Properties

Label 1472.2.a.s
Level $1472$
Weight $2$
Character orbit 1472.a
Self dual yes
Analytic conductor $11.754$
Analytic rank $0$
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] = [1472,2,Mod(1,1472)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1472, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1472.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1472 = 2^{6} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1472.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: \(11.7539791775\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{5}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 23)
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 \(\beta = \sqrt{5}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta q^{3} + ( - \beta + 1) q^{5} + ( - \beta - 1) q^{7} + 2 q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - \beta q^{3} + ( - \beta + 1) q^{5} + ( - \beta - 1) q^{7} + 2 q^{9} + ( - \beta - 3) q^{11} - 3 q^{13} + ( - \beta + 5) q^{15} + ( - \beta + 3) q^{17} - 2 q^{19} + (\beta + 5) q^{21} - q^{23} + ( - 2 \beta + 1) q^{25} + \beta q^{27} + 3 q^{29} - 3 \beta q^{31} + (3 \beta + 5) q^{33} + 4 q^{35} + (\beta - 1) q^{37} + 3 \beta q^{39} + ( - 2 \beta + 1) q^{41} + ( - 2 \beta + 2) q^{45} + \beta q^{47} + (2 \beta - 1) q^{49} + ( - 3 \beta + 5) q^{51} + ( - 2 \beta + 4) q^{53} + (2 \beta + 2) q^{55} + 2 \beta q^{57} + (2 \beta + 2) q^{59} + (4 \beta - 2) q^{61} + ( - 2 \beta - 2) q^{63} + (3 \beta - 3) q^{65} + (\beta - 5) q^{67} + \beta q^{69} + ( - \beta - 10) q^{71} + ( - 2 \beta + 11) q^{73} + ( - \beta + 10) q^{75} + (4 \beta + 8) q^{77} + (4 \beta + 2) q^{79} - 11 q^{81} + (\beta - 11) q^{83} + ( - 4 \beta + 8) q^{85} - 3 \beta q^{87} + ( - 2 \beta - 6) q^{89} + (3 \beta + 3) q^{91} + 15 q^{93} + (2 \beta - 2) q^{95} + (3 \beta + 11) q^{97} + ( - 2 \beta - 6) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{5} - 2 q^{7} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{5} - 2 q^{7} + 4 q^{9} - 6 q^{11} - 6 q^{13} + 10 q^{15} + 6 q^{17} - 4 q^{19} + 10 q^{21} - 2 q^{23} + 2 q^{25} + 6 q^{29} + 10 q^{33} + 8 q^{35} - 2 q^{37} + 2 q^{41} + 4 q^{45} - 2 q^{49} + 10 q^{51} + 8 q^{53} + 4 q^{55} + 4 q^{59} - 4 q^{61} - 4 q^{63} - 6 q^{65} - 10 q^{67} - 20 q^{71} + 22 q^{73} + 20 q^{75} + 16 q^{77} + 4 q^{79} - 22 q^{81} - 22 q^{83} + 16 q^{85} - 12 q^{89} + 6 q^{91} + 30 q^{93} - 4 q^{95} + 22 q^{97} - 12 q^{99}+O(q^{100}) \) 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
1.61803
−0.618034
0 −2.23607 0 −1.23607 0 −3.23607 0 2.00000 0
1.2 0 2.23607 0 3.23607 0 1.23607 0 2.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(23\) \(1\)

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 1472.2.a.s 2
4.b odd 2 1 1472.2.a.t 2
8.b even 2 1 368.2.a.h 2
8.d odd 2 1 23.2.a.a 2
24.f even 2 1 207.2.a.d 2
24.h odd 2 1 3312.2.a.ba 2
40.e odd 2 1 575.2.a.f 2
40.f even 2 1 9200.2.a.bt 2
40.k even 4 2 575.2.b.d 4
56.e even 2 1 1127.2.a.c 2
88.g even 2 1 2783.2.a.c 2
104.h odd 2 1 3887.2.a.i 2
120.m even 2 1 5175.2.a.be 2
136.e odd 2 1 6647.2.a.b 2
152.b even 2 1 8303.2.a.e 2
184.e odd 2 1 8464.2.a.bb 2
184.h even 2 1 529.2.a.a 2
184.j even 22 10 529.2.c.n 20
184.k odd 22 10 529.2.c.o 20
552.h odd 2 1 4761.2.a.w 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
23.2.a.a 2 8.d odd 2 1
207.2.a.d 2 24.f even 2 1
368.2.a.h 2 8.b even 2 1
529.2.a.a 2 184.h even 2 1
529.2.c.n 20 184.j even 22 10
529.2.c.o 20 184.k odd 22 10
575.2.a.f 2 40.e odd 2 1
575.2.b.d 4 40.k even 4 2
1127.2.a.c 2 56.e even 2 1
1472.2.a.s 2 1.a even 1 1 trivial
1472.2.a.t 2 4.b odd 2 1
2783.2.a.c 2 88.g even 2 1
3312.2.a.ba 2 24.h odd 2 1
3887.2.a.i 2 104.h odd 2 1
4761.2.a.w 2 552.h odd 2 1
5175.2.a.be 2 120.m even 2 1
6647.2.a.b 2 136.e odd 2 1
8303.2.a.e 2 152.b even 2 1
8464.2.a.bb 2 184.e odd 2 1
9200.2.a.bt 2 40.f even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1472))\):

\( T_{3}^{2} - 5 \) Copy content Toggle raw display
\( T_{5}^{2} - 2T_{5} - 4 \) Copy content Toggle raw display
\( T_{7}^{2} + 2T_{7} - 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} - 5 \) Copy content Toggle raw display
$5$ \( T^{2} - 2T - 4 \) Copy content Toggle raw display
$7$ \( T^{2} + 2T - 4 \) Copy content Toggle raw display
$11$ \( T^{2} + 6T + 4 \) Copy content Toggle raw display
$13$ \( (T + 3)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} - 6T + 4 \) Copy content Toggle raw display
$19$ \( (T + 2)^{2} \) Copy content Toggle raw display
$23$ \( (T + 1)^{2} \) Copy content Toggle raw display
$29$ \( (T - 3)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} - 45 \) Copy content Toggle raw display
$37$ \( T^{2} + 2T - 4 \) Copy content Toggle raw display
$41$ \( T^{2} - 2T - 19 \) Copy content Toggle raw display
$43$ \( T^{2} \) Copy content Toggle raw display
$47$ \( T^{2} - 5 \) Copy content Toggle raw display
$53$ \( T^{2} - 8T - 4 \) Copy content Toggle raw display
$59$ \( T^{2} - 4T - 16 \) Copy content Toggle raw display
$61$ \( T^{2} + 4T - 76 \) Copy content Toggle raw display
$67$ \( T^{2} + 10T + 20 \) Copy content Toggle raw display
$71$ \( T^{2} + 20T + 95 \) Copy content Toggle raw display
$73$ \( T^{2} - 22T + 101 \) Copy content Toggle raw display
$79$ \( T^{2} - 4T - 76 \) Copy content Toggle raw display
$83$ \( T^{2} + 22T + 116 \) Copy content Toggle raw display
$89$ \( T^{2} + 12T + 16 \) Copy content Toggle raw display
$97$ \( T^{2} - 22T + 76 \) Copy content Toggle raw display
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