# Properties

 Label 325.2.l.c Level $325$ Weight $2$ Character orbit 325.l Analytic conductor $2.595$ Analytic rank $0$ Dimension $60$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [325,2,Mod(66,325)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(325, base_ring=CyclotomicField(10))

chi = DirichletCharacter(H, H._module([2, 0]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("325.66");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$325 = 5^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 325.l (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: $$2.59513806569$$ Analytic rank: $$0$$ Dimension: $$60$$ Relative dimension: $$15$$ over $$\Q(\zeta_{5})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

## $q$-expansion

The algebraic $$q$$-expansion of this newform has not been computed, but we have computed the trace expansion.

 $$\operatorname{Tr}(f)(q) =$$ $$60 q - 3 q^{2} - 6 q^{3} - 9 q^{4} + 2 q^{5} - 2 q^{6} + 10 q^{7} - 25 q^{9}+O(q^{10})$$ 60 * q - 3 * q^2 - 6 * q^3 - 9 * q^4 + 2 * q^5 - 2 * q^6 + 10 * q^7 - 25 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$60 q - 3 q^{2} - 6 q^{3} - 9 q^{4} + 2 q^{5} - 2 q^{6} + 10 q^{7} - 25 q^{9} + 25 q^{10} - q^{11} - 15 q^{13} + 18 q^{14} - 18 q^{15} - 17 q^{16} - 11 q^{17} + 22 q^{18} - 14 q^{19} - 33 q^{20} - 28 q^{21} + 28 q^{22} - 3 q^{23} - 24 q^{24} - 12 q^{25} + 12 q^{26} - 12 q^{27} - 22 q^{28} + 15 q^{29} + 4 q^{30} - 21 q^{31} + 52 q^{32} - 22 q^{33} - 56 q^{34} + 19 q^{35} - 41 q^{36} - 22 q^{37} - 6 q^{38} - q^{39} - 55 q^{40} - 13 q^{41} + 2 q^{42} + 76 q^{43} - 31 q^{44} - 23 q^{45} - 30 q^{46} - 40 q^{47} + 6 q^{48} + 22 q^{49} - 40 q^{50} + 98 q^{51} - 19 q^{52} - 37 q^{53} + 82 q^{54} - 26 q^{55} + 52 q^{56} + 90 q^{57} - 21 q^{58} + 5 q^{59} + 232 q^{60} - 30 q^{61} - 43 q^{62} + 68 q^{64} - 3 q^{65} + 39 q^{66} + 17 q^{67} + 148 q^{68} - 9 q^{69} - 35 q^{70} - 37 q^{71} + 10 q^{72} - 26 q^{73} - 80 q^{74} - 82 q^{75} + 142 q^{76} - 46 q^{77} - 2 q^{78} - 14 q^{79} + 65 q^{80} - 36 q^{81} - 6 q^{82} - 23 q^{83} + 23 q^{84} + 130 q^{85} - 66 q^{86} - 40 q^{87} - 11 q^{88} - 20 q^{89} - 57 q^{90} - 5 q^{91} - 40 q^{92} + 106 q^{93} - 8 q^{94} + 3 q^{95} - 148 q^{96} + 32 q^{97} - 48 q^{98} - 58 q^{99}+O(q^{100})$$ 60 * q - 3 * q^2 - 6 * q^3 - 9 * q^4 + 2 * q^5 - 2 * q^6 + 10 * q^7 - 25 * q^9 + 25 * q^10 - q^11 - 15 * q^13 + 18 * q^14 - 18 * q^15 - 17 * q^16 - 11 * q^17 + 22 * q^18 - 14 * q^19 - 33 * q^20 - 28 * q^21 + 28 * q^22 - 3 * q^23 - 24 * q^24 - 12 * q^25 + 12 * q^26 - 12 * q^27 - 22 * q^28 + 15 * q^29 + 4 * q^30 - 21 * q^31 + 52 * q^32 - 22 * q^33 - 56 * q^34 + 19 * q^35 - 41 * q^36 - 22 * q^37 - 6 * q^38 - q^39 - 55 * q^40 - 13 * q^41 + 2 * q^42 + 76 * q^43 - 31 * q^44 - 23 * q^45 - 30 * q^46 - 40 * q^47 + 6 * q^48 + 22 * q^49 - 40 * q^50 + 98 * q^51 - 19 * q^52 - 37 * q^53 + 82 * q^54 - 26 * q^55 + 52 * q^56 + 90 * q^57 - 21 * q^58 + 5 * q^59 + 232 * q^60 - 30 * q^61 - 43 * q^62 + 68 * q^64 - 3 * q^65 + 39 * q^66 + 17 * q^67 + 148 * q^68 - 9 * q^69 - 35 * q^70 - 37 * q^71 + 10 * q^72 - 26 * q^73 - 80 * q^74 - 82 * q^75 + 142 * q^76 - 46 * q^77 - 2 * q^78 - 14 * q^79 + 65 * q^80 - 36 * q^81 - 6 * q^82 - 23 * q^83 + 23 * q^84 + 130 * q^85 - 66 * q^86 - 40 * q^87 - 11 * q^88 - 20 * q^89 - 57 * q^90 - 5 * q^91 - 40 * q^92 + 106 * q^93 - 8 * q^94 + 3 * q^95 - 148 * q^96 + 32 * q^97 - 48 * q^98 - 58 * 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   $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
66.1 −2.11859 1.53924i 0.771973 + 2.37589i 1.50111 + 4.61993i −2.18599 0.470585i 2.02158 6.22178i −3.55469 2.31251 7.11718i −2.62186 + 1.90489i 3.90686 + 4.36174i
66.2 −2.11550 1.53700i −0.756399 2.32796i 1.49493 + 4.60091i 1.90238 + 1.17514i −1.97790 + 6.08736i 1.79715 2.29298 7.05707i −2.42019 + 1.75837i −2.21830 5.40995i
66.3 −1.71220 1.24398i −0.522792 1.60899i 0.766087 + 2.35777i −1.47820 + 1.67777i −1.10643 + 3.40525i −2.30135 0.313339 0.964358i 0.111515 0.0810203i 4.61809 1.03382i
66.4 −1.60352 1.16503i 0.794864 + 2.44634i 0.595961 + 1.83418i 0.133214 + 2.23210i 1.57547 4.84880i 4.32294 −0.0437495 + 0.134647i −2.92571 + 2.12566i 2.38684 3.73441i
66.5 −1.58418 1.15097i −0.110716 0.340750i 0.566849 + 1.74458i −1.16514 1.90852i −0.216799 + 0.667240i −0.106861 −0.100227 + 0.308467i 2.32320 1.68790i −0.350856 + 4.36448i
66.6 −0.893733 0.649335i 0.209044 + 0.643370i −0.240911 0.741448i 2.03707 0.922134i 0.230934 0.710740i 0.686970 −0.948890 + 2.92038i 2.05683 1.49437i −2.41937 0.498601i
66.7 −0.430798 0.312993i 0.962636 + 2.96269i −0.530412 1.63244i 1.23199 + 1.86607i 0.512600 1.57762i −4.48914 −0.611542 + 1.88213i −5.42381 + 3.94063i 0.0533285 1.18950i
66.8 −0.111048 0.0806808i −0.623987 1.92043i −0.612212 1.88419i −1.64946 1.50973i −0.0856500 + 0.263604i −1.56485 −0.168867 + 0.519718i −0.871658 + 0.633296i 0.0613615 + 0.300732i
66.9 0.410747 + 0.298425i −0.0810801 0.249539i −0.538378 1.65696i −1.91546 + 1.15370i 0.0411653 0.126694i 3.54407 0.587124 1.80698i 2.37136 1.72289i −1.13106 0.0977442i
66.10 0.443042 + 0.321889i 0.814794 + 2.50768i −0.525360 1.61689i 0.585476 2.15806i −0.446206 + 1.37328i 2.12323 0.626157 1.92711i −3.19751 + 2.32313i 0.954045 0.767652i
66.11 0.545716 + 0.396486i −0.905162 2.78580i −0.477429 1.46938i 2.23054 0.157067i 0.610570 1.87914i 2.34324 0.738936 2.27421i −4.51432 + 3.27985i 1.27952 + 0.798666i
66.12 1.34617 + 0.978052i −0.213534 0.657191i 0.237562 + 0.731141i 1.42062 1.72680i 0.355313 1.09354i −2.13212 0.633091 1.94845i 2.04075 1.48269i 3.60130 0.935135i
66.13 1.55873 + 1.13248i 0.447787 + 1.37815i 0.529080 + 1.62834i 0.800256 + 2.08796i −0.862747 + 2.65526i −0.875510 0.171387 0.527474i 0.728277 0.529124i −1.11720 + 4.16084i
66.14 1.82200 + 1.32376i −0.674593 2.07618i 0.949315 + 2.92169i −0.435763 + 2.19320i 1.51927 4.67582i 2.98877 −0.746085 + 2.29621i −1.42841 + 1.03780i −3.69723 + 3.41917i
66.15 2.01610 + 1.46478i 0.623234 + 1.91812i 1.30104 + 4.00418i −1.01154 1.99419i −1.55312 + 4.78002i 0.836184 −1.70207 + 5.23842i −0.863701 + 0.627516i 0.881691 5.50217i
131.1 −0.780630 2.40253i 0.124435 0.0904075i −3.54474 + 2.57541i −1.97204 + 1.05406i −0.314345 0.228385i 3.98052 4.86720 + 3.53623i −0.919740 + 2.83067i 4.07186 + 3.91506i
131.2 −0.668594 2.05772i 0.454228 0.330016i −2.16916 + 1.57599i −0.627399 2.14625i −0.982774 0.714027i −1.81313 1.19242 + 0.866344i −0.829638 + 2.55336i −3.99690 + 2.72598i
131.3 −0.599245 1.84429i −2.63027 + 1.91100i −1.42426 + 1.03479i 1.43620 + 1.71386i 5.10062 + 3.70581i −1.15914 −0.375760 0.273006i 2.33934 7.19975i 2.30021 3.67579i
131.4 −0.344169 1.05924i −0.769833 + 0.559317i 0.614491 0.446454i 2.16018 0.577587i 0.857405 + 0.622941i 3.14196 −2.48648 1.80654i −0.647243 + 1.99201i −1.35527 2.08937i
131.5 −0.316435 0.973887i 1.79577 1.30470i 0.769710 0.559227i 2.19613 + 0.420731i −1.83888 1.33602i −4.19384 −2.44506 1.77644i 0.595492 1.83274i −0.285188 2.27192i
See all 60 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 66.15 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 325.2.l.c 60
25.d even 5 1 inner 325.2.l.c 60
25.d even 5 1 8125.2.a.j 30
25.e even 10 1 8125.2.a.i 30

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
325.2.l.c 60 1.a even 1 1 trivial
325.2.l.c 60 25.d even 5 1 inner
8125.2.a.i 30 25.e even 10 1
8125.2.a.j 30 25.d even 5 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{60} + 3 T_{2}^{59} + 24 T_{2}^{58} + 67 T_{2}^{57} + 356 T_{2}^{56} + 838 T_{2}^{55} + 3836 T_{2}^{54} + \cdots + 25$$ acting on $$S_{2}^{\mathrm{new}}(325, [\chi])$$.