Properties

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

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(2.51528766367\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 35)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
64.1
1.00000i
1.00000i
2.00000i 0 −2.00000 2.00000 + 1.00000i 0 1.00000i 0 0 2.00000 4.00000i
64.2 2.00000i 0 −2.00000 2.00000 1.00000i 0 1.00000i 0 0 2.00000 + 4.00000i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 315.2.d.a 2
3.b odd 2 1 35.2.b.a 2
4.b odd 2 1 5040.2.t.p 2
5.b even 2 1 inner 315.2.d.a 2
5.c odd 4 1 1575.2.a.a 1
5.c odd 4 1 1575.2.a.k 1
7.b odd 2 1 2205.2.d.b 2
12.b even 2 1 560.2.g.b 2
15.d odd 2 1 35.2.b.a 2
15.e even 4 1 175.2.a.a 1
15.e even 4 1 175.2.a.c 1
20.d odd 2 1 5040.2.t.p 2
21.c even 2 1 245.2.b.a 2
21.g even 6 2 245.2.j.d 4
21.h odd 6 2 245.2.j.e 4
24.f even 2 1 2240.2.g.g 2
24.h odd 2 1 2240.2.g.h 2
35.c odd 2 1 2205.2.d.b 2
60.h even 2 1 560.2.g.b 2
60.l odd 4 1 2800.2.a.l 1
60.l odd 4 1 2800.2.a.w 1
105.g even 2 1 245.2.b.a 2
105.k odd 4 1 1225.2.a.a 1
105.k odd 4 1 1225.2.a.i 1
105.o odd 6 2 245.2.j.e 4
105.p even 6 2 245.2.j.d 4
120.i odd 2 1 2240.2.g.h 2
120.m even 2 1 2240.2.g.g 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.2.b.a 2 3.b odd 2 1
35.2.b.a 2 15.d odd 2 1
175.2.a.a 1 15.e even 4 1
175.2.a.c 1 15.e even 4 1
245.2.b.a 2 21.c even 2 1
245.2.b.a 2 105.g even 2 1
245.2.j.d 4 21.g even 6 2
245.2.j.d 4 105.p even 6 2
245.2.j.e 4 21.h odd 6 2
245.2.j.e 4 105.o odd 6 2
315.2.d.a 2 1.a even 1 1 trivial
315.2.d.a 2 5.b even 2 1 inner
560.2.g.b 2 12.b even 2 1
560.2.g.b 2 60.h even 2 1
1225.2.a.a 1 105.k odd 4 1
1225.2.a.i 1 105.k odd 4 1
1575.2.a.a 1 5.c odd 4 1
1575.2.a.k 1 5.c odd 4 1
2205.2.d.b 2 7.b odd 2 1
2205.2.d.b 2 35.c odd 2 1
2240.2.g.g 2 24.f even 2 1
2240.2.g.g 2 120.m even 2 1
2240.2.g.h 2 24.h odd 2 1
2240.2.g.h 2 120.i odd 2 1
2800.2.a.l 1 60.l odd 4 1
2800.2.a.w 1 60.l odd 4 1
5040.2.t.p 2 4.b odd 2 1
5040.2.t.p 2 20.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(315, [\chi])\):

\( T_{2}^{2} + 4 \)
\( T_{11} - 3 \)
\( T_{29} + 5 \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - 2 T + 2 T^{2} )( 1 + 2 T + 2 T^{2} ) \)
$3$ \( \)
$5$ \( 1 - 4 T + 5 T^{2} \)
$7$ \( 1 + T^{2} \)
$11$ \( ( 1 - 3 T + 11 T^{2} )^{2} \)
$13$ \( 1 - 25 T^{2} + 169 T^{4} \)
$17$ \( 1 + 15 T^{2} + 289 T^{4} \)
$19$ \( ( 1 + 19 T^{2} )^{2} \)
$23$ \( 1 - 10 T^{2} + 529 T^{4} \)
$29$ \( ( 1 + 5 T + 29 T^{2} )^{2} \)
$31$ \( ( 1 - 2 T + 31 T^{2} )^{2} \)
$37$ \( ( 1 - 12 T + 37 T^{2} )( 1 + 12 T + 37 T^{2} ) \)
$41$ \( ( 1 + 2 T + 41 T^{2} )^{2} \)
$43$ \( 1 - 70 T^{2} + 1849 T^{4} \)
$47$ \( 1 - 85 T^{2} + 2209 T^{4} \)
$53$ \( 1 - 70 T^{2} + 2809 T^{4} \)
$59$ \( ( 1 - 10 T + 59 T^{2} )^{2} \)
$61$ \( ( 1 + 8 T + 61 T^{2} )^{2} \)
$67$ \( 1 - 130 T^{2} + 4489 T^{4} \)
$71$ \( ( 1 - 8 T + 71 T^{2} )^{2} \)
$73$ \( ( 1 - 16 T + 73 T^{2} )( 1 + 16 T + 73 T^{2} ) \)
$79$ \( ( 1 - 5 T + 79 T^{2} )^{2} \)
$83$ \( 1 - 150 T^{2} + 6889 T^{4} \)
$89$ \( ( 1 + 89 T^{2} )^{2} \)
$97$ \( 1 - 145 T^{2} + 9409 T^{4} \)
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