Properties

Label 68.6.i.b
Level $68$
Weight $6$
Character orbit 68.i
Analytic conductor $10.906$
Analytic rank $0$
Dimension $336$
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] = [68,6,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, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("68.3");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 68 = 2^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 6 \)
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: \(10.9060997473\)
Analytic rank: \(0\)
Dimension: \(336\)
Relative dimension: \(42\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 336 q - 8 q^{2} - 8 q^{4} - 16 q^{5} - 8 q^{6} - 8 q^{8} - 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 336 q - 8 q^{2} - 8 q^{4} - 16 q^{5} - 8 q^{6} - 8 q^{8} - 16 q^{9} - 104 q^{10} - 8 q^{12} - 16 q^{13} - 8 q^{14} - 16 q^{17} - 16 q^{18} - 10504 q^{20} - 16 q^{21} - 8 q^{22} + 20872 q^{24} - 24944 q^{25} - 39048 q^{26} - 33176 q^{28} + 34240 q^{29} + 10936 q^{30} + 37152 q^{32} + 67512 q^{34} + 25912 q^{36} - 16 q^{37} - 31248 q^{38} - 120264 q^{40} + 19792 q^{41} - 90296 q^{42} + 19352 q^{44} + 73856 q^{45} + 117672 q^{46} - 1952 q^{48} - 16 q^{49} - 16 q^{52} + 109648 q^{53} - 162024 q^{54} + 102032 q^{56} - 308096 q^{57} + 233456 q^{58} + 293168 q^{60} - 50288 q^{61} + 10128 q^{62} - 117560 q^{64} + 518656 q^{65} - 432288 q^{66} - 285632 q^{68} + 252704 q^{69} - 271760 q^{70} - 291376 q^{72} - 486176 q^{73} + 159240 q^{74} + 343328 q^{76} - 229488 q^{77} + 789248 q^{78} + 338536 q^{80} - 212096 q^{81} + 176000 q^{82} - 8240 q^{85} - 2336 q^{86} - 531912 q^{88} - 16 q^{89} - 1830792 q^{90} - 821080 q^{92} - 16 q^{93} - 247784 q^{94} + 401424 q^{96} - 16 q^{97} + 1495120 q^{98}+O(q^{100}) \) Copy content Toggle raw display

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 −5.65491 + 0.148136i 2.81032 14.1284i 31.9561 1.67539i 52.9005 + 79.1711i −13.7992 + 80.3112i −67.2962 44.9659i −180.461 + 14.2081i 32.7887 + 13.5815i −310.876 439.870i
3.2 −5.52731 1.20368i −2.20628 + 11.0917i 29.1023 + 13.3062i 19.3762 + 28.9985i 25.5456 58.6516i 61.4747 + 41.0761i −144.841 108.577i 106.345 + 44.0494i −72.1934 183.606i
3.3 −5.45006 1.51553i 0.798157 4.01261i 27.4063 + 16.5195i −17.6385 26.3979i −10.4312 + 20.6593i 40.6237 + 27.1439i −124.330 131.567i 209.039 + 86.5867i 56.1240 + 170.602i
3.4 −5.38279 + 1.73940i −3.80789 + 19.1436i 25.9489 18.7257i 8.73287 + 13.0697i −12.8013 109.669i −126.682 84.6465i −107.106 + 145.932i −127.474 52.8014i −69.7407 55.1613i
3.5 −5.27063 + 2.05438i 2.66275 13.3865i 23.5590 21.6557i −46.2821 69.2661i 13.4667 + 76.0257i −190.959 127.595i −79.6819 + 162.539i 52.3939 + 21.7023i 386.235 + 269.995i
3.6 −5.17956 + 2.27423i −2.66275 + 13.3865i 21.6557 23.5590i −46.2821 69.2661i −16.6522 75.3920i 190.959 + 127.595i −58.5886 + 171.276i 52.3939 + 21.7023i 397.248 + 253.512i
3.7 −5.09611 2.45554i 5.95993 29.9626i 19.9406 + 25.0274i −10.4932 15.7042i −103.947 + 138.058i −33.7942 22.5806i −40.1634 176.508i −637.733 264.158i 14.9122 + 105.797i
3.8 −5.03616 + 2.57627i 3.80789 19.1436i 18.7257 25.9489i 8.73287 + 13.0697i 30.1418 + 106.220i 126.682 + 84.6465i −27.4543 + 178.925i −127.474 52.8014i −77.6510 43.3227i
3.9 −5.02704 2.59401i −5.46638 + 27.4813i 18.5423 + 26.0804i −42.6300 63.8003i 98.7665 123.970i −87.0442 58.1611i −25.5600 179.206i −500.839 207.455i 48.8043 + 431.309i
3.10 −4.10338 + 3.89388i −2.81032 + 14.1284i 1.67539 31.9561i 52.9005 + 79.1711i −43.4825 68.9172i 67.2962 + 44.9659i 117.558 + 137.652i 32.7887 + 13.5815i −525.353 118.881i
3.11 −3.93418 4.06475i −3.88366 + 19.5245i −1.04441 + 31.9830i 41.6932 + 62.3983i 94.6412 61.0268i 56.6616 + 37.8601i 134.112 121.582i −141.620 58.6609i 89.6048 414.959i
3.12 −3.63352 4.33561i 0.619775 3.11582i −5.59506 + 31.5071i −48.1454 72.0546i −15.7610 + 8.63429i −62.2966 41.6253i 156.932 90.2236i 215.179 + 89.1299i −137.464 + 470.552i
3.13 −3.56884 4.38900i 1.15115 5.78724i −6.52670 + 31.3273i 36.5596 + 54.7153i −29.5085 + 15.6013i −203.040 135.667i 160.789 83.1567i 192.336 + 79.6681i 109.670 355.730i
3.14 −3.05727 + 4.75953i 2.20628 11.0917i −13.3062 29.1023i 19.3762 + 28.9985i 46.0460 + 44.4112i −61.4747 41.0761i 179.194 + 25.6427i 106.345 + 44.0494i −197.257 + 3.56512i
3.15 −2.98386 4.80589i 2.84546 14.3051i −14.1932 + 28.6802i 8.63595 + 12.9246i −77.2393 + 29.0094i 163.604 + 109.317i 180.184 17.3667i 27.9632 + 11.5827i 36.3458 80.0686i
3.16 −2.78213 + 4.92542i −0.798157 + 4.01261i −16.5195 27.4063i −17.6385 26.3979i −17.5432 15.0949i −40.6237 27.1439i 180.947 5.11734i 209.039 + 86.5867i 179.093 13.4346i
3.17 −1.86716 + 5.33982i −5.95993 + 29.9626i −25.0274 19.9406i −10.4932 15.7042i −148.867 87.7699i 33.7942 + 22.5806i 153.209 96.4098i −637.733 264.158i 103.450 26.7097i
3.18 −1.72041 + 5.38889i 5.46638 27.4813i −26.0804 18.5423i −42.6300 63.8003i 138.690 + 76.7370i 87.0442 + 58.1611i 144.791 108.644i −500.839 207.455i 417.155 119.966i
3.19 −1.35886 5.49122i −3.94635 + 19.8396i −28.3070 + 14.9236i −16.9624 25.3860i 114.306 5.28894i 71.9049 + 48.0453i 120.414 + 135.161i −153.534 63.5959i −116.351 + 127.640i
3.20 −0.326342 5.64743i −1.90149 + 9.55941i −31.7870 + 3.68599i −4.87594 7.29735i 54.6067 + 7.61888i −106.546 71.1916i 31.1898 + 178.312i 136.736 + 56.6379i −39.6201 + 29.9180i
See next 80 embeddings (of 336 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 3.42
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 68.6.i.b 336
4.b odd 2 1 inner 68.6.i.b 336
17.e odd 16 1 inner 68.6.i.b 336
68.i even 16 1 inner 68.6.i.b 336
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
68.6.i.b 336 1.a even 1 1 trivial
68.6.i.b 336 4.b odd 2 1 inner
68.6.i.b 336 17.e odd 16 1 inner
68.6.i.b 336 68.i even 16 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{336} + 8 T_{3}^{334} + 53056 T_{3}^{332} + 47428896 T_{3}^{330} + 1783506176 T_{3}^{328} - 68136284889824 T_{3}^{326} + \cdots + 92\!\cdots\!64 \) acting on \(S_{6}^{\mathrm{new}}(68, [\chi])\). Copy content Toggle raw display