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Properties

Label 2624.2.d.r
Level $2624$
Weight $2$
Character orbit 2624.d
Analytic conductor $20.953$
Analytic rank $0$
Dimension $10$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2624,2,Mod(2049,2624)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2624, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("2624.2049");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2624 = 2^{6} \cdot 41 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2624.d (of order \(2\), degree \(1\), not 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: \(20.9527454904\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} + 18x^{8} + 60x^{6} + 72x^{4} + 30x^{2} + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: no (minimal twist has level 1312)
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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{5} q^{3} - \beta_{7} q^{5} - \beta_1 q^{7} + ( - \beta_{9} - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{5} q^{3} - \beta_{7} q^{5} - \beta_1 q^{7} + ( - \beta_{9} - 1) q^{9} - \beta_{2} q^{11} + ( - \beta_{5} - \beta_{3} - \beta_1) q^{13} + \beta_{3} q^{15} + (\beta_{8} + \beta_{2} + \beta_1) q^{17} + ( - \beta_{8} - \beta_{3} - \beta_1) q^{19} + (\beta_{7} + \beta_{4}) q^{21} + ( - \beta_{6} + \beta_{4}) q^{23} + ( - \beta_{9} + \beta_{7} + \beta_{6} - 1) q^{25} + ( - \beta_{8} - \beta_{3} + \cdots - \beta_1) q^{27}+ \cdots + (2 \beta_{8} + 3 \beta_{5} + \cdots + 4 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q - 6 q^{9} - 4 q^{21} - 8 q^{23} - 2 q^{25} + 12 q^{33} + 8 q^{37} + 24 q^{39} - 6 q^{41} - 8 q^{43} + 8 q^{45} + 10 q^{49} - 24 q^{51} + 4 q^{57} - 16 q^{59} + 20 q^{61} + 8 q^{73} - 4 q^{77} - 2 q^{81} + 32 q^{83} - 32 q^{87} - 32 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} + 18x^{8} + 60x^{6} + 72x^{4} + 30x^{2} + 2 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu^{9} + 17\nu^{7} + 43\nu^{5} + 29\nu^{3} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{9} + 17\nu^{7} + 43\nu^{5} + 29\nu^{3} + 2\nu \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{9} + 18\nu^{7} + 60\nu^{5} + 72\nu^{3} + 30\nu \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( 2\nu^{8} + 34\nu^{6} + 86\nu^{4} + 58\nu^{2} + 2 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( 4\nu^{9} + 65\nu^{7} + 125\nu^{5} + 49\nu^{3} - 4\nu \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( -6\nu^{8} - 98\nu^{6} - 196\nu^{4} - 94\nu^{2} - 2 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( 6\nu^{8} + 99\nu^{6} + 212\nu^{4} + 122\nu^{2} + 12 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( 20\nu^{9} + 328\nu^{7} + 674\nu^{5} + 342\nu^{3} + 14\nu \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( 20\nu^{8} + 329\nu^{6} + 690\nu^{4} + 370\nu^{2} + 26 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{2} - \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{9} - \beta_{7} + 2\beta_{6} - \beta_{4} - 8 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -2\beta_{8} + 9\beta_{5} + 3\beta_{3} - 13\beta_{2} + 14\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -16\beta_{9} + 20\beta_{7} - 29\beta_{6} + 13\beta_{4} + 92 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 31\beta_{8} - 139\beta_{5} - 45\beta_{3} + 180\beta_{2} - 199\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 114\beta_{9} - 145\beta_{7} + 205\beta_{6} - 90\beta_{4} - 634 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -441\beta_{8} + 1976\beta_{5} + 638\beta_{3} - 2531\beta_{2} + 2809\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( -3217\beta_{9} + 4099\beta_{7} - 5781\beta_{6} + 2531\beta_{4} + 17830 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 3111\beta_{8} - 13938\beta_{5} - 4499\beta_{3} + 17832\beta_{2} - 19800\beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2624\mathbb{Z}\right)^\times\).

\(n\) \(129\) \(575\) \(1477\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2049.1
− 1.23914i
− 0.851160i
0.285622i
− 1.25043i
3.75432i
− 3.75432i
1.25043i
− 0.285622i
0.851160i
1.23914i
0 − 2.96883i 0 0.543653 0 1.07494i 0 −5.81398 0
2049.2 0 − 2.12305i 0 1.10677 0 − 3.94037i 0 −1.50733 0
2049.3 0 − 2.05666i 0 −3.40468 0 0.596615i 0 −1.22983 0
2049.4 0 − 0.526277i 0 3.03918 0 0.968322i 0 2.72303 0
2049.5 0 − 0.414594i 0 −1.28492 0 3.46762i 0 2.82811 0
2049.6 0 0.414594i 0 −1.28492 0 − 3.46762i 0 2.82811 0
2049.7 0 0.526277i 0 3.03918 0 − 0.968322i 0 2.72303 0
2049.8 0 2.05666i 0 −3.40468 0 − 0.596615i 0 −1.22983 0
2049.9 0 2.12305i 0 1.10677 0 3.94037i 0 −1.50733 0
2049.10 0 2.96883i 0 0.543653 0 − 1.07494i 0 −5.81398 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2049.10
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2624.2.d.r 10
4.b odd 2 1 2624.2.d.s 10
8.b even 2 1 1312.2.d.f ✓ 10
8.d odd 2 1 1312.2.d.g yes 10
41.b even 2 1 inner 2624.2.d.r 10
164.d odd 2 1 2624.2.d.s 10
328.c odd 2 1 1312.2.d.g yes 10
328.g even 2 1 1312.2.d.f ✓ 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1312.2.d.f ✓ 10 8.b even 2 1
1312.2.d.f ✓ 10 328.g even 2 1
1312.2.d.g yes 10 8.d odd 2 1
1312.2.d.g yes 10 328.c odd 2 1
2624.2.d.r 10 1.a even 1 1 trivial
2624.2.d.r 10 41.b even 2 1 inner
2624.2.d.s 10 4.b odd 2 1
2624.2.d.s 10 164.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(2624, [\chi])\):

\( T_{3}^{10} + 18T_{3}^{8} + 104T_{3}^{6} + 212T_{3}^{4} + 80T_{3}^{2} + 8 \) Copy content Toggle raw display
\( T_{7}^{10} + 30T_{7}^{8} + 256T_{7}^{6} + 508T_{7}^{4} + 352T_{7}^{2} + 72 \) Copy content Toggle raw display
\( T_{23}^{5} + 4T_{23}^{4} - 64T_{23}^{3} - 256T_{23}^{2} + 448T_{23} + 640 \) Copy content Toggle raw display
\( T_{31}^{5} - 112T_{31}^{3} + 96T_{31}^{2} + 3008T_{31} - 6016 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} \) Copy content Toggle raw display
$3$ \( T^{10} + 18 T^{8} + \cdots + 8 \) Copy content Toggle raw display
$5$ \( (T^{5} - 12 T^{3} + 4 T^{2} + \cdots - 8)^{2} \) Copy content Toggle raw display
$7$ \( T^{10} + 30 T^{8} + \cdots + 72 \) Copy content Toggle raw display
$11$ \( T^{10} + 46 T^{8} + \cdots + 8 \) Copy content Toggle raw display
$13$ \( T^{10} + 72 T^{8} + \cdots + 18432 \) Copy content Toggle raw display
$17$ \( T^{10} + 112 T^{8} + \cdots + 1179648 \) Copy content Toggle raw display
$19$ \( T^{10} + 82 T^{8} + \cdots + 29768 \) Copy content Toggle raw display
$23$ \( (T^{5} + 4 T^{4} + \cdots + 640)^{2} \) Copy content Toggle raw display
$29$ \( T^{10} + 184 T^{8} + \cdots + 1968128 \) Copy content Toggle raw display
$31$ \( (T^{5} - 112 T^{3} + \cdots - 6016)^{2} \) Copy content Toggle raw display
$37$ \( (T^{5} - 4 T^{4} + \cdots + 2200)^{2} \) Copy content Toggle raw display
$41$ \( T^{10} + \cdots + 115856201 \) Copy content Toggle raw display
$43$ \( (T^{5} + 4 T^{4} + \cdots + 14720)^{2} \) Copy content Toggle raw display
$47$ \( T^{10} + 98 T^{8} + \cdots + 200 \) Copy content Toggle raw display
$53$ \( T^{10} + \cdots + 469895168 \) Copy content Toggle raw display
$59$ \( (T^{5} + 8 T^{4} + \cdots - 1920)^{2} \) Copy content Toggle raw display
$61$ \( (T^{5} - 10 T^{4} + \cdots + 4448)^{2} \) Copy content Toggle raw display
$67$ \( T^{10} + 350 T^{8} + \cdots + 14688200 \) Copy content Toggle raw display
$71$ \( T^{10} + \cdots + 193887432 \) Copy content Toggle raw display
$73$ \( (T^{5} - 4 T^{4} + \cdots - 21704)^{2} \) Copy content Toggle raw display
$79$ \( T^{10} + \cdots + 924328008 \) Copy content Toggle raw display
$83$ \( (T^{5} - 16 T^{4} + \cdots - 4096)^{2} \) Copy content Toggle raw display
$89$ \( T^{10} + 368 T^{8} + \cdots + 81920000 \) Copy content Toggle raw display
$97$ \( T^{10} + 288 T^{8} + \cdots + 2097152 \) Copy content Toggle raw display
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