Properties

Label 2601.2.a.bi
Level $2601$
Weight $2$
Character orbit 2601.a
Self dual yes
Analytic conductor $20.769$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2601 = 3^{2} \cdot 17^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2601.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(20.7690895657\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.6.3418281.1
Defining polynomial: \(x^{6} - 9 x^{4} - 4 x^{3} + 18 x^{2} + 12 x - 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 867)
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.

\(f(q)\) \(=\) \( q + \beta_{4} q^{2} + ( 1 + \beta_{1} + \beta_{2} - \beta_{4} + 2 \beta_{5} ) q^{4} + ( 1 + \beta_{3} + \beta_{4} + \beta_{5} ) q^{5} + ( 1 - 2 \beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} ) q^{7} + ( -1 - 2 \beta_{1} - 2 \beta_{2} - \beta_{3} + 2 \beta_{4} - 3 \beta_{5} ) q^{8} +O(q^{10})\) \( q + \beta_{4} q^{2} + ( 1 + \beta_{1} + \beta_{2} - \beta_{4} + 2 \beta_{5} ) q^{4} + ( 1 + \beta_{3} + \beta_{4} + \beta_{5} ) q^{5} + ( 1 - 2 \beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} ) q^{7} + ( -1 - 2 \beta_{1} - 2 \beta_{2} - \beta_{3} + 2 \beta_{4} - 3 \beta_{5} ) q^{8} + ( 2 + \beta_{1} - \beta_{2} - \beta_{3} - \beta_{5} ) q^{10} + ( 1 + \beta_{1} - \beta_{2} - \beta_{4} - 2 \beta_{5} ) q^{11} + ( 2 - \beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} - 2 \beta_{5} ) q^{13} + ( 1 - \beta_{3} + 3 \beta_{5} ) q^{14} + ( 1 + 2 \beta_{2} + \beta_{3} - 3 \beta_{4} + 3 \beta_{5} ) q^{16} + ( 2 + \beta_{1} - 2 \beta_{2} + \beta_{3} + \beta_{4} ) q^{19} + ( -1 - \beta_{1} + 5 \beta_{2} + 2 \beta_{5} ) q^{20} + ( -3 + 4 \beta_{2} + 3 \beta_{3} + \beta_{5} ) q^{22} + ( 2 - 2 \beta_{2} - 2 \beta_{3} + \beta_{4} - 2 \beta_{5} ) q^{23} + ( 3 - 2 \beta_{1} - 4 \beta_{2} + \beta_{4} - 4 \beta_{5} ) q^{25} + ( 2 + \beta_{1} + 3 \beta_{2} + \beta_{3} + 5 \beta_{5} ) q^{26} + ( -1 - \beta_{3} + 3 \beta_{4} + \beta_{5} ) q^{28} + ( 1 + \beta_{1} + \beta_{2} + 2 \beta_{5} ) q^{29} + ( 5 + \beta_{1} + 2 \beta_{4} + \beta_{5} ) q^{31} + ( -6 + \beta_{1} - 5 \beta_{2} - \beta_{3} + 2 \beta_{4} - 5 \beta_{5} ) q^{32} + ( 1 - \beta_{1} - 5 \beta_{2} + 2 \beta_{4} + 2 \beta_{5} ) q^{35} + ( 2 \beta_{1} + \beta_{4} - \beta_{5} ) q^{37} + ( 1 + 4 \beta_{2} + \beta_{3} + \beta_{5} ) q^{38} + ( \beta_{1} - 7 \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} ) q^{40} + ( 3 - 2 \beta_{1} + 2 \beta_{2} + \beta_{3} + \beta_{5} ) q^{41} + ( -3 \beta_{1} + 2 \beta_{2} + \beta_{3} - \beta_{5} ) q^{43} + ( -1 + 4 \beta_{1} - 6 \beta_{2} - \beta_{3} - 3 \beta_{4} - 3 \beta_{5} ) q^{44} + ( 3 - \beta_{1} + 7 \beta_{2} + 2 \beta_{3} + \beta_{4} + 8 \beta_{5} ) q^{46} + ( -5 - \beta_{1} + 3 \beta_{2} - 2 \beta_{4} ) q^{47} + ( 5 + \beta_{2} + 3 \beta_{4} - \beta_{5} ) q^{49} + ( -3 + \beta_{1} + 5 \beta_{2} + 2 \beta_{3} - 2 \beta_{4} + 4 \beta_{5} ) q^{50} + ( -1 + \beta_{1} - 5 \beta_{2} - 2 \beta_{3} + 4 \beta_{4} - 2 \beta_{5} ) q^{52} + ( -4 + \beta_{1} + \beta_{2} - \beta_{3} + 3 \beta_{5} ) q^{53} + ( -3 + \beta_{1} + 3 \beta_{2} + 4 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} ) q^{55} + ( 8 + \beta_{1} + 3 \beta_{2} + \beta_{3} - 2 \beta_{4} + \beta_{5} ) q^{56} + ( 2 - \beta_{1} - \beta_{2} - \beta_{3} + 3 \beta_{4} - \beta_{5} ) q^{58} + ( 1 + \beta_{1} + 3 \beta_{2} + 2 \beta_{3} - \beta_{4} ) q^{59} + ( 2 + 2 \beta_{1} + \beta_{2} + 2 \beta_{3} - 3 \beta_{4} + 3 \beta_{5} ) q^{61} + ( 7 + \beta_{1} + 3 \beta_{2} + 4 \beta_{4} + 4 \beta_{5} ) q^{62} + ( 1 + \beta_{1} + 11 \beta_{2} + 4 \beta_{3} - 6 \beta_{4} + 6 \beta_{5} ) q^{64} + ( 2 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} + 3 \beta_{4} + 4 \beta_{5} ) q^{65} + ( -1 + 3 \beta_{1} - \beta_{2} + 5 \beta_{5} ) q^{67} + ( -5 \beta_{1} + 3 \beta_{2} - 3 \beta_{3} + \beta_{4} + \beta_{5} ) q^{70} + ( 4 + 3 \beta_{1} + 3 \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} ) q^{71} + ( 1 + \beta_{1} + 3 \beta_{2} - \beta_{3} - 4 \beta_{4} - \beta_{5} ) q^{73} + ( 5 + 2 \beta_{1} + 6 \beta_{2} + 3 \beta_{3} - 2 \beta_{4} + 5 \beta_{5} ) q^{74} + ( -1 + 2 \beta_{1} - 2 \beta_{2} - 3 \beta_{3} - \beta_{4} - 3 \beta_{5} ) q^{76} + ( -7 - 6 \beta_{1} - \beta_{3} - 3 \beta_{4} - 5 \beta_{5} ) q^{77} + ( 3 + 4 \beta_{1} + 4 \beta_{2} + 4 \beta_{3} - 2 \beta_{4} + 4 \beta_{5} ) q^{79} + ( -4 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} - \beta_{4} + 2 \beta_{5} ) q^{80} + ( -1 + 2 \beta_{1} - 8 \beta_{2} - 3 \beta_{3} + 3 \beta_{4} - 5 \beta_{5} ) q^{82} + ( -3 + \beta_{1} + 7 \beta_{2} + 2 \beta_{3} - 4 \beta_{4} + 2 \beta_{5} ) q^{83} + ( -2 + 4 \beta_{1} - 8 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} - 4 \beta_{5} ) q^{86} + ( -4 - 7 \beta_{1} + 7 \beta_{2} + \beta_{3} + \beta_{5} ) q^{88} + ( \beta_{1} - 5 \beta_{2} + 3 \beta_{3} + \beta_{5} ) q^{89} + ( 7 - 5 \beta_{1} + \beta_{2} - 3 \beta_{3} + \beta_{4} - 2 \beta_{5} ) q^{91} + ( 3 + 2 \beta_{1} - 14 \beta_{2} - 5 \beta_{3} + 6 \beta_{4} - 7 \beta_{5} ) q^{92} + ( -4 + \beta_{1} - 7 \beta_{2} - \beta_{3} - 3 \beta_{4} - 5 \beta_{5} ) q^{94} + ( 9 - 5 \beta_{1} - 3 \beta_{2} + 2 \beta_{3} + 4 \beta_{4} - 6 \beta_{5} ) q^{95} + ( -7 + 2 \beta_{1} - \beta_{3} - 3 \beta_{4} - 2 \beta_{5} ) q^{97} + ( 10 + 5 \beta_{1} + 3 \beta_{2} + \beta_{3} + \beta_{4} + 7 \beta_{5} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q - 3q^{2} + 9q^{4} + 3q^{5} + 3q^{7} - 12q^{8} + O(q^{10}) \) \( 6q - 3q^{2} + 9q^{4} + 3q^{5} + 3q^{7} - 12q^{8} + 12q^{10} + 9q^{11} + 9q^{13} + 6q^{14} + 15q^{16} + 9q^{19} - 6q^{20} - 18q^{22} + 9q^{23} + 15q^{25} + 12q^{26} - 15q^{28} + 6q^{29} + 24q^{31} - 42q^{32} - 3q^{37} + 6q^{38} - 3q^{40} + 18q^{41} + 3q^{44} + 15q^{46} - 24q^{47} + 21q^{49} - 12q^{50} - 18q^{52} - 24q^{53} - 24q^{55} + 54q^{56} + 3q^{58} + 9q^{59} + 21q^{61} + 30q^{62} + 24q^{64} - 9q^{65} - 6q^{67} - 3q^{70} + 27q^{71} + 18q^{73} + 36q^{74} - 3q^{76} - 33q^{77} + 24q^{79} + 3q^{80} - 15q^{82} - 6q^{83} - 6q^{86} - 24q^{88} + 39q^{91} - 15q^{94} + 42q^{95} - 33q^{97} + 57q^{98} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6} - 9 x^{4} - 4 x^{3} + 18 x^{2} + 12 x - 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{3} - \nu^{2} - 4 \nu + 1 \)\()/2\)
\(\beta_{3}\)\(=\)\( \nu^{2} - \nu - 3 \)
\(\beta_{4}\)\(=\)\((\)\( \nu^{4} - \nu^{3} - 6 \nu^{2} + 3 \nu + 4 \)\()/2\)
\(\beta_{5}\)\(=\)\((\)\( \nu^{5} - \nu^{4} - 8 \nu^{3} + 3 \nu^{2} + 14 \nu + 2 \)\()/2\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{3} + \beta_{1} + 3\)
\(\nu^{3}\)\(=\)\(\beta_{3} + 2 \beta_{2} + 5 \beta_{1} + 2\)
\(\nu^{4}\)\(=\)\(2 \beta_{4} + 7 \beta_{3} + 2 \beta_{2} + 8 \beta_{1} + 16\)
\(\nu^{5}\)\(=\)\(2 \beta_{5} + 2 \beta_{4} + 12 \beta_{3} + 18 \beta_{2} + 31 \beta_{1} + 21\)

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
−1.60714
1.71374
−0.857616
−2.06104
2.73700
0.0750494
−2.74819 0 5.55252 0.973936 0 −1.60333 −9.76298 0 −2.67656
1.2 −2.44395 0 3.97290 −2.87349 0 −1.56252 −4.82168 0 7.02266
1.3 −0.907065 0 −1.17723 −3.19333 0 3.56234 2.88196 0 2.89656
1.4 −0.435433 0 −1.81040 4.22078 0 2.90981 1.65917 0 −1.83787
1.5 1.43915 0 0.0711653 2.31394 0 −4.44173 −2.77589 0 3.33012
1.6 2.09548 0 2.39104 1.55815 0 4.13541 0.819422 0 3.26508
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.6
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(17\) \(1\)

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 2601.2.a.bi 6
3.b odd 2 1 867.2.a.p yes 6
17.b even 2 1 2601.2.a.bh 6
51.c odd 2 1 867.2.a.o 6
51.f odd 4 2 867.2.d.g 12
51.g odd 8 4 867.2.e.k 24
51.i even 16 8 867.2.h.m 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
867.2.a.o 6 51.c odd 2 1
867.2.a.p yes 6 3.b odd 2 1
867.2.d.g 12 51.f odd 4 2
867.2.e.k 24 51.g odd 8 4
867.2.h.m 48 51.i even 16 8
2601.2.a.bh 6 17.b even 2 1
2601.2.a.bi 6 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2601))\):

\( T_{2}^{6} + 3 T_{2}^{5} - 6 T_{2}^{4} - 19 T_{2}^{3} + 6 T_{2}^{2} + 24 T_{2} + 8 \)
\( T_{5}^{6} - 3 T_{5}^{5} - 18 T_{5}^{4} + 51 T_{5}^{3} + 60 T_{5}^{2} - 228 T_{5} + 136 \)
\( T_{7}^{6} - 3 T_{7}^{5} - 27 T_{7}^{4} + 75 T_{7}^{3} + 171 T_{7}^{2} - 297 T_{7} - 477 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 8 + 24 T + 6 T^{2} - 19 T^{3} - 6 T^{4} + 3 T^{5} + T^{6} \)
$3$ \( T^{6} \)
$5$ \( 136 - 228 T + 60 T^{2} + 51 T^{3} - 18 T^{4} - 3 T^{5} + T^{6} \)
$7$ \( -477 - 297 T + 171 T^{2} + 75 T^{3} - 27 T^{4} - 3 T^{5} + T^{6} \)
$11$ \( 296 - 432 T - 282 T^{2} + 199 T^{3} - 6 T^{4} - 9 T^{5} + T^{6} \)
$13$ \( -109 + 321 T - 315 T^{2} + 103 T^{3} + 9 T^{4} - 9 T^{5} + T^{6} \)
$17$ \( T^{6} \)
$19$ \( 333 - 783 T - 171 T^{2} + 207 T^{3} - 9 T^{4} - 9 T^{5} + T^{6} \)
$23$ \( 1576 - 2340 T - 570 T^{2} + 477 T^{3} - 36 T^{4} - 9 T^{5} + T^{6} \)
$29$ \( 136 - 132 T - 36 T^{2} + 57 T^{3} - 3 T^{4} - 6 T^{5} + T^{6} \)
$31$ \( -127 + 198 T + 381 T^{2} - 506 T^{3} + 186 T^{4} - 24 T^{5} + T^{6} \)
$37$ \( -109 + 153 T + 237 T^{2} - 281 T^{3} - 75 T^{4} + 3 T^{5} + T^{6} \)
$41$ \( -1592 + 4680 T - 2910 T^{2} + 423 T^{3} + 63 T^{4} - 18 T^{5} + T^{6} \)
$43$ \( -11736 - 5292 T + 2340 T^{2} + 297 T^{3} - 117 T^{4} + T^{6} \)
$47$ \( 20312 - 4212 T - 3348 T^{2} + 31 T^{3} + 171 T^{4} + 24 T^{5} + T^{6} \)
$53$ \( -5608 - 7896 T - 2898 T^{2} + 95 T^{3} + 171 T^{4} + 24 T^{5} + T^{6} \)
$59$ \( 1432 - 4356 T + 732 T^{2} + 441 T^{3} - 60 T^{4} - 9 T^{5} + T^{6} \)
$61$ \( 31841 - 10773 T - 3159 T^{2} + 985 T^{3} + 45 T^{4} - 21 T^{5} + T^{6} \)
$67$ \( -14552 + 4860 T + 3882 T^{2} - 191 T^{3} - 153 T^{4} + 6 T^{5} + T^{6} \)
$71$ \( -8704 + 9408 T - 2448 T^{2} - 301 T^{3} + 207 T^{4} - 27 T^{5} + T^{6} \)
$73$ \( 92429 - 33639 T - 4524 T^{2} + 1789 T^{3} - 30 T^{4} - 18 T^{5} + T^{6} \)
$79$ \( 31419 - 25272 T + 27 T^{2} + 1968 T^{3} + 9 T^{4} - 24 T^{5} + T^{6} \)
$83$ \( -205336 + 45960 T + 10962 T^{2} - 1579 T^{3} - 243 T^{4} + 6 T^{5} + T^{6} \)
$89$ \( -39896 - 1440 T + 15750 T^{2} - 585 T^{3} - 327 T^{4} + T^{6} \)
$97$ \( -17389 - 27306 T - 7488 T^{2} + 409 T^{3} + 324 T^{4} + 33 T^{5} + T^{6} \)
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