# Properties

 Label 315.2.bz.d Level 315 Weight 2 Character orbit 315.bz Analytic conductor 2.515 Analytic rank 0 Dimension 32 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.bz (of order $$12$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.51528766367$$ Analytic rank: $$0$$ Dimension: $$32$$ Relative dimension: $$8$$ over $$\Q(\zeta_{12})$$ Coefficient ring index: multiple of None Twist minimal: no (minimal twist has level 105) 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.

 $$\operatorname{Tr}(f)(q) =$$ $$32q + 12q^{5} + 8q^{7} + 24q^{8} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$32q + 12q^{5} + 8q^{7} + 24q^{8} - 12q^{10} + 8q^{11} - 8q^{22} + 8q^{23} + 12q^{25} - 24q^{26} - 24q^{28} + 24q^{31} - 24q^{32} - 44q^{35} + 4q^{37} - 12q^{38} + 12q^{40} + 40q^{43} - 40q^{46} + 60q^{47} - 72q^{50} - 108q^{52} + 24q^{53} + 48q^{56} + 4q^{58} - 24q^{61} + 4q^{65} + 8q^{67} - 132q^{68} + 4q^{70} + 16q^{71} + 36q^{73} - 60q^{77} + 12q^{80} + 12q^{82} - 72q^{85} + 16q^{86} - 32q^{88} - 24q^{91} + 56q^{92} + 12q^{95} + 72q^{98} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
73.1 −2.17399 0.582519i 0 2.65485 + 1.53278i 1.96540 + 1.06640i 0 −0.660211 + 2.56205i −1.69581 1.69581i 0 −3.65156 3.46322i
73.2 −1.72112 0.461174i 0 1.01754 + 0.587476i 1.95031 1.09375i 0 1.68076 2.04329i 1.03952 + 1.03952i 0 −3.86114 + 0.983040i
73.3 −1.49657 0.401003i 0 0.346853 + 0.200256i −1.61490 + 1.54664i 0 −2.44171 + 1.01885i 1.75234 + 1.75234i 0 3.03701 1.66707i
73.4 −0.648264 0.173702i 0 −1.34198 0.774791i −2.13259 + 0.672361i 0 2.57939 + 0.588837i 1.68450 + 1.68450i 0 1.49927 0.0654328i
73.5 0.394487 + 0.105703i 0 −1.58760 0.916603i 2.18897 + 0.456535i 0 −0.605712 2.57548i −1.10697 1.10697i 0 0.815263 + 0.411477i
73.6 0.969545 + 0.259789i 0 −0.859523 0.496246i −0.803857 2.08658i 0 2.42328 + 1.06195i −2.12394 2.12394i 0 −0.237305 2.23187i
73.7 2.24814 + 0.602389i 0 2.95923 + 1.70851i 2.22726 0.198269i 0 −2.59417 0.519864i 2.33208 + 2.33208i 0 5.12664 + 0.895939i
73.8 2.42777 + 0.650518i 0 3.73883 + 2.15861i −0.780598 2.09539i 0 1.61838 2.09305i 4.11829 + 4.11829i 0 −0.532020 5.59492i
82.1 −2.17399 + 0.582519i 0 2.65485 1.53278i 1.96540 1.06640i 0 −0.660211 2.56205i −1.69581 + 1.69581i 0 −3.65156 + 3.46322i
82.2 −1.72112 + 0.461174i 0 1.01754 0.587476i 1.95031 + 1.09375i 0 1.68076 + 2.04329i 1.03952 1.03952i 0 −3.86114 0.983040i
82.3 −1.49657 + 0.401003i 0 0.346853 0.200256i −1.61490 1.54664i 0 −2.44171 1.01885i 1.75234 1.75234i 0 3.03701 + 1.66707i
82.4 −0.648264 + 0.173702i 0 −1.34198 + 0.774791i −2.13259 0.672361i 0 2.57939 0.588837i 1.68450 1.68450i 0 1.49927 + 0.0654328i
82.5 0.394487 0.105703i 0 −1.58760 + 0.916603i 2.18897 0.456535i 0 −0.605712 + 2.57548i −1.10697 + 1.10697i 0 0.815263 0.411477i
82.6 0.969545 0.259789i 0 −0.859523 + 0.496246i −0.803857 + 2.08658i 0 2.42328 1.06195i −2.12394 + 2.12394i 0 −0.237305 + 2.23187i
82.7 2.24814 0.602389i 0 2.95923 1.70851i 2.22726 + 0.198269i 0 −2.59417 + 0.519864i 2.33208 2.33208i 0 5.12664 0.895939i
82.8 2.42777 0.650518i 0 3.73883 2.15861i −0.780598 + 2.09539i 0 1.61838 + 2.09305i 4.11829 4.11829i 0 −0.532020 + 5.59492i
208.1 −0.650518 2.42777i 0 −3.73883 + 2.15861i 1.42436 1.72371i 0 2.09305 1.61838i 4.11829 + 4.11829i 0 −5.11135 2.33671i
208.2 −0.602389 2.24814i 0 −2.95923 + 1.70851i 1.28534 + 1.82973i 0 0.519864 + 2.59417i 2.33208 + 2.33208i 0 3.33923 3.99183i
208.3 −0.259789 0.969545i 0 0.859523 0.496246i 1.40510 1.73945i 0 −1.06195 2.42328i −2.12394 2.12394i 0 −2.05151 0.910421i
208.4 −0.105703 0.394487i 0 1.58760 0.916603i 0.699113 + 2.12397i 0 2.57548 + 0.605712i −1.10697 1.10697i 0 0.763981 0.500300i
See all 32 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 262.8 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
7.d odd 6 1 inner
35.k 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.bz.d 32
3.b odd 2 1 105.2.u.a 32
5.c odd 4 1 inner 315.2.bz.d 32
7.d odd 6 1 inner 315.2.bz.d 32
15.d odd 2 1 525.2.bc.e 32
15.e even 4 1 105.2.u.a 32
15.e even 4 1 525.2.bc.e 32
21.c even 2 1 735.2.v.b 32
21.g even 6 1 105.2.u.a 32
21.g even 6 1 735.2.m.c 32
21.h odd 6 1 735.2.m.c 32
21.h odd 6 1 735.2.v.b 32
35.k even 12 1 inner 315.2.bz.d 32
105.k odd 4 1 735.2.v.b 32
105.p even 6 1 525.2.bc.e 32
105.w odd 12 1 105.2.u.a 32
105.w odd 12 1 525.2.bc.e 32
105.w odd 12 1 735.2.m.c 32
105.x even 12 1 735.2.m.c 32
105.x even 12 1 735.2.v.b 32

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.2.u.a 32 3.b odd 2 1
105.2.u.a 32 15.e even 4 1
105.2.u.a 32 21.g even 6 1
105.2.u.a 32 105.w odd 12 1
315.2.bz.d 32 1.a even 1 1 trivial
315.2.bz.d 32 5.c odd 4 1 inner
315.2.bz.d 32 7.d odd 6 1 inner
315.2.bz.d 32 35.k even 12 1 inner
525.2.bc.e 32 15.d odd 2 1
525.2.bc.e 32 15.e even 4 1
525.2.bc.e 32 105.p even 6 1
525.2.bc.e 32 105.w odd 12 1
735.2.m.c 32 21.g even 6 1
735.2.m.c 32 21.h odd 6 1
735.2.m.c 32 105.w odd 12 1
735.2.m.c 32 105.x even 12 1
735.2.v.b 32 21.c even 2 1
735.2.v.b 32 21.h odd 6 1
735.2.v.b 32 105.k odd 4 1
735.2.v.b 32 105.x even 12 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{32} - \cdots$$ acting on $$S_{2}^{\mathrm{new}}(315, [\chi])$$.

## Hecke Characteristic Polynomials

There are no characteristic polynomials of Hecke operators in the database