Properties

Label 22.4.c.a
Level 22
Weight 4
Character orbit 22.c
Analytic conductor 1.298
Analytic rank 0
Dimension 4
CM No
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 22 = 2 \cdot 11 \)
Weight: \( k \) = \( 4 \)
Character orbit: \([\chi]\) = 22.c (of order \(5\) and degree \(4\))

Newform invariants

Self dual: No
Analytic conductor: \(1.29804202013\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{10})\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{10}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( -2 \zeta_{10} q^{2} \) \( + ( -3 + 3 \zeta_{10} + 8 \zeta_{10}^{3} ) q^{3} \) \( + 4 \zeta_{10}^{2} q^{4} \) \( + ( 1 - \zeta_{10}^{3} ) q^{5} \) \( + ( 16 - 10 \zeta_{10} + 10 \zeta_{10}^{2} - 16 \zeta_{10}^{3} ) q^{6} \) \( + ( 11 \zeta_{10} - 3 \zeta_{10}^{2} + 11 \zeta_{10}^{3} ) q^{7} \) \( -8 \zeta_{10}^{3} q^{8} \) \( + ( -39 - 7 \zeta_{10} - 39 \zeta_{10}^{2} ) q^{9} \) \(+O(q^{10})\) \( q\) \( -2 \zeta_{10} q^{2} \) \( + ( -3 + 3 \zeta_{10} + 8 \zeta_{10}^{3} ) q^{3} \) \( + 4 \zeta_{10}^{2} q^{4} \) \( + ( 1 - \zeta_{10}^{3} ) q^{5} \) \( + ( 16 - 10 \zeta_{10} + 10 \zeta_{10}^{2} - 16 \zeta_{10}^{3} ) q^{6} \) \( + ( 11 \zeta_{10} - 3 \zeta_{10}^{2} + 11 \zeta_{10}^{3} ) q^{7} \) \( -8 \zeta_{10}^{3} q^{8} \) \( + ( -39 - 7 \zeta_{10} - 39 \zeta_{10}^{2} ) q^{9} \) \( + ( -2 - 2 \zeta_{10}^{2} + 2 \zeta_{10}^{3} ) q^{10} \) \( + ( 11 + 22 \zeta_{10} - 22 \zeta_{10}^{3} ) q^{11} \) \( + ( -32 - 12 \zeta_{10}^{2} + 12 \zeta_{10}^{3} ) q^{12} \) \( + ( 29 + 4 \zeta_{10} + 29 \zeta_{10}^{2} ) q^{13} \) \( + ( 22 - 22 \zeta_{10} - 16 \zeta_{10}^{3} ) q^{14} \) \( + ( 8 \zeta_{10} + 3 \zeta_{10}^{2} + 8 \zeta_{10}^{3} ) q^{15} \) \( + ( -16 + 16 \zeta_{10} - 16 \zeta_{10}^{2} + 16 \zeta_{10}^{3} ) q^{16} \) \( + ( 77 - 104 \zeta_{10} + 104 \zeta_{10}^{2} - 77 \zeta_{10}^{3} ) q^{17} \) \( + ( 78 \zeta_{10} + 14 \zeta_{10}^{2} + 78 \zeta_{10}^{3} ) q^{18} \) \( + ( 9 - 9 \zeta_{10} + 32 \zeta_{10}^{3} ) q^{19} \) \( + ( 4 + 4 \zeta_{10}^{2} ) q^{20} \) \( + ( -97 - 79 \zeta_{10}^{2} + 79 \zeta_{10}^{3} ) q^{21} \) \( + ( -44 + 22 \zeta_{10} - 88 \zeta_{10}^{2} + 44 \zeta_{10}^{3} ) q^{22} \) \( + ( -44 + 24 \zeta_{10}^{2} - 24 \zeta_{10}^{3} ) q^{23} \) \( + ( 24 + 40 \zeta_{10} + 24 \zeta_{10}^{2} ) q^{24} \) \( + ( 1 - \zeta_{10} + 123 \zeta_{10}^{3} ) q^{25} \) \( + ( -58 \zeta_{10} - 8 \zeta_{10}^{2} - 58 \zeta_{10}^{3} ) q^{26} \) \( + ( 269 - 17 \zeta_{10} + 17 \zeta_{10}^{2} - 269 \zeta_{10}^{3} ) q^{27} \) \( + ( -32 - 12 \zeta_{10} + 12 \zeta_{10}^{2} + 32 \zeta_{10}^{3} ) q^{28} \) \( + ( -159 \zeta_{10} + 107 \zeta_{10}^{2} - 159 \zeta_{10}^{3} ) q^{29} \) \( + ( 16 - 16 \zeta_{10} - 22 \zeta_{10}^{3} ) q^{30} \) \( + ( -143 + 202 \zeta_{10} - 143 \zeta_{10}^{2} ) q^{31} \) \( + 32 q^{32} \) \( + ( -143 + 253 \zeta_{10} - 44 \zeta_{10}^{2} + 264 \zeta_{10}^{3} ) q^{33} \) \( + ( -154 + 54 \zeta_{10}^{2} - 54 \zeta_{10}^{3} ) q^{34} \) \( + ( 8 + 11 \zeta_{10} + 8 \zeta_{10}^{2} ) q^{35} \) \( + ( 156 - 156 \zeta_{10} - 184 \zeta_{10}^{3} ) q^{36} \) \( + ( -79 \zeta_{10} - 97 \zeta_{10}^{2} - 79 \zeta_{10}^{3} ) q^{37} \) \( + ( 64 - 82 \zeta_{10} + 82 \zeta_{10}^{2} - 64 \zeta_{10}^{3} ) q^{38} \) \( + ( -351 + 107 \zeta_{10} - 107 \zeta_{10}^{2} + 351 \zeta_{10}^{3} ) q^{39} \) \( + ( -8 \zeta_{10} - 8 \zeta_{10}^{3} ) q^{40} \) \( + ( 129 - 129 \zeta_{10} - 40 \zeta_{10}^{3} ) q^{41} \) \( + ( 158 + 36 \zeta_{10} + 158 \zeta_{10}^{2} ) q^{42} \) \( + ( 256 - 68 \zeta_{10}^{2} + 68 \zeta_{10}^{3} ) q^{43} \) \( + ( 88 + 44 \zeta_{10}^{2} + 88 \zeta_{10}^{3} ) q^{44} \) \( + ( -85 - 46 \zeta_{10}^{2} + 46 \zeta_{10}^{3} ) q^{45} \) \( + ( -48 + 136 \zeta_{10} - 48 \zeta_{10}^{2} ) q^{46} \) \( + ( 21 - 21 \zeta_{10} + 188 \zeta_{10}^{3} ) q^{47} \) \( + ( -48 \zeta_{10} - 80 \zeta_{10}^{2} - 48 \zeta_{10}^{3} ) q^{48} \) \( + ( 158 - 213 \zeta_{10} + 213 \zeta_{10}^{2} - 158 \zeta_{10}^{3} ) q^{49} \) \( + ( 246 - 248 \zeta_{10} + 248 \zeta_{10}^{2} - 246 \zeta_{10}^{3} ) q^{50} \) \( + ( 96 \zeta_{10} + 439 \zeta_{10}^{2} + 96 \zeta_{10}^{3} ) q^{51} \) \( + ( -116 + 116 \zeta_{10} + 132 \zeta_{10}^{3} ) q^{52} \) \( + ( -223 + 424 \zeta_{10} - 223 \zeta_{10}^{2} ) q^{53} \) \( + ( -538 - 504 \zeta_{10}^{2} + 504 \zeta_{10}^{3} ) q^{54} \) \( + ( 33 - 22 \zeta_{10} + 22 \zeta_{10}^{2} - 55 \zeta_{10}^{3} ) q^{55} \) \( + ( 64 + 88 \zeta_{10}^{2} - 88 \zeta_{10}^{3} ) q^{56} \) \( + ( -51 - 178 \zeta_{10} - 51 \zeta_{10}^{2} ) q^{57} \) \( + ( -318 + 318 \zeta_{10} + 104 \zeta_{10}^{3} ) q^{58} \) \( + ( -301 \zeta_{10} + 225 \zeta_{10}^{2} - 301 \zeta_{10}^{3} ) q^{59} \) \( + ( -44 + 12 \zeta_{10} - 12 \zeta_{10}^{2} + 44 \zeta_{10}^{3} ) q^{60} \) \( + ( 429 + 24 \zeta_{10} - 24 \zeta_{10}^{2} - 429 \zeta_{10}^{3} ) q^{61} \) \( + ( 286 \zeta_{10} - 404 \zeta_{10}^{2} + 286 \zeta_{10}^{3} ) q^{62} \) \( + ( 389 - 389 \zeta_{10} - 797 \zeta_{10}^{3} ) q^{63} \) \( -64 \zeta_{10} q^{64} \) \( + ( 62 + 33 \zeta_{10}^{2} - 33 \zeta_{10}^{3} ) q^{65} \) \( + ( 528 - 242 \zeta_{10} + 22 \zeta_{10}^{2} - 440 \zeta_{10}^{3} ) q^{66} \) \( + ( -80 - 116 \zeta_{10}^{2} + 116 \zeta_{10}^{3} ) q^{67} \) \( + ( -108 + 416 \zeta_{10} - 108 \zeta_{10}^{2} ) q^{68} \) \( + ( 12 - 12 \zeta_{10} - 280 \zeta_{10}^{3} ) q^{69} \) \( + ( -16 \zeta_{10} - 22 \zeta_{10}^{2} - 16 \zeta_{10}^{3} ) q^{70} \) \( + ( -783 + 342 \zeta_{10} - 342 \zeta_{10}^{2} + 783 \zeta_{10}^{3} ) q^{71} \) \( + ( -368 + 56 \zeta_{10} - 56 \zeta_{10}^{2} + 368 \zeta_{10}^{3} ) q^{72} \) \( + ( -51 \zeta_{10} + 51 \zeta_{10}^{2} - 51 \zeta_{10}^{3} ) q^{73} \) \( + ( -158 + 158 \zeta_{10} + 352 \zeta_{10}^{3} ) q^{74} \) \( + ( -364 - 617 \zeta_{10} - 364 \zeta_{10}^{2} ) q^{75} \) \( + ( -128 + 36 \zeta_{10}^{2} - 36 \zeta_{10}^{3} ) q^{76} \) \( + ( -66 + 363 \zeta_{10} + 209 \zeta_{10}^{2} + 55 \zeta_{10}^{3} ) q^{77} \) \( + ( 702 + 488 \zeta_{10}^{2} - 488 \zeta_{10}^{3} ) q^{78} \) \( + ( 537 - 934 \zeta_{10} + 537 \zeta_{10}^{2} ) q^{79} \) \( + ( -16 + 16 \zeta_{10} + 16 \zeta_{10}^{3} ) q^{80} \) \( + ( 1014 \zeta_{10} + 652 \zeta_{10}^{2} + 1014 \zeta_{10}^{3} ) q^{81} \) \( + ( -80 - 178 \zeta_{10} + 178 \zeta_{10}^{2} + 80 \zeta_{10}^{3} ) q^{82} \) \( + ( -63 + 538 \zeta_{10} - 538 \zeta_{10}^{2} + 63 \zeta_{10}^{3} ) q^{83} \) \( + ( -316 \zeta_{10} - 72 \zeta_{10}^{2} - 316 \zeta_{10}^{3} ) q^{84} \) \( + ( 77 - 77 \zeta_{10} - 50 \zeta_{10}^{3} ) q^{85} \) \( + ( 136 - 648 \zeta_{10} + 136 \zeta_{10}^{2} ) q^{86} \) \( + ( 893 + 951 \zeta_{10}^{2} - 951 \zeta_{10}^{3} ) q^{87} \) \( + ( 176 - 352 \zeta_{10} + 176 \zeta_{10}^{2} - 264 \zeta_{10}^{3} ) q^{88} \) \( + ( -38 - 940 \zeta_{10}^{2} + 940 \zeta_{10}^{3} ) q^{89} \) \( + ( 92 + 78 \zeta_{10} + 92 \zeta_{10}^{2} ) q^{90} \) \( + ( -276 + 276 \zeta_{10} + 583 \zeta_{10}^{3} ) q^{91} \) \( + ( 96 \zeta_{10} - 272 \zeta_{10}^{2} + 96 \zeta_{10}^{3} ) q^{92} \) \( + ( -43 + 581 \zeta_{10} - 581 \zeta_{10}^{2} + 43 \zeta_{10}^{3} ) q^{93} \) \( + ( 376 - 418 \zeta_{10} + 418 \zeta_{10}^{2} - 376 \zeta_{10}^{3} ) q^{94} \) \( + ( 32 \zeta_{10} - 9 \zeta_{10}^{2} + 32 \zeta_{10}^{3} ) q^{95} \) \( + ( -96 + 96 \zeta_{10} + 256 \zeta_{10}^{3} ) q^{96} \) \( + ( 665 + 24 \zeta_{10} + 665 \zeta_{10}^{2} ) q^{97} \) \( + ( -316 + 110 \zeta_{10}^{2} - 110 \zeta_{10}^{3} ) q^{98} \) \( + ( -1441 - 781 \zeta_{10} - 737 \zeta_{10}^{2} + 154 \zeta_{10}^{3} ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(4q \) \(\mathstrut -\mathstrut 2q^{2} \) \(\mathstrut -\mathstrut q^{3} \) \(\mathstrut -\mathstrut 4q^{4} \) \(\mathstrut +\mathstrut 3q^{5} \) \(\mathstrut +\mathstrut 28q^{6} \) \(\mathstrut +\mathstrut 25q^{7} \) \(\mathstrut -\mathstrut 8q^{8} \) \(\mathstrut -\mathstrut 124q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(4q \) \(\mathstrut -\mathstrut 2q^{2} \) \(\mathstrut -\mathstrut q^{3} \) \(\mathstrut -\mathstrut 4q^{4} \) \(\mathstrut +\mathstrut 3q^{5} \) \(\mathstrut +\mathstrut 28q^{6} \) \(\mathstrut +\mathstrut 25q^{7} \) \(\mathstrut -\mathstrut 8q^{8} \) \(\mathstrut -\mathstrut 124q^{9} \) \(\mathstrut -\mathstrut 4q^{10} \) \(\mathstrut +\mathstrut 44q^{11} \) \(\mathstrut -\mathstrut 104q^{12} \) \(\mathstrut +\mathstrut 91q^{13} \) \(\mathstrut +\mathstrut 50q^{14} \) \(\mathstrut +\mathstrut 13q^{15} \) \(\mathstrut -\mathstrut 16q^{16} \) \(\mathstrut +\mathstrut 23q^{17} \) \(\mathstrut +\mathstrut 142q^{18} \) \(\mathstrut +\mathstrut 59q^{19} \) \(\mathstrut +\mathstrut 12q^{20} \) \(\mathstrut -\mathstrut 230q^{21} \) \(\mathstrut -\mathstrut 22q^{22} \) \(\mathstrut -\mathstrut 224q^{23} \) \(\mathstrut +\mathstrut 112q^{24} \) \(\mathstrut +\mathstrut 126q^{25} \) \(\mathstrut -\mathstrut 108q^{26} \) \(\mathstrut +\mathstrut 773q^{27} \) \(\mathstrut -\mathstrut 120q^{28} \) \(\mathstrut -\mathstrut 425q^{29} \) \(\mathstrut +\mathstrut 26q^{30} \) \(\mathstrut -\mathstrut 227q^{31} \) \(\mathstrut +\mathstrut 128q^{32} \) \(\mathstrut -\mathstrut 11q^{33} \) \(\mathstrut -\mathstrut 724q^{34} \) \(\mathstrut +\mathstrut 35q^{35} \) \(\mathstrut +\mathstrut 284q^{36} \) \(\mathstrut -\mathstrut 61q^{37} \) \(\mathstrut +\mathstrut 28q^{38} \) \(\mathstrut -\mathstrut 839q^{39} \) \(\mathstrut -\mathstrut 16q^{40} \) \(\mathstrut +\mathstrut 347q^{41} \) \(\mathstrut +\mathstrut 510q^{42} \) \(\mathstrut +\mathstrut 1160q^{43} \) \(\mathstrut +\mathstrut 396q^{44} \) \(\mathstrut -\mathstrut 248q^{45} \) \(\mathstrut -\mathstrut 8q^{46} \) \(\mathstrut +\mathstrut 251q^{47} \) \(\mathstrut -\mathstrut 16q^{48} \) \(\mathstrut +\mathstrut 48q^{49} \) \(\mathstrut +\mathstrut 242q^{50} \) \(\mathstrut -\mathstrut 247q^{51} \) \(\mathstrut -\mathstrut 216q^{52} \) \(\mathstrut -\mathstrut 245q^{53} \) \(\mathstrut -\mathstrut 1144q^{54} \) \(\mathstrut +\mathstrut 33q^{55} \) \(\mathstrut +\mathstrut 80q^{56} \) \(\mathstrut -\mathstrut 331q^{57} \) \(\mathstrut -\mathstrut 850q^{58} \) \(\mathstrut -\mathstrut 827q^{59} \) \(\mathstrut -\mathstrut 108q^{60} \) \(\mathstrut +\mathstrut 1335q^{61} \) \(\mathstrut +\mathstrut 976q^{62} \) \(\mathstrut +\mathstrut 370q^{63} \) \(\mathstrut -\mathstrut 64q^{64} \) \(\mathstrut +\mathstrut 182q^{65} \) \(\mathstrut +\mathstrut 1408q^{66} \) \(\mathstrut -\mathstrut 88q^{67} \) \(\mathstrut +\mathstrut 92q^{68} \) \(\mathstrut -\mathstrut 244q^{69} \) \(\mathstrut -\mathstrut 10q^{70} \) \(\mathstrut -\mathstrut 1665q^{71} \) \(\mathstrut -\mathstrut 992q^{72} \) \(\mathstrut -\mathstrut 153q^{73} \) \(\mathstrut -\mathstrut 122q^{74} \) \(\mathstrut -\mathstrut 1709q^{75} \) \(\mathstrut -\mathstrut 584q^{76} \) \(\mathstrut -\mathstrut 55q^{77} \) \(\mathstrut +\mathstrut 1832q^{78} \) \(\mathstrut +\mathstrut 677q^{79} \) \(\mathstrut -\mathstrut 32q^{80} \) \(\mathstrut +\mathstrut 1376q^{81} \) \(\mathstrut -\mathstrut 596q^{82} \) \(\mathstrut +\mathstrut 887q^{83} \) \(\mathstrut -\mathstrut 560q^{84} \) \(\mathstrut +\mathstrut 181q^{85} \) \(\mathstrut -\mathstrut 240q^{86} \) \(\mathstrut +\mathstrut 1670q^{87} \) \(\mathstrut -\mathstrut 88q^{88} \) \(\mathstrut +\mathstrut 1728q^{89} \) \(\mathstrut +\mathstrut 354q^{90} \) \(\mathstrut -\mathstrut 245q^{91} \) \(\mathstrut +\mathstrut 464q^{92} \) \(\mathstrut +\mathstrut 1033q^{93} \) \(\mathstrut +\mathstrut 292q^{94} \) \(\mathstrut +\mathstrut 73q^{95} \) \(\mathstrut -\mathstrut 32q^{96} \) \(\mathstrut +\mathstrut 2019q^{97} \) \(\mathstrut -\mathstrut 1484q^{98} \) \(\mathstrut -\mathstrut 5654q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/22\mathbb{Z}\right)^\times\).

\(n\) \(13\)
\(\chi(n)\) \(\zeta_{10}^{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3.1
0.809017 0.587785i
−0.309017 0.951057i
−0.309017 + 0.951057i
0.809017 + 0.587785i
−1.61803 + 1.17557i −3.04508 9.37181i 1.23607 3.80423i 1.30902 + 0.951057i 15.9443 + 11.5842i 4.57295 14.0741i 2.47214 + 7.60845i −56.7148 + 41.2057i −3.23607
5.1 0.618034 + 1.90211i 2.54508 + 1.84911i −3.23607 + 2.35114i 0.190983 0.587785i −1.94427 + 5.98385i 7.92705 5.75934i −6.47214 4.70228i −5.28522 16.2662i 1.23607
9.1 0.618034 1.90211i 2.54508 1.84911i −3.23607 2.35114i 0.190983 + 0.587785i −1.94427 5.98385i 7.92705 + 5.75934i −6.47214 + 4.70228i −5.28522 + 16.2662i 1.23607
15.1 −1.61803 1.17557i −3.04508 + 9.37181i 1.23607 + 3.80423i 1.30902 0.951057i 15.9443 11.5842i 4.57295 + 14.0741i 2.47214 7.60845i −56.7148 41.2057i −3.23607
\(n\): 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
11.c Even 1 yes

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{3}^{4} \) \(\mathstrut +\mathstrut T_{3}^{3} \) \(\mathstrut +\mathstrut 76 T_{3}^{2} \) \(\mathstrut -\mathstrut 434 T_{3} \) \(\mathstrut +\mathstrut 961 \) acting on \(S_{4}^{\mathrm{new}}(22, [\chi])\).