Properties

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

Related objects

Downloads

Learn more about

Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(2.51528766367\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 105)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -2 \zeta_{6} q^{2} + ( -2 + 2 \zeta_{6} ) q^{4} + \zeta_{6} q^{5} + ( -2 + 3 \zeta_{6} ) q^{7} +O(q^{10})\) \( q -2 \zeta_{6} q^{2} + ( -2 + 2 \zeta_{6} ) q^{4} + \zeta_{6} q^{5} + ( -2 + 3 \zeta_{6} ) q^{7} + ( 2 - 2 \zeta_{6} ) q^{10} + ( -6 + 6 \zeta_{6} ) q^{11} -3 q^{13} + ( 6 - 2 \zeta_{6} ) q^{14} + 4 \zeta_{6} q^{16} + ( -4 + 4 \zeta_{6} ) q^{17} -\zeta_{6} q^{19} -2 q^{20} + 12 q^{22} -4 \zeta_{6} q^{23} + ( -1 + \zeta_{6} ) q^{25} + 6 \zeta_{6} q^{26} + ( -2 - 4 \zeta_{6} ) q^{28} + 8 q^{29} + ( -1 + \zeta_{6} ) q^{31} + ( 8 - 8 \zeta_{6} ) q^{32} + 8 q^{34} + ( -3 + \zeta_{6} ) q^{35} -7 \zeta_{6} q^{37} + ( -2 + 2 \zeta_{6} ) q^{38} + 6 q^{41} + q^{43} -12 \zeta_{6} q^{44} + ( -8 + 8 \zeta_{6} ) q^{46} + 2 \zeta_{6} q^{47} + ( -5 - 3 \zeta_{6} ) q^{49} + 2 q^{50} + ( 6 - 6 \zeta_{6} ) q^{52} + ( 4 - 4 \zeta_{6} ) q^{53} -6 q^{55} -16 \zeta_{6} q^{58} + ( -8 + 8 \zeta_{6} ) q^{59} + 14 \zeta_{6} q^{61} + 2 q^{62} -8 q^{64} -3 \zeta_{6} q^{65} + ( -7 + 7 \zeta_{6} ) q^{67} -8 \zeta_{6} q^{68} + ( 2 + 4 \zeta_{6} ) q^{70} -6 q^{71} + ( -1 + \zeta_{6} ) q^{73} + ( -14 + 14 \zeta_{6} ) q^{74} + 2 q^{76} + ( -6 - 12 \zeta_{6} ) q^{77} + \zeta_{6} q^{79} + ( -4 + 4 \zeta_{6} ) q^{80} -12 \zeta_{6} q^{82} -2 q^{83} -4 q^{85} -2 \zeta_{6} q^{86} -12 \zeta_{6} q^{89} + ( 6 - 9 \zeta_{6} ) q^{91} + 8 q^{92} + ( 4 - 4 \zeta_{6} ) q^{94} + ( 1 - \zeta_{6} ) q^{95} -6 q^{97} + ( -6 + 16 \zeta_{6} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{2} - 2q^{4} + q^{5} - q^{7} + O(q^{10}) \) \( 2q - 2q^{2} - 2q^{4} + q^{5} - q^{7} + 2q^{10} - 6q^{11} - 6q^{13} + 10q^{14} + 4q^{16} - 4q^{17} - q^{19} - 4q^{20} + 24q^{22} - 4q^{23} - q^{25} + 6q^{26} - 8q^{28} + 16q^{29} - q^{31} + 8q^{32} + 16q^{34} - 5q^{35} - 7q^{37} - 2q^{38} + 12q^{41} + 2q^{43} - 12q^{44} - 8q^{46} + 2q^{47} - 13q^{49} + 4q^{50} + 6q^{52} + 4q^{53} - 12q^{55} - 16q^{58} - 8q^{59} + 14q^{61} + 4q^{62} - 16q^{64} - 3q^{65} - 7q^{67} - 8q^{68} + 8q^{70} - 12q^{71} - q^{73} - 14q^{74} + 4q^{76} - 24q^{77} + q^{79} - 4q^{80} - 12q^{82} - 4q^{83} - 8q^{85} - 2q^{86} - 12q^{89} + 3q^{91} + 16q^{92} + 4q^{94} + q^{95} - 12q^{97} + 4q^{98} + O(q^{100}) \)

Character values

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

\(n\) \(127\) \(136\) \(281\)
\(\chi(n)\) \(1\) \(-\zeta_{6}\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
46.1
0.500000 + 0.866025i
0.500000 0.866025i
−1.00000 1.73205i 0 −1.00000 + 1.73205i 0.500000 + 0.866025i 0 −0.500000 + 2.59808i 0 0 1.00000 1.73205i
226.1 −1.00000 + 1.73205i 0 −1.00000 1.73205i 0.500000 0.866025i 0 −0.500000 2.59808i 0 0 1.00000 + 1.73205i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 315.2.j.a 2
3.b odd 2 1 105.2.i.b 2
7.c even 3 1 inner 315.2.j.a 2
7.c even 3 1 2205.2.a.k 1
7.d odd 6 1 2205.2.a.m 1
12.b even 2 1 1680.2.bg.l 2
15.d odd 2 1 525.2.i.a 2
15.e even 4 2 525.2.r.d 4
21.c even 2 1 735.2.i.f 2
21.g even 6 1 735.2.a.a 1
21.g even 6 1 735.2.i.f 2
21.h odd 6 1 105.2.i.b 2
21.h odd 6 1 735.2.a.b 1
84.n even 6 1 1680.2.bg.l 2
105.o odd 6 1 525.2.i.a 2
105.o odd 6 1 3675.2.a.o 1
105.p even 6 1 3675.2.a.p 1
105.x even 12 2 525.2.r.d 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.2.i.b 2 3.b odd 2 1
105.2.i.b 2 21.h odd 6 1
315.2.j.a 2 1.a even 1 1 trivial
315.2.j.a 2 7.c even 3 1 inner
525.2.i.a 2 15.d odd 2 1
525.2.i.a 2 105.o odd 6 1
525.2.r.d 4 15.e even 4 2
525.2.r.d 4 105.x even 12 2
735.2.a.a 1 21.g even 6 1
735.2.a.b 1 21.h odd 6 1
735.2.i.f 2 21.c even 2 1
735.2.i.f 2 21.g even 6 1
1680.2.bg.l 2 12.b even 2 1
1680.2.bg.l 2 84.n even 6 1
2205.2.a.k 1 7.c even 3 1
2205.2.a.m 1 7.d odd 6 1
3675.2.a.o 1 105.o odd 6 1
3675.2.a.p 1 105.p even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} + 2 T_{2} + 4 \) acting on \(S_{2}^{\mathrm{new}}(315, [\chi])\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 + 2 T + 2 T^{2} + 4 T^{3} + 4 T^{4} \)
$3$ \( \)
$5$ \( 1 - T + T^{2} \)
$7$ \( 1 + T + 7 T^{2} \)
$11$ \( 1 + 6 T + 25 T^{2} + 66 T^{3} + 121 T^{4} \)
$13$ \( ( 1 + 3 T + 13 T^{2} )^{2} \)
$17$ \( 1 + 4 T - T^{2} + 68 T^{3} + 289 T^{4} \)
$19$ \( ( 1 - 7 T + 19 T^{2} )( 1 + 8 T + 19 T^{2} ) \)
$23$ \( 1 + 4 T - 7 T^{2} + 92 T^{3} + 529 T^{4} \)
$29$ \( ( 1 - 8 T + 29 T^{2} )^{2} \)
$31$ \( 1 + T - 30 T^{2} + 31 T^{3} + 961 T^{4} \)
$37$ \( 1 + 7 T + 12 T^{2} + 259 T^{3} + 1369 T^{4} \)
$41$ \( ( 1 - 6 T + 41 T^{2} )^{2} \)
$43$ \( ( 1 - T + 43 T^{2} )^{2} \)
$47$ \( 1 - 2 T - 43 T^{2} - 94 T^{3} + 2209 T^{4} \)
$53$ \( 1 - 4 T - 37 T^{2} - 212 T^{3} + 2809 T^{4} \)
$59$ \( 1 + 8 T + 5 T^{2} + 472 T^{3} + 3481 T^{4} \)
$61$ \( ( 1 - 13 T + 61 T^{2} )( 1 - T + 61 T^{2} ) \)
$67$ \( 1 + 7 T - 18 T^{2} + 469 T^{3} + 4489 T^{4} \)
$71$ \( ( 1 + 6 T + 71 T^{2} )^{2} \)
$73$ \( 1 + T - 72 T^{2} + 73 T^{3} + 5329 T^{4} \)
$79$ \( 1 - T - 78 T^{2} - 79 T^{3} + 6241 T^{4} \)
$83$ \( ( 1 + 2 T + 83 T^{2} )^{2} \)
$89$ \( 1 + 12 T + 55 T^{2} + 1068 T^{3} + 7921 T^{4} \)
$97$ \( ( 1 + 6 T + 97 T^{2} )^{2} \)
show more
show less