Properties

Label 3800.2.a.bc
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: \(x^{6} - 2 x^{5} - 10 x^{4} + 16 x^{3} + 15 x^{2} - 14 x - 3\)
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.

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

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

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - \nu - 4 \)
\(\beta_{3}\)\(=\)\((\)\( \nu^{4} - \nu^{3} - 9 \nu^{2} + 6 \nu + 5 \)\()/2\)
\(\beta_{4}\)\(=\)\((\)\( \nu^{5} - 2 \nu^{4} - 9 \nu^{3} + 15 \nu^{2} + 6 \nu - 8 \)\()/2\)
\(\beta_{5}\)\(=\)\((\)\( \nu^{5} - 2 \nu^{4} - 10 \nu^{3} + 15 \nu^{2} + 15 \nu - 7 \)\()/2\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + \beta_{1} + 4\)
\(\nu^{3}\)\(=\)\(-2 \beta_{5} + 2 \beta_{4} + 9 \beta_{1} + 1\)
\(\nu^{4}\)\(=\)\(-2 \beta_{5} + 2 \beta_{4} + 2 \beta_{3} + 9 \beta_{2} + 12 \beta_{1} + 32\)
\(\nu^{5}\)\(=\)\(-22 \beta_{5} + 24 \beta_{4} + 4 \beta_{3} + 3 \beta_{2} + 84 \beta_{1} + 21\)

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
−2.77008
−1.08999
−0.185519
0.848258
1.93590
3.26143
0 −2.77008 0 0 0 −2.31077 0 4.67334 0
1.2 0 −1.08999 0 0 0 4.19727 0 −1.81192 0
1.3 0 −0.185519 0 0 0 −4.45651 0 −2.96558 0
1.4 0 0.848258 0 0 0 1.74484 0 −2.28046 0
1.5 0 1.93590 0 0 0 −1.24708 0 0.747704 0
1.6 0 3.26143 0 0 0 4.07225 0 7.63693 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.bc yes 6
4.b odd 2 1 7600.2.a.ch 6
5.b even 2 1 3800.2.a.ba 6
5.c odd 4 2 3800.2.d.q 12
20.d odd 2 1 7600.2.a.cl 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3800.2.a.ba 6 5.b even 2 1
3800.2.a.bc yes 6 1.a even 1 1 trivial
3800.2.d.q 12 5.c odd 4 2
7600.2.a.ch 6 4.b odd 2 1
7600.2.a.cl 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(3800))\):

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \)
$3$ \( -3 - 14 T + 15 T^{2} + 16 T^{3} - 10 T^{4} - 2 T^{5} + T^{6} \)
$5$ \( T^{6} \)
$7$ \( -383 - 154 T + 223 T^{2} + 48 T^{3} - 30 T^{4} - 2 T^{5} + T^{6} \)
$11$ \( -88 - 208 T + 10 T^{2} + 139 T^{3} - 40 T^{4} - 3 T^{5} + T^{6} \)
$13$ \( 825 - 1621 T + 883 T^{2} - 15 T^{3} - 65 T^{4} + T^{5} + T^{6} \)
$17$ \( 3147 + 1000 T - 989 T^{2} - 342 T^{3} + 22 T^{4} + 14 T^{5} + T^{6} \)
$19$ \( ( 1 + T )^{6} \)
$23$ \( -2487 - 3914 T - 659 T^{2} + 602 T^{3} - 26 T^{4} - 12 T^{5} + T^{6} \)
$29$ \( 21951 - 12285 T + 315 T^{2} + 705 T^{3} - 67 T^{4} - 9 T^{5} + T^{6} \)
$31$ \( 11000 - 13500 T + 3220 T^{2} + 599 T^{3} - 136 T^{4} - 5 T^{5} + T^{6} \)
$37$ \( -7365 + 6089 T + 582 T^{2} - 737 T^{3} - 90 T^{4} + 8 T^{5} + T^{6} \)
$41$ \( -113472 - 13200 T + 8392 T^{2} + 469 T^{3} - 180 T^{4} - 3 T^{5} + T^{6} \)
$43$ \( 6120 + 1776 T - 3142 T^{2} + 695 T^{3} + 13 T^{4} - 15 T^{5} + T^{6} \)
$47$ \( 1791 - 2115 T + 452 T^{2} + 191 T^{3} - 52 T^{4} - 4 T^{5} + T^{6} \)
$53$ \( 9285 - 43846 T + 11490 T^{2} + 1891 T^{3} - 192 T^{4} - 13 T^{5} + T^{6} \)
$59$ \( -32328 - 728 T + 5938 T^{2} + 255 T^{3} - 193 T^{4} + T^{6} \)
$61$ \( 12424 + 17080 T + 6598 T^{2} + 443 T^{3} - 132 T^{4} - 9 T^{5} + T^{6} \)
$67$ \( -136033 + 3443 T + 9859 T^{2} - 185 T^{3} - 199 T^{4} + 3 T^{5} + T^{6} \)
$71$ \( -27576 + 44984 T - 22606 T^{2} + 3491 T^{3} - 75 T^{4} - 19 T^{5} + T^{6} \)
$73$ \( -741033 - 51021 T + 27869 T^{2} + 821 T^{3} - 305 T^{4} - 3 T^{5} + T^{6} \)
$79$ \( -553536 + 208816 T + 15944 T^{2} - 4505 T^{3} - 285 T^{4} + 16 T^{5} + T^{6} \)
$83$ \( -71160 + 41588 T - 5676 T^{2} - 937 T^{3} + 326 T^{4} - 31 T^{5} + T^{6} \)
$89$ \( 3240 - 1836 T - 1030 T^{2} + 727 T^{3} - 67 T^{4} - 14 T^{5} + T^{6} \)
$97$ \( -288792 + 44292 T + 13744 T^{2} - 1473 T^{3} - 208 T^{4} + 11 T^{5} + T^{6} \)
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