Properties

Label 152.1.k.a
Level $152$
Weight $1$
Character orbit 152.k
Analytic conductor $0.076$
Analytic rank $0$
Dimension $2$
Projective image $D_{3}$
CM discriminant -8
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 152 = 2^{3} \cdot 19 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 152.k (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.0758578819202\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{3}\)
Projective field: Galois closure of 3.1.2888.1
Artin image: $C_3\times S_3$
Artin field: Galois closure of 6.0.184832.1

$q$-expansion

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

\(f(q)\) \(=\) \( q + \zeta_{6}^{2} q^{2} - \zeta_{6}^{2} q^{3} - \zeta_{6} q^{4} + \zeta_{6} q^{6} + q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q + \zeta_{6}^{2} q^{2} - \zeta_{6}^{2} q^{3} - \zeta_{6} q^{4} + \zeta_{6} q^{6} + q^{8} - q^{11} - q^{12} + \zeta_{6}^{2} q^{16} + \zeta_{6}^{2} q^{17} - \zeta_{6} q^{19} - \zeta_{6}^{2} q^{22} - \zeta_{6}^{2} q^{24} - \zeta_{6} q^{25} + q^{27} - \zeta_{6} q^{32} + \zeta_{6}^{2} q^{33} - 2 \zeta_{6} q^{34} + q^{38} - \zeta_{6}^{2} q^{41} + \zeta_{6}^{2} q^{43} + \zeta_{6} q^{44} + \zeta_{6} q^{48} + q^{49} + q^{50} + 2 \zeta_{6} q^{51} + \zeta_{6}^{2} q^{54} - q^{57} - \zeta_{6}^{2} q^{59} + q^{64} - \zeta_{6} q^{66} + \zeta_{6} q^{67} + 2 q^{68} - \zeta_{6}^{2} q^{73} - q^{75} + \zeta_{6}^{2} q^{76} - \zeta_{6}^{2} q^{81} + \zeta_{6} q^{82} - q^{83} - 2 \zeta_{6} q^{86} - q^{88} - \zeta_{6} q^{89} - q^{96} - \zeta_{6}^{2} q^{97} + \zeta_{6}^{2} q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{2} + q^{3} - q^{4} + q^{6} + 2 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - q^{2} + q^{3} - q^{4} + q^{6} + 2 q^{8} - 2 q^{11} - 2 q^{12} - q^{16} - 2 q^{17} - q^{19} + q^{22} + q^{24} - q^{25} + 2 q^{27} - q^{32} - q^{33} - 2 q^{34} + 2 q^{38} + q^{41} - 2 q^{43} + q^{44} + q^{48} + 2 q^{49} + 2 q^{50} + 2 q^{51} - q^{54} - 2 q^{57} + q^{59} + 2 q^{64} - q^{66} + q^{67} + 4 q^{68} + q^{73} - 2 q^{75} - q^{76} + q^{81} + q^{82} - 2 q^{83} - 2 q^{86} - 2 q^{88} - 2 q^{89} - 2 q^{96} + q^{97} - q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/152\mathbb{Z}\right)^\times\).

\(n\) \(39\) \(77\) \(97\)
\(\chi(n)\) \(-1\) \(-1\) \(-\zeta_{6}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
11.1
0.500000 + 0.866025i
0.500000 0.866025i
−0.500000 + 0.866025i 0.500000 0.866025i −0.500000 0.866025i 0 0.500000 + 0.866025i 0 1.00000 0 0
83.1 −0.500000 0.866025i 0.500000 + 0.866025i −0.500000 + 0.866025i 0 0.500000 0.866025i 0 1.00000 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 152.1.k.a 2
3.b odd 2 1 1368.1.bz.a 2
4.b odd 2 1 608.1.o.a 2
5.b even 2 1 3800.1.bd.c 2
5.c odd 4 2 3800.1.bn.b 4
8.b even 2 1 608.1.o.a 2
8.d odd 2 1 CM 152.1.k.a 2
19.b odd 2 1 2888.1.k.a 2
19.c even 3 1 inner 152.1.k.a 2
19.c even 3 1 2888.1.f.b 1
19.d odd 6 1 2888.1.f.a 1
19.d odd 6 1 2888.1.k.a 2
19.e even 9 6 2888.1.u.c 6
19.f odd 18 6 2888.1.u.d 6
24.f even 2 1 1368.1.bz.a 2
40.e odd 2 1 3800.1.bd.c 2
40.k even 4 2 3800.1.bn.b 4
57.h odd 6 1 1368.1.bz.a 2
76.g odd 6 1 608.1.o.a 2
95.i even 6 1 3800.1.bd.c 2
95.m odd 12 2 3800.1.bn.b 4
152.b even 2 1 2888.1.k.a 2
152.k odd 6 1 inner 152.1.k.a 2
152.k odd 6 1 2888.1.f.b 1
152.o even 6 1 2888.1.f.a 1
152.o even 6 1 2888.1.k.a 2
152.p even 6 1 608.1.o.a 2
152.u odd 18 6 2888.1.u.c 6
152.v even 18 6 2888.1.u.d 6
456.u even 6 1 1368.1.bz.a 2
760.bm odd 6 1 3800.1.bd.c 2
760.bw even 12 2 3800.1.bn.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
152.1.k.a 2 1.a even 1 1 trivial
152.1.k.a 2 8.d odd 2 1 CM
152.1.k.a 2 19.c even 3 1 inner
152.1.k.a 2 152.k odd 6 1 inner
608.1.o.a 2 4.b odd 2 1
608.1.o.a 2 8.b even 2 1
608.1.o.a 2 76.g odd 6 1
608.1.o.a 2 152.p even 6 1
1368.1.bz.a 2 3.b odd 2 1
1368.1.bz.a 2 24.f even 2 1
1368.1.bz.a 2 57.h odd 6 1
1368.1.bz.a 2 456.u even 6 1
2888.1.f.a 1 19.d odd 6 1
2888.1.f.a 1 152.o even 6 1
2888.1.f.b 1 19.c even 3 1
2888.1.f.b 1 152.k odd 6 1
2888.1.k.a 2 19.b odd 2 1
2888.1.k.a 2 19.d odd 6 1
2888.1.k.a 2 152.b even 2 1
2888.1.k.a 2 152.o even 6 1
2888.1.u.c 6 19.e even 9 6
2888.1.u.c 6 152.u odd 18 6
2888.1.u.d 6 19.f odd 18 6
2888.1.u.d 6 152.v even 18 6
3800.1.bd.c 2 5.b even 2 1
3800.1.bd.c 2 40.e odd 2 1
3800.1.bd.c 2 95.i even 6 1
3800.1.bd.c 2 760.bm odd 6 1
3800.1.bn.b 4 5.c odd 4 2
3800.1.bn.b 4 40.k even 4 2
3800.1.bn.b 4 95.m odd 12 2
3800.1.bn.b 4 760.bw even 12 2

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(152, [\chi])\).

Hecke characteristic polynomials

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