Properties

Label 370.2.a.e
Level $370$
Weight $2$
Character orbit 370.a
Self dual yes
Analytic conductor $2.954$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 370 = 2 \cdot 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 370.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(2.95446487479\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{3}) \)
Defining polynomial: \(x^{2} - 3\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{3}\). We also show the integral \(q\)-expansion of the trace form.

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

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
−1.73205
1.73205
−1.00000 −2.73205 1.00000 1.00000 2.73205 −1.26795 −1.00000 4.46410 −1.00000
1.2 −1.00000 0.732051 1.00000 1.00000 −0.732051 −4.73205 −1.00000 −2.46410 −1.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(5\) \(-1\)
\(37\) \(-1\)

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 370.2.a.e 2
3.b odd 2 1 3330.2.a.bd 2
4.b odd 2 1 2960.2.a.q 2
5.b even 2 1 1850.2.a.x 2
5.c odd 4 2 1850.2.b.l 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
370.2.a.e 2 1.a even 1 1 trivial
1850.2.a.x 2 5.b even 2 1
1850.2.b.l 4 5.c odd 4 2
2960.2.a.q 2 4.b odd 2 1
3330.2.a.bd 2 3.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(370))\):

\( T_{3}^{2} + 2 T_{3} - 2 \)
\( T_{7}^{2} + 6 T_{7} + 6 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T )^{2} \)
$3$ \( -2 + 2 T + T^{2} \)
$5$ \( ( -1 + T )^{2} \)
$7$ \( 6 + 6 T + T^{2} \)
$11$ \( -8 + 4 T + T^{2} \)
$13$ \( -8 + 4 T + T^{2} \)
$17$ \( -8 - 4 T + T^{2} \)
$19$ \( -26 - 2 T + T^{2} \)
$23$ \( ( 8 + T )^{2} \)
$29$ \( -44 + 4 T + T^{2} \)
$31$ \( -2 + 2 T + T^{2} \)
$37$ \( ( -1 + T )^{2} \)
$41$ \( ( 2 + T )^{2} \)
$43$ \( -48 + T^{2} \)
$47$ \( 6 + 6 T + T^{2} \)
$53$ \( ( 6 + T )^{2} \)
$59$ \( -2 + 10 T + T^{2} \)
$61$ \( -44 - 4 T + T^{2} \)
$67$ \( -50 - 10 T + T^{2} \)
$71$ \( -32 + 8 T + T^{2} \)
$73$ \( -12 - 12 T + T^{2} \)
$79$ \( 46 - 14 T + T^{2} \)
$83$ \( 46 + 14 T + T^{2} \)
$89$ \( ( 2 + T )^{2} \)
$97$ \( ( 2 + T )^{2} \)
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