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: \(\Q(\sqrt{17}) \)
Defining polynomial: \( x^{2} - x - 4 \) Copy content Toggle raw display
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.

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

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:

Atkin-Lehner signs

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

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} - T_{2} - 4 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(315))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - T - 4 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( (T + 1)^{2} \) Copy content Toggle raw display
$7$ \( (T + 1)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + T - 4 \) Copy content Toggle raw display
$13$ \( T^{2} - 5T + 2 \) Copy content Toggle raw display
$17$ \( T^{2} - 5T + 2 \) Copy content Toggle raw display
$19$ \( T^{2} + 6T - 8 \) Copy content Toggle raw display
$23$ \( T^{2} - 2T - 16 \) Copy content Toggle raw display
$29$ \( T^{2} + T - 38 \) Copy content Toggle raw display
$31$ \( T^{2} \) Copy content Toggle raw display
$37$ \( (T - 6)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} + 2T - 16 \) Copy content Toggle raw display
$43$ \( T^{2} - 10T + 8 \) Copy content Toggle raw display
$47$ \( T^{2} - 5T - 32 \) Copy content Toggle raw display
$53$ \( T^{2} - 2T - 16 \) Copy content Toggle raw display
$59$ \( (T - 4)^{2} \) Copy content Toggle raw display
$61$ \( T^{2} - 6T - 144 \) Copy content Toggle raw display
$67$ \( T^{2} - 4T - 64 \) Copy content Toggle raw display
$71$ \( (T + 8)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 8T - 52 \) Copy content Toggle raw display
$79$ \( T^{2} + 9T + 16 \) Copy content Toggle raw display
$83$ \( (T + 4)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} + 6T - 8 \) Copy content Toggle raw display
$97$ \( T^{2} + 9T - 86 \) Copy content Toggle raw display
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