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 251.14
Character \(\chi\) \(=\) 920.251
Dual form 920.2.bb.b.11.14

$q$-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+(-0.953750 - 1.04420i) q^{2} +(-0.132869 - 0.924126i) q^{3} +(-0.180721 + 1.99182i) q^{4} +(0.959493 + 0.281733i) q^{5} +(-0.838252 + 1.02013i) q^{6} +(-2.80604 + 3.23834i) q^{7} +(2.25223 - 1.71099i) q^{8} +(2.04212 - 0.599622i) q^{9} +(-0.620931 - 1.27061i) q^{10} +(-3.09409 - 4.81450i) q^{11} +(1.86470 - 0.0976428i) q^{12} +(-2.73900 + 2.37336i) q^{13} +(6.05774 - 0.158494i) q^{14} +(0.132869 - 0.924126i) q^{15} +(-3.93468 - 0.719926i) q^{16} +(4.43451 + 2.02517i) q^{17} +(-2.57380 - 1.56050i) q^{18} +(-3.44622 + 1.57383i) q^{19} +(-0.734560 + 1.86022i) q^{20} +(3.36547 + 2.16286i) q^{21} +(-2.07633 + 7.82269i) q^{22} +(1.78810 - 4.45002i) q^{23} +(-1.88042 - 1.85400i) q^{24} +(0.841254 + 0.540641i) q^{25} +(5.09059 + 0.596482i) q^{26} +(-1.98899 - 4.35529i) q^{27} +(-5.94308 - 6.17435i) q^{28} +(-2.94715 - 1.34592i) q^{29} +(-1.09170 + 0.742643i) q^{30} +(-4.96005 - 0.713147i) q^{31} +(3.00095 + 4.79523i) q^{32} +(-4.03810 + 3.49903i) q^{33} +(-2.11472 - 6.56203i) q^{34} +(-3.60472 + 2.31661i) q^{35} +(0.825283 + 4.17590i) q^{36} +(-6.20384 + 1.82161i) q^{37} +(4.93023 + 2.09751i) q^{38} +(2.55721 + 2.21583i) q^{39} +(2.64304 - 1.00716i) q^{40} +(-11.1848 - 3.28416i) q^{41} +(-0.951357 - 5.57706i) q^{42} +(-4.00647 + 0.576043i) q^{43} +(10.1488 - 5.29279i) q^{44} +2.12834 q^{45} +(-6.35213 + 2.37707i) q^{46} -2.40525i q^{47} +(-0.142504 + 3.73180i) q^{48} +(-1.61679 - 11.2451i) q^{49} +(-0.237807 - 1.39408i) q^{50} +(1.28230 - 4.36713i) q^{51} +(-4.23230 - 5.88450i) q^{52} +(-6.44475 + 7.43764i) q^{53} +(-2.65080 + 6.23077i) q^{54} +(-1.61236 - 5.49118i) q^{55} +(-0.779068 + 12.0946i) q^{56} +(1.91232 + 2.97563i) q^{57} +(1.40543 + 4.36109i) q^{58} +(-3.78950 - 4.37331i) q^{59} +(1.81668 + 0.431660i) q^{60} +(0.158817 - 1.10460i) q^{61} +(3.98597 + 5.85946i) q^{62} +(-3.78850 + 8.29565i) q^{63} +(2.14504 - 7.70706i) q^{64} +(-3.29670 + 1.50555i) q^{65} +(7.50504 + 0.879392i) q^{66} +(-4.57690 + 7.12180i) q^{67} +(-4.83518 + 8.46674i) q^{68} +(-4.34997 - 1.06116i) q^{69} +(5.85702 + 1.55459i) q^{70} +(1.73600 - 2.70127i) q^{71} +(3.57338 - 4.84453i) q^{72} +(5.43154 + 11.8934i) q^{73} +(7.81905 + 4.74071i) q^{74} +(0.387844 - 0.849259i) q^{75} +(-2.51199 - 7.14866i) q^{76} +(24.2731 + 3.48995i) q^{77} +(-0.125158 - 4.78360i) q^{78} +(10.6141 + 12.2494i) q^{79} +(-3.57247 - 1.79929i) q^{80} +(1.61085 - 1.03523i) q^{81} +(7.23820 + 14.8115i) q^{82} +(1.16726 + 3.97533i) q^{83} +(-4.91623 + 6.31253i) q^{84} +(3.68432 + 3.19248i) q^{85} +(4.42268 + 3.63417i) q^{86} +(-0.852212 + 2.90237i) q^{87} +(-15.2061 - 5.54939i) q^{88} +(-4.72278 + 0.679033i) q^{89} +(-2.02990 - 2.22242i) q^{90} -15.5295i q^{91} +(8.54049 + 4.36578i) q^{92} +4.67847i q^{93} +(-2.51157 + 2.29401i) q^{94} +(-3.75002 + 0.539172i) q^{95} +(4.03267 - 3.41040i) q^{96} +(4.85957 - 16.5502i) q^{97} +(-10.2001 + 12.4132i) q^{98} +(-9.20539 - 7.97652i) 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 \(\left(\mathbb{Z}/920\mathbb{Z}\right)^\times\).

\(n\) \(231\) \(281\) \(461\) \(737\)
\(\chi(n)\) \(-1\) \(e\left(\frac{13}{22}\right)\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.953750 − 1.04420i −0.674403 − 0.738363i
\(3\) −0.132869 − 0.924126i −0.0767121 − 0.533545i −0.991550 − 0.129724i \(-0.958591\pi\)
0.914838 − 0.403821i \(-0.132318\pi\)
\(4\) −0.180721 + 1.99182i −0.0903604 + 0.995909i
\(5\) 0.959493 + 0.281733i 0.429098 + 0.125995i
\(6\) −0.838252 + 1.02013i −0.342215 + 0.416466i
\(7\) −2.80604 + 3.23834i −1.06058 + 1.22398i −0.0868645 + 0.996220i \(0.527685\pi\)
−0.973718 + 0.227757i \(0.926861\pi\)
\(8\) 2.25223 − 1.71099i 0.796282 − 0.604926i
\(9\) 2.04212 − 0.599622i 0.680708 − 0.199874i
\(10\) −0.620931 − 1.27061i −0.196356 − 0.401802i
\(11\) −3.09409 − 4.81450i −0.932904 − 1.45163i −0.891778 − 0.452472i \(-0.850542\pi\)
−0.0411251 − 0.999154i \(-0.513094\pi\)
\(12\) 1.86470 − 0.0976428i 0.538294 − 0.0281870i
\(13\) −2.73900 + 2.37336i −0.759662 + 0.658250i −0.945975 − 0.324240i \(-0.894891\pi\)
0.186313 + 0.982490i \(0.440346\pi\)
\(14\) 6.05774 − 0.158494i 1.61900 − 0.0423594i
\(15\) 0.132869 − 0.924126i 0.0343067 − 0.238608i
\(16\) −3.93468 − 0.719926i −0.983670 − 0.179981i
\(17\) 4.43451 + 2.02517i 1.07553 + 0.491176i 0.872809 − 0.488061i \(-0.162296\pi\)
0.202716 + 0.979238i \(0.435023\pi\)
\(18\) −2.57380 − 1.56050i −0.606651 − 0.367814i
\(19\) −3.44622 + 1.57383i −0.790616 + 0.361062i −0.769445 − 0.638713i \(-0.779467\pi\)
−0.0211717 + 0.999776i \(0.506740\pi\)
\(20\) −0.734560 + 1.86022i −0.164253 + 0.415958i
\(21\) 3.36547 + 2.16286i 0.734406 + 0.471974i
\(22\) −2.07633 + 7.82269i −0.442674 + 1.66780i
\(23\) 1.78810 − 4.45002i 0.372845 − 0.927894i
\(24\) −1.88042 − 1.85400i −0.383839 − 0.378447i
\(25\) 0.841254 + 0.540641i 0.168251 + 0.108128i
\(26\) 5.09059 + 0.596482i 0.998346 + 0.116980i
\(27\) −1.98899 − 4.35529i −0.382782 − 0.838175i
\(28\) −5.94308 − 6.17435i −1.12314 − 1.16684i
\(29\) −2.94715 − 1.34592i −0.547271 − 0.249930i 0.122538 − 0.992464i \(-0.460897\pi\)
−0.669809 + 0.742533i \(0.733624\pi\)
\(30\) −1.09170 + 0.742643i −0.199316 + 0.135587i
\(31\) −4.96005 − 0.713147i −0.890851 − 0.128085i −0.318334 − 0.947978i \(-0.603124\pi\)
−0.572516 + 0.819893i \(0.694033\pi\)
\(32\) 3.00095 + 4.79523i 0.530499 + 0.847686i
\(33\) −4.03810 + 3.49903i −0.702942 + 0.609103i
\(34\) −2.11472 − 6.56203i −0.362672 − 1.12538i
\(35\) −3.60472 + 2.31661i −0.609309 + 0.391579i
\(36\) 0.825283 + 4.17590i 0.137547 + 0.695984i
\(37\) −6.20384 + 1.82161i −1.01991 + 0.299471i −0.748600 − 0.663022i \(-0.769274\pi\)
−0.271306 + 0.962493i \(0.587455\pi\)
\(38\) 4.93023 + 2.09751i 0.799790 + 0.340260i
\(39\) 2.55721 + 2.21583i 0.409481 + 0.354818i
\(40\) 2.64304 − 1.00716i 0.417901 − 0.159245i
\(41\) −11.1848 − 3.28416i −1.74678 − 0.512900i −0.756740 − 0.653716i \(-0.773209\pi\)
−0.990036 + 0.140817i \(0.955027\pi\)
\(42\) −0.951357 − 5.57706i −0.146798 − 0.860559i
\(43\) −4.00647 + 0.576043i −0.610981 + 0.0878458i −0.440858 − 0.897577i \(-0.645326\pi\)
−0.170123 + 0.985423i \(0.554417\pi\)
\(44\) 10.1488 − 5.29279i 1.52999 − 0.797918i
\(45\) 2.12834 0.317274
\(46\) −6.35213 + 2.37707i −0.936570 + 0.350480i
\(47\) − 2.40525i − 0.350842i −0.984494 − 0.175421i \(-0.943871\pi\)
0.984494 − 0.175421i \(-0.0561286\pi\)
\(48\) −0.142504 + 3.73180i −0.0205687 + 0.538639i
\(49\) −1.61679 − 11.2451i −0.230971 − 1.60644i
\(50\) −0.237807 − 1.39408i −0.0336310 − 0.197152i
\(51\) 1.28230 − 4.36713i 0.179558 − 0.611520i
\(52\) −4.23230 − 5.88450i −0.586914 − 0.816034i
\(53\) −6.44475 + 7.43764i −0.885255 + 1.02164i 0.114347 + 0.993441i \(0.463522\pi\)
−0.999602 + 0.0281979i \(0.991023\pi\)
\(54\) −2.65080 + 6.23077i −0.360728 + 0.847900i
\(55\) −1.61236 − 5.49118i −0.217410 − 0.740431i
\(56\) −0.779068 + 12.0946i −0.104107 + 1.61620i
\(57\) 1.91232 + 2.97563i 0.253293 + 0.394131i
\(58\) 1.40543 + 4.36109i 0.184542 + 0.572639i
\(59\) −3.78950 − 4.37331i −0.493350 − 0.569357i 0.453408 − 0.891303i \(-0.350208\pi\)
−0.946758 + 0.321947i \(0.895663\pi\)
\(60\) 1.81668 + 0.431660i 0.234532 + 0.0557271i
\(61\) 0.158817 − 1.10460i 0.0203345 − 0.141429i −0.977125 − 0.212666i \(-0.931785\pi\)
0.997459 + 0.0712365i \(0.0226945\pi\)
\(62\) 3.98597 + 5.85946i 0.506219 + 0.744152i
\(63\) −3.78850 + 8.29565i −0.477306 + 1.04515i
\(64\) 2.14504 − 7.70706i 0.268130 − 0.963383i
\(65\) −3.29670 + 1.50555i −0.408905 + 0.186741i
\(66\) 7.50504 + 0.879392i 0.923806 + 0.108246i
\(67\) −4.57690 + 7.12180i −0.559158 + 0.870066i −0.999616 − 0.0277067i \(-0.991180\pi\)
0.440458 + 0.897773i \(0.354816\pi\)
\(68\) −4.83518 + 8.46674i −0.586352 + 1.02674i
\(69\) −4.34997 − 1.06116i −0.523674 − 0.127749i
\(70\) 5.85702 + 1.55459i 0.700047 + 0.185809i
\(71\) 1.73600 − 2.70127i 0.206026 − 0.320582i −0.722829 − 0.691027i \(-0.757158\pi\)
0.928854 + 0.370445i \(0.120795\pi\)
\(72\) 3.57338 − 4.84453i 0.421127 − 0.570934i
\(73\) 5.43154 + 11.8934i 0.635714 + 1.39202i 0.903520 + 0.428547i \(0.140974\pi\)
−0.267805 + 0.963473i \(0.586298\pi\)
\(74\) 7.81905 + 4.74071i 0.908946 + 0.551096i
\(75\) 0.387844 − 0.849259i 0.0447843 − 0.0980640i
\(76\) −2.51199 − 7.14866i −0.288145 − 0.820008i
\(77\) 24.2731 + 3.48995i 2.76618 + 0.397716i
\(78\) −0.125158 − 4.78360i −0.0141713 − 0.541636i
\(79\) 10.6141 + 12.2494i 1.19418 + 1.37816i 0.907454 + 0.420151i \(0.138023\pi\)
0.286730 + 0.958011i \(0.407432\pi\)
\(80\) −3.57247 − 1.79929i −0.399414 − 0.201167i
\(81\) 1.61085 − 1.03523i 0.178983 − 0.115026i
\(82\) 7.23820 + 14.8115i 0.799325 + 1.63566i
\(83\) 1.16726 + 3.97533i 0.128124 + 0.436349i 0.998421 − 0.0561782i \(-0.0178915\pi\)
−0.870297 + 0.492527i \(0.836073\pi\)
\(84\) −4.91623 + 6.31253i −0.536405 + 0.688754i
\(85\) 3.68432 + 3.19248i 0.399621 + 0.346273i
\(86\) 4.42268 + 3.63417i 0.476910 + 0.391882i
\(87\) −0.852212 + 2.90237i −0.0913667 + 0.311166i
\(88\) −15.2061 − 5.54939i −1.62098 − 0.591567i
\(89\) −4.72278 + 0.679033i −0.500613 + 0.0719773i −0.387996 − 0.921661i \(-0.626833\pi\)
−0.112617 + 0.993638i \(0.535923\pi\)
\(90\) −2.02990 − 2.22242i −0.213970 − 0.234263i
\(91\) − 15.5295i − 1.62794i
\(92\) 8.54049 + 4.36578i 0.890407 + 0.455164i
\(93\) 4.67847i 0.485134i
\(94\) −2.51157 + 2.29401i −0.259048 + 0.236609i
\(95\) −3.75002 + 0.539172i −0.384744 + 0.0553178i
\(96\) 4.03267 − 3.41040i 0.411583 − 0.348073i
\(97\) 4.85957 − 16.5502i 0.493414 − 1.68042i −0.216626 − 0.976255i \(-0.569505\pi\)
0.710041 − 0.704161i \(-0.248677\pi\)
\(98\) −10.2001 + 12.4132i −1.03037 + 1.25393i
\(99\) −9.20539 − 7.97652i −0.925177 − 0.801670i
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.251.14 yes 480
8.3 odd 2 920.2.bb.a.251.5 yes 480
23.11 odd 22 920.2.bb.a.11.5 ✓ 480
184.11 even 22 inner 920.2.bb.b.11.14 yes 480
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
920.2.bb.a.11.5 ✓ 480 23.11 odd 22
920.2.bb.a.251.5 yes 480 8.3 odd 2
920.2.bb.b.11.14 yes 480 184.11 even 22 inner
920.2.bb.b.251.14 yes 480 1.1 even 1 trivial