Properties

Label 1305.2.d.d
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: \( x^{10} + 17x^{8} + 96x^{6} + 201x^{4} + 121x^{2} + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{4} \)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{2} - 1) q^{4} + q^{5} - \beta_{5} q^{7} + (\beta_{9} - \beta_{8} + \cdots - 2 \beta_1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (\beta_{2} - 1) q^{4} + q^{5} - \beta_{5} q^{7} + (\beta_{9} - \beta_{8} + \cdots - 2 \beta_1) q^{8}+ \cdots + (\beta_{9} - 4 \beta_{8} + \cdots - 4 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 14 q^{4} + 10 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 10 q - 14 q^{4} + 10 q^{5} - 4 q^{13} + 30 q^{16} - 14 q^{20} + 14 q^{22} - 2 q^{23} + 10 q^{25} + 10 q^{28} + 10 q^{34} + 4 q^{38} + 14 q^{49} - 6 q^{52} + 26 q^{53} + 50 q^{58} + 12 q^{59} + 4 q^{62} - 36 q^{64} - 4 q^{65} - 20 q^{67} + 80 q^{71} - 20 q^{74} + 30 q^{80} + 68 q^{82} - 78 q^{83} + 96 q^{86} - 108 q^{88} + 56 q^{91} - 116 q^{92} - 50 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} + 17x^{8} + 96x^{6} + 201x^{4} + 121x^{2} + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{8} - 20\nu^{6} - 116\nu^{4} - 189\nu^{2} - 48 ) / 20 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{9} - 15\nu^{7} - 76\nu^{5} - 169\nu^{3} - 183\nu ) / 20 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{8} + 20\nu^{6} + 136\nu^{4} + 329\nu^{2} + 128 ) / 20 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{9} + 20\nu^{7} + 136\nu^{5} + 349\nu^{3} + 228\nu ) / 20 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{8} + 15\nu^{6} + 71\nu^{4} + 114\nu^{2} + 38 ) / 5 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -3\nu^{9} - 50\nu^{7} - 268\nu^{5} - 487\nu^{3} - 174\nu ) / 20 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( -4\nu^{9} - 65\nu^{7} - 344\nu^{5} - 636\nu^{3} - 237\nu ) / 20 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{9} - \beta_{8} - \beta_{4} - 6\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{5} + \beta_{3} - 7\beta_{2} + 17 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -10\beta_{9} + 11\beta_{8} + \beta_{6} + 8\beta_{4} + 39\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -\beta_{7} - 9\beta_{5} - 13\beta_{3} + 48\beta_{2} - 110 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 84\beta_{9} - 96\beta_{8} - 8\beta_{6} - 56\beta_{4} - 261\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 20\beta_{7} + 64\beta_{5} + 124\beta_{3} - 337\beta_{2} + 747 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( -669\beta_{9} + 773\beta_{8} + 44\beta_{6} + 381\beta_{4} + 1782\beta_1 \) 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
2.72674i
2.37667i
1.74703i
0.825137i
0.428173i
0.428173i
0.825137i
1.74703i
2.37667i
2.72674i
2.72674i 0 −5.43512 1.00000 0 −1.78170 9.36669i 0 2.72674i
811.2 2.37667i 0 −3.64857 1.00000 0 −1.11968 3.91810i 0 2.37667i
811.3 1.74703i 0 −1.05213 1.00000 0 4.55534 1.65596i 0 1.74703i
811.4 0.825137i 0 1.31915 1.00000 0 1.95267 2.73875i 0 0.825137i
811.5 0.428173i 0 1.81667 1.00000 0 −3.60663 1.63419i 0 0.428173i
811.6 0.428173i 0 1.81667 1.00000 0 −3.60663 1.63419i 0 0.428173i
811.7 0.825137i 0 1.31915 1.00000 0 1.95267 2.73875i 0 0.825137i
811.8 1.74703i 0 −1.05213 1.00000 0 4.55534 1.65596i 0 1.74703i
811.9 2.37667i 0 −3.64857 1.00000 0 −1.11968 3.91810i 0 2.37667i
811.10 2.72674i 0 −5.43512 1.00000 0 −1.78170 9.36669i 0 2.72674i
\(n\): 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.d 10
3.b odd 2 1 435.2.d.a 10
15.d odd 2 1 2175.2.d.f 10
15.e even 4 1 2175.2.f.d 10
15.e even 4 1 2175.2.f.e 10
29.b even 2 1 inner 1305.2.d.d 10
87.d odd 2 1 435.2.d.a 10
435.b odd 2 1 2175.2.d.f 10
435.p even 4 1 2175.2.f.d 10
435.p even 4 1 2175.2.f.e 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
435.2.d.a 10 3.b odd 2 1
435.2.d.a 10 87.d odd 2 1
1305.2.d.d 10 1.a even 1 1 trivial
1305.2.d.d 10 29.b even 2 1 inner
2175.2.d.f 10 15.d odd 2 1
2175.2.d.f 10 435.b odd 2 1
2175.2.f.d 10 15.e even 4 1
2175.2.f.d 10 435.p even 4 1
2175.2.f.e 10 15.e even 4 1
2175.2.f.e 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 \(S_{2}^{\mathrm{new}}(1305, [\chi])\):

\( T_{2}^{10} + 17T_{2}^{8} + 96T_{2}^{6} + 201T_{2}^{4} + 121T_{2}^{2} + 16 \) Copy content Toggle raw display
\( T_{23}^{5} + T_{23}^{4} - 84T_{23}^{3} - 128T_{23}^{2} + 1728T_{23} + 3328 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} + 17 T^{8} + \cdots + 16 \) Copy content Toggle raw display
$3$ \( T^{10} \) Copy content Toggle raw display
$5$ \( (T - 1)^{10} \) Copy content Toggle raw display
$7$ \( (T^{5} - 21 T^{3} + \cdots + 64)^{2} \) Copy content Toggle raw display
$11$ \( T^{10} + 59 T^{8} + \cdots + 35344 \) Copy content Toggle raw display
$13$ \( (T^{5} + 2 T^{4} + \cdots - 368)^{2} \) Copy content Toggle raw display
$17$ \( T^{10} + 62 T^{8} + \cdots + 64 \) Copy content Toggle raw display
$19$ \( (T^{2} + 4)^{5} \) Copy content Toggle raw display
$23$ \( (T^{5} + T^{4} - 84 T^{3} + \cdots + 3328)^{2} \) Copy content Toggle raw display
$29$ \( T^{10} + 37 T^{8} + \cdots + 20511149 \) Copy content Toggle raw display
$31$ \( (T^{2} + 4)^{5} \) Copy content Toggle raw display
$37$ \( T^{10} + \cdots + 666052864 \) Copy content Toggle raw display
$41$ \( T^{10} + 161 T^{8} + \cdots + 65536 \) Copy content Toggle raw display
$43$ \( T^{10} + 233 T^{8} + \cdots + 16777216 \) Copy content Toggle raw display
$47$ \( T^{10} + 94 T^{8} + \cdots + 861184 \) Copy content Toggle raw display
$53$ \( (T^{5} - 13 T^{4} + \cdots - 784)^{2} \) Copy content Toggle raw display
$59$ \( (T^{5} - 6 T^{4} + \cdots - 21376)^{2} \) Copy content Toggle raw display
$61$ \( T^{10} + 168 T^{8} + \cdots + 4194304 \) Copy content Toggle raw display
$67$ \( (T^{5} + 10 T^{4} + \cdots + 3616)^{2} \) Copy content Toggle raw display
$71$ \( (T - 8)^{10} \) Copy content Toggle raw display
$73$ \( T^{10} + 297 T^{8} + \cdots + 1183744 \) Copy content Toggle raw display
$79$ \( T^{10} + \cdots + 11097358336 \) Copy content Toggle raw display
$83$ \( (T^{5} + 39 T^{4} + \cdots + 4352)^{2} \) Copy content Toggle raw display
$89$ \( T^{10} + 310 T^{8} + \cdots + 16384 \) Copy content Toggle raw display
$97$ \( T^{10} + \cdots + 536015104 \) Copy content Toggle raw display
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