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Properties

Label 261.2.x.a
Level $261$
Weight $2$
Character orbit 261.x
Analytic conductor $2.084$
Analytic rank $0$
Dimension $672$
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(2,261)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(261, base_ring=CyclotomicField(84))
 
chi = DirichletCharacter(H, H._module([14, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("261.2");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 261 = 3^{2} \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 261.x (of order \(84\), degree \(24\), 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: \(672\)
Relative dimension: \(28\) over \(\Q(\zeta_{84})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{84}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 672 q - 36 q^{2} - 24 q^{3} - 14 q^{4} - 42 q^{5} - 28 q^{6} - 10 q^{7} - 28 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 672 q - 36 q^{2} - 24 q^{3} - 14 q^{4} - 42 q^{5} - 28 q^{6} - 10 q^{7} - 28 q^{9} - 56 q^{10} - 48 q^{11} - 10 q^{12} - 14 q^{13} - 24 q^{14} - 20 q^{15} - 54 q^{16} - 50 q^{18} - 48 q^{19} - 30 q^{20} - 80 q^{21} - 14 q^{22} - 30 q^{23} - 16 q^{24} + 30 q^{25} + 36 q^{27} - 168 q^{30} - 12 q^{31} + 24 q^{32} - 70 q^{33} - 14 q^{34} + 128 q^{36} - 48 q^{37} - 42 q^{38} - 44 q^{39} - 2 q^{40} - 24 q^{41} - 28 q^{42} - 12 q^{43} + 24 q^{45} - 64 q^{46} - 42 q^{47} - 28 q^{48} + 22 q^{49} - 66 q^{50} - 28 q^{51} + 22 q^{52} - 36 q^{54} + 76 q^{55} - 42 q^{56} - 42 q^{58} - 132 q^{59} - 20 q^{60} - 28 q^{63} - 308 q^{64} - 66 q^{65} - 46 q^{66} - 14 q^{67} + 60 q^{68} - 64 q^{69} - 14 q^{70} + 56 q^{72} - 48 q^{73} + 66 q^{74} + 24 q^{75} - 28 q^{76} + 30 q^{77} + 72 q^{78} - 12 q^{79} + 140 q^{81} - 136 q^{82} + 246 q^{83} + 238 q^{84} + 34 q^{85} + 118 q^{87} + 8 q^{88} - 48 q^{90} - 56 q^{91} + 462 q^{92} + 28 q^{93} + 26 q^{94} - 246 q^{95} + 126 q^{96} - 36 q^{97} - 16 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2.1 −2.20170 − 1.62493i 0.491207 + 1.66094i 1.61758 + 5.24407i 0.577008 − 1.47019i 1.61742 − 4.45507i −1.61063 − 0.496813i 3.15227 − 9.00866i −2.51743 + 1.63173i −3.65936 + 2.29933i
2.2 −2.09871 − 1.54892i 0.154284 − 1.72517i 1.41592 + 4.59031i 0.316022 − 0.805210i −2.99594 + 3.38165i 0.806416 + 0.248746i 2.41541 − 6.90284i −2.95239 − 0.532331i −1.91044 + 1.20041i
2.3 −1.87887 − 1.38667i −1.39195 + 1.03077i 1.01780 + 3.29961i −0.932500 + 2.37597i 4.04463 − 0.00651310i 0.824688 + 0.254382i 1.12065 − 3.20264i 0.875028 − 2.86955i 5.04674 − 3.17108i
2.4 −1.81062 − 1.33630i −1.71999 − 0.204042i 0.903135 + 2.92789i 1.51600 − 3.86270i 2.84158 + 2.66786i 2.72479 + 0.840488i 0.790819 − 2.26003i 2.91673 + 0.701902i −7.90661 + 4.96805i
2.5 −1.79882 − 1.32759i 1.58696 − 0.693948i 0.883741 + 2.86502i −1.11921 + 2.85170i −3.77592 − 0.858538i −0.451867 − 0.139383i 0.737082 − 2.10646i 2.03687 − 2.20253i 5.79913 − 3.64384i
2.6 −1.45392 − 1.07304i 1.56552 + 0.741045i 0.372954 + 1.20909i 0.540145 − 1.37627i −1.48097 − 2.75729i 3.98072 + 1.22789i −0.438482 + 1.25311i 1.90171 + 2.32024i −2.26212 + 1.42138i
2.7 −1.29500 − 0.955756i 1.12961 + 1.31300i 0.174054 + 0.564270i −0.633409 + 1.61390i −0.207940 − 2.77998i −4.42538 − 1.36505i −0.749268 + 2.14128i −0.447956 + 2.96637i 2.36276 − 1.48462i
2.8 −1.20822 − 0.891704i 0.979461 − 1.42852i 0.0751402 + 0.243598i 1.10182 − 2.80739i −2.45721 + 0.852566i −2.93685 − 0.905898i −0.865489 + 2.47342i −1.08131 − 2.79835i −3.83459 + 2.40943i
2.9 −1.13935 − 0.840879i −1.68635 − 0.395264i 0.00153258 + 0.00496850i 0.00108456 − 0.00276341i 1.58897 + 1.86836i −2.00585 − 0.618723i −0.932952 + 2.66622i 2.68753 + 1.33310i −0.00355938 + 0.00223651i
2.10 −1.13739 − 0.839433i −0.756704 − 1.55801i −0.000498633 − 0.00161653i −1.18474 + 3.01866i −0.447178 + 2.40727i 4.21652 + 1.30062i −0.934565 + 2.67083i −1.85480 + 2.35791i 3.88147 − 2.43889i
2.11 −0.810991 − 0.598538i −0.370120 + 1.69204i −0.290053 − 0.940327i 1.04194 − 2.65481i 1.31292 − 1.15070i −1.09049 − 0.336372i −0.993399 + 2.83897i −2.72602 − 1.25252i −2.43401 + 1.52939i
2.12 −0.583916 − 0.430950i −1.28220 + 1.16446i −0.434270 − 1.40787i −0.323698 + 0.824768i 1.25052 − 0.127381i −0.374294 − 0.115454i −0.832527 + 2.37922i 0.288078 − 2.98614i 0.544446 − 0.342098i
2.13 −0.397396 − 0.293291i 1.36641 − 1.06438i −0.517607 − 1.67804i 0.0814783 − 0.207603i −0.855182 + 0.0222244i 2.51455 + 0.775634i −0.612714 + 1.75104i 0.734172 − 2.90878i −0.0932674 + 0.0586038i
2.14 −0.202514 − 0.149462i 0.356874 + 1.69489i −0.570837 − 1.85061i −1.37930 + 3.51439i 0.181050 − 0.396578i 2.32547 + 0.717312i −0.327254 + 0.935237i −2.74528 + 1.20972i 0.804596 − 0.505561i
2.15 −0.0309399 − 0.0228347i −1.45710 − 0.936412i −0.589074 − 1.90973i 0.319150 − 0.813182i 0.0236998 + 0.0622449i −1.11047 − 0.342536i −0.0507833 + 0.145130i 1.24626 + 2.72889i −0.0284432 + 0.0178721i
2.16 0.0255759 + 0.0188759i 0.0530423 − 1.73124i −0.589213 − 1.91018i −1.13734 + 2.89789i 0.0340352 − 0.0432768i −4.48793 − 1.38434i 0.0419840 − 0.119983i −2.99437 − 0.183658i −0.0837888 + 0.0526480i
2.17 0.253664 + 0.187213i 1.71133 + 0.267109i −0.560213 − 1.81617i 0.760941 − 1.93885i 0.384097 + 0.388139i −2.32357 − 0.716725i 0.406156 − 1.16073i 2.85731 + 0.914223i 0.556000 − 0.349358i
2.18 0.547220 + 0.403867i 0.593978 + 1.62702i −0.453169 − 1.46914i 0.745589 − 1.89973i −0.332062 + 1.13022i 2.64496 + 0.815862i 0.794609 − 2.27086i −2.29438 + 1.93283i 1.17524 − 0.738451i
2.19 0.762374 + 0.562658i −1.70238 − 0.319240i −0.324880 − 1.05323i −0.663401 + 1.69032i −1.11823 − 1.20124i 3.32276 + 1.02494i 0.970824 − 2.77446i 2.79617 + 1.08693i −1.45683 + 0.915387i
2.20 0.912527 + 0.673475i 1.72940 − 0.0957406i −0.210374 − 0.682017i −1.48285 + 3.77823i 1.64261 + 1.07734i 0.633936 + 0.195543i 1.01651 − 2.90503i 2.98167 − 0.331148i −3.89768 + 2.44908i
See next 80 embeddings (of 672 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2.28
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.d odd 6 1 inner
29.f odd 28 1 inner
261.x even 84 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 261.2.x.a ✓ 672
3.b odd 2 1 783.2.be.a 672
9.c even 3 1 783.2.be.a 672
9.d odd 6 1 inner 261.2.x.a ✓ 672
29.f odd 28 1 inner 261.2.x.a ✓ 672
87.k even 28 1 783.2.be.a 672
261.w odd 84 1 783.2.be.a 672
261.x even 84 1 inner 261.2.x.a ✓ 672
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
261.2.x.a ✓ 672 1.a even 1 1 trivial
261.2.x.a ✓ 672 9.d odd 6 1 inner
261.2.x.a ✓ 672 29.f odd 28 1 inner
261.2.x.a ✓ 672 261.x even 84 1 inner
783.2.be.a 672 3.b odd 2 1
783.2.be.a 672 9.c even 3 1
783.2.be.a 672 87.k even 28 1
783.2.be.a 672 261.w odd 84 1

Hecke kernels

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