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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 19.8
Character \(\chi\) \(=\) 230.19
Dual form 230.3.i.a.109.8

$q$-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} +(0.540708 + 1.84148i) q^{3} +(1.30972 - 1.51150i) q^{4} +(4.67844 + 1.76413i) q^{5} +(-1.77742 - 2.05125i) q^{6} +(-0.0656966 - 0.456930i) q^{7} +(-0.796860 + 2.71386i) q^{8} +(4.47258 - 2.87436i) q^{9} +(-7.05482 + 0.479116i) q^{10} +(0.696679 + 0.318163i) q^{11} +(3.49158 + 1.59455i) q^{12} +(20.8158 + 2.99285i) q^{13} +(0.352953 + 0.549205i) q^{14} +(-0.718944 + 9.56916i) q^{15} +(-0.569259 - 3.95929i) q^{16} +(-2.37645 - 2.74257i) q^{17} +(-4.06495 + 6.32519i) q^{18} +(-22.1306 - 19.1762i) q^{19} +(8.79394 - 4.76095i) q^{20} +(0.805906 - 0.368045i) q^{21} -1.08313 q^{22} +(-9.91399 + 20.7536i) q^{23} -5.42839 q^{24} +(18.7757 + 16.5068i) q^{25} +(-28.5359 + 8.37891i) q^{26} +(20.7655 + 17.9934i) q^{27} +(-0.776693 - 0.499151i) q^{28} +(26.9700 + 31.1250i) q^{29} +(-4.69688 - 12.7323i) q^{30} +(35.8114 + 10.5152i) q^{31} +(3.05833 + 4.75885i) q^{32} +(-0.209191 + 1.45496i) q^{33} +(4.66832 + 2.13195i) q^{34} +(0.498726 - 2.25362i) q^{35} +(1.51325 - 10.5249i) q^{36} +(-57.6318 + 37.0377i) q^{37} +(39.7348 + 11.6672i) q^{38} +(5.74396 + 39.9501i) q^{39} +(-8.51566 + 11.2909i) q^{40} +(-30.9951 - 19.9193i) q^{41} +(-0.820508 + 0.946917i) q^{42} +(-31.1513 + 9.14684i) q^{43} +(1.39336 - 0.636325i) q^{44} +(25.9955 - 5.55729i) q^{45} +(0.561045 - 32.5221i) q^{46} -60.7751i q^{47} +(6.98316 - 3.18910i) q^{48} +(46.8107 - 13.7449i) q^{49} +(-33.8508 - 10.2041i) q^{50} +(3.76543 - 5.85913i) q^{51} +(31.7865 - 27.5432i) q^{52} +(11.2608 + 78.3204i) q^{53} +(-37.2840 - 10.9476i) q^{54} +(2.69809 + 2.71754i) q^{55} +(1.29239 + 0.185818i) q^{56} +(23.3466 - 51.1218i) q^{57} +(-52.9801 - 24.1952i) q^{58} +(-8.45263 + 58.7893i) q^{59} +(13.5222 + 13.6196i) q^{60} +(30.8479 - 105.058i) q^{61} +(-52.2458 + 7.51182i) q^{62} +(-1.60721 - 1.85482i) q^{63} +(-6.73003 - 4.32513i) q^{64} +(92.1056 + 50.7236i) q^{65} +(-0.585659 - 1.99457i) q^{66} +(-45.4327 - 99.4838i) q^{67} -7.25789 q^{68} +(-43.5780 - 7.03480i) q^{69} +(0.682400 + 3.19208i) q^{70} +(-33.8025 - 74.0172i) q^{71} +(4.23656 + 14.4284i) q^{72} +(-72.5846 - 62.8949i) q^{73} +(52.3792 - 81.5037i) q^{74} +(-20.2448 + 43.5005i) q^{75} +(-57.9697 + 8.33479i) q^{76} +(0.0996085 - 0.339236i) q^{77} +(-30.8592 - 48.0179i) q^{78} +(-21.5246 - 3.09477i) q^{79} +(4.32145 - 19.5275i) q^{80} +(-2.02928 + 4.44349i) q^{81} +(51.5748 + 7.41534i) q^{82} +(75.4015 - 48.4576i) q^{83} +(0.499213 - 1.70016i) q^{84} +(-6.27984 - 17.0233i) q^{85} +(34.6998 - 30.0676i) q^{86} +(-42.7334 + 66.4944i) q^{87} +(-1.41860 + 1.63715i) q^{88} +(5.79451 + 19.7343i) q^{89} +(-30.1761 + 22.4209i) q^{90} -9.70796i q^{91} +(18.3845 + 42.1665i) q^{92} +71.6318i q^{93} +(35.7045 + 78.1819i) q^{94} +(-69.7072 - 128.756i) q^{95} +(-7.10968 + 8.20501i) q^{96} +(-58.1078 - 37.3436i) q^{97} +(-52.1430 + 45.1822i) q^{98} +(4.03047 - 0.579494i) 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
\(3\) 0.540708 + 1.84148i 0.180236 + 0.613828i 0.999200 + 0.0399834i \(0.0127305\pi\)
−0.818964 + 0.573845i \(0.805451\pi\)
\(4\) 1.30972 − 1.51150i 0.327430 − 0.377875i
\(5\) 4.67844 + 1.76413i 0.935689 + 0.352826i
\(6\) −1.77742 − 2.05125i −0.296237 − 0.341875i
\(7\) −0.0656966 − 0.456930i −0.00938523 − 0.0652757i 0.984591 − 0.174875i \(-0.0559522\pi\)
−0.993976 + 0.109600i \(0.965043\pi\)
\(8\) −0.796860 + 2.71386i −0.0996075 + 0.339232i
\(9\) 4.47258 − 2.87436i 0.496954 − 0.319373i
\(10\) −7.05482 + 0.479116i −0.705482 + 0.0479116i
\(11\) 0.696679 + 0.318163i 0.0633344 + 0.0289239i 0.446830 − 0.894619i \(-0.352553\pi\)
−0.383496 + 0.923543i \(0.625280\pi\)
\(12\) 3.49158 + 1.59455i 0.290965 + 0.132879i
\(13\) 20.8158 + 2.99285i 1.60121 + 0.230220i 0.884292 − 0.466935i \(-0.154642\pi\)
0.716921 + 0.697155i \(0.245551\pi\)
\(14\) 0.352953 + 0.549205i 0.0252109 + 0.0392289i
\(15\) −0.718944 + 9.56916i −0.0479296 + 0.637944i
\(16\) −0.569259 − 3.95929i −0.0355787 − 0.247455i
\(17\) −2.37645 − 2.74257i −0.139791 − 0.161328i 0.681537 − 0.731784i \(-0.261312\pi\)
−0.821328 + 0.570456i \(0.806766\pi\)
\(18\) −4.06495 + 6.32519i −0.225831 + 0.351399i
\(19\) −22.1306 − 19.1762i −1.16477 − 1.00928i −0.999736 − 0.0229870i \(-0.992682\pi\)
−0.165030 − 0.986288i \(-0.552772\pi\)
\(20\) 8.79394 − 4.76095i 0.439697 − 0.238047i
\(21\) 0.805906 − 0.368045i 0.0383765 − 0.0175260i
\(22\) −1.08313 −0.0492333
\(23\) −9.91399 + 20.7536i −0.431043 + 0.902331i
\(24\) −5.42839 −0.226183
\(25\) 18.7757 + 16.5068i 0.751027 + 0.660271i
\(26\) −28.5359 + 8.37891i −1.09754 + 0.322266i
\(27\) 20.7655 + 17.9934i 0.769094 + 0.666424i
\(28\) −0.776693 − 0.499151i −0.0277391 − 0.0178268i
\(29\) 26.9700 + 31.1250i 0.930000 + 1.07328i 0.997143 + 0.0755340i \(0.0240661\pi\)
−0.0671431 + 0.997743i \(0.521388\pi\)
\(30\) −4.69688 − 12.7323i −0.156563 − 0.424409i
\(31\) 35.8114 + 10.5152i 1.15521 + 0.339199i 0.802568 − 0.596560i \(-0.203466\pi\)
0.352639 + 0.935760i \(0.385284\pi\)
\(32\) 3.05833 + 4.75885i 0.0955727 + 0.148714i
\(33\) −0.209191 + 1.45496i −0.00633912 + 0.0440896i
\(34\) 4.66832 + 2.13195i 0.137304 + 0.0627045i
\(35\) 0.498726 − 2.25362i 0.0142493 − 0.0643891i
\(36\) 1.51325 − 10.5249i 0.0420348 − 0.292359i
\(37\) −57.6318 + 37.0377i −1.55762 + 1.00102i −0.574400 + 0.818575i \(0.694765\pi\)
−0.983216 + 0.182444i \(0.941599\pi\)
\(38\) 39.7348 + 11.6672i 1.04565 + 0.307031i
\(39\) 5.74396 + 39.9501i 0.147281 + 1.02436i
\(40\) −8.51566 + 11.2909i −0.212892 + 0.282271i
\(41\) −30.9951 − 19.9193i −0.755978 − 0.485837i 0.105005 − 0.994472i \(-0.466514\pi\)
−0.860983 + 0.508634i \(0.830151\pi\)
\(42\) −0.820508 + 0.946917i −0.0195359 + 0.0225456i
\(43\) −31.1513 + 9.14684i −0.724449 + 0.212717i −0.623109 − 0.782135i \(-0.714131\pi\)
−0.101339 + 0.994852i \(0.532313\pi\)
\(44\) 1.39336 − 0.636325i 0.0316672 − 0.0144619i
\(45\) 25.9955 − 5.55729i 0.577677 − 0.123495i
\(46\) 0.561045 − 32.5221i 0.0121966 − 0.707002i
\(47\) − 60.7751i − 1.29309i −0.762877 − 0.646543i \(-0.776214\pi\)
0.762877 − 0.646543i \(-0.223786\pi\)
\(48\) 6.98316 − 3.18910i 0.145482 − 0.0664396i
\(49\) 46.8107 − 13.7449i 0.955320 − 0.280507i
\(50\) −33.8508 − 10.2041i −0.677016 − 0.204082i
\(51\) 3.76543 − 5.85913i 0.0738321 − 0.114885i
\(52\) 31.7865 − 27.5432i 0.611280 − 0.529677i
\(53\) 11.2608 + 78.3204i 0.212467 + 1.47774i 0.764880 + 0.644173i \(0.222798\pi\)
−0.552412 + 0.833571i \(0.686293\pi\)
\(54\) −37.2840 − 10.9476i −0.690444 − 0.202733i
\(55\) 2.69809 + 2.71754i 0.0490562 + 0.0494098i
\(56\) 1.29239 + 0.185818i 0.0230784 + 0.00331818i
\(57\) 23.3466 − 51.1218i 0.409589 − 0.896874i
\(58\) −52.9801 − 24.1952i −0.913450 − 0.417159i
\(59\) −8.45263 + 58.7893i −0.143265 + 0.996429i 0.783662 + 0.621187i \(0.213349\pi\)
−0.926927 + 0.375242i \(0.877560\pi\)
\(60\) 13.5222 + 13.6196i 0.225369 + 0.226994i
\(61\) 30.8479 − 105.058i 0.505703 − 1.72227i −0.170321 − 0.985389i \(-0.554481\pi\)
0.676025 − 0.736879i \(-0.263701\pi\)
\(62\) −52.2458 + 7.51182i −0.842675 + 0.121158i
\(63\) −1.60721 − 1.85482i −0.0255113 − 0.0294416i
\(64\) −6.73003 − 4.32513i −0.105157 − 0.0675801i
\(65\) 92.1056 + 50.7236i 1.41701 + 0.780363i
\(66\) −0.585659 − 1.99457i −0.00887363 − 0.0302208i
\(67\) −45.4327 − 99.4838i −0.678101 − 1.48483i −0.864643 − 0.502387i \(-0.832455\pi\)
0.186542 − 0.982447i \(-0.440272\pi\)
\(68\) −7.25789 −0.106734
\(69\) −43.5780 − 7.03480i −0.631566 − 0.101954i
\(70\) 0.682400 + 3.19208i 0.00974857 + 0.0456012i
\(71\) −33.8025 − 74.0172i −0.476092 − 1.04250i −0.983520 − 0.180801i \(-0.942131\pi\)
0.507428 − 0.861694i \(-0.330596\pi\)
\(72\) 4.23656 + 14.4284i 0.0588411 + 0.200395i
\(73\) −72.5846 − 62.8949i −0.994310 − 0.861575i −0.00393606 − 0.999992i \(-0.501253\pi\)
−0.990374 + 0.138418i \(0.955798\pi\)
\(74\) 52.3792 − 81.5037i 0.707827 − 1.10140i
\(75\) −20.2448 + 43.5005i −0.269931 + 0.580006i
\(76\) −57.9697 + 8.33479i −0.762760 + 0.109668i
\(77\) 0.0996085 − 0.339236i 0.00129362 − 0.00440566i
\(78\) −30.8592 − 48.0179i −0.395631 − 0.615614i
\(79\) −21.5246 − 3.09477i −0.272463 − 0.0391742i 0.00472905 − 0.999989i \(-0.498495\pi\)
−0.277192 + 0.960815i \(0.589404\pi\)
\(80\) 4.32145 − 19.5275i 0.0540181 − 0.244094i
\(81\) −2.02928 + 4.44349i −0.0250528 + 0.0548580i
\(82\) 51.5748 + 7.41534i 0.628961 + 0.0904310i
\(83\) 75.4015 − 48.4576i 0.908452 − 0.583827i −0.000833557 − 1.00000i \(-0.500265\pi\)
0.909285 + 0.416173i \(0.136629\pi\)
\(84\) 0.499213 − 1.70016i 0.00594301 − 0.0202400i
\(85\) −6.27984 − 17.0233i −0.0738805 − 0.200275i
\(86\) 34.6998 − 30.0676i 0.403486 − 0.349623i
\(87\) −42.7334 + 66.4944i −0.491188 + 0.764304i
\(88\) −1.41860 + 1.63715i −0.0161205 + 0.0186040i
\(89\) 5.79451 + 19.7343i 0.0651069 + 0.221734i 0.985620 − 0.168979i \(-0.0540469\pi\)
−0.920513 + 0.390712i \(0.872229\pi\)
\(90\) −30.1761 + 22.4209i −0.335290 + 0.249122i
\(91\) − 9.70796i − 0.106681i
\(92\) 18.3845 + 42.1665i 0.199832 + 0.458331i
\(93\) 71.6318i 0.770235i
\(94\) 35.7045 + 78.1819i 0.379835 + 0.831722i
\(95\) −69.7072 − 128.756i −0.733760 − 1.35533i
\(96\) −7.10968 + 8.20501i −0.0740592 + 0.0854688i
\(97\) −58.1078 − 37.3436i −0.599050 − 0.384986i 0.205687 − 0.978618i \(-0.434057\pi\)
−0.804737 + 0.593632i \(0.797694\pi\)
\(98\) −52.1430 + 45.1822i −0.532072 + 0.461043i
\(99\) 4.03047 − 0.579494i 0.0407118 − 0.00585347i
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.19.8 ✓ 240
5.4 even 2 inner 230.3.i.a.19.17 yes 240
23.17 odd 22 inner 230.3.i.a.109.17 yes 240
115.109 odd 22 inner 230.3.i.a.109.8 yes 240
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
230.3.i.a.19.8 ✓ 240 1.1 even 1 trivial
230.3.i.a.19.17 yes 240 5.4 even 2 inner
230.3.i.a.109.8 yes 240 115.109 odd 22 inner
230.3.i.a.109.17 yes 240 23.17 odd 22 inner