# Properties

 Label 720.6.f.n Level $720$ Weight $6$ Character orbit 720.f Analytic conductor $115.476$ Analytic rank $0$ Dimension $8$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$720 = 2^{4} \cdot 3^{2} \cdot 5$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 720.f (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$115.476350265$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: $$\mathbb{Q}[x]/(x^{8} + \cdots)$$ Defining polynomial: $$x^{8} + 41 x^{6} + 460 x^{4} + 969 x^{2} + 9$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{31}\cdot 3^{2}\cdot 5^{3}$$ Twist minimal: no (minimal twist has level 40) 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.

 $$f(q)$$ $$=$$ $$q + ( -1 + \beta_{1} ) q^{5} + ( -\beta_{1} + \beta_{6} ) q^{7} +O(q^{10})$$ $$q + ( -1 + \beta_{1} ) q^{5} + ( -\beta_{1} + \beta_{6} ) q^{7} + ( -92 - \beta_{1} + \beta_{4} ) q^{11} + ( -2 \beta_{1} + \beta_{2} - \beta_{3} - 4 \beta_{6} + \beta_{7} ) q^{13} + ( 9 \beta_{1} - 14 \beta_{2} - \beta_{3} - 2 \beta_{7} ) q^{17} + ( -172 + 19 \beta_{1} + 4 \beta_{2} + \beta_{4} + 4 \beta_{7} ) q^{19} + ( \beta_{1} + 6 \beta_{2} - 10 \beta_{3} + 9 \beta_{6} - 4 \beta_{7} ) q^{23} + ( -267 + \beta_{1} - 25 \beta_{2} + 9 \beta_{3} - 4 \beta_{4} + \beta_{5} + 8 \beta_{6} + 3 \beta_{7} ) q^{25} + ( -734 - 26 \beta_{1} - 4 \beta_{2} + 4 \beta_{4} + 2 \beta_{5} - 4 \beta_{7} ) q^{29} + ( -528 + 16 \beta_{1} + 4 \beta_{2} + 4 \beta_{5} + 4 \beta_{7} ) q^{31} + ( 2404 + 15 \beta_{1} + 65 \beta_{2} + 34 \beta_{3} + \beta_{4} - 4 \beta_{5} + 18 \beta_{6} - 12 \beta_{7} ) q^{35} + ( 5 \beta_{1} + 137 \beta_{2} + 26 \beta_{3} + 16 \beta_{6} + \beta_{7} ) q^{37} + ( -2950 + 34 \beta_{1} + 9 \beta_{2} + 16 \beta_{4} - 5 \beta_{5} + 9 \beta_{7} ) q^{41} + ( 54 \beta_{1} - 81 \beta_{2} + 108 \beta_{3} + 54 \beta_{6} ) q^{43} + ( -81 \beta_{1} - 168 \beta_{2} + 50 \beta_{3} - 9 \beta_{6} + 28 \beta_{7} ) q^{47} + ( -5625 - 72 \beta_{1} - 9 \beta_{2} + 24 \beta_{4} + 3 \beta_{5} - 9 \beta_{7} ) q^{49} + ( 70 \beta_{1} - 39 \beta_{2} + 191 \beta_{3} + 36 \beta_{6} + 17 \beta_{7} ) q^{53} + ( -1876 - 204 \beta_{1} - 220 \beta_{2} - 125 \beta_{3} - 15 \beta_{4} - 10 \beta_{5} + 20 \beta_{6} - 80 \beta_{7} ) q^{55} + ( 11460 - 57 \beta_{1} - 12 \beta_{2} + 13 \beta_{4} - 16 \beta_{5} - 12 \beta_{7} ) q^{59} + ( 15482 - 263 \beta_{1} - 53 \beta_{2} - 8 \beta_{4} + 6 \beta_{5} - 53 \beta_{7} ) q^{61} + ( 9008 - 50 \beta_{1} - 95 \beta_{2} - 182 \beta_{3} + 12 \beta_{4} + 7 \beta_{5} - 144 \beta_{6} + 41 \beta_{7} ) q^{65} + ( -214 \beta_{1} - 33 \beta_{2} - 124 \beta_{3} + 10 \beta_{6} + 16 \beta_{7} ) q^{67} + ( -15704 + 458 \beta_{1} + 84 \beta_{2} - 2 \beta_{4} - 36 \beta_{5} + 84 \beta_{7} ) q^{71} + ( -103 \beta_{1} - 388 \beta_{2} - 311 \beta_{3} - 8 \beta_{6} - 40 \beta_{7} ) q^{73} + ( 217 \beta_{1} + 804 \beta_{2} - 111 \beta_{3} - 468 \beta_{6} + 28 \beta_{7} ) q^{77} + ( -5408 - 628 \beta_{1} - 124 \beta_{2} + 28 \beta_{4} - 20 \beta_{5} - 124 \beta_{7} ) q^{79} + ( 302 \beta_{1} - 615 \beta_{2} - 236 \beta_{3} + 342 \beta_{6} - 176 \beta_{7} ) q^{83} + ( -36720 + 85 \beta_{1} - 90 \beta_{2} + 125 \beta_{3} - 80 \beta_{4} + 30 \beta_{5} - 60 \beta_{6} - 10 \beta_{7} ) q^{85} + ( 5238 - 270 \beta_{1} - 36 \beta_{2} + 48 \beta_{4} + 42 \beta_{5} - 36 \beta_{7} ) q^{89} + ( 60952 + 770 \beta_{1} + 140 \beta_{2} - 110 \beta_{4} + 40 \beta_{5} + 140 \beta_{7} ) q^{91} + ( 55324 - 224 \beta_{1} - 720 \beta_{2} + 55 \beta_{3} - 95 \beta_{4} + 10 \beta_{5} + 180 \beta_{6} - 20 \beta_{7} ) q^{95} + ( 107 \beta_{1} - 486 \beta_{2} + 49 \beta_{3} - 488 \beta_{6} + 86 \beta_{7} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q - 8q^{5} + O(q^{10})$$ $$8q - 8q^{5} - 736q^{11} - 1376q^{19} - 2136q^{25} - 5872q^{29} - 4224q^{31} + 19232q^{35} - 23600q^{41} - 45000q^{49} - 15008q^{55} + 91680q^{59} + 123856q^{61} + 72064q^{65} - 125632q^{71} - 43264q^{79} - 293760q^{85} + 41904q^{89} + 487616q^{91} + 442592q^{95} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + 41 x^{6} + 460 x^{4} + 969 x^{2} + 9$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-176 \nu^{7} - 66 \nu^{6} - 7372 \nu^{5} - 2232 \nu^{4} - 86042 \nu^{3} - 19752 \nu^{2} - 203676 \nu - 29412$$$$)/639$$ $$\beta_{2}$$ $$=$$ $$($$$$-182 \nu^{7} - 7420 \nu^{5} - 82106 \nu^{3} - 160458 \nu$$$$)/213$$ $$\beta_{3}$$ $$=$$ $$($$$$-656 \nu^{7} + 66 \nu^{6} - 26548 \nu^{5} + 2232 \nu^{4} - 295142 \nu^{3} + 19752 \nu^{2} - 646692 \nu + 29412$$$$)/639$$ $$\beta_{4}$$ $$=$$ $$($$$$-176 \nu^{7} - 282 \nu^{6} - 7372 \nu^{5} - 17592 \nu^{4} - 86042 \nu^{3} - 259752 \nu^{2} - 203676 \nu - 390564$$$$)/639$$ $$\beta_{5}$$ $$=$$ $$($$$$-176 \nu^{7} + 1710 \nu^{6} - 7372 \nu^{5} + 59688 \nu^{4} - 86042 \nu^{3} + 457848 \nu^{2} - 203676 \nu - 144180$$$$)/639$$ $$\beta_{6}$$ $$=$$ $$($$$$974 \nu^{7} - 66 \nu^{6} + 40168 \nu^{5} - 2232 \nu^{4} + 454172 \nu^{3} - 19752 \nu^{2} + 991374 \nu - 29412$$$$)/639$$ $$\beta_{7}$$ $$=$$ $$($$$$1426 \nu^{7} - 330 \nu^{6} + 59120 \nu^{5} - 11160 \nu^{4} + 676528 \nu^{3} - 98760 \nu^{2} + 1499754 \nu - 147060$$$$)/639$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-24 \beta_{7} + 72 \beta_{6} - 9 \beta_{3} + 64 \beta_{2} + 39 \beta_{1}$$$$)/3840$$ $$\nu^{2}$$ $$=$$ $$($$$$-34 \beta_{7} - 13 \beta_{5} - 3 \beta_{4} - 34 \beta_{2} - 154 \beta_{1} - 19680$$$$)/1920$$ $$\nu^{3}$$ $$=$$ $$($$$$6 \beta_{7} - 18 \beta_{6} - 3 \beta_{3} - 8 \beta_{2} - 15 \beta_{1}$$$$)/48$$ $$\nu^{4}$$ $$=$$ $$($$$$790 \beta_{7} + 283 \beta_{5} - 87 \beta_{4} + 790 \beta_{2} + 3754 \beta_{1} + 365280$$$$)/1920$$ $$\nu^{5}$$ $$=$$ $$($$$$-9048 \beta_{7} + 30888 \beta_{6} + 10329 \beta_{3} + 11104 \beta_{2} + 24681 \beta_{1}$$$$)/3840$$ $$\nu^{6}$$ $$=$$ $$($$$$-230 \beta_{7} - 71 \beta_{5} + 48 \beta_{4} - 230 \beta_{2} - 1127 \beta_{1} - 91488$$$$)/24$$ $$\nu^{7}$$ $$=$$ $$($$$$173496 \beta_{7} - 673128 \beta_{6} - 304899 \beta_{3} - 224896 \beta_{2} - 499251 \beta_{1}$$$$)/3840$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/720\mathbb{Z}\right)^\times$$.

 $$n$$ $$181$$ $$271$$ $$577$$ $$641$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
289.1
 0.0965878i − 0.0965878i − 3.98753i 3.98753i 1.64654i − 1.64654i − 4.73066i 4.73066i
0 0 0 −46.7401 30.6653i 0 179.876i 0 0 0
289.2 0 0 0 −46.7401 + 30.6653i 0 179.876i 0 0 0
289.3 0 0 0 −23.4238 50.7575i 0 10.2635i 0 0 0
289.4 0 0 0 −23.4238 + 50.7575i 0 10.2635i 0 0 0
289.5 0 0 0 13.1588 54.3309i 0 146.828i 0 0 0
289.6 0 0 0 13.1588 + 54.3309i 0 146.828i 0 0 0
289.7 0 0 0 53.0051 17.7613i 0 188.968i 0 0 0
289.8 0 0 0 53.0051 + 17.7613i 0 188.968i 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 289.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 720.6.f.n 8
3.b odd 2 1 80.6.c.d 8
4.b odd 2 1 360.6.f.b 8
5.b even 2 1 inner 720.6.f.n 8
12.b even 2 1 40.6.c.a 8
15.d odd 2 1 80.6.c.d 8
15.e even 4 1 400.6.a.z 4
15.e even 4 1 400.6.a.ba 4
20.d odd 2 1 360.6.f.b 8
24.f even 2 1 320.6.c.j 8
24.h odd 2 1 320.6.c.i 8
60.h even 2 1 40.6.c.a 8
60.l odd 4 1 200.6.a.j 4
60.l odd 4 1 200.6.a.k 4
120.i odd 2 1 320.6.c.i 8
120.m even 2 1 320.6.c.j 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
40.6.c.a 8 12.b even 2 1
40.6.c.a 8 60.h even 2 1
80.6.c.d 8 3.b odd 2 1
80.6.c.d 8 15.d odd 2 1
200.6.a.j 4 60.l odd 4 1
200.6.a.k 4 60.l odd 4 1
320.6.c.i 8 24.h odd 2 1
320.6.c.i 8 120.i odd 2 1
320.6.c.j 8 24.f even 2 1
320.6.c.j 8 120.m even 2 1
360.6.f.b 8 4.b odd 2 1
360.6.f.b 8 20.d odd 2 1
400.6.a.z 4 15.e even 4 1
400.6.a.ba 4 15.e even 4 1
720.6.f.n 8 1.a even 1 1 trivial
720.6.f.n 8 5.b 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 $$S_{6}^{\mathrm{new}}(720, [\chi])$$:

 $$T_{7}^{8} + 89728 T_{7}^{6} + 2632172640 T_{7}^{4} +$$25184325932032'>$$25\!\cdots\!32$$$$T_{7}^{2} +$$2623778518100224'>$$26\!\cdots\!24$$ $$T_{11}^{4} + 368 T_{11}^{3} - 468000 T_{11}^{2} - 126641408 T_{11} + 37397137664$$