↑

Properties

Label 52.3.k.a
Level $52$
Weight $3$
Character orbit 52.k
Analytic conductor $1.417$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [52,3,Mod(33,52)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(52, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([0, 11]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("52.33");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 52 = 2^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 52.k (of order \(12\), degree \(4\), 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: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{12})\)
Coefficient field: 8.0.44991500544.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 38x^{6} + 555x^{4} - 3674x^{2} + 9409 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{5} - \beta_{4} + \beta_1) q^{3} + (\beta_{7} - \beta_{6} + \cdots - \beta_{2}) q^{5}+ \cdots + (\beta_{7} - 2 \beta_{6} - 2 \beta_{5} + \cdots - 3) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{5} - \beta_{4} + \beta_1) q^{3} + (\beta_{7} - \beta_{6} + \cdots - \beta_{2}) q^{5}+ \cdots + (8 \beta_{7} - 10 \beta_{6} + \cdots - 36) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 6 q^{5} + 4 q^{7} - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 6 q^{5} + 4 q^{7} - 6 q^{9} + 24 q^{11} + 18 q^{13} - 60 q^{15} - 54 q^{17} - 50 q^{19} - 54 q^{21} - 24 q^{23} + 36 q^{27} + 108 q^{29} + 176 q^{31} + 114 q^{33} - 30 q^{35} + 104 q^{37} + 120 q^{39} - 168 q^{41} - 198 q^{43} + 6 q^{45} - 96 q^{47} - 162 q^{49} - 72 q^{53} + 126 q^{55} + 318 q^{57} - 18 q^{61} + 180 q^{63} - 72 q^{65} - 326 q^{67} - 498 q^{69} - 162 q^{71} - 98 q^{73} - 240 q^{75} + 192 q^{79} + 240 q^{81} + 408 q^{83} + 318 q^{85} + 36 q^{87} + 54 q^{89} + 280 q^{91} - 66 q^{93} + 312 q^{95} + 182 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 38x^{6} + 555x^{4} - 3674x^{2} + 9409 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{4} - 19\nu^{2} + 2\nu + 97 ) / 4 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{4} + 19\nu^{2} + 2\nu - 97 ) / 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{7} - 38\nu^{5} + 458\nu^{3} - 1831\nu - 194 ) / 388 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 11\nu^{7} - 97\nu^{6} - 321\nu^{5} + 2716\nu^{4} + 3389\nu^{3} - 25414\nu^{2} - 12478\nu + 78279 ) / 5044 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -11\nu^{7} - 97\nu^{6} + 321\nu^{5} + 2716\nu^{4} - 3389\nu^{3} - 25414\nu^{2} + 12478\nu + 78279 ) / 5044 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{6} - 28\nu^{4} + 288\nu^{2} - 1067 ) / 26 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -84\nu^{7} + 97\nu^{6} + 2222\nu^{5} - 2716\nu^{4} - 19460\nu^{3} + 27936\nu^{2} + 54476\nu - 100977 ) / 5044 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_{2} + \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{6} + \beta_{5} + \beta_{4} + 10 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{7} - \beta_{6} - 10\beta_{5} + 10\beta_{4} - 4\beta_{3} + 9\beta_{2} + 9\beta _1 - 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 19\beta_{6} + 19\beta_{5} + 19\beta_{4} - 2\beta_{2} + 2\beta _1 + 93 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 34\beta_{7} - 17\beta_{6} - 196\beta_{5} + 196\beta_{4} - 112\beta_{3} + 74\beta_{2} + 74\beta _1 - 73 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 270\beta_{6} + 244\beta_{5} + 244\beta_{4} - 56\beta_{2} + 56\beta _1 + 791 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 376\beta_{7} - 188\beta_{6} - 2868\beta_{5} + 2868\beta_{4} - 2036\beta_{3} + 521\beta_{2} + 521\beta _1 - 1206 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(27\) \(41\)
\(\chi(n)\) \(1\) \(-\beta_{4}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
33.1
−2.83160 + 0.500000i
2.83160 + 0.500000i
−3.38852 + 0.500000i
3.38852 + 0.500000i
−2.83160 − 0.500000i
2.83160 − 0.500000i
−3.38852 − 0.500000i
3.38852 − 0.500000i
0 −0.982786 + 1.70224i 0 5.05105 + 5.05105i 0 0.383239 + 0.102689i 0 2.56826 + 4.44836i 0
33.2 0 1.84881 − 3.20224i 0 −2.68502 − 2.68502i 0 3.21484 + 0.861413i 0 −2.33621 − 4.04644i 0
37.1 0 −2.12727 − 3.68454i 0 −0.923296 − 0.923296i 0 −2.49330 − 9.30511i 0 −4.55057 + 7.88181i 0
37.2 0 1.26125 + 2.18454i 0 1.55727 + 1.55727i 0 0.895221 + 3.34101i 0 1.31852 − 2.28374i 0
41.1 0 −0.982786 − 1.70224i 0 5.05105 − 5.05105i 0 0.383239 − 0.102689i 0 2.56826 − 4.44836i 0
41.2 0 1.84881 + 3.20224i 0 −2.68502 + 2.68502i 0 3.21484 − 0.861413i 0 −2.33621 + 4.04644i 0
45.1 0 −2.12727 + 3.68454i 0 −0.923296 + 0.923296i 0 −2.49330 + 9.30511i 0 −4.55057 − 7.88181i 0
45.2 0 1.26125 − 2.18454i 0 1.55727 − 1.55727i 0 0.895221 − 3.34101i 0 1.31852 + 2.28374i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 33.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.f odd 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 52.3.k.a ✓ 8
3.b odd 2 1 468.3.cd.b 8
4.b odd 2 1 208.3.bd.e 8
13.c even 3 1 676.3.g.d 8
13.e even 6 1 676.3.g.c 8
13.f odd 12 1 inner 52.3.k.a ✓ 8
13.f odd 12 1 676.3.g.c 8
13.f odd 12 1 676.3.g.d 8
39.k even 12 1 468.3.cd.b 8
52.l even 12 1 208.3.bd.e 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
52.3.k.a ✓ 8 1.a even 1 1 trivial
52.3.k.a ✓ 8 13.f odd 12 1 inner
208.3.bd.e 8 4.b odd 2 1
208.3.bd.e 8 52.l even 12 1
468.3.cd.b 8 3.b odd 2 1
468.3.cd.b 8 39.k even 12 1
676.3.g.c 8 13.e even 6 1
676.3.g.c 8 13.f odd 12 1
676.3.g.d 8 13.c even 3 1
676.3.g.d 8 13.f odd 12 1

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} + 21 T^{6} + \cdots + 6084 \) Copy content Toggle raw display
$5$ \( T^{8} - 6 T^{7} + \cdots + 6084 \) Copy content Toggle raw display
$7$ \( T^{8} - 4 T^{7} + \cdots + 1936 \) Copy content Toggle raw display
$11$ \( T^{8} - 24 T^{7} + \cdots + 16451136 \) Copy content Toggle raw display
$13$ \( T^{8} - 18 T^{7} + \cdots + 815730721 \) Copy content Toggle raw display
$17$ \( T^{8} + 54 T^{7} + \cdots + 229613409 \) Copy content Toggle raw display
$19$ \( T^{8} + 50 T^{7} + \cdots + 64609444 \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 96160769604 \) Copy content Toggle raw display
$29$ \( T^{8} + \cdots + 69062263209 \) Copy content Toggle raw display
$31$ \( T^{8} + \cdots + 206628430096 \) Copy content Toggle raw display
$37$ \( T^{8} + \cdots + 45522062881 \) Copy content Toggle raw display
$41$ \( T^{8} + \cdots + 60225577281 \) Copy content Toggle raw display
$43$ \( T^{8} + \cdots + 67422688721424 \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 351853021584 \) Copy content Toggle raw display
$53$ \( (T^{4} + 36 T^{3} + \cdots - 613164)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 48466826323344 \) Copy content Toggle raw display
$61$ \( T^{8} + \cdots + 5645219185089 \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots + 178293048380164 \) Copy content Toggle raw display
$71$ \( T^{8} + \cdots + 59610474691524 \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots + 574467612546916 \) Copy content Toggle raw display
$79$ \( (T^{4} - 96 T^{3} + \cdots + 11795424)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 246401334374976 \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots + 26147207272356 \) Copy content Toggle raw display
$97$ \( T^{8} + \cdots + 151521790564 \) Copy content Toggle raw display
show more
show less