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Properties

Label 6003.2.a.t.1.5
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 1.5
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)
 
\(f(q)\) \(=\) \(q-1.96070 q^{2} +1.84436 q^{4} -2.84151 q^{5} -2.21458 q^{7} +0.305158 q^{8} +5.57136 q^{10} +1.29670 q^{11} -0.673949 q^{13} +4.34214 q^{14} -4.28705 q^{16} +1.19266 q^{17} -5.61616 q^{19} -5.24077 q^{20} -2.54244 q^{22} +1.00000 q^{23} +3.07417 q^{25} +1.32142 q^{26} -4.08449 q^{28} +1.00000 q^{29} -4.05642 q^{31} +7.79533 q^{32} -2.33846 q^{34} +6.29275 q^{35} +3.39113 q^{37} +11.0116 q^{38} -0.867109 q^{40} +9.19828 q^{41} +9.58517 q^{43} +2.39158 q^{44} -1.96070 q^{46} -4.53489 q^{47} -2.09562 q^{49} -6.02754 q^{50} -1.24301 q^{52} +3.31678 q^{53} -3.68457 q^{55} -0.675798 q^{56} -1.96070 q^{58} +7.04726 q^{59} -13.0676 q^{61} +7.95344 q^{62} -6.71023 q^{64} +1.91503 q^{65} +4.14971 q^{67} +2.19971 q^{68} -12.3382 q^{70} -4.40083 q^{71} -6.12240 q^{73} -6.64901 q^{74} -10.3582 q^{76} -2.87164 q^{77} +1.11103 q^{79} +12.1817 q^{80} -18.0351 q^{82} -2.93238 q^{83} -3.38897 q^{85} -18.7937 q^{86} +0.395697 q^{88} +13.3205 q^{89} +1.49252 q^{91} +1.84436 q^{92} +8.89157 q^{94} +15.9584 q^{95} +0.608559 q^{97} +4.10890 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 22 q - 3 q^{2} + 17 q^{4} - 6 q^{7} - 6 q^{8} - 12 q^{10} - 28 q^{13} - q^{14} + 3 q^{16} - 10 q^{17} - 8 q^{19} - 11 q^{22} + 22 q^{23} + 11 q^{26} - 21 q^{28} + 22 q^{29} - 18 q^{31} + 5 q^{32} - 33 q^{34}+ \cdots - 28 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.96070 −1.38643 −0.693214 − 0.720732i \(-0.743806\pi\)
−0.693214 + 0.720732i \(0.743806\pi\)
\(3\) 0 0
\(4\) 1.84436 0.922182
\(5\) −2.84151 −1.27076 −0.635381 − 0.772199i \(-0.719157\pi\)
−0.635381 + 0.772199i \(0.719157\pi\)
\(6\) 0 0
\(7\) −2.21458 −0.837033 −0.418517 − 0.908209i \(-0.637450\pi\)
−0.418517 + 0.908209i \(0.637450\pi\)
\(8\) 0.305158 0.107890
\(9\) 0 0
\(10\) 5.57136 1.76182
\(11\) 1.29670 0.390968 0.195484 − 0.980707i \(-0.437372\pi\)
0.195484 + 0.980707i \(0.437372\pi\)
\(12\) 0 0
\(13\) −0.673949 −0.186920 −0.0934600 − 0.995623i \(-0.529793\pi\)
−0.0934600 + 0.995623i \(0.529793\pi\)
\(14\) 4.34214 1.16049
\(15\) 0 0
\(16\) −4.28705 −1.07176
\(17\) 1.19266 0.289264 0.144632 − 0.989486i \(-0.453800\pi\)
0.144632 + 0.989486i \(0.453800\pi\)
\(18\) 0 0
\(19\) −5.61616 −1.28844 −0.644218 − 0.764842i \(-0.722817\pi\)
−0.644218 + 0.764842i \(0.722817\pi\)
\(20\) −5.24077 −1.17187
\(21\) 0 0
\(22\) −2.54244 −0.542049
\(23\) 1.00000 0.208514
\(24\) 0 0
\(25\) 3.07417 0.614834
\(26\) 1.32142 0.259151
\(27\) 0 0
\(28\) −4.08449 −0.771897
\(29\) 1.00000 0.185695
\(30\) 0 0
\(31\) −4.05642 −0.728555 −0.364277 − 0.931290i \(-0.618684\pi\)
−0.364277 + 0.931290i \(0.618684\pi\)
\(32\) 7.79533 1.37803
\(33\) 0 0
\(34\) −2.33846 −0.401043
\(35\) 6.29275 1.06367
\(36\) 0 0
\(37\) 3.39113 0.557499 0.278749 − 0.960364i \(-0.410080\pi\)
0.278749 + 0.960364i \(0.410080\pi\)
\(38\) 11.0116 1.78632
\(39\) 0 0
\(40\) −0.867109 −0.137102
\(41\) 9.19828 1.43653 0.718265 − 0.695770i \(-0.244937\pi\)
0.718265 + 0.695770i \(0.244937\pi\)
\(42\) 0 0
\(43\) 9.58517 1.46172 0.730862 − 0.682525i \(-0.239118\pi\)
0.730862 + 0.682525i \(0.239118\pi\)
\(44\) 2.39158 0.360544
\(45\) 0 0
\(46\) −1.96070 −0.289090
\(47\) −4.53489 −0.661481 −0.330741 − 0.943722i \(-0.607299\pi\)
−0.330741 + 0.943722i \(0.607299\pi\)
\(48\) 0 0
\(49\) −2.09562 −0.299375
\(50\) −6.02754 −0.852423
\(51\) 0 0
\(52\) −1.24301 −0.172374
\(53\) 3.31678 0.455595 0.227797 − 0.973709i \(-0.426848\pi\)
0.227797 + 0.973709i \(0.426848\pi\)
\(54\) 0 0
\(55\) −3.68457 −0.496828
\(56\) −0.675798 −0.0903073
\(57\) 0 0
\(58\) −1.96070 −0.257453
\(59\) 7.04726 0.917475 0.458738 − 0.888572i \(-0.348302\pi\)
0.458738 + 0.888572i \(0.348302\pi\)
\(60\) 0 0
\(61\) −13.0676 −1.67314 −0.836568 − 0.547864i \(-0.815441\pi\)
−0.836568 + 0.547864i \(0.815441\pi\)
\(62\) 7.95344 1.01009
\(63\) 0 0
\(64\) −6.71023 −0.838779
\(65\) 1.91503 0.237531
\(66\) 0 0
\(67\) 4.14971 0.506967 0.253484 − 0.967340i \(-0.418424\pi\)
0.253484 + 0.967340i \(0.418424\pi\)
\(68\) 2.19971 0.266753
\(69\) 0 0
\(70\) −12.3382 −1.47470
\(71\) −4.40083 −0.522282 −0.261141 − 0.965301i \(-0.584099\pi\)
−0.261141 + 0.965301i \(0.584099\pi\)
\(72\) 0 0
\(73\) −6.12240 −0.716573 −0.358287 − 0.933612i \(-0.616639\pi\)
−0.358287 + 0.933612i \(0.616639\pi\)
\(74\) −6.64901 −0.772932
\(75\) 0 0
\(76\) −10.3582 −1.18817
\(77\) −2.87164 −0.327254
\(78\) 0 0
\(79\) 1.11103 0.125001 0.0625005 − 0.998045i \(-0.480092\pi\)
0.0625005 + 0.998045i \(0.480092\pi\)
\(80\) 12.1817 1.36195
\(81\) 0 0
\(82\) −18.0351 −1.99164
\(83\) −2.93238 −0.321871 −0.160935 − 0.986965i \(-0.551451\pi\)
−0.160935 + 0.986965i \(0.551451\pi\)
\(84\) 0 0
\(85\) −3.38897 −0.367585
\(86\) −18.7937 −2.02657
\(87\) 0 0
\(88\) 0.395697 0.0421815
\(89\) 13.3205 1.41197 0.705986 − 0.708225i \(-0.250504\pi\)
0.705986 + 0.708225i \(0.250504\pi\)
\(90\) 0 0
\(91\) 1.49252 0.156458
\(92\) 1.84436 0.192288
\(93\) 0 0
\(94\) 8.89157 0.917096
\(95\) 15.9584 1.63730
\(96\) 0 0
\(97\) 0.608559 0.0617898 0.0308949 − 0.999523i \(-0.490164\pi\)
0.0308949 + 0.999523i \(0.490164\pi\)
\(98\) 4.10890 0.415062
\(99\) 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.1.5 ✓ 22
3.2 odd 2 6003.2.a.u.1.18 yes 22
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
6003.2.a.t.1.5 ✓ 22 1.1 even 1 trivial
6003.2.a.u.1.18 yes 22 3.2 odd 2