Properties

Label 201.4.j.a
Level 201
Weight 4
Character orbit 201.j
Analytic conductor 11.859
Analytic rank 0
Dimension 20
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.j (of order \(22\) and degree \(10\))

Newform invariants

Self dual: No
Analytic conductor: \(11.8593839112\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(2\) over \(\Q(\zeta_{22})\)
Coefficient field: \(\Q(\zeta_{33})\)
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 2^{10}\cdot 3^{10} \)
Sato-Tate group: $\mathrm{U}(1)[D_{22}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{33}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 3 \zeta_{33}^{8} - 3 \zeta_{33}^{19} ) q^{3} + ( 8 - 8 \zeta_{33}^{2} + 8 \zeta_{33}^{3} - 8 \zeta_{33}^{5} + 8 \zeta_{33}^{6} - 8 \zeta_{33}^{8} + 8 \zeta_{33}^{9} + 8 \zeta_{33}^{12} - 8 \zeta_{33}^{13} + 8 \zeta_{33}^{15} - 8 \zeta_{33}^{16} + 8 \zeta_{33}^{18} - 8 \zeta_{33}^{19} ) q^{4} + ( -18 \zeta_{33}^{4} - \zeta_{33}^{6} + \zeta_{33}^{15} + 18 \zeta_{33}^{17} ) q^{7} + ( 27 \zeta_{33}^{5} + 27 \zeta_{33}^{16} ) q^{9} +O(q^{10})\) \( q + ( 3 \zeta_{33}^{8} - 3 \zeta_{33}^{19} ) q^{3} + ( 8 - 8 \zeta_{33}^{2} + 8 \zeta_{33}^{3} - 8 \zeta_{33}^{5} + 8 \zeta_{33}^{6} - 8 \zeta_{33}^{8} + 8 \zeta_{33}^{9} + 8 \zeta_{33}^{12} - 8 \zeta_{33}^{13} + 8 \zeta_{33}^{15} - 8 \zeta_{33}^{16} + 8 \zeta_{33}^{18} - 8 \zeta_{33}^{19} ) q^{4} + ( -18 \zeta_{33}^{4} - \zeta_{33}^{6} + \zeta_{33}^{15} + 18 \zeta_{33}^{17} ) q^{7} + ( 27 \zeta_{33}^{5} + 27 \zeta_{33}^{16} ) q^{9} + ( 48 \zeta_{33}^{7} + 24 \zeta_{33}^{18} ) q^{12} + ( 36 \zeta_{33} + 17 \zeta_{33}^{2} + 53 \zeta_{33}^{12} + 53 \zeta_{33}^{13} ) q^{13} + 64 \zeta_{33}^{9} q^{16} + ( -90 + 90 \zeta_{33} - 163 \zeta_{33}^{3} + 90 \zeta_{33}^{4} - 90 \zeta_{33}^{6} + 90 \zeta_{33}^{7} - 73 \zeta_{33}^{9} + 90 \zeta_{33}^{10} - 90 \zeta_{33}^{11} + 90 \zeta_{33}^{13} - 180 \zeta_{33}^{14} + 90 \zeta_{33}^{16} - 90 \zeta_{33}^{17} + 90 \zeta_{33}^{19} ) q^{19} + ( -60 \zeta_{33} - 111 \zeta_{33}^{3} - 111 \zeta_{33}^{12} - 60 \zeta_{33}^{14} ) q^{21} -125 \zeta_{33}^{18} q^{25} + ( -81 \zeta_{33}^{2} + 81 \zeta_{33}^{13} ) q^{27} + ( -152 \zeta_{33}^{3} - 152 \zeta_{33}^{5} - 144 \zeta_{33}^{14} - 8 \zeta_{33}^{16} ) q^{28} + ( 109 \zeta_{33}^{2} + 109 \zeta_{33}^{6} + 199 \zeta_{33}^{13} - 90 \zeta_{33}^{17} ) q^{31} + 216 \zeta_{33}^{15} q^{36} + ( -252 \zeta_{33}^{4} + 252 \zeta_{33}^{7} - 181 \zeta_{33}^{15} + 71 \zeta_{33}^{18} ) q^{37} + ( -477 + 210 \zeta_{33} + 267 \zeta_{33}^{2} - 477 \zeta_{33}^{3} + 210 \zeta_{33}^{4} + 267 \zeta_{33}^{5} - 477 \zeta_{33}^{6} + 210 \zeta_{33}^{7} + 267 \zeta_{33}^{8} - 210 \zeta_{33}^{9} + 420 \zeta_{33}^{10} - 210 \zeta_{33}^{11} - 267 \zeta_{33}^{12} + 477 \zeta_{33}^{13} - 210 \zeta_{33}^{14} - 267 \zeta_{33}^{15} + 477 \zeta_{33}^{16} - 210 \zeta_{33}^{17} - 267 \zeta_{33}^{18} + 477 \zeta_{33}^{19} ) q^{39} + ( 197 \zeta_{33}^{5} - 323 \zeta_{33}^{8} + 323 \zeta_{33}^{16} - 197 \zeta_{33}^{19} ) q^{43} + ( 192 \zeta_{33}^{6} + 384 \zeta_{33}^{17} ) q^{48} + ( 343 + 360 \zeta_{33} - 343 \zeta_{33}^{2} + 343 \zeta_{33}^{3} - 343 \zeta_{33}^{5} + 343 \zeta_{33}^{6} - 20 \zeta_{33}^{8} + 343 \zeta_{33}^{9} + 380 \zeta_{33}^{12} - 343 \zeta_{33}^{13} + 343 \zeta_{33}^{15} - 343 \zeta_{33}^{16} + 343 \zeta_{33}^{18} - 380 \zeta_{33}^{19} ) q^{49} + ( -136 - 288 \zeta_{33} + 288 \zeta_{33}^{11} + 136 \zeta_{33}^{12} ) q^{52} + ( 321 - 489 \zeta_{33}^{6} - 168 \zeta_{33}^{11} - 168 \zeta_{33}^{17} ) q^{57} + ( 361 - 361 \zeta_{33}^{2} + 361 \zeta_{33}^{3} - 182 \zeta_{33}^{5} + 361 \zeta_{33}^{6} - 361 \zeta_{33}^{8} + 361 \zeta_{33}^{9} - 540 \zeta_{33}^{10} + 361 \zeta_{33}^{12} - 361 \zeta_{33}^{13} + 361 \zeta_{33}^{15} - 722 \zeta_{33}^{16} + 361 \zeta_{33}^{18} - 361 \zeta_{33}^{19} ) q^{61} + ( 513 - 486 \zeta_{33} + 486 \zeta_{33}^{3} - 486 \zeta_{33}^{4} + 486 \zeta_{33}^{6} - 486 \zeta_{33}^{7} - 27 \zeta_{33}^{9} - 486 \zeta_{33}^{10} - 486 \zeta_{33}^{13} + 486 \zeta_{33}^{14} - 486 \zeta_{33}^{16} + 486 \zeta_{33}^{17} - 486 \zeta_{33}^{19} ) q^{63} + ( 512 \zeta_{33}^{8} + 512 \zeta_{33}^{19} ) q^{64} + ( -378 \zeta_{33}^{7} + 251 \zeta_{33}^{18} ) q^{67} + ( -487 + 216 \zeta_{33}^{4} + 216 \zeta_{33}^{11} - 487 \zeta_{33}^{15} ) q^{73} + ( 750 \zeta_{33}^{4} + 375 \zeta_{33}^{15} ) q^{75} + ( 136 \zeta_{33}^{2} - 584 \zeta_{33}^{8} - 584 \zeta_{33}^{13} + 136 \zeta_{33}^{19} ) q^{76} + ( -127 + 757 \zeta_{33}^{3} + 630 \zeta_{33}^{11} + 630 \zeta_{33}^{14} ) q^{79} + ( -729 + 729 \zeta_{33}^{2} - 729 \zeta_{33}^{3} + 729 \zeta_{33}^{5} - 729 \zeta_{33}^{6} + 729 \zeta_{33}^{8} - 729 \zeta_{33}^{9} - 729 \zeta_{33}^{12} + 729 \zeta_{33}^{13} - 729 \zeta_{33}^{15} + 729 \zeta_{33}^{16} - 729 \zeta_{33}^{18} + 729 \zeta_{33}^{19} ) q^{81} + ( 408 - 408 \zeta_{33}^{2} - 480 \zeta_{33}^{11} - 888 \zeta_{33}^{13} ) q^{84} + ( -701 \zeta_{33}^{5} - 359 \zeta_{33}^{6} - 990 \zeta_{33}^{7} - 971 \zeta_{33}^{8} - 971 \zeta_{33}^{16} - 990 \zeta_{33}^{17} - 359 \zeta_{33}^{18} - 701 \zeta_{33}^{19} ) q^{91} + ( -867 + 867 \zeta_{33}^{2} + 867 \zeta_{33}^{5} - 867 \zeta_{33}^{6} + 867 \zeta_{33}^{8} - 867 \zeta_{33}^{9} + 924 \zeta_{33}^{10} - 867 \zeta_{33}^{12} + 867 \zeta_{33}^{13} + 924 \zeta_{33}^{14} - 867 \zeta_{33}^{15} + 867 \zeta_{33}^{16} - 867 \zeta_{33}^{18} + 867 \zeta_{33}^{19} ) q^{93} + ( 1061 \zeta_{33}^{3} + 1061 \zeta_{33}^{8} + 792 \zeta_{33}^{14} + 269 \zeta_{33}^{19} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20q + 16q^{4} + 54q^{9} + O(q^{10}) \) \( 20q + 16q^{4} + 54q^{9} - 128q^{16} + 112q^{19} + 324q^{21} + 250q^{25} - 432q^{36} + 220q^{37} - 648q^{39} + 1258q^{49} - 6160q^{52} + 8910q^{57} + 5940q^{63} + 1024q^{64} - 880q^{67} - 10710q^{73} - 896q^{76} - 9724q^{79} - 1458q^{81} + 11664q^{84} - 3888q^{91} - 1620q^{93} + O(q^{100}) \)

Character Values

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

\(n\) \(68\) \(136\)
\(\chi(n)\) \(-1\) \(1 + \zeta_{33} + \zeta_{33}^{2} + \zeta_{33}^{3} + \zeta_{33}^{4} - \zeta_{33}^{5} + \zeta_{33}^{6} - \zeta_{33}^{7} + \zeta_{33}^{8} - \zeta_{33}^{9}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
5.1
0.928368 0.371662i
−0.786053 0.618159i
0.235759 0.971812i
0.723734 + 0.690079i
−0.327068 + 0.945001i
0.981929 0.189251i
−0.327068 0.945001i
0.981929 + 0.189251i
0.580057 0.814576i
−0.995472 0.0950560i
0.580057 + 0.814576i
−0.995472 + 0.0950560i
−0.888835 0.458227i
0.0475819 + 0.998867i
0.928368 + 0.371662i
−0.786053 + 0.618159i
0.235759 + 0.971812i
0.723734 0.690079i
−0.888835 + 0.458227i
0.0475819 0.998867i
0 −4.72659 + 2.15856i 1.13852 + 7.91857i 0 0 18.3144 + 15.8695i 0 17.6812 20.4052i 0
5.2 0 4.72659 2.15856i 1.13852 + 7.91857i 0 0 11.6079 + 10.0583i 0 17.6812 20.4052i 0
8.1 0 −3.92699 + 3.40276i 7.67594 + 2.25386i 0 0 −24.0323 3.45532i 0 3.84250 26.7252i 0
8.2 0 3.92699 3.40276i 7.67594 + 2.25386i 0 0 35.1868 + 5.05910i 0 3.84250 26.7252i 0
53.1 0 −2.80925 + 4.37128i 5.23889 6.04600i 0 0 4.82246 2.20234i 0 −11.2162 24.5601i 0
53.2 0 2.80925 4.37128i 5.23889 6.04600i 0 0 −32.3206 + 14.7603i 0 −11.2162 24.5601i 0
110.1 0 −2.80925 4.37128i 5.23889 + 6.04600i 0 0 4.82246 + 2.20234i 0 −11.2162 + 24.5601i 0
110.2 0 2.80925 + 4.37128i 5.23889 + 6.04600i 0 0 −32.3206 14.7603i 0 −11.2162 + 24.5601i 0
119.1 0 −1.46393 4.98567i −3.32332 + 7.27706i 0 0 −2.83365 4.40925i 0 −22.7138 + 14.5973i 0
119.2 0 1.46393 + 4.98567i −3.32332 + 7.27706i 0 0 −16.8377 26.2000i 0 −22.7138 + 14.5973i 0
125.1 0 −1.46393 + 4.98567i −3.32332 7.27706i 0 0 −2.83365 + 4.40925i 0 −22.7138 14.5973i 0
125.2 0 1.46393 4.98567i −3.32332 7.27706i 0 0 −16.8377 + 26.2000i 0 −22.7138 14.5973i 0
137.1 0 −5.14326 0.739490i −6.73003 4.32513i 0 0 10.4355 35.5401i 0 25.9063 + 7.60678i 0
137.2 0 5.14326 + 0.739490i −6.73003 4.32513i 0 0 −4.34287 + 14.7905i 0 25.9063 + 7.60678i 0
161.1 0 −4.72659 2.15856i 1.13852 7.91857i 0 0 18.3144 15.8695i 0 17.6812 + 20.4052i 0
161.2 0 4.72659 + 2.15856i 1.13852 7.91857i 0 0 11.6079 10.0583i 0 17.6812 + 20.4052i 0
176.1 0 −3.92699 3.40276i 7.67594 2.25386i 0 0 −24.0323 + 3.45532i 0 3.84250 + 26.7252i 0
176.2 0 3.92699 + 3.40276i 7.67594 2.25386i 0 0 35.1868 5.05910i 0 3.84250 + 26.7252i 0
179.1 0 −5.14326 + 0.739490i −6.73003 + 4.32513i 0 0 10.4355 + 35.5401i 0 25.9063 7.60678i 0
179.2 0 5.14326 0.739490i −6.73003 + 4.32513i 0 0 −4.34287 14.7905i 0 25.9063 7.60678i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 179.2
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.f Odd 1 yes
201.j Even 1 yes

Hecke kernels

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