Properties

Label 152.4.a.a
Level $152$
Weight $4$
Character orbit 152.a
Self dual yes
Analytic conductor $8.968$
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] = [152,4,Mod(1,152)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(152, 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("152.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 152 = 2^{3} \cdot 19 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 152.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: \(8.96829032087\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{57}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 14 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
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 \(\beta = \frac{1}{2}(1 + \sqrt{57})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta q^{3} + (\beta - 3) q^{5} + (4 \beta - 5) q^{7} + (\beta - 13) q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - \beta q^{3} + (\beta - 3) q^{5} + (4 \beta - 5) q^{7} + (\beta - 13) q^{9} + (7 \beta - 11) q^{11} + ( - 7 \beta - 26) q^{13} + (2 \beta - 14) q^{15} + ( - 10 \beta - 47) q^{17} - 19 q^{19} + (\beta - 56) q^{21} + ( - 33 \beta + 6) q^{23} + ( - 5 \beta - 102) q^{25} + (39 \beta - 14) q^{27} + ( - 9 \beta - 64) q^{29} + ( - 28 \beta + 16) q^{31} + (4 \beta - 98) q^{33} + ( - 13 \beta + 71) q^{35} + (28 \beta - 90) q^{37} + (33 \beta + 98) q^{39} + (90 \beta - 150) q^{41} + ( - 23 \beta + 45) q^{43} + ( - 15 \beta + 53) q^{45} + ( - 45 \beta + 159) q^{47} + ( - 24 \beta - 94) q^{49} + (57 \beta + 140) q^{51} + ( - 3 \beta + 106) q^{53} + ( - 25 \beta + 131) q^{55} + 19 \beta q^{57} + (63 \beta + 368) q^{59} + ( - 53 \beta + 101) q^{61} + ( - 53 \beta + 121) q^{63} + ( - 12 \beta - 20) q^{65} + (5 \beta + 98) q^{67} + (27 \beta + 462) q^{69} + ( - 100 \beta + 446) q^{71} + ( - 72 \beta - 87) q^{73} + (107 \beta + 70) q^{75} + ( - 51 \beta + 447) q^{77} + (114 \beta - 184) q^{79} + ( - 52 \beta - 195) q^{81} + ( - 154 \beta + 264) q^{83} + ( - 27 \beta + 1) q^{85} + (73 \beta + 126) q^{87} + ( - 196 \beta - 184) q^{89} + ( - 97 \beta - 262) q^{91} + (12 \beta + 392) q^{93} + ( - 19 \beta + 57) q^{95} + (374 \beta - 276) q^{97} + ( - 95 \beta + 241) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{3} - 5 q^{5} - 6 q^{7} - 25 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - q^{3} - 5 q^{5} - 6 q^{7} - 25 q^{9} - 15 q^{11} - 59 q^{13} - 26 q^{15} - 104 q^{17} - 38 q^{19} - 111 q^{21} - 21 q^{23} - 209 q^{25} + 11 q^{27} - 137 q^{29} + 4 q^{31} - 192 q^{33} + 129 q^{35} - 152 q^{37} + 229 q^{39} - 210 q^{41} + 67 q^{43} + 91 q^{45} + 273 q^{47} - 212 q^{49} + 337 q^{51} + 209 q^{53} + 237 q^{55} + 19 q^{57} + 799 q^{59} + 149 q^{61} + 189 q^{63} - 52 q^{65} + 201 q^{67} + 951 q^{69} + 792 q^{71} - 246 q^{73} + 247 q^{75} + 843 q^{77} - 254 q^{79} - 442 q^{81} + 374 q^{83} - 25 q^{85} + 325 q^{87} - 564 q^{89} - 621 q^{91} + 796 q^{93} + 95 q^{95} - 178 q^{97} + 387 q^{99}+O(q^{100}) \) 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
4.27492
−3.27492
0 −4.27492 0 1.27492 0 12.0997 0 −8.72508 0
1.2 0 3.27492 0 −6.27492 0 −18.0997 0 −16.2749 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(19\) \(1\)

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 152.4.a.a 2
3.b odd 2 1 1368.4.a.a 2
4.b odd 2 1 304.4.a.e 2
8.b even 2 1 1216.4.a.m 2
8.d odd 2 1 1216.4.a.k 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
152.4.a.a 2 1.a even 1 1 trivial
304.4.a.e 2 4.b odd 2 1
1216.4.a.k 2 8.d odd 2 1
1216.4.a.m 2 8.b even 2 1
1368.4.a.a 2 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} + T_{3} - 14 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(152))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + T - 14 \) Copy content Toggle raw display
$5$ \( T^{2} + 5T - 8 \) Copy content Toggle raw display
$7$ \( T^{2} + 6T - 219 \) Copy content Toggle raw display
$11$ \( T^{2} + 15T - 642 \) Copy content Toggle raw display
$13$ \( T^{2} + 59T + 172 \) Copy content Toggle raw display
$17$ \( T^{2} + 104T + 1279 \) Copy content Toggle raw display
$19$ \( (T + 19)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 21T - 15408 \) Copy content Toggle raw display
$29$ \( T^{2} + 137T + 3538 \) Copy content Toggle raw display
$31$ \( T^{2} - 4T - 11168 \) Copy content Toggle raw display
$37$ \( T^{2} + 152T - 5396 \) Copy content Toggle raw display
$41$ \( T^{2} + 210T - 104400 \) Copy content Toggle raw display
$43$ \( T^{2} - 67T - 6416 \) Copy content Toggle raw display
$47$ \( T^{2} - 273T - 10224 \) Copy content Toggle raw display
$53$ \( T^{2} - 209T + 10792 \) Copy content Toggle raw display
$59$ \( T^{2} - 799T + 103042 \) Copy content Toggle raw display
$61$ \( T^{2} - 149T - 34478 \) Copy content Toggle raw display
$67$ \( T^{2} - 201T + 9744 \) Copy content Toggle raw display
$71$ \( T^{2} - 792T + 14316 \) Copy content Toggle raw display
$73$ \( T^{2} + 246T - 58743 \) Copy content Toggle raw display
$79$ \( T^{2} + 254T - 169064 \) Copy content Toggle raw display
$83$ \( T^{2} - 374T - 302984 \) Copy content Toggle raw display
$89$ \( T^{2} + 564T - 467904 \) Copy content Toggle raw display
$97$ \( T^{2} + 178 T - 1985312 \) Copy content Toggle raw display
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