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Properties

Label 960.3.bg.c
Level $960$
Weight $3$
Character orbit 960.bg
Analytic conductor $26.158$
Analytic rank $0$
Dimension $4$
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] = [960,3,Mod(193,960)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(960, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 3]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("960.193");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 960 = 2^{6} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 960.bg (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: \(26.1581053786\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
Coefficient field: \(\Q(i, \sqrt{6})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 120)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{3} + (2 \beta_{3} - 3 \beta_{2} - \beta_1 - 1) q^{5} + ( - 2 \beta_{3} + 3 \beta_{2} - 3) q^{7} + 3 \beta_{2} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{3} + (2 \beta_{3} - 3 \beta_{2} - \beta_1 - 1) q^{5} + ( - 2 \beta_{3} + 3 \beta_{2} - 3) q^{7} + 3 \beta_{2} q^{9} + (\beta_{3} - \beta_1 + 4) q^{11} + ( - 12 \beta_{2} + 2 \beta_1 - 12) q^{13} + (3 \beta_{3} + 3 \beta_{2} + \beta_1 + 6) q^{15} + ( - 14 \beta_{3} + 6 \beta_{2} - 6) q^{17} + ( - 6 \beta_{3} + 2 \beta_{2} - 6 \beta_1) q^{19} + ( - 3 \beta_{3} + 3 \beta_1 - 6) q^{21} + ( - 14 \beta_{2} - 6 \beta_1 - 14) q^{23} + (2 \beta_{3} - 3 \beta_{2} + 14 \beta_1 + 4) q^{25} - 3 \beta_{3} q^{27} + (9 \beta_{3} + 22 \beta_{2} + 9 \beta_1) q^{29} + (10 \beta_{3} - 10 \beta_1 - 20) q^{31} + (3 \beta_{2} - 4 \beta_1 + 3) q^{33} + ( - 7 \beta_{3} + 18 \beta_{2} + \cdots + 6) q^{35}+ \cdots + ( - 3 \beta_{3} + 12 \beta_{2} - 3 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{5} - 12 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{5} - 12 q^{7} + 16 q^{11} - 48 q^{13} + 24 q^{15} - 24 q^{17} - 24 q^{21} - 56 q^{23} + 16 q^{25} - 80 q^{31} + 12 q^{33} + 24 q^{35} - 112 q^{37} - 56 q^{41} + 8 q^{43} + 36 q^{45} - 16 q^{47} - 168 q^{51} + 120 q^{53} + 20 q^{55} - 72 q^{57} + 24 q^{61} - 36 q^{63} - 144 q^{65} + 8 q^{67} + 272 q^{71} + 108 q^{73} + 24 q^{75} - 72 q^{77} - 36 q^{81} - 272 q^{83} - 72 q^{85} + 108 q^{87} + 336 q^{91} + 120 q^{93} + 96 q^{95} - 148 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} ) / 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} ) / 3 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 3\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 3\beta_{3} \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/960\mathbb{Z}\right)^\times\).

\(n\) \(511\) \(577\) \(641\) \(901\)
\(\chi(n)\) \(1\) \(-\beta_{2}\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
193.1
1.22474 + 1.22474i
−1.22474 − 1.22474i
1.22474 − 1.22474i
−1.22474 + 1.22474i
0 −1.22474 − 1.22474i 0 −4.67423 − 1.77526i 0 −0.550510 + 0.550510i 0 3.00000i 0
193.2 0 1.22474 + 1.22474i 0 2.67423 − 4.22474i 0 −5.44949 + 5.44949i 0 3.00000i 0
577.1 0 −1.22474 + 1.22474i 0 −4.67423 + 1.77526i 0 −0.550510 − 0.550510i 0 − 3.00000i 0
577.2 0 1.22474 − 1.22474i 0 2.67423 + 4.22474i 0 −5.44949 − 5.44949i 0 − 3.00000i 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 960.3.bg.c 4
4.b odd 2 1 960.3.bg.d 4
5.c odd 4 1 inner 960.3.bg.c 4
8.b even 2 1 120.3.u.a ✓ 4
8.d odd 2 1 240.3.bg.c 4
20.e even 4 1 960.3.bg.d 4
24.f even 2 1 720.3.bh.g 4
24.h odd 2 1 360.3.v.b 4
40.e odd 2 1 1200.3.bg.e 4
40.f even 2 1 600.3.u.e 4
40.i odd 4 1 120.3.u.a ✓ 4
40.i odd 4 1 600.3.u.e 4
40.k even 4 1 240.3.bg.c 4
40.k even 4 1 1200.3.bg.e 4
120.i odd 2 1 1800.3.v.n 4
120.q odd 4 1 720.3.bh.g 4
120.w even 4 1 360.3.v.b 4
120.w even 4 1 1800.3.v.n 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
120.3.u.a ✓ 4 8.b even 2 1
120.3.u.a ✓ 4 40.i odd 4 1
240.3.bg.c 4 8.d odd 2 1
240.3.bg.c 4 40.k even 4 1
360.3.v.b 4 24.h odd 2 1
360.3.v.b 4 120.w even 4 1
600.3.u.e 4 40.f even 2 1
600.3.u.e 4 40.i odd 4 1
720.3.bh.g 4 24.f even 2 1
720.3.bh.g 4 120.q odd 4 1
960.3.bg.c 4 1.a even 1 1 trivial
960.3.bg.c 4 5.c odd 4 1 inner
960.3.bg.d 4 4.b odd 2 1
960.3.bg.d 4 20.e even 4 1
1200.3.bg.e 4 40.e odd 2 1
1200.3.bg.e 4 40.k even 4 1
1800.3.v.n 4 120.i odd 2 1
1800.3.v.n 4 120.w even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{4} + 12T_{7}^{3} + 72T_{7}^{2} + 72T_{7} + 36 \) acting on \(S_{3}^{\mathrm{new}}(960, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} + 9 \) Copy content Toggle raw display
$5$ \( T^{4} + 4 T^{3} + \cdots + 625 \) Copy content Toggle raw display
$7$ \( T^{4} + 12 T^{3} + \cdots + 36 \) Copy content Toggle raw display
$11$ \( (T^{2} - 8 T + 10)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} + 48 T^{3} + \cdots + 76176 \) Copy content Toggle raw display
$17$ \( T^{4} + 24 T^{3} + \cdots + 266256 \) Copy content Toggle raw display
$19$ \( T^{4} + 440 T^{2} + 44944 \) Copy content Toggle raw display
$23$ \( T^{4} + 56 T^{3} + \cdots + 80656 \) Copy content Toggle raw display
$29$ \( T^{4} + 1940T^{2} + 4 \) Copy content Toggle raw display
$31$ \( (T^{2} + 40 T - 200)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} + 112 T^{3} + \cdots + 2131600 \) Copy content Toggle raw display
$41$ \( (T^{2} + 28 T - 980)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} - 8 T^{3} + \cdots + 5494336 \) Copy content Toggle raw display
$47$ \( T^{4} + 16 T^{3} + \cdots + 18490000 \) Copy content Toggle raw display
$53$ \( T^{4} - 120 T^{3} + \cdots + 360000 \) Copy content Toggle raw display
$59$ \( T^{4} + 5324 T^{2} + 6916900 \) Copy content Toggle raw display
$61$ \( (T^{2} - 12 T - 2868)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} - 8 T^{3} + \cdots + 192210496 \) Copy content Toggle raw display
$71$ \( (T^{2} - 136 T + 4240)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} - 108 T^{3} + \cdots + 11168964 \) Copy content Toggle raw display
$79$ \( T^{4} + 25520 T^{2} + 151979584 \) Copy content Toggle raw display
$83$ \( T^{4} + 272 T^{3} + \cdots + 60777616 \) Copy content Toggle raw display
$89$ \( T^{4} + 7880 T^{2} + 14047504 \) Copy content Toggle raw display
$97$ \( (T^{2} + 74 T + 2738)^{2} \) Copy content Toggle raw display
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