# Properties

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

# 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: $$2$$ Coefficient field: $$\Q(\sqrt{2})$$ Defining polynomial: $$x^{2} - 2$$ x^2 - 2 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 105) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = 2\sqrt{2}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta + 1) q^{2} + (2 \beta + 1) q^{4} + 5 q^{5} + ( - 5 \beta + 9) q^{8}+O(q^{10})$$ q + (b + 1) * q^2 + (2*b + 1) * q^4 + 5 * q^5 + (-5*b + 9) * q^8 $$q + (\beta + 1) q^{2} + (2 \beta + 1) q^{4} + 5 q^{5} + ( - 5 \beta + 9) q^{8} + (5 \beta + 5) q^{10} + ( - 20 \beta + 8) q^{11} + (2 \beta + 38) q^{13} + ( - 12 \beta - 39) q^{16} + ( - 2 \beta - 62) q^{17} + (16 \beta + 48) q^{19} + (10 \beta + 5) q^{20} + ( - 12 \beta - 152) q^{22} + (34 \beta + 8) q^{23} + 25 q^{25} + (40 \beta + 54) q^{26} + (54 \beta - 94) q^{29} + ( - 18 \beta + 60) q^{31} + ( - 11 \beta - 207) q^{32} + ( - 64 \beta - 78) q^{34} + ( - 66 \beta - 66) q^{37} + (64 \beta + 176) q^{38} + ( - 25 \beta + 45) q^{40} + (80 \beta + 50) q^{41} + (62 \beta - 268) q^{43} + ( - 4 \beta - 312) q^{44} + (42 \beta + 280) q^{46} + ( - 42 \beta - 464) q^{47} + (25 \beta + 25) q^{50} + (78 \beta + 70) q^{52} + ( - 64 \beta - 442) q^{53} + ( - 100 \beta + 40) q^{55} + ( - 40 \beta + 338) q^{58} + ( - 204 \beta + 52) q^{59} + (42 \beta + 234) q^{61} + (42 \beta - 84) q^{62} + ( - 122 \beta + 17) q^{64} + (10 \beta + 190) q^{65} + (38 \beta - 844) q^{67} + ( - 126 \beta - 94) q^{68} + (150 \beta + 68) q^{71} + ( - 322 \beta - 254) q^{73} + ( - 132 \beta - 594) q^{74} + (112 \beta + 304) q^{76} + ( - 232 \beta - 216) q^{79} + ( - 60 \beta - 195) q^{80} + (130 \beta + 690) q^{82} + ( - 84 \beta - 292) q^{83} + ( - 10 \beta - 310) q^{85} + ( - 206 \beta + 228) q^{86} + ( - 220 \beta + 872) q^{88} + (112 \beta - 702) q^{89} + (50 \beta + 552) q^{92} + ( - 506 \beta - 800) q^{94} + (80 \beta + 240) q^{95} + ( - 46 \beta + 594) q^{97}+O(q^{100})$$ q + (b + 1) * q^2 + (2*b + 1) * q^4 + 5 * q^5 + (-5*b + 9) * q^8 + (5*b + 5) * q^10 + (-20*b + 8) * q^11 + (2*b + 38) * q^13 + (-12*b - 39) * q^16 + (-2*b - 62) * q^17 + (16*b + 48) * q^19 + (10*b + 5) * q^20 + (-12*b - 152) * q^22 + (34*b + 8) * q^23 + 25 * q^25 + (40*b + 54) * q^26 + (54*b - 94) * q^29 + (-18*b + 60) * q^31 + (-11*b - 207) * q^32 + (-64*b - 78) * q^34 + (-66*b - 66) * q^37 + (64*b + 176) * q^38 + (-25*b + 45) * q^40 + (80*b + 50) * q^41 + (62*b - 268) * q^43 + (-4*b - 312) * q^44 + (42*b + 280) * q^46 + (-42*b - 464) * q^47 + (25*b + 25) * q^50 + (78*b + 70) * q^52 + (-64*b - 442) * q^53 + (-100*b + 40) * q^55 + (-40*b + 338) * q^58 + (-204*b + 52) * q^59 + (42*b + 234) * q^61 + (42*b - 84) * q^62 + (-122*b + 17) * q^64 + (10*b + 190) * q^65 + (38*b - 844) * q^67 + (-126*b - 94) * q^68 + (150*b + 68) * q^71 + (-322*b - 254) * q^73 + (-132*b - 594) * q^74 + (112*b + 304) * q^76 + (-232*b - 216) * q^79 + (-60*b - 195) * q^80 + (130*b + 690) * q^82 + (-84*b - 292) * q^83 + (-10*b - 310) * q^85 + (-206*b + 228) * q^86 + (-220*b + 872) * q^88 + (112*b - 702) * q^89 + (50*b + 552) * q^92 + (-506*b - 800) * q^94 + (80*b + 240) * q^95 + (-46*b + 594) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{2} + 2 q^{4} + 10 q^{5} + 18 q^{8}+O(q^{10})$$ 2 * q + 2 * q^2 + 2 * q^4 + 10 * q^5 + 18 * q^8 $$2 q + 2 q^{2} + 2 q^{4} + 10 q^{5} + 18 q^{8} + 10 q^{10} + 16 q^{11} + 76 q^{13} - 78 q^{16} - 124 q^{17} + 96 q^{19} + 10 q^{20} - 304 q^{22} + 16 q^{23} + 50 q^{25} + 108 q^{26} - 188 q^{29} + 120 q^{31} - 414 q^{32} - 156 q^{34} - 132 q^{37} + 352 q^{38} + 90 q^{40} + 100 q^{41} - 536 q^{43} - 624 q^{44} + 560 q^{46} - 928 q^{47} + 50 q^{50} + 140 q^{52} - 884 q^{53} + 80 q^{55} + 676 q^{58} + 104 q^{59} + 468 q^{61} - 168 q^{62} + 34 q^{64} + 380 q^{65} - 1688 q^{67} - 188 q^{68} + 136 q^{71} - 508 q^{73} - 1188 q^{74} + 608 q^{76} - 432 q^{79} - 390 q^{80} + 1380 q^{82} - 584 q^{83} - 620 q^{85} + 456 q^{86} + 1744 q^{88} - 1404 q^{89} + 1104 q^{92} - 1600 q^{94} + 480 q^{95} + 1188 q^{97}+O(q^{100})$$ 2 * q + 2 * q^2 + 2 * q^4 + 10 * q^5 + 18 * q^8 + 10 * q^10 + 16 * q^11 + 76 * q^13 - 78 * q^16 - 124 * q^17 + 96 * q^19 + 10 * q^20 - 304 * q^22 + 16 * q^23 + 50 * q^25 + 108 * q^26 - 188 * q^29 + 120 * q^31 - 414 * q^32 - 156 * q^34 - 132 * q^37 + 352 * q^38 + 90 * q^40 + 100 * q^41 - 536 * q^43 - 624 * q^44 + 560 * q^46 - 928 * q^47 + 50 * q^50 + 140 * q^52 - 884 * q^53 + 80 * q^55 + 676 * q^58 + 104 * q^59 + 468 * q^61 - 168 * q^62 + 34 * q^64 + 380 * q^65 - 1688 * q^67 - 188 * q^68 + 136 * q^71 - 508 * q^73 - 1188 * q^74 + 608 * q^76 - 432 * q^79 - 390 * q^80 + 1380 * q^82 - 584 * q^83 - 620 * q^85 + 456 * q^86 + 1744 * q^88 - 1404 * q^89 + 1104 * q^92 - 1600 * q^94 + 480 * q^95 + 1188 * q^97

## 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
 −1.41421 1.41421
−1.82843 0 −4.65685 5.00000 0 0 23.1421 0 −9.14214
1.2 3.82843 0 6.65685 5.00000 0 0 −5.14214 0 19.1421
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

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

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2205.4.a.bb 2
3.b odd 2 1 735.4.a.o 2
7.b odd 2 1 315.4.a.k 2
21.c even 2 1 105.4.a.e 2
35.c odd 2 1 1575.4.a.q 2
84.h odd 2 1 1680.4.a.bo 2
105.g even 2 1 525.4.a.l 2
105.k odd 4 2 525.4.d.l 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.4.a.e 2 21.c even 2 1
315.4.a.k 2 7.b odd 2 1
525.4.a.l 2 105.g even 2 1
525.4.d.l 4 105.k odd 4 2
735.4.a.o 2 3.b odd 2 1
1575.4.a.q 2 35.c odd 2 1
1680.4.a.bo 2 84.h odd 2 1
2205.4.a.bb 2 1.a even 1 1 trivial

## Hecke kernels

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

 $$T_{2}^{2} - 2T_{2} - 7$$ T2^2 - 2*T2 - 7 $$T_{11}^{2} - 16T_{11} - 3136$$ T11^2 - 16*T11 - 3136 $$T_{13}^{2} - 76T_{13} + 1412$$ T13^2 - 76*T13 + 1412 $$T_{17}^{2} + 124T_{17} + 3812$$ T17^2 + 124*T17 + 3812

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - 2T - 7$$
$3$ $$T^{2}$$
$5$ $$(T - 5)^{2}$$
$7$ $$T^{2}$$
$11$ $$T^{2} - 16T - 3136$$
$13$ $$T^{2} - 76T + 1412$$
$17$ $$T^{2} + 124T + 3812$$
$19$ $$T^{2} - 96T + 256$$
$23$ $$T^{2} - 16T - 9184$$
$29$ $$T^{2} + 188T - 14492$$
$31$ $$T^{2} - 120T + 1008$$
$37$ $$T^{2} + 132T - 30492$$
$41$ $$T^{2} - 100T - 48700$$
$43$ $$T^{2} + 536T + 41072$$
$47$ $$T^{2} + 928T + 201184$$
$53$ $$T^{2} + 884T + 162596$$
$59$ $$T^{2} - 104T - 330224$$
$61$ $$T^{2} - 468T + 40644$$
$67$ $$T^{2} + 1688 T + 700784$$
$71$ $$T^{2} - 136T - 175376$$
$73$ $$T^{2} + 508T - 764956$$
$79$ $$T^{2} + 432T - 383936$$
$83$ $$T^{2} + 584T + 28816$$
$89$ $$T^{2} + 1404 T + 392452$$
$97$ $$T^{2} - 1188 T + 335908$$