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Properties

Label 240.8.f.e
Level $240$
Weight $8$
Character orbit 240.f
Analytic conductor $74.972$
Analytic rank $0$
Dimension $8$
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] = [240,8,Mod(49,240)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(240, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 1]))
 
N = Newforms(chi, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("240.49");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 240 = 2^{4} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 240.f (of order \(2\), degree \(1\), 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: \(74.9724061162\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 162x^{6} + 7361x^{4} + 87300x^{2} + 160000 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{15}\cdot 3^{12}\cdot 5^{3} \)
Twist minimal: no (minimal twist has level 15)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{3} + (\beta_{2} + 3 \beta_1 - 56) q^{5} + (\beta_{6} - \beta_{3} + \cdots + 11 \beta_1) q^{7}+ \cdots - 729 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{3} + (\beta_{2} + 3 \beta_1 - 56) q^{5} + (\beta_{6} - \beta_{3} + \cdots + 11 \beta_1) q^{7}+ \cdots + (2187 \beta_{5} - 729 \beta_{4} + \cdots + 976860) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 444 q^{5} - 5832 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 444 q^{5} - 5832 q^{9} - 10752 q^{11} + 17496 q^{15} + 30464 q^{19} + 64152 q^{21} + 127616 q^{25} + 240072 q^{29} - 233728 q^{31} - 593520 q^{35} + 454896 q^{39} + 507648 q^{41} + 323676 q^{45} - 3267160 q^{49} - 264384 q^{51} - 3525696 q^{55} - 1091424 q^{59} - 6433520 q^{61} - 2555592 q^{65} - 5940864 q^{69} + 1381824 q^{71} - 7768224 q^{75} + 14380160 q^{79} + 4251528 q^{81} - 1452008 q^{85} + 45778896 q^{89} - 24075648 q^{91} + 25774656 q^{95} + 7838208 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 162x^{6} + 7361x^{4} + 87300x^{2} + 160000 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 9\nu^{7} + 1458\nu^{5} + 62649\nu^{3} + 494100\nu ) / 20000 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 3\nu^{7} - 40\nu^{6} + 586\nu^{5} - 5480\nu^{4} + 35983\nu^{3} - 152440\nu^{2} + 721700\nu + 254000 ) / 5000 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 57\nu^{7} + 9234\nu^{5} + 428777\nu^{3} + 5081300\nu ) / 20000 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 39 \nu^{7} + 480 \nu^{6} - 3518 \nu^{5} + 65760 \nu^{4} + 87321 \nu^{3} + 1829280 \nu^{2} + \cdots - 3028000 ) / 20000 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{6} - 242\nu^{4} - 13441\nu^{2} - 81275 ) / 125 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 9\nu^{7} + 4\nu^{6} + 1348\nu^{5} + 548\nu^{4} + 52639\nu^{3} + 15244\nu^{2} + 464000\nu - 25400 ) / 500 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 39 \nu^{7} - 688 \nu^{6} + 5758 \nu^{5} - 98816 \nu^{4} + 216519 \nu^{3} - 3657328 \nu^{2} + \cdots - 24471200 ) / 4000 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( 12\beta_{6} + 12\beta_{4} - 75\beta_{3} + 48\beta_{2} - 17\beta _1 - 12 ) / 4320 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -14\beta_{7} + 7\beta_{6} + 19\beta_{5} - 31\beta_{4} - 55\beta_{3} + 196\beta_{2} - 24\beta _1 - 87590 ) / 2160 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -732\beta_{6} - 732\beta_{4} + 7275\beta_{3} - 2928\beta_{2} - 16063\beta _1 + 732 ) / 4320 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 214 \beta_{7} - 107 \beta_{6} - 719 \beta_{5} + 431 \beta_{4} + 755 \beta_{3} - 2696 \beta_{2} + \cdots + 1038490 ) / 360 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 43692\beta_{6} + 65292\beta_{4} - 637575\beta_{3} + 239568\beta_{2} + 2253803\beta _1 - 65292 ) / 4320 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 122554 \beta_{7} + 61277 \beta_{6} + 518609 \beta_{5} - 209141 \beta_{4} - 357005 \beta_{3} + \cdots - 506144290 ) / 2160 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 2641452 \beta_{6} - 6140652 \beta_{4} + 56763375 \beta_{3} - 21063408 \beta_{2} - 242768243 \beta _1 + 6140652 ) / 4320 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(31\) \(97\) \(161\) \(181\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
49.1
− 7.26440i
9.60626i
1.49423i
− 3.83609i
7.26440i
− 9.60626i
− 1.49423i
3.83609i
0 − 27.0000i 0 −272.800 + 60.8703i 0 − 1505.28i 0 −729.000 0
49.2 0 − 27.0000i 0 −238.301 + 146.073i 0 923.886i 0 −729.000 0
49.3 0 − 27.0000i 0 57.4967 + 273.531i 0 1233.23i 0 −729.000 0
49.4 0 − 27.0000i 0 231.605 − 156.474i 0 536.160i 0 −729.000 0
49.5 0 27.0000i 0 −272.800 − 60.8703i 0 1505.28i 0 −729.000 0
49.6 0 27.0000i 0 −238.301 − 146.073i 0 − 923.886i 0 −729.000 0
49.7 0 27.0000i 0 57.4967 − 273.531i 0 − 1233.23i 0 −729.000 0
49.8 0 27.0000i 0 231.605 + 156.474i 0 − 536.160i 0 −729.000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 49.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 240.8.f.e 8
4.b odd 2 1 15.8.b.a ✓ 8
5.b even 2 1 inner 240.8.f.e 8
12.b even 2 1 45.8.b.d 8
20.d odd 2 1 15.8.b.a ✓ 8
20.e even 4 1 75.8.a.i 4
20.e even 4 1 75.8.a.j 4
60.h even 2 1 45.8.b.d 8
60.l odd 4 1 225.8.a.z 4
60.l odd 4 1 225.8.a.bb 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
15.8.b.a ✓ 8 4.b odd 2 1
15.8.b.a ✓ 8 20.d odd 2 1
45.8.b.d 8 12.b even 2 1
45.8.b.d 8 60.h even 2 1
75.8.a.i 4 20.e even 4 1
75.8.a.j 4 20.e even 4 1
225.8.a.z 4 60.l odd 4 1
225.8.a.bb 4 60.l odd 4 1
240.8.f.e 8 1.a even 1 1 trivial
240.8.f.e 8 5.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{8} + 4927752T_{7}^{6} + 8012197400976T_{7}^{4} + 4861216797360192000T_{7}^{2} + 845565955456658250240000 \) acting on \(S_{8}^{\mathrm{new}}(240, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( (T^{2} + 729)^{4} \) Copy content Toggle raw display
$5$ \( T^{8} + \cdots + 37\!\cdots\!25 \) Copy content Toggle raw display
$7$ \( T^{8} + \cdots + 84\!\cdots\!00 \) Copy content Toggle raw display
$11$ \( (T^{4} + \cdots - 36924996384576)^{2} \) Copy content Toggle raw display
$13$ \( T^{8} + \cdots + 56\!\cdots\!96 \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 49\!\cdots\!76 \) Copy content Toggle raw display
$19$ \( (T^{4} + \cdots + 23\!\cdots\!00)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 38\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( (T^{4} + \cdots - 98\!\cdots\!00)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} + \cdots + 64\!\cdots\!00)^{2} \) Copy content Toggle raw display
$37$ \( T^{8} + \cdots + 29\!\cdots\!00 \) Copy content Toggle raw display
$41$ \( (T^{4} + \cdots + 11\!\cdots\!00)^{2} \) Copy content Toggle raw display
$43$ \( T^{8} + \cdots + 34\!\cdots\!56 \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 40\!\cdots\!16 \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots + 40\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( (T^{4} + \cdots + 69\!\cdots\!00)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} + \cdots - 76\!\cdots\!44)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots + 30\!\cdots\!96 \) Copy content Toggle raw display
$71$ \( (T^{4} + \cdots + 64\!\cdots\!56)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots + 20\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( (T^{4} + \cdots - 18\!\cdots\!00)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 87\!\cdots\!36 \) Copy content Toggle raw display
$89$ \( (T^{4} + \cdots - 29\!\cdots\!00)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} + \cdots + 13\!\cdots\!96 \) Copy content Toggle raw display
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