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 discriminant -151
Inner twists 2

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Newspace parameters

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

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: \( 1 \)
Twist minimal: yes
Projective image \(D_{7}\)
Projective field Galois closure of 7.1.3442951.1
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 root \(\beta\) of the polynomial \(x^{3} - x^{2} - 2 x + 1\). We also show the integral \(q\)-expansion of the trace form.

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

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 Parity Ord Mult Type
1.a even 1 1 trivial
151.b odd 2 1 CM by \(\Q(\sqrt{-151}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 151.1.b.a 3
3.b odd 2 1 1359.1.d.b 3
4.b odd 2 1 2416.1.e.a 3
5.b even 2 1 3775.1.d.d 3
5.c odd 4 2 3775.1.c.c 6
151.b odd 2 1 CM 151.1.b.a 3
453.b even 2 1 1359.1.d.b 3
604.d even 2 1 2416.1.e.a 3
755.c odd 2 1 3775.1.d.d 3
755.f even 4 2 3775.1.c.c 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
151.1.b.a 3 1.a even 1 1 trivial
151.1.b.a 3 151.b odd 2 1 CM
1359.1.d.b 3 3.b odd 2 1
1359.1.d.b 3 453.b even 2 1
2416.1.e.a 3 4.b odd 2 1
2416.1.e.a 3 604.d even 2 1
3775.1.c.c 6 5.c odd 4 2
3775.1.c.c 6 755.f even 4 2
3775.1.d.d 3 5.b even 2 1
3775.1.d.d 3 755.c odd 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(151, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \)
$3$ \( ( 1 - T )^{3}( 1 + T )^{3} \)
$5$ \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \)
$7$ \( ( 1 - T )^{3}( 1 + T )^{3} \)
$11$ \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \)
$13$ \( ( 1 - T )^{3}( 1 + T )^{3} \)
$17$ \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \)
$19$ \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \)
$23$ \( ( 1 - T )^{3}( 1 + T )^{3} \)
$29$ \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \)
$31$ \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \)
$37$ \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \)
$41$ \( ( 1 - T )^{3}( 1 + T )^{3} \)
$43$ \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \)
$47$ \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \)
$53$ \( ( 1 - T )^{3}( 1 + T )^{3} \)
$59$ \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \)
$61$ \( ( 1 - T )^{3}( 1 + T )^{3} \)
$67$ \( ( 1 - T )^{3}( 1 + T )^{3} \)
$71$ \( ( 1 - T )^{3}( 1 + T )^{3} \)
$73$ \( ( 1 - T )^{3}( 1 + T )^{3} \)
$79$ \( ( 1 - T )^{3}( 1 + T )^{3} \)
$83$ \( ( 1 - T )^{3}( 1 + T )^{3} \)
$89$ \( ( 1 - T )^{3}( 1 + T )^{3} \)
$97$ \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \)
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