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Properties

Label 225.4.h.b
Level $225$
Weight $4$
Character orbit 225.h
Analytic conductor $13.275$
Analytic rank $0$
Dimension $28$
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] = [225,4,Mod(46,225)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(225, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([0, 6]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("225.46");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 225 = 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 225.h (of order \(5\), 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: \(13.2754297513\)
Analytic rank: \(0\)
Dimension: \(28\)
Relative dimension: \(7\) over \(\Q(\zeta_{5})\)
Twist minimal: no (minimal twist has level 25)
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 28 q + q^{2} - 31 q^{4} + 20 q^{5} - 16 q^{7} - 100 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 28 q + q^{2} - 31 q^{4} + 20 q^{5} - 16 q^{7} - 100 q^{8} - 25 q^{10} + 89 q^{11} + 33 q^{13} + 17 q^{14} - 207 q^{16} + 191 q^{17} - 115 q^{19} + 225 q^{20} + 808 q^{22} - 433 q^{23} + 90 q^{25} - 586 q^{26} - 13 q^{28} + 5 q^{29} - 639 q^{31} + 1386 q^{32} - 777 q^{34} + 1030 q^{35} + 699 q^{37} + 2355 q^{38} + 410 q^{40} - 341 q^{41} - 172 q^{43} - 548 q^{44} - 1239 q^{46} - 2319 q^{47} + 1344 q^{49} - 2335 q^{50} + 2344 q^{52} + 927 q^{53} + 1225 q^{55} + 2910 q^{56} + 2410 q^{58} + 1905 q^{59} + 1391 q^{61} + 3832 q^{62} - 3596 q^{64} - 1215 q^{65} - 3611 q^{67} - 3622 q^{68} + 560 q^{70} + 3719 q^{71} + 4593 q^{73} - 4848 q^{74} + 3520 q^{76} - 1368 q^{77} + 775 q^{79} - 9500 q^{80} - 6762 q^{82} + 2447 q^{83} - 8185 q^{85} - 3891 q^{86} - 10960 q^{88} + 5075 q^{89} + 376 q^{91} + 8456 q^{92} + 3573 q^{94} - 3265 q^{95} + 7439 q^{97} - 7082 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
46.1 −1.44735 + 4.45449i 0 −11.2755 − 8.19213i −5.51525 − 9.72533i 0 13.4350 22.4976 − 16.3455i 0 51.3039 − 10.4917i
46.2 −0.906232 + 2.78910i 0 −0.485661 − 0.352853i 10.1222 + 4.74784i 0 −23.0864 −17.5561 + 12.7553i 0 −22.4152 + 23.9290i
46.3 −0.591509 + 1.82048i 0 3.50788 + 2.54862i 4.63720 + 10.1733i 0 14.5331 −19.1034 + 13.8794i 0 −21.2632 + 2.42430i
46.4 0.132779 − 0.408651i 0 6.32277 + 4.59376i −11.1601 + 0.672002i 0 −16.1597 5.49773 − 3.99433i 0 −1.20721 + 4.64982i
46.5 0.624983 − 1.92350i 0 3.16288 + 2.29797i 9.66664 − 5.61749i 0 24.6755 19.4867 − 14.1579i 0 −4.76375 − 22.1046i
46.6 1.41896 − 4.36711i 0 −10.5860 − 7.69120i −1.19043 − 11.1168i 0 −20.8866 −18.8904 + 13.7247i 0 −50.2373 − 10.5755i
46.7 1.57739 − 4.85470i 0 −14.6078 − 10.6132i −6.03231 + 9.41335i 0 −5.45530 −41.5290 + 30.1725i 0 36.1837 + 44.1336i
91.1 −4.29718 − 3.12208i 0 6.24620 + 19.2238i 9.58545 + 5.75493i 0 −9.63602 20.0464 − 61.6963i 0 −23.2230 − 54.6565i
91.2 −2.27143 − 1.65029i 0 −0.0362106 − 0.111445i −9.59405 − 5.74058i 0 −35.1773 −7.04252 + 21.6747i 0 12.3186 + 28.8722i
91.3 −1.92104 − 1.39571i 0 −0.729774 − 2.24601i 10.2730 + 4.41186i 0 25.3460 −7.60304 + 23.3997i 0 −13.5772 − 22.8136i
91.4 −0.269925 − 0.196112i 0 −2.43774 − 7.50258i 0.131697 − 11.1796i 0 30.0089 −1.63816 + 5.04172i 0 −2.22799 + 2.99181i
91.5 1.51671 + 1.10196i 0 −1.38603 − 4.26575i −2.00572 + 10.9990i 0 −5.91678 7.23313 − 22.2613i 0 −15.1625 + 14.4720i
91.6 2.81638 + 2.04622i 0 1.27284 + 3.91740i −9.28856 − 6.22275i 0 12.5082 4.17503 − 12.8494i 0 −13.4270 − 36.5321i
91.7 4.11746 + 2.99151i 0 5.53220 + 17.0263i 10.3703 − 4.17822i 0 −12.1888 −15.5740 + 47.9320i 0 55.1983 + 13.8191i
136.1 −4.29718 + 3.12208i 0 6.24620 − 19.2238i 9.58545 − 5.75493i 0 −9.63602 20.0464 + 61.6963i 0 −23.2230 + 54.6565i
136.2 −2.27143 + 1.65029i 0 −0.0362106 + 0.111445i −9.59405 + 5.74058i 0 −35.1773 −7.04252 − 21.6747i 0 12.3186 − 28.8722i
136.3 −1.92104 + 1.39571i 0 −0.729774 + 2.24601i 10.2730 − 4.41186i 0 25.3460 −7.60304 − 23.3997i 0 −13.5772 + 22.8136i
136.4 −0.269925 + 0.196112i 0 −2.43774 + 7.50258i 0.131697 + 11.1796i 0 30.0089 −1.63816 − 5.04172i 0 −2.22799 − 2.99181i
136.5 1.51671 − 1.10196i 0 −1.38603 + 4.26575i −2.00572 − 10.9990i 0 −5.91678 7.23313 + 22.2613i 0 −15.1625 − 14.4720i
136.6 2.81638 − 2.04622i 0 1.27284 − 3.91740i −9.28856 + 6.22275i 0 12.5082 4.17503 + 12.8494i 0 −13.4270 + 36.5321i
See all 28 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 46.7
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
25.d even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 225.4.h.b 28
3.b odd 2 1 25.4.d.a ✓ 28
15.d odd 2 1 125.4.d.a 28
15.e even 4 2 125.4.e.b 56
25.d even 5 1 inner 225.4.h.b 28
75.h odd 10 1 125.4.d.a 28
75.h odd 10 1 625.4.a.d 14
75.j odd 10 1 25.4.d.a ✓ 28
75.j odd 10 1 625.4.a.c 14
75.l even 20 2 125.4.e.b 56
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
25.4.d.a ✓ 28 3.b odd 2 1
25.4.d.a ✓ 28 75.j odd 10 1
125.4.d.a 28 15.d odd 2 1
125.4.d.a 28 75.h odd 10 1
125.4.e.b 56 15.e even 4 2
125.4.e.b 56 75.l even 20 2
225.4.h.b 28 1.a even 1 1 trivial
225.4.h.b 28 25.d even 5 1 inner
625.4.a.c 14 75.j odd 10 1
625.4.a.d 14 75.h odd 10 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{28} - T_{2}^{27} + 44 T_{2}^{26} + 11 T_{2}^{25} + 1345 T_{2}^{24} - 409 T_{2}^{23} + \cdots + 44173950976 \) acting on \(S_{4}^{\mathrm{new}}(225, [\chi])\). Copy content Toggle raw display