Properties

Label 297.1.h.a
Level $297$
Weight $1$
Character orbit 297.h
Analytic conductor $0.148$
Analytic rank $0$
Dimension $2$
Projective image $D_{3}$
CM discriminant -11
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 297 = 3^{3} \cdot 11 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 297.h (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.148222308752\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
Defining polynomial: \(x^{2} - x + 1\)
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: $C_6\times S_3$
Artin field: Galois closure of 12.0.941480149401.1

$q$-expansion

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

\(f(q)\) \(=\) \( q + \zeta_{6}^{2} q^{4} + \zeta_{6}^{2} q^{5} +O(q^{10})\) \( q + \zeta_{6}^{2} q^{4} + \zeta_{6}^{2} q^{5} + \zeta_{6} q^{11} -\zeta_{6} q^{16} -\zeta_{6} q^{20} -2 \zeta_{6}^{2} q^{23} -\zeta_{6}^{2} q^{31} - q^{37} - q^{44} -\zeta_{6} q^{47} + \zeta_{6}^{2} q^{49} + q^{53} - q^{55} + \zeta_{6}^{2} q^{59} + q^{64} -\zeta_{6}^{2} q^{67} + q^{71} + q^{80} -2 q^{89} + 2 \zeta_{6} q^{92} + \zeta_{6} q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{4} - q^{5} + O(q^{10}) \) \( 2 q - q^{4} - q^{5} + q^{11} - q^{16} - q^{20} + 2 q^{23} + q^{31} - 2 q^{37} - 2 q^{44} - q^{47} - q^{49} + 2 q^{53} - 2 q^{55} - q^{59} + 2 q^{64} + q^{67} + 2 q^{71} + 2 q^{80} - 4 q^{89} + 2 q^{92} + q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/297\mathbb{Z}\right)^\times\).

\(n\) \(56\) \(244\)
\(\chi(n)\) \(\zeta_{6}^{2}\) \(-1\)

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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
10.1
0.500000 + 0.866025i
0.500000 0.866025i
0 0 −0.500000 + 0.866025i −0.500000 + 0.866025i 0 0 0 0 0
208.1 0 0 −0.500000 0.866025i −0.500000 0.866025i 0 0 0 0 0
\(n\): e.g. 2-40 or 990-1000
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
99.h odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 297.1.h.a 2
3.b odd 2 1 99.1.h.a 2
9.c even 3 1 inner 297.1.h.a 2
9.c even 3 1 891.1.c.b 1
9.d odd 6 1 99.1.h.a 2
9.d odd 6 1 891.1.c.a 1
11.b odd 2 1 CM 297.1.h.a 2
11.c even 5 4 3267.1.w.a 8
11.d odd 10 4 3267.1.w.a 8
12.b even 2 1 1584.1.bf.b 2
15.d odd 2 1 2475.1.y.a 2
15.e even 4 2 2475.1.t.a 4
33.d even 2 1 99.1.h.a 2
33.f even 10 4 1089.1.s.a 8
33.h odd 10 4 1089.1.s.a 8
36.h even 6 1 1584.1.bf.b 2
45.h odd 6 1 2475.1.y.a 2
45.l even 12 2 2475.1.t.a 4
99.g even 6 1 99.1.h.a 2
99.g even 6 1 891.1.c.a 1
99.h odd 6 1 inner 297.1.h.a 2
99.h odd 6 1 891.1.c.b 1
99.m even 15 4 3267.1.w.a 8
99.n odd 30 4 1089.1.s.a 8
99.o odd 30 4 3267.1.w.a 8
99.p even 30 4 1089.1.s.a 8
132.d odd 2 1 1584.1.bf.b 2
165.d even 2 1 2475.1.y.a 2
165.l odd 4 2 2475.1.t.a 4
396.o odd 6 1 1584.1.bf.b 2
495.r even 6 1 2475.1.y.a 2
495.bd odd 12 2 2475.1.t.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
99.1.h.a 2 3.b odd 2 1
99.1.h.a 2 9.d odd 6 1
99.1.h.a 2 33.d even 2 1
99.1.h.a 2 99.g even 6 1
297.1.h.a 2 1.a even 1 1 trivial
297.1.h.a 2 9.c even 3 1 inner
297.1.h.a 2 11.b odd 2 1 CM
297.1.h.a 2 99.h odd 6 1 inner
891.1.c.a 1 9.d odd 6 1
891.1.c.a 1 99.g even 6 1
891.1.c.b 1 9.c even 3 1
891.1.c.b 1 99.h odd 6 1
1089.1.s.a 8 33.f even 10 4
1089.1.s.a 8 33.h odd 10 4
1089.1.s.a 8 99.n odd 30 4
1089.1.s.a 8 99.p even 30 4
1584.1.bf.b 2 12.b even 2 1
1584.1.bf.b 2 36.h even 6 1
1584.1.bf.b 2 132.d odd 2 1
1584.1.bf.b 2 396.o odd 6 1
2475.1.t.a 4 15.e even 4 2
2475.1.t.a 4 45.l even 12 2
2475.1.t.a 4 165.l odd 4 2
2475.1.t.a 4 495.bd odd 12 2
2475.1.y.a 2 15.d odd 2 1
2475.1.y.a 2 45.h odd 6 1
2475.1.y.a 2 165.d even 2 1
2475.1.y.a 2 495.r even 6 1
3267.1.w.a 8 11.c even 5 4
3267.1.w.a 8 11.d odd 10 4
3267.1.w.a 8 99.m even 15 4
3267.1.w.a 8 99.o odd 30 4

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(297, [\chi])\).

Hecke characteristic polynomials

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