Properties

Label 306.2.b.d
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: \( 2 \)
Twist minimal: no (minimal twist has level 34)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

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

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 2.82843i 0 0 1.00000 0 2.82843i
271.2 1.00000 0 1.00000 2.82843i 0 0 1.00000 0 2.82843i
\(n\): 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.d 2
3.b odd 2 1 34.2.b.a 2
4.b odd 2 1 2448.2.c.n 2
12.b even 2 1 272.2.b.a 2
15.d odd 2 1 850.2.b.f 2
15.e even 4 2 850.2.d.i 4
17.b even 2 1 inner 306.2.b.d 2
17.c even 4 2 5202.2.a.u 2
21.c even 2 1 1666.2.b.c 2
24.f even 2 1 1088.2.b.a 2
24.h odd 2 1 1088.2.b.b 2
51.c odd 2 1 34.2.b.a 2
51.f odd 4 2 578.2.a.d 2
51.g odd 8 2 578.2.c.a 2
51.g odd 8 2 578.2.c.d 2
51.i even 16 8 578.2.d.g 8
68.d odd 2 1 2448.2.c.n 2
204.h even 2 1 272.2.b.a 2
204.l even 4 2 4624.2.a.s 2
255.h odd 2 1 850.2.b.f 2
255.o even 4 2 850.2.d.i 4
357.c even 2 1 1666.2.b.c 2
408.b odd 2 1 1088.2.b.b 2
408.h even 2 1 1088.2.b.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
34.2.b.a 2 3.b odd 2 1
34.2.b.a 2 51.c odd 2 1
272.2.b.a 2 12.b even 2 1
272.2.b.a 2 204.h even 2 1
306.2.b.d 2 1.a even 1 1 trivial
306.2.b.d 2 17.b even 2 1 inner
578.2.a.d 2 51.f odd 4 2
578.2.c.a 2 51.g odd 8 2
578.2.c.d 2 51.g odd 8 2
578.2.d.g 8 51.i even 16 8
850.2.b.f 2 15.d odd 2 1
850.2.b.f 2 255.h odd 2 1
850.2.d.i 4 15.e even 4 2
850.2.d.i 4 255.o even 4 2
1088.2.b.a 2 24.f even 2 1
1088.2.b.a 2 408.h even 2 1
1088.2.b.b 2 24.h odd 2 1
1088.2.b.b 2 408.b odd 2 1
1666.2.b.c 2 21.c even 2 1
1666.2.b.c 2 357.c even 2 1
2448.2.c.n 2 4.b odd 2 1
2448.2.c.n 2 68.d odd 2 1
4624.2.a.s 2 204.l even 4 2
5202.2.a.u 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 \(S_{2}^{\mathrm{new}}(306, [\chi])\):

\( T_{5}^{2} + 8 \)
\( T_{47} \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -1 + T )^{2} \)
$3$ \( T^{2} \)
$5$ \( 8 + T^{2} \)
$7$ \( T^{2} \)
$11$ \( 8 + T^{2} \)
$13$ \( ( -2 + T )^{2} \)
$17$ \( 17 - 6 T + T^{2} \)
$19$ \( ( 4 + T )^{2} \)
$23$ \( 32 + T^{2} \)
$29$ \( 8 + T^{2} \)
$31$ \( T^{2} \)
$37$ \( 72 + T^{2} \)
$41$ \( 32 + T^{2} \)
$43$ \( ( 4 + T )^{2} \)
$47$ \( T^{2} \)
$53$ \( ( 6 + T )^{2} \)
$59$ \( ( 12 + T )^{2} \)
$61$ \( 72 + T^{2} \)
$67$ \( ( 4 + T )^{2} \)
$71$ \( 32 + T^{2} \)
$73$ \( T^{2} \)
$79$ \( 288 + T^{2} \)
$83$ \( ( -12 + T )^{2} \)
$89$ \( ( 6 + T )^{2} \)
$97$ \( 288 + T^{2} \)
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