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Properties

Label 121.4.g.a.4.14
Level $121$
Weight $4$
Character 121.4
Analytic conductor $7.139$
Analytic rank $0$
Dimension $1280$
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] = [121,4,Mod(4,121)] 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("121.4"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(121, base_ring=CyclotomicField(110)) chi = DirichletCharacter(H, H._module([2])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 121 = 11^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 121.g (of order \(55\), degree \(40\), 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: \(7.13923111069\)
Analytic rank: \(0\)
Dimension: \(1280\)
Relative dimension: \(32\) over \(\Q(\zeta_{55})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{55}]$

Embedding invariants

Embedding label 4.14
Character \(\chi\) \(=\) 121.4
Dual form 121.4.g.a.91.14

$q$-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+(-0.641211 - 0.0366657i) q^{2} +(1.82234 - 5.60858i) q^{3} +(-7.53805 - 0.864910i) q^{4} +(-9.80678 - 18.5859i) q^{5} +(-1.37414 + 3.52946i) q^{6} +(10.6511 + 4.50120i) q^{7} +(9.86457 + 1.70713i) q^{8} +(-6.29178 - 4.57125i) q^{9} +(5.60674 + 12.2771i) q^{10} +(-17.1311 + 32.2106i) q^{11} +(-18.5878 + 40.7016i) q^{12} +(-54.0125 + 30.5022i) q^{13} +(-6.66457 - 3.27675i) q^{14} +(-122.112 + 21.1323i) q^{15} +(52.8599 + 12.2920i) q^{16} +(60.6488 - 21.6395i) q^{17} +(3.86675 + 3.16182i) q^{18} +(1.94730 - 68.1642i) q^{19} +(57.8488 + 148.583i) q^{20} +(44.6553 - 51.5349i) q^{21} +(12.1657 - 20.0257i) q^{22} +(-98.6562 - 113.855i) q^{23} +(27.5512 - 52.2153i) q^{24} +(-178.708 + 261.352i) q^{25} +(35.7518 - 17.5780i) q^{26} +(91.7115 - 66.6323i) q^{27} +(-76.3955 - 43.1425i) q^{28} +(-117.760 - 172.219i) q^{29} +(79.0742 - 9.07292i) q^{30} +(-31.3064 + 154.503i) q^{31} +(-110.289 - 32.3838i) q^{32} +(149.437 + 154.780i) q^{33} +(-39.6821 + 11.6517i) q^{34} +(-20.7942 - 242.103i) q^{35} +(43.4740 + 39.9001i) q^{36} +(-155.672 + 142.875i) q^{37} +(-3.74792 + 43.6362i) q^{38} +(72.6453 + 358.519i) q^{39} +(-65.0111 - 200.084i) q^{40} +(51.9308 + 197.565i) q^{41} +(-30.5230 + 31.4074i) q^{42} +(-45.2839 - 314.957i) q^{43} +(156.994 - 227.988i) q^{44} +(-23.2587 + 161.768i) q^{45} +(59.0848 + 76.6226i) q^{46} +(114.751 - 93.8313i) q^{47} +(165.269 - 274.069i) q^{48} +(-145.864 - 150.091i) q^{49} +(124.172 - 161.029i) q^{50} +(-10.8440 - 379.588i) q^{51} +(433.530 - 183.211i) q^{52} +(-158.440 + 36.8436i) q^{53} +(-61.2495 + 39.3627i) q^{54} +(766.665 + 2.51489i) q^{55} +(97.3846 + 62.5853i) q^{56} +(-378.756 - 135.140i) q^{57} +(69.1945 + 114.746i) q^{58} +(-75.2197 + 286.165i) q^{59} +(938.762 - 53.6803i) q^{60} +(-181.660 + 10.3877i) q^{61} +(25.7390 - 97.9212i) q^{62} +(-46.4384 - 77.0095i) q^{63} +(-339.382 - 121.091i) q^{64} +(1096.60 + 704.742i) q^{65} +(-90.1456 - 104.726i) q^{66} +(59.6693 - 38.3471i) q^{67} +(-475.890 + 110.663i) q^{68} +(-818.352 + 345.838i) q^{69} +(4.45660 + 156.001i) q^{70} +(270.516 - 350.812i) q^{71} +(-54.2620 - 55.8343i) q^{72} +(419.760 - 696.094i) q^{73} +(105.057 - 85.9050i) q^{74} +(1140.15 + 1478.57i) q^{75} +(-73.6348 + 512.141i) q^{76} +(-327.452 + 265.969i) q^{77} +(-33.4356 - 232.550i) q^{78} +(-668.390 + 687.757i) q^{79} +(-289.926 - 1102.99i) q^{80} +(-271.471 - 835.501i) q^{81} +(-26.0547 - 128.585i) q^{82} +(118.503 - 1379.70i) q^{83} +(-381.187 + 349.850i) q^{84} +(-996.959 - 915.000i) q^{85} +(17.4884 + 203.614i) q^{86} +(-1180.50 + 346.626i) q^{87} +(-223.979 + 288.499i) q^{88} +(-90.7550 - 26.6481i) q^{89} +(20.8450 - 102.874i) q^{90} +(-712.590 + 81.7621i) q^{91} +(645.201 + 943.576i) q^{92} +(809.493 + 457.141i) q^{93} +(-77.0198 + 55.9582i) q^{94} +(-1285.99 + 632.279i) q^{95} +(-382.611 + 559.551i) q^{96} +(334.473 - 633.897i) q^{97} +(88.0264 + 101.588i) q^{98} +(255.028 - 124.352i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 1280 q - 37 q^{2} - 32 q^{3} + 73 q^{4} - 33 q^{5} + 73 q^{6} - 9 q^{7} - 91 q^{8} - 2748 q^{9} + 48 q^{10} + 43 q^{11} + 323 q^{12} - 287 q^{13} - 120 q^{14} + 632 q^{15} + 785 q^{16} - 13 q^{17} + 443 q^{18}+ \cdots + 13103 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/121\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(e\left(\frac{1}{55}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.641211 − 0.0366657i −0.226702 − 0.0129633i −0.0566314 − 0.998395i \(-0.518036\pi\)
−0.170071 + 0.985432i \(0.554400\pi\)
\(3\) 1.82234 − 5.60858i 0.350709 − 1.07937i −0.607747 − 0.794131i \(-0.707926\pi\)
0.958456 − 0.285241i \(-0.0920735\pi\)
\(4\) −7.53805 − 0.864910i −0.942256 − 0.108114i
\(5\) −9.80678 − 18.5859i −0.877145 − 1.66237i −0.740814 − 0.671711i \(-0.765560\pi\)
−0.136331 − 0.990663i \(-0.543531\pi\)
\(6\) −1.37414 + 3.52946i −0.0934987 + 0.240150i
\(7\) 10.6511 + 4.50120i 0.575106 + 0.243042i 0.657369 − 0.753569i \(-0.271669\pi\)
−0.0822628 + 0.996611i \(0.526215\pi\)
\(8\) 9.86457 + 1.70713i 0.435957 + 0.0754453i
\(9\) −6.29178 − 4.57125i −0.233029 − 0.169305i
\(10\) 5.60674 + 12.2771i 0.177301 + 0.388235i
\(11\) −17.1311 + 32.2106i −0.469566 + 0.882897i
\(12\) −18.5878 + 40.7016i −0.447153 + 0.979128i
\(13\) −54.0125 + 30.5022i −1.15234 + 0.650754i −0.944856 − 0.327485i \(-0.893799\pi\)
−0.207479 + 0.978239i \(0.566526\pi\)
\(14\) −6.66457 − 3.27675i −0.127227 − 0.0625534i
\(15\) −122.112 + 21.1323i −2.10194 + 0.363755i
\(16\) 52.8599 + 12.2920i 0.825936 + 0.192063i
\(17\) 60.6488 − 21.6395i 0.865265 − 0.308726i 0.134134 − 0.990963i \(-0.457175\pi\)
0.731131 + 0.682237i \(0.238993\pi\)
\(18\) 3.86675 + 3.16182i 0.0506334 + 0.0414027i
\(19\) 1.94730 − 68.1642i 0.0235126 − 0.823050i −0.899195 − 0.437549i \(-0.855847\pi\)
0.922707 − 0.385501i \(-0.125971\pi\)
\(20\) 57.8488 + 148.583i 0.646769 + 1.66121i
\(21\) 44.6553 − 51.5349i 0.464028 − 0.535516i
\(22\) 12.1657 − 20.0257i 0.117897 − 0.194068i
\(23\) −98.6562 − 113.855i −0.894402 − 1.03220i −0.999289 − 0.0377109i \(-0.987993\pi\)
0.104887 − 0.994484i \(-0.466552\pi\)
\(24\) 27.5512 − 52.2153i 0.234327 − 0.444100i
\(25\) −178.708 + 261.352i −1.42966 + 2.09081i
\(26\) 35.7518 − 17.5780i 0.269673 − 0.132589i
\(27\) 91.7115 − 66.6323i 0.653699 − 0.474940i
\(28\) −76.3955 − 43.1425i −0.515621 − 0.291185i
\(29\) −117.760 − 172.219i −0.754052 − 1.10277i −0.991339 − 0.131326i \(-0.958076\pi\)
0.237287 − 0.971440i \(-0.423742\pi\)
\(30\) 79.0742 − 9.07292i 0.481230 − 0.0552160i
\(31\) −31.3064 + 154.503i −0.181381 + 0.895148i 0.781227 + 0.624247i \(0.214594\pi\)
−0.962607 + 0.270901i \(0.912678\pi\)
\(32\) −110.289 − 32.3838i −0.609267 − 0.178897i
\(33\) 149.437 + 154.780i 0.788293 + 0.816476i
\(34\) −39.6821 + 11.6517i −0.200160 + 0.0587722i
\(35\) −20.7942 − 242.103i −0.100425 − 1.16922i
\(36\) 43.4740 + 39.9001i 0.201269 + 0.184723i
\(37\) −155.672 + 142.875i −0.691686 + 0.634824i −0.942888 − 0.333111i \(-0.891902\pi\)
0.251202 + 0.967935i \(0.419174\pi\)
\(38\) −3.74792 + 43.6362i −0.0159998 + 0.186282i
\(39\) 72.6453 + 358.519i 0.298271 + 1.47202i
\(40\) −65.0111 − 200.084i −0.256979 − 0.790900i
\(41\) 51.9308 + 197.565i 0.197811 + 0.752550i 0.989854 + 0.142085i \(0.0453807\pi\)
−0.792044 + 0.610464i \(0.790983\pi\)
\(42\) −30.5230 + 31.4074i −0.112138 + 0.115387i
\(43\) −45.2839 − 314.957i −0.160598 − 1.11699i −0.897509 − 0.440996i \(-0.854625\pi\)
0.736910 − 0.675990i \(-0.236284\pi\)
\(44\) 156.994 − 227.988i 0.537905 − 0.781149i
\(45\) −23.2587 + 161.768i −0.0770489 + 0.535887i
\(46\) 59.0848 + 76.6226i 0.189382 + 0.245595i
\(47\) 114.751 − 93.8313i 0.356130 − 0.291206i −0.437974 − 0.898988i \(-0.644304\pi\)
0.794105 + 0.607781i \(0.207940\pi\)
\(48\) 165.269 − 274.069i 0.496971 − 0.824133i
\(49\) −145.864 − 150.091i −0.425260 − 0.437582i
\(50\) 124.172 − 161.029i 0.351211 − 0.455459i
\(51\) −10.8440 − 379.588i −0.0297737 − 1.04222i
\(52\) 433.530 − 183.211i 1.15615 − 0.488593i
\(53\) −158.440 + 36.8436i −0.410630 + 0.0954878i −0.426722 − 0.904383i \(-0.640332\pi\)
0.0160928 + 0.999871i \(0.494877\pi\)
\(54\) −61.2495 + 39.3627i −0.154352 + 0.0991959i
\(55\) 766.665 + 2.51489i 1.87958 + 0.00616559i
\(56\) 97.3846 + 62.5853i 0.232385 + 0.149345i
\(57\) −378.756 − 135.140i −0.880130 − 0.314030i
\(58\) 69.1945 + 114.746i 0.156650 + 0.259774i
\(59\) −75.2197 + 286.165i −0.165979 + 0.631450i 0.831172 + 0.556016i \(0.187671\pi\)
−0.997151 + 0.0754341i \(0.975966\pi\)
\(60\) 938.762 − 53.6803i 2.01989 − 0.115502i
\(61\) −181.660 + 10.3877i −0.381298 + 0.0218034i −0.246687 − 0.969095i \(-0.579342\pi\)
−0.134611 + 0.990899i \(0.542978\pi\)
\(62\) 25.7390 − 97.9212i 0.0527234 − 0.200581i
\(63\) −46.4384 − 77.0095i −0.0928681 − 0.154004i
\(64\) −339.382 − 121.091i −0.662855 − 0.236506i
\(65\) 1096.60 + 704.742i 2.09256 + 1.34481i
\(66\) −90.1456 − 104.726i −0.168124 − 0.195316i
\(67\) 59.6693 − 38.3471i 0.108802 − 0.0699231i −0.485108 − 0.874454i \(-0.661220\pi\)
0.593910 + 0.804531i \(0.297583\pi\)
\(68\) −475.890 + 110.663i −0.848679 + 0.197352i
\(69\) −818.352 + 345.838i −1.42780 + 0.603392i
\(70\) 4.45660 + 156.001i 0.00760951 + 0.266368i
\(71\) 270.516 − 350.812i 0.452174 − 0.586390i −0.509539 − 0.860448i \(-0.670184\pi\)
0.961713 + 0.274057i \(0.0883658\pi\)
\(72\) −54.2620 − 55.8343i −0.0888172 − 0.0913908i
\(73\) 419.760 − 696.094i 0.673003 − 1.11605i −0.313069 − 0.949730i \(-0.601357\pi\)
0.986072 − 0.166320i \(-0.0531883\pi\)
\(74\) 105.057 − 85.9050i 0.165036 − 0.134949i
\(75\) 1140.15 + 1478.57i 1.75537 + 2.27640i
\(76\) −73.6348 + 512.141i −0.111138 + 0.772982i
\(77\) −327.452 + 265.969i −0.484631 + 0.393636i
\(78\) −33.4356 − 232.550i −0.0485363 − 0.337578i
\(79\) −668.390 + 687.757i −0.951896 + 0.979478i −0.999812 − 0.0193736i \(-0.993833\pi\)
0.0479164 + 0.998851i \(0.484742\pi\)
\(80\) −289.926 − 1102.99i −0.405184 − 1.54148i
\(81\) −271.471 − 835.501i −0.372388 − 1.14609i
\(82\) −26.0547 − 128.585i −0.0350886 − 0.173169i
\(83\) 118.503 − 1379.70i 0.156715 − 1.82460i −0.322120 − 0.946699i \(-0.604395\pi\)
0.478835 − 0.877905i \(-0.341059\pi\)
\(84\) −381.187 + 349.850i −0.495129 + 0.454426i
\(85\) −996.959 − 915.000i −1.27218 − 1.16760i
\(86\) 17.4884 + 203.614i 0.0219282 + 0.255305i
\(87\) −1180.50 + 346.626i −1.45475 + 0.427152i
\(88\) −223.979 + 288.499i −0.271321 + 0.349479i
\(89\) −90.7550 − 26.6481i −0.108090 − 0.0317381i 0.227240 − 0.973839i \(-0.427030\pi\)
−0.335330 + 0.942101i \(0.608848\pi\)
\(90\) 20.8450 − 102.874i 0.0244140 − 0.120488i
\(91\) −712.590 + 81.7621i −0.820876 + 0.0941867i
\(92\) 645.201 + 943.576i 0.731161 + 1.06929i
\(93\) 809.493 + 457.141i 0.902586 + 0.509714i
\(94\) −77.0198 + 55.9582i −0.0845105 + 0.0614005i
\(95\) −1285.99 + 632.279i −1.38884 + 0.682847i
\(96\) −382.611 + 559.551i −0.406771 + 0.594884i
\(97\) 334.473 − 633.897i 0.350109 − 0.663531i −0.644795 − 0.764356i \(-0.723057\pi\)
0.994904 + 0.100825i \(0.0321481\pi\)
\(98\) 88.0264 + 101.588i 0.0907348 + 0.104714i
\(99\) 255.028 − 124.352i 0.258902 − 0.126241i
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 121.4.g.a.4.14 ✓ 1280
121.91 even 55 inner 121.4.g.a.91.14 yes 1280
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
121.4.g.a.4.14 ✓ 1280 1.1 even 1 trivial
121.4.g.a.91.14 yes 1280 121.91 even 55 inner