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Properties

Label 456.2.bg.b
Level $456$
Weight $2$
Character orbit 456.bg
Analytic conductor $3.641$
Analytic rank $0$
Dimension $12$
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] = [456,2,Mod(25,456)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(456, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 14]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("456.25");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 456 = 2^{3} \cdot 3 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 456.bg (of order \(9\), degree \(6\), 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: \(3.64117833217\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(2\) over \(\Q(\zeta_{9})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 6 x^{11} - 15 x^{10} + 130 x^{9} + 117 x^{8} - 1314 x^{7} - 740 x^{6} + 7431 x^{5} + \cdots + 30837 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{4} q^{3} + (\beta_{11} + \beta_{9} + \beta_{6} + 1) q^{5} + ( - \beta_{10} + \beta_{9} - \beta_{4}) q^{7} - \beta_{10} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{4} q^{3} + (\beta_{11} + \beta_{9} + \beta_{6} + 1) q^{5} + ( - \beta_{10} + \beta_{9} - \beta_{4}) q^{7} - \beta_{10} q^{9} + (\beta_{11} - \beta_{8} - \beta_{5} + \cdots - 1) q^{11}+ \cdots + (\beta_{10} - \beta_{9} + \beta_{8} + \cdots - 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 3 q^{5} - 6 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 3 q^{5} - 6 q^{7} - 3 q^{11} - 3 q^{13} + 3 q^{15} + 3 q^{17} - 6 q^{21} - 12 q^{23} + 9 q^{25} - 6 q^{27} + 9 q^{29} - 21 q^{31} - 6 q^{33} - 6 q^{35} + 30 q^{37} + 6 q^{39} + 15 q^{41} + 3 q^{43} - 3 q^{45} - 18 q^{47} + 24 q^{49} - 6 q^{51} + 24 q^{53} + 60 q^{55} - 9 q^{57} - 21 q^{59} - 24 q^{61} - 6 q^{63} - 21 q^{65} + 15 q^{67} - 21 q^{71} - 3 q^{73} + 18 q^{75} - 6 q^{77} - 63 q^{79} - 12 q^{83} + 9 q^{85} - 15 q^{87} - 15 q^{89} - 3 q^{93} + 12 q^{95} - 9 q^{97} - 6 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 6 x^{11} - 15 x^{10} + 130 x^{9} + 117 x^{8} - 1314 x^{7} - 740 x^{6} + 7431 x^{5} + \cdots + 30837 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 226339 \nu^{11} - 56586205 \nu^{10} + 203701556 \nu^{9} + 1677350116 \nu^{8} + \cdots + 1107403281211 ) / 79719451993 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 226339 \nu^{11} - 54096476 \nu^{10} + 349711849 \nu^{9} + 1001630830 \nu^{8} + \cdots + 1075539815538 ) / 79719451993 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 13065842 \nu^{11} + 124947624 \nu^{10} - 113137009 \nu^{9} - 1955364614 \nu^{8} + \cdots - 175764475474 ) / 79719451993 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 13065842 \nu^{11} - 18776638 \nu^{10} - 417717921 \nu^{9} + 493181455 \nu^{8} + \cdots - 75870776038 ) / 79719451993 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 14362101 \nu^{11} - 231327984 \nu^{10} + 659024556 \nu^{9} + 3739241955 \nu^{8} + \cdots + 1472238317224 ) / 79719451993 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 14362101 \nu^{11} - 92370246 \nu^{10} + 959466594 \nu^{9} + 1596644404 \nu^{8} + \cdots + 1286282896578 ) / 79719451993 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 16148519 \nu^{11} + 98329541 \nu^{10} + 12461778 \nu^{9} - 919336886 \nu^{8} + \cdots - 89768863081 ) / 79719451993 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 16148519 \nu^{11} + 79304168 \nu^{10} + 107588643 \nu^{9} - 953142826 \nu^{8} + \cdots + 228835027814 ) / 79719451993 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 7938 \nu^{11} - 43659 \nu^{10} - 134545 \nu^{9} + 932895 \nu^{8} + 1223622 \nu^{7} + \cdots + 39342241 ) / 23803957 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 30404053 \nu^{11} + 123649485 \nu^{10} + 680071441 \nu^{9} - 2586105975 \nu^{8} + \cdots + 18486113909 ) / 79719451993 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 43469895 \nu^{11} - 229571736 \nu^{10} - 662061297 \nu^{9} + 4575276529 \nu^{8} + \cdots + 237271730635 ) / 79719451993 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( -\beta_{9} + \beta_{4} - \beta_{3} - \beta_{2} + \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{11} - \beta_{10} - \beta_{9} - \beta_{8} + \beta_{7} + \beta_{4} - 2\beta_{3} - \beta_{2} + \beta _1 + 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( - 3 \beta_{11} + 2 \beta_{10} - 3 \beta_{9} - \beta_{8} + 2 \beta_{7} + \beta_{6} - \beta_{5} + \cdots + 7 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 22 \beta_{11} - 12 \beta_{10} - 5 \beta_{9} - 13 \beta_{8} + 15 \beta_{7} + 5 \beta_{6} + \beta_{5} + \cdots + 26 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 74 \beta_{11} + 21 \beta_{10} + 40 \beta_{9} - 34 \beta_{8} + 31 \beta_{7} + 26 \beta_{6} - 11 \beta_{5} + \cdots + 74 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 291 \beta_{11} - 31 \beta_{10} + 132 \beta_{9} - 158 \beta_{8} + 144 \beta_{7} + 111 \beta_{6} + \cdots + 133 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 1051 \beta_{11} + 295 \beta_{10} + 1053 \beta_{9} - 508 \beta_{8} + 325 \beta_{7} + 392 \beta_{6} + \cdots + 509 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 3098 \beta_{11} + 1096 \beta_{10} + 3585 \beta_{9} - 1776 \beta_{8} + 1114 \beta_{7} + 1464 \beta_{6} + \cdots + 540 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 10608 \beta_{11} + 5804 \beta_{10} + 15725 \beta_{9} - 5680 \beta_{8} + 2594 \beta_{7} + 4288 \beta_{6} + \cdots + 1946 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 26258 \beta_{11} + 26117 \beta_{10} + 52685 \beta_{9} - 17623 \beta_{8} + 7050 \beta_{7} + \cdots - 2323 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 75137 \beta_{11} + 97644 \beta_{10} + 181579 \beta_{9} - 53292 \beta_{8} + 15029 \beta_{7} + \cdots - 16589 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/456\mathbb{Z}\right)^\times\).

\(n\) \(97\) \(229\) \(305\) \(343\)
\(\chi(n)\) \(\beta_{3} + \beta_{10}\) \(1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
25.1
2.32042 + 0.342020i
−1.32042 + 0.342020i
2.32042 − 0.342020i
−1.32042 − 0.342020i
−2.35218 − 0.642788i
3.35218 − 0.642788i
2.90855 − 0.984808i
−1.90855 − 0.984808i
−2.35218 + 0.642788i
3.35218 + 0.642788i
2.90855 + 0.984808i
−1.90855 + 0.984808i
0 −0.939693 − 0.342020i 0 −0.0661133 − 0.374947i 0 −0.673648 + 1.16679i 0 0.766044 + 0.642788i 0
25.2 0 −0.939693 − 0.342020i 0 0.566113 + 3.21059i 0 −0.673648 + 1.16679i 0 0.766044 + 0.642788i 0
73.1 0 −0.939693 + 0.342020i 0 −0.0661133 + 0.374947i 0 −0.673648 − 1.16679i 0 0.766044 − 0.642788i 0
73.2 0 −0.939693 + 0.342020i 0 0.566113 − 3.21059i 0 −0.673648 − 1.16679i 0 0.766044 − 0.642788i 0
169.1 0 0.766044 + 0.642788i 0 −2.43018 + 0.884512i 0 0.439693 + 0.761570i 0 0.173648 + 0.984808i 0
169.2 0 0.766044 + 0.642788i 0 2.93018 − 1.06650i 0 0.439693 + 0.761570i 0 0.173648 + 0.984808i 0
289.1 0 0.173648 + 0.984808i 0 −1.59505 + 1.33841i 0 −1.26604 + 2.19285i 0 −0.939693 + 0.342020i 0
289.2 0 0.173648 + 0.984808i 0 2.09505 − 1.75796i 0 −1.26604 + 2.19285i 0 −0.939693 + 0.342020i 0
313.1 0 0.766044 − 0.642788i 0 −2.43018 − 0.884512i 0 0.439693 − 0.761570i 0 0.173648 − 0.984808i 0
313.2 0 0.766044 − 0.642788i 0 2.93018 + 1.06650i 0 0.439693 − 0.761570i 0 0.173648 − 0.984808i 0
385.1 0 0.173648 − 0.984808i 0 −1.59505 − 1.33841i 0 −1.26604 − 2.19285i 0 −0.939693 − 0.342020i 0
385.2 0 0.173648 − 0.984808i 0 2.09505 + 1.75796i 0 −1.26604 − 2.19285i 0 −0.939693 − 0.342020i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 25.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.e even 9 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 456.2.bg.b ✓ 12
4.b odd 2 1 912.2.bo.i 12
19.e even 9 1 inner 456.2.bg.b ✓ 12
19.e even 9 1 8664.2.a.bk 6
19.f odd 18 1 8664.2.a.bh 6
76.l odd 18 1 912.2.bo.i 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
456.2.bg.b ✓ 12 1.a even 1 1 trivial
456.2.bg.b ✓ 12 19.e even 9 1 inner
912.2.bo.i 12 4.b odd 2 1
912.2.bo.i 12 76.l odd 18 1
8664.2.a.bh 6 19.f odd 18 1
8664.2.a.bk 6 19.e even 9 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{12} - 3 T_{5}^{11} + 20 T_{5}^{9} - 60 T_{5}^{8} - 9 T_{5}^{7} + 835 T_{5}^{6} - 651 T_{5}^{5} + \cdots + 3249 \) acting on \(S_{2}^{\mathrm{new}}(456, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( (T^{6} + T^{3} + 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{12} - 3 T^{11} + \cdots + 3249 \) Copy content Toggle raw display
$7$ \( (T^{6} + 3 T^{5} + 9 T^{4} + \cdots + 9)^{2} \) Copy content Toggle raw display
$11$ \( T^{12} + 3 T^{11} + \cdots + 29241 \) Copy content Toggle raw display
$13$ \( T^{12} + 3 T^{11} + \cdots + 11881 \) Copy content Toggle raw display
$17$ \( T^{12} - 3 T^{11} + \cdots + 11881 \) Copy content Toggle raw display
$19$ \( T^{12} - 18 T^{10} + \cdots + 47045881 \) Copy content Toggle raw display
$23$ \( T^{12} + 12 T^{11} + \cdots + 2601 \) Copy content Toggle raw display
$29$ \( T^{12} - 9 T^{11} + \cdots + 25281 \) Copy content Toggle raw display
$31$ \( T^{12} + 21 T^{11} + \cdots + 5329 \) Copy content Toggle raw display
$37$ \( (T^{6} - 15 T^{5} + \cdots + 91783)^{2} \) Copy content Toggle raw display
$41$ \( T^{12} - 15 T^{11} + \cdots + 16818201 \) Copy content Toggle raw display
$43$ \( T^{12} - 3 T^{11} + \cdots + 3916441 \) Copy content Toggle raw display
$47$ \( T^{12} + 18 T^{11} + \cdots + 7306209 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 207619281 \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 472758049 \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 748514881 \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 763085376 \) Copy content Toggle raw display
$71$ \( T^{12} + 21 T^{11} + \cdots + 106929 \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 43683926049 \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 42961267441 \) Copy content Toggle raw display
$83$ \( T^{12} + 12 T^{11} + \cdots + 1371241 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 7937206281 \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 300987196129 \) Copy content Toggle raw display
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