LMFDB - Embedded newform 6003.2.a.w.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:	$0$
Dimension:	$30$
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)

$f(q)$	$=$	$q-2.64113 q^{2} +4.97557 q^{4} -1.42245 q^{5} +2.00224 q^{7} -7.85888 q^{8} +O(q^{10})$	\(q-2.64113 q^{2} +4.97557 q^{4} -1.42245 q^{5} +2.00224 q^{7} -7.85888 q^{8} +3.75688 q^{10} -1.91463 q^{11} +5.67607 q^{13} -5.28817 q^{14} +10.8052 q^{16} +5.11233 q^{17} -0.0723365 q^{19} -7.07750 q^{20} +5.05678 q^{22} -1.00000 q^{23} -2.97664 q^{25} -14.9913 q^{26} +9.96229 q^{28} +1.00000 q^{29} +10.4736 q^{31} -12.8202 q^{32} -13.5023 q^{34} -2.84808 q^{35} +0.0359099 q^{37} +0.191050 q^{38} +11.1789 q^{40} +11.1841 q^{41} -11.2287 q^{43} -9.52637 q^{44} +2.64113 q^{46} +8.46632 q^{47} -2.99104 q^{49} +7.86169 q^{50} +28.2417 q^{52} +9.11389 q^{53} +2.72346 q^{55} -15.7354 q^{56} -2.64113 q^{58} +6.17774 q^{59} +5.91056 q^{61} -27.6621 q^{62} +12.2494 q^{64} -8.07393 q^{65} -1.78553 q^{67} +25.4368 q^{68} +7.52216 q^{70} -11.9074 q^{71} -5.99221 q^{73} -0.0948427 q^{74} -0.359916 q^{76} -3.83354 q^{77} +7.00625 q^{79} -15.3698 q^{80} -29.5386 q^{82} -7.42232 q^{83} -7.27203 q^{85} +29.6565 q^{86} +15.0468 q^{88} +12.4212 q^{89} +11.3649 q^{91} -4.97557 q^{92} -22.3607 q^{94} +0.102895 q^{95} -1.32074 q^{97} +7.89973 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 30 q + q^{2} + 37 q^{4} + 10 q^{7} + 6 q^{8}+O(q^{10}) $ 30 * q + q^2 + 37 * q^4 + 10 * q^7 + 6 * q^8	$ 30 q + q^{2} + 37 q^{4} + 10 q^{7} + 6 q^{8} + 8 q^{10} + 36 q^{13} + 7 q^{14} + 47 q^{16} + 18 q^{17} + 16 q^{19} + 25 q^{22} - 30 q^{23} + 56 q^{25} + 11 q^{26} + 27 q^{28} + 30 q^{29} + 14 q^{31} - 7 q^{32} + 3 q^{34} - 22 q^{35} + 40 q^{37} + 6 q^{38} + 30 q^{40} + 14 q^{41} + 34 q^{43} + 5 q^{44} - q^{46} - 2 q^{47} + 74 q^{49} - 21 q^{50} + 71 q^{52} + 16 q^{53} + 22 q^{55} + 14 q^{56} + q^{58} - 32 q^{59} + 46 q^{61} + 20 q^{62} + 68 q^{64} + 12 q^{65} + 14 q^{67} + 27 q^{68} - 32 q^{71} + 50 q^{73} - 26 q^{74} + 56 q^{76} + 34 q^{77} + 16 q^{79} + 2 q^{80} + 38 q^{82} - 14 q^{83} + 38 q^{85} + 10 q^{86} + 40 q^{88} - 2 q^{89} + 32 q^{91} - 37 q^{92} + 29 q^{94} - 28 q^{95} + 56 q^{97} + 8 q^{98}+O(q^{100}) $ 30 * q + q^2 + 37 * q^4 + 10 * q^7 + 6 * q^8 + 8 * q^10 + 36 * q^13 + 7 * q^14 + 47 * q^16 + 18 * q^17 + 16 * q^19 + 25 * q^22 - 30 * q^23 + 56 * q^25 + 11 * q^26 + 27 * q^28 + 30 * q^29 + 14 * q^31 - 7 * q^32 + 3 * q^34 - 22 * q^35 + 40 * q^37 + 6 * q^38 + 30 * q^40 + 14 * q^41 + 34 * q^43 + 5 * q^44 - q^46 - 2 * q^47 + 74 * q^49 - 21 * q^50 + 71 * q^52 + 16 * q^53 + 22 * q^55 + 14 * q^56 + q^58 - 32 * q^59 + 46 * q^61 + 20 * q^62 + 68 * q^64 + 12 * q^65 + 14 * q^67 + 27 * q^68 - 32 * q^71 + 50 * q^73 - 26 * q^74 + 56 * q^76 + 34 * q^77 + 16 * q^79 + 2 * q^80 + 38 * q^82 - 14 * q^83 + 38 * q^85 + 10 * q^86 + 40 * q^88 - 2 * q^89 + 32 * q^91 - 37 * q^92 + 29 * q^94 - 28 * q^95 + 56 * q^97 + 8 * 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 $

$2$	−2.64113	−1.86756	−0.933781	−	0.357845i	$-0.883512\pi$
$2$	−2.64113	−1.86756	−0.933781	+	0.357845i	$0.883512\pi$
$3$	0	0
$3$	0	0
$4$	4.97557	2.48779
$5$	−1.42245	−0.636139	−0.318069	−	0.948067i	$-0.603034\pi$
$5$	−1.42245	−0.636139	−0.318069	+	0.948067i	$0.603034\pi$
$6$	0	0
$7$	2.00224	0.756775	0.378388	−	0.925647i	$-0.376479\pi$
$7$	2.00224	0.756775	0.378388	+	0.925647i	$0.376479\pi$
$8$	−7.85888	−2.77853
$9$	0	0
$10$	3.75688	1.18803
$11$	−1.91463	−0.577282	−0.288641	−	0.957437i	$-0.593203\pi$
$11$	−1.91463	−0.577282	−0.288641	+	0.957437i	$0.593203\pi$
$12$	0	0
$13$	5.67607	1.57426	0.787130	−	0.616787i	$-0.211566\pi$
$13$	5.67607	1.57426	0.787130	+	0.616787i	$0.211566\pi$
$14$	−5.28817	−1.41332
$15$	0	0
$16$	10.8052	2.70130
$17$	5.11233	1.23992	0.619961	−	0.784633i	$-0.287149\pi$
$17$	5.11233	1.23992	0.619961	+	0.784633i	$0.287149\pi$
$18$	0	0
$19$	−0.0723365	−0.0165951	−0.00829757	−	0.999966i	$-0.502641\pi$
$19$	−0.0723365	−0.0165951	−0.00829757	+	0.999966i	$0.502641\pi$
$20$	−7.07750	−1.58258
$21$	0	0
$22$	5.05678	1.07811
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	−2.97664	−0.595328
$26$	−14.9913	−2.94003
$27$	0	0
$28$	9.96229	1.88270
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	0	0
$31$	10.4736	1.88111	0.940554	−	0.339643i	$-0.110306\pi$
$31$	10.4736	1.88111	0.940554	+	0.339643i	$0.110306\pi$
$32$	−12.8202	−2.26631
$33$	0	0
$34$	−13.5023	−2.31563
$35$	−2.84808	−0.481414
$36$	0	0
$37$	0.0359099	0.00590355	0.00295177	−	0.999996i	$-0.499060\pi$
$37$	0.0359099	0.00590355	0.00295177	+	0.999996i	$0.499060\pi$
$38$	0.191050	0.0309924
$39$	0	0
$40$	11.1789	1.76753
$41$	11.1841	1.74666	0.873328	−	0.487132i	$-0.161957\pi$
$41$	11.1841	1.74666	0.873328	+	0.487132i	$0.161957\pi$
$42$	0	0
$43$	−11.2287	−1.71236	−0.856181	−	0.516675i	$-0.827169\pi$
$43$	−11.2287	−1.71236	−0.856181	+	0.516675i	$0.827169\pi$
$44$	−9.52637	−1.43615
$45$	0	0
$46$	2.64113	0.389414
$47$	8.46632	1.23494	0.617470	−	0.786595i	$-0.288158\pi$
$47$	8.46632	1.23494	0.617470	+	0.786595i	$0.288158\pi$
$48$	0	0
$49$	−2.99104	−0.427292
$50$	7.86169	1.11181
$51$	0	0
$52$	28.2417	3.91642
$53$	9.11389	1.25189	0.625944	−	0.779868i	$-0.284714\pi$
$53$	9.11389	1.25189	0.625944	+	0.779868i	$0.284714\pi$
$54$	0	0
$55$	2.72346	0.367231
$56$	−15.7354	−2.10273
$57$	0	0
$58$	−2.64113	−0.346798
$59$	6.17774	0.804273	0.402136	−	0.915580i	$-0.368268\pi$
$59$	6.17774	0.804273	0.402136	+	0.915580i	$0.368268\pi$
$60$	0	0
$61$	5.91056	0.756769	0.378385	−	0.925648i	$-0.376480\pi$
$61$	5.91056	0.756769	0.378385	+	0.925648i	$0.376480\pi$
$62$	−27.6621	−3.51309
$63$	0	0
$64$	12.2494	1.53117
$65$	−8.07393	−1.00145
$66$	0	0
$67$	−1.78553	−0.218138	−0.109069	−	0.994034i	$-0.534787\pi$
$67$	−1.78553	−0.218138	−0.109069	+	0.994034i	$0.534787\pi$
$68$	25.4368	3.08466
$69$	0	0
$70$	7.52216	0.899070
$71$	−11.9074	−1.41315	−0.706575	−	0.707638i	$-0.749761\pi$
$71$	−11.9074	−1.41315	−0.706575	+	0.707638i	$0.749761\pi$
$72$	0	0
$73$	−5.99221	−0.701335	−0.350667	−	0.936500i	$-0.614045\pi$
$73$	−5.99221	−0.701335	−0.350667	+	0.936500i	$0.614045\pi$
$74$	−0.0948427	−0.0110252
$75$	0	0
$76$	−0.359916	−0.0412852
$77$	−3.83354	−0.436872
$78$	0	0
$79$	7.00625	0.788265	0.394132	−	0.919054i	$-0.371045\pi$
$79$	7.00625	0.788265	0.394132	+	0.919054i	$0.371045\pi$
$80$	−15.3698	−1.71840
$81$	0	0
$82$	−29.5386	−3.26199
$83$	−7.42232	−0.814705	−0.407353	−	0.913271i	$-0.633548\pi$
$83$	−7.42232	−0.814705	−0.407353	+	0.913271i	$0.633548\pi$
$84$	0	0
$85$	−7.27203	−0.788762
$86$	29.6565	3.19794
$87$	0	0
$88$	15.0468	1.60400
$89$	12.4212	1.31665	0.658323	−	0.752735i	$-0.271266\pi$
$89$	12.4212	1.31665	0.658323	+	0.752735i	$0.271266\pi$
$90$	0	0
$91$	11.3649	1.19136
$92$	−4.97557	−0.518739
$93$	0	0
$94$	−22.3607	−2.30633
$95$	0.102895	0.0105568
$96$	0	0
$97$	−1.32074	−0.134101	−0.0670505	−	0.997750i	$-0.521359\pi$
$97$	−1.32074	−0.134101	−0.0670505	+	0.997750i	$0.521359\pi$
$98$	7.89973	0.797993
$99$	0	0
$100$	−14.8105	−1.48105
$101$	−18.6078	−1.85154	−0.925771	−	0.378085i	$-0.876583\pi$
$101$	−18.6078	−1.85154	−0.925771	+	0.378085i	$0.876583\pi$
$102$	0	0
$103$	−8.54195	−0.841663	−0.420832	−	0.907139i	$-0.638262\pi$
$103$	−8.54195	−0.841663	−0.420832	+	0.907139i	$0.638262\pi$
$104$	−44.6076	−4.37413
$105$	0	0
$106$	−24.0710	−2.33798
$107$	9.02049	0.872043	0.436022	−	0.899936i	$-0.356387\pi$
$107$	9.02049	0.872043	0.436022	+	0.899936i	$0.356387\pi$
$108$	0	0
$109$	8.62220	0.825857	0.412928	−	0.910763i	$-0.364506\pi$
$109$	8.62220	0.825857	0.412928	+	0.910763i	$0.364506\pi$
$110$	−7.19301	−0.685827
$111$	0	0
$112$	21.6346	2.04427
$113$	1.31799	0.123986	0.0619932	−	0.998077i	$-0.480254\pi$
$113$	1.31799	0.123986	0.0619932	+	0.998077i	$0.480254\pi$
$114$	0	0
$115$	1.42245	0.132644
$116$	4.97557	0.461970
$117$	0	0
$118$	−16.3162	−1.50203
$119$	10.2361	0.938342
$120$	0	0
$121$	−7.33421	−0.666746
$122$	−15.6106	−1.41331
$123$	0	0
$124$	52.1120	4.67980
$125$	11.3464	1.01485
$126$	0	0
$127$	10.1134	0.897420	0.448710	−	0.893677i	$-0.351884\pi$
$127$	10.1134	0.897420	0.448710	+	0.893677i	$0.351884\pi$
$128$	−6.71182	−0.593246
$129$	0	0
$130$	21.3243	1.87026
$131$	−6.05518	−0.529043	−0.264522	−	0.964380i	$-0.585214\pi$
$131$	−6.05518	−0.529043	−0.264522	+	0.964380i	$0.585214\pi$
$132$	0	0
$133$	−0.144835	−0.0125588
$134$	4.71583	0.407386
$135$	0	0
$136$	−40.1772	−3.44516
$137$	−9.33113	−0.797212	−0.398606	−	0.917122i	$-0.630506\pi$
$137$	−9.33113	−0.797212	−0.398606	+	0.917122i	$0.630506\pi$
$138$	0	0
$139$	11.4009	0.967014	0.483507	−	0.875341i	$-0.339363\pi$
$139$	11.4009	0.967014	0.483507	+	0.875341i	$0.339363\pi$
$140$	−14.1708	−1.19766
$141$	0	0
$142$	31.4490	2.63914
$143$	−10.8676	−0.908791
$144$	0	0
$145$	−1.42245	−0.118128
$146$	15.8262	1.30979
$147$	0	0
$148$	0.178672	0.0146868
$149$	10.5427	0.863695	0.431847	−	0.901947i	$-0.357862\pi$
$149$	10.5427	0.863695	0.431847	+	0.901947i	$0.357862\pi$
$150$	0	0
$151$	−6.18425	−0.503267	−0.251633	−	0.967823i	$-0.580968\pi$
$151$	−6.18425	−0.503267	−0.251633	+	0.967823i	$0.580968\pi$
$152$	0.568484	0.0461102
$153$	0	0
$154$	10.1249	0.815886
$155$	−14.8981	−1.19665
$156$	0	0
$157$	14.7373	1.17616	0.588081	−	0.808802i	$-0.299884\pi$
$157$	14.7373	1.17616	0.588081	+	0.808802i	$0.299884\pi$
$158$	−18.5044	−1.47213
$159$	0	0
$160$	18.2360	1.44168
$161$	−2.00224	−0.157798
$162$	0	0
$163$	15.0597	1.17957	0.589784	−	0.807561i	$-0.299213\pi$
$163$	15.0597	1.17957	0.589784	+	0.807561i	$0.299213\pi$
$164$	55.6471	4.34531
$165$	0	0
$166$	19.6033	1.52151
$167$	−17.2851	−1.33756	−0.668782	−	0.743458i	$-0.733184\pi$
$167$	−17.2851	−1.33756	−0.668782	+	0.743458i	$0.733184\pi$
$168$	0	0
$169$	19.2178	1.47829
$170$	19.2064	1.47306
$171$	0	0
$172$	−55.8693	−4.25999
$173$	14.2351	1.08227	0.541136	−	0.840935i	$-0.317994\pi$
$173$	14.2351	1.08227	0.541136	+	0.840935i	$0.317994\pi$
$174$	0	0
$175$	−5.95994	−0.450529
$176$	−20.6879	−1.55941
$177$	0	0
$178$	−32.8061	−2.45892
$179$	8.66810	0.647884	0.323942	−	0.946077i	$-0.394992\pi$
$179$	8.66810	0.647884	0.323942	+	0.946077i	$0.394992\pi$
$180$	0	0
$181$	−15.2191	−1.13123	−0.565613	−	0.824671i	$-0.691360\pi$
$181$	−15.2191	−1.13123	−0.565613	+	0.824671i	$0.691360\pi$
$182$	−30.0161	−2.22494
$183$	0	0
$184$	7.85888	0.579364
$185$	−0.0510800	−0.00375547
$186$	0	0
$187$	−9.78820	−0.715784
$188$	42.1248	3.07227
$189$	0	0
$190$	−0.271759	−0.0197155
$191$	−1.94422	−0.140679	−0.0703393	−	0.997523i	$-0.522408\pi$
$191$	−1.94422	−0.140679	−0.0703393	+	0.997523i	$0.522408\pi$
$192$	0	0
$193$	19.2847	1.38814	0.694072	−	0.719905i	$-0.255815\pi$
$193$	19.2847	1.38814	0.694072	+	0.719905i	$0.255815\pi$
$194$	3.48825	0.250442
$195$	0	0
$196$	−14.8821	−1.06301
$197$	−14.1792	−1.01023	−0.505115	−	0.863052i	$-0.668550\pi$
$197$	−14.1792	−1.01023	−0.505115	+	0.863052i	$0.668550\pi$
$198$	0	0
$199$	−24.2579	−1.71960	−0.859800	−	0.510632i	$-0.829412\pi$
$199$	−24.2579	−1.71960	−0.859800	+	0.510632i	$0.829412\pi$
$200$	23.3930	1.65414
$201$	0	0
$202$	49.1455	3.45787
$203$	2.00224	0.140530
$204$	0	0
$205$	−15.9088	−1.11112
$206$	22.5604	1.57186
$207$	0	0
$208$	61.3311	4.25254
$209$	0.138497	0.00958007
$210$	0	0
$211$	6.23830	0.429462	0.214731	−	0.976673i	$-0.431112\pi$
$211$	6.23830	0.429462	0.214731	+	0.976673i	$0.431112\pi$
$212$	45.3468	3.11443
$213$	0	0
$214$	−23.8243	−1.62859
$215$	15.9723	1.08930
$216$	0	0
$217$	20.9706	1.42358
$218$	−22.7724	−1.54234
$219$	0	0
$220$	13.5508	0.913593
$221$	29.0179	1.95196
$222$	0	0
$223$	−23.4777	−1.57218	−0.786090	−	0.618112i	$-0.787898\pi$
$223$	−23.4777	−1.57218	−0.786090	+	0.618112i	$0.787898\pi$
$224$	−25.6690	−1.71508
$225$	0	0
$226$	−3.48100	−0.231552
$227$	−13.3128	−0.883599	−0.441799	−	0.897114i	$-0.645660\pi$
$227$	−13.3128	−0.883599	−0.441799	+	0.897114i	$0.645660\pi$
$228$	0	0
$229$	−25.6700	−1.69632	−0.848161	−	0.529738i	$-0.822290\pi$
$229$	−25.6700	−1.69632	−0.848161	+	0.529738i	$0.822290\pi$
$230$	−3.75688	−0.247721
$231$	0	0
$232$	−7.85888	−0.515961
$233$	10.0205	0.656468	0.328234	−	0.944596i	$-0.393547\pi$
$233$	10.0205	0.656468	0.328234	+	0.944596i	$0.393547\pi$
$234$	0	0
$235$	−12.0429	−0.785593
$236$	30.7378	2.00086
$237$	0	0
$238$	−27.0349	−1.75241
$239$	2.24951	0.145508	0.0727542	−	0.997350i	$-0.476821\pi$
$239$	2.24951	0.145508	0.0727542	+	0.997350i	$0.476821\pi$
$240$	0	0
$241$	22.9132	1.47597	0.737986	−	0.674816i	$-0.235777\pi$
$241$	22.9132	1.47597	0.737986	+	0.674816i	$0.235777\pi$
$242$	19.3706	1.24519
$243$	0	0
$244$	29.4084	1.88268
$245$	4.25460	0.271817
$246$	0	0
$247$	−0.410587	−0.0261250
$248$	−82.3105	−5.22672
$249$	0	0
$250$	−29.9672	−1.89529
$251$	12.2960	0.776118	0.388059	−	0.921635i	$-0.373146\pi$
$251$	12.2960	0.776118	0.388059	+	0.921635i	$0.373146\pi$
$252$	0	0
$253$	1.91463	0.120372
$254$	−26.7108	−1.67599
$255$	0	0
$256$	−6.77191	−0.423245
$257$	−18.4338	−1.14987	−0.574935	−	0.818199i	$-0.694973\pi$
$257$	−18.4338	−1.14987	−0.574935	+	0.818199i	$0.694973\pi$
$258$	0	0
$259$	0.0719001	0.00446766
$260$	−40.1724	−2.49139
$261$	0	0
$262$	15.9925	0.988021
$263$	−16.6009	−1.02366	−0.511828	−	0.859088i	$-0.671031\pi$
$263$	−16.6009	−1.02366	−0.511828	+	0.859088i	$0.671031\pi$
$264$	0	0
$265$	−12.9640	−0.796375
$266$	0.382528	0.0234543
$267$	0	0
$268$	−8.88405	−0.542680
$269$	−1.10443	−0.0673382	−0.0336691	−	0.999433i	$-0.510719\pi$
$269$	−1.10443	−0.0673382	−0.0336691	+	0.999433i	$0.510719\pi$
$270$	0	0
$271$	−26.4168	−1.60471	−0.802354	−	0.596849i	$-0.796419\pi$
$271$	−26.4168	−1.60471	−0.802354	+	0.596849i	$0.796419\pi$
$272$	55.2397	3.34940
$273$	0	0
$274$	24.6447	1.48884
$275$	5.69915	0.343672
$276$	0	0
$277$	0.251166	0.0150911	0.00754554	−	0.999972i	$-0.497598\pi$
$277$	0.251166	0.0150911	0.00754554	+	0.999972i	$0.497598\pi$
$278$	−30.1113	−1.80596
$279$	0	0
$280$	22.3827	1.33763
$281$	−22.6985	−1.35408	−0.677039	−	0.735948i	$-0.736737\pi$
$281$	−22.6985	−1.35408	−0.677039	+	0.735948i	$0.736737\pi$
$282$	0	0
$283$	17.7334	1.05414	0.527070	−	0.849822i	$-0.323290\pi$
$283$	17.7334	1.05414	0.527070	+	0.849822i	$0.323290\pi$
$284$	−59.2462	−3.51561
$285$	0	0
$286$	28.7027	1.69722
$287$	22.3932	1.32183
$288$	0	0
$289$	9.13589	0.537405
$290$	3.75688	0.220611
$291$	0	0
$292$	−29.8147	−1.74477
$293$	33.6235	1.96431	0.982153	−	0.188084i	$-0.0602277\pi$
$293$	33.6235	1.96431	0.982153	+	0.188084i	$0.0602277\pi$
$294$	0	0
$295$	−8.78752	−0.511629
$296$	−0.282211	−0.0164032
$297$	0	0
$298$	−27.8447	−1.61300
$299$	−5.67607	−0.328256
$300$	0	0
$301$	−22.4826	−1.29587
$302$	16.3334	0.939882
$303$	0	0
$304$	−0.781610	−0.0448284
$305$	−8.40747	−0.481410
$306$	0	0
$307$	25.1348	1.43452	0.717260	−	0.696806i	$-0.245396\pi$
$307$	25.1348	1.43452	0.717260	+	0.696806i	$0.245396\pi$
$308$	−19.0741	−1.08685
$309$	0	0
$310$	39.3479	2.23481
$311$	−13.6167	−0.772134	−0.386067	−	0.922471i	$-0.626167\pi$
$311$	−13.6167	−0.772134	−0.386067	+	0.922471i	$0.626167\pi$
$312$	0	0
$313$	30.3238	1.71400	0.857002	−	0.515313i	$-0.172324\pi$
$313$	30.3238	1.71400	0.857002	+	0.515313i	$0.172324\pi$
$314$	−38.9231	−2.19656
$315$	0	0
$316$	34.8601	1.96103
$317$	13.1576	0.739003	0.369501	−	0.929230i	$-0.379529\pi$
$317$	13.1576	0.739003	0.369501	+	0.929230i	$0.379529\pi$
$318$	0	0
$319$	−1.91463	−0.107198
$320$	−17.4241	−0.974036
$321$	0	0
$322$	5.28817	0.294698
$323$	−0.369808	−0.0205767
$324$	0	0
$325$	−16.8956	−0.937200
$326$	−39.7747	−2.20292
$327$	0	0
$328$	−87.8942	−4.85314
$329$	16.9516	0.934571
$330$	0	0
$331$	−20.1494	−1.10751	−0.553755	−	0.832680i	$-0.686805\pi$
$331$	−20.1494	−1.10751	−0.553755	+	0.832680i	$0.686805\pi$
$332$	−36.9303	−2.02681
$333$	0	0
$334$	45.6523	2.49799
$335$	2.53983	0.138766
$336$	0	0
$337$	5.35053	0.291462	0.145731	−	0.989324i	$-0.453447\pi$
$337$	5.35053	0.291462	0.145731	+	0.989324i	$0.453447\pi$
$338$	−50.7567	−2.76080
$339$	0	0
$340$	−36.1825	−1.96227
$341$	−20.0530	−1.08593
$342$	0	0
$343$	−20.0044	−1.08014
$344$	88.2451	4.75786
$345$	0	0
$346$	−37.5967	−2.02121
$347$	9.62976	0.516953	0.258476	−	0.966018i	$-0.416780\pi$
$347$	9.62976	0.516953	0.258476	+	0.966018i	$0.416780\pi$
$348$	0	0
$349$	14.3087	0.765927	0.382963	−	0.923764i	$-0.374904\pi$
$349$	14.3087	0.765927	0.382963	+	0.923764i	$0.374904\pi$
$350$	15.7410	0.841391
$351$	0	0
$352$	24.5458	1.30830
$353$	−27.9824	−1.48935	−0.744675	−	0.667427i	$-0.767396\pi$
$353$	−27.9824	−1.48935	−0.744675	+	0.667427i	$0.767396\pi$
$354$	0	0
$355$	16.9377	0.898959
$356$	61.8027	3.27554
$357$	0	0
$358$	−22.8936	−1.20996
$359$	−26.1126	−1.37817	−0.689086	−	0.724680i	$-0.741988\pi$
$359$	−26.1126	−1.37817	−0.689086	+	0.724680i	$0.741988\pi$
$360$	0	0
$361$	−18.9948	−0.999725
$362$	40.1956	2.11263
$363$	0	0
$364$	56.5467	2.96385
$365$	8.52361	0.446146
$366$	0	0
$367$	30.3151	1.58243	0.791217	−	0.611536i	$-0.209448\pi$
$367$	30.3151	1.58243	0.791217	+	0.611536i	$0.209448\pi$
$368$	−10.8052	−0.563259
$369$	0	0
$370$	0.134909	0.00701358
$371$	18.2482	0.947398
$372$	0	0
$373$	9.11693	0.472057	0.236028	−	0.971746i	$-0.424154\pi$
$373$	9.11693	0.472057	0.236028	+	0.971746i	$0.424154\pi$
$374$	25.8519	1.33677
$375$	0	0
$376$	−66.5358	−3.43132
$377$	5.67607	0.292333
$378$	0	0
$379$	−16.3425	−0.839458	−0.419729	−	0.907649i	$-0.637875\pi$
$379$	−16.3425	−0.839458	−0.419729	+	0.907649i	$0.637875\pi$
$380$	0.511962	0.0262631
$381$	0	0
$382$	5.13493	0.262726
$383$	−18.0074	−0.920135	−0.460068	−	0.887884i	$-0.652175\pi$
$383$	−18.0074	−0.920135	−0.460068	+	0.887884i	$0.652175\pi$
$384$	0	0
$385$	5.45301	0.277911
$386$	−50.9335	−2.59245
$387$	0	0
$388$	−6.57145	−0.333615
$389$	28.1719	1.42837	0.714187	−	0.699955i	$-0.246797\pi$
$389$	28.1719	1.42837	0.714187	+	0.699955i	$0.246797\pi$
$390$	0	0
$391$	−5.11233	−0.258542
$392$	23.5062	1.18724
$393$	0	0
$394$	37.4492	1.88667
$395$	−9.96604	−0.501446
$396$	0	0
$397$	−4.36601	−0.219124	−0.109562	−	0.993980i	$-0.534945\pi$
$397$	−4.36601	−0.219124	−0.109562	+	0.993980i	$0.534945\pi$
$398$	64.0684	3.21146
$399$	0	0
$400$	−32.1631	−1.60816
$401$	18.4178	0.919740	0.459870	−	0.887986i	$-0.347896\pi$
$401$	18.4178	0.919740	0.459870	+	0.887986i	$0.347896\pi$
$402$	0	0
$403$	59.4487	2.96135
$404$	−92.5843	−4.60624
$405$	0	0
$406$	−5.28817	−0.262448
$407$	−0.0687540	−0.00340801
$408$	0	0
$409$	0.867159	0.0428783	0.0214391	−	0.999770i	$-0.493175\pi$
$409$	0.867159	0.0428783	0.0214391	+	0.999770i	$0.493175\pi$
$410$	42.0171	2.07508
$411$	0	0
$412$	−42.5011	−2.09388
$413$	12.3693	0.608653
$414$	0	0
$415$	10.5579	0.518266
$416$	−72.7682	−3.56775
$417$	0	0
$418$	−0.365790	−0.0178914
$419$	−13.4433	−0.656748	−0.328374	−	0.944548i	$-0.606501\pi$
$419$	−13.4433	−0.656748	−0.328374	+	0.944548i	$0.606501\pi$
$420$	0	0
$421$	−16.9398	−0.825595	−0.412798	−	0.910823i	$-0.635448\pi$
$421$	−16.9398	−0.825595	−0.412798	+	0.910823i	$0.635448\pi$
$422$	−16.4762	−0.802048
$423$	0	0
$424$	−71.6250	−3.47842
$425$	−15.2175	−0.738159
$426$	0	0
$427$	11.8343	0.572704
$428$	44.8821	2.16946
$429$	0	0
$430$	−42.1849	−2.03434
$431$	14.8106	0.713400	0.356700	−	0.934219i	$-0.383902\pi$
$431$	14.8106	0.713400	0.356700	+	0.934219i	$0.383902\pi$
$432$	0	0
$433$	18.8677	0.906724	0.453362	−	0.891327i	$-0.350225\pi$
$433$	18.8677	0.906724	0.453362	+	0.891327i	$0.350225\pi$
$434$	−55.3861	−2.65862
$435$	0	0
$436$	42.9004	2.05456
$437$	0.0723365	0.00346033
$438$	0	0
$439$	12.1800	0.581320	0.290660	−	0.956826i	$-0.406125\pi$
$439$	12.1800	0.581320	0.290660	+	0.956826i	$0.406125\pi$
$440$	−21.4033	−1.02036
$441$	0	0
$442$	−76.6402	−3.64540
$443$	−7.88398	−0.374579	−0.187290	−	0.982305i	$-0.559970\pi$
$443$	−7.88398	−0.374579	−0.187290	+	0.982305i	$0.559970\pi$
$444$	0	0
$445$	−17.6686	−0.837570
$446$	62.0076	2.93614
$447$	0	0
$448$	24.5261	1.15875
$449$	−18.8495	−0.889562	−0.444781	−	0.895639i	$-0.646719\pi$
$449$	−18.8495	−0.889562	−0.444781	+	0.895639i	$0.646719\pi$
$450$	0	0
$451$	−21.4133	−1.00831
$452$	6.55778	0.308452
$453$	0	0
$454$	35.1607	1.65018
$455$	−16.1659	−0.757870
$456$	0	0
$457$	3.89055	0.181992	0.0909962	−	0.995851i	$-0.470995\pi$
$457$	3.89055	0.181992	0.0909962	+	0.995851i	$0.470995\pi$
$458$	67.7979	3.16799
$459$	0	0
$460$	7.07750	0.329990
$461$	−39.4437	−1.83708	−0.918538	−	0.395332i	$-0.870629\pi$
$461$	−39.4437	−1.83708	−0.918538	+	0.395332i	$0.870629\pi$
$462$	0	0
$463$	−25.3764	−1.17934	−0.589671	−	0.807644i	$-0.700742\pi$
$463$	−25.3764	−1.17934	−0.589671	+	0.807644i	$0.700742\pi$
$464$	10.8052	0.501618
$465$	0	0
$466$	−26.4656	−1.22599
$467$	−0.973326	−0.0450402	−0.0225201	−	0.999746i	$-0.507169\pi$
$467$	−0.973326	−0.0450402	−0.0225201	+	0.999746i	$0.507169\pi$
$468$	0	0
$469$	−3.57506	−0.165081
$470$	31.8069	1.46714
$471$	0	0
$472$	−48.5501	−2.23470
$473$	21.4988	0.988516
$474$	0	0
$475$	0.215320	0.00987954
$476$	50.9305	2.33439
$477$	0	0
$478$	−5.94124	−0.271746
$479$	27.1912	1.24240	0.621199	−	0.783653i	$-0.286646\pi$
$479$	27.1912	1.24240	0.621199	+	0.783653i	$0.286646\pi$
$480$	0	0
$481$	0.203827	0.00929371
$482$	−60.5169	−2.75647
$483$	0	0
$484$	−36.4919	−1.65872
$485$	1.87869	0.0853069
$486$	0	0
$487$	−20.7644	−0.940924	−0.470462	−	0.882420i	$-0.655913\pi$
$487$	−20.7644	−0.940924	−0.470462	+	0.882420i	$0.655913\pi$
$488$	−46.4504	−2.10271
$489$	0	0
$490$	−11.2370	−0.507635
$491$	16.8514	0.760492	0.380246	−	0.924885i	$-0.375839\pi$
$491$	16.8514	0.760492	0.380246	+	0.924885i	$0.375839\pi$
$492$	0	0
$493$	5.11233	0.230248
$494$	1.08442	0.0487901
$495$	0	0
$496$	113.169	5.08143
$497$	−23.8415	−1.06944
$498$	0	0
$499$	24.8096	1.11063	0.555315	−	0.831640i	$-0.312598\pi$
$499$	24.8096	1.11063	0.555315	+	0.831640i	$0.312598\pi$
$500$	56.4547	2.52473
$501$	0	0
$502$	−32.4754	−1.44945
$503$	18.7903	0.837816	0.418908	−	0.908029i	$-0.362413\pi$
$503$	18.7903	0.837816	0.418908	+	0.908029i	$0.362413\pi$
$504$	0	0
$505$	26.4686	1.17784
$506$	−5.05678	−0.224801
$507$	0	0
$508$	50.3200	2.23259
$509$	−15.0700	−0.667966	−0.333983	−	0.942579i	$-0.608393\pi$
$509$	−15.0700	−0.667966	−0.333983	+	0.942579i	$0.608393\pi$
$510$	0	0
$511$	−11.9978	−0.530753
$512$	31.3091	1.38368
$513$	0	0
$514$	48.6861	2.14745
$515$	12.1505	0.535415
$516$	0	0
$517$	−16.2098	−0.712908
$518$	−0.189898	−0.00834362
$519$	0	0
$520$	63.4520	2.78256
$521$	−16.3705	−0.717205	−0.358602	−	0.933490i	$-0.616747\pi$
$521$	−16.3705	−0.717205	−0.358602	+	0.933490i	$0.616747\pi$
$522$	0	0
$523$	38.0420	1.66346	0.831731	−	0.555178i	$-0.187350\pi$
$523$	38.0420	1.66346	0.831731	+	0.555178i	$0.187350\pi$
$524$	−30.1280	−1.31615
$525$	0	0
$526$	43.8452	1.91174
$527$	53.5443	2.33243
$528$	0	0
$529$	1.00000	0.0434783
$530$	34.2397	1.48728
$531$	0	0
$532$	−0.720637	−0.0312436
$533$	63.4815	2.74969
$534$	0	0
$535$	−12.8312	−0.554740
$536$	14.0323	0.606103
$537$	0	0
$538$	2.91694	0.125758
$539$	5.72673	0.246668
$540$	0	0
$541$	22.3057	0.958998	0.479499	−	0.877542i	$-0.340818\pi$
$541$	22.3057	0.958998	0.479499	+	0.877542i	$0.340818\pi$
$542$	69.7703	2.99689
$543$	0	0
$544$	−65.5409	−2.81004
$545$	−12.2646	−0.525359
$546$	0	0
$547$	17.9549	0.767696	0.383848	−	0.923396i	$-0.374599\pi$
$547$	17.9549	0.767696	0.383848	+	0.923396i	$0.374599\pi$
$548$	−46.4277	−1.98329
$549$	0	0
$550$	−15.0522	−0.641828
$551$	−0.0723365	−0.00308164
$552$	0	0
$553$	14.0282	0.596539
$554$	−0.663361	−0.0281835
$555$	0	0
$556$	56.7261	2.40572
$557$	−32.4211	−1.37372	−0.686862	−	0.726788i	$-0.741012\pi$
$557$	−32.4211	−1.37372	−0.686862	+	0.726788i	$0.741012\pi$
$558$	0	0
$559$	−63.7350	−2.69570
$560$	−30.7741	−1.30044
$561$	0	0
$562$	59.9496	2.52882
$563$	16.1496	0.680624	0.340312	−	0.940313i	$-0.389467\pi$
$563$	16.1496	0.680624	0.340312	+	0.940313i	$0.389467\pi$
$564$	0	0
$565$	−1.87478	−0.0788726
$566$	−46.8362	−1.96867
$567$	0	0
$568$	93.5789	3.92648
$569$	38.4637	1.61248	0.806241	−	0.591588i	$-0.201499\pi$
$569$	38.4637	1.61248	0.806241	+	0.591588i	$0.201499\pi$
$570$	0	0
$571$	−24.7938	−1.03759	−0.518793	−	0.854900i	$-0.673618\pi$
$571$	−24.7938	−1.03759	−0.518793	+	0.854900i	$0.673618\pi$
$572$	−54.0723	−2.26088
$573$	0	0
$574$	−59.1433	−2.46859
$575$	2.97664	0.124134
$576$	0	0
$577$	−40.5890	−1.68974	−0.844872	−	0.534969i	$-0.820323\pi$
$577$	−40.5890	−1.68974	−0.844872	+	0.534969i	$0.820323\pi$
$578$	−24.1291	−1.00364
$579$	0	0
$580$	−7.07750	−0.293877
$581$	−14.8613	−0.616549
$582$	0	0
$583$	−17.4497	−0.722692
$584$	47.0920	1.94868
$585$	0	0
$586$	−88.8041	−3.66846
$587$	45.0257	1.85841	0.929205	−	0.369565i	$-0.120493\pi$
$587$	45.0257	1.85841	0.929205	+	0.369565i	$0.120493\pi$
$588$	0	0
$589$	−0.757622	−0.0312173
$590$	23.2090	0.955499
$591$	0	0
$592$	0.388013	0.0159472
$593$	−24.5871	−1.00967	−0.504835	−	0.863216i	$-0.668447\pi$
$593$	−24.5871	−1.00967	−0.504835	+	0.863216i	$0.668447\pi$
$594$	0	0
$595$	−14.5603	−0.596915
$596$	52.4562	2.14869
$597$	0	0
$598$	14.9913	0.613038
$599$	−0.957342	−0.0391159	−0.0195580	−	0.999809i	$-0.506226\pi$
$599$	−0.957342	−0.0391159	−0.0195580	+	0.999809i	$0.506226\pi$
$600$	0	0
$601$	29.3909	1.19888	0.599440	−	0.800420i	$-0.295390\pi$
$601$	29.3909	1.19888	0.599440	+	0.800420i	$0.295390\pi$
$602$	59.3794	2.42012
$603$	0	0
$604$	−30.7702	−1.25202
$605$	10.4325	0.424143
$606$	0	0
$607$	33.5061	1.35997	0.679984	−	0.733227i	$-0.261987\pi$
$607$	33.5061	1.35997	0.679984	+	0.733227i	$0.261987\pi$
$608$	0.927366	0.0376097
$609$	0	0
$610$	22.2052	0.899064
$611$	48.0554	1.94411
$612$	0	0
$613$	−2.73383	−0.110418	−0.0552092	−	0.998475i	$-0.517583\pi$
$613$	−2.73383	−0.110418	−0.0552092	+	0.998475i	$0.517583\pi$
$614$	−66.3844	−2.67905
$615$	0	0
$616$	30.1273	1.21386
$617$	25.4376	1.02408	0.512040	−	0.858961i	$-0.328890\pi$
$617$	25.4376	1.02408	0.512040	+	0.858961i	$0.328890\pi$
$618$	0	0
$619$	−14.1463	−0.568587	−0.284293	−	0.958737i	$-0.591759\pi$
$619$	−14.1463	−0.568587	−0.284293	+	0.958737i	$0.591759\pi$
$620$	−74.1267	−2.97700
$621$	0	0
$622$	35.9636	1.44201
$623$	24.8702	0.996405
$624$	0	0
$625$	−1.25644	−0.0502576
$626$	−80.0892	−3.20101
$627$	0	0
$628$	73.3264	2.92604
$629$	0.183583	0.00731993
$630$	0	0
$631$	17.4744	0.695646	0.347823	−	0.937560i	$-0.386921\pi$
$631$	17.4744	0.695646	0.347823	+	0.937560i	$0.386921\pi$
$632$	−55.0613	−2.19022
$633$	0	0
$634$	−34.7509	−1.38013
$635$	−14.3858	−0.570884
$636$	0	0
$637$	−16.9774	−0.672668
$638$	5.05678	0.200200
$639$	0	0
$640$	9.54722	0.377387
$641$	33.4534	1.32133	0.660665	−	0.750681i	$-0.270275\pi$
$641$	33.4534	1.32133	0.660665	+	0.750681i	$0.270275\pi$
$642$	0	0
$643$	29.2195	1.15230	0.576152	−	0.817343i	$-0.304554\pi$
$643$	29.2195	1.15230	0.576152	+	0.817343i	$0.304554\pi$
$644$	−9.96229	−0.392569
$645$	0	0
$646$	0.976711	0.0384282
$647$	6.73060	0.264607	0.132304	−	0.991209i	$-0.457763\pi$
$647$	6.73060	0.264607	0.132304	+	0.991209i	$0.457763\pi$
$648$	0	0
$649$	−11.8281	−0.464292
$650$	44.6235	1.75028
$651$	0	0
$652$	74.9308	2.93452
$653$	36.0174	1.40947	0.704734	−	0.709471i	$-0.251066\pi$
$653$	36.0174	1.40947	0.704734	+	0.709471i	$0.251066\pi$
$654$	0	0
$655$	8.61318	0.336545
$656$	120.846	4.71824
$657$	0	0
$658$	−44.7714	−1.74537
$659$	40.4999	1.57765	0.788827	−	0.614616i	$-0.210689\pi$
$659$	40.4999	1.57765	0.788827	+	0.614616i	$0.210689\pi$
$660$	0	0
$661$	42.3949	1.64897	0.824484	−	0.565885i	$-0.191465\pi$
$661$	42.3949	1.64897	0.824484	+	0.565885i	$0.191465\pi$
$662$	53.2171	2.06834
$663$	0	0
$664$	58.3311	2.26369
$665$	0.206020	0.00798913
$666$	0	0
$667$	−1.00000	−0.0387202
$668$	−86.0035	−3.32758
$669$	0	0
$670$	−6.70803	−0.259154
$671$	−11.3165	−0.436869
$672$	0	0
$673$	−12.4923	−0.481542	−0.240771	−	0.970582i	$-0.577400\pi$
$673$	−12.4923	−0.481542	−0.240771	+	0.970582i	$0.577400\pi$
$674$	−14.1314	−0.544323
$675$	0	0
$676$	95.6196	3.67768
$677$	14.4936	0.557033	0.278516	−	0.960431i	$-0.410157\pi$
$677$	14.4936	0.557033	0.278516	+	0.960431i	$0.410157\pi$
$678$	0	0
$679$	−2.64444	−0.101484
$680$	57.1500	2.19160
$681$	0	0
$682$	52.9625	2.02804
$683$	41.3109	1.58072	0.790359	−	0.612644i	$-0.209894\pi$
$683$	41.3109	1.58072	0.790359	+	0.612644i	$0.209894\pi$
$684$	0	0
$685$	13.2731	0.507138
$686$	52.8344	2.01723
$687$	0	0
$688$	−121.328	−4.62560
$689$	51.7311	1.97080
$690$	0	0
$691$	9.61896	0.365923	0.182961	−	0.983120i	$-0.441432\pi$
$691$	9.61896	0.365923	0.182961	+	0.983120i	$0.441432\pi$
$692$	70.8276	2.69246
$693$	0	0
$694$	−25.4335	−0.965441
$695$	−16.2172	−0.615155
$696$	0	0
$697$	57.1766	2.16572
$698$	−37.7911	−1.43042
$699$	0	0
$700$	−29.6541	−1.12082
$701$	24.5483	0.927178	0.463589	−	0.886050i	$-0.346561\pi$
$701$	24.5483	0.927178	0.463589	+	0.886050i	$0.346561\pi$
$702$	0	0
$703$	−0.00259759	−9.79701e−5 0
$704$	−23.4529	−0.883915
$705$	0	0
$706$	73.9051	2.78145
$707$	−37.2572	−1.40120
$708$	0	0
$709$	21.7180	0.815638	0.407819	−	0.913063i	$-0.366289\pi$
$709$	21.7180	0.815638	0.407819	+	0.913063i	$0.366289\pi$
$710$	−44.7347	−1.67886
$711$	0	0
$712$	−97.6169	−3.65835
$713$	−10.4736	−0.392238
$714$	0	0
$715$	15.4586	0.578117
$716$	43.1288	1.61180
$717$	0	0
$718$	68.9669	2.57382
$719$	−19.2656	−0.718487	−0.359243	−	0.933244i	$-0.616965\pi$
$719$	−19.2656	−0.718487	−0.359243	+	0.933244i	$0.616965\pi$
$720$	0	0
$721$	−17.1030	−0.636950
$722$	50.1677	1.86705
$723$	0	0
$724$	−75.7237	−2.81425
$725$	−2.97664	−0.110550
$726$	0	0
$727$	9.72094	0.360530	0.180265	−	0.983618i	$-0.442305\pi$
$727$	9.72094	0.360530	0.180265	+	0.983618i	$0.442305\pi$
$728$	−89.3150	−3.31024
$729$	0	0
$730$	−22.5120	−0.833205
$731$	−57.4049	−2.12320
$732$	0	0
$733$	50.4032	1.86168	0.930842	−	0.365422i	$-0.119075\pi$
$733$	50.4032	1.86168	0.930842	+	0.365422i	$0.119075\pi$
$734$	−80.0661	−2.95529
$735$	0	0
$736$	12.8202	0.472557
$737$	3.41863	0.125927
$738$	0	0
$739$	−13.3839	−0.492336	−0.246168	−	0.969227i	$-0.579171\pi$
$739$	−13.3839	−0.492336	−0.246168	+	0.969227i	$0.579171\pi$
$740$	−0.254152	−0.00934282
$741$	0	0
$742$	−48.1958	−1.76932
$743$	40.2914	1.47815	0.739075	−	0.673624i	$-0.235263\pi$
$743$	40.2914	1.47815	0.739075	+	0.673624i	$0.235263\pi$
$744$	0	0
$745$	−14.9965	−0.549430
$746$	−24.0790	−0.881595
$747$	0	0
$748$	−48.7019	−1.78072
$749$	18.0612	0.659941
$750$	0	0
$751$	−40.0989	−1.46323	−0.731614	−	0.681719i	$-0.761233\pi$
$751$	−40.0989	−1.46323	−0.731614	+	0.681719i	$0.761233\pi$
$752$	91.4802	3.33594
$753$	0	0
$754$	−14.9913	−0.545949
$755$	8.79678	0.320148
$756$	0	0
$757$	−27.1943	−0.988395	−0.494197	−	0.869350i	$-0.664538\pi$
$757$	−27.1943	−0.988395	−0.494197	+	0.869350i	$0.664538\pi$
$758$	43.1627	1.56774
$759$	0	0
$760$	−0.808640	−0.0293325
$761$	−7.30253	−0.264717	−0.132358	−	0.991202i	$-0.542255\pi$
$761$	−7.30253	−0.264717	−0.132358	+	0.991202i	$0.542255\pi$
$762$	0	0
$763$	17.2637	0.624988
$764$	−9.67359	−0.349978
$765$	0	0
$766$	47.5599	1.71841
$767$	35.0653	1.26613
$768$	0	0
$769$	−24.9274	−0.898905	−0.449452	−	0.893304i	$-0.648381\pi$
$769$	−24.9274	−0.898905	−0.449452	+	0.893304i	$0.648381\pi$
$770$	−14.4021	−0.519017
$771$	0	0
$772$	95.9526	3.45341
$773$	10.3413	0.371950	0.185975	−	0.982554i	$-0.440456\pi$
$773$	10.3413	0.371950	0.185975	+	0.982554i	$0.440456\pi$
$774$	0	0
$775$	−31.1760	−1.11988
$776$	10.3796	0.372604
$777$	0	0
$778$	−74.4058	−2.66758
$779$	−0.809016	−0.0289860
$780$	0	0
$781$	22.7982	0.815785
$782$	13.5023	0.482842
$783$	0	0
$784$	−32.3188	−1.15424
$785$	−20.9630	−0.748202
$786$	0	0
$787$	−6.34761	−0.226268	−0.113134	−	0.993580i	$-0.536089\pi$
$787$	−6.34761	−0.226268	−0.113134	+	0.993580i	$0.536089\pi$
$788$	−70.5499	−2.51324
$789$	0	0
$790$	26.3216	0.936481
$791$	2.63894	0.0938299
$792$	0	0
$793$	33.5488	1.19135
$794$	11.5312	0.409227
$795$	0	0
$796$	−120.697	−4.27800
$797$	−11.8123	−0.418411	−0.209206	−	0.977872i	$-0.567088\pi$
$797$	−11.8123	−0.418411	−0.209206	+	0.977872i	$0.567088\pi$
$798$	0	0
$799$	43.2826	1.53123
$800$	38.1610	1.34919
$801$	0	0
$802$	−48.6438	−1.71767
$803$	11.4728	0.404868
$804$	0	0
$805$	2.84808	0.100382
$806$	−157.012	−5.53051
$807$	0	0
$808$	146.236	5.14457
$809$	−26.4901	−0.931344	−0.465672	−	0.884957i	$-0.654187\pi$
$809$	−26.4901	−0.931344	−0.465672	+	0.884957i	$0.654187\pi$
$810$	0	0
$811$	−27.4715	−0.964654	−0.482327	−	0.875991i	$-0.660208\pi$
$811$	−27.4715	−0.964654	−0.482327	+	0.875991i	$0.660208\pi$
$812$	9.96229	0.349608
$813$	0	0
$814$	0.181588	0.00636467
$815$	−21.4217	−0.750369
$816$	0	0
$817$	0.812246	0.0284169
$818$	−2.29028	−0.0800778
$819$	0	0
$820$	−79.1552	−2.76422
$821$	11.2160	0.391440	0.195720	−	0.980660i	$-0.437296\pi$
$821$	11.2160	0.391440	0.195720	+	0.980660i	$0.437296\pi$
$822$	0	0
$823$	−3.19331	−0.111312	−0.0556559	−	0.998450i	$-0.517725\pi$
$823$	−3.19331	−0.111312	−0.0556559	+	0.998450i	$0.517725\pi$
$824$	67.1302	2.33859
$825$	0	0
$826$	−32.6689	−1.13670
$827$	46.6075	1.62070	0.810351	−	0.585945i	$-0.199277\pi$
$827$	46.6075	1.62070	0.810351	+	0.585945i	$0.199277\pi$
$828$	0	0
$829$	10.7922	0.374829	0.187414	−	0.982281i	$-0.439989\pi$
$829$	10.7922	0.374829	0.187414	+	0.982281i	$0.439989\pi$
$830$	−27.8847	−0.967893
$831$	0	0
$832$	69.5282	2.41046
$833$	−15.2912	−0.529808
$834$	0	0
$835$	24.5872	0.850877
$836$	0.689104	0.0238332
$837$	0	0
$838$	35.5055	1.22652
$839$	−11.9224	−0.411608	−0.205804	−	0.978593i	$-0.565981\pi$
$839$	−11.9224	−0.411608	−0.205804	+	0.978593i	$0.565981\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	44.7402	1.54185
$843$	0	0
$844$	31.0391	1.06841
$845$	−27.3364	−0.940399
$846$	0	0
$847$	−14.6848	−0.504577
$848$	98.4773	3.38172
$849$	0	0
$850$	40.1915	1.37856
$851$	−0.0359099	−0.00123097
$852$	0	0
$853$	−42.2918	−1.44804	−0.724021	−	0.689778i	$-0.757708\pi$
$853$	−42.2918	−1.44804	−0.724021	+	0.689778i	$0.757708\pi$
$854$	−31.2561	−1.06956
$855$	0	0
$856$	−70.8909	−2.42300
$857$	−0.248795	−0.00849866	−0.00424933	−	0.999991i	$-0.501353\pi$
$857$	−0.248795	−0.00849866	−0.00424933	+	0.999991i	$0.501353\pi$
$858$	0	0
$859$	−36.9796	−1.26173	−0.630863	−	0.775894i	$-0.717299\pi$
$859$	−36.9796	−1.26173	−0.630863	+	0.775894i	$0.717299\pi$
$860$	79.4713	2.70995
$861$	0	0
$862$	−39.1166	−1.33232
$863$	10.2539	0.349048	0.174524	−	0.984653i	$-0.444161\pi$
$863$	10.2539	0.349048	0.174524	+	0.984653i	$0.444161\pi$
$864$	0	0
$865$	−20.2487	−0.688475
$866$	−49.8321	−1.69336
$867$	0	0
$868$	104.341	3.54155
$869$	−13.4144	−0.455051
$870$	0	0
$871$	−10.1348	−0.343405
$872$	−67.7608	−2.29467
$873$	0	0
$874$	−0.191050	−0.00646237
$875$	22.7181	0.768013
$876$	0	0
$877$	9.30495	0.314206	0.157103	−	0.987582i	$-0.449785\pi$
$877$	9.30495	0.314206	0.157103	+	0.987582i	$0.449785\pi$
$878$	−32.1690	−1.08565
$879$	0	0
$880$	29.4275	0.992001
$881$	58.3711	1.96657	0.983287	−	0.182063i	$-0.0582774\pi$
$881$	58.3711	1.96657	0.983287	+	0.182063i	$0.0582774\pi$
$882$	0	0
$883$	7.64348	0.257223	0.128612	−	0.991695i	$-0.458948\pi$
$883$	7.64348	0.257223	0.128612	+	0.991695i	$0.458948\pi$
$884$	144.381	4.85606
$885$	0	0
$886$	20.8226	0.699550
$887$	−29.3923	−0.986897	−0.493448	−	0.869775i	$-0.664264\pi$
$887$	−29.3923	−0.986897	−0.493448	+	0.869775i	$0.664264\pi$
$888$	0	0
$889$	20.2495	0.679145
$890$	46.6650	1.56421
$891$	0	0
$892$	−116.815	−3.91125
$893$	−0.612424	−0.0204940
$894$	0	0
$895$	−12.3299	−0.412144
$896$	−13.4387	−0.448954
$897$	0	0
$898$	49.7840	1.66131
$899$	10.4736	0.349313
$900$	0	0
$901$	46.5932	1.55224
$902$	56.5553	1.88309
$903$	0	0
$904$	−10.3580	−0.344501
$905$	21.6484	0.719616
$906$	0	0
$907$	35.6615	1.18412	0.592061	−	0.805893i	$-0.298315\pi$
$907$	35.6615	1.18412	0.592061	+	0.805893i	$0.298315\pi$
$908$	−66.2386	−2.19821
$909$	0	0
$910$	42.6963	1.41537
$911$	5.81698	0.192725	0.0963626	−	0.995346i	$-0.469279\pi$
$911$	5.81698	0.192725	0.0963626	+	0.995346i	$0.469279\pi$
$912$	0	0
$913$	14.2110	0.470314
$914$	−10.2755	−0.339882
$915$	0	0
$916$	−127.723	−4.22009
$917$	−12.1239	−0.400367
$918$	0	0
$919$	4.19454	0.138365	0.0691826	−	0.997604i	$-0.477961\pi$
$919$	4.19454	0.138365	0.0691826	+	0.997604i	$0.477961\pi$
$920$	−11.1789	−0.368556
$921$	0	0
$922$	104.176	3.43085
$923$	−67.5873	−2.22466
$924$	0	0
$925$	−0.106891	−0.00351454
$926$	67.0224	2.20249
$927$	0	0
$928$	−12.8202	−0.420842
$929$	2.00482	0.0657762	0.0328881	−	0.999459i	$-0.489530\pi$
$929$	2.00482	0.0657762	0.0328881	+	0.999459i	$0.489530\pi$
$930$	0	0
$931$	0.216362	0.00709096
$932$	49.8580	1.63315
$933$	0	0
$934$	2.57068	0.0841153
$935$	13.9232	0.455338
$936$	0	0
$937$	−41.4043	−1.35262	−0.676310	−	0.736617i	$-0.736422\pi$
$937$	−41.4043	−1.35262	−0.676310	+	0.736617i	$0.736422\pi$
$938$	9.44221	0.308299
$939$	0	0
$940$	−59.9204	−1.95439
$941$	−18.5944	−0.606159	−0.303079	−	0.952965i	$-0.598015\pi$
$941$	−18.5944	−0.606159	−0.303079	+	0.952965i	$0.598015\pi$
$942$	0	0
$943$	−11.1841	−0.364203
$944$	66.7516	2.17258
$945$	0	0
$946$	−56.7811	−1.84611
$947$	−10.9908	−0.357154	−0.178577	−	0.983926i	$-0.557149\pi$
$947$	−10.9908	−0.357154	−0.178577	+	0.983926i	$0.557149\pi$
$948$	0	0
$949$	−34.0122	−1.10408
$950$	−0.568687	−0.0184507
$951$	0	0
$952$	−80.4443	−2.60721
$953$	−39.6091	−1.28307	−0.641533	−	0.767095i	$-0.721701\pi$
$953$	−39.6091	−1.28307	−0.641533	+	0.767095i	$0.721701\pi$
$954$	0	0
$955$	2.76555	0.0894911
$956$	11.1926	0.361994
$957$	0	0
$958$	−71.8156	−2.32026
$959$	−18.6831	−0.603310
$960$	0	0
$961$	78.6957	2.53857
$962$	−0.538334	−0.0173566
$963$	0	0
$964$	114.007	3.67190
$965$	−27.4315	−0.883053
$966$	0	0
$967$	−22.6601	−0.728698	−0.364349	−	0.931262i	$-0.618709\pi$
$967$	−22.6601	−0.728698	−0.364349	+	0.931262i	$0.618709\pi$
$968$	57.6387	1.85258
$969$	0	0
$970$	−4.96186	−0.159316
$971$	21.7459	0.697858	0.348929	−	0.937149i	$-0.386545\pi$
$971$	21.7459	0.697858	0.348929	+	0.937149i	$0.386545\pi$
$972$	0	0
$973$	22.8274	0.731812
$974$	54.8415	1.75723
$975$	0	0
$976$	63.8647	2.04426
$977$	−2.60265	−0.0832662	−0.0416331	−	0.999133i	$-0.513256\pi$
$977$	−2.60265	−0.0832662	−0.0416331	+	0.999133i	$0.513256\pi$
$978$	0	0
$979$	−23.7820	−0.760076
$980$	21.1691	0.676222
$981$	0	0
$982$	−44.5067	−1.42027
$983$	26.4521	0.843690	0.421845	−	0.906668i	$-0.361383\pi$
$983$	26.4521	0.843690	0.421845	+	0.906668i	$0.361383\pi$
$984$	0	0
$985$	20.1693	0.642646
$986$	−13.5023	−0.430002
$987$	0	0
$988$	−2.04291	−0.0649936
$989$	11.2287	0.357052
$990$	0	0
$991$	34.2935	1.08937	0.544684	−	0.838641i	$-0.316650\pi$
$991$	34.2935	1.08937	0.544684	+	0.838641i	$0.316650\pi$
$992$	−134.273	−4.26317
$993$	0	0
$994$	62.9685	1.99724
$995$	34.5057	1.09390
$996$	0	0
$997$	43.6796	1.38335	0.691673	−	0.722211i	$-0.256874\pi$
$997$	43.6796	1.38335	0.691673	+	0.722211i	$0.256874\pi$
$998$	−65.5254	−2.07417
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6003.2.a.w.1.1	yes	30
3.2	odd	2		6003.2.a.v.1.30	✓	30

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6003.2.a.v.1.30	✓	30	3.2	odd	2
6003.2.a.w.1.1	yes	30	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)

\(f(q)\)	\(=\)	\(q-2.64113 q^{2} +4.97557 q^{4} -1.42245 q^{5} +2.00224 q^{7} -7.85888 q^{8} +O(q^{10})\)	\(q-2.64113 q^{2} +4.97557 q^{4} -1.42245 q^{5} +2.00224 q^{7} -7.85888 q^{8} +3.75688 q^{10} -1.91463 q^{11} +5.67607 q^{13} -5.28817 q^{14} +10.8052 q^{16} +5.11233 q^{17} -0.0723365 q^{19} -7.07750 q^{20} +5.05678 q^{22} -1.00000 q^{23} -2.97664 q^{25} -14.9913 q^{26} +9.96229 q^{28} +1.00000 q^{29} +10.4736 q^{31} -12.8202 q^{32} -13.5023 q^{34} -2.84808 q^{35} +0.0359099 q^{37} +0.191050 q^{38} +11.1789 q^{40} +11.1841 q^{41} -11.2287 q^{43} -9.52637 q^{44} +2.64113 q^{46} +8.46632 q^{47} -2.99104 q^{49} +7.86169 q^{50} +28.2417 q^{52} +9.11389 q^{53} +2.72346 q^{55} -15.7354 q^{56} -2.64113 q^{58} +6.17774 q^{59} +5.91056 q^{61} -27.6621 q^{62} +12.2494 q^{64} -8.07393 q^{65} -1.78553 q^{67} +25.4368 q^{68} +7.52216 q^{70} -11.9074 q^{71} -5.99221 q^{73} -0.0948427 q^{74} -0.359916 q^{76} -3.83354 q^{77} +7.00625 q^{79} -15.3698 q^{80} -29.5386 q^{82} -7.42232 q^{83} -7.27203 q^{85} +29.6565 q^{86} +15.0468 q^{88} +12.4212 q^{89} +11.3649 q^{91} -4.97557 q^{92} -22.3607 q^{94} +0.102895 q^{95} -1.32074 q^{97} +7.89973 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 30 q + q^{2} + 37 q^{4} + 10 q^{7} + 6 q^{8}+O(q^{10}) \) 30 * q + q^2 + 37 * q^4 + 10 * q^7 + 6 * q^8	\( 30 q + q^{2} + 37 q^{4} + 10 q^{7} + 6 q^{8} + 8 q^{10} + 36 q^{13} + 7 q^{14} + 47 q^{16} + 18 q^{17} + 16 q^{19} + 25 q^{22} - 30 q^{23} + 56 q^{25} + 11 q^{26} + 27 q^{28} + 30 q^{29} + 14 q^{31} - 7 q^{32} + 3 q^{34} - 22 q^{35} + 40 q^{37} + 6 q^{38} + 30 q^{40} + 14 q^{41} + 34 q^{43} + 5 q^{44} - q^{46} - 2 q^{47} + 74 q^{49} - 21 q^{50} + 71 q^{52} + 16 q^{53} + 22 q^{55} + 14 q^{56} + q^{58} - 32 q^{59} + 46 q^{61} + 20 q^{62} + 68 q^{64} + 12 q^{65} + 14 q^{67} + 27 q^{68} - 32 q^{71} + 50 q^{73} - 26 q^{74} + 56 q^{76} + 34 q^{77} + 16 q^{79} + 2 q^{80} + 38 q^{82} - 14 q^{83} + 38 q^{85} + 10 q^{86} + 40 q^{88} - 2 q^{89} + 32 q^{91} - 37 q^{92} + 29 q^{94} - 28 q^{95} + 56 q^{97} + 8 q^{98}+O(q^{100}) \) 30 * q + q^2 + 37 * q^4 + 10 * q^7 + 6 * q^8 + 8 * q^10 + 36 * q^13 + 7 * q^14 + 47 * q^16 + 18 * q^17 + 16 * q^19 + 25 * q^22 - 30 * q^23 + 56 * q^25 + 11 * q^26 + 27 * q^28 + 30 * q^29 + 14 * q^31 - 7 * q^32 + 3 * q^34 - 22 * q^35 + 40 * q^37 + 6 * q^38 + 30 * q^40 + 14 * q^41 + 34 * q^43 + 5 * q^44 - q^46 - 2 * q^47 + 74 * q^49 - 21 * q^50 + 71 * q^52 + 16 * q^53 + 22 * q^55 + 14 * q^56 + q^58 - 32 * q^59 + 46 * q^61 + 20 * q^62 + 68 * q^64 + 12 * q^65 + 14 * q^67 + 27 * q^68 - 32 * q^71 + 50 * q^73 - 26 * q^74 + 56 * q^76 + 34 * q^77 + 16 * q^79 + 2 * q^80 + 38 * q^82 - 14 * q^83 + 38 * q^85 + 10 * q^86 + 40 * q^88 - 2 * q^89 + 32 * q^91 - 37 * q^92 + 29 * q^94 - 28 * q^95 + 56 * q^97 + 8 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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