# Properties

 Label 625.2.e.j Level $625$ Weight $2$ Character orbit 625.e Analytic conductor $4.991$ Analytic rank $0$ Dimension $32$ CM no Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [625,2,Mod(124,625)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(625, base_ring=CyclotomicField(10))

chi = DirichletCharacter(H, H._module([1]))

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

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

chi := DirichletCharacter("625.124");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$625 = 5^{4}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 625.e (of order $$10$$, degree $$4$$, not 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: $$4.99065012633$$ Analytic rank: $$0$$ Dimension: $$32$$ Relative dimension: $$8$$ over $$\Q(\zeta_{10})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$32 q + 6 q^{4} + 14 q^{6} + 24 q^{9}+O(q^{10})$$ 32 * q + 6 * q^4 + 14 * q^6 + 24 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$32 q + 6 q^{4} + 14 q^{6} + 24 q^{9} - 6 q^{11} + 2 q^{14} + 2 q^{16} - 20 q^{19} + 14 q^{21} - 20 q^{24} + 44 q^{26} - 16 q^{31} + 12 q^{34} + 2 q^{36} - 2 q^{39} - 16 q^{41} + 62 q^{44} + 84 q^{46} + 16 q^{49} - 56 q^{51} - 100 q^{54} + 70 q^{56} + 30 q^{59} + 34 q^{61} - 74 q^{64} + 88 q^{66} + 18 q^{69} - 26 q^{71} + 72 q^{74} - 40 q^{76} + 110 q^{79} - 38 q^{81} - 118 q^{84} + 14 q^{86} - 56 q^{91} - 8 q^{94} - 86 q^{96} + 88 q^{99}+O(q^{100})$$ 32 * q + 6 * q^4 + 14 * q^6 + 24 * q^9 - 6 * q^11 + 2 * q^14 + 2 * q^16 - 20 * q^19 + 14 * q^21 - 20 * q^24 + 44 * q^26 - 16 * q^31 + 12 * q^34 + 2 * q^36 - 2 * q^39 - 16 * q^41 + 62 * q^44 + 84 * q^46 + 16 * q^49 - 56 * q^51 - 100 * q^54 + 70 * q^56 + 30 * q^59 + 34 * q^61 - 74 * q^64 + 88 * q^66 + 18 * q^69 - 26 * q^71 + 72 * q^74 - 40 * q^76 + 110 * q^79 - 38 * q^81 - 118 * q^84 + 14 * q^86 - 56 * q^91 - 8 * q^94 - 86 * q^96 + 88 * 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   $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
124.1 −2.53458 0.823534i 0.446178 0.614111i 4.12784 + 2.99905i 0 −1.63661 + 1.18907i 2.04213i −4.85960 6.68866i 0.748993 + 2.30516i 0
124.2 −1.91511 0.622258i −1.77844 + 2.44781i 1.66242 + 1.20782i 0 4.92909 3.58119i 0.369971i −0.0649413 0.0893841i −1.90189 5.85342i 0
124.3 −1.00562 0.326747i −0.404046 + 0.556121i −0.713519 0.518402i 0 0.588029 0.427228i 1.01199i 1.79116 + 2.46533i 0.781033 + 2.40377i 0
124.4 −0.310759 0.100972i 1.00827 1.38777i −1.53166 1.11281i 0 −0.453455 + 0.329455i 3.42409i 0.747732 + 1.02917i 0.0177620 + 0.0546659i 0
124.5 0.310759 + 0.100972i −1.00827 + 1.38777i −1.53166 1.11281i 0 −0.453455 + 0.329455i 3.42409i −0.747732 1.02917i 0.0177620 + 0.0546659i 0
124.6 1.00562 + 0.326747i 0.404046 0.556121i −0.713519 0.518402i 0 0.588029 0.427228i 1.01199i −1.79116 2.46533i 0.781033 + 2.40377i 0
124.7 1.91511 + 0.622258i 1.77844 2.44781i 1.66242 + 1.20782i 0 4.92909 3.58119i 0.369971i 0.0649413 + 0.0893841i −1.90189 5.85342i 0
124.8 2.53458 + 0.823534i −0.446178 + 0.614111i 4.12784 + 2.99905i 0 −1.63661 + 1.18907i 2.04213i 4.85960 + 6.68866i 0.748993 + 2.30516i 0
249.1 −1.45439 + 2.00179i −2.01315 + 0.654112i −1.27390 3.92066i 0 1.61850 4.98124i 0.973070i 4.99460 + 1.62284i 1.19786 0.870294i 0
249.2 −1.36725 + 1.88186i 2.18963 0.711454i −1.05398 3.24381i 0 −1.65491 + 5.09330i 3.59425i 3.12093 + 1.01405i 1.86126 1.35229i 0
249.3 −0.989484 + 1.36191i 0.675574 0.219507i −0.257680 0.793058i 0 −0.369521 + 1.13727i 4.59110i −1.86699 0.606623i −2.01883 + 1.46677i 0
249.4 −0.294474 + 0.405309i −2.94186 + 0.955869i 0.540474 + 1.66341i 0 0.478880 1.47384i 0.0237879i −1.78629 0.580400i 5.31382 3.86071i 0
249.5 0.294474 0.405309i 2.94186 0.955869i 0.540474 + 1.66341i 0 0.478880 1.47384i 0.0237879i 1.78629 + 0.580400i 5.31382 3.86071i 0
249.6 0.989484 1.36191i −0.675574 + 0.219507i −0.257680 0.793058i 0 −0.369521 + 1.13727i 4.59110i 1.86699 + 0.606623i −2.01883 + 1.46677i 0
249.7 1.36725 1.88186i −2.18963 + 0.711454i −1.05398 3.24381i 0 −1.65491 + 5.09330i 3.59425i −3.12093 1.01405i 1.86126 1.35229i 0
249.8 1.45439 2.00179i 2.01315 0.654112i −1.27390 3.92066i 0 1.61850 4.98124i 0.973070i −4.99460 1.62284i 1.19786 0.870294i 0
374.1 −1.45439 2.00179i −2.01315 0.654112i −1.27390 + 3.92066i 0 1.61850 + 4.98124i 0.973070i 4.99460 1.62284i 1.19786 + 0.870294i 0
374.2 −1.36725 1.88186i 2.18963 + 0.711454i −1.05398 + 3.24381i 0 −1.65491 5.09330i 3.59425i 3.12093 1.01405i 1.86126 + 1.35229i 0
374.3 −0.989484 1.36191i 0.675574 + 0.219507i −0.257680 + 0.793058i 0 −0.369521 1.13727i 4.59110i −1.86699 + 0.606623i −2.01883 1.46677i 0
374.4 −0.294474 0.405309i −2.94186 0.955869i 0.540474 1.66341i 0 0.478880 + 1.47384i 0.0237879i −1.78629 + 0.580400i 5.31382 + 3.86071i 0
See all 32 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 124.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
25.d even 5 1 inner
25.e even 10 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 625.2.e.j 32
5.b even 2 1 inner 625.2.e.j 32
5.c odd 4 1 625.2.d.m 16
5.c odd 4 1 625.2.d.q 16
25.d even 5 1 625.2.b.d 16
25.d even 5 1 inner 625.2.e.j 32
25.d even 5 2 625.2.e.k 32
25.e even 10 1 625.2.b.d 16
25.e even 10 1 inner 625.2.e.j 32
25.e even 10 2 625.2.e.k 32
25.f odd 20 1 625.2.a.e 8
25.f odd 20 1 625.2.a.g yes 8
25.f odd 20 1 625.2.d.m 16
25.f odd 20 2 625.2.d.n 16
25.f odd 20 2 625.2.d.p 16
25.f odd 20 1 625.2.d.q 16
75.l even 20 1 5625.2.a.s 8
75.l even 20 1 5625.2.a.be 8
100.l even 20 1 10000.2.a.be 8
100.l even 20 1 10000.2.a.bn 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
625.2.a.e 8 25.f odd 20 1
625.2.a.g yes 8 25.f odd 20 1
625.2.b.d 16 25.d even 5 1
625.2.b.d 16 25.e even 10 1
625.2.d.m 16 5.c odd 4 1
625.2.d.m 16 25.f odd 20 1
625.2.d.n 16 25.f odd 20 2
625.2.d.p 16 25.f odd 20 2
625.2.d.q 16 5.c odd 4 1
625.2.d.q 16 25.f odd 20 1
625.2.e.j 32 1.a even 1 1 trivial
625.2.e.j 32 5.b even 2 1 inner
625.2.e.j 32 25.d even 5 1 inner
625.2.e.j 32 25.e even 10 1 inner
625.2.e.k 32 25.d even 5 2
625.2.e.k 32 25.e even 10 2
5625.2.a.s 8 75.l even 20 1
5625.2.a.be 8 75.l even 20 1
10000.2.a.be 8 100.l even 20 1
10000.2.a.bn 8 100.l even 20 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(625, [\chi])$$:

 $$T_{2}^{32} - 11 T_{2}^{30} + 100 T_{2}^{28} - 790 T_{2}^{26} + 6030 T_{2}^{24} - 27593 T_{2}^{22} + \cdots + 6561$$ T2^32 - 11*T2^30 + 100*T2^28 - 790*T2^26 + 6030*T2^24 - 27593*T2^22 + 143008*T2^20 - 588175*T2^18 + 1874830*T2^16 - 3845835*T2^14 + 11343258*T2^12 - 15386058*T2^10 + 9969075*T2^8 - 979290*T2^6 + 579555*T2^4 - 94041*T2^2 + 6561 $$T_{3}^{32} - 24 T_{3}^{30} + 320 T_{3}^{28} - 3270 T_{3}^{26} + 31080 T_{3}^{24} - 222072 T_{3}^{22} + \cdots + 707281$$ T3^32 - 24*T3^30 + 320*T3^28 - 3270*T3^26 + 31080*T3^24 - 222072*T3^22 + 1084483*T3^20 - 3554460*T3^18 + 9577910*T3^16 - 23842890*T3^14 + 45045628*T3^12 - 16380462*T3^10 + 19441845*T3^8 - 9616695*T3^6 + 4157030*T3^4 - 978924*T3^2 + 707281