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Properties

Label 6003.2.a.t.1.12
Level $6003$
Weight $2$
Character 6003.1
Self dual yes
Analytic conductor $47.934$
Analytic rank $1$
Dimension $22$
CM no
Inner twists $1$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [6003,2,Mod(1,6003)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://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);
 
Copy content sage: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")
 
Level: \( N \) \(=\) \( 6003 = 3^{2} \cdot 23 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6003.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [22,-3,0,17,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content 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 1.12
Character \(\chi\) \(=\) 6003.1

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+0.144466 q^{2} -1.97913 q^{4} -4.08805 q^{5} -0.828594 q^{7} -0.574849 q^{8} -0.590584 q^{10} +0.0328797 q^{11} -3.12976 q^{13} -0.119704 q^{14} +3.87521 q^{16} +1.27064 q^{17} -4.11917 q^{19} +8.09078 q^{20} +0.00474999 q^{22} +1.00000 q^{23} +11.7122 q^{25} -0.452143 q^{26} +1.63989 q^{28} +1.00000 q^{29} +6.85696 q^{31} +1.70953 q^{32} +0.183564 q^{34} +3.38733 q^{35} -3.30243 q^{37} -0.595079 q^{38} +2.35001 q^{40} +6.64755 q^{41} -1.68035 q^{43} -0.0650731 q^{44} +0.144466 q^{46} +5.69373 q^{47} -6.31343 q^{49} +1.69201 q^{50} +6.19419 q^{52} +7.40328 q^{53} -0.134414 q^{55} +0.476316 q^{56} +0.144466 q^{58} -2.70220 q^{59} +7.51934 q^{61} +0.990597 q^{62} -7.50346 q^{64} +12.7946 q^{65} -4.02039 q^{67} -2.51476 q^{68} +0.489354 q^{70} -3.73962 q^{71} +2.01339 q^{73} -0.477089 q^{74} +8.15236 q^{76} -0.0272439 q^{77} +15.8864 q^{79} -15.8421 q^{80} +0.960344 q^{82} +11.5703 q^{83} -5.19443 q^{85} -0.242754 q^{86} -0.0189008 q^{88} -7.36813 q^{89} +2.59330 q^{91} -1.97913 q^{92} +0.822550 q^{94} +16.8394 q^{95} -17.3350 q^{97} -0.912076 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 22 q - 3 q^{2} + 17 q^{4} - 6 q^{7} - 6 q^{8} - 12 q^{10} - 28 q^{13} - q^{14} + 3 q^{16} - 10 q^{17} - 8 q^{19} - 11 q^{22} + 22 q^{23} + 11 q^{26} - 21 q^{28} + 22 q^{29} - 18 q^{31} + 5 q^{32} - 33 q^{34}+ \cdots - 28 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

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



Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.144466 0.102153 0.0510764 − 0.998695i \(-0.483735\pi\)
0.0510764 + 0.998695i \(0.483735\pi\)
\(3\) 0 0
\(4\) −1.97913 −0.989565
\(5\) −4.08805 −1.82823 −0.914116 − 0.405452i \(-0.867114\pi\)
−0.914116 + 0.405452i \(0.867114\pi\)
\(6\) 0 0
\(7\) −0.828594 −0.313179 −0.156589 − 0.987664i \(-0.550050\pi\)
−0.156589 + 0.987664i \(0.550050\pi\)
\(8\) −0.574849 −0.203240
\(9\) 0 0
\(10\) −0.590584 −0.186759
\(11\) 0.0328797 0.00991359 0.00495680 − 0.999988i \(-0.498422\pi\)
0.00495680 + 0.999988i \(0.498422\pi\)
\(12\) 0 0
\(13\) −3.12976 −0.868038 −0.434019 − 0.900904i \(-0.642905\pi\)
−0.434019 + 0.900904i \(0.642905\pi\)
\(14\) −0.119704 −0.0319921
\(15\) 0 0
\(16\) 3.87521 0.968803
\(17\) 1.27064 0.308175 0.154087 − 0.988057i \(-0.450756\pi\)
0.154087 + 0.988057i \(0.450756\pi\)
\(18\) 0 0
\(19\) −4.11917 −0.945001 −0.472501 − 0.881330i \(-0.656649\pi\)
−0.472501 + 0.881330i \(0.656649\pi\)
\(20\) 8.09078 1.80915
\(21\) 0 0
\(22\) 0.00474999 0.00101270
\(23\) 1.00000 0.208514
\(24\) 0 0
\(25\) 11.7122 2.34243
\(26\) −0.452143 −0.0886726
\(27\) 0 0
\(28\) 1.63989 0.309911
\(29\) 1.00000 0.185695
\(30\) 0 0
\(31\) 6.85696 1.23155 0.615773 − 0.787924i \(-0.288844\pi\)
0.615773 + 0.787924i \(0.288844\pi\)
\(32\) 1.70953 0.302206
\(33\) 0 0
\(34\) 0.183564 0.0314809
\(35\) 3.38733 0.572564
\(36\) 0 0
\(37\) −3.30243 −0.542917 −0.271458 − 0.962450i \(-0.587506\pi\)
−0.271458 + 0.962450i \(0.587506\pi\)
\(38\) −0.595079 −0.0965346
\(39\) 0 0
\(40\) 2.35001 0.371569
\(41\) 6.64755 1.03817 0.519086 − 0.854722i \(-0.326272\pi\)
0.519086 + 0.854722i \(0.326272\pi\)
\(42\) 0 0
\(43\) −1.68035 −0.256251 −0.128126 − 0.991758i \(-0.540896\pi\)
−0.128126 + 0.991758i \(0.540896\pi\)
\(44\) −0.0650731 −0.00981014
\(45\) 0 0
\(46\) 0.144466 0.0213003
\(47\) 5.69373 0.830515 0.415258 − 0.909704i \(-0.363691\pi\)
0.415258 + 0.909704i \(0.363691\pi\)
\(48\) 0 0
\(49\) −6.31343 −0.901919
\(50\) 1.69201 0.239286
\(51\) 0 0
\(52\) 6.19419 0.858980
\(53\) 7.40328 1.01692 0.508459 − 0.861086i \(-0.330215\pi\)
0.508459 + 0.861086i \(0.330215\pi\)
\(54\) 0 0
\(55\) −0.134414 −0.0181243
\(56\) 0.476316 0.0636504
\(57\) 0 0
\(58\) 0.144466 0.0189693
\(59\) −2.70220 −0.351796 −0.175898 − 0.984408i \(-0.556283\pi\)
−0.175898 + 0.984408i \(0.556283\pi\)
\(60\) 0 0
\(61\) 7.51934 0.962753 0.481376 − 0.876514i \(-0.340137\pi\)
0.481376 + 0.876514i \(0.340137\pi\)
\(62\) 0.990597 0.125806
\(63\) 0 0
\(64\) −7.50346 −0.937932
\(65\) 12.7946 1.58698
\(66\) 0 0
\(67\) −4.02039 −0.491169 −0.245585 − 0.969375i \(-0.578980\pi\)
−0.245585 + 0.969375i \(0.578980\pi\)
\(68\) −2.51476 −0.304959
\(69\) 0 0
\(70\) 0.489354 0.0584890
\(71\) −3.73962 −0.443812 −0.221906 − 0.975068i \(-0.571228\pi\)
−0.221906 + 0.975068i \(0.571228\pi\)
\(72\) 0 0
\(73\) 2.01339 0.235649 0.117825 − 0.993034i \(-0.462408\pi\)
0.117825 + 0.993034i \(0.462408\pi\)
\(74\) −0.477089 −0.0554605
\(75\) 0 0
\(76\) 8.15236 0.935140
\(77\) −0.0272439 −0.00310473
\(78\) 0 0
\(79\) 15.8864 1.78736 0.893682 − 0.448702i \(-0.148113\pi\)
0.893682 + 0.448702i \(0.148113\pi\)
\(80\) −15.8421 −1.77120
\(81\) 0 0
\(82\) 0.960344 0.106052
\(83\) 11.5703 1.27000 0.635002 − 0.772511i \(-0.280999\pi\)
0.635002 + 0.772511i \(0.280999\pi\)
\(84\) 0 0
\(85\) −5.19443 −0.563415
\(86\) −0.242754 −0.0261768
\(87\) 0 0
\(88\) −0.0189008 −0.00201484
\(89\) −7.36813 −0.781020 −0.390510 − 0.920599i \(-0.627701\pi\)
−0.390510 + 0.920599i \(0.627701\pi\)
\(90\) 0 0
\(91\) 2.59330 0.271851
\(92\) −1.97913 −0.206339
\(93\) 0 0
\(94\) 0.822550 0.0848395
\(95\) 16.8394 1.72768
\(96\) 0 0
\(97\) −17.3350 −1.76010 −0.880051 − 0.474879i \(-0.842492\pi\)
−0.880051 + 0.474879i \(0.842492\pi\)
\(98\) −0.912076 −0.0921336
\(99\) 0 0
Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 6003.2.a.t.1.12 ✓ 22
3.2 odd 2 6003.2.a.u.1.11 yes 22
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
6003.2.a.t.1.12 ✓ 22 1.1 even 1 trivial
6003.2.a.u.1.11 yes 22 3.2 odd 2