Properties

 Label 7600.2.a.ch Level $7600$ Weight $2$ Character orbit 7600.a Self dual yes Analytic conductor $60.686$ Analytic rank $1$ Dimension $6$ CM no Inner twists $1$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$7600 = 2^{4} \cdot 5^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 7600.a (trivial)

Newform invariants

 Self dual: yes Analytic conductor: $$60.6863055362$$ Analytic rank: $$1$$ Dimension: $$6$$ Coefficient field: $$\mathbb{Q}[x]/(x^{6} - \cdots)$$ Defining polynomial: $$x^{6} - 2x^{5} - 10x^{4} + 16x^{3} + 15x^{2} - 14x - 3$$ x^6 - 2*x^5 - 10*x^4 + 16*x^3 + 15*x^2 - 14*x - 3 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 3800) 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.

 $$f(q)$$ $$=$$ $$q - \beta_1 q^{3} + ( - \beta_{4} - \beta_1) q^{7} + (\beta_{2} + \beta_1 + 1) q^{9}+O(q^{10})$$ q - b1 * q^3 + (-b4 - b1) * q^7 + (b2 + b1 + 1) * q^9 $$q - \beta_1 q^{3} + ( - \beta_{4} - \beta_1) q^{7} + (\beta_{2} + \beta_1 + 1) q^{9} + ( - \beta_{5} + \beta_{4} - \beta_{3} + \beta_1 - 1) q^{11} + (\beta_{4} + \beta_{3} - \beta_{2}) q^{13} + (\beta_{5} - \beta_{4} + \beta_1 - 3) q^{17} + q^{19} + (\beta_{3} + \beta_{2} + \beta_1 + 3) q^{21} + ( - 2 \beta_{5} + \beta_{4} - \beta_{2} + \beta_1 - 2) q^{23} + (2 \beta_{5} - 2 \beta_{4} - 3 \beta_1 - 1) q^{27} + (\beta_{5} - \beta_{3} - \beta_1 + 1) q^{29} + (2 \beta_{5} + \beta_{3} + \beta_{2} - 2 \beta_1) q^{31} + (\beta_{4} - \beta_{2}) q^{33} + ( - \beta_{5} + 3 \beta_1 - 2) q^{37} + ( - \beta_{5} - 2 \beta_{3} - \beta_{2} + 3 \beta_1 - 4) q^{39} + (\beta_{5} - \beta_{3} - \beta_{2} + \beta_1 - 1) q^{41} + (2 \beta_{5} - \beta_{4} + \beta_{3} - 2 \beta_1 - 2) q^{43} + (\beta_{4} - \beta_{2} - 1) q^{47} + ( - 3 \beta_{5} + 2 \beta_{4} - \beta_{2} + \beta_1 + 4) q^{49} + ( - \beta_{5} + \beta_{4} + \beta_{3} - \beta_{2} + 3 \beta_1 - 6) q^{51} + ( - \beta_{5} - 3 \beta_{3} + 1) q^{53} - \beta_1 q^{57} + ( - \beta_{5} + 4 \beta_1 - 1) q^{59} + ( - \beta_{3} + \beta_{2} + 4 \beta_1) q^{61} + (3 \beta_{5} - \beta_{4} - \beta_{3} - 6 \beta_1 - 3) q^{63} + ( - 2 \beta_{5} + \beta_{3} - \beta_1 + 2) q^{67} + ( - \beta_{3} - 2 \beta_{2} + 3 \beta_1 - 4) q^{69} + (2 \beta_{5} - \beta_{4} + \beta_{3} + 2 \beta_{2} - 2 \beta_1 - 2) q^{71} + (2 \beta_{5} + 3 \beta_{3} + \beta_{2} - \beta_1 + 2) q^{73} + (4 \beta_{5} - 3 \beta_{4} + 2 \beta_{3} + \beta_{2} - 6) q^{77} + (3 \beta_{5} + 2 \beta_{3} + 2 \beta_{2} + \beta_1 + 3) q^{79} + ( - 2 \beta_{5} + 2 \beta_{4} + 2 \beta_{3} + 3 \beta_1 + 5) q^{81} + (\beta_{5} + \beta_{3} - \beta_{2} - \beta_1 - 5) q^{83} + ( - 2 \beta_{5} + 3 \beta_{4} + \beta_{3} + \beta_{2} + 2 \beta_1 + 5) q^{87} + (\beta_{5} + \beta_{4} + 2 \beta_{3} - \beta_{2} - \beta_1 + 3) q^{89} + ( - 2 \beta_{5} - \beta_{2} + \beta_1 - 7) q^{91} + (\beta_{5} - 2 \beta_{4} - \beta_{3} + 3 \beta_{2} - \beta_1 + 7) q^{93} + ( - 4 \beta_{5} + 2 \beta_{4} - \beta_{3} - \beta_{2} + 2 \beta_1 - 2) q^{97} + (\beta_{5} - \beta_{4} + 2 \beta_{3} - \beta_{2} + \beta_1 + 1) q^{99}+O(q^{100})$$ q - b1 * q^3 + (-b4 - b1) * q^7 + (b2 + b1 + 1) * q^9 + (-b5 + b4 - b3 + b1 - 1) * q^11 + (b4 + b3 - b2) * q^13 + (b5 - b4 + b1 - 3) * q^17 + q^19 + (b3 + b2 + b1 + 3) * q^21 + (-2*b5 + b4 - b2 + b1 - 2) * q^23 + (2*b5 - 2*b4 - 3*b1 - 1) * q^27 + (b5 - b3 - b1 + 1) * q^29 + (2*b5 + b3 + b2 - 2*b1) * q^31 + (b4 - b2) * q^33 + (-b5 + 3*b1 - 2) * q^37 + (-b5 - 2*b3 - b2 + 3*b1 - 4) * q^39 + (b5 - b3 - b2 + b1 - 1) * q^41 + (2*b5 - b4 + b3 - 2*b1 - 2) * q^43 + (b4 - b2 - 1) * q^47 + (-3*b5 + 2*b4 - b2 + b1 + 4) * q^49 + (-b5 + b4 + b3 - b2 + 3*b1 - 6) * q^51 + (-b5 - 3*b3 + 1) * q^53 - b1 * q^57 + (-b5 + 4*b1 - 1) * q^59 + (-b3 + b2 + 4*b1) * q^61 + (3*b5 - b4 - b3 - 6*b1 - 3) * q^63 + (-2*b5 + b3 - b1 + 2) * q^67 + (-b3 - 2*b2 + 3*b1 - 4) * q^69 + (2*b5 - b4 + b3 + 2*b2 - 2*b1 - 2) * q^71 + (2*b5 + 3*b3 + b2 - b1 + 2) * q^73 + (4*b5 - 3*b4 + 2*b3 + b2 - 6) * q^77 + (3*b5 + 2*b3 + 2*b2 + b1 + 3) * q^79 + (-2*b5 + 2*b4 + 2*b3 + 3*b1 + 5) * q^81 + (b5 + b3 - b2 - b1 - 5) * q^83 + (-2*b5 + 3*b4 + b3 + b2 + 2*b1 + 5) * q^87 + (b5 + b4 + 2*b3 - b2 - b1 + 3) * q^89 + (-2*b5 - b2 + b1 - 7) * q^91 + (b5 - 2*b4 - b3 + 3*b2 - b1 + 7) * q^93 + (-4*b5 + 2*b4 - b3 - b2 + 2*b1 - 2) * q^97 + (b5 - b4 + 2*b3 - b2 + b1 + 1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q - 2 q^{3} - 2 q^{7} + 6 q^{9}+O(q^{10})$$ 6 * q - 2 * q^3 - 2 * q^7 + 6 * q^9 $$6 q - 2 q^{3} - 2 q^{7} + 6 q^{9} - 3 q^{11} - q^{13} - 14 q^{17} + 6 q^{19} + 15 q^{21} - 12 q^{23} - 8 q^{27} + 9 q^{29} - 5 q^{31} + 2 q^{33} - 8 q^{37} - 12 q^{39} + 3 q^{41} - 15 q^{43} - 4 q^{47} + 22 q^{49} - 33 q^{51} + 13 q^{53} - 2 q^{57} + 9 q^{61} - 21 q^{63} + 3 q^{67} - 11 q^{69} - 19 q^{71} + 3 q^{73} - 36 q^{77} + 16 q^{79} + 26 q^{81} - 31 q^{83} + 25 q^{87} + 14 q^{89} - 42 q^{91} + 39 q^{93} - 11 q^{97} + 6 q^{99}+O(q^{100})$$ 6 * q - 2 * q^3 - 2 * q^7 + 6 * q^9 - 3 * q^11 - q^13 - 14 * q^17 + 6 * q^19 + 15 * q^21 - 12 * q^23 - 8 * q^27 + 9 * q^29 - 5 * q^31 + 2 * q^33 - 8 * q^37 - 12 * q^39 + 3 * q^41 - 15 * q^43 - 4 * q^47 + 22 * q^49 - 33 * q^51 + 13 * q^53 - 2 * q^57 + 9 * q^61 - 21 * q^63 + 3 * q^67 - 11 * q^69 - 19 * q^71 + 3 * q^73 - 36 * q^77 + 16 * q^79 + 26 * q^81 - 31 * q^83 + 25 * q^87 + 14 * q^89 - 42 * q^91 + 39 * q^93 - 11 * q^97 + 6 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - 2x^{5} - 10x^{4} + 16x^{3} + 15x^{2} - 14x - 3$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - \nu - 4$$ v^2 - v - 4 $$\beta_{3}$$ $$=$$ $$( \nu^{4} - \nu^{3} - 9\nu^{2} + 6\nu + 5 ) / 2$$ (v^4 - v^3 - 9*v^2 + 6*v + 5) / 2 $$\beta_{4}$$ $$=$$ $$( \nu^{5} - 2\nu^{4} - 9\nu^{3} + 15\nu^{2} + 6\nu - 8 ) / 2$$ (v^5 - 2*v^4 - 9*v^3 + 15*v^2 + 6*v - 8) / 2 $$\beta_{5}$$ $$=$$ $$( \nu^{5} - 2\nu^{4} - 10\nu^{3} + 15\nu^{2} + 15\nu - 7 ) / 2$$ (v^5 - 2*v^4 - 10*v^3 + 15*v^2 + 15*v - 7) / 2
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + \beta _1 + 4$$ b2 + b1 + 4 $$\nu^{3}$$ $$=$$ $$-2\beta_{5} + 2\beta_{4} + 9\beta _1 + 1$$ -2*b5 + 2*b4 + 9*b1 + 1 $$\nu^{4}$$ $$=$$ $$-2\beta_{5} + 2\beta_{4} + 2\beta_{3} + 9\beta_{2} + 12\beta _1 + 32$$ -2*b5 + 2*b4 + 2*b3 + 9*b2 + 12*b1 + 32 $$\nu^{5}$$ $$=$$ $$-22\beta_{5} + 24\beta_{4} + 4\beta_{3} + 3\beta_{2} + 84\beta _1 + 21$$ -22*b5 + 24*b4 + 4*b3 + 3*b2 + 84*b1 + 21

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
 3.26143 1.93590 0.848258 −0.185519 −1.08999 −2.77008
0 −3.26143 0 0 0 −4.07225 0 7.63693 0
1.2 0 −1.93590 0 0 0 1.24708 0 0.747704 0
1.3 0 −0.848258 0 0 0 −1.74484 0 −2.28046 0
1.4 0 0.185519 0 0 0 4.45651 0 −2.96558 0
1.5 0 1.08999 0 0 0 −4.19727 0 −1.81192 0
1.6 0 2.77008 0 0 0 2.31077 0 4.67334 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
$$2$$ $$1$$
$$5$$ $$-1$$
$$19$$ $$-1$$

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 7600.2.a.ch 6
4.b odd 2 1 3800.2.a.bc yes 6
5.b even 2 1 7600.2.a.cl 6
20.d odd 2 1 3800.2.a.ba 6
20.e even 4 2 3800.2.d.q 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3800.2.a.ba 6 20.d odd 2 1
3800.2.a.bc yes 6 4.b odd 2 1
3800.2.d.q 12 20.e even 4 2
7600.2.a.ch 6 1.a even 1 1 trivial
7600.2.a.cl 6 5.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(7600))$$:

 $$T_{3}^{6} + 2T_{3}^{5} - 10T_{3}^{4} - 16T_{3}^{3} + 15T_{3}^{2} + 14T_{3} - 3$$ T3^6 + 2*T3^5 - 10*T3^4 - 16*T3^3 + 15*T3^2 + 14*T3 - 3 $$T_{7}^{6} + 2T_{7}^{5} - 30T_{7}^{4} - 48T_{7}^{3} + 223T_{7}^{2} + 154T_{7} - 383$$ T7^6 + 2*T7^5 - 30*T7^4 - 48*T7^3 + 223*T7^2 + 154*T7 - 383 $$T_{11}^{6} + 3T_{11}^{5} - 40T_{11}^{4} - 139T_{11}^{3} + 10T_{11}^{2} + 208T_{11} - 88$$ T11^6 + 3*T11^5 - 40*T11^4 - 139*T11^3 + 10*T11^2 + 208*T11 - 88 $$T_{13}^{6} + T_{13}^{5} - 65T_{13}^{4} - 15T_{13}^{3} + 883T_{13}^{2} - 1621T_{13} + 825$$ T13^6 + T13^5 - 65*T13^4 - 15*T13^3 + 883*T13^2 - 1621*T13 + 825

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6}$$
$3$ $$T^{6} + 2 T^{5} - 10 T^{4} - 16 T^{3} + \cdots - 3$$
$5$ $$T^{6}$$
$7$ $$T^{6} + 2 T^{5} - 30 T^{4} - 48 T^{3} + \cdots - 383$$
$11$ $$T^{6} + 3 T^{5} - 40 T^{4} - 139 T^{3} + \cdots - 88$$
$13$ $$T^{6} + T^{5} - 65 T^{4} - 15 T^{3} + \cdots + 825$$
$17$ $$T^{6} + 14 T^{5} + 22 T^{4} + \cdots + 3147$$
$19$ $$(T - 1)^{6}$$
$23$ $$T^{6} + 12 T^{5} - 26 T^{4} + \cdots - 2487$$
$29$ $$T^{6} - 9 T^{5} - 67 T^{4} + \cdots + 21951$$
$31$ $$T^{6} + 5 T^{5} - 136 T^{4} + \cdots + 11000$$
$37$ $$T^{6} + 8 T^{5} - 90 T^{4} + \cdots - 7365$$
$41$ $$T^{6} - 3 T^{5} - 180 T^{4} + \cdots - 113472$$
$43$ $$T^{6} + 15 T^{5} + 13 T^{4} + \cdots + 6120$$
$47$ $$T^{6} + 4 T^{5} - 52 T^{4} + \cdots + 1791$$
$53$ $$T^{6} - 13 T^{5} - 192 T^{4} + \cdots + 9285$$
$59$ $$T^{6} - 193 T^{4} - 255 T^{3} + \cdots - 32328$$
$61$ $$T^{6} - 9 T^{5} - 132 T^{4} + \cdots + 12424$$
$67$ $$T^{6} - 3 T^{5} - 199 T^{4} + \cdots - 136033$$
$71$ $$T^{6} + 19 T^{5} - 75 T^{4} + \cdots - 27576$$
$73$ $$T^{6} - 3 T^{5} - 305 T^{4} + \cdots - 741033$$
$79$ $$T^{6} - 16 T^{5} - 285 T^{4} + \cdots - 553536$$
$83$ $$T^{6} + 31 T^{5} + 326 T^{4} + \cdots - 71160$$
$89$ $$T^{6} - 14 T^{5} - 67 T^{4} + \cdots + 3240$$
$97$ $$T^{6} + 11 T^{5} - 208 T^{4} + \cdots - 288792$$