# Properties

 Label 392.4.i.g Level $392$ Weight $4$ Character orbit 392.i Analytic conductor $23.129$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$392 = 2^{3} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 392.i (of order $$3$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$23.1287487223$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{9}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 8) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 4 - 4 \zeta_{6} ) q^{3} + 2 \zeta_{6} q^{5} + 11 \zeta_{6} q^{9} +O(q^{10})$$ $$q + ( 4 - 4 \zeta_{6} ) q^{3} + 2 \zeta_{6} q^{5} + 11 \zeta_{6} q^{9} + ( 44 - 44 \zeta_{6} ) q^{11} + 22 q^{13} + 8 q^{15} + ( -50 + 50 \zeta_{6} ) q^{17} -44 \zeta_{6} q^{19} + 56 \zeta_{6} q^{23} + ( 121 - 121 \zeta_{6} ) q^{25} + 152 q^{27} + 198 q^{29} + ( 160 - 160 \zeta_{6} ) q^{31} -176 \zeta_{6} q^{33} + 162 \zeta_{6} q^{37} + ( 88 - 88 \zeta_{6} ) q^{39} -198 q^{41} + 52 q^{43} + ( -22 + 22 \zeta_{6} ) q^{45} -528 \zeta_{6} q^{47} + 200 \zeta_{6} q^{51} + ( 242 - 242 \zeta_{6} ) q^{53} + 88 q^{55} -176 q^{57} + ( 668 - 668 \zeta_{6} ) q^{59} -550 \zeta_{6} q^{61} + 44 \zeta_{6} q^{65} + ( -188 + 188 \zeta_{6} ) q^{67} + 224 q^{69} + 728 q^{71} + ( -154 + 154 \zeta_{6} ) q^{73} -484 \zeta_{6} q^{75} + 656 \zeta_{6} q^{79} + ( 311 - 311 \zeta_{6} ) q^{81} + 236 q^{83} -100 q^{85} + ( 792 - 792 \zeta_{6} ) q^{87} -714 \zeta_{6} q^{89} -640 \zeta_{6} q^{93} + ( 88 - 88 \zeta_{6} ) q^{95} -478 q^{97} + 484 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 4q^{3} + 2q^{5} + 11q^{9} + O(q^{10})$$ $$2q + 4q^{3} + 2q^{5} + 11q^{9} + 44q^{11} + 44q^{13} + 16q^{15} - 50q^{17} - 44q^{19} + 56q^{23} + 121q^{25} + 304q^{27} + 396q^{29} + 160q^{31} - 176q^{33} + 162q^{37} + 88q^{39} - 396q^{41} + 104q^{43} - 22q^{45} - 528q^{47} + 200q^{51} + 242q^{53} + 176q^{55} - 352q^{57} + 668q^{59} - 550q^{61} + 44q^{65} - 188q^{67} + 448q^{69} + 1456q^{71} - 154q^{73} - 484q^{75} + 656q^{79} + 311q^{81} + 472q^{83} - 200q^{85} + 792q^{87} - 714q^{89} - 640q^{93} + 88q^{95} - 956q^{97} + 968q^{99} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/392\mathbb{Z}\right)^\times$$.

 $$n$$ $$197$$ $$295$$ $$297$$ $$\chi(n)$$ $$1$$ $$1$$ $$-\zeta_{6}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
177.1
 0.5 − 0.866025i 0.5 + 0.866025i
0 2.00000 + 3.46410i 0 1.00000 1.73205i 0 0 0 5.50000 9.52628i 0
361.1 0 2.00000 3.46410i 0 1.00000 + 1.73205i 0 0 0 5.50000 + 9.52628i 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 392.4.i.g 2
7.b odd 2 1 392.4.i.b 2
7.c even 3 1 8.4.a.a 1
7.c even 3 1 inner 392.4.i.g 2
7.d odd 6 1 392.4.a.e 1
7.d odd 6 1 392.4.i.b 2
21.h odd 6 1 72.4.a.c 1
28.f even 6 1 784.4.a.e 1
28.g odd 6 1 16.4.a.a 1
35.j even 6 1 200.4.a.g 1
35.l odd 12 2 200.4.c.e 2
56.k odd 6 1 64.4.a.b 1
56.p even 6 1 64.4.a.d 1
63.g even 3 1 648.4.i.h 2
63.h even 3 1 648.4.i.h 2
63.j odd 6 1 648.4.i.e 2
63.n odd 6 1 648.4.i.e 2
77.h odd 6 1 968.4.a.a 1
84.n even 6 1 144.4.a.e 1
91.r even 6 1 1352.4.a.a 1
105.o odd 6 1 1800.4.a.d 1
105.x even 12 2 1800.4.f.u 2
112.u odd 12 2 256.4.b.g 2
112.w even 12 2 256.4.b.a 2
119.j even 6 1 2312.4.a.a 1
140.p odd 6 1 400.4.a.g 1
140.w even 12 2 400.4.c.i 2
168.s odd 6 1 576.4.a.k 1
168.v even 6 1 576.4.a.j 1
280.bf even 6 1 1600.4.a.o 1
280.bi odd 6 1 1600.4.a.bm 1
308.n even 6 1 1936.4.a.l 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8.4.a.a 1 7.c even 3 1
16.4.a.a 1 28.g odd 6 1
64.4.a.b 1 56.k odd 6 1
64.4.a.d 1 56.p even 6 1
72.4.a.c 1 21.h odd 6 1
144.4.a.e 1 84.n even 6 1
200.4.a.g 1 35.j even 6 1
200.4.c.e 2 35.l odd 12 2
256.4.b.a 2 112.w even 12 2
256.4.b.g 2 112.u odd 12 2
392.4.a.e 1 7.d odd 6 1
392.4.i.b 2 7.b odd 2 1
392.4.i.b 2 7.d odd 6 1
392.4.i.g 2 1.a even 1 1 trivial
392.4.i.g 2 7.c even 3 1 inner
400.4.a.g 1 140.p odd 6 1
400.4.c.i 2 140.w even 12 2
576.4.a.j 1 168.v even 6 1
576.4.a.k 1 168.s odd 6 1
648.4.i.e 2 63.j odd 6 1
648.4.i.e 2 63.n odd 6 1
648.4.i.h 2 63.g even 3 1
648.4.i.h 2 63.h even 3 1
784.4.a.e 1 28.f even 6 1
968.4.a.a 1 77.h odd 6 1
1352.4.a.a 1 91.r even 6 1
1600.4.a.o 1 280.bf even 6 1
1600.4.a.bm 1 280.bi odd 6 1
1800.4.a.d 1 105.o odd 6 1
1800.4.f.u 2 105.x even 12 2
1936.4.a.l 1 308.n even 6 1
2312.4.a.a 1 119.j even 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(392, [\chi])$$:

 $$T_{3}^{2} - 4 T_{3} + 16$$ $$T_{5}^{2} - 2 T_{5} + 4$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$1 - 4 T - 11 T^{2} - 108 T^{3} + 729 T^{4}$$
$5$ $$1 - 2 T - 121 T^{2} - 250 T^{3} + 15625 T^{4}$$
$7$ 1
$11$ $$1 - 44 T + 605 T^{2} - 58564 T^{3} + 1771561 T^{4}$$
$13$ $$( 1 - 22 T + 2197 T^{2} )^{2}$$
$17$ $$1 + 50 T - 2413 T^{2} + 245650 T^{3} + 24137569 T^{4}$$
$19$ $$1 + 44 T - 4923 T^{2} + 301796 T^{3} + 47045881 T^{4}$$
$23$ $$1 - 56 T - 9031 T^{2} - 681352 T^{3} + 148035889 T^{4}$$
$29$ $$( 1 - 198 T + 24389 T^{2} )^{2}$$
$31$ $$1 - 160 T - 4191 T^{2} - 4766560 T^{3} + 887503681 T^{4}$$
$37$ $$1 - 162 T - 24409 T^{2} - 8205786 T^{3} + 2565726409 T^{4}$$
$41$ $$( 1 + 198 T + 68921 T^{2} )^{2}$$
$43$ $$( 1 - 52 T + 79507 T^{2} )^{2}$$
$47$ $$1 + 528 T + 174961 T^{2} + 54818544 T^{3} + 10779215329 T^{4}$$
$53$ $$1 - 242 T - 90313 T^{2} - 36028234 T^{3} + 22164361129 T^{4}$$
$59$ $$1 - 668 T + 240845 T^{2} - 137193172 T^{3} + 42180533641 T^{4}$$
$61$ $$1 + 550 T + 75519 T^{2} + 124839550 T^{3} + 51520374361 T^{4}$$
$67$ $$1 + 188 T - 265419 T^{2} + 56543444 T^{3} + 90458382169 T^{4}$$
$71$ $$( 1 - 728 T + 357911 T^{2} )^{2}$$
$73$ $$1 + 154 T - 365301 T^{2} + 59908618 T^{3} + 151334226289 T^{4}$$
$79$ $$1 - 656 T - 62703 T^{2} - 323433584 T^{3} + 243087455521 T^{4}$$
$83$ $$( 1 - 236 T + 571787 T^{2} )^{2}$$
$89$ $$1 + 714 T - 195173 T^{2} + 503347866 T^{3} + 496981290961 T^{4}$$
$97$ $$( 1 + 478 T + 912673 T^{2} )^{2}$$