# Properties

 Label 1470.3.f.a Level 1470 Weight 3 Character orbit 1470.f Analytic conductor 40.055 Analytic rank 0 Dimension 8 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1470 = 2 \cdot 3 \cdot 5 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 1470.f (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$40.0545988610$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: 8.0.3317760000.3 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{6}$$ Twist minimal: no (minimal twist has level 210) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{2} -\beta_{5} q^{3} + 2 q^{4} + \beta_{6} q^{5} + \beta_{3} q^{6} + 2 \beta_{1} q^{8} -3 q^{9} +O(q^{10})$$ $$q + \beta_{1} q^{2} -\beta_{5} q^{3} + 2 q^{4} + \beta_{6} q^{5} + \beta_{3} q^{6} + 2 \beta_{1} q^{8} -3 q^{9} -\beta_{7} q^{10} + ( 1 - 3 \beta_{1} + 2 \beta_{2} + \beta_{4} ) q^{11} -2 \beta_{5} q^{12} + ( 4 \beta_{3} - 2 \beta_{5} + \beta_{6} - \beta_{7} ) q^{13} + \beta_{4} q^{15} + 4 q^{16} + ( \beta_{3} - 7 \beta_{5} - 3 \beta_{6} + 2 \beta_{7} ) q^{17} -3 \beta_{1} q^{18} + ( -7 \beta_{3} + 9 \beta_{5} - 2 \beta_{6} - \beta_{7} ) q^{19} + 2 \beta_{6} q^{20} + ( -6 + \beta_{1} + \beta_{2} + 4 \beta_{4} ) q^{22} + ( -3 - \beta_{1} + 2 \beta_{2} + 3 \beta_{4} ) q^{23} + 2 \beta_{3} q^{24} -5 q^{25} + ( 2 \beta_{3} - 8 \beta_{5} + 2 \beta_{6} - \beta_{7} ) q^{26} + 3 \beta_{5} q^{27} + ( 9 - 13 \beta_{1} + 2 \beta_{2} - \beta_{4} ) q^{29} + \beta_{2} q^{30} + ( \beta_{3} + 11 \beta_{5} + 2 \beta_{6} + 5 \beta_{7} ) q^{31} + 4 \beta_{1} q^{32} + ( -3 \beta_{3} - \beta_{5} - 3 \beta_{6} + 6 \beta_{7} ) q^{33} + ( 7 \beta_{3} - 2 \beta_{5} - 4 \beta_{6} + 3 \beta_{7} ) q^{34} -6 q^{36} + ( 24 + 2 \beta_{1} - \beta_{2} - 5 \beta_{4} ) q^{37} + ( -9 \beta_{3} + 14 \beta_{5} + 2 \beta_{6} + 2 \beta_{7} ) q^{38} + ( -6 - 12 \beta_{1} + \beta_{2} + \beta_{4} ) q^{39} -2 \beta_{7} q^{40} + ( -\beta_{3} + 3 \beta_{5} - 19 \beta_{6} - 4 \beta_{7} ) q^{41} + ( -14 + 5 \beta_{1} + 4 \beta_{2} + 9 \beta_{4} ) q^{43} + ( 2 - 6 \beta_{1} + 4 \beta_{2} + 2 \beta_{4} ) q^{44} -3 \beta_{6} q^{45} + ( -2 - 3 \beta_{1} + 3 \beta_{2} + 4 \beta_{4} ) q^{46} + ( 20 \beta_{3} - 2 \beta_{5} - 8 \beta_{6} + 2 \beta_{7} ) q^{47} -4 \beta_{5} q^{48} -5 \beta_{1} q^{50} + ( -21 - 3 \beta_{1} - 2 \beta_{2} - 3 \beta_{4} ) q^{51} + ( 8 \beta_{3} - 4 \beta_{5} + 2 \beta_{6} - 2 \beta_{7} ) q^{52} + ( -8 + 26 \beta_{1} + 8 \beta_{2} - 4 \beta_{4} ) q^{53} -3 \beta_{3} q^{54} + ( -10 \beta_{3} + 5 \beta_{5} + \beta_{6} + 3 \beta_{7} ) q^{55} + ( 27 + 21 \beta_{1} + \beta_{2} - 2 \beta_{4} ) q^{57} + ( -26 + 9 \beta_{1} - \beta_{2} + 4 \beta_{4} ) q^{58} + ( -7 \beta_{3} - 11 \beta_{5} + 21 \beta_{6} - 8 \beta_{7} ) q^{59} + 2 \beta_{4} q^{60} + ( -2 \beta_{3} + 8 \beta_{5} - 16 \beta_{6} + 16 \beta_{7} ) q^{61} + ( -11 \beta_{3} - 2 \beta_{5} - 10 \beta_{6} - 2 \beta_{7} ) q^{62} + 8 q^{64} + ( -5 - 5 \beta_{1} + 4 \beta_{2} + 2 \beta_{4} ) q^{65} + ( \beta_{3} + 6 \beta_{5} - 12 \beta_{6} + 3 \beta_{7} ) q^{66} + ( 30 + 9 \beta_{1} + 8 \beta_{2} + 3 \beta_{4} ) q^{67} + ( 2 \beta_{3} - 14 \beta_{5} - 6 \beta_{6} + 4 \beta_{7} ) q^{68} + ( -\beta_{3} + 3 \beta_{5} - 9 \beta_{6} + 6 \beta_{7} ) q^{69} + ( 1 - 29 \beta_{1} + 6 \beta_{2} - 7 \beta_{4} ) q^{71} -6 \beta_{1} q^{72} + ( 10 \beta_{3} - 2 \beta_{5} + 25 \beta_{6} + 15 \beta_{7} ) q^{73} + ( 4 + 24 \beta_{1} - 5 \beta_{2} - 2 \beta_{4} ) q^{74} + 5 \beta_{5} q^{75} + ( -14 \beta_{3} + 18 \beta_{5} - 4 \beta_{6} - 2 \beta_{7} ) q^{76} + ( -24 - 6 \beta_{1} + \beta_{2} + 2 \beta_{4} ) q^{78} + ( -3 - 13 \beta_{1} + 19 \beta_{2} + 2 \beta_{4} ) q^{79} + 4 \beta_{6} q^{80} + 9 q^{81} + ( -3 \beta_{3} + 2 \beta_{5} + 8 \beta_{6} + 19 \beta_{7} ) q^{82} + ( -15 \beta_{3} + 3 \beta_{5} - 7 \beta_{6} - 34 \beta_{7} ) q^{83} + ( 15 + 10 \beta_{1} + \beta_{2} + 7 \beta_{4} ) q^{85} + ( 10 - 14 \beta_{1} + 9 \beta_{2} + 8 \beta_{4} ) q^{86} + ( -13 \beta_{3} - 9 \beta_{5} + 3 \beta_{6} + 6 \beta_{7} ) q^{87} + ( -12 + 2 \beta_{1} + 2 \beta_{2} + 8 \beta_{4} ) q^{88} + ( 11 \beta_{3} + 41 \beta_{5} - 19 \beta_{6} + 4 \beta_{7} ) q^{89} + 3 \beta_{7} q^{90} + ( -6 - 2 \beta_{1} + 4 \beta_{2} + 6 \beta_{4} ) q^{92} + ( 33 - 3 \beta_{1} - 5 \beta_{2} + 2 \beta_{4} ) q^{93} + ( 2 \beta_{3} - 40 \beta_{5} - 4 \beta_{6} + 8 \beta_{7} ) q^{94} + ( 10 - 5 \beta_{1} - 7 \beta_{2} - 9 \beta_{4} ) q^{95} + 4 \beta_{3} q^{96} + ( 20 \beta_{3} - 36 \beta_{5} - 8 \beta_{6} - 18 \beta_{7} ) q^{97} + ( -3 + 9 \beta_{1} - 6 \beta_{2} - 3 \beta_{4} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + 16q^{4} - 24q^{9} + O(q^{10})$$ $$8q + 16q^{4} - 24q^{9} + 8q^{11} + 32q^{16} - 48q^{22} - 24q^{23} - 40q^{25} + 72q^{29} - 48q^{36} + 192q^{37} - 48q^{39} - 112q^{43} + 16q^{44} - 16q^{46} - 168q^{51} - 64q^{53} + 216q^{57} - 208q^{58} + 64q^{64} - 40q^{65} + 240q^{67} + 8q^{71} + 32q^{74} - 192q^{78} - 24q^{79} + 72q^{81} + 120q^{85} + 80q^{86} - 96q^{88} - 48q^{92} + 264q^{93} + 80q^{95} - 24q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 4 x^{6} + 7 x^{4} - 36 x^{2} + 81$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-4 \nu^{7} + 7 \nu^{5} + 35 \nu^{3} + 81 \nu$$$$)/189$$ $$\beta_{2}$$ $$=$$ $$($$$$-10 \nu^{7} + 49 \nu^{5} - 133 \nu^{3} + 801 \nu$$$$)/189$$ $$\beta_{3}$$ $$=$$ $$($$$$2 \nu^{7} + \nu^{5} + 5 \nu^{3} - 63 \nu$$$$)/27$$ $$\beta_{4}$$ $$=$$ $$($$$$-\nu^{6} + 4 \nu^{4} + 2 \nu^{2} + 18$$$$)/9$$ $$\beta_{5}$$ $$=$$ $$($$$$-8 \nu^{6} + 14 \nu^{4} - 56 \nu^{2} + 225$$$$)/63$$ $$\beta_{6}$$ $$=$$ $$($$$$-\nu^{6} + 22$$$$)/7$$ $$\beta_{7}$$ $$=$$ $$($$$$4 \nu^{7} - 7 \nu^{5} + 19 \nu^{3} - 81 \nu$$$$)/27$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{7} - \beta_{3} + \beta_{2} + \beta_{1}$$$$)/4$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{6} - 2 \beta_{5} + \beta_{4} + 2$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$($$$$\beta_{7} + 7 \beta_{1}$$$$)/2$$ $$\nu^{4}$$ $$=$$ $$($$$$-4 \beta_{6} + \beta_{5} + 4 \beta_{4} + 1$$$$)/2$$ $$\nu^{5}$$ $$=$$ $$($$$$-5 \beta_{7} + 19 \beta_{3} + 5 \beta_{2} + 19 \beta_{1}$$$$)/4$$ $$\nu^{6}$$ $$=$$ $$-7 \beta_{6} + 22$$ $$\nu^{7}$$ $$=$$ $$($$$$29 \beta_{7} + 13 \beta_{3} + 29 \beta_{2} - 13 \beta_{1}$$$$)/4$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1470\mathbb{Z}\right)^\times$$.

 $$n$$ $$491$$ $$1081$$ $$1177$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
391.1
 1.01575 − 1.40294i −1.72286 + 0.178197i −1.72286 − 0.178197i 1.01575 + 1.40294i −1.01575 + 1.40294i 1.72286 − 0.178197i 1.72286 + 0.178197i −1.01575 − 1.40294i
−1.41421 1.73205i 2.00000 2.23607i 2.44949i 0 −2.82843 −3.00000 3.16228i
391.2 −1.41421 1.73205i 2.00000 2.23607i 2.44949i 0 −2.82843 −3.00000 3.16228i
391.3 −1.41421 1.73205i 2.00000 2.23607i 2.44949i 0 −2.82843 −3.00000 3.16228i
391.4 −1.41421 1.73205i 2.00000 2.23607i 2.44949i 0 −2.82843 −3.00000 3.16228i
391.5 1.41421 1.73205i 2.00000 2.23607i 2.44949i 0 2.82843 −3.00000 3.16228i
391.6 1.41421 1.73205i 2.00000 2.23607i 2.44949i 0 2.82843 −3.00000 3.16228i
391.7 1.41421 1.73205i 2.00000 2.23607i 2.44949i 0 2.82843 −3.00000 3.16228i
391.8 1.41421 1.73205i 2.00000 2.23607i 2.44949i 0 2.82843 −3.00000 3.16228i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 391.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1470.3.f.a 8
7.b odd 2 1 inner 1470.3.f.a 8
7.c even 3 1 210.3.o.a 8
7.d odd 6 1 210.3.o.a 8
21.g even 6 1 630.3.v.b 8
21.h odd 6 1 630.3.v.b 8
35.i odd 6 1 1050.3.p.b 8
35.j even 6 1 1050.3.p.b 8
35.k even 12 2 1050.3.q.c 16
35.l odd 12 2 1050.3.q.c 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
210.3.o.a 8 7.c even 3 1
210.3.o.a 8 7.d odd 6 1
630.3.v.b 8 21.g even 6 1
630.3.v.b 8 21.h odd 6 1
1050.3.p.b 8 35.i odd 6 1
1050.3.p.b 8 35.j even 6 1
1050.3.q.c 16 35.k even 12 2
1050.3.q.c 16 35.l odd 12 2
1470.3.f.a 8 1.a even 1 1 trivial
1470.3.f.a 8 7.b odd 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{11}^{4} - 4 T_{11}^{3} - 300 T_{11}^{2} + 2048 T_{11} + 4744$$ acting on $$S_{3}^{\mathrm{new}}(1470, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 - 2 T^{2} )^{4}$$
$3$ $$( 1 + 3 T^{2} )^{4}$$
$5$ $$( 1 + 5 T^{2} )^{4}$$
$7$ 1
$11$ $$( 1 - 4 T + 184 T^{2} + 596 T^{3} + 19990 T^{4} + 72116 T^{5} + 2693944 T^{6} - 7086244 T^{7} + 214358881 T^{8} )^{2}$$
$13$ $$1 - 860 T^{2} + 357498 T^{4} - 95438800 T^{6} + 18541152803 T^{8} - 2725827566800 T^{10} + 291622101296058 T^{12} - 20036353205333660 T^{14} + 665416609183179841 T^{16}$$
$17$ $$1 - 1360 T^{2} + 841708 T^{4} - 329902000 T^{6} + 101801249638 T^{8} - 27553744942000 T^{10} + 5871550844149228 T^{12} - 792366242632474960 T^{14} + 48661191875666868481 T^{16}$$
$19$ $$1 - 620 T^{2} + 144858 T^{4} - 89040400 T^{6} + 54471994403 T^{8} - 11603833968400 T^{10} + 2460204974993178 T^{12} - 1372255249821019820 T^{14} +$$$$28\!\cdots\!81$$$$T^{16}$$
$23$ $$( 1 + 12 T + 1656 T^{2} + 17508 T^{3} + 1191350 T^{4} + 9261732 T^{5} + 463416696 T^{6} + 1776430668 T^{7} + 78310985281 T^{8} )^{2}$$
$29$ $$( 1 - 36 T + 2904 T^{2} - 82956 T^{3} + 3490070 T^{4} - 69765996 T^{5} + 2053944024 T^{6} - 21413639556 T^{7} + 500246412961 T^{8} )^{2}$$
$31$ $$1 - 5132 T^{2} + 12557754 T^{4} - 19776576400 T^{6} + 22219492569155 T^{8} - 18264083613504400 T^{10} + 10710395836988867514 T^{12} -$$$$40\!\cdots\!52$$$$T^{14} +$$$$72\!\cdots\!81$$$$T^{16}$$
$37$ $$( 1 - 96 T + 8106 T^{2} - 412320 T^{3} + 18472115 T^{4} - 564466080 T^{5} + 15191949066 T^{6} - 246309735264 T^{7} + 3512479453921 T^{8} )^{2}$$
$41$ $$1 - 5456 T^{2} + 19963884 T^{4} - 49821420976 T^{6} + 96897341176550 T^{8} - 140783428358562736 T^{10} +$$$$15\!\cdots\!64$$$$T^{12} -$$$$12\!\cdots\!36$$$$T^{14} +$$$$63\!\cdots\!41$$$$T^{16}$$
$43$ $$( 1 + 56 T + 5082 T^{2} + 180688 T^{3} + 11078435 T^{4} + 334092112 T^{5} + 17374346682 T^{6} + 353996330744 T^{7} + 11688200277601 T^{8} )^{2}$$
$47$ $$1 - 6584 T^{2} + 27573660 T^{4} - 81178339336 T^{6} + 201485105703494 T^{8} - 396124400069431816 T^{10} +$$$$65\!\cdots\!60$$$$T^{12} -$$$$76\!\cdots\!44$$$$T^{14} +$$$$56\!\cdots\!21$$$$T^{16}$$
$53$ $$( 1 + 32 T + 4596 T^{2} + 359008 T^{3} + 10001030 T^{4} + 1008453472 T^{5} + 36264650676 T^{6} + 709259556128 T^{7} + 62259690411361 T^{8} )^{2}$$
$59$ $$1 - 13840 T^{2} + 111830188 T^{4} - 618404804080 T^{6} + 2488320205143718 T^{8} - 7493434255171632880 T^{10} +$$$$16\!\cdots\!48$$$$T^{12} -$$$$24\!\cdots\!40$$$$T^{14} +$$$$21\!\cdots\!41$$$$T^{16}$$
$61$ $$1 - 13544 T^{2} + 85388316 T^{4} - 376885307608 T^{6} + 1453385293662086 T^{8} - 5218294044376458328 T^{10} +$$$$16\!\cdots\!96$$$$T^{12} -$$$$35\!\cdots\!24$$$$T^{14} +$$$$36\!\cdots\!61$$$$T^{16}$$
$67$ $$( 1 - 120 T + 18922 T^{2} - 1509840 T^{3} + 130500723 T^{4} - 6777671760 T^{5} + 381299511562 T^{6} - 10855005860280 T^{7} + 406067677556641 T^{8} )^{2}$$
$71$ $$( 1 - 4 T + 13176 T^{2} - 338828 T^{3} + 79144886 T^{4} - 1708031948 T^{5} + 334824308856 T^{6} - 512401135684 T^{7} + 645753531245761 T^{8} )^{2}$$
$73$ $$1 - 18684 T^{2} + 172678810 T^{4} - 1136790148176 T^{6} + 6403316119844739 T^{8} - 32282840594327758416 T^{10} +$$$$13\!\cdots\!10$$$$T^{12} -$$$$42\!\cdots\!64$$$$T^{14} +$$$$65\!\cdots\!61$$$$T^{16}$$
$79$ $$( 1 + 12 T + 2562 T^{2} + 208608 T^{3} + 62977883 T^{4} + 1301922528 T^{5} + 99790107522 T^{6} + 2917049466252 T^{7} + 1517108809906561 T^{8} )^{2}$$
$83$ $$1 - 2384 T^{2} + 54393900 T^{4} - 170213772976 T^{6} + 3204332319622694 T^{8} - 8078059876516133296 T^{10} +$$$$12\!\cdots\!00$$$$T^{12} -$$$$25\!\cdots\!24$$$$T^{14} +$$$$50\!\cdots\!81$$$$T^{16}$$
$89$ $$1 - 32432 T^{2} + 572475564 T^{4} - 6995892105040 T^{6} + 63446455115698790 T^{8} -$$$$43\!\cdots\!40$$$$T^{10} +$$$$22\!\cdots\!84$$$$T^{12} -$$$$80\!\cdots\!72$$$$T^{14} +$$$$15\!\cdots\!61$$$$T^{16}$$
$97$ $$1 - 35880 T^{2} + 724155484 T^{4} - 9811041724440 T^{6} + 103976225413471686 T^{8} -$$$$86\!\cdots\!40$$$$T^{10} +$$$$56\!\cdots\!24$$$$T^{12} -$$$$24\!\cdots\!80$$$$T^{14} +$$$$61\!\cdots\!21$$$$T^{16}$$