Properties

Label 315.3.h.b
Level $315$
Weight $3$
Character orbit 315.h
Analytic conductor $8.583$
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: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 315.h (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(8.58312832735\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-5}) \)
Defining polynomial: \( x^{2} + 5 \) Copy content Toggle raw display
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 \(\beta = \sqrt{-5}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{2} - 3 q^{4} + \beta q^{5} + 7 q^{7} - 7 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} - 3 q^{4} + \beta q^{5} + 7 q^{7} - 7 q^{8} + \beta q^{10} - 2 q^{11} + 6 \beta q^{13} + 7 q^{14} + 5 q^{16} + 12 \beta q^{17} + 6 \beta q^{19} - 3 \beta q^{20} - 2 q^{22} - 26 q^{23} - 5 q^{25} + 6 \beta q^{26} - 21 q^{28} + 22 q^{29} + 24 \beta q^{31} + 33 q^{32} + 12 \beta q^{34} + 7 \beta q^{35} + 14 q^{37} + 6 \beta q^{38} - 7 \beta q^{40} - 12 \beta q^{41} - 34 q^{43} + 6 q^{44} - 26 q^{46} - 12 \beta q^{47} + 49 q^{49} - 5 q^{50} - 18 \beta q^{52} + 34 q^{53} - 2 \beta q^{55} - 49 q^{56} + 22 q^{58} - 18 \beta q^{59} - 42 \beta q^{61} + 24 \beta q^{62} + 13 q^{64} - 30 q^{65} + 14 q^{67} - 36 \beta q^{68} + 7 \beta q^{70} - 62 q^{71} + 24 \beta q^{73} + 14 q^{74} - 18 \beta q^{76} - 14 q^{77} + 38 q^{79} + 5 \beta q^{80} - 12 \beta q^{82} + 18 \beta q^{83} - 60 q^{85} - 34 q^{86} + 14 q^{88} - 12 \beta q^{89} + 42 \beta q^{91} + 78 q^{92} - 12 \beta q^{94} - 30 q^{95} + 12 \beta q^{97} + 49 q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} - 6 q^{4} + 14 q^{7} - 14 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} - 6 q^{4} + 14 q^{7} - 14 q^{8} - 4 q^{11} + 14 q^{14} + 10 q^{16} - 4 q^{22} - 52 q^{23} - 10 q^{25} - 42 q^{28} + 44 q^{29} + 66 q^{32} + 28 q^{37} - 68 q^{43} + 12 q^{44} - 52 q^{46} + 98 q^{49} - 10 q^{50} + 68 q^{53} - 98 q^{56} + 44 q^{58} + 26 q^{64} - 60 q^{65} + 28 q^{67} - 124 q^{71} + 28 q^{74} - 28 q^{77} + 76 q^{79} - 120 q^{85} - 68 q^{86} + 28 q^{88} + 156 q^{92} - 60 q^{95} + 98 q^{98}+O(q^{100}) \) Copy content Toggle raw display

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
181.1
2.23607i
2.23607i
1.00000 0 −3.00000 2.23607i 0 7.00000 −7.00000 0 2.23607i
181.2 1.00000 0 −3.00000 2.23607i 0 7.00000 −7.00000 0 2.23607i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 315.3.h.b 2
3.b odd 2 1 35.3.d.a 2
7.b odd 2 1 inner 315.3.h.b 2
12.b even 2 1 560.3.f.a 2
15.d odd 2 1 175.3.d.f 2
15.e even 4 2 175.3.c.d 4
21.c even 2 1 35.3.d.a 2
21.g even 6 2 245.3.h.b 4
21.h odd 6 2 245.3.h.b 4
84.h odd 2 1 560.3.f.a 2
105.g even 2 1 175.3.d.f 2
105.k odd 4 2 175.3.c.d 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.3.d.a 2 3.b odd 2 1
35.3.d.a 2 21.c even 2 1
175.3.c.d 4 15.e even 4 2
175.3.c.d 4 105.k odd 4 2
175.3.d.f 2 15.d odd 2 1
175.3.d.f 2 105.g even 2 1
245.3.h.b 4 21.g even 6 2
245.3.h.b 4 21.h odd 6 2
315.3.h.b 2 1.a even 1 1 trivial
315.3.h.b 2 7.b odd 2 1 inner
560.3.f.a 2 12.b even 2 1
560.3.f.a 2 84.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} - 1 \) acting on \(S_{3}^{\mathrm{new}}(315, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + 5 \) Copy content Toggle raw display
$7$ \( (T - 7)^{2} \) Copy content Toggle raw display
$11$ \( (T + 2)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 180 \) Copy content Toggle raw display
$17$ \( T^{2} + 720 \) Copy content Toggle raw display
$19$ \( T^{2} + 180 \) Copy content Toggle raw display
$23$ \( (T + 26)^{2} \) Copy content Toggle raw display
$29$ \( (T - 22)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 2880 \) Copy content Toggle raw display
$37$ \( (T - 14)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} + 720 \) Copy content Toggle raw display
$43$ \( (T + 34)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 720 \) Copy content Toggle raw display
$53$ \( (T - 34)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} + 1620 \) Copy content Toggle raw display
$61$ \( T^{2} + 8820 \) Copy content Toggle raw display
$67$ \( (T - 14)^{2} \) Copy content Toggle raw display
$71$ \( (T + 62)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 2880 \) Copy content Toggle raw display
$79$ \( (T - 38)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 1620 \) Copy content Toggle raw display
$89$ \( T^{2} + 720 \) Copy content Toggle raw display
$97$ \( T^{2} + 720 \) Copy content Toggle raw display
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