Properties

Label 3549.2.a.z
Level $3549$
Weight $2$
Character orbit 3549.a
Self dual yes
Analytic conductor $28.339$
Analytic rank $1$
Dimension $6$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3549 = 3 \cdot 7 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3549.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(28.3389076774\)
Analytic rank: \(1\)
Dimension: \(6\)
Coefficient field: 6.6.121819537.1
Defining polynomial: \(x^{6} - 3 x^{5} - 25 x^{4} + 55 x^{3} + 224 x^{2} - 252 x - 728\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6} - 3 x^{5} - 25 x^{4} + 55 x^{3} + 224 x^{2} - 252 x - 728\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{2} - \nu - 10 \)\()/2\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{3} - \nu^{2} - 10 \nu \)\()/2\)
\(\beta_{4}\)\(=\)\((\)\( \nu^{4} - 2 \nu^{3} - 17 \nu^{2} + 18 \nu + 76 \)\()/4\)
\(\beta_{5}\)\(=\)\((\)\( \nu^{5} - 3 \nu^{4} - 17 \nu^{3} + 39 \nu^{2} + 72 \nu - 92 \)\()/4\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(2 \beta_{2} + \beta_{1} + 10\)
\(\nu^{3}\)\(=\)\(2 \beta_{3} + 2 \beta_{2} + 11 \beta_{1} + 10\)
\(\nu^{4}\)\(=\)\(4 \beta_{4} + 4 \beta_{3} + 38 \beta_{2} + 21 \beta_{1} + 114\)
\(\nu^{5}\)\(=\)\(4 \beta_{5} + 12 \beta_{4} + 46 \beta_{3} + 70 \beta_{2} + 139 \beta_{1} + 214\)

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
−3.06987
4.06987
−2.55940
3.55940
−2.07801
3.07801
−1.24698 −1.00000 −0.445042 −4.06987 1.24698 1.00000 3.04892 1.00000 5.07504
1.2 −1.24698 −1.00000 −0.445042 3.06987 1.24698 1.00000 3.04892 1.00000 −3.82806
1.3 0.445042 −1.00000 −1.80194 −3.55940 −0.445042 1.00000 −1.69202 1.00000 −1.58408
1.4 0.445042 −1.00000 −1.80194 2.55940 −0.445042 1.00000 −1.69202 1.00000 1.13904
1.5 1.80194 −1.00000 1.24698 −3.07801 −1.80194 1.00000 −1.35690 1.00000 −5.54638
1.6 1.80194 −1.00000 1.24698 2.07801 −1.80194 1.00000 −1.35690 1.00000 3.74444
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.6
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)
\(7\) \(-1\)
\(13\) \(-1\)

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 3549.2.a.z yes 6
13.b even 2 1 3549.2.a.y 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3549.2.a.y 6 13.b even 2 1
3549.2.a.z yes 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(3549))\):

\( T_{2}^{3} - T_{2}^{2} - 2 T_{2} + 1 \)
\( T_{5}^{6} + 3 T_{5}^{5} - 25 T_{5}^{4} - 55 T_{5}^{3} + 224 T_{5}^{2} + 252 T_{5} - 728 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - 2 T - T^{2} + T^{3} )^{2} \)
$3$ \( ( 1 + T )^{6} \)
$5$ \( -728 + 252 T + 224 T^{2} - 55 T^{3} - 25 T^{4} + 3 T^{5} + T^{6} \)
$7$ \( ( -1 + T )^{6} \)
$11$ \( -281 + 434 T + 426 T^{2} - 42 T^{3} - 47 T^{4} + T^{6} \)
$13$ \( T^{6} \)
$17$ \( -8 - 192 T - 886 T^{2} - 451 T^{3} - 34 T^{4} + 9 T^{5} + T^{6} \)
$19$ \( 64 + 256 T - 484 T^{2} - 231 T^{3} + 6 T^{4} + 11 T^{5} + T^{6} \)
$23$ \( 2353 + 1439 T - 323 T^{2} - 259 T^{3} - T^{4} + 11 T^{5} + T^{6} \)
$29$ \( 12649 + 10696 T + 322 T^{2} - 734 T^{3} - 67 T^{4} + 10 T^{5} + T^{6} \)
$31$ \( 4472 + 4676 T + 132 T^{2} - 665 T^{3} - 96 T^{4} + 7 T^{5} + T^{6} \)
$37$ \( 7267 + 2053 T - 3208 T^{2} + 483 T^{3} + 80 T^{4} - 20 T^{5} + T^{6} \)
$41$ \( -1856 - 1776 T + 356 T^{2} + 431 T^{3} - 45 T^{4} - 10 T^{5} + T^{6} \)
$43$ \( -108352 + 21792 T + 5988 T^{2} - 903 T^{3} - 128 T^{4} + 9 T^{5} + T^{6} \)
$47$ \( -302184 - 156456 T - 24054 T^{2} + 263 T^{3} + 401 T^{4} + 36 T^{5} + T^{6} \)
$53$ \( -39403 + 16569 T + 2744 T^{2} - 909 T^{3} - 78 T^{4} + 12 T^{5} + T^{6} \)
$59$ \( 498568 + 82236 T - 32312 T^{2} + 69 T^{3} + 495 T^{4} - 41 T^{5} + T^{6} \)
$61$ \( -7384 - 20204 T - 11518 T^{2} - 1429 T^{3} + 93 T^{4} + 24 T^{5} + T^{6} \)
$67$ \( -39733 + 27105 T + 75 T^{2} - 1683 T^{3} - 89 T^{4} + 15 T^{5} + T^{6} \)
$71$ \( 16457 + 5425 T - 5509 T^{2} - 2421 T^{3} - 149 T^{4} + 13 T^{5} + T^{6} \)
$73$ \( 58024 + 1432 T - 6574 T^{2} - 651 T^{3} + 146 T^{4} + 25 T^{5} + T^{6} \)
$79$ \( -91141 + 14004 T + 8802 T^{2} - 1480 T^{3} - 125 T^{4} + 16 T^{5} + T^{6} \)
$83$ \( -218792 - 22512 T + 13342 T^{2} + 519 T^{3} - 225 T^{4} - 2 T^{5} + T^{6} \)
$89$ \( -51064 - 48572 T + 27840 T^{2} - 2043 T^{3} - 265 T^{4} + 15 T^{5} + T^{6} \)
$97$ \( 11752 + 10848 T - 1058 T^{2} - 1089 T^{3} - 87 T^{4} + 10 T^{5} + T^{6} \)
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