↑

Properties

Label 185.2.v.a.139.7
Level $185$
Weight $2$
Character 185.139
Analytic conductor $1.477$
Analytic rank $0$
Dimension $96$
Inner twists $4$

Related objects

Downloads

Learn more

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(4,185)] 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("185.4"); 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([9, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 185 = 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 185.v (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: \(96\)
Relative dimension: \(16\) over \(\Q(\zeta_{18})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label 139.7
Character \(\chi\) \(=\) 185.139
Dual form 185.2.v.a.4.7

$q$-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+(-0.0448121 - 0.254142i) q^{2} +(-2.87442 - 0.506838i) q^{3} +(1.81681 - 0.661263i) q^{4} +(0.528279 + 2.17277i) q^{5} +0.753224i q^{6} +(1.46076 - 1.74086i) q^{7} +(-0.507532 - 0.879071i) q^{8} +(5.18634 + 1.88767i) q^{9} +(0.528518 - 0.231624i) q^{10} +(-2.91973 - 5.05711i) q^{11} +(-5.55742 + 0.979923i) q^{12} +(5.26665 - 1.91691i) q^{13} +(-0.507886 - 0.293228i) q^{14} +(-0.417255 - 6.51320i) q^{15} +(2.76148 - 2.31716i) q^{16} +(0.761179 + 0.277047i) q^{17} +(0.247326 - 1.40266i) q^{18} +(1.20334 + 0.212182i) q^{19} +(2.39655 + 3.59817i) q^{20} +(-5.08117 + 4.26361i) q^{21} +(-1.15439 + 0.968645i) q^{22} +(-0.941906 + 1.63143i) q^{23} +(1.01331 + 2.78406i) q^{24} +(-4.44184 + 2.29566i) q^{25} +(-0.723176 - 1.25258i) q^{26} +(-6.36780 - 3.67645i) q^{27} +(1.50275 - 4.12876i) q^{28} +(3.21766 - 1.85772i) q^{29} +(-1.63658 + 0.397912i) q^{30} +7.93817i q^{31} +(-2.26780 - 1.90291i) q^{32} +(5.82939 + 16.0161i) q^{33} +(0.0362991 - 0.205863i) q^{34} +(4.55418 + 2.25423i) q^{35} +10.6708 q^{36} +(-6.07741 - 0.255019i) q^{37} -0.315328i q^{38} +(-16.1101 + 2.84065i) q^{39} +(1.64190 - 1.56714i) q^{40} +(-4.08538 + 1.48696i) q^{41} +(1.31126 + 1.10028i) q^{42} +3.50333 q^{43} +(-8.64866 - 7.25708i) q^{44} +(-1.36164 + 12.2659i) q^{45} +(0.456823 + 0.166270i) q^{46} +(3.40089 + 1.96351i) q^{47} +(-9.11208 + 5.26086i) q^{48} +(0.318744 + 1.80769i) q^{49} +(0.782471 + 1.02599i) q^{50} +(-2.04753 - 1.18214i) q^{51} +(8.30091 - 6.96529i) q^{52} +(-1.08785 - 1.29645i) q^{53} +(-0.648987 + 1.78308i) q^{54} +(9.44551 - 9.01546i) q^{55} +(-2.27173 - 0.400567i) q^{56} +(-3.35137 - 1.21980i) q^{57} +(-0.616314 - 0.734495i) q^{58} +(0.225926 + 0.269249i) q^{59} +(-5.06501 - 11.5573i) q^{60} +(-4.51727 - 12.4111i) q^{61} +(2.01742 - 0.355726i) q^{62} +(10.8622 - 6.27128i) q^{63} +(3.22287 - 5.58218i) q^{64} +(6.94725 + 10.4306i) q^{65} +(3.80914 - 2.19921i) q^{66} +(-6.39559 + 7.62196i) q^{67} +1.56611 q^{68} +(3.53431 - 4.21202i) q^{69} +(0.368812 - 1.25843i) q^{70} +(-1.24711 + 7.07270i) q^{71} +(-0.972834 - 5.51721i) q^{72} -1.41548i q^{73} +(0.207530 + 1.55595i) q^{74} +(13.9313 - 4.34739i) q^{75} +(2.32655 - 0.410233i) q^{76} +(-13.0688 - 2.30438i) q^{77} +(1.44386 + 3.96697i) q^{78} +(-3.62772 + 4.32335i) q^{79} +(6.49348 + 4.77595i) q^{80} +(3.75657 + 3.15213i) q^{81} +(0.560972 + 0.971633i) q^{82} +(-1.01957 + 2.80125i) q^{83} +(-6.41214 + 11.1061i) q^{84} +(-0.199843 + 1.80022i) q^{85} +(-0.156992 - 0.890343i) q^{86} +(-10.1905 + 3.70903i) q^{87} +(-2.96371 + 5.13329i) q^{88} +(8.46364 + 10.0866i) q^{89} +(3.17830 - 0.203611i) q^{90} +(4.35624 - 11.9687i) q^{91} +(-0.632456 + 3.58684i) q^{92} +(4.02337 - 22.8177i) q^{93} +(0.346608 - 0.952299i) q^{94} +(0.174679 + 2.72668i) q^{95} +(5.55415 + 6.61918i) q^{96} +(-0.177750 + 0.307871i) q^{97} +(0.445125 - 0.162012i) q^{98} +(-5.59651 - 31.7394i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 96 q - 12 q^{4} - 12 q^{5} - 6 q^{9} - 3 q^{10} - 30 q^{11} - 36 q^{14} - 21 q^{15} - 18 q^{19} - 39 q^{20} - 24 q^{21} + 96 q^{24} + 36 q^{25} + 48 q^{26} - 18 q^{29} - 30 q^{30} - 54 q^{34} + 6 q^{35}+ \cdots - 24 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.0448121 − 0.254142i −0.0316869 − 0.179705i 0.964857 − 0.262777i \(-0.0846382\pi\)
−0.996544 + 0.0830711i \(0.973527\pi\)
\(3\) −2.87442 − 0.506838i −1.65955 − 0.292623i −0.736251 − 0.676709i \(-0.763406\pi\)
−0.923297 + 0.384086i \(0.874517\pi\)
\(4\) 1.81681 − 0.661263i 0.908403 − 0.330632i
\(5\) 0.528279 + 2.17277i 0.236254 + 0.971691i
\(6\) 0.753224i 0.307502i
\(7\) 1.46076 − 1.74086i 0.552115 − 0.657985i −0.415743 − 0.909482i \(-0.636478\pi\)
0.967858 + 0.251497i \(0.0809229\pi\)
\(8\) −0.507532 − 0.879071i −0.179440 − 0.310799i
\(9\) 5.18634 + 1.88767i 1.72878 + 0.629224i
\(10\) 0.528518 − 0.231624i 0.167132 − 0.0732460i
\(11\) −2.91973 − 5.05711i −0.880331 − 1.52478i −0.850974 − 0.525208i \(-0.823987\pi\)
−0.0293569 − 0.999569i \(-0.509346\pi\)
\(12\) −5.55742 + 0.979923i −1.60429 + 0.282879i
\(13\) 5.26665 − 1.91691i 1.46071 − 0.531654i 0.515147 − 0.857102i \(-0.327737\pi\)
0.945560 + 0.325448i \(0.105515\pi\)
\(14\) −0.507886 − 0.293228i −0.135738 − 0.0783686i
\(15\) −0.417255 − 6.51320i −0.107735 − 1.68170i
\(16\) 2.76148 − 2.31716i 0.690370 − 0.579289i
\(17\) 0.761179 + 0.277047i 0.184613 + 0.0671936i 0.432673 − 0.901551i \(-0.357571\pi\)
−0.248060 + 0.968745i \(0.579793\pi\)
\(18\) 0.247326 − 1.40266i 0.0582953 − 0.330609i
\(19\) 1.20334 + 0.212182i 0.276066 + 0.0486779i 0.309967 − 0.950747i \(-0.399682\pi\)
−0.0339011 + 0.999425i \(0.510793\pi\)
\(20\) 2.39655 + 3.59817i 0.535885 + 0.804574i
\(21\) −5.08117 + 4.26361i −1.10880 + 0.930396i
\(22\) −1.15439 + 0.968645i −0.246116 + 0.206516i
\(23\) −0.941906 + 1.63143i −0.196401 + 0.340176i −0.947359 − 0.320174i \(-0.896259\pi\)
0.750958 + 0.660350i \(0.229592\pi\)
\(24\) 1.01331 + 2.78406i 0.206842 + 0.568293i
\(25\) −4.44184 + 2.29566i −0.888369 + 0.459131i
\(26\) −0.723176 − 1.25258i −0.141826 − 0.245651i
\(27\) −6.36780 − 3.67645i −1.22548 − 0.707534i
\(28\) 1.50275 − 4.12876i 0.283992 − 0.780262i
\(29\) 3.21766 − 1.85772i 0.597505 − 0.344970i −0.170554 − 0.985348i \(-0.554556\pi\)
0.768059 + 0.640379i \(0.221223\pi\)
\(30\) −1.63658 + 0.397912i −0.298797 + 0.0726485i
\(31\) 7.93817i 1.42574i 0.701297 + 0.712869i \(0.252605\pi\)
−0.701297 + 0.712869i \(0.747395\pi\)
\(32\) −2.26780 − 1.90291i −0.400895 − 0.336391i
\(33\) 5.82939 + 16.0161i 1.01477 + 2.78805i
\(34\) 0.0362991 − 0.205863i 0.00622525 − 0.0353051i
\(35\) 4.55418 + 2.25423i 0.769798 + 0.381034i
\(36\) 10.6708 1.77847
\(37\) −6.07741 − 0.255019i −0.999121 − 0.0419249i
\(38\) − 0.315328i − 0.0511530i
\(39\) −16.1101 + 2.84065i −2.57969 + 0.454869i
\(40\) 1.64190 − 1.56714i 0.259607 − 0.247787i
\(41\) −4.08538 + 1.48696i −0.638029 + 0.232224i −0.640722 − 0.767773i \(-0.721365\pi\)
0.00269318 + 0.999996i \(0.499143\pi\)
\(42\) 1.31126 + 1.10028i 0.202332 + 0.169777i
\(43\) 3.50333 0.534253 0.267126 − 0.963661i \(-0.413926\pi\)
0.267126 + 0.963661i \(0.413926\pi\)
\(44\) −8.64866 − 7.25708i −1.30383 − 1.09405i
\(45\) −1.36164 + 12.2659i −0.202982 + 1.82850i
\(46\) 0.456823 + 0.166270i 0.0673549 + 0.0245152i
\(47\) 3.40089 + 1.96351i 0.496071 + 0.286407i 0.727090 − 0.686542i \(-0.240872\pi\)
−0.231018 + 0.972949i \(0.574206\pi\)
\(48\) −9.11208 + 5.26086i −1.31522 + 0.759340i
\(49\) 0.318744 + 1.80769i 0.0455348 + 0.258241i
\(50\) 0.782471 + 1.02599i 0.110658 + 0.145096i
\(51\) −2.04753 − 1.18214i −0.286712 − 0.165533i
\(52\) 8.30091 − 6.96529i 1.15113 − 0.965911i
\(53\) −1.08785 − 1.29645i −0.149428 − 0.178082i 0.686138 − 0.727471i \(-0.259305\pi\)
−0.835566 + 0.549390i \(0.814860\pi\)
\(54\) −0.648987 + 1.78308i −0.0883159 + 0.242646i
\(55\) 9.44551 − 9.01546i 1.27363 − 1.21564i
\(56\) −2.27173 − 0.400567i −0.303572 − 0.0535280i
\(57\) −3.35137 − 1.21980i −0.443900 − 0.161567i
\(58\) −0.616314 − 0.734495i −0.0809260 − 0.0964439i
\(59\) 0.225926 + 0.269249i 0.0294131 + 0.0350532i 0.780550 − 0.625093i \(-0.214939\pi\)
−0.751137 + 0.660146i \(0.770494\pi\)
\(60\) −5.06501 − 11.5573i −0.653890 − 1.49204i
\(61\) −4.51727 − 12.4111i −0.578377 − 1.58908i −0.790916 − 0.611925i \(-0.790395\pi\)
0.212539 − 0.977153i \(-0.431827\pi\)
\(62\) 2.01742 − 0.355726i 0.256213 − 0.0451773i
\(63\) 10.8622 − 6.27128i 1.36851 − 0.790107i
\(64\) 3.22287 − 5.58218i 0.402859 − 0.697772i
\(65\) 6.94725 + 10.4306i 0.861701 + 1.29375i
\(66\) 3.80914 − 2.19921i 0.468872 − 0.270704i
\(67\) −6.39559 + 7.62196i −0.781345 + 0.931171i −0.998993 − 0.0448564i \(-0.985717\pi\)
0.217648 + 0.976027i \(0.430161\pi\)
\(68\) 1.56611 0.189919
\(69\) 3.53431 − 4.21202i 0.425480 − 0.507068i
\(70\) 0.368812 − 1.25843i 0.0440814 − 0.150411i
\(71\) −1.24711 + 7.07270i −0.148004 + 0.839375i 0.816901 + 0.576778i \(0.195690\pi\)
−0.964906 + 0.262597i \(0.915421\pi\)
\(72\) −0.972834 − 5.51721i −0.114650 − 0.650210i
\(73\) − 1.41548i − 0.165669i −0.996563 − 0.0828345i \(-0.973603\pi\)
0.996563 − 0.0828345i \(-0.0263973\pi\)
\(74\) 0.207530 + 1.55595i 0.0241249 + 0.180876i
\(75\) 13.9313 − 4.34739i 1.60864 − 0.501993i
\(76\) 2.32655 − 0.410233i 0.266873 − 0.0470570i
\(77\) −13.0688 − 2.30438i −1.48932 − 0.262608i
\(78\) 1.44386 + 3.96697i 0.163485 + 0.449171i
\(79\) −3.62772 + 4.32335i −0.408151 + 0.486415i −0.930487 − 0.366324i \(-0.880616\pi\)
0.522336 + 0.852739i \(0.325061\pi\)
\(80\) 6.49348 + 4.77595i 0.725993 + 0.533968i
\(81\) 3.75657 + 3.15213i 0.417396 + 0.350237i
\(82\) 0.560972 + 0.971633i 0.0619490 + 0.107299i
\(83\) −1.01957 + 2.80125i −0.111913 + 0.307477i −0.982987 − 0.183673i \(-0.941201\pi\)
0.871075 + 0.491150i \(0.163423\pi\)
\(84\) −6.41214 + 11.1061i −0.699621 + 1.21178i
\(85\) −0.199843 + 1.80022i −0.0216760 + 0.195262i
\(86\) −0.156992 − 0.890343i −0.0169288 − 0.0960082i
\(87\) −10.1905 + 3.70903i −1.09253 + 0.397650i
\(88\) −2.96371 + 5.13329i −0.315932 + 0.547211i
\(89\) 8.46364 + 10.0866i 0.897144 + 1.06918i 0.997244 + 0.0741976i \(0.0236396\pi\)
−0.100099 + 0.994977i \(0.531916\pi\)
\(90\) 3.17830 − 0.203611i 0.335023 − 0.0214625i
\(91\) 4.35624 − 11.9687i 0.456658 − 1.25466i
\(92\) −0.632456 + 3.58684i −0.0659381 + 0.373954i
\(93\) 4.02337 − 22.8177i 0.417204 − 2.36608i
\(94\) 0.346608 − 0.952299i 0.0357499 − 0.0982221i
\(95\) 0.174679 + 2.72668i 0.0179217 + 0.279751i
\(96\) 5.55415 + 6.61918i 0.566868 + 0.675567i
\(97\) −0.177750 + 0.307871i −0.0180477 + 0.0312596i −0.874908 − 0.484289i \(-0.839078\pi\)
0.856860 + 0.515548i \(0.172412\pi\)
\(98\) 0.445125 − 0.162012i 0.0449644 − 0.0163657i
\(99\) −5.59651 − 31.7394i −0.562470 − 3.18993i
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.v.a.139.7 yes 96
5.2 odd 4 925.2.bb.e.176.10 96
5.3 odd 4 925.2.bb.e.176.7 96
5.4 even 2 inner 185.2.v.a.139.10 yes 96
37.4 even 18 inner 185.2.v.a.4.10 yes 96
185.4 even 18 inner 185.2.v.a.4.7 ✓ 96
185.78 odd 36 925.2.bb.e.226.7 96
185.152 odd 36 925.2.bb.e.226.10 96
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
185.2.v.a.4.7 ✓ 96 185.4 even 18 inner
185.2.v.a.4.10 yes 96 37.4 even 18 inner
185.2.v.a.139.7 yes 96 1.1 even 1 trivial
185.2.v.a.139.10 yes 96 5.4 even 2 inner
925.2.bb.e.176.7 96 5.3 odd 4
925.2.bb.e.176.10 96 5.2 odd 4
925.2.bb.e.226.7 96 185.78 odd 36
925.2.bb.e.226.10 96 185.152 odd 36