Properties

Label 1911.2.a.u
Level $1911$
Weight $2$
Character orbit 1911.a
Self dual yes
Analytic conductor $15.259$
Analytic rank $0$
Dimension $5$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1911 = 3 \cdot 7^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1911.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(15.2594118263\)
Analytic rank: \(0\)
Dimension: \(5\)
Coefficient field: 5.5.375116.1
Defining polynomial: \(x^{5} - x^{4} - 6 x^{3} + 7 x^{2} + 2 x - 2\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 273)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3,\beta_4\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{3} q^{2} + q^{3} + ( 1 - \beta_{4} ) q^{4} + ( 1 - \beta_{2} ) q^{5} + \beta_{3} q^{6} + ( -\beta_{1} + \beta_{3} - \beta_{4} ) q^{8} + q^{9} +O(q^{10})\) \( q + \beta_{3} q^{2} + q^{3} + ( 1 - \beta_{4} ) q^{4} + ( 1 - \beta_{2} ) q^{5} + \beta_{3} q^{6} + ( -\beta_{1} + \beta_{3} - \beta_{4} ) q^{8} + q^{9} + ( -1 - 2 \beta_{1} + \beta_{2} + 2 \beta_{3} - \beta_{4} ) q^{10} + ( -\beta_{3} + \beta_{4} ) q^{11} + ( 1 - \beta_{4} ) q^{12} + q^{13} + ( 1 - \beta_{2} ) q^{15} + ( -\beta_{1} - \beta_{2} + 2 \beta_{3} ) q^{16} + ( 3 + \beta_{1} ) q^{17} + \beta_{3} q^{18} + ( 1 + \beta_{2} + \beta_{3} ) q^{19} + ( 3 + \beta_{1} - \beta_{2} - 2 \beta_{4} ) q^{20} + ( -3 + \beta_{1} - 2 \beta_{3} + 2 \beta_{4} ) q^{22} + ( \beta_{1} + 2 \beta_{4} ) q^{23} + ( -\beta_{1} + \beta_{3} - \beta_{4} ) q^{24} + ( 3 - 2 \beta_{3} - \beta_{4} ) q^{25} + \beta_{3} q^{26} + q^{27} + ( -2 - \beta_{2} + 2 \beta_{3} ) q^{29} + ( -1 - 2 \beta_{1} + \beta_{2} + 2 \beta_{3} - \beta_{4} ) q^{30} + ( 1 + \beta_{1} + \beta_{2} - 2 \beta_{3} - \beta_{4} ) q^{31} + ( 4 - \beta_{3} - \beta_{4} ) q^{32} + ( -\beta_{3} + \beta_{4} ) q^{33} + ( 1 + \beta_{2} + 3 \beta_{3} ) q^{34} + ( 1 - \beta_{4} ) q^{36} + ( 4 + 2 \beta_{1} - 2 \beta_{2} + \beta_{4} ) q^{37} + ( 4 + 2 \beta_{1} - \beta_{2} ) q^{38} + q^{39} + ( 2 + 4 \beta_{3} - \beta_{4} ) q^{40} + ( 1 - \beta_{1} + \beta_{2} + 2 \beta_{3} - \beta_{4} ) q^{41} + ( 2 + \beta_{1} - 2 \beta_{3} - 2 \beta_{4} ) q^{43} + ( -5 + 2 \beta_{1} + \beta_{2} - 5 \beta_{3} + 2 \beta_{4} ) q^{44} + ( 1 - \beta_{2} ) q^{45} + ( 1 + 2 \beta_{1} + \beta_{2} - 4 \beta_{3} + 2 \beta_{4} ) q^{46} + ( -1 - \beta_{1} + \beta_{2} + \beta_{3} + 3 \beta_{4} ) q^{47} + ( -\beta_{1} - \beta_{2} + 2 \beta_{3} ) q^{48} + ( -6 - \beta_{1} + 5 \beta_{3} + \beta_{4} ) q^{50} + ( 3 + \beta_{1} ) q^{51} + ( 1 - \beta_{4} ) q^{52} + ( -3 \beta_{2} - 2 \beta_{3} + 3 \beta_{4} ) q^{53} + \beta_{3} q^{54} + ( -1 + \beta_{1} - \beta_{2} - 2 \beta_{3} + 3 \beta_{4} ) q^{55} + ( 1 + \beta_{2} + \beta_{3} ) q^{57} + ( 5 - 2 \beta_{1} + \beta_{2} - \beta_{3} - 3 \beta_{4} ) q^{58} + ( 4 + 3 \beta_{1} - 2 \beta_{2} - 3 \beta_{3} + 3 \beta_{4} ) q^{59} + ( 3 + \beta_{1} - \beta_{2} - 2 \beta_{4} ) q^{60} + ( 5 + \beta_{1} ) q^{61} + ( -4 + \beta_{1} + 2 \beta_{3} + 2 \beta_{4} ) q^{62} + ( -3 + \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{64} + ( 1 - \beta_{2} ) q^{65} + ( -3 + \beta_{1} - 2 \beta_{3} + 2 \beta_{4} ) q^{66} + ( 4 - 2 \beta_{1} - \beta_{3} - \beta_{4} ) q^{67} + ( 4 - \beta_{2} - 2 \beta_{4} ) q^{68} + ( \beta_{1} + 2 \beta_{4} ) q^{69} + ( -6 + \beta_{1} + 2 \beta_{2} - \beta_{3} - \beta_{4} ) q^{71} + ( -\beta_{1} + \beta_{3} - \beta_{4} ) q^{72} + ( 3 - \beta_{1} + \beta_{2} + 2 \beta_{3} + \beta_{4} ) q^{73} + ( -3 \beta_{1} + 4 \beta_{2} + 4 \beta_{3} - \beta_{4} ) q^{74} + ( 3 - 2 \beta_{3} - \beta_{4} ) q^{75} + ( -1 - 2 \beta_{1} + \beta_{2} + 3 \beta_{3} - \beta_{4} ) q^{76} + \beta_{3} q^{78} + ( 1 + \beta_{1} + 3 \beta_{2} - 2 \beta_{3} + \beta_{4} ) q^{79} + ( 6 - 3 \beta_{1} + 2 \beta_{2} + 4 \beta_{3} - \beta_{4} ) q^{80} + q^{81} + ( 6 + \beta_{1} - 2 \beta_{2} + 2 \beta_{3} - 2 \beta_{4} ) q^{82} + ( 1 - 4 \beta_{1} - \beta_{2} + 2 \beta_{3} - \beta_{4} ) q^{83} + ( 2 - \beta_{1} - 2 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} ) q^{85} + ( -5 - 2 \beta_{1} + \beta_{2} + 6 \beta_{3} ) q^{86} + ( -2 - \beta_{2} + 2 \beta_{3} ) q^{87} + ( -6 + 2 \beta_{1} + \beta_{2} - 6 \beta_{3} + 4 \beta_{4} ) q^{88} + ( 5 - \beta_{1} + \beta_{2} ) q^{89} + ( -1 - 2 \beta_{1} + \beta_{2} + 2 \beta_{3} - \beta_{4} ) q^{90} + ( -9 + 2 \beta_{1} + \beta_{2} - 4 \beta_{3} + 3 \beta_{4} ) q^{92} + ( 1 + \beta_{1} + \beta_{2} - 2 \beta_{3} - \beta_{4} ) q^{93} + ( 3 + 5 \beta_{1} - 2 \beta_{2} - 8 \beta_{3} + 3 \beta_{4} ) q^{94} + ( -7 - 2 \beta_{1} - \beta_{2} + 4 \beta_{3} ) q^{95} + ( 4 - \beta_{3} - \beta_{4} ) q^{96} + ( 2 + 2 \beta_{1} + 2 \beta_{2} - 4 \beta_{3} - 3 \beta_{4} ) q^{97} + ( -\beta_{3} + \beta_{4} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5q + 5q^{3} + 6q^{4} + 3q^{5} + 3q^{8} + 5q^{9} + O(q^{10}) \) \( 5q + 5q^{3} + 6q^{4} + 3q^{5} + 3q^{8} + 5q^{9} + 2q^{10} - q^{11} + 6q^{12} + 5q^{13} + 3q^{15} + 13q^{17} + 7q^{19} + 13q^{20} - 19q^{22} - 4q^{23} + 3q^{24} + 16q^{25} + 5q^{27} - 12q^{29} + 2q^{30} + 6q^{31} + 21q^{32} - q^{33} + 7q^{34} + 6q^{36} + 11q^{37} + 14q^{38} + 5q^{39} + 11q^{40} + 10q^{41} + 10q^{43} - 29q^{44} + 3q^{45} + q^{46} - 4q^{47} - 29q^{50} + 13q^{51} + 6q^{52} - 9q^{53} - 12q^{55} + 7q^{57} + 34q^{58} + 7q^{59} + 13q^{60} + 23q^{61} - 24q^{62} - 13q^{64} + 3q^{65} - 19q^{66} + 25q^{67} + 20q^{68} - 4q^{69} - 27q^{71} + 3q^{72} + 18q^{73} + 15q^{74} + 16q^{75} + 2q^{76} + 8q^{79} + 41q^{80} + 5q^{81} + 26q^{82} + 12q^{83} + 10q^{85} - 19q^{86} - 12q^{87} - 36q^{88} + 29q^{89} + 2q^{90} - 50q^{92} + 6q^{93} - 2q^{94} - 33q^{95} + 21q^{96} + 13q^{97} - q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{5} - x^{4} - 6 x^{3} + 7 x^{2} + 2 x - 2\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu^{4} - 5 \nu^{2} + 2 \nu \)
\(\beta_{2}\)\(=\)\( -\nu^{4} + 6 \nu^{2} - 3 \)
\(\beta_{3}\)\(=\)\( \nu^{4} - 6 \nu^{2} + 2 \nu + 3 \)
\(\beta_{4}\)\(=\)\( -\nu^{4} + \nu^{3} + 6 \nu^{2} - 6 \nu - 2 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{3} + \beta_{2}\)\()/2\)
\(\nu^{2}\)\(=\)\(-\beta_{3} + \beta_{1} + 3\)
\(\nu^{3}\)\(=\)\(\beta_{4} + 3 \beta_{3} + 2 \beta_{2} - 1\)
\(\nu^{4}\)\(=\)\(-6 \beta_{3} - \beta_{2} + 6 \beta_{1} + 15\)

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
−2.44025
1.32173
−0.562376
2.17362
0.507274
−2.14957 1.00000 2.62066 3.73093 −2.14957 0 −1.33415 1.00000 −8.01989
1.2 −1.78646 1.00000 1.19144 −3.42992 −1.78646 0 1.44447 1.00000 6.12741
1.3 0.0776754 1.00000 −1.99397 2.20243 0.0776754 0 −0.310233 1.00000 0.171074
1.4 1.32155 1.00000 −0.253495 −2.02568 1.32155 0 −2.97812 1.00000 −2.67705
1.5 2.53680 1.00000 4.43536 2.52225 2.53680 0 6.17804 1.00000 6.39846
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.5
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(7\) \(-1\)
\(13\) \(-1\)

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 1911.2.a.u 5
3.b odd 2 1 5733.2.a.bp 5
7.b odd 2 1 1911.2.a.t 5
7.d odd 6 2 273.2.i.e 10
21.c even 2 1 5733.2.a.bq 5
21.g even 6 2 819.2.j.g 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
273.2.i.e 10 7.d odd 6 2
819.2.j.g 10 21.g even 6 2
1911.2.a.t 5 7.b odd 2 1
1911.2.a.u 5 1.a even 1 1 trivial
5733.2.a.bp 5 3.b odd 2 1
5733.2.a.bq 5 21.c even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1911))\):

\( T_{2}^{5} - 8 T_{2}^{3} - T_{2}^{2} + 13 T_{2} - 1 \)
\( T_{5}^{5} - 3 T_{5}^{4} - 16 T_{5}^{3} + 47 T_{5}^{2} + 48 T_{5} - 144 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -1 + 13 T - T^{2} - 8 T^{3} + T^{5} \)
$3$ \( ( -1 + T )^{5} \)
$5$ \( -144 + 48 T + 47 T^{2} - 16 T^{3} - 3 T^{4} + T^{5} \)
$7$ \( T^{5} \)
$11$ \( -1 - 12 T + 38 T^{2} - 23 T^{3} + T^{4} + T^{5} \)
$13$ \( ( -1 + T )^{5} \)
$17$ \( 1 + 22 T - 82 T^{2} + 55 T^{3} - 13 T^{4} + T^{5} \)
$19$ \( -19 - 139 T + 106 T^{2} - 6 T^{3} - 7 T^{4} + T^{5} \)
$23$ \( 1108 + 408 T - 143 T^{2} - 47 T^{3} + 4 T^{4} + T^{5} \)
$29$ \( -251 - 537 T - 231 T^{2} + 2 T^{3} + 12 T^{4} + T^{5} \)
$31$ \( -1376 + 120 T + 251 T^{2} - 39 T^{3} - 6 T^{4} + T^{5} \)
$37$ \( 7456 - 5640 T + 1351 T^{2} - 74 T^{3} - 11 T^{4} + T^{5} \)
$41$ \( -224 - 392 T + 279 T^{2} - 13 T^{3} - 10 T^{4} + T^{5} \)
$43$ \( -76 - 688 T + 413 T^{2} - 37 T^{3} - 10 T^{4} + T^{5} \)
$47$ \( -18392 + 7532 T - 178 T^{2} - 171 T^{3} + 4 T^{4} + T^{5} \)
$53$ \( 84029 + 8722 T - 1960 T^{2} - 209 T^{3} + 9 T^{4} + T^{5} \)
$59$ \( -71807 + 11494 T + 1400 T^{2} - 225 T^{3} - 7 T^{4} + T^{5} \)
$61$ \( -1051 + 1506 T - 804 T^{2} + 199 T^{3} - 23 T^{4} + T^{5} \)
$67$ \( 11303 - 4320 T + 28 T^{2} + 177 T^{3} - 25 T^{4} + T^{5} \)
$71$ \( 761 - 1624 T + 206 T^{2} + 205 T^{3} + 27 T^{4} + T^{5} \)
$73$ \( 7468 - 4280 T + 571 T^{2} + 61 T^{3} - 18 T^{4} + T^{5} \)
$79$ \( 18572 + 11488 T + 1059 T^{2} - 229 T^{3} - 8 T^{4} + T^{5} \)
$83$ \( -17248 - 2856 T + 2103 T^{2} - 145 T^{3} - 12 T^{4} + T^{5} \)
$89$ \( -2144 + 2984 T - 1421 T^{2} + 302 T^{3} - 29 T^{4} + T^{5} \)
$97$ \( -14308 - 6804 T + 3039 T^{2} - 186 T^{3} - 13 T^{4} + T^{5} \)
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