# Properties

 Label 2700.3.p.b Level 2700 Weight 3 Character orbit 2700.p Analytic conductor 73.570 Analytic rank 0 Dimension 4 CM no Inner twists 2

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$73.5696713773$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\sqrt{-3}, \sqrt{-11})$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$3^{2}$$ 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 $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 2 \beta_{1} + \beta_{2} - \beta_{3} ) q^{7} +O(q^{10})$$ $$q + ( 2 \beta_{1} + \beta_{2} - \beta_{3} ) q^{7} + ( 5 - 7 \beta_{2} - 2 \beta_{3} ) q^{11} + ( -1 + \beta_{1} + 2 \beta_{2} - 2 \beta_{3} ) q^{13} + ( -8 + \beta_{1} - 15 \beta_{2} - \beta_{3} ) q^{17} + ( -\beta_{1} - \beta_{2} - \beta_{3} ) q^{19} + ( 33 - \beta_{1} + 16 \beta_{2} ) q^{23} + ( 14 - 7 \beta_{2} + 7 \beta_{3} ) q^{29} + ( -3 + 3 \beta_{1} + 2 \beta_{2} - 6 \beta_{3} ) q^{31} + ( 14 - 4 \beta_{1} - 4 \beta_{2} - 4 \beta_{3} ) q^{37} + ( 6 + 8 \beta_{1} + 7 \beta_{2} ) q^{41} + ( 23 + 23 \beta_{2} ) q^{43} + ( -12 + 15 \beta_{2} + 3 \beta_{3} ) q^{47} + ( -1 + \beta_{1} + 25 \beta_{2} - 2 \beta_{3} ) q^{49} + ( 8 + 8 \beta_{1} + 24 \beta_{2} - 8 \beta_{3} ) q^{53} + ( -42 - 8 \beta_{1} - 25 \beta_{2} ) q^{59} + ( 22 - 6 \beta_{1} + 19 \beta_{2} + 3 \beta_{3} ) q^{61} + ( -6 + 6 \beta_{1} + 55 \beta_{2} - 12 \beta_{3} ) q^{67} + ( -16 + 2 \beta_{1} - 30 \beta_{2} - 2 \beta_{3} ) q^{71} + ( -20 + 3 \beta_{1} + 3 \beta_{2} + 3 \beta_{3} ) q^{73} + ( -93 + 17 \beta_{1} - 38 \beta_{2} ) q^{77} + ( 44 - 10 \beta_{1} + 39 \beta_{2} + 5 \beta_{3} ) q^{79} + ( -12 + 15 \beta_{2} + 3 \beta_{3} ) q^{83} + ( -64 + 8 \beta_{1} - 120 \beta_{2} - 8 \beta_{3} ) q^{89} + ( -77 - 3 \beta_{1} - 3 \beta_{2} - 3 \beta_{3} ) q^{91} + ( 97 + 4 \beta_{1} + 99 \beta_{2} - 2 \beta_{3} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + q^{7} + O(q^{10})$$ $$4q + q^{7} + 36q^{11} - 5q^{13} + 2q^{19} + 99q^{23} + 63q^{29} - 7q^{31} + 64q^{37} + 18q^{41} + 46q^{43} - 81q^{47} - 51q^{49} - 126q^{59} + 41q^{61} - 116q^{67} - 86q^{73} - 279q^{77} + 83q^{79} - 81q^{83} - 302q^{91} + 196q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} - 2 x^{2} - 3 x + 9$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{3} + 2 \nu^{2} + 16 \nu - 9$$$$)/6$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{3} + 2 \nu^{2} - 2 \nu - 9$$$$)/6$$ $$\beta_{3}$$ $$=$$ $$($$$$-4 \nu^{3} + \nu^{2} + 8 \nu + 12$$$$)/3$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-\beta_{2} + \beta_{1}$$$$)/3$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{3} + 8 \beta_{2} + 8$$$$)/3$$ $$\nu^{3}$$ $$=$$ $$($$$$-2 \beta_{3} + 2 \beta_{1} + 11$$$$)/3$$

## Character values

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1601.1
 −1.18614 + 1.26217i 1.68614 − 0.396143i −1.18614 − 1.26217i 1.68614 + 0.396143i
0 0 0 0 0 −4.05842 + 7.02939i 0 0 0
1601.2 0 0 0 0 0 4.55842 7.89542i 0 0 0
2501.1 0 0 0 0 0 −4.05842 7.02939i 0 0 0
2501.2 0 0 0 0 0 4.55842 + 7.89542i 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.d 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.p.b 4
3.b odd 2 1 900.3.p.a 4
5.b even 2 1 108.3.g.a 4
5.c odd 4 2 2700.3.u.b 8
9.c even 3 1 900.3.p.a 4
9.d odd 6 1 inner 2700.3.p.b 4
15.d odd 2 1 36.3.g.a 4
15.e even 4 2 900.3.u.a 8
20.d odd 2 1 432.3.q.b 4
40.e odd 2 1 1728.3.q.h 4
40.f even 2 1 1728.3.q.g 4
45.h odd 6 1 108.3.g.a 4
45.h odd 6 1 324.3.c.b 4
45.j even 6 1 36.3.g.a 4
45.j even 6 1 324.3.c.b 4
45.k odd 12 2 900.3.u.a 8
45.l even 12 2 2700.3.u.b 8
60.h even 2 1 144.3.q.b 4
120.i odd 2 1 576.3.q.d 4
120.m even 2 1 576.3.q.g 4
180.n even 6 1 432.3.q.b 4
180.n even 6 1 1296.3.e.e 4
180.p odd 6 1 144.3.q.b 4
180.p odd 6 1 1296.3.e.e 4
360.z odd 6 1 576.3.q.g 4
360.bd even 6 1 1728.3.q.h 4
360.bh odd 6 1 1728.3.q.g 4
360.bk even 6 1 576.3.q.d 4

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7}^{4} - T_{7}^{3} + 75 T_{7}^{2} + 74 T_{7} + 5476$$ acting on $$S_{3}^{\mathrm{new}}(2700, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ 1
$7$ $$1 - T - 23 T^{2} + 74 T^{3} - 1874 T^{4} + 3626 T^{5} - 55223 T^{6} - 117649 T^{7} + 5764801 T^{8}$$
$11$ $$1 - 36 T + 683 T^{2} - 9036 T^{3} + 100632 T^{4} - 1093356 T^{5} + 9999803 T^{6} - 63776196 T^{7} + 214358881 T^{8}$$
$13$ $$1 + 5 T - 245 T^{2} - 340 T^{3} + 40114 T^{4} - 57460 T^{5} - 6997445 T^{6} + 24134045 T^{7} + 815730721 T^{8}$$
$17$ $$1 - 769 T^{2} + 298176 T^{4} - 64227649 T^{6} + 6975757441 T^{8}$$
$19$ $$( 1 - T + 648 T^{2} - 361 T^{3} + 130321 T^{4} )^{2}$$
$23$ $$1 - 99 T + 5117 T^{2} - 183150 T^{3} + 4870902 T^{4} - 96886350 T^{5} + 1431946397 T^{6} - 14655553011 T^{7} + 78310985281 T^{8}$$
$29$ $$1 - 63 T + 2123 T^{2} - 50400 T^{3} + 1045362 T^{4} - 42386400 T^{5} + 1501557563 T^{6} - 37473869223 T^{7} + 500246412961 T^{8}$$
$31$ $$1 + 7 T - 1217 T^{2} - 4592 T^{3} + 632146 T^{4} - 4412912 T^{5} - 1123925057 T^{6} + 6212525767 T^{7} + 852891037441 T^{8}$$
$37$ $$( 1 - 32 T + 1806 T^{2} - 43808 T^{3} + 1874161 T^{4} )^{2}$$
$41$ $$1 - 18 T + 1913 T^{2} - 32490 T^{3} + 613812 T^{4} - 54615690 T^{5} + 5405680793 T^{6} - 85501876338 T^{7} + 7984925229121 T^{8}$$
$43$ $$( 1 - 23 T - 1320 T^{2} - 42527 T^{3} + 3418801 T^{4} )^{2}$$
$47$ $$1 + 81 T + 6929 T^{2} + 384102 T^{3} + 22437966 T^{4} + 848481318 T^{5} + 33811309649 T^{6} + 873116441649 T^{7} + 23811286661761 T^{8}$$
$53$ $$1 - 7204 T^{2} + 26018214 T^{4} - 56843025124 T^{6} + 62259690411361 T^{8}$$
$59$ $$1 + 126 T + 11993 T^{2} + 844326 T^{3} + 51207492 T^{4} + 2939098806 T^{5} + 145323510473 T^{6} + 5314747238766 T^{7} + 146830437604321 T^{8}$$
$61$ $$1 - 41 T - 5513 T^{2} + 10168 T^{3} + 31652794 T^{4} + 37835128 T^{5} - 76332121433 T^{6} - 2112335348801 T^{7} + 191707312997281 T^{8}$$
$67$ $$1 + 116 T + 3787 T^{2} + 80156 T^{3} + 12934456 T^{4} + 359820284 T^{5} + 76312295227 T^{6} + 10493172331604 T^{7} + 406067677556641 T^{8}$$
$71$ $$1 - 18616 T^{2} + 137194926 T^{4} - 473063853496 T^{6} + 645753531245761 T^{8}$$
$73$ $$( 1 + 43 T + 10452 T^{2} + 229147 T^{3} + 28398241 T^{4} )^{2}$$
$79$ $$1 - 83 T - 5459 T^{2} + 11122 T^{3} + 70528774 T^{4} + 69412402 T^{5} - 212628492179 T^{6} - 20176258808243 T^{7} + 1517108809906561 T^{8}$$
$83$ $$1 + 81 T + 16289 T^{2} + 1142262 T^{3} + 166474326 T^{4} + 7869042918 T^{5} + 773048590769 T^{6} + 26482170242889 T^{7} + 2252292232139041 T^{8}$$
$89$ $$1 - 6916 T^{2} + 69013446 T^{4} - 433925338756 T^{6} + 3936588805702081 T^{8}$$
$97$ $$1 - 196 T + 10291 T^{2} - 1824172 T^{3} + 341030200 T^{4} - 17163634348 T^{5} + 911054830771 T^{6} - 163262512966084 T^{7} + 7837433594376961 T^{8}$$