Properties

Label 151.1.b.a
Level 151
Weight 1
Character orbit 151.b
Self dual Yes
Analytic conductor 0.075
Analytic rank 0
Dimension 3
Projective image \(D_{7}\)
CM disc. -151
Inner twists 2

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) = \( 151 \)
Weight: \( k \) = \( 1 \)
Character orbit: \([\chi]\) = 151.b (of order \(2\) and degree \(1\))

Newform invariants

Self dual: Yes
Analytic conductor: \(0.0753588169076\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: \(\Q(\zeta_{14})^+\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Projective image \(D_{7}\)
Projective field Galois closure of 7.1.3442951.1
Artin image size \(14\)
Artin image $D_7$
Artin field Galois closure of 7.1.3442951.1

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta_{1} q^{2} + ( 1 + \beta_{2} ) q^{4} + ( -1 + \beta_{1} - \beta_{2} ) q^{5} + ( -1 - \beta_{2} ) q^{8} + q^{9} +O(q^{10})\) \( q -\beta_{1} q^{2} + ( 1 + \beta_{2} ) q^{4} + ( -1 + \beta_{1} - \beta_{2} ) q^{5} + ( -1 - \beta_{2} ) q^{8} + q^{9} + ( -1 + \beta_{1} ) q^{10} + \beta_{2} q^{11} + \beta_{1} q^{16} + \beta_{2} q^{17} -\beta_{1} q^{18} -\beta_{1} q^{19} - q^{20} + ( -1 - \beta_{2} ) q^{22} + ( 1 - \beta_{1} ) q^{25} -\beta_{1} q^{29} + ( -1 + \beta_{1} - \beta_{2} ) q^{31} - q^{32} + ( -1 - \beta_{2} ) q^{34} + ( 1 + \beta_{2} ) q^{36} + \beta_{2} q^{37} + ( 2 + \beta_{2} ) q^{38} + q^{40} + ( -1 + \beta_{1} - \beta_{2} ) q^{43} + ( 1 + \beta_{1} ) q^{44} + ( -1 + \beta_{1} - \beta_{2} ) q^{45} -\beta_{1} q^{47} + q^{49} + ( 2 - \beta_{1} + \beta_{2} ) q^{50} + ( -\beta_{1} + \beta_{2} ) q^{55} + ( 2 + \beta_{2} ) q^{58} + ( -1 + \beta_{1} - \beta_{2} ) q^{59} + ( -1 + \beta_{1} ) q^{62} + ( 1 + \beta_{1} ) q^{68} + ( -1 - \beta_{2} ) q^{72} + ( -1 - \beta_{2} ) q^{74} + ( -1 - \beta_{1} - \beta_{2} ) q^{76} + ( 1 - \beta_{1} ) q^{80} + q^{81} + ( -\beta_{1} + \beta_{2} ) q^{85} + ( -1 + \beta_{1} ) q^{86} + ( -1 - \beta_{1} ) q^{88} + ( -1 + \beta_{1} ) q^{90} + ( 2 + \beta_{2} ) q^{94} + ( -1 + \beta_{1} ) q^{95} -\beta_{1} q^{97} -\beta_{1} q^{98} + \beta_{2} q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q - q^{2} + 2q^{4} - q^{5} - 2q^{8} + 3q^{9} + O(q^{10}) \) \( 3q - q^{2} + 2q^{4} - q^{5} - 2q^{8} + 3q^{9} - 2q^{10} - q^{11} + q^{16} - q^{17} - q^{18} - q^{19} - 3q^{20} - 2q^{22} + 2q^{25} - q^{29} - q^{31} - 3q^{32} - 2q^{34} + 2q^{36} - q^{37} + 5q^{38} + 3q^{40} - q^{43} + 4q^{44} - q^{45} - q^{47} + 3q^{49} + 4q^{50} - 2q^{55} + 5q^{58} - q^{59} - 2q^{62} + 4q^{68} - 2q^{72} - 2q^{74} - 3q^{76} + 2q^{80} + 3q^{81} - 2q^{85} - 2q^{86} - 4q^{88} - 2q^{90} + 5q^{94} - 2q^{95} - q^{97} - q^{98} - q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of \(\nu = \zeta_{14} + \zeta_{14}^{-1}\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 2 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 2\)

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/151\mathbb{Z}\right)^\times\).

\(n\) \(6\)
\(\chi(n)\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
150.1
1.80194
0.445042
−1.24698
−1.80194 0 2.24698 −0.445042 0 0 −2.24698 1.00000 0.801938
150.2 −0.445042 0 −0.801938 1.24698 0 0 0.801938 1.00000 −0.554958
150.3 1.24698 0 0.554958 −1.80194 0 0 −0.554958 1.00000 −2.24698
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
151.b Odd 1 CM by \(\Q(\sqrt{-151}) \) yes

Hecke kernels

There are no other newforms in \(S_{1}^{\mathrm{new}}(151, [\chi])\).