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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 176.12
Character \(\chi\) \(=\) 185.176
Dual form 185.2.w.a.41.12

$q$-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 \(\left(\mathbb{Z}/185\mathbb{Z}\right)^\times\).

\(n\) \(76\) \(112\)
\(\chi(n)\) \(e\left(\frac{17}{18}\right)\) \(1\)

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 0.762776 − 0.646663i \(-0.223836\pi\)
0.937947 + 0.346780i \(0.112725\pi\)
\(3\) −0.541060 + 3.06850i −0.312381 + 1.77160i 0.274162 + 0.961683i \(0.411599\pi\)
−0.586543 + 0.809918i \(0.699512\pi\)
\(4\) 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 −0.150201 − 0.988656i \(-0.547992\pi\)
−0.999718 + 0.0237594i \(0.992436\pi\)
\(8\) 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 0.955155 − 0.296107i \(-0.0956886\pi\)
−0.734014 + 0.679135i \(0.762355\pi\)
\(12\) 2.14518 + 12.1659i 0.619260 + 3.51200i
\(13\) −0.385414 − 1.05891i −0.106894 − 0.293690i 0.874701 − 0.484663i \(-0.161058\pi\)
−0.981595 + 0.190973i \(0.938836\pi\)
\(14\) −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 −0.852256 − 0.523124i \(-0.824766\pi\)
0.989124 + 0.147084i \(0.0469886\pi\)
\(18\) −16.1351 − 2.84505i −3.80307 − 0.670585i
\(19\) 7.43945 + 1.31178i 1.70673 + 0.300942i 0.940037 − 0.341074i \(-0.110791\pi\)
0.766691 + 0.642016i \(0.221902\pi\)
\(20\) 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 0.134978 − 0.990849i \(-0.456904\pi\)
−0.790611 + 0.612319i \(0.790237\pi\)
\(24\) 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 −0.908004 − 0.418961i \(-0.862394\pi\)
−0.0911708 + 0.995835i \(0.529061\pi\)
\(30\) 5.82943 + 4.89147i 1.06430 + 0.893057i
\(31\) 8.99026i 1.61470i 0.590074 + 0.807349i \(0.299099\pi\)
−0.590074 + 0.807349i \(0.700901\pi\)
\(32\) −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 −0.141854 − 0.989888i \(-0.545306\pi\)
0.527621 + 0.849480i \(0.323084\pi\)
\(42\) 19.4268 − 23.1520i 2.99762 − 3.57243i
\(43\) − 4.53873i − 0.692150i −0.938207 − 0.346075i \(-0.887514\pi\)
0.938207 − 0.346075i \(-0.112486\pi\)
\(44\) 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 −0.943011 − 0.332762i \(-0.892019\pi\)
0.759686 + 0.650290i \(0.225353\pi\)
\(48\) 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 −0.585462 − 0.810700i \(-0.699087\pi\)
0.696719 + 0.717344i \(0.254643\pi\)
\(54\) 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 0.977426 + 0.211276i \(0.0677620\pi\)
0.0383383 + 0.999265i \(0.487794\pi\)
\(60\) 10.6985 + 6.17680i 1.38117 + 0.797421i
\(61\) −4.20357 − 11.5492i −0.538213 − 1.47873i −0.849075 − 0.528272i \(-0.822840\pi\)
0.310863 − 0.950455i \(-0.399382\pi\)
\(62\) 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 0.801386 − 0.598147i \(-0.204096\pi\)
−0.449900 + 0.893079i \(0.648541\pi\)
\(68\) − 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 −0.979668 − 0.200627i \(-0.935702\pi\)
0.989205 + 0.146539i \(0.0468133\pi\)
\(72\) −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 −0.793773 − 0.608214i \(-0.791886\pi\)
0.736811 + 0.676099i \(0.236331\pi\)
\(80\) − 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 0.756055 − 0.654508i \(-0.227124\pi\)
0.158462 + 0.987365i \(0.449346\pi\)
\(84\) 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 0.711648 − 0.702537i \(-0.247949\pi\)
−0.815440 + 0.578842i \(0.803505\pi\)
\(90\) −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 0.506702 − 0.862121i \(-0.330864\pi\)
−0.493268 + 0.869877i \(0.664198\pi\)
\(98\) 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.176.12 yes 72
5.2 odd 4 925.2.ba.b.324.12 72
5.3 odd 4 925.2.ba.c.324.1 72
5.4 even 2 925.2.bb.b.176.1 72
37.2 odd 36 6845.2.a.u.1.33 36
37.4 even 18 inner 185.2.w.a.41.12 ✓ 72
37.35 odd 36 6845.2.a.t.1.4 36
185.4 even 18 925.2.bb.b.226.1 72
185.78 odd 36 925.2.ba.b.374.12 72
185.152 odd 36 925.2.ba.c.374.1 72
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
185.2.w.a.41.12 ✓ 72 37.4 even 18 inner
185.2.w.a.176.12 yes 72 1.1 even 1 trivial
925.2.ba.b.324.12 72 5.2 odd 4
925.2.ba.b.374.12 72 185.78 odd 36
925.2.ba.c.324.1 72 5.3 odd 4
925.2.ba.c.374.1 72 185.152 odd 36
925.2.bb.b.176.1 72 5.4 even 2
925.2.bb.b.226.1 72 185.4 even 18
6845.2.a.t.1.4 36 37.35 odd 36
6845.2.a.u.1.33 36 37.2 odd 36