Properties

Label 400.4.c.l
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: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 100)
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.

\(f(q)\) \(=\) \( q + i q^{3} - 26 i q^{7} + 26 q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + i q^{3} - 26 i q^{7} + 26 q^{9} - 45 q^{11} - 44 i q^{13} + 117 i q^{17} - 91 q^{19} + 26 q^{21} - 18 i q^{23} + 53 i q^{27} - 144 q^{29} - 26 q^{31} - 45 i q^{33} - 214 i q^{37} + 44 q^{39} - 459 q^{41} - 460 i q^{43} + 468 i q^{47} - 333 q^{49} - 117 q^{51} - 558 i q^{53} - 91 i q^{57} - 72 q^{59} - 118 q^{61} - 676 i q^{63} - 251 i q^{67} + 18 q^{69} - 108 q^{71} - 299 i q^{73} + 1170 i q^{77} - 898 q^{79} + 649 q^{81} + 927 i q^{83} - 144 i q^{87} - 351 q^{89} - 1144 q^{91} - 26 i q^{93} + 386 i q^{97} - 1170 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 52 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 52 q^{9} - 90 q^{11} - 182 q^{19} + 52 q^{21} - 288 q^{29} - 52 q^{31} + 88 q^{39} - 918 q^{41} - 666 q^{49} - 234 q^{51} - 144 q^{59} - 236 q^{61} + 36 q^{69} - 216 q^{71} - 1796 q^{79} + 1298 q^{81} - 702 q^{89} - 2288 q^{91} - 2340 q^{99}+O(q^{100}) \) 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 1.00000i 0 0 0 26.0000i 0 26.0000 0
49.2 0 1.00000i 0 0 0 26.0000i 0 26.0000 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.l 2
4.b odd 2 1 100.4.c.b 2
5.b even 2 1 inner 400.4.c.l 2
5.c odd 4 1 400.4.a.i 1
5.c odd 4 1 400.4.a.l 1
12.b even 2 1 900.4.d.a 2
20.d odd 2 1 100.4.c.b 2
20.e even 4 1 100.4.a.b 1
20.e even 4 1 100.4.a.c yes 1
40.i odd 4 1 1600.4.a.y 1
40.i odd 4 1 1600.4.a.bd 1
40.k even 4 1 1600.4.a.x 1
40.k even 4 1 1600.4.a.bc 1
60.h even 2 1 900.4.d.a 2
60.l odd 4 1 900.4.a.c 1
60.l odd 4 1 900.4.a.p 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
100.4.a.b 1 20.e even 4 1
100.4.a.c yes 1 20.e even 4 1
100.4.c.b 2 4.b odd 2 1
100.4.c.b 2 20.d odd 2 1
400.4.a.i 1 5.c odd 4 1
400.4.a.l 1 5.c odd 4 1
400.4.c.l 2 1.a even 1 1 trivial
400.4.c.l 2 5.b even 2 1 inner
900.4.a.c 1 60.l odd 4 1
900.4.a.p 1 60.l odd 4 1
900.4.d.a 2 12.b even 2 1
900.4.d.a 2 60.h even 2 1
1600.4.a.x 1 40.k even 4 1
1600.4.a.y 1 40.i odd 4 1
1600.4.a.bc 1 40.k even 4 1
1600.4.a.bd 1 40.i odd 4 1

Hecke kernels

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

\( T_{3}^{2} + 1 \) Copy content Toggle raw display
\( T_{7}^{2} + 676 \) Copy content Toggle raw display
\( T_{11} + 45 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 1 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 676 \) Copy content Toggle raw display
$11$ \( (T + 45)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 1936 \) Copy content Toggle raw display
$17$ \( T^{2} + 13689 \) Copy content Toggle raw display
$19$ \( (T + 91)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 324 \) Copy content Toggle raw display
$29$ \( (T + 144)^{2} \) Copy content Toggle raw display
$31$ \( (T + 26)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 45796 \) Copy content Toggle raw display
$41$ \( (T + 459)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 211600 \) Copy content Toggle raw display
$47$ \( T^{2} + 219024 \) Copy content Toggle raw display
$53$ \( T^{2} + 311364 \) Copy content Toggle raw display
$59$ \( (T + 72)^{2} \) Copy content Toggle raw display
$61$ \( (T + 118)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 63001 \) Copy content Toggle raw display
$71$ \( (T + 108)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 89401 \) Copy content Toggle raw display
$79$ \( (T + 898)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 859329 \) Copy content Toggle raw display
$89$ \( (T + 351)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 148996 \) Copy content Toggle raw display
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