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Properties

Label 230.4.g.b
Level $230$
Weight $4$
Character orbit 230.g
Analytic conductor $13.570$
Analytic rank $0$
Dimension $60$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 230 = 2 \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 230.g (of order \(11\), degree \(10\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(13.5704393013\)
Analytic rank: \(0\)
Dimension: \(60\)
Relative dimension: \(6\) over \(\Q(\zeta_{11})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{11}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 60q + 12q^{2} - 3q^{3} - 24q^{4} - 30q^{5} + 6q^{6} + 100q^{7} + 48q^{8} + 69q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 60q + 12q^{2} - 3q^{3} - 24q^{4} - 30q^{5} + 6q^{6} + 100q^{7} + 48q^{8} + 69q^{9} + 60q^{10} - 51q^{11} + 120q^{12} + 184q^{13} + 20q^{14} - 15q^{15} - 96q^{16} - 334q^{17} - 138q^{18} + 258q^{19} - 120q^{20} + 716q^{21} + 344q^{22} - 288q^{23} + 288q^{24} - 150q^{25} + 336q^{26} - 813q^{27} - 40q^{28} + 1462q^{29} + 30q^{30} - 481q^{31} + 192q^{32} + 269q^{33} - 168q^{34} - 380q^{35} + 12q^{36} - 787q^{37} - 978q^{38} + 2065q^{39} + 240q^{40} - 1843q^{41} + 724q^{42} + 128q^{43} - 204q^{44} + 2380q^{45} + 92q^{46} - 1838q^{47} + 480q^{48} - 606q^{49} + 300q^{50} + 2320q^{51} - 12q^{52} - 3284q^{53} + 636q^{54} - 200q^{55} - 800q^{56} - 2710q^{57} - 856q^{58} - 2708q^{59} - 60q^{60} + 437q^{61} + 962q^{62} + 6063q^{63} - 384q^{64} - 15q^{65} + 1090q^{66} + 2804q^{67} + 600q^{68} + 7900q^{69} - 120q^{70} + 2676q^{71} + 1384q^{72} + 876q^{73} + 1574q^{74} - 75q^{75} + 20q^{76} + 9310q^{77} - 2612q^{78} - 1233q^{79} - 480q^{80} - 6030q^{81} - 1132q^{82} - 9923q^{83} - 964q^{84} + 1300q^{85} + 5222q^{86} - 5719q^{87} - 1176q^{88} - 1827q^{89} + 190q^{90} - 7626q^{91} + 564q^{92} + 230q^{93} - 1142q^{94} + 2445q^{95} + 96q^{96} + 939q^{97} - 5916q^{98} - 10589q^{99} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
31.1 0.284630 − 1.97964i −3.91607 − 8.57500i −3.83797 − 1.12693i −3.27430 − 3.77875i −18.0901 + 5.31172i 13.7433 + 8.83226i −3.32332 + 7.27706i −40.5137 + 46.7553i −8.41254 + 5.40641i
31.2 0.284630 − 1.97964i −1.76914 − 3.87387i −3.83797 − 1.12693i −3.27430 − 3.77875i −8.17242 + 2.39964i −26.5942 − 17.0910i −3.32332 + 7.27706i 5.80424 − 6.69844i −8.41254 + 5.40641i
31.3 0.284630 − 1.97964i −0.871814 − 1.90901i −3.83797 − 1.12693i −3.27430 − 3.77875i −4.02729 + 1.18252i 25.9987 + 16.7083i −3.32332 + 7.27706i 14.7970 − 17.0766i −8.41254 + 5.40641i
31.4 0.284630 − 1.97964i −0.654798 − 1.43381i −3.83797 − 1.12693i −3.27430 − 3.77875i −3.02480 + 0.888162i 7.64693 + 4.91439i −3.32332 + 7.27706i 16.0542 − 18.5275i −8.41254 + 5.40641i
31.5 0.284630 − 1.97964i 2.61500 + 5.72604i −3.83797 − 1.12693i −3.27430 − 3.77875i 12.0798 − 3.54696i −5.58110 − 3.58676i −3.32332 + 7.27706i −8.26814 + 9.54194i −8.41254 + 5.40641i
31.6 0.284630 − 1.97964i 2.99586 + 6.56001i −3.83797 − 1.12693i −3.27430 − 3.77875i 13.8392 − 4.06355i 2.05971 + 1.32369i −3.32332 + 7.27706i −16.3774 + 18.9005i −8.41254 + 5.40641i
41.1 1.91899 + 0.563465i −4.61458 + 5.32551i 3.36501 + 2.16256i −0.711574 + 4.94911i −11.8561 + 7.61942i 10.2472 + 22.4383i 5.23889 + 6.04600i −3.22420 − 22.4248i −4.15415 + 9.09632i
41.2 1.91899 + 0.563465i −2.52693 + 2.91624i 3.36501 + 2.16256i −0.711574 + 4.94911i −6.49235 + 4.17238i −13.4966 − 29.5535i 5.23889 + 6.04600i 1.72346 + 11.9869i −4.15415 + 9.09632i
41.3 1.91899 + 0.563465i 0.460964 − 0.531981i 3.36501 + 2.16256i −0.711574 + 4.94911i 1.18434 − 0.761126i −1.15601 − 2.53131i 5.23889 + 6.04600i 3.77198 + 26.2347i −4.15415 + 9.09632i
41.4 1.91899 + 0.563465i 1.94747 − 2.24751i 3.36501 + 2.16256i −0.711574 + 4.94911i 5.00357 − 3.21560i 8.06153 + 17.6523i 5.23889 + 6.04600i 2.58388 + 17.9713i −4.15415 + 9.09632i
41.5 1.91899 + 0.563465i 3.92901 − 4.53431i 3.36501 + 2.16256i −0.711574 + 4.94911i 10.0946 − 6.48743i −9.52105 − 20.8482i 5.23889 + 6.04600i −1.28042 − 8.90551i −4.15415 + 9.09632i
41.6 1.91899 + 0.563465i 6.53866 − 7.54601i 3.36501 + 2.16256i −0.711574 + 4.94911i 16.7995 − 10.7964i 14.9302 + 32.6926i 5.23889 + 6.04600i −10.3458 − 71.9564i −4.15415 + 9.09632i
71.1 −1.68251 − 1.08128i −1.10005 − 7.65101i 1.66166 + 3.63853i −4.79746 − 1.40866i −6.42205 + 14.0623i −11.9276 + 13.7651i 1.13852 − 7.91857i −31.4215 + 9.22619i 6.54861 + 7.55750i
71.2 −1.68251 − 1.08128i −0.719442 − 5.00383i 1.66166 + 3.63853i −4.79746 − 1.40866i −4.20008 + 9.19689i 9.79868 − 11.3083i 1.13852 − 7.91857i 1.38563 − 0.406859i 6.54861 + 7.55750i
71.3 −1.68251 − 1.08128i −0.259790 − 1.80688i 1.66166 + 3.63853i −4.79746 − 1.40866i −1.51665 + 3.32099i −3.50130 + 4.04072i 1.13852 − 7.91857i 22.7090 − 6.66796i 6.54861 + 7.55750i
71.4 −1.68251 − 1.08128i −0.161442 − 1.12285i 1.66166 + 3.63853i −4.79746 − 1.40866i −0.942491 + 2.06377i 20.3265 − 23.4580i 1.13852 − 7.91857i 24.6716 − 7.24423i 6.54861 + 7.55750i
71.5 −1.68251 − 1.08128i 0.839469 + 5.83864i 1.66166 + 3.63853i −4.79746 − 1.40866i 4.90080 − 10.7312i 5.10412 − 5.89047i 1.13852 − 7.91857i −7.47864 + 2.19593i 6.54861 + 7.55750i
71.6 −1.68251 − 1.08128i 1.10986 + 7.71925i 1.66166 + 3.63853i −4.79746 − 1.40866i 6.47933 − 14.1878i −5.92245 + 6.83487i 1.13852 − 7.91857i −32.4487 + 9.52780i 6.54861 + 7.55750i
81.1 −1.68251 + 1.08128i −1.10005 + 7.65101i 1.66166 − 3.63853i −4.79746 + 1.40866i −6.42205 − 14.0623i −11.9276 − 13.7651i 1.13852 + 7.91857i −31.4215 − 9.22619i 6.54861 − 7.55750i
81.2 −1.68251 + 1.08128i −0.719442 + 5.00383i 1.66166 − 3.63853i −4.79746 + 1.40866i −4.20008 − 9.19689i 9.79868 + 11.3083i 1.13852 + 7.91857i 1.38563 + 0.406859i 6.54861 − 7.55750i
See all 60 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 211.6
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
23.c even 11 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 230.4.g.b ✓ 60
23.c even 11 1 inner 230.4.g.b ✓ 60
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
230.4.g.b ✓ 60 1.a even 1 1 trivial
230.4.g.b ✓ 60 23.c even 11 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(27\!\cdots\!43\)\( T_{3}^{46} - \)\(41\!\cdots\!13\)\( T_{3}^{45} + \)\(15\!\cdots\!36\)\( T_{3}^{44} - \)\(23\!\cdots\!20\)\( T_{3}^{43} + \)\(92\!\cdots\!95\)\( T_{3}^{42} - \)\(20\!\cdots\!27\)\( T_{3}^{41} + \)\(57\!\cdots\!15\)\( T_{3}^{40} + \)\(18\!\cdots\!48\)\( T_{3}^{39} + \)\(17\!\cdots\!21\)\( T_{3}^{38} - \)\(87\!\cdots\!75\)\( T_{3}^{37} + \)\(60\!\cdots\!15\)\( T_{3}^{36} - \)\(20\!\cdots\!45\)\( T_{3}^{35} + \)\(18\!\cdots\!85\)\( T_{3}^{34} - \)\(12\!\cdots\!29\)\( T_{3}^{33} + \)\(45\!\cdots\!60\)\( T_{3}^{32} + \)\(26\!\cdots\!95\)\( T_{3}^{31} + \)\(32\!\cdots\!00\)\( T_{3}^{30} + \)\(73\!\cdots\!48\)\( T_{3}^{29} + \)\(15\!\cdots\!25\)\( T_{3}^{28} + \)\(96\!\cdots\!65\)\( T_{3}^{27} + \)\(40\!\cdots\!07\)\( T_{3}^{26} - \)\(18\!\cdots\!85\)\( T_{3}^{25} + \)\(53\!\cdots\!72\)\( T_{3}^{24} - \)\(15\!\cdots\!58\)\( T_{3}^{23} + \)\(33\!\cdots\!80\)\( T_{3}^{22} - \)\(16\!\cdots\!34\)\( T_{3}^{21} + \)\(55\!\cdots\!27\)\( T_{3}^{20} - \)\(54\!\cdots\!73\)\( T_{3}^{19} + \)\(12\!\cdots\!92\)\( T_{3}^{18} + \)\(14\!\cdots\!82\)\( T_{3}^{17} + \)\(19\!\cdots\!06\)\( T_{3}^{16} + \)\(78\!\cdots\!44\)\( T_{3}^{15} + \)\(15\!\cdots\!63\)\( T_{3}^{14} + \)\(20\!\cdots\!63\)\( T_{3}^{13} + \)\(23\!\cdots\!15\)\( T_{3}^{12} - \)\(22\!\cdots\!97\)\( T_{3}^{11} + \)\(52\!\cdots\!15\)\( T_{3}^{10} - \)\(76\!\cdots\!83\)\( T_{3}^{9} + \)\(17\!\cdots\!36\)\( T_{3}^{8} - \)\(14\!\cdots\!18\)\( T_{3}^{7} + \)\(16\!\cdots\!08\)\( T_{3}^{6} - \)\(12\!\cdots\!18\)\( T_{3}^{5} + \)\(59\!\cdots\!82\)\( T_{3}^{4} + \)\(10\!\cdots\!73\)\( T_{3}^{3} - \)\(74\!\cdots\!99\)\( T_{3}^{2} - \)\(44\!\cdots\!58\)\( T_{3} + \)\(52\!\cdots\!61\)\( \)">\(T_{3}^{60} + \cdots\) acting on \(S_{4}^{\mathrm{new}}(230, [\chi])\).