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 151.6
Character \(\chi\) \(=\) 185.151
Dual form 185.2.w.a.136.6

$q$-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+(-0.301512 + 0.359328i) q^{2} +(-2.39156 + 2.00676i) q^{3} +(0.309089 + 1.75293i) q^{4} +(-0.342020 + 0.939693i) q^{5} -1.46442i q^{6} +(1.06448 + 0.387441i) q^{7} +(-1.53553 - 0.886536i) q^{8} +(1.17154 - 6.64413i) q^{9} +(-0.234535 - 0.406226i) q^{10} +(0.690789 - 1.19648i) q^{11} +(-4.25691 - 3.57197i) q^{12} +(-6.56428 + 1.15746i) q^{13} +(-0.460174 + 0.265681i) q^{14} +(-1.06777 - 2.93368i) q^{15} +(-2.56372 + 0.933117i) q^{16} +(3.42415 + 0.603770i) q^{17} +(2.03419 + 2.42425i) q^{18} +(3.70029 + 4.40983i) q^{19} +(-1.75293 - 0.309089i) q^{20} +(-3.32328 + 1.20957i) q^{21} +(0.221648 + 0.608974i) q^{22} +(2.26680 - 1.30874i) q^{23} +(5.45136 - 0.961222i) q^{24} +(-0.766044 - 0.642788i) q^{25} +(1.56330 - 2.70772i) q^{26} +(5.84841 + 10.1297i) q^{27} +(-0.350136 + 1.98572i) q^{28} +(-6.17967 - 3.56783i) q^{29} +(1.37610 + 0.500860i) q^{30} +2.69515i q^{31} +(1.65055 - 4.53485i) q^{32} +(0.748985 + 4.24770i) q^{33} +(-1.24938 + 1.04835i) q^{34} +(-0.728150 + 0.867776i) q^{35} +12.0088 q^{36} +(-5.05847 + 3.37814i) q^{37} -2.70026 q^{38} +(13.3761 - 15.9410i) q^{39} +(1.35825 - 1.13971i) q^{40} +(1.38180 + 7.83657i) q^{41} +(0.567375 - 1.55885i) q^{42} +10.8290i q^{43} +(2.31087 + 0.841086i) q^{44} +(5.84275 + 3.37331i) q^{45} +(-0.213202 + 1.20913i) q^{46} +(0.933248 + 1.61643i) q^{47} +(4.25875 - 7.37636i) q^{48} +(-4.37929 - 3.67466i) q^{49} +(0.461944 - 0.0814531i) q^{50} +(-9.40068 + 5.42748i) q^{51} +(-4.05790 - 11.1490i) q^{52} +(1.92330 - 0.700025i) q^{53} +(-5.40327 - 0.952742i) q^{54} +(0.888061 + 1.05835i) q^{55} +(-1.29106 - 1.53863i) q^{56} +(-17.6989 - 3.12080i) q^{57} +(3.14527 - 1.14479i) q^{58} +(2.08921 + 5.74005i) q^{59} +(4.81251 - 2.77850i) q^{60} +(6.18979 - 1.09143i) q^{61} +(-0.968445 - 0.812622i) q^{62} +(3.82129 - 6.61867i) q^{63} +(-1.59641 - 2.76507i) q^{64} +(1.15746 - 6.56428i) q^{65} +(-1.75215 - 1.01160i) q^{66} +(5.97892 + 2.17615i) q^{67} +6.18892i q^{68} +(-2.79487 + 7.67884i) q^{69} +(-0.0922702 - 0.523290i) q^{70} +(-3.10854 + 2.60837i) q^{71} +(-7.68919 + 9.16362i) q^{72} -0.109760 q^{73} +(0.311329 - 2.83620i) q^{74} +3.12196 q^{75} +(-6.58642 + 7.84939i) q^{76} +(1.19890 - 1.00600i) q^{77} +(1.69500 + 9.61285i) q^{78} +(-0.920146 + 2.52808i) q^{79} -2.72825i q^{80} +(-15.2955 - 5.56709i) q^{81} +(-3.23253 - 1.86630i) q^{82} +(2.40459 - 13.6371i) q^{83} +(-3.14749 - 5.45161i) q^{84} +(-1.73849 + 3.01115i) q^{85} +(-3.89117 - 3.26508i) q^{86} +(21.9388 - 3.86841i) q^{87} +(-2.12145 + 1.22482i) q^{88} +(4.36882 + 12.0032i) q^{89} +(-2.97379 + 1.08237i) q^{90} +(-7.43602 - 1.31117i) q^{91} +(2.99477 + 3.56903i) q^{92} +(-5.40851 - 6.44562i) q^{93} +(-0.862216 - 0.152032i) q^{94} +(-5.40946 + 1.96888i) q^{95} +(5.15295 + 14.1576i) q^{96} +(3.43944 - 1.98576i) q^{97} +(2.64082 - 0.465648i) q^{98} +(-7.14029 - 5.99142i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.301512 + 0.359328i −0.213201 + 0.254084i −0.862037 − 0.506845i \(-0.830812\pi\)
0.648836 + 0.760928i \(0.275256\pi\)
\(3\) −2.39156 + 2.00676i −1.38077 + 1.15860i −0.411840 + 0.911256i \(0.635114\pi\)
−0.968927 + 0.247345i \(0.920442\pi\)
\(4\) 0.309089 + 1.75293i 0.154545 + 0.876466i
\(5\) −0.342020 + 0.939693i −0.152956 + 0.420243i
\(6\) − 1.46442i − 0.597846i
\(7\) 1.06448 + 0.387441i 0.402337 + 0.146439i 0.535259 − 0.844688i \(-0.320214\pi\)
−0.132922 + 0.991126i \(0.542436\pi\)
\(8\) −1.53553 − 0.886536i −0.542890 − 0.313438i
\(9\) 1.17154 − 6.64413i 0.390513 − 2.21471i
\(10\) −0.234535 − 0.406226i −0.0741665 − 0.128460i
\(11\) 0.690789 − 1.19648i 0.208281 − 0.360753i −0.742892 − 0.669411i \(-0.766547\pi\)
0.951173 + 0.308658i \(0.0998799\pi\)
\(12\) −4.25691 − 3.57197i −1.22886 − 1.03114i
\(13\) −6.56428 + 1.15746i −1.82060 + 0.321022i −0.976559 − 0.215250i \(-0.930943\pi\)
−0.844045 + 0.536272i \(0.819832\pi\)
\(14\) −0.460174 + 0.265681i −0.122987 + 0.0710063i
\(15\) −1.06777 − 2.93368i −0.275698 − 0.757473i
\(16\) −2.56372 + 0.933117i −0.640930 + 0.233279i
\(17\) 3.42415 + 0.603770i 0.830478 + 0.146436i 0.572697 − 0.819767i \(-0.305897\pi\)
0.257782 + 0.966203i \(0.417008\pi\)
\(18\) 2.03419 + 2.42425i 0.479463 + 0.571402i
\(19\) 3.70029 + 4.40983i 0.848905 + 1.01169i 0.999733 + 0.0231252i \(0.00736162\pi\)
−0.150828 + 0.988560i \(0.548194\pi\)
\(20\) −1.75293 − 0.309089i −0.391967 − 0.0691144i
\(21\) −3.32328 + 1.20957i −0.725198 + 0.263951i
\(22\) 0.221648 + 0.608974i 0.0472556 + 0.129834i
\(23\) 2.26680 − 1.30874i 0.472661 − 0.272891i −0.244692 − 0.969601i \(-0.578687\pi\)
0.717353 + 0.696710i \(0.245354\pi\)
\(24\) 5.45136 − 0.961222i 1.11275 − 0.196209i
\(25\) −0.766044 − 0.642788i −0.153209 − 0.128558i
\(26\) 1.56330 − 2.70772i 0.306589 − 0.531028i
\(27\) 5.84841 + 10.1297i 1.12553 + 1.94947i
\(28\) −0.350136 + 1.98572i −0.0661696 + 0.375266i
\(29\) −6.17967 − 3.56783i −1.14754 − 0.662530i −0.199250 − 0.979949i \(-0.563851\pi\)
−0.948286 + 0.317418i \(0.897184\pi\)
\(30\) 1.37610 + 0.500860i 0.251241 + 0.0914441i
\(31\) 2.69515i 0.484064i 0.970268 + 0.242032i \(0.0778139\pi\)
−0.970268 + 0.242032i \(0.922186\pi\)
\(32\) 1.65055 − 4.53485i 0.291779 − 0.801655i
\(33\) 0.748985 + 4.24770i 0.130381 + 0.739430i
\(34\) −1.24938 + 1.04835i −0.214266 + 0.179791i
\(35\) −0.728150 + 0.867776i −0.123080 + 0.146681i
\(36\) 12.0088 2.00147
\(37\) −5.05847 + 3.37814i −0.831608 + 0.555363i
\(38\) −2.70026 −0.438040
\(39\) 13.3761 − 15.9410i 2.14189 − 2.55261i
\(40\) 1.35825 − 1.13971i 0.214758 − 0.180204i
\(41\) 1.38180 + 7.83657i 0.215801 + 1.22387i 0.879512 + 0.475877i \(0.157869\pi\)
−0.663711 + 0.747989i \(0.731020\pi\)
\(42\) 0.567375 − 1.55885i 0.0875478 − 0.240536i
\(43\) 10.8290i 1.65141i 0.564105 + 0.825703i \(0.309221\pi\)
−0.564105 + 0.825703i \(0.690779\pi\)
\(44\) 2.31087 + 0.841086i 0.348376 + 0.126799i
\(45\) 5.84275 + 3.37331i 0.870986 + 0.502864i
\(46\) −0.213202 + 1.20913i −0.0314349 + 0.178276i
\(47\) 0.933248 + 1.61643i 0.136128 + 0.235781i 0.926028 − 0.377455i \(-0.123201\pi\)
−0.789900 + 0.613236i \(0.789867\pi\)
\(48\) 4.25875 − 7.37636i 0.614697 − 1.06469i
\(49\) −4.37929 − 3.67466i −0.625613 − 0.524952i
\(50\) 0.461944 − 0.0814531i 0.0653287 − 0.0115192i
\(51\) −9.40068 + 5.42748i −1.31636 + 0.760000i
\(52\) −4.05790 − 11.1490i −0.562729 − 1.54609i
\(53\) 1.92330 − 0.700025i 0.264186 − 0.0961558i −0.206531 − 0.978440i \(-0.566217\pi\)
0.470717 + 0.882284i \(0.343995\pi\)
\(54\) −5.40327 − 0.952742i −0.735292 − 0.129652i
\(55\) 0.888061 + 1.05835i 0.119746 + 0.142708i
\(56\) −1.29106 − 1.53863i −0.172525 − 0.205608i
\(57\) −17.6989 − 3.12080i −2.34428 − 0.413360i
\(58\) 3.14527 − 1.14479i 0.412994 − 0.150318i
\(59\) 2.08921 + 5.74005i 0.271992 + 0.747291i 0.998209 + 0.0598256i \(0.0190545\pi\)
−0.726217 + 0.687465i \(0.758723\pi\)
\(60\) 4.81251 − 2.77850i 0.621292 − 0.358703i
\(61\) 6.18979 − 1.09143i 0.792521 − 0.139743i 0.237290 − 0.971439i \(-0.423741\pi\)
0.555231 + 0.831696i \(0.312630\pi\)
\(62\) −0.968445 − 0.812622i −0.122993 − 0.103203i
\(63\) 3.82129 − 6.61867i 0.481437 − 0.833874i
\(64\) −1.59641 − 2.76507i −0.199552 − 0.345634i
\(65\) 1.15746 − 6.56428i 0.143565 − 0.814199i
\(66\) −1.75215 − 1.01160i −0.215675 − 0.124520i
\(67\) 5.97892 + 2.17615i 0.730442 + 0.265859i 0.680352 − 0.732886i \(-0.261827\pi\)
0.0500900 + 0.998745i \(0.484049\pi\)
\(68\) 6.18892i 0.750517i
\(69\) −2.79487 + 7.67884i −0.336463 + 0.924424i
\(70\) −0.0922702 − 0.523290i −0.0110284 − 0.0625451i
\(71\) −3.10854 + 2.60837i −0.368915 + 0.309557i −0.808332 − 0.588727i \(-0.799629\pi\)
0.439417 + 0.898283i \(0.355185\pi\)
\(72\) −7.68919 + 9.16362i −0.906179 + 1.07994i
\(73\) −0.109760 −0.0128464 −0.00642322 − 0.999979i \(-0.502045\pi\)
−0.00642322 + 0.999979i \(0.502045\pi\)
\(74\) 0.311329 − 2.83620i 0.0361913 − 0.329702i
\(75\) 3.12196 0.360493
\(76\) −6.58642 + 7.84939i −0.755514 + 0.900386i
\(77\) 1.19890 − 1.00600i 0.136627 − 0.114644i
\(78\) 1.69500 + 9.61285i 0.191921 + 1.08844i
\(79\) −0.920146 + 2.52808i −0.103525 + 0.284431i −0.980631 − 0.195865i \(-0.937249\pi\)
0.877106 + 0.480296i \(0.159471\pi\)
\(80\) − 2.72825i − 0.305028i
\(81\) −15.2955 − 5.56709i −1.69950 − 0.618566i
\(82\) −3.23253 − 1.86630i −0.356973 − 0.206099i
\(83\) 2.40459 − 13.6371i 0.263939 − 1.49687i −0.508102 − 0.861297i \(-0.669653\pi\)
0.772041 − 0.635573i \(-0.219236\pi\)
\(84\) −3.14749 − 5.45161i −0.343419 − 0.594819i
\(85\) −1.73849 + 3.01115i −0.188565 + 0.326605i
\(86\) −3.89117 − 3.26508i −0.419595 − 0.352082i
\(87\) 21.9388 − 3.86841i 2.35209 − 0.414737i
\(88\) −2.12145 + 1.22482i −0.226147 + 0.130566i
\(89\) 4.36882 + 12.0032i 0.463093 + 1.27234i 0.923147 + 0.384448i \(0.125608\pi\)
−0.460053 + 0.887891i \(0.652170\pi\)
\(90\) −2.97379 + 1.08237i −0.313465 + 0.114092i
\(91\) −7.43602 − 1.31117i −0.779507 − 0.137448i
\(92\) 2.99477 + 3.56903i 0.312227 + 0.372097i
\(93\) −5.40851 − 6.44562i −0.560837 − 0.668379i
\(94\) −0.862216 − 0.152032i −0.0889308 − 0.0156809i
\(95\) −5.40946 + 1.96888i −0.554999 + 0.202003i
\(96\) 5.15295 + 14.1576i 0.525921 + 1.44495i
\(97\) 3.43944 − 1.98576i 0.349222 − 0.201623i −0.315121 − 0.949052i \(-0.602045\pi\)
0.664343 + 0.747428i \(0.268712\pi\)
\(98\) 2.64082 − 0.465648i 0.266763 − 0.0470376i
\(99\) −7.14029 − 5.99142i −0.717627 − 0.602160i
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.151.6 yes 72
5.2 odd 4 925.2.ba.b.299.5 72
5.3 odd 4 925.2.ba.c.299.8 72
5.4 even 2 925.2.bb.b.151.7 72
37.5 odd 36 6845.2.a.t.1.23 36
37.25 even 18 inner 185.2.w.a.136.6 ✓ 72
37.32 odd 36 6845.2.a.u.1.14 36
185.62 odd 36 925.2.ba.c.99.8 72
185.99 even 18 925.2.bb.b.876.7 72
185.173 odd 36 925.2.ba.b.99.5 72
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
185.2.w.a.136.6 ✓ 72 37.25 even 18 inner
185.2.w.a.151.6 yes 72 1.1 even 1 trivial
925.2.ba.b.99.5 72 185.173 odd 36
925.2.ba.b.299.5 72 5.2 odd 4
925.2.ba.c.99.8 72 185.62 odd 36
925.2.ba.c.299.8 72 5.3 odd 4
925.2.bb.b.151.7 72 5.4 even 2
925.2.bb.b.876.7 72 185.99 even 18
6845.2.a.t.1.23 36 37.5 odd 36
6845.2.a.u.1.14 36 37.32 odd 36