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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 19.5
Character \(\chi\) \(=\) 230.19
Dual form 230.3.i.a.109.5

$q$-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} +(-0.424421 - 1.44544i) q^{3} +(1.30972 - 1.51150i) q^{4} +(3.70278 + 3.35997i) q^{5} +(1.39516 + 1.61010i) q^{6} +(1.33136 + 9.25983i) q^{7} +(-0.796860 + 2.71386i) q^{8} +(5.66211 - 3.63882i) q^{9} +(-6.73725 - 2.14698i) q^{10} +(-11.1673 - 5.09994i) q^{11} +(-2.74066 - 1.25162i) q^{12} +(-16.2447 - 2.33563i) q^{13} +(-7.15270 - 11.1298i) q^{14} +(3.28511 - 6.77820i) q^{15} +(-0.569259 - 3.95929i) q^{16} +(16.9201 + 19.5268i) q^{17} +(-5.14606 + 8.00743i) q^{18} +(22.5651 + 19.5528i) q^{19} +(9.92821 - 1.19613i) q^{20} +(12.8195 - 5.85447i) q^{21} +17.3619 q^{22} +(2.98448 + 22.8055i) q^{23} +4.26093 q^{24} +(2.42122 + 24.8825i) q^{25} +(22.2695 - 6.53892i) q^{26} +(-17.9094 - 15.5186i) q^{27} +(15.7399 + 10.1154i) q^{28} +(9.51860 + 10.9851i) q^{29} +(-0.243912 + 10.6495i) q^{30} +(-22.7320 - 6.67472i) q^{31} +(3.05833 + 4.75885i) q^{32} +(-2.63204 + 18.3062i) q^{33} +(-33.2380 - 15.1793i) q^{34} +(-26.1830 + 38.7605i) q^{35} +(1.91572 - 13.3241i) q^{36} +(37.0350 - 23.8009i) q^{37} +(-40.5151 - 11.8963i) q^{38} +(3.51855 + 24.4720i) q^{39} +(-12.0691 + 7.37140i) q^{40} +(32.0940 + 20.6255i) q^{41} +(-13.0518 + 15.0626i) q^{42} +(5.57263 - 1.63627i) q^{43} +(-22.3346 + 10.1999i) q^{44} +(33.1919 + 5.55076i) q^{45} +(-17.2372 - 27.5840i) q^{46} +52.2193i q^{47} +(-5.48132 + 2.50323i) q^{48} +(-36.9568 + 10.8515i) q^{49} +(-17.7328 - 30.5867i) q^{50} +(21.0437 - 32.7447i) q^{51} +(-24.8063 + 21.4948i) q^{52} +(-4.31324 - 29.9993i) q^{53} +(32.1559 + 9.44182i) q^{54} +(-24.2145 - 56.4058i) q^{55} +(-26.1908 - 3.76566i) q^{56} +(18.6854 - 40.9152i) q^{57} +(-18.6984 - 8.53928i) q^{58} +(0.933436 - 6.49219i) q^{59} +(-5.94267 - 13.8430i) q^{60} +(6.66826 - 22.7100i) q^{61} +(33.1641 - 4.76827i) q^{62} +(41.2331 + 47.5856i) q^{63} +(-6.73003 - 4.32513i) q^{64} +(-52.3028 - 63.2299i) q^{65} +(-7.36875 - 25.0957i) q^{66} +(-35.9742 - 78.7725i) q^{67} +51.6754 q^{68} +(31.6975 - 13.9930i) q^{69} +(10.9109 - 65.2442i) q^{70} +(18.9338 + 41.4593i) q^{71} +(5.36331 + 18.2658i) q^{72} +(-26.9913 - 23.3881i) q^{73} +(-33.6596 + 52.3753i) q^{74} +(34.9386 - 14.0604i) q^{75} +(59.1081 - 8.49846i) q^{76} +(32.3568 - 110.197i) q^{77} +(-18.9033 - 29.4141i) q^{78} +(-17.9264 - 2.57743i) q^{79} +(11.1952 - 16.5731i) q^{80} +(10.3336 - 22.6275i) q^{81} +(-53.4033 - 7.67824i) q^{82} +(-107.016 + 68.7750i) q^{83} +(7.94095 - 27.0444i) q^{84} +(-2.95810 + 129.155i) q^{85} +(-6.20742 + 5.37876i) q^{86} +(11.8384 - 18.4209i) q^{87} +(22.7393 - 26.2425i) q^{88} +(17.1774 + 58.5008i) q^{89} +(-45.9595 + 12.3592i) q^{90} -153.532i q^{91} +(38.3794 + 25.3579i) q^{92} +35.6907i q^{93} +(-30.6781 - 67.1757i) q^{94} +(17.8570 + 148.218i) q^{95} +(5.58063 - 6.44039i) q^{96} +(-129.040 - 82.9290i) q^{97} +(41.1667 - 35.6711i) q^{98} +(-81.7882 + 11.7594i) 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
\(3\) −0.424421 − 1.44544i −0.141474 − 0.481815i 0.858021 − 0.513615i \(-0.171694\pi\)
−0.999494 + 0.0318005i \(0.989876\pi\)
\(4\) 1.30972 − 1.51150i 0.327430 − 0.377875i
\(5\) 3.70278 + 3.35997i 0.740557 + 0.671994i
\(6\) 1.39516 + 1.61010i 0.232526 + 0.268350i
\(7\) 1.33136 + 9.25983i 0.190195 + 1.32283i 0.831490 + 0.555539i \(0.187488\pi\)
−0.641296 + 0.767294i \(0.721603\pi\)
\(8\) −0.796860 + 2.71386i −0.0996075 + 0.339232i
\(9\) 5.66211 − 3.63882i 0.629123 − 0.404313i
\(10\) −6.73725 − 2.14698i −0.673725 − 0.214698i
\(11\) −11.1673 − 5.09994i −1.01521 − 0.463631i −0.162886 − 0.986645i \(-0.552080\pi\)
−0.852324 + 0.523014i \(0.824807\pi\)
\(12\) −2.74066 − 1.25162i −0.228388 − 0.104301i
\(13\) −16.2447 − 2.33563i −1.24959 − 0.179664i −0.514415 − 0.857541i \(-0.671991\pi\)
−0.735175 + 0.677878i \(0.762900\pi\)
\(14\) −7.15270 − 11.1298i −0.510907 − 0.794987i
\(15\) 3.28511 − 6.77820i 0.219007 − 0.451880i
\(16\) −0.569259 − 3.95929i −0.0355787 − 0.247455i
\(17\) 16.9201 + 19.5268i 0.995301 + 1.14864i 0.988889 + 0.148659i \(0.0474956\pi\)
0.00641209 + 0.999979i \(0.497959\pi\)
\(18\) −5.14606 + 8.00743i −0.285892 + 0.444857i
\(19\) 22.5651 + 19.5528i 1.18764 + 1.02909i 0.998892 + 0.0470684i \(0.0149879\pi\)
0.188746 + 0.982026i \(0.439558\pi\)
\(20\) 9.92821 − 1.19613i 0.496410 − 0.0598065i
\(21\) 12.8195 − 5.85447i 0.610453 − 0.278784i
\(22\) 17.3619 0.789178
\(23\) 2.98448 + 22.8055i 0.129760 + 0.991545i
\(24\) 4.26093 0.177539
\(25\) 2.42122 + 24.8825i 0.0968486 + 0.995299i
\(26\) 22.2695 − 6.53892i 0.856520 − 0.251497i
\(27\) −17.9094 − 15.5186i −0.663312 − 0.574763i
\(28\) 15.7399 + 10.1154i 0.562141 + 0.361266i
\(29\) 9.51860 + 10.9851i 0.328228 + 0.378795i 0.895746 − 0.444566i \(-0.146642\pi\)
−0.567519 + 0.823361i \(0.692096\pi\)
\(30\) −0.243912 + 10.6495i −0.00813039 + 0.354984i
\(31\) −22.7320 − 6.67472i −0.733290 − 0.215314i −0.106293 − 0.994335i \(-0.533898\pi\)
−0.626997 + 0.779021i \(0.715716\pi\)
\(32\) 3.05833 + 4.75885i 0.0955727 + 0.148714i
\(33\) −2.63204 + 18.3062i −0.0797587 + 0.554734i
\(34\) −33.2380 − 15.1793i −0.977588 − 0.446450i
\(35\) −26.1830 + 38.7605i −0.748086 + 1.10744i
\(36\) 1.91572 − 13.3241i 0.0532144 − 0.370114i
\(37\) 37.0350 − 23.8009i 1.00094 − 0.643268i 0.0659105 − 0.997826i \(-0.479005\pi\)
0.935034 + 0.354557i \(0.115368\pi\)
\(38\) −40.5151 − 11.8963i −1.06619 − 0.313060i
\(39\) 3.51855 + 24.4720i 0.0902192 + 0.627488i
\(40\) −12.0691 + 7.37140i −0.301727 + 0.184285i
\(41\) 32.0940 + 20.6255i 0.782780 + 0.503062i 0.869955 − 0.493131i \(-0.164148\pi\)
−0.0871753 + 0.996193i \(0.527784\pi\)
\(42\) −13.0518 + 15.0626i −0.310757 + 0.358632i
\(43\) 5.57263 − 1.63627i 0.129596 − 0.0380528i −0.216291 − 0.976329i \(-0.569396\pi\)
0.345887 + 0.938276i \(0.387578\pi\)
\(44\) −22.3346 + 10.1999i −0.507605 + 0.231815i
\(45\) 33.1919 + 5.55076i 0.737597 + 0.123350i
\(46\) −17.2372 − 27.5840i −0.374722 − 0.599653i
\(47\) 52.2193i 1.11105i 0.831500 + 0.555525i \(0.187483\pi\)
−0.831500 + 0.555525i \(0.812517\pi\)
\(48\) −5.48132 + 2.50323i −0.114194 + 0.0521507i
\(49\) −36.9568 + 10.8515i −0.754221 + 0.221459i
\(50\) −17.7328 − 30.5867i −0.354656 − 0.611735i
\(51\) 21.0437 − 32.7447i 0.412622 − 0.642052i
\(52\) −24.8063 + 21.4948i −0.477044 + 0.413361i
\(53\) −4.31324 − 29.9993i −0.0813820 − 0.566024i −0.989190 − 0.146638i \(-0.953155\pi\)
0.907808 − 0.419386i \(-0.137754\pi\)
\(54\) 32.1559 + 9.44182i 0.595479 + 0.174848i
\(55\) −24.2145 − 56.4058i −0.440263 − 1.02556i
\(56\) −26.1908 − 3.76566i −0.467692 − 0.0672440i
\(57\) 18.6854 − 40.9152i 0.327813 − 0.717811i
\(58\) −18.6984 − 8.53928i −0.322387 − 0.147229i
\(59\) 0.933436 − 6.49219i 0.0158209 − 0.110037i −0.980381 − 0.197113i \(-0.936843\pi\)
0.996202 + 0.0870763i \(0.0277524\pi\)
\(60\) −5.94267 − 13.8430i −0.0990446 − 0.230717i
\(61\) 6.66826 − 22.7100i 0.109316 − 0.372295i −0.886605 − 0.462527i \(-0.846943\pi\)
0.995921 + 0.0902320i \(0.0287609\pi\)
\(62\) 33.1641 − 4.76827i 0.534904 − 0.0769076i
\(63\) 41.2331 + 47.5856i 0.654494 + 0.755327i
\(64\) −6.73003 − 4.32513i −0.105157 − 0.0675801i
\(65\) −52.3028 − 63.2299i −0.804659 − 0.972768i
\(66\) −7.36875 − 25.0957i −0.111648 − 0.380237i
\(67\) −35.9742 − 78.7725i −0.536928 − 1.17571i −0.962624 − 0.270843i \(-0.912698\pi\)
0.425695 − 0.904867i \(-0.360030\pi\)
\(68\) 51.6754 0.759933
\(69\) 31.6975 − 13.9930i 0.459383 − 0.202798i
\(70\) 10.9109 − 65.2442i 0.155871 − 0.932060i
\(71\) 18.9338 + 41.4593i 0.266674 + 0.583934i 0.994839 − 0.101468i \(-0.0323538\pi\)
−0.728165 + 0.685402i \(0.759627\pi\)
\(72\) 5.36331 + 18.2658i 0.0744905 + 0.253691i
\(73\) −26.9913 − 23.3881i −0.369744 − 0.320385i 0.450093 − 0.892982i \(-0.351391\pi\)
−0.819837 + 0.572596i \(0.805936\pi\)
\(74\) −33.6596 + 52.3753i −0.454859 + 0.707775i
\(75\) 34.9386 − 14.0604i 0.465848 − 0.187472i
\(76\) 59.1081 − 8.49846i 0.777738 − 0.111822i
\(77\) 32.3568 − 110.197i 0.420219 − 1.43113i
\(78\) −18.9033 − 29.4141i −0.242350 − 0.377104i
\(79\) −17.9264 − 2.57743i −0.226917 − 0.0326257i 0.0279185 − 0.999610i \(-0.491112\pi\)
−0.254835 + 0.966985i \(0.582021\pi\)
\(80\) 11.1952 − 16.5731i 0.139940 − 0.207163i
\(81\) 10.3336 − 22.6275i 0.127576 − 0.279352i
\(82\) −53.4033 − 7.67824i −0.651260 − 0.0936371i
\(83\) −107.016 + 68.7750i −1.28935 + 0.828614i −0.992010 − 0.126158i \(-0.959735\pi\)
−0.297339 + 0.954772i \(0.596099\pi\)
\(84\) 7.94095 − 27.0444i 0.0945352 − 0.321957i
\(85\) −2.95810 + 129.155i −0.0348011 + 1.51947i
\(86\) −6.20742 + 5.37876i −0.0721793 + 0.0625437i
\(87\) 11.8384 − 18.4209i 0.136073 − 0.211734i
\(88\) 22.7393 − 26.2425i 0.258401 − 0.298210i
\(89\) 17.1774 + 58.5008i 0.193004 + 0.657312i 0.997952 + 0.0639666i \(0.0203751\pi\)
−0.804948 + 0.593346i \(0.797807\pi\)
\(90\) −45.9595 + 12.3592i −0.510661 + 0.137324i
\(91\) − 153.532i − 1.68717i
\(92\) 38.3794 + 25.3579i 0.417167 + 0.275629i
\(93\) 35.6907i 0.383771i
\(94\) −30.6781 − 67.1757i −0.326363 − 0.714635i
\(95\) 17.8570 + 148.218i 0.187968 + 1.56019i
\(96\) 5.58063 − 6.44039i 0.0581316 − 0.0670874i
\(97\) −129.040 − 82.9290i −1.33031 − 0.854938i −0.334152 − 0.942519i \(-0.608450\pi\)
−0.996157 + 0.0875812i \(0.972086\pi\)
\(98\) 41.1667 − 35.6711i 0.420068 − 0.363991i
\(99\) −81.7882 + 11.7594i −0.826144 + 0.118782i
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.19.5 ✓ 240
5.4 even 2 inner 230.3.i.a.19.20 yes 240
23.17 odd 22 inner 230.3.i.a.109.20 yes 240
115.109 odd 22 inner 230.3.i.a.109.5 yes 240
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
230.3.i.a.19.5 ✓ 240 1.1 even 1 trivial
230.3.i.a.19.20 yes 240 5.4 even 2 inner
230.3.i.a.109.5 yes 240 115.109 odd 22 inner
230.3.i.a.109.20 yes 240 23.17 odd 22 inner