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Properties

Label 671.2.cm.a
Level $671$
Weight $2$
Character orbit 671.cm
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(49,671)]
 
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, 19]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("671.49");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 671 = 11 \cdot 61 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 671.cm (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.

\(\operatorname{Tr}(f)(q) = \) \( 480 q - 4 q^{2} - 12 q^{3} - 66 q^{4} - 10 q^{5} - 22 q^{6} - 3 q^{7} + 436 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 480 q - 4 q^{2} - 12 q^{3} - 66 q^{4} - 10 q^{5} - 22 q^{6} - 3 q^{7} + 436 q^{9} + 32 q^{10} - 5 q^{11} - 12 q^{12} + q^{13} - 19 q^{14} - 32 q^{15} - 8 q^{17} - 44 q^{18} + 29 q^{19} - 6 q^{20} - 51 q^{21} - 11 q^{22} - 30 q^{23} - 110 q^{24} + 46 q^{25} + 38 q^{26} - 84 q^{27} - 60 q^{28} + 16 q^{29} + 28 q^{30} - 4 q^{31} - 18 q^{32} - 40 q^{33} - 4 q^{34} - 23 q^{35} - 78 q^{36} + 15 q^{38} - 37 q^{39} - 25 q^{40} + 27 q^{41} - 123 q^{42} - 67 q^{43} + 82 q^{44} - 163 q^{45} - 24 q^{46} + 18 q^{47} + 22 q^{48} + 197 q^{49} - 115 q^{50} + 4 q^{51} + 23 q^{52} + 25 q^{53} - 56 q^{54} - 42 q^{55} - 75 q^{56} - 11 q^{57} + 8 q^{58} + 10 q^{59} - 16 q^{60} - 17 q^{61} + 33 q^{62} - 99 q^{63} + 48 q^{64} - 13 q^{65} + 156 q^{66} + 8 q^{67} - 201 q^{68} - 55 q^{69} + 93 q^{70} + 75 q^{71} - 25 q^{72} - 36 q^{73} + 81 q^{74} - 145 q^{75} + 6 q^{76} - 70 q^{77} - 135 q^{78} - 20 q^{79} - 51 q^{80} + 240 q^{81} - 161 q^{82} + 23 q^{83} + 55 q^{84} + 85 q^{85} + 121 q^{86} + 147 q^{87} - 32 q^{88} + 60 q^{89} - 120 q^{90} - 107 q^{91} - 93 q^{92} - 83 q^{93} - 80 q^{94} - 29 q^{95} - 96 q^{96} - 32 q^{97} + 75 q^{98} - 60 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
49.1 −2.67029 − 0.280658i 0.716284 5.09537 + 1.08305i −0.0189705 + 0.180492i −1.91268 − 0.201031i −0.956412 + 0.552185i −8.19495 − 2.66270i −2.48694 0.101313 − 0.476642i
49.2 −2.62565 − 0.275967i 0.193694 4.86158 + 1.03336i 0.126605 − 1.20457i −0.508573 − 0.0534532i 0.0965073 − 0.0557185i −7.45785 − 2.42320i −2.96248 −0.664842 + 3.12783i
49.3 −2.58616 − 0.271816i 3.33812 4.65803 + 0.990096i 0.0520140 − 0.494880i −8.63291 − 0.907356i 3.02554 − 1.74680i −6.83103 − 2.21954i 8.14306 −0.269033 + 1.26570i
49.4 −2.53985 − 0.266949i 2.19174 4.42326 + 0.940193i −0.399622 + 3.80215i −5.56669 − 0.585082i −3.48639 + 2.01287i −6.12573 − 1.99037i 1.80374 2.02995 − 9.55019i
49.5 −2.52780 − 0.265682i −1.30712 4.36288 + 0.927359i −0.260643 + 2.47986i 3.30414 + 0.347280i 3.92086 − 2.26371i −5.94746 − 1.93245i −1.29143 1.31771 − 6.19933i
49.6 −2.47929 − 0.260584i −2.43125 4.12270 + 0.876307i 0.265391 − 2.52503i 6.02778 + 0.633545i −1.24508 + 0.718846i −5.25116 − 1.70620i 2.91096 −1.31596 + 6.19113i
49.7 −2.23955 − 0.235386i −1.65929 3.00389 + 0.638496i 0.173546 − 1.65118i 3.71606 + 0.390574i −3.10981 + 1.79545i −2.29372 − 0.745274i −0.246764 −0.777329 + 3.65704i
49.8 −2.19143 − 0.230328i −3.37929 2.79300 + 0.593670i 0.0160972 − 0.153154i 7.40547 + 0.778346i 3.06483 − 1.76948i −1.79261 − 0.582455i 8.41962 −0.0705515 + 0.331919i
49.9 −2.15322 − 0.226313i 2.38802 2.62885 + 0.558779i 0.372689 − 3.54590i −5.14195 − 0.540440i −3.39475 + 1.95996i −1.41580 − 0.460022i 2.70266 −1.60496 + 7.55077i
49.10 −2.03500 − 0.213887i 1.45238 2.13919 + 0.454698i 0.448523 − 4.26741i −2.95560 − 0.310646i 1.36100 − 0.785776i −0.363869 − 0.118228i −0.890585 −1.82549 + 8.58825i
49.11 −2.00889 − 0.211142i 2.27197 2.03475 + 0.432499i −0.156529 + 1.48928i −4.56412 − 0.479708i 2.37850 − 1.37323i −0.154084 − 0.0500649i 2.16183 0.628900 − 2.95874i
49.12 −1.97766 − 0.207860i −0.542315 1.91163 + 0.406330i −0.353967 + 3.36777i 1.07251 + 0.112726i 0.449845 − 0.259718i 0.0863502 + 0.0280569i −2.70589 1.40005 − 6.58673i
49.13 −1.90929 − 0.200675i 0.523749 1.64883 + 0.350470i −0.0464501 + 0.441943i −0.999989 − 0.105103i 0.328236 − 0.189507i 0.573924 + 0.186479i −2.72569 0.177374 − 0.834478i
49.14 −1.82449 − 0.191762i −1.76842 1.33570 + 0.283911i 0.255126 − 2.42736i 3.22647 + 0.339116i 0.544391 − 0.314304i 1.10698 + 0.359679i 0.127318 −0.930951 + 4.37978i
49.15 −1.71985 − 0.180764i −2.76624 0.968918 + 0.205950i −0.300142 + 2.85566i 4.75752 + 0.500036i −2.14973 + 1.24115i 1.66020 + 0.539433i 4.65209 1.03240 − 4.85706i
49.16 −1.51685 − 0.159427i 0.0130955 0.319111 + 0.0678292i −0.0269956 + 0.256846i −0.0198638 − 0.00208777i −3.86178 + 2.22960i 2.42788 + 0.788865i −2.99983 0.0818965 − 0.385293i
49.17 −1.32339 − 0.139093i 2.85816 −0.224293 − 0.0476749i 0.0517325 − 0.492202i −3.78244 − 0.397551i −1.92561 + 1.11175i 2.82129 + 0.916692i 5.16905 −0.136924 + 0.644177i
49.18 −1.19794 − 0.125909i −1.69854 −0.537078 − 0.114159i 0.176643 − 1.68065i 2.03476 + 0.213862i 3.41729 − 1.97298i 2.92019 + 0.948828i −0.114947 −0.423218 + 1.99108i
49.19 −1.06340 − 0.111768i −2.16183 −0.837959 − 0.178114i −0.187713 + 1.78597i 2.29890 + 0.241624i 1.65627 − 0.956246i 2.90504 + 0.943904i 1.67350 0.399229 − 1.87822i
49.20 −1.00028 − 0.105134i −0.793255 −0.966784 − 0.205496i 0.341239 − 3.24667i 0.793478 + 0.0833979i −0.766461 + 0.442516i 2.85858 + 0.928810i −2.37075 −0.682670 + 3.21171i
See next 80 embeddings (of 480 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 49.60
Significant digits:
Format:

Inner twists

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

Hecke kernels

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