# Properties

 Label 3381.2.a.bd Level $3381$ Weight $2$ Character orbit 3381.a Self dual yes Analytic conductor $26.997$ Analytic rank $1$ Dimension $6$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3381 = 3 \cdot 7^{2} \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3381.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$26.9974209234$$ Analytic rank: $$1$$ Dimension: $$6$$ Coefficient field: 6.6.7997584.1 Defining polynomial: $$x^{6} - x^{5} - 7 x^{4} + 5 x^{3} + 12 x^{2} - 4 x - 2$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 483) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta_{1} q^{2} + q^{3} + ( 1 + \beta_{2} ) q^{4} + ( -\beta_{3} - \beta_{5} ) q^{5} -\beta_{1} q^{6} -\beta_{3} q^{8} + q^{9} +O(q^{10})$$ $$q -\beta_{1} q^{2} + q^{3} + ( 1 + \beta_{2} ) q^{4} + ( -\beta_{3} - \beta_{5} ) q^{5} -\beta_{1} q^{6} -\beta_{3} q^{8} + q^{9} + ( -1 + 2 \beta_{1} + 2 \beta_{3} + \beta_{4} ) q^{10} + ( -2 - 2 \beta_{4} ) q^{11} + ( 1 + \beta_{2} ) q^{12} + ( -\beta_{1} + \beta_{4} + \beta_{5} ) q^{13} + ( -\beta_{3} - \beta_{5} ) q^{15} + ( -3 + \beta_{1} - 2 \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} ) q^{16} + ( -3 + \beta_{1} + \beta_{3} - \beta_{4} ) q^{17} -\beta_{1} q^{18} + ( -1 + \beta_{1} - 2 \beta_{2} + 2 \beta_{5} ) q^{19} + ( -2 - 2 \beta_{1} - \beta_{2} - \beta_{3} - 3 \beta_{4} ) q^{20} + ( -4 + 4 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} ) q^{22} - q^{23} -\beta_{3} q^{24} + ( -1 + 3 \beta_{1} - \beta_{2} + 2 \beta_{3} + 3 \beta_{4} + \beta_{5} ) q^{25} + ( 5 - 2 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} - \beta_{4} + \beta_{5} ) q^{26} + q^{27} + ( -2 + \beta_{1} + \beta_{3} + 2 \beta_{4} - 2 \beta_{5} ) q^{29} + ( -1 + 2 \beta_{1} + 2 \beta_{3} + \beta_{4} ) q^{30} + ( 2 + 2 \beta_{1} + \beta_{2} ) q^{31} + ( 2 \beta_{1} + \beta_{3} - 2 \beta_{4} ) q^{32} + ( -2 - 2 \beta_{4} ) q^{33} + ( -4 + 3 \beta_{1} - 2 \beta_{2} - \beta_{5} ) q^{34} + ( 1 + \beta_{2} ) q^{36} + ( -2 + \beta_{1} - \beta_{2} + \beta_{3} + 2 \beta_{5} ) q^{37} + ( -3 + \beta_{1} - \beta_{2} + 2 \beta_{5} ) q^{38} + ( -\beta_{1} + \beta_{4} + \beta_{5} ) q^{39} + ( 1 + 3 \beta_{1} - \beta_{2} + \beta_{3} + 2 \beta_{4} + \beta_{5} ) q^{40} + ( -2 - 3 \beta_{1} + 2 \beta_{2} + \beta_{3} - 2 \beta_{5} ) q^{41} + ( -7 - \beta_{1} - 2 \beta_{2} + 2 \beta_{5} ) q^{43} + ( -2 + 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} - 2 \beta_{5} ) q^{44} + ( -\beta_{3} - \beta_{5} ) q^{45} + \beta_{1} q^{46} + ( -2 + 3 \beta_{3} + 2 \beta_{5} ) q^{47} + ( -3 + \beta_{1} - 2 \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} ) q^{48} + ( -1 - 4 \beta_{1} - 5 \beta_{3} - 5 \beta_{4} - \beta_{5} ) q^{50} + ( -3 + \beta_{1} + \beta_{3} - \beta_{4} ) q^{51} + ( 2 - 3 \beta_{1} + \beta_{2} + \beta_{4} + \beta_{5} ) q^{52} + ( -2 + 2 \beta_{1} - \beta_{2} - 3 \beta_{3} + \beta_{4} - 2 \beta_{5} ) q^{53} -\beta_{1} q^{54} + ( 2 - 2 \beta_{1} + 4 \beta_{2} + 2 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} ) q^{55} + ( -1 + \beta_{1} - 2 \beta_{2} + 2 \beta_{5} ) q^{57} + ( 2 + \beta_{1} + \beta_{2} - \beta_{3} - 3 \beta_{4} - 3 \beta_{5} ) q^{58} + ( -\beta_{1} + \beta_{3} + 4 \beta_{5} ) q^{59} + ( -2 - 2 \beta_{1} - \beta_{2} - \beta_{3} - 3 \beta_{4} ) q^{60} + ( -2 \beta_{1} + 4 \beta_{3} + 2 \beta_{4} ) q^{61} + ( -6 - 3 \beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{62} + ( -3 - \beta_{1} - \beta_{3} - \beta_{4} - 3 \beta_{5} ) q^{64} + ( -5 + 3 \beta_{1} - 2 \beta_{2} + \beta_{3} + \beta_{4} ) q^{65} + ( -4 + 4 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} ) q^{66} + ( -1 + 2 \beta_{2} + 2 \beta_{4} + \beta_{5} ) q^{67} + ( -3 + 5 \beta_{1} - 3 \beta_{2} + \beta_{3} + 2 \beta_{4} - \beta_{5} ) q^{68} - q^{69} + ( -7 + 3 \beta_{1} - 2 \beta_{2} + 3 \beta_{3} + 3 \beta_{4} + \beta_{5} ) q^{71} -\beta_{3} q^{72} + ( 5 - \beta_{1} + 5 \beta_{2} - \beta_{3} - 3 \beta_{4} - 3 \beta_{5} ) q^{73} + ( -2 - \beta_{2} - 2 \beta_{3} - \beta_{4} + \beta_{5} ) q^{74} + ( -1 + 3 \beta_{1} - \beta_{2} + 2 \beta_{3} + 3 \beta_{4} + \beta_{5} ) q^{75} + ( -1 + 3 \beta_{2} - \beta_{3} - 2 \beta_{5} ) q^{76} + ( 5 - 2 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} - \beta_{4} + \beta_{5} ) q^{78} + ( 3 - 4 \beta_{1} + 3 \beta_{2} - 5 \beta_{3} - 7 \beta_{4} + \beta_{5} ) q^{79} + ( \beta_{2} - \beta_{3} + 3 \beta_{4} ) q^{80} + q^{81} + ( 10 + \beta_{1} + 3 \beta_{2} - \beta_{3} - \beta_{4} - 3 \beta_{5} ) q^{82} + ( -4 - 3 \beta_{2} + 3 \beta_{4} + \beta_{5} ) q^{83} + ( 1 - 6 \beta_{1} + 3 \beta_{2} - 4 \beta_{4} + 2 \beta_{5} ) q^{85} + ( 3 + 7 \beta_{1} + \beta_{2} + 2 \beta_{5} ) q^{86} + ( -2 + \beta_{1} + \beta_{3} + 2 \beta_{4} - 2 \beta_{5} ) q^{87} + ( 2 \beta_{2} + 2 \beta_{3} - 2 \beta_{4} ) q^{88} + ( -5 + 3 \beta_{1} - \beta_{2} + 2 \beta_{3} - 2 \beta_{5} ) q^{89} + ( -1 + 2 \beta_{1} + 2 \beta_{3} + \beta_{4} ) q^{90} + ( -1 - \beta_{2} ) q^{92} + ( 2 + 2 \beta_{1} + \beta_{2} ) q^{93} + ( 3 - 3 \beta_{1} - 5 \beta_{3} - 3 \beta_{4} - \beta_{5} ) q^{94} + ( -1 + 2 \beta_{1} + 2 \beta_{2} - 3 \beta_{3} + 3 \beta_{4} - \beta_{5} ) q^{95} + ( 2 \beta_{1} + \beta_{3} - 2 \beta_{4} ) q^{96} + ( 4 \beta_{1} + 2 \beta_{2} + 4 \beta_{4} - 2 \beta_{5} ) q^{97} + ( -2 - 2 \beta_{4} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6q - q^{2} + 6q^{3} + 3q^{4} - 3q^{5} - q^{6} - 3q^{8} + 6q^{9} + O(q^{10})$$ $$6q - q^{2} + 6q^{3} + 3q^{4} - 3q^{5} - q^{6} - 3q^{8} + 6q^{9} + 3q^{10} - 14q^{11} + 3q^{12} - 3q^{15} - 7q^{16} - 15q^{17} - q^{18} + q^{19} - 17q^{20} - 6q^{22} - 6q^{23} - 3q^{24} + 9q^{25} + 15q^{26} + 6q^{27} - 6q^{29} + 3q^{30} + 11q^{31} + 3q^{32} - 14q^{33} - 15q^{34} + 3q^{36} - 5q^{37} - 14q^{38} + 17q^{40} - 18q^{41} - 37q^{43} - 10q^{44} - 3q^{45} + q^{46} - 3q^{47} - 7q^{48} - 30q^{50} - 15q^{51} + 7q^{52} - 15q^{53} - q^{54} + 2q^{55} + q^{57} + 4q^{58} + 2q^{59} - 17q^{60} + 12q^{61} - 36q^{62} - 23q^{64} - 17q^{65} - 6q^{66} - 10q^{67} + q^{68} - 6q^{69} - 21q^{71} - 3q^{72} + 8q^{73} - 16q^{74} + 9q^{75} - 18q^{76} + 15q^{78} - 17q^{79} - 3q^{80} + 6q^{81} + 48q^{82} - 12q^{83} - 13q^{85} + 22q^{86} - 6q^{87} - 2q^{88} - 18q^{89} + 3q^{90} - 3q^{92} + 11q^{93} - 3q^{94} - 16q^{95} + 3q^{96} + 2q^{97} - 14q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - x^{5} - 7 x^{4} + 5 x^{3} + 12 x^{2} - 4 x - 2$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - 3$$ $$\beta_{3}$$ $$=$$ $$\nu^{3} - 4 \nu$$ $$\beta_{4}$$ $$=$$ $$($$$$\nu^{5} - 7 \nu^{3} + 10 \nu$$$$)/2$$ $$\beta_{5}$$ $$=$$ $$($$$$-\nu^{5} + 2 \nu^{4} + 5 \nu^{3} - 8 \nu^{2} - 4 \nu + 2$$$$)/2$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} + 3$$ $$\nu^{3}$$ $$=$$ $$\beta_{3} + 4 \beta_{1}$$ $$\nu^{4}$$ $$=$$ $$\beta_{5} + \beta_{4} + \beta_{3} + 4 \beta_{2} + \beta_{1} + 11$$ $$\nu^{5}$$ $$=$$ $$2 \beta_{4} + 7 \beta_{3} + 18 \beta_{1}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 2.27267 1.83264 0.593528 −0.289957 −1.36549 −2.04340
−2.27267 1.00000 3.16503 −4.15120 −2.27267 0 −2.64772 1.00000 9.43432
1.2 −1.83264 1.00000 1.35859 0.943490 −1.83264 0 1.17548 1.00000 −1.72908
1.3 −0.593528 1.00000 −1.64772 3.15120 −0.593528 0 2.16503 1.00000 −1.87033
1.4 0.289957 1.00000 −1.91593 −2.32621 0.289957 0 −1.13545 1.00000 −0.674501
1.5 1.36549 1.00000 −0.135449 1.32621 1.36549 0 −2.91593 1.00000 1.81092
1.6 2.04340 1.00000 2.17548 −1.94349 2.04340 0 0.358585 1.00000 −3.97133
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.6 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$-1$$
$$7$$ $$-1$$
$$23$$ $$1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3381.2.a.bd 6
7.b odd 2 1 3381.2.a.bc 6
7.d odd 6 2 483.2.i.f 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
483.2.i.f 12 7.d odd 6 2
3381.2.a.bc 6 7.b odd 2 1
3381.2.a.bd 6 1.a even 1 1 trivial

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(3381))$$:

 $$T_{2}^{6} + T_{2}^{5} - 7 T_{2}^{4} - 5 T_{2}^{3} + 12 T_{2}^{2} + 4 T_{2} - 2$$ $$T_{5}^{6} + 3 T_{5}^{5} - 15 T_{5}^{4} - 35 T_{5}^{3} + 52 T_{5}^{2} + 70 T_{5} - 74$$ $$T_{11}^{6} + 14 T_{11}^{5} + 52 T_{11}^{4} - 24 T_{11}^{3} - 320 T_{11}^{2} - 128 T_{11} + 256$$ $$T_{13}^{6} - 31 T_{13}^{4} - 52 T_{13}^{3} + 47 T_{13}^{2} + 4 T_{13} - 1$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-2 + 4 T + 12 T^{2} - 5 T^{3} - 7 T^{4} + T^{5} + T^{6}$$
$3$ $$( -1 + T )^{6}$$
$5$ $$-74 + 70 T + 52 T^{2} - 35 T^{3} - 15 T^{4} + 3 T^{5} + T^{6}$$
$7$ $$T^{6}$$
$11$ $$256 - 128 T - 320 T^{2} - 24 T^{3} + 52 T^{4} + 14 T^{5} + T^{6}$$
$13$ $$-1 + 4 T + 47 T^{2} - 52 T^{3} - 31 T^{4} + T^{6}$$
$17$ $$-18 - 90 T - 98 T^{2} + 23 T^{3} + 61 T^{4} + 15 T^{5} + T^{6}$$
$19$ $$-4 - 76 T + 74 T^{2} + 157 T^{3} - 65 T^{4} - T^{5} + T^{6}$$
$23$ $$( 1 + T )^{6}$$
$29$ $$-4768 + 4128 T + 712 T^{2} - 414 T^{3} - 66 T^{4} + 6 T^{5} + T^{6}$$
$31$ $$-72 - 12 T + 194 T^{2} + 143 T^{3} + T^{4} - 11 T^{5} + T^{6}$$
$37$ $$424 + 764 T + 270 T^{2} - 111 T^{3} - 39 T^{4} + 5 T^{5} + T^{6}$$
$41$ $$52928 - 10208 T - 12904 T^{2} - 2198 T^{3} - 26 T^{4} + 18 T^{5} + T^{6}$$
$43$ $$-19796 - 4508 T + 6042 T^{2} + 2959 T^{3} + 503 T^{4} + 37 T^{5} + T^{6}$$
$47$ $$-4768 + 768 T + 2060 T^{2} - 179 T^{3} - 113 T^{4} + 3 T^{5} + T^{6}$$
$53$ $$73928 + 6660 T - 10542 T^{2} - 2457 T^{3} - 91 T^{4} + 15 T^{5} + T^{6}$$
$59$ $$-76192 - 22768 T + 8736 T^{2} + 594 T^{3} - 202 T^{4} - 2 T^{5} + T^{6}$$
$61$ $$99648 - 51648 T + 112 T^{2} + 2144 T^{3} - 164 T^{4} - 12 T^{5} + T^{6}$$
$67$ $$33799 + 18502 T + 539 T^{2} - 884 T^{3} - 75 T^{4} + 10 T^{5} + T^{6}$$
$71$ $$-3208 - 8332 T - 5362 T^{2} - 773 T^{3} + 77 T^{4} + 21 T^{5} + T^{6}$$
$73$ $$55359 - 74052 T + 11143 T^{2} + 1564 T^{3} - 215 T^{4} - 8 T^{5} + T^{6}$$
$79$ $$-120548 + 249252 T + 2574 T^{2} - 5899 T^{3} - 299 T^{4} + 17 T^{5} + T^{6}$$
$83$ $$-2224 + 8664 T - 1612 T^{2} - 1222 T^{3} - 80 T^{4} + 12 T^{5} + T^{6}$$
$89$ $$5224 + 2224 T - 1240 T^{2} - 686 T^{3} - 28 T^{4} + 18 T^{5} + T^{6}$$
$97$ $$1152 + 4992 T + 5632 T^{2} + 680 T^{3} - 228 T^{4} - 2 T^{5} + T^{6}$$