Properties

Label 370.2.e.b
Level $370$
Weight $2$
Character orbit 370.e
Analytic conductor $2.954$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 370 = 2 \cdot 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 370.e (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(2.95446487479\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/370\mathbb{Z}\right)^\times\).

\(n\) \(261\) \(297\)
\(\chi(n)\) \(-\zeta_{6}\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
121.1
0.500000 + 0.866025i
0.500000 0.866025i
0.500000 0.866025i −0.500000 0.866025i −0.500000 0.866025i 0.500000 + 0.866025i −1.00000 −1.00000 1.73205i −1.00000 1.00000 1.73205i 1.00000
211.1 0.500000 + 0.866025i −0.500000 + 0.866025i −0.500000 + 0.866025i 0.500000 0.866025i −1.00000 −1.00000 + 1.73205i −1.00000 1.00000 + 1.73205i 1.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
37.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 370.2.e.b 2
37.c even 3 1 inner 370.2.e.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
370.2.e.b 2 1.a even 1 1 trivial
370.2.e.b 2 37.c even 3 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(370, [\chi])\):

\( T_{3}^{2} + T_{3} + 1 \)
\( T_{7}^{2} + 2 T_{7} + 4 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - T + T^{2} \)
$3$ \( 1 + T + T^{2} \)
$5$ \( 1 - T + T^{2} \)
$7$ \( 4 + 2 T + T^{2} \)
$11$ \( T^{2} \)
$13$ \( 25 + 5 T + T^{2} \)
$17$ \( T^{2} \)
$19$ \( 4 + 2 T + T^{2} \)
$23$ \( T^{2} \)
$29$ \( ( -6 + T )^{2} \)
$31$ \( ( 1 + T )^{2} \)
$37$ \( 37 - 11 T + T^{2} \)
$41$ \( 81 - 9 T + T^{2} \)
$43$ \( ( 1 + T )^{2} \)
$47$ \( ( 6 + T )^{2} \)
$53$ \( 81 - 9 T + T^{2} \)
$59$ \( 144 - 12 T + T^{2} \)
$61$ \( 100 - 10 T + T^{2} \)
$67$ \( 16 - 4 T + T^{2} \)
$71$ \( 144 - 12 T + T^{2} \)
$73$ \( ( -8 + T )^{2} \)
$79$ \( 16 - 4 T + T^{2} \)
$83$ \( 144 - 12 T + T^{2} \)
$89$ \( 36 + 6 T + T^{2} \)
$97$ \( ( -2 + T )^{2} \)
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