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Properties

Label 451.2.e.a
Level $451$
Weight $2$
Character orbit 451.e
Analytic conductor $3.601$
Analytic rank $0$
Dimension $68$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [451,2,Mod(155,451)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(451, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("451.155");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 451 = 11 \cdot 41 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 451.e (of order \(4\), 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: \(3.60125313116\)
Analytic rank: \(0\)
Dimension: \(68\)
Relative dimension: \(34\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 68 q - 64 q^{4} + 4 q^{6}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 68 q - 64 q^{4} + 4 q^{6} + 12 q^{12} + 4 q^{13} + 4 q^{14} + 12 q^{15} + 56 q^{16} + 24 q^{17} + 24 q^{23} - 68 q^{25} - 40 q^{26} - 36 q^{27} - 32 q^{28} - 12 q^{29} - 32 q^{30} + 8 q^{31} - 8 q^{34} + 28 q^{35} - 24 q^{37} + 60 q^{38} + 16 q^{41} + 96 q^{45} - 24 q^{47} - 16 q^{48} - 64 q^{51} - 12 q^{52} - 4 q^{53} + 20 q^{54} + 28 q^{56} + 24 q^{57} - 24 q^{58} - 16 q^{59} - 48 q^{60} + 60 q^{63} - 136 q^{64} - 4 q^{65} - 24 q^{66} + 12 q^{67} - 104 q^{68} - 44 q^{69} + 108 q^{70} + 8 q^{71} - 104 q^{72} + 52 q^{75} + 44 q^{81} - 56 q^{82} - 16 q^{83} + 8 q^{85} + 80 q^{86} - 80 q^{89} - 56 q^{92} - 20 q^{93} + 48 q^{94} - 92 q^{95} + 44 q^{96} - 36 q^{97} + 112 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
155.1 − 2.71553i −1.32009 + 1.32009i −5.37410 − 1.70297i 3.58474 + 3.58474i −0.176561 + 0.176561i 9.16247i − 0.485268i −4.62447
155.2 − 2.69615i −0.113427 + 0.113427i −5.26922 3.71342i 0.305818 + 0.305818i 2.57489 − 2.57489i 8.81432i 2.97427i 10.0119
155.3 − 2.50201i 1.43873 − 1.43873i −4.26003 − 2.06563i −3.59971 − 3.59971i 0.276584 − 0.276584i 5.65461i − 1.13988i −5.16822
155.4 − 2.28124i −1.97944 + 1.97944i −3.20408 0.813373i 4.51558 + 4.51558i −2.20608 + 2.20608i 2.74680i − 4.83635i 1.85550
155.5 − 2.18323i 2.43917 − 2.43917i −2.76651 2.01122i −5.32527 − 5.32527i 0.480718 − 0.480718i 1.67348i − 8.89907i 4.39097
155.6 − 1.93069i 0.0341765 − 0.0341765i −1.72754 − 3.01638i −0.0659841 − 0.0659841i 1.97711 − 1.97711i − 0.526025i 2.99766i −5.82368
155.7 − 1.81465i −0.206711 + 0.206711i −1.29296 1.39086i 0.375108 + 0.375108i −3.12067 + 3.12067i − 1.28303i 2.91454i 2.52392
155.8 − 1.73973i −2.09978 + 2.09978i −1.02666 4.37589i 3.65305 + 3.65305i 1.28262 − 1.28262i − 1.69335i − 5.81817i 7.61286
155.9 − 1.61353i 1.27103 − 1.27103i −0.603472 2.15500i −2.05084 − 2.05084i 1.46605 − 1.46605i − 2.25334i − 0.231029i 3.47716
155.10 − 1.40609i 1.64640 − 1.64640i 0.0229010 − 2.35718i −2.31500 − 2.31500i −3.24598 + 3.24598i − 2.84439i − 2.42128i −3.31441
155.11 − 1.36244i −1.70164 + 1.70164i 0.143752 − 3.96241i 2.31838 + 2.31838i −1.76276 + 1.76276i − 2.92074i − 2.79114i −5.39856
155.12 − 0.905987i −1.70369 + 1.70369i 1.17919 − 0.399422i 1.54352 + 1.54352i 0.631667 − 0.631667i − 2.88030i − 2.80515i −0.361871
155.13 − 0.891232i −0.968258 + 0.968258i 1.20570 0.102170i 0.862943 + 0.862943i 2.45953 − 2.45953i − 2.85703i 1.12495i 0.0910568
155.14 − 0.786399i 0.902426 − 0.902426i 1.38158 4.08328i −0.709667 − 0.709667i −1.56732 + 1.56732i − 2.65927i 1.37125i 3.21109
155.15 − 0.614233i 1.12576 − 1.12576i 1.62272 0.823753i −0.691481 − 0.691481i 2.47870 − 2.47870i − 2.22519i 0.465312i 0.505976
155.16 − 0.207473i 0.168267 − 0.168267i 1.95695 − 2.06076i −0.0349109 − 0.0349109i −0.792444 + 0.792444i − 0.820963i 2.94337i −0.427553
155.17 − 0.0118674i 2.02298 − 2.02298i 1.99986 − 0.196042i −0.0240075 − 0.0240075i −1.26397 + 1.26397i − 0.0474680i − 5.18491i −0.00232651
155.18 0.0650163i −0.0112732 + 0.0112732i 1.99577 1.27940i −0.000732945 0 0.000732945i −1.57656 + 1.57656i 0.259790i 2.99975i −0.0831820
155.19 0.387717i 0.492945 − 0.492945i 1.84968 − 4.13307i 0.191123 + 0.191123i 1.62071 − 1.62071i 1.49258i 2.51401i 1.60246
155.20 0.460393i −0.896776 + 0.896776i 1.78804 − 2.81457i −0.412869 − 0.412869i −2.55862 + 2.55862i 1.74399i 1.39159i 1.29581
See all 68 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 155.34
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
41.c even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 451.2.e.a ✓ 68
41.c even 4 1 inner 451.2.e.a ✓ 68
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
451.2.e.a ✓ 68 1.a even 1 1 trivial
451.2.e.a ✓ 68 41.c even 4 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(451, [\chi])\).