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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 13.7
Character \(\chi\) \(=\) 230.13
Dual form 230.3.k.b.177.7

$q$-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} +(0.161368 + 0.741796i) q^{3} +(1.97964 + 0.284630i) q^{4} +(4.91560 - 0.914803i) q^{5} +(0.152788 + 1.06267i) q^{6} +(2.14811 - 1.17296i) q^{7} +(2.76379 + 0.601225i) q^{8} +(7.66247 - 3.49933i) q^{9} +(7.02629 - 0.794501i) q^{10} +(-7.11457 + 8.21065i) q^{11} +(0.108314 + 1.51442i) q^{12} +(-18.1470 - 9.90900i) q^{13} +(3.14848 - 1.43786i) q^{14} +(1.47182 + 3.49875i) q^{15} +(3.83797 + 1.12693i) q^{16} +(5.70029 + 4.26718i) q^{17} +(11.1618 - 4.16314i) q^{18} +(22.8352 + 3.28320i) q^{19} +(9.99152 - 0.411856i) q^{20} +(1.21673 + 1.40418i) q^{21} +(-10.8642 + 10.8642i) q^{22} +(21.0019 + 9.37664i) q^{23} +2.14719i q^{24} +(23.3263 - 8.99361i) q^{25} +(-24.5986 - 15.8086i) q^{26} +(7.92671 + 10.5888i) q^{27} +(4.58635 - 1.71062i) q^{28} +(-44.1820 + 6.35242i) q^{29} +(1.72317 + 5.08387i) q^{30} +(-10.2239 + 6.57049i) q^{31} +(5.30019 + 1.97687i) q^{32} +(-7.23868 - 3.95262i) q^{33} +(7.61038 + 6.59443i) q^{34} +(9.48623 - 7.73089i) q^{35} +(16.1650 - 4.74646i) q^{36} +(-42.4332 - 15.8268i) q^{37} +(31.8803 + 6.93513i) q^{38} +(4.42212 - 15.0603i) q^{39} +(14.1357 + 0.427064i) q^{40} +(-5.99857 + 13.1350i) q^{41} +(1.57467 + 2.10351i) q^{42} +(-6.53521 - 30.0419i) q^{43} +(-16.4213 + 14.2291i) q^{44} +(34.4644 - 24.2110i) q^{45} +(28.6795 + 15.3456i) q^{46} +(-26.8848 + 26.8848i) q^{47} +(-0.216627 + 3.02884i) q^{48} +(-23.2529 + 36.1821i) q^{49} +(33.8116 - 10.3331i) q^{50} +(-2.24554 + 4.91704i) q^{51} +(-33.1041 - 24.7814i) q^{52} +(-33.9911 - 62.2501i) q^{53} +(10.1132 + 15.7365i) q^{54} +(-27.4612 + 46.8687i) q^{55} +(6.64213 - 1.95031i) q^{56} +(1.24940 + 17.4688i) q^{57} +(-62.9645 + 4.50331i) q^{58} +(-30.4408 - 103.672i) q^{59} +(1.91782 + 7.34520i) q^{60} +(-40.9743 + 26.3326i) q^{61} +(-15.0848 + 8.23693i) q^{62} +(12.3553 - 16.5047i) q^{63} +(7.27706 + 3.32332i) q^{64} +(-98.2681 - 32.1078i) q^{65} +(-9.81219 - 6.30591i) q^{66} +(-17.6477 - 1.26219i) q^{67} +(10.0700 + 10.0700i) q^{68} +(-3.56653 + 17.0922i) q^{69} +(14.1613 - 9.94821i) q^{70} +(41.1867 + 47.5320i) q^{71} +(23.2813 - 5.06454i) q^{72} +(53.0677 - 39.7260i) q^{73} +(-58.2600 - 26.6065i) q^{74} +(10.4355 + 15.8521i) q^{75} +(44.2710 + 12.9991i) q^{76} +(-5.65214 + 25.9825i) q^{77} +(7.75730 - 20.7981i) q^{78} +(-4.29273 - 14.6197i) q^{79} +(19.8969 + 2.02855i) q^{80} +(43.0715 - 49.7072i) q^{81} +(-9.78683 + 17.9232i) q^{82} +(-21.5644 + 57.8163i) q^{83} +(2.00902 + 3.12610i) q^{84} +(31.9240 + 15.7611i) q^{85} +(-6.18774 - 43.0367i) q^{86} +(-11.8418 - 31.7490i) q^{87} +(-24.5996 + 18.4150i) q^{88} +(-57.3134 + 89.1814i) q^{89} +(51.0585 - 30.6752i) q^{90} -50.6045 q^{91} +(38.9073 + 24.5402i) q^{92} +(-6.52377 - 6.52377i) q^{93} +(-40.6364 + 35.2117i) q^{94} +(115.252 - 4.75076i) q^{95} +(-0.611153 + 4.25066i) q^{96} +(-21.3737 - 57.3051i) q^{97} +(-36.4511 + 48.6929i) q^{98} +(-25.7833 + 87.8100i) 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
\(3\) 0.161368 + 0.741796i 0.0537893 + 0.247265i 0.996189 − 0.0872244i \(-0.0277997\pi\)
−0.942399 + 0.334490i \(0.891436\pi\)
\(4\) 1.97964 + 0.284630i 0.494911 + 0.0711574i
\(5\) 4.91560 − 0.914803i 0.983120 − 0.182961i
\(6\) 0.152788 + 1.06267i 0.0254647 + 0.177111i
\(7\) 2.14811 − 1.17296i 0.306873 − 0.167565i −0.318432 − 0.947946i \(-0.603156\pi\)
0.625305 + 0.780380i \(0.284975\pi\)
\(8\) 2.76379 + 0.601225i 0.345474 + 0.0751532i
\(9\) 7.66247 − 3.49933i 0.851385 − 0.388815i
\(10\) 7.02629 − 0.794501i 0.702629 − 0.0794501i
\(11\) −7.11457 + 8.21065i −0.646779 + 0.746422i −0.980558 − 0.196229i \(-0.937130\pi\)
0.333779 + 0.942651i \(0.391676\pi\)
\(12\) 0.108314 + 1.51442i 0.00902613 + 0.126202i
\(13\) −18.1470 − 9.90900i −1.39592 − 0.762231i −0.407391 − 0.913254i \(-0.633561\pi\)
−0.988530 + 0.151023i \(0.951743\pi\)
\(14\) 3.14848 − 1.43786i 0.224892 − 0.102705i
\(15\) 1.47182 + 3.49875i 0.0981211 + 0.233250i
\(16\) 3.83797 + 1.12693i 0.239873 + 0.0704331i
\(17\) 5.70029 + 4.26718i 0.335311 + 0.251011i 0.753644 − 0.657283i \(-0.228294\pi\)
−0.418333 + 0.908294i \(0.637385\pi\)
\(18\) 11.1618 − 4.16314i 0.620100 − 0.231285i
\(19\) 22.8352 + 3.28320i 1.20185 + 0.172800i 0.714012 − 0.700134i \(-0.246876\pi\)
0.487839 + 0.872934i \(0.337785\pi\)
\(20\) 9.99152 − 0.411856i 0.499576 − 0.0205928i
\(21\) 1.21673 + 1.40418i 0.0579395 + 0.0668658i
\(22\) −10.8642 + 10.8642i −0.493829 + 0.493829i
\(23\) 21.0019 + 9.37664i 0.913125 + 0.407680i
\(24\) 2.14719i 0.0894661i
\(25\) 23.3263 − 8.99361i 0.933051 − 0.359744i
\(26\) −24.5986 − 15.8086i −0.946100 − 0.608022i
\(27\) 7.92671 + 10.5888i 0.293582 + 0.392179i
\(28\) 4.58635 − 1.71062i 0.163798 − 0.0610936i
\(29\) −44.1820 + 6.35242i −1.52352 + 0.219049i −0.852632 − 0.522512i \(-0.824995\pi\)
−0.670886 + 0.741561i \(0.734086\pi\)
\(30\) 1.72317 + 5.08387i 0.0574391 + 0.169462i
\(31\) −10.2239 + 6.57049i −0.329803 + 0.211951i −0.695054 − 0.718958i \(-0.744619\pi\)
0.365251 + 0.930909i \(0.380983\pi\)
\(32\) 5.30019 + 1.97687i 0.165631 + 0.0617771i
\(33\) −7.23868 − 3.95262i −0.219354 − 0.119776i
\(34\) 7.61038 + 6.59443i 0.223835 + 0.193954i
\(35\) 9.48623 − 7.73089i 0.271035 − 0.220882i
\(36\) 16.1650 − 4.74646i 0.449027 − 0.131846i
\(37\) −42.4332 − 15.8268i −1.14684 − 0.427751i −0.297048 − 0.954862i \(-0.596002\pi\)
−0.849796 + 0.527111i \(0.823275\pi\)
\(38\) 31.8803 + 6.93513i 0.838955 + 0.182503i
\(39\) 4.42212 − 15.0603i 0.113388 − 0.386163i
\(40\) 14.1357 + 0.427064i 0.353392 + 0.0106766i
\(41\) −5.99857 + 13.1350i −0.146307 + 0.320367i −0.968570 − 0.248740i \(-0.919984\pi\)
0.822264 + 0.569107i \(0.192711\pi\)
\(42\) 1.57467 + 2.10351i 0.0374921 + 0.0500835i
\(43\) −6.53521 − 30.0419i −0.151982 − 0.698648i −0.988563 − 0.150806i \(-0.951813\pi\)
0.836582 − 0.547842i \(-0.184550\pi\)
\(44\) −16.4213 + 14.2291i −0.373211 + 0.323389i
\(45\) 34.4644 − 24.2110i 0.765876 − 0.538021i
\(46\) 28.6795 + 15.3456i 0.623466 + 0.333601i
\(47\) −26.8848 + 26.8848i −0.572018 + 0.572018i −0.932692 − 0.360674i \(-0.882547\pi\)
0.360674 + 0.932692i \(0.382547\pi\)
\(48\) −0.216627 + 3.02884i −0.00451306 + 0.0631009i
\(49\) −23.2529 + 36.1821i −0.474548 + 0.738411i
\(50\) 33.8116 − 10.3331i 0.676233 − 0.206662i
\(51\) −2.24554 + 4.91704i −0.0440301 + 0.0964125i
\(52\) −33.1041 − 24.7814i −0.636618 − 0.476566i
\(53\) −33.9911 − 62.2501i −0.641342 − 1.17453i −0.972929 − 0.231105i \(-0.925766\pi\)
0.331587 − 0.943425i \(-0.392416\pi\)
\(54\) 10.1132 + 15.7365i 0.187282 + 0.291416i
\(55\) −27.4612 + 46.8687i −0.499295 + 0.852158i
\(56\) 6.64213 − 1.95031i 0.118610 − 0.0348269i
\(57\) 1.24940 + 17.4688i 0.0219192 + 0.306471i
\(58\) −62.9645 + 4.50331i −1.08559 + 0.0776433i
\(59\) −30.4408 − 103.672i −0.515946 − 1.75715i −0.643630 − 0.765336i \(-0.722573\pi\)
0.127684 − 0.991815i \(-0.459246\pi\)
\(60\) 1.91782 + 7.34520i 0.0319637 + 0.122420i
\(61\) −40.9743 + 26.3326i −0.671710 + 0.431682i −0.831541 − 0.555463i \(-0.812541\pi\)
0.159832 + 0.987144i \(0.448905\pi\)
\(62\) −15.0848 + 8.23693i −0.243303 + 0.132854i
\(63\) 12.3553 − 16.5047i 0.196115 − 0.261979i
\(64\) 7.27706 + 3.32332i 0.113704 + 0.0519269i
\(65\) −98.2681 − 32.1078i −1.51182 − 0.493966i
\(66\) −9.81219 − 6.30591i −0.148670 − 0.0955441i
\(67\) −17.6477 − 1.26219i −0.263398 − 0.0188386i −0.0609843 − 0.998139i \(-0.519424\pi\)
−0.202414 + 0.979300i \(0.564879\pi\)
\(68\) 10.0700 + 10.0700i 0.148088 + 0.148088i
\(69\) −3.56653 + 17.0922i −0.0516888 + 0.247713i
\(70\) 14.1613 − 9.94821i 0.202305 − 0.142117i
\(71\) 41.1867 + 47.5320i 0.580095 + 0.669465i 0.967625 − 0.252392i \(-0.0812173\pi\)
−0.387530 + 0.921857i \(0.626672\pi\)
\(72\) 23.2813 − 5.06454i 0.323352 − 0.0703409i
\(73\) 53.0677 − 39.7260i 0.726955 − 0.544192i −0.170468 − 0.985363i \(-0.554528\pi\)
0.897423 + 0.441172i \(0.145437\pi\)
\(74\) −58.2600 − 26.6065i −0.787298 − 0.359547i
\(75\) 10.4355 + 15.8521i 0.139140 + 0.211361i
\(76\) 44.2710 + 12.9991i 0.582513 + 0.171041i
\(77\) −5.65214 + 25.9825i −0.0734044 + 0.337434i
\(78\) 7.75730 − 20.7981i 0.0994526 − 0.266643i
\(79\) −4.29273 − 14.6197i −0.0543384 − 0.185060i 0.927852 − 0.372949i \(-0.121653\pi\)
−0.982190 + 0.187890i \(0.939835\pi\)
\(80\) 19.8969 + 2.02855i 0.248711 + 0.0253569i
\(81\) 43.0715 − 49.7072i 0.531747 − 0.613669i
\(82\) −9.78683 + 17.9232i −0.119352 + 0.218576i
\(83\) −21.5644 + 57.8163i −0.259812 + 0.696582i 0.739904 + 0.672713i \(0.234871\pi\)
−0.999715 + 0.0238688i \(0.992402\pi\)
\(84\) 2.00902 + 3.12610i 0.0239169 + 0.0372154i
\(85\) 31.9240 + 15.7611i 0.375576 + 0.185425i
\(86\) −6.18774 − 43.0367i −0.0719505 − 0.500427i
\(87\) −11.8418 − 31.7490i −0.136112 − 0.364931i
\(88\) −24.5996 + 18.4150i −0.279541 + 0.209262i
\(89\) −57.3134 + 89.1814i −0.643971 + 1.00204i 0.353801 + 0.935321i \(0.384889\pi\)
−0.997772 + 0.0667173i \(0.978747\pi\)
\(90\) 51.0585 − 30.6752i 0.567317 − 0.340835i
\(91\) −50.6045 −0.556094
\(92\) 38.9073 + 24.5402i 0.422906 + 0.266741i
\(93\) −6.52377 − 6.52377i −0.0701480 − 0.0701480i
\(94\) −40.6364 + 35.2117i −0.432302 + 0.374592i
\(95\) 115.252 − 4.75076i 1.21318 − 0.0500080i
\(96\) −0.611153 + 4.25066i −0.00636617 + 0.0442777i
\(97\) −21.3737 − 57.3051i −0.220347 − 0.590774i 0.778995 − 0.627030i \(-0.215730\pi\)
−0.999343 + 0.0362557i \(0.988457\pi\)
\(98\) −36.4511 + 48.6929i −0.371950 + 0.496867i
\(99\) −25.7833 + 87.8100i −0.260438 + 0.886970i
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.13.7 ✓ 240
5.2 odd 4 inner 230.3.k.b.197.7 yes 240
23.16 even 11 inner 230.3.k.b.223.7 yes 240
115.62 odd 44 inner 230.3.k.b.177.7 yes 240
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
230.3.k.b.13.7 ✓ 240 1.1 even 1 trivial
230.3.k.b.177.7 yes 240 115.62 odd 44 inner
230.3.k.b.197.7 yes 240 5.2 odd 4 inner
230.3.k.b.223.7 yes 240 23.16 even 11 inner