Properties

Label 3700.1.t.a
Level 3700
Weight 1
Character orbit 3700.t
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.t (of order \(4\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.84654054674\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 148)
Projective image \(S_{4}\)
Projective field Galois closure of 4.0.202612.1

$q$-expansion

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

\(f(q)\) \(=\) \( q - q^{3} + i q^{7} +O(q^{10})\) \( q - q^{3} + i q^{7} -i q^{11} + ( -1 + i ) q^{17} + ( -1 - i ) q^{19} -i q^{21} + ( 1 - i ) q^{23} + q^{27} + ( 1 - i ) q^{29} + i q^{33} + q^{37} + i q^{41} -i q^{47} + ( 1 - i ) q^{51} + i q^{53} + ( 1 + i ) q^{57} + ( -1 + i ) q^{69} + q^{71} - q^{73} + q^{77} + ( 1 + i ) q^{79} - q^{81} -i q^{83} + ( -1 + i ) q^{87} + ( 1 - i ) q^{89} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{3} + O(q^{10}) \) \( 2q - 2q^{3} - 2q^{17} - 2q^{19} + 2q^{23} + 2q^{27} + 2q^{29} + 2q^{37} + 2q^{51} + 2q^{57} - 2q^{69} + 2q^{71} - 2q^{73} + 2q^{77} + 2q^{79} - 2q^{81} - 2q^{87} + 2q^{89} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
549.1
1.00000i
1.00000i
0 −1.00000 0 0 0 1.00000i 0 0 0
1449.1 0 −1.00000 0 0 0 1.00000i 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
185.j 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.t.a 2
5.b even 2 1 3700.1.t.b 2
5.c odd 4 1 148.1.f.a 2
5.c odd 4 1 3700.1.j.c 2
15.e even 4 1 1332.1.o.a 2
20.e even 4 1 592.1.k.b 2
37.d odd 4 1 3700.1.t.b 2
40.i odd 4 1 2368.1.k.a 2
40.k even 4 1 2368.1.k.b 2
185.f even 4 1 148.1.f.a 2
185.j odd 4 1 inner 3700.1.t.a 2
185.k even 4 1 3700.1.j.c 2
555.u odd 4 1 1332.1.o.a 2
740.p odd 4 1 592.1.k.b 2
1480.bc odd 4 1 2368.1.k.b 2
1480.bk even 4 1 2368.1.k.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
148.1.f.a 2 5.c odd 4 1
148.1.f.a 2 185.f even 4 1
592.1.k.b 2 20.e even 4 1
592.1.k.b 2 740.p odd 4 1
1332.1.o.a 2 15.e even 4 1
1332.1.o.a 2 555.u odd 4 1
2368.1.k.a 2 40.i odd 4 1
2368.1.k.a 2 1480.bk even 4 1
2368.1.k.b 2 40.k even 4 1
2368.1.k.b 2 1480.bc odd 4 1
3700.1.j.c 2 5.c odd 4 1
3700.1.j.c 2 185.k even 4 1
3700.1.t.a 2 1.a even 1 1 trivial
3700.1.t.a 2 185.j odd 4 1 inner
3700.1.t.b 2 5.b even 2 1
3700.1.t.b 2 37.d odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} + 1 \) acting on \(S_{1}^{\mathrm{new}}(3700, [\chi])\).

Hecke characteristic polynomials

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