# Properties

 Label 1875.2.a.g Level $1875$ Weight $2$ Character orbit 1875.a Self dual yes Analytic conductor $14.972$ Analytic rank $1$ Dimension $4$ CM no Inner twists $1$

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Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1875,2,Mod(1,1875)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1875, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1875.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1875 = 3 \cdot 5^{4}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1875.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: $$14.9719503790$$ Analytic rank: $$1$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{15})^+$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{3} - 4x^{2} + 4x + 1$$ x^4 - x^3 - 4*x^2 + 4*x + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes 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 a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - \beta_{3} - \beta_{2}) q^{2} - q^{3} + \beta_1 q^{4} + (\beta_{3} + \beta_{2}) q^{6} + (2 \beta_{3} + \beta_{2} + 2) q^{7} + (\beta_{2} - 1) q^{8} + q^{9}+O(q^{10})$$ q + (-b3 - b2) * q^2 - q^3 + b1 * q^4 + (b3 + b2) * q^6 + (2*b3 + b2 + 2) * q^7 + (b2 - 1) * q^8 + q^9 $$q + ( - \beta_{3} - \beta_{2}) q^{2} - q^{3} + \beta_1 q^{4} + (\beta_{3} + \beta_{2}) q^{6} + (2 \beta_{3} + \beta_{2} + 2) q^{7} + (\beta_{2} - 1) q^{8} + q^{9} + ( - \beta_{2} - \beta_1 - 1) q^{11} - \beta_1 q^{12} + ( - 2 \beta_{3} + \beta_{2} + 2 \beta_1) q^{13} + ( - \beta_{3} - 2 \beta_{2} - 2 \beta_1 - 2) q^{14} + (\beta_{2} - 2 \beta_1 - 2) q^{16} + (2 \beta_{3} - 2 \beta_{2} - \beta_1) q^{17} + ( - \beta_{3} - \beta_{2}) q^{18} + ( - 2 \beta_{3} + \beta_{2} + 2 \beta_1 - 4) q^{19} + ( - 2 \beta_{3} - \beta_{2} - 2) q^{21} + (4 \beta_{3} + 2 \beta_{2} + 3) q^{22} + ( - 2 \beta_1 + 3) q^{23} + ( - \beta_{2} + 1) q^{24} + ( - 7 \beta_{3} - 2 \beta_{2} + 2 \beta_1 - 4) q^{26} - q^{27} + (3 \beta_{3} + 2 \beta_{2} + \beta_1 + 2) q^{28} + (\beta_{3} - \beta_{2} - \beta_1 - 6) q^{29} + (2 \beta_{3} + \beta_{2} - 3 \beta_1 - 1) q^{31} + (5 \beta_{3} + 2 \beta_{2} + 2) q^{32} + (\beta_{2} + \beta_1 + 1) q^{33} + (6 \beta_{3} + \beta_{2} - 2 \beta_1 + 5) q^{34} + \beta_1 q^{36} + ( - 4 \beta_{3} + \beta_{2} - 3 \beta_1 - 4) q^{37} + ( - 3 \beta_{3} + 2 \beta_{2} + 2 \beta_1 - 4) q^{38} + (2 \beta_{3} - \beta_{2} - 2 \beta_1) q^{39} + (2 \beta_{3} + 3 \beta_{2} - 3 \beta_1 + 1) q^{41} + (\beta_{3} + 2 \beta_{2} + 2 \beta_1 + 2) q^{42} + ( - 6 \beta_{3} - 2 \beta_{2} + 3 \beta_1 - 3) q^{43} + ( - \beta_{3} - \beta_{2} - 2 \beta_1 - 2) q^{44} + (\beta_{3} - \beta_{2} + 2) q^{46} + (\beta_{3} + 3 \beta_{2} - 4 \beta_1 - 5) q^{47} + ( - \beta_{2} + 2 \beta_1 + 2) q^{48} + (5 \beta_{3} + 4 \beta_{2} + 3 \beta_1) q^{49} + ( - 2 \beta_{3} + 2 \beta_{2} + \beta_1) q^{51} + ( - \beta_{3} + 3 \beta_1 + 2) q^{52} + (5 \beta_{3} + 6 \beta_{2} + 1) q^{53} + (\beta_{3} + \beta_{2}) q^{54} + ( - \beta_{3} + \beta_{2} + \beta_1 - 1) q^{56} + (2 \beta_{3} - \beta_{2} - 2 \beta_1 + 4) q^{57} + (10 \beta_{3} + 7 \beta_{2} - \beta_1 + 3) q^{58} + (5 \beta_{3} - 3 \beta_{2} + \beta_1 + 4) q^{59} + ( - 3 \beta_{3} - 2 \beta_{2} + \beta_1 - 12) q^{61} + (8 \beta_{3} + 4 \beta_{2} - 2 \beta_1 + 1) q^{62} + (2 \beta_{3} + \beta_{2} + 2) q^{63} + (\beta_{3} - 4 \beta_{2} - \beta_1) q^{64} + ( - 4 \beta_{3} - 2 \beta_{2} - 3) q^{66} + ( - \beta_{3} - 4 \beta_{2} + 2 \beta_1 + 2) q^{67} + (\beta_{2} - 4 \beta_1) q^{68} + (2 \beta_1 - 3) q^{69} + ( - 3 \beta_{3} + \beta_{2} + 6 \beta_1 - 10) q^{71} + (\beta_{2} - 1) q^{72} + (8 \beta_{3} - \beta_1 + 8) q^{73} + (5 \beta_{3} + 7 \beta_{2} + 4 \beta_1 + 1) q^{74} + ( - \beta_{3} - \beta_1 + 2) q^{76} + ( - 6 \beta_{3} - 5 \beta_{2} - 2 \beta_1 - 5) q^{77} + (7 \beta_{3} + 2 \beta_{2} - 2 \beta_1 + 4) q^{78} + ( - 3 \beta_{3} + 4 \beta_{2} + 4 \beta_1 - 1) q^{79} + q^{81} + (4 \beta_{3} + 2 \beta_{2} - 2 \beta_1 - 3) q^{82} + ( - 5 \beta_{3} + \beta_{2} + 8 \beta_1 - 4) q^{83} + ( - 3 \beta_{3} - 2 \beta_{2} - \beta_1 - 2) q^{84} + ( - 7 \beta_{3} + 6 \beta_1 + 1) q^{86} + ( - \beta_{3} + \beta_{2} + \beta_1 + 6) q^{87} + ( - 2 \beta_{3} + \beta_1 - 2) q^{88} + ( - 4 \beta_{3} - \beta_{2} - 4) q^{89} + (7 \beta_{3} + 6 \beta_{2} + \beta_1 + 3) q^{91} + ( - 2 \beta_{2} + 3 \beta_1 - 4) q^{92} + ( - 2 \beta_{3} - \beta_{2} + 3 \beta_1 + 1) q^{93} + (11 \beta_{3} + 9 \beta_{2} - \beta_1 - 2) q^{94} + ( - 5 \beta_{3} - 2 \beta_{2} - 2) q^{96} + (4 \beta_{3} - 3 \beta_{2} - 8 \beta_1 + 8) q^{97} + ( - 5 \beta_{3} - 3 \beta_{2} - 5 \beta_1 - 11) q^{98} + ( - \beta_{2} - \beta_1 - 1) q^{99}+O(q^{100})$$ q + (-b3 - b2) * q^2 - q^3 + b1 * q^4 + (b3 + b2) * q^6 + (2*b3 + b2 + 2) * q^7 + (b2 - 1) * q^8 + q^9 + (-b2 - b1 - 1) * q^11 - b1 * q^12 + (-2*b3 + b2 + 2*b1) * q^13 + (-b3 - 2*b2 - 2*b1 - 2) * q^14 + (b2 - 2*b1 - 2) * q^16 + (2*b3 - 2*b2 - b1) * q^17 + (-b3 - b2) * q^18 + (-2*b3 + b2 + 2*b1 - 4) * q^19 + (-2*b3 - b2 - 2) * q^21 + (4*b3 + 2*b2 + 3) * q^22 + (-2*b1 + 3) * q^23 + (-b2 + 1) * q^24 + (-7*b3 - 2*b2 + 2*b1 - 4) * q^26 - q^27 + (3*b3 + 2*b2 + b1 + 2) * q^28 + (b3 - b2 - b1 - 6) * q^29 + (2*b3 + b2 - 3*b1 - 1) * q^31 + (5*b3 + 2*b2 + 2) * q^32 + (b2 + b1 + 1) * q^33 + (6*b3 + b2 - 2*b1 + 5) * q^34 + b1 * q^36 + (-4*b3 + b2 - 3*b1 - 4) * q^37 + (-3*b3 + 2*b2 + 2*b1 - 4) * q^38 + (2*b3 - b2 - 2*b1) * q^39 + (2*b3 + 3*b2 - 3*b1 + 1) * q^41 + (b3 + 2*b2 + 2*b1 + 2) * q^42 + (-6*b3 - 2*b2 + 3*b1 - 3) * q^43 + (-b3 - b2 - 2*b1 - 2) * q^44 + (b3 - b2 + 2) * q^46 + (b3 + 3*b2 - 4*b1 - 5) * q^47 + (-b2 + 2*b1 + 2) * q^48 + (5*b3 + 4*b2 + 3*b1) * q^49 + (-2*b3 + 2*b2 + b1) * q^51 + (-b3 + 3*b1 + 2) * q^52 + (5*b3 + 6*b2 + 1) * q^53 + (b3 + b2) * q^54 + (-b3 + b2 + b1 - 1) * q^56 + (2*b3 - b2 - 2*b1 + 4) * q^57 + (10*b3 + 7*b2 - b1 + 3) * q^58 + (5*b3 - 3*b2 + b1 + 4) * q^59 + (-3*b3 - 2*b2 + b1 - 12) * q^61 + (8*b3 + 4*b2 - 2*b1 + 1) * q^62 + (2*b3 + b2 + 2) * q^63 + (b3 - 4*b2 - b1) * q^64 + (-4*b3 - 2*b2 - 3) * q^66 + (-b3 - 4*b2 + 2*b1 + 2) * q^67 + (b2 - 4*b1) * q^68 + (2*b1 - 3) * q^69 + (-3*b3 + b2 + 6*b1 - 10) * q^71 + (b2 - 1) * q^72 + (8*b3 - b1 + 8) * q^73 + (5*b3 + 7*b2 + 4*b1 + 1) * q^74 + (-b3 - b1 + 2) * q^76 + (-6*b3 - 5*b2 - 2*b1 - 5) * q^77 + (7*b3 + 2*b2 - 2*b1 + 4) * q^78 + (-3*b3 + 4*b2 + 4*b1 - 1) * q^79 + q^81 + (4*b3 + 2*b2 - 2*b1 - 3) * q^82 + (-5*b3 + b2 + 8*b1 - 4) * q^83 + (-3*b3 - 2*b2 - b1 - 2) * q^84 + (-7*b3 + 6*b1 + 1) * q^86 + (-b3 + b2 + b1 + 6) * q^87 + (-2*b3 + b1 - 2) * q^88 + (-4*b3 - b2 - 4) * q^89 + (7*b3 + 6*b2 + b1 + 3) * q^91 + (-2*b2 + 3*b1 - 4) * q^92 + (-2*b3 - b2 + 3*b1 + 1) * q^93 + (11*b3 + 9*b2 - b1 - 2) * q^94 + (-5*b3 - 2*b2 - 2) * q^96 + (4*b3 - 3*b2 - 8*b1 + 8) * q^97 + (-5*b3 - 3*b2 - 5*b1 - 11) * q^98 + (-b2 - b1 - 1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + q^{2} - 4 q^{3} + q^{4} - q^{6} + 5 q^{7} - 3 q^{8} + 4 q^{9}+O(q^{10})$$ 4 * q + q^2 - 4 * q^3 + q^4 - q^6 + 5 * q^7 - 3 * q^8 + 4 * q^9 $$4 q + q^{2} - 4 q^{3} + q^{4} - q^{6} + 5 q^{7} - 3 q^{8} + 4 q^{9} - 6 q^{11} - q^{12} + 7 q^{13} - 10 q^{14} - 9 q^{16} - 7 q^{17} + q^{18} - 9 q^{19} - 5 q^{21} + 6 q^{22} + 10 q^{23} + 3 q^{24} - 2 q^{26} - 4 q^{27} + 5 q^{28} - 28 q^{29} - 10 q^{31} + 6 q^{33} + 7 q^{34} + q^{36} - 10 q^{37} - 6 q^{38} - 7 q^{39} + 10 q^{42} + q^{43} - 9 q^{44} + 5 q^{46} - 23 q^{47} + 9 q^{48} - 3 q^{49} + 7 q^{51} + 13 q^{52} - q^{54} + 9 q^{57} - 2 q^{58} + 4 q^{59} - 43 q^{61} - 10 q^{62} + 5 q^{63} - 7 q^{64} - 6 q^{66} + 8 q^{67} - 3 q^{68} - 10 q^{69} - 27 q^{71} - 3 q^{72} + 15 q^{73} + 5 q^{74} + 9 q^{76} - 15 q^{77} + 2 q^{78} + 10 q^{79} + 4 q^{81} - 20 q^{82} + 3 q^{83} - 5 q^{84} + 24 q^{86} + 28 q^{87} - 3 q^{88} - 9 q^{89} + 5 q^{91} - 15 q^{92} + 10 q^{93} - 22 q^{94} + 13 q^{97} - 42 q^{98} - 6 q^{99}+O(q^{100})$$ 4 * q + q^2 - 4 * q^3 + q^4 - q^6 + 5 * q^7 - 3 * q^8 + 4 * q^9 - 6 * q^11 - q^12 + 7 * q^13 - 10 * q^14 - 9 * q^16 - 7 * q^17 + q^18 - 9 * q^19 - 5 * q^21 + 6 * q^22 + 10 * q^23 + 3 * q^24 - 2 * q^26 - 4 * q^27 + 5 * q^28 - 28 * q^29 - 10 * q^31 + 6 * q^33 + 7 * q^34 + q^36 - 10 * q^37 - 6 * q^38 - 7 * q^39 + 10 * q^42 + q^43 - 9 * q^44 + 5 * q^46 - 23 * q^47 + 9 * q^48 - 3 * q^49 + 7 * q^51 + 13 * q^52 - q^54 + 9 * q^57 - 2 * q^58 + 4 * q^59 - 43 * q^61 - 10 * q^62 + 5 * q^63 - 7 * q^64 - 6 * q^66 + 8 * q^67 - 3 * q^68 - 10 * q^69 - 27 * q^71 - 3 * q^72 + 15 * q^73 + 5 * q^74 + 9 * q^76 - 15 * q^77 + 2 * q^78 + 10 * q^79 + 4 * q^81 - 20 * q^82 + 3 * q^83 - 5 * q^84 + 24 * q^86 + 28 * q^87 - 3 * q^88 - 9 * q^89 + 5 * q^91 - 15 * q^92 + 10 * q^93 - 22 * q^94 + 13 * q^97 - 42 * q^98 - 6 * q^99

Basis of coefficient ring in terms of $$\nu = \zeta_{15} + \zeta_{15}^{-1}$$:

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - 2$$ v^2 - 2 $$\beta_{3}$$ $$=$$ $$\nu^{3} - 3\nu$$ v^3 - 3*v
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + 2$$ b2 + 2 $$\nu^{3}$$ $$=$$ $$\beta_{3} + 3\beta_1$$ b3 + 3*b1

## 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
 1.82709 −1.95630 −0.209057 1.33826
−1.95630 −1.00000 1.82709 0 1.95630 4.57433 0.338261 1.00000 0
1.2 −0.209057 −1.00000 −1.95630 0 0.209057 0.591023 0.827091 1.00000 0
1.3 1.33826 −1.00000 −0.209057 0 −1.33826 1.27977 −2.95630 1.00000 0
1.4 1.82709 −1.00000 1.33826 0 −1.82709 −1.44512 −1.20906 1.00000 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 1875.2.a.g yes 4
3.b odd 2 1 5625.2.a.j 4
5.b even 2 1 1875.2.a.f 4
5.c odd 4 2 1875.2.b.d 8
15.d odd 2 1 5625.2.a.m 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1875.2.a.f 4 5.b even 2 1
1875.2.a.g yes 4 1.a even 1 1 trivial
1875.2.b.d 8 5.c odd 4 2
5625.2.a.j 4 3.b odd 2 1
5625.2.a.m 4 15.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{4} - T_{2}^{3} - 4T_{2}^{2} + 4T_{2} + 1$$ acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(1875))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} - T^{3} - 4 T^{2} + 4 T + 1$$
$3$ $$(T + 1)^{4}$$
$5$ $$T^{4}$$
$7$ $$T^{4} - 5 T^{3} + 10 T - 5$$
$11$ $$T^{4} + 6 T^{3} + 6 T^{2} - 9 T - 9$$
$13$ $$T^{4} - 7 T^{3} - 6 T^{2} + 92 T - 89$$
$17$ $$T^{4} + 7 T^{3} - 16 T^{2} - 202 T - 359$$
$19$ $$T^{4} + 9 T^{3} + 6 T^{2} - 36 T - 9$$
$23$ $$T^{4} - 10 T^{3} + 20 T^{2} + 10 T - 5$$
$29$ $$T^{4} + 28 T^{3} + 284 T^{2} + \cdots + 1801$$
$31$ $$T^{4} + 10 T^{3} - 125 T - 125$$
$37$ $$T^{4} + 10 T^{3} - 90 T^{2} + \cdots - 1475$$
$41$ $$T^{4} - 70 T^{2} - 135 T + 145$$
$43$ $$T^{4} - T^{3} - 79 T^{2} - 341 T - 419$$
$47$ $$T^{4} + 23 T^{3} + 89 T^{2} + \cdots - 6089$$
$53$ $$T^{4} - 145 T^{2} - 270 T + 2995$$
$59$ $$T^{4} - 4 T^{3} - 154 T^{2} + \cdots + 1531$$
$61$ $$T^{4} + 43 T^{3} + 669 T^{2} + \cdots + 10261$$
$67$ $$T^{4} - 8 T^{3} - 61 T^{2} + 458 T + 151$$
$71$ $$T^{4} + 27 T^{3} + 134 T^{2} + \cdots + 271$$
$73$ $$T^{4} - 15 T^{3} - 60 T^{2} + \cdots + 2745$$
$79$ $$T^{4} - 10 T^{3} - 105 T^{2} + \cdots - 3155$$
$83$ $$T^{4} - 3 T^{3} - 246 T^{2} + \cdots + 13491$$
$89$ $$T^{4} + 9 T^{3} - 4 T^{2} - 96 T + 61$$
$97$ $$T^{4} - 13 T^{3} - 216 T^{2} + \cdots + 14701$$
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