Properties

Label 201.2.e.b
Level 201
Weight 2
Character orbit 201.e
Analytic conductor 1.605
Analytic rank 0
Dimension 10
CM No
Inner twists 2

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) = \( 201 = 3 \cdot 67 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 201.e (of order \(3\) and degree \(2\))

Newform invariants

Self dual: No
Analytic conductor: \(1.60499308063\)
Analytic rank: \(0\)
Dimension: \(10\)
Relative dimension: \(5\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3 \)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{10} - 2 x^{9} + 10 x^{8} - 8 x^{7} + 49 x^{6} - 39 x^{5} + 128 x^{4} - 14 x^{3} + 119 x^{2} - 49 x + 49\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( -24307 \nu^{9} + 181995 \nu^{8} - 604465 \nu^{7} + 1626651 \nu^{6} - 3301924 \nu^{5} + 7319233 \nu^{4} - 12172941 \nu^{3} + 14158031 \nu^{2} - 13105288 \nu + 6437963 \)\()/4414074\)
\(\beta_{3}\)\(=\)\((\)\( 32653 \nu^{9} - 100089 \nu^{8} + 396649 \nu^{7} - 539103 \nu^{6} + 1764532 \nu^{5} - 2509639 \nu^{4} + 5937753 \nu^{3} - 2828441 \nu^{2} + 1453144 \nu - 2618021 \)\()/4414074\)
\(\beta_{4}\)\(=\)\((\)\( -53429 \nu^{9} + 74205 \nu^{8} - 434201 \nu^{7} + 30783 \nu^{6} - 2078918 \nu^{5} + 319199 \nu^{4} - 4329273 \nu^{3} - 5189747 \nu^{2} - 3529610 \nu + 1164877 \)\()/4414074\)
\(\beta_{5}\)\(=\)\((\)\( 33718 \nu^{9} - 85104 \nu^{8} + 337264 \nu^{7} - 351819 \nu^{6} + 1500352 \nu^{5} - 2133904 \nu^{4} + 4154526 \nu^{3} - 2404976 \nu^{2} + 1235584 \nu - 7130123 \)\()/2207037\)
\(\beta_{6}\)\(=\)\((\)\( -31943 \nu^{9} + 110079 \nu^{8} - 436239 \nu^{7} + 663959 \nu^{6} - 1940652 \nu^{5} + 2760129 \nu^{4} - 5655213 \nu^{3} + 3110751 \nu^{2} - 1598184 \nu + 1081311 \)\()/1471358\)
\(\beta_{7}\)\(=\)\((\)\( -112429 \nu^{9} - 45435 \nu^{8} - 824203 \nu^{7} - 1270707 \nu^{6} - 5458108 \nu^{5} - 6300671 \nu^{4} - 13242393 \nu^{3} - 19664257 \nu^{2} - 25007374 \nu - 11599567 \)\()/4414074\)
\(\beta_{8}\)\(=\)\((\)\( -62752 \nu^{9} + 76248 \nu^{8} - 512362 \nu^{7} - 36093 \nu^{6} - 2500291 \nu^{5} - 400286 \nu^{4} - 5308260 \nu^{3} - 6568339 \nu^{2} - 6992440 \nu - 4073797 \)\()/2207037\)
\(\beta_{9}\)\(=\)\((\)\( -117800 \nu^{9} + 300861 \nu^{8} - 1192301 \nu^{7} + 1489527 \nu^{6} - 5304068 \nu^{5} + 7543811 \nu^{4} - 12249042 \nu^{3} + 8502109 \nu^{2} - 4368056 \nu + 12355714 \)\()/2207037\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{8} - \beta_{5} - 3 \beta_{4} + \beta_{3} + \beta_{1}\)
\(\nu^{3}\)\(=\)\(\beta_{6} - \beta_{5} + 5 \beta_{3} - 1\)
\(\nu^{4}\)\(=\)\(-7 \beta_{8} + \beta_{7} + \beta_{6} + 13 \beta_{4} - \beta_{2} - 8 \beta_{1} - 13\)
\(\nu^{5}\)\(=\)\(2 \beta_{9} - 10 \beta_{8} + 2 \beta_{7} + 10 \beta_{5} + 9 \beta_{4} - 29 \beta_{3} - 8 \beta_{2} - 29 \beta_{1}\)
\(\nu^{6}\)\(=\)\(10 \beta_{9} - 12 \beta_{6} + 45 \beta_{5} - 56 \beta_{3} + 65\)
\(\nu^{7}\)\(=\)\(80 \beta_{8} - 22 \beta_{7} - 55 \beta_{6} - 68 \beta_{4} + 55 \beta_{2} + 178 \beta_{1} + 68\)
\(\nu^{8}\)\(=\)\(-77 \beta_{9} + 288 \beta_{8} - 77 \beta_{7} - 288 \beta_{5} - 352 \beta_{4} + 381 \beta_{3} + 102 \beta_{2} + 381 \beta_{1}\)
\(\nu^{9}\)\(=\)\(-179 \beta_{9} + 365 \beta_{6} - 585 \beta_{5} + 1123 \beta_{3} - 490\)

Character Values

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

\(n\) \(68\) \(136\)
\(\chi(n)\) \(1\) \(-\beta_{4}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
37.1
1.28367 + 2.22339i
0.947050 + 1.64034i
0.330147 + 0.571831i
−0.527336 0.913372i
−1.03354 1.79014i
1.28367 2.22339i
0.947050 1.64034i
0.330147 0.571831i
−0.527336 + 0.913372i
−1.03354 + 1.79014i
−1.28367 2.22339i −1.00000 −2.29564 + 3.97616i −1.02393 1.28367 + 2.22339i −0.850574 + 1.47324i 6.65272 1.00000 1.31439 + 2.27659i
37.2 −0.947050 1.64034i −1.00000 −0.793808 + 1.37492i 1.30648 0.947050 + 1.64034i 2.53710 4.39438i −0.781096 1.00000 −1.23731 2.14308i
37.3 −0.330147 0.571831i −1.00000 0.782006 1.35447i 3.22431 0.330147 + 0.571831i −1.02143 + 1.76916i −2.35330 1.00000 −1.06450 1.84376i
37.4 0.527336 + 0.913372i −1.00000 0.443834 0.768743i 0.832996 −0.527336 0.913372i 1.13029 1.95772i 3.04554 1.00000 0.439269 + 0.760836i
37.5 1.03354 + 1.79014i −1.00000 −1.13639 + 1.96829i −3.33986 −1.03354 1.79014i −2.29539 + 3.97573i −0.563865 1.00000 −3.45186 5.97880i
163.1 −1.28367 + 2.22339i −1.00000 −2.29564 3.97616i −1.02393 1.28367 2.22339i −0.850574 1.47324i 6.65272 1.00000 1.31439 2.27659i
163.2 −0.947050 + 1.64034i −1.00000 −0.793808 1.37492i 1.30648 0.947050 1.64034i 2.53710 + 4.39438i −0.781096 1.00000 −1.23731 + 2.14308i
163.3 −0.330147 + 0.571831i −1.00000 0.782006 + 1.35447i 3.22431 0.330147 0.571831i −1.02143 1.76916i −2.35330 1.00000 −1.06450 + 1.84376i
163.4 0.527336 0.913372i −1.00000 0.443834 + 0.768743i 0.832996 −0.527336 + 0.913372i 1.13029 + 1.95772i 3.04554 1.00000 0.439269 0.760836i
163.5 1.03354 1.79014i −1.00000 −1.13639 1.96829i −3.33986 −1.03354 + 1.79014i −2.29539 3.97573i −0.563865 1.00000 −3.45186 + 5.97880i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 163.5
Significant digits:
Format:

Inner twists

Char. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
67.c Even 1 yes

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{2}^{10} + \cdots\) acting on \(S_{2}^{\mathrm{new}}(201, [\chi])\).