Properties

Label 770.2.i.f
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.

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

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 2.50000 0.866025i −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 2.50000 + 0.866025i −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.f 2
7.c even 3 1 inner 770.2.i.f 2
7.c even 3 1 5390.2.a.p 1
7.d odd 6 1 5390.2.a.e 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
770.2.i.f 2 1.a even 1 1 trivial
770.2.i.f 2 7.c even 3 1 inner
5390.2.a.e 1 7.d odd 6 1
5390.2.a.p 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 \(S_{2}^{\mathrm{new}}(770, [\chi])\):

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - T + T^{2} \)
$3$ \( 1 + T + T^{2} \)
$5$ \( 1 + T + T^{2} \)
$7$ \( 7 - 5 T + T^{2} \)
$11$ \( 1 + T + T^{2} \)
$13$ \( ( -2 + T )^{2} \)
$17$ \( T^{2} \)
$19$ \( 16 - 4 T + T^{2} \)
$23$ \( 9 - 3 T + T^{2} \)
$29$ \( ( -3 + T )^{2} \)
$31$ \( 4 + 2 T + T^{2} \)
$37$ \( 100 - 10 T + T^{2} \)
$41$ \( ( 9 + T )^{2} \)
$43$ \( ( 1 + T )^{2} \)
$47$ \( T^{2} \)
$53$ \( 36 + 6 T + T^{2} \)
$59$ \( 36 - 6 T + T^{2} \)
$61$ \( 1 - T + T^{2} \)
$67$ \( 25 + 5 T + T^{2} \)
$71$ \( ( 12 + T )^{2} \)
$73$ \( 256 - 16 T + T^{2} \)
$79$ \( 4 + 2 T + T^{2} \)
$83$ \( ( -9 + T )^{2} \)
$89$ \( 81 + 9 T + T^{2} \)
$97$ \( ( -8 + T )^{2} \)
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