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Properties

Label 1320.2.w.d
Level $1320$
Weight $2$
Character orbit 1320.w
Analytic conductor $10.540$
Analytic rank $0$
Dimension $18$
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] = [1320,2,Mod(661,1320)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1320, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1320.661");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1320 = 2^{3} \cdot 3 \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1320.w (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: \(10.5402530668\)
Analytic rank: \(0\)
Dimension: \(18\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{18} + x^{16} + 16x^{10} - 16x^{9} + 32x^{8} + 128x^{2} + 512 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{11} \)
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 \(1,\beta_1,\ldots,\beta_{17}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + \beta_{11} q^{3} + \beta_{2} q^{4} - \beta_{11} q^{5} - \beta_{6} q^{6} + \beta_{4} q^{7} + \beta_{3} q^{8} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + \beta_{11} q^{3} + \beta_{2} q^{4} - \beta_{11} q^{5} - \beta_{6} q^{6} + \beta_{4} q^{7} + \beta_{3} q^{8} - q^{9} + \beta_{6} q^{10} + \beta_{11} q^{11} + \beta_{16} q^{12} + ( - \beta_{12} - \beta_{5} + \cdots - \beta_1) q^{13}+ \cdots - \beta_{11} q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q - 2 q^{4} - 2 q^{6} - 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 18 q - 2 q^{4} - 2 q^{6} - 18 q^{9} + 2 q^{10} + 18 q^{15} + 2 q^{16} - 2 q^{22} + 2 q^{24} - 18 q^{25} + 32 q^{26} - 36 q^{31} - 18 q^{33} + 2 q^{36} - 40 q^{38} - 2 q^{40} - 36 q^{46} + 10 q^{49} + 2 q^{54} + 18 q^{55} + 28 q^{57} + 32 q^{58} - 2 q^{60} - 36 q^{62} - 2 q^{64} + 8 q^{71} + 28 q^{73} + 8 q^{79} + 18 q^{81} + 32 q^{82} - 48 q^{86} + 2 q^{88} - 28 q^{89} - 2 q^{90} - 68 q^{92} - 36 q^{94} - 28 q^{95} - 2 q^{96} - 52 q^{97} + 100 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{18} + x^{16} + 16x^{10} - 16x^{9} + 32x^{8} + 128x^{2} + 512 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{13} - \nu^{11} + 16\nu^{4} - 16\nu^{3} ) / 32 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{15} + \nu^{13} + 16\nu^{7} - 16\nu^{6} + 32\nu^{5} ) / 64 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{17} + 10 \nu^{16} - 11 \nu^{15} + 2 \nu^{14} + 20 \nu^{13} - 24 \nu^{12} - 16 \nu^{11} + \cdots + 2816 ) / 896 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 3 \nu^{17} - 16 \nu^{16} + 19 \nu^{15} - 20 \nu^{14} - 18 \nu^{13} + 44 \nu^{12} - 8 \nu^{11} + \cdots - 3072 ) / 896 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 2 \nu^{17} - 8 \nu^{16} + 13 \nu^{15} - 10 \nu^{14} - 9 \nu^{13} + 22 \nu^{12} - 4 \nu^{11} + \cdots - 1536 ) / 448 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - \nu^{17} + 4 \nu^{16} - 3 \nu^{15} - 2 \nu^{14} + 15 \nu^{13} - 18 \nu^{12} + 9 \nu^{11} + \cdots + 768 ) / 224 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 2 \nu^{17} + 6 \nu^{16} - 15 \nu^{15} + 18 \nu^{14} - 9 \nu^{13} - 20 \nu^{12} + 24 \nu^{11} + \cdots + 2048 ) / 448 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 11 \nu^{17} - 2 \nu^{16} - 9 \nu^{15} + 22 \nu^{14} - 4 \nu^{13} - 40 \nu^{12} + 48 \nu^{11} + \cdots + 4096 ) / 1792 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 3 \nu^{17} + 2 \nu^{16} - 5 \nu^{15} + 6 \nu^{14} - 10 \nu^{13} - 16 \nu^{12} + 22 \nu^{11} + \cdots + 1280 ) / 448 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 4 \nu^{17} - 2 \nu^{16} + 5 \nu^{15} + 8 \nu^{14} - 11 \nu^{13} + 2 \nu^{12} + 20 \nu^{11} + \cdots + 512 ) / 448 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 4 \nu^{17} + 9 \nu^{16} - 5 \nu^{15} - \nu^{14} + 11 \nu^{13} - 2 \nu^{12} - 20 \nu^{11} + \cdots + 384 ) / 448 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 5 \nu^{17} - 6 \nu^{16} + \nu^{15} + 10 \nu^{14} - 26 \nu^{13} + 20 \nu^{12} + 18 \nu^{11} + \cdots - 256 ) / 448 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 5 \nu^{17} + 6 \nu^{16} - \nu^{15} - 10 \nu^{14} + 12 \nu^{13} + 8 \nu^{12} - 32 \nu^{11} + \cdots + 256 ) / 448 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 3 \nu^{17} + 2 \nu^{16} - 5 \nu^{15} + 6 \nu^{14} + 4 \nu^{13} - 16 \nu^{12} + 36 \nu^{11} + \cdots + 1280 ) / 224 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{17} - \beta_{15} - \beta_{12} - \beta_{9} + \beta_{4} \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{15} - \beta_{12} + \beta_{9} + \beta_{4} - \beta_{3} \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -\beta_{17} - \beta_{15} + 2\beta_{13} - \beta_{12} + 2\beta_{10} + \beta_{9} + 2\beta_{8} - 2\beta_{5} - \beta_{4} \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -\beta_{15} - 2\beta_{13} + \beta_{12} + 2\beta_{10} - \beta_{9} + 2\beta_{8} + 2\beta_{5} - \beta_{4} + \beta_{3} \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( \beta_{17} + \beta_{15} + 4 \beta_{14} - 2 \beta_{13} + \beta_{12} + 8 \beta_{11} - 2 \beta_{10} + \cdots - 8 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( \beta_{15} + 4 \beta_{14} - 2 \beta_{13} - \beta_{12} + 8 \beta_{11} - 2 \beta_{10} + \beta_{9} + \cdots + 8 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - \beta_{17} + 8 \beta_{16} - \beta_{15} - 4 \beta_{14} + 2 \beta_{13} - \beta_{12} + 8 \beta_{11} + \cdots - 8 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 8 \beta_{17} + 8 \beta_{16} - \beta_{15} - 4 \beta_{14} + 2 \beta_{13} + \beta_{12} - 8 \beta_{11} + \cdots - 8 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( \beta_{17} + 8 \beta_{16} + 17 \beta_{15} + 4 \beta_{14} - 2 \beta_{13} + \beta_{12} - 8 \beta_{11} + \cdots + 8 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 8 \beta_{17} - 8 \beta_{16} - 15 \beta_{15} + 4 \beta_{14} - 2 \beta_{13} - 17 \beta_{12} + 8 \beta_{11} + \cdots + 8 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 17 \beta_{17} - 8 \beta_{16} + 15 \beta_{15} - 4 \beta_{14} + 2 \beta_{13} - \beta_{12} + 8 \beta_{11} + \cdots - 8 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 24 \beta_{17} + 8 \beta_{16} - 17 \beta_{15} - 4 \beta_{14} + 66 \beta_{13} + 17 \beta_{12} + \cdots - 8 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( 33 \beta_{17} + 8 \beta_{16} - 15 \beta_{15} + 4 \beta_{14} - 2 \beta_{13} + 33 \beta_{12} - 136 \beta_{11} + \cdots + 8 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( 40 \beta_{17} - 8 \beta_{16} + 49 \beta_{15} + 4 \beta_{14} - 2 \beta_{13} - 17 \beta_{12} + 8 \beta_{11} + \cdots - 248 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1320\mathbb{Z}\right)^\times\).

\(n\) \(661\) \(881\) \(991\) \(1057\) \(1201\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
661.1
−1.34010 − 0.451804i
−1.34010 + 0.451804i
−1.28792 − 0.584176i
−1.28792 + 0.584176i
−0.653525 − 1.25416i
−0.653525 + 1.25416i
−0.620466 − 1.27084i
−0.620466 + 1.27084i
0.0805765 − 1.41192i
0.0805765 + 1.41192i
0.402092 − 1.35585i
0.402092 + 1.35585i
0.815657 − 1.15529i
0.815657 + 1.15529i
1.23024 − 0.697509i
1.23024 + 0.697509i
1.37345 − 0.337095i
1.37345 + 0.337095i
−1.34010 − 0.451804i 1.00000i 1.59175 + 1.21093i − 1.00000i 0.451804 − 1.34010i −1.27461 −1.58600 − 2.34192i −1.00000 −0.451804 + 1.34010i
661.2 −1.34010 + 0.451804i − 1.00000i 1.59175 − 1.21093i 1.00000i 0.451804 + 1.34010i −1.27461 −1.58600 + 2.34192i −1.00000 −0.451804 − 1.34010i
661.3 −1.28792 − 0.584176i − 1.00000i 1.31748 + 1.50474i 1.00000i −0.584176 + 1.28792i 2.17912 −0.817771 − 2.70763i −1.00000 0.584176 − 1.28792i
661.4 −1.28792 + 0.584176i 1.00000i 1.31748 − 1.50474i − 1.00000i −0.584176 − 1.28792i 2.17912 −0.817771 + 2.70763i −1.00000 0.584176 + 1.28792i
661.5 −0.653525 − 1.25416i 1.00000i −1.14581 + 1.63924i − 1.00000i 1.25416 − 0.653525i −1.00849 2.80468 + 0.365740i −1.00000 −1.25416 + 0.653525i
661.6 −0.653525 + 1.25416i − 1.00000i −1.14581 − 1.63924i 1.00000i 1.25416 + 0.653525i −1.00849 2.80468 − 0.365740i −1.00000 −1.25416 − 0.653525i
661.7 −0.620466 − 1.27084i − 1.00000i −1.23004 + 1.57702i 1.00000i −1.27084 + 0.620466i −1.55868 2.76733 + 0.584694i −1.00000 1.27084 − 0.620466i
661.8 −0.620466 + 1.27084i 1.00000i −1.23004 − 1.57702i − 1.00000i −1.27084 − 0.620466i −1.55868 2.76733 − 0.584694i −1.00000 1.27084 + 0.620466i
661.9 0.0805765 − 1.41192i − 1.00000i −1.98701 − 0.227534i 1.00000i −1.41192 − 0.0805765i 1.10929 −0.481366 + 2.78716i −1.00000 1.41192 + 0.0805765i
661.10 0.0805765 + 1.41192i 1.00000i −1.98701 + 0.227534i − 1.00000i −1.41192 + 0.0805765i 1.10929 −0.481366 − 2.78716i −1.00000 1.41192 − 0.0805765i
661.11 0.402092 − 1.35585i 1.00000i −1.67664 − 1.09035i − 1.00000i 1.35585 + 0.402092i 3.45712 −2.15252 + 1.83485i −1.00000 −1.35585 − 0.402092i
661.12 0.402092 + 1.35585i − 1.00000i −1.67664 + 1.09035i 1.00000i 1.35585 − 0.402092i 3.45712 −2.15252 − 1.83485i −1.00000 −1.35585 + 0.402092i
661.13 0.815657 − 1.15529i − 1.00000i −0.669407 − 1.88465i 1.00000i −1.15529 − 0.815657i −2.33992 −2.72333 − 0.763863i −1.00000 1.15529 + 0.815657i
661.14 0.815657 + 1.15529i 1.00000i −0.669407 + 1.88465i − 1.00000i −1.15529 + 0.815657i −2.33992 −2.72333 + 0.763863i −1.00000 1.15529 − 0.815657i
661.15 1.23024 − 0.697509i 1.00000i 1.02696 − 1.71620i − 1.00000i 0.697509 + 1.23024i −4.71834 0.0663415 − 2.82765i −1.00000 −0.697509 − 1.23024i
661.16 1.23024 + 0.697509i − 1.00000i 1.02696 + 1.71620i 1.00000i 0.697509 − 1.23024i −4.71834 0.0663415 + 2.82765i −1.00000 −0.697509 + 1.23024i
661.17 1.37345 − 0.337095i − 1.00000i 1.77273 − 0.925966i 1.00000i −0.337095 − 1.37345i 4.15452 2.12262 − 1.86935i −1.00000 0.337095 + 1.37345i
661.18 1.37345 + 0.337095i 1.00000i 1.77273 + 0.925966i − 1.00000i −0.337095 + 1.37345i 4.15452 2.12262 + 1.86935i −1.00000 0.337095 − 1.37345i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 661.18
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1320.2.w.d ✓ 18
4.b odd 2 1 5280.2.w.d 18
8.b even 2 1 inner 1320.2.w.d ✓ 18
8.d odd 2 1 5280.2.w.d 18
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1320.2.w.d ✓ 18 1.a even 1 1 trivial
1320.2.w.d ✓ 18 8.b even 2 1 inner
5280.2.w.d 18 4.b odd 2 1
5280.2.w.d 18 8.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1320, [\chi])\):

\( T_{7}^{9} - 34T_{7}^{7} + 328T_{7}^{5} + 128T_{7}^{4} - 1080T_{7}^{3} - 784T_{7}^{2} + 896T_{7} + 768 \) Copy content Toggle raw display
\( T_{23}^{9} - 104 T_{23}^{7} + 144 T_{23}^{6} + 2608 T_{23}^{5} - 7104 T_{23}^{4} - 4928 T_{23}^{3} + \cdots + 3072 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{18} + T^{16} + \cdots + 512 \) Copy content Toggle raw display
$3$ \( (T^{2} + 1)^{9} \) Copy content Toggle raw display
$5$ \( (T^{2} + 1)^{9} \) Copy content Toggle raw display
$7$ \( (T^{9} - 34 T^{7} + \cdots + 768)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} + 1)^{9} \) Copy content Toggle raw display
$13$ \( T^{18} + 112 T^{16} + \cdots + 2359296 \) Copy content Toggle raw display
$17$ \( (T^{9} - 34 T^{7} + \cdots + 768)^{2} \) Copy content Toggle raw display
$19$ \( T^{18} + \cdots + 1605124096 \) Copy content Toggle raw display
$23$ \( (T^{9} - 104 T^{7} + \cdots + 3072)^{2} \) Copy content Toggle raw display
$29$ \( T^{18} + \cdots + 893292544 \) Copy content Toggle raw display
$31$ \( (T^{9} + 18 T^{8} + \cdots - 4096)^{2} \) Copy content Toggle raw display
$37$ \( T^{18} + \cdots + 1824271630336 \) Copy content Toggle raw display
$41$ \( (T^{9} - 126 T^{7} + \cdots - 425792)^{2} \) Copy content Toggle raw display
$43$ \( T^{18} + \cdots + 110166016 \) Copy content Toggle raw display
$47$ \( (T^{9} - 144 T^{7} + \cdots - 263168)^{2} \) Copy content Toggle raw display
$53$ \( T^{18} + \cdots + 5578363764736 \) Copy content Toggle raw display
$59$ \( T^{18} + \cdots + 387620798464 \) Copy content Toggle raw display
$61$ \( T^{18} + \cdots + 4820027047936 \) Copy content Toggle raw display
$67$ \( T^{18} + \cdots + 4177543561216 \) Copy content Toggle raw display
$71$ \( (T^{9} - 4 T^{8} + \cdots + 456704)^{2} \) Copy content Toggle raw display
$73$ \( (T^{9} - 14 T^{8} + \cdots + 71281856)^{2} \) Copy content Toggle raw display
$79$ \( (T^{9} - 4 T^{8} + \cdots - 173115392)^{2} \) Copy content Toggle raw display
$83$ \( T^{18} + \cdots + 11034727284736 \) Copy content Toggle raw display
$89$ \( (T^{9} + 14 T^{8} + \cdots + 14635008)^{2} \) Copy content Toggle raw display
$97$ \( (T^{9} + 26 T^{8} + \cdots - 40031744)^{2} \) Copy content Toggle raw display
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