Properties

Label 528.4.a.t
Level $528$
Weight $4$
Character orbit 528.a
Self dual yes
Analytic conductor $31.153$
Analytic rank $0$
Dimension $3$
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: \(3\)
Coefficient field: 3.3.142161.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 84x + 96 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
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 a basis \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 3 q^{3} + ( - \beta_1 + 1) q^{5} + ( - \beta_{2} + 4) q^{7} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 3 q^{3} + ( - \beta_1 + 1) q^{5} + ( - \beta_{2} + 4) q^{7} + 9 q^{9} + 11 q^{11} + ( - 2 \beta_{2} - 3 \beta_1 + 7) q^{13} + ( - 3 \beta_1 + 3) q^{15} + (\beta_{2} + 3 \beta_1 + 37) q^{17} + (5 \beta_{2} + 3 \beta_1 + 7) q^{19} + ( - 3 \beta_{2} + 12) q^{21} + ( - 6 \beta_{2} - 7 \beta_1 - 43) q^{23} + (8 \beta_{2} - 2 \beta_1 + 101) q^{25} + 27 q^{27} + ( - 3 \beta_{2} + 7 \beta_1 + 133) q^{29} + (8 \beta_{2} - 4 \beta_1 - 68) q^{31} + 33 q^{33} + ( - 2 \beta_{2} + 10 \beta_1 - 78) q^{35} + (10 \beta_{2} - 2 \beta_1 + 188) q^{37} + ( - 6 \beta_{2} - 9 \beta_1 + 21) q^{39} + ( - \beta_{2} - \beta_1 + 329) q^{41} + ( - 13 \beta_{2} + 11 \beta_1 - 33) q^{43} + ( - 9 \beta_1 + 9) q^{45} + (22 \beta_{2} + 23 \beta_1 - 45) q^{47} + ( - 22 \beta_{2} - 12 \beta_1 + 77) q^{49} + (3 \beta_{2} + 9 \beta_1 + 111) q^{51} + ( - 10 \beta_{2} + 17 \beta_1 + 403) q^{53} + ( - 11 \beta_1 + 11) q^{55} + (15 \beta_{2} + 9 \beta_1 + 21) q^{57} + (4 \beta_{2} + 26 \beta_1 - 218) q^{59} + (8 \beta_{2} + 33 \beta_1 - 141) q^{61} + ( - 9 \beta_{2} + 36) q^{63} + (20 \beta_{2} + 18 \beta_1 + 518) q^{65} + ( - 8 \beta_{2} - 32 \beta_1 + 356) q^{67} + ( - 18 \beta_{2} - 21 \beta_1 - 129) q^{69} + ( - 26 \beta_{2} + \beta_1 + 141) q^{71} + ( - 2 \beta_{2} - 2 \beta_1 - 152) q^{73} + (24 \beta_{2} - 6 \beta_1 + 303) q^{75} + ( - 11 \beta_{2} + 44) q^{77} + (7 \beta_{2} - 42 \beta_1 + 266) q^{79} + 81 q^{81} + ( - 2 \beta_{2} - 34 \beta_1 + 258) q^{83} + ( - 22 \beta_{2} - 48 \beta_1 - 556) q^{85} + ( - 9 \beta_{2} + 21 \beta_1 + 399) q^{87} + ( - 12 \beta_{2} - 84 \beta_1 + 6) q^{89} + ( - 46 \beta_{2} + 6 \beta_1 + 590) q^{91} + (24 \beta_{2} - 12 \beta_1 - 204) q^{93} + ( - 14 \beta_{2} - 74 \beta_1 - 258) q^{95} + (22 \beta_{2} + 60 \beta_1 - 114) q^{97} + 99 q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 9 q^{3} + 4 q^{5} + 12 q^{7} + 27 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 9 q^{3} + 4 q^{5} + 12 q^{7} + 27 q^{9} + 33 q^{11} + 24 q^{13} + 12 q^{15} + 108 q^{17} + 18 q^{19} + 36 q^{21} - 122 q^{23} + 305 q^{25} + 81 q^{27} + 392 q^{29} - 200 q^{31} + 99 q^{33} - 244 q^{35} + 566 q^{37} + 72 q^{39} + 988 q^{41} - 110 q^{43} + 36 q^{45} - 158 q^{47} + 243 q^{49} + 324 q^{51} + 1192 q^{53} + 44 q^{55} + 54 q^{57} - 680 q^{59} - 456 q^{61} + 108 q^{63} + 1536 q^{65} + 1100 q^{67} - 366 q^{69} + 422 q^{71} - 454 q^{73} + 915 q^{75} + 132 q^{77} + 840 q^{79} + 243 q^{81} + 808 q^{83} - 1620 q^{85} + 1176 q^{87} + 102 q^{89} + 1764 q^{91} - 600 q^{93} - 700 q^{95} - 402 q^{97} + 297 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - x^{2} - 84x + 96 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 2\nu - 1 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} - \nu - 56 ) / 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 4\beta_{2} + \beta _1 + 113 ) / 2 \) 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
9.08381
1.14512
−9.22893
0 3.00000 0 −16.1676 0 −4.71587 0 9.00000 0
1.2 0 3.00000 0 −0.290245 0 31.9169 0 9.00000 0
1.3 0 3.00000 0 20.4579 0 −15.2010 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.t 3
3.b odd 2 1 1584.4.a.bo 3
4.b odd 2 1 264.4.a.h 3
8.b even 2 1 2112.4.a.bo 3
8.d odd 2 1 2112.4.a.bu 3
12.b even 2 1 792.4.a.m 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
264.4.a.h 3 4.b odd 2 1
528.4.a.t 3 1.a even 1 1 trivial
792.4.a.m 3 12.b even 2 1
1584.4.a.bo 3 3.b odd 2 1
2112.4.a.bo 3 8.b even 2 1
2112.4.a.bu 3 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(528))\):

\( T_{5}^{3} - 4T_{5}^{2} - 332T_{5} - 96 \) Copy content Toggle raw display
\( T_{7}^{3} - 12T_{7}^{2} - 564T_{7} - 2288 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} \) Copy content Toggle raw display
$3$ \( (T - 3)^{3} \) Copy content Toggle raw display
$5$ \( T^{3} - 4 T^{2} - 332 T - 96 \) Copy content Toggle raw display
$7$ \( T^{3} - 12 T^{2} - 564 T - 2288 \) Copy content Toggle raw display
$11$ \( (T - 11)^{3} \) Copy content Toggle raw display
$13$ \( T^{3} - 24 T^{2} - 3732 T + 98496 \) Copy content Toggle raw display
$17$ \( T^{3} - 108 T^{2} + 1020 T + 2736 \) Copy content Toggle raw display
$19$ \( T^{3} - 18 T^{2} - 14328 T + 586432 \) Copy content Toggle raw display
$23$ \( T^{3} + 122 T^{2} - 22680 T - 547392 \) Copy content Toggle raw display
$29$ \( T^{3} - 392 T^{2} + 23724 T + 3116928 \) Copy content Toggle raw display
$31$ \( T^{3} + 200 T^{2} - 39552 T - 3244032 \) Copy content Toggle raw display
$37$ \( T^{3} - 566 T^{2} + 39036 T + 9458136 \) Copy content Toggle raw display
$41$ \( T^{3} - 988 T^{2} + \cdots - 35492672 \) Copy content Toggle raw display
$43$ \( T^{3} + 110 T^{2} - 177392 T + 7269792 \) Copy content Toggle raw display
$47$ \( T^{3} + 158 T^{2} + \cdots - 23902464 \) Copy content Toggle raw display
$53$ \( T^{3} - 1192 T^{2} + \cdots + 51257088 \) Copy content Toggle raw display
$59$ \( T^{3} + 680 T^{2} + \cdots - 50438528 \) Copy content Toggle raw display
$61$ \( T^{3} + 456 T^{2} + \cdots - 100312816 \) Copy content Toggle raw display
$67$ \( T^{3} - 1100 T^{2} + \cdots + 116789184 \) Copy content Toggle raw display
$71$ \( T^{3} - 422 T^{2} + \cdots - 22441728 \) Copy content Toggle raw display
$73$ \( T^{3} + 454 T^{2} + 65948 T + 3048104 \) Copy content Toggle raw display
$79$ \( T^{3} - 840 T^{2} - 466284 T + 7864304 \) Copy content Toggle raw display
$83$ \( T^{3} - 808 T^{2} + \cdots + 81625344 \) Copy content Toggle raw display
$89$ \( T^{3} - 102 T^{2} + \cdots + 505352952 \) Copy content Toggle raw display
$97$ \( T^{3} + 402 T^{2} + \cdots - 619299272 \) Copy content Toggle raw display
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