Properties

Label 315.2.bl.h
Level 315
Weight 2
Character orbit 315.bl
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.bl (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(2.51528766367\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 1 + \zeta_{6} ) q^{3} -2 \zeta_{6} q^{4} + \zeta_{6} q^{5} + ( -1 + 3 \zeta_{6} ) q^{7} + 3 \zeta_{6} q^{9} +O(q^{10})\) \( q + ( 1 + \zeta_{6} ) q^{3} -2 \zeta_{6} q^{4} + \zeta_{6} q^{5} + ( -1 + 3 \zeta_{6} ) q^{7} + 3 \zeta_{6} q^{9} + ( 1 + \zeta_{6} ) q^{11} + ( 2 - 4 \zeta_{6} ) q^{12} + ( -1 + 2 \zeta_{6} ) q^{15} + ( -4 + 4 \zeta_{6} ) q^{16} + 6 q^{17} + ( 4 - 8 \zeta_{6} ) q^{19} + ( 2 - 2 \zeta_{6} ) q^{20} + ( -4 + 5 \zeta_{6} ) q^{21} + ( 4 - 2 \zeta_{6} ) q^{23} + ( -1 + \zeta_{6} ) q^{25} + ( -3 + 6 \zeta_{6} ) q^{27} + ( 6 - 4 \zeta_{6} ) q^{28} + ( -5 - 5 \zeta_{6} ) q^{29} + ( -4 + 2 \zeta_{6} ) q^{31} + 3 \zeta_{6} q^{33} + ( -3 + 2 \zeta_{6} ) q^{35} + ( 6 - 6 \zeta_{6} ) q^{36} -8 q^{37} + 6 \zeta_{6} q^{41} + ( -10 + 10 \zeta_{6} ) q^{43} + ( 2 - 4 \zeta_{6} ) q^{44} + ( -3 + 3 \zeta_{6} ) q^{45} + ( 9 - 9 \zeta_{6} ) q^{47} + ( -8 + 4 \zeta_{6} ) q^{48} + ( -8 + 3 \zeta_{6} ) q^{49} + ( 6 + 6 \zeta_{6} ) q^{51} + ( 8 - 16 \zeta_{6} ) q^{53} + ( -1 + 2 \zeta_{6} ) q^{55} + ( 12 - 12 \zeta_{6} ) q^{57} + ( 4 - 2 \zeta_{6} ) q^{60} + ( -2 - 2 \zeta_{6} ) q^{61} + ( -9 + 6 \zeta_{6} ) q^{63} + 8 q^{64} -8 \zeta_{6} q^{67} -12 \zeta_{6} q^{68} + 6 q^{69} + ( -1 + 2 \zeta_{6} ) q^{71} + ( 5 - 10 \zeta_{6} ) q^{73} + ( -2 + \zeta_{6} ) q^{75} + ( -16 + 8 \zeta_{6} ) q^{76} + ( -4 + 5 \zeta_{6} ) q^{77} + ( 1 - \zeta_{6} ) q^{79} -4 q^{80} + ( -9 + 9 \zeta_{6} ) q^{81} + ( -9 + 9 \zeta_{6} ) q^{83} + ( 10 - 2 \zeta_{6} ) q^{84} + 6 \zeta_{6} q^{85} -15 \zeta_{6} q^{87} + 6 q^{89} + ( -4 - 4 \zeta_{6} ) q^{92} -6 q^{93} + ( 8 - 4 \zeta_{6} ) q^{95} + ( -1 - \zeta_{6} ) q^{97} + ( -3 + 6 \zeta_{6} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 3q^{3} - 2q^{4} + q^{5} + q^{7} + 3q^{9} + O(q^{10}) \) \( 2q + 3q^{3} - 2q^{4} + q^{5} + q^{7} + 3q^{9} + 3q^{11} - 4q^{16} + 12q^{17} + 2q^{20} - 3q^{21} + 6q^{23} - q^{25} + 8q^{28} - 15q^{29} - 6q^{31} + 3q^{33} - 4q^{35} + 6q^{36} - 16q^{37} + 6q^{41} - 10q^{43} - 3q^{45} + 9q^{47} - 12q^{48} - 13q^{49} + 18q^{51} + 12q^{57} + 6q^{60} - 6q^{61} - 12q^{63} + 16q^{64} - 8q^{67} - 12q^{68} + 12q^{69} - 3q^{75} - 24q^{76} - 3q^{77} + q^{79} - 8q^{80} - 9q^{81} - 9q^{83} + 18q^{84} + 6q^{85} - 15q^{87} + 12q^{89} - 12q^{92} - 12q^{93} + 12q^{95} - 3q^{97} + O(q^{100}) \)

Character values

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

\(n\) \(127\) \(136\) \(281\)
\(\chi(n)\) \(1\) \(-1\) \(\zeta_{6}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
41.1
0.500000 0.866025i
0.500000 + 0.866025i
0 1.50000 0.866025i −1.00000 + 1.73205i 0.500000 0.866025i 0 0.500000 2.59808i 0 1.50000 2.59808i 0
146.1 0 1.50000 + 0.866025i −1.00000 1.73205i 0.500000 + 0.866025i 0 0.500000 + 2.59808i 0 1.50000 + 2.59808i 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
63.o even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 315.2.bl.h yes 2
3.b odd 2 1 945.2.bl.a 2
7.b odd 2 1 315.2.bl.e 2
9.c even 3 1 945.2.bl.c 2
9.d odd 6 1 315.2.bl.e 2
21.c even 2 1 945.2.bl.c 2
63.l odd 6 1 945.2.bl.a 2
63.o even 6 1 inner 315.2.bl.h yes 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
315.2.bl.e 2 7.b odd 2 1
315.2.bl.e 2 9.d odd 6 1
315.2.bl.h yes 2 1.a even 1 1 trivial
315.2.bl.h yes 2 63.o even 6 1 inner
945.2.bl.a 2 3.b odd 2 1
945.2.bl.a 2 63.l odd 6 1
945.2.bl.c 2 9.c even 3 1
945.2.bl.c 2 21.c even 2 1

Hecke kernels

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

\( T_{2} \)
\( T_{11}^{2} - 3 T_{11} + 3 \)
\( T_{13} \)
\( T_{17} - 6 \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 + 2 T^{2} + 4 T^{4} \)
$3$ \( 1 - 3 T + 3 T^{2} \)
$5$ \( 1 - T + T^{2} \)
$7$ \( 1 - T + 7 T^{2} \)
$11$ \( 1 - 3 T + 14 T^{2} - 33 T^{3} + 121 T^{4} \)
$13$ \( 1 + 13 T^{2} + 169 T^{4} \)
$17$ \( ( 1 - 6 T + 17 T^{2} )^{2} \)
$19$ \( 1 + 10 T^{2} + 361 T^{4} \)
$23$ \( 1 - 6 T + 35 T^{2} - 138 T^{3} + 529 T^{4} \)
$29$ \( 1 + 15 T + 104 T^{2} + 435 T^{3} + 841 T^{4} \)
$31$ \( 1 + 6 T + 43 T^{2} + 186 T^{3} + 961 T^{4} \)
$37$ \( ( 1 + 8 T + 37 T^{2} )^{2} \)
$41$ \( 1 - 6 T - 5 T^{2} - 246 T^{3} + 1681 T^{4} \)
$43$ \( 1 + 10 T + 57 T^{2} + 430 T^{3} + 1849 T^{4} \)
$47$ \( 1 - 9 T + 34 T^{2} - 423 T^{3} + 2209 T^{4} \)
$53$ \( 1 + 86 T^{2} + 2809 T^{4} \)
$59$ \( 1 - 59 T^{2} + 3481 T^{4} \)
$61$ \( 1 + 6 T + 73 T^{2} + 366 T^{3} + 3721 T^{4} \)
$67$ \( 1 + 8 T - 3 T^{2} + 536 T^{3} + 4489 T^{4} \)
$71$ \( 1 - 139 T^{2} + 5041 T^{4} \)
$73$ \( 1 - 71 T^{2} + 5329 T^{4} \)
$79$ \( 1 - T - 78 T^{2} - 79 T^{3} + 6241 T^{4} \)
$83$ \( 1 + 9 T - 2 T^{2} + 747 T^{3} + 6889 T^{4} \)
$89$ \( ( 1 - 6 T + 89 T^{2} )^{2} \)
$97$ \( 1 + 3 T + 100 T^{2} + 291 T^{3} + 9409 T^{4} \)
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