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Properties

Label 240.4.v.c
Level $240$
Weight $4$
Character orbit 240.v
Analytic conductor $14.160$
Analytic rank $0$
Dimension $8$
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] = [240,4,Mod(17,240)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(240, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 0, 2, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("240.17");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 240 = 2^{4} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 240.v (of order \(4\), degree \(2\), not 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: \(14.1604584014\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: 8.0.28356903014400.8
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 209x^{4} + 1600 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 15)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{7} - \beta_{5} - \beta_{4} + \cdots + 1) q^{3}+ \cdots + ( - 3 \beta_{7} - 3 \beta_{6} + \cdots + 9 \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{7} - \beta_{5} - \beta_{4} + \cdots + 1) q^{3}+ \cdots + (30 \beta_{7} + 30 \beta_{6} + \cdots + 540 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 6 q^{3} + 16 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 6 q^{3} + 16 q^{7} + 68 q^{13} - 90 q^{15} - 492 q^{21} - 220 q^{25} - 702 q^{27} - 616 q^{31} - 240 q^{33} - 1156 q^{37} - 548 q^{43} + 180 q^{45} + 852 q^{51} - 460 q^{55} + 684 q^{57} + 16 q^{61} - 1428 q^{63} - 404 q^{67} - 2512 q^{73} + 2910 q^{75} + 288 q^{81} + 4940 q^{85} + 1680 q^{87} + 1304 q^{91} + 3408 q^{93} + 1904 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 209x^{4} + 1600 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{6} - 249\nu^{2} ) / 680 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{7} + 249\nu^{3} + 680\nu ) / 680 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{7} + 249\nu^{3} - 680\nu ) / 680 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -9\nu^{7} + 40\nu^{5} - 1561\nu^{3} + 7240\nu ) / 2720 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 13\nu^{7} + 40\nu^{5} + 2557\nu^{3} + 4520\nu ) / 2720 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -9\nu^{6} - 20\nu^{5} - 40\nu^{4} - 1561\nu^{2} - 2260\nu - 4520 ) / 1360 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -9\nu^{6} - 20\nu^{5} + 40\nu^{4} - 1561\nu^{2} - 3620\nu + 4520 ) / 1360 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{3} + \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2\beta_{7} + 2\beta_{6} + 2\beta_{5} + 2\beta_{4} - \beta_{3} - \beta_{2} - 18\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -4\beta_{5} + 4\beta_{4} + 13\beta_{3} + 9\beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 34\beta_{7} - 34\beta_{6} - 17\beta_{3} + 17\beta_{2} - 226 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 68\beta_{5} + 68\beta_{4} + 113\beta_{3} - 181\beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -498\beta_{7} - 498\beta_{6} - 498\beta_{5} - 498\beta_{4} + 249\beta_{3} + 249\beta_{2} + 3122\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 996\beta_{5} - 996\beta_{4} - 2557\beta_{3} - 1561\beta_{2} ) / 2 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/240\mathbb{Z}\right)^\times\).

\(n\) \(31\) \(97\) \(161\) \(181\)
\(\chi(n)\) \(1\) \(-\beta_{1}\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
17.1
1.18766 + 1.18766i
−1.18766 − 1.18766i
−2.66260 − 2.66260i
2.66260 + 2.66260i
1.18766 − 1.18766i
−1.18766 + 1.18766i
−2.66260 + 2.66260i
2.66260 − 2.66260i
0 −5.11173 − 0.932827i 0 −2.48157 − 10.9015i 0 13.3578 + 13.3578i 0 25.2597 + 9.53673i 0
17.2 0 0.932827 + 5.11173i 0 2.48157 + 10.9015i 0 13.3578 + 13.3578i 0 −25.2597 + 9.53673i 0
17.3 0 2.80471 − 4.37420i 0 −9.55729 − 5.80157i 0 −9.35782 − 9.35782i 0 −11.2672 − 24.5367i 0
17.4 0 4.37420 − 2.80471i 0 9.55729 + 5.80157i 0 −9.35782 − 9.35782i 0 11.2672 − 24.5367i 0
113.1 0 −5.11173 + 0.932827i 0 −2.48157 + 10.9015i 0 13.3578 − 13.3578i 0 25.2597 − 9.53673i 0
113.2 0 0.932827 − 5.11173i 0 2.48157 − 10.9015i 0 13.3578 − 13.3578i 0 −25.2597 − 9.53673i 0
113.3 0 2.80471 + 4.37420i 0 −9.55729 + 5.80157i 0 −9.35782 + 9.35782i 0 −11.2672 + 24.5367i 0
113.4 0 4.37420 + 2.80471i 0 9.55729 − 5.80157i 0 −9.35782 + 9.35782i 0 11.2672 + 24.5367i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 17.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.c odd 4 1 inner
15.e even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 240.4.v.c 8
3.b odd 2 1 inner 240.4.v.c 8
4.b odd 2 1 15.4.e.a ✓ 8
5.c odd 4 1 inner 240.4.v.c 8
12.b even 2 1 15.4.e.a ✓ 8
15.e even 4 1 inner 240.4.v.c 8
20.d odd 2 1 75.4.e.c 8
20.e even 4 1 15.4.e.a ✓ 8
20.e even 4 1 75.4.e.c 8
60.h even 2 1 75.4.e.c 8
60.l odd 4 1 15.4.e.a ✓ 8
60.l odd 4 1 75.4.e.c 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
15.4.e.a ✓ 8 4.b odd 2 1
15.4.e.a ✓ 8 12.b even 2 1
15.4.e.a ✓ 8 20.e even 4 1
15.4.e.a ✓ 8 60.l odd 4 1
75.4.e.c 8 20.d odd 2 1
75.4.e.c 8 20.e even 4 1
75.4.e.c 8 60.h even 2 1
75.4.e.c 8 60.l odd 4 1
240.4.v.c 8 1.a even 1 1 trivial
240.4.v.c 8 3.b odd 2 1 inner
240.4.v.c 8 5.c odd 4 1 inner
240.4.v.c 8 15.e even 4 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{4} - 8T_{7}^{3} + 32T_{7}^{2} + 2000T_{7} + 62500 \) acting on \(S_{4}^{\mathrm{new}}(240, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} - 6 T^{7} + \cdots + 531441 \) Copy content Toggle raw display
$5$ \( T^{8} + 110 T^{6} + \cdots + 244140625 \) Copy content Toggle raw display
$7$ \( (T^{4} - 8 T^{3} + \cdots + 62500)^{2} \) Copy content Toggle raw display
$11$ \( (T^{4} + 1990 T^{2} + 961000)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} - 34 T^{3} + \cdots + 6400)^{2} \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 32321044225600 \) Copy content Toggle raw display
$19$ \( (T^{4} + 2772 T^{2} + 876096)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 227401574425600 \) Copy content Toggle raw display
$29$ \( (T^{4} - 18390 T^{2} + 40401000)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} + 154 T - 23096)^{4} \) Copy content Toggle raw display
$37$ \( (T^{4} + 578 T^{3} + \cdots + 1695792400)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} + 160240 T^{2} + 1201216000)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} + 274 T^{3} + \cdots + 193210000)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 82\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots + 14\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( (T^{4} - 709410 T^{2} + 94361796000)^{2} \) Copy content Toggle raw display
$61$ \( (T - 2)^{8} \) Copy content Toggle raw display
$67$ \( (T^{4} + 202 T^{3} + \cdots + 80906113600)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} + 568060 T^{2} + 15888196000)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} + 1256 T^{3} + \cdots + 37974316900)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} + 348072 T^{2} + 797271696)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 15\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( (T^{4} + \cdots + 3249684036000)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} - 952 T^{3} + \cdots + 615926736100)^{2} \) Copy content Toggle raw display
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