# Properties

 Label 2205.4.a.bx Level $2205$ Weight $4$ Character orbit 2205.a Self dual yes Analytic conductor $130.099$ Analytic rank $1$ Dimension $6$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2205 = 3^{2} \cdot 5 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 2205.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$130.099211563$$ Analytic rank: $$1$$ Dimension: $$6$$ Coefficient field: $$\mathbb{Q}[x]/(x^{6} - \cdots)$$ Defining polynomial: $$x^{6} - 34x^{4} + 241x^{2} - 64$$ x^6 - 34*x^4 + 241*x^2 - 64 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{9}$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{2} + (\beta_{2} + 3) q^{4} - 5 q^{5} + (\beta_{3} + 4 \beta_1) q^{8}+O(q^{10})$$ q + b1 * q^2 + (b2 + 3) * q^4 - 5 * q^5 + (b3 + 4*b1) * q^8 $$q + \beta_1 q^{2} + (\beta_{2} + 3) q^{4} - 5 q^{5} + (\beta_{3} + 4 \beta_1) q^{8} - 5 \beta_1 q^{10} + \beta_{5} q^{11} + ( - \beta_{5} - \beta_{3} - 3 \beta_1) q^{13} + (\beta_{4} + \beta_{2} + 16) q^{16} + ( - \beta_{4} - 7) q^{17} + ( - 2 \beta_{5} + 3 \beta_{3} - \beta_1) q^{19} + ( - 5 \beta_{2} - 15) q^{20} + (\beta_{4} + 2 \beta_{2} - 5) q^{22} + (\beta_{5} - 4 \beta_{3} + 6 \beta_1) q^{23} + 25 q^{25} + ( - 2 \beta_{4} - 10 \beta_{2} - 24) q^{26} + (2 \beta_{5} - \beta_{3} + 17 \beta_1) q^{29} + (2 \beta_{5} - 8 \beta_{3} - 34 \beta_1) q^{31} + (2 \beta_{5} - 3 \beta_1) q^{32} + ( - 2 \beta_{5} - 7 \beta_{3} - 11 \beta_1) q^{34} + ( - 3 \beta_{4} - 8 \beta_{2} + 67) q^{37} + (\beta_{4} + 10 \beta_{2} - 13) q^{38} + ( - 5 \beta_{3} - 20 \beta_1) q^{40} + (2 \beta_{4} - 20 \beta_{2} - 96) q^{41} + ( - \beta_{4} - 4 \beta_{2} - 53) q^{43} + ( - 6 \beta_{5} + 9 \beta_{3} + 17 \beta_1) q^{44} + ( - 3 \beta_{4} - 12 \beta_{2} + 77) q^{46} + (5 \beta_{4} - 20 \beta_{2} - 15) q^{47} + 25 \beta_1 q^{50} + (4 \beta_{5} - 16 \beta_{3} - 98 \beta_1) q^{52} + ( - 7 \beta_{5} + 14 \beta_{3} - 10 \beta_1) q^{53} - 5 \beta_{5} q^{55} + (\beta_{4} + 16 \beta_{2} + 181) q^{58} + (6 \beta_{4} + 40 \beta_{2} - 38) q^{59} + (15 \beta_{3} - 75 \beta_1) q^{61} + ( - 6 \beta_{4} - 70 \beta_{2} - 352) q^{62} + ( - 6 \beta_{4} - 7 \beta_{2} - 171) q^{64} + (5 \beta_{5} + 5 \beta_{3} + 15 \beta_1) q^{65} + (11 \beta_{4} + 8 \beta_{2} + 179) q^{67} + ( - \beta_{4} - 50 \beta_{2} - 27) q^{68} + ( - 5 \beta_{5} - \beta_{3} - 43 \beta_1) q^{71} + ( - 19 \beta_{5} + 21 \beta_{3} - 137 \beta_1) q^{73} + ( - 6 \beta_{5} - 29 \beta_{3} - 17 \beta_1) q^{74} + (18 \beta_{5} - 7 \beta_{3} + 89 \beta_1) q^{76} + (2 \beta_{4} - 4 \beta_{2} + 234) q^{79} + ( - 5 \beta_{4} - 5 \beta_{2} - 80) q^{80} + (4 \beta_{5} - 6 \beta_{3} - 268 \beta_1) q^{82} + (2 \beta_{4} - 60 \beta_{2} - 466) q^{83} + (5 \beta_{4} + 35) q^{85} + ( - 2 \beta_{5} - 11 \beta_{3} - 93 \beta_1) q^{86} + ( - 5 \beta_{4} + 34 \beta_{2} + 221) q^{88} + (2 \beta_{4} - 80 \beta_{2} - 36) q^{89} + ( - 14 \beta_{5} - \beta_{3} - 91 \beta_1) q^{92} + (10 \beta_{5} + 15 \beta_{3} - 175 \beta_1) q^{94} + (10 \beta_{5} - 15 \beta_{3} + 5 \beta_1) q^{95} + (11 \beta_{5} + \beta_{3} + 203 \beta_1) q^{97}+O(q^{100})$$ q + b1 * q^2 + (b2 + 3) * q^4 - 5 * q^5 + (b3 + 4*b1) * q^8 - 5*b1 * q^10 + b5 * q^11 + (-b5 - b3 - 3*b1) * q^13 + (b4 + b2 + 16) * q^16 + (-b4 - 7) * q^17 + (-2*b5 + 3*b3 - b1) * q^19 + (-5*b2 - 15) * q^20 + (b4 + 2*b2 - 5) * q^22 + (b5 - 4*b3 + 6*b1) * q^23 + 25 * q^25 + (-2*b4 - 10*b2 - 24) * q^26 + (2*b5 - b3 + 17*b1) * q^29 + (2*b5 - 8*b3 - 34*b1) * q^31 + (2*b5 - 3*b1) * q^32 + (-2*b5 - 7*b3 - 11*b1) * q^34 + (-3*b4 - 8*b2 + 67) * q^37 + (b4 + 10*b2 - 13) * q^38 + (-5*b3 - 20*b1) * q^40 + (2*b4 - 20*b2 - 96) * q^41 + (-b4 - 4*b2 - 53) * q^43 + (-6*b5 + 9*b3 + 17*b1) * q^44 + (-3*b4 - 12*b2 + 77) * q^46 + (5*b4 - 20*b2 - 15) * q^47 + 25*b1 * q^50 + (4*b5 - 16*b3 - 98*b1) * q^52 + (-7*b5 + 14*b3 - 10*b1) * q^53 - 5*b5 * q^55 + (b4 + 16*b2 + 181) * q^58 + (6*b4 + 40*b2 - 38) * q^59 + (15*b3 - 75*b1) * q^61 + (-6*b4 - 70*b2 - 352) * q^62 + (-6*b4 - 7*b2 - 171) * q^64 + (5*b5 + 5*b3 + 15*b1) * q^65 + (11*b4 + 8*b2 + 179) * q^67 + (-b4 - 50*b2 - 27) * q^68 + (-5*b5 - b3 - 43*b1) * q^71 + (-19*b5 + 21*b3 - 137*b1) * q^73 + (-6*b5 - 29*b3 - 17*b1) * q^74 + (18*b5 - 7*b3 + 89*b1) * q^76 + (2*b4 - 4*b2 + 234) * q^79 + (-5*b4 - 5*b2 - 80) * q^80 + (4*b5 - 6*b3 - 268*b1) * q^82 + (2*b4 - 60*b2 - 466) * q^83 + (5*b4 + 35) * q^85 + (-2*b5 - 11*b3 - 93*b1) * q^86 + (-5*b4 + 34*b2 + 221) * q^88 + (2*b4 - 80*b2 - 36) * q^89 + (-14*b5 - b3 - 91*b1) * q^92 + (10*b5 + 15*b3 - 175*b1) * q^94 + (10*b5 - 15*b3 + 5*b1) * q^95 + (11*b5 + b3 + 203*b1) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q + 20 q^{4} - 30 q^{5}+O(q^{10})$$ 6 * q + 20 * q^4 - 30 * q^5 $$6 q + 20 q^{4} - 30 q^{5} + 100 q^{16} - 44 q^{17} - 100 q^{20} - 24 q^{22} + 150 q^{25} - 168 q^{26} + 380 q^{37} - 56 q^{38} - 612 q^{41} - 328 q^{43} + 432 q^{46} - 120 q^{47} + 1120 q^{58} - 136 q^{59} - 2264 q^{62} - 1052 q^{64} + 1112 q^{67} - 264 q^{68} + 1400 q^{79} - 500 q^{80} - 2912 q^{83} + 220 q^{85} + 1384 q^{88} - 372 q^{89}+O(q^{100})$$ 6 * q + 20 * q^4 - 30 * q^5 + 100 * q^16 - 44 * q^17 - 100 * q^20 - 24 * q^22 + 150 * q^25 - 168 * q^26 + 380 * q^37 - 56 * q^38 - 612 * q^41 - 328 * q^43 + 432 * q^46 - 120 * q^47 + 1120 * q^58 - 136 * q^59 - 2264 * q^62 - 1052 * q^64 + 1112 * q^67 - 264 * q^68 + 1400 * q^79 - 500 * q^80 - 2912 * q^83 + 220 * q^85 + 1384 * q^88 - 372 * q^89

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - 34x^{4} + 241x^{2} - 64$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - 11$$ v^2 - 11 $$\beta_{3}$$ $$=$$ $$\nu^{3} - 20\nu$$ v^3 - 20*v $$\beta_{4}$$ $$=$$ $$\nu^{4} - 25\nu^{2} + 59$$ v^4 - 25*v^2 + 59 $$\beta_{5}$$ $$=$$ $$( \nu^{5} - 32\nu^{3} + 195\nu ) / 2$$ (v^5 - 32*v^3 + 195*v) / 2
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + 11$$ b2 + 11 $$\nu^{3}$$ $$=$$ $$\beta_{3} + 20\beta_1$$ b3 + 20*b1 $$\nu^{4}$$ $$=$$ $$\beta_{4} + 25\beta_{2} + 216$$ b4 + 25*b2 + 216 $$\nu^{5}$$ $$=$$ $$2\beta_{5} + 32\beta_{3} + 445\beta_1$$ 2*b5 + 32*b3 + 445*b1

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 −4.91092 −3.09945 −0.525584 0.525584 3.09945 4.91092
−4.91092 0 16.1171 −5.00000 0 0 −39.8626 0 24.5546
1.2 −3.09945 0 1.60662 −5.00000 0 0 19.8160 0 15.4973
1.3 −0.525584 0 −7.72376 −5.00000 0 0 8.26415 0 2.62792
1.4 0.525584 0 −7.72376 −5.00000 0 0 −8.26415 0 −2.62792
1.5 3.09945 0 1.60662 −5.00000 0 0 −19.8160 0 −15.4973
1.6 4.91092 0 16.1171 −5.00000 0 0 39.8626 0 −24.5546
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.6 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$1$$
$$5$$ $$1$$
$$7$$ $$-1$$

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
21.c even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2205.4.a.bx 6
3.b odd 2 1 2205.4.a.by yes 6
7.b odd 2 1 2205.4.a.by yes 6
21.c even 2 1 inner 2205.4.a.bx 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2205.4.a.bx 6 1.a even 1 1 trivial
2205.4.a.bx 6 21.c even 2 1 inner
2205.4.a.by yes 6 3.b odd 2 1
2205.4.a.by yes 6 7.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(2205))$$:

 $$T_{2}^{6} - 34T_{2}^{4} + 241T_{2}^{2} - 64$$ T2^6 - 34*T2^4 + 241*T2^2 - 64 $$T_{11}^{6} - 3512T_{11}^{4} + 2814992T_{11}^{2} - 335622400$$ T11^6 - 3512*T11^4 + 2814992*T11^2 - 335622400 $$T_{13}^{6} - 6744T_{13}^{4} + 14726800T_{13}^{2} - 10404000000$$ T13^6 - 6744*T13^4 + 14726800*T13^2 - 10404000000 $$T_{17}^{3} + 22T_{17}^{2} - 5860T_{17} - 216600$$ T17^3 + 22*T17^2 - 5860*T17 - 216600

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6} - 34 T^{4} + 241 T^{2} + \cdots - 64$$
$3$ $$T^{6}$$
$5$ $$(T + 5)^{6}$$
$7$ $$T^{6}$$
$11$ $$T^{6} - 3512 T^{4} + \cdots - 335622400$$
$13$ $$T^{6} - 6744 T^{4} + \cdots - 10404000000$$
$17$ $$(T^{3} + 22 T^{2} - 5860 T - 216600)^{2}$$
$19$ $$T^{6} - 19176 T^{4} + \cdots - 23592960000$$
$23$ $$T^{6} - 23824 T^{4} + \cdots - 183759683584$$
$29$ $$T^{6} - 21856 T^{4} + \cdots - 53084160000$$
$31$ $$T^{6} - 127496 T^{4} + \cdots - 19948728960000$$
$37$ $$(T^{3} - 190 T^{2} - 52804 T - 193800)^{2}$$
$41$ $$(T^{3} + 306 T^{2} - 48260 T - 15501000)^{2}$$
$43$ $$(T^{3} + 164 T^{2} + 400 T - 369600)^{2}$$
$47$ $$(T^{3} + 60 T^{2} - 201200 T - 17640000)^{2}$$
$53$ $$T^{6} + \cdots - 379858852641024$$
$59$ $$(T^{3} + 68 T^{2} - 460240 T - 68808000)^{2}$$
$61$ $$T^{6} - 554400 T^{4} + \cdots - 82301184000000$$
$67$ $$(T^{3} - 556 T^{2} - 639936 T + 377458560)^{2}$$
$71$ $$T^{6} - 153944 T^{4} + \cdots - 16881580038400$$
$73$ $$T^{6} - 1972824 T^{4} + \cdots - 87\!\cdots\!00$$
$79$ $$(T^{3} - 700 T^{2} + 137408 T - 6055424)^{2}$$
$83$ $$(T^{3} + 1456 T^{2} + 170000 T - 185865600)^{2}$$
$89$ $$(T^{3} + 186 T^{2} - 926900 T - 95685000)^{2}$$
$97$ $$T^{6} - 1787864 T^{4} + \cdots - 34\!\cdots\!00$$