Properties

Label 306.2.b.b
Level $306$
Weight $2$
Character orbit 306.b
Analytic conductor $2.443$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 306 = 2 \cdot 3^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 306.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(2.44342230185\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-2}) \)
Defining polynomial: \(x^{2} + 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{-2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{2} + q^{4} + \beta q^{5} -3 \beta q^{7} - q^{8} +O(q^{10})\) \( q - q^{2} + q^{4} + \beta q^{5} -3 \beta q^{7} - q^{8} -\beta q^{10} -2 \beta q^{11} + 2 q^{13} + 3 \beta q^{14} + q^{16} + ( 3 + 2 \beta ) q^{17} + 2 q^{19} + \beta q^{20} + 2 \beta q^{22} -5 \beta q^{23} + 3 q^{25} -2 q^{26} -3 \beta q^{28} -5 \beta q^{29} + 3 \beta q^{31} - q^{32} + ( -3 - 2 \beta ) q^{34} + 6 q^{35} -3 \beta q^{37} -2 q^{38} -\beta q^{40} + 4 \beta q^{41} -4 q^{43} -2 \beta q^{44} + 5 \beta q^{46} + 12 q^{47} -11 q^{49} -3 q^{50} + 2 q^{52} -6 q^{53} + 4 q^{55} + 3 \beta q^{56} + 5 \beta q^{58} -6 q^{59} + 9 \beta q^{61} -3 \beta q^{62} + q^{64} + 2 \beta q^{65} -10 q^{67} + ( 3 + 2 \beta ) q^{68} -6 q^{70} + \beta q^{71} + 3 \beta q^{74} + 2 q^{76} -12 q^{77} + 9 \beta q^{79} + \beta q^{80} -4 \beta q^{82} + 6 q^{83} + ( -4 + 3 \beta ) q^{85} + 4 q^{86} + 2 \beta q^{88} -6 q^{89} -6 \beta q^{91} -5 \beta q^{92} -12 q^{94} + 2 \beta q^{95} -6 \beta q^{97} + 11 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} + 2 q^{4} - 2 q^{8} + O(q^{10}) \) \( 2 q - 2 q^{2} + 2 q^{4} - 2 q^{8} + 4 q^{13} + 2 q^{16} + 6 q^{17} + 4 q^{19} + 6 q^{25} - 4 q^{26} - 2 q^{32} - 6 q^{34} + 12 q^{35} - 4 q^{38} - 8 q^{43} + 24 q^{47} - 22 q^{49} - 6 q^{50} + 4 q^{52} - 12 q^{53} + 8 q^{55} - 12 q^{59} + 2 q^{64} - 20 q^{67} + 6 q^{68} - 12 q^{70} + 4 q^{76} - 24 q^{77} + 12 q^{83} - 8 q^{85} + 8 q^{86} - 12 q^{89} - 24 q^{94} + 22 q^{98} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/306\mathbb{Z}\right)^\times\).

\(n\) \(37\) \(137\)
\(\chi(n)\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
271.1
1.41421i
1.41421i
−1.00000 0 1.00000 1.41421i 0 4.24264i −1.00000 0 1.41421i
271.2 −1.00000 0 1.00000 1.41421i 0 4.24264i −1.00000 0 1.41421i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
17.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 306.2.b.b 2
3.b odd 2 1 306.2.b.c yes 2
4.b odd 2 1 2448.2.c.k 2
12.b even 2 1 2448.2.c.i 2
17.b even 2 1 inner 306.2.b.b 2
17.c even 4 2 5202.2.a.bd 2
51.c odd 2 1 306.2.b.c yes 2
51.f odd 4 2 5202.2.a.t 2
68.d odd 2 1 2448.2.c.k 2
204.h even 2 1 2448.2.c.i 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
306.2.b.b 2 1.a even 1 1 trivial
306.2.b.b 2 17.b even 2 1 inner
306.2.b.c yes 2 3.b odd 2 1
306.2.b.c yes 2 51.c odd 2 1
2448.2.c.i 2 12.b even 2 1
2448.2.c.i 2 204.h even 2 1
2448.2.c.k 2 4.b odd 2 1
2448.2.c.k 2 68.d odd 2 1
5202.2.a.t 2 51.f odd 4 2
5202.2.a.bd 2 17.c even 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(306, [\chi])\):

\( T_{5}^{2} + 2 \)
\( T_{47} - 12 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T )^{2} \)
$3$ \( T^{2} \)
$5$ \( 2 + T^{2} \)
$7$ \( 18 + T^{2} \)
$11$ \( 8 + T^{2} \)
$13$ \( ( -2 + T )^{2} \)
$17$ \( 17 - 6 T + T^{2} \)
$19$ \( ( -2 + T )^{2} \)
$23$ \( 50 + T^{2} \)
$29$ \( 50 + T^{2} \)
$31$ \( 18 + T^{2} \)
$37$ \( 18 + T^{2} \)
$41$ \( 32 + T^{2} \)
$43$ \( ( 4 + T )^{2} \)
$47$ \( ( -12 + T )^{2} \)
$53$ \( ( 6 + T )^{2} \)
$59$ \( ( 6 + T )^{2} \)
$61$ \( 162 + T^{2} \)
$67$ \( ( 10 + T )^{2} \)
$71$ \( 2 + T^{2} \)
$73$ \( T^{2} \)
$79$ \( 162 + T^{2} \)
$83$ \( ( -6 + T )^{2} \)
$89$ \( ( 6 + T )^{2} \)
$97$ \( 72 + T^{2} \)
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