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Properties

Label 810.3.j.a
Level $810$
Weight $3$
Character orbit 810.j
Analytic conductor $22.071$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [810,3,Mod(269,810)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(810, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([1, 3]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("810.269");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 810 = 2 \cdot 3^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 810.j (of order \(6\), degree \(2\), not 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: \(22.0709014132\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: 8.0.81622204416.3
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 14x^{6} + 165x^{4} - 434x^{2} + 961 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 270)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{5} - \beta_1) q^{2} - 2 \beta_{4} q^{4} + ( - \beta_{5} - 3 \beta_{4} - \beta_{3}) q^{5} + ( - 2 \beta_{7} - \beta_{6} - 2 \beta_{2}) q^{7} + 2 \beta_1 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{5} - \beta_1) q^{2} - 2 \beta_{4} q^{4} + ( - \beta_{5} - 3 \beta_{4} - \beta_{3}) q^{5} + ( - 2 \beta_{7} - \beta_{6} - 2 \beta_{2}) q^{7} + 2 \beta_1 q^{8} + ( - \beta_{7} + \beta_{6} - \beta_{3} + \cdots + 2) q^{10}+ \cdots + ( - 45 \beta_1 - 132) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 8 q^{4} - 12 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 8 q^{4} - 12 q^{5} + 16 q^{10} - 16 q^{16} - 24 q^{17} - 24 q^{19} - 24 q^{20} - 60 q^{23} + 12 q^{25} + 68 q^{31} - 56 q^{34} + 144 q^{35} + 24 q^{38} - 16 q^{40} + 224 q^{46} + 240 q^{47} + 180 q^{49} - 96 q^{50} + 408 q^{53} - 176 q^{55} + 196 q^{61} - 240 q^{62} + 64 q^{64} + 24 q^{65} + 24 q^{68} - 80 q^{70} + 24 q^{76} + 312 q^{77} - 180 q^{79} + 96 q^{80} - 108 q^{83} - 20 q^{85} + 912 q^{91} - 120 q^{92} - 112 q^{94} + 60 q^{95} - 1056 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 14x^{6} + 165x^{4} - 434x^{2} + 961 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{6} + 721 ) / 495 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{7} + 1711\nu ) / 495 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{7} + 721\nu ) / 495 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -14\nu^{6} + 165\nu^{4} - 2310\nu^{2} + 6076 ) / 5115 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -67\nu^{6} + 1155\nu^{4} - 11055\nu^{2} + 29078 ) / 15345 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -67\nu^{7} + 1155\nu^{5} - 11055\nu^{3} + 29078\nu ) / 15345 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -182\nu^{7} + 2145\nu^{5} - 24915\nu^{3} + 12493\nu ) / 15345 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{3} + \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 3\beta_{5} - 7\beta_{4} - 3\beta _1 + 7 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -7\beta_{7} + 13\beta_{6} - 13\beta_{3} ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 42\beta_{5} - 67\beta_{4} \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -67\beta_{7} + 151\beta_{6} - 67\beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 495\beta _1 - 721 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 1711\beta_{3} - 721\beta_{2} ) / 2 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/810\mathbb{Z}\right)^\times\).

\(n\) \(487\) \(731\)
\(\chi(n)\) \(-1\) \(\beta_{4}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
269.1
1.43806 − 0.830265i
−1.43806 + 0.830265i
−2.90379 + 1.67650i
2.90379 − 1.67650i
1.43806 + 0.830265i
−1.43806 − 0.830265i
−2.90379 − 1.67650i
2.90379 + 1.67650i
−0.707107 + 1.22474i 0 −1.00000 − 1.73205i −4.24083 − 2.64865i 0 −11.8534 − 6.84358i 2.82843 0 6.24264 − 3.32106i
269.2 −0.707107 + 1.22474i 0 −1.00000 − 1.73205i −0.173381 − 4.99699i 0 11.8534 + 6.84358i 2.82843 0 6.24264 + 3.32106i
269.3 0.707107 − 1.22474i 0 −1.00000 − 1.73205i −4.89947 + 0.997601i 0 −0.704577 − 0.406788i −2.82843 0 −2.24264 + 6.70601i
269.4 0.707107 − 1.22474i 0 −1.00000 − 1.73205i 3.31368 − 3.74426i 0 0.704577 + 0.406788i −2.82843 0 −2.24264 − 6.70601i
539.1 −0.707107 − 1.22474i 0 −1.00000 + 1.73205i −4.24083 + 2.64865i 0 −11.8534 + 6.84358i 2.82843 0 6.24264 + 3.32106i
539.2 −0.707107 − 1.22474i 0 −1.00000 + 1.73205i −0.173381 + 4.99699i 0 11.8534 − 6.84358i 2.82843 0 6.24264 − 3.32106i
539.3 0.707107 + 1.22474i 0 −1.00000 + 1.73205i −4.89947 − 0.997601i 0 −0.704577 + 0.406788i −2.82843 0 −2.24264 − 6.70601i
539.4 0.707107 + 1.22474i 0 −1.00000 + 1.73205i 3.31368 + 3.74426i 0 0.704577 − 0.406788i −2.82843 0 −2.24264 + 6.70601i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 269.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner
15.d odd 2 1 inner
45.h odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 810.3.j.a 8
3.b odd 2 1 810.3.j.f 8
5.b even 2 1 810.3.j.f 8
9.c even 3 1 270.3.b.d yes 4
9.c even 3 1 inner 810.3.j.a 8
9.d odd 6 1 270.3.b.a ✓ 4
9.d odd 6 1 810.3.j.f 8
15.d odd 2 1 inner 810.3.j.a 8
36.f odd 6 1 2160.3.c.m 4
36.h even 6 1 2160.3.c.g 4
45.h odd 6 1 270.3.b.d yes 4
45.h odd 6 1 inner 810.3.j.a 8
45.j even 6 1 270.3.b.a ✓ 4
45.j even 6 1 810.3.j.f 8
45.k odd 12 2 1350.3.d.o 8
45.l even 12 2 1350.3.d.o 8
180.n even 6 1 2160.3.c.m 4
180.p odd 6 1 2160.3.c.g 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
270.3.b.a ✓ 4 9.d odd 6 1
270.3.b.a ✓ 4 45.j even 6 1
270.3.b.d yes 4 9.c even 3 1
270.3.b.d yes 4 45.h odd 6 1
810.3.j.a 8 1.a even 1 1 trivial
810.3.j.a 8 9.c even 3 1 inner
810.3.j.a 8 15.d odd 2 1 inner
810.3.j.a 8 45.h odd 6 1 inner
810.3.j.f 8 3.b odd 2 1
810.3.j.f 8 5.b even 2 1
810.3.j.f 8 9.d odd 6 1
810.3.j.f 8 45.j even 6 1
1350.3.d.o 8 45.k odd 12 2
1350.3.d.o 8 45.l even 12 2
2160.3.c.g 4 36.h even 6 1
2160.3.c.g 4 180.p odd 6 1
2160.3.c.m 4 36.f odd 6 1
2160.3.c.m 4 180.n even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(810, [\chi])\):

\( T_{7}^{8} - 188T_{7}^{6} + 35220T_{7}^{4} - 23312T_{7}^{2} + 15376 \) Copy content Toggle raw display
\( T_{17}^{2} + 6T_{17} - 89 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} + 2 T^{2} + 4)^{2} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} + 12 T^{7} + \cdots + 390625 \) Copy content Toggle raw display
$7$ \( T^{8} - 188 T^{6} + \cdots + 15376 \) Copy content Toggle raw display
$11$ \( T^{8} + \cdots + 1284218896 \) Copy content Toggle raw display
$13$ \( T^{8} - 324 T^{6} + \cdots + 100881936 \) Copy content Toggle raw display
$17$ \( (T^{2} + 6 T - 89)^{4} \) Copy content Toggle raw display
$19$ \( (T^{2} + 6 T - 9)^{4} \) Copy content Toggle raw display
$23$ \( (T^{4} + 30 T^{3} + \cdots + 27889)^{2} \) Copy content Toggle raw display
$29$ \( T^{8} + \cdots + 661886382096 \) Copy content Toggle raw display
$31$ \( (T^{4} - 34 T^{3} + \cdots + 25921)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} + 5408 T^{2} + 2293504)^{2} \) Copy content Toggle raw display
$41$ \( T^{8} + \cdots + 4302835216 \) Copy content Toggle raw display
$43$ \( T^{8} + \cdots + 4302835216 \) Copy content Toggle raw display
$47$ \( (T^{4} - 120 T^{3} + \cdots + 10291264)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} - 102 T + 2439)^{4} \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 5416585950736 \) Copy content Toggle raw display
$61$ \( (T^{4} - 98 T^{3} + \cdots + 1560001)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots + 17\!\cdots\!56 \) Copy content Toggle raw display
$71$ \( (T^{4} + 15876 T^{2} + 53524476)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} + 8036 T^{2} + 35836)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} + 90 T^{3} + \cdots + 34963569)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} + 54 T^{3} + \cdots + 35988001)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} + 23548 T^{2} + 87702844)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} + \cdots + 70\!\cdots\!76 \) Copy content Toggle raw display
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