Properties

Label 1900.2.c.a
Level $1900$
Weight $2$
Character orbit 1900.c
Analytic conductor $15.172$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(15.1715763840\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 380)
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.

\(f(q)\) \(=\) \( q + 2 i q^{3} -2 i q^{7} - q^{9} +O(q^{10})\) \( q + 2 i q^{3} -2 i q^{7} - q^{9} + 6 i q^{13} -2 i q^{17} + q^{19} + 4 q^{21} -2 i q^{23} + 4 i q^{27} + 2 q^{29} + 4 q^{31} + 10 i q^{37} -12 q^{39} -10 q^{41} + 6 i q^{43} + 6 i q^{47} + 3 q^{49} + 4 q^{51} + 6 i q^{53} + 2 i q^{57} + 4 q^{59} + 2 q^{61} + 2 i q^{63} + 2 i q^{67} + 4 q^{69} + 12 q^{71} -6 i q^{73} -8 q^{79} -11 q^{81} -2 i q^{83} + 4 i q^{87} -2 q^{89} + 12 q^{91} + 8 i q^{93} + 18 i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{9} + O(q^{10}) \) \( 2 q - 2 q^{9} + 2 q^{19} + 8 q^{21} + 4 q^{29} + 8 q^{31} - 24 q^{39} - 20 q^{41} + 6 q^{49} + 8 q^{51} + 8 q^{59} + 4 q^{61} + 8 q^{69} + 24 q^{71} - 16 q^{79} - 22 q^{81} - 4 q^{89} + 24 q^{91} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1749.1
1.00000i
1.00000i
0 2.00000i 0 0 0 2.00000i 0 −1.00000 0
1749.2 0 2.00000i 0 0 0 2.00000i 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
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 1900.2.c.a 2
5.b even 2 1 inner 1900.2.c.a 2
5.c odd 4 1 380.2.a.b 1
5.c odd 4 1 1900.2.a.a 1
15.e even 4 1 3420.2.a.f 1
20.e even 4 1 1520.2.a.b 1
20.e even 4 1 7600.2.a.r 1
40.i odd 4 1 6080.2.a.f 1
40.k even 4 1 6080.2.a.v 1
95.g even 4 1 7220.2.a.b 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
380.2.a.b 1 5.c odd 4 1
1520.2.a.b 1 20.e even 4 1
1900.2.a.a 1 5.c odd 4 1
1900.2.c.a 2 1.a even 1 1 trivial
1900.2.c.a 2 5.b even 2 1 inner
3420.2.a.f 1 15.e even 4 1
6080.2.a.f 1 40.i odd 4 1
6080.2.a.v 1 40.k even 4 1
7220.2.a.b 1 95.g even 4 1
7600.2.a.r 1 20.e even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1900, [\chi])\):

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( 4 + T^{2} \)
$5$ \( T^{2} \)
$7$ \( 4 + T^{2} \)
$11$ \( T^{2} \)
$13$ \( 36 + T^{2} \)
$17$ \( 4 + T^{2} \)
$19$ \( ( -1 + T )^{2} \)
$23$ \( 4 + T^{2} \)
$29$ \( ( -2 + T )^{2} \)
$31$ \( ( -4 + T )^{2} \)
$37$ \( 100 + T^{2} \)
$41$ \( ( 10 + T )^{2} \)
$43$ \( 36 + T^{2} \)
$47$ \( 36 + T^{2} \)
$53$ \( 36 + T^{2} \)
$59$ \( ( -4 + T )^{2} \)
$61$ \( ( -2 + T )^{2} \)
$67$ \( 4 + T^{2} \)
$71$ \( ( -12 + T )^{2} \)
$73$ \( 36 + T^{2} \)
$79$ \( ( 8 + T )^{2} \)
$83$ \( 4 + T^{2} \)
$89$ \( ( 2 + T )^{2} \)
$97$ \( 324 + T^{2} \)
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