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: \( x^{2} - 2 \) Copy content Toggle raw display
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.

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

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 \(S_{2}^{\mathrm{new}}(\Gamma_0(1900))\):

\( T_{3}^{2} - 4T_{3} + 2 \) Copy content Toggle raw display
\( T_{7}^{2} - 4T_{7} - 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

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