Properties

 Label 3626.2.a.bg Level $3626$ Weight $2$ Character orbit 3626.a Self dual yes Analytic conductor $28.954$ Analytic rank $0$ Dimension $8$ CM no Inner twists $2$

Related objects

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3626,2,Mod(1,3626)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3626, 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("3626.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$3626 = 2 \cdot 7^{2} \cdot 37$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3626.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: $$28.9537557729$$ 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: $$x^{8} - 21x^{6} + 151x^{4} - 406x^{2} + 242$$ x^8 - 21*x^6 + 151*x^4 - 406*x^2 + 242 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{2}$$ 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.

 $$f(q)$$ $$=$$ $$q + q^{2} + \beta_{4} q^{3} + q^{4} - \beta_{2} q^{5} + \beta_{4} q^{6} + q^{8} + ( - \beta_{6} + 2) q^{9}+O(q^{10})$$ q + q^2 + b4 * q^3 + q^4 - b2 * q^5 + b4 * q^6 + q^8 + (-b6 + 2) * q^9 $$q + q^{2} + \beta_{4} q^{3} + q^{4} - \beta_{2} q^{5} + \beta_{4} q^{6} + q^{8} + ( - \beta_{6} + 2) q^{9} - \beta_{2} q^{10} - \beta_{7} q^{11} + \beta_{4} q^{12} + (\beta_{4} + \beta_{3}) q^{13} + ( - \beta_{7} + \beta_1 - 1) q^{15} + q^{16} + \beta_{3} q^{17} + ( - \beta_{6} + 2) q^{18} + ( - \beta_{5} + \beta_{4}) q^{19} - \beta_{2} q^{20} - \beta_{7} q^{22} + ( - \beta_1 + 1) q^{23} + \beta_{4} q^{24} + (2 \beta_{7} + \beta_1 + 6) q^{25} + (\beta_{4} + \beta_{3}) q^{26} + (3 \beta_{4} - \beta_{3} + \beta_{2}) q^{27} + (\beta_{7} + \beta_{6} - \beta_1 + 2) q^{29} + ( - \beta_{7} + \beta_1 - 1) q^{30} + ( - \beta_{5} - \beta_{4} + \cdots + \beta_{2}) q^{31}+ \cdots + ( - 5 \beta_{7} - 2 \beta_{6} + \cdots - 5) q^{99}+O(q^{100})$$ q + q^2 + b4 * q^3 + q^4 - b2 * q^5 + b4 * q^6 + q^8 + (-b6 + 2) * q^9 - b2 * q^10 - b7 * q^11 + b4 * q^12 + (b4 + b3) * q^13 + (-b7 + b1 - 1) * q^15 + q^16 + b3 * q^17 + (-b6 + 2) * q^18 + (-b5 + b4) * q^19 - b2 * q^20 - b7 * q^22 + (-b1 + 1) * q^23 + b4 * q^24 + (2*b7 + b1 + 6) * q^25 + (b4 + b3) * q^26 + (3*b4 - b3 + b2) * q^27 + (b7 + b6 - b1 + 2) * q^29 + (-b7 + b1 - 1) * q^30 + (-b5 - b4 - b3 + b2) * q^31 + q^32 + (2*b5 - b4 - b3 - 3*b2) * q^33 + b3 * q^34 + (-b6 + 2) * q^36 + q^37 + (-b5 + b4) * q^38 + (b7 + 4) * q^39 - b2 * q^40 + (b5 - b3) * q^41 + (-b7 + b6 - 2*b1 - 3) * q^43 - b7 * q^44 + (3*b5 - 4*b4 - b3 - 3*b2) * q^45 + (-b1 + 1) * q^46 + (-2*b4 - 2*b3) * q^47 + b4 * q^48 + (2*b7 + b1 + 6) * q^50 + (b7 + b6 - 1) * q^51 + (b4 + b3) * q^52 + (-2*b7 - 2*b1 + 4) * q^53 + (3*b4 - b3 + b2) * q^54 + (-b5 + 4*b4 - b3 + 2*b2) * q^55 + (2*b7 + 4) * q^57 + (b7 + b6 - b1 + 2) * q^58 + (b5 - 3*b4) * q^59 + (-b7 + b1 - 1) * q^60 + (2*b3 - b2) * q^61 + (-b5 - b4 - b3 + b2) * q^62 + q^64 + (b7 + 3*b1 + 1) * q^65 + (2*b5 - b4 - b3 - 3*b2) * q^66 + (b7 + 2*b6 + 2) * q^67 + b3 * q^68 + (-b5 + 3*b4 + 3*b2) * q^69 + (b7 - b6 + 5) * q^71 + (-b6 + 2) * q^72 + (-3*b5 + b4 + b2) * q^73 + q^74 + (-3*b5 + 6*b4 + 2*b3 + 3*b2) * q^75 + (-b5 + b4) * q^76 + (b7 + 4) * q^78 + (b1 + 3) * q^79 - b2 * q^80 + (-b6 - b1 + 11) * q^81 + (b5 - b3) * q^82 + (b5 + b4 - b3 - 2*b2) * q^83 + (2*b7 + 2*b1 + 2) * q^85 + (-b7 + b6 - 2*b1 - 3) * q^86 + (-3*b5 + b4 + 2*b3 + 5*b2) * q^87 - b7 * q^88 + (-2*b4 + b3 + 2*b2) * q^89 + (3*b5 - 4*b4 - b3 - 3*b2) * q^90 + (-b1 + 1) * q^92 + (2*b7 + b6 - b1 - 4) * q^93 + (-2*b4 - 2*b3) * q^94 + (2*b6 + 2*b1 + 4) * q^95 + b4 * q^96 + (3*b3 + 2*b2) * q^97 + (-5*b7 - 2*b6 + 3*b1 - 5) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q + 8 q^{2} + 8 q^{4} + 8 q^{8} + 18 q^{9}+O(q^{10})$$ 8 * q + 8 * q^2 + 8 * q^4 + 8 * q^8 + 18 * q^9 $$8 q + 8 q^{2} + 8 q^{4} + 8 q^{8} + 18 q^{9} + 2 q^{11} - 8 q^{15} + 8 q^{16} + 18 q^{18} + 2 q^{22} + 10 q^{23} + 42 q^{25} + 14 q^{29} - 8 q^{30} + 8 q^{32} + 18 q^{36} + 8 q^{37} + 30 q^{39} - 20 q^{43} + 2 q^{44} + 10 q^{46} + 42 q^{50} - 12 q^{51} + 40 q^{53} + 28 q^{57} + 14 q^{58} - 8 q^{60} + 8 q^{64} + 10 q^{67} + 40 q^{71} + 18 q^{72} + 8 q^{74} + 30 q^{78} + 22 q^{79} + 92 q^{81} + 8 q^{85} - 20 q^{86} + 2 q^{88} + 10 q^{92} - 36 q^{93} + 24 q^{95} - 32 q^{99}+O(q^{100})$$ 8 * q + 8 * q^2 + 8 * q^4 + 8 * q^8 + 18 * q^9 + 2 * q^11 - 8 * q^15 + 8 * q^16 + 18 * q^18 + 2 * q^22 + 10 * q^23 + 42 * q^25 + 14 * q^29 - 8 * q^30 + 8 * q^32 + 18 * q^36 + 8 * q^37 + 30 * q^39 - 20 * q^43 + 2 * q^44 + 10 * q^46 + 42 * q^50 - 12 * q^51 + 40 * q^53 + 28 * q^57 + 14 * q^58 - 8 * q^60 + 8 * q^64 + 10 * q^67 + 40 * q^71 + 18 * q^72 + 8 * q^74 + 30 * q^78 + 22 * q^79 + 92 * q^81 + 8 * q^85 - 20 * q^86 + 2 * q^88 + 10 * q^92 - 36 * q^93 + 24 * q^95 - 32 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 21x^{6} + 151x^{4} - 406x^{2} + 242$$ :

 $$\beta_{1}$$ $$=$$ $$\nu^{4} - 8\nu^{2} + 7$$ v^4 - 8*v^2 + 7 $$\beta_{2}$$ $$=$$ $$( -2\nu^{7} + 31\nu^{5} - 126\nu^{3} + 64\nu ) / 11$$ (-2*v^7 + 31*v^5 - 126*v^3 + 64*v) / 11 $$\beta_{3}$$ $$=$$ $$( -3\nu^{7} + 41\nu^{5} - 145\nu^{3} + 52\nu ) / 11$$ (-3*v^7 + 41*v^5 - 145*v^3 + 52*v) / 11 $$\beta_{4}$$ $$=$$ $$( -2\nu^{7} + 31\nu^{5} - 137\nu^{3} + 130\nu ) / 11$$ (-2*v^7 + 31*v^5 - 137*v^3 + 130*v) / 11 $$\beta_{5}$$ $$=$$ $$( -2\nu^{7} + 31\nu^{5} - 137\nu^{3} + 152\nu ) / 11$$ (-2*v^7 + 31*v^5 - 137*v^3 + 152*v) / 11 $$\beta_{6}$$ $$=$$ $$-\nu^{6} + 15\nu^{4} - 60\nu^{2} + 35$$ -v^6 + 15*v^4 - 60*v^2 + 35 $$\beta_{7}$$ $$=$$ $$\nu^{6} - 15\nu^{4} + 62\nu^{2} - 46$$ v^6 - 15*v^4 + 62*v^2 - 46
 $$\nu$$ $$=$$ $$( \beta_{5} - \beta_{4} ) / 2$$ (b5 - b4) / 2 $$\nu^{2}$$ $$=$$ $$( \beta_{7} + \beta_{6} + 11 ) / 2$$ (b7 + b6 + 11) / 2 $$\nu^{3}$$ $$=$$ $$3\beta_{5} - 4\beta_{4} + \beta_{2}$$ 3*b5 - 4*b4 + b2 $$\nu^{4}$$ $$=$$ $$4\beta_{7} + 4\beta_{6} + \beta _1 + 37$$ 4*b7 + 4*b6 + b1 + 37 $$\nu^{5}$$ $$=$$ $$20\beta_{5} - 28\beta_{4} - 2\beta_{3} + 11\beta_{2}$$ 20*b5 - 28*b4 - 2*b3 + 11*b2 $$\nu^{6}$$ $$=$$ $$30\beta_{7} + 29\beta_{6} + 15\beta _1 + 260$$ 30*b7 + 29*b6 + 15*b1 + 260 $$\nu^{7}$$ $$=$$ $$137\beta_{5} - 198\beta_{4} - 31\beta_{3} + 102\beta_{2}$$ 137*b5 - 198*b4 - 31*b3 + 102*b2

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
 2.38022 −0.904109 2.81330 −2.56953 2.56953 −2.81330 0.904109 −2.38022
1.00000 −3.21136 1.00000 4.00765 −3.21136 0 1.00000 7.31284 4.00765
1.2 1.00000 −3.09330 1.00000 −1.59233 −3.09330 0 1.00000 6.56849 −1.59233
1.3 1.00000 −1.02018 1.00000 −4.36631 −1.02018 0 1.00000 −1.95923 −4.36631
1.4 1.00000 −0.279098 1.00000 1.82718 −0.279098 0 1.00000 −2.92210 1.82718
1.5 1.00000 0.279098 1.00000 −1.82718 0.279098 0 1.00000 −2.92210 −1.82718
1.6 1.00000 1.02018 1.00000 4.36631 1.02018 0 1.00000 −1.95923 4.36631
1.7 1.00000 3.09330 1.00000 1.59233 3.09330 0 1.00000 6.56849 1.59233
1.8 1.00000 3.21136 1.00000 −4.00765 3.21136 0 1.00000 7.31284 −4.00765
 $$n$$: 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
$$2$$ $$-1$$
$$7$$ $$-1$$
$$37$$ $$-1$$

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3626.2.a.bg 8
7.b odd 2 1 inner 3626.2.a.bg 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3626.2.a.bg 8 1.a even 1 1 trivial
3626.2.a.bg 8 7.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(3626))$$:

 $$T_{3}^{8} - 21T_{3}^{6} + 121T_{3}^{4} - 112T_{3}^{2} + 8$$ T3^8 - 21*T3^6 + 121*T3^4 - 112*T3^2 + 8 $$T_{5}^{8} - 41T_{5}^{6} + 521T_{5}^{4} - 2096T_{5}^{2} + 2592$$ T5^8 - 41*T5^6 + 521*T5^4 - 2096*T5^2 + 2592 $$T_{11}^{4} - T_{11}^{3} - 31T_{11}^{2} + 48T_{11} + 64$$ T11^4 - T11^3 - 31*T11^2 + 48*T11 + 64

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T - 1)^{8}$$
$3$ $$T^{8} - 21 T^{6} + \cdots + 8$$
$5$ $$T^{8} - 41 T^{6} + \cdots + 2592$$
$7$ $$T^{8}$$
$11$ $$(T^{4} - T^{3} - 31 T^{2} + \cdots + 64)^{2}$$
$13$ $$T^{8} - 53 T^{6} + \cdots + 288$$
$17$ $$T^{8} - 44 T^{6} + \cdots + 128$$
$19$ $$T^{8} - 84 T^{6} + \cdots + 61952$$
$23$ $$(T^{4} - 5 T^{3} - 33 T^{2} + \cdots + 16)^{2}$$
$29$ $$(T^{4} - 7 T^{3} + \cdots - 236)^{2}$$
$31$ $$T^{8} - 133 T^{6} + \cdots + 349448$$
$37$ $$(T - 1)^{8}$$
$41$ $$T^{8} - 141 T^{6} + \cdots + 10368$$
$43$ $$(T^{4} + 10 T^{3} + \cdots - 864)^{2}$$
$47$ $$T^{8} - 212 T^{6} + \cdots + 73728$$
$53$ $$(T^{4} - 20 T^{3} + \cdots - 8448)^{2}$$
$59$ $$T^{8} - 224 T^{6} + \cdots + 2957312$$
$61$ $$T^{8} - 233 T^{6} + \cdots + 3338528$$
$67$ $$(T^{4} - 5 T^{3} + \cdots + 3504)^{2}$$
$71$ $$(T^{4} - 20 T^{3} + \cdots - 384)^{2}$$
$73$ $$T^{8} - 625 T^{6} + \cdots + 538970112$$
$79$ $$(T^{4} - 11 T^{3} + \cdots - 96)^{2}$$
$83$ $$T^{8} - 256 T^{6} + \cdots + 80000$$
$89$ $$T^{8} - 268 T^{6} + \cdots + 839808$$
$97$ $$T^{8} - 512 T^{6} + \cdots + 16866432$$