↑

Properties

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

Related objects

Downloads

Learn more

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
\(\operatorname{Tr}(f)(q) = \) \( 24 q - 2 q^{2} - 2 q^{3} + 2 q^{4} + q^{5} - 4 q^{6} + 4 q^{8} + 2 q^{9} - q^{10} + 6 q^{11} - 2 q^{12} - 2 q^{13} + 9 q^{15} + 2 q^{16} + 7 q^{17} + 12 q^{18} + 22 q^{19} + 5 q^{20} + 5 q^{22} - 42 q^{23} + 2 q^{24} + 29 q^{25} + 34 q^{26} + 4 q^{27} + 14 q^{28} + 6 q^{29} - 6 q^{30} + 2 q^{31} - 2 q^{32} - 20 q^{33} + 7 q^{34} - 28 q^{35} - 4 q^{36} + 9 q^{37} + 6 q^{38} - 8 q^{39} - 15 q^{40} - 28 q^{41} - 7 q^{42} - 16 q^{43} - 22 q^{44} - 13 q^{45} + 7 q^{46} + 40 q^{47} - 24 q^{48} - 14 q^{49} - 12 q^{50} - 14 q^{51} - 13 q^{52} - 34 q^{53} + 2 q^{54} - 13 q^{55} - 21 q^{56} - 5 q^{57} + 24 q^{58} - 6 q^{59} - 15 q^{60} + 13 q^{61} - 24 q^{62} + 14 q^{63} - 4 q^{64} - 59 q^{65} - 8 q^{66} - 27 q^{67} - 7 q^{68} - 21 q^{69} - 14 q^{70} - 4 q^{71} - 2 q^{72} + 31 q^{73} + 40 q^{74} - q^{75} + 5 q^{76} - 14 q^{77} + 5 q^{78} + 40 q^{79} - 6 q^{80} + 2 q^{81} - 77 q^{82} + 13 q^{83} + 35 q^{85} + 20 q^{86} + 24 q^{87} + 22 q^{88} + 9 q^{89} + 2 q^{90} - 7 q^{91} - 7 q^{92} - 2 q^{93} - 5 q^{94} - 52 q^{95} + 2 q^{96} - 18 q^{97} + 21 q^{98} - 12 q^{99}+O(q^{100}) \) 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 −1.14747 − 0.172954i 0.623490 + 0.781831i 2.36688 − 1.18233i −0.623490 + 0.781831i −0.733052 + 0.680173i 1.14747 − 0.172954i
25.2 −0.955573 + 0.294755i −0.365341 − 0.930874i 0.826239 − 0.563320i 3.24239 + 0.488712i 0.623490 + 0.781831i −0.296027 + 2.62914i −0.623490 + 0.781831i −0.733052 + 0.680173i −3.24239 + 0.488712i
37.1 −0.826239 + 0.563320i 0.733052 − 0.680173i 0.365341 − 0.930874i −2.26109 − 0.697454i −0.222521 + 0.974928i −2.27170 + 1.35623i 0.222521 + 0.974928i 0.0747301 − 0.997204i 2.26109 − 0.697454i
37.2 −0.826239 + 0.563320i 0.733052 − 0.680173i 0.365341 − 0.930874i 2.66126 + 0.820891i −0.222521 + 0.974928i 0.800630 + 2.52170i 0.222521 + 0.974928i 0.0747301 − 0.997204i −2.66126 + 0.820891i
109.1 0.733052 − 0.680173i 0.988831 − 0.149042i 0.0747301 − 0.997204i 0.357141 − 0.909982i 0.623490 − 0.781831i −1.62228 − 2.09002i −0.623490 − 0.781831i 0.955573 − 0.294755i −0.357141 − 0.909982i
109.2 0.733052 − 0.680173i 0.988831 − 0.149042i 0.0747301 − 0.997204i 0.651816 − 1.66080i 0.623490 − 0.781831i 1.92190 + 1.81832i −0.623490 − 0.781831i 0.955573 − 0.294755i −0.651816 − 1.66080i
121.1 −0.0747301 + 0.997204i −0.955573 + 0.294755i −0.988831 − 0.149042i −2.93629 − 2.72448i −0.222521 − 0.974928i 2.36506 + 1.18596i 0.222521 − 0.974928i 0.826239 − 0.563320i 2.93629 − 2.72448i
121.2 −0.0747301 + 0.997204i −0.955573 + 0.294755i −0.988831 − 0.149042i −0.457838 − 0.424812i −0.222521 − 0.974928i −1.06155 − 2.42345i 0.222521 − 0.974928i 0.826239 − 0.563320i 0.457838 − 0.424812i
151.1 −0.826239 − 0.563320i 0.733052 + 0.680173i 0.365341 + 0.930874i −2.26109 + 0.697454i −0.222521 − 0.974928i −2.27170 − 1.35623i 0.222521 − 0.974928i 0.0747301 + 0.997204i 2.26109 + 0.697454i
151.2 −0.826239 − 0.563320i 0.733052 + 0.680173i 0.365341 + 0.930874i 2.66126 − 0.820891i −0.222521 − 0.974928i 0.800630 − 2.52170i 0.222521 − 0.974928i 0.0747301 + 0.997204i −2.66126 − 0.820891i
163.1 −0.365341 − 0.930874i −0.0747301 − 0.997204i −0.733052 + 0.680173i −1.94585 + 1.32666i −0.900969 + 0.433884i −1.66390 + 2.05705i 0.900969 + 0.433884i −0.988831 + 0.149042i 1.94585 + 1.32666i
163.2 −0.365341 − 0.930874i −0.0747301 − 0.997204i −0.733052 + 0.680173i 2.34942 − 1.60181i −0.900969 + 0.433884i −1.71046 − 2.01849i 0.900969 + 0.433884i −0.988831 + 0.149042i −2.34942 − 1.60181i
193.1 −0.365341 + 0.930874i −0.0747301 + 0.997204i −0.733052 − 0.680173i −1.94585 − 1.32666i −0.900969 − 0.433884i −1.66390 − 2.05705i 0.900969 − 0.433884i −0.988831 − 0.149042i 1.94585 − 1.32666i
193.2 −0.365341 + 0.930874i −0.0747301 + 0.997204i −0.733052 − 0.680173i 2.34942 + 1.60181i −0.900969 − 0.433884i −1.71046 + 2.01849i 0.900969 − 0.433884i −0.988831 − 0.149042i −2.34942 + 1.60181i
205.1 0.733052 + 0.680173i 0.988831 + 0.149042i 0.0747301 + 0.997204i 0.357141 + 0.909982i 0.623490 + 0.781831i −1.62228 + 2.09002i −0.623490 + 0.781831i 0.955573 + 0.294755i −0.357141 + 0.909982i
205.2 0.733052 + 0.680173i 0.988831 + 0.149042i 0.0747301 + 0.997204i 0.651816 + 1.66080i 0.623490 + 0.781831i 1.92190 − 1.81832i −0.623490 + 0.781831i 0.955573 + 0.294755i −0.651816 + 1.66080i
235.1 0.988831 + 0.149042i −0.826239 + 0.563320i 0.955573 + 0.294755i −0.139635 − 1.86330i −0.900969 + 0.433884i 2.53282 − 0.764745i 0.900969 + 0.433884i 0.365341 − 0.930874i 0.139635 − 1.86330i
235.2 0.988831 + 0.149042i −0.826239 + 0.563320i 0.955573 + 0.294755i 0.126149 + 1.68335i −0.900969 + 0.433884i −1.36136 + 2.26863i 0.900969 + 0.433884i 0.365341 − 0.930874i −0.126149 + 1.68335i
247.1 −0.955573 − 0.294755i −0.365341 + 0.930874i 0.826239 + 0.563320i −1.14747 + 0.172954i 0.623490 − 0.781831i 2.36688 + 1.18233i −0.623490 − 0.781831i −0.733052 − 0.680173i 1.14747 + 0.172954i
247.2 −0.955573 − 0.294755i −0.365341 + 0.930874i 0.826239 + 0.563320i 3.24239 − 0.488712i 0.623490 − 0.781831i −0.296027 − 2.62914i −0.623490 − 0.781831i −0.733052 − 0.680173i −3.24239 − 0.488712i
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.a ✓ 24
3.b odd 2 1 882.2.z.c 24
49.g even 21 1 inner 294.2.m.a ✓ 24
147.n odd 42 1 882.2.z.c 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
294.2.m.a ✓ 24 1.a even 1 1 trivial
294.2.m.a ✓ 24 49.g even 21 1 inner
882.2.z.c 24 3.b odd 2 1
882.2.z.c 24 147.n odd 42 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{24} - T_{5}^{23} - 19 T_{5}^{22} - 26 T_{5}^{21} + 313 T_{5}^{20} + 172 T_{5}^{19} + \cdots + 5340721 \) acting on \(S_{2}^{\mathrm{new}}(294, [\chi])\). Copy content Toggle raw display