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Properties

Label 1407.2.i.h
Level $1407$
Weight $2$
Character orbit 1407.i
Analytic conductor $11.235$
Analytic rank $0$
Dimension $38$
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: \(38\)
Relative dimension: \(19\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 38 q + 5 q^{2} - 19 q^{3} - 13 q^{4} - 2 q^{5} - 10 q^{6} + 2 q^{7} - 30 q^{8} - 19 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 38 q + 5 q^{2} - 19 q^{3} - 13 q^{4} - 2 q^{5} - 10 q^{6} + 2 q^{7} - 30 q^{8} - 19 q^{9} + 9 q^{10} + 12 q^{11} - 13 q^{12} - 44 q^{13} - q^{14} + 4 q^{15} - 5 q^{16} + 2 q^{17} + 5 q^{18} + 14 q^{19} + 54 q^{20} - q^{21} - 18 q^{22} + 14 q^{23} + 15 q^{24} - 11 q^{25} + q^{26} + 38 q^{27} - 6 q^{28} - 52 q^{29} + 9 q^{30} + 13 q^{31} + 10 q^{32} + 12 q^{33} - 2 q^{34} + 12 q^{35} + 26 q^{36} + 16 q^{37} + 8 q^{38} + 22 q^{39} + 24 q^{40} + 28 q^{41} + 2 q^{42} - 54 q^{43} + 25 q^{44} - 2 q^{45} + q^{46} - 8 q^{47} + 10 q^{48} - 4 q^{49} - 104 q^{50} + 2 q^{51} + 59 q^{52} + 8 q^{53} + 5 q^{54} - 58 q^{55} + 18 q^{56} - 28 q^{57} + 28 q^{58} - 12 q^{59} - 27 q^{60} + 19 q^{61} + 12 q^{62} - q^{63} - 30 q^{64} + 19 q^{65} + 9 q^{66} - 19 q^{67} - 9 q^{68} - 28 q^{69} + 50 q^{70} - 120 q^{71} + 15 q^{72} + 50 q^{73} - 32 q^{74} - 11 q^{75} - 26 q^{76} + 45 q^{77} - 2 q^{78} + 56 q^{79} - 18 q^{80} - 19 q^{81} - 36 q^{82} + 36 q^{83} - 76 q^{85} + 17 q^{86} + 26 q^{87} - 63 q^{88} + 2 q^{89} - 18 q^{90} + 18 q^{91} - 98 q^{92} + 13 q^{93} + 50 q^{94} + 87 q^{95} + 10 q^{96} - 18 q^{97} + 41 q^{98} - 24 q^{99}+O(q^{100}) \) 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 −1.21666 − 2.10732i −0.500000 + 0.866025i −1.96053 + 3.39574i −0.0798710 − 0.138341i 2.43332 0.241557 − 2.63470i 4.67457 −0.500000 − 0.866025i −0.194352 + 0.336628i
403.2 −0.980819 − 1.69883i −0.500000 + 0.866025i −0.924012 + 1.60044i 0.491105 + 0.850619i 1.96164 1.48678 + 2.18849i −0.298122 −0.500000 − 0.866025i 0.963370 − 1.66861i
403.3 −0.953041 − 1.65072i −0.500000 + 0.866025i −0.816574 + 1.41435i −0.207893 − 0.360081i 1.90608 −2.58477 + 0.564758i −0.699249 −0.500000 − 0.866025i −0.396260 + 0.686343i
403.4 −0.812379 − 1.40708i −0.500000 + 0.866025i −0.319920 + 0.554118i −1.71297 − 2.96694i 1.62476 2.14209 + 1.55288i −2.20993 −0.500000 − 0.866025i −2.78316 + 4.82057i
403.5 −0.569613 − 0.986599i −0.500000 + 0.866025i 0.351081 − 0.608091i 1.47480 + 2.55443i 1.13923 2.28412 − 1.33522i −3.07838 −0.500000 − 0.866025i 1.68013 − 2.91008i
403.6 −0.481622 − 0.834194i −0.500000 + 0.866025i 0.536081 − 0.928519i −0.0469780 − 0.0813682i 0.963244 −1.50150 + 2.17842i −2.95924 −0.500000 − 0.866025i −0.0452512 + 0.0783774i
403.7 −0.304636 − 0.527644i −0.500000 + 0.866025i 0.814394 − 1.41057i 0.0160155 + 0.0277397i 0.609271 −0.156582 − 2.64111i −2.21092 −0.500000 − 0.866025i 0.00975779 − 0.0169010i
403.8 −0.134027 − 0.232141i −0.500000 + 0.866025i 0.964074 − 1.66982i −1.23111 − 2.13235i 0.268054 −1.42441 − 2.22958i −1.05295 −0.500000 − 0.866025i −0.330004 + 0.571583i
403.9 −0.0294378 − 0.0509878i −0.500000 + 0.866025i 0.998267 − 1.72905i 1.62638 + 2.81697i 0.0588757 −2.63719 + 0.212666i −0.235299 −0.500000 − 0.866025i 0.0957540 − 0.165851i
403.10 0.0511794 + 0.0886453i −0.500000 + 0.866025i 0.994761 − 1.72298i 1.02303 + 1.77193i −0.102359 1.33274 + 2.28556i 0.408363 −0.500000 − 0.866025i −0.104716 + 0.181373i
403.11 0.308565 + 0.534450i −0.500000 + 0.866025i 0.809576 − 1.40223i 1.42238 + 2.46364i −0.617129 2.64135 + 0.152582i 2.23348 −0.500000 − 0.866025i −0.877793 + 1.52038i
403.12 0.347017 + 0.601051i −0.500000 + 0.866025i 0.759159 − 1.31490i −0.992316 − 1.71874i −0.694034 −1.74510 + 1.98862i 2.44183 −0.500000 − 0.866025i 0.688701 − 1.19286i
403.13 0.737718 + 1.27777i −0.500000 + 0.866025i −0.0884559 + 0.153210i −1.74808 − 3.02777i −1.47544 −0.839821 − 2.50892i 2.68985 −0.500000 − 0.866025i 2.57918 − 4.46727i
403.14 0.857214 + 1.48474i −0.500000 + 0.866025i −0.469632 + 0.813426i −1.30702 − 2.26383i −1.71443 1.98453 + 1.74975i 1.81856 −0.500000 − 0.866025i 2.24079 − 3.88117i
403.15 0.955259 + 1.65456i −0.500000 + 0.866025i −0.825039 + 1.42901i 0.648118 + 1.12257i −1.91052 2.32016 − 1.27156i 0.668533 −0.500000 − 0.866025i −1.23824 + 2.14470i
403.16 0.974551 + 1.68797i −0.500000 + 0.866025i −0.899498 + 1.55798i 1.96554 + 3.40442i −1.94910 −1.71946 − 2.01083i 0.391775 −0.500000 − 0.866025i −3.83104 + 6.63555i
403.17 1.17565 + 2.03628i −0.500000 + 0.866025i −1.76429 + 3.05584i −1.28942 − 2.23335i −2.35129 2.05400 − 1.66766i −3.59415 −0.500000 − 0.866025i 3.03181 − 5.25126i
403.18 1.20334 + 2.08424i −0.500000 + 0.866025i −1.89603 + 3.28403i 0.373904 + 0.647621i −2.40667 −2.52049 + 0.804440i −4.31291 −0.500000 − 0.866025i −0.899864 + 1.55861i
403.19 1.37175 + 2.37594i −0.500000 + 0.866025i −2.76340 + 4.78636i −1.42561 − 2.46923i −2.74350 −0.357999 + 2.62142i −9.67581 −0.500000 − 0.866025i 3.91117 − 6.77434i
604.1 −1.21666 + 2.10732i −0.500000 − 0.866025i −1.96053 − 3.39574i −0.0798710 + 0.138341i 2.43332 0.241557 + 2.63470i 4.67457 −0.500000 + 0.866025i −0.194352 − 0.336628i
See all 38 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 403.19
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.h ✓ 38
7.c even 3 1 inner 1407.2.i.h ✓ 38
7.c even 3 1 9849.2.a.bn 19
7.d odd 6 1 9849.2.a.bm 19
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1407.2.i.h ✓ 38 1.a even 1 1 trivial
1407.2.i.h ✓ 38 7.c even 3 1 inner
9849.2.a.bm 19 7.d odd 6 1
9849.2.a.bn 19 7.c even 3 1

Hecke kernels

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

\( T_{2}^{38} - 5 T_{2}^{37} + 38 T_{2}^{36} - 125 T_{2}^{35} + 612 T_{2}^{34} - 1636 T_{2}^{33} + 6366 T_{2}^{32} + \cdots + 1 \) Copy content Toggle raw display
\( T_{5}^{38} + 2 T_{5}^{37} + 55 T_{5}^{36} + 102 T_{5}^{35} + 1753 T_{5}^{34} + 3049 T_{5}^{33} + \cdots + 2209 \) Copy content Toggle raw display
\( T_{11}^{38} - 12 T_{11}^{37} + 179 T_{11}^{36} - 1326 T_{11}^{35} + 12508 T_{11}^{34} + \cdots + 20139738503824 \) Copy content Toggle raw display