# Properties

 Label 1680.2.t.i Level $1680$ Weight $2$ Character orbit 1680.t Analytic conductor $13.415$ Analytic rank $0$ Dimension $6$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1680 = 2^{4} \cdot 3 \cdot 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1680.t (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$13.4148675396$$ 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: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{3}$$ Twist minimal: no (minimal twist has level 840) 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.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{3} + \beta_{5} q^{5} - \beta_1 q^{7} - q^{9}+O(q^{10})$$ q + b1 * q^3 + b5 * q^5 - b1 * q^7 - q^9 $$q + \beta_1 q^{3} + \beta_{5} q^{5} - \beta_1 q^{7} - q^{9} + (\beta_{5} + \beta_{4} + \beta_{3} - \beta_{2}) q^{11} + ( - \beta_{4} + \beta_{3}) q^{13} + \beta_{3} q^{15} + (\beta_{5} + \beta_{2}) q^{17} + ( - \beta_{5} + \beta_{2}) q^{19} + q^{21} + (\beta_{4} - \beta_{3} + 2 \beta_1) q^{23} + ( - \beta_{5} - 2 \beta_{4} + \beta_{2} + 2 \beta_1 - 1) q^{25} - \beta_1 q^{27} + ( - 2 \beta_{5} - 2 \beta_{4} - 2 \beta_{3} + 2 \beta_{2} - 2) q^{29} + ( - \beta_{5} + \beta_{2}) q^{31} + ( - \beta_{5} - \beta_{4} + \beta_{3} - \beta_{2}) q^{33} - \beta_{3} q^{35} + ( - 2 \beta_{4} + 2 \beta_{3}) q^{37} + ( - \beta_{5} + \beta_{2}) q^{39} + ( - \beta_{5} + \beta_{2} + 4) q^{41} + (2 \beta_{5} + 2 \beta_{2} + 4 \beta_1) q^{43} - \beta_{5} q^{45} + (2 \beta_{5} + 2 \beta_{4} - 2 \beta_{3} + 2 \beta_{2} - 4 \beta_1) q^{47} - q^{49} + (\beta_{4} + \beta_{3}) q^{51} + (2 \beta_{5} - \beta_{4} + \beta_{3} + 2 \beta_{2} - 4 \beta_1) q^{53} + ( - \beta_{5} - \beta_{4} - \beta_{3} + 3 \beta_{2} - 4 \beta_1 + 2) q^{55} + (\beta_{4} - \beta_{3}) q^{57} + (2 \beta_{5} + 2 \beta_{4} + 2 \beta_{3} - 2 \beta_{2} + 4) q^{59} + (2 \beta_{5} - 2 \beta_{4} - 2 \beta_{3} - 2 \beta_{2} + 2) q^{61} + \beta_1 q^{63} + (\beta_{4} - \beta_{3} + 2 \beta_{2} + 4 \beta_1 - 2) q^{65} + (\beta_{5} - \beta_{2} - 2) q^{69} + ( - 3 \beta_{5} - \beta_{4} - \beta_{3} + 3 \beta_{2}) q^{71} + (2 \beta_{5} + 3 \beta_{4} - 3 \beta_{3} + 2 \beta_{2} - 4 \beta_1) q^{73} + (\beta_{4} - \beta_{3} + 2 \beta_{2} - \beta_1 - 2) q^{75} + (\beta_{5} + \beta_{4} - \beta_{3} + \beta_{2}) q^{77} + ( - 2 \beta_{5} + 2 \beta_{2} - 8) q^{79} + q^{81} + ( - 2 \beta_{5} - 2 \beta_{4} + 2 \beta_{3} - 2 \beta_{2} + 4 \beta_1) q^{83} + ( - \beta_{5} - 2 \beta_{4} + \beta_{2} + 2 \beta_1 - 6) q^{85} + (2 \beta_{5} + 2 \beta_{4} - 2 \beta_{3} + 2 \beta_{2} - 2 \beta_1) q^{87} + (\beta_{5} + 2 \beta_{4} + 2 \beta_{3} - \beta_{2}) q^{89} + (\beta_{5} - \beta_{2}) q^{91} + (\beta_{4} - \beta_{3}) q^{93} + (\beta_{5} + 2 \beta_{4} - \beta_{2} - 2 \beta_1 - 4) q^{95} + ( - 2 \beta_{5} - \beta_{4} + \beta_{3} - 2 \beta_{2} - 4 \beta_1) q^{97} + ( - \beta_{5} - \beta_{4} - \beta_{3} + \beta_{2}) q^{99}+O(q^{100})$$ q + b1 * q^3 + b5 * q^5 - b1 * q^7 - q^9 + (b5 + b4 + b3 - b2) * q^11 + (-b4 + b3) * q^13 + b3 * q^15 + (b5 + b2) * q^17 + (-b5 + b2) * q^19 + q^21 + (b4 - b3 + 2*b1) * q^23 + (-b5 - 2*b4 + b2 + 2*b1 - 1) * q^25 - b1 * q^27 + (-2*b5 - 2*b4 - 2*b3 + 2*b2 - 2) * q^29 + (-b5 + b2) * q^31 + (-b5 - b4 + b3 - b2) * q^33 - b3 * q^35 + (-2*b4 + 2*b3) * q^37 + (-b5 + b2) * q^39 + (-b5 + b2 + 4) * q^41 + (2*b5 + 2*b2 + 4*b1) * q^43 - b5 * q^45 + (2*b5 + 2*b4 - 2*b3 + 2*b2 - 4*b1) * q^47 - q^49 + (b4 + b3) * q^51 + (2*b5 - b4 + b3 + 2*b2 - 4*b1) * q^53 + (-b5 - b4 - b3 + 3*b2 - 4*b1 + 2) * q^55 + (b4 - b3) * q^57 + (2*b5 + 2*b4 + 2*b3 - 2*b2 + 4) * q^59 + (2*b5 - 2*b4 - 2*b3 - 2*b2 + 2) * q^61 + b1 * q^63 + (b4 - b3 + 2*b2 + 4*b1 - 2) * q^65 + (b5 - b2 - 2) * q^69 + (-3*b5 - b4 - b3 + 3*b2) * q^71 + (2*b5 + 3*b4 - 3*b3 + 2*b2 - 4*b1) * q^73 + (b4 - b3 + 2*b2 - b1 - 2) * q^75 + (b5 + b4 - b3 + b2) * q^77 + (-2*b5 + 2*b2 - 8) * q^79 + q^81 + (-2*b5 - 2*b4 + 2*b3 - 2*b2 + 4*b1) * q^83 + (-b5 - 2*b4 + b2 + 2*b1 - 6) * q^85 + (2*b5 + 2*b4 - 2*b3 + 2*b2 - 2*b1) * q^87 + (b5 + 2*b4 + 2*b3 - b2) * q^89 + (b5 - b2) * q^91 + (b4 - b3) * q^93 + (b5 + 2*b4 - b2 - 2*b1 - 4) * q^95 + (-2*b5 - b4 + b3 - 2*b2 - 4*b1) * q^97 + (-b5 - b4 - b3 + b2) * 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 $$6 q - 2 q^{5} - 6 q^{9} - 4 q^{11} + 4 q^{19} + 6 q^{21} - 2 q^{25} - 4 q^{29} + 4 q^{31} + 4 q^{39} + 28 q^{41} + 2 q^{45} - 6 q^{49} + 20 q^{55} + 16 q^{59} + 4 q^{61} - 8 q^{65} - 16 q^{69} + 12 q^{71} - 8 q^{75} - 40 q^{79} + 6 q^{81} - 32 q^{85} - 4 q^{89} - 4 q^{91} - 28 q^{95} + 4 q^{99}+O(q^{100})$$ 6 * q - 2 * q^5 - 6 * q^9 - 4 * q^11 + 4 * q^19 + 6 * q^21 - 2 * q^25 - 4 * q^29 + 4 * q^31 + 4 * q^39 + 28 * q^41 + 2 * q^45 - 6 * q^49 + 20 * q^55 + 16 * q^59 + 4 * q^61 - 8 * q^65 - 16 * q^69 + 12 * q^71 - 8 * q^75 - 40 * q^79 + 6 * q^81 - 32 * q^85 - 4 * q^89 - 4 * q^91 - 28 * q^95 + 4 * q^99

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

 $$\beta_{1}$$ $$=$$ $$( -7\nu^{5} + 10\nu^{4} - 5\nu^{3} - 30\nu^{2} - 32\nu + 13 ) / 23$$ (-7*v^5 + 10*v^4 - 5*v^3 - 30*v^2 - 32*v + 13) / 23 $$\beta_{2}$$ $$=$$ $$( -9\nu^{5} + 3\nu^{4} + 10\nu^{3} - 32\nu^{2} - 74\nu - 3 ) / 23$$ (-9*v^5 + 3*v^4 + 10*v^3 - 32*v^2 - 74*v - 3) / 23 $$\beta_{3}$$ $$=$$ $$( -10\nu^{5} + 11\nu^{4} - 17\nu^{3} - 10\nu^{2} - 72\nu - 11 ) / 23$$ (-10*v^5 + 11*v^4 - 17*v^3 - 10*v^2 - 72*v - 11) / 23 $$\beta_{4}$$ $$=$$ $$( 12\nu^{5} - 27\nu^{4} + 25\nu^{3} + 12\nu^{2} + 68\nu - 65 ) / 23$$ (12*v^5 - 27*v^4 + 25*v^3 + 12*v^2 + 68*v - 65) / 23 $$\beta_{5}$$ $$=$$ $$( -19\nu^{5} + 37\nu^{4} - 30\nu^{3} - 42\nu^{2} - 54\nu + 55 ) / 23$$ (-19*v^5 + 37*v^4 - 30*v^3 - 42*v^2 - 54*v + 55) / 23
 $$\nu$$ $$=$$ $$( \beta_{5} + \beta_{4} - \beta _1 + 1 ) / 2$$ (b5 + b4 - b1 + 1) / 2 $$\nu^{2}$$ $$=$$ $$( \beta_{5} + \beta_{2} - 4\beta_1 ) / 2$$ (b5 + b2 - 4*b1) / 2 $$\nu^{3}$$ $$=$$ $$( \beta_{5} - \beta_{4} - 3\beta_{3} + 3\beta_{2} - 4\beta _1 - 4 ) / 2$$ (b5 - b4 - 3*b3 + 3*b2 - 4*b1 - 4) / 2 $$\nu^{4}$$ $$=$$ $$( -\beta_{5} - 5\beta_{4} - 5\beta_{3} + \beta_{2} - 14 ) / 2$$ (-b5 - 5*b4 - 5*b3 + b2 - 14) / 2 $$\nu^{5}$$ $$=$$ $$( -11\beta_{5} - 11\beta_{4} - 5\beta_{3} - 5\beta_{2} + 18\beta _1 - 18 ) / 2$$ (-11*b5 - 11*b4 - 5*b3 - 5*b2 + 18*b1 - 18) / 2

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1680\mathbb{Z}\right)^\times$$.

 $$n$$ $$241$$ $$337$$ $$421$$ $$1121$$ $$1471$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1009.1
 −0.854638 − 0.854638i 1.45161 + 1.45161i 0.403032 + 0.403032i −0.854638 + 0.854638i 1.45161 − 1.45161i 0.403032 − 0.403032i
0 1.00000i 0 −2.17009 0.539189i 0 1.00000i 0 −1.00000 0
1009.2 0 1.00000i 0 −0.311108 + 2.21432i 0 1.00000i 0 −1.00000 0
1009.3 0 1.00000i 0 1.48119 1.67513i 0 1.00000i 0 −1.00000 0
1009.4 0 1.00000i 0 −2.17009 + 0.539189i 0 1.00000i 0 −1.00000 0
1009.5 0 1.00000i 0 −0.311108 2.21432i 0 1.00000i 0 −1.00000 0
1009.6 0 1.00000i 0 1.48119 + 1.67513i 0 1.00000i 0 −1.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1009.6 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 1680.2.t.i 6
3.b odd 2 1 5040.2.t.ba 6
4.b odd 2 1 840.2.t.e 6
5.b even 2 1 inner 1680.2.t.i 6
5.c odd 4 1 8400.2.a.dh 3
5.c odd 4 1 8400.2.a.dk 3
12.b even 2 1 2520.2.t.j 6
15.d odd 2 1 5040.2.t.ba 6
20.d odd 2 1 840.2.t.e 6
20.e even 4 1 4200.2.a.bo 3
20.e even 4 1 4200.2.a.bq 3
60.h even 2 1 2520.2.t.j 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
840.2.t.e 6 4.b odd 2 1
840.2.t.e 6 20.d odd 2 1
1680.2.t.i 6 1.a even 1 1 trivial
1680.2.t.i 6 5.b even 2 1 inner
2520.2.t.j 6 12.b even 2 1
2520.2.t.j 6 60.h even 2 1
4200.2.a.bo 3 20.e even 4 1
4200.2.a.bq 3 20.e even 4 1
5040.2.t.ba 6 3.b odd 2 1
5040.2.t.ba 6 15.d odd 2 1
8400.2.a.dh 3 5.c odd 4 1
8400.2.a.dk 3 5.c odd 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1680, [\chi])$$:

 $$T_{11}^{3} + 2T_{11}^{2} - 20T_{11} - 8$$ T11^3 + 2*T11^2 - 20*T11 - 8 $$T_{13}^{6} + 28T_{13}^{4} + 176T_{13}^{2} + 64$$ T13^6 + 28*T13^4 + 176*T13^2 + 64 $$T_{19}^{3} - 2T_{19}^{2} - 12T_{19} + 8$$ T19^3 - 2*T19^2 - 12*T19 + 8

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6}$$
$3$ $$(T^{2} + 1)^{3}$$
$5$ $$T^{6} + 2 T^{5} + 3 T^{4} + 12 T^{3} + \cdots + 125$$
$7$ $$(T^{2} + 1)^{3}$$
$11$ $$(T^{3} + 2 T^{2} - 20 T - 8)^{2}$$
$13$ $$T^{6} + 28 T^{4} + 176 T^{2} + \cdots + 64$$
$17$ $$T^{6} + 32 T^{4} + 256 T^{2} + \cdots + 256$$
$19$ $$(T^{3} - 2 T^{2} - 12 T + 8)^{2}$$
$23$ $$T^{6} + 48 T^{4} + 320 T^{2} + \cdots + 256$$
$29$ $$(T^{3} + 2 T^{2} - 84 T - 104)^{2}$$
$31$ $$(T^{3} - 2 T^{2} - 12 T + 8)^{2}$$
$37$ $$T^{6} + 112 T^{4} + 2816 T^{2} + \cdots + 4096$$
$41$ $$(T^{3} - 14 T^{2} + 52 T - 40)^{2}$$
$43$ $$T^{6} + 176 T^{4} + 7936 T^{2} + \cdots + 102400$$
$47$ $$T^{6} + 192 T^{4} + 8192 T^{2} + \cdots + 65536$$
$53$ $$T^{6} + 252 T^{4} + 14000 T^{2} + \cdots + 222784$$
$59$ $$(T^{3} - 8 T^{2} - 64 T + 256)^{2}$$
$61$ $$(T^{3} - 2 T^{2} - 148 T - 536)^{2}$$
$67$ $$T^{6}$$
$71$ $$(T^{3} - 6 T^{2} - 100 T - 200)^{2}$$
$73$ $$T^{6} + 284 T^{4} + 24496 T^{2} + \cdots + 577600$$
$79$ $$(T^{3} + 20 T^{2} + 80 T - 64)^{2}$$
$83$ $$T^{6} + 192 T^{4} + 8192 T^{2} + \cdots + 65536$$
$89$ $$(T^{3} + 2 T^{2} - 60 T - 200)^{2}$$
$97$ $$T^{6} + 188 T^{4} + 9648 T^{2} + \cdots + 118336$$