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Properties

Label 110.3.h.a.61.1
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,-6] 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: \( x^{16} - 8 x^{15} - 28 x^{14} + 336 x^{13} + 362 x^{12} - 6904 x^{11} - 3132 x^{10} + 87908 x^{9} + \cdots + 24267881 \) 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 61.1
Root \(-1.96901 - 0.437016i\) of defining polynomial
Character \(\chi\) \(=\) 110.61
Dual form 110.3.h.a.101.1

$q$-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} +(-4.29912 - 3.12349i) q^{3} +(1.61803 - 1.17557i) q^{4} +(0.690983 - 2.12663i) q^{5} +(7.14732 + 2.32230i) q^{6} +(-2.06296 - 2.83942i) q^{7} +(-1.66251 + 2.28825i) q^{8} +(5.94506 + 18.2970i) q^{9} +3.16228i q^{10} +(-6.71814 + 8.71014i) q^{11} -10.6280 q^{12} +(-16.7761 + 5.45088i) q^{13} +(4.01554 + 2.91746i) q^{14} +(-9.61312 + 6.98434i) q^{15} +(1.23607 - 3.80423i) q^{16} +(26.2113 + 8.51656i) q^{17} +(-15.9922 - 22.0113i) q^{18} +(-5.64958 + 7.77599i) q^{19} +(-1.38197 - 4.25325i) q^{20} +18.6506i q^{21} +(5.22941 - 14.6510i) q^{22} -17.4290 q^{23} +(14.2946 - 4.64461i) q^{24} +(-4.04508 - 2.93893i) q^{25} +(20.1816 - 14.6628i) q^{26} +(16.8130 - 51.7451i) q^{27} +(-6.67587 - 2.16912i) q^{28} +(7.46021 + 10.2681i) q^{29} +(9.87735 - 13.5950i) q^{30} +(-13.7032 - 42.1741i) q^{31} +5.65685i q^{32} +(56.0881 - 16.4619i) q^{33} -38.9760 q^{34} +(-7.46385 + 2.42515i) q^{35} +(31.1287 + 22.6164i) q^{36} +(-56.3632 + 40.9503i) q^{37} +(4.20044 - 12.9276i) q^{38} +(89.1481 + 28.9660i) q^{39} +(3.71748 + 5.11667i) q^{40} +(-7.27651 + 10.0153i) q^{41} +(-8.15062 - 25.0850i) q^{42} +5.67418i q^{43} +(-0.630798 + 21.9910i) q^{44} +43.0189 q^{45} +(23.4419 - 7.61674i) q^{46} +(-15.5972 - 11.3321i) q^{47} +(-17.1965 + 12.4940i) q^{48} +(11.3353 - 34.8866i) q^{49} +(6.72499 + 2.18508i) q^{50} +(-86.0840 - 118.484i) q^{51} +(-20.7364 + 28.5412i) q^{52} +(-1.11850 - 3.44239i) q^{53} +76.9446i q^{54} +(13.8811 + 20.3055i) q^{55} +9.92696 q^{56} +(48.5765 - 15.7834i) q^{57} +(-14.5213 - 10.5503i) q^{58} +(-49.6546 + 36.0762i) q^{59} +(-7.34377 + 22.6018i) q^{60} +(-84.0721 - 27.3167i) q^{61} +(36.8616 + 50.7356i) q^{62} +(39.6884 - 54.6264i) q^{63} +(-2.47214 - 7.60845i) q^{64} +39.4429i q^{65} +(-68.2443 + 46.6526i) q^{66} -71.7758 q^{67} +(52.4225 - 17.0331i) q^{68} +(74.9292 + 54.4392i) q^{69} +(8.97902 - 6.52364i) q^{70} +(-7.40796 + 22.7994i) q^{71} +(-51.7518 - 16.8152i) q^{72} +(-37.0988 - 51.0621i) q^{73} +(57.9124 - 79.7096i) q^{74} +(8.21058 + 25.2696i) q^{75} +19.2233i q^{76} +(38.5909 + 1.10696i) q^{77} -132.563 q^{78} +(-43.4034 + 14.1026i) q^{79} +(-7.23607 - 5.25731i) q^{80} +(-93.8273 + 68.1695i) q^{81} +(5.41005 - 16.6504i) q^{82} +(123.130 + 40.0072i) q^{83} +(21.9251 + 30.1773i) q^{84} +(36.2231 - 49.8568i) q^{85} +(-2.47971 - 7.63175i) q^{86} -67.4457i q^{87} +(-8.76198 - 29.8534i) q^{88} +3.64047 q^{89} +(-57.8602 + 18.7999i) q^{90} +(50.0856 + 36.3893i) q^{91} +(-28.2007 + 20.4890i) q^{92} +(-72.8189 + 224.114i) q^{93} +(25.9305 + 8.42534i) q^{94} +(12.6329 + 17.3876i) q^{95} +(17.6691 - 24.3195i) q^{96} +(49.9012 + 153.580i) q^{97} +51.8761i q^{98} +(-199.309 - 71.1396i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 6 q^{3} + 8 q^{4} + 20 q^{5} + 20 q^{6} + 10 q^{7} - 10 q^{9} - 10 q^{11} - 48 q^{12} + 40 q^{13} - 4 q^{14} - 30 q^{15} - 16 q^{16} + 70 q^{17} + 30 q^{19} - 40 q^{20} + 40 q^{22} - 12 q^{23} + 40 q^{24}+ \cdots - 552 q^{99}+O(q^{100}) \) 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
\(3\) −4.29912 − 3.12349i −1.43304 − 1.04116i −0.989441 − 0.144938i \(-0.953702\pi\)
−0.443599 − 0.896226i \(-0.646298\pi\)
\(4\) 1.61803 − 1.17557i 0.404508 − 0.293893i
\(5\) 0.690983 − 2.12663i 0.138197 − 0.425325i
\(6\) 7.14732 + 2.32230i 1.19122 + 0.387051i
\(7\) −2.06296 − 2.83942i −0.294708 − 0.405631i 0.635828 − 0.771831i \(-0.280659\pi\)
−0.930536 + 0.366200i \(0.880659\pi\)
\(8\) −1.66251 + 2.28825i −0.207813 + 0.286031i
\(9\) 5.94506 + 18.2970i 0.660562 + 2.03300i
\(10\) 3.16228i 0.316228i
\(11\) −6.71814 + 8.71014i −0.610740 + 0.791831i
\(12\) −10.6280 −0.885667
\(13\) −16.7761 + 5.45088i −1.29047 + 0.419298i −0.872255 − 0.489051i \(-0.837343\pi\)
−0.418212 + 0.908349i \(0.637343\pi\)
\(14\) 4.01554 + 2.91746i 0.286824 + 0.208390i
\(15\) −9.61312 + 6.98434i −0.640875 + 0.465623i
\(16\) 1.23607 − 3.80423i 0.0772542 − 0.237764i
\(17\) 26.2113 + 8.51656i 1.54184 + 0.500974i 0.951883 − 0.306461i \(-0.0991449\pi\)
0.589956 + 0.807435i \(0.299145\pi\)
\(18\) −15.9922 − 22.0113i −0.888454 − 1.22285i
\(19\) −5.64958 + 7.77599i −0.297347 + 0.409262i −0.931383 − 0.364041i \(-0.881397\pi\)
0.634036 + 0.773303i \(0.281397\pi\)
\(20\) −1.38197 − 4.25325i −0.0690983 − 0.212663i
\(21\) 18.6506i 0.888124i
\(22\) 5.22941 − 14.6510i 0.237700 − 0.665957i
\(23\) −17.4290 −0.757781 −0.378890 − 0.925442i \(-0.623694\pi\)
−0.378890 + 0.925442i \(0.623694\pi\)
\(24\) 14.2946 − 4.64461i 0.595610 − 0.193525i
\(25\) −4.04508 − 2.93893i −0.161803 − 0.117557i
\(26\) 20.1816 − 14.6628i 0.776217 − 0.563955i
\(27\) 16.8130 − 51.7451i 0.622704 − 1.91649i
\(28\) −6.67587 − 2.16912i −0.238424 − 0.0774686i
\(29\) 7.46021 + 10.2681i 0.257249 + 0.354072i 0.918033 − 0.396503i \(-0.129776\pi\)
−0.660785 + 0.750575i \(0.729776\pi\)
\(30\) 9.87735 − 13.5950i 0.329245 − 0.453167i
\(31\) −13.7032 − 42.1741i −0.442039 − 1.36046i −0.885698 − 0.464261i \(-0.846320\pi\)
0.443659 − 0.896195i \(-0.353680\pi\)
\(32\) 5.65685i 0.176777i
\(33\) 56.0881 − 16.4619i 1.69964 − 0.498844i
\(34\) −38.9760 −1.14635
\(35\) −7.46385 + 2.42515i −0.213253 + 0.0692900i
\(36\) 31.1287 + 22.6164i 0.864687 + 0.628232i
\(37\) −56.3632 + 40.9503i −1.52333 + 1.10676i −0.563524 + 0.826100i \(0.690555\pi\)
−0.959806 + 0.280664i \(0.909445\pi\)
\(38\) 4.20044 − 12.9276i 0.110538 − 0.340201i
\(39\) 89.1481 + 28.9660i 2.28585 + 0.742717i
\(40\) 3.71748 + 5.11667i 0.0929370 + 0.127917i
\(41\) −7.27651 + 10.0153i −0.177476 + 0.244274i −0.888482 − 0.458911i \(-0.848240\pi\)
0.711006 + 0.703185i \(0.248240\pi\)
\(42\) −8.15062 − 25.0850i −0.194062 − 0.597262i
\(43\) 5.67418i 0.131958i 0.997821 + 0.0659788i \(0.0210170\pi\)
−0.997821 + 0.0659788i \(0.978983\pi\)
\(44\) −0.630798 + 21.9910i −0.0143363 + 0.499794i
\(45\) 43.0189 0.955975
\(46\) 23.4419 − 7.61674i 0.509607 − 0.165581i
\(47\) −15.5972 − 11.3321i −0.331856 − 0.241108i 0.409362 − 0.912372i \(-0.365751\pi\)
−0.741218 + 0.671265i \(0.765751\pi\)
\(48\) −17.1965 + 12.4940i −0.358260 + 0.260291i
\(49\) 11.3353 − 34.8866i 0.231333 − 0.711971i
\(50\) 6.72499 + 2.18508i 0.134500 + 0.0437016i
\(51\) −86.0840 − 118.484i −1.68792 − 2.32322i
\(52\) −20.7364 + 28.5412i −0.398776 + 0.548868i
\(53\) −1.11850 − 3.44239i −0.0211038 − 0.0649508i 0.939950 − 0.341312i \(-0.110871\pi\)
−0.961054 + 0.276361i \(0.910871\pi\)
\(54\) 76.9446i 1.42490i
\(55\) 13.8811 + 20.3055i 0.252384 + 0.369192i
\(56\) 9.92696 0.177267
\(57\) 48.5765 − 15.7834i 0.852219 − 0.276903i
\(58\) −14.5213 − 10.5503i −0.250367 − 0.181902i
\(59\) −49.6546 + 36.0762i −0.841604 + 0.611461i −0.922818 − 0.385235i \(-0.874120\pi\)
0.0812141 + 0.996697i \(0.474120\pi\)
\(60\) −7.34377 + 22.6018i −0.122396 + 0.376697i
\(61\) −84.0721 − 27.3167i −1.37823 − 0.447815i −0.476144 − 0.879367i \(-0.657966\pi\)
−0.902088 + 0.431553i \(0.857966\pi\)
\(62\) 36.8616 + 50.7356i 0.594541 + 0.818316i
\(63\) 39.6884 − 54.6264i 0.629975 − 0.867086i
\(64\) −2.47214 − 7.60845i −0.0386271 − 0.118882i
\(65\) 39.4429i 0.606814i
\(66\) −68.2443 + 46.6526i −1.03400 + 0.706857i
\(67\) −71.7758 −1.07128 −0.535641 − 0.844446i \(-0.679930\pi\)
−0.535641 + 0.844446i \(0.679930\pi\)
\(68\) 52.4225 − 17.0331i 0.770920 − 0.250487i
\(69\) 74.9292 + 54.4392i 1.08593 + 0.788974i
\(70\) 8.97902 − 6.52364i 0.128272 − 0.0931949i
\(71\) −7.40796 + 22.7994i −0.104338 + 0.321118i −0.989574 − 0.144023i \(-0.953996\pi\)
0.885237 + 0.465140i \(0.153996\pi\)
\(72\) −51.7518 − 16.8152i −0.718775 − 0.233544i
\(73\) −37.0988 − 51.0621i −0.508203 − 0.699481i 0.475412 − 0.879763i \(-0.342299\pi\)
−0.983615 + 0.180282i \(0.942299\pi\)
\(74\) 57.9124 − 79.7096i 0.782600 − 1.07716i
\(75\) 8.21058 + 25.2696i 0.109474 + 0.336928i
\(76\) 19.2233i 0.252938i
\(77\) 38.5909 + 1.10696i 0.501181 + 0.0143761i
\(78\) −132.563 −1.69952
\(79\) −43.4034 + 14.1026i −0.549411 + 0.178514i −0.570551 − 0.821262i \(-0.693270\pi\)
0.0211403 + 0.999777i \(0.493270\pi\)
\(80\) −7.23607 − 5.25731i −0.0904508 − 0.0657164i
\(81\) −93.8273 + 68.1695i −1.15836 + 0.841599i
\(82\) 5.41005 − 16.6504i 0.0659763 − 0.203054i
\(83\) 123.130 + 40.0072i 1.48349 + 0.482015i 0.935153 − 0.354243i \(-0.115262\pi\)
0.548336 + 0.836258i \(0.315262\pi\)
\(84\) 21.9251 + 30.1773i 0.261013 + 0.359254i
\(85\) 36.2231 − 49.8568i 0.426154 − 0.586551i
\(86\) −2.47971 − 7.63175i −0.0288338 − 0.0887413i
\(87\) − 67.4457i − 0.775238i
\(88\) −8.76198 − 29.8534i −0.0995679 − 0.339244i
\(89\) 3.64047 0.0409041 0.0204521 − 0.999791i \(-0.493489\pi\)
0.0204521 + 0.999791i \(0.493489\pi\)
\(90\) −57.8602 + 18.7999i −0.642892 + 0.208888i
\(91\) 50.0856 + 36.3893i 0.550391 + 0.399883i
\(92\) −28.2007 + 20.4890i −0.306529 + 0.222706i
\(93\) −72.8189 + 224.114i −0.782999 + 2.40982i
\(94\) 25.9305 + 8.42534i 0.275857 + 0.0896313i
\(95\) 12.6329 + 17.3876i 0.132977 + 0.183028i
\(96\) 17.6691 − 24.3195i 0.184054 − 0.253328i
\(97\) 49.9012 + 153.580i 0.514445 + 1.58330i 0.784289 + 0.620396i \(0.213028\pi\)
−0.269843 + 0.962904i \(0.586972\pi\)
\(98\) 51.8761i 0.529348i
\(99\) −199.309 − 71.1396i −2.01323 − 0.718582i
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.a.61.1 ✓ 16
11.2 odd 10 inner 110.3.h.a.101.1 yes 16
11.3 even 5 1210.3.d.f.241.16 16
11.8 odd 10 1210.3.d.f.241.8 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
110.3.h.a.61.1 ✓ 16 1.1 even 1 trivial
110.3.h.a.101.1 yes 16 11.2 odd 10 inner
1210.3.d.f.241.8 16 11.8 odd 10
1210.3.d.f.241.16 16 11.3 even 5