# Properties

 Label 770.2.m.a Level $770$ Weight $2$ Character orbit 770.m Analytic conductor $6.148$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$770 = 2 \cdot 5 \cdot 7 \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 770.m (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$6.14848095564$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(i)$$ Coefficient field: $$\Q(\zeta_{8})$$ Defining polynomial: $$x^{4} + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{8}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\zeta_{8}^{3} q^{2} -\zeta_{8}^{2} q^{4} + ( -2 - \zeta_{8}^{2} ) q^{5} + \zeta_{8}^{3} q^{7} -\zeta_{8} q^{8} + 3 \zeta_{8}^{2} q^{9} +O(q^{10})$$ $$q -\zeta_{8}^{3} q^{2} -\zeta_{8}^{2} q^{4} + ( -2 - \zeta_{8}^{2} ) q^{5} + \zeta_{8}^{3} q^{7} -\zeta_{8} q^{8} + 3 \zeta_{8}^{2} q^{9} + ( -\zeta_{8} + 2 \zeta_{8}^{3} ) q^{10} + ( -3 + \zeta_{8} + \zeta_{8}^{3} ) q^{11} -2 \zeta_{8} q^{13} + \zeta_{8}^{2} q^{14} - q^{16} + 3 \zeta_{8} q^{18} + ( -3 \zeta_{8} + 3 \zeta_{8}^{3} ) q^{19} + ( -1 + 2 \zeta_{8}^{2} ) q^{20} + ( 1 + \zeta_{8}^{2} + 3 \zeta_{8}^{3} ) q^{22} + ( -2 + 2 \zeta_{8}^{2} ) q^{23} + ( 3 + 4 \zeta_{8}^{2} ) q^{25} -2 q^{26} + \zeta_{8} q^{28} + ( -5 \zeta_{8} + 5 \zeta_{8}^{3} ) q^{29} + \zeta_{8}^{3} q^{32} + ( \zeta_{8} - 2 \zeta_{8}^{3} ) q^{35} + 3 q^{36} + ( 6 + 6 \zeta_{8}^{2} ) q^{37} + ( -3 + 3 \zeta_{8}^{2} ) q^{38} + ( 2 \zeta_{8} + \zeta_{8}^{3} ) q^{40} + ( -\zeta_{8} - \zeta_{8}^{3} ) q^{41} -8 \zeta_{8} q^{43} + ( \zeta_{8} + 3 \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{44} + ( 3 - 6 \zeta_{8}^{2} ) q^{45} + ( 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{46} + ( 7 + 7 \zeta_{8}^{2} ) q^{47} -\zeta_{8}^{2} q^{49} + ( 4 \zeta_{8} - 3 \zeta_{8}^{3} ) q^{50} + 2 \zeta_{8}^{3} q^{52} + ( -4 + 4 \zeta_{8}^{2} ) q^{53} + ( 6 - \zeta_{8} + 3 \zeta_{8}^{2} - 3 \zeta_{8}^{3} ) q^{55} + q^{56} + ( -5 + 5 \zeta_{8}^{2} ) q^{58} -4 \zeta_{8}^{2} q^{59} + ( -2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{61} -3 \zeta_{8} q^{63} + \zeta_{8}^{2} q^{64} + ( 4 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{65} + ( -3 - 3 \zeta_{8}^{2} ) q^{67} + ( 1 - 2 \zeta_{8}^{2} ) q^{70} -8 q^{71} -3 \zeta_{8}^{3} q^{72} -6 \zeta_{8} q^{73} + ( 6 \zeta_{8} - 6 \zeta_{8}^{3} ) q^{74} + ( 3 \zeta_{8} + 3 \zeta_{8}^{3} ) q^{76} + ( -1 - \zeta_{8}^{2} - 3 \zeta_{8}^{3} ) q^{77} + ( 3 \zeta_{8} - 3 \zeta_{8}^{3} ) q^{79} + ( 2 + \zeta_{8}^{2} ) q^{80} -9 q^{81} + ( -1 - \zeta_{8}^{2} ) q^{82} -12 \zeta_{8} q^{83} -8 q^{86} + ( 1 + 3 \zeta_{8} - \zeta_{8}^{2} ) q^{88} -14 \zeta_{8}^{2} q^{89} + ( -6 \zeta_{8} - 3 \zeta_{8}^{3} ) q^{90} + 2 q^{91} + ( 2 + 2 \zeta_{8}^{2} ) q^{92} + ( 7 \zeta_{8} - 7 \zeta_{8}^{3} ) q^{94} + ( 9 \zeta_{8} - 3 \zeta_{8}^{3} ) q^{95} + ( 6 + 6 \zeta_{8}^{2} ) q^{97} -\zeta_{8} q^{98} + ( -3 \zeta_{8} - 9 \zeta_{8}^{2} + 3 \zeta_{8}^{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 8 q^{5} + O(q^{10})$$ $$4 q - 8 q^{5} - 12 q^{11} - 4 q^{16} - 4 q^{20} + 4 q^{22} - 8 q^{23} + 12 q^{25} - 8 q^{26} + 12 q^{36} + 24 q^{37} - 12 q^{38} + 12 q^{45} + 28 q^{47} - 16 q^{53} + 24 q^{55} + 4 q^{56} - 20 q^{58} - 12 q^{67} + 4 q^{70} - 32 q^{71} - 4 q^{77} + 8 q^{80} - 36 q^{81} - 4 q^{82} - 32 q^{86} + 4 q^{88} + 8 q^{91} + 8 q^{92} + 24 q^{97} + O(q^{100})$$

## Character values

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

 $$n$$ $$211$$ $$617$$ $$661$$ $$\chi(n)$$ $$-1$$ $$\zeta_{8}^{2}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
43.1
 −0.707107 + 0.707107i 0.707107 − 0.707107i −0.707107 − 0.707107i 0.707107 + 0.707107i
−0.707107 0.707107i 0 1.00000i −2.00000 + 1.00000i 0 0.707107 + 0.707107i 0.707107 0.707107i 3.00000i 2.12132 + 0.707107i
43.2 0.707107 + 0.707107i 0 1.00000i −2.00000 + 1.00000i 0 −0.707107 0.707107i −0.707107 + 0.707107i 3.00000i −2.12132 0.707107i
197.1 −0.707107 + 0.707107i 0 1.00000i −2.00000 1.00000i 0 0.707107 0.707107i 0.707107 + 0.707107i 3.00000i 2.12132 0.707107i
197.2 0.707107 0.707107i 0 1.00000i −2.00000 1.00000i 0 −0.707107 + 0.707107i −0.707107 0.707107i 3.00000i −2.12132 + 0.707107i
 $$n$$: e.g. 2-40 or 990-1000 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.c odd 4 1 inner
11.b odd 2 1 inner
55.e even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 770.2.m.a 4
5.c odd 4 1 inner 770.2.m.a 4
11.b odd 2 1 inner 770.2.m.a 4
55.e even 4 1 inner 770.2.m.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
770.2.m.a 4 1.a even 1 1 trivial
770.2.m.a 4 5.c odd 4 1 inner
770.2.m.a 4 11.b odd 2 1 inner
770.2.m.a 4 55.e even 4 1 inner

## Hecke kernels

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

 $$T_{3}$$ $$T_{17}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T^{4}$$
$3$ $$T^{4}$$
$5$ $$( 5 + 4 T + T^{2} )^{2}$$
$7$ $$1 + T^{4}$$
$11$ $$( 11 + 6 T + T^{2} )^{2}$$
$13$ $$16 + T^{4}$$
$17$ $$T^{4}$$
$19$ $$( -18 + T^{2} )^{2}$$
$23$ $$( 8 + 4 T + T^{2} )^{2}$$
$29$ $$( -50 + T^{2} )^{2}$$
$31$ $$T^{4}$$
$37$ $$( 72 - 12 T + T^{2} )^{2}$$
$41$ $$( 2 + T^{2} )^{2}$$
$43$ $$4096 + T^{4}$$
$47$ $$( 98 - 14 T + T^{2} )^{2}$$
$53$ $$( 32 + 8 T + T^{2} )^{2}$$
$59$ $$( 16 + T^{2} )^{2}$$
$61$ $$( 8 + T^{2} )^{2}$$
$67$ $$( 18 + 6 T + T^{2} )^{2}$$
$71$ $$( 8 + T )^{4}$$
$73$ $$1296 + T^{4}$$
$79$ $$( -18 + T^{2} )^{2}$$
$83$ $$20736 + T^{4}$$
$89$ $$( 196 + T^{2} )^{2}$$
$97$ $$( 72 - 12 T + T^{2} )^{2}$$