Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [110,2,Mod(7,110)] 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("110.7"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(110, base_ring=CyclotomicField(20)) chi = DirichletCharacter(H, H._module([5, 14])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 110 = 2 \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 110.k (of order \(20\), degree \(8\), 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: \(0.878354422234\)
Analytic rank: \(0\)
Dimension: \(48\)
Relative dimension: \(6\) over \(\Q(\zeta_{20})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{20}]$

Embedding invariants

Embedding label 13.1
Character \(\chi\) \(=\) 110.13
Dual form 110.2.k.a.17.1

$q$-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+(-0.891007 + 0.453990i) q^{2} +(-2.20142 - 0.348671i) q^{3} +(0.587785 - 0.809017i) q^{4} +(2.18748 + 0.463629i) q^{5} +(2.11978 - 0.688757i) q^{6} +(0.620489 + 3.91761i) q^{7} +(-0.156434 + 0.987688i) q^{8} +(1.87153 + 0.608097i) q^{9} +(-2.15954 + 0.579997i) q^{10} +(2.83922 + 1.71431i) q^{11} +(-1.57605 + 1.57605i) q^{12} +(1.35296 + 2.65534i) q^{13} +(-2.33142 - 3.20892i) q^{14} +(-4.65391 - 1.78335i) q^{15} +(-0.309017 - 0.951057i) q^{16} +(2.11841 - 4.15761i) q^{17} +(-1.94362 + 0.307838i) q^{18} +(-2.59693 + 1.88678i) q^{19} +(1.66085 - 1.49719i) q^{20} -8.84068i q^{21} +(-3.30804 - 0.238480i) q^{22} +(-5.14200 - 5.14200i) q^{23} +(0.688757 - 2.11978i) q^{24} +(4.57010 + 2.02835i) q^{25} +(-2.41100 - 1.75169i) q^{26} +(2.04980 + 1.04442i) q^{27} +(3.53413 + 1.80073i) q^{28} +(-0.0660203 - 0.0479665i) q^{29} +(4.95629 - 0.523850i) q^{30} +(-0.600717 + 1.84882i) q^{31} +(0.707107 + 0.707107i) q^{32} +(-5.65259 - 4.76387i) q^{33} +4.66620i q^{34} +(-0.459014 + 8.85736i) q^{35} +(1.59202 - 1.15667i) q^{36} +(-5.78725 + 0.916611i) q^{37} +(1.45730 - 2.86011i) q^{38} +(-2.05261 - 6.31728i) q^{39} +(-0.800117 + 2.08802i) q^{40} +(1.64181 + 2.25975i) q^{41} +(4.01358 + 7.87710i) q^{42} +(2.07213 - 2.07213i) q^{43} +(3.05575 - 1.28933i) q^{44} +(3.81199 + 2.19789i) q^{45} +(6.91597 + 2.24713i) q^{46} +(1.92981 - 12.1843i) q^{47} +(0.348671 + 2.20142i) q^{48} +(-8.30529 + 2.69855i) q^{49} +(-4.99284 + 0.267505i) q^{50} +(-6.11316 + 8.41404i) q^{51} +(2.94347 + 0.466200i) q^{52} +(-3.28846 + 1.67555i) q^{53} -2.30054 q^{54} +(5.41592 + 5.06634i) q^{55} -3.96645 q^{56} +(6.37480 - 3.24812i) q^{57} +(0.0806009 + 0.0127659i) q^{58} +(1.48475 - 2.04358i) q^{59} +(-4.17826 + 2.71686i) q^{60} +(1.70602 - 0.554319i) q^{61} +(-0.304102 - 1.92003i) q^{62} +(-1.22102 + 7.70925i) q^{63} +(-0.951057 - 0.309017i) q^{64} +(1.72848 + 6.43577i) q^{65} +(7.19925 + 1.67841i) q^{66} +(2.34748 - 2.34748i) q^{67} +(-2.11841 - 4.15761i) q^{68} +(9.52685 + 13.1126i) q^{69} +(-3.61217 - 8.10035i) q^{70} +(-1.98584 - 6.11179i) q^{71} +(-0.893382 + 1.75336i) q^{72} +(7.72040 - 1.22279i) q^{73} +(4.74035 - 3.44406i) q^{74} +(-9.35350 - 6.05873i) q^{75} +3.20998i q^{76} +(-4.95428 + 12.1867i) q^{77} +(4.69687 + 4.69687i) q^{78} +(3.80642 - 11.7150i) q^{79} +(-0.235030 - 2.22368i) q^{80} +(-8.92437 - 6.48393i) q^{81} +(-2.48877 - 1.26809i) q^{82} +(13.4709 + 6.86376i) q^{83} +(-7.15226 - 5.19642i) q^{84} +(6.56156 - 8.11252i) q^{85} +(-0.905554 + 2.78701i) q^{86} +(0.128614 + 0.128614i) q^{87} +(-2.13735 + 2.53609i) q^{88} -5.99094i q^{89} +(-4.39433 - 0.227727i) q^{90} +(-9.56310 + 6.94800i) q^{91} +(-7.18235 + 1.13757i) q^{92} +(1.96706 - 3.86058i) q^{93} +(3.81210 + 11.7324i) q^{94} +(-6.55548 + 2.92327i) q^{95} +(-1.31009 - 1.80319i) q^{96} +(-4.98351 - 9.78069i) q^{97} +(6.17495 - 6.17495i) q^{98} +(4.27122 + 4.93489i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 48 q - 4 q^{3} - 8 q^{5} - 20 q^{7} + 12 q^{11} - 16 q^{12} - 16 q^{15} + 12 q^{16} - 20 q^{17} - 4 q^{20} - 4 q^{22} - 8 q^{23} - 20 q^{25} + 8 q^{26} + 8 q^{27} - 20 q^{28} + 16 q^{31} - 104 q^{33} - 4 q^{36}+ \cdots + 72 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/110\mathbb{Z}\right)^\times\).

\(n\) \(67\) \(101\)
\(\chi(n)\) \(e\left(\frac{3}{4}\right)\) \(e\left(\frac{1}{10}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.891007 + 0.453990i −0.630037 + 0.321020i
\(3\) −2.20142 − 0.348671i −1.27099 − 0.201306i −0.515744 − 0.856743i \(-0.672484\pi\)
−0.755250 + 0.655437i \(0.772484\pi\)
\(4\) 0.587785 − 0.809017i 0.293893 − 0.404508i
\(5\) 2.18748 + 0.463629i 0.978269 + 0.207341i
\(6\) 2.11978 − 0.688757i 0.865396 − 0.281184i
\(7\) 0.620489 + 3.91761i 0.234523 + 1.48072i 0.771017 + 0.636815i \(0.219748\pi\)
−0.536494 + 0.843904i \(0.680252\pi\)
\(8\) −0.156434 + 0.987688i −0.0553079 + 0.349201i
\(9\) 1.87153 + 0.608097i 0.623843 + 0.202699i
\(10\) −2.15954 + 0.579997i −0.682906 + 0.183411i
\(11\) 2.83922 + 1.71431i 0.856056 + 0.516883i
\(12\) −1.57605 + 1.57605i −0.454965 + 0.454965i
\(13\) 1.35296 + 2.65534i 0.375245 + 0.736460i 0.998979 − 0.0451771i \(-0.0143852\pi\)
−0.623734 + 0.781637i \(0.714385\pi\)
\(14\) −2.33142 − 3.20892i −0.623098 − 0.857621i
\(15\) −4.65391 − 1.78335i −1.20163 − 0.460460i
\(16\) −0.309017 − 0.951057i −0.0772542 − 0.237764i
\(17\) 2.11841 − 4.15761i 0.513790 − 1.00837i −0.477742 − 0.878500i \(-0.658544\pi\)
0.991531 − 0.129869i \(-0.0414555\pi\)
\(18\) −1.94362 + 0.307838i −0.458115 + 0.0725582i
\(19\) −2.59693 + 1.88678i −0.595776 + 0.432856i −0.844377 − 0.535750i \(-0.820029\pi\)
0.248601 + 0.968606i \(0.420029\pi\)
\(20\) 1.66085 − 1.49719i 0.371377 − 0.334782i
\(21\) − 8.84068i − 1.92919i
\(22\) −3.30804 − 0.238480i −0.705276 − 0.0508440i
\(23\) −5.14200 − 5.14200i −1.07218 − 1.07218i −0.997184 − 0.0749964i \(-0.976105\pi\)
−0.0749964 − 0.997184i \(-0.523895\pi\)
\(24\) 0.688757 − 2.11978i 0.140592 − 0.432698i
\(25\) 4.57010 + 2.02835i 0.914019 + 0.405671i
\(26\) −2.41100 − 1.75169i −0.472836 − 0.343536i
\(27\) 2.04980 + 1.04442i 0.394483 + 0.200999i
\(28\) 3.53413 + 1.80073i 0.667888 + 0.340306i
\(29\) −0.0660203 − 0.0479665i −0.0122597 − 0.00890716i 0.581639 − 0.813447i \(-0.302412\pi\)
−0.593898 + 0.804540i \(0.702412\pi\)
\(30\) 4.95629 − 0.523850i 0.904890 − 0.0956415i
\(31\) −0.600717 + 1.84882i −0.107892 + 0.332057i −0.990398 − 0.138243i \(-0.955854\pi\)
0.882506 + 0.470300i \(0.155854\pi\)
\(32\) 0.707107 + 0.707107i 0.125000 + 0.125000i
\(33\) −5.65259 − 4.76387i −0.983990 − 0.829283i
\(34\) 4.66620i 0.800246i
\(35\) −0.459014 + 8.85736i −0.0775875 + 1.49717i
\(36\) 1.59202 − 1.15667i 0.265336 − 0.192778i
\(37\) −5.78725 + 0.916611i −0.951419 + 0.150690i −0.612804 − 0.790235i \(-0.709958\pi\)
−0.338615 + 0.940925i \(0.609958\pi\)
\(38\) 1.45730 − 2.86011i 0.236405 − 0.463971i
\(39\) −2.05261 − 6.31728i −0.328680 − 1.01157i
\(40\) −0.800117 + 2.08802i −0.126510 + 0.330144i
\(41\) 1.64181 + 2.25975i 0.256407 + 0.352914i 0.917742 − 0.397177i \(-0.130010\pi\)
−0.661335 + 0.750090i \(0.730010\pi\)
\(42\) 4.01358 + 7.87710i 0.619309 + 1.21546i
\(43\) 2.07213 − 2.07213i 0.315997 − 0.315997i −0.531231 − 0.847227i \(-0.678270\pi\)
0.847227 + 0.531231i \(0.178270\pi\)
\(44\) 3.05575 − 1.28933i 0.460672 − 0.194374i
\(45\) 3.81199 + 2.19789i 0.568259 + 0.327642i
\(46\) 6.91597 + 2.24713i 1.01970 + 0.331322i
\(47\) 1.92981 − 12.1843i 0.281492 − 1.77727i −0.290367 − 0.956915i \(-0.593777\pi\)
0.571858 − 0.820352i \(-0.306223\pi\)
\(48\) 0.348671 + 2.20142i 0.0503264 + 0.317748i
\(49\) −8.30529 + 2.69855i −1.18647 + 0.385507i
\(50\) −4.99284 + 0.267505i −0.706094 + 0.0378309i
\(51\) −6.11316 + 8.41404i −0.856013 + 1.17820i
\(52\) 2.94347 + 0.466200i 0.408186 + 0.0646503i
\(53\) −3.28846 + 1.67555i −0.451704 + 0.230155i −0.665016 − 0.746830i \(-0.731575\pi\)
0.213311 + 0.976984i \(0.431575\pi\)
\(54\) −2.30054 −0.313064
\(55\) 5.41592 + 5.06634i 0.730282 + 0.683146i
\(56\) −3.96645 −0.530039
\(57\) 6.37480 − 3.24812i 0.844363 − 0.430225i
\(58\) 0.0806009 + 0.0127659i 0.0105834 + 0.00167625i
\(59\) 1.48475 − 2.04358i 0.193298 − 0.266052i −0.701356 − 0.712811i \(-0.747422\pi\)
0.894654 + 0.446759i \(0.147422\pi\)
\(60\) −4.17826 + 2.71686i −0.539411 + 0.350745i
\(61\) 1.70602 − 0.554319i 0.218433 − 0.0709733i −0.197756 − 0.980251i \(-0.563365\pi\)
0.416189 + 0.909278i \(0.363365\pi\)
\(62\) −0.304102 − 1.92003i −0.0386210 − 0.243844i
\(63\) −1.22102 + 7.70925i −0.153835 + 0.971274i
\(64\) −0.951057 − 0.309017i −0.118882 − 0.0386271i
\(65\) 1.72848 + 6.43577i 0.214392 + 0.798259i
\(66\) 7.19925 + 1.67841i 0.886166 + 0.206599i
\(67\) 2.34748 − 2.34748i 0.286790 − 0.286790i −0.549020 − 0.835809i \(-0.684999\pi\)
0.835809 + 0.549020i \(0.184999\pi\)
\(68\) −2.11841 − 4.15761i −0.256895 − 0.504184i
\(69\) 9.52685 + 13.1126i 1.14690 + 1.57857i
\(70\) −3.61217 − 8.10035i −0.431737 − 0.968177i
\(71\) −1.98584 − 6.11179i −0.235676 − 0.725336i −0.997031 − 0.0770011i \(-0.975466\pi\)
0.761355 − 0.648335i \(-0.224534\pi\)
\(72\) −0.893382 + 1.75336i −0.105286 + 0.206636i
\(73\) 7.72040 − 1.22279i 0.903604 − 0.143117i 0.312694 − 0.949854i \(-0.398769\pi\)
0.590911 + 0.806737i \(0.298769\pi\)
\(74\) 4.74035 − 3.44406i 0.551054 − 0.400364i
\(75\) −9.35350 − 6.05873i −1.08005 − 0.699602i
\(76\) 3.20998i 0.368210i
\(77\) −4.95428 + 12.1867i −0.564593 + 1.38880i
\(78\) 4.69687 + 4.69687i 0.531816 + 0.531816i
\(79\) 3.80642 − 11.7150i 0.428256 − 1.31803i −0.471587 − 0.881820i \(-0.656319\pi\)
0.899842 − 0.436215i \(-0.143681\pi\)
\(80\) −0.235030 − 2.22368i −0.0262771 − 0.248615i
\(81\) −8.92437 − 6.48393i −0.991596 − 0.720437i
\(82\) −2.48877 − 1.26809i −0.274838 − 0.140037i
\(83\) 13.4709 + 6.86376i 1.47862 + 0.753396i 0.992700 − 0.120606i \(-0.0384838\pi\)
0.485922 + 0.874002i \(0.338484\pi\)
\(84\) −7.15226 − 5.19642i −0.780375 − 0.566976i
\(85\) 6.56156 − 8.11252i 0.711701 − 0.879926i
\(86\) −0.905554 + 2.78701i −0.0976484 + 0.300531i
\(87\) 0.128614 + 0.128614i 0.0137889 + 0.0137889i
\(88\) −2.13735 + 2.53609i −0.227842 + 0.270348i
\(89\) − 5.99094i − 0.635038i −0.948252 − 0.317519i \(-0.897150\pi\)
0.948252 − 0.317519i \(-0.102850\pi\)
\(90\) −4.39433 − 0.227727i −0.463203 − 0.0240045i
\(91\) −9.56310 + 6.94800i −1.00249 + 0.728349i
\(92\) −7.18235 + 1.13757i −0.748812 + 0.118600i
\(93\) 1.96706 − 3.86058i 0.203975 − 0.400323i
\(94\) 3.81210 + 11.7324i 0.393188 + 1.21011i
\(95\) −6.55548 + 2.92327i −0.672578 + 0.299921i
\(96\) −1.31009 − 1.80319i −0.133711 − 0.184037i
\(97\) −4.98351 − 9.78069i −0.505999 − 0.993078i −0.992825 − 0.119579i \(-0.961845\pi\)
0.486826 − 0.873499i \(-0.338155\pi\)
\(98\) 6.17495 − 6.17495i 0.623764 − 0.623764i
\(99\) 4.27122 + 4.93489i 0.429273 + 0.495976i
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 110.2.k.a.13.1 ✓ 48
3.2 odd 2 990.2.bh.c.343.4 48
4.3 odd 2 880.2.cm.c.673.5 48
5.2 odd 4 inner 110.2.k.a.57.1 yes 48
5.3 odd 4 550.2.bh.b.57.6 48
5.4 even 2 550.2.bh.b.343.6 48
11.6 odd 10 inner 110.2.k.a.83.1 yes 48
15.2 even 4 990.2.bh.c.937.5 48
20.7 even 4 880.2.cm.c.497.5 48
33.17 even 10 990.2.bh.c.523.5 48
44.39 even 10 880.2.cm.c.193.5 48
55.17 even 20 inner 110.2.k.a.17.1 yes 48
55.28 even 20 550.2.bh.b.457.6 48
55.39 odd 10 550.2.bh.b.193.6 48
165.17 odd 20 990.2.bh.c.127.4 48
220.127 odd 20 880.2.cm.c.17.5 48
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
110.2.k.a.13.1 ✓ 48 1.1 even 1 trivial
110.2.k.a.17.1 yes 48 55.17 even 20 inner
110.2.k.a.57.1 yes 48 5.2 odd 4 inner
110.2.k.a.83.1 yes 48 11.6 odd 10 inner
550.2.bh.b.57.6 48 5.3 odd 4
550.2.bh.b.193.6 48 55.39 odd 10
550.2.bh.b.343.6 48 5.4 even 2
550.2.bh.b.457.6 48 55.28 even 20
880.2.cm.c.17.5 48 220.127 odd 20
880.2.cm.c.193.5 48 44.39 even 10
880.2.cm.c.497.5 48 20.7 even 4
880.2.cm.c.673.5 48 4.3 odd 2
990.2.bh.c.127.4 48 165.17 odd 20
990.2.bh.c.343.4 48 3.2 odd 2
990.2.bh.c.523.5 48 33.17 even 10
990.2.bh.c.937.5 48 15.2 even 4