# Properties

 Label 336.2.y.a Level 336 Weight 2 Character orbit 336.y Analytic conductor 2.683 Analytic rank 0 Dimension 120 CM no Inner twists 8

# Related objects

## Newspace parameters

 Level: $$N$$ = $$336 = 2^{4} \cdot 3 \cdot 7$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 336.y (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.68297350792$$ Analytic rank: $$0$$ Dimension: $$120$$ Relative dimension: $$60$$ over $$\Q(i)$$ Coefficient ring index: multiple of None Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$120q - 8q^{4} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$120q - 8q^{4} - 8q^{15} + 8q^{16} - 32q^{18} - 8q^{21} + 16q^{22} - 16q^{30} - 8q^{36} - 8q^{37} - 24q^{42} - 8q^{43} - 24q^{46} - 8q^{49} - 56q^{51} + 40q^{58} - 96q^{60} + 24q^{63} - 56q^{64} + 24q^{67} - 80q^{70} - 8q^{72} + 64q^{78} - 16q^{79} - 8q^{81} - 48q^{84} - 48q^{85} - 24q^{88} - 8q^{91} + 8q^{93} - 40q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
125.1 −1.41405 0.0212996i −1.55654 + 0.759733i 1.99909 + 0.0602376i −0.700186 + 0.700186i 2.21721 1.04115i 2.03996 1.68481i −2.82554 0.127759i 1.84561 2.36510i 1.00501 0.975187i
125.2 −1.41405 0.0212996i 1.55654 0.759733i 1.99909 + 0.0602376i 0.700186 0.700186i −2.21721 + 1.04115i −2.03996 1.68481i −2.82554 0.127759i 1.84561 2.36510i −1.00501 + 0.975187i
125.3 −1.37838 + 0.316322i −0.367327 1.69265i 1.79988 0.872025i −0.461355 + 0.461355i 1.04174 + 2.21693i −2.58125 + 0.580663i −2.20509 + 1.77133i −2.73014 + 1.24351i 0.489988 0.781861i
125.4 −1.37838 + 0.316322i 0.367327 + 1.69265i 1.79988 0.872025i 0.461355 0.461355i −1.04174 2.21693i 2.58125 + 0.580663i −2.20509 + 1.77133i −2.73014 + 1.24351i −0.489988 + 0.781861i
125.5 −1.37255 0.340756i −1.24999 1.19897i 1.76777 + 0.935408i 2.22287 2.22287i 1.30712 + 2.07158i 1.16923 2.37337i −2.10760 1.88627i 0.124963 + 2.99740i −3.80846 + 2.29354i
125.6 −1.37255 0.340756i 1.24999 + 1.19897i 1.76777 + 0.935408i −2.22287 + 2.22287i −1.30712 2.07158i −1.16923 2.37337i −2.10760 1.88627i 0.124963 + 2.99740i 3.80846 2.29354i
125.7 −1.28455 0.591550i −1.44974 + 0.947768i 1.30014 + 1.51975i 0.639017 0.639017i 2.42291 0.359865i −1.10656 + 2.40323i −0.771085 2.72129i 1.20347 2.74803i −1.19886 + 0.442839i
125.8 −1.28455 0.591550i 1.44974 0.947768i 1.30014 + 1.51975i −0.639017 + 0.639017i −2.42291 + 0.359865i 1.10656 + 2.40323i −0.771085 2.72129i 1.20347 2.74803i 1.19886 0.442839i
125.9 −1.22270 + 0.710642i −1.68333 0.407913i 0.989977 1.73780i 2.35722 2.35722i 2.34809 0.697491i −0.422823 + 2.61175i 0.0245110 + 2.82832i 2.66721 + 1.37331i −1.20703 + 4.55731i
125.10 −1.22270 + 0.710642i 1.68333 + 0.407913i 0.989977 1.73780i −2.35722 + 2.35722i −2.34809 + 0.697491i 0.422823 + 2.61175i 0.0245110 + 2.82832i 2.66721 + 1.37331i 1.20703 4.55731i
125.11 −1.18025 + 0.779110i −0.970819 + 1.43440i 0.785975 1.83909i −1.77648 + 1.77648i 0.0282512 2.44933i −2.61801 0.382105i 0.505204 + 2.78294i −1.11502 2.78509i 0.712616 3.48076i
125.12 −1.18025 + 0.779110i 0.970819 1.43440i 0.785975 1.83909i 1.77648 1.77648i −0.0282512 + 2.44933i 2.61801 0.382105i 0.505204 + 2.78294i −1.11502 2.78509i −0.712616 + 3.48076i
125.13 −1.15231 0.819870i −1.57036 0.730740i 0.655627 + 1.88949i −2.81393 + 2.81393i 1.21042 + 2.12953i −2.59104 0.535248i 0.793648 2.71480i 1.93204 + 2.29504i 5.54958 0.935458i
125.14 −1.15231 0.819870i 1.57036 + 0.730740i 0.655627 + 1.88949i 2.81393 2.81393i −1.21042 2.12953i 2.59104 0.535248i 0.793648 2.71480i 1.93204 + 2.29504i −5.54958 + 0.935458i
125.15 −0.957005 + 1.04122i −1.52023 0.830004i −0.168282 1.99291i −1.24321 + 1.24321i 2.31908 0.788576i 1.10620 2.40340i 2.23610 + 1.73200i 1.62219 + 2.52359i −0.104698 2.48421i
125.16 −0.957005 + 1.04122i 1.52023 + 0.830004i −0.168282 1.99291i 1.24321 1.24321i −2.31908 + 0.788576i −1.10620 2.40340i 2.23610 + 1.73200i 1.62219 + 2.52359i 0.104698 + 2.48421i
125.17 −0.881724 1.10570i −0.285007 + 1.70844i −0.445126 + 1.94984i −2.11567 + 2.11567i 2.14031 1.19124i 2.51356 + 0.825850i 2.54840 1.22704i −2.83754 0.973835i 4.20472 + 0.473850i
125.18 −0.881724 1.10570i 0.285007 1.70844i −0.445126 + 1.94984i 2.11567 2.11567i −2.14031 + 1.19124i −2.51356 + 0.825850i 2.54840 1.22704i −2.83754 0.973835i −4.20472 0.473850i
125.19 −0.621008 + 1.27057i −0.254609 1.71324i −1.22870 1.57807i −0.914067 + 0.914067i 2.33490 + 0.740435i 0.812405 + 2.51794i 2.76808 0.581151i −2.87035 + 0.872410i −0.593743 1.72903i
125.20 −0.621008 + 1.27057i 0.254609 + 1.71324i −1.22870 1.57807i 0.914067 0.914067i −2.33490 0.740435i −0.812405 + 2.51794i 2.76808 0.581151i −2.87035 + 0.872410i 0.593743 + 1.72903i
See next 80 embeddings (of 120 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 293.60 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
7.b odd 2 1 inner
16.e even 4 1 inner
21.c even 2 1 inner
48.i odd 4 1 inner
112.l odd 4 1 inner
336.y even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 336.2.y.a 120
3.b odd 2 1 inner 336.2.y.a 120
7.b odd 2 1 inner 336.2.y.a 120
16.e even 4 1 inner 336.2.y.a 120
21.c even 2 1 inner 336.2.y.a 120
48.i odd 4 1 inner 336.2.y.a 120
112.l odd 4 1 inner 336.2.y.a 120
336.y even 4 1 inner 336.2.y.a 120

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
336.2.y.a 120 1.a even 1 1 trivial
336.2.y.a 120 3.b odd 2 1 inner
336.2.y.a 120 7.b odd 2 1 inner
336.2.y.a 120 16.e even 4 1 inner
336.2.y.a 120 21.c even 2 1 inner
336.2.y.a 120 48.i odd 4 1 inner
336.2.y.a 120 112.l odd 4 1 inner
336.2.y.a 120 336.y even 4 1 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(336, [\chi])$$.

## Hecke Characteristic Polynomials

There are no characteristic polynomials of Hecke operators in the database