Properties

Label 200.8.c.a
Level $200$
Weight $8$
Character orbit 200.c
Analytic conductor $62.477$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 200 = 2^{3} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 200.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(62.4770050968\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 8)
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.

\(f(q)\) \(=\) \( q + 84 i q^{3} -456 i q^{7} -4869 q^{9} +O(q^{10})\) \( q + 84 i q^{3} -456 i q^{7} -4869 q^{9} -2524 q^{11} + 10778 i q^{13} -11150 i q^{17} -4124 q^{19} + 38304 q^{21} -81704 i q^{23} -225288 i q^{27} -99798 q^{29} -40480 q^{31} -212016 i q^{33} -419442 i q^{37} -905352 q^{39} + 141402 q^{41} + 690428 i q^{43} -682032 i q^{47} + 615607 q^{49} + 936600 q^{51} -1813118 i q^{53} -346416 i q^{57} + 966028 q^{59} + 1887670 q^{61} + 2220264 i q^{63} + 2965868 i q^{67} + 6863136 q^{69} -2548232 q^{71} + 1680326 i q^{73} + 1150944 i q^{77} -4038064 q^{79} + 8275689 q^{81} + 5385764 i q^{83} -8383032 i q^{87} + 6473046 q^{89} + 4914768 q^{91} -3400320 i q^{93} -6065758 i q^{97} + 12289356 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 9738q^{9} + O(q^{10}) \) \( 2q - 9738q^{9} - 5048q^{11} - 8248q^{19} + 76608q^{21} - 199596q^{29} - 80960q^{31} - 1810704q^{39} + 282804q^{41} + 1231214q^{49} + 1873200q^{51} + 1932056q^{59} + 3775340q^{61} + 13726272q^{69} - 5096464q^{71} - 8076128q^{79} + 16551378q^{81} + 12946092q^{89} + 9829536q^{91} + 24578712q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/200\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(151\) \(177\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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 84.0000i 0 0 0 456.000i 0 −4869.00 0
49.2 0 84.0000i 0 0 0 456.000i 0 −4869.00 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 200.8.c.a 2
4.b odd 2 1 400.8.c.b 2
5.b even 2 1 inner 200.8.c.a 2
5.c odd 4 1 8.8.a.a 1
5.c odd 4 1 200.8.a.i 1
15.e even 4 1 72.8.a.d 1
20.d odd 2 1 400.8.c.b 2
20.e even 4 1 16.8.a.c 1
20.e even 4 1 400.8.a.b 1
35.f even 4 1 392.8.a.d 1
40.i odd 4 1 64.8.a.g 1
40.k even 4 1 64.8.a.a 1
60.l odd 4 1 144.8.a.g 1
80.i odd 4 1 256.8.b.e 2
80.j even 4 1 256.8.b.c 2
80.s even 4 1 256.8.b.c 2
80.t odd 4 1 256.8.b.e 2
120.q odd 4 1 576.8.a.k 1
120.w even 4 1 576.8.a.j 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8.8.a.a 1 5.c odd 4 1
16.8.a.c 1 20.e even 4 1
64.8.a.a 1 40.k even 4 1
64.8.a.g 1 40.i odd 4 1
72.8.a.d 1 15.e even 4 1
144.8.a.g 1 60.l odd 4 1
200.8.a.i 1 5.c odd 4 1
200.8.c.a 2 1.a even 1 1 trivial
200.8.c.a 2 5.b even 2 1 inner
256.8.b.c 2 80.j even 4 1
256.8.b.c 2 80.s even 4 1
256.8.b.e 2 80.i odd 4 1
256.8.b.e 2 80.t odd 4 1
392.8.a.d 1 35.f even 4 1
400.8.a.b 1 20.e even 4 1
400.8.c.b 2 4.b odd 2 1
400.8.c.b 2 20.d odd 2 1
576.8.a.j 1 120.w even 4 1
576.8.a.k 1 120.q odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} + 7056 \) acting on \(S_{8}^{\mathrm{new}}(200, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( 7056 + T^{2} \)
$5$ \( T^{2} \)
$7$ \( 207936 + T^{2} \)
$11$ \( ( 2524 + T )^{2} \)
$13$ \( 116165284 + T^{2} \)
$17$ \( 124322500 + T^{2} \)
$19$ \( ( 4124 + T )^{2} \)
$23$ \( 6675543616 + T^{2} \)
$29$ \( ( 99798 + T )^{2} \)
$31$ \( ( 40480 + T )^{2} \)
$37$ \( 175931591364 + T^{2} \)
$41$ \( ( -141402 + T )^{2} \)
$43$ \( 476690823184 + T^{2} \)
$47$ \( 465167649024 + T^{2} \)
$53$ \( 3287396881924 + T^{2} \)
$59$ \( ( -966028 + T )^{2} \)
$61$ \( ( -1887670 + T )^{2} \)
$67$ \( 8796372993424 + T^{2} \)
$71$ \( ( 2548232 + T )^{2} \)
$73$ \( 2823495466276 + T^{2} \)
$79$ \( ( 4038064 + T )^{2} \)
$83$ \( 29006453863696 + T^{2} \)
$89$ \( ( -6473046 + T )^{2} \)
$97$ \( 36793420114564 + T^{2} \)
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