Properties

Label 528.2.y.g
Level $528$
Weight $2$
Character orbit 528.y
Analytic conductor $4.216$
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] = [528,2,Mod(49,528)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(528, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("528.49");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 528 = 2^{4} \cdot 3 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 528.y (of order \(5\), degree \(4\), 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: \(4.21610122672\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{10})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} + x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 66)
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-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 \(\zeta_{10}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \zeta_{10}^{2} q^{3} + (\zeta_{10}^{2} - 3 \zeta_{10} + 1) q^{5} + ( - \zeta_{10}^{3} - \zeta_{10} + 1) q^{7} + (\zeta_{10}^{3} - \zeta_{10}^{2} + \zeta_{10} - 1) q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - \zeta_{10}^{2} q^{3} + (\zeta_{10}^{2} - 3 \zeta_{10} + 1) q^{5} + ( - \zeta_{10}^{3} - \zeta_{10} + 1) q^{7} + (\zeta_{10}^{3} - \zeta_{10}^{2} + \zeta_{10} - 1) q^{9} + ( - 2 \zeta_{10}^{3} + 4 \zeta_{10}^{2} - 2 \zeta_{10} + 3) q^{11} + ( - 2 \zeta_{10}^{2} + 2 \zeta_{10}) q^{13} + (2 \zeta_{10}^{3} - \zeta_{10} + 1) q^{15} - 4 \zeta_{10} q^{17} + ( - 6 \zeta_{10}^{3} + 2 \zeta_{10}^{2} - 6 \zeta_{10}) q^{19} + (\zeta_{10}^{3} - \zeta_{10}^{2} - 1) q^{21} + (6 \zeta_{10}^{3} - 6 \zeta_{10}^{2} - 4) q^{23} + ( - 5 \zeta_{10}^{3} + 5 \zeta_{10}^{2} - 5 \zeta_{10}) q^{25} + \zeta_{10} q^{27} + ( - 2 \zeta_{10}^{3} + 3 \zeta_{10} - 3) q^{29} + (5 \zeta_{10}^{3} - 4 \zeta_{10}^{2} + 4 \zeta_{10} - 5) q^{31} + ( - 2 \zeta_{10}^{3} + \zeta_{10}^{2} - 4 \zeta_{10} + 2) q^{33} + (\zeta_{10}^{3} + \zeta_{10}^{2} - \zeta_{10} - 1) q^{35} + ( - 4 \zeta_{10}^{3} - 8 \zeta_{10} + 8) q^{37} + ( - 2 \zeta_{10}^{2} + 2 \zeta_{10} - 2) q^{39} + (6 \zeta_{10}^{3} - 2 \zeta_{10}^{2} + 6 \zeta_{10}) q^{41} + ( - 2 \zeta_{10}^{3} + 2 \zeta_{10}^{2} + 8) q^{43} + (\zeta_{10}^{3} - \zeta_{10}^{2} + 2) q^{45} - 4 \zeta_{10}^{2} q^{47} + ( - \zeta_{10}^{2} + 6 \zeta_{10} - 1) q^{49} + 4 \zeta_{10}^{3} q^{51} + (3 \zeta_{10}^{3} - 2 \zeta_{10}^{2} + 2 \zeta_{10} - 3) q^{53} + ( - 6 \zeta_{10}^{3} + 3 \zeta_{10}^{2} - \zeta_{10} - 5) q^{55} + (4 \zeta_{10}^{3} + 2 \zeta_{10}^{2} - 2 \zeta_{10} - 4) q^{57} + ( - \zeta_{10}^{3} - 7 \zeta_{10} + 7) q^{59} + ( - 6 \zeta_{10}^{2} - 6) q^{61} + (\zeta_{10}^{3} + \zeta_{10}) q^{63} + (6 \zeta_{10}^{3} - 6 \zeta_{10}^{2} + 2) q^{65} + ( - 8 \zeta_{10}^{3} + 8 \zeta_{10}^{2} + 8) q^{67} + (6 \zeta_{10}^{3} - 2 \zeta_{10}^{2} + 6 \zeta_{10}) q^{69} + 6 \zeta_{10} q^{71} + (9 \zeta_{10} - 9) q^{73} + (5 \zeta_{10}^{2} - 5 \zeta_{10}) q^{75} + ( - 5 \zeta_{10}^{3} + 2 \zeta_{10}^{2} - 3 \zeta_{10} + 3) q^{77} + (2 \zeta_{10}^{3} - 5 \zeta_{10}^{2} + 5 \zeta_{10} - 2) q^{79} - \zeta_{10}^{3} q^{81} + ( - \zeta_{10}^{2} + \zeta_{10} - 1) q^{83} + ( - 4 \zeta_{10}^{3} + 12 \zeta_{10}^{2} - 4 \zeta_{10}) q^{85} + ( - 3 \zeta_{10}^{3} + 3 \zeta_{10}^{2} - 2) q^{87} + ( - 2 \zeta_{10}^{3} + 2 \zeta_{10}^{2} + 10) q^{89} - 2 \zeta_{10}^{2} q^{91} + (\zeta_{10}^{2} + 4 \zeta_{10} + 1) q^{93} + (2 \zeta_{10}^{3} + 14 \zeta_{10} - 14) q^{95} + ( - 8 \zeta_{10}^{3} + 9 \zeta_{10}^{2} - 9 \zeta_{10} + 8) q^{97} + (3 \zeta_{10}^{3} - \zeta_{10}^{2} - \zeta_{10} - 1) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + q^{3} + 2 q^{7} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + q^{3} + 2 q^{7} - q^{9} + 4 q^{11} + 4 q^{13} + 5 q^{15} - 4 q^{17} - 14 q^{19} - 2 q^{21} - 4 q^{23} - 15 q^{25} + q^{27} - 11 q^{29} - 7 q^{31} + q^{33} - 5 q^{35} + 20 q^{37} - 4 q^{39} + 14 q^{41} + 28 q^{43} + 10 q^{45} + 4 q^{47} + 3 q^{49} + 4 q^{51} - 5 q^{53} - 30 q^{55} - 16 q^{57} + 20 q^{59} - 18 q^{61} + 2 q^{63} + 20 q^{65} + 16 q^{67} + 14 q^{69} + 6 q^{71} - 27 q^{73} - 10 q^{75} + 2 q^{77} + 4 q^{79} - q^{81} - 2 q^{83} - 20 q^{85} - 14 q^{87} + 36 q^{89} + 2 q^{91} + 7 q^{93} - 40 q^{95} + 6 q^{97} - q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/528\mathbb{Z}\right)^\times\).

\(n\) \(133\) \(145\) \(353\) \(463\)
\(\chi(n)\) \(1\) \(-\zeta_{10}^{3}\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
49.1
−0.309017 + 0.951057i
−0.309017 0.951057i
0.809017 + 0.587785i
0.809017 0.587785i
0 0.809017 + 0.587785i 0 1.11803 3.44095i 0 0.500000 0.363271i 0 0.309017 + 0.951057i 0
97.1 0 0.809017 0.587785i 0 1.11803 + 3.44095i 0 0.500000 + 0.363271i 0 0.309017 0.951057i 0
289.1 0 −0.309017 0.951057i 0 −1.11803 0.812299i 0 0.500000 1.53884i 0 −0.809017 + 0.587785i 0
433.1 0 −0.309017 + 0.951057i 0 −1.11803 + 0.812299i 0 0.500000 + 1.53884i 0 −0.809017 0.587785i 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
11.c even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 528.2.y.g 4
4.b odd 2 1 66.2.e.b 4
11.c even 5 1 inner 528.2.y.g 4
11.c even 5 1 5808.2.a.bz 2
11.d odd 10 1 5808.2.a.by 2
12.b even 2 1 198.2.f.a 4
44.c even 2 1 726.2.e.c 4
44.g even 10 1 726.2.a.m 2
44.g even 10 2 726.2.e.a 4
44.g even 10 1 726.2.e.c 4
44.h odd 10 1 66.2.e.b 4
44.h odd 10 1 726.2.a.k 2
44.h odd 10 2 726.2.e.j 4
132.n odd 10 1 2178.2.a.o 2
132.o even 10 1 198.2.f.a 4
132.o even 10 1 2178.2.a.v 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
66.2.e.b 4 4.b odd 2 1
66.2.e.b 4 44.h odd 10 1
198.2.f.a 4 12.b even 2 1
198.2.f.a 4 132.o even 10 1
528.2.y.g 4 1.a even 1 1 trivial
528.2.y.g 4 11.c even 5 1 inner
726.2.a.k 2 44.h odd 10 1
726.2.a.m 2 44.g even 10 1
726.2.e.a 4 44.g even 10 2
726.2.e.c 4 44.c even 2 1
726.2.e.c 4 44.g even 10 1
726.2.e.j 4 44.h odd 10 2
2178.2.a.o 2 132.n odd 10 1
2178.2.a.v 2 132.o even 10 1
5808.2.a.by 2 11.d odd 10 1
5808.2.a.bz 2 11.c even 5 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(528, [\chi])\):

\( T_{5}^{4} + 10T_{5}^{2} + 25T_{5} + 25 \) Copy content Toggle raw display
\( T_{7}^{4} - 2T_{7}^{3} + 4T_{7}^{2} - 3T_{7} + 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} - T^{3} + T^{2} - T + 1 \) Copy content Toggle raw display
$5$ \( T^{4} + 10 T^{2} + 25 T + 25 \) Copy content Toggle raw display
$7$ \( T^{4} - 2 T^{3} + 4 T^{2} - 3 T + 1 \) Copy content Toggle raw display
$11$ \( T^{4} - 4 T^{3} + 6 T^{2} - 44 T + 121 \) Copy content Toggle raw display
$13$ \( T^{4} - 4 T^{3} + 16 T^{2} - 24 T + 16 \) Copy content Toggle raw display
$17$ \( T^{4} + 4 T^{3} + 16 T^{2} + 64 T + 256 \) Copy content Toggle raw display
$19$ \( T^{4} + 14 T^{3} + 136 T^{2} + \cdots + 1936 \) Copy content Toggle raw display
$23$ \( (T^{2} + 2 T - 44)^{2} \) Copy content Toggle raw display
$29$ \( T^{4} + 11 T^{3} + 46 T^{2} - 4 T + 1 \) Copy content Toggle raw display
$31$ \( T^{4} + 7 T^{3} + 24 T^{2} + 38 T + 361 \) Copy content Toggle raw display
$37$ \( T^{4} - 20 T^{3} + 240 T^{2} + \cdots + 6400 \) Copy content Toggle raw display
$41$ \( T^{4} - 14 T^{3} + 136 T^{2} + \cdots + 1936 \) Copy content Toggle raw display
$43$ \( (T^{2} - 14 T + 44)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} - 4 T^{3} + 16 T^{2} - 64 T + 256 \) Copy content Toggle raw display
$53$ \( T^{4} + 5 T^{3} + 10 T^{2} + 25 \) Copy content Toggle raw display
$59$ \( T^{4} - 20 T^{3} + 190 T^{2} + \cdots + 3025 \) Copy content Toggle raw display
$61$ \( T^{4} + 18 T^{3} + 144 T^{2} + \cdots + 1296 \) Copy content Toggle raw display
$67$ \( (T^{2} - 8 T - 64)^{2} \) Copy content Toggle raw display
$71$ \( T^{4} - 6 T^{3} + 36 T^{2} + \cdots + 1296 \) Copy content Toggle raw display
$73$ \( T^{4} + 27 T^{3} + 324 T^{2} + \cdots + 6561 \) Copy content Toggle raw display
$79$ \( T^{4} - 4 T^{3} + 46 T^{2} + 11 T + 1 \) Copy content Toggle raw display
$83$ \( T^{4} + 2 T^{3} + 4 T^{2} + 3 T + 1 \) Copy content Toggle raw display
$89$ \( (T^{2} - 18 T + 76)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} - 6 T^{3} + 76 T^{2} + \cdots + 5041 \) Copy content Toggle raw display
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