Properties

Label 315.2.p.d
Level 315
Weight 2
Character orbit 315.p
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.p (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(2.51528766367\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: 8.0.40960000.1
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{1} q^{2} -\beta_{3} q^{4} + \beta_{5} q^{5} + ( -1 - \beta_{3} - \beta_{7} ) q^{7} -3 \beta_{6} q^{8} +O(q^{10})\) \( q + \beta_{1} q^{2} -\beta_{3} q^{4} + \beta_{5} q^{5} + ( -1 - \beta_{3} - \beta_{7} ) q^{7} -3 \beta_{6} q^{8} + \beta_{7} q^{10} + ( 3 \beta_{1} - 3 \beta_{6} ) q^{11} + 2 \beta_{2} q^{13} + ( -\beta_{1} + \beta_{4} - \beta_{6} ) q^{14} + q^{16} + ( -\beta_{4} + \beta_{5} ) q^{17} + ( -\beta_{2} + \beta_{7} ) q^{19} + \beta_{4} q^{20} + ( 3 + 3 \beta_{3} ) q^{22} -2 \beta_{6} q^{23} -5 q^{25} + 2 \beta_{5} q^{26} + ( -1 - \beta_{2} + \beta_{3} ) q^{28} + ( 4 \beta_{1} + 4 \beta_{6} ) q^{29} + ( -3 \beta_{2} - 3 \beta_{7} ) q^{31} -5 \beta_{1} q^{32} + ( -\beta_{2} + \beta_{7} ) q^{34} + ( 5 \beta_{1} + \beta_{4} - \beta_{5} ) q^{35} + ( 1 + \beta_{3} ) q^{37} + ( -\beta_{4} - \beta_{5} ) q^{38} + 3 \beta_{2} q^{40} -4 \beta_{5} q^{41} + ( -2 + 2 \beta_{3} ) q^{43} + ( -3 \beta_{1} - 3 \beta_{6} ) q^{44} + 2 q^{46} + ( -4 \beta_{4} + 4 \beta_{5} ) q^{47} + ( -2 \beta_{2} - 3 \beta_{3} + 2 \beta_{7} ) q^{49} -5 \beta_{1} q^{50} -2 \beta_{7} q^{52} + 10 \beta_{6} q^{53} + ( 3 \beta_{2} + 3 \beta_{7} ) q^{55} + ( -3 \beta_{1} - 3 \beta_{5} + 3 \beta_{6} ) q^{56} + ( -4 + 4 \beta_{3} ) q^{58} + ( 3 \beta_{4} - 3 \beta_{5} ) q^{62} -7 \beta_{3} q^{64} + 10 \beta_{6} q^{65} + ( 4 + 4 \beta_{3} ) q^{67} + ( \beta_{4} + \beta_{5} ) q^{68} + ( \beta_{2} + 5 \beta_{3} - \beta_{7} ) q^{70} + ( -3 \beta_{1} + 3 \beta_{6} ) q^{71} + 2 \beta_{2} q^{73} + ( \beta_{1} + \beta_{6} ) q^{74} + ( \beta_{2} + \beta_{7} ) q^{76} + ( -6 \beta_{1} + 3 \beta_{4} - 3 \beta_{5} ) q^{77} -12 \beta_{3} q^{79} + \beta_{5} q^{80} -4 \beta_{7} q^{82} + ( -2 \beta_{4} - 2 \beta_{5} ) q^{83} + ( -5 - 5 \beta_{3} ) q^{85} + ( -2 \beta_{1} + 2 \beta_{6} ) q^{86} + ( 9 - 9 \beta_{3} ) q^{88} -6 \beta_{4} q^{89} + ( 10 - 2 \beta_{2} - 2 \beta_{7} ) q^{91} -2 \beta_{1} q^{92} + ( -4 \beta_{2} + 4 \beta_{7} ) q^{94} + ( -5 \beta_{1} - 5 \beta_{6} ) q^{95} -2 \beta_{7} q^{97} + ( -2 \beta_{4} - 2 \beta_{5} - 3 \beta_{6} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - 8q^{7} + O(q^{10}) \) \( 8q - 8q^{7} + 8q^{16} + 24q^{22} - 40q^{25} - 8q^{28} + 8q^{37} - 16q^{43} + 16q^{46} - 32q^{58} + 32q^{67} - 40q^{85} + 72q^{88} + 80q^{91} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} + 7 x^{4} + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{5} + 5 \nu \)\()/3\)
\(\beta_{2}\)\(=\)\((\)\( \nu^{5} + 11 \nu \)\()/3\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{6} + 8 \nu^{2} \)\()/3\)
\(\beta_{4}\)\(=\)\((\)\( 2 \nu^{4} + 7 \)\()/3\)
\(\beta_{5}\)\(=\)\( \nu^{6} + 6 \nu^{2} \)
\(\beta_{6}\)\(=\)\((\)\( 2 \nu^{7} + 13 \nu^{3} \)\()/3\)
\(\beta_{7}\)\(=\)\((\)\( 4 \nu^{7} + 29 \nu^{3} \)\()/3\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{2} - \beta_{1}\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(-\beta_{5} + 3 \beta_{3}\)\()/2\)
\(\nu^{3}\)\(=\)\(\beta_{7} - 2 \beta_{6}\)
\(\nu^{4}\)\(=\)\((\)\(3 \beta_{4} - 7\)\()/2\)
\(\nu^{5}\)\(=\)\((\)\(-5 \beta_{2} + 11 \beta_{1}\)\()/2\)
\(\nu^{6}\)\(=\)\(4 \beta_{5} - 9 \beta_{3}\)
\(\nu^{7}\)\(=\)\((\)\(-13 \beta_{7} + 29 \beta_{6}\)\()/2\)

Character values

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

\(n\) \(127\) \(136\) \(281\)
\(\chi(n)\) \(\beta_{3}\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
118.1
−0.437016 + 0.437016i
1.14412 1.14412i
0.437016 0.437016i
−1.14412 + 1.14412i
1.14412 + 1.14412i
−0.437016 0.437016i
−1.14412 1.14412i
0.437016 + 0.437016i
−0.707107 + 0.707107i 0 1.00000i 2.23607i 0 −2.58114 0.581139i −2.12132 2.12132i 0 1.58114 + 1.58114i
118.2 −0.707107 + 0.707107i 0 1.00000i 2.23607i 0 0.581139 + 2.58114i −2.12132 2.12132i 0 −1.58114 1.58114i
118.3 0.707107 0.707107i 0 1.00000i 2.23607i 0 0.581139 + 2.58114i 2.12132 + 2.12132i 0 −1.58114 1.58114i
118.4 0.707107 0.707107i 0 1.00000i 2.23607i 0 −2.58114 0.581139i 2.12132 + 2.12132i 0 1.58114 + 1.58114i
307.1 −0.707107 0.707107i 0 1.00000i 2.23607i 0 0.581139 2.58114i −2.12132 + 2.12132i 0 −1.58114 + 1.58114i
307.2 −0.707107 0.707107i 0 1.00000i 2.23607i 0 −2.58114 + 0.581139i −2.12132 + 2.12132i 0 1.58114 1.58114i
307.3 0.707107 + 0.707107i 0 1.00000i 2.23607i 0 −2.58114 + 0.581139i 2.12132 2.12132i 0 1.58114 1.58114i
307.4 0.707107 + 0.707107i 0 1.00000i 2.23607i 0 0.581139 2.58114i 2.12132 2.12132i 0 −1.58114 + 1.58114i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 307.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.c odd 4 1 inner
7.b odd 2 1 inner
15.e even 4 1 inner
21.c even 2 1 inner
35.f even 4 1 inner
105.k odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 315.2.p.d 8
3.b odd 2 1 inner 315.2.p.d 8
5.c odd 4 1 inner 315.2.p.d 8
7.b odd 2 1 inner 315.2.p.d 8
15.e even 4 1 inner 315.2.p.d 8
21.c even 2 1 inner 315.2.p.d 8
35.f even 4 1 inner 315.2.p.d 8
105.k odd 4 1 inner 315.2.p.d 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
315.2.p.d 8 1.a even 1 1 trivial
315.2.p.d 8 3.b odd 2 1 inner
315.2.p.d 8 5.c odd 4 1 inner
315.2.p.d 8 7.b odd 2 1 inner
315.2.p.d 8 15.e even 4 1 inner
315.2.p.d 8 21.c even 2 1 inner
315.2.p.d 8 35.f even 4 1 inner
315.2.p.d 8 105.k odd 4 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(315, [\chi])\):

\( T_{2}^{4} + 1 \)
\( T_{17}^{4} + 100 \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T^{4} + 16 T^{8} )^{2} \)
$3$ \( \)
$5$ \( ( 1 + 5 T^{2} )^{4} \)
$7$ \( ( 1 + 4 T + 8 T^{2} + 28 T^{3} + 49 T^{4} )^{2} \)
$11$ \( ( 1 + 4 T^{2} + 121 T^{4} )^{4} \)
$13$ \( ( 1 - 8 T + 32 T^{2} - 104 T^{3} + 169 T^{4} )^{2}( 1 + 8 T + 32 T^{2} + 104 T^{3} + 169 T^{4} )^{2} \)
$17$ \( ( 1 - 2 T^{4} + 83521 T^{8} )^{2} \)
$19$ \( ( 1 + 28 T^{2} + 361 T^{4} )^{4} \)
$23$ \( ( 1 + 706 T^{4} + 279841 T^{8} )^{2} \)
$29$ \( ( 1 - 26 T^{2} + 841 T^{4} )^{4} \)
$31$ \( ( 1 + 28 T^{2} + 961 T^{4} )^{4} \)
$37$ \( ( 1 - 2 T + 2 T^{2} - 74 T^{3} + 1369 T^{4} )^{4} \)
$41$ \( ( 1 - 2 T^{2} + 1681 T^{4} )^{4} \)
$43$ \( ( 1 + 4 T + 8 T^{2} + 172 T^{3} + 1849 T^{4} )^{4} \)
$47$ \( ( 1 - 62 T^{4} + 4879681 T^{8} )^{2} \)
$53$ \( ( 1 - 5582 T^{4} + 7890481 T^{8} )^{2} \)
$59$ \( ( 1 + 59 T^{2} )^{8} \)
$61$ \( ( 1 - 61 T^{2} )^{8} \)
$67$ \( ( 1 - 8 T + 32 T^{2} - 536 T^{3} + 4489 T^{4} )^{4} \)
$71$ \( ( 1 + 124 T^{2} + 5041 T^{4} )^{4} \)
$73$ \( ( 1 + 5218 T^{4} + 28398241 T^{8} )^{2} \)
$79$ \( ( 1 - 14 T^{2} + 6241 T^{4} )^{4} \)
$83$ \( ( 1 + 2098 T^{4} + 47458321 T^{8} )^{2} \)
$89$ \( ( 1 - 2 T^{2} + 7921 T^{4} )^{4} \)
$97$ \( ( 1 + 11458 T^{4} + 88529281 T^{8} )^{2} \)
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