# Properties

 Label 2175.2.a.bd Level $2175$ Weight $2$ Character orbit 2175.a Self dual yes Analytic conductor $17.367$ Analytic rank $0$ Dimension $8$ CM no Inner twists $1$

# Learn more

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2175,2,Mod(1,2175)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2175, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("2175.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2175 = 3 \cdot 5^{2} \cdot 29$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2175.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: $$17.3674624396$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: $$\mathbb{Q}[x]/(x^{8} - \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{8} - 2x^{7} - 12x^{6} + 23x^{5} + 36x^{4} - 62x^{3} - 15x^{2} + 14x + 2$$ x^8 - 2*x^7 - 12*x^6 + 23*x^5 + 36*x^4 - 62*x^3 - 15*x^2 + 14*x + 2 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 $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{2} + q^{3} + (\beta_{2} + 2) q^{4} + \beta_1 q^{6} + ( - \beta_{5} - \beta_{4} + \beta_1) q^{7} + (\beta_{6} - \beta_{5} + 2 \beta_1) q^{8} + q^{9}+O(q^{10})$$ q + b1 * q^2 + q^3 + (b2 + 2) * q^4 + b1 * q^6 + (-b5 - b4 + b1) * q^7 + (b6 - b5 + 2*b1) * q^8 + q^9 $$q + \beta_1 q^{2} + q^{3} + (\beta_{2} + 2) q^{4} + \beta_1 q^{6} + ( - \beta_{5} - \beta_{4} + \beta_1) q^{7} + (\beta_{6} - \beta_{5} + 2 \beta_1) q^{8} + q^{9} + (\beta_{4} + \beta_{3} - \beta_1 + 1) q^{11} + (\beta_{2} + 2) q^{12} + (\beta_{5} + \beta_{3} - 1) q^{13} + (\beta_{7} - \beta_{6} - \beta_{5} + \cdots + 2) q^{14}+ \cdots + (\beta_{4} + \beta_{3} - \beta_1 + 1) q^{99}+O(q^{100})$$ q + b1 * q^2 + q^3 + (b2 + 2) * q^4 + b1 * q^6 + (-b5 - b4 + b1) * q^7 + (b6 - b5 + 2*b1) * q^8 + q^9 + (b4 + b3 - b1 + 1) * q^11 + (b2 + 2) * q^12 + (b5 + b3 - 1) * q^13 + (b7 - b6 - b5 - 2*b4 - b3 + b2 + b1 + 2) * q^14 + (b7 + b6 - b3 + b2 + 4) * q^16 + (b6 + b2 - b1 + 2) * q^17 + b1 * q^18 + (-b7 - b6 + b5 + b4 + b3) * q^19 + (-b5 - b4 + b1) * q^21 + (2*b5 + b4 - b1 - 1) * q^22 + (-b6 - b4 - b3 - b2 + 2*b1 + 1) * q^23 + (b6 - b5 + 2*b1) * q^24 + (-b7 - b6 + 3*b5 + b4 + b3 + b2 - 4*b1 + 3) * q^26 + q^27 + (-b6 - 5*b5 - 2*b4 - b3 - 2*b2 + 6*b1 - 2) * q^28 + q^29 + (-b7 + b6 - b5 + b4 - b3 - 2*b2) * q^31 + (-b7 + b6 - 2*b5 + b4 - b2 + 3*b1 - 1) * q^32 + (b4 + b3 - b1 + 1) * q^33 + (2*b6 + b4 + 3*b1 - 3) * q^34 + (b2 + 2) * q^36 + (2*b7 - 2*b5) * q^37 + (-b6 + 4*b5 + b4 + b3 + b2 - 2*b1 + 3) * q^38 + (b5 + b3 - 1) * q^39 + (-b7 + b5 - b2 - 2*b1) * q^41 + (b7 - b6 - b5 - 2*b4 - b3 + b2 + b1 + 2) * q^42 + (-b7 + b6 + b5 + b4 - b3 - 2*b2 - 2) * q^43 + (-2*b7 + b6 + 2*b5 + b4 - b2 - b1 - 3) * q^44 + (-2*b6 - 2*b5 - 2*b4 + 2*b1 + 4) * q^46 + (-b7 - 2*b6 - b4 + b2 - b1 + 3) * q^47 + (b7 + b6 - b3 + b2 + 4) * q^48 + (b7 - b5 + b4 - b3 + b2 + b1 + 5) * q^49 + (b6 + b2 - b1 + 2) * q^51 + (-2*b7 + 3*b5 + 3*b4 + b3 - 3*b2 + b1 - 9) * q^52 + (-b6 + 2*b5 - b4 - b3 - b2 + 2*b1 - 1) * q^53 + b1 * q^54 + (3*b7 - 2*b6 - 4*b5 - 4*b4 - 3*b3 + 2*b2 + 10) * q^56 + (-b7 - b6 + b5 + b4 + b3) * q^57 + b1 * q^58 + (-b6 + 2*b5 + b4 + 3*b3 + b2 - 2*b1 + 3) * q^59 + (b7 - b6 + b5 - b4 + b3 - 2*b1 + 2) * q^61 + (2*b7 + b6 + 2*b5 + b4 - b3 + b2 - 2*b1 - 1) * q^62 + (-b5 - b4 + b1) * q^63 + (b7 - b6 + 2*b5 + 3*b2 - 2*b1 + 4) * q^64 + (2*b5 + b4 - b1 - 1) * q^66 + (-b7 - 2*b6 + 3*b5 - b4 - b3 - b2 - b1 - 4) * q^67 + (b6 + 2*b5 + 3*b4 + 3*b2 - 3*b1 + 11) * q^68 + (-b6 - b4 - b3 - b2 + 2*b1 + 1) * q^69 + (b6 - 2*b5 - b4 - b3 - b2 + 3) * q^71 + (b6 - b5 + 2*b1) * q^72 + (2*b7 + 2*b6 + 2*b2 + 2) * q^73 + (-6*b5 - 2*b4 - 2*b3 - 2*b2 + 2*b1 - 2) * q^74 + (-2*b7 + 2*b6 + 2*b5 + 2*b4 + 2*b3 - 2*b2 - 2) * q^76 + (b5 - b4 + 2*b3 + 2*b2 - 5*b1 - 1) * q^77 + (-b7 - b6 + 3*b5 + b4 + b3 + b2 - 4*b1 + 3) * q^78 + (2*b6 - 2*b3 + 2*b1) * q^79 + q^81 + (-b6 + 4*b5 + b4 + b3 - b2 - 3*b1 - 7) * q^82 + (-2*b6 - 2*b5 + 2*b3 + 2*b1 + 6) * q^83 + (-b6 - 5*b5 - 2*b4 - b3 - 2*b2 + 6*b1 - 2) * q^84 + (b6 + 4*b5 + 3*b4 + b3 + b2 - 6*b1 + 1) * q^86 + q^87 + (b6 + 4*b5 + 2*b4 + 2*b3 + 2*b2 - 6*b1 + 2) * q^88 + (b7 + 2*b6 - b3 - b2 - 4*b1 + 4) * q^89 + (b7 - 2*b6 + 4*b5 + 2*b4 + b3 - b2 - 6*b1 + 3) * q^91 + (2*b7 - 2*b6 - 4*b5 - 4*b4 + 2*b2 + 4*b1) * q^92 + (-b7 + b6 - b5 + b4 - b3 - 2*b2) * q^93 + (b7 - 2*b6 - b5 - 3*b4 - 2*b2 + 7*b1 - 7) * q^94 + (-b7 + b6 - 2*b5 + b4 - b2 + 3*b1 - 1) * q^96 + (-b7 + b6 + b5 + 3*b4 + b3 - 2*b1 - 4) * q^97 + (3*b6 - 6*b5 - b3 - b2 + 10*b1 + 2) * q^98 + (b4 + b3 - b1 + 1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q + 2 q^{2} + 8 q^{3} + 12 q^{4} + 2 q^{6} - 2 q^{7} + 3 q^{8} + 8 q^{9}+O(q^{10})$$ 8 * q + 2 * q^2 + 8 * q^3 + 12 * q^4 + 2 * q^6 - 2 * q^7 + 3 * q^8 + 8 * q^9 $$8 q + 2 q^{2} + 8 q^{3} + 12 q^{4} + 2 q^{6} - 2 q^{7} + 3 q^{8} + 8 q^{9} + 6 q^{11} + 12 q^{12} - 6 q^{13} + 9 q^{14} + 32 q^{16} + 12 q^{17} + 2 q^{18} - 2 q^{21} - 3 q^{22} + 14 q^{23} + 3 q^{24} + 18 q^{26} + 8 q^{27} - 14 q^{28} + 8 q^{29} + 8 q^{31} - 2 q^{32} + 6 q^{33} - 13 q^{34} + 12 q^{36} - 4 q^{37} + 26 q^{38} - 6 q^{39} + 2 q^{41} + 9 q^{42} - 2 q^{43} - 15 q^{44} + 24 q^{46} + 12 q^{47} + 32 q^{48} + 38 q^{49} + 12 q^{51} - 49 q^{52} + 4 q^{53} + 2 q^{54} + 58 q^{56} + 2 q^{58} + 18 q^{59} + 12 q^{61} - 4 q^{62} - 2 q^{63} + 21 q^{64} - 3 q^{66} - 26 q^{67} + 81 q^{68} + 14 q^{69} + 24 q^{71} + 3 q^{72} + 14 q^{73} - 22 q^{74} - 26 q^{77} + 18 q^{78} + 10 q^{79} + 8 q^{81} - 48 q^{82} + 40 q^{83} - 14 q^{84} + 8 q^{86} + 8 q^{87} + 10 q^{88} + 34 q^{89} + 26 q^{91} - 18 q^{92} + 8 q^{93} - 43 q^{94} - 2 q^{96} - 30 q^{97} + 29 q^{98} + 6 q^{99}+O(q^{100})$$ 8 * q + 2 * q^2 + 8 * q^3 + 12 * q^4 + 2 * q^6 - 2 * q^7 + 3 * q^8 + 8 * q^9 + 6 * q^11 + 12 * q^12 - 6 * q^13 + 9 * q^14 + 32 * q^16 + 12 * q^17 + 2 * q^18 - 2 * q^21 - 3 * q^22 + 14 * q^23 + 3 * q^24 + 18 * q^26 + 8 * q^27 - 14 * q^28 + 8 * q^29 + 8 * q^31 - 2 * q^32 + 6 * q^33 - 13 * q^34 + 12 * q^36 - 4 * q^37 + 26 * q^38 - 6 * q^39 + 2 * q^41 + 9 * q^42 - 2 * q^43 - 15 * q^44 + 24 * q^46 + 12 * q^47 + 32 * q^48 + 38 * q^49 + 12 * q^51 - 49 * q^52 + 4 * q^53 + 2 * q^54 + 58 * q^56 + 2 * q^58 + 18 * q^59 + 12 * q^61 - 4 * q^62 - 2 * q^63 + 21 * q^64 - 3 * q^66 - 26 * q^67 + 81 * q^68 + 14 * q^69 + 24 * q^71 + 3 * q^72 + 14 * q^73 - 22 * q^74 - 26 * q^77 + 18 * q^78 + 10 * q^79 + 8 * q^81 - 48 * q^82 + 40 * q^83 - 14 * q^84 + 8 * q^86 + 8 * q^87 + 10 * q^88 + 34 * q^89 + 26 * q^91 - 18 * q^92 + 8 * q^93 - 43 * q^94 - 2 * q^96 - 30 * q^97 + 29 * q^98 + 6 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 2x^{7} - 12x^{6} + 23x^{5} + 36x^{4} - 62x^{3} - 15x^{2} + 14x + 2$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - 4$$ v^2 - 4 $$\beta_{3}$$ $$=$$ $$\nu^{6} - 11\nu^{4} + 2\nu^{3} + 28\nu^{2} - 10\nu - 4$$ v^6 - 11*v^4 + 2*v^3 + 28*v^2 - 10*v - 4 $$\beta_{4}$$ $$=$$ $$\nu^{6} + \nu^{5} - 10\nu^{4} - 8\nu^{3} + 22\nu^{2} + 11\nu - 3$$ v^6 + v^5 - 10*v^4 - 8*v^3 + 22*v^2 + 11*v - 3 $$\beta_{5}$$ $$=$$ $$\nu^{7} - 11\nu^{5} + 2\nu^{4} + 29\nu^{3} - 11\nu^{2} - 8\nu + 2$$ v^7 - 11*v^5 + 2*v^4 + 29*v^3 - 11*v^2 - 8*v + 2 $$\beta_{6}$$ $$=$$ $$\nu^{7} - 11\nu^{5} + 2\nu^{4} + 30\nu^{3} - 11\nu^{2} - 14\nu + 2$$ v^7 - 11*v^5 + 2*v^4 + 30*v^3 - 11*v^2 - 14*v + 2 $$\beta_{7}$$ $$=$$ $$-\nu^{7} + \nu^{6} + 11\nu^{5} - 12\nu^{4} - 28\nu^{3} + 32\nu^{2} + 4\nu - 2$$ -v^7 + v^6 + 11*v^5 - 12*v^4 - 28*v^3 + 32*v^2 + 4*v - 2
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + 4$$ b2 + 4 $$\nu^{3}$$ $$=$$ $$\beta_{6} - \beta_{5} + 6\beta_1$$ b6 - b5 + 6*b1 $$\nu^{4}$$ $$=$$ $$\beta_{7} + \beta_{6} - \beta_{3} + 7\beta_{2} + 24$$ b7 + b6 - b3 + 7*b2 + 24 $$\nu^{5}$$ $$=$$ $$-\beta_{7} + 9\beta_{6} - 10\beta_{5} + \beta_{4} - \beta_{2} + 39\beta _1 - 1$$ -b7 + 9*b6 - 10*b5 + b4 - b2 + 39*b1 - 1 $$\nu^{6}$$ $$=$$ $$11\beta_{7} + 9\beta_{6} + 2\beta_{5} - 10\beta_{3} + 49\beta_{2} - 2\beta _1 + 156$$ 11*b7 + 9*b6 + 2*b5 - 10*b3 + 49*b2 - 2*b1 + 156 $$\nu^{7}$$ $$=$$ $$-13\beta_{7} + 68\beta_{6} - 80\beta_{5} + 11\beta_{4} + 2\beta_{3} - 14\beta_{2} + 263\beta _1 - 17$$ -13*b7 + 68*b6 - 80*b5 + 11*b4 + 2*b3 - 14*b2 + 263*b1 - 17

## 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
 −2.72810 −1.98486 −0.485464 −0.135002 0.510732 1.64893 2.57789 2.59587
−2.72810 1.00000 5.44251 0 −2.72810 −4.61695 −9.39150 1.00000 0
1.2 −1.98486 1.00000 1.93969 0 −1.98486 2.16633 0.119715 1.00000 0
1.3 −0.485464 1.00000 −1.76432 0 −0.485464 1.94912 1.82745 1.00000 0
1.4 −0.135002 1.00000 −1.98177 0 −0.135002 1.12340 0.537546 1.00000 0
1.5 0.510732 1.00000 −1.73915 0 0.510732 −4.82343 −1.90970 1.00000 0
1.6 1.64893 1.00000 0.718971 0 1.64893 1.44112 −2.11233 1.00000 0
1.7 2.57789 1.00000 4.64553 0 2.57789 4.69867 6.81989 1.00000 0
1.8 2.59587 1.00000 4.73855 0 2.59587 −3.93826 7.10893 1.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$-1$$
$$5$$ $$-1$$
$$29$$ $$-1$$

## 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 2175.2.a.bd yes 8
3.b odd 2 1 6525.2.a.by 8
5.b even 2 1 2175.2.a.bc 8
5.c odd 4 2 2175.2.c.p 16
15.d odd 2 1 6525.2.a.bz 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2175.2.a.bc 8 5.b even 2 1
2175.2.a.bd yes 8 1.a even 1 1 trivial
2175.2.c.p 16 5.c odd 4 2
6525.2.a.by 8 3.b odd 2 1
6525.2.a.bz 8 15.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(2175))$$:

 $$T_{2}^{8} - 2T_{2}^{7} - 12T_{2}^{6} + 23T_{2}^{5} + 36T_{2}^{4} - 62T_{2}^{3} - 15T_{2}^{2} + 14T_{2} + 2$$ T2^8 - 2*T2^7 - 12*T2^6 + 23*T2^5 + 36*T2^4 - 62*T2^3 - 15*T2^2 + 14*T2 + 2 $$T_{7}^{8} + 2T_{7}^{7} - 45T_{7}^{6} - 44T_{7}^{5} + 667T_{7}^{4} - 270T_{7}^{3} - 3428T_{7}^{2} + 5898T_{7} - 2817$$ T7^8 + 2*T7^7 - 45*T7^6 - 44*T7^5 + 667*T7^4 - 270*T7^3 - 3428*T7^2 + 5898*T7 - 2817

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8} - 2 T^{7} + \cdots + 2$$
$3$ $$(T - 1)^{8}$$
$5$ $$T^{8}$$
$7$ $$T^{8} + 2 T^{7} + \cdots - 2817$$
$11$ $$T^{8} - 6 T^{7} + \cdots - 1140$$
$13$ $$T^{8} + 6 T^{7} + \cdots - 10503$$
$17$ $$T^{8} - 12 T^{7} + \cdots + 18954$$
$19$ $$T^{8} - 99 T^{6} + \cdots + 205760$$
$23$ $$T^{8} - 14 T^{7} + \cdots - 14976$$
$29$ $$(T - 1)^{8}$$
$31$ $$T^{8} - 8 T^{7} + \cdots + 1225792$$
$37$ $$T^{8} + 4 T^{7} + \cdots + 20929536$$
$41$ $$T^{8} - 2 T^{7} + \cdots - 2160$$
$43$ $$T^{8} + 2 T^{7} + \cdots + 486720$$
$47$ $$T^{8} - 12 T^{7} + \cdots + 2390796$$
$53$ $$T^{8} - 4 T^{7} + \cdots + 10368$$
$59$ $$T^{8} - 18 T^{7} + \cdots + 10368$$
$61$ $$T^{8} - 12 T^{7} + \cdots + 386112$$
$67$ $$T^{8} + 26 T^{7} + \cdots - 48194129$$
$71$ $$T^{8} - 24 T^{7} + \cdots - 16256$$
$73$ $$T^{8} - 14 T^{7} + \cdots + 19650816$$
$79$ $$T^{8} - 10 T^{7} + \cdots + 2032128$$
$83$ $$T^{8} - 40 T^{7} + \cdots - 144320256$$
$89$ $$T^{8} - 34 T^{7} + \cdots - 12296640$$
$97$ $$T^{8} + 30 T^{7} + \cdots - 9904960$$
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