# 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 $4$

# Related objects

## 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: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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

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