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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [230,3,Mod(19,230)] 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("230.19"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(230, base_ring=CyclotomicField(22)) chi = DirichletCharacter(H, H._module([11, 15])) N = Newforms(chi, 3, names="a")
 
Level: \( N \) \(=\) \( 230 = 2 \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 230.i (of order \(22\), degree \(10\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(6.26704608029\)
Analytic rank: \(0\)
Dimension: \(240\)
Relative dimension: \(24\) over \(\Q(\zeta_{22})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{22}]$

Embedding invariants

Embedding label 19.6
Character \(\chi\) \(=\) 230.19
Dual form 230.3.i.a.109.6

$q$-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+(-1.28641 + 0.587486i) q^{2} +(-0.00972304 - 0.0331137i) q^{3} +(1.30972 - 1.51150i) q^{4} +(-4.99585 + 0.203788i) q^{5} +(0.0319617 + 0.0368857i) q^{6} +(0.869641 + 6.04849i) q^{7} +(-0.796860 + 2.71386i) q^{8} +(7.57028 - 4.86512i) q^{9} +(6.30700 - 3.19714i) q^{10} +(-0.789357 - 0.360487i) q^{11} +(-0.0627857 - 0.0286733i) q^{12} +(-12.0743 - 1.73602i) q^{13} +(-4.67212 - 7.26996i) q^{14} +(0.0553230 + 0.163449i) q^{15} +(-0.569259 - 3.95929i) q^{16} +(-19.6346 - 22.6595i) q^{17} +(-6.88032 + 10.7060i) q^{18} +(-7.05521 - 6.11337i) q^{19} +(-6.23514 + 7.81812i) q^{20} +(0.191832 - 0.0876067i) q^{21} +1.22722 q^{22} +(-20.5841 + 10.2614i) q^{23} +0.0976136 q^{24} +(24.9169 - 2.03619i) q^{25} +(16.5524 - 4.86022i) q^{26} +(-0.469447 - 0.406778i) q^{27} +(10.2813 + 6.60737i) q^{28} +(-13.1874 - 15.2191i) q^{29} +(-0.167192 - 0.177762i) q^{30} +(-43.4091 - 12.7460i) q^{31} +(3.05833 + 4.75885i) q^{32} +(-0.00426210 + 0.0296435i) q^{33} +(38.5703 + 17.6145i) q^{34} +(-5.57720 - 30.0401i) q^{35} +(2.56133 - 17.8144i) q^{36} +(-11.9429 + 7.67524i) q^{37} +(12.6674 + 3.71949i) q^{38} +(0.0599128 + 0.416703i) q^{39} +(3.42794 - 13.7204i) q^{40} +(-45.5878 - 29.2975i) q^{41} +(-0.195308 + 0.225397i) q^{42} +(61.6500 - 18.1021i) q^{43} +(-1.57871 + 0.720974i) q^{44} +(-36.8285 + 25.8481i) q^{45} +(20.4513 - 25.2932i) q^{46} -58.7319i q^{47} +(-0.125571 + 0.0573466i) q^{48} +(11.1872 - 3.28487i) q^{49} +(-30.8573 + 17.2577i) q^{50} +(-0.559431 + 0.870491i) q^{51} +(-18.4379 + 15.9765i) q^{52} +(-0.467135 - 3.24899i) q^{53} +(0.842880 + 0.247492i) q^{54} +(4.01697 + 1.64008i) q^{55} +(-17.1077 - 2.45972i) q^{56} +(-0.133838 + 0.293064i) q^{57} +(25.9055 + 11.8307i) q^{58} +(-15.7619 + 109.627i) q^{59} +(0.319511 + 0.130452i) q^{60} +(-26.5531 + 90.4315i) q^{61} +(63.3301 - 9.10550i) q^{62} +(36.0101 + 41.5578i) q^{63} +(-6.73003 - 4.32513i) q^{64} +(60.6750 + 6.21228i) q^{65} +(-0.0119323 - 0.0406378i) q^{66} +(16.7830 + 36.7496i) q^{67} -59.9656 q^{68} +(0.539931 + 0.581843i) q^{69} +(24.8227 + 35.3675i) q^{70} +(-31.2064 - 68.3326i) q^{71} +(7.17079 + 24.4215i) q^{72} +(66.0949 + 57.2715i) q^{73} +(10.8544 - 16.8898i) q^{74} +(-0.309694 - 0.805293i) q^{75} +(-18.4807 + 2.65713i) q^{76} +(1.49394 - 5.08791i) q^{77} +(-0.321879 - 0.500854i) q^{78} +(66.1445 + 9.51014i) q^{79} +(3.65079 + 19.6640i) q^{80} +(33.6353 - 73.6509i) q^{81} +(75.8566 + 10.9065i) q^{82} +(-25.0249 + 16.0825i) q^{83} +(0.118829 - 0.404694i) q^{84} +(102.709 + 109.202i) q^{85} +(-68.6727 + 59.5052i) q^{86} +(-0.375739 + 0.584661i) q^{87} +(1.60732 - 1.85494i) q^{88} +(4.94644 + 16.8460i) q^{89} +(32.1913 - 54.8876i) q^{90} -74.5408i q^{91} +(-11.4494 + 44.5523i) q^{92} +1.56136i q^{93} +(34.5042 + 75.5536i) q^{94} +(36.4926 + 29.1037i) q^{95} +(0.127847 - 0.147543i) q^{96} +(-112.999 - 72.6201i) q^{97} +(-12.4616 + 10.7980i) q^{98} +(-7.72947 + 1.11133i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 240 q + 48 q^{4} - 8 q^{6} + 96 q^{9} + 154 q^{15} - 96 q^{16} + 44 q^{20} + 16 q^{24} - 84 q^{25} + 32 q^{26} - 100 q^{29} - 352 q^{30} + 124 q^{31} + 28 q^{35} - 192 q^{36} + 72 q^{39} + 116 q^{41} - 148 q^{46}+ \cdots - 3300 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/230\mathbb{Z}\right)^\times\).

\(n\) \(47\) \(51\)
\(\chi(n)\) \(-1\) \(e\left(\frac{15}{22}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.28641 + 0.587486i −0.643207 + 0.293743i
\(3\) −0.00972304 − 0.0331137i −0.00324101 − 0.0110379i 0.957857 − 0.287247i \(-0.0927400\pi\)
−0.961098 + 0.276209i \(0.910922\pi\)
\(4\) 1.30972 − 1.51150i 0.327430 − 0.377875i
\(5\) −4.99585 + 0.203788i −0.999169 + 0.0407576i
\(6\) 0.0319617 + 0.0368857i 0.00532694 + 0.00614762i
\(7\) 0.869641 + 6.04849i 0.124234 + 0.864070i 0.952675 + 0.303992i \(0.0983196\pi\)
−0.828440 + 0.560078i \(0.810771\pi\)
\(8\) −0.796860 + 2.71386i −0.0996075 + 0.339232i
\(9\) 7.57028 − 4.86512i 0.841142 − 0.540569i
\(10\) 6.30700 − 3.19714i 0.630700 − 0.319714i
\(11\) −0.789357 − 0.360487i −0.0717597 − 0.0327716i 0.379212 − 0.925310i \(-0.376195\pi\)
−0.450972 + 0.892538i \(0.648922\pi\)
\(12\) −0.0627857 − 0.0286733i −0.00523214 − 0.00238944i
\(13\) −12.0743 − 1.73602i −0.928790 − 0.133540i −0.338716 − 0.940889i \(-0.609992\pi\)
−0.590074 + 0.807349i \(0.700902\pi\)
\(14\) −4.67212 − 7.26996i −0.333723 − 0.519283i
\(15\) 0.0553230 + 0.163449i 0.00368820 + 0.0108966i
\(16\) −0.569259 − 3.95929i −0.0355787 − 0.247455i
\(17\) −19.6346 − 22.6595i −1.15497 − 1.33291i −0.933850 − 0.357665i \(-0.883573\pi\)
−0.221124 − 0.975246i \(-0.570973\pi\)
\(18\) −6.88032 + 10.7060i −0.382240 + 0.594777i
\(19\) −7.05521 − 6.11337i −0.371327 − 0.321756i 0.449129 − 0.893467i \(-0.351735\pi\)
−0.820455 + 0.571711i \(0.806280\pi\)
\(20\) −6.23514 + 7.81812i −0.311757 + 0.390906i
\(21\) 0.191832 − 0.0876067i 0.00913485 − 0.00417175i
\(22\) 1.22722 0.0557828
\(23\) −20.5841 + 10.2614i −0.894960 + 0.446146i
\(24\) 0.0976136 0.00406723
\(25\) 24.9169 − 2.03619i 0.996678 − 0.0814476i
\(26\) 16.5524 − 4.86022i 0.636631 − 0.186932i
\(27\) −0.469447 − 0.406778i −0.0173869 − 0.0150659i
\(28\) 10.2813 + 6.60737i 0.367188 + 0.235978i
\(29\) −13.1874 − 15.2191i −0.454740 − 0.524797i 0.481365 − 0.876521i \(-0.340141\pi\)
−0.936104 + 0.351723i \(0.885596\pi\)
\(30\) −0.167192 − 0.177762i −0.00557308 − 0.00592540i
\(31\) −43.4091 − 12.7460i −1.40029 − 0.411163i −0.507508 − 0.861647i \(-0.669433\pi\)
−0.892784 + 0.450484i \(0.851251\pi\)
\(32\) 3.05833 + 4.75885i 0.0955727 + 0.148714i
\(33\) −0.00426210 + 0.0296435i −0.000129154 + 0.000898289i
\(34\) 38.5703 + 17.6145i 1.13442 + 0.518072i
\(35\) −5.57720 − 30.0401i −0.159349 − 0.858288i
\(36\) 2.56133 − 17.8144i 0.0711480 − 0.494845i
\(37\) −11.9429 + 7.67524i −0.322781 + 0.207439i −0.691989 − 0.721908i \(-0.743265\pi\)
0.369208 + 0.929347i \(0.379629\pi\)
\(38\) 12.6674 + 3.71949i 0.333354 + 0.0978814i
\(39\) 0.0599128 + 0.416703i 0.00153623 + 0.0106847i
\(40\) 3.42794 − 13.7204i 0.0856984 − 0.343010i
\(41\) −45.5878 − 29.2975i −1.11190 − 0.714573i −0.150191 − 0.988657i \(-0.547989\pi\)
−0.961706 + 0.274084i \(0.911625\pi\)
\(42\) −0.195308 + 0.225397i −0.00465018 + 0.00536659i
\(43\) 61.6500 − 18.1021i 1.43372 − 0.420978i 0.529597 − 0.848250i \(-0.322343\pi\)
0.904123 + 0.427271i \(0.140525\pi\)
\(44\) −1.57871 + 0.720974i −0.0358799 + 0.0163858i
\(45\) −36.8285 + 25.8481i −0.818411 + 0.574403i
\(46\) 20.4513 − 25.2932i 0.444592 − 0.549852i
\(47\) − 58.7319i − 1.24962i −0.780779 − 0.624808i \(-0.785177\pi\)
0.780779 − 0.624808i \(-0.214823\pi\)
\(48\) −0.125571 + 0.0573466i −0.00261607 + 0.00119472i
\(49\) 11.1872 − 3.28487i 0.228311 − 0.0670382i
\(50\) −30.8573 + 17.2577i −0.617145 + 0.345154i
\(51\) −0.559431 + 0.870491i −0.0109692 + 0.0170685i
\(52\) −18.4379 + 15.9765i −0.354575 + 0.307241i
\(53\) −0.467135 − 3.24899i −0.00881386 − 0.0613018i 0.984940 − 0.172897i \(-0.0553129\pi\)
−0.993754 + 0.111596i \(0.964404\pi\)
\(54\) 0.842880 + 0.247492i 0.0156089 + 0.00458318i
\(55\) 4.01697 + 1.64008i 0.0730358 + 0.0298196i
\(56\) −17.1077 − 2.45972i −0.305495 − 0.0439235i
\(57\) −0.133838 + 0.293064i −0.00234803 + 0.00514148i
\(58\) 25.9055 + 11.8307i 0.446647 + 0.203977i
\(59\) −15.7619 + 109.627i −0.267151 + 1.85808i 0.207942 + 0.978141i \(0.433323\pi\)
−0.475093 + 0.879936i \(0.657586\pi\)
\(60\) 0.319511 + 0.130452i 0.00532518 + 0.00217420i
\(61\) −26.5531 + 90.4315i −0.435296 + 1.48248i 0.391604 + 0.920134i \(0.371920\pi\)
−0.826900 + 0.562349i \(0.809898\pi\)
\(62\) 63.3301 − 9.10550i 1.02145 − 0.146863i
\(63\) 36.0101 + 41.5578i 0.571588 + 0.659648i
\(64\) −6.73003 − 4.32513i −0.105157 − 0.0675801i
\(65\) 60.6750 + 6.21228i 0.933461 + 0.0955736i
\(66\) −0.0119323 − 0.0406378i −0.000180793 − 0.000615724i
\(67\) 16.7830 + 36.7496i 0.250493 + 0.548502i 0.992551 − 0.121834i \(-0.0388774\pi\)
−0.742058 + 0.670336i \(0.766150\pi\)
\(68\) −59.9656 −0.881847
\(69\) 0.539931 + 0.581843i 0.00782509 + 0.00843250i
\(70\) 24.8227 + 35.3675i 0.354610 + 0.505249i
\(71\) −31.2064 − 68.3326i −0.439527 − 0.962431i −0.991685 − 0.128691i \(-0.958922\pi\)
0.552157 − 0.833740i \(-0.313805\pi\)
\(72\) 7.17079 + 24.4215i 0.0995943 + 0.339187i
\(73\) 66.0949 + 57.2715i 0.905409 + 0.784542i 0.977077 − 0.212887i \(-0.0682866\pi\)
−0.0716675 + 0.997429i \(0.522832\pi\)
\(74\) 10.8544 − 16.8898i 0.146681 − 0.228241i
\(75\) −0.309694 − 0.805293i −0.00412926 − 0.0107372i
\(76\) −18.4807 + 2.65713i −0.243167 + 0.0349622i
\(77\) 1.49394 − 5.08791i 0.0194019 − 0.0660767i
\(78\) −0.321879 − 0.500854i −0.00412666 − 0.00642121i
\(79\) 66.1445 + 9.51014i 0.837272 + 0.120381i 0.547595 − 0.836743i \(-0.315543\pi\)
0.289676 + 0.957125i \(0.406452\pi\)
\(80\) 3.65079 + 19.6640i 0.0456348 + 0.245800i
\(81\) 33.6353 − 73.6509i 0.415250 − 0.909271i
\(82\) 75.8566 + 10.9065i 0.925081 + 0.133007i
\(83\) −25.0249 + 16.0825i −0.301504 + 0.193765i −0.682642 − 0.730753i \(-0.739169\pi\)
0.381137 + 0.924519i \(0.375533\pi\)
\(84\) 0.118829 − 0.404694i 0.00141463 − 0.00481779i
\(85\) 102.709 + 109.202i 1.20834 + 1.28473i
\(86\) −68.6727 + 59.5052i −0.798519 + 0.691921i
\(87\) −0.375739 + 0.584661i −0.00431884 + 0.00672024i
\(88\) 1.60732 − 1.85494i 0.0182650 − 0.0210789i
\(89\) 4.94644 + 16.8460i 0.0555780 + 0.189281i 0.982603 − 0.185718i \(-0.0594611\pi\)
−0.927025 + 0.374999i \(0.877643\pi\)
\(90\) 32.1913 − 54.8876i 0.357681 − 0.609862i
\(91\) − 74.5408i − 0.819129i
\(92\) −11.4494 + 44.5523i −0.124450 + 0.484265i
\(93\) 1.56136i 0.0167888i
\(94\) 34.5042 + 75.5536i 0.367065 + 0.803761i
\(95\) 36.4926 + 29.1037i 0.384132 + 0.306355i
\(96\) 0.127847 − 0.147543i 0.00133174 − 0.00153690i
\(97\) −112.999 − 72.6201i −1.16494 − 0.748661i −0.192380 − 0.981320i \(-0.561621\pi\)
−0.972558 + 0.232660i \(0.925257\pi\)
\(98\) −12.4616 + 10.7980i −0.127159 + 0.110184i
\(99\) −7.72947 + 1.11133i −0.0780754 + 0.0112256i
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 230.3.i.a.19.6 ✓ 240
5.4 even 2 inner 230.3.i.a.19.19 yes 240
23.17 odd 22 inner 230.3.i.a.109.19 yes 240
115.109 odd 22 inner 230.3.i.a.109.6 yes 240
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
230.3.i.a.19.6 ✓ 240 1.1 even 1 trivial
230.3.i.a.19.19 yes 240 5.4 even 2 inner
230.3.i.a.109.6 yes 240 115.109 odd 22 inner
230.3.i.a.109.19 yes 240 23.17 odd 22 inner