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Properties

Label 2912.2.k.i
Level $2912$
Weight $2$
Character orbit 2912.k
Analytic conductor $23.252$
Analytic rank $0$
Dimension $14$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2912,2,Mod(1793,2912)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2912, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2912.1793");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2912 = 2^{5} \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2912.k (of order \(2\), degree \(1\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(23.2524370686\)
Analytic rank: \(0\)
Dimension: \(14\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{14} + 37x^{12} + 544x^{10} + 4060x^{8} + 16288x^{6} + 34160x^{4} + 33216x^{2} + 10816 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{13}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{4} q^{3} + (\beta_{13} + \beta_{3}) q^{5} - \beta_{3} q^{7} + ( - \beta_{2} + 2) q^{9} + ( - \beta_{7} - \beta_{6} - \beta_1) q^{11} + (\beta_{7} + \beta_{5} + \beta_{3} + \beta_1) q^{13} + (\beta_{13} - \beta_{8} - \beta_1) q^{15}+ \cdots + (\beta_{13} - 2 \beta_{11} + \cdots - 5 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q - 2 q^{3} + 32 q^{9} - 6 q^{13} + 36 q^{17} - 18 q^{23} - 18 q^{25} - 14 q^{27} - 16 q^{29} + 8 q^{35} + 34 q^{39} + 60 q^{43} - 14 q^{49} + 12 q^{51} + 4 q^{53} - 60 q^{55} + 6 q^{61} + 20 q^{65}+ \cdots + 68 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} + 37x^{12} + 544x^{10} + 4060x^{8} + 16288x^{6} + 34160x^{4} + 33216x^{2} + 10816 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + 5 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -5\nu^{13} - 315\nu^{11} - 6126\nu^{9} - 50408\nu^{7} - 176912\nu^{5} - 197008\nu^{3} + 31520\nu ) / 76544 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 5\nu^{12} + 131\nu^{10} + 1158\nu^{8} + 3672\nu^{6} + 1008\nu^{4} - 7600\nu^{2} - 2080 ) / 2944 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -9\nu^{12} - 291\nu^{10} - 3538\nu^{8} - 19784\nu^{6} - 48624\nu^{4} - 37840\nu^{2} - 3616 ) / 2944 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 38 \nu^{13} + 299 \nu^{12} - 1198 \nu^{11} + 9269 \nu^{10} - 14744 \nu^{9} + 107042 \nu^{8} + \cdots + 487968 ) / 76544 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 38 \nu^{13} - 299 \nu^{12} - 1198 \nu^{11} - 9269 \nu^{10} - 14744 \nu^{9} - 107042 \nu^{8} + \cdots - 487968 ) / 76544 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 97\nu^{13} + 3719\nu^{11} + 56174\nu^{9} + 423928\nu^{7} + 1656272\nu^{5} + 3014416\nu^{3} + 1646560\nu ) / 76544 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( -25\nu^{12} - 839\nu^{10} - 10758\nu^{8} - 65832\nu^{6} - 196400\nu^{4} - 263760\nu^{2} - 125024 ) / 2944 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 7\nu^{12} + 234\nu^{10} + 3015\nu^{8} + 18812\nu^{6} + 58120\nu^{4} + 79520\nu^{2} + 33520 ) / 736 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 155\nu^{13} + 4981\nu^{11} + 60738\nu^{9} + 347512\nu^{7} + 910768\nu^{5} + 787440\nu^{3} - 211680\nu ) / 76544 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 217\nu^{13} + 6495\nu^{11} + 69246\nu^{9} + 298984\nu^{7} + 314448\nu^{5} - 849456\nu^{3} - 1329696\nu ) / 76544 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 331 \nu^{13} + 11285 \nu^{11} + 148162 \nu^{9} + 938312 \nu^{7} + 2920496 \nu^{5} + \cdots + 1740576 \nu ) / 76544 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{13} - 2\beta_{11} - \beta_{8} - \beta_{7} - \beta_{6} - 8\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{10} - 2\beta_{9} + \beta_{7} - \beta_{6} + \beta_{5} + 2\beta_{4} - 12\beta_{2} + 36 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -11\beta_{13} - 2\beta_{12} + 28\beta_{11} + 11\beta_{8} + 17\beta_{7} + 17\beta_{6} + 8\beta_{3} + 74\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 17\beta_{10} + 34\beta_{9} - 21\beta_{7} + 21\beta_{6} - 9\beta_{5} - 38\beta_{4} + 132\beta_{2} - 296 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 107\beta_{13} + 42\beta_{12} - 336\beta_{11} - 99\beta_{8} - 217\beta_{7} - 217\beta_{6} - 132\beta_{3} - 718\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( -205\beta_{10} - 434\beta_{9} + 321\beta_{7} - 321\beta_{6} + 53\beta_{5} + 550\beta_{4} - 1432\beta_{2} + 2612 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 1027 \beta_{13} - 642 \beta_{12} + 3848 \beta_{11} + 843 \beta_{8} + 2505 \beta_{7} + \cdots + 7110 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 2189 \beta_{10} + 5010 \beta_{9} - 4289 \beta_{7} + 4289 \beta_{6} - 149 \beta_{5} - 7206 \beta_{4} + \cdots - 24132 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 9955 \beta_{13} + 8578 \beta_{12} - 43112 \beta_{11} - 7019 \beta_{8} - 27657 \beta_{7} + \cdots - 71286 \beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 22157 \beta_{10} - 55314 \beta_{9} + 53249 \beta_{7} - 53249 \beta_{6} - 1963 \beta_{5} + \cdots + 230260 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 97811 \beta_{13} - 106498 \beta_{12} + 477000 \beta_{11} + 57627 \beta_{8} + 298873 \beta_{7} + \cdots + 721654 \beta_1 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2912\mathbb{Z}\right)^\times\).

\(n\) \(1093\) \(1249\) \(2017\) \(2367\)
\(\chi(n)\) \(1\) \(1\) \(-1\) \(1\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1793.1
− 3.28037i
3.28037i
− 2.63494i
2.63494i
1.28547i
− 1.28547i
− 0.789835i
0.789835i
− 1.80938i
1.80938i
2.18844i
− 2.18844i
2.99280i
− 2.99280i
0 −3.28037 0 − 0.760790i 0 1.00000i 0 7.76086 0
1793.2 0 −3.28037 0 0.760790i 0 − 1.00000i 0 7.76086 0
1793.3 0 −2.63494 0 − 1.68812i 0 1.00000i 0 3.94291 0
1793.4 0 −2.63494 0 1.68812i 0 − 1.00000i 0 3.94291 0
1793.5 0 −1.28547 0 − 1.57070i 0 − 1.00000i 0 −1.34758 0
1793.6 0 −1.28547 0 1.57070i 0 1.00000i 0 −1.34758 0
1793.7 0 −0.789835 0 − 2.82104i 0 1.00000i 0 −2.37616 0
1793.8 0 −0.789835 0 2.82104i 0 − 1.00000i 0 −2.37616 0
1793.9 0 1.80938 0 − 4.01229i 0 − 1.00000i 0 0.273853 0
1793.10 0 1.80938 0 4.01229i 0 1.00000i 0 0.273853 0
1793.11 0 2.18844 0 − 0.616007i 0 1.00000i 0 1.78927 0
1793.12 0 2.18844 0 0.616007i 0 − 1.00000i 0 1.78927 0
1793.13 0 2.99280 0 − 3.69704i 0 1.00000i 0 5.95684 0
1793.14 0 2.99280 0 3.69704i 0 − 1.00000i 0 5.95684 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1793.14
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2912.2.k.i ✓ 14
4.b odd 2 1 2912.2.k.j yes 14
13.b even 2 1 inner 2912.2.k.i ✓ 14
52.b odd 2 1 2912.2.k.j yes 14
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2912.2.k.i ✓ 14 1.a even 1 1 trivial
2912.2.k.i ✓ 14 13.b even 2 1 inner
2912.2.k.j yes 14 4.b odd 2 1
2912.2.k.j yes 14 52.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(2912, [\chi])\):

\( T_{3}^{7} + T_{3}^{6} - 18T_{3}^{5} - 14T_{3}^{4} + 96T_{3}^{3} + 60T_{3}^{2} - 144T_{3} - 104 \) Copy content Toggle raw display
\( T_{5}^{14} + 44T_{5}^{12} + 706T_{5}^{10} + 5092T_{5}^{8} + 16929T_{5}^{6} + 25288T_{5}^{4} + 14548T_{5}^{2} + 2704 \) Copy content Toggle raw display
\( T_{23}^{7} + 9T_{23}^{6} - 74T_{23}^{5} - 586T_{23}^{4} + 2161T_{23}^{3} + 8313T_{23}^{2} - 33216T_{23} + 27488 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{14} \) Copy content Toggle raw display
$3$ \( (T^{7} + T^{6} - 18 T^{5} + \cdots - 104)^{2} \) Copy content Toggle raw display
$5$ \( T^{14} + 44 T^{12} + \cdots + 2704 \) Copy content Toggle raw display
$7$ \( (T^{2} + 1)^{7} \) Copy content Toggle raw display
$11$ \( T^{14} + 105 T^{12} + \cdots + 3564544 \) Copy content Toggle raw display
$13$ \( T^{14} + 6 T^{13} + \cdots + 62748517 \) Copy content Toggle raw display
$17$ \( (T^{7} - 18 T^{6} + \cdots - 10208)^{2} \) Copy content Toggle raw display
$19$ \( T^{14} + 200 T^{12} + \cdots + 1517824 \) Copy content Toggle raw display
$23$ \( (T^{7} + 9 T^{6} + \cdots + 27488)^{2} \) Copy content Toggle raw display
$29$ \( (T^{7} + 8 T^{6} + \cdots - 1888)^{2} \) Copy content Toggle raw display
$31$ \( T^{14} + 189 T^{12} + \cdots + 937024 \) Copy content Toggle raw display
$37$ \( T^{14} + \cdots + 1465970944 \) Copy content Toggle raw display
$41$ \( T^{14} + 233 T^{12} + \cdots + 200704 \) Copy content Toggle raw display
$43$ \( (T^{7} - 30 T^{6} + \cdots - 691808)^{2} \) Copy content Toggle raw display
$47$ \( T^{14} + \cdots + 87204452416 \) Copy content Toggle raw display
$53$ \( (T^{7} - 2 T^{6} + \cdots + 5055704)^{2} \) Copy content Toggle raw display
$59$ \( T^{14} + 208 T^{12} + \cdots + 4096 \) Copy content Toggle raw display
$61$ \( (T^{7} - 3 T^{6} + \cdots + 24544)^{2} \) Copy content Toggle raw display
$67$ \( T^{14} + \cdots + 12451751919616 \) Copy content Toggle raw display
$71$ \( T^{14} + \cdots + 13153337344 \) Copy content Toggle raw display
$73$ \( T^{14} + 133 T^{12} + \cdots + 78074896 \) Copy content Toggle raw display
$79$ \( (T^{7} + 13 T^{6} + \cdots + 85852)^{2} \) Copy content Toggle raw display
$83$ \( T^{14} + \cdots + 292683664 \) Copy content Toggle raw display
$89$ \( T^{14} + \cdots + 1277496709696 \) Copy content Toggle raw display
$97$ \( T^{14} + \cdots + 120384784 \) Copy content Toggle raw display
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