# Properties

 Label 315.2.cc.a Level 315 Weight 2 Character orbit 315.cc Analytic conductor 2.515 Analytic rank 0 Dimension 144 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.cc (of order $$12$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.51528766367$$ Analytic rank: $$0$$ Dimension: $$144$$ Relative dimension: $$36$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$144q + 4q^{3} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$144q + 4q^{3} - 12q^{11} - 16q^{12} + 16q^{15} + 72q^{16} - 64q^{18} - 48q^{20} - 4q^{21} - 24q^{23} - 12q^{25} - 32q^{27} - 20q^{30} - 60q^{32} - 16q^{33} - 16q^{36} - 24q^{37} + 72q^{38} + 48q^{41} - 40q^{42} + 40q^{45} - 48q^{46} + 12q^{47} + 104q^{48} - 76q^{51} - 24q^{55} - 4q^{57} - 92q^{60} + 8q^{63} - 72q^{65} - 80q^{66} - 12q^{67} - 64q^{72} - 108q^{75} - 24q^{76} + 72q^{78} + 32q^{81} - 96q^{82} + 120q^{83} - 48q^{85} - 144q^{86} + 116q^{87} - 48q^{88} + 252q^{90} + 24q^{91} + 156q^{92} - 44q^{93} + 120q^{95} - 96q^{96} - 60q^{97} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
92.1 −0.713154 2.66153i 1.69514 + 0.355654i −4.84309 + 2.79616i 1.19095 + 1.89252i −0.262317 4.76530i 0.965926 0.258819i 6.99916 + 6.99916i 2.74702 + 1.20577i 4.18768 4.51940i
92.2 −0.679005 2.53408i 0.297629 + 1.70629i −4.22847 + 2.44131i 0.0454078 2.23561i 4.12178 1.91279i −0.965926 + 0.258819i 5.34747 + 5.34747i −2.82283 + 1.01568i −5.69604 + 1.40292i
92.3 −0.646610 2.41318i −1.26505 + 1.18307i −3.67328 + 2.12077i −2.08010 + 0.820487i 3.67295 + 2.28781i 0.965926 0.258819i 3.95984 + 3.95984i 0.200708 2.99328i 3.32499 + 4.48911i
92.4 −0.626308 2.33741i −1.18606 1.26224i −3.33919 + 1.92788i −0.366266 + 2.20587i −2.20755 + 3.56287i −0.965926 + 0.258819i 3.17541 + 3.17541i −0.186522 + 2.99420i 5.38542 0.525436i
92.5 −0.616561 2.30104i 1.18411 1.26408i −3.18258 + 1.83746i −2.21436 + 0.310826i −3.63877 1.94529i −0.965926 + 0.258819i 2.82137 + 2.82137i −0.195791 2.99360i 2.08051 + 4.90368i
92.6 −0.540409 2.01684i 1.29595 1.14913i −2.04353 + 1.17983i −0.0970501 2.23396i −3.01795 1.99273i 0.965926 0.258819i 0.531022 + 0.531022i 0.358997 2.97844i −4.45308 + 1.40299i
92.7 −0.522977 1.95178i 0.00508642 1.73204i −1.80388 + 1.04147i 2.08954 + 0.796124i −3.38322 + 0.895892i 0.965926 0.258819i 0.118505 + 0.118505i −2.99995 0.0176198i 0.461075 4.49468i
92.8 −0.484811 1.80934i 0.413768 + 1.68190i −1.30661 + 0.754373i 1.61436 + 1.54720i 2.84253 1.56405i −0.965926 + 0.258819i −0.650680 0.650680i −2.65759 + 1.39183i 2.01675 3.67103i
92.9 −0.372518 1.39026i −1.46969 + 0.916517i −0.0619937 + 0.0357921i −0.217588 + 2.22546i 1.82168 + 1.70183i −0.965926 + 0.258819i −1.96262 1.96262i 1.31999 2.69400i 3.17501 0.526519i
92.10 −0.352695 1.31628i −0.672895 1.59600i 0.123861 0.0715111i −0.441766 2.19200i −1.86345 + 1.44862i −0.965926 + 0.258819i −2.06498 2.06498i −2.09442 + 2.14788i −2.72946 + 1.35459i
92.11 −0.303557 1.13289i −0.471331 + 1.66669i 0.540756 0.312205i −0.436585 2.19303i 2.03125 + 0.0280322i 0.965926 0.258819i −2.17651 2.17651i −2.55569 1.57112i −2.35194 + 1.16031i
92.12 −0.292475 1.09153i −1.63819 0.562423i 0.626149 0.361507i −2.16876 0.544513i −0.134772 + 1.95264i 0.965926 0.258819i −2.17584 2.17584i 2.36736 + 1.84272i 0.0399539 + 2.52653i
92.13 −0.270870 1.01090i 1.36595 + 1.06499i 0.783502 0.452355i 2.05901 0.872052i 0.706600 1.66931i 0.965926 0.258819i −2.14957 2.14957i 0.731614 + 2.90942i −1.43928 1.84524i
92.14 −0.231820 0.865164i 1.63235 0.579155i 1.03728 0.598875i −0.767760 + 2.10013i −0.879477 1.27799i 0.965926 0.258819i −2.02528 2.02528i 2.32916 1.89077i 1.99494 + 0.177387i
92.15 −0.220022 0.821133i 0.953832 1.44575i 1.10620 0.638666i 1.96491 + 1.06730i −1.39702 0.465126i −0.965926 + 0.258819i −1.97004 1.97004i −1.18041 2.75801i 0.444073 1.84828i
92.16 −0.165403 0.617291i −1.49090 + 0.881605i 1.37836 0.795797i 1.71710 1.43233i 0.790805 + 0.774498i −0.965926 + 0.258819i −1.62300 1.62300i 1.44555 2.62876i −1.16818 0.823041i
92.17 −0.0549668 0.205139i 1.73204 0.00719422i 1.69299 0.977448i −1.03445 1.98240i −0.0966803 0.354913i −0.965926 + 0.258819i −0.593915 0.593915i 2.99990 0.0249213i −0.349807 + 0.321173i
92.18 −0.0242815 0.0906199i −0.235889 + 1.71591i 1.72443 0.995599i −0.436480 + 2.19305i 0.161224 0.0202887i 0.965926 0.258819i −0.264770 0.264770i −2.88871 0.809530i 0.209333 0.0136969i
92.19 0.0573303 + 0.213960i −1.05605 1.37287i 1.68956 0.975467i 1.86575 1.23247i 0.233194 0.304659i 0.965926 0.258819i 0.618832 + 0.618832i −0.769518 + 2.89963i 0.370664 + 0.328536i
92.20 0.0867243 + 0.323660i −1.73132 + 0.0504811i 1.63482 0.943862i −2.20059 + 0.396767i −0.166486 0.555979i −0.965926 + 0.258819i 0.921139 + 0.921139i 2.99490 0.174797i −0.319262 0.677831i
See next 80 embeddings (of 144 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 302.36 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
9.d odd 6 1 inner
45.l 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.cc.a 144
3.b odd 2 1 945.2.cf.a 144
5.c odd 4 1 inner 315.2.cc.a 144
9.c even 3 1 945.2.cf.a 144
9.d odd 6 1 inner 315.2.cc.a 144
15.e even 4 1 945.2.cf.a 144
45.k odd 12 1 945.2.cf.a 144
45.l even 12 1 inner 315.2.cc.a 144

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
315.2.cc.a 144 1.a even 1 1 trivial
315.2.cc.a 144 5.c odd 4 1 inner
315.2.cc.a 144 9.d odd 6 1 inner
315.2.cc.a 144 45.l even 12 1 inner
945.2.cf.a 144 3.b odd 2 1
945.2.cf.a 144 9.c even 3 1
945.2.cf.a 144 15.e even 4 1
945.2.cf.a 144 45.k 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