# Properties

 Label 1840.4.a.z Level $1840$ Weight $4$ Character orbit 1840.a Self dual yes Analytic conductor $108.564$ Analytic rank $1$ Dimension $9$ 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: $$9$$ Coefficient field: $$\mathbb{Q}[x]/(x^{9} - \cdots)$$ Defining polynomial: $$x^{9} - 3x^{8} - 148x^{7} + 278x^{6} + 6502x^{5} - 4928x^{4} - 87343x^{3} + 42737x^{2} + 286800x + 53104$$ x^9 - 3*x^8 - 148*x^7 + 278*x^6 + 6502*x^5 - 4928*x^4 - 87343*x^3 + 42737*x^2 + 286800*x + 53104 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{5}$$ Twist minimal: no (minimal twist has level 920) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{8}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{3} - 5 q^{5} + (\beta_{3} - 3) q^{7} + (\beta_{4} - \beta_{3} + 2 \beta_1 + 6) q^{9}+O(q^{10})$$ q + b1 * q^3 - 5 * q^5 + (b3 - 3) * q^7 + (b4 - b3 + 2*b1 + 6) * q^9 $$q + \beta_1 q^{3} - 5 q^{5} + (\beta_{3} - 3) q^{7} + (\beta_{4} - \beta_{3} + 2 \beta_1 + 6) q^{9} + (\beta_{7} + \beta_{6} - \beta_{5} - \beta_{4} + \beta_{3} - 2) q^{11} + (\beta_{8} - \beta_{7} + \beta_{6} + \beta_{5} - \beta_{3} - \beta_{2} - \beta_1 + 4) q^{13} - 5 \beta_1 q^{15} + ( - \beta_{6} + \beta_{5} - \beta_{4} + \beta_{2} - 3 \beta_1 - 14) q^{17} + ( - 2 \beta_{7} - 2 \beta_{6} - \beta_{4} + \beta_{2} + 2 \beta_1 - 13) q^{19} + ( - 2 \beta_{8} - \beta_{6} - 2 \beta_{5} + \beta_{4} - 3 \beta_{3} - 8 \beta_1 + 8) q^{21} + 23 q^{23} + 25 q^{25} + ( - \beta_{8} + 3 \beta_{7} - \beta_{6} + 3 \beta_{5} + 2 \beta_{4} + 2 \beta_{3} + \cdots + 36) q^{27}+ \cdots + ( - 20 \beta_{8} + 4 \beta_{7} + \beta_{6} + 4 \beta_{5} + 9 \beta_{4} + 4 \beta_{3} + \cdots - 346) q^{99}+O(q^{100})$$ q + b1 * q^3 - 5 * q^5 + (b3 - 3) * q^7 + (b4 - b3 + 2*b1 + 6) * q^9 + (b7 + b6 - b5 - b4 + b3 - 2) * q^11 + (b8 - b7 + b6 + b5 - b3 - b2 - b1 + 4) * q^13 - 5*b1 * q^15 + (-b6 + b5 - b4 + b2 - 3*b1 - 14) * q^17 + (-2*b7 - 2*b6 - b4 + b2 + 2*b1 - 13) * q^19 + (-2*b8 - b6 - 2*b5 + b4 - 3*b3 - 8*b1 + 8) * q^21 + 23 * q^23 + 25 * q^25 + (-b8 + 3*b7 - b6 + 3*b5 + 2*b4 + 2*b3 + 12*b1 + 36) * q^27 + (b8 - 6*b6 + b5 + 3*b4 + b2 - 3*b1 + 28) * q^29 + (-3*b8 + b5 - 4*b2 + 13*b1 - 21) * q^31 + (-4*b8 + 6*b7 + 2*b6 - 4*b5 - 3*b4 + 2*b3 + 3*b2 - 10*b1 + 7) * q^33 + (-5*b3 + 15) * q^35 + (-2*b8 + b7 + 4*b6 - 4*b4 + b3 + b2 - 17*b1 + 26) * q^37 + (4*b8 - 4*b7 - 6*b6 + 3*b5 - b4 - 4*b3 + 4*b2 + 6*b1 - 4) * q^39 + (-3*b7 - 5*b6 - 2*b4 - 5*b3 - 5*b2 - 29*b1 - 45) * q^41 + (-3*b8 - 4*b7 - 2*b6 + b5 + 4*b4 + b2 - 9*b1 - 12) * q^43 + (-5*b4 + 5*b3 - 10*b1 - 30) * q^45 + (2*b8 + 3*b7 + 4*b6 - 8*b5 - b4 - 9*b3 + 7*b2 + 19*b1 + 10) * q^47 + (5*b8 - 8*b7 + 4*b6 + 5*b5 + 5*b4 - 5*b2 - 38*b1 + 99) * q^49 + (10*b8 - 12*b7 + 5*b6 - b4 - 8*b3 - 7*b2 - 55*b1 - 80) * q^51 + (2*b8 + 9*b7 + 3*b6 + 2*b5 - 10*b4 + 7*b3 + 7*b2 - 24*b1 - 19) * q^53 + (-5*b7 - 5*b6 + 5*b5 + 5*b4 - 5*b3 + 10) * q^55 + (12*b8 - 19*b7 + 2*b6 + 3*b5 - 5*b3 - 5*b2 - 71*b1 + 108) * q^57 + (-3*b8 + 5*b7 - 6*b6 + b5 + 2*b4 - 7*b3 - 4*b2 - 10*b1 - 88) * q^59 + (-5*b8 + 5*b7 + 6*b6 - 5*b5 + b4 + b3 + 10*b2 - 7*b1 + 220) * q^61 + (-7*b8 + 13*b7 + 6*b6 + 3*b5 - 13*b4 + 14*b3 + 2*b2 + 19*b1 - 247) * q^63 + (-5*b8 + 5*b7 - 5*b6 - 5*b5 + 5*b3 + 5*b2 + 5*b1 - 20) * q^65 + (-b8 - 3*b7 + 11*b6 - 5*b5 + 4*b4 - 9*b3 - 4*b2 - 23*b1 - 167) * q^67 + 23*b1 * q^69 + (-3*b8 + 17*b7 + 17*b6 - 20*b5 - 12*b4 - 5*b3 - 3*b2 - 17*b1 - 136) * q^71 + (12*b8 - 11*b7 + 7*b6 + 12*b5 + b4 - 9*b3 - 21*b2 - 50*b1 + 77) * q^73 + 25*b1 * q^75 + (9*b8 - 5*b7 - 12*b6 + 16*b5 + 8*b4 - 7*b3 - 6*b2 - 8*b1 + 224) * q^77 + (6*b7 - b6 - 10*b5 - 10*b4 + 2*b3 + 16*b2 + 9*b1 - 299) * q^79 + (-9*b8 + 10*b7 + 3*b6 - 7*b5 + 3*b4 - 5*b3 - 11*b2 + 59*b1 + 177) * q^81 + (6*b8 + 9*b7 - 3*b6 + 14*b5 - 2*b4 + 31*b3 - 3*b2 - 56*b1 - 261) * q^83 + (5*b6 - 5*b5 + 5*b4 - 5*b2 + 15*b1 + 70) * q^85 + (12*b8 - 13*b7 + 18*b5 + 7*b4 - 3*b3 - 17*b2 + 59*b1 - 190) * q^87 + (-11*b8 + 6*b7 + 2*b6 - 21*b5 + 10*b4 + 2*b3 + 27*b2 - 47*b1 + 122) * q^89 + (-6*b8 + 21*b7 + 9*b6 - 5*b5 - b4 + 33*b3 + 16*b2 - 62*b1 - 252) * q^91 + (-22*b8 + 22*b7 - 2*b6 - 13*b5 + 23*b4 + 8*b3 + 10*b2 - 8*b1 + 476) * q^93 + (10*b7 + 10*b6 + 5*b4 - 5*b2 - 10*b1 + 65) * q^95 + (-8*b8 + 25*b7 + 27*b6 + 7*b5 - 5*b4 + 3*b3 - 16*b2 - 6*b1 + 252) * q^97 + (-20*b8 + 4*b7 + b6 + 4*b5 + 9*b4 + 4*b3 - 3*b2 - 29*b1 - 346) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$9 q + 3 q^{3} - 45 q^{5} - 25 q^{7} + 62 q^{9}+O(q^{10})$$ 9 * q + 3 * q^3 - 45 * q^5 - 25 * q^7 + 62 * q^9 $$9 q + 3 q^{3} - 45 q^{5} - 25 q^{7} + 62 q^{9} - 22 q^{11} + 23 q^{13} - 15 q^{15} - 135 q^{17} - 102 q^{19} + 54 q^{21} + 207 q^{23} + 225 q^{25} + 363 q^{27} + 280 q^{29} - 168 q^{31} + 28 q^{33} + 125 q^{35} + 153 q^{37} + 5 q^{39} - 502 q^{41} - 110 q^{43} - 310 q^{45} + 153 q^{47} + 764 q^{49} - 924 q^{51} - 273 q^{53} + 110 q^{55} + 748 q^{57} - 827 q^{59} + 1976 q^{61} - 2237 q^{63} - 115 q^{65} - 1613 q^{67} + 69 q^{69} - 1370 q^{71} + 425 q^{73} + 75 q^{75} + 2006 q^{77} - 2624 q^{79} + 1729 q^{81} - 2505 q^{83} + 675 q^{85} - 1591 q^{87} + 1120 q^{89} - 2392 q^{91} + 4401 q^{93} + 510 q^{95} + 2026 q^{97} - 3206 q^{99}+O(q^{100})$$ 9 * q + 3 * q^3 - 45 * q^5 - 25 * q^7 + 62 * q^9 - 22 * q^11 + 23 * q^13 - 15 * q^15 - 135 * q^17 - 102 * q^19 + 54 * q^21 + 207 * q^23 + 225 * q^25 + 363 * q^27 + 280 * q^29 - 168 * q^31 + 28 * q^33 + 125 * q^35 + 153 * q^37 + 5 * q^39 - 502 * q^41 - 110 * q^43 - 310 * q^45 + 153 * q^47 + 764 * q^49 - 924 * q^51 - 273 * q^53 + 110 * q^55 + 748 * q^57 - 827 * q^59 + 1976 * q^61 - 2237 * q^63 - 115 * q^65 - 1613 * q^67 + 69 * q^69 - 1370 * q^71 + 425 * q^73 + 75 * q^75 + 2006 * q^77 - 2624 * q^79 + 1729 * q^81 - 2505 * q^83 + 675 * q^85 - 1591 * q^87 + 1120 * q^89 - 2392 * q^91 + 4401 * q^93 + 510 * q^95 + 2026 * q^97 - 3206 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{9} - 3x^{8} - 148x^{7} + 278x^{6} + 6502x^{5} - 4928x^{4} - 87343x^{3} + 42737x^{2} + 286800x + 53104$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( 163 \nu^{8} + 4127 \nu^{7} - 40716 \nu^{6} - 544206 \nu^{5} + 2205470 \nu^{4} + 18403540 \nu^{3} - 5808013 \nu^{2} - 98808921 \nu - 161016460 ) / 2417192$$ (163*v^8 + 4127*v^7 - 40716*v^6 - 544206*v^5 + 2205470*v^4 + 18403540*v^3 - 5808013*v^2 - 98808921*v - 161016460) / 2417192 $$\beta_{3}$$ $$=$$ $$( - 514256 \nu^{8} - 5815223 \nu^{7} + 100384680 \nu^{6} + 832440392 \nu^{5} - 5143221570 \nu^{4} - 31446688712 \nu^{3} + \cdots + 74795096684 ) / 2501793720$$ (-514256*v^8 - 5815223*v^7 + 100384680*v^6 + 832440392*v^5 - 5143221570*v^4 - 31446688712*v^3 + 42594569976*v^2 + 203410599449*v + 74795096684) / 2501793720 $$\beta_{4}$$ $$=$$ $$( - 514256 \nu^{8} - 5815223 \nu^{7} + 100384680 \nu^{6} + 832440392 \nu^{5} - 5143221570 \nu^{4} - 31446688712 \nu^{3} + \cdots - 7764096076 ) / 2501793720$$ (-514256*v^8 - 5815223*v^7 + 100384680*v^6 + 832440392*v^5 - 5143221570*v^4 - 31446688712*v^3 + 45096363696*v^2 + 198407012009*v - 7764096076) / 2501793720 $$\beta_{5}$$ $$=$$ $$( - 569393 \nu^{8} + 3148951 \nu^{7} + 79478880 \nu^{6} - 360717754 \nu^{5} - 3337353690 \nu^{4} + 11088924784 \nu^{3} + \cdots - 139395895828 ) / 2501793720$$ (-569393*v^8 + 3148951*v^7 + 79478880*v^6 - 360717754*v^5 - 3337353690*v^4 + 11088924784*v^3 + 48012273123*v^2 - 116428273513*v - 139395895828) / 2501793720 $$\beta_{6}$$ $$=$$ $$( 1230451 \nu^{8} - 4616552 \nu^{7} - 147932640 \nu^{6} + 312182138 \nu^{5} + 3965373360 \nu^{4} + 3213171712 \nu^{3} + 7066734219 \nu^{2} + \cdots - 170844106264 ) / 2501793720$$ (1230451*v^8 - 4616552*v^7 - 147932640*v^6 + 312182138*v^5 + 3965373360*v^4 + 3213171712*v^3 + 7066734219*v^2 - 84342328444*v - 170844106264) / 2501793720 $$\beta_{7}$$ $$=$$ $$( 1015897 \nu^{8} + 226071 \nu^{7} - 153500780 \nu^{6} - 184830594 \nu^{5} + 6448742270 \nu^{4} + 13127463324 \nu^{3} - 57548382767 \nu^{2} + \cdots + 14256990172 ) / 833931240$$ (1015897*v^8 + 226071*v^7 - 153500780*v^6 - 184830594*v^5 + 6448742270*v^4 + 13127463324*v^3 - 57548382767*v^2 - 96502268313*v + 14256990172) / 833931240 $$\beta_{8}$$ $$=$$ $$( 1382473 \nu^{8} - 2387616 \nu^{7} - 197866340 \nu^{6} + 90650274 \nu^{5} + 7829453240 \nu^{4} + 6637407996 \nu^{3} - 68527830803 \nu^{2} + \cdots + 35031968488 ) / 833931240$$ (1382473*v^8 - 2387616*v^7 - 197866340*v^6 + 90650274*v^5 + 7829453240*v^4 + 6637407996*v^3 - 68527830803*v^2 - 54903099492*v + 35031968488) / 833931240
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{4} - \beta_{3} + 2\beta _1 + 33$$ b4 - b3 + 2*b1 + 33 $$\nu^{3}$$ $$=$$ $$-\beta_{8} + 3\beta_{7} - \beta_{6} + 3\beta_{5} + 2\beta_{4} + 2\beta_{3} + 66\beta _1 + 36$$ -b8 + 3*b7 - b6 + 3*b5 + 2*b4 + 2*b3 + 66*b1 + 36 $$\nu^{4}$$ $$=$$ $$-9\beta_{8} + 10\beta_{7} + 3\beta_{6} - 7\beta_{5} + 84\beta_{4} - 86\beta_{3} - 11\beta_{2} + 221\beta _1 + 2121$$ -9*b8 + 10*b7 + 3*b6 - 7*b5 + 84*b4 - 86*b3 - 11*b2 + 221*b1 + 2121 $$\nu^{5}$$ $$=$$ $$- 179 \beta_{8} + 390 \beta_{7} - 69 \beta_{6} + 221 \beta_{5} + 225 \beta_{4} + 247 \beta_{3} + 43 \beta_{2} + 5136 \beta _1 + 4632$$ -179*b8 + 390*b7 - 69*b6 + 221*b5 + 225*b4 + 247*b3 + 43*b2 + 5136*b1 + 4632 $$\nu^{6}$$ $$=$$ $$- 1566 \beta_{8} + 1799 \beta_{7} + 596 \beta_{6} - 1068 \beta_{5} + 6866 \beta_{4} - 6586 \beta_{3} - 1099 \beta_{2} + 21512 \beta _1 + 161221$$ -1566*b8 + 1799*b7 + 596*b6 - 1068*b5 + 6866*b4 - 6586*b3 - 1099*b2 + 21512*b1 + 161221 $$\nu^{7}$$ $$=$$ $$- 22114 \beta_{8} + 41284 \beta_{7} - 3210 \beta_{6} + 13650 \beta_{5} + 22456 \beta_{4} + 22652 \beta_{3} + 4826 \beta_{2} + 423129 \beta _1 + 478470$$ -22114*b8 + 41284*b7 - 3210*b6 + 13650*b5 + 22456*b4 + 22652*b3 + 4826*b2 + 423129*b1 + 478470 $$\nu^{8}$$ $$=$$ $$- 194214 \beta_{8} + 232172 \beta_{7} + 72094 \beta_{6} - 118534 \beta_{5} + 570969 \beta_{4} - 491817 \beta_{3} - 89482 \beta_{2} + 2043274 \beta _1 + 13022959$$ -194214*b8 + 232172*b7 + 72094*b6 - 118534*b5 + 570969*b4 - 491817*b3 - 89482*b2 + 2043274*b1 + 13022959

## 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
 −8.72630 −5.18969 −4.91244 −1.70964 −0.192858 2.99459 3.15779 7.95723 9.62132
0 −8.72630 0 −5.00000 0 −28.9030 0 49.1484 0
1.2 0 −5.18969 0 −5.00000 0 −12.0220 0 −0.0670810 0
1.3 0 −4.91244 0 −5.00000 0 33.9952 0 −2.86795 0
1.4 0 −1.70964 0 −5.00000 0 −20.8693 0 −24.0771 0
1.5 0 −0.192858 0 −5.00000 0 11.9366 0 −26.9628 0
1.6 0 2.99459 0 −5.00000 0 22.8972 0 −18.0324 0
1.7 0 3.15779 0 −5.00000 0 −11.8460 0 −17.0284 0
1.8 0 7.95723 0 −5.00000 0 −21.6920 0 36.3175 0
1.9 0 9.62132 0 −5.00000 0 1.50324 0 65.5699 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.9 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.z 9
4.b odd 2 1 920.4.a.e 9

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
920.4.a.e 9 4.b odd 2 1
1840.4.a.z 9 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 $$S_{4}^{\mathrm{new}}(\Gamma_0(1840))$$:

 $$T_{3}^{9} - 3 T_{3}^{8} - 148 T_{3}^{7} + 278 T_{3}^{6} + 6502 T_{3}^{5} - 4928 T_{3}^{4} - 87343 T_{3}^{3} + 42737 T_{3}^{2} + 286800 T_{3} + 53104$$ T3^9 - 3*T3^8 - 148*T3^7 + 278*T3^6 + 6502*T3^5 - 4928*T3^4 - 87343*T3^3 + 42737*T3^2 + 286800*T3 + 53104 $$T_{7}^{9} + 25 T_{7}^{8} - 1613 T_{7}^{7} - 47585 T_{7}^{6} + 521969 T_{7}^{5} + 22839351 T_{7}^{4} + 51541174 T_{7}^{3} - 2543169384 T_{7}^{2} - 13686546952 T_{7} + 26025813504$$ T7^9 + 25*T7^8 - 1613*T7^7 - 47585*T7^6 + 521969*T7^5 + 22839351*T7^4 + 51541174*T7^3 - 2543169384*T7^2 - 13686546952*T7 + 26025813504

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{9}$$
$3$ $$T^{9} - 3 T^{8} - 148 T^{7} + \cdots + 53104$$
$5$ $$(T + 5)^{9}$$
$7$ $$T^{9} + 25 T^{8} + \cdots + 26025813504$$
$11$ $$T^{9} + 22 T^{8} + \cdots + 2775829222400$$
$13$ $$T^{9} + \cdots - 189030103898988$$
$17$ $$T^{9} + \cdots - 287072829439872$$
$19$ $$T^{9} + \cdots + 501210422257408$$
$23$ $$(T - 23)^{9}$$
$29$ $$T^{9} - 280 T^{8} + \cdots - 28\!\cdots\!00$$
$31$ $$T^{9} + 168 T^{8} + \cdots + 26\!\cdots\!52$$
$37$ $$T^{9} - 153 T^{8} + \cdots - 37\!\cdots\!68$$
$41$ $$T^{9} + 502 T^{8} + \cdots - 24\!\cdots\!18$$
$43$ $$T^{9} + 110 T^{8} + \cdots - 25\!\cdots\!08$$
$47$ $$T^{9} - 153 T^{8} + \cdots - 52\!\cdots\!24$$
$53$ $$T^{9} + 273 T^{8} + \cdots - 87\!\cdots\!00$$
$59$ $$T^{9} + 827 T^{8} + \cdots - 28\!\cdots\!52$$
$61$ $$T^{9} - 1976 T^{8} + \cdots + 80\!\cdots\!00$$
$67$ $$T^{9} + 1613 T^{8} + \cdots - 71\!\cdots\!08$$
$71$ $$T^{9} + 1370 T^{8} + \cdots + 78\!\cdots\!00$$
$73$ $$T^{9} - 425 T^{8} + \cdots - 92\!\cdots\!44$$
$79$ $$T^{9} + 2624 T^{8} + \cdots + 93\!\cdots\!52$$
$83$ $$T^{9} + 2505 T^{8} + \cdots - 15\!\cdots\!00$$
$89$ $$T^{9} - 1120 T^{8} + \cdots + 77\!\cdots\!00$$
$97$ $$T^{9} - 2026 T^{8} + \cdots - 12\!\cdots\!84$$