Properties

Label 1305.2.d.c
Level $1305$
Weight $2$
Character orbit 1305.d
Analytic conductor $10.420$
Analytic rank $0$
Dimension $10$
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] = [1305,2,Mod(811,1305)]
 
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, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1305.811");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1305 = 3^{2} \cdot 5 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1305.d (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: \(10.4204774638\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} + 17x^{8} + 104x^{6} + 273x^{4} + 281x^{2} + 64 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 435)
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 a basis \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{2} - 1) q^{4} - q^{5} + (\beta_{9} + \beta_{2} + 1) q^{7} + (\beta_{3} - \beta_1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (\beta_{2} - 1) q^{4} - q^{5} + (\beta_{9} + \beta_{2} + 1) q^{7} + (\beta_{3} - \beta_1) q^{8} - \beta_1 q^{10} + ( - \beta_{7} + \beta_{6} + \cdots + \beta_1) q^{11}+ \cdots + ( - 2 \beta_{7} - \beta_{6} + \cdots + \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 14 q^{4} - 10 q^{5} + 8 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 10 q - 14 q^{4} - 10 q^{5} + 8 q^{7} - 4 q^{13} - 2 q^{16} + 14 q^{20} - 26 q^{22} + 10 q^{23} + 10 q^{25} + 34 q^{28} - 16 q^{29} - 6 q^{34} - 8 q^{35} + 36 q^{38} + 14 q^{49} - 14 q^{52} + 38 q^{53} - 6 q^{58} + 12 q^{59} + 28 q^{62} + 28 q^{64} + 4 q^{65} + 20 q^{67} - 32 q^{71} + 60 q^{74} + 2 q^{80} + 12 q^{82} - 2 q^{83} + 92 q^{88} - 24 q^{91} - 4 q^{92} + 14 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} + 17x^{8} + 104x^{6} + 273x^{4} + 281x^{2} + 64 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} + 5\nu \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{9} - 9\nu^{7} - 16\nu^{5} + 23\nu^{3} + 47\nu ) / 8 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( \nu^{4} + 7\nu^{2} + 8 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( \nu^{5} + 8\nu^{3} + 13\nu \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{9} + 13\nu^{7} + 56\nu^{5} + 89\nu^{3} + 37\nu ) / 8 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( \nu^{6} + 9\nu^{4} + 20\nu^{2} + 8 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( \nu^{8} + 12\nu^{6} + 46\nu^{4} + 61\nu^{2} + 16 ) / 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} - 5\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{5} - 7\beta_{2} + 13 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{6} - 8\beta_{3} + 27\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( \beta_{8} - 9\beta_{5} + 43\beta_{2} - 65 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 2\beta_{7} - 10\beta_{6} + 2\beta_{4} + 52\beta_{3} - 151\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 2\beta_{9} - 12\beta_{8} + 62\beta_{5} - 255\beta_{2} + 349 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( -18\beta_{7} + 74\beta_{6} - 26\beta_{4} - 317\beta_{3} + 859\beta_1 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1305\mathbb{Z}\right)^\times\).

\(n\) \(146\) \(262\) \(901\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
811.1
2.43027i
2.38193i
1.78296i
1.38924i
0.557941i
0.557941i
1.38924i
1.78296i
2.38193i
2.43027i
2.43027i 0 −3.90620 −1.00000 0 0.524070 4.63257i 0 2.43027i
811.2 2.38193i 0 −3.67357 −1.00000 0 −4.05549 3.98632i 0 2.38193i
811.3 1.78296i 0 −1.17894 −1.00000 0 2.60402 1.46391i 0 1.78296i
811.4 1.38924i 0 0.0700071 −1.00000 0 0.680765 2.87574i 0 1.38924i
811.5 0.557941i 0 1.68870 −1.00000 0 4.24664 2.05808i 0 0.557941i
811.6 0.557941i 0 1.68870 −1.00000 0 4.24664 2.05808i 0 0.557941i
811.7 1.38924i 0 0.0700071 −1.00000 0 0.680765 2.87574i 0 1.38924i
811.8 1.78296i 0 −1.17894 −1.00000 0 2.60402 1.46391i 0 1.78296i
811.9 2.38193i 0 −3.67357 −1.00000 0 −4.05549 3.98632i 0 2.38193i
811.10 2.43027i 0 −3.90620 −1.00000 0 0.524070 4.63257i 0 2.43027i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 811.10
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1305.2.d.c 10
3.b odd 2 1 435.2.d.b 10
15.d odd 2 1 2175.2.d.g 10
15.e even 4 1 2175.2.f.c 10
15.e even 4 1 2175.2.f.f 10
29.b even 2 1 inner 1305.2.d.c 10
87.d odd 2 1 435.2.d.b 10
435.b odd 2 1 2175.2.d.g 10
435.p even 4 1 2175.2.f.c 10
435.p even 4 1 2175.2.f.f 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
435.2.d.b 10 3.b odd 2 1
435.2.d.b 10 87.d odd 2 1
1305.2.d.c 10 1.a even 1 1 trivial
1305.2.d.c 10 29.b even 2 1 inner
2175.2.d.g 10 15.d odd 2 1
2175.2.d.g 10 435.b odd 2 1
2175.2.f.c 10 15.e even 4 1
2175.2.f.c 10 435.p even 4 1
2175.2.f.f 10 15.e even 4 1
2175.2.f.f 10 435.p even 4 1

Hecke kernels

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

\( T_{2}^{10} + 17T_{2}^{8} + 104T_{2}^{6} + 273T_{2}^{4} + 281T_{2}^{2} + 64 \) Copy content Toggle raw display
\( T_{23}^{5} - 5T_{23}^{4} - 16T_{23}^{3} + 84T_{23}^{2} + 32T_{23} - 256 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} + 17 T^{8} + \cdots + 64 \) Copy content Toggle raw display
$3$ \( T^{10} \) Copy content Toggle raw display
$5$ \( (T + 1)^{10} \) Copy content Toggle raw display
$7$ \( (T^{5} - 4 T^{4} - 13 T^{3} + \cdots + 16)^{2} \) Copy content Toggle raw display
$11$ \( T^{10} + 99 T^{8} + \cdots + 1600 \) Copy content Toggle raw display
$13$ \( (T^{5} + 2 T^{4} - 27 T^{3} + \cdots + 32)^{2} \) Copy content Toggle raw display
$17$ \( T^{10} + 118 T^{8} + \cdots + 118336 \) Copy content Toggle raw display
$19$ \( T^{10} + 124 T^{8} + \cdots + 4096 \) Copy content Toggle raw display
$23$ \( (T^{5} - 5 T^{4} + \cdots - 256)^{2} \) Copy content Toggle raw display
$29$ \( T^{10} + 16 T^{9} + \cdots + 20511149 \) Copy content Toggle raw display
$31$ \( T^{10} + 220 T^{8} + \cdots + 54169600 \) Copy content Toggle raw display
$37$ \( T^{10} + 217 T^{8} + \cdots + 13191424 \) Copy content Toggle raw display
$41$ \( T^{10} + 113 T^{8} + \cdots + 102400 \) Copy content Toggle raw display
$43$ \( T^{10} + 193 T^{8} + \cdots + 65536 \) Copy content Toggle raw display
$47$ \( T^{10} + 262 T^{8} + \cdots + 20647936 \) Copy content Toggle raw display
$53$ \( (T^{5} - 19 T^{4} + \cdots - 592)^{2} \) Copy content Toggle raw display
$59$ \( (T^{5} - 6 T^{4} + \cdots + 1280)^{2} \) Copy content Toggle raw display
$61$ \( T^{10} + 216 T^{8} + \cdots + 1638400 \) Copy content Toggle raw display
$67$ \( (T^{5} - 10 T^{4} - 65 T^{3} + \cdots + 8)^{2} \) Copy content Toggle raw display
$71$ \( (T^{5} + 16 T^{4} + \cdots + 4096)^{2} \) Copy content Toggle raw display
$73$ \( T^{10} + \cdots + 750321664 \) Copy content Toggle raw display
$79$ \( T^{10} + 100 T^{8} + \cdots + 65536 \) Copy content Toggle raw display
$83$ \( (T^{5} + T^{4} - 212 T^{3} + \cdots - 8768)^{2} \) Copy content Toggle raw display
$89$ \( T^{10} + 294 T^{8} + \cdots + 5456896 \) Copy content Toggle raw display
$97$ \( T^{10} + \cdots + 101929216 \) Copy content Toggle raw display
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