Properties

Label 2366.2.a.t
Level 2366
Weight 2
Character orbit 2366.a
Self dual yes
Analytic conductor 18.893
Analytic rank 0
Dimension 2
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: \(2\)
Coefficient field: \(\Q(\sqrt{3}) \)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 182)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{3}\). We also show the integral \(q\)-expansion of the trace form.

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

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
−1.73205
1.73205
1.00000 −1.73205 1.00000 0.267949 −1.73205 −1.00000 1.00000 0 0.267949
1.2 1.00000 1.73205 1.00000 3.73205 1.73205 −1.00000 1.00000 0 3.73205
\(n\): e.g. 2-40 or 990-1000
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.t 2
13.b even 2 1 2366.2.a.r 2
13.c even 3 2 182.2.g.e 4
13.d odd 4 2 2366.2.d.l 4
39.i odd 6 2 1638.2.r.v 4
52.j odd 6 2 1456.2.s.l 4
91.g even 3 2 1274.2.h.o 4
91.h even 3 2 1274.2.e.o 4
91.m odd 6 2 1274.2.h.p 4
91.n odd 6 2 1274.2.g.l 4
91.v odd 6 2 1274.2.e.p 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
182.2.g.e 4 13.c even 3 2
1274.2.e.o 4 91.h even 3 2
1274.2.e.p 4 91.v odd 6 2
1274.2.g.l 4 91.n odd 6 2
1274.2.h.o 4 91.g even 3 2
1274.2.h.p 4 91.m odd 6 2
1456.2.s.l 4 52.j odd 6 2
1638.2.r.v 4 39.i odd 6 2
2366.2.a.r 2 13.b even 2 1
2366.2.a.t 2 1.a even 1 1 trivial
2366.2.d.l 4 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(2366))\):

\( T_{3}^{2} - 3 \)
\( T_{5}^{2} - 4 T_{5} + 1 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - T )^{2} \)
$3$ \( 1 + 3 T^{2} + 9 T^{4} \)
$5$ \( 1 - 4 T + 11 T^{2} - 20 T^{3} + 25 T^{4} \)
$7$ \( ( 1 + T )^{2} \)
$11$ \( 1 + 4 T + 14 T^{2} + 44 T^{3} + 121 T^{4} \)
$13$ 1
$17$ \( 1 + 22 T^{2} + 289 T^{4} \)
$19$ \( 1 + 26 T^{2} + 361 T^{4} \)
$23$ \( 1 - 10 T + 59 T^{2} - 230 T^{3} + 529 T^{4} \)
$29$ \( 1 - 4 T + 14 T^{2} - 116 T^{3} + 841 T^{4} \)
$31$ \( 1 - 8 T + 66 T^{2} - 248 T^{3} + 961 T^{4} \)
$37$ \( 1 + 12 T + 98 T^{2} + 444 T^{3} + 1369 T^{4} \)
$41$ \( 1 - 12 T + 106 T^{2} - 492 T^{3} + 1681 T^{4} \)
$43$ \( ( 1 - 4 T + 43 T^{2} )^{2} \)
$47$ \( 1 + 4 T + 50 T^{2} + 188 T^{3} + 2209 T^{4} \)
$53$ \( 1 + 58 T^{2} + 2809 T^{4} \)
$59$ \( 1 - 16 T + 155 T^{2} - 944 T^{3} + 3481 T^{4} \)
$61$ \( 1 + 4 T + 99 T^{2} + 244 T^{3} + 3721 T^{4} \)
$67$ \( 1 + 4 T + 90 T^{2} + 268 T^{3} + 4489 T^{4} \)
$71$ \( 1 - 2 T + 131 T^{2} - 142 T^{3} + 5041 T^{4} \)
$73$ \( 1 + 8 T + 150 T^{2} + 584 T^{3} + 5329 T^{4} \)
$79$ \( 1 - 16 T + 174 T^{2} - 1264 T^{3} + 6241 T^{4} \)
$83$ \( 1 + 58 T^{2} + 6889 T^{4} \)
$89$ \( 1 + 24 T + 310 T^{2} + 2136 T^{3} + 7921 T^{4} \)
$97$ \( 1 + 20 T + 246 T^{2} + 1940 T^{3} + 9409 T^{4} \)
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