Properties

Label 315.2.a.f
Level 315
Weight 2
Character orbit 315.a
Self dual yes
Analytic conductor 2.515
Analytic rank 0
Dimension 2
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 315 = 3^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 315.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(2.51528766367\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{2}) \)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

\(f(q)\) \(=\) \( q + ( 1 + \beta ) q^{2} + ( 1 + 2 \beta ) q^{4} + q^{5} - q^{7} + ( 3 + \beta ) q^{8} +O(q^{10})\) \( q + ( 1 + \beta ) q^{2} + ( 1 + 2 \beta ) q^{4} + q^{5} - q^{7} + ( 3 + \beta ) q^{8} + ( 1 + \beta ) q^{10} + ( 2 - 2 \beta ) q^{11} + ( -2 - 2 \beta ) q^{13} + ( -1 - \beta ) q^{14} + 3 q^{16} + ( 2 - 4 \beta ) q^{17} + 2 \beta q^{19} + ( 1 + 2 \beta ) q^{20} -2 q^{22} + ( 2 + 4 \beta ) q^{23} + q^{25} + ( -6 - 4 \beta ) q^{26} + ( -1 - 2 \beta ) q^{28} + 8 q^{29} -6 \beta q^{31} + ( -3 + \beta ) q^{32} + ( -6 - 2 \beta ) q^{34} - q^{35} -6 q^{37} + ( 4 + 2 \beta ) q^{38} + ( 3 + \beta ) q^{40} + ( -2 + 4 \beta ) q^{41} + ( -4 - 4 \beta ) q^{43} + ( -6 + 2 \beta ) q^{44} + ( 10 + 6 \beta ) q^{46} -4 q^{47} + q^{49} + ( 1 + \beta ) q^{50} + ( -10 - 6 \beta ) q^{52} + ( 8 + 2 \beta ) q^{53} + ( 2 - 2 \beta ) q^{55} + ( -3 - \beta ) q^{56} + ( 8 + 8 \beta ) q^{58} -4 q^{59} + 6 q^{61} + ( -12 - 6 \beta ) q^{62} + ( -7 - 2 \beta ) q^{64} + ( -2 - 2 \beta ) q^{65} + ( -4 + 8 \beta ) q^{67} -14 q^{68} + ( -1 - \beta ) q^{70} + ( 2 - 6 \beta ) q^{71} + ( 2 + 10 \beta ) q^{73} + ( -6 - 6 \beta ) q^{74} + ( 8 + 2 \beta ) q^{76} + ( -2 + 2 \beta ) q^{77} -4 \beta q^{79} + 3 q^{80} + ( 6 + 2 \beta ) q^{82} -8 q^{83} + ( 2 - 4 \beta ) q^{85} + ( -12 - 8 \beta ) q^{86} + ( 2 - 4 \beta ) q^{88} + ( 6 + 8 \beta ) q^{89} + ( 2 + 2 \beta ) q^{91} + ( 18 + 8 \beta ) q^{92} + ( -4 - 4 \beta ) q^{94} + 2 \beta q^{95} + ( -6 - 2 \beta ) q^{97} + ( 1 + \beta ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{2} + 2q^{4} + 2q^{5} - 2q^{7} + 6q^{8} + O(q^{10}) \) \( 2q + 2q^{2} + 2q^{4} + 2q^{5} - 2q^{7} + 6q^{8} + 2q^{10} + 4q^{11} - 4q^{13} - 2q^{14} + 6q^{16} + 4q^{17} + 2q^{20} - 4q^{22} + 4q^{23} + 2q^{25} - 12q^{26} - 2q^{28} + 16q^{29} - 6q^{32} - 12q^{34} - 2q^{35} - 12q^{37} + 8q^{38} + 6q^{40} - 4q^{41} - 8q^{43} - 12q^{44} + 20q^{46} - 8q^{47} + 2q^{49} + 2q^{50} - 20q^{52} + 16q^{53} + 4q^{55} - 6q^{56} + 16q^{58} - 8q^{59} + 12q^{61} - 24q^{62} - 14q^{64} - 4q^{65} - 8q^{67} - 28q^{68} - 2q^{70} + 4q^{71} + 4q^{73} - 12q^{74} + 16q^{76} - 4q^{77} + 6q^{80} + 12q^{82} - 16q^{83} + 4q^{85} - 24q^{86} + 4q^{88} + 12q^{89} + 4q^{91} + 36q^{92} - 8q^{94} - 12q^{97} + 2q^{98} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−1.41421
1.41421
−0.414214 0 −1.82843 1.00000 0 −1.00000 1.58579 0 −0.414214
1.2 2.41421 0 3.82843 1.00000 0 −1.00000 4.41421 0 2.41421
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 315.2.a.f yes 2
3.b odd 2 1 315.2.a.c 2
4.b odd 2 1 5040.2.a.bx 2
5.b even 2 1 1575.2.a.m 2
5.c odd 4 2 1575.2.d.j 4
7.b odd 2 1 2205.2.a.y 2
12.b even 2 1 5040.2.a.bu 2
15.d odd 2 1 1575.2.a.u 2
15.e even 4 2 1575.2.d.h 4
21.c even 2 1 2205.2.a.p 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
315.2.a.c 2 3.b odd 2 1
315.2.a.f yes 2 1.a even 1 1 trivial
1575.2.a.m 2 5.b even 2 1
1575.2.a.u 2 15.d odd 2 1
1575.2.d.h 4 15.e even 4 2
1575.2.d.j 4 5.c odd 4 2
2205.2.a.p 2 21.c even 2 1
2205.2.a.y 2 7.b odd 2 1
5040.2.a.bu 2 12.b even 2 1
5040.2.a.bx 2 4.b odd 2 1

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)
\(5\) \(-1\)
\(7\) \(1\)

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} - 2 T_{2} - 1 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(315))\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 - 2 T + 3 T^{2} - 4 T^{3} + 4 T^{4} \)
$3$ \( \)
$5$ \( ( 1 - T )^{2} \)
$7$ \( ( 1 + T )^{2} \)
$11$ \( 1 - 4 T + 18 T^{2} - 44 T^{3} + 121 T^{4} \)
$13$ \( 1 + 4 T + 22 T^{2} + 52 T^{3} + 169 T^{4} \)
$17$ \( 1 - 4 T + 6 T^{2} - 68 T^{3} + 289 T^{4} \)
$19$ \( 1 + 30 T^{2} + 361 T^{4} \)
$23$ \( 1 - 4 T + 18 T^{2} - 92 T^{3} + 529 T^{4} \)
$29$ \( ( 1 - 8 T + 29 T^{2} )^{2} \)
$31$ \( 1 - 10 T^{2} + 961 T^{4} \)
$37$ \( ( 1 + 6 T + 37 T^{2} )^{2} \)
$41$ \( 1 + 4 T + 54 T^{2} + 164 T^{3} + 1681 T^{4} \)
$43$ \( 1 + 8 T + 70 T^{2} + 344 T^{3} + 1849 T^{4} \)
$47$ \( ( 1 + 4 T + 47 T^{2} )^{2} \)
$53$ \( 1 - 16 T + 162 T^{2} - 848 T^{3} + 2809 T^{4} \)
$59$ \( ( 1 + 4 T + 59 T^{2} )^{2} \)
$61$ \( ( 1 - 6 T + 61 T^{2} )^{2} \)
$67$ \( 1 + 8 T + 22 T^{2} + 536 T^{3} + 4489 T^{4} \)
$71$ \( 1 - 4 T + 74 T^{2} - 284 T^{3} + 5041 T^{4} \)
$73$ \( 1 - 4 T - 50 T^{2} - 292 T^{3} + 5329 T^{4} \)
$79$ \( 1 + 126 T^{2} + 6241 T^{4} \)
$83$ \( ( 1 + 8 T + 83 T^{2} )^{2} \)
$89$ \( 1 - 12 T + 86 T^{2} - 1068 T^{3} + 7921 T^{4} \)
$97$ \( 1 + 12 T + 222 T^{2} + 1164 T^{3} + 9409 T^{4} \)
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