# Properties

 Label 2535.2.a.bd Level $2535$ Weight $2$ Character orbit 2535.a Self dual yes Analytic conductor $20.242$ Analytic rank $1$ Dimension $3$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2535,2,Mod(1,2535)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2535, 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("2535.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2535 = 3 \cdot 5 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2535.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: $$20.2420769124$$ Analytic rank: $$1$$ Dimension: $$3$$ Coefficient field: $$\Q(\zeta_{14})^+$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - x^{2} - 2x + 1$$ x^3 - x^2 - 2*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 $$1,\beta_1,\beta_2$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{2} - q^{3} + \beta_{2} q^{4} - q^{5} - \beta_1 q^{6} + (\beta_1 - 1) q^{7} + (\beta_{2} - 2 \beta_1 + 1) q^{8} + q^{9}+O(q^{10})$$ q + b1 * q^2 - q^3 + b2 * q^4 - q^5 - b1 * q^6 + (b1 - 1) * q^7 + (b2 - 2*b1 + 1) * q^8 + q^9 $$q + \beta_1 q^{2} - q^{3} + \beta_{2} q^{4} - q^{5} - \beta_1 q^{6} + (\beta_1 - 1) q^{7} + (\beta_{2} - 2 \beta_1 + 1) q^{8} + q^{9} - \beta_1 q^{10} + q^{11} - \beta_{2} q^{12} + (\beta_{2} - \beta_1 + 2) q^{14} + q^{15} + ( - 3 \beta_{2} + \beta_1 - 3) q^{16} + (2 \beta_1 - 4) q^{17} + \beta_1 q^{18} + ( - 3 \beta_{2} + 1) q^{19} - \beta_{2} q^{20} + ( - \beta_1 + 1) q^{21} + \beta_1 q^{22} + ( - \beta_{2} + \beta_1 + 1) q^{23} + ( - \beta_{2} + 2 \beta_1 - 1) q^{24} + q^{25} - q^{27} + q^{28} + ( - 3 \beta_{2} - 2 \beta_1) q^{29} + \beta_1 q^{30} + (4 \beta_{2} - 2 \beta_1 + 1) q^{31} + ( - 4 \beta_{2} + \beta_1 - 3) q^{32} - q^{33} + (2 \beta_{2} - 4 \beta_1 + 4) q^{34} + ( - \beta_1 + 1) q^{35} + \beta_{2} q^{36} + (\beta_{2} - 4 \beta_1) q^{37} + ( - 3 \beta_{2} + \beta_1 - 3) q^{38} + ( - \beta_{2} + 2 \beta_1 - 1) q^{40} + (\beta_1 + 7) q^{41} + ( - \beta_{2} + \beta_1 - 2) q^{42} + ( - \beta_1 - 5) q^{43} + \beta_{2} q^{44} - q^{45} + (\beta_1 + 1) q^{46} + ( - 4 \beta_{2} - \beta_1 - 5) q^{47} + (3 \beta_{2} - \beta_1 + 3) q^{48} + (\beta_{2} - 2 \beta_1 - 4) q^{49} + \beta_1 q^{50} + ( - 2 \beta_1 + 4) q^{51} + (6 \beta_{2} - 7 \beta_1 + 6) q^{53} - \beta_1 q^{54} - q^{55} + ( - 2 \beta_{2} + 3 \beta_1 - 4) q^{56} + (3 \beta_{2} - 1) q^{57} + ( - 5 \beta_{2} - 7) q^{58} + (2 \beta_{2} - 7 \beta_1 + 2) q^{59} + \beta_{2} q^{60} + ( - 7 \beta_{2} + 7 \beta_1 - 8) q^{61} + (2 \beta_{2} + \beta_1) q^{62} + (\beta_1 - 1) q^{63} + (3 \beta_{2} - 5 \beta_1 + 4) q^{64} - \beta_1 q^{66} + (3 \beta_{2} - \beta_1 + 8) q^{67} + ( - 2 \beta_{2} + 2) q^{68} + (\beta_{2} - \beta_1 - 1) q^{69} + ( - \beta_{2} + \beta_1 - 2) q^{70} + (3 \beta_{2} - 6 \beta_1 + 1) q^{71} + (\beta_{2} - 2 \beta_1 + 1) q^{72} + (7 \beta_{2} - 10 \beta_1 + 5) q^{73} + ( - 3 \beta_{2} - 7) q^{74} - q^{75} + (4 \beta_{2} - 3 \beta_1 - 3) q^{76} + (\beta_1 - 1) q^{77} + ( - 2 \beta_{2} - 7 \beta_1 + 1) q^{79} + (3 \beta_{2} - \beta_1 + 3) q^{80} + q^{81} + (\beta_{2} + 7 \beta_1 + 2) q^{82} + (5 \beta_{2} + 2 \beta_1 - 8) q^{83} - q^{84} + ( - 2 \beta_1 + 4) q^{85} + ( - \beta_{2} - 5 \beta_1 - 2) q^{86} + (3 \beta_{2} + 2 \beta_1) q^{87} + (\beta_{2} - 2 \beta_1 + 1) q^{88} + ( - 7 \beta_{2} + 4 \beta_1 - 5) q^{89} - \beta_1 q^{90} + (3 \beta_{2} - \beta_1) q^{92} + ( - 4 \beta_{2} + 2 \beta_1 - 1) q^{93} + ( - 5 \beta_{2} - 5 \beta_1 - 6) q^{94} + (3 \beta_{2} - 1) q^{95} + (4 \beta_{2} - \beta_1 + 3) q^{96} + ( - 12 \beta_{2} + 6 \beta_1 - 9) q^{97} + ( - \beta_{2} - 4 \beta_1 - 3) q^{98} + q^{99}+O(q^{100})$$ q + b1 * q^2 - q^3 + b2 * q^4 - q^5 - b1 * q^6 + (b1 - 1) * q^7 + (b2 - 2*b1 + 1) * q^8 + q^9 - b1 * q^10 + q^11 - b2 * q^12 + (b2 - b1 + 2) * q^14 + q^15 + (-3*b2 + b1 - 3) * q^16 + (2*b1 - 4) * q^17 + b1 * q^18 + (-3*b2 + 1) * q^19 - b2 * q^20 + (-b1 + 1) * q^21 + b1 * q^22 + (-b2 + b1 + 1) * q^23 + (-b2 + 2*b1 - 1) * q^24 + q^25 - q^27 + q^28 + (-3*b2 - 2*b1) * q^29 + b1 * q^30 + (4*b2 - 2*b1 + 1) * q^31 + (-4*b2 + b1 - 3) * q^32 - q^33 + (2*b2 - 4*b1 + 4) * q^34 + (-b1 + 1) * q^35 + b2 * q^36 + (b2 - 4*b1) * q^37 + (-3*b2 + b1 - 3) * q^38 + (-b2 + 2*b1 - 1) * q^40 + (b1 + 7) * q^41 + (-b2 + b1 - 2) * q^42 + (-b1 - 5) * q^43 + b2 * q^44 - q^45 + (b1 + 1) * q^46 + (-4*b2 - b1 - 5) * q^47 + (3*b2 - b1 + 3) * q^48 + (b2 - 2*b1 - 4) * q^49 + b1 * q^50 + (-2*b1 + 4) * q^51 + (6*b2 - 7*b1 + 6) * q^53 - b1 * q^54 - q^55 + (-2*b2 + 3*b1 - 4) * q^56 + (3*b2 - 1) * q^57 + (-5*b2 - 7) * q^58 + (2*b2 - 7*b1 + 2) * q^59 + b2 * q^60 + (-7*b2 + 7*b1 - 8) * q^61 + (2*b2 + b1) * q^62 + (b1 - 1) * q^63 + (3*b2 - 5*b1 + 4) * q^64 - b1 * q^66 + (3*b2 - b1 + 8) * q^67 + (-2*b2 + 2) * q^68 + (b2 - b1 - 1) * q^69 + (-b2 + b1 - 2) * q^70 + (3*b2 - 6*b1 + 1) * q^71 + (b2 - 2*b1 + 1) * q^72 + (7*b2 - 10*b1 + 5) * q^73 + (-3*b2 - 7) * q^74 - q^75 + (4*b2 - 3*b1 - 3) * q^76 + (b1 - 1) * q^77 + (-2*b2 - 7*b1 + 1) * q^79 + (3*b2 - b1 + 3) * q^80 + q^81 + (b2 + 7*b1 + 2) * q^82 + (5*b2 + 2*b1 - 8) * q^83 - q^84 + (-2*b1 + 4) * q^85 + (-b2 - 5*b1 - 2) * q^86 + (3*b2 + 2*b1) * q^87 + (b2 - 2*b1 + 1) * q^88 + (-7*b2 + 4*b1 - 5) * q^89 - b1 * q^90 + (3*b2 - b1) * q^92 + (-4*b2 + 2*b1 - 1) * q^93 + (-5*b2 - 5*b1 - 6) * q^94 + (3*b2 - 1) * q^95 + (4*b2 - b1 + 3) * q^96 + (-12*b2 + 6*b1 - 9) * q^97 + (-b2 - 4*b1 - 3) * q^98 + q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + q^{2} - 3 q^{3} - q^{4} - 3 q^{5} - q^{6} - 2 q^{7} + 3 q^{9}+O(q^{10})$$ 3 * q + q^2 - 3 * q^3 - q^4 - 3 * q^5 - q^6 - 2 * q^7 + 3 * q^9 $$3 q + q^{2} - 3 q^{3} - q^{4} - 3 q^{5} - q^{6} - 2 q^{7} + 3 q^{9} - q^{10} + 3 q^{11} + q^{12} + 4 q^{14} + 3 q^{15} - 5 q^{16} - 10 q^{17} + q^{18} + 6 q^{19} + q^{20} + 2 q^{21} + q^{22} + 5 q^{23} + 3 q^{25} - 3 q^{27} + 3 q^{28} + q^{29} + q^{30} - 3 q^{31} - 4 q^{32} - 3 q^{33} + 6 q^{34} + 2 q^{35} - q^{36} - 5 q^{37} - 5 q^{38} + 22 q^{41} - 4 q^{42} - 16 q^{43} - q^{44} - 3 q^{45} + 4 q^{46} - 12 q^{47} + 5 q^{48} - 15 q^{49} + q^{50} + 10 q^{51} + 5 q^{53} - q^{54} - 3 q^{55} - 7 q^{56} - 6 q^{57} - 16 q^{58} - 3 q^{59} - q^{60} - 10 q^{61} - q^{62} - 2 q^{63} + 4 q^{64} - q^{66} + 20 q^{67} + 8 q^{68} - 5 q^{69} - 4 q^{70} - 6 q^{71} - 2 q^{73} - 18 q^{74} - 3 q^{75} - 16 q^{76} - 2 q^{77} - 2 q^{79} + 5 q^{80} + 3 q^{81} + 12 q^{82} - 27 q^{83} - 3 q^{84} + 10 q^{85} - 10 q^{86} - q^{87} - 4 q^{89} - q^{90} - 4 q^{92} + 3 q^{93} - 18 q^{94} - 6 q^{95} + 4 q^{96} - 9 q^{97} - 12 q^{98} + 3 q^{99}+O(q^{100})$$ 3 * q + q^2 - 3 * q^3 - q^4 - 3 * q^5 - q^6 - 2 * q^7 + 3 * q^9 - q^10 + 3 * q^11 + q^12 + 4 * q^14 + 3 * q^15 - 5 * q^16 - 10 * q^17 + q^18 + 6 * q^19 + q^20 + 2 * q^21 + q^22 + 5 * q^23 + 3 * q^25 - 3 * q^27 + 3 * q^28 + q^29 + q^30 - 3 * q^31 - 4 * q^32 - 3 * q^33 + 6 * q^34 + 2 * q^35 - q^36 - 5 * q^37 - 5 * q^38 + 22 * q^41 - 4 * q^42 - 16 * q^43 - q^44 - 3 * q^45 + 4 * q^46 - 12 * q^47 + 5 * q^48 - 15 * q^49 + q^50 + 10 * q^51 + 5 * q^53 - q^54 - 3 * q^55 - 7 * q^56 - 6 * q^57 - 16 * q^58 - 3 * q^59 - q^60 - 10 * q^61 - q^62 - 2 * q^63 + 4 * q^64 - q^66 + 20 * q^67 + 8 * q^68 - 5 * q^69 - 4 * q^70 - 6 * q^71 - 2 * q^73 - 18 * q^74 - 3 * q^75 - 16 * q^76 - 2 * q^77 - 2 * q^79 + 5 * q^80 + 3 * q^81 + 12 * q^82 - 27 * q^83 - 3 * q^84 + 10 * q^85 - 10 * q^86 - q^87 - 4 * q^89 - q^90 - 4 * q^92 + 3 * q^93 - 18 * q^94 - 6 * q^95 + 4 * q^96 - 9 * q^97 - 12 * q^98 + 3 * q^99

Basis of coefficient ring in terms of $$\nu = \zeta_{14} + \zeta_{14}^{-1}$$:

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

## 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
 −1.24698 0.445042 1.80194
−1.24698 −1.00000 −0.445042 −1.00000 1.24698 −2.24698 3.04892 1.00000 1.24698
1.2 0.445042 −1.00000 −1.80194 −1.00000 −0.445042 −0.554958 −1.69202 1.00000 −0.445042
1.3 1.80194 −1.00000 1.24698 −1.00000 −1.80194 0.801938 −1.35690 1.00000 −1.80194
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$1$$
$$5$$ $$1$$
$$13$$ $$1$$

## 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 2535.2.a.bd yes 3
3.b odd 2 1 7605.2.a.bq 3
13.b even 2 1 2535.2.a.v 3
39.d odd 2 1 7605.2.a.bz 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2535.2.a.v 3 13.b even 2 1
2535.2.a.bd yes 3 1.a even 1 1 trivial
7605.2.a.bq 3 3.b odd 2 1
7605.2.a.bz 3 39.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 $$S_{2}^{\mathrm{new}}(\Gamma_0(2535))$$:

 $$T_{2}^{3} - T_{2}^{2} - 2T_{2} + 1$$ T2^3 - T2^2 - 2*T2 + 1 $$T_{7}^{3} + 2T_{7}^{2} - T_{7} - 1$$ T7^3 + 2*T7^2 - T7 - 1 $$T_{11} - 1$$ T11 - 1

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3} - T^{2} - 2T + 1$$
$3$ $$(T + 1)^{3}$$
$5$ $$(T + 1)^{3}$$
$7$ $$T^{3} + 2T^{2} - T - 1$$
$11$ $$(T - 1)^{3}$$
$13$ $$T^{3}$$
$17$ $$T^{3} + 10 T^{2} + 24 T + 8$$
$19$ $$T^{3} - 6 T^{2} - 9 T + 41$$
$23$ $$T^{3} - 5 T^{2} + 6 T - 1$$
$29$ $$T^{3} - T^{2} - 44 T + 127$$
$31$ $$T^{3} + 3 T^{2} - 25 T + 29$$
$37$ $$T^{3} + 5 T^{2} - 22 T - 97$$
$41$ $$T^{3} - 22 T^{2} + 159 T - 377$$
$43$ $$T^{3} + 16 T^{2} + 83 T + 139$$
$47$ $$T^{3} + 12 T^{2} - T - 41$$
$53$ $$T^{3} - 5 T^{2} - 92 T + 83$$
$59$ $$T^{3} + 3 T^{2} - 88 T - 377$$
$61$ $$T^{3} + 10 T^{2} - 81 T - 433$$
$67$ $$T^{3} - 20 T^{2} + 117 T - 169$$
$71$ $$T^{3} + 6 T^{2} - 51 T - 307$$
$73$ $$T^{3} + 2 T^{2} - 183 T - 743$$
$79$ $$T^{3} + 2 T^{2} - 155 T + 223$$
$83$ $$T^{3} + 27 T^{2} + 152 T - 377$$
$89$ $$T^{3} + 4 T^{2} - 81 T - 421$$
$97$ $$T^{3} + 9 T^{2} - 225 T - 2241$$