Properties

Label 2366.2.d.f
Level 2366
Weight 2
Character orbit 2366.d
Analytic conductor 18.893
Analytic rank 0
Dimension 2
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) \(=\) \( 2366 = 2 \cdot 7 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2366.d (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(18.8926051182\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 182)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2366\mathbb{Z}\right)^\times\).

\(n\) \(339\) \(2199\)
\(\chi(n)\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
337.1
1.00000i
1.00000i
1.00000i 1.00000 −1.00000 3.00000i 1.00000i 1.00000i 1.00000i −2.00000 3.00000
337.2 1.00000i 1.00000 −1.00000 3.00000i 1.00000i 1.00000i 1.00000i −2.00000 3.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2366.2.d.f 2
13.b even 2 1 inner 2366.2.d.f 2
13.d odd 4 1 2366.2.a.f 1
13.d odd 4 1 2366.2.a.n 1
13.f odd 12 2 182.2.g.b 2
39.k even 12 2 1638.2.r.c 2
52.l even 12 2 1456.2.s.e 2
91.w even 12 2 1274.2.h.i 2
91.x odd 12 2 1274.2.e.d 2
91.ba even 12 2 1274.2.e.i 2
91.bc even 12 2 1274.2.g.g 2
91.bd odd 12 2 1274.2.h.j 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
182.2.g.b 2 13.f odd 12 2
1274.2.e.d 2 91.x odd 12 2
1274.2.e.i 2 91.ba even 12 2
1274.2.g.g 2 91.bc even 12 2
1274.2.h.i 2 91.w even 12 2
1274.2.h.j 2 91.bd odd 12 2
1456.2.s.e 2 52.l even 12 2
1638.2.r.c 2 39.k even 12 2
2366.2.a.f 1 13.d odd 4 1
2366.2.a.n 1 13.d odd 4 1
2366.2.d.f 2 1.a even 1 1 trivial
2366.2.d.f 2 13.b even 2 1 inner

Hecke kernels

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

\( T_{3} - 1 \)
\( T_{5}^{2} + 9 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T^{2} \)
$3$ \( ( 1 - T + 3 T^{2} )^{2} \)
$5$ \( 1 - T^{2} + 25 T^{4} \)
$7$ \( 1 + T^{2} \)
$11$ \( ( 1 - 11 T^{2} )^{2} \)
$13$ 1
$17$ \( ( 1 + 6 T + 17 T^{2} )^{2} \)
$19$ \( 1 - 22 T^{2} + 361 T^{4} \)
$23$ \( ( 1 + 3 T + 23 T^{2} )^{2} \)
$29$ \( ( 1 - 6 T + 29 T^{2} )^{2} \)
$31$ \( 1 + 38 T^{2} + 961 T^{4} \)
$37$ \( 1 - 10 T^{2} + 1369 T^{4} \)
$41$ \( ( 1 - 41 T^{2} )^{2} \)
$43$ \( ( 1 + 8 T + 43 T^{2} )^{2} \)
$47$ \( 1 - 58 T^{2} + 2209 T^{4} \)
$53$ \( ( 1 - 12 T + 53 T^{2} )^{2} \)
$59$ \( 1 - 109 T^{2} + 3481 T^{4} \)
$61$ \( ( 1 - 11 T + 61 T^{2} )^{2} \)
$67$ \( 1 - 130 T^{2} + 4489 T^{4} \)
$71$ \( 1 - 133 T^{2} + 5041 T^{4} \)
$73$ \( 1 - 142 T^{2} + 5329 T^{4} \)
$79$ \( ( 1 + 4 T + 79 T^{2} )^{2} \)
$83$ \( ( 1 - 83 T^{2} )^{2} \)
$89$ \( 1 - 142 T^{2} + 7921 T^{4} \)
$97$ \( 1 - 190 T^{2} + 9409 T^{4} \)
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