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Properties

Label 3840.2.k.x.1921.1
Level $3840$
Weight $2$
Character 3840.1921
Analytic conductor $30.663$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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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] = [3840,2,Mod(1921,3840)] 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("3840.1921"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3840, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 3840 = 2^{8} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3840.k (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,0,0,0,8,0,-2,0,0,0,0,0,-2,0,8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(30.6625543762\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1920)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1921.1
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 3840.1921
Dual form 3840.2.k.x.1921.2

$q$-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)
 
\(f(q)\) \(=\) \(q-1.00000i q^{3} -1.00000i q^{5} +4.00000 q^{7} -1.00000 q^{9} -6.00000i q^{11} -4.00000i q^{13} -1.00000 q^{15} +4.00000 q^{17} -8.00000i q^{19} -4.00000i q^{21} -1.00000 q^{25} +1.00000i q^{27} +2.00000i q^{29} -2.00000 q^{31} -6.00000 q^{33} -4.00000i q^{35} -4.00000i q^{37} -4.00000 q^{39} -6.00000 q^{41} +12.0000i q^{43} +1.00000i q^{45} +9.00000 q^{49} -4.00000i q^{51} +14.0000i q^{53} -6.00000 q^{55} -8.00000 q^{57} +6.00000i q^{59} -6.00000i q^{61} -4.00000 q^{63} -4.00000 q^{65} -4.00000i q^{67} +8.00000 q^{71} +2.00000 q^{73} +1.00000i q^{75} -24.0000i q^{77} +6.00000 q^{79} +1.00000 q^{81} +12.0000i q^{83} -4.00000i q^{85} +2.00000 q^{87} -6.00000 q^{89} -16.0000i q^{91} +2.00000i q^{93} -8.00000 q^{95} -14.0000 q^{97} +6.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 8 q^{7} - 2 q^{9} - 2 q^{15} + 8 q^{17} - 2 q^{25} - 4 q^{31} - 12 q^{33} - 8 q^{39} - 12 q^{41} + 18 q^{49} - 12 q^{55} - 16 q^{57} - 8 q^{63} - 8 q^{65} + 16 q^{71} + 4 q^{73} + 12 q^{79} + 2 q^{81}+ \cdots - 28 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3840\mathbb{Z}\right)^\times\).

\(n\) \(511\) \(1537\) \(2561\) \(2821\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) − 1.00000i − 0.577350i
\(4\) 0 0
\(5\) − 1.00000i − 0.447214i
\(6\) 0 0
\(7\) 4.00000 1.51186 0.755929 − 0.654654i \(-0.227186\pi\)
0.755929 + 0.654654i \(0.227186\pi\)
\(8\) 0 0
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) − 6.00000i − 1.80907i −0.426401 − 0.904534i \(-0.640219\pi\)
0.426401 − 0.904534i \(-0.359781\pi\)
\(12\) 0 0
\(13\) − 4.00000i − 1.10940i −0.832050 − 0.554700i \(-0.812833\pi\)
0.832050 − 0.554700i \(-0.187167\pi\)
\(14\) 0 0
\(15\) −1.00000 −0.258199
\(16\) 0 0
\(17\) 4.00000 0.970143 0.485071 − 0.874475i \(-0.338794\pi\)
0.485071 + 0.874475i \(0.338794\pi\)
\(18\) 0 0
\(19\) − 8.00000i − 1.83533i −0.397360 − 0.917663i \(-0.630073\pi\)
0.397360 − 0.917663i \(-0.369927\pi\)
\(20\) 0 0
\(21\) − 4.00000i − 0.872872i
\(22\) 0 0
\(23\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) 0 0
\(25\) −1.00000 −0.200000
\(26\) 0 0
\(27\) 1.00000i 0.192450i
\(28\) 0 0
\(29\) 2.00000i 0.371391i 0.982607 + 0.185695i \(0.0594537\pi\)
−0.982607 + 0.185695i \(0.940546\pi\)
\(30\) 0 0
\(31\) −2.00000 −0.359211 −0.179605 − 0.983739i \(-0.557482\pi\)
−0.179605 + 0.983739i \(0.557482\pi\)
\(32\) 0 0
\(33\) −6.00000 −1.04447
\(34\) 0 0
\(35\) − 4.00000i − 0.676123i
\(36\) 0 0
\(37\) − 4.00000i − 0.657596i −0.944400 − 0.328798i \(-0.893356\pi\)
0.944400 − 0.328798i \(-0.106644\pi\)
\(38\) 0 0
\(39\) −4.00000 −0.640513
\(40\) 0 0
\(41\) −6.00000 −0.937043 −0.468521 − 0.883452i \(-0.655213\pi\)
−0.468521 + 0.883452i \(0.655213\pi\)
\(42\) 0 0
\(43\) 12.0000i 1.82998i 0.403473 + 0.914991i \(0.367803\pi\)
−0.403473 + 0.914991i \(0.632197\pi\)
\(44\) 0 0
\(45\) 1.00000i 0.149071i
\(46\) 0 0
\(47\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 0 0
\(49\) 9.00000 1.28571
\(50\) 0 0
\(51\) − 4.00000i − 0.560112i
\(52\) 0 0
\(53\) 14.0000i 1.92305i 0.274721 + 0.961524i \(0.411414\pi\)
−0.274721 + 0.961524i \(0.588586\pi\)
\(54\) 0 0
\(55\) −6.00000 −0.809040
\(56\) 0 0
\(57\) −8.00000 −1.05963
\(58\) 0 0
\(59\) 6.00000i 0.781133i 0.920575 + 0.390567i \(0.127721\pi\)
−0.920575 + 0.390567i \(0.872279\pi\)
\(60\) 0 0
\(61\) − 6.00000i − 0.768221i −0.923287 − 0.384111i \(-0.874508\pi\)
0.923287 − 0.384111i \(-0.125492\pi\)
\(62\) 0 0
\(63\) −4.00000 −0.503953
\(64\) 0 0
\(65\) −4.00000 −0.496139
\(66\) 0 0
\(67\) − 4.00000i − 0.488678i −0.969690 − 0.244339i \(-0.921429\pi\)
0.969690 − 0.244339i \(-0.0785709\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 8.00000 0.949425 0.474713 − 0.880141i \(-0.342552\pi\)
0.474713 + 0.880141i \(0.342552\pi\)
\(72\) 0 0
\(73\) 2.00000 0.234082 0.117041 − 0.993127i \(-0.462659\pi\)
0.117041 + 0.993127i \(0.462659\pi\)
\(74\) 0 0
\(75\) 1.00000i 0.115470i
\(76\) 0 0
\(77\) − 24.0000i − 2.73505i
\(78\) 0 0
\(79\) 6.00000 0.675053 0.337526 − 0.941316i \(-0.390410\pi\)
0.337526 + 0.941316i \(0.390410\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 12.0000i 1.31717i 0.752506 + 0.658586i \(0.228845\pi\)
−0.752506 + 0.658586i \(0.771155\pi\)
\(84\) 0 0
\(85\) − 4.00000i − 0.433861i
\(86\) 0 0
\(87\) 2.00000 0.214423
\(88\) 0 0
\(89\) −6.00000 −0.635999 −0.317999 − 0.948091i \(-0.603011\pi\)
−0.317999 + 0.948091i \(0.603011\pi\)
\(90\) 0 0
\(91\) − 16.0000i − 1.67726i
\(92\) 0 0
\(93\) 2.00000i 0.207390i
\(94\) 0 0
\(95\) −8.00000 −0.820783
\(96\) 0 0
\(97\) −14.0000 −1.42148 −0.710742 − 0.703452i \(-0.751641\pi\)
−0.710742 + 0.703452i \(0.751641\pi\)
\(98\) 0 0
\(99\) 6.00000i 0.603023i
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 3840.2.k.x.1921.1 2
4.3 odd 2 3840.2.k.e.1921.2 2
8.3 odd 2 3840.2.k.e.1921.1 2
8.5 even 2 inner 3840.2.k.x.1921.2 2
16.3 odd 4 1920.2.a.f.1.1 ✓ 1
16.5 even 4 1920.2.a.g.1.1 yes 1
16.11 odd 4 1920.2.a.x.1.1 yes 1
16.13 even 4 1920.2.a.m.1.1 yes 1
48.5 odd 4 5760.2.a.a.1.1 1
48.11 even 4 5760.2.a.x.1.1 1
48.29 odd 4 5760.2.a.z.1.1 1
48.35 even 4 5760.2.a.bu.1.1 1
80.19 odd 4 9600.2.a.bf.1.1 1
80.29 even 4 9600.2.a.y.1.1 1
80.59 odd 4 9600.2.a.a.1.1 1
80.69 even 4 9600.2.a.cd.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1920.2.a.f.1.1 ✓ 1 16.3 odd 4
1920.2.a.g.1.1 yes 1 16.5 even 4
1920.2.a.m.1.1 yes 1 16.13 even 4
1920.2.a.x.1.1 yes 1 16.11 odd 4
3840.2.k.e.1921.1 2 8.3 odd 2
3840.2.k.e.1921.2 2 4.3 odd 2
3840.2.k.x.1921.1 2 1.1 even 1 trivial
3840.2.k.x.1921.2 2 8.5 even 2 inner
5760.2.a.a.1.1 1 48.5 odd 4
5760.2.a.x.1.1 1 48.11 even 4
5760.2.a.z.1.1 1 48.29 odd 4
5760.2.a.bu.1.1 1 48.35 even 4
9600.2.a.a.1.1 1 80.59 odd 4
9600.2.a.y.1.1 1 80.29 even 4
9600.2.a.bf.1.1 1 80.19 odd 4
9600.2.a.cd.1.1 1 80.69 even 4