# Properties

 Label 2700.3.u.b Level 2700 Weight 3 Character orbit 2700.u Analytic conductor 73.570 Analytic rank 0 Dimension 8 CM no Inner twists 4

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2700 = 2^{2} \cdot 3^{3} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 2700.u (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$73.5696713773$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(\zeta_{6})$$ Coefficient field: 8.0.303595776.1 Coefficient ring: $$\Z[a_1, \ldots, a_{17}]$$ Coefficient ring index: $$3^{4}$$ Twist minimal: no (minimal twist has level 36) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( -\beta_{3} + \beta_{4} - 2 \beta_{7} ) q^{7} +O(q^{10})$$ $$q + ( -\beta_{3} + \beta_{4} - 2 \beta_{7} ) q^{7} + ( 12 - 5 \beta_{2} - 2 \beta_{5} ) q^{11} + ( -3 \beta_{1} + \beta_{3} - 2 \beta_{4} + \beta_{7} ) q^{13} + ( -15 \beta_{1} + 7 \beta_{3} + \beta_{4} - \beta_{7} ) q^{17} + ( -\beta_{2} + 2 \beta_{5} - \beta_{6} ) q^{19} + ( -17 \beta_{1} + 33 \beta_{3} - \beta_{7} ) q^{23} + ( -21 + 14 \beta_{2} - 7 \beta_{5} ) q^{29} + ( -2 + 5 \beta_{2} - 3 \beta_{5} - 3 \beta_{6} ) q^{31} + ( 4 \beta_{1} - 18 \beta_{3} + 4 \beta_{4} + 4 \beta_{7} ) q^{37} + ( 7 - \beta_{2} + 8 \beta_{5} - 8 \beta_{6} ) q^{41} + ( -23 \beta_{1} + 23 \beta_{3} ) q^{43} + ( 12 \beta_{1} + 15 \beta_{3} - 3 \beta_{4} ) q^{47} + ( 25 - 26 \beta_{2} + \beta_{5} + \beta_{6} ) q^{49} + ( -24 \beta_{1} + 16 \beta_{3} - 8 \beta_{4} + 8 \beta_{7} ) q^{53} + ( 25 + 17 \beta_{2} + 8 \beta_{5} - 8 \beta_{6} ) q^{59} + ( -3 + 22 \beta_{2} - 3 \beta_{5} + 6 \beta_{6} ) q^{61} + ( 61 \beta_{1} - 6 \beta_{3} + 12 \beta_{4} - 6 \beta_{7} ) q^{67} + ( 16 - 30 \beta_{2} - 2 \beta_{6} ) q^{71} + ( 3 \beta_{1} - 23 \beta_{3} + 3 \beta_{4} + 3 \beta_{7} ) q^{73} + ( -55 \beta_{1} + 93 \beta_{3} - 17 \beta_{7} ) q^{77} + ( 5 - 44 \beta_{2} + 5 \beta_{5} - 10 \beta_{6} ) q^{79} + ( -12 \beta_{1} - 15 \beta_{3} + 3 \beta_{4} ) q^{83} + ( -64 + 120 \beta_{2} + 8 \beta_{6} ) q^{89} + ( -77 + 3 \beta_{2} - 6 \beta_{5} + 3 \beta_{6} ) q^{91} + ( 97 \beta_{1} - 99 \beta_{3} + 2 \beta_{4} - 4 \beta_{7} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + O(q^{10})$$ $$8q + 72q^{11} - 4q^{19} - 126q^{29} - 14q^{31} + 36q^{41} + 102q^{49} + 252q^{59} + 82q^{61} - 166q^{79} - 604q^{91} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + 5 x^{6} + 16 x^{4} + 45 x^{2} + 81$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{7} + 32 \nu^{5} + 16 \nu^{3} + 45 \nu$$$$)/432$$ $$\beta_{2}$$ $$=$$ $$($$$$5 \nu^{6} + 16 \nu^{4} + 80 \nu^{2} + 225$$$$)/144$$ $$\beta_{3}$$ $$=$$ $$($$$$5 \nu^{7} + 16 \nu^{5} + 8 \nu^{3} + 81 \nu$$$$)/216$$ $$\beta_{4}$$ $$=$$ $$($$$$-\nu^{7} + 131 \nu$$$$)/48$$ $$\beta_{5}$$ $$=$$ $$($$$$13 \nu^{6} + 128 \nu^{4} + 208 \nu^{2} + 585$$$$)/144$$ $$\beta_{6}$$ $$=$$ $$($$$$-3 \nu^{6} - 7$$$$)/16$$ $$\beta_{7}$$ $$=$$ $$($$$$23 \nu^{7} + 88 \nu^{5} + 368 \nu^{3} + 1035 \nu$$$$)/216$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{4} + \beta_{3} - \beta_{1}$$$$)/3$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{6} - \beta_{5} + 8 \beta_{2} - 8$$$$)/3$$ $$\nu^{3}$$ $$=$$ $$($$$$2 \beta_{7} - 2 \beta_{4} - 11 \beta_{3}$$$$)/3$$ $$\nu^{4}$$ $$=$$ $$($$$$5 \beta_{5} - 13 \beta_{2}$$$$)/3$$ $$\nu^{5}$$ $$=$$ $$($$$$-\beta_{7} + 46 \beta_{1}$$$$)/3$$ $$\nu^{6}$$ $$=$$ $$($$$$-16 \beta_{6} - 7$$$$)/3$$ $$\nu^{7}$$ $$=$$ $$($$$$-13 \beta_{4} + 131 \beta_{3} - 131 \beta_{1}$$$$)/3$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2700\mathbb{Z}\right)^\times$$.

 $$n$$ $$1001$$ $$1351$$ $$2377$$ $$\chi(n)$$ $$1 - \beta_{2}$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
449.1
 −0.396143 − 1.68614i −1.26217 − 1.18614i 1.26217 + 1.18614i 0.396143 + 1.68614i −0.396143 + 1.68614i −1.26217 + 1.18614i 1.26217 − 1.18614i 0.396143 − 1.68614i
0 0 0 0 0 −7.89542 + 4.55842i 0 0 0
449.2 0 0 0 0 0 −7.02939 + 4.05842i 0 0 0
449.3 0 0 0 0 0 7.02939 4.05842i 0 0 0
449.4 0 0 0 0 0 7.89542 4.55842i 0 0 0
2249.1 0 0 0 0 0 −7.89542 4.55842i 0 0 0
2249.2 0 0 0 0 0 −7.02939 4.05842i 0 0 0
2249.3 0 0 0 0 0 7.02939 + 4.05842i 0 0 0
2249.4 0 0 0 0 0 7.89542 + 4.55842i 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 2249.4 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
9.d odd 6 1 inner
45.h odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2700.3.u.b 8
3.b odd 2 1 900.3.u.a 8
5.b even 2 1 inner 2700.3.u.b 8
5.c odd 4 1 108.3.g.a 4
5.c odd 4 1 2700.3.p.b 4
9.c even 3 1 900.3.u.a 8
9.d odd 6 1 inner 2700.3.u.b 8
15.d odd 2 1 900.3.u.a 8
15.e even 4 1 36.3.g.a 4
15.e even 4 1 900.3.p.a 4
20.e even 4 1 432.3.q.b 4
40.i odd 4 1 1728.3.q.g 4
40.k even 4 1 1728.3.q.h 4
45.h odd 6 1 inner 2700.3.u.b 8
45.j even 6 1 900.3.u.a 8
45.k odd 12 1 36.3.g.a 4
45.k odd 12 1 324.3.c.b 4
45.k odd 12 1 900.3.p.a 4
45.l even 12 1 108.3.g.a 4
45.l even 12 1 324.3.c.b 4
45.l even 12 1 2700.3.p.b 4
60.l odd 4 1 144.3.q.b 4
120.q odd 4 1 576.3.q.g 4
120.w even 4 1 576.3.q.d 4
180.v odd 12 1 432.3.q.b 4
180.v odd 12 1 1296.3.e.e 4
180.x even 12 1 144.3.q.b 4
180.x even 12 1 1296.3.e.e 4
360.bo even 12 1 576.3.q.g 4
360.br even 12 1 1728.3.q.g 4
360.bt odd 12 1 1728.3.q.h 4
360.bu odd 12 1 576.3.q.d 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
36.3.g.a 4 15.e even 4 1
36.3.g.a 4 45.k odd 12 1
108.3.g.a 4 5.c odd 4 1
108.3.g.a 4 45.l even 12 1
144.3.q.b 4 60.l odd 4 1
144.3.q.b 4 180.x even 12 1
324.3.c.b 4 45.k odd 12 1
324.3.c.b 4 45.l even 12 1
432.3.q.b 4 20.e even 4 1
432.3.q.b 4 180.v odd 12 1
576.3.q.d 4 120.w even 4 1
576.3.q.d 4 360.bu odd 12 1
576.3.q.g 4 120.q odd 4 1
576.3.q.g 4 360.bo even 12 1
900.3.p.a 4 15.e even 4 1
900.3.p.a 4 45.k odd 12 1
900.3.u.a 8 3.b odd 2 1
900.3.u.a 8 9.c even 3 1
900.3.u.a 8 15.d odd 2 1
900.3.u.a 8 45.j even 6 1
1296.3.e.e 4 180.v odd 12 1
1296.3.e.e 4 180.x even 12 1
1728.3.q.g 4 40.i odd 4 1
1728.3.q.g 4 360.br even 12 1
1728.3.q.h 4 40.k even 4 1
1728.3.q.h 4 360.bt odd 12 1
2700.3.p.b 4 5.c odd 4 1
2700.3.p.b 4 45.l even 12 1
2700.3.u.b 8 1.a even 1 1 trivial
2700.3.u.b 8 5.b even 2 1 inner
2700.3.u.b 8 9.d odd 6 1 inner
2700.3.u.b 8 45.h odd 6 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7}^{8} - 149 T_{7}^{6} + 16725 T_{7}^{4} - 815924 T_{7}^{2} + 29986576$$ acting on $$S_{3}^{\mathrm{new}}(2700, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ 1
$7$ $$1 + 47 T^{2} - 3071 T^{4} + 22466 T^{6} + 16809790 T^{8} + 53940866 T^{10} - 17703703871 T^{12} + 650540498447 T^{14} + 33232930569601 T^{16}$$
$11$ $$( 1 - 36 T + 683 T^{2} - 9036 T^{3} + 100632 T^{4} - 1093356 T^{5} + 9999803 T^{6} - 63776196 T^{7} + 214358881 T^{8} )^{2}$$
$13$ $$1 + 515 T^{2} + 143653 T^{4} + 33191750 T^{6} + 6388929238 T^{8} + 947989571750 T^{10} + 117182165263813 T^{12} + 11998513838077715 T^{14} + 665416609183179841 T^{16}$$
$17$ $$( 1 + 769 T^{2} + 298176 T^{4} + 64227649 T^{6} + 6975757441 T^{8} )^{2}$$
$19$ $$( 1 + T + 648 T^{2} + 361 T^{3} + 130321 T^{4} )^{4}$$
$23$ $$1 - 433 T^{2} - 338207 T^{4} + 14715938 T^{6} + 145578189886 T^{8} + 4118122805858 T^{10} - 26485323398931167 T^{12} - 9489032379064798993 T^{14} +$$$$61\!\cdots\!61$$$$T^{16}$$
$29$ $$( 1 + 63 T + 2123 T^{2} + 50400 T^{3} + 1045362 T^{4} + 42386400 T^{5} + 1501557563 T^{6} + 37473869223 T^{7} + 500246412961 T^{8} )^{2}$$
$31$ $$( 1 + 7 T - 1217 T^{2} - 4592 T^{3} + 632146 T^{4} - 4412912 T^{5} - 1123925057 T^{6} + 6212525767 T^{7} + 852891037441 T^{8} )^{2}$$
$37$ $$( 1 - 2588 T^{2} + 4206246 T^{4} - 4850328668 T^{6} + 3512479453921 T^{8} )^{2}$$
$41$ $$( 1 - 18 T + 1913 T^{2} - 32490 T^{3} + 613812 T^{4} - 54615690 T^{5} + 5405680793 T^{6} - 85501876338 T^{7} + 7984925229121 T^{8} )^{2}$$
$43$ $$( 1 + 3169 T^{2} + 6623760 T^{4} + 10834180369 T^{6} + 11688200277601 T^{8} )^{2}$$
$47$ $$1 - 7297 T^{2} + 30662449 T^{4} - 93579632206 T^{6} + 226390414696510 T^{8} - 456638753262606286 T^{10} +$$$$73\!\cdots\!89$$$$T^{12} -$$$$84\!\cdots\!77$$$$T^{14} +$$$$56\!\cdots\!21$$$$T^{16}$$
$53$ $$( 1 + 7204 T^{2} + 26018214 T^{4} + 56843025124 T^{6} + 62259690411361 T^{8} )^{2}$$
$59$ $$( 1 - 126 T + 11993 T^{2} - 844326 T^{3} + 51207492 T^{4} - 2939098806 T^{5} + 145323510473 T^{6} - 5314747238766 T^{7} + 146830437604321 T^{8} )^{2}$$
$61$ $$( 1 - 41 T - 5513 T^{2} + 10168 T^{3} + 31652794 T^{4} + 37835128 T^{5} - 76332121433 T^{6} - 2112335348801 T^{7} + 191707312997281 T^{8} )^{2}$$
$67$ $$1 + 5882 T^{2} + 21614089 T^{4} - 160686869974 T^{6} - 934674659122220 T^{8} - 3238020559957340854 T^{10} +$$$$87\!\cdots\!49$$$$T^{12} +$$$$48\!\cdots\!02$$$$T^{14} +$$$$16\!\cdots\!81$$$$T^{16}$$
$71$ $$( 1 - 18616 T^{2} + 137194926 T^{4} - 473063853496 T^{6} + 645753531245761 T^{8} )^{2}$$
$73$ $$( 1 - 19055 T^{2} + 146334144 T^{4} - 541128482255 T^{6} + 806460091894081 T^{8} )^{2}$$
$79$ $$( 1 + 83 T - 5459 T^{2} - 11122 T^{3} + 70528774 T^{4} - 69412402 T^{5} - 212628492179 T^{6} + 20176258808243 T^{7} + 1517108809906561 T^{8} )^{2}$$
$83$ $$1 - 26017 T^{2} + 413233729 T^{4} - 4389950344606 T^{6} + 35135533688955790 T^{8} -$$$$20\!\cdots\!26$$$$T^{10} +$$$$93\!\cdots\!89$$$$T^{12} -$$$$27\!\cdots\!37$$$$T^{14} +$$$$50\!\cdots\!81$$$$T^{16}$$
$89$ $$( 1 - 6916 T^{2} + 69013446 T^{4} - 433925338756 T^{6} + 3936588805702081 T^{8} )^{2}$$
$97$ $$1 + 17834 T^{2} + 72889657 T^{4} + 1214554912058 T^{6} + 24110047553298004 T^{8} +$$$$10\!\cdots\!98$$$$T^{10} +$$$$57\!\cdots\!77$$$$T^{12} +$$$$12\!\cdots\!94$$$$T^{14} +$$$$61\!\cdots\!21$$$$T^{16}$$