Properties

Label 960.2.h.b
Level $960$
Weight $2$
Character orbit 960.h
Analytic conductor $7.666$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [960,2,Mod(191,960)]
 
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([1, 0, 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("960.191");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 960 = 2^{6} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 960.h (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: \(7.66563859404\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
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 primitive root of unity \(\zeta_{8}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\zeta_{8}^{3} - \zeta_{8}^{2} - 1) q^{3} - \zeta_{8}^{2} q^{5} + ( - \zeta_{8}^{3} + 2 \zeta_{8}^{2} - \zeta_{8}) q^{7} + ( - 2 \zeta_{8}^{3} + \zeta_{8}^{2} + 2 \zeta_{8}) q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + (\zeta_{8}^{3} - \zeta_{8}^{2} - 1) q^{3} - \zeta_{8}^{2} q^{5} + ( - \zeta_{8}^{3} + 2 \zeta_{8}^{2} - \zeta_{8}) q^{7} + ( - 2 \zeta_{8}^{3} + \zeta_{8}^{2} + 2 \zeta_{8}) q^{9} + (2 \zeta_{8}^{3} - 2 \zeta_{8}) q^{11} + 2 q^{13} + (\zeta_{8}^{2} + \zeta_{8} - 1) q^{15} + (4 \zeta_{8}^{3} - 2 \zeta_{8}^{2} + 4 \zeta_{8}) q^{17} + (2 \zeta_{8}^{3} + 2 \zeta_{8}) q^{19} + (2 \zeta_{8}^{3} - \zeta_{8}^{2} - 2 \zeta_{8} + 3) q^{21} + (\zeta_{8}^{3} - \zeta_{8} + 6) q^{23} - q^{25} + (\zeta_{8}^{2} - 5 \zeta_{8} - 1) q^{27} - 8 \zeta_{8}^{2} q^{29} + ( - 4 \zeta_{8}^{3} - 4 \zeta_{8}) q^{31} + ( - 2 \zeta_{8}^{2} + 4 \zeta_{8} + 2) q^{33} + (\zeta_{8}^{3} - \zeta_{8} + 2) q^{35} + ( - 4 \zeta_{8}^{3} + 4 \zeta_{8} + 6) q^{37} + (2 \zeta_{8}^{3} - 2 \zeta_{8}^{2} - 2) q^{39} - 2 \zeta_{8}^{2} q^{41} + (7 \zeta_{8}^{3} + 2 \zeta_{8}^{2} + 7 \zeta_{8}) q^{43} + ( - 2 \zeta_{8}^{3} - 2 \zeta_{8} + 1) q^{45} + ( - 3 \zeta_{8}^{3} + 3 \zeta_{8} - 2) q^{47} + ( - 4 \zeta_{8}^{3} + 4 \zeta_{8} + 1) q^{49} + ( - 8 \zeta_{8}^{3} - 2 \zeta_{8}^{2} + 2 \zeta_{8} - 6) q^{51} + (4 \zeta_{8}^{3} + 2 \zeta_{8}^{2} + 4 \zeta_{8}) q^{53} + (2 \zeta_{8}^{3} + 2 \zeta_{8}) q^{55} + ( - 4 \zeta_{8}^{3} - 2 \zeta_{8}^{2} - 2) q^{57} + ( - 4 \zeta_{8}^{3} + 4 \zeta_{8} + 4) q^{59} + ( - 4 \zeta_{8}^{3} + 4 \zeta_{8}) q^{61} + (3 \zeta_{8}^{3} - 4 \zeta_{8}^{2} + 5 \zeta_{8} - 2) q^{63} - 2 \zeta_{8}^{2} q^{65} + ( - 5 \zeta_{8}^{3} - 6 \zeta_{8}^{2} - 5 \zeta_{8}) q^{67} + (6 \zeta_{8}^{3} - 7 \zeta_{8}^{2} + 2 \zeta_{8} - 5) q^{69} + ( - 2 \zeta_{8}^{3} + 2 \zeta_{8} + 4) q^{71} + (4 \zeta_{8}^{3} - 4 \zeta_{8} + 10) q^{73} + ( - \zeta_{8}^{3} + \zeta_{8}^{2} + 1) q^{75} + ( - 4 \zeta_{8}^{3} + 4 \zeta_{8}^{2} - 4 \zeta_{8}) q^{77} + ( - 6 \zeta_{8}^{3} - 4 \zeta_{8}^{2} - 6 \zeta_{8}) q^{79} + (4 \zeta_{8}^{3} + 4 \zeta_{8} + 7) q^{81} + ( - 5 \zeta_{8}^{3} + 5 \zeta_{8} + 2) q^{83} + ( - 4 \zeta_{8}^{3} + 4 \zeta_{8} - 2) q^{85} + (8 \zeta_{8}^{2} + 8 \zeta_{8} - 8) q^{87} + ( - 8 \zeta_{8}^{3} - 4 \zeta_{8}^{2} - 8 \zeta_{8}) q^{89} + ( - 2 \zeta_{8}^{3} + 4 \zeta_{8}^{2} - 2 \zeta_{8}) q^{91} + (8 \zeta_{8}^{3} + 4 \zeta_{8}^{2} + 4) q^{93} + ( - 2 \zeta_{8}^{3} + 2 \zeta_{8}) q^{95} + (12 \zeta_{8}^{3} - 12 \zeta_{8} + 2) q^{97} + ( - 2 \zeta_{8}^{3} - 2 \zeta_{8} - 8) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{3}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{3} + 8 q^{13} - 4 q^{15} + 12 q^{21} + 24 q^{23} - 4 q^{25} - 4 q^{27} + 8 q^{33} + 8 q^{35} + 24 q^{37} - 8 q^{39} + 4 q^{45} - 8 q^{47} + 4 q^{49} - 24 q^{51} - 8 q^{57} + 16 q^{59} - 8 q^{63} - 20 q^{69} + 16 q^{71} + 40 q^{73} + 4 q^{75} + 28 q^{81} + 8 q^{83} - 8 q^{85} - 32 q^{87} + 16 q^{93} + 8 q^{97} - 32 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
191.1
0.707107 + 0.707107i
0.707107 0.707107i
−0.707107 0.707107i
−0.707107 + 0.707107i
0 −1.70711 0.292893i 0 1.00000i 0 0.585786i 0 2.82843 + 1.00000i 0
191.2 0 −1.70711 + 0.292893i 0 1.00000i 0 0.585786i 0 2.82843 1.00000i 0
191.3 0 −0.292893 1.70711i 0 1.00000i 0 3.41421i 0 −2.82843 + 1.00000i 0
191.4 0 −0.292893 + 1.70711i 0 1.00000i 0 3.41421i 0 −2.82843 1.00000i 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
12.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 960.2.h.b 4
3.b odd 2 1 960.2.h.f 4
4.b odd 2 1 960.2.h.f 4
8.b even 2 1 480.2.h.d yes 4
8.d odd 2 1 480.2.h.b 4
12.b even 2 1 inner 960.2.h.b 4
24.f even 2 1 480.2.h.d yes 4
24.h odd 2 1 480.2.h.b 4
40.e odd 2 1 2400.2.h.e 4
40.f even 2 1 2400.2.h.b 4
40.i odd 4 1 2400.2.o.b 4
40.i odd 4 1 2400.2.o.j 4
40.k even 4 1 2400.2.o.c 4
40.k even 4 1 2400.2.o.i 4
120.i odd 2 1 2400.2.h.e 4
120.m even 2 1 2400.2.h.b 4
120.q odd 4 1 2400.2.o.b 4
120.q odd 4 1 2400.2.o.j 4
120.w even 4 1 2400.2.o.c 4
120.w even 4 1 2400.2.o.i 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
480.2.h.b 4 8.d odd 2 1
480.2.h.b 4 24.h odd 2 1
480.2.h.d yes 4 8.b even 2 1
480.2.h.d yes 4 24.f even 2 1
960.2.h.b 4 1.a even 1 1 trivial
960.2.h.b 4 12.b even 2 1 inner
960.2.h.f 4 3.b odd 2 1
960.2.h.f 4 4.b odd 2 1
2400.2.h.b 4 40.f even 2 1
2400.2.h.b 4 120.m even 2 1
2400.2.h.e 4 40.e odd 2 1
2400.2.h.e 4 120.i odd 2 1
2400.2.o.b 4 40.i odd 4 1
2400.2.o.b 4 120.q odd 4 1
2400.2.o.c 4 40.k even 4 1
2400.2.o.c 4 120.w even 4 1
2400.2.o.i 4 40.k even 4 1
2400.2.o.i 4 120.w even 4 1
2400.2.o.j 4 40.i odd 4 1
2400.2.o.j 4 120.q odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(960, [\chi])\):

\( T_{7}^{4} + 12T_{7}^{2} + 4 \) Copy content Toggle raw display
\( T_{11}^{2} - 8 \) Copy content Toggle raw display
\( T_{23}^{2} - 12T_{23} + 34 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} + 4 T^{3} + 8 T^{2} + 12 T + 9 \) Copy content Toggle raw display
$5$ \( (T^{2} + 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{4} + 12T^{2} + 4 \) Copy content Toggle raw display
$11$ \( (T^{2} - 8)^{2} \) Copy content Toggle raw display
$13$ \( (T - 2)^{4} \) Copy content Toggle raw display
$17$ \( T^{4} + 72T^{2} + 784 \) Copy content Toggle raw display
$19$ \( (T^{2} + 8)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} - 12 T + 34)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 64)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} + 32)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} - 12 T + 4)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} + 4)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} + 204T^{2} + 8836 \) Copy content Toggle raw display
$47$ \( (T^{2} + 4 T - 14)^{2} \) Copy content Toggle raw display
$53$ \( T^{4} + 72T^{2} + 784 \) Copy content Toggle raw display
$59$ \( (T^{2} - 8 T - 16)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} - 32)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} + 172T^{2} + 196 \) Copy content Toggle raw display
$71$ \( (T^{2} - 8 T + 8)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} - 20 T + 68)^{2} \) Copy content Toggle raw display
$79$ \( T^{4} + 176T^{2} + 3136 \) Copy content Toggle raw display
$83$ \( (T^{2} - 4 T - 46)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} + 288 T^{2} + 12544 \) Copy content Toggle raw display
$97$ \( (T^{2} - 4 T - 284)^{2} \) Copy content Toggle raw display
show more
show less