Properties

Label 370.2.d.a
Level $370$
Weight $2$
Character orbit 370.d
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.d (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.

\(f(q)\) \(=\) \( q + i q^{2} - q^{4} + i q^{5} -2 q^{7} -i q^{8} -3 q^{9} +O(q^{10})\) \( q + i q^{2} - q^{4} + i q^{5} -2 q^{7} -i q^{8} -3 q^{9} - q^{10} -4 q^{11} + 2 i q^{13} -2 i q^{14} + q^{16} + 6 i q^{17} -3 i q^{18} + 2 i q^{19} -i q^{20} -4 i q^{22} -8 i q^{23} - q^{25} -2 q^{26} + 2 q^{28} -2 i q^{29} + 8 i q^{31} + i q^{32} -6 q^{34} -2 i q^{35} + 3 q^{36} + ( -6 + i ) q^{37} -2 q^{38} + q^{40} -2 q^{41} -4 i q^{43} + 4 q^{44} -3 i q^{45} + 8 q^{46} + 6 q^{47} -3 q^{49} -i q^{50} -2 i q^{52} + 12 q^{53} -4 i q^{55} + 2 i q^{56} + 2 q^{58} + 2 i q^{59} + 14 i q^{61} -8 q^{62} + 6 q^{63} - q^{64} -2 q^{65} + 16 q^{67} -6 i q^{68} + 2 q^{70} -12 q^{71} + 3 i q^{72} + 6 q^{73} + ( -1 - 6 i ) q^{74} -2 i q^{76} + 8 q^{77} + i q^{80} + 9 q^{81} -2 i q^{82} -12 q^{83} -6 q^{85} + 4 q^{86} + 4 i q^{88} + 4 i q^{89} + 3 q^{90} -4 i q^{91} + 8 i q^{92} + 6 i q^{94} -2 q^{95} + 10 i q^{97} -3 i q^{98} + 12 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{4} - 4q^{7} - 6q^{9} + O(q^{10}) \) \( 2q - 2q^{4} - 4q^{7} - 6q^{9} - 2q^{10} - 8q^{11} + 2q^{16} - 2q^{25} - 4q^{26} + 4q^{28} - 12q^{34} + 6q^{36} - 12q^{37} - 4q^{38} + 2q^{40} - 4q^{41} + 8q^{44} + 16q^{46} + 12q^{47} - 6q^{49} + 24q^{53} + 4q^{58} - 16q^{62} + 12q^{63} - 2q^{64} - 4q^{65} + 32q^{67} + 4q^{70} - 24q^{71} + 12q^{73} - 2q^{74} + 16q^{77} + 18q^{81} - 24q^{83} - 12q^{85} + 8q^{86} + 6q^{90} - 4q^{95} + 24q^{99} + O(q^{100}) \)

Character values

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
221.1
1.00000i
1.00000i
1.00000i 0 −1.00000 1.00000i 0 −2.00000 1.00000i −3.00000 −1.00000
221.2 1.00000i 0 −1.00000 1.00000i 0 −2.00000 1.00000i −3.00000 −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.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.d.a 2
3.b odd 2 1 3330.2.h.c 2
4.b odd 2 1 2960.2.p.e 2
5.b even 2 1 1850.2.d.c 2
5.c odd 4 1 1850.2.c.a 2
5.c odd 4 1 1850.2.c.d 2
37.b even 2 1 inner 370.2.d.a 2
111.d odd 2 1 3330.2.h.c 2
148.b odd 2 1 2960.2.p.e 2
185.d even 2 1 1850.2.d.c 2
185.h odd 4 1 1850.2.c.a 2
185.h odd 4 1 1850.2.c.d 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
370.2.d.a 2 1.a even 1 1 trivial
370.2.d.a 2 37.b even 2 1 inner
1850.2.c.a 2 5.c odd 4 1
1850.2.c.a 2 185.h odd 4 1
1850.2.c.d 2 5.c odd 4 1
1850.2.c.d 2 185.h odd 4 1
1850.2.d.c 2 5.b even 2 1
1850.2.d.c 2 185.d even 2 1
2960.2.p.e 2 4.b odd 2 1
2960.2.p.e 2 148.b odd 2 1
3330.2.h.c 2 3.b odd 2 1
3330.2.h.c 2 111.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 \(S_{2}^{\mathrm{new}}(370, [\chi])\):

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T^{2} \)
$3$ \( T^{2} \)
$5$ \( 1 + T^{2} \)
$7$ \( ( 2 + T )^{2} \)
$11$ \( ( 4 + T )^{2} \)
$13$ \( 4 + T^{2} \)
$17$ \( 36 + T^{2} \)
$19$ \( 4 + T^{2} \)
$23$ \( 64 + T^{2} \)
$29$ \( 4 + T^{2} \)
$31$ \( 64 + T^{2} \)
$37$ \( 37 + 12 T + T^{2} \)
$41$ \( ( 2 + T )^{2} \)
$43$ \( 16 + T^{2} \)
$47$ \( ( -6 + T )^{2} \)
$53$ \( ( -12 + T )^{2} \)
$59$ \( 4 + T^{2} \)
$61$ \( 196 + T^{2} \)
$67$ \( ( -16 + T )^{2} \)
$71$ \( ( 12 + T )^{2} \)
$73$ \( ( -6 + T )^{2} \)
$79$ \( T^{2} \)
$83$ \( ( 12 + T )^{2} \)
$89$ \( 16 + T^{2} \)
$97$ \( 100 + T^{2} \)
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