# Properties

 Label 294.2.m.b Level $294$ Weight $2$ Character orbit 294.m Analytic conductor $2.348$ Analytic rank $0$ Dimension $24$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$294 = 2 \cdot 3 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 294.m (of order $$21$$, degree $$12$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.34760181943$$ Analytic rank: $$0$$ Dimension: $$24$$ Relative dimension: $$2$$ over $$\Q(\zeta_{21})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{21}]$

## $q$-expansion

The dimension is sufficiently large that we do not compute an algebraic $$q$$-expansion, but we have computed the trace expansion.

 $$\operatorname{Tr}(f)(q) =$$ $$24 q + 2 q^{2} + 2 q^{3} + 2 q^{4} + q^{5} - 4 q^{6} - 4 q^{8} + 2 q^{9}+O(q^{10})$$ 24 * q + 2 * q^2 + 2 * q^3 + 2 * q^4 + q^5 - 4 * q^6 - 4 * q^8 + 2 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$24 q + 2 q^{2} + 2 q^{3} + 2 q^{4} + q^{5} - 4 q^{6} - 4 q^{8} + 2 q^{9} + q^{10} + 8 q^{11} + 2 q^{12} - 2 q^{13} + 5 q^{15} + 2 q^{16} + 9 q^{17} - 12 q^{18} - 4 q^{19} + 5 q^{20} + 14 q^{21} + 5 q^{22} - 38 q^{23} + 2 q^{24} - 19 q^{25} - 20 q^{26} - 4 q^{27} - 14 q^{28} - 14 q^{29} + 8 q^{30} + 4 q^{31} + 2 q^{32} - 6 q^{33} + 17 q^{34} + 42 q^{35} - 4 q^{36} + 7 q^{37} - 18 q^{38} - 6 q^{39} - 13 q^{40} + 2 q^{41} - 7 q^{42} + 4 q^{43} - 6 q^{44} - 13 q^{45} + 11 q^{46} - 8 q^{47} + 24 q^{48} - 42 q^{49} - 4 q^{50} - 40 q^{51} + 15 q^{52} - 10 q^{53} + 2 q^{54} + 33 q^{55} - 7 q^{56} + q^{57} - 28 q^{58} + 14 q^{59} - 13 q^{60} + 7 q^{61} - 8 q^{62} - 14 q^{63} - 4 q^{64} + 23 q^{65} + 8 q^{66} + 23 q^{67} + 9 q^{68} + 13 q^{69} + 28 q^{70} + 36 q^{71} + 2 q^{72} + 15 q^{73} - 14 q^{74} + 9 q^{75} + q^{76} - 28 q^{77} + 19 q^{78} + 38 q^{79} + 8 q^{80} + 2 q^{81} - q^{82} + 27 q^{83} - 47 q^{85} + 26 q^{86} - 28 q^{87} - 6 q^{88} - 121 q^{89} - 2 q^{90} + 21 q^{91} - q^{92} + 46 q^{93} - q^{94} + 44 q^{95} + 2 q^{96} + 14 q^{97} - 21 q^{98} - 44 q^{99}+O(q^{100})$$ 24 * q + 2 * q^2 + 2 * q^3 + 2 * q^4 + q^5 - 4 * q^6 - 4 * q^8 + 2 * q^9 + q^10 + 8 * q^11 + 2 * q^12 - 2 * q^13 + 5 * q^15 + 2 * q^16 + 9 * q^17 - 12 * q^18 - 4 * q^19 + 5 * q^20 + 14 * q^21 + 5 * q^22 - 38 * q^23 + 2 * q^24 - 19 * q^25 - 20 * q^26 - 4 * q^27 - 14 * q^28 - 14 * q^29 + 8 * q^30 + 4 * q^31 + 2 * q^32 - 6 * q^33 + 17 * q^34 + 42 * q^35 - 4 * q^36 + 7 * q^37 - 18 * q^38 - 6 * q^39 - 13 * q^40 + 2 * q^41 - 7 * q^42 + 4 * q^43 - 6 * q^44 - 13 * q^45 + 11 * q^46 - 8 * q^47 + 24 * q^48 - 42 * q^49 - 4 * q^50 - 40 * q^51 + 15 * q^52 - 10 * q^53 + 2 * q^54 + 33 * q^55 - 7 * q^56 + q^57 - 28 * q^58 + 14 * q^59 - 13 * q^60 + 7 * q^61 - 8 * q^62 - 14 * q^63 - 4 * q^64 + 23 * q^65 + 8 * q^66 + 23 * q^67 + 9 * q^68 + 13 * q^69 + 28 * q^70 + 36 * q^71 + 2 * q^72 + 15 * q^73 - 14 * q^74 + 9 * q^75 + q^76 - 28 * q^77 + 19 * q^78 + 38 * q^79 + 8 * q^80 + 2 * q^81 - q^82 + 27 * q^83 - 47 * q^85 + 26 * q^86 - 28 * q^87 - 6 * q^88 - 121 * q^89 - 2 * q^90 + 21 * q^91 - q^92 + 46 * q^93 - q^94 + 44 * q^95 + 2 * q^96 + 14 * q^97 - 21 * q^98 - 44 * q^99

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
25.1 0.955573 0.294755i 0.365341 + 0.930874i 0.826239 0.563320i −0.0494904 0.00745948i 0.623490 + 0.781831i −0.154679 + 2.64123i 0.623490 0.781831i −0.733052 + 0.680173i −0.0494904 + 0.00745948i
25.2 0.955573 0.294755i 0.365341 + 0.930874i 0.826239 0.563320i 1.22261 + 0.184279i 0.623490 + 0.781831i 0.792174 2.52437i 0.623490 0.781831i −0.733052 + 0.680173i 1.22261 0.184279i
37.1 0.826239 0.563320i −0.733052 + 0.680173i 0.365341 0.930874i −3.12319 0.963376i −0.222521 + 0.974928i −2.56883 + 0.633346i −0.222521 0.974928i 0.0747301 0.997204i −3.12319 + 0.963376i
37.2 0.826239 0.563320i −0.733052 + 0.680173i 0.365341 0.930874i 1.32658 + 0.409195i −0.222521 + 0.974928i 0.662652 2.56142i −0.222521 0.974928i 0.0747301 0.997204i 1.32658 0.409195i
109.1 −0.733052 + 0.680173i −0.988831 + 0.149042i 0.0747301 0.997204i −1.49873 + 3.81871i 0.623490 0.781831i −0.683647 2.55590i 0.623490 + 0.781831i 0.955573 0.294755i −1.49873 3.81871i
109.2 −0.733052 + 0.680173i −0.988831 + 0.149042i 0.0747301 0.997204i 0.380567 0.969668i 0.623490 0.781831i −2.32432 + 1.26394i 0.623490 + 0.781831i 0.955573 0.294755i 0.380567 + 0.969668i
121.1 0.0747301 0.997204i 0.955573 0.294755i −0.988831 0.149042i −1.21552 1.12784i −0.222521 0.974928i 0.254582 2.63347i −0.222521 + 0.974928i 0.826239 0.563320i −1.21552 + 1.12784i
121.2 0.0747301 0.997204i 0.955573 0.294755i −0.988831 0.149042i 1.71019 + 1.58683i −0.222521 0.974928i 1.81916 + 1.92111i −0.222521 + 0.974928i 0.826239 0.563320i 1.71019 1.58683i
151.1 0.826239 + 0.563320i −0.733052 0.680173i 0.365341 + 0.930874i −3.12319 + 0.963376i −0.222521 0.974928i −2.56883 0.633346i −0.222521 + 0.974928i 0.0747301 + 0.997204i −3.12319 0.963376i
151.2 0.826239 + 0.563320i −0.733052 0.680173i 0.365341 + 0.930874i 1.32658 0.409195i −0.222521 0.974928i 0.662652 + 2.56142i −0.222521 + 0.974928i 0.0747301 + 0.997204i 1.32658 + 0.409195i
163.1 0.365341 + 0.930874i 0.0747301 + 0.997204i −0.733052 + 0.680173i −0.549960 + 0.374956i −0.900969 + 0.433884i −0.0527428 + 2.64523i −0.900969 0.433884i −0.988831 + 0.149042i −0.549960 0.374956i
163.2 0.365341 + 0.930874i 0.0747301 + 0.997204i −0.733052 + 0.680173i 2.56909 1.75158i −0.900969 + 0.433884i 1.92409 1.81601i −0.900969 0.433884i −0.988831 + 0.149042i 2.56909 + 1.75158i
193.1 0.365341 0.930874i 0.0747301 0.997204i −0.733052 0.680173i −0.549960 0.374956i −0.900969 0.433884i −0.0527428 2.64523i −0.900969 + 0.433884i −0.988831 0.149042i −0.549960 + 0.374956i
193.2 0.365341 0.930874i 0.0747301 0.997204i −0.733052 0.680173i 2.56909 + 1.75158i −0.900969 0.433884i 1.92409 + 1.81601i −0.900969 + 0.433884i −0.988831 0.149042i 2.56909 1.75158i
205.1 −0.733052 0.680173i −0.988831 0.149042i 0.0747301 + 0.997204i −1.49873 3.81871i 0.623490 + 0.781831i −0.683647 + 2.55590i 0.623490 0.781831i 0.955573 + 0.294755i −1.49873 + 3.81871i
205.2 −0.733052 0.680173i −0.988831 0.149042i 0.0747301 + 0.997204i 0.380567 + 0.969668i 0.623490 + 0.781831i −2.32432 1.26394i 0.623490 0.781831i 0.955573 + 0.294755i 0.380567 0.969668i
235.1 −0.988831 0.149042i 0.826239 0.563320i 0.955573 + 0.294755i −0.189189 2.52456i −0.900969 + 0.433884i 2.49030 + 0.893546i −0.900969 0.433884i 0.365341 0.930874i −0.189189 + 2.52456i
235.2 −0.988831 0.149042i 0.826239 0.563320i 0.955573 + 0.294755i −0.0829648 1.10709i −0.900969 + 0.433884i −2.15873 1.52966i −0.900969 0.433884i 0.365341 0.930874i −0.0829648 + 1.10709i
247.1 0.955573 + 0.294755i 0.365341 0.930874i 0.826239 + 0.563320i −0.0494904 + 0.00745948i 0.623490 0.781831i −0.154679 2.64123i 0.623490 + 0.781831i −0.733052 0.680173i −0.0494904 0.00745948i
247.2 0.955573 + 0.294755i 0.365341 0.930874i 0.826239 + 0.563320i 1.22261 0.184279i 0.623490 0.781831i 0.792174 + 2.52437i 0.623490 + 0.781831i −0.733052 0.680173i 1.22261 + 0.184279i
See all 24 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 289.2 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
49.g even 21 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 294.2.m.b 24
3.b odd 2 1 882.2.z.a 24
49.g even 21 1 inner 294.2.m.b 24
147.n odd 42 1 882.2.z.a 24

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
294.2.m.b 24 1.a even 1 1 trivial
294.2.m.b 24 49.g even 21 1 inner
882.2.z.a 24 3.b odd 2 1
882.2.z.a 24 147.n odd 42 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{24} - T_{5}^{23} + 5 T_{5}^{22} - 28 T_{5}^{21} + 35 T_{5}^{20} + 476 T_{5}^{19} - 741 T_{5}^{18} - 112 T_{5}^{17} + 6640 T_{5}^{16} - 33035 T_{5}^{15} + 104328 T_{5}^{14} - 150654 T_{5}^{13} + 144502 T_{5}^{12} - 381570 T_{5}^{11} + \cdots + 729$$ acting on $$S_{2}^{\mathrm{new}}(294, [\chi])$$.