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Properties

Label 4680.2.l.j
Level $4680$
Weight $2$
Character orbit 4680.l
Analytic conductor $37.370$
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] = [4680,2,Mod(2809,4680)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4680, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("4680.2809");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4680 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4680.l (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: \(37.3699881460\)
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: \( x^{18} - 8 x^{15} + 150 x^{14} - 40 x^{13} + 32 x^{12} - 976 x^{11} + 6361 x^{10} - 4696 x^{9} + \cdots + 128 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
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.

\(f(q)\) \(=\) \( q + \beta_{9} q^{5} + ( - \beta_{2} - \beta_1) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{9} q^{5} + ( - \beta_{2} - \beta_1) q^{7} + ( - \beta_{13} - 1) q^{11} + \beta_{2} q^{13} + (\beta_{17} - \beta_{15} + \cdots - \beta_1) q^{17}+ \cdots + ( - \beta_{17} + \beta_{14} + \cdots + \beta_{2}) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q+O(q^{10}) \) Copy content Toggle raw display \( 18 q - 12 q^{11} - 12 q^{19} - 10 q^{25} + 20 q^{29} + 4 q^{31} + 12 q^{35} - 28 q^{41} - 6 q^{49} + 20 q^{55} + 12 q^{59} + 2 q^{65} - 56 q^{71} - 12 q^{79} - 28 q^{85} + 12 q^{89} + 12 q^{91} + 52 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{18} - 8 x^{15} + 150 x^{14} - 40 x^{13} + 32 x^{12} - 976 x^{11} + 6361 x^{10} - 4696 x^{9} + \cdots + 128 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 11\!\cdots\!95 \nu^{17} + \cdots - 68\!\cdots\!16 ) / 13\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 44\!\cdots\!73 \nu^{17} + \cdots + 32\!\cdots\!64 ) / 13\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 13\!\cdots\!27 \nu^{17} + \cdots + 27\!\cdots\!48 ) / 27\!\cdots\!16 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 19\!\cdots\!05 \nu^{17} + \cdots - 62\!\cdots\!24 ) / 27\!\cdots\!16 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 62\!\cdots\!23 \nu^{17} + \cdots - 36\!\cdots\!12 ) / 82\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 32\!\cdots\!37 \nu^{17} + \cdots - 21\!\cdots\!84 ) / 41\!\cdots\!24 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 66\!\cdots\!05 \nu^{17} + \cdots + 53\!\cdots\!88 ) / 82\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 34\!\cdots\!44 \nu^{17} + \cdots + 43\!\cdots\!52 ) / 41\!\cdots\!24 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 72\!\cdots\!19 \nu^{17} + \cdots - 57\!\cdots\!52 ) / 82\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 36\!\cdots\!76 \nu^{17} + \cdots + 43\!\cdots\!16 ) / 41\!\cdots\!24 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 78\!\cdots\!33 \nu^{17} + \cdots + 71\!\cdots\!84 ) / 82\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 13\!\cdots\!70 \nu^{17} + \cdots - 13\!\cdots\!40 ) / 13\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 98\!\cdots\!89 \nu^{17} + \cdots + 29\!\cdots\!88 ) / 82\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 42\!\cdots\!75 \nu^{17} + \cdots + 45\!\cdots\!20 ) / 27\!\cdots\!16 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 64\!\cdots\!95 \nu^{17} + \cdots + 40\!\cdots\!04 ) / 41\!\cdots\!24 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 25\!\cdots\!00 \nu^{17} + \cdots + 15\!\cdots\!48 ) / 13\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 77\!\cdots\!18 \nu^{17} + \cdots + 66\!\cdots\!52 ) / 41\!\cdots\!24 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{9} + \beta_{8} - \beta_{7} + \beta_{6} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{15} - 2\beta_{11} + \beta_{10} - 8\beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - \beta_{17} + \beta_{13} + \beta_{12} - \beta_{11} + 2 \beta_{10} + 7 \beta_{9} + 7 \beta_{8} + \cdots + 4 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 6 \beta_{16} - 11 \beta_{15} + 18 \beta_{13} - 2 \beta_{12} + 11 \beta_{10} + 2 \beta_{9} + 4 \beta_{8} + \cdots - 60 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 11 \beta_{17} - 24 \beta_{15} + 11 \beta_{13} + 15 \beta_{12} + 11 \beta_{11} + 55 \beta_{9} - 53 \beta_{8} + \cdots + 38 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 24 \beta_{17} - 105 \beta_{15} + 88 \beta_{14} + 162 \beta_{11} - 105 \beta_{10} + 60 \beta_{9} + \cdots - 20 \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 105 \beta_{17} - 109 \beta_{13} - 177 \beta_{12} + 109 \beta_{11} - 254 \beta_{10} - 423 \beta_{9} + \cdots - 340 ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 1002 \beta_{16} + 975 \beta_{15} - 1482 \beta_{13} + 390 \beta_{12} - 975 \beta_{10} - 454 \beta_{9} + \cdots + 4332 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 975 \beta_{17} + 8 \beta_{16} + 2588 \beta_{15} - 8 \beta_{14} - 1039 \beta_{13} - 1899 \beta_{12} + \cdots - 3066 ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 2588 \beta_{17} + 8973 \beta_{15} - 10396 \beta_{14} - 13706 \beta_{11} + 8973 \beta_{10} + \cdots + 1096 \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 8973 \beta_{17} - 160 \beta_{16} - 160 \beta_{14} + 9713 \beta_{13} + 19421 \beta_{12} - 9713 \beta_{11} + \cdots + 27968 ) / 2 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 103166 \beta_{16} - 82331 \beta_{15} + 127586 \beta_{13} - 44874 \beta_{12} + 82331 \beta_{10} + \cdots - 347164 ) / 2 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 82331 \beta_{17} - 2032 \beta_{16} - 252672 \beta_{15} + 2032 \beta_{14} + 89859 \beta_{13} + \cdots + 257574 ) / 2 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 252672 \beta_{17} - 755233 \beta_{15} + 999808 \beta_{14} + 1192130 \beta_{11} - 755233 \beta_{10} + \cdots - 29356 \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 755233 \beta_{17} + 20416 \beta_{16} + 20416 \beta_{14} - 826613 \beta_{13} - 1892185 \beta_{12} + \cdots - 2390396 ) / 2 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( ( 9561682 \beta_{16} + 6935687 \beta_{15} - 11162298 \beta_{13} + 4537870 \beta_{12} - 6935687 \beta_{10} + \cdots + 29237772 ) / 2 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( ( - 6935687 \beta_{17} + 170136 \beta_{16} + 23500548 \beta_{15} - 170136 \beta_{14} - 7580503 \beta_{13} + \cdots - 22321586 ) / 2 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4680\mathbb{Z}\right)^\times\).

\(n\) \(937\) \(1081\) \(2081\) \(2341\) \(3511\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2809.1
2.15978 + 2.15978i
2.15978 − 2.15978i
1.72930 − 1.72930i
1.72930 + 1.72930i
0.739827 − 0.739827i
0.739827 + 0.739827i
0.691618 + 0.691618i
0.691618 − 0.691618i
0.326284 + 0.326284i
0.326284 − 0.326284i
0.115242 − 0.115242i
0.115242 + 0.115242i
−1.66112 − 1.66112i
−1.66112 + 1.66112i
−1.92536 + 1.92536i
−1.92536 − 1.92536i
−2.17557 + 2.17557i
−2.17557 − 2.17557i
0 0 0 −2.15978 − 0.579083i 0 2.65747i 0 0 0
2809.2 0 0 0 −2.15978 + 0.579083i 0 − 2.65747i 0 0 0
2809.3 0 0 0 −1.72930 − 1.41758i 0 3.77891i 0 0 0
2809.4 0 0 0 −1.72930 + 1.41758i 0 − 3.77891i 0 0 0
2809.5 0 0 0 −0.739827 − 2.11013i 0 − 4.89023i 0 0 0
2809.6 0 0 0 −0.739827 + 2.11013i 0 4.89023i 0 0 0
2809.7 0 0 0 −0.691618 − 2.12642i 0 − 1.64257i 0 0 0
2809.8 0 0 0 −0.691618 + 2.12642i 0 1.64257i 0 0 0
2809.9 0 0 0 −0.326284 − 2.21213i 0 2.14079i 0 0 0
2809.10 0 0 0 −0.326284 + 2.21213i 0 − 2.14079i 0 0 0
2809.11 0 0 0 −0.115242 − 2.23310i 0 0.942230i 0 0 0
2809.12 0 0 0 −0.115242 + 2.23310i 0 − 0.942230i 0 0 0
2809.13 0 0 0 1.66112 − 1.49689i 0 3.43127i 0 0 0
2809.14 0 0 0 1.66112 + 1.49689i 0 − 3.43127i 0 0 0
2809.15 0 0 0 1.92536 − 1.13710i 0 0.885499i 0 0 0
2809.16 0 0 0 1.92536 + 1.13710i 0 − 0.885499i 0 0 0
2809.17 0 0 0 2.17557 − 0.516615i 0 − 0.129455i 0 0 0
2809.18 0 0 0 2.17557 + 0.516615i 0 0.129455i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2809.18
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4680.2.l.j ✓ 18
3.b odd 2 1 4680.2.l.k yes 18
5.b even 2 1 inner 4680.2.l.j ✓ 18
15.d odd 2 1 4680.2.l.k yes 18
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4680.2.l.j ✓ 18 1.a even 1 1 trivial
4680.2.l.j ✓ 18 5.b even 2 1 inner
4680.2.l.k yes 18 3.b odd 2 1
4680.2.l.k yes 18 15.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 \(S_{2}^{\mathrm{new}}(4680, [\chi])\):

\( T_{7}^{18} + 66 T_{7}^{16} + 1681 T_{7}^{14} + 21344 T_{7}^{12} + 145120 T_{7}^{10} + 529040 T_{7}^{8} + \cdots + 4096 \) Copy content Toggle raw display
\( T_{11}^{9} + 6 T_{11}^{8} - 37 T_{11}^{7} - 214 T_{11}^{6} + 276 T_{11}^{5} + 1384 T_{11}^{4} + \cdots + 992 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{18} \) Copy content Toggle raw display
$3$ \( T^{18} \) Copy content Toggle raw display
$5$ \( T^{18} + 5 T^{16} + \cdots + 1953125 \) Copy content Toggle raw display
$7$ \( T^{18} + 66 T^{16} + \cdots + 4096 \) Copy content Toggle raw display
$11$ \( (T^{9} + 6 T^{8} + \cdots + 992)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + 1)^{9} \) Copy content Toggle raw display
$17$ \( T^{18} + 162 T^{16} + \cdots + 2359296 \) Copy content Toggle raw display
$19$ \( (T^{9} + 6 T^{8} + \cdots + 41728)^{2} \) Copy content Toggle raw display
$23$ \( T^{18} + 218 T^{16} + \cdots + 65536 \) Copy content Toggle raw display
$29$ \( (T^{9} - 10 T^{8} + \cdots - 221184)^{2} \) Copy content Toggle raw display
$31$ \( (T^{9} - 2 T^{8} + \cdots + 5529600)^{2} \) Copy content Toggle raw display
$37$ \( T^{18} + \cdots + 390335053824 \) Copy content Toggle raw display
$41$ \( (T^{9} + 14 T^{8} + \cdots - 193184)^{2} \) Copy content Toggle raw display
$43$ \( T^{18} + \cdots + 13931406950400 \) Copy content Toggle raw display
$47$ \( T^{18} + \cdots + 1719926784 \) Copy content Toggle raw display
$53$ \( T^{18} + \cdots + 17738517970944 \) Copy content Toggle raw display
$59$ \( (T^{9} - 6 T^{8} + \cdots - 1124352)^{2} \) Copy content Toggle raw display
$61$ \( (T^{9} - 327 T^{7} + \cdots - 4468736)^{2} \) Copy content Toggle raw display
$67$ \( T^{18} + \cdots + 554986148724736 \) Copy content Toggle raw display
$71$ \( (T^{9} + 28 T^{8} + \cdots - 6561216)^{2} \) Copy content Toggle raw display
$73$ \( T^{18} + \cdots + 37991345618944 \) Copy content Toggle raw display
$79$ \( (T^{9} + 6 T^{8} + \cdots + 251648)^{2} \) Copy content Toggle raw display
$83$ \( T^{18} + \cdots + 37\!\cdots\!76 \) Copy content Toggle raw display
$89$ \( (T^{9} - 6 T^{8} + \cdots - 319511520)^{2} \) Copy content Toggle raw display
$97$ \( T^{18} + \cdots + 26\!\cdots\!76 \) Copy content Toggle raw display
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