# Properties

 Label 1840.4.a.v Level $1840$ Weight $4$ Character orbit 1840.a Self dual yes Analytic conductor $108.564$ Analytic rank $0$ Dimension $8$ 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: $$0$$ Dimension: $$8$$ Coefficient field: $$\mathbb{Q}[x]/(x^{8} - \cdots)$$ Defining polynomial: $$x^{8} - 2x^{7} - 49x^{6} + 31x^{5} + 750x^{4} + 249x^{3} - 2892x^{2} - 620x + 2400$$ x^8 - 2*x^7 - 49*x^6 + 31*x^5 + 750*x^4 + 249*x^3 - 2892*x^2 - 620*x + 2400 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{5}$$ 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 a basis $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{3} q^{3} + 5 q^{5} + (\beta_{6} + \beta_{3} - 1) q^{7} + ( - \beta_{7} - \beta_{5} - \beta_{4} + 2 \beta_{3} + \beta_1 + 20) q^{9}+O(q^{10})$$ q + b3 * q^3 + 5 * q^5 + (b6 + b3 - 1) * q^7 + (-b7 - b5 - b4 + 2*b3 + b1 + 20) * q^9 $$q + \beta_{3} q^{3} + 5 q^{5} + (\beta_{6} + \beta_{3} - 1) q^{7} + ( - \beta_{7} - \beta_{5} - \beta_{4} + 2 \beta_{3} + \beta_1 + 20) q^{9} + ( - \beta_{7} + \beta_{5} + \beta_{2} - 6) q^{11} + ( - \beta_{7} + \beta_{4} + 3 \beta_{3} - \beta_{2} - \beta_1 + 3) q^{13} + 5 \beta_{3} q^{15} + ( - \beta_{7} - \beta_{6} + 2 \beta_{5} + \beta_{4} - 3 \beta_{3} + \beta_{2} + 2 \beta_1 + 7) q^{17} + ( - 3 \beta_{7} - \beta_{6} + \beta_{5} - 2 \beta_{4} - 2 \beta_{3} - 2 \beta_{2} + \cdots - 22) q^{19}+ \cdots + ( - 5 \beta_{7} + 45 \beta_{6} - 26 \beta_{5} + 30 \beta_{4} - 3 \beta_{3} + \cdots - 162) q^{99}+O(q^{100})$$ q + b3 * q^3 + 5 * q^5 + (b6 + b3 - 1) * q^7 + (-b7 - b5 - b4 + 2*b3 + b1 + 20) * q^9 + (-b7 + b5 + b2 - 6) * q^11 + (-b7 + b4 + 3*b3 - b2 - b1 + 3) * q^13 + 5*b3 * q^15 + (-b7 - b6 + 2*b5 + b4 - 3*b3 + b2 + 2*b1 + 7) * q^17 + (-3*b7 - b6 + b5 - 2*b4 - 2*b3 - 2*b2 + b1 - 22) * q^19 + (-5*b7 + 6*b6 + b4 + 2*b3 + b2 + 4*b1 + 36) * q^21 - 23 * q^23 + 25 * q^25 + (-b7 + 3*b6 - 4*b5 + 2*b4 + 24*b3 + 3*b2 - 8*b1 + 61) * q^27 + (6*b7 - 8*b6 - b5 - 2*b4 - b3 + 4*b2 + 3*b1 + 27) * q^29 + (-3*b7 - b6 + 6*b5 - 4*b4 + 11*b3 - 2*b2 - 2*b1 + 5) * q^31 + (-b7 + 11*b6 - b5 + 2*b4 - 4*b3 - 2*b2 - 4*b1 - 3) * q^33 + (5*b6 + 5*b3 - 5) * q^35 + (2*b7 + 3*b6 - b5 - 3*b4 - b3 - 8*b2 + 4*b1 - 37) * q^37 + (3*b7 - 5*b6 - 9*b5 + 7*b3 + 8*b2 - b1 + 132) * q^39 + (-2*b7 + b6 + 2*b5 - 3*b4 + 21*b3 - 7*b2 + 12*b1 + 81) * q^41 + (9*b7 - 13*b6 - 6*b5 + b4 + 4*b3 + 4*b2 + 13*b1 + 46) * q^43 + (-5*b7 - 5*b5 - 5*b4 + 10*b3 + 5*b1 + 100) * q^45 + (-5*b7 + 2*b6 - 6*b5 - 4*b3 - 10*b2 - 4*b1 - 12) * q^47 + (-2*b7 - 2*b6 - 7*b5 - 6*b4 + 50*b3 + 4*b2 + 2*b1 + 111) * q^49 + (7*b7 + 17*b6 + 6*b5 + 2*b4 - 13*b3 - 9*b2 - 12*b1 - 109) * q^51 + (7*b6 + 10*b5 - 3*b4 + 26*b3 + 7*b2 - 15*b1 + 176) * q^53 + (-5*b7 + 5*b5 + 5*b2 - 30) * q^55 + (26*b7 - 17*b6 - 7*b5 + 8*b4 - 11*b3 + 20*b2 - 13*b1 - 150) * q^57 + (5*b7 + 6*b6 + 5*b5 - 16*b4 + 9*b3 - 16*b2 + 4*b1 + 102) * q^59 + (8*b7 - 3*b6 + 6*b5 - b4 - 7*b3 + 12*b2 - b1 + 178) * q^61 + (-22*b7 + 52*b6 - 2*b5 + 25*b4 + 68*b3 + 6*b2 - 22*b1 + 4) * q^63 + (-5*b7 + 5*b4 + 15*b3 - 5*b2 - 5*b1 + 15) * q^65 + (19*b7 - 10*b5 + 8*b4 - 24*b3 + 3*b2 + b1 + 201) * q^67 - 23*b3 * q^69 + (7*b7 - 28*b6 - 8*b5 - 4*b4 - 37*b3 + 23*b2 + 12*b1 - 188) * q^71 + (9*b7 + 18*b6 + 19*b5 + 2*b4 - 27*b3 - 19*b2 - 6*b1 + 16) * q^73 + 25*b3 * q^75 + (-9*b7 - 10*b6 - 13*b5 - 5*b4 + 79*b3 - 16*b2 + 20*b1 + 64) * q^77 + (-6*b7 - 16*b6 + 11*b5 - 6*b4 - 29*b3 + 5*b2 + 11*b1 - 97) * q^79 + (-32*b7 + 21*b6 - 14*b5 + 16*b4 + 112*b3 + 3*b2 - 10*b1 + 528) * q^81 + (12*b7 - 21*b6 + 18*b5 - 13*b4 + 58*b3 - 5*b2 + 12*b1 - 183) * q^83 + (-5*b7 - 5*b6 + 10*b5 + 5*b4 - 15*b3 + 5*b2 + 10*b1 + 35) * q^85 + (b7 - 30*b6 + 20*b5 - 32*b4 + 8*b3 - 48*b2 - 3*b1 + 121) * q^87 + (-3*b7 - 7*b6 - 4*b5 + b4 - 42*b3 - 18*b2 + 4*b1 + 173) * q^89 + (-27*b7 + 14*b6 - 9*b5 + 6*b4 + 15*b2 + 19*b1 - 349) * q^91 + (25*b7 - 37*b6 - 21*b5 - 10*b4 - 11*b3 + 30*b2 + 30*b1 + 471) * q^93 + (-15*b7 - 5*b6 + 5*b5 - 10*b4 - 10*b3 - 10*b2 + 5*b1 - 110) * q^95 + (23*b7 - 14*b6 + 33*b5 + 12*b4 + 58*b3 - 15*b2 - 2*b1 - 108) * q^97 + (-5*b7 + 45*b6 - 26*b5 + 30*b4 - 3*b3 + b2 + 27*b1 - 162) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q + 40 q^{5} - 11 q^{7} + 158 q^{9}+O(q^{10})$$ 8 * q + 40 * q^5 - 11 * q^7 + 158 * q^9 $$8 q + 40 q^{5} - 11 q^{7} + 158 q^{9} - 41 q^{11} + 28 q^{13} + 71 q^{17} - 177 q^{19} + 292 q^{21} - 184 q^{23} + 200 q^{25} + 495 q^{27} + 225 q^{29} + 36 q^{31} - 53 q^{33} - 55 q^{35} - 348 q^{37} + 1077 q^{39} + 620 q^{41} + 390 q^{43} + 790 q^{45} - 123 q^{47} + 881 q^{49} - 957 q^{51} + 1406 q^{53} - 205 q^{55} - 1142 q^{57} + 676 q^{59} + 1447 q^{61} + 58 q^{63} + 140 q^{65} + 1582 q^{67} - 1396 q^{71} + 17 q^{73} + 488 q^{77} - 708 q^{79} + 4316 q^{81} - 1486 q^{83} + 355 q^{85} + 803 q^{87} + 1360 q^{89} - 2693 q^{91} + 3833 q^{93} - 885 q^{95} - 855 q^{97} - 1319 q^{99}+O(q^{100})$$ 8 * q + 40 * q^5 - 11 * q^7 + 158 * q^9 - 41 * q^11 + 28 * q^13 + 71 * q^17 - 177 * q^19 + 292 * q^21 - 184 * q^23 + 200 * q^25 + 495 * q^27 + 225 * q^29 + 36 * q^31 - 53 * q^33 - 55 * q^35 - 348 * q^37 + 1077 * q^39 + 620 * q^41 + 390 * q^43 + 790 * q^45 - 123 * q^47 + 881 * q^49 - 957 * q^51 + 1406 * q^53 - 205 * q^55 - 1142 * q^57 + 676 * q^59 + 1447 * q^61 + 58 * q^63 + 140 * q^65 + 1582 * q^67 - 1396 * q^71 + 17 * q^73 + 488 * q^77 - 708 * q^79 + 4316 * q^81 - 1486 * q^83 + 355 * q^85 + 803 * q^87 + 1360 * q^89 - 2693 * q^91 + 3833 * q^93 - 885 * q^95 - 855 * q^97 - 1319 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 2x^{7} - 49x^{6} + 31x^{5} + 750x^{4} + 249x^{3} - 2892x^{2} - 620x + 2400$$ :

 $$\beta_{1}$$ $$=$$ $$4\nu - 1$$ 4*v - 1 $$\beta_{2}$$ $$=$$ $$( -11\nu^{7} - 68\nu^{6} + 819\nu^{5} + 3349\nu^{4} - 15140\nu^{3} - 42139\nu^{2} + 52362\nu + 38800 ) / 1840$$ (-11*v^7 - 68*v^6 + 819*v^5 + 3349*v^4 - 15140*v^3 - 42139*v^2 + 52362*v + 38800) / 1840 $$\beta_{3}$$ $$=$$ $$( -79\nu^{7} + 348\nu^{6} + 2871\nu^{5} - 8399\nu^{4} - 33460\nu^{3} + 36929\nu^{2} + 71618\nu - 30800 ) / 3680$$ (-79*v^7 + 348*v^6 + 2871*v^5 - 8399*v^4 - 33460*v^3 + 36929*v^2 + 71618*v - 30800) / 3680 $$\beta_{4}$$ $$=$$ $$( -5\nu^{7} + 36\nu^{6} + 113\nu^{5} - 945\nu^{4} - 860\nu^{3} + 6631\nu^{2} + 3586\nu - 7488 ) / 184$$ (-5*v^7 + 36*v^6 + 113*v^5 - 945*v^4 - 860*v^3 + 6631*v^2 + 3586*v - 7488) / 184 $$\beta_{5}$$ $$=$$ $$( -143\nu^{7} + 956\nu^{6} + 3287\nu^{5} - 22703\nu^{4} - 23860\nu^{3} + 129313\nu^{2} + 21986\nu - 161680 ) / 3680$$ (-143*v^7 + 956*v^6 + 3287*v^5 - 22703*v^4 - 23860*v^3 + 129313*v^2 + 21986*v - 161680) / 3680 $$\beta_{6}$$ $$=$$ $$( 29\nu^{7} - 172\nu^{6} - 821\nu^{5} + 4469\nu^{4} + 7012\nu^{3} - 28211\nu^{2} - 6686\nu + 32096 ) / 368$$ (29*v^7 - 172*v^6 - 821*v^5 + 4469*v^4 + 7012*v^3 - 28211*v^2 - 6686*v + 32096) / 368 $$\beta_{7}$$ $$=$$ $$( 3\nu^{7} - 16\nu^{6} - 87\nu^{5} + 363\nu^{4} + 800\nu^{3} - 1553\nu^{2} - 866\nu + 480 ) / 40$$ (3*v^7 - 16*v^6 - 87*v^5 + 363*v^4 + 800*v^3 - 1553*v^2 - 866*v + 480) / 40
 $$\nu$$ $$=$$ $$( \beta _1 + 1 ) / 4$$ (b1 + 1) / 4 $$\nu^{2}$$ $$=$$ $$( \beta_{7} - \beta_{6} + \beta_{5} - \beta_{4} - \beta_{3} + \beta_{2} + 2\beta _1 + 51 ) / 4$$ (b7 - b6 + b5 - b4 - b3 + b2 + 2*b1 + 51) / 4 $$\nu^{3}$$ $$=$$ $$( 2\beta_{7} - 2\beta_{6} + 4\beta_{5} - 6\beta_{4} + 25\beta _1 + 107 ) / 4$$ (2*b7 - 2*b6 + 4*b5 - 6*b4 + 25*b1 + 107) / 4 $$\nu^{4}$$ $$=$$ $$( 13\beta_{7} - 9\beta_{6} + 20\beta_{5} - 19\beta_{4} - 2\beta_{3} + 8\beta_{2} + 45\beta _1 + 594 ) / 2$$ (13*b7 - 9*b6 + 20*b5 - 19*b4 - 2*b3 + 8*b2 + 45*b1 + 594) / 2 $$\nu^{5}$$ $$=$$ $$( 107\beta_{7} - 51\beta_{6} + 205\beta_{5} - 231\beta_{4} + 111\beta_{3} - 11\beta_{2} + 705\beta _1 + 4636 ) / 4$$ (107*b7 - 51*b6 + 205*b5 - 231*b4 + 111*b3 - 11*b2 + 705*b1 + 4636) / 4 $$\nu^{6}$$ $$=$$ $$202\beta_{7} - 74\beta_{6} + 381\beta_{5} - 340\beta_{4} + 167\beta_{3} + 28\beta_{2} + 833\beta _1 + 8573$$ 202*b7 - 74*b6 + 381*b5 - 340*b4 + 167*b3 + 28*b2 + 833*b1 + 8573 $$\nu^{7}$$ $$=$$ $$( 4304 \beta_{7} - 864 \beta_{6} + 8684 \beta_{5} - 8272 \beta_{4} + 6748 \beta_{3} - 1140 \beta_{2} + 21983 \beta _1 + 171103 ) / 4$$ (4304*b7 - 864*b6 + 8684*b5 - 8272*b4 + 6748*b3 - 1140*b2 + 21983*b1 + 171103) / 4

## 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
 −1.16661 4.85207 −3.57019 −3.81977 1.76920 6.03341 −3.05234 0.954235
0 −8.59146 0 5.00000 0 −9.86011 0 46.8131 0
1.2 0 −8.45088 0 5.00000 0 −16.9190 0 44.4174 0
1.3 0 −4.96142 0 5.00000 0 18.4259 0 −2.38431 0
1.4 0 −1.39521 0 5.00000 0 −13.5587 0 −25.0534 0
1.5 0 0.0181662 0 5.00000 0 −10.0918 0 −26.9997 0
1.6 0 4.21852 0 5.00000 0 19.9088 0 −9.20409 0
1.7 0 8.94300 0 5.00000 0 −32.7645 0 52.9772 0
1.8 0 10.2193 0 5.00000 0 33.8594 0 77.4338 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.8 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.v 8
4.b odd 2 1 115.4.a.f 8
12.b even 2 1 1035.4.a.r 8
20.d odd 2 1 575.4.a.n 8
20.e even 4 2 575.4.b.k 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
115.4.a.f 8 4.b odd 2 1
575.4.a.n 8 20.d odd 2 1
575.4.b.k 16 20.e even 4 2
1035.4.a.r 8 12.b even 2 1
1840.4.a.v 8 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 $$S_{4}^{\mathrm{new}}(\Gamma_0(1840))$$:

 $$T_{3}^{8} - 187T_{3}^{6} - 165T_{3}^{5} + 10290T_{3}^{4} + 15441T_{3}^{3} - 137140T_{3}^{2} - 191280T_{3} + 3520$$ T3^8 - 187*T3^6 - 165*T3^5 + 10290*T3^4 + 15441*T3^3 - 137140*T3^2 - 191280*T3 + 3520 $$T_{7}^{8} + 11 T_{7}^{7} - 1752 T_{7}^{6} - 22539 T_{7}^{5} + 783793 T_{7}^{4} + 12717810 T_{7}^{3} - 76746972 T_{7}^{2} - 2135252960 T_{7} - 9289584128$$ T7^8 + 11*T7^7 - 1752*T7^6 - 22539*T7^5 + 783793*T7^4 + 12717810*T7^3 - 76746972*T7^2 - 2135252960*T7 - 9289584128

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$T^{8} - 187 T^{6} - 165 T^{5} + \cdots + 3520$$
$5$ $$(T - 5)^{8}$$
$7$ $$T^{8} + 11 T^{7} + \cdots - 9289584128$$
$11$ $$T^{8} + 41 T^{7} + \cdots - 11345758080$$
$13$ $$T^{8} - 28 T^{7} + \cdots - 5251923133176$$
$17$ $$T^{8} - 71 T^{7} + \cdots + 52834929586560$$
$19$ $$T^{8} + 177 T^{7} + \cdots - 54772847726720$$
$23$ $$(T + 23)^{8}$$
$29$ $$T^{8} - 225 T^{7} + \cdots + 12\!\cdots\!00$$
$31$ $$T^{8} - 36 T^{7} + \cdots + 74\!\cdots\!00$$
$37$ $$T^{8} + 348 T^{7} + \cdots - 15\!\cdots\!64$$
$41$ $$T^{8} - 620 T^{7} + \cdots - 18\!\cdots\!22$$
$43$ $$T^{8} - 390 T^{7} + \cdots + 10\!\cdots\!00$$
$47$ $$T^{8} + 123 T^{7} + \cdots - 30\!\cdots\!80$$
$53$ $$T^{8} - 1406 T^{7} + \cdots - 62\!\cdots\!56$$
$59$ $$T^{8} - 676 T^{7} + \cdots + 10\!\cdots\!00$$
$61$ $$T^{8} - 1447 T^{7} + \cdots - 94\!\cdots\!32$$
$67$ $$T^{8} - 1582 T^{7} + \cdots - 27\!\cdots\!80$$
$71$ $$T^{8} + 1396 T^{7} + \cdots + 30\!\cdots\!20$$
$73$ $$T^{8} - 17 T^{7} + \cdots - 59\!\cdots\!16$$
$79$ $$T^{8} + 708 T^{7} + \cdots + 11\!\cdots\!40$$
$83$ $$T^{8} + 1486 T^{7} + \cdots + 84\!\cdots\!48$$
$89$ $$T^{8} - 1360 T^{7} + \cdots + 13\!\cdots\!20$$
$97$ $$T^{8} + 855 T^{7} + \cdots + 12\!\cdots\!44$$