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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 11.37
Character \(\chi\) \(=\) 920.11
Dual form 920.2.bb.b.251.37

$q$-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+(1.05918 + 0.937093i) q^{2} +(-0.429309 + 2.98591i) q^{3} +(0.243714 + 1.98510i) q^{4} +(0.959493 - 0.281733i) q^{5} +(-3.25279 + 2.76031i) q^{6} +(1.41433 + 1.63223i) q^{7} +(-1.60208 + 2.33095i) q^{8} +(-5.85288 - 1.71856i) q^{9} +(1.28028 + 0.600729i) q^{10} +(0.260861 - 0.405908i) q^{11} +(-6.03195 - 0.124512i) q^{12} +(0.398969 + 0.345709i) q^{13} +(-0.0315190 + 3.05418i) q^{14} +(0.429309 + 2.98591i) q^{15} +(-3.88121 + 0.967590i) q^{16} +(1.00123 - 0.457246i) q^{17} +(-4.58879 - 7.30495i) q^{18} +(-2.30954 - 1.05473i) q^{19} +(0.793108 + 1.83602i) q^{20} +(-5.48088 + 3.52235i) q^{21} +(0.656672 - 0.185477i) q^{22} +(4.73300 + 0.773739i) q^{23} +(-6.27222 - 5.78438i) q^{24} +(0.841254 - 0.540641i) q^{25} +(0.0986179 + 0.740038i) q^{26} +(3.88472 - 8.50636i) q^{27} +(-2.89544 + 3.20539i) q^{28} +(-7.04860 + 3.21899i) q^{29} +(-2.34336 + 3.56491i) q^{30} +(4.41035 - 0.634113i) q^{31} +(-5.01761 - 2.61220i) q^{32} +(1.10002 + 0.953169i) q^{33} +(1.48896 + 0.453940i) q^{34} +(1.81690 + 1.16765i) q^{35} +(1.98508 - 12.0374i) q^{36} +(-3.42102 - 1.00450i) q^{37} +(-1.45783 - 3.28140i) q^{38} +(-1.20354 + 1.04287i) q^{39} +(-0.880482 + 2.68789i) q^{40} +(5.07992 - 1.49160i) q^{41} +(-9.10599 - 1.40530i) q^{42} +(9.40966 + 1.35291i) q^{43} +(0.869342 + 0.418909i) q^{44} -6.09997 q^{45} +(4.28803 + 5.25479i) q^{46} -1.55715i q^{47} +(-1.22290 - 12.0043i) q^{48} +(0.332374 - 2.31171i) q^{49} +(1.39767 + 0.215698i) q^{50} +(0.935459 + 3.18588i) q^{51} +(-0.589030 + 0.876246i) q^{52} +(0.419139 + 0.483712i) q^{53} +(12.0859 - 5.36940i) q^{54} +(0.135937 - 0.462959i) q^{55} +(-6.07053 + 0.681778i) q^{56} +(4.14084 - 6.44327i) q^{57} +(-10.4822 - 3.19571i) q^{58} +(7.29413 - 8.41787i) q^{59} +(-5.82269 + 1.57993i) q^{60} +(1.47679 + 10.2713i) q^{61} +(5.26557 + 3.46127i) q^{62} +(-5.47285 - 11.9839i) q^{63} +(-2.86666 - 7.46875i) q^{64} +(0.480205 + 0.219302i) q^{65} +(0.271904 + 2.04039i) q^{66} +(-0.346109 - 0.538557i) q^{67} +(1.15169 + 1.87610i) q^{68} +(-4.34224 + 13.8002i) q^{69} +(0.830220 + 2.93935i) q^{70} +(3.06304 + 4.76619i) q^{71} +(13.3827 - 10.8895i) q^{72} +(5.17201 - 11.3251i) q^{73} +(-2.68216 - 4.26976i) q^{74} +(1.25315 + 2.74401i) q^{75} +(1.53087 - 4.84170i) q^{76} +(1.03148 - 0.148304i) q^{77} +(-2.25203 - 0.0232408i) q^{78} +(-1.22567 + 1.41450i) q^{79} +(-3.45139 + 2.02186i) q^{80} +(8.33661 + 5.35761i) q^{81} +(6.77831 + 3.18049i) q^{82} +(-2.13633 + 7.27566i) q^{83} +(-8.32796 - 10.0216i) q^{84} +(0.831851 - 0.720803i) q^{85} +(8.69871 + 10.2507i) q^{86} +(-6.58558 - 22.4284i) q^{87} +(0.528230 + 1.25835i) q^{88} +(-17.3851 - 2.49959i) q^{89} +(-6.46095 - 5.71624i) q^{90} +1.14016i q^{91} +(-0.382448 + 9.58404i) q^{92} +13.4412i q^{93} +(1.45920 - 1.64930i) q^{94} +(-2.51314 - 0.361335i) q^{95} +(9.95391 - 13.8607i) q^{96} +(4.34339 + 14.7922i) q^{97} +(2.51833 - 2.13705i) q^{98} +(-2.22437 + 1.92743i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.05918 + 0.937093i 0.748952 + 0.662625i
\(3\) −0.429309 + 2.98591i −0.247862 + 1.72392i 0.362664 + 0.931920i \(0.381867\pi\)
−0.610525 + 0.791997i \(0.709042\pi\)
\(4\) 0.243714 + 1.98510i 0.121857 + 0.992548i
\(5\) 0.959493 − 0.281733i 0.429098 − 0.125995i
\(6\) −3.25279 + 2.76031i −1.32795 + 1.12689i
\(7\) 1.41433 + 1.63223i 0.534568 + 0.616925i 0.957218 − 0.289369i \(-0.0934453\pi\)
−0.422650 + 0.906293i \(0.638900\pi\)
\(8\) −1.60208 + 2.33095i −0.566422 + 0.824116i
\(9\) −5.85288 − 1.71856i −1.95096 − 0.572854i
\(10\) 1.28028 + 0.600729i 0.404861 + 0.189967i
\(11\) 0.260861 − 0.405908i 0.0786526 − 0.122386i −0.799699 − 0.600402i \(-0.795007\pi\)
0.878351 + 0.478016i \(0.158644\pi\)
\(12\) −6.03195 − 0.124512i −1.74127 − 0.0359436i
\(13\) 0.398969 + 0.345709i 0.110654 + 0.0958823i 0.708431 − 0.705780i \(-0.249404\pi\)
−0.597776 + 0.801663i \(0.703949\pi\)
\(14\) −0.0315190 + 3.05418i −0.00842382 + 0.816265i
\(15\) 0.429309 + 2.98591i 0.110847 + 0.770959i
\(16\) −3.88121 + 0.967590i −0.970302 + 0.241898i
\(17\) 1.00123 − 0.457246i 0.242834 − 0.110898i −0.290281 − 0.956941i \(-0.593749\pi\)
0.533115 + 0.846043i \(0.321021\pi\)
\(18\) −4.58879 − 7.30495i −1.08159 − 1.72179i
\(19\) −2.30954 − 1.05473i −0.529844 − 0.241972i 0.132485 − 0.991185i \(-0.457705\pi\)
−0.662329 + 0.749213i \(0.730432\pi\)
\(20\) 0.793108 + 1.83602i 0.177344 + 0.410547i
\(21\) −5.48088 + 3.52235i −1.19603 + 0.768639i
\(22\) 0.656672 − 0.185477i 0.140003 − 0.0395439i
\(23\) 4.73300 + 0.773739i 0.986900 + 0.161336i
\(24\) −6.27222 − 5.78438i −1.28031 − 1.18073i
\(25\) 0.841254 − 0.540641i 0.168251 − 0.108128i
\(26\) 0.0986179 + 0.740038i 0.0193406 + 0.145133i
\(27\) 3.88472 − 8.50636i 0.747615 − 1.63705i
\(28\) −2.89544 + 3.20539i −0.547186 + 0.605761i
\(29\) −7.04860 + 3.21899i −1.30889 + 0.597751i −0.942965 − 0.332892i \(-0.891976\pi\)
−0.365928 + 0.930643i \(0.619248\pi\)
\(30\) −2.34336 + 3.56491i −0.427837 + 0.650861i
\(31\) 4.41035 − 0.634113i 0.792123 − 0.113890i 0.265636 − 0.964073i \(-0.414418\pi\)
0.526487 + 0.850183i \(0.323509\pi\)
\(32\) −5.01761 − 2.61220i −0.886996 − 0.461776i
\(33\) 1.10002 + 0.953169i 0.191488 + 0.165925i
\(34\) 1.48896 + 0.453940i 0.255355 + 0.0778500i
\(35\) 1.81690 + 1.16765i 0.307111 + 0.197369i
\(36\) 1.98508 − 12.0374i 0.330847 − 2.00623i
\(37\) −3.42102 − 1.00450i −0.562413 − 0.165139i −0.0118450 − 0.999930i \(-0.503770\pi\)
−0.550568 + 0.834791i \(0.685589\pi\)
\(38\) −1.45783 − 3.28140i −0.236491 − 0.532313i
\(39\) −1.20354 + 1.04287i −0.192720 + 0.166993i
\(40\) −0.880482 + 2.68789i −0.139216 + 0.424993i
\(41\) 5.07992 − 1.49160i 0.793351 − 0.232949i 0.140147 − 0.990131i \(-0.455243\pi\)
0.653204 + 0.757182i \(0.273424\pi\)
\(42\) −9.10599 − 1.40530i −1.40508 − 0.216843i
\(43\) 9.40966 + 1.35291i 1.43496 + 0.206316i 0.815521 − 0.578728i \(-0.196451\pi\)
0.619440 + 0.785044i \(0.287360\pi\)
\(44\) 0.869342 + 0.418909i 0.131058 + 0.0631529i
\(45\) −6.09997 −0.909330
\(46\) 4.28803 + 5.25479i 0.632235 + 0.774777i
\(47\) − 1.55715i − 0.227134i −0.993530 − 0.113567i \(-0.963772\pi\)
0.993530 − 0.113567i \(-0.0362276\pi\)
\(48\) −1.22290 − 12.0043i −0.176510 − 1.73268i
\(49\) 0.332374 − 2.31171i 0.0474821 − 0.330245i
\(50\) 1.39767 + 0.215698i 0.197660 + 0.0305043i
\(51\) 0.935459 + 3.18588i 0.130990 + 0.446112i
\(52\) −0.589030 + 0.876246i −0.0816838 + 0.121513i
\(53\) 0.419139 + 0.483712i 0.0575732 + 0.0664430i 0.783806 − 0.621006i \(-0.213276\pi\)
−0.726232 + 0.687449i \(0.758730\pi\)
\(54\) 12.0859 − 5.36940i 1.64468 − 0.730682i
\(55\) 0.135937 − 0.462959i 0.0183297 − 0.0624254i
\(56\) −6.07053 + 0.681778i −0.811208 + 0.0911064i
\(57\) 4.14084 − 6.44327i 0.548467 − 0.853432i
\(58\) −10.4822 − 3.19571i −1.37638 − 0.419618i
\(59\) 7.29413 − 8.41787i 0.949615 − 1.09591i −0.0456737 − 0.998956i \(-0.514543\pi\)
0.995289 − 0.0969575i \(-0.0309111\pi\)
\(60\) −5.82269 + 1.57993i −0.751706 + 0.203968i
\(61\) 1.47679 + 10.2713i 0.189084 + 1.31511i 0.834386 + 0.551181i \(0.185823\pi\)
−0.645301 + 0.763928i \(0.723268\pi\)
\(62\) 5.26557 + 3.46127i 0.668728 + 0.439582i
\(63\) −5.47285 − 11.9839i −0.689514 − 1.50982i
\(64\) −2.86666 − 7.46875i −0.358333 − 0.933594i
\(65\) 0.480205 + 0.219302i 0.0595622 + 0.0272011i
\(66\) 0.271904 + 2.04039i 0.0334690 + 0.251155i
\(67\) −0.346109 − 0.538557i −0.0422840 − 0.0657952i 0.819468 − 0.573125i \(-0.194269\pi\)
−0.861752 + 0.507330i \(0.830633\pi\)
\(68\) 1.15169 + 1.87610i 0.139663 + 0.227510i
\(69\) −4.34224 + 13.8002i −0.522744 + 1.66134i
\(70\) 0.830220 + 2.93935i 0.0992303 + 0.351319i
\(71\) 3.06304 + 4.76619i 0.363516 + 0.565642i 0.974046 − 0.226351i \(-0.0726797\pi\)
−0.610529 + 0.791994i \(0.709043\pi\)
\(72\) 13.3827 − 10.8895i 1.57716 − 1.28334i
\(73\) 5.17201 − 11.3251i 0.605338 − 1.32551i −0.320379 − 0.947289i \(-0.603810\pi\)
0.925717 − 0.378216i \(-0.123463\pi\)
\(74\) −2.68216 − 4.26976i −0.311794 − 0.496350i
\(75\) 1.25315 + 2.74401i 0.144701 + 0.316851i
\(76\) 1.53087 − 4.84170i 0.175603 − 0.555382i
\(77\) 1.03148 − 0.148304i 0.117548 − 0.0169009i
\(78\) −2.25203 − 0.0232408i −0.254992 − 0.00263150i
\(79\) −1.22567 + 1.41450i −0.137899 + 0.159143i −0.820498 − 0.571649i \(-0.806304\pi\)
0.682600 + 0.730793i \(0.260849\pi\)
\(80\) −3.45139 + 2.02186i −0.385877 + 0.226051i
\(81\) 8.33661 + 5.35761i 0.926290 + 0.595291i
\(82\) 6.77831 + 3.18049i 0.748539 + 0.351226i
\(83\) −2.13633 + 7.27566i −0.234492 + 0.798607i 0.755212 + 0.655480i \(0.227534\pi\)
−0.989704 + 0.143127i \(0.954284\pi\)
\(84\) −8.32796 − 10.0216i −0.908655 − 1.09345i
\(85\) 0.831851 − 0.720803i 0.0902269 − 0.0781820i
\(86\) 8.69871 + 10.2507i 0.938006 + 1.10536i
\(87\) −6.58558 − 22.4284i −0.706049 − 2.40458i
\(88\) 0.528230 + 1.25835i 0.0563096 + 0.134141i
\(89\) −17.3851 − 2.49959i −1.84281 − 0.264956i −0.869415 − 0.494082i \(-0.835504\pi\)
−0.973397 + 0.229126i \(0.926413\pi\)
\(90\) −6.46095 − 5.71624i −0.681044 − 0.602545i
\(91\) 1.14016i 0.119521i
\(92\) −0.382448 + 9.58404i −0.0398730 + 0.999205i
\(93\) 13.4412i 1.39378i
\(94\) 1.45920 − 1.64930i 0.150505 − 0.170112i
\(95\) −2.51314 − 0.361335i −0.257842 − 0.0370721i
\(96\) 9.95391 − 13.8607i 1.01592 − 1.41465i
\(97\) 4.34339 + 14.7922i 0.441005 + 1.50192i 0.817732 + 0.575600i \(0.195231\pi\)
−0.376727 + 0.926324i \(0.622951\pi\)
\(98\) 2.51833 − 2.13705i 0.254390 − 0.215875i
\(99\) −2.22437 + 1.92743i −0.223557 + 0.193714i
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.11.37 yes 480
8.3 odd 2 920.2.bb.a.11.19 ✓ 480
23.21 odd 22 920.2.bb.a.251.19 yes 480
184.67 even 22 inner 920.2.bb.b.251.37 yes 480
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
920.2.bb.a.11.19 ✓ 480 8.3 odd 2
920.2.bb.a.251.19 yes 480 23.21 odd 22
920.2.bb.b.11.37 yes 480 1.1 even 1 trivial
920.2.bb.b.251.37 yes 480 184.67 even 22 inner