↑

Properties

Label 845.2.u.a
Level $845$
Weight $2$
Character orbit 845.u
Analytic conductor $6.747$
Analytic rank $0$
Dimension $372$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [845,2,Mod(66,845)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(845, base_ring=CyclotomicField(26))
 
chi = DirichletCharacter(H, H._module([0, 12]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("845.66");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 845 = 5 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 845.u (of order \(13\), degree \(12\), 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: \(6.74735897080\)
Analytic rank: \(0\)
Dimension: \(372\)
Relative dimension: \(31\) over \(\Q(\zeta_{13})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{13}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 372 q - q^{2} - 4 q^{3} - 37 q^{4} - 31 q^{5} - 16 q^{6} + q^{7} - 9 q^{8} - 39 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 372 q - q^{2} - 4 q^{3} - 37 q^{4} - 31 q^{5} - 16 q^{6} + q^{7} - 9 q^{8} - 39 q^{9} - q^{10} - 14 q^{11} - 20 q^{12} + 67 q^{13} - 10 q^{14} - 4 q^{15} - 13 q^{16} - 12 q^{17} + 2 q^{18} + 80 q^{19} - 37 q^{20} - 22 q^{21} + 54 q^{23} + 128 q^{24} - 31 q^{25} - 41 q^{26} - 40 q^{27} - 76 q^{28} - 2 q^{29} - 16 q^{30} - 16 q^{31} + 204 q^{32} - 27 q^{33} + 25 q^{34} - 12 q^{35} - 107 q^{36} - 36 q^{37} + 17 q^{38} - 52 q^{39} - 9 q^{40} - 28 q^{41} - 102 q^{42} - 36 q^{43} - 88 q^{44} - 39 q^{45} - 66 q^{46} - 3 q^{47} + 237 q^{48} + 52 q^{49} - q^{50} + 42 q^{51} + 80 q^{52} - 36 q^{53} - 16 q^{54} - q^{55} - 84 q^{56} + 71 q^{57} - 50 q^{58} + 92 q^{59} + 19 q^{60} + 14 q^{61} + 81 q^{62} - 81 q^{63} - 135 q^{64} - 11 q^{65} + 14 q^{66} - 10 q^{67} - 61 q^{68} - 8 q^{69} - 10 q^{70} + 19 q^{71} - 20 q^{72} - 68 q^{73} - 128 q^{74} - 4 q^{75} + 17 q^{76} - 104 q^{77} + 95 q^{78} - 74 q^{79} + 416 q^{80} + 73 q^{81} + 75 q^{82} - 12 q^{83} - 182 q^{84} - 12 q^{85} - 98 q^{86} - 132 q^{87} - 158 q^{88} + 88 q^{89} + 2 q^{90} - 48 q^{91} + 35 q^{92} + 65 q^{93} - 46 q^{94} - 24 q^{95} - 67 q^{96} - 80 q^{97} - 56 q^{98} + 116 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
66.1 −0.321428 − 2.64719i −1.64567 − 1.45794i −4.96244 + 1.22313i 0.885456 + 0.464723i −3.33048 + 4.82503i 0.503513 + 1.32765i 2.94173 + 7.75670i 0.221041 + 1.82044i 0.945603 − 2.49335i
66.2 −0.318984 − 2.62707i 0.202057 + 0.179007i −4.85787 + 1.19736i 0.885456 + 0.464723i 0.405811 − 0.587918i 0.471060 + 1.24208i 2.81829 + 7.43123i −0.352826 − 2.90579i 0.938414 − 2.47439i
66.3 −0.307050 − 2.52879i 2.28462 + 2.02400i −4.35859 + 1.07430i 0.885456 + 0.464723i 4.41676 − 6.39879i −0.0542133 − 0.142949i 2.24836 + 5.92844i 0.761318 + 6.27002i 0.903306 − 2.38182i
66.4 −0.277541 − 2.28576i −1.66884 − 1.47846i −3.20576 + 0.790150i 0.885456 + 0.464723i −2.91623 + 4.22489i −1.73037 − 4.56262i 1.06283 + 2.80246i 0.237563 + 1.95651i 0.816493 − 2.15292i
66.5 −0.258552 − 2.12937i 1.73405 + 1.53623i −2.52548 + 0.622476i 0.885456 + 0.464723i 2.82287 − 4.08963i 0.677710 + 1.78698i 0.457189 + 1.20551i 0.285303 + 2.34968i 0.760631 − 2.00562i
66.6 −0.253450 − 2.08735i −0.953925 − 0.845104i −2.35090 + 0.579444i 0.885456 + 0.464723i −1.52225 + 2.20536i 1.16790 + 3.07950i 0.314094 + 0.828198i −0.165838 − 1.36579i 0.745620 − 1.96604i
66.7 −0.211763 − 1.74403i −1.51934 − 1.34602i −1.05490 + 0.260010i 0.885456 + 0.464723i −2.02575 + 2.93481i 1.28972 + 3.40072i −0.569112 − 1.50063i 0.135022 + 1.11201i 0.622983 − 1.64267i
66.8 −0.202882 − 1.67088i 0.410347 + 0.363536i −0.808797 + 0.199351i 0.885456 + 0.464723i 0.524173 − 0.759395i −0.505882 − 1.33390i −0.696527 − 1.83659i −0.325384 − 2.67978i 0.596854 − 1.57378i
66.9 −0.189708 − 1.56238i −2.45442 − 2.17443i −0.463171 + 0.114161i 0.885456 + 0.464723i −2.93167 + 4.24725i −0.680263 − 1.79371i −0.849965 − 2.24117i 0.934438 + 7.69579i 0.558098 − 1.47158i
66.10 −0.159853 − 1.31651i 2.20927 + 1.95724i 0.234236 − 0.0577341i 0.885456 + 0.464723i 2.22357 − 3.22140i −1.48881 − 3.92566i −1.05399 − 2.77915i 0.688466 + 5.67003i 0.470270 − 1.24000i
66.11 −0.124688 − 1.02690i −0.0276287 − 0.0244769i 0.902914 − 0.222548i 0.885456 + 0.464723i −0.0216903 + 0.0314238i −0.412038 − 1.08646i −1.07475 − 2.83388i −0.361446 − 2.97677i 0.366817 − 0.967217i
66.12 −0.105874 − 0.871954i 1.83780 + 1.62815i 1.19279 − 0.293996i 0.885456 + 0.464723i 1.22509 − 1.77485i 0.828647 + 2.18496i −1.00558 − 2.65149i 0.365028 + 3.00628i 0.311470 − 0.821279i
66.13 −0.0972629 − 0.801032i 0.458017 + 0.405767i 1.30969 − 0.322810i 0.885456 + 0.464723i 0.280485 − 0.406352i 1.36633 + 3.60272i −0.958238 − 2.52667i −0.316478 − 2.60643i 0.286136 − 0.754479i
66.14 −0.0359872 − 0.296381i −1.10172 − 0.976039i 1.85534 − 0.457300i 0.885456 + 0.464723i −0.249632 + 0.361654i −1.05078 − 2.77068i −0.414043 − 1.09174i −0.100475 − 0.827483i 0.105870 − 0.279156i
66.15 −0.0155175 − 0.127798i −2.15025 − 1.90496i 1.92579 − 0.474665i 0.885456 + 0.464723i −0.210084 + 0.304358i 0.0965176 + 0.254496i −0.181846 − 0.479489i 0.633112 + 5.21415i 0.0456507 − 0.120371i
66.16 0.00835807 + 0.0688349i 1.54982 + 1.37302i 1.93722 − 0.477481i 0.885456 + 0.464723i −0.0815582 + 0.118157i −1.87012 − 4.93111i 0.0982356 + 0.259026i 0.155146 + 1.27774i −0.0245885 + 0.0648344i
66.17 0.0248221 + 0.204428i −0.0268110 − 0.0237524i 1.90071 − 0.468483i 0.885456 + 0.464723i 0.00419016 − 0.00607050i 1.28229 + 3.38111i 0.288998 + 0.762025i −0.361455 − 2.97685i −0.0730236 + 0.192547i
66.18 0.0488709 + 0.402488i 2.17777 + 1.92934i 1.78228 − 0.439291i 0.885456 + 0.464723i −0.670106 + 0.970816i 0.162628 + 0.428815i 0.551456 + 1.45407i 0.658738 + 5.42520i −0.143772 + 0.379097i
66.19 0.0884364 + 0.728340i 0.0409502 + 0.0362787i 1.41923 − 0.349808i 0.885456 + 0.464723i −0.0228018 + 0.0330340i −0.382125 − 1.00758i 0.900630 + 2.37477i −0.361249 − 2.97516i −0.260170 + 0.686011i
66.20 0.131377 + 1.08199i −1.92598 − 1.70627i 0.788447 − 0.194335i 0.885456 + 0.464723i 1.59313 − 2.30805i 1.46572 + 3.86478i 1.08684 + 2.86577i 0.436433 + 3.59435i −0.386496 + 1.01911i
See next 80 embeddings (of 372 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 66.31
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
169.g even 13 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 845.2.u.a ✓ 372
169.g even 13 1 inner 845.2.u.a ✓ 372
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
845.2.u.a ✓ 372 1.a even 1 1 trivial
845.2.u.a ✓ 372 169.g even 13 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{372} + T_{2}^{371} + 50 T_{2}^{370} + 54 T_{2}^{369} + 1336 T_{2}^{368} + 1447 T_{2}^{367} + \cdots + 36\!\cdots\!81 \) acting on \(S_{2}^{\mathrm{new}}(845, [\chi])\). Copy content Toggle raw display