# Properties

 Label 25.4.d.a Level $25$ Weight $4$ Character orbit 25.d Analytic conductor $1.475$ Analytic rank $0$ Dimension $28$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [25,4,Mod(6,25)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(25, base_ring=CyclotomicField(10))

chi = DirichletCharacter(H, H._module([4]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("25.6");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$25 = 5^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 25.d (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: $$1.47504775014$$ Analytic rank: $$0$$ Dimension: $$28$$ Relative dimension: $$7$$ over $$\Q(\zeta_{5})$$ Twist minimal: yes 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.

 $$\operatorname{Tr}(f)(q) =$$ $$28 q - q^{2} - 7 q^{3} - 31 q^{4} - 20 q^{5} + q^{6} - 16 q^{7} + 100 q^{8} - 34 q^{9}+O(q^{10})$$ 28 * q - q^2 - 7 * q^3 - 31 * q^4 - 20 * q^5 + q^6 - 16 * q^7 + 100 * q^8 - 34 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$28 q - q^{2} - 7 q^{3} - 31 q^{4} - 20 q^{5} + q^{6} - 16 q^{7} + 100 q^{8} - 34 q^{9} - 25 q^{10} - 89 q^{11} + 139 q^{12} + 33 q^{13} - 17 q^{14} + 225 q^{15} - 207 q^{16} - 191 q^{17} - 552 q^{18} - 115 q^{19} - 225 q^{20} - 144 q^{21} + 808 q^{22} + 433 q^{23} + 780 q^{24} + 90 q^{25} + 586 q^{26} + 35 q^{27} - 13 q^{28} - 5 q^{29} + 675 q^{30} - 639 q^{31} - 1386 q^{32} + 251 q^{33} - 777 q^{34} - 1030 q^{35} + 673 q^{36} + 699 q^{37} - 2355 q^{38} - 1133 q^{39} + 410 q^{40} + 341 q^{41} - 2407 q^{42} - 172 q^{43} + 548 q^{44} + 470 q^{45} - 1239 q^{46} + 2319 q^{47} + 4738 q^{48} + 1344 q^{49} + 2335 q^{50} + 2006 q^{51} + 2344 q^{52} - 927 q^{53} + 1615 q^{54} + 1225 q^{55} - 2910 q^{56} - 770 q^{57} + 2410 q^{58} - 1905 q^{59} - 12030 q^{60} + 1391 q^{61} - 3832 q^{62} - 6142 q^{63} - 3596 q^{64} + 1215 q^{65} + 3632 q^{66} - 3611 q^{67} + 3622 q^{68} + 2687 q^{69} + 560 q^{70} - 3719 q^{71} + 9025 q^{72} + 4593 q^{73} + 4848 q^{74} + 3815 q^{75} + 3520 q^{76} + 1368 q^{77} - 3679 q^{78} + 775 q^{79} + 9500 q^{80} - 3712 q^{81} - 6762 q^{82} - 2447 q^{83} - 7612 q^{84} - 8185 q^{85} + 3891 q^{86} - 85 q^{87} - 10960 q^{88} - 5075 q^{89} + 685 q^{90} + 376 q^{91} - 8456 q^{92} + 4366 q^{93} + 3573 q^{94} + 3265 q^{95} - 7754 q^{96} + 7439 q^{97} + 7082 q^{98} + 6572 q^{99}+O(q^{100})$$ 28 * q - q^2 - 7 * q^3 - 31 * q^4 - 20 * q^5 + q^6 - 16 * q^7 + 100 * q^8 - 34 * q^9 - 25 * q^10 - 89 * q^11 + 139 * q^12 + 33 * q^13 - 17 * q^14 + 225 * q^15 - 207 * q^16 - 191 * q^17 - 552 * q^18 - 115 * q^19 - 225 * q^20 - 144 * q^21 + 808 * q^22 + 433 * q^23 + 780 * q^24 + 90 * q^25 + 586 * q^26 + 35 * q^27 - 13 * q^28 - 5 * q^29 + 675 * q^30 - 639 * q^31 - 1386 * q^32 + 251 * q^33 - 777 * q^34 - 1030 * q^35 + 673 * q^36 + 699 * q^37 - 2355 * q^38 - 1133 * q^39 + 410 * q^40 + 341 * q^41 - 2407 * q^42 - 172 * q^43 + 548 * q^44 + 470 * q^45 - 1239 * q^46 + 2319 * q^47 + 4738 * q^48 + 1344 * q^49 + 2335 * q^50 + 2006 * q^51 + 2344 * q^52 - 927 * q^53 + 1615 * q^54 + 1225 * q^55 - 2910 * q^56 - 770 * q^57 + 2410 * q^58 - 1905 * q^59 - 12030 * q^60 + 1391 * q^61 - 3832 * q^62 - 6142 * q^63 - 3596 * q^64 + 1215 * q^65 + 3632 * q^66 - 3611 * q^67 + 3622 * q^68 + 2687 * q^69 + 560 * q^70 - 3719 * q^71 + 9025 * q^72 + 4593 * q^73 + 4848 * q^74 + 3815 * q^75 + 3520 * q^76 + 1368 * q^77 - 3679 * q^78 + 775 * q^79 + 9500 * q^80 - 3712 * q^81 - 6762 * q^82 - 2447 * q^83 - 7612 * q^84 - 8185 * q^85 + 3891 * q^86 - 85 * q^87 - 10960 * q^88 - 5075 * q^89 + 685 * q^90 + 376 * q^91 - 8456 * q^92 + 4366 * q^93 + 3573 * q^94 + 3265 * q^95 - 7754 * q^96 + 7439 * q^97 + 7082 * q^98 + 6572 * q^99

## 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 −1.57739 4.85470i 6.28151 4.56378i −14.6078 + 10.6132i 6.03231 + 9.41335i −32.0642 23.2960i −5.45530 41.5290 + 30.1725i 10.2858 31.6563i 36.1837 44.1336i
6.2 −1.41896 4.36711i −7.07217 + 5.13823i −10.5860 + 7.69120i 1.19043 11.1168i 32.4743 + 23.5940i −20.8866 18.8904 + 13.7247i 15.2707 46.9983i −50.2373 + 10.5755i
6.3 −0.624983 1.92350i 1.69858 1.23409i 3.16288 2.29797i −9.66664 5.61749i −3.43535 2.49593i 24.6755 −19.4867 14.1579i −6.98127 + 21.4861i −4.76375 + 22.1046i
6.4 −0.132779 0.408651i 0.0617670 0.0448763i 6.32277 4.59376i 11.1601 + 0.672002i −0.0265401 0.0192825i −16.1597 −5.49773 3.99433i −8.34166 + 25.6730i −1.20721 4.64982i
6.5 0.591509 + 1.82048i −7.58563 + 5.51128i 3.50788 2.54862i −4.63720 + 10.1733i −14.5201 10.5495i 14.5331 19.1034 + 13.8794i 18.8241 57.9345i −21.2632 2.42430i
6.6 0.906232 + 2.78910i 5.49244 3.99049i −0.485661 + 0.352853i −10.1222 + 4.74784i 16.1073 + 11.7026i −23.0864 17.5561 + 12.7553i 5.89942 18.1565i −22.4152 23.9290i
6.7 1.44735 + 4.45449i −1.18551 + 0.861326i −11.2755 + 8.19213i 5.51525 9.72533i −5.55262 4.03421i 13.4350 −22.4976 16.3455i −7.67990 + 23.6363i 51.3039 + 10.4917i
11.1 −4.11746 + 2.99151i −1.18297 + 3.64082i 5.53220 17.0263i −10.3703 4.17822i −6.02069 18.5298i −12.1888 15.5740 + 47.9320i 9.98733 + 7.25622i 55.1983 13.8191i
11.2 −2.81638 + 2.04622i 2.63646 8.11420i 1.27284 3.91740i 9.28856 6.22275i 9.17815 + 28.2474i 12.5082 −4.17503 12.8494i −37.0458 26.9153i −13.4270 + 36.5321i
11.3 −1.51671 + 1.10196i −0.496278 + 1.52739i −1.38603 + 4.26575i 2.00572 + 10.9990i −0.930402 2.86348i −5.91678 −7.23313 22.2613i 19.7568 + 14.3542i −15.1625 14.4720i
11.4 0.269925 0.196112i −2.60870 + 8.02875i −2.43774 + 7.50258i −0.131697 11.1796i 0.870380 + 2.67876i 30.0089 1.63816 + 5.04172i −35.8121 26.0190i −2.22799 2.99181i
11.5 1.92104 1.39571i 1.98691 6.11509i −0.729774 + 2.24601i −10.2730 + 4.41186i −4.71799 14.5205i 25.3460 7.60304 + 23.3997i −11.6030 8.43010i −13.5772 + 22.8136i
11.6 2.27143 1.65029i 0.411392 1.26614i −0.0362106 + 0.111445i 9.59405 5.74058i −1.15504 3.55485i −35.1773 7.04252 + 21.6747i 20.4096 + 14.8284i 12.3186 28.8722i
11.7 4.29718 3.12208i −1.93780 + 5.96393i 6.24620 19.2238i −9.58545 + 5.75493i 10.2928 + 31.6780i −9.63602 −20.0464 61.6963i −9.96993 7.24358i −23.2230 + 54.6565i
16.1 −4.11746 2.99151i −1.18297 3.64082i 5.53220 + 17.0263i −10.3703 + 4.17822i −6.02069 + 18.5298i −12.1888 15.5740 47.9320i 9.98733 7.25622i 55.1983 + 13.8191i
16.2 −2.81638 2.04622i 2.63646 + 8.11420i 1.27284 + 3.91740i 9.28856 + 6.22275i 9.17815 28.2474i 12.5082 −4.17503 + 12.8494i −37.0458 + 26.9153i −13.4270 36.5321i
16.3 −1.51671 1.10196i −0.496278 1.52739i −1.38603 4.26575i 2.00572 10.9990i −0.930402 + 2.86348i −5.91678 −7.23313 + 22.2613i 19.7568 14.3542i −15.1625 + 14.4720i
16.4 0.269925 + 0.196112i −2.60870 8.02875i −2.43774 7.50258i −0.131697 + 11.1796i 0.870380 2.67876i 30.0089 1.63816 5.04172i −35.8121 + 26.0190i −2.22799 + 2.99181i
16.5 1.92104 + 1.39571i 1.98691 + 6.11509i −0.729774 2.24601i −10.2730 4.41186i −4.71799 + 14.5205i 25.3460 7.60304 23.3997i −11.6030 + 8.43010i −13.5772 22.8136i
16.6 2.27143 + 1.65029i 0.411392 + 1.26614i −0.0362106 0.111445i 9.59405 + 5.74058i −1.15504 + 3.55485i −35.1773 7.04252 21.6747i 20.4096 14.8284i 12.3186 + 28.8722i
See all 28 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 6.7 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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 25.4.d.a 28
3.b odd 2 1 225.4.h.b 28
5.b even 2 1 125.4.d.a 28
5.c odd 4 2 125.4.e.b 56
25.d even 5 1 inner 25.4.d.a 28
25.d even 5 1 625.4.a.c 14
25.e even 10 1 125.4.d.a 28
25.e even 10 1 625.4.a.d 14
25.f odd 20 2 125.4.e.b 56
75.j odd 10 1 225.4.h.b 28

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

## Hecke kernels

This newform subspace is the entire newspace $$S_{4}^{\mathrm{new}}(25, [\chi])$$.