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Properties

Label 240.6.f.f
Level $240$
Weight $6$
Character orbit 240.f
Analytic conductor $38.492$
Analytic rank $0$
Dimension $8$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [240,6,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, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("240.49");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 240 = 2^{4} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 6 \)
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: \(38.4921167551\)
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: \( x^{8} + 142x^{6} + 5761x^{4} + 52020x^{2} + 129600 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{10}\cdot 3^{2}\cdot 5^{3} \)
Twist minimal: no (minimal twist has level 120)
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.

\(f(q)\) \(=\) \( q + 9 \beta_1 q^{3} + (\beta_{2} - 10 \beta_1 - 8) q^{5} + (\beta_{5} - \beta_{4} - 2 \beta_1) q^{7} - 81 q^{9} + (2 \beta_{7} + 2 \beta_{6} + 3 \beta_{4} + \cdots - 86) q^{11} + ( - 5 \beta_{7} + 7 \beta_{5} + \cdots - 5) q^{13}+ \cdots + ( - 162 \beta_{7} - 162 \beta_{6} + \cdots + 6966) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 66 q^{5} - 648 q^{9} - 684 q^{11} + 738 q^{15} + 3040 q^{19} + 108 q^{21} - 564 q^{25} - 8004 q^{29} - 6024 q^{31} + 4476 q^{35} - 3996 q^{39} - 37800 q^{41} + 5346 q^{45} - 34368 q^{49} - 11124 q^{51}+ \cdots + 55404 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 142x^{6} + 5761x^{4} + 52020x^{2} + 129600 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{7} - 142\nu^{5} - 5401\nu^{3} - 26460\nu ) / 10800 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -5\nu^{7} + 12\nu^{6} - 674\nu^{5} + 1272\nu^{4} - 25169\nu^{3} + 31980\nu^{2} - 160740\nu + 82080 ) / 2160 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 19\nu^{7} + 2698\nu^{5} - 2700\nu^{4} + 111619\nu^{3} - 218700\nu^{2} + 979740\nu - 1933200 ) / 10800 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 8\nu^{7} - 30\nu^{6} + 1046\nu^{5} - 3855\nu^{4} + 36368\nu^{3} - 134625\nu^{2} + 163530\nu - 688500 ) / 2700 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -22\nu^{7} - 30\nu^{6} - 2854\nu^{5} - 3855\nu^{4} - 97402\nu^{3} - 134625\nu^{2} - 325170\nu - 688500 ) / 2700 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 5\nu^{7} + 60\nu^{6} + 674\nu^{5} + 8520\nu^{4} + 25169\nu^{3} + 317580\nu^{2} + 160740\nu + 1343520 ) / 2160 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 89 \nu^{7} - 300 \nu^{6} - 11738 \nu^{5} - 37200 \nu^{4} - 416789 \nu^{3} - 1236900 \nu^{2} + \cdots - 5929200 ) / 10800 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -4\beta_{7} + 5\beta_{5} + 6\beta_{4} - 3\beta_{3} + 2\beta_{2} + \beta _1 - 4 ) / 120 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -3\beta_{7} - 5\beta_{6} - 10\beta_{4} - 4\beta_{3} - 10\beta_{2} - 4\beta _1 - 1423 ) / 40 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 212\beta_{7} - 265\beta_{5} - 390\beta_{4} + 231\beta_{3} - 250\beta_{2} + 2611\beta _1 + 212 ) / 120 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 243\beta_{7} + 405\beta_{6} + 730\beta_{4} + 244\beta_{3} + 650\beta_{2} + 244\beta _1 + 86623 ) / 40 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -13252\beta_{7} + 17465\beta_{5} + 24630\beta_{4} - 15591\beta_{3} + 17930\beta_{2} - 313811\beta _1 - 13252 ) / 120 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -18483\beta_{7} - 29605\beta_{6} - 50730\beta_{4} - 13764\beta_{3} - 38650\beta_{2} - 13764\beta _1 - 5664063 ) / 40 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 842612 \beta_{7} - 1181065 \beta_{5} - 1549830 \beta_{4} + 1045671 \beta_{3} - 1248730 \beta_{2} + \cdots + 842612 ) / 120 \) 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
− 2.61388i
2.11144i
8.33154i
− 7.82911i
2.61388i
− 2.11144i
− 8.33154i
7.82911i
0 − 9.00000i 0 −45.4388 + 32.5625i 0 − 242.714i 0 −81.0000 0
49.2 0 − 9.00000i 0 −24.7939 − 50.1025i 0 27.6635i 0 −81.0000 0
49.3 0 − 9.00000i 0 −18.3731 + 52.7961i 0 122.878i 0 −81.0000 0
49.4 0 − 9.00000i 0 55.6058 + 5.74389i 0 98.1720i 0 −81.0000 0
49.5 0 9.00000i 0 −45.4388 − 32.5625i 0 242.714i 0 −81.0000 0
49.6 0 9.00000i 0 −24.7939 + 50.1025i 0 − 27.6635i 0 −81.0000 0
49.7 0 9.00000i 0 −18.3731 − 52.7961i 0 − 122.878i 0 −81.0000 0
49.8 0 9.00000i 0 55.6058 − 5.74389i 0 − 98.1720i 0 −81.0000 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 240.6.f.f 8
3.b odd 2 1 720.6.f.o 8
4.b odd 2 1 120.6.f.b ✓ 8
5.b even 2 1 inner 240.6.f.f 8
12.b even 2 1 360.6.f.c 8
15.d odd 2 1 720.6.f.o 8
20.d odd 2 1 120.6.f.b ✓ 8
20.e even 4 1 600.6.a.v 4
20.e even 4 1 600.6.a.w 4
60.h even 2 1 360.6.f.c 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
120.6.f.b ✓ 8 4.b odd 2 1
120.6.f.b ✓ 8 20.d odd 2 1
240.6.f.f 8 1.a even 1 1 trivial
240.6.f.f 8 5.b even 2 1 inner
360.6.f.c 8 12.b even 2 1
360.6.f.c 8 60.h even 2 1
600.6.a.v 4 20.e even 4 1
600.6.a.w 4 20.e even 4 1
720.6.f.o 8 3.b odd 2 1
720.6.f.o 8 15.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{8} + 84412T_{7}^{6} + 1666776048T_{7}^{4} + 9799162809664T_{7}^{2} + 6560352016000000 \) acting on \(S_{6}^{\mathrm{new}}(240, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( (T^{2} + 81)^{4} \) Copy content Toggle raw display
$5$ \( T^{8} + \cdots + 95367431640625 \) Copy content Toggle raw display
$7$ \( T^{8} + \cdots + 65\!\cdots\!00 \) Copy content Toggle raw display
$11$ \( (T^{4} + 342 T^{3} + \cdots - 7954368992)^{2} \) Copy content Toggle raw display
$13$ \( T^{8} + \cdots + 28\!\cdots\!64 \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 18\!\cdots\!96 \) Copy content Toggle raw display
$19$ \( (T^{4} + \cdots - 1452392480768)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 91\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( (T^{4} + \cdots - 318720439844864)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} + \cdots - 51134602752000)^{2} \) Copy content Toggle raw display
$37$ \( T^{8} + \cdots + 12\!\cdots\!00 \) Copy content Toggle raw display
$41$ \( (T^{4} + \cdots + 39\!\cdots\!92)^{2} \) Copy content Toggle raw display
$43$ \( T^{8} + \cdots + 22\!\cdots\!96 \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 41\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots + 25\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( (T^{4} + \cdots - 69\!\cdots\!32)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} + \cdots - 15\!\cdots\!28)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots + 34\!\cdots\!24 \) Copy content Toggle raw display
$71$ \( (T^{4} + \cdots - 12\!\cdots\!32)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots + 35\!\cdots\!16 \) Copy content Toggle raw display
$79$ \( (T^{4} + \cdots + 12\!\cdots\!56)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 13\!\cdots\!44 \) Copy content Toggle raw display
$89$ \( (T^{4} + \cdots - 17\!\cdots\!40)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} + \cdots + 18\!\cdots\!96 \) Copy content Toggle raw display
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