# Properties

 Label 2700.3.g.s Level $2700$ Weight $3$ Character orbit 2700.g 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.g (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$73.5696713773$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: 8.0.488455618816.6 Defining polynomial: $$x^{8} - 4x^{7} + 14x^{5} + 105x^{4} - 238x^{3} - 426x^{2} + 548x + 3140$$ x^8 - 4*x^7 + 14*x^5 + 105*x^4 - 238*x^3 - 426*x^2 + 548*x + 3140 Coefficient ring: $$\Z[a_1, \ldots, a_{17}]$$ Coefficient ring index: $$2^{8}\cdot 3^{4}$$ Twist minimal: no (minimal twist has level 540) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{7}+O(q^{10})$$ q + b1 * q^7 $$q + \beta_1 q^{7} + (\beta_{6} - \beta_{3}) q^{11} + ( - \beta_{5} + \beta_1) q^{13} - \beta_{2} q^{17} + (\beta_{7} - 1) q^{19} + (2 \beta_{4} - \beta_{2}) q^{23} + 3 \beta_{6} q^{29} + ( - 2 \beta_{7} + 3) q^{31} + (2 \beta_{5} + 2 \beta_1) q^{37} - \beta_{6} q^{41} + ( - 3 \beta_{5} - 5 \beta_1) q^{43} + (6 \beta_{4} + 4 \beta_{2}) q^{47} + (\beta_{7} - 19) q^{49} + (19 \beta_{4} + 6 \beta_{2}) q^{53} + ( - 3 \beta_{6} + 8 \beta_{3}) q^{59} + ( - \beta_{7} + 9) q^{61} + (2 \beta_{5} + 12 \beta_1) q^{67} + ( - \beta_{6} + 14 \beta_{3}) q^{71} + (4 \beta_{5} - 9 \beta_1) q^{73} + (15 \beta_{4} + \beta_{2}) q^{77} + (3 \beta_{7} + 3) q^{79} + (19 \beta_{4} - 10 \beta_{2}) q^{83} + ( - 3 \beta_{6} - 14 \beta_{3}) q^{89} + ( - 2 \beta_{7} + 20) q^{91} + (\beta_{5} - 26 \beta_1) q^{97}+O(q^{100})$$ q + b1 * q^7 + (b6 - b3) * q^11 + (-b5 + b1) * q^13 - b2 * q^17 + (b7 - 1) * q^19 + (2*b4 - b2) * q^23 + 3*b6 * q^29 + (-2*b7 + 3) * q^31 + (2*b5 + 2*b1) * q^37 - b6 * q^41 + (-3*b5 - 5*b1) * q^43 + (6*b4 + 4*b2) * q^47 + (b7 - 19) * q^49 + (19*b4 + 6*b2) * q^53 + (-3*b6 + 8*b3) * q^59 + (-b7 + 9) * q^61 + (2*b5 + 12*b1) * q^67 + (-b6 + 14*b3) * q^71 + (4*b5 - 9*b1) * q^73 + (15*b4 + b2) * q^77 + (3*b7 + 3) * q^79 + (19*b4 - 10*b2) * q^83 + (-3*b6 - 14*b3) * q^89 + (-2*b7 + 20) * q^91 + (b5 - 26*b1) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q+O(q^{10})$$ 8 * q $$8 q - 12 q^{19} + 32 q^{31} - 156 q^{49} + 76 q^{61} + 12 q^{79} + 168 q^{91}+O(q^{100})$$ 8 * q - 12 * q^19 + 32 * q^31 - 156 * q^49 + 76 * q^61 + 12 * q^79 + 168 * q^91

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 4x^{7} + 14x^{5} + 105x^{4} - 238x^{3} - 426x^{2} + 548x + 3140$$ :

 $$\beta_{1}$$ $$=$$ $$( 107\nu^{7} + 1001\nu^{6} - 4543\nu^{5} - 13153\nu^{4} + 51884\nu^{3} + 72002\nu^{2} - 258088\nu - 983740 ) / 137550$$ (107*v^7 + 1001*v^6 - 4543*v^5 - 13153*v^4 + 51884*v^3 + 72002*v^2 - 258088*v - 983740) / 137550 $$\beta_{2}$$ $$=$$ $$( 39\nu^{7} + 322\nu^{6} - 756\nu^{5} - 1666\nu^{4} + 6013\nu^{3} + 70294\nu^{2} - 52076\nu - 180730 ) / 45850$$ (39*v^7 + 322*v^6 - 756*v^5 - 1666*v^4 + 6013*v^3 + 70294*v^2 - 52076*v - 180730) / 45850 $$\beta_{3}$$ $$=$$ $$( 73\nu^{7} + 1120\nu^{6} - 4025\nu^{5} + 5887\nu^{4} + 4648\nu^{3} + 61978\nu^{2} - 65216\nu + 73420 ) / 68775$$ (73*v^7 + 1120*v^6 - 4025*v^5 + 5887*v^4 + 4648*v^3 + 61978*v^2 - 65216*v + 73420) / 68775 $$\beta_{4}$$ $$=$$ $$( -78\nu^{7} + 273\nu^{6} - 1239\nu^{5} + 2415\nu^{4} - 5607\nu^{3} + 6132\nu^{2} - 46236\nu + 22170 ) / 45850$$ (-78*v^7 + 273*v^6 - 1239*v^5 + 2415*v^4 - 5607*v^3 + 6132*v^2 - 46236*v + 22170) / 45850 $$\beta_{5}$$ $$=$$ $$( 107\nu^{7} - 571\nu^{6} + 173\nu^{5} + 4139\nu^{4} + 9440\nu^{3} - 38038\nu^{2} - 126040\nu + 226700 ) / 19650$$ (107*v^7 - 571*v^6 + 173*v^5 + 4139*v^4 + 9440*v^3 - 38038*v^2 - 126040*v + 226700) / 19650 $$\beta_{6}$$ $$=$$ $$( 584\nu^{7} - 2044\nu^{6} + 812\nu^{5} + 3080\nu^{4} + 70196\nu^{3} - 109396\nu^{2} + 72488\nu - 17860 ) / 68775$$ (584*v^7 - 2044*v^6 + 812*v^5 + 3080*v^4 + 70196*v^3 - 109396*v^2 + 72488*v - 17860) / 68775 $$\beta_{7}$$ $$=$$ $$( -24\nu^{7} + 84\nu^{6} + 42\nu^{5} - 315\nu^{4} - 1302\nu^{3} + 2310\nu^{2} + 8628\nu - 5170 ) / 917$$ (-24*v^7 + 84*v^6 + 42*v^5 - 315*v^4 - 1302*v^3 + 2310*v^2 + 8628*v - 5170) / 917
 $$\nu$$ $$=$$ $$( 3\beta_{6} - 3\beta_{5} + 4\beta_{4} - 3\beta _1 + 12 ) / 24$$ (3*b6 - 3*b5 + 4*b4 - 3*b1 + 12) / 24 $$\nu^{2}$$ $$=$$ $$( 2\beta_{6} - \beta_{5} + 8\beta_{4} - 4\beta_{3} + 12\beta_{2} - 5\beta _1 + 24 ) / 12$$ (2*b6 - b5 + 8*b4 - 4*b3 + 12*b2 - 5*b1 + 24) / 12 $$\nu^{3}$$ $$=$$ $$( 3\beta_{7} + 9\beta_{6} + 3\beta_{5} + 11\beta_{4} - 3\beta_{3} + 9\beta_{2} + 18 ) / 6$$ (3*b7 + 9*b6 + 3*b5 + 11*b4 - 3*b3 + 9*b2 + 18) / 6 $$\nu^{4}$$ $$=$$ $$( 2\beta_{7} + 5\beta_{6} + 4\beta_{5} + 8\beta_{4} + 4\beta_{3} + 8\beta_{2} - 12\beta _1 - 96 ) / 2$$ (2*b7 + 5*b6 + 4*b5 + 8*b4 + 4*b3 + 8*b2 - 12*b1 - 96) / 2 $$\nu^{5}$$ $$=$$ $$( 25\beta_{7} - 13\beta_{6} + 68\beta_{5} - 209\beta_{4} + 35\beta_{3} + 45\beta_{2} - 47\beta _1 - 742 ) / 6$$ (25*b7 - 13*b6 + 68*b5 - 209*b4 + 35*b3 + 45*b2 - 47*b1 - 742) / 6 $$\nu^{6}$$ $$=$$ $$( 30\beta_{7} - 62\beta_{6} + 76\beta_{5} - 428\beta_{4} + 229\beta_{3} - 132\beta_{2} + 50\beta _1 - 924 ) / 3$$ (30*b7 - 62*b6 + 76*b5 - 428*b4 + 229*b3 - 132*b2 + 50*b1 - 924) / 3 $$\nu^{7}$$ $$=$$ $$( -217\beta_{7} - 776\beta_{6} + 13\beta_{5} - 3529\beta_{4} + 1477\beta_{3} - 1071\beta_{2} + 230\beta _1 - 4022 ) / 6$$ (-217*b7 - 776*b6 + 13*b5 - 3529*b4 + 1477*b3 - 1071*b2 + 230*b1 - 4022) / 6

## Character values

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

 $$n$$ $$1001$$ $$1351$$ $$2377$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
701.1
 2.67945 − 1.15831i 2.67945 + 1.15831i −1.67945 − 1.15831i −1.67945 + 1.15831i 2.67945 − 2.15831i 2.67945 + 2.15831i −1.67945 − 2.15831i −1.67945 + 2.15831i
0 0 0 0 0 −7.15439 0 0 0
701.2 0 0 0 0 0 −7.15439 0 0 0
701.3 0 0 0 0 0 −2.79549 0 0 0
701.4 0 0 0 0 0 −2.79549 0 0 0
701.5 0 0 0 0 0 2.79549 0 0 0
701.6 0 0 0 0 0 2.79549 0 0 0
701.7 0 0 0 0 0 7.15439 0 0 0
701.8 0 0 0 0 0 7.15439 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 701.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.b even 2 1 inner
15.d odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2700.3.g.s 8
3.b odd 2 1 inner 2700.3.g.s 8
5.b even 2 1 inner 2700.3.g.s 8
5.c odd 4 1 540.3.b.a 4
5.c odd 4 1 540.3.b.b yes 4
15.d odd 2 1 inner 2700.3.g.s 8
15.e even 4 1 540.3.b.a 4
15.e even 4 1 540.3.b.b yes 4
20.e even 4 1 2160.3.c.h 4
20.e even 4 1 2160.3.c.l 4
45.k odd 12 2 1620.3.t.a 8
45.k odd 12 2 1620.3.t.d 8
45.l even 12 2 1620.3.t.a 8
45.l even 12 2 1620.3.t.d 8
60.l odd 4 1 2160.3.c.h 4
60.l odd 4 1 2160.3.c.l 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
540.3.b.a 4 5.c odd 4 1
540.3.b.a 4 15.e even 4 1
540.3.b.b yes 4 5.c odd 4 1
540.3.b.b yes 4 15.e even 4 1
1620.3.t.a 8 45.k odd 12 2
1620.3.t.a 8 45.l even 12 2
1620.3.t.d 8 45.k odd 12 2
1620.3.t.d 8 45.l even 12 2
2160.3.c.h 4 20.e even 4 1
2160.3.c.h 4 60.l odd 4 1
2160.3.c.l 4 20.e even 4 1
2160.3.c.l 4 60.l odd 4 1
2700.3.g.s 8 1.a even 1 1 trivial
2700.3.g.s 8 3.b odd 2 1 inner
2700.3.g.s 8 5.b even 2 1 inner
2700.3.g.s 8 15.d 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 $$S_{3}^{\mathrm{new}}(2700, [\chi])$$:

 $$T_{7}^{4} - 59T_{7}^{2} + 400$$ T7^4 - 59*T7^2 + 400 $$T_{11}^{4} + 355T_{11}^{2} + 8464$$ T11^4 + 355*T11^2 + 8464 $$T_{13}^{4} - 540T_{13}^{2} + 5184$$ T13^4 - 540*T13^2 + 5184

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$T^{8}$$
$5$ $$T^{8}$$
$7$ $$(T^{4} - 59 T^{2} + 400)^{2}$$
$11$ $$(T^{4} + 355 T^{2} + 8464)^{2}$$
$13$ $$(T^{4} - 540 T^{2} + 5184)^{2}$$
$17$ $$(T^{4} + 109 T^{2} + 2500)^{2}$$
$19$ $$(T^{2} + 3 T - 468)^{4}$$
$23$ $$(T^{4} + 217 T^{2} + 16)^{2}$$
$29$ $$(T^{2} + 1584)^{4}$$
$31$ $$(T^{2} - 8 T - 1865)^{4}$$
$37$ $$(T^{2} - 1216)^{4}$$
$41$ $$(T^{2} + 176)^{4}$$
$43$ $$(T^{4} - 6620 T^{2} + 9684544)^{2}$$
$47$ $$(T^{4} + 1960 T^{2} + 478864)^{2}$$
$53$ $$(T^{4} + 8370 T^{2} + 178929)^{2}$$
$59$ $$(T^{4} + 6880 T^{2} + 4129024)^{2}$$
$61$ $$(T^{2} - 19 T - 380)^{4}$$
$67$ $$(T^{4} - 11372 T^{2} + 541696)^{2}$$
$71$ $$(T^{4} + 16956 T^{2} + 68558400)^{2}$$
$73$ $$(T^{4} - 11795 T^{2} + 6091024)^{2}$$
$79$ $$(T^{2} - 3 T - 4230)^{4}$$
$83$ $$(T^{4} + 20818 T^{2} + 1681)^{2}$$
$89$ $$(T^{4} + 24700 T^{2} + 19430464)^{2}$$
$97$ $$(T^{4} - 39515 T^{2} + \cdots + 266603584)^{2}$$