Properties

Label 3800.2.a.ba
Level $3800$
Weight $2$
Character orbit 3800.a
Self dual yes
Analytic conductor $30.343$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3800 = 2^{3} \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3800.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(30.3431527681\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
Defining polynomial: \( x^{6} - 2x^{5} - 10x^{4} + 16x^{3} + 15x^{2} - 14x - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q - \beta_1 q^{3} + ( - \beta_{4} - \beta_1) q^{7} + (\beta_{2} + \beta_1 + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{3} + ( - \beta_{4} - \beta_1) q^{7} + (\beta_{2} + \beta_1 + 1) q^{9} + (\beta_{5} - \beta_{4} + \beta_{3} - \beta_1 + 1) q^{11} + ( - \beta_{4} - \beta_{3} + \beta_{2}) q^{13} + ( - \beta_{5} + \beta_{4} - \beta_1 + 3) q^{17} - q^{19} + (\beta_{3} + \beta_{2} + \beta_1 + 3) q^{21} + ( - 2 \beta_{5} + \beta_{4} - \beta_{2} + \beta_1 - 2) q^{23} + (2 \beta_{5} - 2 \beta_{4} - 3 \beta_1 - 1) q^{27} + (\beta_{5} - \beta_{3} - \beta_1 + 1) q^{29} + ( - 2 \beta_{5} - \beta_{3} - \beta_{2} + 2 \beta_1) q^{31} + ( - \beta_{4} + \beta_{2}) q^{33} + (\beta_{5} - 3 \beta_1 + 2) q^{37} + (\beta_{5} + 2 \beta_{3} + \beta_{2} - 3 \beta_1 + 4) q^{39} + (\beta_{5} - \beta_{3} - \beta_{2} + \beta_1 - 1) q^{41} + (2 \beta_{5} - \beta_{4} + \beta_{3} - 2 \beta_1 - 2) q^{43} + (\beta_{4} - \beta_{2} - 1) q^{47} + ( - 3 \beta_{5} + 2 \beta_{4} - \beta_{2} + \beta_1 + 4) q^{49} + (\beta_{5} - \beta_{4} - \beta_{3} + \beta_{2} - 3 \beta_1 + 6) q^{51} + (\beta_{5} + 3 \beta_{3} - 1) q^{53} + \beta_1 q^{57} + (\beta_{5} - 4 \beta_1 + 1) q^{59} + ( - \beta_{3} + \beta_{2} + 4 \beta_1) q^{61} + (3 \beta_{5} - \beta_{4} - \beta_{3} - 6 \beta_1 - 3) q^{63} + ( - 2 \beta_{5} + \beta_{3} - \beta_1 + 2) q^{67} + ( - \beta_{3} - 2 \beta_{2} + 3 \beta_1 - 4) q^{69} + ( - 2 \beta_{5} + \beta_{4} - \beta_{3} - 2 \beta_{2} + 2 \beta_1 + 2) q^{71} + ( - 2 \beta_{5} - 3 \beta_{3} - \beta_{2} + \beta_1 - 2) q^{73} + ( - 4 \beta_{5} + 3 \beta_{4} - 2 \beta_{3} - \beta_{2} + 6) q^{77} + ( - 3 \beta_{5} - 2 \beta_{3} - 2 \beta_{2} - \beta_1 - 3) q^{79} + ( - 2 \beta_{5} + 2 \beta_{4} + 2 \beta_{3} + 3 \beta_1 + 5) q^{81} + (\beta_{5} + \beta_{3} - \beta_{2} - \beta_1 - 5) q^{83} + ( - 2 \beta_{5} + 3 \beta_{4} + \beta_{3} + \beta_{2} + 2 \beta_1 + 5) q^{87} + (\beta_{5} + \beta_{4} + 2 \beta_{3} - \beta_{2} - \beta_1 + 3) q^{89} + (2 \beta_{5} + \beta_{2} - \beta_1 + 7) q^{91} + ( - \beta_{5} + 2 \beta_{4} + \beta_{3} - 3 \beta_{2} + \beta_1 - 7) q^{93} + (4 \beta_{5} - 2 \beta_{4} + \beta_{3} + \beta_{2} - 2 \beta_1 + 2) q^{97} + ( - \beta_{5} + \beta_{4} - 2 \beta_{3} + \beta_{2} - \beta_1 - 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 2 q^{3} - 2 q^{7} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 2 q^{3} - 2 q^{7} + 6 q^{9} + 3 q^{11} + q^{13} + 14 q^{17} - 6 q^{19} + 15 q^{21} - 12 q^{23} - 8 q^{27} + 9 q^{29} + 5 q^{31} - 2 q^{33} + 8 q^{37} + 12 q^{39} + 3 q^{41} - 15 q^{43} - 4 q^{47} + 22 q^{49} + 33 q^{51} - 13 q^{53} + 2 q^{57} + 9 q^{61} - 21 q^{63} + 3 q^{67} - 11 q^{69} + 19 q^{71} - 3 q^{73} + 36 q^{77} - 16 q^{79} + 26 q^{81} - 31 q^{83} + 25 q^{87} + 14 q^{89} + 42 q^{91} - 39 q^{93} + 11 q^{97} - 6 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 2x^{5} - 10x^{4} + 16x^{3} + 15x^{2} - 14x - 3 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{4} - \nu^{3} - 9\nu^{2} + 6\nu + 5 ) / 2 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{5} - 2\nu^{4} - 9\nu^{3} + 15\nu^{2} + 6\nu - 8 ) / 2 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{5} - 2\nu^{4} - 10\nu^{3} + 15\nu^{2} + 15\nu - 7 ) / 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -2\beta_{5} + 2\beta_{4} + 9\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -2\beta_{5} + 2\beta_{4} + 2\beta_{3} + 9\beta_{2} + 12\beta _1 + 32 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -22\beta_{5} + 24\beta_{4} + 4\beta_{3} + 3\beta_{2} + 84\beta _1 + 21 \) 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
3.26143
1.93590
0.848258
−0.185519
−1.08999
−2.77008
0 −3.26143 0 0 0 −4.07225 0 7.63693 0
1.2 0 −1.93590 0 0 0 1.24708 0 0.747704 0
1.3 0 −0.848258 0 0 0 −1.74484 0 −2.28046 0
1.4 0 0.185519 0 0 0 4.45651 0 −2.96558 0
1.5 0 1.08999 0 0 0 −4.19727 0 −1.81192 0
1.6 0 2.77008 0 0 0 2.31077 0 4.67334 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

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 3800.2.a.ba 6
4.b odd 2 1 7600.2.a.cl 6
5.b even 2 1 3800.2.a.bc yes 6
5.c odd 4 2 3800.2.d.q 12
20.d odd 2 1 7600.2.a.ch 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3800.2.a.ba 6 1.a even 1 1 trivial
3800.2.a.bc yes 6 5.b even 2 1
3800.2.d.q 12 5.c odd 4 2
7600.2.a.ch 6 20.d odd 2 1
7600.2.a.cl 6 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(3800))\):

\( T_{3}^{6} + 2T_{3}^{5} - 10T_{3}^{4} - 16T_{3}^{3} + 15T_{3}^{2} + 14T_{3} - 3 \) Copy content Toggle raw display
\( T_{7}^{6} + 2T_{7}^{5} - 30T_{7}^{4} - 48T_{7}^{3} + 223T_{7}^{2} + 154T_{7} - 383 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( T^{6} + 2 T^{5} - 10 T^{4} - 16 T^{3} + \cdots - 3 \) Copy content Toggle raw display
$5$ \( T^{6} \) Copy content Toggle raw display
$7$ \( T^{6} + 2 T^{5} - 30 T^{4} - 48 T^{3} + \cdots - 383 \) Copy content Toggle raw display
$11$ \( T^{6} - 3 T^{5} - 40 T^{4} + 139 T^{3} + \cdots - 88 \) Copy content Toggle raw display
$13$ \( T^{6} - T^{5} - 65 T^{4} + 15 T^{3} + \cdots + 825 \) Copy content Toggle raw display
$17$ \( T^{6} - 14 T^{5} + 22 T^{4} + \cdots + 3147 \) Copy content Toggle raw display
$19$ \( (T + 1)^{6} \) Copy content Toggle raw display
$23$ \( T^{6} + 12 T^{5} - 26 T^{4} + \cdots - 2487 \) Copy content Toggle raw display
$29$ \( T^{6} - 9 T^{5} - 67 T^{4} + \cdots + 21951 \) Copy content Toggle raw display
$31$ \( T^{6} - 5 T^{5} - 136 T^{4} + \cdots + 11000 \) Copy content Toggle raw display
$37$ \( T^{6} - 8 T^{5} - 90 T^{4} + \cdots - 7365 \) Copy content Toggle raw display
$41$ \( T^{6} - 3 T^{5} - 180 T^{4} + \cdots - 113472 \) Copy content Toggle raw display
$43$ \( T^{6} + 15 T^{5} + 13 T^{4} + \cdots + 6120 \) Copy content Toggle raw display
$47$ \( T^{6} + 4 T^{5} - 52 T^{4} + \cdots + 1791 \) Copy content Toggle raw display
$53$ \( T^{6} + 13 T^{5} - 192 T^{4} + \cdots + 9285 \) Copy content Toggle raw display
$59$ \( T^{6} - 193 T^{4} + 255 T^{3} + \cdots - 32328 \) Copy content Toggle raw display
$61$ \( T^{6} - 9 T^{5} - 132 T^{4} + \cdots + 12424 \) Copy content Toggle raw display
$67$ \( T^{6} - 3 T^{5} - 199 T^{4} + \cdots - 136033 \) Copy content Toggle raw display
$71$ \( T^{6} - 19 T^{5} - 75 T^{4} + \cdots - 27576 \) Copy content Toggle raw display
$73$ \( T^{6} + 3 T^{5} - 305 T^{4} + \cdots - 741033 \) Copy content Toggle raw display
$79$ \( T^{6} + 16 T^{5} - 285 T^{4} + \cdots - 553536 \) Copy content Toggle raw display
$83$ \( T^{6} + 31 T^{5} + 326 T^{4} + \cdots - 71160 \) Copy content Toggle raw display
$89$ \( T^{6} - 14 T^{5} - 67 T^{4} + \cdots + 3240 \) Copy content Toggle raw display
$97$ \( T^{6} - 11 T^{5} - 208 T^{4} + \cdots - 288792 \) Copy content Toggle raw display
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