Properties

Label 370.2.a.f
Level $370$
Weight $2$
Character orbit 370.a
Self dual yes
Analytic conductor $2.954$
Analytic rank $0$
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: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{33}) \)
Defining polynomial: \(x^{2} - x - 8\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
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 \(\beta = \frac{1}{2}(1 + \sqrt{33})\). We also show the integral \(q\)-expansion of the trace form.

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

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 −4.37228 1.00000 1.00000 1.00000
1.2 1.00000 2.00000 1.00000 1.00000 2.00000 1.37228 1.00000 1.00000 1.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

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

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -1 + T )^{2} \)
$3$ \( ( -2 + T )^{2} \)
$5$ \( ( -1 + T )^{2} \)
$7$ \( -6 + 3 T + T^{2} \)
$11$ \( -8 + T + T^{2} \)
$13$ \( -32 - 2 T + T^{2} \)
$17$ \( -2 + 5 T + T^{2} \)
$19$ \( ( 2 + T )^{2} \)
$23$ \( -32 - 2 T + T^{2} \)
$29$ \( -74 + T + T^{2} \)
$31$ \( 22 + 11 T + T^{2} \)
$37$ \( ( -1 + T )^{2} \)
$41$ \( -2 - 5 T + T^{2} \)
$43$ \( 12 - 9 T + T^{2} \)
$47$ \( -24 + 6 T + T^{2} \)
$53$ \( -6 + 3 T + T^{2} \)
$59$ \( 16 - 14 T + T^{2} \)
$61$ \( -2 + 5 T + T^{2} \)
$67$ \( -32 + 2 T + T^{2} \)
$71$ \( -32 + 2 T + T^{2} \)
$73$ \( -24 - 6 T + T^{2} \)
$79$ \( -32 - 2 T + T^{2} \)
$83$ \( -8 - 10 T + T^{2} \)
$89$ \( ( -10 + T )^{2} \)
$97$ \( -2 - 17 T + T^{2} \)
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