Properties

Label 2366.2.a.z
Level 2366
Weight 2
Character orbit 2366.a
Self dual yes
Analytic conductor 18.893
Analytic rank 1
Dimension 3
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: \(1\)
Dimension: \(3\)
Coefficient field: \(\Q(\zeta_{14})^+\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
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 root \(\beta\) of the polynomial \(x^{3} - x^{2} - 2 x + 1\). We also show the integral \(q\)-expansion of the trace form.

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

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
1.80194
−1.24698
0.445042
1.00000 −3.24698 1.00000 3.60388 −3.24698 −1.00000 1.00000 7.54288 3.60388
1.2 1.00000 −1.55496 1.00000 −2.49396 −1.55496 −1.00000 1.00000 −0.582105 −2.49396
1.3 1.00000 −0.198062 1.00000 0.890084 −0.198062 −1.00000 1.00000 −2.96077 0.890084
\(n\): 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.z yes 3
13.b even 2 1 2366.2.a.u 3
13.d odd 4 2 2366.2.d.m 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2366.2.a.u 3 13.b even 2 1
2366.2.a.z yes 3 1.a even 1 1 trivial
2366.2.d.m 6 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(2366))\):

\( T_{3}^{3} + 5 T_{3}^{2} + 6 T_{3} + 1 \)
\( T_{5}^{3} - 2 T_{5}^{2} - 8 T_{5} + 8 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - T )^{3} \)
$3$ \( 1 + 5 T + 15 T^{2} + 31 T^{3} + 45 T^{4} + 45 T^{5} + 27 T^{6} \)
$5$ \( 1 - 2 T + 7 T^{2} - 12 T^{3} + 35 T^{4} - 50 T^{5} + 125 T^{6} \)
$7$ \( ( 1 + T )^{3} \)
$11$ \( 1 - 3 T + 15 T^{2} - 53 T^{3} + 165 T^{4} - 363 T^{5} + 1331 T^{6} \)
$13$ 1
$17$ \( 1 + 7 T + 65 T^{2} + 245 T^{3} + 1105 T^{4} + 2023 T^{5} + 4913 T^{6} \)
$19$ \( 1 + 11 T + 53 T^{2} + 207 T^{3} + 1007 T^{4} + 3971 T^{5} + 6859 T^{6} \)
$23$ \( 1 + 14 T + 125 T^{2} + 700 T^{3} + 2875 T^{4} + 7406 T^{5} + 12167 T^{6} \)
$29$ \( 1 - 2 T + 79 T^{2} - 108 T^{3} + 2291 T^{4} - 1682 T^{5} + 24389 T^{6} \)
$31$ \( 1 + 14 T + 149 T^{2} + 924 T^{3} + 4619 T^{4} + 13454 T^{5} + 29791 T^{6} \)
$37$ \( 1 + 8 T + 123 T^{2} + 584 T^{3} + 4551 T^{4} + 10952 T^{5} + 50653 T^{6} \)
$41$ \( 1 + 9 T + 59 T^{2} + 205 T^{3} + 2419 T^{4} + 15129 T^{5} + 68921 T^{6} \)
$43$ \( 1 - 19 T + 219 T^{2} - 1663 T^{3} + 9417 T^{4} - 35131 T^{5} + 79507 T^{6} \)
$47$ \( 1 - 14 T + 169 T^{2} - 1260 T^{3} + 7943 T^{4} - 30926 T^{5} + 103823 T^{6} \)
$53$ \( 1 + 14 T + 215 T^{2} + 1540 T^{3} + 11395 T^{4} + 39326 T^{5} + 148877 T^{6} \)
$59$ \( 1 + 9 T - 13 T^{2} - 745 T^{3} - 767 T^{4} + 31329 T^{5} + 205379 T^{6} \)
$61$ \( 1 + 4 T + 123 T^{2} + 256 T^{3} + 7503 T^{4} + 14884 T^{5} + 226981 T^{6} \)
$67$ \( 1 - 3 T + 141 T^{2} - 151 T^{3} + 9447 T^{4} - 13467 T^{5} + 300763 T^{6} \)
$71$ \( 1 + 8 T + 169 T^{2} + 792 T^{3} + 11999 T^{4} + 40328 T^{5} + 357911 T^{6} \)
$73$ \( 1 + 13 T + 203 T^{2} + 1731 T^{3} + 14819 T^{4} + 69277 T^{5} + 389017 T^{6} \)
$79$ \( 1 + 16 T + 201 T^{2} + 2296 T^{3} + 15879 T^{4} + 99856 T^{5} + 493039 T^{6} \)
$83$ \( 1 + 21 T + 333 T^{2} + 3577 T^{3} + 27639 T^{4} + 144669 T^{5} + 571787 T^{6} \)
$89$ \( 1 + 25 T + 431 T^{2} + 4547 T^{3} + 38359 T^{4} + 198025 T^{5} + 704969 T^{6} \)
$97$ \( 1 + 31 T + 609 T^{2} + 7093 T^{3} + 59073 T^{4} + 291679 T^{5} + 912673 T^{6} \)
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