Properties

Label 69.4.g.a
Level $69$
Weight $4$
Character orbit 69.g
Analytic conductor $4.071$
Analytic rank $0$
Dimension $220$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 69 = 3 \cdot 23 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 69.g (of order \(22\), degree \(10\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(4.07113179040\)
Analytic rank: \(0\)
Dimension: \(220\)
Relative dimension: \(22\) over \(\Q(\zeta_{22})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{22}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 220q - 5q^{3} + 50q^{4} - 32q^{6} - 22q^{7} - 5q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 220q - 5q^{3} + 50q^{4} - 32q^{6} - 22q^{7} - 5q^{9} - 22q^{10} + 4q^{12} + 30q^{13} + 363q^{15} + 162q^{16} - 78q^{18} - 22q^{19} - 671q^{21} - 1258q^{24} - 632q^{25} - 269q^{27} - 22q^{28} + 1485q^{30} - 138q^{31} + 957q^{33} + 1892q^{34} - 518q^{36} + 1166q^{37} - 401q^{39} - 2332q^{40} - 11q^{42} - 1738q^{43} - 4326q^{46} + 1706q^{48} - 3212q^{49} - 11q^{51} - 346q^{52} - 2307q^{54} - 1030q^{55} - 2123q^{57} + 8656q^{58} - 451q^{60} + 506q^{61} + 2079q^{63} + 2524q^{64} + 5742q^{66} + 1430q^{67} + 4875q^{69} + 8092q^{70} + 6445q^{72} - 114q^{73} + 9349q^{75} + 1034q^{76} + 6684q^{78} - 1870q^{79} - 5497q^{81} - 4220q^{82} - 16753q^{84} - 958q^{85} - 3785q^{87} + 1386q^{88} - 18700q^{90} - 14746q^{93} - 6900q^{94} - 10367q^{96} + 12650q^{97} - 4631q^{99} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
5.1 −1.55934 + 5.31062i 3.37983 3.94674i −19.0411 12.2370i 1.69511 11.7897i 15.6894 + 24.1033i 10.2590 4.68513i 61.2142 53.0424i −4.15356 26.6786i 59.9675 + 27.3862i
5.2 −1.48758 + 5.06624i −5.07448 + 1.11787i −16.7239 10.7478i −2.59342 + 18.0376i 1.88530 27.3715i 2.05975 0.940657i 47.4055 41.0771i 24.5007 11.3452i −87.5251 39.9714i
5.3 −1.20747 + 4.11226i −2.48738 + 4.56212i −8.72270 5.60574i 2.62405 18.2506i −15.7572 15.7374i −25.3847 + 11.5928i 7.67232 6.64810i −14.6259 22.6954i 71.8830 + 32.8279i
5.4 −1.17465 + 4.00049i 4.31812 + 2.89031i −7.89411 5.07323i −0.800687 + 5.56890i −16.6349 + 13.8795i −5.95353 + 2.71889i 4.36018 3.77812i 10.2923 + 24.9614i −21.3378 9.74465i
5.5 −1.00406 + 3.41950i −0.818289 5.13132i −3.95481 2.54160i −1.20932 + 8.41101i 18.3681 + 2.35399i −15.7004 + 7.17015i −8.88523 + 7.69910i −25.6608 + 8.39780i −27.5472 12.5804i
5.6 −0.978191 + 3.33141i −4.71623 2.18110i −3.41142 2.19239i 1.70238 11.8403i 11.8795 13.5782i 22.3835 10.2222i −10.3513 + 8.96943i 17.4856 + 20.5731i 37.7798 + 17.2535i
5.7 −0.698369 + 2.37843i 4.35121 2.84024i 1.56084 + 1.00309i −0.0326676 + 0.227208i 3.71656 + 12.3326i 16.1821 7.39011i −18.4629 + 15.9982i 10.8660 24.7170i −0.517584 0.236373i
5.8 −0.579088 + 1.97219i −1.85709 + 4.85296i 3.17583 + 2.04098i −1.18604 + 8.24906i −8.49555 6.47283i 14.0781 6.42924i −18.2916 + 15.8497i −20.1024 18.0248i −15.5819 7.11602i
5.9 −0.222958 + 0.759325i 4.12739 + 3.15668i 6.20316 + 3.98653i 2.75670 19.1733i −3.31718 + 2.43022i 15.1346 6.91176i −9.19481 + 7.96735i 7.07070 + 26.0577i 13.9441 + 6.36807i
5.10 −0.201291 + 0.685535i −5.17034 0.517272i 6.30059 + 4.04914i 0.256284 1.78249i 1.39535 3.44033i −23.2362 + 10.6116i −8.36380 + 7.24728i 26.4649 + 5.34895i 1.17037 + 0.534491i
5.11 −0.0243207 + 0.0828288i 0.514506 5.17062i 6.72376 + 4.32110i 2.06531 14.3646i 0.415763 + 0.168369i −11.4770 + 5.24135i −1.04336 + 0.904078i −26.4706 5.32063i 1.13957 + 0.520424i
5.12 0.0243207 0.0828288i 5.04477 1.24512i 6.72376 + 4.32110i −2.06531 + 14.3646i 0.0195602 0.448134i −11.4770 + 5.24135i 1.04336 0.904078i 23.8993 12.5627i 1.13957 + 0.520424i
5.13 0.201291 0.685535i 1.24782 + 5.04410i 6.30059 + 4.04914i −0.256284 + 1.78249i 3.70908 + 0.159907i −23.2362 + 10.6116i 8.36380 7.24728i −23.8859 + 12.5883i 1.17037 + 0.534491i
5.14 0.222958 0.759325i −3.71194 3.63614i 6.20316 + 3.98653i −2.75670 + 19.1733i −3.58862 + 2.00786i 15.1346 6.91176i 9.19481 7.96735i 0.557026 + 26.9943i 13.9441 + 6.36807i
5.15 0.579088 1.97219i −4.53927 + 2.52884i 3.17583 + 2.04098i 1.18604 8.24906i 2.35872 + 10.4167i 14.0781 6.42924i 18.2916 15.8497i 14.2100 22.9582i −15.5819 7.11602i
5.16 0.698369 2.37843i 2.19209 4.71113i 1.56084 + 1.00309i 0.0326676 0.227208i −9.67418 8.50383i 16.1821 7.39011i 18.4629 15.9982i −17.3895 20.6545i −0.517584 0.236373i
5.17 0.978191 3.33141i 2.83009 + 4.35782i −3.41142 2.19239i −1.70238 + 11.8403i 17.2861 5.16540i 22.3835 10.2222i 10.3513 8.96943i −10.9812 + 24.6660i 37.7798 + 17.2535i
5.18 1.00406 3.41950i 5.19554 + 0.0796979i −3.95481 2.54160i 1.20932 8.41101i 5.48914 17.6861i −15.7004 + 7.17015i 8.88523 7.69910i 26.9873 + 0.828147i −27.5472 12.5804i
5.19 1.17465 4.00049i −3.47542 3.86283i −7.89411 5.07323i 0.800687 5.56890i −19.5356 + 9.36591i −5.95353 + 2.71889i −4.36018 + 3.77812i −2.84293 + 26.8499i −21.3378 9.74465i
5.20 1.20747 4.11226i −4.16170 + 3.11132i −8.72270 5.60574i −2.62405 + 18.2506i 7.76943 + 20.8708i −25.3847 + 11.5928i −7.67232 + 6.64810i 7.63942 25.8967i 71.8830 + 32.8279i
See next 80 embeddings (of 220 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 65.22
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
23.d odd 22 1 inner
69.g even 22 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 69.4.g.a 220
3.b odd 2 1 inner 69.4.g.a 220
23.d odd 22 1 inner 69.4.g.a 220
69.g even 22 1 inner 69.4.g.a 220
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
69.4.g.a 220 1.a even 1 1 trivial
69.4.g.a 220 3.b odd 2 1 inner
69.4.g.a 220 23.d odd 22 1 inner
69.4.g.a 220 69.g even 22 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(69, [\chi])\).