Properties

Label 370.2.h.a
Level $370$
Weight $2$
Character orbit 370.h
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.h (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: \(\Z[a_1, \ldots, a_{9}]\)
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.

\(f(q)\) \(=\) \( q + q^{2} + q^{4} + ( -1 + 2 i ) q^{5} + ( -2 + 2 i ) q^{7} + q^{8} + 3 i q^{9} +O(q^{10})\) \( q + q^{2} + q^{4} + ( -1 + 2 i ) q^{5} + ( -2 + 2 i ) q^{7} + q^{8} + 3 i q^{9} + ( -1 + 2 i ) q^{10} + 4 q^{13} + ( -2 + 2 i ) q^{14} + q^{16} + 2 i q^{17} + 3 i q^{18} + ( -2 - 2 i ) q^{19} + ( -1 + 2 i ) q^{20} + 4 q^{23} + ( -3 - 4 i ) q^{25} + 4 q^{26} + ( -2 + 2 i ) q^{28} + ( 7 - 7 i ) q^{29} + ( -4 - 4 i ) q^{31} + q^{32} + 2 i q^{34} + ( -2 - 6 i ) q^{35} + 3 i q^{36} + ( -1 + 6 i ) q^{37} + ( -2 - 2 i ) q^{38} + ( -1 + 2 i ) q^{40} + 4 q^{43} + ( -6 - 3 i ) q^{45} + 4 q^{46} + ( -2 + 2 i ) q^{47} -i q^{49} + ( -3 - 4 i ) q^{50} + 4 q^{52} + ( -1 - i ) q^{53} + ( -2 + 2 i ) q^{56} + ( 7 - 7 i ) q^{58} + ( -2 - 2 i ) q^{59} + ( 1 + i ) q^{61} + ( -4 - 4 i ) q^{62} + ( -6 - 6 i ) q^{63} + q^{64} + ( -4 + 8 i ) q^{65} + 2 i q^{68} + ( -2 - 6 i ) q^{70} + 12 q^{71} + 3 i q^{72} + ( 5 - 5 i ) q^{73} + ( -1 + 6 i ) q^{74} + ( -2 - 2 i ) q^{76} + ( -12 - 12 i ) q^{79} + ( -1 + 2 i ) q^{80} -9 q^{81} + ( 4 + 4 i ) q^{83} + ( -4 - 2 i ) q^{85} + 4 q^{86} + ( 7 - 7 i ) q^{89} + ( -6 - 3 i ) q^{90} + ( -8 + 8 i ) q^{91} + 4 q^{92} + ( -2 + 2 i ) q^{94} + ( 6 - 2 i ) q^{95} + 12 i q^{97} -i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{2} + 2q^{4} - 2q^{5} - 4q^{7} + 2q^{8} + O(q^{10}) \) \( 2q + 2q^{2} + 2q^{4} - 2q^{5} - 4q^{7} + 2q^{8} - 2q^{10} + 8q^{13} - 4q^{14} + 2q^{16} - 4q^{19} - 2q^{20} + 8q^{23} - 6q^{25} + 8q^{26} - 4q^{28} + 14q^{29} - 8q^{31} + 2q^{32} - 4q^{35} - 2q^{37} - 4q^{38} - 2q^{40} + 8q^{43} - 12q^{45} + 8q^{46} - 4q^{47} - 6q^{50} + 8q^{52} - 2q^{53} - 4q^{56} + 14q^{58} - 4q^{59} + 2q^{61} - 8q^{62} - 12q^{63} + 2q^{64} - 8q^{65} - 4q^{70} + 24q^{71} + 10q^{73} - 2q^{74} - 4q^{76} - 24q^{79} - 2q^{80} - 18q^{81} + 8q^{83} - 8q^{85} + 8q^{86} + 14q^{89} - 12q^{90} - 16q^{91} + 8q^{92} - 4q^{94} + 12q^{95} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
117.1
1.00000i
1.00000i
1.00000 0 1.00000 −1.00000 2.00000i 0 −2.00000 2.00000i 1.00000 3.00000i −1.00000 2.00000i
253.1 1.00000 0 1.00000 −1.00000 + 2.00000i 0 −2.00000 + 2.00000i 1.00000 3.00000i −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.k 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.h.a yes 2
5.c odd 4 1 370.2.g.b 2
37.d odd 4 1 370.2.g.b 2
185.k even 4 1 inner 370.2.h.a yes 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
370.2.g.b 2 5.c odd 4 1
370.2.g.b 2 37.d odd 4 1
370.2.h.a yes 2 1.a even 1 1 trivial
370.2.h.a yes 2 185.k even 4 1 inner

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -1 + T )^{2} \)
$3$ \( T^{2} \)
$5$ \( 5 + 2 T + T^{2} \)
$7$ \( 8 + 4 T + T^{2} \)
$11$ \( T^{2} \)
$13$ \( ( -4 + T )^{2} \)
$17$ \( 4 + T^{2} \)
$19$ \( 8 + 4 T + T^{2} \)
$23$ \( ( -4 + T )^{2} \)
$29$ \( 98 - 14 T + T^{2} \)
$31$ \( 32 + 8 T + T^{2} \)
$37$ \( 37 + 2 T + T^{2} \)
$41$ \( T^{2} \)
$43$ \( ( -4 + T )^{2} \)
$47$ \( 8 + 4 T + T^{2} \)
$53$ \( 2 + 2 T + T^{2} \)
$59$ \( 8 + 4 T + T^{2} \)
$61$ \( 2 - 2 T + T^{2} \)
$67$ \( T^{2} \)
$71$ \( ( -12 + T )^{2} \)
$73$ \( 50 - 10 T + T^{2} \)
$79$ \( 288 + 24 T + T^{2} \)
$83$ \( 32 - 8 T + T^{2} \)
$89$ \( 98 - 14 T + T^{2} \)
$97$ \( 144 + T^{2} \)
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