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Properties

Label 407.2.k.f
Level $407$
Weight $2$
Character orbit 407.k
Analytic conductor $3.250$
Analytic rank $0$
Dimension $32$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [407,2,Mod(122,407)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(407, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("407.122");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 407 = 11 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 407.k (of order \(6\), 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: \(3.24991136227\)
Analytic rank: \(0\)
Dimension: \(32\)
Relative dimension: \(16\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 32 q + 6 q^{2} + 3 q^{3} + 18 q^{4} + 9 q^{5} - q^{7} - 17 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 32 q + 6 q^{2} + 3 q^{3} + 18 q^{4} + 9 q^{5} - q^{7} - 17 q^{9} + 12 q^{10} + 32 q^{11} - q^{12} - 18 q^{13} - 18 q^{15} - 22 q^{16} - 9 q^{17} - 15 q^{18} + 3 q^{19} - 27 q^{20} + 19 q^{21} + 6 q^{22} + 60 q^{24} + 21 q^{25} - 32 q^{26} - 18 q^{27} + 41 q^{28} + 17 q^{30} - 69 q^{32} + 3 q^{33} - 16 q^{34} - 27 q^{35} - 36 q^{36} + 2 q^{37} + 26 q^{38} - 42 q^{39} + 23 q^{40} + 15 q^{41} - 18 q^{42} + 18 q^{44} - q^{46} - 34 q^{47} + 78 q^{48} - 9 q^{49} - 51 q^{50} + 27 q^{52} - 19 q^{53} - 57 q^{54} + 9 q^{55} - 6 q^{57} - 7 q^{58} + 3 q^{59} - 15 q^{61} - 8 q^{62} + 38 q^{63} - 24 q^{64} + 6 q^{65} - 9 q^{67} - 33 q^{69} - 102 q^{70} - 6 q^{71} - 48 q^{72} - 6 q^{73} + 56 q^{74} + 108 q^{75} + 33 q^{76} - q^{77} + 38 q^{78} + 18 q^{79} + 8 q^{81} - 31 q^{83} + 32 q^{84} + 38 q^{85} + 45 q^{86} - 15 q^{87} + 54 q^{89} - 109 q^{90} + 24 q^{91} - 63 q^{92} + 30 q^{93} - 84 q^{94} - 6 q^{95} + 135 q^{96} - 24 q^{98} - 17 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
122.1 −2.12131 + 1.22474i 1.17001 − 2.02651i 1.99997 − 3.46404i −3.70235 − 2.13755i 5.73180i −1.91773 + 3.32161i 4.89879i −1.23783 − 2.14398i 10.4718
122.2 −1.84049 + 1.06261i −0.863969 + 1.49644i 1.25826 − 2.17937i 2.39616 + 1.38342i − 3.67223i 0.232258 − 0.402283i 1.09772i 0.00711469 + 0.0123230i −5.88013
122.3 −1.82587 + 1.05416i 0.843332 − 1.46069i 1.22253 − 2.11748i −0.0176035 − 0.0101634i 3.55604i 0.916045 − 1.58664i 0.938320i 0.0775826 + 0.134377i 0.0428555
122.4 −1.38483 + 0.799534i −0.0892787 + 0.154635i 0.278510 − 0.482394i −1.29102 − 0.745371i − 0.285526i 0.506041 − 0.876489i − 2.30742i 1.48406 + 2.57046i 2.38380
122.5 −0.984490 + 0.568396i 1.16052 − 2.01008i −0.353853 + 0.612891i 3.56207 + 2.05656i 2.63854i 1.21506 − 2.10455i − 3.07810i −1.19361 − 2.06740i −4.67577
122.6 −0.700838 + 0.404629i −1.50628 + 2.60896i −0.672551 + 1.16489i −0.613248 − 0.354059i − 2.43794i 2.58683 − 4.48053i − 2.70705i −3.03777 − 5.26157i 0.573050
122.7 −0.339653 + 0.196099i −0.295301 + 0.511476i −0.923091 + 1.59884i 3.46181 + 1.99868i − 0.231632i −1.19245 + 2.06538i − 1.50846i 1.32559 + 2.29600i −1.56775
122.8 −0.0597273 + 0.0344836i 1.38683 − 2.40205i −0.997622 + 1.72793i −2.16653 − 1.25085i 0.191291i 1.10411 − 1.91237i − 0.275541i −2.34658 − 4.06439i 0.172535
122.9 0.407923 − 0.235514i −1.05955 + 1.83520i −0.889066 + 1.53991i −0.263792 − 0.152300i 0.998159i −0.399350 + 0.691695i 1.77961i −0.745302 − 1.29090i −0.143476
122.10 0.575504 − 0.332267i 0.636759 − 1.10290i −0.779197 + 1.34961i 0.746730 + 0.431125i − 0.846297i −1.31982 + 2.28600i 2.36468i 0.689076 + 1.19351i 0.572995
122.11 1.18144 − 0.682102i 0.240208 − 0.416052i −0.0694726 + 0.120330i 1.33009 + 0.767929i − 0.655385i 1.61585 − 2.79873i 2.91796i 1.38460 + 2.39820i 2.09522
122.12 1.58357 − 0.914272i −0.944712 + 1.63629i 0.671787 − 1.16357i 0.234055 + 0.135132i 3.45489i 0.270070 − 0.467775i 1.20031i −0.284961 − 0.493567i 0.494189
122.13 1.69017 − 0.975821i 1.66483 − 2.88357i 0.904452 − 1.56656i 2.24559 + 1.29649i − 6.49830i −2.13364 + 3.69558i 0.372952i −4.04331 − 7.00322i 5.06056
122.14 2.16876 − 1.25214i 0.817866 − 1.41659i 2.13569 − 3.69912i −1.35904 − 0.784642i − 4.09632i −0.995536 + 1.72432i − 5.68815i 0.162189 + 0.280920i −3.92991
122.15 2.25430 − 1.30152i −0.335375 + 0.580887i 2.38791 − 4.13597i −2.81973 − 1.62797i 1.74599i 0.919890 − 1.59330i − 7.22554i 1.27505 + 2.20845i −8.47534
122.16 2.39554 − 1.38307i −1.32588 + 2.29649i 2.82575 − 4.89435i 2.75681 + 1.59164i 7.33512i −1.90763 + 3.30411i − 10.1006i −2.01590 − 3.49164i 8.80540
397.1 −2.12131 − 1.22474i 1.17001 + 2.02651i 1.99997 + 3.46404i −3.70235 + 2.13755i − 5.73180i −1.91773 − 3.32161i − 4.89879i −1.23783 + 2.14398i 10.4718
397.2 −1.84049 − 1.06261i −0.863969 − 1.49644i 1.25826 + 2.17937i 2.39616 − 1.38342i 3.67223i 0.232258 + 0.402283i − 1.09772i 0.00711469 − 0.0123230i −5.88013
397.3 −1.82587 − 1.05416i 0.843332 + 1.46069i 1.22253 + 2.11748i −0.0176035 + 0.0101634i − 3.55604i 0.916045 + 1.58664i − 0.938320i 0.0775826 − 0.134377i 0.0428555
397.4 −1.38483 − 0.799534i −0.0892787 − 0.154635i 0.278510 + 0.482394i −1.29102 + 0.745371i 0.285526i 0.506041 + 0.876489i 2.30742i 1.48406 − 2.57046i 2.38380
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 122.16
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
37.e even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 407.2.k.f ✓ 32
37.e even 6 1 inner 407.2.k.f ✓ 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
407.2.k.f ✓ 32 1.a even 1 1 trivial
407.2.k.f ✓ 32 37.e even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{32} - 6 T_{2}^{31} - 7 T_{2}^{30} + 114 T_{2}^{29} - 19 T_{2}^{28} - 1341 T_{2}^{27} + 1276 T_{2}^{26} + \cdots + 144 \) acting on \(S_{2}^{\mathrm{new}}(407, [\chi])\). Copy content Toggle raw display