Properties

Label 4680.2.l.h
Level $4680$
Weight $2$
Character orbit 4680.l
Analytic conductor $37.370$
Analytic rank $0$
Dimension $10$
CM no
Inner twists $2$

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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: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} + 13x^{8} + 56x^{6} + 97x^{4} + 61x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 1560)
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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{2} q^{5} - \beta_{9} q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{2} q^{5} - \beta_{9} q^{7} + (\beta_{4} - 1) q^{11} + \beta_{6} q^{13} + ( - \beta_{9} - \beta_{8} + 2 \beta_{6} + \beta_{5} - \beta_{3} + \beta_{2}) q^{17} + (\beta_{8} + \beta_{5} - \beta_{4} - \beta_1 - 1) q^{19} + ( - \beta_{9} - \beta_{8} + \beta_{5} - \beta_{3} + \beta_{2}) q^{23} + (\beta_{9} + \beta_{8} + \beta_{7} - \beta_{4} - \beta_{2} - \beta_1 + 1) q^{25} + ( - \beta_{4} - \beta_{3} - \beta_{2} - \beta_1 + 1) q^{29} + (2 \beta_{8} + 2 \beta_{5}) q^{31} + (2 \beta_{9} + \beta_{8} - \beta_{7} - 3 \beta_{6} - \beta_{5} - \beta_{4} - \beta_{3} - \beta_{2} - \beta_1 - 1) q^{35} + ( - 2 \beta_{9} - \beta_{8} - \beta_{7} - \beta_{6} + \beta_{5} - 2 \beta_{3} + 2 \beta_{2}) q^{37} + ( - \beta_{8} - \beta_{5} - 3 \beta_{4} - \beta_{3} - \beta_{2} - 2 \beta_1 + 1) q^{41} + ( - 4 \beta_{9} - \beta_{8} + 2 \beta_{6} + \beta_{5} - \beta_{3} + \beta_{2}) q^{43} + ( - 2 \beta_{9} - 2 \beta_{8} + 2 \beta_{7} + 4 \beta_{6} + 2 \beta_{5} - \beta_{3} + \beta_{2}) q^{47} + (\beta_{4} - \beta_{3} - \beta_{2}) q^{49} + ( - 4 \beta_{9} - 3 \beta_{8} + \beta_{7} + \beta_{6} + 3 \beta_{5} - 2 \beta_{3} + 2 \beta_{2}) q^{53} + (\beta_{9} + 3 \beta_{8} - \beta_{7} + \beta_{3} - \beta_1 + 1) q^{55} + ( - 2 \beta_{8} - 2 \beta_{5} + 2 \beta_{4} - \beta_{3} - \beta_{2} + 2 \beta_1 + 4) q^{59} + ( - \beta_{8} - \beta_{5} - 3 \beta_{4} - 2 \beta_{3} - 2 \beta_{2} - 3) q^{61} + \beta_{5} q^{65} + (4 \beta_{9} + 2 \beta_{8} - 4 \beta_{6} - 2 \beta_{5} + 2 \beta_{3} - 2 \beta_{2}) q^{67} + ( - \beta_{8} - \beta_{5} - 3 \beta_{4} - 3 \beta_{3} - 3 \beta_{2} - 2 \beta_1 - 1) q^{71} + ( - \beta_{9} + \beta_{8} - \beta_{7} + 3 \beta_{6} - \beta_{5} - 2 \beta_{3} + 2 \beta_{2}) q^{73} + (4 \beta_{9} + \beta_{8} - \beta_{7} + \beta_{6} - \beta_{5} + 2 \beta_{3} - 2 \beta_{2}) q^{77} + ( - \beta_{8} - \beta_{5} + 3 \beta_{4} + 2 \beta_{3} + 2 \beta_{2} + 4 \beta_1 - 3) q^{79} + (2 \beta_{9} - 2 \beta_{7} - \beta_{3} + \beta_{2}) q^{83} + ( - \beta_{7} + 3 \beta_{6} + 2 \beta_{5} + \beta_{4} - 2 \beta_{3} - \beta_1 + 3) q^{85} + ( - \beta_{8} - \beta_{5} + \beta_{4} + \beta_{3} + \beta_{2} + 2 \beta_1 + 1) q^{89} + \beta_1 q^{91} + ( - \beta_{9} - 2 \beta_{8} + \beta_{7} - 3 \beta_{6} + 2 \beta_{4} - \beta_{3} + 3 \beta_{2} + \cdots - 4) q^{95}+ \cdots + (3 \beta_{9} + 2 \beta_{8} - 2 \beta_{7} - 4 \beta_{6} - 2 \beta_{5}) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 2 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 10 q - 2 q^{5} - 10 q^{11} - 12 q^{19} + 6 q^{25} + 4 q^{29} - 16 q^{35} + 2 q^{41} - 4 q^{49} + 10 q^{55} + 40 q^{59} - 38 q^{61} - 26 q^{71} - 14 q^{79} + 24 q^{85} + 18 q^{89} + 2 q^{91} - 32 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} + 13x^{8} + 56x^{6} + 97x^{4} + 61x^{2} + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( -\nu^{8} - 11\nu^{6} - 34\nu^{4} - 29\nu^{2} - 1 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 3\nu^{9} + 3\nu^{8} + 34\nu^{7} + 34\nu^{6} + 110\nu^{5} + 110\nu^{4} + 93\nu^{3} + 97\nu^{2} - 16\nu ) / 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -3\nu^{9} + 3\nu^{8} - 34\nu^{7} + 34\nu^{6} - 110\nu^{5} + 110\nu^{4} - 93\nu^{3} + 97\nu^{2} + 16\nu ) / 4 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -7\nu^{8} - 80\nu^{6} - 266\nu^{4} - 257\nu^{2} - 12 ) / 2 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 5\nu^{9} + 5\nu^{8} + 56\nu^{7} + 58\nu^{6} + 178\nu^{5} + 198\nu^{4} + 151\nu^{3} + 203\nu^{2} - 14\nu + 20 ) / 4 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -2\nu^{9} - 23\nu^{7} - 78\nu^{5} - 82\nu^{3} - 13\nu ) / 2 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( \nu^{9} + 12\nu^{7} + 44\nu^{5} + 54\nu^{3} + 14\nu \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -5\nu^{9} + 5\nu^{8} - 56\nu^{7} + 58\nu^{6} - 178\nu^{5} + 198\nu^{4} - 151\nu^{3} + 203\nu^{2} + 14\nu + 20 ) / 4 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 5\nu^{9} + 57\nu^{7} + 188\nu^{5} + 177\nu^{3} + 5\nu ) / 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{9} + \beta_{8} - \beta_{7} - \beta_{6} - \beta_{5} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{8} + \beta_{5} + \beta_{4} - \beta _1 - 5 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{9} - 2\beta_{8} + 2\beta_{7} + 3\beta_{6} + 2\beta_{5} + \beta_{3} - \beta_{2} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -6\beta_{8} - 6\beta_{5} - 7\beta_{4} - 3\beta_{3} - 3\beta_{2} + 5\beta _1 + 23 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 3\beta_{9} + 20\beta_{8} - 21\beta_{7} - 41\beta_{6} - 20\beta_{5} - 15\beta_{3} + 15\beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 19\beta_{8} + 19\beta_{5} + 23\beta_{4} + 14\beta_{3} + 14\beta_{2} - 12\beta _1 - 64 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 7\beta_{9} - 111\beta_{8} + 125\beta_{7} + 273\beta_{6} + 111\beta_{5} + 98\beta_{3} - 98\beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( -243\beta_{8} - 243\beta_{5} - 297\beta_{4} - 206\beta_{3} - 206\beta_{2} + 121\beta _1 + 769 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( -61\beta_{9} + 327\beta_{8} - 388\beta_{7} - 891\beta_{6} - 327\beta_{5} - 312\beta_{3} + 312\beta_{2} \) Copy content Toggle raw display

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
1.28447i
1.28447i
1.84576i
1.84576i
1.23118i
1.23118i
0.271831i
0.271831i
2.52064i
2.52064i
0 0 0 −2.23081 0.153266i 0 3.71215i 0 0 0
2809.2 0 0 0 −2.23081 + 0.153266i 0 3.71215i 0 0 0
2809.3 0 0 0 −1.49436 1.66340i 0 3.42163i 0 0 0
2809.4 0 0 0 −1.49436 + 1.66340i 0 3.42163i 0 0 0
2809.5 0 0 0 −0.782984 2.09450i 0 2.13051i 0 0 0
2809.6 0 0 0 −0.782984 + 2.09450i 0 2.13051i 0 0 0
2809.7 0 0 0 1.64514 1.51444i 0 0.961637i 0 0 0
2809.8 0 0 0 1.64514 + 1.51444i 0 0.961637i 0 0 0
2809.9 0 0 0 1.86302 1.23660i 0 2.45939i 0 0 0
2809.10 0 0 0 1.86302 + 1.23660i 0 2.45939i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2809.10
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.h 10
3.b odd 2 1 1560.2.l.f 10
5.b even 2 1 inner 4680.2.l.h 10
12.b even 2 1 3120.2.l.q 10
15.d odd 2 1 1560.2.l.f 10
15.e even 4 1 7800.2.a.bz 5
15.e even 4 1 7800.2.a.ca 5
60.h even 2 1 3120.2.l.q 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1560.2.l.f 10 3.b odd 2 1
1560.2.l.f 10 15.d odd 2 1
3120.2.l.q 10 12.b even 2 1
3120.2.l.q 10 60.h even 2 1
4680.2.l.h 10 1.a even 1 1 trivial
4680.2.l.h 10 5.b even 2 1 inner
7800.2.a.bz 5 15.e even 4 1
7800.2.a.ca 5 15.e even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(4680, [\chi])\):

\( T_{7}^{10} + 37T_{7}^{8} + 492T_{7}^{6} + 2832T_{7}^{4} + 6656T_{7}^{2} + 4096 \) Copy content Toggle raw display
\( T_{11}^{5} + 5T_{11}^{4} - 10T_{11}^{3} - 66T_{11}^{2} + 8T_{11} + 184 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} \) Copy content Toggle raw display
$3$ \( T^{10} \) Copy content Toggle raw display
$5$ \( T^{10} + 2 T^{9} - T^{8} - 4 T^{7} + \cdots + 3125 \) Copy content Toggle raw display
$7$ \( T^{10} + 37 T^{8} + 492 T^{6} + \cdots + 4096 \) Copy content Toggle raw display
$11$ \( (T^{5} + 5 T^{4} - 10 T^{3} - 66 T^{2} + \cdots + 184)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + 1)^{5} \) Copy content Toggle raw display
$17$ \( T^{10} + 61 T^{8} + 388 T^{6} + \cdots + 256 \) Copy content Toggle raw display
$19$ \( (T^{5} + 6 T^{4} - 28 T^{3} - 160 T^{2} + \cdots + 128)^{2} \) Copy content Toggle raw display
$23$ \( T^{10} + 53 T^{8} + 860 T^{6} + \cdots + 1024 \) Copy content Toggle raw display
$29$ \( (T^{5} - 2 T^{4} - 24 T^{3} + 24 T^{2} + \cdots - 64)^{2} \) Copy content Toggle raw display
$31$ \( (T^{5} - 88 T^{3} + 64 T^{2} + 1664 T - 1024)^{2} \) Copy content Toggle raw display
$37$ \( T^{10} + 225 T^{8} + 16256 T^{6} + \cdots + 135424 \) Copy content Toggle raw display
$41$ \( (T^{5} - T^{4} - 186 T^{3} - 226 T^{2} + \cdots + 29816)^{2} \) Copy content Toggle raw display
$43$ \( T^{10} + 376 T^{8} + \cdots + 409010176 \) Copy content Toggle raw display
$47$ \( T^{10} + 240 T^{8} + \cdots + 40960000 \) Copy content Toggle raw display
$53$ \( T^{10} + 369 T^{8} + 44832 T^{6} + \cdots + 891136 \) Copy content Toggle raw display
$59$ \( (T^{5} - 20 T^{4} - 106 T^{3} + \cdots - 45088)^{2} \) Copy content Toggle raw display
$61$ \( (T^{5} + 19 T^{4} - 104 T^{3} + \cdots - 62912)^{2} \) Copy content Toggle raw display
$67$ \( T^{10} + 368 T^{8} + \cdots + 16777216 \) Copy content Toggle raw display
$71$ \( (T^{5} + 13 T^{4} - 210 T^{3} - 3422 T^{2} + \cdots - 3104)^{2} \) Copy content Toggle raw display
$73$ \( T^{10} + 328 T^{8} + \cdots + 190219264 \) Copy content Toggle raw display
$79$ \( (T^{5} + 7 T^{4} - 192 T^{3} - 300 T^{2} + \cdots + 2752)^{2} \) Copy content Toggle raw display
$83$ \( T^{10} + 304 T^{8} + 29460 T^{6} + \cdots + 1048576 \) Copy content Toggle raw display
$89$ \( (T^{5} - 9 T^{4} - 10 T^{3} + 206 T^{2} + \cdots - 200)^{2} \) Copy content Toggle raw display
$97$ \( T^{10} + 293 T^{8} + 28692 T^{6} + \cdots + 2408704 \) Copy content Toggle raw display
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