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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 $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [294,2,Mod(25,294)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(294, base_ring=CyclotomicField(42))
 
chi = DirichletCharacter(H, H._module([0, 16]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("294.25");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
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

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
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.

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

Display 100 coefficients 10 coefficients

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
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 25.2
Significant digits:
Format:

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 \( T_{5}^{24} - T_{5}^{23} + 5 T_{5}^{22} - 28 T_{5}^{21} + 35 T_{5}^{20} + 476 T_{5}^{19} - 741 T_{5}^{18} + \cdots + 729 \) acting on \(S_{2}^{\mathrm{new}}(294, [\chi])\). Copy content Toggle raw display