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Properties

Label 495.2.n.a
Level $495$
Weight $2$
Character orbit 495.n
Analytic conductor $3.953$
Analytic rank $0$
Dimension $8$
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] = [495,2,Mod(91,495)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(495, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([0, 0, 8]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("495.91");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 495 = 3^{2} \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 495.n (of order \(5\), degree \(4\), 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: \(3.95259490005\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{5})\)
Coefficient field: 8.0.13140625.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 3x^{7} + 5x^{6} - 3x^{5} + 4x^{4} + 3x^{3} + 5x^{2} + 3x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 165)
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( - \beta_{7} + \beta_{6} + \beta_{5} + \cdots - 1) q^{2}+ \cdots + ( - \beta_{7} + 2 \beta_{6} + \cdots - 2 \beta_1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{7} + \beta_{6} + \beta_{5} + \cdots - 1) q^{2}+ \cdots + ( - 2 \beta_{7} + 7 \beta_{5} + \cdots - 4) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 4 q^{2} + 2 q^{4} + 2 q^{5} + 3 q^{7} + q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 4 q^{2} + 2 q^{4} + 2 q^{5} + 3 q^{7} + q^{8} - 6 q^{10} - 3 q^{11} - 4 q^{13} + 4 q^{14} - 12 q^{16} + 2 q^{19} + 3 q^{20} + 9 q^{22} + 6 q^{23} - 2 q^{25} - 2 q^{26} - 11 q^{28} - 10 q^{29} + 19 q^{31} - 12 q^{32} - 6 q^{34} - 3 q^{35} - q^{37} + 20 q^{38} - q^{40} + 9 q^{41} - 17 q^{44} - 22 q^{46} + 19 q^{47} + q^{49} - 4 q^{50} - 2 q^{52} - 25 q^{53} + 3 q^{55} + 16 q^{56} - 12 q^{58} - 13 q^{59} + 13 q^{61} - 35 q^{62} + 39 q^{64} + 14 q^{65} + 2 q^{67} - 19 q^{68} - 4 q^{70} + 11 q^{71} - 7 q^{73} + 43 q^{74} - 38 q^{76} + 7 q^{77} - 22 q^{79} - 13 q^{80} - 35 q^{82} + 21 q^{83} + 10 q^{85} - 20 q^{86} + 59 q^{88} + 20 q^{89} - 11 q^{91} + 28 q^{92} - 35 q^{94} - 2 q^{95} + 31 q^{97} - 22 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 3x^{7} + 5x^{6} - 3x^{5} + 4x^{4} + 3x^{3} + 5x^{2} + 3x + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{7} + 2\nu^{6} - 3\nu^{5} - 4\nu^{3} - 7\nu^{2} - 12\nu - 7 ) / 8 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{7} - 7\nu^{5} + 20\nu^{4} - 16\nu^{3} + 19\nu^{2} + 6\nu + 9 ) / 8 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{7} + 4\nu^{6} - 9\nu^{5} + 12\nu^{4} - 16\nu^{3} + 13\nu^{2} - 10\nu - 1 ) / 8 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -3\nu^{7} + 10\nu^{6} - 17\nu^{5} + 8\nu^{4} - 4\nu^{3} - 13\nu^{2} - 8\nu - 5 ) / 8 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 3\nu^{7} - 12\nu^{6} + 23\nu^{5} - 20\nu^{4} + 16\nu^{3} + \nu^{2} + 6\nu - 1 ) / 8 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -5\nu^{7} + 18\nu^{6} - 35\nu^{5} + 32\nu^{4} - 28\nu^{3} - 11\nu^{2} - 12\nu - 7 ) / 8 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{6} + \beta_{5} + \beta_{4} - \beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{7} + 3\beta_{6} + 2\beta_{5} + \beta_{4} - 3\beta_{2} - 2\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 3\beta_{7} + 4\beta_{6} + \beta_{4} - \beta_{3} - 5\beta_{2} - 4\beta_1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 4\beta_{7} - 6\beta_{5} - 4\beta_{3} - 6\beta_{2} - 6\beta _1 - 1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -16\beta_{6} - 16\beta_{5} - 6\beta_{4} - 6\beta_{3} - 7\beta _1 - 6 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -16\beta_{7} - 51\beta_{6} - 29\beta_{5} - 23\beta_{4} + 29\beta_{2} - 16 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/495\mathbb{Z}\right)^\times\).

\(n\) \(46\) \(56\) \(397\)
\(\chi(n)\) \(-\beta_{3}\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
91.1
0.418926 − 1.28932i
−0.227943 + 0.701538i
0.418926 + 1.28932i
−0.227943 − 0.701538i
1.69513 − 1.23158i
−0.386111 + 0.280526i
1.69513 + 1.23158i
−0.386111 − 0.280526i
−1.90578 + 1.38463i 0 1.09676 − 3.37549i 0.809017 + 0.587785i 0 0.0598032 − 0.184055i 1.12773 + 3.47080i 0 −2.35567
91.2 −0.212253 + 0.154211i 0 −0.596764 + 1.83665i 0.809017 + 0.587785i 0 −0.986854 + 3.03722i −0.318714 − 0.980901i 0 −0.262360
136.1 −1.90578 − 1.38463i 0 1.09676 + 3.37549i 0.809017 − 0.587785i 0 0.0598032 + 0.184055i 1.12773 − 3.47080i 0 −2.35567
136.2 −0.212253 − 0.154211i 0 −0.596764 − 1.83665i 0.809017 − 0.587785i 0 −0.986854 − 3.03722i −0.318714 + 0.980901i 0 −0.262360
181.1 −0.338464 − 1.04169i 0 0.647481 − 0.470423i −0.309017 + 0.951057i 0 0.570387 − 0.414410i −2.48141 − 1.80285i 0 1.09529
181.2 0.456498 + 1.40496i 0 −0.147481 + 0.107152i −0.309017 + 0.951057i 0 1.85666 − 1.34895i 2.17239 + 1.57833i 0 −1.47726
361.1 −0.338464 + 1.04169i 0 0.647481 + 0.470423i −0.309017 − 0.951057i 0 0.570387 + 0.414410i −2.48141 + 1.80285i 0 1.09529
361.2 0.456498 − 1.40496i 0 −0.147481 − 0.107152i −0.309017 − 0.951057i 0 1.85666 + 1.34895i 2.17239 − 1.57833i 0 −1.47726
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 91.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.c even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 495.2.n.a 8
3.b odd 2 1 165.2.m.d ✓ 8
11.c even 5 1 inner 495.2.n.a 8
11.c even 5 1 5445.2.a.bt 4
11.d odd 10 1 5445.2.a.bf 4
15.d odd 2 1 825.2.n.g 8
15.e even 4 2 825.2.bx.f 16
33.f even 10 1 1815.2.a.w 4
33.h odd 10 1 165.2.m.d ✓ 8
33.h odd 10 1 1815.2.a.p 4
165.o odd 10 1 825.2.n.g 8
165.o odd 10 1 9075.2.a.di 4
165.r even 10 1 9075.2.a.cm 4
165.v even 20 2 825.2.bx.f 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
165.2.m.d ✓ 8 3.b odd 2 1
165.2.m.d ✓ 8 33.h odd 10 1
495.2.n.a 8 1.a even 1 1 trivial
495.2.n.a 8 11.c even 5 1 inner
825.2.n.g 8 15.d odd 2 1
825.2.n.g 8 165.o odd 10 1
825.2.bx.f 16 15.e even 4 2
825.2.bx.f 16 165.v even 20 2
1815.2.a.p 4 33.h odd 10 1
1815.2.a.w 4 33.f even 10 1
5445.2.a.bf 4 11.d odd 10 1
5445.2.a.bt 4 11.c even 5 1
9075.2.a.cm 4 165.r even 10 1
9075.2.a.di 4 165.o odd 10 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{8} + 4T_{2}^{7} + 9T_{2}^{6} + 13T_{2}^{5} + 24T_{2}^{4} + 21T_{2}^{3} + 21T_{2}^{2} + 7T_{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(495, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} + 4 T^{7} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( (T^{4} - T^{3} + T^{2} + \cdots + 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{8} - 3 T^{7} + \cdots + 1 \) Copy content Toggle raw display
$11$ \( T^{8} + 3 T^{7} + \cdots + 14641 \) Copy content Toggle raw display
$13$ \( T^{8} + 4 T^{7} + \cdots + 121 \) Copy content Toggle raw display
$17$ \( T^{8} + 47 T^{6} + \cdots + 1 \) Copy content Toggle raw display
$19$ \( T^{8} - 2 T^{7} + \cdots + 1 \) Copy content Toggle raw display
$23$ \( (T^{4} - 3 T^{3} + \cdots + 449)^{2} \) Copy content Toggle raw display
$29$ \( T^{8} + 10 T^{7} + \cdots + 249001 \) Copy content Toggle raw display
$31$ \( T^{8} - 19 T^{7} + \cdots + 1560001 \) Copy content Toggle raw display
$37$ \( T^{8} + T^{7} + \cdots + 12538681 \) Copy content Toggle raw display
$41$ \( T^{8} - 9 T^{7} + \cdots + 1681 \) Copy content Toggle raw display
$43$ \( (T^{4} - 110 T^{2} + \cdots + 275)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} - 19 T^{7} + \cdots + 961 \) Copy content Toggle raw display
$53$ \( T^{8} + 25 T^{7} + \cdots + 410881 \) Copy content Toggle raw display
$59$ \( T^{8} + 13 T^{7} + \cdots + 346921 \) Copy content Toggle raw display
$61$ \( T^{8} - 13 T^{7} + \cdots + 1324801 \) Copy content Toggle raw display
$67$ \( (T^{4} - T^{3} - 100 T^{2} + \cdots + 619)^{2} \) Copy content Toggle raw display
$71$ \( T^{8} - 11 T^{7} + \cdots + 78961 \) Copy content Toggle raw display
$73$ \( T^{8} + 7 T^{7} + \cdots + 866761 \) Copy content Toggle raw display
$79$ \( T^{8} + 22 T^{7} + \cdots + 28561 \) Copy content Toggle raw display
$83$ \( T^{8} - 21 T^{7} + \cdots + 1681 \) Copy content Toggle raw display
$89$ \( (T^{4} - 10 T^{3} + \cdots + 209)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} - 31 T^{7} + \cdots + 1343281 \) Copy content Toggle raw display
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