Properties

Label 2366.2.d.b
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: \(\Q(\sqrt{-1}) \)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 14)
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.

\(f(q)\) \(=\) \( q -i q^{2} -2 q^{3} - q^{4} + 2 i q^{6} -i q^{7} + i q^{8} + q^{9} +O(q^{10})\) \( q -i q^{2} -2 q^{3} - q^{4} + 2 i q^{6} -i q^{7} + i q^{8} + q^{9} + 2 q^{12} - q^{14} + q^{16} -6 q^{17} -i q^{18} + 2 i q^{19} + 2 i q^{21} -2 i q^{24} + 5 q^{25} + 4 q^{27} + i q^{28} -6 q^{29} -4 i q^{31} -i q^{32} + 6 i q^{34} - q^{36} -2 i q^{37} + 2 q^{38} + 6 i q^{41} + 2 q^{42} -8 q^{43} + 12 i q^{47} -2 q^{48} - q^{49} -5 i q^{50} + 12 q^{51} + 6 q^{53} -4 i q^{54} + q^{56} -4 i q^{57} + 6 i q^{58} + 6 i q^{59} + 8 q^{61} -4 q^{62} -i q^{63} - q^{64} -4 i q^{67} + 6 q^{68} + i q^{72} -2 i q^{73} -2 q^{74} -10 q^{75} -2 i q^{76} + 8 q^{79} -11 q^{81} + 6 q^{82} -6 i q^{83} -2 i q^{84} + 8 i q^{86} + 12 q^{87} + 6 i q^{89} + 8 i q^{93} + 12 q^{94} + 2 i q^{96} -10 i q^{97} + i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 4q^{3} - 2q^{4} + 2q^{9} + O(q^{10}) \) \( 2q - 4q^{3} - 2q^{4} + 2q^{9} + 4q^{12} - 2q^{14} + 2q^{16} - 12q^{17} + 10q^{25} + 8q^{27} - 12q^{29} - 2q^{36} + 4q^{38} + 4q^{42} - 16q^{43} - 4q^{48} - 2q^{49} + 24q^{51} + 12q^{53} + 2q^{56} + 16q^{61} - 8q^{62} - 2q^{64} + 12q^{68} - 4q^{74} - 20q^{75} + 16q^{79} - 22q^{81} + 12q^{82} + 24q^{87} + 24q^{94} + O(q^{100}) \)

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 −2.00000 −1.00000 0 2.00000i 1.00000i 1.00000i 1.00000 0
337.2 1.00000i −2.00000 −1.00000 0 2.00000i 1.00000i 1.00000i 1.00000 0
\(n\): 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.b 2
13.b even 2 1 inner 2366.2.d.b 2
13.d odd 4 1 14.2.a.a 1
13.d odd 4 1 2366.2.a.j 1
39.f even 4 1 126.2.a.b 1
52.f even 4 1 112.2.a.c 1
65.f even 4 1 350.2.c.d 2
65.g odd 4 1 350.2.a.f 1
65.k even 4 1 350.2.c.d 2
91.i even 4 1 98.2.a.a 1
91.z odd 12 2 98.2.c.b 2
91.bb even 12 2 98.2.c.a 2
104.j odd 4 1 448.2.a.g 1
104.m even 4 1 448.2.a.a 1
117.y odd 12 2 1134.2.f.l 2
117.z even 12 2 1134.2.f.f 2
143.g even 4 1 1694.2.a.e 1
156.l odd 4 1 1008.2.a.h 1
195.j odd 4 1 3150.2.g.j 2
195.n even 4 1 3150.2.a.i 1
195.u odd 4 1 3150.2.g.j 2
208.l even 4 1 1792.2.b.g 2
208.m odd 4 1 1792.2.b.c 2
208.r odd 4 1 1792.2.b.c 2
208.s even 4 1 1792.2.b.g 2
221.g odd 4 1 4046.2.a.f 1
247.i even 4 1 5054.2.a.c 1
260.l odd 4 1 2800.2.g.h 2
260.s odd 4 1 2800.2.g.h 2
260.u even 4 1 2800.2.a.g 1
273.o odd 4 1 882.2.a.i 1
273.cb odd 12 2 882.2.g.d 2
273.cd even 12 2 882.2.g.c 2
299.g even 4 1 7406.2.a.a 1
312.w odd 4 1 4032.2.a.r 1
312.y even 4 1 4032.2.a.w 1
364.p odd 4 1 784.2.a.b 1
364.bw odd 12 2 784.2.i.i 2
364.ce even 12 2 784.2.i.c 2
455.n odd 4 1 2450.2.c.c 2
455.u even 4 1 2450.2.a.t 1
455.w odd 4 1 2450.2.c.c 2
728.x odd 4 1 3136.2.a.z 1
728.ba even 4 1 3136.2.a.e 1
1092.u even 4 1 7056.2.a.bd 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
14.2.a.a 1 13.d odd 4 1
98.2.a.a 1 91.i even 4 1
98.2.c.a 2 91.bb even 12 2
98.2.c.b 2 91.z odd 12 2
112.2.a.c 1 52.f even 4 1
126.2.a.b 1 39.f even 4 1
350.2.a.f 1 65.g odd 4 1
350.2.c.d 2 65.f even 4 1
350.2.c.d 2 65.k even 4 1
448.2.a.a 1 104.m even 4 1
448.2.a.g 1 104.j odd 4 1
784.2.a.b 1 364.p odd 4 1
784.2.i.c 2 364.ce even 12 2
784.2.i.i 2 364.bw odd 12 2
882.2.a.i 1 273.o odd 4 1
882.2.g.c 2 273.cd even 12 2
882.2.g.d 2 273.cb odd 12 2
1008.2.a.h 1 156.l odd 4 1
1134.2.f.f 2 117.z even 12 2
1134.2.f.l 2 117.y odd 12 2
1694.2.a.e 1 143.g even 4 1
1792.2.b.c 2 208.m odd 4 1
1792.2.b.c 2 208.r odd 4 1
1792.2.b.g 2 208.l even 4 1
1792.2.b.g 2 208.s even 4 1
2366.2.a.j 1 13.d odd 4 1
2366.2.d.b 2 1.a even 1 1 trivial
2366.2.d.b 2 13.b even 2 1 inner
2450.2.a.t 1 455.u even 4 1
2450.2.c.c 2 455.n odd 4 1
2450.2.c.c 2 455.w odd 4 1
2800.2.a.g 1 260.u even 4 1
2800.2.g.h 2 260.l odd 4 1
2800.2.g.h 2 260.s odd 4 1
3136.2.a.e 1 728.ba even 4 1
3136.2.a.z 1 728.x odd 4 1
3150.2.a.i 1 195.n even 4 1
3150.2.g.j 2 195.j odd 4 1
3150.2.g.j 2 195.u odd 4 1
4032.2.a.r 1 312.w odd 4 1
4032.2.a.w 1 312.y even 4 1
4046.2.a.f 1 221.g odd 4 1
5054.2.a.c 1 247.i even 4 1
7056.2.a.bd 1 1092.u even 4 1
7406.2.a.a 1 299.g even 4 1

Hecke kernels

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

\( T_{3} + 2 \)
\( T_{5} \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 + T^{2} \)
$3$ \( ( 1 + 2 T + 3 T^{2} )^{2} \)
$5$ \( ( 1 - 5 T^{2} )^{2} \)
$7$ \( 1 + T^{2} \)
$11$ \( ( 1 - 11 T^{2} )^{2} \)
$13$ \( \)
$17$ \( ( 1 + 6 T + 17 T^{2} )^{2} \)
$19$ \( 1 - 34 T^{2} + 361 T^{4} \)
$23$ \( ( 1 + 23 T^{2} )^{2} \)
$29$ \( ( 1 + 6 T + 29 T^{2} )^{2} \)
$31$ \( 1 - 46 T^{2} + 961 T^{4} \)
$37$ \( ( 1 - 12 T + 37 T^{2} )( 1 + 12 T + 37 T^{2} ) \)
$41$ \( 1 - 46 T^{2} + 1681 T^{4} \)
$43$ \( ( 1 + 8 T + 43 T^{2} )^{2} \)
$47$ \( 1 + 50 T^{2} + 2209 T^{4} \)
$53$ \( ( 1 - 6 T + 53 T^{2} )^{2} \)
$59$ \( 1 - 82 T^{2} + 3481 T^{4} \)
$61$ \( ( 1 - 8 T + 61 T^{2} )^{2} \)
$67$ \( 1 - 118 T^{2} + 4489 T^{4} \)
$71$ \( ( 1 - 71 T^{2} )^{2} \)
$73$ \( 1 - 142 T^{2} + 5329 T^{4} \)
$79$ \( ( 1 - 8 T + 79 T^{2} )^{2} \)
$83$ \( 1 - 130 T^{2} + 6889 T^{4} \)
$89$ \( 1 - 142 T^{2} + 7921 T^{4} \)
$97$ \( 1 - 94 T^{2} + 9409 T^{4} \)
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