# Properties

 Label 2700.3.u.b Level $2700$ Weight $3$ Character orbit 2700.u Analytic conductor $73.570$ Analytic rank $0$ Dimension $8$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2700 = 2^{2} \cdot 3^{3} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 2700.u (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$73.5696713773$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(\zeta_{6})$$ Coefficient field: 8.0.303595776.1 Defining polynomial: $$x^{8} + 5x^{6} + 16x^{4} + 45x^{2} + 81$$ x^8 + 5*x^6 + 16*x^4 + 45*x^2 + 81 Coefficient ring: $$\Z[a_1, \ldots, a_{17}]$$ Coefficient ring index: $$3^{4}$$ Twist minimal: no (minimal twist has level 36) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

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.

 $$f(q)$$ $$=$$ $$q + ( - 2 \beta_{7} + \beta_{4} - \beta_{3}) q^{7}+O(q^{10})$$ q + (-2*b7 + b4 - b3) * q^7 $$q + ( - 2 \beta_{7} + \beta_{4} - \beta_{3}) q^{7} + ( - 2 \beta_{5} - 5 \beta_{2} + 12) q^{11} + (\beta_{7} - 2 \beta_{4} + \beta_{3} - 3 \beta_1) q^{13} + ( - \beta_{7} + \beta_{4} + 7 \beta_{3} - 15 \beta_1) q^{17} + ( - \beta_{6} + 2 \beta_{5} - \beta_{2}) q^{19} + ( - \beta_{7} + 33 \beta_{3} - 17 \beta_1) q^{23} + ( - 7 \beta_{5} + 14 \beta_{2} - 21) q^{29} + ( - 3 \beta_{6} - 3 \beta_{5} + 5 \beta_{2} - 2) q^{31} + (4 \beta_{7} + 4 \beta_{4} - 18 \beta_{3} + 4 \beta_1) q^{37} + ( - 8 \beta_{6} + 8 \beta_{5} - \beta_{2} + 7) q^{41} + (23 \beta_{3} - 23 \beta_1) q^{43} + ( - 3 \beta_{4} + 15 \beta_{3} + 12 \beta_1) q^{47} + (\beta_{6} + \beta_{5} - 26 \beta_{2} + 25) q^{49} + (8 \beta_{7} - 8 \beta_{4} + 16 \beta_{3} - 24 \beta_1) q^{53} + ( - 8 \beta_{6} + 8 \beta_{5} + 17 \beta_{2} + 25) q^{59} + (6 \beta_{6} - 3 \beta_{5} + 22 \beta_{2} - 3) q^{61} + ( - 6 \beta_{7} + 12 \beta_{4} - 6 \beta_{3} + 61 \beta_1) q^{67} + ( - 2 \beta_{6} - 30 \beta_{2} + 16) q^{71} + (3 \beta_{7} + 3 \beta_{4} - 23 \beta_{3} + 3 \beta_1) q^{73} + ( - 17 \beta_{7} + 93 \beta_{3} - 55 \beta_1) q^{77} + ( - 10 \beta_{6} + 5 \beta_{5} - 44 \beta_{2} + 5) q^{79} + (3 \beta_{4} - 15 \beta_{3} - 12 \beta_1) q^{83} + (8 \beta_{6} + 120 \beta_{2} - 64) q^{89} + (3 \beta_{6} - 6 \beta_{5} + 3 \beta_{2} - 77) q^{91} + ( - 4 \beta_{7} + 2 \beta_{4} - 99 \beta_{3} + 97 \beta_1) q^{97}+O(q^{100})$$ q + (-2*b7 + b4 - b3) * q^7 + (-2*b5 - 5*b2 + 12) * q^11 + (b7 - 2*b4 + b3 - 3*b1) * q^13 + (-b7 + b4 + 7*b3 - 15*b1) * q^17 + (-b6 + 2*b5 - b2) * q^19 + (-b7 + 33*b3 - 17*b1) * q^23 + (-7*b5 + 14*b2 - 21) * q^29 + (-3*b6 - 3*b5 + 5*b2 - 2) * q^31 + (4*b7 + 4*b4 - 18*b3 + 4*b1) * q^37 + (-8*b6 + 8*b5 - b2 + 7) * q^41 + (23*b3 - 23*b1) * q^43 + (-3*b4 + 15*b3 + 12*b1) * q^47 + (b6 + b5 - 26*b2 + 25) * q^49 + (8*b7 - 8*b4 + 16*b3 - 24*b1) * q^53 + (-8*b6 + 8*b5 + 17*b2 + 25) * q^59 + (6*b6 - 3*b5 + 22*b2 - 3) * q^61 + (-6*b7 + 12*b4 - 6*b3 + 61*b1) * q^67 + (-2*b6 - 30*b2 + 16) * q^71 + (3*b7 + 3*b4 - 23*b3 + 3*b1) * q^73 + (-17*b7 + 93*b3 - 55*b1) * q^77 + (-10*b6 + 5*b5 - 44*b2 + 5) * q^79 + (3*b4 - 15*b3 - 12*b1) * q^83 + (8*b6 + 120*b2 - 64) * q^89 + (3*b6 - 6*b5 + 3*b2 - 77) * q^91 + (-4*b7 + 2*b4 - 99*b3 + 97*b1) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q+O(q^{10})$$ 8 * q $$8 q + 72 q^{11} - 4 q^{19} - 126 q^{29} - 14 q^{31} + 36 q^{41} + 102 q^{49} + 252 q^{59} + 82 q^{61} - 166 q^{79} - 604 q^{91}+O(q^{100})$$ 8 * q + 72 * q^11 - 4 * q^19 - 126 * q^29 - 14 * q^31 + 36 * q^41 + 102 * q^49 + 252 * q^59 + 82 * q^61 - 166 * q^79 - 604 * q^91

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + 5x^{6} + 16x^{4} + 45x^{2} + 81$$ :

 $$\beta_{1}$$ $$=$$ $$( \nu^{7} + 32\nu^{5} + 16\nu^{3} + 45\nu ) / 432$$ (v^7 + 32*v^5 + 16*v^3 + 45*v) / 432 $$\beta_{2}$$ $$=$$ $$( 5\nu^{6} + 16\nu^{4} + 80\nu^{2} + 225 ) / 144$$ (5*v^6 + 16*v^4 + 80*v^2 + 225) / 144 $$\beta_{3}$$ $$=$$ $$( 5\nu^{7} + 16\nu^{5} + 8\nu^{3} + 81\nu ) / 216$$ (5*v^7 + 16*v^5 + 8*v^3 + 81*v) / 216 $$\beta_{4}$$ $$=$$ $$( -\nu^{7} + 131\nu ) / 48$$ (-v^7 + 131*v) / 48 $$\beta_{5}$$ $$=$$ $$( 13\nu^{6} + 128\nu^{4} + 208\nu^{2} + 585 ) / 144$$ (13*v^6 + 128*v^4 + 208*v^2 + 585) / 144 $$\beta_{6}$$ $$=$$ $$( -3\nu^{6} - 7 ) / 16$$ (-3*v^6 - 7) / 16 $$\beta_{7}$$ $$=$$ $$( 23\nu^{7} + 88\nu^{5} + 368\nu^{3} + 1035\nu ) / 216$$ (23*v^7 + 88*v^5 + 368*v^3 + 1035*v) / 216
 $$\nu$$ $$=$$ $$( \beta_{4} + \beta_{3} - \beta_1 ) / 3$$ (b4 + b3 - b1) / 3 $$\nu^{2}$$ $$=$$ $$( \beta_{6} - \beta_{5} + 8\beta_{2} - 8 ) / 3$$ (b6 - b5 + 8*b2 - 8) / 3 $$\nu^{3}$$ $$=$$ $$( 2\beta_{7} - 2\beta_{4} - 11\beta_{3} ) / 3$$ (2*b7 - 2*b4 - 11*b3) / 3 $$\nu^{4}$$ $$=$$ $$( 5\beta_{5} - 13\beta_{2} ) / 3$$ (5*b5 - 13*b2) / 3 $$\nu^{5}$$ $$=$$ $$( -\beta_{7} + 46\beta_1 ) / 3$$ (-b7 + 46*b1) / 3 $$\nu^{6}$$ $$=$$ $$( -16\beta_{6} - 7 ) / 3$$ (-16*b6 - 7) / 3 $$\nu^{7}$$ $$=$$ $$( -13\beta_{4} + 131\beta_{3} - 131\beta_1 ) / 3$$ (-13*b4 + 131*b3 - 131*b1) / 3

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2700\mathbb{Z}\right)^\times$$.

 $$n$$ $$1001$$ $$1351$$ $$2377$$ $$\chi(n)$$ $$1 - \beta_{2}$$ $$1$$ $$-1$$

## 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.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
449.1
 −0.396143 − 1.68614i −1.26217 − 1.18614i 1.26217 + 1.18614i 0.396143 + 1.68614i −0.396143 + 1.68614i −1.26217 + 1.18614i 1.26217 − 1.18614i 0.396143 − 1.68614i
0 0 0 0 0 −7.89542 + 4.55842i 0 0 0
449.2 0 0 0 0 0 −7.02939 + 4.05842i 0 0 0
449.3 0 0 0 0 0 7.02939 4.05842i 0 0 0
449.4 0 0 0 0 0 7.89542 4.55842i 0 0 0
2249.1 0 0 0 0 0 −7.89542 4.55842i 0 0 0
2249.2 0 0 0 0 0 −7.02939 4.05842i 0 0 0
2249.3 0 0 0 0 0 7.02939 + 4.05842i 0 0 0
2249.4 0 0 0 0 0 7.89542 + 4.55842i 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 2249.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
9.d odd 6 1 inner
45.h odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2700.3.u.b 8
3.b odd 2 1 900.3.u.a 8
5.b even 2 1 inner 2700.3.u.b 8
5.c odd 4 1 108.3.g.a 4
5.c odd 4 1 2700.3.p.b 4
9.c even 3 1 900.3.u.a 8
9.d odd 6 1 inner 2700.3.u.b 8
15.d odd 2 1 900.3.u.a 8
15.e even 4 1 36.3.g.a 4
15.e even 4 1 900.3.p.a 4
20.e even 4 1 432.3.q.b 4
40.i odd 4 1 1728.3.q.g 4
40.k even 4 1 1728.3.q.h 4
45.h odd 6 1 inner 2700.3.u.b 8
45.j even 6 1 900.3.u.a 8
45.k odd 12 1 36.3.g.a 4
45.k odd 12 1 324.3.c.b 4
45.k odd 12 1 900.3.p.a 4
45.l even 12 1 108.3.g.a 4
45.l even 12 1 324.3.c.b 4
45.l even 12 1 2700.3.p.b 4
60.l odd 4 1 144.3.q.b 4
120.q odd 4 1 576.3.q.g 4
120.w even 4 1 576.3.q.d 4
180.v odd 12 1 432.3.q.b 4
180.v odd 12 1 1296.3.e.e 4
180.x even 12 1 144.3.q.b 4
180.x even 12 1 1296.3.e.e 4
360.bo even 12 1 576.3.q.g 4
360.br even 12 1 1728.3.q.g 4
360.bt odd 12 1 1728.3.q.h 4
360.bu odd 12 1 576.3.q.d 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
36.3.g.a 4 15.e even 4 1
36.3.g.a 4 45.k odd 12 1
108.3.g.a 4 5.c odd 4 1
108.3.g.a 4 45.l even 12 1
144.3.q.b 4 60.l odd 4 1
144.3.q.b 4 180.x even 12 1
324.3.c.b 4 45.k odd 12 1
324.3.c.b 4 45.l even 12 1
432.3.q.b 4 20.e even 4 1
432.3.q.b 4 180.v odd 12 1
576.3.q.d 4 120.w even 4 1
576.3.q.d 4 360.bu odd 12 1
576.3.q.g 4 120.q odd 4 1
576.3.q.g 4 360.bo even 12 1
900.3.p.a 4 15.e even 4 1
900.3.p.a 4 45.k odd 12 1
900.3.u.a 8 3.b odd 2 1
900.3.u.a 8 9.c even 3 1
900.3.u.a 8 15.d odd 2 1
900.3.u.a 8 45.j even 6 1
1296.3.e.e 4 180.v odd 12 1
1296.3.e.e 4 180.x even 12 1
1728.3.q.g 4 40.i odd 4 1
1728.3.q.g 4 360.br even 12 1
1728.3.q.h 4 40.k even 4 1
1728.3.q.h 4 360.bt odd 12 1
2700.3.p.b 4 5.c odd 4 1
2700.3.p.b 4 45.l even 12 1
2700.3.u.b 8 1.a even 1 1 trivial
2700.3.u.b 8 5.b even 2 1 inner
2700.3.u.b 8 9.d odd 6 1 inner
2700.3.u.b 8 45.h odd 6 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7}^{8} - 149T_{7}^{6} + 16725T_{7}^{4} - 815924T_{7}^{2} + 29986576$$ acting on $$S_{3}^{\mathrm{new}}(2700, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$T^{8}$$
$5$ $$T^{8}$$
$7$ $$T^{8} - 149 T^{6} + \cdots + 29986576$$
$11$ $$(T^{4} - 36 T^{3} + 441 T^{2} - 324 T + 81)^{2}$$
$13$ $$T^{8} - 161 T^{6} + \cdots + 21381376$$
$17$ $$(T^{4} - 387 T^{2} + 20736)^{2}$$
$19$ $$(T^{2} + T - 74)^{4}$$
$23$ $$T^{8} + 1683 T^{6} + \cdots + 393460125696$$
$29$ $$(T^{4} + 63 T^{3} + 441 T^{2} + \cdots + 777924)^{2}$$
$31$ $$(T^{4} + 7 T^{3} + 705 T^{2} + \cdots + 430336)^{2}$$
$37$ $$(T^{4} + 2888 T^{2} + 868624)^{2}$$
$41$ $$(T^{4} - 18 T^{3} - 1449 T^{2} + \cdots + 2424249)^{2}$$
$43$ $$(T^{4} - 529 T^{2} + 279841)^{2}$$
$47$ $$T^{8} + 1539 T^{6} + \cdots + 11019960576$$
$53$ $$(T^{4} - 4032 T^{2} + 1327104)^{2}$$
$59$ $$(T^{4} - 126 T^{3} + 5031 T^{2} + \cdots + 68121)^{2}$$
$61$ $$(T^{4} - 41 T^{3} + 1929 T^{2} + \cdots + 61504)^{2}$$
$67$ $$T^{8} - 12074 T^{6} + \cdots + 227988105361$$
$71$ $$(T^{4} + 1548 T^{2} + 331776)^{2}$$
$73$ $$(T^{4} + 2261 T^{2} + 42436)^{2}$$
$79$ $$(T^{4} + 83 T^{3} + 7023 T^{2} + \cdots + 17956)^{2}$$
$83$ $$T^{8} + 1539 T^{6} + \cdots + 11019960576$$
$89$ $$(T^{4} + 24768 T^{2} + 84934656)^{2}$$
$97$ $$T^{8} - 19802 T^{6} + \cdots + 75\!\cdots\!01$$