# Properties

 Label 252.2.bi.c Level 252 Weight 2 Character orbit 252.bi Analytic conductor 2.012 Analytic rank 0 Dimension 80 CM no Inner twists 8

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$2.01223013094$$ Analytic rank: $$0$$ Dimension: $$80$$ Relative dimension: $$40$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$80q - 6q^{2} - 2q^{4} - 24q^{8} - 32q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$80q - 6q^{2} - 2q^{4} - 24q^{8} - 32q^{9} + 10q^{14} - 18q^{16} - 22q^{18} + 4q^{21} - 14q^{22} + 32q^{25} + 28q^{28} + 8q^{29} - 16q^{30} - 16q^{32} + 34q^{36} - 16q^{37} - 8q^{42} - 84q^{44} + 24q^{46} - 24q^{49} - 12q^{50} - 48q^{53} + 32q^{56} + 72q^{57} - 14q^{58} + 56q^{60} - 8q^{64} - 40q^{65} - 22q^{70} - 100q^{72} + 64q^{74} - 12q^{77} - 24q^{78} - 104q^{81} + 62q^{84} + 40q^{85} - 52q^{86} + 6q^{88} + 30q^{92} + 48q^{93} + 20q^{98} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
139.1 −1.41265 0.0664123i −0.758625 1.55708i 1.99118 + 0.187635i 2.66647 + 1.53948i 0.968265 + 2.24999i 1.15806 + 2.37885i −2.80038 0.397302i −1.84898 + 2.36247i −3.66455 2.35184i
139.2 −1.41265 0.0664123i 0.758625 + 1.55708i 1.99118 + 0.187635i −2.66647 1.53948i −0.968265 2.24999i −1.48111 2.19233i −2.80038 0.397302i −1.84898 + 2.36247i 3.66455 + 2.35184i
139.3 −1.38071 + 0.306010i −1.38170 + 1.04446i 1.81272 0.845022i 1.82425 + 1.05323i 1.58811 1.86491i 0.655553 2.56325i −2.24425 + 1.72144i 0.818200 2.88627i −2.84106 0.895969i
139.4 −1.38071 + 0.306010i 1.38170 1.04446i 1.81272 0.845022i −1.82425 1.05323i −1.58811 + 1.86491i 2.54762 + 0.713899i −2.24425 + 1.72144i 0.818200 2.88627i 2.84106 + 0.895969i
139.5 −1.33347 + 0.471009i −1.61200 0.633613i 1.55630 1.25616i −3.16954 1.82994i 2.44799 + 0.0856414i −2.26193 + 1.37246i −1.48363 + 2.40808i 2.19707 + 2.04277i 5.08842 + 0.947288i
139.6 −1.33347 + 0.471009i 1.61200 + 0.633613i 1.55630 1.25616i 3.16954 + 1.82994i −2.44799 0.0856414i −2.31955 + 1.27266i −1.48363 + 2.40808i 2.19707 + 2.04277i −5.08842 0.947288i
139.7 −1.28849 0.582915i −1.60547 + 0.649977i 1.32042 + 1.50216i −1.35495 0.782280i 2.44751 + 0.0983632i 2.11799 + 1.58559i −0.825716 2.70522i 2.15506 2.08703i 1.28984 + 1.79778i
139.8 −1.28849 0.582915i 1.60547 0.649977i 1.32042 + 1.50216i 1.35495 + 0.782280i −2.44751 0.0983632i −0.314165 2.62703i −0.825716 2.70522i 2.15506 2.08703i −1.28984 1.79778i
139.9 −1.13617 0.842090i −1.59413 0.677299i 0.581768 + 1.91352i 1.43245 + 0.827025i 1.24086 + 2.11193i −2.30627 1.29658i 0.950366 2.66398i 2.08253 + 2.15941i −0.931077 2.14589i
139.10 −1.13617 0.842090i 1.59413 + 0.677299i 0.581768 + 1.91352i −1.43245 0.827025i −1.24086 2.11193i −0.0302588 + 2.64558i 0.950366 2.66398i 2.08253 + 2.15941i 0.931077 + 2.14589i
139.11 −1.07529 + 0.918559i −0.476707 1.66516i 0.312499 1.97544i 0.0753642 + 0.0435115i 2.04214 + 1.35264i 0.0425387 2.64541i 1.47853 + 2.41122i −2.54550 + 1.58759i −0.121006 + 0.0224389i
139.12 −1.07529 + 0.918559i 0.476707 + 1.66516i 0.312499 1.97544i −0.0753642 0.0435115i −2.04214 1.35264i 2.31226 + 1.28587i 1.47853 + 2.41122i −2.54550 + 1.58759i 0.121006 0.0224389i
139.13 −0.801205 1.16536i −0.420223 + 1.68030i −0.716141 + 1.86739i 2.25662 + 1.30286i 2.29485 0.856554i −2.51837 + 0.811054i 2.74996 0.661599i −2.64683 1.41220i −0.289711 3.67364i
139.14 −0.801205 1.16536i 0.420223 1.68030i −0.716141 + 1.86739i −2.25662 1.30286i −2.29485 + 0.856554i −1.96158 + 1.77545i 2.74996 0.661599i −2.64683 1.41220i 0.289711 + 3.67364i
139.15 −0.608631 1.27655i −0.420223 + 1.68030i −1.25914 + 1.55389i −2.25662 1.30286i 2.40074 0.486250i 1.96158 1.77545i 2.74996 + 0.661599i −2.64683 1.41220i −0.289711 + 3.67364i
139.16 −0.608631 1.27655i 0.420223 1.68030i −1.25914 + 1.55389i 2.25662 + 1.30286i −2.40074 + 0.486250i 2.51837 0.811054i 2.74996 + 0.661599i −2.64683 1.41220i 0.289711 3.67364i
139.17 −0.557437 + 1.29972i −0.677739 + 1.59395i −1.37853 1.44902i −1.39056 0.802839i −1.69389 1.76939i −2.55603 0.683151i 2.65176 0.983961i −2.08134 2.16056i 1.81861 1.35980i
139.18 −0.557437 + 1.29972i 0.677739 1.59395i −1.37853 1.44902i 1.39056 + 0.802839i 1.69389 + 1.76939i −0.686390 + 2.55517i 2.65176 0.983961i −2.08134 2.16056i −1.81861 + 1.35980i
139.19 −0.161186 1.40500i −1.59413 0.677299i −1.94804 + 0.452933i −1.43245 0.827025i −0.694651 + 2.34893i 0.0302588 2.64558i 0.950366 + 2.66398i 2.08253 + 2.15941i −0.931077 + 2.14589i
139.20 −0.161186 1.40500i 1.59413 + 0.677299i −1.94804 + 0.452933i 1.43245 + 0.827025i 0.694651 2.34893i 2.30627 + 1.29658i 0.950366 + 2.66398i 2.08253 + 2.15941i 0.931077 2.14589i
See all 80 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 223.40 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
7.b odd 2 1 inner
9.c even 3 1 inner
28.d even 2 1 inner
36.f odd 6 1 inner
63.l odd 6 1 inner
252.bi 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.bi.c 80
3.b odd 2 1 756.2.bi.c 80
4.b odd 2 1 inner 252.2.bi.c 80
7.b odd 2 1 inner 252.2.bi.c 80
9.c even 3 1 inner 252.2.bi.c 80
9.d odd 6 1 756.2.bi.c 80
12.b even 2 1 756.2.bi.c 80
21.c even 2 1 756.2.bi.c 80
28.d even 2 1 inner 252.2.bi.c 80
36.f odd 6 1 inner 252.2.bi.c 80
36.h even 6 1 756.2.bi.c 80
63.l odd 6 1 inner 252.2.bi.c 80
63.o even 6 1 756.2.bi.c 80
84.h odd 2 1 756.2.bi.c 80
252.s odd 6 1 756.2.bi.c 80
252.bi even 6 1 inner 252.2.bi.c 80

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
252.2.bi.c 80 1.a even 1 1 trivial
252.2.bi.c 80 4.b odd 2 1 inner
252.2.bi.c 80 7.b odd 2 1 inner
252.2.bi.c 80 9.c even 3 1 inner
252.2.bi.c 80 28.d even 2 1 inner
252.2.bi.c 80 36.f odd 6 1 inner
252.2.bi.c 80 63.l odd 6 1 inner
252.2.bi.c 80 252.bi even 6 1 inner
756.2.bi.c 80 3.b odd 2 1
756.2.bi.c 80 9.d odd 6 1
756.2.bi.c 80 12.b even 2 1
756.2.bi.c 80 21.c even 2 1
756.2.bi.c 80 36.h even 6 1
756.2.bi.c 80 63.o even 6 1
756.2.bi.c 80 84.h odd 2 1
756.2.bi.c 80 252.s odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{40} - \cdots$$ acting on $$S_{2}^{\mathrm{new}}(252, [\chi])$$.

## Hecke Characteristic Polynomials

There are no characteristic polynomials of Hecke operators in the database