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Properties

Label 294.2.m.c
Level $294$
Weight $2$
Character orbit 294.m
Analytic conductor $2.348$
Analytic rank $0$
Dimension $36$
Inner twists $2$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [294,2,Mod(25,294)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://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);
 
Copy content sage: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")
 
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

Copy content comment:select newform
 
Copy content sage:traces = [36,-3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content 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 algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 36 q - 3 q^{2} + 3 q^{3} + 3 q^{4} + 2 q^{5} + 6 q^{6} - 5 q^{7} + 6 q^{8} + 3 q^{9} - 2 q^{10} + 9 q^{11} + 3 q^{12} + 10 q^{13} - 4 q^{14} - 4 q^{15} + 3 q^{16} + 7 q^{17} + 18 q^{18} + 16 q^{19} + 3 q^{20}+ \cdots + 10 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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.

Copy content comment:embeddings in the coefficient field
 
Copy content 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 −2.14348 − 0.323077i −0.623490 − 0.781831i 2.52421 − 0.792709i −0.623490 + 0.781831i −0.733052 + 0.680173i 2.14348 − 0.323077i
25.2 −0.955573 + 0.294755i 0.365341 + 0.930874i 0.826239 − 0.563320i −1.08148 − 0.163007i −0.623490 − 0.781831i −1.37377 + 2.26114i −0.623490 + 0.781831i −0.733052 + 0.680173i 1.08148 − 0.163007i
25.3 −0.955573 + 0.294755i 0.365341 + 0.930874i 0.826239 − 0.563320i 3.87015 + 0.583331i −0.623490 − 0.781831i 2.64466 − 0.0758986i −0.623490 + 0.781831i −0.733052 + 0.680173i −3.87015 + 0.583331i
37.1 −0.826239 + 0.563320i −0.733052 + 0.680173i 0.365341 − 0.930874i −2.59837 − 0.801491i 0.222521 − 0.974928i 2.59324 − 0.524514i 0.222521 + 0.974928i 0.0747301 − 0.997204i 2.59837 − 0.801491i
37.2 −0.826239 + 0.563320i −0.733052 + 0.680173i 0.365341 − 0.930874i −0.203744 − 0.0628466i 0.222521 − 0.974928i −1.93486 − 1.80452i 0.222521 + 0.974928i 0.0747301 − 0.997204i 0.203744 − 0.0628466i
37.3 −0.826239 + 0.563320i −0.733052 + 0.680173i 0.365341 − 0.930874i 3.05947 + 0.943720i 0.222521 − 0.974928i −1.76838 + 1.96795i 0.222521 + 0.974928i 0.0747301 − 0.997204i −3.05947 + 0.943720i
109.1 0.733052 − 0.680173i −0.988831 + 0.149042i 0.0747301 − 0.997204i −1.05865 + 2.69740i −0.623490 + 0.781831i 0.494639 + 2.59910i −0.623490 − 0.781831i 0.955573 − 0.294755i 1.05865 + 2.69740i
109.2 0.733052 − 0.680173i −0.988831 + 0.149042i 0.0747301 − 0.997204i 0.00944986 − 0.0240779i −0.623490 + 0.781831i 2.05870 − 1.66185i −0.623490 − 0.781831i 0.955573 − 0.294755i −0.00944986 − 0.0240779i
109.3 0.733052 − 0.680173i −0.988831 + 0.149042i 0.0747301 − 0.997204i 1.35994 − 3.46506i −0.623490 + 0.781831i −2.55254 + 0.696071i −0.623490 − 0.781831i 0.955573 − 0.294755i −1.35994 − 3.46506i
121.1 −0.0747301 + 0.997204i 0.955573 − 0.294755i −0.988831 − 0.149042i −3.03776 − 2.81863i 0.222521 + 0.974928i −2.64177 + 0.145096i 0.222521 − 0.974928i 0.826239 − 0.563320i 3.03776 − 2.81863i
121.2 −0.0747301 + 0.997204i 0.955573 − 0.294755i −0.988831 − 0.149042i −0.0373053 − 0.0346142i 0.222521 + 0.974928i 0.835928 − 2.51022i 0.222521 − 0.974928i 0.826239 − 0.563320i 0.0373053 − 0.0346142i
121.3 −0.0747301 + 0.997204i 0.955573 − 0.294755i −0.988831 − 0.149042i 0.892287 + 0.827921i 0.222521 + 0.974928i 0.278782 + 2.63102i 0.222521 − 0.974928i 0.826239 − 0.563320i −0.892287 + 0.827921i
151.1 −0.826239 − 0.563320i −0.733052 − 0.680173i 0.365341 + 0.930874i −2.59837 + 0.801491i 0.222521 + 0.974928i 2.59324 + 0.524514i 0.222521 − 0.974928i 0.0747301 + 0.997204i 2.59837 + 0.801491i
151.2 −0.826239 − 0.563320i −0.733052 − 0.680173i 0.365341 + 0.930874i −0.203744 + 0.0628466i 0.222521 + 0.974928i −1.93486 + 1.80452i 0.222521 − 0.974928i 0.0747301 + 0.997204i 0.203744 + 0.0628466i
151.3 −0.826239 − 0.563320i −0.733052 − 0.680173i 0.365341 + 0.930874i 3.05947 − 0.943720i 0.222521 + 0.974928i −1.76838 − 1.96795i 0.222521 − 0.974928i 0.0747301 + 0.997204i −3.05947 − 0.943720i
163.1 −0.365341 − 0.930874i 0.0747301 + 0.997204i −0.733052 + 0.680173i −1.01252 + 0.690323i 0.900969 − 0.433884i −2.64531 + 0.0485591i 0.900969 + 0.433884i −0.988831 + 0.149042i 1.01252 + 0.690323i
163.2 −0.365341 − 0.930874i 0.0747301 + 0.997204i −0.733052 + 0.680173i −0.172293 + 0.117467i 0.900969 − 0.433884i −0.0217470 − 2.64566i 0.900969 + 0.433884i −0.988831 + 0.149042i 0.172293 + 0.117467i
163.3 −0.365341 − 0.930874i 0.0747301 + 0.997204i −0.733052 + 0.680173i 3.22240 − 2.19700i 0.900969 − 0.433884i 0.294655 + 2.62929i 0.900969 + 0.433884i −0.988831 + 0.149042i −3.22240 − 2.19700i
193.1 −0.365341 + 0.930874i 0.0747301 − 0.997204i −0.733052 − 0.680173i −1.01252 − 0.690323i 0.900969 + 0.433884i −2.64531 − 0.0485591i 0.900969 − 0.433884i −0.988831 − 0.149042i 1.01252 − 0.690323i
193.2 −0.365341 + 0.930874i 0.0747301 − 0.997204i −0.733052 − 0.680173i −0.172293 − 0.117467i 0.900969 + 0.433884i −0.0217470 + 2.64566i 0.900969 − 0.433884i −0.988831 − 0.149042i 0.172293 − 0.117467i
See all 36 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 25.3
Significant digits:
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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.c ✓ 36
3.b odd 2 1 882.2.z.f 36
49.g even 21 1 inner 294.2.m.c ✓ 36
147.n odd 42 1 882.2.z.f 36
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
294.2.m.c ✓ 36 1.a even 1 1 trivial
294.2.m.c ✓ 36 49.g even 21 1 inner
882.2.z.f 36 3.b odd 2 1
882.2.z.f 36 147.n odd 42 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{36} - 2 T_{5}^{35} + 6 T_{5}^{34} - 44 T_{5}^{33} - 22 T_{5}^{32} - 613 T_{5}^{31} + 3398 T_{5}^{30} + \cdots + 729 \) acting on \(S_{2}^{\mathrm{new}}(294, [\chi])\). Copy content Toggle raw display