Properties

Label 2541.2.a.bn
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

Related objects

Downloads

Learn more about

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: 4.4.725.1
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 231)
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.

\(f(q)\) \(=\) \( q + ( 1 - \beta_{1} - \beta_{2} ) q^{2} - q^{3} + ( \beta_{1} + \beta_{3} ) q^{4} -2 \beta_{2} q^{5} + ( -1 + \beta_{1} + \beta_{2} ) q^{6} - q^{7} + ( -2 - \beta_{1} ) q^{8} + q^{9} +O(q^{10})\) \( q + ( 1 - \beta_{1} - \beta_{2} ) q^{2} - q^{3} + ( \beta_{1} + \beta_{3} ) q^{4} -2 \beta_{2} q^{5} + ( -1 + \beta_{1} + \beta_{2} ) q^{6} - q^{7} + ( -2 - \beta_{1} ) q^{8} + q^{9} + ( 2 + 2 \beta_{1} ) q^{10} + ( -\beta_{1} - \beta_{3} ) q^{12} + ( 2 \beta_{1} + 2 \beta_{2} ) q^{13} + ( -1 + \beta_{1} + \beta_{2} ) q^{14} + 2 \beta_{2} q^{15} + ( -2 + \beta_{1} + 3 \beta_{2} - \beta_{3} ) q^{16} + ( 2 - 4 \beta_{1} + 2 \beta_{3} ) q^{17} + ( 1 - \beta_{1} - \beta_{2} ) q^{18} + ( -2 + 2 \beta_{1} - 2 \beta_{3} ) q^{19} + ( 2 - 4 \beta_{1} - 2 \beta_{3} ) q^{20} + q^{21} + ( -3 - 4 \beta_{1} + 3 \beta_{3} ) q^{23} + ( 2 + \beta_{1} ) q^{24} + ( 3 + 4 \beta_{2} - 4 \beta_{3} ) q^{25} + ( -2 - 4 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} ) q^{26} - q^{27} + ( -\beta_{1} - \beta_{3} ) q^{28} + ( 2 + 2 \beta_{1} - 4 \beta_{2} - \beta_{3} ) q^{29} + ( -2 - 2 \beta_{1} ) q^{30} + ( -4 - 2 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} ) q^{31} + ( -1 + 2 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} ) q^{32} + ( 2 - 2 \beta_{1} + 6 \beta_{3} ) q^{34} + 2 \beta_{2} q^{35} + ( \beta_{1} + \beta_{3} ) q^{36} + ( -3 + 4 \beta_{1} + 4 \beta_{2} - \beta_{3} ) q^{37} + ( -2 + 4 \beta_{1} + 2 \beta_{2} - 4 \beta_{3} ) q^{38} + ( -2 \beta_{1} - 2 \beta_{2} ) q^{39} + ( -2 + 2 \beta_{1} + 4 \beta_{2} + 2 \beta_{3} ) q^{40} + ( -2 - 2 \beta_{1} + 4 \beta_{2} + 6 \beta_{3} ) q^{41} + ( 1 - \beta_{1} - \beta_{2} ) q^{42} + ( 6 \beta_{1} - 5 \beta_{3} ) q^{43} -2 \beta_{2} q^{45} + ( -3 + \beta_{1} + 4 \beta_{2} + 7 \beta_{3} ) q^{46} + ( 2 + 4 \beta_{1} + 2 \beta_{2} - 6 \beta_{3} ) q^{47} + ( 2 - \beta_{1} - 3 \beta_{2} + \beta_{3} ) q^{48} + q^{49} + ( -1 + \beta_{1} + \beta_{2} - 4 \beta_{3} ) q^{50} + ( -2 + 4 \beta_{1} - 2 \beta_{3} ) q^{51} + ( 8 \beta_{1} + 4 \beta_{2} + 2 \beta_{3} ) q^{52} + ( -2 - 2 \beta_{1} + 5 \beta_{3} ) q^{53} + ( -1 + \beta_{1} + \beta_{2} ) q^{54} + ( 2 + \beta_{1} ) q^{56} + ( 2 - 2 \beta_{1} + 2 \beta_{3} ) q^{57} + ( 6 + 2 \beta_{1} - 3 \beta_{2} - 3 \beta_{3} ) q^{58} + ( -6 + 2 \beta_{2} + 2 \beta_{3} ) q^{59} + ( -2 + 4 \beta_{1} + 2 \beta_{3} ) q^{60} + ( 2 + 2 \beta_{1} + 4 \beta_{2} - 2 \beta_{3} ) q^{61} + ( -6 + 8 \beta_{1} + 8 \beta_{2} ) q^{62} - q^{63} + ( 1 - \beta_{1} - 5 \beta_{2} - 2 \beta_{3} ) q^{64} + ( -4 - 4 \beta_{1} - 4 \beta_{2} ) q^{65} + ( -8 - 2 \beta_{1} + 3 \beta_{3} ) q^{67} + ( -2 - 4 \beta_{1} - 6 \beta_{2} + 4 \beta_{3} ) q^{68} + ( 3 + 4 \beta_{1} - 3 \beta_{3} ) q^{69} + ( -2 - 2 \beta_{1} ) q^{70} + ( -6 + 4 \beta_{1} + 4 \beta_{2} - 3 \beta_{3} ) q^{71} + ( -2 - \beta_{1} ) q^{72} + ( -2 + 2 \beta_{2} ) q^{73} + ( -7 - 3 \beta_{1} - 5 \beta_{3} ) q^{74} + ( -3 - 4 \beta_{2} + 4 \beta_{3} ) q^{75} + ( 2 \beta_{2} - 4 \beta_{3} ) q^{76} + ( 2 + 4 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{78} + ( -4 - 2 \beta_{1} - \beta_{3} ) q^{79} + ( -10 - 2 \beta_{2} + 4 \beta_{3} ) q^{80} + q^{81} + ( -6 - 12 \beta_{1} - 2 \beta_{2} + 8 \beta_{3} ) q^{82} + ( 2 + 6 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} ) q^{83} + ( \beta_{1} + \beta_{3} ) q^{84} + ( -8 + 4 \beta_{1} - 4 \beta_{2} + 8 \beta_{3} ) q^{85} + ( 4 \beta_{1} - \beta_{2} - 11 \beta_{3} ) q^{86} + ( -2 - 2 \beta_{1} + 4 \beta_{2} + \beta_{3} ) q^{87} + ( -2 - 4 \beta_{1} - 6 \beta_{2} + 4 \beta_{3} ) q^{89} + ( 2 + 2 \beta_{1} ) q^{90} + ( -2 \beta_{1} - 2 \beta_{2} ) q^{91} + ( -1 - 8 \beta_{1} - 5 \beta_{2} ) q^{92} + ( 4 + 2 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} ) q^{93} + ( 4 \beta_{1} - 10 \beta_{3} ) q^{94} + ( 4 + 4 \beta_{2} - 4 \beta_{3} ) q^{95} + ( 1 - 2 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} ) q^{96} + ( 8 + 4 \beta_{1} + 4 \beta_{2} - 6 \beta_{3} ) q^{97} + ( 1 - \beta_{1} - \beta_{2} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + q^{2} - 4q^{3} + 3q^{4} - 4q^{5} - q^{6} - 4q^{7} - 9q^{8} + 4q^{9} + O(q^{10}) \) \( 4q + q^{2} - 4q^{3} + 3q^{4} - 4q^{5} - q^{6} - 4q^{7} - 9q^{8} + 4q^{9} + 10q^{10} - 3q^{12} + 6q^{13} - q^{14} + 4q^{15} - 3q^{16} + 8q^{17} + q^{18} - 10q^{19} + 4q^{21} - 10q^{23} + 9q^{24} + 12q^{25} - 20q^{26} - 4q^{27} - 3q^{28} - 10q^{30} - 18q^{31} - 2q^{32} + 18q^{34} + 4q^{35} + 3q^{36} - 2q^{37} - 8q^{38} - 6q^{39} + 6q^{40} + 10q^{41} + q^{42} - 4q^{43} - 4q^{45} + 11q^{46} + 4q^{47} + 3q^{48} + 4q^{49} - 9q^{50} - 8q^{51} + 20q^{52} - q^{54} + 9q^{56} + 10q^{57} + 14q^{58} - 16q^{59} + 14q^{61} - 4q^{63} - 11q^{64} - 28q^{65} - 28q^{67} - 16q^{68} + 10q^{69} - 10q^{70} - 18q^{71} - 9q^{72} - 4q^{73} - 41q^{74} - 12q^{75} - 4q^{76} + 20q^{78} - 20q^{79} - 36q^{80} + 4q^{81} - 24q^{82} + 6q^{83} + 3q^{84} - 20q^{85} - 20q^{86} - 16q^{89} + 10q^{90} - 6q^{91} - 22q^{92} + 18q^{93} - 16q^{94} + 16q^{95} + 2q^{96} + 32q^{97} + q^{98} + O(q^{100}) \)

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

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

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
2.09529
−1.35567
0.737640
−0.477260
−2.39026 −1.00000 3.71333 −2.58993 2.39026 −1.00000 −4.09529 1.00000 6.19059
1.2 0.162147 −1.00000 −1.97371 −4.38705 −0.162147 −1.00000 −0.644326 1.00000 −0.711349
1.3 1.45589 −1.00000 0.119606 2.38705 −1.45589 −1.00000 −2.73764 1.00000 3.47528
1.4 1.77222 −1.00000 1.14077 0.589926 −1.77222 −1.00000 −1.52274 1.00000 1.04548
\(n\): 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.bn 4
3.b odd 2 1 7623.2.a.ci 4
11.b odd 2 1 2541.2.a.bm 4
11.c even 5 2 231.2.j.f 8
33.d even 2 1 7623.2.a.cl 4
33.h odd 10 2 693.2.m.f 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
231.2.j.f 8 11.c even 5 2
693.2.m.f 8 33.h odd 10 2
2541.2.a.bm 4 11.b odd 2 1
2541.2.a.bn 4 1.a even 1 1 trivial
7623.2.a.ci 4 3.b odd 2 1
7623.2.a.cl 4 33.d even 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(2541))\):

\( T_{2}^{4} - T_{2}^{3} - 5 T_{2}^{2} + 7 T_{2} - 1 \)
\( T_{5}^{4} + 4 T_{5}^{3} - 8 T_{5}^{2} - 24 T_{5} + 16 \)
\( T_{13}^{4} - 6 T_{13}^{3} - 8 T_{13}^{2} + 16 T_{13} + 16 \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 - T + 3 T^{2} + T^{3} + 3 T^{4} + 2 T^{5} + 12 T^{6} - 8 T^{7} + 16 T^{8} \)
$3$ \( ( 1 + T )^{4} \)
$5$ \( 1 + 4 T + 12 T^{2} + 36 T^{3} + 86 T^{4} + 180 T^{5} + 300 T^{6} + 500 T^{7} + 625 T^{8} \)
$7$ \( ( 1 + T )^{4} \)
$11$ \( \)
$13$ \( 1 - 6 T + 44 T^{2} - 218 T^{3} + 822 T^{4} - 2834 T^{5} + 7436 T^{6} - 13182 T^{7} + 28561 T^{8} \)
$17$ \( 1 - 8 T + 48 T^{2} - 264 T^{3} + 1358 T^{4} - 4488 T^{5} + 13872 T^{6} - 39304 T^{7} + 83521 T^{8} \)
$19$ \( 1 + 10 T + 100 T^{2} + 570 T^{3} + 3062 T^{4} + 10830 T^{5} + 36100 T^{6} + 68590 T^{7} + 130321 T^{8} \)
$23$ \( 1 + 10 T + 83 T^{2} + 500 T^{3} + 2869 T^{4} + 11500 T^{5} + 43907 T^{6} + 121670 T^{7} + 279841 T^{8} \)
$29$ \( 1 + 37 T^{2} + 493 T^{4} + 31117 T^{6} + 707281 T^{8} \)
$31$ \( 1 + 18 T + 192 T^{2} + 1362 T^{3} + 8238 T^{4} + 42222 T^{5} + 184512 T^{6} + 536238 T^{7} + 923521 T^{8} \)
$37$ \( 1 + 2 T + 71 T^{2} + 4 T^{3} + 2797 T^{4} + 148 T^{5} + 97199 T^{6} + 101306 T^{7} + 1874161 T^{8} \)
$41$ \( 1 - 10 T + 60 T^{2} + 290 T^{3} - 2938 T^{4} + 11890 T^{5} + 100860 T^{6} - 689210 T^{7} + 2825761 T^{8} \)
$43$ \( 1 + 4 T + 69 T^{2} + 392 T^{3} + 4097 T^{4} + 16856 T^{5} + 127581 T^{6} + 318028 T^{7} + 3418801 T^{8} \)
$47$ \( 1 - 4 T + 108 T^{2} + 4 T^{3} + 4758 T^{4} + 188 T^{5} + 238572 T^{6} - 415292 T^{7} + 4879681 T^{8} \)
$53$ \( 1 + 161 T^{2} - 20 T^{3} + 11657 T^{4} - 1060 T^{5} + 452249 T^{6} + 7890481 T^{8} \)
$59$ \( 1 + 16 T + 308 T^{2} + 2936 T^{3} + 29398 T^{4} + 173224 T^{5} + 1072148 T^{6} + 3286064 T^{7} + 12117361 T^{8} \)
$61$ \( 1 - 14 T + 260 T^{2} - 2306 T^{3} + 24294 T^{4} - 140666 T^{5} + 967460 T^{6} - 3177734 T^{7} + 13845841 T^{8} \)
$67$ \( 1 + 28 T + 541 T^{2} + 6696 T^{3} + 64817 T^{4} + 448632 T^{5} + 2428549 T^{6} + 8421364 T^{7} + 20151121 T^{8} \)
$71$ \( 1 + 18 T + 327 T^{2} + 3632 T^{3} + 36173 T^{4} + 257872 T^{5} + 1648407 T^{6} + 6442398 T^{7} + 25411681 T^{8} \)
$73$ \( 1 + 4 T + 284 T^{2} + 852 T^{3} + 30822 T^{4} + 62196 T^{5} + 1513436 T^{6} + 1556068 T^{7} + 28398241 T^{8} \)
$79$ \( 1 + 20 T + 445 T^{2} + 5040 T^{3} + 57977 T^{4} + 398160 T^{5} + 2777245 T^{6} + 9860780 T^{7} + 38950081 T^{8} \)
$83$ \( 1 - 6 T + 212 T^{2} - 982 T^{3} + 24278 T^{4} - 81506 T^{5} + 1460468 T^{6} - 3430722 T^{7} + 47458321 T^{8} \)
$89$ \( 1 + 16 T + 308 T^{2} + 3096 T^{3} + 38678 T^{4} + 275544 T^{5} + 2439668 T^{6} + 11279504 T^{7} + 62742241 T^{8} \)
$97$ \( 1 - 32 T + 656 T^{2} - 8944 T^{3} + 99982 T^{4} - 867568 T^{5} + 6172304 T^{6} - 29205536 T^{7} + 88529281 T^{8} \)
show more
show less