Properties

Label 370.2.b.a
Level $370$
Weight $2$
Character orbit 370.b
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.b (of order \(2\), degree \(1\), 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: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + i q^{2} - q^{4} + ( -2 - i ) q^{5} -2 i q^{7} -i q^{8} + 3 q^{9} +O(q^{10})\) \( q + i q^{2} - q^{4} + ( -2 - i ) q^{5} -2 i q^{7} -i q^{8} + 3 q^{9} + ( 1 - 2 i ) q^{10} -2 i q^{13} + 2 q^{14} + q^{16} -6 i q^{17} + 3 i q^{18} + 6 q^{19} + ( 2 + i ) q^{20} -4 i q^{23} + ( 3 + 4 i ) q^{25} + 2 q^{26} + 2 i q^{28} -4 q^{31} + i q^{32} + 6 q^{34} + ( -2 + 4 i ) q^{35} -3 q^{36} -i q^{37} + 6 i q^{38} + ( -1 + 2 i ) q^{40} -10 q^{41} + 4 i q^{43} + ( -6 - 3 i ) q^{45} + 4 q^{46} -2 i q^{47} + 3 q^{49} + ( -4 + 3 i ) q^{50} + 2 i q^{52} -2 i q^{53} -2 q^{56} + 6 q^{59} -4 i q^{62} -6 i q^{63} - q^{64} + ( -2 + 4 i ) q^{65} + 8 i q^{67} + 6 i q^{68} + ( -4 - 2 i ) q^{70} -3 i q^{72} -8 i q^{73} + q^{74} -6 q^{76} -4 q^{79} + ( -2 - i ) q^{80} + 9 q^{81} -10 i q^{82} + 12 i q^{83} + ( -6 + 12 i ) q^{85} -4 q^{86} -6 q^{89} + ( 3 - 6 i ) q^{90} -4 q^{91} + 4 i q^{92} + 2 q^{94} + ( -12 - 6 i ) q^{95} -10 i q^{97} + 3 i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{4} - 4q^{5} + 6q^{9} + O(q^{10}) \) \( 2q - 2q^{4} - 4q^{5} + 6q^{9} + 2q^{10} + 4q^{14} + 2q^{16} + 12q^{19} + 4q^{20} + 6q^{25} + 4q^{26} - 8q^{31} + 12q^{34} - 4q^{35} - 6q^{36} - 2q^{40} - 20q^{41} - 12q^{45} + 8q^{46} + 6q^{49} - 8q^{50} - 4q^{56} + 12q^{59} - 2q^{64} - 4q^{65} - 8q^{70} + 2q^{74} - 12q^{76} - 8q^{79} - 4q^{80} + 18q^{81} - 12q^{85} - 8q^{86} - 12q^{89} + 6q^{90} - 8q^{91} + 4q^{94} - 24q^{95} + O(q^{100}) \)

Character values

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

\(n\) \(261\) \(297\)
\(\chi(n)\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
149.1
1.00000i
1.00000i
1.00000i 0 −1.00000 −2.00000 + 1.00000i 0 2.00000i 1.00000i 3.00000 1.00000 + 2.00000i
149.2 1.00000i 0 −1.00000 −2.00000 1.00000i 0 2.00000i 1.00000i 3.00000 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
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 370.2.b.a 2
3.b odd 2 1 3330.2.d.f 2
5.b even 2 1 inner 370.2.b.a 2
5.c odd 4 1 1850.2.a.e 1
5.c odd 4 1 1850.2.a.j 1
15.d odd 2 1 3330.2.d.f 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
370.2.b.a 2 1.a even 1 1 trivial
370.2.b.a 2 5.b even 2 1 inner
1850.2.a.e 1 5.c odd 4 1
1850.2.a.j 1 5.c odd 4 1
3330.2.d.f 2 3.b odd 2 1
3330.2.d.f 2 15.d odd 2 1

Hecke kernels

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

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T^{2} \)
$3$ \( T^{2} \)
$5$ \( 5 + 4 T + T^{2} \)
$7$ \( 4 + T^{2} \)
$11$ \( T^{2} \)
$13$ \( 4 + T^{2} \)
$17$ \( 36 + T^{2} \)
$19$ \( ( -6 + T )^{2} \)
$23$ \( 16 + T^{2} \)
$29$ \( T^{2} \)
$31$ \( ( 4 + T )^{2} \)
$37$ \( 1 + T^{2} \)
$41$ \( ( 10 + T )^{2} \)
$43$ \( 16 + T^{2} \)
$47$ \( 4 + T^{2} \)
$53$ \( 4 + T^{2} \)
$59$ \( ( -6 + T )^{2} \)
$61$ \( T^{2} \)
$67$ \( 64 + T^{2} \)
$71$ \( T^{2} \)
$73$ \( 64 + T^{2} \)
$79$ \( ( 4 + T )^{2} \)
$83$ \( 144 + T^{2} \)
$89$ \( ( 6 + T )^{2} \)
$97$ \( 100 + T^{2} \)
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