Properties

Label 400.4.c.c
Level $400$
Weight $4$
Character orbit 400.c
Analytic conductor $23.601$
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: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 400.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(23.6007640023\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 10)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + 4 \beta q^{3} - 2 \beta q^{7} - 37 q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + 4 \beta q^{3} - 2 \beta q^{7} - 37 q^{9} - 12 q^{11} - 29 \beta q^{13} - 33 \beta q^{17} - 100 q^{19} + 32 q^{21} - 66 \beta q^{23} - 40 \beta q^{27} + 90 q^{29} - 152 q^{31} - 48 \beta q^{33} + 17 \beta q^{37} + 464 q^{39} - 438 q^{41} - 16 \beta q^{43} - 102 \beta q^{47} + 327 q^{49} + 528 q^{51} + 111 \beta q^{53} - 400 \beta q^{57} + 420 q^{59} + 902 q^{61} + 74 \beta q^{63} - 512 \beta q^{67} + 1056 q^{69} - 432 q^{71} + 181 \beta q^{73} + 24 \beta q^{77} - 160 q^{79} - 359 q^{81} - 36 \beta q^{83} + 360 \beta q^{87} - 810 q^{89} - 232 q^{91} - 608 \beta q^{93} - 553 \beta q^{97} + 444 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 74 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 74 q^{9} - 24 q^{11} - 200 q^{19} + 64 q^{21} + 180 q^{29} - 304 q^{31} + 928 q^{39} - 876 q^{41} + 654 q^{49} + 1056 q^{51} + 840 q^{59} + 1804 q^{61} + 2112 q^{69} - 864 q^{71} - 320 q^{79} - 718 q^{81} - 1620 q^{89} - 464 q^{91} + 888 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
49.1
1.00000i
1.00000i
0 8.00000i 0 0 0 4.00000i 0 −37.0000 0
49.2 0 8.00000i 0 0 0 4.00000i 0 −37.0000 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.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 400.4.c.c 2
4.b odd 2 1 50.4.b.a 2
5.b even 2 1 inner 400.4.c.c 2
5.c odd 4 1 80.4.a.f 1
5.c odd 4 1 400.4.a.b 1
12.b even 2 1 450.4.c.d 2
15.e even 4 1 720.4.a.j 1
20.d odd 2 1 50.4.b.a 2
20.e even 4 1 10.4.a.a 1
20.e even 4 1 50.4.a.c 1
40.i odd 4 1 320.4.a.b 1
40.i odd 4 1 1600.4.a.bx 1
40.k even 4 1 320.4.a.m 1
40.k even 4 1 1600.4.a.d 1
60.h even 2 1 450.4.c.d 2
60.l odd 4 1 90.4.a.a 1
60.l odd 4 1 450.4.a.q 1
80.i odd 4 1 1280.4.d.g 2
80.j even 4 1 1280.4.d.j 2
80.s even 4 1 1280.4.d.j 2
80.t odd 4 1 1280.4.d.g 2
140.j odd 4 1 490.4.a.o 1
140.j odd 4 1 2450.4.a.b 1
140.w even 12 2 490.4.e.i 2
140.x odd 12 2 490.4.e.a 2
180.v odd 12 2 810.4.e.w 2
180.x even 12 2 810.4.e.c 2
220.i odd 4 1 1210.4.a.b 1
260.p even 4 1 1690.4.a.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
10.4.a.a 1 20.e even 4 1
50.4.a.c 1 20.e even 4 1
50.4.b.a 2 4.b odd 2 1
50.4.b.a 2 20.d odd 2 1
80.4.a.f 1 5.c odd 4 1
90.4.a.a 1 60.l odd 4 1
320.4.a.b 1 40.i odd 4 1
320.4.a.m 1 40.k even 4 1
400.4.a.b 1 5.c odd 4 1
400.4.c.c 2 1.a even 1 1 trivial
400.4.c.c 2 5.b even 2 1 inner
450.4.a.q 1 60.l odd 4 1
450.4.c.d 2 12.b even 2 1
450.4.c.d 2 60.h even 2 1
490.4.a.o 1 140.j odd 4 1
490.4.e.a 2 140.x odd 12 2
490.4.e.i 2 140.w even 12 2
720.4.a.j 1 15.e even 4 1
810.4.e.c 2 180.x even 12 2
810.4.e.w 2 180.v odd 12 2
1210.4.a.b 1 220.i odd 4 1
1280.4.d.g 2 80.i odd 4 1
1280.4.d.g 2 80.t odd 4 1
1280.4.d.j 2 80.j even 4 1
1280.4.d.j 2 80.s even 4 1
1600.4.a.d 1 40.k even 4 1
1600.4.a.bx 1 40.i odd 4 1
1690.4.a.a 1 260.p even 4 1
2450.4.a.b 1 140.j odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(400, [\chi])\):

\( T_{3}^{2} + 64 \) Copy content Toggle raw display
\( T_{7}^{2} + 16 \) Copy content Toggle raw display
\( T_{11} + 12 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 64 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 16 \) Copy content Toggle raw display
$11$ \( (T + 12)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 3364 \) Copy content Toggle raw display
$17$ \( T^{2} + 4356 \) Copy content Toggle raw display
$19$ \( (T + 100)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 17424 \) Copy content Toggle raw display
$29$ \( (T - 90)^{2} \) Copy content Toggle raw display
$31$ \( (T + 152)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 1156 \) Copy content Toggle raw display
$41$ \( (T + 438)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 1024 \) Copy content Toggle raw display
$47$ \( T^{2} + 41616 \) Copy content Toggle raw display
$53$ \( T^{2} + 49284 \) Copy content Toggle raw display
$59$ \( (T - 420)^{2} \) Copy content Toggle raw display
$61$ \( (T - 902)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 1048576 \) Copy content Toggle raw display
$71$ \( (T + 432)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 131044 \) Copy content Toggle raw display
$79$ \( (T + 160)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 5184 \) Copy content Toggle raw display
$89$ \( (T + 810)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 1223236 \) Copy content Toggle raw display
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