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Properties

Label 1680.2.v.a
Level $1680$
Weight $2$
Character orbit 1680.v
Analytic conductor $13.415$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1680,2,Mod(239,1680)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1680, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 1, 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("1680.239");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1680 = 2^{4} \cdot 3 \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1680.v (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: \(13.4148675396\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 8x^{10} + 55x^{8} - 272x^{6} + 1375x^{4} - 5000x^{2} + 15625 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{5} \)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{3} q^{3} + (\beta_{4} + \beta_{3} + \beta_{2} - \beta_1) q^{5} - q^{7} + \beta_{7} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{3} q^{3} + (\beta_{4} + \beta_{3} + \beta_{2} - \beta_1) q^{5} - q^{7} + \beta_{7} q^{9} - \beta_{11} q^{11} + ( - \beta_{9} - \beta_{8} + \cdots + \beta_{3}) q^{13}+ \cdots + (\beta_{10} - \beta_{9} + \cdots - 5 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 2 q^{3} - 12 q^{7} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 2 q^{3} - 12 q^{7} + 6 q^{9} + 12 q^{15} + 2 q^{21} + 16 q^{25} + 4 q^{27} + 8 q^{43} - 4 q^{45} + 12 q^{49} - 16 q^{55} + 32 q^{61} - 6 q^{63} - 24 q^{67} + 36 q^{69} + 18 q^{75} + 22 q^{81} + 8 q^{85} + 22 q^{87}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 8x^{10} + 55x^{8} - 272x^{6} + 1375x^{4} - 5000x^{2} + 15625 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 7\nu^{11} - 181\nu^{9} + 760\nu^{7} - 6904\nu^{5} + 18625\nu^{3} - 61875\nu ) / 100000 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 5 \nu^{11} - 8 \nu^{10} - 15 \nu^{9} - 36 \nu^{8} + 200 \nu^{7} + 360 \nu^{6} - 360 \nu^{5} + \cdots - 12500 ) / 40000 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 5 \nu^{11} + 8 \nu^{10} - 15 \nu^{9} + 36 \nu^{8} + 200 \nu^{7} - 360 \nu^{6} - 360 \nu^{5} + \cdots + 12500 ) / 40000 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 34 \nu^{11} - 125 \nu^{10} - 722 \nu^{9} + 375 \nu^{8} + 1720 \nu^{7} - 5000 \nu^{6} + \cdots + 190625 ) / 200000 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 8 \nu^{11} + 25 \nu^{10} - 36 \nu^{9} - 75 \nu^{8} + 360 \nu^{7} + 1000 \nu^{6} - 824 \nu^{5} + \cdots - 38125 ) / 40000 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 37 \nu^{11} + 200 \nu^{10} + 271 \nu^{9} - 1100 \nu^{8} + 40 \nu^{7} + 7000 \nu^{6} + \cdots - 337500 ) / 200000 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 31 \nu^{11} - 45 \nu^{10} - 173 \nu^{9} - 265 \nu^{8} + 480 \nu^{7} - 600 \nu^{6} - 2432 \nu^{5} + \cdots - 209375 ) / 100000 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 31 \nu^{11} - 115 \nu^{10} - 173 \nu^{9} + 1045 \nu^{8} + 480 \nu^{7} - 4200 \nu^{6} + \cdots + 321875 ) / 100000 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 37 \nu^{11} + 200 \nu^{10} - 271 \nu^{9} - 1100 \nu^{8} - 40 \nu^{7} + 7000 \nu^{6} - 3064 \nu^{5} + \cdots - 337500 ) / 200000 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 9\nu^{11} - 47\nu^{9} + 295\nu^{7} - 1073\nu^{5} + 2450\nu^{3} - 13750\nu ) / 12500 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{8} - \beta_{7} + \beta_{6} - \beta_{5} - \beta_{3} + 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -2\beta_{11} + \beta_{10} - \beta_{7} + 4\beta_{4} + 4\beta_{3} + \beta_{2} - \beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{10} - 4\beta_{9} - 4\beta_{8} - 7\beta_{7} + 3\beta_{4} + 5\beta_{3} - 5 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -2\beta_{11} + 3\beta_{10} - 3\beta_{7} + 8\beta_{6} + 8\beta_{5} + 9\beta_{4} + 9\beta_{3} - 24\beta_{2} - 5\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 15\beta_{10} + 8\beta_{9} - 17\beta_{8} - 8\beta_{7} + 7\beta_{6} - 7\beta_{5} - 31\beta_{4} + 40\beta_{3} - 14 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 24 \beta_{11} - 56 \beta_{10} + 56 \beta_{7} + 24 \beta_{6} + 24 \beta_{5} + 27 \beta_{4} + \cdots - 3 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 40 \beta_{10} + 184 \beta_{9} + 56 \beta_{8} + 24 \beta_{7} + 128 \beta_{6} - 128 \beta_{5} - 88 \beta_{4} + \cdots - 143 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 8 \beta_{11} - 216 \beta_{10} + 216 \beta_{7} - 248 \beta_{6} - 248 \beta_{5} + 184 \beta_{4} + \cdots - 271 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 392 \beta_{10} - 56 \beta_{9} - 455 \beta_{8} + 103 \beta_{7} - 111 \beta_{6} + 111 \beta_{5} + \cdots - 734 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 950 \beta_{11} + 793 \beta_{10} - 793 \beta_{7} - 1128 \beta_{6} - 1128 \beta_{5} + 60 \beta_{4} + \cdots - 113 \beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1680\mathbb{Z}\right)^\times\).

\(n\) \(241\) \(337\) \(421\) \(1121\) \(1471\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
239.1
1.20750 + 1.88200i
−1.20750 + 1.88200i
1.20750 − 1.88200i
−1.20750 − 1.88200i
2.15605 + 0.592845i
−2.15605 + 0.592845i
2.15605 − 0.592845i
−2.15605 − 0.592845i
1.84212 + 1.26752i
−1.84212 + 1.26752i
1.84212 − 1.26752i
−1.84212 − 1.26752i
0 −1.64854 − 0.531349i 0 −1.20750 + 1.88200i 0 −1.00000 0 2.43534 + 1.75189i 0
239.2 0 −1.64854 − 0.531349i 0 1.20750 + 1.88200i 0 −1.00000 0 2.43534 + 1.75189i 0
239.3 0 −1.64854 + 0.531349i 0 −1.20750 − 1.88200i 0 −1.00000 0 2.43534 − 1.75189i 0
239.4 0 −1.64854 + 0.531349i 0 1.20750 − 1.88200i 0 −1.00000 0 2.43534 − 1.75189i 0
239.5 0 −0.393401 − 1.68678i 0 −2.15605 + 0.592845i 0 −1.00000 0 −2.69047 + 1.32716i 0
239.6 0 −0.393401 − 1.68678i 0 2.15605 + 0.592845i 0 −1.00000 0 −2.69047 + 1.32716i 0
239.7 0 −0.393401 + 1.68678i 0 −2.15605 − 0.592845i 0 −1.00000 0 −2.69047 − 1.32716i 0
239.8 0 −0.393401 + 1.68678i 0 2.15605 − 0.592845i 0 −1.00000 0 −2.69047 − 1.32716i 0
239.9 0 1.54194 − 0.788944i 0 −1.84212 + 1.26752i 0 −1.00000 0 1.75513 − 2.43300i 0
239.10 0 1.54194 − 0.788944i 0 1.84212 + 1.26752i 0 −1.00000 0 1.75513 − 2.43300i 0
239.11 0 1.54194 + 0.788944i 0 −1.84212 − 1.26752i 0 −1.00000 0 1.75513 + 2.43300i 0
239.12 0 1.54194 + 0.788944i 0 1.84212 − 1.26752i 0 −1.00000 0 1.75513 + 2.43300i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 239.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
20.d odd 2 1 inner
60.h even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1680.2.v.a ✓ 12
3.b odd 2 1 inner 1680.2.v.a ✓ 12
4.b odd 2 1 1680.2.v.b yes 12
5.b even 2 1 1680.2.v.b yes 12
12.b even 2 1 1680.2.v.b yes 12
15.d odd 2 1 1680.2.v.b yes 12
20.d odd 2 1 inner 1680.2.v.a ✓ 12
60.h even 2 1 inner 1680.2.v.a ✓ 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1680.2.v.a ✓ 12 1.a even 1 1 trivial
1680.2.v.a ✓ 12 3.b odd 2 1 inner
1680.2.v.a ✓ 12 20.d odd 2 1 inner
1680.2.v.a ✓ 12 60.h even 2 1 inner
1680.2.v.b yes 12 4.b odd 2 1
1680.2.v.b yes 12 5.b even 2 1
1680.2.v.b yes 12 12.b even 2 1
1680.2.v.b yes 12 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 \(S_{2}^{\mathrm{new}}(1680, [\chi])\):

\( T_{11}^{6} - 49T_{11}^{4} + 604T_{11}^{2} - 92 \) Copy content Toggle raw display
\( T_{43}^{3} - 2T_{43}^{2} - 40T_{43} + 64 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( (T^{6} + T^{5} - T^{4} + \cdots + 27)^{2} \) Copy content Toggle raw display
$5$ \( T^{12} - 8 T^{10} + \cdots + 15625 \) Copy content Toggle raw display
$7$ \( (T + 1)^{12} \) Copy content Toggle raw display
$11$ \( (T^{6} - 49 T^{4} + \cdots - 92)^{2} \) Copy content Toggle raw display
$13$ \( (T^{6} + 41 T^{4} + \cdots + 184)^{2} \) Copy content Toggle raw display
$17$ \( (T^{6} - 51 T^{4} + \cdots - 1472)^{2} \) Copy content Toggle raw display
$19$ \( (T^{6} + 78 T^{4} + \cdots + 2944)^{2} \) Copy content Toggle raw display
$23$ \( (T^{6} + 34 T^{4} + \cdots + 32)^{2} \) Copy content Toggle raw display
$29$ \( (T^{6} + 99 T^{4} + \cdots + 8192)^{2} \) Copy content Toggle raw display
$31$ \( (T^{6} + 70 T^{4} + \cdots + 11776)^{2} \) Copy content Toggle raw display
$37$ \( (T^{6} + 144 T^{4} + \cdots + 47104)^{2} \) Copy content Toggle raw display
$41$ \( (T^{6} + 230 T^{4} + \cdots + 270848)^{2} \) Copy content Toggle raw display
$43$ \( (T^{3} - 2 T^{2} - 40 T + 64)^{4} \) Copy content Toggle raw display
$47$ \( (T^{6} + 91 T^{4} + \cdots + 2048)^{2} \) Copy content Toggle raw display
$53$ \( (T^{6} - 82 T^{4} + \cdots - 1472)^{2} \) Copy content Toggle raw display
$59$ \( (T^{6} - 324 T^{4} + \cdots - 376832)^{2} \) Copy content Toggle raw display
$61$ \( (T^{3} - 8 T^{2} + \cdots + 128)^{4} \) Copy content Toggle raw display
$67$ \( (T^{3} + 6 T^{2} + \cdots - 736)^{4} \) Copy content Toggle raw display
$71$ \( (T^{6} - 112 T^{4} + \cdots - 5888)^{2} \) Copy content Toggle raw display
$73$ \( (T^{6} + 366 T^{4} + \cdots + 753664)^{2} \) Copy content Toggle raw display
$79$ \( (T^{6} + 483 T^{4} + \cdots + 3014656)^{2} \) Copy content Toggle raw display
$83$ \( (T^{6} + 228 T^{4} + \cdots + 512)^{2} \) Copy content Toggle raw display
$89$ \( (T^{6} + 294 T^{4} + \cdots + 32768)^{2} \) Copy content Toggle raw display
$97$ \( (T^{6} + 361 T^{4} + \cdots + 406456)^{2} \) Copy content Toggle raw display
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