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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 41.12
Character \(\chi\) \(=\) 185.41
Dual form 185.2.w.a.176.12

$q$-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+(2.40518 + 0.424099i) q^{2} +(-0.541060 - 3.06850i) q^{3} +(3.72567 + 1.35603i) q^{4} +(0.642788 + 0.766044i) q^{5} -7.60978i q^{6} +(-3.04240 + 2.55288i) q^{7} +(4.15566 + 2.39927i) q^{8} +(-6.30389 + 2.29443i) q^{9} +(1.22114 + 2.11508i) q^{10} +(0.733441 - 1.27036i) q^{11} +(2.14518 - 12.1659i) q^{12} +(-0.385414 + 1.05891i) q^{13} +(-8.40020 + 4.84986i) q^{14} +(2.00282 - 2.38687i) q^{15} +(2.90321 + 2.43609i) q^{16} +(0.564319 + 1.55045i) q^{17} +(-16.1351 + 2.84505i) q^{18} +(7.43945 - 1.31178i) q^{19} +(1.35603 + 3.72567i) q^{20} +(9.47963 + 7.95435i) q^{21} +(2.30282 - 2.74439i) q^{22} +(-3.14430 + 1.81536i) q^{23} +(5.11371 - 14.0498i) q^{24} +(-0.173648 + 0.984808i) q^{25} +(-1.37608 + 2.38343i) q^{26} +(5.77748 + 10.0069i) q^{27} +(-14.7967 + 5.38557i) q^{28} +(-5.38072 - 3.10656i) q^{29} +(5.82943 - 4.89147i) q^{30} -8.99026i q^{31} +(-0.219263 - 0.261307i) q^{32} +(-4.29493 - 1.56323i) q^{33} +(0.699746 + 3.96846i) q^{34} +(-3.91123 - 0.689656i) q^{35} -26.5975 q^{36} +(-4.62280 + 3.95345i) q^{37} +18.4496 q^{38} +(3.45782 + 0.609706i) q^{39} +(0.833258 + 4.72564i) q^{40} +(2.47011 + 0.899047i) q^{41} +(19.4268 + 23.1520i) q^{42} +4.53873i q^{43} +(4.45520 - 3.73836i) q^{44} +(-5.80970 - 3.35423i) q^{45} +(-8.33252 + 3.03279i) q^{46} +(-1.25681 - 2.17686i) q^{47} +(5.90432 - 10.2266i) q^{48} +(1.52348 - 8.64007i) q^{49} +(-0.835312 + 2.29500i) q^{50} +(4.45225 - 2.57051i) q^{51} +(-2.87185 + 3.42253i) q^{52} +(0.809961 + 0.679638i) q^{53} +(9.65200 + 26.5186i) q^{54} +(1.44460 - 0.254721i) q^{55} +(-18.7682 + 3.30934i) q^{56} +(-8.05038 - 22.1182i) q^{57} +(-11.6241 - 9.75381i) q^{58} +(7.80224 - 9.29834i) q^{59} +(10.6985 - 6.17680i) q^{60} +(-4.20357 + 11.5492i) q^{61} +(3.81276 - 21.6232i) q^{62} +(13.3216 - 23.0736i) q^{63} +(-4.20642 - 7.28574i) q^{64} +(-1.05891 + 0.385414i) q^{65} +(-9.66714 - 5.58133i) q^{66} +(2.87704 - 2.41412i) q^{67} +6.54171i q^{68} +(7.27171 + 8.66609i) q^{69} +(-9.11475 - 3.31750i) q^{70} +(0.0803616 + 0.455753i) q^{71} +(-31.7018 - 5.58988i) q^{72} +0.804587 q^{73} +(-12.7953 + 7.54825i) q^{74} +3.11584 q^{75} +(29.4957 + 5.20090i) q^{76} +(1.01164 + 5.73732i) q^{77} +(8.05811 + 2.93291i) q^{78} +(-0.506288 - 0.603371i) q^{79} +3.78988i q^{80} +(12.1633 - 10.2062i) q^{81} +(5.55979 + 3.20994i) q^{82} +(8.33165 - 3.03247i) q^{83} +(24.5316 + 42.4899i) q^{84} +(-0.824980 + 1.42891i) q^{85} +(-1.92487 + 10.9165i) q^{86} +(-6.62120 + 18.1916i) q^{87} +(6.09586 - 3.51945i) q^{88} +(-0.979173 + 1.16693i) q^{89} +(-12.5509 - 10.5314i) q^{90} +(-1.53070 - 4.20555i) q^{91} +(-14.1763 + 2.49967i) q^{92} +(-27.5866 + 4.86427i) q^{93} +(-2.09966 - 5.76877i) q^{94} +(5.78687 + 4.85576i) q^{95} +(-0.683188 + 0.814192i) q^{96} +(0.132311 - 0.0763897i) q^{97} +(7.32849 - 20.1349i) q^{98} +(-1.70879 + 9.69102i) 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 \(\left(\mathbb{Z}/185\mathbb{Z}\right)^\times\).

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

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.40518 + 0.424099i 1.70072 + 0.299883i 0.937947 − 0.346780i \(-0.112725\pi\)
0.762776 + 0.646663i \(0.223836\pi\)
\(3\) −0.541060 − 3.06850i −0.312381 − 1.77160i −0.586543 − 0.809918i \(-0.699512\pi\)
0.274162 − 0.961683i \(-0.411599\pi\)
\(4\) 3.72567 + 1.35603i 1.86283 + 0.678016i
\(5\) 0.642788 + 0.766044i 0.287463 + 0.342585i
\(6\) − 7.60978i − 3.10668i
\(7\) −3.04240 + 2.55288i −1.14992 + 0.964896i −0.999718 − 0.0237594i \(-0.992436\pi\)
−0.150201 + 0.988656i \(0.547992\pi\)
\(8\) 4.15566 + 2.39927i 1.46925 + 0.848270i
\(9\) −6.30389 + 2.29443i −2.10130 + 0.764809i
\(10\) 1.22114 + 2.11508i 0.386160 + 0.668848i
\(11\) 0.733441 − 1.27036i 0.221141 − 0.383027i −0.734014 − 0.679135i \(-0.762355\pi\)
0.955155 + 0.296107i \(0.0956886\pi\)
\(12\) 2.14518 − 12.1659i 0.619260 − 3.51200i
\(13\) −0.385414 + 1.05891i −0.106894 + 0.293690i −0.981595 − 0.190973i \(-0.938836\pi\)
0.874701 + 0.484663i \(0.161058\pi\)
\(14\) −8.40020 + 4.84986i −2.24505 + 1.29618i
\(15\) 2.00282 − 2.38687i 0.517127 − 0.616288i
\(16\) 2.90321 + 2.43609i 0.725803 + 0.609021i
\(17\) 0.564319 + 1.55045i 0.136868 + 0.376041i 0.989124 − 0.147084i \(-0.0469886\pi\)
−0.852256 + 0.523124i \(0.824766\pi\)
\(18\) −16.1351 + 2.84505i −3.80307 + 0.670585i
\(19\) 7.43945 − 1.31178i 1.70673 − 0.300942i 0.766691 − 0.642016i \(-0.221902\pi\)
0.940037 + 0.341074i \(0.110791\pi\)
\(20\) 1.35603 + 3.72567i 0.303218 + 0.833085i
\(21\) 9.47963 + 7.95435i 2.06862 + 1.73578i
\(22\) 2.30282 − 2.74439i 0.490963 − 0.585106i
\(23\) −3.14430 + 1.81536i −0.655633 + 0.378530i −0.790611 − 0.612319i \(-0.790237\pi\)
0.134978 + 0.990849i \(0.456904\pi\)
\(24\) 5.11371 − 14.0498i 1.04383 − 2.86790i
\(25\) −0.173648 + 0.984808i −0.0347296 + 0.196962i
\(26\) −1.37608 + 2.38343i −0.269871 + 0.467430i
\(27\) 5.77748 + 10.0069i 1.11188 + 1.92583i
\(28\) −14.7967 + 5.38557i −2.79632 + 1.01778i
\(29\) −5.38072 − 3.10656i −0.999175 − 0.576874i −0.0911708 − 0.995835i \(-0.529061\pi\)
−0.908004 + 0.418961i \(0.862394\pi\)
\(30\) 5.82943 − 4.89147i 1.06430 − 0.893057i
\(31\) − 8.99026i − 1.61470i −0.590074 − 0.807349i \(-0.700901\pi\)
0.590074 − 0.807349i \(-0.299099\pi\)
\(32\) −0.219263 − 0.261307i −0.0387606 − 0.0461931i
\(33\) −4.29493 − 1.56323i −0.747652 − 0.272123i
\(34\) 0.699746 + 3.96846i 0.120005 + 0.680585i
\(35\) −3.91123 − 0.689656i −0.661119 − 0.116573i
\(36\) −26.5975 −4.43292
\(37\) −4.62280 + 3.95345i −0.759983 + 0.649943i
\(38\) 18.4496 2.99292
\(39\) 3.45782 + 0.609706i 0.553694 + 0.0976311i
\(40\) 0.833258 + 4.72564i 0.131750 + 0.747189i
\(41\) 2.47011 + 0.899047i 0.385766 + 0.140407i 0.527621 − 0.849480i \(-0.323084\pi\)
−0.141854 + 0.989888i \(0.545306\pi\)
\(42\) 19.4268 + 23.1520i 2.99762 + 3.57243i
\(43\) 4.53873i 0.692150i 0.938207 + 0.346075i \(0.112486\pi\)
−0.938207 + 0.346075i \(0.887514\pi\)
\(44\) 4.45520 − 3.73836i 0.671647 − 0.563579i
\(45\) −5.80970 − 3.35423i −0.866058 − 0.500019i
\(46\) −8.33252 + 3.03279i −1.22856 + 0.447161i
\(47\) −1.25681 − 2.17686i −0.183325 − 0.317528i 0.759686 − 0.650290i \(-0.225353\pi\)
−0.943011 + 0.332762i \(0.892019\pi\)
\(48\) 5.90432 − 10.2266i 0.852216 − 1.47608i
\(49\) 1.52348 − 8.64007i 0.217640 − 1.23430i
\(50\) −0.835312 + 2.29500i −0.118131 + 0.324562i
\(51\) 4.45225 − 2.57051i 0.623439 − 0.359943i
\(52\) −2.87185 + 3.42253i −0.398253 + 0.474620i
\(53\) 0.809961 + 0.679638i 0.111257 + 0.0933554i 0.696719 − 0.717344i \(-0.254643\pi\)
−0.585462 + 0.810700i \(0.699087\pi\)
\(54\) 9.65200 + 26.5186i 1.31347 + 3.60873i
\(55\) 1.44460 − 0.254721i 0.194789 − 0.0343466i
\(56\) −18.7682 + 3.30934i −2.50801 + 0.442229i
\(57\) −8.05038 − 22.1182i −1.06630 − 2.92963i
\(58\) −11.6241 − 9.75381i −1.52632 − 1.28074i
\(59\) 7.80224 − 9.29834i 1.01576 − 1.21054i 0.0383383 − 0.999265i \(-0.487794\pi\)
0.977426 − 0.211276i \(-0.0677620\pi\)
\(60\) 10.6985 − 6.17680i 1.38117 − 0.797421i
\(61\) −4.20357 + 11.5492i −0.538213 + 1.47873i 0.310863 + 0.950455i \(0.399382\pi\)
−0.849075 + 0.528272i \(0.822840\pi\)
\(62\) 3.81276 − 21.6232i 0.484221 − 2.74615i
\(63\) 13.3216 − 23.0736i 1.67836 − 2.90700i
\(64\) −4.20642 − 7.28574i −0.525803 − 0.910717i
\(65\) −1.05891 + 0.385414i −0.131342 + 0.0478047i
\(66\) −9.66714 − 5.58133i −1.18994 − 0.687014i
\(67\) 2.87704 − 2.41412i 0.351486 − 0.294932i −0.449900 − 0.893079i \(-0.648541\pi\)
0.801386 + 0.598147i \(0.204096\pi\)
\(68\) 6.54171i 0.793299i
\(69\) 7.27171 + 8.66609i 0.875411 + 1.04327i
\(70\) −9.11475 − 3.31750i −1.08942 − 0.396517i
\(71\) 0.0803616 + 0.455753i 0.00953716 + 0.0540879i 0.989205 − 0.146539i \(-0.0468133\pi\)
−0.979668 + 0.200627i \(0.935702\pi\)
\(72\) −31.7018 − 5.58988i −3.73609 − 0.658773i
\(73\) 0.804587 0.0941698 0.0470849 − 0.998891i \(-0.485007\pi\)
0.0470849 + 0.998891i \(0.485007\pi\)
\(74\) −12.7953 + 7.54825i −1.48743 + 0.877466i
\(75\) 3.11584 0.359786
\(76\) 29.4957 + 5.20090i 3.38339 + 0.596584i
\(77\) 1.01164 + 5.73732i 0.115288 + 0.653828i
\(78\) 8.05811 + 2.93291i 0.912401 + 0.332087i
\(79\) −0.506288 − 0.603371i −0.0569619 − 0.0678846i 0.736811 − 0.676099i \(-0.236331\pi\)
−0.793773 + 0.608214i \(0.791886\pi\)
\(80\) 3.78988i 0.423721i
\(81\) 12.1633 − 10.2062i 1.35148 − 1.13402i
\(82\) 5.55979 + 3.20994i 0.613976 + 0.354479i
\(83\) 8.33165 − 3.03247i 0.914517 − 0.332857i 0.158462 − 0.987365i \(-0.449346\pi\)
0.756055 + 0.654508i \(0.227124\pi\)
\(84\) 24.5316 + 42.4899i 2.67661 + 4.63603i
\(85\) −0.824980 + 1.42891i −0.0894816 + 0.154987i
\(86\) −1.92487 + 10.9165i −0.207564 + 1.17715i
\(87\) −6.62120 + 18.1916i −0.709867 + 1.95034i
\(88\) 6.09586 − 3.51945i 0.649821 − 0.375174i
\(89\) −0.979173 + 1.16693i −0.103792 + 0.123695i −0.815440 − 0.578842i \(-0.803505\pi\)
0.711648 + 0.702537i \(0.247949\pi\)
\(90\) −12.5509 − 10.5314i −1.32298 − 1.11011i
\(91\) −1.53070 − 4.20555i −0.160461 − 0.440862i
\(92\) −14.1763 + 2.49967i −1.47798 + 0.260608i
\(93\) −27.5866 + 4.86427i −2.86060 + 0.504401i
\(94\) −2.09966 − 5.76877i −0.216564 − 0.595003i
\(95\) 5.78687 + 4.85576i 0.593720 + 0.498190i
\(96\) −0.683188 + 0.814192i −0.0697276 + 0.0830982i
\(97\) 0.132311 − 0.0763897i 0.0134341 − 0.00775620i −0.493268 − 0.869877i \(-0.664198\pi\)
0.506702 + 0.862121i \(0.330864\pi\)
\(98\) 7.32849 − 20.1349i 0.740289 − 2.03393i
\(99\) −1.70879 + 9.69102i −0.171740 + 0.973984i
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.41.12 ✓ 72
5.2 odd 4 925.2.ba.c.374.1 72
5.3 odd 4 925.2.ba.b.374.12 72
5.4 even 2 925.2.bb.b.226.1 72
37.18 odd 36 6845.2.a.t.1.4 36
37.19 odd 36 6845.2.a.u.1.33 36
37.28 even 18 inner 185.2.w.a.176.12 yes 72
185.28 odd 36 925.2.ba.c.324.1 72
185.102 odd 36 925.2.ba.b.324.12 72
185.139 even 18 925.2.bb.b.176.1 72
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
185.2.w.a.41.12 ✓ 72 1.1 even 1 trivial
185.2.w.a.176.12 yes 72 37.28 even 18 inner
925.2.ba.b.324.12 72 185.102 odd 36
925.2.ba.b.374.12 72 5.3 odd 4
925.2.ba.c.324.1 72 185.28 odd 36
925.2.ba.c.374.1 72 5.2 odd 4
925.2.bb.b.176.1 72 185.139 even 18
925.2.bb.b.226.1 72 5.4 even 2
6845.2.a.t.1.4 36 37.18 odd 36
6845.2.a.u.1.33 36 37.19 odd 36