# Properties

 Label 1520.2.g.f Level $1520$ Weight $2$ Character orbit 1520.g Analytic conductor $12.137$ Analytic rank $0$ Dimension $16$ CM no Inner twists $8$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1520 = 2^{4} \cdot 5 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1520.g (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$12.1372611072$$ Analytic rank: $$0$$ Dimension: $$16$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ Defining polynomial: $$x^{16} - 32 x^{14} + 380 x^{12} - 1752 x^{10} + 1904 x^{8} + 7824 x^{6} + 7352 x^{4} + 2992 x^{2} + 484$$ Coefficient ring: $$\Z[a_1, \ldots, a_{19}]$$ Coefficient ring index: $$2^{15}$$ 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.

 $$f(q)$$ $$=$$ $$q + \beta_{4} q^{3} + ( -\beta_{1} + \beta_{2} ) q^{5} + \beta_{8} q^{7} + ( -2 + \beta_{2} ) q^{9} +O(q^{10})$$ $$q + \beta_{4} q^{3} + ( -\beta_{1} + \beta_{2} ) q^{5} + \beta_{8} q^{7} + ( -2 + \beta_{2} ) q^{9} -\beta_{6} q^{11} + \beta_{3} q^{13} + ( -\beta_{4} + \beta_{7} + \beta_{9} ) q^{15} + ( -3 \beta_{1} + \beta_{10} ) q^{17} + ( \beta_{6} - \beta_{13} ) q^{19} + \beta_{11} q^{21} + ( -\beta_{8} + \beta_{14} ) q^{23} + ( 1 - 2 \beta_{10} ) q^{25} + \beta_{7} q^{27} + \beta_{15} q^{29} + ( -\beta_{9} + \beta_{12} - \beta_{13} ) q^{31} + \beta_{5} q^{33} + ( -\beta_{8} - \beta_{12} - \beta_{13} + \beta_{14} ) q^{35} + ( \beta_{3} - \beta_{5} ) q^{37} + ( \beta_{6} - 3 \beta_{12} - 3 \beta_{13} ) q^{39} -\beta_{11} q^{41} + ( \beta_{8} + \beta_{14} ) q^{43} + ( 3 + 2 \beta_{1} - 2 \beta_{2} - \beta_{10} ) q^{45} -3 \beta_{8} q^{47} + ( 1 - 2 \beta_{2} ) q^{49} + ( 4 \beta_{9} - \beta_{12} + \beta_{13} ) q^{51} + ( \beta_{3} + \beta_{5} ) q^{53} + ( -\beta_{6} - \beta_{12} - \beta_{13} - \beta_{14} ) q^{55} + ( \beta_{1} - \beta_{3} - \beta_{5} + 2 \beta_{10} ) q^{57} + ( \beta_{9} - \beta_{12} + \beta_{13} ) q^{59} + ( 3 + 3 \beta_{2} ) q^{61} + ( -3 \beta_{8} + \beta_{14} ) q^{63} + ( -\beta_{3} - \beta_{5} + \beta_{11} ) q^{65} + ( -\beta_{4} + \beta_{7} ) q^{67} + ( -\beta_{11} - \beta_{15} ) q^{69} + ( \beta_{9} + 2 \beta_{12} - 2 \beta_{13} ) q^{71} + ( 6 \beta_{1} + 2 \beta_{10} ) q^{73} + ( \beta_{4} - 2 \beta_{9} + 2 \beta_{12} - 2 \beta_{13} ) q^{75} + ( -\beta_{1} - 3 \beta_{10} ) q^{77} + ( 3 \beta_{9} - 2 \beta_{12} + 2 \beta_{13} ) q^{79} + ( -8 - \beta_{2} ) q^{81} + ( -\beta_{8} - 2 \beta_{14} ) q^{83} + ( -6 + 3 \beta_{1} + 2 \beta_{2} - 3 \beta_{10} ) q^{85} + ( -2 \beta_{8} + 4 \beta_{14} ) q^{87} -\beta_{15} q^{89} + ( -5 \beta_{9} + \beta_{12} - \beta_{13} ) q^{91} + ( -3 \beta_{1} + 5 \beta_{10} ) q^{93} + ( -\beta_{7} - \beta_{8} + \beta_{9} + 2 \beta_{12} + \beta_{13} + \beta_{14} ) q^{95} + ( -\beta_{3} + \beta_{5} ) q^{97} + ( \beta_{6} - \beta_{12} - \beta_{13} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16 q - 32 q^{9} + O(q^{10})$$ $$16 q - 32 q^{9} + 16 q^{25} + 48 q^{45} + 16 q^{49} + 48 q^{61} - 128 q^{81} - 96 q^{85} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} - 32 x^{14} + 380 x^{12} - 1752 x^{10} + 1904 x^{8} + 7824 x^{6} + 7352 x^{4} + 2992 x^{2} + 484$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$273231 \nu^{15} - 8704991 \nu^{13} + 102633488 \nu^{11} - 465108276 \nu^{9} + 463305394 \nu^{7} + 2175968394 \nu^{5} + 2314047260 \nu^{3} + 1434400396 \nu$$$$)/ 371999496$$ $$\beta_{2}$$ $$=$$ $$($$$$-9418 \nu^{14} + 299253 \nu^{12} - 3506282 \nu^{10} + 15542327 \nu^{8} - 12340648 \nu^{6} - 87304466 \nu^{4} - 69461172 \nu^{2} - 18646288$$$$)/768594$$ $$\beta_{3}$$ $$=$$ $$($$$$-417883 \nu^{14} + 13275704 \nu^{12} - 155512084 \nu^{10} + 689037518 \nu^{8} - 546462062 \nu^{6} - 3869146772 \nu^{4} - 3078554636 \nu^{2} - 977505188$$$$)/33818136$$ $$\beta_{4}$$ $$=$$ $$($$$$607847 \nu^{14} - 19751712 \nu^{12} + 240757978 \nu^{10} - 1184238544 \nu^{8} + 1745285738 \nu^{6} + 3885107176 \nu^{4} + 2555420376 \nu^{2} + 574684088$$$$)/33818136$$ $$\beta_{5}$$ $$=$$ $$($$$$-716283 \nu^{14} + 22762217 \nu^{12} - 266747237 \nu^{10} + 1182820446 \nu^{8} - 940027588 \nu^{6} - 6645946230 \nu^{4} - 5287396490 \nu^{2} - 1394895040$$$$)/16909068$$ $$\beta_{6}$$ $$=$$ $$($$$$-920545 \nu^{14} + 29979829 \nu^{12} - 366750576 \nu^{10} + 1818589610 \nu^{8} - 2755021050 \nu^{6} - 5797901006 \nu^{4} - 3190168300 \nu^{2} - 566216376$$$$)/16909068$$ $$\beta_{7}$$ $$=$$ $$($$$$-353982 \nu^{14} + 11529208 \nu^{12} - 141053133 \nu^{10} + 699541134 \nu^{8} - 1060054206 \nu^{6} - 2228864424 \nu^{4} - 1222168758 \nu^{2} - 215694204$$$$)/5636356$$ $$\beta_{8}$$ $$=$$ $$($$$$-13061002 \nu^{15} + 424638381 \nu^{13} - 5180449592 \nu^{11} + 25531213262 \nu^{9} - 37894771792 \nu^{7} - 82991136530 \nu^{5} - 53288307768 \nu^{3} - 11340893344 \nu$$$$)/ 371999496$$ $$\beta_{9}$$ $$=$$ $$($$$$-1212203 \nu^{15} + 39394852 \nu^{13} - 480280280 \nu^{11} + 2363301958 \nu^{9} - 3487097926 \nu^{7} - 7742464084 \nu^{5} - 5054759548 \nu^{3} - 1093754068 \nu$$$$)/33818136$$ $$\beta_{10}$$ $$=$$ $$($$$$-113 \nu^{15} + 3627 \nu^{13} - 43270 \nu^{11} + 201430 \nu^{9} - 225338 \nu^{7} - 910798 \nu^{5} - 655920 \nu^{3} - 162272 \nu$$$$)/2904$$ $$\beta_{11}$$ $$=$$ $$($$$$478877 \nu^{15} - 15361992 \nu^{13} + 183103598 \nu^{11} - 850654440 \nu^{9} + 943754254 \nu^{7} + 3855966728 \nu^{5} + 2878061792 \nu^{3} + 828700840 \nu$$$$)/11272712$$ $$\beta_{12}$$ $$=$$ $$($$$$1775295 \nu^{15} + 604356 \nu^{14} - 57983887 \nu^{13} - 19643140 \nu^{12} + 712771771 \nu^{11} + 239522302 \nu^{10} - 3575281818 \nu^{9} - 1179063414 \nu^{8} + 5663689412 \nu^{7} + 1741812188 \nu^{6} + 10563796026 \nu^{5} + 3857356908 \nu^{4} + 5328528790 \nu^{3} + 2499339172 \nu^{2} + 819989720 \nu + 552888116$$$$)/33818136$$ $$\beta_{13}$$ $$=$$ $$($$$$-1775295 \nu^{15} + 604356 \nu^{14} + 57983887 \nu^{13} - 19643140 \nu^{12} - 712771771 \nu^{11} + 239522302 \nu^{10} + 3575281818 \nu^{9} - 1179063414 \nu^{8} - 5663689412 \nu^{7} + 1741812188 \nu^{6} - 10563796026 \nu^{5} + 3857356908 \nu^{4} - 5328528790 \nu^{3} + 2499339172 \nu^{2} - 819989720 \nu + 552888116$$$$)/33818136$$ $$\beta_{14}$$ $$=$$ $$($$$$16985579 \nu^{15} - 554706168 \nu^{13} + 6817424848 \nu^{11} - 34181649619 \nu^{9} + 54072323630 \nu^{7} + 101208945526 \nu^{5} + 51378699528 \nu^{3} + 7996450550 \nu$$$$)/ 185999748$$ $$\beta_{15}$$ $$=$$ $$($$$$2258416 \nu^{15} - 72492744 \nu^{13} + 864912425 \nu^{11} - 4027203638 \nu^{9} + 4509531406 \nu^{7} + 18205941992 \nu^{5} + 13062002142 \nu^{3} + 3174857356 \nu$$$$)/16909068$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{9} - \beta_{8} + \beta_{1}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$-\beta_{13} - \beta_{12} + 2 \beta_{4} + 2 \beta_{3} - 2 \beta_{2} + 8$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$($$$$2 \beta_{14} + 2 \beta_{13} - 2 \beta_{12} - 3 \beta_{11} - 3 \beta_{10} + 9 \beta_{9} - 10 \beta_{8} + 13 \beta_{1}$$$$)/2$$ $$\nu^{4}$$ $$=$$ $$-11 \beta_{13} - 11 \beta_{12} - 4 \beta_{7} + 4 \beta_{6} + 2 \beta_{5} + 20 \beta_{4} + 8 \beta_{3} - 15 \beta_{2} + 33$$ $$\nu^{5}$$ $$=$$ $$($$$$-10 \beta_{15} + 27 \beta_{14} + 25 \beta_{13} - 25 \beta_{12} - 50 \beta_{11} - 85 \beta_{10} + 72 \beta_{9} - 78 \beta_{8} + 209 \beta_{1}$$$$)/2$$ $$\nu^{6}$$ $$=$$ $$-173 \beta_{13} - 173 \beta_{12} - 95 \beta_{7} + 101 \beta_{6} + 21 \beta_{5} + 318 \beta_{4} + 50 \beta_{3} - 123 \beta_{2} + 193$$ $$\nu^{7}$$ $$=$$ $$($$$$-217 \beta_{15} + 156 \beta_{14} + 141 \beta_{13} - 141 \beta_{12} - 714 \beta_{11} - 1470 \beta_{10} + 290 \beta_{9} - 312 \beta_{8} + 2876 \beta_{1}$$$$)/2$$ $$\nu^{8}$$ $$=$$ $$-2344 \beta_{13} - 2344 \beta_{12} - 1488 \beta_{7} + 1604 \beta_{6} + 40 \beta_{5} + 4320 \beta_{4} - 56 \beta_{3} - 82 \beta_{2} - 308$$ $$\nu^{9}$$ $$=$$ $$-1566 \beta_{15} - 709 \beta_{14} - 671 \beta_{13} + 671 \beta_{12} - 4476 \beta_{11} - 9927 \beta_{10} - 2373 \beta_{9} + 2585 \beta_{8} + 17732 \beta_{1}$$ $$\nu^{10}$$ $$=$$ $$-27531 \beta_{13} - 27531 \beta_{12} - 18430 \beta_{7} + 19958 \beta_{6} - 2946 \beta_{5} + 50798 \beta_{4} - 9434 \beta_{3} + 19696 \beta_{2} - 37886$$ $$\nu^{11}$$ $$=$$ $$-17721 \beta_{15} - 30436 \beta_{14} - 28199 \beta_{13} + 28199 \beta_{12} - 48213 \beta_{11} - 109879 \beta_{10} - 80721 \beta_{9} + 87570 \beta_{8} + 189767 \beta_{1}$$ $$\nu^{12}$$ $$=$$ $$-268398 \beta_{13} - 268398 \beta_{12} - 183056 \beta_{7} + 198540 \beta_{6} - 76356 \beta_{5} + 495432 \beta_{4} - 216504 \beta_{3} + 482274 \beta_{2} - 856818$$ $$\nu^{13}$$ $$=$$ $$-152620 \beta_{15} - 601371 \beta_{14} - 555451 \beta_{13} + 555451 \beta_{12} - 407862 \beta_{11} - 938899 \beta_{10} - 1534878 \beta_{9} + 1663812 \beta_{8} + 1601437 \beta_{1}$$ $$\nu^{14}$$ $$=$$ $$-1814468 \beta_{13} - 1814468 \beta_{12} - 1249170 \beta_{7} + 1355870 \beta_{6} - 1309442 \beta_{5} + 3350008 \beta_{4} - 3606552 \beta_{3} + 8163506 \beta_{2} - 14218802$$ $$\nu^{15}$$ $$=$$ $$-647799 \beta_{15} - 9181848 \beta_{14} - 8473777 \beta_{13} + 8473777 \beta_{12} - 1686196 \beta_{11} - 3939864 \beta_{10} - 23191980 \beta_{9} + 25134720 \beta_{8} + 6596386 \beta_{1}$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1520\mathbb{Z}\right)^\times$$.

 $$n$$ $$191$$ $$401$$ $$1141$$ $$1217$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1519.1
 −0.141739 + 0.707107i −3.52761 + 0.707107i 3.52761 − 0.707107i 0.141739 − 0.707107i −0.213388 + 0.707107i −2.34315 + 0.707107i 2.34315 − 0.707107i 0.213388 − 0.707107i 2.34315 + 0.707107i 0.213388 + 0.707107i −0.213388 − 0.707107i −2.34315 − 0.707107i 3.52761 + 0.707107i 0.141739 + 0.707107i −0.141739 − 0.707107i −3.52761 − 0.707107i
0 2.59462i 0 −1.73205 1.41421i 0 −3.38587 0 −3.73205 0
1519.2 0 2.59462i 0 −1.73205 1.41421i 0 3.38587 0 −3.73205 0
1519.3 0 2.59462i 0 −1.73205 + 1.41421i 0 −3.38587 0 −3.73205 0
1519.4 0 2.59462i 0 −1.73205 + 1.41421i 0 3.38587 0 −3.73205 0
1519.5 0 1.80775i 0 1.73205 1.41421i 0 −2.12976 0 −0.267949 0
1519.6 0 1.80775i 0 1.73205 1.41421i 0 2.12976 0 −0.267949 0
1519.7 0 1.80775i 0 1.73205 + 1.41421i 0 −2.12976 0 −0.267949 0
1519.8 0 1.80775i 0 1.73205 + 1.41421i 0 2.12976 0 −0.267949 0
1519.9 0 1.80775i 0 1.73205 1.41421i 0 −2.12976 0 −0.267949 0
1519.10 0 1.80775i 0 1.73205 1.41421i 0 2.12976 0 −0.267949 0
1519.11 0 1.80775i 0 1.73205 + 1.41421i 0 −2.12976 0 −0.267949 0
1519.12 0 1.80775i 0 1.73205 + 1.41421i 0 2.12976 0 −0.267949 0
1519.13 0 2.59462i 0 −1.73205 1.41421i 0 −3.38587 0 −3.73205 0
1519.14 0 2.59462i 0 −1.73205 1.41421i 0 3.38587 0 −3.73205 0
1519.15 0 2.59462i 0 −1.73205 + 1.41421i 0 −3.38587 0 −3.73205 0
1519.16 0 2.59462i 0 −1.73205 + 1.41421i 0 3.38587 0 −3.73205 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1519.16 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
5.b even 2 1 inner
19.b odd 2 1 inner
20.d odd 2 1 inner
76.d even 2 1 inner
95.d odd 2 1 inner
380.d even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1520.2.g.f 16
4.b odd 2 1 inner 1520.2.g.f 16
5.b even 2 1 inner 1520.2.g.f 16
19.b odd 2 1 inner 1520.2.g.f 16
20.d odd 2 1 inner 1520.2.g.f 16
76.d even 2 1 inner 1520.2.g.f 16
95.d odd 2 1 inner 1520.2.g.f 16
380.d even 2 1 inner 1520.2.g.f 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1520.2.g.f 16 1.a even 1 1 trivial
1520.2.g.f 16 4.b odd 2 1 inner
1520.2.g.f 16 5.b even 2 1 inner
1520.2.g.f 16 19.b odd 2 1 inner
1520.2.g.f 16 20.d odd 2 1 inner
1520.2.g.f 16 76.d even 2 1 inner
1520.2.g.f 16 95.d odd 2 1 inner
1520.2.g.f 16 380.d even 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1520, [\chi])$$:

 $$T_{3}^{4} + 10 T_{3}^{2} + 22$$ $$T_{7}^{4} - 16 T_{7}^{2} + 52$$ $$T_{31}^{4} - 60 T_{31}^{2} + 792$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{16}$$
$3$ $$( 22 + 10 T^{2} + T^{4} )^{4}$$
$5$ $$( 25 - 2 T^{2} + T^{4} )^{4}$$
$7$ $$( 52 - 16 T^{2} + T^{4} )^{4}$$
$11$ $$( 52 + 20 T^{2} + T^{4} )^{4}$$
$13$ $$( 286 - 46 T^{2} + T^{4} )^{4}$$
$17$ $$( 144 + 48 T^{2} + T^{4} )^{4}$$
$19$ $$( 130321 + 7220 T^{2} + 390 T^{4} + 20 T^{6} + T^{8} )^{2}$$
$23$ $$( 468 - 48 T^{2} + T^{4} )^{4}$$
$29$ $$( 4576 + 152 T^{2} + T^{4} )^{4}$$
$31$ $$( 792 - 60 T^{2} + T^{4} )^{4}$$
$37$ $$( 286 - 106 T^{2} + T^{4} )^{4}$$
$41$ $$( 1144 + 92 T^{2} + T^{4} )^{4}$$
$43$ $$( 52 - 64 T^{2} + T^{4} )^{4}$$
$47$ $$( 4212 - 144 T^{2} + T^{4} )^{4}$$
$53$ $$( 2574 - 138 T^{2} + T^{4} )^{4}$$
$59$ $$( 792 - 60 T^{2} + T^{4} )^{4}$$
$61$ $$( -18 - 6 T + T^{2} )^{8}$$
$67$ $$( 198 + 30 T^{2} + T^{4} )^{4}$$
$71$ $$( 792 - 276 T^{2} + T^{4} )^{4}$$
$73$ $$( 2304 + 192 T^{2} + T^{4} )^{4}$$
$79$ $$( 10648 - 308 T^{2} + T^{4} )^{4}$$
$83$ $$( 468 - 192 T^{2} + T^{4} )^{4}$$
$89$ $$( 4576 + 152 T^{2} + T^{4} )^{4}$$
$97$ $$( 286 - 106 T^{2} + T^{4} )^{4}$$