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Properties

Label 624.4.bv.c
Level $624$
Weight $4$
Character orbit 624.bv
Analytic conductor $36.817$
Analytic rank $0$
Dimension $4$
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] = [624,4,Mod(49,624)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(624, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 5]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("624.49");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 624 = 2^{4} \cdot 3 \cdot 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 624.bv (of order \(6\), degree \(2\), 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: \(36.8171918436\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{-17})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 17x^{2} + 289 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 39)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (3 \beta_{2} - 3) q^{3} + (2 \beta_{3} - 6 \beta_{2} + 3) q^{5} + (3 \beta_{3} - 11 \beta_{2} + \cdots + 22) q^{7}+ \cdots - 9 \beta_{2} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (3 \beta_{2} - 3) q^{3} + (2 \beta_{3} - 6 \beta_{2} + 3) q^{5} + (3 \beta_{3} - 11 \beta_{2} + \cdots + 22) q^{7}+ \cdots + (9 \beta_{3} - 378 \beta_{2} + 189) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 6 q^{3} + 66 q^{7} - 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 6 q^{3} + 66 q^{7} - 18 q^{9} + 126 q^{11} + 40 q^{13} + 54 q^{15} - 72 q^{17} - 222 q^{19} + 138 q^{23} + 120 q^{25} + 108 q^{27} - 6 q^{29} - 378 q^{33} - 402 q^{35} + 492 q^{37} - 168 q^{39} + 180 q^{41} - 470 q^{43} - 162 q^{45} + 346 q^{49} + 432 q^{51} - 2268 q^{53} + 446 q^{55} - 2160 q^{59} - 160 q^{61} - 594 q^{63} - 804 q^{65} + 498 q^{67} + 414 q^{69} + 1314 q^{71} - 180 q^{75} + 2976 q^{77} - 8 q^{79} - 162 q^{81} - 852 q^{85} - 18 q^{87} - 252 q^{89} + 1668 q^{91} - 1404 q^{93} + 54 q^{95} - 336 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 17x^{2} + 289 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} ) / 17 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} ) / 17 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 17\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 17\beta_{3} \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/624\mathbb{Z}\right)^\times\).

\(n\) \(79\) \(145\) \(209\) \(469\)
\(\chi(n)\) \(1\) \(\beta_{2}\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
49.1
3.57071 − 2.06155i
−3.57071 + 2.06155i
−3.57071 − 2.06155i
3.57071 + 2.06155i
0 −1.50000 − 2.59808i 0 − 3.05006i 0 5.78786 + 3.34162i 0 −4.50000 + 7.79423i 0
49.2 0 −1.50000 − 2.59808i 0 13.4424i 0 27.2121 + 15.7109i 0 −4.50000 + 7.79423i 0
433.1 0 −1.50000 + 2.59808i 0 − 13.4424i 0 27.2121 − 15.7109i 0 −4.50000 − 7.79423i 0
433.2 0 −1.50000 + 2.59808i 0 3.05006i 0 5.78786 − 3.34162i 0 −4.50000 − 7.79423i 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.e even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 624.4.bv.c 4
4.b odd 2 1 39.4.j.b ✓ 4
12.b even 2 1 117.4.q.d 4
13.e even 6 1 inner 624.4.bv.c 4
52.i odd 6 1 39.4.j.b ✓ 4
52.i odd 6 1 507.4.b.e 4
52.j odd 6 1 507.4.b.e 4
52.l even 12 2 507.4.a.k 4
156.r even 6 1 117.4.q.d 4
156.v odd 12 2 1521.4.a.z 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.4.j.b ✓ 4 4.b odd 2 1
39.4.j.b ✓ 4 52.i odd 6 1
117.4.q.d 4 12.b even 2 1
117.4.q.d 4 156.r even 6 1
507.4.a.k 4 52.l even 12 2
507.4.b.e 4 52.i odd 6 1
507.4.b.e 4 52.j odd 6 1
624.4.bv.c 4 1.a even 1 1 trivial
624.4.bv.c 4 13.e even 6 1 inner
1521.4.a.z 4 156.v odd 12 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} + 190T_{5}^{2} + 1681 \) acting on \(S_{4}^{\mathrm{new}}(624, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( (T^{2} + 3 T + 9)^{2} \) Copy content Toggle raw display
$5$ \( T^{4} + 190T^{2} + 1681 \) Copy content Toggle raw display
$7$ \( T^{4} - 66 T^{3} + \cdots + 44100 \) Copy content Toggle raw display
$11$ \( T^{4} - 126 T^{3} + \cdots + 1705636 \) Copy content Toggle raw display
$13$ \( T^{4} - 40 T^{3} + \cdots + 4826809 \) Copy content Toggle raw display
$17$ \( T^{4} + 72 T^{3} + \cdots + 1550025 \) Copy content Toggle raw display
$19$ \( T^{4} + 222 T^{3} + \cdots + 7452900 \) Copy content Toggle raw display
$23$ \( T^{4} - 138 T^{3} + \cdots + 5116644 \) Copy content Toggle raw display
$29$ \( T^{4} + 6 T^{3} + \cdots + 608855625 \) Copy content Toggle raw display
$31$ \( T^{4} + 96480 T^{2} + 137733696 \) Copy content Toggle raw display
$37$ \( T^{4} - 492 T^{3} + \cdots + 203148009 \) Copy content Toggle raw display
$41$ \( T^{4} - 180 T^{3} + \cdots + 7198489 \) Copy content Toggle raw display
$43$ \( T^{4} + \cdots + 1914062500 \) Copy content Toggle raw display
$47$ \( T^{4} + \cdots + 4779509956 \) Copy content Toggle raw display
$53$ \( (T^{2} + 1134 T + 304965)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} + \cdots + 150320594944 \) Copy content Toggle raw display
$61$ \( T^{4} + \cdots + 144110585161 \) Copy content Toggle raw display
$67$ \( T^{4} - 498 T^{3} + \cdots + 173448900 \) Copy content Toggle raw display
$71$ \( T^{4} + \cdots + 19312660900 \) Copy content Toggle raw display
$73$ \( (T^{2} + 151875)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} + 4 T - 809672)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + \cdots + 798145692100 \) Copy content Toggle raw display
$89$ \( T^{4} + \cdots + 980686167616 \) Copy content Toggle raw display
$97$ \( T^{4} + \cdots + 7495877379600 \) Copy content Toggle raw display
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