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Properties

Label 240.4.v.b
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: 8.0.370150560000.7
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 3x^{6} + 131x^{4} + 705x^{2} + 1600 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{8} \)
Twist minimal: no (minimal twist has level 60)
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.

\(f(q)\) \(=\) \( q + \beta_{3} q^{3} + (\beta_{7} - \beta_{6} + \cdots + \beta_{2}) q^{5}+ \cdots + (3 \beta_{2} - 12 \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{3} q^{3} + (\beta_{7} - \beta_{6} + \cdots + \beta_{2}) q^{5}+ \cdots + ( - 15 \beta_{7} + 135 \beta_{6} + \cdots + 780 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 80 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 80 q^{7} + 120 q^{13} + 120 q^{15} + 312 q^{21} + 40 q^{25} + 448 q^{31} - 600 q^{33} + 600 q^{37} - 480 q^{43} + 1560 q^{45} + 480 q^{51} + 2400 q^{55} - 1560 q^{57} - 528 q^{61} + 960 q^{63} - 2080 q^{67} + 2600 q^{73} - 3120 q^{75} + 3528 q^{81} - 3560 q^{85} + 2400 q^{87} + 1152 q^{91} - 6240 q^{93} - 120 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 3x^{6} + 131x^{4} + 705x^{2} + 1600 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -3\nu^{7} - 14\nu^{5} - 283\nu^{3} - 2320\nu ) / 3200 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{6} - 18\nu^{4} - 81\nu^{2} - 1440 ) / 160 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 6\nu^{7} - 55\nu^{6} + 28\nu^{5} + 10\nu^{4} + 966\nu^{3} - 6655\nu^{2} + 1840\nu - 17600 ) / 3200 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 3\nu^{7} + 14\nu^{5} + 443\nu^{3} + 3760\nu ) / 640 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -6\nu^{7} - 55\nu^{6} - 28\nu^{5} + 10\nu^{4} - 966\nu^{3} - 6655\nu^{2} - 1840\nu - 17600 ) / 3200 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -13\nu^{7} + 10\nu^{6} + 6\nu^{5} - 20\nu^{4} - 1693\nu^{3} + 610\nu^{2} - 3120\nu + 1600 ) / 1600 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -13\nu^{7} - 10\nu^{6} + 6\nu^{5} + 20\nu^{4} - 1693\nu^{3} - 610\nu^{2} - 3120\nu - 1600 ) / 1600 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{5} + \beta_{4} - \beta_{3} + \beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 5\beta_{7} - 5\beta_{6} - 2\beta_{5} - 2\beta_{3} + \beta_{2} - 3 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -9\beta_{5} + 7\beta_{4} + 9\beta_{3} + 71\beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 11\beta_{7} - 11\beta_{6} + 2\beta_{5} + 2\beta_{3} - 33\beta_{2} - 253 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 48\beta_{7} + 48\beta_{6} - 167\beta_{5} - 55\beta_{4} + 167\beta_{3} - 439\beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -603\beta_{7} + 603\beta_{6} + 126\beta_{5} + 126\beta_{3} - 127\beta_{2} - 963 ) / 4 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -224\beta_{7} - 224\beta_{6} + 855\beta_{5} - 1177\beta_{4} - 855\beta_{3} - 9689\beta_1 ) / 4 \) Copy content Toggle raw display

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

Character values

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

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

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
2.52950 + 2.26556i
0.593004 − 1.76556i
−0.593004 − 1.76556i
−2.52950 + 2.26556i
2.52950 − 2.26556i
0.593004 + 1.76556i
−0.593004 + 1.76556i
−2.52950 − 2.26556i
0 −5.05899 + 1.18601i 0 8.06226 + 7.74597i 0 3.75500 + 3.75500i 0 24.1868 − 12.0000i 0
17.2 0 −1.18601 + 5.05899i 0 −8.06226 − 7.74597i 0 3.75500 + 3.75500i 0 −24.1868 − 12.0000i 0
17.3 0 1.18601 − 5.05899i 0 −8.06226 + 7.74597i 0 16.2450 + 16.2450i 0 −24.1868 − 12.0000i 0
17.4 0 5.05899 − 1.18601i 0 8.06226 − 7.74597i 0 16.2450 + 16.2450i 0 24.1868 − 12.0000i 0
113.1 0 −5.05899 − 1.18601i 0 8.06226 − 7.74597i 0 3.75500 − 3.75500i 0 24.1868 + 12.0000i 0
113.2 0 −1.18601 − 5.05899i 0 −8.06226 + 7.74597i 0 3.75500 − 3.75500i 0 −24.1868 + 12.0000i 0
113.3 0 1.18601 + 5.05899i 0 −8.06226 − 7.74597i 0 16.2450 − 16.2450i 0 −24.1868 + 12.0000i 0
113.4 0 5.05899 + 1.18601i 0 8.06226 + 7.74597i 0 16.2450 − 16.2450i 0 24.1868 + 12.0000i 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.b 8
3.b odd 2 1 inner 240.4.v.b 8
4.b odd 2 1 60.4.i.b ✓ 8
5.c odd 4 1 inner 240.4.v.b 8
12.b even 2 1 60.4.i.b ✓ 8
15.e even 4 1 inner 240.4.v.b 8
20.d odd 2 1 300.4.i.f 8
20.e even 4 1 60.4.i.b ✓ 8
20.e even 4 1 300.4.i.f 8
60.h even 2 1 300.4.i.f 8
60.l odd 4 1 60.4.i.b ✓ 8
60.l odd 4 1 300.4.i.f 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
60.4.i.b ✓ 8 4.b odd 2 1
60.4.i.b ✓ 8 12.b even 2 1
60.4.i.b ✓ 8 20.e even 4 1
60.4.i.b ✓ 8 60.l odd 4 1
240.4.v.b 8 1.a even 1 1 trivial
240.4.v.b 8 3.b odd 2 1 inner
240.4.v.b 8 5.c odd 4 1 inner
240.4.v.b 8 15.e even 4 1 inner
300.4.i.f 8 20.d odd 2 1
300.4.i.f 8 20.e even 4 1
300.4.i.f 8 60.h even 2 1
300.4.i.f 8 60.l odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{4} - 40T_{7}^{3} + 800T_{7}^{2} - 4880T_{7} + 14884 \) 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
$3$ \( T^{8} - 882 T^{4} + 531441 \) Copy content Toggle raw display
$5$ \( (T^{4} - 10 T^{2} + 15625)^{2} \) Copy content Toggle raw display
$7$ \( (T^{4} - 40 T^{3} + \cdots + 14884)^{2} \) Copy content Toggle raw display
$11$ \( (T^{4} + 5080 T^{2} + 211600)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} - 60 T^{3} + \cdots + 19044)^{2} \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 58873394410000 \) Copy content Toggle raw display
$19$ \( (T^{4} + 13208 T^{2} + 1430416)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 12\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( (T^{4} - 73480 T^{2} + 126787600)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} - 112 T - 59264)^{4} \) Copy content Toggle raw display
$37$ \( (T^{4} - 300 T^{3} + \cdots + 196616484)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} + 200320 T^{2} + 8434585600)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} + 240 T^{3} + \cdots + 9916574724)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 18\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots + 41\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( (T^{2} - 50960)^{4} \) Copy content Toggle raw display
$61$ \( (T^{2} + 132 T - 58044)^{4} \) Copy content Toggle raw display
$67$ \( (T^{4} + 1040 T^{3} + \cdots + 16610569924)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} + 77080 T^{2} + 1329331600)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} - 1300 T^{3} + \cdots + 44100840004)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} + 1003992 T^{2} + 94246544016)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} + 376345440900)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} - 457480 T^{2} + 41314627600)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} + 60 T^{3} + \cdots + 380952324)^{2} \) Copy content Toggle raw display
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