Properties

Label 201.2.a.e
Level 201
Weight 2
Character orbit 201.a
Self dual yes
Analytic conductor 1.605
Analytic rank 0
Dimension 5
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: \(5\)
Coefficient field: 5.5.1025428.1
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
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 \(1,\beta_1,\beta_2,\beta_3,\beta_4\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{5} - 8 x^{3} + 13 x - 2\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 3 \)
\(\beta_{3}\)\(=\)\((\)\( \nu^{4} - \nu^{3} - 5 \nu^{2} + 5 \nu \)\()/2\)
\(\beta_{4}\)\(=\)\((\)\( \nu^{4} + \nu^{3} - 7 \nu^{2} - 5 \nu + 8 \)\()/2\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 3\)
\(\nu^{3}\)\(=\)\(\beta_{4} - \beta_{3} + \beta_{2} + 5 \beta_{1} - 1\)
\(\nu^{4}\)\(=\)\(\beta_{4} + \beta_{3} + 6 \beta_{2} + 14\)

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
2.44016
1.37452
0.156184
−1.63326
−2.33760
−2.44016 1.00000 3.95438 1.05143 −2.44016 1.76320 −4.76901 1.00000 −2.56565
1.2 −1.37452 1.00000 −0.110700 −3.96568 −1.37452 3.17518 2.90120 1.00000 5.45090
1.3 −0.156184 1.00000 −1.97561 2.52637 −0.156184 0.828315 0.620924 1.00000 −0.394577
1.4 1.63326 1.00000 0.667538 −0.873745 1.63326 4.38246 −2.17626 1.00000 −1.42705
1.5 2.33760 1.00000 3.46438 −1.73837 2.33760 −3.14916 3.42315 1.00000 −4.06362
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.5
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.e 5
3.b odd 2 1 603.2.a.k 5
4.b odd 2 1 3216.2.a.y 5
5.b even 2 1 5025.2.a.x 5
7.b odd 2 1 9849.2.a.bb 5
12.b even 2 1 9648.2.a.cd 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
201.2.a.e 5 1.a even 1 1 trivial
603.2.a.k 5 3.b odd 2 1
3216.2.a.y 5 4.b odd 2 1
5025.2.a.x 5 5.b even 2 1
9648.2.a.cd 5 12.b even 2 1
9849.2.a.bb 5 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 \( T_{2}^{5} - 8 T_{2}^{3} + 13 T_{2} + 2 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(201))\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 + 2 T^{2} + 5 T^{4} + 2 T^{5} + 10 T^{6} + 16 T^{8} + 32 T^{10} \)
$3$ \( ( 1 - T )^{5} \)
$5$ \( 1 + 3 T + 16 T^{2} + 41 T^{3} + 125 T^{4} + 276 T^{5} + 625 T^{6} + 1025 T^{7} + 2000 T^{8} + 1875 T^{9} + 3125 T^{10} \)
$7$ \( 1 - 7 T + 38 T^{2} - 133 T^{3} + 425 T^{4} - 1112 T^{5} + 2975 T^{6} - 6517 T^{7} + 13034 T^{8} - 16807 T^{9} + 16807 T^{10} \)
$11$ \( 1 + 35 T^{2} - 4 T^{3} + 606 T^{4} - 120 T^{5} + 6666 T^{6} - 484 T^{7} + 46585 T^{8} + 161051 T^{10} \)
$13$ \( 1 - 10 T + 85 T^{2} - 484 T^{3} + 2382 T^{4} - 9236 T^{5} + 30966 T^{6} - 81796 T^{7} + 186745 T^{8} - 285610 T^{9} + 371293 T^{10} \)
$17$ \( 1 + 5 T + 39 T^{2} + 244 T^{3} + 1180 T^{4} + 4838 T^{5} + 20060 T^{6} + 70516 T^{7} + 191607 T^{8} + 417605 T^{9} + 1419857 T^{10} \)
$19$ \( 1 - 5 T + 49 T^{2} - 132 T^{3} + 808 T^{4} - 1422 T^{5} + 15352 T^{6} - 47652 T^{7} + 336091 T^{8} - 651605 T^{9} + 2476099 T^{10} \)
$23$ \( 1 + 2 T + 101 T^{2} + 192 T^{3} + 4335 T^{4} + 6712 T^{5} + 99705 T^{6} + 101568 T^{7} + 1228867 T^{8} + 559682 T^{9} + 6436343 T^{10} \)
$29$ \( 1 - 3 T + 47 T^{2} - 124 T^{3} + 1932 T^{4} - 4194 T^{5} + 56028 T^{6} - 104284 T^{7} + 1146283 T^{8} - 2121843 T^{9} + 20511149 T^{10} \)
$31$ \( 1 - 9 T + 154 T^{2} - 943 T^{3} + 9185 T^{4} - 41200 T^{5} + 284735 T^{6} - 906223 T^{7} + 4587814 T^{8} - 8311689 T^{9} + 28629151 T^{10} \)
$37$ \( 1 - 8 T + 117 T^{2} - 746 T^{3} + 6797 T^{4} - 34118 T^{5} + 251489 T^{6} - 1021274 T^{7} + 5926401 T^{8} - 14993288 T^{9} + 69343957 T^{10} \)
$41$ \( 1 + 7 T + 190 T^{2} + 1019 T^{3} + 14951 T^{4} + 60056 T^{5} + 612991 T^{6} + 1712939 T^{7} + 13094990 T^{8} + 19780327 T^{9} + 115856201 T^{10} \)
$43$ \( 1 - T + 124 T^{2} + 33 T^{3} + 8725 T^{4} + 480 T^{5} + 375175 T^{6} + 61017 T^{7} + 9858868 T^{8} - 3418801 T^{9} + 147008443 T^{10} \)
$47$ \( 1 + 5 T + 189 T^{2} + 692 T^{3} + 15424 T^{4} + 42974 T^{5} + 724928 T^{6} + 1528628 T^{7} + 19622547 T^{8} + 24398405 T^{9} + 229345007 T^{10} \)
$53$ \( 1 + 15 T + 168 T^{2} + 1247 T^{3} + 8491 T^{4} + 46324 T^{5} + 450023 T^{6} + 3502823 T^{7} + 25011336 T^{8} + 118357215 T^{9} + 418195493 T^{10} \)
$59$ \( 1 + 6 T + 191 T^{2} + 1132 T^{3} + 18867 T^{4} + 91308 T^{5} + 1113153 T^{6} + 3940492 T^{7} + 39227389 T^{8} + 72704166 T^{9} + 714924299 T^{10} \)
$61$ \( 1 - 6 T + 209 T^{2} - 420 T^{3} + 16170 T^{4} - 2732 T^{5} + 986370 T^{6} - 1562820 T^{7} + 47439029 T^{8} - 83075046 T^{9} + 844596301 T^{10} \)
$67$ \( ( 1 + T )^{5} \)
$71$ \( 1 - 22 T + 375 T^{2} - 4100 T^{3} + 42078 T^{4} - 349772 T^{5} + 2987538 T^{6} - 20668100 T^{7} + 134216625 T^{8} - 559056982 T^{9} + 1804229351 T^{10} \)
$73$ \( 1 + 81 T^{2} + 534 T^{3} + 11057 T^{4} - 874 T^{5} + 807161 T^{6} + 2845686 T^{7} + 31510377 T^{8} + 2073071593 T^{10} \)
$79$ \( 1 - 28 T + 371 T^{2} - 2912 T^{3} + 17042 T^{4} - 111624 T^{5} + 1346318 T^{6} - 18173792 T^{7} + 182917469 T^{8} - 1090602268 T^{9} + 3077056399 T^{10} \)
$83$ \( 1 - 9 T + 186 T^{2} - 169 T^{3} + 5585 T^{4} + 99852 T^{5} + 463555 T^{6} - 1164241 T^{7} + 106352382 T^{8} - 427124889 T^{9} + 3939040643 T^{10} \)
$89$ \( 1 + 11 T + 365 T^{2} + 3632 T^{3} + 59750 T^{4} + 469970 T^{5} + 5317750 T^{6} + 28769072 T^{7} + 257313685 T^{8} + 690164651 T^{9} + 5584059449 T^{10} \)
$97$ \( 1 + 14 T + 309 T^{2} + 1468 T^{3} + 20994 T^{4} - 15492 T^{5} + 2036418 T^{6} + 13812412 T^{7} + 282015957 T^{8} + 1239409934 T^{9} + 8587340257 T^{10} \)
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