Properties

Label 315.2.d.c
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: \(\Q(\sqrt{-1}) \)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 105)
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.

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

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
1.00000i 0 1.00000 −1.00000 + 2.00000i 0 1.00000i 3.00000i 0 2.00000 + 1.00000i
64.2 1.00000i 0 1.00000 −1.00000 2.00000i 0 1.00000i 3.00000i 0 2.00000 1.00000i
\(n\): 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.c 2
3.b odd 2 1 105.2.d.a 2
4.b odd 2 1 5040.2.t.e 2
5.b even 2 1 inner 315.2.d.c 2
5.c odd 4 1 1575.2.a.e 1
5.c odd 4 1 1575.2.a.i 1
7.b odd 2 1 2205.2.d.f 2
12.b even 2 1 1680.2.t.f 2
15.d odd 2 1 105.2.d.a 2
15.e even 4 1 525.2.a.b 1
15.e even 4 1 525.2.a.c 1
20.d odd 2 1 5040.2.t.e 2
21.c even 2 1 735.2.d.a 2
21.g even 6 2 735.2.q.b 4
21.h odd 6 2 735.2.q.a 4
35.c odd 2 1 2205.2.d.f 2
60.h even 2 1 1680.2.t.f 2
60.l odd 4 1 8400.2.a.bj 1
60.l odd 4 1 8400.2.a.ch 1
105.g even 2 1 735.2.d.a 2
105.k odd 4 1 3675.2.a.d 1
105.k odd 4 1 3675.2.a.l 1
105.o odd 6 2 735.2.q.a 4
105.p even 6 2 735.2.q.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.2.d.a 2 3.b odd 2 1
105.2.d.a 2 15.d odd 2 1
315.2.d.c 2 1.a even 1 1 trivial
315.2.d.c 2 5.b even 2 1 inner
525.2.a.b 1 15.e even 4 1
525.2.a.c 1 15.e even 4 1
735.2.d.a 2 21.c even 2 1
735.2.d.a 2 105.g even 2 1
735.2.q.a 4 21.h odd 6 2
735.2.q.a 4 105.o odd 6 2
735.2.q.b 4 21.g even 6 2
735.2.q.b 4 105.p even 6 2
1575.2.a.e 1 5.c odd 4 1
1575.2.a.i 1 5.c odd 4 1
1680.2.t.f 2 12.b even 2 1
1680.2.t.f 2 60.h even 2 1
2205.2.d.f 2 7.b odd 2 1
2205.2.d.f 2 35.c odd 2 1
3675.2.a.d 1 105.k odd 4 1
3675.2.a.l 1 105.k odd 4 1
5040.2.t.e 2 4.b odd 2 1
5040.2.t.e 2 20.d odd 2 1
8400.2.a.bj 1 60.l odd 4 1
8400.2.a.ch 1 60.l odd 4 1

Hecke kernels

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

\( T_{2}^{2} + 1 \)
\( T_{11} - 6 \)
\( T_{29} + 2 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T^{2} \)
$3$ \( T^{2} \)
$5$ \( 5 + 2 T + T^{2} \)
$7$ \( 1 + T^{2} \)
$11$ \( ( -6 + T )^{2} \)
$13$ \( 4 + T^{2} \)
$17$ \( 16 + T^{2} \)
$19$ \( ( -6 + T )^{2} \)
$23$ \( T^{2} \)
$29$ \( ( 2 + T )^{2} \)
$31$ \( ( 10 + T )^{2} \)
$37$ \( 16 + T^{2} \)
$41$ \( ( 2 + T )^{2} \)
$43$ \( 16 + T^{2} \)
$47$ \( T^{2} \)
$53$ \( 36 + T^{2} \)
$59$ \( ( 8 + T )^{2} \)
$61$ \( ( 2 + T )^{2} \)
$67$ \( 256 + T^{2} \)
$71$ \( ( 10 + T )^{2} \)
$73$ \( 36 + T^{2} \)
$79$ \( ( 4 + T )^{2} \)
$83$ \( 64 + T^{2} \)
$89$ \( ( -6 + T )^{2} \)
$97$ \( 4 + T^{2} \)
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