Properties

Label 201.4.d.a
Level 201
Weight 4
Character orbit 201.d
Analytic conductor 11.859
Analytic rank 0
Dimension 2
CM disc. -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 \(2\) and degree \(1\))

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: \( 2\cdot 3 \)
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. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
3.b Odd 1 CM by \(\Q(\sqrt{-3}) \) yes
67.b Odd 1 yes
201.d Even 1 yes

Hecke kernels

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