Properties

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

Related objects

Downloads

Learn more

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: \(2\)
Coefficient field: \(\Q(\sqrt{33}) \)
Defining polynomial: \(x^{2} - x - 8\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
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 \(\beta = \sqrt{33}\). We also show the integral \(q\)-expansion of the trace form.

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

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
3.37228
−2.37228
1.00000 2.00000 1.00000 1.00000 2.00000 1.00000 1.00000 1.00000 1.00000
1.2 1.00000 2.00000 1.00000 1.00000 2.00000 1.00000 1.00000 1.00000 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.k 2
3.b odd 2 1 6930.2.a.bo 2
4.b odd 2 1 6160.2.a.r 2
5.b even 2 1 3850.2.a.bc 2
5.c odd 4 2 3850.2.c.y 4
7.b odd 2 1 5390.2.a.bq 2
11.b odd 2 1 8470.2.a.bu 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
770.2.a.k 2 1.a even 1 1 trivial
3850.2.a.bc 2 5.b even 2 1
3850.2.c.y 4 5.c odd 4 2
5390.2.a.bq 2 7.b odd 2 1
6160.2.a.r 2 4.b odd 2 1
6930.2.a.bo 2 3.b odd 2 1
8470.2.a.bu 2 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(770))\):

\( T_{3} - 2 \)
\( T_{13}^{2} - 2 T_{13} - 32 \)
\( T_{17}^{2} + 2 T_{17} - 32 \)
\( T_{19}^{2} + 2 T_{19} - 32 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -1 + T )^{2} \)
$3$ \( ( -2 + T )^{2} \)
$5$ \( ( -1 + T )^{2} \)
$7$ \( ( -1 + T )^{2} \)
$11$ \( ( 1 + T )^{2} \)
$13$ \( -32 - 2 T + T^{2} \)
$17$ \( -32 + 2 T + T^{2} \)
$19$ \( -32 + 2 T + T^{2} \)
$23$ \( -32 + 2 T + T^{2} \)
$29$ \( -24 - 6 T + T^{2} \)
$31$ \( -32 + 2 T + T^{2} \)
$37$ \( -8 + 10 T + T^{2} \)
$41$ \( ( 4 + T )^{2} \)
$43$ \( ( -4 + T )^{2} \)
$47$ \( -32 + 2 T + T^{2} \)
$53$ \( 16 + 14 T + T^{2} \)
$59$ \( -24 + 6 T + T^{2} \)
$61$ \( 16 - 14 T + T^{2} \)
$67$ \( ( 4 + T )^{2} \)
$71$ \( ( 4 + T )^{2} \)
$73$ \( -8 - 10 T + T^{2} \)
$79$ \( -32 - 2 T + T^{2} \)
$83$ \( ( -8 + T )^{2} \)
$89$ \( -116 - 8 T + T^{2} \)
$97$ \( 88 + 22 T + T^{2} \)
show more
show less