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Properties

Label 9.8.c.a.7.4
Level $9$
Weight $8$
Character 9.7
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 7.4
Root \(0.500000 - 1.48508i\) of defining polynomial
Character \(\chi\) \(=\) 9.7
Dual form 9.8.c.a.4.4

$q$-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 \(\left(\mathbb{Z}/9\mathbb{Z}\right)^\times\).

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

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 −0.841359 − 0.540476i \(-0.818244\pi\)
0.888746 + 0.458400i \(0.151577\pi\)
\(3\) −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 0.939008 − 0.343895i \(-0.111746\pi\)
−0.767326 + 0.641257i \(0.778413\pi\)
\(6\) 25.6198 + 43.1048i 0.0484224 + 0.0814696i
\(7\) −189.000 + 327.358i −0.208266 + 0.360728i −0.951169 − 0.308672i \(-0.900115\pi\)
0.742902 + 0.669400i \(0.233449\pi\)
\(8\) 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 −0.154308 − 0.988023i \(-0.549315\pi\)
0.932807 − 0.360377i \(-0.117352\pi\)
\(12\) −5931.73 + 75.1254i −0.990938 + 0.0125502i
\(13\) 4826.64 + 8359.99i 0.609317 + 1.05537i 0.991353 + 0.131220i \(0.0418896\pi\)
−0.382036 + 0.924147i \(0.624777\pi\)
\(14\) 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 −0.998925 − 0.0463526i \(-0.985240\pi\)
0.459320 − 0.888271i \(-0.348093\pi\)
\(24\) −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 0.0387428 + 0.999249i \(0.487665\pi\)
−0.884747 + 0.466072i \(0.845669\pi\)
\(30\) −2353.26 + 4197.85i −0.0159127 + 0.0283859i
\(31\) −57729.1 − 99989.7i −0.348040 − 0.602822i 0.637862 − 0.770151i \(-0.279819\pi\)
−0.985901 + 0.167329i \(0.946486\pi\)
\(32\) 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 −0.975875 − 0.218328i \(-0.929940\pi\)
0.298860 − 0.954297i \(-0.403394\pi\)
\(42\) −18952.8 + 240.038i −0.0394732 + 0.000499928i
\(43\) −33874.2 + 58671.8i −0.0649725 + 0.112536i −0.896682 − 0.442676i \(-0.854029\pi\)
0.831709 + 0.555211i \(0.187363\pi\)
\(44\) 871875. 1.54301
\(45\) 100310. − 184373.i 0.164096 − 0.301615i
\(46\) −67522.1 −0.102281
\(47\) −151740. + 262822.i −0.213186 + 0.369249i −0.952710 − 0.303881i \(-0.901717\pi\)
0.739524 + 0.673130i \(0.235051\pi\)
\(48\) −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 −0.999995 − 0.00318395i \(-0.998987\pi\)
0.497240 − 0.867613i \(-0.334347\pi\)
\(60\) −290888. − 489412.i −0.173858 − 0.292513i
\(61\) 1.12706e6 − 1.95213e6i 0.635760 − 1.10117i −0.350593 − 0.936528i \(-0.614020\pi\)
0.986353 − 0.164641i \(-0.0526467\pi\)
\(62\) −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 −0.990549 − 0.137162i \(-0.956202\pi\)
0.376489 − 0.926421i \(-0.377131\pi\)
\(68\) 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 −0.883042 − 0.469293i \(-0.844509\pi\)
0.847941 + 0.530090i \(0.177842\pi\)
\(80\) −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 −0.673970 − 0.738758i \(-0.735412\pi\)
0.976769 + 0.214296i \(0.0687458\pi\)
\(84\) 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 0.448101 + 0.893983i \(0.352100\pi\)
−0.998262 + 0.0589243i \(0.981233\pi\)
\(98\) 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.7.4 yes 12
3.2 odd 2 27.8.c.a.19.3 12
4.3 odd 2 144.8.i.c.97.4 12
9.2 odd 6 81.8.a.c.1.4 6
9.4 even 3 inner 9.8.c.a.4.4 ✓ 12
9.5 odd 6 27.8.c.a.10.3 12
9.7 even 3 81.8.a.e.1.3 6
12.11 even 2 432.8.i.c.289.3 12
36.23 even 6 432.8.i.c.145.3 12
36.31 odd 6 144.8.i.c.49.4 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
9.8.c.a.4.4 ✓ 12 9.4 even 3 inner
9.8.c.a.7.4 yes 12 1.1 even 1 trivial
27.8.c.a.10.3 12 9.5 odd 6
27.8.c.a.19.3 12 3.2 odd 2
81.8.a.c.1.4 6 9.2 odd 6
81.8.a.e.1.3 6 9.7 even 3
144.8.i.c.49.4 12 36.31 odd 6
144.8.i.c.97.4 12 4.3 odd 2
432.8.i.c.145.3 12 36.23 even 6
432.8.i.c.289.3 12 12.11 even 2