Properties

Label 108.6.i.a
Level 108
Weight 6
Character orbit 108.i
Analytic conductor 17.321
Analytic rank 0
Dimension 90
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 108 = 2^{2} \cdot 3^{3} \)
Weight: \( k \) = \( 6 \)
Character orbit: \([\chi]\) = 108.i (of order \(9\), degree \(6\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(17.3214525398\)
Analytic rank: \(0\)
Dimension: \(90\)
Relative dimension: \(15\) over \(\Q(\zeta_{9})\)
Coefficient ring index: multiple of None
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 90q - 87q^{5} + 330q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 90q - 87q^{5} + 330q^{9} - 1257q^{11} + 531q^{15} - 3468q^{17} + 12894q^{21} + 8106q^{23} + 4959q^{25} - 17415q^{27} + 3468q^{29} - 6651q^{31} + 33624q^{33} - 8229q^{35} - 10545q^{39} + 68673q^{41} + 9459q^{43} - 53469q^{45} - 57087q^{47} - 5490q^{49} + 42831q^{51} - 4146q^{53} + 24624q^{57} + 5388q^{59} + 70110q^{61} - 98115q^{63} - 172425q^{65} - 15039q^{67} + 251037q^{69} + 67812q^{71} - 27009q^{73} - 75273q^{75} + 23991q^{77} - 216180q^{79} + 177822q^{81} - 76725q^{83} - 53100q^{85} - 201483q^{87} - 98814q^{89} - 90999q^{91} + 21765q^{93} - 143490q^{95} - 71739q^{97} + 13635q^{99} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
13.1 0 −15.3241 + 2.85854i 0 −63.0232 52.8828i 0 −226.162 82.3161i 0 226.658 87.6091i 0
13.2 0 −14.7517 5.03867i 0 8.41214 + 7.05862i 0 −17.8438 6.49461i 0 192.224 + 148.658i 0
13.3 0 −14.2968 + 6.21296i 0 82.0644 + 68.8602i 0 4.01162 + 1.46011i 0 165.798 177.651i 0
13.4 0 −12.9624 + 8.65895i 0 −29.6003 24.8376i 0 188.011 + 68.4304i 0 93.0451 224.481i 0
13.5 0 −11.5186 10.5034i 0 0.194482 + 0.163190i 0 83.4487 + 30.3729i 0 22.3564 + 241.969i 0
13.6 0 −4.93759 + 14.7858i 0 −1.72634 1.44857i 0 22.1444 + 8.05989i 0 −194.240 146.012i 0
13.7 0 −2.03335 15.4553i 0 58.1486 + 48.7924i 0 −131.626 47.9080i 0 −234.731 + 62.8520i 0
13.8 0 −1.96043 + 15.4647i 0 25.7256 + 21.5863i 0 −179.693 65.4027i 0 −235.313 60.6348i 0
13.9 0 0.571582 15.5780i 0 −56.0600 47.0399i 0 −23.6165 8.59570i 0 −242.347 17.8082i 0
13.10 0 5.03917 + 14.7515i 0 −66.1309 55.4904i 0 74.2607 + 27.0287i 0 −192.214 + 148.671i 0
13.11 0 9.06551 12.6813i 0 40.6723 + 34.1282i 0 234.829 + 85.4708i 0 −78.6332 229.926i 0
13.12 0 11.6047 + 10.4082i 0 50.4250 + 42.3116i 0 105.429 + 38.3730i 0 26.3374 + 241.569i 0
13.13 0 13.5505 7.70608i 0 23.4643 + 19.6889i 0 −110.864 40.3512i 0 124.233 208.843i 0
13.14 0 14.3467 + 6.09679i 0 −19.6970 16.5278i 0 −125.630 45.7255i 0 168.658 + 174.938i 0
13.15 0 15.0066 4.21923i 0 −56.3178 47.2563i 0 103.300 + 37.5982i 0 207.396 126.633i 0
25.1 0 −15.3241 2.85854i 0 −63.0232 + 52.8828i 0 −226.162 + 82.3161i 0 226.658 + 87.6091i 0
25.2 0 −14.7517 + 5.03867i 0 8.41214 7.05862i 0 −17.8438 + 6.49461i 0 192.224 148.658i 0
25.3 0 −14.2968 6.21296i 0 82.0644 68.8602i 0 4.01162 1.46011i 0 165.798 + 177.651i 0
25.4 0 −12.9624 8.65895i 0 −29.6003 + 24.8376i 0 188.011 68.4304i 0 93.0451 + 224.481i 0
25.5 0 −11.5186 + 10.5034i 0 0.194482 0.163190i 0 83.4487 30.3729i 0 22.3564 241.969i 0
See all 90 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 97.15
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
27.e even 9 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 108.6.i.a 90
3.b odd 2 1 324.6.i.a 90
27.e even 9 1 inner 108.6.i.a 90
27.f odd 18 1 324.6.i.a 90
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
108.6.i.a 90 1.a even 1 1 trivial
108.6.i.a 90 27.e even 9 1 inner
324.6.i.a 90 3.b odd 2 1
324.6.i.a 90 27.f odd 18 1

Hecke kernels

This newform subspace is the entire newspace \(S_{6}^{\mathrm{new}}(108, [\chi])\).

Hecke Characteristic Polynomials

There are no characteristic polynomials of Hecke operators in the database