Properties

Label 117.2.i.a
Level $117$
Weight $2$
Character orbit 117.i
Analytic conductor $0.934$
Analytic rank $0$
Dimension $12$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [117,2,Mod(8,117)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(117, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("117.8");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 117 = 3^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 117.i (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: \(0.934249703649\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(i)\)
Coefficient field: 12.0.26525057735983104.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 16x^{10} + 103x^{8} - 260x^{6} + 259x^{4} + 356x^{2} + 169 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{5}\cdot 3^{2} \)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{9} q^{2} + ( - 2 \beta_{4} + \beta_{3}) q^{4} - \beta_{8} q^{5} + (\beta_{4} + \beta_{2} - 1) q^{7} + ( - \beta_{7} + \beta_{6}) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{9} q^{2} + ( - 2 \beta_{4} + \beta_{3}) q^{4} - \beta_{8} q^{5} + (\beta_{4} + \beta_{2} - 1) q^{7} + ( - \beta_{7} + \beta_{6}) q^{8} + ( - \beta_{5} + \beta_{4} + \cdots - \beta_{2}) q^{10}+ \cdots + ( - 2 \beta_{11} + 2 \beta_{10} + \cdots + 5 \beta_{6}) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 8 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 8 q^{7} - 4 q^{13} - 20 q^{16} - 8 q^{19} + 16 q^{22} - 12 q^{34} + 12 q^{37} + 96 q^{40} - 72 q^{46} + 40 q^{52} - 80 q^{55} - 92 q^{58} - 8 q^{61} + 64 q^{67} + 88 q^{70} + 4 q^{73} + 48 q^{76} - 32 q^{79} + 24 q^{85} + 64 q^{91} + 16 q^{94} + 12 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 16x^{10} + 103x^{8} - 260x^{6} + 259x^{4} + 356x^{2} + 169 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 81\nu^{10} - 1627\nu^{8} + 10208\nu^{6} - 16433\nu^{4} - 19201\nu^{2} - 33817 ) / 72651 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 871\nu^{10} - 20784\nu^{8} + 200656\nu^{6} - 908896\nu^{4} + 1694415\nu^{2} - 122962 ) / 435906 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -138\nu^{10} + 1875\nu^{8} - 10216\nu^{6} + 15440\nu^{4} - 30072\nu^{2} - 13243 ) / 48434 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 34\nu^{10} - 531\nu^{8} + 3334\nu^{6} - 8476\nu^{4} + 11310\nu^{2} + 5219 ) / 7146 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 4327\nu^{10} - 74058\nu^{8} + 507040\nu^{6} - 1384012\nu^{4} + 1133487\nu^{2} + 1453232 ) / 435906 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -9766\nu^{11} + 171648\nu^{9} - 1365595\nu^{7} + 5617729\nu^{5} - 13810209\nu^{3} + 7912201\nu ) / 5666778 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -13135\nu^{11} + 241113\nu^{9} - 1780306\nu^{7} + 5551390\nu^{5} - 6855987\nu^{3} + 5997421\nu ) / 5666778 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -51325\nu^{11} + 794745\nu^{9} - 4933876\nu^{7} + 11860576\nu^{5} - 13472367\nu^{3} - 10663541\nu ) / 5666778 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( -52882\nu^{11} + 893289\nu^{9} - 6176809\nu^{7} + 18058495\nu^{5} - 21689553\nu^{3} - 16977236\nu ) / 5666778 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 8121\nu^{11} - 124554\nu^{9} + 763338\nu^{7} - 1713036\nu^{5} + 1501179\nu^{3} + 4063884\nu ) / 629642 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 101\nu^{11} - 1668\nu^{9} + 11144\nu^{7} - 30914\nu^{5} + 39861\nu^{3} + 9592\nu ) / 4758 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{11} + \beta_{10} + 3\beta_{8} + 3\beta_{7} ) / 6 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{5} - 2\beta_{3} + \beta_{2} + 2\beta _1 + 4 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 8\beta_{11} + 10\beta_{10} + 12\beta_{9} + 21\beta_{8} + 3\beta_{7} - 6\beta_{6} ) / 6 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -8\beta_{4} - 10\beta_{3} + 2\beta_{2} + 5\beta _1 + 6 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 23\beta_{11} + 89\beta_{10} + 78\beta_{9} + 129\beta_{8} - 81\beta_{7} - 42\beta_{6} ) / 6 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 33\beta_{5} - 170\beta_{4} - 152\beta_{3} + 23\beta_{2} + 2\beta _1 - 20 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -164\beta_{11} + 608\beta_{10} + 294\beta_{9} + 531\beta_{8} - 1065\beta_{7} - 420\beta_{6} ) / 6 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 167\beta_{5} - 568\beta_{4} - 452\beta_{3} + 29\beta_{2} - 270\beta _1 - 383 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( -3293\beta_{11} + 3079\beta_{10} - 84\beta_{9} - 345\beta_{8} - 8733\beta_{7} - 3504\beta_{6} ) / 6 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 2313\beta_{5} - 4640\beta_{4} - 3534\beta_{3} - 685\beta_{2} - 6802\beta _1 - 8648 ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( -32218\beta_{11} + 6964\beta_{10} - 14688\beta_{9} - 33375\beta_{8} - 52977\beta_{7} - 22014\beta_{6} ) / 6 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/117\mathbb{Z}\right)^\times\).

\(n\) \(28\) \(92\)
\(\chi(n)\) \(\beta_{4}\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
8.1
−1.64111 0.707107i
0.248859 + 0.707107i
2.59708 0.707107i
−2.59708 + 0.707107i
−0.248859 0.707107i
1.64111 + 0.707107i
−1.64111 + 0.707107i
0.248859 0.707107i
2.59708 + 0.707107i
−2.59708 0.707107i
−0.248859 + 0.707107i
1.64111 0.707107i
−1.77776 + 1.77776i 0 4.32088i 2.34822 2.34822i 0 1.51414 1.51414i 4.12598 + 4.12598i 0 8.34916i
8.2 −1.47512 + 1.47512i 0 2.35194i −0.955965 + 0.955965i 0 −3.08613 + 3.08613i 0.519151 + 0.519151i 0 2.82032i
8.3 −0.404460 + 0.404460i 0 1.67282i −1.88997 + 1.88997i 0 −0.428007 + 0.428007i −1.48551 1.48551i 0 1.52884i
8.4 0.404460 0.404460i 0 1.67282i 1.88997 1.88997i 0 −0.428007 + 0.428007i 1.48551 + 1.48551i 0 1.52884i
8.5 1.47512 1.47512i 0 2.35194i 0.955965 0.955965i 0 −3.08613 + 3.08613i −0.519151 0.519151i 0 2.82032i
8.6 1.77776 1.77776i 0 4.32088i −2.34822 + 2.34822i 0 1.51414 1.51414i −4.12598 4.12598i 0 8.34916i
44.1 −1.77776 1.77776i 0 4.32088i 2.34822 + 2.34822i 0 1.51414 + 1.51414i 4.12598 4.12598i 0 8.34916i
44.2 −1.47512 1.47512i 0 2.35194i −0.955965 0.955965i 0 −3.08613 3.08613i 0.519151 0.519151i 0 2.82032i
44.3 −0.404460 0.404460i 0 1.67282i −1.88997 1.88997i 0 −0.428007 0.428007i −1.48551 + 1.48551i 0 1.52884i
44.4 0.404460 + 0.404460i 0 1.67282i 1.88997 + 1.88997i 0 −0.428007 0.428007i 1.48551 1.48551i 0 1.52884i
44.5 1.47512 + 1.47512i 0 2.35194i 0.955965 + 0.955965i 0 −3.08613 3.08613i −0.519151 + 0.519151i 0 2.82032i
44.6 1.77776 + 1.77776i 0 4.32088i −2.34822 2.34822i 0 1.51414 + 1.51414i −4.12598 + 4.12598i 0 8.34916i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 8.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
13.d odd 4 1 inner
39.f even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 117.2.i.a 12
3.b odd 2 1 inner 117.2.i.a 12
4.b odd 2 1 1872.2.bi.f 12
12.b even 2 1 1872.2.bi.f 12
13.b even 2 1 1521.2.i.g 12
13.d odd 4 1 inner 117.2.i.a 12
13.d odd 4 1 1521.2.i.g 12
39.d odd 2 1 1521.2.i.g 12
39.f even 4 1 inner 117.2.i.a 12
39.f even 4 1 1521.2.i.g 12
52.f even 4 1 1872.2.bi.f 12
156.l odd 4 1 1872.2.bi.f 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
117.2.i.a 12 1.a even 1 1 trivial
117.2.i.a 12 3.b odd 2 1 inner
117.2.i.a 12 13.d odd 4 1 inner
117.2.i.a 12 39.f even 4 1 inner
1521.2.i.g 12 13.b even 2 1
1521.2.i.g 12 13.d odd 4 1
1521.2.i.g 12 39.d odd 2 1
1521.2.i.g 12 39.f even 4 1
1872.2.bi.f 12 4.b odd 2 1
1872.2.bi.f 12 12.b even 2 1
1872.2.bi.f 12 52.f even 4 1
1872.2.bi.f 12 156.l odd 4 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(117, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} + 59 T^{8} + \cdots + 81 \) Copy content Toggle raw display
$3$ \( T^{12} \) Copy content Toggle raw display
$5$ \( T^{12} + 176 T^{8} + \cdots + 20736 \) Copy content Toggle raw display
$7$ \( (T^{6} + 4 T^{5} + 8 T^{4} + \cdots + 32)^{2} \) Copy content Toggle raw display
$11$ \( T^{12} + 176 T^{8} + \cdots + 20736 \) Copy content Toggle raw display
$13$ \( (T^{6} + 2 T^{5} + \cdots + 2197)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} - 18)^{6} \) Copy content Toggle raw display
$19$ \( (T^{6} + 4 T^{5} + 8 T^{4} + \cdots + 32)^{2} \) Copy content Toggle raw display
$23$ \( (T^{6} - 96 T^{4} + \cdots - 10368)^{2} \) Copy content Toggle raw display
$29$ \( (T^{6} + 166 T^{4} + \cdots + 159048)^{2} \) Copy content Toggle raw display
$31$ \( (T^{6} - 104 T^{3} + \cdots + 5408)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} - 2 T + 2)^{6} \) Copy content Toggle raw display
$41$ \( T^{12} + \cdots + 136048896 \) Copy content Toggle raw display
$43$ \( (T^{6} + 192 T^{4} + \cdots + 186624)^{2} \) Copy content Toggle raw display
$47$ \( T^{12} + 176 T^{8} + \cdots + 20736 \) Copy content Toggle raw display
$53$ \( (T^{6} + 214 T^{4} + \cdots + 20808)^{2} \) Copy content Toggle raw display
$59$ \( T^{12} + 52880 T^{8} + \cdots + 1679616 \) Copy content Toggle raw display
$61$ \( (T^{3} + 2 T^{2} - 44 T - 72)^{4} \) Copy content Toggle raw display
$67$ \( (T^{6} - 32 T^{5} + \cdots + 119072)^{2} \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 592240896 \) Copy content Toggle raw display
$73$ \( (T^{6} - 2 T^{5} + \cdots + 40328)^{2} \) Copy content Toggle raw display
$79$ \( (T^{3} + 8 T^{2} + \cdots + 288)^{4} \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 58594980096 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 11019960576 \) Copy content Toggle raw display
$97$ \( (T^{6} - 6 T^{5} + \cdots + 1352)^{2} \) Copy content Toggle raw display
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