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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 11.19
Character \(\chi\) \(=\) 920.11
Dual form 920.2.bb.a.251.19

$q$-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+(-0.412411 - 1.35274i) q^{2} +(-0.429309 + 2.98591i) q^{3} +(-1.65983 + 1.11577i) q^{4} +(-0.959493 + 0.281733i) q^{5} +(4.21623 - 0.650679i) q^{6} +(-1.41433 - 1.63223i) q^{7} +(2.19389 + 1.78517i) q^{8} +(-5.85288 - 1.71856i) q^{9} +(0.776818 + 1.18176i) q^{10} +(0.260861 - 0.405908i) q^{11} +(-2.61902 - 5.43513i) q^{12} +(-0.398969 - 0.345709i) q^{13} +(-1.62470 + 2.58638i) q^{14} +(-0.429309 - 2.98591i) q^{15} +(1.51009 - 3.70400i) q^{16} +(1.00123 - 0.457246i) q^{17} +(0.0890221 + 8.62620i) q^{18} +(-2.30954 - 1.05473i) q^{19} +(1.27825 - 1.53821i) 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} +(-0.303116 + 0.682277i) q^{26} +(3.88472 - 8.50636i) q^{27} +(4.16876 + 1.13115i) q^{28} +(7.04860 - 3.21899i) q^{29} +(-3.86212 + 1.81217i) q^{30} +(-4.41035 + 0.634113i) q^{31} +(-5.63334 - 0.515200i) q^{32} +(1.10002 + 0.953169i) q^{33} +(-1.03145 - 1.16583i) q^{34} +(1.81690 + 1.16765i) q^{35} +(11.6323 - 3.67797i) q^{36} +(3.42102 + 1.00450i) q^{37} +(-0.474301 + 3.55920i) q^{38} +(1.20354 - 1.04287i) q^{39} +(-2.60796 - 1.09477i) q^{40} +(5.07992 - 1.49160i) q^{41} +(-7.02521 - 5.96157i) q^{42} +(9.40966 + 1.35291i) q^{43} +(0.0199156 + 0.964802i) q^{44} +6.09997 q^{45} +(0.905274 + 6.72164i) q^{46} +1.55715i q^{47} +(10.4115 + 6.09917i) q^{48} +(0.332374 - 2.31171i) q^{49} +(-1.07829 - 0.915034i) q^{50} +(0.935459 + 3.18588i) q^{51} +(1.04796 + 0.128659i) q^{52} +(-0.419139 - 0.483712i) q^{53} +(-13.1090 - 1.74692i) q^{54} +(-0.135937 + 0.462959i) q^{55} +(-0.189087 - 6.10576i) q^{56} +(4.14084 - 6.44327i) q^{57} +(-7.26139 - 8.20741i) q^{58} +(7.29413 - 8.41787i) q^{59} +(4.04419 + 4.47710i) q^{60} +(-1.47679 - 10.2713i) q^{61} +(2.67667 + 5.70456i) q^{62} +(5.47285 + 11.9839i) q^{63} +(1.62632 + 7.83295i) q^{64} +(0.480205 + 0.219302i) q^{65} +(0.835735 - 1.88114i) 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} +(-9.77266 - 14.2187i) q^{72} +(5.17201 - 11.3251i) q^{73} +(-0.0520336 - 5.04204i) q^{74} +(1.25315 + 2.74401i) q^{75} +(5.01029 - 0.826246i) q^{76} +(-1.03148 + 0.148304i) q^{77} +(-1.90709 - 1.19799i) q^{78} +(1.22567 - 1.41450i) q^{79} +(-0.405388 + 3.97940i) q^{80} +(8.33661 + 5.35761i) q^{81} +(-4.11277 - 6.25668i) q^{82} +(-2.13633 + 7.27566i) q^{83} +(-5.16720 + 11.9619i) q^{84} +(-0.831851 + 0.720803i) q^{85} +(-2.05052 - 13.2868i) q^{86} +(6.58558 + 22.4284i) q^{87} +(1.29692 - 0.424836i) q^{88} +(-17.3851 - 2.49959i) q^{89} +(-2.51570 - 8.25170i) q^{90} +1.14016i q^{91} +(8.71932 - 3.99669i) q^{92} -13.4412i q^{93} +(2.10643 - 0.642187i) q^{94} +(2.51314 + 0.361335i) q^{95} +(3.95679 - 16.5995i) q^{96} +(4.34339 + 14.7922i) q^{97} +(-3.26423 + 0.503760i) q^{98} +(-2.22437 + 1.92743i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 480 q + 2 q^{2} - 4 q^{4} - 48 q^{5} + 5 q^{6} - q^{8} - 48 q^{9} + 2 q^{10} + 3 q^{12} + 32 q^{16} + 7 q^{18} - 4 q^{20} + 8 q^{21} + 4 q^{23} + 2 q^{24} - 48 q^{25} + 7 q^{26} + 12 q^{27} + 5 q^{30}+ \cdots - 98 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.412411 − 1.35274i −0.291619 − 0.956535i
\(3\) −0.429309 + 2.98591i −0.247862 + 1.72392i 0.362664 + 0.931920i \(0.381867\pi\)
−0.610525 + 0.791997i \(0.709042\pi\)
\(4\) −1.65983 + 1.11577i −0.829917 + 0.557887i
\(5\) −0.959493 + 0.281733i −0.429098 + 0.125995i
\(6\) 4.21623 − 0.650679i 1.72127 − 0.265638i
\(7\) −1.41433 − 1.63223i −0.534568 − 0.616925i 0.422650 − 0.906293i \(-0.361100\pi\)
−0.957218 + 0.289369i \(0.906555\pi\)
\(8\) 2.19389 + 1.78517i 0.775658 + 0.631154i
\(9\) −5.85288 − 1.71856i −1.95096 − 0.572854i
\(10\) 0.776818 + 1.18176i 0.245651 + 0.373705i
\(11\) 0.260861 − 0.405908i 0.0786526 − 0.122386i −0.799699 − 0.600402i \(-0.795007\pi\)
0.878351 + 0.478016i \(0.158644\pi\)
\(12\) −2.61902 − 5.43513i −0.756046 − 1.56899i
\(13\) −0.398969 − 0.345709i −0.110654 − 0.0958823i 0.597776 − 0.801663i \(-0.296051\pi\)
−0.708431 + 0.705780i \(0.750596\pi\)
\(14\) −1.62470 + 2.58638i −0.434219 + 0.691240i
\(15\) −0.429309 − 2.98591i −0.110847 − 0.770959i
\(16\) 1.51009 − 3.70400i 0.377524 − 0.926000i
\(17\) 1.00123 − 0.457246i 0.242834 − 0.110898i −0.290281 − 0.956941i \(-0.593749\pi\)
0.533115 + 0.846043i \(0.321021\pi\)
\(18\) 0.0890221 + 8.62620i 0.0209827 + 2.03322i
\(19\) −2.30954 − 1.05473i −0.529844 − 0.241972i 0.132485 − 0.991185i \(-0.457705\pi\)
−0.662329 + 0.749213i \(0.730432\pi\)
\(20\) 1.27825 − 1.53821i 0.285825 − 0.343954i
\(21\) 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
\(26\) −0.303116 + 0.682277i −0.0594459 + 0.133806i
\(27\) 3.88472 − 8.50636i 0.747615 − 1.63705i
\(28\) 4.16876 + 1.13115i 0.787821 + 0.213767i
\(29\) 7.04860 − 3.21899i 1.30889 − 0.597751i 0.365928 − 0.930643i \(-0.380752\pi\)
0.942965 + 0.332892i \(0.108024\pi\)
\(30\) −3.86212 + 1.81217i −0.705124 + 0.330855i
\(31\) −4.41035 + 0.634113i −0.792123 + 0.113890i −0.526487 − 0.850183i \(-0.676491\pi\)
−0.265636 + 0.964073i \(0.585582\pi\)
\(32\) −5.63334 − 0.515200i −0.995844 − 0.0910753i
\(33\) 1.10002 + 0.953169i 0.191488 + 0.165925i
\(34\) −1.03145 − 1.16583i −0.176893 − 0.199939i
\(35\) 1.81690 + 1.16765i 0.307111 + 0.197369i
\(36\) 11.6323 − 3.67797i 1.93872 − 0.612995i
\(37\) 3.42102 + 1.00450i 0.562413 + 0.165139i 0.550568 − 0.834791i \(-0.314411\pi\)
0.0118450 + 0.999930i \(0.496230\pi\)
\(38\) −0.474301 + 3.55920i −0.0769417 + 0.577378i
\(39\) 1.20354 − 1.04287i 0.192720 − 0.166993i
\(40\) −2.60796 − 1.09477i −0.412355 − 0.173098i
\(41\) 5.07992 − 1.49160i 0.793351 − 0.232949i 0.140147 − 0.990131i \(-0.455243\pi\)
0.653204 + 0.757182i \(0.273424\pi\)
\(42\) −7.02521 − 5.96157i −1.08401 − 0.919890i
\(43\) 9.40966 + 1.35291i 1.43496 + 0.206316i 0.815521 − 0.578728i \(-0.196451\pi\)
0.619440 + 0.785044i \(0.287360\pi\)
\(44\) 0.0199156 + 0.964802i 0.00300238 + 0.145449i
\(45\) 6.09997 0.909330
\(46\) 0.905274 + 6.72164i 0.133475 + 0.991052i
\(47\) 1.55715i 0.227134i 0.993530 + 0.113567i \(0.0362276\pi\)
−0.993530 + 0.113567i \(0.963772\pi\)
\(48\) 10.4115 + 6.09917i 1.50277 + 0.880339i
\(49\) 0.332374 − 2.31171i 0.0474821 − 0.330245i
\(50\) −1.07829 − 0.915034i −0.152493 − 0.129405i
\(51\) 0.935459 + 3.18588i 0.130990 + 0.446112i
\(52\) 1.04796 + 0.128659i 0.145325 + 0.0178418i
\(53\) −0.419139 − 0.483712i −0.0575732 − 0.0664430i 0.726232 − 0.687449i \(-0.241270\pi\)
−0.783806 + 0.621006i \(0.786724\pi\)
\(54\) −13.1090 − 1.74692i −1.78391 − 0.237725i
\(55\) −0.135937 + 0.462959i −0.0183297 + 0.0624254i
\(56\) −0.189087 − 6.10576i −0.0252679 − 0.815917i
\(57\) 4.14084 − 6.44327i 0.548467 − 0.853432i
\(58\) −7.26139 − 8.20741i −0.953468 − 1.07769i
\(59\) 7.29413 − 8.41787i 0.949615 − 1.09591i −0.0456737 − 0.998956i \(-0.514543\pi\)
0.995289 − 0.0969575i \(-0.0309111\pi\)
\(60\) 4.04419 + 4.47710i 0.522102 + 0.577992i
\(61\) −1.47679 − 10.2713i −0.189084 − 1.31511i −0.834386 − 0.551181i \(-0.814177\pi\)
0.645301 − 0.763928i \(-0.276732\pi\)
\(62\) 2.67667 + 5.70456i 0.339938 + 0.724480i
\(63\) 5.47285 + 11.9839i 0.689514 + 1.50982i
\(64\) 1.62632 + 7.83295i 0.203290 + 0.979119i
\(65\) 0.480205 + 0.219302i 0.0595622 + 0.0272011i
\(66\) 0.835735 − 1.88114i 0.102872 − 0.231552i
\(67\) −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 0.610529 − 0.791994i \(-0.290957\pi\)
−0.974046 + 0.226351i \(0.927320\pi\)
\(72\) −9.77266 − 14.2187i −1.15172 − 1.67569i
\(73\) 5.17201 − 11.3251i 0.605338 − 1.32551i −0.320379 − 0.947289i \(-0.603810\pi\)
0.925717 − 0.378216i \(-0.123463\pi\)
\(74\) −0.0520336 − 5.04204i −0.00604878 − 0.586125i
\(75\) 1.25315 + 2.74401i 0.144701 + 0.316851i
\(76\) 5.01029 − 0.826246i 0.574720 − 0.0947769i
\(77\) −1.03148 + 0.148304i −0.117548 + 0.0169009i
\(78\) −1.90709 − 1.19799i −0.215935 − 0.135645i
\(79\) 1.22567 − 1.41450i 0.137899 − 0.159143i −0.682600 − 0.730793i \(-0.739151\pi\)
0.820498 + 0.571649i \(0.193696\pi\)
\(80\) −0.405388 + 3.97940i −0.0453238 + 0.444911i
\(81\) 8.33661 + 5.35761i 0.926290 + 0.595291i
\(82\) −4.11277 − 6.25668i −0.454180 − 0.690935i
\(83\) −2.13633 + 7.27566i −0.234492 + 0.798607i 0.755212 + 0.655480i \(0.227534\pi\)
−0.989704 + 0.143127i \(0.954284\pi\)
\(84\) −5.16720 + 11.9619i −0.563788 + 1.30515i
\(85\) −0.831851 + 0.720803i −0.0902269 + 0.0781820i
\(86\) −2.05052 − 13.2868i −0.221113 − 1.43275i
\(87\) 6.58558 + 22.4284i 0.706049 + 2.40458i
\(88\) 1.29692 − 0.424836i 0.138252 − 0.0452877i
\(89\) −17.3851 − 2.49959i −1.84281 − 0.264956i −0.869415 − 0.494082i \(-0.835504\pi\)
−0.973397 + 0.229126i \(0.926413\pi\)
\(90\) −2.51570 − 8.25170i −0.265178 − 0.869806i
\(91\) 1.14016i 0.119521i
\(92\) 8.71932 − 3.99669i 0.909052 − 0.416683i
\(93\) − 13.4412i − 1.39378i
\(94\) 2.10643 − 0.642187i 0.217261 − 0.0662365i
\(95\) 2.51314 + 0.361335i 0.257842 + 0.0370721i
\(96\) 3.95679 − 16.5995i 0.403838 − 1.69418i
\(97\) 4.34339 + 14.7922i 0.441005 + 1.50192i 0.817732 + 0.575600i \(0.195231\pi\)
−0.376727 + 0.926324i \(0.622951\pi\)
\(98\) −3.26423 + 0.503760i −0.329737 + 0.0508875i
\(99\) −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.a.11.19 ✓ 480
8.3 odd 2 920.2.bb.b.11.37 yes 480
23.21 odd 22 920.2.bb.b.251.37 yes 480
184.67 even 22 inner 920.2.bb.a.251.19 yes 480
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
920.2.bb.a.11.19 ✓ 480 1.1 even 1 trivial
920.2.bb.a.251.19 yes 480 184.67 even 22 inner
920.2.bb.b.11.37 yes 480 8.3 odd 2
920.2.bb.b.251.37 yes 480 23.21 odd 22