Properties

Label 29.3.f.a
Level 29
Weight 3
Character orbit 29.f
Analytic conductor 0.790
Analytic rank 0
Dimension 48
CM No

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Newspace parameters

Level: \( N \) = \( 29 \)
Weight: \( k \) = \( 3 \)
Character orbit: \([\chi]\) = 29.f (of order \(28\) and degree \(12\))

Newform invariants

Self dual: No
Analytic conductor: \(0.790192766645\)
Analytic rank: \(0\)
Dimension: \(48\)
Relative dimension: \(4\) over \(\Q(\zeta_{28})\)
Sato-Tate group: $\mathrm{SU}(2)[C_{28}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \(48q \) \(\mathstrut -\mathstrut 16q^{2} \) \(\mathstrut -\mathstrut 12q^{3} \) \(\mathstrut -\mathstrut 14q^{4} \) \(\mathstrut -\mathstrut 14q^{5} \) \(\mathstrut -\mathstrut 14q^{6} \) \(\mathstrut -\mathstrut 10q^{7} \) \(\mathstrut +\mathstrut 28q^{8} \) \(\mathstrut -\mathstrut 14q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \(48q \) \(\mathstrut -\mathstrut 16q^{2} \) \(\mathstrut -\mathstrut 12q^{3} \) \(\mathstrut -\mathstrut 14q^{4} \) \(\mathstrut -\mathstrut 14q^{5} \) \(\mathstrut -\mathstrut 14q^{6} \) \(\mathstrut -\mathstrut 10q^{7} \) \(\mathstrut +\mathstrut 28q^{8} \) \(\mathstrut -\mathstrut 14q^{9} \) \(\mathstrut -\mathstrut 20q^{10} \) \(\mathstrut -\mathstrut 8q^{11} \) \(\mathstrut -\mathstrut 68q^{12} \) \(\mathstrut -\mathstrut 14q^{13} \) \(\mathstrut +\mathstrut 26q^{14} \) \(\mathstrut -\mathstrut 4q^{15} \) \(\mathstrut +\mathstrut 18q^{16} \) \(\mathstrut -\mathstrut 26q^{17} \) \(\mathstrut -\mathstrut 34q^{18} \) \(\mathstrut +\mathstrut 2q^{19} \) \(\mathstrut +\mathstrut 46q^{20} \) \(\mathstrut +\mathstrut 218q^{21} \) \(\mathstrut +\mathstrut 154q^{22} \) \(\mathstrut +\mathstrut 56q^{23} \) \(\mathstrut +\mathstrut 154q^{24} \) \(\mathstrut -\mathstrut 34q^{25} \) \(\mathstrut +\mathstrut 110q^{26} \) \(\mathstrut +\mathstrut 126q^{27} \) \(\mathstrut -\mathstrut 170q^{29} \) \(\mathstrut +\mathstrut 24q^{30} \) \(\mathstrut -\mathstrut 88q^{31} \) \(\mathstrut -\mathstrut 132q^{32} \) \(\mathstrut -\mathstrut 224q^{33} \) \(\mathstrut -\mathstrut 224q^{34} \) \(\mathstrut -\mathstrut 210q^{35} \) \(\mathstrut -\mathstrut 434q^{36} \) \(\mathstrut -\mathstrut 56q^{37} \) \(\mathstrut -\mathstrut 294q^{38} \) \(\mathstrut -\mathstrut 232q^{39} \) \(\mathstrut -\mathstrut 492q^{40} \) \(\mathstrut -\mathstrut 34q^{41} \) \(\mathstrut -\mathstrut 14q^{42} \) \(\mathstrut +\mathstrut 176q^{43} \) \(\mathstrut +\mathstrut 126q^{44} \) \(\mathstrut +\mathstrut 114q^{45} \) \(\mathstrut +\mathstrut 744q^{46} \) \(\mathstrut +\mathstrut 208q^{47} \) \(\mathstrut +\mathstrut 640q^{48} \) \(\mathstrut +\mathstrut 506q^{49} \) \(\mathstrut +\mathstrut 732q^{50} \) \(\mathstrut +\mathstrut 322q^{51} \) \(\mathstrut +\mathstrut 690q^{52} \) \(\mathstrut -\mathstrut 14q^{53} \) \(\mathstrut -\mathstrut 36q^{54} \) \(\mathstrut +\mathstrut 284q^{55} \) \(\mathstrut +\mathstrut 332q^{56} \) \(\mathstrut -\mathstrut 508q^{58} \) \(\mathstrut -\mathstrut 44q^{59} \) \(\mathstrut -\mathstrut 316q^{60} \) \(\mathstrut -\mathstrut 30q^{61} \) \(\mathstrut -\mathstrut 504q^{62} \) \(\mathstrut -\mathstrut 686q^{63} \) \(\mathstrut -\mathstrut 896q^{64} \) \(\mathstrut -\mathstrut 554q^{65} \) \(\mathstrut -\mathstrut 608q^{66} \) \(\mathstrut -\mathstrut 574q^{67} \) \(\mathstrut -\mathstrut 796q^{68} \) \(\mathstrut -\mathstrut 806q^{69} \) \(\mathstrut -\mathstrut 1066q^{70} \) \(\mathstrut +\mathstrut 224q^{71} \) \(\mathstrut +\mathstrut 748q^{72} \) \(\mathstrut -\mathstrut 22q^{73} \) \(\mathstrut +\mathstrut 820q^{74} \) \(\mathstrut +\mathstrut 768q^{75} \) \(\mathstrut +\mathstrut 514q^{76} \) \(\mathstrut +\mathstrut 436q^{77} \) \(\mathstrut +\mathstrut 282q^{78} \) \(\mathstrut +\mathstrut 564q^{79} \) \(\mathstrut +\mathstrut 1162q^{80} \) \(\mathstrut +\mathstrut 670q^{81} \) \(\mathstrut -\mathstrut 18q^{82} \) \(\mathstrut -\mathstrut 126q^{83} \) \(\mathstrut +\mathstrut 572q^{84} \) \(\mathstrut +\mathstrut 38q^{85} \) \(\mathstrut -\mathstrut 118q^{87} \) \(\mathstrut -\mathstrut 384q^{88} \) \(\mathstrut -\mathstrut 160q^{89} \) \(\mathstrut -\mathstrut 828q^{90} \) \(\mathstrut -\mathstrut 434q^{91} \) \(\mathstrut -\mathstrut 1022q^{92} \) \(\mathstrut -\mathstrut 406q^{93} \) \(\mathstrut -\mathstrut 2q^{94} \) \(\mathstrut -\mathstrut 642q^{95} \) \(\mathstrut -\mathstrut 1176q^{96} \) \(\mathstrut +\mathstrut 604q^{97} \) \(\mathstrut -\mathstrut 102q^{98} \) \(\mathstrut +\mathstrut 316q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2.1 −3.56136 0.401269i 1.78374 + 1.12080i 8.62256 + 1.96804i 5.24106 4.17960i −5.90279 4.70732i 2.11977 + 9.28734i −16.3872 5.73413i −1.97942 4.11031i −20.3424 + 12.7820i
2.2 −1.68783 0.190173i −3.78594 2.37886i −1.08711 0.248126i 0.141728 0.113024i 5.93762 + 4.73509i −1.55116 6.79606i 8.20045 + 2.86946i 4.76938 + 9.90371i −0.260707 + 0.163813i
2.3 0.415096 + 0.0467701i 2.68233 + 1.68542i −3.72959 0.851255i 0.738700 0.589093i 1.03460 + 0.825065i −0.577468 2.53005i −3.08546 1.07965i 0.449302 + 0.932984i 0.334184 0.209981i
2.4 2.58169 + 0.290886i −2.11401 1.32832i 2.68078 + 0.611870i −2.49104 + 1.98654i −5.07133 4.04425i 1.30161 + 5.70272i −3.06598 1.07283i −1.20034 2.49253i −7.00893 + 4.40400i
3.1 −3.24379 2.03821i −2.23035 + 0.780434i 4.63233 + 9.61914i −5.12808 + 1.17045i 8.82547 + 2.01436i −6.56728 3.16264i 2.86375 25.4165i −2.67109 + 2.13013i 19.0200 + 6.65539i
3.2 −1.29187 0.811733i 2.15095 0.752648i −0.725528 1.50658i 3.36294 0.767569i −3.38968 0.773673i −0.255710 0.123143i −0.968958 + 8.59974i −2.97640 + 2.37360i −4.96753 1.73821i
3.3 1.44470 + 0.907767i 1.22940 0.430186i −0.472410 0.980970i −8.15497 + 1.86132i 2.16663 + 0.494518i 7.53715 + 3.62970i 0.972146 8.62804i −5.71012 + 4.55367i −13.4711 4.71376i
3.4 2.05804 + 1.29315i −2.93183 + 1.02589i 0.827740 + 1.71882i 5.87720 1.34143i −7.36043 1.67997i −9.36468 4.50979i 0.569384 5.05343i 0.506667 0.404053i 13.8301 + 4.83938i
8.1 −2.32673 0.814157i 0.184193 1.63476i 1.62348 + 1.29468i −3.83207 7.95737i −1.75952 + 3.65367i 2.23777 + 2.80607i 2.52264 + 4.01476i 6.13584 + 1.40047i 2.43763 + 21.6345i
8.2 −2.20133 0.770280i −0.552425 + 4.90291i 1.12521 + 0.897324i 2.64264 + 5.48750i 4.99268 10.3674i −3.22936 4.04949i 3.17747 + 5.05691i −14.9590 3.41428i −1.59042 14.1154i
8.3 −0.0310749 0.0108736i 0.529545 4.69985i −3.12648 2.49328i 3.79007 + 7.87017i −0.0675598 + 0.140289i 3.62612 + 4.54701i 0.140107 + 0.222980i −13.0338 2.97487i −0.0321993 0.285777i
8.4 1.65381 + 0.578694i −0.186386 + 1.65422i −0.727117 0.579856i −0.825315 1.71379i −1.26553 + 2.62791i −1.24782 1.56472i −4.59573 7.31406i 6.07265 + 1.38604i −0.373160 3.31188i
10.1 −3.24379 + 2.03821i −2.23035 0.780434i 4.63233 9.61914i −5.12808 1.17045i 8.82547 2.01436i −6.56728 + 3.16264i 2.86375 + 25.4165i −2.67109 2.13013i 19.0200 6.65539i
10.2 −1.29187 + 0.811733i 2.15095 + 0.752648i −0.725528 + 1.50658i 3.36294 + 0.767569i −3.38968 + 0.773673i −0.255710 + 0.123143i −0.968958 8.59974i −2.97640 2.37360i −4.96753 + 1.73821i
10.3 1.44470 0.907767i 1.22940 + 0.430186i −0.472410 + 0.980970i −8.15497 1.86132i 2.16663 0.494518i 7.53715 3.62970i 0.972146 + 8.62804i −5.71012 4.55367i −13.4711 + 4.71376i
10.4 2.05804 1.29315i −2.93183 1.02589i 0.827740 1.71882i 5.87720 + 1.34143i −7.36043 + 1.67997i −9.36468 + 4.50979i 0.569384 + 5.05343i 0.506667 + 0.404053i 13.8301 4.83938i
11.1 −2.32673 + 0.814157i 0.184193 + 1.63476i 1.62348 1.29468i −3.83207 + 7.95737i −1.75952 3.65367i 2.23777 2.80607i 2.52264 4.01476i 6.13584 1.40047i 2.43763 21.6345i
11.2 −2.20133 + 0.770280i −0.552425 4.90291i 1.12521 0.897324i 2.64264 5.48750i 4.99268 + 10.3674i −3.22936 + 4.04949i 3.17747 5.05691i −14.9590 + 3.41428i −1.59042 + 14.1154i
11.3 −0.0310749 + 0.0108736i 0.529545 + 4.69985i −3.12648 + 2.49328i 3.79007 7.87017i −0.0675598 0.140289i 3.62612 4.54701i 0.140107 0.222980i −13.0338 + 2.97487i −0.0321993 + 0.285777i
11.4 1.65381 0.578694i −0.186386 1.65422i −0.727117 + 0.579856i −0.825315 + 1.71379i −1.26553 2.62791i −1.24782 + 1.56472i −4.59573 + 7.31406i 6.07265 1.38604i −0.373160 + 3.31188i
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 27.4
Significant digits:
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Inner twists

This newform does not have CM; other inner twists have not been computed.

Hecke kernels

There are no other newforms in \(S_{3}^{\mathrm{new}}(29, [\chi])\).