# Properties

 Label 2100.4.k.g Level $2100$ Weight $4$ Character orbit 2100.k Analytic conductor $123.904$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2100 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 2100.k (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$123.904011012$$ 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 84) 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.

 $$f(q)$$ $$=$$ $$q + 3 i q^{3} + 7 i q^{7} -9 q^{9} +O(q^{10})$$ $$q + 3 i q^{3} + 7 i q^{7} -9 q^{9} + 4 q^{11} + 54 i q^{13} + 14 i q^{17} -92 q^{19} -21 q^{21} -152 i q^{23} -27 i q^{27} + 106 q^{29} -144 q^{31} + 12 i q^{33} -158 i q^{37} -162 q^{39} -390 q^{41} -508 i q^{43} + 528 i q^{47} -49 q^{49} -42 q^{51} + 606 i q^{53} -276 i q^{57} + 364 q^{59} + 678 q^{61} -63 i q^{63} -844 i q^{67} + 456 q^{69} -8 q^{71} -422 i q^{73} + 28 i q^{77} -384 q^{79} + 81 q^{81} -548 i q^{83} + 318 i q^{87} -1194 q^{89} -378 q^{91} -432 i q^{93} + 1502 i q^{97} -36 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 18q^{9} + O(q^{10})$$ $$2q - 18q^{9} + 8q^{11} - 184q^{19} - 42q^{21} + 212q^{29} - 288q^{31} - 324q^{39} - 780q^{41} - 98q^{49} - 84q^{51} + 728q^{59} + 1356q^{61} + 912q^{69} - 16q^{71} - 768q^{79} + 162q^{81} - 2388q^{89} - 756q^{91} - 72q^{99} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2100\mathbb{Z}\right)^\times$$.

 $$n$$ $$701$$ $$1051$$ $$1177$$ $$1501$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1849.1
 − 1.00000i 1.00000i
0 3.00000i 0 0 0 7.00000i 0 −9.00000 0
1849.2 0 3.00000i 0 0 0 7.00000i 0 −9.00000 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 2100.4.k.g 2
5.b even 2 1 inner 2100.4.k.g 2
5.c odd 4 1 84.4.a.b 1
5.c odd 4 1 2100.4.a.g 1
15.e even 4 1 252.4.a.a 1
20.e even 4 1 336.4.a.e 1
35.f even 4 1 588.4.a.a 1
35.k even 12 2 588.4.i.h 2
35.l odd 12 2 588.4.i.a 2
40.i odd 4 1 1344.4.a.b 1
40.k even 4 1 1344.4.a.p 1
60.l odd 4 1 1008.4.a.d 1
105.k odd 4 1 1764.4.a.l 1
105.w odd 12 2 1764.4.k.c 2
105.x even 12 2 1764.4.k.n 2
140.j odd 4 1 2352.4.a.v 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
84.4.a.b 1 5.c odd 4 1
252.4.a.a 1 15.e even 4 1
336.4.a.e 1 20.e even 4 1
588.4.a.a 1 35.f even 4 1
588.4.i.a 2 35.l odd 12 2
588.4.i.h 2 35.k even 12 2
1008.4.a.d 1 60.l odd 4 1
1344.4.a.b 1 40.i odd 4 1
1344.4.a.p 1 40.k even 4 1
1764.4.a.l 1 105.k odd 4 1
1764.4.k.c 2 105.w odd 12 2
1764.4.k.n 2 105.x even 12 2
2100.4.a.g 1 5.c odd 4 1
2100.4.k.g 2 1.a even 1 1 trivial
2100.4.k.g 2 5.b even 2 1 inner
2352.4.a.v 1 140.j odd 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(2100, [\chi])$$:

 $$T_{11} - 4$$ $$T_{13}^{2} + 2916$$ $$T_{17}^{2} + 196$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$9 + T^{2}$$
$5$ $$T^{2}$$
$7$ $$49 + T^{2}$$
$11$ $$( -4 + T )^{2}$$
$13$ $$2916 + T^{2}$$
$17$ $$196 + T^{2}$$
$19$ $$( 92 + T )^{2}$$
$23$ $$23104 + T^{2}$$
$29$ $$( -106 + T )^{2}$$
$31$ $$( 144 + T )^{2}$$
$37$ $$24964 + T^{2}$$
$41$ $$( 390 + T )^{2}$$
$43$ $$258064 + T^{2}$$
$47$ $$278784 + T^{2}$$
$53$ $$367236 + T^{2}$$
$59$ $$( -364 + T )^{2}$$
$61$ $$( -678 + T )^{2}$$
$67$ $$712336 + T^{2}$$
$71$ $$( 8 + T )^{2}$$
$73$ $$178084 + T^{2}$$
$79$ $$( 384 + T )^{2}$$
$83$ $$300304 + T^{2}$$
$89$ $$( 1194 + T )^{2}$$
$97$ $$2256004 + T^{2}$$