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Properties

Label 1407.2.i.g
Level $1407$
Weight $2$
Character orbit 1407.i
Analytic conductor $11.235$
Analytic rank $0$
Dimension $34$
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] = [1407,2,Mod(403,1407)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1407, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 4, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1407.403");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1407 = 3 \cdot 7 \cdot 67 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1407.i (of order \(3\), degree \(2\), 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: \(11.2349515644\)
Analytic rank: \(0\)
Dimension: \(34\)
Relative dimension: \(17\) over \(\Q(\zeta_{3})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 34 q + 5 q^{2} + 17 q^{3} - 15 q^{4} + 2 q^{5} + 10 q^{6} - 2 q^{7} - 6 q^{8} - 17 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 34 q + 5 q^{2} + 17 q^{3} - 15 q^{4} + 2 q^{5} + 10 q^{6} - 2 q^{7} - 6 q^{8} - 17 q^{9} - 5 q^{10} + 14 q^{11} + 15 q^{12} + 44 q^{13} - 9 q^{14} + 4 q^{15} - 7 q^{16} + 6 q^{17} + 5 q^{18} - 18 q^{19} - 18 q^{20} - q^{21} - 18 q^{22} + 20 q^{23} - 3 q^{24} - 7 q^{25} + q^{26} - 34 q^{27} + 4 q^{28} - 60 q^{29} + 5 q^{30} - 17 q^{31} + 46 q^{32} - 14 q^{33} + 18 q^{34} - 8 q^{35} + 30 q^{36} + 8 q^{37} + 4 q^{38} + 22 q^{39} - 34 q^{40} - 44 q^{41} + 6 q^{42} + 18 q^{43} + 27 q^{44} + 2 q^{45} + 41 q^{46} + 4 q^{47} - 14 q^{48} + 4 q^{50} - 6 q^{51} - 45 q^{52} + 42 q^{53} - 5 q^{54} + 54 q^{55} - 12 q^{56} - 36 q^{57} + 12 q^{58} + 24 q^{59} - 9 q^{60} - 45 q^{61} - 52 q^{62} + q^{63} - 34 q^{64} + 41 q^{65} - 9 q^{66} + 17 q^{67} - 23 q^{68} + 40 q^{69} + 26 q^{70} - 80 q^{71} + 3 q^{72} - 24 q^{73} + 20 q^{74} + 7 q^{75} - 26 q^{76} + 33 q^{77} + 2 q^{78} + 50 q^{79} + 26 q^{80} - 17 q^{81} - 24 q^{82} - 28 q^{83} + 26 q^{84} - 28 q^{85} - 7 q^{86} - 30 q^{87} + 13 q^{88} + 30 q^{89} + 10 q^{90} + 22 q^{91} + 2 q^{92} + 17 q^{93} + 40 q^{94} + 27 q^{95} - 46 q^{96} + 26 q^{97} - 65 q^{98} - 28 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
403.1 −1.14021 − 1.97491i 0.500000 − 0.866025i −1.60018 + 2.77159i −0.244200 − 0.422967i −2.28043 0.452440 − 2.60678i 2.73733 −0.500000 − 0.866025i −0.556881 + 0.964547i
403.2 −1.12732 − 1.95257i 0.500000 − 0.866025i −1.54169 + 2.67028i −1.06827 − 1.85030i −2.25463 −2.06809 + 1.65015i 2.44262 −0.500000 − 0.866025i −2.40856 + 4.17175i
403.3 −1.06442 − 1.84363i 0.500000 − 0.866025i −1.26599 + 2.19275i 0.105609 + 0.182921i −2.12884 1.93721 − 1.80200i 1.13248 −0.500000 − 0.866025i 0.224826 − 0.389410i
403.4 −0.898918 − 1.55697i 0.500000 − 0.866025i −0.616107 + 1.06713i 1.96648 + 3.40604i −1.79784 −0.844488 + 2.50736i −1.38035 −0.500000 − 0.866025i 3.53540 − 6.12349i
403.5 −0.558593 − 0.967512i 0.500000 − 0.866025i 0.375947 − 0.651159i −1.59172 − 2.75694i −1.11719 2.49075 − 0.892280i −3.07438 −0.500000 − 0.866025i −1.77825 + 3.08002i
403.6 −0.355152 − 0.615141i 0.500000 − 0.866025i 0.747734 − 1.29511i 1.13644 + 1.96838i −0.710304 −1.47329 − 2.19759i −2.48285 −0.500000 − 0.866025i 0.807221 − 1.39815i
403.7 0.0928522 + 0.160825i 0.500000 − 0.866025i 0.982757 − 1.70218i 0.625600 + 1.08357i 0.185704 −2.55838 + 0.674325i 0.736413 −0.500000 − 0.866025i −0.116177 + 0.201224i
403.8 0.217012 + 0.375875i 0.500000 − 0.866025i 0.905812 − 1.56891i 0.753035 + 1.30429i 0.434023 0.778121 + 2.52874i 1.65433 −0.500000 − 0.866025i −0.326835 + 0.566094i
403.9 0.231611 + 0.401162i 0.500000 − 0.866025i 0.892712 − 1.54622i 0.00385934 + 0.00668457i 0.463222 −1.61382 − 2.09656i 1.75349 −0.500000 − 0.866025i −0.00178773 + 0.00309644i
403.10 0.324600 + 0.562224i 0.500000 − 0.866025i 0.789270 − 1.36706i −2.03114 − 3.51804i 0.649200 −1.36055 − 2.26912i 2.32319 −0.500000 − 0.866025i 1.31862 − 2.28391i
403.11 0.615166 + 1.06550i 0.500000 − 0.866025i 0.243141 − 0.421133i 1.88685 + 3.26811i 1.23033 2.06214 + 1.65759i 3.05895 −0.500000 − 0.866025i −2.32145 + 4.02086i
403.12 0.680933 + 1.17941i 0.500000 − 0.866025i 0.0726608 − 0.125852i −1.10628 − 1.91613i 1.36187 −0.132308 + 2.64244i 2.92164 −0.500000 − 0.866025i 1.50660 − 2.60951i
403.13 0.803186 + 1.39116i 0.500000 − 0.866025i −0.290216 + 0.502668i −0.752571 − 1.30349i 1.60637 −2.37384 + 1.16829i 2.28036 −0.500000 − 0.866025i 1.20891 − 2.09389i
403.14 0.872206 + 1.51070i 0.500000 − 0.866025i −0.521485 + 0.903239i −0.223106 − 0.386432i 1.74441 2.40138 + 1.11058i 1.66945 −0.500000 − 0.866025i 0.389189 − 0.674096i
403.15 1.21459 + 2.10374i 0.500000 − 0.866025i −1.95047 + 3.37832i −0.488409 − 0.845950i 2.42919 2.52951 + 0.775607i −4.61774 −0.500000 − 0.866025i 1.18644 − 2.05497i
403.16 1.26856 + 2.19722i 0.500000 − 0.866025i −2.21850 + 3.84256i 1.82544 + 3.16175i 2.53713 1.35499 − 2.27244i −6.18299 −0.500000 − 0.866025i −4.63136 + 8.02176i
403.17 1.32390 + 2.29305i 0.500000 − 0.866025i −2.50540 + 4.33948i 0.202395 + 0.350559i 2.64779 −2.58178 − 0.578291i −7.97195 −0.500000 − 0.866025i −0.535900 + 0.928207i
604.1 −1.14021 + 1.97491i 0.500000 + 0.866025i −1.60018 − 2.77159i −0.244200 + 0.422967i −2.28043 0.452440 + 2.60678i 2.73733 −0.500000 + 0.866025i −0.556881 − 0.964547i
604.2 −1.12732 + 1.95257i 0.500000 + 0.866025i −1.54169 − 2.67028i −1.06827 + 1.85030i −2.25463 −2.06809 − 1.65015i 2.44262 −0.500000 + 0.866025i −2.40856 − 4.17175i
604.3 −1.06442 + 1.84363i 0.500000 + 0.866025i −1.26599 − 2.19275i 0.105609 − 0.182921i −2.12884 1.93721 + 1.80200i 1.13248 −0.500000 + 0.866025i 0.224826 + 0.389410i
See all 34 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 403.17
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1407.2.i.g ✓ 34
7.c even 3 1 inner 1407.2.i.g ✓ 34
7.c even 3 1 9849.2.a.bk 17
7.d odd 6 1 9849.2.a.bl 17
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1407.2.i.g ✓ 34 1.a even 1 1 trivial
1407.2.i.g ✓ 34 7.c even 3 1 inner
9849.2.a.bk 17 7.c even 3 1
9849.2.a.bl 17 7.d odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1407, [\chi])\):

\( T_{2}^{34} - 5 T_{2}^{33} + 37 T_{2}^{32} - 128 T_{2}^{31} + 614 T_{2}^{30} - 1795 T_{2}^{29} + \cdots + 841 \) Copy content Toggle raw display
\( T_{5}^{34} - 2 T_{5}^{33} + 48 T_{5}^{32} - 56 T_{5}^{31} + 1346 T_{5}^{30} - 1103 T_{5}^{29} + 23940 T_{5}^{28} + \cdots + 9 \) Copy content Toggle raw display
\( T_{11}^{34} - 14 T_{11}^{33} + 179 T_{11}^{32} - 1468 T_{11}^{31} + 11776 T_{11}^{30} + \cdots + 87828064164 \) Copy content Toggle raw display