# Properties

 Label 252.2.n.b Level 252 Weight 2 Character orbit 252.n Analytic conductor 2.012 Analytic rank 0 Dimension 84 CM no Inner twists 4

# Learn more about

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$2.01223013094$$ Analytic rank: $$0$$ Dimension: $$84$$ Relative dimension: $$42$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$84q - q^{2} + q^{4} - 16q^{8} - 14q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$84q - q^{2} + q^{4} - 16q^{8} - 14q^{9} - 18q^{10} + 9q^{12} - 18q^{13} - 25q^{14} - 7q^{16} + 6q^{17} - 13q^{18} + 24q^{20} + 4q^{21} + 6q^{22} + 6q^{24} - 32q^{25} - 30q^{26} - 4q^{28} + 10q^{29} + 14q^{30} + 9q^{32} - 6q^{33} + 24q^{34} - 38q^{36} + 2q^{37} - 6q^{41} + 7q^{42} - 13q^{44} - 18q^{45} + 10q^{46} - 9q^{48} + 2q^{49} - 17q^{50} - 2q^{53} - 42q^{54} - 32q^{56} + 6q^{57} + 26q^{58} + 8q^{60} - 24q^{61} - 8q^{64} + 50q^{65} + 27q^{66} + 18q^{69} - 4q^{70} - 7q^{72} + 30q^{73} + 46q^{74} + 46q^{77} + 15q^{78} + 3q^{80} - 26q^{81} - 18q^{82} + 29q^{84} - 50q^{85} + 18q^{86} - 2q^{88} - 102q^{89} + 39q^{90} + 28q^{92} - 24q^{93} + 3q^{94} - 30q^{96} - 6q^{97} + 21q^{98} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
31.1 −1.41347 0.0457258i −0.962720 1.43985i 1.99582 + 0.129265i 0.724862i 1.29494 + 2.07921i −1.25991 + 2.32651i −2.81513 0.273973i −1.14634 + 2.77235i 0.0331449 1.02457i
31.2 −1.41219 + 0.0755732i −1.69164 + 0.371957i 1.98858 0.213448i 1.05993i 2.36081 0.653118i 0.349222 2.62260i −2.79212 + 0.451713i 2.72330 1.25843i −0.0801021 1.49682i
31.3 −1.39808 0.213015i −0.709178 + 1.58021i 1.90925 + 0.595623i 2.93241i 1.32809 2.05819i −2.24678 + 1.39713i −2.54241 1.23943i −1.99413 2.24130i −0.624647 + 4.09975i
31.4 −1.38558 + 0.283143i 0.385748 + 1.68855i 1.83966 0.784633i 3.94623i −1.01258 2.23040i 2.37341 + 1.16916i −2.32683 + 1.60806i −2.70240 + 1.30271i −1.11734 5.46781i
31.5 −1.33749 + 0.459470i 1.58493 0.698557i 1.57778 1.22907i 2.69249i −1.79887 + 1.66254i 1.51663 + 2.16791i −1.54554 + 2.36882i 2.02404 2.21433i 1.23712 + 3.60119i
31.6 −1.32994 0.480902i 1.67381 + 0.445360i 1.53747 + 1.27914i 0.815110i −2.01189 1.39724i −0.448419 2.60747i −1.42959 2.44055i 2.60331 + 1.49090i −0.391988 + 1.08405i
31.7 −1.26565 0.630973i 1.11108 1.32872i 1.20375 + 1.59718i 2.00240i −2.24463 + 0.980638i 2.64523 + 0.0525529i −0.515742 2.78101i −0.531008 2.95263i 1.26346 2.53433i
31.8 −1.18174 + 0.776846i −0.238791 1.71551i 0.793020 1.83606i 0.121070i 1.61488 + 1.84178i 0.910048 2.48431i 0.489193 + 2.78580i −2.88596 + 0.819299i −0.0940527 0.143073i
31.9 −1.16223 + 0.805738i 1.39154 + 1.03131i 0.701573 1.87291i 0.0505362i −2.44827 0.0774077i −2.64573 0.0112150i 0.693684 + 2.74204i 0.872785 + 2.87023i −0.0407190 0.0587349i
31.10 −1.13008 0.850244i −1.49040 0.882451i 0.554170 + 1.92169i 4.05112i 0.933971 + 2.26444i 2.21649 1.44470i 1.00765 2.64285i 1.44256 + 2.63040i −3.44444 + 4.57810i
31.11 −0.998009 + 1.00199i −1.26661 + 1.18139i −0.00795539 1.99998i 1.65619i 0.0803514 2.44817i 2.45603 + 0.983840i 2.01190 + 1.98803i 0.208620 2.99274i 1.65948 + 1.65289i
31.12 −0.989784 1.01011i −0.187304 + 1.72189i −0.0406542 + 1.99959i 2.71053i 1.92470 1.51511i −1.53391 2.15572i 2.06005 1.93809i −2.92983 0.645035i 2.73794 2.68284i
31.13 −0.964964 1.03385i 0.422109 1.67983i −0.137689 + 1.99525i 0.594537i −2.14401 + 1.18458i −2.59555 + 0.512964i 2.19566 1.78300i −2.64365 1.41814i −0.614662 + 0.573707i
31.14 −0.836845 + 1.14004i −1.73205 0.00306272i −0.599381 1.90807i 3.10969i 1.45295 1.97204i −2.27748 + 1.34650i 2.67687 + 0.913443i 2.99998 + 0.0106095i −3.54517 2.60233i
31.15 −0.795200 1.16947i 0.815267 + 1.52818i −0.735314 + 1.85992i 1.91789i 1.13886 2.16864i 1.85299 + 1.88850i 2.75984 0.619084i −1.67068 + 2.49175i −2.24291 + 1.52511i
31.16 −0.568881 + 1.29475i 1.73205 + 0.00306272i −1.35275 1.47312i 3.10969i −0.989294 + 2.24082i 2.27748 1.34650i 2.67687 0.913443i 2.99998 + 0.0106095i −4.02627 1.76904i
31.17 −0.368742 + 1.36529i 1.26661 1.18139i −1.72806 1.00688i 1.65619i 1.14590 + 2.16493i −2.45603 0.983840i 2.01190 1.98803i 0.208620 2.99274i 2.26119 + 0.610706i
31.18 −0.290988 1.38395i −1.20973 1.23958i −1.83065 + 0.805427i 2.88398i −1.36351 + 2.03491i 2.24777 + 1.39554i 1.64737 + 2.29917i −0.0731219 + 2.99911i 3.99129 0.839202i
31.19 −0.121544 1.40898i 1.71229 0.260902i −1.97045 + 0.342507i 2.56839i −0.575725 2.38087i −0.477445 2.60232i 0.722083 + 2.73470i 2.86386 0.893480i −3.61882 + 0.312174i
31.20 −0.116673 + 1.40939i −1.39154 1.03131i −1.97277 0.328876i 0.0505362i 1.61588 1.84090i 2.64573 + 0.0112150i 0.693684 2.74204i 0.872785 + 2.87023i −0.0712254 0.00589621i
See all 84 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 187.42 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.k odd 6 1 inner
252.n 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.n.b 84
3.b odd 2 1 756.2.n.b 84
4.b odd 2 1 inner 252.2.n.b 84
7.d odd 6 1 252.2.bj.b yes 84
9.c even 3 1 252.2.bj.b yes 84
9.d odd 6 1 756.2.bj.b 84
12.b even 2 1 756.2.n.b 84
21.g even 6 1 756.2.bj.b 84
28.f even 6 1 252.2.bj.b yes 84
36.f odd 6 1 252.2.bj.b yes 84
36.h even 6 1 756.2.bj.b 84
63.k odd 6 1 inner 252.2.n.b 84
63.s even 6 1 756.2.n.b 84
84.j odd 6 1 756.2.bj.b 84
252.n even 6 1 inner 252.2.n.b 84
252.bn odd 6 1 756.2.n.b 84

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{42} + \cdots$$ acting on $$S_{2}^{\mathrm{new}}(252, [\chi])$$.

## Hecke Characteristic Polynomials

There are no characteristic polynomials of Hecke operators in the database