Properties

Label 315.2.bl.c
Level 315
Weight 2
Character orbit 315.bl
Analytic conductor 2.515
Analytic rank 1
Dimension 2
CM no
Inner twists 2

Related objects

Downloads

Learn more about

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: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Coefficient ring: \(\Z[a_1, a_2]\)
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.

\(f(q)\) \(=\) \( q + ( -1 - \zeta_{6} ) q^{2} + ( -1 + 2 \zeta_{6} ) q^{3} + \zeta_{6} q^{4} + \zeta_{6} q^{5} + ( 3 - 3 \zeta_{6} ) q^{6} + ( -3 + \zeta_{6} ) q^{7} + ( -1 + 2 \zeta_{6} ) q^{8} -3 q^{9} +O(q^{10})\) \( q + ( -1 - \zeta_{6} ) q^{2} + ( -1 + 2 \zeta_{6} ) q^{3} + \zeta_{6} q^{4} + \zeta_{6} q^{5} + ( 3 - 3 \zeta_{6} ) q^{6} + ( -3 + \zeta_{6} ) q^{7} + ( -1 + 2 \zeta_{6} ) q^{8} -3 q^{9} + ( 1 - 2 \zeta_{6} ) q^{10} + ( -3 - 3 \zeta_{6} ) q^{11} + ( -2 + \zeta_{6} ) q^{12} + ( 6 - 3 \zeta_{6} ) q^{13} + ( 4 + \zeta_{6} ) q^{14} + ( -2 + \zeta_{6} ) q^{15} + ( 5 - 5 \zeta_{6} ) q^{16} -3 q^{17} + ( 3 + 3 \zeta_{6} ) q^{18} + ( 2 - 4 \zeta_{6} ) q^{19} + ( -1 + \zeta_{6} ) q^{20} + ( 1 - 5 \zeta_{6} ) q^{21} + 9 \zeta_{6} q^{22} + ( -4 + 2 \zeta_{6} ) q^{23} -3 q^{24} + ( -1 + \zeta_{6} ) q^{25} -9 q^{26} + ( 3 - 6 \zeta_{6} ) q^{27} + ( -1 - 2 \zeta_{6} ) q^{28} + ( -4 - 4 \zeta_{6} ) q^{29} + 3 q^{30} + ( -8 + 4 \zeta_{6} ) q^{31} + ( -6 + 3 \zeta_{6} ) q^{32} + ( 9 - 9 \zeta_{6} ) q^{33} + ( 3 + 3 \zeta_{6} ) q^{34} + ( -1 - 2 \zeta_{6} ) q^{35} -3 \zeta_{6} q^{36} -2 q^{37} + ( -6 + 6 \zeta_{6} ) q^{38} + 9 \zeta_{6} q^{39} + ( -2 + \zeta_{6} ) q^{40} + ( -6 + 9 \zeta_{6} ) q^{42} + ( -4 + 4 \zeta_{6} ) q^{43} + ( 3 - 6 \zeta_{6} ) q^{44} -3 \zeta_{6} q^{45} + 6 q^{46} + ( -3 + 3 \zeta_{6} ) q^{47} + ( 5 + 5 \zeta_{6} ) q^{48} + ( 8 - 5 \zeta_{6} ) q^{49} + ( 2 - \zeta_{6} ) q^{50} + ( 3 - 6 \zeta_{6} ) q^{51} + ( 3 + 3 \zeta_{6} ) q^{52} + ( -4 + 8 \zeta_{6} ) q^{53} + ( -9 + 9 \zeta_{6} ) q^{54} + ( 3 - 6 \zeta_{6} ) q^{55} + ( 1 - 5 \zeta_{6} ) q^{56} + 6 q^{57} + 12 \zeta_{6} q^{58} + 6 \zeta_{6} q^{59} + ( -1 - \zeta_{6} ) q^{60} + ( 2 + 2 \zeta_{6} ) q^{61} + 12 q^{62} + ( 9 - 3 \zeta_{6} ) q^{63} - q^{64} + ( 3 + 3 \zeta_{6} ) q^{65} + ( -18 + 9 \zeta_{6} ) q^{66} -14 \zeta_{6} q^{67} -3 \zeta_{6} q^{68} -6 \zeta_{6} q^{69} + ( -1 + 5 \zeta_{6} ) q^{70} + ( 3 - 6 \zeta_{6} ) q^{71} + ( 3 - 6 \zeta_{6} ) q^{72} + ( -5 + 10 \zeta_{6} ) q^{73} + ( 2 + 2 \zeta_{6} ) q^{74} + ( -1 - \zeta_{6} ) q^{75} + ( 4 - 2 \zeta_{6} ) q^{76} + ( 12 + 3 \zeta_{6} ) q^{77} + ( 9 - 18 \zeta_{6} ) q^{78} + ( -5 + 5 \zeta_{6} ) q^{79} + 5 q^{80} + 9 q^{81} + ( 9 - 9 \zeta_{6} ) q^{83} + ( 5 - 4 \zeta_{6} ) q^{84} -3 \zeta_{6} q^{85} + ( 8 - 4 \zeta_{6} ) q^{86} + ( 12 - 12 \zeta_{6} ) q^{87} + ( 9 - 9 \zeta_{6} ) q^{88} + 6 q^{89} + ( -3 + 6 \zeta_{6} ) q^{90} + ( -15 + 12 \zeta_{6} ) q^{91} + ( -2 - 2 \zeta_{6} ) q^{92} -12 \zeta_{6} q^{93} + ( 6 - 3 \zeta_{6} ) q^{94} + ( 4 - 2 \zeta_{6} ) q^{95} -9 \zeta_{6} q^{96} + ( 7 + 7 \zeta_{6} ) q^{97} + ( -13 + 2 \zeta_{6} ) q^{98} + ( 9 + 9 \zeta_{6} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 3q^{2} + q^{4} + q^{5} + 3q^{6} - 5q^{7} - 6q^{9} + O(q^{10}) \) \( 2q - 3q^{2} + q^{4} + q^{5} + 3q^{6} - 5q^{7} - 6q^{9} - 9q^{11} - 3q^{12} + 9q^{13} + 9q^{14} - 3q^{15} + 5q^{16} - 6q^{17} + 9q^{18} - q^{20} - 3q^{21} + 9q^{22} - 6q^{23} - 6q^{24} - q^{25} - 18q^{26} - 4q^{28} - 12q^{29} + 6q^{30} - 12q^{31} - 9q^{32} + 9q^{33} + 9q^{34} - 4q^{35} - 3q^{36} - 4q^{37} - 6q^{38} + 9q^{39} - 3q^{40} - 3q^{42} - 4q^{43} - 3q^{45} + 12q^{46} - 3q^{47} + 15q^{48} + 11q^{49} + 3q^{50} + 9q^{52} - 9q^{54} - 3q^{56} + 12q^{57} + 12q^{58} + 6q^{59} - 3q^{60} + 6q^{61} + 24q^{62} + 15q^{63} - 2q^{64} + 9q^{65} - 27q^{66} - 14q^{67} - 3q^{68} - 6q^{69} + 3q^{70} + 6q^{74} - 3q^{75} + 6q^{76} + 27q^{77} - 5q^{79} + 10q^{80} + 18q^{81} + 9q^{83} + 6q^{84} - 3q^{85} + 12q^{86} + 12q^{87} + 9q^{88} + 12q^{89} - 18q^{91} - 6q^{92} - 12q^{93} + 9q^{94} + 6q^{95} - 9q^{96} + 21q^{97} - 24q^{98} + 27q^{99} + O(q^{100}) \)

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
−1.50000 + 0.866025i 1.73205i 0.500000 0.866025i 0.500000 0.866025i 1.50000 + 2.59808i −2.50000 0.866025i 1.73205i −3.00000 1.73205i
146.1 −1.50000 0.866025i 1.73205i 0.500000 + 0.866025i 0.500000 + 0.866025i 1.50000 2.59808i −2.50000 + 0.866025i 1.73205i −3.00000 1.73205i
\(n\): 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.c yes 2
3.b odd 2 1 945.2.bl.e 2
7.b odd 2 1 315.2.bl.b 2
9.c even 3 1 945.2.bl.h 2
9.d odd 6 1 315.2.bl.b 2
21.c even 2 1 945.2.bl.h 2
63.l odd 6 1 945.2.bl.e 2
63.o even 6 1 inner 315.2.bl.c yes 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
315.2.bl.b 2 7.b odd 2 1
315.2.bl.b 2 9.d odd 6 1
315.2.bl.c yes 2 1.a even 1 1 trivial
315.2.bl.c yes 2 63.o even 6 1 inner
945.2.bl.e 2 3.b odd 2 1
945.2.bl.e 2 63.l odd 6 1
945.2.bl.h 2 9.c even 3 1
945.2.bl.h 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 \(S_{2}^{\mathrm{new}}(315, [\chi])\):

\( T_{2}^{2} + 3 T_{2} + 3 \)
\( T_{11}^{2} + 9 T_{11} + 27 \)
\( T_{13}^{2} - 9 T_{13} + 27 \)
\( T_{17} + 3 \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 + 3 T + 5 T^{2} + 6 T^{3} + 4 T^{4} \)
$3$ \( 1 + 3 T^{2} \)
$5$ \( 1 - T + T^{2} \)
$7$ \( 1 + 5 T + 7 T^{2} \)
$11$ \( 1 + 9 T + 38 T^{2} + 99 T^{3} + 121 T^{4} \)
$13$ \( ( 1 - 7 T + 13 T^{2} )( 1 - 2 T + 13 T^{2} ) \)
$17$ \( ( 1 + 3 T + 17 T^{2} )^{2} \)
$19$ \( ( 1 - 8 T + 19 T^{2} )( 1 + 8 T + 19 T^{2} ) \)
$23$ \( 1 + 6 T + 35 T^{2} + 138 T^{3} + 529 T^{4} \)
$29$ \( 1 + 12 T + 77 T^{2} + 348 T^{3} + 841 T^{4} \)
$31$ \( 1 + 12 T + 79 T^{2} + 372 T^{3} + 961 T^{4} \)
$37$ \( ( 1 + 2 T + 37 T^{2} )^{2} \)
$41$ \( 1 - 41 T^{2} + 1681 T^{4} \)
$43$ \( 1 + 4 T - 27 T^{2} + 172 T^{3} + 1849 T^{4} \)
$47$ \( 1 + 3 T - 38 T^{2} + 141 T^{3} + 2209 T^{4} \)
$53$ \( 1 - 58 T^{2} + 2809 T^{4} \)
$59$ \( 1 - 6 T - 23 T^{2} - 354 T^{3} + 3481 T^{4} \)
$61$ \( 1 - 6 T + 73 T^{2} - 366 T^{3} + 3721 T^{4} \)
$67$ \( 1 + 14 T + 129 T^{2} + 938 T^{3} + 4489 T^{4} \)
$71$ \( 1 - 115 T^{2} + 5041 T^{4} \)
$73$ \( 1 - 71 T^{2} + 5329 T^{4} \)
$79$ \( 1 + 5 T - 54 T^{2} + 395 T^{3} + 6241 T^{4} \)
$83$ \( 1 - 9 T - 2 T^{2} - 747 T^{3} + 6889 T^{4} \)
$89$ \( ( 1 - 6 T + 89 T^{2} )^{2} \)
$97$ \( 1 - 21 T + 244 T^{2} - 2037 T^{3} + 9409 T^{4} \)
show more
show less