Properties

Label 484.2.e.a
Level $484$
Weight $2$
Character orbit 484.e
Analytic conductor $3.865$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 484 = 2^{2} \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 484.e (of order \(5\), degree \(4\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(3.86475945783\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{10})\)
Defining polynomial: \( x^{4} - x^{3} + x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 44)
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{10}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \zeta_{10}^{2} q^{3} + 3 \zeta_{10} q^{5} - 2 \zeta_{10}^{3} q^{7} + ( - 2 \zeta_{10}^{3} + 2 \zeta_{10}^{2} - 2 \zeta_{10} + 2) q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + \zeta_{10}^{2} q^{3} + 3 \zeta_{10} q^{5} - 2 \zeta_{10}^{3} q^{7} + ( - 2 \zeta_{10}^{3} + 2 \zeta_{10}^{2} - 2 \zeta_{10} + 2) q^{9} + ( - 4 \zeta_{10}^{3} + 4 \zeta_{10}^{2} - 4 \zeta_{10} + 4) q^{13} + 3 \zeta_{10}^{3} q^{15} - 6 \zeta_{10} q^{17} + 8 \zeta_{10}^{2} q^{19} + 2 q^{21} - 3 q^{23} + 4 \zeta_{10}^{2} q^{25} + 5 \zeta_{10} q^{27} + (5 \zeta_{10}^{3} - 5 \zeta_{10}^{2} + 5 \zeta_{10} - 5) q^{31} + ( - 6 \zeta_{10}^{3} + 6 \zeta_{10}^{2} - 6 \zeta_{10} + 6) q^{35} + \zeta_{10}^{3} q^{37} + 4 \zeta_{10} q^{39} - 10 q^{43} + 6 q^{45} + 3 \zeta_{10} q^{49} - 6 \zeta_{10}^{3} q^{51} + ( - 6 \zeta_{10}^{3} + 6 \zeta_{10}^{2} - 6 \zeta_{10} + 6) q^{53} + (8 \zeta_{10}^{3} - 8 \zeta_{10}^{2} + 8 \zeta_{10} - 8) q^{57} - 3 \zeta_{10}^{3} q^{59} + 4 \zeta_{10} q^{61} - 4 \zeta_{10}^{2} q^{63} + 12 q^{65} - q^{67} - 3 \zeta_{10}^{2} q^{69} - 15 \zeta_{10} q^{71} + 4 \zeta_{10}^{3} q^{73} + (4 \zeta_{10}^{3} - 4 \zeta_{10}^{2} + 4 \zeta_{10} - 4) q^{75} + (2 \zeta_{10}^{3} - 2 \zeta_{10}^{2} + 2 \zeta_{10} - 2) q^{79} - \zeta_{10}^{3} q^{81} - 6 \zeta_{10} q^{83} - 18 \zeta_{10}^{2} q^{85} - 9 q^{89} - 8 \zeta_{10}^{2} q^{91} - 5 \zeta_{10} q^{93} + 24 \zeta_{10}^{3} q^{95} + ( - 7 \zeta_{10}^{3} + 7 \zeta_{10}^{2} - 7 \zeta_{10} + 7) q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - q^{3} + 3 q^{5} - 2 q^{7} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - q^{3} + 3 q^{5} - 2 q^{7} + 2 q^{9} + 4 q^{13} + 3 q^{15} - 6 q^{17} - 8 q^{19} + 8 q^{21} - 12 q^{23} - 4 q^{25} + 5 q^{27} - 5 q^{31} + 6 q^{35} + q^{37} + 4 q^{39} - 40 q^{43} + 24 q^{45} + 3 q^{49} - 6 q^{51} + 6 q^{53} - 8 q^{57} - 3 q^{59} + 4 q^{61} + 4 q^{63} + 48 q^{65} - 4 q^{67} + 3 q^{69} - 15 q^{71} + 4 q^{73} - 4 q^{75} - 2 q^{79} - q^{81} - 6 q^{83} + 18 q^{85} - 36 q^{89} + 8 q^{91} - 5 q^{93} + 24 q^{95} + 7 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/484\mathbb{Z}\right)^\times\).

\(n\) \(243\) \(365\)
\(\chi(n)\) \(1\) \(-\zeta_{10}^{3}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
9.1
−0.309017 0.951057i
0.809017 0.587785i
0.809017 + 0.587785i
−0.309017 + 0.951057i
0 −0.809017 + 0.587785i 0 −0.927051 2.85317i 0 −1.61803 1.17557i 0 −0.618034 + 1.90211i 0
81.1 0 0.309017 0.951057i 0 2.42705 1.76336i 0 0.618034 + 1.90211i 0 1.61803 + 1.17557i 0
245.1 0 0.309017 + 0.951057i 0 2.42705 + 1.76336i 0 0.618034 1.90211i 0 1.61803 1.17557i 0
269.1 0 −0.809017 0.587785i 0 −0.927051 + 2.85317i 0 −1.61803 + 1.17557i 0 −0.618034 1.90211i 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
11.c even 5 3 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 484.2.e.a 4
11.b odd 2 1 484.2.e.b 4
11.c even 5 1 44.2.a.a 1
11.c even 5 3 inner 484.2.e.a 4
11.d odd 10 1 484.2.a.a 1
11.d odd 10 3 484.2.e.b 4
33.f even 10 1 4356.2.a.j 1
33.h odd 10 1 396.2.a.c 1
44.g even 10 1 1936.2.a.c 1
44.h odd 10 1 176.2.a.a 1
55.j even 10 1 1100.2.a.b 1
55.k odd 20 2 1100.2.b.c 2
77.j odd 10 1 2156.2.a.a 1
77.m even 15 2 2156.2.i.b 2
77.p odd 30 2 2156.2.i.c 2
88.k even 10 1 7744.2.a.bc 1
88.l odd 10 1 704.2.a.i 1
88.o even 10 1 704.2.a.f 1
88.p odd 10 1 7744.2.a.m 1
99.m even 15 2 3564.2.i.j 2
99.n odd 30 2 3564.2.i.a 2
132.o even 10 1 1584.2.a.p 1
143.n even 10 1 7436.2.a.d 1
165.o odd 10 1 9900.2.a.h 1
165.v even 20 2 9900.2.c.g 2
176.v odd 20 2 2816.2.c.k 2
176.w even 20 2 2816.2.c.e 2
220.n odd 10 1 4400.2.a.v 1
220.v even 20 2 4400.2.b.k 2
264.t odd 10 1 6336.2.a.j 1
264.w even 10 1 6336.2.a.i 1
308.t even 10 1 8624.2.a.w 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
44.2.a.a 1 11.c even 5 1
176.2.a.a 1 44.h odd 10 1
396.2.a.c 1 33.h odd 10 1
484.2.a.a 1 11.d odd 10 1
484.2.e.a 4 1.a even 1 1 trivial
484.2.e.a 4 11.c even 5 3 inner
484.2.e.b 4 11.b odd 2 1
484.2.e.b 4 11.d odd 10 3
704.2.a.f 1 88.o even 10 1
704.2.a.i 1 88.l odd 10 1
1100.2.a.b 1 55.j even 10 1
1100.2.b.c 2 55.k odd 20 2
1584.2.a.p 1 132.o even 10 1
1936.2.a.c 1 44.g even 10 1
2156.2.a.a 1 77.j odd 10 1
2156.2.i.b 2 77.m even 15 2
2156.2.i.c 2 77.p odd 30 2
2816.2.c.e 2 176.w even 20 2
2816.2.c.k 2 176.v odd 20 2
3564.2.i.a 2 99.n odd 30 2
3564.2.i.j 2 99.m even 15 2
4356.2.a.j 1 33.f even 10 1
4400.2.a.v 1 220.n odd 10 1
4400.2.b.k 2 220.v even 20 2
6336.2.a.i 1 264.w even 10 1
6336.2.a.j 1 264.t odd 10 1
7436.2.a.d 1 143.n even 10 1
7744.2.a.m 1 88.p odd 10 1
7744.2.a.bc 1 88.k even 10 1
8624.2.a.w 1 308.t even 10 1
9900.2.a.h 1 165.o odd 10 1
9900.2.c.g 2 165.v even 20 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(484, [\chi])\):

\( T_{3}^{4} + T_{3}^{3} + T_{3}^{2} + T_{3} + 1 \) Copy content Toggle raw display
\( T_{7}^{4} + 2T_{7}^{3} + 4T_{7}^{2} + 8T_{7} + 16 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} + T^{3} + T^{2} + T + 1 \) Copy content Toggle raw display
$5$ \( T^{4} - 3 T^{3} + 9 T^{2} - 27 T + 81 \) Copy content Toggle raw display
$7$ \( T^{4} + 2 T^{3} + 4 T^{2} + 8 T + 16 \) Copy content Toggle raw display
$11$ \( T^{4} \) Copy content Toggle raw display
$13$ \( T^{4} - 4 T^{3} + 16 T^{2} - 64 T + 256 \) Copy content Toggle raw display
$17$ \( T^{4} + 6 T^{3} + 36 T^{2} + \cdots + 1296 \) Copy content Toggle raw display
$19$ \( T^{4} + 8 T^{3} + 64 T^{2} + \cdots + 4096 \) Copy content Toggle raw display
$23$ \( (T + 3)^{4} \) Copy content Toggle raw display
$29$ \( T^{4} \) Copy content Toggle raw display
$31$ \( T^{4} + 5 T^{3} + 25 T^{2} + 125 T + 625 \) Copy content Toggle raw display
$37$ \( T^{4} - T^{3} + T^{2} - T + 1 \) Copy content Toggle raw display
$41$ \( T^{4} \) Copy content Toggle raw display
$43$ \( (T + 10)^{4} \) Copy content Toggle raw display
$47$ \( T^{4} \) Copy content Toggle raw display
$53$ \( T^{4} - 6 T^{3} + 36 T^{2} + \cdots + 1296 \) Copy content Toggle raw display
$59$ \( T^{4} + 3 T^{3} + 9 T^{2} + 27 T + 81 \) Copy content Toggle raw display
$61$ \( T^{4} - 4 T^{3} + 16 T^{2} - 64 T + 256 \) Copy content Toggle raw display
$67$ \( (T + 1)^{4} \) Copy content Toggle raw display
$71$ \( T^{4} + 15 T^{3} + 225 T^{2} + \cdots + 50625 \) Copy content Toggle raw display
$73$ \( T^{4} - 4 T^{3} + 16 T^{2} - 64 T + 256 \) Copy content Toggle raw display
$79$ \( T^{4} + 2 T^{3} + 4 T^{2} + 8 T + 16 \) Copy content Toggle raw display
$83$ \( T^{4} + 6 T^{3} + 36 T^{2} + \cdots + 1296 \) Copy content Toggle raw display
$89$ \( (T + 9)^{4} \) Copy content Toggle raw display
$97$ \( T^{4} - 7 T^{3} + 49 T^{2} + \cdots + 2401 \) Copy content Toggle raw display
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