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Properties

Label 261.2.e.b
Level $261$
Weight $2$
Character orbit 261.e
Analytic conductor $2.084$
Analytic rank $0$
Dimension $34$
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] = [261,2,Mod(88,261)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(261, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([4, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("261.88");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 261 = 3^{2} \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 261.e (of order \(3\), degree \(2\), 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: \(2.08409549276\)
Analytic rank: \(0\)
Dimension: \(34\)
Relative dimension: \(17\) over \(\Q(\zeta_{3})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 34 q - q^{2} - 2 q^{3} - 23 q^{4} + q^{5} - q^{6} - 9 q^{7} + 6 q^{8} - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 34 q - q^{2} - 2 q^{3} - 23 q^{4} + q^{5} - q^{6} - 9 q^{7} + 6 q^{8} - 6 q^{9} + 20 q^{10} - 3 q^{11} - 16 q^{12} - 9 q^{13} - 8 q^{14} + 20 q^{15} - 35 q^{16} + 2 q^{17} - 24 q^{18} + 48 q^{19} + 4 q^{20} - 7 q^{22} + 8 q^{23} + 31 q^{24} - 26 q^{25} - 12 q^{26} - 5 q^{27} + 48 q^{28} - 17 q^{29} - 24 q^{30} - 29 q^{31} - 7 q^{32} + 12 q^{33} - 25 q^{34} - 22 q^{35} - 13 q^{36} + 20 q^{37} + 23 q^{38} - 13 q^{39} - 36 q^{40} - 3 q^{41} + 102 q^{42} - 27 q^{43} + 46 q^{44} + 12 q^{45} + 44 q^{46} - 6 q^{47} - 73 q^{48} - 42 q^{49} + 37 q^{50} + 21 q^{51} - 28 q^{52} - 12 q^{53} - 3 q^{54} + 58 q^{55} - 26 q^{56} - 59 q^{57} - q^{58} + 60 q^{60} - 39 q^{61} - 12 q^{62} + 57 q^{63} + 130 q^{64} + 49 q^{65} - 87 q^{66} - 31 q^{67} - 76 q^{68} - 7 q^{70} - 2 q^{71} + 38 q^{72} + 18 q^{73} - 14 q^{74} - 20 q^{75} - 40 q^{76} + 33 q^{77} + 99 q^{78} - 29 q^{79} - 74 q^{80} + 26 q^{81} + 22 q^{82} - 24 q^{83} - 74 q^{84} - 7 q^{85} - q^{86} + q^{87} - q^{88} + 46 q^{89} - 5 q^{90} + 130 q^{91} + 29 q^{92} - 44 q^{93} - 14 q^{94} - 32 q^{95} + 117 q^{96} - 14 q^{97} + 86 q^{98} - 2 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
88.1 −1.34972 + 2.33778i 1.73091 + 0.0629887i −2.64347 − 4.57862i −0.422925 − 0.732527i −2.48348 + 3.96145i 1.76494 − 3.05697i 8.87286 2.99206 + 0.218055i 2.28331
88.2 −1.34496 + 2.32954i 0.0357543 + 1.73168i −2.61783 − 4.53422i 0.704342 + 1.21996i −4.08211 − 2.24575i −2.59523 + 4.49507i 8.70368 −2.99744 + 0.123830i −3.78925
88.3 −1.11327 + 1.92825i 0.335866 − 1.69917i −1.47876 − 2.56128i −0.877415 − 1.51973i 2.90252 + 2.53928i −0.994822 + 1.72308i 2.13196 −2.77439 − 1.14139i 3.90721
88.4 −1.10654 + 1.91659i −1.69962 − 0.333624i −1.44888 − 2.50953i −1.58641 − 2.74774i 2.52012 − 2.88830i −1.23991 + 2.14758i 1.98681 2.77739 + 1.13406i 7.02172
88.5 −0.814303 + 1.41041i 1.68245 − 0.411525i −0.326178 − 0.564957i 1.83529 + 3.17881i −0.789606 + 2.70806i −0.253590 + 0.439231i −2.19478 2.66130 − 1.38474i −5.97791
88.6 −0.626036 + 1.08433i −0.941284 − 1.45395i 0.216158 + 0.374397i −0.133753 − 0.231667i 2.16584 − 0.110432i 1.84812 − 3.20103i −3.04543 −1.22797 + 2.73717i 0.334937
88.7 −0.388144 + 0.672285i −0.783898 + 1.54451i 0.698689 + 1.21016i −1.63386 − 2.82992i −0.734084 − 1.12649i −1.21627 + 2.10664i −2.63734 −1.77101 − 2.42147i 2.53668
88.8 −0.349776 + 0.605830i −1.39011 + 1.03324i 0.755313 + 1.30824i 1.70335 + 2.95028i −0.139740 − 1.20358i −1.05931 + 1.83478i −2.45587 0.864830 − 2.87264i −2.38316
88.9 −0.0978309 + 0.169448i −1.73144 − 0.0459643i 0.980858 + 1.69890i 0.187362 + 0.324520i 0.177177 − 0.288892i 1.43771 − 2.49018i −0.775156 2.99577 + 0.159169i −0.0733190
88.10 0.140131 − 0.242715i 1.60426 + 0.652955i 0.960726 + 1.66403i −0.682674 − 1.18243i 0.383289 − 0.297878i −1.33514 + 2.31253i 1.09904 2.14730 + 2.09502i −0.382656
88.11 0.597489 − 1.03488i −0.451058 − 1.67229i 0.286014 + 0.495390i 1.35305 + 2.34355i −2.00012 − 0.532382i 1.19262 − 2.06568i 3.07352 −2.59309 + 1.50860i 3.23373
88.12 0.599120 − 1.03771i −1.56466 − 0.742848i 0.282111 + 0.488630i 1.18848 + 2.05851i −1.70828 + 1.17861i −2.33213 + 4.03937i 3.07255 1.89635 + 2.32462i 2.84818
88.13 0.622253 − 1.07777i 0.771217 + 1.55088i 0.225603 + 0.390756i −1.90664 − 3.30241i 2.15139 + 0.133841i 2.40913 − 4.17274i 3.05054 −1.81045 + 2.39213i −4.74566
88.14 0.973669 − 1.68644i 0.738428 − 1.56676i −0.896064 − 1.55203i −1.26998 − 2.19966i −1.92326 − 2.77082i −2.23621 + 3.87322i 0.404796 −1.90945 − 2.31387i −4.94615
88.15 1.09449 − 1.89571i 0.891156 + 1.48521i −1.39581 − 2.41761i 1.19173 + 2.06414i 3.79088 − 0.0638318i −0.325653 + 0.564048i −1.73282 −1.41168 + 2.64710i 5.21734
88.16 1.29776 − 2.24779i −0.995529 + 1.41736i −2.36839 − 4.10217i −0.999300 − 1.73084i 1.89398 + 4.07715i 0.411724 − 0.713127i −7.10338 −1.01784 − 2.82206i −5.18742
88.17 1.36567 − 2.36541i 0.767565 − 1.55269i −2.73010 − 4.72867i 1.84935 + 3.20317i −2.62450 − 3.93606i 0.0240123 − 0.0415906i −9.45097 −1.82169 − 2.38358i 10.1024
175.1 −1.34972 − 2.33778i 1.73091 − 0.0629887i −2.64347 + 4.57862i −0.422925 + 0.732527i −2.48348 − 3.96145i 1.76494 + 3.05697i 8.87286 2.99206 − 0.218055i 2.28331
175.2 −1.34496 − 2.32954i 0.0357543 − 1.73168i −2.61783 + 4.53422i 0.704342 − 1.21996i −4.08211 + 2.24575i −2.59523 − 4.49507i 8.70368 −2.99744 − 0.123830i −3.78925
175.3 −1.11327 − 1.92825i 0.335866 + 1.69917i −1.47876 + 2.56128i −0.877415 + 1.51973i 2.90252 − 2.53928i −0.994822 − 1.72308i 2.13196 −2.77439 + 1.14139i 3.90721
See all 34 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 88.17
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 261.2.e.b ✓ 34
3.b odd 2 1 783.2.e.b 34
9.c even 3 1 inner 261.2.e.b ✓ 34
9.c even 3 1 2349.2.a.j 17
9.d odd 6 1 783.2.e.b 34
9.d odd 6 1 2349.2.a.i 17
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
261.2.e.b ✓ 34 1.a even 1 1 trivial
261.2.e.b ✓ 34 9.c even 3 1 inner
783.2.e.b 34 3.b odd 2 1
783.2.e.b 34 9.d odd 6 1
2349.2.a.i 17 9.d odd 6 1
2349.2.a.j 17 9.c even 3 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{34} + T_{2}^{33} + 29 T_{2}^{32} + 24 T_{2}^{31} + 493 T_{2}^{30} + 363 T_{2}^{29} + 5549 T_{2}^{28} + \cdots + 13689 \) acting on \(S_{2}^{\mathrm{new}}(261, [\chi])\). Copy content Toggle raw display