Properties

Label 3332.2.b.b
Level $3332$
Weight $2$
Character orbit 3332.b
Analytic conductor $26.606$
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] = [3332,2,Mod(2549,3332)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3332, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3332.2549");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3332 = 2^{2} \cdot 7^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3332.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: \(26.6061539535\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.980441344.2
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 8x^{6} + 18x^{4} + 9x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 476)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{7} + \beta_{4}) q^{3} + \beta_{3} q^{5} + ( - \beta_{5} - \beta_{2} - \beta_1 - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{7} + \beta_{4}) q^{3} + \beta_{3} q^{5} + ( - \beta_{5} - \beta_{2} - \beta_1 - 1) q^{9} + ( - \beta_{7} - \beta_{6} + 2 \beta_{4}) q^{11} + ( - \beta_{5} - \beta_1 - 2) q^{13} + ( - 2 \beta_{5} - 2 \beta_{2} + 1) q^{15} + (\beta_{7} - \beta_{4} + \beta_{3} + \beta_{2} - \beta_1) q^{17} + (\beta_{5} - \beta_{2} - 2) q^{19} + (\beta_{7} - 2 \beta_{6} + \beta_{3}) q^{23} + (\beta_{5} - \beta_{2} - \beta_1 - 2) q^{25} + ( - \beta_{7} + 2 \beta_{6} - \beta_{4} - 3 \beta_{3}) q^{27} + (\beta_{7} - \beta_{6} + 2 \beta_{4}) q^{29} + (2 \beta_{4} - \beta_{3}) q^{31} + (\beta_{5} - 3 \beta_{2}) q^{33} + ( - \beta_{7} - 2 \beta_{4} - \beta_{3}) q^{37} + ( - 5 \beta_{7} + \beta_{6} - 2 \beta_{4} - 2 \beta_{3}) q^{39} + (\beta_{7} - 2 \beta_{6} - \beta_{4}) q^{41} + ( - \beta_{5} + 3 \beta_{2} + \beta_1 - 1) q^{43} + ( - 3 \beta_{7} + 2 \beta_{6} - 3 \beta_{4} - 3 \beta_{3}) q^{45} + (2 \beta_{5} + 3 \beta_{2} + \beta_1 - 4) q^{47} + ( - \beta_{7} - 3 \beta_{5} + 2 \beta_{4} + \beta_{3} - \beta_{2} - \beta_1 - 1) q^{51} + (\beta_{5} + 3 \beta_{2} - \beta_1 + 2) q^{53} + ( - 3 \beta_{5} + 2 \beta_{2} - \beta_1 + 2) q^{55} + (\beta_{6} - 6 \beta_{4} + \beta_{3}) q^{57} + ( - 2 \beta_{5} + 4 \beta_{2} + 4) q^{59} + (\beta_{7} - 2 \beta_{6} + \beta_{4} + 2 \beta_{3}) q^{61} + ( - \beta_{7} + 2 \beta_{6} - 4 \beta_{4} - 3 \beta_{3}) q^{65} + ( - \beta_{5} + \beta_{2} - 3 \beta_1) q^{67} + ( - 3 \beta_{5} - 4 \beta_{2} - 3 \beta_1 - 4) q^{69} + (3 \beta_{7} - 4 \beta_{4} - \beta_{3}) q^{71} + ( - 5 \beta_{7} - 2 \beta_{6} + 3 \beta_{4} - 2 \beta_{3}) q^{73} + ( - \beta_{7} + 2 \beta_{6} - 7 \beta_{4} + \beta_{3}) q^{75} + ( - \beta_{7} + 3 \beta_{6} + 4 \beta_{4} - 4 \beta_{3}) q^{79} + (4 \beta_{5} + 6 \beta_{2}) q^{81} + ( - \beta_{5} + 2 \beta_{2} - 3 \beta_1) q^{83} + (2 \beta_{7} + \beta_{6} + \beta_{5} + 2 \beta_{4} - \beta_{3} - 3 \beta_{2} - \beta_1 - 6) q^{85} + ( - \beta_{5} - 3 \beta_{2} - 2 \beta_1 - 6) q^{87} + ( - 4 \beta_{5} - \beta_{2} + \beta_1 + 2) q^{89} + (2 \beta_{5} - 3) q^{93} + ( - \beta_{7} - \beta_{6} + 8 \beta_{4} - 4 \beta_{3}) q^{95} + ( - 2 \beta_{6} - 4 \beta_{4} - \beta_{3}) q^{97} + ( - \beta_{7} - 4 \beta_{4} - \beta_{3}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 12 q^{9} - 20 q^{13} - 12 q^{19} - 12 q^{25} + 4 q^{33} - 12 q^{43} - 24 q^{47} - 20 q^{51} + 20 q^{53} + 4 q^{55} + 24 q^{59} - 4 q^{67} - 44 q^{69} + 16 q^{81} - 4 q^{83} - 44 q^{85} - 52 q^{87} - 16 q^{93}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 8x^{6} + 18x^{4} + 9x^{2} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( -\nu^{4} - 3\nu^{2} + 1 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{4} + 5\nu^{2} + 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{5} + 6\nu^{3} + 8\nu \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \nu^{7} + 8\nu^{5} + 17\nu^{3} + 5\nu \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( 2\nu^{6} + 15\nu^{4} + 29\nu^{2} + 7 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( -2\nu^{7} - 15\nu^{5} - 30\nu^{3} - 8\nu \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( 2\nu^{7} + 15\nu^{5} + 30\nu^{3} + 10\nu \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{7} + \beta_{6} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{2} + \beta _1 - 4 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -3\beta_{7} - 4\beta_{6} - 2\beta_{4} + \beta_{3} ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -3\beta_{2} - 5\beta _1 + 14 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 5\beta_{7} + 8\beta_{6} + 6\beta_{4} - 2\beta_{3} \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( \beta_{5} + 8\beta_{2} + 23\beta _1 - 54 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -34\beta_{7} - 65\beta_{6} - 60\beta_{4} + 15\beta_{3} ) / 2 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3332\mathbb{Z}\right)^\times\).

\(n\) \(785\) \(885\) \(1667\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2549.1
0.396339i
1.76401i
0.693822i
2.06150i
2.06150i
0.693822i
1.76401i
0.396339i
0 3.23925i 0 2.80694i 0 0 0 −7.49277 0
2549.2 0 1.87576i 0 1.74199i 0 0 0 −0.518489 0
2549.3 0 1.82479i 0 3.70737i 0 0 0 −0.329851 0
2549.4 0 0.811721i 0 1.15845i 0 0 0 2.34111 0
2549.5 0 0.811721i 0 1.15845i 0 0 0 2.34111 0
2549.6 0 1.82479i 0 3.70737i 0 0 0 −0.329851 0
2549.7 0 1.87576i 0 1.74199i 0 0 0 −0.518489 0
2549.8 0 3.23925i 0 2.80694i 0 0 0 −7.49277 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2549.8
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3332.2.b.b 8
7.b odd 2 1 476.2.b.a 8
17.b even 2 1 inner 3332.2.b.b 8
21.c even 2 1 4284.2.d.e 8
28.d even 2 1 1904.2.c.f 8
119.d odd 2 1 476.2.b.a 8
119.f odd 4 1 8092.2.a.q 4
119.f odd 4 1 8092.2.a.r 4
357.c even 2 1 4284.2.d.e 8
476.e even 2 1 1904.2.c.f 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
476.2.b.a 8 7.b odd 2 1
476.2.b.a 8 119.d odd 2 1
1904.2.c.f 8 28.d even 2 1
1904.2.c.f 8 476.e even 2 1
3332.2.b.b 8 1.a even 1 1 trivial
3332.2.b.b 8 17.b even 2 1 inner
4284.2.d.e 8 21.c even 2 1
4284.2.d.e 8 357.c even 2 1
8092.2.a.q 4 119.f odd 4 1
8092.2.a.r 4 119.f odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(3332, [\chi])\):

\( T_{3}^{8} + 18T_{3}^{6} + 95T_{3}^{4} + 178T_{3}^{2} + 81 \) Copy content Toggle raw display
\( T_{13}^{4} + 10T_{13}^{3} + 20T_{13}^{2} - 24T_{13} - 16 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} + 18 T^{6} + 95 T^{4} + 178 T^{2} + \cdots + 81 \) Copy content Toggle raw display
$5$ \( T^{8} + 26 T^{6} + 207 T^{4} + \cdots + 441 \) Copy content Toggle raw display
$7$ \( T^{8} \) Copy content Toggle raw display
$11$ \( T^{8} + 48 T^{6} + 608 T^{4} + \cdots + 2304 \) Copy content Toggle raw display
$13$ \( (T^{4} + 10 T^{3} + 20 T^{2} - 24 T - 16)^{2} \) Copy content Toggle raw display
$17$ \( T^{8} + 4 T^{6} - 64 T^{5} + \cdots + 83521 \) Copy content Toggle raw display
$19$ \( (T^{4} + 6 T^{3} - 8 T^{2} - 32 T - 16)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} + 124 T^{6} + 3984 T^{4} + \cdots + 256 \) Copy content Toggle raw display
$29$ \( T^{8} + 64 T^{6} + 1184 T^{4} + \cdots + 256 \) Copy content Toggle raw display
$31$ \( T^{8} + 34 T^{6} + 47 T^{4} + 18 T^{2} + \cdots + 1 \) Copy content Toggle raw display
$37$ \( T^{8} + 60 T^{6} + 528 T^{4} + \cdots + 256 \) Copy content Toggle raw display
$41$ \( T^{8} + 138 T^{6} + 4559 T^{4} + \cdots + 196249 \) Copy content Toggle raw display
$43$ \( (T^{4} + 6 T^{3} - 101 T^{2} - 314 T + 2429)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} + 12 T^{3} - 56 T^{2} - 1160 T - 3664)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} - 10 T^{3} - 41 T^{2} + 350 T + 557)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} - 12 T^{3} - 124 T^{2} + 1312 T - 1104)^{2} \) Copy content Toggle raw display
$61$ \( T^{8} + 178 T^{6} + 5831 T^{4} + \cdots + 47961 \) Copy content Toggle raw display
$67$ \( (T^{4} + 2 T^{3} - 101 T^{2} - 586 T - 787)^{2} \) Copy content Toggle raw display
$71$ \( T^{8} + 244 T^{6} + 7904 T^{4} + \cdots + 256 \) Copy content Toggle raw display
$73$ \( T^{8} + 514 T^{6} + \cdots + 154828249 \) Copy content Toggle raw display
$79$ \( T^{8} + 480 T^{6} + \cdots + 16128256 \) Copy content Toggle raw display
$83$ \( (T^{4} + 2 T^{3} - 112 T^{2} - 432 T - 368)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} - 240 T^{2} - 1576 T - 1776)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} + 242 T^{6} + 10791 T^{4} + \cdots + 978121 \) Copy content Toggle raw display
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