Properties

Label 5577.2.a.y
Level $5577$
Weight $2$
Character orbit 5577.a
Self dual yes
Analytic conductor $44.533$
Analytic rank $0$
Dimension $7$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 5577 = 3 \cdot 11 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5577.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(44.5325692073\)
Analytic rank: \(0\)
Dimension: \(7\)
Coefficient field: \(\mathbb{Q}[x]/(x^{7} - \cdots)\)
Defining polynomial: \(x^{7} - 3 x^{6} - 7 x^{5} + 21 x^{4} + 13 x^{3} - 33 x^{2} - 7 x + 7\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 429)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{6}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{1} q^{2} + q^{3} + ( 1 + \beta_{1} + \beta_{2} ) q^{4} + ( 1 - \beta_{6} ) q^{5} + \beta_{1} q^{6} + ( 1 + \beta_{4} ) q^{7} + ( 1 + 2 \beta_{1} + \beta_{2} + \beta_{3} ) q^{8} + q^{9} +O(q^{10})\) \( q + \beta_{1} q^{2} + q^{3} + ( 1 + \beta_{1} + \beta_{2} ) q^{4} + ( 1 - \beta_{6} ) q^{5} + \beta_{1} q^{6} + ( 1 + \beta_{4} ) q^{7} + ( 1 + 2 \beta_{1} + \beta_{2} + \beta_{3} ) q^{8} + q^{9} + ( \beta_{1} - \beta_{5} - \beta_{6} ) q^{10} + q^{11} + ( 1 + \beta_{1} + \beta_{2} ) q^{12} + ( \beta_{1} + \beta_{3} + \beta_{5} ) q^{14} + ( 1 - \beta_{6} ) q^{15} + ( 1 + 3 \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} ) q^{16} + ( -1 + \beta_{1} - \beta_{2} + \beta_{6} ) q^{17} + \beta_{1} q^{18} + ( 1 + \beta_{1} - \beta_{5} ) q^{19} + ( \beta_{1} + \beta_{2} - \beta_{4} - 2 \beta_{5} - \beta_{6} ) q^{20} + ( 1 + \beta_{4} ) q^{21} + \beta_{1} q^{22} + ( 2 - 2 \beta_{1} - \beta_{3} + \beta_{4} ) q^{23} + ( 1 + 2 \beta_{1} + \beta_{2} + \beta_{3} ) q^{24} + ( 1 + 2 \beta_{1} + \beta_{3} + \beta_{4} - 2 \beta_{6} ) q^{25} + q^{27} + ( 1 + \beta_{1} + 2 \beta_{2} + \beta_{5} + 2 \beta_{6} ) q^{28} + ( -2 + \beta_{1} - \beta_{5} + \beta_{6} ) q^{29} + ( \beta_{1} - \beta_{5} - \beta_{6} ) q^{30} + ( -1 - 2 \beta_{1} - 2 \beta_{2} + \beta_{6} ) q^{31} + ( 4 + 2 \beta_{1} + 2 \beta_{2} + \beta_{4} + \beta_{5} ) q^{32} + q^{33} + ( 5 - 2 \beta_{1} + \beta_{2} - \beta_{3} + \beta_{5} + \beta_{6} ) q^{34} + ( 1 - 2 \beta_{1} - 2 \beta_{2} - \beta_{3} + 2 \beta_{4} - 2 \beta_{6} ) q^{35} + ( 1 + \beta_{1} + \beta_{2} ) q^{36} + ( 1 - \beta_{1} - \beta_{3} - \beta_{4} + \beta_{5} + 2 \beta_{6} ) q^{37} + ( 2 + 2 \beta_{1} + \beta_{2} - \beta_{4} - \beta_{5} - 2 \beta_{6} ) q^{38} + ( -1 + \beta_{1} + \beta_{2} - 2 \beta_{4} - 2 \beta_{5} - 3 \beta_{6} ) q^{40} + ( 1 - \beta_{1} + \beta_{2} - \beta_{3} + 2 \beta_{6} ) q^{41} + ( \beta_{1} + \beta_{3} + \beta_{5} ) q^{42} + ( -3 + \beta_{1} - \beta_{2} - 2 \beta_{4} - \beta_{6} ) q^{43} + ( 1 + \beta_{1} + \beta_{2} ) q^{44} + ( 1 - \beta_{6} ) q^{45} + ( -5 - 3 \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} ) q^{46} + ( 1 + \beta_{1} - 2 \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} ) q^{47} + ( 1 + 3 \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} ) q^{48} + ( 3 - \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} + 2 \beta_{5} - 2 \beta_{6} ) q^{49} + ( 5 + 3 \beta_{1} + 3 \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} - 2 \beta_{6} ) q^{50} + ( -1 + \beta_{1} - \beta_{2} + \beta_{6} ) q^{51} + ( 2 - \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} - 2 \beta_{6} ) q^{53} + \beta_{1} q^{54} + ( 1 - \beta_{6} ) q^{55} + ( 4 \beta_{1} + \beta_{2} + \beta_{4} + \beta_{5} + 4 \beta_{6} ) q^{56} + ( 1 + \beta_{1} - \beta_{5} ) q^{57} + ( 2 - \beta_{1} + \beta_{2} - \beta_{4} - \beta_{6} ) q^{58} + ( 2 - 4 \beta_{1} - \beta_{2} - \beta_{3} + \beta_{5} ) q^{59} + ( \beta_{1} + \beta_{2} - \beta_{4} - 2 \beta_{5} - \beta_{6} ) q^{60} + ( -1 - \beta_{3} ) q^{61} + ( -2 - 7 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} + \beta_{5} + \beta_{6} ) q^{62} + ( 1 + \beta_{4} ) q^{63} + ( 1 + 4 \beta_{1} + \beta_{3} - \beta_{4} + 2 \beta_{5} + 2 \beta_{6} ) q^{64} + \beta_{1} q^{66} + ( 1 + 3 \beta_{2} + \beta_{3} + \beta_{5} - \beta_{6} ) q^{67} + ( -4 + 3 \beta_{1} - \beta_{2} + \beta_{3} + 2 \beta_{5} + \beta_{6} ) q^{68} + ( 2 - 2 \beta_{1} - \beta_{3} + \beta_{4} ) q^{69} + ( -1 - 5 \beta_{1} - 3 \beta_{2} - \beta_{4} - 2 \beta_{6} ) q^{70} + ( 5 + \beta_{1} + 3 \beta_{2} - \beta_{4} ) q^{71} + ( 1 + 2 \beta_{1} + \beta_{2} + \beta_{3} ) q^{72} + ( 5 - 4 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} - \beta_{4} ) q^{73} + ( -1 - 2 \beta_{2} - \beta_{3} + 2 \beta_{5} + 4 \beta_{6} ) q^{74} + ( 1 + 2 \beta_{1} + \beta_{3} + \beta_{4} - 2 \beta_{6} ) q^{75} + ( 1 + 4 \beta_{1} + 2 \beta_{2} - \beta_{4} - 2 \beta_{5} - 4 \beta_{6} ) q^{76} + ( 1 + \beta_{4} ) q^{77} + ( 1 - \beta_{1} - \beta_{3} - 2 \beta_{4} + \beta_{5} + \beta_{6} ) q^{79} + ( -1 - \beta_{2} - \beta_{3} - 3 \beta_{5} - 5 \beta_{6} ) q^{80} + q^{81} + ( -4 + 2 \beta_{1} - 2 \beta_{2} + \beta_{3} - \beta_{4} + 2 \beta_{5} + 2 \beta_{6} ) q^{82} + ( 4 + \beta_{1} + \beta_{4} - \beta_{5} + 2 \beta_{6} ) q^{83} + ( 1 + \beta_{1} + 2 \beta_{2} + \beta_{5} + 2 \beta_{6} ) q^{84} + ( -5 - \beta_{1} - \beta_{2} - \beta_{3} ) q^{85} + ( 5 - 4 \beta_{1} + \beta_{2} - 3 \beta_{3} - 3 \beta_{5} - \beta_{6} ) q^{86} + ( -2 + \beta_{1} - \beta_{5} + \beta_{6} ) q^{87} + ( 1 + 2 \beta_{1} + \beta_{2} + \beta_{3} ) q^{88} + ( 6 - \beta_{1} + \beta_{2} - 2 \beta_{3} - \beta_{4} + \beta_{6} ) q^{89} + ( \beta_{1} - \beta_{5} - \beta_{6} ) q^{90} + ( 2 - 7 \beta_{1} + \beta_{2} - 2 \beta_{3} + 2 \beta_{6} ) q^{92} + ( -1 - 2 \beta_{1} - 2 \beta_{2} + \beta_{6} ) q^{93} + ( 7 - 2 \beta_{1} + 2 \beta_{2} - 3 \beta_{3} + 2 \beta_{4} + 2 \beta_{6} ) q^{94} + ( 1 + 6 \beta_{1} + 3 \beta_{2} - \beta_{5} - 2 \beta_{6} ) q^{95} + ( 4 + 2 \beta_{1} + 2 \beta_{2} + \beta_{4} + \beta_{5} ) q^{96} + ( 3 - \beta_{1} - \beta_{2} - \beta_{4} - 2 \beta_{5} ) q^{97} + ( 3 \beta_{4} + \beta_{5} + 2 \beta_{6} ) q^{98} + q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7q + 3q^{2} + 7q^{3} + 9q^{4} + 6q^{5} + 3q^{6} + 6q^{7} + 15q^{8} + 7q^{9} + O(q^{10}) \) \( 7q + 3q^{2} + 7q^{3} + 9q^{4} + 6q^{5} + 3q^{6} + 6q^{7} + 15q^{8} + 7q^{9} + 7q^{11} + 9q^{12} + 8q^{14} + 6q^{15} + 17q^{16} - 2q^{17} + 3q^{18} + 8q^{19} - 2q^{20} + 6q^{21} + 3q^{22} + 4q^{23} + 15q^{24} + 13q^{25} + 7q^{27} + 12q^{28} - 12q^{29} - 10q^{31} + 33q^{32} + 7q^{33} + 28q^{34} - 4q^{35} + 9q^{36} + 6q^{37} + 16q^{38} - 10q^{40} + 2q^{41} + 8q^{42} - 16q^{43} + 9q^{44} + 6q^{45} - 26q^{46} + 18q^{47} + 17q^{48} + 23q^{49} + 39q^{50} - 2q^{51} + 10q^{53} + 3q^{54} + 6q^{55} + 16q^{56} + 8q^{57} + 10q^{58} + 2q^{59} - 2q^{60} - 10q^{61} - 36q^{62} + 6q^{63} + 29q^{64} + 3q^{66} + 8q^{67} - 10q^{68} + 4q^{69} - 20q^{70} + 36q^{71} + 15q^{72} + 20q^{73} + 13q^{75} + 10q^{76} + 6q^{77} + 6q^{79} - 20q^{80} + 7q^{81} - 10q^{82} + 30q^{83} + 12q^{84} - 40q^{85} + 6q^{86} - 12q^{87} + 15q^{88} + 34q^{89} - 12q^{92} - 10q^{93} + 32q^{94} + 18q^{95} + 33q^{96} + 16q^{97} + q^{98} + 7q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{7} - 3 x^{6} - 7 x^{5} + 21 x^{4} + 13 x^{3} - 33 x^{2} - 7 x + 7\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - \nu - 3 \)
\(\beta_{3}\)\(=\)\( \nu^{3} - \nu^{2} - 5 \nu + 2 \)
\(\beta_{4}\)\(=\)\( \nu^{4} - \nu^{3} - 6 \nu^{2} + 3 \nu + 4 \)
\(\beta_{5}\)\(=\)\( \nu^{5} - \nu^{4} - 7 \nu^{3} + 4 \nu^{2} + 9 \nu - 2 \)
\(\beta_{6}\)\(=\)\((\)\( \nu^{6} - 2 \nu^{5} - 7 \nu^{4} + 12 \nu^{3} + 11 \nu^{2} - 14 \nu - 3 \)\()/2\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + \beta_{1} + 3\)
\(\nu^{3}\)\(=\)\(\beta_{3} + \beta_{2} + 6 \beta_{1} + 1\)
\(\nu^{4}\)\(=\)\(\beta_{4} + \beta_{3} + 7 \beta_{2} + 9 \beta_{1} + 15\)
\(\nu^{5}\)\(=\)\(\beta_{5} + \beta_{4} + 8 \beta_{3} + 10 \beta_{2} + 38 \beta_{1} + 12\)
\(\nu^{6}\)\(=\)\(2 \beta_{6} + 2 \beta_{5} + 9 \beta_{4} + 11 \beta_{3} + 46 \beta_{2} + 70 \beta_{1} + 87\)

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
−2.15754
−1.36814
−0.584778
0.409068
1.42819
2.53441
2.73878
−2.15754 1.00000 2.65498 0.710210 −2.15754 2.30964 −1.41315 1.00000 −1.53231
1.2 −1.36814 1.00000 −0.128197 2.18365 −1.36814 −4.27070 2.91167 1.00000 −2.98753
1.3 −0.584778 1.00000 −1.65803 −1.95350 −0.584778 1.51078 2.13914 1.00000 1.14237
1.4 0.409068 1.00000 −1.83266 4.13953 0.409068 5.18273 −1.56782 1.00000 1.69335
1.5 1.42819 1.00000 0.0397381 0.0606573 1.42819 −1.70646 −2.79963 1.00000 0.0866304
1.6 2.53441 1.00000 4.42325 3.70100 2.53441 −0.957295 6.14151 1.00000 9.37985
1.7 2.73878 1.00000 5.50093 −2.84154 2.73878 3.93129 9.58828 1.00000 −7.78236
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.7
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(11\) \(-1\)
\(13\) \(-1\)

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 5577.2.a.y 7
13.b even 2 1 5577.2.a.x 7
13.d odd 4 2 429.2.b.b 14
39.f even 4 2 1287.2.b.c 14
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
429.2.b.b 14 13.d odd 4 2
1287.2.b.c 14 39.f even 4 2
5577.2.a.x 7 13.b even 2 1
5577.2.a.y 7 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(5577))\):

\( T_{2}^{7} - 3 T_{2}^{6} - 7 T_{2}^{5} + 21 T_{2}^{4} + 13 T_{2}^{3} - 33 T_{2}^{2} - 7 T_{2} + 7 \)
\( T_{5}^{7} - 6 T_{5}^{6} - 6 T_{5}^{5} + 74 T_{5}^{4} - 32 T_{5}^{3} - 198 T_{5}^{2} + 144 T_{5} - 8 \)
\( T_{7}^{7} - 6 T_{7}^{6} - 18 T_{7}^{5} + 136 T_{7}^{4} - 16 T_{7}^{3} - 524 T_{7}^{2} + 160 T_{7} + 496 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 7 - 7 T - 33 T^{2} + 13 T^{3} + 21 T^{4} - 7 T^{5} - 3 T^{6} + T^{7} \)
$3$ \( ( -1 + T )^{7} \)
$5$ \( -8 + 144 T - 198 T^{2} - 32 T^{3} + 74 T^{4} - 6 T^{5} - 6 T^{6} + T^{7} \)
$7$ \( 496 + 160 T - 524 T^{2} - 16 T^{3} + 136 T^{4} - 18 T^{5} - 6 T^{6} + T^{7} \)
$11$ \( ( -1 + T )^{7} \)
$13$ \( T^{7} \)
$17$ \( -1256 - 2672 T + 134 T^{2} + 698 T^{3} - 32 T^{4} - 50 T^{5} + 2 T^{6} + T^{7} \)
$19$ \( -80 - 496 T - 636 T^{2} + 244 T^{3} + 168 T^{4} - 22 T^{5} - 8 T^{6} + T^{7} \)
$23$ \( -1024 - 704 T + 2044 T^{2} + 1448 T^{3} + 48 T^{4} - 76 T^{5} - 4 T^{6} + T^{7} \)
$29$ \( 104 + 3040 T + 1170 T^{2} - 790 T^{3} - 358 T^{4} + 2 T^{5} + 12 T^{6} + T^{7} \)
$31$ \( 392 - 5712 T + 4414 T^{2} + 460 T^{3} - 594 T^{4} - 62 T^{5} + 10 T^{6} + T^{7} \)
$37$ \( 125824 - 8832 T - 27896 T^{2} + 4752 T^{3} + 792 T^{4} - 136 T^{5} - 6 T^{6} + T^{7} \)
$41$ \( 39760 - 39088 T - 3924 T^{2} + 4420 T^{3} + 180 T^{4} - 130 T^{5} - 2 T^{6} + T^{7} \)
$43$ \( -692704 - 35104 T + 63862 T^{2} + 2962 T^{3} - 1840 T^{4} - 90 T^{5} + 16 T^{6} + T^{7} \)
$47$ \( -122368 + 1344 T + 50456 T^{2} - 20120 T^{3} + 2736 T^{4} - 44 T^{5} - 18 T^{6} + T^{7} \)
$53$ \( 11456 + 19584 T - 8048 T^{2} - 2688 T^{3} + 1248 T^{4} - 88 T^{5} - 10 T^{6} + T^{7} \)
$59$ \( 145408 - 20224 T - 24224 T^{2} + 4496 T^{3} + 720 T^{4} - 160 T^{5} - 2 T^{6} + T^{7} \)
$61$ \( 32 + 192 T + 56 T^{2} - 248 T^{3} - 116 T^{4} + 12 T^{5} + 10 T^{6} + T^{7} \)
$67$ \( 500296 - 272720 T - 81874 T^{2} + 14516 T^{3} + 1610 T^{4} - 222 T^{5} - 8 T^{6} + T^{7} \)
$71$ \( -593536 + 190272 T + 115384 T^{2} - 33248 T^{3} + 1120 T^{4} + 352 T^{5} - 36 T^{6} + T^{7} \)
$73$ \( -555776 + 328608 T + 5588 T^{2} - 27624 T^{3} + 4544 T^{4} - 114 T^{5} - 20 T^{6} + T^{7} \)
$79$ \( 55424 - 44976 T - 5158 T^{2} + 6730 T^{3} + 560 T^{4} - 154 T^{5} - 6 T^{6} + T^{7} \)
$83$ \( 35840 - 106624 T + 68976 T^{2} - 16000 T^{3} + 704 T^{4} + 232 T^{5} - 30 T^{6} + T^{7} \)
$89$ \( -87880 + 19184 T + 24834 T^{2} - 8640 T^{3} - 212 T^{4} + 330 T^{5} - 34 T^{6} + T^{7} \)
$97$ \( -216704 + 129920 T + 30376 T^{2} - 29136 T^{3} + 5216 T^{4} - 212 T^{5} - 16 T^{6} + T^{7} \)
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