Properties

Label 2254.4.a.b.1.1
Level $2254$
Weight $4$
Character 2254.1
Self dual yes
Analytic conductor $132.990$
Analytic rank $0$
Dimension $1$
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] = [2254,4,Mod(1,2254)] 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("2254.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2254, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 2254 = 2 \cdot 7^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2254.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,2,9,4,20] 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: \(132.990305153\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 46)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 2254.1

$q$-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)
 
\(f(q)\) \(=\) \(q+2.00000 q^{2} +9.00000 q^{3} +4.00000 q^{4} +20.0000 q^{5} +18.0000 q^{6} +8.00000 q^{8} +54.0000 q^{9} +40.0000 q^{10} -52.0000 q^{11} +36.0000 q^{12} -43.0000 q^{13} +180.000 q^{15} +16.0000 q^{16} +50.0000 q^{17} +108.000 q^{18} +74.0000 q^{19} +80.0000 q^{20} -104.000 q^{22} -23.0000 q^{23} +72.0000 q^{24} +275.000 q^{25} -86.0000 q^{26} +243.000 q^{27} -7.00000 q^{29} +360.000 q^{30} +273.000 q^{31} +32.0000 q^{32} -468.000 q^{33} +100.000 q^{34} +216.000 q^{36} -4.00000 q^{37} +148.000 q^{38} -387.000 q^{39} +160.000 q^{40} -123.000 q^{41} -152.000 q^{43} -208.000 q^{44} +1080.00 q^{45} -46.0000 q^{46} -75.0000 q^{47} +144.000 q^{48} +550.000 q^{50} +450.000 q^{51} -172.000 q^{52} +86.0000 q^{53} +486.000 q^{54} -1040.00 q^{55} +666.000 q^{57} -14.0000 q^{58} +444.000 q^{59} +720.000 q^{60} -262.000 q^{61} +546.000 q^{62} +64.0000 q^{64} -860.000 q^{65} -936.000 q^{66} +764.000 q^{67} +200.000 q^{68} -207.000 q^{69} -21.0000 q^{71} +432.000 q^{72} -681.000 q^{73} -8.00000 q^{74} +2475.00 q^{75} +296.000 q^{76} -774.000 q^{78} +426.000 q^{79} +320.000 q^{80} +729.000 q^{81} -246.000 q^{82} -902.000 q^{83} +1000.00 q^{85} -304.000 q^{86} -63.0000 q^{87} -416.000 q^{88} +1272.00 q^{89} +2160.00 q^{90} -92.0000 q^{92} +2457.00 q^{93} -150.000 q^{94} +1480.00 q^{95} +288.000 q^{96} +342.000 q^{97} -2808.00 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.00000 0.707107
\(3\) 9.00000 1.73205 0.866025 − 0.500000i \(-0.166667\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(4\) 4.00000 0.500000
\(5\) 20.0000 1.78885 0.894427 − 0.447214i \(-0.147584\pi\)
0.894427 + 0.447214i \(0.147584\pi\)
\(6\) 18.0000 1.22474
\(7\) 0 0
\(8\) 8.00000 0.353553
\(9\) 54.0000 2.00000
\(10\) 40.0000 1.26491
\(11\) −52.0000 −1.42533 −0.712663 − 0.701506i \(-0.752511\pi\)
−0.712663 + 0.701506i \(0.752511\pi\)
\(12\) 36.0000 0.866025
\(13\) −43.0000 −0.917389 −0.458694 − 0.888594i \(-0.651683\pi\)
−0.458694 + 0.888594i \(0.651683\pi\)
\(14\) 0 0
\(15\) 180.000 3.09839
\(16\) 16.0000 0.250000
\(17\) 50.0000 0.713340 0.356670 − 0.934230i \(-0.383912\pi\)
0.356670 + 0.934230i \(0.383912\pi\)
\(18\) 108.000 1.41421
\(19\) 74.0000 0.893514 0.446757 − 0.894655i \(-0.352579\pi\)
0.446757 + 0.894655i \(0.352579\pi\)
\(20\) 80.0000 0.894427
\(21\) 0 0
\(22\) −104.000 −1.00786
\(23\) −23.0000 −0.208514
\(24\) 72.0000 0.612372
\(25\) 275.000 2.20000
\(26\) −86.0000 −0.648692
\(27\) 243.000 1.73205
\(28\) 0 0
\(29\) −7.00000 −0.0448230 −0.0224115 − 0.999749i \(-0.507134\pi\)
−0.0224115 + 0.999749i \(0.507134\pi\)
\(30\) 360.000 2.19089
\(31\) 273.000 1.58169 0.790843 − 0.612019i \(-0.209643\pi\)
0.790843 + 0.612019i \(0.209643\pi\)
\(32\) 32.0000 0.176777
\(33\) −468.000 −2.46874
\(34\) 100.000 0.504408
\(35\) 0 0
\(36\) 216.000 1.00000
\(37\) −4.00000 −0.0177729 −0.00888643 − 0.999961i \(-0.502829\pi\)
−0.00888643 + 0.999961i \(0.502829\pi\)
\(38\) 148.000 0.631810
\(39\) −387.000 −1.58896
\(40\) 160.000 0.632456
\(41\) −123.000 −0.468521 −0.234261 − 0.972174i \(-0.575267\pi\)
−0.234261 + 0.972174i \(0.575267\pi\)
\(42\) 0 0
\(43\) −152.000 −0.539065 −0.269532 − 0.962991i \(-0.586869\pi\)
−0.269532 + 0.962991i \(0.586869\pi\)
\(44\) −208.000 −0.712663
\(45\) 1080.00 3.57771
\(46\) −46.0000 −0.147442
\(47\) −75.0000 −0.232763 −0.116382 − 0.993205i \(-0.537130\pi\)
−0.116382 + 0.993205i \(0.537130\pi\)
\(48\) 144.000 0.433013
\(49\) 0 0
\(50\) 550.000 1.55563
\(51\) 450.000 1.23554
\(52\) −172.000 −0.458694
\(53\) 86.0000 0.222887 0.111443 − 0.993771i \(-0.464453\pi\)
0.111443 + 0.993771i \(0.464453\pi\)
\(54\) 486.000 1.22474
\(55\) −1040.00 −2.54970
\(56\) 0 0
\(57\) 666.000 1.54761
\(58\) −14.0000 −0.0316947
\(59\) 444.000 0.979727 0.489863 − 0.871799i \(-0.337047\pi\)
0.489863 + 0.871799i \(0.337047\pi\)
\(60\) 720.000 1.54919
\(61\) −262.000 −0.549929 −0.274964 − 0.961454i \(-0.588666\pi\)
−0.274964 + 0.961454i \(0.588666\pi\)
\(62\) 546.000 1.11842
\(63\) 0 0
\(64\) 64.0000 0.125000
\(65\) −860.000 −1.64107
\(66\) −936.000 −1.74566
\(67\) 764.000 1.39310 0.696548 − 0.717510i \(-0.254718\pi\)
0.696548 + 0.717510i \(0.254718\pi\)
\(68\) 200.000 0.356670
\(69\) −207.000 −0.361158
\(70\) 0 0
\(71\) −21.0000 −0.0351020 −0.0175510 − 0.999846i \(-0.505587\pi\)
−0.0175510 + 0.999846i \(0.505587\pi\)
\(72\) 432.000 0.707107
\(73\) −681.000 −1.09185 −0.545925 − 0.837834i \(-0.683822\pi\)
−0.545925 + 0.837834i \(0.683822\pi\)
\(74\) −8.00000 −0.0125673
\(75\) 2475.00 3.81051
\(76\) 296.000 0.446757
\(77\) 0 0
\(78\) −774.000 −1.12357
\(79\) 426.000 0.606693 0.303346 − 0.952880i \(-0.401896\pi\)
0.303346 + 0.952880i \(0.401896\pi\)
\(80\) 320.000 0.447214
\(81\) 729.000 1.00000
\(82\) −246.000 −0.331295
\(83\) −902.000 −1.19286 −0.596430 − 0.802665i \(-0.703415\pi\)
−0.596430 + 0.802665i \(0.703415\pi\)
\(84\) 0 0
\(85\) 1000.00 1.27606
\(86\) −304.000 −0.381176
\(87\) −63.0000 −0.0776357
\(88\) −416.000 −0.503929
\(89\) 1272.00 1.51496 0.757482 − 0.652856i \(-0.226430\pi\)
0.757482 + 0.652856i \(0.226430\pi\)
\(90\) 2160.00 2.52982
\(91\) 0 0
\(92\) −92.0000 −0.104257
\(93\) 2457.00 2.73956
\(94\) −150.000 −0.164588
\(95\) 1480.00 1.59837
\(96\) 288.000 0.306186
\(97\) 342.000 0.357988 0.178994 − 0.983850i \(-0.442716\pi\)
0.178994 + 0.983850i \(0.442716\pi\)
\(98\) 0 0
\(99\) −2808.00 −2.85065
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 2254.4.a.b.1.1 1
7.6 odd 2 46.4.a.b.1.1 ✓ 1
21.20 even 2 414.4.a.b.1.1 1
28.27 even 2 368.4.a.e.1.1 1
35.13 even 4 1150.4.b.a.599.1 2
35.27 even 4 1150.4.b.a.599.2 2
35.34 odd 2 1150.4.a.d.1.1 1
56.13 odd 2 1472.4.a.j.1.1 1
56.27 even 2 1472.4.a.a.1.1 1
161.160 even 2 1058.4.a.b.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
46.4.a.b.1.1 ✓ 1 7.6 odd 2
368.4.a.e.1.1 1 28.27 even 2
414.4.a.b.1.1 1 21.20 even 2
1058.4.a.b.1.1 1 161.160 even 2
1150.4.a.d.1.1 1 35.34 odd 2
1150.4.b.a.599.1 2 35.13 even 4
1150.4.b.a.599.2 2 35.27 even 4
1472.4.a.a.1.1 1 56.27 even 2
1472.4.a.j.1.1 1 56.13 odd 2
2254.4.a.b.1.1 1 1.1 even 1 trivial