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Properties

Label 261.2.l.a
Level $261$
Weight $2$
Character orbit 261.l
Analytic conductor $2.084$
Analytic rank $0$
Dimension $112$
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] = [261,2,Mod(41,261)]
 
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([10, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("261.41");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 261 = 3^{2} \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 261.l (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: \(2.08409549276\)
Analytic rank: \(0\)
Dimension: \(112\)
Relative dimension: \(28\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 112 q - 6 q^{2} - 4 q^{3} - 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 112 q - 6 q^{2} - 4 q^{3} - 4 q^{7} + 6 q^{11} - 18 q^{12} - 18 q^{14} - 8 q^{15} + 40 q^{16} + 22 q^{18} - 8 q^{19} - 12 q^{20} + 24 q^{21} - 12 q^{23} - 96 q^{24} - 44 q^{25} + 20 q^{27} - 42 q^{29} + 28 q^{30} - 2 q^{31} - 66 q^{32} + 12 q^{36} - 8 q^{37} - 12 q^{39} - 12 q^{40} - 18 q^{41} - 2 q^{43} - 52 q^{45} + 8 q^{46} - 36 q^{49} + 24 q^{50} - 36 q^{52} + 8 q^{54} + 36 q^{55} + 84 q^{56} + 28 q^{58} + 48 q^{59} - 36 q^{60} - 14 q^{61} + 24 q^{65} + 18 q^{66} - 102 q^{68} + 36 q^{69} - 8 q^{73} + 144 q^{74} + 18 q^{75} + 14 q^{76} - 72 q^{77} + 12 q^{78} - 2 q^{79} - 56 q^{81} + 80 q^{82} - 120 q^{83} - 14 q^{84} - 48 q^{85} - 76 q^{87} - 36 q^{88} + 160 q^{90} - 40 q^{94} + 204 q^{95} + 22 q^{97} - 54 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
41.1 −2.58894 − 0.693705i 1.11469 − 1.32570i 4.48934 + 2.59192i 1.87884 − 3.25425i −3.80551 + 2.65888i −0.557243 − 0.965174i −6.03415 − 6.03415i −0.514937 − 2.95548i −7.12170 + 7.12170i
41.2 −2.53829 − 0.680131i 1.14019 + 1.30383i 4.24826 + 2.45274i −0.621203 + 1.07596i −2.00736 − 4.08496i 2.07180 + 3.58847i −5.39881 − 5.39881i −0.399921 + 2.97322i 2.30858 − 2.30858i
41.3 −2.50426 − 0.671015i −1.40115 − 1.01822i 4.08901 + 2.36079i −1.37767 + 2.38620i 2.82561 + 3.49008i −1.08986 − 1.88769i −4.98934 − 4.98934i 0.926458 + 2.85336i 5.05123 − 5.05123i
41.4 −2.28686 − 0.612761i −1.11376 + 1.32648i 3.12219 + 1.80260i 0.548233 − 0.949568i 3.35982 − 2.35100i 0.0705617 + 0.122216i −2.68724 − 2.68724i −0.519088 − 2.95475i −1.83559 + 1.83559i
41.5 −2.04933 − 0.549115i 0.850029 − 1.50912i 2.16616 + 1.25063i −1.19443 + 2.06881i −2.57067 + 2.62592i 0.190553 + 0.330047i −0.752008 − 0.752008i −1.55490 − 2.56559i 3.58379 − 3.58379i
41.6 −1.80586 − 0.483880i −1.12121 − 1.32018i 1.29495 + 0.747640i 0.957523 − 1.65848i 1.38595 + 2.92660i 2.42523 + 4.20063i 0.667230 + 0.667230i −0.485769 + 2.96041i −2.53166 + 2.53166i
41.7 −1.78585 − 0.478518i 1.70849 + 0.284740i 1.22825 + 0.709129i −1.14932 + 1.99067i −2.91485 − 1.32605i −1.59562 − 2.76369i 0.760534 + 0.760534i 2.83785 + 0.972949i 3.00508 − 3.00508i
41.8 −1.45552 − 0.390005i −1.72711 − 0.130672i 0.234384 + 0.135322i 1.15386 − 1.99855i 2.46289 + 0.863780i −1.71329 − 2.96750i 1.84266 + 1.84266i 2.96585 + 0.451370i −2.45892 + 2.45892i
41.9 −1.27699 − 0.342169i −0.592542 + 1.62754i −0.218425 − 0.126108i −1.41847 + 2.45685i 1.31356 − 1.87561i −0.670736 − 1.16175i 2.10542 + 2.10542i −2.29779 − 1.92877i 2.65203 − 2.65203i
41.10 −1.04682 − 0.280495i 1.72928 − 0.0978759i −0.714895 − 0.412745i 1.56226 − 2.70591i −1.83770 − 0.382596i 0.431040 + 0.746582i 2.16525 + 2.16525i 2.98084 − 0.338510i −2.39439 + 2.39439i
41.11 −0.983600 − 0.263555i 0.777862 + 1.54756i −0.834044 − 0.481535i 0.359924 − 0.623407i −0.357240 − 1.72718i 0.867014 + 1.50171i 2.13354 + 2.13354i −1.78986 + 2.40757i −0.518323 + 0.518323i
41.12 −0.726652 − 0.194706i 0.184317 − 1.72222i −1.24194 − 0.717034i 0.993651 − 1.72105i −0.469260 + 1.21556i −0.796821 − 1.38013i 1.82674 + 1.82674i −2.93205 − 0.634868i −1.05714 + 1.05714i
41.13 −0.466927 − 0.125113i −0.517906 − 1.65281i −1.52968 − 0.883163i −1.57483 + 2.72769i 0.0350373 + 0.836538i 0.454558 + 0.787318i 1.28738 + 1.28738i −2.46355 + 1.71200i 1.07660 − 1.07660i
41.14 −0.0776028 − 0.0207936i −1.64649 + 0.537659i −1.72646 − 0.996773i 0.0760639 − 0.131746i 0.138952 − 0.00748738i 0.766676 + 1.32792i 0.226870 + 0.226870i 2.42185 − 1.77050i −0.00864225 + 0.00864225i
41.15 0.188778 + 0.0505829i 1.73139 + 0.0476698i −1.69897 − 0.980902i −1.89314 + 3.27902i 0.324438 + 0.0965779i 2.26829 + 3.92879i −0.547502 − 0.547502i 2.99546 + 0.165070i −0.523246 + 0.523246i
41.16 0.220291 + 0.0590268i 1.37820 − 1.04907i −1.68701 − 0.973994i −0.486226 + 0.842168i 0.365530 − 0.149750i −2.17228 − 3.76249i −0.636670 − 0.636670i 0.798892 − 2.89167i −0.156822 + 0.156822i
41.17 0.469013 + 0.125672i −0.397710 + 1.68577i −1.52787 − 0.882117i 2.09952 − 3.63648i −0.398385 + 0.740668i −1.45885 − 2.52681i −1.29242 − 1.29242i −2.68365 − 1.34090i 1.44170 − 1.44170i
41.18 0.703401 + 0.188476i −1.41578 − 0.997788i −1.27280 − 0.734852i −0.0672512 + 0.116482i −0.807798 − 0.968684i 0.205252 + 0.355506i −1.78664 − 1.78664i 1.00884 + 2.82529i −0.0692586 + 0.0692586i
41.19 0.984157 + 0.263704i 1.58668 + 0.694591i −0.833025 − 0.480947i 0.783948 − 1.35784i 1.37837 + 1.10200i 0.259733 + 0.449871i −2.13391 − 2.13391i 2.03509 + 2.20418i 1.12960 − 1.12960i
41.20 1.05023 + 0.281409i −1.42135 + 0.989831i −0.708253 − 0.408910i −1.55901 + 2.70029i −1.77130 + 0.639572i −1.67740 − 2.90535i −2.16641 − 2.16641i 1.04047 − 2.81379i −2.39721 + 2.39721i
See next 80 embeddings (of 112 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 41.28
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.d odd 6 1 inner
29.c odd 4 1 inner
261.l even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 261.2.l.a ✓ 112
3.b odd 2 1 783.2.m.a 112
9.c even 3 1 783.2.m.a 112
9.d odd 6 1 inner 261.2.l.a ✓ 112
29.c odd 4 1 inner 261.2.l.a ✓ 112
87.f even 4 1 783.2.m.a 112
261.l even 12 1 inner 261.2.l.a ✓ 112
261.m odd 12 1 783.2.m.a 112
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
261.2.l.a ✓ 112 1.a even 1 1 trivial
261.2.l.a ✓ 112 9.d odd 6 1 inner
261.2.l.a ✓ 112 29.c odd 4 1 inner
261.2.l.a ✓ 112 261.l even 12 1 inner
783.2.m.a 112 3.b odd 2 1
783.2.m.a 112 9.c even 3 1
783.2.m.a 112 87.f even 4 1
783.2.m.a 112 261.m odd 12 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(261, [\chi])\).