Properties

Label 528.4.a.r
Level $528$
Weight $4$
Character orbit 528.a
Self dual yes
Analytic conductor $31.153$
Analytic rank $1$
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: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 4 \) 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 \(\beta = \sqrt{17}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 3 q^{3} + ( - \beta - 3) q^{5} + ( - \beta - 5) q^{7} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 3 q^{3} + ( - \beta - 3) q^{5} + ( - \beta - 5) q^{7} + 9 q^{9} - 11 q^{11} + (17 \beta - 1) q^{13} + ( - 3 \beta - 9) q^{15} + ( - 18 \beta - 52) q^{17} + (30 \beta - 2) q^{19} + ( - 3 \beta - 15) q^{21} + ( - 13 \beta + 51) q^{23} + (6 \beta - 99) q^{25} + 27 q^{27} + (26 \beta - 196) q^{29} + ( - 40 \beta + 32) q^{31} - 33 q^{33} + (8 \beta + 32) q^{35} + ( - 80 \beta - 82) q^{37} + (51 \beta - 3) q^{39} + (20 \beta - 366) q^{41} + ( - 68 \beta - 84) q^{43} + ( - 9 \beta - 27) q^{45} + (45 \beta + 157) q^{47} + (10 \beta - 301) q^{49} + ( - 54 \beta - 156) q^{51} + ( - 37 \beta - 191) q^{53} + (11 \beta + 33) q^{55} + (90 \beta - 6) q^{57} + ( - 58 \beta - 254) q^{59} + (151 \beta - 3) q^{61} + ( - 9 \beta - 45) q^{63} + ( - 50 \beta - 286) q^{65} + (136 \beta - 108) q^{67} + ( - 39 \beta + 153) q^{69} + (63 \beta + 439) q^{71} + ( - 224 \beta + 130) q^{73} + (18 \beta - 297) q^{75} + (11 \beta + 55) q^{77} + (97 \beta - 59) q^{79} + 81 q^{81} + ( - 212 \beta - 248) q^{83} + (106 \beta + 462) q^{85} + (78 \beta - 588) q^{87} + (168 \beta - 878) q^{89} + ( - 84 \beta - 284) q^{91} + ( - 120 \beta + 96) q^{93} + ( - 88 \beta - 504) q^{95} + (38 \beta + 984) q^{97} - 99 q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 6 q^{3} - 6 q^{5} - 10 q^{7} + 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 6 q^{3} - 6 q^{5} - 10 q^{7} + 18 q^{9} - 22 q^{11} - 2 q^{13} - 18 q^{15} - 104 q^{17} - 4 q^{19} - 30 q^{21} + 102 q^{23} - 198 q^{25} + 54 q^{27} - 392 q^{29} + 64 q^{31} - 66 q^{33} + 64 q^{35} - 164 q^{37} - 6 q^{39} - 732 q^{41} - 168 q^{43} - 54 q^{45} + 314 q^{47} - 602 q^{49} - 312 q^{51} - 382 q^{53} + 66 q^{55} - 12 q^{57} - 508 q^{59} - 6 q^{61} - 90 q^{63} - 572 q^{65} - 216 q^{67} + 306 q^{69} + 878 q^{71} + 260 q^{73} - 594 q^{75} + 110 q^{77} - 118 q^{79} + 162 q^{81} - 496 q^{83} + 924 q^{85} - 1176 q^{87} - 1756 q^{89} - 568 q^{91} + 192 q^{93} - 1008 q^{95} + 1968 q^{97} - 198 q^{99}+O(q^{100}) \) 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
2.56155
−1.56155
0 3.00000 0 −7.12311 0 −9.12311 0 9.00000 0
1.2 0 3.00000 0 1.12311 0 −0.876894 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.r 2
3.b odd 2 1 1584.4.a.bf 2
4.b odd 2 1 264.4.a.e 2
8.b even 2 1 2112.4.a.be 2
8.d odd 2 1 2112.4.a.bl 2
12.b even 2 1 792.4.a.h 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
264.4.a.e 2 4.b odd 2 1
528.4.a.r 2 1.a even 1 1 trivial
792.4.a.h 2 12.b even 2 1
1584.4.a.bf 2 3.b odd 2 1
2112.4.a.be 2 8.b even 2 1
2112.4.a.bl 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(528))\):

\( T_{5}^{2} + 6T_{5} - 8 \) Copy content Toggle raw display
\( T_{7}^{2} + 10T_{7} + 8 \) 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
$5$ \( T^{2} + 6T - 8 \) Copy content Toggle raw display
$7$ \( T^{2} + 10T + 8 \) Copy content Toggle raw display
$11$ \( (T + 11)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 2T - 4912 \) Copy content Toggle raw display
$17$ \( T^{2} + 104T - 2804 \) Copy content Toggle raw display
$19$ \( T^{2} + 4T - 15296 \) Copy content Toggle raw display
$23$ \( T^{2} - 102T - 272 \) Copy content Toggle raw display
$29$ \( T^{2} + 392T + 26924 \) Copy content Toggle raw display
$31$ \( T^{2} - 64T - 26176 \) Copy content Toggle raw display
$37$ \( T^{2} + 164T - 102076 \) Copy content Toggle raw display
$41$ \( T^{2} + 732T + 127156 \) Copy content Toggle raw display
$43$ \( T^{2} + 168T - 71552 \) Copy content Toggle raw display
$47$ \( T^{2} - 314T - 9776 \) Copy content Toggle raw display
$53$ \( T^{2} + 382T + 13208 \) Copy content Toggle raw display
$59$ \( T^{2} + 508T + 7328 \) Copy content Toggle raw display
$61$ \( T^{2} + 6T - 387608 \) Copy content Toggle raw display
$67$ \( T^{2} + 216T - 302768 \) Copy content Toggle raw display
$71$ \( T^{2} - 878T + 125248 \) Copy content Toggle raw display
$73$ \( T^{2} - 260T - 836092 \) Copy content Toggle raw display
$79$ \( T^{2} + 118T - 156472 \) Copy content Toggle raw display
$83$ \( T^{2} + 496T - 702544 \) Copy content Toggle raw display
$89$ \( T^{2} + 1756 T + 291076 \) Copy content Toggle raw display
$97$ \( T^{2} - 1968 T + 943708 \) Copy content Toggle raw display
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