# Properties

 Label 315.2.bx.a Level 315 Weight 2 Character orbit 315.bx Analytic conductor 2.515 Analytic rank 0 Dimension 176 CM no Inner twists 4

# Learn more about

## Newspace parameters

 Level: $$N$$ = $$315 = 3^{2} \cdot 5 \cdot 7$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 315.bx (of order $$12$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.51528766367$$ Analytic rank: $$0$$ Dimension: $$176$$ Relative dimension: $$44$$ over $$\Q(\zeta_{12})$$ Coefficient ring index: multiple of None Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$176q - 6q^{2} - 2q^{3} - 24q^{6} - 2q^{7} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$176q - 6q^{2} - 2q^{3} - 24q^{6} - 2q^{7} - 4q^{10} - 22q^{12} - 4q^{13} - 14q^{15} + 68q^{16} - 18q^{17} - 10q^{18} - 12q^{20} + 20q^{21} + 4q^{22} - 4q^{25} - 32q^{27} - 4q^{28} - 20q^{30} + 4q^{31} - 90q^{32} + 32q^{33} + 8q^{36} - 4q^{37} - 36q^{40} - 36q^{41} + 14q^{42} - 4q^{43} - 68q^{45} + 4q^{46} - 6q^{47} + 38q^{48} + 36q^{50} + 20q^{51} - 52q^{52} + 4q^{55} - 96q^{56} + 32q^{57} - 12q^{58} - 74q^{60} - 8q^{61} + 14q^{63} - 78q^{65} - 92q^{66} + 2q^{67} - 42q^{70} - 46q^{72} - 4q^{73} + 54q^{75} - 24q^{76} + 42q^{77} + 54q^{78} + 36q^{80} + 20q^{81} - 8q^{82} - 12q^{83} - 4q^{85} - 28q^{87} + 12q^{88} - 24q^{90} - 16q^{91} + 72q^{92} + 4q^{93} - 66q^{95} - 4q^{97} + 12q^{98} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
2.1 −2.69307 0.721605i 1.50080 0.864638i 4.99984 + 2.88666i 1.13456 + 1.92686i −4.66568 + 1.24554i 1.58627 2.11749i −7.43896 7.43896i 1.50480 2.59530i −1.66500 6.00786i
2.2 −2.59467 0.695240i −0.295616 + 1.70664i 4.51691 + 2.60784i 1.82199 1.29628i 1.95355 4.22264i −2.60972 0.435150i −6.10795 6.10795i −2.82522 1.00902i −5.62870 + 2.09669i
2.3 −2.48085 0.664741i −0.0323957 1.73175i 3.98067 + 2.29824i −0.385521 2.20258i −1.07079 + 4.31774i −0.368703 + 2.61993i −4.71549 4.71549i −2.99790 + 0.112202i −0.507729 + 5.72054i
2.4 −2.41637 0.647465i −1.67924 0.424464i 3.68760 + 2.12903i −2.17829 0.505032i 3.78283 + 2.11291i 0.757887 2.53488i −3.99432 3.99432i 2.63966 + 1.42555i 4.93657 + 2.63071i
2.5 −2.27659 0.610010i −1.55968 + 0.753259i 3.07869 + 1.77748i −0.123575 + 2.23265i 4.01024 0.763441i 0.379075 + 2.61845i −2.59147 2.59147i 1.86520 2.34969i 1.64327 5.00744i
2.6 −2.11283 0.566131i 1.70063 0.328428i 2.41150 + 1.39228i −2.23606 + 0.00531389i −3.77907 0.268867i −1.66975 + 2.05230i −1.21347 1.21347i 2.78427 1.11707i 4.72743 + 1.25468i
2.7 −2.07435 0.555820i 1.52422 + 0.822645i 2.26194 + 1.30593i 0.0290922 2.23588i −2.70453 2.55365i 2.57272 0.617326i −0.929129 0.929129i 1.64651 + 2.50779i −1.30309 + 4.62182i
2.8 −1.95085 0.522728i −1.45630 0.937654i 1.80051 + 1.03953i 2.23387 + 0.0991923i 2.35088 + 2.59047i −1.87306 1.86860i −0.112896 0.112896i 1.24161 + 2.73101i −4.30608 1.36121i
2.9 −1.93868 0.519467i 0.966001 + 1.43765i 1.75657 + 1.01416i −1.10599 + 1.94340i −1.12595 3.28895i −1.57987 2.12227i −0.0401862 0.0401862i −1.13368 + 2.77755i 3.15369 3.19309i
2.10 −1.62572 0.435610i 0.558438 1.63956i 0.721151 + 0.416356i 1.01641 + 1.99171i −1.62207 + 2.42220i −2.57976 + 0.587246i 1.38920 + 1.38920i −2.37629 1.83118i −0.784783 3.68072i
2.11 −1.58231 0.423978i −0.361579 1.69389i 0.591884 + 0.341725i −1.51691 + 1.64286i −0.146042 + 2.83355i 2.64543 0.0411436i 1.52500 + 1.52500i −2.73852 + 1.22495i 3.09675 1.95638i
2.12 −1.47675 0.395694i −0.550393 + 1.64228i 0.292161 + 0.168679i 2.03723 + 0.921779i 1.46263 2.20744i 2.43080 1.04461i 1.79741 + 1.79741i −2.39414 1.80779i −2.64374 2.16736i
2.13 −1.45905 0.390953i −1.73094 + 0.0619698i 0.243946 + 0.140842i 1.00295 1.99852i 2.54977 + 0.586299i 0.279016 + 2.63100i 1.83534 + 1.83534i 2.99232 0.214532i −2.24469 + 2.52385i
2.14 −1.17797 0.315636i 1.34464 1.09176i −0.444064 0.256381i 2.21223 0.325607i −1.92855 + 0.861638i 2.06273 + 1.65685i 2.16684 + 2.16684i 0.616135 2.93605i −2.70872 0.314706i
2.15 −0.983057 0.263409i 1.53646 0.799565i −0.835034 0.482107i −1.82992 1.28507i −1.72104 + 0.381301i −0.244874 2.63439i 2.13319 + 2.13319i 1.72139 2.45699i 1.46042 + 1.74531i
2.16 −0.904631 0.242395i 0.776514 + 1.54823i −0.972449 0.561443i 0.304672 2.21521i −0.327174 1.58880i −1.94982 + 1.78835i 2.06809 + 2.06809i −1.79405 + 2.40445i −0.812573 + 1.93010i
2.17 −0.839010 0.224812i −0.882497 1.49037i −1.07865 0.622761i −1.58499 1.57728i 0.405371 + 1.44883i −2.60439 0.466002i 1.99339 + 1.99339i −1.44240 + 2.63049i 0.975229 + 1.67968i
2.18 −0.816718 0.218839i 0.0813481 + 1.73014i −1.11291 0.642541i −2.21384 + 0.314524i 0.312183 1.43084i 1.44664 + 2.21523i 1.96408 + 1.96408i −2.98676 + 0.281487i 1.87691 + 0.227596i
2.19 −0.580282 0.155486i −1.43439 + 0.970830i −1.41950 0.819548i −1.41555 + 1.73096i 0.983304 0.340327i −1.77488 1.96209i 1.54587 + 1.54587i 1.11498 2.78511i 1.09056 0.784343i
2.20 −0.534455 0.143207i 1.61596 + 0.623443i −1.46692 0.846925i 2.22680 + 0.203348i −0.774376 0.564618i −1.18425 2.36592i 1.44521 + 1.44521i 2.22264 + 2.01491i −1.16101 0.427574i
See next 80 embeddings (of 176 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 158.44 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.c odd 4 1 inner
63.n odd 6 1 inner
315.bx even 12 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 315.2.bx.a yes 176
3.b odd 2 1 945.2.ca.a 176
5.c odd 4 1 inner 315.2.bx.a yes 176
7.c even 3 1 315.2.bv.a 176
9.c even 3 1 945.2.by.a 176
9.d odd 6 1 315.2.bv.a 176
15.e even 4 1 945.2.ca.a 176
21.h odd 6 1 945.2.by.a 176
35.l odd 12 1 315.2.bv.a 176
45.k odd 12 1 945.2.by.a 176
45.l even 12 1 315.2.bv.a 176
63.g even 3 1 945.2.ca.a 176
63.n odd 6 1 inner 315.2.bx.a yes 176
105.x even 12 1 945.2.by.a 176
315.bx even 12 1 inner 315.2.bx.a yes 176
315.ch odd 12 1 945.2.ca.a 176

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
315.2.bv.a 176 7.c even 3 1
315.2.bv.a 176 9.d odd 6 1
315.2.bv.a 176 35.l odd 12 1
315.2.bv.a 176 45.l even 12 1
315.2.bx.a yes 176 1.a even 1 1 trivial
315.2.bx.a yes 176 5.c odd 4 1 inner
315.2.bx.a yes 176 63.n odd 6 1 inner
315.2.bx.a yes 176 315.bx even 12 1 inner
945.2.by.a 176 9.c even 3 1
945.2.by.a 176 21.h odd 6 1
945.2.by.a 176 45.k odd 12 1
945.2.by.a 176 105.x even 12 1
945.2.ca.a 176 3.b odd 2 1
945.2.ca.a 176 15.e even 4 1
945.2.ca.a 176 63.g even 3 1
945.2.ca.a 176 315.ch odd 12 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(315, [\chi])$$.

## Hecke Characteristic Polynomials

There are no characteristic polynomials of Hecke operators in the database