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Properties

Label 671.2.m.b
Level $671$
Weight $2$
Character orbit 671.m
Analytic conductor $5.358$
Analytic rank $0$
Dimension $236$
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] = [671,2,Mod(20,671)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(671, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([6, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("671.20");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 671 = 11 \cdot 61 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 671.m (of order \(5\), degree \(4\), 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: \(5.35796197563\)
Analytic rank: \(0\)
Dimension: \(236\)
Relative dimension: \(59\) over \(\Q(\zeta_{5})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 236 q - 8 q^{2} - 2 q^{3} + 228 q^{4} + 4 q^{5} - 16 q^{6} - 9 q^{7} - 18 q^{8} - 59 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 236 q - 8 q^{2} - 2 q^{3} + 228 q^{4} + 4 q^{5} - 16 q^{6} - 9 q^{7} - 18 q^{8} - 59 q^{9} + 16 q^{10} - 15 q^{11} - 17 q^{12} - 8 q^{13} - 16 q^{14} + 15 q^{15} + 220 q^{16} - q^{17} + 3 q^{18} + 2 q^{19} - 29 q^{20} - 4 q^{22} - 10 q^{23} - 8 q^{24} - 63 q^{25} - 20 q^{26} + 10 q^{27} - 55 q^{28} - 31 q^{29} + 44 q^{30} + 13 q^{31} - 64 q^{32} + 18 q^{33} + 11 q^{34} - 28 q^{35} - 82 q^{36} + 5 q^{37} - 36 q^{38} - 5 q^{39} - 29 q^{40} + 19 q^{41} - 80 q^{42} + 28 q^{43} - 53 q^{44} + 66 q^{45} - 7 q^{46} - 8 q^{47} + 30 q^{48} - 84 q^{49} + 18 q^{50} + 24 q^{51} - 35 q^{52} + 21 q^{53} - 56 q^{54} + 21 q^{55} - 25 q^{56} - 11 q^{57} - 98 q^{58} + 8 q^{59} + 16 q^{60} - 10 q^{61} + 54 q^{62} - 68 q^{63} + 190 q^{64} + 32 q^{65} - 80 q^{66} + 29 q^{67} + 42 q^{69} + 130 q^{70} + 6 q^{71} - 198 q^{72} - 38 q^{73} + 18 q^{74} + 39 q^{75} + 19 q^{76} + 52 q^{77} + 15 q^{78} - 10 q^{79} - 128 q^{80} - 79 q^{81} - 64 q^{82} + 12 q^{83} + 146 q^{84} + 24 q^{85} + 108 q^{86} + 59 q^{87} - 21 q^{88} - 64 q^{89} - 179 q^{90} + 39 q^{91} - 104 q^{92} + 31 q^{93} - 104 q^{94} + 64 q^{95} + 56 q^{96} - 2 q^{97} - 7 q^{98} + 97 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
20.1 −2.80087 1.30096 + 0.945205i 5.84486 −1.66163 − 1.20724i −3.64382 − 2.64739i 2.27993 − 1.65647i −10.7690 −0.127959 − 0.393818i 4.65400 + 3.38133i
20.2 −2.78676 −1.77599 − 1.29033i 5.76604 −2.23648 − 1.62490i 4.94925 + 3.59584i −2.20360 + 1.60101i −10.4950 0.562128 + 1.73005i 6.23253 + 4.52820i
20.3 −2.56444 1.87532 + 1.36250i 4.57637 3.36912 + 2.44781i −4.80915 − 3.49405i 0.342862 − 0.249104i −6.60696 0.733365 + 2.25707i −8.63992 − 6.27727i
20.4 −2.54127 −1.44226 − 1.04786i 4.45806 2.32606 + 1.68998i 3.66518 + 2.66291i −1.57596 + 1.14500i −6.24661 0.0550462 + 0.169415i −5.91114 − 4.29470i
20.5 −2.42121 −0.461153 − 0.335048i 3.86228 1.11147 + 0.807528i 1.11655 + 0.811222i 3.00867 − 2.18593i −4.50898 −0.826645 − 2.54415i −2.69110 − 1.95520i
20.6 −2.39295 1.45839 + 1.05958i 3.72619 0.569802 + 0.413985i −3.48985 − 2.53553i −4.00231 + 2.90785i −4.13069 0.0771386 + 0.237408i −1.36351 − 0.990645i
20.7 −2.34854 1.51480 + 1.10056i 3.51564 −1.68956 − 1.22753i −3.55756 − 2.58472i −1.15635 + 0.840134i −3.55953 0.156315 + 0.481088i 3.96799 + 2.88291i
20.8 −2.22018 −0.121433 − 0.0882261i 2.92918 1.41936 + 1.03122i 0.269602 + 0.195877i −0.194823 + 0.141547i −2.06294 −0.920089 − 2.83174i −3.15122 − 2.28949i
20.9 −2.16016 −2.04100 − 1.48288i 2.66628 −2.19103 − 1.59187i 4.40889 + 3.20325i 3.13197 − 2.27551i −1.43927 1.03972 + 3.19994i 4.73296 + 3.43870i
20.10 −2.04387 −2.65041 − 1.92564i 2.17740 0.439414 + 0.319253i 5.41710 + 3.93575i −0.263542 + 0.191474i −0.362581 2.38956 + 7.35431i −0.898105 − 0.652512i
20.11 −1.83966 2.10951 + 1.53265i 1.38437 1.67802 + 1.21916i −3.88079 − 2.81956i 0.0344013 − 0.0249940i 1.13256 1.17397 + 3.61310i −3.08700 − 2.24284i
20.12 −1.83788 −0.159683 − 0.116017i 1.37780 −2.48081 − 1.80241i 0.293479 + 0.213225i 2.69557 − 1.95845i 1.14352 −0.915012 − 2.81612i 4.55943 + 3.31262i
20.13 −1.79446 −1.05890 − 0.769335i 1.22010 −2.87521 − 2.08896i 1.90016 + 1.38054i −0.836913 + 0.608053i 1.39950 −0.397661 − 1.22387i 5.15946 + 3.74857i
20.14 −1.73946 2.56544 + 1.86390i 1.02573 −3.40153 − 2.47135i −4.46249 − 3.24219i −0.565524 + 0.410877i 1.69471 2.18031 + 6.71032i 5.91683 + 4.29883i
20.15 −1.69401 1.55813 + 1.13204i 0.869656 −0.796648 − 0.578798i −2.63947 − 1.91769i 2.36680 − 1.71958i 1.91481 0.219180 + 0.674567i 1.34953 + 0.980488i
20.16 −1.46455 −2.13886 − 1.55397i 0.144901 2.41578 + 1.75516i 3.13247 + 2.27587i 0.673602 − 0.489401i 2.71688 1.23284 + 3.79431i −3.53802 − 2.57052i
20.17 −1.39233 −0.184756 − 0.134233i −0.0614310 2.10198 + 1.52718i 0.257240 + 0.186896i −3.39105 + 2.46374i 2.87018 −0.910935 − 2.80357i −2.92664 − 2.12633i
20.18 −1.38437 −1.91157 − 1.38883i −0.0835159 −1.31902 − 0.958323i 2.64632 + 1.92266i −3.46717 + 2.51905i 2.88436 0.798175 + 2.45653i 1.82601 + 1.32668i
20.19 −1.37587 −0.310644 − 0.225696i −0.106979 0.737195 + 0.535604i 0.427405 + 0.310528i −1.73309 + 1.25916i 2.89893 −0.881490 − 2.71295i −1.01429 − 0.736922i
20.20 −1.26163 0.326456 + 0.237184i −0.408290 1.71708 + 1.24753i −0.411867 − 0.299239i 2.10545 − 1.52970i 3.03837 −0.876734 − 2.69831i −2.16632 − 1.57393i
See next 80 embeddings (of 236 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 20.59
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
671.m even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 671.2.m.b yes 236
11.c even 5 1 671.2.h.b ✓ 236
61.e even 5 1 671.2.h.b ✓ 236
671.m even 5 1 inner 671.2.m.b yes 236
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
671.2.h.b ✓ 236 11.c even 5 1
671.2.h.b ✓ 236 61.e even 5 1
671.2.m.b yes 236 1.a even 1 1 trivial
671.2.m.b yes 236 671.m even 5 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{118} + 4 T_{2}^{117} - 167 T_{2}^{116} - 681 T_{2}^{115} + 13522 T_{2}^{114} + 56299 T_{2}^{113} + \cdots + 10109 \) acting on \(S_{2}^{\mathrm{new}}(671, [\chi])\). Copy content Toggle raw display