↑

Properties

Label 245.4.s.a
Level $245$
Weight $4$
Character orbit 245.s
Analytic conductor $14.455$
Analytic rank $0$
Dimension $984$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [245,4,Mod(13,245)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(245, base_ring=CyclotomicField(28))
 
chi = DirichletCharacter(H, H._module([21, 22]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("245.13");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 245 = 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 245.s (of order \(28\), degree \(12\), 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: \(14.4554679514\)
Analytic rank: \(0\)
Dimension: \(984\)
Relative dimension: \(82\) over \(\Q(\zeta_{28})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{28}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 984 q - 10 q^{2} - 14 q^{3} - 14 q^{5} + 28 q^{6} - 4 q^{7} - 126 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 984 q - 10 q^{2} - 14 q^{3} - 14 q^{5} + 28 q^{6} - 4 q^{7} - 126 q^{8} - 14 q^{10} + 36 q^{11} - 14 q^{12} - 14 q^{13} - 262 q^{15} + 2524 q^{16} - 462 q^{17} - 164 q^{18} - 14 q^{20} - 496 q^{21} + 862 q^{22} + 102 q^{23} + 214 q^{25} + 1036 q^{26} - 14 q^{27} + 514 q^{28} - 860 q^{30} + 1446 q^{32} - 14 q^{33} - 104 q^{35} + 2972 q^{36} - 346 q^{37} - 126 q^{38} - 14 q^{40} + 196 q^{41} - 2690 q^{42} - 738 q^{43} + 3234 q^{45} - 1756 q^{46} + 70 q^{47} - 11056 q^{50} - 1700 q^{51} - 14 q^{52} + 1390 q^{53} - 1106 q^{55} - 2004 q^{56} - 3864 q^{57} - 1386 q^{58} - 290 q^{60} + 6020 q^{61} + 98 q^{62} - 1844 q^{63} - 1858 q^{65} - 3360 q^{66} - 3888 q^{67} - 11342 q^{70} - 244 q^{71} + 7308 q^{72} - 14 q^{73} + 10150 q^{75} - 1820 q^{76} + 4010 q^{77} - 10226 q^{78} + 1404 q^{81} + 12166 q^{82} + 6510 q^{83} + 46 q^{85} + 10676 q^{86} + 6622 q^{87} + 5012 q^{88} + 13930 q^{90} + 7884 q^{91} + 3658 q^{92} - 10940 q^{93} - 6636 q^{95} + 11816 q^{96} - 12072 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
13.1 −4.73481 − 2.97508i 5.92807 + 0.667933i 10.0963 + 20.9652i 9.19520 − 6.35990i −26.0812 − 20.7990i 13.9761 − 12.1519i 9.56023 − 84.8494i 8.37284 + 1.91105i −62.4588 + 2.75647i
13.2 −4.62468 − 2.90588i 4.62710 + 0.521349i 9.47246 + 19.6698i −11.0371 + 1.78401i −19.8839 − 15.8569i 2.54795 + 18.3442i 8.45859 − 75.0721i −5.18482 − 1.18340i 56.2271 + 23.8219i
13.3 −4.53454 − 2.84924i −8.89819 − 1.00259i 8.97279 + 18.6322i 9.66406 + 5.62192i 37.4926 + 29.8993i 17.5428 + 5.93719i 7.60319 − 67.4802i 51.8496 + 11.8343i −27.8038 − 53.0280i
13.4 −4.53116 − 2.84712i −2.73948 − 0.308665i 8.95426 + 18.5937i 1.41115 + 11.0909i 11.5342 + 9.19823i −17.1097 − 7.08923i 7.57195 − 67.2029i −18.9136 − 4.31690i 25.1830 − 54.2724i
13.5 −4.39700 − 2.76282i −5.74692 − 0.647522i 8.22938 + 17.0885i −6.64767 − 8.98936i 23.4802 + 18.7249i −17.6761 − 5.52777i 6.37638 − 56.5919i 6.28476 + 1.43446i 4.39384 + 57.8925i
13.6 −4.22151 − 2.65255i −1.91523 − 0.215795i 7.31407 + 15.1878i 8.80027 − 6.89604i 7.51276 + 5.99123i −7.88293 + 16.7589i 4.94429 − 43.8818i −22.7015 − 5.18147i −55.4426 + 5.76854i
13.7 −4.05412 − 2.54737i −1.02566 − 0.115564i 6.47569 + 13.4469i −10.3611 − 4.20088i 3.86375 + 3.08124i 16.3615 − 8.67773i 3.71237 − 32.9482i −25.2844 − 5.77101i 31.3039 + 43.4244i
13.8 −3.97521 − 2.49779i −4.39514 − 0.495213i 6.09229 + 12.6508i −6.02668 + 9.41696i 16.2347 + 12.9467i 14.4459 − 11.5895i 3.17562 − 28.1844i −7.25103 − 1.65500i 47.4790 − 22.3810i
13.9 −3.92611 − 2.46694i 5.38471 + 0.606712i 5.85748 + 12.1632i 9.01924 + 6.60706i −19.6443 − 15.6658i −18.4196 + 1.92869i 2.85543 − 25.3427i 2.30399 + 0.525871i −19.1113 − 48.1899i
13.10 −3.84825 − 2.41801i 9.49222 + 1.06952i 5.49115 + 11.4025i −9.20513 + 6.34551i −33.9423 − 27.0681i −9.65689 − 15.8033i 2.36917 − 21.0269i 62.6352 + 14.2961i 50.7672 − 2.16096i
13.11 −3.80488 − 2.39076i −8.95302 − 1.00876i 5.29029 + 10.9854i −10.6264 − 3.47562i 31.6534 + 25.2428i 5.58610 + 17.6577i 2.10955 − 18.7228i 52.8158 + 12.0549i 32.1227 + 38.6295i
13.12 −3.72699 − 2.34183i 3.96746 + 0.447025i 4.93527 + 10.2482i −3.31558 − 10.6774i −13.7398 − 10.9572i −9.11254 − 16.1233i 1.66312 − 14.7606i −10.7821 − 2.46095i −12.6475 + 47.5591i
13.13 −3.65674 − 2.29768i 6.79992 + 0.766168i 4.62132 + 9.59628i 4.51987 + 10.2260i −23.1051 − 18.4257i 17.2423 + 6.76040i 1.28190 − 11.3771i 19.3289 + 4.41170i 6.96806 − 47.7790i
13.14 −3.58884 − 2.25502i 9.31526 + 1.04958i 4.32361 + 8.97807i 1.46878 − 11.0834i −31.0642 − 24.7729i −6.58724 + 17.3092i 0.932472 − 8.27592i 59.3493 + 13.5461i −30.2646 + 36.4646i
13.15 −3.46930 − 2.17990i −2.35902 − 0.265797i 3.81297 + 7.91771i 4.93486 − 10.0323i 7.60471 + 6.06456i 15.2803 + 10.4648i 0.361498 − 3.20838i −20.8287 − 4.75403i −38.9899 + 24.0475i
13.16 −3.43366 − 2.15751i −8.68653 − 0.978736i 3.66411 + 7.60860i 9.75148 − 5.46888i 27.7150 + 22.1020i −12.0567 − 14.0583i 0.202009 − 1.79288i 48.1748 + 10.9956i −45.2825 − 2.26064i
13.17 −3.29403 − 2.06977i −0.763810 − 0.0860607i 3.09558 + 6.42804i 10.8054 + 2.87115i 2.33788 + 1.86440i 4.97154 − 17.8405i −0.376954 + 3.34556i −25.7471 − 5.87660i −29.6506 − 31.8224i
13.18 −3.29339 − 2.06938i −0.786663 − 0.0886356i 3.09305 + 6.42278i −3.92569 + 10.4685i 2.40737 + 1.91981i 2.03213 + 18.4084i −0.379426 + 3.36749i −25.7121 − 5.86861i 34.5920 − 26.3531i
13.19 −2.81364 − 1.76792i −8.12264 − 0.915201i 1.31992 + 2.74084i −8.89181 + 6.77758i 21.2361 + 16.9352i −9.38626 − 15.9655i −1.84461 + 16.3713i 38.8166 + 8.85963i 37.0006 − 3.34960i
13.20 −2.70548 − 1.69997i 1.67406 + 0.188622i 0.958681 + 1.99072i −10.7105 + 3.20701i −4.20850 − 3.35617i −18.3635 + 2.40436i −2.07156 + 18.3856i −23.5561 − 5.37654i 34.4289 + 9.53101i
See next 80 embeddings (of 984 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 13.82
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
49.f odd 14 1 inner
245.s even 28 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 245.4.s.a ✓ 984
5.c odd 4 1 inner 245.4.s.a ✓ 984
49.f odd 14 1 inner 245.4.s.a ✓ 984
245.s even 28 1 inner 245.4.s.a ✓ 984
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
245.4.s.a ✓ 984 1.a even 1 1 trivial
245.4.s.a ✓ 984 5.c odd 4 1 inner
245.4.s.a ✓ 984 49.f odd 14 1 inner
245.4.s.a ✓ 984 245.s even 28 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(245, [\chi])\).