# Properties

 Label 315.2.u.a Level 315 Weight 2 Character orbit 315.u Analytic conductor 2.515 Analytic rank 0 Dimension 88 CM no Inner twists 4

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$2.51528766367$$ Analytic rank: $$0$$ Dimension: $$88$$ Relative dimension: $$44$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$88q - 38q^{4} - 6q^{5} + 12q^{6} - 6q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$88q - 38q^{4} - 6q^{5} + 12q^{6} - 6q^{9} - 6q^{10} - 12q^{14} - 6q^{15} - 26q^{16} - 12q^{19} + 6q^{20} - 12q^{21} - 42q^{24} - 2q^{25} + 12q^{26} + 6q^{29} - 18q^{30} - 6q^{31} + 12q^{34} - 36q^{36} - 6q^{41} + 84q^{44} - 12q^{45} - 18q^{46} + 10q^{49} + 30q^{50} - 6q^{51} - 48q^{54} - 90q^{56} - 6q^{59} + 54q^{60} + 12q^{61} - 8q^{64} + 54q^{65} + 78q^{66} - 60q^{69} - 30q^{70} + 12q^{75} + 48q^{76} + 8q^{79} + 69q^{80} + 42q^{81} + 120q^{84} - 7q^{85} - 72q^{89} - 33q^{90} + 20q^{91} - 6q^{94} - 93q^{95} - 12q^{96} + 30q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
59.1 −1.35415 + 2.34546i −1.70299 + 0.315967i −2.66745 4.62015i −2.11290 + 0.731888i 1.56501 4.42215i −1.60604 2.10253i 9.03188 2.80033 1.07618i 1.14457 5.94680i
59.2 −1.28800 + 2.23088i 1.68461 + 0.402583i −2.31788 4.01469i 0.488384 + 2.18208i −3.06790 + 3.23965i −2.15932 + 1.52884i 6.78974 2.67585 + 1.35639i −5.49700 1.72099i
59.3 −1.26586 + 2.19253i −0.586821 1.62961i −2.20480 3.81882i 1.14318 1.92175i 4.31581 + 0.776236i −1.80040 + 1.93870i 6.10041 −2.31128 + 1.91258i 2.76640 + 4.93913i
59.4 −1.23016 + 2.13070i 0.771307 1.55083i −2.02658 3.51014i −2.13577 + 0.662193i 2.35553 + 3.55119i 2.39641 + 1.12126i 5.05144 −1.81017 2.39234i 1.21640 5.36528i
59.5 −1.14897 + 1.99008i −1.60485 + 0.651500i −1.64028 2.84105i 1.90726 1.16720i 0.547395 3.94234i 2.44187 + 1.01847i 2.94267 2.15110 2.09112i 0.131420 + 5.13669i
59.6 −1.08083 + 1.87205i −0.102923 + 1.72899i −1.33638 2.31468i −1.85798 1.24416i −3.12551 2.06142i −0.599782 + 2.57687i 1.45428 −2.97881 0.355906i 4.33727 2.13351i
59.7 −1.03259 + 1.78850i −1.10896 1.33049i −1.13248 1.96151i 1.44626 + 1.70539i 3.52468 0.609527i −0.162008 2.64079i 0.547180 −0.540404 + 2.95093i −4.54346 + 0.825662i
59.8 −1.02260 + 1.77120i 1.55827 0.756177i −1.09143 1.89041i −0.00270873 2.23607i −0.254147 + 3.53327i −1.69883 2.02829i 0.373988 1.85639 2.35665i 3.96329 + 2.28181i
59.9 −1.00105 + 1.73387i 0.113913 + 1.72830i −1.00421 1.73934i 0.439789 + 2.19239i −3.11068 1.53261i 2.40459 1.10361i 0.0168431 −2.97405 + 0.393751i −4.24158 1.43216i
59.10 −0.851246 + 1.47440i 1.72149 0.191018i −0.449240 0.778107i 2.23006 + 0.163754i −1.18377 + 2.70076i 2.57388 + 0.612504i −1.87533 2.92702 0.657668i −2.13977 + 3.14861i
59.11 −0.754840 + 1.30742i −0.856257 1.50560i −0.139566 0.241735i −1.87872 1.21261i 2.61479 + 0.0169972i 1.79617 1.94262i −2.59796 −1.53365 + 2.57836i 3.00352 1.54095i
59.12 −0.686620 + 1.18926i 0.795887 + 1.53836i 0.0571066 + 0.0989115i 2.03494 0.926840i −2.37599 0.109754i −2.64486 0.0687428i −2.90332 −1.73313 + 2.44873i −0.294973 + 3.05646i
59.13 −0.665500 + 1.15268i −1.68054 0.419267i 0.114220 + 0.197834i −1.38997 + 1.75157i 1.60168 1.65810i 1.25852 + 2.32726i −2.96605 2.64843 + 1.40919i −1.09398 2.76786i
59.14 −0.603467 + 1.04524i −1.29781 + 1.14703i 0.271654 + 0.470519i −0.171693 2.22947i −0.415733 2.04872i 0.716364 2.54692i −3.06961 0.368633 2.97727i 2.43393 + 1.16595i
59.15 −0.601450 + 1.04174i 0.375988 1.69075i 0.276517 + 0.478941i 1.70560 + 1.44600i 1.53519 + 1.40858i −0.552728 + 2.58737i −3.07104 −2.71727 1.27140i −2.53219 + 0.907096i
59.16 −0.466107 + 0.807321i −1.01831 + 1.40109i 0.565488 + 0.979455i −1.03266 + 1.98333i −0.656489 1.47516i −2.46700 0.955995i −2.91874 −0.926102 2.85348i −1.11985 1.75814i
59.17 −0.413705 + 0.716558i 1.72082 + 0.196965i 0.657696 + 1.13916i −2.23428 + 0.0894268i −0.853047 + 1.15158i −1.49622 + 2.18205i −2.74319 2.92241 + 0.677881i 0.860253 1.63799i
59.18 −0.293005 + 0.507499i 1.21352 + 1.23587i 0.828297 + 1.43465i −0.212444 2.22595i −0.982768 + 0.253746i 2.62667 + 0.317199i −2.14280 −0.0547336 + 2.99950i 1.19192 + 0.544399i
59.19 −0.252360 + 0.437100i 1.57415 0.722528i 0.872629 + 1.51144i −1.06742 + 1.96485i −0.0814357 + 0.870398i 1.93769 1.80149i −1.89030 1.95591 2.27474i −0.589461 0.962416i
59.20 −0.243540 + 0.421824i −0.0380306 1.73163i 0.881376 + 1.52659i −2.06439 0.859249i 0.739706 + 0.405680i −2.62390 0.339358i −1.83276 −2.99711 + 0.131710i 0.865213 0.661546i
See all 88 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 299.44 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.b even 2 1 inner
63.s even 6 1 inner
315.u 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.u.a 88
3.b odd 2 1 945.2.u.a 88
5.b even 2 1 inner 315.2.u.a 88
7.d odd 6 1 315.2.bq.a yes 88
9.c even 3 1 945.2.bq.a 88
9.d odd 6 1 315.2.bq.a yes 88
15.d odd 2 1 945.2.u.a 88
21.g even 6 1 945.2.bq.a 88
35.i odd 6 1 315.2.bq.a yes 88
45.h odd 6 1 315.2.bq.a yes 88
45.j even 6 1 945.2.bq.a 88
63.k odd 6 1 945.2.u.a 88
63.s even 6 1 inner 315.2.u.a 88
105.p even 6 1 945.2.bq.a 88
315.u even 6 1 inner 315.2.u.a 88
315.bn odd 6 1 945.2.u.a 88

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
315.2.u.a 88 1.a even 1 1 trivial
315.2.u.a 88 5.b even 2 1 inner
315.2.u.a 88 63.s even 6 1 inner
315.2.u.a 88 315.u even 6 1 inner
315.2.bq.a yes 88 7.d odd 6 1
315.2.bq.a yes 88 9.d odd 6 1
315.2.bq.a yes 88 35.i odd 6 1
315.2.bq.a yes 88 45.h odd 6 1
945.2.u.a 88 3.b odd 2 1
945.2.u.a 88 15.d odd 2 1
945.2.u.a 88 63.k odd 6 1
945.2.u.a 88 315.bn odd 6 1
945.2.bq.a 88 9.c even 3 1
945.2.bq.a 88 21.g even 6 1
945.2.bq.a 88 45.j even 6 1
945.2.bq.a 88 105.p even 6 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