# Properties

 Label 770.2.n.a Level $770$ Weight $2$ Character orbit 770.n Analytic conductor $6.148$ Analytic rank $0$ Dimension $4$ 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.n (of order $$5$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$6.14848095564$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{10})$$ Defining polynomial: $$x^{4} - x^{3} + x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{10}$$. We also show the integral $$q$$-expansion of the trace form.

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

## Character values

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

 $$n$$ $$211$$ $$617$$ $$661$$ $$\chi(n)$$ $$-\zeta_{10}^{3}$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
71.1
 −0.309017 + 0.951057i −0.309017 − 0.951057i 0.809017 + 0.587785i 0.809017 − 0.587785i
0.309017 + 0.951057i −1.30902 0.951057i −0.809017 + 0.587785i 0.309017 0.951057i 0.500000 1.53884i −0.809017 + 0.587785i −0.809017 0.587785i −0.118034 0.363271i 1.00000
141.1 0.309017 0.951057i −1.30902 + 0.951057i −0.809017 0.587785i 0.309017 + 0.951057i 0.500000 + 1.53884i −0.809017 0.587785i −0.809017 + 0.587785i −0.118034 + 0.363271i 1.00000
421.1 −0.809017 + 0.587785i −0.190983 0.587785i 0.309017 0.951057i −0.809017 0.587785i 0.500000 + 0.363271i 0.309017 0.951057i 0.309017 + 0.951057i 2.11803 1.53884i 1.00000
631.1 −0.809017 0.587785i −0.190983 + 0.587785i 0.309017 + 0.951057i −0.809017 + 0.587785i 0.500000 0.363271i 0.309017 + 0.951057i 0.309017 0.951057i 2.11803 + 1.53884i 1.00000
 $$n$$: 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
11.c even 5 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 770.2.n.a 4
11.c even 5 1 inner 770.2.n.a 4
11.c even 5 1 8470.2.a.cc 2
11.d odd 10 1 8470.2.a.bq 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
770.2.n.a 4 1.a even 1 1 trivial
770.2.n.a 4 11.c even 5 1 inner
8470.2.a.bq 2 11.d odd 10 1
8470.2.a.cc 2 11.c even 5 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{4} + 3 T_{3}^{3} + 4 T_{3}^{2} + 2 T_{3} + 1$$ acting on $$S_{2}^{\mathrm{new}}(770, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T + T^{2} + T^{3} + T^{4}$$
$3$ $$1 + 2 T + 4 T^{2} + 3 T^{3} + T^{4}$$
$5$ $$1 + T + T^{2} + T^{3} + T^{4}$$
$7$ $$1 + T + T^{2} + T^{3} + T^{4}$$
$11$ $$121 - 11 T - 9 T^{2} - T^{3} + T^{4}$$
$13$ $$361 + 133 T + 24 T^{2} + 2 T^{3} + T^{4}$$
$17$ $$121 + 66 T + 16 T^{2} + T^{3} + T^{4}$$
$19$ $$16 + 8 T + 24 T^{2} - 8 T^{3} + T^{4}$$
$23$ $$( -4 - 8 T + T^{2} )^{2}$$
$29$ $$25 - 25 T + 40 T^{2} - 10 T^{3} + T^{4}$$
$31$ $$16 + 32 T + 24 T^{2} - 2 T^{3} + T^{4}$$
$37$ $$1296 - 648 T + 144 T^{2} - 12 T^{3} + T^{4}$$
$41$ $$6400 + 1600 T + 240 T^{2} + 20 T^{3} + T^{4}$$
$43$ $$( -4 - 2 T + T^{2} )^{2}$$
$47$ $$625 + 125 T + 150 T^{2} - 20 T^{3} + T^{4}$$
$53$ $$256 - 192 T + 64 T^{2} - 8 T^{3} + T^{4}$$
$59$ $$1936 + 528 T + 64 T^{2} + 2 T^{3} + T^{4}$$
$61$ $$400 + 400 T + 160 T^{2} + 10 T^{3} + T^{4}$$
$67$ $$( -4 - 2 T + T^{2} )^{2}$$
$71$ $$3721 + 1037 T + 184 T^{2} + 18 T^{3} + T^{4}$$
$73$ $$2401 + 1029 T + 196 T^{2} + 14 T^{3} + T^{4}$$
$79$ $$25 + 25 T + 40 T^{2} + 10 T^{3} + T^{4}$$
$83$ $$24025 + 2325 T + 160 T^{2} + 10 T^{3} + T^{4}$$
$89$ $$( 4 - 14 T + T^{2} )^{2}$$
$97$ $$361 - 38 T + 24 T^{2} - 7 T^{3} + T^{4}$$