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Properties

Label 145.4.m.a
Level $145$
Weight $4$
Character orbit 145.m
Analytic conductor $8.555$
Analytic rank $0$
Dimension $84$
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] = [145,4,Mod(6,145)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(145, base_ring=CyclotomicField(14))
 
chi = DirichletCharacter(H, H._module([0, 3]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("145.6");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 145 = 5 \cdot 29 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 145.m (of order \(14\), degree \(6\), 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: \(8.55527695083\)
Analytic rank: \(0\)
Dimension: \(84\)
Relative dimension: \(14\) over \(\Q(\zeta_{14})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{14}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 84 q + 38 q^{4} - 70 q^{5} - 102 q^{6} + 10 q^{7} - 21 q^{8} + 204 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 84 q + 38 q^{4} - 70 q^{5} - 102 q^{6} + 10 q^{7} - 21 q^{8} + 204 q^{9} + 35 q^{10} - 70 q^{11} + 38 q^{13} - 378 q^{14} - 90 q^{16} + 365 q^{20} - 56 q^{21} - 229 q^{22} + 378 q^{23} + 944 q^{24} - 350 q^{25} - 840 q^{27} + 1184 q^{28} - 478 q^{29} + 330 q^{30} - 154 q^{31} - 658 q^{32} - 1130 q^{33} - 110 q^{34} + 50 q^{35} + 684 q^{36} + 924 q^{37} + 524 q^{38} + 952 q^{39} + 105 q^{40} + 1673 q^{42} - 1834 q^{43} + 1099 q^{44} + 1020 q^{45} + 1386 q^{47} - 2541 q^{48} - 1706 q^{49} + 24 q^{51} - 1675 q^{52} + 1454 q^{53} - 3777 q^{54} - 630 q^{55} - 5670 q^{56} + 2668 q^{57} - 2276 q^{58} - 2628 q^{59} - 1365 q^{60} - 812 q^{61} - 783 q^{62} + 156 q^{63} + 3285 q^{64} - 930 q^{65} + 8190 q^{66} - 3638 q^{67} + 9478 q^{68} + 2912 q^{69} + 552 q^{71} - 1855 q^{72} + 1344 q^{73} + 5477 q^{74} + 518 q^{76} - 2366 q^{77} - 3620 q^{78} + 7350 q^{79} + 950 q^{80} - 2046 q^{81} - 502 q^{82} - 1486 q^{83} + 4550 q^{84} + 1806 q^{86} - 7708 q^{87} - 12116 q^{88} - 7518 q^{89} + 5670 q^{90} - 2616 q^{91} + 1377 q^{92} + 6882 q^{93} - 12448 q^{94} - 5471 q^{96} + 9240 q^{97} + 4438 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
6.1 −4.19855 − 3.34824i −1.67091 − 0.381374i 4.63701 + 20.3161i 3.11745 − 3.90916i 5.73847 + 7.19582i 3.73404 − 16.3599i 29.9141 − 62.1172i −21.6797 − 10.4404i −26.1776 + 5.97486i
6.2 −3.56768 − 2.84513i 5.19257 + 1.18517i 2.85342 + 12.5016i 3.11745 − 3.90916i −15.1535 − 19.0018i −0.803171 + 3.51892i 9.54941 − 19.8295i 1.23195 + 0.593278i −22.2441 + 5.07708i
6.3 −2.64031 − 2.10558i −7.84143 − 1.78976i 0.757622 + 3.31936i 3.11745 − 3.90916i 16.9354 + 21.2363i −1.70562 + 7.47280i −6.73329 + 13.9818i 33.9587 + 16.3536i −16.4621 + 3.75736i
6.4 −2.57419 − 2.05285i −4.30470 − 0.982519i 0.632096 + 2.76939i 3.11745 − 3.90916i 9.06413 + 11.3661i 6.21730 − 27.2398i −7.37053 + 15.3051i −6.76109 − 3.25597i −16.0498 + 3.66326i
6.5 −2.18837 − 1.74517i 3.36689 + 0.768472i −0.0368043 − 0.161250i 3.11745 − 3.90916i −6.02691 − 7.55751i −7.16035 + 31.3716i −9.91652 + 20.5919i −13.5807 − 6.54014i −13.6443 + 3.11422i
6.6 −0.650602 − 0.518838i 2.21115 + 0.504681i −1.62608 − 7.12431i 3.11745 − 3.90916i −1.17673 − 1.47558i 3.15049 − 13.8032i −5.52689 + 11.4767i −19.6917 − 9.48301i −4.05644 + 0.925856i
6.7 −0.575541 − 0.458979i 9.89056 + 2.25745i −1.65958 − 7.27110i 3.11745 − 3.90916i −4.65630 − 5.83881i 1.61957 − 7.09579i −4.93733 + 10.2525i 68.4008 + 32.9401i −3.58844 + 0.819038i
6.8 0.791697 + 0.631357i −7.79761 − 1.77975i −1.55200 − 6.79974i 3.11745 − 3.90916i −5.04968 − 6.33210i −0.159037 + 0.696788i 6.57922 − 13.6619i 33.3090 + 16.0408i 4.93615 − 1.12664i
6.9 1.74286 + 1.38989i −2.16633 − 0.494450i −0.674380 − 2.95465i 3.11745 − 3.90916i −3.08838 − 3.87271i −3.99317 + 17.4952i 10.6690 − 22.1544i −19.8777 − 9.57258i 10.8666 − 2.48023i
6.10 2.05666 + 1.64013i 6.77737 + 1.54689i −0.240351 − 1.05305i 3.11745 − 3.90916i 11.4016 + 14.2972i −1.66064 + 7.27573i 10.3637 − 21.5204i 19.2137 + 9.25281i 12.8231 − 2.92678i
6.11 2.31952 + 1.84975i 0.890247 + 0.203193i 0.178406 + 0.781648i 3.11745 − 3.90916i 1.68909 + 2.11805i 7.30598 − 32.0096i 9.26584 − 19.2407i −23.5749 − 11.3531i 14.4620 − 3.30085i
6.12 3.40239 + 2.71332i −8.41155 − 1.91988i 2.43401 + 10.6641i 3.11745 − 3.90916i −23.4101 − 29.3554i 4.27709 − 18.7392i −5.54816 + 11.5209i 42.7421 + 20.5835i 21.2136 − 4.84186i
6.13 3.67920 + 2.93407i −3.67557 − 0.838925i 3.14762 + 13.7906i 3.11745 − 3.90916i −11.0617 − 13.8709i −6.98700 + 30.6121i −12.5475 + 26.0551i −11.5201 − 5.54781i 22.9395 − 5.23578i
6.14 3.92738 + 3.13198i 5.84729 + 1.33461i 3.83485 + 16.8016i 3.11745 − 3.90916i 18.7846 + 23.5551i 0.0817442 − 0.358145i −20.1251 + 41.7901i 8.08348 + 3.89280i 24.4868 − 5.58896i
51.1 −5.10783 + 1.16583i −2.38151 + 4.94525i 17.5230 − 8.43863i −1.11260 − 4.87464i 6.39901 − 28.0359i −5.25164 − 2.52905i −46.8971 + 37.3992i −1.94967 − 2.44481i 11.3660 + 23.6017i
51.2 −4.72714 + 1.07894i 1.92846 − 4.00449i 13.9740 − 6.72952i −1.11260 − 4.87464i −4.79551 + 21.0105i 18.1161 + 8.72426i −28.4693 + 22.7035i 4.51724 + 5.66444i 10.5189 + 21.8427i
51.3 −3.40276 + 0.776658i 3.78879 − 7.86751i 3.76784 − 1.81449i −1.11260 − 4.87464i −6.78199 + 29.7139i −12.7868 − 6.15781i 10.4186 − 8.30857i −30.7085 − 38.5073i 7.57186 + 15.7231i
51.4 −3.09893 + 0.707310i −3.67857 + 7.63862i 1.89531 − 0.912732i −1.11260 − 4.87464i 5.99673 − 26.2734i 7.62973 + 3.67428i 14.6533 − 11.6856i −27.9825 − 35.0889i 6.89576 + 14.3192i
51.5 −2.70318 + 0.616982i −0.904155 + 1.87750i −0.281259 + 0.135447i −1.11260 − 4.87464i 1.28571 − 5.63305i 10.4805 + 5.04713i 18.0190 − 14.3696i 14.1267 + 17.7144i 6.01513 + 12.4906i
51.6 −0.781696 + 0.178417i 0.624486 − 1.29676i −6.62854 + 3.19213i −1.11260 − 4.87464i −0.256794 + 1.12509i −15.8375 − 7.62692i 9.62694 − 7.67723i 15.5426 + 19.4898i 1.73944 + 3.61198i
See all 84 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 6.14
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
29.e even 14 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 145.4.m.a ✓ 84
29.e even 14 1 inner 145.4.m.a ✓ 84
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
145.4.m.a ✓ 84 1.a even 1 1 trivial
145.4.m.a ✓ 84 29.e even 14 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{84} - 75 T_{2}^{82} + 7 T_{2}^{81} + 3721 T_{2}^{80} + 413 T_{2}^{79} - 152881 T_{2}^{78} + \cdots + 13\!\cdots\!76 \) acting on \(S_{4}^{\mathrm{new}}(145, [\chi])\). Copy content Toggle raw display