# Properties

 Label 3344.2.a.ba Level $3344$ Weight $2$ Character orbit 3344.a Self dual yes Analytic conductor $26.702$ Analytic rank $1$ Dimension $7$ 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: $$1$$ Dimension: $$7$$ Coefficient field: $$\mathbb{Q}[x]/(x^{7} - \cdots)$$ Defining polynomial: $$x^{7} - x^{6} - 14x^{5} + 10x^{4} + 59x^{3} - 27x^{2} - 66x + 30$$ x^7 - x^6 - 14*x^5 + 10*x^4 + 59*x^3 - 27*x^2 - 66*x + 30 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 209) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{6}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{2} q^{3} - \beta_{4} q^{5} + (\beta_{5} - \beta_{2} - 2) q^{7} + ( - \beta_{6} + 2) q^{9}+O(q^{10})$$ q + b2 * q^3 - b4 * q^5 + (b5 - b2 - 2) * q^7 + (-b6 + 2) * q^9 $$q + \beta_{2} q^{3} - \beta_{4} q^{5} + (\beta_{5} - \beta_{2} - 2) q^{7} + ( - \beta_{6} + 2) q^{9} + q^{11} + ( - \beta_{5} + \beta_{4} - \beta_{3} - \beta_{2} + \beta_1) q^{13} + ( - \beta_{5} + \beta_{4} - \beta_1 - 1) q^{15} + (\beta_{6} + \beta_{5} - \beta_{3}) q^{17} - q^{19} + (\beta_{6} + \beta_{4} + \beta_{3} - \beta_{2} + \beta_1 - 3) q^{21} + ( - 2 \beta_{5} + \beta_{3} + \beta_{2} - 1) q^{23} + (\beta_{6} + 2 \beta_{3}) q^{25} + (\beta_{6} - \beta_{4} + 2 \beta_{3} + 2 \beta_{2} - 3 \beta_1) q^{27} + (\beta_{3} - 3) q^{29} + ( - \beta_{5} + \beta_{4} + \beta_1 - 3) q^{31} + \beta_{2} q^{33} + (\beta_{5} + \beta_{4} - \beta_{3} - 2 \beta_{2} - 2 \beta_1 - 1) q^{35} + ( - \beta_{6} - \beta_{5} + 2 \beta_{4} - \beta_{2} + 1) q^{37} + (2 \beta_{6} + \beta_{5} - \beta_{4} - \beta_{3} - 2 \beta_{2} - 5) q^{39} + (\beta_{6} + \beta_{4} - \beta_{2} - \beta_1 - 2) q^{41} + (2 \beta_{5} - \beta_{4} - \beta_{3} - \beta_{2} - 1) q^{43} + ( - \beta_{6} - \beta_{3} - \beta_{2}) q^{45} + ( - 2 \beta_{6} - 2 \beta_{5} + 2 \beta_1) q^{47} + ( - 2 \beta_{5} - \beta_{4} - \beta_{3} + 3 \beta_{2} + 4) q^{49} + ( - \beta_{6} - \beta_{5} + 2 \beta_{4} - \beta_{3} - 2 \beta_{2} + 4 \beta_1 + 4) q^{51} + (2 \beta_{4} + 2 \beta_{3}) q^{53} - \beta_{4} q^{55} - \beta_{2} q^{57} + ( - \beta_{6} - \beta_{5} + 4 \beta_{4} - 2 \beta_{3} - 3 \beta_{2} + 2 \beta_1 + 3) q^{59} + (\beta_{6} + \beta_{5} - \beta_{3} - 2 \beta_1 + 2) q^{61} + (\beta_{6} + \beta_{4} - 2 \beta_{3} - 4 \beta_{2} + 4 \beta_1 - 1) q^{63} + ( - \beta_{6} + \beta_{4} - \beta_{3} + \beta_{2} + 3 \beta_1 - 1) q^{65} + ( - \beta_{5} - \beta_{4} + 2 \beta_{3} + 2 \beta_{2} - 3 \beta_1 - 1) q^{67} + ( - \beta_{6} + \beta_{5} - 2 \beta_{4} - 2 \beta_{3} - 3 \beta_{2} - 2 \beta_1 - 1) q^{69} + (2 \beta_{5} - 2 \beta_{4} - 2 \beta_{3} - \beta_{2} - 2) q^{71} + ( - 2 \beta_{6} - 2 \beta_{4} + 2 \beta_{2}) q^{73} + ( - \beta_{6} + 2 \beta_{5} + \beta_{4} - 2 \beta_{3} - 3 \beta_{2} + 3 \beta_1 - 4) q^{75} + (\beta_{5} - \beta_{2} - 2) q^{77} + (\beta_{6} + \beta_{5} + \beta_{3} - 4 \beta_1 - 8) q^{79} + ( - 3 \beta_{6} - 2 \beta_{5} - \beta_{4} - 2 \beta_{3} + 2 \beta_1 + 2) q^{81} + ( - \beta_{6} + \beta_{5} + \beta_{4} + 2 \beta_{3} - \beta_{2} - 2 \beta_1 + 1) q^{83} + (\beta_{6} + \beta_{5} - \beta_{3} - 2 \beta_{2} - 4 \beta_1 - 2) q^{85} + (\beta_{5} - 3 \beta_{2} - 2) q^{87} + (\beta_{6} - \beta_{5} + 2 \beta_{4} + 4 \beta_{3} + \beta_{2} - 1) q^{89} + ( - 3 \beta_{6} - \beta_{5} - 2 \beta_{4} + 4 \beta_{3} + 4 \beta_{2} - 2 \beta_1 - 1) q^{91} + (\beta_{6} + 2 \beta_{5} - \beta_{4} - \beta_{3} - 5 \beta_{2} - 2) q^{93} + \beta_{4} q^{95} + (\beta_{6} + \beta_{5} - 2 \beta_{4} - 2 \beta_{3} + \beta_{2} - 4 \beta_1 - 3) q^{97} + ( - \beta_{6} + 2) q^{99}+O(q^{100})$$ q + b2 * q^3 - b4 * q^5 + (b5 - b2 - 2) * q^7 + (-b6 + 2) * q^9 + q^11 + (-b5 + b4 - b3 - b2 + b1) * q^13 + (-b5 + b4 - b1 - 1) * q^15 + (b6 + b5 - b3) * q^17 - q^19 + (b6 + b4 + b3 - b2 + b1 - 3) * q^21 + (-2*b5 + b3 + b2 - 1) * q^23 + (b6 + 2*b3) * q^25 + (b6 - b4 + 2*b3 + 2*b2 - 3*b1) * q^27 + (b3 - 3) * q^29 + (-b5 + b4 + b1 - 3) * q^31 + b2 * q^33 + (b5 + b4 - b3 - 2*b2 - 2*b1 - 1) * q^35 + (-b6 - b5 + 2*b4 - b2 + 1) * q^37 + (2*b6 + b5 - b4 - b3 - 2*b2 - 5) * q^39 + (b6 + b4 - b2 - b1 - 2) * q^41 + (2*b5 - b4 - b3 - b2 - 1) * q^43 + (-b6 - b3 - b2) * q^45 + (-2*b6 - 2*b5 + 2*b1) * q^47 + (-2*b5 - b4 - b3 + 3*b2 + 4) * q^49 + (-b6 - b5 + 2*b4 - b3 - 2*b2 + 4*b1 + 4) * q^51 + (2*b4 + 2*b3) * q^53 - b4 * q^55 - b2 * q^57 + (-b6 - b5 + 4*b4 - 2*b3 - 3*b2 + 2*b1 + 3) * q^59 + (b6 + b5 - b3 - 2*b1 + 2) * q^61 + (b6 + b4 - 2*b3 - 4*b2 + 4*b1 - 1) * q^63 + (-b6 + b4 - b3 + b2 + 3*b1 - 1) * q^65 + (-b5 - b4 + 2*b3 + 2*b2 - 3*b1 - 1) * q^67 + (-b6 + b5 - 2*b4 - 2*b3 - 3*b2 - 2*b1 - 1) * q^69 + (2*b5 - 2*b4 - 2*b3 - b2 - 2) * q^71 + (-2*b6 - 2*b4 + 2*b2) * q^73 + (-b6 + 2*b5 + b4 - 2*b3 - 3*b2 + 3*b1 - 4) * q^75 + (b5 - b2 - 2) * q^77 + (b6 + b5 + b3 - 4*b1 - 8) * q^79 + (-3*b6 - 2*b5 - b4 - 2*b3 + 2*b1 + 2) * q^81 + (-b6 + b5 + b4 + 2*b3 - b2 - 2*b1 + 1) * q^83 + (b6 + b5 - b3 - 2*b2 - 4*b1 - 2) * q^85 + (b5 - 3*b2 - 2) * q^87 + (b6 - b5 + 2*b4 + 4*b3 + b2 - 1) * q^89 + (-3*b6 - b5 - 2*b4 + 4*b3 + 4*b2 - 2*b1 - 1) * q^91 + (b6 + 2*b5 - b4 - b3 - 5*b2 - 2) * q^93 + b4 * q^95 + (b6 + b5 - 2*b4 - 2*b3 + b2 - 4*b1 - 3) * q^97 + (-b6 + 2) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$7 q - 2 q^{3} + 2 q^{5} - 10 q^{7} + 11 q^{9}+O(q^{10})$$ 7 * q - 2 * q^3 + 2 * q^5 - 10 * q^7 + 11 * q^9 $$7 q - 2 q^{3} + 2 q^{5} - 10 q^{7} + 11 q^{9} + 7 q^{11} - 4 q^{13} - 12 q^{15} + 2 q^{17} - 7 q^{19} - 14 q^{21} - 10 q^{23} + 9 q^{25} + 4 q^{27} - 18 q^{29} - 24 q^{31} - 2 q^{33} - 8 q^{35} - 24 q^{39} - 12 q^{41} - 2 q^{43} - 4 q^{45} - 8 q^{47} + 17 q^{49} + 24 q^{51} + 2 q^{53} + 2 q^{55} + 2 q^{57} + 10 q^{59} + 14 q^{61} - 14 q^{65} - 8 q^{67} - 6 q^{69} - 10 q^{71} - 6 q^{73} - 26 q^{75} - 10 q^{77} - 52 q^{79} - q^{81} + 10 q^{83} - 12 q^{85} - 6 q^{87} - 12 q^{91} + 2 q^{93} - 2 q^{95} - 24 q^{97} + 11 q^{99}+O(q^{100})$$ 7 * q - 2 * q^3 + 2 * q^5 - 10 * q^7 + 11 * q^9 + 7 * q^11 - 4 * q^13 - 12 * q^15 + 2 * q^17 - 7 * q^19 - 14 * q^21 - 10 * q^23 + 9 * q^25 + 4 * q^27 - 18 * q^29 - 24 * q^31 - 2 * q^33 - 8 * q^35 - 24 * q^39 - 12 * q^41 - 2 * q^43 - 4 * q^45 - 8 * q^47 + 17 * q^49 + 24 * q^51 + 2 * q^53 + 2 * q^55 + 2 * q^57 + 10 * q^59 + 14 * q^61 - 14 * q^65 - 8 * q^67 - 6 * q^69 - 10 * q^71 - 6 * q^73 - 26 * q^75 - 10 * q^77 - 52 * q^79 - q^81 + 10 * q^83 - 12 * q^85 - 6 * q^87 - 12 * q^91 + 2 * q^93 - 2 * q^95 - 24 * q^97 + 11 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{7} - x^{6} - 14x^{5} + 10x^{4} + 59x^{3} - 27x^{2} - 66x + 30$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{4} - 7\nu^{2} - 2\nu + 4 ) / 2$$ (v^4 - 7*v^2 - 2*v + 4) / 2 $$\beta_{3}$$ $$=$$ $$( -\nu^{4} + 9\nu^{2} + 2\nu - 12 ) / 2$$ (-v^4 + 9*v^2 + 2*v - 12) / 2 $$\beta_{4}$$ $$=$$ $$( \nu^{5} - 9\nu^{3} + 14\nu - 6 ) / 2$$ (v^5 - 9*v^3 + 14*v - 6) / 2 $$\beta_{5}$$ $$=$$ $$( \nu^{6} - 10\nu^{4} + 23\nu^{2} - 4\nu - 10 ) / 4$$ (v^6 - 10*v^4 + 23*v^2 - 4*v - 10) / 4 $$\beta_{6}$$ $$=$$ $$( -\nu^{6} + 12\nu^{4} + 4\nu^{3} - 41\nu^{2} - 20\nu + 34 ) / 4$$ (-v^6 + 12*v^4 + 4*v^3 - 41*v^2 - 20*v + 34) / 4
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{3} + \beta_{2} + 4$$ b3 + b2 + 4 $$\nu^{3}$$ $$=$$ $$\beta_{6} + \beta_{5} + \beta_{3} + 5\beta_1$$ b6 + b5 + b3 + 5*b1 $$\nu^{4}$$ $$=$$ $$7\beta_{3} + 9\beta_{2} + 2\beta _1 + 24$$ 7*b3 + 9*b2 + 2*b1 + 24 $$\nu^{5}$$ $$=$$ $$9\beta_{6} + 9\beta_{5} + 2\beta_{4} + 9\beta_{3} + 31\beta _1 + 6$$ 9*b6 + 9*b5 + 2*b4 + 9*b3 + 31*b1 + 6 $$\nu^{6}$$ $$=$$ $$4\beta_{5} + 47\beta_{3} + 67\beta_{2} + 24\beta _1 + 158$$ 4*b5 + 47*b3 + 67*b2 + 24*b1 + 158

## 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
 1.19313 −2.03821 −1.45416 2.61330 0.456669 2.78328 −2.55401
0 −3.16232 0 1.08235 0 1.30958 0 7.00029 0
1.2 0 −1.87275 0 −3.24760 0 −1.92338 0 0.507178 0
1.3 0 −1.71116 0 2.59296 0 2.00933 0 −0.0719365 0
1.4 0 −1.19599 0 4.07680 0 −3.61829 0 −1.56960 0
1.5 0 0.835165 0 0.221953 0 −4.69915 0 −2.30250 0
1.6 0 2.10880 0 −2.97131 0 1.34513 0 1.44705 0
1.7 0 2.99825 0 0.244850 0 −4.42321 0 5.98952 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.7 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.ba 7
4.b odd 2 1 209.2.a.d 7
12.b even 2 1 1881.2.a.p 7
20.d odd 2 1 5225.2.a.n 7
44.c even 2 1 2299.2.a.q 7
76.d even 2 1 3971.2.a.i 7

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
209.2.a.d 7 4.b odd 2 1
1881.2.a.p 7 12.b even 2 1
2299.2.a.q 7 44.c even 2 1
3344.2.a.ba 7 1.a even 1 1 trivial
3971.2.a.i 7 76.d even 2 1
5225.2.a.n 7 20.d odd 2 1

## Hecke kernels

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

 $$T_{3}^{7} + 2T_{3}^{6} - 14T_{3}^{5} - 28T_{3}^{4} + 46T_{3}^{3} + 100T_{3}^{2} - 17T_{3} - 64$$ T3^7 + 2*T3^6 - 14*T3^5 - 28*T3^4 + 46*T3^3 + 100*T3^2 - 17*T3 - 64 $$T_{5}^{7} - 2T_{5}^{6} - 20T_{5}^{5} + 34T_{5}^{4} + 88T_{5}^{3} - 156T_{5}^{2} + 57T_{5} - 6$$ T5^7 - 2*T5^6 - 20*T5^5 + 34*T5^4 + 88*T5^3 - 156*T5^2 + 57*T5 - 6 $$T_{7}^{7} + 10T_{7}^{6} + 17T_{7}^{5} - 86T_{7}^{4} - 185T_{7}^{3} + 316T_{7}^{2} + 394T_{7} - 512$$ T7^7 + 10*T7^6 + 17*T7^5 - 86*T7^4 - 185*T7^3 + 316*T7^2 + 394*T7 - 512

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{7}$$
$3$ $$T^{7} + 2 T^{6} - 14 T^{5} - 28 T^{4} + \cdots - 64$$
$5$ $$T^{7} - 2 T^{6} - 20 T^{5} + 34 T^{4} + \cdots - 6$$
$7$ $$T^{7} + 10 T^{6} + 17 T^{5} + \cdots - 512$$
$11$ $$(T - 1)^{7}$$
$13$ $$T^{7} + 4 T^{6} - 51 T^{5} + \cdots - 5716$$
$17$ $$T^{7} - 2 T^{6} - 70 T^{5} + \cdots - 17088$$
$19$ $$(T + 1)^{7}$$
$23$ $$T^{7} + 10 T^{6} - 51 T^{5} + \cdots - 1920$$
$29$ $$T^{7} + 18 T^{6} + 117 T^{5} + \cdots - 276$$
$31$ $$T^{7} + 24 T^{6} + 214 T^{5} + 904 T^{4} + \cdots - 4$$
$37$ $$T^{7} - 121 T^{5} - 194 T^{4} + \cdots - 8992$$
$41$ $$T^{7} + 12 T^{6} - 5 T^{5} + \cdots - 1824$$
$43$ $$T^{7} + 2 T^{6} - 89 T^{5} + \cdots - 4976$$
$47$ $$T^{7} + 8 T^{6} - 152 T^{5} + \cdots - 79872$$
$53$ $$T^{7} - 2 T^{6} - 160 T^{5} + \cdots + 768$$
$59$ $$T^{7} - 10 T^{6} - 345 T^{5} + \cdots + 6552192$$
$61$ $$T^{7} - 14 T^{6} - 34 T^{5} + \cdots - 36544$$
$67$ $$T^{7} + 8 T^{6} - 170 T^{5} + \cdots - 13544$$
$71$ $$T^{7} + 10 T^{6} - 134 T^{5} + \cdots - 39756$$
$73$ $$T^{7} + 6 T^{6} - 220 T^{5} + \cdots + 67328$$
$79$ $$T^{7} + 52 T^{6} + 970 T^{5} + \cdots + 203264$$
$83$ $$T^{7} - 10 T^{6} - 219 T^{5} + \cdots - 576936$$
$89$ $$T^{7} - 401 T^{5} - 698 T^{4} + \cdots - 8199552$$
$97$ $$T^{7} + 24 T^{6} - 189 T^{5} + \cdots - 17393056$$