↑

Properties

Label 117.2.e.b
Level $117$
Weight $2$
Character orbit 117.e
Analytic conductor $0.934$
Analytic rank $0$
Dimension $10$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [117,2,Mod(40,117)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(117, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([2, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("117.40");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 117 = 3^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 117.e (of order \(3\), 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: \(10\)
Relative dimension: \(5\) over \(\Q(\zeta_{3})\)
Coefficient field: 10.0.487558322307.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} + 13x^{8} + 43x^{6} + 48x^{4} + 21x^{2} + 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 3 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{9} q^{2} + (\beta_{9} - \beta_{5} + \beta_1) q^{3} + (\beta_{9} + \beta_{8} - \beta_{6} + \beta_1) q^{4} + ( - \beta_{9} + \beta_{5} + \cdots - \beta_1) q^{5}+ \cdots + ( - \beta_{9} + \beta_{8} - \beta_{6} + \cdots - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{9} q^{2} + (\beta_{9} - \beta_{5} + \beta_1) q^{3} + (\beta_{9} + \beta_{8} - \beta_{6} + \beta_1) q^{4} + ( - \beta_{9} + \beta_{5} + \cdots - \beta_1) q^{5}+ \cdots + (3 \beta_{9} + \beta_{8} + 2 \beta_{6} + \cdots + 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 2 q^{2} + q^{3} - 4 q^{4} - q^{5} + 13 q^{6} + 2 q^{7} + 24 q^{8} - 13 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q - 2 q^{2} + q^{3} - 4 q^{4} - q^{5} + 13 q^{6} + 2 q^{7} + 24 q^{8} - 13 q^{9} - 4 q^{10} - 11 q^{11} - 19 q^{12} - 5 q^{13} + 5 q^{14} - 10 q^{15} - 10 q^{16} + 14 q^{17} - 10 q^{18} + 6 q^{19} + q^{20} + 8 q^{21} + 7 q^{22} - 18 q^{23} + 18 q^{24} + 8 q^{25} + 4 q^{26} - 11 q^{27} - 38 q^{28} - 4 q^{29} + 5 q^{30} + 16 q^{31} - 31 q^{32} + 19 q^{33} - 13 q^{34} + 22 q^{35} - 5 q^{36} + 10 q^{37} - 24 q^{38} - 2 q^{39} - 18 q^{40} - 12 q^{41} + 59 q^{42} + 2 q^{44} - 14 q^{45} + 12 q^{46} - 15 q^{47} + 29 q^{48} - 3 q^{49} + 5 q^{50} - 7 q^{51} - 4 q^{52} - 6 q^{53} - 11 q^{54} - 34 q^{55} + 24 q^{56} + 12 q^{57} + 2 q^{58} - 3 q^{59} + 19 q^{60} + q^{61} - 8 q^{62} - 14 q^{63} + 124 q^{64} - q^{65} - 11 q^{66} + 4 q^{67} + 16 q^{68} + 25 q^{70} + 30 q^{71} - 66 q^{72} - 10 q^{73} - 11 q^{74} + 14 q^{75} - 9 q^{76} - q^{77} - 2 q^{78} - 13 q^{79} + 64 q^{80} - 25 q^{81} - 42 q^{82} - 26 q^{83} - 17 q^{84} + 7 q^{85} + 6 q^{86} - 13 q^{87} - 3 q^{88} + 58 q^{89} - 26 q^{90} - 4 q^{91} + 6 q^{92} + 58 q^{93} - 12 q^{94} - 12 q^{95} - 67 q^{96} + 16 q^{97} - 78 q^{98} + 17 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} + 13x^{8} + 43x^{6} + 48x^{4} + 21x^{2} + 3 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( -\nu^{8} - 12\nu^{6} - 31\nu^{4} - 17\nu^{2} - 3 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{9} - 3\nu^{8} + 12\nu^{7} - 38\nu^{6} + 31\nu^{5} - 116\nu^{4} + 17\nu^{3} - 102\nu^{2} + 3\nu - 24 ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{9} + 3\nu^{8} + 12\nu^{7} + 38\nu^{6} + 31\nu^{5} + 116\nu^{4} + 17\nu^{3} + 102\nu^{2} + 3\nu + 24 ) / 2 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -6\nu^{8} - \nu^{7} - 74\nu^{6} - 12\nu^{5} - 209\nu^{4} - 31\nu^{3} - 152\nu^{2} - 15\nu - 30 ) / 2 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -6\nu^{8} + \nu^{7} - 74\nu^{6} + 12\nu^{5} - 209\nu^{4} + 31\nu^{3} - 152\nu^{2} + 15\nu - 30 ) / 2 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( 4\nu^{8} + 49\nu^{6} + 135\nu^{4} + 88\nu^{2} + 13 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -4\nu^{9} - 49\nu^{7} - 135\nu^{5} - 88\nu^{3} - 12\nu + 1 ) / 2 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -5\nu^{9} + 4\nu^{8} - 62\nu^{7} + 49\nu^{6} - 178\nu^{5} + 135\nu^{4} - 136\nu^{3} + 88\nu^{2} - 33\nu + 13 ) / 2 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( -10\nu^{9} + \nu^{8} - 123\nu^{7} + 12\nu^{6} - 344\nu^{5} + 31\nu^{4} - 240\nu^{3} + 17\nu^{2} - 42\nu + 3 ) / 2 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -2\beta_{8} + 2\beta_{7} + \beta_{6} - \beta_{5} + \beta_{4} - \beta_{3} - \beta_{2} - 1 ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{5} + \beta_{4} + \beta_{3} - \beta_{2} - 3\beta _1 - 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{9} + 2\beta_{8} - 6\beta_{7} - \beta_{6} + 2\beta_{5} - 2\beta_{4} + 3\beta_{3} + 3\beta_{2} + \beta _1 + 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -2\beta_{6} - 11\beta_{5} - 11\beta_{4} - 10\beta_{3} + 10\beta_{2} + 28\beta _1 + 20 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 22 \beta_{9} - 12 \beta_{8} + 58 \beta_{7} + 6 \beta_{6} - 16 \beta_{5} + 16 \beta_{4} - 24 \beta_{3} + \cdots - 29 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 23\beta_{6} + 101\beta_{5} + 101\beta_{4} + 90\beta_{3} - 90\beta_{2} - 244\beta _1 - 161 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 202 \beta_{9} + 92 \beta_{8} - 520 \beta_{7} - 46 \beta_{6} + 136 \beta_{5} - 136 \beta_{4} + 200 \beta_{3} + \cdots + 260 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( -214\beta_{6} - 888\beta_{5} - 888\beta_{4} - 787\beta_{3} + 787\beta_{2} + 2110\beta _1 + 1360 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 1776 \beta_{9} - 764 \beta_{8} + 4542 \beta_{7} + 382 \beta_{6} - 1169 \beta_{5} + 1169 \beta_{4} + \cdots - 2271 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(28\) \(92\)
\(\chi(n)\) \(1\) \(-1 + \beta_{7}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
40.1
− 0.858204i
0.775434i
0.534580i
2.93761i
− 1.65737i
0.858204i
− 0.775434i
− 0.534580i
− 2.93761i
1.65737i
−1.39763 − 2.42077i 0.413993 + 1.68185i −2.90675 + 5.03463i −0.413993 + 0.717057i 3.49275 − 3.35278i 1.24323 + 2.15333i 10.6597 −2.65722 + 1.39255i 2.31444
40.2 −0.754070 − 1.30609i −1.32157 + 1.11957i −0.137244 + 0.237713i 1.32157 − 2.28903i 2.45882 + 0.881858i −0.171546 − 0.297126i −2.60231 0.493121 − 2.95919i −3.98624
40.3 −0.200060 − 0.346514i 0.771753 − 1.55061i 0.919952 − 1.59340i −0.771753 + 1.33672i −0.691705 + 0.0427920i 0.0370405 + 0.0641560i −1.53642 −1.80879 − 2.39338i 0.617587
40.4 0.565535 + 0.979535i 1.19379 + 1.25494i 0.360341 − 0.624129i −1.19379 + 2.06770i −0.554128 + 1.87907i −2.04404 − 3.54039i 3.07728 −0.149744 + 2.99626i −2.70051
40.5 0.786226 + 1.36178i −0.557959 − 1.63972i −0.236304 + 0.409291i 0.557959 − 0.966413i 1.79426 − 2.04901i 1.93532 + 3.35208i 2.40175 −2.37736 + 1.82979i 1.75473
79.1 −1.39763 + 2.42077i 0.413993 − 1.68185i −2.90675 − 5.03463i −0.413993 − 0.717057i 3.49275 + 3.35278i 1.24323 − 2.15333i 10.6597 −2.65722 − 1.39255i 2.31444
79.2 −0.754070 + 1.30609i −1.32157 − 1.11957i −0.137244 − 0.237713i 1.32157 + 2.28903i 2.45882 − 0.881858i −0.171546 + 0.297126i −2.60231 0.493121 + 2.95919i −3.98624
79.3 −0.200060 + 0.346514i 0.771753 + 1.55061i 0.919952 + 1.59340i −0.771753 − 1.33672i −0.691705 − 0.0427920i 0.0370405 − 0.0641560i −1.53642 −1.80879 + 2.39338i 0.617587
79.4 0.565535 − 0.979535i 1.19379 − 1.25494i 0.360341 + 0.624129i −1.19379 − 2.06770i −0.554128 − 1.87907i −2.04404 + 3.54039i 3.07728 −0.149744 − 2.99626i −2.70051
79.5 0.786226 − 1.36178i −0.557959 + 1.63972i −0.236304 − 0.409291i 0.557959 + 0.966413i 1.79426 + 2.04901i 1.93532 − 3.35208i 2.40175 −2.37736 − 1.82979i 1.75473
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 40.5
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 117.2.e.b ✓ 10
3.b odd 2 1 351.2.e.b 10
9.c even 3 1 inner 117.2.e.b ✓ 10
9.c even 3 1 1053.2.a.k 5
9.d odd 6 1 351.2.e.b 10
9.d odd 6 1 1053.2.a.j 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
117.2.e.b ✓ 10 1.a even 1 1 trivial
117.2.e.b ✓ 10 9.c even 3 1 inner
351.2.e.b 10 3.b odd 2 1
351.2.e.b 10 9.d odd 6 1
1053.2.a.j 5 9.d odd 6 1
1053.2.a.k 5 9.c even 3 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{10} + 2T_{2}^{9} + 9T_{2}^{8} + 2T_{2}^{7} + 31T_{2}^{6} + 9T_{2}^{5} + 60T_{2}^{4} - 6T_{2}^{3} + 54T_{2}^{2} + 18T_{2} + 9 \) acting on \(S_{2}^{\mathrm{new}}(117, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} + 2 T^{9} + \cdots + 9 \) Copy content Toggle raw display
$3$ \( T^{10} - T^{9} + \cdots + 243 \) Copy content Toggle raw display
$5$ \( T^{10} + T^{9} + \cdots + 81 \) Copy content Toggle raw display
$7$ \( T^{10} - 2 T^{9} + \cdots + 1 \) Copy content Toggle raw display
$11$ \( T^{10} + 11 T^{9} + \cdots + 84681 \) Copy content Toggle raw display
$13$ \( (T^{2} + T + 1)^{5} \) Copy content Toggle raw display
$17$ \( (T^{5} - 7 T^{4} + \cdots - 339)^{2} \) Copy content Toggle raw display
$19$ \( (T^{5} - 3 T^{4} - 33 T^{3} + \cdots - 45)^{2} \) Copy content Toggle raw display
$23$ \( T^{10} + 18 T^{9} + \cdots + 50625 \) Copy content Toggle raw display
$29$ \( T^{10} + 4 T^{9} + \cdots + 225 \) Copy content Toggle raw display
$31$ \( T^{10} - 16 T^{9} + \cdots + 5004169 \) Copy content Toggle raw display
$37$ \( (T^{5} - 5 T^{4} + \cdots - 1223)^{2} \) Copy content Toggle raw display
$41$ \( T^{10} + 12 T^{9} + \cdots + 2025 \) Copy content Toggle raw display
$43$ \( T^{10} + 69 T^{8} + \cdots + 136161 \) Copy content Toggle raw display
$47$ \( T^{10} + \cdots + 316590849 \) Copy content Toggle raw display
$53$ \( (T^{5} + 3 T^{4} + \cdots + 2997)^{2} \) Copy content Toggle raw display
$59$ \( T^{10} + 3 T^{9} + \cdots + 729 \) Copy content Toggle raw display
$61$ \( T^{10} - T^{9} + \cdots + 28058209 \) Copy content Toggle raw display
$67$ \( T^{10} - 4 T^{9} + \cdots + 1050625 \) Copy content Toggle raw display
$71$ \( (T^{5} - 15 T^{4} + \cdots - 7911)^{2} \) Copy content Toggle raw display
$73$ \( (T^{5} + 5 T^{4} + \cdots + 82975)^{2} \) Copy content Toggle raw display
$79$ \( T^{10} + \cdots + 218005225 \) Copy content Toggle raw display
$83$ \( T^{10} + 26 T^{9} + \cdots + 136161 \) Copy content Toggle raw display
$89$ \( (T^{5} - 29 T^{4} + \cdots + 41901)^{2} \) Copy content Toggle raw display
$97$ \( T^{10} + \cdots + 674388961 \) Copy content Toggle raw display
show more
show less