Properties

Label 315.2.a.e
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: \(\Q(\sqrt{17}) \)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 35)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{17})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{2} + ( 2 + \beta ) q^{4} - q^{5} - q^{7} + ( 4 + \beta ) q^{8} +O(q^{10})\) \( q + \beta q^{2} + ( 2 + \beta ) q^{4} - q^{5} - q^{7} + ( 4 + \beta ) q^{8} -\beta q^{10} + ( -1 + \beta ) q^{11} + ( 3 - \beta ) q^{13} -\beta q^{14} + 3 \beta q^{16} + ( 3 - \beta ) q^{17} + ( -2 - 2 \beta ) q^{19} + ( -2 - \beta ) q^{20} + 4 q^{22} + ( 2 - 2 \beta ) q^{23} + q^{25} + ( -4 + 2 \beta ) q^{26} + ( -2 - \beta ) q^{28} + ( 1 - 3 \beta ) q^{29} + ( 4 + \beta ) q^{32} + ( -4 + 2 \beta ) q^{34} + q^{35} + 6 q^{37} + ( -8 - 4 \beta ) q^{38} + ( -4 - \beta ) q^{40} -2 \beta q^{41} + ( 6 - 2 \beta ) q^{43} + ( 2 + 2 \beta ) q^{44} -8 q^{46} + ( 1 + 3 \beta ) q^{47} + q^{49} + \beta q^{50} + 2 q^{52} + 2 \beta q^{53} + ( 1 - \beta ) q^{55} + ( -4 - \beta ) q^{56} + ( -12 - 2 \beta ) q^{58} + 4 q^{59} + 6 \beta q^{61} + ( 4 - \beta ) q^{64} + ( -3 + \beta ) q^{65} + 4 \beta q^{67} + 2 q^{68} + \beta q^{70} -8 q^{71} + ( -2 - 4 \beta ) q^{73} + 6 \beta q^{74} + ( -12 - 8 \beta ) q^{76} + ( 1 - \beta ) q^{77} + ( -5 + \beta ) q^{79} -3 \beta q^{80} + ( -8 - 2 \beta ) q^{82} -4 q^{83} + ( -3 + \beta ) q^{85} + ( -8 + 4 \beta ) q^{86} + 4 \beta q^{88} + ( -4 + 2 \beta ) q^{89} + ( -3 + \beta ) q^{91} + ( -4 - 4 \beta ) q^{92} + ( 12 + 4 \beta ) q^{94} + ( 2 + 2 \beta ) q^{95} + ( -7 + 5 \beta ) q^{97} + \beta q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + q^{2} + 5q^{4} - 2q^{5} - 2q^{7} + 9q^{8} + O(q^{10}) \) \( 2q + q^{2} + 5q^{4} - 2q^{5} - 2q^{7} + 9q^{8} - q^{10} - q^{11} + 5q^{13} - q^{14} + 3q^{16} + 5q^{17} - 6q^{19} - 5q^{20} + 8q^{22} + 2q^{23} + 2q^{25} - 6q^{26} - 5q^{28} - q^{29} + 9q^{32} - 6q^{34} + 2q^{35} + 12q^{37} - 20q^{38} - 9q^{40} - 2q^{41} + 10q^{43} + 6q^{44} - 16q^{46} + 5q^{47} + 2q^{49} + q^{50} + 4q^{52} + 2q^{53} + q^{55} - 9q^{56} - 26q^{58} + 8q^{59} + 6q^{61} + 7q^{64} - 5q^{65} + 4q^{67} + 4q^{68} + q^{70} - 16q^{71} - 8q^{73} + 6q^{74} - 32q^{76} + q^{77} - 9q^{79} - 3q^{80} - 18q^{82} - 8q^{83} - 5q^{85} - 12q^{86} + 4q^{88} - 6q^{89} - 5q^{91} - 12q^{92} + 28q^{94} + 6q^{95} - 9q^{97} + q^{98} + O(q^{100}) \)

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
−1.56155
2.56155
−1.56155 0 0.438447 −1.00000 0 −1.00000 2.43845 0 1.56155
1.2 2.56155 0 4.56155 −1.00000 0 −1.00000 6.56155 0 −2.56155
\(n\): 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.e 2
3.b odd 2 1 35.2.a.b 2
4.b odd 2 1 5040.2.a.bt 2
5.b even 2 1 1575.2.a.p 2
5.c odd 4 2 1575.2.d.e 4
7.b odd 2 1 2205.2.a.x 2
12.b even 2 1 560.2.a.i 2
15.d odd 2 1 175.2.a.f 2
15.e even 4 2 175.2.b.b 4
21.c even 2 1 245.2.a.d 2
21.g even 6 2 245.2.e.h 4
21.h odd 6 2 245.2.e.i 4
24.f even 2 1 2240.2.a.bd 2
24.h odd 2 1 2240.2.a.bh 2
33.d even 2 1 4235.2.a.m 2
39.d odd 2 1 5915.2.a.l 2
60.h even 2 1 2800.2.a.bi 2
60.l odd 4 2 2800.2.g.t 4
84.h odd 2 1 3920.2.a.bs 2
105.g even 2 1 1225.2.a.s 2
105.k odd 4 2 1225.2.b.f 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.2.a.b 2 3.b odd 2 1
175.2.a.f 2 15.d odd 2 1
175.2.b.b 4 15.e even 4 2
245.2.a.d 2 21.c even 2 1
245.2.e.h 4 21.g even 6 2
245.2.e.i 4 21.h odd 6 2
315.2.a.e 2 1.a even 1 1 trivial
560.2.a.i 2 12.b even 2 1
1225.2.a.s 2 105.g even 2 1
1225.2.b.f 4 105.k odd 4 2
1575.2.a.p 2 5.b even 2 1
1575.2.d.e 4 5.c odd 4 2
2205.2.a.x 2 7.b odd 2 1
2240.2.a.bd 2 24.f even 2 1
2240.2.a.bh 2 24.h odd 2 1
2800.2.a.bi 2 60.h even 2 1
2800.2.g.t 4 60.l odd 4 2
3920.2.a.bs 2 84.h odd 2 1
4235.2.a.m 2 33.d even 2 1
5040.2.a.bt 2 4.b odd 2 1
5915.2.a.l 2 39.d 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 \( T_{2}^{2} - T_{2} - 4 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(315))\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 - T - 2 T^{3} + 4 T^{4} \)
$3$ \( \)
$5$ \( ( 1 + T )^{2} \)
$7$ \( ( 1 + T )^{2} \)
$11$ \( 1 + T + 18 T^{2} + 11 T^{3} + 121 T^{4} \)
$13$ \( 1 - 5 T + 28 T^{2} - 65 T^{3} + 169 T^{4} \)
$17$ \( 1 - 5 T + 36 T^{2} - 85 T^{3} + 289 T^{4} \)
$19$ \( 1 + 6 T + 30 T^{2} + 114 T^{3} + 361 T^{4} \)
$23$ \( 1 - 2 T + 30 T^{2} - 46 T^{3} + 529 T^{4} \)
$29$ \( 1 + T + 20 T^{2} + 29 T^{3} + 841 T^{4} \)
$31$ \( ( 1 + 31 T^{2} )^{2} \)
$37$ \( ( 1 - 6 T + 37 T^{2} )^{2} \)
$41$ \( 1 + 2 T + 66 T^{2} + 82 T^{3} + 1681 T^{4} \)
$43$ \( 1 - 10 T + 94 T^{2} - 430 T^{3} + 1849 T^{4} \)
$47$ \( 1 - 5 T + 62 T^{2} - 235 T^{3} + 2209 T^{4} \)
$53$ \( 1 - 2 T + 90 T^{2} - 106 T^{3} + 2809 T^{4} \)
$59$ \( ( 1 - 4 T + 59 T^{2} )^{2} \)
$61$ \( 1 - 6 T - 22 T^{2} - 366 T^{3} + 3721 T^{4} \)
$67$ \( 1 - 4 T + 70 T^{2} - 268 T^{3} + 4489 T^{4} \)
$71$ \( ( 1 + 8 T + 71 T^{2} )^{2} \)
$73$ \( 1 + 8 T + 94 T^{2} + 584 T^{3} + 5329 T^{4} \)
$79$ \( 1 + 9 T + 174 T^{2} + 711 T^{3} + 6241 T^{4} \)
$83$ \( ( 1 + 4 T + 83 T^{2} )^{2} \)
$89$ \( 1 + 6 T + 170 T^{2} + 534 T^{3} + 7921 T^{4} \)
$97$ \( 1 + 9 T + 108 T^{2} + 873 T^{3} + 9409 T^{4} \)
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