# Properties

 Label 1840.3.k.b Level $1840$ Weight $3$ Character orbit 1840.k Analytic conductor $50.136$ Analytic rank $0$ Dimension $10$ CM no Inner twists $2$

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## Newspace parameters

 Level: $$N$$ $$=$$ $$1840 = 2^{4} \cdot 5 \cdot 23$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 1840.k (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$50.1363686423$$ Analytic rank: $$0$$ Dimension: $$10$$ Coefficient field: $$\mathbb{Q}[x]/(x^{10} - \cdots)$$ Defining polynomial: $$x^{10} - 2 x^{9} - 10 x^{8} + 34 x^{7} + 346 x^{6} - 968 x^{5} + 165 x^{4} + 6972 x^{3} + 19344 x^{2} - 89080 x + 225444$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{2}\cdot 5$$ Twist minimal: no (minimal twist has level 115) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{9}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta_{5} q^{3} -\beta_{3} q^{5} + ( \beta_{2} + \beta_{3} ) q^{7} + ( -2 + \beta_{7} ) q^{9} +O(q^{10})$$ $$q -\beta_{5} q^{3} -\beta_{3} q^{5} + ( \beta_{2} + \beta_{3} ) q^{7} + ( -2 + \beta_{7} ) q^{9} + ( -\beta_{2} - \beta_{8} ) q^{11} + ( -\beta_{4} + \beta_{5} + \beta_{6} - \beta_{7} ) q^{13} -\beta_{2} q^{15} + ( -\beta_{1} + 2 \beta_{2} + 2 \beta_{8} ) q^{17} + ( \beta_{1} + \beta_{2} + 2 \beta_{3} + \beta_{9} ) q^{19} + ( -\beta_{1} + \beta_{2} + 7 \beta_{3} ) q^{21} + ( -5 - 2 \beta_{2} + 3 \beta_{3} + 2 \beta_{4} - 3 \beta_{5} + \beta_{6} + 2 \beta_{7} + \beta_{8} ) q^{23} -5 q^{25} + ( -3 - \beta_{4} + 5 \beta_{5} + 2 \beta_{6} - \beta_{7} ) q^{27} + ( -3 - \beta_{4} + 8 \beta_{5} - \beta_{6} - \beta_{7} ) q^{29} + ( -\beta_{6} - 4 \beta_{7} ) q^{31} + ( \beta_{1} + \beta_{2} - 6 \beta_{3} - \beta_{9} ) q^{33} + ( 5 - 5 \beta_{5} ) q^{35} + ( -\beta_{1} - \beta_{2} + 10 \beta_{3} - \beta_{8} + 4 \beta_{9} ) q^{37} + ( -6 + 2 \beta_{4} + 4 \beta_{5} - \beta_{6} + \beta_{7} ) q^{39} + ( -6 + 9 \beta_{4} - 8 \beta_{5} + 2 \beta_{6} - \beta_{7} ) q^{41} + ( -2 \beta_{1} - 3 \beta_{2} - \beta_{3} + \beta_{8} - 3 \beta_{9} ) q^{43} + ( \beta_{1} + 2 \beta_{3} ) q^{45} + ( -13 - 2 \beta_{4} - \beta_{5} + \beta_{6} + 2 \beta_{7} ) q^{47} + ( 9 + 10 \beta_{5} - 5 \beta_{7} ) q^{49} + ( -\beta_{1} + 4 \beta_{2} + 9 \beta_{3} - \beta_{8} + 4 \beta_{9} ) q^{51} + ( -3 \beta_{1} - 19 \beta_{3} + 2 \beta_{8} + \beta_{9} ) q^{53} + ( -5 \beta_{4} + 5 \beta_{5} ) q^{55} + ( -3 \beta_{1} - 3 \beta_{2} + 7 \beta_{3} + 2 \beta_{8} ) q^{57} + ( 23 - 6 \beta_{4} - 6 \beta_{5} + 7 \beta_{6} - \beta_{7} ) q^{59} + ( 7 \beta_{2} - 4 \beta_{3} - 3 \beta_{8} + 4 \beta_{9} ) q^{61} + ( 4 \beta_{2} - 5 \beta_{3} - \beta_{8} + 2 \beta_{9} ) q^{63} + ( -\beta_{1} + \beta_{2} + \beta_{8} - \beta_{9} ) q^{65} + ( \beta_{1} + 3 \beta_{2} - 18 \beta_{3} - 5 \beta_{8} + 4 \beta_{9} ) q^{67} + ( 10 + 2 \beta_{1} + 2 \beta_{2} - 15 \beta_{3} - \beta_{4} - 6 \beta_{5} + 8 \beta_{6} + 2 \beta_{7} + \beta_{9} ) q^{69} + ( -18 + 11 \beta_{4} - 8 \beta_{5} - 10 \beta_{6} - 7 \beta_{7} ) q^{71} + ( 21 + 13 \beta_{5} - 7 \beta_{6} - 4 \beta_{7} ) q^{73} + 5 \beta_{5} q^{75} + ( 30 + 5 \beta_{4} + 5 \beta_{6} + 5 \beta_{7} ) q^{77} + ( -2 \beta_{1} + 3 \beta_{2} + 5 \beta_{3} + 7 \beta_{8} + 3 \beta_{9} ) q^{79} + ( -19 + 3 \beta_{4} + 6 \beta_{5} + \beta_{6} - 11 \beta_{7} ) q^{81} + ( -3 \beta_{1} + 6 \beta_{2} + \beta_{3} - 5 \beta_{9} ) q^{83} + ( 10 \beta_{4} - 10 \beta_{5} + 5 \beta_{7} ) q^{85} + ( -49 + 9 \beta_{5} - 5 \beta_{6} - 8 \beta_{7} ) q^{87} + ( -4 \beta_{1} + 5 \beta_{2} - 9 \beta_{3} - 5 \beta_{8} + 11 \beta_{9} ) q^{89} + ( -5 \beta_{2} - 6 \beta_{3} + \beta_{8} ) q^{91} + ( 15 + 3 \beta_{4} + 25 \beta_{5} - 10 \beta_{6} + 3 \beta_{7} ) q^{93} + ( 10 - 5 \beta_{5} + 5 \beta_{6} - 5 \beta_{7} ) q^{95} + ( 2 \beta_{1} - 9 \beta_{2} - 10 \beta_{3} + 9 \beta_{8} - 4 \beta_{9} ) q^{97} + ( -\beta_{1} - 4 \beta_{2} + 13 \beta_{3} + 9 \beta_{8} - 4 \beta_{9} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$10q + 2q^{3} - 16q^{9} + O(q^{10})$$ $$10q + 2q^{3} - 16q^{9} - 2q^{13} - 44q^{23} - 50q^{25} - 40q^{27} - 46q^{29} - 16q^{31} + 60q^{35} - 72q^{39} - 84q^{41} - 112q^{47} + 50q^{49} + 10q^{55} + 262q^{59} + 124q^{69} - 236q^{71} + 168q^{73} - 10q^{75} + 300q^{77} - 258q^{81} - 540q^{87} + 100q^{93} + 90q^{95} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{10} - 2 x^{9} - 10 x^{8} + 34 x^{7} + 346 x^{6} - 968 x^{5} + 165 x^{4} + 6972 x^{3} + 19344 x^{2} - 89080 x + 225444$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$18413999298 \nu^{9} + 6940583162233 \nu^{8} + 2184040960024 \nu^{7} - 55776313196156 \nu^{6} - 6517813050410 \nu^{5} + 1539918810175326 \nu^{4} + 491894487341618 \nu^{3} + 2768080292176325 \nu^{2} + 10197120293765962 \nu + 64951601035622166$$$$)/ 5791009908414440$$ $$\beta_{2}$$ $$=$$ $$($$$$-3178911074 \nu^{9} + 78987068911 \nu^{8} + 430137087328 \nu^{7} - 1072934217052 \nu^{6} - 7088796581670 \nu^{5} + 35789989530522 \nu^{4} + 107377324329566 \nu^{3} - 108581674614685 \nu^{2} - 326238370122746 \nu + 3418363084066042$$$$)/ 251783039496280$$ $$\beta_{3}$$ $$=$$ $$($$$$-27126561061 \nu^{9} + 36976343259 \nu^{8} + 207292153512 \nu^{7} - 662996871978 \nu^{6} - 9673453676400 \nu^{5} + 18624554785488 \nu^{4} - 28895174126101 \nu^{3} - 146401588536315 \nu^{2} - 1283490894627774 \nu + 1692482235453388$$$$)/ 1447752477103610$$ $$\beta_{4}$$ $$=$$ $$($$$$-61680118787 \nu^{9} - 90970570937 \nu^{8} + 725904561704 \nu^{7} - 1238323970566 \nu^{6} - 24941366319520 \nu^{5} - 30214028316584 \nu^{4} + 56554416235853 \nu^{3} - 216158506360285 \nu^{2} - 1283642065245148 \nu - 4897259103221374$$$$)/ 1447752477103610$$ $$\beta_{5}$$ $$=$$ $$($$$$-468602611879 \nu^{9} - 217704046159 \nu^{8} + 6293663580928 \nu^{7} + 12414624463218 \nu^{6} - 196255517839040 \nu^{5} - 75828656056898 \nu^{4} + 775382511035361 \nu^{3} + 912032566785975 \nu^{2} - 14729541411749116 \nu + 20117555613991862$$$$)/ 5791009908414440$$ $$\beta_{6}$$ $$=$$ $$($$$$-498310598539 \nu^{9} - 877397075979 \nu^{8} + 7538944599648 \nu^{7} + 12765303556778 \nu^{6} - 218633353705920 \nu^{5} - 345681207607138 \nu^{4} + 1232761568987581 \nu^{3} + 1218611249635355 \nu^{2} - 21178202332536396 \nu - 7220328774106298$$$$)/ 5791009908414440$$ $$\beta_{7}$$ $$=$$ $$($$$$-41025221057 \nu^{9} + 13161849303 \nu^{8} + 712152635904 \nu^{7} - 310444293146 \nu^{6} - 13274870104560 \nu^{5} + 9777674573986 \nu^{4} + 59463180686023 \nu^{3} - 125586360875135 \nu^{2} - 362194258629668 \nu + 1565531822973306$$$$)/ 251783039496280$$ $$\beta_{8}$$ $$=$$ $$($$$$236841500106 \nu^{9} - 458127503999 \nu^{8} - 2135744859512 \nu^{7} + 6675104951458 \nu^{6} + 76977727981220 \nu^{5} - 220963040069008 \nu^{4} - 219938936410664 \nu^{3} + 754716139973685 \nu^{2} + 6060410005984514 \nu - 19230564880585838$$$$)/ 1447752477103610$$ $$\beta_{9}$$ $$=$$ $$($$$$1050188941786 \nu^{9} - 2989519571129 \nu^{8} - 19681413465312 \nu^{7} + 51027227704468 \nu^{6} + 493331069170170 \nu^{5} - 1437169367927798 \nu^{4} - 3806813263174894 \nu^{3} + 11000674301597555 \nu^{2} + 32402773549134054 \nu - 139788745884593038$$$$)/ 5791009908414440$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-\beta_{6} + \beta_{5} + \beta_{4} - 2 \beta_{3} + 1$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$2 \beta_{9} - 2 \beta_{8} + \beta_{6} - \beta_{5} + \beta_{4} - 2 \beta_{3} + 2 \beta_{2} + 5$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$($$$$-3 \beta_{9} - 3 \beta_{8} - 2 \beta_{7} + 7 \beta_{6} - 5 \beta_{5} - 5 \beta_{4} - 35 \beta_{3} - 3 \beta_{2} - 9$$$$)/2$$ $$\nu^{4}$$ $$=$$ $$($$$$12 \beta_{9} - 20 \beta_{8} - 19 \beta_{6} + 11 \beta_{5} - 13 \beta_{4} - 4 \beta_{3} + 20 \beta_{2} - 8 \beta_{1} - 259$$$$)/2$$ $$\nu^{5}$$ $$=$$ $$($$$$-5 \beta_{9} - 35 \beta_{8} + 64 \beta_{7} + 198 \beta_{6} - 270 \beta_{5} - 272 \beta_{4} - 5 \beta_{3} - 45 \beta_{2} - 102$$$$)/2$$ $$\nu^{6}$$ $$=$$ $$-94 \beta_{9} + 116 \beta_{8} + \beta_{7} - 172 \beta_{6} + 395 \beta_{5} - 332 \beta_{4} + 6 \beta_{3} - 110 \beta_{2} - 8 \beta_{1} - 1868$$ $$\nu^{7}$$ $$=$$ $$($$$$658 \beta_{9} + 798 \beta_{8} + 1534 \beta_{7} + 1163 \beta_{6} - 2001 \beta_{5} - 1831 \beta_{4} + 6802 \beta_{3} + 980 \beta_{2} + 14 \beta_{1} - 2261$$$$)/2$$ $$\nu^{8}$$ $$=$$ $$($$$$-4984 \beta_{9} + 6968 \beta_{8} - 240 \beta_{7} + 1923 \beta_{6} + 3469 \beta_{5} - 3745 \beta_{4} + 4808 \beta_{3} - 7208 \beta_{2} + 3312 \beta_{1} + 6511$$$$)/2$$ $$\nu^{9}$$ $$=$$ $$($$$$5253 \beta_{9} + 22665 \beta_{8} - 9346 \beta_{7} - 29275 \beta_{6} + 37373 \beta_{5} + 37817 \beta_{4} + 93241 \beta_{3} + 25221 \beta_{2} - 480 \beta_{1} + 1529$$$$)/2$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1840\mathbb{Z}\right)^\times$$.

 $$n$$ $$737$$ $$1151$$ $$1201$$ $$1381$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
321.1
 2.24810 − 2.23607i 2.24810 + 2.23607i −2.67869 − 2.23607i −2.67869 + 2.23607i 3.44206 − 2.23607i 3.44206 + 2.23607i 1.32878 − 2.23607i 1.32878 + 2.23607i −3.34025 − 2.23607i −3.34025 + 2.23607i
0 −3.98928 0 2.23607i 0 6.68423i 0 6.91435 0
321.2 0 −3.98928 0 2.23607i 0 6.68423i 0 6.91435 0
321.3 0 −1.23330 0 2.23607i 0 0.521669i 0 −7.47898 0
321.4 0 −1.23330 0 2.23607i 0 0.521669i 0 −7.47898 0
321.5 0 0.0436725 0 2.23607i 0 2.33372i 0 −8.99809 0
321.6 0 0.0436725 0 2.23607i 0 2.33372i 0 −8.99809 0
321.7 0 2.60299 0 2.23607i 0 8.05652i 0 −2.22446 0
321.8 0 2.60299 0 2.23607i 0 8.05652i 0 −2.22446 0
321.9 0 3.57592 0 2.23607i 0 10.2321i 0 3.78719 0
321.10 0 3.57592 0 2.23607i 0 10.2321i 0 3.78719 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 321.10 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
23.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1840.3.k.b 10
4.b odd 2 1 115.3.d.b 10
12.b even 2 1 1035.3.g.b 10
20.d odd 2 1 575.3.d.g 10
20.e even 4 2 575.3.c.d 20
23.b odd 2 1 inner 1840.3.k.b 10
92.b even 2 1 115.3.d.b 10
276.h odd 2 1 1035.3.g.b 10
460.g even 2 1 575.3.d.g 10
460.k odd 4 2 575.3.c.d 20

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
115.3.d.b 10 4.b odd 2 1
115.3.d.b 10 92.b even 2 1
575.3.c.d 20 20.e even 4 2
575.3.c.d 20 460.k odd 4 2
575.3.d.g 10 20.d odd 2 1
575.3.d.g 10 460.g even 2 1
1035.3.g.b 10 12.b even 2 1
1035.3.g.b 10 276.h odd 2 1
1840.3.k.b 10 1.a even 1 1 trivial
1840.3.k.b 10 23.b odd 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{5} - T_{3}^{4} - 18 T_{3}^{3} + 19 T_{3}^{2} + 45 T_{3} - 2$$ acting on $$S_{3}^{\mathrm{new}}(1840, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{10}$$
$3$ $$( -2 + 45 T + 19 T^{2} - 18 T^{3} - T^{4} + T^{5} )^{2}$$
$5$ $$( 5 + T^{2} )^{5}$$
$7$ $$450000 + 1757500 T^{2} + 386125 T^{4} + 15600 T^{6} + 220 T^{8} + T^{10}$$
$11$ $$5056200000 + 388480000 T^{2} + 10118000 T^{4} + 114825 T^{6} + 575 T^{8} + T^{10}$$
$13$ $$( 2272 - 1085 T - 949 T^{2} - 148 T^{3} + T^{4} + T^{5} )^{2}$$
$17$ $$32080050000 + 30754945000 T^{2} + 430109125 T^{4} + 1704000 T^{6} + 2320 T^{8} + T^{10}$$
$19$ $$1154881800000 + 35217400000 T^{2} + 344392000 T^{4} + 1234425 T^{6} + 1855 T^{8} + T^{10}$$
$23$ $$41426511213649 + 3445683352364 T + 170685380017 T^{2} + 4996840896 T^{3} - 22946962 T^{4} - 6276056 T^{5} - 43378 T^{6} + 17856 T^{7} + 1153 T^{8} + 44 T^{9} + T^{10}$$
$29$ $$( 482044 + 94614 T - 5435 T^{2} - 927 T^{3} + 23 T^{4} + T^{5} )^{2}$$
$31$ $$( -4585671 + 586314 T + 555 T^{2} - 1527 T^{3} + 8 T^{4} + T^{5} )^{2}$$
$37$ $$632002759200000 + 102193665040000 T^{2} + 129636326000 T^{4} + 61415400 T^{6} + 12845 T^{8} + T^{10}$$
$41$ $$( 137382991 + 1594244 T - 249405 T^{2} - 4937 T^{3} + 42 T^{4} + T^{5} )^{2}$$
$43$ $$5504562000000000 + 24695692000000 T^{2} + 40842568000 T^{4} + 30042000 T^{6} + 9340 T^{8} + T^{10}$$
$47$ $$( 75512 - 34280 T - 2864 T^{2} + 587 T^{3} + 56 T^{4} + T^{5} )^{2}$$
$53$ $$550998028800000 + 33240624480000 T^{2} + 179247762000 T^{4} + 101500200 T^{6} + 17925 T^{8} + T^{10}$$
$59$ $$( -547216 - 66912 T + 35216 T^{2} + 3780 T^{3} - 131 T^{4} + T^{5} )^{2}$$
$61$ $$19935444817800000 + 103449292480000 T^{2} + 143730422000 T^{4} + 74379825 T^{6} + 14975 T^{8} + T^{10}$$
$67$ $$13709523571200000 + 236641310560000 T^{2} + 295832330000 T^{4} + 123774300 T^{6} + 19385 T^{8} + T^{10}$$
$71$ $$( 2383066649 + 7330364 T - 1357435 T^{2} - 10577 T^{3} + 118 T^{4} + T^{5} )^{2}$$
$73$ $$( 181129672 - 16403800 T + 474996 T^{2} - 2693 T^{3} - 84 T^{4} + T^{5} )^{2}$$
$79$ $$322541393155200000 + 2480964607840000 T^{2} + 1695058696000 T^{4} + 371552400 T^{6} + 32620 T^{8} + T^{10}$$
$83$ $$150005056320000000 + 1424982143200000 T^{2} + 1238883858000 T^{4} + 348091400 T^{6} + 34125 T^{8} + T^{10}$$
$89$ $$13383985162320000000 + 48285335288800000 T^{2} + 19307364488000 T^{4} + 1978333200 T^{6} + 76220 T^{8} + T^{10}$$
$97$ $$920856337216200000 + 27319495966000000 T^{2} + 9938367406000 T^{4} + 1234128825 T^{6} + 60235 T^{8} + T^{10}$$
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