# Properties

 Label 252.2.o.a Level 252 Weight 2 Character orbit 252.o Analytic conductor 2.012 Analytic rank 0 Dimension 88 CM no Inner twists 4

# Related objects

## Newspace parameters

 Level: $$N$$ = $$252 = 2^{2} \cdot 3^{2} \cdot 7$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 252.o (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.01223013094$$ Analytic rank: $$0$$ Dimension: $$88$$ Relative dimension: $$44$$ over $$\Q(\zeta_{6})$$ Coefficient ring index: multiple of None Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$88q - 3q^{2} + q^{4} - 6q^{6} - 2q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$88q - 3q^{2} + q^{4} - 6q^{6} - 2q^{9} + 2q^{10} + 3q^{12} - 4q^{13} - 3q^{14} + q^{16} + 5q^{18} - 6q^{20} - 6q^{22} - 14q^{24} - 60q^{25} - 6q^{26} - 24q^{29} + 22q^{30} + 27q^{32} - 26q^{33} - 4q^{34} + 2q^{36} - 4q^{37} + 8q^{40} - 12q^{41} - 13q^{42} - 57q^{44} + 42q^{45} - 6q^{46} - 43q^{48} - 2q^{49} + 9q^{50} + 14q^{52} - 22q^{54} - 66q^{56} - 28q^{57} - 10q^{58} + 32q^{60} + 2q^{61} - 8q^{64} + 18q^{65} - 93q^{66} - 6q^{69} + 30q^{70} + 53q^{72} - 4q^{73} - 6q^{76} - 30q^{77} + 55q^{78} + 87q^{80} + 26q^{81} - 4q^{82} - 7q^{84} - 14q^{85} - 18q^{88} + 60q^{89} + 41q^{90} + 24q^{92} - 30q^{93} + 9q^{94} - 20q^{96} - 4q^{97} - 57q^{98} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
95.1 −1.41411 0.0168321i −0.830122 1.52016i 1.99943 + 0.0476050i 1.70405i 1.14830 + 2.16366i 2.51318 + 0.826989i −2.82662 0.100974i −1.62179 + 2.52384i 0.0286827 2.40972i
95.2 −1.40966 + 0.113368i 1.56103 0.750453i 1.97430 0.319620i 1.54547i −2.11545 + 1.23486i 0.867509 2.49949i −2.74686 + 0.674378i 1.87364 2.34296i 0.175206 + 2.17859i
95.3 −1.34631 0.432946i −0.902384 + 1.47841i 1.62512 + 1.16576i 3.26102i 1.85496 1.59972i 0.260575 + 2.63289i −1.68320 2.27307i −1.37140 2.66819i 1.41185 4.39036i
95.4 −1.33435 + 0.468511i −0.316645 + 1.70286i 1.56099 1.25032i 2.87282i −0.375293 2.42057i −2.24795 + 1.39525i −1.49713 + 2.39971i −2.79947 1.07841i 1.34595 + 3.83335i
95.5 −1.33050 + 0.479354i −1.71695 0.228179i 1.54044 1.27556i 1.07889i 2.39378 0.519437i −2.47353 0.938963i −1.43811 + 2.43554i 2.89587 + 0.783546i −0.517168 1.43545i
95.6 −1.31468 0.521159i 1.73204 + 0.00507424i 1.45679 + 1.37032i 0.202205i −2.27444 0.909340i −0.546403 + 2.58871i −1.20106 2.56075i 2.99995 + 0.0175776i 0.105381 0.265836i
95.7 −1.30585 + 0.542910i 0.773010 + 1.54999i 1.41050 1.41792i 2.45799i −1.85094 1.60438i 2.25199 1.38871i −1.07210 + 2.61737i −1.80491 + 2.39631i −1.33447 3.20977i
95.8 −1.28779 0.584462i −0.698604 1.58491i 1.31681 + 1.50533i 3.64364i −0.0266661 + 2.44934i −2.62804 + 0.305637i −0.815968 2.70817i −2.02391 + 2.21445i −2.12957 + 4.69225i
95.9 −1.15005 0.823029i 0.698604 + 1.58491i 0.645248 + 1.89305i 3.64364i 0.500998 2.39771i 2.62804 0.305637i 0.815968 2.70817i −2.02391 + 2.21445i −2.99882 + 4.19038i
95.10 −1.12610 + 0.855508i 0.649599 1.60562i 0.536211 1.92678i 1.62292i 0.642108 + 2.36383i 1.09603 + 2.40805i 1.04455 + 2.62848i −2.15604 2.08602i 1.38842 + 1.82757i
95.11 −1.10868 0.877970i −1.73204 0.00507424i 0.458336 + 1.94677i 0.202205i 1.91582 + 1.52631i 0.546403 2.58871i 1.20106 2.56075i 2.99995 + 0.0175776i 0.177530 0.224181i
95.12 −1.04810 0.949468i 0.902384 1.47841i 0.197022 + 1.99027i 3.26102i −2.34949 + 0.692737i −0.260575 2.63289i 1.68320 2.27307i −1.37140 2.66819i 3.09624 3.41787i
95.13 −1.01599 + 0.983752i 1.71855 0.215869i 0.0644638 1.99896i 3.68619i −1.53366 + 1.90994i −2.28903 + 1.32677i 1.90099 + 2.09434i 2.90680 0.741961i −3.62630 3.74513i
95.14 −0.834187 + 1.14199i −1.27532 + 1.17199i −0.608265 1.90526i 1.08703i −0.274541 2.43406i 2.19009 1.48443i 2.68319 + 0.894712i 0.252880 2.98932i 1.24137 + 0.906782i
95.15 −0.721634 1.21624i 0.830122 + 1.52016i −0.958489 + 1.75536i 1.70405i 1.24984 2.10663i −2.51318 0.826989i 2.82662 0.100974i −1.62179 + 2.52384i 2.07253 1.22970i
95.16 −0.695328 + 1.23147i 1.57321 + 0.724578i −1.03304 1.71255i 3.37581i −1.98619 + 1.43354i −2.01379 1.71600i 2.82726 0.0813731i 1.94997 + 2.27983i 4.15721 + 2.34729i
95.17 −0.612119 + 1.27488i −1.50043 0.865277i −1.25062 1.56075i 1.15027i 2.02156 1.38321i −0.422424 + 2.61181i 2.75529 0.639023i 1.50259 + 2.59658i 1.46646 + 0.704103i
95.18 −0.606652 1.27749i −1.56103 + 0.750453i −1.26395 + 1.54998i 1.54547i 1.90570 + 1.53893i −0.867509 + 2.49949i 2.74686 + 0.674378i 1.87364 2.34296i −1.97432 + 0.937562i
95.19 −0.261434 1.38984i 0.316645 1.70286i −1.86330 + 0.726701i 2.87282i −2.44948 + 0.00509913i 2.24795 1.39525i 1.49713 + 2.39971i −2.79947 1.07841i −3.99275 + 0.751051i
95.20 −0.250116 1.39192i 1.71695 + 0.228179i −1.87488 + 0.696282i 1.07889i −0.111830 2.44694i 2.47353 + 0.938963i 1.43811 + 2.43554i 2.89587 + 0.783546i 1.50172 0.269846i
See all 88 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 191.44 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
63.n odd 6 1 inner
252.o even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 252.2.o.a 88
3.b odd 2 1 756.2.o.a 88
4.b odd 2 1 inner 252.2.o.a 88
7.c even 3 1 252.2.bb.a yes 88
9.c even 3 1 756.2.bb.a 88
9.d odd 6 1 252.2.bb.a yes 88
12.b even 2 1 756.2.o.a 88
21.h odd 6 1 756.2.bb.a 88
28.g odd 6 1 252.2.bb.a yes 88
36.f odd 6 1 756.2.bb.a 88
36.h even 6 1 252.2.bb.a yes 88
63.g even 3 1 756.2.o.a 88
63.n odd 6 1 inner 252.2.o.a 88
84.n even 6 1 756.2.bb.a 88
252.o even 6 1 inner 252.2.o.a 88
252.bl odd 6 1 756.2.o.a 88

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
252.2.o.a 88 1.a even 1 1 trivial
252.2.o.a 88 4.b odd 2 1 inner
252.2.o.a 88 63.n odd 6 1 inner
252.2.o.a 88 252.o even 6 1 inner
252.2.bb.a yes 88 7.c even 3 1
252.2.bb.a yes 88 9.d odd 6 1
252.2.bb.a yes 88 28.g odd 6 1
252.2.bb.a yes 88 36.h even 6 1
756.2.o.a 88 3.b odd 2 1
756.2.o.a 88 12.b even 2 1
756.2.o.a 88 63.g even 3 1
756.2.o.a 88 252.bl odd 6 1
756.2.bb.a 88 9.c even 3 1
756.2.bb.a 88 21.h odd 6 1
756.2.bb.a 88 36.f odd 6 1
756.2.bb.a 88 84.n even 6 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(252, [\chi])$$.

## Hecke Characteristic Polynomials

There are no characteristic polynomials of Hecke operators in the database