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Properties

Label 402.2.i.d
Level $402$
Weight $2$
Character orbit 402.i
Analytic conductor $3.210$
Analytic rank $0$
Dimension $30$
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(25,402)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(402, base_ring=CyclotomicField(22))
 
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("402.25");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 402 = 2 \cdot 3 \cdot 67 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 402.i (of order \(11\), degree \(10\), 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: \(30\)
Relative dimension: \(3\) over \(\Q(\zeta_{11})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{11}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 30 q + 3 q^{2} + 3 q^{3} - 3 q^{4} - q^{5} - 3 q^{6} - 3 q^{7} + 3 q^{8} - 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 30 q + 3 q^{2} + 3 q^{3} - 3 q^{4} - q^{5} - 3 q^{6} - 3 q^{7} + 3 q^{8} - 3 q^{9} + q^{10} + 15 q^{11} + 3 q^{12} - 16 q^{13} - 8 q^{14} - 10 q^{15} - 3 q^{16} + 9 q^{17} + 3 q^{18} - 24 q^{19} - q^{20} + 3 q^{21} - 4 q^{22} + 11 q^{23} - 3 q^{24} - 10 q^{25} + 5 q^{26} + 3 q^{27} + 8 q^{28} - 20 q^{29} - 12 q^{30} - 23 q^{31} + 3 q^{32} - 4 q^{33} + 13 q^{34} + 68 q^{35} - 3 q^{36} - 4 q^{37} + 2 q^{38} + 16 q^{39} + q^{40} + 34 q^{41} + 8 q^{42} - 23 q^{43} + 4 q^{44} - q^{45} - 11 q^{46} - 5 q^{47} + 3 q^{48} + 10 q^{49} + 21 q^{50} + 2 q^{51} + 28 q^{52} + 17 q^{53} - 3 q^{54} + 3 q^{56} - 20 q^{57} - 13 q^{58} - 35 q^{59} - 10 q^{60} - 44 q^{61} - 10 q^{62} + 8 q^{63} - 3 q^{64} - 39 q^{65} - 18 q^{66} - 12 q^{67} + 20 q^{68} + 11 q^{69} - 68 q^{70} - 32 q^{71} + 3 q^{72} + 32 q^{73} + 15 q^{74} + 10 q^{75} - 24 q^{76} - 76 q^{77} + 6 q^{78} + 16 q^{79} - q^{80} - 3 q^{81} - 12 q^{82} - 3 q^{83} - 8 q^{84} + 77 q^{85} - 10 q^{86} + 20 q^{87} - 15 q^{88} + 37 q^{89} + q^{90} - 11 q^{91} - 11 q^{92} + 12 q^{93} + 16 q^{94} + 13 q^{95} + 30 q^{96} - 66 q^{97} + 12 q^{98} + 4 q^{99}+O(q^{100}) \) 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 0.142315 + 0.989821i 0.654861 − 0.755750i −0.959493 + 0.281733i −2.68097 + 1.72295i 0.841254 + 0.540641i 0.192304 + 1.33750i −0.415415 − 0.909632i −0.142315 − 0.989821i −2.08696 − 2.40848i
25.2 0.142315 + 0.989821i 0.654861 − 0.755750i −0.959493 + 0.281733i −0.704731 + 0.452903i 0.841254 + 0.540641i −0.473416 − 3.29268i −0.415415 − 0.909632i −0.142315 − 0.989821i −0.548587 − 0.633103i
25.3 0.142315 + 0.989821i 0.654861 − 0.755750i −0.959493 + 0.281733i 2.30500 − 1.48133i 0.841254 + 0.540641i 0.578288 + 4.02208i −0.415415 − 0.909632i −0.142315 − 0.989821i 1.79429 + 2.07072i
91.1 −0.415415 + 0.909632i −0.841254 + 0.540641i −0.654861 − 0.755750i −0.570008 + 3.96449i −0.142315 − 0.989821i −0.128063 + 0.280419i 0.959493 − 0.281733i 0.415415 − 0.909632i −3.36944 − 2.16541i
91.2 −0.415415 + 0.909632i −0.841254 + 0.540641i −0.654861 − 0.755750i 0.0686591 − 0.477534i −0.142315 − 0.989821i 0.183415 − 0.401622i 0.959493 − 0.281733i 0.415415 − 0.909632i 0.405858 + 0.260829i
91.3 −0.415415 + 0.909632i −0.841254 + 0.540641i −0.654861 − 0.755750i 0.525425 − 3.65441i −0.142315 − 0.989821i −1.81202 + 3.96777i 0.959493 − 0.281733i 0.415415 − 0.909632i 3.10590 + 1.99604i
193.1 0.142315 − 0.989821i 0.654861 + 0.755750i −0.959493 − 0.281733i −2.68097 − 1.72295i 0.841254 − 0.540641i 0.192304 − 1.33750i −0.415415 + 0.909632i −0.142315 + 0.989821i −2.08696 + 2.40848i
193.2 0.142315 − 0.989821i 0.654861 + 0.755750i −0.959493 − 0.281733i −0.704731 − 0.452903i 0.841254 − 0.540641i −0.473416 + 3.29268i −0.415415 + 0.909632i −0.142315 + 0.989821i −0.548587 + 0.633103i
193.3 0.142315 − 0.989821i 0.654861 + 0.755750i −0.959493 − 0.281733i 2.30500 + 1.48133i 0.841254 − 0.540641i 0.578288 − 4.02208i −0.415415 + 0.909632i −0.142315 + 0.989821i 1.79429 − 2.07072i
223.1 0.959493 − 0.281733i 0.142315 + 0.989821i 0.841254 − 0.540641i −1.07319 + 2.34996i 0.415415 + 0.909632i 3.57575 − 1.04994i 0.654861 − 0.755750i −0.959493 + 0.281733i −0.367659 + 2.55712i
223.2 0.959493 − 0.281733i 0.142315 + 0.989821i 0.841254 − 0.540641i −0.927019 + 2.02989i 0.415415 + 0.909632i −4.61223 + 1.35427i 0.654861 − 0.755750i −0.959493 + 0.281733i −0.317582 + 2.20883i
223.3 0.959493 − 0.281733i 0.142315 + 0.989821i 0.841254 − 0.540641i 0.787619 − 1.72464i 0.415415 + 0.909632i 1.63829 − 0.481044i 0.654861 − 0.755750i −0.959493 + 0.281733i 0.269826 − 1.87668i
241.1 0.654861 + 0.755750i −0.415415 + 0.909632i −0.142315 + 0.989821i −2.50916 − 0.736756i −0.959493 + 0.281733i −2.25892 − 2.60693i −0.841254 + 0.540641i −0.654861 − 0.755750i −1.08635 − 2.37877i
241.2 0.654861 + 0.755750i −0.415415 + 0.909632i −0.142315 + 0.989821i 0.542248 + 0.159218i −0.959493 + 0.281733i 1.96503 + 2.26777i −0.841254 + 0.540641i −0.654861 − 0.755750i 0.234768 + 0.514070i
241.3 0.654861 + 0.755750i −0.415415 + 0.909632i −0.142315 + 0.989821i 4.18307 + 1.22826i −0.959493 + 0.281733i 0.0333362 + 0.0384720i −0.841254 + 0.540641i −0.654861 − 0.755750i 1.81107 + 3.96570i
265.1 0.959493 + 0.281733i 0.142315 − 0.989821i 0.841254 + 0.540641i −1.07319 − 2.34996i 0.415415 − 0.909632i 3.57575 + 1.04994i 0.654861 + 0.755750i −0.959493 − 0.281733i −0.367659 − 2.55712i
265.2 0.959493 + 0.281733i 0.142315 − 0.989821i 0.841254 + 0.540641i −0.927019 − 2.02989i 0.415415 − 0.909632i −4.61223 − 1.35427i 0.654861 + 0.755750i −0.959493 − 0.281733i −0.317582 − 2.20883i
265.3 0.959493 + 0.281733i 0.142315 − 0.989821i 0.841254 + 0.540641i 0.787619 + 1.72464i 0.415415 − 0.909632i 1.63829 + 0.481044i 0.654861 + 0.755750i −0.959493 − 0.281733i 0.269826 + 1.87668i
277.1 −0.841254 − 0.540641i 0.959493 + 0.281733i 0.415415 + 0.909632i −2.39503 + 2.76401i −0.654861 − 0.755750i −0.456232 − 0.293203i 0.142315 − 0.989821i 0.841254 + 0.540641i 3.50916 − 1.03038i
277.2 −0.841254 − 0.540641i 0.959493 + 0.281733i 0.415415 + 0.909632i −0.207890 + 0.239917i −0.654861 − 0.755750i −1.95051 − 1.25352i 0.142315 − 0.989821i 0.841254 + 0.540641i 0.304597 − 0.0894377i
See all 30 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
67.e even 11 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 402.2.i.d ✓ 30
67.e even 11 1 inner 402.2.i.d ✓ 30
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
402.2.i.d ✓ 30 1.a even 1 1 trivial
402.2.i.d ✓ 30 67.e even 11 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{30} + T_{5}^{29} + 13 T_{5}^{28} - 42 T_{5}^{27} - 41 T_{5}^{26} - 756 T_{5}^{25} + \cdots + 197346304 \) acting on \(S_{2}^{\mathrm{new}}(402, [\chi])\). Copy content Toggle raw display