# Properties

 Label 76.2.d.a Level $76$ Weight $2$ Character orbit 76.d Analytic conductor $0.607$ Analytic rank $0$ Dimension $8$ CM no Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [76,2,Mod(75,76)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(76, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 1]))

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

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

chi := DirichletCharacter("76.75");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$76 = 2^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 76.d (of order $$2$$, degree $$1$$, minimal)

## 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: no Analytic conductor: $$0.606863055362$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: 8.0.14453810176.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{8} + 3x^{6} + 6x^{4} + 12x^{2} + 16$$ x^8 + 3*x^6 + 6*x^4 + 12*x^2 + 16 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{2} + \beta_{4} q^{3} + (\beta_{2} - 1) q^{4} + ( - \beta_{6} + \beta_{2} - 1) q^{5} + (\beta_{6} - \beta_{5} - \beta_{2}) q^{6} + ( - \beta_{6} + \beta_{5}) q^{7} + (\beta_{3} - \beta_1) q^{8} + (\beta_{6} - \beta_{2} + 1) q^{9}+O(q^{10})$$ q + b1 * q^2 + b4 * q^3 + (b2 - 1) * q^4 + (-b6 + b2 - 1) * q^5 + (b6 - b5 - b2) * q^6 + (-b6 + b5) * q^7 + (b3 - b1) * q^8 + (b6 - b2 + 1) * q^9 $$q + \beta_1 q^{2} + \beta_{4} q^{3} + (\beta_{2} - 1) q^{4} + ( - \beta_{6} + \beta_{2} - 1) q^{5} + (\beta_{6} - \beta_{5} - \beta_{2}) q^{6} + ( - \beta_{6} + \beta_{5}) q^{7} + (\beta_{3} - \beta_1) q^{8} + (\beta_{6} - \beta_{2} + 1) q^{9} + ( - \beta_{7} + \beta_{3} - \beta_1) q^{10} + ( - \beta_{5} - \beta_{2}) q^{11} + (\beta_{7} - 2 \beta_{4} - \beta_{3}) q^{12} + (\beta_{7} - \beta_{4} - 2 \beta_{3} + \beta_1) q^{13} + ( - \beta_{7} + 2 \beta_{4}) q^{14} + (\beta_{7} - 2 \beta_{4} - \beta_1) q^{15} + (\beta_{6} + \beta_{5} - \beta_{2}) q^{16} - q^{17} + (\beta_{7} - \beta_{3} + \beta_1) q^{18} + ( - \beta_{7} + \beta_{5} + \cdots + \beta_1) q^{19}+ \cdots + (2 \beta_{6} - \beta_{5} + \beta_{2}) q^{99}+O(q^{100})$$ q + b1 * q^2 + b4 * q^3 + (b2 - 1) * q^4 + (-b6 + b2 - 1) * q^5 + (b6 - b5 - b2) * q^6 + (-b6 + b5) * q^7 + (b3 - b1) * q^8 + (b6 - b2 + 1) * q^9 + (-b7 + b3 - b1) * q^10 + (-b5 - b2) * q^11 + (b7 - 2*b4 - b3) * q^12 + (b7 - b4 - 2*b3 + b1) * q^13 + (-b7 + 2*b4) * q^14 + (b7 - 2*b4 - b1) * q^15 + (b6 + b5 - b2) * q^16 - q^17 + (b7 - b3 + b1) * q^18 + (-b7 + b5 + b4 + b2 + b1) * q^19 + (b6 + b5 + 3) * q^20 + (-2*b7 + b4 + 2*b3 - 4*b1) * q^21 + (-2*b4 - b3) * q^22 + (b6 + b2) * q^23 + (-3*b6 + b5 + b2 - 2) * q^24 + (b6 - b2) * q^25 + (-3*b6 - b5 + b2 - 2) * q^26 + (-b7 - b4 + b1) * q^27 + (2*b6 - 2*b5 - b2 + 3) * q^28 + (b4 + 2*b3 + 2*b1) * q^29 + (-2*b6 + 2*b5 - 2) * q^30 + (b7 - b1) * q^31 + (b7 + 2*b4 - b3) * q^32 + (b7 + 3*b1) * q^33 - b1 * q^34 + (2*b6 - 3*b5 - b2) * q^35 + (-b6 - b5 - 3) * q^36 + (-b7 - 3*b1) * q^37 + (b6 - b5 + 2*b4 + b3 + b2 + 2) * q^38 + (3*b6 - 2*b5 + b2) * q^39 + (b7 + 2*b4 + 3*b1) * q^40 + (b7 + 3*b1) * q^41 + (3*b6 + b5 - 3*b2 + 8) * q^42 + (-2*b6 - b5 - 3*b2) * q^43 + (-3*b6 + b5 + 2*b2 + 1) * q^44 + (-b6 + b2 - 5) * q^45 + (b7 + b3) * q^46 + (3*b5 + 3*b2) * q^47 + (-3*b7 + 2*b4 + b3 - 2*b1) * q^48 + (-4*b6 + 4*b2 - 4) * q^49 + (b7 - b3) * q^50 - b4 * q^51 + (-3*b7 - 2*b4 + b3 - 2*b1) * q^52 + (2*b7 - b4 - 2*b3 + 4*b1) * q^53 + (-b6 + b5 + 3*b2 + 2) * q^54 + (-2*b6 + b5 - b2) * q^55 + (2*b7 - 4*b4 - b3 + 3*b1) * q^56 + (-b7 + 3*b6 - 3*b2 - 3*b1 + 4) * q^57 + (3*b6 + b5 + b2 - 4) * q^58 + (-3*b7 + b4 + 3*b1) * q^59 + (-2*b7 + 4*b4 - 2*b1) * q^60 + (3*b6 - 3*b2 + 7) * q^61 + (-2*b2 - 2) * q^62 + (-2*b6 + 3*b5 + b2) * q^63 + (b6 - 3*b5 - 3*b2 - 2) * q^64 + (-2*b7 - 6*b1) * q^65 + (2*b2 - 6) * q^66 + (2*b7 + b4 - 2*b1) * q^67 + (-b2 + 1) * q^68 + (b7 - b4 - 2*b3 + b1) * q^69 + (2*b7 - 6*b4 - b3) * q^70 + (2*b7 + 2*b4 - 2*b1) * q^71 + (-b7 - 2*b4 - 3*b1) * q^72 + (-2*b6 + 2*b2 + 3) * q^73 + (-2*b2 + 6) * q^74 + (-b7 + b4 + b1) * q^75 + (b7 + 3*b6 - b5 - 2*b4 + b3 - 2*b2 + 2*b1 - 1) * q^76 + (3*b6 - 3*b2 + 5) * q^77 + (3*b7 - 4*b4 + b3) * q^78 + (3*b7 - 4*b4 - 3*b1) * q^79 + (2*b6 - 2*b5 - 6) * q^80 + (-2*b6 + 2*b2 - 7) * q^81 + (2*b2 - 6) * q^82 + (-4*b6 + 2*b5 - 2*b2) * q^83 + (3*b7 + 2*b4 - 3*b3 + 8*b1) * q^84 + (b6 - b2 + 1) * q^85 + (-2*b7 - 2*b4 - 3*b3) * q^86 + (-b6 - 2*b5 - 3*b2) * q^87 + (-3*b7 + 2*b4 + 2*b3 + b1) * q^88 + (-3*b7 + 2*b4 + 4*b3 - 5*b1) * q^89 + (-b7 + b3 - 5*b1) * q^90 + (-b7 + 7*b4 + b1) * q^91 + (b6 + b5 - b2 - 4) * q^92 + (-2*b6 + 2*b2) * q^93 + (6*b4 + 3*b3) * q^94 + (2*b6 - b5 - 4*b4 + b2) * q^95 + (3*b6 - b5 - b2 + 10) * q^96 + (-2*b4 - 4*b3 - 4*b1) * q^97 + (-4*b7 + 4*b3 - 4*b1) * q^98 + (2*b6 - b5 + b2) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q - 6 q^{4} - 4 q^{5} - 2 q^{6} + 4 q^{9}+O(q^{10})$$ 8 * q - 6 * q^4 - 4 * q^5 - 2 * q^6 + 4 * q^9 $$8 q - 6 q^{4} - 4 q^{5} - 2 q^{6} + 4 q^{9} - 6 q^{16} - 8 q^{17} + 20 q^{20} - 10 q^{24} - 4 q^{25} - 6 q^{26} + 22 q^{28} - 16 q^{30} - 20 q^{36} + 18 q^{38} + 50 q^{42} + 16 q^{44} - 36 q^{45} - 16 q^{49} + 22 q^{54} + 20 q^{57} - 38 q^{58} + 44 q^{61} - 20 q^{62} - 18 q^{64} - 44 q^{66} + 6 q^{68} + 32 q^{73} + 44 q^{74} - 16 q^{76} + 28 q^{77} - 48 q^{80} - 48 q^{81} - 44 q^{82} + 4 q^{85} - 38 q^{92} + 8 q^{93} + 74 q^{96}+O(q^{100})$$ 8 * q - 6 * q^4 - 4 * q^5 - 2 * q^6 + 4 * q^9 - 6 * q^16 - 8 * q^17 + 20 * q^20 - 10 * q^24 - 4 * q^25 - 6 * q^26 + 22 * q^28 - 16 * q^30 - 20 * q^36 + 18 * q^38 + 50 * q^42 + 16 * q^44 - 36 * q^45 - 16 * q^49 + 22 * q^54 + 20 * q^57 - 38 * q^58 + 44 * q^61 - 20 * q^62 - 18 * q^64 - 44 * q^66 + 6 * q^68 + 32 * q^73 + 44 * q^74 - 16 * q^76 + 28 * q^77 - 48 * q^80 - 48 * q^81 - 44 * q^82 + 4 * q^85 - 38 * q^92 + 8 * q^93 + 74 * q^96

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + 3x^{6} + 6x^{4} + 12x^{2} + 16$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} + 1$$ v^2 + 1 $$\beta_{3}$$ $$=$$ $$\nu^{3} + \nu$$ v^3 + v $$\beta_{4}$$ $$=$$ $$( -\nu^{7} + \nu^{5} - 2\nu^{3} - 4\nu ) / 8$$ (-v^7 + v^5 - 2*v^3 - 4*v) / 8 $$\beta_{5}$$ $$=$$ $$( -\nu^{6} + \nu^{4} - 2\nu^{2} - 4 ) / 4$$ (-v^6 + v^4 - 2*v^2 - 4) / 4 $$\beta_{6}$$ $$=$$ $$( \nu^{6} + 3\nu^{4} + 6\nu^{2} + 8 ) / 4$$ (v^6 + 3*v^4 + 6*v^2 + 8) / 4 $$\beta_{7}$$ $$=$$ $$( \nu^{7} + 3\nu^{5} + 6\nu^{3} + 8\nu ) / 4$$ (v^7 + 3*v^5 + 6*v^3 + 8*v) / 4
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} - 1$$ b2 - 1 $$\nu^{3}$$ $$=$$ $$\beta_{3} - \beta_1$$ b3 - b1 $$\nu^{4}$$ $$=$$ $$\beta_{6} + \beta_{5} - \beta_{2}$$ b6 + b5 - b2 $$\nu^{5}$$ $$=$$ $$\beta_{7} + 2\beta_{4} - \beta_{3}$$ b7 + 2*b4 - b3 $$\nu^{6}$$ $$=$$ $$\beta_{6} - 3\beta_{5} - 3\beta_{2} - 2$$ b6 - 3*b5 - 3*b2 - 2 $$\nu^{7}$$ $$=$$ $$\beta_{7} - 6\beta_{4} - 3\beta_{3} - 2\beta_1$$ b7 - 6*b4 - 3*b3 - 2*b1

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/76\mathbb{Z}\right)^\times$$.

 $$n$$ $$21$$ $$39$$ $$\chi(n)$$ $$-1$$ $$-1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
75.1
 −1.06789 − 0.927153i −1.06789 + 0.927153i −0.331077 − 1.37491i −0.331077 + 1.37491i 0.331077 − 1.37491i 0.331077 + 1.37491i 1.06789 − 0.927153i 1.06789 + 0.927153i
−1.06789 0.927153i 1.19935 0.280776 + 1.98019i 1.56155 −1.28078 1.11198i 0.868210i 1.53610 2.37495i −1.56155 −1.66757 1.44780i
75.2 −1.06789 + 0.927153i 1.19935 0.280776 1.98019i 1.56155 −1.28078 + 1.11198i 0.868210i 1.53610 + 2.37495i −1.56155 −1.66757 + 1.44780i
75.3 −0.331077 1.37491i −2.35829 −1.78078 + 0.910404i −2.56155 0.780776 + 3.24245i 4.15286i 1.84130 + 2.14700i 2.56155 0.848071 + 3.52191i
75.4 −0.331077 + 1.37491i −2.35829 −1.78078 0.910404i −2.56155 0.780776 3.24245i 4.15286i 1.84130 2.14700i 2.56155 0.848071 3.52191i
75.5 0.331077 1.37491i 2.35829 −1.78078 0.910404i −2.56155 0.780776 3.24245i 4.15286i −1.84130 + 2.14700i 2.56155 −0.848071 + 3.52191i
75.6 0.331077 + 1.37491i 2.35829 −1.78078 + 0.910404i −2.56155 0.780776 + 3.24245i 4.15286i −1.84130 2.14700i 2.56155 −0.848071 3.52191i
75.7 1.06789 0.927153i −1.19935 0.280776 1.98019i 1.56155 −1.28078 + 1.11198i 0.868210i −1.53610 2.37495i −1.56155 1.66757 1.44780i
75.8 1.06789 + 0.927153i −1.19935 0.280776 + 1.98019i 1.56155 −1.28078 1.11198i 0.868210i −1.53610 + 2.37495i −1.56155 1.66757 + 1.44780i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 75.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
19.b odd 2 1 inner
76.d even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 76.2.d.a 8
3.b odd 2 1 684.2.f.b 8
4.b odd 2 1 inner 76.2.d.a 8
8.b even 2 1 1216.2.h.d 8
8.d odd 2 1 1216.2.h.d 8
12.b even 2 1 684.2.f.b 8
19.b odd 2 1 inner 76.2.d.a 8
57.d even 2 1 684.2.f.b 8
76.d even 2 1 inner 76.2.d.a 8
152.b even 2 1 1216.2.h.d 8
152.g odd 2 1 1216.2.h.d 8
228.b odd 2 1 684.2.f.b 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
76.2.d.a 8 1.a even 1 1 trivial
76.2.d.a 8 4.b odd 2 1 inner
76.2.d.a 8 19.b odd 2 1 inner
76.2.d.a 8 76.d even 2 1 inner
684.2.f.b 8 3.b odd 2 1
684.2.f.b 8 12.b even 2 1
684.2.f.b 8 57.d even 2 1
684.2.f.b 8 228.b odd 2 1
1216.2.h.d 8 8.b even 2 1
1216.2.h.d 8 8.d odd 2 1
1216.2.h.d 8 152.b even 2 1
1216.2.h.d 8 152.g odd 2 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(76, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8} + 3 T^{6} + \cdots + 16$$
$3$ $$(T^{4} - 7 T^{2} + 8)^{2}$$
$5$ $$(T^{2} + T - 4)^{4}$$
$7$ $$(T^{4} + 18 T^{2} + 13)^{2}$$
$11$ $$(T^{4} + 15 T^{2} + 52)^{2}$$
$13$ $$(T^{4} + 41 T^{2} + 416)^{2}$$
$17$ $$(T + 1)^{8}$$
$19$ $$T^{8} - 16 T^{6} + \cdots + 130321$$
$23$ $$(T^{4} + 19 T^{2} + 52)^{2}$$
$29$ $$(T^{4} + 73 T^{2} + 104)^{2}$$
$31$ $$(T^{4} - 20 T^{2} + 32)^{2}$$
$37$ $$(T^{4} + 44 T^{2} + 416)^{2}$$
$41$ $$(T^{4} + 44 T^{2} + 416)^{2}$$
$43$ $$(T^{4} + 123 T^{2} + 208)^{2}$$
$47$ $$(T^{4} + 135 T^{2} + 4212)^{2}$$
$53$ $$(T^{4} + 97 T^{2} + 104)^{2}$$
$59$ $$(T^{4} - 175 T^{2} + 5408)^{2}$$
$61$ $$(T^{2} - 11 T - 8)^{4}$$
$67$ $$(T^{4} - 95 T^{2} + 8)^{2}$$
$71$ $$(T^{4} - 124 T^{2} + 512)^{2}$$
$73$ $$(T^{2} - 8 T - 1)^{4}$$
$79$ $$(T^{4} - 244 T^{2} + 11552)^{2}$$
$83$ $$(T^{4} + 236 T^{2} + 13312)^{2}$$
$89$ $$(T^{4} + 232 T^{2} + 6656)^{2}$$
$97$ $$(T^{4} + 292 T^{2} + 1664)^{2}$$