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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [925,2,Mod(99,925)] 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("925.99"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(925, base_ring=CyclotomicField(18)) chi = DirichletCharacter(H, H._module([9, 5])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 925 = 5^{2} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 925.ba (of order \(18\), degree \(6\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [72,0,0,6,0,0,6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(7.38616218697\)
Analytic rank: \(0\)
Dimension: \(72\)
Relative dimension: \(12\) over \(\Q(\zeta_{18})\)
Twist minimal: no (minimal twist has level 185)
Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label 299.6
Character \(\chi\) \(=\) 925.299
Dual form 925.2.ba.c.99.6

$q$-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+(-0.637595 - 0.535006i) q^{2} +(-0.702462 - 0.837162i) q^{3} +(-0.227000 - 1.28738i) q^{4} +0.909592i q^{6} +(-0.257425 + 0.707269i) q^{7} +(-1.37634 + 2.38390i) q^{8} +(0.313558 - 1.77827i) q^{9} +(-1.05763 + 1.83188i) q^{11} +(-0.918288 + 1.09437i) q^{12} +(0.905788 + 5.13698i) q^{13} +(0.542525 - 0.313227i) q^{14} +(-0.303862 + 0.110597i) q^{16} +(-0.0962871 + 0.546071i) q^{17} +(-1.15131 + 0.966063i) q^{18} +(2.17681 + 2.59422i) q^{19} +(0.772930 - 0.281323i) q^{21} +(1.65441 - 0.602155i) q^{22} +(3.01833 + 5.22791i) q^{23} +(2.96254 - 0.522375i) q^{24} +(2.17079 - 3.75991i) q^{26} +(-4.54824 + 2.62593i) q^{27} +(0.968960 + 0.170854i) q^{28} +(-0.774719 - 0.447284i) q^{29} -5.25941i q^{31} +(5.42627 + 1.97500i) q^{32} +(2.27653 - 0.401413i) q^{33} +(0.353543 - 0.296658i) q^{34} -2.36049 q^{36} +(-5.89661 - 1.49332i) q^{37} -2.81867i q^{38} +(3.66420 - 4.36683i) q^{39} +(1.46584 + 8.31319i) q^{41} +(-0.643326 - 0.234151i) q^{42} -0.284447 q^{43} +(2.59841 + 0.945743i) q^{44} +(0.872486 - 4.94811i) q^{46} +(-2.08485 + 1.20369i) q^{47} +(0.306039 + 0.176692i) q^{48} +(4.92835 + 4.13538i) q^{49} +(0.524788 - 0.302987i) q^{51} +(6.40764 - 2.33219i) q^{52} +(-0.872086 - 2.39604i) q^{53} +(4.30482 + 0.759056i) q^{54} +(-1.33175 - 1.58712i) q^{56} +(0.642656 - 3.64468i) q^{57} +(0.254657 + 0.699665i) q^{58} +(-1.62011 - 4.45122i) q^{59} +(15.1133 - 2.66489i) q^{61} +(-2.81382 + 3.35337i) q^{62} +(1.17700 + 0.679541i) q^{63} +(-2.07976 - 3.60225i) q^{64} +(-1.66626 - 0.962016i) q^{66} +(2.25851 - 6.20521i) q^{67} +0.724859 q^{68} +(2.25634 - 6.19924i) q^{69} +(-6.19327 + 5.19677i) q^{71} +(3.80766 + 3.19500i) q^{72} +4.72457i q^{73} +(2.96072 + 4.10685i) q^{74} +(2.84562 - 3.39127i) q^{76} +(-1.02337 - 1.21960i) q^{77} +(-4.67255 + 0.823897i) q^{78} +(-1.50681 + 4.13992i) q^{79} +(0.302869 + 0.110235i) q^{81} +(3.51299 - 6.08468i) q^{82} +(11.7571 + 2.07309i) q^{83} +(-0.537626 - 0.931195i) q^{84} +(0.181362 + 0.152181i) q^{86} +(0.169762 + 0.962766i) q^{87} +(-2.91134 - 5.04259i) q^{88} +(2.17254 + 5.96900i) q^{89} +(-3.86640 - 0.681750i) q^{91} +(6.04515 - 5.07248i) q^{92} +(-4.40298 + 3.69454i) q^{93} +(1.97327 + 0.347942i) q^{94} +(-2.15836 - 5.93003i) q^{96} +(-3.46995 - 6.01013i) q^{97} +(-0.929841 - 5.27339i) q^{98} +(2.92595 + 2.45516i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 72 q + 6 q^{4} + 6 q^{7} + 18 q^{8} + 42 q^{12} + 18 q^{13} + 18 q^{14} - 18 q^{16} - 12 q^{17} - 24 q^{18} + 18 q^{19} - 24 q^{22} + 24 q^{23} + 30 q^{24} + 48 q^{28} - 18 q^{29} - 24 q^{33} - 66 q^{34}+ \cdots + 12 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/925\mathbb{Z}\right)^\times\).

\(n\) \(76\) \(852\)
\(\chi(n)\) \(e\left(\frac{13}{18}\right)\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.637595 − 0.535006i −0.450848 − 0.378306i 0.388903 − 0.921279i \(-0.372854\pi\)
−0.839750 + 0.542973i \(0.817299\pi\)
\(3\) −0.702462 − 0.837162i −0.405567 − 0.483336i 0.524142 − 0.851631i \(-0.324386\pi\)
−0.929709 + 0.368295i \(0.879942\pi\)
\(4\) −0.227000 − 1.28738i −0.113500 − 0.643691i
\(5\) 0 0
\(6\) 0.909592i 0.371339i
\(7\) −0.257425 + 0.707269i −0.0972974 + 0.267322i −0.978787 − 0.204881i \(-0.934319\pi\)
0.881489 + 0.472204i \(0.156541\pi\)
\(8\) −1.37634 + 2.38390i −0.486611 + 0.842835i
\(9\) 0.313558 − 1.77827i 0.104519 − 0.592758i
\(10\) 0 0
\(11\) −1.05763 + 1.83188i −0.318889 + 0.552332i −0.980256 − 0.197730i \(-0.936643\pi\)
0.661368 + 0.750062i \(0.269976\pi\)
\(12\) −0.918288 + 1.09437i −0.265087 + 0.315918i
\(13\) 0.905788 + 5.13698i 0.251220 + 1.42474i 0.805591 + 0.592472i \(0.201848\pi\)
−0.554371 + 0.832270i \(0.687041\pi\)
\(14\) 0.542525 − 0.313227i 0.144996 − 0.0837135i
\(15\) 0 0
\(16\) −0.303862 + 0.110597i −0.0759655 + 0.0276492i
\(17\) −0.0962871 + 0.546071i −0.0233530 + 0.132442i −0.994255 − 0.107034i \(-0.965865\pi\)
0.970902 + 0.239476i \(0.0769756\pi\)
\(18\) −1.15131 + 0.966063i −0.271366 + 0.227703i
\(19\) 2.17681 + 2.59422i 0.499394 + 0.595155i 0.955581 − 0.294729i \(-0.0952295\pi\)
−0.456187 + 0.889884i \(0.650785\pi\)
\(20\) 0 0
\(21\) 0.772930 − 0.281323i 0.168667 − 0.0613898i
\(22\) 1.65441 − 0.602155i 0.352721 − 0.128380i
\(23\) 3.01833 + 5.22791i 0.629366 + 1.09009i 0.987679 + 0.156492i \(0.0500186\pi\)
−0.358313 + 0.933601i \(0.616648\pi\)
\(24\) 2.96254 − 0.522375i 0.604726 − 0.106629i
\(25\) 0 0
\(26\) 2.17079 − 3.75991i 0.425726 − 0.737380i
\(27\) −4.54824 + 2.62593i −0.875309 + 0.505360i
\(28\) 0.968960 + 0.170854i 0.183116 + 0.0322883i
\(29\) −0.774719 − 0.447284i −0.143862 − 0.0830586i 0.426341 − 0.904562i \(-0.359802\pi\)
−0.570203 + 0.821504i \(0.693136\pi\)
\(30\) 0 0
\(31\) − 5.25941i − 0.944618i −0.881433 − 0.472309i \(-0.843421\pi\)
0.881433 − 0.472309i \(-0.156579\pi\)
\(32\) 5.42627 + 1.97500i 0.959238 + 0.349134i
\(33\) 2.27653 − 0.401413i 0.396293 − 0.0698771i
\(34\) 0.353543 − 0.296658i 0.0606322 − 0.0508764i
\(35\) 0 0
\(36\) −2.36049 −0.393416
\(37\) −5.89661 − 1.49332i −0.969397 − 0.245500i
\(38\) − 2.81867i − 0.457248i
\(39\) 3.66420 − 4.36683i 0.586742 − 0.699252i
\(40\) 0 0
\(41\) 1.46584 + 8.31319i 0.228926 + 1.29830i 0.855037 + 0.518568i \(0.173535\pi\)
−0.626111 + 0.779734i \(0.715354\pi\)
\(42\) −0.643326 − 0.234151i −0.0992673 − 0.0361303i
\(43\) −0.284447 −0.0433778 −0.0216889 − 0.999765i \(-0.506904\pi\)
−0.0216889 + 0.999765i \(0.506904\pi\)
\(44\) 2.59841 + 0.945743i 0.391725 + 0.142576i
\(45\) 0 0
\(46\) 0.872486 − 4.94811i 0.128641 − 0.729559i
\(47\) −2.08485 + 1.20369i −0.304107 + 0.175576i −0.644287 − 0.764784i \(-0.722846\pi\)
0.340179 + 0.940361i \(0.389512\pi\)
\(48\) 0.306039 + 0.176692i 0.0441729 + 0.0255033i
\(49\) 4.92835 + 4.13538i 0.704050 + 0.590768i
\(50\) 0 0
\(51\) 0.524788 − 0.302987i 0.0734850 − 0.0424266i
\(52\) 6.40764 − 2.33219i 0.888580 − 0.323417i
\(53\) −0.872086 − 2.39604i −0.119790 − 0.329121i 0.865276 − 0.501295i \(-0.167143\pi\)
−0.985066 + 0.172174i \(0.944921\pi\)
\(54\) 4.30482 + 0.759056i 0.585812 + 0.103294i
\(55\) 0 0
\(56\) −1.33175 − 1.58712i −0.177963 − 0.212088i
\(57\) 0.642656 − 3.64468i 0.0851219 − 0.482750i
\(58\) 0.254657 + 0.699665i 0.0334381 + 0.0918706i
\(59\) −1.62011 − 4.45122i −0.210921 − 0.579500i 0.788445 − 0.615105i \(-0.210886\pi\)
−0.999366 + 0.0356052i \(0.988664\pi\)
\(60\) 0 0
\(61\) 15.1133 − 2.66489i 1.93506 − 0.341204i 0.935150 − 0.354253i \(-0.115265\pi\)
0.999915 + 0.0130489i \(0.00415372\pi\)
\(62\) −2.81382 + 3.35337i −0.357355 + 0.425879i
\(63\) 1.17700 + 0.679541i 0.148288 + 0.0856141i
\(64\) −2.07976 − 3.60225i −0.259970 − 0.450282i
\(65\) 0 0
\(66\) −1.66626 − 0.962016i −0.205103 − 0.118416i
\(67\) 2.25851 − 6.20521i 0.275921 − 0.758087i −0.721893 − 0.692004i \(-0.756728\pi\)
0.997814 − 0.0660821i \(-0.0210499\pi\)
\(68\) 0.724859 0.0879021
\(69\) 2.25634 − 6.19924i 0.271631 − 0.746301i
\(70\) 0 0
\(71\) −6.19327 + 5.19677i −0.735006 + 0.616743i −0.931491 − 0.363764i \(-0.881492\pi\)
0.196486 + 0.980507i \(0.437047\pi\)
\(72\) 3.80766 + 3.19500i 0.448737 + 0.376535i
\(73\) 4.72457i 0.552969i 0.961018 + 0.276485i \(0.0891695\pi\)
−0.961018 + 0.276485i \(0.910831\pi\)
\(74\) 2.96072 + 4.10685i 0.344176 + 0.477412i
\(75\) 0 0
\(76\) 2.84562 − 3.39127i 0.326414 − 0.389006i
\(77\) −1.02337 − 1.21960i −0.116624 − 0.138987i
\(78\) −4.67255 + 0.823897i −0.529063 + 0.0932880i
\(79\) −1.50681 + 4.13992i −0.169529 + 0.465778i −0.995141 − 0.0984611i \(-0.968608\pi\)
0.825612 + 0.564239i \(0.190830\pi\)
\(80\) 0 0
\(81\) 0.302869 + 0.110235i 0.0336522 + 0.0122484i
\(82\) 3.51299 − 6.08468i 0.387945 − 0.671940i
\(83\) 11.7571 + 2.07309i 1.29051 + 0.227551i 0.776436 − 0.630196i \(-0.217025\pi\)
0.514072 + 0.857747i \(0.328137\pi\)
\(84\) −0.537626 − 0.931195i −0.0586598 − 0.101602i
\(85\) 0 0
\(86\) 0.181362 + 0.152181i 0.0195568 + 0.0164101i
\(87\) 0.169762 + 0.962766i 0.0182004 + 0.103219i
\(88\) −2.91134 − 5.04259i −0.310350 − 0.537541i
\(89\) 2.17254 + 5.96900i 0.230288 + 0.632712i 0.999984 − 0.00569911i \(-0.00181409\pi\)
−0.769695 + 0.638411i \(0.779592\pi\)
\(90\) 0 0
\(91\) −3.86640 − 0.681750i −0.405308 − 0.0714668i
\(92\) 6.04515 − 5.07248i 0.630250 − 0.528843i
\(93\) −4.40298 + 3.69454i −0.456568 + 0.383106i
\(94\) 1.97327 + 0.347942i 0.203528 + 0.0358874i
\(95\) 0 0
\(96\) −2.15836 − 5.93003i −0.220286 − 0.605231i
\(97\) −3.46995 − 6.01013i −0.352320 − 0.610236i 0.634335 − 0.773058i \(-0.281274\pi\)
−0.986656 + 0.162822i \(0.947940\pi\)
\(98\) −0.929841 − 5.27339i −0.0939281 − 0.532693i
\(99\) 2.92595 + 2.45516i 0.294069 + 0.246753i
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 925.2.ba.c.299.6 72
5.2 odd 4 185.2.w.a.151.8 yes 72
5.3 odd 4 925.2.bb.b.151.5 72
5.4 even 2 925.2.ba.b.299.7 72
37.25 even 18 925.2.ba.b.99.7 72
185.32 even 36 6845.2.a.u.1.21 36
185.42 even 36 6845.2.a.t.1.16 36
185.62 odd 36 185.2.w.a.136.8 ✓ 72
185.99 even 18 inner 925.2.ba.c.99.6 72
185.173 odd 36 925.2.bb.b.876.5 72
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
185.2.w.a.136.8 ✓ 72 185.62 odd 36
185.2.w.a.151.8 yes 72 5.2 odd 4
925.2.ba.b.99.7 72 37.25 even 18
925.2.ba.b.299.7 72 5.4 even 2
925.2.ba.c.99.6 72 185.99 even 18 inner
925.2.ba.c.299.6 72 1.1 even 1 trivial
925.2.bb.b.151.5 72 5.3 odd 4
925.2.bb.b.876.5 72 185.173 odd 36
6845.2.a.t.1.16 36 185.42 even 36
6845.2.a.u.1.21 36 185.32 even 36