Properties

Label 2254.2.a.x
Level $2254$
Weight $2$
Character orbit 2254.a
Self dual yes
Analytic conductor $17.998$
Analytic rank $1$
Dimension $4$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 2254 = 2 \cdot 7^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2254.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(17.9982806156\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: 4.4.1957.1
Defining polynomial: \(x^{4} - 4 x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 322)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - 4 x^{2} - x + 1\):

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

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
2.06150
−0.693822
−1.76401
0.396339
1.00000 −2.57641 1.00000 −1.32664 −2.57641 0 1.00000 3.63791 −1.32664
1.2 1.00000 −1.74747 1.00000 −4.26608 −1.74747 0 1.00000 0.0536460 −4.26608
1.3 1.00000 0.197126 1.00000 0.308875 0.197126 0 1.00000 −2.96114 0.308875
1.4 1.00000 1.12676 1.00000 −1.71616 1.12676 0 1.00000 −1.73042 −1.71616
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(7\) \(-1\)
\(23\) \(1\)

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 2254.2.a.x 4
7.b odd 2 1 2254.2.a.z 4
7.d odd 6 2 322.2.e.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
322.2.e.a 8 7.d odd 6 2
2254.2.a.x 4 1.a even 1 1 trivial
2254.2.a.z 4 7.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(2254))\):

\( T_{3}^{4} + 3 T_{3}^{3} - T_{3}^{2} - 5 T_{3} + 1 \)
\( T_{5}^{4} + 7 T_{5}^{3} + 13 T_{5}^{2} + 5 T_{5} - 3 \)
\( T_{11}^{4} - 2 T_{11}^{3} - 12 T_{11}^{2} + 29 T_{11} - 9 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -1 + T )^{4} \)
$3$ \( 1 - 5 T - T^{2} + 3 T^{3} + T^{4} \)
$5$ \( -3 + 5 T + 13 T^{2} + 7 T^{3} + T^{4} \)
$7$ \( T^{4} \)
$11$ \( -9 + 29 T - 12 T^{2} - 2 T^{3} + T^{4} \)
$13$ \( 343 - 42 T^{2} + T^{3} + T^{4} \)
$17$ \( -423 - 266 T - 36 T^{2} + 5 T^{3} + T^{4} \)
$19$ \( 101 - 283 T - 13 T^{2} + 11 T^{3} + T^{4} \)
$23$ \( ( 1 + T )^{4} \)
$29$ \( -21 + 32 T - 11 T^{2} - 2 T^{3} + T^{4} \)
$31$ \( -707 - 461 T - 66 T^{2} + 6 T^{3} + T^{4} \)
$37$ \( -1367 - 663 T - 60 T^{2} + 8 T^{3} + T^{4} \)
$41$ \( 417 - 194 T - 34 T^{2} + 9 T^{3} + T^{4} \)
$43$ \( 1021 + 285 T - 84 T^{2} - 4 T^{3} + T^{4} \)
$47$ \( 687 - 251 T - 35 T^{2} + 11 T^{3} + T^{4} \)
$53$ \( -933 + 607 T - 105 T^{2} - T^{3} + T^{4} \)
$59$ \( -81 - 81 T + 18 T^{2} + 12 T^{3} + T^{4} \)
$61$ \( -12473 - 2696 T - 30 T^{2} + 21 T^{3} + T^{4} \)
$67$ \( 5827 - 440 T - 202 T^{2} + 3 T^{3} + T^{4} \)
$71$ \( -189 + 424 T - 76 T^{2} - 11 T^{3} + T^{4} \)
$73$ \( -1493 + 555 T + 2 T^{2} - 16 T^{3} + T^{4} \)
$79$ \( -53 + 2 T + 102 T^{2} + 21 T^{3} + T^{4} \)
$83$ \( 5913 - 352 T - 155 T^{2} + 4 T^{3} + T^{4} \)
$89$ \( -3999 - 829 T + 137 T^{2} + 27 T^{3} + T^{4} \)
$97$ \( -23 - 194 T - 33 T^{2} + 6 T^{3} + T^{4} \)
show more
show less