Properties

Label 315.2.b.a
Level 315
Weight 2
Character orbit 315.b
Analytic conductor 2.515
Analytic rank 0
Dimension 4
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 315 = 3^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 315.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(2.51528766367\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{3})\)
Coefficient ring: \(\Z[a_1, a_2]\)
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 a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{1} q^{2} + \beta_{2} q^{4} - q^{5} + ( 1 + \beta_{1} + \beta_{3} ) q^{7} + \beta_{3} q^{8} +O(q^{10})\) \( q + \beta_{1} q^{2} + \beta_{2} q^{4} - q^{5} + ( 1 + \beta_{1} + \beta_{3} ) q^{7} + \beta_{3} q^{8} -\beta_{1} q^{10} + ( 3 \beta_{1} - \beta_{3} ) q^{11} + ( -4 \beta_{1} + 2 \beta_{3} ) q^{13} + ( -3 + \beta_{1} + \beta_{2} ) q^{14} + ( -1 + 2 \beta_{2} ) q^{16} + 2 \beta_{2} q^{17} + ( 4 \beta_{1} - 2 \beta_{3} ) q^{19} -\beta_{2} q^{20} + ( -5 + 3 \beta_{2} ) q^{22} + ( -\beta_{1} + \beta_{3} ) q^{23} + q^{25} + ( 6 - 4 \beta_{2} ) q^{26} + ( -3 \beta_{1} + \beta_{2} + 3 \beta_{3} ) q^{28} + ( -\beta_{1} + \beta_{3} ) q^{29} + ( 2 \beta_{1} - 4 \beta_{3} ) q^{31} + ( -5 \beta_{1} + 4 \beta_{3} ) q^{32} + ( -4 \beta_{1} + 2 \beta_{3} ) q^{34} + ( -1 - \beta_{1} - \beta_{3} ) q^{35} + ( 2 - 2 \beta_{2} ) q^{37} + ( -6 + 4 \beta_{2} ) q^{38} -\beta_{3} q^{40} -6 \beta_{2} q^{41} + ( 2 - 4 \beta_{2} ) q^{43} + ( -5 \beta_{1} + \beta_{3} ) q^{44} + ( 1 - \beta_{2} ) q^{46} + ( 6 + 2 \beta_{2} ) q^{47} + ( -5 + 2 \beta_{1} + 2 \beta_{3} ) q^{49} + \beta_{1} q^{50} + 6 \beta_{1} q^{52} + ( -\beta_{1} - 5 \beta_{3} ) q^{53} + ( -3 \beta_{1} + \beta_{3} ) q^{55} + ( -3 - \beta_{2} + \beta_{3} ) q^{56} + ( 1 - \beta_{2} ) q^{58} + ( 6 + 2 \beta_{2} ) q^{59} + ( 4 \beta_{1} - 8 \beta_{3} ) q^{61} + 2 \beta_{2} q^{62} + ( 4 - \beta_{2} ) q^{64} + ( 4 \beta_{1} - 2 \beta_{3} ) q^{65} + ( -4 - 4 \beta_{2} ) q^{67} + 6 q^{68} + ( 3 - \beta_{1} - \beta_{2} ) q^{70} + ( -3 \beta_{1} - 7 \beta_{3} ) q^{71} + ( 4 \beta_{1} - 2 \beta_{3} ) q^{73} + ( 6 \beta_{1} - 2 \beta_{3} ) q^{74} -6 \beta_{1} q^{76} + ( -6 + 3 \beta_{1} + 4 \beta_{2} - \beta_{3} ) q^{77} + ( -4 - 4 \beta_{2} ) q^{79} + ( 1 - 2 \beta_{2} ) q^{80} + ( 12 \beta_{1} - 6 \beta_{3} ) q^{82} + ( -6 - 2 \beta_{2} ) q^{83} -2 \beta_{2} q^{85} + ( 10 \beta_{1} - 4 \beta_{3} ) q^{86} + ( -1 + \beta_{2} ) q^{88} + ( 6 + 4 \beta_{2} ) q^{89} + ( 6 - 4 \beta_{1} - 6 \beta_{2} + 2 \beta_{3} ) q^{91} + ( \beta_{1} + \beta_{3} ) q^{92} + ( 2 \beta_{1} + 2 \beta_{3} ) q^{94} + ( -4 \beta_{1} + 2 \beta_{3} ) q^{95} + ( -8 \beta_{1} + 10 \beta_{3} ) q^{97} + ( -6 - 5 \beta_{1} + 2 \beta_{2} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 4q^{5} + 4q^{7} + O(q^{10}) \) \( 4q - 4q^{5} + 4q^{7} - 12q^{14} - 4q^{16} - 20q^{22} + 4q^{25} + 24q^{26} - 4q^{35} + 8q^{37} - 24q^{38} + 8q^{43} + 4q^{46} + 24q^{47} - 20q^{49} - 12q^{56} + 4q^{58} + 24q^{59} + 16q^{64} - 16q^{67} + 24q^{68} + 12q^{70} - 24q^{77} - 16q^{79} + 4q^{80} - 24q^{83} - 4q^{88} + 24q^{89} + 24q^{91} - 24q^{98} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 4 x^{2} + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} + 2 \)
\(\beta_{3}\)\(=\)\( \nu^{3} + 4 \nu \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} - 2\)
\(\nu^{3}\)\(=\)\(\beta_{3} - 4 \beta_{1}\)

Character values

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

\(n\) \(127\) \(136\) \(281\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
251.1
1.93185i
0.517638i
0.517638i
1.93185i
1.93185i 0 −1.73205 −1.00000 0 1.00000 2.44949i 0.517638i 0 1.93185i
251.2 0.517638i 0 1.73205 −1.00000 0 1.00000 2.44949i 1.93185i 0 0.517638i
251.3 0.517638i 0 1.73205 −1.00000 0 1.00000 + 2.44949i 1.93185i 0 0.517638i
251.4 1.93185i 0 −1.73205 −1.00000 0 1.00000 + 2.44949i 0.517638i 0 1.93185i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
21.c even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 315.2.b.a 4
3.b odd 2 1 315.2.b.b yes 4
4.b odd 2 1 5040.2.f.a 4
5.b even 2 1 1575.2.b.b 4
5.c odd 4 2 1575.2.g.c 8
7.b odd 2 1 315.2.b.b yes 4
12.b even 2 1 5040.2.f.c 4
15.d odd 2 1 1575.2.b.c 4
15.e even 4 2 1575.2.g.a 8
21.c even 2 1 inner 315.2.b.a 4
28.d even 2 1 5040.2.f.c 4
35.c odd 2 1 1575.2.b.c 4
35.f even 4 2 1575.2.g.a 8
84.h odd 2 1 5040.2.f.a 4
105.g even 2 1 1575.2.b.b 4
105.k odd 4 2 1575.2.g.c 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
315.2.b.a 4 1.a even 1 1 trivial
315.2.b.a 4 21.c even 2 1 inner
315.2.b.b yes 4 3.b odd 2 1
315.2.b.b yes 4 7.b odd 2 1
1575.2.b.b 4 5.b even 2 1
1575.2.b.b 4 105.g even 2 1
1575.2.b.c 4 15.d odd 2 1
1575.2.b.c 4 35.c odd 2 1
1575.2.g.a 8 15.e even 4 2
1575.2.g.a 8 35.f even 4 2
1575.2.g.c 8 5.c odd 4 2
1575.2.g.c 8 105.k odd 4 2
5040.2.f.a 4 4.b odd 2 1
5040.2.f.a 4 84.h odd 2 1
5040.2.f.c 4 12.b even 2 1
5040.2.f.c 4 28.d even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{47}^{2} - 12 T_{47} + 24 \) acting on \(S_{2}^{\mathrm{new}}(315, [\chi])\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 - 4 T^{2} + 9 T^{4} - 16 T^{6} + 16 T^{8} \)
$3$ \( \)
$5$ \( ( 1 + T )^{4} \)
$7$ \( ( 1 - 2 T + 7 T^{2} )^{2} \)
$11$ \( 1 - 16 T^{2} + 114 T^{4} - 1936 T^{6} + 14641 T^{8} \)
$13$ \( 1 - 4 T^{2} - 90 T^{4} - 676 T^{6} + 28561 T^{8} \)
$17$ \( ( 1 + 22 T^{2} + 289 T^{4} )^{2} \)
$19$ \( 1 - 28 T^{2} + 486 T^{4} - 10108 T^{6} + 130321 T^{8} \)
$23$ \( ( 1 - 44 T^{2} + 529 T^{4} )^{2} \)
$29$ \( ( 1 - 56 T^{2} + 841 T^{4} )^{2} \)
$31$ \( 1 - 76 T^{2} + 2934 T^{4} - 73036 T^{6} + 923521 T^{8} \)
$37$ \( ( 1 - 4 T + 66 T^{2} - 148 T^{3} + 1369 T^{4} )^{2} \)
$41$ \( ( 1 - 26 T^{2} + 1681 T^{4} )^{2} \)
$43$ \( ( 1 - 4 T + 42 T^{2} - 172 T^{3} + 1849 T^{4} )^{2} \)
$47$ \( ( 1 - 12 T + 118 T^{2} - 564 T^{3} + 2209 T^{4} )^{2} \)
$53$ \( 1 - 88 T^{2} + 5826 T^{4} - 247192 T^{6} + 7890481 T^{8} \)
$59$ \( ( 1 - 12 T + 142 T^{2} - 708 T^{3} + 3481 T^{4} )^{2} \)
$61$ \( 1 - 52 T^{2} + 1206 T^{4} - 193492 T^{6} + 13845841 T^{8} \)
$67$ \( ( 1 + 8 T + 102 T^{2} + 536 T^{3} + 4489 T^{4} )^{2} \)
$71$ \( 1 + 32 T^{2} + 5538 T^{4} + 161312 T^{6} + 25411681 T^{8} \)
$73$ \( 1 - 244 T^{2} + 25110 T^{4} - 1300276 T^{6} + 28398241 T^{8} \)
$79$ \( ( 1 + 8 T + 126 T^{2} + 632 T^{3} + 6241 T^{4} )^{2} \)
$83$ \( ( 1 + 12 T + 190 T^{2} + 996 T^{3} + 6889 T^{4} )^{2} \)
$89$ \( ( 1 - 12 T + 166 T^{2} - 1068 T^{3} + 7921 T^{4} )^{2} \)
$97$ \( 1 - 52 T^{2} + 15606 T^{4} - 489268 T^{6} + 88529281 T^{8} \)
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