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Properties

Label 103.2.e.a
Level $103$
Weight $2$
Character orbit 103.e
Analytic conductor $0.822$
Analytic rank $0$
Dimension $112$
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] = [103,2,Mod(8,103)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(103, base_ring=CyclotomicField(34))
 
chi = DirichletCharacter(H, H._module([10]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("103.8");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 103 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 103.e (of order \(17\), degree \(16\), 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: \(0.822459140819\)
Analytic rank: \(0\)
Dimension: \(112\)
Relative dimension: \(7\) over \(\Q(\zeta_{17})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{17}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 112 q - 15 q^{2} - 11 q^{3} - 19 q^{4} - 11 q^{5} - 5 q^{6} - 5 q^{7} - 5 q^{8} - 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 112 q - 15 q^{2} - 11 q^{3} - 19 q^{4} - 11 q^{5} - 5 q^{6} - 5 q^{7} - 5 q^{8} - 8 q^{9} - 59 q^{10} - q^{11} - 41 q^{12} + q^{13} - 21 q^{14} + 13 q^{15} - q^{16} - 11 q^{17} + 19 q^{18} - 12 q^{19} + 31 q^{20} - 7 q^{21} + 23 q^{22} - 22 q^{23} + 73 q^{24} - 52 q^{25} + 18 q^{26} + 13 q^{27} - 50 q^{28} + 7 q^{29} - 13 q^{30} + 31 q^{31} + 34 q^{32} + 13 q^{33} - 91 q^{34} + 23 q^{35} - 53 q^{36} - 30 q^{37} + 15 q^{38} - 105 q^{39} + 75 q^{40} + 11 q^{41} + 57 q^{42} + 37 q^{43} + 83 q^{44} - 4 q^{45} - 56 q^{46} - 154 q^{47} - 9 q^{48} + 20 q^{49} + 12 q^{50} + 51 q^{51} + 113 q^{52} + 27 q^{53} + 95 q^{54} + 12 q^{55} + 8 q^{56} - 40 q^{57} - 13 q^{58} - 9 q^{59} - 84 q^{60} + 29 q^{61} + 41 q^{62} + 103 q^{63} - 57 q^{64} + 47 q^{65} - 3 q^{66} + 10 q^{67} - 105 q^{68} - 35 q^{69} + 143 q^{70} + 11 q^{71} + 135 q^{72} - 40 q^{73} + 97 q^{74} - 117 q^{75} + 131 q^{76} - 19 q^{77} + 81 q^{78} + 77 q^{79} - 29 q^{80} + 104 q^{81} - 162 q^{82} + 73 q^{83} - 163 q^{84} + 55 q^{85} - 99 q^{86} + 75 q^{87} - 63 q^{88} + 54 q^{89} + 107 q^{90} - 113 q^{91} - 11 q^{92} - 197 q^{93} + 33 q^{94} - 146 q^{95} + 49 q^{96} - 142 q^{97} - 2 q^{98} + 102 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
8.1 −2.39943 + 0.448531i 0.0268231 + 0.289468i 3.69113 − 1.42995i −1.02301 − 1.35469i −0.194195 − 0.682526i 0.635463 + 0.393462i −4.06448 + 2.51662i 2.86585 − 0.535720i 3.06227 + 2.79162i
8.2 −1.95331 + 0.365137i −0.294344 − 3.17647i 1.81715 − 0.703966i 1.08771 + 1.44036i 1.73479 + 6.09716i −4.06977 − 2.51990i 0.0866052 − 0.0536236i −7.05442 + 1.31870i −2.65056 − 2.41630i
8.3 −1.02397 + 0.191413i −0.0614087 − 0.662705i −0.853072 + 0.330482i 1.15988 + 1.53593i 0.189731 + 0.666835i 3.28073 + 2.03134i 2.58161 − 1.59847i 2.51351 − 0.469857i −1.48168 − 1.35073i
8.4 0.0797342 − 0.0149049i 0.284285 + 3.06792i −1.85881 + 0.720106i −1.51372 − 2.00449i 0.0683942 + 0.240381i 3.32177 + 2.05675i −0.275409 + 0.170526i −6.38239 + 1.19308i −0.150572 − 0.137264i
8.5 0.283205 − 0.0529401i 0.135418 + 1.46139i −1.78754 + 0.692498i 1.97447 + 2.61462i 0.115717 + 0.406704i −3.42841 − 2.12278i −0.959492 + 0.594092i 0.831589 − 0.155451i 0.697599 + 0.635945i
8.6 1.28051 − 0.239368i −0.240301 − 2.59326i −0.282547 + 0.109459i 0.235695 + 0.312111i −0.928452 − 3.26317i 1.40985 + 0.872944i −2.55073 + 1.57935i −3.71835 + 0.695081i 0.376518 + 0.343242i
8.7 1.80079 − 0.336626i 0.0591724 + 0.638572i 1.26458 − 0.489899i −0.551389 − 0.730157i 0.321517 + 1.13001i −0.608782 − 0.376942i −1.00284 + 0.620934i 2.54465 − 0.475677i −1.23872 − 1.12925i
9.1 −1.08551 + 2.17999i 2.69872 − 1.67098i −2.36876 − 3.13675i 0.369204 − 0.336574i 0.713237 + 7.69705i 1.23546 + 0.230948i 4.62172 − 0.863950i 3.15372 − 6.33353i 0.332954 + 1.17021i
9.2 −0.782612 + 1.57170i −0.156591 + 0.0969570i −0.652480 − 0.864023i −1.96201 + 1.78861i −0.0298371 − 0.321993i 1.29375 + 0.241845i −1.58312 + 0.295936i −1.32209 + 2.65512i −1.27566 − 4.48347i
9.3 −0.521502 + 1.04732i −2.66663 + 1.65111i 0.380359 + 0.503677i 0.746722 − 0.680727i −0.338581 − 3.65387i −2.90961 − 0.543901i −3.02597 + 0.565653i 3.04756 − 6.12032i 0.323520 + 1.13706i
9.4 −0.0973686 + 0.195542i 1.76009 − 1.08980i 1.17651 + 1.55796i −0.0971128 + 0.0885300i 0.0417250 + 0.450285i −3.92208 − 0.733163i −0.848650 + 0.158640i 0.573034 − 1.15081i −0.00785564 − 0.0276097i
9.5 0.258380 − 0.518897i −1.24680 + 0.771983i 1.00278 + 1.32789i 0.295386 − 0.269280i 0.0784327 + 0.846424i 2.64368 + 0.494190i 2.08773 − 0.390264i −0.378673 + 0.760478i −0.0634067 − 0.222851i
9.6 0.912658 − 1.83286i −0.480518 + 0.297524i −1.32118 − 1.74952i 1.42563 − 1.29963i 0.106773 + 1.15226i −2.13391 − 0.398897i −0.387107 + 0.0723628i −1.19484 + 2.39956i −1.08094 − 3.79910i
9.7 0.918586 − 1.84477i 1.60306 − 0.992571i −1.35410 − 1.79312i −2.94362 + 2.68347i −0.358518 − 3.86903i 0.512164 + 0.0957400i −0.500295 + 0.0935213i 0.247379 − 0.496804i 2.24641 + 7.89530i
13.1 −2.39943 − 0.448531i 0.0268231 − 0.289468i 3.69113 + 1.42995i −1.02301 + 1.35469i −0.194195 + 0.682526i 0.635463 − 0.393462i −4.06448 − 2.51662i 2.86585 + 0.535720i 3.06227 − 2.79162i
13.2 −1.95331 − 0.365137i −0.294344 + 3.17647i 1.81715 + 0.703966i 1.08771 − 1.44036i 1.73479 − 6.09716i −4.06977 + 2.51990i 0.0866052 + 0.0536236i −7.05442 − 1.31870i −2.65056 + 2.41630i
13.3 −1.02397 − 0.191413i −0.0614087 + 0.662705i −0.853072 − 0.330482i 1.15988 − 1.53593i 0.189731 − 0.666835i 3.28073 − 2.03134i 2.58161 + 1.59847i 2.51351 + 0.469857i −1.48168 + 1.35073i
13.4 0.0797342 + 0.0149049i 0.284285 − 3.06792i −1.85881 − 0.720106i −1.51372 + 2.00449i 0.0683942 − 0.240381i 3.32177 − 2.05675i −0.275409 − 0.170526i −6.38239 − 1.19308i −0.150572 + 0.137264i
13.5 0.283205 + 0.0529401i 0.135418 − 1.46139i −1.78754 − 0.692498i 1.97447 − 2.61462i 0.115717 − 0.406704i −3.42841 + 2.12278i −0.959492 − 0.594092i 0.831589 + 0.155451i 0.697599 − 0.635945i
13.6 1.28051 + 0.239368i −0.240301 + 2.59326i −0.282547 − 0.109459i 0.235695 − 0.312111i −0.928452 + 3.26317i 1.40985 − 0.872944i −2.55073 − 1.57935i −3.71835 − 0.695081i 0.376518 − 0.343242i
See next 80 embeddings (of 112 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 8.7
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
103.e even 17 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 103.2.e.a ✓ 112
3.b odd 2 1 927.2.u.a 112
103.e even 17 1 inner 103.2.e.a ✓ 112
309.l odd 34 1 927.2.u.a 112
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
103.2.e.a ✓ 112 1.a even 1 1 trivial
103.2.e.a ✓ 112 103.e even 17 1 inner
927.2.u.a 112 3.b odd 2 1
927.2.u.a 112 309.l odd 34 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(103, [\chi])\).