# Properties

 Label 1840.2.i.b Level $1840$ Weight $2$ Character orbit 1840.i Analytic conductor $14.692$ Analytic rank $0$ Dimension $16$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1840 = 2^{4} \cdot 5 \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1840.i (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$14.6924739719$$ Analytic rank: $$0$$ Dimension: $$16$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ Defining polynomial: $$x^{16} - 8 x^{14} + 33 x^{12} - 98 x^{10} + 272 x^{8} - 882 x^{6} + 2673 x^{4} - 5832 x^{2} + 6561$$ Coefficient ring: $$\Z[a_1, \ldots, a_{23}]$$ Coefficient ring index: $$2^{10}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{15}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{7} q^{3} + \beta_{2} q^{5} + \beta_{13} q^{7} + ( -1 - \beta_{4} ) q^{9} +O(q^{10})$$ $$q + \beta_{7} q^{3} + \beta_{2} q^{5} + \beta_{13} q^{7} + ( -1 - \beta_{4} ) q^{9} + ( -\beta_{1} - \beta_{6} ) q^{11} + ( -1 + \beta_{3} ) q^{13} + \beta_{1} q^{15} + ( \beta_{2} - \beta_{5} + \beta_{12} ) q^{17} + ( \beta_{1} - \beta_{6} ) q^{19} + ( -2 \beta_{2} + \beta_{5} - \beta_{8} ) q^{21} + ( -\beta_{9} - \beta_{10} - \beta_{15} ) q^{23} - q^{25} + ( -\beta_{7} - \beta_{10} + \beta_{14} ) q^{27} + ( 2 + \beta_{3} + \beta_{4} - \beta_{11} ) q^{29} + ( -\beta_{7} + \beta_{15} ) q^{31} + ( 4 \beta_{2} - \beta_{5} + \beta_{8} ) q^{33} + \beta_{15} q^{35} + ( \beta_{2} + \beta_{8} + \beta_{12} ) q^{37} + ( -2 \beta_{7} + \beta_{9} - \beta_{10} + \beta_{14} - 2 \beta_{15} ) q^{39} + ( -2 - \beta_{3} - \beta_{4} - \beta_{11} ) q^{41} + ( -\beta_{1} - \beta_{6} - \beta_{10} + 2 \beta_{13} - \beta_{14} ) q^{43} + ( -\beta_{2} - \beta_{8} ) q^{45} + ( -\beta_{7} - 2 \beta_{9} ) q^{47} + ( 1 + \beta_{3} - \beta_{11} ) q^{49} + ( -2 \beta_{1} - 2 \beta_{10} + 2 \beta_{13} - 2 \beta_{14} ) q^{51} + ( -\beta_{2} - \beta_{5} - \beta_{12} ) q^{53} + ( \beta_{7} + \beta_{9} ) q^{55} + ( -4 \beta_{2} - \beta_{5} - \beta_{8} ) q^{57} + ( 3 \beta_{7} + \beta_{9} - \beta_{10} + \beta_{14} + \beta_{15} ) q^{59} + ( 2 \beta_{2} + \beta_{5} + \beta_{8} ) q^{61} + ( -3 \beta_{1} - \beta_{6} + \beta_{13} ) q^{63} -\beta_{5} q^{65} + ( 2 \beta_{1} - 2 \beta_{6} + \beta_{13} ) q^{67} + ( -1 - \beta_{3} + \beta_{5} + \beta_{8} + \beta_{11} - \beta_{12} ) q^{69} + ( \beta_{7} + 2 \beta_{9} + \beta_{15} ) q^{71} + ( -3 + \beta_{3} - 2 \beta_{11} ) q^{73} -\beta_{7} q^{75} + ( -4 - 2 \beta_{3} - 2 \beta_{4} + 2 \beta_{11} ) q^{77} + ( \beta_{1} - \beta_{6} ) q^{79} + ( 3 - 2 \beta_{3} + 2 \beta_{11} ) q^{81} + ( 3 \beta_{1} - \beta_{6} + \beta_{13} ) q^{83} + ( -\beta_{3} + \beta_{11} ) q^{85} + ( 3 \beta_{7} - \beta_{10} + \beta_{14} - 2 \beta_{15} ) q^{87} + ( 2 \beta_{2} - \beta_{5} + 3 \beta_{8} + 2 \beta_{12} ) q^{89} + ( 3 \beta_{1} - \beta_{6} - 2 \beta_{10} + 2 \beta_{13} - 2 \beta_{14} ) q^{91} + ( 5 + \beta_{3} + 2 \beta_{4} ) q^{93} + ( -\beta_{7} + \beta_{9} ) q^{95} + ( -2 \beta_{2} + \beta_{5} - 3 \beta_{8} ) q^{97} + ( 2 \beta_{1} - 2 \beta_{6} + 2 \beta_{13} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16q - 16q^{9} + O(q^{10})$$ $$16q - 16q^{9} - 8q^{13} - 16q^{25} + 36q^{29} - 44q^{41} + 20q^{49} - 20q^{69} - 48q^{73} - 72q^{77} + 40q^{81} - 4q^{85} + 88q^{93} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} - 8 x^{14} + 33 x^{12} - 98 x^{10} + 272 x^{8} - 882 x^{6} + 2673 x^{4} - 5832 x^{2} + 6561$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-5 \nu^{15} - 41 \nu^{13} + 78 \nu^{11} + 328 \nu^{9} + 1232 \nu^{7} - 8550 \nu^{5} + 23733 \nu^{3} - 28431 \nu$$$$)/69984$$ $$\beta_{2}$$ $$=$$ $$($$$$-5 \nu^{14} + 7 \nu^{12} - 90 \nu^{10} + 184 \nu^{8} - 232 \nu^{6} + 834 \nu^{4} - 2619 \nu^{2} + 8505$$$$)/15552$$ $$\beta_{3}$$ $$=$$ $$($$$$-4 \nu^{14} + 5 \nu^{12} + 3 \nu^{10} + 149 \nu^{8} - 386 \nu^{6} + 477 \nu^{4} + 567 \nu^{2} + 729$$$$)/2916$$ $$\beta_{4}$$ $$=$$ $$($$$$\nu^{14} - 8 \nu^{12} + 33 \nu^{10} - 98 \nu^{8} + 272 \nu^{6} - 882 \nu^{4} + 1944 \nu^{2} - 4374$$$$)/729$$ $$\beta_{5}$$ $$=$$ $$($$$$\nu^{14} - 8 \nu^{12} + 33 \nu^{10} - 98 \nu^{8} + 272 \nu^{6} - 882 \nu^{4} + 3402 \nu^{2} - 5832$$$$)/729$$ $$\beta_{6}$$ $$=$$ $$($$$$-\nu^{15} + 8 \nu^{13} - 33 \nu^{11} + 98 \nu^{9} - 272 \nu^{7} + 882 \nu^{5} - 2673 \nu^{3} + 8019 \nu$$$$)/2187$$ $$\beta_{7}$$ $$=$$ $$($$$$\nu^{15} - 8 \nu^{13} + 33 \nu^{11} - 98 \nu^{9} + 272 \nu^{7} - 882 \nu^{5} + 2673 \nu^{3} - 3645 \nu$$$$)/2187$$ $$\beta_{8}$$ $$=$$ $$($$$$13 \nu^{14} - 41 \nu^{12} + 168 \nu^{10} - 410 \nu^{8} + 1736 \nu^{6} - 6480 \nu^{4} + 14661 \nu^{2} - 22599$$$$)/5832$$ $$\beta_{9}$$ $$=$$ $$($$$$5 \nu^{15} - 13 \nu^{13} + 111 \nu^{11} - 166 \nu^{9} + 415 \nu^{7} - 2007 \nu^{5} + 5913 \nu^{3} - 13122 \nu$$$$)/8748$$ $$\beta_{10}$$ $$=$$ $$($$$$-145 \nu^{15} + 1403 \nu^{13} - 4866 \nu^{11} + 9512 \nu^{9} - 25832 \nu^{7} + 85770 \nu^{5} - 322623 \nu^{3} + 499365 \nu$$$$)/139968$$ $$\beta_{11}$$ $$=$$ $$($$$$-14 \nu^{14} + 67 \nu^{12} - 183 \nu^{10} + 535 \nu^{8} - 1342 \nu^{6} + 5859 \nu^{4} - 14661 \nu^{2} + 19683$$$$)/2916$$ $$\beta_{12}$$ $$=$$ $$($$$$149 \nu^{14} - 895 \nu^{12} + 2946 \nu^{10} - 7312 \nu^{8} + 22600 \nu^{6} - 71370 \nu^{4} + 211491 \nu^{2} - 333153$$$$)/23328$$ $$\beta_{13}$$ $$=$$ $$($$$$229 \nu^{15} - 1175 \nu^{13} + 3354 \nu^{11} - 8456 \nu^{9} + 24632 \nu^{7} - 113346 \nu^{5} + 243243 \nu^{3} - 257337 \nu$$$$)/139968$$ $$\beta_{14}$$ $$=$$ $$($$$$239 \nu^{15} - 1093 \nu^{13} + 3198 \nu^{11} - 9112 \nu^{9} + 22168 \nu^{7} - 96246 \nu^{5} + 335745 \nu^{3} - 480411 \nu$$$$)/139968$$ $$\beta_{15}$$ $$=$$ $$($$$$2 \nu^{15} - 10 \nu^{13} + 27 \nu^{11} - 70 \nu^{9} + 253 \nu^{7} - 771 \nu^{5} + 2016 \nu^{3} - 2835 \nu$$$$)/972$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{7} + \beta_{6}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{5} - \beta_{4} + 2$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$\beta_{14} - \beta_{13} + \beta_{7} + \beta_{6} + \beta_{1}$$ $$\nu^{4}$$ $$=$$ $$($$$$2 \beta_{12} + 2 \beta_{11} - 2 \beta_{8} + \beta_{5} - 3 \beta_{4} - 2 \beta_{3} - 4 \beta_{2}$$$$)/2$$ $$\nu^{5}$$ $$=$$ $$($$$$4 \beta_{15} + 2 \beta_{14} - 8 \beta_{13} - 2 \beta_{10} + \beta_{7} + 3 \beta_{6} - 4 \beta_{1}$$$$)/2$$ $$\nu^{6}$$ $$=$$ $$3 \beta_{12} + 5 \beta_{11} + 3 \beta_{8} - \beta_{5} - \beta_{4} - 3 \beta_{3} + 10 \beta_{2} + 2$$ $$\nu^{7}$$ $$=$$ $$($$$$24 \beta_{15} - 10 \beta_{14} - 20 \beta_{13} + 6 \beta_{10} + 4 \beta_{9} + 17 \beta_{7} + 5 \beta_{6} + 16 \beta_{1}$$$$)/2$$ $$\nu^{8}$$ $$=$$ $$($$$$14 \beta_{12} + 14 \beta_{11} + 26 \beta_{8} - 17 \beta_{5} - 15 \beta_{4} + 18 \beta_{3} + 36 \beta_{2} - 44$$$$)/2$$ $$\nu^{9}$$ $$=$$ $$18 \beta_{15} - 8 \beta_{14} - 13 \beta_{13} - 3 \beta_{10} + 22 \beta_{9} - 21 \beta_{7} + 9 \beta_{6} + 57 \beta_{1}$$ $$\nu^{10}$$ $$=$$ $$($$$$20 \beta_{12} + 32 \beta_{11} + 12 \beta_{8} - 37 \beta_{5} + 23 \beta_{4} + 56 \beta_{3} - 296 \beta_{2} + 106$$$$)/2$$ $$\nu^{11}$$ $$=$$ $$($$$$-4 \beta_{15} - 92 \beta_{14} - 4 \beta_{13} - 64 \beta_{10} + 220 \beta_{9} + 25 \beta_{7} + 71 \beta_{6} - 12 \beta_{1}$$$$)/2$$ $$\nu^{12}$$ $$=$$ $$-48 \beta_{12} + 34 \beta_{11} + 104 \beta_{8} + 56 \beta_{5} + 6 \beta_{4} - 14 \beta_{3} - 416 \beta_{2} + 203$$ $$\nu^{13}$$ $$=$$ $$($$$$44 \beta_{15} + 32 \beta_{14} - 236 \beta_{13} + 332 \beta_{10} + 700 \beta_{9} + 613 \beta_{7} + 197 \beta_{6} + 76 \beta_{1}$$$$)/2$$ $$\nu^{14}$$ $$=$$ $$($$$$76 \beta_{12} - 96 \beta_{11} + 420 \beta_{8} - 67 \beta_{5} - 833 \beta_{4} - 440 \beta_{3} - 2328 \beta_{2} - 790$$$$)/2$$ $$\nu^{15}$$ $$=$$ $$506 \beta_{15} + 431 \beta_{14} - 287 \beta_{13} + 392 \beta_{10} + 782 \beta_{9} - 553 \beta_{7} + 291 \beta_{6} - 525 \beta_{1}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1471.1
 0.739948 − 1.56604i −0.739948 − 1.56604i −1.38594 − 1.03883i 1.38594 − 1.03883i 1.59950 − 0.664536i −1.59950 − 0.664536i −1.72432 − 0.163515i 1.72432 − 0.163515i 1.72432 + 0.163515i −1.72432 + 0.163515i −1.59950 + 0.664536i 1.59950 + 0.664536i 1.38594 + 1.03883i −1.38594 + 1.03883i −0.739948 + 1.56604i 0.739948 + 1.56604i
0 3.13208i 0 1.00000i 0 −1.01363 0 −6.80991 0
1471.2 0 3.13208i 0 1.00000i 0 1.01363 0 −6.80991 0
1471.3 0 2.07767i 0 1.00000i 0 −3.88693 0 −1.31671 0
1471.4 0 2.07767i 0 1.00000i 0 3.88693 0 −1.31671 0
1471.5 0 1.32907i 0 1.00000i 0 3.37473 0 1.23357 0
1471.6 0 1.32907i 0 1.00000i 0 −3.37473 0 1.23357 0
1471.7 0 0.327030i 0 1.00000i 0 2.33999 0 2.89305 0
1471.8 0 0.327030i 0 1.00000i 0 −2.33999 0 2.89305 0
1471.9 0 0.327030i 0 1.00000i 0 −2.33999 0 2.89305 0
1471.10 0 0.327030i 0 1.00000i 0 2.33999 0 2.89305 0
1471.11 0 1.32907i 0 1.00000i 0 −3.37473 0 1.23357 0
1471.12 0 1.32907i 0 1.00000i 0 3.37473 0 1.23357 0
1471.13 0 2.07767i 0 1.00000i 0 3.88693 0 −1.31671 0
1471.14 0 2.07767i 0 1.00000i 0 −3.88693 0 −1.31671 0
1471.15 0 3.13208i 0 1.00000i 0 1.01363 0 −6.80991 0
1471.16 0 3.13208i 0 1.00000i 0 −1.01363 0 −6.80991 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1471.16 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1840.2.i.b 16
4.b odd 2 1 inner 1840.2.i.b 16
23.b odd 2 1 inner 1840.2.i.b 16
92.b even 2 1 inner 1840.2.i.b 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1840.2.i.b 16 1.a even 1 1 trivial
1840.2.i.b 16 4.b odd 2 1 inner
1840.2.i.b 16 23.b odd 2 1 inner
1840.2.i.b 16 92.b even 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{8} + 16 T_{3}^{6} + 69 T_{3}^{4} + 82 T_{3}^{2} + 8$$ acting on $$S_{2}^{\mathrm{new}}(1840, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{16}$$
$3$ $$( 8 + 82 T^{2} + 69 T^{4} + 16 T^{6} + T^{8} )^{2}$$
$5$ $$( 1 + T^{2} )^{8}$$
$7$ $$( 968 - 1268 T^{2} + 350 T^{4} - 33 T^{6} + T^{8} )^{2}$$
$11$ $$( 3200 - 2448 T^{2} + 580 T^{4} - 44 T^{6} + T^{8} )^{2}$$
$13$ $$( 128 - 78 T - 35 T^{2} + 2 T^{3} + T^{4} )^{4}$$
$17$ $$( 179776 + 51120 T^{2} + 4300 T^{4} + 117 T^{6} + T^{8} )^{2}$$
$19$ $$( 2048 - 4656 T^{2} + 868 T^{4} - 52 T^{6} + T^{8} )^{2}$$
$23$ $$78310985281 + 3256789558 T^{2} + 255214992 T^{4} + 9755818 T^{6} + 450974 T^{8} + 18442 T^{10} + 912 T^{12} + 22 T^{14} + T^{16}$$
$29$ $$( -110 + 153 T - 19 T^{2} - 9 T^{3} + T^{4} )^{4}$$
$31$ $$( 10368 + 5931 T^{2} + 1063 T^{4} + 61 T^{6} + T^{8} )^{2}$$
$37$ $$( 55696 + 27572 T^{2} + 2976 T^{4} + 101 T^{6} + T^{8} )^{2}$$
$41$ $$( -3244 - 1147 T - 69 T^{2} + 11 T^{3} + T^{4} )^{4}$$
$43$ $$( 850208 - 243080 T^{2} + 11420 T^{4} - 186 T^{6} + T^{8} )^{2}$$
$47$ $$( 2312 + 78066 T^{2} + 5869 T^{4} + 136 T^{6} + T^{8} )^{2}$$
$53$ $$( 331776 + 96768 T^{2} + 7396 T^{4} + 189 T^{6} + T^{8} )^{2}$$
$59$ $$( 4147200 + 1271808 T^{2} + 34664 T^{4} + 323 T^{6} + T^{8} )^{2}$$
$61$ $$( 270400 + 70944 T^{2} + 6196 T^{4} + 192 T^{6} + T^{8} )^{2}$$
$67$ $$( 1555848 - 333396 T^{2} + 14798 T^{4} - 217 T^{6} + T^{8} )^{2}$$
$71$ $$( 2592 + 4779 T^{2} + 1655 T^{4} + 109 T^{6} + T^{8} )^{2}$$
$73$ $$( 4244 - 820 T - 103 T^{2} + 12 T^{3} + T^{4} )^{4}$$
$79$ $$( 2048 - 4656 T^{2} + 868 T^{4} - 52 T^{6} + T^{8} )^{2}$$
$83$ $$( 2562848 - 352896 T^{2} + 15322 T^{4} - 233 T^{6} + T^{8} )^{2}$$
$89$ $$( 34857216 + 2892240 T^{2} + 63940 T^{4} + 492 T^{6} + T^{8} )^{2}$$
$97$ $$( 97298496 + 4271904 T^{2} + 67636 T^{4} + 448 T^{6} + T^{8} )^{2}$$