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Properties

Label 17.11.e.a
Level $17$
Weight $11$
Character orbit 17.e
Analytic conductor $10.801$
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] = [17,11,Mod(3,17)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(17, base_ring=CyclotomicField(16))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 11, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("17.3");
 
S:= CuspForms(chi, 11);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 17 \)
Weight: \( k \) \(=\) \( 11 \)
Character orbit: \([\chi]\) \(=\) 17.e (of order \(16\), degree \(8\), 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: \(10.8010732955\)
Analytic rank: \(0\)
Dimension: \(112\)
Relative dimension: \(14\) over \(\Q(\zeta_{16})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{16}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 112 q - 8 q^{2} - 8 q^{3} - 8 q^{4} - 8 q^{5} - 8 q^{6} - 8 q^{7} - 8 q^{8} - 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 112 q - 8 q^{2} - 8 q^{3} - 8 q^{4} - 8 q^{5} - 8 q^{6} - 8 q^{7} - 8 q^{8} - 8 q^{9} - 30344 q^{10} + 458192 q^{11} - 270344 q^{12} - 826592 q^{13} + 1008120 q^{14} + 4565536 q^{15} - 2602240 q^{17} - 10126352 q^{18} + 1756816 q^{19} + 22335480 q^{20} + 12521392 q^{21} - 13960712 q^{22} - 14875792 q^{23} - 43778936 q^{24} + 44963040 q^{25} + 54577112 q^{26} - 10392632 q^{27} - 173302808 q^{28} - 50527608 q^{29} + 106080184 q^{30} + 162662712 q^{31} + 184358944 q^{32} - 378854728 q^{34} - 285891920 q^{35} - 384220360 q^{36} + 80037240 q^{37} + 813232016 q^{38} + 27771912 q^{39} + 761916984 q^{40} + 401786496 q^{41} - 880232312 q^{42} - 826499880 q^{43} - 1294131048 q^{44} + 632144552 q^{45} - 264367896 q^{46} + 1104027712 q^{47} + 2951599424 q^{48} + 848673752 q^{49} - 999270376 q^{51} - 3352526864 q^{52} - 763872224 q^{53} - 6671964840 q^{54} + 673064216 q^{55} + 9343853232 q^{56} + 8439888512 q^{57} + 2446156112 q^{58} - 2429006824 q^{59} - 17331774640 q^{60} - 6471192120 q^{61} - 6559331344 q^{62} + 7833307888 q^{63} + 14603881096 q^{64} + 11862114112 q^{65} + 2307132288 q^{66} - 3297863904 q^{68} - 23771220544 q^{69} - 22978325936 q^{70} - 1679848920 q^{71} + 15609317104 q^{72} + 14979081272 q^{73} + 10828279528 q^{74} + 22044104152 q^{75} - 6515833984 q^{76} - 4883972424 q^{77} - 25569823968 q^{78} - 10256870184 q^{79} - 5313987096 q^{80} + 1801151256 q^{81} + 34648177728 q^{82} + 55524121520 q^{83} - 61020827920 q^{85} - 14085650336 q^{86} + 5308640296 q^{87} + 2642840376 q^{88} + 14551869616 q^{89} - 5750978056 q^{90} - 14605964088 q^{91} - 30421976856 q^{92} - 20170086888 q^{93} + 2181844760 q^{94} + 9583306680 q^{95} + 49757899760 q^{96} - 4299869672 q^{97} + 78989351920 q^{98} + 19670705720 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3.1 −53.0389 + 21.9694i 195.324 + 38.8524i 1606.39 − 1606.39i −130.877 − 195.872i −11213.4 + 2230.48i −7359.09 + 11013.7i −27413.1 + 66181.0i −17912.0 − 7419.41i 11244.8 + 7513.53i
3.2 −48.2907 + 20.0027i −351.084 − 69.8350i 1207.81 − 1207.81i −95.4986 − 142.924i 18351.0 − 3650.24i 8809.03 − 13183.6i −13683.8 + 33035.5i 63829.2 + 26438.9i 7470.55 + 4991.66i
3.3 −32.7462 + 13.5639i −12.9719 − 2.58028i 164.255 − 164.255i −1061.55 − 1588.72i 459.780 − 91.4559i 4030.08 − 6031.44i 10738.7 − 25925.4i −54392.5 − 22530.1i 56311.1 + 37625.8i
3.4 −29.3837 + 12.1711i 384.312 + 76.4443i −8.81107 + 8.81107i 2956.30 + 4424.42i −12222.9 + 2431.29i 10978.5 − 16430.5i 12614.9 − 30455.1i 87297.5 + 36159.8i −140717. − 94024.3i
3.5 −25.8553 + 10.7096i −234.015 − 46.5485i −170.276 + 170.276i 2938.84 + 4398.29i 6549.06 − 1302.69i −18411.5 + 27554.8i 13545.6 − 32702.0i −1957.74 − 810.924i −123089. − 82245.2i
3.6 −12.1874 + 5.04821i 296.534 + 58.9843i −601.028 + 601.028i −2420.01 − 3621.81i −3911.76 + 778.097i −5362.08 + 8024.92i 9460.25 − 22839.1i 29899.1 + 12384.6i 47777.4 + 31923.9i
3.7 −1.64785 + 0.682561i −385.955 − 76.7712i −721.828 + 721.828i −2376.67 − 3556.94i 688.396 − 136.930i −5637.89 + 8437.70i 1395.71 − 3369.55i 88513.3 + 36663.4i 6344.21 + 4239.07i
3.8 2.24950 − 0.931775i −83.0370 − 16.5171i −719.885 + 719.885i 1077.87 + 1613.15i −202.182 + 40.2166i 12379.3 − 18526.9i −1902.75 + 4593.65i −47931.8 − 19854.0i 3927.77 + 2624.45i
3.9 16.4700 − 6.82208i 288.623 + 57.4107i −499.359 + 499.359i 1390.71 + 2081.35i 5145.27 − 1023.46i −10751.3 + 16090.5i −11803.6 + 28496.3i 25453.2 + 10543.0i 37104.1 + 24792.2i
3.10 25.3239 − 10.4895i −45.2880 − 9.00834i −192.809 + 192.809i −76.9350 − 115.141i −1241.36 + 246.922i −5345.24 + 7999.72i −13601.4 + 32836.8i −52584.3 − 21781.1i −3156.06 − 2108.82i
3.11 38.4079 − 15.9091i 356.539 + 70.9200i 497.988 − 497.988i −1404.47 − 2101.94i 14822.2 − 2948.31i 14501.2 − 21702.6i −5086.73 + 12280.5i 67536.1 + 27974.4i −87382.6 − 58387.2i
3.12 38.4781 − 15.9382i −443.238 − 88.1655i 502.463 − 502.463i 2631.40 + 3938.17i −18460.2 + 3671.95i 7889.67 − 11807.7i −4995.17 + 12059.4i 134133. + 55559.5i 164018. + 109594.i
3.13 46.3668 − 19.2057i −156.643 − 31.1583i 1056.94 − 1056.94i −2022.63 − 3027.08i −7861.46 + 1563.74i −1776.04 + 2658.04i 9040.87 − 21826.6i −30987.9 − 12835.6i −151920. − 101510.i
3.14 55.0519 − 22.8033i 153.095 + 30.4526i 1786.65 − 1786.65i 2503.91 + 3747.36i 9122.61 − 1814.60i −6067.01 + 9079.92i 34266.5 − 82726.7i −32043.3 − 13272.8i 223297. + 149202.i
5.1 −24.4033 − 58.9149i 80.3820 + 120.300i −2151.36 + 2151.36i 257.489 − 1294.49i 5125.88 − 7671.42i 2871.54 + 14436.2i 118919. + 49257.7i 14586.2 − 35214.2i −82548.0 + 16419.8i
5.2 −17.3119 − 41.7946i −177.419 − 265.526i −723.013 + 723.013i −470.383 + 2364.78i −8026.10 + 12011.9i 1565.92 + 7872.42i −62.9219 − 26.0631i −16429.5 + 39664.3i 106978. − 21279.3i
5.3 −15.4528 − 37.3064i 126.432 + 189.218i −428.902 + 428.902i −566.679 + 2848.89i 5105.34 − 7640.68i −4533.97 − 22793.8i −15573.2 − 6450.65i 2778.43 − 6707.71i 115039. − 22882.6i
5.4 −14.9348 − 36.0558i −82.1455 − 122.939i −352.897 + 352.897i 1058.86 − 5323.27i −3205.85 + 4797.90i −2754.26 − 13846.6i −18926.7 − 7839.71i 14230.9 − 34356.3i −207749. + 41323.8i
5.5 −10.7844 − 26.0358i 205.409 + 307.417i 162.517 − 162.517i 159.991 − 804.328i 5788.63 − 8663.30i 5407.64 + 27186.0i −32644.6 − 13521.8i −29715.0 + 71738.4i −22666.7 + 4508.69i
5.6 −2.65297 − 6.40483i −48.4034 − 72.4408i 690.094 − 690.094i 245.270 − 1233.05i −335.559 + 502.199i 2683.92 + 13493.0i −12809.3 − 5305.78i 19692.3 − 47541.4i −8548.19 + 1700.34i
See next 80 embeddings (of 112 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 3.14
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
17.e odd 16 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 17.11.e.a ✓ 112
17.e odd 16 1 inner 17.11.e.a ✓ 112
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
17.11.e.a ✓ 112 1.a even 1 1 trivial
17.11.e.a ✓ 112 17.e odd 16 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{11}^{\mathrm{new}}(17, [\chi])\).