Properties

Label 315.2.p.c
Level 315
Weight 2
Character orbit 315.p
Analytic conductor 2.515
Analytic rank 0
Dimension 4
CM no
Inner twists 4

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Newspace parameters

Level: \( N \) = \( 315 = 3^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 315.p (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(2.51528766367\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
Coefficient field: \(\Q(i, \sqrt{10})\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 35)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 1 + \beta_{2} ) q^{2} + \beta_{1} q^{5} + ( 1 + \beta_{2} + \beta_{3} ) q^{7} + ( 2 - 2 \beta_{2} ) q^{8} +O(q^{10})\) \( q + ( 1 + \beta_{2} ) q^{2} + \beta_{1} q^{5} + ( 1 + \beta_{2} + \beta_{3} ) q^{7} + ( 2 - 2 \beta_{2} ) q^{8} + ( \beta_{1} + \beta_{3} ) q^{10} + q^{11} -\beta_{1} q^{13} + ( -\beta_{1} + 2 \beta_{2} + \beta_{3} ) q^{14} + 4 q^{16} + \beta_{3} q^{17} + ( \beta_{1} - \beta_{3} ) q^{19} + ( 1 + \beta_{2} ) q^{22} + ( -2 + 2 \beta_{2} ) q^{23} + 5 \beta_{2} q^{25} + ( -\beta_{1} - \beta_{3} ) q^{26} -3 \beta_{2} q^{29} + ( -\beta_{1} - \beta_{3} ) q^{31} + ( -\beta_{1} + \beta_{3} ) q^{34} + ( -5 + \beta_{1} + \beta_{3} ) q^{35} + ( -6 - 6 \beta_{2} ) q^{37} + 2 \beta_{1} q^{38} + ( 2 \beta_{1} - 2 \beta_{3} ) q^{40} + ( -3 \beta_{1} - 3 \beta_{3} ) q^{41} + ( -3 + 3 \beta_{2} ) q^{43} -4 q^{46} -3 \beta_{3} q^{47} + ( -2 \beta_{1} - 3 \beta_{2} + 2 \beta_{3} ) q^{49} + ( -5 + 5 \beta_{2} ) q^{50} + ( -1 + \beta_{2} ) q^{53} + \beta_{1} q^{55} + ( 4 + 2 \beta_{1} + 2 \beta_{3} ) q^{56} + ( 3 - 3 \beta_{2} ) q^{58} + ( -3 \beta_{1} + 3 \beta_{3} ) q^{59} + ( -2 \beta_{1} - 2 \beta_{3} ) q^{61} -2 \beta_{3} q^{62} -8 \beta_{2} q^{64} -5 \beta_{2} q^{65} + ( -1 - \beta_{2} ) q^{67} + ( -5 - 5 \beta_{2} + 2 \beta_{3} ) q^{70} + 6 q^{71} -12 \beta_{2} q^{74} + ( 1 + \beta_{2} + \beta_{3} ) q^{77} + 13 \beta_{2} q^{79} + 4 \beta_{1} q^{80} -6 \beta_{3} q^{82} + 2 \beta_{1} q^{83} -5 q^{85} -6 q^{86} + ( 2 - 2 \beta_{2} ) q^{88} + ( 2 \beta_{1} - 2 \beta_{3} ) q^{89} + ( 5 - \beta_{1} - \beta_{3} ) q^{91} + ( 3 \beta_{1} - 3 \beta_{3} ) q^{94} + ( 5 + 5 \beta_{2} ) q^{95} + \beta_{3} q^{97} + ( 3 - 4 \beta_{1} - 3 \beta_{2} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 4q^{2} + 4q^{7} + 8q^{8} + O(q^{10}) \) \( 4q + 4q^{2} + 4q^{7} + 8q^{8} + 4q^{11} + 16q^{16} + 4q^{22} - 8q^{23} - 20q^{35} - 24q^{37} - 12q^{43} - 16q^{46} - 20q^{50} - 4q^{53} + 16q^{56} + 12q^{58} - 4q^{67} - 20q^{70} + 24q^{71} + 4q^{77} - 20q^{85} - 24q^{86} + 8q^{88} + 20q^{91} + 20q^{95} + 12q^{98} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 25\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} \)\(/5\)
\(\beta_{3}\)\(=\)\( \nu^{3} \)\(/5\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(5 \beta_{2}\)
\(\nu^{3}\)\(=\)\(5 \beta_{3}\)

Character values

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

\(n\) \(127\) \(136\) \(281\)
\(\chi(n)\) \(\beta_{2}\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
118.1
−1.58114 + 1.58114i
1.58114 1.58114i
−1.58114 1.58114i
1.58114 + 1.58114i
1.00000 1.00000i 0 0 −1.58114 + 1.58114i 0 2.58114 + 0.581139i 2.00000 + 2.00000i 0 3.16228i
118.2 1.00000 1.00000i 0 0 1.58114 1.58114i 0 −0.581139 2.58114i 2.00000 + 2.00000i 0 3.16228i
307.1 1.00000 + 1.00000i 0 0 −1.58114 1.58114i 0 2.58114 0.581139i 2.00000 2.00000i 0 3.16228i
307.2 1.00000 + 1.00000i 0 0 1.58114 + 1.58114i 0 −0.581139 + 2.58114i 2.00000 2.00000i 0 3.16228i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
7.b odd 2 1 inner
35.f even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 315.2.p.c 4
3.b odd 2 1 35.2.f.a 4
5.c odd 4 1 inner 315.2.p.c 4
7.b odd 2 1 inner 315.2.p.c 4
12.b even 2 1 560.2.bj.a 4
15.d odd 2 1 175.2.f.c 4
15.e even 4 1 35.2.f.a 4
15.e even 4 1 175.2.f.c 4
21.c even 2 1 35.2.f.a 4
21.g even 6 2 245.2.l.c 8
21.h odd 6 2 245.2.l.c 8
35.f even 4 1 inner 315.2.p.c 4
60.l odd 4 1 560.2.bj.a 4
84.h odd 2 1 560.2.bj.a 4
105.g even 2 1 175.2.f.c 4
105.k odd 4 1 35.2.f.a 4
105.k odd 4 1 175.2.f.c 4
105.w odd 12 2 245.2.l.c 8
105.x even 12 2 245.2.l.c 8
420.w even 4 1 560.2.bj.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.2.f.a 4 3.b odd 2 1
35.2.f.a 4 15.e even 4 1
35.2.f.a 4 21.c even 2 1
35.2.f.a 4 105.k odd 4 1
175.2.f.c 4 15.d odd 2 1
175.2.f.c 4 15.e even 4 1
175.2.f.c 4 105.g even 2 1
175.2.f.c 4 105.k odd 4 1
245.2.l.c 8 21.g even 6 2
245.2.l.c 8 21.h odd 6 2
245.2.l.c 8 105.w odd 12 2
245.2.l.c 8 105.x even 12 2
315.2.p.c 4 1.a even 1 1 trivial
315.2.p.c 4 5.c odd 4 1 inner
315.2.p.c 4 7.b odd 2 1 inner
315.2.p.c 4 35.f even 4 1 inner
560.2.bj.a 4 12.b even 2 1
560.2.bj.a 4 60.l odd 4 1
560.2.bj.a 4 84.h odd 2 1
560.2.bj.a 4 420.w even 4 1

Hecke kernels

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

\( T_{2}^{2} - 2 T_{2} + 2 \)
\( T_{17}^{4} + 25 \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - 2 T + 2 T^{2} - 4 T^{3} + 4 T^{4} )^{2} \)
$3$ 1
$5$ \( 1 + 25 T^{4} \)
$7$ \( 1 - 4 T + 8 T^{2} - 28 T^{3} + 49 T^{4} \)
$11$ \( ( 1 - T + 11 T^{2} )^{4} \)
$13$ \( 1 + 103 T^{4} + 28561 T^{8} \)
$17$ \( 1 + 263 T^{4} + 83521 T^{8} \)
$19$ \( ( 1 + 28 T^{2} + 361 T^{4} )^{2} \)
$23$ \( ( 1 + 4 T + 8 T^{2} + 92 T^{3} + 529 T^{4} )^{2} \)
$29$ \( ( 1 - 49 T^{2} + 841 T^{4} )^{2} \)
$31$ \( ( 1 - 52 T^{2} + 961 T^{4} )^{2} \)
$37$ \( ( 1 + 12 T + 72 T^{2} + 444 T^{3} + 1369 T^{4} )^{2} \)
$41$ \( ( 1 + 8 T^{2} + 1681 T^{4} )^{2} \)
$43$ \( ( 1 + 6 T + 18 T^{2} + 258 T^{3} + 1849 T^{4} )^{2} \)
$47$ \( 1 - 2017 T^{4} + 4879681 T^{8} \)
$53$ \( ( 1 + 2 T + 2 T^{2} + 106 T^{3} + 2809 T^{4} )^{2} \)
$59$ \( ( 1 + 28 T^{2} + 3481 T^{4} )^{2} \)
$61$ \( ( 1 - 82 T^{2} + 3721 T^{4} )^{2} \)
$67$ \( ( 1 + 2 T + 2 T^{2} + 134 T^{3} + 4489 T^{4} )^{2} \)
$71$ \( ( 1 - 6 T + 71 T^{2} )^{4} \)
$73$ \( ( 1 + 5329 T^{4} )^{2} \)
$79$ \( ( 1 + 11 T^{2} + 6241 T^{4} )^{2} \)
$83$ \( 1 + 7538 T^{4} + 47458321 T^{8} \)
$89$ \( ( 1 + 138 T^{2} + 7921 T^{4} )^{2} \)
$97$ \( 1 + 16903 T^{4} + 88529281 T^{8} \)
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