# Properties

 Label 1840.4.a.h 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: $$\Q(\sqrt{109})$$ Defining polynomial: $$x^{2} - x - 27$$ x^2 - x - 27 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 115) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \frac{1}{2}(1 + \sqrt{109})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta + 1) q^{3} + 5 q^{5} + ( - 5 \beta + 2) q^{7} + (3 \beta + 1) q^{9}+O(q^{10})$$ q + (b + 1) * q^3 + 5 * q^5 + (-5*b + 2) * q^7 + (3*b + 1) * q^9 $$q + (\beta + 1) q^{3} + 5 q^{5} + ( - 5 \beta + 2) q^{7} + (3 \beta + 1) q^{9} + (5 \beta + 11) q^{11} + ( - 3 \beta - 6) q^{13} + (5 \beta + 5) q^{15} + (7 \beta - 43) q^{17} + ( - 13 \beta + 42) q^{19} + ( - 8 \beta - 133) q^{21} + 23 q^{23} + 25 q^{25} + ( - 20 \beta + 55) q^{27} + (10 \beta - 220) q^{29} + (17 \beta + 144) q^{31} + (21 \beta + 146) q^{33} + ( - 25 \beta + 10) q^{35} + ( - 28 \beta - 20) q^{37} + ( - 12 \beta - 87) q^{39} + ( - 11 \beta - 291) q^{41} + ( - 8 \beta - 320) q^{43} + (15 \beta + 5) q^{45} + (74 \beta - 228) q^{47} + (5 \beta + 336) q^{49} + ( - 29 \beta + 146) q^{51} + ( - 44 \beta - 210) q^{53} + (25 \beta + 55) q^{55} + (16 \beta - 309) q^{57} + (54 \beta - 18) q^{59} + ( - 57 \beta + 25) q^{61} + ( - 14 \beta - 403) q^{63} + ( - 15 \beta - 30) q^{65} + (76 \beta - 68) q^{67} + (23 \beta + 23) q^{69} + ( - 97 \beta + 563) q^{71} + (62 \beta + 6) q^{73} + (25 \beta + 25) q^{75} + ( - 70 \beta - 653) q^{77} + ( - 172 \beta - 260) q^{79} + ( - 66 \beta - 512) q^{81} + (144 \beta + 658) q^{83} + (35 \beta - 215) q^{85} + ( - 200 \beta + 50) q^{87} + (180 \beta - 200) q^{89} + (39 \beta + 393) q^{91} + (178 \beta + 603) q^{93} + ( - 65 \beta + 210) q^{95} + ( - 203 \beta + 771) q^{97} + (53 \beta + 416) q^{99}+O(q^{100})$$ q + (b + 1) * q^3 + 5 * q^5 + (-5*b + 2) * q^7 + (3*b + 1) * q^9 + (5*b + 11) * q^11 + (-3*b - 6) * q^13 + (5*b + 5) * q^15 + (7*b - 43) * q^17 + (-13*b + 42) * q^19 + (-8*b - 133) * q^21 + 23 * q^23 + 25 * q^25 + (-20*b + 55) * q^27 + (10*b - 220) * q^29 + (17*b + 144) * q^31 + (21*b + 146) * q^33 + (-25*b + 10) * q^35 + (-28*b - 20) * q^37 + (-12*b - 87) * q^39 + (-11*b - 291) * q^41 + (-8*b - 320) * q^43 + (15*b + 5) * q^45 + (74*b - 228) * q^47 + (5*b + 336) * q^49 + (-29*b + 146) * q^51 + (-44*b - 210) * q^53 + (25*b + 55) * q^55 + (16*b - 309) * q^57 + (54*b - 18) * q^59 + (-57*b + 25) * q^61 + (-14*b - 403) * q^63 + (-15*b - 30) * q^65 + (76*b - 68) * q^67 + (23*b + 23) * q^69 + (-97*b + 563) * q^71 + (62*b + 6) * q^73 + (25*b + 25) * q^75 + (-70*b - 653) * q^77 + (-172*b - 260) * q^79 + (-66*b - 512) * q^81 + (144*b + 658) * q^83 + (35*b - 215) * q^85 + (-200*b + 50) * q^87 + (180*b - 200) * q^89 + (39*b + 393) * q^91 + (178*b + 603) * q^93 + (-65*b + 210) * q^95 + (-203*b + 771) * q^97 + (53*b + 416) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 3 q^{3} + 10 q^{5} - q^{7} + 5 q^{9}+O(q^{10})$$ 2 * q + 3 * q^3 + 10 * q^5 - q^7 + 5 * q^9 $$2 q + 3 q^{3} + 10 q^{5} - q^{7} + 5 q^{9} + 27 q^{11} - 15 q^{13} + 15 q^{15} - 79 q^{17} + 71 q^{19} - 274 q^{21} + 46 q^{23} + 50 q^{25} + 90 q^{27} - 430 q^{29} + 305 q^{31} + 313 q^{33} - 5 q^{35} - 68 q^{37} - 186 q^{39} - 593 q^{41} - 648 q^{43} + 25 q^{45} - 382 q^{47} + 677 q^{49} + 263 q^{51} - 464 q^{53} + 135 q^{55} - 602 q^{57} + 18 q^{59} - 7 q^{61} - 820 q^{63} - 75 q^{65} - 60 q^{67} + 69 q^{69} + 1029 q^{71} + 74 q^{73} + 75 q^{75} - 1376 q^{77} - 692 q^{79} - 1090 q^{81} + 1460 q^{83} - 395 q^{85} - 100 q^{87} - 220 q^{89} + 825 q^{91} + 1384 q^{93} + 355 q^{95} + 1339 q^{97} + 885 q^{99}+O(q^{100})$$ 2 * q + 3 * q^3 + 10 * q^5 - q^7 + 5 * q^9 + 27 * q^11 - 15 * q^13 + 15 * q^15 - 79 * q^17 + 71 * q^19 - 274 * q^21 + 46 * q^23 + 50 * q^25 + 90 * q^27 - 430 * q^29 + 305 * q^31 + 313 * q^33 - 5 * q^35 - 68 * q^37 - 186 * q^39 - 593 * q^41 - 648 * q^43 + 25 * q^45 - 382 * q^47 + 677 * q^49 + 263 * q^51 - 464 * q^53 + 135 * q^55 - 602 * q^57 + 18 * q^59 - 7 * q^61 - 820 * q^63 - 75 * q^65 - 60 * q^67 + 69 * q^69 + 1029 * q^71 + 74 * q^73 + 75 * q^75 - 1376 * q^77 - 692 * q^79 - 1090 * q^81 + 1460 * q^83 - 395 * q^85 - 100 * q^87 - 220 * q^89 + 825 * q^91 + 1384 * q^93 + 355 * q^95 + 1339 * q^97 + 885 * 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
 −4.72015 5.72015
0 −3.72015 0 5.00000 0 25.6008 0 −13.1605 0
1.2 0 6.72015 0 5.00000 0 −26.6008 0 18.1605 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.h 2
4.b odd 2 1 115.4.a.c 2
12.b even 2 1 1035.4.a.g 2
20.d odd 2 1 575.4.a.h 2
20.e even 4 2 575.4.b.f 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
115.4.a.c 2 4.b odd 2 1
575.4.a.h 2 20.d odd 2 1
575.4.b.f 4 20.e even 4 2
1035.4.a.g 2 12.b even 2 1
1840.4.a.h 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 $$S_{4}^{\mathrm{new}}(\Gamma_0(1840))$$:

 $$T_{3}^{2} - 3T_{3} - 25$$ T3^2 - 3*T3 - 25 $$T_{7}^{2} + T_{7} - 681$$ T7^2 + T7 - 681

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} - 3T - 25$$
$5$ $$(T - 5)^{2}$$
$7$ $$T^{2} + T - 681$$
$11$ $$T^{2} - 27T - 499$$
$13$ $$T^{2} + 15T - 189$$
$17$ $$T^{2} + 79T + 225$$
$19$ $$T^{2} - 71T - 3345$$
$23$ $$(T - 23)^{2}$$
$29$ $$T^{2} + 430T + 43500$$
$31$ $$T^{2} - 305T + 15381$$
$37$ $$T^{2} + 68T - 20208$$
$41$ $$T^{2} + 593T + 84615$$
$43$ $$T^{2} + 648T + 103232$$
$47$ $$T^{2} + 382T - 112740$$
$53$ $$T^{2} + 464T + 1068$$
$59$ $$T^{2} - 18T - 79380$$
$61$ $$T^{2} + 7T - 88523$$
$67$ $$T^{2} + 60T - 156496$$
$71$ $$T^{2} - 1029T + 8315$$
$73$ $$T^{2} - 74T - 103380$$
$79$ $$T^{2} + 692T - 686448$$
$83$ $$T^{2} - 1460T - 32156$$
$89$ $$T^{2} + 220T - 870800$$
$97$ $$T^{2} - 1339 T - 674715$$