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Properties

Label 135.4.b.c
Level $135$
Weight $4$
Character orbit 135.b
Analytic conductor $7.965$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [135,4,Mod(109,135)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(135, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("135.109");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 135 = 3^{3} \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 135.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: \(7.96525785077\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 255x^{8} + 1289x^{4} + 1600 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4}\cdot 3^{18}\cdot 5 \)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{7} q^{2} + ( - \beta_1 - 5) q^{4} - \beta_{3} q^{5} + \beta_{8} q^{7} + ( - 3 \beta_{7} + \beta_{3} + \beta_{2}) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{7} q^{2} + ( - \beta_1 - 5) q^{4} - \beta_{3} q^{5} + \beta_{8} q^{7} + ( - 3 \beta_{7} + \beta_{3} + \beta_{2}) q^{8} + (\beta_{8} + \beta_{6} + \beta_{5} + \cdots - 3) q^{10}+ \cdots + ( - 2 \beta_{9} - 545 \beta_{7} + \cdots + 57 \beta_{2}) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 60 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 60 q^{4} - 36 q^{10} + 84 q^{16} - 348 q^{25} + 252 q^{31} + 1068 q^{34} + 1320 q^{40} - 1668 q^{46} - 2868 q^{49} + 684 q^{55} + 792 q^{61} + 2268 q^{64} - 5652 q^{70} + 1824 q^{76} + 2196 q^{79} + 3816 q^{85} - 3384 q^{91} - 5148 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 255x^{8} + 1289x^{4} + 1600 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -51\nu^{8} - 12807\nu^{4} - 26996 ) / 967 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 21\nu^{11} - 4680\nu^{9} + 7435\nu^{7} - 1170680\nu^{5} + 512989\nu^{3} - 1355560\nu ) / 154720 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 249\nu^{11} + 3480\nu^{9} + 66055\nu^{7} + 878440\nu^{5} + 954721\nu^{3} + 2654360\nu ) / 154720 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 513\nu^{10} + 131895\nu^{6} + 993897\nu^{2} ) / 38680 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 1533\nu^{10} + 12600\nu^{8} + 388035\nu^{6} + 3184560\nu^{4} + 1205037\nu^{2} + 8410200 ) / 38680 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -1533\nu^{10} - 388035\nu^{6} - 1205037\nu^{2} ) / 19340 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 435\nu^{11} + 696\nu^{9} + 109805\nu^{7} + 175688\nu^{5} + 277643\nu^{3} + 438040\nu ) / 30944 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 1617\nu^{10} + 408105\nu^{6} + 1013553\nu^{2} ) / 9670 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( -174\nu^{11} - 21\nu^{9} - 43922\nu^{7} - 5501\nu^{5} - 113378\nu^{3} - 66235\nu ) / 1934 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 3621\nu^{11} - 5448\nu^{9} + 915099\nu^{7} - 1376280\nu^{5} + 2588781\nu^{3} - 3866280\nu ) / 30944 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( -4503\nu^{11} + 7896\nu^{9} - 1134537\nu^{7} + 1991016\nu^{5} - 2318799\nu^{3} + 4094520\nu ) / 30944 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -5\beta_{11} - 5\beta_{10} - 3\beta_{9} - 33\beta_{7} + 33\beta_{3} - 12\beta_{2} ) / 270 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 6\beta_{8} + 15\beta_{6} + 14\beta_{4} ) / 54 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 2\beta_{11} + \beta_{10} - 5\beta_{9} - 19\beta_{7} - 5\beta_{3} - 5\beta_{2} ) / 9 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 17\beta_{6} + 34\beta_{5} + 210\beta _1 - 1530 ) / 18 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 61\beta_{11} + 27\beta_{10} - 5\beta_{9} + 425\beta_{7} - 467\beta_{3} + 328\beta_{2} ) / 18 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -289\beta_{8} - 694\beta_{6} - 504\beta_{4} ) / 9 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -2881\beta_{11} - 1261\beta_{10} + 7545\beta_{9} + 27843\beta_{7} + 9111\beta_{3} + 7806\beta_{2} ) / 54 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( -1423\beta_{6} - 2846\beta_{5} - 17692\beta _1 + 124894 ) / 6 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( -22760\beta_{11} - 9953\beta_{10} + 2082\beta_{9} - 158244\beta_{7} + 174747\beta_{3} - 123438\beta_{2} ) / 27 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 48244\beta_{8} + 115725\beta_{6} + 83826\beta_{4} ) / 6 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 719507 \beta_{11} + 314621 \beta_{10} - 1885401 \beta_{9} - 6953595 \beta_{7} + \cdots - 1951284 \beta_{2} ) / 54 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/135\mathbb{Z}\right)^\times\).

\(n\) \(56\) \(82\)
\(\chi(n)\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
109.1
−0.930240 + 0.930240i
0.930240 + 0.930240i
−0.855023 − 0.855023i
0.855023 − 0.855023i
2.81133 + 2.81133i
−2.81133 + 2.81133i
2.81133 − 2.81133i
−2.81133 − 2.81133i
−0.855023 + 0.855023i
0.855023 + 0.855023i
−0.930240 − 0.930240i
0.930240 − 0.930240i
− 4.92742i 0 −16.2795 −7.57192 − 8.22594i 0 34.7816i 40.7965i 0 −40.5327 + 37.3100i
109.2 − 4.92742i 0 −16.2795 7.57192 − 8.22594i 0 − 34.7816i 40.7965i 0 −40.5327 − 37.3100i
109.3 − 3.62699i 0 −5.15507 −2.20385 + 10.9610i 0 22.4519i − 10.3185i 0 39.7554 + 7.99335i
109.4 − 3.62699i 0 −5.15507 2.20385 + 10.9610i 0 − 22.4519i − 10.3185i 0 39.7554 − 7.99335i
109.5 − 1.25118i 0 6.43456 −9.04484 − 6.57198i 0 − 5.67029i − 18.0602i 0 −8.22271 + 11.3167i
109.6 − 1.25118i 0 6.43456 9.04484 − 6.57198i 0 5.67029i − 18.0602i 0 −8.22271 − 11.3167i
109.7 1.25118i 0 6.43456 −9.04484 + 6.57198i 0 5.67029i 18.0602i 0 −8.22271 − 11.3167i
109.8 1.25118i 0 6.43456 9.04484 + 6.57198i 0 − 5.67029i 18.0602i 0 −8.22271 + 11.3167i
109.9 3.62699i 0 −5.15507 −2.20385 − 10.9610i 0 − 22.4519i 10.3185i 0 39.7554 − 7.99335i
109.10 3.62699i 0 −5.15507 2.20385 − 10.9610i 0 22.4519i 10.3185i 0 39.7554 + 7.99335i
109.11 4.92742i 0 −16.2795 −7.57192 + 8.22594i 0 − 34.7816i − 40.7965i 0 −40.5327 − 37.3100i
109.12 4.92742i 0 −16.2795 7.57192 + 8.22594i 0 34.7816i − 40.7965i 0 −40.5327 + 37.3100i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 109.12
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 135.4.b.c ✓ 12
3.b odd 2 1 inner 135.4.b.c ✓ 12
5.b even 2 1 inner 135.4.b.c ✓ 12
5.c odd 4 1 675.4.a.bb 6
5.c odd 4 1 675.4.a.bc 6
15.d odd 2 1 inner 135.4.b.c ✓ 12
15.e even 4 1 675.4.a.bb 6
15.e even 4 1 675.4.a.bc 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
135.4.b.c ✓ 12 1.a even 1 1 trivial
135.4.b.c ✓ 12 3.b odd 2 1 inner
135.4.b.c ✓ 12 5.b even 2 1 inner
135.4.b.c ✓ 12 15.d odd 2 1 inner
675.4.a.bb 6 5.c odd 4 1
675.4.a.bb 6 15.e even 4 1
675.4.a.bc 6 5.c odd 4 1
675.4.a.bc 6 15.e even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{6} + 39T_{2}^{4} + 378T_{2}^{2} + 500 \) acting on \(S_{4}^{\mathrm{new}}(135, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{6} + 39 T^{4} + \cdots + 500)^{2} \) Copy content Toggle raw display
$3$ \( T^{12} \) Copy content Toggle raw display
$5$ \( T^{12} + \cdots + 3814697265625 \) Copy content Toggle raw display
$7$ \( (T^{6} + 1746 T^{4} + \cdots + 19607184)^{2} \) Copy content Toggle raw display
$11$ \( (T^{6} - 3357 T^{4} + \cdots - 192820500)^{2} \) Copy content Toggle raw display
$13$ \( (T^{6} + 10251 T^{4} + \cdots + 16621655625)^{2} \) Copy content Toggle raw display
$17$ \( (T^{6} + 16113 T^{4} + \cdots + 76335368000)^{2} \) Copy content Toggle raw display
$19$ \( (T^{3} - 5943 T - 119558)^{4} \) Copy content Toggle raw display
$23$ \( (T^{6} + 22989 T^{4} + \cdots + 307148112500)^{2} \) Copy content Toggle raw display
$29$ \( (T^{6} + \cdots - 4585850680500)^{2} \) Copy content Toggle raw display
$31$ \( (T^{3} - 63 T^{2} + \cdots + 186100)^{4} \) Copy content Toggle raw display
$37$ \( (T^{6} + \cdots + 12916620360900)^{2} \) Copy content Toggle raw display
$41$ \( (T^{6} + \cdots - 967871758050000)^{2} \) Copy content Toggle raw display
$43$ \( (T^{6} + \cdots + 9123033630096)^{2} \) Copy content Toggle raw display
$47$ \( (T^{6} + \cdots + 869738053832000)^{2} \) Copy content Toggle raw display
$53$ \( (T^{6} + \cdots + 108910045472000)^{2} \) Copy content Toggle raw display
$59$ \( (T^{6} + \cdots - 35\!\cdots\!00)^{2} \) Copy content Toggle raw display
$61$ \( (T^{3} - 198 T^{2} + \cdots + 19133944)^{4} \) Copy content Toggle raw display
$67$ \( (T^{6} + \cdots + 1109946531600)^{2} \) Copy content Toggle raw display
$71$ \( (T^{6} + \cdots - 74\!\cdots\!00)^{2} \) Copy content Toggle raw display
$73$ \( (T^{6} + \cdots + 26\!\cdots\!00)^{2} \) Copy content Toggle raw display
$79$ \( (T^{3} - 549 T^{2} + \cdots + 285462565)^{4} \) Copy content Toggle raw display
$83$ \( (T^{6} + \cdots + 14\!\cdots\!00)^{2} \) Copy content Toggle raw display
$89$ \( (T^{6} + \cdots - 35\!\cdots\!00)^{2} \) Copy content Toggle raw display
$97$ \( (T^{6} + \cdots + 66\!\cdots\!24)^{2} \) Copy content Toggle raw display
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