# Properties

 Label 315.2.be.b Level 315 Weight 2 Character orbit 315.be 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.be (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.

 $$\operatorname{Tr}(f)(q) =$$ $$30q + 3q^{2} - q^{3} + 15q^{4} + 30q^{5} + q^{6} + 6q^{7} - 5q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$30q + 3q^{2} - q^{3} + 15q^{4} + 30q^{5} + q^{6} + 6q^{7} - 5q^{9} + 3q^{10} - 18q^{12} + 12q^{13} - 9q^{14} - q^{15} - 21q^{16} + 3q^{17} - 22q^{18} + 15q^{20} - 10q^{21} + 15q^{22} + 2q^{24} + 30q^{25} - 24q^{26} + 5q^{27} + 27q^{28} + q^{30} + 6q^{31} + 9q^{32} - 17q^{33} - 48q^{34} + 6q^{35} + 21q^{36} - 3q^{37} - 60q^{38} + 12q^{39} + 18q^{41} - 47q^{42} + 12q^{43} - 15q^{44} - 5q^{45} + 9q^{46} - 30q^{47} + 40q^{48} - 24q^{49} + 3q^{50} + 33q^{51} + 30q^{53} + 13q^{54} + 72q^{56} - 21q^{57} + 15q^{59} - 18q^{60} - 30q^{61} - 12q^{62} + 10q^{63} - 138q^{64} + 12q^{65} + 44q^{66} - 6q^{67} - 42q^{68} - 32q^{69} - 9q^{70} - 137q^{72} + 6q^{73} - q^{75} + 54q^{76} - 21q^{77} - 18q^{78} - 12q^{79} - 21q^{80} - 17q^{81} + 6q^{82} + 6q^{83} - 12q^{84} + 3q^{85} - 47q^{87} + 96q^{88} + 3q^{89} - 22q^{90} + 15q^{91} - 3q^{92} - 18q^{93} + 3q^{94} + 60q^{96} - 36q^{97} - 24q^{98} - q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
236.1 −2.43144 + 1.40379i −1.45528 0.939239i 2.94126 5.09441i 1.00000 4.85691 + 0.240797i 0.979355 2.45782i 10.9005i 1.23566 + 2.73371i −2.43144 + 1.40379i
236.2 −2.02825 + 1.17101i 0.587320 + 1.62943i 1.74252 3.01813i 1.00000 −3.09931 2.61714i 2.45764 + 0.979787i 3.47799i −2.31011 + 1.91400i −2.02825 + 1.17101i
236.3 −1.57665 + 0.910280i −0.582275 1.63124i 0.657218 1.13833i 1.00000 2.40293 + 2.04187i −0.602558 + 2.57622i 1.24811i −2.32191 + 1.89966i −1.57665 + 0.910280i
236.4 −1.16645 + 0.673451i −1.58477 + 0.698930i −0.0929278 + 0.160956i 1.00000 1.37786 1.88253i 0.696317 2.55248i 2.94413i 2.02299 2.21529i −1.16645 + 0.673451i
236.5 −1.16637 + 0.673405i 1.48755 0.887234i −0.0930506 + 0.161168i 1.00000 −1.13757 + 2.03657i 2.59059 0.537423i 2.94426i 1.42563 2.63962i −1.16637 + 0.673405i
236.6 −0.804799 + 0.464651i 1.68100 + 0.417423i −0.568200 + 0.984151i 1.00000 −1.54682 + 0.445136i −2.32336 + 1.26570i 2.91466i 2.65152 + 1.40338i −0.804799 + 0.464651i
236.7 −0.0228655 + 0.0132014i −0.396980 1.68594i −0.999651 + 1.73145i 1.00000 0.0313339 + 0.0333092i −1.43459 2.22305i 0.105593i −2.68481 + 1.33857i −0.0228655 + 0.0132014i
236.8 0.441540 0.254923i 1.05462 1.37396i −0.870028 + 1.50693i 1.00000 0.115404 0.875506i 2.05349 + 1.66829i 1.90685i −0.775539 2.89802i 0.441540 0.254923i
236.9 0.600029 0.346427i −0.431152 + 1.67753i −0.759977 + 1.31632i 1.00000 0.322438 + 1.15593i 2.55145 0.700092i 2.43881i −2.62822 1.44654i 0.600029 0.346427i
236.10 0.794909 0.458941i −1.63482 0.572153i −0.578746 + 1.00242i 1.00000 −1.56212 + 0.295477i −2.55889 + 0.672387i 2.89821i 2.34528 + 1.87074i 0.794909 0.458941i
236.11 1.07014 0.617846i 1.67621 + 0.436249i −0.236533 + 0.409687i 1.00000 2.06332 0.568792i −0.432012 2.61024i 3.05595i 2.61937 + 1.46249i 1.07014 0.617846i
236.12 1.52284 0.879212i 0.575251 + 1.63373i 0.546028 0.945749i 1.00000 2.31241 + 1.98215i −1.21730 + 2.34908i 1.59655i −2.33817 + 1.87962i 1.52284 0.879212i
236.13 1.87276 1.08124i 0.865496 1.50031i 1.33817 2.31777i 1.00000 −0.00132154 3.74553i −1.75124 + 1.98322i 1.46255i −1.50183 2.59702i 1.87276 1.08124i
236.14 2.03201 1.17318i −0.909209 1.47423i 1.75271 3.03578i 1.00000 −3.57705 1.92897i 0.498797 2.59831i 3.53226i −1.34668 + 2.68076i 2.03201 1.17318i
236.15 2.36259 1.36404i −1.43297 + 0.972928i 2.72121 4.71328i 1.00000 −2.05841 + 4.25326i 1.49231 + 2.18472i 9.39122i 1.10682 2.78836i 2.36259 1.36404i
311.1 −2.43144 1.40379i −1.45528 + 0.939239i 2.94126 + 5.09441i 1.00000 4.85691 0.240797i 0.979355 + 2.45782i 10.9005i 1.23566 2.73371i −2.43144 1.40379i
311.2 −2.02825 1.17101i 0.587320 1.62943i 1.74252 + 3.01813i 1.00000 −3.09931 + 2.61714i 2.45764 0.979787i 3.47799i −2.31011 1.91400i −2.02825 1.17101i
311.3 −1.57665 0.910280i −0.582275 + 1.63124i 0.657218 + 1.13833i 1.00000 2.40293 2.04187i −0.602558 2.57622i 1.24811i −2.32191 1.89966i −1.57665 0.910280i
311.4 −1.16645 0.673451i −1.58477 0.698930i −0.0929278 0.160956i 1.00000 1.37786 + 1.88253i 0.696317 + 2.55248i 2.94413i 2.02299 + 2.21529i −1.16645 0.673451i
311.5 −1.16637 0.673405i 1.48755 + 0.887234i −0.0930506 0.161168i 1.00000 −1.13757 2.03657i 2.59059 + 0.537423i 2.94426i 1.42563 + 2.63962i −1.16637 0.673405i
See all 30 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 311.15 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.s 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.be.b yes 30
3.b odd 2 1 945.2.be.b 30
7.d odd 6 1 315.2.t.b 30
9.c even 3 1 945.2.t.b 30
9.d odd 6 1 315.2.t.b 30
21.g even 6 1 945.2.t.b 30
63.k odd 6 1 945.2.be.b 30
63.s even 6 1 inner 315.2.be.b yes 30

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