Properties

Label 400.4.c.k
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: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 5)
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.

\(f(q)\) \(=\) \( q + 2 i q^{3} -6 i q^{7} + 23 q^{9} +O(q^{10})\) \( q + 2 i q^{3} -6 i q^{7} + 23 q^{9} -32 q^{11} + 38 i q^{13} + 26 i q^{17} + 100 q^{19} + 12 q^{21} -78 i q^{23} + 100 i q^{27} + 50 q^{29} + 108 q^{31} -64 i q^{33} + 266 i q^{37} -76 q^{39} + 22 q^{41} + 442 i q^{43} + 514 i q^{47} + 307 q^{49} -52 q^{51} -2 i q^{53} + 200 i q^{57} + 500 q^{59} -518 q^{61} -138 i q^{63} -126 i q^{67} + 156 q^{69} -412 q^{71} + 878 i q^{73} + 192 i q^{77} + 600 q^{79} + 421 q^{81} + 282 i q^{83} + 100 i q^{87} + 150 q^{89} + 228 q^{91} + 216 i q^{93} + 386 i q^{97} -736 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 46q^{9} + O(q^{10}) \) \( 2q + 46q^{9} - 64q^{11} + 200q^{19} + 24q^{21} + 100q^{29} + 216q^{31} - 152q^{39} + 44q^{41} + 614q^{49} - 104q^{51} + 1000q^{59} - 1036q^{61} + 312q^{69} - 824q^{71} + 1200q^{79} + 842q^{81} + 300q^{89} + 456q^{91} - 1472q^{99} + O(q^{100}) \)

Character values

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

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

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 2.00000i 0 0 0 6.00000i 0 23.0000 0
49.2 0 2.00000i 0 0 0 6.00000i 0 23.0000 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.k 2
4.b odd 2 1 25.4.b.a 2
5.b even 2 1 inner 400.4.c.k 2
5.c odd 4 1 80.4.a.d 1
5.c odd 4 1 400.4.a.m 1
12.b even 2 1 225.4.b.c 2
15.e even 4 1 720.4.a.u 1
20.d odd 2 1 25.4.b.a 2
20.e even 4 1 5.4.a.a 1
20.e even 4 1 25.4.a.c 1
40.i odd 4 1 320.4.a.h 1
40.i odd 4 1 1600.4.a.s 1
40.k even 4 1 320.4.a.g 1
40.k even 4 1 1600.4.a.bi 1
60.h even 2 1 225.4.b.c 2
60.l odd 4 1 45.4.a.d 1
60.l odd 4 1 225.4.a.b 1
80.i odd 4 1 1280.4.d.l 2
80.j even 4 1 1280.4.d.e 2
80.s even 4 1 1280.4.d.e 2
80.t odd 4 1 1280.4.d.l 2
140.j odd 4 1 245.4.a.a 1
140.j odd 4 1 1225.4.a.k 1
140.w even 12 2 245.4.e.f 2
140.x odd 12 2 245.4.e.g 2
180.v odd 12 2 405.4.e.c 2
180.x even 12 2 405.4.e.l 2
220.i odd 4 1 605.4.a.d 1
260.p even 4 1 845.4.a.b 1
340.r even 4 1 1445.4.a.a 1
380.j odd 4 1 1805.4.a.h 1
420.w even 4 1 2205.4.a.q 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5.4.a.a 1 20.e even 4 1
25.4.a.c 1 20.e even 4 1
25.4.b.a 2 4.b odd 2 1
25.4.b.a 2 20.d odd 2 1
45.4.a.d 1 60.l odd 4 1
80.4.a.d 1 5.c odd 4 1
225.4.a.b 1 60.l odd 4 1
225.4.b.c 2 12.b even 2 1
225.4.b.c 2 60.h even 2 1
245.4.a.a 1 140.j odd 4 1
245.4.e.f 2 140.w even 12 2
245.4.e.g 2 140.x odd 12 2
320.4.a.g 1 40.k even 4 1
320.4.a.h 1 40.i odd 4 1
400.4.a.m 1 5.c odd 4 1
400.4.c.k 2 1.a even 1 1 trivial
400.4.c.k 2 5.b even 2 1 inner
405.4.e.c 2 180.v odd 12 2
405.4.e.l 2 180.x even 12 2
605.4.a.d 1 220.i odd 4 1
720.4.a.u 1 15.e even 4 1
845.4.a.b 1 260.p even 4 1
1225.4.a.k 1 140.j odd 4 1
1280.4.d.e 2 80.j even 4 1
1280.4.d.e 2 80.s even 4 1
1280.4.d.l 2 80.i odd 4 1
1280.4.d.l 2 80.t odd 4 1
1445.4.a.a 1 340.r even 4 1
1600.4.a.s 1 40.i odd 4 1
1600.4.a.bi 1 40.k even 4 1
1805.4.a.h 1 380.j odd 4 1
2205.4.a.q 1 420.w even 4 1

Hecke kernels

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

\( T_{3}^{2} + 4 \)
\( T_{7}^{2} + 36 \)
\( T_{11} + 32 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 - 50 T^{2} + 729 T^{4} \)
$5$ 1
$7$ \( 1 - 650 T^{2} + 117649 T^{4} \)
$11$ \( ( 1 + 32 T + 1331 T^{2} )^{2} \)
$13$ \( 1 - 2950 T^{2} + 4826809 T^{4} \)
$17$ \( 1 - 9150 T^{2} + 24137569 T^{4} \)
$19$ \( ( 1 - 100 T + 6859 T^{2} )^{2} \)
$23$ \( 1 - 18250 T^{2} + 148035889 T^{4} \)
$29$ \( ( 1 - 50 T + 24389 T^{2} )^{2} \)
$31$ \( ( 1 - 108 T + 29791 T^{2} )^{2} \)
$37$ \( 1 - 30550 T^{2} + 2565726409 T^{4} \)
$41$ \( ( 1 - 22 T + 68921 T^{2} )^{2} \)
$43$ \( 1 + 36350 T^{2} + 6321363049 T^{4} \)
$47$ \( 1 + 56550 T^{2} + 10779215329 T^{4} \)
$53$ \( 1 - 297750 T^{2} + 22164361129 T^{4} \)
$59$ \( ( 1 - 500 T + 205379 T^{2} )^{2} \)
$61$ \( ( 1 + 518 T + 226981 T^{2} )^{2} \)
$67$ \( 1 - 585650 T^{2} + 90458382169 T^{4} \)
$71$ \( ( 1 + 412 T + 357911 T^{2} )^{2} \)
$73$ \( 1 - 7150 T^{2} + 151334226289 T^{4} \)
$79$ \( ( 1 - 600 T + 493039 T^{2} )^{2} \)
$83$ \( 1 - 1064050 T^{2} + 326940373369 T^{4} \)
$89$ \( ( 1 - 150 T + 704969 T^{2} )^{2} \)
$97$ \( 1 - 1676350 T^{2} + 832972004929 T^{4} \)
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