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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 13.8
Character \(\chi\) \(=\) 230.13
Dual form 230.3.k.b.177.8

$q$-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+(1.41061 + 0.100889i) q^{2} +(0.302791 + 1.39191i) q^{3} +(1.97964 + 0.284630i) q^{4} +(-2.63202 + 4.25117i) q^{5} +(0.286693 + 1.99399i) q^{6} +(-5.20021 + 2.83953i) q^{7} +(2.76379 + 0.601225i) q^{8} +(6.34096 - 2.89582i) q^{9} +(-4.14165 + 5.73121i) q^{10} +(-12.2025 + 14.0824i) q^{11} +(0.203240 + 2.84167i) q^{12} +(4.43840 + 2.42355i) q^{13} +(-7.62194 + 3.48083i) q^{14} +(-6.71420 - 2.37632i) q^{15} +(3.83797 + 1.12693i) q^{16} +(3.49894 + 2.61927i) q^{17} +(9.23678 - 3.44514i) q^{18} +(7.14287 + 1.02699i) q^{19} +(-6.42047 + 7.66665i) q^{20} +(-5.52695 - 6.37844i) q^{21} +(-18.6337 + 18.6337i) q^{22} +(-20.7644 + 9.89133i) q^{23} +4.02899i q^{24} +(-11.1449 - 22.3783i) q^{25} +(6.01634 + 3.86647i) q^{26} +(13.6335 + 18.2123i) q^{27} +(-11.1028 + 4.14112i) q^{28} +(29.0844 - 4.18171i) q^{29} +(-9.23138 - 4.02944i) q^{30} +(33.9744 - 21.8340i) q^{31} +(5.30019 + 1.97687i) q^{32} +(-23.2962 - 12.7207i) q^{33} +(4.67138 + 4.04778i) q^{34} +(1.61573 - 29.5807i) q^{35} +(13.3771 - 3.92786i) q^{36} +(24.1808 + 9.01895i) q^{37} +(9.97219 + 2.16932i) q^{38} +(-2.02945 + 6.91168i) q^{39} +(-9.83026 + 10.1669i) q^{40} +(-0.860004 + 1.88315i) q^{41} +(-7.15286 - 9.55510i) q^{42} +(1.03806 + 4.77191i) q^{43} +(-28.1648 + 24.4049i) q^{44} +(-4.37890 + 34.5784i) q^{45} +(-30.2884 + 11.8579i) q^{46} +(21.6665 - 21.6665i) q^{47} +(-0.406480 + 5.68334i) q^{48} +(-7.51215 + 11.6891i) q^{49} +(-13.4634 - 32.6915i) q^{50} +(-2.58634 + 5.66330i) q^{51} +(8.09663 + 6.06106i) q^{52} +(-24.5027 - 44.8733i) q^{53} +(17.3942 + 27.0659i) q^{54} +(-27.7496 - 88.9399i) q^{55} +(-16.0795 + 4.72136i) q^{56} +(0.733323 + 10.2532i) q^{57} +(41.4486 - 2.96447i) q^{58} +(-0.851055 - 2.89843i) q^{59} +(-12.6154 - 6.61532i) q^{60} +(78.2787 - 50.3067i) q^{61} +(50.1274 - 27.3716i) q^{62} +(-24.7515 + 33.0642i) q^{63} +(7.27706 + 3.32332i) q^{64} +(-21.9849 + 12.4896i) q^{65} +(-31.5785 - 20.2943i) q^{66} +(49.0512 + 3.50821i) q^{67} +(6.18112 + 6.18112i) q^{68} +(-20.0551 - 25.9072i) q^{69} +(5.26352 - 41.5638i) q^{70} +(56.5829 + 65.3002i) q^{71} +(19.2661 - 4.19108i) q^{72} +(-51.8388 + 38.8061i) q^{73} +(33.1997 + 15.1618i) q^{74} +(27.7740 - 22.2887i) q^{75} +(13.8480 + 4.06614i) q^{76} +(23.4680 - 107.881i) q^{77} +(-3.56008 + 9.54494i) q^{78} +(-36.6732 - 124.897i) q^{79} +(-14.8924 + 13.3498i) q^{80} +(19.8630 - 22.9231i) q^{81} +(-1.40312 + 2.56962i) q^{82} +(-4.21574 + 11.3028i) q^{83} +(-9.12589 - 14.2002i) q^{84} +(-20.3443 + 7.98061i) q^{85} +(0.982872 + 6.83603i) q^{86} +(14.6271 + 39.2167i) q^{87} +(-42.1917 + 31.5843i) q^{88} +(3.24075 - 5.04271i) q^{89} +(-9.66550 + 48.3348i) q^{90} -29.9623 q^{91} +(-43.9215 + 13.6711i) q^{92} +(40.6781 + 40.6781i) q^{93} +(32.7490 - 28.3771i) q^{94} +(-23.1661 + 27.6625i) q^{95} +(-1.14677 + 7.97596i) q^{96} +(-23.1227 - 61.9945i) q^{97} +(-11.7760 + 15.7309i) q^{98} +(-36.5953 + 124.632i) 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
\(3\) 0.302791 + 1.39191i 0.100930 + 0.463970i 0.999675 + 0.0254971i \(0.00811685\pi\)
−0.898744 + 0.438473i \(0.855520\pi\)
\(4\) 1.97964 + 0.284630i 0.494911 + 0.0711574i
\(5\) −2.63202 + 4.25117i −0.526404 + 0.850235i
\(6\) 0.286693 + 1.99399i 0.0477821 + 0.332332i
\(7\) −5.20021 + 2.83953i −0.742887 + 0.405647i −0.805658 − 0.592381i \(-0.798188\pi\)
0.0627710 + 0.998028i \(0.480006\pi\)
\(8\) 2.76379 + 0.601225i 0.345474 + 0.0751532i
\(9\) 6.34096 − 2.89582i 0.704551 − 0.321758i
\(10\) −4.14165 + 5.73121i −0.414165 + 0.573121i
\(11\) −12.2025 + 14.0824i −1.10932 + 1.28022i −0.152889 + 0.988243i \(0.548858\pi\)
−0.956426 + 0.291975i \(0.905688\pi\)
\(12\) 0.203240 + 2.84167i 0.0169367 + 0.236806i
\(13\) 4.43840 + 2.42355i 0.341415 + 0.186427i 0.640804 − 0.767704i \(-0.278601\pi\)
−0.299389 + 0.954131i \(0.596783\pi\)
\(14\) −7.62194 + 3.48083i −0.544425 + 0.248630i
\(15\) −6.71420 − 2.37632i −0.447614 − 0.158421i
\(16\) 3.83797 + 1.12693i 0.239873 + 0.0704331i
\(17\) 3.49894 + 2.61927i 0.205820 + 0.154075i 0.697215 − 0.716862i \(-0.254422\pi\)
−0.491396 + 0.870937i \(0.663513\pi\)
\(18\) 9.23678 − 3.44514i 0.513154 − 0.191397i
\(19\) 7.14287 + 1.02699i 0.375940 + 0.0540521i 0.327695 − 0.944783i \(-0.393728\pi\)
0.0482451 + 0.998836i \(0.484637\pi\)
\(20\) −6.42047 + 7.66665i −0.321023 + 0.383333i
\(21\) −5.52695 − 6.37844i −0.263188 − 0.303735i
\(22\) −18.6337 + 18.6337i −0.846986 + 0.846986i
\(23\) −20.7644 + 9.89133i −0.902801 + 0.430058i
\(24\) 4.02899i 0.167875i
\(25\) −11.1449 − 22.3783i −0.445798 − 0.895134i
\(26\) 6.01634 + 3.86647i 0.231398 + 0.148710i
\(27\) 13.6335 + 18.2123i 0.504946 + 0.674529i
\(28\) −11.1028 + 4.14112i −0.396527 + 0.147897i
\(29\) 29.0844 − 4.18171i 1.00291 − 0.144197i 0.378755 − 0.925497i \(-0.376352\pi\)
0.624155 + 0.781300i \(0.285443\pi\)
\(30\) −9.23138 − 4.02944i −0.307713 − 0.134315i
\(31\) 33.9744 − 21.8340i 1.09595 − 0.704323i 0.137761 − 0.990466i \(-0.456010\pi\)
0.958187 + 0.286143i \(0.0923732\pi\)
\(32\) 5.30019 + 1.97687i 0.165631 + 0.0617771i
\(33\) −23.2962 − 12.7207i −0.705946 − 0.385476i
\(34\) 4.67138 + 4.04778i 0.137394 + 0.119052i
\(35\) 1.61573 − 29.5807i 0.0461636 − 0.845162i
\(36\) 13.3771 − 3.92786i 0.371585 − 0.109107i
\(37\) 24.1808 + 9.01895i 0.653534 + 0.243756i 0.654289 − 0.756244i \(-0.272968\pi\)
−0.000755643 1.00000i \(0.500241\pi\)
\(38\) 9.97219 + 2.16932i 0.262426 + 0.0570873i
\(39\) −2.02945 + 6.91168i −0.0520372 + 0.177223i
\(40\) −9.83026 + 10.1669i −0.245757 + 0.254173i
\(41\) −0.860004 + 1.88315i −0.0209757 + 0.0459304i −0.919830 − 0.392317i \(-0.871673\pi\)
0.898854 + 0.438248i \(0.144401\pi\)
\(42\) −7.15286 − 9.55510i −0.170306 − 0.227502i
\(43\) 1.03806 + 4.77191i 0.0241410 + 0.110975i 0.987641 − 0.156735i \(-0.0500969\pi\)
−0.963500 + 0.267710i \(0.913733\pi\)
\(44\) −28.1648 + 24.4049i −0.640109 + 0.554658i
\(45\) −4.37890 + 34.5784i −0.0973090 + 0.768408i
\(46\) −30.2884 + 11.8579i −0.658444 + 0.257781i
\(47\) 21.6665 − 21.6665i 0.460990 − 0.460990i −0.437990 − 0.898980i \(-0.644309\pi\)
0.898980 + 0.437990i \(0.144309\pi\)
\(48\) −0.406480 + 5.68334i −0.00846834 + 0.118403i
\(49\) −7.51215 + 11.6891i −0.153309 + 0.238554i
\(50\) −13.4634 − 32.6915i −0.269269 − 0.653830i
\(51\) −2.58634 + 5.66330i −0.0507126 + 0.111045i
\(52\) 8.09663 + 6.06106i 0.155704 + 0.116559i
\(53\) −24.5027 − 44.8733i −0.462315 − 0.846667i −0.999994 − 0.00351698i \(-0.998881\pi\)
0.537679 − 0.843150i \(-0.319301\pi\)
\(54\) 17.3942 + 27.0659i 0.322115 + 0.501221i
\(55\) −27.7496 − 88.9399i −0.504538 − 1.61709i
\(56\) −16.0795 + 4.72136i −0.287133 + 0.0843100i
\(57\) 0.733323 + 10.2532i 0.0128653 + 0.179881i
\(58\) 41.4486 − 2.96447i 0.714632 − 0.0511115i
\(59\) −0.851055 − 2.89843i −0.0144247 − 0.0491259i 0.951966 − 0.306203i \(-0.0990587\pi\)
−0.966391 + 0.257077i \(0.917241\pi\)
\(60\) −12.6154 − 6.61532i −0.210256 − 0.110255i
\(61\) 78.2787 − 50.3067i 1.28326 − 0.824700i 0.291972 − 0.956427i \(-0.405689\pi\)
0.991286 + 0.131727i \(0.0420523\pi\)
\(62\) 50.1274 − 27.3716i 0.808507 − 0.441478i
\(63\) −24.7515 + 33.0642i −0.392882 + 0.524828i
\(64\) 7.27706 + 3.32332i 0.113704 + 0.0519269i
\(65\) −21.9849 + 12.4896i −0.338229 + 0.192147i
\(66\) −31.5785 − 20.2943i −0.478463 − 0.307489i
\(67\) 49.0512 + 3.50821i 0.732107 + 0.0523613i 0.432415 − 0.901675i \(-0.357662\pi\)
0.299692 + 0.954036i \(0.403116\pi\)
\(68\) 6.18112 + 6.18112i 0.0908989 + 0.0908989i
\(69\) −20.0551 − 25.9072i −0.290654 − 0.375467i
\(70\) 5.26352 − 41.5638i 0.0751932 − 0.593769i
\(71\) 56.5829 + 65.3002i 0.796942 + 0.919721i 0.998209 − 0.0598158i \(-0.0190513\pi\)
−0.201267 + 0.979536i \(0.564506\pi\)
\(72\) 19.2661 − 4.19108i 0.267585 − 0.0582095i
\(73\) −51.8388 + 38.8061i −0.710121 + 0.531590i −0.892170 − 0.451700i \(-0.850818\pi\)
0.182049 + 0.983289i \(0.441727\pi\)
\(74\) 33.1997 + 15.1618i 0.448645 + 0.204889i
\(75\) 27.7740 − 22.2887i 0.370321 − 0.297183i
\(76\) 13.8480 + 4.06614i 0.182211 + 0.0535019i
\(77\) 23.4680 − 107.881i 0.304779 − 1.40105i
\(78\) −3.56008 + 9.54494i −0.0456420 + 0.122371i
\(79\) −36.6732 − 124.897i −0.464218 − 1.58098i −0.775940 − 0.630807i \(-0.782724\pi\)
0.311723 − 0.950173i \(-0.399094\pi\)
\(80\) −14.8924 + 13.3498i −0.186155 + 0.166872i
\(81\) 19.8630 − 22.9231i 0.245222 − 0.283002i
\(82\) −1.40312 + 2.56962i −0.0171112 + 0.0313368i
\(83\) −4.21574 + 11.3028i −0.0507921 + 0.136179i −0.959868 − 0.280451i \(-0.909516\pi\)
0.909076 + 0.416630i \(0.136789\pi\)
\(84\) −9.12589 − 14.2002i −0.108642 − 0.169050i
\(85\) −20.3443 + 7.98061i −0.239344 + 0.0938896i
\(86\) 0.982872 + 6.83603i 0.0114287 + 0.0794887i
\(87\) 14.6271 + 39.2167i 0.168127 + 0.450766i
\(88\) −42.1917 + 31.5843i −0.479452 + 0.358913i
\(89\) 3.24075 − 5.04271i 0.0364129 − 0.0566596i −0.822570 − 0.568664i \(-0.807460\pi\)
0.858983 + 0.512004i \(0.171097\pi\)
\(90\) −9.66550 + 48.3348i −0.107394 + 0.537053i
\(91\) −29.9623 −0.329256
\(92\) −43.9215 + 13.6711i −0.477408 + 0.148599i
\(93\) 40.6781 + 40.6781i 0.437399 + 0.437399i
\(94\) 32.7490 − 28.3771i 0.348393 − 0.301884i
\(95\) −23.1661 + 27.6625i −0.243853 + 0.291184i
\(96\) −1.14677 + 7.97596i −0.0119455 + 0.0830829i
\(97\) −23.1227 − 61.9945i −0.238379 − 0.639118i 0.761581 − 0.648070i \(-0.224423\pi\)
−0.999960 + 0.00895124i \(0.997151\pi\)
\(98\) −11.7760 + 15.7309i −0.120164 + 0.160520i
\(99\) −36.5953 + 124.632i −0.369649 + 1.25891i
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.13.8 ✓ 240
5.2 odd 4 inner 230.3.k.b.197.8 yes 240
23.16 even 11 inner 230.3.k.b.223.8 yes 240
115.62 odd 44 inner 230.3.k.b.177.8 yes 240
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
230.3.k.b.13.8 ✓ 240 1.1 even 1 trivial
230.3.k.b.177.8 yes 240 115.62 odd 44 inner
230.3.k.b.197.8 yes 240 5.2 odd 4 inner
230.3.k.b.223.8 yes 240 23.16 even 11 inner