Properties

Label 770.2.a.l
Level $770$
Weight $2$
Character orbit 770.a
Self dual yes
Analytic conductor $6.148$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 770 = 2 \cdot 5 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 770.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(6.14848095564\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.892.1
Defining polynomial: \(x^{3} - x^{2} - 8 x + 10\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{3} - x^{2} - 8 x + 10\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu^{2} + \nu - 6 \)
\(\beta_{2}\)\(=\)\( -\nu^{2} + \nu + 5 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{2} + \beta_{1} + 1\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(-\beta_{2} + \beta_{1} + 11\)\()/2\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
1.31955
−2.91729
2.59774
−1.00000 −2.93923 1.00000 −1.00000 2.93923 −1.00000 −1.00000 5.63910 1.00000
1.2 −1.00000 −0.406728 1.00000 −1.00000 0.406728 −1.00000 −1.00000 −2.83457 1.00000
1.3 −1.00000 3.34596 1.00000 −1.00000 −3.34596 −1.00000 −1.00000 8.19547 1.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(5\) \(1\)
\(7\) \(1\)
\(11\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 770.2.a.l 3
3.b odd 2 1 6930.2.a.cl 3
4.b odd 2 1 6160.2.a.bi 3
5.b even 2 1 3850.2.a.bu 3
5.c odd 4 2 3850.2.c.z 6
7.b odd 2 1 5390.2.a.bz 3
11.b odd 2 1 8470.2.a.cl 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
770.2.a.l 3 1.a even 1 1 trivial
3850.2.a.bu 3 5.b even 2 1
3850.2.c.z 6 5.c odd 4 2
5390.2.a.bz 3 7.b odd 2 1
6160.2.a.bi 3 4.b odd 2 1
6930.2.a.cl 3 3.b odd 2 1
8470.2.a.cl 3 11.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(770))\):

\( T_{3}^{3} - 10 T_{3} - 4 \)
\( T_{13}^{3} - 40 T_{13} - 32 \)
\( T_{17}^{3} + 8 T_{17}^{2} - 12 T_{17} - 128 \)
\( T_{19}^{3} + 2 T_{19}^{2} - 30 T_{19} - 56 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T )^{3} \)
$3$ \( -4 - 10 T + T^{3} \)
$5$ \( ( 1 + T )^{3} \)
$7$ \( ( 1 + T )^{3} \)
$11$ \( ( -1 + T )^{3} \)
$13$ \( -32 - 40 T + T^{3} \)
$17$ \( -128 - 12 T + 8 T^{2} + T^{3} \)
$19$ \( -56 - 30 T + 2 T^{2} + T^{3} \)
$23$ \( 56 - 30 T - 2 T^{2} + T^{3} \)
$29$ \( 428 - 82 T - 4 T^{2} + T^{3} \)
$31$ \( ( -6 + T )^{3} \)
$37$ \( 124 - 50 T - 4 T^{2} + T^{3} \)
$41$ \( 160 + 98 T + 18 T^{2} + T^{3} \)
$43$ \( 16 - 28 T - 8 T^{2} + T^{3} \)
$47$ \( -64 - 48 T - 2 T^{2} + T^{3} \)
$53$ \( 428 - 82 T - 4 T^{2} + T^{3} \)
$59$ \( 608 - 64 T - 10 T^{2} + T^{3} \)
$61$ \( -64 + 84 T - 20 T^{2} + T^{3} \)
$67$ \( ( -8 + T )^{3} \)
$71$ \( 32 - 104 T - 8 T^{2} + T^{3} \)
$73$ \( -784 - 140 T + 4 T^{2} + T^{3} \)
$79$ \( -2272 - 262 T + 6 T^{2} + T^{3} \)
$83$ \( 160 + 152 T + 24 T^{2} + T^{3} \)
$89$ \( -40 - 28 T + 6 T^{2} + T^{3} \)
$97$ \( 100 - 10 T - 8 T^{2} + T^{3} \)
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