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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 91.10
Character \(\chi\) \(=\) 121.91
Dual form 121.4.g.a.4.10

$q$-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+(-2.47271 + 0.141395i) q^{2} +(-1.86923 - 5.75291i) q^{3} +(-1.85357 + 0.212677i) q^{4} +(5.25998 - 9.96877i) q^{5} +(5.43550 + 13.9610i) q^{6} +(20.5753 - 8.69518i) q^{7} +(24.0770 - 4.16669i) q^{8} +(-7.75847 + 5.63686i) q^{9} +(-11.5969 + 25.3936i) q^{10} +(-13.7221 - 33.8039i) q^{11} +(4.68826 + 10.2659i) q^{12} +(-14.8919 - 8.40982i) q^{13} +(-49.6472 + 24.4099i) q^{14} +(-67.1816 - 11.6262i) q^{15} +(-44.4084 + 10.3267i) q^{16} +(-21.3129 - 7.60443i) q^{17} +(18.3874 - 15.0353i) q^{18} +(0.515755 + 18.0538i) q^{19} +(-7.62959 + 19.5964i) q^{20} +(-88.4826 - 102.114i) q^{21} +(38.7103 + 81.6470i) q^{22} +(-63.4542 + 73.2300i) q^{23} +(-68.9762 - 130.724i) q^{24} +(-1.15357 - 1.68705i) q^{25} +(38.0123 + 18.6894i) q^{26} +(-85.1996 - 61.9012i) q^{27} +(-36.2883 + 20.4930i) q^{28} +(109.000 - 159.407i) q^{29} +(167.764 + 19.2491i) q^{30} +(10.3946 + 51.2994i) q^{31} +(-79.2122 + 23.2588i) q^{32} +(-168.821 + 142.129i) q^{33} +(53.7758 + 15.7900i) q^{34} +(21.5452 - 250.847i) q^{35} +(13.1820 - 12.0983i) q^{36} +(-127.153 - 116.700i) q^{37} +(-3.82802 - 44.5688i) q^{38} +(-20.5446 + 101.391i) q^{39} +(85.1079 - 261.935i) q^{40} +(18.9616 - 72.1373i) q^{41} +(233.230 + 239.988i) q^{42} +(-33.8365 + 235.338i) q^{43} +(32.6240 + 59.7394i) q^{44} +(15.3831 + 106.992i) q^{45} +(146.549 - 190.048i) q^{46} +(416.160 + 340.292i) q^{47} +(142.418 + 236.174i) q^{48} +(108.686 - 111.835i) q^{49} +(3.09099 + 4.00847i) q^{50} +(-3.90879 + 136.826i) q^{51} +(29.3916 + 12.4210i) q^{52} +(220.489 + 51.2725i) q^{53} +(219.426 + 141.017i) q^{54} +(-409.161 - 41.0159i) q^{55} +(459.161 - 295.085i) q^{56} +(102.898 - 36.7138i) q^{57} +(-246.985 + 409.579i) q^{58} +(-19.2978 - 73.4163i) q^{59} +(126.998 + 7.26201i) q^{60} +(-481.329 - 27.5234i) q^{61} +(-32.9563 - 125.379i) q^{62} +(-110.619 + 183.441i) q^{63} +(536.114 - 191.285i) q^{64} +(-162.166 + 104.218i) q^{65} +(397.349 - 375.314i) q^{66} +(-440.588 - 283.149i) q^{67} +(41.1221 + 9.56254i) q^{68} +(539.896 + 228.162i) q^{69} +(-17.8067 + 623.316i) q^{70} +(200.921 + 260.559i) q^{71} +(-163.314 + 168.046i) q^{72} +(237.375 + 393.642i) q^{73} +(330.913 + 270.586i) q^{74} +(-7.54913 + 9.78989i) q^{75} +(-4.79561 - 33.3542i) q^{76} +(-576.266 - 576.209i) q^{77} +(36.4645 - 253.616i) q^{78} +(-330.434 - 340.008i) q^{79} +(-130.642 + 497.015i) q^{80} +(-276.867 + 852.110i) q^{81} +(-36.6866 + 181.055i) q^{82} +(-115.533 - 1345.12i) q^{83} +(185.726 + 170.457i) q^{84} +(-187.912 + 172.464i) q^{85} +(50.3922 - 586.706i) q^{86} +(-1120.80 - 329.097i) q^{87} +(-471.237 - 756.722i) q^{88} +(1102.81 - 323.815i) q^{89} +(-53.1661 - 262.385i) q^{90} +(-379.529 - 43.5469i) q^{91} +(102.042 - 149.232i) q^{92} +(275.691 - 155.690i) q^{93} +(-1077.16 - 782.600i) q^{94} +(182.687 + 89.8210i) q^{95} +(281.872 + 412.224i) q^{96} +(-800.798 - 1517.68i) q^{97} +(-252.936 + 291.903i) q^{98} +(297.010 + 184.917i) 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 \(\left(\mathbb{Z}/121\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(e\left(\frac{54}{55}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.47271 + 0.141395i −0.874234 + 0.0499905i −0.488443 − 0.872596i \(-0.662435\pi\)
−0.385791 + 0.922586i \(0.626071\pi\)
\(3\) −1.86923 − 5.75291i −0.359734 − 1.10715i −0.953213 − 0.302299i \(-0.902246\pi\)
0.593479 − 0.804849i \(-0.297754\pi\)
\(4\) −1.85357 + 0.212677i −0.231696 + 0.0265846i
\(5\) 5.25998 − 9.96877i 0.470467 − 0.891634i −0.528618 − 0.848860i \(-0.677290\pi\)
0.999085 − 0.0427739i \(-0.0136195\pi\)
\(6\) 5.43550 + 13.9610i 0.369839 + 0.949923i
\(7\) 20.5753 − 8.69518i 1.11096 − 0.469496i 0.244957 − 0.969534i \(-0.421226\pi\)
0.866003 + 0.500038i \(0.166681\pi\)
\(8\) 24.0770 − 4.16669i 1.06406 − 0.184144i
\(9\) −7.75847 + 5.63686i −0.287351 + 0.208773i
\(10\) −11.5969 + 25.3936i −0.366725 + 0.803015i
\(11\) −13.7221 − 33.8039i −0.376123 − 0.926570i
\(12\) 4.68826 + 10.2659i 0.112782 + 0.246958i
\(13\) −14.8919 − 8.40982i −0.317712 − 0.179420i 0.324624 − 0.945843i \(-0.394762\pi\)
−0.642337 + 0.766423i \(0.722035\pi\)
\(14\) −49.6472 + 24.4099i −0.947769 + 0.465987i
\(15\) −67.1816 − 11.6262i −1.15641 − 0.200125i
\(16\) −44.4084 + 10.3267i −0.693881 + 0.161355i
\(17\) −21.3129 − 7.60443i −0.304067 − 0.108491i 0.179609 − 0.983738i \(-0.442517\pi\)
−0.483676 + 0.875247i \(0.660699\pi\)
\(18\) 18.3874 − 15.0353i 0.240775 − 0.196881i
\(19\) 0.515755 + 18.0538i 0.00622749 + 0.217991i 0.996746 + 0.0806095i \(0.0256867\pi\)
−0.990518 + 0.137381i \(0.956132\pi\)
\(20\) −7.62959 + 19.5964i −0.0853014 + 0.219095i
\(21\) −88.4826 − 102.114i −0.919451 − 1.06110i
\(22\) 38.7103 + 81.6470i 0.375139 + 0.791236i
\(23\) −63.4542 + 73.2300i −0.575266 + 0.663892i −0.966580 − 0.256365i \(-0.917475\pi\)
0.391314 + 0.920257i \(0.372020\pi\)
\(24\) −68.9762 − 130.724i −0.586654 − 1.11183i
\(25\) −1.15357 − 1.68705i −0.00922859 − 0.0134964i
\(26\) 38.0123 + 18.6894i 0.286724 + 0.140973i
\(27\) −85.1996 − 61.9012i −0.607284 − 0.441218i
\(28\) −36.2883 + 20.4930i −0.244923 + 0.138315i
\(29\) 109.000 − 159.407i 0.697957 − 1.02073i −0.299813 − 0.953998i \(-0.596924\pi\)
0.997771 − 0.0667324i \(-0.0212574\pi\)
\(30\) 167.764 + 19.2491i 1.02098 + 0.117147i
\(31\) 10.3946 + 51.2994i 0.0602234 + 0.297214i 0.998862 − 0.0476872i \(-0.0151851\pi\)
−0.938639 + 0.344901i \(0.887912\pi\)
\(32\) −79.2122 + 23.2588i −0.437590 + 0.128488i
\(33\) −168.821 + 142.129i −0.890545 + 0.749743i
\(34\) 53.7758 + 15.7900i 0.271249 + 0.0796459i
\(35\) 21.5452 − 250.847i 0.104052 − 1.21145i
\(36\) 13.1820 − 12.0983i 0.0610278 − 0.0560108i
\(37\) −127.153 − 116.700i −0.564968 − 0.518523i 0.342013 − 0.939695i \(-0.388891\pi\)
−0.906981 + 0.421172i \(0.861619\pi\)
\(38\) −3.82802 − 44.5688i −0.0163417 − 0.190263i
\(39\) −20.5446 + 101.391i −0.0843529 + 0.416298i
\(40\) 85.1079 − 261.935i 0.336418 − 1.03539i
\(41\) 18.9616 − 72.1373i 0.0722268 − 0.274779i −0.921225 − 0.389030i \(-0.872810\pi\)
0.993452 + 0.114251i \(0.0364467\pi\)
\(42\) 233.230 + 239.988i 0.856861 + 0.881689i
\(43\) −33.8365 + 235.338i −0.120000 + 0.834620i 0.837551 + 0.546359i \(0.183986\pi\)
−0.957552 + 0.288262i \(0.906923\pi\)
\(44\) 32.6240 + 59.7394i 0.111779 + 0.204683i
\(45\) 15.3831 + 106.992i 0.0509597 + 0.354432i
\(46\) 146.549 − 190.048i 0.469728 − 0.609155i
\(47\) 416.160 + 340.292i 1.29156 + 1.05610i 0.994874 + 0.101125i \(0.0322441\pi\)
0.296683 + 0.954976i \(0.404120\pi\)
\(48\) 142.418 + 236.174i 0.428257 + 0.710184i
\(49\) 108.686 − 111.835i 0.316869 − 0.326050i
\(50\) 3.09099 + 4.00847i 0.00874264 + 0.0113377i
\(51\) −3.90879 + 136.826i −0.0107322 + 0.375675i
\(52\) 29.3916 + 12.4210i 0.0783824 + 0.0331247i
\(53\) 220.489 + 51.2725i 0.571443 + 0.132883i 0.502254 − 0.864720i \(-0.332504\pi\)
0.0691891 + 0.997604i \(0.477959\pi\)
\(54\) 219.426 + 141.017i 0.552965 + 0.355369i
\(55\) −409.161 − 41.0159i −1.00311 − 0.100556i
\(56\) 459.161 − 295.085i 1.09568 − 0.704150i
\(57\) 102.898 − 36.7138i 0.239108 − 0.0853134i
\(58\) −246.985 + 409.579i −0.559151 + 0.927248i
\(59\) −19.2978 − 73.4163i −0.0425823 − 0.162000i 0.942458 − 0.334325i \(-0.108508\pi\)
−0.985040 + 0.172325i \(0.944872\pi\)
\(60\) 126.998 + 7.26201i 0.273256 + 0.0156254i
\(61\) −481.329 − 27.5234i −1.01029 − 0.0577706i −0.455904 − 0.890029i \(-0.650684\pi\)
−0.554388 + 0.832258i \(0.687048\pi\)
\(62\) −32.9563 − 125.379i −0.0675072 − 0.256824i
\(63\) −110.619 + 183.441i −0.221217 + 0.366848i
\(64\) 536.114 − 191.285i 1.04710 − 0.373604i
\(65\) −162.166 + 104.218i −0.309450 + 0.198872i
\(66\) 397.349 − 375.314i 0.741065 − 0.699969i
\(67\) −440.588 − 283.149i −0.803379 − 0.516300i 0.0733379 − 0.997307i \(-0.476635\pi\)
−0.876717 + 0.481007i \(0.840271\pi\)
\(68\) 41.1221 + 9.56254i 0.0733352 + 0.0170534i
\(69\) 539.896 + 228.162i 0.941969 + 0.398079i
\(70\) −17.8067 + 623.316i −0.0304044 + 1.06429i
\(71\) 200.921 + 260.559i 0.335844 + 0.435531i 0.929294 − 0.369340i \(-0.120416\pi\)
−0.593450 + 0.804871i \(0.702234\pi\)
\(72\) −163.314 + 168.046i −0.267316 + 0.275061i
\(73\) 237.375 + 393.642i 0.380584 + 0.631127i 0.986378 − 0.164494i \(-0.0525991\pi\)
−0.605794 + 0.795621i \(0.707145\pi\)
\(74\) 330.913 + 270.586i 0.519836 + 0.425067i
\(75\) −7.54913 + 9.78989i −0.0116226 + 0.0150725i
\(76\) −4.79561 − 33.3542i −0.00723807 − 0.0503419i
\(77\) −576.266 − 576.209i −0.852878 − 0.852794i
\(78\) 36.4645 − 253.616i 0.0529332 − 0.368159i
\(79\) −330.434 − 340.008i −0.470591 − 0.484227i 0.439730 − 0.898130i \(-0.355074\pi\)
−0.910321 + 0.413904i \(0.864165\pi\)
\(80\) −130.642 + 497.015i −0.182578 + 0.694600i
\(81\) −276.867 + 852.110i −0.379791 + 1.16888i
\(82\) −36.6866 + 181.055i −0.0494068 + 0.243832i
\(83\) −115.533 − 1345.12i −0.152787 − 1.77887i −0.525372 − 0.850872i \(-0.676074\pi\)
0.372585 − 0.927998i \(-0.378471\pi\)
\(84\) 185.726 + 170.457i 0.241242 + 0.221410i
\(85\) −187.912 + 172.464i −0.239788 + 0.220075i
\(86\) 50.3922 − 586.706i 0.0631852 − 0.735652i
\(87\) −1120.80 − 329.097i −1.38118 − 0.405551i
\(88\) −471.237 − 756.722i −0.570841 − 0.916669i
\(89\) 1102.81 − 323.815i 1.31346 − 0.385666i 0.451331 − 0.892357i \(-0.350949\pi\)
0.862128 + 0.506690i \(0.169131\pi\)
\(90\) −53.1661 − 262.385i −0.0622689 − 0.307309i
\(91\) −379.529 − 43.5469i −0.437203 − 0.0501643i
\(92\) 102.042 − 149.232i 0.115637 − 0.169114i
\(93\) 275.691 − 155.690i 0.307396 − 0.173594i
\(94\) −1077.16 − 782.600i −1.18192 − 0.858714i
\(95\) 182.687 + 89.8210i 0.197298 + 0.0970047i
\(96\) 281.872 + 412.224i 0.299671 + 0.438255i
\(97\) −800.798 − 1517.68i −0.838234 − 1.58863i −0.808071 − 0.589086i \(-0.799488\pi\)
−0.0301638 − 0.999545i \(-0.509603\pi\)
\(98\) −252.936 + 291.903i −0.260718 + 0.300884i
\(99\) 297.010 + 184.917i 0.301522 + 0.187726i
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.91.10 yes 1280
121.4 even 55 inner 121.4.g.a.4.10 ✓ 1280
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
121.4.g.a.4.10 ✓ 1280 121.4 even 55 inner
121.4.g.a.91.10 yes 1280 1.1 even 1 trivial