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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 11.3
Character \(\chi\) \(=\) 920.11
Dual form 920.2.bb.a.251.3

$q$-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+(-1.41066 + 0.100200i) q^{2} +(0.472161 - 3.28395i) q^{3} +(1.97992 - 0.282697i) q^{4} +(-0.959493 + 0.281733i) q^{5} +(-0.337006 + 4.67985i) q^{6} +(-2.50066 - 2.88591i) q^{7} +(-2.76467 + 0.597177i) q^{8} +(-7.68293 - 2.25591i) q^{9} +(1.32529 - 0.493570i) q^{10} +(3.41038 - 5.30666i) q^{11} +(0.00647816 - 6.63544i) q^{12} +(-2.21330 - 1.91783i) q^{13} +(3.81675 + 3.82047i) q^{14} +(0.472161 + 3.28395i) q^{15} +(3.84017 - 1.11943i) q^{16} +(2.31222 - 1.05596i) q^{17} +(11.0640 + 2.41249i) q^{18} +(4.58611 + 2.09441i) q^{19} +(-1.82007 + 0.829053i) q^{20} +(-10.6579 + 6.84943i) q^{21} +(-4.27916 + 7.82761i) q^{22} +(-4.07093 - 2.53526i) q^{23} +(0.655734 + 9.36100i) q^{24} +(0.841254 - 0.540641i) q^{25} +(3.31438 + 2.48364i) q^{26} +(-6.90119 + 15.1115i) q^{27} +(-5.76694 - 5.00695i) q^{28} +(-2.30440 + 1.05238i) q^{29} +(-0.995111 - 4.58523i) q^{30} +(7.20895 - 1.03649i) q^{31} +(-5.30500 + 1.96393i) q^{32} +(-15.8166 - 13.7051i) q^{33} +(-3.15595 + 1.72128i) q^{34} +(3.21242 + 2.06450i) q^{35} +(-15.8493 - 2.29458i) q^{36} +(4.44531 + 1.30526i) q^{37} +(-6.67930 - 2.49497i) q^{38} +(-7.34310 + 6.36284i) q^{39} +(2.48443 - 1.35188i) q^{40} +(-1.04437 + 0.306656i) q^{41} +(14.3484 - 10.7301i) q^{42} +(2.20221 + 0.316629i) q^{43} +(5.25211 - 11.4709i) q^{44} +8.00728 q^{45} +(5.99672 + 3.16849i) q^{46} +3.74592i q^{47} +(-1.86299 - 13.1395i) q^{48} +(-1.07900 + 7.50461i) q^{49} +(-1.13255 + 0.846954i) q^{50} +(-2.37597 - 8.09181i) q^{51} +(-4.92432 - 3.17146i) q^{52} +(2.63107 + 3.03642i) q^{53} +(8.22106 - 22.0087i) q^{54} +(-1.77718 + 6.05252i) q^{55} +(8.63689 + 6.48525i) q^{56} +(9.04332 - 14.0717i) q^{57} +(3.14527 - 1.71545i) q^{58} +(-6.58513 + 7.59965i) q^{59} +(1.86320 + 6.36848i) q^{60} +(0.283392 + 1.97104i) q^{61} +(-10.0655 + 2.18447i) q^{62} +(12.7020 + 27.8135i) q^{63} +(7.28676 - 3.30199i) q^{64} +(2.66396 + 1.21659i) q^{65} +(23.6851 + 17.7485i) q^{66} +(-4.44821 - 6.92155i) q^{67} +(4.27950 - 2.74437i) q^{68} +(-10.2478 + 12.1717i) q^{69} +(-4.73849 - 2.59042i) q^{70} +(-0.130034 - 0.202337i) q^{71} +(22.5879 + 1.64877i) q^{72} +(1.49297 - 3.26915i) q^{73} +(-6.40161 - 1.39586i) q^{74} +(-1.37823 - 3.01791i) q^{75} +(9.67222 + 2.85028i) q^{76} +(-23.8428 + 3.42807i) q^{77} +(9.72106 - 9.71157i) q^{78} +(6.19095 - 7.14474i) q^{79} +(-3.36923 + 2.15599i) q^{80} +(26.1585 + 16.8111i) q^{81} +(1.44253 - 0.537233i) q^{82} +(0.562599 - 1.91603i) q^{83} +(-19.1655 + 16.5743i) q^{84} +(-1.92106 + 1.66461i) q^{85} +(-3.13829 - 0.225995i) q^{86} +(2.36793 + 8.06442i) q^{87} +(-6.25956 + 16.7078i) q^{88} +(-6.65719 - 0.957159i) q^{89} +(-11.2955 + 0.802332i) q^{90} +11.1832i q^{91} +(-8.77682 - 3.86878i) q^{92} -24.1633i q^{93} +(-0.375342 - 5.28421i) q^{94} +(-4.99041 - 0.717512i) q^{95} +(3.94463 + 18.3486i) q^{96} +(-2.22710 - 7.58480i) q^{97} +(0.770138 - 10.6946i) q^{98} +(-38.1731 + 33.0772i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.41066 + 0.100200i −0.997487 + 0.0708523i
\(3\) 0.472161 − 3.28395i 0.272602 − 1.89599i −0.148394 − 0.988928i \(-0.547410\pi\)
0.420996 − 0.907063i \(-0.361681\pi\)
\(4\) 1.97992 − 0.282697i 0.989960 − 0.141348i
\(5\) −0.959493 + 0.281733i −0.429098 + 0.125995i
\(6\) −0.337006 + 4.67985i −0.137582 + 1.91054i
\(7\) −2.50066 − 2.88591i −0.945160 − 1.09077i −0.995754 − 0.0920532i \(-0.970657\pi\)
0.0505944 − 0.998719i \(-0.483888\pi\)
\(8\) −2.76467 + 0.597177i −0.977457 + 0.211134i
\(9\) −7.68293 − 2.25591i −2.56098 − 0.751971i
\(10\) 1.32529 − 0.493570i 0.419093 − 0.156081i
\(11\) 3.41038 − 5.30666i 1.02827 − 1.60002i 0.253957 − 0.967215i \(-0.418268\pi\)
0.774313 − 0.632803i \(-0.218096\pi\)
\(12\) 0.00647816 − 6.63544i 0.00187008 − 1.91549i
\(13\) −2.21330 − 1.91783i −0.613858 − 0.531911i 0.291494 − 0.956573i \(-0.405848\pi\)
−0.905352 + 0.424662i \(0.860393\pi\)
\(14\) 3.81675 + 3.82047i 1.02007 + 1.02106i
\(15\) 0.472161 + 3.28395i 0.121911 + 0.847913i
\(16\) 3.84017 − 1.11943i 0.960041 − 0.279859i
\(17\) 2.31222 − 1.05596i 0.560796 − 0.256107i −0.114788 − 0.993390i \(-0.536619\pi\)
0.675584 + 0.737283i \(0.263892\pi\)
\(18\) 11.0640 + 2.41249i 2.60782 + 0.568630i
\(19\) 4.58611 + 2.09441i 1.05213 + 0.480490i 0.864961 − 0.501838i \(-0.167343\pi\)
0.187165 + 0.982328i \(0.440070\pi\)
\(20\) −1.82007 + 0.829053i −0.406981 + 0.185382i
\(21\) −10.6579 + 6.84943i −2.32575 + 1.49467i
\(22\) −4.27916 + 7.82761i −0.912320 + 1.66885i
\(23\) −4.07093 − 2.53526i −0.848847 − 0.528639i
\(24\) 0.655734 + 9.36100i 0.133851 + 1.91081i
\(25\) 0.841254 − 0.540641i 0.168251 − 0.108128i
\(26\) 3.31438 + 2.48364i 0.650002 + 0.487081i
\(27\) −6.90119 + 15.1115i −1.32814 + 2.90821i
\(28\) −5.76694 − 5.00695i −1.08985 − 0.946224i
\(29\) −2.30440 + 1.05238i −0.427916 + 0.195423i −0.617718 − 0.786400i \(-0.711943\pi\)
0.189802 + 0.981822i \(0.439215\pi\)
\(30\) −0.995111 − 4.58523i −0.181682 − 0.837144i
\(31\) 7.20895 − 1.03649i 1.29477 − 0.186159i 0.539728 − 0.841840i \(-0.318527\pi\)
0.755039 + 0.655680i \(0.227618\pi\)
\(32\) −5.30500 + 1.96393i −0.937800 + 0.347176i
\(33\) −15.8166 − 13.7051i −2.75331 − 2.38576i
\(34\) −3.15595 + 1.72128i −0.541241 + 0.295197i
\(35\) 3.21242 + 2.06450i 0.542998 + 0.348964i
\(36\) −15.8493 − 2.29458i −2.64155 − 0.382431i
\(37\) 4.44531 + 1.30526i 0.730804 + 0.214584i 0.625905 − 0.779899i \(-0.284730\pi\)
0.104899 + 0.994483i \(0.466548\pi\)
\(38\) −6.67930 − 2.49497i −1.08353 − 0.404737i
\(39\) −7.34310 + 6.36284i −1.17584 + 1.01887i
\(40\) 2.48443 − 1.35188i 0.392823 − 0.213752i
\(41\) −1.04437 + 0.306656i −0.163104 + 0.0478916i −0.362265 − 0.932075i \(-0.617996\pi\)
0.199161 + 0.979967i \(0.436178\pi\)
\(42\) 14.3484 − 10.7301i 2.21400 − 1.65570i
\(43\) 2.20221 + 0.316629i 0.335833 + 0.0482855i 0.308169 − 0.951332i \(-0.400284\pi\)
0.0276646 + 0.999617i \(0.491193\pi\)
\(44\) 5.25211 − 11.4709i 0.791786 − 1.72930i
\(45\) 8.00728 1.19365
\(46\) 5.99672 + 3.16849i 0.884169 + 0.467168i
\(47\) 3.74592i 0.546398i 0.961958 + 0.273199i \(0.0880818\pi\)
−0.961958 + 0.273199i \(0.911918\pi\)
\(48\) −1.86299 − 13.1395i −0.268900 − 1.89652i
\(49\) −1.07900 + 7.50461i −0.154143 + 1.07209i
\(50\) −1.13255 + 0.846954i −0.160167 + 0.119777i
\(51\) −2.37597 − 8.09181i −0.332702 − 1.13308i
\(52\) −4.92432 − 3.17146i −0.682880 − 0.439803i
\(53\) 2.63107 + 3.03642i 0.361406 + 0.417084i 0.907110 − 0.420893i \(-0.138283\pi\)
−0.545704 + 0.837978i \(0.683738\pi\)
\(54\) 8.22106 − 22.0087i 1.11874 − 2.99500i
\(55\) −1.77718 + 6.05252i −0.239635 + 0.816122i
\(56\) 8.63689 + 6.48525i 1.15415 + 0.866628i
\(57\) 9.04332 − 14.0717i 1.19782 − 1.86384i
\(58\) 3.14527 − 1.71545i 0.412994 − 0.225250i
\(59\) −6.58513 + 7.59965i −0.857311 + 0.989390i −1.00000 0.000377513i \(-0.999880\pi\)
0.142688 + 0.989768i \(0.454425\pi\)
\(60\) 1.86320 + 6.36848i 0.240539 + 0.822168i
\(61\) 0.283392 + 1.97104i 0.0362847 + 0.252365i 0.999888 − 0.0149710i \(-0.00476560\pi\)
−0.963603 + 0.267336i \(0.913857\pi\)
\(62\) −10.0655 + 2.18447i −1.27832 + 0.277429i
\(63\) 12.7020 + 27.8135i 1.60030 + 3.50418i
\(64\) 7.28676 − 3.30199i 0.910845 − 0.412749i
\(65\) 2.66396 + 1.21659i 0.330423 + 0.150899i
\(66\) 23.6851 + 17.7485i 2.91543 + 2.18468i
\(67\) −4.44821 − 6.92155i −0.543435 − 0.845602i 0.455570 − 0.890200i \(-0.349436\pi\)
−0.999005 + 0.0445982i \(0.985799\pi\)
\(68\) 4.27950 − 2.74437i 0.518966 − 0.332803i
\(69\) −10.2478 + 12.1717i −1.23369 + 1.46530i
\(70\) −4.73849 − 2.59042i −0.566358 − 0.309614i
\(71\) −0.130034 − 0.202337i −0.0154322 − 0.0240130i 0.833452 − 0.552592i \(-0.186361\pi\)
−0.848884 + 0.528579i \(0.822725\pi\)
\(72\) 22.5879 + 1.64877i 2.66201 + 0.194310i
\(73\) 1.49297 − 3.26915i 0.174739 − 0.382626i −0.801916 − 0.597436i \(-0.796186\pi\)
0.976656 + 0.214811i \(0.0689134\pi\)
\(74\) −6.40161 − 1.39586i −0.744172 − 0.162265i
\(75\) −1.37823 − 3.01791i −0.159144 − 0.348478i
\(76\) 9.67222 + 2.85028i 1.10948 + 0.326950i
\(77\) −23.8428 + 3.42807i −2.71714 + 0.390665i
\(78\) 9.72106 − 9.71157i 1.10069 − 1.09962i
\(79\) 6.19095 − 7.14474i 0.696537 − 0.803846i −0.291744 − 0.956497i \(-0.594235\pi\)
0.988280 + 0.152650i \(0.0487808\pi\)
\(80\) −3.36923 + 2.15599i −0.376691 + 0.241047i
\(81\) 26.1585 + 16.8111i 2.90650 + 1.86790i
\(82\) 1.44253 − 0.537233i 0.159301 − 0.0593275i
\(83\) 0.562599 − 1.91603i 0.0617532 − 0.210312i −0.922837 − 0.385192i \(-0.874135\pi\)
0.984590 + 0.174879i \(0.0559536\pi\)
\(84\) −19.1655 + 16.5743i −2.09113 + 1.80840i
\(85\) −1.92106 + 1.66461i −0.208369 + 0.180552i
\(86\) −3.13829 − 0.225995i −0.338410 − 0.0243696i
\(87\) 2.36793 + 8.06442i 0.253869 + 0.864597i
\(88\) −6.25956 + 16.7078i −0.667271 + 1.78105i
\(89\) −6.65719 − 0.957159i −0.705661 − 0.101459i −0.219864 − 0.975531i \(-0.570561\pi\)
−0.485797 + 0.874072i \(0.661470\pi\)
\(90\) −11.2955 + 0.802332i −1.19066 + 0.0845732i
\(91\) 11.1832i 1.17232i
\(92\) −8.77682 − 3.86878i −0.915046 − 0.403349i
\(93\) − 24.1633i − 2.50561i
\(94\) −0.375342 − 5.28421i −0.0387136 − 0.545025i
\(95\) −4.99041 − 0.717512i −0.512005 − 0.0736152i
\(96\) 3.94463 + 18.3486i 0.402597 + 1.87270i
\(97\) −2.22710 − 7.58480i −0.226127 − 0.770119i −0.991901 − 0.127010i \(-0.959462\pi\)
0.765774 − 0.643110i \(-0.222356\pi\)
\(98\) 0.770138 − 10.6946i 0.0777957 − 1.08031i
\(99\) −38.1731 + 33.0772i −3.83654 + 3.32438i
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.11.3 ✓ 480
8.3 odd 2 920.2.bb.b.11.17 yes 480
23.21 odd 22 920.2.bb.b.251.17 yes 480
184.67 even 22 inner 920.2.bb.a.251.3 yes 480
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
920.2.bb.a.11.3 ✓ 480 1.1 even 1 trivial
920.2.bb.a.251.3 yes 480 184.67 even 22 inner
920.2.bb.b.11.17 yes 480 8.3 odd 2
920.2.bb.b.251.17 yes 480 23.21 odd 22