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Properties

Label 68.5.f.c
Level $68$
Weight $5$
Character orbit 68.f
Analytic conductor $7.029$
Analytic rank $0$
Dimension $64$
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] = [68,5,Mod(47,68)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(68, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 1]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("68.47");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 68 = 2^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 68.f (of order \(4\), degree \(2\), 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: \(7.02915748970\)
Analytic rank: \(0\)
Dimension: \(64\)
Relative dimension: \(32\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 64 q + 60 q^{4} - 8 q^{5} + 2 q^{6}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 64 q + 60 q^{4} - 8 q^{5} + 2 q^{6} - 182 q^{10} - 70 q^{12} - 8 q^{13} - 820 q^{14} - 1980 q^{16} + 40 q^{17} - 1236 q^{18} - 666 q^{20} - 8 q^{21} - 550 q^{22} + 122 q^{24} + 2900 q^{28} + 672 q^{29} + 6008 q^{30} + 2712 q^{33} - 1978 q^{34} + 5720 q^{37} - 5168 q^{38} - 7814 q^{40} + 9792 q^{41} + 1734 q^{44} + 7680 q^{45} + 11232 q^{46} + 4646 q^{48} - 23356 q^{50} - 5516 q^{52} + 17552 q^{54} + 3668 q^{56} + 7064 q^{57} - 1146 q^{58} - 15144 q^{61} - 15508 q^{62} + 4764 q^{64} - 28232 q^{65} - 38358 q^{68} - 43464 q^{69} + 74556 q^{72} - 23456 q^{73} + 48118 q^{74} - 6332 q^{78} - 51662 q^{80} + 19032 q^{81} - 37672 q^{82} + 23272 q^{84} - 7680 q^{85} + 148 q^{86} - 13398 q^{88} - 8 q^{89} + 27586 q^{90} + 35832 q^{92} + 17942 q^{96} + 30840 q^{97} + 50464 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
47.1 −3.98502 − 0.345886i 3.51617 − 3.51617i 15.7607 + 2.75672i 13.4656 + 13.4656i −15.2282 + 12.7958i −4.62414 − 4.62414i −61.8533 − 16.4370i 56.2731i −49.0030 − 58.3181i
47.2 −3.92597 − 0.766025i 9.54379 − 9.54379i 14.8264 + 6.01478i −33.6627 − 33.6627i −44.7794 + 30.1578i 1.58620 + 1.58620i −53.6005 − 34.9712i − 101.168i 106.372 + 157.945i
47.3 −3.90521 + 0.865634i −9.42696 + 9.42696i 14.5014 − 6.76097i −17.7572 − 17.7572i 28.6540 − 44.9746i −25.6088 − 25.6088i −50.7783 + 38.9559i − 96.7350i 84.7170 + 53.9744i
47.4 −3.76902 − 1.33960i −5.76676 + 5.76676i 12.4110 + 10.0979i −7.67744 − 7.67744i 29.4601 − 14.0099i 54.5230 + 54.5230i −33.2500 − 54.6849i 14.4890i 18.6517 + 39.2211i
47.5 −3.51705 + 1.90535i 2.07801 − 2.07801i 8.73927 − 13.4024i −2.97668 − 2.97668i −3.34913 + 11.2678i −25.0243 − 25.0243i −5.20009 + 63.7884i 72.3637i 16.1408 + 4.79751i
47.6 −3.37555 − 2.14608i −6.85220 + 6.85220i 6.78872 + 14.4884i 23.2592 + 23.2592i 37.8353 − 8.42462i −36.1271 − 36.1271i 8.17751 − 63.4754i − 12.9053i −28.5966 − 128.429i
47.7 −3.29849 + 2.26274i −8.74819 + 8.74819i 5.76001 − 14.9272i 32.8116 + 32.8116i 9.06089 − 48.6507i 52.2468 + 52.2468i 14.7771 + 62.2707i − 72.0618i −182.473 − 33.9845i
47.8 −3.25824 + 2.32032i 12.1530 − 12.1530i 5.23226 − 15.1203i 17.2087 + 17.2087i −11.3986 + 67.7960i 27.6148 + 27.6148i 18.0359 + 61.4061i − 214.389i −95.9996 − 16.1404i
47.9 −2.49249 − 3.12850i 6.29457 − 6.29457i −3.57497 + 15.5955i 9.38077 + 9.38077i −35.3817 − 4.00337i 67.5840 + 67.5840i 57.7010 − 27.6874i 1.75685i 5.96621 − 52.7292i
47.10 −2.49175 − 3.12909i −3.79141 + 3.79141i −3.58239 + 15.5938i −21.4859 − 21.4859i 21.3109 + 2.41642i −13.9370 − 13.9370i 57.7208 − 27.6462i 52.2504i −13.6938 + 120.769i
47.11 −2.41073 + 3.19193i 0.888075 − 0.888075i −4.37679 − 15.3897i −26.7949 − 26.7949i 0.693764 + 4.97558i 56.9418 + 56.9418i 59.6741 + 23.1300i 79.4226i 150.123 − 20.9322i
47.12 −2.39446 − 3.20415i 7.92868 − 7.92868i −4.53315 + 15.3444i 1.66273 + 1.66273i −44.3895 − 6.41979i −51.8270 − 51.8270i 60.0202 − 22.2166i − 44.7278i 1.34630 − 9.30897i
47.13 −1.73734 + 3.60301i −6.27640 + 6.27640i −9.96333 − 12.5193i 2.37435 + 2.37435i −11.7097 − 33.5181i −40.6243 − 40.6243i 62.4167 − 14.1478i 2.21365i −12.6798 + 4.42975i
47.14 −1.23798 + 3.80361i 3.17299 − 3.17299i −12.9348 − 9.41754i 20.3915 + 20.3915i 8.14071 + 15.9969i −15.9772 − 15.9772i 51.8336 − 37.5403i 60.8643i −102.805 + 52.3170i
47.15 −0.952257 − 3.88500i −11.6878 + 11.6878i −14.1864 + 7.39904i 1.72347 + 1.72347i 56.5368 + 34.2772i 9.22067 + 9.22067i 42.2544 + 48.0684i − 192.209i 5.05448 − 8.33685i
47.16 −0.257060 + 3.99173i 9.28543 − 9.28543i −15.8678 − 2.05223i −13.9231 − 13.9231i 34.6780 + 39.4518i −23.1941 − 23.1941i 12.2710 − 62.8126i − 91.4383i 59.1563 − 51.9981i
47.17 0.257060 + 3.99173i −9.28543 + 9.28543i −15.8678 + 2.05223i −13.9231 − 13.9231i −39.4518 − 34.6780i 23.1941 + 23.1941i −12.2710 − 62.8126i − 91.4383i 51.9981 − 59.1563i
47.18 0.952257 − 3.88500i 11.6878 − 11.6878i −14.1864 − 7.39904i 1.72347 + 1.72347i −34.2772 − 56.5368i −9.22067 − 9.22067i −42.2544 + 48.0684i − 192.209i 8.33685 − 5.05448i
47.19 1.23798 + 3.80361i −3.17299 + 3.17299i −12.9348 + 9.41754i 20.3915 + 20.3915i −15.9969 − 8.14071i 15.9772 + 15.9772i −51.8336 − 37.5403i 60.8643i −52.3170 + 102.805i
47.20 1.73734 + 3.60301i 6.27640 − 6.27640i −9.96333 + 12.5193i 2.37435 + 2.37435i 33.5181 + 11.7097i 40.6243 + 40.6243i −62.4167 − 14.1478i 2.21365i −4.42975 + 12.6798i
See all 64 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 47.32
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
17.c even 4 1 inner
68.f odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 68.5.f.c ✓ 64
4.b odd 2 1 inner 68.5.f.c ✓ 64
17.c even 4 1 inner 68.5.f.c ✓ 64
68.f odd 4 1 inner 68.5.f.c ✓ 64
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
68.5.f.c ✓ 64 1.a even 1 1 trivial
68.5.f.c ✓ 64 4.b odd 2 1 inner
68.5.f.c ✓ 64 17.c even 4 1 inner
68.5.f.c ✓ 64 68.f odd 4 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{5}^{\mathrm{new}}(68, [\chi])\):

\( T_{3}^{64} + 323292 T_{3}^{60} + 43687087696 T_{3}^{56} + \cdots + 43\!\cdots\!00 \) Copy content Toggle raw display
\( T_{5}^{32} + 4 T_{5}^{31} + 8 T_{5}^{30} - 15064 T_{5}^{29} + 7752216 T_{5}^{28} + 9459344 T_{5}^{27} + \cdots + 41\!\cdots\!04 \) Copy content Toggle raw display