# Properties

 Label 1680.4.a.bd Level 1680 Weight 4 Character orbit 1680.a Self dual yes Analytic conductor 99.123 Analytic rank 0 Dimension 2 CM no Inner twists 1

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1680 = 2^{4} \cdot 3 \cdot 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1680.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$99.1232088096$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{5})$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 105) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \frac{1}{2}(1 + \sqrt{5})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -3 q^{3} -5 q^{5} + 7 q^{7} + 9 q^{9} +O(q^{10})$$ $$q -3 q^{3} -5 q^{5} + 7 q^{7} + 9 q^{9} + ( 48 - 4 \beta ) q^{11} + ( -34 + 76 \beta ) q^{13} + 15 q^{15} + ( 22 - 88 \beta ) q^{17} + ( 28 + 52 \beta ) q^{19} -21 q^{21} + ( 180 - 40 \beta ) q^{23} + 25 q^{25} -27 q^{27} + ( -106 - 24 \beta ) q^{29} + ( -72 + 204 \beta ) q^{31} + ( -144 + 12 \beta ) q^{33} -35 q^{35} + ( 126 - 48 \beta ) q^{37} + ( 102 - 228 \beta ) q^{39} + ( -58 + 160 \beta ) q^{41} + ( -196 + 256 \beta ) q^{43} -45 q^{45} + ( -32 - 336 \beta ) q^{47} + 49 q^{49} + ( -66 + 264 \beta ) q^{51} + ( 94 - 172 \beta ) q^{53} + ( -240 + 20 \beta ) q^{55} + ( -84 - 156 \beta ) q^{57} + ( 268 - 72 \beta ) q^{59} + ( -258 - 168 \beta ) q^{61} + 63 q^{63} + ( 170 - 380 \beta ) q^{65} + ( -532 + 328 \beta ) q^{67} + ( -540 + 120 \beta ) q^{69} + ( 508 - 276 \beta ) q^{71} + ( 90 + 244 \beta ) q^{73} -75 q^{75} + ( 336 - 28 \beta ) q^{77} + ( 688 - 968 \beta ) q^{79} + 81 q^{81} + ( -220 - 168 \beta ) q^{83} + ( -110 + 440 \beta ) q^{85} + ( 318 + 72 \beta ) q^{87} + ( -778 + 224 \beta ) q^{89} + ( -238 + 532 \beta ) q^{91} + ( 216 - 612 \beta ) q^{93} + ( -140 - 260 \beta ) q^{95} + ( -1310 + 172 \beta ) q^{97} + ( 432 - 36 \beta ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 6q^{3} - 10q^{5} + 14q^{7} + 18q^{9} + O(q^{10})$$ $$2q - 6q^{3} - 10q^{5} + 14q^{7} + 18q^{9} + 92q^{11} + 8q^{13} + 30q^{15} - 44q^{17} + 108q^{19} - 42q^{21} + 320q^{23} + 50q^{25} - 54q^{27} - 236q^{29} + 60q^{31} - 276q^{33} - 70q^{35} + 204q^{37} - 24q^{39} + 44q^{41} - 136q^{43} - 90q^{45} - 400q^{47} + 98q^{49} + 132q^{51} + 16q^{53} - 460q^{55} - 324q^{57} + 464q^{59} - 684q^{61} + 126q^{63} - 40q^{65} - 736q^{67} - 960q^{69} + 740q^{71} + 424q^{73} - 150q^{75} + 644q^{77} + 408q^{79} + 162q^{81} - 608q^{83} + 220q^{85} + 708q^{87} - 1332q^{89} + 56q^{91} - 180q^{93} - 540q^{95} - 2448q^{97} + 828q^{99} + O(q^{100})$$

## 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
 1.61803 −0.618034
0 −3.00000 0 −5.00000 0 7.00000 0 9.00000 0
1.2 0 −3.00000 0 −5.00000 0 7.00000 0 9.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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 1680.4.a.bd 2
4.b odd 2 1 105.4.a.d 2
12.b even 2 1 315.4.a.l 2
20.d odd 2 1 525.4.a.o 2
20.e even 4 2 525.4.d.k 4
28.d even 2 1 735.4.a.m 2
60.h even 2 1 1575.4.a.n 2
84.h odd 2 1 2205.4.a.be 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.4.a.d 2 4.b odd 2 1
315.4.a.l 2 12.b even 2 1
525.4.a.o 2 20.d odd 2 1
525.4.d.k 4 20.e even 4 2
735.4.a.m 2 28.d even 2 1
1575.4.a.n 2 60.h even 2 1
1680.4.a.bd 2 1.a even 1 1 trivial
2205.4.a.be 2 84.h odd 2 1

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$3$$ $$1$$
$$5$$ $$1$$
$$7$$ $$-1$$

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(1680))$$:

 $$T_{11}^{2} - 92 T_{11} + 2096$$ $$T_{13}^{2} - 8 T_{13} - 7204$$ $$T_{17}^{2} + 44 T_{17} - 9196$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$( 1 + 3 T )^{2}$$
$5$ $$( 1 + 5 T )^{2}$$
$7$ $$( 1 - 7 T )^{2}$$
$11$ $$1 - 92 T + 4758 T^{2} - 122452 T^{3} + 1771561 T^{4}$$
$13$ $$1 - 8 T - 2810 T^{2} - 17576 T^{3} + 4826809 T^{4}$$
$17$ $$1 + 44 T + 630 T^{2} + 216172 T^{3} + 24137569 T^{4}$$
$19$ $$1 - 108 T + 13254 T^{2} - 740772 T^{3} + 47045881 T^{4}$$
$23$ $$1 - 320 T + 47934 T^{2} - 3893440 T^{3} + 148035889 T^{4}$$
$29$ $$1 + 236 T + 61982 T^{2} + 5755804 T^{3} + 594823321 T^{4}$$
$31$ $$1 - 60 T + 8462 T^{2} - 1787460 T^{3} + 887503681 T^{4}$$
$37$ $$1 - 204 T + 108830 T^{2} - 10333212 T^{3} + 2565726409 T^{4}$$
$41$ $$1 - 44 T + 106326 T^{2} - 3032524 T^{3} + 4750104241 T^{4}$$
$43$ $$1 + 136 T + 81718 T^{2} + 10812952 T^{3} + 6321363049 T^{4}$$
$47$ $$1 + 400 T + 106526 T^{2} + 41529200 T^{3} + 10779215329 T^{4}$$
$53$ $$1 - 16 T + 260838 T^{2} - 2382032 T^{3} + 22164361129 T^{4}$$
$59$ $$1 - 464 T + 458102 T^{2} - 95295856 T^{3} + 42180533641 T^{4}$$
$61$ $$1 + 684 T + 535646 T^{2} + 155255004 T^{3} + 51520374361 T^{4}$$
$67$ $$1 + 736 T + 602470 T^{2} + 221361568 T^{3} + 90458382169 T^{4}$$
$71$ $$1 - 740 T + 757502 T^{2} - 264854140 T^{3} + 128100283921 T^{4}$$
$73$ $$1 - 424 T + 748558 T^{2} - 164943208 T^{3} + 151334226289 T^{4}$$
$79$ $$1 - 408 T - 143586 T^{2} - 201159912 T^{3} + 243087455521 T^{4}$$
$83$ $$1 + 608 T + 1200710 T^{2} + 347646496 T^{3} + 326940373369 T^{4}$$
$89$ $$1 + 1332 T + 1790774 T^{2} + 939018708 T^{3} + 496981290961 T^{4}$$
$97$ $$1 + 2448 T + 3286542 T^{2} + 2234223504 T^{3} + 832972004929 T^{4}$$