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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 151.11
Character \(\chi\) \(=\) 185.151
Dual form 185.2.w.a.136.11

$q$-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+(1.29158 - 1.53924i) q^{2} +(0.583938 - 0.489982i) q^{3} +(-0.353796 - 2.00647i) q^{4} +(0.342020 - 0.939693i) q^{5} -1.53167i q^{6} +(-1.63114 - 0.593686i) q^{7} +(-0.0651292 - 0.0376024i) q^{8} +(-0.420044 + 2.38219i) q^{9} +(-1.00467 - 1.74014i) q^{10} +(-0.381991 + 0.661628i) q^{11} +(-1.18973 - 0.998303i) q^{12} +(-0.0386391 + 0.00681312i) q^{13} +(-3.02057 + 1.74393i) q^{14} +(-0.260714 - 0.716306i) q^{15} +(3.68711 - 1.34200i) q^{16} +(-4.30304 - 0.758742i) q^{17} +(3.12424 + 3.72332i) q^{18} +(2.89648 + 3.45189i) q^{19} +(-2.00647 - 0.353796i) q^{20} +(-1.24338 + 0.452553i) q^{21} +(0.525034 + 1.44252i) q^{22} +(1.58218 - 0.913472i) q^{23} +(-0.0564559 + 0.00995470i) q^{24} +(-0.766044 - 0.642788i) q^{25} +(-0.0394183 + 0.0682745i) q^{26} +(2.06536 + 3.57731i) q^{27} +(-0.614127 + 3.48288i) q^{28} +(-4.07808 - 2.35448i) q^{29} +(-1.43930 - 0.523862i) q^{30} +6.00718i q^{31} +(2.74797 - 7.54998i) q^{32} +(0.101127 + 0.573518i) q^{33} +(-6.72559 + 5.64344i) q^{34} +(-1.11577 + 1.32972i) q^{35} +4.92840 q^{36} +(-1.93907 - 5.76541i) q^{37} +9.05431 q^{38} +(-0.0192245 + 0.0229109i) q^{39} +(-0.0576102 + 0.0483407i) q^{40} +(1.01550 + 5.75920i) q^{41} +(-0.909331 + 2.49837i) q^{42} +3.43382i q^{43} +(1.46269 + 0.532374i) q^{44} +(2.09486 + 1.20947i) q^{45} +(0.637453 - 3.61517i) q^{46} +(-1.81560 - 3.14471i) q^{47} +(1.49549 - 2.59026i) q^{48} +(-3.05416 - 2.56274i) q^{49} +(-1.97881 + 0.348917i) q^{50} +(-2.88448 + 1.66535i) q^{51} +(0.0273407 + 0.0751179i) q^{52} +(11.2190 - 4.08338i) q^{53} +(8.17392 + 1.44128i) q^{54} +(0.491078 + 0.585244i) q^{55} +(0.0839109 + 0.100001i) q^{56} +(3.38273 + 0.596466i) q^{57} +(-8.89125 + 3.23615i) q^{58} +(-2.59304 - 7.12432i) q^{59} +(-1.34501 + 0.776542i) q^{60} +(-5.71230 + 1.00723i) q^{61} +(9.24649 + 7.75873i) q^{62} +(2.09942 - 3.63630i) q^{63} +(-4.14828 - 7.18504i) q^{64} +(-0.00681312 + 0.0386391i) q^{65} +(1.01339 + 0.585084i) q^{66} +(0.268567 + 0.0977503i) q^{67} +8.90238i q^{68} +(0.476310 - 1.30865i) q^{69} +(0.605659 + 3.43486i) q^{70} +(-4.33845 + 3.64039i) q^{71} +(0.116933 - 0.139355i) q^{72} -4.13245 q^{73} +(-11.3788 - 4.46177i) q^{74} -0.762276 q^{75} +(5.90137 - 7.03298i) q^{76} +(1.01588 - 0.852425i) q^{77} +(0.0104354 + 0.0591823i) q^{78} +(-0.799712 + 2.19719i) q^{79} -3.92374i q^{80} +(-3.86030 - 1.40504i) q^{81} +(10.1764 + 5.87534i) q^{82} +(0.460892 - 2.61385i) q^{83} +(1.34794 + 2.33470i) q^{84} +(-2.18471 + 3.78403i) q^{85} +(5.28548 + 4.43504i) q^{86} +(-3.53499 + 0.623315i) q^{87} +(0.0497576 - 0.0287276i) q^{88} +(-5.87216 - 16.1336i) q^{89} +(4.56733 - 1.66237i) q^{90} +(0.0670707 + 0.0118264i) q^{91} +(-2.39263 - 2.85142i) q^{92} +(2.94341 + 3.50782i) q^{93} +(-7.18544 - 1.26699i) q^{94} +(4.23437 - 1.54118i) q^{95} +(-2.09471 - 5.75518i) q^{96} +(-12.8808 + 7.43673i) q^{97} +(-7.88935 + 1.39111i) q^{98} +(-1.41567 - 1.18789i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.29158 − 1.53924i 0.913282 − 1.08841i −0.0824941 − 0.996592i \(-0.526289\pi\)
0.995776 − 0.0918155i \(-0.0292670\pi\)
\(3\) 0.583938 − 0.489982i 0.337137 − 0.282891i −0.458464 − 0.888713i \(-0.651600\pi\)
0.795600 + 0.605822i \(0.207156\pi\)
\(4\) −0.353796 − 2.00647i −0.176898 − 1.00324i
\(5\) 0.342020 − 0.939693i 0.152956 − 0.420243i
\(6\) − 1.53167i − 0.625301i
\(7\) −1.63114 − 0.593686i −0.616513 − 0.224392i 0.0148376 − 0.999890i \(-0.495277\pi\)
−0.631351 + 0.775498i \(0.717499\pi\)
\(8\) −0.0651292 − 0.0376024i −0.0230267 − 0.0132945i
\(9\) −0.420044 + 2.38219i −0.140015 + 0.794062i
\(10\) −1.00467 − 1.74014i −0.317704 − 0.550279i
\(11\) −0.381991 + 0.661628i −0.115175 + 0.199488i −0.917850 − 0.396928i \(-0.870076\pi\)
0.802675 + 0.596417i \(0.203409\pi\)
\(12\) −1.18973 − 0.998303i −0.343446 − 0.288185i
\(13\) −0.0386391 + 0.00681312i −0.0107166 + 0.00188962i −0.179004 − 0.983848i \(-0.557287\pi\)
0.168287 + 0.985738i \(0.446176\pi\)
\(14\) −3.02057 + 1.74393i −0.807280 + 0.466084i
\(15\) −0.260714 − 0.716306i −0.0673160 − 0.184949i
\(16\) 3.68711 − 1.34200i 0.921778 − 0.335500i
\(17\) −4.30304 − 0.758742i −1.04364 − 0.184022i −0.374553 − 0.927206i \(-0.622204\pi\)
−0.669088 + 0.743184i \(0.733315\pi\)
\(18\) 3.12424 + 3.72332i 0.736390 + 0.877595i
\(19\) 2.89648 + 3.45189i 0.664498 + 0.791918i 0.988024 − 0.154301i \(-0.0493127\pi\)
−0.323526 + 0.946219i \(0.604868\pi\)
\(20\) −2.00647 − 0.353796i −0.448661 − 0.0791111i
\(21\) −1.24338 + 0.452553i −0.271328 + 0.0987552i
\(22\) 0.525034 + 1.44252i 0.111938 + 0.307546i
\(23\) 1.58218 − 0.913472i 0.329907 − 0.190472i −0.325893 − 0.945407i \(-0.605665\pi\)
0.655800 + 0.754935i \(0.272331\pi\)
\(24\) −0.0564559 + 0.00995470i −0.0115240 + 0.00203199i
\(25\) −0.766044 − 0.642788i −0.153209 − 0.128558i
\(26\) −0.0394183 + 0.0682745i −0.00773057 + 0.0133897i
\(27\) 2.06536 + 3.57731i 0.397479 + 0.688454i
\(28\) −0.614127 + 3.48288i −0.116059 + 0.658203i
\(29\) −4.07808 − 2.35448i −0.757280 − 0.437216i 0.0710385 − 0.997474i \(-0.477369\pi\)
−0.828318 + 0.560258i \(0.810702\pi\)
\(30\) −1.43930 − 0.523862i −0.262779 − 0.0956436i
\(31\) 6.00718i 1.07892i 0.842011 + 0.539461i \(0.181372\pi\)
−0.842011 + 0.539461i \(0.818628\pi\)
\(32\) 2.74797 − 7.54998i 0.485777 − 1.33466i
\(33\) 0.101127 + 0.573518i 0.0176039 + 0.0998367i
\(34\) −6.72559 + 5.64344i −1.15343 + 0.967842i
\(35\) −1.11577 + 1.32972i −0.188599 + 0.224763i
\(36\) 4.92840 0.821401
\(37\) −1.93907 − 5.76541i −0.318782 − 0.947828i
\(38\) 9.05431 1.46880
\(39\) −0.0192245 + 0.0229109i −0.00307839 + 0.00366868i
\(40\) −0.0576102 + 0.0483407i −0.00910897 + 0.00764334i
\(41\) 1.01550 + 5.75920i 0.158595 + 0.899436i 0.955425 + 0.295233i \(0.0953973\pi\)
−0.796830 + 0.604203i \(0.793492\pi\)
\(42\) −0.909331 + 2.49837i −0.140313 + 0.385506i
\(43\) 3.43382i 0.523653i 0.965115 + 0.261827i \(0.0843248\pi\)
−0.965115 + 0.261827i \(0.915675\pi\)
\(44\) 1.46269 + 0.532374i 0.220508 + 0.0802584i
\(45\) 2.09486 + 1.20947i 0.312283 + 0.180297i
\(46\) 0.637453 − 3.61517i 0.0939873 − 0.533028i
\(47\) −1.81560 − 3.14471i −0.264832 − 0.458703i 0.702688 − 0.711499i \(-0.251983\pi\)
−0.967520 + 0.252796i \(0.918650\pi\)
\(48\) 1.49549 − 2.59026i 0.215855 − 0.373872i
\(49\) −3.05416 − 2.56274i −0.436308 − 0.366106i
\(50\) −1.97881 + 0.348917i −0.279846 + 0.0493444i
\(51\) −2.88448 + 1.66535i −0.403907 + 0.233196i
\(52\) 0.0273407 + 0.0751179i 0.00379147 + 0.0104170i
\(53\) 11.2190 − 4.08338i 1.54105 − 0.560895i 0.574750 − 0.818329i \(-0.305099\pi\)
0.966296 + 0.257434i \(0.0828771\pi\)
\(54\) 8.17392 + 1.44128i 1.11233 + 0.196134i
\(55\) 0.491078 + 0.585244i 0.0662170 + 0.0789143i
\(56\) 0.0839109 + 0.100001i 0.0112131 + 0.0133632i
\(57\) 3.38273 + 0.596466i 0.448053 + 0.0790039i
\(58\) −8.89125 + 3.23615i −1.16748 + 0.424927i
\(59\) −2.59304 − 7.12432i −0.337585 − 0.927507i −0.986078 − 0.166286i \(-0.946822\pi\)
0.648492 − 0.761221i \(-0.275400\pi\)
\(60\) −1.34501 + 0.776542i −0.173640 + 0.100251i
\(61\) −5.71230 + 1.00723i −0.731385 + 0.128963i −0.526925 − 0.849912i \(-0.676655\pi\)
−0.204460 + 0.978875i \(0.565544\pi\)
\(62\) 9.24649 + 7.75873i 1.17431 + 0.985359i
\(63\) 2.09942 − 3.63630i 0.264502 − 0.458131i
\(64\) −4.14828 − 7.18504i −0.518535 − 0.898130i
\(65\) −0.00681312 + 0.0386391i −0.000845063 + 0.00479259i
\(66\) 1.01339 + 0.585084i 0.124740 + 0.0720188i
\(67\) 0.268567 + 0.0977503i 0.0328106 + 0.0119421i 0.358373 − 0.933578i \(-0.383331\pi\)
−0.325563 + 0.945520i \(0.605554\pi\)
\(68\) 8.90238i 1.07957i
\(69\) 0.476310 − 1.30865i 0.0573410 − 0.157543i
\(70\) 0.605659 + 3.43486i 0.0723901 + 0.410545i
\(71\) −4.33845 + 3.64039i −0.514879 + 0.432035i −0.862842 − 0.505473i \(-0.831318\pi\)
0.347963 + 0.937508i \(0.386874\pi\)
\(72\) 0.116933 − 0.139355i 0.0137807 − 0.0164232i
\(73\) −4.13245 −0.483667 −0.241834 − 0.970318i \(-0.577749\pi\)
−0.241834 + 0.970318i \(0.577749\pi\)
\(74\) −11.3788 − 4.46177i −1.32276 − 0.518670i
\(75\) −0.762276 −0.0880201
\(76\) 5.90137 − 7.03298i 0.676933 − 0.806738i
\(77\) 1.01588 − 0.852425i 0.115770 − 0.0971428i
\(78\) 0.0104354 + 0.0591823i 0.00118158 + 0.00670108i
\(79\) −0.799712 + 2.19719i −0.0899746 + 0.247203i −0.976516 − 0.215447i \(-0.930879\pi\)
0.886541 + 0.462650i \(0.153101\pi\)
\(80\) − 3.92374i − 0.438688i
\(81\) −3.86030 − 1.40504i −0.428923 − 0.156115i
\(82\) 10.1764 + 5.87534i 1.12379 + 0.648823i
\(83\) 0.460892 − 2.61385i 0.0505895 − 0.286907i −0.949009 − 0.315250i \(-0.897912\pi\)
0.999598 + 0.0283424i \(0.00902288\pi\)
\(84\) 1.34794 + 2.33470i 0.147072 + 0.254736i
\(85\) −2.18471 + 3.78403i −0.236965 + 0.410436i
\(86\) 5.28548 + 4.43504i 0.569948 + 0.478243i
\(87\) −3.53499 + 0.623315i −0.378991 + 0.0668264i
\(88\) 0.0497576 − 0.0287276i 0.00530418 − 0.00306237i
\(89\) −5.87216 − 16.1336i −0.622448 − 1.71016i −0.700916 − 0.713244i \(-0.747225\pi\)
0.0784679 − 0.996917i \(-0.474997\pi\)
\(90\) 4.56733 − 1.66237i 0.481439 − 0.175229i
\(91\) 0.0670707 + 0.0118264i 0.00703092 + 0.00123974i
\(92\) −2.39263 − 2.85142i −0.249449 − 0.297281i
\(93\) 2.94341 + 3.50782i 0.305217 + 0.363744i
\(94\) −7.18544 − 1.26699i −0.741122 − 0.130680i
\(95\) 4.23437 − 1.54118i 0.434437 − 0.158122i
\(96\) −2.09471 − 5.75518i −0.213791 − 0.587385i
\(97\) −12.8808 + 7.43673i −1.30785 + 0.755086i −0.981736 − 0.190246i \(-0.939072\pi\)
−0.326111 + 0.945332i \(0.605738\pi\)
\(98\) −7.88935 + 1.39111i −0.796945 + 0.140523i
\(99\) −1.41567 − 1.18789i −0.142280 − 0.119387i
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.151.11 yes 72
5.2 odd 4 925.2.ba.c.299.11 72
5.3 odd 4 925.2.ba.b.299.2 72
5.4 even 2 925.2.bb.b.151.2 72
37.5 odd 36 6845.2.a.u.1.5 36
37.25 even 18 inner 185.2.w.a.136.11 ✓ 72
37.32 odd 36 6845.2.a.t.1.32 36
185.62 odd 36 925.2.ba.b.99.2 72
185.99 even 18 925.2.bb.b.876.2 72
185.173 odd 36 925.2.ba.c.99.11 72
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
185.2.w.a.136.11 ✓ 72 37.25 even 18 inner
185.2.w.a.151.11 yes 72 1.1 even 1 trivial
925.2.ba.b.99.2 72 185.62 odd 36
925.2.ba.b.299.2 72 5.3 odd 4
925.2.ba.c.99.11 72 185.173 odd 36
925.2.ba.c.299.11 72 5.2 odd 4
925.2.bb.b.151.2 72 5.4 even 2
925.2.bb.b.876.2 72 185.99 even 18
6845.2.a.t.1.32 36 37.32 odd 36
6845.2.a.u.1.5 36 37.5 odd 36