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Properties

Label 671.2.ce.a
Level $671$
Weight $2$
Character orbit 671.ce
Analytic conductor $5.358$
Analytic rank $0$
Dimension $480$
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(5,671)]
 
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([12, 11]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("671.5");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 671 = 11 \cdot 61 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 671.ce (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: \(480\)
Relative dimension: \(60\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 480 q - 4 q^{2} - 2 q^{3} - 66 q^{4} + 3 q^{6} + 2 q^{7} - 20 q^{8} - 114 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 480 q - 4 q^{2} - 2 q^{3} - 66 q^{4} + 3 q^{6} + 2 q^{7} - 20 q^{8} - 114 q^{9} + 32 q^{10} + 5 q^{11} - 12 q^{12} + q^{13} + 11 q^{14} + 13 q^{15} + 70 q^{16} + 12 q^{17} + 11 q^{18} - 16 q^{19} + 4 q^{20} - 51 q^{21} + 29 q^{22} - 30 q^{23} + 40 q^{24} - 214 q^{25} - 57 q^{26} + 31 q^{27} - 10 q^{28} + 26 q^{29} - 27 q^{30} - 14 q^{31} - 18 q^{32} - 5 q^{33} - 4 q^{34} + 37 q^{35} + 167 q^{36} - 30 q^{37} - 62 q^{39} - 50 q^{40} - 158 q^{41} - 103 q^{42} - 67 q^{43} - 53 q^{44} - 163 q^{45} + 41 q^{46} + 3 q^{47} - 23 q^{48} - 38 q^{49} + 115 q^{50} + 9 q^{51} + 43 q^{52} - 56 q^{54} + 3 q^{55} - 75 q^{56} - 36 q^{57} + 98 q^{58} - 15 q^{59} - 16 q^{60} + 8 q^{61} + 33 q^{62} + 26 q^{63} + 148 q^{64} - 13 q^{65} + 46 q^{66} + 8 q^{67} + 124 q^{68} + 45 q^{69} - 97 q^{70} - 15 q^{71} + 25 q^{72} - 31 q^{73} - 39 q^{74} + 170 q^{75} + 6 q^{76} + 45 q^{77} - 135 q^{78} + 15 q^{79} + 79 q^{80} - 10 q^{81} - 31 q^{82} + 28 q^{83} - 180 q^{84} - 184 q^{86} + 147 q^{87} + 53 q^{88} + 60 q^{89} - 15 q^{90} - 27 q^{91} + 62 q^{92} + 47 q^{93} - 50 q^{94} - 74 q^{95} - 231 q^{96} + 58 q^{97} + 75 q^{98} + 5 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
5.1 −2.08105 + 1.87379i −0.389752 − 1.19953i 0.610641 − 5.80986i −0.236259 + 0.409213i 3.05877 + 1.76598i 2.50122 − 2.25211i 6.32369 + 8.70381i 1.14008 − 0.828315i −0.275111 − 1.29429i
5.2 −1.99028 + 1.79205i −0.586324 − 1.80452i 0.540689 − 5.14431i −0.886570 + 1.53558i 4.40074 + 2.54077i −2.71172 + 2.44165i 4.99436 + 6.87414i −0.485465 + 0.352711i −0.987328 − 4.64501i
5.3 −1.86535 + 1.67956i 0.0777543 + 0.239303i 0.449520 − 4.27689i 1.91125 − 3.31038i −0.546964 − 0.315790i 0.480967 − 0.433065i 3.39405 + 4.67151i 2.37583 − 1.72614i 1.99486 + 9.38507i
5.4 −1.85814 + 1.67308i 0.439436 + 1.35244i 0.444443 − 4.22860i −1.45399 + 2.51838i −3.07928 − 1.77782i −1.98638 + 1.78855i 3.30957 + 4.55523i 0.791048 − 0.574730i −1.51173 − 7.11214i
5.5 −1.82649 + 1.64458i 0.367387 + 1.13070i 0.422371 − 4.01860i 0.399986 − 0.692797i −2.53056 − 1.46102i −0.638807 + 0.575184i 2.94815 + 4.05778i 1.28354 − 0.932545i 0.408789 + 1.92320i
5.6 −1.74400 + 1.57031i −0.767833 − 2.36315i 0.366625 − 3.48820i 1.50995 − 2.61531i 5.04997 + 2.91560i −3.14365 + 2.83055i 2.07934 + 2.86196i −2.56785 + 1.86565i 1.47348 + 6.93219i
5.7 −1.72779 + 1.55571i 0.686577 + 2.11307i 0.355972 − 3.38685i 0.0779861 − 0.135076i −4.47359 − 2.58283i 0.565483 − 0.509163i 1.92075 + 2.64368i −1.56661 + 1.13821i 0.0753953 + 0.354707i
5.8 −1.67280 + 1.50620i −0.890716 − 2.74134i 0.320579 − 3.05011i −0.268598 + 0.465225i 5.61900 + 3.24413i 0.998787 − 0.899312i 1.41162 + 1.94292i −4.29453 + 3.12016i −0.251410 − 1.18279i
5.9 −1.56057 + 1.40514i 0.244268 + 0.751780i 0.251893 − 2.39660i −1.88276 + 3.26104i −1.43756 − 0.829973i 3.55271 − 3.19888i 0.505831 + 0.696217i 1.92155 − 1.39608i −1.64404 − 7.73462i
5.10 −1.45854 + 1.31328i −0.388209 − 1.19479i 0.193591 − 1.84190i −1.50523 + 2.60713i 2.13530 + 1.23282i −0.285824 + 0.257357i −0.170688 − 0.234931i 1.15025 − 0.835703i −1.22845 − 5.77939i
5.11 −1.43131 + 1.28876i 0.955697 + 2.94133i 0.178698 − 1.70020i 1.69513 − 2.93604i −5.15858 − 2.97831i −3.14418 + 2.83103i −0.328797 − 0.452550i −5.31103 + 3.85869i 1.35760 + 6.38702i
5.12 −1.39883 + 1.25952i −0.693919 − 2.13566i 0.161300 − 1.53466i 0.269762 − 0.467241i 3.66058 + 2.11344i 3.27949 − 2.95286i −0.505493 − 0.695752i −1.65248 + 1.20060i 0.211146 + 0.993363i
5.13 −1.34943 + 1.21503i 0.931765 + 2.86768i 0.135601 − 1.29016i −0.560974 + 0.971636i −4.74167 − 2.73761i −1.01768 + 0.916323i −0.750047 − 1.03235i −4.92835 + 3.58065i −0.423573 − 1.99275i
5.14 −1.31797 + 1.18670i 0.626327 + 1.92764i 0.119716 − 1.13902i 0.217609 − 0.376910i −3.11301 − 1.79730i 3.06842 − 2.76282i −0.890975 − 1.22632i −0.896449 + 0.651308i 0.160478 + 0.754991i
5.15 −1.22432 + 1.10239i −0.700965 − 2.15735i 0.0746570 − 0.710314i 1.43617 − 2.48751i 3.23644 + 1.86856i −0.428115 + 0.385476i −1.24511 − 1.71374i −1.73575 + 1.26109i 0.983867 + 4.62873i
5.16 −1.10712 + 0.996855i −0.222898 − 0.686008i 0.0229369 − 0.218230i −1.11880 + 1.93781i 0.930625 + 0.537297i −0.0579488 + 0.0521773i −1.55919 − 2.14604i 2.00613 − 1.45754i −0.693077 − 3.26067i
5.17 −1.08591 + 0.977758i −0.0575138 − 0.177009i 0.0141333 − 0.134470i 0.418950 − 0.725642i 0.235527 + 0.135982i 1.17626 − 1.05911i −1.60165 − 2.20449i 2.39903 − 1.74299i 0.254561 + 1.19761i
5.18 −1.01649 + 0.915248i 0.0636254 + 0.195819i −0.0134922 + 0.128370i 0.914063 − 1.58320i −0.243897 − 0.140814i −2.84240 + 2.55931i −1.71174 − 2.35601i 2.39275 − 1.73844i 0.519892 + 2.44590i
5.19 −0.914901 + 0.823780i 0.267857 + 0.824378i −0.0506276 + 0.481689i 1.31716 − 2.28138i −0.924168 − 0.533569i 0.534870 − 0.481599i −1.79776 − 2.47440i 1.81920 − 1.32173i 0.674291 + 3.17229i
5.20 −0.827220 + 0.744832i −0.672227 − 2.06890i −0.0795390 + 0.756763i −1.73366 + 3.00279i 2.09706 + 1.21074i −1.90550 + 1.71572i −1.80643 − 2.48634i −1.40141 + 1.01819i −0.802456 − 3.77526i
See next 80 embeddings (of 480 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 5.60
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
671.ce 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.ce.a ✓ 480
11.c even 5 1 671.2.cm.a yes 480
61.k even 30 1 671.2.cm.a yes 480
671.ce even 30 1 inner 671.2.ce.a ✓ 480
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
671.2.ce.a ✓ 480 1.a even 1 1 trivial
671.2.ce.a ✓ 480 671.ce even 30 1 inner
671.2.cm.a yes 480 11.c even 5 1
671.2.cm.a yes 480 61.k even 30 1

Hecke kernels

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