Show commands: Magma / Pari/GP / SageMath

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 11.7
Character \(\chi\) \(=\) 920.11
Dual form 920.2.bb.b.251.7

$q$-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+(-1.29234 + 0.574339i) q^{2} +(0.272959 - 1.89847i) q^{3} +(1.34027 - 1.48448i) q^{4} +(0.959493 - 0.281733i) q^{5} +(0.737611 + 2.61024i) q^{6} +(0.510137 + 0.588729i) q^{7} +(-0.879486 + 2.68822i) q^{8} +(-0.651210 - 0.191212i) q^{9} +(-1.07818 + 0.915168i) q^{10} +(-0.344646 + 0.536279i) q^{11} +(-2.45240 - 2.94967i) q^{12} +(-3.54957 - 3.07572i) q^{13} +(-0.997399 - 0.467845i) q^{14} +(-0.272959 - 1.89847i) q^{15} +(-0.407355 - 3.97920i) q^{16} +(4.90669 - 2.24081i) q^{17} +(0.951403 - 0.126904i) q^{18} +(2.35087 + 1.07360i) q^{19} +(0.867753 - 1.80194i) 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} +(6.35374 + 1.93621i) q^{26} +(1.84953 - 4.04990i) q^{27} +(1.55768 + 0.0317686i) q^{28} +(-5.16378 + 2.35822i) q^{29} +(1.44312 + 2.29669i) q^{30} +(-1.14997 + 0.165341i) q^{31} +(2.81185 + 4.90851i) q^{32} +(0.924037 + 0.800682i) q^{33} +(-5.05411 + 5.71398i) q^{34} +(0.655337 + 0.421160i) q^{35} +(-1.15665 + 0.710431i) q^{36} +(4.79915 + 1.40916i) q^{37} +(-3.65472 - 0.0372649i) q^{38} +(-6.80805 + 5.89921i) q^{39} +(-0.0865026 + 2.82710i) q^{40} +(9.81972 - 2.88333i) q^{41} +(-1.16044 + 1.76583i) q^{42} +(-11.7594 - 1.69075i) q^{43} +(0.334177 + 1.23038i) q^{44} -0.678702 q^{45} +(-1.41392 + 6.63331i) q^{46} -5.76144i q^{47} +(-7.66560 - 0.312808i) q^{48} +(0.909841 - 6.32808i) q^{49} +(-0.776672 + 1.18185i) q^{50} +(-2.91479 - 9.92686i) q^{51} +(-9.32322 + 1.14697i) q^{52} +(-2.42105 - 2.79404i) q^{53} +(-0.0641973 + 6.29609i) q^{54} +(-0.179598 + 0.611654i) q^{55} +(-2.03129 + 0.853579i) q^{56} +(2.67990 - 4.17000i) q^{57} +(5.31892 - 6.01337i) q^{58} +(0.808707 - 0.933298i) q^{59} +(-3.18408 - 2.13926i) q^{60} +(2.05660 + 14.3040i) q^{61} +(1.39119 - 0.874150i) q^{62} +(-0.219634 - 0.480931i) q^{63} +(-6.45301 - 4.72850i) q^{64} +(-4.27232 - 1.95110i) q^{65} +(-1.65403 - 0.504041i) 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} +(1.08675 - 1.58242i) q^{72} +(4.64901 - 10.1799i) q^{73} +(-7.01145 + 0.935232i) q^{74} +(-0.796764 - 1.74467i) q^{75} +(4.74454 - 2.05089i) q^{76} +(-0.491540 + 0.0706727i) q^{77} +(5.41015 - 11.5339i) q^{78} +(-10.7816 + 12.4426i) q^{79} +(-1.51193 - 3.70325i) q^{80} +(-8.89665 - 5.71753i) q^{81} +(-11.0344 + 9.36608i) q^{82} +(-0.377510 + 1.28568i) q^{83} +(0.485494 - 2.94853i) q^{84} +(4.07662 - 3.53241i) q^{85} +(16.1682 - 4.56889i) q^{86} +(3.06751 + 10.4470i) q^{87} +(-1.13852 - 1.39813i) q^{88} +(10.5368 + 1.51496i) q^{89} +(0.877111 - 0.389805i) q^{90} -3.65877i q^{91} +(-1.98250 - 9.38455i) q^{92} +2.22832i q^{93} +(3.30902 + 7.44573i) q^{94} +(2.55811 + 0.367801i) q^{95} +(10.0862 - 3.99840i) q^{96} +(2.86214 + 9.74756i) q^{97} +(2.45864 + 8.70058i) q^{98} +(0.326980 - 0.283330i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 480 q + 2 q^{2} - 4 q^{4} + 48 q^{5} + 5 q^{6} - q^{8} - 48 q^{9} - 2 q^{10} + 3 q^{12} - 56 q^{16} + 7 q^{18} + 4 q^{20} - 8 q^{21} - 4 q^{23} + 2 q^{24} - 48 q^{25} + 7 q^{26} + 12 q^{27} - 5 q^{30}+ \cdots - 98 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.29234 + 0.574339i −0.913820 + 0.406119i
\(3\) 0.272959 − 1.89847i 0.157593 − 1.09608i −0.745459 − 0.666552i \(-0.767770\pi\)
0.903052 − 0.429532i \(-0.141321\pi\)
\(4\) 1.34027 − 1.48448i 0.670135 − 0.742239i
\(5\) 0.959493 − 0.281733i 0.429098 − 0.125995i
\(6\) 0.737611 + 2.61024i 0.301129 + 1.06562i
\(7\) 0.510137 + 0.588729i 0.192814 + 0.222519i 0.843922 − 0.536466i \(-0.180241\pi\)
−0.651108 + 0.758985i \(0.725696\pi\)
\(8\) −0.879486 + 2.68822i −0.310945 + 0.950428i
\(9\) −0.651210 − 0.191212i −0.217070 − 0.0637375i
\(10\) −1.07818 + 0.915168i −0.340950 + 0.289401i
\(11\) −0.344646 + 0.536279i −0.103915 + 0.161694i −0.889317 − 0.457291i \(-0.848820\pi\)
0.785402 + 0.618986i \(0.212456\pi\)
\(12\) −2.45240 − 2.94967i −0.707948 − 0.851495i
\(13\) −3.54957 − 3.07572i −0.984473 − 0.853051i 0.00467239 − 0.999989i \(-0.498513\pi\)
−0.989146 + 0.146938i \(0.953058\pi\)
\(14\) −0.997399 − 0.467845i −0.266566 − 0.125037i
\(15\) −0.272959 − 1.89847i −0.0704777 − 0.490183i
\(16\) −0.407355 − 3.97920i −0.101839 − 0.994801i
\(17\) 4.90669 − 2.24081i 1.19005 − 0.543476i 0.280813 − 0.959763i \(-0.409396\pi\)
0.909233 + 0.416286i \(0.136669\pi\)
\(18\) 0.951403 − 0.126904i 0.224248 − 0.0299116i
\(19\) 2.35087 + 1.07360i 0.539326 + 0.246302i 0.666404 − 0.745591i \(-0.267833\pi\)
−0.127078 + 0.991893i \(0.540560\pi\)
\(20\) 0.867753 − 1.80194i 0.194036 − 0.402927i
\(21\) 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
\(26\) 6.35374 + 1.93621i 1.24607 + 0.379722i
\(27\) 1.84953 − 4.04990i 0.355941 − 0.779403i
\(28\) 1.55768 + 0.0317686i 0.294373 + 0.00600371i
\(29\) −5.16378 + 2.35822i −0.958889 + 0.437910i −0.832472 − 0.554067i \(-0.813075\pi\)
−0.126417 + 0.991977i \(0.540348\pi\)
\(30\) 1.44312 + 2.29669i 0.263477 + 0.419317i
\(31\) −1.14997 + 0.165341i −0.206541 + 0.0296961i −0.244809 − 0.969571i \(-0.578725\pi\)
0.0382676 + 0.999268i \(0.487816\pi\)
\(32\) 2.81185 + 4.90851i 0.497070 + 0.867711i
\(33\) 0.924037 + 0.800682i 0.160854 + 0.139381i
\(34\) −5.05411 + 5.71398i −0.866773 + 0.979940i
\(35\) 0.655337 + 0.421160i 0.110772 + 0.0711890i
\(36\) −1.15665 + 0.710431i −0.192775 + 0.118405i
\(37\) 4.79915 + 1.40916i 0.788975 + 0.231664i 0.651307 − 0.758814i \(-0.274221\pi\)
0.137668 + 0.990478i \(0.456039\pi\)
\(38\) −3.65472 − 0.0372649i −0.592875 − 0.00604517i
\(39\) −6.80805 + 5.89921i −1.09016 + 0.944630i
\(40\) −0.0865026 + 2.82710i −0.0136773 + 0.447004i
\(41\) 9.81972 − 2.88333i 1.53358 − 0.450300i 0.597438 − 0.801915i \(-0.296186\pi\)
0.936145 + 0.351615i \(0.114367\pi\)
\(42\) −1.16044 + 1.76583i −0.179060 + 0.272474i
\(43\) −11.7594 − 1.69075i −1.79330 − 0.257837i −0.836371 − 0.548164i \(-0.815327\pi\)
−0.956928 + 0.290326i \(0.906236\pi\)
\(44\) 0.334177 + 1.23038i 0.0503791 + 0.185486i
\(45\) −0.678702 −0.101175
\(46\) −1.41392 + 6.63331i −0.208472 + 0.978028i
\(47\) − 5.76144i − 0.840393i −0.907433 − 0.420196i \(-0.861961\pi\)
0.907433 − 0.420196i \(-0.138039\pi\)
\(48\) −7.66560 − 0.312808i −1.10643 − 0.0451499i
\(49\) 0.909841 − 6.32808i 0.129977 − 0.904012i
\(50\) −0.776672 + 1.18185i −0.109838 + 0.167140i
\(51\) −2.91479 − 9.92686i −0.408152 − 1.39004i
\(52\) −9.32322 + 1.14697i −1.29290 + 0.159056i
\(53\) −2.42105 − 2.79404i −0.332557 − 0.383791i 0.564703 − 0.825294i \(-0.308991\pi\)
−0.897260 + 0.441503i \(0.854445\pi\)
\(54\) −0.0641973 + 6.29609i −0.00873614 + 0.856789i
\(55\) −0.179598 + 0.611654i −0.0242170 + 0.0824754i
\(56\) −2.03129 + 0.853579i −0.271443 + 0.114064i
\(57\) 2.67990 − 4.17000i 0.354961 − 0.552331i
\(58\) 5.31892 − 6.01337i 0.698409 − 0.789594i
\(59\) 0.808707 − 0.933298i 0.105285 − 0.121505i −0.700661 − 0.713494i \(-0.747112\pi\)
0.805946 + 0.591989i \(0.201657\pi\)
\(60\) −3.18408 − 2.13926i −0.411063 − 0.276178i
\(61\) 2.05660 + 14.3040i 0.263321 + 1.83144i 0.507435 + 0.861690i \(0.330594\pi\)
−0.244114 + 0.969746i \(0.578497\pi\)
\(62\) 1.39119 − 0.874150i 0.176681 − 0.111017i
\(63\) −0.219634 − 0.480931i −0.0276712 − 0.0605916i
\(64\) −6.45301 − 4.72850i −0.806626 − 0.591062i
\(65\) −4.27232 − 1.95110i −0.529916 − 0.242004i
\(66\) −1.65403 − 0.504041i −0.203597 − 0.0620432i
\(67\) −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 0.682564 − 0.730826i \(-0.260865\pi\)
−0.948330 + 0.317285i \(0.897229\pi\)
\(72\) 1.08675 − 1.58242i 0.128075 − 0.186490i
\(73\) 4.64901 − 10.1799i 0.544126 − 1.19147i −0.415346 − 0.909664i \(-0.636339\pi\)
0.959471 − 0.281806i \(-0.0909333\pi\)
\(74\) −7.01145 + 0.935232i −0.815064 + 0.108718i
\(75\) −0.796764 − 1.74467i −0.0920023 − 0.201457i
\(76\) 4.74454 − 2.05089i 0.544236 − 0.235253i
\(77\) −0.491540 + 0.0706727i −0.0560161 + 0.00805391i
\(78\) 5.41015 − 11.5339i 0.612579 − 1.30596i
\(79\) −10.7816 + 12.4426i −1.21302 + 1.39990i −0.321505 + 0.946908i \(0.604189\pi\)
−0.891515 + 0.452991i \(0.850357\pi\)
\(80\) −1.51193 − 3.70325i −0.169038 − 0.414036i
\(81\) −8.89665 − 5.71753i −0.988517 − 0.635281i
\(82\) −11.0344 + 9.36608i −1.21854 + 1.03431i
\(83\) −0.377510 + 1.28568i −0.0414371 + 0.141122i −0.977613 − 0.210409i \(-0.932520\pi\)
0.936176 + 0.351531i \(0.114339\pi\)
\(84\) 0.485494 − 2.94853i 0.0529717 − 0.321712i
\(85\) 4.07662 − 3.53241i 0.442172 − 0.383144i
\(86\) 16.1682 − 4.56889i 1.74347 − 0.492676i
\(87\) 3.06751 + 10.4470i 0.328872 + 1.12003i
\(88\) −1.13852 − 1.39813i −0.121367 − 0.149041i
\(89\) 10.5368 + 1.51496i 1.11690 + 0.160586i 0.675950 − 0.736947i \(-0.263733\pi\)
0.440947 + 0.897533i \(0.354643\pi\)
\(90\) 0.877111 − 0.389805i 0.0924557 − 0.0410890i
\(91\) − 3.65877i − 0.383544i
\(92\) −1.98250 − 9.38455i −0.206690 − 0.978406i
\(93\) 2.22832i 0.231066i
\(94\) 3.30902 + 7.44573i 0.341300 + 0.767968i
\(95\) 2.55811 + 0.367801i 0.262456 + 0.0377355i
\(96\) 10.0862 − 3.99840i 1.02942 − 0.408085i
\(97\) 2.86214 + 9.74756i 0.290606 + 0.989714i 0.967339 + 0.253486i \(0.0815771\pi\)
−0.676733 + 0.736229i \(0.736605\pi\)
\(98\) 2.45864 + 8.70058i 0.248361 + 0.878891i
\(99\) 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.b.11.7 yes 480
8.3 odd 2 920.2.bb.a.11.12 ✓ 480
23.21 odd 22 920.2.bb.a.251.12 yes 480
184.67 even 22 inner 920.2.bb.b.251.7 yes 480
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
920.2.bb.a.11.12 ✓ 480 8.3 odd 2
920.2.bb.a.251.12 yes 480 23.21 odd 22
920.2.bb.b.11.7 yes 480 1.1 even 1 trivial
920.2.bb.b.251.7 yes 480 184.67 even 22 inner