# Properties

 Label 3381.2.a.bb 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: 6.6.62622704.1 Defining polynomial: $$x^{6} - 3 x^{5} - 5 x^{4} + 13 x^{3} + 9 x^{2} - 5 x - 2$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes 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.

 $$f(q)$$ $$=$$ $$q + ( -1 + \beta_{1} ) q^{2} + q^{3} + ( 2 - \beta_{1} + \beta_{2} ) q^{4} + ( -\beta_{1} - \beta_{3} ) q^{5} + ( -1 + \beta_{1} ) q^{6} + ( -3 + 2 \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{8} + q^{9} +O(q^{10})$$ $$q + ( -1 + \beta_{1} ) q^{2} + q^{3} + ( 2 - \beta_{1} + \beta_{2} ) q^{4} + ( -\beta_{1} - \beta_{3} ) q^{5} + ( -1 + \beta_{1} ) q^{6} + ( -3 + 2 \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{8} + q^{9} + ( -2 - \beta_{2} - \beta_{4} ) q^{10} + ( -1 + 2 \beta_{1} + \beta_{3} + \beta_{4} - \beta_{5} ) q^{11} + ( 2 - \beta_{1} + \beta_{2} ) q^{12} + ( 1 - \beta_{1} + \beta_{2} + \beta_{3} ) q^{13} + ( -\beta_{1} - \beta_{3} ) q^{15} + ( 4 - 5 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} + \beta_{4} ) q^{16} + ( -2 - 2 \beta_{2} + \beta_{5} ) q^{17} + ( -1 + \beta_{1} ) q^{18} + ( -4 - \beta_{3} - 2 \beta_{4} + \beta_{5} ) q^{19} + ( 2 - 3 \beta_{1} + 2 \beta_{2} - \beta_{3} - \beta_{5} ) q^{20} + ( 5 - \beta_{1} + \beta_{2} + \beta_{3} + 2 \beta_{5} ) q^{22} - q^{23} + ( -3 + 2 \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{24} + ( \beta_{1} + \beta_{3} + \beta_{4} + \beta_{5} ) q^{25} + ( -5 + 3 \beta_{1} - 2 \beta_{2} + \beta_{3} + \beta_{4} ) q^{26} + q^{27} + ( -2 - 3 \beta_{5} ) q^{29} + ( -2 - \beta_{2} - \beta_{4} ) q^{30} + ( 1 + 2 \beta_{3} + \beta_{4} + \beta_{5} ) q^{31} + ( -11 + 5 \beta_{1} - 4 \beta_{2} + 2 \beta_{3} - 2 \beta_{4} + \beta_{5} ) q^{32} + ( -1 + 2 \beta_{1} + \beta_{3} + \beta_{4} - \beta_{5} ) q^{33} + ( 3 - 5 \beta_{1} + 2 \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} ) q^{34} + ( 2 - \beta_{1} + \beta_{2} ) q^{36} + ( -1 + 2 \beta_{3} + \beta_{4} - \beta_{5} ) q^{37} + ( 6 - 5 \beta_{1} + 2 \beta_{2} - 3 \beta_{3} - 3 \beta_{5} ) q^{38} + ( 1 - \beta_{1} + \beta_{2} + \beta_{3} ) q^{39} + ( -7 + 5 \beta_{1} - 3 \beta_{2} + \beta_{3} + \beta_{5} ) q^{40} + ( 1 - 4 \beta_{1} - \beta_{3} - \beta_{4} + 3 \beta_{5} ) q^{41} + ( -1 + \beta_{1} + \beta_{2} + \beta_{3} - 2 \beta_{4} ) q^{43} + ( -5 + 5 \beta_{1} - 2 \beta_{2} + \beta_{3} + \beta_{4} ) q^{44} + ( -\beta_{1} - \beta_{3} ) q^{45} + ( 1 - \beta_{1} ) q^{46} + ( -2 \beta_{1} + \beta_{2} - 3 \beta_{3} + \beta_{4} + \beta_{5} ) q^{47} + ( 4 - 5 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} + \beta_{4} ) q^{48} + ( 3 + 2 \beta_{1} + 3 \beta_{3} + 2 \beta_{4} ) q^{50} + ( -2 - 2 \beta_{2} + \beta_{5} ) q^{51} + ( 11 - 6 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} + \beta_{4} + \beta_{5} ) q^{52} + ( -1 - \beta_{1} + \beta_{2} - 2 \beta_{3} - 2 \beta_{4} + \beta_{5} ) q^{53} + ( -1 + \beta_{1} ) q^{54} + ( -4 - 3 \beta_{1} - \beta_{2} - \beta_{3} - 3 \beta_{4} + \beta_{5} ) q^{55} + ( -4 - \beta_{3} - 2 \beta_{4} + \beta_{5} ) q^{57} + ( -1 - 5 \beta_{1} - 3 \beta_{3} - 3 \beta_{4} + 3 \beta_{5} ) q^{58} + ( -5 + \beta_{1} - 3 \beta_{2} - \beta_{5} ) q^{59} + ( 2 - 3 \beta_{1} + 2 \beta_{2} - \beta_{3} - \beta_{5} ) q^{60} + ( -5 + \beta_{1} + \beta_{2} - 4 \beta_{5} ) q^{61} + ( -2 + 3 \beta_{1} - \beta_{2} + 3 \beta_{3} + 3 \beta_{4} ) q^{62} + ( 17 - 10 \beta_{1} + 7 \beta_{2} - 3 \beta_{3} + \beta_{4} - 3 \beta_{5} ) q^{64} + ( -1 - 2 \beta_{1} + 3 \beta_{2} - 2 \beta_{5} ) q^{65} + ( 5 - \beta_{1} + \beta_{2} + \beta_{3} + 2 \beta_{5} ) q^{66} + ( 2 - \beta_{1} - \beta_{2} - 3 \beta_{3} - \beta_{4} - \beta_{5} ) q^{67} + ( -14 + 7 \beta_{1} - 4 \beta_{2} + 3 \beta_{3} - 2 \beta_{4} ) q^{68} - q^{69} + ( -2 + \beta_{1} - 2 \beta_{2} + \beta_{3} + 5 \beta_{5} ) q^{71} + ( -3 + 2 \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{72} + ( 1 + 2 \beta_{1} + 2 \beta_{3} + \beta_{4} - 3 \beta_{5} ) q^{73} + ( -2 - \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} + 2 \beta_{5} ) q^{74} + ( \beta_{1} + \beta_{3} + \beta_{4} + \beta_{5} ) q^{75} + ( -13 + 7 \beta_{1} - 7 \beta_{2} + \beta_{3} - 2 \beta_{4} + \beta_{5} ) q^{76} + ( -5 + 3 \beta_{1} - 2 \beta_{2} + \beta_{3} + \beta_{4} ) q^{78} + ( -3 - 3 \beta_{4} - \beta_{5} ) q^{79} + ( 18 - 6 \beta_{1} + 4 \beta_{2} + 2 \beta_{4} + \beta_{5} ) q^{80} + q^{81} + ( -9 + 3 \beta_{1} - 3 \beta_{2} + \beta_{3} + 2 \beta_{4} - 4 \beta_{5} ) q^{82} + ( -2 - 2 \beta_{1} + 2 \beta_{3} + 2 \beta_{4} - 3 \beta_{5} ) q^{83} + ( -1 + 5 \beta_{1} - 3 \beta_{2} + \beta_{3} + 2 \beta_{4} + \beta_{5} ) q^{85} + ( 3 - \beta_{1} + 2 \beta_{2} - 3 \beta_{3} + \beta_{4} - 2 \beta_{5} ) q^{86} + ( -2 - 3 \beta_{5} ) q^{87} + ( 9 - 6 \beta_{1} + 4 \beta_{2} - 2 \beta_{3} + \beta_{4} - 3 \beta_{5} ) q^{88} + ( -2 - \beta_{1} + 2 \beta_{2} - \beta_{3} + \beta_{5} ) q^{89} + ( -2 - \beta_{2} - \beta_{4} ) q^{90} + ( -2 + \beta_{1} - \beta_{2} ) q^{92} + ( 1 + 2 \beta_{3} + \beta_{4} + \beta_{5} ) q^{93} + ( -2 + 4 \beta_{1} - 4 \beta_{2} + 4 \beta_{3} - 2 \beta_{4} ) q^{94} + ( -2 + 9 \beta_{1} + 5 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} ) q^{95} + ( -11 + 5 \beta_{1} - 4 \beta_{2} + 2 \beta_{3} - 2 \beta_{4} + \beta_{5} ) q^{96} + ( -2 - 2 \beta_{1} + \beta_{2} + \beta_{4} - 2 \beta_{5} ) q^{97} + ( -1 + 2 \beta_{1} + \beta_{3} + \beta_{4} - \beta_{5} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6q - 3q^{2} + 6q^{3} + 7q^{4} - 2q^{5} - 3q^{6} - 9q^{8} + 6q^{9} + O(q^{10})$$ $$6q - 3q^{2} + 6q^{3} + 7q^{4} - 2q^{5} - 3q^{6} - 9q^{8} + 6q^{9} - 8q^{10} - 3q^{11} + 7q^{12} - 2q^{15} + 5q^{16} - 8q^{17} - 3q^{18} - 19q^{19} + 24q^{22} - 6q^{23} - 9q^{24} - 20q^{26} + 6q^{27} - 12q^{29} - 8q^{30} + 2q^{31} - 41q^{32} - 3q^{33} - 2q^{34} + 7q^{36} - 10q^{37} + 20q^{38} - 22q^{40} - 3q^{41} - 2q^{43} - 14q^{44} - 2q^{45} + 3q^{46} - 7q^{47} + 5q^{48} + 17q^{50} - 8q^{51} + 44q^{52} - 5q^{53} - 3q^{54} - 24q^{55} - 19q^{57} - 12q^{58} - 21q^{59} - 29q^{61} - 10q^{62} + 59q^{64} - 18q^{65} + 24q^{66} + 16q^{67} - 54q^{68} - 6q^{69} - 6q^{71} - 9q^{72} + 8q^{73} - 16q^{74} - 40q^{76} - 20q^{78} - 12q^{79} + 78q^{80} + 6q^{81} - 44q^{82} - 24q^{83} + 10q^{85} + 12q^{86} - 12q^{87} + 28q^{88} - 18q^{89} - 8q^{90} - 7q^{92} + 2q^{93} + 8q^{94} + 6q^{95} - 41q^{96} - 22q^{97} - 3q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - 3 x^{5} - 5 x^{4} + 13 x^{3} + 9 x^{2} - 5 x - 2$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - \nu - 3$$ $$\beta_{3}$$ $$=$$ $$\nu^{3} - \nu^{2} - 5 \nu$$ $$\beta_{4}$$ $$=$$ $$\nu^{4} - 2 \nu^{3} - 4 \nu^{2} + 5 \nu + 1$$ $$\beta_{5}$$ $$=$$ $$\nu^{5} - 3 \nu^{4} - 4 \nu^{3} + 12 \nu^{2} + 4 \nu - 4$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} + \beta_{1} + 3$$ $$\nu^{3}$$ $$=$$ $$\beta_{3} + \beta_{2} + 6 \beta_{1} + 3$$ $$\nu^{4}$$ $$=$$ $$\beta_{4} + 2 \beta_{3} + 6 \beta_{2} + 11 \beta_{1} + 17$$ $$\nu^{5}$$ $$=$$ $$\beta_{5} + 3 \beta_{4} + 10 \beta_{3} + 10 \beta_{2} + 41 \beta_{1} + 31$$

## 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
 −1.82095 −0.835848 −0.311423 0.584512 2.52648 2.85723
−2.82095 1.00000 5.95777 2.07008 −2.82095 0 −11.1647 1.00000 −5.83961
1.2 −1.83585 1.00000 1.37034 −2.06079 −1.83585 0 1.15597 1.00000 3.78330
1.3 −1.31142 1.00000 −0.280171 −1.11850 −1.31142 0 2.99027 1.00000 1.46683
1.4 −0.415488 1.00000 −1.82737 2.48000 −0.415488 0 1.59023 1.00000 −1.03041
1.5 1.52648 1.00000 0.330154 0.362203 1.52648 0 −2.54899 1.00000 0.552896
1.6 1.85723 1.00000 1.44929 −3.73299 1.85723 0 −1.02280 1.00000 −6.93301
 $$n$$: 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.bb yes 6
7.b odd 2 1 3381.2.a.ba 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3381.2.a.ba 6 7.b odd 2 1
3381.2.a.bb yes 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(3381))$$:

 $$T_{2}^{6} + 3 T_{2}^{5} - 5 T_{2}^{4} - 17 T_{2}^{3} + 3 T_{2}^{2} + 23 T_{2} + 8$$ $$T_{5}^{6} + 2 T_{5}^{5} - 13 T_{5}^{4} - 16 T_{5}^{3} + 41 T_{5}^{2} + 32 T_{5} - 16$$ $$T_{11}^{6} + 3 T_{11}^{5} - 38 T_{11}^{4} - 122 T_{11}^{3} + 181 T_{11}^{2} + 827 T_{11} + 596$$ $$T_{13}^{6} - 41 T_{13}^{4} + 36 T_{13}^{3} + 347 T_{13}^{2} - 480 T_{13} + 64$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$8 + 23 T + 3 T^{2} - 17 T^{3} - 5 T^{4} + 3 T^{5} + T^{6}$$
$3$ $$( -1 + T )^{6}$$
$5$ $$-16 + 32 T + 41 T^{2} - 16 T^{3} - 13 T^{4} + 2 T^{5} + T^{6}$$
$7$ $$T^{6}$$
$11$ $$596 + 827 T + 181 T^{2} - 122 T^{3} - 38 T^{4} + 3 T^{5} + T^{6}$$
$13$ $$64 - 480 T + 347 T^{2} + 36 T^{3} - 41 T^{4} + T^{6}$$
$17$ $$1480 + 1866 T - 195 T^{2} - 326 T^{3} - 30 T^{4} + 8 T^{5} + T^{6}$$
$19$ $$26816 + 3035 T - 3223 T^{2} - 542 T^{3} + 70 T^{4} + 19 T^{5} + T^{6}$$
$23$ $$( 1 + T )^{6}$$
$29$ $$4744 + 2802 T - 3351 T^{2} - 1226 T^{3} - 66 T^{4} + 12 T^{5} + T^{6}$$
$31$ $$-824 - 702 T + 447 T^{2} + 248 T^{3} - 84 T^{4} - 2 T^{5} + T^{6}$$
$37$ $$-128 + 258 T + 115 T^{2} - 284 T^{3} - 24 T^{4} + 10 T^{5} + T^{6}$$
$41$ $$-81584 + 10723 T + 6725 T^{2} - 418 T^{3} - 162 T^{4} + 3 T^{5} + T^{6}$$
$43$ $$392 + 2394 T + 2615 T^{2} - 18 T^{3} - 125 T^{4} + 2 T^{5} + T^{6}$$
$47$ $$256 + 7232 T + 7840 T^{2} - 712 T^{3} - 168 T^{4} + 7 T^{5} + T^{6}$$
$53$ $$674 + 2495 T + 639 T^{2} - 433 T^{3} - 85 T^{4} + 5 T^{5} + T^{6}$$
$59$ $$44476 + 15523 T - 3539 T^{2} - 1321 T^{3} + 29 T^{4} + 21 T^{5} + T^{6}$$
$61$ $$130124 + 8963 T - 16537 T^{2} - 1981 T^{3} + 153 T^{4} + 29 T^{5} + T^{6}$$
$67$ $$12304 - 6992 T - 6165 T^{2} + 1988 T^{3} - 71 T^{4} - 16 T^{5} + T^{6}$$
$71$ $$-1394528 + 57522 T + 38453 T^{2} - 1176 T^{3} - 345 T^{4} + 6 T^{5} + T^{6}$$
$73$ $$-64 + 240 T + 1055 T^{2} + 56 T^{3} - 100 T^{4} - 8 T^{5} + T^{6}$$
$79$ $$-297056 + 66012 T + 12711 T^{2} - 1808 T^{3} - 188 T^{4} + 12 T^{5} + T^{6}$$
$83$ $$612464 + 97334 T - 18863 T^{2} - 3978 T^{3} - 34 T^{4} + 24 T^{5} + T^{6}$$
$89$ $$1864 + 970 T - 1219 T^{2} - 340 T^{3} + 51 T^{4} + 18 T^{5} + T^{6}$$
$97$ $$71272 + 2582 T - 6903 T^{2} - 908 T^{3} + 94 T^{4} + 22 T^{5} + T^{6}$$