Properties

Label 845.4.a.d
Level $845$
Weight $4$
Character orbit 845.a
Self dual yes
Analytic conductor $49.857$
Analytic rank $1$
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] = [845,4,Mod(1,845)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(845, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("845.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 845 = 5 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 845.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: \(49.8566139549\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 65)
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 \(\beta = \frac{1}{2}(1 + \sqrt{17})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta q^{2} + (4 \beta - 2) q^{3} + (\beta - 4) q^{4} + 5 q^{5} + ( - 2 \beta - 16) q^{6} + (2 \beta - 30) q^{7} + (11 \beta - 4) q^{8} + 41 q^{9} - 5 \beta q^{10} + ( - 2 \beta + 44) q^{11} + ( - 14 \beta + 24) q^{12} + \cdots + ( - 82 \beta + 1804) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{2} - 7 q^{4} + 10 q^{5} - 34 q^{6} - 58 q^{7} + 3 q^{8} + 82 q^{9} - 5 q^{10} + 86 q^{11} + 34 q^{12} + 12 q^{14} - 39 q^{16} - 28 q^{17} - 41 q^{18} - 166 q^{19} - 35 q^{20} + 68 q^{21} - 26 q^{22}+ \cdots + 3526 q^{99}+O(q^{100}) \) 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
2.56155
−1.56155
−2.56155 8.24621 −1.43845 5.00000 −21.1231 −24.8769 24.1771 41.0000 −12.8078
1.2 1.56155 −8.24621 −5.56155 5.00000 −12.8769 −33.1231 −21.1771 41.0000 7.80776
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(5\) \( -1 \)
\(13\) \( +1 \)

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 845.4.a.d 2
13.b even 2 1 65.4.a.c 2
39.d odd 2 1 585.4.a.h 2
52.b odd 2 1 1040.4.a.k 2
65.d even 2 1 325.4.a.g 2
65.h odd 4 2 325.4.b.f 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
65.4.a.c 2 13.b even 2 1
325.4.a.g 2 65.d even 2 1
325.4.b.f 4 65.h odd 4 2
585.4.a.h 2 39.d odd 2 1
845.4.a.d 2 1.a even 1 1 trivial
1040.4.a.k 2 52.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} + T_{2} - 4 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(845))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + T - 4 \) Copy content Toggle raw display
$3$ \( T^{2} - 68 \) Copy content Toggle raw display
$5$ \( (T - 5)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 58T + 824 \) Copy content Toggle raw display
$11$ \( T^{2} - 86T + 1832 \) Copy content Toggle raw display
$13$ \( T^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 28T - 4156 \) Copy content Toggle raw display
$19$ \( T^{2} + 166T + 4832 \) Copy content Toggle raw display
$23$ \( T^{2} - 60T - 16508 \) Copy content Toggle raw display
$29$ \( T^{2} - 120T + 268 \) Copy content Toggle raw display
$31$ \( T^{2} - 78T - 65952 \) Copy content Toggle raw display
$37$ \( T^{2} + 360T + 26892 \) Copy content Toggle raw display
$41$ \( T^{2} - 72T - 64052 \) Copy content Toggle raw display
$43$ \( T^{2} + 44T - 6316 \) Copy content Toggle raw display
$47$ \( T^{2} + 362T + 16424 \) Copy content Toggle raw display
$53$ \( T^{2} + 396T + 36 \) Copy content Toggle raw display
$59$ \( T^{2} + 18T - 128592 \) Copy content Toggle raw display
$61$ \( (T - 442)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 1322 T + 249496 \) Copy content Toggle raw display
$71$ \( T^{2} + 634T - 318544 \) Copy content Toggle raw display
$73$ \( T^{2} - 60T - 478908 \) Copy content Toggle raw display
$79$ \( T^{2} + 180T + 7488 \) Copy content Toggle raw display
$83$ \( T^{2} + 714T - 74528 \) Copy content Toggle raw display
$89$ \( T^{2} - 852T + 120276 \) Copy content Toggle raw display
$97$ \( T^{2} + 32T - 2970052 \) Copy content Toggle raw display
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