# Properties

 Label 260.2.m.c Level $260$ Weight $2$ Character orbit 260.m Analytic conductor $2.076$ Analytic rank $0$ Dimension $8$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$260 = 2^{2} \cdot 5 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 260.m (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.07611045255$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(i)$$ Coefficient field: 8.0.31678304256.2 Defining polynomial: $$x^{8} - 2x^{7} + 2x^{6} + 8x^{5} + 32x^{4} - 20x^{3} + 8x^{2} + 8x + 4$$ x^8 - 2*x^7 + 2*x^6 + 8*x^5 + 32*x^4 - 20*x^3 + 8*x^2 + 8*x + 4 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta_{3} q^{3} + ( - \beta_{6} + \beta_{3}) q^{5} + (\beta_{5} + \beta_{4}) q^{7} + (\beta_{5} - \beta_{3} - \beta_1) q^{9}+O(q^{10})$$ q - b3 * q^3 + (-b6 + b3) * q^5 + (b5 + b4) * q^7 + (b5 - b3 - b1) * q^9 $$q - \beta_{3} q^{3} + ( - \beta_{6} + \beta_{3}) q^{5} + (\beta_{5} + \beta_{4}) q^{7} + (\beta_{5} - \beta_{3} - \beta_1) q^{9} + ( - \beta_{7} + \beta_{6} - 2 \beta_{4} + \beta_1 + 2) q^{11} + ( - \beta_{7} - \beta_1 + 2) q^{13} + ( - \beta_{5} - 2 \beta_{4} + \beta_1 + 1) q^{15} + ( - \beta_{7} - \beta_{6} + 2 \beta_{3}) q^{17} + ( - \beta_{5} + \beta_{2} + \beta_1) q^{19} + (\beta_{7} - \beta_{6} - \beta_{4} - 4 \beta_1 + 1) q^{21} + (\beta_{7} - \beta_{6} + 3 \beta_{4} - \beta_1 - 3) q^{23} + (\beta_{5} + \beta_{4} + \beta_{3} - \beta_{2} - \beta_1 + 2) q^{25} + (\beta_{7} - \beta_{6} + \beta_{5} + 2 \beta_{4} - \beta_{2} - 2 \beta_1 - 2) q^{27} + ( - 2 \beta_{7} - \beta_{5} - 5 \beta_{4} + \beta_{3} + \beta_1) q^{29} + (\beta_{7} + \beta_{6} + \beta_{3}) q^{31} + ( - 2 \beta_{3} + \beta_{2} + 2 \beta_1 + 1) q^{33} + (\beta_{7} + \beta_{6} + \beta_{5} + 2 \beta_{4} - \beta_{3} + 3 \beta_1 - 1) q^{35} + ( - \beta_{5} + 3 \beta_{4} + \beta_{3} + \beta_1) q^{37} + (\beta_{4} - \beta_{3} - \beta_{2} - 2 \beta_1 - 4) q^{39} + (\beta_{7} + \beta_{6} + \beta_{5} - \beta_{4} - 2 \beta_{3} + \beta_{2} - 1) q^{41} + ( - 3 \beta_{7} + 3 \beta_{6} + \beta_{4} + 3 \beta_1 - 1) q^{43} + (2 \beta_{7} + \beta_{6} + \beta_{4} - 2 \beta_{3} + \beta_{2} + 3 \beta_1 + 2) q^{45} + (2 \beta_{7} - 3 \beta_{5} - \beta_{4}) q^{47} + (2 \beta_{6} - 2 \beta_{3} - 2 \beta_{2} + 2 \beta_1 - 3) q^{49} + ( - 2 \beta_{5} - 4 \beta_{4} + \beta_{3} + \beta_1) q^{51} + ( - 2 \beta_{7} - 2 \beta_{6} + \beta_{5} - 2 \beta_{4} + \beta_{2} - 2) q^{53} + (2 \beta_{7} - 2 \beta_{6} - \beta_{5} + 3 \beta_{4} + 2 \beta_{3} - \beta_1 - 4) q^{55} + (2 \beta_{6} - 4 \beta_{3} + \beta_{2} + 4 \beta_1 + 1) q^{57} + (\beta_{5} - \beta_{3} + \beta_{2}) q^{59} + (4 \beta_{6} + \beta_{3} - \beta_{2} - \beta_1 + 1) q^{61} + (2 \beta_{3} - \beta_{2} - 2 \beta_1 - 7) q^{63} + ( - 2 \beta_{6} - \beta_{5} + 4 \beta_{4} + \beta_{3} + \beta_{2} + \beta_1 + 3) q^{65} + (\beta_{3} + \beta_{2} - \beta_1 + 3) q^{67} + (3 \beta_{3} - \beta_{2} - 3 \beta_1 - 1) q^{69} + (4 \beta_{7} + 4 \beta_{6} - \beta_{4} - \beta_{3} - 1) q^{71} + ( - 4 \beta_{6} + \beta_{3} - 3 \beta_{2} - \beta_1 - 5) q^{73} + ( - 2 \beta_{6} - \beta_{5} - 3 \beta_{4} + 3 \beta_{3} - \beta_{2} - 4 \beta_1 - 1) q^{75} + ( - \beta_{7} - \beta_{6} + \beta_{5} - 2 \beta_{3} + \beta_{2}) q^{77} + ( - 2 \beta_{7} - 2 \beta_{5} + 2 \beta_{4} - \beta_{3} - \beta_1) q^{79} + ( - 2 \beta_{6} + 3 \beta_{3} + \beta_{2} - 3 \beta_1 - 2) q^{81} + (2 \beta_{7} - \beta_{5} - \beta_{4} - 2 \beta_{3} - 2 \beta_1) q^{83} + (\beta_{5} + 6 \beta_{4} + \beta_{3} - \beta_{2} - \beta_1 + 2) q^{85} + ( - \beta_{7} + \beta_{6} - \beta_{5} + \beta_{2} + 8 \beta_1) q^{87} + (2 \beta_{7} + 2 \beta_{6} + \beta_{5} + 2 \beta_{4} + 2 \beta_{3} + \beta_{2} + 2) q^{89} + ( - \beta_{7} - 3 \beta_{6} + 2 \beta_{5} + 3 \beta_{4} + 5 \beta_{3} - \beta_{2} - \beta_1) q^{91} + ( - \beta_{5} - 5 \beta_{4} + 2 \beta_{3} + 2 \beta_1) q^{93} + ( - 3 \beta_{7} + \beta_{6} - \beta_{5} - 2 \beta_{4} + 4 \beta_{3} - \beta_1 + 1) q^{95} + ( - 3 \beta_{3} - 2 \beta_{2} + 3 \beta_1) q^{97} + ( - 2 \beta_{7} - 2 \beta_{6} + 2 \beta_{5} - \beta_{4} - 5 \beta_{3} + 2 \beta_{2} + \cdots - 1) q^{99}+O(q^{100})$$ q - b3 * q^3 + (-b6 + b3) * q^5 + (b5 + b4) * q^7 + (b5 - b3 - b1) * q^9 + (-b7 + b6 - 2*b4 + b1 + 2) * q^11 + (-b7 - b1 + 2) * q^13 + (-b5 - 2*b4 + b1 + 1) * q^15 + (-b7 - b6 + 2*b3) * q^17 + (-b5 + b2 + b1) * q^19 + (b7 - b6 - b4 - 4*b1 + 1) * q^21 + (b7 - b6 + 3*b4 - b1 - 3) * q^23 + (b5 + b4 + b3 - b2 - b1 + 2) * q^25 + (b7 - b6 + b5 + 2*b4 - b2 - 2*b1 - 2) * q^27 + (-2*b7 - b5 - 5*b4 + b3 + b1) * q^29 + (b7 + b6 + b3) * q^31 + (-2*b3 + b2 + 2*b1 + 1) * q^33 + (b7 + b6 + b5 + 2*b4 - b3 + 3*b1 - 1) * q^35 + (-b5 + 3*b4 + b3 + b1) * q^37 + (b4 - b3 - b2 - 2*b1 - 4) * q^39 + (b7 + b6 + b5 - b4 - 2*b3 + b2 - 1) * q^41 + (-3*b7 + 3*b6 + b4 + 3*b1 - 1) * q^43 + (2*b7 + b6 + b4 - 2*b3 + b2 + 3*b1 + 2) * q^45 + (2*b7 - 3*b5 - b4) * q^47 + (2*b6 - 2*b3 - 2*b2 + 2*b1 - 3) * q^49 + (-2*b5 - 4*b4 + b3 + b1) * q^51 + (-2*b7 - 2*b6 + b5 - 2*b4 + b2 - 2) * q^53 + (2*b7 - 2*b6 - b5 + 3*b4 + 2*b3 - b1 - 4) * q^55 + (2*b6 - 4*b3 + b2 + 4*b1 + 1) * q^57 + (b5 - b3 + b2) * q^59 + (4*b6 + b3 - b2 - b1 + 1) * q^61 + (2*b3 - b2 - 2*b1 - 7) * q^63 + (-2*b6 - b5 + 4*b4 + b3 + b2 + b1 + 3) * q^65 + (b3 + b2 - b1 + 3) * q^67 + (3*b3 - b2 - 3*b1 - 1) * q^69 + (4*b7 + 4*b6 - b4 - b3 - 1) * q^71 + (-4*b6 + b3 - 3*b2 - b1 - 5) * q^73 + (-2*b6 - b5 - 3*b4 + 3*b3 - b2 - 4*b1 - 1) * q^75 + (-b7 - b6 + b5 - 2*b3 + b2) * q^77 + (-2*b7 - 2*b5 + 2*b4 - b3 - b1) * q^79 + (-2*b6 + 3*b3 + b2 - 3*b1 - 2) * q^81 + (2*b7 - b5 - b4 - 2*b3 - 2*b1) * q^83 + (b5 + 6*b4 + b3 - b2 - b1 + 2) * q^85 + (-b7 + b6 - b5 + b2 + 8*b1) * q^87 + (2*b7 + 2*b6 + b5 + 2*b4 + 2*b3 + b2 + 2) * q^89 + (-b7 - 3*b6 + 2*b5 + 3*b4 + 5*b3 - b2 - b1) * q^91 + (-b5 - 5*b4 + 2*b3 + 2*b1) * q^93 + (-3*b7 + b6 - b5 - 2*b4 + 4*b3 - b1 + 1) * q^95 + (-3*b3 - 2*b2 + 3*b1) * q^97 + (-2*b7 - 2*b6 + 2*b5 - b4 - 5*b3 + 2*b2 - 1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q + 2 q^{3} + 2 q^{5}+O(q^{10})$$ 8 * q + 2 * q^3 + 2 * q^5 $$8 q + 2 q^{3} + 2 q^{5} + 14 q^{11} + 14 q^{13} + 10 q^{15} + 2 q^{19} + 4 q^{21} - 22 q^{23} + 12 q^{25} - 16 q^{27} - 6 q^{31} + 16 q^{33} - 4 q^{35} - 34 q^{39} - 8 q^{41} - 14 q^{43} + 22 q^{45} - 24 q^{49} - 8 q^{53} - 30 q^{55} + 16 q^{57} + 2 q^{59} - 12 q^{61} - 64 q^{63} + 32 q^{65} + 20 q^{67} - 20 q^{69} - 22 q^{71} - 28 q^{73} - 14 q^{75} + 8 q^{77} - 20 q^{81} + 12 q^{85} + 12 q^{87} + 4 q^{89} - 6 q^{95} + 12 q^{97} + 10 q^{99}+O(q^{100})$$ 8 * q + 2 * q^3 + 2 * q^5 + 14 * q^11 + 14 * q^13 + 10 * q^15 + 2 * q^19 + 4 * q^21 - 22 * q^23 + 12 * q^25 - 16 * q^27 - 6 * q^31 + 16 * q^33 - 4 * q^35 - 34 * q^39 - 8 * q^41 - 14 * q^43 + 22 * q^45 - 24 * q^49 - 8 * q^53 - 30 * q^55 + 16 * q^57 + 2 * q^59 - 12 * q^61 - 64 * q^63 + 32 * q^65 + 20 * q^67 - 20 * q^69 - 22 * q^71 - 28 * q^73 - 14 * q^75 + 8 * q^77 - 20 * q^81 + 12 * q^85 + 12 * q^87 + 4 * q^89 - 6 * q^95 + 12 * q^97 + 10 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 2x^{7} + 2x^{6} + 8x^{5} + 32x^{4} - 20x^{3} + 8x^{2} + 8x + 4$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( 97\nu^{7} - 173\nu^{6} + 220\nu^{5} + 316\nu^{4} + 4010\nu^{3} - 1148\nu^{2} - 1300\nu - 6130 ) / 2462$$ (97*v^7 - 173*v^6 + 220*v^5 + 316*v^4 + 4010*v^3 - 1148*v^2 - 1300*v - 6130) / 2462 $$\beta_{3}$$ $$=$$ $$( 140\nu^{7} - 237\nu^{6} + 216\nu^{5} + 1116\nu^{4} + 5280\nu^{3} - 1530\nu^{2} + 738\nu + 696 ) / 2462$$ (140*v^7 - 237*v^6 + 216*v^5 + 1116*v^4 + 5280*v^3 - 1530*v^2 + 738*v + 696) / 2462 $$\beta_{4}$$ $$=$$ $$( 174\nu^{7} - 488\nu^{6} + 585\nu^{5} + 1176\nu^{4} + 4452\nu^{3} - 8760\nu^{2} + 2922\nu + 654 ) / 2462$$ (174*v^7 - 488*v^6 + 585*v^5 + 1176*v^4 + 4452*v^3 - 8760*v^2 + 2922*v + 654) / 2462 $$\beta_{5}$$ $$=$$ $$( -382\nu^{7} + 1227\nu^{6} - 1539\nu^{5} - 2412\nu^{4} - 8076\nu^{3} + 22288\nu^{2} - 5566\nu - 1266 ) / 2462$$ (-382*v^7 + 1227*v^6 - 1539*v^5 - 2412*v^4 - 8076*v^3 + 22288*v^2 - 5566*v - 1266) / 2462 $$\beta_{6}$$ $$=$$ $$( 452\nu^{7} - 730\nu^{6} + 416\nu^{5} + 4201\nu^{4} + 15640\nu^{3} - 4588\nu^{2} - 5144\nu + 4076 ) / 2462$$ (452*v^7 - 730*v^6 + 416*v^5 + 4201*v^4 + 15640*v^3 - 4588*v^2 - 5144*v + 4076) / 2462 $$\beta_{7}$$ $$=$$ $$( 605\nu^{7} - 1244\nu^{6} + 1461\nu^{5} + 4471\nu^{4} + 19300\nu^{3} - 11272\nu^{2} + 12070\nu + 2656 ) / 2462$$ (605*v^7 - 1244*v^6 + 1461*v^5 + 4471*v^4 + 19300*v^3 - 11272*v^2 + 12070*v + 2656) / 2462
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$-\beta_{5} - 3\beta_{4} + \beta_{3} + \beta_1$$ -b5 - 3*b4 + b3 + b1 $$\nu^{3}$$ $$=$$ $$-\beta_{7} - \beta_{6} - \beta_{5} - 2\beta_{4} + 8\beta_{3} - \beta_{2} - 2$$ -b7 - b6 - b5 - 2*b4 + 8*b3 - b2 - 2 $$\nu^{4}$$ $$=$$ $$-2\beta_{6} + 12\beta_{3} - 8\beta_{2} - 12\beta _1 - 20$$ -2*b6 + 12*b3 - 8*b2 - 12*b1 - 20 $$\nu^{5}$$ $$=$$ $$8\beta_{7} - 8\beta_{6} + 12\beta_{5} + 26\beta_{4} - 12\beta_{2} - 66\beta _1 - 26$$ 8*b7 - 8*b6 + 12*b5 + 26*b4 - 12*b2 - 66*b1 - 26 $$\nu^{6}$$ $$=$$ $$24\beta_{7} + 66\beta_{5} + 158\beta_{4} - 120\beta_{3} - 120\beta_1$$ 24*b7 + 66*b5 + 158*b4 - 120*b3 - 120*b1 $$\nu^{7}$$ $$=$$ $$66\beta_{7} + 66\beta_{6} + 120\beta_{5} + 270\beta_{4} - 572\beta_{3} + 120\beta_{2} + 270$$ 66*b7 + 66*b6 + 120*b5 + 270*b4 - 572*b3 + 120*b2 + 270

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/260\mathbb{Z}\right)^\times$$.

 $$n$$ $$41$$ $$131$$ $$157$$ $$\chi(n)$$ $$\beta_{4}$$ $$1$$ $$-\beta_{4}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
57.1
 −1.42497 − 1.42497i −0.285451 − 0.285451i 0.575868 + 0.575868i 2.13456 + 2.13456i −1.42497 + 1.42497i −0.285451 + 0.285451i 0.575868 − 0.575868i 2.13456 − 2.13456i
0 −1.42497 + 1.42497i 0 1.72321 1.42497i 0 4.91106i 0 1.06111i 0
57.2 0 −0.285451 + 0.285451i 0 −2.21777 0.285451i 0 1.26613i 0 2.83704i 0
57.3 0 0.575868 0.575868i 0 2.16064 + 0.575868i 0 2.48849i 0 2.33675i 0
57.4 0 2.13456 2.13456i 0 −0.666078 + 2.13456i 0 2.84356i 0 6.11268i 0
73.1 0 −1.42497 1.42497i 0 1.72321 + 1.42497i 0 4.91106i 0 1.06111i 0
73.2 0 −0.285451 0.285451i 0 −2.21777 + 0.285451i 0 1.26613i 0 2.83704i 0
73.3 0 0.575868 + 0.575868i 0 2.16064 0.575868i 0 2.48849i 0 2.33675i 0
73.4 0 2.13456 + 2.13456i 0 −0.666078 2.13456i 0 2.84356i 0 6.11268i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 73.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
65.k even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 260.2.m.c 8
3.b odd 2 1 2340.2.u.g 8
4.b odd 2 1 1040.2.bg.m 8
5.b even 2 1 1300.2.m.c 8
5.c odd 4 1 260.2.r.c yes 8
5.c odd 4 1 1300.2.r.c 8
13.d odd 4 1 260.2.r.c yes 8
15.e even 4 1 2340.2.bp.g 8
20.e even 4 1 1040.2.cd.m 8
39.f even 4 1 2340.2.bp.g 8
52.f even 4 1 1040.2.cd.m 8
65.f even 4 1 1300.2.m.c 8
65.g odd 4 1 1300.2.r.c 8
65.k even 4 1 inner 260.2.m.c 8
195.j odd 4 1 2340.2.u.g 8
260.s odd 4 1 1040.2.bg.m 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
260.2.m.c 8 1.a even 1 1 trivial
260.2.m.c 8 65.k even 4 1 inner
260.2.r.c yes 8 5.c odd 4 1
260.2.r.c yes 8 13.d odd 4 1
1040.2.bg.m 8 4.b odd 2 1
1040.2.bg.m 8 260.s odd 4 1
1040.2.cd.m 8 20.e even 4 1
1040.2.cd.m 8 52.f even 4 1
1300.2.m.c 8 5.b even 2 1
1300.2.m.c 8 65.f even 4 1
1300.2.r.c 8 5.c odd 4 1
1300.2.r.c 8 65.g odd 4 1
2340.2.u.g 8 3.b odd 2 1
2340.2.u.g 8 195.j odd 4 1
2340.2.bp.g 8 15.e even 4 1
2340.2.bp.g 8 39.f even 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{8} - 2T_{3}^{7} + 2T_{3}^{6} + 8T_{3}^{5} + 32T_{3}^{4} - 20T_{3}^{3} + 8T_{3}^{2} + 8T_{3} + 4$$ acting on $$S_{2}^{\mathrm{new}}(260, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$T^{8} - 2 T^{7} + 2 T^{6} + 8 T^{5} + \cdots + 4$$
$5$ $$T^{8} - 2 T^{7} - 4 T^{6} + 10 T^{5} + \cdots + 625$$
$7$ $$T^{8} + 40 T^{6} + 456 T^{4} + \cdots + 1936$$
$11$ $$T^{8} - 14 T^{7} + 98 T^{6} - 332 T^{5} + \cdots + 4$$
$13$ $$T^{8} - 14 T^{7} + 112 T^{6} + \cdots + 28561$$
$17$ $$T^{8} + 8 T^{5} + 360 T^{4} + \cdots + 11664$$
$19$ $$T^{8} - 2 T^{7} + 2 T^{6} + \cdots + 86436$$
$23$ $$T^{8} + 22 T^{7} + 242 T^{6} + \cdots + 6724$$
$29$ $$T^{8} + 156 T^{6} + 8136 T^{4} + \cdots + 810000$$
$31$ $$T^{8} + 6 T^{7} + 18 T^{6} + \cdots + 22500$$
$37$ $$T^{8} + 76 T^{6} + 1848 T^{4} + \cdots + 35344$$
$41$ $$T^{8} + 8 T^{7} + 32 T^{6} - 152 T^{5} + \cdots + 784$$
$43$ $$T^{8} + 14 T^{7} + 98 T^{6} + \cdots + 1085764$$
$47$ $$T^{8} + 400 T^{6} + \cdots + 29637136$$
$53$ $$T^{8} + 8 T^{7} + 32 T^{6} - 928 T^{5} + \cdots + 16$$
$59$ $$T^{8} - 2 T^{7} + 2 T^{6} + \cdots + 86436$$
$61$ $$(T^{4} + 6 T^{3} - 168 T^{2} - 1376 T - 1452)^{2}$$
$67$ $$(T^{4} - 10 T^{3} + 24 T - 12)^{2}$$
$71$ $$T^{8} + 22 T^{7} + 242 T^{6} + \cdots + 46321636$$
$73$ $$(T^{4} + 14 T^{3} - 168 T^{2} + \cdots - 13212)^{2}$$
$79$ $$T^{8} + 340 T^{6} + 25584 T^{4} + \cdots + 440896$$
$83$ $$T^{8} + 160 T^{6} + 4536 T^{4} + \cdots + 11664$$
$89$ $$T^{8} - 4 T^{7} + 8 T^{6} + \cdots + 22391824$$
$97$ $$(T^{4} - 6 T^{3} - 216 T^{2} + 2192 T - 4392)^{2}$$