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Properties

Label 845.2.b.e
Level $845$
Weight $2$
Character orbit 845.b
Analytic conductor $6.747$
Analytic rank $0$
Dimension $6$
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] = [845,2,Mod(339,845)]
 
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([1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("845.339");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 845 = 5 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 845.b (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: \(6.74735897080\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.49843600.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} + 8x^{4} + 10x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 65)
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + ( - \beta_{5} + \beta_1) q^{3} + (\beta_{2} - 1) q^{4} + (\beta_{3} + 1) q^{5} + (\beta_{2} - 2) q^{6} + (\beta_{5} + \beta_1) q^{7} + (\beta_{5} + \beta_{4} - \beta_{3} + \cdots - 1) q^{8}+ \cdots + ( - \beta_{4} - \beta_{3} - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + ( - \beta_{5} + \beta_1) q^{3} + (\beta_{2} - 1) q^{4} + (\beta_{3} + 1) q^{5} + (\beta_{2} - 2) q^{6} + (\beta_{5} + \beta_1) q^{7} + (\beta_{5} + \beta_{4} - \beta_{3} + \cdots - 1) q^{8}+ \cdots + ( - 2 \beta_{2} - 8) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 4 q^{4} + 3 q^{5} - 10 q^{6} - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 4 q^{4} + 3 q^{5} - 10 q^{6} - 6 q^{9} - 7 q^{10} - 22 q^{14} - 4 q^{15} + 16 q^{16} + 12 q^{19} - q^{20} + 4 q^{21} + 32 q^{24} - q^{25} - 18 q^{29} - 4 q^{30} + 8 q^{31} - 8 q^{34} - 10 q^{35} + 2 q^{36} + 35 q^{40} + 14 q^{41} + 2 q^{44} - 29 q^{45} + 10 q^{46} - 6 q^{49} - 31 q^{50} + 12 q^{51} - 22 q^{54} + 26 q^{55} + 16 q^{56} - 4 q^{59} + 48 q^{60} - 6 q^{61} - 6 q^{64} + 2 q^{66} + 24 q^{69} - 10 q^{70} - 12 q^{71} - 8 q^{74} - 2 q^{75} - 10 q^{76} - 52 q^{79} + 33 q^{80} - 14 q^{81} + 90 q^{84} + 21 q^{85} + 2 q^{86} + 20 q^{89} + 31 q^{90} - 56 q^{94} - 20 q^{95} - 6 q^{96} - 52 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} + 8x^{4} + 10x^{2} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{5} + \nu^{4} + 7\nu^{3} + 7\nu^{2} + 5\nu + 3 ) / 2 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{5} + \nu^{4} - 7\nu^{3} + 7\nu^{2} - 5\nu + 5 ) / 2 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( \nu^{5} + 8\nu^{3} + 10\nu \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{5} + \beta_{4} - \beta_{3} - 5\beta _1 - 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{4} + \beta_{3} - 7\beta_{2} + 17 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -7\beta_{5} - 8\beta_{4} + 8\beta_{3} + 30\beta _1 + 8 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/845\mathbb{Z}\right)^\times\).

\(n\) \(171\) \(677\)
\(\chi(n)\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
339.1
− 2.54574i
− 1.18733i
− 0.330837i
0.330837i
1.18733i
2.54574i
− 2.54574i − 2.15293i −4.48079 0.817544 − 2.08125i −5.48079 − 2.93855i 6.31544i −1.63509 −5.29833 − 2.08125i
339.2 − 1.18733i − 0.345110i 0.590239 −1.44045 + 1.71029i −0.409761 − 2.02956i − 3.07548i 2.88090 2.03069 + 1.71029i
339.3 − 0.330837i 2.69180i 1.89055 2.12291 − 0.702335i 0.890547 − 3.35348i − 1.28714i −4.24581 −0.232358 − 0.702335i
339.4 0.330837i − 2.69180i 1.89055 2.12291 + 0.702335i 0.890547 3.35348i 1.28714i −4.24581 −0.232358 + 0.702335i
339.5 1.18733i 0.345110i 0.590239 −1.44045 − 1.71029i −0.409761 2.02956i 3.07548i 2.88090 2.03069 − 1.71029i
339.6 2.54574i 2.15293i −4.48079 0.817544 + 2.08125i −5.48079 2.93855i − 6.31544i −1.63509 −5.29833 + 2.08125i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 339.6
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 845.2.b.e 6
5.b even 2 1 inner 845.2.b.e 6
5.c odd 4 2 4225.2.a.bq 6
13.b even 2 1 845.2.b.d 6
13.c even 3 2 845.2.n.e 12
13.d odd 4 2 845.2.d.d 12
13.e even 6 2 65.2.n.a ✓ 12
13.f odd 12 4 845.2.l.f 24
39.h odd 6 2 585.2.bs.a 12
52.i odd 6 2 1040.2.dh.a 12
65.d even 2 1 845.2.b.d 6
65.g odd 4 2 845.2.d.d 12
65.h odd 4 2 4225.2.a.br 6
65.l even 6 2 65.2.n.a ✓ 12
65.n even 6 2 845.2.n.e 12
65.r odd 12 4 325.2.e.e 12
65.s odd 12 4 845.2.l.f 24
195.y odd 6 2 585.2.bs.a 12
260.w odd 6 2 1040.2.dh.a 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
65.2.n.a ✓ 12 13.e even 6 2
65.2.n.a ✓ 12 65.l even 6 2
325.2.e.e 12 65.r odd 12 4
585.2.bs.a 12 39.h odd 6 2
585.2.bs.a 12 195.y odd 6 2
845.2.b.d 6 13.b even 2 1
845.2.b.d 6 65.d even 2 1
845.2.b.e 6 1.a even 1 1 trivial
845.2.b.e 6 5.b even 2 1 inner
845.2.d.d 12 13.d odd 4 2
845.2.d.d 12 65.g odd 4 2
845.2.l.f 24 13.f odd 12 4
845.2.l.f 24 65.s odd 12 4
845.2.n.e 12 13.c even 3 2
845.2.n.e 12 65.n even 6 2
1040.2.dh.a 12 52.i odd 6 2
1040.2.dh.a 12 260.w odd 6 2
4225.2.a.bq 6 5.c odd 4 2
4225.2.a.br 6 65.h odd 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(845, [\chi])\):

\( T_{2}^{6} + 8T_{2}^{4} + 10T_{2}^{2} + 1 \) Copy content Toggle raw display
\( T_{11}^{3} - 13T_{11} + 8 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} + 8 T^{4} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( T^{6} + 12 T^{4} + \cdots + 4 \) Copy content Toggle raw display
$5$ \( T^{6} - 3 T^{5} + \cdots + 125 \) Copy content Toggle raw display
$7$ \( T^{6} + 24 T^{4} + \cdots + 400 \) Copy content Toggle raw display
$11$ \( (T^{3} - 13 T + 8)^{2} \) Copy content Toggle raw display
$13$ \( T^{6} \) Copy content Toggle raw display
$17$ \( T^{6} + 35 T^{4} + \cdots + 169 \) Copy content Toggle raw display
$19$ \( (T^{3} - 6 T^{2} - T + 10)^{2} \) Copy content Toggle raw display
$23$ \( T^{6} + 12 T^{4} + \cdots + 4 \) Copy content Toggle raw display
$29$ \( (T + 3)^{6} \) Copy content Toggle raw display
$31$ \( (T^{3} - 4 T^{2} - 40 T - 40)^{2} \) Copy content Toggle raw display
$37$ \( T^{6} + 35 T^{4} + \cdots + 169 \) Copy content Toggle raw display
$41$ \( (T^{3} - 7 T^{2} - 29 T - 5)^{2} \) Copy content Toggle raw display
$43$ \( T^{6} + 80 T^{4} + \cdots + 256 \) Copy content Toggle raw display
$47$ \( T^{6} + 236 T^{4} + \cdots + 270400 \) Copy content Toggle raw display
$53$ \( T^{6} + 171 T^{4} + \cdots + 400 \) Copy content Toggle raw display
$59$ \( (T^{3} + 2 T^{2} + \cdots - 136)^{2} \) Copy content Toggle raw display
$61$ \( (T^{3} + 3 T^{2} + \cdots - 115)^{2} \) Copy content Toggle raw display
$67$ \( T^{6} + 100 T^{4} + \cdots + 20164 \) Copy content Toggle raw display
$71$ \( (T^{3} + 6 T^{2} - T - 26)^{2} \) Copy content Toggle raw display
$73$ \( T^{6} + 215 T^{4} + \cdots + 250000 \) Copy content Toggle raw display
$79$ \( (T^{3} + 26 T^{2} + \cdots + 160)^{2} \) Copy content Toggle raw display
$83$ \( T^{6} + 276 T^{4} + \cdots + 640000 \) Copy content Toggle raw display
$89$ \( (T^{3} - 10 T^{2} + \cdots + 1586)^{2} \) Copy content Toggle raw display
$97$ \( T^{6} + 280 T^{4} + \cdots + 204304 \) Copy content Toggle raw display
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