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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 4.18
Character \(\chi\) \(=\) 121.4
Dual form 121.4.g.a.91.18

$q$-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+(0.532391 + 0.0304432i) q^{2} +(-0.448204 + 1.37943i) q^{3} +(-7.66534 - 0.879516i) q^{4} +(-3.29729 - 6.24906i) q^{5} +(-0.280614 + 0.720752i) q^{6} +(17.8801 + 7.55618i) q^{7} +(-8.25779 - 1.42907i) q^{8} +(20.1415 + 14.6337i) q^{9} +(-1.56521 - 3.42733i) q^{10} +(36.3062 + 3.58606i) q^{11} +(4.64887 - 10.1796i) q^{12} +(45.5096 - 25.7004i) q^{13} +(9.28917 + 4.56718i) q^{14} +(10.0980 - 1.74753i) q^{15} +(55.7681 + 12.9683i) q^{16} +(-0.967589 + 0.345235i) q^{17} +(10.2777 + 8.40401i) q^{18} +(-2.82413 + 98.8573i) q^{19} +(19.7787 + 50.8012i) q^{20} +(-18.4372 + 21.2776i) q^{21} +(19.2199 + 3.01447i) q^{22} +(-60.7867 - 70.1516i) q^{23} +(5.67247 - 10.7505i) q^{24} +(42.3767 - 61.9740i) q^{25} +(25.0113 - 12.2972i) q^{26} +(-60.8958 + 44.2434i) q^{27} +(-130.411 - 73.6465i) q^{28} +(51.0483 + 74.6557i) q^{29} +(5.42929 - 0.622953i) q^{30} +(-0.989829 + 4.88500i) q^{31} +(93.6242 + 27.4905i) q^{32} +(-21.2193 + 48.4746i) q^{33} +(-0.525646 + 0.154344i) q^{34} +(-11.7368 - 136.649i) q^{35} +(-141.521 - 129.887i) q^{36} +(243.601 - 223.575i) q^{37} +(-4.51308 + 52.5448i) q^{38} +(15.0544 + 74.2963i) q^{39} +(18.2980 + 56.3155i) q^{40} +(56.2361 + 213.944i) q^{41} +(-10.4635 + 10.7667i) q^{42} +(11.3490 + 78.9339i) q^{43} +(-275.145 - 59.4203i) q^{44} +(25.0343 - 174.117i) q^{45} +(-30.2267 - 39.1986i) q^{46} +(37.4109 - 30.5907i) q^{47} +(-42.8844 + 71.1157i) q^{48} +(23.5517 + 24.2342i) q^{49} +(24.4477 - 31.7043i) q^{50} +(-0.0425504 - 1.48946i) q^{51} +(-371.450 + 156.976i) q^{52} +(-394.294 + 91.6891i) q^{53} +(-33.7673 + 21.7009i) q^{54} +(-97.3026 - 238.704i) q^{55} +(-136.852 - 87.9492i) q^{56} +(-135.101 - 48.2039i) q^{57} +(24.9049 + 41.3001i) q^{58} +(-10.5696 + 40.2108i) q^{59} +(-78.9416 + 4.51405i) q^{60} +(280.457 - 16.0371i) q^{61} +(-0.675691 + 2.57060i) q^{62} +(249.557 + 413.844i) q^{63} +(-382.402 - 136.441i) q^{64} +(-310.662 - 199.650i) q^{65} +(-12.7727 + 25.1615i) q^{66} +(-494.763 + 317.965i) q^{67} +(7.72054 - 1.79533i) q^{68} +(124.014 - 52.4088i) q^{69} +(-2.08852 - 73.1079i) q^{70} +(447.955 - 580.919i) q^{71} +(-145.412 - 149.625i) q^{72} +(139.795 - 231.823i) q^{73} +(136.498 - 111.614i) q^{74} +(66.4954 + 86.2328i) q^{75} +(108.594 - 755.291i) q^{76} +(622.061 + 338.455i) q^{77} +(5.75301 + 40.0130i) q^{78} +(-787.957 + 810.789i) q^{79} +(-102.844 - 391.259i) q^{80} +(173.984 + 535.468i) q^{81} +(23.4264 + 115.614i) q^{82} +(-10.4669 + 121.864i) q^{83} +(160.041 - 146.884i) q^{84} +(5.34782 + 4.90819i) q^{85} +(3.63909 + 42.3692i) q^{86} +(-125.862 + 36.9565i) q^{87} +(-294.684 - 81.4970i) q^{88} +(-77.9510 - 22.8885i) q^{89} +(18.6287 - 91.9363i) q^{90} +(1007.91 - 115.647i) q^{91} +(404.251 + 591.199i) q^{92} +(-6.29487 - 3.55487i) q^{93} +(20.8485 - 15.1473i) q^{94} +(627.078 - 308.313i) q^{95} +(-79.8840 + 116.827i) q^{96} +(73.3736 - 139.058i) q^{97} +(11.8010 + 13.6190i) q^{98} +(678.785 + 603.522i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.532391 + 0.0304432i 0.188229 + 0.0107633i 0.150949 − 0.988542i \(-0.451767\pi\)
0.0372798 + 0.999305i \(0.488131\pi\)
\(3\) −0.448204 + 1.37943i −0.0862569 + 0.265471i −0.984877 − 0.173257i \(-0.944571\pi\)
0.898620 + 0.438728i \(0.144571\pi\)
\(4\) −7.66534 − 0.879516i −0.958168 − 0.109939i
\(5\) −3.29729 − 6.24906i −0.294919 − 0.558933i 0.691659 − 0.722224i \(-0.256880\pi\)
−0.986578 + 0.163291i \(0.947789\pi\)
\(6\) −0.280614 + 0.720752i −0.0190934 + 0.0490410i
\(7\) 17.8801 + 7.55618i 0.965434 + 0.407996i 0.814253 − 0.580511i \(-0.197147\pi\)
0.151181 + 0.988506i \(0.451692\pi\)
\(8\) −8.25779 − 1.42907i −0.364946 − 0.0631564i
\(9\) 20.1415 + 14.6337i 0.745982 + 0.541988i
\(10\) −1.56521 − 3.42733i −0.0494962 − 0.108382i
\(11\) 36.3062 + 3.58606i 0.995157 + 0.0982945i
\(12\) 4.64887 − 10.1796i 0.111834 − 0.244883i
\(13\) 45.5096 − 25.7004i 0.970930 − 0.548309i 0.0772485 − 0.997012i \(-0.475387\pi\)
0.893681 + 0.448703i \(0.148114\pi\)
\(14\) 9.28917 + 4.56718i 0.177331 + 0.0871878i
\(15\) 10.0980 − 1.74753i 0.173820 − 0.0300807i
\(16\) 55.7681 + 12.9683i 0.871376 + 0.202630i
\(17\) −0.967589 + 0.345235i −0.0138044 + 0.00492540i −0.342943 − 0.939356i \(-0.611424\pi\)
0.329138 + 0.944282i \(0.393242\pi\)
\(18\) 10.2777 + 8.40401i 0.134582 + 0.110047i
\(19\) −2.82413 + 98.8573i −0.0341000 + 1.19365i 0.784812 + 0.619734i \(0.212760\pi\)
−0.818912 + 0.573920i \(0.805422\pi\)
\(20\) 19.7787 + 50.8012i 0.221133 + 0.567975i
\(21\) −18.4372 + 21.2776i −0.191587 + 0.221103i
\(22\) 19.2199 + 3.01447i 0.186259 + 0.0292130i
\(23\) −60.7867 − 70.1516i −0.551083 − 0.635983i 0.410052 − 0.912062i \(-0.365510\pi\)
−0.961135 + 0.276079i \(0.910965\pi\)
\(24\) 5.67247 − 10.7505i 0.0482454 − 0.0914352i
\(25\) 42.3767 − 61.9740i 0.339014 − 0.495792i
\(26\) 25.0113 − 12.2972i 0.188659 − 0.0927572i
\(27\) −60.8958 + 44.2434i −0.434052 + 0.315357i
\(28\) −130.411 − 73.6465i −0.880192 − 0.497067i
\(29\) 51.0483 + 74.6557i 0.326877 + 0.478042i 0.953548 − 0.301242i \(-0.0974012\pi\)
−0.626671 + 0.779284i \(0.715583\pi\)
\(30\) 5.42929 − 0.622953i 0.0330416 − 0.00379117i
\(31\) −0.989829 + 4.88500i −0.00573479 + 0.0283023i −0.982848 − 0.184419i \(-0.940960\pi\)
0.977113 + 0.212721i \(0.0682325\pi\)
\(32\) 93.6242 + 27.4905i 0.517205 + 0.151865i
\(33\) −21.2193 + 48.4746i −0.111934 + 0.255707i
\(34\) −0.525646 + 0.154344i −0.00265140 + 0.000778521i
\(35\) −11.7368 − 136.649i −0.0566822 − 0.659939i
\(36\) −141.521 − 129.887i −0.655190 − 0.601328i
\(37\) 243.601 − 223.575i 1.08237 − 0.993394i 0.0823790 − 0.996601i \(-0.473748\pi\)
0.999995 + 0.00320729i \(0.00102091\pi\)
\(38\) −4.51308 + 52.5448i −0.0192663 + 0.224313i
\(39\) 15.0544 + 74.2963i 0.0618111 + 0.305050i
\(40\) 18.2980 + 56.3155i 0.0723293 + 0.222607i
\(41\) 56.2361 + 213.944i 0.214210 + 0.814938i 0.984226 + 0.176914i \(0.0566116\pi\)
−0.770017 + 0.638024i \(0.779752\pi\)
\(42\) −10.4635 + 10.7667i −0.0384419 + 0.0395558i
\(43\) 11.3490 + 78.9339i 0.0402489 + 0.279937i 1.00000 0.000916145i \(-0.000291618\pi\)
−0.959751 + 0.280853i \(0.909383\pi\)
\(44\) −275.145 − 59.4203i −0.942721 − 0.203590i
\(45\) 25.0343 − 174.117i 0.0829308 − 0.576797i
\(46\) −30.2267 − 39.1986i −0.0968843 − 0.125642i
\(47\) 37.4109 − 30.5907i 0.116105 − 0.0949386i −0.573178 − 0.819431i \(-0.694289\pi\)
0.689283 + 0.724493i \(0.257926\pi\)
\(48\) −42.8844 + 71.1157i −0.128955 + 0.213847i
\(49\) 23.5517 + 24.2342i 0.0686639 + 0.0706535i
\(50\) 24.4477 − 31.7043i 0.0691485 − 0.0896734i
\(51\) −0.0425504 − 1.48946i −0.000116828 − 0.00408953i
\(52\) −371.450 + 156.976i −0.990594 + 0.418629i
\(53\) −394.294 + 91.6891i −1.02190 + 0.237632i −0.703778 − 0.710420i \(-0.748505\pi\)
−0.318117 + 0.948051i \(0.603051\pi\)
\(54\) −33.7673 + 21.7009i −0.0850954 + 0.0546875i
\(55\) −97.3026 − 238.704i −0.238551 − 0.585215i
\(56\) −136.852 − 87.9492i −0.326564 − 0.209870i
\(57\) −135.101 − 48.2039i −0.313940 − 0.112013i
\(58\) 24.9049 + 41.3001i 0.0563823 + 0.0934995i
\(59\) −10.5696 + 40.2108i −0.0233227 + 0.0887289i −0.977790 − 0.209586i \(-0.932788\pi\)
0.954467 + 0.298315i \(0.0964247\pi\)
\(60\) −78.9416 + 4.51405i −0.169855 + 0.00971268i
\(61\) 280.457 − 16.0371i 0.588670 − 0.0336614i 0.239783 − 0.970826i \(-0.422924\pi\)
0.348886 + 0.937165i \(0.386560\pi\)
\(62\) −0.675691 + 2.57060i −0.00138408 + 0.00526558i
\(63\) 249.557 + 413.844i 0.499068 + 0.827611i
\(64\) −382.402 − 136.441i −0.746880 − 0.266486i
\(65\) −310.662 − 199.650i −0.592814 − 0.380978i
\(66\) −12.7727 + 25.1615i −0.0238214 + 0.0469267i
\(67\) −494.763 + 317.965i −0.902162 + 0.579785i −0.907431 − 0.420202i \(-0.861959\pi\)
0.00526836 + 0.999986i \(0.498323\pi\)
\(68\) 7.72054 − 1.79533i 0.0137684 − 0.00320171i
\(69\) 124.014 − 52.4088i 0.216370 − 0.0914388i
\(70\) −2.08852 − 73.1079i −0.00356609 − 0.124830i
\(71\) 447.955 − 580.919i 0.748768 − 0.971019i −0.251228 − 0.967928i \(-0.580834\pi\)
0.999995 − 0.00309114i \(-0.000983942\pi\)
\(72\) −145.412 − 149.625i −0.238013 − 0.244910i
\(73\) 139.795 − 231.823i 0.224133 − 0.371683i −0.723874 − 0.689932i \(-0.757640\pi\)
0.948007 + 0.318249i \(0.103095\pi\)
\(74\) 136.498 − 111.614i 0.214426 − 0.175335i
\(75\) 66.4954 + 86.2328i 0.102376 + 0.132764i
\(76\) 108.594 − 755.291i 0.163903 − 1.13997i
\(77\) 622.061 + 338.455i 0.920655 + 0.500917i
\(78\) 5.75301 + 40.0130i 0.00835128 + 0.0580844i
\(79\) −787.957 + 810.789i −1.12218 + 1.15469i −0.135709 + 0.990749i \(0.543331\pi\)
−0.986469 + 0.163946i \(0.947578\pi\)
\(80\) −102.844 − 391.259i −0.143729 − 0.546800i
\(81\) 173.984 + 535.468i 0.238661 + 0.734524i
\(82\) 23.4264 + 115.614i 0.0315490 + 0.155700i
\(83\) −10.4669 + 121.864i −0.0138420 + 0.161160i −0.999989 − 0.00471477i \(-0.998499\pi\)
0.986147 + 0.165875i \(0.0530447\pi\)
\(84\) 160.041 − 146.884i 0.207880 − 0.190790i
\(85\) 5.34782 + 4.90819i 0.00682415 + 0.00626315i
\(86\) 3.63909 + 42.3692i 0.00456295 + 0.0531255i
\(87\) −125.862 + 36.9565i −0.155102 + 0.0455420i
\(88\) −294.684 − 81.4970i −0.356971 − 0.0987228i
\(89\) −77.9510 − 22.8885i −0.0928404 − 0.0272604i 0.234982 − 0.972000i \(-0.424497\pi\)
−0.327823 + 0.944739i \(0.606315\pi\)
\(90\) 18.6287 − 91.9363i 0.0218182 − 0.107677i
\(91\) 1007.91 − 115.647i 1.16108 − 0.133221i
\(92\) 404.251 + 591.199i 0.458110 + 0.669964i
\(93\) −6.29487 − 3.55487i −0.00701879 − 0.00396369i
\(94\) 20.8485 − 15.1473i 0.0228762 − 0.0166205i
\(95\) 627.078 − 308.313i 0.677230 − 0.332971i
\(96\) −79.8840 + 116.827i −0.0849284 + 0.124204i
\(97\) 73.3736 − 139.058i 0.0768037 − 0.145559i −0.843032 − 0.537864i \(-0.819231\pi\)
0.919835 + 0.392305i \(0.128322\pi\)
\(98\) 11.8010 + 13.6190i 0.0121641 + 0.0140381i
\(99\) 678.785 + 603.522i 0.689095 + 0.612689i
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.4.18 ✓ 1280
121.91 even 55 inner 121.4.g.a.91.18 yes 1280
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
121.4.g.a.4.18 ✓ 1280 1.1 even 1 trivial
121.4.g.a.91.18 yes 1280 121.91 even 55 inner