Properties

Label 200.4.f.a
Level $200$
Weight $4$
Character orbit 200.f
Analytic conductor $11.800$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 200 = 2^{3} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 200.f (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(11.8003820011\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{7})\)
Defining polynomial: \( x^{4} - 3x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{4} \)
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 a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + (\beta_{2} - 2 \beta_1) q^{3} + (\beta_{3} + 6) q^{4} + (\beta_{3} + 14) q^{6} - 4 \beta_{2} q^{7} + ( - 8 \beta_{2} - 4 \beta_1) q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + (\beta_{2} - 2 \beta_1) q^{3} + (\beta_{3} + 6) q^{4} + (\beta_{3} + 14) q^{6} - 4 \beta_{2} q^{7} + ( - 8 \beta_{2} - 4 \beta_1) q^{8} + q^{9} + 3 \beta_{3} q^{11} + ( - 8 \beta_{2} - 12 \beta_1) q^{12} + (10 \beta_{2} - 20 \beta_1) q^{13} + (4 \beta_{3} - 8) q^{14} + (12 \beta_{3} + 8) q^{16} - 7 \beta_{2} q^{17} - \beta_1 q^{18} - 7 \beta_{3} q^{19} + 8 \beta_{3} q^{21} + ( - 24 \beta_{2} + 6 \beta_1) q^{22} + 76 \beta_{2} q^{23} + (20 \beta_{3} + 56) q^{24} + (10 \beta_{3} + 140) q^{26} + ( - 26 \beta_{2} + 52 \beta_1) q^{27} + ( - 32 \beta_{2} + 16 \beta_1) q^{28} - 30 \beta_{3} q^{29} + 224 q^{31} + ( - 96 \beta_{2} + 16 \beta_1) q^{32} - 42 \beta_{2} q^{33} + (7 \beta_{3} - 14) q^{34} + (\beta_{3} + 6) q^{36} + ( - 46 \beta_{2} + 92 \beta_1) q^{37} + (56 \beta_{2} - 14 \beta_1) q^{38} + 280 q^{39} - 70 q^{41} + ( - 64 \beta_{2} + 16 \beta_1) q^{42} + ( - 83 \beta_{2} + 166 \beta_1) q^{43} + (18 \beta_{3} - 84) q^{44} + ( - 76 \beta_{3} + 152) q^{46} + 168 \beta_{2} q^{47} + ( - 160 \beta_{2} - 16 \beta_1) q^{48} + 279 q^{49} + 14 \beta_{3} q^{51} + ( - 80 \beta_{2} - 120 \beta_1) q^{52} + (6 \beta_{2} - 12 \beta_1) q^{53} + ( - 26 \beta_{3} - 364) q^{54} + (16 \beta_{3} - 160) q^{56} + 98 \beta_{2} q^{57} + (240 \beta_{2} - 60 \beta_1) q^{58} + 101 \beta_{3} q^{59} - 18 \beta_{3} q^{61} - 224 \beta_1 q^{62} - 4 \beta_{2} q^{63} + (80 \beta_{3} - 288) q^{64} + (42 \beta_{3} - 84) q^{66} + ( - 33 \beta_{2} + 66 \beta_1) q^{67} + ( - 56 \beta_{2} + 28 \beta_1) q^{68} - 152 \beta_{3} q^{69} - 72 q^{71} + ( - 8 \beta_{2} - 4 \beta_1) q^{72} + 147 \beta_{2} q^{73} + ( - 46 \beta_{3} - 644) q^{74} + ( - 42 \beta_{3} + 196) q^{76} + ( - 24 \beta_{2} + 48 \beta_1) q^{77} - 280 \beta_1 q^{78} + 464 q^{79} - 755 q^{81} + 70 \beta_1 q^{82} + ( - 103 \beta_{2} + 206 \beta_1) q^{83} + (48 \beta_{3} - 224) q^{84} + ( - 83 \beta_{3} - 1162) q^{86} + 420 \beta_{2} q^{87} + ( - 144 \beta_{2} + 120 \beta_1) q^{88} - 266 q^{89} + 80 \beta_{3} q^{91} + (608 \beta_{2} - 304 \beta_1) q^{92} + (224 \beta_{2} - 448 \beta_1) q^{93} + ( - 168 \beta_{3} + 336) q^{94} + (176 \beta_{3} - 224) q^{96} + 497 \beta_{2} q^{97} - 279 \beta_1 q^{98} + 3 \beta_{3} q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 24 q^{4} + 56 q^{6} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 24 q^{4} + 56 q^{6} + 4 q^{9} - 32 q^{14} + 32 q^{16} + 224 q^{24} + 560 q^{26} + 896 q^{31} - 56 q^{34} + 24 q^{36} + 1120 q^{39} - 280 q^{41} - 336 q^{44} + 608 q^{46} + 1116 q^{49} - 1456 q^{54} - 640 q^{56} - 1152 q^{64} - 336 q^{66} - 288 q^{71} - 2576 q^{74} + 784 q^{76} + 1856 q^{79} - 3020 q^{81} - 896 q^{84} - 4648 q^{86} - 1064 q^{89} + 1344 q^{94} - 896 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 3x^{2} + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 2\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{3} - \nu \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 4\nu^{2} - 6 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} + 6 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 2\beta_{2} + \beta_1 ) / 2 \) Copy content Toggle raw display

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
149.1
1.32288 + 0.500000i
1.32288 0.500000i
−1.32288 + 0.500000i
−1.32288 0.500000i
−2.64575 1.00000i −5.29150 6.00000 + 5.29150i 0 14.0000 + 5.29150i 8.00000i −10.5830 20.0000i 1.00000 0
149.2 −2.64575 + 1.00000i −5.29150 6.00000 5.29150i 0 14.0000 5.29150i 8.00000i −10.5830 + 20.0000i 1.00000 0
149.3 2.64575 1.00000i 5.29150 6.00000 5.29150i 0 14.0000 5.29150i 8.00000i 10.5830 20.0000i 1.00000 0
149.4 2.64575 + 1.00000i 5.29150 6.00000 + 5.29150i 0 14.0000 + 5.29150i 8.00000i 10.5830 + 20.0000i 1.00000 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
8.b even 2 1 inner
40.f even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 200.4.f.a 4
4.b odd 2 1 800.4.f.a 4
5.b even 2 1 inner 200.4.f.a 4
5.c odd 4 1 8.4.b.a 2
5.c odd 4 1 200.4.d.a 2
8.b even 2 1 inner 200.4.f.a 4
8.d odd 2 1 800.4.f.a 4
15.e even 4 1 72.4.d.b 2
20.d odd 2 1 800.4.f.a 4
20.e even 4 1 32.4.b.a 2
20.e even 4 1 800.4.d.a 2
40.e odd 2 1 800.4.f.a 4
40.f even 2 1 inner 200.4.f.a 4
40.i odd 4 1 8.4.b.a 2
40.i odd 4 1 200.4.d.a 2
40.k even 4 1 32.4.b.a 2
40.k even 4 1 800.4.d.a 2
60.l odd 4 1 288.4.d.a 2
80.i odd 4 1 256.4.a.l 2
80.j even 4 1 256.4.a.j 2
80.s even 4 1 256.4.a.j 2
80.t odd 4 1 256.4.a.l 2
120.q odd 4 1 288.4.d.a 2
120.w even 4 1 72.4.d.b 2
240.z odd 4 1 2304.4.a.v 2
240.bb even 4 1 2304.4.a.bn 2
240.bd odd 4 1 2304.4.a.v 2
240.bf even 4 1 2304.4.a.bn 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8.4.b.a 2 5.c odd 4 1
8.4.b.a 2 40.i odd 4 1
32.4.b.a 2 20.e even 4 1
32.4.b.a 2 40.k even 4 1
72.4.d.b 2 15.e even 4 1
72.4.d.b 2 120.w even 4 1
200.4.d.a 2 5.c odd 4 1
200.4.d.a 2 40.i odd 4 1
200.4.f.a 4 1.a even 1 1 trivial
200.4.f.a 4 5.b even 2 1 inner
200.4.f.a 4 8.b even 2 1 inner
200.4.f.a 4 40.f even 2 1 inner
256.4.a.j 2 80.j even 4 1
256.4.a.j 2 80.s even 4 1
256.4.a.l 2 80.i odd 4 1
256.4.a.l 2 80.t odd 4 1
288.4.d.a 2 60.l odd 4 1
288.4.d.a 2 120.q odd 4 1
800.4.d.a 2 20.e even 4 1
800.4.d.a 2 40.k even 4 1
800.4.f.a 4 4.b odd 2 1
800.4.f.a 4 8.d odd 2 1
800.4.f.a 4 20.d odd 2 1
800.4.f.a 4 40.e odd 2 1
2304.4.a.v 2 240.z odd 4 1
2304.4.a.v 2 240.bd odd 4 1
2304.4.a.bn 2 240.bb even 4 1
2304.4.a.bn 2 240.bf even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} - 28 \) acting on \(S_{4}^{\mathrm{new}}(200, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - 12T^{2} + 64 \) Copy content Toggle raw display
$3$ \( (T^{2} - 28)^{2} \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( (T^{2} + 64)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} + 252)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} - 2800)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} + 196)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} + 1372)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} + 23104)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 25200)^{2} \) Copy content Toggle raw display
$31$ \( (T - 224)^{4} \) Copy content Toggle raw display
$37$ \( (T^{2} - 59248)^{2} \) Copy content Toggle raw display
$41$ \( (T + 70)^{4} \) Copy content Toggle raw display
$43$ \( (T^{2} - 192892)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} + 112896)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} - 1008)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} + 285628)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + 9072)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} - 30492)^{2} \) Copy content Toggle raw display
$71$ \( (T + 72)^{4} \) Copy content Toggle raw display
$73$ \( (T^{2} + 86436)^{2} \) Copy content Toggle raw display
$79$ \( (T - 464)^{4} \) Copy content Toggle raw display
$83$ \( (T^{2} - 297052)^{2} \) Copy content Toggle raw display
$89$ \( (T + 266)^{4} \) Copy content Toggle raw display
$97$ \( (T^{2} + 988036)^{2} \) Copy content Toggle raw display
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