Properties

Label 608.1.o.a
Level $608$
Weight $1$
Character orbit 608.o
Analytic conductor $0.303$
Analytic rank $0$
Dimension $2$
Projective image $D_{3}$
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.o (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.303431527681\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 152)
Projective image: \(D_{3}\)
Projective field: Galois closure of 3.1.2888.1
Artin image: $C_6\times S_3$
Artin field: Galois closure of 12.0.2186423566336.3

$q$-expansion

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

\(f(q)\) \(=\) \( q + \zeta_{6}^{2} q^{3} +O(q^{10})\) \( q + \zeta_{6}^{2} q^{3} + q^{11} + 2 \zeta_{6}^{2} q^{17} + \zeta_{6} q^{19} -\zeta_{6} q^{25} - q^{27} + \zeta_{6}^{2} q^{33} -\zeta_{6}^{2} q^{41} -2 \zeta_{6}^{2} q^{43} + q^{49} -2 \zeta_{6} q^{51} - q^{57} + \zeta_{6}^{2} q^{59} -\zeta_{6} q^{67} -\zeta_{6}^{2} q^{73} + q^{75} -\zeta_{6}^{2} q^{81} + q^{83} -2 \zeta_{6} q^{89} -\zeta_{6}^{2} q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - q^{3} + O(q^{10}) \) \( 2q - q^{3} + 2q^{11} - 2q^{17} + q^{19} - q^{25} - 2q^{27} - q^{33} + q^{41} + 2q^{43} + 2q^{49} - 2q^{51} - 2q^{57} - q^{59} - q^{67} + q^{73} + 2q^{75} + q^{81} + 2q^{83} - 2q^{89} + q^{97} + O(q^{100}) \)

Character values

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

\(n\) \(97\) \(191\) \(229\)
\(\chi(n)\) \(-\zeta_{6}\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
239.1
0.500000 + 0.866025i
0.500000 0.866025i
0 −0.500000 + 0.866025i 0 0 0 0 0 0 0
463.1 0 −0.500000 0.866025i 0 0 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
8.d odd 2 1 CM by \(\Q(\sqrt{-2}) \)
19.c even 3 1 inner
152.k odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 608.1.o.a 2
4.b odd 2 1 152.1.k.a 2
8.b even 2 1 152.1.k.a 2
8.d odd 2 1 CM 608.1.o.a 2
12.b even 2 1 1368.1.bz.a 2
19.c even 3 1 inner 608.1.o.a 2
20.d odd 2 1 3800.1.bd.c 2
20.e even 4 2 3800.1.bn.b 4
24.h odd 2 1 1368.1.bz.a 2
40.f even 2 1 3800.1.bd.c 2
40.i odd 4 2 3800.1.bn.b 4
76.d even 2 1 2888.1.k.a 2
76.f even 6 1 2888.1.f.a 1
76.f even 6 1 2888.1.k.a 2
76.g odd 6 1 152.1.k.a 2
76.g odd 6 1 2888.1.f.b 1
76.k even 18 6 2888.1.u.d 6
76.l odd 18 6 2888.1.u.c 6
152.g odd 2 1 2888.1.k.a 2
152.k odd 6 1 inner 608.1.o.a 2
152.l odd 6 1 2888.1.f.a 1
152.l odd 6 1 2888.1.k.a 2
152.p even 6 1 152.1.k.a 2
152.p even 6 1 2888.1.f.b 1
152.s odd 18 6 2888.1.u.d 6
152.t even 18 6 2888.1.u.c 6
228.m even 6 1 1368.1.bz.a 2
380.p odd 6 1 3800.1.bd.c 2
380.v even 12 2 3800.1.bn.b 4
456.x odd 6 1 1368.1.bz.a 2
760.z even 6 1 3800.1.bd.c 2
760.br odd 12 2 3800.1.bn.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
152.1.k.a 2 4.b odd 2 1
152.1.k.a 2 8.b even 2 1
152.1.k.a 2 76.g odd 6 1
152.1.k.a 2 152.p even 6 1
608.1.o.a 2 1.a even 1 1 trivial
608.1.o.a 2 8.d odd 2 1 CM
608.1.o.a 2 19.c even 3 1 inner
608.1.o.a 2 152.k odd 6 1 inner
1368.1.bz.a 2 12.b even 2 1
1368.1.bz.a 2 24.h odd 2 1
1368.1.bz.a 2 228.m even 6 1
1368.1.bz.a 2 456.x odd 6 1
2888.1.f.a 1 76.f even 6 1
2888.1.f.a 1 152.l odd 6 1
2888.1.f.b 1 76.g odd 6 1
2888.1.f.b 1 152.p even 6 1
2888.1.k.a 2 76.d even 2 1
2888.1.k.a 2 76.f even 6 1
2888.1.k.a 2 152.g odd 2 1
2888.1.k.a 2 152.l odd 6 1
2888.1.u.c 6 76.l odd 18 6
2888.1.u.c 6 152.t even 18 6
2888.1.u.d 6 76.k even 18 6
2888.1.u.d 6 152.s odd 18 6
3800.1.bd.c 2 20.d odd 2 1
3800.1.bd.c 2 40.f even 2 1
3800.1.bd.c 2 380.p odd 6 1
3800.1.bd.c 2 760.z even 6 1
3800.1.bn.b 4 20.e even 4 2
3800.1.bn.b 4 40.i odd 4 2
3800.1.bn.b 4 380.v even 12 2
3800.1.bn.b 4 760.br odd 12 2

Hecke kernels

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

Hecke characteristic polynomials

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