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Properties

Label 45.12.b.c
Level $45$
Weight $12$
Character orbit 45.b
Analytic conductor $34.575$
Analytic rank $0$
Dimension $8$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [45,12,Mod(19,45)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(45, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 12, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("45.19");
 
S:= CuspForms(chi, 12);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 45 = 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 12 \)
Character orbit: \([\chi]\) \(=\) 45.b (of order \(2\), degree \(1\), 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: \(34.5754431252\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 2 x^{7} - 5706 x^{6} + 91460 x^{5} + 8073323 x^{4} - 237717036 x^{3} + 1329030846 x^{2} + \cdots + 2557326711753 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{20}\cdot 3^{12}\cdot 5^{9} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{3} - 1122) q^{4} + (\beta_{2} + 46 \beta_1) q^{5} - \beta_{5} q^{7} + (16 \beta_{4} - 1348 \beta_1) q^{8} + (\beta_{7} + 3 \beta_{5} + \cdots - 144475) q^{10} + ( - \beta_{6} - 27 \beta_{4} + \cdots + 27 \beta_1) q^{11}+ \cdots + ( - 5417280 \beta_{4} + 570418663 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 8976 q^{4} - 1155800 q^{10} + 15740992 q^{16} + 61220032 q^{19} + 184187000 q^{25} + 279698464 q^{31} + 309623120 q^{34} - 62144800 q^{40} - 13964841760 q^{46} - 1596098056 q^{49} + 15819804000 q^{55}+ \cdots - 460752353440 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 2 x^{7} - 5706 x^{6} + 91460 x^{5} + 8073323 x^{4} - 237717036 x^{3} + 1329030846 x^{2} + \cdots + 2557326711753 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 204272349772633 \nu^{7} + \cdots + 99\!\cdots\!87 ) / 17\!\cdots\!25 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 63\!\cdots\!06 \nu^{7} + \cdots + 41\!\cdots\!91 ) / 34\!\cdots\!50 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 522596740 \nu^{7} + 21933629000 \nu^{6} - 2409355769170 \nu^{5} - 59489761647320 \nu^{4} + \cdots + 31\!\cdots\!90 ) / 26\!\cdots\!99 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 21\!\cdots\!59 \nu^{7} + \cdots + 18\!\cdots\!76 ) / 57\!\cdots\!75 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 63224941064 \nu^{7} + 4023377232856 \nu^{6} - 104365227890702 \nu^{5} + \cdots - 17\!\cdots\!44 ) / 40\!\cdots\!85 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 48\!\cdots\!94 \nu^{7} + \cdots + 10\!\cdots\!91 ) / 34\!\cdots\!50 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 2019605839238 \nu^{7} + 116738896580452 \nu^{6} + \cdots - 14\!\cdots\!73 ) / 40\!\cdots\!85 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{6} + 34\beta_{4} + 120\beta_{3} - 261\beta_{2} - 4306\beta _1 + 10800 ) / 43200 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 52 \beta_{7} - 33 \beta_{6} - 184 \beta_{5} - 2682 \beta_{4} - 22346 \beta_{3} + 2133 \beta_{2} + \cdots + 61646400 ) / 43200 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 126 \beta_{7} + 4981 \beta_{6} - 9792 \beta_{5} + 255034 \beta_{4} + 483537 \beta_{3} + \cdots - 1296734400 ) / 43200 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 149248 \beta_{7} - 120009 \beta_{6} + 624584 \beta_{5} - 11835546 \beta_{4} - 70314944 \beta_{3} + \cdots + 173789344800 ) / 43200 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 4788150 \beta_{7} + 14795584 \beta_{6} - 26065200 \beta_{5} + 1108109056 \beta_{4} + \cdots - 6358599457200 ) / 43200 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 321701744 \beta_{7} - 270889605 \beta_{6} + 1728017752 \beta_{5} - 21560898210 \beta_{4} + \cdots + 279669788164800 ) / 21600 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 29883863352 \beta_{7} + 45830071063 \beta_{6} - 145662369216 \beta_{5} + 3896956282462 \beta_{4} + \cdots - 24\!\cdots\!00 ) / 43200 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/45\mathbb{Z}\right)^\times\).

\(n\) \(11\) \(37\)
\(\chi(n)\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1
45.4346 + 3.04491i
−59.8426 + 3.04491i
24.4724 + 17.3199i
−9.06433 + 17.3199i
24.4724 − 17.3199i
−9.06433 − 17.3199i
45.4346 − 3.04491i
−59.8426 − 3.04491i
− 76.5078i 0 −3805.45 −6524.36 − 2502.17i 0 − 35612.9i 134459.i 0 −191435. + 499165.i
19.2 − 76.5078i 0 −3805.45 6524.36 − 2502.17i 0 35612.9i 134459.i 0 −191435. − 499165.i
19.3 − 22.0579i 0 1561.45 −5411.49 − 4420.85i 0 55546.4i − 79616.9i 0 −97514.6 + 119366.i
19.4 − 22.0579i 0 1561.45 5411.49 − 4420.85i 0 − 55546.4i − 79616.9i 0 −97514.6 − 119366.i
19.5 22.0579i 0 1561.45 −5411.49 + 4420.85i 0 − 55546.4i 79616.9i 0 −97514.6 − 119366.i
19.6 22.0579i 0 1561.45 5411.49 + 4420.85i 0 55546.4i 79616.9i 0 −97514.6 + 119366.i
19.7 76.5078i 0 −3805.45 −6524.36 + 2502.17i 0 35612.9i − 134459.i 0 −191435. − 499165.i
19.8 76.5078i 0 −3805.45 6524.36 + 2502.17i 0 − 35612.9i − 134459.i 0 −191435. + 499165.i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 19.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.b even 2 1 inner
15.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 45.12.b.c ✓ 8
3.b odd 2 1 inner 45.12.b.c ✓ 8
5.b even 2 1 inner 45.12.b.c ✓ 8
5.c odd 4 2 225.12.a.z 8
15.d odd 2 1 inner 45.12.b.c ✓ 8
15.e even 4 2 225.12.a.z 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
45.12.b.c ✓ 8 1.a even 1 1 trivial
45.12.b.c ✓ 8 3.b odd 2 1 inner
45.12.b.c ✓ 8 5.b even 2 1 inner
45.12.b.c ✓ 8 15.d odd 2 1 inner
225.12.a.z 8 5.c odd 4 2
225.12.a.z 8 15.e even 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} + 6340T_{2}^{2} + 2848000 \) acting on \(S_{12}^{\mathrm{new}}(45, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} + 6340 T^{2} + 2848000)^{2} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} + \cdots + 56\!\cdots\!25 \) Copy content Toggle raw display
$7$ \( (T^{4} + \cdots + 39\!\cdots\!00)^{2} \) Copy content Toggle raw display
$11$ \( (T^{4} + \cdots + 13\!\cdots\!00)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} + \cdots + 80\!\cdots\!00)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} + \cdots + 35\!\cdots\!00)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} + \cdots + 50861932029616)^{4} \) Copy content Toggle raw display
$23$ \( (T^{4} + \cdots + 12\!\cdots\!00)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} + \cdots + 13\!\cdots\!00)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} + \cdots - 16\!\cdots\!36)^{4} \) Copy content Toggle raw display
$37$ \( (T^{4} + \cdots + 24\!\cdots\!00)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} + \cdots + 50\!\cdots\!00)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} + \cdots + 14\!\cdots\!00)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} + \cdots + 18\!\cdots\!00)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} + \cdots + 18\!\cdots\!00)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} + \cdots + 22\!\cdots\!00)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + \cdots - 10\!\cdots\!96)^{4} \) Copy content Toggle raw display
$67$ \( (T^{4} + \cdots + 73\!\cdots\!00)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} + \cdots + 16\!\cdots\!00)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} + \cdots + 50\!\cdots\!00)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} + \cdots + 51\!\cdots\!56)^{4} \) Copy content Toggle raw display
$83$ \( (T^{4} + \cdots + 29\!\cdots\!00)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} + \cdots + 11\!\cdots\!00)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} + \cdots + 98\!\cdots\!00)^{2} \) Copy content Toggle raw display
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