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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 4.3
Character \(\chi\) \(=\) 121.4
Dual form 121.4.g.a.91.3

$q$-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+(-4.90410 - 0.280426i) q^{2} +(-3.02997 + 9.32529i) q^{3} +(16.0237 + 1.83855i) q^{4} +(-1.45061 - 2.74922i) q^{5} +(17.4743 - 44.8825i) q^{6} +(1.17065 + 0.494721i) q^{7} +(-39.3447 - 6.80888i) q^{8} +(-55.9369 - 40.6405i) q^{9} +(6.34300 + 13.8892i) q^{10} +(-34.3053 - 12.4157i) q^{11} +(-65.6962 + 143.855i) q^{12} +(48.6346 - 27.4652i) q^{13} +(-5.60225 - 2.75444i) q^{14} +(30.0326 - 5.19734i) q^{15} +(65.3639 + 15.1997i) q^{16} +(13.2680 - 4.73403i) q^{17} +(262.923 + 214.991i) q^{18} +(0.624842 - 21.8723i) q^{19} +(-18.1896 - 46.7196i) q^{20} +(-8.16046 + 9.41767i) q^{21} +(164.755 + 70.5079i) q^{22} +(-63.4233 - 73.1944i) q^{23} +(182.708 - 346.271i) q^{24} +(65.1015 - 95.2079i) q^{25} +(-246.211 + 121.054i) q^{26} +(334.293 - 242.878i) q^{27} +(17.8486 + 10.0795i) q^{28} +(83.2118 + 121.693i) q^{29} +(-148.740 + 17.0663i) q^{30} +(-64.1148 + 316.419i) q^{31} +(-9.79132 - 2.87499i) q^{32} +(219.724 - 282.287i) q^{33} +(-66.3953 + 19.4954i) q^{34} +(-0.338066 - 3.93603i) q^{35} +(-821.595 - 754.053i) q^{36} +(267.633 - 245.631i) q^{37} +(-9.19786 + 107.089i) q^{38} +(108.760 + 536.751i) q^{39} +(38.3549 + 118.044i) q^{40} +(-65.8592 - 250.554i) q^{41} +(42.6606 - 43.8968i) q^{42} +(57.5736 + 400.433i) q^{43} +(-526.869 - 262.017i) q^{44} +(-30.5869 + 212.737i) q^{45} +(290.508 + 376.738i) q^{46} +(-71.2365 + 58.2498i) q^{47} +(-339.793 + 563.483i) q^{48} +(-237.924 - 244.818i) q^{49} +(-345.963 + 448.652i) q^{50} +(3.94441 + 138.072i) q^{51} +(829.802 - 350.677i) q^{52} +(-374.521 + 87.0912i) q^{53} +(-1707.51 + 1097.35i) q^{54} +(15.6302 + 112.323i) q^{55} +(-42.6905 - 27.4355i) q^{56} +(202.073 + 72.0993i) q^{57} +(-373.953 - 620.131i) q^{58} +(104.394 - 397.155i) q^{59} +(490.788 - 28.0643i) q^{60} +(167.669 - 9.58765i) q^{61} +(403.158 - 1533.77i) q^{62} +(-45.3769 - 75.2491i) q^{63} +(-458.430 - 163.568i) q^{64} +(-146.058 - 93.8658i) q^{65} +(-1156.71 + 1322.75i) q^{66} +(208.146 - 133.767i) q^{67} +(221.306 - 51.4626i) q^{68} +(874.729 - 369.664i) q^{69} +(0.554141 + 19.3975i) q^{70} +(492.305 - 638.433i) q^{71} +(1924.11 + 1979.86i) q^{72} +(411.970 - 683.176i) q^{73} +(-1381.38 + 1129.55i) q^{74} +(690.586 + 895.567i) q^{75} +(50.2255 - 349.326i) q^{76} +(-34.0172 - 31.5060i) q^{77} +(-382.850 - 2662.78i) q^{78} +(276.364 - 284.372i) q^{79} +(-53.0304 - 201.749i) q^{80} +(675.128 + 2077.83i) q^{81} +(252.718 + 1247.21i) q^{82} +(62.3873 - 726.362i) q^{83} +(-148.075 + 135.902i) q^{84} +(-32.2617 - 29.6095i) q^{85} +(-170.054 - 1979.91i) q^{86} +(-1386.96 + 407.247i) q^{87} +(1265.19 + 722.073i) q^{88} +(-969.120 - 284.559i) q^{89} +(209.658 - 1034.70i) q^{90} +(70.5218 - 8.09163i) q^{91} +(-881.703 - 1289.45i) q^{92} +(-2756.43 - 1556.63i) q^{93} +(365.685 - 265.686i) q^{94} +(-61.0382 + 30.0105i) q^{95} +(56.4775 - 82.5958i) q^{96} +(-129.041 + 244.559i) q^{97} +(1098.15 + 1267.33i) q^{98} +(1414.35 + 2088.68i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −4.90410 − 0.280426i −1.73386 − 0.0991457i −0.838616 − 0.544723i \(-0.816635\pi\)
−0.895244 + 0.445577i \(0.852999\pi\)
\(3\) −3.02997 + 9.32529i −0.583118 + 1.79465i 0.0235809 + 0.999722i \(0.492493\pi\)
−0.606699 + 0.794932i \(0.707507\pi\)
\(4\) 16.0237 + 1.83855i 2.00296 + 0.229818i
\(5\) −1.45061 − 2.74922i −0.129747 − 0.245898i 0.810972 − 0.585085i \(-0.198939\pi\)
−0.940719 + 0.339187i \(0.889848\pi\)
\(6\) 17.4743 − 44.8825i 1.18898 − 3.05386i
\(7\) 1.17065 + 0.494721i 0.0632092 + 0.0267124i 0.420646 − 0.907225i \(-0.361804\pi\)
−0.357436 + 0.933938i \(0.616349\pi\)
\(8\) −39.3447 − 6.80888i −1.73881 − 0.300913i
\(9\) −55.9369 − 40.6405i −2.07174 − 1.50521i
\(10\) 6.34300 + 13.8892i 0.200583 + 0.439216i
\(11\) −34.3053 − 12.4157i −0.940311 − 0.340316i
\(12\) −65.6962 + 143.855i −1.58041 + 3.46061i
\(13\) 48.6346 − 27.4652i 1.03760 − 0.585961i 0.123878 − 0.992297i \(-0.460467\pi\)
0.913723 + 0.406337i \(0.133194\pi\)
\(14\) −5.60225 − 2.75444i −0.106948 − 0.0525826i
\(15\) 30.0326 − 5.19734i 0.516959 − 0.0894632i
\(16\) 65.3639 + 15.1997i 1.02131 + 0.237496i
\(17\) 13.2680 − 4.73403i 0.189292 − 0.0675394i −0.239702 − 0.970846i \(-0.577050\pi\)
0.428995 + 0.903307i \(0.358868\pi\)
\(18\) 262.923 + 214.991i 3.44287 + 2.81522i
\(19\) 0.624842 − 21.8723i 0.00754466 − 0.264098i −0.987059 − 0.160355i \(-0.948736\pi\)
0.994604 − 0.103743i \(-0.0330820\pi\)
\(20\) −18.1896 − 46.7196i −0.203366 − 0.522341i
\(21\) −8.16046 + 9.41767i −0.0847980 + 0.0978621i
\(22\) 164.755 + 70.5079i 1.59663 + 0.683288i
\(23\) −63.4233 − 73.1944i −0.574985 − 0.663569i 0.391533 − 0.920164i \(-0.371945\pi\)
−0.966519 + 0.256595i \(0.917399\pi\)
\(24\) 182.708 − 346.271i 1.55396 − 2.94509i
\(25\) 65.1015 − 95.2079i 0.520812 − 0.761663i
\(26\) −246.211 + 121.054i −1.85715 + 0.913100i
\(27\) 334.293 − 242.878i 2.38277 − 1.73118i
\(28\) 17.8486 + 10.0795i 0.120466 + 0.0680305i
\(29\) 83.2118 + 121.693i 0.532829 + 0.779238i 0.994151 − 0.107995i \(-0.0344430\pi\)
−0.461322 + 0.887233i \(0.652625\pi\)
\(30\) −148.740 + 17.0663i −0.905204 + 0.103862i
\(31\) −64.1148 + 316.419i −0.371463 + 1.83324i 0.161207 + 0.986921i \(0.448461\pi\)
−0.532670 + 0.846323i \(0.678811\pi\)
\(32\) −9.79132 − 2.87499i −0.0540899 − 0.0158822i
\(33\) 219.724 − 282.287i 1.15906 − 1.48909i
\(34\) −66.3953 + 19.4954i −0.334903 + 0.0983363i
\(35\) −0.338066 − 3.93603i −0.00163267 − 0.0190089i
\(36\) −821.595 − 754.053i −3.80368 − 3.49099i
\(37\) 267.633 − 245.631i 1.18915 − 1.09139i 0.195291 − 0.980745i \(-0.437435\pi\)
0.993860 − 0.110647i \(-0.0352923\pi\)
\(38\) −9.19786 + 107.089i −0.0392655 + 0.457160i
\(39\) 108.760 + 536.751i 0.446552 + 2.20382i
\(40\) 38.3549 + 118.044i 0.151611 + 0.466611i
\(41\) −65.8592 − 250.554i −0.250865 − 0.954390i −0.966657 − 0.256076i \(-0.917570\pi\)
0.715791 − 0.698314i \(-0.246066\pi\)
\(42\) 42.6606 − 43.8968i 0.156731 − 0.161272i
\(43\) 57.5736 + 400.433i 0.204184 + 1.42013i 0.791697 + 0.610914i \(0.209198\pi\)
−0.587514 + 0.809214i \(0.699893\pi\)
\(44\) −526.869 − 262.017i −1.80519 − 0.897739i
\(45\) −30.5869 + 212.737i −0.101325 + 0.704731i
\(46\) 290.508 + 376.738i 0.931154 + 1.20754i
\(47\) −71.2365 + 58.2498i −0.221083 + 0.180779i −0.737067 − 0.675819i \(-0.763790\pi\)
0.515984 + 0.856598i \(0.327426\pi\)
\(48\) −339.793 + 563.483i −1.02177 + 1.69441i
\(49\) −237.924 − 244.818i −0.693656 − 0.713755i
\(50\) −345.963 + 448.652i −0.978530 + 1.26898i
\(51\) 3.94441 + 138.072i 0.0108300 + 0.379098i
\(52\) 829.802 − 350.677i 2.21294 − 0.935196i
\(53\) −374.521 + 87.0912i −0.970651 + 0.225715i −0.681702 − 0.731630i \(-0.738760\pi\)
−0.288948 + 0.957345i \(0.593306\pi\)
\(54\) −1707.51 + 1097.35i −4.30302 + 2.76538i
\(55\) 15.6302 + 112.323i 0.0383196 + 0.275375i
\(56\) −42.6905 − 27.4355i −0.101871 − 0.0654683i
\(57\) 202.073 + 72.0993i 0.469564 + 0.167540i
\(58\) −373.953 − 620.131i −0.846593 − 1.40392i
\(59\) 104.394 − 397.155i 0.230354 − 0.876359i −0.747013 − 0.664810i \(-0.768513\pi\)
0.977367 − 0.211550i \(-0.0678510\pi\)
\(60\) 490.788 − 28.0643i 1.05601 − 0.0603847i
\(61\) 167.669 − 9.58765i 0.351931 − 0.0201242i 0.119770 − 0.992802i \(-0.461784\pi\)
0.232161 + 0.972677i \(0.425420\pi\)
\(62\) 403.158 − 1533.77i 0.825824 − 3.14176i
\(63\) −45.3769 − 75.2491i −0.0907452 − 0.150484i
\(64\) −458.430 − 163.568i −0.895371 − 0.319468i
\(65\) −146.058 − 93.8658i −0.278712 − 0.179117i
\(66\) −1156.71 + 1322.75i −2.15729 + 2.46695i
\(67\) 208.146 − 133.767i 0.379539 − 0.243915i −0.336936 − 0.941527i \(-0.609391\pi\)
0.716475 + 0.697613i \(0.245754\pi\)
\(68\) 221.306 − 51.4626i 0.394667 − 0.0917758i
\(69\) 874.729 − 369.664i 1.52616 − 0.644961i
\(70\) 0.554141 + 19.3975i 0.000946178 + 0.0331206i
\(71\) 492.305 − 638.433i 0.822900 − 1.06716i −0.173703 − 0.984798i \(-0.555573\pi\)
0.996603 − 0.0823574i \(-0.0262449\pi\)
\(72\) 1924.11 + 1979.86i 3.14942 + 3.24068i
\(73\) 411.970 − 683.176i 0.660513 − 1.09534i −0.328068 − 0.944654i \(-0.606398\pi\)
0.988582 − 0.150685i \(-0.0481478\pi\)
\(74\) −1381.38 + 1129.55i −2.17003 + 1.77442i
\(75\) 690.586 + 895.567i 1.06323 + 1.37882i
\(76\) 50.2255 − 349.326i 0.0758061 − 0.527243i
\(77\) −34.0172 − 31.5060i −0.0503457 − 0.0466291i
\(78\) −382.850 − 2662.78i −0.555759 − 3.86539i
\(79\) 276.364 − 284.372i 0.393587 − 0.404992i −0.491155 − 0.871072i \(-0.663425\pi\)
0.884742 + 0.466081i \(0.154334\pi\)
\(80\) −53.0304 − 201.749i −0.0741122 − 0.281952i
\(81\) 675.128 + 2077.83i 0.926102 + 2.85025i
\(82\) 252.718 + 1247.21i 0.340342 + 1.67965i
\(83\) 62.3873 − 726.362i 0.0825048 − 0.960586i −0.831671 − 0.555268i \(-0.812616\pi\)
0.914176 − 0.405318i \(-0.132839\pi\)
\(84\) −148.075 + 135.902i −0.192337 + 0.176526i
\(85\) −32.2617 − 29.6095i −0.0411679 − 0.0377835i
\(86\) −170.054 − 1979.91i −0.213226 − 2.48255i
\(87\) −1386.96 + 407.247i −1.70916 + 0.501856i
\(88\) 1265.19 + 722.073i 1.53262 + 0.874696i
\(89\) −969.120 − 284.559i −1.15423 − 0.338913i −0.352042 − 0.935984i \(-0.614513\pi\)
−0.802188 + 0.597072i \(0.796331\pi\)
\(90\) 209.658 − 1034.70i 0.245554 − 1.21186i
\(91\) 70.5218 − 8.09163i 0.0812384 − 0.00932124i
\(92\) −881.703 − 1289.45i −0.999172 − 1.46124i
\(93\) −2756.43 − 1556.63i −3.07343 − 1.73565i
\(94\) 365.685 − 265.686i 0.401251 − 0.291526i
\(95\) −61.0382 + 30.0105i −0.0659199 + 0.0324106i
\(96\) 56.4775 − 82.5958i 0.0600439 − 0.0878114i
\(97\) −129.041 + 244.559i −0.135073 + 0.255992i −0.942659 − 0.333758i \(-0.891683\pi\)
0.807586 + 0.589750i \(0.200774\pi\)
\(98\) 1098.15 + 1267.33i 1.13194 + 1.30632i
\(99\) 1414.35 + 2088.68i 1.43583 + 2.12041i
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.4.3 ✓ 1280
121.91 even 55 inner 121.4.g.a.91.3 yes 1280
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
121.4.g.a.4.3 ✓ 1280 1.1 even 1 trivial
121.4.g.a.91.3 yes 1280 121.91 even 55 inner