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Properties

Label 68.4.i.b
Level $68$
Weight $4$
Character orbit 68.i
Analytic conductor $4.012$
Analytic rank $0$
Dimension $192$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [68,4,Mod(3,68)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(68, base_ring=CyclotomicField(16))
 
chi = DirichletCharacter(H, H._module([8, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("68.3");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 68 = 2^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 68.i (of order \(16\), degree \(8\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(4.01212988039\)
Analytic rank: \(0\)
Dimension: \(192\)
Relative dimension: \(24\) over \(\Q(\zeta_{16})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{16}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 192 q - 8 q^{2} - 8 q^{4} - 16 q^{5} - 8 q^{6} - 8 q^{8} - 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 192 q - 8 q^{2} - 8 q^{4} - 16 q^{5} - 8 q^{6} - 8 q^{8} - 16 q^{9} + 136 q^{10} - 8 q^{12} - 16 q^{13} - 8 q^{14} - 16 q^{17} - 16 q^{18} + 696 q^{20} - 16 q^{21} - 8 q^{22} + 904 q^{24} - 368 q^{25} - 152 q^{26} - 152 q^{28} - 1152 q^{29} - 1352 q^{30} - 1248 q^{32} - 2120 q^{34} - 1736 q^{36} - 16 q^{37} - 528 q^{38} + 568 q^{40} + 1872 q^{41} + 1864 q^{42} + 2456 q^{44} - 448 q^{45} + 1960 q^{46} - 224 q^{48} - 16 q^{49} - 16 q^{52} + 3200 q^{53} - 4392 q^{54} - 2608 q^{56} + 3424 q^{57} - 1296 q^{58} + 1136 q^{60} - 816 q^{61} + 2064 q^{62} + 2632 q^{64} - 7888 q^{65} + 6816 q^{66} + 9216 q^{68} - 4192 q^{69} + 5424 q^{70} + 1424 q^{72} + 528 q^{73} - 3688 q^{74} - 288 q^{76} + 720 q^{77} - 4480 q^{78} - 3992 q^{80} + 9184 q^{81} - 7664 q^{82} - 3072 q^{85} + 11104 q^{86} + 5176 q^{88} - 16 q^{89} + 8328 q^{90} - 1240 q^{92} - 16 q^{93} - 4328 q^{94} - 10800 q^{96} - 16 q^{97} - 6624 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3.1 −2.82840 − 0.0121775i 1.20209 − 6.04333i 7.99970 + 0.0688860i −8.22961 − 12.3165i −3.47359 + 17.0783i 10.5599 + 7.05589i −22.6255 − 0.292254i −10.1320 − 4.19683i 23.1267 + 34.9362i
3.2 −2.82656 + 0.102703i −1.92718 + 9.68857i 7.97890 − 0.580595i 5.30883 + 7.94522i 4.45224 − 27.5833i 24.0722 + 16.0845i −22.4932 + 2.46055i −65.2096 − 27.0107i −15.8217 − 21.9124i
3.3 −2.82033 + 0.213838i −0.00627997 + 0.0315715i 7.90855 − 1.20619i 1.51856 + 2.27269i 0.0109604 − 0.0903851i −12.1586 − 8.12410i −22.0468 + 5.09301i 24.9438 + 10.3321i −4.76884 − 6.08501i
3.4 −2.40643 − 1.48630i 1.14955 − 5.77917i 3.58180 + 7.15337i 11.5468 + 17.2810i −11.3559 + 12.1986i 13.1862 + 8.81073i 2.01275 − 22.5377i −7.13263 − 2.95443i −2.10170 − 58.7474i
3.5 −2.40305 − 1.49177i −0.850520 + 4.27585i 3.54927 + 7.16957i 0.695925 + 1.04153i 8.42241 − 9.00630i −24.4804 − 16.3572i 2.16627 − 22.5235i 7.38521 + 3.05905i −0.118628 − 3.54099i
3.6 −2.14548 + 1.84307i 0.00627997 − 0.0315715i 1.20619 − 7.90855i 1.51856 + 2.27269i 0.0447150 + 0.0793106i 12.1586 + 8.12410i 11.9881 + 19.1907i 24.9438 + 10.3321i −7.44677 − 2.07720i
3.7 −2.07130 + 1.92606i 1.92718 − 9.68857i 0.580595 − 7.97890i 5.30883 + 7.94522i 14.6690 + 23.7798i −24.0722 − 16.0845i 14.1653 + 17.6450i −65.2096 − 27.0107i −26.2992 − 6.23185i
3.8 −1.99137 + 2.00859i −1.20209 + 6.04333i −0.0688860 − 7.99970i −8.22961 − 12.3165i −9.74477 − 14.4490i −10.5599 − 7.05589i 16.2053 + 15.7920i −10.1320 − 4.19683i 41.1270 + 7.99675i
3.9 −1.88006 − 2.11315i −0.641350 + 3.22428i −0.930778 + 7.94567i −7.74711 − 11.5944i 8.01916 − 4.70657i 20.6562 + 13.8021i 18.5403 − 12.9714i 14.9601 + 6.19666i −9.93560 + 38.1688i
3.10 −0.924023 − 2.67323i 1.30103 − 6.54070i −6.29236 + 4.94026i −3.57163 − 5.34532i −18.6870 + 2.56581i −8.61854 − 5.75873i 19.0208 + 12.2560i −16.1433 − 6.68679i −10.9890 + 14.4870i
3.11 −0.650626 + 2.75258i −1.14955 + 5.77917i −7.15337 − 3.58180i 11.5468 + 17.2810i −15.1597 − 6.92430i −13.1862 − 8.81073i 14.5133 − 17.3598i −7.13263 − 2.95443i −55.0798 + 20.5399i
3.12 −0.644373 + 2.75405i 0.850520 − 4.27585i −7.16957 − 3.54927i 0.695925 + 1.04153i 11.2279 + 5.09762i 24.4804 + 16.3572i 14.3947 − 17.4583i 7.38521 + 3.05905i −3.31685 + 1.24548i
3.13 −0.318751 − 2.81041i −0.510458 + 2.56625i −7.79680 + 1.79164i 7.31816 + 10.9524i 7.37491 + 0.616601i 3.91038 + 2.61283i 7.52049 + 21.3411i 18.6197 + 7.71253i 28.4480 − 24.0581i
3.14 0.164820 + 2.82362i 0.641350 − 3.22428i −7.94567 + 0.930778i −7.74711 − 11.5944i 9.20986 + 1.27950i −20.6562 − 13.8021i −3.93777 − 22.2821i 14.9601 + 6.19666i 31.4612 − 23.7859i
3.15 0.615325 − 2.76068i −1.82896 + 9.19480i −7.24275 − 3.39743i −7.91861 − 11.8510i 24.2585 + 10.7070i −6.40398 − 4.27900i −13.8359 + 17.9044i −56.2545 − 23.3014i −37.5895 + 14.5685i
3.16 1.23688 + 2.54365i −1.30103 + 6.54070i −4.94026 + 6.29236i −3.57163 − 5.34532i −18.2464 + 4.78070i 8.61854 + 5.75873i −22.1160 − 4.78339i −16.1433 − 6.68679i 9.17892 − 15.6965i
3.17 1.27916 − 2.52265i 0.324395 − 1.63084i −4.72750 − 6.45374i −2.36912 − 3.54564i −3.69909 − 2.90445i −17.9945 − 12.0236i −22.3277 + 3.67045i 22.3903 + 9.27438i −11.9749 + 1.44102i
3.18 1.62397 − 2.31575i 1.59000 − 7.99347i −2.72542 − 7.52144i 2.22957 + 3.33678i −15.9288 − 16.6632i 22.4666 + 15.0117i −21.8438 − 5.90324i −36.4227 − 15.0868i 11.3479 + 0.255724i
3.19 1.76187 + 2.21265i 0.510458 − 2.56625i −1.79164 + 7.79680i 7.31816 + 10.9524i 6.57756 − 3.39192i −3.91038 − 2.61283i −20.4082 + 9.77264i 18.6197 + 7.71253i −11.3402 + 35.4892i
3.20 2.24272 − 1.72343i −1.05408 + 5.29922i 2.05956 − 7.73034i 5.83314 + 8.72991i 6.76884 + 13.7013i 13.6382 + 9.11275i −8.70370 − 20.8865i −2.02590 − 0.839155i 28.1275 + 9.52571i
See next 80 embeddings (of 192 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 3.24
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
17.e odd 16 1 inner
68.i even 16 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 68.4.i.b ✓ 192
4.b odd 2 1 inner 68.4.i.b ✓ 192
17.e odd 16 1 inner 68.4.i.b ✓ 192
68.i even 16 1 inner 68.4.i.b ✓ 192
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
68.4.i.b ✓ 192 1.a even 1 1 trivial
68.4.i.b ✓ 192 4.b odd 2 1 inner
68.4.i.b ✓ 192 17.e odd 16 1 inner
68.4.i.b ✓ 192 68.i even 16 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{192} + 8 T_{3}^{190} - 2264 T_{3}^{188} + 155856 T_{3}^{186} + 3955616 T_{3}^{184} + \cdots + 84\!\cdots\!24 \) acting on \(S_{4}^{\mathrm{new}}(68, [\chi])\). Copy content Toggle raw display