Properties

Label 525.4.a.t
Level $525$
Weight $4$
Character orbit 525.a
Self dual yes
Analytic conductor $30.976$
Analytic rank $1$
Dimension $4$
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] = [525,4,Mod(1,525)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(525, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("525.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 525 = 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 525.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: \(30.9760027530\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 24x^{2} - 3x + 46 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} - 3 q^{3} + (\beta_{2} + 4) q^{4} + 3 \beta_1 q^{6} - 7 q^{7} + ( - \beta_{3} - 4 \beta_1 - 2) q^{8} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} - 3 q^{3} + (\beta_{2} + 4) q^{4} + 3 \beta_1 q^{6} - 7 q^{7} + ( - \beta_{3} - 4 \beta_1 - 2) q^{8} + 9 q^{9} + (\beta_{3} - 3 \beta_{2} + 5 \beta_1 + 5) q^{11} + ( - 3 \beta_{2} - 12) q^{12} + (3 \beta_{3} - \beta_{2} - 3 \beta_1 - 2) q^{13} + 7 \beta_1 q^{14} + (3 \beta_1 + 18) q^{16} + ( - 3 \beta_{3} - 3 \beta_{2} + \cdots - 24) q^{17}+ \cdots + (9 \beta_{3} - 27 \beta_{2} + \cdots + 45) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 12 q^{3} + 16 q^{4} - 28 q^{7} - 9 q^{8} + 36 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 12 q^{3} + 16 q^{4} - 28 q^{7} - 9 q^{8} + 36 q^{9} + 21 q^{11} - 48 q^{12} - 5 q^{13} + 72 q^{16} - 99 q^{17} + 72 q^{19} + 84 q^{21} - 221 q^{22} - 102 q^{23} + 27 q^{24} + 129 q^{26} - 108 q^{27} - 112 q^{28} - 240 q^{29} + 351 q^{31} - 72 q^{32} - 63 q^{33} - 285 q^{34} + 144 q^{36} - 399 q^{37} - 324 q^{38} + 15 q^{39} + 381 q^{41} - 460 q^{43} - 975 q^{44} + 550 q^{46} - 60 q^{47} - 216 q^{48} + 196 q^{49} + 297 q^{51} - 223 q^{52} - 873 q^{53} + 63 q^{56} - 216 q^{57} - 1408 q^{58} - 855 q^{59} + 687 q^{61} + 477 q^{62} - 252 q^{63} - 1285 q^{64} + 663 q^{66} + 503 q^{67} - 1725 q^{68} + 306 q^{69} - 681 q^{71} - 81 q^{72} - 1228 q^{73} - 3369 q^{74} + 1910 q^{76} - 147 q^{77} - 387 q^{78} + 345 q^{79} + 324 q^{81} + 495 q^{82} - 1509 q^{83} + 336 q^{84} - 540 q^{86} + 720 q^{87} - 2266 q^{88} - 198 q^{89} + 35 q^{91} - 2994 q^{92} - 1053 q^{93} - 1732 q^{94} + 216 q^{96} + 372 q^{97} + 189 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 24x^{2} - 3x + 46 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 12 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - 20\nu - 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 12 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + 20\beta _1 + 2 \) 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
4.75345
1.37627
−1.52801
−4.60171
−4.75345 −3.00000 14.5953 0 14.2604 −7.00000 −31.3504 9.00000 0
1.2 −1.37627 −3.00000 −6.10588 0 4.12881 −7.00000 19.4135 9.00000 0
1.3 1.52801 −3.00000 −5.66519 0 −4.58402 −7.00000 −20.8805 9.00000 0
1.4 4.60171 −3.00000 13.1758 0 −13.8051 −7.00000 23.8174 9.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \( +1 \)
\(5\) \( -1 \)
\(7\) \( +1 \)

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 525.4.a.t 4
3.b odd 2 1 1575.4.a.bj 4
5.b even 2 1 525.4.a.u yes 4
5.c odd 4 2 525.4.d.n 8
15.d odd 2 1 1575.4.a.bk 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
525.4.a.t 4 1.a even 1 1 trivial
525.4.a.u yes 4 5.b even 2 1
525.4.d.n 8 5.c odd 4 2
1575.4.a.bj 4 3.b odd 2 1
1575.4.a.bk 4 15.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 \(S_{4}^{\mathrm{new}}(\Gamma_0(525))\):

\( T_{2}^{4} - 24T_{2}^{2} + 3T_{2} + 46 \) Copy content Toggle raw display
\( T_{11}^{4} - 21T_{11}^{3} - 2643T_{11}^{2} + 61617T_{11} - 304010 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - 24 T^{2} + \cdots + 46 \) Copy content Toggle raw display
$3$ \( (T + 3)^{4} \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( (T + 7)^{4} \) Copy content Toggle raw display
$11$ \( T^{4} - 21 T^{3} + \cdots - 304010 \) Copy content Toggle raw display
$13$ \( T^{4} + 5 T^{3} + \cdots + 545368 \) Copy content Toggle raw display
$17$ \( T^{4} + 99 T^{3} + \cdots - 25684592 \) Copy content Toggle raw display
$19$ \( T^{4} - 72 T^{3} + \cdots + 1017760 \) Copy content Toggle raw display
$23$ \( T^{4} + 102 T^{3} + \cdots - 390917 \) Copy content Toggle raw display
$29$ \( T^{4} + 240 T^{3} + \cdots + 130344085 \) Copy content Toggle raw display
$31$ \( T^{4} - 351 T^{3} + \cdots + 137611224 \) Copy content Toggle raw display
$37$ \( T^{4} + 399 T^{3} + \cdots - 476873082 \) Copy content Toggle raw display
$41$ \( T^{4} - 381 T^{3} + \cdots - 91750400 \) Copy content Toggle raw display
$43$ \( T^{4} + \cdots + 3696646993 \) Copy content Toggle raw display
$47$ \( T^{4} + \cdots + 11323563904 \) Copy content Toggle raw display
$53$ \( T^{4} + \cdots - 2685647720 \) Copy content Toggle raw display
$59$ \( T^{4} + \cdots - 35121553400 \) Copy content Toggle raw display
$61$ \( T^{4} + \cdots + 13537528704 \) Copy content Toggle raw display
$67$ \( T^{4} + \cdots + 120089209012 \) Copy content Toggle raw display
$71$ \( T^{4} + \cdots - 3046275956 \) Copy content Toggle raw display
$73$ \( T^{4} + \cdots - 498004222688 \) Copy content Toggle raw display
$79$ \( T^{4} + \cdots - 14999641370 \) Copy content Toggle raw display
$83$ \( T^{4} + \cdots - 374122465304 \) Copy content Toggle raw display
$89$ \( T^{4} + \cdots - 25670269520 \) Copy content Toggle raw display
$97$ \( T^{4} + \cdots + 190440971632 \) Copy content Toggle raw display
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