Properties

Label 315.2.bb.a
Level 315
Weight 2
Character orbit 315.bb
Analytic conductor 2.515
Analytic rank 0
Dimension 8
CM no
Inner twists 8

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Newspace parameters

Level: \( N \) = \( 315 = 3^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 315.bb (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(2.51528766367\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: 8.0.12745506816.5
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 2 \beta_{4} q^{4} + ( \beta_{1} - \beta_{2} + \beta_{5} ) q^{5} + ( \beta_{3} + \beta_{6} ) q^{7} +O(q^{10})\) \( q + 2 \beta_{4} q^{4} + ( \beta_{1} - \beta_{2} + \beta_{5} ) q^{5} + ( \beta_{3} + \beta_{6} ) q^{7} -2 \beta_{2} q^{11} + \beta_{3} q^{13} + ( -4 + 4 \beta_{4} ) q^{16} + ( -2 \beta_{1} + \beta_{2} ) q^{17} + ( 1 + \beta_{4} ) q^{19} + ( -2 \beta_{2} + 2 \beta_{5} - 2 \beta_{7} ) q^{20} + ( 2 \beta_{1} - \beta_{2} - 2 \beta_{5} + \beta_{7} ) q^{23} + ( -2 \beta_{3} + 2 \beta_{4} - \beta_{6} ) q^{25} + 2 \beta_{3} q^{28} + \beta_{7} q^{29} + ( 6 - 3 \beta_{4} ) q^{31} + ( \beta_{1} + 3 \beta_{2} + 2 \beta_{5} - \beta_{7} ) q^{35} + ( -\beta_{3} + \beta_{6} ) q^{37} + ( 4 \beta_{2} + 2 \beta_{7} ) q^{41} + ( \beta_{3} + 2 \beta_{6} ) q^{43} + ( -4 \beta_{2} - 4 \beta_{7} ) q^{44} + ( 2 \beta_{1} - \beta_{2} + 2 \beta_{5} - \beta_{7} ) q^{47} -7 \beta_{4} q^{49} -2 \beta_{6} q^{52} + ( -4 \beta_{1} + 2 \beta_{2} - 8 \beta_{5} + 4 \beta_{7} ) q^{53} + ( 2 - 2 \beta_{3} - 4 \beta_{4} ) q^{55} + ( 3 \beta_{2} + 6 \beta_{7} ) q^{59} + ( -6 - 6 \beta_{4} ) q^{61} -8 q^{64} + ( -\beta_{1} + 4 \beta_{2} + \beta_{5} + 3 \beta_{7} ) q^{65} + ( -6 \beta_{3} - 3 \beta_{6} ) q^{67} + ( -4 \beta_{1} + 2 \beta_{2} - 4 \beta_{5} + 2 \beta_{7} ) q^{68} -8 \beta_{7} q^{71} -5 \beta_{6} q^{73} + ( -2 + 4 \beta_{4} ) q^{76} + ( 4 \beta_{5} - 2 \beta_{7} ) q^{77} + ( 7 - 7 \beta_{4} ) q^{79} + ( -4 \beta_{1} - 4 \beta_{7} ) q^{80} + ( -8 \beta_{5} + 4 \beta_{7} ) q^{83} + ( -7 + \beta_{3} + 2 \beta_{6} ) q^{85} + ( 7 \beta_{2} - 7 \beta_{7} ) q^{89} + ( 7 - 7 \beta_{4} ) q^{91} + ( 8 \beta_{1} - 4 \beta_{2} + 4 \beta_{5} - 2 \beta_{7} ) q^{92} + ( \beta_{1} - 2 \beta_{2} + 2 \beta_{5} - \beta_{7} ) q^{95} -2 \beta_{3} q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + 8q^{4} + O(q^{10}) \) \( 8q + 8q^{4} - 16q^{16} + 12q^{19} + 8q^{25} + 36q^{31} - 28q^{49} - 72q^{61} - 64q^{64} + 28q^{79} - 56q^{85} + 28q^{91} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - 8 x^{6} + 55 x^{4} - 72 x^{2} + 81\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( -\nu^{7} - 203 \nu \)\()/165\)
\(\beta_{3}\)\(=\)\((\)\( -\nu^{6} - 148 \)\()/55\)
\(\beta_{4}\)\(=\)\((\)\( -8 \nu^{6} + 55 \nu^{4} - 440 \nu^{2} + 576 \)\()/495\)
\(\beta_{5}\)\(=\)\((\)\( -8 \nu^{7} + 55 \nu^{5} - 440 \nu^{3} + 81 \nu \)\()/495\)
\(\beta_{6}\)\(=\)\((\)\( -23 \nu^{6} + 220 \nu^{4} - 1265 \nu^{2} + 1656 \)\()/495\)
\(\beta_{7}\)\(=\)\((\)\( 8 \nu^{7} - 55 \nu^{5} + 341 \nu^{3} - 81 \nu \)\()/297\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{6} - 4 \beta_{4} + \beta_{3} + 4\)
\(\nu^{3}\)\(=\)\(-3 \beta_{7} - 5 \beta_{5}\)
\(\nu^{4}\)\(=\)\(8 \beta_{6} - 23 \beta_{4}\)
\(\nu^{5}\)\(=\)\(-24 \beta_{7} - 31 \beta_{5} - 24 \beta_{2} - 31 \beta_{1}\)
\(\nu^{6}\)\(=\)\(-55 \beta_{3} - 148\)
\(\nu^{7}\)\(=\)\(-165 \beta_{2} - 203 \beta_{1}\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/315\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(136\) \(281\)
\(\chi(n)\) \(-1\) \(1 - \beta_{4}\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
89.1
−1.00781 + 0.581861i
−2.23256 + 1.28897i
2.23256 1.28897i
1.00781 0.581861i
−1.00781 0.581861i
−2.23256 1.28897i
2.23256 + 1.28897i
1.00781 + 0.581861i
0 0 1.00000 + 1.73205i −2.23256 + 0.125246i 0 −1.32288 + 2.29129i 0 0 0
89.2 0 0 1.00000 + 1.73205i −1.00781 1.99607i 0 1.32288 2.29129i 0 0 0
89.3 0 0 1.00000 + 1.73205i 1.00781 + 1.99607i 0 1.32288 2.29129i 0 0 0
89.4 0 0 1.00000 + 1.73205i 2.23256 0.125246i 0 −1.32288 + 2.29129i 0 0 0
269.1 0 0 1.00000 1.73205i −2.23256 0.125246i 0 −1.32288 2.29129i 0 0 0
269.2 0 0 1.00000 1.73205i −1.00781 + 1.99607i 0 1.32288 + 2.29129i 0 0 0
269.3 0 0 1.00000 1.73205i 1.00781 1.99607i 0 1.32288 + 2.29129i 0 0 0
269.4 0 0 1.00000 1.73205i 2.23256 + 0.125246i 0 −1.32288 2.29129i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 269.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.b even 2 1 inner
7.d odd 6 1 inner
15.d odd 2 1 inner
21.g even 6 1 inner
35.i odd 6 1 inner
105.p even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 315.2.bb.a 8
3.b odd 2 1 inner 315.2.bb.a 8
5.b even 2 1 inner 315.2.bb.a 8
5.c odd 4 2 1575.2.bk.d 8
7.c even 3 1 2205.2.g.a 8
7.d odd 6 1 inner 315.2.bb.a 8
7.d odd 6 1 2205.2.g.a 8
15.d odd 2 1 inner 315.2.bb.a 8
15.e even 4 2 1575.2.bk.d 8
21.g even 6 1 inner 315.2.bb.a 8
21.g even 6 1 2205.2.g.a 8
21.h odd 6 1 2205.2.g.a 8
35.i odd 6 1 inner 315.2.bb.a 8
35.i odd 6 1 2205.2.g.a 8
35.j even 6 1 2205.2.g.a 8
35.k even 12 2 1575.2.bk.d 8
105.o odd 6 1 2205.2.g.a 8
105.p even 6 1 inner 315.2.bb.a 8
105.p even 6 1 2205.2.g.a 8
105.w odd 12 2 1575.2.bk.d 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
315.2.bb.a 8 1.a even 1 1 trivial
315.2.bb.a 8 3.b odd 2 1 inner
315.2.bb.a 8 5.b even 2 1 inner
315.2.bb.a 8 7.d odd 6 1 inner
315.2.bb.a 8 15.d odd 2 1 inner
315.2.bb.a 8 21.g even 6 1 inner
315.2.bb.a 8 35.i odd 6 1 inner
315.2.bb.a 8 105.p even 6 1 inner
1575.2.bk.d 8 5.c odd 4 2
1575.2.bk.d 8 15.e even 4 2
1575.2.bk.d 8 35.k even 12 2
1575.2.bk.d 8 105.w odd 12 2
2205.2.g.a 8 7.c even 3 1
2205.2.g.a 8 7.d odd 6 1
2205.2.g.a 8 21.g even 6 1
2205.2.g.a 8 21.h odd 6 1
2205.2.g.a 8 35.i odd 6 1
2205.2.g.a 8 35.j even 6 1
2205.2.g.a 8 105.o odd 6 1
2205.2.g.a 8 105.p even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} \) acting on \(S_{2}^{\mathrm{new}}(315, [\chi])\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - 2 T^{2} + 4 T^{4} )^{4} \)
$3$ 1
$5$ \( 1 - 4 T^{2} - 9 T^{4} - 100 T^{6} + 625 T^{8} \)
$7$ \( ( 1 + 7 T^{2} + 49 T^{4} )^{2} \)
$11$ \( ( 1 - 6 T + 25 T^{2} - 66 T^{3} + 121 T^{4} )^{2}( 1 + 6 T + 25 T^{2} + 66 T^{3} + 121 T^{4} )^{2} \)
$13$ \( ( 1 + 19 T^{2} + 169 T^{4} )^{4} \)
$17$ \( ( 1 + 20 T^{2} + 111 T^{4} + 5780 T^{6} + 83521 T^{8} )^{2} \)
$19$ \( ( 1 - 3 T + 22 T^{2} - 57 T^{3} + 361 T^{4} )^{4} \)
$23$ \( ( 1 - 4 T^{2} - 513 T^{4} - 2116 T^{6} + 279841 T^{8} )^{2} \)
$29$ \( ( 1 - 56 T^{2} + 841 T^{4} )^{4} \)
$31$ \( ( 1 - 9 T + 58 T^{2} - 279 T^{3} + 961 T^{4} )^{4} \)
$37$ \( ( 1 + 53 T^{2} + 1440 T^{4} + 72557 T^{6} + 1874161 T^{8} )^{2} \)
$41$ \( ( 1 + 58 T^{2} + 1681 T^{4} )^{4} \)
$43$ \( ( 1 - 65 T^{2} + 1849 T^{4} )^{4} \)
$47$ \( ( 1 + 80 T^{2} + 4191 T^{4} + 176720 T^{6} + 4879681 T^{8} )^{2} \)
$53$ \( ( 1 + 62 T^{2} + 1035 T^{4} + 174158 T^{6} + 7890481 T^{8} )^{2} \)
$59$ \( ( 1 - 64 T^{2} + 615 T^{4} - 222784 T^{6} + 12117361 T^{8} )^{2} \)
$61$ \( ( 1 + 18 T + 169 T^{2} + 1098 T^{3} + 3721 T^{4} )^{4} \)
$67$ \( ( 1 - 55 T^{2} - 1464 T^{4} - 246895 T^{6} + 20151121 T^{8} )^{2} \)
$71$ \( ( 1 - 14 T^{2} + 5041 T^{4} )^{4} \)
$73$ \( ( 1 + 29 T^{2} - 4488 T^{4} + 154541 T^{6} + 28398241 T^{8} )^{2} \)
$79$ \( ( 1 - 7 T - 30 T^{2} - 553 T^{3} + 6241 T^{4} )^{4} \)
$83$ \( ( 1 + 58 T^{2} + 6889 T^{4} )^{4} \)
$89$ \( ( 1 + 116 T^{2} + 5535 T^{4} + 918836 T^{6} + 62742241 T^{8} )^{2} \)
$97$ \( ( 1 + 166 T^{2} + 9409 T^{4} )^{4} \)
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