Properties

Label 1521.4.a.x
Level $1521$
Weight $4$
Character orbit 1521.a
Self dual yes
Analytic conductor $89.742$
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] = [1521,4,Mod(1,1521)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1521, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("1521.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1521 = 3^{2} \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1521.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: \(89.7419051187\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.5054412.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 29x^{2} + 48 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 39)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{3} + 7) q^{4} - \beta_{2} q^{5} + 6 \beta_1 q^{7} + (2 \beta_{2} + 9 \beta_1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (\beta_{3} + 7) q^{4} - \beta_{2} q^{5} + 6 \beta_1 q^{7} + (2 \beta_{2} + 9 \beta_1) q^{8} + ( - 2 \beta_{3} - 6) q^{10} + ( - \beta_{2} - 2 \beta_1) q^{11} + (6 \beta_{3} + 90) q^{14} + (5 \beta_{3} + 91) q^{16} + 54 q^{17} + ( - 6 \beta_{2} - 6 \beta_1) q^{19} + (4 \beta_{2} - 26 \beta_1) q^{20} + ( - 4 \beta_{3} - 36) q^{22} + (12 \beta_{3} + 36) q^{23} + ( - 8 \beta_{3} + 7) q^{25} + (12 \beta_{2} + 102 \beta_1) q^{28} + (12 \beta_{3} + 18) q^{29} + ( - 6 \beta_{2} + 18 \beta_1) q^{31} + ( - 6 \beta_{2} + 69 \beta_1) q^{32} + 54 \beta_1 q^{34} + ( - 12 \beta_{3} - 36) q^{35} + ( - 24 \beta_{2} + 48 \beta_1) q^{37} + ( - 18 \beta_{3} - 126) q^{38} + ( - 2 \beta_{3} - 318) q^{40} + (13 \beta_{2} - 4 \beta_1) q^{41} + (12 \beta_{3} - 260) q^{43} - 60 \beta_1 q^{44} + (24 \beta_{2} + 156 \beta_1) q^{46} + ( - 21 \beta_{2} + 122 \beta_1) q^{47} + (36 \beta_{3} + 197) q^{49} + ( - 16 \beta_{2} - 73 \beta_1) q^{50} + ( - 36 \beta_{3} + 198) q^{53} + ( - 4 \beta_{3} + 144) q^{55} + (78 \beta_{3} + 882) q^{56} + (24 \beta_{2} + 138 \beta_1) q^{58} + ( - 15 \beta_{2} - 34 \beta_1) q^{59} + (8 \beta_{3} - 566) q^{61} + (6 \beta_{3} + 234) q^{62} + (17 \beta_{3} + 271) q^{64} + ( - 12 \beta_{2} - 150 \beta_1) q^{67} + (54 \beta_{3} + 378) q^{68} + ( - 24 \beta_{2} - 156 \beta_1) q^{70} + (23 \beta_{2} + 6 \beta_1) q^{71} - 54 \beta_{2} q^{73} + 576 q^{74} + (12 \beta_{2} - 258 \beta_1) q^{76} + ( - 24 \beta_{3} - 216) q^{77} + (32 \beta_{3} + 88) q^{79} + ( - 36 \beta_{2} - 130 \beta_1) q^{80} + (22 \beta_{3} + 18) q^{82} + ( - 25 \beta_{2} + 138 \beta_1) q^{83} - 54 \beta_{2} q^{85} + (24 \beta_{2} - 140 \beta_1) q^{86} + ( - 28 \beta_{3} - 612) q^{88} + (15 \beta_{2} - 4 \beta_1) q^{89} + (108 \beta_{3} + 2196) q^{92} + (80 \beta_{3} + 1704) q^{94} + ( - 36 \beta_{3} + 828) q^{95} + ( - 54 \beta_{2} + 336 \beta_1) q^{97} + (72 \beta_{2} + 557 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 26 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 26 q^{4} - 20 q^{10} + 348 q^{14} + 354 q^{16} + 216 q^{17} - 136 q^{22} + 120 q^{23} + 44 q^{25} + 48 q^{29} - 120 q^{35} - 468 q^{38} - 1268 q^{40} - 1064 q^{43} + 716 q^{49} + 864 q^{53} + 584 q^{55} + 3372 q^{56} - 2280 q^{61} + 924 q^{62} + 1050 q^{64} + 1404 q^{68} + 2304 q^{74} - 816 q^{77} + 288 q^{79} + 28 q^{82} - 2392 q^{88} + 8568 q^{92} + 6656 q^{94} + 3384 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 29x^{2} + 48 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} - 25\nu ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{2} - 15 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + 15 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{2} + 25\beta_1 \) 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
−5.21898
−1.32750
1.32750
5.21898
−5.21898 0 19.2377 5.83936 0 −31.3139 −58.6495 0 −30.4755
1.2 −1.32750 0 −6.23774 −15.4241 0 −7.96501 18.9006 0 20.4755
1.3 1.32750 0 −6.23774 15.4241 0 7.96501 −18.9006 0 20.4755
1.4 5.21898 0 19.2377 −5.83936 0 31.3139 58.6495 0 −30.4755
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(13\) \(-1\)

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1521.4.a.x 4
3.b odd 2 1 507.4.a.j 4
13.b even 2 1 inner 1521.4.a.x 4
13.d odd 4 2 117.4.b.d 4
39.d odd 2 1 507.4.a.j 4
39.f even 4 2 39.4.b.a 4
156.l odd 4 2 624.4.c.e 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.4.b.a 4 39.f even 4 2
117.4.b.d 4 13.d odd 4 2
507.4.a.j 4 3.b odd 2 1
507.4.a.j 4 39.d odd 2 1
624.4.c.e 4 156.l odd 4 2
1521.4.a.x 4 1.a even 1 1 trivial
1521.4.a.x 4 13.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1521))\):

\( T_{2}^{4} - 29T_{2}^{2} + 48 \) Copy content Toggle raw display
\( T_{5}^{4} - 272T_{5}^{2} + 8112 \) Copy content Toggle raw display
\( T_{7}^{4} - 1044T_{7}^{2} + 62208 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - 29T^{2} + 48 \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} - 272T^{2} + 8112 \) Copy content Toggle raw display
$7$ \( T^{4} - 1044 T^{2} + 62208 \) Copy content Toggle raw display
$11$ \( T^{4} - 428 T^{2} + 43200 \) Copy content Toggle raw display
$13$ \( T^{4} \) Copy content Toggle raw display
$17$ \( (T - 54)^{4} \) Copy content Toggle raw display
$19$ \( T^{4} - 11556 T^{2} + \cdots + 31492800 \) Copy content Toggle raw display
$23$ \( (T^{2} - 60 T - 22464)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} - 24 T - 23220)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} - 17028 T^{2} + \cdots + 47044800 \) Copy content Toggle raw display
$37$ \( T^{4} - 200448 T^{2} + \cdots + 2293235712 \) Copy content Toggle raw display
$41$ \( T^{4} - 45392 T^{2} + \cdots + 128314800 \) Copy content Toggle raw display
$43$ \( (T^{2} + 532 T + 47392)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} - 500348 T^{2} + \cdots + 62387841792 \) Copy content Toggle raw display
$53$ \( (T^{2} - 432 T - 163620)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} - 104924 T^{2} + \cdots + 2436066048 \) Copy content Toggle raw display
$61$ \( (T^{2} + 1140 T + 314516)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} - 727668 T^{2} + \cdots + 143327232 \) Copy content Toggle raw display
$71$ \( T^{4} - 147692 T^{2} + \cdots + 3298756800 \) Copy content Toggle raw display
$73$ \( T^{4} - 793152 T^{2} + \cdots + 68976790272 \) Copy content Toggle raw display
$79$ \( (T^{2} - 144 T - 160960)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} - 653276 T^{2} + \cdots + 106682723328 \) Copy content Toggle raw display
$89$ \( T^{4} - 60464 T^{2} + \cdots + 249304368 \) Copy content Toggle raw display
$97$ \( T^{4} - 3704256 T^{2} + \cdots + 3383532000000 \) Copy content Toggle raw display
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