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Properties

Label 671.2.ba.a
Level $671$
Weight $2$
Character orbit 671.ba
Analytic conductor $5.358$
Analytic rank $0$
Dimension $240$
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(64,671)]
 
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, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("671.64");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 671 = 11 \cdot 61 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 671.ba (of order \(10\), 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: \(240\)
Relative dimension: \(60\) over \(\Q(\zeta_{10})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 240 q - 5 q^{2} - q^{3} + 57 q^{4} - 5 q^{6} - 5 q^{7} - 5 q^{8} - 57 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 240 q - 5 q^{2} - q^{3} + 57 q^{4} - 5 q^{6} - 5 q^{7} - 5 q^{8} - 57 q^{9} - 3 q^{12} - 10 q^{13} - 2 q^{14} - 24 q^{15} - 67 q^{16} + 7 q^{19} + 18 q^{20} - 9 q^{22} + 35 q^{23} - 54 q^{25} + 35 q^{26} + 11 q^{27} - 35 q^{30} - 15 q^{33} - 24 q^{34} + 15 q^{35} + 111 q^{36} + 5 q^{37} + 5 q^{38} - 26 q^{39} + 15 q^{40} + 18 q^{41} + 48 q^{42} + 25 q^{43} - 5 q^{44} + 4 q^{45} - 17 q^{46} + 3 q^{47} - 63 q^{48} + 71 q^{49} + 30 q^{50} - 108 q^{52} + 10 q^{53} - 120 q^{54} + 15 q^{55} + 3 q^{56} - 21 q^{57} + 3 q^{58} - 15 q^{59} + 4 q^{60} + q^{61} + 125 q^{62} + 10 q^{63} + 47 q^{64} + 44 q^{65} - 65 q^{66} + 10 q^{67} + 30 q^{69} - 113 q^{70} - 65 q^{71} - 170 q^{72} - 14 q^{73} + 3 q^{74} + 4 q^{75} - 12 q^{76} - 13 q^{77} - 85 q^{78} - 45 q^{79} + 44 q^{80} - 24 q^{81} + 104 q^{83} - 110 q^{84} - 50 q^{85} - 148 q^{86} + 40 q^{87} - 105 q^{88} - 60 q^{89} + 5 q^{90} + 75 q^{91} - 15 q^{92} - 70 q^{93} + 144 q^{95} + 240 q^{96} + 30 q^{97} - 145 q^{98} + 55 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
64.1 −1.61291 + 2.21998i 0.810587 + 2.49473i −1.70879 − 5.25911i 0.896853 − 2.76023i −6.84564 − 2.22428i −3.57342 − 1.16108i 9.21175 + 2.99308i −3.13957 + 2.28103i 4.68110 + 6.44298i
64.2 −1.60421 + 2.20800i 0.775621 + 2.38712i −1.68375 − 5.18205i −1.26053 + 3.87952i −6.51501 − 2.11685i 0.727472 + 0.236370i 8.95174 + 2.90860i −2.66969 + 1.93964i −6.54383 − 9.00681i
64.3 −1.59007 + 2.18855i −0.912713 − 2.80904i −1.64338 − 5.05780i −0.0720450 + 0.221732i 7.59901 + 2.46907i 4.42937 + 1.43919i 8.53674 + 2.77375i −4.63062 + 3.36434i −0.370714 − 0.510244i
64.4 −1.51786 + 2.08916i −0.0861304 − 0.265082i −1.44265 − 4.44001i 0.389896 − 1.19998i 0.684533 + 0.222418i −0.963293 − 0.312993i 6.55373 + 2.12944i 2.36420 − 1.71769i 1.91514 + 2.63596i
64.5 −1.47446 + 2.02942i 0.218234 + 0.671655i −1.32648 − 4.08248i −0.444877 + 1.36919i −1.68485 − 0.547439i 1.89525 + 0.615805i 5.46944 + 1.77713i 2.02356 − 1.47020i −2.12271 − 2.92166i
64.6 −1.38250 + 1.90285i 0.552629 + 1.70082i −1.09150 − 3.35928i 0.558608 − 1.71922i −4.00041 − 1.29981i 2.34364 + 0.761494i 3.42735 + 1.11361i −0.160329 + 0.116486i 2.49914 + 3.43977i
64.7 −1.33661 + 1.83968i −0.978099 − 3.01028i −0.979882 − 3.01577i −0.0881129 + 0.271183i 6.84530 + 2.22417i −2.59708 − 0.843841i 2.53242 + 0.822833i −5.67805 + 4.12534i −0.381120 − 0.524566i
64.8 −1.32016 + 1.81705i −0.599198 − 1.84414i −0.940795 − 2.89547i 1.06963 − 3.29197i 4.14193 + 1.34579i −4.18646 − 1.36026i 2.23107 + 0.724919i −0.614768 + 0.446655i 4.56959 + 6.28950i
64.9 −1.30499 + 1.79616i −0.466845 − 1.43680i −0.905175 − 2.78584i −1.21780 + 3.74800i 3.18996 + 1.03648i −1.34910 − 0.438349i 1.96204 + 0.637505i 0.580597 − 0.421828i −5.14280 − 7.07846i
64.10 −1.16944 + 1.60960i −0.637034 − 1.96059i −0.605175 − 1.86254i 1.00031 − 3.07865i 3.90073 + 1.26742i 2.02907 + 0.659286i −0.0787371 − 0.0255832i −1.01104 + 0.734566i 3.78558 + 5.21040i
64.11 −1.16204 + 1.59941i 0.388556 + 1.19585i −0.589750 − 1.81506i −0.482778 + 1.48584i −2.36419 − 0.768170i −3.51853 − 1.14324i −0.172095 − 0.0559170i 1.14796 − 0.834043i −1.81546 − 2.49877i
64.12 −1.14188 + 1.57166i 0.0511602 + 0.157455i −0.548191 − 1.68716i 0.165424 − 0.509124i −0.305884 − 0.0993876i −1.34333 − 0.436474i −0.417587 − 0.135682i 2.40488 − 1.74724i 0.611274 + 0.841346i
64.13 −1.13659 + 1.56438i 0.858082 + 2.64090i −0.537417 − 1.65400i 0.639911 − 1.96944i −5.10666 − 1.65926i 3.43621 + 1.11649i −0.479772 − 0.155887i −3.81102 + 2.76887i 2.35364 + 3.23951i
64.14 −1.13466 + 1.56172i −0.238316 − 0.733462i −0.533490 − 1.64191i −0.739632 + 2.27635i 1.41587 + 0.460044i 4.65041 + 1.51101i −0.502286 − 0.163203i 1.94588 − 1.41376i −2.71580 − 3.73797i
64.15 −1.05367 + 1.45025i 0.619167 + 1.90560i −0.374972 − 1.15405i −0.923190 + 2.84129i −3.41599 − 1.10992i −3.60953 − 1.17281i −1.34099 − 0.435714i −0.820892 + 0.596413i −3.14784 − 4.33262i
64.16 −0.909703 + 1.25210i 0.482388 + 1.48464i −0.122157 − 0.375961i 1.18748 − 3.65468i −2.29774 − 0.746582i −1.82032 − 0.591457i −2.36199 − 0.767458i 0.455599 − 0.331012i 3.49576 + 4.81151i
64.17 −0.879769 + 1.21090i −0.647273 − 1.99210i −0.0742464 − 0.228507i −0.145196 + 0.446867i 2.98168 + 0.968807i 1.14979 + 0.373590i −2.50497 − 0.813915i −1.12245 + 0.815510i −0.413372 − 0.568957i
64.18 −0.698586 + 0.961521i 0.414519 + 1.27576i 0.181533 + 0.558702i −0.631910 + 1.94482i −1.51625 − 0.492658i 3.83798 + 1.24704i −2.92469 − 0.950290i 0.971319 − 0.705705i −1.42854 − 1.96622i
64.19 −0.664033 + 0.913964i −0.781050 − 2.40383i 0.223645 + 0.688309i 0.286455 − 0.881617i 2.71565 + 0.882369i 1.84816 + 0.600502i −2.92645 − 0.950862i −2.74129 + 1.99166i 0.615551 + 0.847233i
64.20 −0.629733 + 0.866753i 0.417837 + 1.28597i 0.263336 + 0.810466i 0.779551 − 2.39921i −1.37775 − 0.447657i 0.708178 + 0.230101i −2.90617 − 0.944270i 0.947918 − 0.688703i 1.58862 + 2.18654i
See next 80 embeddings (of 240 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 64.60
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
671.ba even 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 671.2.ba.a yes 240
11.c even 5 1 671.2.q.a ✓ 240
61.g even 10 1 671.2.q.a ✓ 240
671.ba even 10 1 inner 671.2.ba.a yes 240
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
671.2.q.a ✓ 240 11.c even 5 1
671.2.q.a ✓ 240 61.g even 10 1
671.2.ba.a yes 240 1.a even 1 1 trivial
671.2.ba.a yes 240 671.ba even 10 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(671, [\chi])\).