# Properties

 Label 504.2.t.d Level 504 Weight 2 Character orbit 504.t Analytic conductor 4.024 Analytic rank 0 Dimension 22 CM no Inner twists 2

# Learn more about

## Newspace parameters

 Level: $$N$$ = $$504 = 2^{3} \cdot 3^{2} \cdot 7$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 504.t (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.02446026187$$ Analytic rank: $$0$$ Dimension: $$22$$ Relative dimension: $$11$$ over $$\Q(\zeta_{3})$$ Coefficient ring index: multiple of None Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

The dimension is sufficiently large that we do not compute an algebraic $$q$$-expansion, but we have computed the trace expansion.

 $$\operatorname{Tr}(f)(q) =$$ $$22q + 2q^{3} - 6q^{5} + 7q^{7} - 8q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$22q + 2q^{3} - 6q^{5} + 7q^{7} - 8q^{9} + 6q^{11} - 3q^{13} - q^{15} + 7q^{17} - q^{19} - 15q^{21} - 4q^{23} + 20q^{25} - 4q^{27} + 9q^{29} - 4q^{31} - 31q^{33} + 14q^{35} + 2q^{37} + 8q^{39} + 16q^{41} + 22q^{45} + 5q^{47} - 15q^{49} + 7q^{51} + 11q^{53} + 22q^{55} + 7q^{57} - 19q^{59} - 13q^{61} + 21q^{63} + 13q^{65} + 26q^{67} - 4q^{69} - 48q^{71} - 35q^{73} - 8q^{75} - 4q^{77} + 10q^{79} - 8q^{81} - 28q^{83} - 20q^{85} + 9q^{87} + 6q^{89} - 37q^{91} - 32q^{93} + 12q^{95} - 29q^{97} - 56q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
193.1 0 −1.61774 0.618811i 0 −1.83657 0 2.45061 + 0.997255i 0 2.23415 + 2.00215i 0
193.2 0 −1.40598 1.01154i 0 3.84095 0 0.676469 2.55781i 0 0.953553 + 2.84442i 0
193.3 0 −1.21966 + 1.22980i 0 −0.481387 0 2.53326 0.763277i 0 −0.0248369 2.99990i 0
193.4 0 −0.849966 1.50916i 0 −1.58188 0 −1.80922 + 1.93047i 0 −1.55512 + 2.56547i 0
193.5 0 0.134843 + 1.72679i 0 −3.43592 0 −1.83889 1.90223i 0 −2.96363 + 0.465691i 0
193.6 0 0.371921 + 1.69165i 0 1.68316 0 0.960133 + 2.46539i 0 −2.72335 + 1.25832i 0
193.7 0 0.444471 1.67405i 0 −3.52959 0 1.16715 2.37440i 0 −2.60489 1.48813i 0
193.8 0 0.577666 1.63288i 0 1.85591 0 −2.60465 0.464545i 0 −2.33261 1.88652i 0
193.9 0 1.34414 1.09237i 0 2.66802 0 1.94471 + 1.79391i 0 0.613444 2.93661i 0
193.10 0 1.51940 + 0.831519i 0 −2.52290 0 −1.07705 + 2.41660i 0 1.61715 + 2.52682i 0
193.11 0 1.70090 + 0.327002i 0 0.340200 0 1.09748 2.40739i 0 2.78614 + 1.11240i 0
457.1 0 −1.61774 + 0.618811i 0 −1.83657 0 2.45061 0.997255i 0 2.23415 2.00215i 0
457.2 0 −1.40598 + 1.01154i 0 3.84095 0 0.676469 + 2.55781i 0 0.953553 2.84442i 0
457.3 0 −1.21966 1.22980i 0 −0.481387 0 2.53326 + 0.763277i 0 −0.0248369 + 2.99990i 0
457.4 0 −0.849966 + 1.50916i 0 −1.58188 0 −1.80922 1.93047i 0 −1.55512 2.56547i 0
457.5 0 0.134843 1.72679i 0 −3.43592 0 −1.83889 + 1.90223i 0 −2.96363 0.465691i 0
457.6 0 0.371921 1.69165i 0 1.68316 0 0.960133 2.46539i 0 −2.72335 1.25832i 0
457.7 0 0.444471 + 1.67405i 0 −3.52959 0 1.16715 + 2.37440i 0 −2.60489 + 1.48813i 0
457.8 0 0.577666 + 1.63288i 0 1.85591 0 −2.60465 + 0.464545i 0 −2.33261 + 1.88652i 0
457.9 0 1.34414 + 1.09237i 0 2.66802 0 1.94471 1.79391i 0 0.613444 + 2.93661i 0
See all 22 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 457.11 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
63.g even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 504.2.t.d yes 22
3.b odd 2 1 1512.2.t.d 22
4.b odd 2 1 1008.2.t.k 22
7.c even 3 1 504.2.q.d 22
9.c even 3 1 504.2.q.d 22
9.d odd 6 1 1512.2.q.c 22
12.b even 2 1 3024.2.t.l 22
21.h odd 6 1 1512.2.q.c 22
28.g odd 6 1 1008.2.q.k 22
36.f odd 6 1 1008.2.q.k 22
36.h even 6 1 3024.2.q.k 22
63.g even 3 1 inner 504.2.t.d yes 22
63.n odd 6 1 1512.2.t.d 22
84.n even 6 1 3024.2.q.k 22
252.o even 6 1 3024.2.t.l 22
252.bl odd 6 1 1008.2.t.k 22

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
504.2.q.d 22 7.c even 3 1
504.2.q.d 22 9.c even 3 1
504.2.t.d yes 22 1.a even 1 1 trivial
504.2.t.d yes 22 63.g even 3 1 inner
1008.2.q.k 22 28.g odd 6 1
1008.2.q.k 22 36.f odd 6 1
1008.2.t.k 22 4.b odd 2 1
1008.2.t.k 22 252.bl odd 6 1
1512.2.q.c 22 9.d odd 6 1
1512.2.q.c 22 21.h odd 6 1
1512.2.t.d 22 3.b odd 2 1
1512.2.t.d 22 63.n odd 6 1
3024.2.q.k 22 36.h even 6 1
3024.2.q.k 22 84.n even 6 1
3024.2.t.l 22 12.b even 2 1
3024.2.t.l 22 252.o even 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{11} + \cdots$$ acting on $$S_{2}^{\mathrm{new}}(504, [\chi])$$.

## Hecke Characteristic Polynomials

There are no characteristic polynomials of Hecke operators in the database