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Properties

Label 261.3.m.a
Level $261$
Weight $3$
Character orbit 261.m
Analytic conductor $7.112$
Analytic rank $0$
Dimension $232$
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(70,261)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(261, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([8, 3]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("261.70");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 261 = 3^{2} \cdot 29 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 261.m (of order \(12\), degree \(4\), 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: \(232\)
Relative dimension: \(58\) over \(\Q(\zeta_{12})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 232 q - 2 q^{2} - 4 q^{3} - 4 q^{7} + 12 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 232 q - 2 q^{2} - 4 q^{3} - 4 q^{7} + 12 q^{8} - 24 q^{10} + 10 q^{11} + 30 q^{12} - 10 q^{14} - 8 q^{15} + 412 q^{16} - 44 q^{17} - 26 q^{18} - 8 q^{19} - 132 q^{20} - 60 q^{21} - 4 q^{23} - 72 q^{24} + 496 q^{25} - 240 q^{26} + 128 q^{27} - 2 q^{29} + 112 q^{30} - 2 q^{31} - 26 q^{32} - 324 q^{36} - 8 q^{37} + 228 q^{39} + 96 q^{40} + 154 q^{41} - 2 q^{43} - 28 q^{44} + 44 q^{45} + 56 q^{46} - 212 q^{47} - 516 q^{48} - 648 q^{49} + 64 q^{50} - 132 q^{52} + 32 q^{53} + 344 q^{54} - 60 q^{55} - 12 q^{56} + 40 q^{58} - 64 q^{59} + 492 q^{60} - 98 q^{61} - 60 q^{65} - 498 q^{66} - 130 q^{68} - 252 q^{69} - 408 q^{70} + 456 q^{72} - 8 q^{73} - 136 q^{74} + 294 q^{75} + 14 q^{76} - 28 q^{77} + 106 q^{79} + 304 q^{81} + 176 q^{82} + 464 q^{83} - 1670 q^{84} + 84 q^{85} - 868 q^{87} - 132 q^{88} + 4 q^{89} - 1268 q^{90} + 464 q^{94} + 120 q^{95} + 190 q^{97} + 1044 q^{98} - 510 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
70.1 −3.76534 + 1.00892i −2.93174 − 0.636328i 9.69575 − 5.59784i −1.58836 + 0.917043i 11.6810 − 0.561898i 0.382573 − 0.662637i −19.8343 + 19.8343i 8.19017 + 3.73109i 5.05551 − 5.05551i
70.2 −3.64574 + 0.976872i −0.0954464 + 2.99848i 8.87300 − 5.12283i 3.09326 − 1.78589i −2.58116 − 11.0249i −0.692489 + 1.19943i −16.6688 + 16.6688i −8.98178 − 0.572388i −9.53261 + 9.53261i
70.3 −3.58718 + 0.961182i 2.62948 − 1.44423i 8.47988 − 4.89586i −6.53487 + 3.77291i −8.04426 + 7.70814i −6.23157 + 10.7934i −15.2091 + 15.2091i 4.82838 − 7.59518i 19.8153 − 19.8153i
70.4 −3.43090 + 0.919306i 0.709004 − 2.91502i 7.46182 − 4.30809i −1.59937 + 0.923399i 0.247273 + 10.6529i 6.66564 − 11.5452i −11.5939 + 11.5939i −7.99463 − 4.13351i 4.63840 − 4.63840i
70.5 −3.29486 + 0.882856i 2.86249 + 0.897853i 6.61260 − 3.81778i 6.20506 − 3.58250i −10.2242 − 0.431135i −0.817293 + 1.41559i −8.76902 + 8.76902i 7.38772 + 5.14019i −17.2820 + 17.2820i
70.6 −3.29350 + 0.882491i 2.57724 + 1.53552i 6.60426 − 3.81297i −3.55613 + 2.05313i −9.84324 − 2.78283i 3.81337 − 6.60494i −8.74218 + 8.74218i 4.28437 + 7.91481i 9.90026 − 9.90026i
70.7 −3.24223 + 0.868753i −0.566803 − 2.94597i 6.29323 − 3.63340i 3.99088 − 2.30414i 4.39703 + 9.05910i −4.23897 + 7.34211i −7.75367 + 7.75367i −8.35747 + 3.33957i −10.9376 + 10.9376i
70.8 −3.01928 + 0.809013i −2.46764 + 1.70610i 4.99743 − 2.88527i 3.24126 − 1.87134i 6.07023 − 7.14753i −2.28533 + 3.95831i −3.91335 + 3.91335i 3.17847 − 8.42006i −8.27231 + 8.27231i
70.9 −2.96375 + 0.794135i −1.91233 + 2.31149i 4.68908 − 2.70724i −7.55155 + 4.35989i 3.83205 − 8.36933i 2.82099 − 4.88610i −3.06889 + 3.06889i −1.68595 − 8.84068i 18.9186 − 18.9186i
70.10 −2.74035 + 0.734274i −2.58206 − 1.52741i 3.50625 − 2.02433i 8.42868 − 4.86630i 8.19728 + 2.28968i 4.77859 − 8.27675i −0.0976284 + 0.0976284i 4.33406 + 7.88771i −19.5243 + 19.5243i
70.11 −2.68411 + 0.719205i 0.484117 + 2.96068i 3.22308 − 1.86085i −3.43075 + 1.98074i −3.42876 − 7.59861i −6.15777 + 10.6656i 0.546838 − 0.546838i −8.53126 + 2.86663i 7.78394 − 7.78394i
70.12 −2.53510 + 0.679278i −1.47330 − 2.61331i 2.50121 − 1.44407i −6.56508 + 3.79035i 5.51012 + 5.62423i 0.522619 − 0.905202i 2.06341 − 2.06341i −4.65880 + 7.70037i 14.0684 − 14.0684i
70.13 −2.50783 + 0.671971i 2.63067 − 1.44208i 2.37356 − 1.37038i 3.14479 − 1.81564i −5.62822 + 5.38423i 1.84659 − 3.19839i 2.31180 − 2.31180i 4.84080 − 7.58727i −6.66653 + 6.66653i
70.14 −2.29877 + 0.615955i −2.99979 + 0.0356494i 1.44086 − 0.831880i −0.161013 + 0.0929611i 6.87388 − 1.92968i −1.79125 + 3.10253i 3.93147 − 3.93147i 8.99746 − 0.213881i 0.312874 − 0.312874i
70.15 −2.17910 + 0.583888i 0.0946875 + 2.99851i 0.943445 − 0.544698i 4.82438 − 2.78536i −1.95712 − 6.47875i 4.02245 − 6.96709i 4.64303 − 4.64303i −8.98207 + 0.567842i −8.88647 + 8.88647i
70.16 −2.10165 + 0.563136i 0.829200 − 2.88313i 0.635713 − 0.367029i −0.335376 + 0.193629i −0.119096 + 6.52628i −1.70999 + 2.96179i 5.02470 − 5.02470i −7.62486 − 4.78138i 0.595803 − 0.595803i
70.17 −1.93303 + 0.517954i 2.87313 − 0.863215i 0.00422430 − 0.00243890i −2.82502 + 1.63102i −5.10673 + 3.15677i −0.758176 + 1.31320i 5.65340 − 5.65340i 7.50972 − 4.96025i 4.61605 − 4.61605i
70.18 −1.85790 + 0.497823i 1.10268 + 2.79000i −0.260133 + 0.150188i −0.439239 + 0.253595i −3.43760 − 4.63460i 3.77209 − 6.53345i 5.84885 − 5.84885i −6.56818 + 6.15297i 0.689817 − 0.689817i
70.19 −1.58837 + 0.425601i 2.65808 + 1.39090i −1.12233 + 0.647978i 5.47283 − 3.15974i −4.81397 − 1.07798i −5.92698 + 10.2658i 6.15795 − 6.15795i 5.13077 + 7.39426i −7.34807 + 7.34807i
70.20 −1.48029 + 0.396643i −2.36770 − 1.84228i −1.43017 + 0.825707i −1.67484 + 0.966970i 4.23561 + 1.78798i 5.19876 − 9.00452i 6.12414 − 6.12414i 2.21201 + 8.72393i 2.09571 − 2.09571i
See next 80 embeddings (of 232 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 70.58
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner
29.c odd 4 1 inner
261.m odd 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 261.3.m.a ✓ 232
9.c even 3 1 inner 261.3.m.a ✓ 232
29.c odd 4 1 inner 261.3.m.a ✓ 232
261.m odd 12 1 inner 261.3.m.a ✓ 232
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
261.3.m.a ✓ 232 1.a even 1 1 trivial
261.3.m.a ✓ 232 9.c even 3 1 inner
261.3.m.a ✓ 232 29.c odd 4 1 inner
261.3.m.a ✓ 232 261.m odd 12 1 inner

Hecke kernels

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