LMFDB - Embedded newform 6003.2.a.t.1.1

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:	$22$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
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)

$f(q)$	$=$	$q-2.68779 q^{2} +5.22422 q^{4} -0.687677 q^{5} -3.46312 q^{7} -8.66603 q^{8} +O(q^{10})$	\(q-2.68779 q^{2} +5.22422 q^{4} -0.687677 q^{5} -3.46312 q^{7} -8.66603 q^{8} +1.84833 q^{10} +4.59608 q^{11} -5.14171 q^{13} +9.30814 q^{14} +12.8440 q^{16} +5.97345 q^{17} +2.27986 q^{19} -3.59258 q^{20} -12.3533 q^{22} +1.00000 q^{23} -4.52710 q^{25} +13.8198 q^{26} -18.0921 q^{28} +1.00000 q^{29} +1.71616 q^{31} -17.1900 q^{32} -16.0554 q^{34} +2.38151 q^{35} +1.76379 q^{37} -6.12779 q^{38} +5.95943 q^{40} +0.217203 q^{41} -10.2031 q^{43} +24.0109 q^{44} -2.68779 q^{46} -3.23531 q^{47} +4.99320 q^{49} +12.1679 q^{50} -26.8614 q^{52} -9.45848 q^{53} -3.16062 q^{55} +30.0115 q^{56} -2.68779 q^{58} -4.74278 q^{59} +0.257666 q^{61} -4.61267 q^{62} +20.5151 q^{64} +3.53584 q^{65} -12.3078 q^{67} +31.2066 q^{68} -6.40100 q^{70} +4.10541 q^{71} +3.50091 q^{73} -4.74070 q^{74} +11.9105 q^{76} -15.9168 q^{77} +12.8224 q^{79} -8.83255 q^{80} -0.583797 q^{82} +5.50862 q^{83} -4.10780 q^{85} +27.4238 q^{86} -39.8297 q^{88} +15.5057 q^{89} +17.8064 q^{91} +5.22422 q^{92} +8.69583 q^{94} -1.56781 q^{95} +18.9149 q^{97} -13.4207 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 22 q - 3 q^{2} + 17 q^{4} - 6 q^{7} - 6 q^{8}+O(q^{10}) $ 22 * q - 3 * q^2 + 17 * q^4 - 6 * q^7 - 6 * q^8	$ 22 q - 3 q^{2} + 17 q^{4} - 6 q^{7} - 6 q^{8} - 12 q^{10} - 28 q^{13} - q^{14} + 3 q^{16} - 10 q^{17} - 8 q^{19} - 11 q^{22} + 22 q^{23} + 11 q^{26} - 21 q^{28} + 22 q^{29} - 18 q^{31} + 5 q^{32} - 33 q^{34} + 2 q^{35} - 28 q^{37} + 14 q^{38} - 30 q^{40} - 10 q^{41} - 14 q^{43} + 37 q^{44} - 3 q^{46} - 18 q^{47} + 2 q^{49} + 7 q^{50} - 57 q^{52} + 20 q^{53} - 42 q^{55} - 2 q^{56} - 3 q^{58} - 20 q^{59} - 38 q^{61} + 4 q^{62} - 24 q^{64} + 12 q^{65} - 50 q^{67} + 11 q^{68} - 48 q^{70} + 12 q^{71} - 46 q^{73} - 6 q^{74} - 16 q^{76} - 14 q^{77} - 20 q^{79} - 58 q^{80} - 42 q^{82} + 22 q^{83} - 66 q^{85} + 22 q^{86} - 68 q^{88} - 14 q^{89} - 16 q^{91} + 17 q^{92} - 27 q^{94} - 20 q^{95} - 48 q^{97} - 28 q^{98}+O(q^{100}) $ 22 * q - 3 * q^2 + 17 * q^4 - 6 * q^7 - 6 * q^8 - 12 * q^10 - 28 * q^13 - q^14 + 3 * q^16 - 10 * q^17 - 8 * q^19 - 11 * q^22 + 22 * q^23 + 11 * q^26 - 21 * q^28 + 22 * q^29 - 18 * q^31 + 5 * q^32 - 33 * q^34 + 2 * q^35 - 28 * q^37 + 14 * q^38 - 30 * q^40 - 10 * q^41 - 14 * q^43 + 37 * q^44 - 3 * q^46 - 18 * q^47 + 2 * q^49 + 7 * q^50 - 57 * q^52 + 20 * q^53 - 42 * q^55 - 2 * q^56 - 3 * q^58 - 20 * q^59 - 38 * q^61 + 4 * q^62 - 24 * q^64 + 12 * q^65 - 50 * q^67 + 11 * q^68 - 48 * q^70 + 12 * q^71 - 46 * q^73 - 6 * q^74 - 16 * q^76 - 14 * q^77 - 20 * q^79 - 58 * q^80 - 42 * q^82 + 22 * q^83 - 66 * q^85 + 22 * q^86 - 68 * q^88 - 14 * q^89 - 16 * q^91 + 17 * q^92 - 27 * q^94 - 20 * q^95 - 48 * q^97 - 28 * 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 $

$2$	−2.68779	−1.90056	−0.950278	−	0.311404i	$-0.899201\pi$
$2$	−2.68779	−1.90056	−0.950278	+	0.311404i	$0.899201\pi$
$3$	0	0
$3$	0	0
$4$	5.22422	2.61211
$5$	−0.687677	−0.307539	−0.153769	−	0.988107i	$-0.549141\pi$
$5$	−0.687677	−0.307539	−0.153769	+	0.988107i	$0.549141\pi$
$6$	0	0
$7$	−3.46312	−1.30894	−0.654468	−	0.756089i	$-0.727107\pi$
$7$	−3.46312	−1.30894	−0.654468	+	0.756089i	$0.727107\pi$
$8$	−8.66603	−3.06390
$9$	0	0
$10$	1.84833	0.584494
$11$	4.59608	1.38577	0.692884	−	0.721049i	$-0.256340\pi$
$11$	4.59608	1.38577	0.692884	+	0.721049i	$0.256340\pi$
$12$	0	0
$13$	−5.14171	−1.42605	−0.713027	−	0.701137i	$-0.752676\pi$
$13$	−5.14171	−1.42605	−0.713027	+	0.701137i	$0.752676\pi$
$14$	9.30814	2.48771
$15$	0	0
$16$	12.8440	3.21101
$17$	5.97345	1.44877	0.724387	−	0.689394i	$-0.242123\pi$
$17$	5.97345	1.44877	0.724387	+	0.689394i	$0.242123\pi$
$18$	0	0
$19$	2.27986	0.523036	0.261518	−	0.965199i	$-0.415777\pi$
$19$	2.27986	0.523036	0.261518	+	0.965199i	$0.415777\pi$
$20$	−3.59258	−0.803324
$21$	0	0
$22$	−12.3533	−2.63373
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	−4.52710	−0.905420
$26$	13.8198	2.71029
$27$	0	0
$28$	−18.0921	−3.41909
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	0	0
$31$	1.71616	0.308231	0.154115	−	0.988053i	$-0.450747\pi$
$31$	1.71616	0.308231	0.154115	+	0.988053i	$0.450747\pi$
$32$	−17.1900	−3.03879
$33$	0	0
$34$	−16.0554	−2.75347
$35$	2.38151	0.402548
$36$	0	0
$37$	1.76379	0.289965	0.144983	−	0.989434i	$-0.453687\pi$
$37$	1.76379	0.289965	0.144983	+	0.989434i	$0.453687\pi$
$38$	−6.12779	−0.994059
$39$	0	0
$40$	5.95943	0.942268
$41$	0.217203	0.0339215	0.0169607	−	0.999856i	$-0.494601\pi$
$41$	0.217203	0.0339215	0.0169607	+	0.999856i	$0.494601\pi$
$42$	0	0
$43$	−10.2031	−1.55596	−0.777980	−	0.628289i	$-0.783756\pi$
$43$	−10.2031	−1.55596	−0.777980	+	0.628289i	$0.783756\pi$
$44$	24.0109	3.61978
$45$	0	0
$46$	−2.68779	−0.396293
$47$	−3.23531	−0.471918	−0.235959	−	0.971763i	$-0.575823\pi$
$47$	−3.23531	−0.471918	−0.235959	+	0.971763i	$0.575823\pi$
$48$	0	0
$49$	4.99320	0.713315
$50$	12.1679	1.72080
$51$	0	0
$52$	−26.8614	−3.72501
$53$	−9.45848	−1.29922	−0.649611	−	0.760267i	$-0.725068\pi$
$53$	−9.45848	−1.29922	−0.649611	+	0.760267i	$0.725068\pi$
$54$	0	0
$55$	−3.16062	−0.426177
$56$	30.0115	4.01046
$57$	0	0
$58$	−2.68779	−0.352924
$59$	−4.74278	−0.617458	−0.308729	−	0.951150i	$-0.599904\pi$
$59$	−4.74278	−0.617458	−0.308729	+	0.951150i	$0.599904\pi$
$60$	0	0
$61$	0.257666	0.0329907	0.0164953	−	0.999864i	$-0.494749\pi$
$61$	0.257666	0.0329907	0.0164953	+	0.999864i	$0.494749\pi$
$62$	−4.61267	−0.585810
$63$	0	0
$64$	20.5151	2.56439
$65$	3.53584	0.438567
$66$	0	0
$67$	−12.3078	−1.50364	−0.751818	−	0.659371i	$-0.770823\pi$
$67$	−12.3078	−1.50364	−0.751818	+	0.659371i	$0.770823\pi$
$68$	31.2066	3.78436
$69$	0	0
$70$	−6.40100	−0.765065
$71$	4.10541	0.487222	0.243611	−	0.969873i	$-0.421668\pi$
$71$	4.10541	0.487222	0.243611	+	0.969873i	$0.421668\pi$
$72$	0	0
$73$	3.50091	0.409750	0.204875	−	0.978788i	$-0.434321\pi$
$73$	3.50091	0.409750	0.204875	+	0.978788i	$0.434321\pi$
$74$	−4.74070	−0.551095
$75$	0	0
$76$	11.9105	1.36623
$77$	−15.9168	−1.81388
$78$	0	0
$79$	12.8224	1.44263	0.721316	−	0.692606i	$-0.243537\pi$
$79$	12.8224	1.44263	0.721316	+	0.692606i	$0.243537\pi$
$80$	−8.83255	−0.987509
$81$	0	0
$82$	−0.583797	−0.0644696
$83$	5.50862	0.604649	0.302325	−	0.953205i	$-0.402237\pi$
$83$	5.50862	0.604649	0.302325	+	0.953205i	$0.402237\pi$
$84$	0	0
$85$	−4.10780	−0.445554
$86$	27.4238	2.95719
$87$	0	0
$88$	−39.8297	−4.24586
$89$	15.5057	1.64360	0.821799	−	0.569778i	$-0.192971\pi$
$89$	15.5057	1.64360	0.821799	+	0.569778i	$0.192971\pi$
$90$	0	0
$91$	17.8064	1.86661
$92$	5.22422	0.544663
$93$	0	0
$94$	8.69583	0.896906
$95$	−1.56781	−0.160854
$96$	0	0
$97$	18.9149	1.92051	0.960256	−	0.279120i	$-0.0900429\pi$
$97$	18.9149	1.92051	0.960256	+	0.279120i	$0.0900429\pi$
$98$	−13.4207	−1.35569
$99$	0	0
$100$	−23.6506	−2.36506
$101$	10.8479	1.07940	0.539702	−	0.841856i	$-0.318537\pi$
$101$	10.8479	1.07940	0.539702	+	0.841856i	$0.318537\pi$
$102$	0	0
$103$	3.93386	0.387615	0.193807	−	0.981040i	$-0.437916\pi$
$103$	3.93386	0.387615	0.193807	+	0.981040i	$0.437916\pi$
$104$	44.5582	4.36929
$105$	0	0
$106$	25.4224	2.46924
$107$	4.74779	0.458986	0.229493	−	0.973310i	$-0.426293\pi$
$107$	4.74779	0.458986	0.229493	+	0.973310i	$0.426293\pi$
$108$	0	0
$109$	−12.7331	−1.21961	−0.609806	−	0.792551i	$-0.708753\pi$
$109$	−12.7331	−1.21961	−0.609806	+	0.792551i	$0.708753\pi$
$110$	8.49507	0.809974
$111$	0	0
$112$	−44.4804	−4.20301
$113$	−9.80657	−0.922524	−0.461262	−	0.887264i	$-0.652603\pi$
$113$	−9.80657	−0.922524	−0.461262	+	0.887264i	$0.652603\pi$
$114$	0	0
$115$	−0.687677	−0.0641262
$116$	5.22422	0.485057
$117$	0	0
$118$	12.7476	1.17351
$119$	−20.6868	−1.89635
$120$	0	0
$121$	10.1239	0.920355
$122$	−0.692551	−0.0627006
$123$	0	0
$124$	8.96558	0.805133
$125$	6.55157	0.585990
$126$	0	0
$127$	21.0846	1.87095	0.935476	−	0.353391i	$-0.114972\pi$
$127$	21.0846	1.87095	0.935476	+	0.353391i	$0.114972\pi$
$128$	−20.7603	−1.83497
$129$	0	0
$130$	−9.50359	−0.833520
$131$	2.04581	0.178743	0.0893715	−	0.995998i	$-0.471514\pi$
$131$	2.04581	0.178743	0.0893715	+	0.995998i	$0.471514\pi$
$132$	0	0
$133$	−7.89544	−0.684621
$134$	33.0808	2.85774
$135$	0	0
$136$	−51.7661	−4.43890
$137$	16.0616	1.37224	0.686118	−	0.727491i	$-0.259314\pi$
$137$	16.0616	1.37224	0.686118	+	0.727491i	$0.259314\pi$
$138$	0	0
$139$	−9.16271	−0.777171	−0.388585	−	0.921413i	$-0.627036\pi$
$139$	−9.16271	−0.777171	−0.388585	+	0.921413i	$0.627036\pi$
$140$	12.4415	1.05150
$141$	0	0
$142$	−11.0345	−0.925993
$143$	−23.6317	−1.97618
$144$	0	0
$145$	−0.687677	−0.0571085
$146$	−9.40970	−0.778752
$147$	0	0
$148$	9.21443	0.757422
$149$	−19.0418	−1.55997	−0.779984	−	0.625800i	$-0.784773\pi$
$149$	−19.0418	−1.55997	−0.779984	+	0.625800i	$0.784773\pi$
$150$	0	0
$151$	12.7520	1.03774	0.518870	−	0.854853i	$-0.326353\pi$
$151$	12.7520	1.03774	0.518870	+	0.854853i	$0.326353\pi$
$152$	−19.7574	−1.60253
$153$	0	0
$154$	42.7809	3.44739
$155$	−1.18016	−0.0947929
$156$	0	0
$157$	−17.2736	−1.37858	−0.689290	−	0.724485i	$-0.742078\pi$
$157$	−17.2736	−1.37858	−0.689290	+	0.724485i	$0.742078\pi$
$158$	−34.4639	−2.74180
$159$	0	0
$160$	11.8212	0.934546
$161$	−3.46312	−0.272932
$162$	0	0
$163$	−15.6701	−1.22737	−0.613687	−	0.789549i	$-0.710314\pi$
$163$	−15.6701	−1.22737	−0.613687	+	0.789549i	$0.710314\pi$
$164$	1.13472	0.0886066
$165$	0	0
$166$	−14.8060	−1.14917
$167$	−7.21984	−0.558688	−0.279344	−	0.960191i	$-0.590117\pi$
$167$	−7.21984	−0.558688	−0.279344	+	0.960191i	$0.590117\pi$
$168$	0	0
$169$	13.4372	1.03363
$170$	11.0409	0.846799
$171$	0	0
$172$	−53.3033	−4.06434
$173$	−21.9238	−1.66683	−0.833417	−	0.552644i	$-0.813619\pi$
$173$	−21.9238	−1.66683	−0.833417	+	0.552644i	$0.813619\pi$
$174$	0	0
$175$	15.6779	1.18514
$176$	59.0321	4.44972
$177$	0	0
$178$	−41.6760	−3.12375
$179$	−15.0851	−1.12751	−0.563757	−	0.825941i	$-0.690644\pi$
$179$	−15.0851	−1.12751	−0.563757	+	0.825941i	$0.690644\pi$
$180$	0	0
$181$	−12.6056	−0.936969	−0.468484	−	0.883472i	$-0.655200\pi$
$181$	−12.6056	−0.936969	−0.468484	+	0.883472i	$0.655200\pi$
$182$	−47.8598	−3.54760
$183$	0	0
$184$	−8.66603	−0.638868
$185$	−1.21292	−0.0891755
$186$	0	0
$187$	27.4544	2.00767
$188$	−16.9020	−1.23270
$189$	0	0
$190$	4.21394	0.305712
$191$	−5.43046	−0.392934	−0.196467	−	0.980510i	$-0.562947\pi$
$191$	−5.43046	−0.392934	−0.196467	+	0.980510i	$0.562947\pi$
$192$	0	0
$193$	20.8624	1.50171	0.750856	−	0.660466i	$-0.229641\pi$
$193$	20.8624	1.50171	0.750856	+	0.660466i	$0.229641\pi$
$194$	−50.8392	−3.65004
$195$	0	0
$196$	26.0856	1.86326
$197$	8.43033	0.600636	0.300318	−	0.953839i	$-0.402907\pi$
$197$	8.43033	0.600636	0.300318	+	0.953839i	$0.402907\pi$
$198$	0	0
$199$	−18.0862	−1.28209	−0.641047	−	0.767502i	$-0.721500\pi$
$199$	−18.0862	−1.28209	−0.641047	+	0.767502i	$0.721500\pi$
$200$	39.2320	2.77412
$201$	0	0
$202$	−29.1568	−2.05147
$203$	−3.46312	−0.243063
$204$	0	0
$205$	−0.149366	−0.0104322
$206$	−10.5734	−0.736683
$207$	0	0
$208$	−66.0403	−4.57907
$209$	10.4784	0.724808
$210$	0	0
$211$	−5.36796	−0.369545	−0.184773	−	0.982781i	$-0.559155\pi$
$211$	−5.36796	−0.369545	−0.184773	+	0.982781i	$0.559155\pi$
$212$	−49.4132	−3.39371
$213$	0	0
$214$	−12.7611	−0.872328
$215$	7.01645	0.478518
$216$	0	0
$217$	−5.94326	−0.403455
$218$	34.2240	2.31794
$219$	0	0
$220$	−16.5118	−1.11322
$221$	−30.7137	−2.06603
$222$	0	0
$223$	13.0007	0.870588	0.435294	−	0.900288i	$-0.356644\pi$
$223$	13.0007	0.870588	0.435294	+	0.900288i	$0.356644\pi$
$224$	59.5311	3.97759
$225$	0	0
$226$	26.3580	1.75331
$227$	20.0165	1.32854	0.664271	−	0.747492i	$-0.268742\pi$
$227$	20.0165	1.32854	0.664271	+	0.747492i	$0.268742\pi$
$228$	0	0
$229$	−20.8501	−1.37781	−0.688905	−	0.724851i	$-0.741908\pi$
$229$	−20.8501	−1.37781	−0.688905	+	0.724851i	$0.741908\pi$
$230$	1.84833	0.121875
$231$	0	0
$232$	−8.66603	−0.568953
$233$	26.7760	1.75416	0.877078	−	0.480348i	$-0.159490\pi$
$233$	26.7760	1.75416	0.877078	+	0.480348i	$0.159490\pi$
$234$	0	0
$235$	2.22485	0.145133
$236$	−24.7773	−1.61287
$237$	0	0
$238$	55.6017	3.60412
$239$	−2.28068	−0.147525	−0.0737624	−	0.997276i	$-0.523501\pi$
$239$	−2.28068	−0.147525	−0.0737624	+	0.997276i	$0.523501\pi$
$240$	0	0
$241$	13.3485	0.859853	0.429926	−	0.902864i	$-0.358539\pi$
$241$	13.3485	0.859853	0.429926	+	0.902864i	$0.358539\pi$
$242$	−27.2110	−1.74919
$243$	0	0
$244$	1.34610	0.0861753
$245$	−3.43371	−0.219372
$246$	0	0
$247$	−11.7224	−0.745878
$248$	−14.8723	−0.944390
$249$	0	0
$250$	−17.6092	−1.11371
$251$	6.58893	0.415889	0.207945	−	0.978141i	$-0.433323\pi$
$251$	6.58893	0.415889	0.207945	+	0.978141i	$0.433323\pi$
$252$	0	0
$253$	4.59608	0.288953
$254$	−56.6709	−3.55585
$255$	0	0
$256$	14.7691	0.923067
$257$	6.96581	0.434515	0.217257	−	0.976114i	$-0.430289\pi$
$257$	6.96581	0.434515	0.217257	+	0.976114i	$0.430289\pi$
$258$	0	0
$259$	−6.10822	−0.379546
$260$	18.4720	1.14558
$261$	0	0
$262$	−5.49870	−0.339711
$263$	26.9584	1.66233	0.831164	−	0.556027i	$-0.187675\pi$
$263$	26.9584	1.66233	0.831164	+	0.556027i	$0.187675\pi$
$264$	0	0
$265$	6.50438	0.399561
$266$	21.2213	1.30116
$267$	0	0
$268$	−64.2986	−3.92766
$269$	15.4844	0.944098	0.472049	−	0.881572i	$-0.343515\pi$
$269$	15.4844	0.944098	0.472049	+	0.881572i	$0.343515\pi$
$270$	0	0
$271$	−18.6202	−1.13110	−0.565548	−	0.824716i	$-0.691335\pi$
$271$	−18.6202	−1.13110	−0.565548	+	0.824716i	$0.691335\pi$
$272$	76.7231	4.65202
$273$	0	0
$274$	−43.1702	−2.60801
$275$	−20.8069	−1.25470
$276$	0	0
$277$	14.4460	0.867976	0.433988	−	0.900919i	$-0.357106\pi$
$277$	14.4460	0.867976	0.433988	+	0.900919i	$0.357106\pi$
$278$	24.6274	1.47706
$279$	0	0
$280$	−20.6382	−1.23337
$281$	−21.8021	−1.30061	−0.650303	−	0.759675i	$-0.725358\pi$
$281$	−21.8021	−1.30061	−0.650303	+	0.759675i	$0.725358\pi$
$282$	0	0
$283$	7.24549	0.430700	0.215350	−	0.976537i	$-0.430911\pi$
$283$	7.24549	0.430700	0.215350	+	0.976537i	$0.430911\pi$
$284$	21.4476	1.27268
$285$	0	0
$286$	63.5170	3.75584
$287$	−0.752202	−0.0444011
$288$	0	0
$289$	18.6821	1.09894
$290$	1.84833	0.108538
$291$	0	0
$292$	18.2895	1.07031
$293$	−6.89562	−0.402847	−0.201423	−	0.979504i	$-0.564557\pi$
$293$	−6.89562	−0.402847	−0.201423	+	0.979504i	$0.564557\pi$
$294$	0	0
$295$	3.26150	0.189892
$296$	−15.2851	−0.888426
$297$	0	0
$298$	51.1805	2.96480
$299$	−5.14171	−0.297353
$300$	0	0
$301$	35.3346	2.03665
$302$	−34.2746	−1.97228
$303$	0	0
$304$	29.2826	1.67947
$305$	−0.177191	−0.0101459
$306$	0	0
$307$	3.63268	0.207328	0.103664	−	0.994612i	$-0.466943\pi$
$307$	3.63268	0.207328	0.103664	+	0.994612i	$0.466943\pi$
$308$	−83.1527	−4.73806
$309$	0	0
$310$	3.17203	0.180159
$311$	−28.6581	−1.62505	−0.812527	−	0.582923i	$-0.801909\pi$
$311$	−28.6581	−1.62505	−0.812527	+	0.582923i	$0.801909\pi$
$312$	0	0
$313$	−22.6141	−1.27823	−0.639114	−	0.769112i	$-0.720699\pi$
$313$	−22.6141	−1.27823	−0.639114	+	0.769112i	$0.720699\pi$
$314$	46.4277	2.62007
$315$	0	0
$316$	66.9870	3.76831
$317$	−10.9269	−0.613714	−0.306857	−	0.951756i	$-0.599277\pi$
$317$	−10.9269	−0.613714	−0.306857	+	0.951756i	$0.599277\pi$
$318$	0	0
$319$	4.59608	0.257331
$320$	−14.1078	−0.788648
$321$	0	0
$322$	9.30814	0.518723
$323$	13.6186	0.757761
$324$	0	0
$325$	23.2770	1.29118
$326$	42.1179	2.33269
$327$	0	0
$328$	−1.88229	−0.103932
$329$	11.2043	0.617711
$330$	0	0
$331$	−17.2545	−0.948394	−0.474197	−	0.880419i	$-0.657262\pi$
$331$	−17.2545	−0.948394	−0.474197	+	0.880419i	$0.657262\pi$
$332$	28.7782	1.57941
$333$	0	0
$334$	19.4054	1.06182
$335$	8.46379	0.462426
$336$	0	0
$337$	19.5276	1.06374	0.531868	−	0.846827i	$-0.321490\pi$
$337$	19.5276	1.06374	0.531868	+	0.846827i	$0.321490\pi$
$338$	−36.1164	−1.96447
$339$	0	0
$340$	−21.4601	−1.16384
$341$	7.88759	0.427137
$342$	0	0
$343$	6.94978	0.375253
$344$	88.4205	4.76731
$345$	0	0
$346$	58.9265	3.16791
$347$	−32.7192	−1.75646	−0.878229	−	0.478240i	$-0.841275\pi$
$347$	−32.7192	−1.75646	−0.878229	+	0.478240i	$0.841275\pi$
$348$	0	0
$349$	−1.17233	−0.0627536	−0.0313768	−	0.999508i	$-0.509989\pi$
$349$	−1.17233	−0.0627536	−0.0313768	+	0.999508i	$0.509989\pi$
$350$	−42.1389	−2.25242
$351$	0	0
$352$	−79.0066	−4.21107
$353$	−7.39875	−0.393796	−0.196898	−	0.980424i	$-0.563087\pi$
$353$	−7.39875	−0.393796	−0.196898	+	0.980424i	$0.563087\pi$
$354$	0	0
$355$	−2.82319	−0.149840
$356$	81.0050	4.29326
$357$	0	0
$358$	40.5456	2.14290
$359$	−13.9250	−0.734932	−0.367466	−	0.930037i	$-0.619775\pi$
$359$	−13.9250	−0.734932	−0.367466	+	0.930037i	$0.619775\pi$
$360$	0	0
$361$	−13.8022	−0.726433
$362$	33.8813	1.78076
$363$	0	0
$364$	93.0244	4.87580
$365$	−2.40749	−0.126014
$366$	0	0
$367$	−23.7137	−1.23784	−0.618922	−	0.785452i	$-0.712430\pi$
$367$	−23.7137	−1.23784	−0.618922	+	0.785452i	$0.712430\pi$
$368$	12.8440	0.669541
$369$	0	0
$370$	3.26007	0.169483
$371$	32.7559	1.70060
$372$	0	0
$373$	−4.03710	−0.209033	−0.104517	−	0.994523i	$-0.533330\pi$
$373$	−4.03710	−0.209033	−0.104517	+	0.994523i	$0.533330\pi$
$374$	−73.7917	−3.81568
$375$	0	0
$376$	28.0373	1.44591
$377$	−5.14171	−0.264812
$378$	0	0
$379$	−24.0642	−1.23610	−0.618048	−	0.786141i	$-0.712076\pi$
$379$	−24.0642	−1.23610	−0.618048	+	0.786141i	$0.712076\pi$
$380$	−8.19058	−0.420168
$381$	0	0
$382$	14.5959	0.746794
$383$	1.24183	0.0634545	0.0317273	−	0.999497i	$-0.489899\pi$
$383$	1.24183	0.0634545	0.0317273	+	0.999497i	$0.489899\pi$
$384$	0	0
$385$	10.9456	0.557839
$386$	−56.0739	−2.85409
$387$	0	0
$388$	98.8153	5.01659
$389$	−25.7280	−1.30446	−0.652231	−	0.758020i	$-0.726167\pi$
$389$	−25.7280	−1.30446	−0.652231	+	0.758020i	$0.726167\pi$
$390$	0	0
$391$	5.97345	0.302090
$392$	−43.2712	−2.18553
$393$	0	0
$394$	−22.6590	−1.14154
$395$	−8.81767	−0.443665
$396$	0	0
$397$	−21.6301	−1.08559	−0.542793	−	0.839867i	$-0.682633\pi$
$397$	−21.6301	−1.08559	−0.542793	+	0.839867i	$0.682633\pi$
$398$	48.6118	2.43669
$399$	0	0
$400$	−58.1462	−2.90731
$401$	−5.76638	−0.287959	−0.143980	−	0.989581i	$-0.545990\pi$
$401$	−5.76638	−0.287959	−0.143980	+	0.989581i	$0.545990\pi$
$402$	0	0
$403$	−8.82399	−0.439554
$404$	56.6717	2.81952
$405$	0	0
$406$	9.30814	0.461955
$407$	8.10652	0.401825
$408$	0	0
$409$	−17.2558	−0.853245	−0.426623	−	0.904430i	$-0.640297\pi$
$409$	−17.2558	−0.853245	−0.426623	+	0.904430i	$0.640297\pi$
$410$	0.401464	0.0198269
$411$	0	0
$412$	20.5513	1.01249
$413$	16.4248	0.808213
$414$	0	0
$415$	−3.78815	−0.185953
$416$	88.3861	4.33349
$417$	0	0
$418$	−28.1638	−1.37754
$419$	−0.0725445	−0.00354403	−0.00177202	−	0.999998i	$-0.500564\pi$
$419$	−0.0725445	−0.00354403	−0.00177202	+	0.999998i	$0.500564\pi$
$420$	0	0
$421$	−1.44894	−0.0706170	−0.0353085	−	0.999376i	$-0.511241\pi$
$421$	−1.44894	−0.0706170	−0.0353085	+	0.999376i	$0.511241\pi$
$422$	14.4279	0.702342
$423$	0	0
$424$	81.9674	3.98069
$425$	−27.0424	−1.31175
$426$	0	0
$427$	−0.892327	−0.0431827
$428$	24.8035	1.19892
$429$	0	0
$430$	−18.8587	−0.909449
$431$	−13.1784	−0.634782	−0.317391	−	0.948295i	$-0.602807\pi$
$431$	−13.1784	−0.634782	−0.317391	+	0.948295i	$0.602807\pi$
$432$	0	0
$433$	−38.6226	−1.85608	−0.928042	−	0.372476i	$-0.878509\pi$
$433$	−38.6226	−1.85608	−0.928042	+	0.372476i	$0.878509\pi$
$434$	15.9742	0.766788
$435$	0	0
$436$	−66.5206	−3.18576
$437$	2.27986	0.109061
$438$	0	0
$439$	−14.4449	−0.689417	−0.344708	−	0.938710i	$-0.612022\pi$
$439$	−14.4449	−0.689417	−0.344708	+	0.938710i	$0.612022\pi$
$440$	27.3900	1.30577
$441$	0	0
$442$	82.5521	3.92660
$443$	7.89171	0.374946	0.187473	−	0.982270i	$-0.439970\pi$
$443$	7.89171	0.374946	0.187473	+	0.982270i	$0.439970\pi$
$444$	0	0
$445$	−10.6629	−0.505469
$446$	−34.9430	−1.65460
$447$	0	0
$448$	−71.0463	−3.35662
$449$	−0.0225254	−0.00106304	−0.000531521	−	1.00000i	$-0.500169\pi$
$449$	−0.0225254	−0.00106304	−0.000531521		1.00000i	$0.500169\pi$
$450$	0	0
$451$	0.998283	0.0470073
$452$	−51.2317	−2.40973
$453$	0	0
$454$	−53.8002	−2.52497
$455$	−12.2450	−0.574056
$456$	0	0
$457$	−33.7655	−1.57948	−0.789742	−	0.613439i	$-0.789786\pi$
$457$	−33.7655	−1.57948	−0.789742	+	0.613439i	$0.789786\pi$
$458$	56.0406	2.61860
$459$	0	0
$460$	−3.59258	−0.167505
$461$	16.2703	0.757783	0.378892	−	0.925441i	$-0.376305\pi$
$461$	16.2703	0.757783	0.378892	+	0.925441i	$0.376305\pi$
$462$	0	0
$463$	−16.5955	−0.771260	−0.385630	−	0.922653i	$-0.626016\pi$
$463$	−16.5955	−0.771260	−0.385630	+	0.922653i	$0.626016\pi$
$464$	12.8440	0.596269
$465$	0	0
$466$	−71.9683	−3.33387
$467$	24.7206	1.14393	0.571967	−	0.820277i	$-0.306180\pi$
$467$	24.7206	1.14393	0.571967	+	0.820277i	$0.306180\pi$
$468$	0	0
$469$	42.6234	1.96816
$470$	−5.97992	−0.275833
$471$	0	0
$472$	41.1011	1.89183
$473$	−46.8943	−2.15620
$474$	0	0
$475$	−10.3212	−0.473568
$476$	−108.072	−4.95348
$477$	0	0
$478$	6.12998	0.280379
$479$	17.0459	0.778849	0.389424	−	0.921058i	$-0.372674\pi$
$479$	17.0459	0.778849	0.389424	+	0.921058i	$0.372674\pi$
$480$	0	0
$481$	−9.06890	−0.413506
$482$	−35.8780	−1.63420
$483$	0	0
$484$	52.8895	2.40407
$485$	−13.0073	−0.590631
$486$	0	0
$487$	9.44988	0.428215	0.214108	−	0.976810i	$-0.431316\pi$
$487$	9.44988	0.428215	0.214108	+	0.976810i	$0.431316\pi$
$488$	−2.23294	−0.101080
$489$	0	0
$490$	9.22910	0.416928
$491$	−9.10839	−0.411056	−0.205528	−	0.978651i	$-0.565891\pi$
$491$	−9.10839	−0.411056	−0.205528	+	0.978651i	$0.565891\pi$
$492$	0	0
$493$	5.97345	0.269030
$494$	31.5073	1.41758
$495$	0	0
$496$	22.0424	0.989732
$497$	−14.2175	−0.637743
$498$	0	0
$499$	11.3259	0.507016	0.253508	−	0.967333i	$-0.418416\pi$
$499$	11.3259	0.507016	0.253508	+	0.967333i	$0.418416\pi$
$500$	34.2268	1.53067
$501$	0	0
$502$	−17.7097	−0.790421
$503$	−12.6017	−0.561880	−0.280940	−	0.959725i	$-0.590646\pi$
$503$	−12.6017	−0.561880	−0.280940	+	0.959725i	$0.590646\pi$
$504$	0	0
$505$	−7.45983	−0.331958
$506$	−12.3533	−0.549171
$507$	0	0
$508$	110.150	4.88713
$509$	33.1311	1.46851	0.734254	−	0.678875i	$-0.237532\pi$
$509$	33.1311	1.46851	0.734254	+	0.678875i	$0.237532\pi$
$510$	0	0
$511$	−12.1241	−0.536337
$512$	1.82437	0.0806266
$513$	0	0
$514$	−18.7226	−0.825820
$515$	−2.70522	−0.119206
$516$	0	0
$517$	−14.8697	−0.653969
$518$	16.4176	0.721349
$519$	0	0
$520$	−30.6417	−1.34373
$521$	23.0329	1.00909	0.504546	−	0.863385i	$-0.331660\pi$
$521$	23.0329	1.00909	0.504546	+	0.863385i	$0.331660\pi$
$522$	0	0
$523$	−0.588330	−0.0257259	−0.0128629	−	0.999917i	$-0.504095\pi$
$523$	−0.588330	−0.0257259	−0.0128629	+	0.999917i	$0.504095\pi$
$524$	10.6877	0.466896
$525$	0	0
$526$	−72.4586	−3.15935
$527$	10.2514	0.446557
$528$	0	0
$529$	1.00000	0.0434783
$530$	−17.4824	−0.759387
$531$	0	0
$532$	−41.2475	−1.78831
$533$	−1.11680	−0.0483739
$534$	0	0
$535$	−3.26494	−0.141156
$536$	106.660	4.60700
$537$	0	0
$538$	−41.6187	−1.79431
$539$	22.9491	0.988490
$540$	0	0
$541$	8.17964	0.351670	0.175835	−	0.984420i	$-0.443737\pi$
$541$	8.17964	0.351670	0.175835	+	0.984420i	$0.443737\pi$
$542$	50.0472	2.14971
$543$	0	0
$544$	−102.684	−4.40252
$545$	8.75627	0.375078
$546$	0	0
$547$	−25.2950	−1.08153	−0.540767	−	0.841172i	$-0.681866\pi$
$547$	−25.2950	−1.08153	−0.540767	+	0.841172i	$0.681866\pi$
$548$	83.9093	3.58443
$549$	0	0
$550$	55.9246	2.38463
$551$	2.27986	0.0971254
$552$	0	0
$553$	−44.4055	−1.88831
$554$	−38.8278	−1.64964
$555$	0	0
$556$	−47.8680	−2.03006
$557$	−19.3382	−0.819385	−0.409693	−	0.912224i	$-0.634364\pi$
$557$	−19.3382	−0.819385	−0.409693	+	0.912224i	$0.634364\pi$
$558$	0	0
$559$	52.4615	2.21888
$560$	30.5882	1.29259
$561$	0	0
$562$	58.5996	2.47187
$563$	−33.8682	−1.42737	−0.713687	−	0.700465i	$-0.752976\pi$
$563$	−33.8682	−1.42737	−0.713687	+	0.700465i	$0.752976\pi$
$564$	0	0
$565$	6.74375	0.283712
$566$	−19.4744	−0.818569
$567$	0	0
$568$	−35.5776	−1.49280
$569$	41.3111	1.73185	0.865926	−	0.500171i	$-0.166730\pi$
$569$	41.3111	1.73185	0.865926	+	0.500171i	$0.166730\pi$
$570$	0	0
$571$	6.63809	0.277795	0.138898	−	0.990307i	$-0.455644\pi$
$571$	6.63809	0.277795	0.138898	+	0.990307i	$0.455644\pi$
$572$	−123.457	−5.16200
$573$	0	0
$574$	2.02176	0.0843867
$575$	−4.52710	−0.188793
$576$	0	0
$577$	−20.9868	−0.873694	−0.436847	−	0.899536i	$-0.643905\pi$
$577$	−20.9868	−0.873694	−0.436847	+	0.899536i	$0.643905\pi$
$578$	−50.2135	−2.08861
$579$	0	0
$580$	−3.59258	−0.149174
$581$	−19.0770	−0.791448
$582$	0	0
$583$	−43.4719	−1.80042
$584$	−30.3389	−1.25543
$585$	0	0
$586$	18.5340	0.765632
$587$	−25.7439	−1.06257	−0.531283	−	0.847195i	$-0.678290\pi$
$587$	−25.7439	−1.06257	−0.531283	+	0.847195i	$0.678290\pi$
$588$	0	0
$589$	3.91260	0.161216
$590$	−8.76624	−0.360900
$591$	0	0
$592$	22.6542	0.931081
$593$	30.7966	1.26466	0.632332	−	0.774697i	$-0.282098\pi$
$593$	30.7966	1.26466	0.632332	+	0.774697i	$0.282098\pi$
$594$	0	0
$595$	14.2258	0.583201
$596$	−99.4787	−4.07481
$597$	0	0
$598$	13.8198	0.565135
$599$	−6.10728	−0.249537	−0.124768	−	0.992186i	$-0.539819\pi$
$599$	−6.10728	−0.249537	−0.124768	+	0.992186i	$0.539819\pi$
$600$	0	0
$601$	−11.2448	−0.458685	−0.229343	−	0.973346i	$-0.573658\pi$
$601$	−11.2448	−0.458685	−0.229343	+	0.973346i	$0.573658\pi$
$602$	−94.9720	−3.87077
$603$	0	0
$604$	66.6190	2.71069
$605$	−6.96198	−0.283045
$606$	0	0
$607$	−7.27827	−0.295416	−0.147708	−	0.989031i	$-0.547190\pi$
$607$	−7.27827	−0.295416	−0.147708	+	0.989031i	$0.547190\pi$
$608$	−39.1909	−1.58940
$609$	0	0
$610$	0.476251	0.0192829
$611$	16.6350	0.672981
$612$	0	0
$613$	−15.4067	−0.622269	−0.311134	−	0.950366i	$-0.600709\pi$
$613$	−15.4067	−0.622269	−0.311134	+	0.950366i	$0.600709\pi$
$614$	−9.76389	−0.394039
$615$	0	0
$616$	137.935	5.55756
$617$	18.9132	0.761417	0.380708	−	0.924695i	$-0.375680\pi$
$617$	18.9132	0.761417	0.380708	+	0.924695i	$0.375680\pi$
$618$	0	0
$619$	−7.27806	−0.292530	−0.146265	−	0.989245i	$-0.546725\pi$
$619$	−7.27806	−0.292530	−0.146265	+	0.989245i	$0.546725\pi$
$620$	−6.16543	−0.247609
$621$	0	0
$622$	77.0271	3.08851
$623$	−53.6980	−2.15136
$624$	0	0
$625$	18.1301	0.725206
$626$	60.7821	2.42934
$627$	0	0
$628$	−90.2409	−3.60100
$629$	10.5359	0.420094
$630$	0	0
$631$	−5.52403	−0.219908	−0.109954	−	0.993937i	$-0.535070\pi$
$631$	−5.52403	−0.219908	−0.109954	+	0.993937i	$0.535070\pi$
$632$	−111.119	−4.42009
$633$	0	0
$634$	29.3691	1.16640
$635$	−14.4994	−0.575390
$636$	0	0
$637$	−25.6736	−1.01723
$638$	−12.3533	−0.489071
$639$	0	0
$640$	14.2764	0.564323
$641$	−43.0502	−1.70038	−0.850189	−	0.526477i	$-0.823513\pi$
$641$	−43.0502	−1.70038	−0.850189	+	0.526477i	$0.823513\pi$
$642$	0	0
$643$	−8.00495	−0.315685	−0.157842	−	0.987464i	$-0.550454\pi$
$643$	−8.00495	−0.315685	−0.157842	+	0.987464i	$0.550454\pi$
$644$	−18.0921	−0.712929
$645$	0	0
$646$	−36.6040	−1.44017
$647$	−34.1726	−1.34346	−0.671732	−	0.740794i	$-0.734449\pi$
$647$	−34.1726	−1.34346	−0.671732	+	0.740794i	$0.734449\pi$
$648$	0	0
$649$	−21.7982	−0.855654
$650$	−62.5638	−2.45395
$651$	0	0
$652$	−81.8639	−3.20604
$653$	15.0216	0.587839	0.293920	−	0.955830i	$-0.405040\pi$
$653$	15.0216	0.587839	0.293920	+	0.955830i	$0.405040\pi$
$654$	0	0
$655$	−1.40685	−0.0549703
$656$	2.78977	0.108922
$657$	0	0
$658$	−30.1147	−1.17399
$659$	−43.8463	−1.70801	−0.854005	−	0.520265i	$-0.825833\pi$
$659$	−43.8463	−1.70801	−0.854005	+	0.520265i	$0.825833\pi$
$660$	0	0
$661$	11.7036	0.455218	0.227609	−	0.973753i	$-0.426909\pi$
$661$	11.7036	0.455218	0.227609	+	0.973753i	$0.426909\pi$
$662$	46.3765	1.80247
$663$	0	0
$664$	−47.7378	−1.85259
$665$	5.42951	0.210547
$666$	0	0
$667$	1.00000	0.0387202
$668$	−37.7180	−1.45935
$669$	0	0
$670$	−22.7489	−0.878866
$671$	1.18425	0.0457175
$672$	0	0
$673$	−27.5495	−1.06196	−0.530978	−	0.847386i	$-0.678175\pi$
$673$	−27.5495	−1.06196	−0.530978	+	0.847386i	$0.678175\pi$
$674$	−52.4861	−2.02169
$675$	0	0
$676$	70.1989	2.69996
$677$	−20.1222	−0.773358	−0.386679	−	0.922214i	$-0.626378\pi$
$677$	−20.1222	−0.773358	−0.386679	+	0.922214i	$0.626378\pi$
$678$	0	0
$679$	−65.5044	−2.51383
$680$	35.5983	1.36513
$681$	0	0
$682$	−21.2002	−0.811797
$683$	13.2484	0.506936	0.253468	−	0.967344i	$-0.418429\pi$
$683$	13.2484	0.506936	0.253468	+	0.967344i	$0.418429\pi$
$684$	0	0
$685$	−11.0452	−0.422015
$686$	−18.6795	−0.713188
$687$	0	0
$688$	−131.049	−4.99620
$689$	48.6328	1.85276
$690$	0	0
$691$	40.4059	1.53711	0.768557	−	0.639782i	$-0.220975\pi$
$691$	40.4059	1.53711	0.768557	+	0.639782i	$0.220975\pi$
$692$	−114.535	−4.35395
$693$	0	0
$694$	87.9424	3.33825
$695$	6.30098	0.239010
$696$	0	0
$697$	1.29745	0.0491445
$698$	3.15099	0.119267
$699$	0	0
$700$	81.9048	3.09571
$701$	−21.7925	−0.823093	−0.411546	−	0.911389i	$-0.635011\pi$
$701$	−21.7925	−0.823093	−0.411546	+	0.911389i	$0.635011\pi$
$702$	0	0
$703$	4.02120	0.151662
$704$	94.2890	3.55365
$705$	0	0
$706$	19.8863	0.748431
$707$	−37.5675	−1.41287
$708$	0	0
$709$	−39.1893	−1.47179	−0.735893	−	0.677097i	$-0.763238\pi$
$709$	−39.1893	−1.47179	−0.735893	+	0.677097i	$0.763238\pi$
$710$	7.58816	0.284778
$711$	0	0
$712$	−134.373	−5.03582
$713$	1.71616	0.0642706
$714$	0	0
$715$	16.2510	0.607752
$716$	−78.8079	−2.94519
$717$	0	0
$718$	37.4274	1.39678
$719$	−11.5706	−0.431509	−0.215754	−	0.976448i	$-0.569221\pi$
$719$	−11.5706	−0.431509	−0.215754	+	0.976448i	$0.569221\pi$
$720$	0	0
$721$	−13.6234	−0.507363
$722$	37.0975	1.38063
$723$	0	0
$724$	−65.8546	−2.44747
$725$	−4.52710	−0.168132
$726$	0	0
$727$	−17.8413	−0.661699	−0.330849	−	0.943684i	$-0.607335\pi$
$727$	−17.8413	−0.661699	−0.330849	+	0.943684i	$0.607335\pi$
$728$	−154.310	−5.71913
$729$	0	0
$730$	6.47084	0.239496
$731$	−60.9477	−2.25423
$732$	0	0
$733$	−28.5426	−1.05425	−0.527123	−	0.849789i	$-0.676729\pi$
$733$	−28.5426	−1.05425	−0.527123	+	0.849789i	$0.676729\pi$
$734$	63.7374	2.35259
$735$	0	0
$736$	−17.1900	−0.633632
$737$	−56.5675	−2.08369
$738$	0	0
$739$	51.0985	1.87969	0.939845	−	0.341602i	$-0.110970\pi$
$739$	51.0985	1.87969	0.939845	+	0.341602i	$0.110970\pi$
$740$	−6.33655	−0.232936
$741$	0	0
$742$	−88.0409	−3.23208
$743$	2.05369	0.0753425	0.0376713	−	0.999290i	$-0.488006\pi$
$743$	2.05369	0.0753425	0.0376713	+	0.999290i	$0.488006\pi$
$744$	0	0
$745$	13.0946	0.479750
$746$	10.8509	0.397279
$747$	0	0
$748$	143.428	5.24424
$749$	−16.4422	−0.600783
$750$	0	0
$751$	27.4583	1.00197	0.500984	−	0.865456i	$-0.332971\pi$
$751$	27.4583	1.00197	0.500984	+	0.865456i	$0.332971\pi$
$752$	−41.5544	−1.51533
$753$	0	0
$754$	13.8198	0.503289
$755$	−8.76922	−0.319145
$756$	0	0
$757$	12.5724	0.456951	0.228475	−	0.973550i	$-0.426626\pi$
$757$	12.5724	0.456951	0.228475	+	0.973550i	$0.426626\pi$
$758$	64.6796	2.34927
$759$	0	0
$760$	13.5867	0.492841
$761$	−16.8228	−0.609825	−0.304912	−	0.952380i	$-0.598627\pi$
$761$	−16.8228	−0.609825	−0.304912	+	0.952380i	$0.598627\pi$
$762$	0	0
$763$	44.0963	1.59639
$764$	−28.3699	−1.02639
$765$	0	0
$766$	−3.33778	−0.120599
$767$	24.3860	0.880528
$768$	0	0
$769$	39.6902	1.43127	0.715633	−	0.698477i	$-0.246139\pi$
$769$	39.6902	1.43127	0.715633	+	0.698477i	$0.246139\pi$
$770$	−29.4195	−1.06020
$771$	0	0
$772$	108.990	3.92264
$773$	25.9789	0.934398	0.467199	−	0.884152i	$-0.345263\pi$
$773$	25.9789	0.934398	0.467199	+	0.884152i	$0.345263\pi$
$774$	0	0
$775$	−7.76922	−0.279079
$776$	−163.917	−5.88426
$777$	0	0
$778$	69.1516	2.47920
$779$	0.495194	0.0177422
$780$	0	0
$781$	18.8688	0.675177
$782$	−16.0554	−0.574139
$783$	0	0
$784$	64.1329	2.29046
$785$	11.8786	0.423967
$786$	0	0
$787$	−35.2024	−1.25483	−0.627415	−	0.778685i	$-0.715887\pi$
$787$	−35.2024	−1.25483	−0.627415	+	0.778685i	$0.715887\pi$
$788$	44.0419	1.56893
$789$	0	0
$790$	23.7001	0.843210
$791$	33.9613	1.20753
$792$	0	0
$793$	−1.32484	−0.0470465
$794$	58.1373	2.06322
$795$	0	0
$796$	−94.4861	−3.34897
$797$	33.9033	1.20092	0.600458	−	0.799656i	$-0.294985\pi$
$797$	33.9033	1.20092	0.600458	+	0.799656i	$0.294985\pi$
$798$	0	0
$799$	−19.3259	−0.683702
$800$	77.8209	2.75139
$801$	0	0
$802$	15.4988	0.547283
$803$	16.0904	0.567819
$804$	0	0
$805$	2.38151	0.0839371
$806$	23.7170	0.835397
$807$	0	0
$808$	−94.0080	−3.30719
$809$	−35.0829	−1.23345	−0.616725	−	0.787179i	$-0.711541\pi$
$809$	−35.0829	−1.23345	−0.616725	+	0.787179i	$0.711541\pi$
$810$	0	0
$811$	48.8056	1.71379	0.856897	−	0.515487i	$-0.172389\pi$
$811$	48.8056	1.71379	0.856897	+	0.515487i	$0.172389\pi$
$812$	−18.0921	−0.634908
$813$	0	0
$814$	−21.7886	−0.763691
$815$	10.7759	0.377465
$816$	0	0
$817$	−23.2617	−0.813824
$818$	46.3800	1.62164
$819$	0	0
$820$	−0.780320	−0.0272499
$821$	−48.5225	−1.69345	−0.846724	−	0.532033i	$-0.821428\pi$
$821$	−48.5225	−1.69345	−0.846724	+	0.532033i	$0.821428\pi$
$822$	0	0
$823$	−2.03604	−0.0709717	−0.0354859	−	0.999370i	$-0.511298\pi$
$823$	−2.03604	−0.0709717	−0.0354859	+	0.999370i	$0.511298\pi$
$824$	−34.0909	−1.18761
$825$	0	0
$826$	−44.1465	−1.53605
$827$	−12.7928	−0.444849	−0.222425	−	0.974950i	$-0.571397\pi$
$827$	−12.7928	−0.444849	−0.222425	+	0.974950i	$0.571397\pi$
$828$	0	0
$829$	−47.1565	−1.63781	−0.818907	−	0.573926i	$-0.805420\pi$
$829$	−47.1565	−1.63781	−0.818907	+	0.573926i	$0.805420\pi$
$830$	10.1818	0.353414
$831$	0	0
$832$	−105.483	−3.65696
$833$	29.8266	1.03343
$834$	0	0
$835$	4.96492	0.171818
$836$	54.7416	1.89328
$837$	0	0
$838$	0.194984	0.00673563
$839$	0.628748	0.0217068	0.0108534	−	0.999941i	$-0.496545\pi$
$839$	0.628748	0.0217068	0.0108534	+	0.999941i	$0.496545\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	3.89445	0.134212
$843$	0	0
$844$	−28.0434	−0.965293
$845$	−9.24045	−0.317881
$846$	0	0
$847$	−35.0603	−1.20469
$848$	−121.485	−4.17181
$849$	0	0
$850$	72.6843	2.49305
$851$	1.76379	0.0604620
$852$	0	0
$853$	−26.8985	−0.920986	−0.460493	−	0.887663i	$-0.652327\pi$
$853$	−26.8985	−0.920986	−0.460493	+	0.887663i	$0.652327\pi$
$854$	2.39839	0.0820711
$855$	0	0
$856$	−41.1444	−1.40629
$857$	22.5856	0.771510	0.385755	−	0.922601i	$-0.373941\pi$
$857$	22.5856	0.771510	0.385755	+	0.922601i	$0.373941\pi$
$858$	0	0
$859$	−22.6647	−0.773308	−0.386654	−	0.922225i	$-0.626369\pi$
$859$	−22.6647	−0.773308	−0.386654	+	0.922225i	$0.626369\pi$
$860$	36.6555	1.24994
$861$	0	0
$862$	35.4208	1.20644
$863$	16.0538	0.546478	0.273239	−	0.961946i	$-0.411905\pi$
$863$	16.0538	0.546478	0.273239	+	0.961946i	$0.411905\pi$
$864$	0	0
$865$	15.0765	0.512616
$866$	103.809	3.52759
$867$	0	0
$868$	−31.0489	−1.05387
$869$	58.9327	1.99916
$870$	0	0
$871$	63.2831	2.14427
$872$	110.346	3.73677
$873$	0	0
$874$	−6.12779	−0.207276
$875$	−22.6889	−0.767024
$876$	0	0
$877$	−23.5671	−0.795805	−0.397903	−	0.917428i	$-0.630262\pi$
$877$	−23.5671	−0.795805	−0.397903	+	0.917428i	$0.630262\pi$
$878$	38.8248	1.31028
$879$	0	0
$880$	−40.5950	−1.36846
$881$	−3.08556	−0.103955	−0.0519775	−	0.998648i	$-0.516552\pi$
$881$	−3.08556	−0.103955	−0.0519775	+	0.998648i	$0.516552\pi$
$882$	0	0
$883$	−7.53587	−0.253602	−0.126801	−	0.991928i	$-0.540471\pi$
$883$	−7.53587	−0.253602	−0.126801	+	0.991928i	$0.540471\pi$
$884$	−160.455	−5.39670
$885$	0	0
$886$	−21.2113	−0.712606
$887$	8.48441	0.284879	0.142439	−	0.989804i	$-0.454505\pi$
$887$	8.48441	0.284879	0.142439	+	0.989804i	$0.454505\pi$
$888$	0	0
$889$	−73.0183	−2.44896
$890$	28.6596	0.960672
$891$	0	0
$892$	67.9183	2.27407
$893$	−7.37606	−0.246830
$894$	0	0
$895$	10.3737	0.346754
$896$	71.8953	2.40185
$897$	0	0
$898$	0.0605437	0.00202037
$899$	1.71616	0.0572371
$900$	0	0
$901$	−56.4997	−1.88228
$902$	−2.68318	−0.0893400
$903$	0	0
$904$	84.9840	2.82652
$905$	8.66860	0.288154
$906$	0	0
$907$	46.4640	1.54281	0.771405	−	0.636344i	$-0.219554\pi$
$907$	46.4640	1.54281	0.771405	+	0.636344i	$0.219554\pi$
$908$	104.571	3.47030
$909$	0	0
$910$	32.9121	1.09102
$911$	−23.9669	−0.794060	−0.397030	−	0.917806i	$-0.629959\pi$
$911$	−23.9669	−0.794060	−0.397030	+	0.917806i	$0.629959\pi$
$912$	0	0
$913$	25.3180	0.837904
$914$	90.7547	3.00190
$915$	0	0
$916$	−108.925	−3.59899
$917$	−7.08488	−0.233963
$918$	0	0
$919$	28.2588	0.932172	0.466086	−	0.884739i	$-0.345664\pi$
$919$	28.2588	0.932172	0.466086	+	0.884739i	$0.345664\pi$
$920$	5.95943	0.196477
$921$	0	0
$922$	−43.7311	−1.44021
$923$	−21.1088	−0.694805
$924$	0	0
$925$	−7.98486	−0.262541
$926$	44.6053	1.46582
$927$	0	0
$928$	−17.1900	−0.564290
$929$	11.4877	0.376901	0.188450	−	0.982083i	$-0.439654\pi$
$929$	11.4877	0.376901	0.188450	+	0.982083i	$0.439654\pi$
$930$	0	0
$931$	11.3838	0.373090
$932$	139.884	4.58205
$933$	0	0
$934$	−66.4439	−2.17411
$935$	−18.8798	−0.617434
$936$	0	0
$937$	−21.1049	−0.689467	−0.344734	−	0.938701i	$-0.612031\pi$
$937$	−21.1049	−0.689467	−0.344734	+	0.938701i	$0.612031\pi$
$938$	−114.563	−3.74060
$939$	0	0
$940$	11.6231	0.379103
$941$	−41.1147	−1.34030	−0.670151	−	0.742225i	$-0.733771\pi$
$941$	−41.1147	−1.34030	−0.670151	+	0.742225i	$0.733771\pi$
$942$	0	0
$943$	0.217203	0.00707312
$944$	−60.9165	−1.98266
$945$	0	0
$946$	126.042	4.09798
$947$	44.3513	1.44122	0.720611	−	0.693339i	$-0.243861\pi$
$947$	44.3513	1.44122	0.720611	+	0.693339i	$0.243861\pi$
$948$	0	0
$949$	−18.0006	−0.584326
$950$	27.7411	0.900041
$951$	0	0
$952$	179.272	5.81024
$953$	41.1340	1.33246	0.666231	−	0.745746i	$-0.267907\pi$
$953$	41.1340	1.33246	0.666231	+	0.745746i	$0.267907\pi$
$954$	0	0
$955$	3.73440	0.120842
$956$	−11.9148	−0.385351
$957$	0	0
$958$	−45.8159	−1.48024
$959$	−55.6233	−1.79617
$960$	0	0
$961$	−28.0548	−0.904994
$962$	24.3753	0.785892
$963$	0	0
$964$	69.7355	2.24603
$965$	−14.3466	−0.461834
$966$	0	0
$967$	9.08667	0.292207	0.146104	−	0.989269i	$-0.453327\pi$
$967$	9.08667	0.292207	0.146104	+	0.989269i	$0.453327\pi$
$968$	−87.7341	−2.81988
$969$	0	0
$970$	34.9609	1.12253
$971$	−48.2330	−1.54787	−0.773935	−	0.633265i	$-0.781714\pi$
$971$	−48.2330	−1.54787	−0.773935	+	0.633265i	$0.781714\pi$
$972$	0	0
$973$	31.7316	1.01727
$974$	−25.3993	−0.813846
$975$	0	0
$976$	3.30946	0.105933
$977$	−18.0712	−0.578148	−0.289074	−	0.957307i	$-0.593347\pi$
$977$	−18.0712	−0.578148	−0.289074	+	0.957307i	$0.593347\pi$
$978$	0	0
$979$	71.2652	2.27765
$980$	−17.9385	−0.573023
$981$	0	0
$982$	24.4814	0.781234
$983$	3.85948	0.123098	0.0615491	−	0.998104i	$-0.480396\pi$
$983$	3.85948	0.123098	0.0615491	+	0.998104i	$0.480396\pi$
$984$	0	0
$985$	−5.79734	−0.184719
$986$	−16.0554	−0.511307
$987$	0	0
$988$	−61.2404	−1.94832
$989$	−10.2031	−0.324440
$990$	0	0
$991$	−37.8631	−1.20276	−0.601381	−	0.798963i	$-0.705382\pi$
$991$	−37.8631	−1.20276	−0.601381	+	0.798963i	$0.705382\pi$
$992$	−29.5008	−0.936651
$993$	0	0
$994$	38.2137	1.21207
$995$	12.4374	0.394293
$996$	0	0
$997$	−55.9892	−1.77320	−0.886598	−	0.462541i	$-0.846938\pi$
$997$	−55.9892	−1.77320	−0.886598	+	0.462541i	$0.846938\pi$
$998$	−30.4416	−0.963612
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6003.2.a.t.1.1	✓	22
3.2	odd	2		6003.2.a.u.1.22	yes	22

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6003.2.a.t.1.1	✓	22	1.1	even	1	trivial
6003.2.a.u.1.22	yes	22	3.2	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:	\( N \)	\(=\)	\( 6003 = 3^{2} \cdot 23 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6003.a (trivial)

\(f(q)\)	\(=\)	\(q-2.68779 q^{2} +5.22422 q^{4} -0.687677 q^{5} -3.46312 q^{7} -8.66603 q^{8} +O(q^{10})\)	\(q-2.68779 q^{2} +5.22422 q^{4} -0.687677 q^{5} -3.46312 q^{7} -8.66603 q^{8} +1.84833 q^{10} +4.59608 q^{11} -5.14171 q^{13} +9.30814 q^{14} +12.8440 q^{16} +5.97345 q^{17} +2.27986 q^{19} -3.59258 q^{20} -12.3533 q^{22} +1.00000 q^{23} -4.52710 q^{25} +13.8198 q^{26} -18.0921 q^{28} +1.00000 q^{29} +1.71616 q^{31} -17.1900 q^{32} -16.0554 q^{34} +2.38151 q^{35} +1.76379 q^{37} -6.12779 q^{38} +5.95943 q^{40} +0.217203 q^{41} -10.2031 q^{43} +24.0109 q^{44} -2.68779 q^{46} -3.23531 q^{47} +4.99320 q^{49} +12.1679 q^{50} -26.8614 q^{52} -9.45848 q^{53} -3.16062 q^{55} +30.0115 q^{56} -2.68779 q^{58} -4.74278 q^{59} +0.257666 q^{61} -4.61267 q^{62} +20.5151 q^{64} +3.53584 q^{65} -12.3078 q^{67} +31.2066 q^{68} -6.40100 q^{70} +4.10541 q^{71} +3.50091 q^{73} -4.74070 q^{74} +11.9105 q^{76} -15.9168 q^{77} +12.8224 q^{79} -8.83255 q^{80} -0.583797 q^{82} +5.50862 q^{83} -4.10780 q^{85} +27.4238 q^{86} -39.8297 q^{88} +15.5057 q^{89} +17.8064 q^{91} +5.22422 q^{92} +8.69583 q^{94} -1.56781 q^{95} +18.9149 q^{97} -13.4207 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 22 q - 3 q^{2} + 17 q^{4} - 6 q^{7} - 6 q^{8}+O(q^{10}) \) 22 * q - 3 * q^2 + 17 * q^4 - 6 * q^7 - 6 * q^8	\( 22 q - 3 q^{2} + 17 q^{4} - 6 q^{7} - 6 q^{8} - 12 q^{10} - 28 q^{13} - q^{14} + 3 q^{16} - 10 q^{17} - 8 q^{19} - 11 q^{22} + 22 q^{23} + 11 q^{26} - 21 q^{28} + 22 q^{29} - 18 q^{31} + 5 q^{32} - 33 q^{34} + 2 q^{35} - 28 q^{37} + 14 q^{38} - 30 q^{40} - 10 q^{41} - 14 q^{43} + 37 q^{44} - 3 q^{46} - 18 q^{47} + 2 q^{49} + 7 q^{50} - 57 q^{52} + 20 q^{53} - 42 q^{55} - 2 q^{56} - 3 q^{58} - 20 q^{59} - 38 q^{61} + 4 q^{62} - 24 q^{64} + 12 q^{65} - 50 q^{67} + 11 q^{68} - 48 q^{70} + 12 q^{71} - 46 q^{73} - 6 q^{74} - 16 q^{76} - 14 q^{77} - 20 q^{79} - 58 q^{80} - 42 q^{82} + 22 q^{83} - 66 q^{85} + 22 q^{86} - 68 q^{88} - 14 q^{89} - 16 q^{91} + 17 q^{92} - 27 q^{94} - 20 q^{95} - 48 q^{97} - 28 q^{98}+O(q^{100}) \) 22 * q - 3 * q^2 + 17 * q^4 - 6 * q^7 - 6 * q^8 - 12 * q^10 - 28 * q^13 - q^14 + 3 * q^16 - 10 * q^17 - 8 * q^19 - 11 * q^22 + 22 * q^23 + 11 * q^26 - 21 * q^28 + 22 * q^29 - 18 * q^31 + 5 * q^32 - 33 * q^34 + 2 * q^35 - 28 * q^37 + 14 * q^38 - 30 * q^40 - 10 * q^41 - 14 * q^43 + 37 * q^44 - 3 * q^46 - 18 * q^47 + 2 * q^49 + 7 * q^50 - 57 * q^52 + 20 * q^53 - 42 * q^55 - 2 * q^56 - 3 * q^58 - 20 * q^59 - 38 * q^61 + 4 * q^62 - 24 * q^64 + 12 * q^65 - 50 * q^67 + 11 * q^68 - 48 * q^70 + 12 * q^71 - 46 * q^73 - 6 * q^74 - 16 * q^76 - 14 * q^77 - 20 * q^79 - 58 * q^80 - 42 * q^82 + 22 * q^83 - 66 * q^85 + 22 * q^86 - 68 * q^88 - 14 * q^89 - 16 * q^91 + 17 * q^92 - 27 * q^94 - 20 * q^95 - 48 * q^97 - 28 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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