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Properties

Label 11.6.c.a.3.4
Level $11$
Weight $6$
Character 11.3
Analytic conductor $1.764$
Analytic rank $0$
Dimension $16$
Inner twists $2$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [11,6,Mod(3,11)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("11.3"); S:= CuspForms(chi, 6); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(11, base_ring=CyclotomicField(10)) chi = DirichletCharacter(H, H._module([8])) N = Newforms(chi, 6, names="a")
 
Level: \( N \) \(=\) \( 11 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 11.c (of order \(5\), degree \(4\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(1.76422201794\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(4\) over \(\Q(\zeta_{5})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 5 x^{15} + 86 x^{14} - 146 x^{13} + 7205 x^{12} - 23732 x^{11} + 774165 x^{10} + \cdots + 393784336 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

Embedding invariants

Embedding label 3.4
Root \(-2.13151 - 6.56011i\) of defining polynomial
Character \(\chi\) \(=\) 11.3
Dual form 11.6.c.a.4.4

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(6.38938 - 4.64216i) q^{2} +(3.63629 + 11.1914i) q^{3} +(9.38602 - 28.8872i) q^{4} +(-51.9930 - 37.7751i) q^{5} +(75.1856 + 54.6256i) q^{6} +(-33.5385 + 103.221i) q^{7} +(3.96879 + 12.2147i) q^{8} +(84.5674 - 61.4418i) q^{9} -507.561 q^{10} +(-398.317 + 48.9383i) q^{11} +357.417 q^{12} +(830.047 - 603.064i) q^{13} +(264.877 + 815.208i) q^{14} +(233.693 - 719.233i) q^{15} +(868.394 + 630.925i) q^{16} +(-1047.81 - 761.277i) q^{17} +(255.111 - 785.150i) q^{18} +(105.257 + 323.946i) q^{19} +(-1579.22 + 1147.37i) q^{20} -1277.14 q^{21} +(-2317.82 + 2161.73i) q^{22} +817.577 q^{23} +(-122.267 + 88.8322i) q^{24} +(310.631 + 956.025i) q^{25} +(2503.97 - 7706.41i) q^{26} +(3308.47 + 2403.75i) q^{27} +(2666.97 + 1937.66i) q^{28} +(-1222.58 + 3762.70i) q^{29} +(-1845.64 - 5680.29i) q^{30} +(-3173.87 + 2305.95i) q^{31} +8066.37 q^{32} +(-1996.08 - 4279.75i) q^{33} -10228.8 q^{34} +(5642.94 - 4099.83i) q^{35} +(-981.131 - 3019.61i) q^{36} +(-3790.45 + 11665.8i) q^{37} +(2176.33 + 1581.20i) q^{38} +(9767.39 + 7096.43i) q^{39} +(255.061 - 784.998i) q^{40} +(-4315.18 - 13280.8i) q^{41} +(-8160.10 + 5928.66i) q^{42} -691.044 q^{43} +(-2324.92 + 11965.6i) q^{44} -6717.88 q^{45} +(5223.81 - 3795.32i) q^{46} +(-4530.99 - 13944.9i) q^{47} +(-3903.17 + 12012.7i) q^{48} +(4067.45 + 2955.17i) q^{49} +(6422.76 + 4666.41i) q^{50} +(4709.59 - 14494.6i) q^{51} +(-9630.00 - 29638.1i) q^{52} +(-1158.68 + 841.828i) q^{53} +32297.6 q^{54} +(22558.3 + 12502.0i) q^{55} -1393.92 q^{56} +(-3242.65 + 2355.93i) q^{57} +(9655.55 + 29716.7i) q^{58} +(2260.89 - 6958.29i) q^{59} +(-18583.2 - 13501.5i) q^{60} +(-14069.1 - 10221.8i) q^{61} +(-9574.48 + 29467.2i) q^{62} +(3505.81 + 10789.8i) q^{63} +(23750.5 - 17255.7i) q^{64} -65937.4 q^{65} +(-32621.0 - 18078.8i) q^{66} -16996.6 q^{67} +(-31825.9 + 23122.9i) q^{68} +(2972.95 + 9149.79i) q^{69} +(17022.8 - 52390.8i) q^{70} +(57494.7 + 41772.4i) q^{71} +(1086.12 + 789.114i) q^{72} +(10940.8 - 33672.4i) q^{73} +(29935.9 + 92133.1i) q^{74} +(-9569.67 + 6952.77i) q^{75} +10345.8 q^{76} +(8307.48 - 42755.9i) q^{77} +95350.3 q^{78} +(54310.8 - 39459.1i) q^{79} +(-21317.1 - 65607.3i) q^{80} +(-7021.24 + 21609.1i) q^{81} +(-89222.8 - 64824.1i) q^{82} +(-44144.8 - 32073.0i) q^{83} +(-11987.2 + 36892.9i) q^{84} +(25721.3 + 79162.1i) q^{85} +(-4415.34 + 3207.93i) q^{86} -46555.4 q^{87} +(-2178.60 - 4671.08i) q^{88} +57821.9 q^{89} +(-42923.1 + 31185.5i) q^{90} +(34410.3 + 105904. i) q^{91} +(7673.79 - 23617.5i) q^{92} +(-37347.9 - 27134.8i) q^{93} +(-93684.8 - 68066.0i) q^{94} +(6764.50 - 20819.0i) q^{95} +(29331.7 + 90273.5i) q^{96} +(61580.2 - 44740.6i) q^{97} +39706.8 q^{98} +(-30677.7 + 28611.9i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - q^{2} - 24 q^{3} - 73 q^{4} - 10 q^{5} + 121 q^{6} + 196 q^{7} + 527 q^{8} - 530 q^{9} - 672 q^{10} - 692 q^{11} - 1562 q^{12} + 1162 q^{13} + 560 q^{14} + 1796 q^{15} + 6399 q^{16} - 22 q^{17}+ \cdots + 724964 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/11\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(e\left(\frac{4}{5}\right)\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\). You can download additional coefficients here.



Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 6.38938 − 4.64216i 1.12949 − 0.820625i 0.143873 − 0.989596i \(-0.454044\pi\)
0.985621 + 0.168971i \(0.0540445\pi\)
\(3\) 3.63629 + 11.1914i 0.233268 + 0.717925i 0.997346 + 0.0728019i \(0.0231941\pi\)
−0.764078 + 0.645124i \(0.776806\pi\)
\(4\) 9.38602 − 28.8872i 0.293313 − 0.902725i
\(5\) −51.9930 − 37.7751i −0.930078 − 0.675741i 0.0159338 − 0.999873i \(-0.494928\pi\)
−0.946012 + 0.324132i \(0.894928\pi\)
\(6\) 75.1856 + 54.6256i 0.852622 + 0.619466i
\(7\) −33.5385 + 103.221i −0.258701 + 0.796200i 0.734377 + 0.678742i \(0.237475\pi\)
−0.993078 + 0.117458i \(0.962525\pi\)
\(8\) 3.96879 + 12.2147i 0.0219247 + 0.0674772i
\(9\) 84.5674 − 61.4418i 0.348014 − 0.252847i
\(10\) −507.561 −1.60505
\(11\) −398.317 + 48.9383i −0.992537 + 0.121946i
\(12\) 357.417 0.716510
\(13\) 830.047 − 603.064i 1.36221 − 0.989704i 0.363909 − 0.931434i \(-0.381442\pi\)
0.998301 − 0.0582693i \(-0.0185582\pi\)
\(14\) 264.877 + 815.208i 0.361180 + 1.11160i
\(15\) 233.693 − 719.233i 0.268174 − 0.825356i
\(16\) 868.394 + 630.925i 0.848041 + 0.616138i
\(17\) −1047.81 − 761.277i −0.879346 − 0.638882i 0.0537327 − 0.998555i \(-0.482888\pi\)
−0.933078 + 0.359673i \(0.882888\pi\)
\(18\) 255.111 − 785.150i 0.185587 − 0.571178i
\(19\) 105.257 + 323.946i 0.0668906 + 0.205868i 0.978915 − 0.204268i \(-0.0654813\pi\)
−0.912024 + 0.410136i \(0.865481\pi\)
\(20\) −1579.22 + 1147.37i −0.882813 + 0.641401i
\(21\) −1277.14 −0.631959
\(22\) −2317.82 + 2161.73i −1.02099 + 0.952238i
\(23\) 817.577 0.322262 0.161131 − 0.986933i \(-0.448486\pi\)
0.161131 + 0.986933i \(0.448486\pi\)
\(24\) −122.267 + 88.8322i −0.0433293 + 0.0314806i
\(25\) 310.631 + 956.025i 0.0994021 + 0.305928i
\(26\) 2503.97 − 7706.41i 0.726432 − 2.23573i
\(27\) 3308.47 + 2403.75i 0.873410 + 0.634569i
\(28\) 2666.97 + 1937.66i 0.642869 + 0.467072i
\(29\) −1222.58 + 3762.70i −0.269948 + 0.830816i 0.720563 + 0.693389i \(0.243883\pi\)
−0.990512 + 0.137427i \(0.956117\pi\)
\(30\) −1845.64 − 5680.29i −0.374406 − 1.15230i
\(31\) −3173.87 + 2305.95i −0.593178 + 0.430969i −0.843451 − 0.537206i \(-0.819480\pi\)
0.250273 + 0.968175i \(0.419480\pi\)
\(32\) 8066.37 1.39253
\(33\) −1996.08 − 4279.75i −0.319075 − 0.684121i
\(34\) −10228.8 −1.51750
\(35\) 5642.94 − 4099.83i 0.778637 − 0.565713i
\(36\) −981.131 − 3019.61i −0.126174 − 0.388324i
\(37\) −3790.45 + 11665.8i −0.455183 + 1.40091i 0.415736 + 0.909485i \(0.363524\pi\)
−0.870920 + 0.491425i \(0.836476\pi\)
\(38\) 2176.33 + 1581.20i 0.244493 + 0.177635i
\(39\) 9767.39 + 7096.43i 1.02829 + 0.747099i
\(40\) 255.061 − 784.998i 0.0252055 − 0.0775745i
\(41\) −4315.18 − 13280.8i −0.400903 − 1.23385i −0.924268 − 0.381745i \(-0.875323\pi\)
0.523364 − 0.852109i \(-0.324677\pi\)
\(42\) −8160.10 + 5928.66i −0.713793 + 0.518601i
\(43\) −691.044 −0.0569947 −0.0284974 − 0.999594i \(-0.509072\pi\)
−0.0284974 + 0.999594i \(0.509072\pi\)
\(44\) −2324.92 + 11965.6i −0.181041 + 0.931756i
\(45\) −6717.88 −0.494539
\(46\) 5223.81 − 3795.32i 0.363993 − 0.264456i
\(47\) −4530.99 − 13944.9i −0.299191 − 0.920814i −0.981781 − 0.190013i \(-0.939147\pi\)
0.682591 − 0.730801i \(-0.260853\pi\)
\(48\) −3903.17 + 12012.7i −0.244520 + 0.752555i
\(49\) 4067.45 + 2955.17i 0.242009 + 0.175830i
\(50\) 6422.76 + 4666.41i 0.363326 + 0.263972i
\(51\) 4709.59 − 14494.6i 0.253546 − 0.780335i
\(52\) −9630.00 − 29638.1i −0.493876 − 1.51999i
\(53\) −1158.68 + 841.828i −0.0566595 + 0.0411655i −0.615754 − 0.787938i \(-0.711149\pi\)
0.559095 + 0.829104i \(0.311149\pi\)
\(54\) 32297.6 1.50725
\(55\) 22558.3 + 12502.0i 1.00554 + 0.557279i
\(56\) −1393.92 −0.0593973
\(57\) −3242.65 + 2355.93i −0.132195 + 0.0960449i
\(58\) 9655.55 + 29716.7i 0.376883 + 1.15993i
\(59\) 2260.89 − 6958.29i 0.0845568 − 0.260239i −0.899835 − 0.436231i \(-0.856313\pi\)
0.984392 + 0.175992i \(0.0563132\pi\)
\(60\) −18583.2 − 13501.5i −0.666410 − 0.484175i
\(61\) −14069.1 − 10221.8i −0.484107 − 0.351724i 0.318807 − 0.947820i \(-0.396718\pi\)
−0.802914 + 0.596095i \(0.796718\pi\)
\(62\) −9574.48 + 29467.2i −0.316327 + 0.973554i
\(63\) 3505.81 + 10789.8i 0.111285 + 0.342500i
\(64\) 23750.5 − 17255.7i 0.724807 − 0.526603i
\(65\) −65937.4 −1.93575
\(66\) −32621.0 − 18078.8i −0.921800 − 0.510870i
\(67\) −16996.6 −0.462566 −0.231283 − 0.972886i \(-0.574292\pi\)
−0.231283 + 0.972886i \(0.574292\pi\)
\(68\) −31825.9 + 23122.9i −0.834658 + 0.606415i
\(69\) 2972.95 + 9149.79i 0.0751735 + 0.231360i
\(70\) 17022.8 − 52390.8i 0.415227 − 1.27794i
\(71\) 57494.7 + 41772.4i 1.35357 + 0.983429i 0.998825 + 0.0484678i \(0.0154338\pi\)
0.354749 + 0.934961i \(0.384566\pi\)
\(72\) 1086.12 + 789.114i 0.0246915 + 0.0179394i
\(73\) 10940.8 − 33672.4i 0.240294 − 0.739549i −0.756081 − 0.654478i \(-0.772888\pi\)
0.996375 − 0.0850710i \(-0.0271117\pi\)
\(74\) 29935.9 + 92133.1i 0.635495 + 1.95585i
\(75\) −9569.67 + 6952.77i −0.196446 + 0.142727i
\(76\) 10345.8 0.205462
\(77\) 8307.48 − 42755.9i 0.159677 − 0.821805i
\(78\) 95350.3 1.77454
\(79\) 54310.8 − 39459.1i 0.979081 − 0.711344i 0.0215782 − 0.999767i \(-0.493131\pi\)
0.957503 + 0.288423i \(0.0931309\pi\)
\(80\) −21317.1 − 65607.3i −0.372395 − 1.14611i
\(81\) −7021.24 + 21609.1i −0.118905 + 0.365953i
\(82\) −89222.8 − 64824.1i −1.46535 − 1.06464i
\(83\) −44144.8 − 32073.0i −0.703370 − 0.511028i 0.177658 − 0.984092i \(-0.443148\pi\)
−0.881028 + 0.473064i \(0.843148\pi\)
\(84\) −11987.2 + 36892.9i −0.185362 + 0.570485i
\(85\) 25721.3 + 79162.1i 0.386141 + 1.18842i
\(86\) −4415.34 + 3207.93i −0.0643751 + 0.0467713i
\(87\) −46555.4 −0.659434
\(88\) −2178.60 − 4671.08i −0.0299896 − 0.0643000i
\(89\) 57821.9 0.773779 0.386890 − 0.922126i \(-0.373549\pi\)
0.386890 + 0.922126i \(0.373549\pi\)
\(90\) −42923.1 + 31185.5i −0.558579 + 0.405831i
\(91\) 34410.3 + 105904.i 0.435597 + 1.34063i
\(92\) 7673.79 − 23617.5i 0.0945237 − 0.290914i
\(93\) −37347.9 − 27134.8i −0.447773 − 0.325326i
\(94\) −93684.8 − 68066.0i −1.09358 − 0.794530i
\(95\) 6764.50 − 20819.0i 0.0769001 − 0.236674i
\(96\) 29331.7 + 90273.5i 0.324832 + 0.999729i
\(97\) 61580.2 − 44740.6i 0.664525 − 0.482806i −0.203663 − 0.979041i \(-0.565285\pi\)
0.868188 + 0.496235i \(0.165285\pi\)
\(98\) 39706.8 0.417638
\(99\) −30677.7 + 28611.9i −0.314583 + 0.293399i
Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 11.6.c.a.3.4 ✓ 16
3.2 odd 2 99.6.f.a.91.1 16
11.2 odd 10 121.6.a.i.1.7 8
11.4 even 5 inner 11.6.c.a.4.4 yes 16
11.9 even 5 121.6.a.g.1.2 8
33.2 even 10 1089.6.a.bb.1.2 8
33.20 odd 10 1089.6.a.bg.1.7 8
33.26 odd 10 99.6.f.a.37.1 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
11.6.c.a.3.4 ✓ 16 1.1 even 1 trivial
11.6.c.a.4.4 yes 16 11.4 even 5 inner
99.6.f.a.37.1 16 33.26 odd 10
99.6.f.a.91.1 16 3.2 odd 2
121.6.a.g.1.2 8 11.9 even 5
121.6.a.i.1.7 8 11.2 odd 10
1089.6.a.bb.1.2 8 33.2 even 10
1089.6.a.bg.1.7 8 33.20 odd 10