Properties

 Label 756.2.be.c Level 756 Weight 2 Character orbit 756.be Analytic conductor 6.037 Analytic rank 0 Dimension 28 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.be (of order $$6$$, degree $$2$$, minimal)

Newform invariants

 Self dual: no Analytic conductor: $$6.03669039281$$ Analytic rank: $$0$$ Dimension: $$28$$ Relative dimension: $$14$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$28q - 4q^{4} - 2q^{7} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$28q - 4q^{4} - 2q^{7} - 20q^{10} + 8q^{13} + 12q^{16} + 42q^{19} + 4q^{22} + 6q^{25} - 28q^{28} - 30q^{31} + 24q^{34} + 12q^{37} + 36q^{40} - 12q^{46} - 14q^{49} + 84q^{52} + 28q^{58} + 6q^{61} + 8q^{64} - 24q^{67} + 128q^{70} - 22q^{73} - 48q^{79} - 36q^{82} - 24q^{85} - 16q^{88} - 16q^{91} - 12q^{94} - 4q^{97} + O(q^{100})$$

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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
107.1 −1.39533 + 0.230338i 0 1.89389 0.642794i −0.936239 0.540538i 0 −0.749282 2.53743i −2.49454 + 1.33314i 0 1.43087 + 0.538577i
107.2 −1.38410 0.290278i 0 1.83148 + 0.803549i 2.75822 + 1.59246i 0 −0.944233 + 2.47152i −2.30170 1.64383i 0 −3.35540 3.00478i
107.3 −1.04864 + 0.948869i 0 0.199295 1.99005i −0.479813 0.277020i 0 2.45883 0.976797i 1.67930 + 2.27595i 0 0.766008 0.164785i
107.4 −0.994183 + 1.00578i 0 −0.0232006 1.99987i 1.28692 + 0.743002i 0 −1.36953 + 2.26371i 2.03450 + 1.96490i 0 −2.02673 + 0.555680i
107.5 −0.687910 1.23563i 0 −1.05356 + 1.70000i 0.303357 + 0.175143i 0 −2.63538 0.234036i 2.82533 + 0.132362i 0 0.00772994 0.495319i
107.6 −0.253271 1.39135i 0 −1.87171 + 0.704776i 2.47695 + 1.43007i 0 2.52686 + 0.784194i 1.45464 + 2.42570i 0 1.36239 3.80850i
107.7 −0.109104 + 1.41000i 0 −1.97619 0.307674i 3.44992 + 1.99182i 0 0.212727 2.63719i 0.649430 2.75286i 0 −3.18486 + 4.64707i
107.8 0.109104 1.41000i 0 −1.97619 0.307674i −3.44992 1.99182i 0 0.212727 2.63719i −0.649430 + 2.75286i 0 −3.18486 + 4.64707i
107.9 0.253271 + 1.39135i 0 −1.87171 + 0.704776i −2.47695 1.43007i 0 2.52686 + 0.784194i −1.45464 2.42570i 0 1.36239 3.80850i
107.10 0.687910 + 1.23563i 0 −1.05356 + 1.70000i −0.303357 0.175143i 0 −2.63538 0.234036i −2.82533 0.132362i 0 0.00772994 0.495319i
107.11 0.994183 1.00578i 0 −0.0232006 1.99987i −1.28692 0.743002i 0 −1.36953 + 2.26371i −2.03450 1.96490i 0 −2.02673 + 0.555680i
107.12 1.04864 0.948869i 0 0.199295 1.99005i 0.479813 + 0.277020i 0 2.45883 0.976797i −1.67930 2.27595i 0 0.766008 0.164785i
107.13 1.38410 + 0.290278i 0 1.83148 + 0.803549i −2.75822 1.59246i 0 −0.944233 + 2.47152i 2.30170 + 1.64383i 0 −3.35540 3.00478i
107.14 1.39533 0.230338i 0 1.89389 0.642794i 0.936239 + 0.540538i 0 −0.749282 2.53743i 2.49454 1.33314i 0 1.43087 + 0.538577i
431.1 −1.39533 0.230338i 0 1.89389 + 0.642794i −0.936239 + 0.540538i 0 −0.749282 + 2.53743i −2.49454 1.33314i 0 1.43087 0.538577i
431.2 −1.38410 + 0.290278i 0 1.83148 0.803549i 2.75822 1.59246i 0 −0.944233 2.47152i −2.30170 + 1.64383i 0 −3.35540 + 3.00478i
431.3 −1.04864 0.948869i 0 0.199295 + 1.99005i −0.479813 + 0.277020i 0 2.45883 + 0.976797i 1.67930 2.27595i 0 0.766008 + 0.164785i
431.4 −0.994183 1.00578i 0 −0.0232006 + 1.99987i 1.28692 0.743002i 0 −1.36953 2.26371i 2.03450 1.96490i 0 −2.02673 0.555680i
431.5 −0.687910 + 1.23563i 0 −1.05356 1.70000i 0.303357 0.175143i 0 −2.63538 + 0.234036i 2.82533 0.132362i 0 0.00772994 + 0.495319i
431.6 −0.253271 + 1.39135i 0 −1.87171 0.704776i 2.47695 1.43007i 0 2.52686 0.784194i 1.45464 2.42570i 0 1.36239 + 3.80850i
See all 28 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 431.14 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
28.g odd 6 1 inner
84.n 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.be.c 28
3.b odd 2 1 inner 756.2.be.c 28
4.b odd 2 1 756.2.be.d yes 28
7.c even 3 1 756.2.be.d yes 28
12.b even 2 1 756.2.be.d yes 28
21.h odd 6 1 756.2.be.d yes 28
28.g odd 6 1 inner 756.2.be.c 28
84.n even 6 1 inner 756.2.be.c 28

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

Hecke kernels

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

 $$T_{5}^{28} - \cdots$$ $$T_{19}^{14} - \cdots$$

Hecke Characteristic Polynomials

There are no characteristic polynomials of Hecke operators in the database