Properties

Label 153.4.a.h
Level $153$
Weight $4$
Character orbit 153.a
Self dual yes
Analytic conductor $9.027$
Analytic rank $1$
Dimension $4$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [153,4,Mod(1,153)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(153, 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("153.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 153 = 3^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 153.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: \(9.02729223088\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: 4.4.1506848.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 17x^{2} + 18 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
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.

\(f(q)\) \(=\) \( q + ( - \beta_{2} - 1) q^{2} + ( - \beta_{3} + 2 \beta_{2} + 6) q^{4} + (\beta_{3} + \beta_{2} - 3 \beta_1 - 5) q^{5} + (\beta_{2} + 5 \beta_1 - 6) q^{7} + (3 \beta_{3} - 6 \beta_{2} + 8 \beta_1 - 24) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{2} - 1) q^{2} + ( - \beta_{3} + 2 \beta_{2} + 6) q^{4} + (\beta_{3} + \beta_{2} - 3 \beta_1 - 5) q^{5} + (\beta_{2} + 5 \beta_1 - 6) q^{7} + (3 \beta_{3} - 6 \beta_{2} + 8 \beta_1 - 24) q^{8} + (3 \beta_{3} + 10 \beta_{2} - 5 \beta_1 + 1) q^{10} + ( - 5 \beta_{3} + 6 \beta_{2} + \cdots - 15) q^{11}+ \cdots + (14 \beta_{3} + 255 \beta_{2} + \cdots + 375) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{2} + 26 q^{4} - 22 q^{5} - 24 q^{7} - 102 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{2} + 26 q^{4} - 22 q^{5} - 24 q^{7} - 102 q^{8} - 2 q^{10} - 50 q^{11} + 26 q^{13} - 80 q^{14} + 138 q^{16} + 68 q^{17} + 34 q^{19} - 312 q^{20} - 254 q^{22} - 382 q^{23} + 138 q^{25} + 22 q^{26} + 52 q^{28} - 540 q^{29} - 356 q^{31} - 730 q^{32} - 68 q^{34} - 304 q^{35} - 404 q^{37} - 298 q^{38} + 332 q^{40} + 114 q^{41} + 570 q^{43} + 1368 q^{44} - 290 q^{46} - 496 q^{47} - 224 q^{49} + 1862 q^{50} - 1012 q^{52} - 92 q^{53} - 482 q^{55} + 1428 q^{56} + 1324 q^{58} + 48 q^{59} - 1036 q^{61} + 2564 q^{62} + 2898 q^{64} + 342 q^{65} + 812 q^{67} + 442 q^{68} + 152 q^{70} - 1044 q^{71} - 1212 q^{73} + 1444 q^{74} + 2268 q^{76} + 564 q^{77} + 488 q^{79} + 1000 q^{80} - 938 q^{82} - 1708 q^{83} - 374 q^{85} + 2446 q^{86} - 3868 q^{88} + 8 q^{89} + 716 q^{91} - 1356 q^{92} - 1224 q^{94} - 1010 q^{95} - 76 q^{97} + 1472 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 17x^{2} + 18 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} - 14\nu ) / 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{2} - 9 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + 9 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 3\beta_{2} + 14\beta_1 \) 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
−1.06515
3.98315
−3.98315
1.06515
−5.56787 0 23.0012 −5.10214 0 −6.75787 −83.5248 0 28.4081
1.2 −3.47680 0 4.08814 −7.60718 0 16.3925 13.6008 0 26.4486
1.3 1.47680 0 −5.81906 11.3381 0 −28.3925 −20.4080 0 16.7441
1.4 3.56787 0 4.72971 −20.6288 0 −5.24213 −11.6680 0 −73.6008
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \( +1 \)
\(17\) \( -1 \)

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 153.4.a.h 4
3.b odd 2 1 153.4.a.i yes 4
4.b odd 2 1 2448.4.a.bo 4
12.b even 2 1 2448.4.a.bs 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
153.4.a.h 4 1.a even 1 1 trivial
153.4.a.i yes 4 3.b odd 2 1
2448.4.a.bo 4 4.b odd 2 1
2448.4.a.bs 4 12.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(153))\):

\( T_{2}^{4} + 4T_{2}^{3} - 21T_{2}^{2} - 50T_{2} + 102 \) Copy content Toggle raw display
\( T_{5}^{4} + 22T_{5}^{3} - 77T_{5}^{2} - 2612T_{5} - 9078 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + 4 T^{3} + \cdots + 102 \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} + 22 T^{3} + \cdots - 9078 \) Copy content Toggle raw display
$7$ \( T^{4} + 24 T^{3} + \cdots - 16488 \) Copy content Toggle raw display
$11$ \( T^{4} + 50 T^{3} + \cdots - 669528 \) Copy content Toggle raw display
$13$ \( T^{4} - 26 T^{3} + \cdots + 1030444 \) Copy content Toggle raw display
$17$ \( (T - 17)^{4} \) Copy content Toggle raw display
$19$ \( T^{4} - 34 T^{3} + \cdots + 4368204 \) Copy content Toggle raw display
$23$ \( T^{4} + 382 T^{3} + \cdots + 49041294 \) Copy content Toggle raw display
$29$ \( T^{4} + 540 T^{3} + \cdots - 232426584 \) Copy content Toggle raw display
$31$ \( T^{4} + \cdots - 2282423072 \) Copy content Toggle raw display
$37$ \( T^{4} + \cdots - 6443365976 \) Copy content Toggle raw display
$41$ \( T^{4} + \cdots + 9365221188 \) Copy content Toggle raw display
$43$ \( T^{4} + \cdots - 6091569648 \) Copy content Toggle raw display
$47$ \( T^{4} + 496 T^{3} + \cdots - 153753984 \) Copy content Toggle raw display
$53$ \( T^{4} + \cdots - 2914187328 \) Copy content Toggle raw display
$59$ \( T^{4} + \cdots + 23071976496 \) Copy content Toggle raw display
$61$ \( T^{4} + 1036 T^{3} + \cdots + 531291272 \) Copy content Toggle raw display
$67$ \( T^{4} + \cdots + 1798725056 \) Copy content Toggle raw display
$71$ \( T^{4} + \cdots + 82941464832 \) Copy content Toggle raw display
$73$ \( T^{4} + \cdots - 188519703168 \) Copy content Toggle raw display
$79$ \( T^{4} + \cdots - 23511973896 \) Copy content Toggle raw display
$83$ \( T^{4} + \cdots - 13135194432 \) Copy content Toggle raw display
$89$ \( T^{4} + \cdots + 113358314448 \) Copy content Toggle raw display
$97$ \( T^{4} + \cdots + 896966722688 \) Copy content Toggle raw display
show more
show less