Properties

Label 69.4.e.b
Level $69$
Weight $4$
Character orbit 69.e
Analytic conductor $4.071$
Analytic rank $0$
Dimension $60$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 69 = 3 \cdot 23 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 69.e (of order \(11\), degree \(10\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(4.07113179040\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 60q + 4q^{2} + 18q^{3} - 28q^{4} - 6q^{5} + 21q^{6} - 4q^{7} - 52q^{8} - 54q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 60q + 4q^{2} + 18q^{3} - 28q^{4} - 6q^{5} + 21q^{6} - 4q^{7} - 52q^{8} - 54q^{9} - 78q^{10} + 10q^{11} + 84q^{12} + 50q^{13} - 224q^{14} + 150q^{15} + 260q^{16} - 662q^{17} + 36q^{18} - 4q^{19} - 735q^{20} + 12q^{21} + 622q^{22} - 438q^{23} - 108q^{24} - 754q^{25} - 40q^{26} + 162q^{27} + 672q^{28} + 1302q^{29} + 234q^{30} + 1528q^{31} + 1588q^{32} - 492q^{33} + 29q^{34} + 950q^{35} + 243q^{36} + 316q^{37} + 3122q^{38} - 150q^{39} - 1939q^{40} - 1500q^{41} - 2298q^{42} - 1316q^{43} - 2901q^{44} + 936q^{45} - 1980q^{46} - 1440q^{47} - 2265q^{48} - 2310q^{49} + 195q^{50} - 126q^{51} + 6189q^{52} - 148q^{53} + 189q^{54} - 606q^{55} - 432q^{56} + 1398q^{57} - 2623q^{58} + 5264q^{59} + 753q^{60} + 1482q^{61} - 2299q^{62} - 36q^{63} - 6780q^{64} - 1446q^{65} + 1731q^{66} + 388q^{67} + 5604q^{68} - 138q^{69} + 2984q^{70} - 3316q^{71} - 468q^{72} + 2072q^{73} - 6556q^{74} + 1206q^{75} + 9841q^{76} + 9338q^{77} - 3048q^{78} + 268q^{79} + 7980q^{80} - 486q^{81} + 7742q^{82} - 3494q^{83} + 2604q^{84} - 3842q^{85} - 4792q^{86} - 672q^{87} - 7960q^{88} - 2754q^{89} - 702q^{90} - 5436q^{91} - 17609q^{92} + 2280q^{93} - 10961q^{94} - 2396q^{95} + 6852q^{96} - 5654q^{97} + 14411q^{98} + 1476q^{99} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
4.1 −1.62255 3.55289i 2.87848 0.845198i −4.75146 + 5.48347i 2.37866 + 1.52867i −7.67337 8.85554i −5.05798 35.1790i −2.78946 0.819058i 7.57128 4.86577i 1.57171 10.9315i
4.2 −0.936192 2.04997i 2.87848 0.845198i 1.91295 2.20766i 17.1881 + 11.0461i −4.42744 5.10954i 3.15549 + 21.9469i −23.6153 6.93407i 7.57128 4.86577i 6.55289 45.5764i
4.3 −0.468580 1.02605i 2.87848 0.845198i 4.40568 5.08442i −16.0845 10.3369i −2.21601 2.55741i 1.00849 + 7.01421i −15.9396 4.68029i 7.57128 4.86577i −3.06925 + 21.3471i
4.4 0.675509 + 1.47916i 2.87848 0.845198i 3.50729 4.04763i 3.16001 + 2.03082i 3.19462 + 3.68679i −2.13282 14.8341i 20.8382 + 6.11864i 7.57128 4.86577i −0.869282 + 6.04599i
4.5 1.45666 + 3.18964i 2.87848 0.845198i −2.81307 + 3.24646i −1.51957 0.976567i 6.88885 + 7.95015i 4.44617 + 30.9238i 12.4631 + 3.65950i 7.57128 4.86577i 0.901405 6.26941i
4.6 2.23641 + 4.89704i 2.87848 0.845198i −13.7407 + 15.8576i 9.26837 + 5.95642i 10.5764 + 12.2058i −1.89824 13.2025i −67.0611 19.6909i 7.57128 4.86577i −8.44102 + 58.7086i
13.1 −2.63888 3.04543i −2.52376 1.62192i −1.17243 + 8.15446i −4.73327 + 10.3644i 1.72045 + 11.9660i 10.4190 + 3.05928i 0.807865 0.519183i 3.73874 + 8.18669i 44.0546 12.9356i
13.2 −2.23546 2.57986i −2.52376 1.62192i −0.519874 + 3.61580i 8.61254 18.8588i 1.45744 + 10.1367i −2.20230 0.646654i −12.4835 + 8.02266i 3.73874 + 8.18669i −67.9062 + 19.9391i
13.3 −0.298405 0.344378i −2.52376 1.62192i 1.10897 7.71304i −4.80158 + 10.5140i 0.194549 + 1.35312i −20.2653 5.95042i −6.05385 + 3.89057i 3.73874 + 8.18669i 5.05360 1.48387i
13.4 1.56552 + 1.80671i −2.52376 1.62192i 0.325181 2.26169i 1.13914 2.49437i −1.02066 7.09886i 33.1193 + 9.72471i 20.6842 13.2929i 3.73874 + 8.18669i 6.28995 1.84690i
13.5 1.56668 + 1.80804i −2.52376 1.62192i 0.323978 2.25332i 6.20398 13.5848i −1.02142 7.10410i −24.5496 7.20842i 20.6825 13.2918i 3.73874 + 8.18669i 34.2816 10.0660i
13.6 2.95596 + 3.41136i −2.52376 1.62192i −1.76116 + 12.2491i −6.65677 + 14.5763i −1.92717 13.4038i 1.88458 + 0.553362i −16.6135 + 10.6769i 3.73874 + 8.18669i −69.4021 + 20.3783i
16.1 −2.63888 + 3.04543i −2.52376 + 1.62192i −1.17243 8.15446i −4.73327 10.3644i 1.72045 11.9660i 10.4190 3.05928i 0.807865 + 0.519183i 3.73874 8.18669i 44.0546 + 12.9356i
16.2 −2.23546 + 2.57986i −2.52376 + 1.62192i −0.519874 3.61580i 8.61254 + 18.8588i 1.45744 10.1367i −2.20230 + 0.646654i −12.4835 8.02266i 3.73874 8.18669i −67.9062 19.9391i
16.3 −0.298405 + 0.344378i −2.52376 + 1.62192i 1.10897 + 7.71304i −4.80158 10.5140i 0.194549 1.35312i −20.2653 + 5.95042i −6.05385 3.89057i 3.73874 8.18669i 5.05360 + 1.48387i
16.4 1.56552 1.80671i −2.52376 + 1.62192i 0.325181 + 2.26169i 1.13914 + 2.49437i −1.02066 + 7.09886i 33.1193 9.72471i 20.6842 + 13.2929i 3.73874 8.18669i 6.28995 + 1.84690i
16.5 1.56668 1.80804i −2.52376 + 1.62192i 0.323978 + 2.25332i 6.20398 + 13.5848i −1.02142 + 7.10410i −24.5496 + 7.20842i 20.6825 + 13.2918i 3.73874 8.18669i 34.2816 + 10.0660i
16.6 2.95596 3.41136i −2.52376 + 1.62192i −1.76116 12.2491i −6.65677 14.5763i −1.92717 + 13.4038i 1.88458 0.553362i −16.6135 10.6769i 3.73874 8.18669i −69.4021 20.3783i
25.1 −4.69841 3.01948i 0.426945 + 2.96946i 9.63444 + 21.0965i −6.04862 1.77604i 6.96029 15.2409i 10.0657 11.6164i 12.0753 83.9856i −8.63544 + 2.53559i 23.0562 + 26.6083i
25.2 −2.39601 1.53982i 0.426945 + 2.96946i 0.0464818 + 0.101781i −1.22325 0.359177i 3.54948 7.77227i −1.84765 + 2.13230i −3.19731 + 22.2377i −8.63544 + 2.53559i 2.37784 + 2.74417i
See all 60 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 64.6
Significant digits:
Format:

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 69.4.e.b 60
3.b odd 2 1 207.4.i.b 60
23.c even 11 1 inner 69.4.e.b 60
23.c even 11 1 1587.4.a.w 30
23.d odd 22 1 1587.4.a.v 30
69.h odd 22 1 207.4.i.b 60
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
69.4.e.b 60 1.a even 1 1 trivial
69.4.e.b 60 23.c even 11 1 inner
207.4.i.b 60 3.b odd 2 1
207.4.i.b 60 69.h odd 22 1
1587.4.a.v 30 23.d odd 22 1
1587.4.a.w 30 23.c even 11 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(61\!\cdots\!84\)\( T_{2}^{42} - \)\(17\!\cdots\!12\)\( T_{2}^{41} + \)\(11\!\cdots\!79\)\( T_{2}^{40} - \)\(30\!\cdots\!94\)\( T_{2}^{39} + \)\(13\!\cdots\!30\)\( T_{2}^{38} - \)\(31\!\cdots\!86\)\( T_{2}^{37} + \)\(12\!\cdots\!53\)\( T_{2}^{36} - \)\(16\!\cdots\!62\)\( T_{2}^{35} + \)\(15\!\cdots\!36\)\( T_{2}^{34} + \)\(11\!\cdots\!26\)\( T_{2}^{33} + \)\(22\!\cdots\!79\)\( T_{2}^{32} + \)\(39\!\cdots\!94\)\( T_{2}^{31} + \)\(28\!\cdots\!35\)\( T_{2}^{30} + \)\(43\!\cdots\!55\)\( T_{2}^{29} + \)\(31\!\cdots\!17\)\( T_{2}^{28} + \)\(43\!\cdots\!88\)\( T_{2}^{27} + \)\(29\!\cdots\!78\)\( T_{2}^{26} + \)\(47\!\cdots\!22\)\( T_{2}^{25} + \)\(22\!\cdots\!64\)\( T_{2}^{24} + \)\(30\!\cdots\!34\)\( T_{2}^{23} + \)\(16\!\cdots\!90\)\( T_{2}^{22} + \)\(17\!\cdots\!81\)\( T_{2}^{21} + \)\(96\!\cdots\!11\)\( T_{2}^{20} + \)\(88\!\cdots\!14\)\( T_{2}^{19} + \)\(35\!\cdots\!44\)\( T_{2}^{18} + \)\(50\!\cdots\!68\)\( T_{2}^{17} + \)\(91\!\cdots\!04\)\( T_{2}^{16} + \)\(18\!\cdots\!84\)\( T_{2}^{15} + \)\(16\!\cdots\!32\)\( T_{2}^{14} + \)\(22\!\cdots\!24\)\( T_{2}^{13} + \)\(47\!\cdots\!20\)\( T_{2}^{12} - \)\(51\!\cdots\!68\)\( T_{2}^{11} + \)\(87\!\cdots\!44\)\( T_{2}^{10} - \)\(13\!\cdots\!24\)\( T_{2}^{9} + \)\(80\!\cdots\!48\)\( T_{2}^{8} - \)\(99\!\cdots\!96\)\( T_{2}^{7} + \)\(83\!\cdots\!20\)\( T_{2}^{6} + \)\(33\!\cdots\!92\)\( T_{2}^{5} - \)\(96\!\cdots\!72\)\( T_{2}^{4} - \)\(12\!\cdots\!16\)\( T_{2}^{3} + \)\(17\!\cdots\!28\)\( T_{2}^{2} + \)\(29\!\cdots\!92\)\( T_{2} + \)\(86\!\cdots\!56\)\( \)">\(T_{2}^{60} - \cdots\) acting on \(S_{4}^{\mathrm{new}}(69, [\chi])\).