Properties

Label 315.5.e.b
Level $315$
Weight $5$
Character orbit 315.e
Self dual yes
Analytic conductor $32.562$
Analytic rank $0$
Dimension $1$
CM discriminant -35
Inner twists $2$

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Newspace parameters

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

Newform invariants

Self dual: yes
Analytic conductor: \(32.5615383714\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 35)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

\(f(q)\) \(=\) \( q + 16q^{4} + 25q^{5} - 49q^{7} + O(q^{10}) \) \( q + 16q^{4} + 25q^{5} - 49q^{7} + 73q^{11} - 23q^{13} + 256q^{16} + 263q^{17} + 400q^{20} + 625q^{25} - 784q^{28} + 1153q^{29} - 1225q^{35} + 1168q^{44} - 3457q^{47} + 2401q^{49} - 368q^{52} + 1825q^{55} + 4096q^{64} - 575q^{65} + 4208q^{68} + 10078q^{71} + 9502q^{73} - 3577q^{77} + 12167q^{79} + 6400q^{80} - 6382q^{83} + 6575q^{85} + 1127q^{91} - 3383q^{97} + O(q^{100}) \)

Character values

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
244.1
0
0 0 16.0000 25.0000 0 −49.0000 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
35.c odd 2 1 CM by \(\Q(\sqrt{-35}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 315.5.e.b 1
3.b odd 2 1 35.5.c.b yes 1
5.b even 2 1 315.5.e.a 1
7.b odd 2 1 315.5.e.a 1
12.b even 2 1 560.5.p.a 1
15.d odd 2 1 35.5.c.a 1
15.e even 4 2 175.5.d.c 2
21.c even 2 1 35.5.c.a 1
35.c odd 2 1 CM 315.5.e.b 1
60.h even 2 1 560.5.p.b 1
84.h odd 2 1 560.5.p.b 1
105.g even 2 1 35.5.c.b yes 1
105.k odd 4 2 175.5.d.c 2
420.o odd 2 1 560.5.p.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.5.c.a 1 15.d odd 2 1
35.5.c.a 1 21.c even 2 1
35.5.c.b yes 1 3.b odd 2 1
35.5.c.b yes 1 105.g even 2 1
175.5.d.c 2 15.e even 4 2
175.5.d.c 2 105.k odd 4 2
315.5.e.a 1 5.b even 2 1
315.5.e.a 1 7.b odd 2 1
315.5.e.b 1 1.a even 1 1 trivial
315.5.e.b 1 35.c odd 2 1 CM
560.5.p.a 1 12.b even 2 1
560.5.p.a 1 420.o odd 2 1
560.5.p.b 1 60.h even 2 1
560.5.p.b 1 84.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{5}^{\mathrm{new}}(315, [\chi])\):

\( T_{2} \)
\( T_{13} + 23 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \)
$3$ \( T \)
$5$ \( -25 + T \)
$7$ \( 49 + T \)
$11$ \( -73 + T \)
$13$ \( 23 + T \)
$17$ \( -263 + T \)
$19$ \( T \)
$23$ \( T \)
$29$ \( -1153 + T \)
$31$ \( T \)
$37$ \( T \)
$41$ \( T \)
$43$ \( T \)
$47$ \( 3457 + T \)
$53$ \( T \)
$59$ \( T \)
$61$ \( T \)
$67$ \( T \)
$71$ \( -10078 + T \)
$73$ \( -9502 + T \)
$79$ \( -12167 + T \)
$83$ \( 6382 + T \)
$89$ \( T \)
$97$ \( 3383 + T \)
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