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Properties

Label 671.2.i.b
Level $671$
Weight $2$
Character orbit 671.i
Analytic conductor $5.358$
Analytic rank $0$
Dimension $108$
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(34,671)]
 
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([0, 8]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("671.34");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 671 = 11 \cdot 61 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 671.i (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: \(108\)
Relative dimension: \(27\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 108 q - 2 q^{3} - 34 q^{4} + 4 q^{5} + 6 q^{6} + 10 q^{7} - 18 q^{8} - 25 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 108 q - 2 q^{3} - 34 q^{4} + 4 q^{5} + 6 q^{6} + 10 q^{7} - 18 q^{8} - 25 q^{9} + 6 q^{10} - 108 q^{11} - 15 q^{12} - 26 q^{13} + 2 q^{14} + 8 q^{15} - 40 q^{16} + 4 q^{17} + 50 q^{18} + 8 q^{19} + 8 q^{20} - 64 q^{21} - 3 q^{23} + 71 q^{24} - 7 q^{25} + 29 q^{26} - 8 q^{27} + 42 q^{28} - 40 q^{29} - 36 q^{30} - 7 q^{31} - 6 q^{32} + 2 q^{33} - 4 q^{34} + 2 q^{35} + 30 q^{36} - 32 q^{37} - 3 q^{38} + 10 q^{39} - 154 q^{40} + 10 q^{41} - 36 q^{42} - 29 q^{43} + 34 q^{44} - 23 q^{45} + 45 q^{46} + 40 q^{47} + 140 q^{48} - 15 q^{49} + 6 q^{50} - 25 q^{51} - 7 q^{52} + 15 q^{53} - 2 q^{54} - 4 q^{55} + 35 q^{56} + 20 q^{57} + 77 q^{58} + 22 q^{59} + 72 q^{60} + 20 q^{61} + 80 q^{62} - 6 q^{63} - 30 q^{64} + 56 q^{65} - 6 q^{66} - 4 q^{67} - 109 q^{68} - 46 q^{69} + 41 q^{70} + 7 q^{71} - 264 q^{72} - 16 q^{73} - 44 q^{74} - 6 q^{75} + 22 q^{76} - 10 q^{77} - 60 q^{78} + 79 q^{79} + 57 q^{80} - 25 q^{81} - 28 q^{82} + 2 q^{83} + 87 q^{84} + 30 q^{85} - 10 q^{86} - 38 q^{87} + 18 q^{88} + 18 q^{89} + 62 q^{90} - 30 q^{91} - 6 q^{92} - 24 q^{93} - 16 q^{94} - 10 q^{95} + 65 q^{96} + 16 q^{97} + 9 q^{98} + 25 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
34.1 −2.23817 + 1.62613i 1.62923 − 1.18371i 1.74710 − 5.37701i −1.12969 + 3.47683i −1.72165 + 5.29869i 1.38855 + 1.00884i 3.12359 + 9.61343i 0.326187 − 1.00390i −3.12533 − 9.61877i
34.2 −2.03472 + 1.47831i −0.731310 + 0.531328i 1.33664 − 4.11377i −0.0507246 + 0.156114i 0.702543 − 2.16221i −1.21326 − 0.881484i 1.80734 + 5.56242i −0.674546 + 2.07604i −0.127575 − 0.392635i
34.3 −2.02871 + 1.47394i 0.336907 − 0.244777i 1.32512 − 4.07830i 0.844318 − 2.59854i −0.322699 + 0.993165i 4.02382 + 2.92348i 1.77310 + 5.45706i −0.873461 + 2.68824i 2.11723 + 6.51617i
34.4 −1.81176 + 1.31632i −1.72644 + 1.25433i 0.931737 − 2.86759i −0.221989 + 0.683212i 1.47679 − 4.54509i 0.869564 + 0.631776i 0.702523 + 2.16214i 0.480194 − 1.47789i −0.497135 − 1.53002i
34.5 −1.57226 + 1.14231i 2.15583 − 1.56630i 0.549088 − 1.68992i −0.499508 + 1.53733i −1.60032 + 4.92527i −0.211501 − 0.153664i −0.133991 − 0.412382i 1.26724 − 3.90018i −0.970755 − 2.98768i
34.6 −1.54615 + 1.12334i 1.69000 − 1.22786i 0.510639 − 1.57159i 0.921081 − 2.83480i −1.23369 + 3.79690i −2.08797 − 1.51700i −0.205246 − 0.631683i 0.421418 − 1.29699i 1.76032 + 5.41770i
34.7 −1.18635 + 0.861930i −1.84504 + 1.34050i 0.0464573 − 0.142981i 0.741569 − 2.28232i 1.03344 − 3.18060i −3.77805 − 2.74491i −0.838162 − 2.57960i 0.680185 − 2.09339i 1.08744 + 3.34679i
34.8 −1.03003 + 0.748358i 0.543252 − 0.394695i −0.117119 + 0.360455i 0.241791 − 0.744155i −0.264190 + 0.813094i 0.0279434 + 0.0203021i −0.935984 − 2.88066i −0.787713 + 2.42433i 0.307844 + 0.947446i
34.9 −0.861629 + 0.626010i −2.29217 + 1.66536i −0.267518 + 0.823336i 1.03731 − 3.19250i 0.932468 − 2.86984i 3.79190 + 2.75498i −0.943141 − 2.90269i 1.55357 − 4.78140i 1.10476 + 3.40012i
34.10 −0.852936 + 0.619695i −0.657643 + 0.477806i −0.274555 + 0.844993i −0.817488 + 2.51597i 0.264834 − 0.815076i −0.325438 − 0.236445i −0.941045 − 2.89624i −0.722855 + 2.22472i −0.861867 − 2.65255i
34.11 −0.760459 + 0.552506i −2.70431 + 1.96479i −0.344999 + 1.06180i −1.08340 + 3.33436i 0.970954 − 2.98829i −0.0693778 − 0.0504059i −0.905231 − 2.78601i 2.52581 − 7.77364i −1.01837 − 3.13423i
34.12 −0.427865 + 0.310862i 2.12795 − 1.54604i −0.531601 + 1.63610i −0.491131 + 1.51155i −0.429867 + 1.32300i 3.30098 + 2.39830i −0.608007 − 1.87125i 1.21086 − 3.72663i −0.259744 − 0.799411i
34.13 −0.200295 + 0.145523i −0.573642 + 0.416776i −0.599093 + 1.84382i 1.29845 − 3.99621i 0.0542474 − 0.166956i −0.123158 − 0.0894793i −0.301334 − 0.927412i −0.771687 + 2.37501i 0.321467 + 0.989375i
34.14 0.142884 − 0.103811i 2.73469 − 1.98687i −0.608395 + 1.87245i 0.520069 − 1.60061i 0.184484 − 0.567784i −3.19918 − 2.32434i 0.216605 + 0.666641i 2.60384 − 8.01380i −0.0918515 − 0.282690i
34.15 0.164681 − 0.119648i 1.09278 − 0.793948i −0.605230 + 1.86271i −0.765842 + 2.35702i 0.0849654 − 0.261497i −2.48994 − 1.80905i 0.249004 + 0.766356i −0.363246 + 1.11796i 0.155893 + 0.479788i
34.16 0.298751 − 0.217055i −1.23851 + 0.899829i −0.575895 + 1.77242i −0.511589 + 1.57451i −0.174693 + 0.537649i 2.71359 + 1.97154i 0.440889 + 1.35692i −0.202840 + 0.624276i 0.188918 + 0.581429i
34.17 0.458977 − 0.333466i −1.71323 + 1.24474i −0.518574 + 1.59601i −0.0476904 + 0.146776i −0.371256 + 1.14261i −1.95822 − 1.42273i 0.644828 + 1.98458i 0.458744 − 1.41187i 0.0270560 + 0.0832699i
34.18 0.729261 − 0.529839i −0.384786 + 0.279563i −0.366942 + 1.12933i 0.465877 − 1.43382i −0.132486 + 0.407749i 2.67691 + 1.94489i 0.887873 + 2.73259i −0.857146 + 2.63803i −0.419949 − 1.29247i
34.19 0.970978 − 0.705457i 2.28885 − 1.66295i −0.172905 + 0.532146i 0.537396 − 1.65393i 1.04929 − 3.22937i 2.66365 + 1.93525i 0.949281 + 2.92159i 1.54639 − 4.75931i −0.644980 − 1.98504i
34.20 1.02354 − 0.743648i 0.881334 − 0.640326i −0.123405 + 0.379802i −0.846466 + 2.60515i 0.425906 − 1.31080i −1.71951 − 1.24930i 0.938046 + 2.88701i −0.560320 + 1.72449i 1.07092 + 3.29596i
See next 80 embeddings (of 108 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 34.27
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
61.e 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.i.b ✓ 108
61.e even 5 1 inner 671.2.i.b ✓ 108
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
671.2.i.b ✓ 108 1.a even 1 1 trivial
671.2.i.b ✓ 108 61.e even 5 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{108} + 44 T_{2}^{106} + 6 T_{2}^{105} + 1068 T_{2}^{104} + 246 T_{2}^{103} + 19011 T_{2}^{102} + \cdots + 64304361 \) acting on \(S_{2}^{\mathrm{new}}(671, [\chi])\). Copy content Toggle raw display