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Properties

Label 224.4.u.a
Level $224$
Weight $4$
Character orbit 224.u
Analytic conductor $13.216$
Analytic rank $0$
Dimension $140$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [224,4,Mod(29,224)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(224, base_ring=CyclotomicField(8))
 
chi = DirichletCharacter(H, H._module([0, 3, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("224.29");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 224 = 2^{5} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 224.u (of order \(8\), degree \(4\), 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: \(13.2164278413\)
Analytic rank: \(0\)
Dimension: \(140\)
Relative dimension: \(35\) over \(\Q(\zeta_{8})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{8}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 140 q - 20 q^{6}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 140 q - 20 q^{6} - 120 q^{10} - 92 q^{12} + 56 q^{16} - 360 q^{18} - 824 q^{22} - 244 q^{23} - 456 q^{24} - 20 q^{26} + 264 q^{27} + 168 q^{28} - 580 q^{30} + 620 q^{32} + 1584 q^{33} + 312 q^{34} - 1240 q^{36} - 212 q^{38} - 600 q^{39} + 820 q^{40} + 196 q^{43} - 860 q^{44} + 2680 q^{45} + 1440 q^{46} + 1964 q^{48} + 1196 q^{50} + 1488 q^{51} + 864 q^{52} - 2028 q^{53} - 2144 q^{54} + 288 q^{55} + 140 q^{56} - 4932 q^{58} + 388 q^{60} + 1824 q^{61} - 3420 q^{62} + 7812 q^{63} - 48 q^{64} - 5200 q^{65} - 3440 q^{66} + 460 q^{67} - 2516 q^{68} + 1056 q^{69} + 504 q^{70} - 4888 q^{72} + 2880 q^{74} + 2208 q^{75} - 1504 q^{76} - 1092 q^{77} + 2532 q^{78} + 9500 q^{80} + 3984 q^{82} + 1728 q^{85} - 3500 q^{86} + 1288 q^{87} + 4008 q^{88} - 7980 q^{90} - 3276 q^{92} - 14108 q^{94} + 6080 q^{95} - 7496 q^{96} + 6984 q^{97} + 5312 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
29.1 −2.82701 − 0.0894526i 2.36840 + 0.981025i 7.98400 + 0.505767i 2.98710 + 7.21150i −6.60775 − 2.98523i 4.94975 − 4.94975i −22.5256 − 2.14400i −14.4450 − 14.4450i −7.79949 − 20.6542i
29.2 −2.82267 − 0.180425i 8.94755 + 3.70620i 7.93489 + 1.01856i −5.50821 − 13.2980i −24.5873 − 12.0757i 4.94975 − 4.94975i −22.2138 − 4.30670i 47.2309 + 47.2309i 13.1486 + 38.5296i
29.3 −2.81375 − 0.287765i −8.01711 − 3.32080i 7.83438 + 1.61940i −2.55421 − 6.16641i 21.6025 + 11.6509i 4.94975 − 4.94975i −21.5780 − 6.81103i 34.1545 + 34.1545i 5.41244 + 18.0857i
29.4 −2.67774 − 0.910887i 0.836532 + 0.346503i 6.34057 + 4.87824i 2.18623 + 5.27803i −1.92439 − 1.68983i 4.94975 − 4.94975i −12.5349 − 18.8382i −18.5122 − 18.5122i −1.04647 − 16.1246i
29.5 −2.62799 + 1.04577i −6.58597 − 2.72800i 5.81271 − 5.49658i 7.52198 + 18.1597i 20.1608 + 0.281723i 4.94975 − 4.94975i −9.52759 + 20.5238i 16.8412 + 16.8412i −38.7586 − 39.8572i
29.6 −2.56089 + 1.20077i 7.06644 + 2.92701i 5.11632 − 6.15006i 3.96866 + 9.58119i −21.6110 + 0.989380i 4.94975 − 4.94975i −5.71754 + 21.8931i 22.2752 + 22.2752i −21.6681 − 19.7709i
29.7 −2.33082 − 1.60227i 2.54429 + 1.05388i 2.86545 + 7.46922i −6.41104 − 15.4776i −4.24168 − 6.53304i 4.94975 − 4.94975i 5.28887 − 22.0006i −13.7292 − 13.7292i −9.85637 + 46.3478i
29.8 −2.27517 + 1.68036i 0.692049 + 0.286656i 2.35276 − 7.64621i −5.63604 − 13.6066i −2.05621 + 0.510703i 4.94975 − 4.94975i 7.49548 + 21.3499i −18.6951 − 18.6951i 35.6870 + 21.4867i
29.9 −2.14397 − 1.84483i −6.50322 − 2.69372i 1.19322 + 7.91051i 0.166115 + 0.401037i 8.97325 + 17.7726i 4.94975 − 4.94975i 12.0353 − 19.1612i 15.9438 + 15.9438i 0.383699 − 1.16627i
29.10 −1.91298 + 2.08338i −8.05349 − 3.33586i −0.680979 − 7.97096i −5.74420 − 13.8677i 22.3561 − 10.3971i 4.94975 − 4.94975i 17.9093 + 13.8296i 34.6388 + 34.6388i 39.8804 + 14.5614i
29.11 −1.75433 + 2.21863i −1.70574 − 0.706540i −1.84463 − 7.78443i 3.11491 + 7.52005i 4.55998 − 2.54489i 4.94975 − 4.94975i 20.5069 + 9.56393i −16.6815 − 16.6815i −22.1488 − 6.28185i
29.12 −1.52380 − 2.38286i 2.12512 + 0.880255i −3.35607 + 7.26201i 8.08907 + 19.5287i −1.14074 − 6.40521i 4.94975 − 4.94975i 22.4184 − 3.06880i −15.3506 − 15.3506i 34.2082 − 49.0330i
29.13 −1.25291 + 2.53579i 5.65221 + 2.34122i −4.86044 − 6.35422i −4.14360 − 10.0035i −13.0186 + 11.3995i 4.94975 − 4.94975i 22.2026 − 4.36379i 7.37430 + 7.37430i 30.5584 + 2.02622i
29.14 −1.12766 − 2.59391i −3.03773 − 1.25827i −5.45677 + 5.85010i −3.05117 − 7.36618i 0.161680 + 9.29850i 4.94975 − 4.94975i 21.3280 + 7.55749i −11.4473 − 11.4473i −15.6666 + 16.2210i
29.15 −0.791489 − 2.71543i 8.59902 + 3.56183i −6.74709 + 4.29846i −1.15499 − 2.78839i 2.86587 − 26.1692i 4.94975 − 4.94975i 17.0124 + 14.9191i 42.1647 + 42.1647i −6.65750 + 5.34326i
29.16 −0.461524 + 2.79052i −4.06210 − 1.68257i −7.57399 − 2.57578i 4.51885 + 10.9095i 6.57001 − 10.5588i 4.94975 − 4.94975i 10.6834 − 19.9466i −5.42232 − 5.42232i −32.5286 + 7.57495i
29.17 −0.446664 + 2.79294i 3.90182 + 1.61619i −7.60098 − 2.49501i −1.21986 − 2.94501i −6.25670 + 10.1756i 4.94975 − 4.94975i 10.3635 − 20.1146i −6.47977 − 6.47977i 8.77009 − 2.09157i
29.18 0.0170002 − 2.82838i −6.98661 − 2.89395i −7.99942 − 0.0961657i 3.91954 + 9.46261i −8.30395 + 19.7116i 4.94975 − 4.94975i −0.407984 + 22.6237i 21.3459 + 21.3459i 26.8305 − 10.9251i
29.19 0.153695 − 2.82425i 3.46993 + 1.43729i −7.95276 − 0.868145i 1.61982 + 3.91060i 4.59258 − 9.57903i 4.94975 − 4.94975i −3.67415 + 22.3271i −9.11729 − 9.11729i 11.2935 − 3.97374i
29.20 0.390074 + 2.80140i −5.73255 − 2.37450i −7.69569 + 2.18550i −3.87321 − 9.35077i 4.41581 − 16.9854i 4.94975 − 4.94975i −9.12436 − 20.7062i 8.13200 + 8.13200i 24.6844 − 14.4979i
See next 80 embeddings (of 140 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 29.35
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
32.g even 8 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 224.4.u.a ✓ 140
32.g even 8 1 inner 224.4.u.a ✓ 140
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
224.4.u.a ✓ 140 1.a even 1 1 trivial
224.4.u.a ✓ 140 32.g even 8 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{140} - 88 T_{3}^{137} + 7128 T_{3}^{135} + 8168 T_{3}^{134} + 1462472 T_{3}^{133} + \cdots + 21\!\cdots\!52 \) acting on \(S_{4}^{\mathrm{new}}(224, [\chi])\). Copy content Toggle raw display