Properties

Label 952.2.b.b
Level $952$
Weight $2$
Character orbit 952.b
Analytic conductor $7.602$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [952,2,Mod(477,952)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(952, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 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("952.477");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 952 = 2^{3} \cdot 7 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 952.b (of order \(2\), degree \(1\), minimal)

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: no
Analytic conductor: \(7.60175827243\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-2}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 \(\beta = \sqrt{-2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{2} - 2 q^{4} - q^{7} - 2 \beta q^{8} + 3 q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + \beta q^{2} - 2 q^{4} - q^{7} - 2 \beta q^{8} + 3 q^{9} + \beta q^{11} - 3 \beta q^{13} - \beta q^{14} + 4 q^{16} + q^{17} + 3 \beta q^{18} + 5 \beta q^{19} - 2 q^{22} + 4 q^{23} + 5 q^{25} + 6 q^{26} + 2 q^{28} + 3 \beta q^{29} + 2 q^{31} + 4 \beta q^{32} + \beta q^{34} - 6 q^{36} - 3 \beta q^{37} - 10 q^{38} - 4 q^{41} + 4 \beta q^{43} - 2 \beta q^{44} + 4 \beta q^{46} + 8 q^{47} + q^{49} + 5 \beta q^{50} + 6 \beta q^{52} + 2 \beta q^{53} + 2 \beta q^{56} - 6 q^{58} + 7 \beta q^{59} + 2 \beta q^{61} + 2 \beta q^{62} - 3 q^{63} - 8 q^{64} - 2 \beta q^{67} - 2 q^{68} + 12 q^{71} - 6 \beta q^{72} - 10 q^{73} + 6 q^{74} - 10 \beta q^{76} - \beta q^{77} + 9 q^{81} - 4 \beta q^{82} + \beta q^{83} - 8 q^{86} + 4 q^{88} - 6 q^{89} + 3 \beta q^{91} - 8 q^{92} + 8 \beta q^{94} + 12 q^{97} + \beta q^{98} + 3 \beta q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 4 q^{4} - 2 q^{7} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 4 q^{4} - 2 q^{7} + 6 q^{9} + 8 q^{16} + 2 q^{17} - 4 q^{22} + 8 q^{23} + 10 q^{25} + 12 q^{26} + 4 q^{28} + 4 q^{31} - 12 q^{36} - 20 q^{38} - 8 q^{41} + 16 q^{47} + 2 q^{49} - 12 q^{58} - 6 q^{63} - 16 q^{64} - 4 q^{68} + 24 q^{71} - 20 q^{73} + 12 q^{74} + 18 q^{81} - 16 q^{86} + 8 q^{88} - 12 q^{89} - 16 q^{92} + 24 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/952\mathbb{Z}\right)^\times\).

\(n\) \(239\) \(409\) \(477\) \(785\)
\(\chi(n)\) \(1\) \(1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
477.1
1.41421i
1.41421i
1.41421i 0 −2.00000 0 0 −1.00000 2.82843i 3.00000 0
477.2 1.41421i 0 −2.00000 0 0 −1.00000 2.82843i 3.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 952.2.b.b 2
4.b odd 2 1 3808.2.b.c 2
8.b even 2 1 inner 952.2.b.b 2
8.d odd 2 1 3808.2.b.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
952.2.b.b 2 1.a even 1 1 trivial
952.2.b.b 2 8.b even 2 1 inner
3808.2.b.c 2 4.b odd 2 1
3808.2.b.c 2 8.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} \) acting on \(S_{2}^{\mathrm{new}}(952, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 2 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( (T + 1)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 2 \) Copy content Toggle raw display
$13$ \( T^{2} + 18 \) Copy content Toggle raw display
$17$ \( (T - 1)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} + 50 \) Copy content Toggle raw display
$23$ \( (T - 4)^{2} \) Copy content Toggle raw display
$29$ \( T^{2} + 18 \) Copy content Toggle raw display
$31$ \( (T - 2)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 18 \) Copy content Toggle raw display
$41$ \( (T + 4)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 32 \) Copy content Toggle raw display
$47$ \( (T - 8)^{2} \) Copy content Toggle raw display
$53$ \( T^{2} + 8 \) Copy content Toggle raw display
$59$ \( T^{2} + 98 \) Copy content Toggle raw display
$61$ \( T^{2} + 8 \) Copy content Toggle raw display
$67$ \( T^{2} + 8 \) Copy content Toggle raw display
$71$ \( (T - 12)^{2} \) Copy content Toggle raw display
$73$ \( (T + 10)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 2 \) Copy content Toggle raw display
$89$ \( (T + 6)^{2} \) Copy content Toggle raw display
$97$ \( (T - 12)^{2} \) Copy content Toggle raw display
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