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Properties

Label 624.4.q.f
Level $624$
Weight $4$
Character orbit 624.q
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(289,624)]
 
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, 2]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("624.289");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 624 = 2^{4} \cdot 3 \cdot 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 624.q (of order \(3\), 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 \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{673})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} + 169x^{2} + 168x + 28224 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 78)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( - 3 \beta_{2} + 3) q^{3} + (\beta_{3} + 6) q^{5} + (4 \beta_{2} + \beta_1) q^{7} - 9 \beta_{2} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - 3 \beta_{2} + 3) q^{3} + (\beta_{3} + 6) q^{5} + (4 \beta_{2} + \beta_1) q^{7} - 9 \beta_{2} q^{9} + (2 \beta_{3} - 20 \beta_{2} + \cdots + 18) q^{11}+ \cdots + ( - 18 \beta_{3} - 162) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 6 q^{3} + 26 q^{5} + 9 q^{7} - 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 6 q^{3} + 26 q^{5} + 9 q^{7} - 18 q^{9} + 38 q^{11} + 12 q^{13} + 39 q^{15} - 99 q^{17} - 16 q^{19} + 54 q^{21} - 14 q^{23} + 342 q^{25} - 108 q^{27} - 121 q^{29} + 234 q^{31} - 114 q^{33} - 278 q^{35} - 389 q^{37} + 9 q^{39} + 333 q^{41} + 645 q^{43} - 117 q^{45} - 1252 q^{47} + 309 q^{49} - 594 q^{51} - 1458 q^{53} + 920 q^{55} - 96 q^{57} + 574 q^{59} + 846 q^{61} + 81 q^{63} + 751 q^{65} - 1059 q^{67} + 42 q^{69} + 1346 q^{71} + 2888 q^{73} + 513 q^{75} - 1004 q^{77} + 1198 q^{79} - 162 q^{81} - 624 q^{83} - 1653 q^{85} + 363 q^{87} + 354 q^{89} + 1723 q^{91} + 351 q^{93} - 2796 q^{95} - 295 q^{97} - 684 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} + 169x^{2} + 168x + 28224 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{3} + 169\nu^{2} - 169\nu + 28224 ) / 28392 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} + 337 ) / 169 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + 168\beta_{2} + \beta _1 - 169 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 169\beta_{3} - 337 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
289.1
6.73556 − 11.6663i
−6.23556 + 10.8003i
6.73556 + 11.6663i
−6.23556 − 10.8003i
0 1.50000 + 2.59808i 0 −6.47112 0 8.73556 − 15.1304i 0 −4.50000 + 7.79423i 0
289.2 0 1.50000 + 2.59808i 0 19.4711 0 −4.23556 + 7.33621i 0 −4.50000 + 7.79423i 0
529.1 0 1.50000 − 2.59808i 0 −6.47112 0 8.73556 + 15.1304i 0 −4.50000 − 7.79423i 0
529.2 0 1.50000 − 2.59808i 0 19.4711 0 −4.23556 − 7.33621i 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.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 624.4.q.f 4
4.b odd 2 1 78.4.e.b ✓ 4
12.b even 2 1 234.4.h.g 4
13.c even 3 1 inner 624.4.q.f 4
52.i odd 6 1 1014.4.a.m 2
52.j odd 6 1 78.4.e.b ✓ 4
52.j odd 6 1 1014.4.a.s 2
52.l even 12 2 1014.4.b.j 4
156.p even 6 1 234.4.h.g 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
78.4.e.b ✓ 4 4.b odd 2 1
78.4.e.b ✓ 4 52.j odd 6 1
234.4.h.g 4 12.b even 2 1
234.4.h.g 4 156.p even 6 1
624.4.q.f 4 1.a even 1 1 trivial
624.4.q.f 4 13.c even 3 1 inner
1014.4.a.m 2 52.i odd 6 1
1014.4.a.s 2 52.j odd 6 1
1014.4.b.j 4 52.l even 12 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(624, [\chi])\):

\( T_{5}^{2} - 13T_{5} - 126 \) Copy content Toggle raw display
\( T_{7}^{4} - 9T_{7}^{3} + 229T_{7}^{2} + 1332T_{7} + 21904 \) 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
$5$ \( (T^{2} - 13 T - 126)^{2} \) Copy content Toggle raw display
$7$ \( T^{4} - 9 T^{3} + \cdots + 21904 \) Copy content Toggle raw display
$11$ \( T^{4} - 38 T^{3} + \cdots + 97344 \) Copy content Toggle raw display
$13$ \( T^{4} - 12 T^{3} + \cdots + 4826809 \) Copy content Toggle raw display
$17$ \( T^{4} + 99 T^{3} + \cdots + 876096 \) Copy content Toggle raw display
$19$ \( T^{4} + 16 T^{3} + \cdots + 114575616 \) Copy content Toggle raw display
$23$ \( T^{4} + 14 T^{3} + \cdots + 389376 \) Copy content Toggle raw display
$29$ \( T^{4} + 121 T^{3} + \cdots + 278823204 \) Copy content Toggle raw display
$31$ \( (T^{2} - 117 T - 784)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} + \cdots + 1418426244 \) Copy content Toggle raw display
$41$ \( T^{4} + \cdots + 2160018576 \) Copy content Toggle raw display
$43$ \( T^{4} + \cdots + 6996987904 \) Copy content Toggle raw display
$47$ \( (T^{2} + 626 T + 81144)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} + 729 T + 119232)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} + \cdots + 25787221056 \) Copy content Toggle raw display
$61$ \( T^{4} + \cdots + 31059480169 \) Copy content Toggle raw display
$67$ \( T^{4} + \cdots + 5515141696 \) Copy content Toggle raw display
$71$ \( T^{4} + \cdots + 44089920576 \) Copy content Toggle raw display
$73$ \( (T^{2} - 1444 T + 369859)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} - 599 T + 51844)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} + 312 T - 72576)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} + \cdots + 70471135296 \) Copy content Toggle raw display
$97$ \( T^{4} + \cdots + 101729102500 \) Copy content Toggle raw display
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