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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [920,2,Mod(11,920)] 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("920.11"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(920, base_ring=CyclotomicField(22)) chi = DirichletCharacter(H, H._module([11, 11, 0, 9])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 920 = 2^{3} \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 920.bb (of order \(22\), degree \(10\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [480,2,0,-4,-48] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(7.34623698596\)
Analytic rank: \(0\)
Dimension: \(480\)
Relative dimension: \(48\) over \(\Q(\zeta_{22})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{22}]$

Embedding invariants

Embedding label 11.12
Character \(\chi\) \(=\) 920.11
Dual form 920.2.bb.a.251.12

$q$-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+(-1.05929 + 0.936962i) q^{2} +(0.272959 - 1.89847i) q^{3} +(0.244204 - 1.98504i) q^{4} +(-0.959493 + 0.281733i) q^{5} +(1.48965 + 2.26679i) q^{6} +(-0.510137 - 0.588729i) q^{7} +(1.60122 + 2.33154i) q^{8} +(-0.651210 - 0.191212i) q^{9} +(0.752412 - 1.19745i) q^{10} +(-0.344646 + 0.536279i) q^{11} +(-3.70188 - 1.00545i) q^{12} +(3.54957 + 3.07572i) q^{13} +(1.09200 + 0.145658i) q^{14} +(0.272959 + 1.89847i) q^{15} +(-3.88073 - 0.969508i) q^{16} +(4.90669 - 2.24081i) q^{17} +(0.868981 - 0.407609i) q^{18} +(2.35087 + 1.07360i) q^{19} +(0.324937 + 1.97343i) q^{20} +(-1.25693 + 0.807782i) q^{21} +(-0.137392 - 0.890997i) q^{22} +(-2.81852 + 3.88020i) q^{23} +(4.86344 - 2.40345i) q^{24} +(0.841254 - 0.540641i) q^{25} +(-6.64187 + 0.0677230i) q^{26} +(1.84953 - 4.04990i) q^{27} +(-1.29323 + 0.868869i) q^{28} +(5.16378 - 2.35822i) q^{29} +(-2.06794 - 1.75529i) q^{30} +(1.14997 - 0.165341i) q^{31} +(5.01922 - 2.60910i) q^{32} +(0.924037 + 0.800682i) q^{33} +(-3.09807 + 6.97105i) q^{34} +(0.655337 + 0.421160i) q^{35} +(-0.538591 + 1.24598i) q^{36} +(-4.79915 - 1.40916i) q^{37} +(-3.49618 + 1.06541i) q^{38} +(6.80805 - 5.89921i) q^{39} +(-2.19323 - 1.78599i) q^{40} +(9.81972 - 2.88333i) q^{41} +(0.574600 - 2.03338i) q^{42} +(-11.7594 - 1.69075i) q^{43} +(0.980369 + 0.815095i) q^{44} +0.678702 q^{45} +(-0.649963 - 6.75111i) q^{46} +5.76144i q^{47} +(-2.89986 + 7.10282i) q^{48} +(0.909841 - 6.32808i) q^{49} +(-0.384574 + 1.36092i) q^{50} +(-2.91479 - 9.92686i) q^{51} +(6.97223 - 6.29491i) q^{52} +(2.42105 + 2.79404i) q^{53} +(1.83541 + 6.02296i) q^{54} +(0.179598 - 0.611654i) q^{55} +(0.555808 - 2.13209i) q^{56} +(2.67990 - 4.17000i) q^{57} +(-3.26039 + 7.33631i) q^{58} +(0.808707 - 0.933298i) q^{59} +(3.83519 - 0.0782182i) q^{60} +(-2.05660 - 14.3040i) q^{61} +(-1.06324 + 1.25262i) q^{62} +(0.219634 + 0.480931i) q^{63} +(-2.87220 + 7.46662i) q^{64} +(-4.27232 - 1.95110i) q^{65} +(-1.72903 + 0.0176299i) q^{66} +(-3.92380 - 6.10555i) q^{67} +(-3.24985 - 10.2872i) q^{68} +(6.59711 + 6.41001i) q^{69} +(-1.08880 + 0.167894i) q^{70} +(2.23939 + 3.48456i) q^{71} +(-0.596909 - 1.82450i) q^{72} +(4.64901 - 10.1799i) q^{73} +(6.40403 - 3.00391i) q^{74} +(-0.796764 - 1.74467i) q^{75} +(2.70524 - 4.40437i) q^{76} +(0.491540 - 0.0706727i) q^{77} +(-1.68439 + 12.6279i) q^{78} +(10.7816 - 12.4426i) q^{79} +(3.99667 - 0.163091i) q^{80} +(-8.89665 - 5.71753i) q^{81} +(-7.70039 + 12.2550i) q^{82} +(-0.377510 + 1.28568i) q^{83} +(1.29653 + 2.69232i) q^{84} +(-4.07662 + 3.53241i) q^{85} +(14.0409 - 9.22715i) q^{86} +(-3.06751 - 10.4470i) q^{87} +(-1.80221 + 0.0551434i) q^{88} +(10.5368 + 1.51496i) q^{89} +(-0.718944 + 0.635918i) q^{90} -3.65877i q^{91} +(7.01404 + 6.54242i) q^{92} -2.22832i q^{93} +(-5.39825 - 6.10306i) q^{94} +(-2.55811 - 0.367801i) q^{95} +(-3.58326 - 10.2410i) q^{96} +(2.86214 + 9.74756i) q^{97} +(4.96539 + 7.55578i) q^{98} +(0.326980 - 0.283330i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 480 q + 2 q^{2} - 4 q^{4} - 48 q^{5} + 5 q^{6} - q^{8} - 48 q^{9} + 2 q^{10} + 3 q^{12} + 32 q^{16} + 7 q^{18} - 4 q^{20} + 8 q^{21} + 4 q^{23} + 2 q^{24} - 48 q^{25} + 7 q^{26} + 12 q^{27} + 5 q^{30}+ \cdots - 98 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/920\mathbb{Z}\right)^\times\).

\(n\) \(231\) \(281\) \(461\) \(737\)
\(\chi(n)\) \(-1\) \(e\left(\frac{9}{22}\right)\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.05929 + 0.936962i −0.749033 + 0.662532i
\(3\) 0.272959 − 1.89847i 0.157593 − 1.09608i −0.745459 − 0.666552i \(-0.767770\pi\)
0.903052 − 0.429532i \(-0.141321\pi\)
\(4\) 0.244204 − 1.98504i 0.122102 − 0.992518i
\(5\) −0.959493 + 0.281733i −0.429098 + 0.125995i
\(6\) 1.48965 + 2.26679i 0.608148 + 0.925413i
\(7\) −0.510137 − 0.588729i −0.192814 − 0.222519i 0.651108 − 0.758985i \(-0.274304\pi\)
−0.843922 + 0.536466i \(0.819759\pi\)
\(8\) 1.60122 + 2.33154i 0.566116 + 0.824325i
\(9\) −0.651210 − 0.191212i −0.217070 − 0.0637375i
\(10\) 0.752412 − 1.19745i 0.237933 − 0.378666i
\(11\) −0.344646 + 0.536279i −0.103915 + 0.161694i −0.889317 − 0.457291i \(-0.848820\pi\)
0.785402 + 0.618986i \(0.212456\pi\)
\(12\) −3.70188 − 1.00545i −1.06864 − 0.290248i
\(13\) 3.54957 + 3.07572i 0.984473 + 0.853051i 0.989146 − 0.146938i \(-0.0469418\pi\)
−0.00467239 + 0.999989i \(0.501487\pi\)
\(14\) 1.09200 + 0.145658i 0.291850 + 0.0389288i
\(15\) 0.272959 + 1.89847i 0.0704777 + 0.490183i
\(16\) −3.88073 − 0.969508i −0.970182 − 0.242377i
\(17\) 4.90669 − 2.24081i 1.19005 − 0.543476i 0.280813 − 0.959763i \(-0.409396\pi\)
0.909233 + 0.416286i \(0.136669\pi\)
\(18\) 0.868981 − 0.407609i 0.204821 − 0.0960743i
\(19\) 2.35087 + 1.07360i 0.539326 + 0.246302i 0.666404 − 0.745591i \(-0.267833\pi\)
−0.127078 + 0.991893i \(0.540560\pi\)
\(20\) 0.324937 + 1.97343i 0.0726580 + 0.441272i
\(21\) −1.25693 + 0.807782i −0.274285 + 0.176272i
\(22\) −0.137392 − 0.890997i −0.0292921 − 0.189961i
\(23\) −2.81852 + 3.88020i −0.587702 + 0.809078i
\(24\) 4.86344 − 2.40345i 0.992745 − 0.490603i
\(25\) 0.841254 − 0.540641i 0.168251 − 0.108128i
\(26\) −6.64187 + 0.0677230i −1.30258 + 0.0132816i
\(27\) 1.84953 − 4.04990i 0.355941 − 0.779403i
\(28\) −1.29323 + 0.868869i −0.244397 + 0.164201i
\(29\) 5.16378 − 2.35822i 0.958889 − 0.437910i 0.126417 − 0.991977i \(-0.459652\pi\)
0.832472 + 0.554067i \(0.186925\pi\)
\(30\) −2.06794 − 1.75529i −0.377552 − 0.320470i
\(31\) 1.14997 − 0.165341i 0.206541 − 0.0296961i −0.0382676 − 0.999268i \(-0.512184\pi\)
0.244809 + 0.969571i \(0.421275\pi\)
\(32\) 5.01922 − 2.60910i 0.887281 − 0.461228i
\(33\) 0.924037 + 0.800682i 0.160854 + 0.139381i
\(34\) −3.09807 + 6.97105i −0.531314 + 1.19553i
\(35\) 0.655337 + 0.421160i 0.110772 + 0.0711890i
\(36\) −0.538591 + 1.24598i −0.0897652 + 0.207663i
\(37\) −4.79915 − 1.40916i −0.788975 − 0.231664i −0.137668 − 0.990478i \(-0.543961\pi\)
−0.651307 + 0.758814i \(0.725779\pi\)
\(38\) −3.49618 + 1.06541i −0.567156 + 0.172832i
\(39\) 6.80805 − 5.89921i 1.09016 − 0.944630i
\(40\) −2.19323 − 1.78599i −0.346780 − 0.282389i
\(41\) 9.81972 − 2.88333i 1.53358 − 0.450300i 0.597438 − 0.801915i \(-0.296186\pi\)
0.936145 + 0.351615i \(0.114367\pi\)
\(42\) 0.574600 − 2.03338i 0.0886626 − 0.313757i
\(43\) −11.7594 − 1.69075i −1.79330 − 0.257837i −0.836371 − 0.548164i \(-0.815327\pi\)
−0.956928 + 0.290326i \(0.906236\pi\)
\(44\) 0.980369 + 0.815095i 0.147796 + 0.122880i
\(45\) 0.678702 0.101175
\(46\) −0.649963 − 6.75111i −0.0958319 − 0.995398i
\(47\) 5.76144i 0.840393i 0.907433 + 0.420196i \(0.138039\pi\)
−0.907433 + 0.420196i \(0.861961\pi\)
\(48\) −2.89986 + 7.10282i −0.418559 + 1.02520i
\(49\) 0.909841 − 6.32808i 0.129977 − 0.904012i
\(50\) −0.384574 + 1.36092i −0.0543870 + 0.192463i
\(51\) −2.91479 − 9.92686i −0.408152 − 1.39004i
\(52\) 6.97223 − 6.29491i 0.966874 − 0.872948i
\(53\) 2.42105 + 2.79404i 0.332557 + 0.383791i 0.897260 − 0.441503i \(-0.145555\pi\)
−0.564703 + 0.825294i \(0.691009\pi\)
\(54\) 1.83541 + 6.02296i 0.249768 + 0.819622i
\(55\) 0.179598 − 0.611654i 0.0242170 − 0.0824754i
\(56\) 0.555808 − 2.13209i 0.0742730 − 0.284913i
\(57\) 2.67990 − 4.17000i 0.354961 − 0.552331i
\(58\) −3.26039 + 7.33631i −0.428111 + 0.963304i
\(59\) 0.808707 − 0.933298i 0.105285 − 0.121505i −0.700661 − 0.713494i \(-0.747112\pi\)
0.805946 + 0.591989i \(0.201657\pi\)
\(60\) 3.83519 − 0.0782182i 0.495121 − 0.0100979i
\(61\) −2.05660 − 14.3040i −0.263321 − 1.83144i −0.507435 − 0.861690i \(-0.669406\pi\)
0.244114 − 0.969746i \(-0.421503\pi\)
\(62\) −1.06324 + 1.25262i −0.135032 + 0.159083i
\(63\) 0.219634 + 0.480931i 0.0276712 + 0.0605916i
\(64\) −2.87220 + 7.46662i −0.359025 + 0.933328i
\(65\) −4.27232 − 1.95110i −0.529916 − 0.242004i
\(66\) −1.72903 + 0.0176299i −0.212829 + 0.00217009i
\(67\) −3.92380 − 6.10555i −0.479369 − 0.745912i 0.514379 − 0.857563i \(-0.328023\pi\)
−0.993747 + 0.111651i \(0.964386\pi\)
\(68\) −3.24985 − 10.2872i −0.394102 − 1.24750i
\(69\) 6.59711 + 6.41001i 0.794199 + 0.771675i
\(70\) −1.08880 + 0.167894i −0.130137 + 0.0200672i
\(71\) 2.23939 + 3.48456i 0.265767 + 0.413541i 0.948330 − 0.317285i \(-0.102771\pi\)
−0.682564 + 0.730826i \(0.739135\pi\)
\(72\) −0.596909 − 1.82450i −0.0703464 − 0.215019i
\(73\) 4.64901 − 10.1799i 0.544126 − 1.19147i −0.415346 − 0.909664i \(-0.636339\pi\)
0.959471 − 0.281806i \(-0.0909333\pi\)
\(74\) 6.40403 − 3.00391i 0.744454 − 0.349197i
\(75\) −0.796764 − 1.74467i −0.0920023 − 0.201457i
\(76\) 2.70524 − 4.40437i 0.310312 − 0.505216i
\(77\) 0.491540 − 0.0706727i 0.0560161 − 0.00805391i
\(78\) −1.68439 + 12.6279i −0.190719 + 1.42983i
\(79\) 10.7816 − 12.4426i 1.21302 − 1.39990i 0.321505 − 0.946908i \(-0.395811\pi\)
0.891515 − 0.452991i \(-0.149643\pi\)
\(80\) 3.99667 − 0.163091i 0.446842 − 0.0182341i
\(81\) −8.89665 − 5.71753i −0.988517 − 0.635281i
\(82\) −7.70039 + 12.2550i −0.850366 + 1.35334i
\(83\) −0.377510 + 1.28568i −0.0414371 + 0.141122i −0.977613 − 0.210409i \(-0.932520\pi\)
0.936176 + 0.351531i \(0.114339\pi\)
\(84\) 1.29653 + 2.69232i 0.141463 + 0.293756i
\(85\) −4.07662 + 3.53241i −0.442172 + 0.383144i
\(86\) 14.0409 − 9.22715i 1.51407 − 0.994989i
\(87\) −3.06751 − 10.4470i −0.328872 − 1.12003i
\(88\) −1.80221 + 0.0551434i −0.192116 + 0.00587830i
\(89\) 10.5368 + 1.51496i 1.11690 + 0.160586i 0.675950 − 0.736947i \(-0.263733\pi\)
0.440947 + 0.897533i \(0.354643\pi\)
\(90\) −0.718944 + 0.635918i −0.0757834 + 0.0670316i
\(91\) − 3.65877i − 0.383544i
\(92\) 7.01404 + 6.54242i 0.731264 + 0.682094i
\(93\) − 2.22832i − 0.231066i
\(94\) −5.39825 − 6.10306i −0.556787 − 0.629482i
\(95\) −2.55811 − 0.367801i −0.262456 − 0.0377355i
\(96\) −3.58326 − 10.2410i −0.365715 − 1.04522i
\(97\) 2.86214 + 9.74756i 0.290606 + 0.989714i 0.967339 + 0.253486i \(0.0815771\pi\)
−0.676733 + 0.736229i \(0.736605\pi\)
\(98\) 4.96539 + 7.55578i 0.501580 + 0.763249i
\(99\) 0.326980 − 0.283330i 0.0328627 − 0.0284757i
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 920.2.bb.a.11.12 ✓ 480
8.3 odd 2 920.2.bb.b.11.7 yes 480
23.21 odd 22 920.2.bb.b.251.7 yes 480
184.67 even 22 inner 920.2.bb.a.251.12 yes 480
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
920.2.bb.a.11.12 ✓ 480 1.1 even 1 trivial
920.2.bb.a.251.12 yes 480 184.67 even 22 inner
920.2.bb.b.11.7 yes 480 8.3 odd 2
920.2.bb.b.251.7 yes 480 23.21 odd 22