Properties

Label 3267.1.w.a
Level $3267$
Weight $1$
Character orbit 3267.w
Analytic conductor $1.630$
Analytic rank $0$
Dimension $8$
Projective image $D_{3}$
CM discriminant -11
Inner twists $16$

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Newspace parameters

Level: \( N \) \(=\) \( 3267 = 3^{3} \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3267.w (of order \(30\), degree \(8\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.63044539627\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\Q(\zeta_{15})\)
Defining polynomial: \(x^{8} - x^{7} + x^{5} - x^{4} + x^{3} - x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 99)
Projective image: \(D_{3}\)
Projective field: Galois closure of 3.1.891.1
Artin image: $C_{30}\times S_3$
Artin field: Galois closure of \(\mathbb{Q}[x]/(x^{60} - \cdots)\)

$q$-expansion

The \(q\)-expansion and trace form are shown below.

\(f(q)\) \(=\) \( q -\zeta_{30}^{13} q^{4} -\zeta_{30} q^{5} +O(q^{10})\) \( q -\zeta_{30}^{13} q^{4} -\zeta_{30} q^{5} -\zeta_{30}^{11} q^{16} + \zeta_{30}^{14} q^{20} -2 \zeta_{30}^{10} q^{23} -\zeta_{30}^{4} q^{31} + \zeta_{30}^{3} q^{37} + \zeta_{30}^{2} q^{47} -\zeta_{30} q^{49} -\zeta_{30}^{9} q^{53} -\zeta_{30}^{13} q^{59} -\zeta_{30}^{9} q^{64} -\zeta_{30}^{10} q^{67} + \zeta_{30}^{6} q^{71} + \zeta_{30}^{12} q^{80} -2 q^{89} -2 \zeta_{30}^{8} q^{92} -\zeta_{30}^{14} q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + q^{4} + q^{5} + O(q^{10}) \) \( 8q + q^{4} + q^{5} + q^{16} + q^{20} + 8q^{23} - q^{31} + 2q^{37} + q^{47} + q^{49} - 2q^{53} + q^{59} - 2q^{64} + 4q^{67} - 2q^{71} - 2q^{80} - 16q^{89} - 2q^{92} - q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3267\mathbb{Z}\right)^\times\).

\(n\) \(244\) \(3026\)
\(\chi(n)\) \(\zeta_{30}^{3}\) \(\zeta_{30}^{10}\)

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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
118.1
0.913545 0.406737i
−0.104528 0.994522i
0.669131 + 0.743145i
−0.104528 + 0.994522i
0.913545 + 0.406737i
−0.978148 + 0.207912i
−0.978148 0.207912i
0.669131 0.743145i
0 0 0.669131 + 0.743145i 0.913545 0.406737i 0 0 0 0 0
766.1 0 0 −0.978148 0.207912i −0.104528 0.994522i 0 0 0 0 0
820.1 0 0 −0.104528 0.994522i 0.669131 + 0.743145i 0 0 0 0 0
1207.1 0 0 −0.978148 + 0.207912i −0.104528 + 0.994522i 0 0 0 0 0
1855.1 0 0 0.669131 0.743145i 0.913545 + 0.406737i 0 0 0 0 0
1909.1 0 0 0.913545 + 0.406737i −0.978148 + 0.207912i 0 0 0 0 0
1927.1 0 0 0.913545 0.406737i −0.978148 0.207912i 0 0 0 0 0
3016.1 0 0 −0.104528 + 0.994522i 0.669131 0.743145i 0 0 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 3016.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.b odd 2 1 CM by \(\Q(\sqrt{-11}) \)
9.c even 3 1 inner
11.c even 5 3 inner
11.d odd 10 3 inner
99.h odd 6 1 inner
99.m even 15 3 inner
99.o odd 30 3 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3267.1.w.a 8
3.b odd 2 1 1089.1.s.a 8
9.c even 3 1 inner 3267.1.w.a 8
9.d odd 6 1 1089.1.s.a 8
11.b odd 2 1 CM 3267.1.w.a 8
11.c even 5 1 297.1.h.a 2
11.c even 5 3 inner 3267.1.w.a 8
11.d odd 10 1 297.1.h.a 2
11.d odd 10 3 inner 3267.1.w.a 8
33.d even 2 1 1089.1.s.a 8
33.f even 10 1 99.1.h.a 2
33.f even 10 3 1089.1.s.a 8
33.h odd 10 1 99.1.h.a 2
33.h odd 10 3 1089.1.s.a 8
99.g even 6 1 1089.1.s.a 8
99.h odd 6 1 inner 3267.1.w.a 8
99.m even 15 1 297.1.h.a 2
99.m even 15 1 891.1.c.b 1
99.m even 15 3 inner 3267.1.w.a 8
99.n odd 30 1 99.1.h.a 2
99.n odd 30 1 891.1.c.a 1
99.n odd 30 3 1089.1.s.a 8
99.o odd 30 1 297.1.h.a 2
99.o odd 30 1 891.1.c.b 1
99.o odd 30 3 inner 3267.1.w.a 8
99.p even 30 1 99.1.h.a 2
99.p even 30 1 891.1.c.a 1
99.p even 30 3 1089.1.s.a 8
132.n odd 10 1 1584.1.bf.b 2
132.o even 10 1 1584.1.bf.b 2
165.o odd 10 1 2475.1.y.a 2
165.r even 10 1 2475.1.y.a 2
165.u odd 20 2 2475.1.t.a 4
165.v even 20 2 2475.1.t.a 4
396.ba even 30 1 1584.1.bf.b 2
396.bb odd 30 1 1584.1.bf.b 2
495.bo even 30 1 2475.1.y.a 2
495.bp odd 30 1 2475.1.y.a 2
495.bu odd 60 2 2475.1.t.a 4
495.bv even 60 2 2475.1.t.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
99.1.h.a 2 33.f even 10 1
99.1.h.a 2 33.h odd 10 1
99.1.h.a 2 99.n odd 30 1
99.1.h.a 2 99.p even 30 1
297.1.h.a 2 11.c even 5 1
297.1.h.a 2 11.d odd 10 1
297.1.h.a 2 99.m even 15 1
297.1.h.a 2 99.o odd 30 1
891.1.c.a 1 99.n odd 30 1
891.1.c.a 1 99.p even 30 1
891.1.c.b 1 99.m even 15 1
891.1.c.b 1 99.o odd 30 1
1089.1.s.a 8 3.b odd 2 1
1089.1.s.a 8 9.d odd 6 1
1089.1.s.a 8 33.d even 2 1
1089.1.s.a 8 33.f even 10 3
1089.1.s.a 8 33.h odd 10 3
1089.1.s.a 8 99.g even 6 1
1089.1.s.a 8 99.n odd 30 3
1089.1.s.a 8 99.p even 30 3
1584.1.bf.b 2 132.n odd 10 1
1584.1.bf.b 2 132.o even 10 1
1584.1.bf.b 2 396.ba even 30 1
1584.1.bf.b 2 396.bb odd 30 1
2475.1.t.a 4 165.u odd 20 2
2475.1.t.a 4 165.v even 20 2
2475.1.t.a 4 495.bu odd 60 2
2475.1.t.a 4 495.bv even 60 2
2475.1.y.a 2 165.o odd 10 1
2475.1.y.a 2 165.r even 10 1
2475.1.y.a 2 495.bo even 30 1
2475.1.y.a 2 495.bp odd 30 1
3267.1.w.a 8 1.a even 1 1 trivial
3267.1.w.a 8 9.c even 3 1 inner
3267.1.w.a 8 11.b odd 2 1 CM
3267.1.w.a 8 11.c even 5 3 inner
3267.1.w.a 8 11.d odd 10 3 inner
3267.1.w.a 8 99.h odd 6 1 inner
3267.1.w.a 8 99.m even 15 3 inner
3267.1.w.a 8 99.o odd 30 3 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} \) acting on \(S_{1}^{\mathrm{new}}(3267, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \)
$3$ \( T^{8} \)
$5$ \( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} \)
$7$ \( T^{8} \)
$11$ \( T^{8} \)
$13$ \( T^{8} \)
$17$ \( T^{8} \)
$19$ \( T^{8} \)
$23$ \( ( 4 - 2 T + T^{2} )^{4} \)
$29$ \( T^{8} \)
$31$ \( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} \)
$37$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \)
$41$ \( T^{8} \)
$43$ \( T^{8} \)
$47$ \( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} \)
$53$ \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
$59$ \( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} \)
$61$ \( T^{8} \)
$67$ \( ( 1 - T + T^{2} )^{4} \)
$71$ \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
$73$ \( T^{8} \)
$79$ \( T^{8} \)
$83$ \( T^{8} \)
$89$ \( ( 2 + T )^{8} \)
$97$ \( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} \)
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