# Properties

 Label 384.2.k.a Level $384$ Weight $2$ Character orbit 384.k Analytic conductor $3.066$ Analytic rank $0$ Dimension $12$ CM no Inner twists $4$

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## Newspace parameters

 Level: $$N$$ $$=$$ $$384 = 2^{7} \cdot 3$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 384.k (of order $$4$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.06625543762$$ Analytic rank: $$0$$ Dimension: $$12$$ Relative dimension: $$6$$ over $$\Q(i)$$ Coefficient field: 12.0.163368480538624.2 Defining polynomial: $$x^{12} - 2 x^{10} - 2 x^{8} + 16 x^{6} - 8 x^{4} - 32 x^{2} + 64$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{7}$$ Twist minimal: no (minimal twist has level 48) 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.

 $$f(q)$$ $$=$$ $$q + \beta_{4} q^{3} + \beta_{6} q^{5} + ( 1 - \beta_{11} ) q^{7} + ( -\beta_{3} + \beta_{5} + \beta_{8} + \beta_{9} ) q^{9} +O(q^{10})$$ $$q + \beta_{4} q^{3} + \beta_{6} q^{5} + ( 1 - \beta_{11} ) q^{7} + ( -\beta_{3} + \beta_{5} + \beta_{8} + \beta_{9} ) q^{9} + ( \beta_{2} + \beta_{3} - \beta_{8} + \beta_{10} ) q^{11} + ( \beta_{5} + \beta_{11} ) q^{13} + ( \beta_{1} - \beta_{3} + \beta_{5} - \beta_{6} - \beta_{7} + \beta_{9} ) q^{15} + ( -\beta_{3} - \beta_{4} + \beta_{6} - \beta_{7} - \beta_{9} - \beta_{10} ) q^{17} + ( -1 + \beta_{1} - \beta_{4} + \beta_{5} + \beta_{9} - \beta_{11} ) q^{19} + ( 1 - \beta_{1} + \beta_{2} + 2 \beta_{4} - \beta_{6} + \beta_{8} ) q^{21} + ( 2 \beta_{2} + \beta_{3} + \beta_{4} + \beta_{9} + \beta_{10} ) q^{23} + ( -\beta_{1} + \beta_{3} + \beta_{4} - \beta_{9} - \beta_{10} ) q^{25} + ( 1 + \beta_{1} - \beta_{2} - \beta_{3} - \beta_{5} - 2 \beta_{7} + \beta_{8} - 2 \beta_{10} - \beta_{11} ) q^{27} + ( -2 \beta_{3} - \beta_{7} - 2 \beta_{10} ) q^{29} + ( \beta_{1} + \beta_{3} + \beta_{4} - \beta_{5} - \beta_{9} - \beta_{10} ) q^{31} + ( -1 + \beta_{2} + 2 \beta_{3} + \beta_{4} - \beta_{6} + \beta_{7} + 2 \beta_{9} + \beta_{10} + \beta_{11} ) q^{33} + ( \beta_{2} + \beta_{8} ) q^{35} + ( -2 \beta_{4} + \beta_{5} + 2 \beta_{9} - \beta_{11} ) q^{37} + ( -2 - 2 \beta_{2} - \beta_{4} + \beta_{6} - \beta_{7} - \beta_{10} ) q^{39} + ( \beta_{6} + \beta_{7} - 2 \beta_{8} ) q^{41} + ( 1 + \beta_{1} - \beta_{3} + \beta_{5} + \beta_{10} + \beta_{11} ) q^{43} + ( 1 + \beta_{1} - \beta_{2} + 2 \beta_{3} - \beta_{5} + \beta_{7} + \beta_{8} - \beta_{11} ) q^{45} + ( 2 \beta_{3} - 2 \beta_{4} + 2 \beta_{6} + 2 \beta_{7} - 2 \beta_{8} - 2 \beta_{9} + 2 \beta_{10} ) q^{47} + ( -1 - \beta_{3} + \beta_{4} - \beta_{9} + \beta_{10} ) q^{49} + ( 2 - 2 \beta_{1} - \beta_{2} - 2 \beta_{6} - \beta_{8} ) q^{51} + ( -2 \beta_{2} - 4 \beta_{4} + \beta_{6} - 2 \beta_{8} - 4 \beta_{9} ) q^{53} + ( -2 - \beta_{3} + \beta_{4} - \beta_{9} + \beta_{10} + 2 \beta_{11} ) q^{55} + ( \beta_{1} + \beta_{3} - \beta_{5} - \beta_{6} - \beta_{7} + \beta_{8} - \beta_{9} ) q^{57} + ( \beta_{3} + 4 \beta_{7} + \beta_{10} ) q^{59} + ( -2 \beta_{3} - \beta_{5} + 2 \beta_{10} - \beta_{11} ) q^{61} + ( -3 \beta_{1} - 2 \beta_{3} + 2 \beta_{4} - \beta_{5} + 2 \beta_{8} + 2 \beta_{9} - 2 \beta_{10} ) q^{63} + ( -2 \beta_{2} + \beta_{6} - \beta_{7} ) q^{65} + ( 2 - 2 \beta_{1} + \beta_{4} - 2 \beta_{5} - \beta_{9} + 2 \beta_{11} ) q^{67} + ( -1 + \beta_{1} + \beta_{2} + 2 \beta_{4} - \beta_{5} - 2 \beta_{6} + \beta_{8} + 4 \beta_{9} + \beta_{11} ) q^{69} + ( -\beta_{3} - \beta_{4} - 2 \beta_{6} + 2 \beta_{7} - \beta_{9} - \beta_{10} ) q^{71} + ( 2 \beta_{1} - 2 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} + 2 \beta_{9} + 2 \beta_{10} ) q^{73} + ( -3 - 3 \beta_{1} - \beta_{2} - 2 \beta_{3} + \beta_{5} + \beta_{8} - \beta_{10} + \beta_{11} ) q^{75} + ( 2 \beta_{2} + 4 \beta_{3} - 2 \beta_{8} + 4 \beta_{10} ) q^{77} + ( -5 \beta_{1} - \beta_{3} - \beta_{4} + \beta_{5} + \beta_{9} + \beta_{10} ) q^{79} + ( 1 - \beta_{3} + \beta_{4} - 2 \beta_{6} + 2 \beta_{7} - \beta_{9} + \beta_{10} - 2 \beta_{11} ) q^{81} + ( -\beta_{2} - 3 \beta_{4} + 4 \beta_{6} - \beta_{8} - 3 \beta_{9} ) q^{83} + ( -2 + 2 \beta_{1} + 4 \beta_{4} - 2 \beta_{5} - 4 \beta_{9} + 2 \beta_{11} ) q^{85} + ( 5 - 2 \beta_{2} - \beta_{3} - \beta_{6} + \beta_{7} - \beta_{9} + \beta_{11} ) q^{87} + ( 3 \beta_{3} - 3 \beta_{4} - 2 \beta_{8} - 3 \beta_{9} + 3 \beta_{10} ) q^{89} + ( -5 - 5 \beta_{1} + 2 \beta_{3} - \beta_{5} - 2 \beta_{10} - \beta_{11} ) q^{91} + ( -2 - 2 \beta_{1} - 2 \beta_{3} + \beta_{5} + \beta_{7} + 2 \beta_{10} + \beta_{11} ) q^{93} + ( \beta_{3} - \beta_{4} + 2 \beta_{8} - \beta_{9} + \beta_{10} ) q^{95} + ( -2 + 3 \beta_{3} - 3 \beta_{4} + 3 \beta_{9} - 3 \beta_{10} - 2 \beta_{11} ) q^{97} + ( -5 + 5 \beta_{1} - \beta_{4} - \beta_{5} + 2 \beta_{6} + \beta_{9} + \beta_{11} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12q - 2q^{3} + 8q^{7} + O(q^{10})$$ $$12q - 2q^{3} + 8q^{7} + 4q^{13} - 12q^{19} + 8q^{21} + 10q^{27} - 4q^{33} + 4q^{37} - 20q^{39} + 12q^{43} + 12q^{45} - 20q^{49} + 24q^{51} - 24q^{55} - 12q^{61} + 28q^{67} - 4q^{69} - 34q^{75} - 4q^{81} - 32q^{85} + 60q^{87} - 56q^{91} - 28q^{93} - 8q^{97} - 52q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} - 2 x^{10} - 2 x^{8} + 16 x^{6} - 8 x^{4} - 32 x^{2} + 64$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{8} - 2 \nu^{4} + 4 \nu^{2} + 8$$$$)/8$$ $$\beta_{2}$$ $$=$$ $$($$$$-\nu^{9} - 2 \nu^{7} + 2 \nu^{5} - 24 \nu$$$$)/8$$ $$\beta_{3}$$ $$=$$ $$($$$$2 \nu^{11} - \nu^{10} - 8 \nu^{7} + 2 \nu^{6} + 16 \nu^{5} + 4 \nu^{4} + 24 \nu^{3} - 16 \nu^{2} - 32 \nu$$$$)/32$$ $$\beta_{4}$$ $$=$$ $$($$$$-\nu^{11} + \nu^{10} + 6 \nu^{7} - 10 \nu^{6} - 4 \nu^{5} + 12 \nu^{4} - 8 \nu^{3} + 32 \nu^{2} + 32 \nu - 64$$$$)/32$$ $$\beta_{5}$$ $$=$$ $$($$$$\nu^{10} - 2 \nu^{6} + 12 \nu^{4} + 16 \nu^{2} - 16$$$$)/16$$ $$\beta_{6}$$ $$=$$ $$($$$$-\nu^{11} + 2 \nu^{9} + 2 \nu^{7} - 8 \nu^{5} + 8 \nu^{3} + 16 \nu$$$$)/16$$ $$\beta_{7}$$ $$=$$ $$($$$$-\nu^{11} + 2 \nu^{7} - 4 \nu^{5} - 16 \nu^{3} + 16 \nu$$$$)/16$$ $$\beta_{8}$$ $$=$$ $$($$$$\nu^{11} - 6 \nu^{7} + 12 \nu^{5} + 8 \nu^{3} - 48 \nu$$$$)/16$$ $$\beta_{9}$$ $$=$$ $$($$$$-\nu^{11} - \nu^{10} + 6 \nu^{7} + 10 \nu^{6} - 4 \nu^{5} - 12 \nu^{4} - 8 \nu^{3} - 32 \nu^{2} + 32 \nu + 64$$$$)/32$$ $$\beta_{10}$$ $$=$$ $$($$$$2 \nu^{11} + \nu^{10} - 8 \nu^{7} - 2 \nu^{6} + 16 \nu^{5} - 4 \nu^{4} + 24 \nu^{3} + 16 \nu^{2} - 32 \nu$$$$)/32$$ $$\beta_{11}$$ $$=$$ $$($$$$-3 \nu^{10} + 2 \nu^{8} + 14 \nu^{6} - 24 \nu^{4} - 24 \nu^{2} + 80$$$$)/16$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{10} - \beta_{8} + \beta_{7} + \beta_{3}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{11} + \beta_{10} - \beta_{9} + \beta_{5} + \beta_{4} - \beta_{3} - \beta_{1} + 1$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$($$$$\beta_{10} + \beta_{9} - \beta_{8} - \beta_{7} + \beta_{6} + \beta_{4} + \beta_{3} + \beta_{2}$$$$)/2$$ $$\nu^{4}$$ $$=$$ $$-\beta_{10} + \beta_{5} + \beta_{3} + 1$$ $$\nu^{5}$$ $$=$$ $$\beta_{10} + 2 \beta_{9} + \beta_{8} + \beta_{7} + 2 \beta_{4} + \beta_{3}$$ $$\nu^{6}$$ $$=$$ $$\beta_{11} + \beta_{10} + \beta_{9} + 3 \beta_{5} - \beta_{4} - \beta_{3} - \beta_{1} - 5$$ $$\nu^{7}$$ $$=$$ $$-3 \beta_{10} + 3 \beta_{9} + 3 \beta_{8} - 5 \beta_{7} - \beta_{6} + 3 \beta_{4} - 3 \beta_{3} - \beta_{2}$$ $$\nu^{8}$$ $$=$$ $$-2 \beta_{11} - 4 \beta_{10} + 2 \beta_{9} - 2 \beta_{4} + 4 \beta_{3} + 10 \beta_{1} - 8$$ $$\nu^{9}$$ $$=$$ $$-4 \beta_{10} - 2 \beta_{9} + 8 \beta_{8} + 2 \beta_{6} - 2 \beta_{4} - 4 \beta_{3} - 6 \beta_{2}$$ $$\nu^{10}$$ $$=$$ $$-6 \beta_{11} + 6 \beta_{10} + 10 \beta_{9} + 2 \beta_{5} - 10 \beta_{4} - 6 \beta_{3} + 6 \beta_{1} - 14$$ $$\nu^{11}$$ $$=$$ $$-10 \beta_{10} - 10 \beta_{9} + 2 \beta_{8} - 14 \beta_{7} - 10 \beta_{6} - 10 \beta_{4} - 10 \beta_{3} - 10 \beta_{2}$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/384\mathbb{Z}\right)^\times$$.

 $$n$$ $$127$$ $$133$$ $$257$$ $$\chi(n)$$ $$-1$$ $$-\beta_{1}$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
95.1
 0.204810 − 1.39930i −1.35164 − 0.416001i −0.204810 + 1.39930i 1.27715 − 0.607364i 1.35164 + 0.416001i −1.27715 + 0.607364i 0.204810 + 1.39930i −1.35164 + 0.416001i −0.204810 − 1.39930i 1.27715 + 0.607364i 1.35164 − 0.416001i −1.27715 − 0.607364i
0 −1.52878 + 0.814141i 0 2.08397 + 2.08397i 0 1.14637 0 1.67435 2.48929i 0
95.2 0 −1.43726 0.966579i 0 −1.57184 1.57184i 0 −2.24914 0 1.13145 + 2.77846i 0
95.3 0 −0.814141 + 1.52878i 0 −2.08397 2.08397i 0 1.14637 0 −1.67435 2.48929i 0
95.4 0 0.0835731 1.73003i 0 −0.431733 0.431733i 0 3.10278 0 −2.98603 0.289169i 0
95.5 0 0.966579 + 1.43726i 0 1.57184 + 1.57184i 0 −2.24914 0 −1.13145 + 2.77846i 0
95.6 0 1.73003 0.0835731i 0 0.431733 + 0.431733i 0 3.10278 0 2.98603 0.289169i 0
287.1 0 −1.52878 0.814141i 0 2.08397 2.08397i 0 1.14637 0 1.67435 + 2.48929i 0
287.2 0 −1.43726 + 0.966579i 0 −1.57184 + 1.57184i 0 −2.24914 0 1.13145 2.77846i 0
287.3 0 −0.814141 1.52878i 0 −2.08397 + 2.08397i 0 1.14637 0 −1.67435 + 2.48929i 0
287.4 0 0.0835731 + 1.73003i 0 −0.431733 + 0.431733i 0 3.10278 0 −2.98603 + 0.289169i 0
287.5 0 0.966579 1.43726i 0 1.57184 1.57184i 0 −2.24914 0 −1.13145 2.77846i 0
287.6 0 1.73003 + 0.0835731i 0 0.431733 0.431733i 0 3.10278 0 2.98603 + 0.289169i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 287.6 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
16.f odd 4 1 inner
48.k even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 384.2.k.a 12
3.b odd 2 1 inner 384.2.k.a 12
4.b odd 2 1 384.2.k.b 12
8.b even 2 1 192.2.k.a 12
8.d odd 2 1 48.2.k.a 12
12.b even 2 1 384.2.k.b 12
16.e even 4 1 48.2.k.a 12
16.e even 4 1 384.2.k.b 12
16.f odd 4 1 192.2.k.a 12
16.f odd 4 1 inner 384.2.k.a 12
24.f even 2 1 48.2.k.a 12
24.h odd 2 1 192.2.k.a 12
48.i odd 4 1 48.2.k.a 12
48.i odd 4 1 384.2.k.b 12
48.k even 4 1 192.2.k.a 12
48.k even 4 1 inner 384.2.k.a 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
48.2.k.a 12 8.d odd 2 1
48.2.k.a 12 16.e even 4 1
48.2.k.a 12 24.f even 2 1
48.2.k.a 12 48.i odd 4 1
192.2.k.a 12 8.b even 2 1
192.2.k.a 12 16.f odd 4 1
192.2.k.a 12 24.h odd 2 1
192.2.k.a 12 48.k even 4 1
384.2.k.a 12 1.a even 1 1 trivial
384.2.k.a 12 3.b odd 2 1 inner
384.2.k.a 12 16.f odd 4 1 inner
384.2.k.a 12 48.k even 4 1 inner
384.2.k.b 12 4.b odd 2 1
384.2.k.b 12 12.b even 2 1
384.2.k.b 12 16.e even 4 1
384.2.k.b 12 48.i odd 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7}^{3} - 2 T_{7}^{2} - 6 T_{7} + 8$$ acting on $$S_{2}^{\mathrm{new}}(384, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{12}$$
$3$ $$729 + 486 T + 162 T^{2} - 54 T^{3} - 45 T^{4} - 60 T^{5} - 28 T^{6} - 20 T^{7} - 5 T^{8} - 2 T^{9} + 2 T^{10} + 2 T^{11} + T^{12}$$
$5$ $$256 + 1856 T^{4} + 100 T^{8} + T^{12}$$
$7$ $$( 8 - 6 T - 2 T^{2} + T^{3} )^{4}$$
$11$ $$65536 + 12288 T^{4} + 356 T^{8} + T^{12}$$
$13$ $$( 8 + 56 T + 196 T^{2} + 32 T^{3} + 2 T^{4} - 2 T^{5} + T^{6} )^{2}$$
$17$ $$( 512 + 288 T^{2} + 40 T^{4} + T^{6} )^{2}$$
$19$ $$( 128 + 128 T + 64 T^{2} - 32 T^{3} + 18 T^{4} + 6 T^{5} + T^{6} )^{2}$$
$23$ $$( 2048 + 640 T^{2} + 52 T^{4} + T^{6} )^{2}$$
$29$ $$71639296 + 548160 T^{4} + 1316 T^{8} + T^{12}$$
$31$ $$( 16 + 68 T^{2} + 36 T^{4} + T^{6} )^{2}$$
$37$ $$( 2312 + 2040 T + 900 T^{2} + 128 T^{3} + 2 T^{4} - 2 T^{5} + T^{6} )^{2}$$
$41$ $$( -128 + 384 T^{2} - 108 T^{4} + T^{6} )^{2}$$
$43$ $$( 128 - 128 T + 64 T^{2} + 32 T^{3} + 18 T^{4} - 6 T^{5} + T^{6} )^{2}$$
$47$ $$( -32768 + 3712 T^{2} - 112 T^{4} + T^{6} )^{2}$$
$53$ $$256 + 87360 T^{4} + 24356 T^{8} + T^{12}$$
$59$ $$4096 + 24418304 T^{4} + 27748 T^{8} + T^{12}$$
$61$ $$( 95048 + 27032 T + 3844 T^{2} + 64 T^{3} + 18 T^{4} + 6 T^{5} + T^{6} )^{2}$$
$67$ $$( 32 + 128 T + 256 T^{2} + 232 T^{3} + 98 T^{4} - 14 T^{5} + T^{6} )^{2}$$
$71$ $$( 147968 + 10176 T^{2} + 196 T^{4} + T^{6} )^{2}$$
$73$ $$( 369664 + 18496 T^{2} + 272 T^{4} + T^{6} )^{2}$$
$79$ $$( 8464 + 1956 T^{2} + 116 T^{4} + T^{6} )^{2}$$
$83$ $$5473632256 + 32944128 T^{4} + 19044 T^{8} + T^{12}$$
$89$ $$( -2048 + 8576 T^{2} - 212 T^{4} + T^{6} )^{2}$$
$97$ $$( -608 - 128 T + 2 T^{2} + T^{3} )^{4}$$
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