Properties

Label 828.2.o.a
Level $828$
Weight $2$
Character orbit 828.o
Analytic conductor $6.612$
Analytic rank $0$
Dimension $132$
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] = [828,2,Mod(47,828)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(828, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("828.47");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 828 = 2^{2} \cdot 3^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 828.o (of order \(6\), 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: \(6.61161328736\)
Analytic rank: \(0\)
Dimension: \(132\)
Relative dimension: \(66\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 132 q - 9 q^{8} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 132 q - 9 q^{8} + 2 q^{9} - 29 q^{12} - 15 q^{14} - 34 q^{18} - 21 q^{20} + 66 q^{23} + 15 q^{24} + 66 q^{25} - 54 q^{26} + 12 q^{27} + 77 q^{30} + 30 q^{32} - 22 q^{33} + 36 q^{36} - 24 q^{39} - 30 q^{41} - 11 q^{42} - 65 q^{48} + 66 q^{49} - 32 q^{51} - 64 q^{54} - 42 q^{56} - 6 q^{57} + 27 q^{58} + 36 q^{59} + 58 q^{60} + 63 q^{62} - 20 q^{63} - 9 q^{64} + 78 q^{66} - 15 q^{68} - 78 q^{70} + 36 q^{72} - 12 q^{73} - 21 q^{74} + 48 q^{75} - 87 q^{76} - 11 q^{78} + 72 q^{80} - 10 q^{81} - 9 q^{82} - 157 q^{84} - 24 q^{86} + 96 q^{87} + 108 q^{88} - 63 q^{90} - 12 q^{93} + 75 q^{94} - 9 q^{96} - 6 q^{97} + 20 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
47.1 −1.41421 0.00214998i −1.34750 1.08823i 1.99999 + 0.00608105i −1.10823 + 0.639837i 1.90331 + 1.54189i −3.10783 1.79431i −2.82840 0.0128998i 0.631500 + 2.93278i 1.56865 0.902482i
47.2 −1.40722 + 0.140428i −1.25401 + 1.19476i 1.96056 0.395228i −0.532014 + 0.307158i 1.59689 1.85740i 1.07017 + 0.617862i −2.70345 + 0.831492i 0.145075 2.99649i 0.705529 0.506950i
47.3 −1.39754 0.216507i 1.65212 + 0.520096i 1.90625 + 0.605156i 2.17762 1.25725i −2.19630 1.08455i 3.49734 + 2.01919i −2.53304 1.25845i 2.45900 + 1.71852i −3.31551 + 1.28559i
47.4 −1.35893 + 0.391560i −1.70231 0.319612i 1.69336 1.06420i −0.811777 + 0.468680i 2.43846 0.232226i 4.54437 + 2.62369i −1.88445 + 2.10923i 2.79570 + 1.08816i 0.919629 0.954761i
47.5 −1.35549 0.403287i −1.22500 + 1.22449i 1.67472 + 1.09331i 2.26529 1.30786i 2.15430 1.16576i 0.222415 + 0.128411i −1.82915 2.15736i 0.00123454 3.00000i −3.59802 + 0.859237i
47.6 −1.35188 + 0.415239i 0.255232 1.71314i 1.65515 1.12271i 2.99840 1.73112i 0.366321 + 2.42194i 1.15385 + 0.666173i −1.77138 + 2.20505i −2.86971 0.874496i −3.33464 + 3.58532i
47.7 −1.35126 0.417239i 0.0496784 + 1.73134i 1.65182 + 1.12760i −3.55240 + 2.05098i 0.655254 2.36022i 3.04028 + 1.75531i −1.76157 2.21289i −2.99506 + 0.172020i 5.65597 1.28921i
47.8 −1.32104 + 0.504830i 1.42989 0.977450i 1.49029 1.33380i −1.85603 + 1.07158i −1.39550 + 2.01310i −3.04212 1.75637i −1.29539 + 2.51435i 1.08918 2.79530i 1.91093 2.35258i
47.9 −1.30862 0.536209i 1.72401 + 0.166709i 1.42496 + 1.40339i −2.66178 + 1.53678i −2.16668 1.14259i −2.35100 1.35735i −1.11222 2.60057i 2.94442 + 0.574816i 4.30729 0.583786i
47.10 −1.25744 + 0.647177i −0.0751077 1.73042i 1.16232 1.62758i −3.62509 + 2.09295i 1.21433 + 2.12730i 1.00949 + 0.582830i −0.408228 + 2.79881i −2.98872 + 0.259936i 3.20384 4.97784i
47.11 −1.25736 0.647332i −0.895324 1.48270i 1.16192 + 1.62786i 2.77988 1.60497i 0.165950 + 2.44386i 0.136435 + 0.0787710i −0.407193 2.79896i −1.39679 + 2.65499i −4.53427 + 0.218519i
47.12 −1.24577 0.669373i 0.632882 1.61228i 1.10388 + 1.66777i 0.439521 0.253757i −1.86764 + 1.58490i −2.35304 1.35853i −0.258822 2.81656i −2.19892 2.04077i −0.717399 + 0.0219199i
47.13 −1.22156 + 0.712591i −0.317113 + 1.70277i 0.984428 1.74095i 2.33026 1.34538i −0.826008 2.30602i −3.25138 1.87718i 0.0380433 + 2.82817i −2.79888 1.07994i −1.88786 + 3.30399i
47.14 −1.17797 + 0.782556i 1.48590 + 0.890002i 0.775213 1.84365i 1.46262 0.844445i −2.44682 + 0.114406i −0.558634 0.322527i 0.529584 + 2.77841i 1.41579 + 2.64491i −1.06209 + 2.13931i
47.15 −1.17633 + 0.785014i −1.35464 1.07932i 0.767507 1.84687i 2.67514 1.54449i 2.44079 + 0.206227i −1.19116 0.687714i 0.546978 + 2.77503i 0.670120 + 2.92420i −1.93440 + 3.91685i
47.16 −1.14034 0.836430i 1.64820 0.532379i 0.600769 + 1.90764i −0.636131 + 0.367271i −2.32482 0.771512i 2.19630 + 1.26803i 0.910522 2.67786i 2.43315 1.75494i 1.03260 + 0.113265i
47.17 −1.00627 0.993688i 1.01905 + 1.40055i 0.0251665 + 1.99984i −1.87865 + 1.08464i 0.366266 2.42195i −1.45711 0.841265i 1.96190 2.03739i −0.923070 + 2.85446i 2.96823 + 0.775351i
47.18 −0.863315 + 1.12013i −1.71951 0.208062i −0.509373 1.93405i −1.75788 + 1.01492i 1.71753 1.74645i −4.07349 2.35183i 2.60613 + 1.09913i 2.91342 + 0.715530i 0.380774 2.84525i
47.19 −0.852594 + 1.12831i 1.71309 + 0.255573i −0.546167 1.92398i −2.82290 + 1.62980i −1.74894 + 1.71500i 3.09894 + 1.78918i 2.63651 + 1.02413i 2.86936 + 0.875640i 0.567866 4.57467i
47.20 −0.828510 1.14611i −1.40140 + 1.01788i −0.627142 + 1.89913i −2.23183 + 1.28855i 2.32768 + 0.762840i −2.37288 1.36998i 2.69621 0.854674i 0.927852 2.85291i 3.32591 + 1.49035i
See next 80 embeddings (of 132 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 47.66
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
36.h even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 828.2.o.a 132
4.b odd 2 1 828.2.o.b yes 132
9.d odd 6 1 828.2.o.b yes 132
36.h even 6 1 inner 828.2.o.a 132
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
828.2.o.a 132 1.a even 1 1 trivial
828.2.o.a 132 36.h even 6 1 inner
828.2.o.b yes 132 4.b odd 2 1
828.2.o.b yes 132 9.d odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{132} - 264 T_{7}^{130} + 36816 T_{7}^{128} + 732 T_{7}^{127} - 3538956 T_{7}^{126} - 177732 T_{7}^{125} + 260801214 T_{7}^{124} + 22853088 T_{7}^{123} - 15593339592 T_{7}^{122} + \cdots + 37\!\cdots\!76 \) acting on \(S_{2}^{\mathrm{new}}(828, [\chi])\). Copy content Toggle raw display