Properties

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

Newform invariants

Self dual: no
Analytic conductor: \(2.51528766367\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
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 \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \zeta_{12}^{3} q^{2} + ( \zeta_{12} + \zeta_{12}^{3} ) q^{3} + q^{4} + ( -1 + 2 \zeta_{12} + \zeta_{12}^{2} ) q^{5} + ( -2 + \zeta_{12}^{2} ) q^{6} + ( 2 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{7} + 3 \zeta_{12}^{3} q^{8} + ( -3 + 3 \zeta_{12}^{2} ) q^{9} +O(q^{10})\) \( q + \zeta_{12}^{3} q^{2} + ( \zeta_{12} + \zeta_{12}^{3} ) q^{3} + q^{4} + ( -1 + 2 \zeta_{12} + \zeta_{12}^{2} ) q^{5} + ( -2 + \zeta_{12}^{2} ) q^{6} + ( 2 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{7} + 3 \zeta_{12}^{3} q^{8} + ( -3 + 3 \zeta_{12}^{2} ) q^{9} + ( -2 - \zeta_{12} + 2 \zeta_{12}^{2} ) q^{10} -6 \zeta_{12}^{2} q^{11} + ( \zeta_{12} + \zeta_{12}^{3} ) q^{12} + ( 4 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{13} + ( 1 + 2 \zeta_{12}^{2} ) q^{14} + ( -2 - 2 \zeta_{12} + 4 \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{15} - q^{16} -2 \zeta_{12} q^{17} -3 \zeta_{12} q^{18} -6 \zeta_{12}^{2} q^{19} + ( -1 + 2 \zeta_{12} + \zeta_{12}^{2} ) q^{20} + ( 4 + \zeta_{12}^{2} ) q^{21} + ( 6 \zeta_{12} - 6 \zeta_{12}^{3} ) q^{22} -3 \zeta_{12} q^{23} + ( -6 + 3 \zeta_{12}^{2} ) q^{24} + ( -4 \zeta_{12} + 3 \zeta_{12}^{2} + 4 \zeta_{12}^{3} ) q^{25} + 4 \zeta_{12}^{2} q^{26} + ( -6 \zeta_{12} + 3 \zeta_{12}^{3} ) q^{27} + ( 2 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{28} + ( -2 + 2 \zeta_{12}^{2} ) q^{29} + ( 1 - 4 \zeta_{12} - 2 \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{30} -4 q^{31} + 5 \zeta_{12}^{3} q^{32} + ( 6 \zeta_{12} - 12 \zeta_{12}^{3} ) q^{33} + ( 2 - 2 \zeta_{12}^{2} ) q^{34} + ( 6 + \zeta_{12} - 2 \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{35} + ( -3 + 3 \zeta_{12}^{2} ) q^{36} + ( -8 \zeta_{12} + 8 \zeta_{12}^{3} ) q^{37} + ( 6 \zeta_{12} - 6 \zeta_{12}^{3} ) q^{38} + ( 4 + 4 \zeta_{12}^{2} ) q^{39} + ( -6 - 3 \zeta_{12} + 6 \zeta_{12}^{2} ) q^{40} -2 \zeta_{12}^{2} q^{41} + ( -\zeta_{12} + 5 \zeta_{12}^{3} ) q^{42} + \zeta_{12} q^{43} -6 \zeta_{12}^{2} q^{44} + ( -6 \zeta_{12} - 3 \zeta_{12}^{2} + 6 \zeta_{12}^{3} ) q^{45} + ( 3 - 3 \zeta_{12}^{2} ) q^{46} + 3 \zeta_{12}^{3} q^{47} + ( -\zeta_{12} - \zeta_{12}^{3} ) q^{48} + ( 3 - 8 \zeta_{12}^{2} ) q^{49} + ( -3 \zeta_{12} - 4 \zeta_{12}^{2} + 3 \zeta_{12}^{3} ) q^{50} + ( 2 - 4 \zeta_{12}^{2} ) q^{51} + ( 4 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{52} + 12 \zeta_{12} q^{53} + ( 3 - 6 \zeta_{12}^{2} ) q^{54} + ( 6 - 12 \zeta_{12}^{3} ) q^{55} + ( 3 + 6 \zeta_{12}^{2} ) q^{56} + ( 6 \zeta_{12} - 12 \zeta_{12}^{3} ) q^{57} -2 \zeta_{12} q^{58} -4 q^{59} + ( -2 - 2 \zeta_{12} + 4 \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{60} + 7 q^{61} -4 \zeta_{12}^{3} q^{62} + ( 3 \zeta_{12} + 6 \zeta_{12}^{3} ) q^{63} -7 q^{64} + ( 8 + 4 \zeta_{12}^{3} ) q^{65} + ( 6 + 6 \zeta_{12}^{2} ) q^{66} + 7 \zeta_{12}^{3} q^{67} -2 \zeta_{12} q^{68} + ( 3 - 6 \zeta_{12}^{2} ) q^{69} + ( -3 + 2 \zeta_{12} + \zeta_{12}^{2} + 4 \zeta_{12}^{3} ) q^{70} + 4 q^{71} -9 \zeta_{12} q^{72} -8 \zeta_{12}^{2} q^{74} + ( -4 - 3 \zeta_{12} - 4 \zeta_{12}^{2} + 6 \zeta_{12}^{3} ) q^{75} -6 \zeta_{12}^{2} q^{76} + ( -18 \zeta_{12} + 6 \zeta_{12}^{3} ) q^{77} + ( -4 \zeta_{12} + 8 \zeta_{12}^{3} ) q^{78} + 14 q^{79} + ( 1 - 2 \zeta_{12} - \zeta_{12}^{2} ) q^{80} -9 \zeta_{12}^{2} q^{81} + ( 2 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{82} + 4 \zeta_{12} q^{83} + ( 4 + \zeta_{12}^{2} ) q^{84} + ( 2 \zeta_{12} - 4 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{85} + ( -1 + \zeta_{12}^{2} ) q^{86} + ( -4 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{87} + ( 18 \zeta_{12} - 18 \zeta_{12}^{3} ) q^{88} -3 \zeta_{12}^{2} q^{89} + ( 3 \zeta_{12} - 6 \zeta_{12}^{2} - 3 \zeta_{12}^{3} ) q^{90} + ( 8 - 12 \zeta_{12}^{2} ) q^{91} -3 \zeta_{12} q^{92} + ( -4 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{93} -3 q^{94} + ( 6 - 12 \zeta_{12}^{3} ) q^{95} + ( -10 + 5 \zeta_{12}^{2} ) q^{96} -2 \zeta_{12} q^{97} + ( 8 \zeta_{12} - 5 \zeta_{12}^{3} ) q^{98} + 18 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 4q^{4} - 2q^{5} - 6q^{6} - 6q^{9} + O(q^{10}) \) \( 4q + 4q^{4} - 2q^{5} - 6q^{6} - 6q^{9} - 4q^{10} - 12q^{11} + 8q^{14} - 4q^{16} - 12q^{19} - 2q^{20} + 18q^{21} - 18q^{24} + 6q^{25} + 8q^{26} - 4q^{29} - 16q^{31} + 4q^{34} + 20q^{35} - 6q^{36} + 24q^{39} - 12q^{40} - 4q^{41} - 12q^{44} - 6q^{45} + 6q^{46} - 4q^{49} - 8q^{50} + 24q^{55} + 24q^{56} - 16q^{59} + 28q^{61} - 28q^{64} + 32q^{65} + 36q^{66} - 10q^{70} + 16q^{71} - 16q^{74} - 24q^{75} - 12q^{76} + 56q^{79} + 2q^{80} - 18q^{81} + 18q^{84} - 8q^{85} - 2q^{86} - 6q^{89} - 12q^{90} + 8q^{91} - 12q^{94} + 24q^{95} - 30q^{96} + 72q^{99} + O(q^{100}) \)

Character values

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

\(n\) \(127\) \(136\) \(281\)
\(\chi(n)\) \(-1\) \(-\zeta_{12}^{2}\) \(-\zeta_{12}^{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
184.1
0.866025 0.500000i
−0.866025 + 0.500000i
−0.866025 0.500000i
0.866025 + 0.500000i
1.00000i 0.866025 1.50000i 1.00000 1.23205 1.86603i −1.50000 0.866025i 1.73205 + 2.00000i 3.00000i −1.50000 2.59808i −1.86603 1.23205i
184.2 1.00000i −0.866025 + 1.50000i 1.00000 −2.23205 + 0.133975i −1.50000 0.866025i −1.73205 2.00000i 3.00000i −1.50000 2.59808i −0.133975 2.23205i
214.1 1.00000i −0.866025 1.50000i 1.00000 −2.23205 0.133975i −1.50000 + 0.866025i −1.73205 + 2.00000i 3.00000i −1.50000 + 2.59808i −0.133975 + 2.23205i
214.2 1.00000i 0.866025 + 1.50000i 1.00000 1.23205 + 1.86603i −1.50000 + 0.866025i 1.73205 2.00000i 3.00000i −1.50000 + 2.59808i −1.86603 + 1.23205i
\(n\): 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
63.h even 3 1 inner
315.r 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.r.a 4
3.b odd 2 1 945.2.r.a 4
5.b even 2 1 inner 315.2.r.a 4
7.c even 3 1 315.2.bo.a yes 4
9.c even 3 1 315.2.bo.a yes 4
9.d odd 6 1 945.2.bo.a 4
15.d odd 2 1 945.2.r.a 4
21.h odd 6 1 945.2.bo.a 4
35.j even 6 1 315.2.bo.a yes 4
45.h odd 6 1 945.2.bo.a 4
45.j even 6 1 315.2.bo.a yes 4
63.h even 3 1 inner 315.2.r.a 4
63.j odd 6 1 945.2.r.a 4
105.o odd 6 1 945.2.bo.a 4
315.r even 6 1 inner 315.2.r.a 4
315.br odd 6 1 945.2.r.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
315.2.r.a 4 1.a even 1 1 trivial
315.2.r.a 4 5.b even 2 1 inner
315.2.r.a 4 63.h even 3 1 inner
315.2.r.a 4 315.r even 6 1 inner
315.2.bo.a yes 4 7.c even 3 1
315.2.bo.a yes 4 9.c even 3 1
315.2.bo.a yes 4 35.j even 6 1
315.2.bo.a yes 4 45.j even 6 1
945.2.r.a 4 3.b odd 2 1
945.2.r.a 4 15.d odd 2 1
945.2.r.a 4 63.j odd 6 1
945.2.r.a 4 315.br odd 6 1
945.2.bo.a 4 9.d odd 6 1
945.2.bo.a 4 21.h odd 6 1
945.2.bo.a 4 45.h odd 6 1
945.2.bo.a 4 105.o odd 6 1

Hecke kernels

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

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - 3 T^{2} + 4 T^{4} )^{2} \)
$3$ \( 1 + 3 T^{2} + 9 T^{4} \)
$5$ \( 1 + 2 T - T^{2} + 10 T^{3} + 25 T^{4} \)
$7$ \( 1 + 2 T^{2} + 49 T^{4} \)
$11$ \( ( 1 + 6 T + 25 T^{2} + 66 T^{3} + 121 T^{4} )^{2} \)
$13$ \( ( 1 - 6 T + 23 T^{2} - 78 T^{3} + 169 T^{4} )( 1 + 6 T + 23 T^{2} + 78 T^{3} + 169 T^{4} ) \)
$17$ \( ( 1 - 8 T + 47 T^{2} - 136 T^{3} + 289 T^{4} )( 1 + 8 T + 47 T^{2} + 136 T^{3} + 289 T^{4} ) \)
$19$ \( ( 1 + 6 T + 17 T^{2} + 114 T^{3} + 361 T^{4} )^{2} \)
$23$ \( 1 + 37 T^{2} + 840 T^{4} + 19573 T^{6} + 279841 T^{8} \)
$29$ \( ( 1 + 2 T - 25 T^{2} + 58 T^{3} + 841 T^{4} )^{2} \)
$31$ \( ( 1 + 4 T + 31 T^{2} )^{4} \)
$37$ \( 1 + 10 T^{2} - 1269 T^{4} + 13690 T^{6} + 1874161 T^{8} \)
$41$ \( ( 1 + 2 T - 37 T^{2} + 82 T^{3} + 1681 T^{4} )^{2} \)
$43$ \( 1 + 85 T^{2} + 5376 T^{4} + 157165 T^{6} + 3418801 T^{8} \)
$47$ \( ( 1 - 85 T^{2} + 2209 T^{4} )^{2} \)
$53$ \( 1 - 38 T^{2} - 1365 T^{4} - 106742 T^{6} + 7890481 T^{8} \)
$59$ \( ( 1 + 4 T + 59 T^{2} )^{4} \)
$61$ \( ( 1 - 7 T + 61 T^{2} )^{4} \)
$67$ \( ( 1 - 85 T^{2} + 4489 T^{4} )^{2} \)
$71$ \( ( 1 - 4 T + 71 T^{2} )^{4} \)
$73$ \( ( 1 + 73 T^{2} + 5329 T^{4} )^{2} \)
$79$ \( ( 1 - 14 T + 79 T^{2} )^{4} \)
$83$ \( 1 + 150 T^{2} + 15611 T^{4} + 1033350 T^{6} + 47458321 T^{8} \)
$89$ \( ( 1 + 3 T - 80 T^{2} + 267 T^{3} + 7921 T^{4} )^{2} \)
$97$ \( 1 + 190 T^{2} + 26691 T^{4} + 1787710 T^{6} + 88529281 T^{8} \)
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