Properties

Label 770.2.i.g
Level $770$
Weight $2$
Character orbit 770.i
Analytic conductor $6.148$
Analytic rank $0$
Dimension $2$
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.i (of order \(3\), degree \(2\), minimal)

Newform invariants

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

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

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

Character values

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

\(n\) \(211\) \(617\) \(661\)
\(\chi(n)\) \(1\) \(1\) \(-\zeta_{6}\)

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
0.500000 + 0.866025i
0.500000 0.866025i
0.500000 + 0.866025i −0.500000 + 0.866025i −0.500000 + 0.866025i 0.500000 + 0.866025i −1.00000 −0.500000 + 2.59808i −1.00000 1.00000 + 1.73205i −0.500000 + 0.866025i
331.1 0.500000 0.866025i −0.500000 0.866025i −0.500000 0.866025i 0.500000 0.866025i −1.00000 −0.500000 2.59808i −1.00000 1.00000 1.73205i −0.500000 0.866025i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

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.g 2
7.c even 3 1 inner 770.2.i.g 2
7.c even 3 1 5390.2.a.m 1
7.d odd 6 1 5390.2.a.g 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
770.2.i.g 2 1.a even 1 1 trivial
770.2.i.g 2 7.c even 3 1 inner
5390.2.a.g 1 7.d odd 6 1
5390.2.a.m 1 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 \(S_{2}^{\mathrm{new}}(770, [\chi])\):

\( T_{3}^{2} + T_{3} + 1 \)
\( T_{13} - 5 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - T + T^{2} \)
$3$ \( 1 + T + T^{2} \)
$5$ \( 1 - T + T^{2} \)
$7$ \( 7 + T + T^{2} \)
$11$ \( 1 + T + T^{2} \)
$13$ \( ( -5 + T )^{2} \)
$17$ \( T^{2} \)
$19$ \( 4 + 2 T + T^{2} \)
$23$ \( 36 + 6 T + T^{2} \)
$29$ \( ( 3 + T )^{2} \)
$31$ \( 64 + 8 T + T^{2} \)
$37$ \( 16 - 4 T + T^{2} \)
$41$ \( T^{2} \)
$43$ \( ( -2 + T )^{2} \)
$47$ \( T^{2} \)
$53$ \( 36 - 6 T + T^{2} \)
$59$ \( 9 + 3 T + T^{2} \)
$61$ \( 169 - 13 T + T^{2} \)
$67$ \( 169 - 13 T + T^{2} \)
$71$ \( ( -12 + T )^{2} \)
$73$ \( 4 + 2 T + T^{2} \)
$79$ \( 121 + 11 T + T^{2} \)
$83$ \( ( 6 + T )^{2} \)
$89$ \( 36 - 6 T + T^{2} \)
$97$ \( ( -5 + T )^{2} \)
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