# Properties

 Label 2640.2.d.i Level $2640$ Weight $2$ Character orbit 2640.d Analytic conductor $21.081$ Analytic rank $0$ Dimension $6$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2640 = 2^{4} \cdot 3 \cdot 5 \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2640.d (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$21.0805061336$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: 6.0.350464.1 Defining polynomial: $$x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 4x^{2} - 4x + 2$$ x^6 - 2*x^5 + 2*x^4 + 2*x^3 + 4*x^2 - 4*x + 2 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 165) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + \beta_{4} q^{3} + ( - \beta_{5} - \beta_1) q^{5} + (3 \beta_{4} + \beta_1) q^{7} - q^{9}+O(q^{10})$$ q + b4 * q^3 + (-b5 - b1) * q^5 + (3*b4 + b1) * q^7 - q^9 $$q + \beta_{4} q^{3} + ( - \beta_{5} - \beta_1) q^{5} + (3 \beta_{4} + \beta_1) q^{7} - q^{9} + q^{11} + (2 \beta_{5} + \beta_{4} - 2 \beta_{3} + \beta_1) q^{13} + ( - \beta_{3} + \beta_{2}) q^{15} + (\beta_{5} - \beta_{4} - \beta_{3} + 3 \beta_1) q^{17} + ( - \beta_{5} - \beta_{3} - 2 \beta_{2} - 2) q^{19} + ( - \beta_{2} - 3) q^{21} - 4 \beta_{4} q^{23} + (\beta_{5} - 2 \beta_{4} - \beta_{3} - 2 \beta_{2} - 1) q^{25} - \beta_{4} q^{27} + ( - 2 \beta_{2} + 2) q^{29} + (\beta_{5} + \beta_{3} - 2) q^{31} + \beta_{4} q^{33} + ( - \beta_{5} - \beta_{4} - 3 \beta_{3} + 4 \beta_{2} + \beta_1 + 2) q^{35} + ( - 2 \beta_{4} - 2 \beta_1) q^{37} + (2 \beta_{5} + 2 \beta_{3} - \beta_{2} - 1) q^{39} + (2 \beta_{2} - 2) q^{41} + ( - 3 \beta_{4} - \beta_1) q^{43} + (\beta_{5} + \beta_1) q^{45} + (2 \beta_{5} + 2 \beta_{4} - 2 \beta_{3} + 2 \beta_1) q^{47} + (\beta_{5} + \beta_{3} - 6 \beta_{2} - 5) q^{49} + (\beta_{5} + \beta_{3} - 3 \beta_{2} + 1) q^{51} + (\beta_{5} - 4 \beta_{4} - \beta_{3} - 2 \beta_1) q^{53} + ( - \beta_{5} - \beta_1) q^{55} + (\beta_{5} - 2 \beta_{4} - \beta_{3} - 2 \beta_1) q^{57} + ( - 3 \beta_{5} - 3 \beta_{3} + 2 \beta_{2} - 4) q^{59} + (2 \beta_{5} + 2 \beta_{3} + 2) q^{61} + ( - 3 \beta_{4} - \beta_1) q^{63} + (\beta_{5} + 7 \beta_{4} + \beta_{3} + 2 \beta_{2} - 3 \beta_1 + 6) q^{65} + ( - 2 \beta_{5} + 4 \beta_{4} + 2 \beta_{3} - 4 \beta_1) q^{67} + 4 q^{69} + (\beta_{5} + \beta_{3} - 2 \beta_{2} + 4) q^{71} + (7 \beta_{4} + 3 \beta_1) q^{73} + (\beta_{5} - \beta_{4} + \beta_{3} - 2 \beta_1 + 2) q^{75} + (3 \beta_{4} + \beta_1) q^{77} + (\beta_{5} + \beta_{3} - 2 \beta_{2} - 2) q^{79} + q^{81} + (\beta_{4} + \beta_1) q^{83} + ( - 2 \beta_{5} + \beta_{4} + 2 \beta_{3} + 2 \beta_{2} + \beta_1 + 8) q^{85} + (2 \beta_{4} - 2 \beta_1) q^{87} + (\beta_{5} + \beta_{3}) q^{89} + (5 \beta_{5} + 5 \beta_{3} - 8 \beta_{2} - 2) q^{91} + ( - \beta_{5} - 2 \beta_{4} + \beta_{3}) q^{93} + (3 \beta_{5} + 2 \beta_{4} - 3 \beta_{3} - 4 \beta_{2} + 4 \beta_1 + 6) q^{95} + ( - 2 \beta_{5} + 4 \beta_{4} + 2 \beta_{3}) q^{97} - q^{99}+O(q^{100})$$ q + b4 * q^3 + (-b5 - b1) * q^5 + (3*b4 + b1) * q^7 - q^9 + q^11 + (2*b5 + b4 - 2*b3 + b1) * q^13 + (-b3 + b2) * q^15 + (b5 - b4 - b3 + 3*b1) * q^17 + (-b5 - b3 - 2*b2 - 2) * q^19 + (-b2 - 3) * q^21 - 4*b4 * q^23 + (b5 - 2*b4 - b3 - 2*b2 - 1) * q^25 - b4 * q^27 + (-2*b2 + 2) * q^29 + (b5 + b3 - 2) * q^31 + b4 * q^33 + (-b5 - b4 - 3*b3 + 4*b2 + b1 + 2) * q^35 + (-2*b4 - 2*b1) * q^37 + (2*b5 + 2*b3 - b2 - 1) * q^39 + (2*b2 - 2) * q^41 + (-3*b4 - b1) * q^43 + (b5 + b1) * q^45 + (2*b5 + 2*b4 - 2*b3 + 2*b1) * q^47 + (b5 + b3 - 6*b2 - 5) * q^49 + (b5 + b3 - 3*b2 + 1) * q^51 + (b5 - 4*b4 - b3 - 2*b1) * q^53 + (-b5 - b1) * q^55 + (b5 - 2*b4 - b3 - 2*b1) * q^57 + (-3*b5 - 3*b3 + 2*b2 - 4) * q^59 + (2*b5 + 2*b3 + 2) * q^61 + (-3*b4 - b1) * q^63 + (b5 + 7*b4 + b3 + 2*b2 - 3*b1 + 6) * q^65 + (-2*b5 + 4*b4 + 2*b3 - 4*b1) * q^67 + 4 * q^69 + (b5 + b3 - 2*b2 + 4) * q^71 + (7*b4 + 3*b1) * q^73 + (b5 - b4 + b3 - 2*b1 + 2) * q^75 + (3*b4 + b1) * q^77 + (b5 + b3 - 2*b2 - 2) * q^79 + q^81 + (b4 + b1) * q^83 + (-2*b5 + b4 + 2*b3 + 2*b2 + b1 + 8) * q^85 + (2*b4 - 2*b1) * q^87 + (b5 + b3) * q^89 + (5*b5 + 5*b3 - 8*b2 - 2) * q^91 + (-b5 - 2*b4 + b3) * q^93 + (3*b5 + 2*b4 - 3*b3 - 4*b2 + 4*b1 + 6) * q^95 + (-2*b5 + 4*b4 + 2*b3) * q^97 - q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q + 2 q^{5} - 6 q^{9}+O(q^{10})$$ 6 * q + 2 * q^5 - 6 * q^9 $$6 q + 2 q^{5} - 6 q^{9} + 6 q^{11} - 4 q^{19} - 16 q^{21} - 2 q^{25} + 16 q^{29} - 16 q^{31} + 12 q^{35} - 12 q^{39} - 16 q^{41} - 2 q^{45} - 22 q^{49} + 8 q^{51} + 2 q^{55} - 16 q^{59} + 4 q^{61} + 28 q^{65} + 24 q^{69} + 24 q^{71} + 8 q^{75} - 12 q^{79} + 6 q^{81} + 44 q^{85} - 4 q^{89} - 16 q^{91} + 44 q^{95} - 6 q^{99}+O(q^{100})$$ 6 * q + 2 * q^5 - 6 * q^9 + 6 * q^11 - 4 * q^19 - 16 * q^21 - 2 * q^25 + 16 * q^29 - 16 * q^31 + 12 * q^35 - 12 * q^39 - 16 * q^41 - 2 * q^45 - 22 * q^49 + 8 * q^51 + 2 * q^55 - 16 * q^59 + 4 * q^61 + 28 * q^65 + 24 * q^69 + 24 * q^71 + 8 * q^75 - 12 * q^79 + 6 * q^81 + 44 * q^85 - 4 * q^89 - 16 * q^91 + 44 * q^95 - 6 * q^99

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

 $$\beta_{1}$$ $$=$$ $$( -3\nu^{5} + \nu^{4} + 11\nu^{3} - 26\nu^{2} + 6\nu - 1 ) / 23$$ (-3*v^5 + v^4 + 11*v^3 - 26*v^2 + 6*v - 1) / 23 $$\beta_{2}$$ $$=$$ $$( -4\nu^{5} + 9\nu^{4} - 16\nu^{3} - 4\nu^{2} + 8\nu - 9 ) / 23$$ (-4*v^5 + 9*v^4 - 16*v^3 - 4*v^2 + 8*v - 9) / 23 $$\beta_{3}$$ $$=$$ $$( 6\nu^{5} - 2\nu^{4} + \nu^{3} + 6\nu^{2} + 80\nu + 2 ) / 23$$ (6*v^5 - 2*v^4 + v^3 + 6*v^2 + 80*v + 2) / 23 $$\beta_{4}$$ $$=$$ $$( 7\nu^{5} - 10\nu^{4} + 5\nu^{3} + 30\nu^{2} + 32\nu - 13 ) / 23$$ (7*v^5 - 10*v^4 + 5*v^3 + 30*v^2 + 32*v - 13) / 23 $$\beta_{5}$$ $$=$$ $$( -16\nu^{5} + 36\nu^{4} - 41\nu^{3} - 16\nu^{2} - 60\nu + 56 ) / 23$$ (-16*v^5 + 36*v^4 - 41*v^3 - 16*v^2 - 60*v + 56) / 23
 $$\nu$$ $$=$$ $$( \beta_{4} + \beta_{2} + \beta _1 + 1 ) / 2$$ (b4 + b2 + b1 + 1) / 2 $$\nu^{2}$$ $$=$$ $$( \beta_{5} + 4\beta_{4} - \beta_{3} + 2\beta_1 ) / 2$$ (b5 + 4*b4 - b3 + 2*b1) / 2 $$\nu^{3}$$ $$=$$ $$\beta_{5} + 2\beta_{4} - 2\beta_{2} + 2\beta _1 - 2$$ b5 + 2*b4 - 2*b2 + 2*b1 - 2 $$\nu^{4}$$ $$=$$ $$2\beta_{5} + 2\beta_{3} - 5\beta_{2} - 7$$ 2*b5 + 2*b3 - 5*b2 - 7 $$\nu^{5}$$ $$=$$ $$-9\beta_{4} + 5\beta_{3} - 8\beta_{2} - 8\beta _1 - 9$$ -9*b4 + 5*b3 - 8*b2 - 8*b1 - 9

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2640\mathbb{Z}\right)^\times$$.

 $$n$$ $$661$$ $$881$$ $$991$$ $$1057$$ $$1201$$ $$\chi(n)$$ $$1$$ $$1$$ $$1$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
529.1
 0.403032 − 0.403032i 1.45161 − 1.45161i −0.854638 + 0.854638i 0.403032 + 0.403032i 1.45161 + 1.45161i −0.854638 − 0.854638i
0 1.00000i 0 −1.48119 1.67513i 0 2.80606i 0 −1.00000 0
529.2 0 1.00000i 0 0.311108 + 2.21432i 0 4.90321i 0 −1.00000 0
529.3 0 1.00000i 0 2.17009 0.539189i 0 0.290725i 0 −1.00000 0
529.4 0 1.00000i 0 −1.48119 + 1.67513i 0 2.80606i 0 −1.00000 0
529.5 0 1.00000i 0 0.311108 2.21432i 0 4.90321i 0 −1.00000 0
529.6 0 1.00000i 0 2.17009 + 0.539189i 0 0.290725i 0 −1.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 529.6 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2640.2.d.i 6
4.b odd 2 1 165.2.c.a 6
5.b even 2 1 inner 2640.2.d.i 6
12.b even 2 1 495.2.c.d 6
20.d odd 2 1 165.2.c.a 6
20.e even 4 1 825.2.a.h 3
20.e even 4 1 825.2.a.n 3
44.c even 2 1 1815.2.c.d 6
60.h even 2 1 495.2.c.d 6
60.l odd 4 1 2475.2.a.y 3
60.l odd 4 1 2475.2.a.be 3
220.g even 2 1 1815.2.c.d 6
220.i odd 4 1 9075.2.a.cc 3
220.i odd 4 1 9075.2.a.ck 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
165.2.c.a 6 4.b odd 2 1
165.2.c.a 6 20.d odd 2 1
495.2.c.d 6 12.b even 2 1
495.2.c.d 6 60.h even 2 1
825.2.a.h 3 20.e even 4 1
825.2.a.n 3 20.e even 4 1
1815.2.c.d 6 44.c even 2 1
1815.2.c.d 6 220.g even 2 1
2475.2.a.y 3 60.l odd 4 1
2475.2.a.be 3 60.l odd 4 1
2640.2.d.i 6 1.a even 1 1 trivial
2640.2.d.i 6 5.b even 2 1 inner
9075.2.a.cc 3 220.i odd 4 1
9075.2.a.ck 3 220.i odd 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7}^{6} + 32T_{7}^{4} + 192T_{7}^{2} + 16$$ acting on $$S_{2}^{\mathrm{new}}(2640, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6}$$
$3$ $$(T^{2} + 1)^{3}$$
$5$ $$T^{6} - 2 T^{5} + 3 T^{4} - 12 T^{3} + \cdots + 125$$
$7$ $$T^{6} + 32 T^{4} + 192 T^{2} + \cdots + 16$$
$11$ $$(T - 1)^{6}$$
$13$ $$T^{6} + 92 T^{4} + 2560 T^{2} + \cdots + 21904$$
$17$ $$T^{6} + 72 T^{4} + 1712 T^{2} + \cdots + 13456$$
$19$ $$(T^{3} + 2 T^{2} - 52 T - 184)^{2}$$
$23$ $$(T^{2} + 16)^{3}$$
$29$ $$(T^{3} - 8 T^{2} + 32)^{2}$$
$31$ $$(T^{3} + 8 T^{2} + 8 T - 16)^{2}$$
$37$ $$T^{6} + 48 T^{4} + 512 T^{2} + \cdots + 1024$$
$41$ $$(T^{3} + 8 T^{2} - 32)^{2}$$
$43$ $$T^{6} + 32 T^{4} + 192 T^{2} + \cdots + 16$$
$47$ $$T^{6} + 96 T^{4} + 2816 T^{2} + \cdots + 25600$$
$53$ $$T^{6} + 128 T^{4} + 5376 T^{2} + \cdots + 73984$$
$59$ $$(T^{3} + 8 T^{2} - 64 T + 80)^{2}$$
$61$ $$(T^{3} - 2 T^{2} - 52 T + 40)^{2}$$
$67$ $$T^{6} + 176 T^{4} + 7936 T^{2} + \cdots + 102400$$
$71$ $$(T^{3} - 12 T^{2} + 32 T + 16)^{2}$$
$73$ $$T^{6} + 204 T^{4} + 6912 T^{2} + \cdots + 8464$$
$79$ $$(T^{3} + 6 T^{2} - 4 T - 8)^{2}$$
$83$ $$T^{6} + 12 T^{4} + 32 T^{2} + 16$$
$89$ $$(T^{3} + 2 T^{2} - 12 T - 8)^{2}$$
$97$ $$T^{6} + 128 T^{4} + 3072 T^{2} + \cdots + 16384$$