Properties

Label 2366.2.a.v
Level 2366
Weight 2
Character orbit 2366.a
Self dual yes
Analytic conductor 18.893
Analytic rank 0
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: \(0\)
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.

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

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
0.445042
−1.24698
−1.00000 −2.80194 1.00000 −0.554958 2.80194 −1.00000 −1.00000 4.85086 0.554958
1.2 −1.00000 −1.44504 1.00000 −2.24698 1.44504 −1.00000 −1.00000 −0.911854 2.24698
1.3 −1.00000 0.246980 1.00000 0.801938 −0.246980 −1.00000 −1.00000 −2.93900 −0.801938
\(n\): 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.v 3
13.b even 2 1 2366.2.a.ba yes 3
13.d odd 4 2 2366.2.d.n 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2366.2.a.v 3 1.a even 1 1 trivial
2366.2.a.ba yes 3 13.b even 2 1
2366.2.d.n 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(2366))\):

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T )^{3} \)
$3$ \( 1 + 4 T + 12 T^{2} + 23 T^{3} + 36 T^{4} + 36 T^{5} + 27 T^{6} \)
$5$ \( 1 + 2 T + 14 T^{2} + 19 T^{3} + 70 T^{4} + 50 T^{5} + 125 T^{6} \)
$7$ \( ( 1 + T )^{3} \)
$11$ \( 1 - 11 T + 71 T^{2} - 283 T^{3} + 781 T^{4} - 1331 T^{5} + 1331 T^{6} \)
$13$ 1
$17$ \( 1 + 2 T - 6 T^{2} - 3 T^{3} - 102 T^{4} + 578 T^{5} + 4913 T^{6} \)
$19$ \( 1 - 13 T + 111 T^{2} - 565 T^{3} + 2109 T^{4} - 4693 T^{5} + 6859 T^{6} \)
$23$ \( 1 + 7 T + 69 T^{2} + 273 T^{3} + 1587 T^{4} + 3703 T^{5} + 12167 T^{6} \)
$29$ \( 1 + 3 T + 41 T^{2} + 175 T^{3} + 1189 T^{4} + 2523 T^{5} + 24389 T^{6} \)
$31$ \( 1 - 2 T + 64 T^{2} - 53 T^{3} + 1984 T^{4} - 1922 T^{5} + 29791 T^{6} \)
$37$ \( 1 + 90 T^{2} + 7 T^{3} + 3330 T^{4} + 50653 T^{6} \)
$41$ \( 1 - 7 T + 25 T^{2} - 231 T^{3} + 1025 T^{4} - 11767 T^{5} + 68921 T^{6} \)
$43$ \( 1 + 17 T + 209 T^{2} + 1533 T^{3} + 8987 T^{4} + 31433 T^{5} + 79507 T^{6} \)
$47$ \( 1 + 5 T + 77 T^{2} + 499 T^{3} + 3619 T^{4} + 11045 T^{5} + 103823 T^{6} \)
$53$ \( 1 + 110 T^{2} + 91 T^{3} + 5830 T^{4} + 148877 T^{6} \)
$59$ \( 1 - 17 T + 271 T^{2} - 2175 T^{3} + 15989 T^{4} - 59177 T^{5} + 205379 T^{6} \)
$61$ \( 1 - 3 T + 137 T^{2} - 367 T^{3} + 8357 T^{4} - 11163 T^{5} + 226981 T^{6} \)
$67$ \( 1 - 21 T + 327 T^{2} - 3017 T^{3} + 21909 T^{4} - 94269 T^{5} + 300763 T^{6} \)
$71$ \( 1 + 16 T + 184 T^{2} + 1431 T^{3} + 13064 T^{4} + 80656 T^{5} + 357911 T^{6} \)
$73$ \( 1 + T + 203 T^{2} + 117 T^{3} + 14819 T^{4} + 5329 T^{5} + 389017 T^{6} \)
$79$ \( 1 - 10 T + 212 T^{2} - 1455 T^{3} + 16748 T^{4} - 62410 T^{5} + 493039 T^{6} \)
$83$ \( 1 - T + 149 T^{2} - 347 T^{3} + 12367 T^{4} - 6889 T^{5} + 571787 T^{6} \)
$89$ \( 1 + 4 T + 158 T^{2} + 473 T^{3} + 14062 T^{4} + 31684 T^{5} + 704969 T^{6} \)
$97$ \( 1 - 19 T + 311 T^{2} - 3225 T^{3} + 30167 T^{4} - 178771 T^{5} + 912673 T^{6} \)
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