Properties

Label 352.1.t.a
Level $352$
Weight $1$
Character orbit 352.t
Analytic conductor $0.176$
Analytic rank $0$
Dimension $4$
Projective image $D_{5}$
CM discriminant -8
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 352 = 2^{5} \cdot 11 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 352.t (of order \(10\), degree \(4\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.175670884447\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{10})\)
Defining polynomial: \(x^{4} - x^{3} + x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 88)
Projective image: \(D_{5}\)
Projective field: Galois closure of 5.1.937024.1

$q$-expansion

The \(q\)-expansion and trace form are shown below.

\(f(q)\) \(=\) \( q + ( \zeta_{10}^{3} - \zeta_{10}^{4} ) q^{3} + ( -\zeta_{10} + \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{9} +O(q^{10})\) \( q + ( \zeta_{10}^{3} - \zeta_{10}^{4} ) q^{3} + ( -\zeta_{10} + \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{9} + \zeta_{10} q^{11} + ( \zeta_{10}^{2} + \zeta_{10}^{4} ) q^{17} + ( -1 - \zeta_{10}^{2} ) q^{19} -\zeta_{10} q^{25} + ( -1 + \zeta_{10} - \zeta_{10}^{2} - \zeta_{10}^{4} ) q^{27} + ( 1 + \zeta_{10}^{4} ) q^{33} + ( -\zeta_{10}^{3} + \zeta_{10}^{4} ) q^{41} + ( -\zeta_{10}^{2} + \zeta_{10}^{3} ) q^{43} -\zeta_{10}^{3} q^{49} + ( -1 + \zeta_{10} - \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{51} + ( 1 - \zeta_{10} - \zeta_{10}^{3} + \zeta_{10}^{4} ) q^{57} + ( -1 + \zeta_{10}^{3} ) q^{59} + ( \zeta_{10} - \zeta_{10}^{4} ) q^{67} + ( -\zeta_{10} + \zeta_{10}^{2} ) q^{73} + ( -1 - \zeta_{10}^{4} ) q^{75} + ( 1 - \zeta_{10} + \zeta_{10}^{2} - \zeta_{10}^{3} + \zeta_{10}^{4} ) q^{81} + ( -1 + \zeta_{10} ) q^{83} + ( \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{89} + ( 1 + \zeta_{10}^{4} ) q^{97} + ( -\zeta_{10}^{2} + \zeta_{10}^{3} - \zeta_{10}^{4} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 2q^{3} - 3q^{9} + O(q^{10}) \) \( 4q + 2q^{3} - 3q^{9} + q^{11} - 2q^{17} - 3q^{19} - q^{25} - q^{27} + 3q^{33} - 2q^{41} + 2q^{43} - q^{49} - q^{51} + q^{57} - 3q^{59} + 2q^{67} - 2q^{73} - 3q^{75} - 3q^{83} - 2q^{89} + 3q^{97} + 3q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/352\mathbb{Z}\right)^\times\).

\(n\) \(133\) \(287\) \(321\)
\(\chi(n)\) \(-1\) \(-1\) \(\zeta_{10}^{4}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
15.1
−0.309017 + 0.951057i
−0.309017 0.951057i
0.809017 0.587785i
0.809017 + 0.587785i
0 0.500000 1.53884i 0 0 0 0 0 −1.30902 0.951057i 0
47.1 0 0.500000 + 1.53884i 0 0 0 0 0 −1.30902 + 0.951057i 0
207.1 0 0.500000 0.363271i 0 0 0 0 0 −0.190983 + 0.587785i 0
335.1 0 0.500000 + 0.363271i 0 0 0 0 0 −0.190983 0.587785i 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
8.d odd 2 1 CM by \(\Q(\sqrt{-2}) \)
11.c even 5 1 inner
88.l odd 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 352.1.t.a 4
3.b odd 2 1 3168.1.ck.a 4
4.b odd 2 1 88.1.l.a 4
8.b even 2 1 88.1.l.a 4
8.d odd 2 1 CM 352.1.t.a 4
11.b odd 2 1 3872.1.t.c 4
11.c even 5 1 inner 352.1.t.a 4
11.c even 5 1 3872.1.f.a 2
11.c even 5 2 3872.1.t.b 4
11.d odd 10 1 3872.1.f.b 2
11.d odd 10 2 3872.1.t.a 4
11.d odd 10 1 3872.1.t.c 4
12.b even 2 1 792.1.bu.a 4
16.e even 4 2 2816.1.v.c 8
16.f odd 4 2 2816.1.v.c 8
20.d odd 2 1 2200.1.cl.a 4
20.e even 4 2 2200.1.dd.a 8
24.f even 2 1 3168.1.ck.a 4
24.h odd 2 1 792.1.bu.a 4
33.h odd 10 1 3168.1.ck.a 4
40.f even 2 1 2200.1.cl.a 4
40.i odd 4 2 2200.1.dd.a 8
44.c even 2 1 968.1.l.b 4
44.g even 10 1 968.1.f.a 2
44.g even 10 1 968.1.l.b 4
44.g even 10 2 968.1.l.c 4
44.h odd 10 1 88.1.l.a 4
44.h odd 10 1 968.1.f.b 2
44.h odd 10 2 968.1.l.a 4
88.b odd 2 1 968.1.l.b 4
88.g even 2 1 3872.1.t.c 4
88.k even 10 1 3872.1.f.b 2
88.k even 10 2 3872.1.t.a 4
88.k even 10 1 3872.1.t.c 4
88.l odd 10 1 inner 352.1.t.a 4
88.l odd 10 1 3872.1.f.a 2
88.l odd 10 2 3872.1.t.b 4
88.o even 10 1 88.1.l.a 4
88.o even 10 1 968.1.f.b 2
88.o even 10 2 968.1.l.a 4
88.p odd 10 1 968.1.f.a 2
88.p odd 10 1 968.1.l.b 4
88.p odd 10 2 968.1.l.c 4
132.o even 10 1 792.1.bu.a 4
176.v odd 20 2 2816.1.v.c 8
176.w even 20 2 2816.1.v.c 8
220.n odd 10 1 2200.1.cl.a 4
220.v even 20 2 2200.1.dd.a 8
264.t odd 10 1 792.1.bu.a 4
264.w even 10 1 3168.1.ck.a 4
440.bd even 10 1 2200.1.cl.a 4
440.bp odd 20 2 2200.1.dd.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
88.1.l.a 4 4.b odd 2 1
88.1.l.a 4 8.b even 2 1
88.1.l.a 4 44.h odd 10 1
88.1.l.a 4 88.o even 10 1
352.1.t.a 4 1.a even 1 1 trivial
352.1.t.a 4 8.d odd 2 1 CM
352.1.t.a 4 11.c even 5 1 inner
352.1.t.a 4 88.l odd 10 1 inner
792.1.bu.a 4 12.b even 2 1
792.1.bu.a 4 24.h odd 2 1
792.1.bu.a 4 132.o even 10 1
792.1.bu.a 4 264.t odd 10 1
968.1.f.a 2 44.g even 10 1
968.1.f.a 2 88.p odd 10 1
968.1.f.b 2 44.h odd 10 1
968.1.f.b 2 88.o even 10 1
968.1.l.a 4 44.h odd 10 2
968.1.l.a 4 88.o even 10 2
968.1.l.b 4 44.c even 2 1
968.1.l.b 4 44.g even 10 1
968.1.l.b 4 88.b odd 2 1
968.1.l.b 4 88.p odd 10 1
968.1.l.c 4 44.g even 10 2
968.1.l.c 4 88.p odd 10 2
2200.1.cl.a 4 20.d odd 2 1
2200.1.cl.a 4 40.f even 2 1
2200.1.cl.a 4 220.n odd 10 1
2200.1.cl.a 4 440.bd even 10 1
2200.1.dd.a 8 20.e even 4 2
2200.1.dd.a 8 40.i odd 4 2
2200.1.dd.a 8 220.v even 20 2
2200.1.dd.a 8 440.bp odd 20 2
2816.1.v.c 8 16.e even 4 2
2816.1.v.c 8 16.f odd 4 2
2816.1.v.c 8 176.v odd 20 2
2816.1.v.c 8 176.w even 20 2
3168.1.ck.a 4 3.b odd 2 1
3168.1.ck.a 4 24.f even 2 1
3168.1.ck.a 4 33.h odd 10 1
3168.1.ck.a 4 264.w even 10 1
3872.1.f.a 2 11.c even 5 1
3872.1.f.a 2 88.l odd 10 1
3872.1.f.b 2 11.d odd 10 1
3872.1.f.b 2 88.k even 10 1
3872.1.t.a 4 11.d odd 10 2
3872.1.t.a 4 88.k even 10 2
3872.1.t.b 4 11.c even 5 2
3872.1.t.b 4 88.l odd 10 2
3872.1.t.c 4 11.b odd 2 1
3872.1.t.c 4 11.d odd 10 1
3872.1.t.c 4 88.g even 2 1
3872.1.t.c 4 88.k even 10 1

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(352, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( 1 - 3 T + 4 T^{2} - 2 T^{3} + T^{4} \)
$5$ \( T^{4} \)
$7$ \( T^{4} \)
$11$ \( 1 - T + T^{2} - T^{3} + T^{4} \)
$13$ \( T^{4} \)
$17$ \( 1 + 3 T + 4 T^{2} + 2 T^{3} + T^{4} \)
$19$ \( 1 + 2 T + 4 T^{2} + 3 T^{3} + T^{4} \)
$23$ \( T^{4} \)
$29$ \( T^{4} \)
$31$ \( T^{4} \)
$37$ \( T^{4} \)
$41$ \( 1 + 3 T + 4 T^{2} + 2 T^{3} + T^{4} \)
$43$ \( ( -1 - T + T^{2} )^{2} \)
$47$ \( T^{4} \)
$53$ \( T^{4} \)
$59$ \( 1 + 2 T + 4 T^{2} + 3 T^{3} + T^{4} \)
$61$ \( T^{4} \)
$67$ \( ( -1 - T + T^{2} )^{2} \)
$71$ \( T^{4} \)
$73$ \( 1 + 3 T + 4 T^{2} + 2 T^{3} + T^{4} \)
$79$ \( T^{4} \)
$83$ \( 1 + 2 T + 4 T^{2} + 3 T^{3} + T^{4} \)
$89$ \( ( -1 + T + T^{2} )^{2} \)
$97$ \( 1 - 2 T + 4 T^{2} - 3 T^{3} + T^{4} \)
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