Properties

Label 429.2.bg.b
Level $429$
Weight $2$
Character orbit 429.bg
Analytic conductor $3.426$
Analytic rank $0$
Dimension $112$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 429 = 3 \cdot 11 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 429.bg (of order \(15\), degree \(8\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(3.42558224671\)
Analytic rank: \(0\)
Dimension: \(112\)
Relative dimension: \(14\) over \(\Q(\zeta_{15})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{15}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 112q + 14q^{3} + 8q^{4} - 6q^{5} - 24q^{8} + 14q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 112q + 14q^{3} + 8q^{4} - 6q^{5} - 24q^{8} + 14q^{9} - 24q^{10} + q^{11} + 104q^{12} - 15q^{13} - 30q^{14} - 2q^{15} + 24q^{16} + 4q^{17} - 28q^{20} + 7q^{22} + 8q^{23} - 18q^{24} - 42q^{25} + 7q^{26} - 28q^{27} - 6q^{28} - 2q^{29} + 6q^{30} + 12q^{31} - 6q^{32} + 6q^{33} + 64q^{34} - 8q^{35} + 8q^{36} + 4q^{37} - 6q^{38} + 12q^{39} - 36q^{40} - 26q^{41} - 20q^{42} - 76q^{43} + 12q^{44} - 2q^{45} + 18q^{46} + 50q^{47} - 6q^{48} + 32q^{49} + 17q^{50} - 18q^{51} - 39q^{52} - 54q^{53} - 36q^{55} + 12q^{56} + 48q^{58} + 20q^{59} + 56q^{60} + 34q^{61} + 33q^{62} - 68q^{64} - 44q^{65} - 4q^{66} - 36q^{67} + 4q^{68} + 3q^{69} - 92q^{70} + 22q^{71} - 18q^{72} - 34q^{73} - 4q^{74} - 4q^{75} + 4q^{76} + 32q^{77} - 18q^{78} - 20q^{79} + 30q^{80} + 14q^{81} - 41q^{82} + 56q^{83} + 9q^{84} + 6q^{85} - 86q^{86} - 52q^{87} - 70q^{88} - 96q^{89} - 12q^{90} - 7q^{91} - 126q^{92} - 6q^{93} - 10q^{94} + 52q^{95} - 88q^{96} - 5q^{97} + 104q^{98} - 2q^{99} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
16.1 −1.60050 + 1.77754i −0.104528 + 0.994522i −0.388975 3.70085i −1.06190 + 3.26819i −1.60050 1.77754i 0.204434 + 1.94506i 3.33076 + 2.41994i −0.978148 0.207912i −4.10976 7.11831i
16.2 −1.53199 + 1.70145i −0.104528 + 0.994522i −0.338874 3.22417i 0.485300 1.49360i −1.53199 1.70145i 0.0971269 + 0.924101i 2.30039 + 1.67133i −0.978148 0.207912i 1.79781 + 3.11390i
16.3 −1.22841 + 1.36429i −0.104528 + 0.994522i −0.143231 1.36275i −0.972545 + 2.99319i −1.22841 1.36429i −0.369323 3.51387i −0.935298 0.679534i −0.978148 0.207912i −2.88888 5.00368i
16.4 −1.12855 + 1.25338i −0.104528 + 0.994522i −0.0882831 0.839957i 0.636771 1.95978i −1.12855 1.25338i −0.222872 2.12049i −1.57654 1.14542i −0.978148 0.207912i 1.73772 + 3.00982i
16.5 −0.802761 + 0.891557i −0.104528 + 0.994522i 0.0586092 + 0.557629i 0.833954 2.56665i −0.802761 0.891557i 0.259398 + 2.46800i −2.48538 1.80573i −0.978148 0.207912i 1.61885 + 2.80392i
16.6 −0.471127 + 0.523240i −0.104528 + 0.994522i 0.157238 + 1.49602i −0.263224 + 0.810122i −0.471127 0.523240i 0.496767 + 4.72642i −1.99610 1.45025i −0.978148 0.207912i −0.299876 0.519400i
16.7 0.0112957 0.0125452i −0.104528 + 0.994522i 0.209027 + 1.98876i 1.04169 3.20599i 0.0112957 + 0.0125452i −0.462856 4.40378i 0.0546248 + 0.0396872i −0.978148 0.207912i −0.0284531 0.0492821i
16.8 0.0581605 0.0645938i −0.104528 + 0.994522i 0.208267 + 1.98153i −0.913448 + 2.81130i 0.0581605 + 0.0645938i −0.0279312 0.265748i 0.280746 + 0.203974i −0.978148 0.207912i 0.128466 + 0.222510i
16.9 0.433437 0.481381i −0.104528 + 0.994522i 0.165197 + 1.57175i −0.543246 + 1.67194i 0.433437 + 0.481381i 0.0903479 + 0.859603i 1.87631 + 1.36322i −0.978148 0.207912i 0.569377 + 0.986190i
16.10 0.682919 0.758458i −0.104528 + 0.994522i 0.100176 + 0.953113i 0.0526100 0.161917i 0.682919 + 0.758458i −0.171511 1.63181i 2.44268 + 1.77471i −0.978148 0.207912i −0.0868789 0.150479i
16.11 0.932419 1.03556i −0.104528 + 0.994522i 0.00608545 + 0.0578992i 0.554150 1.70550i 0.932419 + 1.03556i 0.357902 + 3.40521i 2.32033 + 1.68582i −0.978148 0.207912i −1.24944 2.16409i
16.12 1.38006 1.53272i −0.104528 + 0.994522i −0.235585 2.24144i −1.05420 + 3.24451i 1.38006 + 1.53272i 0.100634 + 0.957465i −0.423465 0.307666i −0.978148 0.207912i 3.51804 + 6.09342i
16.13 1.49760 1.66326i −0.104528 + 0.994522i −0.314550 2.99275i 0.954688 2.93823i 1.49760 + 1.66326i −0.278334 2.64817i −1.82740 1.32769i −0.978148 0.207912i −3.45728 5.98819i
16.14 1.76744 1.96294i −0.104528 + 0.994522i −0.520238 4.94974i 0.0584231 0.179808i 1.76744 + 1.96294i −0.0737832 0.702000i −6.36167 4.62202i −0.978148 0.207912i −0.249693 0.432481i
256.1 −0.265449 2.52558i −0.978148 0.207912i −4.35177 + 0.924997i −0.205708 0.149455i −0.265449 + 2.52558i −0.893413 + 0.189901i 1.92183 + 5.91479i 0.913545 + 0.406737i −0.322856 + 0.559203i
256.2 −0.245821 2.33883i −0.978148 0.207912i −3.45339 + 0.734041i 1.76043 + 1.27903i −0.245821 + 2.33883i 2.44940 0.520637i 1.11227 + 3.42322i 0.913545 + 0.406737i 2.55867 4.43175i
256.3 −0.190759 1.81495i −0.978148 0.207912i −1.30136 + 0.276613i −3.21129 2.33314i −0.190759 + 1.81495i 3.99809 0.849821i −0.377595 1.16212i 0.913545 + 0.406737i −3.62195 + 6.27341i
256.4 −0.176325 1.67762i −0.978148 0.207912i −0.827032 + 0.175791i 0.993113 + 0.721539i −0.176325 + 1.67762i −1.11199 + 0.236360i −0.601801 1.85215i 0.913545 + 0.406737i 1.03536 1.79330i
256.5 −0.121483 1.15584i −0.978148 0.207912i 0.635096 0.134994i −2.64351 1.92062i −0.121483 + 1.15584i −2.84394 + 0.604498i −0.951465 2.92831i 0.913545 + 0.406737i −1.89878 + 3.28879i
256.6 −0.0891258 0.847975i −0.978148 0.207912i 1.24518 0.264670i 1.84578 + 1.34104i −0.0891258 + 0.847975i −2.53940 + 0.539767i −0.862376 2.65412i 0.913545 + 0.406737i 0.972662 1.68470i
See next 80 embeddings (of 112 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 412.14
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.c even 5 1 inner
13.c even 3 1 inner
143.q even 15 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 429.2.bg.b 112
11.c even 5 1 inner 429.2.bg.b 112
13.c even 3 1 inner 429.2.bg.b 112
143.q even 15 1 inner 429.2.bg.b 112
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
429.2.bg.b 112 1.a even 1 1 trivial
429.2.bg.b 112 11.c even 5 1 inner
429.2.bg.b 112 13.c even 3 1 inner
429.2.bg.b 112 143.q even 15 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(59\!\cdots\!70\)\( T_{2}^{82} + 673362357978 T_{2}^{81} + \)\(28\!\cdots\!43\)\( T_{2}^{80} - \)\(43\!\cdots\!01\)\( T_{2}^{79} - \)\(39\!\cdots\!18\)\( T_{2}^{78} + \)\(13\!\cdots\!73\)\( T_{2}^{77} - \)\(32\!\cdots\!51\)\( T_{2}^{76} + \)\(13\!\cdots\!96\)\( T_{2}^{75} + \)\(23\!\cdots\!49\)\( T_{2}^{74} - \)\(12\!\cdots\!52\)\( T_{2}^{73} - \)\(74\!\cdots\!02\)\( T_{2}^{72} - \)\(12\!\cdots\!52\)\( T_{2}^{71} + \)\(42\!\cdots\!83\)\( T_{2}^{70} + \)\(10\!\cdots\!88\)\( T_{2}^{69} + \)\(77\!\cdots\!84\)\( T_{2}^{68} - \)\(22\!\cdots\!47\)\( T_{2}^{67} - \)\(41\!\cdots\!59\)\( T_{2}^{66} - \)\(42\!\cdots\!32\)\( T_{2}^{65} + \)\(93\!\cdots\!53\)\( T_{2}^{64} + \)\(39\!\cdots\!37\)\( T_{2}^{63} + \)\(47\!\cdots\!65\)\( T_{2}^{62} - \)\(11\!\cdots\!73\)\( T_{2}^{61} - \)\(93\!\cdots\!26\)\( T_{2}^{60} + \)\(39\!\cdots\!06\)\( T_{2}^{59} + \)\(29\!\cdots\!00\)\( T_{2}^{58} + \)\(63\!\cdots\!35\)\( T_{2}^{57} - \)\(39\!\cdots\!46\)\( T_{2}^{56} - \)\(21\!\cdots\!19\)\( T_{2}^{55} - \)\(29\!\cdots\!23\)\( T_{2}^{54} + \)\(31\!\cdots\!07\)\( T_{2}^{53} + \)\(33\!\cdots\!83\)\( T_{2}^{52} - \)\(10\!\cdots\!60\)\( T_{2}^{51} - \)\(85\!\cdots\!64\)\( T_{2}^{50} + \)\(31\!\cdots\!35\)\( T_{2}^{49} + \)\(10\!\cdots\!85\)\( T_{2}^{48} - \)\(23\!\cdots\!32\)\( T_{2}^{47} - \)\(25\!\cdots\!63\)\( T_{2}^{46} - \)\(50\!\cdots\!63\)\( T_{2}^{45} - \)\(10\!\cdots\!19\)\( T_{2}^{44} + \)\(11\!\cdots\!91\)\( T_{2}^{43} + \)\(26\!\cdots\!16\)\( T_{2}^{42} - \)\(20\!\cdots\!97\)\( T_{2}^{41} - \)\(24\!\cdots\!27\)\( T_{2}^{40} + \)\(26\!\cdots\!68\)\( T_{2}^{39} - \)\(49\!\cdots\!17\)\( T_{2}^{38} - \)\(10\!\cdots\!65\)\( T_{2}^{37} + \)\(31\!\cdots\!52\)\( T_{2}^{36} - \)\(20\!\cdots\!40\)\( T_{2}^{35} - \)\(24\!\cdots\!27\)\( T_{2}^{34} + \)\(28\!\cdots\!61\)\( T_{2}^{33} + \)\(62\!\cdots\!06\)\( T_{2}^{32} - \)\(81\!\cdots\!50\)\( T_{2}^{31} + \)\(12\!\cdots\!03\)\( T_{2}^{30} - \)\(86\!\cdots\!64\)\( T_{2}^{29} - \)\(80\!\cdots\!71\)\( T_{2}^{28} + \)\(79\!\cdots\!49\)\( T_{2}^{27} + \)\(19\!\cdots\!53\)\( T_{2}^{26} - \)\(24\!\cdots\!54\)\( T_{2}^{25} + \)\(12\!\cdots\!29\)\( T_{2}^{24} - \)\(22\!\cdots\!60\)\( T_{2}^{23} - \)\(12\!\cdots\!69\)\( T_{2}^{22} + \)\(57\!\cdots\!14\)\( T_{2}^{21} + \)\(72\!\cdots\!37\)\( T_{2}^{20} - \)\(49\!\cdots\!86\)\( T_{2}^{19} - \)\(17\!\cdots\!00\)\( T_{2}^{18} + \)\(17\!\cdots\!91\)\( T_{2}^{17} - \)\(98\!\cdots\!41\)\( T_{2}^{16} + \)\(12\!\cdots\!83\)\( T_{2}^{15} + \)\(82\!\cdots\!33\)\( T_{2}^{14} - \)\(10\!\cdots\!03\)\( T_{2}^{13} + \)\(23\!\cdots\!65\)\( T_{2}^{12} + \)\(40\!\cdots\!89\)\( T_{2}^{11} - \)\(32\!\cdots\!18\)\( T_{2}^{10} + \)\(14\!\cdots\!51\)\( T_{2}^{9} + \)\(23\!\cdots\!60\)\( T_{2}^{8} + \)\(12\!\cdots\!02\)\( T_{2}^{7} + 33801426268 T_{2}^{6} + 953593498 T_{2}^{5} + 20268973 T_{2}^{4} + 155018 T_{2}^{3} + 2355 T_{2}^{2} + 99 T_{2} + 1 \)">\(T_{2}^{112} - \cdots\) acting on \(S_{2}^{\mathrm{new}}(429, [\chi])\).