# Properties

 Label 6762.2.a.ct Level $6762$ Weight $2$ Character orbit 6762.a Self dual yes Analytic conductor $53.995$ Analytic rank $0$ Dimension $4$ CM no Inner twists $1$

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## Newspace parameters

 Level: $$N$$ $$=$$ $$6762 = 2 \cdot 3 \cdot 7^{2} \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 6762.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$53.9948418468$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: 4.4.2624.1 Defining polynomial: $$x^{4} - 2x^{3} - 3x^{2} + 2x + 1$$ x^4 - 2*x^3 - 3*x^2 + 2*x + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{2}$$ 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 $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + q^{2} + q^{3} + q^{4} + (\beta_1 + 2) q^{5} + q^{6} + q^{8} + q^{9}+O(q^{10})$$ q + q^2 + q^3 + q^4 + (b1 + 2) * q^5 + q^6 + q^8 + q^9 $$q + q^{2} + q^{3} + q^{4} + (\beta_1 + 2) q^{5} + q^{6} + q^{8} + q^{9} + (\beta_1 + 2) q^{10} + (\beta_{2} + 1) q^{11} + q^{12} + ( - \beta_{3} - \beta_1 + 3) q^{13} + (\beta_1 + 2) q^{15} + q^{16} + ( - \beta_{3} + \beta_1 - 1) q^{17} + q^{18} + (2 \beta_{3} + \beta_{2} - \beta_1 + 1) q^{19} + (\beta_1 + 2) q^{20} + (\beta_{2} + 1) q^{22} + q^{23} + q^{24} + (4 \beta_1 + 1) q^{25} + ( - \beta_{3} - \beta_1 + 3) q^{26} + q^{27} + ( - 2 \beta_{3} - 2 \beta_{2} - 2 \beta_1 + 2) q^{29} + (\beta_1 + 2) q^{30} + (2 \beta_{3} + \beta_{2} - \beta_1 - 1) q^{31} + q^{32} + (\beta_{2} + 1) q^{33} + ( - \beta_{3} + \beta_1 - 1) q^{34} + q^{36} + (\beta_{3} - \beta_1 + 1) q^{37} + (2 \beta_{3} + \beta_{2} - \beta_1 + 1) q^{38} + ( - \beta_{3} - \beta_1 + 3) q^{39} + (\beta_1 + 2) q^{40} + ( - 2 \beta_{2} - \beta_1 - 2) q^{41} + (3 \beta_{3} + \beta_1 + 1) q^{43} + (\beta_{2} + 1) q^{44} + (\beta_1 + 2) q^{45} + q^{46} + ( - 2 \beta_{3} + \beta_{2} + \beta_1 + 3) q^{47} + q^{48} + (4 \beta_1 + 1) q^{50} + ( - \beta_{3} + \beta_1 - 1) q^{51} + ( - \beta_{3} - \beta_1 + 3) q^{52} + ( - \beta_{3} - 2 \beta_{2} + \beta_1 + 1) q^{53} + q^{54} + (\beta_{3} + \beta_{2} + \beta_1 + 2) q^{55} + (2 \beta_{3} + \beta_{2} - \beta_1 + 1) q^{57} + ( - 2 \beta_{3} - 2 \beta_{2} - 2 \beta_1 + 2) q^{58} + ( - 2 \beta_{3} - 2 \beta_{2} - 2 \beta_1 + 4) q^{59} + (\beta_1 + 2) q^{60} + ( - 2 \beta_{2} - \beta_1 + 4) q^{61} + (2 \beta_{3} + \beta_{2} - \beta_1 - 1) q^{62} + q^{64} + ( - 3 \beta_{3} - \beta_{2} + \beta_1 + 4) q^{65} + (\beta_{2} + 1) q^{66} + ( - \beta_{3} + 5 \beta_1 + 5) q^{67} + ( - \beta_{3} + \beta_1 - 1) q^{68} + q^{69} + ( - \beta_{3} + \beta_{2} - \beta_1 - 6) q^{71} + q^{72} + ( - \beta_{3} - \beta_{2} - 4 \beta_1 + 2) q^{73} + (\beta_{3} - \beta_1 + 1) q^{74} + (4 \beta_1 + 1) q^{75} + (2 \beta_{3} + \beta_{2} - \beta_1 + 1) q^{76} + ( - \beta_{3} - \beta_1 + 3) q^{78} + (\beta_{3} - \beta_{2} - 3 \beta_1 - 8) q^{79} + (\beta_1 + 2) q^{80} + q^{81} + ( - 2 \beta_{2} - \beta_1 - 2) q^{82} + (\beta_{2} + \beta_1 - 1) q^{83} + ( - 3 \beta_{3} - \beta_{2} + \beta_1) q^{85} + (3 \beta_{3} + \beta_1 + 1) q^{86} + ( - 2 \beta_{3} - 2 \beta_{2} - 2 \beta_1 + 2) q^{87} + (\beta_{2} + 1) q^{88} + (3 \beta_{3} - 2 \beta_{2} + 3 \beta_1 + 5) q^{89} + (\beta_1 + 2) q^{90} + q^{92} + (2 \beta_{3} + \beta_{2} - \beta_1 - 1) q^{93} + ( - 2 \beta_{3} + \beta_{2} + \beta_1 + 3) q^{94} + (7 \beta_{3} + 3 \beta_{2} - \beta_1) q^{95} + q^{96} + (\beta_{3} - 7 \beta_1 + 1) q^{97} + (\beta_{2} + 1) q^{99}+O(q^{100})$$ q + q^2 + q^3 + q^4 + (b1 + 2) * q^5 + q^6 + q^8 + q^9 + (b1 + 2) * q^10 + (b2 + 1) * q^11 + q^12 + (-b3 - b1 + 3) * q^13 + (b1 + 2) * q^15 + q^16 + (-b3 + b1 - 1) * q^17 + q^18 + (2*b3 + b2 - b1 + 1) * q^19 + (b1 + 2) * q^20 + (b2 + 1) * q^22 + q^23 + q^24 + (4*b1 + 1) * q^25 + (-b3 - b1 + 3) * q^26 + q^27 + (-2*b3 - 2*b2 - 2*b1 + 2) * q^29 + (b1 + 2) * q^30 + (2*b3 + b2 - b1 - 1) * q^31 + q^32 + (b2 + 1) * q^33 + (-b3 + b1 - 1) * q^34 + q^36 + (b3 - b1 + 1) * q^37 + (2*b3 + b2 - b1 + 1) * q^38 + (-b3 - b1 + 3) * q^39 + (b1 + 2) * q^40 + (-2*b2 - b1 - 2) * q^41 + (3*b3 + b1 + 1) * q^43 + (b2 + 1) * q^44 + (b1 + 2) * q^45 + q^46 + (-2*b3 + b2 + b1 + 3) * q^47 + q^48 + (4*b1 + 1) * q^50 + (-b3 + b1 - 1) * q^51 + (-b3 - b1 + 3) * q^52 + (-b3 - 2*b2 + b1 + 1) * q^53 + q^54 + (b3 + b2 + b1 + 2) * q^55 + (2*b3 + b2 - b1 + 1) * q^57 + (-2*b3 - 2*b2 - 2*b1 + 2) * q^58 + (-2*b3 - 2*b2 - 2*b1 + 4) * q^59 + (b1 + 2) * q^60 + (-2*b2 - b1 + 4) * q^61 + (2*b3 + b2 - b1 - 1) * q^62 + q^64 + (-3*b3 - b2 + b1 + 4) * q^65 + (b2 + 1) * q^66 + (-b3 + 5*b1 + 5) * q^67 + (-b3 + b1 - 1) * q^68 + q^69 + (-b3 + b2 - b1 - 6) * q^71 + q^72 + (-b3 - b2 - 4*b1 + 2) * q^73 + (b3 - b1 + 1) * q^74 + (4*b1 + 1) * q^75 + (2*b3 + b2 - b1 + 1) * q^76 + (-b3 - b1 + 3) * q^78 + (b3 - b2 - 3*b1 - 8) * q^79 + (b1 + 2) * q^80 + q^81 + (-2*b2 - b1 - 2) * q^82 + (b2 + b1 - 1) * q^83 + (-3*b3 - b2 + b1) * q^85 + (3*b3 + b1 + 1) * q^86 + (-2*b3 - 2*b2 - 2*b1 + 2) * q^87 + (b2 + 1) * q^88 + (3*b3 - 2*b2 + 3*b1 + 5) * q^89 + (b1 + 2) * q^90 + q^92 + (2*b3 + b2 - b1 - 1) * q^93 + (-2*b3 + b2 + b1 + 3) * q^94 + (7*b3 + 3*b2 - b1) * q^95 + q^96 + (b3 - 7*b1 + 1) * q^97 + (b2 + 1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 4 q^{2} + 4 q^{3} + 4 q^{4} + 8 q^{5} + 4 q^{6} + 4 q^{8} + 4 q^{9}+O(q^{10})$$ 4 * q + 4 * q^2 + 4 * q^3 + 4 * q^4 + 8 * q^5 + 4 * q^6 + 4 * q^8 + 4 * q^9 $$4 q + 4 q^{2} + 4 q^{3} + 4 q^{4} + 8 q^{5} + 4 q^{6} + 4 q^{8} + 4 q^{9} + 8 q^{10} + 4 q^{11} + 4 q^{12} + 12 q^{13} + 8 q^{15} + 4 q^{16} - 4 q^{17} + 4 q^{18} + 4 q^{19} + 8 q^{20} + 4 q^{22} + 4 q^{23} + 4 q^{24} + 4 q^{25} + 12 q^{26} + 4 q^{27} + 8 q^{29} + 8 q^{30} - 4 q^{31} + 4 q^{32} + 4 q^{33} - 4 q^{34} + 4 q^{36} + 4 q^{37} + 4 q^{38} + 12 q^{39} + 8 q^{40} - 8 q^{41} + 4 q^{43} + 4 q^{44} + 8 q^{45} + 4 q^{46} + 12 q^{47} + 4 q^{48} + 4 q^{50} - 4 q^{51} + 12 q^{52} + 4 q^{53} + 4 q^{54} + 8 q^{55} + 4 q^{57} + 8 q^{58} + 16 q^{59} + 8 q^{60} + 16 q^{61} - 4 q^{62} + 4 q^{64} + 16 q^{65} + 4 q^{66} + 20 q^{67} - 4 q^{68} + 4 q^{69} - 24 q^{71} + 4 q^{72} + 8 q^{73} + 4 q^{74} + 4 q^{75} + 4 q^{76} + 12 q^{78} - 32 q^{79} + 8 q^{80} + 4 q^{81} - 8 q^{82} - 4 q^{83} + 4 q^{86} + 8 q^{87} + 4 q^{88} + 20 q^{89} + 8 q^{90} + 4 q^{92} - 4 q^{93} + 12 q^{94} + 4 q^{96} + 4 q^{97} + 4 q^{99}+O(q^{100})$$ 4 * q + 4 * q^2 + 4 * q^3 + 4 * q^4 + 8 * q^5 + 4 * q^6 + 4 * q^8 + 4 * q^9 + 8 * q^10 + 4 * q^11 + 4 * q^12 + 12 * q^13 + 8 * q^15 + 4 * q^16 - 4 * q^17 + 4 * q^18 + 4 * q^19 + 8 * q^20 + 4 * q^22 + 4 * q^23 + 4 * q^24 + 4 * q^25 + 12 * q^26 + 4 * q^27 + 8 * q^29 + 8 * q^30 - 4 * q^31 + 4 * q^32 + 4 * q^33 - 4 * q^34 + 4 * q^36 + 4 * q^37 + 4 * q^38 + 12 * q^39 + 8 * q^40 - 8 * q^41 + 4 * q^43 + 4 * q^44 + 8 * q^45 + 4 * q^46 + 12 * q^47 + 4 * q^48 + 4 * q^50 - 4 * q^51 + 12 * q^52 + 4 * q^53 + 4 * q^54 + 8 * q^55 + 4 * q^57 + 8 * q^58 + 16 * q^59 + 8 * q^60 + 16 * q^61 - 4 * q^62 + 4 * q^64 + 16 * q^65 + 4 * q^66 + 20 * q^67 - 4 * q^68 + 4 * q^69 - 24 * q^71 + 4 * q^72 + 8 * q^73 + 4 * q^74 + 4 * q^75 + 4 * q^76 + 12 * q^78 - 32 * q^79 + 8 * q^80 + 4 * q^81 - 8 * q^82 - 4 * q^83 + 4 * q^86 + 8 * q^87 + 4 * q^88 + 20 * q^89 + 8 * q^90 + 4 * q^92 - 4 * q^93 + 12 * q^94 + 4 * q^96 + 4 * q^97 + 4 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 2x^{3} - 3x^{2} + 2x + 1$$ :

 $$\beta_{1}$$ $$=$$ $$\nu^{3} - 2\nu^{2} - 2\nu + 1$$ v^3 - 2*v^2 - 2*v + 1 $$\beta_{2}$$ $$=$$ $$2\nu^{2} - 4\nu - 3$$ 2*v^2 - 4*v - 3 $$\beta_{3}$$ $$=$$ $$-\nu^{3} + 2\nu^{2} + 4\nu - 2$$ -v^3 + 2*v^2 + 4*v - 2
 $$\nu$$ $$=$$ $$( \beta_{3} + \beta _1 + 1 ) / 2$$ (b3 + b1 + 1) / 2 $$\nu^{2}$$ $$=$$ $$( 2\beta_{3} + \beta_{2} + 2\beta _1 + 5 ) / 2$$ (2*b3 + b2 + 2*b1 + 5) / 2 $$\nu^{3}$$ $$=$$ $$3\beta_{3} + \beta_{2} + 4\beta _1 + 5$$ 3*b3 + b2 + 4*b1 + 5

## 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
 0.814115 −1.22833 −0.360409 2.77462
1.00000 1.00000 1.00000 0.585786 1.00000 0 1.00000 1.00000 0.585786
1.2 1.00000 1.00000 1.00000 0.585786 1.00000 0 1.00000 1.00000 0.585786
1.3 1.00000 1.00000 1.00000 3.41421 1.00000 0 1.00000 1.00000 3.41421
1.4 1.00000 1.00000 1.00000 3.41421 1.00000 0 1.00000 1.00000 3.41421
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$3$$ $$-1$$
$$7$$ $$1$$
$$23$$ $$-1$$

## 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 6762.2.a.ct yes 4
7.b odd 2 1 6762.2.a.ci 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6762.2.a.ci 4 7.b odd 2 1
6762.2.a.ct yes 4 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(6762))$$:

 $$T_{5}^{2} - 4T_{5} + 2$$ T5^2 - 4*T5 + 2 $$T_{11}^{4} - 4T_{11}^{3} - 20T_{11}^{2} + 48T_{11} + 16$$ T11^4 - 4*T11^3 - 20*T11^2 + 48*T11 + 16 $$T_{13}^{4} - 12T_{13}^{3} + 36T_{13}^{2} + 16T_{13} - 112$$ T13^4 - 12*T13^3 + 36*T13^2 + 16*T13 - 112 $$T_{17}^{4} + 4T_{17}^{3} - 12T_{17}^{2} - 48T_{17} - 16$$ T17^4 + 4*T17^3 - 12*T17^2 - 48*T17 - 16

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T - 1)^{4}$$
$3$ $$(T - 1)^{4}$$
$5$ $$(T^{2} - 4 T + 2)^{2}$$
$7$ $$T^{4}$$
$11$ $$T^{4} - 4 T^{3} - 20 T^{2} + 48 T + 16$$
$13$ $$T^{4} - 12 T^{3} + 36 T^{2} + \cdots - 112$$
$17$ $$T^{4} + 4 T^{3} - 12 T^{2} - 48 T - 16$$
$19$ $$T^{4} - 4 T^{3} - 56 T^{2} + 280 T - 292$$
$23$ $$(T - 1)^{4}$$
$29$ $$T^{4} - 8 T^{3} - 104 T^{2} + \cdots + 784$$
$31$ $$T^{4} + 4 T^{3} - 56 T^{2} + 40 T + 28$$
$37$ $$T^{4} - 4 T^{3} - 12 T^{2} + 48 T - 16$$
$41$ $$T^{4} + 8 T^{3} - 84 T^{2} - 656 T - 476$$
$43$ $$T^{4} - 4 T^{3} - 124 T^{2} + \cdots + 3088$$
$47$ $$T^{4} - 12 T^{3} - 56 T^{2} + \cdots + 412$$
$53$ $$T^{4} - 4 T^{3} - 92 T^{2} + \cdots + 1648$$
$59$ $$T^{4} - 16 T^{3} - 32 T^{2} + \cdots - 1024$$
$61$ $$T^{4} - 16 T^{3} - 12 T^{2} + 352 T + 4$$
$67$ $$T^{4} - 20 T^{3} + 36 T^{2} + 560 T + 16$$
$71$ $$T^{4} + 24 T^{3} + 160 T^{2} + \cdots - 1424$$
$73$ $$T^{4} - 8 T^{3} - 68 T^{2} + 464 T - 316$$
$79$ $$T^{4} + 32 T^{3} + 296 T^{2} + \cdots - 5056$$
$83$ $$T^{4} + 4 T^{3} - 24 T^{2} + 8 T + 28$$
$89$ $$T^{4} - 20 T^{3} - 188 T^{2} + \cdots - 9968$$
$97$ $$T^{4} - 4 T^{3} - 204 T^{2} + \cdots + 7952$$
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