Properties

Label 1900.1.g.a
Level $1900$
Weight $1$
Character orbit 1900.g
Analytic conductor $0.948$
Analytic rank $0$
Dimension $2$
Projective image $D_{3}$
CM discriminant -19
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 1900 = 2^{2} \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1900.g (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.948223524003\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 76)
Projective image: \(D_{3}\)
Projective field: Galois closure of 3.1.76.1
Artin image: $C_4\times S_3$
Artin field: Galois closure of \(\mathbb{Q}[x]/(x^{12} + \cdots)\)

$q$-expansion

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

\(f(q)\) \(=\) \( q -i q^{7} - q^{9} +O(q^{10})\) \( q -i q^{7} - q^{9} - q^{11} -i q^{17} - q^{19} -2 i q^{23} + i q^{43} -i q^{47} - q^{61} + i q^{63} + i q^{73} + i q^{77} + q^{81} -2 i q^{83} + q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{9} + O(q^{10}) \) \( 2q - 2q^{9} - 2q^{11} - 2q^{19} - 2q^{61} + 2q^{81} + 2q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1900\mathbb{Z}\right)^\times\).

\(n\) \(77\) \(401\) \(951\)
\(\chi(n)\) \(-1\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
949.1
1.00000i
1.00000i
0 0 0 0 0 1.00000i 0 −1.00000 0
949.2 0 0 0 0 0 1.00000i 0 −1.00000 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
19.b odd 2 1 CM by \(\Q(\sqrt{-19}) \)
5.b even 2 1 inner
95.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1900.1.g.a 2
5.b even 2 1 inner 1900.1.g.a 2
5.c odd 4 1 76.1.c.a 1
5.c odd 4 1 1900.1.e.a 1
15.e even 4 1 684.1.h.a 1
19.b odd 2 1 CM 1900.1.g.a 2
20.e even 4 1 304.1.e.a 1
35.f even 4 1 3724.1.e.c 1
35.k even 12 2 3724.1.bc.b 2
35.l odd 12 2 3724.1.bc.c 2
40.i odd 4 1 1216.1.e.a 1
40.k even 4 1 1216.1.e.b 1
60.l odd 4 1 2736.1.o.b 1
95.d odd 2 1 inner 1900.1.g.a 2
95.g even 4 1 76.1.c.a 1
95.g even 4 1 1900.1.e.a 1
95.l even 12 2 1444.1.h.a 2
95.m odd 12 2 1444.1.h.a 2
95.q odd 36 6 1444.1.j.a 6
95.r even 36 6 1444.1.j.a 6
285.j odd 4 1 684.1.h.a 1
380.j odd 4 1 304.1.e.a 1
665.n odd 4 1 3724.1.e.c 1
665.ca odd 12 2 3724.1.bc.b 2
665.ck even 12 2 3724.1.bc.c 2
760.t even 4 1 1216.1.e.a 1
760.y odd 4 1 1216.1.e.b 1
1140.w even 4 1 2736.1.o.b 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
76.1.c.a 1 5.c odd 4 1
76.1.c.a 1 95.g even 4 1
304.1.e.a 1 20.e even 4 1
304.1.e.a 1 380.j odd 4 1
684.1.h.a 1 15.e even 4 1
684.1.h.a 1 285.j odd 4 1
1216.1.e.a 1 40.i odd 4 1
1216.1.e.a 1 760.t even 4 1
1216.1.e.b 1 40.k even 4 1
1216.1.e.b 1 760.y odd 4 1
1444.1.h.a 2 95.l even 12 2
1444.1.h.a 2 95.m odd 12 2
1444.1.j.a 6 95.q odd 36 6
1444.1.j.a 6 95.r even 36 6
1900.1.e.a 1 5.c odd 4 1
1900.1.e.a 1 95.g even 4 1
1900.1.g.a 2 1.a even 1 1 trivial
1900.1.g.a 2 5.b even 2 1 inner
1900.1.g.a 2 19.b odd 2 1 CM
1900.1.g.a 2 95.d odd 2 1 inner
2736.1.o.b 1 60.l odd 4 1
2736.1.o.b 1 1140.w even 4 1
3724.1.e.c 1 35.f even 4 1
3724.1.e.c 1 665.n odd 4 1
3724.1.bc.b 2 35.k even 12 2
3724.1.bc.b 2 665.ca odd 12 2
3724.1.bc.c 2 35.l odd 12 2
3724.1.bc.c 2 665.ck even 12 2

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(1900, [\chi])\).

Hecke characteristic polynomials

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