Properties

Label 152.1.u.a
Level 152
Weight 1
Character orbit 152.u
Analytic conductor 0.076
Analytic rank 0
Dimension 6
Projective image \(D_{9}\)
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.u (of order \(18\), degree \(6\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.0758578819202\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\Q(\zeta_{18})\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image \(D_{9}\)
Projective field Galois closure of 9.1.69564674215936.1

$q$-expansion

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

\(f(q)\) \(=\) \( q -\zeta_{18}^{5} q^{2} + ( \zeta_{18}^{6} - \zeta_{18}^{7} ) q^{3} -\zeta_{18} q^{4} + ( \zeta_{18}^{2} - \zeta_{18}^{3} ) q^{6} + \zeta_{18}^{6} q^{8} + ( -\zeta_{18}^{3} + \zeta_{18}^{4} - \zeta_{18}^{5} ) q^{9} +O(q^{10})\) \( q -\zeta_{18}^{5} q^{2} + ( \zeta_{18}^{6} - \zeta_{18}^{7} ) q^{3} -\zeta_{18} q^{4} + ( \zeta_{18}^{2} - \zeta_{18}^{3} ) q^{6} + \zeta_{18}^{6} q^{8} + ( -\zeta_{18}^{3} + \zeta_{18}^{4} - \zeta_{18}^{5} ) q^{9} + ( \zeta_{18}^{4} + \zeta_{18}^{8} ) q^{11} + ( -\zeta_{18}^{7} + \zeta_{18}^{8} ) q^{12} + \zeta_{18}^{2} q^{16} + \zeta_{18}^{5} q^{17} + ( 1 - \zeta_{18} + \zeta_{18}^{8} ) q^{18} -\zeta_{18} q^{19} + ( 1 + \zeta_{18}^{4} ) q^{22} + ( -\zeta_{18}^{3} + \zeta_{18}^{4} ) q^{24} -\zeta_{18}^{7} q^{25} + ( 1 - \zeta_{18} + \zeta_{18}^{2} - \zeta_{18}^{3} ) q^{27} -\zeta_{18}^{7} q^{32} + ( -\zeta_{18} + \zeta_{18}^{2} - \zeta_{18}^{5} + \zeta_{18}^{6} ) q^{33} + \zeta_{18} q^{34} + ( \zeta_{18}^{4} - \zeta_{18}^{5} + \zeta_{18}^{6} ) q^{36} + \zeta_{18}^{6} q^{38} + ( -\zeta_{18} - \zeta_{18}^{3} ) q^{41} -\zeta_{18}^{8} q^{43} + ( 1 - \zeta_{18}^{5} ) q^{44} + ( 1 + \zeta_{18}^{8} ) q^{48} + \zeta_{18}^{6} q^{49} -\zeta_{18}^{3} q^{50} + ( -\zeta_{18}^{2} + \zeta_{18}^{3} ) q^{51} + ( -\zeta_{18}^{5} + \zeta_{18}^{6} - \zeta_{18}^{7} + \zeta_{18}^{8} ) q^{54} + ( -\zeta_{18}^{7} + \zeta_{18}^{8} ) q^{57} + ( -\zeta_{18}^{3} - \zeta_{18}^{7} ) q^{59} -\zeta_{18}^{3} q^{64} + ( -\zeta_{18} + \zeta_{18}^{2} + \zeta_{18}^{6} - \zeta_{18}^{7} ) q^{66} + ( -\zeta_{18}^{3} - \zeta_{18}^{5} ) q^{67} -\zeta_{18}^{6} q^{68} + ( 1 - \zeta_{18} + \zeta_{18}^{2} ) q^{72} + ( 1 + \zeta_{18}^{4} ) q^{73} + ( \zeta_{18}^{4} - \zeta_{18}^{5} ) q^{75} + \zeta_{18}^{2} q^{76} + ( 1 - \zeta_{18} + \zeta_{18}^{6} - \zeta_{18}^{7} + \zeta_{18}^{8} ) q^{81} + ( \zeta_{18}^{6} + \zeta_{18}^{8} ) q^{82} + ( \zeta_{18}^{2} + \zeta_{18}^{4} ) q^{83} -\zeta_{18}^{4} q^{86} + ( -\zeta_{18} - \zeta_{18}^{5} ) q^{88} + \zeta_{18}^{7} q^{89} + ( \zeta_{18}^{4} - \zeta_{18}^{5} ) q^{96} + ( \zeta_{18}^{4} + \zeta_{18}^{6} ) q^{97} + \zeta_{18}^{2} q^{98} + ( 1 + \zeta_{18}^{2} - \zeta_{18}^{3} + \zeta_{18}^{4} - \zeta_{18}^{7} + \zeta_{18}^{8} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q - 3q^{3} - 3q^{6} - 3q^{8} - 3q^{9} + O(q^{10}) \) \( 6q - 3q^{3} - 3q^{6} - 3q^{8} - 3q^{9} + 6q^{18} + 6q^{22} - 3q^{24} + 3q^{27} - 3q^{33} - 3q^{36} - 3q^{38} - 3q^{41} + 6q^{44} + 6q^{48} - 3q^{49} - 3q^{50} + 3q^{51} - 3q^{54} - 3q^{59} - 3q^{64} - 3q^{66} - 3q^{67} + 3q^{68} + 6q^{72} + 6q^{73} + 3q^{81} - 3q^{82} - 3q^{97} + 3q^{99} + O(q^{100}) \)

Character values

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

\(n\) \(39\) \(77\) \(97\)
\(\chi(n)\) \(-1\) \(-1\) \(-\zeta_{18}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
35.1
−0.173648 0.984808i
−0.766044 + 0.642788i
−0.766044 0.642788i
0.939693 0.342020i
0.939693 + 0.342020i
−0.173648 + 0.984808i
0.766044 + 0.642788i −1.43969 + 0.524005i 0.173648 + 0.984808i 0 −1.43969 0.524005i 0 −0.500000 + 0.866025i 1.03209 0.866025i 0
43.1 −0.939693 + 0.342020i −0.326352 + 1.85083i 0.766044 0.642788i 0 −0.326352 1.85083i 0 −0.500000 + 0.866025i −2.37939 0.866025i 0
99.1 −0.939693 0.342020i −0.326352 1.85083i 0.766044 + 0.642788i 0 −0.326352 + 1.85083i 0 −0.500000 0.866025i −2.37939 + 0.866025i 0
123.1 0.173648 + 0.984808i 0.266044 0.223238i −0.939693 + 0.342020i 0 0.266044 + 0.223238i 0 −0.500000 0.866025i −0.152704 + 0.866025i 0
131.1 0.173648 0.984808i 0.266044 + 0.223238i −0.939693 0.342020i 0 0.266044 0.223238i 0 −0.500000 + 0.866025i −0.152704 0.866025i 0
139.1 0.766044 0.642788i −1.43969 0.524005i 0.173648 0.984808i 0 −1.43969 + 0.524005i 0 −0.500000 0.866025i 1.03209 + 0.866025i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 139.1
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 152.1.u.a 6
3.b odd 2 1 1368.1.eh.a 6
4.b odd 2 1 608.1.bg.a 6
5.b even 2 1 3800.1.cv.c 6
5.c odd 4 2 3800.1.cq.b 12
8.b even 2 1 608.1.bg.a 6
8.d odd 2 1 CM 152.1.u.a 6
19.b odd 2 1 2888.1.u.e 6
19.c even 3 1 2888.1.u.b 6
19.c even 3 1 2888.1.u.g 6
19.d odd 6 1 2888.1.u.a 6
19.d odd 6 1 2888.1.u.f 6
19.e even 9 1 inner 152.1.u.a 6
19.e even 9 1 2888.1.f.d 3
19.e even 9 2 2888.1.k.b 6
19.e even 9 1 2888.1.u.b 6
19.e even 9 1 2888.1.u.g 6
19.f odd 18 1 2888.1.f.c 3
19.f odd 18 2 2888.1.k.c 6
19.f odd 18 1 2888.1.u.a 6
19.f odd 18 1 2888.1.u.e 6
19.f odd 18 1 2888.1.u.f 6
24.f even 2 1 1368.1.eh.a 6
40.e odd 2 1 3800.1.cv.c 6
40.k even 4 2 3800.1.cq.b 12
57.l odd 18 1 1368.1.eh.a 6
76.l odd 18 1 608.1.bg.a 6
95.p even 18 1 3800.1.cv.c 6
95.q odd 36 2 3800.1.cq.b 12
152.b even 2 1 2888.1.u.e 6
152.k odd 6 1 2888.1.u.b 6
152.k odd 6 1 2888.1.u.g 6
152.o even 6 1 2888.1.u.a 6
152.o even 6 1 2888.1.u.f 6
152.t even 18 1 608.1.bg.a 6
152.u odd 18 1 inner 152.1.u.a 6
152.u odd 18 1 2888.1.f.d 3
152.u odd 18 2 2888.1.k.b 6
152.u odd 18 1 2888.1.u.b 6
152.u odd 18 1 2888.1.u.g 6
152.v even 18 1 2888.1.f.c 3
152.v even 18 2 2888.1.k.c 6
152.v even 18 1 2888.1.u.a 6
152.v even 18 1 2888.1.u.e 6
152.v even 18 1 2888.1.u.f 6
456.bu even 18 1 1368.1.eh.a 6
760.bz odd 18 1 3800.1.cv.c 6
760.cp even 36 2 3800.1.cq.b 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
152.1.u.a 6 1.a even 1 1 trivial
152.1.u.a 6 8.d odd 2 1 CM
152.1.u.a 6 19.e even 9 1 inner
152.1.u.a 6 152.u odd 18 1 inner
608.1.bg.a 6 4.b odd 2 1
608.1.bg.a 6 8.b even 2 1
608.1.bg.a 6 76.l odd 18 1
608.1.bg.a 6 152.t even 18 1
1368.1.eh.a 6 3.b odd 2 1
1368.1.eh.a 6 24.f even 2 1
1368.1.eh.a 6 57.l odd 18 1
1368.1.eh.a 6 456.bu even 18 1
2888.1.f.c 3 19.f odd 18 1
2888.1.f.c 3 152.v even 18 1
2888.1.f.d 3 19.e even 9 1
2888.1.f.d 3 152.u odd 18 1
2888.1.k.b 6 19.e even 9 2
2888.1.k.b 6 152.u odd 18 2
2888.1.k.c 6 19.f odd 18 2
2888.1.k.c 6 152.v even 18 2
2888.1.u.a 6 19.d odd 6 1
2888.1.u.a 6 19.f odd 18 1
2888.1.u.a 6 152.o even 6 1
2888.1.u.a 6 152.v even 18 1
2888.1.u.b 6 19.c even 3 1
2888.1.u.b 6 19.e even 9 1
2888.1.u.b 6 152.k odd 6 1
2888.1.u.b 6 152.u odd 18 1
2888.1.u.e 6 19.b odd 2 1
2888.1.u.e 6 19.f odd 18 1
2888.1.u.e 6 152.b even 2 1
2888.1.u.e 6 152.v even 18 1
2888.1.u.f 6 19.d odd 6 1
2888.1.u.f 6 19.f odd 18 1
2888.1.u.f 6 152.o even 6 1
2888.1.u.f 6 152.v even 18 1
2888.1.u.g 6 19.c even 3 1
2888.1.u.g 6 19.e even 9 1
2888.1.u.g 6 152.k odd 6 1
2888.1.u.g 6 152.u odd 18 1
3800.1.cq.b 12 5.c odd 4 2
3800.1.cq.b 12 40.k even 4 2
3800.1.cq.b 12 95.q odd 36 2
3800.1.cq.b 12 760.cp even 36 2
3800.1.cv.c 6 5.b even 2 1
3800.1.cv.c 6 40.e odd 2 1
3800.1.cv.c 6 95.p even 18 1
3800.1.cv.c 6 760.bz odd 18 1

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T^{3} + T^{6} \)
$3$ \( ( 1 + T + T^{2} )^{3}( 1 + T^{3} + T^{6} ) \)
$5$ \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
$7$ \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \)
$11$ \( ( 1 + T^{3} + T^{6} )^{2} \)
$13$ \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
$17$ \( ( 1 + T^{3} + T^{6} )^{2} \)
$19$ \( 1 + T^{3} + T^{6} \)
$23$ \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
$29$ \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
$31$ \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \)
$37$ \( ( 1 - T )^{6}( 1 + T )^{6} \)
$41$ \( ( 1 + T + T^{2} )^{3}( 1 + T^{3} + T^{6} ) \)
$43$ \( ( 1 + T^{3} + T^{6} )^{2} \)
$47$ \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
$53$ \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
$59$ \( ( 1 + T + T^{2} )^{3}( 1 + T^{3} + T^{6} ) \)
$61$ \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
$67$ \( ( 1 + T + T^{2} )^{3}( 1 + T^{3} + T^{6} ) \)
$71$ \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
$73$ \( ( 1 - T )^{6}( 1 + T^{3} + T^{6} ) \)
$79$ \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
$83$ \( ( 1 + T^{3} + T^{6} )^{2} \)
$89$ \( ( 1 + T^{3} + T^{6} )^{2} \)
$97$ \( ( 1 + T + T^{2} )^{3}( 1 + T^{3} + T^{6} ) \)
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