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Properties

Label 671.2.bn.a
Level $671$
Weight $2$
Character orbit 671.bn
Analytic conductor $5.358$
Analytic rank $0$
Dimension $208$
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(12,671)]
 
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, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("671.12");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 671 = 11 \cdot 61 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 671.bn (of order \(15\), 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: \(208\)
Relative dimension: \(26\) over \(\Q(\zeta_{15})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{15}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 208 q - 2 q^{3} + 30 q^{4} - 10 q^{5} - 12 q^{6} - 10 q^{7} + 18 q^{8} - 60 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 208 q - 2 q^{3} + 30 q^{4} - 10 q^{5} - 12 q^{6} - 10 q^{7} + 18 q^{8} - 60 q^{9} - 75 q^{10} - 208 q^{11} - 25 q^{12} - 17 q^{14} - 8 q^{15} + 42 q^{16} + 8 q^{17} - 5 q^{18} - 19 q^{19} - 26 q^{20} + 50 q^{21} + 18 q^{23} - 41 q^{24} - 6 q^{25} - 32 q^{26} - 8 q^{27} + 24 q^{28} + 40 q^{29} - 12 q^{30} - 13 q^{31} + 12 q^{32} + 2 q^{33} - 38 q^{34} + 40 q^{35} - 94 q^{36} + 66 q^{37} + 15 q^{38} - 58 q^{39} + 157 q^{40} + 20 q^{41} + 42 q^{42} - 3 q^{43} - 30 q^{44} - 25 q^{45} + 90 q^{46} - 34 q^{47} - 58 q^{48} + 56 q^{49} - 84 q^{50} + 37 q^{51} + 98 q^{52} + 18 q^{53} + 92 q^{54} + 10 q^{55} + 76 q^{56} + 16 q^{57} - 119 q^{58} - 7 q^{59} - 60 q^{60} - 7 q^{61} + 34 q^{62} + 30 q^{63} + 38 q^{64} - 95 q^{65} + 12 q^{66} - 43 q^{67} + 40 q^{68} + 100 q^{69} + 43 q^{70} - 46 q^{71} - 156 q^{72} - 28 q^{73} - 4 q^{74} - 47 q^{75} - 121 q^{76} + 10 q^{77} - 108 q^{78} + 38 q^{79} - 45 q^{80} - 78 q^{81} - 2 q^{82} - 62 q^{83} - 14 q^{84} + 54 q^{85} + 7 q^{86} + 23 q^{87} - 18 q^{88} + 6 q^{89} + 121 q^{90} + 32 q^{91} - 39 q^{92} + 70 q^{93} + 73 q^{94} - 74 q^{95} + 220 q^{96} - 49 q^{97} - 63 q^{98} + 60 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
12.1 −2.44202 − 1.08726i −0.804668 + 0.584626i 3.44307 + 3.82391i 0.901907 − 0.191706i 2.60065 − 0.552786i 0.248700 − 2.36623i −2.59838 − 7.99699i −0.621347 + 1.91231i −2.41091 − 0.512455i
12.2 −2.20902 − 0.983518i 1.12937 − 0.820537i 2.57419 + 2.85893i 2.38909 − 0.507816i −3.30182 + 0.701823i −0.394204 + 3.75060i −1.38018 − 4.24775i −0.324850 + 0.999785i −5.77698 − 1.22794i
12.3 −2.03902 − 0.907829i 1.68678 − 1.22552i 1.99518 + 2.21587i −2.26004 + 0.480387i −4.55193 + 0.967543i −0.0297354 + 0.282913i −0.677129 − 2.08399i 0.416282 − 1.28118i 5.04437 + 1.07221i
12.4 −1.99672 − 0.888998i −1.99478 + 1.44929i 1.85832 + 2.06388i −3.85055 + 0.818461i 5.27143 − 1.12048i 0.354102 − 3.36906i −0.524944 − 1.61561i 0.951640 − 2.92885i 8.41610 + 1.78890i
12.5 −1.49894 − 0.667370i −2.51072 + 1.82415i 0.463172 + 0.514404i 3.30317 − 0.702110i 4.98080 − 1.05870i 0.509023 − 4.84303i 0.663098 + 2.04080i 2.04917 − 6.30669i −5.41981 − 1.15202i
12.6 −1.44644 − 0.643999i 2.40560 − 1.74777i 0.339207 + 0.376728i 3.57398 − 0.759672i −4.60513 + 0.978850i −0.174203 + 1.65744i 0.730520 + 2.24831i 1.80516 − 5.55570i −5.65879 − 1.20281i
12.7 −1.42037 − 0.632389i −1.92311 + 1.39722i 0.279269 + 0.310160i −1.56369 + 0.332372i 3.61511 − 0.768416i −0.418438 + 3.98117i 0.760388 + 2.34023i 0.819074 − 2.52085i 2.43120 + 0.516767i
12.8 −1.38609 − 0.617125i −0.378565 + 0.275044i 0.202128 + 0.224486i 1.57109 − 0.333945i 0.694460 − 0.147612i 0.221426 − 2.10673i 0.796087 + 2.45010i −0.859388 + 2.64493i −2.38374 − 0.506680i
12.9 −1.19189 − 0.530662i −0.109504 + 0.0795593i −0.199270 − 0.221312i −3.14218 + 0.667891i 0.172735 − 0.0367161i 0.123197 − 1.17214i 0.926403 + 2.85118i −0.921390 + 2.83575i 4.09955 + 0.871386i
12.10 −0.813144 − 0.362035i −0.721028 + 0.523858i −0.808127 − 0.897516i 2.56393 − 0.544979i 0.775955 − 0.164934i −0.373697 + 3.55549i 0.882302 + 2.71545i −0.681596 + 2.09774i −2.28214 − 0.485085i
12.11 −0.715109 − 0.318387i 2.66094 − 1.93328i −0.928250 − 1.03093i −1.74442 + 0.370789i −2.51839 + 0.535301i −0.0560194 + 0.532989i 0.819354 + 2.52171i 2.41595 − 7.43552i 1.36551 + 0.290248i
12.12 −0.198793 − 0.0885084i −1.66665 + 1.21089i −1.30658 − 1.45110i 1.53322 − 0.325895i 0.438493 − 0.0932046i −0.00883956 + 0.0841028i 0.265792 + 0.818023i 0.384416 − 1.18311i −0.333637 − 0.0709168i
12.13 −0.179359 − 0.0798556i 1.57061 − 1.14111i −1.31247 − 1.45764i −2.31593 + 0.492266i −0.372826 + 0.0792466i −0.365330 + 3.47588i 0.240342 + 0.739696i 0.237615 − 0.731304i 0.454693 + 0.0966479i
12.14 0.0935857 + 0.0416670i −2.70186 + 1.96301i −1.33124 − 1.47849i −2.66315 + 0.566070i −0.334648 + 0.0711317i 0.0559884 − 0.532694i −0.126293 − 0.388691i 2.51956 − 7.75441i −0.272819 − 0.0579895i
12.15 0.120219 + 0.0535248i 0.641778 − 0.466279i −1.32667 − 1.47342i −0.796736 + 0.169352i 0.102111 − 0.0217044i 0.505967 − 4.81395i −0.161957 − 0.498453i −0.732588 + 2.25467i −0.104847 − 0.0222859i
12.16 0.289359 + 0.128831i 1.73028 − 1.25712i −1.27113 − 1.41173i 3.50322 − 0.744632i 0.662627 − 0.140846i 0.141363 − 1.34498i −0.381695 − 1.17474i 0.486458 − 1.49717i 1.10962 + 0.235857i
12.17 0.452954 + 0.201668i 0.0191029 − 0.0138791i −1.17376 − 1.30360i −1.23520 + 0.262550i 0.0114517 − 0.00243414i −0.0107478 + 0.102258i −0.575200 − 1.77028i −0.926879 + 2.85264i −0.612437 − 0.130177i
12.18 1.17380 + 0.522608i −1.59666 + 1.16004i −0.233583 − 0.259421i −0.745917 + 0.158550i −2.48040 + 0.527226i −0.251949 + 2.39714i −0.932703 − 2.87057i 0.276580 − 0.851224i −0.958413 − 0.203717i
12.19 1.29079 + 0.574697i 2.31449 − 1.68157i −0.00239807 − 0.00266332i −1.90367 + 0.404638i 3.95391 − 0.840430i 0.240172 − 2.28508i −0.874813 − 2.69240i 1.60211 − 4.93078i −2.68979 − 0.571732i
12.20 1.70626 + 0.759675i 0.825279 − 0.599601i 0.995952 + 1.10612i −4.16889 + 0.886124i 1.86364 − 0.396129i −0.513479 + 4.88543i −0.295260 − 0.908717i −0.605486 + 1.86349i −7.78637 − 1.65504i
See next 80 embeddings (of 208 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 12.26
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
61.i even 15 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 671.2.bn.a ✓ 208
61.i even 15 1 inner 671.2.bn.a ✓ 208
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
671.2.bn.a ✓ 208 1.a even 1 1 trivial
671.2.bn.a ✓ 208 61.i even 15 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{208} - 41 T_{2}^{206} - 6 T_{2}^{205} + 748 T_{2}^{204} + 234 T_{2}^{203} - 7056 T_{2}^{202} + \cdots + 140920641 \) acting on \(S_{2}^{\mathrm{new}}(671, [\chi])\). Copy content Toggle raw display