Properties

Label 315.2.bs.b
Level 315
Weight 2
Character orbit 315.bs
Analytic conductor 2.515
Analytic rank 0
Dimension 4
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.bs (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(2.51528766367\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( -1 + \zeta_{12} + \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{2} + ( -\zeta_{12} - \zeta_{12}^{3} ) q^{3} + ( 1 - 2 \zeta_{12}^{2} ) q^{4} + ( 2 - \zeta_{12} - 2 \zeta_{12}^{2} ) q^{5} + ( -1 + 2 \zeta_{12} - \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{6} + ( -2 \zeta_{12} + 3 \zeta_{12}^{3} ) q^{7} + ( 1 + \zeta_{12} - \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{8} + ( -3 + 3 \zeta_{12}^{2} ) q^{9} +O(q^{10})\) \( q + ( -1 + \zeta_{12} + \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{2} + ( -\zeta_{12} - \zeta_{12}^{3} ) q^{3} + ( 1 - 2 \zeta_{12}^{2} ) q^{4} + ( 2 - \zeta_{12} - 2 \zeta_{12}^{2} ) q^{5} + ( -1 + 2 \zeta_{12} - \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{6} + ( -2 \zeta_{12} + 3 \zeta_{12}^{3} ) q^{7} + ( 1 + \zeta_{12} - \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{8} + ( -3 + 3 \zeta_{12}^{2} ) q^{9} + ( -1 + \zeta_{12} + 2 \zeta_{12}^{2} - 3 \zeta_{12}^{3} ) q^{10} + ( -2 \zeta_{12} + 2 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{11} + ( -3 \zeta_{12} + 3 \zeta_{12}^{3} ) q^{12} + ( 2 - 2 \zeta_{12} ) q^{13} + ( -2 - \zeta_{12} + 3 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{14} + ( -1 - 4 \zeta_{12} + 2 \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{15} + ( 1 - 4 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{16} + ( -2 + 2 \zeta_{12} + 2 \zeta_{12}^{2} ) q^{17} + ( -3 \zeta_{12}^{2} + 3 \zeta_{12}^{3} ) q^{18} + ( -\zeta_{12} - 5 \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{19} + ( -2 - \zeta_{12} - 2 \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{20} + ( 4 + \zeta_{12}^{2} ) q^{21} + ( -4 + 6 \zeta_{12} - 2 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{22} + ( -3 + 3 \zeta_{12} + 2 \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{23} + ( -1 - 2 \zeta_{12} - \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{24} + ( -4 \zeta_{12} - 3 \zeta_{12}^{2} + 4 \zeta_{12}^{3} ) q^{25} + ( -4 + 4 \zeta_{12} + 2 \zeta_{12}^{2} - 4 \zeta_{12}^{3} ) q^{26} + ( 6 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{27} + ( 4 \zeta_{12} + \zeta_{12}^{3} ) q^{28} + ( 4 + 4 \zeta_{12}^{2} ) q^{29} + ( -5 + 3 \zeta_{12} + \zeta_{12}^{2} - 3 \zeta_{12}^{3} ) q^{30} + ( 3 - 6 \zeta_{12}^{2} - 5 \zeta_{12}^{3} ) q^{31} + ( -5 + \zeta_{12} + \zeta_{12}^{2} - 5 \zeta_{12}^{3} ) q^{32} + ( -6 + 2 \zeta_{12} + 6 \zeta_{12}^{2} - 4 \zeta_{12}^{3} ) q^{33} + ( 2 - 2 \zeta_{12} - 2 \zeta_{12}^{2} + 4 \zeta_{12}^{3} ) q^{34} + ( 3 + 2 \zeta_{12} - \zeta_{12}^{2} + 4 \zeta_{12}^{3} ) q^{35} + ( 3 + 3 \zeta_{12}^{2} ) q^{36} + ( 3 \zeta_{12} - 3 \zeta_{12}^{2} - 3 \zeta_{12}^{3} ) q^{37} + ( 4 - 3 \zeta_{12} - \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{38} + ( -2 - 2 \zeta_{12} + 4 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{39} + ( -1 - \zeta_{12} - 2 \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{40} + ( 4 + 6 \zeta_{12} - 2 \zeta_{12}^{2} - 6 \zeta_{12}^{3} ) q^{41} + ( -5 + 5 \zeta_{12} + 4 \zeta_{12}^{2} - 4 \zeta_{12}^{3} ) q^{42} + ( 3 \zeta_{12}^{2} + 3 \zeta_{12}^{3} ) q^{43} + ( 4 - 6 \zeta_{12} - 2 \zeta_{12}^{2} + 6 \zeta_{12}^{3} ) q^{44} + ( 3 \zeta_{12} + 6 \zeta_{12}^{2} - 3 \zeta_{12}^{3} ) q^{45} + ( 4 - 3 \zeta_{12} - 4 \zeta_{12}^{2} + 6 \zeta_{12}^{3} ) q^{46} + ( 3 + 9 \zeta_{12} - 9 \zeta_{12}^{2} - 3 \zeta_{12}^{3} ) q^{47} + ( -\zeta_{12} + 6 \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{48} + ( 3 - 8 \zeta_{12}^{2} ) q^{49} + ( -1 - 3 \zeta_{12} + 4 \zeta_{12}^{2} - 4 \zeta_{12}^{3} ) q^{50} + ( 2 + 4 \zeta_{12} - 4 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{51} + ( 2 - 2 \zeta_{12} - 4 \zeta_{12}^{2} + 4 \zeta_{12}^{3} ) q^{52} + ( 3 - 3 \zeta_{12} + \zeta_{12}^{2} + 4 \zeta_{12}^{3} ) q^{53} + ( 6 - 3 \zeta_{12} - 3 \zeta_{12}^{2} + 6 \zeta_{12}^{3} ) q^{54} + ( 2 - 8 \zeta_{12} + 4 \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{55} + ( -2 + \zeta_{12} + 3 \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{56} + ( -3 - 5 \zeta_{12} + 3 \zeta_{12}^{2} + 10 \zeta_{12}^{3} ) q^{57} + ( -8 + 8 \zeta_{12} + 4 \zeta_{12}^{2} - 4 \zeta_{12}^{3} ) q^{58} + ( -4 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{59} + ( 3 + 6 \zeta_{12}^{3} ) q^{60} + ( -6 + 12 \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{61} + ( 3 + 2 \zeta_{12} - 2 \zeta_{12}^{2} - 3 \zeta_{12}^{3} ) q^{62} + ( -3 \zeta_{12} - 6 \zeta_{12}^{3} ) q^{63} + ( 1 - 2 \zeta_{12}^{2} + 4 \zeta_{12}^{3} ) q^{64} + ( 4 - 6 \zeta_{12} - 2 \zeta_{12}^{2} + 4 \zeta_{12}^{3} ) q^{65} + ( 2 + 2 \zeta_{12} - 10 \zeta_{12}^{2} + 8 \zeta_{12}^{3} ) q^{66} + ( 2 + 3 \zeta_{12} + 3 \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{67} + ( 2 + 2 \zeta_{12} + 2 \zeta_{12}^{2} - 4 \zeta_{12}^{3} ) q^{68} + ( 1 + 5 \zeta_{12} - 5 \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{69} + ( -4 \zeta_{12} + 7 \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{70} -6 q^{71} + ( 3 \zeta_{12}^{2} + 3 \zeta_{12}^{3} ) q^{72} + ( 1 + \zeta_{12} - 5 \zeta_{12}^{2} + 4 \zeta_{12}^{3} ) q^{73} + ( 6 - 3 \zeta_{12} - 3 \zeta_{12}^{2} + 3 \zeta_{12}^{3} ) q^{74} + ( 4 - 3 \zeta_{12} + 4 \zeta_{12}^{2} + 6 \zeta_{12}^{3} ) q^{75} + ( -10 - 3 \zeta_{12} + 5 \zeta_{12}^{2} + 3 \zeta_{12}^{3} ) q^{76} + ( 8 - 6 \zeta_{12} + 2 \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{77} + ( -4 + 6 \zeta_{12} - 4 \zeta_{12}^{2} ) q^{78} + ( 2 - 4 \zeta_{12}^{2} - 8 \zeta_{12}^{3} ) q^{79} + ( 4 - 5 \zeta_{12} + 8 \zeta_{12}^{3} ) q^{80} -9 \zeta_{12}^{2} q^{81} + ( 4 + 2 \zeta_{12} - 2 \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{82} + ( 1 + \zeta_{12} - \zeta_{12}^{2} ) q^{83} + ( 6 - 9 \zeta_{12}^{2} ) q^{84} + ( 6 \zeta_{12} + 2 \zeta_{12}^{2} - 6 \zeta_{12}^{3} ) q^{85} + ( -3 + 3 \zeta_{12}^{2} ) q^{86} -12 \zeta_{12}^{3} q^{87} + ( -2 \zeta_{12} - 2 \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{88} + ( 3 \zeta_{12} - 4 \zeta_{12}^{2} + 3 \zeta_{12}^{3} ) q^{89} + ( -3 + 6 \zeta_{12} - 3 \zeta_{12}^{2} + 3 \zeta_{12}^{3} ) q^{90} + ( 6 - 4 \zeta_{12} - 2 \zeta_{12}^{2} + 6 \zeta_{12}^{3} ) q^{91} + ( 1 + \zeta_{12} + 4 \zeta_{12}^{2} - 5 \zeta_{12}^{3} ) q^{92} + ( -10 - 9 \zeta_{12} + 5 \zeta_{12}^{2} + 9 \zeta_{12}^{3} ) q^{93} + ( 15 - 12 \zeta_{12} + 6 \zeta_{12}^{3} ) q^{94} + ( -11 - 4 \zeta_{12} + 2 \zeta_{12}^{2} + 7 \zeta_{12}^{3} ) q^{95} + ( -9 + 6 \zeta_{12} + 3 \zeta_{12}^{2} + 3 \zeta_{12}^{3} ) q^{96} + ( 5 - 5 \zeta_{12} - 3 \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{97} + ( 5 - 5 \zeta_{12} + 3 \zeta_{12}^{2} - 3 \zeta_{12}^{3} ) q^{98} + ( -6 + 12 \zeta_{12} - 6 \zeta_{12}^{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 2q^{2} + 4q^{5} - 6q^{6} + 2q^{8} - 6q^{9} + O(q^{10}) \) \( 4q - 2q^{2} + 4q^{5} - 6q^{6} + 2q^{8} - 6q^{9} + 4q^{11} + 8q^{13} - 2q^{14} + 4q^{16} - 4q^{17} - 6q^{18} - 10q^{19} - 12q^{20} + 18q^{21} - 20q^{22} - 8q^{23} - 6q^{24} - 6q^{25} - 12q^{26} + 24q^{29} - 18q^{30} - 18q^{32} - 12q^{33} + 4q^{34} + 10q^{35} + 18q^{36} - 6q^{37} + 14q^{38} - 8q^{40} + 12q^{41} - 12q^{42} + 6q^{43} + 12q^{44} + 12q^{45} + 8q^{46} - 6q^{47} + 12q^{48} - 4q^{49} + 4q^{50} + 14q^{53} + 18q^{54} + 16q^{55} - 2q^{56} - 6q^{57} - 24q^{58} + 12q^{60} + 8q^{62} + 12q^{65} - 12q^{66} + 14q^{67} + 12q^{68} - 6q^{69} + 14q^{70} - 24q^{71} + 6q^{72} - 6q^{73} + 18q^{74} + 24q^{75} - 30q^{76} + 36q^{77} - 24q^{78} + 16q^{80} - 18q^{81} + 12q^{82} + 2q^{83} + 6q^{84} + 4q^{85} - 6q^{86} - 4q^{88} - 8q^{89} - 18q^{90} + 20q^{91} + 12q^{92} - 30q^{93} + 60q^{94} - 40q^{95} - 30q^{96} + 14q^{97} + 26q^{98} - 24q^{99} + O(q^{100}) \)

Character values

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

\(n\) \(127\) \(136\) \(281\)
\(\chi(n)\) \(\zeta_{12}^{3}\) \(\zeta_{12}^{2}\) \(-\zeta_{12}^{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
52.1
0.866025 + 0.500000i
0.866025 0.500000i
−0.866025 0.500000i
−0.866025 + 0.500000i
0.366025 + 0.366025i −0.866025 1.50000i 1.73205i 0.133975 2.23205i 0.232051 0.866025i −1.73205 + 2.00000i 1.36603 1.36603i −1.50000 + 2.59808i 0.866025 0.767949i
103.1 0.366025 0.366025i −0.866025 + 1.50000i 1.73205i 0.133975 + 2.23205i 0.232051 + 0.866025i −1.73205 2.00000i 1.36603 + 1.36603i −1.50000 2.59808i 0.866025 + 0.767949i
178.1 −1.36603 + 1.36603i 0.866025 + 1.50000i 1.73205i 1.86603 1.23205i −3.23205 0.866025i 1.73205 2.00000i −0.366025 0.366025i −1.50000 + 2.59808i −0.866025 + 4.23205i
292.1 −1.36603 1.36603i 0.866025 1.50000i 1.73205i 1.86603 + 1.23205i −3.23205 + 0.866025i 1.73205 + 2.00000i −0.366025 + 0.366025i −1.50000 2.59808i −0.866025 4.23205i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
315.bs even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 315.2.bs.b 4
3.b odd 2 1 945.2.bv.a 4
5.c odd 4 1 315.2.bs.c yes 4
7.d odd 6 1 315.2.cg.c yes 4
9.c even 3 1 315.2.cg.a yes 4
9.d odd 6 1 945.2.cj.d 4
15.e even 4 1 945.2.bv.d 4
21.g even 6 1 945.2.cj.a 4
35.k even 12 1 315.2.cg.a yes 4
45.k odd 12 1 315.2.cg.c yes 4
45.l even 12 1 945.2.cj.a 4
63.i even 6 1 945.2.bv.d 4
63.t odd 6 1 315.2.bs.c yes 4
105.w odd 12 1 945.2.cj.d 4
315.bs even 12 1 inner 315.2.bs.b 4
315.bu odd 12 1 945.2.bv.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
315.2.bs.b 4 1.a even 1 1 trivial
315.2.bs.b 4 315.bs even 12 1 inner
315.2.bs.c yes 4 5.c odd 4 1
315.2.bs.c yes 4 63.t odd 6 1
315.2.cg.a yes 4 9.c even 3 1
315.2.cg.a yes 4 35.k even 12 1
315.2.cg.c yes 4 7.d odd 6 1
315.2.cg.c yes 4 45.k odd 12 1
945.2.bv.a 4 3.b odd 2 1
945.2.bv.a 4 315.bu odd 12 1
945.2.bv.d 4 15.e even 4 1
945.2.bv.d 4 63.i even 6 1
945.2.cj.a 4 21.g even 6 1
945.2.cj.a 4 45.l even 12 1
945.2.cj.d 4 9.d odd 6 1
945.2.cj.d 4 105.w odd 12 1

Hecke kernels

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

\( T_{2}^{4} + 2 T_{2}^{3} + 2 T_{2}^{2} - 2 T_{2} + 1 \)
\( T_{11}^{4} - 4 T_{11}^{3} + 24 T_{11}^{2} + 32 T_{11} + 64 \)
\( T_{13}^{4} - 8 T_{13}^{3} + 20 T_{13}^{2} - 16 T_{13} + 16 \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 + 2 T + 2 T^{2} + 2 T^{3} + T^{4} + 4 T^{5} + 8 T^{6} + 16 T^{7} + 16 T^{8} \)
$3$ \( 1 + 3 T^{2} + 9 T^{4} \)
$5$ \( 1 - 4 T + 11 T^{2} - 20 T^{3} + 25 T^{4} \)
$7$ \( 1 + 2 T^{2} + 49 T^{4} \)
$11$ \( 1 - 4 T + 2 T^{2} + 32 T^{3} - 101 T^{4} + 352 T^{5} + 242 T^{6} - 5324 T^{7} + 14641 T^{8} \)
$13$ \( 1 - 8 T + 20 T^{2} + 36 T^{3} - 361 T^{4} + 468 T^{5} + 3380 T^{6} - 17576 T^{7} + 28561 T^{8} \)
$17$ \( 1 + 4 T + 8 T^{2} - 104 T^{3} - 497 T^{4} - 1768 T^{5} + 2312 T^{6} + 19652 T^{7} + 83521 T^{8} \)
$19$ \( 1 + 10 T + 40 T^{2} + 220 T^{3} + 1339 T^{4} + 4180 T^{5} + 14440 T^{6} + 68590 T^{7} + 130321 T^{8} \)
$23$ \( 1 + 8 T + 17 T^{2} - 208 T^{3} - 1604 T^{4} - 4784 T^{5} + 8993 T^{6} + 97336 T^{7} + 279841 T^{8} \)
$29$ \( ( 1 - 12 T + 77 T^{2} - 348 T^{3} + 841 T^{4} )^{2} \)
$31$ \( 1 - 20 T^{2} - 678 T^{4} - 19220 T^{6} + 923521 T^{8} \)
$37$ \( 1 + 6 T + 18 T^{2} - 336 T^{3} - 2377 T^{4} - 12432 T^{5} + 24642 T^{6} + 303918 T^{7} + 1874161 T^{8} \)
$41$ \( 1 - 12 T + 106 T^{2} - 696 T^{3} + 3651 T^{4} - 28536 T^{5} + 178186 T^{6} - 827052 T^{7} + 2825761 T^{8} \)
$43$ \( 1 - 6 T + 45 T^{2} - 366 T^{3} + 1328 T^{4} - 15738 T^{5} + 83205 T^{6} - 477042 T^{7} + 3418801 T^{8} \)
$47$ \( 1 + 6 T + 18 T^{2} - 420 T^{3} - 3889 T^{4} - 19740 T^{5} + 39762 T^{6} + 622938 T^{7} + 4879681 T^{8} \)
$53$ \( 1 - 14 T + 74 T^{2} - 8 T^{3} - 2537 T^{4} - 424 T^{5} + 207866 T^{6} - 2084278 T^{7} + 7890481 T^{8} \)
$59$ \( ( 1 + 106 T^{2} + 3481 T^{4} )^{2} \)
$61$ \( 1 - 26 T^{2} + 7179 T^{4} - 96746 T^{6} + 13845841 T^{8} \)
$67$ \( 1 - 14 T + 98 T^{2} - 1092 T^{3} + 12047 T^{4} - 73164 T^{5} + 439922 T^{6} - 4210682 T^{7} + 20151121 T^{8} \)
$71$ \( ( 1 + 6 T + 71 T^{2} )^{4} \)
$73$ \( 1 + 6 T + 90 T^{2} + 984 T^{3} + 6095 T^{4} + 71832 T^{5} + 479610 T^{6} + 2334102 T^{7} + 28398241 T^{8} \)
$79$ \( 1 - 164 T^{2} + 16134 T^{4} - 1023524 T^{6} + 38950081 T^{8} \)
$83$ \( 1 - 2 T + 2 T^{2} + 328 T^{3} - 7217 T^{4} + 27224 T^{5} + 13778 T^{6} - 1143574 T^{7} + 47458321 T^{8} \)
$89$ \( 1 + 8 T - 103 T^{2} - 88 T^{3} + 14272 T^{4} - 7832 T^{5} - 815863 T^{6} + 5639752 T^{7} + 62742241 T^{8} \)
$97$ \( 1 - 14 T + 50 T^{2} + 1500 T^{3} - 23089 T^{4} + 145500 T^{5} + 470450 T^{6} - 12777422 T^{7} + 88529281 T^{8} \)
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