Properties

Label 3700.1.j.a
Level $3700$
Weight $1$
Character orbit 3700.j
Analytic conductor $1.847$
Analytic rank $0$
Dimension $2$
Projective image $S_{4}$
CM/RM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 3700 = 2^{2} \cdot 5^{2} \cdot 37 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3700.j (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.84654054674\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 740)
Projective image: \(S_{4}\)
Projective field: Galois closure of 4.0.5065300.1

$q$-expansion

The \(q\)-expansion and trace form are shown below.

\(f(q)\) \(=\) \( q -i q^{3} - q^{7} +O(q^{10})\) \( q -i q^{3} - q^{7} -i q^{11} + i q^{21} -i q^{27} + ( 1 - i ) q^{31} - q^{33} - q^{37} -i q^{41} + ( -1 - i ) q^{43} + q^{47} - q^{53} + ( -1 + i ) q^{61} - q^{71} + i q^{73} + i q^{77} + ( -1 - i ) q^{79} - q^{81} - q^{83} + ( 1 - i ) q^{89} + ( -1 - i ) q^{93} + ( -1 - i ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{7} + O(q^{10}) \) \( 2 q - 2 q^{7} + 2 q^{31} - 2 q^{33} - 2 q^{37} - 2 q^{43} + 2 q^{47} - 2 q^{53} - 2 q^{61} - 2 q^{71} - 2 q^{79} - 2 q^{81} - 2 q^{83} + 2 q^{89} - 2 q^{93} - 2 q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3700\mathbb{Z}\right)^\times\).

\(n\) \(1001\) \(1777\) \(1851\)
\(\chi(n)\) \(-i\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
401.1
1.00000i
1.00000i
0 1.00000i 0 0 0 −1.00000 0 0 0
1301.1 0 1.00000i 0 0 0 −1.00000 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
37.d odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3700.1.j.a 2
5.b even 2 1 3700.1.j.b 2
5.c odd 4 1 740.1.t.a 2
5.c odd 4 1 740.1.t.b yes 2
20.e even 4 1 2960.1.cj.a 2
20.e even 4 1 2960.1.cj.b 2
37.d odd 4 1 inner 3700.1.j.a 2
185.f even 4 1 740.1.t.b yes 2
185.j odd 4 1 3700.1.j.b 2
185.k even 4 1 740.1.t.a 2
740.p odd 4 1 2960.1.cj.a 2
740.s odd 4 1 2960.1.cj.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
740.1.t.a 2 5.c odd 4 1
740.1.t.a 2 185.k even 4 1
740.1.t.b yes 2 5.c odd 4 1
740.1.t.b yes 2 185.f even 4 1
2960.1.cj.a 2 20.e even 4 1
2960.1.cj.a 2 740.p odd 4 1
2960.1.cj.b 2 20.e even 4 1
2960.1.cj.b 2 740.s odd 4 1
3700.1.j.a 2 1.a even 1 1 trivial
3700.1.j.a 2 37.d odd 4 1 inner
3700.1.j.b 2 5.b even 2 1
3700.1.j.b 2 185.j odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(3700, [\chi])\):

\( T_{7} + 1 \)
\( T_{17} \)

Hecke characteristic polynomials

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