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Properties

Label 121.4.c.e.27.2
Level $121$
Weight $4$
Character 121.27
Analytic conductor $7.139$
Analytic rank $0$
Dimension $8$
Inner twists $8$

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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(3,121)] 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("121.3"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(121, base_ring=CyclotomicField(10)) chi = DirichletCharacter(H, H._module([8])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 121 = 11^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 121.c (of order \(5\), degree \(4\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,0,10] 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: \(7.13923111069\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{5})\)
Coefficient field: 8.0.1827904000000.7
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 26x^{6} + 676x^{4} + 17576x^{2} + 456976 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

Embedding invariants

Embedding label 27.2
Root \(-4.12519 - 2.99713i\) of defining polynomial
Character \(\chi\) \(=\) 121.27
Dual form 121.4.c.e.9.2

$q$-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+(1.57568 + 4.84946i) q^{2} +(4.04508 + 2.93893i) q^{3} +(-14.5623 + 10.5801i) q^{4} +(1.54508 - 4.75528i) q^{5} +(-7.87842 + 24.2473i) q^{6} +(-16.5008 + 11.9885i) q^{7} +(-41.2519 - 29.9713i) q^{8} +(-0.618034 - 1.90211i) q^{9} +25.4951 q^{10} -90.0000 q^{12} +(18.9082 + 58.1935i) q^{13} +(-84.1378 - 61.1297i) q^{14} +(20.2254 - 14.6946i) q^{15} +(35.8460 - 110.323i) q^{16} +(-6.30273 + 19.3978i) q^{17} +(8.25039 - 5.99426i) q^{18} +(82.5039 + 59.9426i) q^{19} +(27.8115 + 85.5951i) q^{20} -101.980 q^{21} +35.0000 q^{23} +(-78.7842 - 242.473i) q^{24} +(80.9017 + 58.7785i) q^{25} +(-252.413 + 183.389i) q^{26} +(44.8075 - 137.903i) q^{27} +(113.449 - 349.161i) q^{28} +(165.008 - 119.885i) q^{29} +(103.130 + 74.9282i) q^{30} +(4.63525 + 14.2658i) q^{31} +183.565 q^{32} -104.000 q^{34} +(31.5137 + 96.9891i) q^{35} +(29.1246 + 21.1603i) q^{36} +(214.390 - 155.763i) q^{37} +(-160.689 + 494.549i) q^{38} +(-94.5410 + 290.967i) q^{39} +(-206.260 + 149.856i) q^{40} +(82.5039 + 59.9426i) q^{41} +(-160.689 - 494.549i) q^{42} -448.714 q^{43} -10.0000 q^{45} +(55.1489 + 169.731i) q^{46} +(-307.426 - 223.358i) q^{47} +(469.230 - 340.915i) q^{48} +(22.5582 - 69.4271i) q^{49} +(-157.568 + 484.946i) q^{50} +(-82.5039 + 59.9426i) q^{51} +(-891.042 - 647.380i) q^{52} +(157.599 + 485.039i) q^{53} +739.358 q^{54} +1040.00 q^{56} +(157.568 + 484.946i) q^{57} +(841.378 + 611.297i) q^{58} +(-16.9894 + 12.3435i) q^{59} +(-139.058 + 427.975i) q^{60} +(63.0273 - 193.978i) q^{61} +(-61.8779 + 44.9569i) q^{62} +(33.0015 + 23.9770i) q^{63} +(2.47214 + 7.60845i) q^{64} +305.941 q^{65} +585.000 q^{67} +(-113.449 - 349.161i) q^{68} +(141.578 + 102.862i) q^{69} +(-420.689 + 305.648i) q^{70} +(96.7223 - 297.681i) q^{71} +(-31.5137 + 96.9891i) q^{72} +(-379.518 + 275.736i) q^{73} +(1093.18 + 794.239i) q^{74} +(154.508 + 475.528i) q^{75} -1835.65 q^{76} -1560.00 q^{78} +(-189.082 - 581.935i) q^{79} +(-469.230 - 340.915i) q^{80} +(542.850 - 394.404i) q^{81} +(-160.689 + 494.549i) q^{82} +(201.688 - 620.730i) q^{83} +(1485.07 - 1078.97i) q^{84} +(82.5039 + 59.9426i) q^{85} +(-707.031 - 2176.02i) q^{86} +1019.80 q^{87} -185.000 q^{89} +(-15.7568 - 48.4946i) q^{90} +(-1009.65 - 733.556i) q^{91} +(-509.681 + 370.305i) q^{92} +(-23.1763 + 71.3292i) q^{93} +(598.760 - 1842.79i) q^{94} +(412.519 - 299.713i) q^{95} +(742.535 + 539.483i) q^{96} +(242.578 + 746.579i) q^{97} +372.228 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 10 q^{3} - 36 q^{4} - 10 q^{5} + 4 q^{9} - 720 q^{12} - 208 q^{14} + 50 q^{15} - 232 q^{16} - 180 q^{20} + 280 q^{23} + 200 q^{25} - 624 q^{26} - 290 q^{27} - 30 q^{31} - 832 q^{34} + 72 q^{36} + 530 q^{37}+ \cdots - 1570 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.57568 + 4.84946i 0.557088 + 1.71454i 0.690364 + 0.723462i \(0.257451\pi\)
−0.133275 + 0.991079i \(0.542549\pi\)
\(3\) 4.04508 + 2.93893i 0.778477 + 0.565597i 0.904522 − 0.426428i \(-0.140228\pi\)
−0.126045 + 0.992025i \(0.540228\pi\)
\(4\) −14.5623 + 10.5801i −1.82029 + 1.32252i
\(5\) 1.54508 − 4.75528i 0.138197 − 0.425325i −0.857877 − 0.513855i \(-0.828217\pi\)
0.996074 + 0.0885298i \(0.0282169\pi\)
\(6\) −7.87842 + 24.2473i −0.536058 + 1.64982i
\(7\) −16.5008 + 11.9885i −0.890958 + 0.647319i −0.936128 − 0.351660i \(-0.885617\pi\)
0.0451696 + 0.998979i \(0.485617\pi\)
\(8\) −41.2519 − 29.9713i −1.82310 − 1.32456i
\(9\) −0.618034 − 1.90211i −0.0228901 − 0.0704486i
\(10\) 25.4951 0.806226
\(11\) 0 0
\(12\) −90.0000 −2.16506
\(13\) 18.9082 + 58.1935i 0.403399 + 1.24154i 0.922224 + 0.386655i \(0.126370\pi\)
−0.518825 + 0.854881i \(0.673630\pi\)
\(14\) −84.1378 − 61.1297i −1.60620 − 1.16697i
\(15\) 20.2254 − 14.6946i 0.348145 − 0.252942i
\(16\) 35.8460 − 110.323i 0.560093 − 1.72379i
\(17\) −6.30273 + 19.3978i −0.0899199 + 0.276745i −0.985896 − 0.167356i \(-0.946477\pi\)
0.895977 + 0.444101i \(0.146477\pi\)
\(18\) 8.25039 − 5.99426i 0.108035 − 0.0784922i
\(19\) 82.5039 + 59.9426i 0.996194 + 0.723777i 0.961269 − 0.275613i \(-0.0888807\pi\)
0.0349252 + 0.999390i \(0.488881\pi\)
\(20\) 27.8115 + 85.5951i 0.310942 + 0.956982i
\(21\) −101.980 −1.05971
\(22\) 0 0
\(23\) 35.0000 0.317305 0.158652 − 0.987335i \(-0.449285\pi\)
0.158652 + 0.987335i \(0.449285\pi\)
\(24\) −78.7842 − 242.473i −0.670073 − 2.06227i
\(25\) 80.9017 + 58.7785i 0.647214 + 0.470228i
\(26\) −252.413 + 183.389i −1.90394 + 1.38329i
\(27\) 44.8075 − 137.903i 0.319378 − 0.982944i
\(28\) 113.449 − 349.161i 0.765710 − 2.35661i
\(29\) 165.008 − 119.885i 1.05659 − 0.767659i 0.0831371 − 0.996538i \(-0.473506\pi\)
0.973455 + 0.228879i \(0.0735060\pi\)
\(30\) 103.130 + 74.9282i 0.627628 + 0.455999i
\(31\) 4.63525 + 14.2658i 0.0268554 + 0.0826523i 0.963586 − 0.267399i \(-0.0861642\pi\)
−0.936731 + 0.350051i \(0.886164\pi\)
\(32\) 183.565 1.01406
\(33\) 0 0
\(34\) −104.000 −0.524584
\(35\) 31.5137 + 96.9891i 0.152194 + 0.468404i
\(36\) 29.1246 + 21.1603i 0.134836 + 0.0979642i
\(37\) 214.390 − 155.763i 0.952579 − 0.692089i 0.00116345 − 0.999999i \(-0.499630\pi\)
0.951415 + 0.307910i \(0.0996297\pi\)
\(38\) −160.689 + 494.549i −0.685978 + 2.11122i
\(39\) −94.5410 + 290.967i −0.388171 + 1.19467i
\(40\) −206.260 + 149.856i −0.815313 + 0.592360i
\(41\) 82.5039 + 59.9426i 0.314267 + 0.228328i 0.733725 − 0.679446i \(-0.237780\pi\)
−0.419458 + 0.907775i \(0.637780\pi\)
\(42\) −160.689 − 494.549i −0.590353 − 1.81692i
\(43\) −448.714 −1.59135 −0.795677 − 0.605721i \(-0.792885\pi\)
−0.795677 + 0.605721i \(0.792885\pi\)
\(44\) 0 0
\(45\) −10.0000 −0.0331269
\(46\) 55.1489 + 169.731i 0.176767 + 0.544032i
\(47\) −307.426 − 223.358i −0.954101 − 0.693195i −0.00232792 − 0.999997i \(-0.500741\pi\)
−0.951773 + 0.306802i \(0.900741\pi\)
\(48\) 469.230 − 340.915i 1.41099 − 1.02514i
\(49\) 22.5582 − 69.4271i 0.0657675 − 0.202411i
\(50\) −157.568 + 484.946i −0.445671 + 1.37163i
\(51\) −82.5039 + 59.9426i −0.226527 + 0.164581i
\(52\) −891.042 − 647.380i −2.37626 − 1.72645i
\(53\) 157.599 + 485.039i 0.408450 + 1.25708i 0.917980 + 0.396627i \(0.129819\pi\)
−0.509530 + 0.860453i \(0.670181\pi\)
\(54\) 739.358 1.86322
\(55\) 0 0
\(56\) 1040.00 2.48171
\(57\) 157.568 + 484.946i 0.366148 + 1.12689i
\(58\) 841.378 + 611.297i 1.90480 + 1.38392i
\(59\) −16.9894 + 12.3435i −0.0374886 + 0.0272370i −0.606372 − 0.795181i \(-0.707376\pi\)
0.568883 + 0.822418i \(0.307376\pi\)
\(60\) −139.058 + 427.975i −0.299204 + 0.920857i
\(61\) 63.0273 − 193.978i 0.132292 − 0.407154i −0.862867 − 0.505431i \(-0.831333\pi\)
0.995159 + 0.0982778i \(0.0313334\pi\)
\(62\) −61.8779 + 44.9569i −0.126750 + 0.0920893i
\(63\) 33.0015 + 23.9770i 0.0659969 + 0.0479495i
\(64\) 2.47214 + 7.60845i 0.00482839 + 0.0148603i
\(65\) 305.941 0.583805
\(66\) 0 0
\(67\) 585.000 1.06670 0.533352 − 0.845894i \(-0.320932\pi\)
0.533352 + 0.845894i \(0.320932\pi\)
\(68\) −113.449 − 349.161i −0.202320 − 0.622676i
\(69\) 141.578 + 102.862i 0.247014 + 0.179466i
\(70\) −420.689 + 305.648i −0.718313 + 0.521885i
\(71\) 96.7223 − 297.681i 0.161674 − 0.497580i −0.837102 − 0.547047i \(-0.815752\pi\)
0.998776 + 0.0494663i \(0.0157520\pi\)
\(72\) −31.5137 + 96.9891i −0.0515823 + 0.158754i
\(73\) −379.518 + 275.736i −0.608482 + 0.442088i −0.848880 − 0.528586i \(-0.822722\pi\)
0.240397 + 0.970675i \(0.422722\pi\)
\(74\) 1093.18 + 794.239i 1.71729 + 1.24768i
\(75\) 154.508 + 475.528i 0.237881 + 0.732124i
\(76\) −1835.65 −2.77057
\(77\) 0 0
\(78\) −1560.00 −2.26455
\(79\) −189.082 − 581.935i −0.269283 − 0.828769i −0.990676 − 0.136242i \(-0.956497\pi\)
0.721392 − 0.692527i \(-0.243503\pi\)
\(80\) −469.230 − 340.915i −0.655769 − 0.476444i
\(81\) 542.850 − 394.404i 0.744651 − 0.541020i
\(82\) −160.689 + 494.549i −0.216404 + 0.666022i
\(83\) 201.688 − 620.730i 0.266724 − 0.820892i −0.724567 − 0.689204i \(-0.757960\pi\)
0.991291 − 0.131688i \(-0.0420396\pi\)
\(84\) 1485.07 − 1078.97i 1.92898 − 1.40149i
\(85\) 82.5039 + 59.9426i 0.105280 + 0.0764904i
\(86\) −707.031 − 2176.02i −0.886524 − 2.72844i
\(87\) 1019.80 1.25672
\(88\) 0 0
\(89\) −185.000 −0.220337 −0.110168 − 0.993913i \(-0.535139\pi\)
−0.110168 + 0.993913i \(0.535139\pi\)
\(90\) −15.7568 − 48.4946i −0.0184546 − 0.0567975i
\(91\) −1009.65 − 733.556i −1.16308 − 0.845028i
\(92\) −509.681 + 370.305i −0.577586 + 0.419641i
\(93\) −23.1763 + 71.3292i −0.0258416 + 0.0795322i
\(94\) 598.760 − 1842.79i 0.656993 − 2.02202i
\(95\) 412.519 − 299.713i 0.445511 − 0.323683i
\(96\) 742.535 + 539.483i 0.789423 + 0.573550i
\(97\) 242.578 + 746.579i 0.253919 + 0.781481i 0.994041 + 0.109009i \(0.0347676\pi\)
−0.740122 + 0.672472i \(0.765232\pi\)
\(98\) 372.228 0.383681
\(99\) 0 0
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.c.e.27.2 8
11.2 odd 10 inner 121.4.c.e.9.1 8
11.3 even 5 121.4.a.d.1.2 yes 2
11.4 even 5 inner 121.4.c.e.3.1 8
11.5 even 5 inner 121.4.c.e.81.1 8
11.6 odd 10 inner 121.4.c.e.81.2 8
11.7 odd 10 inner 121.4.c.e.3.2 8
11.8 odd 10 121.4.a.d.1.1 ✓ 2
11.9 even 5 inner 121.4.c.e.9.2 8
11.10 odd 2 inner 121.4.c.e.27.1 8
33.8 even 10 1089.4.a.r.1.2 2
33.14 odd 10 1089.4.a.r.1.1 2
44.3 odd 10 1936.4.a.ba.1.1 2
44.19 even 10 1936.4.a.ba.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
121.4.a.d.1.1 ✓ 2 11.8 odd 10
121.4.a.d.1.2 yes 2 11.3 even 5
121.4.c.e.3.1 8 11.4 even 5 inner
121.4.c.e.3.2 8 11.7 odd 10 inner
121.4.c.e.9.1 8 11.2 odd 10 inner
121.4.c.e.9.2 8 11.9 even 5 inner
121.4.c.e.27.1 8 11.10 odd 2 inner
121.4.c.e.27.2 8 1.1 even 1 trivial
121.4.c.e.81.1 8 11.5 even 5 inner
121.4.c.e.81.2 8 11.6 odd 10 inner
1089.4.a.r.1.1 2 33.14 odd 10
1089.4.a.r.1.2 2 33.8 even 10
1936.4.a.ba.1.1 2 44.3 odd 10
1936.4.a.ba.1.2 2 44.19 even 10