# Properties

 Label 2205.4.a.bh Level $2205$ Weight $4$ Character orbit 2205.a Self dual yes Analytic conductor $130.099$ Analytic rank $0$ 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: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{17})$$ Defining polynomial: $$x^{2} - x - 4$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ 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 $$\beta = \frac{1}{2}(1 + \sqrt{17})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 4 - \beta ) q^{2} + ( 12 - 7 \beta ) q^{4} -5 q^{5} + ( 44 - 25 \beta ) q^{8} +O(q^{10})$$ $$q + ( 4 - \beta ) q^{2} + ( 12 - 7 \beta ) q^{4} -5 q^{5} + ( 44 - 25 \beta ) q^{8} + ( -20 + 5 \beta ) q^{10} + ( 18 - 10 \beta ) q^{11} + ( 4 - 22 \beta ) q^{13} + ( 180 - 63 \beta ) q^{16} + ( 22 - 28 \beta ) q^{17} + ( -74 - 26 \beta ) q^{19} + ( -60 + 35 \beta ) q^{20} + ( 112 - 48 \beta ) q^{22} + ( -120 + 56 \beta ) q^{23} + 25 q^{25} + ( 104 - 70 \beta ) q^{26} + ( 58 - 84 \beta ) q^{29} + ( -174 + 18 \beta ) q^{31} + ( 620 - 169 \beta ) q^{32} + ( 200 - 106 \beta ) q^{34} + ( -54 - 24 \beta ) q^{37} + ( -192 - 4 \beta ) q^{38} + ( -220 + 125 \beta ) q^{40} + ( 170 - 140 \beta ) q^{41} + ( 148 + 68 \beta ) q^{43} + ( 496 - 176 \beta ) q^{44} + ( -704 + 288 \beta ) q^{46} + ( 200 - 108 \beta ) q^{47} + ( 100 - 25 \beta ) q^{50} + ( 664 - 138 \beta ) q^{52} + ( -124 + 214 \beta ) q^{53} + ( -90 + 50 \beta ) q^{55} + ( 568 - 310 \beta ) q^{58} + ( 200 - 36 \beta ) q^{59} + ( -270 - 252 \beta ) q^{61} + ( -768 + 228 \beta ) q^{62} + ( 1716 - 623 \beta ) q^{64} + ( -20 + 110 \beta ) q^{65} + ( -380 - 28 \beta ) q^{67} + ( 1048 - 294 \beta ) q^{68} + ( -62 - 330 \beta ) q^{71} + ( -300 - 178 \beta ) q^{73} + ( -120 - 18 \beta ) q^{74} + ( -160 + 388 \beta ) q^{76} + ( 248 - 88 \beta ) q^{79} + ( -900 + 315 \beta ) q^{80} + ( 1240 - 590 \beta ) q^{82} + ( 436 + 264 \beta ) q^{83} + ( -110 + 140 \beta ) q^{85} + ( 320 + 56 \beta ) q^{86} + ( 1792 - 640 \beta ) q^{88} + ( -346 + 728 \beta ) q^{89} + ( -3008 + 1120 \beta ) q^{92} + ( 1232 - 524 \beta ) q^{94} + ( 370 + 130 \beta ) q^{95} + ( 176 + 146 \beta ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 7 q^{2} + 17 q^{4} - 10 q^{5} + 63 q^{8} + O(q^{10})$$ $$2 q + 7 q^{2} + 17 q^{4} - 10 q^{5} + 63 q^{8} - 35 q^{10} + 26 q^{11} - 14 q^{13} + 297 q^{16} + 16 q^{17} - 174 q^{19} - 85 q^{20} + 176 q^{22} - 184 q^{23} + 50 q^{25} + 138 q^{26} + 32 q^{29} - 330 q^{31} + 1071 q^{32} + 294 q^{34} - 132 q^{37} - 388 q^{38} - 315 q^{40} + 200 q^{41} + 364 q^{43} + 816 q^{44} - 1120 q^{46} + 292 q^{47} + 175 q^{50} + 1190 q^{52} - 34 q^{53} - 130 q^{55} + 826 q^{58} + 364 q^{59} - 792 q^{61} - 1308 q^{62} + 2809 q^{64} + 70 q^{65} - 788 q^{67} + 1802 q^{68} - 454 q^{71} - 778 q^{73} - 258 q^{74} + 68 q^{76} + 408 q^{79} - 1485 q^{80} + 1890 q^{82} + 1136 q^{83} - 80 q^{85} + 696 q^{86} + 2944 q^{88} + 36 q^{89} - 4896 q^{92} + 1940 q^{94} + 870 q^{95} + 498 q^{97} + O(q^{100})$$

## 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
 2.56155 −1.56155
1.43845 0 −5.93087 −5.00000 0 0 −20.0388 0 −7.19224
1.2 5.56155 0 22.9309 −5.00000 0 0 83.0388 0 −27.8078
 $$n$$: 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.bh 2
3.b odd 2 1 735.4.a.k 2
7.b odd 2 1 315.4.a.m 2
21.c even 2 1 105.4.a.c 2
35.c odd 2 1 1575.4.a.m 2
84.h odd 2 1 1680.4.a.bk 2
105.g even 2 1 525.4.a.p 2
105.k odd 4 2 525.4.d.i 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.4.a.c 2 21.c even 2 1
315.4.a.m 2 7.b odd 2 1
525.4.a.p 2 105.g even 2 1
525.4.d.i 4 105.k odd 4 2
735.4.a.k 2 3.b odd 2 1
1575.4.a.m 2 35.c odd 2 1
1680.4.a.bk 2 84.h odd 2 1
2205.4.a.bh 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 $$S_{4}^{\mathrm{new}}(\Gamma_0(2205))$$:

 $$T_{2}^{2} - 7 T_{2} + 8$$ $$T_{11}^{2} - 26 T_{11} - 256$$ $$T_{13}^{2} + 14 T_{13} - 2008$$ $$T_{17}^{2} - 16 T_{17} - 3268$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$8 - 7 T + T^{2}$$
$3$ $$T^{2}$$
$5$ $$( 5 + T )^{2}$$
$7$ $$T^{2}$$
$11$ $$-256 - 26 T + T^{2}$$
$13$ $$-2008 + 14 T + T^{2}$$
$17$ $$-3268 - 16 T + T^{2}$$
$19$ $$4696 + 174 T + T^{2}$$
$23$ $$-4864 + 184 T + T^{2}$$
$29$ $$-29732 - 32 T + T^{2}$$
$31$ $$25848 + 330 T + T^{2}$$
$37$ $$1908 + 132 T + T^{2}$$
$41$ $$-73300 - 200 T + T^{2}$$
$43$ $$13472 - 364 T + T^{2}$$
$47$ $$-28256 - 292 T + T^{2}$$
$53$ $$-194344 + 34 T + T^{2}$$
$59$ $$27616 - 364 T + T^{2}$$
$61$ $$-113076 + 792 T + T^{2}$$
$67$ $$151904 + 788 T + T^{2}$$
$71$ $$-411296 + 454 T + T^{2}$$
$73$ $$16664 + 778 T + T^{2}$$
$79$ $$8704 - 408 T + T^{2}$$
$83$ $$26416 - 1136 T + T^{2}$$
$89$ $$-2252108 - 36 T + T^{2}$$
$97$ $$-28592 - 498 T + T^{2}$$