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Properties

Label 6003.2.a.t.1.4
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 1.4
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)
 
\(f(q)\) \(=\) \(q-2.18238 q^{2} +2.76278 q^{4} +0.541552 q^{5} +0.936842 q^{7} -1.66467 q^{8} -1.18187 q^{10} -3.69864 q^{11} -4.89549 q^{13} -2.04454 q^{14} -1.89261 q^{16} +7.10752 q^{17} -1.09532 q^{19} +1.49619 q^{20} +8.07183 q^{22} +1.00000 q^{23} -4.70672 q^{25} +10.6838 q^{26} +2.58829 q^{28} +1.00000 q^{29} -5.91418 q^{31} +7.45974 q^{32} -15.5113 q^{34} +0.507349 q^{35} +10.3636 q^{37} +2.39040 q^{38} -0.901507 q^{40} -0.971796 q^{41} -1.94722 q^{43} -10.2185 q^{44} -2.18238 q^{46} +6.84150 q^{47} -6.12233 q^{49} +10.2718 q^{50} -13.5252 q^{52} +10.6007 q^{53} -2.00300 q^{55} -1.55953 q^{56} -2.18238 q^{58} -5.79346 q^{59} -3.13155 q^{61} +12.9070 q^{62} -12.4948 q^{64} -2.65117 q^{65} +12.7930 q^{67} +19.6365 q^{68} -1.10723 q^{70} +11.6056 q^{71} +9.09037 q^{73} -22.6173 q^{74} -3.02612 q^{76} -3.46504 q^{77} +12.3322 q^{79} -1.02495 q^{80} +2.12083 q^{82} +3.23404 q^{83} +3.84909 q^{85} +4.24957 q^{86} +6.15702 q^{88} -18.7802 q^{89} -4.58630 q^{91} +2.76278 q^{92} -14.9307 q^{94} -0.593172 q^{95} -13.0946 q^{97} +13.3612 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 22 q - 3 q^{2} + 17 q^{4} - 6 q^{7} - 6 q^{8} - 12 q^{10} - 28 q^{13} - q^{14} + 3 q^{16} - 10 q^{17} - 8 q^{19} - 11 q^{22} + 22 q^{23} + 11 q^{26} - 21 q^{28} + 22 q^{29} - 18 q^{31} + 5 q^{32} - 33 q^{34}+ \cdots - 28 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.18238 −1.54318 −0.771588 − 0.636123i \(-0.780537\pi\)
−0.771588 + 0.636123i \(0.780537\pi\)
\(3\) 0 0
\(4\) 2.76278 1.38139
\(5\) 0.541552 0.242190 0.121095 − 0.992641i \(-0.461360\pi\)
0.121095 + 0.992641i \(0.461360\pi\)
\(6\) 0 0
\(7\) 0.936842 0.354093 0.177046 − 0.984203i \(-0.443346\pi\)
0.177046 + 0.984203i \(0.443346\pi\)
\(8\) −1.66467 −0.588550
\(9\) 0 0
\(10\) −1.18187 −0.373741
\(11\) −3.69864 −1.11518 −0.557590 − 0.830116i \(-0.688274\pi\)
−0.557590 + 0.830116i \(0.688274\pi\)
\(12\) 0 0
\(13\) −4.89549 −1.35777 −0.678883 − 0.734247i \(-0.737536\pi\)
−0.678883 + 0.734247i \(0.737536\pi\)
\(14\) −2.04454 −0.546427
\(15\) 0 0
\(16\) −1.89261 −0.473153
\(17\) 7.10752 1.72383 0.861914 − 0.507055i \(-0.169266\pi\)
0.861914 + 0.507055i \(0.169266\pi\)
\(18\) 0 0
\(19\) −1.09532 −0.251283 −0.125642 − 0.992076i \(-0.540099\pi\)
−0.125642 + 0.992076i \(0.540099\pi\)
\(20\) 1.49619 0.334558
\(21\) 0 0
\(22\) 8.07183 1.72092
\(23\) 1.00000 0.208514
\(24\) 0 0
\(25\) −4.70672 −0.941344
\(26\) 10.6838 2.09527
\(27\) 0 0
\(28\) 2.58829 0.489140
\(29\) 1.00000 0.185695
\(30\) 0 0
\(31\) −5.91418 −1.06222 −0.531109 − 0.847303i \(-0.678225\pi\)
−0.531109 + 0.847303i \(0.678225\pi\)
\(32\) 7.45974 1.31871
\(33\) 0 0
\(34\) −15.5113 −2.66017
\(35\) 0.507349 0.0857576
\(36\) 0 0
\(37\) 10.3636 1.70377 0.851883 − 0.523732i \(-0.175461\pi\)
0.851883 + 0.523732i \(0.175461\pi\)
\(38\) 2.39040 0.387774
\(39\) 0 0
\(40\) −0.901507 −0.142541
\(41\) −0.971796 −0.151769 −0.0758845 − 0.997117i \(-0.524178\pi\)
−0.0758845 + 0.997117i \(0.524178\pi\)
\(42\) 0 0
\(43\) −1.94722 −0.296948 −0.148474 − 0.988916i \(-0.547436\pi\)
−0.148474 + 0.988916i \(0.547436\pi\)
\(44\) −10.2185 −1.54050
\(45\) 0 0
\(46\) −2.18238 −0.321774
\(47\) 6.84150 0.997935 0.498968 − 0.866621i \(-0.333713\pi\)
0.498968 + 0.866621i \(0.333713\pi\)
\(48\) 0 0
\(49\) −6.12233 −0.874618
\(50\) 10.2718 1.45266
\(51\) 0 0
\(52\) −13.5252 −1.87560
\(53\) 10.6007 1.45612 0.728061 − 0.685513i \(-0.240422\pi\)
0.728061 + 0.685513i \(0.240422\pi\)
\(54\) 0 0
\(55\) −2.00300 −0.270085
\(56\) −1.55953 −0.208401
\(57\) 0 0
\(58\) −2.18238 −0.286560
\(59\) −5.79346 −0.754244 −0.377122 − 0.926164i \(-0.623086\pi\)
−0.377122 + 0.926164i \(0.623086\pi\)
\(60\) 0 0
\(61\) −3.13155 −0.400954 −0.200477 − 0.979698i \(-0.564249\pi\)
−0.200477 + 0.979698i \(0.564249\pi\)
\(62\) 12.9070 1.63919
\(63\) 0 0
\(64\) −12.4948 −1.56184
\(65\) −2.65117 −0.328837
\(66\) 0 0
\(67\) 12.7930 1.56292 0.781459 − 0.623957i \(-0.214476\pi\)
0.781459 + 0.623957i \(0.214476\pi\)
\(68\) 19.6365 2.38128
\(69\) 0 0
\(70\) −1.10723 −0.132339
\(71\) 11.6056 1.37733 0.688666 − 0.725079i \(-0.258197\pi\)
0.688666 + 0.725079i \(0.258197\pi\)
\(72\) 0 0
\(73\) 9.09037 1.06395 0.531974 − 0.846761i \(-0.321451\pi\)
0.531974 + 0.846761i \(0.321451\pi\)
\(74\) −22.6173 −2.62921
\(75\) 0 0
\(76\) −3.02612 −0.347120
\(77\) −3.46504 −0.394877
\(78\) 0 0
\(79\) 12.3322 1.38748 0.693741 − 0.720224i \(-0.255961\pi\)
0.693741 + 0.720224i \(0.255961\pi\)
\(80\) −1.02495 −0.114593
\(81\) 0 0
\(82\) 2.12083 0.234206
\(83\) 3.23404 0.354982 0.177491 − 0.984122i \(-0.443202\pi\)
0.177491 + 0.984122i \(0.443202\pi\)
\(84\) 0 0
\(85\) 3.84909 0.417493
\(86\) 4.24957 0.458243
\(87\) 0 0
\(88\) 6.15702 0.656340
\(89\) −18.7802 −1.99070 −0.995350 − 0.0963207i \(-0.969293\pi\)
−0.995350 + 0.0963207i \(0.969293\pi\)
\(90\) 0 0
\(91\) −4.58630 −0.480775
\(92\) 2.76278 0.288040
\(93\) 0 0
\(94\) −14.9307 −1.53999
\(95\) −0.593172 −0.0608582
\(96\) 0 0
\(97\) −13.0946 −1.32956 −0.664778 − 0.747041i \(-0.731474\pi\)
−0.664778 + 0.747041i \(0.731474\pi\)
\(98\) 13.3612 1.34969
\(99\) 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.1.4 ✓ 22
3.2 odd 2 6003.2.a.u.1.19 yes 22
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
6003.2.a.t.1.4 ✓ 22 1.1 even 1 trivial
6003.2.a.u.1.19 yes 22 3.2 odd 2