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Properties

Label 9.8.c.a.4.4
Level $9$
Weight $8$
Character 9.4
Analytic conductor $2.811$
Analytic rank $0$
Dimension $12$
Inner twists $2$

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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] = [9,8,Mod(4,9)] 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("9.4"); S:= CuspForms(chi, 8); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(9, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([2])) N = Newforms(chi, 8, names="a")
 
Level: \( N \) \(=\) \( 9 = 3^{2} \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 9.c (of order \(3\), degree \(2\), 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: \(2.81146522936\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 6 x^{11} + 375 x^{10} - 1820 x^{9} + 50808 x^{8} - 192378 x^{7} + 3002887 x^{6} + \cdots + 754412211 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{8}\cdot 3^{15} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 4.4
Root \(0.500000 + 1.48508i\) of defining polynomial
Character \(\chi\) \(=\) 9.4
Dual form 9.8.c.a.7.4

$q$-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+(0.536120 + 0.928588i) q^{2} +(-22.8679 - 40.7929i) q^{3} +(63.4251 - 109.856i) q^{4} +(47.9866 - 83.1153i) q^{5} +(25.6198 - 43.1048i) q^{6} +(-189.000 - 327.358i) q^{7} +273.261 q^{8} +(-1141.12 + 1865.70i) q^{9} +102.906 q^{10} +(3436.63 + 5952.41i) q^{11} +(-5931.73 - 75.1254i) q^{12} +(4826.64 - 8359.99i) q^{13} +(202.654 - 351.007i) q^{14} +(-4487.87 - 56.8389i) q^{15} +(-7971.92 - 13807.8i) q^{16} +21431.3 q^{17} +(-2344.24 - 59.3891i) q^{18} +5518.94 q^{19} +(-6087.12 - 10543.2i) q^{20} +(-9031.83 + 15195.9i) q^{21} +(-3684.89 + 6382.42i) q^{22} +(-31486.4 + 54536.1i) q^{23} +(-6248.91 - 11147.1i) q^{24} +(34457.1 + 59681.4i) q^{25} +10350.6 q^{26} +(102202. + 3884.83i) q^{27} -47949.5 q^{28} +(-111113. - 192454. i) q^{29} +(-2353.26 - 4197.85i) q^{30} +(-57729.1 + 99989.7i) q^{31} +(26036.5 - 45096.6i) q^{32} +(164228. - 276309. i) q^{33} +(11489.7 + 19900.8i) q^{34} -36277.9 q^{35} +(132582. + 243690. i) q^{36} +81737.7 q^{37} +(2958.82 + 5124.82i) q^{38} +(-451403. - 5717.02i) q^{39} +(13112.9 - 22712.2i) q^{40} +(-298773. + 517491. i) q^{41} +(-18952.8 - 240.038i) q^{42} +(-33874.2 - 58671.8i) q^{43} +871875. q^{44} +(100310. + 184373. i) q^{45} -67522.1 q^{46} +(-151740. - 262822. i) q^{47} +(-380957. + 640952. i) q^{48} +(340329. - 589468. i) q^{49} +(-36946.3 + 63992.8i) q^{50} +(-490089. - 874243. i) q^{51} +(-612261. - 1.06047e6i) q^{52} +846755. q^{53} +(51185.2 + 96986.3i) q^{54} +659649. q^{55} +(-51646.4 - 89454.2i) q^{56} +(-126207. - 225134. i) q^{57} +(119140. - 206357. i) q^{58} +(-793119. + 1.37372e6i) q^{59} +(-290888. + 489412. i) q^{60} +(1.12706e6 + 1.95213e6i) q^{61} -123799. q^{62} +(826422. + 20936.6i) q^{63} -1.98498e6 q^{64} +(-463228. - 802335. i) q^{65} +(344623. + 4364.66i) q^{66} +(-1.51172e6 + 2.61838e6i) q^{67} +(1.35928e6 - 2.35434e6i) q^{68} +(2.94471e6 + 37294.8i) q^{69} +(-19449.3 - 33687.3i) q^{70} -4.41675e6 q^{71} +(-311823. + 509822. i) q^{72} +2.21484e6 q^{73} +(43821.3 + 75900.6i) q^{74} +(1.64661e6 - 2.77039e6i) q^{75} +(350040. - 606287. i) q^{76} +(1.29905e6 - 2.25002e6i) q^{77} +(-236698. - 422232. i) q^{78} +(-153821. - 266426. i) q^{79} -1.53018e6 q^{80} +(-2.17868e6 - 4.25795e6i) q^{81} -640714. q^{82} +(1.57735e6 + 2.73204e6i) q^{83} +(1.09650e6 + 1.95600e6i) q^{84} +(1.02841e6 - 1.78127e6i) q^{85} +(36321.3 - 62910.3i) q^{86} +(-5.30981e6 + 8.93363e6i) q^{87} +(939096. + 1.62656e6i) q^{88} +1.93441e6 q^{89} +(-117428. + 191992. i) q^{90} -3.64895e6 q^{91} +(3.99406e6 + 6.91792e6i) q^{92} +(5.39901e6 + 68378.5i) q^{93} +(162702. - 281808. i) q^{94} +(264836. - 458709. i) q^{95} +(-2.43502e6 - 30839.5i) q^{96} +(-4.94528e6 - 8.56548e6i) q^{97} +729830. q^{98} +(-1.50270e7 - 380695. i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 9 q^{2} + 24 q^{3} - 321 q^{4} - 180 q^{5} - 1233 q^{6} - 84 q^{7} + 5922 q^{8} + 990 q^{9} + 252 q^{10} - 8460 q^{11} + 8052 q^{12} - 1848 q^{13} - 16272 q^{14} - 1188 q^{15} - 12417 q^{16} + 30564 q^{17}+ \cdots - 49382676 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/9\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(e\left(\frac{1}{3}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.536120 + 0.928588i 0.0473868 + 0.0820763i 0.888746 − 0.458400i \(-0.151577\pi\)
−0.841359 + 0.540476i \(0.818244\pi\)
\(3\) −22.8679 − 40.7929i −0.488993 − 0.872288i
\(4\) 63.4251 − 109.856i 0.495509 − 0.858247i
\(5\) 47.9866 − 83.1153i 0.171682 − 0.297362i −0.767326 − 0.641257i \(-0.778413\pi\)
0.939008 + 0.343895i \(0.111746\pi\)
\(6\) 25.6198 − 43.1048i 0.0484224 − 0.0814696i
\(7\) −189.000 − 327.358i −0.208266 − 0.360728i 0.742902 − 0.669400i \(-0.233449\pi\)
−0.951169 + 0.308672i \(0.900115\pi\)
\(8\) 273.261 0.188696
\(9\) −1141.12 + 1865.70i −0.521773 + 0.853085i
\(10\) 102.906 0.0325419
\(11\) 3436.63 + 5952.41i 0.778499 + 1.34840i 0.932807 + 0.360377i \(0.117352\pi\)
−0.154308 + 0.988023i \(0.549315\pi\)
\(12\) −5931.73 − 75.1254i −0.990938 − 0.0125502i
\(13\) 4826.64 − 8359.99i 0.609317 − 1.05537i −0.382036 − 0.924147i \(-0.624777\pi\)
0.991353 − 0.131220i \(-0.0418896\pi\)
\(14\) 202.654 − 351.007i 0.0197382 − 0.0341875i
\(15\) −4487.87 − 56.8389i −0.343337 − 0.00434836i
\(16\) −7971.92 − 13807.8i −0.486567 − 0.842759i
\(17\) 21431.3 1.05798 0.528989 − 0.848629i \(-0.322571\pi\)
0.528989 + 0.848629i \(0.322571\pi\)
\(18\) −2344.24 − 59.3891i −0.0947432 − 0.00240023i
\(19\) 5518.94 0.184594 0.0922972 − 0.995732i \(-0.470579\pi\)
0.0922972 + 0.995732i \(0.470579\pi\)
\(20\) −6087.12 − 10543.2i −0.170140 − 0.294691i
\(21\) −9031.83 + 15195.9i −0.212818 + 0.358062i
\(22\) −3684.89 + 6382.42i −0.0737812 + 0.127793i
\(23\) −31486.4 + 54536.1i −0.539605 + 0.934623i 0.459320 + 0.888271i \(0.348093\pi\)
−0.998925 + 0.0463526i \(0.985240\pi\)
\(24\) −6248.91 − 11147.1i −0.0922709 − 0.164597i
\(25\) 34457.1 + 59681.4i 0.441050 + 0.763922i
\(26\) 10350.6 0.115494
\(27\) 102202. + 3884.83i 0.999278 + 0.0379839i
\(28\) −47949.5 −0.412792
\(29\) −111113. − 192454.i −0.846004 − 1.46532i −0.884747 − 0.466072i \(-0.845669\pi\)
0.0387428 − 0.999249i \(-0.487665\pi\)
\(30\) −2353.26 − 4197.85i −0.0159127 − 0.0283859i
\(31\) −57729.1 + 99989.7i −0.348040 + 0.602822i −0.985901 − 0.167329i \(-0.946486\pi\)
0.637862 + 0.770151i \(0.279819\pi\)
\(32\) 26036.5 − 45096.6i 0.140462 − 0.243287i
\(33\) 164228. − 276309.i 0.795513 − 1.33843i
\(34\) 11489.7 + 19900.8i 0.0501342 + 0.0868350i
\(35\) −36277.9 −0.143023
\(36\) 132582. + 243690.i 0.473614 + 0.870521i
\(37\) 81737.7 0.265287 0.132644 − 0.991164i \(-0.457653\pi\)
0.132644 + 0.991164i \(0.457653\pi\)
\(38\) 2958.82 + 5124.82i 0.00874734 + 0.0151508i
\(39\) −451403. − 5717.02i −1.21854 − 0.0154328i
\(40\) 13112.9 − 22712.2i 0.0323957 − 0.0561111i
\(41\) −298773. + 517491.i −0.677015 + 1.17262i 0.298860 + 0.954297i \(0.403394\pi\)
−0.975875 + 0.218328i \(0.929940\pi\)
\(42\) −18952.8 − 240.038i −0.0394732 − 0.000499928i
\(43\) −33874.2 − 58671.8i −0.0649725 − 0.112536i 0.831709 − 0.555211i \(-0.187363\pi\)
−0.896682 + 0.442676i \(0.854029\pi\)
\(44\) 871875. 1.54301
\(45\) 100310. + 184373.i 0.164096 + 0.301615i
\(46\) −67522.1 −0.102281
\(47\) −151740. − 262822.i −0.213186 − 0.369249i 0.739524 − 0.673130i \(-0.235051\pi\)
−0.952710 + 0.303881i \(0.901717\pi\)
\(48\) −380957. + 640952.i −0.497201 + 0.836530i
\(49\) 340329. − 589468.i 0.413250 − 0.715770i
\(50\) −36946.3 + 63992.8i −0.0417999 + 0.0723996i
\(51\) −490089. − 874243.i −0.517343 − 0.922861i
\(52\) −612261. − 1.06047e6i −0.603844 − 1.04589i
\(53\) 846755. 0.781254 0.390627 − 0.920549i \(-0.372258\pi\)
0.390627 + 0.920549i \(0.372258\pi\)
\(54\) 51185.2 + 96986.3i 0.0442350 + 0.0838170i
\(55\) 659649. 0.534618
\(56\) −51646.4 − 89454.2i −0.0392990 − 0.0680679i
\(57\) −126207. − 225134.i −0.0902653 − 0.161019i
\(58\) 119140. − 206357.i 0.0801788 − 0.138874i
\(59\) −793119. + 1.37372e6i −0.502755 + 0.870797i 0.497240 + 0.867613i \(0.334347\pi\)
−0.999995 + 0.00318395i \(0.998987\pi\)
\(60\) −290888. + 489412.i −0.173858 + 0.292513i
\(61\) 1.12706e6 + 1.95213e6i 0.635760 + 1.10117i 0.986353 + 0.164641i \(0.0526467\pi\)
−0.350593 + 0.936528i \(0.614020\pi\)
\(62\) −123799. −0.0659699
\(63\) 826422. + 20936.6i 0.416399 + 0.0105491i
\(64\) −1.98498e6 −0.946510
\(65\) −463228. − 802335.i −0.209218 − 0.362376i
\(66\) 344623. + 4364.66i 0.147550 + 0.00186873i
\(67\) −1.51172e6 + 2.61838e6i −0.614060 + 1.06358i 0.376489 + 0.926421i \(0.377131\pi\)
−0.990549 + 0.137162i \(0.956202\pi\)
\(68\) 1.35928e6 − 2.35434e6i 0.524238 − 0.908006i
\(69\) 2.94471e6 + 37294.8i 1.07912 + 0.0136671i
\(70\) −19449.3 − 33687.3i −0.00677738 − 0.0117388i
\(71\) −4.41675e6 −1.46453 −0.732266 − 0.681018i \(-0.761537\pi\)
−0.732266 + 0.681018i \(0.761537\pi\)
\(72\) −311823. + 509822.i −0.0984564 + 0.160974i
\(73\) 2.21484e6 0.666366 0.333183 − 0.942862i \(-0.391877\pi\)
0.333183 + 0.942862i \(0.391877\pi\)
\(74\) 43821.3 + 75900.6i 0.0125711 + 0.0217738i
\(75\) 1.64661e6 − 2.77039e6i 0.450689 − 0.758275i
\(76\) 350040. − 606287.i 0.0914682 − 0.158428i
\(77\) 1.29905e6 − 2.25002e6i 0.324271 − 0.561653i
\(78\) −236698. − 422232.i −0.0564758 − 0.100744i
\(79\) −153821. − 266426.i −0.0351011 − 0.0607969i 0.847941 − 0.530090i \(-0.177842\pi\)
−0.883042 + 0.469293i \(0.844509\pi\)
\(80\) −1.53018e6 −0.334140
\(81\) −2.17868e6 − 4.25795e6i −0.455507 − 0.890232i
\(82\) −640714. −0.128326
\(83\) 1.57735e6 + 2.73204e6i 0.302798 + 0.524462i 0.976769 − 0.214296i \(-0.0687458\pi\)
−0.673970 + 0.738758i \(0.735412\pi\)
\(84\) 1.09650e6 + 1.95600e6i 0.201852 + 0.360073i
\(85\) 1.02841e6 − 1.78127e6i 0.181636 − 0.314603i
\(86\) 36321.3 − 62910.3i 0.00615768 − 0.0106654i
\(87\) −5.30981e6 + 8.93363e6i −0.864493 + 1.45449i
\(88\) 939096. + 1.62656e6i 0.146900 + 0.254438i
\(89\) 1.93441e6 0.290859 0.145430 − 0.989369i \(-0.453544\pi\)
0.145430 + 0.989369i \(0.453544\pi\)
\(90\) −117428. + 191992.i −0.0169795 + 0.0277610i
\(91\) −3.64895e6 −0.507601
\(92\) 3.99406e6 + 6.91792e6i 0.534758 + 0.926229i
\(93\) 5.39901e6 + 68378.5i 0.696023 + 0.00881514i
\(94\) 162702. − 281808.i 0.0202044 − 0.0349950i
\(95\) 264836. − 458709.i 0.0316916 − 0.0548914i
\(96\) −2.43502e6 − 30839.5i −0.280901 − 0.00355761i
\(97\) −4.94528e6 − 8.56548e6i −0.550161 − 0.952907i −0.998262 − 0.0589243i \(-0.981233\pi\)
0.448101 − 0.893983i \(-0.352100\pi\)
\(98\) 729830. 0.0783304
\(99\) −1.50270e7 − 380695.i −1.55650 − 0.0394325i
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 9.8.c.a.4.4 ✓ 12
3.2 odd 2 27.8.c.a.10.3 12
4.3 odd 2 144.8.i.c.49.4 12
9.2 odd 6 27.8.c.a.19.3 12
9.4 even 3 81.8.a.e.1.3 6
9.5 odd 6 81.8.a.c.1.4 6
9.7 even 3 inner 9.8.c.a.7.4 yes 12
12.11 even 2 432.8.i.c.145.3 12
36.7 odd 6 144.8.i.c.97.4 12
36.11 even 6 432.8.i.c.289.3 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
9.8.c.a.4.4 ✓ 12 1.1 even 1 trivial
9.8.c.a.7.4 yes 12 9.7 even 3 inner
27.8.c.a.10.3 12 3.2 odd 2
27.8.c.a.19.3 12 9.2 odd 6
81.8.a.c.1.4 6 9.5 odd 6
81.8.a.e.1.3 6 9.4 even 3
144.8.i.c.49.4 12 4.3 odd 2
144.8.i.c.97.4 12 36.7 odd 6
432.8.i.c.145.3 12 12.11 even 2
432.8.i.c.289.3 12 36.11 even 6