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Properties

Label 2624.2.g.c
Level $2624$
Weight $2$
Character orbit 2624.g
Analytic conductor $20.953$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2624,2,Mod(737,2624)]
 
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, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2624.737");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2624 = 2^{6} \cdot 41 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2624.g (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: \(20.9527454904\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\Q(\zeta_{40})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - x^{12} + x^{8} - x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{43}]\)
Coefficient ring index: \( 2^{29} \)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{7} q^{3} + \beta_{2} q^{5} - \beta_{10} q^{7} + ( - \beta_{6} + 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{7} q^{3} + \beta_{2} q^{5} - \beta_{10} q^{7} + ( - \beta_{6} + 2) q^{9} + ( - \beta_{9} + \beta_{7}) q^{11} - \beta_{12} q^{13} + (\beta_{10} - 3 \beta_{3}) q^{15} + \beta_{15} q^{17} - \beta_{7} q^{19} + (\beta_{14} - \beta_{2}) q^{21} + \beta_{4} q^{23} + ( - 2 \beta_{6} - 5) q^{25} - \beta_{9} q^{27} + ( - 2 \beta_{12} + \beta_{8}) q^{29} + \beta_{4} q^{31} + ( - 5 \beta_{6} + 5) q^{33} + ( - \beta_{9} + 2 \beta_{7}) q^{35} - \beta_{2} q^{37} - \beta_{4} q^{39} + (\beta_{13} - 2 \beta_{6} + 1) q^{41} - \beta_1 q^{43} + ( - \beta_{14} + \beta_{2}) q^{45} + ( - 2 \beta_{10} - 3 \beta_{3}) q^{47} + 3 \beta_{6} q^{49} + (\beta_{5} - 5 \beta_1) q^{51} + ( - 2 \beta_{12} - 3 \beta_{8}) q^{53} + (5 \beta_{10} + 5 \beta_{3}) q^{55} + (\beta_{6} - 5) q^{57} + ( - \beta_{5} + 4 \beta_1) q^{59} + ( - \beta_{14} + 2 \beta_{2}) q^{61} + ( - 4 \beta_{10} + \beta_{3}) q^{63} - 2 \beta_{13} q^{65} + ( - \beta_{9} - 3 \beta_{7}) q^{67} + (4 \beta_{12} - 2 \beta_{8}) q^{69} + ( - 2 \beta_{10} + \beta_{3}) q^{71} + ( - 3 \beta_{6} - 3) q^{73} + ( - 2 \beta_{9} - 3 \beta_{7}) q^{75} + (2 \beta_{14} - 3 \beta_{2}) q^{77} + (2 \beta_{10} - 3 \beta_{3}) q^{79} + ( - \beta_{6} - 6) q^{81} + ( - \beta_{5} + 4 \beta_1) q^{83} + (2 \beta_{12} + 4 \beta_{8}) q^{85} + (\beta_{11} - 3 \beta_{4}) q^{87} + (\beta_{15} - 2 \beta_{13}) q^{89} + (\beta_{5} - 3 \beta_1) q^{91} + (4 \beta_{12} - 2 \beta_{8}) q^{93} + ( - \beta_{10} + 3 \beta_{3}) q^{95} + ( - \beta_{15} - 2 \beta_{13}) q^{97} + ( - 2 \beta_{9} + 7 \beta_{7}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 32 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 32 q^{9} - 80 q^{25} + 80 q^{33} + 16 q^{41} - 80 q^{57} - 48 q^{73} - 96 q^{81}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( 2\zeta_{40}^{10} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( 2\zeta_{40}^{12} + 2\zeta_{40}^{8} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -\zeta_{40}^{15} - \zeta_{40}^{9} - \zeta_{40}^{7} + \zeta_{40}^{3} - \zeta_{40} \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( -4\zeta_{40}^{14} + 4\zeta_{40}^{6} \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( 4\zeta_{40}^{14} - 2\zeta_{40}^{10} + 4\zeta_{40}^{6} \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( -2\zeta_{40}^{12} + 2\zeta_{40}^{8} + 1 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( -\zeta_{40}^{15} + 2\zeta_{40}^{11} - \zeta_{40}^{9} - \zeta_{40}^{7} + \zeta_{40}^{3} + \zeta_{40} \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( 2\zeta_{40}^{9} - 2\zeta_{40}^{7} - 2\zeta_{40}^{5} + 2\zeta_{40}^{3} + 2\zeta_{40} \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( -4\zeta_{40}^{13} + 2\zeta_{40}^{9} + 2\zeta_{40}^{7} - 2\zeta_{40}^{5} + 2\zeta_{40}^{3} + 2\zeta_{40} \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( -\zeta_{40}^{15} + \zeta_{40}^{9} + \zeta_{40}^{7} - 2\zeta_{40}^{5} - \zeta_{40}^{3} + \zeta_{40} \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( -4\zeta_{40}^{14} + 4\zeta_{40}^{10} - 4\zeta_{40}^{6} + 8\zeta_{40}^{2} \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( -2\zeta_{40}^{15} + 2\zeta_{40}^{5} \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( 4\zeta_{40}^{13} - 2\zeta_{40}^{9} + 2\zeta_{40}^{7} + 2\zeta_{40}^{5} + 2\zeta_{40}^{3} - 2\zeta_{40} \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( 4\zeta_{40}^{12} - 4\zeta_{40}^{8} + 8\zeta_{40}^{4} - 4 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( 2\zeta_{40}^{15} - 4\zeta_{40}^{11} - 2\zeta_{40}^{9} + 2\zeta_{40}^{7} - 2\zeta_{40}^{3} + 2\zeta_{40} \) Copy content Toggle raw display

Display \(\zeta_{40}^j\) in terms of \(\beta_i\)

\(\zeta_{40}\)\(=\) \( ( \beta_{15} + \beta_{12} + \beta_{8} + 2\beta_{7} - 2\beta_{3} ) / 8 \) Copy content Toggle raw display
\(\zeta_{40}^{2}\)\(=\) \( ( \beta_{11} + \beta_{5} - \beta_1 ) / 8 \) Copy content Toggle raw display
\(\zeta_{40}^{3}\)\(=\) \( ( \beta_{13} - \beta_{10} + \beta_{9} + \beta_{8} + \beta_{3} ) / 8 \) Copy content Toggle raw display
\(\zeta_{40}^{4}\)\(=\) \( ( \beta_{14} + 2\beta_{6} + 2 ) / 8 \) Copy content Toggle raw display
\(\zeta_{40}^{5}\)\(=\) \( ( \beta_{12} - \beta_{10} - \beta_{3} ) / 4 \) Copy content Toggle raw display
\(\zeta_{40}^{6}\)\(=\) \( ( \beta_{5} + \beta_{4} + \beta_1 ) / 8 \) Copy content Toggle raw display
\(\zeta_{40}^{7}\)\(=\) \( ( \beta_{13} + \beta_{10} + \beta_{9} - \beta_{8} - \beta_{3} ) / 8 \) Copy content Toggle raw display
\(\zeta_{40}^{8}\)\(=\) \( ( \beta_{6} + \beta_{2} - 1 ) / 4 \) Copy content Toggle raw display
\(\zeta_{40}^{9}\)\(=\) \( ( -\beta_{15} + \beta_{12} + \beta_{8} - 2\beta_{7} - 2\beta_{3} ) / 8 \) Copy content Toggle raw display
\(\zeta_{40}^{10}\)\(=\) \( ( \beta_1 ) / 2 \) Copy content Toggle raw display
\(\zeta_{40}^{11}\)\(=\) \( ( -\beta_{15} - \beta_{12} - \beta_{8} + 2\beta_{7} - 2\beta_{3} ) / 8 \) Copy content Toggle raw display
\(\zeta_{40}^{12}\)\(=\) \( ( -\beta_{6} + \beta_{2} + 1 ) / 4 \) Copy content Toggle raw display
\(\zeta_{40}^{13}\)\(=\) \( ( \beta_{13} + \beta_{10} - \beta_{9} + \beta_{8} - \beta_{3} ) / 8 \) Copy content Toggle raw display
\(\zeta_{40}^{14}\)\(=\) \( ( \beta_{5} - \beta_{4} + \beta_1 ) / 8 \) Copy content Toggle raw display
\(\zeta_{40}^{15}\)\(=\) \( ( -\beta_{12} - \beta_{10} - \beta_{3} ) / 4 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\zeta_{40}^j\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
737.1
−0.891007 − 0.453990i
−0.453990 + 0.891007i
−0.891007 + 0.453990i
−0.453990 − 0.891007i
0.156434 + 0.987688i
−0.987688 + 0.156434i
0.156434 − 0.987688i
−0.987688 − 0.156434i
0.987688 − 0.156434i
−0.156434 − 0.987688i
0.987688 + 0.156434i
−0.156434 + 0.987688i
0.453990 − 0.891007i
0.891007 + 0.453990i
0.453990 + 0.891007i
0.891007 − 0.453990i
0 −2.68999 0 − 2.35114i 0 − 3.70246i 0 4.23607 0
737.2 0 −2.68999 0 − 2.35114i 0 − 3.70246i 0 4.23607 0
737.3 0 −2.68999 0 2.35114i 0 3.70246i 0 4.23607 0
737.4 0 −2.68999 0 2.35114i 0 3.70246i 0 4.23607 0
737.5 0 −1.66251 0 − 3.80423i 0 0.540182i 0 −0.236068 0
737.6 0 −1.66251 0 − 3.80423i 0 0.540182i 0 −0.236068 0
737.7 0 −1.66251 0 3.80423i 0 − 0.540182i 0 −0.236068 0
737.8 0 −1.66251 0 3.80423i 0 − 0.540182i 0 −0.236068 0
737.9 0 1.66251 0 − 3.80423i 0 − 0.540182i 0 −0.236068 0
737.10 0 1.66251 0 − 3.80423i 0 − 0.540182i 0 −0.236068 0
737.11 0 1.66251 0 3.80423i 0 0.540182i 0 −0.236068 0
737.12 0 1.66251 0 3.80423i 0 0.540182i 0 −0.236068 0
737.13 0 2.68999 0 − 2.35114i 0 3.70246i 0 4.23607 0
737.14 0 2.68999 0 − 2.35114i 0 3.70246i 0 4.23607 0
737.15 0 2.68999 0 2.35114i 0 − 3.70246i 0 4.23607 0
737.16 0 2.68999 0 2.35114i 0 − 3.70246i 0 4.23607 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 737.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
8.b even 2 1 inner
8.d odd 2 1 inner
41.b even 2 1 inner
164.d odd 2 1 inner
328.c odd 2 1 inner
328.g 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.g.c ✓ 16
4.b odd 2 1 inner 2624.2.g.c ✓ 16
8.b even 2 1 inner 2624.2.g.c ✓ 16
8.d odd 2 1 inner 2624.2.g.c ✓ 16
41.b even 2 1 inner 2624.2.g.c ✓ 16
164.d odd 2 1 inner 2624.2.g.c ✓ 16
328.c odd 2 1 inner 2624.2.g.c ✓ 16
328.g even 2 1 inner 2624.2.g.c ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2624.2.g.c ✓ 16 1.a even 1 1 trivial
2624.2.g.c ✓ 16 4.b odd 2 1 inner
2624.2.g.c ✓ 16 8.b even 2 1 inner
2624.2.g.c ✓ 16 8.d odd 2 1 inner
2624.2.g.c ✓ 16 41.b even 2 1 inner
2624.2.g.c ✓ 16 164.d odd 2 1 inner
2624.2.g.c ✓ 16 328.c odd 2 1 inner
2624.2.g.c ✓ 16 328.g even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} - 10T_{3}^{2} + 20 \) acting on \(S_{2}^{\mathrm{new}}(2624, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( (T^{4} - 10 T^{2} + 20)^{4} \) Copy content Toggle raw display
$5$ \( (T^{4} + 20 T^{2} + 80)^{4} \) Copy content Toggle raw display
$7$ \( (T^{4} + 14 T^{2} + 4)^{4} \) Copy content Toggle raw display
$11$ \( (T^{4} - 50 T^{2} + 500)^{4} \) Copy content Toggle raw display
$13$ \( (T^{2} - 8)^{8} \) Copy content Toggle raw display
$17$ \( (T^{4} + 40 T^{2} + 320)^{4} \) Copy content Toggle raw display
$19$ \( (T^{4} - 10 T^{2} + 20)^{4} \) Copy content Toggle raw display
$23$ \( (T^{4} - 80 T^{2} + 1280)^{4} \) Copy content Toggle raw display
$29$ \( (T^{4} - 120 T^{2} + 1600)^{4} \) Copy content Toggle raw display
$31$ \( (T^{4} - 80 T^{2} + 1280)^{4} \) Copy content Toggle raw display
$37$ \( (T^{4} + 20 T^{2} + 80)^{4} \) Copy content Toggle raw display
$41$ \( (T^{4} - 4 T^{3} + \cdots + 1681)^{4} \) Copy content Toggle raw display
$43$ \( (T^{2} + 4)^{8} \) Copy content Toggle raw display
$47$ \( (T^{4} + 86 T^{2} + 1444)^{4} \) Copy content Toggle raw display
$53$ \( (T^{4} - 184 T^{2} + 7744)^{4} \) Copy content Toggle raw display
$59$ \( (T^{4} + 168 T^{2} + 1936)^{4} \) Copy content Toggle raw display
$61$ \( (T^{4} + 160 T^{2} + 1280)^{4} \) Copy content Toggle raw display
$67$ \( (T^{4} - 130 T^{2} + 2420)^{4} \) Copy content Toggle raw display
$71$ \( (T^{4} + 70 T^{2} + 100)^{4} \) Copy content Toggle raw display
$73$ \( (T^{2} + 6 T - 36)^{8} \) Copy content Toggle raw display
$79$ \( (T^{4} + 134 T^{2} + 3364)^{4} \) Copy content Toggle raw display
$83$ \( (T^{4} + 168 T^{2} + 1936)^{4} \) Copy content Toggle raw display
$89$ \( (T^{4} + 200 T^{2} + 320)^{4} \) Copy content Toggle raw display
$97$ \( (T^{4} + 200 T^{2} + 8000)^{4} \) Copy content Toggle raw display
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