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Properties

Label 960.4.f.r
Level $960$
Weight $4$
Character orbit 960.f
Analytic conductor $56.642$
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] = [960,4,Mod(769,960)]
 
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, 0, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("960.769");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 960 = 2^{6} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 960.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: \(56.6418336055\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.34177656384.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} + 75x^{4} + 1389x^{2} + 100 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 480)
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.

\(f(q)\) \(=\) \( q + 3 \beta_1 q^{3} + ( - \beta_{2} + \beta_1 - 2) q^{5} + (\beta_{5} + \beta_{3} + 10 \beta_1) q^{7} - 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 3 \beta_1 q^{3} + ( - \beta_{2} + \beta_1 - 2) q^{5} + (\beta_{5} + \beta_{3} + 10 \beta_1) q^{7} - 9 q^{9} + (\beta_{5} + 2 \beta_{4} - \beta_{3} + \cdots - 14) q^{11}+ \cdots + ( - 9 \beta_{5} - 18 \beta_{4} + \cdots + 126) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 10 q^{5} - 54 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 10 q^{5} - 54 q^{9} - 96 q^{11} - 12 q^{15} + 176 q^{19} - 192 q^{21} - 138 q^{25} + 116 q^{29} - 112 q^{31} - 176 q^{35} - 456 q^{39} + 724 q^{41} + 90 q^{45} + 218 q^{49} - 96 q^{51} - 864 q^{55} + 1632 q^{59} + 1684 q^{61} - 984 q^{65} + 120 q^{69} + 608 q^{71} - 360 q^{75} + 880 q^{79} + 486 q^{81} - 2936 q^{85} + 1180 q^{89} - 1888 q^{91} - 2608 q^{95} + 864 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} + 75x^{4} + 1389x^{2} + 100 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{5} + 65\nu^{3} + 1009\nu ) / 270 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{5} - 2\nu^{4} - 83\nu^{3} - 94\nu^{2} - 1621\nu - 470 ) / 54 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{5} + 2\nu^{4} + 83\nu^{3} + 58\nu^{2} + 1729\nu - 430 ) / 54 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{5} - 2\nu^{4} + 83\nu^{3} - 94\nu^{2} + 1621\nu - 470 ) / 54 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{5} - 2\nu^{4} + 83\nu^{3} - 58\nu^{2} + 1729\nu + 430 ) / 54 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{5} - \beta_{4} + \beta_{3} + \beta_{2} ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 3\beta_{5} - 3\beta_{4} - 3\beta_{3} - 3\beta_{2} - 100 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -17\beta_{5} + 20\beta_{4} - 17\beta_{3} - 20\beta_{2} - 30\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -141\beta_{5} + 87\beta_{4} + 141\beta_{3} + 87\beta_{2} + 3760 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 1201\beta_{5} - 1591\beta_{4} + 1201\beta_{3} + 1591\beta_{2} + 4980\beta_1 ) / 4 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
769.1
5.74427i
− 6.47542i
− 0.268842i
− 5.74427i
6.47542i
0.268842i
0 − 3.00000i 0 −10.4098 − 4.07873i 0 6.81960i 0 −9.00000 0
769.2 0 − 3.00000i 0 −1.16828 + 11.1191i 0 − 11.6634i 0 −9.00000 0
769.3 0 − 3.00000i 0 6.57808 − 9.04040i 0 − 27.1562i 0 −9.00000 0
769.4 0 3.00000i 0 −10.4098 + 4.07873i 0 − 6.81960i 0 −9.00000 0
769.5 0 3.00000i 0 −1.16828 − 11.1191i 0 11.6634i 0 −9.00000 0
769.6 0 3.00000i 0 6.57808 + 9.04040i 0 27.1562i 0 −9.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 769.6
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 960.4.f.r 6
4.b odd 2 1 960.4.f.s 6
5.b even 2 1 inner 960.4.f.r 6
8.b even 2 1 480.4.f.e yes 6
8.d odd 2 1 480.4.f.d ✓ 6
20.d odd 2 1 960.4.f.s 6
24.f even 2 1 1440.4.f.j 6
24.h odd 2 1 1440.4.f.i 6
40.e odd 2 1 480.4.f.d ✓ 6
40.f even 2 1 480.4.f.e yes 6
40.i odd 4 1 2400.4.a.bf 3
40.i odd 4 1 2400.4.a.bx 3
40.k even 4 1 2400.4.a.be 3
40.k even 4 1 2400.4.a.bw 3
120.i odd 2 1 1440.4.f.i 6
120.m even 2 1 1440.4.f.j 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
480.4.f.d ✓ 6 8.d odd 2 1
480.4.f.d ✓ 6 40.e odd 2 1
480.4.f.e yes 6 8.b even 2 1
480.4.f.e yes 6 40.f even 2 1
960.4.f.r 6 1.a even 1 1 trivial
960.4.f.r 6 5.b even 2 1 inner
960.4.f.s 6 4.b odd 2 1
960.4.f.s 6 20.d odd 2 1
1440.4.f.i 6 24.h odd 2 1
1440.4.f.i 6 120.i odd 2 1
1440.4.f.j 6 24.f even 2 1
1440.4.f.j 6 120.m even 2 1
2400.4.a.be 3 40.k even 4 1
2400.4.a.bf 3 40.i odd 4 1
2400.4.a.bw 3 40.k even 4 1
2400.4.a.bx 3 40.i odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(960, [\chi])\):

\( T_{7}^{6} + 920T_{7}^{4} + 140944T_{7}^{2} + 4665600 \) Copy content Toggle raw display
\( T_{11}^{3} + 48T_{11}^{2} - 564T_{11} - 34560 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( (T^{2} + 9)^{3} \) Copy content Toggle raw display
$5$ \( T^{6} + 10 T^{5} + \cdots + 1953125 \) Copy content Toggle raw display
$7$ \( T^{6} + 920 T^{4} + \cdots + 4665600 \) Copy content Toggle raw display
$11$ \( (T^{3} + 48 T^{2} + \cdots - 34560)^{2} \) Copy content Toggle raw display
$13$ \( T^{6} + 2808 T^{4} + \cdots + 5683456 \) Copy content Toggle raw display
$17$ \( T^{6} + \cdots + 2647719936 \) Copy content Toggle raw display
$19$ \( (T^{3} - 88 T^{2} + \cdots + 92160)^{2} \) Copy content Toggle raw display
$23$ \( T^{6} + \cdots + 2456416751616 \) Copy content Toggle raw display
$29$ \( (T^{3} - 58 T^{2} + \cdots + 578400)^{2} \) Copy content Toggle raw display
$31$ \( (T^{3} + 56 T^{2} + \cdots - 374400)^{2} \) Copy content Toggle raw display
$37$ \( T^{6} + \cdots + 6698323724544 \) Copy content Toggle raw display
$41$ \( (T^{3} - 362 T^{2} + \cdots + 2021736)^{2} \) Copy content Toggle raw display
$43$ \( T^{6} + \cdots + 1312546000896 \) Copy content Toggle raw display
$47$ \( T^{6} + \cdots + 785319466696704 \) Copy content Toggle raw display
$53$ \( T^{6} + \cdots + 27936721670400 \) Copy content Toggle raw display
$59$ \( (T^{3} - 816 T^{2} + \cdots + 97507584)^{2} \) Copy content Toggle raw display
$61$ \( (T^{3} - 842 T^{2} + \cdots + 15055560)^{2} \) Copy content Toggle raw display
$67$ \( T^{6} + \cdots + 12\!\cdots\!00 \) Copy content Toggle raw display
$71$ \( (T^{3} - 304 T^{2} + \cdots + 137352960)^{2} \) Copy content Toggle raw display
$73$ \( T^{6} + \cdots + 135556654694400 \) Copy content Toggle raw display
$79$ \( (T^{3} - 440 T^{2} + \cdots + 640296576)^{2} \) Copy content Toggle raw display
$83$ \( T^{6} + \cdots + 16\!\cdots\!04 \) Copy content Toggle raw display
$89$ \( (T^{3} - 590 T^{2} + \cdots - 2789032)^{2} \) Copy content Toggle raw display
$97$ \( T^{6} + \cdots + 67\!\cdots\!84 \) Copy content Toggle raw display
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