# Properties

 Label 3380.2.f.i Level $3380$ Weight $2$ Character orbit 3380.f Analytic conductor $26.989$ 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] = [3380,2,Mod(3041,3380)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3380, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 1]))

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

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

chi := DirichletCharacter("3380.3041");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$3380 = 2^{2} \cdot 5 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3380.f (of order $$2$$, degree $$1$$, not 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: $$26.9894358832$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: 8.0.22581504.2 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{8} - 4x^{7} + 5x^{6} + 2x^{5} - 11x^{4} + 4x^{3} + 20x^{2} - 32x + 16$$ x^8 - 4*x^7 + 5*x^6 + 2*x^5 - 11*x^4 + 4*x^3 + 20*x^2 - 32*x + 16 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{6}$$ Twist minimal: no (minimal twist has level 260) 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.

 $$f(q)$$ $$=$$ $$q + ( - \beta_{4} + \beta_{2}) q^{3} - \beta_{5} q^{5} + (\beta_{5} - \beta_{3}) q^{7} + ( - \beta_1 + 1) q^{9}+O(q^{10})$$ q + (-b4 + b2) * q^3 - b5 * q^5 + (b5 - b3) * q^7 + (-b1 + 1) * q^9 $$q + ( - \beta_{4} + \beta_{2}) q^{3} - \beta_{5} q^{5} + (\beta_{5} - \beta_{3}) q^{7} + ( - \beta_1 + 1) q^{9} + \beta_{6} q^{11} + ( - \beta_{6} + \beta_{3}) q^{15} + ( - \beta_{4} - \beta_{2} - \beta_1 - 1) q^{17} + (\beta_{7} + \beta_{6} - \beta_{5} - 2 \beta_{3}) q^{19} + (\beta_{6} + 2 \beta_{5} - 2 \beta_{3}) q^{21} + ( - \beta_{4} + \beta_{2} - \beta_1 + 1) q^{23} - q^{25} + ( - \beta_{4} - \beta_{2} - \beta_1 + 1) q^{27} + ( - 2 \beta_{4} + \beta_1) q^{29} + ( - 2 \beta_{6} + 2 \beta_{5} + 4 \beta_{3}) q^{31} + ( - \beta_{7} + 2 \beta_{5} + \beta_{3}) q^{33} + (\beta_{2} + 1) q^{35} + (2 \beta_{6} + 3 \beta_{5} - 3 \beta_{3}) q^{37} + (\beta_{6} - 6 \beta_{5}) q^{41} + ( - \beta_{4} - \beta_{2} + 2 \beta_1 - 2) q^{43} + (\beta_{7} - \beta_{5}) q^{45} + 2 \beta_{6} q^{47} + ( - 4 \beta_{2} - \beta_1 + 3) q^{49} + ( - 2 \beta_{4} - 2 \beta_{2} - 2 \beta_1 + 1) q^{51} + ( - 4 \beta_{2} - 3 \beta_1 + 5) q^{53} - \beta_{4} q^{55} + ( - 4 \beta_{6} + 5 \beta_{5} + \beta_{3}) q^{57} + (2 \beta_{7} + 3 \beta_{6} + 2 \beta_{5} - 2 \beta_{3}) q^{59} + (4 \beta_{4} - 2 \beta_{2} + \beta_1 + 2) q^{61} + ( - \beta_{7} + 2 \beta_{6} + 3 \beta_{5}) q^{63} + ( - 4 \beta_{6} - 5 \beta_{5} - \beta_{3}) q^{67} + ( - 4 \beta_{4} + 2 \beta_{2} - 2 \beta_1 + 5) q^{69} + ( - 3 \beta_{7} + 3 \beta_{6} - 3 \beta_{5}) q^{71} + (\beta_{7} - 2 \beta_{6} + 7 \beta_{5} + 2 \beta_{3}) q^{73} + (\beta_{4} - \beta_{2}) q^{75} + (\beta_{4} + \beta_{2} + \beta_1 + 1) q^{77} + ( - 6 \beta_{4} + 2 \beta_{2} - \beta_1 - 3) q^{79} + ( - 4 \beta_{4} + \beta_1 - 2) q^{81} + ( - 2 \beta_{7} + 10 \beta_{5} + 2 \beta_{3}) q^{83} + (\beta_{7} - \beta_{6} + \beta_{5} - \beta_{3}) q^{85} + (3 \beta_{4} - 3 \beta_{2} - \beta_1 + 3) q^{87} + (2 \beta_{7} - \beta_{6} + 2 \beta_{5} - 2 \beta_{3}) q^{89} + (2 \beta_{7} + 2 \beta_{6} - 12 \beta_{5}) q^{93} + ( - \beta_{4} + 2 \beta_{2} + \beta_1 - 1) q^{95} + ( - 2 \beta_{7} + 4 \beta_{6} - 3 \beta_{5} - 3 \beta_{3}) q^{97} + ( - \beta_{7} + 2 \beta_{6} - \beta_{5} - 2 \beta_{3}) q^{99}+O(q^{100})$$ q + (-b4 + b2) * q^3 - b5 * q^5 + (b5 - b3) * q^7 + (-b1 + 1) * q^9 + b6 * q^11 + (-b6 + b3) * q^15 + (-b4 - b2 - b1 - 1) * q^17 + (b7 + b6 - b5 - 2*b3) * q^19 + (b6 + 2*b5 - 2*b3) * q^21 + (-b4 + b2 - b1 + 1) * q^23 - q^25 + (-b4 - b2 - b1 + 1) * q^27 + (-2*b4 + b1) * q^29 + (-2*b6 + 2*b5 + 4*b3) * q^31 + (-b7 + 2*b5 + b3) * q^33 + (b2 + 1) * q^35 + (2*b6 + 3*b5 - 3*b3) * q^37 + (b6 - 6*b5) * q^41 + (-b4 - b2 + 2*b1 - 2) * q^43 + (b7 - b5) * q^45 + 2*b6 * q^47 + (-4*b2 - b1 + 3) * q^49 + (-2*b4 - 2*b2 - 2*b1 + 1) * q^51 + (-4*b2 - 3*b1 + 5) * q^53 - b4 * q^55 + (-4*b6 + 5*b5 + b3) * q^57 + (2*b7 + 3*b6 + 2*b5 - 2*b3) * q^59 + (4*b4 - 2*b2 + b1 + 2) * q^61 + (-b7 + 2*b6 + 3*b5) * q^63 + (-4*b6 - 5*b5 - b3) * q^67 + (-4*b4 + 2*b2 - 2*b1 + 5) * q^69 + (-3*b7 + 3*b6 - 3*b5) * q^71 + (b7 - 2*b6 + 7*b5 + 2*b3) * q^73 + (b4 - b2) * q^75 + (b4 + b2 + b1 + 1) * q^77 + (-6*b4 + 2*b2 - b1 - 3) * q^79 + (-4*b4 + b1 - 2) * q^81 + (-2*b7 + 10*b5 + 2*b3) * q^83 + (b7 - b6 + b5 - b3) * q^85 + (3*b4 - 3*b2 - b1 + 3) * q^87 + (2*b7 - b6 + 2*b5 - 2*b3) * q^89 + (2*b7 + 2*b6 - 12*b5) * q^93 + (-b4 + 2*b2 + b1 - 1) * q^95 + (-2*b7 + 4*b6 - 3*b5 - 3*b3) * q^97 + (-b7 + 2*b6 - b5 - 2*b3) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q + 4 q^{3} + 8 q^{9}+O(q^{10})$$ 8 * q + 4 * q^3 + 8 * q^9 $$8 q + 4 q^{3} + 8 q^{9} - 12 q^{17} + 12 q^{23} - 8 q^{25} + 4 q^{27} + 12 q^{35} - 20 q^{43} + 8 q^{49} + 24 q^{53} + 8 q^{61} + 48 q^{69} - 4 q^{75} + 12 q^{77} - 16 q^{79} - 16 q^{81} + 12 q^{87}+O(q^{100})$$ 8 * q + 4 * q^3 + 8 * q^9 - 12 * q^17 + 12 * q^23 - 8 * q^25 + 4 * q^27 + 12 * q^35 - 20 * q^43 + 8 * q^49 + 24 * q^53 + 8 * q^61 + 48 * q^69 - 4 * q^75 + 12 * q^77 - 16 * q^79 - 16 * q^81 + 12 * q^87

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 4x^{7} + 5x^{6} + 2x^{5} - 11x^{4} + 4x^{3} + 20x^{2} - 32x + 16$$ :

 $$\beta_{1}$$ $$=$$ $$( -\nu^{7} + 2\nu^{6} + \nu^{5} - 4\nu^{4} + \nu^{3} + 6\nu^{2} - 10\nu + 2 ) / 2$$ (-v^7 + 2*v^6 + v^5 - 4*v^4 + v^3 + 6*v^2 - 10*v + 2) / 2 $$\beta_{2}$$ $$=$$ $$( 3\nu^{7} - 8\nu^{6} + 3\nu^{5} + 10\nu^{4} - 13\nu^{3} - 8\nu^{2} + 32\nu - 24 ) / 8$$ (3*v^7 - 8*v^6 + 3*v^5 + 10*v^4 - 13*v^3 - 8*v^2 + 32*v - 24) / 8 $$\beta_{3}$$ $$=$$ $$( -5\nu^{7} + 12\nu^{6} - 5\nu^{5} - 18\nu^{4} + 19\nu^{3} + 28\nu^{2} - 64\nu + 40 ) / 8$$ (-5*v^7 + 12*v^6 - 5*v^5 - 18*v^4 + 19*v^3 + 28*v^2 - 64*v + 40) / 8 $$\beta_{4}$$ $$=$$ $$( -3\nu^{7} + 9\nu^{6} - 5\nu^{5} - 13\nu^{4} + 21\nu^{3} + 13\nu^{2} - 54\nu + 40 ) / 4$$ (-3*v^7 + 9*v^6 - 5*v^5 - 13*v^4 + 21*v^3 + 13*v^2 - 54*v + 40) / 4 $$\beta_{5}$$ $$=$$ $$( -4\nu^{7} + 11\nu^{6} - 6\nu^{5} - 17\nu^{4} + 24\nu^{3} + 15\nu^{2} - 62\nu + 48 ) / 4$$ (-4*v^7 + 11*v^6 - 6*v^5 - 17*v^4 + 24*v^3 + 15*v^2 - 62*v + 48) / 4 $$\beta_{6}$$ $$=$$ $$( 3\nu^{7} - 7\nu^{6} + 3\nu^{5} + 11\nu^{4} - 15\nu^{3} - 11\nu^{2} + 40\nu - 30 ) / 2$$ (3*v^7 - 7*v^6 + 3*v^5 + 11*v^4 - 15*v^3 - 11*v^2 + 40*v - 30) / 2 $$\beta_{7}$$ $$=$$ $$( 5\nu^{7} - 15\nu^{6} + 11\nu^{5} + 19\nu^{4} - 35\nu^{3} - 11\nu^{2} + 90\nu - 80 ) / 4$$ (5*v^7 - 15*v^6 + 11*v^5 + 19*v^4 - 35*v^3 - 11*v^2 + 90*v - 80) / 4
 $$\nu$$ $$=$$ $$( \beta_{7} + \beta_{5} - 2\beta_{2} - \beta _1 + 3 ) / 4$$ (b7 + b5 - 2*b2 - b1 + 3) / 4 $$\nu^{2}$$ $$=$$ $$( \beta_{7} - \beta_{5} + 2\beta_{4} + 2\beta_{3} - \beta _1 + 3 ) / 4$$ (b7 - b5 + 2*b4 + 2*b3 - b1 + 3) / 4 $$\nu^{3}$$ $$=$$ $$( \beta_{7} - \beta_{6} - \beta_{5} + 2\beta_{4} - \beta_{3} - \beta_{2} - \beta_1 ) / 2$$ (b7 - b6 - b5 + 2*b4 - b3 - b2 - b1) / 2 $$\nu^{4}$$ $$=$$ $$( -\beta_{7} - 2\beta_{6} - 11\beta_{5} + 6\beta_{4} + 2\beta_{3} - 4\beta_{2} - \beta _1 + 1 ) / 4$$ (-b7 - 2*b6 - 11*b5 + 6*b4 + 2*b3 - 4*b2 - b1 + 1) / 4 $$\nu^{5}$$ $$=$$ $$( \beta_{7} - 2\beta_{6} - 7\beta_{5} + 8\beta_{4} - 6\beta_{3} - 4\beta_{2} + 3\beta _1 + 9 ) / 4$$ (b7 - 2*b6 - 7*b5 + 8*b4 - 6*b3 - 4*b2 + 3*b1 + 9) / 4 $$\nu^{6}$$ $$=$$ $$( 3\beta_{6} - 2\beta_{5} + 4\beta_{4} - 8\beta_{2} + \beta _1 + 4 ) / 2$$ (3*b6 - 2*b5 + 4*b4 - 8*b2 + b1 + 4) / 2 $$\nu^{7}$$ $$=$$ $$( 3\beta_{7} + 16\beta_{6} + 11\beta_{5} + 16\beta_{4} - 4\beta_{3} - 2\beta_{2} + 5\beta _1 + 17 ) / 4$$ (3*b7 + 16*b6 + 11*b5 + 16*b4 - 4*b3 - 2*b2 + 5*b1 + 17) / 4

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/3380\mathbb{Z}\right)^\times$$.

 $$n$$ $$677$$ $$1691$$ $$1861$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
3041.1
 1.40994 − 0.109843i 1.40994 + 0.109843i 0.665665 + 1.24775i 0.665665 − 1.24775i −1.27597 + 0.609843i −1.27597 − 0.609843i 1.20036 − 0.747754i 1.20036 + 0.747754i
0 −2.33225 0 1.00000i 0 0.399804i 0 2.43937 0
3041.2 0 −2.33225 0 1.00000i 0 0.399804i 0 2.43937 0
3041.3 0 −0.0947876 0 1.00000i 0 0.826838i 0 −2.99102 0
3041.4 0 −0.0947876 0 1.00000i 0 0.826838i 0 −2.99102 0
3041.5 0 1.60020 0 1.00000i 0 4.33225i 0 −0.439374 0
3041.6 0 1.60020 0 1.00000i 0 4.33225i 0 −0.439374 0
3041.7 0 2.82684 0 1.00000i 0 2.09479i 0 4.99102 0
3041.8 0 2.82684 0 1.00000i 0 2.09479i 0 4.99102 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 3041.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3380.2.f.i 8
13.b even 2 1 inner 3380.2.f.i 8
13.c even 3 1 260.2.x.a 8
13.d odd 4 1 3380.2.a.p 4
13.d odd 4 1 3380.2.a.q 4
13.e even 6 1 260.2.x.a 8
39.h odd 6 1 2340.2.dj.d 8
39.i odd 6 1 2340.2.dj.d 8
52.i odd 6 1 1040.2.da.c 8
52.j odd 6 1 1040.2.da.c 8
65.l even 6 1 1300.2.y.b 8
65.n even 6 1 1300.2.y.b 8
65.q odd 12 1 1300.2.ba.b 8
65.q odd 12 1 1300.2.ba.c 8
65.r odd 12 1 1300.2.ba.b 8
65.r odd 12 1 1300.2.ba.c 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
260.2.x.a 8 13.c even 3 1
260.2.x.a 8 13.e even 6 1
1040.2.da.c 8 52.i odd 6 1
1040.2.da.c 8 52.j odd 6 1
1300.2.y.b 8 65.l even 6 1
1300.2.y.b 8 65.n even 6 1
1300.2.ba.b 8 65.q odd 12 1
1300.2.ba.b 8 65.r odd 12 1
1300.2.ba.c 8 65.q odd 12 1
1300.2.ba.c 8 65.r odd 12 1
2340.2.dj.d 8 39.h odd 6 1
2340.2.dj.d 8 39.i odd 6 1
3380.2.a.p 4 13.d odd 4 1
3380.2.a.q 4 13.d odd 4 1
3380.2.f.i 8 1.a even 1 1 trivial
3380.2.f.i 8 13.b even 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(3380, [\chi])$$:

 $$T_{3}^{4} - 2T_{3}^{3} - 6T_{3}^{2} + 10T_{3} + 1$$ T3^4 - 2*T3^3 - 6*T3^2 + 10*T3 + 1 $$T_{19}^{4} + 30T_{19}^{2} + 33$$ T19^4 + 30*T19^2 + 33

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$(T^{4} - 2 T^{3} - 6 T^{2} + 10 T + 1)^{2}$$
$5$ $$(T^{2} + 1)^{4}$$
$7$ $$T^{8} + 24 T^{6} + 102 T^{4} + 72 T^{2} + \cdots + 9$$
$11$ $$(T^{2} + 3)^{4}$$
$13$ $$T^{8}$$
$17$ $$(T^{4} + 6 T^{3} - 18 T^{2} - 54 T - 27)^{2}$$
$19$ $$(T^{4} + 30 T^{2} + 33)^{2}$$
$23$ $$(T^{4} - 6 T^{3} - 18 T^{2} + 30 T - 3)^{2}$$
$29$ $$(T^{4} - 42 T^{2} - 96 T - 39)^{2}$$
$31$ $$(T^{4} + 96 T^{2} + 2112)^{2}$$
$37$ $$T^{8} + 192 T^{6} + 9774 T^{4} + \cdots + 42849$$
$41$ $$(T^{4} + 78 T^{2} + 1089)^{2}$$
$43$ $$(T^{4} + 10 T^{3} - 66 T^{2} - 914 T - 2243)^{2}$$
$47$ $$(T^{2} + 12)^{4}$$
$53$ $$(T^{4} - 12 T^{3} - 156 T^{2} + 1920 T - 624)^{2}$$
$59$ $$T^{8} + 228 T^{6} + 14958 T^{4} + \cdots + 558009$$
$61$ $$(T^{4} - 4 T^{3} - 102 T^{2} + 644 T - 971)^{2}$$
$67$ $$T^{8} + 336 T^{6} + 31278 T^{4} + \cdots + 1083681$$
$71$ $$T^{8} + 468 T^{6} + \cdots + 45198729$$
$73$ $$T^{8} + 264 T^{6} + 21168 T^{4} + \cdots + 2509056$$
$79$ $$(T^{4} + 8 T^{3} - 180 T^{2} - 1504 T - 368)^{2}$$
$83$ $$T^{8} + 480 T^{6} + 63936 T^{4} + \cdots + 331776$$
$89$ $$T^{8} + 228 T^{6} + 9774 T^{4} + \cdots + 13689$$
$97$ $$T^{8} + 408 T^{6} + \cdots + 12981609$$