Properties

Label 370.2.g.a
Level $370$
Weight $2$
Character orbit 370.g
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.g (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(2.95446487479\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

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

Character values

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

\(n\) \(261\) \(297\)
\(\chi(n)\) \(i\) \(i\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
43.1
1.00000i
1.00000i
1.00000i −1.00000 + 1.00000i −1.00000 2.00000 1.00000i 1.00000 + 1.00000i 1.00000 1.00000i 1.00000i 1.00000i −1.00000 2.00000i
327.1 1.00000i −1.00000 1.00000i −1.00000 2.00000 + 1.00000i 1.00000 1.00000i 1.00000 + 1.00000i 1.00000i 1.00000i −1.00000 + 2.00000i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
185.f even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 370.2.g.a 2
5.c odd 4 1 370.2.h.b yes 2
37.d odd 4 1 370.2.h.b yes 2
185.f even 4 1 inner 370.2.g.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
370.2.g.a 2 1.a even 1 1 trivial
370.2.g.a 2 185.f even 4 1 inner
370.2.h.b yes 2 5.c odd 4 1
370.2.h.b yes 2 37.d odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} + 2 T_{3} + 2 \) acting on \(S_{2}^{\mathrm{new}}(370, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T^{2} \)
$3$ \( 2 + 2 T + T^{2} \)
$5$ \( 5 - 4 T + T^{2} \)
$7$ \( 2 - 2 T + T^{2} \)
$11$ \( 4 + T^{2} \)
$13$ \( 4 + T^{2} \)
$17$ \( ( -4 + T )^{2} \)
$19$ \( 50 - 10 T + T^{2} \)
$23$ \( T^{2} \)
$29$ \( 18 - 6 T + T^{2} \)
$31$ \( 98 - 14 T + T^{2} \)
$37$ \( 37 + 12 T + T^{2} \)
$41$ \( T^{2} \)
$43$ \( 16 + T^{2} \)
$47$ \( 98 + 14 T + T^{2} \)
$53$ \( 2 - 2 T + T^{2} \)
$59$ \( 2 - 2 T + T^{2} \)
$61$ \( 18 + 6 T + T^{2} \)
$67$ \( 18 + 6 T + T^{2} \)
$71$ \( ( 8 + T )^{2} \)
$73$ \( 162 + 18 T + T^{2} \)
$79$ \( 2 - 2 T + T^{2} \)
$83$ \( 50 + 10 T + T^{2} \)
$89$ \( 50 + 10 T + T^{2} \)
$97$ \( ( -8 + T )^{2} \)
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