Properties

Label 3549.2.a.bb
Level $3549$
Weight $2$
Character orbit 3549.a
Self dual yes
Analytic conductor $28.339$
Analytic rank $1$
Dimension $8$
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: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Defining polynomial: \(x^{8} - 2 x^{7} - 9 x^{6} + 14 x^{5} + 25 x^{4} - 24 x^{3} - 16 x^{2} + 8 x + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 273)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - 2 x^{7} - 9 x^{6} + 14 x^{5} + 25 x^{4} - 24 x^{3} - 16 x^{2} + 8 x + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - \nu - 3 \)
\(\beta_{3}\)\(=\)\( \nu^{3} - \nu^{2} - 4 \nu + 1 \)
\(\beta_{4}\)\(=\)\( \nu^{4} - \nu^{3} - 5 \nu^{2} + 2 \nu + 2 \)
\(\beta_{5}\)\(=\)\( \nu^{5} - \nu^{4} - 6 \nu^{3} + 3 \nu^{2} + 6 \nu - 1 \)
\(\beta_{6}\)\(=\)\( \nu^{6} - 2 \nu^{5} - 7 \nu^{4} + 10 \nu^{3} + 14 \nu^{2} - 7 \nu - 4 \)
\(\beta_{7}\)\(=\)\( \nu^{7} - 2 \nu^{6} - 9 \nu^{5} + 13 \nu^{4} + 25 \nu^{3} - 18 \nu^{2} - 13 \nu + 3 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + \beta_{1} + 3\)
\(\nu^{3}\)\(=\)\(\beta_{3} + \beta_{2} + 5 \beta_{1} + 2\)
\(\nu^{4}\)\(=\)\(\beta_{4} + \beta_{3} + 6 \beta_{2} + 8 \beta_{1} + 15\)
\(\nu^{5}\)\(=\)\(\beta_{5} + \beta_{4} + 7 \beta_{3} + 9 \beta_{2} + 29 \beta_{1} + 19\)
\(\nu^{6}\)\(=\)\(\beta_{6} + 2 \beta_{5} + 9 \beta_{4} + 11 \beta_{3} + 36 \beta_{2} + 57 \beta_{1} + 85\)
\(\nu^{7}\)\(=\)\(\beta_{7} + 2 \beta_{6} + 13 \beta_{5} + 14 \beta_{4} + 47 \beta_{3} + 68 \beta_{2} + 177 \beta_{1} + 147\)

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
2.60802
2.45308
1.10207
0.485989
−0.106359
−0.775848
−1.77930
−1.98765
−2.60802 1.00000 4.80176 1.50528 −2.60802 −1.00000 −7.30704 1.00000 −3.92579
1.2 −2.45308 1.00000 4.01758 −0.0682999 −2.45308 −1.00000 −4.94928 1.00000 0.167545
1.3 −1.10207 1.00000 −0.785435 −3.28432 −1.10207 −1.00000 3.06975 1.00000 3.61956
1.4 −0.485989 1.00000 −1.76381 −1.06536 −0.485989 −1.00000 1.82917 1.00000 0.517753
1.5 0.106359 1.00000 −1.98869 1.41292 0.106359 −1.00000 −0.424234 1.00000 0.150278
1.6 0.775848 1.00000 −1.39806 3.03444 0.775848 −1.00000 −2.63638 1.00000 2.35426
1.7 1.77930 1.00000 1.16591 −0.681820 1.77930 −1.00000 −1.48409 1.00000 −1.21316
1.8 1.98765 1.00000 1.95074 −2.85284 1.98765 −1.00000 −0.0979034 1.00000 −5.67044
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.8
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.bb 8
13.b even 2 1 3549.2.a.bd 8
13.f odd 12 2 273.2.bd.a 16
39.k even 12 2 819.2.ct.b 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
273.2.bd.a 16 13.f odd 12 2
819.2.ct.b 16 39.k even 12 2
3549.2.a.bb 8 1.a even 1 1 trivial
3549.2.a.bd 8 13.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3549))\):

\(T_{2}^{8} + \cdots\)
\(T_{5}^{8} + \cdots\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - 8 T - 16 T^{2} + 24 T^{3} + 25 T^{4} - 14 T^{5} - 9 T^{6} + 2 T^{7} + T^{8} \)
$3$ \( ( -1 + T )^{8} \)
$5$ \( -3 - 48 T - 56 T^{2} + 58 T^{3} + 59 T^{4} - 24 T^{5} - 15 T^{6} + 2 T^{7} + T^{8} \)
$7$ \( ( 1 + T )^{8} \)
$11$ \( -12 - 144 T - 35 T^{2} + 400 T^{3} + 101 T^{4} - 138 T^{5} - 12 T^{6} + 8 T^{7} + T^{8} \)
$13$ \( T^{8} \)
$17$ \( -53699 - 4370 T + 43050 T^{2} - 15808 T^{3} - 443 T^{4} + 798 T^{5} - 55 T^{6} - 10 T^{7} + T^{8} \)
$19$ \( 159172 + 164096 T + 41957 T^{2} - 8114 T^{3} - 5253 T^{4} - 592 T^{5} + 67 T^{6} + 18 T^{7} + T^{8} \)
$23$ \( 38032 + 40200 T - 27643 T^{2} - 2918 T^{3} + 2858 T^{4} - 2 T^{5} - 97 T^{6} + 2 T^{7} + T^{8} \)
$29$ \( -50387 + 51334 T + 78809 T^{2} + 21728 T^{3} - 1627 T^{4} - 1120 T^{5} - 58 T^{6} + 12 T^{7} + T^{8} \)
$31$ \( -2816 + 9632 T + 3573 T^{2} - 6884 T^{3} - 4532 T^{4} - 810 T^{5} + 17 T^{6} + 16 T^{7} + T^{8} \)
$37$ \( -127643 + 171632 T + 134702 T^{2} + 2404 T^{3} - 8869 T^{4} - 1004 T^{5} + 119 T^{6} + 24 T^{7} + T^{8} \)
$41$ \( 832 + 4032 T - 7872 T^{2} + 848 T^{3} + 2060 T^{4} + 120 T^{5} - 83 T^{6} - 4 T^{7} + T^{8} \)
$43$ \( 125556 - 104868 T - 79643 T^{2} + 21524 T^{3} + 5106 T^{4} - 918 T^{5} - 115 T^{6} + 10 T^{7} + T^{8} \)
$47$ \( 1156 - 4080 T - 7439 T^{2} + 5014 T^{3} + 1050 T^{4} - 478 T^{5} - 61 T^{6} + 10 T^{7} + T^{8} \)
$53$ \( -76707 + 89910 T - 5832 T^{2} - 19170 T^{3} + 2754 T^{4} + 1026 T^{5} - 168 T^{6} - 6 T^{7} + T^{8} \)
$59$ \( -62348 - 63016 T - 3859 T^{2} + 11644 T^{3} + 2256 T^{4} - 520 T^{5} - 107 T^{6} + 6 T^{7} + T^{8} \)
$61$ \( -322283 + 394238 T - 27976 T^{2} - 56546 T^{3} + 9194 T^{4} + 1402 T^{5} - 232 T^{6} - 6 T^{7} + T^{8} \)
$67$ \( -76592 - 251776 T + 131593 T^{2} + 27600 T^{3} - 8962 T^{4} - 1708 T^{5} + 69 T^{6} + 24 T^{7} + T^{8} \)
$71$ \( 91909428 + 36254088 T + 2884885 T^{2} - 616474 T^{3} - 121193 T^{4} - 4616 T^{5} + 436 T^{6} + 42 T^{7} + T^{8} \)
$73$ \( -4160451 + 1577508 T + 719170 T^{2} - 69698 T^{3} - 42295 T^{4} - 3330 T^{5} + 189 T^{6} + 32 T^{7} + T^{8} \)
$79$ \( 75240852 + 5078148 T - 6074987 T^{2} + 70920 T^{3} + 99754 T^{4} - 974 T^{5} - 563 T^{6} + 2 T^{7} + T^{8} \)
$83$ \( 1463172 + 707736 T - 452555 T^{2} - 71404 T^{3} + 21819 T^{4} + 1410 T^{5} - 340 T^{6} - 2 T^{7} + T^{8} \)
$89$ \( -269692704 - 102273384 T - 3217523 T^{2} + 1633816 T^{3} + 102907 T^{4} - 7900 T^{5} - 593 T^{6} + 12 T^{7} + T^{8} \)
$97$ \( -970544 - 3544216 T - 2628447 T^{2} - 480428 T^{3} + 24664 T^{4} + 14622 T^{5} + 1505 T^{6} + 64 T^{7} + T^{8} \)
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