# Properties

 Label 2025.2.b.o Level $2025$ Weight $2$ Character orbit 2025.b Analytic conductor $16.170$ Analytic rank $0$ Dimension $8$ 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] = [2025,2,Mod(649,2025)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2025, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1]))

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

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

chi := DirichletCharacter("2025.649");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2025 = 3^{4} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2025.b (of order $$2$$, degree $$1$$, 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: $$16.1697064093$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: 8.0.34810603776.6 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{8} + 12x^{6} + 42x^{4} + 49x^{2} + 9$$ x^8 + 12*x^6 + 42*x^4 + 49*x^2 + 9 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 225) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{2} + ( - \beta_{6} + \beta_{4} - 1) q^{4} + ( - \beta_{7} + \beta_{2}) q^{7} + (\beta_{7} + \beta_{3} - 2 \beta_{2} - \beta_1) q^{8}+O(q^{10})$$ q + b1 * q^2 + (-b6 + b4 - 1) * q^4 + (-b7 + b2) * q^7 + (b7 + b3 - 2*b2 - b1) * q^8 $$q + \beta_1 q^{2} + ( - \beta_{6} + \beta_{4} - 1) q^{4} + ( - \beta_{7} + \beta_{2}) q^{7} + (\beta_{7} + \beta_{3} - 2 \beta_{2} - \beta_1) q^{8} + (\beta_{5} + \beta_{4}) q^{11} + (2 \beta_{2} - \beta_1) q^{13} + ( - \beta_{6} - \beta_{5} + 2 \beta_{4}) q^{14} + (3 \beta_{6} + 2 \beta_{5} - 2 \beta_{4} + 1) q^{16} + (\beta_{7} - 2 \beta_{3} + \beta_{2}) q^{17} + ( - 2 \beta_{6} - \beta_{4} + 1) q^{19} + (\beta_{7} - 2 \beta_1) q^{22} + ( - \beta_{7} + 5 \beta_{3} + 2 \beta_{2} + \beta_1) q^{23} + ( - 3 \beta_{6} - 2 \beta_{5} + \beta_{4} + 3) q^{26} + ( - \beta_{3} - 2 \beta_{2} - 2 \beta_1) q^{28} + ( - 2 \beta_{6} - \beta_{5} + 2 \beta_{4}) q^{29} + ( - \beta_{6} - 2 \beta_{5} - 1) q^{31} + (3 \beta_{2} + 2 \beta_1) q^{32} + ( - 5 \beta_{6} - \beta_{5}) q^{34} + (\beta_{7} - 3 \beta_{3} - 2 \beta_{2} - 4 \beta_1) q^{37} + ( - \beta_{7} + 5 \beta_{3} - \beta_{2}) q^{38} + ( - \beta_{6} + \beta_{5} - 2 \beta_{4} + 3) q^{41} + ( - 2 \beta_{7} + \beta_{3} - 2 \beta_1) q^{43} + (\beta_{6} + 2 \beta_{5} - \beta_{4} + 6) q^{44} + (\beta_{6} - 2 \beta_{5} + 4 \beta_{4} - 3) q^{46} + ( - 2 \beta_{7} - 5 \beta_{3} + \beta_{2}) q^{47} + ( - \beta_{6} + \beta_{5} - 3 \beta_{4} + 3) q^{49} + (\beta_{7} + 3 \beta_{3} - 2 \beta_{2} - \beta_1) q^{52} + (6 \beta_{3} + 3 \beta_{2} - \beta_1) q^{53} + (3 \beta_{6} + 6) q^{56} + (2 \beta_{7} + \beta_{3} - 5 \beta_{2} - 3 \beta_1) q^{58} + (\beta_{6} + 3 \beta_{5} - 3 \beta_{4} + 6) q^{59} + ( - 2 \beta_{6} - \beta_{5} + 3 \beta_{4} - 4) q^{61} - 3 \beta_{2} q^{62} + ( - 2 \beta_{6} + \beta_{5} + \beta_{4} - 4) q^{64} + ( - \beta_{7} + 3 \beta_{3} + \beta_{2} - 3 \beta_1) q^{67} + (2 \beta_{7} + 5 \beta_{3} - 4 \beta_{2} - 4 \beta_1) q^{68} + (\beta_{6} + 3 \beta_{4}) q^{71} + (2 \beta_{7} - \beta_{3} + \beta_{2} - 3 \beta_1) q^{73} + (4 \beta_{6} + 2 \beta_{5} - 7 \beta_{4} + 12) q^{74} + (4 \beta_{6} + \beta_{5} - 2 \beta_{4} + 2) q^{76} + ( - 2 \beta_{7} - 5 \beta_{3} - 2 \beta_{2} - \beta_1) q^{77} + (\beta_{5} + 2 \beta_{4} + 1) q^{79} + ( - 2 \beta_{7} + 5 \beta_{3} + 2 \beta_{2} + 3 \beta_1) q^{82} + (9 \beta_{3} + 3 \beta_1) q^{83} + (5 \beta_{6} + 6) q^{86} + (\beta_{7} + \beta_{3} + 4 \beta_{2} + 2 \beta_1) q^{88} + ( - 3 \beta_{6} - 3 \beta_{5} + 3 \beta_{4} - 3) q^{89} + ( - \beta_{6} + 3 \beta_{5} - 2 \beta_{4} - 2) q^{91} + (2 \beta_{7} + 2 \beta_{3} - \beta_{2} - 2 \beta_1) q^{92} + ( - 5 \beta_{6} - \beta_{5} + 3 \beta_{4}) q^{94} + ( - \beta_{7} - 5 \beta_{3} + 2 \beta_{2} - 2 \beta_1) q^{97} + ( - 3 \beta_{7} + 6 \beta_{3} + 3 \beta_{2} + 4 \beta_1) q^{98}+O(q^{100})$$ q + b1 * q^2 + (-b6 + b4 - 1) * q^4 + (-b7 + b2) * q^7 + (b7 + b3 - 2*b2 - b1) * q^8 + (b5 + b4) * q^11 + (2*b2 - b1) * q^13 + (-b6 - b5 + 2*b4) * q^14 + (3*b6 + 2*b5 - 2*b4 + 1) * q^16 + (b7 - 2*b3 + b2) * q^17 + (-2*b6 - b4 + 1) * q^19 + (b7 - 2*b1) * q^22 + (-b7 + 5*b3 + 2*b2 + b1) * q^23 + (-3*b6 - 2*b5 + b4 + 3) * q^26 + (-b3 - 2*b2 - 2*b1) * q^28 + (-2*b6 - b5 + 2*b4) * q^29 + (-b6 - 2*b5 - 1) * q^31 + (3*b2 + 2*b1) * q^32 + (-5*b6 - b5) * q^34 + (b7 - 3*b3 - 2*b2 - 4*b1) * q^37 + (-b7 + 5*b3 - b2) * q^38 + (-b6 + b5 - 2*b4 + 3) * q^41 + (-2*b7 + b3 - 2*b1) * q^43 + (b6 + 2*b5 - b4 + 6) * q^44 + (b6 - 2*b5 + 4*b4 - 3) * q^46 + (-2*b7 - 5*b3 + b2) * q^47 + (-b6 + b5 - 3*b4 + 3) * q^49 + (b7 + 3*b3 - 2*b2 - b1) * q^52 + (6*b3 + 3*b2 - b1) * q^53 + (3*b6 + 6) * q^56 + (2*b7 + b3 - 5*b2 - 3*b1) * q^58 + (b6 + 3*b5 - 3*b4 + 6) * q^59 + (-2*b6 - b5 + 3*b4 - 4) * q^61 - 3*b2 * q^62 + (-2*b6 + b5 + b4 - 4) * q^64 + (-b7 + 3*b3 + b2 - 3*b1) * q^67 + (2*b7 + 5*b3 - 4*b2 - 4*b1) * q^68 + (b6 + 3*b4) * q^71 + (2*b7 - b3 + b2 - 3*b1) * q^73 + (4*b6 + 2*b5 - 7*b4 + 12) * q^74 + (4*b6 + b5 - 2*b4 + 2) * q^76 + (-2*b7 - 5*b3 - 2*b2 - b1) * q^77 + (b5 + 2*b4 + 1) * q^79 + (-2*b7 + 5*b3 + 2*b2 + 3*b1) * q^82 + (9*b3 + 3*b1) * q^83 + (5*b6 + 6) * q^86 + (b7 + b3 + 4*b2 + 2*b1) * q^88 + (-3*b6 - 3*b5 + 3*b4 - 3) * q^89 + (-b6 + 3*b5 - 2*b4 - 2) * q^91 + (2*b7 + 2*b3 - b2 - 2*b1) * q^92 + (-5*b6 - b5 + 3*b4) * q^94 + (-b7 - 5*b3 + 2*b2 - 2*b1) * q^97 + (-3*b7 + 6*b3 + 3*b2 + 4*b1) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q - 8 q^{4}+O(q^{10})$$ 8 * q - 8 * q^4 $$8 q - 8 q^{4} + 2 q^{11} + 6 q^{14} + 8 q^{16} - 4 q^{19} + 20 q^{26} + 2 q^{29} - 8 q^{31} - 18 q^{34} + 10 q^{41} + 44 q^{44} + 6 q^{49} + 60 q^{56} + 34 q^{59} - 26 q^{61} - 38 q^{64} + 16 q^{71} + 80 q^{74} + 22 q^{76} + 14 q^{79} + 68 q^{86} - 18 q^{89} - 34 q^{91} - 6 q^{94}+O(q^{100})$$ 8 * q - 8 * q^4 + 2 * q^11 + 6 * q^14 + 8 * q^16 - 4 * q^19 + 20 * q^26 + 2 * q^29 - 8 * q^31 - 18 * q^34 + 10 * q^41 + 44 * q^44 + 6 * q^49 + 60 * q^56 + 34 * q^59 - 26 * q^61 - 38 * q^64 + 16 * q^71 + 80 * q^74 + 22 * q^76 + 14 * q^79 + 68 * q^86 - 18 * q^89 - 34 * q^91 - 6 * q^94

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + 12x^{6} + 42x^{4} + 49x^{2} + 9$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{5} + 8\nu^{3} + 10\nu ) / 3$$ (v^5 + 8*v^3 + 10*v) / 3 $$\beta_{3}$$ $$=$$ $$( \nu^{7} + 10\nu^{5} + 23\nu^{3} + 11\nu ) / 3$$ (v^7 + 10*v^5 + 23*v^3 + 11*v) / 3 $$\beta_{4}$$ $$=$$ $$( -2\nu^{6} - 19\nu^{4} - 35\nu^{2} ) / 3$$ (-2*v^6 - 19*v^4 - 35*v^2) / 3 $$\beta_{5}$$ $$=$$ $$( \nu^{6} + 11\nu^{4} + 31\nu^{2} + 18 ) / 3$$ (v^6 + 11*v^4 + 31*v^2 + 18) / 3 $$\beta_{6}$$ $$=$$ $$( -2\nu^{6} - 19\nu^{4} - 38\nu^{2} - 9 ) / 3$$ (-2*v^6 - 19*v^4 - 38*v^2 - 9) / 3 $$\beta_{7}$$ $$=$$ $$( -\nu^{7} - 8\nu^{5} - 4\nu^{3} + 24\nu ) / 3$$ (-v^7 - 8*v^5 - 4*v^3 + 24*v) / 3
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$-\beta_{6} + \beta_{4} - 3$$ -b6 + b4 - 3 $$\nu^{3}$$ $$=$$ $$\beta_{7} + \beta_{3} - 2\beta_{2} - 5\beta_1$$ b7 + b3 - 2*b2 - 5*b1 $$\nu^{4}$$ $$=$$ $$9\beta_{6} + 2\beta_{5} - 8\beta_{4} + 15$$ 9*b6 + 2*b5 - 8*b4 + 15 $$\nu^{5}$$ $$=$$ $$-8\beta_{7} - 8\beta_{3} + 19\beta_{2} + 30\beta_1$$ -8*b7 - 8*b3 + 19*b2 + 30*b1 $$\nu^{6}$$ $$=$$ $$-68\beta_{6} - 19\beta_{5} + 57\beta_{4} - 90$$ -68*b6 - 19*b5 + 57*b4 - 90 $$\nu^{7}$$ $$=$$ $$57\beta_{7} + 60\beta_{3} - 144\beta_{2} - 196\beta_1$$ 57*b7 + 60*b3 - 144*b2 - 196*b1

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2025\mathbb{Z}\right)^\times$$.

 $$n$$ $$326$$ $$1702$$ $$\chi(n)$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
649.1
 − 2.63372i − 1.63372i − 1.47325i − 0.473255i 0.473255i 1.47325i 1.63372i 2.63372i
2.63372i 0 −4.93650 0 0 1.79743i 7.73393i 0 0
649.2 1.63372i 0 −0.669052 0 0 0.505348i 2.17440i 0 0
649.3 1.47325i 0 −0.170479 0 0 3.86583i 2.69535i 0 0
649.4 0.473255i 0 1.77603 0 0 2.56305i 1.78702i 0 0
649.5 0.473255i 0 1.77603 0 0 2.56305i 1.78702i 0 0
649.6 1.47325i 0 −0.170479 0 0 3.86583i 2.69535i 0 0
649.7 1.63372i 0 −0.669052 0 0 0.505348i 2.17440i 0 0
649.8 2.63372i 0 −4.93650 0 0 1.79743i 7.73393i 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 649.8 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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2025.2.b.o 8
3.b odd 2 1 2025.2.b.n 8
5.b even 2 1 inner 2025.2.b.o 8
5.c odd 4 1 2025.2.a.p 4
5.c odd 4 1 2025.2.a.z 4
9.c even 3 2 675.2.k.c 16
9.d odd 6 2 225.2.k.c 16
15.d odd 2 1 2025.2.b.n 8
15.e even 4 1 2025.2.a.q 4
15.e even 4 1 2025.2.a.y 4
45.h odd 6 2 225.2.k.c 16
45.j even 6 2 675.2.k.c 16
45.k odd 12 2 675.2.e.c 8
45.k odd 12 2 675.2.e.e 8
45.l even 12 2 225.2.e.c 8
45.l even 12 2 225.2.e.e yes 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
225.2.e.c 8 45.l even 12 2
225.2.e.e yes 8 45.l even 12 2
225.2.k.c 16 9.d odd 6 2
225.2.k.c 16 45.h odd 6 2
675.2.e.c 8 45.k odd 12 2
675.2.e.e 8 45.k odd 12 2
675.2.k.c 16 9.c even 3 2
675.2.k.c 16 45.j even 6 2
2025.2.a.p 4 5.c odd 4 1
2025.2.a.q 4 15.e even 4 1
2025.2.a.y 4 15.e even 4 1
2025.2.a.z 4 5.c odd 4 1
2025.2.b.n 8 3.b odd 2 1
2025.2.b.n 8 15.d odd 2 1
2025.2.b.o 8 1.a even 1 1 trivial
2025.2.b.o 8 5.b even 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(2025, [\chi])$$:

 $$T_{2}^{8} + 12T_{2}^{6} + 42T_{2}^{4} + 49T_{2}^{2} + 9$$ T2^8 + 12*T2^6 + 42*T2^4 + 49*T2^2 + 9 $$T_{11}^{4} - T_{11}^{3} - 25T_{11}^{2} - 41T_{11} - 9$$ T11^4 - T11^3 - 25*T11^2 - 41*T11 - 9

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8} + 12 T^{6} + 42 T^{4} + 49 T^{2} + \cdots + 9$$
$3$ $$T^{8}$$
$5$ $$T^{8}$$
$7$ $$T^{8} + 25 T^{6} + 174 T^{4} + \cdots + 81$$
$11$ $$(T^{4} - T^{3} - 25 T^{2} - 41 T - 9)^{2}$$
$13$ $$T^{8} + 64 T^{6} + 1134 T^{4} + \cdots + 11449$$
$17$ $$T^{8} + 81 T^{6} + 2214 T^{4} + \cdots + 91809$$
$19$ $$(T^{4} + 2 T^{3} - 27 T^{2} - 80 T - 25)^{2}$$
$23$ $$T^{8} + 111 T^{6} + 3033 T^{4} + \cdots + 59049$$
$29$ $$(T^{4} - T^{3} - 40 T^{2} + 142 T - 129)^{2}$$
$31$ $$(T^{4} + 4 T^{3} - 42 T^{2} - 27 T + 243)^{2}$$
$37$ $$T^{8} + 199 T^{6} + 9513 T^{4} + \cdots + 418609$$
$41$ $$(T^{4} - 5 T^{3} - 25 T^{2} + 161 T - 207)^{2}$$
$43$ $$T^{8} + 196 T^{6} + 12678 T^{4} + \cdots + 452929$$
$47$ $$T^{8} + 186 T^{6} + 7887 T^{4} + \cdots + 145161$$
$53$ $$T^{8} + 228 T^{6} + 13614 T^{4} + \cdots + 221841$$
$59$ $$(T^{4} - 17 T^{3} + 2 T^{2} + 914 T - 2313)^{2}$$
$61$ $$(T^{4} + 13 T^{3} - 3 T^{2} - 91 T - 1)^{2}$$
$67$ $$T^{8} + 217 T^{6} + 11982 T^{4} + \cdots + 59049$$
$71$ $$(T^{4} - 8 T^{3} - 40 T^{2} + 263 T + 381)^{2}$$
$73$ $$T^{8} + 196 T^{6} + 8478 T^{4} + \cdots + 12769$$
$79$ $$(T^{4} - 7 T^{3} - 33 T^{2} + 69 T + 207)^{2}$$
$83$ $$T^{8} + 324 T^{6} + 27702 T^{4} + \cdots + 531441$$
$89$ $$(T^{4} + 9 T^{3} - 99 T^{2} - 405 T + 2025)^{2}$$
$97$ $$T^{8} + 199 T^{6} + 10773 T^{4} + \cdots + 908209$$
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