# Properties

 Label 4900.2.e.t Level $4900$ Weight $2$ Character orbit 4900.e Analytic conductor $39.127$ Analytic rank $0$ Dimension $6$ CM no Inner twists $2$

# Learn more

## Newspace parameters

 Level: $$N$$ $$=$$ $$4900 = 2^{2} \cdot 5^{2} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4900.e (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$39.1266969904$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: 6.0.4227136.2 Defining polynomial: $$x^{6} + 9 x^{4} + 22 x^{2} + 9$$ Coefficient ring: $$\Z[a_1, \ldots, a_{17}]$$ Coefficient ring index: $$3^{2}$$ Twist minimal: no (minimal twist has level 700) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{3} + ( -2 - \beta_{4} - \beta_{5} ) q^{9} +O(q^{10})$$ $$q + \beta_{1} q^{3} + ( -2 - \beta_{4} - \beta_{5} ) q^{9} + ( 1 + \beta_{5} ) q^{11} + ( -\beta_{1} + 3 \beta_{2} ) q^{13} + ( -3 \beta_{2} - \beta_{3} ) q^{17} + ( 1 - \beta_{4} ) q^{19} + ( -\beta_{1} - \beta_{2} + \beta_{3} ) q^{23} + ( -3 \beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{27} + ( -1 + \beta_{4} + 2 \beta_{5} ) q^{29} + ( -\beta_{4} - 2 \beta_{5} ) q^{31} + ( 2 \beta_{1} + 5 \beta_{2} + \beta_{3} ) q^{33} + ( -3 \beta_{1} - \beta_{2} ) q^{37} + ( 5 + \beta_{4} - 2 \beta_{5} ) q^{39} + ( 3 - \beta_{4} + 3 \beta_{5} ) q^{41} + ( -\beta_{1} - 3 \beta_{2} - \beta_{3} ) q^{43} + ( \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{47} + ( -3 + 6 \beta_{5} ) q^{51} + ( -3 \beta_{1} + 6 \beta_{2} - \beta_{3} ) q^{53} + ( -2 \beta_{1} + 3 \beta_{2} ) q^{57} + ( 2 - \beta_{5} ) q^{59} + ( 6 + \beta_{4} - 2 \beta_{5} ) q^{61} + ( -2 \beta_{1} - 6 \beta_{2} ) q^{67} + ( 8 + \beta_{4} - \beta_{5} ) q^{69} + ( -7 + 2 \beta_{5} ) q^{71} + 4 \beta_{2} q^{73} + ( 1 + \beta_{4} - 3 \beta_{5} ) q^{79} + ( 6 + 5 \beta_{5} ) q^{81} + ( -4 \beta_{1} + 11 \beta_{2} - \beta_{3} ) q^{83} + ( 4 \beta_{1} + 7 \beta_{2} + 2 \beta_{3} ) q^{87} + ( 4 + \beta_{4} + \beta_{5} ) q^{89} + ( -5 \beta_{1} - 7 \beta_{2} - 2 \beta_{3} ) q^{93} + ( \beta_{1} - 6 \beta_{2} + 2 \beta_{3} ) q^{97} + ( -4 - 2 \beta_{4} - 7 \beta_{5} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q - 16 q^{9} + O(q^{10})$$ $$6 q - 16 q^{9} + 8 q^{11} + 4 q^{19} - 6 q^{31} + 28 q^{39} + 22 q^{41} - 6 q^{51} + 10 q^{59} + 34 q^{61} + 48 q^{69} - 38 q^{71} + 2 q^{79} + 46 q^{81} + 28 q^{89} - 42 q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} + 9 x^{4} + 22 x^{2} + 9$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu^{3} + 5 \nu$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{5} + 6 \nu^{3} + 7 \nu$$$$)/3$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{5} + 9 \nu^{3} + 13 \nu$$$$)/3$$ $$\beta_{4}$$ $$=$$ $$-2 \nu^{4} - 9 \nu^{2} - 2$$ $$\beta_{5}$$ $$=$$ $$\nu^{4} + 6 \nu^{2} + 6$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-\beta_{3} + \beta_{2} + \beta_{1}$$$$)/3$$ $$\nu^{2}$$ $$=$$ $$($$$$2 \beta_{5} + \beta_{4} - 10$$$$)/3$$ $$\nu^{3}$$ $$=$$ $$($$$$5 \beta_{3} - 5 \beta_{2} - 2 \beta_{1}$$$$)/3$$ $$\nu^{4}$$ $$=$$ $$-3 \beta_{5} - 2 \beta_{4} + 14$$ $$\nu^{5}$$ $$=$$ $$($$$$-23 \beta_{3} + 32 \beta_{2} + 5 \beta_{1}$$$$)/3$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/4900\mathbb{Z}\right)^\times$$.

 $$n$$ $$101$$ $$1177$$ $$2451$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
2549.1
 − 0.713538i − 1.91223i − 2.19869i 2.19869i 1.91223i 0.713538i
0 3.20440i 0 0 0 0 0 −7.26819 0
2549.2 0 2.56885i 0 0 0 0 0 −3.59899 0
2549.3 0 0.364448i 0 0 0 0 0 2.86718 0
2549.4 0 0.364448i 0 0 0 0 0 2.86718 0
2549.5 0 2.56885i 0 0 0 0 0 −3.59899 0
2549.6 0 3.20440i 0 0 0 0 0 −7.26819 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 2549.6 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 4900.2.e.t 6
5.b even 2 1 inner 4900.2.e.t 6
5.c odd 4 1 4900.2.a.ba 3
5.c odd 4 1 4900.2.a.bc 3
7.b odd 2 1 4900.2.e.s 6
7.c even 3 2 700.2.r.d 12
35.c odd 2 1 4900.2.e.s 6
35.f even 4 1 4900.2.a.bb 3
35.f even 4 1 4900.2.a.bd 3
35.j even 6 2 700.2.r.d 12
35.l odd 12 2 700.2.i.d 6
35.l odd 12 2 700.2.i.e yes 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
700.2.i.d 6 35.l odd 12 2
700.2.i.e yes 6 35.l odd 12 2
700.2.r.d 12 7.c even 3 2
700.2.r.d 12 35.j even 6 2
4900.2.a.ba 3 5.c odd 4 1
4900.2.a.bb 3 35.f even 4 1
4900.2.a.bc 3 5.c odd 4 1
4900.2.a.bd 3 35.f even 4 1
4900.2.e.s 6 7.b odd 2 1
4900.2.e.s 6 35.c odd 2 1
4900.2.e.t 6 1.a even 1 1 trivial
4900.2.e.t 6 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 $$S_{2}^{\mathrm{new}}(4900, [\chi])$$:

 $$T_{3}^{6} + 17 T_{3}^{4} + 70 T_{3}^{2} + 9$$ $$T_{11}^{3} - 4 T_{11}^{2} - 3 T_{11} + 9$$ $$T_{19}^{3} - 2 T_{19}^{2} - 23 T_{19} - 21$$ $$T_{31}^{3} + 3 T_{31}^{2} - 42 T_{31} + 37$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6}$$
$3$ $$9 + 70 T^{2} + 17 T^{4} + T^{6}$$
$5$ $$T^{6}$$
$7$ $$T^{6}$$
$11$ $$( 9 - 3 T - 4 T^{2} + T^{3} )^{2}$$
$13$ $$9 + 217 T^{2} + 38 T^{4} + T^{6}$$
$17$ $$6561 + 1701 T^{2} + 82 T^{4} + T^{6}$$
$19$ $$( -21 - 23 T - 2 T^{2} + T^{3} )^{2}$$
$23$ $$6561 + 1782 T^{2} + 81 T^{4} + T^{6}$$
$29$ $$( -81 - 45 T + T^{3} )^{2}$$
$31$ $$( 37 - 42 T + 3 T^{2} + T^{3} )^{2}$$
$37$ $$22201 + 5757 T^{2} + 162 T^{4} + T^{6}$$
$41$ $$( 873 - 78 T - 11 T^{2} + T^{3} )^{2}$$
$43$ $$5041 + 1758 T^{2} + 93 T^{4} + T^{6}$$
$47$ $$81 + 513 T^{2} + 58 T^{4} + T^{6}$$
$53$ $$301401 + 15597 T^{2} + 226 T^{4} + T^{6}$$
$59$ $$( 9 - 5 T^{2} + T^{3} )^{2}$$
$61$ $$( -1 + 26 T - 17 T^{2} + T^{3} )^{2}$$
$67$ $$5184 + 7120 T^{2} + 200 T^{4} + T^{6}$$
$71$ $$( 45 + 87 T + 19 T^{2} + T^{3} )^{2}$$
$73$ $$( 16 + T^{2} )^{3}$$
$79$ $$( -449 - 118 T - T^{2} + T^{3} )^{2}$$
$83$ $$962361 + 71577 T^{2} + 526 T^{4} + T^{6}$$
$89$ $$( 45 + 39 T - 14 T^{2} + T^{3} )^{2}$$
$97$ $$25 + 474 T^{2} + 261 T^{4} + T^{6}$$
show more
show less