Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [185,2,Mod(21,185)] 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("185.21"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(185, base_ring=CyclotomicField(18)) chi = DirichletCharacter(H, H._module([0, 11])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 185 = 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 185.w (of order \(18\), degree \(6\), 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: \(1.47723243739\)
Analytic rank: \(0\)
Dimension: \(72\)
Relative dimension: \(12\) over \(\Q(\zeta_{18})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label 176.5
Character \(\chi\) \(=\) 185.176
Dual form 185.2.w.a.41.5

$q$-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+(-0.540933 + 0.0953810i) q^{2} +(0.128393 - 0.728150i) q^{3} +(-1.59587 + 0.580851i) q^{4} +(-0.642788 + 0.766044i) q^{5} +0.406126i q^{6} +(1.81619 + 1.52397i) q^{7} +(1.75924 - 1.01569i) q^{8} +(2.30536 + 0.839082i) q^{9} +(0.274639 - 0.475688i) q^{10} +(2.36489 + 4.09611i) q^{11} +(0.218048 + 1.23661i) q^{12} +(0.296686 + 0.815139i) q^{13} +(-1.12779 - 0.651133i) q^{14} +(0.475266 + 0.566400i) q^{15} +(1.74719 - 1.46606i) q^{16} +(0.493197 - 1.35505i) q^{17} +(-1.32708 - 0.234000i) q^{18} +(-6.76229 - 1.19237i) q^{19} +(0.580851 - 1.59587i) q^{20} +(1.34286 - 1.12679i) q^{21} +(-1.66994 - 1.99016i) q^{22} +(5.98365 + 3.45466i) q^{23} +(-0.513706 - 1.41140i) q^{24} +(-0.173648 - 0.984808i) q^{25} +(-0.238236 - 0.412637i) q^{26} +(2.01604 - 3.49189i) q^{27} +(-3.78361 - 1.37712i) q^{28} +(-8.47779 + 4.89465i) q^{29} +(-0.311111 - 0.261053i) q^{30} -2.80209i q^{31} +(-3.41678 + 4.07196i) q^{32} +(3.28622 - 1.19609i) q^{33} +(-0.137540 + 0.780031i) q^{34} +(-2.33485 + 0.411697i) q^{35} -4.16645 q^{36} +(5.23691 + 3.09431i) q^{37} +3.77167 q^{38} +(0.631636 - 0.111374i) q^{39} +(-0.352747 + 2.00053i) q^{40} +(-0.948571 + 0.345252i) q^{41} +(-0.618923 + 0.737603i) q^{42} -5.15867i q^{43} +(-6.15330 - 5.16323i) q^{44} +(-2.12463 + 1.22666i) q^{45} +(-3.56626 - 1.29801i) q^{46} +(6.40700 - 11.0972i) q^{47} +(-0.843189 - 1.46045i) q^{48} +(-0.239457 - 1.35803i) q^{49} +(0.187864 + 0.516152i) q^{50} +(-0.923355 - 0.533099i) q^{51} +(-0.946948 - 1.12853i) q^{52} +(-0.536972 + 0.450573i) q^{53} +(-0.757484 + 2.08117i) q^{54} +(-4.65793 - 0.821318i) q^{55} +(4.74299 + 0.836317i) q^{56} +(-1.73645 + 4.77087i) q^{57} +(4.11906 - 3.45630i) q^{58} +(3.21523 + 3.83177i) q^{59} +(-1.08746 - 0.627845i) q^{60} +(-3.88070 - 10.6621i) q^{61} +(0.267267 + 1.51574i) q^{62} +(2.90824 + 5.03722i) q^{63} +(-0.820931 + 1.42189i) q^{64} +(-0.815139 - 0.296686i) q^{65} +(-1.66354 + 0.960445i) q^{66} +(-4.78653 - 4.01637i) q^{67} +2.44896i q^{68} +(3.28377 - 3.91344i) q^{69} +(1.22373 - 0.445401i) q^{70} +(-1.27272 + 7.21798i) q^{71} +(4.90792 - 0.865399i) q^{72} -9.09139 q^{73} +(-3.12796 - 1.17431i) q^{74} -0.739383 q^{75} +(11.4844 - 2.02500i) q^{76} +(-1.94724 + 11.0433i) q^{77} +(-0.331049 + 0.120492i) q^{78} +(7.63743 - 9.10193i) q^{79} +2.28079i q^{80} +(3.35426 + 2.81456i) q^{81} +(0.480183 - 0.277234i) q^{82} +(14.7369 + 5.36381i) q^{83} +(-1.48854 + 2.57822i) q^{84} +(0.721005 + 1.24882i) q^{85} +(0.492039 + 2.79049i) q^{86} +(2.47556 + 6.80154i) q^{87} +(8.32080 + 4.80402i) q^{88} +(0.00507885 + 0.00605274i) q^{89} +(1.03228 - 0.866188i) q^{90} +(-0.703404 + 1.93259i) q^{91} +(-11.5558 - 2.03760i) q^{92} +(-2.04034 - 0.359768i) q^{93} +(-2.40729 + 6.61397i) q^{94} +(5.26012 - 4.41377i) q^{95} +(2.52631 + 3.01074i) q^{96} +(-9.50450 - 5.48742i) q^{97} +(0.259060 + 0.711762i) q^{98} +(2.01495 + 11.4274i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 72 q + 6 q^{2} - 6 q^{4} - 6 q^{7} - 18 q^{8} + 6 q^{10} + 30 q^{12} - 30 q^{13} - 18 q^{14} - 18 q^{16} - 30 q^{18} - 18 q^{19} + 30 q^{22} - 30 q^{24} - 24 q^{27} + 30 q^{28} + 18 q^{29} + 24 q^{30} - 36 q^{32}+ \cdots - 12 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/185\mathbb{Z}\right)^\times\).

\(n\) \(76\) \(112\)
\(\chi(n)\) \(e\left(\frac{17}{18}\right)\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.540933 + 0.0953810i −0.382497 + 0.0674446i −0.361591 − 0.932337i \(-0.617766\pi\)
−0.0209063 + 0.999781i \(0.506655\pi\)
\(3\) 0.128393 − 0.728150i 0.0741275 − 0.420398i −0.925050 − 0.379845i \(-0.875977\pi\)
0.999177 − 0.0405522i \(-0.0129117\pi\)
\(4\) −1.59587 + 0.580851i −0.797937 + 0.290425i
\(5\) −0.642788 + 0.766044i −0.287463 + 0.342585i
\(6\) 0.406126i 0.165800i
\(7\) 1.81619 + 1.52397i 0.686456 + 0.576005i 0.917885 − 0.396847i \(-0.129896\pi\)
−0.231429 + 0.972852i \(0.574340\pi\)
\(8\) 1.75924 − 1.01569i 0.621984 − 0.359102i
\(9\) 2.30536 + 0.839082i 0.768453 + 0.279694i
\(10\) 0.274639 − 0.475688i 0.0868484 − 0.150426i
\(11\) 2.36489 + 4.09611i 0.713042 + 1.23502i 0.963710 + 0.266951i \(0.0860163\pi\)
−0.250668 + 0.968073i \(0.580650\pi\)
\(12\) 0.218048 + 1.23661i 0.0629451 + 0.356979i
\(13\) 0.296686 + 0.815139i 0.0822859 + 0.226079i 0.974012 − 0.226497i \(-0.0727274\pi\)
−0.891726 + 0.452576i \(0.850505\pi\)
\(14\) −1.12779 − 0.651133i −0.301416 − 0.174023i
\(15\) 0.475266 + 0.566400i 0.122713 + 0.146244i
\(16\) 1.74719 − 1.46606i 0.436797 − 0.366516i
\(17\) 0.493197 − 1.35505i 0.119618 − 0.328647i −0.865405 − 0.501074i \(-0.832939\pi\)
0.985022 + 0.172427i \(0.0551608\pi\)
\(18\) −1.32708 − 0.234000i −0.312795 − 0.0551542i
\(19\) −6.76229 − 1.19237i −1.55137 − 0.273549i −0.668700 − 0.743533i \(-0.733149\pi\)
−0.882675 + 0.469984i \(0.844260\pi\)
\(20\) 0.580851 − 1.59587i 0.129882 − 0.356848i
\(21\) 1.34286 − 1.12679i 0.293036 − 0.245887i
\(22\) −1.66994 − 1.99016i −0.356032 − 0.424303i
\(23\) 5.98365 + 3.45466i 1.24768 + 0.720347i 0.970646 − 0.240514i \(-0.0773161\pi\)
0.277031 + 0.960861i \(0.410649\pi\)
\(24\) −0.513706 − 1.41140i −0.104860 − 0.288100i
\(25\) −0.173648 − 0.984808i −0.0347296 − 0.196962i
\(26\) −0.238236 − 0.412637i −0.0467219 − 0.0809248i
\(27\) 2.01604 − 3.49189i 0.387988 − 0.672014i
\(28\) −3.78361 − 1.37712i −0.715035 − 0.260251i
\(29\) −8.47779 + 4.89465i −1.57429 + 0.908914i −0.578651 + 0.815575i \(0.696421\pi\)
−0.995634 + 0.0933391i \(0.970246\pi\)
\(30\) −0.311111 − 0.261053i −0.0568008 − 0.0476616i
\(31\) − 2.80209i − 0.503271i −0.967822 − 0.251635i \(-0.919032\pi\)
0.967822 − 0.251635i \(-0.0809684\pi\)
\(32\) −3.41678 + 4.07196i −0.604007 + 0.719828i
\(33\) 3.28622 − 1.19609i 0.572057 − 0.208212i
\(34\) −0.137540 + 0.780031i −0.0235880 + 0.133774i
\(35\) −2.33485 + 0.411697i −0.394662 + 0.0695895i
\(36\) −4.16645 −0.694408
\(37\) 5.23691 + 3.09431i 0.860943 + 0.508702i
\(38\) 3.77167 0.611846
\(39\) 0.631636 − 0.111374i 0.101143 − 0.0178342i
\(40\) −0.352747 + 2.00053i −0.0557742 + 0.316311i
\(41\) −0.948571 + 0.345252i −0.148142 + 0.0539193i −0.415027 − 0.909809i \(-0.636228\pi\)
0.266885 + 0.963728i \(0.414006\pi\)
\(42\) −0.618923 + 0.737603i −0.0955019 + 0.113815i
\(43\) − 5.15867i − 0.786690i −0.919391 − 0.393345i \(-0.871318\pi\)
0.919391 − 0.393345i \(-0.128682\pi\)
\(44\) −6.15330 − 5.16323i −0.927645 − 0.778387i
\(45\) −2.12463 + 1.22666i −0.316721 + 0.182859i
\(46\) −3.56626 − 1.29801i −0.525816 − 0.191382i
\(47\) 6.40700 − 11.0972i 0.934557 − 1.61870i 0.159135 − 0.987257i \(-0.449130\pi\)
0.775422 − 0.631443i \(-0.217537\pi\)
\(48\) −0.843189 − 1.46045i −0.121704 − 0.210797i
\(49\) −0.239457 − 1.35803i −0.0342081 − 0.194004i
\(50\) 0.187864 + 0.516152i 0.0265680 + 0.0729949i
\(51\) −0.923355 − 0.533099i −0.129296 − 0.0746488i
\(52\) −0.946948 − 1.12853i −0.131318 − 0.156499i
\(53\) −0.536972 + 0.450573i −0.0737588 + 0.0618909i −0.678922 − 0.734211i \(-0.737552\pi\)
0.605163 + 0.796102i \(0.293108\pi\)
\(54\) −0.757484 + 2.08117i −0.103081 + 0.283211i
\(55\) −4.65793 − 0.821318i −0.628075 − 0.110747i
\(56\) 4.74299 + 0.836317i 0.633809 + 0.111758i
\(57\) −1.73645 + 4.77087i −0.229999 + 0.631917i
\(58\) 4.11906 − 3.45630i 0.540859 − 0.453834i
\(59\) 3.21523 + 3.83177i 0.418588 + 0.498853i 0.933594 − 0.358333i \(-0.116655\pi\)
−0.515006 + 0.857187i \(0.672210\pi\)
\(60\) −1.08746 − 0.627845i −0.140390 − 0.0810544i
\(61\) −3.88070 − 10.6621i −0.496872 − 1.36515i −0.894282 − 0.447504i \(-0.852313\pi\)
0.397410 − 0.917641i \(-0.369909\pi\)
\(62\) 0.267267 + 1.51574i 0.0339429 + 0.192500i
\(63\) 2.90824 + 5.03722i 0.366404 + 0.634630i
\(64\) −0.820931 + 1.42189i −0.102616 + 0.177737i
\(65\) −0.815139 − 0.296686i −0.101106 − 0.0367994i
\(66\) −1.66354 + 0.960445i −0.204768 + 0.118223i
\(67\) −4.78653 − 4.01637i −0.584768 − 0.490678i 0.301741 − 0.953390i \(-0.402432\pi\)
−0.886509 + 0.462712i \(0.846877\pi\)
\(68\) 2.44896i 0.296980i
\(69\) 3.28377 − 3.91344i 0.395319 − 0.471123i
\(70\) 1.22373 − 0.445401i 0.146264 − 0.0532356i
\(71\) −1.27272 + 7.21798i −0.151045 + 0.856616i 0.811269 + 0.584673i \(0.198777\pi\)
−0.962313 + 0.271943i \(0.912334\pi\)
\(72\) 4.90792 − 0.865399i 0.578404 − 0.101988i
\(73\) −9.09139 −1.06407 −0.532033 − 0.846723i \(-0.678572\pi\)
−0.532033 + 0.846723i \(0.678572\pi\)
\(74\) −3.12796 − 1.17431i −0.363617 − 0.136511i
\(75\) −0.739383 −0.0853766
\(76\) 11.4844 − 2.02500i 1.31735 − 0.232284i
\(77\) −1.94724 + 11.0433i −0.221908 + 1.25851i
\(78\) −0.331049 + 0.120492i −0.0374840 + 0.0136430i
\(79\) 7.63743 − 9.10193i 0.859278 − 1.02405i −0.140147 − 0.990131i \(-0.544757\pi\)
0.999425 − 0.0339168i \(-0.0107981\pi\)
\(80\) 2.28079i 0.255000i
\(81\) 3.35426 + 2.81456i 0.372696 + 0.312729i
\(82\) 0.480183 − 0.277234i 0.0530273 − 0.0306153i
\(83\) 14.7369 + 5.36381i 1.61759 + 0.588755i 0.982921 − 0.184029i \(-0.0589142\pi\)
0.634669 + 0.772784i \(0.281136\pi\)
\(84\) −1.48854 + 2.57822i −0.162413 + 0.281307i
\(85\) 0.721005 + 1.24882i 0.0782040 + 0.135453i
\(86\) 0.492039 + 2.79049i 0.0530580 + 0.300907i
\(87\) 2.47556 + 6.80154i 0.265408 + 0.729202i
\(88\) 8.32080 + 4.80402i 0.887000 + 0.512110i
\(89\) 0.00507885 + 0.00605274i 0.000538357 + 0.000641589i 0.766314 − 0.642467i \(-0.222089\pi\)
−0.765775 + 0.643108i \(0.777644\pi\)
\(90\) 1.03228 − 0.866188i 0.108812 − 0.0913043i
\(91\) −0.703404 + 1.93259i −0.0737368 + 0.202590i
\(92\) −11.5558 − 2.03760i −1.20477 − 0.212434i
\(93\) −2.04034 − 0.359768i −0.211574 − 0.0373062i
\(94\) −2.40729 + 6.61397i −0.248293 + 0.682179i
\(95\) 5.26012 − 4.41377i 0.539677 − 0.452843i
\(96\) 2.52631 + 3.01074i 0.257840 + 0.307282i
\(97\) −9.50450 − 5.48742i −0.965035 − 0.557163i −0.0673162 − 0.997732i \(-0.521444\pi\)
−0.897719 + 0.440568i \(0.854777\pi\)
\(98\) 0.259060 + 0.711762i 0.0261690 + 0.0718989i
\(99\) 2.01495 + 11.4274i 0.202510 + 1.14849i
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 185.2.w.a.176.5 yes 72
5.2 odd 4 925.2.ba.c.324.6 72
5.3 odd 4 925.2.ba.b.324.7 72
5.4 even 2 925.2.bb.b.176.8 72
37.2 odd 36 6845.2.a.t.1.17 36
37.4 even 18 inner 185.2.w.a.41.5 ✓ 72
37.35 odd 36 6845.2.a.u.1.20 36
185.4 even 18 925.2.bb.b.226.8 72
185.78 odd 36 925.2.ba.c.374.6 72
185.152 odd 36 925.2.ba.b.374.7 72
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
185.2.w.a.41.5 ✓ 72 37.4 even 18 inner
185.2.w.a.176.5 yes 72 1.1 even 1 trivial
925.2.ba.b.324.7 72 5.3 odd 4
925.2.ba.b.374.7 72 185.152 odd 36
925.2.ba.c.324.6 72 5.2 odd 4
925.2.ba.c.374.6 72 185.78 odd 36
925.2.bb.b.176.8 72 5.4 even 2
925.2.bb.b.226.8 72 185.4 even 18
6845.2.a.t.1.17 36 37.2 odd 36
6845.2.a.u.1.20 36 37.35 odd 36