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Properties

Label 110.3.h.b.61.4
Level $110$
Weight $3$
Character 110.61
Analytic conductor $2.997$
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] = [110,3,Mod(41,110)] 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("110.41"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(110, base_ring=CyclotomicField(10)) chi = DirichletCharacter(H, H._module([0, 3])) N = Newforms(chi, 3, names="a")
 
Level: \( N \) \(=\) \( 110 = 2 \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 110.h (of order \(10\), degree \(4\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [16,0,14] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(2.99728290796\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(4\) over \(\Q(\zeta_{10})\)
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: \( x^{16} - 8 x^{15} + 92 x^{14} - 504 x^{13} + 3078 x^{12} - 12280 x^{11} + 49836 x^{10} - 147672 x^{9} + \cdots + 339856 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

Embedding invariants

Embedding label 61.4
Root \(0.500000 - 1.09141i\) of defining polynomial
Character \(\chi\) \(=\) 110.61
Dual form 110.3.h.b.101.4

$q$-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+(1.34500 - 0.437016i) q^{2} +(4.46995 + 3.24761i) q^{3} +(1.61803 - 1.17557i) q^{4} +(-0.690983 + 2.12663i) q^{5} +(7.43133 + 2.41458i) q^{6} +(-6.89431 - 9.48921i) q^{7} +(1.66251 - 2.28825i) q^{8} +(6.65234 + 20.4738i) q^{9} +3.16228i q^{10} +(-7.06953 - 8.42744i) q^{11} +11.0503 q^{12} +(-11.5517 + 3.75339i) q^{13} +(-13.4198 - 9.75003i) q^{14} +(-9.99511 + 7.26187i) q^{15} +(1.23607 - 3.80423i) q^{16} +(9.90261 + 3.21755i) q^{17} +(17.8947 + 24.6300i) q^{18} +(2.98177 - 4.10405i) q^{19} +(1.38197 + 4.25325i) q^{20} -64.8063i q^{21} +(-13.1914 - 8.24539i) q^{22} +30.9230 q^{23} +(14.8627 - 4.82917i) q^{24} +(-4.04508 - 2.93893i) q^{25} +(-13.8968 + 10.0966i) q^{26} +(-21.3889 + 65.8283i) q^{27} +(-22.3105 - 7.24911i) q^{28} +(-9.06130 - 12.4718i) q^{29} +(-10.2698 + 14.1352i) q^{30} +(4.06761 + 12.5188i) q^{31} -5.65685i q^{32} +(-4.23139 - 60.6293i) q^{33} +14.7251 q^{34} +(24.9439 - 8.10475i) q^{35} +(34.8321 + 25.3070i) q^{36} +(-5.25425 + 3.81744i) q^{37} +(2.21693 - 6.82302i) q^{38} +(-63.8252 - 20.7381i) q^{39} +(3.71748 + 5.11667i) q^{40} +(-28.6013 + 39.3663i) q^{41} +(-28.3214 - 87.1643i) q^{42} +35.6140i q^{43} +(-21.3458 - 5.32516i) q^{44} -48.1368 q^{45} +(41.5914 - 13.5139i) q^{46} +(-27.4119 - 19.9159i) q^{47} +(17.8798 - 12.9904i) q^{48} +(-27.3717 + 84.2413i) q^{49} +(-6.72499 - 2.18508i) q^{50} +(33.8148 + 46.5421i) q^{51} +(-14.2787 + 19.6530i) q^{52} +(-17.1632 - 52.8230i) q^{53} +97.8862i q^{54} +(22.8070 - 9.21103i) q^{55} -33.1755 q^{56} +(26.6567 - 8.66129i) q^{57} +(-17.6378 - 12.8146i) q^{58} +(-43.5697 + 31.6552i) q^{59} +(-7.63559 + 23.4999i) q^{60} +(41.6541 + 13.5342i) q^{61} +(10.9418 + 15.0602i) q^{62} +(148.417 - 204.278i) q^{63} +(-2.47214 - 7.60845i) q^{64} -27.1597i q^{65} +(-32.1872 - 79.6970i) q^{66} +117.778 q^{67} +(19.8052 - 6.43511i) q^{68} +(138.224 + 100.426i) q^{69} +(30.0075 - 21.8017i) q^{70} +(28.7042 - 88.3424i) q^{71} +(57.9086 + 18.8156i) q^{72} +(-0.746670 - 1.02770i) q^{73} +(-5.39867 + 7.43064i) q^{74} +(-8.53684 - 26.2737i) q^{75} -10.1458i q^{76} +(-31.2302 + 125.186i) q^{77} -94.9075 q^{78} +(-106.814 + 34.7061i) q^{79} +(7.23607 + 5.25731i) q^{80} +(-152.648 + 110.905i) q^{81} +(-21.2650 + 65.4469i) q^{82} +(123.042 + 39.9788i) q^{83} +(-76.1844 - 104.859i) q^{84} +(-13.6851 + 18.8359i) q^{85} +(15.5639 + 47.9008i) q^{86} -85.1759i q^{87} +(-31.0372 + 2.16612i) q^{88} -4.92279 q^{89} +(-64.7438 + 21.0365i) q^{90} +(115.258 + 83.7398i) q^{91} +(50.0345 - 36.3522i) q^{92} +(-22.4742 + 69.1684i) q^{93} +(-45.5725 - 14.8074i) q^{94} +(6.66743 + 9.17694i) q^{95} +(18.3712 - 25.2859i) q^{96} +(-5.54993 - 17.0809i) q^{97} +125.266i q^{98} +(125.513 - 200.802i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 14 q^{3} + 8 q^{4} - 20 q^{5} + 20 q^{6} - 10 q^{7} - 10 q^{9} - 10 q^{11} + 32 q^{12} - 100 q^{13} - 4 q^{14} - 30 q^{15} - 16 q^{16} + 10 q^{17} - 80 q^{18} + 30 q^{19} + 40 q^{20} - 40 q^{22}+ \cdots - 232 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/110\mathbb{Z}\right)^\times\).

\(n\) \(67\) \(101\)
\(\chi(n)\) \(1\) \(e\left(\frac{9}{10}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.34500 − 0.437016i 0.672499 − 0.218508i
\(3\) 4.46995 + 3.24761i 1.48998 + 1.08254i 0.974165 + 0.225838i \(0.0725121\pi\)
0.515818 + 0.856698i \(0.327488\pi\)
\(4\) 1.61803 − 1.17557i 0.404508 − 0.293893i
\(5\) −0.690983 + 2.12663i −0.138197 + 0.425325i
\(6\) 7.43133 + 2.41458i 1.23855 + 0.402431i
\(7\) −6.89431 − 9.48921i −0.984902 − 1.35560i −0.934147 − 0.356890i \(-0.883837\pi\)
−0.0507552 − 0.998711i \(-0.516163\pi\)
\(8\) 1.66251 − 2.28825i 0.207813 − 0.286031i
\(9\) 6.65234 + 20.4738i 0.739148 + 2.27486i
\(10\) 3.16228i 0.316228i
\(11\) −7.06953 − 8.42744i −0.642684 − 0.766131i
\(12\) 11.0503 0.920860
\(13\) −11.5517 + 3.75339i −0.888595 + 0.288722i −0.717521 − 0.696536i \(-0.754723\pi\)
−0.171073 + 0.985258i \(0.554723\pi\)
\(14\) −13.4198 − 9.75003i −0.958555 − 0.696431i
\(15\) −9.99511 + 7.26187i −0.666341 + 0.484125i
\(16\) 1.23607 − 3.80423i 0.0772542 − 0.237764i
\(17\) 9.90261 + 3.21755i 0.582507 + 0.189268i 0.585423 − 0.810728i \(-0.300928\pi\)
−0.00291653 + 0.999996i \(0.500928\pi\)
\(18\) 17.8947 + 24.6300i 0.994152 + 1.36833i
\(19\) 2.98177 − 4.10405i 0.156935 − 0.216003i −0.723308 − 0.690525i \(-0.757379\pi\)
0.880243 + 0.474523i \(0.157379\pi\)
\(20\) 1.38197 + 4.25325i 0.0690983 + 0.212663i
\(21\) − 64.8063i − 3.08601i
\(22\) −13.1914 − 8.24539i −0.599610 − 0.374790i
\(23\) 30.9230 1.34448 0.672240 − 0.740333i \(-0.265332\pi\)
0.672240 + 0.740333i \(0.265332\pi\)
\(24\) 14.8627 − 4.82917i 0.619277 − 0.201215i
\(25\) −4.04508 − 2.93893i −0.161803 − 0.117557i
\(26\) −13.8968 + 10.0966i −0.534491 + 0.388330i
\(27\) −21.3889 + 65.8283i −0.792182 + 2.43809i
\(28\) −22.3105 − 7.24911i −0.796802 − 0.258897i
\(29\) −9.06130 − 12.4718i −0.312458 − 0.430062i 0.623688 − 0.781674i \(-0.285634\pi\)
−0.936146 + 0.351612i \(0.885634\pi\)
\(30\) −10.2698 + 14.1352i −0.342328 + 0.471174i
\(31\) 4.06761 + 12.5188i 0.131213 + 0.403833i 0.994982 − 0.100056i \(-0.0319022\pi\)
−0.863769 + 0.503889i \(0.831902\pi\)
\(32\) − 5.65685i − 0.176777i
\(33\) −4.23139 − 60.6293i −0.128224 − 1.83725i
\(34\) 14.7251 0.433091
\(35\) 24.9439 − 8.10475i 0.712682 − 0.231564i
\(36\) 34.8321 + 25.3070i 0.967558 + 0.702972i
\(37\) −5.25425 + 3.81744i −0.142007 + 0.103174i −0.656521 − 0.754308i \(-0.727972\pi\)
0.514514 + 0.857482i \(0.327972\pi\)
\(38\) 2.21693 − 6.82302i 0.0583403 − 0.179553i
\(39\) −63.8252 − 20.7381i −1.63654 − 0.531745i
\(40\) 3.71748 + 5.11667i 0.0929370 + 0.127917i
\(41\) −28.6013 + 39.3663i −0.697593 + 0.960155i 0.302382 + 0.953187i \(0.402218\pi\)
−0.999976 + 0.00696805i \(0.997782\pi\)
\(42\) −28.3214 − 87.1643i −0.674319 − 2.07534i
\(43\) 35.6140i 0.828234i 0.910224 + 0.414117i \(0.135909\pi\)
−0.910224 + 0.414117i \(0.864091\pi\)
\(44\) −21.3458 − 5.32516i −0.485132 − 0.121026i
\(45\) −48.1368 −1.06971
\(46\) 41.5914 − 13.5139i 0.904161 − 0.293780i
\(47\) −27.4119 − 19.9159i −0.583232 − 0.423743i 0.256656 − 0.966503i \(-0.417379\pi\)
−0.839888 + 0.542760i \(0.817379\pi\)
\(48\) 17.8798 − 12.9904i 0.372496 − 0.270634i
\(49\) −27.3717 + 84.2413i −0.558605 + 1.71921i
\(50\) −6.72499 − 2.18508i −0.134500 − 0.0437016i
\(51\) 33.8148 + 46.5421i 0.663036 + 0.912591i
\(52\) −14.2787 + 19.6530i −0.274591 + 0.377942i
\(53\) −17.1632 − 52.8230i −0.323834 − 0.996660i −0.971964 − 0.235130i \(-0.924448\pi\)
0.648129 − 0.761530i \(-0.275552\pi\)
\(54\) 97.8862i 1.81271i
\(55\) 22.8070 − 9.21103i 0.414672 − 0.167473i
\(56\) −33.1755 −0.592419
\(57\) 26.6567 − 8.66129i 0.467661 − 0.151952i
\(58\) −17.6378 − 12.8146i −0.304100 − 0.220942i
\(59\) −43.5697 + 31.6552i −0.738469 + 0.536529i −0.892231 − 0.451579i \(-0.850861\pi\)
0.153762 + 0.988108i \(0.450861\pi\)
\(60\) −7.63559 + 23.4999i −0.127260 + 0.391665i
\(61\) 41.6541 + 13.5342i 0.682854 + 0.221873i 0.629844 − 0.776721i \(-0.283119\pi\)
0.0530093 + 0.998594i \(0.483119\pi\)
\(62\) 10.9418 + 15.0602i 0.176481 + 0.242906i
\(63\) 148.417 − 204.278i 2.35582 − 3.24251i
\(64\) −2.47214 − 7.60845i −0.0386271 − 0.118882i
\(65\) − 27.1597i − 0.417842i
\(66\) −32.1872 − 79.6970i −0.487685 − 1.20753i
\(67\) 117.778 1.75788 0.878938 − 0.476937i \(-0.158253\pi\)
0.878938 + 0.476937i \(0.158253\pi\)
\(68\) 19.8052 − 6.43511i 0.291253 − 0.0946340i
\(69\) 138.224 + 100.426i 2.00325 + 1.45545i
\(70\) 30.0075 − 21.8017i 0.428679 − 0.311453i
\(71\) 28.7042 − 88.3424i 0.404284 − 1.24426i −0.517207 − 0.855860i \(-0.673028\pi\)
0.921491 − 0.388399i \(-0.126972\pi\)
\(72\) 57.9086 + 18.8156i 0.804286 + 0.261328i
\(73\) −0.746670 − 1.02770i −0.0102284 − 0.0140781i 0.803872 − 0.594802i \(-0.202770\pi\)
−0.814101 + 0.580724i \(0.802770\pi\)
\(74\) −5.39867 + 7.43064i −0.0729551 + 0.100414i
\(75\) −8.53684 − 26.2737i −0.113825 − 0.350316i
\(76\) − 10.1458i − 0.133497i
\(77\) −31.2302 + 125.186i −0.405587 + 1.62579i
\(78\) −94.9075 −1.21676
\(79\) −106.814 + 34.7061i −1.35208 + 0.439318i −0.893392 − 0.449279i \(-0.851681\pi\)
−0.458689 + 0.888597i \(0.651681\pi\)
\(80\) 7.23607 + 5.25731i 0.0904508 + 0.0657164i
\(81\) −152.648 + 110.905i −1.88454 + 1.36920i
\(82\) −21.2650 + 65.4469i −0.259329 + 0.798132i
\(83\) 123.042 + 39.9788i 1.48244 + 0.481673i 0.934840 − 0.355070i \(-0.115543\pi\)
0.547597 + 0.836742i \(0.315543\pi\)
\(84\) −76.1844 − 104.859i −0.906957 − 1.24832i
\(85\) −13.6851 + 18.8359i −0.161001 + 0.221599i
\(86\) 15.5639 + 47.9008i 0.180976 + 0.556986i
\(87\) − 85.1759i − 0.979033i
\(88\) −31.0372 + 2.16612i −0.352695 + 0.0246150i
\(89\) −4.92279 −0.0553122 −0.0276561 − 0.999617i \(-0.508804\pi\)
−0.0276561 + 0.999617i \(0.508804\pi\)
\(90\) −64.7438 + 21.0365i −0.719375 + 0.233739i
\(91\) 115.258 + 83.7398i 1.26657 + 0.920217i
\(92\) 50.0345 − 36.3522i 0.543854 − 0.395133i
\(93\) −22.4742 + 69.1684i −0.241658 + 0.743747i
\(94\) −45.5725 − 14.8074i −0.484814 − 0.157526i
\(95\) 6.66743 + 9.17694i 0.0701835 + 0.0965993i
\(96\) 18.3712 − 25.2859i 0.191367 − 0.263394i
\(97\) −5.54993 − 17.0809i −0.0572158 − 0.176092i 0.918364 − 0.395736i \(-0.129510\pi\)
−0.975580 + 0.219644i \(0.929510\pi\)
\(98\) 125.266i 1.27823i
\(99\) 125.513 − 200.802i 1.26781 − 2.02830i
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 110.3.h.b.61.4 ✓ 16
11.2 odd 10 inner 110.3.h.b.101.4 yes 16
11.3 even 5 1210.3.d.c.241.1 16
11.8 odd 10 1210.3.d.c.241.9 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
110.3.h.b.61.4 ✓ 16 1.1 even 1 trivial
110.3.h.b.101.4 yes 16 11.2 odd 10 inner
1210.3.d.c.241.1 16 11.3 even 5
1210.3.d.c.241.9 16 11.8 odd 10