# Properties

 Label 756.2.bf.b Level 756 Weight 2 Character orbit 756.bf Analytic conductor 6.037 Analytic rank 0 Dimension 32 CM no Inner twists 4

# Related objects

## Newspace parameters

 Level: $$N$$ = $$756 = 2^{2} \cdot 3^{3} \cdot 7$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 756.bf (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$6.03669039281$$ Analytic rank: $$0$$ Dimension: $$32$$ Relative dimension: $$16$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$32q + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$32q - 6q^{11} - 17q^{14} - 4q^{16} + 8q^{20} + 2q^{22} + 14q^{25} + 15q^{26} - 13q^{28} + 15q^{32} + 6q^{35} + 4q^{37} - q^{38} - 15q^{40} - 42q^{44} - 9q^{46} - 4q^{47} + 14q^{49} - 9q^{52} + 45q^{56} + 10q^{58} - 16q^{59} - 42q^{64} - 49q^{68} - 33q^{70} + 36q^{73} - 54q^{74} - 15q^{80} - 51q^{82} + 20q^{83} + 16q^{85} + 78q^{86} - 2q^{88} - 27q^{94} + 24q^{95} - 46q^{98} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
271.1 −1.40043 0.196974i 0 1.92240 + 0.551696i −1.71130 0.988019i 0 −2.27191 + 1.35588i −2.58352 1.15127i 0 2.20194 + 1.72073i
271.2 −1.35565 0.402746i 0 1.67559 + 1.09197i 2.11960 + 1.22375i 0 −1.64814 2.06969i −1.83173 2.15517i 0 −2.38058 2.51264i
271.3 −1.19041 + 0.763500i 0 0.834135 1.81775i 1.13024 + 0.652543i 0 2.50492 + 0.851694i 0.394894 + 2.80072i 0 −1.84366 + 0.0861452i
271.4 −1.02662 0.972657i 0 0.107877 + 1.99709i −2.11960 1.22375i 0 1.64814 + 2.06969i 1.83173 2.15517i 0 0.985722 + 3.31796i
271.5 −0.953127 + 1.04477i 0 −0.183097 1.99160i 0.703801 + 0.406340i 0 −2.60258 + 0.476010i 2.25528 + 1.70695i 0 −1.09534 + 0.348018i
271.6 −0.870799 1.11432i 0 −0.483419 + 1.94070i 1.71130 + 0.988019i 0 2.27191 1.35588i 2.58352 1.15127i 0 −0.389228 2.76730i
271.7 −0.370117 + 1.36492i 0 −1.72603 1.01036i −3.53919 2.04335i 0 2.14499 + 1.54888i 2.01790 1.98194i 0 4.09893 4.07444i
271.8 0.0660073 1.41267i 0 −1.99129 0.186493i −1.13024 0.652543i 0 −2.50492 0.851694i −0.394894 + 2.80072i 0 −0.996433 + 1.55358i
271.9 0.145354 + 1.40672i 0 −1.95774 + 0.408946i 3.03704 + 1.75344i 0 0.151085 + 2.64143i −0.859840 2.69456i 0 −2.02516 + 4.52715i
271.10 0.428236 1.34782i 0 −1.63323 1.15437i −0.703801 0.406340i 0 2.60258 0.476010i −2.25528 + 1.70695i 0 −0.849065 + 0.774587i
271.11 0.434237 + 1.34590i 0 −1.62288 + 1.16888i −0.421763 0.243505i 0 −0.250520 2.63386i −2.27790 1.67665i 0 0.144587 0.673389i
271.12 0.996999 1.00299i 0 −0.0119854 1.99996i 3.53919 + 2.04335i 0 −2.14499 1.54888i −2.01790 1.98194i 0 5.57803 1.51256i
271.13 1.10700 + 0.880081i 0 0.450916 + 1.94851i −2.03190 1.17312i 0 −2.03407 + 1.69191i −1.21568 + 2.55385i 0 −1.21688 3.08688i
271.14 1.29094 0.577482i 0 1.33303 1.49098i −3.03704 1.75344i 0 −0.151085 2.64143i 0.859840 2.69456i 0 −4.93321 0.509738i
271.15 1.31567 + 0.518653i 0 1.46200 + 1.36476i 2.03190 + 1.17312i 0 2.03407 1.69191i 1.21568 + 2.55385i 0 2.06488 + 2.59729i
271.16 1.38270 0.296888i 0 1.82372 0.821014i 0.421763 + 0.243505i 0 0.250520 + 2.63386i 2.27790 1.67665i 0 0.655465 + 0.211478i
703.1 −1.40043 + 0.196974i 0 1.92240 0.551696i −1.71130 + 0.988019i 0 −2.27191 1.35588i −2.58352 + 1.15127i 0 2.20194 1.72073i
703.2 −1.35565 + 0.402746i 0 1.67559 1.09197i 2.11960 1.22375i 0 −1.64814 + 2.06969i −1.83173 + 2.15517i 0 −2.38058 + 2.51264i
703.3 −1.19041 0.763500i 0 0.834135 + 1.81775i 1.13024 0.652543i 0 2.50492 0.851694i 0.394894 2.80072i 0 −1.84366 0.0861452i
703.4 −1.02662 + 0.972657i 0 0.107877 1.99709i −2.11960 + 1.22375i 0 1.64814 2.06969i 1.83173 + 2.15517i 0 0.985722 3.31796i
See all 32 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 703.16 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
12.b even 2 1 inner
21.g even 6 1 inner
28.f even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 756.2.bf.b 32
3.b odd 2 1 756.2.bf.c yes 32
4.b odd 2 1 756.2.bf.c yes 32
7.d odd 6 1 756.2.bf.c yes 32
12.b even 2 1 inner 756.2.bf.b 32
21.g even 6 1 inner 756.2.bf.b 32
28.f even 6 1 inner 756.2.bf.b 32
84.j odd 6 1 756.2.bf.c yes 32

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
756.2.bf.b 32 1.a even 1 1 trivial
756.2.bf.b 32 12.b even 2 1 inner
756.2.bf.b 32 21.g even 6 1 inner
756.2.bf.b 32 28.f even 6 1 inner
756.2.bf.c yes 32 3.b odd 2 1
756.2.bf.c yes 32 4.b odd 2 1
756.2.bf.c yes 32 7.d odd 6 1
756.2.bf.c yes 32 84.j odd 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(756, [\chi])$$:

 $$T_{5}^{32} - \cdots$$ $$T_{11}^{16} + \cdots$$ $$T_{19}^{32} + \cdots$$

## Hecke Characteristic Polynomials

There are no characteristic polynomials of Hecke operators in the database