−	0.996365i	$-0.527148\pi$
$601$	−4.17667	−0.170370	−0.0851850	+	0.996365i	$0.527148\pi$
$602$	10.6798	0.435277
$603$	0	0
$604$	10.1331	0.412311
$605$	4.09212	0.166368
$606$	0	0
$607$	−19.0944	−0.775017	−0.387508	−	0.921866i	$-0.626664\pi$
$607$	−19.0944	−0.775017	−0.387508	+	0.921866i	$0.626664\pi$
$608$	−0.598385	−0.0242677
$609$	0	0
$610$	2.04736	0.0828950
$611$	−6.45051	−0.260960
$612$	0	0
$613$	18.3726	0.742062	0.371031	−	0.928620i	$-0.379004\pi$
$613$	18.3726	0.742062	0.371031	+	0.928620i	$0.379004\pi$
$614$	−8.81795	−0.355864
$615$	0	0
$616$	38.2762	1.54219
$617$	36.9471	1.48743	0.743716	−	0.668495i	$-0.233061\pi$
$617$	36.9471	1.48743	0.743716	+	0.668495i	$0.233061\pi$
$618$	0	0
$619$	−35.4201	−1.42365	−0.711827	−	0.702355i	$-0.752132\pi$
$619$	−35.4201	−1.42365	−0.711827	+	0.702355i	$0.752132\pi$
$620$	−2.60524	−0.104629
$621$	0	0
$622$	2.93703	0.117764
$623$	−14.9319	−0.598233
$624$	0	0
$625$	23.8453	0.953813
$626$	8.01023	0.320153
$627$	0	0
$628$	−8.39018	−0.334805
$629$	0.570314	0.0227399
$630$	0	0
$631$	−29.4582	−1.17271	−0.586355	−	0.810054i	$-0.699438\pi$
$631$	−29.4582	−1.17271	−0.586355	+	0.810054i	$0.699438\pi$
$632$	−26.1839	−1.04154
$633$	0	0
$634$	−17.4073	−0.691334
$635$	−2.61850	−0.103912
$636$	0	0
$637$	−0.164476	−0.00651678
$638$	4.32356	0.171171
$639$	0	0
$640$	−2.10547	−0.0832262
$641$	−14.9498	−0.590481	−0.295241	−	0.955423i	$-0.595400\pi$
$641$	−14.9498	−0.590481	−0.295241	+	0.955423i	$0.595400\pi$
$642$	0	0
$643$	25.7271	1.01458	0.507289	−	0.861776i	$-0.330648\pi$
$643$	25.7271	1.01458	0.507289	+	0.861776i	$0.330648\pi$
$644$	3.44317	0.135680
$645$	0	0
$646$	0.195760	0.00770206
$647$	−25.3141	−0.995201	−0.497600	−	0.867406i	$-0.665785\pi$
$647$	−25.3141	−0.995201	−0.497600	+	0.867406i	$0.665785\pi$
$648$	0	0
$649$	1.64279	0.0644852
$650$	2.18206	0.0855873
$651$	0	0
$652$	−27.0981	−1.06124
$653$	−17.9366	−0.701914	−0.350957	−	0.936391i	$-0.614144\pi$
$653$	−17.9366	−0.701914	−0.350957	+	0.936391i	$0.614144\pi$
$654$	0	0
$655$	−0.740869	−0.0289481
$656$	−0.0846587	−0.00330537
$657$	0	0
$658$	28.6180	1.11565
$659$	−0.907905	−0.0353670	−0.0176835	−	0.999844i	$-0.505629\pi$
$659$	−0.907905	−0.0353670	−0.0176835	+	0.999844i	$0.505629\pi$
$660$	0	0
$661$	−21.1348	−0.822048	−0.411024	−	0.911624i	$-0.634829\pi$
$661$	−21.1348	−0.822048	−0.411024	+	0.911624i	$0.634829\pi$
$662$	−1.39788	−0.0543303
$663$	0	0
$664$	−18.6304	−0.723000
$665$	−0.0786669	−0.00305057
$666$	0	0
$667$	1.00000	0.0387202
$668$	23.7125	0.917462
$669$	0	0
$670$	−0.785528	−0.0303476
$671$	−43.7682	−1.68965
$672$	0	0
$673$	−9.47393	−0.365193	−0.182597	−	0.983188i	$-0.558450\pi$
$673$	−9.47393	−0.365193	−0.182597	+	0.983188i	$0.558450\pi$
$674$	21.7163	0.836480
$675$	0	0
$676$	16.2043	0.623242
$677$	44.9991	1.72945	0.864727	−	0.502242i	$-0.167491\pi$
$677$	44.9991	1.72945	0.864727	+	0.502242i	$0.167491\pi$
$678$	0	0
$679$	1.58785	0.0609359
$680$	1.70466	0.0653707
$681$	0	0
$682$	−31.8105	−1.21809
$683$	15.2628	0.584016	0.292008	−	0.956416i	$-0.405677\pi$
$683$	15.2628	0.584016	0.292008	+	0.956416i	$0.405677\pi$
$684$	0	0
$685$	3.17525	0.121320
$686$	−15.4150	−0.588546
$687$	0	0
$688$	0.769998	0.0293559
$689$	0.542167	0.0206549
$690$	0	0
$691$	35.3108	1.34329	0.671644	−	0.740874i	$-0.265589\pi$
$691$	35.3108	1.34329	0.671644	+	0.740874i	$0.265589\pi$
$692$	−12.9633	−0.492790
$693$	0	0
$694$	26.3041	0.998489
$695$	3.51080	0.133172
$696$	0	0
$697$	1.11794	0.0423452
$698$	6.01967	0.227848
$699$	0	0
$700$	16.9494	0.640628
$701$	−20.2694	−0.765565	−0.382782	−	0.923839i	$-0.625034\pi$
$701$	−20.2694	−0.765565	−0.382782	+	0.923839i	$0.625034\pi$
$702$	0	0
$703$	0.0271545	0.00102415
$704$	−23.0587	−0.869058
$705$	0	0
$706$	−13.9742	−0.525926
$707$	−33.4704	−1.25879
$708$	0	0
$709$	−14.0026	−0.525879	−0.262939	−	0.964812i	$-0.584692\pi$
$709$	−14.0026	−0.525879	−0.262939	+	0.964812i	$0.584692\pi$
$710$	3.35053	0.125743
$711$	0	0
$712$	15.4059	0.577362
$713$	−7.35747	−0.275539
$714$	0	0
$715$	0.733253	0.0274221
$716$	−6.60087	−0.246686
$717$	0	0
$718$	−20.7123	−0.772975
$719$	−29.9013	−1.11513	−0.557566	−	0.830133i	$-0.688265\pi$
$719$	−29.9013	−1.11513	−0.557566	+	0.830133i	$0.688265\pi$
$720$	0	0
$721$	36.9721	1.37691
$722$	−16.1915	−0.602585
$723$	0	0
$724$	−14.4237	−0.536053
$725$	4.92262	0.182822
$726$	0	0
$727$	17.5958	0.652593	0.326297	−	0.945267i	$-0.394199\pi$
$727$	17.5958	0.652593	0.326297	+	0.945267i	$0.394199\pi$
$728$	3.92425	0.145442
$729$	0	0
$730$	−0.113755	−0.00421026
$731$	−10.1681	−0.376080
$732$	0	0
$733$	−26.8913	−0.993254	−0.496627	−	0.867964i	$-0.665428\pi$
$733$	−26.8913	−0.993254	−0.496627	+	0.867964i	$0.665428\pi$
$734$	17.9003	0.660714
$735$	0	0
$736$	−5.72330	−0.210964
$737$	16.7930	0.618577
$738$	0	0
$739$	−25.8478	−0.950828	−0.475414	−	0.879762i	$-0.657702\pi$
$739$	−25.8478	−0.950828	−0.475414	+	0.879762i	$0.657702\pi$
$740$	0.0919660	0.00338074
$741$	0	0
$742$	−2.40535	−0.0883032
$743$	−40.1155	−1.47170	−0.735848	−	0.677147i	$-0.763216\pi$
$743$	−40.1155	−1.47170	−0.735848	+	0.677147i	$0.763216\pi$
$744$	0	0
$745$	−2.97315	−0.108928
$746$	7.56039	0.276805
$747$	0	0
$748$	−14.1734	−0.518232
$749$	13.4997	0.493270
$750$	0	0
$751$	17.9390	0.654603	0.327302	−	0.944920i	$-0.393861\pi$
$751$	17.9390	0.654603	0.327302	+	0.944920i	$0.393861\pi$
$752$	2.06331	0.0752413
$753$	0	0
$754$	0.443271	0.0161430
$755$	−2.21433	−0.0805878
$756$	0	0
$757$	−39.6003	−1.43930	−0.719648	−	0.694339i	$-0.755697\pi$
$757$	−39.6003	−1.43930	−0.719648	+	0.694339i	$0.755697\pi$
$758$	−15.3580	−0.557829
$759$	0	0
$760$	0.0811644	0.00294414
$761$	−4.42374	−0.160361	−0.0801803	−	0.996780i	$-0.525550\pi$
$761$	−4.42374	−0.160361	−0.0801803	+	0.996780i	$0.525550\pi$
$762$	0	0
$763$	22.0425	0.797992
$764$	−19.1388	−0.692416
$765$	0	0
$766$	−19.7895	−0.715025
$767$	0.168426	0.00608153
$768$	0	0
$769$	−22.4905	−0.811029	−0.405514	−	0.914089i	$-0.632908\pi$
$769$	−22.4905	−0.811029	−0.405514	+	0.914089i	$0.632908\pi$
$770$	−3.25311	−0.117234
$771$	0	0
$772$	−14.5982	−0.525400
$773$	24.2864	0.873521	0.436760	−	0.899578i	$-0.356126\pi$
$773$	24.2864	0.873521	0.436760	+	0.899578i	$0.356126\pi$
$774$	0	0
$775$	−36.2180	−1.30099
$776$	−1.63826	−0.0588100
$777$	0	0
$778$	10.8692	0.389678
$779$	0.0532290	0.00190713
$780$	0	0
$781$	−71.6273	−2.56303
$782$	1.87236	0.0669554
$783$	0	0
$784$	0.0526107	0.00187895
$785$	1.83346	0.0654388
$786$	0	0
$787$	−33.4845	−1.19359	−0.596797	−	0.802393i	$-0.703560\pi$
$787$	−33.4845	−1.19359	−0.596797	+	0.802393i	$0.703560\pi$
$788$	−31.8679	−1.13525
$789$	0	0
$790$	2.22539	0.0791757
$791$	−40.3264	−1.43384
$792$	0	0
$793$	−4.48732	−0.159349
$794$	0.228153	0.00809684
$795$	0	0
$796$	7.32303	0.259558
$797$	−31.4213	−1.11300	−0.556500	−	0.830847i	$-0.687856\pi$
$797$	−31.4213	−1.11300	−0.556500	+	0.830847i	$0.687856\pi$
$798$	0	0
$799$	−27.2467	−0.963919
$800$	−28.1737	−0.996089
$801$	0	0
$802$	21.3984	0.755603
$803$	2.43185	0.0858180
$804$	0	0
$805$	−0.752416	−0.0265192
$806$	−3.26135	−0.114876
$807$	0	0
$808$	34.5331	1.21487
$809$	−33.6559	−1.18328	−0.591639	−	0.806203i	$-0.701519\pi$
$809$	−33.6559	−1.18328	−0.591639	+	0.806203i	$0.701519\pi$
$810$	0	0
$811$	−2.12467	−0.0746072	−0.0373036	−	0.999304i	$-0.511877\pi$
$811$	−2.12467	−0.0746072	−0.0373036	+	0.999304i	$0.511877\pi$
$812$	3.44317	0.120832
$813$	0	0
$814$	1.12292	0.0393584
$815$	5.92159	0.207424
$816$	0	0
$817$	−0.484135	−0.0169377
$818$	−21.0639	−0.736481
$819$	0	0
$820$	0.180274	0.00629545
$821$	33.4063	1.16589	0.582944	−	0.812512i	$-0.301901\pi$
$821$	33.4063	1.16589	0.582944	+	0.812512i	$0.301901\pi$
$822$	0	0
$823$	21.5526	0.751276	0.375638	−	0.926766i	$-0.377424\pi$
$823$	21.5526	0.751276	0.375638	+	0.926766i	$0.377424\pi$
$824$	−38.1459	−1.32888
$825$	0	0
$826$	−0.747232	−0.0259995
$827$	−37.6143	−1.30798	−0.653988	−	0.756505i	$-0.726905\pi$
$827$	−37.6143	−1.30798	−0.653988	+	0.756505i	$0.726905\pi$
$828$	0	0
$829$	27.0749	0.940349	0.470174	−	0.882574i	$-0.344191\pi$
$829$	27.0749	0.940349	0.470174	+	0.882574i	$0.344191\pi$
$830$	1.58341	0.0549609
$831$	0	0
$832$	−2.36408	−0.0819598
$833$	−0.694740	−0.0240713
$834$	0	0
$835$	−5.18174	−0.179322
$836$	−0.674843	−0.0233399
$837$	0	0
$838$	12.0219	0.415291
$839$	−36.0003	−1.24287	−0.621434	−	0.783467i	$-0.713449\pi$
$839$	−36.0003	−1.24287	−0.621434	+	0.783467i	$0.713449\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	16.1268	0.555765
$843$	0	0
$844$	−4.48733	−0.154460
$845$	−3.54103	−0.121815
$846$	0	0
$847$	39.7912	1.36724
$848$	−0.173422	−0.00595534
$849$	0	0
$850$	9.21692	0.316138
$851$	0.259721	0.00890313
$852$	0	0
$853$	−8.31131	−0.284574	−0.142287	−	0.989825i	$-0.545446\pi$
$853$	−8.31131	−0.284574	−0.142287	+	0.989825i	$0.545446\pi$
$854$	19.9082	0.681245
$855$	0	0
$856$	−13.9283	−0.476061
$857$	−13.7510	−0.469726	−0.234863	−	0.972029i	$-0.575464\pi$
$857$	−13.7510	−0.469726	−0.234863	+	0.972029i	$0.575464\pi$
$858$	0	0
$859$	−28.3847	−0.968474	−0.484237	−	0.874937i	$-0.660903\pi$
$859$	−28.3847	−0.968474	−0.484237	+	0.874937i	$0.660903\pi$
$860$	−1.63965	−0.0559117
$861$	0	0
$862$	5.99606	0.204226
$863$	−13.2385	−0.450643	−0.225322	−	0.974284i	$-0.572343\pi$
$863$	−13.2385	−0.450643	−0.225322	+	0.974284i	$0.572343\pi$
$864$	0	0
$865$	2.83279	0.0963177
$866$	−17.4448	−0.592798
$867$	0	0
$868$	−25.3330	−0.859859
$869$	−47.5742	−1.61384
$870$	0	0
$871$	1.72169	0.0583373
$872$	−22.7423	−0.770152
$873$	0	0
$874$	0.0891492	0.00301552
$875$	−7.46593	−0.252395
$876$	0	0
$877$	51.8672	1.75143	0.875715	−	0.482829i	$-0.160391\pi$
$877$	51.8672	1.75143	0.875715	+	0.482829i	$0.160391\pi$
$878$	−24.4321	−0.824543
$879$	0	0
$880$	−0.234545	−0.00790649
$881$	−53.8584	−1.81454	−0.907268	−	0.420553i	$-0.861836\pi$
$881$	−53.8584	−1.81454	−0.907268	+	0.420553i	$0.861836\pi$
$882$	0	0
$883$	7.91168	0.266249	0.133125	−	0.991099i	$-0.457499\pi$
$883$	7.91168	0.266249	0.133125	+	0.991099i	$0.457499\pi$
$884$	−1.45312	−0.0488738
$885$	0	0
$886$	31.9189	1.07234
$887$	−23.3214	−0.783054	−0.391527	−	0.920167i	$-0.628053\pi$
$887$	−23.3214	−0.783054	−0.391527	+	0.920167i	$0.628053\pi$
$888$	0	0
$889$	−25.4620	−0.853967
$890$	−1.30936	−0.0438898
$891$	0	0
$892$	28.6063	0.957811
$893$	−1.29730	−0.0434126
$894$	0	0
$895$	1.44245	0.0482158
$896$	−20.4733	−0.683967
$897$	0	0
$898$	−3.55174	−0.118523
$899$	−7.35747	−0.245385
$900$	0	0
$901$	2.29009	0.0762941
$902$	2.20118	0.0732913
$903$	0	0
$904$	41.6067	1.38382
$905$	3.15193	0.104774
$906$	0	0
$907$	−23.9008	−0.793612	−0.396806	−	0.917902i	$-0.629881\pi$
$907$	−23.9008	−0.793612	−0.396806	+	0.917902i	$0.629881\pi$
$908$	−17.2039	−0.570931
$909$	0	0
$910$	−0.333524	−0.0110562
$911$	29.0376	0.962058	0.481029	−	0.876705i	$-0.340263\pi$
$911$	29.0376	0.962058	0.481029	+	0.876705i	$0.340263\pi$
$912$	0	0
$913$	−33.8500	−1.12027
$914$	13.3502	0.441587
$915$	0	0
$916$	7.77846	0.257008
$917$	−7.20411	−0.237901
$918$	0	0
$919$	43.7999	1.44483	0.722413	−	0.691462i	$-0.243033\pi$
$919$	43.7999	1.44483	0.722413	+	0.691462i	$0.243033\pi$
$920$	0.776303	0.0255940
$921$	0	0
$922$	19.3028	0.635703
$923$	−7.34356	−0.241716
$924$	0	0
$925$	1.27851	0.0420372
$926$	−3.45150	−0.113423
$927$	0	0
$928$	−5.72330	−0.187877
$929$	26.4998	0.869431	0.434715	−	0.900568i	$-0.356849\pi$
$929$	26.4998	0.869431	0.434715	+	0.900568i	$0.356849\pi$
$930$	0	0
$931$	−0.0330789	−0.00108412
$932$	−22.1319	−0.724954
$933$	0	0
$934$	2.13995	0.0700213
$935$	3.09723	0.101290
$936$	0	0
$937$	−20.4056	−0.666621	−0.333310	−	0.942817i	$-0.608166\pi$
$937$	−20.4056	−0.666621	−0.333310	+	0.942817i	$0.608166\pi$
$938$	−7.63837	−0.249402
$939$	0	0
$940$	−4.39367	−0.143306
$941$	50.4396	1.64428	0.822142	−	0.569282i	$-0.192779\pi$
$941$	50.4396	1.64428	0.822142	+	0.569282i	$0.192779\pi$
$942$	0	0
$943$	0.509113	0.0165790
$944$	−0.0538743	−0.00175346
$945$	0	0
$946$	−20.0205	−0.650921
$947$	−33.4712	−1.08767	−0.543833	−	0.839193i	$-0.683028\pi$
$947$	−33.4712	−1.08767	−0.543833	+	0.839193i	$0.683028\pi$
$948$	0	0
$949$	0.249324	0.00809340
$950$	0.438848	0.0142381
$951$	0	0
$952$	16.5759	0.537227
$953$	17.1268	0.554790	0.277395	−	0.960756i	$-0.410529\pi$
$953$	17.1268	0.554790	0.277395	+	0.960756i	$0.410529\pi$
$954$	0	0
$955$	4.18228	0.135335
$956$	2.10961	0.0682297
$957$	0	0
$958$	−13.0474	−0.421542
$959$	30.8757	0.997029
$960$	0	0
$961$	23.1323	0.746204
$962$	0.115127	0.00371184
$963$	0	0
$964$	−25.2230	−0.812378
$965$	3.19006	0.102692
$966$	0	0
$967$	50.4031	1.62085	0.810427	−	0.585839i	$-0.199235\pi$
$967$	50.4031	1.62085	0.810427	+	0.585839i	$0.199235\pi$
$968$	−41.0545	−1.31954
$969$	0	0
$970$	0.139236	0.00447061
$971$	−24.2480	−0.778156	−0.389078	−	0.921205i	$-0.627206\pi$
$971$	−24.2480	−0.778156	−0.389078	+	0.921205i	$0.627206\pi$
$972$	0	0
$973$	34.1386	1.09443
$974$	−10.6270	−0.340512
$975$	0	0
$976$	1.43535	0.0459445
$977$	17.1751	0.549480	0.274740	−	0.961519i	$-0.411408\pi$
$977$	17.1751	0.549480	0.274740	+	0.961519i	$0.411408\pi$
$978$	0	0
$979$	27.9914	0.894608
$980$	−0.112030	−0.00357868
$981$	0	0
$982$	−5.90937	−0.188576
$983$	34.0769	1.08689	0.543443	−	0.839446i	$-0.317121\pi$
$983$	34.0769	1.08689	0.543443	+	0.839446i	$0.317121\pi$
$984$	0	0
$985$	6.96390	0.221888
$986$	1.87236	0.0596281
$987$	0	0
$988$	−0.0691880	−0.00220116
$989$	−4.63055	−0.147243
$990$	0	0
$991$	−52.6162	−1.67141	−0.835705	−	0.549179i	$-0.814940\pi$
$991$	−52.6162	−1.67141	−0.835705	+	0.549179i	$0.814940\pi$
$992$	42.1090	1.33696
$993$	0	0
$994$	32.5801	1.03338
$995$	−1.60026	−0.0507316
$996$	0	0
$997$	−15.7721	−0.499506	−0.249753	−	0.968310i	$-0.580349\pi$
$997$	−15.7721	−0.499506	−0.249753	+	0.968310i	$0.580349\pi$
$998$	23.4948	0.743715
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6003.2.a.s.1.13		20
3.2	odd	2		2001.2.a.o.1.8	✓	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2001.2.a.o.1.8	✓	20	3.2	odd	2
6003.2.a.s.1.13		20	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:	\( N \)	\(=\)	\( 6003 = 3^{2} \cdot 23 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6003.a (trivial)

\(f(q)\)	\(=\)	\(q+0.852674 q^{2} -1.27295 q^{4} +0.278169 q^{5} +2.70488 q^{7} -2.79076 q^{8} +O(q^{10})\)	\(q+0.852674 q^{2} -1.27295 q^{4} +0.278169 q^{5} +2.70488 q^{7} -2.79076 q^{8} +0.237188 q^{10} -5.07059 q^{11} -0.519860 q^{13} +2.30638 q^{14} +0.166287 q^{16} -2.19587 q^{17} -0.104552 q^{19} -0.354095 q^{20} -4.32356 q^{22} -1.00000 q^{23} -4.92262 q^{25} -0.443271 q^{26} -3.44317 q^{28} -1.00000 q^{29} +7.35747 q^{31} +5.72330 q^{32} -1.87236 q^{34} +0.752416 q^{35} -0.259721 q^{37} -0.0891492 q^{38} -0.776303 q^{40} -0.509113 q^{41} +4.63055 q^{43} +6.45459 q^{44} -0.852674 q^{46} +12.4082 q^{47} +0.316385 q^{49} -4.19739 q^{50} +0.661754 q^{52} -1.04291 q^{53} -1.41048 q^{55} -7.54867 q^{56} -0.852674 q^{58} -0.323984 q^{59} +8.63178 q^{61} +6.27352 q^{62} +4.54754 q^{64} -0.144609 q^{65} -3.31184 q^{67} +2.79522 q^{68} +0.641565 q^{70} +14.1260 q^{71} -0.479599 q^{73} -0.221458 q^{74} +0.133090 q^{76} -13.7153 q^{77} +9.38238 q^{79} +0.0462559 q^{80} -0.434108 q^{82} +6.67576 q^{83} -0.610823 q^{85} +3.94835 q^{86} +14.1508 q^{88} -5.52034 q^{89} -1.40616 q^{91} +1.27295 q^{92} +10.5801 q^{94} -0.0290833 q^{95} +0.587030 q^{97} +0.269774 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 2 q^{2} + 30 q^{4} + q^{5} + 9 q^{7} - 6 q^{8}+O(q^{10}) \) 20 * q - 2 * q^2 + 30 * q^4 + q^5 + 9 * q^7 - 6 * q^8	\( 20 q - 2 q^{2} + 30 q^{4} + q^{5} + 9 q^{7} - 6 q^{8} + 7 q^{10} + 21 q^{13} + q^{14} + 58 q^{16} + 4 q^{17} + 7 q^{19} + 20 q^{20} + 7 q^{22} - 20 q^{23} + 47 q^{25} - 8 q^{26} + 11 q^{28} - 20 q^{29} + 28 q^{31} - 14 q^{32} + 16 q^{34} - 9 q^{35} + 14 q^{37} + 20 q^{38} + 34 q^{40} - 7 q^{41} + 3 q^{43} + q^{44} + 2 q^{46} - 3 q^{47} + 35 q^{49} + 24 q^{50} + 73 q^{52} + 19 q^{53} + 29 q^{55} + 30 q^{56} + 2 q^{58} - 20 q^{59} + 15 q^{61} - 12 q^{62} + 82 q^{64} + 28 q^{65} + 20 q^{67} + 23 q^{68} - 24 q^{70} - 63 q^{71} + 19 q^{73} - 16 q^{74} - 44 q^{76} + 7 q^{77} + 32 q^{79} + 56 q^{80} - 20 q^{82} + 21 q^{83} + 4 q^{85} + 6 q^{86} + 55 q^{88} + 13 q^{89} + 70 q^{91} - 30 q^{92} - 12 q^{94} - 9 q^{95} - 9 q^{97} - 31 q^{98}+O(q^{100}) \) 20 * q - 2 * q^2 + 30 * q^4 + q^5 + 9 * q^7 - 6 * q^8 + 7 * q^10 + 21 * q^13 + q^14 + 58 * q^16 + 4 * q^17 + 7 * q^19 + 20 * q^20 + 7 * q^22 - 20 * q^23 + 47 * q^25 - 8 * q^26 + 11 * q^28 - 20 * q^29 + 28 * q^31 - 14 * q^32 + 16 * q^34 - 9 * q^35 + 14 * q^37 + 20 * q^38 + 34 * q^40 - 7 * q^41 + 3 * q^43 + q^44 + 2 * q^46 - 3 * q^47 + 35 * q^49 + 24 * q^50 + 73 * q^52 + 19 * q^53 + 29 * q^55 + 30 * q^56 + 2 * q^58 - 20 * q^59 + 15 * q^61 - 12 * q^62 + 82 * q^64 + 28 * q^65 + 20 * q^67 + 23 * q^68 - 24 * q^70 - 63 * q^71 + 19 * q^73 - 16 * q^74 - 44 * q^76 + 7 * q^77 + 32 * q^79 + 56 * q^80 - 20 * q^82 + 21 * q^83 + 4 * q^85 + 6 * q^86 + 55 * q^88 + 13 * q^89 + 70 * q^91 - 30 * q^92 - 12 * q^94 - 9 * q^95 - 9 * q^97 - 31 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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