Properties

Label 624.4.a.q
Level $624$
Weight $4$
Character orbit 624.a
Self dual yes
Analytic conductor $36.817$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [624,4,Mod(1,624)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(624, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("624.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 624 = 2^{4} \cdot 3 \cdot 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 624.a (trivial)

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: yes
Analytic conductor: \(36.8171918436\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{113}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 28 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 312)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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 \(\beta = \sqrt{113}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 3 q^{3} + (\beta + 3) q^{5} + (\beta + 5) q^{7} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 3 q^{3} + (\beta + 3) q^{5} + (\beta + 5) q^{7} + 9 q^{9} + (4 \beta + 8) q^{11} + 13 q^{13} + (3 \beta + 9) q^{15} + 2 q^{17} + ( - 5 \beta + 35) q^{19} + (3 \beta + 15) q^{21} + 64 q^{23} + (6 \beta - 3) q^{25} + 27 q^{27} + ( - 6 \beta - 40) q^{29} + ( - 15 \beta + 125) q^{31} + (12 \beta + 24) q^{33} + (8 \beta + 128) q^{35} + ( - 18 \beta - 76) q^{37} + 39 q^{39} + ( - 7 \beta - 73) q^{41} + ( - 8 \beta + 252) q^{43} + (9 \beta + 27) q^{45} + (6 \beta + 262) q^{47} + (10 \beta - 205) q^{49} + 6 q^{51} + ( - 8 \beta - 26) q^{53} + (20 \beta + 476) q^{55} + ( - 15 \beta + 105) q^{57} + ( - 10 \beta + 82) q^{59} + ( - 46 \beta - 152) q^{61} + (9 \beta + 45) q^{63} + (13 \beta + 39) q^{65} + ( - 7 \beta + 457) q^{67} + 192 q^{69} - 48 \beta q^{71} + ( - 6 \beta - 228) q^{73} + (18 \beta - 9) q^{75} + (28 \beta + 492) q^{77} + (36 \beta + 412) q^{79} + 81 q^{81} + ( - 50 \beta - 414) q^{83} + (2 \beta + 6) q^{85} + ( - 18 \beta - 120) q^{87} + ( - 9 \beta + 413) q^{89} + (13 \beta + 65) q^{91} + ( - 45 \beta + 375) q^{93} + (20 \beta - 460) q^{95} + (122 \beta + 276) q^{97} + (36 \beta + 72) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 6 q^{3} + 6 q^{5} + 10 q^{7} + 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 6 q^{3} + 6 q^{5} + 10 q^{7} + 18 q^{9} + 16 q^{11} + 26 q^{13} + 18 q^{15} + 4 q^{17} + 70 q^{19} + 30 q^{21} + 128 q^{23} - 6 q^{25} + 54 q^{27} - 80 q^{29} + 250 q^{31} + 48 q^{33} + 256 q^{35} - 152 q^{37} + 78 q^{39} - 146 q^{41} + 504 q^{43} + 54 q^{45} + 524 q^{47} - 410 q^{49} + 12 q^{51} - 52 q^{53} + 952 q^{55} + 210 q^{57} + 164 q^{59} - 304 q^{61} + 90 q^{63} + 78 q^{65} + 914 q^{67} + 384 q^{69} - 456 q^{73} - 18 q^{75} + 984 q^{77} + 824 q^{79} + 162 q^{81} - 828 q^{83} + 12 q^{85} - 240 q^{87} + 826 q^{89} + 130 q^{91} + 750 q^{93} - 920 q^{95} + 552 q^{97} + 144 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−4.81507
5.81507
0 3.00000 0 −7.63015 0 −5.63015 0 9.00000 0
1.2 0 3.00000 0 13.6301 0 15.6301 0 9.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( +1 \)
\(3\) \( -1 \)
\(13\) \( -1 \)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 624.4.a.q 2
3.b odd 2 1 1872.4.a.x 2
4.b odd 2 1 312.4.a.c 2
8.b even 2 1 2496.4.a.v 2
8.d odd 2 1 2496.4.a.be 2
12.b even 2 1 936.4.a.d 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
312.4.a.c 2 4.b odd 2 1
624.4.a.q 2 1.a even 1 1 trivial
936.4.a.d 2 12.b even 2 1
1872.4.a.x 2 3.b odd 2 1
2496.4.a.v 2 8.b even 2 1
2496.4.a.be 2 8.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 \(S_{4}^{\mathrm{new}}(\Gamma_0(624))\):

\( T_{5}^{2} - 6T_{5} - 104 \) Copy content Toggle raw display
\( T_{7}^{2} - 10T_{7} - 88 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( (T - 3)^{2} \) Copy content Toggle raw display
$5$ \( T^{2} - 6T - 104 \) Copy content Toggle raw display
$7$ \( T^{2} - 10T - 88 \) Copy content Toggle raw display
$11$ \( T^{2} - 16T - 1744 \) Copy content Toggle raw display
$13$ \( (T - 13)^{2} \) Copy content Toggle raw display
$17$ \( (T - 2)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} - 70T - 1600 \) Copy content Toggle raw display
$23$ \( (T - 64)^{2} \) Copy content Toggle raw display
$29$ \( T^{2} + 80T - 2468 \) Copy content Toggle raw display
$31$ \( T^{2} - 250T - 9800 \) Copy content Toggle raw display
$37$ \( T^{2} + 152T - 30836 \) Copy content Toggle raw display
$41$ \( T^{2} + 146T - 208 \) Copy content Toggle raw display
$43$ \( T^{2} - 504T + 56272 \) Copy content Toggle raw display
$47$ \( T^{2} - 524T + 64576 \) Copy content Toggle raw display
$53$ \( T^{2} + 52T - 6556 \) Copy content Toggle raw display
$59$ \( T^{2} - 164T - 4576 \) Copy content Toggle raw display
$61$ \( T^{2} + 304T - 216004 \) Copy content Toggle raw display
$67$ \( T^{2} - 914T + 203312 \) Copy content Toggle raw display
$71$ \( T^{2} - 260352 \) Copy content Toggle raw display
$73$ \( T^{2} + 456T + 47916 \) Copy content Toggle raw display
$79$ \( T^{2} - 824T + 23296 \) Copy content Toggle raw display
$83$ \( T^{2} + 828T - 111104 \) Copy content Toggle raw display
$89$ \( T^{2} - 826T + 161416 \) Copy content Toggle raw display
$97$ \( T^{2} - 552 T - 1605716 \) Copy content Toggle raw display
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