LMFDB - Embedded newform 6003.2.a.m.1.10

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:	$1$
Dimension:	$11$
Coefficient field:	$\mathbb{Q}[x]/(x^{11} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{11} - 2 x^{10} - 18 x^{9} + 30 x^{8} + 124 x^{7} - 152 x^{6} - 408 x^{5} + 285 x^{4} + 634 x^{3} + \cdots - 108 $ x^11 - 2x^10 - 18x^9 + 30x^8 + 124x^7 - 152x^6 - 408x^5 + 285x^4 + 634x^3 - 93x^2 - 369x - 108
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 3 $
Twist minimal:	no (minimal twist has level 2001)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.10
Root			$-1.96728$ 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)

$f(q)$	$=$	$q+1.96728 q^{2} +1.87020 q^{4} +3.90206 q^{5} -0.839519 q^{7} -0.255353 q^{8} +O(q^{10})$	\(q+1.96728 q^{2} +1.87020 q^{4} +3.90206 q^{5} -0.839519 q^{7} -0.255353 q^{8} +7.67646 q^{10} -4.08430 q^{11} -6.57808 q^{13} -1.65157 q^{14} -4.24275 q^{16} -0.379112 q^{17} -6.25665 q^{19} +7.29764 q^{20} -8.03497 q^{22} -1.00000 q^{23} +10.2261 q^{25} -12.9409 q^{26} -1.57007 q^{28} +1.00000 q^{29} +3.23356 q^{31} -7.83599 q^{32} -0.745820 q^{34} -3.27585 q^{35} -7.47652 q^{37} -12.3086 q^{38} -0.996402 q^{40} +9.84358 q^{41} -6.99632 q^{43} -7.63846 q^{44} -1.96728 q^{46} +9.65140 q^{47} -6.29521 q^{49} +20.1176 q^{50} -12.3023 q^{52} -10.9060 q^{53} -15.9372 q^{55} +0.214373 q^{56} +1.96728 q^{58} +11.3965 q^{59} +2.37385 q^{61} +6.36133 q^{62} -6.93009 q^{64} -25.6681 q^{65} -11.1611 q^{67} -0.709015 q^{68} -6.44453 q^{70} -4.24762 q^{71} -8.99243 q^{73} -14.7084 q^{74} -11.7012 q^{76} +3.42884 q^{77} +1.79119 q^{79} -16.5555 q^{80} +19.3651 q^{82} +6.12986 q^{83} -1.47932 q^{85} -13.7637 q^{86} +1.04294 q^{88} -4.65351 q^{89} +5.52242 q^{91} -1.87020 q^{92} +18.9870 q^{94} -24.4138 q^{95} -2.57912 q^{97} -12.3845 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 11 q - 2 q^{2} + 18 q^{4} - 2 q^{5} + 3 q^{7} - 18 q^{8}+O(q^{10}) $ 11 * q - 2 * q^2 + 18 * q^4 - 2 * q^5 + 3 * q^7 - 18 * q^8	$ 11 q - 2 q^{2} + 18 q^{4} - 2 q^{5} + 3 q^{7} - 18 q^{8} + 14 q^{10} - 11 q^{11} - 5 q^{13} - 17 q^{14} + 20 q^{16} - 15 q^{17} - 6 q^{19} - 21 q^{20} - 10 q^{22} - 11 q^{23} + 3 q^{25} + 5 q^{26} + 7 q^{28} + 11 q^{29} + 35 q^{31} - 28 q^{32} + 28 q^{34} - 15 q^{35} - 28 q^{37} + 2 q^{38} - q^{40} - 10 q^{41} - 6 q^{43} - 18 q^{44} + 2 q^{46} - 15 q^{47} + 22 q^{49} - 15 q^{50} - 36 q^{52} + 7 q^{53} - 12 q^{55} - 56 q^{56} - 2 q^{58} + 20 q^{59} - 20 q^{61} + 11 q^{62} + 36 q^{64} - 11 q^{65} - 39 q^{67} - 35 q^{68} + 38 q^{70} - 49 q^{71} - 3 q^{73} - 37 q^{74} - 18 q^{76} - 25 q^{77} + 41 q^{79} - 51 q^{80} - 19 q^{82} - 13 q^{83} - 62 q^{86} - 40 q^{88} - 34 q^{89} + 2 q^{91} - 18 q^{92} - 14 q^{94} - 25 q^{95} - 11 q^{97} - 53 q^{98}+O(q^{100}) $ 11 * q - 2 * q^2 + 18 * q^4 - 2 * q^5 + 3 * q^7 - 18 * q^8 + 14 * q^10 - 11 * q^11 - 5 * q^13 - 17 * q^14 + 20 * q^16 - 15 * q^17 - 6 * q^19 - 21 * q^20 - 10 * q^22 - 11 * q^23 + 3 * q^25 + 5 * q^26 + 7 * q^28 + 11 * q^29 + 35 * q^31 - 28 * q^32 + 28 * q^34 - 15 * q^35 - 28 * q^37 + 2 * q^38 - q^40 - 10 * q^41 - 6 * q^43 - 18 * q^44 + 2 * q^46 - 15 * q^47 + 22 * q^49 - 15 * q^50 - 36 * q^52 + 7 * q^53 - 12 * q^55 - 56 * q^56 - 2 * q^58 + 20 * q^59 - 20 * q^61 + 11 * q^62 + 36 * q^64 - 11 * q^65 - 39 * q^67 - 35 * q^68 + 38 * q^70 - 49 * q^71 - 3 * q^73 - 37 * q^74 - 18 * q^76 - 25 * q^77 + 41 * q^79 - 51 * q^80 - 19 * q^82 - 13 * q^83 - 62 * q^86 - 40 * q^88 - 34 * q^89 + 2 * q^91 - 18 * q^92 - 14 * q^94 - 25 * q^95 - 11 * q^97 - 53 * 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 $

$2$	1.96728	1.39108	0.695539	−	0.718488i	$-0.255165\pi$
$2$	1.96728	1.39108	0.695539	+	0.718488i	$0.255165\pi$
$3$	0	0
$3$	0	0
$4$	1.87020	0.935100
$5$	3.90206	1.74505	0.872527	−	0.488565i	$-0.162480\pi$
$5$	3.90206	1.74505	0.872527	+	0.488565i	$0.162480\pi$
$6$	0	0
$7$	−0.839519	−0.317308	−0.158654	−	0.987334i	$-0.550715\pi$
$7$	−0.839519	−0.317308	−0.158654	+	0.987334i	$0.550715\pi$
$8$	−0.255353	−0.0902808
$9$	0	0
$10$	7.67646	2.42751
$11$	−4.08430	−1.23146	−0.615731	−	0.787956i	$-0.711139\pi$
$11$	−4.08430	−1.23146	−0.615731	+	0.787956i	$0.711139\pi$
$12$	0	0
$13$	−6.57808	−1.82443	−0.912215	−	0.409712i	$-0.865629\pi$
$13$	−6.57808	−1.82443	−0.912215	+	0.409712i	$0.865629\pi$
$14$	−1.65157	−0.441401
$15$	0	0
$16$	−4.24275	−1.06069
$17$	−0.379112	−0.0919481	−0.0459741	−	0.998943i	$-0.514639\pi$
$17$	−0.379112	−0.0919481	−0.0459741	+	0.998943i	$0.514639\pi$
$18$	0	0
$19$	−6.25665	−1.43537	−0.717687	−	0.696366i	$-0.754799\pi$
$19$	−6.25665	−1.43537	−0.717687	+	0.696366i	$0.754799\pi$
$20$	7.29764	1.63180
$21$	0	0
$22$	−8.03497	−1.71306
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	10.2261	2.04522
$26$	−12.9409	−2.53793
$27$	0	0
$28$	−1.57007	−0.296715
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	0	0
$31$	3.23356	0.580765	0.290383	−	0.956911i	$-0.406217\pi$
$31$	3.23356	0.580765	0.290383	+	0.956911i	$0.406217\pi$
$32$	−7.83599	−1.38522
$33$	0	0
$34$	−0.745820	−0.127907
$35$	−3.27585	−0.553720
$36$	0	0
$37$	−7.47652	−1.22913	−0.614566	−	0.788865i	$-0.710669\pi$
$37$	−7.47652	−1.22913	−0.614566	+	0.788865i	$0.710669\pi$
$38$	−12.3086	−1.99672
$39$	0	0
$40$	−0.996402	−0.157545
$41$	9.84358	1.53731	0.768655	−	0.639664i	$-0.220927\pi$
$41$	9.84358	1.53731	0.768655	+	0.639664i	$0.220927\pi$
$42$	0	0
$43$	−6.99632	−1.06693	−0.533465	−	0.845822i	$-0.679110\pi$
$43$	−6.99632	−1.06693	−0.533465	+	0.845822i	$0.679110\pi$
$44$	−7.63846	−1.15154
$45$	0	0
$46$	−1.96728	−0.290060
$47$	9.65140	1.40780	0.703900	−	0.710299i	$-0.251440\pi$
$47$	9.65140	1.40780	0.703900	+	0.710299i	$0.251440\pi$
$48$	0	0
$49$	−6.29521	−0.899316
$50$	20.1176	2.84506
$51$	0	0
$52$	−12.3023	−1.70602
$53$	−10.9060	−1.49805	−0.749027	−	0.662540i	$-0.769479\pi$
$53$	−10.9060	−1.49805	−0.749027	+	0.662540i	$0.769479\pi$
$54$	0	0
$55$	−15.9372	−2.14897
$56$	0.214373	0.0286468
$57$	0	0
$58$	1.96728	0.258317
$59$	11.3965	1.48370	0.741850	−	0.670566i	$-0.233949\pi$
$59$	11.3965	1.48370	0.741850	+	0.670566i	$0.233949\pi$
$60$	0	0
$61$	2.37385	0.303940	0.151970	−	0.988385i	$-0.451438\pi$
$61$	2.37385	0.303940	0.151970	+	0.988385i	$0.451438\pi$
$62$	6.36133	0.807890
$63$	0	0
$64$	−6.93009	−0.866262
$65$	−25.6681	−3.18373
$66$	0	0
$67$	−11.1611	−1.36355	−0.681774	−	0.731563i	$-0.738791\pi$
$67$	−11.1611	−1.36355	−0.681774	+	0.731563i	$0.738791\pi$
$68$	−0.709015	−0.0859807
$69$	0	0
$70$	−6.44453	−0.770268
$71$	−4.24762	−0.504100	−0.252050	−	0.967714i	$-0.581105\pi$
$71$	−4.24762	−0.504100	−0.252050	+	0.967714i	$0.581105\pi$
$72$	0	0
$73$	−8.99243	−1.05248	−0.526242	−	0.850335i	$-0.676400\pi$
$73$	−8.99243	−1.05248	−0.526242	+	0.850335i	$0.676400\pi$
$74$	−14.7084	−1.70982
$75$	0	0
$76$	−11.7012	−1.34222
$77$	3.42884	0.390753
$78$	0	0
$79$	1.79119	0.201524	0.100762	−	0.994911i	$-0.467872\pi$
$79$	1.79119	0.201524	0.100762	+	0.994911i	$0.467872\pi$
$80$	−16.5555	−1.85096
$81$	0	0
$82$	19.3651	2.13852
$83$	6.12986	0.672839	0.336420	−	0.941712i	$-0.390784\pi$
$83$	6.12986	0.672839	0.336420	+	0.941712i	$0.390784\pi$
$84$	0	0
$85$	−1.47932	−0.160455
$86$	−13.7637	−1.48418
$87$	0	0
$88$	1.04294	0.111177
$89$	−4.65351	−0.493271	−0.246636	−	0.969108i	$-0.579325\pi$
$89$	−4.65351	−0.493271	−0.246636	+	0.969108i	$0.579325\pi$
$90$	0	0
$91$	5.52242	0.578907
$92$	−1.87020	−0.194982
$93$	0	0
$94$	18.9870	1.95836
$95$	−24.4138	−2.50480
$96$	0	0
$97$	−2.57912	−0.261870	−0.130935	−	0.991391i	$-0.541798\pi$
$97$	−2.57912	−0.261870	−0.130935	+	0.991391i	$0.541798\pi$
$98$	−12.3845	−1.25102
$99$	0	0
$100$	19.1248	1.91248
$101$	−17.5570	−1.74699	−0.873495	−	0.486834i	$-0.838152\pi$
$101$	−17.5570	−1.74699	−0.873495	+	0.486834i	$0.838152\pi$
$102$	0	0
$103$	10.5650	1.04100	0.520500	−	0.853862i	$-0.325746\pi$
$103$	10.5650	1.04100	0.520500	+	0.853862i	$0.325746\pi$
$104$	1.67973	0.164711
$105$	0	0
$106$	−21.4552	−2.08391
$107$	10.3172	0.997405	0.498703	−	0.866773i	$-0.333810\pi$
$107$	10.3172	0.997405	0.498703	+	0.866773i	$0.333810\pi$
$108$	0	0
$109$	5.06258	0.484907	0.242454	−	0.970163i	$-0.422048\pi$
$109$	5.06258	0.484907	0.242454	+	0.970163i	$0.422048\pi$
$110$	−31.3529	−2.98939
$111$	0	0
$112$	3.56187	0.336565
$113$	−3.10215	−0.291826	−0.145913	−	0.989297i	$-0.546612\pi$
$113$	−3.10215	−0.291826	−0.145913	+	0.989297i	$0.546612\pi$
$114$	0	0
$115$	−3.90206	−0.363869
$116$	1.87020	0.173644
$117$	0	0
$118$	22.4202	2.06394
$119$	0.318271	0.0291759
$120$	0	0
$121$	5.68150	0.516500
$122$	4.67003	0.422805
$123$	0	0
$124$	6.04741	0.543074
$125$	20.3925	1.82396
$126$	0	0
$127$	−10.0701	−0.893578	−0.446789	−	0.894639i	$-0.647433\pi$
$127$	−10.0701	−0.893578	−0.446789	+	0.894639i	$0.647433\pi$
$128$	2.03852	0.180181
$129$	0	0
$130$	−50.4963	−4.42882
$131$	16.4759	1.43950	0.719752	−	0.694231i	$-0.244256\pi$
$131$	16.4759	1.43950	0.719752	+	0.694231i	$0.244256\pi$
$132$	0	0
$133$	5.25257	0.455456
$134$	−21.9571	−1.89680
$135$	0	0
$136$	0.0968072	0.00830115
$137$	−19.6782	−1.68122	−0.840610	−	0.541640i	$-0.817804\pi$
$137$	−19.6782	−1.68122	−0.840610	+	0.541640i	$0.817804\pi$
$138$	0	0
$139$	19.0807	1.61840	0.809202	−	0.587530i	$-0.199900\pi$
$139$	19.0807	1.61840	0.809202	+	0.587530i	$0.199900\pi$
$140$	−6.12650	−0.517784
$141$	0	0
$142$	−8.35628	−0.701243
$143$	26.8668	2.24672
$144$	0	0
$145$	3.90206	0.324049
$146$	−17.6907	−1.46409
$147$	0	0
$148$	−13.9826	−1.14936
$149$	−0.870886	−0.0713457	−0.0356729	−	0.999364i	$-0.511357\pi$
$149$	−0.870886	−0.0713457	−0.0356729	+	0.999364i	$0.511357\pi$
$150$	0	0
$151$	11.5494	0.939873	0.469937	−	0.882700i	$-0.344277\pi$
$151$	11.5494	0.939873	0.469937	+	0.882700i	$0.344277\pi$
$152$	1.59765	0.129587
$153$	0	0
$154$	6.74551	0.543568
$155$	12.6176	1.01347
$156$	0	0
$157$	−6.51319	−0.519810	−0.259905	−	0.965634i	$-0.583691\pi$
$157$	−6.51319	−0.519810	−0.259905	+	0.965634i	$0.583691\pi$
$158$	3.52377	0.280336
$159$	0	0
$160$	−30.5765	−2.41728
$161$	0.839519	0.0661633
$162$	0	0
$163$	1.58258	0.123957	0.0619786	−	0.998077i	$-0.480259\pi$
$163$	1.58258	0.123957	0.0619786	+	0.998077i	$0.480259\pi$
$164$	18.4095	1.43754
$165$	0	0
$166$	12.0592	0.935972
$167$	−6.04190	−0.467536	−0.233768	−	0.972292i	$-0.575106\pi$
$167$	−6.04190	−0.467536	−0.233768	+	0.972292i	$0.575106\pi$
$168$	0	0
$169$	30.2711	2.32854
$170$	−2.91024	−0.223205
$171$	0	0
$172$	−13.0845	−0.997686
$173$	2.34652	0.178403	0.0892013	−	0.996014i	$-0.471569\pi$
$173$	2.34652	0.178403	0.0892013	+	0.996014i	$0.471569\pi$
$174$	0	0
$175$	−8.58499	−0.648964
$176$	17.3287	1.30620
$177$	0	0
$178$	−9.15477	−0.686179
$179$	−8.16175	−0.610038	−0.305019	−	0.952346i	$-0.598663\pi$
$179$	−8.16175	−0.610038	−0.305019	+	0.952346i	$0.598663\pi$
$180$	0	0
$181$	−3.58883	−0.266756	−0.133378	−	0.991065i	$-0.542582\pi$
$181$	−3.58883	−0.266756	−0.133378	+	0.991065i	$0.542582\pi$
$182$	10.8642	0.805305
$183$	0	0
$184$	0.255353	0.0188248
$185$	−29.1738	−2.14490
$186$	0	0
$187$	1.54841	0.113231
$188$	18.0500	1.31643
$189$	0	0
$190$	−48.0289	−3.48438
$191$	−25.9036	−1.87432	−0.937159	−	0.348902i	$-0.886555\pi$
$191$	−25.9036	−1.87432	−0.937159	+	0.348902i	$0.886555\pi$
$192$	0	0
$193$	−12.9762	−0.934050	−0.467025	−	0.884244i	$-0.654674\pi$
$193$	−12.9762	−0.934050	−0.467025	+	0.884244i	$0.654674\pi$
$194$	−5.07386	−0.364282
$195$	0	0
$196$	−11.7733	−0.840950
$197$	10.6793	0.760867	0.380433	−	0.924808i	$-0.375775\pi$
$197$	10.6793	0.760867	0.380433	+	0.924808i	$0.375775\pi$
$198$	0	0
$199$	3.77912	0.267895	0.133947	−	0.990988i	$-0.457235\pi$
$199$	3.77912	0.267895	0.133947	+	0.990988i	$0.457235\pi$
$200$	−2.61126	−0.184644
$201$	0	0
$202$	−34.5396	−2.43020
$203$	−0.839519	−0.0589226
$204$	0	0
$205$	38.4103	2.68269
$206$	20.7843	1.44811
$207$	0	0
$208$	27.9091	1.93515
$209$	25.5540	1.76761
$210$	0	0
$211$	19.8083	1.36366	0.681829	−	0.731512i	$-0.261185\pi$
$211$	19.8083	1.36366	0.681829	+	0.731512i	$0.261185\pi$
$212$	−20.3964	−1.40083
$213$	0	0
$214$	20.2969	1.38747
$215$	−27.3001	−1.86185
$216$	0	0
$217$	−2.71464	−0.184282
$218$	9.95953	0.674544
$219$	0	0
$220$	−29.8057	−2.00950
$221$	2.49383	0.167753
$222$	0	0
$223$	21.9181	1.46775	0.733874	−	0.679286i	$-0.237710\pi$
$223$	21.9181	1.46775	0.733874	+	0.679286i	$0.237710\pi$
$224$	6.57845	0.439541
$225$	0	0
$226$	−6.10281	−0.405953
$227$	12.5521	0.833113	0.416556	−	0.909110i	$-0.363237\pi$
$227$	12.5521	0.833113	0.416556	+	0.909110i	$0.363237\pi$
$228$	0	0
$229$	4.66018	0.307953	0.153977	−	0.988074i	$-0.450792\pi$
$229$	4.66018	0.307953	0.153977	+	0.988074i	$0.450792\pi$
$230$	−7.67646	−0.506171
$231$	0	0
$232$	−0.255353	−0.0167647
$233$	−13.1954	−0.864459	−0.432229	−	0.901764i	$-0.642273\pi$
$233$	−13.1954	−0.864459	−0.432229	+	0.901764i	$0.642273\pi$
$234$	0	0
$235$	37.6603	2.45669
$236$	21.3138	1.38741
$237$	0	0
$238$	0.626130	0.0405860
$239$	4.52134	0.292461	0.146231	−	0.989251i	$-0.453286\pi$
$239$	4.52134	0.292461	0.146231	+	0.989251i	$0.453286\pi$
$240$	0	0
$241$	0.464697	0.0299338	0.0149669	−	0.999888i	$-0.495236\pi$
$241$	0.464697	0.0299338	0.0149669	+	0.999888i	$0.495236\pi$
$242$	11.1771	0.718492
$243$	0	0
$244$	4.43958	0.284215
$245$	−24.5643	−1.56936
$246$	0	0
$247$	41.1567	2.61874
$248$	−0.825699	−0.0524319
$249$	0	0
$250$	40.1178	2.53727
$251$	8.29616	0.523649	0.261825	−	0.965115i	$-0.415676\pi$
$251$	8.29616	0.523649	0.261825	+	0.965115i	$0.415676\pi$
$252$	0	0
$253$	4.08430	0.256778
$254$	−19.8108	−1.24304
$255$	0	0
$256$	17.8705	1.11691
$257$	9.38027	0.585125	0.292563	−	0.956246i	$-0.405492\pi$
$257$	9.38027	0.585125	0.292563	+	0.956246i	$0.405492\pi$
$258$	0	0
$259$	6.27668	0.390014
$260$	−48.0044	−2.97711
$261$	0	0
$262$	32.4127	2.00246
$263$	19.7330	1.21679	0.608394	−	0.793635i	$-0.291814\pi$
$263$	19.7330	1.21679	0.608394	+	0.793635i	$0.291814\pi$
$264$	0	0
$265$	−42.5559	−2.61419
$266$	10.3333	0.633575
$267$	0	0
$268$	−20.8735	−1.27505
$269$	−23.5286	−1.43457	−0.717283	−	0.696782i	$-0.754615\pi$
$269$	−23.5286	−1.43457	−0.717283	+	0.696782i	$0.754615\pi$
$270$	0	0
$271$	−9.85432	−0.598607	−0.299304	−	0.954158i	$-0.596754\pi$
$271$	−9.85432	−0.598607	−0.299304	+	0.954158i	$0.596754\pi$
$272$	1.60848	0.0975282
$273$	0	0
$274$	−38.7126	−2.33871
$275$	−41.7664	−2.51861
$276$	0	0
$277$	−27.2188	−1.63542	−0.817709	−	0.575632i	$-0.804756\pi$
$277$	−27.2188	−1.63542	−0.817709	+	0.575632i	$0.804756\pi$
$278$	37.5371	2.25133
$279$	0	0
$280$	0.836498	0.0499903
$281$	19.1639	1.14322	0.571611	−	0.820525i	$-0.306319\pi$
$281$	19.1639	1.14322	0.571611	+	0.820525i	$0.306319\pi$
$282$	0	0
$283$	−18.5391	−1.10203	−0.551017	−	0.834494i	$-0.685760\pi$
$283$	−18.5391	−1.10203	−0.551017	+	0.834494i	$0.685760\pi$
$284$	−7.94391	−0.471384
$285$	0	0
$286$	52.8546	3.12536
$287$	−8.26387	−0.487801
$288$	0	0
$289$	−16.8563	−0.991546
$290$	7.67646	0.450777
$291$	0	0
$292$	−16.8177	−0.984179
$293$	−13.4465	−0.785550	−0.392775	−	0.919635i	$-0.628485\pi$
$293$	−13.4465	−0.785550	−0.392775	+	0.919635i	$0.628485\pi$
$294$	0	0
$295$	44.4699	2.58914
$296$	1.90915	0.110967
$297$	0	0
$298$	−1.71328	−0.0992476
$299$	6.57808	0.380420
$300$	0	0
$301$	5.87354	0.338545
$302$	22.7208	1.30744
$303$	0	0
$304$	26.5454	1.52248
$305$	9.26291	0.530393
$306$	0	0
$307$	20.2033	1.15306	0.576532	−	0.817074i	$-0.304406\pi$
$307$	20.2033	1.15306	0.576532	+	0.817074i	$0.304406\pi$
$308$	6.41263	0.365393
$309$	0	0
$310$	24.8223	1.40981
$311$	7.95378	0.451017	0.225509	−	0.974241i	$-0.427596\pi$
$311$	7.95378	0.451017	0.225509	+	0.974241i	$0.427596\pi$
$312$	0	0
$313$	5.26012	0.297319	0.148660	−	0.988888i	$-0.452504\pi$
$313$	5.26012	0.297319	0.148660	+	0.988888i	$0.452504\pi$
$314$	−12.8133	−0.723096
$315$	0	0
$316$	3.34988	0.188446
$317$	−32.0143	−1.79810	−0.899052	−	0.437841i	$-0.855743\pi$
$317$	−32.0143	−1.79810	−0.899052	+	0.437841i	$0.855743\pi$
$318$	0	0
$319$	−4.08430	−0.228677
$320$	−27.0417	−1.51167
$321$	0	0
$322$	1.65157	0.0920384
$323$	2.37197	0.131980
$324$	0	0
$325$	−67.2680	−3.73135
$326$	3.11338	0.172434
$327$	0	0
$328$	−2.51358	−0.138789
$329$	−8.10253	−0.446707
$330$	0	0
$331$	−27.8202	−1.52914	−0.764569	−	0.644542i	$-0.777048\pi$
$331$	−27.8202	−1.52914	−0.764569	+	0.644542i	$0.777048\pi$
$332$	11.4641	0.629172
$333$	0	0
$334$	−11.8861	−0.650380
$335$	−43.5514	−2.37947
$336$	0	0
$337$	−20.1838	−1.09948	−0.549742	−	0.835334i	$-0.685274\pi$
$337$	−20.1838	−1.09948	−0.549742	+	0.835334i	$0.685274\pi$
$338$	59.5518	3.23919
$339$	0	0
$340$	−2.76662	−0.150041
$341$	−13.2068	−0.715190
$342$	0	0
$343$	11.1616	0.602668
$344$	1.78653	0.0963232
$345$	0	0
$346$	4.61627	0.248172
$347$	−1.48365	−0.0796466	−0.0398233	−	0.999207i	$-0.512680\pi$
$347$	−1.48365	−0.0796466	−0.0398233	+	0.999207i	$0.512680\pi$
$348$	0	0
$349$	12.5241	0.670398	0.335199	−	0.942147i	$-0.391197\pi$
$349$	12.5241	0.670398	0.335199	+	0.942147i	$0.391197\pi$
$350$	−16.8891	−0.902760
$351$	0	0
$352$	32.0045	1.70585
$353$	−22.5177	−1.19850	−0.599248	−	0.800564i	$-0.704534\pi$
$353$	−22.5177	−1.19850	−0.599248	+	0.800564i	$0.704534\pi$
$354$	0	0
$355$	−16.5745	−0.879683
$356$	−8.70300	−0.461258
$357$	0	0
$358$	−16.0565	−0.848610
$359$	31.5736	1.66639	0.833196	−	0.552978i	$-0.186508\pi$
$359$	31.5736	1.66639	0.833196	+	0.552978i	$0.186508\pi$
$360$	0	0
$361$	20.1456	1.06030
$362$	−7.06024	−0.371078
$363$	0	0
$364$	10.3280	0.541336
$365$	−35.0890	−1.83664
$366$	0	0
$367$	13.4332	0.701207	0.350604	−	0.936524i	$-0.385976\pi$
$367$	13.4332	0.701207	0.350604	+	0.936524i	$0.385976\pi$
$368$	4.24275	0.221169
$369$	0	0
$370$	−57.3932	−2.98373
$371$	9.15578	0.475345
$372$	0	0
$373$	6.88479	0.356481	0.178240	−	0.983987i	$-0.442960\pi$
$373$	6.88479	0.356481	0.178240	+	0.983987i	$0.442960\pi$
$374$	3.04615	0.157513
$375$	0	0
$376$	−2.46451	−0.127097
$377$	−6.57808	−0.338788
$378$	0	0
$379$	3.53749	0.181708	0.0908542	−	0.995864i	$-0.471040\pi$
$379$	3.53749	0.181708	0.0908542	+	0.995864i	$0.471040\pi$
$380$	−45.6587	−2.34224
$381$	0	0
$382$	−50.9597	−2.60732
$383$	−34.7172	−1.77396	−0.886982	−	0.461804i	$-0.847202\pi$
$383$	−34.7172	−1.77396	−0.886982	+	0.461804i	$0.847202\pi$
$384$	0	0
$385$	13.3796	0.681886
$386$	−25.5279	−1.29934
$387$	0	0
$388$	−4.82348	−0.244875
$389$	34.0537	1.72659	0.863294	−	0.504701i	$-0.168397\pi$
$389$	34.0537	1.72659	0.863294	+	0.504701i	$0.168397\pi$
$390$	0	0
$391$	0.379112	0.0191725
$392$	1.60750	0.0811909
$393$	0	0
$394$	21.0092	1.05843
$395$	6.98933	0.351671
$396$	0	0
$397$	−18.1742	−0.912135	−0.456068	−	0.889945i	$-0.650743\pi$
$397$	−18.1742	−0.912135	−0.456068	+	0.889945i	$0.650743\pi$
$398$	7.43460	0.372663
$399$	0	0
$400$	−43.3867	−2.16934
$401$	−8.34533	−0.416746	−0.208373	−	0.978049i	$-0.566817\pi$
$401$	−8.34533	−0.416746	−0.208373	+	0.978049i	$0.566817\pi$
$402$	0	0
$403$	−21.2706	−1.05957
$404$	−32.8352	−1.63361
$405$	0	0
$406$	−1.65157	−0.0819660
$407$	30.5363	1.51363
$408$	0	0
$409$	24.6842	1.22056	0.610278	−	0.792188i	$-0.291058\pi$
$409$	24.6842	1.22056	0.610278	+	0.792188i	$0.291058\pi$
$410$	75.5638	3.73183
$411$	0	0
$412$	19.7586	0.973439
$413$	−9.56758	−0.470790
$414$	0	0
$415$	23.9191	1.17414
$416$	51.5457	2.52724
$417$	0	0
$418$	50.2720	2.45888
$419$	0.233296	0.0113972	0.00569862	−	0.999984i	$-0.498186\pi$
$419$	0.233296	0.0113972	0.00569862	+	0.999984i	$0.498186\pi$
$420$	0	0
$421$	15.0075	0.731421	0.365710	−	0.930729i	$-0.380826\pi$
$421$	15.0075	0.731421	0.365710	+	0.930729i	$0.380826\pi$
$422$	38.9685	1.89696
$423$	0	0
$424$	2.78487	0.135245
$425$	−3.87683	−0.188054
$426$	0	0
$427$	−1.99289	−0.0964428
$428$	19.2953	0.932674
$429$	0	0
$430$	−53.7070	−2.58998
$431$	−33.8478	−1.63039	−0.815195	−	0.579187i	$-0.803370\pi$
$431$	−33.8478	−1.63039	−0.815195	+	0.579187i	$0.803370\pi$
$432$	0	0
$433$	−25.3136	−1.21649	−0.608247	−	0.793748i	$-0.708127\pi$
$433$	−25.3136	−1.21649	−0.608247	+	0.793748i	$0.708127\pi$
$434$	−5.34046	−0.256350
$435$	0	0
$436$	9.46804	0.453437
$437$	6.25665	0.299296
$438$	0	0
$439$	−1.60242	−0.0764793	−0.0382397	−	0.999269i	$-0.512175\pi$
$439$	−1.60242	−0.0764793	−0.0382397	+	0.999269i	$0.512175\pi$
$440$	4.06960	0.194011
$441$	0	0
$442$	4.90606	0.233357
$443$	−12.6508	−0.601055	−0.300528	−	0.953773i	$-0.597163\pi$
$443$	−12.6508	−0.601055	−0.300528	+	0.953773i	$0.597163\pi$
$444$	0	0
$445$	−18.1583	−0.860785
$446$	43.1192	2.04175
$447$	0	0
$448$	5.81794	0.274872
$449$	4.38659	0.207016	0.103508	−	0.994629i	$-0.466993\pi$
$449$	4.38659	0.207016	0.103508	+	0.994629i	$0.466993\pi$
$450$	0	0
$451$	−40.2041	−1.89314
$452$	−5.80165	−0.272886
$453$	0	0
$454$	24.6935	1.15893
$455$	21.5488	1.01022
$456$	0	0
$457$	−11.9937	−0.561040	−0.280520	−	0.959848i	$-0.590507\pi$
$457$	−11.9937	−0.561040	−0.280520	+	0.959848i	$0.590507\pi$
$458$	9.16789	0.428387
$459$	0	0
$460$	−7.29764	−0.340254
$461$	−6.56938	−0.305967	−0.152983	−	0.988229i	$-0.548888\pi$
$461$	−6.56938	−0.305967	−0.152983	+	0.988229i	$0.548888\pi$
$462$	0	0
$463$	−25.2208	−1.17211	−0.586054	−	0.810272i	$-0.699319\pi$
$463$	−25.2208	−1.17211	−0.586054	+	0.810272i	$0.699319\pi$
$464$	−4.24275	−0.196965
$465$	0	0
$466$	−25.9591	−1.20253
$467$	−18.1238	−0.838671	−0.419336	−	0.907831i	$-0.637737\pi$
$467$	−18.1238	−0.838671	−0.419336	+	0.907831i	$0.637737\pi$
$468$	0	0
$469$	9.36997	0.432665
$470$	74.0885	3.41745
$471$	0	0
$472$	−2.91013	−0.133950
$473$	28.5751	1.31388
$474$	0	0
$475$	−63.9810	−2.93565
$476$	0.595231	0.0272824
$477$	0	0
$478$	8.89475	0.406837
$479$	29.4997	1.34787	0.673937	−	0.738789i	$-0.264602\pi$
$479$	29.4997	1.34787	0.673937	+	0.738789i	$0.264602\pi$
$480$	0	0
$481$	49.1811	2.24247
$482$	0.914190	0.0416402
$483$	0	0
$484$	10.6255	0.482979
$485$	−10.0639	−0.456978
$486$	0	0
$487$	−1.65098	−0.0748129	−0.0374064	−	0.999300i	$-0.511910\pi$
$487$	−1.65098	−0.0748129	−0.0374064	+	0.999300i	$0.511910\pi$
$488$	−0.606169	−0.0274400
$489$	0	0
$490$	−48.3249	−2.18310
$491$	−30.3853	−1.37127	−0.685636	−	0.727945i	$-0.740476\pi$
$491$	−30.3853	−1.37127	−0.685636	+	0.727945i	$0.740476\pi$
$492$	0	0
$493$	−0.379112	−0.0170743
$494$	80.9668	3.64287
$495$	0	0
$496$	−13.7192	−0.616010
$497$	3.56596	0.159955
$498$	0	0
$499$	−29.1755	−1.30607	−0.653037	−	0.757326i	$-0.726505\pi$
$499$	−29.1755	−1.30607	−0.653037	+	0.757326i	$0.726505\pi$
$500$	38.1381	1.70559
$501$	0	0
$502$	16.3209	0.728437
$503$	−33.6484	−1.50031	−0.750155	−	0.661262i	$-0.770021\pi$
$503$	−33.6484	−1.50031	−0.750155	+	0.661262i	$0.770021\pi$
$504$	0	0
$505$	−68.5086	−3.04859
$506$	8.03497	0.357198
$507$	0	0
$508$	−18.8331	−0.835585
$509$	6.99105	0.309873	0.154937	−	0.987924i	$-0.450483\pi$
$509$	6.99105	0.309873	0.154937	+	0.987924i	$0.450483\pi$
$510$	0	0
$511$	7.54931	0.333962
$512$	31.0793	1.37353
$513$	0	0
$514$	18.4536	0.813955
$515$	41.2252	1.81660
$516$	0	0
$517$	−39.4192	−1.73365
$518$	12.3480	0.542540
$519$	0	0
$520$	6.55441	0.287430
$521$	−8.97472	−0.393189	−0.196595	−	0.980485i	$-0.562988\pi$
$521$	−8.97472	−0.393189	−0.196595	+	0.980485i	$0.562988\pi$
$522$	0	0
$523$	−29.4473	−1.28764	−0.643819	−	0.765177i	$-0.722651\pi$
$523$	−29.4473	−1.28764	−0.643819	+	0.765177i	$0.722651\pi$
$524$	30.8132	1.34608
$525$	0	0
$526$	38.8203	1.69265
$527$	−1.22588	−0.0534003
$528$	0	0
$529$	1.00000	0.0434783
$530$	−83.7194	−3.63654
$531$	0	0
$532$	9.82336	0.425897
$533$	−64.7518	−2.80471
$534$	0	0
$535$	40.2585	1.74053
$536$	2.85002	0.123102
$537$	0	0
$538$	−46.2875	−1.99560
$539$	25.7115	1.10747
$540$	0	0
$541$	−1.59606	−0.0686201	−0.0343101	−	0.999411i	$-0.510923\pi$
$541$	−1.59606	−0.0686201	−0.0343101	+	0.999411i	$0.510923\pi$
$542$	−19.3862	−0.832710
$543$	0	0
$544$	2.97071	0.127368
$545$	19.7545	0.846190
$546$	0	0
$547$	13.6372	0.583086	0.291543	−	0.956558i	$-0.405831\pi$
$547$	13.6372	0.583086	0.291543	+	0.956558i	$0.405831\pi$
$548$	−36.8022	−1.57211
$549$	0	0
$550$	−82.1663	−3.50358
$551$	−6.25665	−0.266542
$552$	0	0
$553$	−1.50374	−0.0639454
$554$	−53.5470	−2.27499
$555$	0	0
$556$	35.6847	1.51337
$557$	1.68424	0.0713636	0.0356818	−	0.999363i	$-0.488640\pi$
$557$	1.68424	0.0713636	0.0356818	+	0.999363i	$0.488640\pi$
$558$	0	0
$559$	46.0224	1.94654
$560$	13.8986	0.587324
$561$	0	0
$562$	37.7008	1.59031
$563$	−0.563021	−0.0237285	−0.0118643	−	0.999930i	$-0.503777\pi$
$563$	−0.563021	−0.0237285	−0.0118643	+	0.999930i	$0.503777\pi$
$564$	0	0
$565$	−12.1048	−0.509252
$566$	−36.4716	−1.53302
$567$	0	0
$568$	1.08464	0.0455106
$569$	30.3716	1.27324	0.636622	−	0.771176i	$-0.280331\pi$
$569$	30.3716	1.27324	0.636622	+	0.771176i	$0.280331\pi$
$570$	0	0
$571$	9.24952	0.387080	0.193540	−	0.981092i	$-0.438003\pi$
$571$	9.24952	0.387080	0.193540	+	0.981092i	$0.438003\pi$
$572$	50.2464	2.10091
$573$	0	0
$574$	−16.2574	−0.678569
$575$	−10.2261	−0.426457
$576$	0	0
$577$	35.0432	1.45887	0.729433	−	0.684052i	$-0.239784\pi$
$577$	35.0432	1.45887	0.729433	+	0.684052i	$0.239784\pi$
$578$	−33.1611	−1.37932
$579$	0	0
$580$	7.29764	0.303018
$581$	−5.14613	−0.213497
$582$	0	0
$583$	44.5433	1.84480
$584$	2.29624	0.0950192
$585$	0	0
$586$	−26.4530	−1.09276
$587$	5.85702	0.241745	0.120873	−	0.992668i	$-0.461431\pi$
$587$	5.85702	0.241745	0.120873	+	0.992668i	$0.461431\pi$
$588$	0	0
$589$	−20.2313	−0.833615
$590$	87.4848	3.60169
$591$	0	0
$592$	31.7210	1.30373
$593$	13.3743	0.549217	0.274608	−	0.961556i	$-0.411452\pi$
$593$	13.3743	0.549217	0.274608	+	0.961556i	$0.411452\pi$
$594$	0	0
$595$	1.24191	0.0509135
$596$	−1.62873	−0.0667154
$597$	0	0
$598$	12.9409	0.529194
$599$	−17.1311	−0.699958	−0.349979	−	0.936758i	$-0.613811\pi$
$599$	−17.1311	−0.699958	−0.349979	+	0.936758i	$0.613811\pi$
$600$	0	0
$601$	39.8948	1.62734	0.813671	−	0.581325i	$-0.197466\pi$
$601$	39.8948	1.62734	0.813671	+	0.581325i	$0.197466\pi$
$602$	11.5549	0.470943
$603$	0	0
$604$	21.5996	0.878876
$605$	22.1696	0.901320
$606$	0	0
$607$	29.2826	1.18854	0.594272	−	0.804264i	$-0.297440\pi$
$607$	29.2826	1.18854	0.594272	+	0.804264i	$0.297440\pi$
$608$	49.0270	1.98831
$609$	0	0
$610$	18.2228	0.737818
$611$	−63.4876	−2.56843
$612$	0	0
$613$	23.7708	0.960093	0.480046	−	0.877243i	$-0.340620\pi$
$613$	23.7708	0.960093	0.480046	+	0.877243i	$0.340620\pi$
$614$	39.7456	1.60400
$615$	0	0
$616$	−0.875565	−0.0352775
$617$	12.5181	0.503958	0.251979	−	0.967733i	$-0.418919\pi$
$617$	12.5181	0.503958	0.251979	+	0.967733i	$0.418919\pi$
$618$	0	0
$619$	−27.2765	−1.09633	−0.548167	−	0.836369i	$-0.684674\pi$
$619$	−27.2765	−1.09633	−0.548167	+	0.836369i	$0.684674\pi$
$620$	23.5974	0.947693
$621$	0	0
$622$	15.6473	0.627401
$623$	3.90671	0.156519
$624$	0	0
$625$	28.4424	1.13769
$626$	10.3481	0.413595
$627$	0	0
$628$	−12.1810	−0.486074
$629$	2.83444	0.113016
$630$	0	0
$631$	−17.2183	−0.685448	−0.342724	−	0.939436i	$-0.611350\pi$
$631$	−17.2183	−0.685448	−0.342724	+	0.939436i	$0.611350\pi$
$632$	−0.457385	−0.0181938
$633$	0	0
$634$	−62.9813	−2.50131
$635$	−39.2942	−1.55934
$636$	0	0
$637$	41.4104	1.64074
$638$	−8.03497	−0.318107
$639$	0	0
$640$	7.95443	0.314426
$641$	−31.7850	−1.25543	−0.627715	−	0.778443i	$-0.716010\pi$
$641$	−31.7850	−1.25543	−0.627715	+	0.778443i	$0.716010\pi$
$642$	0	0
$643$	27.2635	1.07517	0.537584	−	0.843210i	$-0.319337\pi$
$643$	27.2635	1.07517	0.537584	+	0.843210i	$0.319337\pi$
$644$	1.57007	0.0618693
$645$	0	0
$646$	4.66633	0.183594
$647$	6.76480	0.265952	0.132976	−	0.991119i	$-0.457547\pi$
$647$	6.76480	0.265952	0.132976	+	0.991119i	$0.457547\pi$
$648$	0	0
$649$	−46.5468	−1.82712
$650$	−132.335	−5.19061
$651$	0	0
$652$	2.95974	0.115912
$653$	9.63997	0.377241	0.188621	−	0.982050i	$-0.439598\pi$
$653$	9.63997	0.377241	0.188621	+	0.982050i	$0.439598\pi$
$654$	0	0
$655$	64.2899	2.51201
$656$	−41.7639	−1.63061
$657$	0	0
$658$	−15.9400	−0.621404
$659$	−37.9893	−1.47985	−0.739927	−	0.672687i	$-0.765140\pi$
$659$	−37.9893	−1.47985	−0.739927	+	0.672687i	$0.765140\pi$
$660$	0	0
$661$	−11.8372	−0.460414	−0.230207	−	0.973142i	$-0.573940\pi$
$661$	−11.8372	−0.460414	−0.230207	+	0.973142i	$0.573940\pi$
$662$	−54.7302	−2.12715
$663$	0	0
$664$	−1.56528	−0.0607445
$665$	20.4959	0.794795
$666$	0	0
$667$	−1.00000	−0.0387202
$668$	−11.2996	−0.437193
$669$	0	0
$670$	−85.6779	−3.31003
$671$	−9.69551	−0.374291
$672$	0	0
$673$	−11.9904	−0.462196	−0.231098	−	0.972931i	$-0.574232\pi$
$673$	−11.9904	−0.462196	−0.231098	+	0.972931i	$0.574232\pi$
$674$	−39.7073	−1.52947
$675$	0	0
$676$	56.6130	2.17742
$677$	36.3152	1.39571	0.697853	−	0.716241i	$-0.254139\pi$
$677$	36.3152	1.39571	0.697853	+	0.716241i	$0.254139\pi$
$678$	0	0
$679$	2.16522	0.0830936
$680$	0.377748	0.0144860
$681$	0	0
$682$	−25.9816	−0.994886
$683$	46.4937	1.77903	0.889516	−	0.456904i	$-0.151042\pi$
$683$	46.4937	1.77903	0.889516	+	0.456904i	$0.151042\pi$
$684$	0	0
$685$	−76.7855	−2.93382
$686$	21.9580	0.838359
$687$	0	0
$688$	29.6837	1.13168
$689$	71.7404	2.73309
$690$	0	0
$691$	19.1780	0.729564	0.364782	−	0.931093i	$-0.381143\pi$
$691$	19.1780	0.729564	0.364782	+	0.931093i	$0.381143\pi$
$692$	4.38846	0.166824
$693$	0	0
$694$	−2.91876	−0.110795
$695$	74.4541	2.82420
$696$	0	0
$697$	−3.73182	−0.141353
$698$	24.6384	0.932576
$699$	0	0
$700$	−16.0556	−0.606846
$701$	−16.6639	−0.629387	−0.314694	−	0.949193i	$-0.601902\pi$
$701$	−16.6639	−0.629387	−0.314694	+	0.949193i	$0.601902\pi$
$702$	0	0
$703$	46.7779	1.76426
$704$	28.3046	1.06677
$705$	0	0
$706$	−44.2987	−1.66720
$707$	14.7394	0.554334
$708$	0	0
$709$	−34.2240	−1.28531	−0.642655	−	0.766155i	$-0.722167\pi$
$709$	−34.2240	−1.28531	−0.642655	+	0.766155i	$0.722167\pi$
$710$	−32.6067	−1.22371
$711$	0	0
$712$	1.18829	0.0445329
$713$	−3.23356	−0.121098
$714$	0	0
$715$	104.836	3.92064
$716$	−15.2641	−0.570446
$717$	0	0
$718$	62.1142	2.31808
$719$	19.1950	0.715851	0.357926	−	0.933750i	$-0.383484\pi$
$719$	19.1950	0.715851	0.357926	+	0.933750i	$0.383484\pi$
$720$	0	0
$721$	−8.86951	−0.330318
$722$	39.6321	1.47495
$723$	0	0
$724$	−6.71183	−0.249443
$725$	10.2261	0.379787
$726$	0	0
$727$	−1.78597	−0.0662380	−0.0331190	−	0.999451i	$-0.510544\pi$
$727$	−1.78597	−0.0662380	−0.0331190	+	0.999451i	$0.510544\pi$
$728$	−1.41016	−0.0522641
$729$	0	0
$730$	−69.0300	−2.55492
$731$	2.65239	0.0981021
$732$	0	0
$733$	−12.2076	−0.450898	−0.225449	−	0.974255i	$-0.572385\pi$
$733$	−12.2076	−0.450898	−0.225449	+	0.974255i	$0.572385\pi$
$734$	26.4269	0.975435
$735$	0	0
$736$	7.83599	0.288838
$737$	45.5854	1.67916
$738$	0	0
$739$	13.4944	0.496399	0.248200	−	0.968709i	$-0.420161\pi$
$739$	13.4944	0.496399	0.248200	+	0.968709i	$0.420161\pi$
$740$	−54.5609	−2.00570
$741$	0	0
$742$	18.0120	0.661242
$743$	−29.3204	−1.07566	−0.537830	−	0.843053i	$-0.680756\pi$
$743$	−29.3204	−1.07566	−0.537830	+	0.843053i	$0.680756\pi$
$744$	0	0
$745$	−3.39825	−0.124502
$746$	13.5443	0.495893
$747$	0	0
$748$	2.89583	0.105882
$749$	−8.66151	−0.316485
$750$	0	0
$751$	−20.9802	−0.765577	−0.382789	−	0.923836i	$-0.625036\pi$
$751$	−20.9802	−0.765577	−0.382789	+	0.923836i	$0.625036\pi$
$752$	−40.9485	−1.49324
$753$	0	0
$754$	−12.9409	−0.471281
$755$	45.0663	1.64013
$756$	0	0
$757$	−47.6763	−1.73283	−0.866413	−	0.499328i	$-0.833580\pi$
$757$	−47.6763	−1.73283	−0.866413	+	0.499328i	$0.833580\pi$
$758$	6.95923	0.252771
$759$	0	0
$760$	6.23413	0.226136
$761$	−35.1401	−1.27383	−0.636913	−	0.770935i	$-0.719789\pi$
$761$	−35.1401	−1.27383	−0.636913	+	0.770935i	$0.719789\pi$
$762$	0	0
$763$	−4.25013	−0.153865
$764$	−48.4449	−1.75268
$765$	0	0
$766$	−68.2985	−2.46772
$767$	−74.9671	−2.70691
$768$	0	0
$769$	7.10129	0.256079	0.128040	−	0.991769i	$-0.459132\pi$
$769$	7.10129	0.256079	0.128040	+	0.991769i	$0.459132\pi$
$770$	26.3214	0.948557
$771$	0	0
$772$	−24.2682	−0.873430
$773$	20.1935	0.726311	0.363155	−	0.931729i	$-0.381699\pi$
$773$	20.1935	0.726311	0.363155	+	0.931729i	$0.381699\pi$
$774$	0	0
$775$	33.0667	1.18779
$776$	0.658586	0.0236419
$777$	0	0
$778$	66.9932	2.40182
$779$	−61.5878	−2.20661
$780$	0	0
$781$	17.3486	0.620780
$782$	0.745820	0.0266705
$783$	0	0
$784$	26.7090	0.953893
$785$	−25.4149	−0.907096
$786$	0	0
$787$	19.3983	0.691476	0.345738	−	0.938331i	$-0.387629\pi$
$787$	19.3983	0.691476	0.345738	+	0.938331i	$0.387629\pi$
$788$	19.9724	0.711487
$789$	0	0
$790$	13.7500	0.489202
$791$	2.60431	0.0925988
$792$	0	0
$793$	−15.6154	−0.554518
$794$	−35.7537	−1.26885
$795$	0	0
$796$	7.06772	0.250509
$797$	15.2083	0.538704	0.269352	−	0.963042i	$-0.413190\pi$
$797$	15.2083	0.538704	0.269352	+	0.963042i	$0.413190\pi$
$798$	0	0
$799$	−3.65896	−0.129445
$800$	−80.1314	−2.83307
$801$	0	0
$802$	−16.4176	−0.579727
$803$	36.7278	1.29610
$804$	0	0
$805$	3.27585	0.115459
$806$	−41.8453	−1.47394
$807$	0	0
$808$	4.48323	0.157720
$809$	25.7908	0.906755	0.453377	−	0.891319i	$-0.350219\pi$
$809$	25.7908	0.906755	0.453377	+	0.891319i	$0.350219\pi$
$810$	0	0
$811$	−28.2524	−0.992076	−0.496038	−	0.868301i	$-0.665212\pi$
$811$	−28.2524	−0.992076	−0.496038	+	0.868301i	$0.665212\pi$
$812$	−1.57007	−0.0550986
$813$	0	0
$814$	60.0736	2.10558
$815$	6.17532	0.216312
$816$	0	0
$817$	43.7735	1.53144
$818$	48.5608	1.69789
$819$	0	0
$820$	71.8349	2.50858
$821$	−48.0439	−1.67674	−0.838372	−	0.545099i	$-0.816492\pi$
$821$	−48.0439	−1.67674	−0.838372	+	0.545099i	$0.816492\pi$
$822$	0	0
$823$	−11.9942	−0.418092	−0.209046	−	0.977906i	$-0.567036\pi$
$823$	−11.9942	−0.418092	−0.209046	+	0.977906i	$0.567036\pi$
$824$	−2.69780	−0.0939823
$825$	0	0
$826$	−18.8221	−0.654906
$827$	−14.4329	−0.501880	−0.250940	−	0.968003i	$-0.580740\pi$
$827$	−14.4329	−0.501880	−0.250940	+	0.968003i	$0.580740\pi$
$828$	0	0
$829$	11.6772	0.405568	0.202784	−	0.979224i	$-0.435001\pi$
$829$	11.6772	0.405568	0.202784	+	0.979224i	$0.435001\pi$
$830$	47.0556	1.63332
$831$	0	0
$832$	45.5867	1.58043
$833$	2.38659	0.0826904
$834$	0	0
$835$	−23.5759	−0.815877
$836$	47.7911	1.65289
$837$	0	0
$838$	0.458958	0.0158545
$839$	−38.6936	−1.33585	−0.667925	−	0.744228i	$-0.732817\pi$
$839$	−38.6936	−1.33585	−0.667925	+	0.744228i	$0.732817\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	29.5240	1.01746
$843$	0	0
$844$	37.0454	1.27516
$845$	118.120	4.06344
$846$	0	0
$847$	−4.76972	−0.163890
$848$	46.2714	1.58897
$849$	0	0
$850$	−7.62682	−0.261598
$851$	7.47652	0.256292
$852$	0	0
$853$	−4.48283	−0.153489	−0.0767446	−	0.997051i	$-0.524453\pi$
$853$	−4.48283	−0.153489	−0.0767446	+	0.997051i	$0.524453\pi$
$854$	−3.92058	−0.134159
$855$	0	0
$856$	−2.63453	−0.0900465
$857$	−4.53684	−0.154976	−0.0774878	−	0.996993i	$-0.524690\pi$
$857$	−4.53684	−0.154976	−0.0774878	+	0.996993i	$0.524690\pi$
$858$	0	0
$859$	36.2457	1.23669	0.618343	−	0.785908i	$-0.287804\pi$
$859$	36.2457	1.23669	0.618343	+	0.785908i	$0.287804\pi$
$860$	−51.0566	−1.74102
$861$	0	0
$862$	−66.5881	−2.26800
$863$	−39.8579	−1.35678	−0.678390	−	0.734702i	$-0.737322\pi$
$863$	−39.8579	−1.35678	−0.678390	+	0.734702i	$0.737322\pi$
$864$	0	0
$865$	9.15626	0.311322
$866$	−49.7990	−1.69224
$867$	0	0
$868$	−5.07691	−0.172322
$869$	−7.31575	−0.248170
$870$	0	0
$871$	73.4187	2.48770
$872$	−1.29274	−0.0437778
$873$	0	0
$874$	12.3086	0.416344
$875$	−17.1199	−0.578758
$876$	0	0
$877$	−49.1279	−1.65893	−0.829466	−	0.558557i	$-0.811355\pi$
$877$	−49.1279	−1.65893	−0.829466	+	0.558557i	$0.811355\pi$
$878$	−3.15241	−0.106389
$879$	0	0
$880$	67.6175	2.27939
$881$	34.2531	1.15402	0.577008	−	0.816738i	$-0.304220\pi$
$881$	34.2531	1.15402	0.577008	+	0.816738i	$0.304220\pi$
$882$	0	0
$883$	−6.47062	−0.217754	−0.108877	−	0.994055i	$-0.534725\pi$
$883$	−6.47062	−0.217754	−0.108877	+	0.994055i	$0.534725\pi$
$884$	4.66395	0.156866
$885$	0	0
$886$	−24.8876	−0.836115
$887$	−5.70034	−0.191399	−0.0956993	−	0.995410i	$-0.530509\pi$
$887$	−5.70034	−0.191399	−0.0956993	+	0.995410i	$0.530509\pi$
$888$	0	0
$889$	8.45405	0.283540
$890$	−35.7225	−1.19742
$891$	0	0
$892$	40.9913	1.37249
$893$	−60.3854	−2.02072
$894$	0	0
$895$	−31.8476	−1.06455
$896$	−1.71137	−0.0571730
$897$	0	0
$898$	8.62967	0.287976
$899$	3.23356	0.107845
$900$	0	0
$901$	4.13459	0.137743
$902$	−79.0929	−2.63350
$903$	0	0
$904$	0.792143	0.0263463
$905$	−14.0038	−0.465503
$906$	0	0
$907$	−10.6632	−0.354066	−0.177033	−	0.984205i	$-0.556650\pi$
$907$	−10.6632	−0.354066	−0.177033	+	0.984205i	$0.556650\pi$
$908$	23.4750	0.779044
$909$	0	0
$910$	42.3926	1.40530
$911$	−3.27255	−0.108424	−0.0542122	−	0.998529i	$-0.517265\pi$
$911$	−3.27255	−0.108424	−0.0542122	+	0.998529i	$0.517265\pi$
$912$	0	0
$913$	−25.0362	−0.828576
$914$	−23.5949	−0.780451
$915$	0	0
$916$	8.71546	0.287967
$917$	−13.8318	−0.456766
$918$	0	0
$919$	43.3072	1.42857	0.714286	−	0.699854i	$-0.246752\pi$
$919$	43.3072	1.42857	0.714286	+	0.699854i	$0.246752\pi$
$920$	0.996402	0.0328504
$921$	0	0
$922$	−12.9238	−0.425624
$923$	27.9412	0.919696
$924$	0	0
$925$	−76.4555	−2.51384
$926$	−49.6164	−1.63050
$927$	0	0
$928$	−7.83599	−0.257229
$929$	−6.72374	−0.220599	−0.110299	−	0.993898i	$-0.535181\pi$
$929$	−6.72374	−0.220599	−0.110299	+	0.993898i	$0.535181\pi$
$930$	0	0
$931$	39.3869	1.29085
$932$	−24.6780	−0.808355
$933$	0	0
$934$	−35.6547	−1.16666
$935$	6.04197	0.197594
$936$	0	0
$937$	−34.4674	−1.12600	−0.563000	−	0.826457i	$-0.690353\pi$
$937$	−34.4674	−1.12600	−0.563000	+	0.826457i	$0.690353\pi$
$938$	18.4334	0.601871
$939$	0	0
$940$	70.4324	2.29725
$941$	20.1814	0.657895	0.328947	−	0.944348i	$-0.393306\pi$
$941$	20.1814	0.657895	0.328947	+	0.944348i	$0.393306\pi$
$942$	0	0
$943$	−9.84358	−0.320551
$944$	−48.3526	−1.57374
$945$	0	0
$946$	56.2153	1.82772
$947$	−54.5419	−1.77237	−0.886186	−	0.463329i	$-0.846655\pi$
$947$	−54.5419	−1.77237	−0.886186	+	0.463329i	$0.846655\pi$
$948$	0	0
$949$	59.1529	1.92018
$950$	−125.869	−4.08372
$951$	0	0
$952$	−0.0812714	−0.00263402
$953$	40.3802	1.30804	0.654021	−	0.756476i	$-0.273081\pi$
$953$	40.3802	1.30804	0.654021	+	0.756476i	$0.273081\pi$
$954$	0	0
$955$	−101.077	−3.27079
$956$	8.45581	0.273481
$957$	0	0
$958$	58.0342	1.87500
$959$	16.5202	0.533465
$960$	0	0
$961$	−20.5441	−0.662712
$962$	96.7531	3.11945
$963$	0	0
$964$	0.869076	0.0279911
$965$	−50.6341	−1.62997
$966$	0	0
$967$	31.3292	1.00748	0.503740	−	0.863855i	$-0.331957\pi$
$967$	31.3292	1.00748	0.503740	+	0.863855i	$0.331957\pi$
$968$	−1.45079	−0.0466300
$969$	0	0
$970$	−19.7985	−0.635692
$971$	−13.1463	−0.421886	−0.210943	−	0.977498i	$-0.567654\pi$
$971$	−13.1463	−0.421886	−0.210943	+	0.977498i	$0.567654\pi$
$972$	0	0
$973$	−16.0186	−0.513533
$974$	−3.24794	−0.104071
$975$	0	0
$976$	−10.0717	−0.322386
$977$	57.3837	1.83587	0.917934	−	0.396732i	$-0.129856\pi$
$977$	57.3837	1.83587	0.917934	+	0.396732i	$0.129856\pi$
$978$	0	0
$979$	19.0063	0.607445
$980$	−45.9401	−1.46750
$981$	0	0
$982$	−59.7765	−1.90755
$983$	−50.1387	−1.59918	−0.799588	−	0.600549i	$-0.794949\pi$
$983$	−50.1387	−1.59918	−0.799588	+	0.600549i	$0.794949\pi$
$984$	0	0
$985$	41.6712	1.32775
$986$	−0.745820	−0.0237517
$987$	0	0
$988$	76.9713	2.44878
$989$	6.99632	0.222470
$990$	0	0
$991$	−3.14635	−0.0999472	−0.0499736	−	0.998751i	$-0.515914\pi$
$991$	−3.14635	−0.0999472	−0.0499736	+	0.998751i	$0.515914\pi$
$992$	−25.3382	−0.804487
$993$	0	0
$994$	7.01525	0.222510
$995$	14.7464	0.467491
$996$	0	0
$997$	24.3387	0.770813	0.385406	−	0.922747i	$-0.374061\pi$
$997$	24.3387	0.770813	0.385406	+	0.922747i	$0.374061\pi$
$998$	−57.3964	−1.81685
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6003.2.a.m.1.10		11
3.2	odd	2		2001.2.a.l.1.2	✓	11

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2001.2.a.l.1.2	✓	11	3.2	odd	2
6003.2.a.m.1.10		11	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)

\(f(q)\)	\(=\)	\(q+1.96728 q^{2} +1.87020 q^{4} +3.90206 q^{5} -0.839519 q^{7} -0.255353 q^{8} +O(q^{10})\)	\(q+1.96728 q^{2} +1.87020 q^{4} +3.90206 q^{5} -0.839519 q^{7} -0.255353 q^{8} +7.67646 q^{10} -4.08430 q^{11} -6.57808 q^{13} -1.65157 q^{14} -4.24275 q^{16} -0.379112 q^{17} -6.25665 q^{19} +7.29764 q^{20} -8.03497 q^{22} -1.00000 q^{23} +10.2261 q^{25} -12.9409 q^{26} -1.57007 q^{28} +1.00000 q^{29} +3.23356 q^{31} -7.83599 q^{32} -0.745820 q^{34} -3.27585 q^{35} -7.47652 q^{37} -12.3086 q^{38} -0.996402 q^{40} +9.84358 q^{41} -6.99632 q^{43} -7.63846 q^{44} -1.96728 q^{46} +9.65140 q^{47} -6.29521 q^{49} +20.1176 q^{50} -12.3023 q^{52} -10.9060 q^{53} -15.9372 q^{55} +0.214373 q^{56} +1.96728 q^{58} +11.3965 q^{59} +2.37385 q^{61} +6.36133 q^{62} -6.93009 q^{64} -25.6681 q^{65} -11.1611 q^{67} -0.709015 q^{68} -6.44453 q^{70} -4.24762 q^{71} -8.99243 q^{73} -14.7084 q^{74} -11.7012 q^{76} +3.42884 q^{77} +1.79119 q^{79} -16.5555 q^{80} +19.3651 q^{82} +6.12986 q^{83} -1.47932 q^{85} -13.7637 q^{86} +1.04294 q^{88} -4.65351 q^{89} +5.52242 q^{91} -1.87020 q^{92} +18.9870 q^{94} -24.4138 q^{95} -2.57912 q^{97} -12.3845 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 11 q - 2 q^{2} + 18 q^{4} - 2 q^{5} + 3 q^{7} - 18 q^{8}+O(q^{10}) \) 11 * q - 2 * q^2 + 18 * q^4 - 2 * q^5 + 3 * q^7 - 18 * q^8	\( 11 q - 2 q^{2} + 18 q^{4} - 2 q^{5} + 3 q^{7} - 18 q^{8} + 14 q^{10} - 11 q^{11} - 5 q^{13} - 17 q^{14} + 20 q^{16} - 15 q^{17} - 6 q^{19} - 21 q^{20} - 10 q^{22} - 11 q^{23} + 3 q^{25} + 5 q^{26} + 7 q^{28} + 11 q^{29} + 35 q^{31} - 28 q^{32} + 28 q^{34} - 15 q^{35} - 28 q^{37} + 2 q^{38} - q^{40} - 10 q^{41} - 6 q^{43} - 18 q^{44} + 2 q^{46} - 15 q^{47} + 22 q^{49} - 15 q^{50} - 36 q^{52} + 7 q^{53} - 12 q^{55} - 56 q^{56} - 2 q^{58} + 20 q^{59} - 20 q^{61} + 11 q^{62} + 36 q^{64} - 11 q^{65} - 39 q^{67} - 35 q^{68} + 38 q^{70} - 49 q^{71} - 3 q^{73} - 37 q^{74} - 18 q^{76} - 25 q^{77} + 41 q^{79} - 51 q^{80} - 19 q^{82} - 13 q^{83} - 62 q^{86} - 40 q^{88} - 34 q^{89} + 2 q^{91} - 18 q^{92} - 14 q^{94} - 25 q^{95} - 11 q^{97} - 53 q^{98}+O(q^{100}) \) 11 * q - 2 * q^2 + 18 * q^4 - 2 * q^5 + 3 * q^7 - 18 * q^8 + 14 * q^10 - 11 * q^11 - 5 * q^13 - 17 * q^14 + 20 * q^16 - 15 * q^17 - 6 * q^19 - 21 * q^20 - 10 * q^22 - 11 * q^23 + 3 * q^25 + 5 * q^26 + 7 * q^28 + 11 * q^29 + 35 * q^31 - 28 * q^32 + 28 * q^34 - 15 * q^35 - 28 * q^37 + 2 * q^38 - q^40 - 10 * q^41 - 6 * q^43 - 18 * q^44 + 2 * q^46 - 15 * q^47 + 22 * q^49 - 15 * q^50 - 36 * q^52 + 7 * q^53 - 12 * q^55 - 56 * q^56 - 2 * q^58 + 20 * q^59 - 20 * q^61 + 11 * q^62 + 36 * q^64 - 11 * q^65 - 39 * q^67 - 35 * q^68 + 38 * q^70 - 49 * q^71 - 3 * q^73 - 37 * q^74 - 18 * q^76 - 25 * q^77 + 41 * q^79 - 51 * q^80 - 19 * q^82 - 13 * q^83 - 62 * q^86 - 40 * q^88 - 34 * q^89 + 2 * q^91 - 18 * q^92 - 14 * q^94 - 25 * q^95 - 11 * q^97 - 53 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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