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: \( x^{2} - 3 \) Copy content Toggle raw display
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.

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

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 \(S_{2}^{\mathrm{new}}(\Gamma_0(370))\):

\( T_{3}^{2} + 2T_{3} - 2 \) Copy content Toggle raw display
\( T_{7}^{2} + 6T_{7} + 6 \) Copy content Toggle raw display

Hecke characteristic polynomials

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