# Properties

 Label 8100.2.d.c Level $8100$ Weight $2$ Character orbit 8100.d Analytic conductor $64.679$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

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## Newspace parameters

 Level: $$N$$ $$=$$ $$8100 = 2^{2} \cdot 3^{4} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 8100.d (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$64.6788256372$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 36) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + i q^{7} +O(q^{10})$$ $$q + i q^{7} -3 q^{11} -i q^{13} + 6 i q^{17} + 4 q^{19} + 3 i q^{23} + 3 q^{29} + 5 q^{31} -2 i q^{37} -3 q^{41} -i q^{43} -9 i q^{47} + 6 q^{49} + 6 i q^{53} -3 q^{59} -13 q^{61} + 7 i q^{67} + 12 q^{71} -10 i q^{73} -3 i q^{77} -11 q^{79} + 9 i q^{83} + 6 q^{89} + q^{91} -11 i q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + O(q^{10})$$ $$2q - 6q^{11} + 8q^{19} + 6q^{29} + 10q^{31} - 6q^{41} + 12q^{49} - 6q^{59} - 26q^{61} + 24q^{71} - 22q^{79} + 12q^{89} + 2q^{91} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/8100\mathbb{Z}\right)^\times$$.

 $$n$$ $$4051$$ $$6401$$ $$7777$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
649.1
 − 1.00000i 1.00000i
0 0 0 0 0 1.00000i 0 0 0
649.2 0 0 0 0 0 1.00000i 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
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 8100.2.d.c 2
3.b odd 2 1 8100.2.d.h 2
5.b even 2 1 inner 8100.2.d.c 2
5.c odd 4 1 324.2.a.a 1
5.c odd 4 1 8100.2.a.g 1
9.c even 3 2 2700.2.s.b 4
9.d odd 6 2 900.2.s.b 4
15.d odd 2 1 8100.2.d.h 2
15.e even 4 1 324.2.a.c 1
15.e even 4 1 8100.2.a.j 1
20.e even 4 1 1296.2.a.b 1
40.i odd 4 1 5184.2.a.ba 1
40.k even 4 1 5184.2.a.bb 1
45.h odd 6 2 900.2.s.b 4
45.j even 6 2 2700.2.s.b 4
45.k odd 12 2 108.2.e.a 2
45.k odd 12 2 2700.2.i.b 2
45.l even 12 2 36.2.e.a 2
45.l even 12 2 900.2.i.b 2
60.l odd 4 1 1296.2.a.k 1
120.q odd 4 1 5184.2.a.f 1
120.w even 4 1 5184.2.a.e 1
180.v odd 12 2 144.2.i.a 2
180.x even 12 2 432.2.i.c 2
315.bs even 12 2 5292.2.i.a 2
315.bt odd 12 2 5292.2.i.c 2
315.bu odd 12 2 1764.2.i.c 2
315.bv even 12 2 1764.2.i.a 2
315.bw odd 12 2 1764.2.l.a 2
315.bx even 12 2 1764.2.l.c 2
315.cb even 12 2 5292.2.j.a 2
315.cf odd 12 2 1764.2.j.b 2
315.cg even 12 2 5292.2.l.c 2
315.ch odd 12 2 5292.2.l.a 2
360.bo even 12 2 1728.2.i.c 2
360.br even 12 2 576.2.i.f 2
360.bt odd 12 2 576.2.i.e 2
360.bu odd 12 2 1728.2.i.d 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
36.2.e.a 2 45.l even 12 2
108.2.e.a 2 45.k odd 12 2
144.2.i.a 2 180.v odd 12 2
324.2.a.a 1 5.c odd 4 1
324.2.a.c 1 15.e even 4 1
432.2.i.c 2 180.x even 12 2
576.2.i.e 2 360.bt odd 12 2
576.2.i.f 2 360.br even 12 2
900.2.i.b 2 45.l even 12 2
900.2.s.b 4 9.d odd 6 2
900.2.s.b 4 45.h odd 6 2
1296.2.a.b 1 20.e even 4 1
1296.2.a.k 1 60.l odd 4 1
1728.2.i.c 2 360.bo even 12 2
1728.2.i.d 2 360.bu odd 12 2
1764.2.i.a 2 315.bv even 12 2
1764.2.i.c 2 315.bu odd 12 2
1764.2.j.b 2 315.cf odd 12 2
1764.2.l.a 2 315.bw odd 12 2
1764.2.l.c 2 315.bx even 12 2
2700.2.i.b 2 45.k odd 12 2
2700.2.s.b 4 9.c even 3 2
2700.2.s.b 4 45.j even 6 2
5184.2.a.e 1 120.w even 4 1
5184.2.a.f 1 120.q odd 4 1
5184.2.a.ba 1 40.i odd 4 1
5184.2.a.bb 1 40.k even 4 1
5292.2.i.a 2 315.bs even 12 2
5292.2.i.c 2 315.bt odd 12 2
5292.2.j.a 2 315.cb even 12 2
5292.2.l.a 2 315.ch odd 12 2
5292.2.l.c 2 315.cg even 12 2
8100.2.a.g 1 5.c odd 4 1
8100.2.a.j 1 15.e even 4 1
8100.2.d.c 2 1.a even 1 1 trivial
8100.2.d.c 2 5.b even 2 1 inner
8100.2.d.h 2 3.b odd 2 1
8100.2.d.h 2 15.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(8100, [\chi])$$:

 $$T_{7}^{2} + 1$$ $$T_{11} + 3$$ $$T_{29} - 3$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$1 + T^{2}$$
$11$ $$( 3 + T )^{2}$$
$13$ $$1 + T^{2}$$
$17$ $$36 + T^{2}$$
$19$ $$( -4 + T )^{2}$$
$23$ $$9 + T^{2}$$
$29$ $$( -3 + T )^{2}$$
$31$ $$( -5 + T )^{2}$$
$37$ $$4 + T^{2}$$
$41$ $$( 3 + T )^{2}$$
$43$ $$1 + T^{2}$$
$47$ $$81 + T^{2}$$
$53$ $$36 + T^{2}$$
$59$ $$( 3 + T )^{2}$$
$61$ $$( 13 + T )^{2}$$
$67$ $$49 + T^{2}$$
$71$ $$( -12 + T )^{2}$$
$73$ $$100 + T^{2}$$
$79$ $$( 11 + T )^{2}$$
$83$ $$81 + T^{2}$$
$89$ $$( -6 + T )^{2}$$
$97$ $$121 + T^{2}$$
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