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Properties

Label 36.3.f.c.31.1
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 31.1
Root \(1.93353 + 0.511345i\) of defining polynomial
Character \(\chi\) \(=\) 36.31
Dual form 36.3.f.c.7.1

$q$-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+(-1.93353 - 0.511345i) q^{2} +(-2.76570 + 1.16229i) q^{3} +(3.47705 + 1.97740i) q^{4} +(-4.03104 + 6.98197i) q^{5} +(5.94188 - 0.833101i) q^{6} +(-3.90254 + 2.25313i) q^{7} +(-5.71184 - 5.60133i) q^{8} +(6.29815 - 6.42910i) q^{9} +(11.3643 - 11.4386i) q^{10} +(3.25842 - 1.88125i) q^{11} +(-11.9148 - 1.42753i) q^{12} +(-3.52605 + 6.10730i) q^{13} +(8.69780 - 2.36095i) q^{14} +(3.03354 - 23.9953i) q^{15} +(8.17979 + 13.7510i) q^{16} +0.517890 q^{17} +(-15.4651 + 9.21031i) q^{18} +16.4164i q^{19} +(-27.8223 + 16.3057i) q^{20} +(8.17444 - 10.7674i) q^{21} +(-7.26222 + 1.97127i) q^{22} +(27.7049 + 15.9954i) q^{23} +(22.3076 + 8.85273i) q^{24} +(-19.9986 - 34.6387i) q^{25} +(9.94065 - 10.0056i) q^{26} +(-9.94627 + 25.1012i) q^{27} +(-18.0247 + 0.117384i) q^{28} +(9.48394 + 16.4267i) q^{29} +(-18.1353 + 44.8443i) q^{30} +(-13.1355 - 7.58377i) q^{31} +(-8.78432 - 30.7707i) q^{32} +(-6.82524 + 8.99021i) q^{33} +(-1.00135 - 0.264820i) q^{34} -36.3299i q^{35} +(34.6119 - 9.90037i) q^{36} +0.592061 q^{37} +(8.39446 - 31.7416i) q^{38} +(2.65351 - 20.9892i) q^{39} +(62.1330 - 17.3007i) q^{40} +(12.3766 - 21.4369i) q^{41} +(-21.3114 + 16.6391i) q^{42} +(27.8686 - 16.0900i) q^{43} +(15.0497 - 0.0980099i) q^{44} +(19.4997 + 69.8895i) q^{45} +(-45.3890 - 45.0944i) q^{46} +(-52.4682 + 30.2925i) q^{47} +(-38.6056 - 28.5239i) q^{48} +(-14.3468 + 24.8493i) q^{49} +(20.9556 + 77.2010i) q^{50} +(-1.43233 + 0.601940i) q^{51} +(-24.3368 + 14.2630i) q^{52} -0.664765 q^{53} +(32.0668 - 43.4479i) q^{54} +30.3336i q^{55} +(34.9113 + 8.98987i) q^{56} +(-19.0807 - 45.4029i) q^{57} +(-9.93776 - 36.6110i) q^{58} +(-30.5921 - 17.6623i) q^{59} +(57.9960 - 77.4343i) q^{60} +(33.7750 + 58.5000i) q^{61} +(21.5199 + 21.3802i) q^{62} +(-10.0932 + 39.2804i) q^{63} +(1.25029 + 63.9878i) q^{64} +(-28.4273 - 49.2376i) q^{65} +(17.7939 - 13.8928i) q^{66} +(-74.4692 - 42.9948i) q^{67} +(1.80073 + 1.02407i) q^{68} +(-95.2148 - 12.0373i) q^{69} +(-18.5771 + 70.2449i) q^{70} +56.4434i q^{71} +(-71.9855 + 1.44402i) q^{72} +131.921 q^{73} +(-1.14477 - 0.302748i) q^{74} +(95.5705 + 72.5557i) q^{75} +(-32.4618 + 57.0808i) q^{76} +(-8.47743 + 14.6833i) q^{77} +(-15.8634 + 39.2264i) q^{78} +(126.869 - 73.2481i) q^{79} +(-128.982 + 1.68005i) q^{80} +(-1.66664 - 80.9829i) q^{81} +(-34.8921 + 35.1201i) q^{82} +(87.1029 - 50.2889i) q^{83} +(49.7144 - 21.2746i) q^{84} +(-2.08764 + 3.61589i) q^{85} +(-62.1122 + 16.8599i) q^{86} +(-45.3223 - 34.4081i) q^{87} +(-29.1491 - 7.50608i) q^{88} -25.8362 q^{89} +(-1.96553 - 145.104i) q^{90} -31.7786i q^{91} +(64.7021 + 110.401i) q^{92} +(45.1433 + 5.70713i) q^{93} +(116.939 - 31.7420i) q^{94} +(-114.619 - 66.1754i) q^{95} +(60.0593 + 74.8924i) q^{96} +(-48.2534 - 83.5773i) q^{97} +(40.4465 - 40.7107i) q^{98} +(8.42728 - 32.7971i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.93353 − 0.511345i −0.966763 − 0.255672i
\(3\) −2.76570 + 1.16229i −0.921899 + 0.387431i
\(4\) 3.47705 + 1.97740i 0.869263 + 0.494350i
\(5\) −4.03104 + 6.98197i −0.806209 + 1.39639i 0.109263 + 0.994013i \(0.465151\pi\)
−0.915472 + 0.402382i \(0.868182\pi\)
\(6\) 5.94188 − 0.833101i 0.990313 − 0.138850i
\(7\) −3.90254 + 2.25313i −0.557506 + 0.321876i −0.752144 − 0.658999i \(-0.770980\pi\)
0.194638 + 0.980875i \(0.437647\pi\)
\(8\) −5.71184 − 5.60133i −0.713980 − 0.700166i
\(9\) 6.29815 − 6.42910i 0.699794 − 0.714344i
\(10\) 11.3643 − 11.4386i 1.13643 − 1.14386i
\(11\) 3.25842 − 1.88125i 0.296220 − 0.171023i −0.344523 − 0.938778i \(-0.611959\pi\)
0.640744 + 0.767755i \(0.278626\pi\)
\(12\) −11.9148 − 1.42753i −0.992899 − 0.118961i
\(13\) −3.52605 + 6.10730i −0.271235 + 0.469792i −0.969178 − 0.246361i \(-0.920765\pi\)
0.697944 + 0.716153i \(0.254099\pi\)
\(14\) 8.69780 − 2.36095i 0.621271 − 0.168639i
\(15\) 3.03354 − 23.9953i 0.202236 − 1.59968i
\(16\) 8.17979 + 13.7510i 0.511237 + 0.859440i
\(17\) 0.517890 0.0304641 0.0152321 − 0.999884i \(-0.495151\pi\)
0.0152321 + 0.999884i \(0.495151\pi\)
\(18\) −15.4651 + 9.21031i −0.859174 + 0.511684i
\(19\) 16.4164i 0.864023i 0.901868 + 0.432012i \(0.142196\pi\)
−0.901868 + 0.432012i \(0.857804\pi\)
\(20\) −27.8223 + 16.3057i −1.39111 + 0.815286i
\(21\) 8.17444 − 10.7674i 0.389259 − 0.512733i
\(22\) −7.26222 + 1.97127i −0.330101 + 0.0896033i
\(23\) 27.7049 + 15.9954i 1.20456 + 0.695454i 0.961566 − 0.274573i \(-0.0885366\pi\)
0.242996 + 0.970027i \(0.421870\pi\)
\(24\) 22.3076 + 8.85273i 0.929483 + 0.368864i
\(25\) −19.9986 − 34.6387i −0.799946 − 1.38555i
\(26\) 9.94065 − 10.0056i 0.382333 − 0.384831i
\(27\) −9.94627 + 25.1012i −0.368380 + 0.929675i
\(28\) −18.0247 + 0.117384i −0.643739 + 0.00419230i
\(29\) 9.48394 + 16.4267i 0.327032 + 0.566437i 0.981922 − 0.189288i \(-0.0606180\pi\)
−0.654889 + 0.755725i \(0.727285\pi\)
\(30\) −18.1353 + 44.8443i −0.604510 + 1.49481i
\(31\) −13.1355 − 7.58377i −0.423725 − 0.244638i 0.272945 − 0.962030i \(-0.412002\pi\)
−0.696670 + 0.717392i \(0.745336\pi\)
\(32\) −8.78432 − 30.7707i −0.274510 − 0.961584i
\(33\) −6.82524 + 8.99021i −0.206826 + 0.272431i
\(34\) −1.00135 − 0.264820i −0.0294516 − 0.00778884i
\(35\) − 36.3299i − 1.03800i
\(36\) 34.6119 − 9.90037i 0.961441 − 0.275010i
\(37\) 0.592061 0.0160017 0.00800083 − 0.999968i \(-0.497453\pi\)
0.00800083 + 0.999968i \(0.497453\pi\)
\(38\) 8.39446 − 31.7416i 0.220907 − 0.835306i
\(39\) 2.65351 − 20.9892i 0.0680387 − 0.538185i
\(40\) 62.1330 − 17.3007i 1.55333 − 0.432518i
\(41\) 12.3766 − 21.4369i 0.301868 − 0.522850i −0.674691 − 0.738100i \(-0.735723\pi\)
0.976559 + 0.215250i \(0.0690565\pi\)
\(42\) −21.3114 + 16.6391i −0.507413 + 0.396168i
\(43\) 27.8686 − 16.0900i 0.648107 − 0.374185i −0.139623 − 0.990205i \(-0.544589\pi\)
0.787731 + 0.616020i \(0.211256\pi\)
\(44\) 15.0497 − 0.0980099i 0.342039 − 0.00222750i
\(45\) 19.4997 + 69.8895i 0.433326 + 1.55310i
\(46\) −45.3890 − 45.0944i −0.986718 − 0.980313i
\(47\) −52.4682 + 30.2925i −1.11634 + 0.644521i −0.940465 − 0.339890i \(-0.889610\pi\)
−0.175879 + 0.984412i \(0.556277\pi\)
\(48\) −38.6056 − 28.5239i −0.804282 − 0.594247i
\(49\) −14.3468 + 24.8493i −0.292791 + 0.507129i
\(50\) 20.9556 + 77.2010i 0.419112 + 1.54402i
\(51\) −1.43233 + 0.601940i −0.0280848 + 0.0118027i
\(52\) −24.3368 + 14.2630i −0.468016 + 0.274288i
\(53\) −0.664765 −0.0125427 −0.00627137 − 0.999980i \(-0.501996\pi\)
−0.00627137 + 0.999980i \(0.501996\pi\)
\(54\) 32.0668 − 43.4479i 0.593829 − 0.804591i
\(55\) 30.3336i 0.551521i
\(56\) 34.9113 + 8.98987i 0.623415 + 0.160533i
\(57\) −19.0807 − 45.4029i −0.334749 − 0.796542i
\(58\) −9.93776 − 36.6110i −0.171341 − 0.631224i
\(59\) −30.5921 − 17.6623i −0.518510 − 0.299362i 0.217815 − 0.975990i \(-0.430107\pi\)
−0.736325 + 0.676628i \(0.763440\pi\)
\(60\) 57.9960 − 77.4343i 0.966600 − 1.29057i
\(61\) 33.7750 + 58.5000i 0.553688 + 0.959016i 0.998004 + 0.0631460i \(0.0201134\pi\)
−0.444316 + 0.895870i \(0.646553\pi\)
\(62\) 21.5199 + 21.3802i 0.347095 + 0.344842i
\(63\) −10.0932 + 39.2804i −0.160209 + 0.623499i
\(64\) 1.25029 + 63.9878i 0.0195357 + 0.999809i
\(65\) −28.4273 − 49.2376i −0.437343 − 0.757501i
\(66\) 17.7939 − 13.8928i 0.269604 − 0.210496i
\(67\) −74.4692 − 42.9948i −1.11148 − 0.641714i −0.172269 − 0.985050i \(-0.555110\pi\)
−0.939213 + 0.343336i \(0.888443\pi\)
\(68\) 1.80073 + 1.02407i 0.0264813 + 0.0150599i
\(69\) −95.2148 − 12.0373i −1.37992 − 0.174454i
\(70\) −18.5771 + 70.2449i −0.265388 + 1.00350i
\(71\) 56.4434i 0.794977i 0.917607 + 0.397489i \(0.130118\pi\)
−0.917607 + 0.397489i \(0.869882\pi\)
\(72\) −71.9855 + 1.44402i −0.999799 + 0.0200558i
\(73\) 131.921 1.80713 0.903567 − 0.428447i \(-0.140939\pi\)
0.903567 + 0.428447i \(0.140939\pi\)
\(74\) −1.14477 − 0.302748i −0.0154698 − 0.00409118i
\(75\) 95.5705 + 72.5557i 1.27427 + 0.967410i
\(76\) −32.4618 + 57.0808i −0.427129 + 0.751063i
\(77\) −8.47743 + 14.6833i −0.110096 + 0.190693i
\(78\) −15.8634 + 39.2264i −0.203377 + 0.502902i
\(79\) 126.869 − 73.2481i 1.60594 − 0.927191i 0.615677 − 0.787999i \(-0.288883\pi\)
0.990265 − 0.139192i \(-0.0444505\pi\)
\(80\) −128.982 + 1.68005i −1.61228 + 0.0210006i
\(81\) −1.66664 − 80.9829i −0.0205758 − 0.999788i
\(82\) −34.8921 + 35.1201i −0.425513 + 0.428293i
\(83\) 87.1029 − 50.2889i 1.04943 − 0.605890i 0.126942 − 0.991910i \(-0.459484\pi\)
0.922491 + 0.386020i \(0.126150\pi\)
\(84\) 49.7144 − 21.2746i 0.591838 − 0.253269i
\(85\) −2.08764 + 3.61589i −0.0245604 + 0.0425399i
\(86\) −62.1122 + 16.8599i −0.722235 + 0.196045i
\(87\) −45.3223 − 34.4081i −0.520946 − 0.395495i
\(88\) −29.1491 − 7.50608i −0.331240 − 0.0852964i
\(89\) −25.8362 −0.290295 −0.145147 − 0.989410i \(-0.546366\pi\)
−0.145147 + 0.989410i \(0.546366\pi\)
\(90\) −1.96553 − 145.104i −0.0218392 − 1.61227i
\(91\) − 31.7786i − 0.349216i
\(92\) 64.7021 + 110.401i 0.703284 + 1.20001i
\(93\) 45.1433 + 5.70713i 0.485412 + 0.0613670i
\(94\) 116.939 − 31.7420i 1.24403 − 0.337681i
\(95\) −114.619 − 66.1754i −1.20652 − 0.696583i
\(96\) 60.0593 + 74.8924i 0.625618 + 0.780129i
\(97\) −48.2534 − 83.5773i −0.497457 − 0.861621i 0.502538 − 0.864555i \(-0.332400\pi\)
−0.999996 + 0.00293363i \(0.999066\pi\)
\(98\) 40.4465 − 40.7107i 0.412719 − 0.415415i
\(99\) 8.42728 − 32.7971i 0.0851241 − 0.331284i
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.31.1 yes 16
3.2 odd 2 108.3.f.c.91.8 16
4.3 odd 2 inner 36.3.f.c.31.5 yes 16
8.3 odd 2 576.3.o.g.319.1 16
8.5 even 2 576.3.o.g.319.8 16
9.2 odd 6 108.3.f.c.19.4 16
9.4 even 3 324.3.d.i.163.5 8
9.5 odd 6 324.3.d.g.163.4 8
9.7 even 3 inner 36.3.f.c.7.5 yes 16
12.11 even 2 108.3.f.c.91.4 16
24.5 odd 2 1728.3.o.g.1279.1 16
24.11 even 2 1728.3.o.g.1279.2 16
36.7 odd 6 inner 36.3.f.c.7.1 ✓ 16
36.11 even 6 108.3.f.c.19.8 16
36.23 even 6 324.3.d.g.163.3 8
36.31 odd 6 324.3.d.i.163.6 8
72.11 even 6 1728.3.o.g.127.1 16
72.29 odd 6 1728.3.o.g.127.2 16
72.43 odd 6 576.3.o.g.511.8 16
72.61 even 6 576.3.o.g.511.1 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
36.3.f.c.7.1 ✓ 16 36.7 odd 6 inner
36.3.f.c.7.5 yes 16 9.7 even 3 inner
36.3.f.c.31.1 yes 16 1.1 even 1 trivial
36.3.f.c.31.5 yes 16 4.3 odd 2 inner
108.3.f.c.19.4 16 9.2 odd 6
108.3.f.c.19.8 16 36.11 even 6
108.3.f.c.91.4 16 12.11 even 2
108.3.f.c.91.8 16 3.2 odd 2
324.3.d.g.163.3 8 36.23 even 6
324.3.d.g.163.4 8 9.5 odd 6
324.3.d.i.163.5 8 9.4 even 3
324.3.d.i.163.6 8 36.31 odd 6
576.3.o.g.319.1 16 8.3 odd 2
576.3.o.g.319.8 16 8.5 even 2
576.3.o.g.511.1 16 72.61 even 6
576.3.o.g.511.8 16 72.43 odd 6
1728.3.o.g.127.1 16 72.11 even 6
1728.3.o.g.127.2 16 72.29 odd 6
1728.3.o.g.1279.1 16 24.5 odd 2
1728.3.o.g.1279.2 16 24.11 even 2