Properties

Label 1900.2.a.e
Level $1900$
Weight $2$
Character orbit 1900.a
Self dual yes
Analytic conductor $15.172$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1900 = 2^{2} \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1900.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(15.1715763840\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{2}) \)
Defining polynomial: \(x^{2} - 2\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 380)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 2 + \beta ) q^{3} + ( 2 - 2 \beta ) q^{7} + ( 3 + 4 \beta ) q^{9} +O(q^{10})\) \( q + ( 2 + \beta ) q^{3} + ( 2 - 2 \beta ) q^{7} + ( 3 + 4 \beta ) q^{9} -2 q^{11} + ( 2 + 3 \beta ) q^{13} + ( 2 - 2 \beta ) q^{17} - q^{19} -2 \beta q^{21} + 6 q^{23} + ( 8 + 8 \beta ) q^{27} + ( 2 - 6 \beta ) q^{29} + ( -4 - 2 \beta ) q^{31} + ( -4 - 2 \beta ) q^{33} + ( 6 - 3 \beta ) q^{37} + ( 10 + 8 \beta ) q^{39} + ( -2 + 4 \beta ) q^{41} + ( -2 - 2 \beta ) q^{43} + ( 2 + 2 \beta ) q^{47} + ( 5 - 8 \beta ) q^{49} -2 \beta q^{51} + ( -2 - 5 \beta ) q^{53} + ( -2 - \beta ) q^{57} + ( 8 + 4 \beta ) q^{59} + ( -8 - 4 \beta ) q^{61} + ( -10 + 2 \beta ) q^{63} + ( 2 + \beta ) q^{67} + ( 12 + 6 \beta ) q^{69} + ( 8 - 2 \beta ) q^{71} + ( -6 + 6 \beta ) q^{73} + ( -4 + 4 \beta ) q^{77} + ( -4 + 4 \beta ) q^{79} + ( 23 + 12 \beta ) q^{81} + ( 2 + 8 \beta ) q^{83} + ( -8 - 10 \beta ) q^{87} + ( 2 - 6 \beta ) q^{89} + ( -8 + 2 \beta ) q^{91} + ( -12 - 8 \beta ) q^{93} + ( 6 + 3 \beta ) q^{97} + ( -6 - 8 \beta ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 4q^{3} + 4q^{7} + 6q^{9} + O(q^{10}) \) \( 2q + 4q^{3} + 4q^{7} + 6q^{9} - 4q^{11} + 4q^{13} + 4q^{17} - 2q^{19} + 12q^{23} + 16q^{27} + 4q^{29} - 8q^{31} - 8q^{33} + 12q^{37} + 20q^{39} - 4q^{41} - 4q^{43} + 4q^{47} + 10q^{49} - 4q^{53} - 4q^{57} + 16q^{59} - 16q^{61} - 20q^{63} + 4q^{67} + 24q^{69} + 16q^{71} - 12q^{73} - 8q^{77} - 8q^{79} + 46q^{81} + 4q^{83} - 16q^{87} + 4q^{89} - 16q^{91} - 24q^{93} + 12q^{97} - 12q^{99} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−1.41421
1.41421
0 0.585786 0 0 0 4.82843 0 −2.65685 0
1.2 0 3.41421 0 0 0 −0.828427 0 8.65685 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(5\) \(1\)
\(19\) \(1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1900.2.a.e 2
4.b odd 2 1 7600.2.a.u 2
5.b even 2 1 380.2.a.c 2
5.c odd 4 2 1900.2.c.d 4
15.d odd 2 1 3420.2.a.g 2
20.d odd 2 1 1520.2.a.o 2
40.e odd 2 1 6080.2.a.y 2
40.f even 2 1 6080.2.a.bl 2
95.d odd 2 1 7220.2.a.m 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
380.2.a.c 2 5.b even 2 1
1520.2.a.o 2 20.d odd 2 1
1900.2.a.e 2 1.a even 1 1 trivial
1900.2.c.d 4 5.c odd 4 2
3420.2.a.g 2 15.d odd 2 1
6080.2.a.y 2 40.e odd 2 1
6080.2.a.bl 2 40.f even 2 1
7220.2.a.m 2 95.d odd 2 1
7600.2.a.u 2 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1900))\):

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( 2 - 4 T + T^{2} \)
$5$ \( T^{2} \)
$7$ \( -4 - 4 T + T^{2} \)
$11$ \( ( 2 + T )^{2} \)
$13$ \( -14 - 4 T + T^{2} \)
$17$ \( -4 - 4 T + T^{2} \)
$19$ \( ( 1 + T )^{2} \)
$23$ \( ( -6 + T )^{2} \)
$29$ \( -68 - 4 T + T^{2} \)
$31$ \( 8 + 8 T + T^{2} \)
$37$ \( 18 - 12 T + T^{2} \)
$41$ \( -28 + 4 T + T^{2} \)
$43$ \( -4 + 4 T + T^{2} \)
$47$ \( -4 - 4 T + T^{2} \)
$53$ \( -46 + 4 T + T^{2} \)
$59$ \( 32 - 16 T + T^{2} \)
$61$ \( 32 + 16 T + T^{2} \)
$67$ \( 2 - 4 T + T^{2} \)
$71$ \( 56 - 16 T + T^{2} \)
$73$ \( -36 + 12 T + T^{2} \)
$79$ \( -16 + 8 T + T^{2} \)
$83$ \( -124 - 4 T + T^{2} \)
$89$ \( -68 - 4 T + T^{2} \)
$97$ \( 18 - 12 T + T^{2} \)
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