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Properties

Label 3920.2.k.d
Level $3920$
Weight $2$
Character orbit 3920.k
Analytic conductor $31.301$
Analytic rank $0$
Dimension $12$
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] = [3920,2,Mod(2351,3920)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3920, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 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("3920.2351");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3920 = 2^{4} \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3920.k (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: \(31.3013575923\)
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: \( x^{12} - 6 x^{11} + 43 x^{10} - 160 x^{9} + 572 x^{8} - 1394 x^{7} + 3039 x^{6} - 4844 x^{5} + \cdots + 657 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: no (minimal twist has level 560)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} + \beta_{6} q^{5} + (\beta_{4} + \beta_{2} + 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} + \beta_{6} q^{5} + (\beta_{4} + \beta_{2} + 2) q^{9} + (\beta_{9} - \beta_{8} + \cdots - 2 \beta_{6}) q^{11}+ \cdots + (2 \beta_{10} + \beta_{9} + \cdots - 8 \beta_{6}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 4 q^{3} + 20 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 4 q^{3} + 20 q^{9} - 4 q^{19} - 12 q^{25} + 20 q^{27} + 16 q^{29} - 8 q^{31} + 8 q^{37} - 24 q^{47} + 24 q^{53} + 24 q^{55} + 16 q^{57} - 48 q^{59} + 4 q^{65} + 4 q^{75} + 12 q^{81} + 36 q^{83} - 16 q^{85} + 60 q^{87} - 56 q^{93}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 6 x^{11} + 43 x^{10} - 160 x^{9} + 572 x^{8} - 1394 x^{7} + 3039 x^{6} - 4844 x^{5} + \cdots + 657 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 8 \nu^{10} - 40 \nu^{9} + 295 \nu^{8} - 940 \nu^{7} + 3336 \nu^{6} - 6886 \nu^{5} + 14489 \nu^{4} + \cdots + 6831 ) / 306 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 13 \nu^{10} - 65 \nu^{9} + 473 \nu^{8} - 1502 \nu^{7} + 5217 \nu^{6} - 10667 \nu^{5} + 21715 \nu^{4} + \cdots + 9513 ) / 204 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 19 \nu^{10} - 95 \nu^{9} + 707 \nu^{8} - 2258 \nu^{7} + 8025 \nu^{6} - 16571 \nu^{5} + 34099 \nu^{4} + \cdots + 12609 ) / 306 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 13 \nu^{10} + 65 \nu^{9} - 507 \nu^{8} + 1638 \nu^{7} - 6169 \nu^{6} + 13047 \nu^{5} - 28821 \nu^{4} + \cdots - 16143 ) / 204 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 50 \nu^{10} - 250 \nu^{9} + 1831 \nu^{8} - 5824 \nu^{7} + 20340 \nu^{6} - 41686 \nu^{5} + 84755 \nu^{4} + \cdots + 35847 ) / 306 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 1690 \nu^{11} + 9295 \nu^{10} - 64491 \nu^{9} + 220497 \nu^{8} - 735856 \nu^{7} + 1611575 \nu^{6} + \cdots + 994725 ) / 617508 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 200 \nu^{11} + 1100 \nu^{10} - 8050 \nu^{9} + 27975 \nu^{8} - 99908 \nu^{7} + 226828 \nu^{6} + \cdots + 90261 ) / 6054 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 200 \nu^{11} + 1100 \nu^{10} - 8050 \nu^{9} + 27975 \nu^{8} - 99908 \nu^{7} + 226828 \nu^{6} + \cdots + 93288 ) / 3027 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 24686 \nu^{11} - 135773 \nu^{10} + 966873 \nu^{9} - 3332631 \nu^{8} + 11497040 \nu^{7} + \cdots - 7364763 ) / 308754 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 27934 \nu^{11} + 153637 \nu^{10} - 1105677 \nu^{9} + 3823269 \nu^{8} - 13407232 \nu^{7} + \cdots + 12449925 ) / 205836 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 19178 \nu^{11} + 105479 \nu^{10} - 762329 \nu^{9} + 2639388 \nu^{8} - 9284864 \nu^{7} + \cdots + 8000478 ) / 102918 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{8} + 2\beta_{7} + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{8} + 2\beta_{7} + 2\beta_{5} - 2\beta_{3} - 4\beta_{2} + 2\beta _1 - 9 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 5 \beta_{11} + \beta_{10} - 4 \beta_{9} + 15 \beta_{8} - 16 \beta_{7} + 5 \beta_{6} + 3 \beta_{5} + \cdots - 14 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 10 \beta_{11} + 2 \beta_{10} - 8 \beta_{9} + 31 \beta_{8} - 34 \beta_{7} + 10 \beta_{6} - 22 \beta_{5} + \cdots + 61 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 64 \beta_{11} - 12 \beta_{10} + 49 \beta_{9} - 161 \beta_{8} + 138 \beta_{7} - 110 \beta_{6} + \cdots + 176 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 217 \beta_{11} - 41 \beta_{10} + 167 \beta_{9} - 561 \beta_{8} + 500 \beta_{7} - 355 \beta_{6} + \cdots - 471 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 613 \beta_{11} + 100 \beta_{10} - 472 \beta_{9} + 1479 \beta_{8} - 1157 \beta_{7} + 1250 \beta_{6} + \cdots - 2281 ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 3488 \beta_{11} + 596 \beta_{10} - 2686 \beta_{9} + 8607 \beta_{8} - 7042 \beta_{7} + 6680 \beta_{6} + \cdots + 3419 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 4248 \beta_{11} - 608 \beta_{10} + 3289 \beta_{9} - 10142 \beta_{8} + 7558 \beta_{7} - 9386 \beta_{6} + \cdots + 29839 ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 48969 \beta_{11} - 7807 \beta_{10} + 37799 \beta_{9} - 119346 \beta_{8} + 94278 \beta_{7} - 99565 \beta_{6} + \cdots - 14424 ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 4234 \beta_{11} - 71 \beta_{10} - 3344 \beta_{9} + 9184 \beta_{8} - 3589 \beta_{7} + 14983 \beta_{6} + \cdots - 379961 ) / 2 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3920\mathbb{Z}\right)^\times\).

\(n\) \(981\) \(1471\) \(3041\) \(3137\)
\(\chi(n)\) \(1\) \(-1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2351.1
0.500000 + 3.52469i
0.500000 − 3.52469i
0.500000 + 1.58132i
0.500000 − 1.58132i
0.500000 + 0.777299i
0.500000 − 0.777299i
0.500000 + 0.694987i
0.500000 − 0.694987i
0.500000 − 2.22465i
0.500000 + 2.22465i
0.500000 − 2.35365i
0.500000 + 2.35365i
0 −2.65867 0 − 1.00000i 0 0 0 4.06851 0
2351.2 0 −2.65867 0 1.00000i 0 0 0 4.06851 0
2351.3 0 −2.44735 0 − 1.00000i 0 0 0 2.98952 0
2351.4 0 −2.44735 0 1.00000i 0 0 0 2.98952 0
2351.5 0 −1.64332 0 − 1.00000i 0 0 0 −0.299485 0
2351.6 0 −1.64332 0 1.00000i 0 0 0 −0.299485 0
2351.7 0 0.171039 0 − 1.00000i 0 0 0 −2.97075 0
2351.8 0 0.171039 0 1.00000i 0 0 0 −2.97075 0
2351.9 0 1.35862 0 − 1.00000i 0 0 0 −1.15414 0
2351.10 0 1.35862 0 1.00000i 0 0 0 −1.15414 0
2351.11 0 3.21968 0 − 1.00000i 0 0 0 7.36634 0
2351.12 0 3.21968 0 1.00000i 0 0 0 7.36634 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2351.12
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 3920.2.k.d 12
4.b odd 2 1 3920.2.k.e 12
7.b odd 2 1 3920.2.k.e 12
7.c even 3 1 560.2.bs.c yes 12
7.d odd 6 1 560.2.bs.b ✓ 12
28.d even 2 1 inner 3920.2.k.d 12
28.f even 6 1 560.2.bs.c yes 12
28.g odd 6 1 560.2.bs.b ✓ 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
560.2.bs.b ✓ 12 7.d odd 6 1
560.2.bs.b ✓ 12 28.g odd 6 1
560.2.bs.c yes 12 7.c even 3 1
560.2.bs.c yes 12 28.f even 6 1
3920.2.k.d 12 1.a even 1 1 trivial
3920.2.k.d 12 28.d even 2 1 inner
3920.2.k.e 12 4.b odd 2 1
3920.2.k.e 12 7.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{6} + 2T_{3}^{5} - 12T_{3}^{4} - 26T_{3}^{3} + 21T_{3}^{2} + 44T_{3} - 8 \) acting on \(S_{2}^{\mathrm{new}}(3920, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( (T^{6} + 2 T^{5} - 12 T^{4} + \cdots - 8)^{2} \) Copy content Toggle raw display
$5$ \( (T^{2} + 1)^{6} \) Copy content Toggle raw display
$7$ \( T^{12} \) Copy content Toggle raw display
$11$ \( T^{12} + 86 T^{10} + \cdots + 82944 \) Copy content Toggle raw display
$13$ \( T^{12} + 106 T^{10} + \cdots + 2108304 \) Copy content Toggle raw display
$17$ \( T^{12} + 112 T^{10} + \cdots + 11943936 \) Copy content Toggle raw display
$19$ \( (T^{6} + 2 T^{5} + \cdots - 396)^{2} \) Copy content Toggle raw display
$23$ \( T^{12} + 134 T^{10} + \cdots + 9801 \) Copy content Toggle raw display
$29$ \( (T^{6} - 8 T^{5} + \cdots + 37476)^{2} \) Copy content Toggle raw display
$31$ \( (T^{6} + 4 T^{5} + \cdots + 10384)^{2} \) Copy content Toggle raw display
$37$ \( (T^{6} - 4 T^{5} - 55 T^{4} + \cdots + 64)^{2} \) Copy content Toggle raw display
$41$ \( T^{12} + 234 T^{10} + \cdots + 77951241 \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots + 394896384 \) Copy content Toggle raw display
$47$ \( (T^{6} + 12 T^{5} + \cdots + 1188)^{2} \) Copy content Toggle raw display
$53$ \( (T^{6} - 12 T^{5} + \cdots + 324)^{2} \) Copy content Toggle raw display
$59$ \( (T^{6} + 24 T^{5} + \cdots - 137664)^{2} \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 742671504 \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 75475473984 \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 14561731584 \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 12230590464 \) Copy content Toggle raw display
$79$ \( T^{12} + 240 T^{10} + \cdots + 22581504 \) Copy content Toggle raw display
$83$ \( (T^{6} - 18 T^{5} + \cdots - 3564)^{2} \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 264722598144 \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 642318336 \) Copy content Toggle raw display
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