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Properties

Label 671.2.cf.a
Level $671$
Weight $2$
Character orbit 671.cf
Analytic conductor $5.358$
Analytic rank $0$
Dimension $400$
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(45,671)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(671, base_ring=CyclotomicField(30))
 
chi = DirichletCharacter(H, H._module([0, 17]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("671.45");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 671 = 11 \cdot 61 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 671.cf (of order \(30\), degree \(8\), 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: \(400\)
Relative dimension: \(50\) over \(\Q(\zeta_{30})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{30}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 400 q - 4 q^{3} - 38 q^{4} + 16 q^{5} - 32 q^{6} + 14 q^{7} - 76 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 400 q - 4 q^{3} - 38 q^{4} + 16 q^{5} - 32 q^{6} + 14 q^{7} - 76 q^{9} - 12 q^{10} - 94 q^{12} + 4 q^{13} - 34 q^{14} - 32 q^{15} + 70 q^{16} - 76 q^{18} + 44 q^{19} + 32 q^{20} + 60 q^{21} - 2 q^{22} - 20 q^{23} - 30 q^{24} + 30 q^{25} + 56 q^{26} - 16 q^{27} - 30 q^{29} - 92 q^{30} - 46 q^{31} + 90 q^{32} - 20 q^{34} + 24 q^{35} + 34 q^{36} - 10 q^{38} - 4 q^{39} - 36 q^{40} - 4 q^{41} + 32 q^{42} + 40 q^{43} + 64 q^{45} - 188 q^{46} + 88 q^{47} - 52 q^{48} - 74 q^{49} + 88 q^{51} + 52 q^{52} - 50 q^{53} + 84 q^{54} - 140 q^{56} - 164 q^{57} + 24 q^{58} + 62 q^{59} + 424 q^{60} + 10 q^{61} - 68 q^{62} + 2 q^{63} + 168 q^{64} + 182 q^{65} + 8 q^{66} - 74 q^{67} - 24 q^{68} - 40 q^{69} - 6 q^{70} + 24 q^{71} - 84 q^{73} - 48 q^{74} - 38 q^{75} + 24 q^{76} - 14 q^{77} + 100 q^{78} - 112 q^{79} - 92 q^{80} - 88 q^{81} + 66 q^{82} - 78 q^{83} - 140 q^{84} + 30 q^{85} - 50 q^{86} - 6 q^{87} + 24 q^{88} + 70 q^{89} + 24 q^{90} - 36 q^{91} + 54 q^{92} - 12 q^{93} - 140 q^{94} - 50 q^{95} - 196 q^{96} + 32 q^{97} - 10 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
45.1 −0.551892 + 2.59645i 0.485490 + 1.49419i −4.60987 − 2.05245i −0.201554 − 1.91766i −4.14751 + 0.435921i 2.73231 − 2.46019i 4.75273 − 6.54157i 0.430162 − 0.312531i 5.09034 + 0.535016i
45.2 −0.526907 + 2.47890i −0.980115 − 3.01648i −4.04023 − 1.79883i 0.0926318 + 0.881332i 7.99400 − 0.840203i −1.47113 + 1.32461i 3.60872 − 4.96697i −5.71150 + 4.14965i −2.23355 − 0.234755i
45.3 −0.513914 + 2.41778i −0.293851 − 0.904381i −3.75445 − 1.67159i −0.118472 − 1.12718i 2.33761 − 0.245692i −0.982534 + 0.884677i 3.06522 − 4.21892i 1.69549 − 1.23185i 2.78616 + 0.292837i
45.4 −0.505113 + 2.37637i 1.04936 + 3.22959i −3.56490 − 1.58719i 0.344206 + 3.27490i −8.20475 + 0.862354i 2.67976 − 2.41286i 2.71643 − 3.73885i −6.90208 + 5.01465i −7.95624 − 0.836235i
45.5 −0.480576 + 2.26093i 0.535666 + 1.64861i −3.05378 − 1.35963i −0.423022 − 4.02478i −3.98483 + 0.418822i −0.694722 + 0.625531i 1.82434 − 2.51099i −0.00392617 + 0.00285253i 9.30307 + 0.977792i
45.6 −0.467452 + 2.19919i −0.279122 − 0.859048i −2.79083 − 1.24256i −0.161081 − 1.53259i 2.01969 − 0.212277i −3.07385 + 2.76770i 1.39414 − 1.91888i 1.76700 − 1.28380i 3.44574 + 0.362162i
45.7 −0.457294 + 2.15140i 0.707185 + 2.17649i −2.59230 − 1.15417i −0.0309215 − 0.294198i −5.00589 + 0.526140i −1.46510 + 1.31918i 1.08289 − 1.49047i −1.80995 + 1.31501i 0.647077 + 0.0680105i
45.8 −0.456564 + 2.14796i 0.368877 + 1.13529i −2.57821 − 1.14789i 0.407555 + 3.87763i −2.60697 + 0.274004i −3.14870 + 2.83510i 1.06126 − 1.46070i 1.27425 − 0.925794i −8.51508 − 0.894971i
45.9 −0.408414 + 1.92144i 0.0939759 + 0.289228i −1.69802 − 0.756007i −0.00468843 − 0.0446075i −0.594114 + 0.0624439i 2.08520 − 1.87752i −0.163131 + 0.224531i 2.35223 − 1.70899i 0.0876251 + 0.00920977i
45.10 −0.402123 + 1.89184i −0.649835 − 1.99999i −1.59026 − 0.708031i 0.0623526 + 0.593245i 4.04497 − 0.425143i 1.27807 − 1.15078i −0.294715 + 0.405640i −1.15061 + 0.835970i −1.14740 − 0.120596i
45.11 −0.390913 + 1.83910i −0.600025 − 1.84669i −1.40239 − 0.624386i −0.367755 − 3.49895i 3.63080 − 0.381613i 1.78699 − 1.60901i −0.513771 + 0.707145i −0.623168 + 0.452758i 6.57869 + 0.691448i
45.12 −0.362627 + 1.70603i 0.516876 + 1.59078i −0.951942 − 0.423832i 0.244050 + 2.32198i −2.90135 + 0.304944i 0.333751 − 0.300511i −0.982092 + 1.35173i 0.163627 − 0.118882i −4.04986 − 0.425657i
45.13 −0.362014 + 1.70314i −0.698406 − 2.14947i −0.942547 − 0.419649i 0.436154 + 4.14972i 3.91369 − 0.411346i −1.45515 + 1.31023i −0.990956 + 1.36393i −1.70541 + 1.23906i −7.22546 − 0.759427i
45.14 −0.293891 + 1.38265i 1.01972 + 3.13838i 0.00174932 0.000778849i −0.266633 − 2.53685i −4.63895 + 0.487574i −1.08541 + 0.977305i −1.66330 + 2.28934i −6.38252 + 4.63718i 3.58593 + 0.376896i
45.15 −0.291879 + 1.37318i 0.0345383 + 0.106298i 0.0266517 + 0.0118661i −0.0120071 − 0.114240i −0.156048 + 0.0164013i 2.60206 − 2.34290i −1.67441 + 2.30463i 2.41694 − 1.75601i 0.160378 + 0.0168564i
45.16 −0.277211 + 1.30417i −0.924432 − 2.84511i 0.203069 + 0.0904119i −0.416084 − 3.95878i 3.96678 − 0.416925i −3.45619 + 3.11197i −1.74160 + 2.39711i −4.81302 + 3.49687i 5.27828 + 0.554769i
45.17 −0.218156 + 1.02634i −0.486806 − 1.49824i 0.821307 + 0.365669i 0.156896 + 1.49276i 1.64390 − 0.172781i −0.756789 + 0.681416i −1.78797 + 2.46092i 0.419321 − 0.304654i −1.56631 − 0.164626i
45.18 −0.204729 + 0.963174i 0.732110 + 2.25320i 0.941301 + 0.419094i 0.176114 + 1.67561i −2.32011 + 0.243853i −1.83943 + 1.65623i −1.75395 + 2.41410i −2.11389 + 1.53583i −1.64996 − 0.173418i
45.19 −0.188529 + 0.886958i 0.758318 + 2.33386i 1.07594 + 0.479039i −0.133520 − 1.27035i −2.21300 + 0.232596i 3.29259 − 2.96466i −1.69371 + 2.33119i −2.44482 + 1.77626i 1.15192 + 0.121072i
45.20 −0.141522 + 0.665811i −0.400082 − 1.23133i 1.40382 + 0.625019i 0.436677 + 4.15470i 0.876450 − 0.0921186i 2.75932 − 2.48450i −1.41501 + 1.94759i 1.07095 − 0.778093i −2.82805 − 0.297240i
See next 80 embeddings (of 400 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 45.50
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
61.k even 30 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 671.2.cf.a ✓ 400
61.k even 30 1 inner 671.2.cf.a ✓ 400
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
671.2.cf.a ✓ 400 1.a even 1 1 trivial
671.2.cf.a ✓ 400 61.k even 30 1 inner

Hecke kernels

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