Properties

Label 400.2.q.a
Level $400$
Weight $2$
Character orbit 400.q
Analytic conductor $3.194$
Analytic rank $1$
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: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \(x^{2} + 1\)
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.

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

Character values

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

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

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.a 2
4.b odd 2 1 1600.2.q.b 2
5.b even 2 1 400.2.q.b 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.b 2
16.f odd 4 1 1600.2.q.a 2
20.d odd 2 1 1600.2.q.a 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 1 128.2.e.b 2
80.i odd 4 1 400.2.l.c 2
80.j even 4 1 64.2.e.a 2
80.k odd 4 1 1600.2.q.b 2
80.q even 4 1 inner 400.2.q.a 2
80.s even 4 1 128.2.e.a 2
80.s even 4 1 1600.2.l.a 2
80.t odd 4 1 16.2.e.a 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.a.e 2
160.v odd 8 2 1024.2.b.e 2
160.ba even 8 2 1024.2.b.b 2
160.bb odd 8 2 1024.2.a.b 2
240.z odd 4 1 1152.2.k.a 2
240.bb even 4 1 1152.2.k.b 2
240.bd odd 4 1 576.2.k.a 2
240.bf even 4 1 144.2.k.a 2
480.bq odd 8 2 9216.2.a.s 2
480.cb even 8 2 9216.2.a.d 2
560.r even 4 1 784.2.m.b 2
560.cg odd 12 2 784.2.x.f 4
560.cz even 12 2 784.2.x.c 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.t odd 4 1
64.2.e.a 2 20.e even 4 1
64.2.e.a 2 80.j even 4 1
128.2.e.a 2 40.k even 4 1
128.2.e.a 2 80.s even 4 1
128.2.e.b 2 40.i odd 4 1
128.2.e.b 2 80.i odd 4 1
144.2.k.a 2 15.e even 4 1
144.2.k.a 2 240.bf even 4 1
400.2.l.c 2 5.c odd 4 1
400.2.l.c 2 80.i odd 4 1
400.2.q.a 2 1.a even 1 1 trivial
400.2.q.a 2 80.q even 4 1 inner
400.2.q.b 2 5.b even 2 1
400.2.q.b 2 16.e even 4 1
576.2.k.a 2 60.l odd 4 1
576.2.k.a 2 240.bd odd 4 1
784.2.m.b 2 35.f even 4 1
784.2.m.b 2 560.r even 4 1
784.2.x.c 4 35.k even 12 2
784.2.x.c 4 560.cz even 12 2
784.2.x.f 4 35.l odd 12 2
784.2.x.f 4 560.cg odd 12 2
1024.2.a.b 2 160.bb odd 8 2
1024.2.a.e 2 160.u even 8 2
1024.2.b.b 2 160.ba even 8 2
1024.2.b.e 2 160.v odd 8 2
1152.2.k.a 2 120.q odd 4 1
1152.2.k.a 2 240.z odd 4 1
1152.2.k.b 2 120.w even 4 1
1152.2.k.b 2 240.bb even 4 1
1600.2.l.a 2 20.e even 4 1
1600.2.l.a 2 80.s even 4 1
1600.2.q.a 2 16.f odd 4 1
1600.2.q.a 2 20.d odd 2 1
1600.2.q.b 2 4.b odd 2 1
1600.2.q.b 2 80.k odd 4 1
9216.2.a.d 2 480.cb even 8 2
9216.2.a.s 2 480.bq odd 8 2

Hecke kernels

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

Hecke characteristic polynomials

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