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Properties

Label 121.4.g.a.4.11
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 4.11
Character \(\chi\) \(=\) 121.4
Dual form 121.4.g.a.91.11

$q$-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+(-2.45967 - 0.140649i) q^{2} +(0.290500 - 0.894067i) q^{3} +(-1.91764 - 0.220029i) q^{4} +(-4.41327 - 8.36407i) q^{5} +(-0.840285 + 2.15826i) q^{6} +(-21.5269 - 9.09733i) q^{7} +(24.1067 + 4.17183i) q^{8} +(21.1285 + 15.3507i) q^{9} +(9.67880 + 21.1936i) q^{10} +(-32.0953 + 17.3462i) q^{11} +(-0.753795 + 1.65058i) q^{12} +(43.4196 - 24.5202i) q^{13} +(51.6695 + 25.4042i) q^{14} +(-8.76010 + 1.51600i) q^{15} +(-43.6674 - 10.1544i) q^{16} +(-50.8208 + 18.1328i) q^{17} +(-49.8101 - 40.7295i) q^{18} +(-4.31783 + 151.144i) q^{19} +(6.62273 + 17.0103i) q^{20} +(-14.3872 + 16.6037i) q^{21} +(81.3837 - 38.1519i) q^{22} +(74.4947 + 85.9715i) q^{23} +(10.7329 - 20.3411i) q^{24} +(20.0746 - 29.3582i) q^{25} +(-110.247 + 54.2048i) q^{26} +(40.3970 - 29.3501i) q^{27} +(39.2791 + 22.1819i) q^{28} +(66.9815 + 97.9574i) q^{29} +(21.7602 - 2.49675i) q^{30} +(-57.1562 + 282.077i) q^{31} +(-81.8128 - 24.0224i) q^{32} +(6.18501 + 33.7344i) q^{33} +(127.553 - 37.4529i) q^{34} +(18.9131 + 220.201i) q^{35} +(-37.1392 - 34.0861i) q^{36} +(-102.362 + 93.9467i) q^{37} +(31.8787 - 371.157i) q^{38} +(-9.30930 - 45.9432i) q^{39} +(-71.4958 - 220.042i) q^{40} +(-98.3009 - 373.975i) q^{41} +(37.7231 - 38.8161i) q^{42} +(0.0582202 + 0.404930i) q^{43} +(65.3639 - 26.2019i) q^{44} +(35.1491 - 244.467i) q^{45} +(-171.141 - 221.939i) q^{46} +(6.30007 - 5.15154i) q^{47} +(-21.7641 + 36.0917i) q^{48} +(141.595 + 145.698i) q^{49} +(-53.5062 + 69.3881i) q^{50} +(1.44852 + 50.7048i) q^{51} +(-88.6584 + 37.4674i) q^{52} +(309.208 - 71.9033i) q^{53} +(-103.491 + 66.5099i) q^{54} +(286.730 + 191.894i) q^{55} +(-480.989 - 309.113i) q^{56} +(133.878 + 47.7677i) q^{57} +(-150.975 - 250.364i) q^{58} +(66.3651 - 252.479i) q^{59} +(17.1323 - 0.979660i) q^{60} +(-621.174 + 35.5200i) q^{61} +(180.259 - 685.778i) q^{62} +(-315.179 - 522.666i) q^{63} +(535.656 + 191.122i) q^{64} +(-396.711 - 254.951i) q^{65} +(-10.4684 - 83.8456i) q^{66} +(-71.8261 + 46.1598i) q^{67} +(101.446 - 23.5902i) q^{68} +(98.5050 - 41.6286i) q^{69} +(-15.5489 - 544.283i) q^{70} +(394.259 - 511.284i) q^{71} +(445.297 + 458.200i) q^{72} +(-603.113 + 1000.15i) q^{73} +(264.990 - 216.681i) q^{74} +(-20.4165 - 26.4766i) q^{75} +(41.5360 - 288.889i) q^{76} +(848.716 - 81.4285i) q^{77} +(16.4360 + 114.315i) q^{78} +(-678.144 + 697.794i) q^{79} +(107.784 + 410.051i) q^{80} +(203.394 + 625.983i) q^{81} +(189.189 + 933.683i) q^{82} +(-42.5526 + 495.431i) q^{83} +(31.2427 - 28.6743i) q^{84} +(375.950 + 345.044i) q^{85} +(-0.0862496 - 1.00419i) q^{86} +(107.039 - 31.4294i) q^{87} +(-846.077 + 284.264i) q^{88} +(-304.099 - 89.2914i) q^{89} +(-120.839 + 596.366i) q^{90} +(-1157.76 + 132.840i) q^{91} +(-123.938 - 181.253i) q^{92} +(235.592 + 133.045i) q^{93} +(-16.2207 + 11.7850i) q^{94} +(1283.23 - 630.923i) q^{95} +(-45.2443 + 66.1676i) q^{96} +(319.663 - 605.829i) q^{97} +(-327.785 - 378.284i) q^{98} +(-944.403 - 126.187i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.45967 − 0.140649i −0.869626 − 0.0497270i −0.383423 − 0.923573i \(-0.625255\pi\)
−0.486203 + 0.873846i \(0.661618\pi\)
\(3\) 0.290500 − 0.894067i 0.0559068 − 0.172063i −0.919204 − 0.393782i \(-0.871167\pi\)
0.975111 + 0.221718i \(0.0711666\pi\)
\(4\) −1.91764 − 0.220029i −0.239705 − 0.0275036i
\(5\) −4.41327 − 8.36407i −0.394735 − 0.748106i 0.604073 − 0.796929i \(-0.293544\pi\)
−0.998807 + 0.0488237i \(0.984453\pi\)
\(6\) −0.840285 + 2.15826i −0.0571742 + 0.146851i
\(7\) −21.5269 − 9.09733i −1.16234 − 0.491210i −0.279162 − 0.960244i \(-0.590057\pi\)
−0.883180 + 0.469034i \(0.844602\pi\)
\(8\) 24.1067 + 4.17183i 1.06538 + 0.184370i
\(9\) 21.1285 + 15.3507i 0.782537 + 0.568546i
\(10\) 9.67880 + 21.1936i 0.306071 + 0.670201i
\(11\) −32.0953 + 17.3462i −0.879736 + 0.475462i
\(12\) −0.753795 + 1.65058i −0.0181335 + 0.0397068i
\(13\) 43.4196 − 24.5202i 0.926342 − 0.523129i 0.0467754 − 0.998905i \(-0.485105\pi\)
0.879566 + 0.475776i \(0.157833\pi\)
\(14\) 51.6695 + 25.4042i 0.986376 + 0.484968i
\(15\) −8.76010 + 1.51600i −0.150790 + 0.0260952i
\(16\) −43.6674 − 10.1544i −0.682302 − 0.158663i
\(17\) −50.8208 + 18.1328i −0.725050 + 0.258697i −0.672683 − 0.739931i \(-0.734858\pi\)
−0.0523667 + 0.998628i \(0.516676\pi\)
\(18\) −49.8101 − 40.7295i −0.652242 − 0.533336i
\(19\) −4.31783 + 151.144i −0.0521357 + 1.82499i 0.382026 + 0.924151i \(0.375226\pi\)
−0.434162 + 0.900835i \(0.642955\pi\)
\(20\) 6.62273 + 17.0103i 0.0740443 + 0.190181i
\(21\) −14.3872 + 16.6037i −0.149502 + 0.172534i
\(22\) 81.3837 − 38.1519i 0.788685 − 0.369728i
\(23\) 74.4947 + 85.9715i 0.675357 + 0.779404i 0.985205 − 0.171382i \(-0.0548233\pi\)
−0.309847 + 0.950786i \(0.600278\pi\)
\(24\) 10.7329 − 20.3411i 0.0912851 − 0.173004i
\(25\) 20.0746 − 29.3582i 0.160597 − 0.234865i
\(26\) −110.247 + 54.2048i −0.831585 + 0.408863i
\(27\) 40.3970 − 29.3501i 0.287941 − 0.209201i
\(28\) 39.2791 + 22.1819i 0.265109 + 0.149714i
\(29\) 66.9815 + 97.9574i 0.428902 + 0.627249i 0.978024 − 0.208493i \(-0.0668559\pi\)
−0.549122 + 0.835742i \(0.685038\pi\)
\(30\) 21.7602 − 2.49675i 0.132428 − 0.0151947i
\(31\) −57.1562 + 282.077i −0.331147 + 1.63427i 0.375582 + 0.926789i \(0.377443\pi\)
−0.706729 + 0.707484i \(0.749830\pi\)
\(32\) −81.8128 − 24.0224i −0.451956 − 0.132706i
\(33\) 6.18501 + 33.7344i 0.0326264 + 0.177952i
\(34\) 127.553 − 37.4529i 0.643386 − 0.188915i
\(35\) 18.9131 + 220.201i 0.0913399 + 1.06345i
\(36\) −37.1392 − 34.0861i −0.171941 − 0.157806i
\(37\) −102.362 + 93.9467i −0.454815 + 0.417425i −0.870565 − 0.492054i \(-0.836246\pi\)
0.415750 + 0.909479i \(0.363519\pi\)
\(38\) 31.8787 − 371.157i 0.136090 − 1.58446i
\(39\) −9.30930 − 45.9432i −0.0382226 − 0.188636i
\(40\) −71.4958 − 220.042i −0.282612 − 0.869791i
\(41\) −98.3009 − 373.975i −0.374439 − 1.42452i −0.841484 − 0.540282i \(-0.818318\pi\)
0.467045 − 0.884234i \(-0.345319\pi\)
\(42\) 37.7231 − 38.8161i 0.138590 − 0.142606i
\(43\) 0.0582202 + 0.404930i 0.000206477 + 0.00143608i 0.989924 − 0.141597i \(-0.0452236\pi\)
−0.989718 + 0.143033i \(0.954315\pi\)
\(44\) 65.3639 − 26.2019i 0.223954 − 0.0897748i
\(45\) 35.1491 − 244.467i 0.116438 − 0.809845i
\(46\) −171.141 − 221.939i −0.548551 − 0.711374i
\(47\) 6.30007 − 5.15154i 0.0195523 − 0.0159879i −0.623186 − 0.782074i \(-0.714162\pi\)
0.642738 + 0.766086i \(0.277798\pi\)
\(48\) −21.7641 + 36.0917i −0.0654453 + 0.108529i
\(49\) 141.595 + 145.698i 0.412813 + 0.424775i
\(50\) −53.5062 + 69.3881i −0.151338 + 0.196259i
\(51\) 1.44852 + 50.7048i 0.00397712 + 0.139217i
\(52\) −88.6584 + 37.4674i −0.236437 + 0.0999190i
\(53\) 309.208 − 71.9033i 0.801378 − 0.186352i 0.194245 − 0.980953i \(-0.437774\pi\)
0.607133 + 0.794600i \(0.292320\pi\)
\(54\) −103.491 + 66.5099i −0.260804 + 0.167608i
\(55\) 286.730 + 191.894i 0.702958 + 0.470454i
\(56\) −480.989 − 309.113i −1.14777 − 0.737624i
\(57\) 133.878 + 47.7677i 0.311099 + 0.111000i
\(58\) −150.975 − 250.364i −0.341793 − 0.566800i
\(59\) 66.3651 − 252.479i 0.146441 − 0.557118i −0.852943 − 0.522005i \(-0.825184\pi\)
0.999383 − 0.0351135i \(-0.0111793\pi\)
\(60\) 17.1323 − 0.979660i 0.0368628 − 0.00210789i
\(61\) −621.174 + 35.5200i −1.30382 + 0.0745553i −0.695149 − 0.718866i \(-0.744662\pi\)
−0.608673 + 0.793421i \(0.708298\pi\)
\(62\) 180.259 − 685.778i 0.369241 − 1.40474i
\(63\) −315.179 − 522.666i −0.630300 − 1.04523i
\(64\) 535.656 + 191.122i 1.04620 + 0.373284i
\(65\) −396.711 − 254.951i −0.757015 − 0.486504i
\(66\) −10.4684 − 83.8456i −0.0195238 − 0.156374i
\(67\) −71.8261 + 46.1598i −0.130969 + 0.0841690i −0.604485 − 0.796616i \(-0.706621\pi\)
0.473516 + 0.880785i \(0.342985\pi\)
\(68\) 101.446 − 23.5902i 0.180913 − 0.0420696i
\(69\) 98.5050 − 41.6286i 0.171864 − 0.0726303i
\(70\) −15.5489 − 544.283i −0.0265493 − 0.929347i
\(71\) 394.259 − 511.284i 0.659013 − 0.854624i −0.337212 − 0.941429i \(-0.609484\pi\)
0.996225 + 0.0868050i \(0.0276657\pi\)
\(72\) 445.297 + 458.200i 0.728872 + 0.749992i
\(73\) −603.113 + 1000.15i −0.966973 + 1.60355i −0.182664 + 0.983175i \(0.558472\pi\)
−0.784310 + 0.620370i \(0.786983\pi\)
\(74\) 264.990 − 216.681i 0.416276 − 0.340387i
\(75\) −20.4165 − 26.4766i −0.0314333 − 0.0407634i
\(76\) 41.5360 − 288.889i 0.0626909 − 0.436025i
\(77\) 848.716 − 81.4285i 1.25611 − 0.120515i
\(78\) 16.4360 + 114.315i 0.0238591 + 0.165943i
\(79\) −678.144 + 697.794i −0.965787 + 0.993771i −0.999984 − 0.00564404i \(-0.998203\pi\)
0.0341973 + 0.999415i \(0.489113\pi\)
\(80\) 107.784 + 410.051i 0.150632 + 0.573064i
\(81\) 203.394 + 625.983i 0.279004 + 0.858687i
\(82\) 189.189 + 933.683i 0.254785 + 1.25742i
\(83\) −42.5526 + 495.431i −0.0562742 + 0.655188i 0.912798 + 0.408411i \(0.133917\pi\)
−0.969072 + 0.246777i \(0.920628\pi\)
\(84\) 31.2427 − 28.6743i 0.0405817 − 0.0372455i
\(85\) 375.950 + 345.044i 0.479735 + 0.440297i
\(86\) −0.0862496 − 1.00419i −0.000108146 − 0.00125912i
\(87\) 107.039 − 31.4294i 0.131905 − 0.0387308i
\(88\) −846.077 + 284.264i −1.02491 + 0.344348i
\(89\) −304.099 − 89.2914i −0.362184 − 0.106347i 0.0955770 − 0.995422i \(-0.469530\pi\)
−0.457761 + 0.889075i \(0.651349\pi\)
\(90\) −120.839 + 596.366i −0.141529 + 0.698472i
\(91\) −1157.76 + 132.840i −1.33369 + 0.153027i
\(92\) −123.938 − 181.253i −0.140450 − 0.205402i
\(93\) 235.592 + 133.045i 0.262685 + 0.148345i
\(94\) −16.2207 + 11.7850i −0.0177982 + 0.0129312i
\(95\) 1283.23 − 630.923i 1.38586 − 0.681383i
\(96\) −45.2443 + 66.1676i −0.0481013 + 0.0703459i
\(97\) 319.663 − 605.829i 0.334607 − 0.634151i −0.658339 − 0.752722i \(-0.728740\pi\)
0.992946 + 0.118571i \(0.0378314\pi\)
\(98\) −327.785 − 378.284i −0.337870 − 0.389923i
\(99\) −944.403 − 126.187i −0.958748 − 0.128104i
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.4.11 ✓ 1280
121.91 even 55 inner 121.4.g.a.91.11 yes 1280
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
121.4.g.a.4.11 ✓ 1280 1.1 even 1 trivial
121.4.g.a.91.11 yes 1280 121.91 even 55 inner