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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [230,3,Mod(3,230)] 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("230.3"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(230, base_ring=CyclotomicField(44)) chi = DirichletCharacter(H, H._module([33, 32])) N = Newforms(chi, 3, names="a")
 
Level: \( N \) \(=\) \( 230 = 2 \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 230.k (of order \(44\), degree \(20\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [240,24] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == 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: \(12\) over \(\Q(\zeta_{44})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{44}]$

Embedding invariants

Embedding label 13.6
Character \(\chi\) \(=\) 230.13
Dual form 230.3.k.b.177.6

$q$-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+(1.41061 + 0.100889i) q^{2} +(0.0578351 + 0.265863i) q^{3} +(1.97964 + 0.284630i) q^{4} +(-0.330642 - 4.98906i) q^{5} +(0.0547601 + 0.380865i) q^{6} +(-4.77713 + 2.60851i) q^{7} +(2.76379 + 0.601225i) q^{8} +(8.11935 - 3.70798i) q^{9} +(0.0369324 - 7.07097i) q^{10} +(12.2267 - 14.1104i) q^{11} +(0.0388201 + 0.542776i) q^{12} +(7.23245 + 3.94921i) q^{13} +(-7.00183 + 3.19763i) q^{14} +(1.30728 - 0.376448i) q^{15} +(3.83797 + 1.12693i) q^{16} +(-11.1442 - 8.34248i) q^{17} +(11.8273 - 4.41137i) q^{18} +(27.0736 + 3.89260i) q^{19} +(0.765479 - 9.97066i) q^{20} +(-0.969793 - 1.11920i) q^{21} +(18.6707 - 18.6707i) q^{22} +(-20.2227 + 10.9564i) q^{23} +0.769562i q^{24} +(-24.7814 + 3.29919i) q^{25} +(9.80373 + 6.30048i) q^{26} +(2.92287 + 3.90450i) q^{27} +(-10.1995 + 3.80420i) q^{28} +(-50.8646 + 7.31323i) q^{29} +(1.88205 - 0.399131i) q^{30} +(24.2824 - 15.6053i) q^{31} +(5.30019 + 1.97687i) q^{32} +(4.45856 + 2.43456i) q^{33} +(-14.8785 - 12.8923i) q^{34} +(14.5935 + 22.9709i) q^{35} +(17.1288 - 5.02947i) q^{36} +(48.4332 + 18.0646i) q^{37} +(37.7976 + 8.22237i) q^{38} +(-0.631663 + 2.15125i) q^{39} +(2.08572 - 13.9875i) q^{40} +(-8.69524 + 19.0399i) q^{41} +(-1.25508 - 1.67660i) q^{42} +(5.57051 + 25.6072i) q^{43} +(28.2207 - 24.4534i) q^{44} +(-21.1839 - 39.2819i) q^{45} +(-29.6317 + 13.4149i) q^{46} +(-19.9708 + 19.9708i) q^{47} +(-0.0776403 + 1.08555i) q^{48} +(-10.4748 + 16.2991i) q^{49} +(-35.2897 + 2.15371i) q^{50} +(1.57343 - 3.44533i) q^{51} +(13.1936 + 9.87660i) q^{52} +(-40.8076 - 74.7336i) q^{53} +(3.72911 + 5.80261i) q^{54} +(-74.4400 - 56.3342i) q^{55} +(-14.7713 + 4.33724i) q^{56} +(0.530905 + 7.42302i) q^{57} +(-72.4880 + 5.18444i) q^{58} +(19.5603 + 66.6162i) q^{59} +(2.69511 - 0.373141i) q^{60} +(-39.7930 + 25.5734i) q^{61} +(35.8274 - 19.5632i) q^{62} +(-29.1149 + 38.8929i) q^{63} +(7.27706 + 3.32332i) q^{64} +(17.3115 - 37.3889i) q^{65} +(6.04367 + 3.88403i) q^{66} +(61.4972 + 4.39836i) q^{67} +(-19.6871 - 19.6871i) q^{68} +(-4.08248 - 4.74281i) q^{69} +(18.2683 + 33.8753i) q^{70} +(-55.5312 - 64.0864i) q^{71} +(24.6695 - 5.36652i) q^{72} +(-92.6110 + 69.3277i) q^{73} +(66.4978 + 30.3685i) q^{74} +(-2.31036 - 6.39765i) q^{75} +(52.4882 + 15.4119i) q^{76} +(-21.6015 + 99.3004i) q^{77} +(-1.10807 + 2.97084i) q^{78} +(28.8399 + 98.2198i) q^{79} +(4.35332 - 19.5205i) q^{80} +(51.7384 - 59.7093i) q^{81} +(-14.1865 + 25.9807i) q^{82} +(-6.07420 + 16.2856i) q^{83} +(-1.60129 - 2.49165i) q^{84} +(-37.9363 + 58.3576i) q^{85} +(5.27434 + 36.6838i) q^{86} +(-4.88608 - 13.1001i) q^{87} +(42.2755 - 31.6471i) q^{88} +(60.3002 - 93.8290i) q^{89} +(-25.9192 - 57.5486i) q^{90} -44.8519 q^{91} +(-43.1522 + 15.9337i) q^{92} +(5.55326 + 5.55326i) q^{93} +(-30.1859 + 26.1562i) q^{94} +(10.4687 - 136.359i) q^{95} +(-0.219040 + 1.52346i) q^{96} +(26.6519 + 71.4565i) q^{97} +(-16.4202 + 21.9349i) q^{98} +(46.9519 - 159.903i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 240 q + 24 q^{2} + 8 q^{3} - 4 q^{5} - 16 q^{6} + 50 q^{7} - 48 q^{8} + 16 q^{10} - 24 q^{11} - 28 q^{12} - 8 q^{13} + 8 q^{15} + 96 q^{16} + 44 q^{17} + 200 q^{18} - 24 q^{20} + 24 q^{21} + 24 q^{22} - 40 q^{23}+ \cdots - 776 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(47\) \(51\)
\(\chi(n)\) \(e\left(\frac{3}{4}\right)\) \(e\left(\frac{7}{11}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.41061 + 0.100889i 0.705305 + 0.0504444i
\(3\) 0.0578351 + 0.265863i 0.0192784 + 0.0886212i 0.985781 − 0.168036i \(-0.0537425\pi\)
−0.966502 + 0.256657i \(0.917379\pi\)
\(4\) 1.97964 + 0.284630i 0.494911 + 0.0711574i
\(5\) −0.330642 − 4.98906i −0.0661285 − 0.997811i
\(6\) 0.0547601 + 0.380865i 0.00912668 + 0.0634774i
\(7\) −4.77713 + 2.60851i −0.682447 + 0.372644i −0.782770 − 0.622311i \(-0.786194\pi\)
0.100324 + 0.994955i \(0.468012\pi\)
\(8\) 2.76379 + 0.601225i 0.345474 + 0.0751532i
\(9\) 8.11935 − 3.70798i 0.902150 − 0.411998i
\(10\) 0.0369324 − 7.07097i 0.00369324 − 0.707097i
\(11\) 12.2267 − 14.1104i 1.11152 − 1.28276i 0.156023 − 0.987753i \(-0.450133\pi\)
0.955495 − 0.295007i \(-0.0953218\pi\)
\(12\) 0.0388201 + 0.542776i 0.00323501 + 0.0452314i
\(13\) 7.23245 + 3.94921i 0.556342 + 0.303786i 0.732713 − 0.680538i \(-0.238254\pi\)
−0.176371 + 0.984324i \(0.556436\pi\)
\(14\) −7.00183 + 3.19763i −0.500131 + 0.228402i
\(15\) 1.30728 − 0.376448i 0.0871523 − 0.0250965i
\(16\) 3.83797 + 1.12693i 0.239873 + 0.0704331i
\(17\) −11.1442 − 8.34248i −0.655544 − 0.490734i 0.218972 − 0.975731i \(-0.429730\pi\)
−0.874516 + 0.484997i \(0.838821\pi\)
\(18\) 11.8273 − 4.41137i 0.657074 − 0.245076i
\(19\) 27.0736 + 3.89260i 1.42493 + 0.204874i 0.811258 − 0.584688i \(-0.198783\pi\)
0.613671 + 0.789562i \(0.289692\pi\)
\(20\) 0.765479 − 9.97066i 0.0382740 − 0.498533i
\(21\) −0.969793 − 1.11920i −0.0461806 − 0.0532952i
\(22\) 18.6707 − 18.6707i 0.848667 − 0.848667i
\(23\) −20.2227 + 10.9564i −0.879248 + 0.476364i
\(24\) 0.769562i 0.0320651i
\(25\) −24.7814 + 3.29919i −0.991254 + 0.131967i
\(26\) 9.80373 + 6.30048i 0.377067 + 0.242326i
\(27\) 2.92287 + 3.90450i 0.108254 + 0.144611i
\(28\) −10.1995 + 3.80420i −0.364267 + 0.135864i
\(29\) −50.8646 + 7.31323i −1.75395 + 0.252180i −0.942967 − 0.332886i \(-0.891978\pi\)
−0.810986 + 0.585066i \(0.801069\pi\)
\(30\) 1.88205 − 0.399131i 0.0627350 − 0.0133044i
\(31\) 24.2824 − 15.6053i 0.783303 − 0.503398i −0.0868254 − 0.996224i \(-0.527672\pi\)
0.870128 + 0.492825i \(0.164036\pi\)
\(32\) 5.30019 + 1.97687i 0.165631 + 0.0617771i
\(33\) 4.45856 + 2.43456i 0.135108 + 0.0737745i
\(34\) −14.8785 − 12.8923i −0.437604 − 0.379186i
\(35\) 14.5935 + 22.9709i 0.416958 + 0.656310i
\(36\) 17.1288 − 5.02947i 0.475800 − 0.139708i
\(37\) 48.4332 + 18.0646i 1.30900 + 0.488233i 0.904567 − 0.426332i \(-0.140195\pi\)
0.404438 + 0.914566i \(0.367467\pi\)
\(38\) 37.7976 + 8.22237i 0.994675 + 0.216378i
\(39\) −0.631663 + 2.15125i −0.0161965 + 0.0551602i
\(40\) 2.08572 − 13.9875i 0.0521430 − 0.349687i
\(41\) −8.69524 + 19.0399i −0.212079 + 0.464388i −0.985537 − 0.169459i \(-0.945798\pi\)
0.773458 + 0.633847i \(0.218525\pi\)
\(42\) −1.25508 − 1.67660i −0.0298830 − 0.0399190i
\(43\) 5.57051 + 25.6072i 0.129547 + 0.595517i 0.995448 + 0.0953025i \(0.0303818\pi\)
−0.865902 + 0.500214i \(0.833255\pi\)
\(44\) 28.2207 − 24.4534i 0.641380 − 0.555759i
\(45\) −21.1839 − 39.2819i −0.470754 − 0.872930i
\(46\) −29.6317 + 13.4149i −0.644168 + 0.291629i
\(47\) −19.9708 + 19.9708i −0.424911 + 0.424911i −0.886890 − 0.461980i \(-0.847139\pi\)
0.461980 + 0.886890i \(0.347139\pi\)
\(48\) −0.0776403 + 1.08555i −0.00161751 + 0.0226157i
\(49\) −10.4748 + 16.2991i −0.213771 + 0.332634i
\(50\) −35.2897 + 2.15371i −0.705794 + 0.0430741i
\(51\) 1.57343 − 3.44533i 0.0308516 − 0.0675556i
\(52\) 13.1936 + 9.87660i 0.253723 + 0.189935i
\(53\) −40.8076 − 74.7336i −0.769955 − 1.41007i −0.907709 − 0.419601i \(-0.862170\pi\)
0.137754 − 0.990467i \(-0.456012\pi\)
\(54\) 3.72911 + 5.80261i 0.0690576 + 0.107456i
\(55\) −74.4400 − 56.3342i −1.35346 − 1.02426i
\(56\) −14.7713 + 4.33724i −0.263773 + 0.0774506i
\(57\) 0.530905 + 7.42302i 0.00931412 + 0.130228i
\(58\) −72.4880 + 5.18444i −1.24979 + 0.0893870i
\(59\) 19.5603 + 66.6162i 0.331530 + 1.12909i 0.941598 + 0.336739i \(0.109324\pi\)
−0.610068 + 0.792349i \(0.708858\pi\)
\(60\) 2.69511 − 0.373141i 0.0449184 − 0.00621901i
\(61\) −39.7930 + 25.5734i −0.652345 + 0.419236i −0.824522 − 0.565830i \(-0.808556\pi\)
0.172178 + 0.985066i \(0.444920\pi\)
\(62\) 35.8274 − 19.5632i 0.577861 − 0.315536i
\(63\) −29.1149 + 38.8929i −0.462141 + 0.617348i
\(64\) 7.27706 + 3.32332i 0.113704 + 0.0519269i
\(65\) 17.3115 − 37.3889i 0.266331 − 0.575213i
\(66\) 6.04367 + 3.88403i 0.0915708 + 0.0588490i
\(67\) 61.4972 + 4.39836i 0.917868 + 0.0656472i 0.522274 − 0.852778i \(-0.325084\pi\)
0.395595 + 0.918425i \(0.370539\pi\)
\(68\) −19.6871 − 19.6871i −0.289516 − 0.289516i
\(69\) −4.08248 − 4.74281i −0.0591664 − 0.0687364i
\(70\) 18.2683 + 33.8753i 0.260975 + 0.483932i
\(71\) −55.5312 − 64.0864i −0.782129 − 0.902625i 0.215132 − 0.976585i \(-0.430982\pi\)
−0.997261 + 0.0739600i \(0.976436\pi\)
\(72\) 24.6695 − 5.36652i 0.342632 − 0.0745350i
\(73\) −92.6110 + 69.3277i −1.26864 + 0.949695i −0.999909 − 0.0135194i \(-0.995697\pi\)
−0.268735 + 0.963214i \(0.586606\pi\)
\(74\) 66.4978 + 30.3685i 0.898619 + 0.410385i
\(75\) −2.31036 − 6.39765i −0.0308049 − 0.0853020i
\(76\) 52.4882 + 15.4119i 0.690634 + 0.202788i
\(77\) −21.6015 + 99.3004i −0.280539 + 1.28962i
\(78\) −1.10807 + 2.97084i −0.0142060 + 0.0380877i
\(79\) 28.8399 + 98.2198i 0.365062 + 1.24329i 0.913411 + 0.407039i \(0.133439\pi\)
−0.548348 + 0.836250i \(0.684743\pi\)
\(80\) 4.35332 − 19.5205i 0.0544165 − 0.244006i
\(81\) 51.7384 − 59.7093i 0.638746 − 0.737152i
\(82\) −14.1865 + 25.9807i −0.173006 + 0.316837i
\(83\) −6.07420 + 16.2856i −0.0731832 + 0.196212i −0.968391 − 0.249437i \(-0.919754\pi\)
0.895208 + 0.445649i \(0.147027\pi\)
\(84\) −1.60129 − 2.49165i −0.0190629 − 0.0296625i
\(85\) −37.9363 + 58.3576i −0.446310 + 0.686560i
\(86\) 5.27434 + 36.6838i 0.0613295 + 0.426556i
\(87\) −4.88608 − 13.1001i −0.0561618 − 0.150576i
\(88\) 42.2755 − 31.6471i 0.480404 − 0.359626i
\(89\) 60.3002 − 93.8290i 0.677531 − 1.05426i −0.316858 − 0.948473i \(-0.602628\pi\)
0.994389 − 0.105785i \(-0.0337357\pi\)
\(90\) −25.9192 − 57.5486i −0.287991 − 0.639429i
\(91\) −44.8519 −0.492878
\(92\) −43.1522 + 15.9337i −0.469046 + 0.173193i
\(93\) 5.55326 + 5.55326i 0.0597125 + 0.0597125i
\(94\) −30.1859 + 26.1562i −0.321126 + 0.278257i
\(95\) 10.4687 − 136.359i 0.110197 − 1.43536i
\(96\) −0.219040 + 1.52346i −0.00228167 + 0.0158694i
\(97\) 26.6519 + 71.4565i 0.274762 + 0.736665i 0.998895 + 0.0470042i \(0.0149674\pi\)
−0.724133 + 0.689660i \(0.757760\pi\)
\(98\) −16.4202 + 21.9349i −0.167553 + 0.223825i
\(99\) 46.9519 − 159.903i 0.474261 − 1.61519i
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.k.b.13.6 ✓ 240
5.2 odd 4 inner 230.3.k.b.197.6 yes 240
23.16 even 11 inner 230.3.k.b.223.6 yes 240
115.62 odd 44 inner 230.3.k.b.177.6 yes 240
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
230.3.k.b.13.6 ✓ 240 1.1 even 1 trivial
230.3.k.b.177.6 yes 240 115.62 odd 44 inner
230.3.k.b.197.6 yes 240 5.2 odd 4 inner
230.3.k.b.223.6 yes 240 23.16 even 11 inner