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Properties

Label 360.2.b.d
Level $360$
Weight $2$
Character orbit 360.b
Analytic conductor $2.875$
Analytic rank $0$
Dimension $6$
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] = [360,2,Mod(251,360)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(360, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("360.251");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 360 = 2^{3} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 360.b (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: \(2.87461447277\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.2580992.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 2x^{5} + x^{4} + 2x^{2} - 8x + 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{4} q^{2} + (\beta_{5} + \beta_{2}) q^{4} + q^{5} + \beta_{2} q^{7} + (\beta_{5} - \beta_{3} - \beta_{2}) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{4} q^{2} + (\beta_{5} + \beta_{2}) q^{4} + q^{5} + \beta_{2} q^{7} + (\beta_{5} - \beta_{3} - \beta_{2}) q^{8} - \beta_{4} q^{10} + ( - 2 \beta_{4} - \beta_{3} + \beta_{2} + \cdots - 1) q^{11}+ \cdots - 5 \beta_{4} q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 2 q^{2} + 2 q^{4} + 6 q^{5} + 2 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 2 q^{2} + 2 q^{4} + 6 q^{5} + 2 q^{8} + 2 q^{10} - 6 q^{16} + 16 q^{19} + 2 q^{20} - 20 q^{22} + 4 q^{23} + 6 q^{25} + 20 q^{26} - 8 q^{28} - 12 q^{29} + 22 q^{32} - 4 q^{34} - 20 q^{38} + 2 q^{40} - 16 q^{43} - 12 q^{44} - 8 q^{46} + 8 q^{47} + 30 q^{49} + 2 q^{50} - 4 q^{52} + 8 q^{53} + 12 q^{56} - 20 q^{58} + 4 q^{62} + 14 q^{64} - 44 q^{68} - 48 q^{71} - 12 q^{73} + 4 q^{74} - 4 q^{76} - 12 q^{77} - 6 q^{80} + 16 q^{82} - 40 q^{86} - 8 q^{88} - 12 q^{91} - 32 q^{92} + 44 q^{94} + 16 q^{95} + 4 q^{97} + 10 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 2x^{5} + x^{4} + 2x^{2} - 8x + 8 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{5} + \nu^{3} + 2\nu^{2} + 6\nu - 8 ) / 4 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{5} + \nu^{3} - 2\nu^{2} + 2\nu - 4 ) / 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{5} - \nu^{3} - 2\nu^{2} + 2\nu + 4 ) / 4 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{5} - 2\nu^{4} + \nu^{3} + 2\nu - 8 ) / 4 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -3\nu^{5} + 2\nu^{4} + \nu^{3} - 6\nu + 16 ) / 4 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{3} + \beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{3} - 2\beta_{2} + \beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 2\beta_{5} + 2\beta_{4} - \beta_{3} + 2\beta_{2} + \beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -4\beta_{4} - \beta_{3} + 2\beta_{2} + \beta _1 - 3 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -2\beta_{5} - 2\beta_{4} - 3\beta_{3} + 2\beta_{2} - \beta _1 + 7 ) / 2 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/360\mathbb{Z}\right)^\times\).

\(n\) \(181\) \(217\) \(271\) \(281\)
\(\chi(n)\) \(-1\) \(1\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
251.1
−1.06244 + 0.933389i
−1.06244 − 0.933389i
0.681664 + 1.23909i
0.681664 − 1.23909i
1.38078 + 0.305697i
1.38078 − 0.305697i
−1.06244 − 0.933389i 0 0.257569 + 1.98335i 1.00000 0 1.41421i 1.57758 − 2.34760i 0 −1.06244 − 0.933389i
251.2 −1.06244 + 0.933389i 0 0.257569 − 1.98335i 1.00000 0 − 1.41421i 1.57758 + 2.34760i 0 −1.06244 + 0.933389i
251.3 0.681664 − 1.23909i 0 −1.07067 − 1.68928i 1.00000 0 − 1.41421i −2.82300 + 0.175128i 0 0.681664 − 1.23909i
251.4 0.681664 + 1.23909i 0 −1.07067 + 1.68928i 1.00000 0 1.41421i −2.82300 − 0.175128i 0 0.681664 + 1.23909i
251.5 1.38078 − 0.305697i 0 1.81310 − 0.844199i 1.00000 0 1.41421i 2.24542 − 1.71991i 0 1.38078 − 0.305697i
251.6 1.38078 + 0.305697i 0 1.81310 + 0.844199i 1.00000 0 − 1.41421i 2.24542 + 1.71991i 0 1.38078 + 0.305697i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 251.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
24.f even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 360.2.b.d yes 6
3.b odd 2 1 360.2.b.c ✓ 6
4.b odd 2 1 1440.2.b.d 6
5.b even 2 1 1800.2.b.d 6
5.c odd 4 2 1800.2.m.e 12
8.b even 2 1 1440.2.b.c 6
8.d odd 2 1 360.2.b.c ✓ 6
12.b even 2 1 1440.2.b.c 6
15.d odd 2 1 1800.2.b.e 6
15.e even 4 2 1800.2.m.d 12
20.d odd 2 1 7200.2.b.e 6
20.e even 4 2 7200.2.m.e 12
24.f even 2 1 inner 360.2.b.d yes 6
24.h odd 2 1 1440.2.b.d 6
40.e odd 2 1 1800.2.b.e 6
40.f even 2 1 7200.2.b.d 6
40.i odd 4 2 7200.2.m.d 12
40.k even 4 2 1800.2.m.d 12
60.h even 2 1 7200.2.b.d 6
60.l odd 4 2 7200.2.m.d 12
120.i odd 2 1 7200.2.b.e 6
120.m even 2 1 1800.2.b.d 6
120.q odd 4 2 1800.2.m.e 12
120.w even 4 2 7200.2.m.e 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
360.2.b.c ✓ 6 3.b odd 2 1
360.2.b.c ✓ 6 8.d odd 2 1
360.2.b.d yes 6 1.a even 1 1 trivial
360.2.b.d yes 6 24.f even 2 1 inner
1440.2.b.c 6 8.b even 2 1
1440.2.b.c 6 12.b even 2 1
1440.2.b.d 6 4.b odd 2 1
1440.2.b.d 6 24.h odd 2 1
1800.2.b.d 6 5.b even 2 1
1800.2.b.d 6 120.m even 2 1
1800.2.b.e 6 15.d odd 2 1
1800.2.b.e 6 40.e odd 2 1
1800.2.m.d 12 15.e even 4 2
1800.2.m.d 12 40.k even 4 2
1800.2.m.e 12 5.c odd 4 2
1800.2.m.e 12 120.q odd 4 2
7200.2.b.d 6 40.f even 2 1
7200.2.b.d 6 60.h even 2 1
7200.2.b.e 6 20.d odd 2 1
7200.2.b.e 6 120.i odd 2 1
7200.2.m.d 12 40.i odd 4 2
7200.2.m.d 12 60.l odd 4 2
7200.2.m.e 12 20.e even 4 2
7200.2.m.e 12 120.w even 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(360, [\chi])\):

\( T_{7}^{2} + 2 \) Copy content Toggle raw display
\( T_{23}^{3} - 2T_{23}^{2} - 32T_{23} + 32 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} - 2 T^{5} + \cdots + 8 \) Copy content Toggle raw display
$3$ \( T^{6} \) Copy content Toggle raw display
$5$ \( (T - 1)^{6} \) Copy content Toggle raw display
$7$ \( (T^{2} + 2)^{3} \) Copy content Toggle raw display
$11$ \( T^{6} + 46 T^{4} + \cdots + 8 \) Copy content Toggle raw display
$13$ \( T^{6} + 46 T^{4} + \cdots + 2312 \) Copy content Toggle raw display
$17$ \( T^{6} + 80 T^{4} + \cdots + 15488 \) Copy content Toggle raw display
$19$ \( (T^{3} - 8 T^{2} - 4 T + 16)^{2} \) Copy content Toggle raw display
$23$ \( (T^{3} - 2 T^{2} - 32 T + 32)^{2} \) Copy content Toggle raw display
$29$ \( (T^{3} + 6 T^{2} - 28 T - 8)^{2} \) Copy content Toggle raw display
$31$ \( T^{6} + 80 T^{4} + \cdots + 15488 \) Copy content Toggle raw display
$37$ \( T^{6} + 78 T^{4} + \cdots + 5000 \) Copy content Toggle raw display
$41$ \( T^{6} + 134 T^{4} + \cdots + 49928 \) Copy content Toggle raw display
$43$ \( (T^{3} + 8 T^{2} - 56 T + 64)^{2} \) Copy content Toggle raw display
$47$ \( (T^{3} - 4 T^{2} + \cdots + 352)^{2} \) Copy content Toggle raw display
$53$ \( (T^{3} - 4 T^{2} + \cdots + 352)^{2} \) Copy content Toggle raw display
$59$ \( T^{6} + 110 T^{4} + \cdots + 8 \) Copy content Toggle raw display
$61$ \( T^{6} + 184 T^{4} + \cdots + 512 \) Copy content Toggle raw display
$67$ \( (T^{3} - 40 T + 64)^{2} \) Copy content Toggle raw display
$71$ \( (T^{3} + 24 T^{2} + \cdots + 128)^{2} \) Copy content Toggle raw display
$73$ \( (T^{3} + 6 T^{2} + \cdots - 824)^{2} \) Copy content Toggle raw display
$79$ \( T^{6} + 336 T^{4} + \cdots + 881792 \) Copy content Toggle raw display
$83$ \( T^{6} + 312 T^{4} + \cdots + 320000 \) Copy content Toggle raw display
$89$ \( T^{6} + 118 T^{4} + \cdots + 200 \) Copy content Toggle raw display
$97$ \( (T^{3} - 2 T^{2} + \cdots + 1208)^{2} \) Copy content Toggle raw display
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