Properties

Label 468.2.n.d
Level $468$
Weight $2$
Character orbit 468.n
Analytic conductor $3.737$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [468,2,Mod(307,468)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(468, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("468.307");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 468 = 2^{2} \cdot 3^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 468.n (of order \(4\), degree \(2\), 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: \(3.73699881460\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 156)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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 \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - i + 1) q^{2} - 2 i q^{4} + ( - 2 i - 2) q^{5} + ( - 2 i - 2) q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q + ( - i + 1) q^{2} - 2 i q^{4} + ( - 2 i - 2) q^{5} + ( - 2 i - 2) q^{8} - 4 q^{10} + ( - i - 1) q^{11} + (2 i - 3) q^{13} - 4 q^{16} - 2 i q^{17} + ( - 2 i + 2) q^{19} + (4 i - 4) q^{20} - 2 q^{22} + 6 q^{23} + 3 i q^{25} + (5 i - 1) q^{26} - 10 q^{29} + ( - 6 i + 6) q^{31} + (4 i - 4) q^{32} + ( - 2 i - 2) q^{34} + ( - 3 i + 3) q^{37} - 4 i q^{38} + 8 i q^{40} + (4 i + 4) q^{41} + 4 q^{43} + (2 i - 2) q^{44} + ( - 6 i + 6) q^{46} + ( - 5 i - 5) q^{47} - 7 i q^{49} + (3 i + 3) q^{50} + (6 i + 4) q^{52} + 6 q^{53} + 4 i q^{55} + (10 i - 10) q^{58} + (7 i + 7) q^{59} - 8 q^{61} - 12 i q^{62} + 8 i q^{64} + (2 i + 10) q^{65} + ( - 8 i + 8) q^{67} - 4 q^{68} + ( - 9 i + 9) q^{71} + ( - 5 i + 5) q^{73} - 6 i q^{74} + ( - 4 i - 4) q^{76} + 16 i q^{79} + (8 i + 8) q^{80} + 8 q^{82} + (5 i - 5) q^{83} + (4 i - 4) q^{85} + ( - 4 i + 4) q^{86} + 4 i q^{88} + (2 i - 2) q^{89} - 12 i q^{92} - 10 q^{94} - 8 q^{95} + (5 i + 5) q^{97} + ( - 7 i - 7) q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} - 4 q^{5} - 4 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} - 4 q^{5} - 4 q^{8} - 8 q^{10} - 2 q^{11} - 6 q^{13} - 8 q^{16} + 4 q^{19} - 8 q^{20} - 4 q^{22} + 12 q^{23} - 2 q^{26} - 20 q^{29} + 12 q^{31} - 8 q^{32} - 4 q^{34} + 6 q^{37} + 8 q^{41} + 8 q^{43} - 4 q^{44} + 12 q^{46} - 10 q^{47} + 6 q^{50} + 8 q^{52} + 12 q^{53} - 20 q^{58} + 14 q^{59} - 16 q^{61} + 20 q^{65} + 16 q^{67} - 8 q^{68} + 18 q^{71} + 10 q^{73} - 8 q^{76} + 16 q^{80} + 16 q^{82} - 10 q^{83} - 8 q^{85} + 8 q^{86} - 4 q^{89} - 20 q^{94} - 16 q^{95} + 10 q^{97} - 14 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/468\mathbb{Z}\right)^\times\).

\(n\) \(145\) \(209\) \(235\)
\(\chi(n)\) \(i\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
307.1
1.00000i
1.00000i
1.00000 1.00000i 0 2.00000i −2.00000 2.00000i 0 0 −2.00000 2.00000i 0 −4.00000
343.1 1.00000 + 1.00000i 0 2.00000i −2.00000 + 2.00000i 0 0 −2.00000 + 2.00000i 0 −4.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
52.f even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 468.2.n.d 2
3.b odd 2 1 156.2.k.c 2
4.b odd 2 1 468.2.n.b 2
12.b even 2 1 156.2.k.d yes 2
13.d odd 4 1 468.2.n.b 2
39.f even 4 1 156.2.k.d yes 2
52.f even 4 1 inner 468.2.n.d 2
156.l odd 4 1 156.2.k.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
156.2.k.c 2 3.b odd 2 1
156.2.k.c 2 156.l odd 4 1
156.2.k.d yes 2 12.b even 2 1
156.2.k.d yes 2 39.f even 4 1
468.2.n.b 2 4.b odd 2 1
468.2.n.b 2 13.d odd 4 1
468.2.n.d 2 1.a even 1 1 trivial
468.2.n.d 2 52.f even 4 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(468, [\chi])\):

\( T_{5}^{2} + 4T_{5} + 8 \) Copy content Toggle raw display
\( T_{7} \) Copy content Toggle raw display
\( T_{11}^{2} + 2T_{11} + 2 \) Copy content Toggle raw display
\( T_{17}^{2} + 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - 2T + 2 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + 4T + 8 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 2T + 2 \) Copy content Toggle raw display
$13$ \( T^{2} + 6T + 13 \) Copy content Toggle raw display
$17$ \( T^{2} + 4 \) Copy content Toggle raw display
$19$ \( T^{2} - 4T + 8 \) Copy content Toggle raw display
$23$ \( (T - 6)^{2} \) Copy content Toggle raw display
$29$ \( (T + 10)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} - 12T + 72 \) Copy content Toggle raw display
$37$ \( T^{2} - 6T + 18 \) Copy content Toggle raw display
$41$ \( T^{2} - 8T + 32 \) Copy content Toggle raw display
$43$ \( (T - 4)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 10T + 50 \) Copy content Toggle raw display
$53$ \( (T - 6)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} - 14T + 98 \) Copy content Toggle raw display
$61$ \( (T + 8)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} - 16T + 128 \) Copy content Toggle raw display
$71$ \( T^{2} - 18T + 162 \) Copy content Toggle raw display
$73$ \( T^{2} - 10T + 50 \) Copy content Toggle raw display
$79$ \( T^{2} + 256 \) Copy content Toggle raw display
$83$ \( T^{2} + 10T + 50 \) Copy content Toggle raw display
$89$ \( T^{2} + 4T + 8 \) Copy content Toggle raw display
$97$ \( T^{2} - 10T + 50 \) Copy content Toggle raw display
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