Properties

Label 69.4.e.a
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.

\(\operatorname{Tr}(f)(q) = \) \( 60q - 18q^{3} - 28q^{4} + 22q^{5} - 33q^{6} + 24q^{7} + 16q^{8} - 54q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 60q - 18q^{3} - 28q^{4} + 22q^{5} - 33q^{6} + 24q^{7} + 16q^{8} - 54q^{9} + 58q^{10} - 10q^{11} - 84q^{12} + 14q^{13} + 68q^{14} - 66q^{15} + 292q^{16} + 742q^{17} - 160q^{19} - 37q^{20} + 72q^{21} - 1346q^{22} - 530q^{23} - 216q^{24} - 370q^{25} - 104q^{26} - 162q^{27} + 856q^{28} - 398q^{29} + 174q^{30} - 628q^{31} + 560q^{32} + 432q^{33} + 2469q^{34} + 1006q^{35} + 243q^{36} + 812q^{37} - 1716q^{38} + 42q^{39} + 1485q^{40} + 1136q^{41} - 456q^{42} - 888q^{43} - 2921q^{44} - 792q^{45} - 2164q^{46} - 2712q^{47} - 1071q^{48} + 2266q^{49} - 2953q^{50} - 414q^{51} - 3455q^{52} - 1216q^{53} + 297q^{54} + 3894q^{55} + 6282q^{56} + 1962q^{57} + 4297q^{58} - 1292q^{59} + 2661q^{60} - 150q^{61} + 3163q^{62} + 216q^{63} + 1316q^{64} + 1270q^{65} - 1827q^{66} - 472q^{67} - 8128q^{68} - 138q^{69} - 11776q^{70} + 2108q^{71} + 144q^{72} - 2432q^{73} + 10590q^{74} - 54q^{75} + 3049q^{76} + 2238q^{77} + 2856q^{78} + 4640q^{79} + 9182q^{80} - 486q^{81} - 3834q^{82} - 186q^{83} - 2052q^{84} - 402q^{85} - 7184q^{86} + 720q^{87} - 1124q^{88} - 8642q^{89} + 522q^{90} - 9676q^{91} - 409q^{92} - 1224q^{93} - 869q^{94} - 3064q^{95} + 96q^{96} - 638q^{97} - 7063q^{98} + 1296q^{99} + O(q^{100}) \)

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 −2.04147 4.47020i −2.87848 + 0.845198i −10.5762 + 12.2056i 16.4166 + 10.5503i 9.65454 + 11.1419i 0.635011 + 4.41660i 38.4307 + 11.2843i 7.57128 4.86577i 13.6479 94.9235i
4.2 −1.25209 2.74169i −2.87848 + 0.845198i −0.710260 + 0.819684i −9.32369 5.99197i 5.92138 + 6.83364i 2.45322 + 17.0626i −19.9992 5.87229i 7.57128 4.86577i −4.75405 + 33.0652i
4.3 −0.464261 1.01659i −2.87848 + 0.845198i 4.42097 5.10207i 6.11072 + 3.92712i 2.19559 + 2.53384i −1.36340 9.48266i −15.8177 4.64450i 7.57128 4.86577i 1.15530 8.03531i
4.4 0.700410 + 1.53368i −2.87848 + 0.845198i 3.37727 3.89758i −14.3333 9.21148i −3.31238 3.82269i −3.63633 25.2912i 21.2851 + 6.24988i 7.57128 4.86577i 4.08828 28.4346i
4.5 0.718480 + 1.57325i −2.87848 + 0.845198i 3.27998 3.78529i 5.04169 + 3.24010i −3.39784 3.92132i 4.06727 + 28.2885i 21.5877 + 6.33873i 7.57128 4.86577i −1.47514 + 10.2598i
4.6 2.08583 + 4.56734i −2.87848 + 0.845198i −11.2710 + 13.0074i −5.38817 3.46277i −9.86434 11.3841i 1.35014 + 9.39044i −44.3773 13.0304i 7.57128 4.86577i 4.57681 31.8324i
13.1 −3.15951 3.64626i 2.52376 + 1.62192i −2.17425 + 15.1222i 0.670517 1.46823i −2.05988 14.3268i 29.0868 + 8.54065i 29.5387 18.9834i 3.73874 + 8.18669i −7.47204 + 2.19399i
13.2 −2.24664 2.59277i 2.52376 + 1.62192i −0.536504 + 3.73147i −5.18705 + 11.3581i −1.46473 10.1874i −25.5228 7.49416i −12.2087 + 7.84605i 3.73874 + 8.18669i 41.1022 12.0687i
13.3 −0.456676 0.527032i 2.52376 + 1.62192i 1.06931 7.43721i 3.32762 7.28646i −0.297736 2.07080i −13.2035 3.87690i −9.10125 + 5.84902i 3.73874 + 8.18669i −5.35984 + 1.57379i
13.4 0.280926 + 0.324206i 2.52376 + 1.62192i 1.11233 7.73642i −9.21494 + 20.1779i 0.183153 + 1.27386i 29.8306 + 8.75906i 5.70777 3.66816i 3.73874 + 8.18669i −9.13053 + 2.68096i
13.5 1.38600 + 1.59953i 2.52376 + 1.62192i 0.501022 3.48468i 6.21106 13.6003i 0.903619 + 6.28481i 10.6195 + 3.11816i 20.5122 13.1824i 3.73874 + 8.18669i 30.3627 8.91528i
13.6 2.90770 + 3.35566i 2.52376 + 1.62192i −1.66725 + 11.5960i −1.16276 + 2.54608i 1.89571 + 13.1850i −5.53916 1.62644i −13.8774 + 8.91848i 3.73874 + 8.18669i −11.9247 + 3.50142i
16.1 −3.15951 + 3.64626i 2.52376 1.62192i −2.17425 15.1222i 0.670517 + 1.46823i −2.05988 + 14.3268i 29.0868 8.54065i 29.5387 + 18.9834i 3.73874 8.18669i −7.47204 2.19399i
16.2 −2.24664 + 2.59277i 2.52376 1.62192i −0.536504 3.73147i −5.18705 11.3581i −1.46473 + 10.1874i −25.5228 + 7.49416i −12.2087 7.84605i 3.73874 8.18669i 41.1022 + 12.0687i
16.3 −0.456676 + 0.527032i 2.52376 1.62192i 1.06931 + 7.43721i 3.32762 + 7.28646i −0.297736 + 2.07080i −13.2035 + 3.87690i −9.10125 5.84902i 3.73874 8.18669i −5.35984 1.57379i
16.4 0.280926 0.324206i 2.52376 1.62192i 1.11233 + 7.73642i −9.21494 20.1779i 0.183153 1.27386i 29.8306 8.75906i 5.70777 + 3.66816i 3.73874 8.18669i −9.13053 2.68096i
16.5 1.38600 1.59953i 2.52376 1.62192i 0.501022 + 3.48468i 6.21106 + 13.6003i 0.903619 6.28481i 10.6195 3.11816i 20.5122 + 13.1824i 3.73874 8.18669i 30.3627 + 8.91528i
16.6 2.90770 3.35566i 2.52376 1.62192i −1.66725 11.5960i −1.16276 2.54608i 1.89571 13.1850i −5.53916 + 1.62644i −13.8774 8.91848i 3.73874 8.18669i −11.9247 3.50142i
25.1 −3.93223 2.52709i −0.426945 2.96946i 5.75294 + 12.5972i −6.57189 1.92968i −5.82527 + 12.7556i −12.7631 + 14.7294i 3.89062 27.0599i −8.63544 + 2.53559i 20.9657 + 24.1957i
25.2 −3.39636 2.18271i −0.426945 2.96946i 3.44773 + 7.54946i 16.0504 + 4.71284i −5.03142 + 11.0173i 23.9549 27.6454i 0.172065 1.19674i −8.63544 + 2.53559i −44.2263 51.0399i
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.a 60
3.b odd 2 1 207.4.i.c 60
23.c even 11 1 inner 69.4.e.a 60
23.c even 11 1 1587.4.a.t 30
23.d odd 22 1 1587.4.a.u 30
69.h odd 22 1 207.4.i.c 60
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
69.4.e.a 60 1.a even 1 1 trivial
69.4.e.a 60 23.c even 11 1 inner
207.4.i.c 60 3.b odd 2 1
207.4.i.c 60 69.h odd 22 1
1587.4.a.t 30 23.c even 11 1
1587.4.a.u 30 23.d odd 22 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(57\!\cdots\!84\)\( T_{2}^{42} - \)\(12\!\cdots\!72\)\( T_{2}^{41} + \)\(12\!\cdots\!79\)\( T_{2}^{40} - \)\(27\!\cdots\!42\)\( T_{2}^{39} + \)\(16\!\cdots\!98\)\( T_{2}^{38} - \)\(66\!\cdots\!78\)\( T_{2}^{37} + \)\(17\!\cdots\!37\)\( T_{2}^{36} - \)\(86\!\cdots\!90\)\( T_{2}^{35} + \)\(40\!\cdots\!64\)\( T_{2}^{34} - \)\(83\!\cdots\!74\)\( T_{2}^{33} + \)\(45\!\cdots\!03\)\( T_{2}^{32} - \)\(89\!\cdots\!94\)\( T_{2}^{31} + \)\(34\!\cdots\!41\)\( T_{2}^{30} - \)\(85\!\cdots\!81\)\( T_{2}^{29} + \)\(25\!\cdots\!55\)\( T_{2}^{28} - \)\(34\!\cdots\!36\)\( T_{2}^{27} + \)\(13\!\cdots\!18\)\( T_{2}^{26} - \)\(40\!\cdots\!50\)\( T_{2}^{25} + \)\(15\!\cdots\!30\)\( T_{2}^{24} - \)\(57\!\cdots\!78\)\( T_{2}^{23} + \)\(24\!\cdots\!78\)\( T_{2}^{22} - \)\(85\!\cdots\!53\)\( T_{2}^{21} + \)\(24\!\cdots\!67\)\( T_{2}^{20} - \)\(58\!\cdots\!42\)\( T_{2}^{19} + \)\(11\!\cdots\!12\)\( T_{2}^{18} - \)\(18\!\cdots\!60\)\( T_{2}^{17} + \)\(25\!\cdots\!20\)\( T_{2}^{16} - \)\(30\!\cdots\!64\)\( T_{2}^{15} + \)\(39\!\cdots\!20\)\( T_{2}^{14} - \)\(58\!\cdots\!32\)\( T_{2}^{13} + \)\(81\!\cdots\!12\)\( T_{2}^{12} - \)\(98\!\cdots\!92\)\( T_{2}^{11} + \)\(10\!\cdots\!40\)\( T_{2}^{10} - \)\(11\!\cdots\!52\)\( T_{2}^{9} + \)\(11\!\cdots\!64\)\( T_{2}^{8} - \)\(73\!\cdots\!60\)\( T_{2}^{7} + \)\(37\!\cdots\!84\)\( T_{2}^{6} - \)\(29\!\cdots\!12\)\( T_{2}^{5} + \)\(21\!\cdots\!80\)\( T_{2}^{4} - \)\(86\!\cdots\!04\)\( T_{2}^{3} + \)\(22\!\cdots\!48\)\( T_{2}^{2} - \)\(30\!\cdots\!56\)\( T_{2} + \)\(86\!\cdots\!64\)\( \)">\(T_{2}^{60} + \cdots\) acting on \(S_{4}^{\mathrm{new}}(69, [\chi])\).