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Properties

Label 1680.2.d.d
Level $1680$
Weight $2$
Character orbit 1680.d
Analytic conductor $13.415$
Analytic rank $0$
Dimension $12$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1680,2,Mod(1231,1680)]
 
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, 0, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1680.1231");
 
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.d (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: \( x^{12} - 6 x^{11} + 35 x^{10} - 120 x^{9} + 328 x^{8} - 658 x^{7} + 1045 x^{6} - 1270 x^{5} + 1183 x^{4} + \cdots + 13 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{14} \)
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.

\(f(q)\) \(=\) \( q + q^{3} + \beta_1 q^{5} + \beta_{8} q^{7} + q^{9} - \beta_{7} q^{11} + (\beta_{6} - \beta_{5} - 2 \beta_1) q^{13} + \beta_1 q^{15} + (\beta_{11} + \beta_1) q^{17} + (\beta_{4} - \beta_{3} - 1) q^{19}+ \cdots - \beta_{7} q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 12 q^{3} - 4 q^{7} + 12 q^{9} - 16 q^{19} - 4 q^{21} - 12 q^{25} + 12 q^{27} + 8 q^{29} + 32 q^{31} + 4 q^{35} - 16 q^{37} + 24 q^{47} - 4 q^{49} + 16 q^{53} - 16 q^{57} + 24 q^{59} - 4 q^{63} + 16 q^{65}+ \cdots + 32 q^{93}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 6 x^{11} + 35 x^{10} - 120 x^{9} + 328 x^{8} - 658 x^{7} + 1045 x^{6} - 1270 x^{5} + 1183 x^{4} + \cdots + 13 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 36 \nu^{11} + 198 \nu^{10} - 1172 \nu^{9} + 3789 \nu^{8} - 10216 \nu^{7} + 19460 \nu^{6} + \cdots + 1156 ) / 99 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 8 \nu^{10} - 40 \nu^{9} + 232 \nu^{8} - 688 \nu^{7} + 1710 \nu^{6} - 2890 \nu^{5} + 3904 \nu^{4} + \cdots + 272 ) / 9 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 4 \nu^{10} - 20 \nu^{9} + 122 \nu^{8} - 368 \nu^{7} + 996 \nu^{6} - 1784 \nu^{5} + 2684 \nu^{4} + \cdots + 232 ) / 3 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 16 \nu^{10} - 80 \nu^{9} + 476 \nu^{8} - 1424 \nu^{7} + 3708 \nu^{6} - 6476 \nu^{5} + 9380 \nu^{4} + \cdots + 781 ) / 9 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 200 \nu^{11} + 935 \nu^{10} - 5625 \nu^{9} + 15924 \nu^{8} - 40742 \nu^{7} + 67108 \nu^{6} + \cdots - 1499 ) / 99 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 200 \nu^{11} - 1265 \nu^{10} + 7275 \nu^{9} - 25626 \nu^{8} + 69650 \nu^{7} - 140764 \nu^{6} + \cdots - 10447 ) / 99 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 452 \nu^{11} - 2486 \nu^{10} + 14478 \nu^{9} - 46506 \nu^{8} + 122264 \nu^{7} - 228298 \nu^{6} + \cdots - 10234 ) / 99 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 614 \nu^{11} - 3355 \nu^{10} + 19631 \nu^{9} - 62803 \nu^{8} + 165728 \nu^{7} - 308630 \nu^{6} + \cdots - 11036 ) / 99 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 614 \nu^{11} + 3399 \nu^{10} - 19851 \nu^{9} + 64211 \nu^{8} - 170040 \nu^{7} + 321104 \nu^{6} + \cdots + 14644 ) / 99 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 800 \nu^{11} - 4400 \nu^{10} + 25800 \nu^{9} - 83100 \nu^{8} + 220784 \nu^{7} - 415744 \nu^{6} + \cdots - 17698 ) / 99 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 328 \nu^{11} + 1804 \nu^{10} - 10578 \nu^{9} + 34071 \nu^{8} - 90532 \nu^{7} + 170492 \nu^{6} + \cdots + 7604 ) / 33 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{10} + 2\beta_{6} - 2\beta_{5} + 2 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{10} + 4\beta_{9} + 4\beta_{8} + 2\beta_{6} - 2\beta_{5} + 2\beta_{4} - 4\beta_{3} - 8 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 6 \beta_{11} + 14 \beta_{10} + 7 \beta_{9} + 5 \beta_{8} + 3 \beta_{7} - 15 \beta_{6} + 15 \beta_{5} + \cdots - 13 ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 12 \beta_{11} + 29 \beta_{10} - 44 \beta_{9} - 48 \beta_{8} + 6 \beta_{7} - 28 \beta_{6} + 36 \beta_{5} + \cdots + 56 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 72 \beta_{11} - 154 \beta_{10} - 138 \beta_{9} - 112 \beta_{8} - 35 \beta_{7} + 156 \beta_{6} + \cdots + 162 ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 246 \beta_{11} - 535 \beta_{10} + 407 \beta_{9} + 495 \beta_{8} - 120 \beta_{7} + 477 \beta_{6} + \cdots - 515 ) / 4 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 655 \beta_{11} + 1401 \beta_{10} + 2146 \beta_{9} + 1900 \beta_{8} + 322 \beta_{7} - 1498 \beta_{6} + \cdots - 2385 ) / 4 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 3796 \beta_{11} + 8169 \beta_{10} - 2928 \beta_{9} - 4332 \beta_{8} + 1862 \beta_{7} - 7426 \beta_{6} + \cdots + 4174 ) / 4 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 4370 \beta_{11} - 9371 \beta_{10} - 29832 \beta_{9} - 28188 \beta_{8} - 2154 \beta_{7} + 11476 \beta_{6} + \cdots + 33800 ) / 4 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 52102 \beta_{11} - 112014 \beta_{10} + 6865 \beta_{9} + 26251 \beta_{8} - 25605 \beta_{7} + \cdots - 19131 ) / 4 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 2533 \beta_{11} + 5385 \beta_{10} + 381155 \beta_{9} + 380155 \beta_{8} + 1274 \beta_{7} - 39735 \beta_{6} + \cdots - 443144 ) / 4 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1231.1
0.500000 − 3.50753i
0.500000 − 1.60175i
0.500000 + 0.280541i
0.500000 + 1.18724i
0.500000 + 0.189815i
0.500000 + 1.45168i
0.500000 + 3.50753i
0.500000 + 1.60175i
0.500000 − 0.280541i
0.500000 − 1.18724i
0.500000 − 0.189815i
0.500000 − 1.45168i
0 1.00000 0 − 1.00000i 0 −2.64150 − 0.149926i 0 1.00000 0
1231.2 0 1.00000 0 − 1.00000i 0 −2.46778 + 0.953976i 0 1.00000 0
1231.3 0 1.00000 0 − 1.00000i 0 −0.585484 − 2.58016i 0 1.00000 0
1231.4 0 1.00000 0 − 1.00000i 0 0.321212 + 2.62618i 0 1.00000 0
1231.5 0 1.00000 0 − 1.00000i 0 1.05584 + 2.42594i 0 1.00000 0
1231.6 0 1.00000 0 − 1.00000i 0 2.31771 − 1.27602i 0 1.00000 0
1231.7 0 1.00000 0 1.00000i 0 −2.64150 + 0.149926i 0 1.00000 0
1231.8 0 1.00000 0 1.00000i 0 −2.46778 − 0.953976i 0 1.00000 0
1231.9 0 1.00000 0 1.00000i 0 −0.585484 + 2.58016i 0 1.00000 0
1231.10 0 1.00000 0 1.00000i 0 0.321212 − 2.62618i 0 1.00000 0
1231.11 0 1.00000 0 1.00000i 0 1.05584 − 2.42594i 0 1.00000 0
1231.12 0 1.00000 0 1.00000i 0 2.31771 + 1.27602i 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1231.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
28.d 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.d.d yes 12
3.b odd 2 1 5040.2.d.f 12
4.b odd 2 1 1680.2.d.c ✓ 12
7.b odd 2 1 1680.2.d.c ✓ 12
12.b even 2 1 5040.2.d.g 12
21.c even 2 1 5040.2.d.g 12
28.d even 2 1 inner 1680.2.d.d yes 12
84.h odd 2 1 5040.2.d.f 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1680.2.d.c ✓ 12 4.b odd 2 1
1680.2.d.c ✓ 12 7.b odd 2 1
1680.2.d.d yes 12 1.a even 1 1 trivial
1680.2.d.d yes 12 28.d even 2 1 inner
5040.2.d.f 12 3.b odd 2 1
5040.2.d.f 12 84.h odd 2 1
5040.2.d.g 12 12.b even 2 1
5040.2.d.g 12 21.c even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1680, [\chi])\):

\( T_{11}^{12} + 96T_{11}^{10} + 3360T_{11}^{8} + 51840T_{11}^{6} + 334080T_{11}^{4} + 718848T_{11}^{2} + 331776 \) Copy content Toggle raw display
\( T_{19}^{6} + 8T_{19}^{5} - 40T_{19}^{4} - 304T_{19}^{3} + 656T_{19}^{2} + 2624T_{19} - 5312 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( (T - 1)^{12} \) Copy content Toggle raw display
$5$ \( (T^{2} + 1)^{6} \) Copy content Toggle raw display
$7$ \( T^{12} + 4 T^{11} + \cdots + 117649 \) Copy content Toggle raw display
$11$ \( T^{12} + 96 T^{10} + \cdots + 331776 \) Copy content Toggle raw display
$13$ \( T^{12} + 88 T^{10} + \cdots + 331776 \) Copy content Toggle raw display
$17$ \( T^{12} + 112 T^{10} + \cdots + 36864 \) Copy content Toggle raw display
$19$ \( (T^{6} + 8 T^{5} + \cdots - 5312)^{2} \) Copy content Toggle raw display
$23$ \( T^{12} + 184 T^{10} + \cdots + 36864 \) Copy content Toggle raw display
$29$ \( (T^{6} - 4 T^{5} + \cdots + 576)^{2} \) Copy content Toggle raw display
$31$ \( (T^{6} - 16 T^{5} + \cdots - 192)^{2} \) Copy content Toggle raw display
$37$ \( (T^{6} + 8 T^{5} + \cdots + 12352)^{2} \) Copy content Toggle raw display
$41$ \( T^{12} + 232 T^{10} + \cdots + 50466816 \) Copy content Toggle raw display
$43$ \( T^{12} + 208 T^{10} + \cdots + 6230016 \) Copy content Toggle raw display
$47$ \( (T^{6} - 12 T^{5} + \cdots + 5184)^{2} \) Copy content Toggle raw display
$53$ \( (T^{6} - 8 T^{5} + \cdots + 576)^{2} \) Copy content Toggle raw display
$59$ \( (T^{6} - 12 T^{5} + \cdots - 1728)^{2} \) Copy content Toggle raw display
$61$ \( T^{12} + 376 T^{10} + \cdots + 6230016 \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 178259595264 \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 9475854336 \) Copy content Toggle raw display
$73$ \( T^{12} + 280 T^{10} + \cdots + 6230016 \) Copy content Toggle raw display
$79$ \( T^{12} + 456 T^{10} + \cdots + 2985984 \) Copy content Toggle raw display
$83$ \( (T^{6} - 8 T^{5} + \cdots - 294912)^{2} \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 732893184 \) Copy content Toggle raw display
$97$ \( T^{12} + 472 T^{10} + \cdots + 331776 \) Copy content Toggle raw display
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