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Properties

Label 36.3.f.c.31.6
Level $36$
Weight $3$
Character 36.31
Analytic conductor $0.981$
Analytic rank $0$
Dimension $16$
Inner twists $4$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [36,3,Mod(7,36)] 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("36.7"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(36, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([3, 4])) N = Newforms(chi, 3, names="a")
 
Level: \( N \) \(=\) \( 36 = 2^{2} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 36.f (of order \(6\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [16,-3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(0.980928951697\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 3 x^{15} + 7 x^{14} - 30 x^{13} + 76 x^{12} - 144 x^{11} + 424 x^{10} - 912 x^{9} + \cdots + 65536 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{8}\cdot 3^{3} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 31.6
Root \(-0.710719 - 1.86946i\) of defining polynomial
Character \(\chi\) \(=\) 36.31
Dual form 36.3.f.c.7.6

$q$-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+(0.710719 + 1.86946i) q^{2} +(2.32245 - 1.89900i) q^{3} +(-2.98976 + 2.65732i) q^{4} +(1.35609 - 2.34881i) q^{5} +(5.20072 + 2.99207i) q^{6} +(-10.0431 + 5.79837i) q^{7} +(-7.09263 - 3.70062i) q^{8} +(1.78756 - 8.82069i) q^{9} +(5.35481 + 0.865806i) q^{10} +(8.54822 - 4.93532i) q^{11} +(-1.89731 + 11.8491i) q^{12} +(0.296185 - 0.513008i) q^{13} +(-17.9776 - 14.6541i) q^{14} +(-1.31096 - 8.03023i) q^{15} +(1.87730 - 15.8895i) q^{16} -8.87968 q^{17} +(17.7604 - 2.92725i) q^{18} +14.0989i q^{19} +(2.18718 + 10.6259i) q^{20} +(-12.3134 + 32.5383i) q^{21} +(15.3018 + 12.4729i) q^{22} +(18.2754 + 10.5513i) q^{23} +(-23.4998 + 4.87441i) q^{24} +(8.82205 + 15.2802i) q^{25} +(1.16955 + 0.189102i) q^{26} +(-12.5990 - 23.8802i) q^{27} +(14.6182 - 44.0234i) q^{28} +(10.1764 + 17.6260i) q^{29} +(14.0805 - 8.15802i) q^{30} +(-14.3357 - 8.27670i) q^{31} +(31.0390 - 7.78342i) q^{32} +(10.4806 - 27.6952i) q^{33} +(-6.31095 - 16.6002i) q^{34} +31.4524i q^{35} +(18.0950 + 31.1219i) q^{36} -40.6557 q^{37} +(-26.3573 + 10.0203i) q^{38} +(-0.286328 - 1.75389i) q^{39} +(-18.3103 + 11.6409i) q^{40} +(21.2177 - 36.7502i) q^{41} +(-69.5804 + 0.106123i) q^{42} +(32.2385 - 18.6129i) q^{43} +(-12.4424 + 37.4708i) q^{44} +(-18.2941 - 16.1603i) q^{45} +(-6.73658 + 41.6642i) q^{46} +(-1.57134 + 0.907211i) q^{47} +(-25.8143 - 40.4676i) q^{48} +(42.7423 - 74.0318i) q^{49} +(-22.2958 + 27.3524i) q^{50} +(-20.6226 + 16.8625i) q^{51} +(0.477704 + 2.32083i) q^{52} -21.1005 q^{53} +(35.6888 - 40.5254i) q^{54} -26.7709i q^{55} +(92.6894 - 3.96007i) q^{56} +(26.7738 + 32.7440i) q^{57} +(-25.7186 + 31.5515i) q^{58} +(-76.6879 - 44.2758i) q^{59} +(25.2583 + 20.5248i) q^{60} +(36.4925 + 63.2069i) q^{61} +(5.28433 - 32.6823i) q^{62} +(33.1930 + 98.9519i) q^{63} +(36.6108 + 52.4943i) q^{64} +(-0.803307 - 1.39137i) q^{65} +(59.2238 - 0.0903273i) q^{66} +(38.3110 + 22.1189i) q^{67} +(26.5481 - 23.5961i) q^{68} +(62.4808 - 10.2002i) q^{69} +(-58.7990 + 22.3538i) q^{70} -111.798i q^{71} +(-45.3206 + 55.9468i) q^{72} -76.2003 q^{73} +(-28.8948 - 76.0042i) q^{74} +(49.5060 + 18.7345i) q^{75} +(-37.4652 - 42.1522i) q^{76} +(-57.2337 + 99.1316i) q^{77} +(3.07534 - 1.78181i) q^{78} +(8.30434 - 4.79451i) q^{79} +(-34.7757 - 25.9570i) q^{80} +(-74.6092 - 31.5351i) q^{81} +(83.7828 + 13.5466i) q^{82} +(-73.6244 + 42.5070i) q^{83} +(-49.6505 - 130.002i) q^{84} +(-12.0416 + 20.8567i) q^{85} +(57.7086 + 47.0400i) q^{86} +(57.1060 + 21.6106i) q^{87} +(-78.8931 + 3.37063i) q^{88} +64.7845 q^{89} +(17.2091 - 45.6855i) q^{90} +6.86958i q^{91} +(-82.6773 + 17.0178i) q^{92} +(-49.0114 + 8.00125i) q^{93} +(-2.81277 - 2.29278i) q^{94} +(33.1157 + 19.1193i) q^{95} +(57.3058 - 77.0198i) q^{96} +(-3.59139 - 6.22047i) q^{97} +(168.777 + 27.2892i) q^{98} +(-28.2524 - 84.2235i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 3 q^{2} - 5 q^{4} + 6 q^{5} + 9 q^{6} - 54 q^{8} + 18 q^{9} + 20 q^{10} - 36 q^{12} - 46 q^{13} - 12 q^{14} - 17 q^{16} + 12 q^{17} + 48 q^{18} + 36 q^{20} - 66 q^{21} + 33 q^{22} + 129 q^{24} - 30 q^{25}+ \cdots + 1170 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/36\mathbb{Z}\right)^\times\).

\(n\) \(19\) \(29\)
\(\chi(n)\) \(-1\) \(e\left(\frac{1}{3}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.710719 + 1.86946i 0.355359 + 0.934730i
\(3\) 2.32245 − 1.89900i 0.774151 − 0.633001i
\(4\) −2.98976 + 2.65732i −0.747439 + 0.664330i
\(5\) 1.35609 − 2.34881i 0.271218 − 0.469763i −0.697956 − 0.716140i \(-0.745907\pi\)
0.969174 + 0.246378i \(0.0792403\pi\)
\(6\) 5.20072 + 2.99207i 0.866787 + 0.498679i
\(7\) −10.0431 + 5.79837i −1.43473 + 0.828339i −0.997476 − 0.0710013i \(-0.977381\pi\)
−0.437249 + 0.899340i \(0.644047\pi\)
\(8\) −7.09263 − 3.70062i −0.886579 − 0.462578i
\(9\) 1.78756 − 8.82069i 0.198618 − 0.980077i
\(10\) 5.35481 + 0.865806i 0.535481 + 0.0865806i
\(11\) 8.54822 − 4.93532i 0.777111 − 0.448665i −0.0582943 − 0.998299i \(-0.518566\pi\)
0.835406 + 0.549634i \(0.185233\pi\)
\(12\) −1.89731 + 11.8491i −0.158109 + 0.987422i
\(13\) 0.296185 − 0.513008i 0.0227835 − 0.0394622i −0.854409 − 0.519601i \(-0.826080\pi\)
0.877192 + 0.480139i \(0.159414\pi\)
\(14\) −17.9776 − 14.6541i −1.28412 − 1.04672i
\(15\) −1.31096 − 8.03023i −0.0873972 − 0.535348i
\(16\) 1.87730 − 15.8895i 0.117331 − 0.993093i
\(17\) −8.87968 −0.522334 −0.261167 − 0.965294i \(-0.584107\pi\)
−0.261167 + 0.965294i \(0.584107\pi\)
\(18\) 17.7604 − 2.92725i 0.986688 − 0.162625i
\(19\) 14.0989i 0.742046i 0.928624 + 0.371023i \(0.120993\pi\)
−0.928624 + 0.371023i \(0.879007\pi\)
\(20\) 2.18718 + 10.6259i 0.109359 + 0.531297i
\(21\) −12.3134 + 32.5383i −0.586354 + 1.54944i
\(22\) 15.3018 + 12.4729i 0.695535 + 0.566952i
\(23\) 18.2754 + 10.5513i 0.794583 + 0.458753i 0.841574 − 0.540142i \(-0.181630\pi\)
−0.0469902 + 0.998895i \(0.514963\pi\)
\(24\) −23.4998 + 4.87441i −0.979158 + 0.203101i
\(25\) 8.82205 + 15.2802i 0.352882 + 0.611209i
\(26\) 1.16955 + 0.189102i 0.0449828 + 0.00727316i
\(27\) −12.5990 − 23.8802i −0.466630 − 0.884453i
\(28\) 14.6182 − 44.0234i 0.522080 − 1.57226i
\(29\) 10.1764 + 17.6260i 0.350910 + 0.607793i 0.986409 − 0.164308i \(-0.0525391\pi\)
−0.635499 + 0.772101i \(0.719206\pi\)
\(30\) 14.0805 − 8.15802i 0.469349 − 0.271934i
\(31\) −14.3357 − 8.27670i −0.462441 − 0.266990i 0.250629 − 0.968083i \(-0.419362\pi\)
−0.713070 + 0.701093i \(0.752696\pi\)
\(32\) 31.0390 − 7.78342i 0.969968 − 0.243232i
\(33\) 10.4806 − 27.6952i 0.317595 − 0.839247i
\(34\) −6.31095 − 16.6002i −0.185616 − 0.488241i
\(35\) 31.4524i 0.898641i
\(36\) 18.0950 + 31.1219i 0.502639 + 0.864496i
\(37\) −40.6557 −1.09880 −0.549401 − 0.835559i \(-0.685144\pi\)
−0.549401 + 0.835559i \(0.685144\pi\)
\(38\) −26.3573 + 10.0203i −0.693613 + 0.263693i
\(39\) −0.286328 − 1.75389i −0.00734176 − 0.0449716i
\(40\) −18.3103 + 11.6409i −0.457758 + 0.291022i
\(41\) 21.2177 − 36.7502i 0.517506 − 0.896346i −0.482288 − 0.876013i \(-0.660194\pi\)
0.999793 − 0.0203330i \(-0.00647263\pi\)
\(42\) −69.5804 + 0.106123i −1.65668 + 0.00252674i
\(43\) 32.2385 − 18.6129i 0.749732 − 0.432858i −0.0758649 − 0.997118i \(-0.524172\pi\)
0.825597 + 0.564260i \(0.190838\pi\)
\(44\) −12.4424 + 37.4708i −0.282782 + 0.851609i
\(45\) −18.2941 − 16.1603i −0.406535 − 0.359118i
\(46\) −6.73658 + 41.6642i −0.146447 + 0.905743i
\(47\) −1.57134 + 0.907211i −0.0334327 + 0.0193024i −0.516623 − 0.856213i \(-0.672811\pi\)
0.483191 + 0.875515i \(0.339478\pi\)
\(48\) −25.8143 − 40.4676i −0.537797 − 0.843074i
\(49\) 42.7423 − 74.0318i 0.872291 − 1.51085i
\(50\) −22.2958 + 27.3524i −0.445916 + 0.547048i
\(51\) −20.6226 + 16.8625i −0.404365 + 0.330638i
\(52\) 0.477704 + 2.32083i 0.00918662 + 0.0446313i
\(53\) −21.1005 −0.398122 −0.199061 − 0.979987i \(-0.563789\pi\)
−0.199061 + 0.979987i \(0.563789\pi\)
\(54\) 35.6888 − 40.5254i 0.660903 − 0.750471i
\(55\) − 26.7709i − 0.486744i
\(56\) 92.6894 − 3.96007i 1.65517 − 0.0707155i
\(57\) 26.7738 + 32.7440i 0.469716 + 0.574456i
\(58\) −25.7186 + 31.5515i −0.443423 + 0.543991i
\(59\) −76.6879 − 44.2758i −1.29980 − 0.750437i −0.319427 − 0.947611i \(-0.603490\pi\)
−0.980369 + 0.197174i \(0.936824\pi\)
\(60\) 25.2583 + 20.5248i 0.420972 + 0.342080i
\(61\) 36.4925 + 63.2069i 0.598238 + 1.03618i 0.993081 + 0.117431i \(0.0374657\pi\)
−0.394843 + 0.918749i \(0.629201\pi\)
\(62\) 5.28433 − 32.6823i 0.0852311 − 0.527134i
\(63\) 33.1930 + 98.9519i 0.526873 + 1.57066i
\(64\) 36.6108 + 52.4943i 0.572043 + 0.820223i
\(65\) −0.803307 − 1.39137i −0.0123586 − 0.0214057i
\(66\) 59.2238 − 0.0903273i 0.897330 − 0.00136860i
\(67\) 38.3110 + 22.1189i 0.571807 + 0.330133i 0.757871 − 0.652405i \(-0.226240\pi\)
−0.186064 + 0.982538i \(0.559573\pi\)
\(68\) 26.5481 − 23.5961i 0.390413 − 0.347002i
\(69\) 62.4808 − 10.2002i 0.905519 − 0.147829i
\(70\) −58.7990 + 22.3538i −0.839986 + 0.319341i
\(71\) − 111.798i − 1.57462i −0.616557 − 0.787310i \(-0.711473\pi\)
0.616557 − 0.787310i \(-0.288527\pi\)
\(72\) −45.3206 + 55.9468i −0.629453 + 0.777039i
\(73\) −76.2003 −1.04384 −0.521920 − 0.852995i \(-0.674784\pi\)
−0.521920 + 0.852995i \(0.674784\pi\)
\(74\) −28.8948 − 76.0042i −0.390470 − 1.02708i
\(75\) 49.5060 + 18.7345i 0.660080 + 0.249793i
\(76\) −37.4652 − 42.1522i −0.492964 − 0.554635i
\(77\) −57.2337 + 99.1316i −0.743294 + 1.28742i
\(78\) 3.07534 − 1.78181i 0.0394274 − 0.0228437i
\(79\) 8.30434 − 4.79451i 0.105118 − 0.0606901i −0.446519 − 0.894774i \(-0.647337\pi\)
0.551637 + 0.834084i \(0.314003\pi\)
\(80\) −34.7757 − 25.9570i −0.434696 − 0.324462i
\(81\) −74.6092 − 31.5351i −0.921102 − 0.389322i
\(82\) 83.7828 + 13.5466i 1.02174 + 0.165203i
\(83\) −73.6244 + 42.5070i −0.887041 + 0.512133i −0.872973 − 0.487768i \(-0.837811\pi\)
−0.0140672 + 0.999901i \(0.504478\pi\)
\(84\) −49.6505 − 130.002i −0.591077 − 1.54765i
\(85\) −12.0416 + 20.8567i −0.141666 + 0.245373i
\(86\) 57.7086 + 47.0400i 0.671030 + 0.546977i
\(87\) 57.1060 + 21.6106i 0.656391 + 0.248397i
\(88\) −78.8931 + 3.37063i −0.896513 + 0.0383026i
\(89\) 64.7845 0.727916 0.363958 − 0.931415i \(-0.381425\pi\)
0.363958 + 0.931415i \(0.381425\pi\)
\(90\) 17.2091 − 45.6855i 0.191212 − 0.507616i
\(91\) 6.86958i 0.0754898i
\(92\) −82.6773 + 17.0178i −0.898666 + 0.184976i
\(93\) −49.0114 + 8.00125i −0.527004 + 0.0860350i
\(94\) −2.81277 − 2.29278i −0.0299231 − 0.0243912i
\(95\) 33.1157 + 19.1193i 0.348586 + 0.201256i
\(96\) 57.3058 − 77.0198i 0.596935 − 0.802289i
\(97\) −3.59139 − 6.22047i −0.0370246 − 0.0641285i 0.846919 − 0.531721i \(-0.178455\pi\)
−0.883944 + 0.467593i \(0.845121\pi\)
\(98\) 168.777 + 27.2892i 1.72222 + 0.278461i
\(99\) −28.2524 − 84.2235i −0.285378 − 0.850742i
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 36.3.f.c.31.6 yes 16
3.2 odd 2 108.3.f.c.91.3 16
4.3 odd 2 inner 36.3.f.c.31.7 yes 16
8.3 odd 2 576.3.o.g.319.6 16
8.5 even 2 576.3.o.g.319.3 16
9.2 odd 6 108.3.f.c.19.2 16
9.4 even 3 324.3.d.i.163.1 8
9.5 odd 6 324.3.d.g.163.8 8
9.7 even 3 inner 36.3.f.c.7.7 yes 16
12.11 even 2 108.3.f.c.91.2 16
24.5 odd 2 1728.3.o.g.1279.5 16
24.11 even 2 1728.3.o.g.1279.6 16
36.7 odd 6 inner 36.3.f.c.7.6 ✓ 16
36.11 even 6 108.3.f.c.19.3 16
36.23 even 6 324.3.d.g.163.7 8
36.31 odd 6 324.3.d.i.163.2 8
72.11 even 6 1728.3.o.g.127.5 16
72.29 odd 6 1728.3.o.g.127.6 16
72.43 odd 6 576.3.o.g.511.3 16
72.61 even 6 576.3.o.g.511.6 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
36.3.f.c.7.6 ✓ 16 36.7 odd 6 inner
36.3.f.c.7.7 yes 16 9.7 even 3 inner
36.3.f.c.31.6 yes 16 1.1 even 1 trivial
36.3.f.c.31.7 yes 16 4.3 odd 2 inner
108.3.f.c.19.2 16 9.2 odd 6
108.3.f.c.19.3 16 36.11 even 6
108.3.f.c.91.2 16 12.11 even 2
108.3.f.c.91.3 16 3.2 odd 2
324.3.d.g.163.7 8 36.23 even 6
324.3.d.g.163.8 8 9.5 odd 6
324.3.d.i.163.1 8 9.4 even 3
324.3.d.i.163.2 8 36.31 odd 6
576.3.o.g.319.3 16 8.5 even 2
576.3.o.g.319.6 16 8.3 odd 2
576.3.o.g.511.3 16 72.43 odd 6
576.3.o.g.511.6 16 72.61 even 6
1728.3.o.g.127.5 16 72.11 even 6
1728.3.o.g.127.6 16 72.29 odd 6
1728.3.o.g.1279.5 16 24.5 odd 2
1728.3.o.g.1279.6 16 24.11 even 2