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Properties

Label 960.3.l.f
Level $960$
Weight $3$
Character orbit 960.l
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(641,960)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(960, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("960.641");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 960 = 2^{6} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 960.l (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: \(26.1581053786\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{-5})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 4x^{2} + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 30)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{2} - \beta_1 + 1) q^{3} - \beta_{2} q^{5} + (\beta_{3} - 5 \beta_1 - 2) q^{7} + (2 \beta_{3} - \beta_{2} - 2 \beta_1 - 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{2} - \beta_1 + 1) q^{3} - \beta_{2} q^{5} + (\beta_{3} - 5 \beta_1 - 2) q^{7} + (2 \beta_{3} - \beta_{2} - 2 \beta_1 - 2) q^{9} + (2 \beta_{3} + 2 \beta_1) q^{11} + 10 q^{13} + (\beta_{3} - \beta_{2} - 5) q^{15} + ( - 4 \beta_{3} + 6 \beta_{2} - 4 \beta_1) q^{17} + (2 \beta_{3} - 10 \beta_1 + 8) q^{19} + (6 \beta_{3} + 5 \beta_{2} + 2 \beta_1 + 13) q^{21} + (\beta_{3} + 6 \beta_{2} + \beta_1) q^{23} - 5 q^{25} + (5 \beta_{3} - \beta_{2} + 10 \beta_1 + 7) q^{27} - 12 \beta_{2} q^{29} - 8 q^{31} + ( - 6 \beta_{2} + 12 \beta_1 + 6) q^{33} + (5 \beta_{3} + 2 \beta_{2} + 5 \beta_1) q^{35} + ( - 4 \beta_{3} + 20 \beta_1 + 22) q^{37} + ( - 10 \beta_{2} - 10 \beta_1 + 10) q^{39} + (8 \beta_{3} - 6 \beta_{2} + 8 \beta_1) q^{41} + ( - 3 \beta_{3} + 15 \beta_1 + 14) q^{43} + (2 \beta_{3} + 2 \beta_{2} + 10 \beta_1 - 5) q^{45} + (5 \beta_{3} + 30 \beta_{2} + 5 \beta_1) q^{47} + ( - 4 \beta_{3} + 20 \beta_1 + 45) q^{49} + ( - 6 \beta_{3} + 18 \beta_{2} + \cdots + 18) q^{51}+ \cdots + ( - 6 \beta_{3} - 24 \beta_{2} + \cdots - 48) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{3} - 8 q^{7} - 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{3} - 8 q^{7} - 8 q^{9} + 40 q^{13} - 20 q^{15} + 32 q^{19} + 52 q^{21} - 20 q^{25} + 28 q^{27} - 32 q^{31} + 24 q^{33} + 88 q^{37} + 40 q^{39} + 56 q^{43} - 20 q^{45} + 180 q^{49} + 72 q^{51} + 152 q^{57} + 64 q^{61} + 256 q^{63} - 328 q^{67} + 132 q^{69} + 200 q^{73} - 20 q^{75} + 112 q^{79} + 28 q^{81} + 120 q^{85} - 240 q^{87} - 80 q^{91} - 32 q^{93} + 296 q^{97} - 192 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 4x^{2} + 9 \) : Copy content Toggle raw display

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

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

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

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

Character values

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

\(n\) \(511\) \(577\) \(641\) \(901\)
\(\chi(n)\) \(1\) \(1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
641.1
1.58114 + 0.707107i
1.58114 − 0.707107i
−1.58114 − 0.707107i
−1.58114 + 0.707107i
0 −0.581139 − 2.94317i 0 − 2.23607i 0 −11.4868 0 −8.32456 + 3.42079i 0
641.2 0 −0.581139 + 2.94317i 0 2.23607i 0 −11.4868 0 −8.32456 − 3.42079i 0
641.3 0 2.58114 − 1.52896i 0 − 2.23607i 0 7.48683 0 4.32456 − 7.89292i 0
641.4 0 2.58114 + 1.52896i 0 2.23607i 0 7.48683 0 4.32456 + 7.89292i 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
3.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 960.3.l.f 4
3.b odd 2 1 inner 960.3.l.f 4
4.b odd 2 1 960.3.l.e 4
8.b even 2 1 240.3.l.c 4
8.d odd 2 1 30.3.d.a ✓ 4
12.b even 2 1 960.3.l.e 4
24.f even 2 1 30.3.d.a ✓ 4
24.h odd 2 1 240.3.l.c 4
40.e odd 2 1 150.3.d.c 4
40.f even 2 1 1200.3.l.u 4
40.i odd 4 2 1200.3.c.k 8
40.k even 4 2 150.3.b.b 8
72.l even 6 2 810.3.h.a 8
72.p odd 6 2 810.3.h.a 8
120.i odd 2 1 1200.3.l.u 4
120.m even 2 1 150.3.d.c 4
120.q odd 4 2 150.3.b.b 8
120.w even 4 2 1200.3.c.k 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
30.3.d.a ✓ 4 8.d odd 2 1
30.3.d.a ✓ 4 24.f even 2 1
150.3.b.b 8 40.k even 4 2
150.3.b.b 8 120.q odd 4 2
150.3.d.c 4 40.e odd 2 1
150.3.d.c 4 120.m even 2 1
240.3.l.c 4 8.b even 2 1
240.3.l.c 4 24.h odd 2 1
810.3.h.a 8 72.l even 6 2
810.3.h.a 8 72.p odd 6 2
960.3.l.e 4 4.b odd 2 1
960.3.l.e 4 12.b even 2 1
960.3.l.f 4 1.a even 1 1 trivial
960.3.l.f 4 3.b odd 2 1 inner
1200.3.c.k 8 40.i odd 4 2
1200.3.c.k 8 120.w even 4 2
1200.3.l.u 4 40.f even 2 1
1200.3.l.u 4 120.i odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{2} + 4T_{7} - 86 \) 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
$3$ \( T^{4} - 4 T^{3} + \cdots + 81 \) Copy content Toggle raw display
$5$ \( (T^{2} + 5)^{2} \) Copy content Toggle raw display
$7$ \( (T^{2} + 4 T - 86)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} + 72)^{2} \) Copy content Toggle raw display
$13$ \( (T - 10)^{4} \) Copy content Toggle raw display
$17$ \( T^{4} + 936 T^{2} + 11664 \) Copy content Toggle raw display
$19$ \( (T^{2} - 16 T - 296)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} + 396 T^{2} + 26244 \) Copy content Toggle raw display
$29$ \( (T^{2} + 720)^{2} \) Copy content Toggle raw display
$31$ \( (T + 8)^{4} \) Copy content Toggle raw display
$37$ \( (T^{2} - 44 T - 956)^{2} \) Copy content Toggle raw display
$41$ \( T^{4} + 2664 T^{2} + 944784 \) Copy content Toggle raw display
$43$ \( (T^{2} - 28 T - 614)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + 9900 T^{2} + 16402500 \) Copy content Toggle raw display
$53$ \( T^{4} + 936 T^{2} + 11664 \) Copy content Toggle raw display
$59$ \( T^{4} + 6624 T^{2} + 3504384 \) Copy content Toggle raw display
$61$ \( (T^{2} - 32 T - 1184)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} + 164 T + 5914)^{2} \) Copy content Toggle raw display
$71$ \( T^{4} + 5040 T^{2} + 1166400 \) Copy content Toggle raw display
$73$ \( (T^{2} - 100 T + 1060)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} - 56 T + 424)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + 684T^{2} + 324 \) Copy content Toggle raw display
$89$ \( T^{4} + 3744 T^{2} + 186624 \) Copy content Toggle raw display
$97$ \( (T^{2} - 148 T + 4036)^{2} \) Copy content Toggle raw display
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