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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 91.7
Character \(\chi\) \(=\) 121.91
Dual form 121.4.g.a.4.7

$q$-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+(-3.63025 + 0.207585i) q^{2} +(-1.34595 - 4.14240i) q^{3} +(5.18774 - 0.595238i) q^{4} +(-8.43183 + 15.9801i) q^{5} +(5.74602 + 14.7585i) q^{6} +(11.1935 - 4.73039i) q^{7} +(9.95414 - 1.72263i) q^{8} +(6.49556 - 4.71930i) q^{9} +(27.2924 - 59.7620i) q^{10} +(23.4307 + 27.9643i) q^{11} +(-9.44815 - 20.6886i) q^{12} +(-55.7872 - 31.5045i) q^{13} +(-39.6531 + 19.4961i) q^{14} +(77.5448 + 13.4196i) q^{15} +(-76.4670 + 17.7816i) q^{16} +(-72.3898 - 25.8286i) q^{17} +(-22.6008 + 18.4806i) q^{18} +(0.259590 + 9.08683i) q^{19} +(-34.2302 + 87.9196i) q^{20} +(-34.6610 - 40.0009i) q^{21} +(-90.8643 - 96.6533i) q^{22} +(13.1250 - 15.1471i) q^{23} +(-20.5336 - 38.9154i) q^{24} +(-113.712 - 166.299i) q^{25} +(209.061 + 102.788i) q^{26} +(-123.433 - 89.6792i) q^{27} +(55.2531 - 31.2029i) q^{28} +(97.9186 - 143.201i) q^{29} +(-284.292 - 32.6195i) q^{30} +(-32.5145 - 160.466i) q^{31} +(196.360 - 57.6564i) q^{32} +(84.3027 - 134.698i) q^{33} +(268.154 + 78.7373i) q^{34} +(-18.7892 + 218.758i) q^{35} +(30.8882 - 28.3489i) q^{36} +(-109.442 - 100.445i) q^{37} +(-2.82866 - 32.9335i) q^{38} +(-55.4175 + 273.496i) q^{39} +(-56.4038 + 173.593i) q^{40} +(29.2065 - 111.113i) q^{41} +(134.132 + 138.018i) q^{42} +(23.2604 - 161.780i) q^{43} +(138.198 + 131.125i) q^{44} +(20.6454 + 143.592i) q^{45} +(-44.5027 + 57.7122i) q^{46} +(-113.491 - 92.8015i) q^{47} +(176.579 + 292.824i) q^{48} +(-136.133 + 140.077i) q^{49} +(447.325 + 580.101i) q^{50} +(-9.55965 + 334.631i) q^{51} +(-308.162 - 130.230i) q^{52} +(230.828 + 53.6769i) q^{53} +(466.707 + 299.935i) q^{54} +(-644.436 + 138.635i) q^{55} +(103.272 - 66.3692i) q^{56} +(37.2919 - 13.3057i) q^{57} +(-325.742 + 540.183i) q^{58} +(-84.7819 - 322.544i) q^{59} +(410.270 + 23.4601i) q^{60} +(518.696 + 29.6601i) q^{61} +(151.346 + 575.780i) q^{62} +(50.3836 - 83.5518i) q^{63} +(-109.333 + 39.0099i) q^{64} +(973.833 - 625.844i) q^{65} +(-278.078 + 506.487i) q^{66} +(-489.884 - 314.830i) q^{67} +(-390.914 - 90.9032i) q^{68} +(-80.4108 - 33.9819i) q^{69} +(22.7984 - 798.047i) q^{70} +(-618.716 - 802.365i) q^{71} +(56.5280 - 58.1660i) q^{72} +(-498.051 - 825.924i) q^{73} +(418.154 + 341.923i) q^{74} +(-535.826 + 694.872i) q^{75} +(6.75551 + 46.9856i) q^{76} +(394.553 + 202.180i) q^{77} +(144.406 - 1004.36i) q^{78} +(828.703 + 852.715i) q^{79} +(360.605 - 1371.88i) q^{80} +(-138.364 + 425.840i) q^{81} +(-82.9615 + 409.431i) q^{82} +(97.1921 + 1131.59i) q^{83} +(-203.622 - 186.883i) q^{84} +(1023.12 - 939.013i) q^{85} +(-50.8579 + 592.128i) q^{86} +(-724.991 - 212.876i) q^{87} +(281.405 + 237.998i) q^{88} +(-1006.12 + 295.424i) q^{89} +(-104.755 - 516.988i) q^{90} +(-773.480 - 88.7486i) q^{91} +(59.0731 - 86.3917i) q^{92} +(-620.950 + 350.666i) q^{93} +(431.266 + 313.333i) q^{94} +(-147.397 - 72.4703i) q^{95} +(-503.126 - 735.798i) q^{96} +(-492.663 - 933.700i) q^{97} +(465.117 - 536.774i) q^{98} +(284.167 + 71.0669i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −3.63025 + 0.207585i −1.28349 + 0.0733924i −0.685475 − 0.728096i \(-0.740406\pi\)
−0.598011 + 0.801488i \(0.704042\pi\)
\(3\) −1.34595 − 4.14240i −0.259028 − 0.797205i −0.993009 − 0.118036i \(-0.962340\pi\)
0.733982 − 0.679169i \(-0.237660\pi\)
\(4\) 5.18774 − 0.595238i 0.648468 − 0.0744048i
\(5\) −8.43183 + 15.9801i −0.754166 + 1.42930i 0.143968 + 0.989582i \(0.454014\pi\)
−0.898134 + 0.439721i \(0.855077\pi\)
\(6\) 5.74602 + 14.7585i 0.390967 + 1.00419i
\(7\) 11.1935 − 4.73039i 0.604390 − 0.255417i −0.0655301 − 0.997851i \(-0.520874\pi\)
0.669920 + 0.742433i \(0.266328\pi\)
\(8\) 9.95414 − 1.72263i 0.439915 − 0.0761303i
\(9\) 6.49556 − 4.71930i 0.240576 − 0.174789i
\(10\) 27.2924 − 59.7620i 0.863061 − 1.88984i
\(11\) 23.4307 + 27.9643i 0.642239 + 0.766504i
\(12\) −9.44815 − 20.6886i −0.227287 − 0.497689i
\(13\) −55.7872 − 31.5045i −1.19020 − 0.672136i −0.236129 − 0.971722i \(-0.575879\pi\)
−0.954069 + 0.299586i \(0.903152\pi\)
\(14\) −39.6531 + 19.4961i −0.756980 + 0.372182i
\(15\) 77.5448 + 13.4196i 1.33480 + 0.230996i
\(16\) −76.4670 + 17.7816i −1.19480 + 0.277838i
\(17\) −72.3898 − 25.8286i −1.03277 − 0.368492i −0.235347 − 0.971911i \(-0.575623\pi\)
−0.797424 + 0.603419i \(0.793804\pi\)
\(18\) −22.6008 + 18.4806i −0.295948 + 0.241995i
\(19\) 0.259590 + 9.08683i 0.00313442 + 0.109719i 0.999653 + 0.0263465i \(0.00838732\pi\)
−0.996518 + 0.0833725i \(0.973431\pi\)
\(20\) −34.2302 + 87.9196i −0.382706 + 0.982971i
\(21\) −34.6610 − 40.0009i −0.360174 − 0.415663i
\(22\) −90.8643 − 96.6533i −0.880561 − 0.936662i
\(23\) 13.1250 − 15.1471i 0.118989 − 0.137321i −0.693129 − 0.720813i \(-0.743769\pi\)
0.812119 + 0.583492i \(0.198314\pi\)
\(24\) −20.5336 − 38.9154i −0.174642 − 0.330983i
\(25\) −113.712 − 166.299i −0.909699 − 1.33039i
\(26\) 209.061 + 102.788i 1.57693 + 0.775326i
\(27\) −123.433 − 89.6792i −0.879802 − 0.639213i
\(28\) 55.2531 − 31.2029i 0.372923 − 0.210599i
\(29\) 97.9186 − 143.201i 0.627001 − 0.916960i −0.372940 − 0.927856i \(-0.621650\pi\)
0.999941 + 0.0108960i \(0.00346836\pi\)
\(30\) −284.292 − 32.6195i −1.73015 − 0.198516i
\(31\) −32.5145 − 160.466i −0.188380 − 0.929692i −0.957023 − 0.290014i \(-0.906340\pi\)
0.768642 − 0.639679i \(-0.220933\pi\)
\(32\) 196.360 − 57.6564i 1.08474 − 0.318510i
\(33\) 84.3027 − 134.698i 0.444703 − 0.710542i
\(34\) 268.154 + 78.7373i 1.35259 + 0.397157i
\(35\) −18.7892 + 218.758i −0.0907415 + 1.05648i
\(36\) 30.8882 − 28.3489i 0.143001 − 0.131245i
\(37\) −109.442 − 100.445i −0.486277 − 0.446301i 0.395154 − 0.918615i \(-0.370691\pi\)
−0.881430 + 0.472315i \(0.843419\pi\)
\(38\) −2.82866 − 32.9335i −0.0120755 − 0.140593i
\(39\) −55.4175 + 273.496i −0.227536 + 1.12293i
\(40\) −56.4038 + 173.593i −0.222956 + 0.686187i
\(41\) 29.2065 − 111.113i 0.111251 − 0.423243i −0.888085 − 0.459680i \(-0.847964\pi\)
0.999336 + 0.0364364i \(0.0116006\pi\)
\(42\) 134.132 + 138.018i 0.492784 + 0.507063i
\(43\) 23.2604 − 161.780i 0.0824925 − 0.573748i −0.906092 − 0.423081i \(-0.860949\pi\)
0.988585 − 0.150667i \(-0.0481422\pi\)
\(44\) 138.198 + 131.125i 0.473503 + 0.449268i
\(45\) 20.6454 + 143.592i 0.0683919 + 0.475676i
\(46\) −44.5027 + 57.7122i −0.142643 + 0.184983i
\(47\) −113.491 − 92.8015i −0.352222 − 0.288010i 0.440304 − 0.897849i \(-0.354871\pi\)
−0.792526 + 0.609838i \(0.791234\pi\)
\(48\) 176.579 + 292.824i 0.530979 + 0.880531i
\(49\) −136.133 + 140.077i −0.396888 + 0.408389i
\(50\) 447.325 + 580.101i 1.26523 + 1.64077i
\(51\) −9.55965 + 334.631i −0.0262474 + 0.918780i
\(52\) −308.162 − 130.230i −0.821816 − 0.347302i
\(53\) 230.828 + 53.6769i 0.598240 + 0.139115i 0.514672 − 0.857387i \(-0.327914\pi\)
0.0835683 + 0.996502i \(0.473368\pi\)
\(54\) 466.707 + 299.935i 1.17613 + 0.755851i
\(55\) −644.436 + 138.635i −1.57992 + 0.339884i
\(56\) 103.272 − 66.3692i 0.246435 − 0.158374i
\(57\) 37.2919 − 13.3057i 0.0866567 − 0.0309190i
\(58\) −325.742 + 540.183i −0.737449 + 1.22292i
\(59\) −84.7819 − 322.544i −0.187079 − 0.711723i −0.992837 − 0.119477i \(-0.961878\pi\)
0.805758 − 0.592245i \(-0.201758\pi\)
\(60\) 410.270 + 23.4601i 0.882761 + 0.0504781i
\(61\) 518.696 + 29.6601i 1.08873 + 0.0622556i 0.592219 − 0.805777i \(-0.298252\pi\)
0.496507 + 0.868033i \(0.334616\pi\)
\(62\) 151.346 + 575.780i 0.310016 + 1.17942i
\(63\) 50.3836 − 83.5518i 0.100758 − 0.167088i
\(64\) −109.333 + 39.0099i −0.213541 + 0.0761911i
\(65\) 973.833 − 625.844i 1.85829 − 1.19425i
\(66\) −278.078 + 506.487i −0.518622 + 0.944609i
\(67\) −489.884 − 314.830i −0.893267 − 0.574068i 0.0115191 − 0.999934i \(-0.496333\pi\)
−0.904786 + 0.425866i \(0.859970\pi\)
\(68\) −390.914 − 90.9032i −0.697137 − 0.162112i
\(69\) −80.4108 − 33.9819i −0.140295 − 0.0592890i
\(70\) 22.7984 − 798.047i 0.0389275 − 1.36264i
\(71\) −618.716 − 802.365i −1.03420 − 1.34117i −0.938316 − 0.345779i \(-0.887615\pi\)
−0.0958815 − 0.995393i \(-0.530567\pi\)
\(72\) 56.5280 − 58.1660i 0.0925263 − 0.0952073i
\(73\) −498.051 − 825.924i −0.798527 − 1.32421i −0.942358 − 0.334605i \(-0.891397\pi\)
0.143832 − 0.989602i \(-0.454058\pi\)
\(74\) 418.154 + 341.923i 0.656884 + 0.537132i
\(75\) −535.826 + 694.872i −0.824958 + 1.06982i
\(76\) 6.75551 + 46.9856i 0.0101962 + 0.0709161i
\(77\) 394.553 + 202.180i 0.583941 + 0.299228i
\(78\) 144.406 − 1004.36i 0.209624 − 1.45797i
\(79\) 828.703 + 852.715i 1.18021 + 1.21440i 0.971658 + 0.236393i \(0.0759653\pi\)
0.208550 + 0.978012i \(0.433126\pi\)
\(80\) 360.605 − 1371.88i 0.503960 − 1.91726i
\(81\) −138.364 + 425.840i −0.189799 + 0.584142i
\(82\) −82.9615 + 409.431i −0.111726 + 0.551392i
\(83\) 97.1921 + 1131.59i 0.128533 + 1.49648i 0.722152 + 0.691734i \(0.243153\pi\)
−0.593620 + 0.804746i \(0.702302\pi\)
\(84\) −203.622 − 186.883i −0.264488 − 0.242745i
\(85\) 1023.12 − 939.013i 1.30557 − 1.19824i
\(86\) −50.8579 + 592.128i −0.0637692 + 0.742452i
\(87\) −724.991 − 212.876i −0.893416 − 0.262330i
\(88\) 281.405 + 237.998i 0.340885 + 0.288303i
\(89\) −1006.12 + 295.424i −1.19830 + 0.351852i −0.819202 − 0.573505i \(-0.805583\pi\)
−0.379096 + 0.925357i \(0.623765\pi\)
\(90\) −104.755 − 516.988i −0.122691 − 0.605504i
\(91\) −773.480 − 88.7486i −0.891019 − 0.102235i
\(92\) 59.0731 − 86.3917i 0.0669435 − 0.0979017i
\(93\) −620.950 + 350.666i −0.692360 + 0.390994i
\(94\) 431.266 + 313.333i 0.473210 + 0.343807i
\(95\) −147.397 − 72.4703i −0.159186 − 0.0782663i
\(96\) −503.126 − 735.798i −0.534897 − 0.782261i
\(97\) −492.663 − 933.700i −0.515694 − 0.977349i −0.995017 − 0.0997103i \(-0.968208\pi\)
0.479322 − 0.877639i \(-0.340883\pi\)
\(98\) 465.117 − 536.774i 0.479428 − 0.553290i
\(99\) 284.167 + 71.0669i 0.288484 + 0.0721464i
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.91.7 yes 1280
121.4 even 55 inner 121.4.g.a.4.7 ✓ 1280
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
121.4.g.a.4.7 ✓ 1280 121.4 even 55 inner
121.4.g.a.91.7 yes 1280 1.1 even 1 trivial