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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 4.12
Character \(\chi\) \(=\) 121.4
Dual form 121.4.g.a.91.12

$q$-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+(-1.85939 - 0.106324i) q^{2} +(-2.64809 + 8.14999i) q^{3} +(-4.50182 - 0.516536i) q^{4} +(-2.36375 - 4.47981i) q^{5} +(5.79038 - 14.8725i) q^{6} +(-23.5797 - 9.96488i) q^{7} +(22.9969 + 3.97978i) q^{8} +(-37.5664 - 27.2936i) q^{9} +(3.91883 + 8.58103i) q^{10} +(31.6555 + 18.1363i) q^{11} +(16.1310 - 35.3220i) q^{12} +(-31.3742 + 17.7178i) q^{13} +(42.7844 + 21.0357i) q^{14} +(42.7698 - 7.40160i) q^{15} +(-7.02839 - 1.63438i) q^{16} +(92.8960 - 33.1452i) q^{17} +(66.9488 + 54.7437i) q^{18} +(-2.20777 + 77.2820i) q^{19} +(8.32720 + 21.3882i) q^{20} +(143.655 - 165.787i) q^{21} +(-56.9317 - 37.0883i) q^{22} +(-4.93772 - 5.69843i) q^{23} +(-93.3331 + 176.886i) q^{24} +(56.0741 - 82.0057i) q^{25} +(60.2208 - 29.6086i) q^{26} +(134.737 - 97.8919i) q^{27} +(101.005 + 57.0399i) q^{28} +(-167.675 - 245.216i) q^{29} +(-80.3127 + 9.21503i) q^{30} +(19.8073 - 97.7528i) q^{31} +(-166.252 - 48.8161i) q^{32} +(-231.638 + 209.966i) q^{33} +(-176.254 + 51.7529i) q^{34} +(11.0959 + 129.187i) q^{35} +(155.019 + 142.275i) q^{36} +(277.607 - 254.785i) q^{37} +(12.3220 - 143.463i) q^{38} +(-61.3184 - 302.618i) q^{39} +(-36.5304 - 112.429i) q^{40} +(49.6510 + 188.892i) q^{41} +(-284.738 + 292.988i) q^{42} +(-21.8282 - 151.819i) q^{43} +(-133.140 - 97.9978i) q^{44} +(-33.4724 + 232.806i) q^{45} +(8.57527 + 11.1206i) q^{46} +(-137.975 + 112.822i) q^{47} +(31.9320 - 52.9533i) q^{48} +(217.655 + 223.962i) q^{49} +(-112.983 + 146.519i) q^{50} +(24.1361 + 844.872i) q^{51} +(150.393 - 63.5567i) q^{52} +(399.934 - 93.0007i) q^{53} +(-260.936 + 167.694i) q^{54} +(6.42147 - 184.680i) q^{55} +(-502.604 - 323.004i) q^{56} +(-624.001 - 222.643i) q^{57} +(285.700 + 473.781i) q^{58} +(47.0889 - 179.145i) q^{59} +(-196.365 + 11.2286i) q^{60} +(-869.738 + 49.7334i) q^{61} +(-47.2229 + 179.655i) q^{62} +(613.829 + 1017.92i) q^{63} +(358.308 + 127.844i) q^{64} +(153.533 + 98.6699i) q^{65} +(453.030 - 365.780i) q^{66} +(405.476 - 260.584i) q^{67} +(-435.322 + 101.230i) q^{68} +(59.5176 - 25.1524i) q^{69} +(-6.89593 - 241.389i) q^{70} +(-293.147 + 380.160i) q^{71} +(-755.291 - 777.176i) q^{72} +(237.142 - 393.256i) q^{73} +(-543.269 + 444.229i) q^{74} +(519.856 + 674.162i) q^{75} +(49.8579 - 346.770i) q^{76} +(-565.703 - 743.094i) q^{77} +(81.8394 + 569.205i) q^{78} +(194.827 - 200.472i) q^{79} +(9.29165 + 35.3491i) q^{80} +(53.5969 + 164.954i) q^{81} +(-72.2370 - 356.504i) q^{82} +(99.2631 - 1155.70i) q^{83} +(-732.344 + 672.139i) q^{84} +(-368.067 - 337.809i) q^{85} +(24.4453 + 284.611i) q^{86} +(2442.53 - 717.191i) q^{87} +(655.802 + 543.062i) q^{88} +(-1403.29 - 412.044i) q^{89} +(86.9911 - 429.318i) q^{90} +(916.352 - 105.142i) q^{91} +(19.2853 + 28.2038i) q^{92} +(744.232 + 420.287i) q^{93} +(268.545 - 195.110i) q^{94} +(351.427 - 172.785i) q^{95} +(838.102 - 1225.68i) q^{96} +(-337.088 + 638.853i) q^{97} +(-380.894 - 439.575i) q^{98} +(-694.180 - 1545.31i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.85939 − 0.106324i −0.657394 − 0.0375912i −0.274791 − 0.961504i \(-0.588609\pi\)
−0.382603 + 0.923913i \(0.624972\pi\)
\(3\) −2.64809 + 8.14999i −0.509625 + 1.56847i 0.283227 + 0.959053i \(0.408595\pi\)
−0.792853 + 0.609413i \(0.791405\pi\)
\(4\) −4.50182 − 0.516536i −0.562728 − 0.0645670i
\(5\) −2.36375 − 4.47981i −0.211420 − 0.400686i 0.755733 − 0.654880i \(-0.227281\pi\)
−0.967153 + 0.254194i \(0.918190\pi\)
\(6\) 5.79038 − 14.8725i 0.393985 − 1.01194i
\(7\) −23.5797 − 9.96488i −1.27319 − 0.538053i −0.355246 − 0.934773i \(-0.615603\pi\)
−0.917940 + 0.396720i \(0.870148\pi\)
\(8\) 22.9969 + 3.97978i 1.01633 + 0.175883i
\(9\) −37.5664 − 27.2936i −1.39135 − 1.01087i
\(10\) 3.91883 + 8.58103i 0.123924 + 0.271356i
\(11\) 31.6555 + 18.1363i 0.867682 + 0.497119i
\(12\) 16.1310 − 35.3220i 0.388051 − 0.849714i
\(13\) −31.3742 + 17.7178i −0.669358 + 0.378004i −0.788639 − 0.614857i \(-0.789214\pi\)
0.119281 + 0.992861i \(0.461941\pi\)
\(14\) 42.7844 + 21.0357i 0.816759 + 0.401573i
\(15\) 42.7698 − 7.40160i 0.736207 − 0.127406i
\(16\) −7.02839 − 1.63438i −0.109819 − 0.0255372i
\(17\) 92.8960 − 33.1452i 1.32533 − 0.472876i 0.423862 − 0.905727i \(-0.360674\pi\)
0.901466 + 0.432850i \(0.142492\pi\)
\(18\) 66.9488 + 54.7437i 0.876665 + 0.716846i
\(19\) −2.20777 + 77.2820i −0.0266577 + 0.933143i 0.870694 + 0.491826i \(0.163670\pi\)
−0.897351 + 0.441317i \(0.854511\pi\)
\(20\) 8.32720 + 21.3882i 0.0931010 + 0.239128i
\(21\) 143.655 − 165.787i 1.49277 − 1.72274i
\(22\) −56.9317 − 37.0883i −0.551722 − 0.359420i
\(23\) −4.93772 − 5.69843i −0.0447646 − 0.0516611i 0.732926 − 0.680309i \(-0.238154\pi\)
−0.777690 + 0.628648i \(0.783609\pi\)
\(24\) −93.3331 + 176.886i −0.793814 + 1.50445i
\(25\) 56.0741 − 82.0057i 0.448593 − 0.656046i
\(26\) 60.2208 − 29.6086i 0.454241 − 0.223335i
\(27\) 134.737 − 97.8919i 0.960373 − 0.697752i
\(28\) 101.005 + 57.0399i 0.681716 + 0.384983i
\(29\) −167.675 − 245.216i −1.07367 − 1.57019i −0.791843 − 0.610725i \(-0.790878\pi\)
−0.281826 − 0.959466i \(-0.590940\pi\)
\(30\) −80.3127 + 9.21503i −0.488768 + 0.0560809i
\(31\) 19.8073 − 97.7528i 0.114758 − 0.566352i −0.880903 − 0.473297i \(-0.843064\pi\)
0.995661 − 0.0930554i \(-0.0296634\pi\)
\(32\) −166.252 − 48.8161i −0.918423 − 0.269673i
\(33\) −231.638 + 209.966i −1.22191 + 1.10759i
\(34\) −176.254 + 51.7529i −0.889039 + 0.261045i
\(35\) 11.0959 + 129.187i 0.0535871 + 0.623903i
\(36\) 155.019 + 142.275i 0.717682 + 0.658683i
\(37\) 277.607 − 254.785i 1.23347 − 1.13207i 0.246842 − 0.969056i \(-0.420607\pi\)
0.986624 − 0.163010i \(-0.0521203\pi\)
\(38\) 12.3220 − 143.463i 0.0526026 − 0.612441i
\(39\) −61.3184 − 302.618i −0.251764 − 1.24250i
\(40\) −36.5304 − 112.429i −0.144399 − 0.444415i
\(41\) 49.6510 + 188.892i 0.189127 + 0.719512i 0.992310 + 0.123780i \(0.0395017\pi\)
−0.803183 + 0.595732i \(0.796862\pi\)
\(42\) −284.738 + 292.988i −1.04610 + 1.07641i
\(43\) −21.8282 − 151.819i −0.0774133 − 0.538421i −0.991215 − 0.132261i \(-0.957776\pi\)
0.913802 − 0.406161i \(-0.133133\pi\)
\(44\) −133.140 − 97.9978i −0.456171 − 0.335766i
\(45\) −33.4724 + 232.806i −0.110884 + 0.771214i
\(46\) 8.57527 + 11.1206i 0.0274860 + 0.0356444i
\(47\) −137.975 + 112.822i −0.428207 + 0.350143i −0.822425 − 0.568874i \(-0.807379\pi\)
0.394218 + 0.919017i \(0.371016\pi\)
\(48\) 31.9320 − 52.9533i 0.0960206 − 0.159232i
\(49\) 217.655 + 223.962i 0.634564 + 0.652950i
\(50\) −112.983 + 146.519i −0.319564 + 0.414417i
\(51\) 24.1361 + 844.872i 0.0662691 + 2.31972i
\(52\) 150.393 − 63.5567i 0.401073 − 0.169495i
\(53\) 399.934 − 93.0007i 1.03651 − 0.241031i 0.326484 − 0.945203i \(-0.394136\pi\)
0.710029 + 0.704172i \(0.248682\pi\)
\(54\) −260.936 + 167.694i −0.657573 + 0.422597i
\(55\) 6.42147 − 184.680i 0.0157431 − 0.452769i
\(56\) −502.604 − 323.004i −1.19934 − 0.770771i
\(57\) −624.001 − 222.643i −1.45002 − 0.517365i
\(58\) 285.700 + 473.781i 0.646798 + 1.07259i
\(59\) 47.0889 − 179.145i 0.103906 − 0.395299i −0.894782 − 0.446503i \(-0.852669\pi\)
0.998688 + 0.0512034i \(0.0163057\pi\)
\(60\) −196.365 + 11.2286i −0.422511 + 0.0241600i
\(61\) −869.738 + 49.7334i −1.82555 + 0.104389i −0.935903 − 0.352259i \(-0.885414\pi\)
−0.889648 + 0.456648i \(0.849050\pi\)
\(62\) −47.2229 + 179.655i −0.0967309 + 0.368003i
\(63\) 613.829 + 1017.92i 1.22754 + 2.03565i
\(64\) 358.308 + 127.844i 0.699820 + 0.249695i
\(65\) 153.533 + 98.6699i 0.292977 + 0.188285i
\(66\) 453.030 − 365.780i 0.844910 − 0.682187i
\(67\) 405.476 − 260.584i 0.739355 − 0.475154i −0.115966 − 0.993253i \(-0.536996\pi\)
0.855321 + 0.518099i \(0.173360\pi\)
\(68\) −435.322 + 101.230i −0.776331 + 0.180528i
\(69\) 59.5176 − 25.1524i 0.103842 − 0.0438839i
\(70\) −6.89593 − 241.389i −0.0117746 − 0.412165i
\(71\) −293.147 + 380.160i −0.490002 + 0.635446i −0.970371 − 0.241621i \(-0.922321\pi\)
0.480369 + 0.877067i \(0.340503\pi\)
\(72\) −755.291 − 777.176i −1.23628 − 1.27210i
\(73\) 237.142 − 393.256i 0.380211 − 0.630509i −0.606103 − 0.795386i \(-0.707268\pi\)
0.986314 + 0.164877i \(0.0527227\pi\)
\(74\) −543.269 + 444.229i −0.853430 + 0.697846i
\(75\) 519.856 + 674.162i 0.800371 + 1.03794i
\(76\) 49.8579 − 346.770i 0.0752513 − 0.523384i
\(77\) −565.703 − 743.094i −0.837244 − 1.09978i
\(78\) 81.8394 + 569.205i 0.118801 + 0.826279i
\(79\) 194.827 − 200.472i 0.277465 − 0.285505i −0.564047 − 0.825742i \(-0.690757\pi\)
0.841513 + 0.540237i \(0.181666\pi\)
\(80\) 9.29165 + 35.3491i 0.0129855 + 0.0494019i
\(81\) 53.5969 + 164.954i 0.0735211 + 0.226275i
\(82\) −72.2370 − 356.504i −0.0972834 − 0.480113i
\(83\) 99.2631 − 1155.70i 0.131272 − 1.52837i −0.573661 − 0.819093i \(-0.694477\pi\)
0.704932 − 0.709275i \(-0.250977\pi\)
\(84\) −732.344 + 672.139i −0.951253 + 0.873052i
\(85\) −368.067 − 337.809i −0.469676 − 0.431065i
\(86\) 24.4453 + 284.611i 0.0306512 + 0.356865i
\(87\) 2442.53 − 717.191i 3.00996 − 0.883804i
\(88\) 655.802 + 543.062i 0.794417 + 0.657848i
\(89\) −1403.29 − 412.044i −1.67133 − 0.490748i −0.697229 − 0.716848i \(-0.745584\pi\)
−0.974104 + 0.226100i \(0.927402\pi\)
\(90\) 86.9911 − 429.318i 0.101885 − 0.502823i
\(91\) 916.352 − 105.142i 1.05560 − 0.121119i
\(92\) 19.2853 + 28.2038i 0.0218547 + 0.0319614i
\(93\) 744.232 + 420.287i 0.829820 + 0.468621i
\(94\) 268.545 − 195.110i 0.294663 − 0.214085i
\(95\) 351.427 − 172.785i 0.379533 − 0.186604i
\(96\) 838.102 − 1225.68i 0.891025 − 1.30308i
\(97\) −337.088 + 638.853i −0.352847 + 0.668719i −0.995218 − 0.0976744i \(-0.968860\pi\)
0.642372 + 0.766393i \(0.277951\pi\)
\(98\) −380.894 − 439.575i −0.392613 − 0.453100i
\(99\) −694.180 − 1545.31i −0.704724 − 1.56878i
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.4.12 ✓ 1280
121.91 even 55 inner 121.4.g.a.91.12 yes 1280
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
121.4.g.a.4.12 ✓ 1280 1.1 even 1 trivial
121.4.g.a.91.12 yes 1280 121.91 even 55 inner