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Properties

Label 402.2.m.b
Level $402$
Weight $2$
Character orbit 402.m
Analytic conductor $3.210$
Analytic rank $0$
Dimension $60$
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] = [402,2,Mod(19,402)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(402, base_ring=CyclotomicField(66))
 
chi = DirichletCharacter(H, H._module([0, 10]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("402.19");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 402 = 2 \cdot 3 \cdot 67 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 402.m (of order \(33\), degree \(20\), 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: \(3.20998616126\)
Analytic rank: \(0\)
Dimension: \(60\)
Relative dimension: \(3\) over \(\Q(\zeta_{33})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{33}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 60 q - 3 q^{2} + 6 q^{3} + 3 q^{4} + 4 q^{5} + 3 q^{6} - 3 q^{7} + 6 q^{8} - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 60 q - 3 q^{2} + 6 q^{3} + 3 q^{4} + 4 q^{5} + 3 q^{6} - 3 q^{7} + 6 q^{8} - 6 q^{9} + 2 q^{10} - 12 q^{11} - 3 q^{12} + 19 q^{13} + 5 q^{14} + 7 q^{15} + 3 q^{16} + 24 q^{17} - 3 q^{18} + 21 q^{19} - 2 q^{20} + 3 q^{21} - 2 q^{22} - 26 q^{23} - 6 q^{24} + 10 q^{25} - 8 q^{26} + 6 q^{27} + 19 q^{28} - 16 q^{29} + 9 q^{30} + 23 q^{31} - 3 q^{32} + q^{33} + 20 q^{34} - 68 q^{35} + 3 q^{36} + 13 q^{37} + q^{38} - 19 q^{39} - 4 q^{40} + 26 q^{41} - 5 q^{42} + 20 q^{43} - q^{44} + 4 q^{45} + 26 q^{46} + 41 q^{47} - 3 q^{48} + 44 q^{49} - 6 q^{50} - 35 q^{51} + 17 q^{52} - 65 q^{53} + 3 q^{54} + 39 q^{55} + 3 q^{56} - 10 q^{57} - 32 q^{58} - 16 q^{59} - 20 q^{60} - 22 q^{61} - 20 q^{62} - 14 q^{63} - 6 q^{64} - 54 q^{65} - 42 q^{66} + 42 q^{67} + 40 q^{68} - 29 q^{69} - 37 q^{70} - 40 q^{71} + 6 q^{72} - 11 q^{73} - 57 q^{74} - 10 q^{75} - 9 q^{76} - 113 q^{77} - 36 q^{78} + 5 q^{79} - 2 q^{80} - 6 q^{81} - 3 q^{82} - 33 q^{83} + 14 q^{84} - 131 q^{85} + 10 q^{86} - 17 q^{87} - 21 q^{88} + 11 q^{89} + 2 q^{90} - 25 q^{91} + 8 q^{92} - 12 q^{93} - 28 q^{94} + 200 q^{95} - 30 q^{96} + 51 q^{97} - q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1 0.888835 + 0.458227i 0.959493 − 0.281733i 0.580057 + 0.814576i −2.03671 − 2.35049i 0.981929 + 0.189251i −0.225927 − 4.74280i 0.142315 + 0.989821i 0.841254 − 0.540641i −0.733243 − 3.02247i
19.2 0.888835 + 0.458227i 0.959493 − 0.281733i 0.580057 + 0.814576i 0.793793 + 0.916086i 0.981929 + 0.189251i 0.147686 + 3.10031i 0.142315 + 0.989821i 0.841254 − 0.540641i 0.285777 + 1.17799i
19.3 0.888835 + 0.458227i 0.959493 − 0.281733i 0.580057 + 0.814576i 1.83546 + 2.11823i 0.981929 + 0.189251i −0.138877 − 2.91539i 0.142315 + 0.989821i 0.841254 − 0.540641i 0.660790 + 2.72381i
49.1 −0.580057 + 0.814576i −0.841254 − 0.540641i −0.327068 − 0.945001i −0.103625 − 0.720728i 0.928368 − 0.371662i 2.65737 + 0.253748i 0.959493 + 0.281733i 0.415415 + 0.909632i 0.647196 + 0.333653i
49.2 −0.580057 + 0.814576i −0.841254 − 0.540641i −0.327068 − 0.945001i −0.000162911 − 0.00113307i 0.928368 − 0.371662i −4.42777 − 0.422801i 0.959493 + 0.281733i 0.415415 + 0.909632i 0.00101747 0.000524542i
49.3 −0.580057 + 0.814576i −0.841254 − 0.540641i −0.327068 − 0.945001i 0.529444 + 3.68236i 0.928368 − 0.371662i 0.341525 + 0.0326117i 0.959493 + 0.281733i 0.415415 + 0.909632i −3.30667 − 1.70471i
55.1 −0.928368 − 0.371662i 0.654861 − 0.755750i 0.723734 + 0.690079i −3.38201 + 2.17348i −0.888835 + 0.458227i 2.18713 − 1.71998i −0.415415 − 0.909632i −0.142315 − 0.989821i 3.94755 − 0.760828i
55.2 −0.928368 − 0.371662i 0.654861 − 0.755750i 0.723734 + 0.690079i −0.518636 + 0.333307i −0.888835 + 0.458227i −2.33617 + 1.83719i −0.415415 − 0.909632i −0.142315 − 0.989821i 0.605362 − 0.116674i
55.3 −0.928368 − 0.371662i 0.654861 − 0.755750i 0.723734 + 0.690079i 1.73685 − 1.11621i −0.888835 + 0.458227i 2.12944 − 1.67461i −0.415415 − 0.909632i −0.142315 − 0.989821i −2.02729 + 0.390728i
73.1 0.327068 − 0.945001i −0.415415 + 0.909632i −0.786053 − 0.618159i −3.48581 − 1.02353i 0.723734 + 0.690079i 2.43937 − 0.470150i −0.841254 + 0.540641i −0.654861 − 0.755750i −2.10733 + 2.95933i
73.2 0.327068 − 0.945001i −0.415415 + 0.909632i −0.786053 − 0.618159i 0.0606451 + 0.0178070i 0.723734 + 0.690079i −4.40937 + 0.849837i −0.841254 + 0.540641i −0.654861 − 0.755750i 0.0366627 − 0.0514855i
73.3 0.327068 − 0.945001i −0.415415 + 0.909632i −0.786053 − 0.618159i 2.50035 + 0.734168i 0.723734 + 0.690079i 1.06571 − 0.205399i −0.841254 + 0.540641i −0.654861 − 0.755750i 1.51157 − 2.12271i
103.1 0.786053 + 0.618159i 0.654861 + 0.755750i 0.235759 + 0.971812i −1.63481 − 1.05063i 0.0475819 + 0.998867i 2.79995 − 1.12093i −0.415415 + 0.909632i −0.142315 + 0.989821i −0.635593 − 1.83643i
103.2 0.786053 + 0.618159i 0.654861 + 0.755750i 0.235759 + 0.971812i 1.06109 + 0.681922i 0.0475819 + 0.998867i −3.89895 + 1.56090i −0.415415 + 0.909632i −0.142315 + 0.989821i 0.412538 + 1.19195i
103.3 0.786053 + 0.618159i 0.654861 + 0.755750i 0.235759 + 0.971812i 1.29445 + 0.831895i 0.0475819 + 0.998867i 2.59143 − 1.03745i −0.415415 + 0.909632i −0.142315 + 0.989821i 0.503266 + 1.45409i
121.1 0.786053 − 0.618159i 0.654861 − 0.755750i 0.235759 − 0.971812i −1.63481 + 1.05063i 0.0475819 − 0.998867i 2.79995 + 1.12093i −0.415415 − 0.909632i −0.142315 − 0.989821i −0.635593 + 1.83643i
121.2 0.786053 − 0.618159i 0.654861 − 0.755750i 0.235759 − 0.971812i 1.06109 − 0.681922i 0.0475819 − 0.998867i −3.89895 − 1.56090i −0.415415 − 0.909632i −0.142315 − 0.989821i 0.412538 − 1.19195i
121.3 0.786053 − 0.618159i 0.654861 − 0.755750i 0.235759 − 0.971812i 1.29445 − 0.831895i 0.0475819 − 0.998867i 2.59143 + 1.03745i −0.415415 − 0.909632i −0.142315 − 0.989821i 0.503266 − 1.45409i
127.1 0.888835 − 0.458227i 0.959493 + 0.281733i 0.580057 − 0.814576i −2.03671 + 2.35049i 0.981929 − 0.189251i −0.225927 + 4.74280i 0.142315 − 0.989821i 0.841254 + 0.540641i −0.733243 + 3.02247i
127.2 0.888835 − 0.458227i 0.959493 + 0.281733i 0.580057 − 0.814576i 0.793793 − 0.916086i 0.981929 − 0.189251i 0.147686 − 3.10031i 0.142315 − 0.989821i 0.841254 + 0.540641i 0.285777 − 1.17799i
See all 60 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 19.3
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
67.g even 33 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 402.2.m.b ✓ 60
67.g even 33 1 inner 402.2.m.b ✓ 60
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
402.2.m.b ✓ 60 1.a even 1 1 trivial
402.2.m.b ✓ 60 67.g even 33 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{60} - 4 T_{5}^{59} + 18 T_{5}^{58} + 26 T_{5}^{57} + 6 T_{5}^{56} - 19 T_{5}^{55} + 4534 T_{5}^{54} + \cdots + 4489 \) acting on \(S_{2}^{\mathrm{new}}(402, [\chi])\). Copy content Toggle raw display