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Properties

Label 5054.2.a.w.1.4
Level $5054$
Weight $2$
Character 5054.1
Self dual yes
Analytic conductor $40.356$
Analytic rank $0$
Dimension $4$
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] = [5054,2,Mod(1,5054)] 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("5054.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(5054, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 5054 = 2 \cdot 7 \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5054.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,-4,-1,4,1,1,4,-4,9] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(40.3563931816\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.151572.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 10x^{2} + 8x + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 266)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(-3.02917\) of defining polynomial
Character \(\chi\) \(=\) 5054.1

$q$-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)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +3.02917 q^{3} +1.00000 q^{4} -3.36893 q^{5} -3.02917 q^{6} +1.00000 q^{7} -1.00000 q^{8} +6.17589 q^{9} +3.36893 q^{10} +5.02917 q^{11} +3.02917 q^{12} -4.83613 q^{13} -1.00000 q^{14} -10.2051 q^{15} +1.00000 q^{16} -0.146715 q^{17} -6.17589 q^{18} -3.36893 q^{20} +3.02917 q^{21} -5.02917 q^{22} +4.56197 q^{23} -3.02917 q^{24} +6.34967 q^{25} +4.83613 q^{26} +9.62031 q^{27} +1.00000 q^{28} +7.52555 q^{29} +10.2051 q^{30} -1.46721 q^{31} -1.00000 q^{32} +15.2342 q^{33} +0.146715 q^{34} -3.36893 q^{35} +6.17589 q^{36} -4.73785 q^{37} -14.6495 q^{39} +3.36893 q^{40} -0.970827 q^{41} -3.02917 q^{42} +4.53279 q^{43} +5.02917 q^{44} -20.8061 q^{45} -4.56197 q^{46} -7.52555 q^{47} +3.02917 q^{48} +1.00000 q^{49} -6.34967 q^{50} -0.444424 q^{51} -4.83613 q^{52} -2.53279 q^{53} -9.62031 q^{54} -16.9429 q^{55} -1.00000 q^{56} -7.52555 q^{58} +2.54481 q^{59} -10.2051 q^{60} +13.0412 q^{61} +1.46721 q^{62} +6.17589 q^{63} +1.00000 q^{64} +16.2926 q^{65} -15.2342 q^{66} +0.882458 q^{67} -0.146715 q^{68} +13.8190 q^{69} +3.36893 q^{70} +1.29132 q^{71} -6.17589 q^{72} +9.85539 q^{73} +4.73785 q^{74} +19.2342 q^{75} +5.02917 q^{77} +14.6495 q^{78} +16.4101 q^{79} -3.36893 q^{80} +10.6139 q^{81} +0.970827 q^{82} -11.2243 q^{83} +3.02917 q^{84} +0.494271 q^{85} -4.53279 q^{86} +22.7962 q^{87} -5.02917 q^{88} +1.94165 q^{89} +20.8061 q^{90} -4.83613 q^{91} +4.56197 q^{92} -4.44442 q^{93} +7.52555 q^{94} -3.02917 q^{96} +11.0875 q^{97} -1.00000 q^{98} +31.0596 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{2} - q^{3} + 4 q^{4} + q^{5} + q^{6} + 4 q^{7} - 4 q^{8} + 9 q^{9} - q^{10} + 7 q^{11} - q^{12} - 5 q^{13} - 4 q^{14} - 12 q^{15} + 4 q^{16} + 2 q^{17} - 9 q^{18} + q^{20} - q^{21} - 7 q^{22}+ \cdots + 14 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) 3.02917 1.74889 0.874447 − 0.485121i \(-0.161225\pi\)
0.874447 + 0.485121i \(0.161225\pi\)
\(4\) 1.00000 0.500000
\(5\) −3.36893 −1.50663 −0.753315 − 0.657660i \(-0.771546\pi\)
−0.753315 + 0.657660i \(0.771546\pi\)
\(6\) −3.02917 −1.23665
\(7\) 1.00000 0.377964
\(8\) −1.00000 −0.353553
\(9\) 6.17589 2.05863
\(10\) 3.36893 1.06535
\(11\) 5.02917 1.51635 0.758176 − 0.652050i \(-0.226091\pi\)
0.758176 + 0.652050i \(0.226091\pi\)
\(12\) 3.02917 0.874447
\(13\) −4.83613 −1.34130 −0.670651 − 0.741773i \(-0.733985\pi\)
−0.670651 + 0.741773i \(0.733985\pi\)
\(14\) −1.00000 −0.267261
\(15\) −10.2051 −2.63494
\(16\) 1.00000 0.250000
\(17\) −0.146715 −0.0355835 −0.0177918 − 0.999842i \(-0.505664\pi\)
−0.0177918 + 0.999842i \(0.505664\pi\)
\(18\) −6.17589 −1.45567
\(19\) 0 0
\(20\) −3.36893 −0.753315
\(21\) 3.02917 0.661020
\(22\) −5.02917 −1.07222
\(23\) 4.56197 0.951236 0.475618 − 0.879652i \(-0.342225\pi\)
0.475618 + 0.879652i \(0.342225\pi\)
\(24\) −3.02917 −0.618327
\(25\) 6.34967 1.26993
\(26\) 4.83613 0.948444
\(27\) 9.62031 1.85143
\(28\) 1.00000 0.188982
\(29\) 7.52555 1.39746 0.698730 − 0.715385i \(-0.253749\pi\)
0.698730 + 0.715385i \(0.253749\pi\)
\(30\) 10.2051 1.86318
\(31\) −1.46721 −0.263518 −0.131759 − 0.991282i \(-0.542063\pi\)
−0.131759 + 0.991282i \(0.542063\pi\)
\(32\) −1.00000 −0.176777
\(33\) 15.2342 2.65194
\(34\) 0.146715 0.0251613
\(35\) −3.36893 −0.569453
\(36\) 6.17589 1.02931
\(37\) −4.73785 −0.778898 −0.389449 − 0.921048i \(-0.627335\pi\)
−0.389449 + 0.921048i \(0.627335\pi\)
\(38\) 0 0
\(39\) −14.6495 −2.34579
\(40\) 3.36893 0.532674
\(41\) −0.970827 −0.151618 −0.0758089 − 0.997122i \(-0.524154\pi\)
−0.0758089 + 0.997122i \(0.524154\pi\)
\(42\) −3.02917 −0.467411
\(43\) 4.53279 0.691244 0.345622 − 0.938374i \(-0.387668\pi\)
0.345622 + 0.938374i \(0.387668\pi\)
\(44\) 5.02917 0.758176
\(45\) −20.8061 −3.10159
\(46\) −4.56197 −0.672625
\(47\) −7.52555 −1.09771 −0.548857 − 0.835916i \(-0.684937\pi\)
−0.548857 + 0.835916i \(0.684937\pi\)
\(48\) 3.02917 0.437223
\(49\) 1.00000 0.142857
\(50\) −6.34967 −0.897978
\(51\) −0.444424 −0.0622318
\(52\) −4.83613 −0.670651
\(53\) −2.53279 −0.347906 −0.173953 − 0.984754i \(-0.555654\pi\)
−0.173953 + 0.984754i \(0.555654\pi\)
\(54\) −9.62031 −1.30916
\(55\) −16.9429 −2.28458
\(56\) −1.00000 −0.133631
\(57\) 0 0
\(58\) −7.52555 −0.988153
\(59\) 2.54481 0.331307 0.165653 − 0.986184i \(-0.447027\pi\)
0.165653 + 0.986184i \(0.447027\pi\)
\(60\) −10.2051 −1.31747
\(61\) 13.0412 1.66975 0.834877 − 0.550437i \(-0.185539\pi\)
0.834877 + 0.550437i \(0.185539\pi\)
\(62\) 1.46721 0.186335
\(63\) 6.17589 0.778089
\(64\) 1.00000 0.125000
\(65\) 16.2926 2.02085
\(66\) −15.2342 −1.87520
\(67\) 0.882458 0.107809 0.0539047 − 0.998546i \(-0.482833\pi\)
0.0539047 + 0.998546i \(0.482833\pi\)
\(68\) −0.146715 −0.0177918
\(69\) 13.8190 1.66361
\(70\) 3.36893 0.402664
\(71\) 1.29132 0.153251 0.0766257 − 0.997060i \(-0.475585\pi\)
0.0766257 + 0.997060i \(0.475585\pi\)
\(72\) −6.17589 −0.727835
\(73\) 9.85539 1.15349 0.576743 − 0.816925i \(-0.304323\pi\)
0.576743 + 0.816925i \(0.304323\pi\)
\(74\) 4.73785 0.550764
\(75\) 19.2342 2.22098
\(76\) 0 0
\(77\) 5.02917 0.573127
\(78\) 14.6495 1.65873
\(79\) 16.4101 1.84628 0.923141 − 0.384461i \(-0.125613\pi\)
0.923141 + 0.384461i \(0.125613\pi\)
\(80\) −3.36893 −0.376657
\(81\) 10.6139 1.17932
\(82\) 0.970827 0.107210
\(83\) −11.2243 −1.23203 −0.616015 − 0.787735i \(-0.711254\pi\)
−0.616015 + 0.787735i \(0.711254\pi\)
\(84\) 3.02917 0.330510
\(85\) 0.494271 0.0536112
\(86\) −4.53279 −0.488784
\(87\) 22.7962 2.44401
\(88\) −5.02917 −0.536112
\(89\) 1.94165 0.205815 0.102907 − 0.994691i \(-0.467185\pi\)
0.102907 + 0.994691i \(0.467185\pi\)
\(90\) 20.8061 2.19316
\(91\) −4.83613 −0.506965
\(92\) 4.56197 0.475618
\(93\) −4.44442 −0.460865
\(94\) 7.52555 0.776201
\(95\) 0 0
\(96\) −3.02917 −0.309164
\(97\) 11.0875 1.12577 0.562883 − 0.826536i \(-0.309692\pi\)
0.562883 + 0.826536i \(0.309692\pi\)
\(98\) −1.00000 −0.101015
\(99\) 31.0596 3.12161
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 5054.2.a.w.1.4 4
19.7 even 3 266.2.f.d.239.1 yes 8
19.11 even 3 266.2.f.d.197.1 ✓ 8
19.18 odd 2 5054.2.a.x.1.1 4
57.11 odd 6 2394.2.o.v.1261.1 8
57.26 odd 6 2394.2.o.v.505.1 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
266.2.f.d.197.1 ✓ 8 19.11 even 3
266.2.f.d.239.1 yes 8 19.7 even 3
2394.2.o.v.505.1 8 57.26 odd 6
2394.2.o.v.1261.1 8 57.11 odd 6
5054.2.a.w.1.4 4 1.1 even 1 trivial
5054.2.a.x.1.1 4 19.18 odd 2