Properties

Label 2541.2.a.bl
Level 2541
Weight 2
Character orbit 2541.a
Self dual yes
Analytic conductor 20.290
Analytic rank 1
Dimension 4
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 2541 = 3 \cdot 7 \cdot 11^{2} \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 2541.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(20.2899871536\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: 4.4.7488.1
Coefficient ring: \(\Z[a_1, a_2]\)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{3} q^{2} + q^{3} + ( 1 - \beta_{2} ) q^{4} + ( -1 - \beta_{3} ) q^{5} + \beta_{3} q^{6} + q^{7} + ( \beta_{1} + \beta_{2} + \beta_{3} ) q^{8} + q^{9} +O(q^{10})\) \( q + \beta_{3} q^{2} + q^{3} + ( 1 - \beta_{2} ) q^{4} + ( -1 - \beta_{3} ) q^{5} + \beta_{3} q^{6} + q^{7} + ( \beta_{1} + \beta_{2} + \beta_{3} ) q^{8} + q^{9} + ( -3 + \beta_{2} - \beta_{3} ) q^{10} + ( 1 - \beta_{2} ) q^{12} + ( -3 + \beta_{1} ) q^{13} + \beta_{3} q^{14} + ( -1 - \beta_{3} ) q^{15} + ( -2 \beta_{1} - 2 \beta_{3} ) q^{16} + ( -1 - 3 \beta_{1} + \beta_{2} - 2 \beta_{3} ) q^{17} + \beta_{3} q^{18} + ( -5 + \beta_{2} - \beta_{3} ) q^{19} + ( -1 - \beta_{1} - 3 \beta_{3} ) q^{20} + q^{21} + ( -2 + 3 \beta_{1} - \beta_{2} ) q^{23} + ( \beta_{1} + \beta_{2} + \beta_{3} ) q^{24} + ( -1 - \beta_{2} + 2 \beta_{3} ) q^{25} + ( -1 - \beta_{1} - 3 \beta_{3} ) q^{26} + q^{27} + ( 1 - \beta_{2} ) q^{28} + ( -\beta_{1} - 3 \beta_{2} + 2 \beta_{3} ) q^{29} + ( -3 + \beta_{2} - \beta_{3} ) q^{30} + ( -\beta_{1} + 3 \beta_{2} - \beta_{3} ) q^{31} + ( -4 - 2 \beta_{3} ) q^{32} + ( -3 + 2 \beta_{1} + \beta_{2} - 3 \beta_{3} ) q^{34} + ( -1 - \beta_{3} ) q^{35} + ( 1 - \beta_{2} ) q^{36} + ( 1 - 2 \beta_{1} + \beta_{2} + 2 \beta_{3} ) q^{37} + ( -3 - \beta_{1} - 7 \beta_{3} ) q^{38} + ( -3 + \beta_{1} ) q^{39} + ( -2 + \beta_{1} + \beta_{2} + \beta_{3} ) q^{40} + ( 1 + \beta_{1} + 3 \beta_{3} ) q^{41} + \beta_{3} q^{42} + ( -4 + \beta_{1} + 3 \beta_{3} ) q^{43} + ( -1 - \beta_{3} ) q^{45} + ( -3 - 2 \beta_{1} + \beta_{2} ) q^{46} + ( -2 + \beta_{1} + 4 \beta_{2} ) q^{47} + ( -2 \beta_{1} - 2 \beta_{3} ) q^{48} + q^{49} + ( 6 + \beta_{1} - \beta_{2} + \beta_{3} ) q^{50} + ( -1 - 3 \beta_{1} + \beta_{2} - 2 \beta_{3} ) q^{51} + ( -2 - \beta_{1} + 3 \beta_{2} - \beta_{3} ) q^{52} + ( -2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} ) q^{53} + \beta_{3} q^{54} + ( \beta_{1} + \beta_{2} + \beta_{3} ) q^{56} + ( -5 + \beta_{2} - \beta_{3} ) q^{57} + ( 7 + 4 \beta_{1} + \beta_{2} + 6 \beta_{3} ) q^{58} + ( -1 + 2 \beta_{1} + 3 \beta_{3} ) q^{59} + ( -1 - \beta_{1} - 3 \beta_{3} ) q^{60} + ( 5 - 4 \beta_{1} - 3 \beta_{2} + \beta_{3} ) q^{61} + ( -2 - 2 \beta_{1} - 2 \beta_{2} - 6 \beta_{3} ) q^{62} + q^{63} + ( -6 + 4 \beta_{1} + 2 \beta_{2} ) q^{64} + ( 4 + 3 \beta_{3} ) q^{65} + ( -1 + 4 \beta_{1} - 2 \beta_{2} ) q^{67} + ( -9 + 3 \beta_{1} - \beta_{3} ) q^{68} + ( -2 + 3 \beta_{1} - \beta_{2} ) q^{69} + ( -3 + \beta_{2} - \beta_{3} ) q^{70} + ( -2 - 2 \beta_{1} + 2 \beta_{2} - 3 \beta_{3} ) q^{71} + ( \beta_{1} + \beta_{2} + \beta_{3} ) q^{72} + ( -9 - \beta_{1} - 2 \beta_{3} ) q^{73} + ( 8 + \beta_{1} - 3 \beta_{2} - \beta_{3} ) q^{74} + ( -1 - \beta_{2} + 2 \beta_{3} ) q^{75} + ( -10 + \beta_{1} + 5 \beta_{2} - \beta_{3} ) q^{76} + ( -1 - \beta_{1} - 3 \beta_{3} ) q^{78} + ( -9 + 6 \beta_{1} + 3 \beta_{2} ) q^{79} + ( 4 - 2 \beta_{2} + 2 \beta_{3} ) q^{80} + q^{81} + ( 8 - \beta_{1} - 3 \beta_{2} + \beta_{3} ) q^{82} + ( -2 + 4 \beta_{1} + \beta_{2} + 3 \beta_{3} ) q^{83} + ( 1 - \beta_{2} ) q^{84} + ( 4 + \beta_{1} - 2 \beta_{2} + 5 \beta_{3} ) q^{85} + ( 8 - \beta_{1} - 3 \beta_{2} - 4 \beta_{3} ) q^{86} + ( -\beta_{1} - 3 \beta_{2} + 2 \beta_{3} ) q^{87} + ( 5 + 4 \beta_{1} + 5 \beta_{3} ) q^{89} + ( -3 + \beta_{2} - \beta_{3} ) q^{90} + ( -3 + \beta_{1} ) q^{91} + ( 6 - 5 \beta_{1} + \beta_{2} - 5 \beta_{3} ) q^{92} + ( -\beta_{1} + 3 \beta_{2} - \beta_{3} ) q^{93} + ( -1 - 5 \beta_{1} - 4 \beta_{2} - 10 \beta_{3} ) q^{94} + ( 8 + \beta_{1} - \beta_{2} + 8 \beta_{3} ) q^{95} + ( -4 - 2 \beta_{3} ) q^{96} + ( -1 - \beta_{1} + 2 \beta_{3} ) q^{97} + \beta_{3} q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 2q^{2} + 4q^{3} + 4q^{4} - 2q^{5} - 2q^{6} + 4q^{7} + 4q^{9} + O(q^{10}) \) \( 4q - 2q^{2} + 4q^{3} + 4q^{4} - 2q^{5} - 2q^{6} + 4q^{7} + 4q^{9} - 10q^{10} + 4q^{12} - 10q^{13} - 2q^{14} - 2q^{15} - 6q^{17} - 2q^{18} - 18q^{19} + 4q^{21} - 2q^{23} - 8q^{25} + 4q^{27} + 4q^{28} - 6q^{29} - 10q^{30} - 12q^{32} - 2q^{34} - 2q^{35} + 4q^{36} - 4q^{37} - 10q^{39} - 8q^{40} - 2q^{42} - 20q^{43} - 2q^{45} - 16q^{46} - 6q^{47} + 4q^{49} + 24q^{50} - 6q^{51} - 8q^{52} - 2q^{54} - 18q^{57} + 24q^{58} - 6q^{59} + 10q^{61} + 4q^{63} - 16q^{64} + 10q^{65} + 4q^{67} - 28q^{68} - 2q^{69} - 10q^{70} - 6q^{71} - 34q^{73} + 36q^{74} - 8q^{75} - 36q^{76} - 24q^{79} + 12q^{80} + 4q^{81} + 28q^{82} - 6q^{83} + 4q^{84} + 8q^{85} + 38q^{86} - 6q^{87} + 18q^{89} - 10q^{90} - 10q^{91} + 24q^{92} + 6q^{94} + 18q^{95} - 12q^{96} - 10q^{97} - 2q^{98} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - 2 x^{3} - 4 x^{2} + 2 x + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 2 \nu - 2 \)
\(\beta_{3}\)\(=\)\( \nu^{3} - 2 \nu^{2} - 4 \nu + 1 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 2 \beta_{1} + 2\)
\(\nu^{3}\)\(=\)\(\beta_{3} + 2 \beta_{2} + 8 \beta_{1} + 3\)

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
0.698857
3.05896
−1.43091
−0.326909
−2.43091 1.00000 3.90931 1.43091 −2.43091 1.00000 −4.64136 1.00000 −3.47841
1.2 −1.32691 1.00000 −0.239314 0.326909 −1.32691 1.00000 2.97136 1.00000 −0.433778
1.3 −0.301143 1.00000 −1.90931 −0.698857 −0.301143 1.00000 1.17726 1.00000 0.210456
1.4 2.05896 1.00000 2.23931 −3.05896 2.05896 1.00000 0.492737 1.00000 −6.29827
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

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 2541.2.a.bl 4
3.b odd 2 1 7623.2.a.cn 4
11.b odd 2 1 2541.2.a.bp yes 4
33.d even 2 1 7623.2.a.cg 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2541.2.a.bl 4 1.a even 1 1 trivial
2541.2.a.bp yes 4 11.b odd 2 1
7623.2.a.cg 4 33.d even 2 1
7623.2.a.cn 4 3.b odd 2 1

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(7\) \(-1\)
\(11\) \(1\)

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2541))\):

\( T_{2}^{4} + 2 T_{2}^{3} - 4 T_{2}^{2} - 8 T_{2} - 2 \)
\( T_{5}^{4} + 2 T_{5}^{3} - 4 T_{5}^{2} - 2 T_{5} + 1 \)
\( T_{13}^{4} + 10 T_{13}^{3} + 32 T_{13}^{2} + 32 T_{13} - 2 \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 + 2 T + 4 T^{2} + 4 T^{3} + 6 T^{4} + 8 T^{5} + 16 T^{6} + 16 T^{7} + 16 T^{8} \)
$3$ \( ( 1 - T )^{4} \)
$5$ \( 1 + 2 T + 16 T^{2} + 28 T^{3} + 111 T^{4} + 140 T^{5} + 400 T^{6} + 250 T^{7} + 625 T^{8} \)
$7$ \( ( 1 - T )^{4} \)
$11$ \( \)
$13$ \( 1 + 10 T + 84 T^{2} + 422 T^{3} + 1844 T^{4} + 5486 T^{5} + 14196 T^{6} + 21970 T^{7} + 28561 T^{8} \)
$17$ \( 1 + 6 T + 28 T^{2} + 24 T^{3} + 111 T^{4} + 408 T^{5} + 8092 T^{6} + 29478 T^{7} + 83521 T^{8} \)
$19$ \( 1 + 18 T + 184 T^{2} + 1278 T^{3} + 6468 T^{4} + 24282 T^{5} + 66424 T^{6} + 123462 T^{7} + 130321 T^{8} \)
$23$ \( 1 + 2 T + 28 T^{2} + 190 T^{3} + 516 T^{4} + 4370 T^{5} + 14812 T^{6} + 24334 T^{7} + 279841 T^{8} \)
$29$ \( 1 + 6 T + 20 T^{2} + 126 T^{3} + 1404 T^{4} + 3654 T^{5} + 16820 T^{6} + 146334 T^{7} + 707281 T^{8} \)
$31$ \( 1 + 28 T^{2} - 216 T^{3} + 606 T^{4} - 6696 T^{5} + 26908 T^{6} + 923521 T^{8} \)
$37$ \( 1 + 4 T + 72 T^{2} + 284 T^{3} + 4082 T^{4} + 10508 T^{5} + 98568 T^{6} + 202612 T^{7} + 1874161 T^{8} \)
$41$ \( 1 + 124 T^{2} - 48 T^{3} + 6810 T^{4} - 1968 T^{5} + 208444 T^{6} + 2825761 T^{8} \)
$43$ \( 1 + 20 T + 282 T^{2} + 2632 T^{3} + 19943 T^{4} + 113176 T^{5} + 521418 T^{6} + 1590140 T^{7} + 3418801 T^{8} \)
$47$ \( 1 + 6 T + 44 T^{2} + 504 T^{3} + 4371 T^{4} + 23688 T^{5} + 97196 T^{6} + 622938 T^{7} + 4879681 T^{8} \)
$53$ \( 1 + 148 T^{2} - 192 T^{3} + 9942 T^{4} - 10176 T^{5} + 415732 T^{6} + 7890481 T^{8} \)
$59$ \( 1 + 6 T + 208 T^{2} + 936 T^{3} + 17751 T^{4} + 55224 T^{5} + 724048 T^{6} + 1232274 T^{7} + 12117361 T^{8} \)
$61$ \( 1 - 10 T + 108 T^{2} - 314 T^{3} + 3596 T^{4} - 19154 T^{5} + 401868 T^{6} - 2269810 T^{7} + 13845841 T^{8} \)
$67$ \( 1 - 4 T + 130 T^{2} + 56 T^{3} + 8299 T^{4} + 3752 T^{5} + 583570 T^{6} - 1203052 T^{7} + 20151121 T^{8} \)
$71$ \( 1 + 6 T + 220 T^{2} + 1014 T^{3} + 20916 T^{4} + 71994 T^{5} + 1109020 T^{6} + 2147466 T^{7} + 25411681 T^{8} \)
$73$ \( 1 + 34 T + 708 T^{2} + 9614 T^{3} + 96764 T^{4} + 701822 T^{5} + 3772932 T^{6} + 13226578 T^{7} + 28398241 T^{8} \)
$79$ \( 1 + 24 T + 280 T^{2} + 1584 T^{3} + 8754 T^{4} + 125136 T^{5} + 1747480 T^{6} + 11832936 T^{7} + 38950081 T^{8} \)
$83$ \( 1 + 6 T + 260 T^{2} + 1188 T^{3} + 30039 T^{4} + 98604 T^{5} + 1791140 T^{6} + 3430722 T^{7} + 47458321 T^{8} \)
$89$ \( 1 - 18 T + 352 T^{2} - 4068 T^{3} + 48255 T^{4} - 362052 T^{5} + 2788192 T^{6} - 12689442 T^{7} + 62742241 T^{8} \)
$97$ \( 1 + 10 T + 388 T^{2} + 2758 T^{3} + 56428 T^{4} + 267526 T^{5} + 3650692 T^{6} + 9126730 T^{7} + 88529281 T^{8} \)
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