# Properties

 Label 4004.2.m.a Level 4004 Weight 2 Character orbit 4004.m Analytic conductor 31.972 Analytic rank 0 Dimension 2 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ = $$4004 = 2^{2} \cdot 7 \cdot 11 \cdot 13$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 4004.m (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$31.9721009693$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: yes 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.

 $$f(q)$$ $$=$$ $$q + 2 q^{3} + i q^{5} + i q^{7} + q^{9} +O(q^{10})$$ $$q + 2 q^{3} + i q^{5} + i q^{7} + q^{9} -i q^{11} + ( 2 + 3 i ) q^{13} + 2 i q^{15} -4 q^{17} + 5 i q^{19} + 2 i q^{21} -9 q^{23} + 4 q^{25} -4 q^{27} + 5 q^{29} + 7 i q^{31} -2 i q^{33} - q^{35} + 2 i q^{37} + ( 4 + 6 i ) q^{39} -6 i q^{41} -11 q^{43} + i q^{45} -i q^{47} - q^{49} -8 q^{51} - q^{53} + q^{55} + 10 i q^{57} + 4 i q^{59} -6 q^{61} + i q^{63} + ( -3 + 2 i ) q^{65} + 10 i q^{67} -18 q^{69} + 4 i q^{71} + 3 i q^{73} + 8 q^{75} + q^{77} + 11 q^{79} -11 q^{81} -9 i q^{83} -4 i q^{85} + 10 q^{87} -15 i q^{89} + ( -3 + 2 i ) q^{91} + 14 i q^{93} -5 q^{95} + 7 i q^{97} -i q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 4q^{3} + 2q^{9} + O(q^{10})$$ $$2q + 4q^{3} + 2q^{9} + 4q^{13} - 8q^{17} - 18q^{23} + 8q^{25} - 8q^{27} + 10q^{29} - 2q^{35} + 8q^{39} - 22q^{43} - 2q^{49} - 16q^{51} - 2q^{53} + 2q^{55} - 12q^{61} - 6q^{65} - 36q^{69} + 16q^{75} + 2q^{77} + 22q^{79} - 22q^{81} + 20q^{87} - 6q^{91} - 10q^{95} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/4004\mathbb{Z}\right)^\times$$.

 $$n$$ $$365$$ $$925$$ $$2003$$ $$3433$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
2157.1
 − 1.00000i 1.00000i
0 2.00000 0 1.00000i 0 1.00000i 0 1.00000 0
2157.2 0 2.00000 0 1.00000i 0 1.00000i 0 1.00000 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
13.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4004.2.m.a 2
13.b even 2 1 inner 4004.2.m.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4004.2.m.a 2 1.a even 1 1 trivial
4004.2.m.a 2 13.b even 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3} - 2$$ acting on $$S_{2}^{\mathrm{new}}(4004, [\chi])$$.

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ 
$3$ $$( 1 - 2 T + 3 T^{2} )^{2}$$
$5$ $$1 - 9 T^{2} + 25 T^{4}$$
$7$ $$1 + T^{2}$$
$11$ $$1 + T^{2}$$
$13$ $$1 - 4 T + 13 T^{2}$$
$17$ $$( 1 + 4 T + 17 T^{2} )^{2}$$
$19$ $$1 - 13 T^{2} + 361 T^{4}$$
$23$ $$( 1 + 9 T + 23 T^{2} )^{2}$$
$29$ $$( 1 - 5 T + 29 T^{2} )^{2}$$
$31$ $$1 - 13 T^{2} + 961 T^{4}$$
$37$ $$( 1 - 12 T + 37 T^{2} )( 1 + 12 T + 37 T^{2} )$$
$41$ $$1 - 46 T^{2} + 1681 T^{4}$$
$43$ $$( 1 + 11 T + 43 T^{2} )^{2}$$
$47$ $$1 - 93 T^{2} + 2209 T^{4}$$
$53$ $$( 1 + T + 53 T^{2} )^{2}$$
$59$ $$1 - 102 T^{2} + 3481 T^{4}$$
$61$ $$( 1 + 6 T + 61 T^{2} )^{2}$$
$67$ $$1 - 34 T^{2} + 4489 T^{4}$$
$71$ $$1 - 126 T^{2} + 5041 T^{4}$$
$73$ $$1 - 137 T^{2} + 5329 T^{4}$$
$79$ $$( 1 - 11 T + 79 T^{2} )^{2}$$
$83$ $$1 - 85 T^{2} + 6889 T^{4}$$
$89$ $$1 + 47 T^{2} + 7921 T^{4}$$
$97$ $$1 - 145 T^{2} + 9409 T^{4}$$