Properties

Label 960.2.w.g
Level $960$
Weight $2$
Character orbit 960.w
Analytic conductor $7.666$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 960 = 2^{6} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 960.w (of order \(4\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(7.66563859404\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(i)\)
Coefficient field: 12.0.426337261060096.1
Defining polynomial: \(x^{12} - 4 x^{9} - 3 x^{8} + 4 x^{7} + 8 x^{6} + 8 x^{5} - 12 x^{4} - 32 x^{3} + 64\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: no (minimal twist has level 60)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{1} q^{3} -\beta_{10} q^{5} + ( \beta_{2} + \beta_{9} ) q^{7} + \beta_{7} q^{9} +O(q^{10})\) \( q + \beta_{1} q^{3} -\beta_{10} q^{5} + ( \beta_{2} + \beta_{9} ) q^{7} + \beta_{7} q^{9} + ( -\beta_{1} + \beta_{2} - \beta_{3} - \beta_{6} ) q^{11} + ( -\beta_{8} + \beta_{10} ) q^{13} -\beta_{3} q^{15} + ( -2 + \beta_{4} + \beta_{5} - 2 \beta_{7} ) q^{17} + ( \beta_{2} + \beta_{3} + \beta_{9} + \beta_{11} ) q^{19} + ( \beta_{5} + \beta_{10} ) q^{21} + ( 2 \beta_{1} + \beta_{2} + \beta_{3} - \beta_{9} + \beta_{11} ) q^{23} + ( -2 - \beta_{4} - \beta_{5} + \beta_{7} - 2 \beta_{8} ) q^{25} + \beta_{6} q^{27} + ( \beta_{4} + \beta_{8} ) q^{29} + ( -2 \beta_{1} - 2 \beta_{6} - 2 \beta_{9} + 2 \beta_{11} ) q^{31} + ( 1 - \beta_{4} + \beta_{5} - \beta_{7} ) q^{33} + ( -\beta_{1} + \beta_{2} + \beta_{3} + 3 \beta_{6} + 2 \beta_{11} ) q^{35} + ( -\beta_{4} - \beta_{5} ) q^{37} + ( \beta_{2} + \beta_{3} ) q^{39} + ( 2 - \beta_{4} - \beta_{5} + \beta_{8} - \beta_{10} ) q^{41} + ( 2 \beta_{2} - 2 \beta_{9} ) q^{43} -\beta_{4} q^{45} + ( \beta_{2} - \beta_{3} + 2 \beta_{6} + \beta_{9} + \beta_{11} ) q^{47} + ( \beta_{4} - \beta_{5} + \beta_{7} + \beta_{8} + \beta_{10} ) q^{49} + ( -2 \beta_{1} - 2 \beta_{6} - \beta_{9} + \beta_{11} ) q^{51} + ( -\beta_{4} + \beta_{5} ) q^{53} + ( -2 \beta_{1} + 2 \beta_{2} + \beta_{3} + 6 \beta_{6} - \beta_{11} ) q^{55} + ( \beta_{4} + \beta_{5} + \beta_{8} + \beta_{10} ) q^{57} + ( 3 \beta_{1} - 3 \beta_{6} - \beta_{9} - \beta_{11} ) q^{59} + ( 2 + \beta_{4} - 3 \beta_{5} - \beta_{8} - 3 \beta_{10} ) q^{61} + ( \beta_{3} + \beta_{11} ) q^{63} + ( 2 + \beta_{4} + \beta_{5} + 4 \beta_{7} + 2 \beta_{8} ) q^{65} + ( -2 \beta_{3} + 2 \beta_{11} ) q^{67} + ( \beta_{4} + \beta_{5} + 2 \beta_{7} + \beta_{8} - \beta_{10} ) q^{69} + ( -4 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} - 4 \beta_{6} + 2 \beta_{9} - 2 \beta_{11} ) q^{71} + ( 3 + 2 \beta_{4} - 2 \beta_{5} - 3 \beta_{7} ) q^{73} + ( -2 \beta_{1} + 2 \beta_{2} + \beta_{6} + \beta_{9} - \beta_{11} ) q^{75} + ( -4 + \beta_{4} + \beta_{5} - 4 \beta_{7} + \beta_{8} + \beta_{10} ) q^{77} + ( 6 \beta_{1} - 6 \beta_{6} + 2 \beta_{9} + 2 \beta_{11} ) q^{79} - q^{81} + ( 6 \beta_{1} - \beta_{2} + \beta_{3} + \beta_{9} + \beta_{11} ) q^{83} + ( -4 + 2 \beta_{4} - 2 \beta_{5} + 2 \beta_{7} + \beta_{8} + \beta_{10} ) q^{85} + ( -\beta_{2} - \beta_{9} ) q^{87} + ( -\beta_{4} + \beta_{5} - 2 \beta_{7} - \beta_{8} - \beta_{10} ) q^{89} + ( -2 \beta_{1} - 2 \beta_{6} + 2 \beta_{9} - 2 \beta_{11} ) q^{91} + ( 2 - 2 \beta_{7} + 2 \beta_{8} - 2 \beta_{10} ) q^{93} + ( -4 \beta_{1} - \beta_{2} + \beta_{3} + 2 \beta_{6} - \beta_{9} + 3 \beta_{11} ) q^{95} + ( -1 - 4 \beta_{4} - 4 \beta_{5} - \beta_{7} - 2 \beta_{8} - 2 \beta_{10} ) q^{97} + ( \beta_{1} - \beta_{6} + \beta_{9} + \beta_{11} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12q + O(q^{10}) \) \( 12q + 4q^{13} - 20q^{17} - 20q^{25} + 8q^{33} - 4q^{37} + 16q^{41} - 4q^{45} - 4q^{53} + 32q^{61} + 20q^{65} + 44q^{73} - 48q^{77} - 12q^{81} - 44q^{85} + 16q^{93} - 20q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{12} - 4 x^{9} - 3 x^{8} + 4 x^{7} + 8 x^{6} + 8 x^{5} - 12 x^{4} - 32 x^{3} + 64\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{11} + \nu^{10} + 4 \nu^{9} - 2 \nu^{8} - 3 \nu^{7} + 5 \nu^{6} + 24 \nu^{5} + 2 \nu^{4} - 8 \nu^{3} - 24 \nu^{2} - 48 \nu \)\()/160\)
\(\beta_{2}\)\(=\)\((\)\( -4 \nu^{11} + 5 \nu^{10} + 8 \nu^{9} - 6 \nu^{8} + 4 \nu^{7} - 27 \nu^{6} + 4 \nu^{5} + 18 \nu^{4} + 60 \nu^{3} + 104 \nu^{2} + 16 \nu - 32 \)\()/160\)
\(\beta_{3}\)\(=\)\((\)\( 3 \nu^{11} - \nu^{10} - 22 \nu^{9} - 2 \nu^{8} + 19 \nu^{7} + 67 \nu^{6} + 22 \nu^{5} - 110 \nu^{4} - 172 \nu^{3} + 40 \nu^{2} + 272 \nu + 352 \)\()/160\)
\(\beta_{4}\)\(=\)\((\)\( -5 \nu^{11} + 8 \nu^{10} + 8 \nu^{9} + 12 \nu^{8} - \nu^{7} - 44 \nu^{6} + 112 \nu^{4} + 76 \nu^{3} + 96 \nu^{2} - 352 \nu - 64 \)\()/160\)
\(\beta_{5}\)\(=\)\((\)\( 3 \nu^{11} - 8 \nu^{10} + 16 \nu^{9} + 20 \nu^{8} + 23 \nu^{7} - 12 \nu^{6} - 88 \nu^{5} - 8 \nu^{4} + 124 \nu^{3} + 16 \nu^{2} + 160 \nu - 512 \)\()/160\)
\(\beta_{6}\)\(=\)\((\)\( -\nu^{11} + 9 \nu^{10} + 6 \nu^{9} + 2 \nu^{8} - 17 \nu^{7} - 35 \nu^{6} + 26 \nu^{5} + 38 \nu^{4} + 28 \nu^{3} - 56 \nu^{2} - 192 \nu \)\()/160\)
\(\beta_{7}\)\(=\)\((\)\( 3 \nu^{11} + \nu^{10} - 4 \nu^{8} - 5 \nu^{7} + \nu^{6} + 12 \nu^{5} + 8 \nu^{4} + 12 \nu^{3} - 36 \nu^{2} - 16 \nu - 64 \)\()/80\)
\(\beta_{8}\)\(=\)\((\)\( 3 \nu^{11} + 2 \nu^{10} + 6 \nu^{9} - 17 \nu^{7} - 2 \nu^{6} - 18 \nu^{5} + 12 \nu^{4} + 44 \nu^{3} - 24 \nu^{2} - 40 \nu - 192 \)\()/80\)
\(\beta_{9}\)\(=\)\((\)\( -4 \nu^{11} - 7 \nu^{10} - 24 \nu^{9} + 6 \nu^{8} + 28 \nu^{7} + 49 \nu^{6} + 4 \nu^{5} - 130 \nu^{4} - 44 \nu^{3} + 144 \nu + 384 \)\()/160\)
\(\beta_{10}\)\(=\)\((\)\( -5 \nu^{11} - 2 \nu^{10} - 2 \nu^{9} + 12 \nu^{8} - \nu^{7} - 14 \nu^{6} - 10 \nu^{5} - 8 \nu^{4} + 36 \nu^{3} + 96 \nu^{2} + 8 \nu + 96 \)\()/80\)
\(\beta_{11}\)\(=\)\((\)\( 7 \nu^{11} + 15 \nu^{10} + 6 \nu^{9} - 22 \nu^{8} - 57 \nu^{7} + 11 \nu^{6} + 98 \nu^{5} + 126 \nu^{4} + 20 \nu^{3} - 352 \nu^{2} - 208 \nu - 64 \)\()/160\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{11} + \beta_{10} - \beta_{9} - \beta_{8} + \beta_{5} - \beta_{4} + \beta_{3} + \beta_{2} - 2 \beta_{1}\)\()/4\)
\(\nu^{2}\)\(=\)\((\)\(-\beta_{11} + \beta_{10} - \beta_{9} + \beta_{8} + 2 \beta_{7} - 2 \beta_{6} - \beta_{5} + \beta_{4} + \beta_{3} + \beta_{2} + 2 \beta_{1}\)\()/4\)
\(\nu^{3}\)\(=\)\((\)\(\beta_{11} + \beta_{10} + 3 \beta_{9} + \beta_{8} + 4 \beta_{7} - 2 \beta_{6} + \beta_{5} + \beta_{4} - \beta_{3} + 3 \beta_{2} + 4\)\()/4\)
\(\nu^{4}\)\(=\)\((\)\(3 \beta_{11} + \beta_{10} - 3 \beta_{9} - 3 \beta_{8} - 6 \beta_{6} + \beta_{5} + 3 \beta_{4} + \beta_{3} - \beta_{2} - 6 \beta_{1} + 2\)\()/4\)
\(\nu^{5}\)\(=\)\((\)\(\beta_{11} + 5 \beta_{10} - \beta_{9} - 5 \beta_{8} + 8 \beta_{7} + \beta_{5} - \beta_{4} + \beta_{3} + \beta_{2} + 14 \beta_{1} - 8\)\()/4\)
\(\nu^{6}\)\(=\)\((\)\(5 \beta_{11} - \beta_{10} + 5 \beta_{9} + 7 \beta_{8} - 10 \beta_{7} - 14 \beta_{6} + \beta_{5} + 7 \beta_{4} + 3 \beta_{3} + 3 \beta_{2} + 14 \beta_{1}\)\()/4\)
\(\nu^{7}\)\(=\)\((\)\(-\beta_{11} - 9 \beta_{10} + 5 \beta_{9} - 9 \beta_{8} + 4 \beta_{7} + 2 \beta_{6} + 7 \beta_{5} + 7 \beta_{4} + \beta_{3} + 5 \beta_{2} + 4\)\()/4\)
\(\nu^{8}\)\(=\)\((\)\(5 \beta_{11} + 15 \beta_{10} - 5 \beta_{9} - 5 \beta_{8} + 14 \beta_{6} + 15 \beta_{5} + 5 \beta_{4} + 15 \beta_{3} - 15 \beta_{2} + 14 \beta_{1} - 2\)\()/4\)
\(\nu^{9}\)\(=\)\((\)\(-\beta_{11} - 5 \beta_{10} - 7 \beta_{9} + 5 \beta_{8} - 32 \beta_{7} + 7 \beta_{5} - 7 \beta_{4} - \beta_{3} + 7 \beta_{2} + 66 \beta_{1} + 32\)\()/4\)
\(\nu^{10}\)\(=\)\((\)\(11 \beta_{11} - 15 \beta_{10} + 11 \beta_{9} + 17 \beta_{8} - 22 \beta_{7} + 38 \beta_{6} + 15 \beta_{5} + 17 \beta_{4} + 37 \beta_{3} + 37 \beta_{2} - 38 \beta_{1}\)\()/4\)
\(\nu^{11}\)\(=\)\((\)\(-23 \beta_{11} + \beta_{10} - 21 \beta_{9} + \beta_{8} + 100 \beta_{7} + 14 \beta_{6} + 9 \beta_{5} + 9 \beta_{4} + 23 \beta_{3} - 21 \beta_{2} + 100\)\()/4\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/960\mathbb{Z}\right)^\times\).

\(n\) \(511\) \(577\) \(641\) \(901\)
\(\chi(n)\) \(-1\) \(\beta_{7}\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
127.1
−0.0912546 1.41127i
1.19252 + 0.760198i
−0.394157 + 1.35818i
1.41127 + 0.0912546i
−0.760198 1.19252i
−1.35818 + 0.394157i
−0.0912546 + 1.41127i
1.19252 0.760198i
−0.394157 1.35818i
1.41127 0.0912546i
−0.760198 + 1.19252i
−1.35818 0.394157i
0 −0.707107 0.707107i 0 −1.32001 + 1.80487i 0 −1.86678 + 1.86678i 0 1.00000i 0
127.2 0 −0.707107 0.707107i 0 −0.432320 2.19388i 0 −0.611393 + 0.611393i 0 1.00000i 0
127.3 0 −0.707107 0.707107i 0 1.75233 + 1.38900i 0 2.47817 2.47817i 0 1.00000i 0
127.4 0 0.707107 + 0.707107i 0 −1.32001 + 1.80487i 0 1.86678 1.86678i 0 1.00000i 0
127.5 0 0.707107 + 0.707107i 0 −0.432320 2.19388i 0 0.611393 0.611393i 0 1.00000i 0
127.6 0 0.707107 + 0.707107i 0 1.75233 + 1.38900i 0 −2.47817 + 2.47817i 0 1.00000i 0
703.1 0 −0.707107 + 0.707107i 0 −1.32001 1.80487i 0 −1.86678 1.86678i 0 1.00000i 0
703.2 0 −0.707107 + 0.707107i 0 −0.432320 + 2.19388i 0 −0.611393 0.611393i 0 1.00000i 0
703.3 0 −0.707107 + 0.707107i 0 1.75233 1.38900i 0 2.47817 + 2.47817i 0 1.00000i 0
703.4 0 0.707107 0.707107i 0 −1.32001 1.80487i 0 1.86678 + 1.86678i 0 1.00000i 0
703.5 0 0.707107 0.707107i 0 −0.432320 + 2.19388i 0 0.611393 + 0.611393i 0 1.00000i 0
703.6 0 0.707107 0.707107i 0 1.75233 1.38900i 0 −2.47817 2.47817i 0 1.00000i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 703.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
5.c odd 4 1 inner
20.e even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 960.2.w.g 12
4.b odd 2 1 inner 960.2.w.g 12
5.c odd 4 1 inner 960.2.w.g 12
8.b even 2 1 60.2.j.a 12
8.d odd 2 1 60.2.j.a 12
20.e even 4 1 inner 960.2.w.g 12
24.f even 2 1 180.2.k.e 12
24.h odd 2 1 180.2.k.e 12
40.e odd 2 1 300.2.j.d 12
40.f even 2 1 300.2.j.d 12
40.i odd 4 1 60.2.j.a 12
40.i odd 4 1 300.2.j.d 12
40.k even 4 1 60.2.j.a 12
40.k even 4 1 300.2.j.d 12
120.i odd 2 1 900.2.k.n 12
120.m even 2 1 900.2.k.n 12
120.q odd 4 1 180.2.k.e 12
120.q odd 4 1 900.2.k.n 12
120.w even 4 1 180.2.k.e 12
120.w even 4 1 900.2.k.n 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
60.2.j.a 12 8.b even 2 1
60.2.j.a 12 8.d odd 2 1
60.2.j.a 12 40.i odd 4 1
60.2.j.a 12 40.k even 4 1
180.2.k.e 12 24.f even 2 1
180.2.k.e 12 24.h odd 2 1
180.2.k.e 12 120.q odd 4 1
180.2.k.e 12 120.w even 4 1
300.2.j.d 12 40.e odd 2 1
300.2.j.d 12 40.f even 2 1
300.2.j.d 12 40.i odd 4 1
300.2.j.d 12 40.k even 4 1
900.2.k.n 12 120.i odd 2 1
900.2.k.n 12 120.m even 2 1
900.2.k.n 12 120.q odd 4 1
900.2.k.n 12 120.w even 4 1
960.2.w.g 12 1.a even 1 1 trivial
960.2.w.g 12 4.b odd 2 1 inner
960.2.w.g 12 5.c odd 4 1 inner
960.2.w.g 12 20.e even 4 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{12} + 200 T_{7}^{8} + 7440 T_{7}^{4} + 4096 \) acting on \(S_{2}^{\mathrm{new}}(960, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \)
$3$ \( ( 1 + T^{4} )^{3} \)
$5$ \( ( 125 + 25 T^{2} - 8 T^{3} + 5 T^{4} + T^{6} )^{2} \)
$7$ \( 4096 + 7440 T^{4} + 200 T^{8} + T^{12} \)
$11$ \( ( 128 + 260 T^{2} + 36 T^{4} + T^{6} )^{2} \)
$13$ \( ( 32 + 96 T + 144 T^{2} + 32 T^{3} + 2 T^{4} - 2 T^{5} + T^{6} )^{2} \)
$17$ \( ( 800 - 160 T + 16 T^{2} + 80 T^{3} + 50 T^{4} + 10 T^{5} + T^{6} )^{2} \)
$19$ \( ( -512 + 400 T^{2} - 40 T^{4} + T^{6} )^{2} \)
$23$ \( 65536 + 37120 T^{4} + 4640 T^{8} + T^{12} \)
$29$ \( ( 64 + 100 T^{2} + 20 T^{4} + T^{6} )^{2} \)
$31$ \( ( 32768 + 3648 T^{2} + 112 T^{4} + T^{6} )^{2} \)
$37$ \( ( 32 - 96 T + 144 T^{2} - 32 T^{3} + 2 T^{4} + 2 T^{5} + T^{6} )^{2} \)
$41$ \( ( 64 - 20 T - 4 T^{2} + T^{3} )^{4} \)
$43$ \( 15352201216 + 24350720 T^{4} + 9600 T^{8} + T^{12} \)
$47$ \( 40960000 + 901376 T^{4} + 4896 T^{8} + T^{12} \)
$53$ \( ( 128 + 256 T + 256 T^{2} - 16 T^{3} + 2 T^{4} + 2 T^{5} + T^{6} )^{2} \)
$59$ \( ( -512 + 1860 T^{2} - 100 T^{4} + T^{6} )^{2} \)
$61$ \( ( -176 - 100 T - 8 T^{2} + T^{3} )^{4} \)
$67$ \( 15352201216 + 24350720 T^{4} + 9600 T^{8} + T^{12} \)
$71$ \( ( 204800 + 15616 T^{2} + 256 T^{4} + T^{6} )^{2} \)
$73$ \( ( 55112 + 4648 T + 196 T^{2} - 640 T^{3} + 242 T^{4} - 22 T^{5} + T^{6} )^{2} \)
$79$ \( ( -2048 + 12864 T^{2} - 304 T^{4} + T^{6} )^{2} \)
$83$ \( 65536 + 10264832 T^{4} + 16672 T^{8} + T^{12} \)
$89$ \( ( 1024 + 1040 T^{2} + 72 T^{4} + T^{6} )^{2} \)
$97$ \( ( 35912 - 47704 T + 31684 T^{2} - 2048 T^{3} + 50 T^{4} + 10 T^{5} + T^{6} )^{2} \)
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