Properties

Label 608.1.bg.a
Level $608$
Weight $1$
Character orbit 608.bg
Analytic conductor $0.303$
Analytic rank $0$
Dimension $6$
Projective image $D_{9}$
CM discriminant -8
Inner twists $4$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 608 = 2^{5} \cdot 19 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 608.bg (of order \(18\), degree \(6\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.303431527681\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\Q(\zeta_{18})\)
Defining polynomial: \( x^{6} - x^{3} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 152)
Projective image: \(D_{9}\)
Projective field: Galois closure of 9.1.69564674215936.1

$q$-expansion

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

\(f(q)\) \(=\) \( q + ( - \zeta_{18}^{6} + \zeta_{18}) q^{3} + ( - \zeta_{18}^{7} - \zeta_{18}^{3} + \zeta_{18}^{2}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \zeta_{18}^{6} + \zeta_{18}) q^{3} + ( - \zeta_{18}^{7} - \zeta_{18}^{3} + \zeta_{18}^{2}) q^{9} + (\zeta_{18}^{7} + \zeta_{18}^{5}) q^{11} - \zeta_{18}^{2} q^{17} - \zeta_{18}^{4} q^{19} - \zeta_{18} q^{25} + ( - \zeta_{18}^{8} + \zeta_{18}^{4} + \zeta_{18}^{3} - 1) q^{27} + (\zeta_{18}^{8} + \zeta_{18}^{6} + \zeta_{18}^{4} + \zeta_{18}^{2}) q^{33} + (\zeta_{18}^{4} - \zeta_{18}^{3}) q^{41} - \zeta_{18}^{5} q^{43} + \zeta_{18}^{6} q^{49} + (\zeta_{18}^{8} - \zeta_{18}^{3}) q^{51} + ( - \zeta_{18}^{5} - \zeta_{18}) q^{57} + (\zeta_{18}^{3} + \zeta_{18}) q^{59} + (\zeta_{18}^{3} - \zeta_{18}^{2}) q^{67} + ( - \zeta_{18}^{7} + 1) q^{73} + (\zeta_{18}^{7} - \zeta_{18}^{2}) q^{75} + (\zeta_{18}^{6} + \zeta_{18}^{5} + \zeta_{18}^{4} + \zeta_{18} - 1) q^{81} + ( - \zeta_{18}^{8} + \zeta_{18}^{7}) q^{83} + \zeta_{18} q^{89} + ( - \zeta_{18}^{7} + \zeta_{18}^{6}) q^{97} + ( - \zeta_{18}^{8} + \zeta_{18}^{7} + \zeta_{18}^{5} + \zeta_{18}^{3} + \zeta_{18} - 1) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 3 q^{3} - 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 3 q^{3} - 3 q^{9} - 3 q^{27} - 3 q^{33} - 3 q^{41} - 3 q^{49} - 3 q^{51} + 3 q^{59} + 3 q^{67} + 6 q^{73} + 3 q^{81} - 3 q^{97} - 3 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/608\mathbb{Z}\right)^\times\).

\(n\) \(97\) \(191\) \(229\)
\(\chi(n)\) \(\zeta_{18}^{4}\) \(-1\) \(-1\)

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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
47.1
−0.766044 0.642788i
0.939693 + 0.342020i
−0.173648 + 0.984808i
−0.766044 + 0.642788i
−0.173648 0.984808i
0.939693 0.342020i
0 −0.266044 + 0.223238i 0 0 0 0 0 −0.152704 + 0.866025i 0
111.1 0 1.43969 0.524005i 0 0 0 0 0 1.03209 0.866025i 0
175.1 0 0.326352 + 1.85083i 0 0 0 0 0 −2.37939 + 0.866025i 0
207.1 0 −0.266044 0.223238i 0 0 0 0 0 −0.152704 0.866025i 0
271.1 0 0.326352 1.85083i 0 0 0 0 0 −2.37939 0.866025i 0
367.1 0 1.43969 + 0.524005i 0 0 0 0 0 1.03209 + 0.866025i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 367.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.d odd 2 1 CM by \(\Q(\sqrt{-2}) \)
19.e even 9 1 inner
152.u odd 18 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 608.1.bg.a 6
4.b odd 2 1 152.1.u.a 6
8.b even 2 1 152.1.u.a 6
8.d odd 2 1 CM 608.1.bg.a 6
12.b even 2 1 1368.1.eh.a 6
19.e even 9 1 inner 608.1.bg.a 6
20.d odd 2 1 3800.1.cv.c 6
20.e even 4 2 3800.1.cq.b 12
24.h odd 2 1 1368.1.eh.a 6
40.f even 2 1 3800.1.cv.c 6
40.i odd 4 2 3800.1.cq.b 12
76.d even 2 1 2888.1.u.e 6
76.f even 6 1 2888.1.u.a 6
76.f even 6 1 2888.1.u.f 6
76.g odd 6 1 2888.1.u.b 6
76.g odd 6 1 2888.1.u.g 6
76.k even 18 1 2888.1.f.c 3
76.k even 18 2 2888.1.k.c 6
76.k even 18 1 2888.1.u.a 6
76.k even 18 1 2888.1.u.e 6
76.k even 18 1 2888.1.u.f 6
76.l odd 18 1 152.1.u.a 6
76.l odd 18 1 2888.1.f.d 3
76.l odd 18 2 2888.1.k.b 6
76.l odd 18 1 2888.1.u.b 6
76.l odd 18 1 2888.1.u.g 6
152.g odd 2 1 2888.1.u.e 6
152.l odd 6 1 2888.1.u.a 6
152.l odd 6 1 2888.1.u.f 6
152.p even 6 1 2888.1.u.b 6
152.p even 6 1 2888.1.u.g 6
152.s odd 18 1 2888.1.f.c 3
152.s odd 18 2 2888.1.k.c 6
152.s odd 18 1 2888.1.u.a 6
152.s odd 18 1 2888.1.u.e 6
152.s odd 18 1 2888.1.u.f 6
152.t even 18 1 152.1.u.a 6
152.t even 18 1 2888.1.f.d 3
152.t even 18 2 2888.1.k.b 6
152.t even 18 1 2888.1.u.b 6
152.t even 18 1 2888.1.u.g 6
152.u odd 18 1 inner 608.1.bg.a 6
228.v even 18 1 1368.1.eh.a 6
380.ba odd 18 1 3800.1.cv.c 6
380.bj even 36 2 3800.1.cq.b 12
456.bh odd 18 1 1368.1.eh.a 6
760.cj even 18 1 3800.1.cv.c 6
760.cq odd 36 2 3800.1.cq.b 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
152.1.u.a 6 4.b odd 2 1
152.1.u.a 6 8.b even 2 1
152.1.u.a 6 76.l odd 18 1
152.1.u.a 6 152.t even 18 1
608.1.bg.a 6 1.a even 1 1 trivial
608.1.bg.a 6 8.d odd 2 1 CM
608.1.bg.a 6 19.e even 9 1 inner
608.1.bg.a 6 152.u odd 18 1 inner
1368.1.eh.a 6 12.b even 2 1
1368.1.eh.a 6 24.h odd 2 1
1368.1.eh.a 6 228.v even 18 1
1368.1.eh.a 6 456.bh odd 18 1
2888.1.f.c 3 76.k even 18 1
2888.1.f.c 3 152.s odd 18 1
2888.1.f.d 3 76.l odd 18 1
2888.1.f.d 3 152.t even 18 1
2888.1.k.b 6 76.l odd 18 2
2888.1.k.b 6 152.t even 18 2
2888.1.k.c 6 76.k even 18 2
2888.1.k.c 6 152.s odd 18 2
2888.1.u.a 6 76.f even 6 1
2888.1.u.a 6 76.k even 18 1
2888.1.u.a 6 152.l odd 6 1
2888.1.u.a 6 152.s odd 18 1
2888.1.u.b 6 76.g odd 6 1
2888.1.u.b 6 76.l odd 18 1
2888.1.u.b 6 152.p even 6 1
2888.1.u.b 6 152.t even 18 1
2888.1.u.e 6 76.d even 2 1
2888.1.u.e 6 76.k even 18 1
2888.1.u.e 6 152.g odd 2 1
2888.1.u.e 6 152.s odd 18 1
2888.1.u.f 6 76.f even 6 1
2888.1.u.f 6 76.k even 18 1
2888.1.u.f 6 152.l odd 6 1
2888.1.u.f 6 152.s odd 18 1
2888.1.u.g 6 76.g odd 6 1
2888.1.u.g 6 76.l odd 18 1
2888.1.u.g 6 152.p even 6 1
2888.1.u.g 6 152.t even 18 1
3800.1.cq.b 12 20.e even 4 2
3800.1.cq.b 12 40.i odd 4 2
3800.1.cq.b 12 380.bj even 36 2
3800.1.cq.b 12 760.cq odd 36 2
3800.1.cv.c 6 20.d odd 2 1
3800.1.cv.c 6 40.f even 2 1
3800.1.cv.c 6 380.ba odd 18 1
3800.1.cv.c 6 760.cj even 18 1

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(608, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( T^{6} - 3 T^{5} + 6 T^{4} - 8 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$5$ \( T^{6} \) Copy content Toggle raw display
$7$ \( T^{6} \) Copy content Toggle raw display
$11$ \( T^{6} + 3 T^{4} - 2 T^{3} + 9 T^{2} + \cdots + 1 \) Copy content Toggle raw display
$13$ \( T^{6} \) Copy content Toggle raw display
$17$ \( T^{6} - T^{3} + 1 \) Copy content Toggle raw display
$19$ \( T^{6} - T^{3} + 1 \) Copy content Toggle raw display
$23$ \( T^{6} \) Copy content Toggle raw display
$29$ \( T^{6} \) Copy content Toggle raw display
$31$ \( T^{6} \) Copy content Toggle raw display
$37$ \( T^{6} \) Copy content Toggle raw display
$41$ \( T^{6} + 3 T^{5} + 6 T^{4} + 8 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$43$ \( T^{6} + T^{3} + 1 \) Copy content Toggle raw display
$47$ \( T^{6} \) Copy content Toggle raw display
$53$ \( T^{6} \) Copy content Toggle raw display
$59$ \( T^{6} - 3 T^{5} + 6 T^{4} - 8 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$61$ \( T^{6} \) Copy content Toggle raw display
$67$ \( T^{6} - 3 T^{5} + 6 T^{4} - 8 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$71$ \( T^{6} \) Copy content Toggle raw display
$73$ \( T^{6} - 6 T^{5} + 15 T^{4} - 19 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$79$ \( T^{6} \) Copy content Toggle raw display
$83$ \( T^{6} + 3 T^{4} - 2 T^{3} + 9 T^{2} + \cdots + 1 \) Copy content Toggle raw display
$89$ \( T^{6} - T^{3} + 1 \) Copy content Toggle raw display
$97$ \( T^{6} + 3 T^{5} + 6 T^{4} + 8 T^{3} + \cdots + 1 \) Copy content Toggle raw display
show more
show less