Properties

Label 201.2.a.d
Level 201
Weight 2
Character orbit 201.a
Self dual yes
Analytic conductor 1.605
Analytic rank 0
Dimension 3
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 201 = 3 \cdot 67 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 201.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(1.60499308063\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.148.1
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 1 + \beta_{2} ) q^{2} - q^{3} + ( 2 - \beta_{1} + \beta_{2} ) q^{4} + \beta_{1} q^{5} + ( -1 - \beta_{2} ) q^{6} + ( \beta_{1} + \beta_{2} ) q^{7} + ( 4 - 3 \beta_{1} ) q^{8} + q^{9} +O(q^{10})\) \( q + ( 1 + \beta_{2} ) q^{2} - q^{3} + ( 2 - \beta_{1} + \beta_{2} ) q^{4} + \beta_{1} q^{5} + ( -1 - \beta_{2} ) q^{6} + ( \beta_{1} + \beta_{2} ) q^{7} + ( 4 - 3 \beta_{1} ) q^{8} + q^{9} + ( -1 + 2 \beta_{1} ) q^{10} + ( 3 + \beta_{1} - \beta_{2} ) q^{11} + ( -2 + \beta_{1} - \beta_{2} ) q^{12} + ( -3 + \beta_{1} - \beta_{2} ) q^{13} + ( 2 + \beta_{1} ) q^{14} -\beta_{1} q^{15} + ( 3 - 4 \beta_{1} + 2 \beta_{2} ) q^{16} + ( 1 - 3 \beta_{1} - \beta_{2} ) q^{17} + ( 1 + \beta_{2} ) q^{18} + ( -1 + \beta_{1} - 3 \beta_{2} ) q^{19} + ( -3 + 2 \beta_{1} - \beta_{2} ) q^{20} + ( -\beta_{1} - \beta_{2} ) q^{21} + ( -1 + 3 \beta_{1} + 3 \beta_{2} ) q^{22} + ( 2 - 3 \beta_{1} - 2 \beta_{2} ) q^{23} + ( -4 + 3 \beta_{1} ) q^{24} + ( -3 + \beta_{1} + \beta_{2} ) q^{25} + ( -7 + 3 \beta_{1} - 3 \beta_{2} ) q^{26} - q^{27} + q^{28} + 4 \beta_{1} q^{29} + ( 1 - 2 \beta_{1} ) q^{30} + ( 5 - 4 \beta_{1} - 2 \beta_{2} ) q^{31} + ( 5 - 4 \beta_{1} + 3 \beta_{2} ) q^{32} + ( -3 - \beta_{1} + \beta_{2} ) q^{33} + ( 1 - 5 \beta_{1} + \beta_{2} ) q^{34} + ( 1 + 2 \beta_{1} + \beta_{2} ) q^{35} + ( 2 - \beta_{1} + \beta_{2} ) q^{36} + ( -2 - 3 \beta_{1} + \beta_{2} ) q^{37} + ( -11 + 5 \beta_{1} - \beta_{2} ) q^{38} + ( 3 - \beta_{1} + \beta_{2} ) q^{39} + ( -6 + \beta_{1} - 3 \beta_{2} ) q^{40} + ( 1 - 2 \beta_{1} + 3 \beta_{2} ) q^{41} + ( -2 - \beta_{1} ) q^{42} - q^{43} + ( -1 + \beta_{1} + \beta_{2} ) q^{44} + \beta_{1} q^{45} + ( -1 - 4 \beta_{1} + 2 \beta_{2} ) q^{46} + ( 5 + 3 \beta_{1} + 3 \beta_{2} ) q^{47} + ( -3 + 4 \beta_{1} - 2 \beta_{2} ) q^{48} + ( -4 + 2 \beta_{1} ) q^{49} + ( -1 + \beta_{1} - 3 \beta_{2} ) q^{50} + ( -1 + 3 \beta_{1} + \beta_{2} ) q^{51} + ( -13 + 7 \beta_{1} - 5 \beta_{2} ) q^{52} + ( 3 - 2 \beta_{1} - 5 \beta_{2} ) q^{53} + ( -1 - \beta_{2} ) q^{54} + ( 3 + 3 \beta_{1} + \beta_{2} ) q^{55} + ( -3 - 2 \beta_{1} + \beta_{2} ) q^{56} + ( 1 - \beta_{1} + 3 \beta_{2} ) q^{57} + ( -4 + 8 \beta_{1} ) q^{58} + ( 5 + 5 \beta_{2} ) q^{59} + ( 3 - 2 \beta_{1} + \beta_{2} ) q^{60} + ( 1 - 5 \beta_{1} - \beta_{2} ) q^{61} + ( 3 - 6 \beta_{1} + 5 \beta_{2} ) q^{62} + ( \beta_{1} + \beta_{2} ) q^{63} + ( 12 - 3 \beta_{1} + \beta_{2} ) q^{64} + ( 3 - 3 \beta_{1} + \beta_{2} ) q^{65} + ( 1 - 3 \beta_{1} - 3 \beta_{2} ) q^{66} + q^{67} + ( 7 - 5 \beta_{1} + 3 \beta_{2} ) q^{68} + ( -2 + 3 \beta_{1} + 2 \beta_{2} ) q^{69} + ( 2 + 3 \beta_{1} + \beta_{2} ) q^{70} + ( 5 + 3 \beta_{1} - \beta_{2} ) q^{71} + ( 4 - 3 \beta_{1} ) q^{72} + ( -7 + 2 \beta_{1} - 2 \beta_{2} ) q^{73} + ( 4 - 7 \beta_{1} - 2 \beta_{2} ) q^{74} + ( 3 - \beta_{1} - \beta_{2} ) q^{75} + ( -17 + 9 \beta_{1} - 5 \beta_{2} ) q^{76} + ( -1 + 5 \beta_{1} + 5 \beta_{2} ) q^{77} + ( 7 - 3 \beta_{1} + 3 \beta_{2} ) q^{78} + ( 10 - 2 \beta_{1} ) q^{79} + ( -10 + \beta_{1} - 4 \beta_{2} ) q^{80} + q^{81} + ( 12 - 7 \beta_{1} + \beta_{2} ) q^{82} + ( -3 + 2 \beta_{1} + 3 \beta_{2} ) q^{83} - q^{84} + ( -5 - 3 \beta_{1} - 3 \beta_{2} ) q^{85} + ( -1 - \beta_{2} ) q^{86} -4 \beta_{1} q^{87} + ( 3 - 5 \beta_{1} - 7 \beta_{2} ) q^{88} + ( -3 + 3 \beta_{1} - 5 \beta_{2} ) q^{89} + ( -1 + 2 \beta_{1} ) q^{90} + ( -1 - \beta_{1} - \beta_{2} ) q^{91} + ( 5 - 4 \beta_{1} + 3 \beta_{2} ) q^{92} + ( -5 + 4 \beta_{1} + 2 \beta_{2} ) q^{93} + ( 11 + 3 \beta_{1} + 5 \beta_{2} ) q^{94} + ( 5 - 3 \beta_{1} + \beta_{2} ) q^{95} + ( -5 + 4 \beta_{1} - 3 \beta_{2} ) q^{96} + ( -5 + 7 \beta_{1} + 7 \beta_{2} ) q^{97} + ( -6 + 4 \beta_{1} - 4 \beta_{2} ) q^{98} + ( 3 + \beta_{1} - \beta_{2} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q + 3q^{2} - 3q^{3} + 5q^{4} + q^{5} - 3q^{6} + q^{7} + 9q^{8} + 3q^{9} + O(q^{10}) \) \( 3q + 3q^{2} - 3q^{3} + 5q^{4} + q^{5} - 3q^{6} + q^{7} + 9q^{8} + 3q^{9} - q^{10} + 10q^{11} - 5q^{12} - 8q^{13} + 7q^{14} - q^{15} + 5q^{16} + 3q^{18} - 2q^{19} - 7q^{20} - q^{21} + 3q^{23} - 9q^{24} - 8q^{25} - 18q^{26} - 3q^{27} + 3q^{28} + 4q^{29} + q^{30} + 11q^{31} + 11q^{32} - 10q^{33} - 2q^{34} + 5q^{35} + 5q^{36} - 9q^{37} - 28q^{38} + 8q^{39} - 17q^{40} + q^{41} - 7q^{42} - 3q^{43} - 2q^{44} + q^{45} - 7q^{46} + 18q^{47} - 5q^{48} - 10q^{49} - 2q^{50} - 32q^{52} + 7q^{53} - 3q^{54} + 12q^{55} - 11q^{56} + 2q^{57} - 4q^{58} + 15q^{59} + 7q^{60} - 2q^{61} + 3q^{62} + q^{63} + 33q^{64} + 6q^{65} + 3q^{67} + 16q^{68} - 3q^{69} + 9q^{70} + 18q^{71} + 9q^{72} - 19q^{73} + 5q^{74} + 8q^{75} - 42q^{76} + 2q^{77} + 18q^{78} + 28q^{79} - 29q^{80} + 3q^{81} + 29q^{82} - 7q^{83} - 3q^{84} - 18q^{85} - 3q^{86} - 4q^{87} + 4q^{88} - 6q^{89} - q^{90} - 4q^{91} + 11q^{92} - 11q^{93} + 36q^{94} + 12q^{95} - 11q^{96} - 8q^{97} - 14q^{98} + 10q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{3} - x^{2} - 3 x + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - \nu - 2 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + \beta_{1} + 2\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
0.311108
2.17009
−1.48119
−1.21432 −1.00000 −0.525428 0.311108 1.21432 −1.90321 3.06668 1.00000 −0.377784
1.2 1.53919 −1.00000 0.369102 2.17009 −1.53919 2.70928 −2.51026 1.00000 3.34017
1.3 2.67513 −1.00000 5.15633 −1.48119 −2.67513 0.193937 8.44358 1.00000 −3.96239
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 201.2.a.d 3
3.b odd 2 1 603.2.a.i 3
4.b odd 2 1 3216.2.a.u 3
5.b even 2 1 5025.2.a.m 3
7.b odd 2 1 9849.2.a.ba 3
12.b even 2 1 9648.2.a.bn 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
201.2.a.d 3 1.a even 1 1 trivial
603.2.a.i 3 3.b odd 2 1
3216.2.a.u 3 4.b odd 2 1
5025.2.a.m 3 5.b even 2 1
9648.2.a.bn 3 12.b even 2 1
9849.2.a.ba 3 7.b odd 2 1

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)
\(67\) \(-1\)

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{3} - 3 T_{2}^{2} - T_{2} + 5 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(201))\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 - 3 T + 5 T^{2} - 7 T^{3} + 10 T^{4} - 12 T^{5} + 8 T^{6} \)
$3$ \( ( 1 + T )^{3} \)
$5$ \( 1 - T + 12 T^{2} - 9 T^{3} + 60 T^{4} - 25 T^{5} + 125 T^{6} \)
$7$ \( 1 - T + 16 T^{2} - 13 T^{3} + 112 T^{4} - 49 T^{5} + 343 T^{6} \)
$11$ \( 1 - 10 T + 57 T^{2} - 216 T^{3} + 627 T^{4} - 1210 T^{5} + 1331 T^{6} \)
$13$ \( 1 + 8 T + 51 T^{2} + 212 T^{3} + 663 T^{4} + 1352 T^{5} + 2197 T^{6} \)
$17$ \( 1 + 23 T^{2} + 52 T^{3} + 391 T^{4} + 4913 T^{6} \)
$19$ \( 1 + 2 T + 13 T^{2} + 56 T^{3} + 247 T^{4} + 722 T^{5} + 6859 T^{6} \)
$23$ \( 1 - 3 T + 38 T^{2} - 43 T^{3} + 874 T^{4} - 1587 T^{5} + 12167 T^{6} \)
$29$ \( 1 - 4 T + 39 T^{2} - 168 T^{3} + 1131 T^{4} - 3364 T^{5} + 24389 T^{6} \)
$31$ \( 1 - 11 T + 80 T^{2} - 387 T^{3} + 2480 T^{4} - 10571 T^{5} + 29791 T^{6} \)
$37$ \( 1 + 9 T + 98 T^{2} + 497 T^{3} + 3626 T^{4} + 12321 T^{5} + 50653 T^{6} \)
$41$ \( 1 - T + 62 T^{2} - 179 T^{3} + 2542 T^{4} - 1681 T^{5} + 68921 T^{6} \)
$43$ \( ( 1 + T + 43 T^{2} )^{3} \)
$47$ \( 1 - 18 T + 201 T^{2} - 1640 T^{3} + 9447 T^{4} - 39762 T^{5} + 103823 T^{6} \)
$53$ \( 1 - 7 T + 82 T^{2} - 873 T^{3} + 4346 T^{4} - 19663 T^{5} + 148877 T^{6} \)
$59$ \( 1 - 15 T + 152 T^{2} - 1145 T^{3} + 8968 T^{4} - 52215 T^{5} + 205379 T^{6} \)
$61$ \( 1 + 2 T + 107 T^{2} + 360 T^{3} + 6527 T^{4} + 7442 T^{5} + 226981 T^{6} \)
$67$ \( ( 1 - T )^{3} \)
$71$ \( 1 - 18 T + 281 T^{2} - 2456 T^{3} + 19951 T^{4} - 90738 T^{5} + 357911 T^{6} \)
$73$ \( 1 + 19 T + 302 T^{2} + 2871 T^{3} + 22046 T^{4} + 101251 T^{5} + 389017 T^{6} \)
$79$ \( 1 - 28 T + 485 T^{2} - 5112 T^{3} + 38315 T^{4} - 174748 T^{5} + 493039 T^{6} \)
$83$ \( 1 + 7 T + 228 T^{2} + 1137 T^{3} + 18924 T^{4} + 48223 T^{5} + 571787 T^{6} \)
$89$ \( 1 + 6 T + 119 T^{2} + 1184 T^{3} + 10591 T^{4} + 47526 T^{5} + 704969 T^{6} \)
$97$ \( 1 + 8 T + 51 T^{2} + 620 T^{3} + 4947 T^{4} + 75272 T^{5} + 912673 T^{6} \)
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