Properties

Label 108.4.l.a
Level 108
Weight 4
Character orbit 108.l
Analytic conductor 6.372
Analytic rank 0
Dimension 312
CM no
Inner twists 4

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Newspace parameters

Level: \( N \) \(=\) \( 108 = 2^{2} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 108.l (of order \(18\), degree \(6\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(6.37220628062\)
Analytic rank: \(0\)
Dimension: \(312\)
Relative dimension: \(52\) over \(\Q(\zeta_{18})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 312q - 6q^{2} - 6q^{4} - 12q^{5} - 6q^{6} - 9q^{8} - 12q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 312q - 6q^{2} - 6q^{4} - 12q^{5} - 6q^{6} - 9q^{8} - 12q^{9} - 3q^{10} - 123q^{12} - 12q^{13} + 69q^{14} - 6q^{16} - 18q^{17} + 351q^{18} + 225q^{20} - 12q^{21} - 6q^{22} - 300q^{24} - 12q^{25} - 12q^{28} - 96q^{29} - 207q^{30} - 696q^{32} + 858q^{33} - 30q^{34} - 1056q^{36} - 6q^{37} - 900q^{38} - 381q^{40} + 138q^{41} + 2574q^{42} + 2655q^{44} - 672q^{45} - 3q^{46} - 435q^{48} - 12q^{49} - 2829q^{50} + 1371q^{52} - 4458q^{54} - 2925q^{56} + 660q^{57} + 885q^{58} + 966q^{60} - 12q^{61} + 1872q^{62} - 3q^{64} - 708q^{65} + 3093q^{66} + 2211q^{68} - 1572q^{69} - 1011q^{70} - 4524q^{72} - 6q^{73} - 5883q^{74} - 198q^{76} - 996q^{77} - 2976q^{78} + 444q^{81} - 12q^{82} + 6324q^{84} - 762q^{85} + 8322q^{86} + 1530q^{88} + 4212q^{89} - 1104q^{90} - 3255q^{92} + 7404q^{93} + 2019q^{94} + 582q^{96} - 66q^{97} + 2898q^{98} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
11.1 −2.82000 + 0.218140i −2.56535 + 4.51874i 7.90483 1.23031i 2.35563 6.47204i 6.24857 13.3025i −6.19890 1.09303i −22.0233 + 5.19383i −13.8380 23.1843i −5.23107 + 18.7650i
11.2 −2.81863 + 0.235239i −0.238051 5.19070i 7.88933 1.32610i 5.65263 15.5305i 1.89203 + 14.5746i 28.8083 + 5.07968i −21.9251 + 5.59366i −26.8867 + 2.47130i −12.2793 + 45.1044i
11.3 −2.76593 + 0.591311i −5.08011 + 1.09199i 7.30070 3.27105i −6.87889 + 18.8996i 13.4055 6.02429i 22.1047 + 3.89766i −18.2590 + 13.3645i 24.6151 11.0949i 7.85096 56.3424i
11.4 −2.75026 0.660376i 2.16188 4.72507i 7.12781 + 3.63240i −3.60998 + 9.91833i −9.06605 + 11.5675i −12.1874 2.14897i −17.2045 14.6971i −17.6525 20.4301i 16.4782 24.8940i
11.5 −2.73813 0.708980i −4.61237 2.39291i 6.99469 + 3.88256i 0.577330 1.58620i 10.9327 + 9.82218i −27.0016 4.76111i −16.3997 15.5900i 15.5480 + 22.0740i −2.70539 + 3.93391i
11.6 −2.66507 + 0.947300i 5.14914 0.697370i 6.20524 5.04925i −1.80167 + 4.95006i −13.0622 + 6.73633i 14.7819 + 2.60644i −11.7543 + 19.3349i 26.0273 7.18172i 0.112404 14.8990i
11.7 −2.63308 1.03289i 5.16641 + 0.555160i 5.86626 + 5.43939i 6.86537 18.8624i −13.0302 6.79813i −18.0159 3.17669i −9.82807 20.3816i 26.3836 + 5.73637i −37.5600 + 42.5752i
11.8 −2.61271 + 1.08339i 2.79011 + 4.38352i 5.65252 5.66118i −1.04436 + 2.86937i −12.0388 8.43010i −28.5377 5.03197i −8.63512 + 20.9149i −11.4306 + 24.4610i −0.380029 8.62829i
11.9 −2.60460 1.10274i 2.16828 + 4.72213i 5.56791 + 5.74442i −0.836765 + 2.29899i −0.440216 14.6903i 22.6981 + 4.00229i −8.16758 21.1019i −17.5971 + 20.4779i 4.71464 5.06522i
11.10 −2.33178 + 1.60087i −4.92086 1.66888i 2.87442 7.46577i 3.67143 10.0872i 14.1460 3.98618i −6.04634 1.06613i 5.24922 + 22.0101i 21.4296 + 16.4247i 7.58728 + 29.3985i
11.11 −2.10858 + 1.88517i −1.35776 5.01562i 0.892234 7.95009i −5.18482 + 14.2452i 12.3183 + 8.01624i −9.21752 1.62530i 13.1060 + 18.4454i −23.3130 + 13.6200i −15.9220 39.8114i
11.12 −2.10574 1.88834i −3.74172 3.60549i 0.868318 + 7.95274i −1.60571 + 4.41165i 1.07071 + 14.6579i 23.1356 + 4.07942i 13.1890 18.3861i 1.00093 + 26.9814i 11.7119 6.25768i
11.13 −2.04068 1.95848i −2.83464 + 4.35486i 0.328747 + 7.99324i −6.07198 + 16.6826i 14.3135 3.33532i −22.8171 4.02326i 14.9837 16.9555i −10.9297 24.6889i 45.0635 22.1521i
11.14 −1.88673 + 2.10719i 2.88986 4.31841i −0.880530 7.95139i 3.28651 9.02962i 3.64735 + 14.2372i −18.0956 3.19074i 18.4164 + 13.1467i −10.2974 24.9592i 12.8264 + 23.9617i
11.15 −1.88377 2.10984i −4.43306 + 2.71072i −0.902841 + 7.94889i 5.32629 14.6339i 14.0700 + 4.24668i 6.67869 + 1.17763i 18.4716 13.0690i 12.3040 24.0335i −40.9086 + 16.3292i
11.16 −1.76579 2.20953i 4.72349 2.16532i −1.76401 + 7.80309i −1.80584 + 4.96151i −13.1250 6.61318i 9.40209 + 1.65784i 20.3560 9.88097i 17.6228 20.4558i 14.1513 4.77091i
11.17 −1.74755 + 2.22397i −2.88986 + 4.31841i −1.89211 7.77303i 3.28651 9.02962i −4.55384 13.9736i 18.0956 + 3.19074i 20.5936 + 9.37578i −10.2974 24.9592i 14.3383 + 23.0889i
11.18 −1.49038 + 2.40390i 1.35776 + 5.01562i −3.55752 7.16548i −5.18482 + 14.2452i −14.0807 4.21127i 9.21752 + 1.62530i 22.5272 + 2.12738i −23.3130 + 13.6200i −26.5167 33.6946i
11.19 −1.17164 + 2.57435i 4.92086 + 1.66888i −5.25451 6.03242i 3.67143 10.0872i −10.0618 + 10.7127i 6.04634 + 1.06613i 21.6859 6.45911i 21.4296 + 16.4247i 21.6662 + 21.2701i
11.20 −1.11352 2.60001i −0.629751 5.15785i −5.52015 + 5.79033i 4.68740 12.8785i −12.7092 + 7.38072i −21.3342 3.76180i 21.2017 + 7.90483i −26.2068 + 6.49632i −38.7038 + 2.15317i
See next 80 embeddings (of 312 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 95.52
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
27.f odd 18 1 inner
108.l even 18 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 108.4.l.a 312
4.b odd 2 1 inner 108.4.l.a 312
27.f odd 18 1 inner 108.4.l.a 312
108.l even 18 1 inner 108.4.l.a 312
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
108.4.l.a 312 1.a even 1 1 trivial
108.4.l.a 312 4.b odd 2 1 inner
108.4.l.a 312 27.f odd 18 1 inner
108.4.l.a 312 108.l even 18 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(108, [\chi])\).

Hecke characteristic polynomials

There are no characteristic polynomials of Hecke operators in the database