Properties

Label 400.3.p.b
Level 400
Weight 3
Character orbit 400.p
Analytic conductor 10.899
Analytic rank 0
Dimension 2
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) \(=\) \( 400 = 2^{4} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 400.p (of order \(4\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(10.8992105744\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 10)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( -2 + 2 i ) q^{3} + ( 2 + 2 i ) q^{7} + i q^{9} +O(q^{10})\) \( q + ( -2 + 2 i ) q^{3} + ( 2 + 2 i ) q^{7} + i q^{9} + 8 q^{11} + ( -3 + 3 i ) q^{13} + ( -7 - 7 i ) q^{17} + 20 i q^{19} -8 q^{21} + ( -2 + 2 i ) q^{23} + ( -20 - 20 i ) q^{27} + 40 i q^{29} -52 q^{31} + ( -16 + 16 i ) q^{33} + ( 3 + 3 i ) q^{37} -12 i q^{39} -8 q^{41} + ( -42 + 42 i ) q^{43} + ( -18 - 18 i ) q^{47} -41 i q^{49} + 28 q^{51} + ( -53 + 53 i ) q^{53} + ( -40 - 40 i ) q^{57} + 20 i q^{59} -48 q^{61} + ( -2 + 2 i ) q^{63} + ( 62 + 62 i ) q^{67} -8 i q^{69} + 28 q^{71} + ( 47 - 47 i ) q^{73} + ( 16 + 16 i ) q^{77} + 71 q^{81} + ( 18 - 18 i ) q^{83} + ( -80 - 80 i ) q^{87} + 80 i q^{89} -12 q^{91} + ( 104 - 104 i ) q^{93} + ( 63 + 63 i ) q^{97} + 8 i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 4q^{3} + 4q^{7} + O(q^{10}) \) \( 2q - 4q^{3} + 4q^{7} + 16q^{11} - 6q^{13} - 14q^{17} - 16q^{21} - 4q^{23} - 40q^{27} - 104q^{31} - 32q^{33} + 6q^{37} - 16q^{41} - 84q^{43} - 36q^{47} + 56q^{51} - 106q^{53} - 80q^{57} - 96q^{61} - 4q^{63} + 124q^{67} + 56q^{71} + 94q^{73} + 32q^{77} + 142q^{81} + 36q^{83} - 160q^{87} - 24q^{91} + 208q^{93} + 126q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/400\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(177\) \(351\)
\(\chi(n)\) \(1\) \(i\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
193.1
1.00000i
1.00000i
0 −2.00000 2.00000i 0 0 0 2.00000 2.00000i 0 1.00000i 0
257.1 0 −2.00000 + 2.00000i 0 0 0 2.00000 + 2.00000i 0 1.00000i 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
5.c odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 400.3.p.b 2
4.b odd 2 1 50.3.c.c 2
5.b even 2 1 80.3.p.c 2
5.c odd 4 1 80.3.p.c 2
5.c odd 4 1 inner 400.3.p.b 2
12.b even 2 1 450.3.g.b 2
15.d odd 2 1 720.3.bh.c 2
15.e even 4 1 720.3.bh.c 2
20.d odd 2 1 10.3.c.a 2
20.e even 4 1 10.3.c.a 2
20.e even 4 1 50.3.c.c 2
40.e odd 2 1 320.3.p.h 2
40.f even 2 1 320.3.p.a 2
40.i odd 4 1 320.3.p.a 2
40.k even 4 1 320.3.p.h 2
60.h even 2 1 90.3.g.b 2
60.l odd 4 1 90.3.g.b 2
60.l odd 4 1 450.3.g.b 2
140.c even 2 1 490.3.f.b 2
140.j odd 4 1 490.3.f.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
10.3.c.a 2 20.d odd 2 1
10.3.c.a 2 20.e even 4 1
50.3.c.c 2 4.b odd 2 1
50.3.c.c 2 20.e even 4 1
80.3.p.c 2 5.b even 2 1
80.3.p.c 2 5.c odd 4 1
90.3.g.b 2 60.h even 2 1
90.3.g.b 2 60.l odd 4 1
320.3.p.a 2 40.f even 2 1
320.3.p.a 2 40.i odd 4 1
320.3.p.h 2 40.e odd 2 1
320.3.p.h 2 40.k even 4 1
400.3.p.b 2 1.a even 1 1 trivial
400.3.p.b 2 5.c odd 4 1 inner
450.3.g.b 2 12.b even 2 1
450.3.g.b 2 60.l odd 4 1
490.3.f.b 2 140.c even 2 1
490.3.f.b 2 140.j odd 4 1
720.3.bh.c 2 15.d odd 2 1
720.3.bh.c 2 15.e even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(400, [\chi])\):

\( T_{3}^{2} + 4 T_{3} + 8 \)
\( T_{7}^{2} - 4 T_{7} + 8 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 + 4 T + 8 T^{2} + 36 T^{3} + 81 T^{4} \)
$5$ 1
$7$ \( 1 - 4 T + 8 T^{2} - 196 T^{3} + 2401 T^{4} \)
$11$ \( ( 1 - 8 T + 121 T^{2} )^{2} \)
$13$ \( 1 + 6 T + 18 T^{2} + 1014 T^{3} + 28561 T^{4} \)
$17$ \( ( 1 - 16 T + 289 T^{2} )( 1 + 30 T + 289 T^{2} ) \)
$19$ \( 1 - 322 T^{2} + 130321 T^{4} \)
$23$ \( 1 + 4 T + 8 T^{2} + 2116 T^{3} + 279841 T^{4} \)
$29$ \( ( 1 - 42 T + 841 T^{2} )( 1 + 42 T + 841 T^{2} ) \)
$31$ \( ( 1 + 52 T + 961 T^{2} )^{2} \)
$37$ \( 1 - 6 T + 18 T^{2} - 8214 T^{3} + 1874161 T^{4} \)
$41$ \( ( 1 + 8 T + 1681 T^{2} )^{2} \)
$43$ \( 1 + 84 T + 3528 T^{2} + 155316 T^{3} + 3418801 T^{4} \)
$47$ \( 1 + 36 T + 648 T^{2} + 79524 T^{3} + 4879681 T^{4} \)
$53$ \( ( 1 + 53 T )^{2}( 1 + 2809 T^{2} ) \)
$59$ \( 1 - 6562 T^{2} + 12117361 T^{4} \)
$61$ \( ( 1 + 48 T + 3721 T^{2} )^{2} \)
$67$ \( 1 - 124 T + 7688 T^{2} - 556636 T^{3} + 20151121 T^{4} \)
$71$ \( ( 1 - 28 T + 5041 T^{2} )^{2} \)
$73$ \( 1 - 94 T + 4418 T^{2} - 500926 T^{3} + 28398241 T^{4} \)
$79$ \( ( 1 - 79 T )^{2}( 1 + 79 T )^{2} \)
$83$ \( 1 - 36 T + 648 T^{2} - 248004 T^{3} + 47458321 T^{4} \)
$89$ \( 1 - 9442 T^{2} + 62742241 T^{4} \)
$97$ \( 1 - 126 T + 7938 T^{2} - 1185534 T^{3} + 88529281 T^{4} \)
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