# Properties

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

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## 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.

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

## 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 0.500000 + 0.363271i −0.809017 + 0.587785i −0.309017 + 0.951057i 0.190983 0.587785i −0.809017 + 0.587785i 0.809017 + 0.587785i −0.809017 2.48990i 1.00000
141.1 −0.309017 + 0.951057i 0.500000 0.363271i −0.809017 0.587785i −0.309017 0.951057i 0.190983 + 0.587785i −0.809017 0.587785i 0.809017 0.587785i −0.809017 + 2.48990i 1.00000
421.1 0.809017 0.587785i 0.500000 + 1.53884i 0.309017 0.951057i 0.809017 + 0.587785i 1.30902 + 0.951057i 0.309017 0.951057i −0.309017 0.951057i 0.309017 0.224514i 1.00000
631.1 0.809017 + 0.587785i 0.500000 1.53884i 0.309017 + 0.951057i 0.809017 0.587785i 1.30902 0.951057i 0.309017 + 0.951057i −0.309017 + 0.951057i 0.309017 + 0.224514i 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.d 4
11.c even 5 1 inner 770.2.n.d 4
11.c even 5 1 8470.2.a.bp 2
11.d odd 10 1 8470.2.a.cb 2

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

## Hecke kernels

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - T + T^{2} - T^{3} + T^{4}$$
$3$ $$1 - 3 T + 4 T^{2} - 2 T^{3} + T^{4}$$
$5$ $$1 - T + T^{2} - T^{3} + T^{4}$$
$7$ $$1 + T + T^{2} + T^{3} + T^{4}$$
$11$ $$121 - 11 T + 21 T^{2} - T^{3} + T^{4}$$
$13$ $$16 + 8 T + 24 T^{2} - 8 T^{3} + T^{4}$$
$17$ $$1 - T + 6 T^{2} + 4 T^{3} + T^{4}$$
$19$ $$81 + 108 T + 54 T^{2} - 3 T^{3} + T^{4}$$
$23$ $$( -4 - 8 T + T^{2} )^{2}$$
$29$ $$400 - 200 T + 60 T^{2} - 10 T^{3} + T^{4}$$
$31$ $$256 + 192 T + 64 T^{2} + 8 T^{3} + T^{4}$$
$37$ $$5776 - 1064 T + 96 T^{2} - 4 T^{3} + T^{4}$$
$41$ $$121 - 77 T + 34 T^{2} - 8 T^{3} + T^{4}$$
$43$ $$( -19 - 7 T + T^{2} )^{2}$$
$47$ $$1936 + 88 T + 64 T^{2} + 12 T^{3} + T^{4}$$
$53$ $$400 - 200 T + 40 T^{2} + T^{4}$$
$59$ $$2025 + 90 T^{2} + 15 T^{3} + T^{4}$$
$61$ $$16 - 32 T + 184 T^{2} + 22 T^{3} + T^{4}$$
$67$ $$( 19 + 9 T + T^{2} )^{2}$$
$71$ $$5776 + 1368 T + 244 T^{2} + 22 T^{3} + T^{4}$$
$73$ $$121 - 77 T + 34 T^{2} - 8 T^{3} + T^{4}$$
$79$ $$256 + 64 T + 16 T^{2} + 4 T^{3} + T^{4}$$
$83$ $$361 + 38 T + 64 T^{2} - 13 T^{3} + T^{4}$$
$89$ $$( 121 + 23 T + T^{2} )^{2}$$
$97$ $$7921 - 1068 T + 94 T^{2} - 7 T^{3} + T^{4}$$
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