Properties

Label 3042.3.d.a
Level $3042$
Weight $3$
Character orbit 3042.d
Analytic conductor $82.888$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 3042 = 2 \cdot 3^{2} \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 3042.d (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(82.8884964184\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
Defining polynomial: \(x^{4} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{2} \)
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 a primitive root of unity \(\zeta_{8}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( \zeta_{8} - \zeta_{8}^{3} ) q^{2} + 2 q^{4} + ( -3 \zeta_{8} + 3 \zeta_{8}^{3} ) q^{5} + 4 \zeta_{8}^{2} q^{7} + ( 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{8} +O(q^{10})\) \( q + ( \zeta_{8} - \zeta_{8}^{3} ) q^{2} + 2 q^{4} + ( -3 \zeta_{8} + 3 \zeta_{8}^{3} ) q^{5} + 4 \zeta_{8}^{2} q^{7} + ( 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{8} -6 q^{10} + ( -12 \zeta_{8} + 12 \zeta_{8}^{3} ) q^{11} + ( 4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{14} + 4 q^{16} + ( -9 \zeta_{8} - 9 \zeta_{8}^{3} ) q^{17} -16 \zeta_{8}^{2} q^{19} + ( -6 \zeta_{8} + 6 \zeta_{8}^{3} ) q^{20} -24 q^{22} + ( -12 \zeta_{8} - 12 \zeta_{8}^{3} ) q^{23} -7 q^{25} + 8 \zeta_{8}^{2} q^{28} + ( -3 \zeta_{8} - 3 \zeta_{8}^{3} ) q^{29} + 44 \zeta_{8}^{2} q^{31} + ( 4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{32} -18 \zeta_{8}^{2} q^{34} + ( -12 \zeta_{8} - 12 \zeta_{8}^{3} ) q^{35} + 34 \zeta_{8}^{2} q^{37} + ( -16 \zeta_{8} - 16 \zeta_{8}^{3} ) q^{38} -12 q^{40} + ( 33 \zeta_{8} - 33 \zeta_{8}^{3} ) q^{41} + 40 q^{43} + ( -24 \zeta_{8} + 24 \zeta_{8}^{3} ) q^{44} -24 \zeta_{8}^{2} q^{46} + ( 60 \zeta_{8} - 60 \zeta_{8}^{3} ) q^{47} + 33 q^{49} + ( -7 \zeta_{8} + 7 \zeta_{8}^{3} ) q^{50} + ( -27 \zeta_{8} - 27 \zeta_{8}^{3} ) q^{53} + 72 q^{55} + ( 8 \zeta_{8} + 8 \zeta_{8}^{3} ) q^{56} -6 \zeta_{8}^{2} q^{58} + ( -24 \zeta_{8} + 24 \zeta_{8}^{3} ) q^{59} + 50 q^{61} + ( 44 \zeta_{8} + 44 \zeta_{8}^{3} ) q^{62} + 8 q^{64} + 8 \zeta_{8}^{2} q^{67} + ( -18 \zeta_{8} - 18 \zeta_{8}^{3} ) q^{68} -24 \zeta_{8}^{2} q^{70} + ( -36 \zeta_{8} + 36 \zeta_{8}^{3} ) q^{71} + 16 \zeta_{8}^{2} q^{73} + ( 34 \zeta_{8} + 34 \zeta_{8}^{3} ) q^{74} -32 \zeta_{8}^{2} q^{76} + ( -48 \zeta_{8} - 48 \zeta_{8}^{3} ) q^{77} -76 q^{79} + ( -12 \zeta_{8} + 12 \zeta_{8}^{3} ) q^{80} + 66 q^{82} + ( 84 \zeta_{8} - 84 \zeta_{8}^{3} ) q^{83} + 54 \zeta_{8}^{2} q^{85} + ( 40 \zeta_{8} - 40 \zeta_{8}^{3} ) q^{86} -48 q^{88} + ( -9 \zeta_{8} + 9 \zeta_{8}^{3} ) q^{89} + ( -24 \zeta_{8} - 24 \zeta_{8}^{3} ) q^{92} + 120 q^{94} + ( 48 \zeta_{8} + 48 \zeta_{8}^{3} ) q^{95} + 176 \zeta_{8}^{2} q^{97} + ( 33 \zeta_{8} - 33 \zeta_{8}^{3} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 8q^{4} + O(q^{10}) \) \( 4q + 8q^{4} - 24q^{10} + 16q^{16} - 96q^{22} - 28q^{25} - 48q^{40} + 160q^{43} + 132q^{49} + 288q^{55} + 200q^{61} + 32q^{64} - 304q^{79} + 264q^{82} - 192q^{88} + 480q^{94} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3042\mathbb{Z}\right)^\times\).

\(n\) \(677\) \(847\)
\(\chi(n)\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3041.1
−0.707107 + 0.707107i
−0.707107 0.707107i
0.707107 0.707107i
0.707107 + 0.707107i
−1.41421 0 2.00000 4.24264 0 4.00000i −2.82843 0 −6.00000
3041.2 −1.41421 0 2.00000 4.24264 0 4.00000i −2.82843 0 −6.00000
3041.3 1.41421 0 2.00000 −4.24264 0 4.00000i 2.82843 0 −6.00000
3041.4 1.41421 0 2.00000 −4.24264 0 4.00000i 2.82843 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
13.b even 2 1 inner
39.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3042.3.d.a 4
3.b odd 2 1 inner 3042.3.d.a 4
13.b even 2 1 inner 3042.3.d.a 4
13.d odd 4 1 18.3.b.a 2
13.d odd 4 1 3042.3.c.e 2
39.d odd 2 1 inner 3042.3.d.a 4
39.f even 4 1 18.3.b.a 2
39.f even 4 1 3042.3.c.e 2
52.f even 4 1 144.3.e.b 2
65.f even 4 1 450.3.b.b 4
65.g odd 4 1 450.3.d.f 2
65.k even 4 1 450.3.b.b 4
91.i even 4 1 882.3.b.a 2
91.z odd 12 2 882.3.s.b 4
91.bb even 12 2 882.3.s.d 4
104.j odd 4 1 576.3.e.c 2
104.m even 4 1 576.3.e.f 2
117.y odd 12 2 162.3.d.b 4
117.z even 12 2 162.3.d.b 4
143.g even 4 1 2178.3.c.d 2
156.l odd 4 1 144.3.e.b 2
195.j odd 4 1 450.3.b.b 4
195.n even 4 1 450.3.d.f 2
195.u odd 4 1 450.3.b.b 4
208.l even 4 1 2304.3.h.c 4
208.m odd 4 1 2304.3.h.f 4
208.r odd 4 1 2304.3.h.f 4
208.s even 4 1 2304.3.h.c 4
260.l odd 4 1 3600.3.c.b 4
260.s odd 4 1 3600.3.c.b 4
260.u even 4 1 3600.3.l.d 2
273.o odd 4 1 882.3.b.a 2
273.cb odd 12 2 882.3.s.d 4
273.cd even 12 2 882.3.s.b 4
312.w odd 4 1 576.3.e.f 2
312.y even 4 1 576.3.e.c 2
429.l odd 4 1 2178.3.c.d 2
468.bs even 12 2 1296.3.q.f 4
468.ch odd 12 2 1296.3.q.f 4
624.s odd 4 1 2304.3.h.c 4
624.u even 4 1 2304.3.h.f 4
624.bm even 4 1 2304.3.h.f 4
624.bo odd 4 1 2304.3.h.c 4
780.u even 4 1 3600.3.c.b 4
780.bb odd 4 1 3600.3.l.d 2
780.bn even 4 1 3600.3.c.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
18.3.b.a 2 13.d odd 4 1
18.3.b.a 2 39.f even 4 1
144.3.e.b 2 52.f even 4 1
144.3.e.b 2 156.l odd 4 1
162.3.d.b 4 117.y odd 12 2
162.3.d.b 4 117.z even 12 2
450.3.b.b 4 65.f even 4 1
450.3.b.b 4 65.k even 4 1
450.3.b.b 4 195.j odd 4 1
450.3.b.b 4 195.u odd 4 1
450.3.d.f 2 65.g odd 4 1
450.3.d.f 2 195.n even 4 1
576.3.e.c 2 104.j odd 4 1
576.3.e.c 2 312.y even 4 1
576.3.e.f 2 104.m even 4 1
576.3.e.f 2 312.w odd 4 1
882.3.b.a 2 91.i even 4 1
882.3.b.a 2 273.o odd 4 1
882.3.s.b 4 91.z odd 12 2
882.3.s.b 4 273.cd even 12 2
882.3.s.d 4 91.bb even 12 2
882.3.s.d 4 273.cb odd 12 2
1296.3.q.f 4 468.bs even 12 2
1296.3.q.f 4 468.ch odd 12 2
2178.3.c.d 2 143.g even 4 1
2178.3.c.d 2 429.l odd 4 1
2304.3.h.c 4 208.l even 4 1
2304.3.h.c 4 208.s even 4 1
2304.3.h.c 4 624.s odd 4 1
2304.3.h.c 4 624.bo odd 4 1
2304.3.h.f 4 208.m odd 4 1
2304.3.h.f 4 208.r odd 4 1
2304.3.h.f 4 624.u even 4 1
2304.3.h.f 4 624.bm even 4 1
3042.3.c.e 2 13.d odd 4 1
3042.3.c.e 2 39.f even 4 1
3042.3.d.a 4 1.a even 1 1 trivial
3042.3.d.a 4 3.b odd 2 1 inner
3042.3.d.a 4 13.b even 2 1 inner
3042.3.d.a 4 39.d odd 2 1 inner
3600.3.c.b 4 260.l odd 4 1
3600.3.c.b 4 260.s odd 4 1
3600.3.c.b 4 780.u even 4 1
3600.3.c.b 4 780.bn even 4 1
3600.3.l.d 2 260.u even 4 1
3600.3.l.d 2 780.bb odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} - 18 \) acting on \(S_{3}^{\mathrm{new}}(3042, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -2 + T^{2} )^{2} \)
$3$ \( T^{4} \)
$5$ \( ( -18 + T^{2} )^{2} \)
$7$ \( ( 16 + T^{2} )^{2} \)
$11$ \( ( -288 + T^{2} )^{2} \)
$13$ \( T^{4} \)
$17$ \( ( 162 + T^{2} )^{2} \)
$19$ \( ( 256 + T^{2} )^{2} \)
$23$ \( ( 288 + T^{2} )^{2} \)
$29$ \( ( 18 + T^{2} )^{2} \)
$31$ \( ( 1936 + T^{2} )^{2} \)
$37$ \( ( 1156 + T^{2} )^{2} \)
$41$ \( ( -2178 + T^{2} )^{2} \)
$43$ \( ( -40 + T )^{4} \)
$47$ \( ( -7200 + T^{2} )^{2} \)
$53$ \( ( 1458 + T^{2} )^{2} \)
$59$ \( ( -1152 + T^{2} )^{2} \)
$61$ \( ( -50 + T )^{4} \)
$67$ \( ( 64 + T^{2} )^{2} \)
$71$ \( ( -2592 + T^{2} )^{2} \)
$73$ \( ( 256 + T^{2} )^{2} \)
$79$ \( ( 76 + T )^{4} \)
$83$ \( ( -14112 + T^{2} )^{2} \)
$89$ \( ( -162 + T^{2} )^{2} \)
$97$ \( ( 30976 + T^{2} )^{2} \)
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