Properties

Label 528.4.a.q
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: \(\Q(\sqrt{137}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 34 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 264)
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 \(\beta = \sqrt{137}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 3 q^{3} + ( - \beta - 3) q^{5} + ( - 2 \beta - 8) q^{7} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 3 q^{3} + ( - \beta - 3) q^{5} + ( - 2 \beta - 8) q^{7} + 9 q^{9} + 11 q^{11} + (\beta + 11) q^{13} + ( - 3 \beta - 9) q^{15} + ( - 3 \beta - 11) q^{17} + ( - 3 \beta + 31) q^{19} + ( - 6 \beta - 24) q^{21} + (5 \beta + 105) q^{23} + (6 \beta + 21) q^{25} + 27 q^{27} + ( - 7 \beta - 107) q^{29} + (4 \beta + 164) q^{31} + 33 q^{33} + (14 \beta + 298) q^{35} + (2 \beta - 100) q^{37} + (3 \beta + 33) q^{39} + (29 \beta + 57) q^{41} + (17 \beta + 303) q^{43} + ( - 9 \beta - 27) q^{45} + ( - 21 \beta + 191) q^{47} + (32 \beta + 269) q^{49} + ( - 9 \beta - 33) q^{51} + (29 \beta - 1) q^{53} + ( - 11 \beta - 33) q^{55} + ( - 9 \beta + 93) q^{57} + (2 \beta + 542) q^{59} + ( - 55 \beta - 177) q^{61} + ( - 18 \beta - 72) q^{63} + ( - 14 \beta - 170) q^{65} + (8 \beta - 228) q^{67} + (15 \beta + 315) q^{69} + ( - 27 \beta + 17) q^{71} + ( - 70 \beta - 56) q^{73} + (18 \beta + 63) q^{75} + ( - 22 \beta - 88) q^{77} + (20 \beta - 410) q^{79} + 81 q^{81} + (82 \beta + 506) q^{83} + (20 \beta + 444) q^{85} + ( - 21 \beta - 321) q^{87} + ( - 12 \beta + 518) q^{89} + ( - 30 \beta - 362) q^{91} + (12 \beta + 492) q^{93} + ( - 22 \beta + 318) q^{95} + (16 \beta + 654) q^{97} + 99 q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 6 q^{3} - 6 q^{5} - 16 q^{7} + 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 6 q^{3} - 6 q^{5} - 16 q^{7} + 18 q^{9} + 22 q^{11} + 22 q^{13} - 18 q^{15} - 22 q^{17} + 62 q^{19} - 48 q^{21} + 210 q^{23} + 42 q^{25} + 54 q^{27} - 214 q^{29} + 328 q^{31} + 66 q^{33} + 596 q^{35} - 200 q^{37} + 66 q^{39} + 114 q^{41} + 606 q^{43} - 54 q^{45} + 382 q^{47} + 538 q^{49} - 66 q^{51} - 2 q^{53} - 66 q^{55} + 186 q^{57} + 1084 q^{59} - 354 q^{61} - 144 q^{63} - 340 q^{65} - 456 q^{67} + 630 q^{69} + 34 q^{71} - 112 q^{73} + 126 q^{75} - 176 q^{77} - 820 q^{79} + 162 q^{81} + 1012 q^{83} + 888 q^{85} - 642 q^{87} + 1036 q^{89} - 724 q^{91} + 984 q^{93} + 636 q^{95} + 1308 q^{97} + 198 q^{99}+O(q^{100}) \) 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
6.35235
−5.35235
0 3.00000 0 −14.7047 0 −31.4094 0 9.00000 0
1.2 0 3.00000 0 8.70470 0 15.4094 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.q 2
3.b odd 2 1 1584.4.a.be 2
4.b odd 2 1 264.4.a.f 2
8.b even 2 1 2112.4.a.bd 2
8.d odd 2 1 2112.4.a.bm 2
12.b even 2 1 792.4.a.i 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
264.4.a.f 2 4.b odd 2 1
528.4.a.q 2 1.a even 1 1 trivial
792.4.a.i 2 12.b even 2 1
1584.4.a.be 2 3.b odd 2 1
2112.4.a.bd 2 8.b even 2 1
2112.4.a.bm 2 8.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 \(S_{4}^{\mathrm{new}}(\Gamma_0(528))\):

\( T_{5}^{2} + 6T_{5} - 128 \) Copy content Toggle raw display
\( T_{7}^{2} + 16T_{7} - 484 \) 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
$5$ \( T^{2} + 6T - 128 \) Copy content Toggle raw display
$7$ \( T^{2} + 16T - 484 \) Copy content Toggle raw display
$11$ \( (T - 11)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} - 22T - 16 \) Copy content Toggle raw display
$17$ \( T^{2} + 22T - 1112 \) Copy content Toggle raw display
$19$ \( T^{2} - 62T - 272 \) Copy content Toggle raw display
$23$ \( T^{2} - 210T + 7600 \) Copy content Toggle raw display
$29$ \( T^{2} + 214T + 4736 \) Copy content Toggle raw display
$31$ \( T^{2} - 328T + 24704 \) Copy content Toggle raw display
$37$ \( T^{2} + 200T + 9452 \) Copy content Toggle raw display
$41$ \( T^{2} - 114T - 111968 \) Copy content Toggle raw display
$43$ \( T^{2} - 606T + 52216 \) Copy content Toggle raw display
$47$ \( T^{2} - 382T - 23936 \) Copy content Toggle raw display
$53$ \( T^{2} + 2T - 115216 \) Copy content Toggle raw display
$59$ \( T^{2} - 1084 T + 293216 \) Copy content Toggle raw display
$61$ \( T^{2} + 354T - 383096 \) Copy content Toggle raw display
$67$ \( T^{2} + 456T + 43216 \) Copy content Toggle raw display
$71$ \( T^{2} - 34T - 99584 \) Copy content Toggle raw display
$73$ \( T^{2} + 112T - 668164 \) Copy content Toggle raw display
$79$ \( T^{2} + 820T + 113300 \) Copy content Toggle raw display
$83$ \( T^{2} - 1012 T - 665152 \) Copy content Toggle raw display
$89$ \( T^{2} - 1036 T + 248596 \) Copy content Toggle raw display
$97$ \( T^{2} - 1308 T + 392644 \) Copy content Toggle raw display
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