Properties

Label 2178.3.c.d
Level $2178$
Weight $3$
Character orbit 2178.c
Analytic conductor $59.346$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 2178 = 2 \cdot 3^{2} \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 2178.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(59.3462015777\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-2}) \)
Defining polynomial: \(x^{2} + 2\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 18)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{-2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{2} -2 q^{4} + 3 \beta q^{5} + 4 q^{7} -2 \beta q^{8} +O(q^{10})\) \( q + \beta q^{2} -2 q^{4} + 3 \beta q^{5} + 4 q^{7} -2 \beta q^{8} -6 q^{10} -8 q^{13} + 4 \beta q^{14} + 4 q^{16} -9 \beta q^{17} + 16 q^{19} -6 \beta q^{20} + 12 \beta q^{23} + 7 q^{25} -8 \beta q^{26} -8 q^{28} + 3 \beta q^{29} + 44 q^{31} + 4 \beta q^{32} + 18 q^{34} + 12 \beta q^{35} -34 q^{37} + 16 \beta q^{38} + 12 q^{40} + 33 \beta q^{41} + 40 q^{43} -24 q^{46} + 60 \beta q^{47} -33 q^{49} + 7 \beta q^{50} + 16 q^{52} -27 \beta q^{53} -8 \beta q^{56} -6 q^{58} -24 \beta q^{59} -50 q^{61} + 44 \beta q^{62} -8 q^{64} -24 \beta q^{65} + 8 q^{67} + 18 \beta q^{68} -24 q^{70} + 36 \beta q^{71} + 16 q^{73} -34 \beta q^{74} -32 q^{76} + 76 q^{79} + 12 \beta q^{80} -66 q^{82} + 84 \beta q^{83} + 54 q^{85} + 40 \beta q^{86} -9 \beta q^{89} -32 q^{91} -24 \beta q^{92} -120 q^{94} + 48 \beta q^{95} + 176 q^{97} -33 \beta q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 4q^{4} + 8q^{7} + O(q^{10}) \) \( 2q - 4q^{4} + 8q^{7} - 12q^{10} - 16q^{13} + 8q^{16} + 32q^{19} + 14q^{25} - 16q^{28} + 88q^{31} + 36q^{34} - 68q^{37} + 24q^{40} + 80q^{43} - 48q^{46} - 66q^{49} + 32q^{52} - 12q^{58} - 100q^{61} - 16q^{64} + 16q^{67} - 48q^{70} + 32q^{73} - 64q^{76} + 152q^{79} - 132q^{82} + 108q^{85} - 64q^{91} - 240q^{94} + 352q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2178\mathbb{Z}\right)^\times\).

\(n\) \(1333\) \(1937\)
\(\chi(n)\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
485.1
1.41421i
1.41421i
1.41421i 0 −2.00000 4.24264i 0 4.00000 2.82843i 0 −6.00000
485.2 1.41421i 0 −2.00000 4.24264i 0 4.00000 2.82843i 0 −6.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2178.3.c.d 2
3.b odd 2 1 inner 2178.3.c.d 2
11.b odd 2 1 18.3.b.a 2
33.d even 2 1 18.3.b.a 2
44.c even 2 1 144.3.e.b 2
55.d odd 2 1 450.3.d.f 2
55.e even 4 2 450.3.b.b 4
77.b even 2 1 882.3.b.a 2
77.h odd 6 2 882.3.s.b 4
77.i even 6 2 882.3.s.d 4
88.b odd 2 1 576.3.e.c 2
88.g even 2 1 576.3.e.f 2
99.g even 6 2 162.3.d.b 4
99.h odd 6 2 162.3.d.b 4
132.d odd 2 1 144.3.e.b 2
143.d odd 2 1 3042.3.c.e 2
143.g even 4 2 3042.3.d.a 4
165.d even 2 1 450.3.d.f 2
165.l odd 4 2 450.3.b.b 4
176.i even 4 2 2304.3.h.c 4
176.l odd 4 2 2304.3.h.f 4
220.g even 2 1 3600.3.l.d 2
220.i odd 4 2 3600.3.c.b 4
231.h odd 2 1 882.3.b.a 2
231.k odd 6 2 882.3.s.d 4
231.l even 6 2 882.3.s.b 4
264.m even 2 1 576.3.e.c 2
264.p odd 2 1 576.3.e.f 2
396.k even 6 2 1296.3.q.f 4
396.o odd 6 2 1296.3.q.f 4
429.e even 2 1 3042.3.c.e 2
429.l odd 4 2 3042.3.d.a 4
528.s odd 4 2 2304.3.h.c 4
528.x even 4 2 2304.3.h.f 4
660.g odd 2 1 3600.3.l.d 2
660.q even 4 2 3600.3.c.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
18.3.b.a 2 11.b odd 2 1
18.3.b.a 2 33.d even 2 1
144.3.e.b 2 44.c even 2 1
144.3.e.b 2 132.d odd 2 1
162.3.d.b 4 99.g even 6 2
162.3.d.b 4 99.h odd 6 2
450.3.b.b 4 55.e even 4 2
450.3.b.b 4 165.l odd 4 2
450.3.d.f 2 55.d odd 2 1
450.3.d.f 2 165.d even 2 1
576.3.e.c 2 88.b odd 2 1
576.3.e.c 2 264.m even 2 1
576.3.e.f 2 88.g even 2 1
576.3.e.f 2 264.p odd 2 1
882.3.b.a 2 77.b even 2 1
882.3.b.a 2 231.h odd 2 1
882.3.s.b 4 77.h odd 6 2
882.3.s.b 4 231.l even 6 2
882.3.s.d 4 77.i even 6 2
882.3.s.d 4 231.k odd 6 2
1296.3.q.f 4 396.k even 6 2
1296.3.q.f 4 396.o odd 6 2
2178.3.c.d 2 1.a even 1 1 trivial
2178.3.c.d 2 3.b odd 2 1 inner
2304.3.h.c 4 176.i even 4 2
2304.3.h.c 4 528.s odd 4 2
2304.3.h.f 4 176.l odd 4 2
2304.3.h.f 4 528.x even 4 2
3042.3.c.e 2 143.d odd 2 1
3042.3.c.e 2 429.e even 2 1
3042.3.d.a 4 143.g even 4 2
3042.3.d.a 4 429.l odd 4 2
3600.3.c.b 4 220.i odd 4 2
3600.3.c.b 4 660.q even 4 2
3600.3.l.d 2 220.g even 2 1
3600.3.l.d 2 660.g odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(2178, [\chi])\):

\( T_{5}^{2} + 18 \)
\( T_{7} - 4 \)
\( T_{13} + 8 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 2 + T^{2} \)
$3$ \( T^{2} \)
$5$ \( 18 + T^{2} \)
$7$ \( ( -4 + T )^{2} \)
$11$ \( T^{2} \)
$13$ \( ( 8 + T )^{2} \)
$17$ \( 162 + T^{2} \)
$19$ \( ( -16 + T )^{2} \)
$23$ \( 288 + T^{2} \)
$29$ \( 18 + T^{2} \)
$31$ \( ( -44 + T )^{2} \)
$37$ \( ( 34 + T )^{2} \)
$41$ \( 2178 + T^{2} \)
$43$ \( ( -40 + T )^{2} \)
$47$ \( 7200 + T^{2} \)
$53$ \( 1458 + T^{2} \)
$59$ \( 1152 + T^{2} \)
$61$ \( ( 50 + T )^{2} \)
$67$ \( ( -8 + T )^{2} \)
$71$ \( 2592 + T^{2} \)
$73$ \( ( -16 + T )^{2} \)
$79$ \( ( -76 + T )^{2} \)
$83$ \( 14112 + T^{2} \)
$89$ \( 162 + T^{2} \)
$97$ \( ( -176 + T )^{2} \)
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