# Properties

 Label 8280.2.a.bw Level $8280$ Weight $2$ Character orbit 8280.a Self dual yes Analytic conductor $66.116$ Analytic rank $1$ Dimension $7$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$8280 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 8280.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$66.1161328736$$ Analytic rank: $$1$$ Dimension: $$7$$ Coefficient field: $$\mathbb{Q}[x]/(x^{7} - \cdots)$$ Defining polynomial: $$x^{7} - 3x^{6} - 18x^{5} + 46x^{4} + 60x^{3} - 76x^{2} - 51x - 7$$ x^7 - 3*x^6 - 18*x^5 + 46*x^4 + 60*x^3 - 76*x^2 - 51*x - 7 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{3}$$ 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 $$1,\beta_1,\ldots,\beta_{6}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + q^{5} + (\beta_{4} - 1) q^{7}+O(q^{10})$$ q + q^5 + (b4 - 1) * q^7 $$q + q^{5} + (\beta_{4} - 1) q^{7} - \beta_{3} q^{11} + ( - \beta_{6} + \beta_{3} + \beta_{2} - 2) q^{13} + (\beta_{6} - \beta_{4}) q^{17} + ( - \beta_{5} - \beta_1 - 1) q^{19} + q^{23} + q^{25} + ( - \beta_{4} - \beta_{2} + \beta_1 + 1) q^{29} + (\beta_{5} - \beta_{4} - \beta_{2} - \beta_1 - 1) q^{31} + (\beta_{4} - 1) q^{35} + (\beta_{4} + \beta_{3} + \beta_{2} + \beta_1 - 3) q^{37} + (\beta_{6} + \beta_{5} - \beta_{4} - \beta_{3} - 2 \beta_{2} + 2 \beta_1 + 2) q^{41} + (\beta_{6} + \beta_{5} + \beta_{3} - \beta_{2} + \beta_1 - 1) q^{43} + (\beta_{5} - 2 \beta_{4} + \beta_{3} - \beta_{2} - 1) q^{47} + (\beta_{5} - \beta_{4} - \beta_{3} - 2 \beta_{2} + 2 \beta_1 + 4) q^{49} + ( - \beta_{4} + \beta_{2} - 2 \beta_1 - 2) q^{53} - \beta_{3} q^{55} + (\beta_{6} + \beta_{5} - \beta_{4} - 2 \beta_{3} - \beta_{2} + \beta_1 - 2) q^{59} + ( - \beta_{6} - \beta_{5} - 2 \beta_{4} + \beta_{3} + 1) q^{61} + ( - \beta_{6} + \beta_{3} + \beta_{2} - 2) q^{65} + ( - \beta_{5} + \beta_{4} - \beta_{2} + 2 \beta_1 - 2) q^{67} + ( - 2 \beta_{6} - 2 \beta_{5} + \beta_{4} + \beta_{3} + 2 \beta_{2} - 2 \beta_1 - 3) q^{71} + (\beta_{5} + \beta_{3} + \beta_{2} - 2 \beta_1 - 1) q^{73} + ( - \beta_{5} + 3 \beta_{3} + \beta_{2} - 2 \beta_1 - 3) q^{77} + (\beta_{6} + 2 \beta_{4} - \beta_{3} - \beta_1) q^{79} + (\beta_{6} - \beta_{4} - 2 \beta_{3} - 2) q^{83} + (\beta_{6} - \beta_{4}) q^{85} + (2 \beta_{6} - \beta_{5} - \beta_{3} - \beta_1 - 1) q^{89} + ( - 3 \beta_{5} + 2 \beta_{2} - \beta_1 - 3) q^{91} + ( - \beta_{5} - \beta_1 - 1) q^{95} + (\beta_{6} - \beta_{5} - \beta_{3} + \beta_{2} - \beta_1 + 1) q^{97}+O(q^{100})$$ q + q^5 + (b4 - 1) * q^7 - b3 * q^11 + (-b6 + b3 + b2 - 2) * q^13 + (b6 - b4) * q^17 + (-b5 - b1 - 1) * q^19 + q^23 + q^25 + (-b4 - b2 + b1 + 1) * q^29 + (b5 - b4 - b2 - b1 - 1) * q^31 + (b4 - 1) * q^35 + (b4 + b3 + b2 + b1 - 3) * q^37 + (b6 + b5 - b4 - b3 - 2*b2 + 2*b1 + 2) * q^41 + (b6 + b5 + b3 - b2 + b1 - 1) * q^43 + (b5 - 2*b4 + b3 - b2 - 1) * q^47 + (b5 - b4 - b3 - 2*b2 + 2*b1 + 4) * q^49 + (-b4 + b2 - 2*b1 - 2) * q^53 - b3 * q^55 + (b6 + b5 - b4 - 2*b3 - b2 + b1 - 2) * q^59 + (-b6 - b5 - 2*b4 + b3 + 1) * q^61 + (-b6 + b3 + b2 - 2) * q^65 + (-b5 + b4 - b2 + 2*b1 - 2) * q^67 + (-2*b6 - 2*b5 + b4 + b3 + 2*b2 - 2*b1 - 3) * q^71 + (b5 + b3 + b2 - 2*b1 - 1) * q^73 + (-b5 + 3*b3 + b2 - 2*b1 - 3) * q^77 + (b6 + 2*b4 - b3 - b1) * q^79 + (b6 - b4 - 2*b3 - 2) * q^83 + (b6 - b4) * q^85 + (2*b6 - b5 - b3 - b1 - 1) * q^89 + (-3*b5 + 2*b2 - b1 - 3) * q^91 + (-b5 - b1 - 1) * q^95 + (b6 - b5 - b3 + b2 - b1 + 1) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$7 q + 7 q^{5} - 6 q^{7}+O(q^{10})$$ 7 * q + 7 * q^5 - 6 * q^7 $$7 q + 7 q^{5} - 6 q^{7} - 2 q^{11} - 6 q^{13} - 4 q^{17} - 8 q^{19} + 7 q^{23} + 7 q^{25} + 2 q^{29} - 8 q^{31} - 6 q^{35} - 16 q^{37} + 2 q^{41} - 10 q^{43} - 8 q^{47} + 19 q^{49} - 10 q^{53} - 2 q^{55} - 24 q^{59} + 8 q^{61} - 6 q^{65} - 20 q^{67} - 8 q^{71} + 2 q^{73} - 12 q^{77} - 2 q^{79} - 22 q^{83} - 4 q^{85} - 16 q^{89} - 20 q^{91} - 8 q^{95} + 4 q^{97}+O(q^{100})$$ 7 * q + 7 * q^5 - 6 * q^7 - 2 * q^11 - 6 * q^13 - 4 * q^17 - 8 * q^19 + 7 * q^23 + 7 * q^25 + 2 * q^29 - 8 * q^31 - 6 * q^35 - 16 * q^37 + 2 * q^41 - 10 * q^43 - 8 * q^47 + 19 * q^49 - 10 * q^53 - 2 * q^55 - 24 * q^59 + 8 * q^61 - 6 * q^65 - 20 * q^67 - 8 * q^71 + 2 * q^73 - 12 * q^77 - 2 * q^79 - 22 * q^83 - 4 * q^85 - 16 * q^89 - 20 * q^91 - 8 * q^95 + 4 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{7} - 3x^{6} - 18x^{5} + 46x^{4} + 60x^{3} - 76x^{2} - 51x - 7$$ :

 $$\beta_{1}$$ $$=$$ $$( -11\nu^{6} + 24\nu^{5} + 238\nu^{4} - 352\nu^{3} - 1172\nu^{2} + 264\nu + 553 ) / 224$$ (-11*v^6 + 24*v^5 + 238*v^4 - 352*v^3 - 1172*v^2 + 264*v + 553) / 224 $$\beta_{2}$$ $$=$$ $$( 5\nu^{6} - 16\nu^{5} - 84\nu^{4} + 244\nu^{3} + 212\nu^{2} - 386\nu - 133 ) / 14$$ (5*v^6 - 16*v^5 - 84*v^4 + 244*v^3 + 212*v^2 - 386*v - 133) / 14 $$\beta_{3}$$ $$=$$ $$( -13\nu^{6} + 40\nu^{5} + 226\nu^{4} - 608\nu^{3} - 652\nu^{2} + 984\nu + 383 ) / 32$$ (-13*v^6 + 40*v^5 + 226*v^4 - 608*v^3 - 652*v^2 + 984*v + 383) / 32 $$\beta_{4}$$ $$=$$ $$( 25\nu^{6} - 80\nu^{5} - 434\nu^{4} + 1248\nu^{3} + 1228\nu^{2} - 2280\nu - 651 ) / 56$$ (25*v^6 - 80*v^5 - 434*v^4 + 1248*v^3 + 1228*v^2 - 2280*v - 651) / 56 $$\beta_{5}$$ $$=$$ $$( 111\nu^{6} - 344\nu^{5} - 1974\nu^{4} + 5344\nu^{3} + 6308\nu^{2} - 9608\nu - 4501 ) / 224$$ (111*v^6 - 344*v^5 - 1974*v^4 + 5344*v^3 + 6308*v^2 - 9608*v - 4501) / 224 $$\beta_{6}$$ $$=$$ $$( -99\nu^{6} + 328\nu^{5} + 1694\nu^{4} - 5072\nu^{3} - 4612\nu^{2} + 8872\nu + 2737 ) / 112$$ (-99*v^6 + 328*v^5 + 1694*v^4 - 5072*v^3 - 4612*v^2 + 8872*v + 2737) / 112
 $$\nu$$ $$=$$ $$( \beta_{3} + \beta_{2} - \beta_1 ) / 2$$ (b3 + b2 - b1) / 2 $$\nu^{2}$$ $$=$$ $$( 2\beta_{5} - 2\beta_{4} + \beta_{3} + \beta_{2} + \beta _1 + 12 ) / 2$$ (2*b5 - 2*b4 + b3 + b2 + b1 + 12) / 2 $$\nu^{3}$$ $$=$$ $$\beta_{6} + \beta_{5} + 2\beta_{4} + 7\beta_{3} + 6\beta_{2} - 4\beta _1 + 2$$ b6 + b5 + 2*b4 + 7*b3 + 6*b2 - 4*b1 + 2 $$\nu^{4}$$ $$=$$ $$( 4\beta_{6} + 28\beta_{5} - 24\beta_{4} + 15\beta_{3} + 21\beta_{2} + 21\beta _1 + 154 ) / 2$$ (4*b6 + 28*b5 - 24*b4 + 15*b3 + 21*b2 + 21*b1 + 154) / 2 $$\nu^{5}$$ $$=$$ $$( 52\beta_{6} + 40\beta_{5} + 78\beta_{4} + 187\beta_{3} + 177\beta_{2} - 83\beta _1 + 88 ) / 2$$ (52*b6 + 40*b5 + 78*b4 + 187*b3 + 177*b2 - 83*b1 + 88) / 2 $$\nu^{6}$$ $$=$$ $$68\beta_{6} + 208\beta_{5} - 132\beta_{4} + 101\beta_{3} + 187\beta_{2} + 179\beta _1 + 1109$$ 68*b6 + 208*b5 - 132*b4 + 101*b3 + 187*b2 + 179*b1 + 1109

## 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
 −3.70204 −0.218266 3.22540 1.37429 4.09437 −0.330805 −1.44295
0 0 0 1.00000 0 −5.18676 0 0 0
1.2 0 0 0 1.00000 0 −2.94234 0 0 0
1.3 0 0 0 1.00000 0 −2.82985 0 0 0
1.4 0 0 0 1.00000 0 −0.958551 0 0 0
1.5 0 0 0 1.00000 0 −0.627340 0 0 0
1.6 0 0 0 1.00000 0 2.34984 0 0 0
1.7 0 0 0 1.00000 0 4.19500 0 0 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
$$2$$ $$1$$
$$3$$ $$1$$
$$5$$ $$-1$$
$$23$$ $$-1$$

## 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 8280.2.a.bw yes 7
3.b odd 2 1 8280.2.a.bv 7

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8280.2.a.bv 7 3.b odd 2 1
8280.2.a.bw yes 7 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(8280))$$:

 $$T_{7}^{7} + 6T_{7}^{6} - 16T_{7}^{5} - 134T_{7}^{4} - 77T_{7}^{3} + 516T_{7}^{2} + 732T_{7} + 256$$ T7^7 + 6*T7^6 - 16*T7^5 - 134*T7^4 - 77*T7^3 + 516*T7^2 + 732*T7 + 256 $$T_{11}^{7} + 2T_{11}^{6} - 50T_{11}^{5} - 128T_{11}^{4} + 556T_{11}^{3} + 1928T_{11}^{2} + 1728T_{11} + 384$$ T11^7 + 2*T11^6 - 50*T11^5 - 128*T11^4 + 556*T11^3 + 1928*T11^2 + 1728*T11 + 384 $$T_{13}^{7} + 6T_{13}^{6} - 58T_{13}^{5} - 248T_{13}^{4} + 1412T_{13}^{3} + 2072T_{13}^{2} - 14368T_{13} + 14336$$ T13^7 + 6*T13^6 - 58*T13^5 - 248*T13^4 + 1412*T13^3 + 2072*T13^2 - 14368*T13 + 14336 $$T_{17}^{7} + 4T_{17}^{6} - 88T_{17}^{5} - 364T_{17}^{4} + 1599T_{17}^{3} + 7840T_{17}^{2} + 6564T_{17} - 2688$$ T17^7 + 4*T17^6 - 88*T17^5 - 364*T17^4 + 1599*T17^3 + 7840*T17^2 + 6564*T17 - 2688

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{7}$$
$3$ $$T^{7}$$
$5$ $$(T - 1)^{7}$$
$7$ $$T^{7} + 6 T^{6} - 16 T^{5} - 134 T^{4} + \cdots + 256$$
$11$ $$T^{7} + 2 T^{6} - 50 T^{5} - 128 T^{4} + \cdots + 384$$
$13$ $$T^{7} + 6 T^{6} - 58 T^{5} + \cdots + 14336$$
$17$ $$T^{7} + 4 T^{6} - 88 T^{5} + \cdots - 2688$$
$19$ $$T^{7} + 8 T^{6} - 74 T^{5} + \cdots - 3072$$
$23$ $$(T - 1)^{7}$$
$29$ $$T^{7} - 2 T^{6} - 98 T^{5} + 102 T^{4} + \cdots + 336$$
$31$ $$T^{7} + 8 T^{6} - 162 T^{5} + \cdots - 294144$$
$37$ $$T^{7} + 16 T^{6} - 40 T^{5} + \cdots + 65536$$
$41$ $$T^{7} - 2 T^{6} - 190 T^{5} + \cdots - 390144$$
$43$ $$T^{7} + 10 T^{6} - 164 T^{5} + \cdots + 248832$$
$47$ $$T^{7} + 8 T^{6} - 162 T^{5} + \cdots + 4096$$
$53$ $$T^{7} + 10 T^{6} - 144 T^{5} + \cdots - 310472$$
$59$ $$T^{7} + 24 T^{6} + 26 T^{5} + \cdots + 19528$$
$61$ $$T^{7} - 8 T^{6} - 278 T^{5} + \cdots + 1757504$$
$67$ $$T^{7} + 20 T^{6} - 176 T^{5} + \cdots - 383968$$
$71$ $$T^{7} + 8 T^{6} - 290 T^{5} + \cdots - 1146496$$
$73$ $$T^{7} - 2 T^{6} - 326 T^{5} + \cdots + 2552736$$
$79$ $$T^{7} + 2 T^{6} - 272 T^{5} + \cdots + 73728$$
$83$ $$T^{7} + 22 T^{6} - 28 T^{5} + \cdots - 300512$$
$89$ $$T^{7} + 16 T^{6} - 276 T^{5} + \cdots - 3662848$$
$97$ $$T^{7} - 4 T^{6} - 240 T^{5} + \cdots - 284928$$