# Properties

 Label 315.2.z.b Level 315 Weight 2 Character orbit 315.z Analytic conductor 2.515 Analytic rank 0 Dimension 80 CM no Inner twists 8

# Learn more about

## Newspace parameters

 Level: $$N$$ = $$315 = 3^{2} \cdot 5 \cdot 7$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 315.z (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.51528766367$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$80q - 52q^{4} - 4q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$80q - 52q^{4} - 4q^{9} - 24q^{11} - 12q^{14} - 34q^{15} - 28q^{16} + 20q^{21} - 22q^{25} - 48q^{29} + 36q^{30} - 40q^{36} + 16q^{39} - 24q^{46} + 20q^{49} - 42q^{50} - 8q^{51} + 24q^{56} + 116q^{60} + 128q^{64} + 90q^{65} - 6q^{70} + 12q^{74} - 32q^{79} + 68q^{81} - 4q^{84} - 2q^{85} + 156q^{86} - 8q^{91} - 24q^{95} - 116q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
104.1 −1.37508 2.38171i −1.72249 0.181752i −2.78170 + 4.81804i 2.23561 + 0.0452693i 1.93568 + 4.35239i 0.187702 2.63908i 9.79992 2.93393 + 0.626130i −2.96633 5.38682i
104.2 −1.37508 2.38171i 1.72249 + 0.181752i −2.78170 + 4.81804i −2.23561 0.0452693i −1.93568 4.35239i 2.37937 + 1.15699i 9.79992 2.93393 + 0.626130i 2.96633 + 5.38682i
104.3 −1.19580 2.07119i −0.122865 + 1.72769i −1.85987 + 3.22139i 0.810771 2.08390i 3.72528 1.81149i −1.82814 + 1.91257i 4.11294 −2.96981 0.424546i −5.28567 + 0.812672i
104.4 −1.19580 2.07119i 0.122865 1.72769i −1.85987 + 3.22139i −0.810771 + 2.08390i −3.72528 + 1.81149i −2.57040 + 0.626932i 4.11294 −2.96981 0.424546i 5.28567 0.812672i
104.5 −1.18498 2.05244i −0.887757 1.48724i −1.80833 + 3.13213i −0.317361 2.21343i −2.00050 + 3.58441i 0.783979 + 2.52693i 3.83142 −1.42378 + 2.64062i −4.16687 + 3.27423i
104.6 −1.18498 2.05244i 0.887757 + 1.48724i −1.80833 + 3.13213i 0.317361 + 2.21343i 2.00050 3.58441i −1.79640 1.94241i 3.83142 −1.42378 + 2.64062i 4.16687 3.27423i
104.7 −1.05869 1.83370i −1.03202 + 1.39102i −1.24164 + 2.15058i −1.87597 1.21685i 3.64330 + 0.419758i 1.87249 1.86916i 1.02327 −0.869874 2.87112i −0.245270 + 4.72824i
104.8 −1.05869 1.83370i 1.03202 1.39102i −1.24164 + 2.15058i 1.87597 + 1.21685i −3.64330 0.419758i 2.55499 0.687047i 1.02327 −0.869874 2.87112i 0.245270 4.72824i
104.9 −0.938003 1.62467i −1.62365 0.603134i −0.759700 + 1.31584i −1.09295 + 1.95076i 0.543094 + 3.20363i 2.27915 + 1.34367i −0.901609 2.27246 + 1.95855i 4.19453 0.0541402i
104.10 −0.938003 1.62467i 1.62365 + 0.603134i −0.759700 + 1.31584i 1.09295 1.95076i −0.543094 3.20363i −0.0240802 2.64564i −0.901609 2.27246 + 1.95855i −4.19453 + 0.0541402i
104.11 −0.804619 1.39364i −1.60125 + 0.660311i −0.294824 + 0.510650i 2.20682 + 0.360486i 2.20863 + 1.70026i −1.06757 + 2.42081i −2.26959 2.12798 2.11464i −1.27326 3.36557i
104.12 −0.804619 1.39364i 1.60125 0.660311i −0.294824 + 0.510650i −2.20682 0.360486i −2.20863 1.70026i −2.63026 0.285863i −2.26959 2.12798 2.11464i 1.27326 + 3.36557i
104.13 −0.639446 1.10755i −0.624436 1.61557i 0.182218 0.315610i 2.10896 0.743174i −1.39004 + 1.72467i −2.47149 0.944330i −3.02386 −2.22016 + 2.01765i −2.17167 1.86056i
104.14 −0.639446 1.10755i 0.624436 + 1.61557i 0.182218 0.315610i −2.10896 + 0.743174i 1.39004 1.72467i −0.417929 + 2.61253i −3.02386 −2.22016 + 2.01765i 2.17167 + 1.86056i
104.15 −0.513079 0.888680i −1.29309 + 1.15236i 0.473499 0.820124i −0.229284 + 2.22428i 1.68753 + 0.557890i −1.40254 2.24341i −3.02409 0.344150 2.98019i 2.09431 0.937473i
104.16 −0.513079 0.888680i 1.29309 1.15236i 0.473499 0.820124i 0.229284 2.22428i −1.68753 0.557890i 1.24158 + 2.33634i −3.02409 0.344150 2.98019i −2.09431 + 0.937473i
104.17 −0.430691 0.745979i −0.0265195 1.73185i 0.629010 1.08948i −2.22929 + 0.173963i −1.28050 + 0.765674i 1.96991 1.76620i −2.80640 −2.99859 + 0.0918554i 1.08991 + 1.58808i
104.18 −0.430691 0.745979i 0.0265195 + 1.73185i 0.629010 1.08948i 2.22929 0.173963i 1.28050 0.765674i 2.51453 0.822893i −2.80640 −2.99859 + 0.0918554i −1.08991 1.58808i
104.19 −0.139034 0.240814i −1.62845 0.590047i 0.961339 1.66509i 0.326583 2.21209i 0.0843182 + 0.474190i 1.72305 2.00776i −1.09077 2.30369 + 1.92172i −0.578109 + 0.228910i
104.20 −0.139034 0.240814i 1.62845 + 0.590047i 0.961339 1.66509i −0.326583 + 2.21209i −0.0843182 0.474190i 2.60030 0.488328i −1.09077 2.30369 + 1.92172i 0.578109 0.228910i
See all 80 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 209.40 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
7.b odd 2 1 inner
9.d odd 6 1 inner
35.c odd 2 1 inner
45.h odd 6 1 inner
63.o even 6 1 inner
315.z even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 315.2.z.b 80
3.b odd 2 1 945.2.z.b 80
5.b even 2 1 inner 315.2.z.b 80
7.b odd 2 1 inner 315.2.z.b 80
9.c even 3 1 945.2.z.b 80
9.d odd 6 1 inner 315.2.z.b 80
15.d odd 2 1 945.2.z.b 80
21.c even 2 1 945.2.z.b 80
35.c odd 2 1 inner 315.2.z.b 80
45.h odd 6 1 inner 315.2.z.b 80
45.j even 6 1 945.2.z.b 80
63.l odd 6 1 945.2.z.b 80
63.o even 6 1 inner 315.2.z.b 80
105.g even 2 1 945.2.z.b 80
315.z even 6 1 inner 315.2.z.b 80
315.bg odd 6 1 945.2.z.b 80

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
315.2.z.b 80 1.a even 1 1 trivial
315.2.z.b 80 5.b even 2 1 inner
315.2.z.b 80 7.b odd 2 1 inner
315.2.z.b 80 9.d odd 6 1 inner
315.2.z.b 80 35.c odd 2 1 inner
315.2.z.b 80 45.h odd 6 1 inner
315.2.z.b 80 63.o even 6 1 inner
315.2.z.b 80 315.z even 6 1 inner
945.2.z.b 80 3.b odd 2 1
945.2.z.b 80 9.c even 3 1
945.2.z.b 80 15.d odd 2 1
945.2.z.b 80 21.c even 2 1
945.2.z.b 80 45.j even 6 1
945.2.z.b 80 63.l odd 6 1
945.2.z.b 80 105.g even 2 1
945.2.z.b 80 315.bg odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{40} + \cdots$$ acting on $$S_{2}^{\mathrm{new}}(315, [\chi])$$.

## Hecke Characteristic Polynomials

There are no characteristic polynomials of Hecke operators in the database