Properties

Label 770.2.i.d
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.

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

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 −1.50000 + 2.59808i −0.500000 + 0.866025i −0.500000 0.866025i −3.00000 0.500000 2.59808i −1.00000 −3.00000 5.19615i 0.500000 0.866025i
331.1 0.500000 0.866025i −1.50000 2.59808i −0.500000 0.866025i −0.500000 + 0.866025i −3.00000 0.500000 + 2.59808i −1.00000 −3.00000 + 5.19615i 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.d 2
7.c even 3 1 inner 770.2.i.d 2
7.c even 3 1 5390.2.a.t 1
7.d odd 6 1 5390.2.a.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
770.2.i.d 2 1.a even 1 1 trivial
770.2.i.d 2 7.c even 3 1 inner
5390.2.a.a 1 7.d odd 6 1
5390.2.a.t 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 \(S_{2}^{\mathrm{new}}(770, [\chi])\):

\( T_{3}^{2} + 3 T_{3} + 9 \)
\( T_{13} + 1 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - T + T^{2} \)
$3$ \( 9 + 3 T + T^{2} \)
$5$ \( 1 + T + T^{2} \)
$7$ \( 7 - T + T^{2} \)
$11$ \( 1 - T + T^{2} \)
$13$ \( ( 1 + T )^{2} \)
$17$ \( 36 - 6 T + T^{2} \)
$19$ \( 16 + 4 T + T^{2} \)
$23$ \( 16 + 4 T + T^{2} \)
$29$ \( ( -1 + T )^{2} \)
$31$ \( 64 + 8 T + T^{2} \)
$37$ \( 64 + 8 T + T^{2} \)
$41$ \( ( 12 + T )^{2} \)
$43$ \( ( -12 + T )^{2} \)
$47$ \( 36 - 6 T + T^{2} \)
$53$ \( 36 - 6 T + T^{2} \)
$59$ \( 225 + 15 T + T^{2} \)
$61$ \( 9 + 3 T + T^{2} \)
$67$ \( 49 - 7 T + T^{2} \)
$71$ \( ( -10 + T )^{2} \)
$73$ \( 16 - 4 T + T^{2} \)
$79$ \( 225 + 15 T + T^{2} \)
$83$ \( ( -10 + T )^{2} \)
$89$ \( 36 + 6 T + T^{2} \)
$97$ \( ( 5 + T )^{2} \)
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