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Properties

Label 402.2.m.d
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.

\(\operatorname{Tr}(f)(q) = \) \( 60 q + 3 q^{2} + 6 q^{3} + 3 q^{4} - 3 q^{6} + q^{7} - 6 q^{8} - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 60 q + 3 q^{2} + 6 q^{3} + 3 q^{4} - 3 q^{6} + q^{7} - 6 q^{8} - 6 q^{9} - 10 q^{11} - 3 q^{12} - 17 q^{13} - 13 q^{14} + 11 q^{15} + 3 q^{16} + 18 q^{17} + 3 q^{18} - 7 q^{19} - q^{21} - 2 q^{22} - 6 q^{23} + 6 q^{24} + 50 q^{25} + 16 q^{26} + 6 q^{27} - 21 q^{28} + 30 q^{29} + 11 q^{30} - 11 q^{31} + 3 q^{32} - q^{33} - 26 q^{34} + 46 q^{35} + 3 q^{36} + 27 q^{37} - 7 q^{38} + 17 q^{39} + 84 q^{41} - 9 q^{42} + 2 q^{43} + q^{44} - 28 q^{46} - 41 q^{47} - 3 q^{48} + 36 q^{49} + 30 q^{50} + 15 q^{51} + q^{52} - 27 q^{53} - 3 q^{54} - 3 q^{55} + q^{56} - 48 q^{57} - 16 q^{58} - 4 q^{59} - 22 q^{60} - 22 q^{61} - 44 q^{62} - 10 q^{63} - 6 q^{64} - 96 q^{65} - 42 q^{66} - 94 q^{67} - 36 q^{68} - 5 q^{69} + 7 q^{70} - 36 q^{71} - 6 q^{72} - 27 q^{73} - 39 q^{74} - 50 q^{75} - 19 q^{76} + 109 q^{77} - 16 q^{78} + 29 q^{79} - 6 q^{81} - 25 q^{82} - 61 q^{83} + 10 q^{84} + 47 q^{85} - 12 q^{86} + 25 q^{87} + 23 q^{88} - 39 q^{89} - 127 q^{91} + 12 q^{92} + 16 q^{94} + 48 q^{95} + 30 q^{96} - 5 q^{97} + 36 q^{98} + q^{99}+O(q^{100}) \) 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 −1.33858 − 1.54480i −0.981929 − 0.189251i −0.163220 − 3.42641i −0.142315 − 0.989821i 0.841254 − 0.540641i 0.481906 + 1.98644i
19.2 −0.888835 − 0.458227i 0.959493 − 0.281733i 0.580057 + 0.814576i −1.16137 − 1.34029i −0.981929 − 0.189251i 0.108143 + 2.27021i −0.142315 − 0.989821i 0.841254 − 0.540641i 0.418109 + 1.72347i
19.3 −0.888835 − 0.458227i 0.959493 − 0.281733i 0.580057 + 0.814576i 1.78277 + 2.05742i −0.981929 − 0.189251i 0.0827295 + 1.73671i −0.142315 − 0.989821i 0.841254 − 0.540641i −0.641821 − 2.64562i
49.1 0.580057 − 0.814576i −0.841254 − 0.540641i −0.327068 − 0.945001i −0.417875 − 2.90638i −0.928368 + 0.371662i −2.45115 − 0.234056i −0.959493 − 0.281733i 0.415415 + 0.909632i −2.60986 − 1.34548i
49.2 0.580057 − 0.814576i −0.841254 − 0.540641i −0.327068 − 0.945001i 0.221350 + 1.53952i −0.928368 + 0.371662i −2.88773 − 0.275745i −0.959493 − 0.281733i 0.415415 + 0.909632i 1.38245 + 0.712705i
49.3 0.580057 − 0.814576i −0.841254 − 0.540641i −0.327068 − 0.945001i 0.337551 + 2.34772i −0.928368 + 0.371662i 2.85783 + 0.272890i −0.959493 − 0.281733i 0.415415 + 0.909632i 2.10819 + 1.08685i
55.1 0.928368 + 0.371662i 0.654861 − 0.755750i 0.723734 + 0.690079i −3.56282 + 2.28968i 0.888835 − 0.458227i −3.24411 + 2.55120i 0.415415 + 0.909632i −0.142315 − 0.989821i −4.15860 + 0.801504i
55.2 0.928368 + 0.371662i 0.654861 − 0.755750i 0.723734 + 0.690079i 0.627443 − 0.403233i 0.888835 − 0.458227i 0.395650 − 0.311142i 0.415415 + 0.909632i −0.142315 − 0.989821i 0.732364 − 0.141152i
55.3 0.928368 + 0.371662i 0.654861 − 0.755750i 0.723734 + 0.690079i 2.45409 − 1.57715i 0.888835 − 0.458227i 0.126785 − 0.0997049i 0.415415 + 0.909632i −0.142315 − 0.989821i 2.86447 − 0.552080i
73.1 −0.327068 + 0.945001i −0.415415 + 0.909632i −0.786053 − 0.618159i −3.50582 − 1.02940i −0.723734 − 0.690079i 4.53427 − 0.873910i 0.841254 − 0.540641i −0.654861 − 0.755750i 2.11942 − 2.97632i
73.2 −0.327068 + 0.945001i −0.415415 + 0.909632i −0.786053 − 0.618159i −1.15828 − 0.340101i −0.723734 − 0.690079i −2.19716 + 0.423469i 0.841254 − 0.540641i −0.654861 − 0.755750i 0.700230 − 0.983336i
73.3 −0.327068 + 0.945001i −0.415415 + 0.909632i −0.786053 − 0.618159i 1.82029 + 0.534486i −0.723734 − 0.690079i 2.21355 − 0.426626i 0.841254 − 0.540641i −0.654861 − 0.755750i −1.10045 + 1.54536i
103.1 −0.786053 − 0.618159i 0.654861 + 0.755750i 0.235759 + 0.971812i −2.31138 − 1.48543i −0.0475819 − 0.998867i −0.164207 + 0.0657384i 0.415415 − 0.909632i −0.142315 + 0.989821i 0.898632 + 2.59643i
103.2 −0.786053 − 0.618159i 0.654861 + 0.755750i 0.235759 + 0.971812i 1.71884 + 1.10463i −0.0475819 − 0.998867i −3.09398 + 1.23864i 0.415415 − 0.909632i −0.142315 + 0.989821i −0.668262 − 1.93082i
103.3 −0.786053 − 0.618159i 0.654861 + 0.755750i 0.235759 + 0.971812i 2.99577 + 1.92527i −0.0475819 − 0.998867i 4.45333 − 1.78284i 0.415415 − 0.909632i −0.142315 + 0.989821i −1.16472 − 3.36523i
121.1 −0.786053 + 0.618159i 0.654861 − 0.755750i 0.235759 − 0.971812i −2.31138 + 1.48543i −0.0475819 + 0.998867i −0.164207 − 0.0657384i 0.415415 + 0.909632i −0.142315 − 0.989821i 0.898632 − 2.59643i
121.2 −0.786053 + 0.618159i 0.654861 − 0.755750i 0.235759 − 0.971812i 1.71884 − 1.10463i −0.0475819 + 0.998867i −3.09398 − 1.23864i 0.415415 + 0.909632i −0.142315 − 0.989821i −0.668262 + 1.93082i
121.3 −0.786053 + 0.618159i 0.654861 − 0.755750i 0.235759 − 0.971812i 2.99577 − 1.92527i −0.0475819 + 0.998867i 4.45333 + 1.78284i 0.415415 + 0.909632i −0.142315 − 0.989821i −1.16472 + 3.36523i
127.1 −0.888835 + 0.458227i 0.959493 + 0.281733i 0.580057 − 0.814576i −1.33858 + 1.54480i −0.981929 + 0.189251i −0.163220 + 3.42641i −0.142315 + 0.989821i 0.841254 + 0.540641i 0.481906 − 1.98644i
127.2 −0.888835 + 0.458227i 0.959493 + 0.281733i 0.580057 − 0.814576i −1.16137 + 1.34029i −0.981929 + 0.189251i 0.108143 − 2.27021i −0.142315 + 0.989821i 0.841254 + 0.540641i 0.418109 − 1.72347i
See all 60 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 19.3
Significant digits:
Format:

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{60} - 10 T_{5}^{58} + 68 T_{5}^{57} + 130 T_{5}^{56} - 1163 T_{5}^{55} + 3496 T_{5}^{54} + \cdots + 713159855732809 \) acting on \(S_{2}^{\mathrm{new}}(402, [\chi])\). Copy content Toggle raw display