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 disc. -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 \(18\) and degree \(6\))

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: \( 1 \)
Projective image \(D_{9}\)
Projective field Galois closure of 9.1.69564674215936.1

$q$-expansion

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

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

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. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
8.d Odd 1 CM by \(\Q(\sqrt{-2}) \) yes
19.e Even 1 yes
152.u Odd 1 yes

Hecke kernels

There are no other newforms in \(S_{1}^{\mathrm{new}}(152, [\chi])\).