Properties

Label 23.4.a.b
Level 23
Weight 4
Character orbit 23.a
Self dual Yes
Analytic conductor 1.357
Analytic rank 0
Dimension 4
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 23 \)
Weight: \( k \) = \( 4 \)
Character orbit: \([\chi]\) = 23.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(1.35704393013\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.334189.1
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( + ( 1 + \beta_{3} ) q^{2} \) \( + ( 1 + \beta_{1} + \beta_{2} ) q^{3} \) \( + ( 6 - 3 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} ) q^{4} \) \( + ( 2 + 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} ) q^{5} \) \( + ( -5 + \beta_{1} + 3 \beta_{2} + \beta_{3} ) q^{6} \) \( + ( 4 - 4 \beta_{1} + 4 \beta_{2} - 2 \beta_{3} ) q^{7} \) \( + ( -15 + 4 \beta_{1} - \beta_{2} + 5 \beta_{3} ) q^{8} \) \( + ( -10 + 5 \beta_{1} + \beta_{2} + 2 \beta_{3} ) q^{9} \) \(+O(q^{10})\) \( q\) \( + ( 1 + \beta_{3} ) q^{2} \) \( + ( 1 + \beta_{1} + \beta_{2} ) q^{3} \) \( + ( 6 - 3 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} ) q^{4} \) \( + ( 2 + 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} ) q^{5} \) \( + ( -5 + \beta_{1} + 3 \beta_{2} + \beta_{3} ) q^{6} \) \( + ( 4 - 4 \beta_{1} + 4 \beta_{2} - 2 \beta_{3} ) q^{7} \) \( + ( -15 + 4 \beta_{1} - \beta_{2} + 5 \beta_{3} ) q^{8} \) \( + ( -10 + 5 \beta_{1} + \beta_{2} + 2 \beta_{3} ) q^{9} \) \( + ( -16 + 4 \beta_{1} - 6 \beta_{2} + 4 \beta_{3} ) q^{10} \) \( + ( 10 - 12 \beta_{1} - 4 \beta_{2} + 2 \beta_{3} ) q^{11} \) \( + ( -16 - 8 \beta_{1} + \beta_{2} - 6 \beta_{3} ) q^{12} \) \( + ( 31 - 11 \beta_{1} - 11 \beta_{2} - 10 \beta_{3} ) q^{13} \) \( + ( -38 + 10 \beta_{1} + 24 \beta_{2} + 18 \beta_{3} ) q^{14} \) \( + ( -10 + 18 \beta_{1} + 2 \beta_{2} - 6 \beta_{3} ) q^{15} \) \( + ( 3 + 8 \beta_{1} - 2 \beta_{2} - 19 \beta_{3} ) q^{16} \) \( + ( 32 - 2 \beta_{1} - 6 \beta_{2} + 10 \beta_{3} ) q^{17} \) \( + ( 6 - 5 \beta_{1} - 5 \beta_{2} - 20 \beta_{3} ) q^{18} \) \( + ( 14 + 32 \beta_{1} + 24 \beta_{2} + 24 \beta_{3} ) q^{19} \) \( + ( 46 - 34 \beta_{1} - 20 \beta_{2} - 22 \beta_{3} ) q^{20} \) \( + ( 64 - 28 \beta_{1} - 8 \beta_{2} + 6 \beta_{3} ) q^{21} \) \( + ( 68 - 10 \beta_{1} - 8 \beta_{2} + 12 \beta_{3} ) q^{22} \) \( -23 q^{23} \) \( + ( -51 + 11 \beta_{1} + 3 \beta_{3} ) q^{24} \) \( + ( 43 + 8 \beta_{1} + 28 \beta_{2} + 16 \beta_{3} ) q^{25} \) \( + ( -33 + 19 \beta_{1} - 13 \beta_{2} + 61 \beta_{3} ) q^{26} \) \( + ( -25 - 9 \beta_{1} - 29 \beta_{2} + 4 \beta_{3} ) q^{27} \) \( + ( 34 + 2 \beta_{1} + 18 \beta_{2} - 62 \beta_{3} ) q^{28} \) \( + ( -31 + 35 \beta_{1} + 23 \beta_{2} - 26 \beta_{3} ) q^{29} \) \( + ( -116 + 20 \beta_{1} + 2 \beta_{2} - 8 \beta_{3} ) q^{30} \) \( + ( -35 - 19 \beta_{1} + 9 \beta_{2} + 12 \beta_{3} ) q^{31} \) \( + ( -122 + 23 \beta_{1} + 30 \beta_{2} + 10 \beta_{3} ) q^{32} \) \( + ( -82 - 54 \beta_{1} + 6 \beta_{2} - 6 \beta_{3} ) q^{33} \) \( + ( 194 - 36 \beta_{1} - 42 \beta_{2} - 2 \beta_{3} ) q^{34} \) \( + ( -212 - 12 \beta_{1} - 48 \beta_{2} - 84 \beta_{3} ) q^{35} \) \( + ( -144 + 15 \beta_{1} + 17 \beta_{2} + 50 \beta_{3} ) q^{36} \) \( + ( 46 + 30 \beta_{1} - 54 \beta_{2} + 10 \beta_{3} ) q^{37} \) \( + ( 174 - 48 \beta_{1} + 16 \beta_{2} - 66 \beta_{3} ) q^{38} \) \( + ( -85 - 13 \beta_{1} + 11 \beta_{2} - 32 \beta_{3} ) q^{39} \) \( + ( 22 + 14 \beta_{1} + 46 \beta_{2} + 94 \beta_{3} ) q^{40} \) \( + ( -49 + 21 \beta_{1} - 7 \beta_{2} - 18 \beta_{3} ) q^{41} \) \( + ( 210 - 26 \beta_{1} - 16 \beta_{2} + 66 \beta_{3} ) q^{42} \) \( + ( -24 + 70 \beta_{1} + 14 \beta_{2} + 28 \beta_{3} ) q^{43} \) \( + ( 194 + 52 \beta_{1} - 14 \beta_{2} + 18 \beta_{3} ) q^{44} \) \( + ( 36 + 40 \beta_{1} + 48 \beta_{2} + 52 \beta_{3} ) q^{45} \) \( + ( -23 - 23 \beta_{3} ) q^{46} \) \( + ( -119 - 71 \beta_{1} + 13 \beta_{2} + 36 \beta_{3} ) q^{47} \) \( + ( 105 + 55 \beta_{1} - 25 \beta_{2} - 23 \beta_{3} ) q^{48} \) \( + ( 365 - 92 \beta_{1} + 8 \beta_{2} + 32 \beta_{3} ) q^{49} \) \( + ( 103 - 20 \beta_{1} + 72 \beta_{2} + 15 \beta_{3} ) q^{50} \) \( + ( -116 + 32 \beta_{1} + 56 \beta_{2} - 2 \beta_{3} ) q^{51} \) \( + ( 558 - 108 \beta_{1} - 105 \beta_{2} - 168 \beta_{3} ) q^{52} \) \( + ( -118 + 14 \beta_{1} - 26 \beta_{2} - 120 \beta_{3} ) q^{53} \) \( + ( 181 - 41 \beta_{1} - 115 \beta_{2} - 57 \beta_{3} ) q^{54} \) \( + ( -176 - 120 \beta_{1} - 108 \beta_{2} - 40 \beta_{3} ) q^{55} \) \( + ( -560 + 124 \beta_{1} + 2 \beta_{2} + 92 \beta_{3} ) q^{56} \) \( + ( 270 + 158 \beta_{1} + 70 \beta_{2} + 72 \beta_{3} ) q^{57} \) \( + ( -519 + 101 \beta_{1} + 109 \beta_{2} + 35 \beta_{3} ) q^{58} \) \( + ( -304 + 36 \beta_{1} + 60 \beta_{2} + 28 \beta_{3} ) q^{59} \) \( + ( -170 - 118 \beta_{1} - 12 \beta_{2} - 62 \beta_{3} ) q^{60} \) \( + ( 206 + 14 \beta_{1} - 26 \beta_{2} + 36 \beta_{3} ) q^{61} \) \( + ( 95 - 27 \beta_{1} + 31 \beta_{2} - 43 \beta_{3} ) q^{62} \) \( + ( -248 + 20 \beta_{1} - 52 \beta_{2} + 44 \beta_{3} ) q^{63} \) \( + ( -189 - 64 \beta_{1} + 93 \beta_{2} + 7 \beta_{3} ) q^{64} \) \( + ( 374 - 134 \beta_{1} - 66 \beta_{2} - 78 \beta_{3} ) q^{65} \) \( + ( -136 + 24 \beta_{1} + 90 \beta_{2} - 4 \beta_{3} ) q^{66} \) \( + ( 110 + 92 \beta_{1} + 84 \beta_{2} + 110 \beta_{3} ) q^{67} \) \( + ( 158 - 20 \beta_{1} - 80 \beta_{2} + 114 \beta_{3} ) q^{68} \) \( + ( -23 - 23 \beta_{1} - 23 \beta_{2} ) q^{69} \) \( + ( -1052 + 204 \beta_{1} - 12 \beta_{2} + 4 \beta_{3} ) q^{70} \) \( + ( 33 - 143 \beta_{1} - 111 \beta_{2} + 68 \beta_{3} ) q^{71} \) \( + ( 358 - 93 \beta_{1} - 7 \beta_{2} - 132 \beta_{3} ) q^{72} \) \( + ( 281 + 127 \beta_{1} - \beta_{2} - 66 \beta_{3} ) q^{73} \) \( + ( 416 - 84 \beta_{1} - 266 \beta_{2} - 68 \beta_{3} ) q^{74} \) \( + ( 355 + 35 \beta_{1} + 55 \beta_{2} + 72 \beta_{3} ) q^{75} \) \( + ( -828 - 42 \beta_{1} + 52 \beta_{2} + 244 \beta_{3} ) q^{76} \) \( + ( 84 + 132 \beta_{1} + 208 \beta_{2} + 36 \beta_{3} ) q^{77} \) \( + ( -543 + 107 \beta_{1} + 121 \beta_{2} + 35 \beta_{3} ) q^{78} \) \( + ( -214 - 98 \beta_{1} - 50 \beta_{2} - 132 \beta_{3} ) q^{79} \) \( + ( 632 + 36 \beta_{1} + 142 \beta_{2} - 52 \beta_{3} ) q^{80} \) \( + ( -203 - 156 \beta_{1} - 24 \beta_{2} - 108 \beta_{3} ) q^{81} \) \( + ( -269 + 47 \beta_{1} - 13 \beta_{2} - 23 \beta_{3} ) q^{82} \) \( + ( 96 - 146 \beta_{1} + 6 \beta_{2} - 22 \beta_{3} ) q^{83} \) \( + ( 662 + 10 \beta_{1} - 106 \beta_{2} - 26 \beta_{3} ) q^{84} \) \( + ( 60 + 28 \beta_{1} - 96 \beta_{2} + 44 \beta_{3} ) q^{85} \) \( + ( 200 - 70 \beta_{1} - 70 \beta_{2} - 164 \beta_{3} ) q^{86} \) \( + ( 517 + 133 \beta_{1} - 71 \beta_{2} + 20 \beta_{3} ) q^{87} \) \( + ( -98 + 12 \beta_{1} - 80 \beta_{2} - 22 \beta_{3} ) q^{88} \) \( + ( 604 + 82 \beta_{1} + 82 \beta_{2} + 160 \beta_{3} ) q^{89} \) \( + ( 432 - 108 \beta_{1} + 48 \beta_{2} - 112 \beta_{3} ) q^{90} \) \( + ( -116 + 56 \beta_{2} - 350 \beta_{3} ) q^{91} \) \( + ( -138 + 69 \beta_{1} + 46 \beta_{2} + 46 \beta_{3} ) q^{92} \) \( + ( -19 - 167 \beta_{1} - 39 \beta_{2} + 30 \beta_{3} ) q^{93} \) \( + ( 355 - 95 \beta_{1} + 51 \beta_{2} - 143 \beta_{3} ) q^{94} \) \( + ( -484 + 540 \beta_{1} + 36 \beta_{2} + 68 \beta_{3} ) q^{95} \) \( + ( 284 - 44 \beta_{1} - 109 \beta_{2} + 70 \beta_{3} ) q^{96} \) \( + ( 388 - 38 \beta_{1} + 150 \beta_{2} + 282 \beta_{3} ) q^{97} \) \( + ( 833 - 88 \beta_{1} + 60 \beta_{2} + 369 \beta_{3} ) q^{98} \) \( + ( -340 - 94 \beta_{1} - 46 \beta_{2} - 48 \beta_{3} ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(4q \) \(\mathstrut +\mathstrut 2q^{2} \) \(\mathstrut +\mathstrut 7q^{3} \) \(\mathstrut +\mathstrut 20q^{4} \) \(\mathstrut +\mathstrut 14q^{5} \) \(\mathstrut -\mathstrut 17q^{6} \) \(\mathstrut +\mathstrut 16q^{7} \) \(\mathstrut -\mathstrut 63q^{8} \) \(\mathstrut -\mathstrut 33q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(4q \) \(\mathstrut +\mathstrut 2q^{2} \) \(\mathstrut +\mathstrut 7q^{3} \) \(\mathstrut +\mathstrut 20q^{4} \) \(\mathstrut +\mathstrut 14q^{5} \) \(\mathstrut -\mathstrut 17q^{6} \) \(\mathstrut +\mathstrut 16q^{7} \) \(\mathstrut -\mathstrut 63q^{8} \) \(\mathstrut -\mathstrut 33q^{9} \) \(\mathstrut -\mathstrut 70q^{10} \) \(\mathstrut +\mathstrut 8q^{11} \) \(\mathstrut -\mathstrut 67q^{12} \) \(\mathstrut +\mathstrut 111q^{13} \) \(\mathstrut -\mathstrut 144q^{14} \) \(\mathstrut +\mathstrut 10q^{15} \) \(\mathstrut +\mathstrut 64q^{16} \) \(\mathstrut +\mathstrut 98q^{17} \) \(\mathstrut +\mathstrut 49q^{18} \) \(\mathstrut +\mathstrut 96q^{19} \) \(\mathstrut +\mathstrut 140q^{20} \) \(\mathstrut +\mathstrut 180q^{21} \) \(\mathstrut +\mathstrut 220q^{22} \) \(\mathstrut -\mathstrut 92q^{23} \) \(\mathstrut -\mathstrut 188q^{24} \) \(\mathstrut +\mathstrut 184q^{25} \) \(\mathstrut -\mathstrut 229q^{26} \) \(\mathstrut -\mathstrut 155q^{27} \) \(\mathstrut +\mathstrut 282q^{28} \) \(\mathstrut +\mathstrut 21q^{29} \) \(\mathstrut -\mathstrut 406q^{30} \) \(\mathstrut -\mathstrut 193q^{31} \) \(\mathstrut -\mathstrut 432q^{32} \) \(\mathstrut -\mathstrut 418q^{33} \) \(\mathstrut +\mathstrut 666q^{34} \) \(\mathstrut -\mathstrut 752q^{35} \) \(\mathstrut -\mathstrut 629q^{36} \) \(\mathstrut +\mathstrut 170q^{37} \) \(\mathstrut +\mathstrut 748q^{38} \) \(\mathstrut -\mathstrut 291q^{39} \) \(\mathstrut -\mathstrut 26q^{40} \) \(\mathstrut -\mathstrut 125q^{41} \) \(\mathstrut +\mathstrut 640q^{42} \) \(\mathstrut +\mathstrut 2q^{43} \) \(\mathstrut +\mathstrut 830q^{44} \) \(\mathstrut +\mathstrut 168q^{45} \) \(\mathstrut -\mathstrut 46q^{46} \) \(\mathstrut -\mathstrut 677q^{47} \) \(\mathstrut +\mathstrut 551q^{48} \) \(\mathstrut +\mathstrut 1220q^{49} \) \(\mathstrut +\mathstrut 414q^{50} \) \(\mathstrut -\mathstrut 340q^{51} \) \(\mathstrut +\mathstrut 2247q^{52} \) \(\mathstrut -\mathstrut 230q^{53} \) \(\mathstrut +\mathstrut 641q^{54} \) \(\mathstrut -\mathstrut 972q^{55} \) \(\mathstrut -\mathstrut 2174q^{56} \) \(\mathstrut +\mathstrut 1322q^{57} \) \(\mathstrut -\mathstrut 1835q^{58} \) \(\mathstrut -\mathstrut 1140q^{59} \) \(\mathstrut -\mathstrut 804q^{60} \) \(\mathstrut +\mathstrut 754q^{61} \) \(\mathstrut +\mathstrut 443q^{62} \) \(\mathstrut -\mathstrut 1092q^{63} \) \(\mathstrut -\mathstrut 805q^{64} \) \(\mathstrut +\mathstrut 1318q^{65} \) \(\mathstrut -\mathstrut 398q^{66} \) \(\mathstrut +\mathstrut 488q^{67} \) \(\mathstrut +\mathstrut 284q^{68} \) \(\mathstrut -\mathstrut 161q^{69} \) \(\mathstrut -\mathstrut 3820q^{70} \) \(\mathstrut -\mathstrut 401q^{71} \) \(\mathstrut +\mathstrut 1503q^{72} \) \(\mathstrut +\mathstrut 1509q^{73} \) \(\mathstrut +\mathstrut 1366q^{74} \) \(\mathstrut +\mathstrut 1401q^{75} \) \(\mathstrut -\mathstrut 3832q^{76} \) \(\mathstrut +\mathstrut 736q^{77} \) \(\mathstrut -\mathstrut 1907q^{78} \) \(\mathstrut -\mathstrut 838q^{79} \) \(\mathstrut +\mathstrut 2846q^{80} \) \(\mathstrut -\mathstrut 932q^{81} \) \(\mathstrut -\mathstrut 949q^{82} \) \(\mathstrut +\mathstrut 142q^{83} \) \(\mathstrut +\mathstrut 2614q^{84} \) \(\mathstrut +\mathstrut 112q^{85} \) \(\mathstrut +\mathstrut 918q^{86} \) \(\mathstrut +\mathstrut 2223q^{87} \) \(\mathstrut -\mathstrut 404q^{88} \) \(\mathstrut +\mathstrut 2342q^{89} \) \(\mathstrut +\mathstrut 1784q^{90} \) \(\mathstrut +\mathstrut 292q^{91} \) \(\mathstrut -\mathstrut 460q^{92} \) \(\mathstrut -\mathstrut 509q^{93} \) \(\mathstrut +\mathstrut 1567q^{94} \) \(\mathstrut -\mathstrut 956q^{95} \) \(\mathstrut +\mathstrut 799q^{96} \) \(\mathstrut +\mathstrut 1062q^{97} \) \(\mathstrut +\mathstrut 2478q^{98} \) \(\mathstrut -\mathstrut 1498q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4}\mathstrut -\mathstrut \) \(2\) \(x^{3}\mathstrut -\mathstrut \) \(16\) \(x^{2}\mathstrut -\mathstrut \) \(5\) \(x\mathstrut +\mathstrut \) \(4\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{3} - \nu^{2} - 20 \nu - 10 \)\()/3\)
\(\beta_{3}\)\(=\)\((\)\( -2 \nu^{3} + 5 \nu^{2} + 28 \nu - 1 \)\()/3\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{3}\mathstrut +\mathstrut \) \(2\) \(\beta_{2}\mathstrut +\mathstrut \) \(4\) \(\beta_{1}\mathstrut +\mathstrut \) \(7\)
\(\nu^{3}\)\(=\)\(\beta_{3}\mathstrut +\mathstrut \) \(5\) \(\beta_{2}\mathstrut +\mathstrut \) \(24\) \(\beta_{1}\mathstrut +\mathstrut \) \(17\)

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
−0.743529
5.22031
−2.83969
0.362907
−5.07751 1.55870 17.7811 10.0635 −7.91434 24.3381 −49.6639 −24.5704 −51.0976
1.2 −0.0323756 6.42170 −7.99895 14.1026 −0.207906 −14.0109 0.517976 14.2382 −0.456580
1.3 2.86845 3.43737 0.228032 −17.9704 9.85995 32.7301 −22.2935 −15.1845 −51.5473
1.4 4.24143 −4.41777 9.98977 7.80430 −18.7377 −27.0572 8.43948 −7.48328 33.1014
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

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

Atkin-Lehner signs

\( p \) Sign
\(23\) \(1\)

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{2}^{4} \) \(\mathstrut -\mathstrut 2 T_{2}^{3} \) \(\mathstrut -\mathstrut 24 T_{2}^{2} \) \(\mathstrut +\mathstrut 61 T_{2} \) \(\mathstrut +\mathstrut 2 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(23))\).