Properties

Label 400.4.c.e
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: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 25)
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.

\(f(q)\) \(=\) \( q + 7 i q^{3} -6 i q^{7} -22 q^{9} +O(q^{10})\) \( q + 7 i q^{3} -6 i q^{7} -22 q^{9} + 43 q^{11} + 28 i q^{13} + 91 i q^{17} -35 q^{19} + 42 q^{21} + 162 i q^{23} + 35 i q^{27} -160 q^{29} -42 q^{31} + 301 i q^{33} -314 i q^{37} -196 q^{39} -203 q^{41} + 92 i q^{43} -196 i q^{47} + 307 q^{49} -637 q^{51} -82 i q^{53} -245 i q^{57} -280 q^{59} -518 q^{61} + 132 i q^{63} -141 i q^{67} -1134 q^{69} -412 q^{71} + 763 i q^{73} -258 i q^{77} + 510 q^{79} -839 q^{81} + 777 i q^{83} -1120 i q^{87} + 945 q^{89} + 168 q^{91} -294 i q^{93} + 1246 i q^{97} -946 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 44 q^{9} + O(q^{10}) \) \( 2 q - 44 q^{9} + 86 q^{11} - 70 q^{19} + 84 q^{21} - 320 q^{29} - 84 q^{31} - 392 q^{39} - 406 q^{41} + 614 q^{49} - 1274 q^{51} - 560 q^{59} - 1036 q^{61} - 2268 q^{69} - 824 q^{71} + 1020 q^{79} - 1678 q^{81} + 1890 q^{89} + 336 q^{91} - 1892 q^{99} + O(q^{100}) \)

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 7.00000i 0 0 0 6.00000i 0 −22.0000 0
49.2 0 7.00000i 0 0 0 6.00000i 0 −22.0000 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.e 2
4.b odd 2 1 25.4.b.b 2
5.b even 2 1 inner 400.4.c.e 2
5.c odd 4 1 400.4.a.c 1
5.c odd 4 1 400.4.a.s 1
12.b even 2 1 225.4.b.f 2
20.d odd 2 1 25.4.b.b 2
20.e even 4 1 25.4.a.a 1
20.e even 4 1 25.4.a.b yes 1
40.i odd 4 1 1600.4.a.h 1
40.i odd 4 1 1600.4.a.bs 1
40.k even 4 1 1600.4.a.i 1
40.k even 4 1 1600.4.a.bt 1
60.h even 2 1 225.4.b.f 2
60.l odd 4 1 225.4.a.c 1
60.l odd 4 1 225.4.a.e 1
140.j odd 4 1 1225.4.a.h 1
140.j odd 4 1 1225.4.a.i 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
25.4.a.a 1 20.e even 4 1
25.4.a.b yes 1 20.e even 4 1
25.4.b.b 2 4.b odd 2 1
25.4.b.b 2 20.d odd 2 1
225.4.a.c 1 60.l odd 4 1
225.4.a.e 1 60.l odd 4 1
225.4.b.f 2 12.b even 2 1
225.4.b.f 2 60.h even 2 1
400.4.a.c 1 5.c odd 4 1
400.4.a.s 1 5.c odd 4 1
400.4.c.e 2 1.a even 1 1 trivial
400.4.c.e 2 5.b even 2 1 inner
1225.4.a.h 1 140.j odd 4 1
1225.4.a.i 1 140.j odd 4 1
1600.4.a.h 1 40.i odd 4 1
1600.4.a.i 1 40.k even 4 1
1600.4.a.bs 1 40.i odd 4 1
1600.4.a.bt 1 40.k even 4 1

Hecke kernels

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

\( T_{3}^{2} + 49 \)
\( T_{7}^{2} + 36 \)
\( T_{11} - 43 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( 49 + T^{2} \)
$5$ \( T^{2} \)
$7$ \( 36 + T^{2} \)
$11$ \( ( -43 + T )^{2} \)
$13$ \( 784 + T^{2} \)
$17$ \( 8281 + T^{2} \)
$19$ \( ( 35 + T )^{2} \)
$23$ \( 26244 + T^{2} \)
$29$ \( ( 160 + T )^{2} \)
$31$ \( ( 42 + T )^{2} \)
$37$ \( 98596 + T^{2} \)
$41$ \( ( 203 + T )^{2} \)
$43$ \( 8464 + T^{2} \)
$47$ \( 38416 + T^{2} \)
$53$ \( 6724 + T^{2} \)
$59$ \( ( 280 + T )^{2} \)
$61$ \( ( 518 + T )^{2} \)
$67$ \( 19881 + T^{2} \)
$71$ \( ( 412 + T )^{2} \)
$73$ \( 582169 + T^{2} \)
$79$ \( ( -510 + T )^{2} \)
$83$ \( 603729 + T^{2} \)
$89$ \( ( -945 + T )^{2} \)
$97$ \( 1552516 + T^{2} \)
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