↑

Properties

Label 210.4.j.b
Level $210$
Weight $4$
Character orbit 210.j
Analytic conductor $12.390$
Analytic rank $0$
Dimension $36$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [210,4,Mod(113,210)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(210, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 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("210.113");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 210 = 2 \cdot 3 \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 210.j (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: \(12.3904011012\)
Analytic rank: \(0\)
Dimension: \(36\)
Relative dimension: \(18\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 36 q - 8 q^{3} + 36 q^{5}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 36 q - 8 q^{3} + 36 q^{5} + 32 q^{12} + 120 q^{13} + 504 q^{14} + 20 q^{15} - 576 q^{16} + 84 q^{17} - 144 q^{18} - 112 q^{21} - 72 q^{22} + 792 q^{23} + 128 q^{24} - 540 q^{25} - 164 q^{27} - 216 q^{29} + 256 q^{30} + 216 q^{31} - 608 q^{33} - 168 q^{35} - 128 q^{36} - 660 q^{37} + 72 q^{38} + 680 q^{39} + 192 q^{40} - 112 q^{42} + 624 q^{43} - 96 q^{44} + 1748 q^{45} + 144 q^{46} + 528 q^{47} + 128 q^{48} - 576 q^{50} - 676 q^{51} - 480 q^{52} + 72 q^{53} + 8 q^{54} - 2136 q^{55} - 1992 q^{57} - 1104 q^{58} + 96 q^{59} - 64 q^{60} + 2184 q^{62} + 560 q^{63} - 5040 q^{65} - 944 q^{66} + 2760 q^{67} - 336 q^{68} - 416 q^{69} + 504 q^{70} - 640 q^{72} - 312 q^{73} + 144 q^{74} + 5616 q^{75} + 252 q^{77} + 3456 q^{78} - 576 q^{80} + 3792 q^{81} - 2496 q^{82} - 648 q^{83} + 3204 q^{85} - 6772 q^{87} - 288 q^{88} + 1584 q^{89} + 648 q^{90} - 1008 q^{91} + 3168 q^{92} + 1600 q^{93} - 5040 q^{95} + 6816 q^{97} - 1720 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
113.1 −1.41421 − 1.41421i −5.19105 − 0.230214i 4.00000i 5.10113 − 9.94879i 7.01568 + 7.66683i −4.94975 + 4.94975i 5.65685 − 5.65685i 26.8940 + 2.39011i −21.2838 + 6.85563i
113.2 −1.41421 − 1.41421i −4.78196 − 2.03294i 4.00000i 6.69236 + 8.95613i 3.88770 + 9.63773i −4.94975 + 4.94975i 5.65685 − 5.65685i 18.7343 + 19.4429i 3.20146 − 22.1303i
113.3 −1.41421 − 1.41421i −2.24537 + 4.68597i 4.00000i −3.20945 + 10.7098i 9.80239 − 3.45153i −4.94975 + 4.94975i 5.65685 − 5.65685i −16.9166 − 21.0435i 19.6848 − 10.6071i
113.4 −1.41421 − 1.41421i −1.90471 + 4.83447i 4.00000i 10.1419 − 4.70553i 9.53064 − 4.14330i −4.94975 + 4.94975i 5.65685 − 5.65685i −19.7441 − 18.4165i −20.9974 − 7.68818i
113.5 −1.41421 − 1.41421i 0.763740 − 5.13972i 4.00000i 9.39583 − 6.05957i −8.34875 + 6.18857i −4.94975 + 4.94975i 5.65685 − 5.65685i −25.8334 − 7.85082i −21.8572 − 4.71818i
113.6 −1.41421 − 1.41421i 1.14516 − 5.06839i 4.00000i −7.86278 + 7.94837i −8.78729 + 5.54828i −4.94975 + 4.94975i 5.65685 − 5.65685i −24.3772 − 11.6083i 22.3604 − 0.121041i
113.7 −1.41421 − 1.41421i 2.77545 + 4.39282i 4.00000i −2.76353 − 10.8334i 2.28731 − 10.1375i −4.94975 + 4.94975i 5.65685 − 5.65685i −11.5938 + 24.3841i −11.4125 + 19.2290i
113.8 −1.41421 − 1.41421i 5.08926 − 1.04856i 4.00000i 6.91071 + 8.78874i −8.68018 − 5.71440i −4.94975 + 4.94975i 5.65685 − 5.65685i 24.8010 − 10.6728i 2.65594 − 22.2024i
113.9 −1.41421 − 1.41421i 5.17791 + 0.434995i 4.00000i −11.1635 − 0.613081i −6.70750 − 7.93785i −4.94975 + 4.94975i 5.65685 − 5.65685i 26.6216 + 4.50473i 14.9206 + 16.6546i
113.10 1.41421 + 1.41421i −4.98081 − 1.48039i 4.00000i 4.96906 + 10.0154i −4.95034 − 9.13751i 4.94975 − 4.94975i −5.65685 + 5.65685i 22.6169 + 14.7470i −7.13662 + 21.1912i
113.11 1.41421 + 1.41421i −4.62340 − 2.37153i 4.00000i −11.1791 − 0.166014i −3.18463 − 9.89233i 4.94975 − 4.94975i −5.65685 + 5.65685i 15.7517 + 21.9291i −15.5749 − 16.0444i
113.12 1.41421 + 1.41421i −3.87832 + 3.45813i 4.00000i 6.21899 − 9.29108i −10.3753 − 0.594244i 4.94975 − 4.94975i −5.65685 + 5.65685i 3.08273 − 26.8234i 21.9345 − 4.34460i
113.13 1.41421 + 1.41421i −3.70752 − 3.64065i 4.00000i 5.12637 − 9.93581i −0.0945669 − 10.3919i 4.94975 − 4.94975i −5.65685 + 5.65685i 0.491364 + 26.9955i 21.3011 − 6.80157i
113.14 1.41421 + 1.41421i 0.388730 + 5.18159i 4.00000i −6.96176 + 8.74836i −6.77813 + 7.87762i 4.94975 − 4.94975i −5.65685 + 5.65685i −26.6978 + 4.02848i −22.2175 + 2.52663i
113.15 1.41421 + 1.41421i 0.510379 − 5.17103i 4.00000i 2.34461 + 10.9317i 8.03472 − 6.59115i 4.94975 − 4.94975i −5.65685 + 5.65685i −26.4790 − 5.27837i −12.1440 + 18.7756i
113.16 1.41421 + 1.41421i 2.80208 − 4.37588i 4.00000i −9.14030 − 6.43854i 10.1512 − 2.22568i 4.94975 − 4.94975i −5.65685 + 5.65685i −11.2966 − 24.5232i −3.82087 − 22.0318i
113.17 1.41421 + 1.41421i 3.47260 + 3.86537i 4.00000i 2.55004 − 10.8856i −0.555456 + 10.3775i 4.94975 − 4.94975i −5.65685 + 5.65685i −2.88211 + 26.8457i 19.0009 − 11.7883i
113.18 1.41421 + 1.41421i 5.18783 − 0.294039i 4.00000i 10.8295 + 2.77895i 7.75253 + 6.92086i 4.94975 − 4.94975i −5.65685 + 5.65685i 26.8271 − 3.05084i 11.3852 + 19.2452i
197.1 −1.41421 + 1.41421i −5.19105 + 0.230214i − 4.00000i 5.10113 + 9.94879i 7.01568 − 7.66683i −4.94975 − 4.94975i 5.65685 + 5.65685i 26.8940 − 2.39011i −21.2838 − 6.85563i
197.2 −1.41421 + 1.41421i −4.78196 + 2.03294i − 4.00000i 6.69236 − 8.95613i 3.88770 − 9.63773i −4.94975 − 4.94975i 5.65685 + 5.65685i 18.7343 − 19.4429i 3.20146 + 22.1303i
See all 36 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 113.18
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
15.e even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 210.4.j.b yes 36
3.b odd 2 1 210.4.j.a ✓ 36
5.c odd 4 1 210.4.j.a ✓ 36
15.e even 4 1 inner 210.4.j.b yes 36
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
210.4.j.a ✓ 36 3.b odd 2 1
210.4.j.a ✓ 36 5.c odd 4 1
210.4.j.b yes 36 1.a even 1 1 trivial
210.4.j.b yes 36 15.e even 4 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{17}^{36} - 84 T_{17}^{35} + 3528 T_{17}^{34} - 668200 T_{17}^{33} + 585587238 T_{17}^{32} + \cdots + 56\!\cdots\!56 \) acting on \(S_{4}^{\mathrm{new}}(210, [\chi])\). Copy content Toggle raw display