# Properties

 Label 1368.2.e.e Level $1368$ Weight $2$ Character orbit 1368.e Analytic conductor $10.924$ Analytic rank $0$ Dimension $12$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1368 = 2^{3} \cdot 3^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1368.e (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$10.9235349965$$ Analytic rank: $$0$$ Dimension: $$12$$ Coefficient field: 12.0.319794774016000000.1 Defining polynomial: $$x^{12} - 2 x^{10} + 2 x^{8} + 8 x^{4} - 32 x^{2} + 64$$ Coefficient ring: $$\Z[a_1, \ldots, a_{29}]$$ Coefficient ring index: $$2^{4}$$ Twist minimal: no (minimal twist has level 152) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q -\beta_{1} q^{2} + \beta_{2} q^{4} + ( \beta_{5} - \beta_{8} ) q^{5} -\beta_{8} q^{7} + ( -\beta_{3} - \beta_{4} ) q^{8} +O(q^{10})$$ $$q -\beta_{1} q^{2} + \beta_{2} q^{4} + ( \beta_{5} - \beta_{8} ) q^{5} -\beta_{8} q^{7} + ( -\beta_{3} - \beta_{4} ) q^{8} + ( -\beta_{4} - \beta_{6} + \beta_{7} + \beta_{9} ) q^{10} + \beta_{10} q^{11} + ( -2 \beta_{1} + \beta_{3} + \beta_{6} - \beta_{9} ) q^{13} + ( -\beta_{4} + \beta_{9} ) q^{14} + ( \beta_{2} + \beta_{5} + \beta_{8} + \beta_{10} - \beta_{11} ) q^{16} + ( -1 - 2 \beta_{10} ) q^{17} + ( -2 - \beta_{1} + \beta_{3} + \beta_{4} - \beta_{7} - \beta_{10} ) q^{19} + ( -\beta_{2} - 2 \beta_{8} + 2 \beta_{10} ) q^{20} + ( -\beta_{1} - \beta_{6} - \beta_{7} ) q^{22} + ( -\beta_{2} - \beta_{5} - 2 \beta_{8} + \beta_{11} ) q^{23} + ( -3 - \beta_{2} + \beta_{10} - \beta_{11} ) q^{25} + ( 3 + 2 \beta_{2} + \beta_{5} - \beta_{10} ) q^{26} + ( -2 + \beta_{5} - \beta_{8} + \beta_{10} + \beta_{11} ) q^{28} + ( -\beta_{4} + \beta_{6} - \beta_{7} ) q^{29} + ( -\beta_{1} - \beta_{4} - \beta_{7} ) q^{31} + ( -2 \beta_{3} - 2 \beta_{6} ) q^{32} + ( 3 \beta_{1} + 2 \beta_{6} + 2 \beta_{7} ) q^{34} + ( -2 - 2 \beta_{2} + \beta_{10} - 2 \beta_{11} ) q^{35} + ( 2 \beta_{1} - \beta_{3} - \beta_{4} - \beta_{7} + \beta_{9} ) q^{37} + ( -1 + 3 \beta_{1} - \beta_{5} + \beta_{6} + \beta_{7} - \beta_{10} + 2 \beta_{11} ) q^{38} + ( -2 \beta_{1} + \beta_{3} - \beta_{4} - 2 \beta_{6} - 2 \beta_{7} + 2 \beta_{9} ) q^{40} + ( -2 \beta_{1} + 3 \beta_{3} + \beta_{4} - \beta_{7} + \beta_{9} ) q^{41} + ( -2 + 3 \beta_{10} ) q^{43} + ( -\beta_{5} + \beta_{8} + \beta_{10} + \beta_{11} ) q^{44} + ( -\beta_{1} + 2 \beta_{3} - \beta_{4} + \beta_{6} - \beta_{7} + \beta_{9} ) q^{46} + ( \beta_{2} - \beta_{5} + 3 \beta_{8} - \beta_{11} ) q^{47} + 2 q^{49} + ( 3 \beta_{1} + \beta_{4} - \beta_{6} - \beta_{7} + \beta_{9} ) q^{50} + ( -2 \beta_{1} - 2 \beta_{3} - 2 \beta_{4} + 2 \beta_{7} ) q^{52} + ( 4 \beta_{1} + \beta_{4} + 3 \beta_{6} + \beta_{7} ) q^{53} + ( -3 \beta_{2} - \beta_{5} - \beta_{8} + 3 \beta_{11} ) q^{55} + ( \beta_{3} - \beta_{4} - 2 \beta_{6} ) q^{56} + ( -1 + \beta_{2} + 2 \beta_{5} + \beta_{8} + \beta_{11} ) q^{58} + ( -\beta_{1} - \beta_{3} - \beta_{4} + \beta_{7} + 2 \beta_{9} ) q^{59} + ( -2 \beta_{2} + \beta_{5} + \beta_{8} + 2 \beta_{11} ) q^{61} + ( \beta_{2} + \beta_{5} + \beta_{8} + \beta_{10} + \beta_{11} ) q^{62} + ( -2 \beta_{5} + 2 \beta_{8} + 2 \beta_{10} - 2 \beta_{11} ) q^{64} + ( -2 \beta_{3} - 4 \beta_{4} + 4 \beta_{7} + 4 \beta_{9} ) q^{65} + ( 2 \beta_{1} - 3 \beta_{3} - 2 \beta_{4} + 2 \beta_{7} ) q^{67} + ( -\beta_{2} + 2 \beta_{5} - 2 \beta_{8} - 2 \beta_{10} - 2 \beta_{11} ) q^{68} + ( 3 \beta_{1} + 2 \beta_{4} - \beta_{6} - \beta_{7} + 2 \beta_{9} ) q^{70} + ( -6 \beta_{1} + 2 \beta_{3} - 2 \beta_{4} + 2 \beta_{6} - 2 \beta_{7} - 2 \beta_{9} ) q^{71} + ( -1 + 2 \beta_{2} - 2 \beta_{10} + 2 \beta_{11} ) q^{73} + ( -4 - \beta_{2} + \beta_{5} + \beta_{8} + \beta_{10} + \beta_{11} ) q^{74} + ( -2 \beta_{2} + 2 \beta_{3} + \beta_{5} + 2 \beta_{6} - \beta_{8} - 2 \beta_{9} - \beta_{10} - \beta_{11} ) q^{76} + ( -2 \beta_{2} - \beta_{5} - \beta_{8} + 2 \beta_{11} ) q^{77} + ( -7 \beta_{1} + 2 \beta_{3} + \beta_{4} - 2 \beta_{6} + \beta_{7} - 2 \beta_{9} ) q^{79} + ( -4 - \beta_{2} - \beta_{5} - \beta_{8} + 3 \beta_{10} + 5 \beta_{11} ) q^{80} + ( -2 - \beta_{2} - \beta_{5} - 3 \beta_{8} - \beta_{10} + 5 \beta_{11} ) q^{82} -2 q^{83} + ( 6 \beta_{2} + \beta_{5} + 3 \beta_{8} - 6 \beta_{11} ) q^{85} + ( -\beta_{1} - 3 \beta_{6} - 3 \beta_{7} ) q^{86} + ( -2 \beta_{1} + \beta_{3} + \beta_{4} - 2 \beta_{7} - 2 \beta_{9} ) q^{88} + ( -2 \beta_{1} + \beta_{3} + 3 \beta_{4} - 3 \beta_{7} - \beta_{9} ) q^{89} + ( \beta_{1} - \beta_{3} - 3 \beta_{4} + 3 \beta_{7} + 2 \beta_{9} ) q^{91} + ( -2 + 2 \beta_{5} - 2 \beta_{8} + 4 \beta_{11} ) q^{92} + ( \beta_{1} - 2 \beta_{3} + 2 \beta_{4} + \beta_{6} - \beta_{7} - 2 \beta_{9} ) q^{94} + ( -2 \beta_{1} + 3 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} - \beta_{5} + 2 \beta_{6} + 2 \beta_{7} + 3 \beta_{8} - 2 \beta_{9} - 3 \beta_{11} ) q^{95} + ( -4 \beta_{1} + 2 \beta_{4} - 2 \beta_{7} + 2 \beta_{9} ) q^{97} -2 \beta_{1} q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12 q + 4 q^{4} + O(q^{10})$$ $$12 q + 4 q^{4} - 12 q^{17} - 24 q^{19} - 4 q^{20} - 44 q^{25} + 44 q^{26} - 20 q^{28} - 40 q^{35} - 4 q^{38} - 24 q^{43} + 4 q^{44} + 24 q^{49} - 4 q^{58} + 8 q^{62} - 8 q^{64} - 12 q^{68} + 4 q^{73} - 48 q^{74} - 12 q^{76} - 32 q^{80} - 8 q^{82} - 24 q^{83} - 8 q^{92} + O(q^{100})$$

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

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

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1368\mathbb{Z}\right)^\times$$.

 $$n$$ $$343$$ $$685$$ $$1009$$ $$1217$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
379.1
 1.37364 + 0.336338i 1.37364 − 0.336338i 1.17117 + 0.792696i 1.17117 − 0.792696i 0.491416 + 1.32609i 0.491416 − 1.32609i −0.491416 + 1.32609i −0.491416 − 1.32609i −1.17117 + 0.792696i −1.17117 − 0.792696i −1.37364 + 0.336338i −1.37364 − 0.336338i
−1.37364 0.336338i 0 1.77375 + 0.924013i 3.04222i 0 2.23607i −2.12571 1.86584i 0 −1.02321 + 4.17890i
379.2 −1.37364 + 0.336338i 0 1.77375 0.924013i 3.04222i 0 2.23607i −2.12571 + 1.86584i 0 −1.02321 4.17890i
379.3 −1.17117 0.792696i 0 0.743268 + 1.85676i 3.51876i 0 2.23607i 0.601353 2.76376i 0 2.78930 4.12105i
379.4 −1.17117 + 0.792696i 0 0.743268 1.85676i 3.51876i 0 2.23607i 0.601353 + 2.76376i 0 2.78930 + 4.12105i
379.5 −0.491416 1.32609i 0 −1.51702 + 1.30332i 2.08884i 0 2.23607i 2.47381 + 1.37123i 0 −2.76999 + 1.02649i
379.6 −0.491416 + 1.32609i 0 −1.51702 1.30332i 2.08884i 0 2.23607i 2.47381 1.37123i 0 −2.76999 1.02649i
379.7 0.491416 1.32609i 0 −1.51702 1.30332i 2.08884i 0 2.23607i −2.47381 + 1.37123i 0 2.76999 + 1.02649i
379.8 0.491416 + 1.32609i 0 −1.51702 + 1.30332i 2.08884i 0 2.23607i −2.47381 1.37123i 0 2.76999 1.02649i
379.9 1.17117 0.792696i 0 0.743268 1.85676i 3.51876i 0 2.23607i −0.601353 2.76376i 0 −2.78930 4.12105i
379.10 1.17117 + 0.792696i 0 0.743268 + 1.85676i 3.51876i 0 2.23607i −0.601353 + 2.76376i 0 −2.78930 + 4.12105i
379.11 1.37364 0.336338i 0 1.77375 0.924013i 3.04222i 0 2.23607i 2.12571 1.86584i 0 1.02321 + 4.17890i
379.12 1.37364 + 0.336338i 0 1.77375 + 0.924013i 3.04222i 0 2.23607i 2.12571 + 1.86584i 0 1.02321 4.17890i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 379.12 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.d odd 2 1 inner
19.b odd 2 1 inner
152.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1368.2.e.e 12
3.b odd 2 1 152.2.b.c 12
4.b odd 2 1 5472.2.e.e 12
8.b even 2 1 5472.2.e.e 12
8.d odd 2 1 inner 1368.2.e.e 12
12.b even 2 1 608.2.b.c 12
19.b odd 2 1 inner 1368.2.e.e 12
24.f even 2 1 152.2.b.c 12
24.h odd 2 1 608.2.b.c 12
57.d even 2 1 152.2.b.c 12
76.d even 2 1 5472.2.e.e 12
152.b even 2 1 inner 1368.2.e.e 12
152.g odd 2 1 5472.2.e.e 12
228.b odd 2 1 608.2.b.c 12
456.l odd 2 1 152.2.b.c 12
456.p even 2 1 608.2.b.c 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
152.2.b.c 12 3.b odd 2 1
152.2.b.c 12 24.f even 2 1
152.2.b.c 12 57.d even 2 1
152.2.b.c 12 456.l odd 2 1
608.2.b.c 12 12.b even 2 1
608.2.b.c 12 24.h odd 2 1
608.2.b.c 12 228.b odd 2 1
608.2.b.c 12 456.p even 2 1
1368.2.e.e 12 1.a even 1 1 trivial
1368.2.e.e 12 8.d odd 2 1 inner
1368.2.e.e 12 19.b odd 2 1 inner
1368.2.e.e 12 152.b even 2 1 inner
5472.2.e.e 12 4.b odd 2 1
5472.2.e.e 12 8.b even 2 1
5472.2.e.e 12 76.d even 2 1
5472.2.e.e 12 152.g odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1368, [\chi])$$:

 $$T_{5}^{6} + 26 T_{5}^{4} + 209 T_{5}^{2} + 500$$ $$T_{7}^{2} + 5$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$64 - 32 T^{2} + 8 T^{4} + 2 T^{8} - 2 T^{10} + T^{12}$$
$3$ $$T^{12}$$
$5$ $$( 500 + 209 T^{2} + 26 T^{4} + T^{6} )^{2}$$
$7$ $$( 5 + T^{2} )^{6}$$
$11$ $$( 4 - 7 T + T^{3} )^{4}$$
$13$ $$( -640 + 464 T^{2} - 51 T^{4} + T^{6} )^{2}$$
$17$ $$( -59 - 25 T + 3 T^{2} + T^{3} )^{4}$$
$19$ $$( 6859 + 4332 T + 1463 T^{2} + 376 T^{3} + 77 T^{4} + 12 T^{5} + T^{6} )^{2}$$
$23$ $$( 320 + 1184 T^{2} + 69 T^{4} + T^{6} )^{2}$$
$29$ $$( -40 + 244 T^{2} - 71 T^{4} + T^{6} )^{2}$$
$31$ $$( -640 + 504 T^{2} - 44 T^{4} + T^{6} )^{2}$$
$37$ $$( -2560 + 1784 T^{2} - 84 T^{4} + T^{6} )^{2}$$
$41$ $$( 70688 + 5640 T^{2} + 136 T^{4} + T^{6} )^{2}$$
$43$ $$( -10 - 51 T + 6 T^{2} + T^{3} )^{4}$$
$47$ $$( 80 + 1609 T^{2} + 106 T^{4} + T^{6} )^{2}$$
$53$ $$( -112360 + 8724 T^{2} - 191 T^{4} + T^{6} )^{2}$$
$59$ $$( 12800 + 2656 T^{2} + 103 T^{4} + T^{6} )^{2}$$
$61$ $$( 50000 + 10809 T^{2} + 214 T^{4} + T^{6} )^{2}$$
$67$ $$( 75272 + 6452 T^{2} + 155 T^{4} + T^{6} )^{2}$$
$71$ $$( -3504640 + 72384 T^{2} - 476 T^{4} + T^{6} )^{2}$$
$73$ $$( 97 - 65 T - T^{2} + T^{3} )^{4}$$
$79$ $$( -1697440 + 51464 T^{2} - 424 T^{4} + T^{6} )^{2}$$
$83$ $$( 2 + T )^{12}$$
$89$ $$( 231200 + 13960 T^{2} + 232 T^{4} + T^{6} )^{2}$$
$97$ $$( 2048 + 1920 T^{2} + 316 T^{4} + T^{6} )^{2}$$