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Properties

Label 2835.2.a.x
Level $2835$
Weight $2$
Character orbit 2835.a
Self dual yes
Analytic conductor $22.638$
Analytic rank $0$
Dimension $8$
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] = [2835,2,Mod(1,2835)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2835, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2835.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2835 = 3^{4} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2835.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: \(22.6375889730\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{7} - 13x^{6} + 11x^{5} + 49x^{4} - 32x^{3} - 55x^{2} + 22x + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 3^{2} \)
Twist minimal: no (minimal twist has level 315)
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 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.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + (\beta_{2} + 1) q^{4} + q^{5} - q^{7} + (\beta_{6} - \beta_{4} + \beta_{3} - \beta_1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + (\beta_{2} + 1) q^{4} + q^{5} - q^{7} + (\beta_{6} - \beta_{4} + \beta_{3} - \beta_1) q^{8} - \beta_1 q^{10} + (\beta_{5} + \beta_{3} + \beta_{2} - \beta_1) q^{11} + ( - \beta_{5} + \beta_{3} + \beta_1) q^{13} + \beta_1 q^{14} + (\beta_{7} + \beta_{6} + \beta_{4} + \cdots + 3) q^{16}+ \cdots - \beta_1 q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - q^{2} + 11 q^{4} + 8 q^{5} - 8 q^{7} - 3 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - q^{2} + 11 q^{4} + 8 q^{5} - 8 q^{7} - 3 q^{8} - q^{10} + 4 q^{11} + 5 q^{13} + q^{14} + 21 q^{16} + 4 q^{17} + 3 q^{19} + 11 q^{20} + 23 q^{22} - 8 q^{23} + 8 q^{25} - 3 q^{26} - 11 q^{28} + 19 q^{29} - 12 q^{32} + 9 q^{34} - 8 q^{35} + 21 q^{37} - 28 q^{38} - 3 q^{40} + 20 q^{41} + 13 q^{43} + 15 q^{44} + 17 q^{46} - 11 q^{47} + 8 q^{49} - q^{50} + 13 q^{52} - 8 q^{53} + 4 q^{55} + 3 q^{56} + 37 q^{58} + 7 q^{59} + 24 q^{61} - 15 q^{62} + 55 q^{64} + 5 q^{65} + 16 q^{67} + 5 q^{68} + q^{70} - 5 q^{71} + 10 q^{73} + 21 q^{74} + 25 q^{76} - 4 q^{77} + 27 q^{79} + 21 q^{80} + 36 q^{82} + 5 q^{83} + 4 q^{85} - 27 q^{86} + 67 q^{88} + 27 q^{89} - 5 q^{91} - 93 q^{92} - 17 q^{94} + 3 q^{95} + 27 q^{97} - q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - x^{7} - 13x^{6} + 11x^{5} + 49x^{4} - 32x^{3} - 55x^{2} + 22x + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{6} - 12\nu^{4} - \nu^{3} + 36\nu^{2} + 3\nu - 16 ) / 3 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{7} - 12\nu^{5} - \nu^{4} + 39\nu^{3} + 6\nu^{2} - 31\nu - 6 ) / 3 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{7} + \nu^{6} - 15\nu^{5} - 13\nu^{4} + 65\nu^{3} + 42\nu^{2} - 67\nu - 22 ) / 6 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{7} - \nu^{6} - 12\nu^{5} + 11\nu^{4} + 37\nu^{3} - 30\nu^{2} - 19\nu + 10 ) / 3 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -2\nu^{7} + \nu^{6} + 24\nu^{5} - 7\nu^{4} - 76\nu^{3} + 3\nu^{2} + 50\nu + 8 ) / 3 \) 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
\(\nu^{3}\)\(=\) \( -\beta_{6} + \beta_{4} - \beta_{3} + 5\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{7} + \beta_{6} + \beta_{4} + 7\beta_{2} + 17 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -9\beta_{6} - 2\beta_{5} + 10\beta_{4} - 8\beta_{3} + 32\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 12\beta_{7} + 11\beta_{6} + 13\beta_{4} + 2\beta_{3} + 48\beta_{2} + 2\beta _1 + 112 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( \beta_{7} - 68\beta_{6} - 24\beta_{5} + 85\beta_{4} - 57\beta_{3} + \beta_{2} + 220\beta _1 + 5 \) Copy content Toggle raw display

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

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
2.77799
2.18071
1.25545
0.882742
−0.439045
−1.04408
−1.95649
−2.65729
−2.77799 0 5.71724 1.00000 0 −1.00000 −10.3265 0 −2.77799
1.2 −2.18071 0 2.75549 1.00000 0 −1.00000 −1.64751 0 −2.18071
1.3 −1.25545 0 −0.423841 1.00000 0 −1.00000 3.04302 0 −1.25545
1.4 −0.882742 0 −1.22077 1.00000 0 −1.00000 2.84311 0 −0.882742
1.5 0.439045 0 −1.80724 1.00000 0 −1.00000 −1.67155 0 0.439045
1.6 1.04408 0 −0.909902 1.00000 0 −1.00000 −3.03816 0 1.04408
1.7 1.95649 0 1.82784 1.00000 0 −1.00000 −0.336819 0 1.95649
1.8 2.65729 0 5.06117 1.00000 0 −1.00000 8.13440 0 2.65729
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.8
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)
\(5\) \(-1\)
\(7\) \(1\)

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 2835.2.a.x 8
3.b odd 2 1 2835.2.a.y 8
9.c even 3 2 315.2.i.f ✓ 16
9.d odd 6 2 945.2.i.f 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
315.2.i.f ✓ 16 9.c even 3 2
945.2.i.f 16 9.d odd 6 2
2835.2.a.x 8 1.a even 1 1 trivial
2835.2.a.y 8 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2835))\):

\( T_{2}^{8} + T_{2}^{7} - 13T_{2}^{6} - 11T_{2}^{5} + 49T_{2}^{4} + 32T_{2}^{3} - 55T_{2}^{2} - 22T_{2} + 16 \) Copy content Toggle raw display
\( T_{11}^{8} - 4T_{11}^{7} - 64T_{11}^{6} + 134T_{11}^{5} + 1600T_{11}^{4} + 97T_{11}^{3} - 14581T_{11}^{2} - 28295T_{11} - 15536 \) Copy content Toggle raw display
\( T_{13}^{8} - 5T_{13}^{7} - 55T_{13}^{6} + 268T_{13}^{5} + 784T_{13}^{4} - 3946T_{13}^{3} - 1876T_{13}^{2} + 12947T_{13} - 1766 \) Copy content Toggle raw display
\( T_{17}^{8} - 4T_{17}^{7} - 88T_{17}^{6} + 416T_{17}^{5} + 1606T_{17}^{4} - 10391T_{17}^{3} + 11243T_{17}^{2} + 8803T_{17} - 13682 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} + T^{7} + \cdots + 16 \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( (T - 1)^{8} \) Copy content Toggle raw display
$7$ \( (T + 1)^{8} \) Copy content Toggle raw display
$11$ \( T^{8} - 4 T^{7} + \cdots - 15536 \) Copy content Toggle raw display
$13$ \( T^{8} - 5 T^{7} + \cdots - 1766 \) Copy content Toggle raw display
$17$ \( T^{8} - 4 T^{7} + \cdots - 13682 \) Copy content Toggle raw display
$19$ \( T^{8} - 3 T^{7} + \cdots + 96256 \) Copy content Toggle raw display
$23$ \( T^{8} + 8 T^{7} + \cdots - 18944 \) Copy content Toggle raw display
$29$ \( T^{8} - 19 T^{7} + \cdots - 425564 \) Copy content Toggle raw display
$31$ \( T^{8} - 108 T^{6} + \cdots - 82944 \) Copy content Toggle raw display
$37$ \( T^{8} - 21 T^{7} + \cdots - 82944 \) Copy content Toggle raw display
$41$ \( T^{8} - 20 T^{7} + \cdots - 28928 \) Copy content Toggle raw display
$43$ \( T^{8} - 13 T^{7} + \cdots - 587648 \) Copy content Toggle raw display
$47$ \( T^{8} + 11 T^{7} + \cdots + 64 \) Copy content Toggle raw display
$53$ \( T^{8} + 8 T^{7} + \cdots - 165056 \) Copy content Toggle raw display
$59$ \( T^{8} - 7 T^{7} + \cdots - 251936 \) Copy content Toggle raw display
$61$ \( T^{8} - 24 T^{7} + \cdots + 165888 \) Copy content Toggle raw display
$67$ \( T^{8} - 16 T^{7} + \cdots - 2336 \) Copy content Toggle raw display
$71$ \( T^{8} + 5 T^{7} + \cdots - 1249607 \) Copy content Toggle raw display
$73$ \( T^{8} - 10 T^{7} + \cdots + 124561 \) Copy content Toggle raw display
$79$ \( T^{8} - 27 T^{7} + \cdots - 128 \) Copy content Toggle raw display
$83$ \( T^{8} - 5 T^{7} + \cdots + 2606137 \) Copy content Toggle raw display
$89$ \( T^{8} - 27 T^{7} + \cdots - 6305216 \) Copy content Toggle raw display
$97$ \( T^{8} - 27 T^{7} + \cdots + 2089342 \) Copy content Toggle raw display
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