Properties

Label 261.3.v.a
Level $261$
Weight $3$
Character orbit 261.v
Analytic conductor $7.112$
Analytic rank $0$
Dimension $696$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [261,3,Mod(5,261)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(261, base_ring=CyclotomicField(42))
 
chi = DirichletCharacter(H, H._module([35, 33]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("261.5");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 261 = 3^{2} \cdot 29 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 261.v (of order \(42\), degree \(12\), 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: \(7.11173489980\)
Analytic rank: \(0\)
Dimension: \(696\)
Relative dimension: \(58\) over \(\Q(\zeta_{42})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{42}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 696 q - 21 q^{2} - 14 q^{3} + 107 q^{4} - 15 q^{5} - 52 q^{6} - 5 q^{7} + 14 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 696 q - 21 q^{2} - 14 q^{3} + 107 q^{4} - 15 q^{5} - 52 q^{6} - 5 q^{7} + 14 q^{9} - 28 q^{10} - 21 q^{11} - 5 q^{13} - 21 q^{14} - 14 q^{15} + 211 q^{16} + 84 q^{18} - 28 q^{19} - 165 q^{20} + 196 q^{21} - 21 q^{22} - 150 q^{23} + 16 q^{24} - 255 q^{25} - 203 q^{27} - 128 q^{28} - 153 q^{29} - 148 q^{30} - 7 q^{31} - 21 q^{32} + 131 q^{33} - 25 q^{34} + 464 q^{36} - 28 q^{37} + 69 q^{38} + 168 q^{39} - 7 q^{40} - 211 q^{42} - 7 q^{43} - 342 q^{45} - 21 q^{47} + 420 q^{48} + 197 q^{49} - 21 q^{50} + 266 q^{51} - 43 q^{52} + 564 q^{54} - 196 q^{55} - 105 q^{56} - 234 q^{57} + 55 q^{58} + 306 q^{59} + 133 q^{60} - 7 q^{61} + 520 q^{63} - 1434 q^{64} - 249 q^{65} - 14 q^{66} - 5 q^{67} - 357 q^{68} - 238 q^{69} + 245 q^{72} - 28 q^{73} - 2469 q^{74} - 7 q^{76} - 21 q^{77} - 712 q^{78} - 7 q^{79} - 6 q^{81} + 512 q^{82} - 825 q^{83} - 1491 q^{84} - 182 q^{85} + 288 q^{86} + 560 q^{87} + 60 q^{88} + 896 q^{90} + 204 q^{91} + 1125 q^{92} + 478 q^{93} + 165 q^{94} - 525 q^{95} + 547 q^{96} - 679 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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 −1.43075 + 3.64550i 0.801822 2.89086i −8.31042 7.71094i 1.02064 + 6.77148i 9.39143 + 7.05916i 0.703203 0.652477i 25.8869 12.4665i −7.71416 4.63591i −26.1457 5.96759i
5.2 −1.38256 + 3.52270i 2.99971 + 0.0418777i −7.56576 7.02000i −0.455445 3.02168i −4.29479 + 10.5092i −0.470271 + 0.436347i 21.5513 10.3786i 8.99649 + 0.251242i 11.2742 + 2.57325i
5.3 −1.37031 + 3.49151i −2.56398 1.55756i −7.38064 6.84823i −0.849898 5.63871i 8.95169 6.81781i −2.57897 + 2.39294i 20.5071 9.87570i 4.14802 + 7.98711i 20.8522 + 4.75938i
5.4 −1.33390 + 3.39871i −1.10049 + 2.79087i −6.83975 6.34636i −0.781471 5.18472i −8.01741 7.46296i 3.86951 3.59038i 17.5348 8.44433i −6.57786 6.14262i 18.6638 + 4.25989i
5.5 −1.25942 + 3.20895i −2.38451 + 1.82047i −5.77903 5.36216i 0.930464 + 6.17322i −2.83871 9.94452i −8.70372 + 8.07587i 12.0617 5.80861i 2.37176 8.68186i −20.9814 4.78888i
5.6 −1.18777 + 3.02639i −2.93094 + 0.640011i −4.81603 4.46863i 0.685670 + 4.54912i 1.54436 9.63034i 7.90119 7.33124i 7.52748 3.62505i 8.18077 3.75166i −14.5818 3.32821i
5.7 −1.14410 + 2.91513i 1.50801 + 2.59344i −4.25679 3.94973i −0.447770 2.97076i −9.28552 + 1.42888i −9.00164 + 8.35230i 5.09826 2.45519i −4.45183 + 7.82184i 9.17244 + 2.09355i
5.8 −1.12715 + 2.87193i 2.45483 + 1.72447i −4.04532 3.75351i 0.472918 + 3.13760i −7.71952 + 5.10638i 4.84953 4.49971i 4.22082 2.03264i 3.05241 + 8.46657i −9.54404 2.17836i
5.9 −1.09182 + 2.78191i −2.10748 2.13507i −3.61477 3.35402i 0.713024 + 4.73060i 8.24057 3.53173i 0.281758 0.261433i 2.50709 1.20735i −0.117015 + 8.99924i −13.9386 3.18140i
5.10 −1.08159 + 2.75585i 1.92322 2.30243i −3.49267 3.24073i −1.29663 8.60259i 4.26502 + 7.79042i −4.96575 + 4.60754i 2.03933 0.982090i −1.60241 8.85620i 25.1099 + 5.73117i
5.11 −1.04968 + 2.67453i 2.14106 2.10139i −3.11908 2.89409i 0.0180756 + 0.119924i 3.37281 + 7.93212i 5.54916 5.14887i 0.659912 0.317797i 0.168307 8.99843i −0.339713 0.0775374i
5.12 −1.02316 + 2.60697i −0.635014 2.93202i −2.81724 2.61402i −0.162656 1.07915i 8.29343 + 1.34447i −1.81026 + 1.67968i −0.395726 + 0.190571i −8.19351 + 3.72375i 2.97974 + 0.680106i
5.13 −0.987218 + 2.51539i 2.99149 0.225795i −2.42038 2.24578i 1.34341 + 8.91298i −2.38529 + 7.74767i −6.98422 + 6.48041i −1.69986 + 0.818611i 8.89803 1.35093i −23.7458 5.41984i
5.14 −0.911365 + 2.32212i −0.687966 + 2.92005i −1.62945 1.51191i −0.126374 0.838435i −6.15372 4.25877i 0.932879 0.865585i −3.99423 + 1.92352i −8.05341 4.01779i 2.06212 + 0.470665i
5.15 −0.799677 + 2.03754i −2.97515 0.385309i −0.579897 0.538066i −0.638382 4.23539i 3.16425 5.75389i 6.14549 5.70218i −6.32829 + 3.04754i 8.70307 + 2.29271i 9.14030 + 2.08621i
5.16 −0.752873 + 1.91829i 2.07135 + 2.17014i −0.180804 0.167762i −1.48108 9.82631i −5.72242 + 2.33960i 7.62135 7.07158i −6.96871 + 3.35595i −0.419050 + 8.99024i 19.9648 + 4.55682i
5.17 −0.738339 + 1.88126i −2.89765 + 0.776937i −0.0617723 0.0573163i −0.901243 5.97935i 0.677828 6.02486i −6.52464 + 6.05398i −7.12984 + 3.43355i 7.79274 4.50258i 11.9141 + 2.71932i
5.18 −0.686937 + 1.75029i 0.153024 + 2.99609i 0.340585 + 0.316017i 1.19231 + 7.91045i −5.34914 1.79029i 2.90380 2.69434i −7.56331 + 3.64230i −8.95317 + 0.916950i −14.6646 3.34710i
5.19 −0.593958 + 1.51338i 0.825786 2.88411i 0.994673 + 0.922922i 0.542859 + 3.60163i 3.87427 + 2.96277i −6.59851 + 6.12253i −7.84657 + 3.77871i −7.63616 4.76331i −5.77307 1.31767i
5.20 −0.524166 + 1.33555i −2.01600 + 2.22165i 1.42325 + 1.32059i 0.215073 + 1.42692i −1.91041 3.85699i −1.72570 + 1.60122i −7.68033 + 3.69865i −0.871474 8.95771i −2.01846 0.460699i
See next 80 embeddings (of 696 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 5.58
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.d odd 6 1 inner
29.e even 14 1 inner
261.v odd 42 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 261.3.v.a 696
9.d odd 6 1 inner 261.3.v.a 696
29.e even 14 1 inner 261.3.v.a 696
261.v odd 42 1 inner 261.3.v.a 696
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
261.3.v.a 696 1.a even 1 1 trivial
261.3.v.a 696 9.d odd 6 1 inner
261.3.v.a 696 29.e even 14 1 inner
261.3.v.a 696 261.v odd 42 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(261, [\chi])\).