# Properties

 Label 225.2.h.c Level $225$ Weight $2$ Character orbit 225.h Analytic conductor $1.797$ Analytic rank $0$ Dimension $8$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [225,2,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, 2, names="a")

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

chi := DirichletCharacter("225.46");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$225 = 3^{2} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ 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: $$1.79663404548$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$2$$ over $$\Q(\zeta_{5})$$ Coefficient field: 8.0.26265625.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{8} - 3x^{7} + 2x^{6} + x^{4} + 8x^{2} - 24x + 16$$ x^8 - 3*x^7 + 2*x^6 + x^4 + 8*x^2 - 24*x + 16 Coefficient ring: $$\Z[a_1, \ldots, a_{4}]$$ Coefficient ring index: $$5$$ Twist minimal: no (minimal twist has level 75) Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta_{7} + \beta_{3}) q^{2} + (\beta_{6} - \beta_{4} + 2 \beta_{3} + \cdots - 1) q^{4}+ \cdots + ( - \beta_{5} + \beta_{3} + \beta_{2} + \cdots - 1) q^{8}+O(q^{10})$$ q + (b7 + b3) * q^2 + (b6 - b4 + 2*b3 + 2*b1 - 1) * q^4 + (b6 + b4 + b1) * q^5 + (b7 - 2*b6 - b3 - b2 + 1) * q^7 + (-b5 + b3 + b2 + 5*b1 - 1) * q^8 $$q + (\beta_{7} + \beta_{3}) q^{2} + (\beta_{6} - \beta_{4} + 2 \beta_{3} + \cdots - 1) q^{4}+ \cdots + (\beta_{7} - 13 \beta_{6} + \cdots - 13 \beta_1) q^{98}+O(q^{100})$$ q + (b7 + b3) * q^2 + (b6 - b4 + 2*b3 + 2*b1 - 1) * q^4 + (b6 + b4 + b1) * q^5 + (b7 - 2*b6 - b3 - b2 + 1) * q^7 + (-b5 + b3 + b2 + 5*b1 - 1) * q^8 + (b7 + b4 - 4*b1 + 1) * q^10 + (-3*b6 - b4 - 3*b3 - b2 - 3*b1) * q^11 + (b7 + b6 + b4 + b3 + 2*b1 - 2) * q^13 + (b6 + 2*b4 + 4*b3 + 2*b2 + b1) * q^14 + (-2*b7 + b6 - 3*b5 + b4 - 2*b3 + 3*b2 + 5*b1 - 2) * q^16 + b5 * q^17 + (2*b5 + b2 - 2*b1) * q^19 + (b7 + 3*b6 + 3*b5 - 2*b4 + 2*b3 - 2*b2 - 3*b1) * q^20 + (-2*b7 - 2*b6 + 2*b5 + b4 - b3 - b1 + 2) * q^22 + (-b7 - 2*b6 - b4 - b3 - b2 - 2*b1) * q^23 + (-b6 - b5 + b4 + b3 - b2 + 3*b1 - 3) * q^25 + (-b7 + 3*b6 - b5 + b4 + 2*b3 + b2 + b1 - 2) * q^26 + (-b7 + 2*b6 + b5 - 3*b4 - b3 - b1 - 2) * q^28 + (4*b6 - b4 + 5*b3 + 5*b1 - 4) * q^29 + (2*b3 - b2 - 5*b1 - 2) * q^31 + (-2*b7 - 7*b6 - 2*b5 + 2*b4 - 9*b3 + 2*b2 + 2*b1 + 1) * q^32 + (3*b6 - b1 + 1) * q^34 + (2*b7 - 2*b6 - 2*b5 + b4 - 4*b3 - b2 - b1 + 3) * q^35 + (-2*b7 - 2*b6 + b5 - 3*b4 - 2*b3 - b2 - 2*b1 + 1) * q^37 + (-b7 + 8*b6 + 2*b5 - 3*b4 - b3 - 2*b2 - b1 - 1) * q^38 + (5*b7 + 6*b6 + b5 - b4 + 9*b3 - 4*b2 + 2*b1 - 2) * q^40 + (-5*b6 - 4*b1 + 4) * q^41 + (-b6 + 2*b5 - 2*b4 - b3 - 2*b1 + 5) * q^43 + (2*b5 - 3*b3 - b2 - 10*b1 + 3) * q^44 + (-b7 - 5*b6 + b5 - 4*b3 - 4*b1 + 5) * q^46 + (2*b7 - 2*b5 + b4 + b3 + b1) * q^47 + (-b7 + b6 - 4*b5 + 4*b4 + b2 + 4*b1 + 1) * q^49 + (-3*b7 - 5*b6 - 2*b5 + 3*b4 - 4*b3 + 4*b2 - 4*b1 + 3) * q^50 - 3*b3 * q^52 + (2*b7 + b6 - 2*b5 + b4 + 3*b3 + 3*b1 - 1) * q^53 + (-b7 - 3*b6 - 2*b5 - 2*b4 - 7*b3 + 2*b2 - 2*b1 + 6) * q^55 + (-2*b5 - 3*b3 + b2 + b1 + 3) * q^56 + (-6*b5 + b3 + b2 + 10*b1 - 1) * q^58 + (-3*b7 - 3*b6 - 3*b4 - 3*b3 + 3*b1 - 3) * q^59 + (-4*b6 - 2*b4 - b3 - 2*b2 - 4*b1) * q^61 + (-b7 - 2*b6 + 5*b5 - 6*b4 - b3 - 5*b2 - 8*b1 + 3) * q^62 + (-2*b7 - 4*b6 + 5*b4 - 10*b3 + 5*b2 - 4*b1) * q^64 + (2*b7 - 2*b6 - b5 + 2*b3 - 2*b2 - 2*b1 - 3) * q^65 + (-2*b5 - b3 + 1) * q^67 + (2*b7 + b5 - b4 + 2*b3 - 2*b2 - b1 + 2) * q^68 + (2*b7 - 2*b6 + 2*b5 + 4*b4 + 9*b3 + b2 + b1 - 4) * q^70 + (-4*b7 + 2*b6 + 4*b5 - b4 - 4*b3 - 4*b1 - 2) * q^71 + (4*b7 + b6 - 2*b4 - b3 - 2*b2 + b1) * q^73 + (-5*b6 - b5 + b4 - 5*b3 + b1 + 7) * q^74 + (5*b7 - b6 + 4*b3 - 5*b2 + 8) * q^76 + (-5*b7 + 6*b6 + b4 + 5*b3 + b2 + 6*b1) * q^77 + (-3*b7 + 10*b6 + 3*b5 + 6*b3 + 6*b1 - 10) * q^79 + (2*b7 + 10*b6 - b5 - 3*b4 + 17*b3 - 2*b2 + 10*b1 - 3) * q^80 + (-b7 + 4*b5 - 4*b4 - b3 + b2 - 4*b1) * q^82 + (4*b2 + 5*b1) * q^83 + (b7 + b4 + b1 - 4) * q^85 + (4*b7 + 6*b6 - b4 + 6*b3 - b2 + 6*b1) * q^86 + (4*b5 - 4*b4 - 4*b2 - 15*b1 + 11) * q^88 + (4*b6 + b4 - 7*b3 + b2 + 4*b1) * q^89 + (3*b6 - 3*b5 + 3*b4 + 3*b2 + 3) * q^91 + (2*b5 - b2 - 7*b1) * q^92 + (7*b3 + b2 + 6*b1 - 7) * q^94 + (b7 + b6 + 3*b4 + 2*b3 + 3*b2 - 8) * q^95 + (4*b7 + 5*b6 - 4*b5 + 4*b4 + 7*b3 + 7*b1 - 5) * q^97 + (b7 - 13*b6 + 3*b4 - 7*b3 + 3*b2 - 13*b1) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q + q^{2} + q^{4} + 5 q^{5} + 4 q^{7}+O(q^{10})$$ 8 * q + q^2 + q^4 + 5 * q^5 + 4 * q^7 $$8 q + q^{2} + q^{4} + 5 q^{5} + 4 q^{7} - 16 q^{11} - 8 q^{13} + 8 q^{14} - 17 q^{16} + q^{17} - 5 q^{19} + 10 q^{20} + 13 q^{22} - 7 q^{23} - 15 q^{25} - 6 q^{26} - 17 q^{28} - 5 q^{29} - 19 q^{31} - 24 q^{32} + 12 q^{34} + 10 q^{35} - q^{37} + 10 q^{38} + 25 q^{40} + 14 q^{41} + 32 q^{43} + 3 q^{44} + 16 q^{46} + q^{47} + 16 q^{49} - 10 q^{50} - 6 q^{52} + 3 q^{53} + 15 q^{55} + 15 q^{56} + 5 q^{58} - 30 q^{59} - 14 q^{61} + 17 q^{62} - 44 q^{64} - 25 q^{65} + 4 q^{67} + 22 q^{68} - 15 q^{70} - 21 q^{71} + 2 q^{73} + 38 q^{74} + 80 q^{76} + 37 q^{77} - 30 q^{79} + 50 q^{80} - 12 q^{82} - 2 q^{83} - 30 q^{85} + 34 q^{86} + 70 q^{88} + 21 q^{91} - 9 q^{92} - 33 q^{94} - 65 q^{95} - 6 q^{97} - 73 q^{98}+O(q^{100})$$ 8 * q + q^2 + q^4 + 5 * q^5 + 4 * q^7 - 16 * q^11 - 8 * q^13 + 8 * q^14 - 17 * q^16 + q^17 - 5 * q^19 + 10 * q^20 + 13 * q^22 - 7 * q^23 - 15 * q^25 - 6 * q^26 - 17 * q^28 - 5 * q^29 - 19 * q^31 - 24 * q^32 + 12 * q^34 + 10 * q^35 - q^37 + 10 * q^38 + 25 * q^40 + 14 * q^41 + 32 * q^43 + 3 * q^44 + 16 * q^46 + q^47 + 16 * q^49 - 10 * q^50 - 6 * q^52 + 3 * q^53 + 15 * q^55 + 15 * q^56 + 5 * q^58 - 30 * q^59 - 14 * q^61 + 17 * q^62 - 44 * q^64 - 25 * q^65 + 4 * q^67 + 22 * q^68 - 15 * q^70 - 21 * q^71 + 2 * q^73 + 38 * q^74 + 80 * q^76 + 37 * q^77 - 30 * q^79 + 50 * q^80 - 12 * q^82 - 2 * q^83 - 30 * q^85 + 34 * q^86 + 70 * q^88 + 21 * q^91 - 9 * q^92 - 33 * q^94 - 65 * q^95 - 6 * q^97 - 73 * q^98

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 3x^{7} + 2x^{6} + x^{4} + 8x^{2} - 24x + 16$$ :

 $$\beta_{1}$$ $$=$$ $$( \nu^{7} - \nu^{6} + \nu^{3} + 2\nu^{2} + 4\nu - 8 ) / 8$$ (v^7 - v^6 + v^3 + 2*v^2 + 4*v - 8) / 8 $$\beta_{2}$$ $$=$$ $$( \nu^{7} - \nu^{6} + \nu^{3} - 2\nu^{2} + 12\nu - 12 ) / 4$$ (v^7 - v^6 + v^3 - 2*v^2 + 12*v - 12) / 4 $$\beta_{3}$$ $$=$$ $$( -7\nu^{7} + 9\nu^{6} + 2\nu^{5} + 4\nu^{4} + \nu^{3} - 4\nu^{2} - 60\nu + 64 ) / 8$$ (-7*v^7 + 9*v^6 + 2*v^5 + 4*v^4 + v^3 - 4*v^2 - 60*v + 64) / 8 $$\beta_{4}$$ $$=$$ $$( 11\nu^{7} - 15\nu^{6} - 4\nu^{5} - 8\nu^{4} + 3\nu^{3} + 2\nu^{2} + 92\nu - 96 ) / 8$$ (11*v^7 - 15*v^6 - 4*v^5 - 8*v^4 + 3*v^3 + 2*v^2 + 92*v - 96) / 8 $$\beta_{5}$$ $$=$$ $$( 11\nu^{7} - 13\nu^{6} - 6\nu^{5} - 8\nu^{4} + 3\nu^{3} + 4\nu^{2} + 96\nu - 88 ) / 8$$ (11*v^7 - 13*v^6 - 6*v^5 - 8*v^4 + 3*v^3 + 4*v^2 + 96*v - 88) / 8 $$\beta_{6}$$ $$=$$ $$( 13\nu^{7} - 21\nu^{6} - 4\nu^{5} - 4\nu^{4} + 5\nu^{3} + 10\nu^{2} + 120\nu - 144 ) / 8$$ (13*v^7 - 21*v^6 - 4*v^5 - 4*v^4 + 5*v^3 + 10*v^2 + 120*v - 144) / 8 $$\beta_{7}$$ $$=$$ $$( 19\nu^{7} - 31\nu^{6} - 4\nu^{5} - 8\nu^{4} + 11\nu^{3} + 18\nu^{2} + 180\nu - 224 ) / 8$$ (19*v^7 - 31*v^6 - 4*v^5 - 8*v^4 + 11*v^3 + 18*v^2 + 180*v - 224) / 8
 $$\nu$$ $$=$$ $$( \beta_{7} - \beta_{6} + \beta_{5} - 2\beta_{4} - \beta_{3} + \beta_{2} - 4\beta _1 + 4 ) / 5$$ (b7 - b6 + b5 - 2*b4 - b3 + b2 - 4*b1 + 4) / 5 $$\nu^{2}$$ $$=$$ $$( 2\beta_{7} - 2\beta_{6} + 2\beta_{5} - 4\beta_{4} - 2\beta_{3} - 3\beta_{2} + 2\beta _1 + 3 ) / 5$$ (2*b7 - 2*b6 + 2*b5 - 4*b4 - 2*b3 - 3*b2 + 2*b1 + 3) / 5 $$\nu^{3}$$ $$=$$ $$( 2\beta_{7} - 2\beta_{6} + 2\beta_{5} + \beta_{4} + 8\beta_{3} + 2\beta_{2} + 7\beta _1 + 3 ) / 5$$ (2*b7 - 2*b6 + 2*b5 + b4 + 8*b3 + 2*b2 + 7*b1 + 3) / 5 $$\nu^{4}$$ $$=$$ $$-\beta_{7} + 2\beta_{6} - \beta_{4} + \beta_{2} + 2\beta _1 + 1$$ -b7 + 2*b6 - b4 + b2 + 2*b1 + 1 $$\nu^{5}$$ $$=$$ $$( 6\beta_{7} - 11\beta_{6} - 9\beta_{5} + 3\beta_{4} - 11\beta_{3} + 6\beta_{2} + 6\beta _1 + 19 ) / 5$$ (6*b7 - 11*b6 - 9*b5 + 3*b4 - 11*b3 + 6*b2 + 6*b1 + 19) / 5 $$\nu^{6}$$ $$=$$ $$( 2\beta_{7} - 7\beta_{6} + 7\beta_{5} - 9\beta_{4} - 7\beta_{3} + 7\beta_{2} + 12\beta _1 - 12 ) / 5$$ (2*b7 - 7*b6 + 7*b5 - 9*b4 - 7*b3 + 7*b2 + 12*b1 - 12) / 5 $$\nu^{7}$$ $$=$$ $$( -8\beta_{7} + 3\beta_{6} - 3\beta_{5} + 6\beta_{4} - 7\beta_{3} + 7\beta_{2} + 57\beta _1 + 3 ) / 5$$ (-8*b7 + 3*b6 - 3*b5 + 6*b4 - 7*b3 + 7*b2 + 57*b1 + 3) / 5

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/225\mathbb{Z}\right)^\times$$.

 $$n$$ $$101$$ $$127$$ $$\chi(n)$$ $$1$$ $$-1 + \beta_{1} + \beta_{3} + \beta_{6}$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
46.1
 −1.21700 + 0.720348i 1.40799 − 0.132563i −0.0272949 − 1.41395i 1.33631 + 0.462894i −0.0272949 + 1.41395i 1.33631 − 0.462894i −1.21700 − 0.720348i 1.40799 + 0.132563i
−0.655837 + 2.01846i 0 −2.02602 1.47199i 2.21700 0.291365i 0 4.35840 0.865884 0.629102i 0 −0.865884 + 4.66602i
46.2 0.346820 1.06740i 0 0.598970 + 0.435177i −0.407987 2.19853i 0 1.11373 2.48822 1.80780i 0 −2.48822 0.327009i
91.1 −1.38048 1.00297i 0 0.281722 + 0.867051i 1.02729 + 1.98612i 0 −3.94243 −0.573870 + 1.76619i 0 0.573870 3.77214i
91.2 2.18949 + 1.59076i 0 1.64533 + 5.06380i −0.336312 2.21063i 0 0.470294 −2.78023 + 8.55667i 0 2.78023 5.37515i
136.1 −1.38048 + 1.00297i 0 0.281722 0.867051i 1.02729 1.98612i 0 −3.94243 −0.573870 1.76619i 0 0.573870 + 3.77214i
136.2 2.18949 1.59076i 0 1.64533 5.06380i −0.336312 + 2.21063i 0 0.470294 −2.78023 8.55667i 0 2.78023 + 5.37515i
181.1 −0.655837 2.01846i 0 −2.02602 + 1.47199i 2.21700 + 0.291365i 0 4.35840 0.865884 + 0.629102i 0 −0.865884 4.66602i
181.2 0.346820 + 1.06740i 0 0.598970 0.435177i −0.407987 + 2.19853i 0 1.11373 2.48822 + 1.80780i 0 −2.48822 + 0.327009i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 46.2 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 225.2.h.c 8
3.b odd 2 1 75.2.g.b 8
15.d odd 2 1 375.2.g.b 8
15.e even 4 2 375.2.i.b 16
25.d even 5 1 inner 225.2.h.c 8
25.d even 5 1 5625.2.a.i 4
25.e even 10 1 5625.2.a.n 4
75.h odd 10 1 375.2.g.b 8
75.h odd 10 1 1875.2.a.e 4
75.j odd 10 1 75.2.g.b 8
75.j odd 10 1 1875.2.a.h 4
75.l even 20 2 375.2.i.b 16
75.l even 20 2 1875.2.b.c 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
75.2.g.b 8 3.b odd 2 1
75.2.g.b 8 75.j odd 10 1
225.2.h.c 8 1.a even 1 1 trivial
225.2.h.c 8 25.d even 5 1 inner
375.2.g.b 8 15.d odd 2 1
375.2.g.b 8 75.h odd 10 1
375.2.i.b 16 15.e even 4 2
375.2.i.b 16 75.l even 20 2
1875.2.a.e 4 75.h odd 10 1
1875.2.a.h 4 75.j odd 10 1
1875.2.b.c 8 75.l even 20 2
5625.2.a.i 4 25.d even 5 1
5625.2.a.n 4 25.e even 10 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{8} - T_{2}^{7} + 2T_{2}^{6} - 3T_{2}^{5} + 25T_{2}^{4} + 43T_{2}^{3} + 82T_{2}^{2} + 11T_{2} + 121$$ acting on $$S_{2}^{\mathrm{new}}(225, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8} - T^{7} + \cdots + 121$$
$3$ $$T^{8}$$
$5$ $$T^{8} - 5 T^{7} + \cdots + 625$$
$7$ $$(T^{4} - 2 T^{3} - 16 T^{2} + \cdots - 9)^{2}$$
$11$ $$T^{8} + 16 T^{7} + \cdots + 11881$$
$13$ $$T^{8} + 8 T^{7} + \cdots + 81$$
$17$ $$T^{8} - T^{7} + \cdots + 121$$
$19$ $$T^{8} + 5 T^{7} + \cdots + 75625$$
$23$ $$T^{8} + 7 T^{7} + \cdots + 361$$
$29$ $$T^{8} + 5 T^{7} + \cdots + 164025$$
$31$ $$T^{8} + 19 T^{7} + \cdots + 505521$$
$37$ $$T^{8} + T^{7} + 77 T^{6} + \cdots + 1$$
$41$ $$(T^{4} - 7 T^{3} + \cdots + 121)^{2}$$
$43$ $$(T^{4} - 16 T^{3} + \cdots - 99)^{2}$$
$47$ $$T^{8} - T^{7} + \cdots + 96721$$
$53$ $$T^{8} - 3 T^{7} + \cdots + 68121$$
$59$ $$T^{8} + 30 T^{7} + \cdots + 13286025$$
$61$ $$T^{8} + 14 T^{7} + \cdots + 81$$
$67$ $$T^{8} - 4 T^{7} + \cdots + 29241$$
$71$ $$T^{8} + 21 T^{7} + \cdots + 829921$$
$73$ $$T^{8} - 2 T^{7} + \cdots + 23707161$$
$79$ $$T^{8} + 30 T^{7} + \cdots + 96924025$$
$83$ $$T^{8} + 2 T^{7} + \cdots + 958441$$
$89$ $$T^{8} + 255 T^{6} + \cdots + 10923025$$
$97$ $$T^{8} + 6 T^{7} + \cdots + 4414201$$