# Properties

 Label 1840.4.a.i Level $1840$ Weight $4$ Character orbit 1840.a Self dual yes Analytic conductor $108.564$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1840 = 2^{4} \cdot 5 \cdot 23$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1840.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$108.563514411$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{73})$$ Defining polynomial: $$x^{2} - x - 18$$ x^2 - x - 18 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 230) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \frac{1}{2}(1 + \sqrt{73})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta + 1) q^{3} + 5 q^{5} + (\beta + 8) q^{7} + (3 \beta - 8) q^{9}+O(q^{10})$$ q + (b + 1) * q^3 + 5 * q^5 + (b + 8) * q^7 + (3*b - 8) * q^9 $$q + (\beta + 1) q^{3} + 5 q^{5} + (\beta + 8) q^{7} + (3 \beta - 8) q^{9} + ( - 10 \beta - 4) q^{11} + (9 \beta - 9) q^{13} + (5 \beta + 5) q^{15} + ( - 11 \beta - 34) q^{17} + ( - 10 \beta - 12) q^{19} + (10 \beta + 26) q^{21} + 23 q^{23} + 25 q^{25} + ( - 29 \beta + 19) q^{27} + ( - 2 \beta - 55) q^{29} + (26 \beta + 33) q^{31} + ( - 24 \beta - 184) q^{33} + (5 \beta + 40) q^{35} + ( - 7 \beta - 242) q^{37} + (9 \beta + 153) q^{39} + ( - 74 \beta - 129) q^{41} + (22 \beta + 166) q^{43} + (15 \beta - 40) q^{45} + (5 \beta + 297) q^{47} + (17 \beta - 261) q^{49} + ( - 56 \beta - 232) q^{51} + (7 \beta - 156) q^{53} + ( - 50 \beta - 20) q^{55} + ( - 32 \beta - 192) q^{57} + ( - 105 \beta - 126) q^{59} + (12 \beta - 92) q^{61} + (19 \beta - 10) q^{63} + (45 \beta - 45) q^{65} + ( - 41 \beta + 286) q^{67} + (23 \beta + 23) q^{69} + (104 \beta - 679) q^{71} + (23 \beta - 183) q^{73} + (25 \beta + 25) q^{75} + ( - 94 \beta - 212) q^{77} + (56 \beta + 16) q^{79} + ( - 120 \beta - 287) q^{81} + ( - 117 \beta - 578) q^{83} + ( - 55 \beta - 170) q^{85} + ( - 59 \beta - 91) q^{87} + ( - 18 \beta + 562) q^{89} + (72 \beta + 90) q^{91} + (85 \beta + 501) q^{93} + ( - 50 \beta - 60) q^{95} + ( - 164 \beta - 1038) q^{97} + (38 \beta - 508) q^{99}+O(q^{100})$$ q + (b + 1) * q^3 + 5 * q^5 + (b + 8) * q^7 + (3*b - 8) * q^9 + (-10*b - 4) * q^11 + (9*b - 9) * q^13 + (5*b + 5) * q^15 + (-11*b - 34) * q^17 + (-10*b - 12) * q^19 + (10*b + 26) * q^21 + 23 * q^23 + 25 * q^25 + (-29*b + 19) * q^27 + (-2*b - 55) * q^29 + (26*b + 33) * q^31 + (-24*b - 184) * q^33 + (5*b + 40) * q^35 + (-7*b - 242) * q^37 + (9*b + 153) * q^39 + (-74*b - 129) * q^41 + (22*b + 166) * q^43 + (15*b - 40) * q^45 + (5*b + 297) * q^47 + (17*b - 261) * q^49 + (-56*b - 232) * q^51 + (7*b - 156) * q^53 + (-50*b - 20) * q^55 + (-32*b - 192) * q^57 + (-105*b - 126) * q^59 + (12*b - 92) * q^61 + (19*b - 10) * q^63 + (45*b - 45) * q^65 + (-41*b + 286) * q^67 + (23*b + 23) * q^69 + (104*b - 679) * q^71 + (23*b - 183) * q^73 + (25*b + 25) * q^75 + (-94*b - 212) * q^77 + (56*b + 16) * q^79 + (-120*b - 287) * q^81 + (-117*b - 578) * q^83 + (-55*b - 170) * q^85 + (-59*b - 91) * q^87 + (-18*b + 562) * q^89 + (72*b + 90) * q^91 + (85*b + 501) * q^93 + (-50*b - 60) * q^95 + (-164*b - 1038) * q^97 + (38*b - 508) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 3 q^{3} + 10 q^{5} + 17 q^{7} - 13 q^{9}+O(q^{10})$$ 2 * q + 3 * q^3 + 10 * q^5 + 17 * q^7 - 13 * q^9 $$2 q + 3 q^{3} + 10 q^{5} + 17 q^{7} - 13 q^{9} - 18 q^{11} - 9 q^{13} + 15 q^{15} - 79 q^{17} - 34 q^{19} + 62 q^{21} + 46 q^{23} + 50 q^{25} + 9 q^{27} - 112 q^{29} + 92 q^{31} - 392 q^{33} + 85 q^{35} - 491 q^{37} + 315 q^{39} - 332 q^{41} + 354 q^{43} - 65 q^{45} + 599 q^{47} - 505 q^{49} - 520 q^{51} - 305 q^{53} - 90 q^{55} - 416 q^{57} - 357 q^{59} - 172 q^{61} - q^{63} - 45 q^{65} + 531 q^{67} + 69 q^{69} - 1254 q^{71} - 343 q^{73} + 75 q^{75} - 518 q^{77} + 88 q^{79} - 694 q^{81} - 1273 q^{83} - 395 q^{85} - 241 q^{87} + 1106 q^{89} + 252 q^{91} + 1087 q^{93} - 170 q^{95} - 2240 q^{97} - 978 q^{99}+O(q^{100})$$ 2 * q + 3 * q^3 + 10 * q^5 + 17 * q^7 - 13 * q^9 - 18 * q^11 - 9 * q^13 + 15 * q^15 - 79 * q^17 - 34 * q^19 + 62 * q^21 + 46 * q^23 + 50 * q^25 + 9 * q^27 - 112 * q^29 + 92 * q^31 - 392 * q^33 + 85 * q^35 - 491 * q^37 + 315 * q^39 - 332 * q^41 + 354 * q^43 - 65 * q^45 + 599 * q^47 - 505 * q^49 - 520 * q^51 - 305 * q^53 - 90 * q^55 - 416 * q^57 - 357 * q^59 - 172 * q^61 - q^63 - 45 * q^65 + 531 * q^67 + 69 * q^69 - 1254 * q^71 - 343 * q^73 + 75 * q^75 - 518 * q^77 + 88 * q^79 - 694 * q^81 - 1273 * q^83 - 395 * q^85 - 241 * q^87 + 1106 * q^89 + 252 * q^91 + 1087 * q^93 - 170 * q^95 - 2240 * q^97 - 978 * q^99

## 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
 −3.77200 4.77200
0 −2.77200 0 5.00000 0 4.22800 0 −19.3160 0
1.2 0 5.77200 0 5.00000 0 12.7720 0 6.31601 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$5$$ $$-1$$
$$23$$ $$-1$$

## 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 1840.4.a.i 2
4.b odd 2 1 230.4.a.f 2
12.b even 2 1 2070.4.a.s 2
20.d odd 2 1 1150.4.a.l 2
20.e even 4 2 1150.4.b.k 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
230.4.a.f 2 4.b odd 2 1
1150.4.a.l 2 20.d odd 2 1
1150.4.b.k 4 20.e even 4 2
1840.4.a.i 2 1.a even 1 1 trivial
2070.4.a.s 2 12.b even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(1840))$$:

 $$T_{3}^{2} - 3T_{3} - 16$$ T3^2 - 3*T3 - 16 $$T_{7}^{2} - 17T_{7} + 54$$ T7^2 - 17*T7 + 54

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} - 3T - 16$$
$5$ $$(T - 5)^{2}$$
$7$ $$T^{2} - 17T + 54$$
$11$ $$T^{2} + 18T - 1744$$
$13$ $$T^{2} + 9T - 1458$$
$17$ $$T^{2} + 79T - 648$$
$19$ $$T^{2} + 34T - 1536$$
$23$ $$(T - 23)^{2}$$
$29$ $$T^{2} + 112T + 3063$$
$31$ $$T^{2} - 92T - 10221$$
$37$ $$T^{2} + 491T + 59376$$
$41$ $$T^{2} + 332T - 72381$$
$43$ $$T^{2} - 354T + 22496$$
$47$ $$T^{2} - 599T + 89244$$
$53$ $$T^{2} + 305T + 22362$$
$59$ $$T^{2} + 357T - 169344$$
$61$ $$T^{2} + 172T + 4768$$
$67$ $$T^{2} - 531T + 39812$$
$71$ $$T^{2} + 1254 T + 195737$$
$73$ $$T^{2} + 343T + 19758$$
$79$ $$T^{2} - 88T - 55296$$
$83$ $$T^{2} + 1273 T + 155308$$
$89$ $$T^{2} - 1106 T + 299896$$
$97$ $$T^{2} + 2240 T + 763548$$