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Properties

Label 671.2.l.b
Level $671$
Weight $2$
Character orbit 671.l
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(9,671)]
 
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, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("671.9");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 671 = 11 \cdot 61 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 671.l (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.

\(\operatorname{Tr}(f)(q) = \) \( 236 q + 2 q^{2} - 6 q^{3} - 62 q^{4} + 4 q^{5} - 8 q^{6} - 4 q^{7} + 2 q^{8} + 210 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 236 q + 2 q^{2} - 6 q^{3} - 62 q^{4} + 4 q^{5} - 8 q^{6} - 4 q^{7} + 2 q^{8} + 210 q^{9} - 34 q^{10} - 20 q^{11} - 10 q^{12} + 5 q^{13} - 6 q^{14} + q^{15} - 70 q^{16} - 10 q^{17} - 10 q^{18} - 42 q^{19} + 16 q^{20} - 62 q^{21} + 21 q^{22} + 7 q^{23} - 8 q^{24} - 53 q^{25} + 44 q^{26} - 10 q^{28} - 37 q^{29} + 12 q^{30} - 3 q^{31} + 46 q^{32} - 21 q^{33} - 11 q^{34} + 7 q^{35} - 84 q^{36} + 40 q^{37} + 7 q^{38} + 14 q^{39} + 56 q^{40} + 25 q^{41} - 76 q^{42} - 18 q^{43} + 27 q^{44} - 30 q^{45} - 11 q^{46} - 21 q^{47} + 31 q^{48} + 196 q^{49} - 17 q^{50} + 12 q^{51} + 41 q^{52} + 10 q^{53} - 56 q^{54} + 41 q^{55} + 35 q^{56} - 186 q^{57} + 84 q^{58} - 10 q^{59} - 52 q^{60} - 27 q^{61} - 34 q^{62} - 16 q^{63} - 90 q^{64} + 20 q^{65} - 22 q^{66} + 49 q^{67} - 58 q^{68} + 21 q^{69} - 35 q^{70} - 84 q^{71} - 58 q^{72} + 28 q^{73} - 27 q^{74} - 60 q^{75} + 31 q^{76} + 42 q^{77} + 67 q^{78} - 26 q^{79} + 52 q^{80} + 164 q^{81} + 14 q^{82} + 4 q^{83} - 33 q^{84} + 30 q^{85} - 10 q^{86} - 101 q^{87} + 44 q^{88} - 3 q^{89} - 12 q^{90} - 2 q^{91} - 70 q^{92} - 42 q^{93} + 67 q^{94} + 45 q^{95} + 154 q^{96} + 23 q^{97} + 73 q^{98} - 44 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
9.1 −2.23304 + 1.62240i 2.83178 1.73625 − 5.34363i 1.45219 − 1.05508i −6.32346 + 4.59426i 1.45894 3.08648 + 9.49922i 5.01896 −1.53104 + 4.71207i
9.2 −2.21142 + 1.60669i −1.67328 1.69088 − 5.20401i −0.250239 + 0.181809i 3.70033 − 2.68844i −2.35718 2.93260 + 9.02561i −0.200127 0.261272 − 0.804113i
9.3 −2.07895 + 1.51045i −0.912140 1.42256 − 4.37819i −2.51371 + 1.82632i 1.89630 − 1.37774i 1.98321 2.06742 + 6.36285i −2.16800 2.46733 − 7.59366i
9.4 −2.04621 + 1.48666i 0.756283 1.35878 − 4.18191i 2.56589 − 1.86423i −1.54751 + 1.12433i −3.96077 1.87355 + 5.76618i −2.42804 −2.47887 + 7.62919i
9.5 −2.03468 + 1.47828i 1.81211 1.33657 − 4.11354i −2.28382 + 1.65929i −3.68706 + 2.67881i 2.05265 1.80712 + 5.56175i 0.283747 2.19394 − 6.75226i
9.6 −2.00855 + 1.45930i −3.17897 1.28670 − 3.96005i −0.702432 + 0.510347i 6.38513 − 4.63907i 0.781889 1.66010 + 5.10925i 7.10585 0.666124 − 2.05012i
9.7 −1.97137 + 1.43229i −2.10168 1.21683 − 3.74503i 3.45938 − 2.51339i 4.14320 − 3.01021i 1.58222 1.45912 + 4.49072i 1.41707 −3.21984 + 9.90964i
9.8 −1.78030 + 1.29346i 0.586665 0.878387 − 2.70340i 1.08185 − 0.786008i −1.04444 + 0.758830i 4.30102 0.572927 + 1.76329i −2.65582 −0.909341 + 2.79866i
9.9 −1.77948 + 1.29287i −0.543356 0.877001 − 2.69913i −0.888707 + 0.645684i 0.966889 − 0.702486i −2.76711 0.569612 + 1.75309i −2.70476 0.746651 − 2.29796i
9.10 −1.69204 + 1.22934i 0.885649 0.733693 − 2.25808i 0.846094 − 0.614723i −1.49856 + 1.08876i −1.09239 0.241899 + 0.744487i −2.21562 −0.675922 + 2.08027i
9.11 −1.60220 + 1.16407i 3.17631 0.593958 − 1.82802i 1.26519 − 0.919215i −5.08908 + 3.69744i −1.19949 −0.0476802 − 0.146745i 7.08896 −0.957061 + 2.94553i
9.12 −1.50215 + 1.09138i −2.33637 0.447324 − 1.37672i −0.160613 + 0.116692i 3.50959 − 2.54987i 3.71476 −0.316968 − 0.975528i 2.45864 0.113910 − 0.350579i
9.13 −1.46816 + 1.06668i 2.72227 0.399657 − 1.23002i −2.69372 + 1.95710i −3.99674 + 2.90381i −2.53381 −0.396299 − 1.21968i 4.41078 1.86721 − 5.74669i
9.14 −1.39754 + 1.01537i −1.04278 0.304107 − 0.935945i −1.75681 + 1.27639i 1.45733 − 1.05881i −4.05807 −0.542296 − 1.66901i −1.91261 1.15919 − 3.56763i
9.15 −1.30900 + 0.951042i −0.981660 0.190958 − 0.587708i 1.67455 − 1.21663i 1.28499 − 0.933600i 1.04402 −0.691013 − 2.12672i −2.03634 −1.03491 + 3.18514i
9.16 −1.12439 + 0.816917i −3.23630 −0.0211356 + 0.0650486i −2.12691 + 1.54529i 3.63886 − 2.64379i −0.283651 −0.888332 − 2.73400i 7.47365 1.12910 − 3.47502i
9.17 −1.09692 + 0.796960i 0.343214 −0.0499429 + 0.153708i −2.35653 + 1.71212i −0.376478 + 0.273528i 1.25261 −0.905689 − 2.78743i −2.88220 1.22044 − 3.75611i
9.18 −1.09537 + 0.795833i 2.17204 −0.0515483 + 0.158649i 3.20402 − 2.32786i −2.37919 + 1.72858i 4.30691 −0.906582 − 2.79017i 1.71776 −1.65700 + 5.09973i
9.19 −1.06778 + 0.775787i −2.86438 −0.0797265 + 0.245373i 2.17092 − 1.57727i 3.05852 − 2.22215i −4.24298 −0.920938 − 2.83436i 5.20466 −1.09444 + 3.36835i
9.20 −0.814801 + 0.591988i 0.660566 −0.304583 + 0.937409i 2.64027 − 1.91827i −0.538230 + 0.391047i −1.96807 −0.929213 − 2.85982i −2.56365 −1.01570 + 3.12601i
See next 80 embeddings (of 236 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 9.59
Significant digits:
Format:

Inner twists

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{236} - 2 T_{2}^{235} + 92 T_{2}^{234} - 178 T_{2}^{233} + 4445 T_{2}^{232} + \cdots + 50\!\cdots\!81 \) acting on \(S_{2}^{\mathrm{new}}(671, [\chi])\). Copy content Toggle raw display