# Properties

 Label 315.2.t.b Level 315 Weight 2 Character orbit 315.t Analytic conductor 2.515 Analytic rank 0 Dimension 30 CM no Inner twists 2

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$2.51528766367$$ Analytic rank: $$0$$ Dimension: $$30$$ Relative dimension: $$15$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$30q + 4q^{3} - 30q^{4} + 15q^{5} - q^{6} - 3q^{7} - 2q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$30q + 4q^{3} - 30q^{4} + 15q^{5} - q^{6} - 3q^{7} - 2q^{9} + 3q^{10} + 9q^{11} + 15q^{12} - 12q^{13} - 27q^{14} - q^{15} + 42q^{16} - 3q^{17} - 4q^{18} - 15q^{20} + 4q^{21} + 15q^{22} + q^{24} - 15q^{25} + 24q^{26} - 5q^{27} + 27q^{28} - 2q^{30} - 25q^{33} + 48q^{34} - 6q^{35} + 21q^{36} - 3q^{37} - 30q^{38} - 3q^{39} + 3q^{40} - 18q^{41} - 16q^{42} + 12q^{43} + 15q^{44} - 7q^{45} + 9q^{46} - 60q^{47} - 40q^{48} - 15q^{49} + 3q^{50} - 48q^{51} - 33q^{52} - 30q^{53} + 35q^{54} + 42q^{56} - 21q^{57} + 30q^{59} + 33q^{60} + 12q^{62} - 47q^{63} - 138q^{64} + 100q^{66} + 12q^{67} - 21q^{68} + 32q^{69} - 18q^{70} + 85q^{72} + 6q^{73} + 54q^{74} - 5q^{75} - 54q^{76} - 9q^{77} - 18q^{78} + 24q^{79} + 21q^{80} - 14q^{81} + 6q^{82} - 6q^{83} - 9q^{84} + 3q^{85} - 60q^{86} - 16q^{87} - 48q^{88} - 3q^{89} + 22q^{90} + 15q^{91} - 3q^{92} + 69q^{93} - 48q^{96} + 36q^{97} + 24q^{98} - q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
101.1 2.80758i 0.0857668 1.72993i −5.88252 0.500000 0.866025i −4.85691 0.240797i 1.63885 2.07705i 10.9005i −2.98529 0.296741i −2.43144 1.40379i
101.2 2.34202i −1.11747 + 1.32335i −3.48504 0.500000 0.866025i 3.09931 + 2.61714i −2.07734 1.63849i 3.47799i −0.502517 2.95761i −2.02825 1.17101i
101.3 1.82056i 1.12156 1.31989i −1.31444 0.500000 0.866025i −2.40293 2.04187i −1.92980 + 1.80994i 1.24811i −0.484202 2.96067i −1.57665 0.910280i
101.4 1.34690i −1.39768 1.02299i 0.185856 0.500000 0.866025i −1.37786 + 1.88253i 1.86235 1.87927i 2.94413i 0.907000 + 2.85961i −1.16645 0.673451i
101.5 1.34681i 1.51214 + 0.844642i 0.186101 0.500000 0.866025i 1.13757 2.03657i −0.829875 2.51223i 2.94426i 1.57316 + 2.55444i −1.16637 0.673405i
101.6 0.929301i 0.479000 + 1.66450i 1.13640 0.500000 0.866025i 1.54682 0.445136i 0.0655497 + 2.64494i 2.91466i −2.54112 + 1.59459i −0.804799 0.464651i
101.7 0.0264028i 1.26158 1.18677i 1.99930 0.500000 0.866025i −0.0313339 0.0333092i 2.64251 + 0.130871i 0.105593i 0.183169 2.99440i −0.0228655 0.0132014i
101.8 0.509846i 1.71720 + 0.226350i 1.74006 0.500000 0.866025i −0.115404 + 0.875506i −2.47153 0.944226i 1.90685i 2.89753 + 0.777376i 0.441540 + 0.254923i
101.9 0.692853i −1.66836 + 0.465377i 1.51995 0.500000 0.866025i −0.322438 1.15593i −0.669425 2.55966i 2.43881i 2.56685 1.55283i 0.600029 + 0.346427i
101.10 0.917882i −0.321911 1.70187i 1.15749 0.500000 0.866025i 1.56212 0.295477i 0.697139 + 2.55225i 2.89821i −2.79275 + 1.09570i 0.794909 + 0.458941i
101.11 1.23569i 0.460303 + 1.66977i 0.473066 0.500000 0.866025i −2.06332 + 0.568792i 2.47654 0.930987i 3.05595i −2.57624 + 1.53720i 1.07014 + 0.617846i
101.12 1.75842i −1.12723 + 1.31505i −1.09206 0.500000 0.866025i −2.31241 1.98215i −1.42571 + 2.22875i 1.59655i −0.458709 2.96472i 1.52284 + 0.879212i
101.13 2.16248i 1.73205 0.000611122i −2.67633 0.500000 0.866025i 0.00132154 + 3.74553i −0.841895 + 2.50823i 1.46255i 3.00000 0.00211699i 1.87276 + 1.08124i
101.14 2.34636i 0.822112 1.52451i −3.50542 0.500000 0.866025i 3.57705 + 1.92897i 2.00080 1.73112i 3.53226i −1.64826 2.50664i 2.03201 + 1.17318i
101.15 2.72808i −1.55907 0.754526i −5.44243 0.500000 0.866025i 2.05841 4.25326i −2.63818 0.200022i 9.39122i 1.86138 + 2.35271i 2.36259 + 1.36404i
131.1 2.72808i −1.55907 + 0.754526i −5.44243 0.500000 + 0.866025i 2.05841 + 4.25326i −2.63818 + 0.200022i 9.39122i 1.86138 2.35271i 2.36259 1.36404i
131.2 2.34636i 0.822112 + 1.52451i −3.50542 0.500000 + 0.866025i 3.57705 1.92897i 2.00080 + 1.73112i 3.53226i −1.64826 + 2.50664i 2.03201 1.17318i
131.3 2.16248i 1.73205 0.000611122i −2.67633 0.500000 + 0.866025i 0.00132154 3.74553i −0.841895 2.50823i 1.46255i 3.00000 + 0.00211699i 1.87276 1.08124i
131.4 1.75842i −1.12723 1.31505i −1.09206 0.500000 + 0.866025i −2.31241 + 1.98215i −1.42571 2.22875i 1.59655i −0.458709 + 2.96472i 1.52284 0.879212i
131.5 1.23569i 0.460303 1.66977i 0.473066 0.500000 + 0.866025i −2.06332 0.568792i 2.47654 + 0.930987i 3.05595i −2.57624 1.53720i 1.07014 0.617846i
See all 30 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 131.15 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
63.i 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.t.b 30
3.b odd 2 1 945.2.t.b 30
7.d odd 6 1 315.2.be.b yes 30
9.c even 3 1 945.2.be.b 30
9.d odd 6 1 315.2.be.b yes 30
21.g even 6 1 945.2.be.b 30
63.i even 6 1 inner 315.2.t.b 30
63.t odd 6 1 945.2.t.b 30

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
315.2.t.b 30 1.a even 1 1 trivial
315.2.t.b 30 63.i even 6 1 inner
315.2.be.b yes 30 7.d odd 6 1
315.2.be.b yes 30 9.d odd 6 1
945.2.t.b 30 3.b odd 2 1
945.2.t.b 30 63.t odd 6 1
945.2.be.b 30 9.c even 3 1
945.2.be.b 30 21.g even 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{30} + \cdots$$ acting on $$S_{2}^{\mathrm{new}}(315, [\chi])$$.

## Hecke Characteristic Polynomials

There are no characteristic polynomials of Hecke operators in the database