Properties

Label 315.2.cg.a
Level 315
Weight 2
Character orbit 315.cg
Analytic conductor 2.515
Analytic rank 1
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.cg (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(2.51528766367\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
Coefficient ring: \(\Z[a_1, a_2]\)
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.

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

Character values

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

\(n\) \(127\) \(136\) \(281\)
\(\chi(n)\) \(\zeta_{12}^{3}\) \(\zeta_{12}^{2}\) \(-1 + \zeta_{12}^{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
157.1
0.866025 + 0.500000i
−0.866025 + 0.500000i
−0.866025 0.500000i
0.866025 0.500000i
−0.500000 + 0.133975i 1.73205i −1.50000 + 0.866025i −2.00000 + 1.00000i −0.232051 0.866025i −0.866025 2.50000i 1.36603 1.36603i −3.00000 0.866025 0.767949i
187.1 −0.500000 + 1.86603i 1.73205i −1.50000 0.866025i −2.00000 + 1.00000i 3.23205 + 0.866025i 0.866025 2.50000i −0.366025 + 0.366025i −3.00000 −0.866025 4.23205i
283.1 −0.500000 1.86603i 1.73205i −1.50000 + 0.866025i −2.00000 1.00000i 3.23205 0.866025i 0.866025 + 2.50000i −0.366025 0.366025i −3.00000 −0.866025 + 4.23205i
313.1 −0.500000 0.133975i 1.73205i −1.50000 0.866025i −2.00000 1.00000i −0.232051 + 0.866025i −0.866025 + 2.50000i 1.36603 + 1.36603i −3.00000 0.866025 + 0.767949i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

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

Hecke kernels

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

\( T_{2}^{4} + 2 T_{2}^{3} + 5 T_{2}^{2} + 4 T_{2} + 1 \)
\( T_{11}^{2} + 4 T_{11} - 8 \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 + 2 T + 5 T^{2} + 8 T^{3} + 13 T^{4} + 16 T^{5} + 20 T^{6} + 16 T^{7} + 16 T^{8} \)
$3$ \( ( 1 + 3 T^{2} )^{2} \)
$5$ \( ( 1 + 4 T + 5 T^{2} )^{2} \)
$7$ \( 1 + 11 T^{2} + 49 T^{4} \)
$11$ \( ( 1 + 4 T + 14 T^{2} + 44 T^{3} + 121 T^{4} )^{2} \)
$13$ \( 1 + 4 T + 20 T^{2} + 84 T^{3} + 263 T^{4} + 1092 T^{5} + 3380 T^{6} + 8788 T^{7} + 28561 T^{8} \)
$17$ \( 1 + 4 T + 8 T^{2} - 104 T^{3} - 497 T^{4} - 1768 T^{5} + 2312 T^{6} + 19652 T^{7} + 83521 T^{8} \)
$19$ \( 1 + 10 T + 40 T^{2} + 220 T^{3} + 1339 T^{4} + 4180 T^{5} + 14440 T^{6} + 68590 T^{7} + 130321 T^{8} \)
$23$ \( 1 - 10 T + 50 T^{2} - 340 T^{3} + 2191 T^{4} - 7820 T^{5} + 26450 T^{6} - 121670 T^{7} + 279841 T^{8} \)
$29$ \( ( 1 + 12 T + 77 T^{2} + 348 T^{3} + 841 T^{4} )^{2} \)
$31$ \( 1 + 18 T + 172 T^{2} + 1152 T^{3} + 6483 T^{4} + 35712 T^{5} + 165292 T^{6} + 536238 T^{7} + 923521 T^{8} \)
$37$ \( 1 + 6 T + 18 T^{2} - 336 T^{3} - 2377 T^{4} - 12432 T^{5} + 24642 T^{6} + 303918 T^{7} + 1874161 T^{8} \)
$41$ \( 1 + 12 T + 106 T^{2} + 696 T^{3} + 3651 T^{4} + 28536 T^{5} + 178186 T^{6} + 827052 T^{7} + 2825761 T^{8} \)
$43$ \( 1 + 12 T + 45 T^{2} - 204 T^{3} - 3316 T^{4} - 8772 T^{5} + 83205 T^{6} + 954084 T^{7} + 3418801 T^{8} \)
$47$ \( 1 - 30 T + 369 T^{2} - 2526 T^{3} + 14864 T^{4} - 118722 T^{5} + 815121 T^{6} - 3114690 T^{7} + 4879681 T^{8} \)
$53$ \( 1 - 14 T + 74 T^{2} - 8 T^{3} - 2537 T^{4} - 424 T^{5} + 207866 T^{6} - 2084278 T^{7} + 7890481 T^{8} \)
$59$ \( 1 - 106 T^{2} + 7755 T^{4} - 368986 T^{6} + 12117361 T^{8} \)
$61$ \( 1 + 36 T + 661 T^{2} + 8244 T^{3} + 75072 T^{4} + 502884 T^{5} + 2459581 T^{6} + 8171316 T^{7} + 13845841 T^{8} \)
$67$ \( 1 - 2 T + 65 T^{2} - 762 T^{3} + 2600 T^{4} - 51054 T^{5} + 291785 T^{6} - 601526 T^{7} + 20151121 T^{8} \)
$71$ \( ( 1 + 6 T + 71 T^{2} )^{4} \)
$73$ \( 1 + 6 T + 90 T^{2} + 984 T^{3} + 6095 T^{4} + 71832 T^{5} + 479610 T^{6} + 2334102 T^{7} + 28398241 T^{8} \)
$79$ \( 1 - 12 T + 154 T^{2} - 1272 T^{3} + 8787 T^{4} - 100488 T^{5} + 961114 T^{6} - 5916468 T^{7} + 38950081 T^{8} \)
$83$ \( 1 - 2 T + 2 T^{2} + 328 T^{3} - 7217 T^{4} + 27224 T^{5} + 13778 T^{6} - 1143574 T^{7} + 47458321 T^{8} \)
$89$ \( 1 + 8 T - 103 T^{2} - 88 T^{3} + 14272 T^{4} - 7832 T^{5} - 815863 T^{6} + 5639752 T^{7} + 62742241 T^{8} \)
$97$ \( 1 - 2 T + 50 T^{2} + 1188 T^{3} - 4465 T^{4} + 115236 T^{5} + 470450 T^{6} - 1825346 T^{7} + 88529281 T^{8} \)
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