# Properties

 Label 2025.4.a.g Level $2025$ Weight $4$ Character orbit 2025.a Self dual yes Analytic conductor $119.479$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2025,4,Mod(1,2025)]

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, 0]))

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

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

chi := DirichletCharacter("2025.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2025 = 3^{4} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 2025.a (trivial)

## 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: yes Analytic conductor: $$119.478867762$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{33})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 8$$ x^2 - x - 8 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 9) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-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 $$\beta = \frac{1}{2}(1 + \sqrt{33})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - \beta - 1) q^{2} + (3 \beta + 1) q^{4} + ( - 3 \beta - 2) q^{7} + (\beta - 17) q^{8}+O(q^{10})$$ q + (-b - 1) * q^2 + (3*b + 1) * q^4 + (-3*b - 2) * q^7 + (b - 17) * q^8 $$q + ( - \beta - 1) q^{2} + (3 \beta + 1) q^{4} + ( - 3 \beta - 2) q^{7} + (\beta - 17) q^{8} + ( - 8 \beta + 37) q^{11} + (15 \beta - 2) q^{13} + (8 \beta + 26) q^{14} + ( - 9 \beta + 1) q^{16} + ( - 9 \beta - 45) q^{17} + ( - 27 \beta - 25) q^{19} + ( - 21 \beta + 27) q^{22} + (19 \beta - 26) q^{23} + ( - 28 \beta - 118) q^{26} + ( - 18 \beta - 74) q^{28} + (\beta - 26) q^{29} + (3 \beta + 20) q^{31} + (9 \beta + 207) q^{32} + (63 \beta + 117) q^{34} + ( - 54 \beta + 52) q^{37} + (79 \beta + 241) q^{38} + (98 \beta + 17) q^{41} + (6 \beta - 47) q^{43} + (79 \beta - 155) q^{44} + ( - 12 \beta - 126) q^{46} + ( - 91 \beta - 154) q^{47} + (21 \beta - 267) q^{49} + (54 \beta + 358) q^{52} + (162 \beta - 108) q^{53} + (46 \beta + 10) q^{56} + (24 \beta + 18) q^{58} + ( - 136 \beta + 467) q^{59} + ( - 105 \beta + 272) q^{61} + ( - 26 \beta - 44) q^{62} + ( - 153 \beta - 287) q^{64} + ( - 66 \beta - 461) q^{67} + ( - 171 \beta - 261) q^{68} + (144 \beta + 612) q^{71} + ( - 243 \beta + 349) q^{73} + (56 \beta + 380) q^{74} + ( - 183 \beta - 673) q^{76} + ( - 71 \beta + 118) q^{77} + (309 \beta - 556) q^{79} + ( - 213 \beta - 801) q^{82} + (107 \beta - 460) q^{83} + (35 \beta - 1) q^{86} + (165 \beta - 693) q^{88} + (72 \beta - 234) q^{89} + ( - 69 \beta - 356) q^{91} + ( - 2 \beta + 430) q^{92} + (336 \beta + 882) q^{94} + ( - 102 \beta - 317) q^{97} + (225 \beta + 99) q^{98}+O(q^{100})$$ q + (-b - 1) * q^2 + (3*b + 1) * q^4 + (-3*b - 2) * q^7 + (b - 17) * q^8 + (-8*b + 37) * q^11 + (15*b - 2) * q^13 + (8*b + 26) * q^14 + (-9*b + 1) * q^16 + (-9*b - 45) * q^17 + (-27*b - 25) * q^19 + (-21*b + 27) * q^22 + (19*b - 26) * q^23 + (-28*b - 118) * q^26 + (-18*b - 74) * q^28 + (b - 26) * q^29 + (3*b + 20) * q^31 + (9*b + 207) * q^32 + (63*b + 117) * q^34 + (-54*b + 52) * q^37 + (79*b + 241) * q^38 + (98*b + 17) * q^41 + (6*b - 47) * q^43 + (79*b - 155) * q^44 + (-12*b - 126) * q^46 + (-91*b - 154) * q^47 + (21*b - 267) * q^49 + (54*b + 358) * q^52 + (162*b - 108) * q^53 + (46*b + 10) * q^56 + (24*b + 18) * q^58 + (-136*b + 467) * q^59 + (-105*b + 272) * q^61 + (-26*b - 44) * q^62 + (-153*b - 287) * q^64 + (-66*b - 461) * q^67 + (-171*b - 261) * q^68 + (144*b + 612) * q^71 + (-243*b + 349) * q^73 + (56*b + 380) * q^74 + (-183*b - 673) * q^76 + (-71*b + 118) * q^77 + (309*b - 556) * q^79 + (-213*b - 801) * q^82 + (107*b - 460) * q^83 + (35*b - 1) * q^86 + (165*b - 693) * q^88 + (72*b - 234) * q^89 + (-69*b - 356) * q^91 + (-2*b + 430) * q^92 + (336*b + 882) * q^94 + (-102*b - 317) * q^97 + (225*b + 99) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 3 q^{2} + 5 q^{4} - 7 q^{7} - 33 q^{8}+O(q^{10})$$ 2 * q - 3 * q^2 + 5 * q^4 - 7 * q^7 - 33 * q^8 $$2 q - 3 q^{2} + 5 q^{4} - 7 q^{7} - 33 q^{8} + 66 q^{11} + 11 q^{13} + 60 q^{14} - 7 q^{16} - 99 q^{17} - 77 q^{19} + 33 q^{22} - 33 q^{23} - 264 q^{26} - 166 q^{28} - 51 q^{29} + 43 q^{31} + 423 q^{32} + 297 q^{34} + 50 q^{37} + 561 q^{38} + 132 q^{41} - 88 q^{43} - 231 q^{44} - 264 q^{46} - 399 q^{47} - 513 q^{49} + 770 q^{52} - 54 q^{53} + 66 q^{56} + 60 q^{58} + 798 q^{59} + 439 q^{61} - 114 q^{62} - 727 q^{64} - 988 q^{67} - 693 q^{68} + 1368 q^{71} + 455 q^{73} + 816 q^{74} - 1529 q^{76} + 165 q^{77} - 803 q^{79} - 1815 q^{82} - 813 q^{83} + 33 q^{86} - 1221 q^{88} - 396 q^{89} - 781 q^{91} + 858 q^{92} + 2100 q^{94} - 736 q^{97} + 423 q^{98}+O(q^{100})$$ 2 * q - 3 * q^2 + 5 * q^4 - 7 * q^7 - 33 * q^8 + 66 * q^11 + 11 * q^13 + 60 * q^14 - 7 * q^16 - 99 * q^17 - 77 * q^19 + 33 * q^22 - 33 * q^23 - 264 * q^26 - 166 * q^28 - 51 * q^29 + 43 * q^31 + 423 * q^32 + 297 * q^34 + 50 * q^37 + 561 * q^38 + 132 * q^41 - 88 * q^43 - 231 * q^44 - 264 * q^46 - 399 * q^47 - 513 * q^49 + 770 * q^52 - 54 * q^53 + 66 * q^56 + 60 * q^58 + 798 * q^59 + 439 * q^61 - 114 * q^62 - 727 * q^64 - 988 * q^67 - 693 * q^68 + 1368 * q^71 + 455 * q^73 + 816 * q^74 - 1529 * q^76 + 165 * q^77 - 803 * q^79 - 1815 * q^82 - 813 * q^83 + 33 * q^86 - 1221 * q^88 - 396 * q^89 - 781 * q^91 + 858 * q^92 + 2100 * q^94 - 736 * q^97 + 423 * q^98

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 3.37228 −2.37228
−4.37228 0 11.1168 0 0 −12.1168 −13.6277 0 0
1.2 1.37228 0 −6.11684 0 0 5.11684 −19.3723 0 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$1$$
$$5$$ $$1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2025.4.a.g 2
3.b odd 2 1 2025.4.a.n 2
5.b even 2 1 81.4.a.d 2
9.c even 3 2 225.4.e.b 4
15.d odd 2 1 81.4.a.a 2
20.d odd 2 1 1296.4.a.u 2
45.h odd 6 2 27.4.c.a 4
45.j even 6 2 9.4.c.a 4
45.k odd 12 4 225.4.k.b 8
60.h even 2 1 1296.4.a.i 2
180.n even 6 2 432.4.i.c 4
180.p odd 6 2 144.4.i.c 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
9.4.c.a 4 45.j even 6 2
27.4.c.a 4 45.h odd 6 2
81.4.a.a 2 15.d odd 2 1
81.4.a.d 2 5.b even 2 1
144.4.i.c 4 180.p odd 6 2
225.4.e.b 4 9.c even 3 2
225.4.k.b 8 45.k odd 12 4
432.4.i.c 4 180.n even 6 2
1296.4.a.i 2 60.h even 2 1
1296.4.a.u 2 20.d odd 2 1
2025.4.a.g 2 1.a even 1 1 trivial
2025.4.a.n 2 3.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(2025))$$:

 $$T_{2}^{2} + 3T_{2} - 6$$ T2^2 + 3*T2 - 6 $$T_{7}^{2} + 7T_{7} - 62$$ T7^2 + 7*T7 - 62 $$T_{11}^{2} - 66T_{11} + 561$$ T11^2 - 66*T11 + 561

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 3T - 6$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 7T - 62$$
$11$ $$T^{2} - 66T + 561$$
$13$ $$T^{2} - 11T - 1826$$
$17$ $$T^{2} + 99T + 1782$$
$19$ $$T^{2} + 77T - 4532$$
$23$ $$T^{2} + 33T - 2706$$
$29$ $$T^{2} + 51T + 642$$
$31$ $$T^{2} - 43T + 388$$
$37$ $$T^{2} - 50T - 23432$$
$41$ $$T^{2} - 132T - 74877$$
$43$ $$T^{2} + 88T + 1639$$
$47$ $$T^{2} + 399T - 28518$$
$53$ $$T^{2} + 54T - 215784$$
$59$ $$T^{2} - 798T + 6609$$
$61$ $$T^{2} - 439T - 42776$$
$67$ $$T^{2} + 988T + 208099$$
$71$ $$T^{2} - 1368 T + 296784$$
$73$ $$T^{2} - 455T - 435398$$
$79$ $$T^{2} + 803T - 626516$$
$83$ $$T^{2} + 813T + 70788$$
$89$ $$T^{2} + 396T - 3564$$
$97$ $$T^{2} + 736T + 49591$$