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Properties

Label 103.6.e.a
Level $103$
Weight $6$
Character orbit 103.e
Analytic conductor $16.520$
Analytic rank $0$
Dimension $688$
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] = [103,6,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, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("103.8");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 103 \)
Weight: \( k \) \(=\) \( 6 \)
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: \(16.5195334407\)
Analytic rank: \(0\)
Dimension: \(688\)
Relative dimension: \(43\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 688 q - 15 q^{2} + 5 q^{3} - 719 q^{4} - 23 q^{5} + 41 q^{6} + 3 q^{7} + 295 q^{8} - 3596 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 688 q - 15 q^{2} + 5 q^{3} - 719 q^{4} - 23 q^{5} + 41 q^{6} + 3 q^{7} + 295 q^{8} - 3596 q^{9} - 5041 q^{10} + 287 q^{11} + 8557 q^{12} + 457 q^{13} - 2615 q^{14} - 2491 q^{15} - 14279 q^{16} - 259 q^{17} + 319 q^{18} + 1212 q^{19} - 875 q^{20} - 13919 q^{21} + 6565 q^{22} + 11726 q^{23} - 3509 q^{24} - 2948 q^{25} + 1780 q^{26} + 3917 q^{27} + 72430 q^{28} - 9557 q^{29} + 51267 q^{30} + 2303 q^{31} + 13594 q^{32} - 6535 q^{33} + 132929 q^{34} + 13447 q^{35} - 181237 q^{36} + 74016 q^{37} - 111161 q^{38} + 42535 q^{39} + 17409 q^{40} + 28239 q^{41} + 14365 q^{42} - 8347 q^{43} - 57287 q^{44} + 152438 q^{45} + 244272 q^{46} - 363010 q^{47} - 211549 q^{48} - 8872 q^{49} - 210658 q^{50} - 24653 q^{51} + 29585 q^{52} - 5885 q^{53} + 80739 q^{54} + 163952 q^{55} + 31284 q^{56} - 204796 q^{57} - 283389 q^{58} + 137079 q^{59} + 461120 q^{60} + 25301 q^{61} - 98747 q^{62} + 57887 q^{63} - 234051 q^{64} + 71023 q^{65} - 400511 q^{66} - 51678 q^{67} + 1210611 q^{68} - 183095 q^{69} + 173227 q^{70} + 57515 q^{71} + 187023 q^{72} - 269490 q^{73} + 293787 q^{74} - 244901 q^{75} - 55945 q^{76} + 227417 q^{77} - 799423 q^{78} + 45373 q^{79} + 210475 q^{80} - 130012 q^{81} - 1960880 q^{82} - 298471 q^{83} + 867671 q^{84} - 375465 q^{85} - 496707 q^{86} - 174037 q^{87} + 463425 q^{88} - 64100 q^{89} + 1357061 q^{90} + 1588335 q^{91} + 449219 q^{92} + 40467 q^{93} + 255023 q^{94} + 97738 q^{95} + 27203 q^{96} + 514132 q^{97} - 614148 q^{98} - 329618 q^{99}+O(q^{100}) \) 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 −10.6504 + 1.99091i 2.48603 + 26.8285i 79.6283 − 30.8482i 53.0243 + 70.2156i −79.8902 − 280.785i 14.6304 + 9.05877i −491.873 + 304.555i −474.726 + 88.7417i −704.523 − 642.257i
8.2 −10.6072 + 1.98283i 0.0919698 + 0.992512i 78.7419 − 30.5048i −22.3980 − 29.6598i −2.94352 − 10.3454i 186.614 + 115.547i −481.156 + 297.919i 237.886 − 44.4686i 296.390 + 270.195i
8.3 −10.5130 + 1.96521i −1.21159 − 13.0751i 76.8212 − 29.7607i 22.9744 + 30.4230i 38.4328 + 135.077i −178.128 − 110.292i −458.153 + 283.676i 69.3716 − 12.9678i −301.316 − 274.686i
8.4 −9.27462 + 1.73373i −2.63289 − 28.4134i 53.1737 − 20.5996i −60.0360 − 79.5005i 73.6802 + 258.959i 13.0294 + 8.06748i −200.747 + 124.297i −561.527 + 104.968i 694.644 + 633.251i
8.5 −9.09571 + 1.70028i 1.80863 + 19.5182i 50.0020 − 19.3709i −45.7811 − 60.6240i −49.6374 − 174.457i −71.3831 − 44.1986i −170.114 + 105.330i −138.828 + 25.9515i 519.490 + 473.578i
8.6 −8.57888 + 1.60367i −2.55628 − 27.5866i 41.1864 − 15.9557i 41.5962 + 55.0823i 66.1699 + 232.563i 103.598 + 64.1452i −90.2972 + 55.9097i −515.625 + 96.3871i −445.183 − 405.838i
8.7 −8.54028 + 1.59646i −0.784691 − 8.46817i 40.5486 − 15.7086i −18.8119 − 24.9110i 20.2205 + 71.0678i −13.1025 − 8.11275i −84.8390 + 52.5301i 167.768 − 31.3613i 200.428 + 182.714i
8.8 −8.31625 + 1.55458i 1.08759 + 11.7369i 36.9042 − 14.2968i 19.4514 + 25.7578i −27.2906 − 95.9164i −38.6783 − 23.9486i −54.4999 + 33.7449i 102.290 − 19.1213i −201.805 − 183.970i
8.9 −8.18181 + 1.52945i −0.372250 − 4.01721i 34.7637 − 13.4675i 56.4223 + 74.7152i 9.18979 + 32.2988i 160.047 + 99.0971i −37.3744 + 23.1413i 222.863 − 41.6603i −575.910 − 525.011i
8.10 −6.49361 + 1.21387i 1.13167 + 12.2127i 10.8544 − 4.20503i 37.0386 + 49.0470i −22.1732 − 77.9307i −94.2311 − 58.3454i 114.352 − 70.8035i 90.9936 − 17.0097i −300.051 − 273.532i
8.11 −6.34798 + 1.18664i 2.68076 + 28.9300i 9.04965 − 3.50585i −8.64145 − 11.4431i −51.3471 − 180.466i 169.172 + 104.747i 122.414 − 75.7956i −590.899 + 110.458i 68.4347 + 62.3865i
8.12 −6.09806 + 1.13992i −1.83490 − 19.8018i 6.04776 − 2.34291i −3.03817 − 4.02319i 33.7619 + 118.661i −20.5665 − 12.7342i 134.575 − 83.3250i −149.881 + 28.0176i 23.1131 + 21.0704i
8.13 −4.79590 + 0.896510i −1.87127 − 20.1942i −7.64215 + 2.96058i 7.72068 + 10.2238i 27.0787 + 95.1719i −46.2979 − 28.6665i 166.739 − 103.240i −165.442 + 30.9266i −46.1934 − 42.1108i
8.14 −4.63424 + 0.866290i −0.556714 − 6.00790i −9.11337 + 3.53054i −46.4538 − 61.5147i 7.78453 + 27.3598i 123.697 + 76.5898i 167.443 − 103.676i 203.078 − 37.9618i 268.568 + 244.832i
8.15 −4.55535 + 0.851543i 1.45636 + 15.7167i −9.81301 + 3.80158i −4.93157 − 6.53046i −20.0176 − 70.3547i 92.4765 + 57.2590i 167.549 − 103.742i −6.02984 + 1.12717i 28.0260 + 25.5491i
8.16 −4.26113 + 0.796543i 0.0344373 + 0.371638i −12.3164 + 4.77140i −38.9124 − 51.5284i −0.442767 − 1.55616i −213.559 − 132.230i 166.621 − 103.168i 238.726 − 44.6255i 206.855 + 188.574i
8.17 −3.13099 + 0.585283i −1.66555 − 17.9741i −20.3786 + 7.89470i 57.9817 + 76.7802i 15.7348 + 55.3020i −143.402 − 88.7905i 145.845 − 90.3032i −81.4327 + 15.2224i −226.478 − 206.462i
8.18 −2.81119 + 0.525503i 2.45514 + 26.4952i −22.2125 + 8.60515i 33.9356 + 44.9380i −20.8252 − 73.1930i −166.533 − 103.113i 135.730 − 84.0407i −457.106 + 85.4479i −119.015 − 108.496i
8.19 −2.15278 + 0.402425i 0.316436 + 3.41488i −25.3666 + 9.82707i 41.5031 + 54.9590i −2.05545 − 7.22416i 74.3445 + 46.0322i 110.239 − 68.2573i 227.301 − 42.4900i −111.464 − 101.613i
8.20 −0.789665 + 0.147614i 1.94432 + 20.9825i −29.2373 + 11.3266i −51.9301 − 68.7665i −4.63267 − 16.2821i −10.4899 − 6.49505i 43.2722 − 26.7930i −197.623 + 36.9422i 51.1582 + 46.6369i
See next 80 embeddings (of 688 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 8.43
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 103.6.e.a ✓ 688
103.e even 17 1 inner 103.6.e.a ✓ 688
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
103.6.e.a ✓ 688 1.a even 1 1 trivial
103.6.e.a ✓ 688 103.e even 17 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{6}^{\mathrm{new}}(103, [\chi])\).