# Properties

 Label 3344.2.a.x Level $3344$ Weight $2$ Character orbit 3344.a Self dual yes Analytic conductor $26.702$ Analytic rank $0$ Dimension $6$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3344 = 2^{4} \cdot 11 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3344.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$26.7019744359$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: $$\mathbb{Q}[x]/(x^{6} - \cdots)$$ Defining polynomial: $$x^{6} - 2x^{5} - 12x^{4} + 28x^{3} + 16x^{2} - 60x + 27$$ x^6 - 2*x^5 - 12*x^4 + 28*x^3 + 16*x^2 - 60*x + 27 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 836) 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.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{3} - \beta_{5} q^{5} - \beta_{4} q^{7} + (\beta_{2} + 2) q^{9}+O(q^{10})$$ q + b1 * q^3 - b5 * q^5 - b4 * q^7 + (b2 + 2) * q^9 $$q + \beta_1 q^{3} - \beta_{5} q^{5} - \beta_{4} q^{7} + (\beta_{2} + 2) q^{9} - q^{11} + \beta_{3} q^{13} + ( - \beta_{5} + \beta_{3} + \beta_{2} + 2) q^{15} + ( - \beta_{5} - \beta_{4} - 2 \beta_1 + 3) q^{17} - q^{19} + ( - \beta_{4} - \beta_{3} - \beta_{2} - \beta_1 + 1) q^{21} + (\beta_{5} + \beta_{2} + 2 \beta_1) q^{23} + (\beta_{2} + 2 \beta_1 + 4) q^{25} + ( - \beta_{5} + \beta_{4} - \beta_{3} + \beta_1 - 3) q^{27} + (\beta_{5} - \beta_{2} + \beta_1) q^{29} + ( - \beta_{5} - \beta_{3} + \beta_{2} + 4) q^{31} - \beta_1 q^{33} + ( - \beta_{4} + 2 \beta_{3} - 3 \beta_{2} - 2 \beta_1 - 4) q^{35} + (2 \beta_{5} + \beta_{4} + \beta_{2} + 2 \beta_1 + 3) q^{37} + ( - 2 \beta_{5} + \beta_{4} - \beta_{2} - 2) q^{39} + ( - \beta_{5} - \beta_{4} + \beta_{3} + 2 \beta_1 - 1) q^{41} + (\beta_{2} + 2 \beta_1 - 2) q^{43} + ( - \beta_{5} + 2 \beta_{4} + 4 \beta_1 - 3) q^{45} + ( - 2 \beta_{3} - 2) q^{47} + ( - \beta_{2} + 2 \beta_1 + 7) q^{49} + ( - \beta_{5} - \beta_{4} - 2 \beta_{2} + 2 \beta_1 - 7) q^{51} + ( - 2 \beta_{2} - 2 \beta_1 + 2) q^{53} + \beta_{5} q^{55} - \beta_1 q^{57} + (\beta_{4} + \beta_{2} - 2 \beta_1 + 3) q^{59} + ( - \beta_{5} + \beta_{4} - 2 \beta_{3} + 2 \beta_{2} + 3) q^{61} + (3 \beta_{5} - \beta_{2} - 2 \beta_1 + 1) q^{63} + ( - \beta_{4} - \beta_{3} + \beta_{2} + 5 \beta_1 + 1) q^{65} + (\beta_{5} - 2 \beta_{4} + \beta_{3} - \beta_{2} + 2) q^{67} + (\beta_{4} - 2 \beta_{3} + \beta_{2} + 2 \beta_1 + 5) q^{69} + ( - 3 \beta_1 + 6) q^{71} + (2 \beta_{5} - 2 \beta_{2} - 2 \beta_1 + 4) q^{73} + ( - \beta_{5} + \beta_{4} - \beta_{3} + 2 \beta_{2} + 6 \beta_1 + 7) q^{75} + \beta_{4} q^{77} + (3 \beta_{5} + \beta_{4} + 2 \beta_{2} + 4 \beta_1 - 1) q^{79} + (\beta_{5} + 2 \beta_{3} + \beta_{2} - 2 \beta_1 + 2) q^{81} + ( - \beta_{5} + \beta_{4} - 2 \beta_{3} + \beta_{2} + 3) q^{83} + ( - \beta_{5} - \beta_{4} - 4 \beta_{2} + 1) q^{85} + (2 \beta_{5} - \beta_{4} - 2 \beta_1 + 6) q^{87} + ( - 2 \beta_{5} + \beta_{4} - \beta_{2} - 3) q^{89} + ( - 4 \beta_{5} - \beta_{4} - \beta_{2} - 7 \beta_1 - 1) q^{91} + (2 \beta_{2} + 6 \beta_1 + 1) q^{93} + \beta_{5} q^{95} + ( - \beta_{4} + \beta_{2} - 7) q^{97} + ( - \beta_{2} - 2) q^{99}+O(q^{100})$$ q + b1 * q^3 - b5 * q^5 - b4 * q^7 + (b2 + 2) * q^9 - q^11 + b3 * q^13 + (-b5 + b3 + b2 + 2) * q^15 + (-b5 - b4 - 2*b1 + 3) * q^17 - q^19 + (-b4 - b3 - b2 - b1 + 1) * q^21 + (b5 + b2 + 2*b1) * q^23 + (b2 + 2*b1 + 4) * q^25 + (-b5 + b4 - b3 + b1 - 3) * q^27 + (b5 - b2 + b1) * q^29 + (-b5 - b3 + b2 + 4) * q^31 - b1 * q^33 + (-b4 + 2*b3 - 3*b2 - 2*b1 - 4) * q^35 + (2*b5 + b4 + b2 + 2*b1 + 3) * q^37 + (-2*b5 + b4 - b2 - 2) * q^39 + (-b5 - b4 + b3 + 2*b1 - 1) * q^41 + (b2 + 2*b1 - 2) * q^43 + (-b5 + 2*b4 + 4*b1 - 3) * q^45 + (-2*b3 - 2) * q^47 + (-b2 + 2*b1 + 7) * q^49 + (-b5 - b4 - 2*b2 + 2*b1 - 7) * q^51 + (-2*b2 - 2*b1 + 2) * q^53 + b5 * q^55 - b1 * q^57 + (b4 + b2 - 2*b1 + 3) * q^59 + (-b5 + b4 - 2*b3 + 2*b2 + 3) * q^61 + (3*b5 - b2 - 2*b1 + 1) * q^63 + (-b4 - b3 + b2 + 5*b1 + 1) * q^65 + (b5 - 2*b4 + b3 - b2 + 2) * q^67 + (b4 - 2*b3 + b2 + 2*b1 + 5) * q^69 + (-3*b1 + 6) * q^71 + (2*b5 - 2*b2 - 2*b1 + 4) * q^73 + (-b5 + b4 - b3 + 2*b2 + 6*b1 + 7) * q^75 + b4 * q^77 + (3*b5 + b4 + 2*b2 + 4*b1 - 1) * q^79 + (b5 + 2*b3 + b2 - 2*b1 + 2) * q^81 + (-b5 + b4 - 2*b3 + b2 + 3) * q^83 + (-b5 - b4 - 4*b2 + 1) * q^85 + (2*b5 - b4 - 2*b1 + 6) * q^87 + (-2*b5 + b4 - b2 - 3) * q^89 + (-4*b5 - b4 - b2 - 7*b1 - 1) * q^91 + (2*b2 + 6*b1 + 1) * q^93 + b5 * q^95 + (-b4 + b2 - 7) * q^97 + (-b2 - 2) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q + 2 q^{3} - 2 q^{7} + 10 q^{9}+O(q^{10})$$ 6 * q + 2 * q^3 - 2 * q^7 + 10 * q^9 $$6 q + 2 q^{3} - 2 q^{7} + 10 q^{9} - 6 q^{11} + 2 q^{13} + 12 q^{15} + 12 q^{17} - 6 q^{19} + 2 q^{21} + 2 q^{23} + 26 q^{25} - 16 q^{27} + 4 q^{29} + 20 q^{31} - 2 q^{33} - 20 q^{35} + 22 q^{37} - 8 q^{39} - 2 q^{41} - 10 q^{43} - 6 q^{45} - 16 q^{47} + 48 q^{49} - 36 q^{51} + 12 q^{53} - 2 q^{57} + 14 q^{59} + 12 q^{61} + 4 q^{63} + 10 q^{65} + 12 q^{67} + 30 q^{69} + 30 q^{71} + 24 q^{73} + 50 q^{75} + 2 q^{77} + 10 q^{81} + 14 q^{83} + 12 q^{85} + 30 q^{87} - 14 q^{89} - 20 q^{91} + 14 q^{93} - 46 q^{97} - 10 q^{99}+O(q^{100})$$ 6 * q + 2 * q^3 - 2 * q^7 + 10 * q^9 - 6 * q^11 + 2 * q^13 + 12 * q^15 + 12 * q^17 - 6 * q^19 + 2 * q^21 + 2 * q^23 + 26 * q^25 - 16 * q^27 + 4 * q^29 + 20 * q^31 - 2 * q^33 - 20 * q^35 + 22 * q^37 - 8 * q^39 - 2 * q^41 - 10 * q^43 - 6 * q^45 - 16 * q^47 + 48 * q^49 - 36 * q^51 + 12 * q^53 - 2 * q^57 + 14 * q^59 + 12 * q^61 + 4 * q^63 + 10 * q^65 + 12 * q^67 + 30 * q^69 + 30 * q^71 + 24 * q^73 + 50 * q^75 + 2 * q^77 + 10 * q^81 + 14 * q^83 + 12 * q^85 + 30 * q^87 - 14 * q^89 - 20 * q^91 + 14 * q^93 - 46 * q^97 - 10 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - 2x^{5} - 12x^{4} + 28x^{3} + 16x^{2} - 60x + 27$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - 5$$ v^2 - 5 $$\beta_{3}$$ $$=$$ $$( \nu^{5} + \nu^{4} - 12\nu^{3} - 5\nu^{2} + 28\nu - 3 ) / 3$$ (v^5 + v^4 - 12*v^3 - 5*v^2 + 28*v - 3) / 3 $$\beta_{4}$$ $$=$$ $$( -\nu^{5} + 2\nu^{4} + 15\nu^{3} - 25\nu^{2} - 43\nu + 48 ) / 3$$ (-v^5 + 2*v^4 + 15*v^3 - 25*v^2 - 43*v + 48) / 3 $$\beta_{5}$$ $$=$$ $$( -2\nu^{5} + \nu^{4} + 24\nu^{3} - 20\nu^{2} - 50\nu + 42 ) / 3$$ (-2*v^5 + v^4 + 24*v^3 - 20*v^2 - 50*v + 42) / 3
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + 5$$ b2 + 5 $$\nu^{3}$$ $$=$$ $$-\beta_{5} + \beta_{4} - \beta_{3} + 7\beta _1 - 3$$ -b5 + b4 - b3 + 7*b1 - 3 $$\nu^{4}$$ $$=$$ $$\beta_{5} + 2\beta_{3} + 10\beta_{2} - 2\beta _1 + 38$$ b5 + 2*b3 + 10*b2 - 2*b1 + 38 $$\nu^{5}$$ $$=$$ $$-13\beta_{5} + 12\beta_{4} - 11\beta_{3} - 5\beta_{2} + 58\beta _1 - 46$$ -13*b5 + 12*b4 - 11*b3 - 5*b2 + 58*b1 - 46

## 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
 −3.25580 −1.64714 0.658537 1.24102 2.18868 2.81471
0 −3.25580 0 −2.84405 0 1.37411 0 7.60023 0
1.2 0 −1.64714 0 1.84900 0 −3.60453 0 −0.286920 0
1.3 0 0.658537 0 −2.39807 0 −4.45908 0 −2.56633 0
1.4 0 1.24102 0 2.83235 0 4.46564 0 −1.45986 0
1.5 0 2.18868 0 −3.62873 0 4.31127 0 1.79031 0
1.6 0 2.81471 0 4.18951 0 −4.08740 0 4.92257 0
 $$n$$: 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
$$2$$ $$-1$$
$$11$$ $$1$$
$$19$$ $$1$$

## 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 3344.2.a.x 6
4.b odd 2 1 836.2.a.d 6
12.b even 2 1 7524.2.a.r 6
44.c even 2 1 9196.2.a.k 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
836.2.a.d 6 4.b odd 2 1
3344.2.a.x 6 1.a even 1 1 trivial
7524.2.a.r 6 12.b even 2 1
9196.2.a.k 6 44.c even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(3344))$$:

 $$T_{3}^{6} - 2T_{3}^{5} - 12T_{3}^{4} + 28T_{3}^{3} + 16T_{3}^{2} - 60T_{3} + 27$$ T3^6 - 2*T3^5 - 12*T3^4 + 28*T3^3 + 16*T3^2 - 60*T3 + 27 $$T_{5}^{6} - 28T_{5}^{4} - 6T_{5}^{3} + 228T_{5}^{2} + 48T_{5} - 543$$ T5^6 - 28*T5^4 - 6*T5^3 + 228*T5^2 + 48*T5 - 543 $$T_{7}^{6} + 2T_{7}^{5} - 43T_{7}^{4} - 78T_{7}^{3} + 547T_{7}^{2} + 760T_{7} - 1738$$ T7^6 + 2*T7^5 - 43*T7^4 - 78*T7^3 + 547*T7^2 + 760*T7 - 1738

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6}$$
$3$ $$T^{6} - 2 T^{5} - 12 T^{4} + 28 T^{3} + \cdots + 27$$
$5$ $$T^{6} - 28 T^{4} - 6 T^{3} + 228 T^{2} + \cdots - 543$$
$7$ $$T^{6} + 2 T^{5} - 43 T^{4} + \cdots - 1738$$
$11$ $$(T + 1)^{6}$$
$13$ $$T^{6} - 2 T^{5} - 49 T^{4} + \cdots + 2462$$
$17$ $$T^{6} - 12 T^{5} - 18 T^{4} + \cdots - 2592$$
$19$ $$(T + 1)^{6}$$
$23$ $$T^{6} - 2 T^{5} - 75 T^{4} + \cdots - 1584$$
$29$ $$T^{6} - 4 T^{5} - 69 T^{4} + \cdots - 2682$$
$31$ $$T^{6} - 20 T^{5} + 88 T^{4} + \cdots - 7593$$
$37$ $$T^{6} - 22 T^{5} + 67 T^{4} + \cdots + 91824$$
$41$ $$T^{6} + 2 T^{5} - 159 T^{4} + \cdots + 24912$$
$43$ $$T^{6} + 10 T^{5} - 25 T^{4} + \cdots + 4900$$
$47$ $$T^{6} + 16 T^{5} - 96 T^{4} + \cdots + 328896$$
$53$ $$T^{6} - 12 T^{5} - 96 T^{4} + \cdots - 31104$$
$59$ $$T^{6} - 14 T^{5} - 81 T^{4} + \cdots + 31536$$
$61$ $$T^{6} - 12 T^{5} - 214 T^{4} + \cdots + 109728$$
$67$ $$T^{6} - 12 T^{5} - 184 T^{4} + \cdots + 253887$$
$71$ $$T^{6} - 30 T^{5} + 252 T^{4} + \cdots + 2187$$
$73$ $$T^{6} - 24 T^{5} - 52 T^{4} + \cdots - 217728$$
$79$ $$T^{6} - 370 T^{4} + 1184 T^{3} + \cdots + 428832$$
$83$ $$T^{6} - 14 T^{5} - 135 T^{4} + \cdots - 177642$$
$89$ $$T^{6} + 14 T^{5} - 173 T^{4} + \cdots + 19776$$
$97$ $$T^{6} + 46 T^{5} + 791 T^{4} + \cdots + 752$$