Properties

Label 1305.2.a.n
Level $1305$
Weight $2$
Character orbit 1305.a
Self dual yes
Analytic conductor $10.420$
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] = [1305,2,Mod(1,1305)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1305, 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("1305.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1305 = 3^{2} \cdot 5 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1305.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: \(10.4204774638\)
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: no (minimal twist has level 145)
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 \(\beta = \sqrt{2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta + 1) q^{2} + (2 \beta + 1) q^{4} - q^{5} + (2 \beta - 2) q^{7} + (\beta + 3) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta + 1) q^{2} + (2 \beta + 1) q^{4} - q^{5} + (2 \beta - 2) q^{7} + (\beta + 3) q^{8} + ( - \beta - 1) q^{10} + (2 \beta + 2) q^{11} - 2 q^{13} + 2 q^{14} + 3 q^{16} + 2 \beta q^{17} + (2 \beta - 2) q^{19} + ( - 2 \beta - 1) q^{20} + (4 \beta + 6) q^{22} + (2 \beta + 6) q^{23} + q^{25} + ( - 2 \beta - 2) q^{26} + ( - 2 \beta + 6) q^{28} - q^{29} + ( - 6 \beta - 2) q^{31} + (\beta - 3) q^{32} + (2 \beta + 4) q^{34} + ( - 2 \beta + 2) q^{35} + 6 \beta q^{37} + 2 q^{38} + ( - \beta - 3) q^{40} + 6 q^{41} - 6 q^{43} + (6 \beta + 10) q^{44} + (8 \beta + 10) q^{46} + ( - 4 \beta + 6) q^{47} + ( - 8 \beta + 5) q^{49} + (\beta + 1) q^{50} + ( - 4 \beta - 2) q^{52} + ( - 4 \beta - 2) q^{53} + ( - 2 \beta - 2) q^{55} + (4 \beta - 2) q^{56} + ( - \beta - 1) q^{58} + (4 \beta + 2) q^{61} + ( - 8 \beta - 14) q^{62} + ( - 2 \beta - 7) q^{64} + 2 q^{65} + ( - 6 \beta - 2) q^{67} + (2 \beta + 8) q^{68} - 2 q^{70} + ( - 8 \beta + 4) q^{71} - 6 \beta q^{73} + (6 \beta + 12) q^{74} + ( - 2 \beta + 6) q^{76} + 4 q^{77} + (6 \beta + 6) q^{79} - 3 q^{80} + (6 \beta + 6) q^{82} + ( - 2 \beta - 10) q^{83} - 2 \beta q^{85} + ( - 6 \beta - 6) q^{86} + (8 \beta + 10) q^{88} + ( - 4 \beta + 2) q^{89} + ( - 4 \beta + 4) q^{91} + (14 \beta + 14) q^{92} + (2 \beta - 2) q^{94} + ( - 2 \beta + 2) q^{95} + (6 \beta - 4) q^{97} + ( - 3 \beta - 11) q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} + 2 q^{4} - 2 q^{5} - 4 q^{7} + 6 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} + 2 q^{4} - 2 q^{5} - 4 q^{7} + 6 q^{8} - 2 q^{10} + 4 q^{11} - 4 q^{13} + 4 q^{14} + 6 q^{16} - 4 q^{19} - 2 q^{20} + 12 q^{22} + 12 q^{23} + 2 q^{25} - 4 q^{26} + 12 q^{28} - 2 q^{29} - 4 q^{31} - 6 q^{32} + 8 q^{34} + 4 q^{35} + 4 q^{38} - 6 q^{40} + 12 q^{41} - 12 q^{43} + 20 q^{44} + 20 q^{46} + 12 q^{47} + 10 q^{49} + 2 q^{50} - 4 q^{52} - 4 q^{53} - 4 q^{55} - 4 q^{56} - 2 q^{58} + 4 q^{61} - 28 q^{62} - 14 q^{64} + 4 q^{65} - 4 q^{67} + 16 q^{68} - 4 q^{70} + 8 q^{71} + 24 q^{74} + 12 q^{76} + 8 q^{77} + 12 q^{79} - 6 q^{80} + 12 q^{82} - 20 q^{83} - 12 q^{86} + 20 q^{88} + 4 q^{89} + 8 q^{91} + 28 q^{92} - 4 q^{94} + 4 q^{95} - 8 q^{97} - 22 q^{98}+O(q^{100}) \) 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
−1.41421
1.41421
−0.414214 0 −1.82843 −1.00000 0 −4.82843 1.58579 0 0.414214
1.2 2.41421 0 3.82843 −1.00000 0 0.828427 4.41421 0 −2.41421
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(5\) \(1\)
\(29\) \(1\)

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 1305.2.a.n 2
3.b odd 2 1 145.2.a.b 2
5.b even 2 1 6525.2.a.p 2
12.b even 2 1 2320.2.a.k 2
15.d odd 2 1 725.2.a.c 2
15.e even 4 2 725.2.b.c 4
21.c even 2 1 7105.2.a.e 2
24.f even 2 1 9280.2.a.w 2
24.h odd 2 1 9280.2.a.be 2
87.d odd 2 1 4205.2.a.d 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
145.2.a.b 2 3.b odd 2 1
725.2.a.c 2 15.d odd 2 1
725.2.b.c 4 15.e even 4 2
1305.2.a.n 2 1.a even 1 1 trivial
2320.2.a.k 2 12.b even 2 1
4205.2.a.d 2 87.d odd 2 1
6525.2.a.p 2 5.b even 2 1
7105.2.a.e 2 21.c even 2 1
9280.2.a.w 2 24.f even 2 1
9280.2.a.be 2 24.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1305))\):

\( T_{2}^{2} - 2T_{2} - 1 \) Copy content Toggle raw display
\( T_{7}^{2} + 4T_{7} - 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - 2T - 1 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( (T + 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 4T - 4 \) Copy content Toggle raw display
$11$ \( T^{2} - 4T - 4 \) Copy content Toggle raw display
$13$ \( (T + 2)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} - 8 \) Copy content Toggle raw display
$19$ \( T^{2} + 4T - 4 \) Copy content Toggle raw display
$23$ \( T^{2} - 12T + 28 \) Copy content Toggle raw display
$29$ \( (T + 1)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 4T - 68 \) Copy content Toggle raw display
$37$ \( T^{2} - 72 \) Copy content Toggle raw display
$41$ \( (T - 6)^{2} \) Copy content Toggle raw display
$43$ \( (T + 6)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} - 12T + 4 \) Copy content Toggle raw display
$53$ \( T^{2} + 4T - 28 \) Copy content Toggle raw display
$59$ \( T^{2} \) Copy content Toggle raw display
$61$ \( T^{2} - 4T - 28 \) Copy content Toggle raw display
$67$ \( T^{2} + 4T - 68 \) Copy content Toggle raw display
$71$ \( T^{2} - 8T - 112 \) Copy content Toggle raw display
$73$ \( T^{2} - 72 \) Copy content Toggle raw display
$79$ \( T^{2} - 12T - 36 \) Copy content Toggle raw display
$83$ \( T^{2} + 20T + 92 \) Copy content Toggle raw display
$89$ \( T^{2} - 4T - 28 \) Copy content Toggle raw display
$97$ \( T^{2} + 8T - 56 \) Copy content Toggle raw display
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