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Properties

Label 261.3.s.a
Level $261$
Weight $3$
Character orbit 261.s
Analytic conductor $7.112$
Analytic rank $0$
Dimension $48$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 261 = 3^{2} \cdot 29 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 261.s (of order \(28\), degree \(12\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(7.11173489980\)
Analytic rank: \(0\)
Dimension: \(48\)
Relative dimension: \(4\) over \(\Q(\zeta_{28})\)
Twist minimal: no (minimal twist has level 29)
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.

\(\operatorname{Tr}(f)(q) = \) \( 48q + 16q^{2} - 14q^{4} + 14q^{5} - 10q^{7} - 28q^{8} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 48q + 16q^{2} - 14q^{4} + 14q^{5} - 10q^{7} - 28q^{8} - 20q^{10} + 8q^{11} - 14q^{13} - 26q^{14} + 18q^{16} + 26q^{17} + 2q^{19} - 46q^{20} + 154q^{22} - 56q^{23} - 34q^{25} - 110q^{26} + 170q^{29} - 88q^{31} + 132q^{32} - 224q^{34} + 210q^{35} - 56q^{37} + 294q^{38} - 492q^{40} + 34q^{41} + 176q^{43} - 126q^{44} + 744q^{46} - 208q^{47} + 506q^{49} - 732q^{50} + 690q^{52} + 14q^{53} + 284q^{55} - 332q^{56} - 508q^{58} + 44q^{59} - 30q^{61} + 504q^{62} - 896q^{64} + 554q^{65} - 574q^{67} + 796q^{68} - 1066q^{70} - 224q^{71} - 22q^{73} - 820q^{74} + 514q^{76} - 436q^{77} + 564q^{79} - 1162q^{80} - 18q^{82} + 126q^{83} + 38q^{85} - 384q^{88} + 160q^{89} - 434q^{91} + 1022q^{92} - 2q^{94} + 642q^{95} + 604q^{97} + 102q^{98} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
10.1 −2.05804 + 1.29315i 0 0.827740 − 1.71882i −5.87720 − 1.34143i 0 −9.36468 + 4.50979i −0.569384 − 5.05343i 0 13.8301 − 4.83938i
10.2 −1.44470 + 0.907767i 0 −0.472410 + 0.980970i 8.15497 + 1.86132i 0 7.53715 − 3.62970i −0.972146 − 8.62804i 0 −13.4711 + 4.71376i
10.3 1.29187 − 0.811733i 0 −0.725528 + 1.50658i −3.36294 − 0.767569i 0 −0.255710 + 0.123143i 0.968958 + 8.59974i 0 −4.96753 + 1.73821i
10.4 3.24379 − 2.03821i 0 4.63233 − 9.61914i 5.12808 + 1.17045i 0 −6.56728 + 3.16264i −2.86375 − 25.4165i 0 19.0200 − 6.65539i
19.1 −1.42815 − 2.27289i 0 −1.39088 + 2.88819i 6.69994 + 1.52922i 0 −4.78081 + 2.30231i −2.11889 + 0.238741i 0 −6.09279 − 17.4122i
19.2 −0.488049 − 0.776726i 0 1.37042 − 2.84571i 1.98497 + 0.453055i 0 9.56374 − 4.60566i −6.52543 + 0.735239i 0 −0.616861 − 1.76289i
19.3 0.448531 + 0.713833i 0 1.42716 − 2.96352i −4.21883 − 0.962920i 0 −10.1429 + 4.88457i 6.10659 − 0.688048i 0 −1.20491 − 3.44344i
19.4 1.63282 + 2.59861i 0 −2.35117 + 4.88225i 4.43970 + 1.01333i 0 10.7635 − 5.18344i −4.32723 + 0.487562i 0 4.61596 + 13.1917i
37.1 −1.65381 − 0.578694i 0 −0.727117 − 0.579856i 0.825315 + 1.71379i 0 −1.24782 − 1.56472i 4.59573 + 7.31406i 0 −0.373160 − 3.31188i
37.2 0.0310749 + 0.0108736i 0 −3.12648 − 2.49328i −3.79007 − 7.87017i 0 3.62612 + 4.54701i −0.140107 − 0.222980i 0 −0.0321993 − 0.285777i
37.3 2.20133 + 0.770280i 0 1.12521 + 0.897324i −2.64264 − 5.48750i 0 −3.22936 − 4.04949i −3.17747 − 5.05691i 0 −1.59042 − 14.1154i
37.4 2.32673 + 0.814157i 0 1.62348 + 1.29468i 3.83207 + 7.95737i 0 2.23777 + 2.80607i −2.52264 − 4.01476i 0 2.43763 + 21.6345i
55.1 −1.42815 + 2.27289i 0 −1.39088 − 2.88819i 6.69994 − 1.52922i 0 −4.78081 − 2.30231i −2.11889 − 0.238741i 0 −6.09279 + 17.4122i
55.2 −0.488049 + 0.776726i 0 1.37042 + 2.84571i 1.98497 − 0.453055i 0 9.56374 + 4.60566i −6.52543 − 0.735239i 0 −0.616861 + 1.76289i
55.3 0.448531 − 0.713833i 0 1.42716 + 2.96352i −4.21883 + 0.962920i 0 −10.1429 − 4.88457i 6.10659 + 0.688048i 0 −1.20491 + 3.44344i
55.4 1.63282 − 2.59861i 0 −2.35117 − 4.88225i 4.43970 − 1.01333i 0 10.7635 + 5.18344i −4.32723 − 0.487562i 0 4.61596 − 13.1917i
73.1 −2.58169 + 0.290886i 0 2.68078 − 0.611870i 2.49104 + 1.98654i 0 1.30161 − 5.70272i 3.06598 − 1.07283i 0 −7.00893 − 4.40400i
73.2 −0.415096 + 0.0467701i 0 −3.72959 + 0.851255i −0.738700 − 0.589093i 0 −0.577468 + 2.53005i 3.08546 − 1.07965i 0 0.334184 + 0.209981i
73.3 1.68783 − 0.190173i 0 −1.08711 + 0.248126i −0.141728 − 0.113024i 0 −1.55116 + 6.79606i −8.20045 + 2.86946i 0 −0.260707 − 0.163813i
73.4 3.56136 − 0.401269i 0 8.62256 − 1.96804i −5.24106 − 4.17960i 0 2.11977 − 9.28734i 16.3872 − 5.73413i 0 −20.3424 − 12.7820i
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 253.4
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
29.f odd 28 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 261.3.s.a 48
3.b odd 2 1 29.3.f.a ✓ 48
29.f odd 28 1 inner 261.3.s.a 48
87.k even 28 1 29.3.f.a ✓ 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
29.3.f.a ✓ 48 3.b odd 2 1
29.3.f.a ✓ 48 87.k even 28 1
261.3.s.a 48 1.a even 1 1 trivial
261.3.s.a 48 29.f odd 28 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(11\!\cdots\!18\)\( T_{2}^{22} + \)\(12\!\cdots\!40\)\( T_{2}^{21} + \)\(56\!\cdots\!63\)\( T_{2}^{20} - \)\(14\!\cdots\!16\)\( T_{2}^{19} + \)\(82\!\cdots\!53\)\( T_{2}^{18} + \)\(94\!\cdots\!60\)\( T_{2}^{17} - \)\(25\!\cdots\!94\)\( T_{2}^{16} + \)\(54\!\cdots\!88\)\( T_{2}^{15} + \)\(97\!\cdots\!09\)\( T_{2}^{14} - \)\(17\!\cdots\!36\)\( T_{2}^{13} + \)\(11\!\cdots\!70\)\( T_{2}^{12} - \)\(95\!\cdots\!52\)\( T_{2}^{11} + \)\(13\!\cdots\!48\)\( T_{2}^{10} - \)\(83\!\cdots\!12\)\( T_{2}^{9} + \)\(42\!\cdots\!26\)\( T_{2}^{8} + \)\(36\!\cdots\!72\)\( T_{2}^{7} - \)\(23\!\cdots\!75\)\( T_{2}^{6} + \)\(96\!\cdots\!48\)\( T_{2}^{5} + \)\(22\!\cdots\!69\)\( T_{2}^{4} + 114085022712 T_{2}^{3} + 311190034388 T_{2}^{2} - 19931804632 T_{2} + 352951369 \)">\(T_{2}^{48} - \cdots\) acting on \(S_{3}^{\mathrm{new}}(261, [\chi])\).