# Properties

 Label 770.2.i.j Level $770$ Weight $2$ Character orbit 770.i Analytic conductor $6.148$ Analytic rank $0$ Dimension $6$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$770 = 2 \cdot 5 \cdot 7 \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 770.i (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$6.14848095564$$ Analytic rank: $$0$$ Dimension: $$6$$ Relative dimension: $$3$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\Q(\zeta_{18})$$ Defining polynomial: $$x^{6} - x^{3} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{18}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 1 - \zeta_{18}^{3} ) q^{2} + ( -\zeta_{18} + \zeta_{18}^{2} + \zeta_{18}^{4} - \zeta_{18}^{5} ) q^{3} -\zeta_{18}^{3} q^{4} + ( -1 + \zeta_{18}^{3} ) q^{5} + ( \zeta_{18}^{4} - \zeta_{18}^{5} ) q^{6} + ( -3 \zeta_{18} + \zeta_{18}^{4} ) q^{7} - q^{8} + ( 1 + \zeta_{18} - \zeta_{18}^{3} - \zeta_{18}^{4} - \zeta_{18}^{5} ) q^{9} +O(q^{10})$$ $$q + ( 1 - \zeta_{18}^{3} ) q^{2} + ( -\zeta_{18} + \zeta_{18}^{2} + \zeta_{18}^{4} - \zeta_{18}^{5} ) q^{3} -\zeta_{18}^{3} q^{4} + ( -1 + \zeta_{18}^{3} ) q^{5} + ( \zeta_{18}^{4} - \zeta_{18}^{5} ) q^{6} + ( -3 \zeta_{18} + \zeta_{18}^{4} ) q^{7} - q^{8} + ( 1 + \zeta_{18} - \zeta_{18}^{3} - \zeta_{18}^{4} - \zeta_{18}^{5} ) q^{9} + \zeta_{18}^{3} q^{10} + \zeta_{18}^{3} q^{11} + ( \zeta_{18} - \zeta_{18}^{2} ) q^{12} + ( 2 + 2 \zeta_{18} + 2 \zeta_{18}^{2} + 2 \zeta_{18}^{4} - 4 \zeta_{18}^{5} ) q^{13} + ( -2 \zeta_{18} + 3 \zeta_{18}^{4} ) q^{14} + ( -\zeta_{18}^{4} + \zeta_{18}^{5} ) q^{15} + ( -1 + \zeta_{18}^{3} ) q^{16} + ( \zeta_{18}^{2} - 4 \zeta_{18}^{3} + \zeta_{18}^{4} ) q^{17} + ( -\zeta_{18}^{2} - \zeta_{18}^{3} - \zeta_{18}^{4} ) q^{18} + ( 2 - 2 \zeta_{18} + 5 \zeta_{18}^{2} - 2 \zeta_{18}^{3} - 3 \zeta_{18}^{4} - 3 \zeta_{18}^{5} ) q^{19} + q^{20} + ( -3 + 2 \zeta_{18}^{2} + \zeta_{18}^{3} - 3 \zeta_{18}^{5} ) q^{21} + q^{22} + ( -2 + 4 \zeta_{18} + 2 \zeta_{18}^{2} + 2 \zeta_{18}^{3} - 6 \zeta_{18}^{4} - 6 \zeta_{18}^{5} ) q^{23} + ( \zeta_{18} - \zeta_{18}^{2} - \zeta_{18}^{4} + \zeta_{18}^{5} ) q^{24} -\zeta_{18}^{3} q^{25} + ( 2 + 4 \zeta_{18} - 2 \zeta_{18}^{2} - 2 \zeta_{18}^{3} - 2 \zeta_{18}^{4} - 2 \zeta_{18}^{5} ) q^{26} + ( 1 + 3 \zeta_{18}^{4} - 3 \zeta_{18}^{5} ) q^{27} + ( \zeta_{18} + 2 \zeta_{18}^{4} ) q^{28} + ( -2 - 5 \zeta_{18} - 5 \zeta_{18}^{2} + 5 \zeta_{18}^{4} ) q^{29} + ( -\zeta_{18} + \zeta_{18}^{2} ) q^{30} + ( -3 \zeta_{18} - 2 \zeta_{18}^{2} - 2 \zeta_{18}^{4} - 3 \zeta_{18}^{5} ) q^{31} + \zeta_{18}^{3} q^{32} + ( -\zeta_{18} + \zeta_{18}^{2} ) q^{33} + ( -4 + \zeta_{18} + \zeta_{18}^{2} - \zeta_{18}^{5} ) q^{34} + ( 2 \zeta_{18} - 3 \zeta_{18}^{4} ) q^{35} + ( -1 - \zeta_{18} - \zeta_{18}^{2} + \zeta_{18}^{5} ) q^{36} + ( -6 - \zeta_{18} - 2 \zeta_{18}^{2} + 6 \zeta_{18}^{3} + 3 \zeta_{18}^{4} + 3 \zeta_{18}^{5} ) q^{37} + ( -5 \zeta_{18} + 2 \zeta_{18}^{2} - 2 \zeta_{18}^{3} + 2 \zeta_{18}^{4} - 5 \zeta_{18}^{5} ) q^{38} + ( -2 \zeta_{18}^{2} + 6 \zeta_{18}^{3} - 2 \zeta_{18}^{4} ) q^{39} + ( 1 - \zeta_{18}^{3} ) q^{40} + ( 8 - 2 \zeta_{18} - 2 \zeta_{18}^{2} + 2 \zeta_{18}^{5} ) q^{41} + ( -2 - \zeta_{18}^{2} + 3 \zeta_{18}^{3} - 2 \zeta_{18}^{5} ) q^{42} + ( 2 + 6 \zeta_{18} + 6 \zeta_{18}^{2} - 6 \zeta_{18}^{4} ) q^{43} + ( 1 - \zeta_{18}^{3} ) q^{44} + ( \zeta_{18}^{2} + \zeta_{18}^{3} + \zeta_{18}^{4} ) q^{45} + ( -2 \zeta_{18} - 4 \zeta_{18}^{2} + 2 \zeta_{18}^{3} - 4 \zeta_{18}^{4} - 2 \zeta_{18}^{5} ) q^{46} + ( 4 + 8 \zeta_{18} - 6 \zeta_{18}^{2} - 4 \zeta_{18}^{3} - 2 \zeta_{18}^{4} - 2 \zeta_{18}^{5} ) q^{47} + ( -\zeta_{18}^{4} + \zeta_{18}^{5} ) q^{48} + ( 8 \zeta_{18}^{2} - 5 \zeta_{18}^{5} ) q^{49} - q^{50} + ( -1 + 5 \zeta_{18} - 5 \zeta_{18}^{2} + \zeta_{18}^{3} ) q^{51} + ( 2 \zeta_{18} - 4 \zeta_{18}^{2} - 2 \zeta_{18}^{3} - 4 \zeta_{18}^{4} + 2 \zeta_{18}^{5} ) q^{52} + ( \zeta_{18} + \zeta_{18}^{2} + \zeta_{18}^{4} + \zeta_{18}^{5} ) q^{53} + ( 1 + 3 \zeta_{18} - 3 \zeta_{18}^{2} - \zeta_{18}^{3} ) q^{54} - q^{55} + ( 3 \zeta_{18} - \zeta_{18}^{4} ) q^{56} + ( -7 + 5 \zeta_{18} + 5 \zeta_{18}^{2} - \zeta_{18}^{4} - 4 \zeta_{18}^{5} ) q^{57} + ( -2 - 5 \zeta_{18}^{2} + 2 \zeta_{18}^{3} + 5 \zeta_{18}^{4} + 5 \zeta_{18}^{5} ) q^{58} + ( 4 \zeta_{18}^{2} - 4 \zeta_{18}^{3} + 4 \zeta_{18}^{4} ) q^{59} + ( -\zeta_{18} + \zeta_{18}^{2} + \zeta_{18}^{4} - \zeta_{18}^{5} ) q^{60} + ( 2 - 3 \zeta_{18} - 2 \zeta_{18}^{2} - 2 \zeta_{18}^{3} + 5 \zeta_{18}^{4} + 5 \zeta_{18}^{5} ) q^{61} + ( -5 \zeta_{18} - 5 \zeta_{18}^{2} + 3 \zeta_{18}^{4} + 2 \zeta_{18}^{5} ) q^{62} + ( -2 - 2 \zeta_{18} - 2 \zeta_{18}^{2} + 3 \zeta_{18}^{3} + 3 \zeta_{18}^{4} + 3 \zeta_{18}^{5} ) q^{63} + q^{64} + ( -2 - 4 \zeta_{18} + 2 \zeta_{18}^{2} + 2 \zeta_{18}^{3} + 2 \zeta_{18}^{4} + 2 \zeta_{18}^{5} ) q^{65} + ( -\zeta_{18} + \zeta_{18}^{2} + \zeta_{18}^{4} - \zeta_{18}^{5} ) q^{66} + ( 2 \zeta_{18} + 2 \zeta_{18}^{2} - 5 \zeta_{18}^{3} + 2 \zeta_{18}^{4} + 2 \zeta_{18}^{5} ) q^{67} + ( -4 + \zeta_{18} + 4 \zeta_{18}^{3} - \zeta_{18}^{4} - \zeta_{18}^{5} ) q^{68} + ( 2 + 2 \zeta_{18} + 2 \zeta_{18}^{2} - 8 \zeta_{18}^{4} + 6 \zeta_{18}^{5} ) q^{69} + ( -\zeta_{18} - 2 \zeta_{18}^{4} ) q^{70} + ( -\zeta_{18} - \zeta_{18}^{2} - \zeta_{18}^{4} + 2 \zeta_{18}^{5} ) q^{71} + ( -1 - \zeta_{18} + \zeta_{18}^{3} + \zeta_{18}^{4} + \zeta_{18}^{5} ) q^{72} + ( -4 \zeta_{18} + 4 \zeta_{18}^{2} - \zeta_{18}^{3} + 4 \zeta_{18}^{4} - 4 \zeta_{18}^{5} ) q^{73} + ( 2 \zeta_{18} + \zeta_{18}^{2} + 6 \zeta_{18}^{3} + \zeta_{18}^{4} + 2 \zeta_{18}^{5} ) q^{74} + ( \zeta_{18} - \zeta_{18}^{2} ) q^{75} + ( -2 - 3 \zeta_{18} - 3 \zeta_{18}^{2} + 5 \zeta_{18}^{4} - 2 \zeta_{18}^{5} ) q^{76} + ( -\zeta_{18} - 2 \zeta_{18}^{4} ) q^{77} + ( 6 - 2 \zeta_{18} - 2 \zeta_{18}^{2} + 2 \zeta_{18}^{5} ) q^{78} + ( -4 - 2 \zeta_{18} + 2 \zeta_{18}^{2} + 4 \zeta_{18}^{3} ) q^{79} -\zeta_{18}^{3} q^{80} + ( -\zeta_{18} - 5 \zeta_{18}^{2} + 3 \zeta_{18}^{3} - 5 \zeta_{18}^{4} - \zeta_{18}^{5} ) q^{81} + ( 8 - 2 \zeta_{18} - 8 \zeta_{18}^{3} + 2 \zeta_{18}^{4} + 2 \zeta_{18}^{5} ) q^{82} + ( -8 + 4 \zeta_{18} + 4 \zeta_{18}^{2} - 4 \zeta_{18}^{4} ) q^{83} + ( 1 - 3 \zeta_{18}^{2} + 2 \zeta_{18}^{3} + \zeta_{18}^{5} ) q^{84} + ( 4 - \zeta_{18} - \zeta_{18}^{2} + \zeta_{18}^{5} ) q^{85} + ( 2 + 6 \zeta_{18}^{2} - 2 \zeta_{18}^{3} - 6 \zeta_{18}^{4} - 6 \zeta_{18}^{5} ) q^{86} + ( -3 \zeta_{18} - 2 \zeta_{18}^{2} + 5 \zeta_{18}^{3} - 2 \zeta_{18}^{4} - 3 \zeta_{18}^{5} ) q^{87} -\zeta_{18}^{3} q^{88} + ( 4 - 3 \zeta_{18} - 5 \zeta_{18}^{2} - 4 \zeta_{18}^{3} + 8 \zeta_{18}^{4} + 8 \zeta_{18}^{5} ) q^{89} + ( 1 + \zeta_{18} + \zeta_{18}^{2} - \zeta_{18}^{5} ) q^{90} + ( -10 - 6 \zeta_{18} - 8 \zeta_{18}^{2} + 8 \zeta_{18}^{3} + 2 \zeta_{18}^{4} - 2 \zeta_{18}^{5} ) q^{91} + ( 2 - 6 \zeta_{18} - 6 \zeta_{18}^{2} + 2 \zeta_{18}^{4} + 4 \zeta_{18}^{5} ) q^{92} + ( -1 - 2 \zeta_{18} + 5 \zeta_{18}^{2} + \zeta_{18}^{3} - 3 \zeta_{18}^{4} - 3 \zeta_{18}^{5} ) q^{93} + ( 6 \zeta_{18} - 8 \zeta_{18}^{2} - 4 \zeta_{18}^{3} - 8 \zeta_{18}^{4} + 6 \zeta_{18}^{5} ) q^{94} + ( 5 \zeta_{18} - 2 \zeta_{18}^{2} + 2 \zeta_{18}^{3} - 2 \zeta_{18}^{4} + 5 \zeta_{18}^{5} ) q^{95} + ( -\zeta_{18} + \zeta_{18}^{2} ) q^{96} + ( 6 - 2 \zeta_{18}^{4} + 2 \zeta_{18}^{5} ) q^{97} + ( 3 \zeta_{18}^{2} - 8 \zeta_{18}^{5} ) q^{98} + ( 1 + \zeta_{18} + \zeta_{18}^{2} - \zeta_{18}^{5} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6q + 3q^{2} - 3q^{4} - 3q^{5} - 6q^{8} + 3q^{9} + O(q^{10})$$ $$6q + 3q^{2} - 3q^{4} - 3q^{5} - 6q^{8} + 3q^{9} + 3q^{10} + 3q^{11} + 12q^{13} - 3q^{16} - 12q^{17} - 3q^{18} + 6q^{19} + 6q^{20} - 15q^{21} + 6q^{22} - 6q^{23} - 3q^{25} + 6q^{26} + 6q^{27} - 12q^{29} + 3q^{32} - 24q^{34} - 6q^{36} - 18q^{37} - 6q^{38} + 18q^{39} + 3q^{40} + 48q^{41} - 3q^{42} + 12q^{43} + 3q^{44} + 3q^{45} + 6q^{46} + 12q^{47} - 6q^{50} - 3q^{51} - 6q^{52} + 3q^{54} - 6q^{55} - 42q^{57} - 6q^{58} - 12q^{59} + 6q^{61} - 3q^{63} + 6q^{64} - 6q^{65} - 15q^{67} - 12q^{68} + 12q^{69} - 3q^{72} - 3q^{73} + 18q^{74} - 12q^{76} + 36q^{78} - 12q^{79} - 3q^{80} + 9q^{81} + 24q^{82} - 48q^{83} + 12q^{84} + 24q^{85} + 6q^{86} + 15q^{87} - 3q^{88} + 12q^{89} + 6q^{90} - 36q^{91} + 12q^{92} - 3q^{93} - 12q^{94} + 6q^{95} + 36q^{97} + 6q^{99} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/770\mathbb{Z}\right)^\times$$.

 $$n$$ $$211$$ $$617$$ $$661$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1 + \zeta_{18}^{3}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
221.1
 −0.766044 − 0.642788i 0.939693 − 0.342020i −0.173648 + 0.984808i −0.766044 + 0.642788i 0.939693 + 0.342020i −0.173648 − 0.984808i
0.500000 + 0.866025i −0.939693 + 1.62760i −0.500000 + 0.866025i −0.500000 0.866025i −1.87939 1.35844 + 2.27038i −1.00000 −0.266044 0.460802i 0.500000 0.866025i
221.2 0.500000 + 0.866025i 0.173648 0.300767i −0.500000 + 0.866025i −0.500000 0.866025i 0.347296 −2.64543 + 0.0412527i −1.00000 1.43969 + 2.49362i 0.500000 0.866025i
221.3 0.500000 + 0.866025i 0.766044 1.32683i −0.500000 + 0.866025i −0.500000 0.866025i 1.53209 1.28699 2.31164i −1.00000 0.326352 + 0.565258i 0.500000 0.866025i
331.1 0.500000 0.866025i −0.939693 1.62760i −0.500000 0.866025i −0.500000 + 0.866025i −1.87939 1.35844 2.27038i −1.00000 −0.266044 + 0.460802i 0.500000 + 0.866025i
331.2 0.500000 0.866025i 0.173648 + 0.300767i −0.500000 0.866025i −0.500000 + 0.866025i 0.347296 −2.64543 0.0412527i −1.00000 1.43969 2.49362i 0.500000 + 0.866025i
331.3 0.500000 0.866025i 0.766044 + 1.32683i −0.500000 0.866025i −0.500000 + 0.866025i 1.53209 1.28699 + 2.31164i −1.00000 0.326352 0.565258i 0.500000 + 0.866025i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 331.3 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 770.2.i.j 6
7.c even 3 1 inner 770.2.i.j 6
7.c even 3 1 5390.2.a.bx 3
7.d odd 6 1 5390.2.a.bv 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
770.2.i.j 6 1.a even 1 1 trivial
770.2.i.j 6 7.c even 3 1 inner
5390.2.a.bv 3 7.d odd 6 1
5390.2.a.bx 3 7.c even 3 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(770, [\chi])$$:

 $$T_{3}^{6} + 3 T_{3}^{4} - 2 T_{3}^{3} + 9 T_{3}^{2} - 3 T_{3} + 1$$ $$T_{13}^{3} - 6 T_{13}^{2} - 24 T_{13} + 136$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 - T + T^{2} )^{3}$$
$3$ $$1 - 3 T + 9 T^{2} - 2 T^{3} + 3 T^{4} + T^{6}$$
$5$ $$( 1 + T + T^{2} )^{3}$$
$7$ $$343 + 37 T^{3} + T^{6}$$
$11$ $$( 1 - T + T^{2} )^{3}$$
$13$ $$( 136 - 24 T - 6 T^{2} + T^{3} )^{2}$$
$17$ $$2601 + 2295 T + 1413 T^{2} + 438 T^{3} + 99 T^{4} + 12 T^{5} + T^{6}$$
$19$ $$72361 - 12105 T + 3639 T^{2} - 268 T^{3} + 81 T^{4} - 6 T^{5} + T^{6}$$
$23$ $$207936 + 32832 T + 7920 T^{2} + 480 T^{3} + 108 T^{4} + 6 T^{5} + T^{6}$$
$29$ $$( -267 - 63 T + 6 T^{2} + T^{3} )^{2}$$
$31$ $$11449 - 6099 T + 3249 T^{2} - 214 T^{3} + 57 T^{4} + T^{6}$$
$37$ $$11449 + 9309 T + 5643 T^{2} + 1352 T^{3} + 237 T^{4} + 18 T^{5} + T^{6}$$
$41$ $$( -408 + 180 T - 24 T^{2} + T^{3} )^{2}$$
$43$ $$( 424 - 96 T - 6 T^{2} + T^{3} )^{2}$$
$47$ $$1617984 - 137376 T + 26928 T^{2} - 1248 T^{3} + 252 T^{4} - 12 T^{5} + T^{6}$$
$53$ $$81 + 81 T + 81 T^{2} + 18 T^{3} + 9 T^{4} + T^{6}$$
$59$ $$36864 + 2304 T^{2} + 384 T^{3} + 144 T^{4} + 12 T^{5} + T^{6}$$
$61$ $$72361 - 12105 T + 3639 T^{2} - 268 T^{3} + 81 T^{4} - 6 T^{5} + T^{6}$$
$67$ $$16129 - 4953 T + 3426 T^{2} + 839 T^{3} + 186 T^{4} + 15 T^{5} + T^{6}$$
$71$ $$( -9 - 9 T + T^{3} )^{2}$$
$73$ $$289 - 765 T + 1974 T^{2} - 169 T^{3} + 54 T^{4} + 3 T^{5} + T^{6}$$
$79$ $$64 + 288 T + 1200 T^{2} + 416 T^{3} + 108 T^{4} + 12 T^{5} + T^{6}$$
$83$ $$( 192 + 144 T + 24 T^{2} + T^{3} )^{2}$$
$89$ $$848241 - 91179 T + 20853 T^{2} - 654 T^{3} + 243 T^{4} - 12 T^{5} + T^{6}$$
$97$ $$( -152 + 96 T - 18 T^{2} + T^{3} )^{2}$$