Properties

Label 528.4.a.n
Level $528$
Weight $4$
Character orbit 528.a
Self dual yes
Analytic conductor $31.153$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [528,4,Mod(1,528)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(528, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("528.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 528 = 2^{4} \cdot 3 \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 528.a (trivial)

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: yes
Analytic conductor: \(31.1530084830\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{97}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 24 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 66)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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 \(\beta = \sqrt{97}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 3 q^{3} + ( - \beta + 5) q^{5} + ( - 3 \beta + 1) q^{7} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 3 q^{3} + ( - \beta + 5) q^{5} + ( - 3 \beta + 1) q^{7} + 9 q^{9} + 11 q^{11} + ( - 7 \beta + 7) q^{13} + (3 \beta - 15) q^{15} + (2 \beta - 40) q^{17} + ( - 10 \beta + 30) q^{19} + (9 \beta - 3) q^{21} + (5 \beta + 101) q^{23} + ( - 10 \beta - 3) q^{25} - 27 q^{27} + (6 \beta - 168) q^{29} + (16 \beta - 64) q^{31} - 33 q^{33} + ( - 16 \beta + 296) q^{35} + (32 \beta + 94) q^{37} + (21 \beta - 21) q^{39} + ( - 24 \beta - 66) q^{41} + (8 \beta - 240) q^{43} + ( - 9 \beta + 45) q^{45} + (3 \beta + 195) q^{47} + ( - 6 \beta + 531) q^{49} + ( - 6 \beta + 120) q^{51} + (35 \beta + 305) q^{53} + ( - 11 \beta + 55) q^{55} + (30 \beta - 90) q^{57} + (18 \beta - 186) q^{59} + (7 \beta + 525) q^{61} + ( - 27 \beta + 9) q^{63} + ( - 42 \beta + 714) q^{65} + ( - 16 \beta - 204) q^{67} + ( - 15 \beta - 303) q^{69} + ( - 15 \beta - 471) q^{71} + ( - 64 \beta + 370) q^{73} + (30 \beta + 9) q^{75} + ( - 33 \beta + 11) q^{77} + (27 \beta + 823) q^{79} + 81 q^{81} + ( - 52 \beta + 176) q^{83} + (50 \beta - 394) q^{85} + ( - 18 \beta + 504) q^{87} + (16 \beta + 1018) q^{89} + ( - 28 \beta + 2044) q^{91} + ( - 48 \beta + 192) q^{93} + ( - 80 \beta + 1120) q^{95} + (70 \beta + 168) q^{97} + 99 q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 6 q^{3} + 10 q^{5} + 2 q^{7} + 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 6 q^{3} + 10 q^{5} + 2 q^{7} + 18 q^{9} + 22 q^{11} + 14 q^{13} - 30 q^{15} - 80 q^{17} + 60 q^{19} - 6 q^{21} + 202 q^{23} - 6 q^{25} - 54 q^{27} - 336 q^{29} - 128 q^{31} - 66 q^{33} + 592 q^{35} + 188 q^{37} - 42 q^{39} - 132 q^{41} - 480 q^{43} + 90 q^{45} + 390 q^{47} + 1062 q^{49} + 240 q^{51} + 610 q^{53} + 110 q^{55} - 180 q^{57} - 372 q^{59} + 1050 q^{61} + 18 q^{63} + 1428 q^{65} - 408 q^{67} - 606 q^{69} - 942 q^{71} + 740 q^{73} + 18 q^{75} + 22 q^{77} + 1646 q^{79} + 162 q^{81} + 352 q^{83} - 788 q^{85} + 1008 q^{87} + 2036 q^{89} + 4088 q^{91} + 384 q^{93} + 2240 q^{95} + 336 q^{97} + 198 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
5.42443
−4.42443
0 −3.00000 0 −4.84886 0 −28.5466 0 9.00000 0
1.2 0 −3.00000 0 14.8489 0 30.5466 0 9.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(1\)
\(11\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 528.4.a.n 2
3.b odd 2 1 1584.4.a.ba 2
4.b odd 2 1 66.4.a.c 2
8.b even 2 1 2112.4.a.bi 2
8.d odd 2 1 2112.4.a.bb 2
12.b even 2 1 198.4.a.h 2
20.d odd 2 1 1650.4.a.s 2
20.e even 4 2 1650.4.c.u 4
44.c even 2 1 726.4.a.o 2
132.d odd 2 1 2178.4.a.bf 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
66.4.a.c 2 4.b odd 2 1
198.4.a.h 2 12.b even 2 1
528.4.a.n 2 1.a even 1 1 trivial
726.4.a.o 2 44.c even 2 1
1584.4.a.ba 2 3.b odd 2 1
1650.4.a.s 2 20.d odd 2 1
1650.4.c.u 4 20.e even 4 2
2112.4.a.bb 2 8.d odd 2 1
2112.4.a.bi 2 8.b even 2 1
2178.4.a.bf 2 132.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(528))\):

\( T_{5}^{2} - 10T_{5} - 72 \) Copy content Toggle raw display
\( T_{7}^{2} - 2T_{7} - 872 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( (T + 3)^{2} \) Copy content Toggle raw display
$5$ \( T^{2} - 10T - 72 \) Copy content Toggle raw display
$7$ \( T^{2} - 2T - 872 \) Copy content Toggle raw display
$11$ \( (T - 11)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} - 14T - 4704 \) Copy content Toggle raw display
$17$ \( T^{2} + 80T + 1212 \) Copy content Toggle raw display
$19$ \( T^{2} - 60T - 8800 \) Copy content Toggle raw display
$23$ \( T^{2} - 202T + 7776 \) Copy content Toggle raw display
$29$ \( T^{2} + 336T + 24732 \) Copy content Toggle raw display
$31$ \( T^{2} + 128T - 20736 \) Copy content Toggle raw display
$37$ \( T^{2} - 188T - 90492 \) Copy content Toggle raw display
$41$ \( T^{2} + 132T - 51516 \) Copy content Toggle raw display
$43$ \( T^{2} + 480T + 51392 \) Copy content Toggle raw display
$47$ \( T^{2} - 390T + 37152 \) Copy content Toggle raw display
$53$ \( T^{2} - 610T - 25800 \) Copy content Toggle raw display
$59$ \( T^{2} + 372T + 3168 \) Copy content Toggle raw display
$61$ \( T^{2} - 1050 T + 270872 \) Copy content Toggle raw display
$67$ \( T^{2} + 408T + 16784 \) Copy content Toggle raw display
$71$ \( T^{2} + 942T + 200016 \) Copy content Toggle raw display
$73$ \( T^{2} - 740T - 260412 \) Copy content Toggle raw display
$79$ \( T^{2} - 1646 T + 606616 \) Copy content Toggle raw display
$83$ \( T^{2} - 352T - 231312 \) Copy content Toggle raw display
$89$ \( T^{2} - 2036 T + 1011492 \) Copy content Toggle raw display
$97$ \( T^{2} - 336T - 447076 \) Copy content Toggle raw display
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