Properties

Label 400.2.q.b
Level $400$
Weight $2$
Character orbit 400.q
Analytic conductor $3.194$
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: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 400.q (of order \(4\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(3.19401608085\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 16)
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.

\(f(q)\) \(=\) \( q + ( - i + 1) q^{2} + (i + 1) q^{3} - 2 i q^{4} + 2 q^{6} + 2 q^{7} + ( - 2 i - 2) q^{8} - i q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + ( - i + 1) q^{2} + (i + 1) q^{3} - 2 i q^{4} + 2 q^{6} + 2 q^{7} + ( - 2 i - 2) q^{8} - i q^{9} + (i + 1) q^{11} + ( - 2 i + 2) q^{12} + (i + 1) q^{13} + ( - 2 i + 2) q^{14} - 4 q^{16} - 2 i q^{17} + ( - i - 1) q^{18} + (3 i - 3) q^{19} + (2 i + 2) q^{21} + 2 q^{22} + 6 q^{23} - 4 i q^{24} + 2 q^{26} + ( - 4 i + 4) q^{27} - 4 i q^{28} + (3 i - 3) q^{29} - 8 q^{31} + (4 i - 4) q^{32} + 2 i q^{33} + ( - 2 i - 2) q^{34} - 2 q^{36} + (3 i - 3) q^{37} + 6 i q^{38} + 2 i q^{39} + 4 q^{42} + ( - 5 i + 5) q^{43} + ( - 2 i + 2) q^{44} + ( - 6 i + 6) q^{46} + 8 i q^{47} + ( - 4 i - 4) q^{48} - 3 q^{49} + ( - 2 i + 2) q^{51} + ( - 2 i + 2) q^{52} + (5 i - 5) q^{53} - 8 i q^{54} + ( - 4 i - 4) q^{56} - 6 q^{57} + 6 i q^{58} + (3 i + 3) q^{59} + (9 i - 9) q^{61} + (8 i - 8) q^{62} - 2 i q^{63} + 8 i q^{64} + (2 i + 2) q^{66} + ( - 5 i - 5) q^{67} - 4 q^{68} + (6 i + 6) q^{69} - 10 i q^{71} + (2 i - 2) q^{72} - 4 q^{73} + 6 i q^{74} + (6 i + 6) q^{76} + (2 i + 2) q^{77} + (2 i + 2) q^{78} + 5 q^{81} + (i + 1) q^{83} + ( - 4 i + 4) q^{84} - 10 i q^{86} - 6 q^{87} - 4 i q^{88} - 4 i q^{89} + (2 i + 2) q^{91} - 12 i q^{92} + ( - 8 i - 8) q^{93} + (8 i + 8) q^{94} - 8 q^{96} - 2 i q^{97} + (3 i - 3) q^{98} + ( - i + 1) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} + 2 q^{3} + 4 q^{6} + 4 q^{7} - 4 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} + 2 q^{3} + 4 q^{6} + 4 q^{7} - 4 q^{8} + 2 q^{11} + 4 q^{12} + 2 q^{13} + 4 q^{14} - 8 q^{16} - 2 q^{18} - 6 q^{19} + 4 q^{21} + 4 q^{22} + 12 q^{23} + 4 q^{26} + 8 q^{27} - 6 q^{29} - 16 q^{31} - 8 q^{32} - 4 q^{34} - 4 q^{36} - 6 q^{37} + 8 q^{42} + 10 q^{43} + 4 q^{44} + 12 q^{46} - 8 q^{48} - 6 q^{49} + 4 q^{51} + 4 q^{52} - 10 q^{53} - 8 q^{56} - 12 q^{57} + 6 q^{59} - 18 q^{61} - 16 q^{62} + 4 q^{66} - 10 q^{67} - 8 q^{68} + 12 q^{69} - 4 q^{72} - 8 q^{73} + 12 q^{76} + 4 q^{77} + 4 q^{78} + 10 q^{81} + 2 q^{83} + 8 q^{84} - 12 q^{87} + 4 q^{91} - 16 q^{93} + 16 q^{94} - 16 q^{96} - 6 q^{98} + 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
149.1
1.00000i
1.00000i
1.00000 1.00000i 1.00000 + 1.00000i 2.00000i 0 2.00000 2.00000 −2.00000 2.00000i 1.00000i 0
349.1 1.00000 + 1.00000i 1.00000 1.00000i 2.00000i 0 2.00000 2.00000 −2.00000 + 2.00000i 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
80.q even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 400.2.q.b 2
4.b odd 2 1 1600.2.q.a 2
5.b even 2 1 400.2.q.a 2
5.c odd 4 1 16.2.e.a 2
5.c odd 4 1 400.2.l.c 2
15.e even 4 1 144.2.k.a 2
16.e even 4 1 400.2.q.a 2
16.f odd 4 1 1600.2.q.b 2
20.d odd 2 1 1600.2.q.b 2
20.e even 4 1 64.2.e.a 2
20.e even 4 1 1600.2.l.a 2
35.f even 4 1 784.2.m.b 2
35.k even 12 2 784.2.x.c 4
35.l odd 12 2 784.2.x.f 4
40.i odd 4 1 128.2.e.b 2
40.k even 4 1 128.2.e.a 2
60.l odd 4 1 576.2.k.a 2
80.i odd 4 1 16.2.e.a 2
80.j even 4 1 128.2.e.a 2
80.j even 4 1 1600.2.l.a 2
80.k odd 4 1 1600.2.q.a 2
80.q even 4 1 inner 400.2.q.b 2
80.s even 4 1 64.2.e.a 2
80.t odd 4 1 128.2.e.b 2
80.t odd 4 1 400.2.l.c 2
120.q odd 4 1 1152.2.k.a 2
120.w even 4 1 1152.2.k.b 2
160.u even 8 2 1024.2.b.b 2
160.v odd 8 2 1024.2.a.b 2
160.ba even 8 2 1024.2.a.e 2
160.bb odd 8 2 1024.2.b.e 2
240.z odd 4 1 576.2.k.a 2
240.bb even 4 1 144.2.k.a 2
240.bd odd 4 1 1152.2.k.a 2
240.bf even 4 1 1152.2.k.b 2
480.br even 8 2 9216.2.a.d 2
480.ca odd 8 2 9216.2.a.s 2
560.bn even 4 1 784.2.m.b 2
560.ch even 12 2 784.2.x.c 4
560.cy odd 12 2 784.2.x.f 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
16.2.e.a 2 5.c odd 4 1
16.2.e.a 2 80.i odd 4 1
64.2.e.a 2 20.e even 4 1
64.2.e.a 2 80.s even 4 1
128.2.e.a 2 40.k even 4 1
128.2.e.a 2 80.j even 4 1
128.2.e.b 2 40.i odd 4 1
128.2.e.b 2 80.t odd 4 1
144.2.k.a 2 15.e even 4 1
144.2.k.a 2 240.bb even 4 1
400.2.l.c 2 5.c odd 4 1
400.2.l.c 2 80.t odd 4 1
400.2.q.a 2 5.b even 2 1
400.2.q.a 2 16.e even 4 1
400.2.q.b 2 1.a even 1 1 trivial
400.2.q.b 2 80.q even 4 1 inner
576.2.k.a 2 60.l odd 4 1
576.2.k.a 2 240.z odd 4 1
784.2.m.b 2 35.f even 4 1
784.2.m.b 2 560.bn even 4 1
784.2.x.c 4 35.k even 12 2
784.2.x.c 4 560.ch even 12 2
784.2.x.f 4 35.l odd 12 2
784.2.x.f 4 560.cy odd 12 2
1024.2.a.b 2 160.v odd 8 2
1024.2.a.e 2 160.ba even 8 2
1024.2.b.b 2 160.u even 8 2
1024.2.b.e 2 160.bb odd 8 2
1152.2.k.a 2 120.q odd 4 1
1152.2.k.a 2 240.bd odd 4 1
1152.2.k.b 2 120.w even 4 1
1152.2.k.b 2 240.bf even 4 1
1600.2.l.a 2 20.e even 4 1
1600.2.l.a 2 80.j even 4 1
1600.2.q.a 2 4.b odd 2 1
1600.2.q.a 2 80.k odd 4 1
1600.2.q.b 2 16.f odd 4 1
1600.2.q.b 2 20.d odd 2 1
9216.2.a.d 2 480.br even 8 2
9216.2.a.s 2 480.ca odd 8 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} - 2T_{3} + 2 \) acting on \(S_{2}^{\mathrm{new}}(400, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - 2T + 2 \) Copy content Toggle raw display
$3$ \( T^{2} - 2T + 2 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( (T - 2)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} - 2T + 2 \) Copy content Toggle raw display
$13$ \( T^{2} - 2T + 2 \) Copy content Toggle raw display
$17$ \( T^{2} + 4 \) Copy content Toggle raw display
$19$ \( T^{2} + 6T + 18 \) Copy content Toggle raw display
$23$ \( (T - 6)^{2} \) Copy content Toggle raw display
$29$ \( T^{2} + 6T + 18 \) Copy content Toggle raw display
$31$ \( (T + 8)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 6T + 18 \) Copy content Toggle raw display
$41$ \( T^{2} \) Copy content Toggle raw display
$43$ \( T^{2} - 10T + 50 \) Copy content Toggle raw display
$47$ \( T^{2} + 64 \) Copy content Toggle raw display
$53$ \( T^{2} + 10T + 50 \) Copy content Toggle raw display
$59$ \( T^{2} - 6T + 18 \) Copy content Toggle raw display
$61$ \( T^{2} + 18T + 162 \) Copy content Toggle raw display
$67$ \( T^{2} + 10T + 50 \) Copy content Toggle raw display
$71$ \( T^{2} + 100 \) Copy content Toggle raw display
$73$ \( (T + 4)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} \) Copy content Toggle raw display
$83$ \( T^{2} - 2T + 2 \) Copy content Toggle raw display
$89$ \( T^{2} + 16 \) Copy content Toggle raw display
$97$ \( T^{2} + 4 \) Copy content Toggle raw display
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