Properties

Label 2366.2.a.bg
Level 2366
Weight 2
Character orbit 2366.a
Self dual yes
Analytic conductor 18.893
Analytic rank 0
Dimension 6
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 2366 = 2 \cdot 7 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2366.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(18.8926051182\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.6.6052921.1
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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6} - x^{5} - 11 x^{4} + 7 x^{3} + 33 x^{2} - 9 x - 27\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{4} - \nu^{3} - 8 \nu^{2} + 4 \nu + 12 \)\()/3\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{5} - 4 \nu^{4} - 8 \nu^{3} + 31 \nu^{2} + 12 \nu - 36 \)\()/9\)
\(\beta_{4}\)\(=\)\((\)\( 4 \nu^{5} - 4 \nu^{4} - 35 \nu^{3} + 19 \nu^{2} + 60 \nu \)\()/9\)
\(\beta_{5}\)\(=\)\((\)\( 4 \nu^{5} - 7 \nu^{4} - 32 \nu^{3} + 52 \nu^{2} + 39 \nu - 72 \)\()/9\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{5} - \beta_{4} + \beta_{2} + \beta_{1} + 4\)
\(\nu^{3}\)\(=\)\(\beta_{5} - 4 \beta_{3} - 3 \beta_{2} + 5 \beta_{1} + 4\)
\(\nu^{4}\)\(=\)\(9 \beta_{5} - 8 \beta_{4} - 4 \beta_{3} + 8 \beta_{2} + 9 \beta_{1} + 24\)
\(\nu^{5}\)\(=\)\(13 \beta_{5} - \beta_{4} - 39 \beta_{3} - 23 \beta_{2} + 33 \beta_{1} + 40\)

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
−2.46434
−1.52377
−1.05140
1.21736
1.96881
2.85334
1.00000 −2.46434 1.00000 3.19361 −2.46434 1.00000 1.00000 3.07299 3.19361
1.2 1.00000 −1.52377 1.00000 −1.45506 −1.52377 1.00000 1.00000 −0.678139 −1.45506
1.3 1.00000 −1.05140 1.00000 2.26985 −1.05140 1.00000 1.00000 −1.89456 2.26985
1.4 1.00000 1.21736 1.00000 −3.44059 1.21736 1.00000 1.00000 −1.51803 −3.44059
1.5 1.00000 1.96881 1.00000 2.90010 1.96881 1.00000 1.00000 0.876202 2.90010
1.6 1.00000 2.85334 1.00000 0.532083 2.85334 1.00000 1.00000 5.14154 0.532083
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.6
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 2366.2.a.bg yes 6
13.b even 2 1 2366.2.a.be 6
13.d odd 4 2 2366.2.d.q 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2366.2.a.be 6 13.b even 2 1
2366.2.a.bg yes 6 1.a even 1 1 trivial
2366.2.d.q 12 13.d odd 4 2

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(7\) \(-1\)
\(13\) \(-1\)

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2366))\):

\( T_{3}^{6} - T_{3}^{5} - 11 T_{3}^{4} + 7 T_{3}^{3} + 33 T_{3}^{2} - 9 T_{3} - 27 \)
\( T_{5}^{6} - 4 T_{5}^{5} - 11 T_{5}^{4} + 57 T_{5}^{3} - 14 T_{5}^{2} - 112 T_{5} + 56 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - T )^{6} \)
$3$ \( 1 - T + 7 T^{2} - 8 T^{3} + 36 T^{4} - 36 T^{5} + 117 T^{6} - 108 T^{7} + 324 T^{8} - 216 T^{9} + 567 T^{10} - 243 T^{11} + 729 T^{12} \)
$5$ \( 1 - 4 T + 19 T^{2} - 43 T^{3} + 141 T^{4} - 257 T^{5} + 766 T^{6} - 1285 T^{7} + 3525 T^{8} - 5375 T^{9} + 11875 T^{10} - 12500 T^{11} + 15625 T^{12} \)
$7$ \( ( 1 - T )^{6} \)
$11$ \( 1 - 6 T + 61 T^{2} - 270 T^{3} + 1565 T^{4} - 5398 T^{5} + 22427 T^{6} - 59378 T^{7} + 189365 T^{8} - 359370 T^{9} + 893101 T^{10} - 966306 T^{11} + 1771561 T^{12} \)
$13$ 1
$17$ \( 1 + 9 T + 101 T^{2} + 638 T^{3} + 4134 T^{4} + 19764 T^{5} + 92011 T^{6} + 335988 T^{7} + 1194726 T^{8} + 3134494 T^{9} + 8435621 T^{10} + 12778713 T^{11} + 24137569 T^{12} \)
$19$ \( 1 - 10 T + 97 T^{2} - 568 T^{3} + 3695 T^{4} - 17488 T^{5} + 89105 T^{6} - 332272 T^{7} + 1333895 T^{8} - 3895912 T^{9} + 12641137 T^{10} - 24760990 T^{11} + 47045881 T^{12} \)
$23$ \( 1 - 21 T + 244 T^{2} - 1974 T^{3} + 13037 T^{4} - 74557 T^{5} + 380660 T^{6} - 1714811 T^{7} + 6896573 T^{8} - 24017658 T^{9} + 68281204 T^{10} - 135163203 T^{11} + 148035889 T^{12} \)
$29$ \( 1 + T + 60 T^{2} - 38 T^{3} + 1813 T^{4} - 5663 T^{5} + 52956 T^{6} - 164227 T^{7} + 1524733 T^{8} - 926782 T^{9} + 42436860 T^{10} + 20511149 T^{11} + 594823321 T^{12} \)
$31$ \( 1 - 20 T + 321 T^{2} - 3425 T^{3} + 31041 T^{4} - 220905 T^{5} + 1365586 T^{6} - 6848055 T^{7} + 29830401 T^{8} - 102034175 T^{9} + 296450241 T^{10} - 572583020 T^{11} + 887503681 T^{12} \)
$37$ \( 1 - 16 T + 179 T^{2} - 1601 T^{3} + 12575 T^{4} - 87987 T^{5} + 557826 T^{6} - 3255519 T^{7} + 17215175 T^{8} - 81095453 T^{9} + 335474819 T^{10} - 1109503312 T^{11} + 2565726409 T^{12} \)
$41$ \( 1 - 2 T + 77 T^{2} + 18 T^{3} + 4625 T^{4} - 618 T^{5} + 215007 T^{6} - 25338 T^{7} + 7774625 T^{8} + 1240578 T^{9} + 217583597 T^{10} - 231712402 T^{11} + 4750104241 T^{12} \)
$43$ \( 1 - 2 T + 25 T^{2} - 136 T^{3} + 5603 T^{4} - 9604 T^{5} + 92961 T^{6} - 412972 T^{7} + 10359947 T^{8} - 10812952 T^{9} + 85470025 T^{10} - 294016886 T^{11} + 6321363049 T^{12} \)
$47$ \( 1 + 5 T + 164 T^{2} + 954 T^{3} + 15413 T^{4} + 75645 T^{5} + 908724 T^{6} + 3555315 T^{7} + 34047317 T^{8} + 99047142 T^{9} + 800267684 T^{10} + 1146725035 T^{11} + 10779215329 T^{12} \)
$53$ \( 1 - 28 T + 551 T^{2} - 7735 T^{3} + 88485 T^{4} - 826301 T^{5} + 6574358 T^{6} - 43793953 T^{7} + 248554365 T^{8} - 1151563595 T^{9} + 4347655031 T^{10} - 11709473804 T^{11} + 22164361129 T^{12} \)
$59$ \( 1 + 12 T + 243 T^{2} + 1492 T^{3} + 18371 T^{4} + 52238 T^{5} + 891263 T^{6} + 3082042 T^{7} + 63949451 T^{8} + 306425468 T^{9} + 2944518723 T^{10} + 8579091588 T^{11} + 42180533641 T^{12} \)
$61$ \( 1 + 27 T + 598 T^{2} + 8766 T^{3} + 111115 T^{4} + 1098335 T^{5} + 9564612 T^{6} + 66998435 T^{7} + 413458915 T^{8} + 1989715446 T^{9} + 8279812918 T^{10} + 22804100127 T^{11} + 51520374361 T^{12} \)
$67$ \( 1 - 16 T + 259 T^{2} - 2960 T^{3} + 34347 T^{4} - 316274 T^{5} + 2882699 T^{6} - 21190358 T^{7} + 154183683 T^{8} - 890258480 T^{9} + 5219140339 T^{10} - 21602001712 T^{11} + 90458382169 T^{12} \)
$71$ \( 1 + 169 T^{2} - 35 T^{3} + 11271 T^{4} + 4445 T^{5} + 581862 T^{6} + 315595 T^{7} + 56817111 T^{8} - 12526885 T^{9} + 4294574089 T^{10} + 128100283921 T^{12} \)
$73$ \( 1 - 38 T + 811 T^{2} - 12294 T^{3} + 150495 T^{4} - 1568896 T^{5} + 14314189 T^{6} - 114529408 T^{7} + 801987855 T^{8} - 4782574998 T^{9} + 23030973451 T^{10} - 78776720534 T^{11} + 151334226289 T^{12} \)
$79$ \( 1 - 6 T + 337 T^{2} - 1579 T^{3} + 54647 T^{4} - 210917 T^{5} + 5415446 T^{6} - 16662443 T^{7} + 341051927 T^{8} - 778508581 T^{9} + 13126177297 T^{10} - 18462338394 T^{11} + 243087455521 T^{12} \)
$83$ \( 1 + 6 T + 265 T^{2} + 1224 T^{3} + 35891 T^{4} + 128808 T^{5} + 3418761 T^{6} + 10691064 T^{7} + 247253099 T^{8} + 699867288 T^{9} + 12576455065 T^{10} + 23634243858 T^{11} + 326940373369 T^{12} \)
$89$ \( 1 + T + 229 T^{2} - 94 T^{3} + 31324 T^{4} + 21644 T^{5} + 3356869 T^{6} + 1926316 T^{7} + 248117404 T^{8} - 66267086 T^{9} + 14367973189 T^{10} + 5584059449 T^{11} + 496981290961 T^{12} \)
$97$ \( 1 - 16 T + 511 T^{2} - 6198 T^{3} + 113769 T^{4} - 1087256 T^{5} + 14310613 T^{6} - 105463832 T^{7} + 1070452521 T^{8} - 5656747254 T^{9} + 45238462591 T^{10} - 137397444112 T^{11} + 832972004929 T^{12} \)
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