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Properties

Label 6003.2.a.t.1.7
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 1.7
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)
 
\(f(q)\) \(=\) \(q-1.35991 q^{2} -0.150655 q^{4} +4.20064 q^{5} +0.870260 q^{7} +2.92469 q^{8} -5.71248 q^{10} -3.24281 q^{11} -1.60184 q^{13} -1.18347 q^{14} -3.67599 q^{16} -7.96215 q^{17} -1.89212 q^{19} -0.632848 q^{20} +4.40992 q^{22} +1.00000 q^{23} +12.6454 q^{25} +2.17835 q^{26} -0.131109 q^{28} +1.00000 q^{29} -6.87026 q^{31} -0.850373 q^{32} +10.8278 q^{34} +3.65565 q^{35} +5.65195 q^{37} +2.57310 q^{38} +12.2856 q^{40} +5.17129 q^{41} +12.5424 q^{43} +0.488546 q^{44} -1.35991 q^{46} +4.23722 q^{47} -6.24265 q^{49} -17.1965 q^{50} +0.241325 q^{52} -6.69371 q^{53} -13.6219 q^{55} +2.54524 q^{56} -1.35991 q^{58} +1.30032 q^{59} -14.9100 q^{61} +9.34291 q^{62} +8.50841 q^{64} -6.72875 q^{65} -0.0645202 q^{67} +1.19954 q^{68} -4.97134 q^{70} -7.40240 q^{71} -4.96764 q^{73} -7.68612 q^{74} +0.285057 q^{76} -2.82209 q^{77} -3.28830 q^{79} -15.4415 q^{80} -7.03247 q^{82} -2.68133 q^{83} -33.4461 q^{85} -17.0565 q^{86} -9.48422 q^{88} -11.1742 q^{89} -1.39402 q^{91} -0.150655 q^{92} -5.76222 q^{94} -7.94809 q^{95} +1.63869 q^{97} +8.48942 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 22 q - 3 q^{2} + 17 q^{4} - 6 q^{7} - 6 q^{8} - 12 q^{10} - 28 q^{13} - q^{14} + 3 q^{16} - 10 q^{17} - 8 q^{19} - 11 q^{22} + 22 q^{23} + 11 q^{26} - 21 q^{28} + 22 q^{29} - 18 q^{31} + 5 q^{32} - 33 q^{34}+ \cdots - 28 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.35991 −0.961599 −0.480799 − 0.876831i \(-0.659653\pi\)
−0.480799 + 0.876831i \(0.659653\pi\)
\(3\) 0 0
\(4\) −0.150655 −0.0753275
\(5\) 4.20064 1.87858 0.939291 − 0.343120i \(-0.111484\pi\)
0.939291 + 0.343120i \(0.111484\pi\)
\(6\) 0 0
\(7\) 0.870260 0.328927 0.164464 − 0.986383i \(-0.447411\pi\)
0.164464 + 0.986383i \(0.447411\pi\)
\(8\) 2.92469 1.03403
\(9\) 0 0
\(10\) −5.71248 −1.80644
\(11\) −3.24281 −0.977745 −0.488872 − 0.872355i \(-0.662592\pi\)
−0.488872 + 0.872355i \(0.662592\pi\)
\(12\) 0 0
\(13\) −1.60184 −0.444270 −0.222135 − 0.975016i \(-0.571303\pi\)
−0.222135 + 0.975016i \(0.571303\pi\)
\(14\) −1.18347 −0.316296
\(15\) 0 0
\(16\) −3.67599 −0.918998
\(17\) −7.96215 −1.93110 −0.965552 − 0.260210i \(-0.916208\pi\)
−0.965552 + 0.260210i \(0.916208\pi\)
\(18\) 0 0
\(19\) −1.89212 −0.434081 −0.217040 − 0.976163i \(-0.569640\pi\)
−0.217040 + 0.976163i \(0.569640\pi\)
\(20\) −0.632848 −0.141509
\(21\) 0 0
\(22\) 4.40992 0.940198
\(23\) 1.00000 0.208514
\(24\) 0 0
\(25\) 12.6454 2.52907
\(26\) 2.17835 0.427210
\(27\) 0 0
\(28\) −0.131109 −0.0247773
\(29\) 1.00000 0.185695
\(30\) 0 0
\(31\) −6.87026 −1.23394 −0.616968 − 0.786988i \(-0.711639\pi\)
−0.616968 + 0.786988i \(0.711639\pi\)
\(32\) −0.850373 −0.150326
\(33\) 0 0
\(34\) 10.8278 1.85695
\(35\) 3.65565 0.617917
\(36\) 0 0
\(37\) 5.65195 0.929175 0.464587 − 0.885527i \(-0.346203\pi\)
0.464587 + 0.885527i \(0.346203\pi\)
\(38\) 2.57310 0.417412
\(39\) 0 0
\(40\) 12.2856 1.94252
\(41\) 5.17129 0.807620 0.403810 − 0.914843i \(-0.367686\pi\)
0.403810 + 0.914843i \(0.367686\pi\)
\(42\) 0 0
\(43\) 12.5424 1.91269 0.956347 − 0.292232i \(-0.0943980\pi\)
0.956347 + 0.292232i \(0.0943980\pi\)
\(44\) 0.488546 0.0736511
\(45\) 0 0
\(46\) −1.35991 −0.200507
\(47\) 4.23722 0.618062 0.309031 − 0.951052i \(-0.399995\pi\)
0.309031 + 0.951052i \(0.399995\pi\)
\(48\) 0 0
\(49\) −6.24265 −0.891807
\(50\) −17.1965 −2.43195
\(51\) 0 0
\(52\) 0.241325 0.0334658
\(53\) −6.69371 −0.919452 −0.459726 − 0.888061i \(-0.652052\pi\)
−0.459726 + 0.888061i \(0.652052\pi\)
\(54\) 0 0
\(55\) −13.6219 −1.83677
\(56\) 2.54524 0.340122
\(57\) 0 0
\(58\) −1.35991 −0.178564
\(59\) 1.30032 0.169287 0.0846435 − 0.996411i \(-0.473025\pi\)
0.0846435 + 0.996411i \(0.473025\pi\)
\(60\) 0 0
\(61\) −14.9100 −1.90902 −0.954512 − 0.298172i \(-0.903623\pi\)
−0.954512 + 0.298172i \(0.903623\pi\)
\(62\) 9.34291 1.18655
\(63\) 0 0
\(64\) 8.50841 1.06355
\(65\) −6.72875 −0.834598
\(66\) 0 0
\(67\) −0.0645202 −0.00788240 −0.00394120 − 0.999992i \(-0.501255\pi\)
−0.00394120 + 0.999992i \(0.501255\pi\)
\(68\) 1.19954 0.145465
\(69\) 0 0
\(70\) −4.97134 −0.594188
\(71\) −7.40240 −0.878503 −0.439251 − 0.898364i \(-0.644756\pi\)
−0.439251 + 0.898364i \(0.644756\pi\)
\(72\) 0 0
\(73\) −4.96764 −0.581419 −0.290709 − 0.956811i \(-0.593891\pi\)
−0.290709 + 0.956811i \(0.593891\pi\)
\(74\) −7.68612 −0.893493
\(75\) 0 0
\(76\) 0.285057 0.0326983
\(77\) −2.82209 −0.321607
\(78\) 0 0
\(79\) −3.28830 −0.369962 −0.184981 − 0.982742i \(-0.559222\pi\)
−0.184981 + 0.982742i \(0.559222\pi\)
\(80\) −15.4415 −1.72641
\(81\) 0 0
\(82\) −7.03247 −0.776606
\(83\) −2.68133 −0.294314 −0.147157 − 0.989113i \(-0.547012\pi\)
−0.147157 + 0.989113i \(0.547012\pi\)
\(84\) 0 0
\(85\) −33.4461 −3.62774
\(86\) −17.0565 −1.83925
\(87\) 0 0
\(88\) −9.48422 −1.01102
\(89\) −11.1742 −1.18446 −0.592232 − 0.805767i \(-0.701753\pi\)
−0.592232 + 0.805767i \(0.701753\pi\)
\(90\) 0 0
\(91\) −1.39402 −0.146133
\(92\) −0.150655 −0.0157069
\(93\) 0 0
\(94\) −5.76222 −0.594328
\(95\) −7.94809 −0.815457
\(96\) 0 0
\(97\) 1.63869 0.166384 0.0831918 − 0.996534i \(-0.473489\pi\)
0.0831918 + 0.996534i \(0.473489\pi\)
\(98\) 8.48942 0.857561
\(99\) 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.1.7 ✓ 22
3.2 odd 2 6003.2.a.u.1.16 yes 22
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
6003.2.a.t.1.7 ✓ 22 1.1 even 1 trivial
6003.2.a.u.1.16 yes 22 3.2 odd 2