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Properties

Label 294.2.m.d
Level $294$
Weight $2$
Character orbit 294.m
Analytic conductor $2.348$
Analytic rank $0$
Dimension $36$
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: \(36\)
Relative dimension: \(3\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 36 q + 3 q^{2} - 3 q^{3} + 3 q^{4} + 6 q^{6} - q^{7} - 6 q^{8} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 36 q + 3 q^{2} - 3 q^{3} + 3 q^{4} + 6 q^{6} - q^{7} - 6 q^{8} + 3 q^{9} + 13 q^{11} - 3 q^{12} + 2 q^{13} - 4 q^{14} + 3 q^{16} + q^{17} - 18 q^{18} - 2 q^{19} + 7 q^{20} - 12 q^{21} - 12 q^{22} + 42 q^{23} - 3 q^{24} + 33 q^{25} + 34 q^{26} + 6 q^{27} - 9 q^{28} - 4 q^{29} - 7 q^{30} - q^{31} + 3 q^{32} + q^{33} + 5 q^{34} - 4 q^{35} - 6 q^{36} + 45 q^{37} - 30 q^{38} - 6 q^{39} - 58 q^{41} - 20 q^{42} - q^{44} - 14 q^{45} - 7 q^{46} - 68 q^{47} - 36 q^{48} - 29 q^{49} + 74 q^{50} - 8 q^{51} - 15 q^{52} - 19 q^{53} - 3 q^{54} + 48 q^{55} - 15 q^{56} + 3 q^{57} - 40 q^{58} - 123 q^{59} - 13 q^{61} + 2 q^{62} + 10 q^{63} - 6 q^{64} + 9 q^{65} - 6 q^{66} - 7 q^{67} + q^{68} + 21 q^{69} - 16 q^{70} - 38 q^{71} + 3 q^{72} + 65 q^{73} + 38 q^{74} + 9 q^{75} - 17 q^{76} + 25 q^{77} + 5 q^{78} - 71 q^{79} + 7 q^{80} + 3 q^{81} + 29 q^{82} + 8 q^{83} + 4 q^{84} + 73 q^{85} - 14 q^{86} + 40 q^{87} - q^{88} + 45 q^{89} - 35 q^{91} - 7 q^{92} - 6 q^{93} + 9 q^{94} - 144 q^{95} - 3 q^{96} + 56 q^{97} + 24 q^{98} + 30 q^{99}+O(q^{100}) \) 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 −2.79734 − 0.421632i −0.623490 − 0.781831i −0.640804 − 2.56698i 0.623490 − 0.781831i −0.733052 + 0.680173i −2.79734 + 0.421632i
25.2 0.955573 − 0.294755i −0.365341 − 0.930874i 0.826239 − 0.563320i 1.51747 + 0.228722i −0.623490 − 0.781831i 2.12911 + 1.57064i 0.623490 − 0.781831i −0.733052 + 0.680173i 1.51747 − 0.228722i
25.3 0.955573 − 0.294755i −0.365341 − 0.930874i 0.826239 − 0.563320i 3.90272 + 0.588241i −0.623490 − 0.781831i −2.05876 − 1.66178i 0.623490 − 0.781831i −0.733052 + 0.680173i 3.90272 − 0.588241i
37.1 0.826239 − 0.563320i 0.733052 − 0.680173i 0.365341 − 0.930874i −4.26921 − 1.31688i 0.222521 − 0.974928i −1.15057 − 2.38248i −0.222521 − 0.974928i 0.0747301 − 0.997204i −4.26921 + 1.31688i
37.2 0.826239 − 0.563320i 0.733052 − 0.680173i 0.365341 − 0.930874i −0.409482 − 0.126309i 0.222521 − 0.974928i 2.34610 + 1.22304i −0.222521 − 0.974928i 0.0747301 − 0.997204i −0.409482 + 0.126309i
37.3 0.826239 − 0.563320i 0.733052 − 0.680173i 0.365341 − 0.930874i 3.02490 + 0.933059i 0.222521 − 0.974928i −2.44171 − 1.01884i −0.222521 − 0.974928i 0.0747301 − 0.997204i 3.02490 − 0.933059i
109.1 −0.733052 + 0.680173i 0.988831 − 0.149042i 0.0747301 − 0.997204i −1.17326 + 2.98942i −0.623490 + 0.781831i 1.68654 + 2.03853i 0.623490 + 0.781831i 0.955573 − 0.294755i −1.17326 − 2.98942i
109.2 −0.733052 + 0.680173i 0.988831 − 0.149042i 0.0747301 − 0.997204i −0.513031 + 1.30718i −0.623490 + 0.781831i −0.0547842 − 2.64518i 0.623490 + 0.781831i 0.955573 − 0.294755i −0.513031 − 1.30718i
109.3 −0.733052 + 0.680173i 0.988831 − 0.149042i 0.0747301 − 0.997204i 1.26635 − 3.22660i −0.623490 + 0.781831i −1.11625 + 2.39875i 0.623490 + 0.781831i 0.955573 − 0.294755i 1.26635 + 3.22660i
121.1 0.0747301 − 0.997204i −0.955573 + 0.294755i −0.988831 − 0.149042i −2.24988 − 2.08758i 0.222521 + 0.974928i 1.57456 + 2.12621i −0.222521 + 0.974928i 0.826239 − 0.563320i −2.24988 + 2.08758i
121.2 0.0747301 − 0.997204i −0.955573 + 0.294755i −0.988831 − 0.149042i 0.738585 + 0.685307i 0.222521 + 0.974928i −1.26941 + 2.32133i −0.222521 + 0.974928i 0.826239 − 0.563320i 0.738585 − 0.685307i
121.3 0.0747301 − 0.997204i −0.955573 + 0.294755i −0.988831 − 0.149042i 0.794619 + 0.737299i 0.222521 + 0.974928i 2.24297 − 1.40324i −0.222521 + 0.974928i 0.826239 − 0.563320i 0.794619 − 0.737299i
151.1 0.826239 + 0.563320i 0.733052 + 0.680173i 0.365341 + 0.930874i −4.26921 + 1.31688i 0.222521 + 0.974928i −1.15057 + 2.38248i −0.222521 + 0.974928i 0.0747301 + 0.997204i −4.26921 − 1.31688i
151.2 0.826239 + 0.563320i 0.733052 + 0.680173i 0.365341 + 0.930874i −0.409482 + 0.126309i 0.222521 + 0.974928i 2.34610 − 1.22304i −0.222521 + 0.974928i 0.0747301 + 0.997204i −0.409482 − 0.126309i
151.3 0.826239 + 0.563320i 0.733052 + 0.680173i 0.365341 + 0.930874i 3.02490 − 0.933059i 0.222521 + 0.974928i −2.44171 + 1.01884i −0.222521 + 0.974928i 0.0747301 + 0.997204i 3.02490 + 0.933059i
163.1 0.365341 + 0.930874i −0.0747301 − 0.997204i −0.733052 + 0.680173i −2.53296 + 1.72694i 0.900969 − 0.433884i −2.53425 + 0.759993i −0.900969 − 0.433884i −0.988831 + 0.149042i −2.53296 − 1.72694i
163.2 0.365341 + 0.930874i −0.0747301 − 0.997204i −0.733052 + 0.680173i 0.374506 − 0.255334i 0.900969 − 0.433884i 1.59832 + 2.10840i −0.900969 − 0.433884i −0.988831 + 0.149042i 0.374506 + 0.255334i
163.3 0.365341 + 0.930874i −0.0747301 − 0.997204i −0.733052 + 0.680173i 2.54357 − 1.73418i 0.900969 − 0.433884i −0.146091 − 2.64171i −0.900969 − 0.433884i −0.988831 + 0.149042i 2.54357 + 1.73418i
193.1 0.365341 − 0.930874i −0.0747301 + 0.997204i −0.733052 − 0.680173i −2.53296 − 1.72694i 0.900969 + 0.433884i −2.53425 − 0.759993i −0.900969 + 0.433884i −0.988831 − 0.149042i −2.53296 + 1.72694i
193.2 0.365341 − 0.930874i −0.0747301 + 0.997204i −0.733052 − 0.680173i 0.374506 + 0.255334i 0.900969 + 0.433884i 1.59832 − 2.10840i −0.900969 + 0.433884i −0.988831 − 0.149042i 0.374506 − 0.255334i
See all 36 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 25.3
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.d ✓ 36
3.b odd 2 1 882.2.z.e 36
49.g even 21 1 inner 294.2.m.d ✓ 36
147.n odd 42 1 882.2.z.e 36
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
294.2.m.d ✓ 36 1.a even 1 1 trivial
294.2.m.d ✓ 36 49.g even 21 1 inner
882.2.z.e 36 3.b odd 2 1
882.2.z.e 36 147.n odd 42 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{36} - 24 T_{5}^{34} - 18 T_{5}^{33} + 56 T_{5}^{32} + 235 T_{5}^{31} + 4260 T_{5}^{30} + \cdots + 439279681 \) acting on \(S_{2}^{\mathrm{new}}(294, [\chi])\). Copy content Toggle raw display