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Properties

Label 117.3.j.b
Level $117$
Weight $3$
Character orbit 117.j
Analytic conductor $3.188$
Analytic rank $0$
Dimension $8$
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] = [117,3,Mod(73,117)]
 
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([0, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("117.73");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 117 = 3^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 117.j (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: \(3.18801909302\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: 8.0.1579585536.10
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 2x^{7} - 4x^{6} + 28x^{5} - 38x^{4} + 8x^{3} + 200x^{2} - 352x + 484 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 39)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{4} q^{2} + ( - \beta_{7} - \beta_{4} - \beta_1 - 1) q^{4} + ( - 3 \beta_{5} - \beta_{3} - \beta_1 + 3) q^{5} + (2 \beta_{6} + 2 \beta_{5} + \cdots + \beta_1) q^{7}+ \cdots + ( - \beta_{7} + \beta_{6} + 4 \beta_{5} + \cdots + 2) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{4} q^{2} + ( - \beta_{7} - \beta_{4} - \beta_1 - 1) q^{4} + ( - 3 \beta_{5} - \beta_{3} - \beta_1 + 3) q^{5} + (2 \beta_{6} + 2 \beta_{5} + \cdots + \beta_1) q^{7}+ \cdots + ( - 29 \beta_{7} + 16 \beta_{6} + \cdots - 113) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 4 q^{2} + 20 q^{5} + 8 q^{7} + 24 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 4 q^{2} + 20 q^{5} + 8 q^{7} + 24 q^{8} + 20 q^{11} + 8 q^{13} + 16 q^{14} + 56 q^{16} - 40 q^{19} - 44 q^{20} - 128 q^{22} + 32 q^{26} + 32 q^{28} - 56 q^{29} + 32 q^{31} - 148 q^{32} + 96 q^{34} - 104 q^{35} - 16 q^{37} + 48 q^{40} + 116 q^{41} + 92 q^{44} - 264 q^{46} - 100 q^{47} - 20 q^{50} + 72 q^{52} + 232 q^{53} - 176 q^{55} - 104 q^{58} - 316 q^{59} + 160 q^{61} + 92 q^{65} + 176 q^{67} - 168 q^{68} + 184 q^{70} - 4 q^{71} - 64 q^{73} + 64 q^{74} + 112 q^{76} + 208 q^{79} + 332 q^{80} + 212 q^{83} - 168 q^{85} + 452 q^{89} - 400 q^{91} + 720 q^{92} + 16 q^{94} + 128 q^{97} - 724 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 2x^{7} - 4x^{6} + 28x^{5} - 38x^{4} + 8x^{3} + 200x^{2} - 352x + 484 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{7} + 46\nu^{6} - 150\nu^{5} + 27\nu^{4} + 720\nu^{3} - 1768\nu^{2} + 2792\nu - 1881 ) / 935 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -118\nu^{7} - 607\nu^{6} + 490\nu^{5} - 384\nu^{4} - 8710\nu^{3} + 19856\nu^{2} - 35024\nu - 91398 ) / 68510 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 3448 \nu^{7} + 21163 \nu^{6} + 45670 \nu^{5} - 200494 \nu^{4} + 126360 \nu^{3} + 635936 \nu^{2} + \cdots + 2421122 ) / 753610 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 634\nu^{7} - 1964\nu^{6} + 4915\nu^{5} + 902\nu^{4} - 39130\nu^{3} + 104652\nu^{2} - 103278\nu + 63754 ) / 68510 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -86\nu^{7} + 227\nu^{6} - 162\nu^{5} - 758\nu^{4} + 1794\nu^{3} - 8124\nu^{2} + 8408\nu - 13662 ) / 8866 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 3777 \nu^{7} + 11052 \nu^{6} + 45875 \nu^{5} - 222466 \nu^{4} + 199615 \nu^{3} + 521764 \nu^{2} + \cdots + 3415148 ) / 376805 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 10422 \nu^{7} - 42767 \nu^{6} + 8395 \nu^{5} + 210416 \nu^{4} - 556790 \nu^{3} + 515236 \nu^{2} + \cdots - 1719828 ) / 753610 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{7} - \beta_{4} + \beta_{3} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -2\beta_{7} - \beta_{6} - 3\beta_{5} + \beta_{3} + 4\beta_{2} + 2 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -4\beta_{5} - 4\beta_{4} + \beta_{3} - \beta_{2} - \beta _1 - 9 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -12\beta_{7} - 5\beta_{6} - 8\beta_{5} + 2\beta_{4} - 3\beta_{3} - \beta_{2} - 4\beta _1 + 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -3\beta_{7} - 2\beta_{6} - 2\beta_{5} - 7\beta_{4} + \beta_{3} - 34\beta_{2} - 12\beta _1 - 58 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -70\beta_{7} - 22\beta_{6} + 4\beta_{5} + 70\beta_{4} - 22\beta_{3} - 22\beta_{2} + 22 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 70\beta_{7} + 37\beta_{6} + 40\beta_{5} + 48\beta_{4} - 11\beta_{3} - 195\beta_{2} + 37\beta _1 - 309 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(28\) \(92\)
\(\chi(n)\) \(-\beta_{5}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
73.1
2.22833 + 1.32913i
0.252411 − 1.79004i
1.11361 + 1.42401i
−2.59436 + 0.0368949i
2.22833 − 1.32913i
0.252411 + 1.79004i
1.11361 − 1.42401i
−2.59436 − 0.0368949i
−2.26523 − 2.26523i 0 6.26250i 2.07379 + 2.07379i 0 7.04857 − 7.04857i 5.12509 − 5.12509i 0 − 9.39521i
73.2 −1.67642 − 1.67642i 0 1.62080i 4.84803 + 4.84803i 0 −8.89220 + 8.89220i −3.98855 + 3.98855i 0 − 16.2547i
73.3 0.676424 + 0.676424i 0 − 3.08490i −1.58008 − 1.58008i 0 3.96400 − 3.96400i 4.79240 − 4.79240i 0 − 2.13761i
73.4 1.26523 + 1.26523i 0 − 0.798403i 4.65826 + 4.65826i 0 1.87963 − 1.87963i 6.07107 − 6.07107i 0 11.7875i
109.1 −2.26523 + 2.26523i 0 − 6.26250i 2.07379 − 2.07379i 0 7.04857 + 7.04857i 5.12509 + 5.12509i 0 9.39521i
109.2 −1.67642 + 1.67642i 0 − 1.62080i 4.84803 − 4.84803i 0 −8.89220 − 8.89220i −3.98855 − 3.98855i 0 16.2547i
109.3 0.676424 − 0.676424i 0 3.08490i −1.58008 + 1.58008i 0 3.96400 + 3.96400i 4.79240 + 4.79240i 0 2.13761i
109.4 1.26523 − 1.26523i 0 0.798403i 4.65826 − 4.65826i 0 1.87963 + 1.87963i 6.07107 + 6.07107i 0 − 11.7875i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 73.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.d odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 117.3.j.b 8
3.b odd 2 1 39.3.g.a ✓ 8
12.b even 2 1 624.3.ba.b 8
13.d odd 4 1 inner 117.3.j.b 8
39.f even 4 1 39.3.g.a ✓ 8
156.l odd 4 1 624.3.ba.b 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.3.g.a ✓ 8 3.b odd 2 1
39.3.g.a ✓ 8 39.f even 4 1
117.3.j.b 8 1.a even 1 1 trivial
117.3.j.b 8 13.d odd 4 1 inner
624.3.ba.b 8 12.b even 2 1
624.3.ba.b 8 156.l odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{8} + 4T_{2}^{7} + 8T_{2}^{6} - 8T_{2}^{5} + 10T_{2}^{4} + 44T_{2}^{3} + 128T_{2}^{2} - 208T_{2} + 169 \) acting on \(S_{3}^{\mathrm{new}}(117, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} + 4 T^{7} + \cdots + 169 \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} - 20 T^{7} + \cdots + 87616 \) Copy content Toggle raw display
$7$ \( T^{8} - 8 T^{7} + \cdots + 3489424 \) Copy content Toggle raw display
$11$ \( T^{8} - 20 T^{7} + \cdots + 602176 \) Copy content Toggle raw display
$13$ \( T^{8} - 8 T^{7} + \cdots + 815730721 \) Copy content Toggle raw display
$17$ \( T^{8} + 336 T^{6} + \cdots + 3504384 \) Copy content Toggle raw display
$19$ \( T^{8} + 40 T^{7} + \cdots + 3671056 \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 8422834176 \) Copy content Toggle raw display
$29$ \( (T^{4} + 28 T^{3} + \cdots - 3824)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} + \cdots + 246909610000 \) Copy content Toggle raw display
$37$ \( T^{8} + \cdots + 12456636124816 \) Copy content Toggle raw display
$41$ \( T^{8} + \cdots + 26504951703616 \) Copy content Toggle raw display
$43$ \( T^{8} + 4704 T^{6} + \cdots + 632623104 \) Copy content Toggle raw display
$47$ \( T^{8} + 100 T^{7} + \cdots + 326019136 \) Copy content Toggle raw display
$53$ \( (T^{4} - 116 T^{3} + \cdots + 304336)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 62781028208704 \) Copy content Toggle raw display
$61$ \( (T^{4} - 80 T^{3} + \cdots + 2335552)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots + 780982247824 \) Copy content Toggle raw display
$71$ \( T^{8} + \cdots + 34773288084544 \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots + 450106810000 \) Copy content Toggle raw display
$79$ \( (T^{4} - 104 T^{3} + \cdots - 5173376)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 31365107407936 \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots + 14\!\cdots\!16 \) Copy content Toggle raw display
$97$ \( T^{8} + \cdots + 190804383734416 \) Copy content Toggle raw display
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