Properties

Label 52.3.g.a
Level $52$
Weight $3$
Character orbit 52.g
Analytic conductor $1.417$
Analytic rank $0$
Dimension $6$
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] = [52,3,Mod(5,52)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(52, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 3]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("52.5");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 52 = 2^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 52.g (of order \(4\), degree \(2\), 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: \(1.41689737467\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(i)\)
Coefficient field: 6.0.20819026944.3
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} + 42x^{4} + 441x^{2} + 36 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{3} q^{3} + ( - \beta_{5} + \beta_{2} - 1) q^{5} + (\beta_{4} - \beta_{3} + \beta_{2} + \beta_1 + 1) q^{7} + (\beta_{5} - \beta_{4} + \beta_{3} + 5) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{3} q^{3} + ( - \beta_{5} + \beta_{2} - 1) q^{5} + (\beta_{4} - \beta_{3} + \beta_{2} + \beta_1 + 1) q^{7} + (\beta_{5} - \beta_{4} + \beta_{3} + 5) q^{9} + (2 \beta_{3} - 3 \beta_{2} - 2 \beta_1 - 3) q^{11} + ( - \beta_{4} - 2 \beta_{3} + \beta_1 - 5) q^{13} + (\beta_{5} - 4 \beta_{3} - 4 \beta_{2} - 4 \beta_1 + 4) q^{15} + (\beta_{5} + \beta_{4} - 2 \beta_{2} + 5 \beta_1) q^{17} + ( - \beta_{3} + 13 \beta_{2} - \beta_1 - 13) q^{19} + (\beta_{4} - 3 \beta_{3} - 10 \beta_{2} + 3 \beta_1 - 10) q^{21} + ( - \beta_{5} - \beta_{4} + 20 \beta_{2}) q^{23} + ( - \beta_{5} - \beta_{4} - 27 \beta_{2} - 7 \beta_1) q^{25} + (3 \beta_{3} + 6) q^{27} + ( - \beta_{5} + \beta_{4} + 6 \beta_{3} + 10) q^{29} + (4 \beta_{5} + 5 \beta_{3} + \beta_{2} + 5 \beta_1 - 1) q^{31} + ( - 4 \beta_{4} - \beta_{3} + 28 \beta_{2} + \beta_1 + 28) q^{33} + ( - 4 \beta_{5} + 4 \beta_{4} + \beta_{3} + 40) q^{35} + (\beta_{4} - 9 \beta_{2} - 9) q^{37} + ( - \beta_{5} + 4 \beta_{4} - 4 \beta_{3} - 18 \beta_{2} - 2 \beta_1 - 32) q^{39} + ( - 4 \beta_{5} + 3 \beta_{3} - 11 \beta_{2} + 3 \beta_1 + 11) q^{41} + ( - 4 \beta_{5} - 4 \beta_{4} + 14 \beta_{2} - \beta_1) q^{43} + (3 \beta_{3} + 51 \beta_{2} + 3 \beta_1 - 51) q^{45} + ( - 5 \beta_{4} + 7 \beta_{3} - 19 \beta_{2} - 7 \beta_1 - 19) q^{47} + (5 \beta_{5} + 5 \beta_{4} + 15 \beta_{2} - 3 \beta_1) q^{49} + (4 \beta_{5} + 4 \beta_{4} - 62 \beta_{2} + 13 \beta_1) q^{51} + (\beta_{5} - \beta_{4} - 4 \beta_{3} + 38) q^{53} + (5 \beta_{5} - 5 \beta_{4} - 16 \beta_{3} + 22) q^{55} + ( - 2 \beta_{5} - 14 \beta_{3} + 14 \beta_{2} - 14 \beta_1 - 14) q^{57} + (2 \beta_{4} + 3 \beta_{3} + 31 \beta_{2} - 3 \beta_1 + 31) q^{59} + (2 \beta_{5} - 2 \beta_{4} - 8 \beta_{3} + 34) q^{61} + ( - 4 \beta_{4} - 7 \beta_{3} - 47 \beta_{2} + 7 \beta_1 - 47) q^{63} + (5 \beta_{5} - \beta_{4} + 19 \beta_{3} - \beta_{2} + 4 \beta_1 - 57) q^{65} + (2 \beta_{5} + 8 \beta_{3} - 13 \beta_{2} + 8 \beta_1 + 13) q^{67} + (\beta_{5} + \beta_{4} - 8 \beta_{2} - 26 \beta_1) q^{69} + (\beta_{5} - 5 \beta_{3} + 35 \beta_{2} - 5 \beta_1 - 35) q^{71} + (6 \beta_{4} - 7 \beta_{3} - 37 \beta_{2} + 7 \beta_1 - 37) q^{73} + ( - 6 \beta_{5} - 6 \beta_{4} + 90 \beta_{2} + 14 \beta_1) q^{75} + ( - \beta_{5} - \beta_{4} - 46 \beta_{2} + 6 \beta_1) q^{77} + ( - 5 \beta_{5} + 5 \beta_{4} - 12 \beta_{3} - 10) q^{79} + ( - 6 \beta_{5} + 6 \beta_{4} - 3) q^{81} + ( - 8 \beta_{5} + 6 \beta_{3} - 13 \beta_{2} + 6 \beta_1 + 13) q^{83} + (7 \beta_{4} + 13 \beta_{3} + 32 \beta_{2} - 13 \beta_1 + 32) q^{85} + (7 \beta_{5} - 7 \beta_{4} + 10 \beta_{3} + 92) q^{87} + ( - 7 \beta_{3} + 39 \beta_{2} + 7 \beta_1 + 39) q^{89} + ( - 6 \beta_{5} - 12 \beta_{4} + 8 \beta_{3} - 17 \beta_{2} - 15 \beta_1 + 5) q^{91} + (6 \beta_{5} + 16 \beta_{3} - 54 \beta_{2} + 16 \beta_1 + 54) q^{93} + (12 \beta_{5} + 12 \beta_{4} - 18 \beta_{2} + 8 \beta_1) q^{95} + ( - 6 \beta_{5} + 8 \beta_{3} + 59 \beta_{2} + 8 \beta_1 - 59) q^{97} + (6 \beta_{4} + 21 \beta_{3} - 3 \beta_{2} - 21 \beta_1 - 3) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 6 q^{5} + 6 q^{7} + 30 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 6 q^{5} + 6 q^{7} + 30 q^{9} - 18 q^{11} - 30 q^{13} + 24 q^{15} - 78 q^{19} - 60 q^{21} + 36 q^{27} + 60 q^{29} - 6 q^{31} + 168 q^{33} + 240 q^{35} - 54 q^{37} - 192 q^{39} + 66 q^{41} - 306 q^{45} - 114 q^{47} + 228 q^{53} + 132 q^{55} - 84 q^{57} + 186 q^{59} + 204 q^{61} - 282 q^{63} - 342 q^{65} + 78 q^{67} - 210 q^{71} - 222 q^{73} - 60 q^{79} - 18 q^{81} + 78 q^{83} + 192 q^{85} + 552 q^{87} + 234 q^{89} + 30 q^{91} + 324 q^{93} - 354 q^{97} - 18 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} + 42x^{4} + 441x^{2} + 36 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} + 21\nu ) / 6 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{4} + 21\nu^{2} ) / 6 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{5} + \nu^{4} + 35\nu^{3} + 27\nu^{2} + 288\nu + 84 ) / 12 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{5} - \nu^{4} + 35\nu^{3} - 27\nu^{2} + 288\nu - 84 ) / 12 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{5} + \beta_{4} - \beta_{3} - 14 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 6\beta_{2} - 21\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 21\beta_{5} - 21\beta_{4} + 27\beta_{3} + 294 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 6\beta_{5} + 6\beta_{4} - 210\beta_{2} + 447\beta_1 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/52\mathbb{Z}\right)^\times\).

\(n\) \(27\) \(41\)
\(\chi(n)\) \(1\) \(-\beta_{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
5.1
4.43242i
0.286838i
4.71926i
4.43242i
0.286838i
4.71926i
0 −4.43242 0 −6.03938 + 6.03938i 0 0.393041 + 0.393041i 0 10.6463 0
5.2 0 −0.286838 0 5.81544 5.81544i 0 8.10228 + 8.10228i 0 −8.91772 0
5.3 0 4.71926 0 −2.77606 + 2.77606i 0 −5.49532 5.49532i 0 13.2714 0
21.1 0 −4.43242 0 −6.03938 6.03938i 0 0.393041 0.393041i 0 10.6463 0
21.2 0 −0.286838 0 5.81544 + 5.81544i 0 8.10228 8.10228i 0 −8.91772 0
21.3 0 4.71926 0 −2.77606 2.77606i 0 −5.49532 + 5.49532i 0 13.2714 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 5.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.d odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 52.3.g.a 6
3.b odd 2 1 468.3.m.c 6
4.b odd 2 1 208.3.t.d 6
5.b even 2 1 1300.3.t.a 6
5.c odd 4 1 1300.3.k.a 6
5.c odd 4 1 1300.3.k.b 6
13.b even 2 1 676.3.g.b 6
13.d odd 4 1 inner 52.3.g.a 6
13.d odd 4 1 676.3.g.b 6
39.f even 4 1 468.3.m.c 6
52.f even 4 1 208.3.t.d 6
65.f even 4 1 1300.3.k.b 6
65.g odd 4 1 1300.3.t.a 6
65.k even 4 1 1300.3.k.a 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
52.3.g.a 6 1.a even 1 1 trivial
52.3.g.a 6 13.d odd 4 1 inner
208.3.t.d 6 4.b odd 2 1
208.3.t.d 6 52.f even 4 1
468.3.m.c 6 3.b odd 2 1
468.3.m.c 6 39.f even 4 1
676.3.g.b 6 13.b even 2 1
676.3.g.b 6 13.d odd 4 1
1300.3.k.a 6 5.c odd 4 1
1300.3.k.a 6 65.k even 4 1
1300.3.k.b 6 5.c odd 4 1
1300.3.k.b 6 65.f even 4 1
1300.3.t.a 6 5.b even 2 1
1300.3.t.a 6 65.g odd 4 1

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(52, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( (T^{3} - 21 T - 6)^{2} \) Copy content Toggle raw display
$5$ \( T^{6} + 6 T^{5} + 18 T^{4} + \cdots + 76050 \) Copy content Toggle raw display
$7$ \( T^{6} - 6 T^{5} + 18 T^{4} + \cdots + 2450 \) Copy content Toggle raw display
$11$ \( T^{6} + 18 T^{5} + 162 T^{4} + \cdots + 596232 \) Copy content Toggle raw display
$13$ \( T^{6} + 30 T^{5} + 225 T^{4} + \cdots + 4826809 \) Copy content Toggle raw display
$17$ \( T^{6} + 1122 T^{4} + \cdots + 14745600 \) Copy content Toggle raw display
$19$ \( T^{6} + 78 T^{5} + 3042 T^{4} + \cdots + 29799200 \) Copy content Toggle raw display
$23$ \( T^{6} + 1500 T^{4} + \cdots + 20358144 \) Copy content Toggle raw display
$29$ \( (T^{3} - 30 T^{2} - 750 T + 20316)^{2} \) Copy content Toggle raw display
$31$ \( T^{6} + 6 T^{5} + 18 T^{4} + \cdots + 56180000 \) Copy content Toggle raw display
$37$ \( T^{6} + 54 T^{5} + 1458 T^{4} + \cdots + 873842 \) Copy content Toggle raw display
$41$ \( T^{6} - 66 T^{5} + \cdots + 1839089952 \) Copy content Toggle raw display
$43$ \( T^{6} + 5238 T^{4} + \cdots + 4493289024 \) Copy content Toggle raw display
$47$ \( T^{6} + 114 T^{5} + \cdots + 7858818450 \) Copy content Toggle raw display
$53$ \( (T^{3} - 114 T^{2} + 3750 T - 38076)^{2} \) Copy content Toggle raw display
$59$ \( T^{6} - 186 T^{5} + \cdots + 1542123648 \) Copy content Toggle raw display
$61$ \( (T^{3} - 102 T^{2} + 1140 T - 2712)^{2} \) Copy content Toggle raw display
$67$ \( T^{6} - 78 T^{5} + 3042 T^{4} + \cdots + 30952712 \) Copy content Toggle raw display
$71$ \( T^{6} + 210 T^{5} + \cdots + 2561418738 \) Copy content Toggle raw display
$73$ \( T^{6} + 222 T^{5} + \cdots + 2377740800 \) Copy content Toggle raw display
$79$ \( (T^{3} + 30 T^{2} - 5034 T - 37452)^{2} \) Copy content Toggle raw display
$83$ \( T^{6} - 78 T^{5} + \cdots + 73972426248 \) Copy content Toggle raw display
$89$ \( T^{6} - 234 T^{5} + \cdots + 2347495200 \) Copy content Toggle raw display
$97$ \( T^{6} + 354 T^{5} + \cdots + 52696863368 \) Copy content Toggle raw display
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