Properties

Label 201.4.d.a
Level 201
Weight 4
Character orbit 201.d
Analytic conductor 11.859
Analytic rank 0
Dimension 2
CM discriminant -3
Inner twists 4

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Newspace parameters

Level: \( N \) = \( 201 = 3 \cdot 67 \)
Weight: \( k \) = \( 4 \)
Character orbit: \([\chi]\) = 201.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(11.8593839112\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2\cdot 3 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 3 - 6 \zeta_{6} ) q^{3} -8 q^{4} + ( 18 - 36 \zeta_{6} ) q^{7} -27 q^{9} +O(q^{10})\) \( q + ( 3 - 6 \zeta_{6} ) q^{3} -8 q^{4} + ( 18 - 36 \zeta_{6} ) q^{7} -27 q^{9} + ( -24 + 48 \zeta_{6} ) q^{12} + ( -36 + 72 \zeta_{6} ) q^{13} + 64 q^{16} -56 q^{19} -162 q^{21} -125 q^{25} + ( -81 + 162 \zeta_{6} ) q^{27} + ( -144 + 288 \zeta_{6} ) q^{28} + ( -90 + 180 \zeta_{6} ) q^{31} + 216 q^{36} -110 q^{37} + 324 q^{39} + ( -126 + 252 \zeta_{6} ) q^{43} + ( 192 - 384 \zeta_{6} ) q^{48} -629 q^{49} + ( 288 - 576 \zeta_{6} ) q^{52} + ( -168 + 336 \zeta_{6} ) q^{57} + ( 540 - 1080 \zeta_{6} ) q^{61} + ( -486 + 972 \zeta_{6} ) q^{63} -512 q^{64} + ( 629 - 378 \zeta_{6} ) q^{67} -1190 q^{73} + ( -375 + 750 \zeta_{6} ) q^{75} + 448 q^{76} + ( 630 - 1260 \zeta_{6} ) q^{79} + 729 q^{81} + 1296 q^{84} + 1944 q^{91} + 810 q^{93} + ( 792 - 1584 \zeta_{6} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 16q^{4} - 54q^{9} + O(q^{10}) \) \( 2q - 16q^{4} - 54q^{9} + 128q^{16} - 112q^{19} - 324q^{21} - 250q^{25} + 432q^{36} - 220q^{37} + 648q^{39} - 1258q^{49} - 1024q^{64} + 880q^{67} - 2380q^{73} + 896q^{76} + 1458q^{81} + 2592q^{84} + 3888q^{91} + 1620q^{93} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/201\mathbb{Z}\right)^\times\).

\(n\) \(68\) \(136\)
\(\chi(n)\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
200.1
0.500000 + 0.866025i
0.500000 0.866025i
0 5.19615i −8.00000 0 0 31.1769i 0 −27.0000 0
200.2 0 5.19615i −8.00000 0 0 31.1769i 0 −27.0000 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
3.b odd 2 1 CM by \(\Q(\sqrt{-3}) \)
67.b odd 2 1 inner
201.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 201.4.d.a 2
3.b odd 2 1 CM 201.4.d.a 2
67.b odd 2 1 inner 201.4.d.a 2
201.d even 2 1 inner 201.4.d.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
201.4.d.a 2 1.a even 1 1 trivial
201.4.d.a 2 3.b odd 2 1 CM
201.4.d.a 2 67.b odd 2 1 inner
201.4.d.a 2 201.d even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} \) acting on \(S_{4}^{\mathrm{new}}(201, [\chi])\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + 8 T^{2} )^{2} \)
$3$ \( 1 + 27 T^{2} \)
$5$ \( ( 1 + 125 T^{2} )^{2} \)
$7$ \( ( 1 - 20 T + 343 T^{2} )( 1 + 20 T + 343 T^{2} ) \)
$11$ \( ( 1 + 1331 T^{2} )^{2} \)
$13$ \( ( 1 - 70 T + 2197 T^{2} )( 1 + 70 T + 2197 T^{2} ) \)
$17$ \( ( 1 - 4913 T^{2} )^{2} \)
$19$ \( ( 1 + 56 T + 6859 T^{2} )^{2} \)
$23$ \( ( 1 - 12167 T^{2} )^{2} \)
$29$ \( ( 1 - 24389 T^{2} )^{2} \)
$31$ \( ( 1 - 308 T + 29791 T^{2} )( 1 + 308 T + 29791 T^{2} ) \)
$37$ \( ( 1 + 110 T + 50653 T^{2} )^{2} \)
$41$ \( ( 1 + 68921 T^{2} )^{2} \)
$43$ \( ( 1 - 520 T + 79507 T^{2} )( 1 + 520 T + 79507 T^{2} ) \)
$47$ \( ( 1 - 103823 T^{2} )^{2} \)
$53$ \( ( 1 + 148877 T^{2} )^{2} \)
$59$ \( ( 1 - 205379 T^{2} )^{2} \)
$61$ \( ( 1 - 182 T + 226981 T^{2} )( 1 + 182 T + 226981 T^{2} ) \)
$67$ \( 1 - 880 T + 300763 T^{2} \)
$71$ \( ( 1 - 357911 T^{2} )^{2} \)
$73$ \( ( 1 + 1190 T + 389017 T^{2} )^{2} \)
$79$ \( ( 1 - 884 T + 493039 T^{2} )( 1 + 884 T + 493039 T^{2} ) \)
$83$ \( ( 1 - 571787 T^{2} )^{2} \)
$89$ \( ( 1 - 704969 T^{2} )^{2} \)
$97$ \( ( 1 - 1330 T + 912673 T^{2} )( 1 + 1330 T + 912673 T^{2} ) \)
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