Properties

Label 1900.2.a.k
Level $1900$
Weight $2$
Character orbit 1900.a
Self dual yes
Analytic conductor $15.172$
Analytic rank $0$
Dimension $6$
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.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(15.1715763840\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.6.56310016.1
Defining polynomial: \( x^{6} - 9x^{4} + 14x^{2} - 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
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 a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{3} q^{3} - \beta_{4} q^{7} + (\beta_{5} - \beta_1 + 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{3} q^{3} - \beta_{4} q^{7} + (\beta_{5} - \beta_1 + 2) q^{9} + ( - \beta_1 + 3) q^{11} + \beta_{2} q^{13} + ( - \beta_{4} + \beta_{3} - \beta_{2}) q^{17} + q^{19} - 2 \beta_{5} q^{21} + ( - \beta_{3} + \beta_{2}) q^{23} + (2 \beta_{4} + 3 \beta_{3} - \beta_{2}) q^{27} + (2 \beta_{5} + 2 \beta_1) q^{29} + (2 \beta_{5} + 2 \beta_1 + 2) q^{31} + (5 \beta_{3} - \beta_{2}) q^{33} + \beta_{3} q^{37} + ( - \beta_{5} + \beta_1 - 1) q^{39} + ( - 2 \beta_{5} + 2 \beta_1) q^{41} + (\beta_{4} - 2 \beta_{3} - 2 \beta_{2}) q^{43} + ( - \beta_{4} - 2 \beta_{3} + 2 \beta_{2}) q^{47} + ( - 3 \beta_{5} - 2 \beta_1 + 1) q^{49} + ( - 2 \beta_1 + 6) q^{51} + (2 \beta_{4} + \beta_{3} + 2 \beta_{2}) q^{53} + \beta_{3} q^{57} + ( - 2 \beta_{5} + 4) q^{59} + ( - \beta_1 + 5) q^{61} + ( - \beta_{4} - 4 \beta_{3}) q^{63} + (4 \beta_{4} + 2 \beta_{3} + \beta_{2}) q^{67} + ( - 2 \beta_{5} + 2 \beta_1 - 6) q^{69} + (2 \beta_{5} + 8) q^{71} + ( - 3 \beta_{4} - \beta_{3} - 3 \beta_{2}) q^{73} + ( - 5 \beta_{4} - \beta_{3} - \beta_{2}) q^{77} + 4 q^{79} + (5 \beta_{5} - \beta_1 + 10) q^{81} + (\beta_{3} - \beta_{2}) q^{83} + (4 \beta_{4} + 2 \beta_{2}) q^{87} + (2 \beta_{5} + 2) q^{89} + 4 q^{91} + (4 \beta_{4} + 2 \beta_{3} + 2 \beta_{2}) q^{93} + (2 \beta_{4} + \beta_{3} + 2 \beta_{2}) q^{97} + (6 \beta_{5} - 3 \beta_1 + 17) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 10 q^{9} + 18 q^{11} + 6 q^{19} + 4 q^{21} - 4 q^{29} + 8 q^{31} - 4 q^{39} + 4 q^{41} + 12 q^{49} + 36 q^{51} + 28 q^{59} + 30 q^{61} - 32 q^{69} + 44 q^{71} + 24 q^{79} + 50 q^{81} + 8 q^{89} + 24 q^{91} + 90 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 9x^{4} + 14x^{2} - 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{5} + 9\nu^{3} - 10\nu ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{5} - 9\nu^{3} + 14\nu ) / 2 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \nu^{5} - 8\nu^{3} + 6\nu \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( \nu^{4} - 8\nu^{2} + 6 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{3} + \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta _1 + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{4} + 2\beta_{3} + 4\beta_{2} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{5} + 8\beta _1 + 18 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 9\beta_{4} + 13\beta_{3} + 29\beta_{2} \) 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
−0.608430
−1.23277
−2.66648
2.66648
1.23277
0.608430
0 −3.28715 0 0 0 1.93210 0 7.80536 0
1.2 0 −1.62236 0 0 0 −4.74397 0 −0.367938 0
1.3 0 −0.750054 0 0 0 −0.872810 0 −2.43742 0
1.4 0 0.750054 0 0 0 0.872810 0 −2.43742 0
1.5 0 1.62236 0 0 0 4.74397 0 −0.367938 0
1.6 0 3.28715 0 0 0 −1.93210 0 7.80536 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.6
Significant digits:
Format:

Atkin-Lehner signs

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

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.a.k 6
4.b odd 2 1 7600.2.a.cj 6
5.b even 2 1 inner 1900.2.a.k 6
5.c odd 4 2 380.2.c.b 6
15.e even 4 2 3420.2.f.c 6
20.d odd 2 1 7600.2.a.cj 6
20.e even 4 2 1520.2.d.i 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
380.2.c.b 6 5.c odd 4 2
1520.2.d.i 6 20.e even 4 2
1900.2.a.k 6 1.a even 1 1 trivial
1900.2.a.k 6 5.b even 2 1 inner
3420.2.f.c 6 15.e even 4 2
7600.2.a.cj 6 4.b odd 2 1
7600.2.a.cj 6 20.d odd 2 1

Hecke kernels

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

\( T_{3}^{6} - 14T_{3}^{4} + 36T_{3}^{2} - 16 \) Copy content Toggle raw display
\( T_{7}^{6} - 27T_{7}^{4} + 104T_{7}^{2} - 64 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( T^{6} - 14 T^{4} + 36 T^{2} - 16 \) Copy content Toggle raw display
$5$ \( T^{6} \) Copy content Toggle raw display
$7$ \( T^{6} - 27 T^{4} + 104 T^{2} + \cdots - 64 \) Copy content Toggle raw display
$11$ \( (T^{3} - 9 T^{2} + 14 T + 28)^{2} \) Copy content Toggle raw display
$13$ \( T^{6} - 26 T^{4} + 108 T^{2} + \cdots - 64 \) Copy content Toggle raw display
$17$ \( T^{6} - 51 T^{4} + 728 T^{2} + \cdots - 3136 \) Copy content Toggle raw display
$19$ \( (T - 1)^{6} \) Copy content Toggle raw display
$23$ \( T^{6} - 44 T^{4} + 448 T^{2} + \cdots - 256 \) Copy content Toggle raw display
$29$ \( (T^{3} + 2 T^{2} - 84 T + 88)^{2} \) Copy content Toggle raw display
$31$ \( (T^{3} - 4 T^{2} - 80 T + 256)^{2} \) Copy content Toggle raw display
$37$ \( T^{6} - 14 T^{4} + 36 T^{2} - 16 \) Copy content Toggle raw display
$41$ \( (T^{3} - 2 T^{2} - 116 T + 488)^{2} \) Copy content Toggle raw display
$43$ \( T^{6} - 227 T^{4} + 12520 T^{2} + \cdots - 3136 \) Copy content Toggle raw display
$47$ \( T^{6} - 243 T^{4} + 14024 T^{2} + \cdots - 118336 \) Copy content Toggle raw display
$53$ \( T^{6} - 114 T^{4} + 3500 T^{2} + \cdots - 23104 \) Copy content Toggle raw display
$59$ \( (T^{3} - 14 T^{2} + 16 T + 128)^{2} \) Copy content Toggle raw display
$61$ \( (T^{3} - 15 T^{2} + 62 T - 44)^{2} \) Copy content Toggle raw display
$67$ \( T^{6} - 378 T^{4} + 35660 T^{2} + \cdots - 222784 \) Copy content Toggle raw display
$71$ \( (T^{3} - 22 T^{2} + 112 T - 32)^{2} \) Copy content Toggle raw display
$73$ \( T^{6} - 251 T^{4} + 16344 T^{2} + \cdots - 118336 \) Copy content Toggle raw display
$79$ \( (T - 4)^{6} \) Copy content Toggle raw display
$83$ \( T^{6} - 44 T^{4} + 448 T^{2} + \cdots - 256 \) Copy content Toggle raw display
$89$ \( (T^{3} - 4 T^{2} - 44 T + 64)^{2} \) Copy content Toggle raw display
$97$ \( T^{6} - 114 T^{4} + 3500 T^{2} + \cdots - 23104 \) Copy content Toggle raw display
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