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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 151.1
Character \(\chi\) \(=\) 185.151
Dual form 185.2.w.a.136.1

$q$-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+(-1.50953 + 1.79899i) q^{2} +(0.670695 - 0.562780i) q^{3} +(-0.610385 - 3.46166i) q^{4} +(-0.342020 + 0.939693i) q^{5} +2.05611i q^{6} +(-4.17567 - 1.51982i) q^{7} +(3.08132 + 1.77900i) q^{8} +(-0.387834 + 2.19951i) q^{9} +(-1.17421 - 2.03379i) q^{10} +(-2.58382 + 4.47531i) q^{11} +(-2.35754 - 1.97821i) q^{12} +(-4.66864 + 0.823207i) q^{13} +(9.03746 - 5.21778i) q^{14} +(0.299449 + 0.822730i) q^{15} +(-1.24561 + 0.453366i) q^{16} +(-1.88224 - 0.331889i) q^{17} +(-3.37146 - 4.01795i) q^{18} +(1.95658 + 2.33176i) q^{19} +(3.46166 + 0.610385i) q^{20} +(-3.65593 + 1.33065i) q^{21} +(-4.15068 - 11.4039i) q^{22} +(8.02299 - 4.63208i) q^{23} +(3.06782 - 0.540939i) q^{24} +(-0.766044 - 0.642788i) q^{25} +(5.56653 - 9.64151i) q^{26} +(2.29102 + 3.96817i) q^{27} +(-2.71234 + 15.3824i) q^{28} +(-2.53625 - 1.46431i) q^{29} +(-1.93211 - 0.703232i) q^{30} -5.61699i q^{31} +(-1.36913 + 3.76165i) q^{32} +(0.785658 + 4.45569i) q^{33} +(3.43836 - 2.88513i) q^{34} +(2.85633 - 3.40404i) q^{35} +7.85071 q^{36} +(-4.05888 + 4.53051i) q^{37} -7.14833 q^{38} +(-2.66795 + 3.17954i) q^{39} +(-2.72559 + 2.28704i) q^{40} +(0.132785 + 0.753063i) q^{41} +(3.12492 - 8.58565i) q^{42} +0.369905i q^{43} +(17.0691 + 6.21265i) q^{44} +(-1.93422 - 1.11672i) q^{45} +(-3.77791 + 21.4256i) q^{46} +(5.66373 + 9.80987i) q^{47} +(-0.580281 + 1.00508i) q^{48} +(9.76406 + 8.19302i) q^{49} +(2.31274 - 0.407798i) q^{50} +(-1.44919 + 0.836689i) q^{51} +(5.69933 + 15.6588i) q^{52} +(-4.35631 + 1.58557i) q^{53} +(-10.5971 - 1.86855i) q^{54} +(-3.32169 - 3.95864i) q^{55} +(-10.1628 - 12.1116i) q^{56} +(2.62453 + 0.462776i) q^{57} +(6.46283 - 2.35228i) q^{58} +(-1.52849 - 4.19950i) q^{59} +(2.66523 - 1.53877i) q^{60} +(-2.71555 + 0.478824i) q^{61} +(10.1049 + 8.47904i) q^{62} +(4.96233 - 8.59501i) q^{63} +(-6.02598 - 10.4373i) q^{64} +(0.823207 - 4.66864i) q^{65} +(-9.20173 - 5.31262i) q^{66} +(-4.24119 - 1.54367i) q^{67} +6.71825i q^{68} +(2.77414 - 7.62190i) q^{69} +(1.81212 + 10.2770i) q^{70} +(-3.35412 + 2.81444i) q^{71} +(-5.10798 + 6.08746i) q^{72} -2.68113 q^{73} +(-2.02333 - 14.1408i) q^{74} -0.875531 q^{75} +(6.87750 - 8.19628i) q^{76} +(17.5908 - 14.7605i) q^{77} +(-1.69261 - 9.59925i) q^{78} +(0.687516 - 1.88893i) q^{79} -1.32555i q^{80} +(-2.52648 - 0.919562i) q^{81} +(-1.55520 - 0.897894i) q^{82} +(0.321200 - 1.82162i) q^{83} +(6.83778 + 11.8434i) q^{84} +(0.955636 - 1.65521i) q^{85} +(-0.665457 - 0.558384i) q^{86} +(-2.52514 + 0.445249i) q^{87} +(-15.9232 + 9.19324i) q^{88} +(2.11794 + 5.81901i) q^{89} +(4.92875 - 1.79392i) q^{90} +(20.7458 + 3.65805i) q^{91} +(-20.9318 - 24.9456i) q^{92} +(-3.16113 - 3.76729i) q^{93} +(-26.1975 - 4.61932i) q^{94} +(-2.86032 + 1.04107i) q^{95} +(1.19871 + 3.29344i) q^{96} +(2.46710 - 1.42438i) q^{97} +(-29.4784 + 5.19783i) q^{98} +(-8.84141 - 7.41882i) 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 \(\left(\mathbb{Z}/185\mathbb{Z}\right)^\times\).

\(n\) \(76\) \(112\)
\(\chi(n)\) \(e\left(\frac{13}{18}\right)\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.50953 + 1.79899i −1.06740 + 1.27208i −0.106761 + 0.994285i \(0.534048\pi\)
−0.960640 + 0.277795i \(0.910396\pi\)
\(3\) 0.670695 − 0.562780i 0.387226 − 0.324921i −0.428305 − 0.903634i \(-0.640889\pi\)
0.815531 + 0.578713i \(0.196445\pi\)
\(4\) −0.610385 − 3.46166i −0.305192 − 1.73083i
\(5\) −0.342020 + 0.939693i −0.152956 + 0.420243i
\(6\) 2.05611i 0.839404i
\(7\) −4.17567 − 1.51982i −1.57826 − 0.574438i −0.603431 − 0.797415i \(-0.706200\pi\)
−0.974824 + 0.222977i \(0.928423\pi\)
\(8\) 3.08132 + 1.77900i 1.08941 + 0.628973i
\(9\) −0.387834 + 2.19951i −0.129278 + 0.733172i
\(10\) −1.17421 − 2.03379i −0.371317 − 0.643141i
\(11\) −2.58382 + 4.47531i −0.779051 + 1.34936i 0.153439 + 0.988158i \(0.450965\pi\)
−0.932489 + 0.361197i \(0.882368\pi\)
\(12\) −2.35754 − 1.97821i −0.680563 − 0.571060i
\(13\) −4.66864 + 0.823207i −1.29485 + 0.228317i −0.778273 − 0.627926i \(-0.783904\pi\)
−0.516575 + 0.856242i \(0.672793\pi\)
\(14\) 9.03746 − 5.21778i 2.41536 − 1.39451i
\(15\) 0.299449 + 0.822730i 0.0773174 + 0.212428i
\(16\) −1.24561 + 0.453366i −0.311403 + 0.113341i
\(17\) −1.88224 − 0.331889i −0.456509 − 0.0804949i −0.0593351 − 0.998238i \(-0.518898\pi\)
−0.397174 + 0.917743i \(0.630009\pi\)
\(18\) −3.37146 − 4.01795i −0.794661 − 0.947040i
\(19\) 1.95658 + 2.33176i 0.448870 + 0.534942i 0.942267 − 0.334862i \(-0.108690\pi\)
−0.493398 + 0.869804i \(0.664245\pi\)
\(20\) 3.46166 + 0.610385i 0.774051 + 0.136486i
\(21\) −3.65593 + 1.33065i −0.797789 + 0.290371i
\(22\) −4.15068 − 11.4039i −0.884928 − 2.43132i
\(23\) 8.02299 − 4.63208i 1.67291 − 0.965855i 0.706913 − 0.707300i \(-0.250087\pi\)
0.965997 − 0.258555i \(-0.0832462\pi\)
\(24\) 3.06782 − 0.540939i 0.626216 − 0.110419i
\(25\) −0.766044 − 0.642788i −0.153209 − 0.128558i
\(26\) 5.56653 − 9.64151i 1.09169 − 1.89086i
\(27\) 2.29102 + 3.96817i 0.440907 + 0.763674i
\(28\) −2.71234 + 15.3824i −0.512584 + 2.90701i
\(29\) −2.53625 − 1.46431i −0.470970 − 0.271915i 0.245676 − 0.969352i \(-0.420990\pi\)
−0.716646 + 0.697437i \(0.754324\pi\)
\(30\) −1.93211 − 0.703232i −0.352754 − 0.128392i
\(31\) − 5.61699i − 1.00884i −0.863458 − 0.504421i \(-0.831706\pi\)
0.863458 − 0.504421i \(-0.168294\pi\)
\(32\) −1.36913 + 3.76165i −0.242030 + 0.664971i
\(33\) 0.785658 + 4.45569i 0.136766 + 0.775636i
\(34\) 3.43836 − 2.88513i 0.589675 − 0.494796i
\(35\) 2.85633 − 3.40404i 0.482807 − 0.575387i
\(36\) 7.85071 1.30845
\(37\) −4.05888 + 4.53051i −0.667276 + 0.744811i
\(38\) −7.14833 −1.15961
\(39\) −2.66795 + 3.17954i −0.427214 + 0.509134i
\(40\) −2.72559 + 2.28704i −0.430954 + 0.361613i
\(41\) 0.132785 + 0.753063i 0.0207376 + 0.117609i 0.993419 − 0.114534i \(-0.0365375\pi\)
−0.972682 + 0.232143i \(0.925426\pi\)
\(42\) 3.12492 − 8.58565i 0.482186 − 1.32479i
\(43\) 0.369905i 0.0564100i 0.999602 + 0.0282050i \(0.00897912\pi\)
−0.999602 + 0.0282050i \(0.991021\pi\)
\(44\) 17.0691 + 6.21265i 2.57327 + 0.936593i
\(45\) −1.93422 − 1.11672i −0.288337 − 0.166471i
\(46\) −3.77791 + 21.4256i −0.557022 + 3.15903i
\(47\) 5.66373 + 9.80987i 0.826140 + 1.43092i 0.901045 + 0.433726i \(0.142801\pi\)
−0.0749052 + 0.997191i \(0.523865\pi\)
\(48\) −0.580281 + 1.00508i −0.0837564 + 0.145070i
\(49\) 9.76406 + 8.19302i 1.39487 + 1.17043i
\(50\) 2.31274 − 0.407798i 0.327071 − 0.0576714i
\(51\) −1.44919 + 0.836689i −0.202927 + 0.117160i
\(52\) 5.69933 + 15.6588i 0.790355 + 2.17148i
\(53\) −4.35631 + 1.58557i −0.598385 + 0.217794i −0.623414 − 0.781892i \(-0.714255\pi\)
0.0250282 + 0.999687i \(0.492032\pi\)
\(54\) −10.5971 − 1.86855i −1.44208 − 0.254277i
\(55\) −3.32169 − 3.95864i −0.447897 − 0.533783i
\(56\) −10.1628 − 12.1116i −1.35807 − 1.61848i
\(57\) 2.62453 + 0.462776i 0.347628 + 0.0612962i
\(58\) 6.46283 − 2.35228i 0.848612 − 0.308869i
\(59\) −1.52849 − 4.19950i −0.198993 − 0.546728i 0.799556 − 0.600592i \(-0.205068\pi\)
−0.998548 + 0.0538642i \(0.982846\pi\)
\(60\) 2.66523 − 1.53877i 0.344080 − 0.198655i
\(61\) −2.71555 + 0.478824i −0.347690 + 0.0613071i −0.344766 − 0.938689i \(-0.612042\pi\)
−0.00292379 + 0.999996i \(0.500931\pi\)
\(62\) 10.1049 + 8.47904i 1.28333 + 1.07684i
\(63\) 4.96233 − 8.59501i 0.625195 − 1.08287i
\(64\) −6.02598 − 10.4373i −0.753248 − 1.30466i
\(65\) 0.823207 − 4.66864i 0.102106 − 0.579074i
\(66\) −9.20173 − 5.31262i −1.13265 − 0.653938i
\(67\) −4.24119 − 1.54367i −0.518144 − 0.188589i 0.0696928 − 0.997569i \(-0.477798\pi\)
−0.587837 + 0.808979i \(0.700020\pi\)
\(68\) 6.71825i 0.814707i
\(69\) 2.77414 − 7.62190i 0.333968 − 0.917568i
\(70\) 1.81212 + 10.2770i 0.216589 + 1.22834i
\(71\) −3.35412 + 2.81444i −0.398061 + 0.334013i −0.819744 − 0.572731i \(-0.805884\pi\)
0.421683 + 0.906744i \(0.361440\pi\)
\(72\) −5.10798 + 6.08746i −0.601982 + 0.717414i
\(73\) −2.68113 −0.313803 −0.156902 − 0.987614i \(-0.550151\pi\)
−0.156902 + 0.987614i \(0.550151\pi\)
\(74\) −2.02333 − 14.1408i −0.235207 − 1.64384i
\(75\) −0.875531 −0.101098
\(76\) 6.87750 − 8.19628i 0.788903 − 0.940178i
\(77\) 17.5908 − 14.7605i 2.00466 − 1.68211i
\(78\) −1.69261 − 9.59925i −0.191650 − 1.08690i
\(79\) 0.687516 − 1.88893i 0.0773516 − 0.212522i −0.894989 − 0.446087i \(-0.852817\pi\)
0.972341 + 0.233566i \(0.0750393\pi\)
\(80\) − 1.32555i − 0.148201i
\(81\) −2.52648 − 0.919562i −0.280720 − 0.102174i
\(82\) −1.55520 − 0.897894i −0.171743 − 0.0991558i
\(83\) 0.321200 − 1.82162i 0.0352563 − 0.199949i −0.962092 − 0.272725i \(-0.912075\pi\)
0.997348 + 0.0727768i \(0.0231861\pi\)
\(84\) 6.83778 + 11.8434i 0.746063 + 1.29222i
\(85\) 0.955636 − 1.65521i 0.103653 − 0.179533i
\(86\) −0.665457 − 0.558384i −0.0717580 − 0.0602121i
\(87\) −2.52514 + 0.445249i −0.270723 + 0.0477358i
\(88\) −15.9232 + 9.19324i −1.69741 + 0.980003i
\(89\) 2.11794 + 5.81901i 0.224502 + 0.616813i 0.999892 − 0.0146684i \(-0.00466927\pi\)
−0.775391 + 0.631482i \(0.782447\pi\)
\(90\) 4.92875 − 1.79392i 0.519536 − 0.189096i
\(91\) 20.7458 + 3.65805i 2.17475 + 0.383468i
\(92\) −20.9318 − 24.9456i −2.18229 − 2.60075i
\(93\) −3.16113 − 3.76729i −0.327794 − 0.390650i
\(94\) −26.1975 − 4.61932i −2.70206 − 0.476446i
\(95\) −2.86032 + 1.04107i −0.293463 + 0.106812i
\(96\) 1.19871 + 3.29344i 0.122343 + 0.336135i
\(97\) 2.46710 − 1.42438i 0.250496 − 0.144624i −0.369496 − 0.929233i \(-0.620469\pi\)
0.619991 + 0.784609i \(0.287136\pi\)
\(98\) −29.4784 + 5.19783i −2.97776 + 0.525060i
\(99\) −8.84141 − 7.41882i −0.888595 − 0.745620i
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.151.1 yes 72
5.2 odd 4 925.2.ba.b.299.1 72
5.3 odd 4 925.2.ba.c.299.12 72
5.4 even 2 925.2.bb.b.151.12 72
37.5 odd 36 6845.2.a.t.1.35 36
37.25 even 18 inner 185.2.w.a.136.1 ✓ 72
37.32 odd 36 6845.2.a.u.1.2 36
185.62 odd 36 925.2.ba.c.99.12 72
185.99 even 18 925.2.bb.b.876.12 72
185.173 odd 36 925.2.ba.b.99.1 72
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
185.2.w.a.136.1 ✓ 72 37.25 even 18 inner
185.2.w.a.151.1 yes 72 1.1 even 1 trivial
925.2.ba.b.99.1 72 185.173 odd 36
925.2.ba.b.299.1 72 5.2 odd 4
925.2.ba.c.99.12 72 185.62 odd 36
925.2.ba.c.299.12 72 5.3 odd 4
925.2.bb.b.151.12 72 5.4 even 2
925.2.bb.b.876.12 72 185.99 even 18
6845.2.a.t.1.35 36 37.5 odd 36
6845.2.a.u.1.2 36 37.32 odd 36