# Properties

 Label 6900.2.f.r Level $6900$ Weight $2$ Character orbit 6900.f Analytic conductor $55.097$ Analytic rank $0$ Dimension $6$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$6900 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 6900.f (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$55.0967773947$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: 6.0.158155776.1 Defining polynomial: $$x^{6} - 2x^{5} + 2x^{4} + 24x^{3} + 81x^{2} + 54x + 18$$ x^6 - 2*x^5 + 2*x^4 + 24*x^3 + 81*x^2 + 54*x + 18 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 1380) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{2} q^{3} + ( - \beta_{5} - \beta_{2}) q^{7} - q^{9}+O(q^{10})$$ q + b2 * q^3 + (-b5 - b2) * q^7 - q^9 $$q + \beta_{2} q^{3} + ( - \beta_{5} - \beta_{2}) q^{7} - q^{9} + (\beta_1 + 1) q^{11} + (\beta_{4} + \beta_{2}) q^{13} + (\beta_{5} - \beta_{4}) q^{17} + ( - \beta_1 - 3) q^{19} + ( - \beta_{3} + 1) q^{21} + \beta_{2} q^{23} - \beta_{2} q^{27} + (\beta_{3} - \beta_1) q^{29} + (\beta_{3} + 3) q^{31} + ( - \beta_{4} + \beta_{2}) q^{33} + (\beta_{5} - 2 \beta_{4} - \beta_{2}) q^{37} + (\beta_1 - 1) q^{39} + (3 \beta_{3} - \beta_1 + 2) q^{41} + (2 \beta_{5} + 6 \beta_{2}) q^{43} + ( - 2 \beta_{5} + \beta_{4} + \beta_{2}) q^{47} + (\beta_{3} - 2 \beta_1 - 4) q^{49} + (\beta_{3} - \beta_1) q^{51} + ( - 3 \beta_{5} + \beta_{4} + 2 \beta_{2}) q^{53} + (\beta_{4} - 3 \beta_{2}) q^{57} + ( - \beta_{3} + \beta_1) q^{59} + (2 \beta_{3} - \beta_1 + 3) q^{61} + (\beta_{5} + \beta_{2}) q^{63} + (\beta_{5} - 3 \beta_{2}) q^{67} - q^{69} + ( - 3 \beta_{3} + \beta_1 - 2) q^{71} + ( - 3 \beta_{4} + 5 \beta_{2}) q^{73} + ( - 4 \beta_{5} + \beta_{4} + 3 \beta_{2}) q^{77} + ( - 2 \beta_{3} - 4) q^{79} + q^{81} + (\beta_{5} + \beta_{4} + 4 \beta_{2}) q^{83} + ( - \beta_{5} + \beta_{4}) q^{87} + (2 \beta_{3} + 2 \beta_1 - 2) q^{89} + (2 \beta_{3} - \beta_1 + 5) q^{91} + ( - \beta_{5} + 3 \beta_{2}) q^{93} + ( - 4 \beta_{5} + 2 \beta_{4} + 6 \beta_{2}) q^{97} + ( - \beta_1 - 1) q^{99}+O(q^{100})$$ q + b2 * q^3 + (-b5 - b2) * q^7 - q^9 + (b1 + 1) * q^11 + (b4 + b2) * q^13 + (b5 - b4) * q^17 + (-b1 - 3) * q^19 + (-b3 + 1) * q^21 + b2 * q^23 - b2 * q^27 + (b3 - b1) * q^29 + (b3 + 3) * q^31 + (-b4 + b2) * q^33 + (b5 - 2*b4 - b2) * q^37 + (b1 - 1) * q^39 + (3*b3 - b1 + 2) * q^41 + (2*b5 + 6*b2) * q^43 + (-2*b5 + b4 + b2) * q^47 + (b3 - 2*b1 - 4) * q^49 + (b3 - b1) * q^51 + (-3*b5 + b4 + 2*b2) * q^53 + (b4 - 3*b2) * q^57 + (-b3 + b1) * q^59 + (2*b3 - b1 + 3) * q^61 + (b5 + b2) * q^63 + (b5 - 3*b2) * q^67 - q^69 + (-3*b3 + b1 - 2) * q^71 + (-3*b4 + 5*b2) * q^73 + (-4*b5 + b4 + 3*b2) * q^77 + (-2*b3 - 4) * q^79 + q^81 + (b5 + b4 + 4*b2) * q^83 + (-b5 + b4) * q^87 + (2*b3 + 2*b1 - 2) * q^89 + (2*b3 - b1 + 5) * q^91 + (-b5 + 3*b2) * q^93 + (-4*b5 + 2*b4 + 6*b2) * q^97 + (-b1 - 1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q - 6 q^{9}+O(q^{10})$$ 6 * q - 6 * q^9 $$6 q - 6 q^{9} + 8 q^{11} - 20 q^{19} + 4 q^{21} + 20 q^{31} - 4 q^{39} + 16 q^{41} - 26 q^{49} + 20 q^{61} - 6 q^{69} - 16 q^{71} - 28 q^{79} + 6 q^{81} - 4 q^{89} + 32 q^{91} - 8 q^{99}+O(q^{100})$$ 6 * q - 6 * q^9 + 8 * q^11 - 20 * q^19 + 4 * q^21 + 20 * q^31 - 4 * q^39 + 16 * q^41 - 26 * q^49 + 20 * q^61 - 6 * q^69 - 16 * q^71 - 28 * q^79 + 6 * q^81 - 4 * q^89 + 32 * q^91 - 8 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - 2x^{5} + 2x^{4} + 24x^{3} + 81x^{2} + 54x + 18$$ :

 $$\beta_{1}$$ $$=$$ $$( 11\nu^{5} - 34\nu^{4} + 121\nu^{3} + 132\nu^{2} + 66\nu + 1203 ) / 681$$ (11*v^5 - 34*v^4 + 121*v^3 + 132*v^2 + 66*v + 1203) / 681 $$\beta_{2}$$ $$=$$ $$( -29\nu^{5} + 69\nu^{4} - 92\nu^{3} - 575\nu^{2} - 2217\nu - 819 ) / 681$$ (-29*v^5 + 69*v^4 - 92*v^3 - 575*v^2 - 2217*v - 819) / 681 $$\beta_{3}$$ $$=$$ $$( 34\nu^{5} - 167\nu^{4} + 374\nu^{3} + 408\nu^{2} + 204\nu - 2163 ) / 681$$ (34*v^5 - 167*v^4 + 374*v^3 + 408*v^2 + 204*v - 2163) / 681 $$\beta_{4}$$ $$=$$ $$( 40\nu^{5} - 103\nu^{4} + 213\nu^{3} + 707\nu^{2} + 3645\nu + 1341 ) / 681$$ (40*v^5 - 103*v^4 + 213*v^3 + 707*v^2 + 3645*v + 1341) / 681 $$\beta_{5}$$ $$=$$ $$( -123\nu^{5} + 277\nu^{4} - 218\nu^{3} - 3292\nu^{2} - 8229\nu - 3051 ) / 681$$ (-123*v^5 + 277*v^4 - 218*v^3 - 3292*v^2 - 8229*v - 3051) / 681
 $$\nu$$ $$=$$ $$( \beta_{4} + \beta_{2} - \beta _1 + 1 ) / 2$$ (b4 + b2 - b1 + 1) / 2 $$\nu^{2}$$ $$=$$ $$-\beta_{5} + 2\beta_{4} + 7\beta_{2}$$ -b5 + 2*b4 + 7*b2 $$\nu^{3}$$ $$=$$ $$( -2\beta_{5} + 13\beta_{4} - 2\beta_{3} + 29\beta_{2} + 13\beta _1 - 29 ) / 2$$ (-2*b5 + 13*b4 - 2*b3 + 29*b2 + 13*b1 - 29) / 2 $$\nu^{4}$$ $$=$$ $$-11\beta_{3} + 34\beta _1 - 95$$ -11*b3 + 34*b1 - 95 $$\nu^{5}$$ $$=$$ $$( 46\beta_{5} - 197\beta_{4} - 46\beta_{3} - 493\beta_{2} + 197\beta _1 - 493 ) / 2$$ (46*b5 - 197*b4 - 46*b3 - 493*b2 + 197*b1 - 493) / 2

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/6900\mathbb{Z}\right)^\times$$.

 $$n$$ $$277$$ $$1201$$ $$3451$$ $$4601$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
6349.1
 −1.42234 + 1.42234i 2.79911 − 2.79911i −0.376763 + 0.376763i −0.376763 − 0.376763i 2.79911 + 2.79911i −1.42234 − 1.42234i
0 1.00000i 0 0 0 3.73549i 0 −1.00000 0
6349.2 0 1.00000i 0 0 0 1.52644i 0 −1.00000 0
6349.3 0 1.00000i 0 0 0 4.20905i 0 −1.00000 0
6349.4 0 1.00000i 0 0 0 4.20905i 0 −1.00000 0
6349.5 0 1.00000i 0 0 0 1.52644i 0 −1.00000 0
6349.6 0 1.00000i 0 0 0 3.73549i 0 −1.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 6349.6 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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 6900.2.f.r 6
5.b even 2 1 inner 6900.2.f.r 6
5.c odd 4 1 1380.2.a.j 3
5.c odd 4 1 6900.2.a.x 3
15.e even 4 1 4140.2.a.s 3
20.e even 4 1 5520.2.a.bv 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1380.2.a.j 3 5.c odd 4 1
4140.2.a.s 3 15.e even 4 1
5520.2.a.bv 3 20.e even 4 1
6900.2.a.x 3 5.c odd 4 1
6900.2.f.r 6 1.a even 1 1 trivial
6900.2.f.r 6 5.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 $$S_{2}^{\mathrm{new}}(6900, [\chi])$$:

 $$T_{7}^{6} + 34T_{7}^{4} + 321T_{7}^{2} + 576$$ T7^6 + 34*T7^4 + 321*T7^2 + 576 $$T_{11}^{3} - 4T_{11}^{2} - 14T_{11} + 48$$ T11^3 - 4*T11^2 - 14*T11 + 48

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6}$$
$3$ $$(T^{2} + 1)^{3}$$
$5$ $$T^{6}$$
$7$ $$T^{6} + 34 T^{4} + 321 T^{2} + \cdots + 576$$
$11$ $$(T^{3} - 4 T^{2} - 14 T + 48)^{2}$$
$13$ $$T^{6} + 40 T^{4} + 276 T^{2} + \cdots + 144$$
$17$ $$T^{6} + 42 T^{4} + 441 T^{2} + \cdots + 324$$
$19$ $$(T^{3} + 10 T^{2} + 14 T - 52)^{2}$$
$23$ $$(T^{2} + 1)^{3}$$
$29$ $$(T^{3} - 21 T + 18)^{2}$$
$31$ $$(T^{3} - 10 T^{2} + 17 T + 4)^{2}$$
$37$ $$T^{6} + 130 T^{4} + 3537 T^{2} + \cdots + 11664$$
$41$ $$(T^{3} - 8 T^{2} - 101 T + 582)^{2}$$
$43$ $$T^{6} + 216 T^{4} + 10128 T^{2} + \cdots + 92416$$
$47$ $$T^{6} + 116 T^{4} + 4228 T^{2} + \cdots + 46656$$
$53$ $$T^{6} + 266 T^{4} + 19513 T^{2} + \cdots + 338724$$
$59$ $$(T^{3} - 21 T - 18)^{2}$$
$61$ $$(T^{3} - 10 T^{2} - 22 T + 292)^{2}$$
$67$ $$T^{6} + 66 T^{4} + 369 T^{2} + \cdots + 16$$
$71$ $$(T^{3} + 8 T^{2} - 101 T - 582)^{2}$$
$73$ $$T^{6} + 456 T^{4} + 58068 T^{2} + \cdots + 2226064$$
$79$ $$(T^{3} + 14 T^{2} - 96)^{2}$$
$83$ $$T^{6} + 134 T^{4} + 4849 T^{2} + \cdots + 51984$$
$89$ $$(T^{3} + 2 T^{2} - 200 T - 912)^{2}$$
$97$ $$T^{6} + 576 T^{4} + 101184 T^{2} + \cdots + 5456896$$