Properties

Label 370.2.e.c
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.

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

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 1.50000 + 2.59808i −0.500000 0.866025i 0.500000 + 0.866025i 3.00000 0 −1.00000 −3.00000 + 5.19615i 1.00000
211.1 0.500000 + 0.866025i 1.50000 2.59808i −0.500000 + 0.866025i 0.500000 0.866025i 3.00000 0 −1.00000 −3.00000 5.19615i 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.c 2
37.c even 3 1 inner 370.2.e.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
370.2.e.c 2 1.a even 1 1 trivial
370.2.e.c 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 \(S_{2}^{\mathrm{new}}(370, [\chi])\):

\( T_{3}^{2} - 3 T_{3} + 9 \)
\( T_{7} \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - T + T^{2} \)
$3$ \( 9 - 3 T + T^{2} \)
$5$ \( 1 - T + T^{2} \)
$7$ \( T^{2} \)
$11$ \( ( -2 + T )^{2} \)
$13$ \( 1 - T + T^{2} \)
$17$ \( 4 - 2 T + T^{2} \)
$19$ \( 4 + 2 T + T^{2} \)
$23$ \( ( 6 + T )^{2} \)
$29$ \( ( -6 + T )^{2} \)
$31$ \( ( -5 + T )^{2} \)
$37$ \( 37 + 11 T + T^{2} \)
$41$ \( 9 + 3 T + T^{2} \)
$43$ \( ( -7 + T )^{2} \)
$47$ \( ( 8 + T )^{2} \)
$53$ \( 121 - 11 T + T^{2} \)
$59$ \( 4 - 2 T + T^{2} \)
$61$ \( 100 + 10 T + T^{2} \)
$67$ \( 16 + 4 T + T^{2} \)
$71$ \( T^{2} \)
$73$ \( ( 2 + T )^{2} \)
$79$ \( 16 - 4 T + T^{2} \)
$83$ \( 144 + 12 T + T^{2} \)
$89$ \( 4 + 2 T + T^{2} \)
$97$ \( ( -18 + T )^{2} \)
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