# Properties

 Label 252.4.i.a Level $252$ Weight $4$ Character orbit 252.i Analytic conductor $14.868$ Analytic rank $0$ Dimension $48$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$252 = 2^{2} \cdot 3^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 252.i (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$14.8684813214$$ Analytic rank: $$0$$ Dimension: $$48$$ Relative dimension: $$24$$ over $$\Q(\zeta_{3})$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$48q + 20q^{5} - 6q^{7} - 44q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$48q + 20q^{5} - 6q^{7} - 44q^{9} + 4q^{11} - 12q^{13} - 26q^{15} + 112q^{17} + 60q^{19} - 80q^{21} + 10q^{23} - 600q^{25} + 194q^{29} + 60q^{31} - 472q^{33} + 394q^{35} - 84q^{37} + 604q^{39} + 210q^{41} + 42q^{43} + 254q^{45} - 132q^{47} - 78q^{49} - 58q^{51} - 468q^{53} + 612q^{55} + 1476q^{57} - 916q^{59} - 804q^{61} - 444q^{63} + 1656q^{65} - 588q^{67} - 28q^{69} - 2228q^{71} - 336q^{73} - 668q^{75} - 1216q^{77} - 768q^{79} - 104q^{81} + 1024q^{83} + 360q^{85} + 2188q^{87} + 2922q^{89} - 120q^{91} - 1292q^{93} + 2428q^{95} - 264q^{97} - 2246q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
25.1 0 −5.19456 0.128780i 0 −5.68089 + 9.83959i 0 18.4438 + 1.68138i 0 26.9668 + 1.33791i 0
25.2 0 −5.19442 0.133994i 0 7.81356 13.5335i 0 6.71488 17.2601i 0 26.9641 + 1.39204i 0
25.3 0 −4.72803 + 2.15539i 0 4.93620 8.54976i 0 −6.20989 + 17.4481i 0 17.7086 20.3815i 0
25.4 0 −4.58842 2.43852i 0 −6.20694 + 10.7507i 0 −18.4402 + 1.72023i 0 15.1073 + 22.3779i 0
25.5 0 −4.33327 + 2.86754i 0 0.863703 1.49598i 0 −16.5772 8.25811i 0 10.5544 24.8517i 0
25.6 0 −4.01985 3.29253i 0 3.23563 5.60427i 0 4.58917 + 17.9427i 0 5.31845 + 26.4710i 0
25.7 0 −2.96592 + 4.26654i 0 −5.16462 + 8.94539i 0 8.74826 16.3238i 0 −9.40665 25.3084i 0
25.8 0 −2.55972 4.52193i 0 6.02006 10.4271i 0 −8.82278 16.2837i 0 −13.8957 + 23.1497i 0
25.9 0 −1.93223 + 4.82354i 0 −7.79077 + 13.4940i 0 0.496367 + 18.5136i 0 −19.5330 18.6403i 0
25.10 0 −1.41439 4.99995i 0 2.84184 4.92220i 0 15.2960 + 10.4419i 0 −22.9990 + 14.1438i 0
25.11 0 −1.19189 + 5.05761i 0 8.50761 14.7356i 0 18.0631 + 4.08939i 0 −24.1588 12.0562i 0
25.12 0 −0.977623 5.10336i 0 −6.36172 + 11.0188i 0 5.55536 17.6674i 0 −25.0885 + 9.97831i 0
25.13 0 0.340754 + 5.18497i 0 4.29795 7.44428i 0 −11.6662 14.3840i 0 −26.7678 + 3.53359i 0
25.14 0 1.24769 + 5.04413i 0 1.32728 2.29891i 0 −14.0290 + 12.0908i 0 −23.8866 + 12.5870i 0
25.15 0 1.30312 5.03010i 0 4.53303 7.85144i 0 −15.6380 + 9.92235i 0 −23.6038 13.1096i 0
25.16 0 2.05679 4.77175i 0 −7.73619 + 13.3995i 0 10.1757 + 15.4743i 0 −18.5392 19.6290i 0
25.17 0 3.26868 + 4.03928i 0 −0.275008 + 0.476327i 0 18.2696 3.03661i 0 −5.63149 + 26.4062i 0
25.18 0 3.59994 + 3.74705i 0 −10.9363 + 18.9422i 0 −13.2752 12.9139i 0 −1.08080 + 26.9784i 0
25.19 0 3.62259 3.72516i 0 10.2794 17.8044i 0 18.5152 0.432294i 0 −0.753703 26.9895i 0
25.20 0 4.15765 3.11672i 0 −4.98332 + 8.63137i 0 −18.3914 + 2.18061i 0 7.57211 25.9165i 0
See all 48 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 121.24 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.h even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 252.4.i.a 48
3.b odd 2 1 756.4.i.a 48
7.c even 3 1 252.4.l.a yes 48
9.c even 3 1 252.4.l.a yes 48
9.d odd 6 1 756.4.l.a 48
21.h odd 6 1 756.4.l.a 48
63.h even 3 1 inner 252.4.i.a 48
63.j odd 6 1 756.4.i.a 48

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
252.4.i.a 48 1.a even 1 1 trivial
252.4.i.a 48 63.h even 3 1 inner
252.4.l.a yes 48 7.c even 3 1
252.4.l.a yes 48 9.c even 3 1
756.4.i.a 48 3.b odd 2 1
756.4.i.a 48 63.j odd 6 1
756.4.l.a 48 9.d odd 6 1
756.4.l.a 48 21.h odd 6 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{4}^{\mathrm{new}}(252, [\chi])$$.