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Properties

Label 9.8.c.a.4.6
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 4.6
Root \(0.500000 + 9.80854i\) of defining polynomial
Character \(\chi\) \(=\) 9.4
Dual form 9.8.c.a.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+(7.74445 + 13.4138i) q^{2} +(-35.7652 + 30.1306i) q^{3} +(-55.9529 + 96.9133i) q^{4} +(52.7641 - 91.3900i) q^{5} +(-681.146 - 246.401i) q^{6} +(761.419 + 1318.82i) q^{7} +249.280 q^{8} +(371.296 - 2155.25i) q^{9} +1634.51 q^{10} +(-1045.08 - 1810.14i) q^{11} +(-918.889 - 5152.02i) q^{12} +(4667.91 - 8085.06i) q^{13} +(-11793.5 + 20427.0i) q^{14} +(866.519 + 4858.39i) q^{15} +(9092.51 + 15748.7i) q^{16} -20641.6 q^{17} +(31785.5 - 11710.8i) q^{18} -9369.08 q^{19} +(5904.61 + 10227.1i) q^{20} +(-66969.0 - 24225.7i) q^{21} +(16187.2 - 28037.1i) q^{22} +(38695.2 - 67022.1i) q^{23} +(-8915.55 + 7510.96i) q^{24} +(33494.4 + 58014.0i) q^{25} +144602. q^{26} +(51659.5 + 88270.3i) q^{27} -170415. q^{28} +(-81478.1 - 141124. i) q^{29} +(-58458.7 + 49248.9i) q^{30} +(-9829.95 + 17026.0i) q^{31} +(-124879. + 216297. i) q^{32} +(91918.2 + 33250.9i) q^{33} +(-159857. - 276881. i) q^{34} +160702. q^{35} +(188098. + 156576. i) q^{36} +105494. q^{37} +(-72558.3 - 125675. i) q^{38} +(76658.9 + 429811. i) q^{39} +(13153.0 - 22781.7i) q^{40} +(-68590.9 + 118803. i) q^{41} +(-193680. - 1.08592e6i) q^{42} +(-6130.74 - 10618.8i) q^{43} +233902. q^{44} +(-177377. - 147653. i) q^{45} +1.19869e6 q^{46} +(-270053. - 467745. i) q^{47} +(-799713. - 289292. i) q^{48} +(-747747. + 1.29514e6i) q^{49} +(-518791. + 898573. i) q^{50} +(738249. - 621942. i) q^{51} +(522367. + 904766. i) q^{52} +17622.4 q^{53} +(-783963. + 1.37655e6i) q^{54} -220572. q^{55} +(189807. + 328755. i) q^{56} +(335087. - 282296. i) q^{57} +(1.26201e6 - 2.18586e6i) q^{58} +(-194345. + 336615. i) q^{59} +(-519327. - 187864. i) q^{60} +(-1.32379e6 - 2.29287e6i) q^{61} -304510. q^{62} +(3.12509e6 - 1.15138e6i) q^{63} -1.54079e6 q^{64} +(-492596. - 853201. i) q^{65} +(265835. + 1.49048e6i) q^{66} +(-300341. + 520206. i) q^{67} +(1.15496e6 - 2.00044e6i) q^{68} +(635473. + 3.56297e6i) q^{69} +(1.24455e6 + 2.15562e6i) q^{70} +1.63676e6 q^{71} +(92556.6 - 537261. i) q^{72} -3.75812e6 q^{73} +(816991. + 1.41507e6i) q^{74} +(-2.94593e6 - 1.06568e6i) q^{75} +(524227. - 907988. i) q^{76} +(1.59149e6 - 2.75655e6i) q^{77} +(-5.17170e6 + 4.35693e6i) q^{78} +(1.30240e6 + 2.25582e6i) q^{79} +1.91903e6 q^{80} +(-4.50725e6 - 1.60047e6i) q^{81} -2.12479e6 q^{82} +(-192939. - 334181. i) q^{83} +(6.09491e6 - 5.13469e6i) q^{84} +(-1.08913e6 + 1.88643e6i) q^{85} +(94958.4 - 164473. i) q^{86} +(7.16623e6 + 2.59235e6i) q^{87} +(-260519. - 451232. i) q^{88} -9.20167e6 q^{89} +(606888. - 3.52279e6i) q^{90} +1.42170e7 q^{91} +(4.33022e6 + 7.50017e6i) q^{92} +(-161433. - 905119. i) q^{93} +(4.18282e6 - 7.24486e6i) q^{94} +(-494350. + 856240. i) q^{95} +(-2.05083e6 - 1.14986e7i) q^{96} +(-1.90585e6 - 3.30104e6i) q^{97} -2.31636e7 q^{98} +(-4.28934e6 + 1.58032e6i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 7.74445 + 13.4138i 0.684519 + 1.18562i 0.973588 + 0.228313i \(0.0733210\pi\)
−0.289069 + 0.957308i \(0.593346\pi\)
\(3\) −35.7652 + 30.1306i −0.764779 + 0.644293i
\(4\) −55.9529 + 96.9133i −0.437132 + 0.757135i
\(5\) 52.7641 − 91.3900i 0.188774 − 0.326967i −0.756067 − 0.654494i \(-0.772882\pi\)
0.944842 + 0.327527i \(0.106215\pi\)
\(6\) −681.146 − 246.401i −1.28739 − 0.465708i
\(7\) 761.419 + 1318.82i 0.839036 + 1.45325i 0.890701 + 0.454589i \(0.150214\pi\)
−0.0516649 + 0.998664i \(0.516453\pi\)
\(8\) 249.280 0.172136
\(9\) 371.296 − 2155.25i 0.169774 − 0.985483i
\(10\) 1634.51 0.516879
\(11\) −1045.08 − 1810.14i −0.236743 − 0.410051i 0.723035 − 0.690811i \(-0.242747\pi\)
−0.959778 + 0.280761i \(0.909413\pi\)
\(12\) −918.889 − 5152.02i −0.153507 − 0.860682i
\(13\) 4667.91 − 8085.06i 0.589279 − 1.02066i −0.405048 − 0.914295i \(-0.632745\pi\)
0.994327 − 0.106366i \(-0.0339214\pi\)
\(14\) −11793.5 + 20427.0i −1.14867 + 1.98956i
\(15\) 866.519 + 4858.39i 0.0662917 + 0.371683i
\(16\) 9092.51 + 15748.7i 0.554963 + 0.961224i
\(17\) −20641.6 −1.01899 −0.509497 − 0.860473i \(-0.670168\pi\)
−0.509497 + 0.860473i \(0.670168\pi\)
\(18\) 31785.5 − 11710.8i 1.28462 − 0.473294i
\(19\) −9369.08 −0.313371 −0.156686 − 0.987649i \(-0.550081\pi\)
−0.156686 + 0.987649i \(0.550081\pi\)
\(20\) 5904.61 + 10227.1i 0.165039 + 0.285856i
\(21\) −66969.0 − 24225.7i −1.57800 − 0.570833i
\(22\) 16187.2 − 28037.1i 0.324110 − 0.561375i
\(23\) 38695.2 − 67022.1i 0.663147 − 1.14860i −0.316637 − 0.948547i \(-0.602554\pi\)
0.979784 − 0.200058i \(-0.0641130\pi\)
\(24\) −8915.55 + 7510.96i −0.131646 + 0.110906i
\(25\) 33494.4 + 58014.0i 0.428728 + 0.742579i
\(26\) 144602. 1.61349
\(27\) 51659.5 + 88270.3i 0.505100 + 0.863061i
\(28\) −170415. −1.46708
\(29\) −81478.1 − 141124.i −0.620366 − 1.07450i −0.989418 − 0.145096i \(-0.953651\pi\)
0.369052 − 0.929409i \(-0.379682\pi\)
\(30\) −58458.7 + 49248.9i −0.395298 + 0.333021i
\(31\) −9829.95 + 17026.0i −0.0592632 + 0.102647i −0.894135 − 0.447798i \(-0.852208\pi\)
0.834872 + 0.550445i \(0.185542\pi\)
\(32\) −124879. + 216297.i −0.673697 + 1.16688i
\(33\) 91918.2 + 33250.9i 0.445249 + 0.161066i
\(34\) −159857. − 276881.i −0.697520 − 1.20814i
\(35\) 160702. 0.633554
\(36\) 188098. + 156576.i 0.671930 + 0.559328i
\(37\) 105494. 0.342390 0.171195 − 0.985237i \(-0.445237\pi\)
0.171195 + 0.985237i \(0.445237\pi\)
\(38\) −72558.3 − 125675.i −0.214509 − 0.371540i
\(39\) 76658.9 + 429811.i 0.206936 + 1.16025i
\(40\) 13153.0 − 22781.7i 0.0324949 − 0.0562829i
\(41\) −68590.9 + 118803.i −0.155426 + 0.269205i −0.933214 − 0.359321i \(-0.883008\pi\)
0.777788 + 0.628526i \(0.216342\pi\)
\(42\) −193680. − 1.08592e6i −0.403378 − 2.26165i
\(43\) −6130.74 − 10618.8i −0.0117591 − 0.0203673i 0.860086 − 0.510149i \(-0.170410\pi\)
−0.871845 + 0.489782i \(0.837076\pi\)
\(44\) 233902. 0.413952
\(45\) −177377. − 147653.i −0.290171 − 0.241544i
\(46\) 1.19869e6 1.81575
\(47\) −270053. − 467745.i −0.379408 − 0.657154i 0.611568 − 0.791192i \(-0.290539\pi\)
−0.990976 + 0.134038i \(0.957206\pi\)
\(48\) −799713. − 289292.i −1.04373 − 0.377565i
\(49\) −747747. + 1.29514e6i −0.907964 + 1.57264i
\(50\) −518791. + 898573.i −0.586945 + 1.01662i
\(51\) 738249. − 621942.i 0.779305 − 0.656530i
\(52\) 522367. + 904766.i 0.515186 + 0.892328i
\(53\) 17622.4 0.0162592 0.00812962 − 0.999967i \(-0.497412\pi\)
0.00812962 + 0.999967i \(0.497412\pi\)
\(54\) −783963. + 1.37655e6i −0.677513 + 1.18964i
\(55\) −220572. −0.178764
\(56\) 189807. + 328755.i 0.144429 + 0.250158i
\(57\) 335087. − 282296.i 0.239660 − 0.201903i
\(58\) 1.26201e6 − 2.18586e6i 0.849304 − 1.47104i
\(59\) −194345. + 336615.i −0.123194 + 0.213379i −0.921026 − 0.389502i \(-0.872647\pi\)
0.797831 + 0.602881i \(0.205981\pi\)
\(60\) −519327. − 187864.i −0.310393 − 0.112283i
\(61\) −1.32379e6 − 2.29287e6i −0.746731 − 1.29338i −0.949381 − 0.314126i \(-0.898289\pi\)
0.202650 − 0.979251i \(-0.435045\pi\)
\(62\) −304510. −0.162267
\(63\) 3.12509e6 − 1.15138e6i 1.57460 − 0.580132i
\(64\) −1.54079e6 −0.734708
\(65\) −492596. − 853201.i −0.222482 − 0.385349i
\(66\) 265835. + 1.49048e6i 0.113817 + 0.638149i
\(67\) −300341. + 520206.i −0.121998 + 0.211307i −0.920556 − 0.390612i \(-0.872264\pi\)
0.798557 + 0.601919i \(0.205597\pi\)
\(68\) 1.15496e6 − 2.00044e6i 0.445435 − 0.771516i
\(69\) 635473. + 3.56297e6i 0.232876 + 1.30569i
\(70\) 1.24455e6 + 2.15562e6i 0.433680 + 0.751156i
\(71\) 1.63676e6 0.542727 0.271364 − 0.962477i \(-0.412525\pi\)
0.271364 + 0.962477i \(0.412525\pi\)
\(72\) 92556.6 − 537261.i 0.0292243 − 0.169637i
\(73\) −3.75812e6 −1.13068 −0.565341 − 0.824857i \(-0.691255\pi\)
−0.565341 + 0.824857i \(0.691255\pi\)
\(74\) 816991. + 1.41507e6i 0.234372 + 0.405945i
\(75\) −2.94593e6 − 1.06568e6i −0.806321 − 0.291683i
\(76\) 524227. − 907988.i 0.136985 − 0.237264i
\(77\) 1.59149e6 − 2.75655e6i 0.397272 − 0.688095i
\(78\) −5.17170e6 + 4.35693e6i −1.23396 + 1.03956i
\(79\) 1.30240e6 + 2.25582e6i 0.297200 + 0.514765i 0.975494 − 0.220025i \(-0.0706139\pi\)
−0.678294 + 0.734790i \(0.737281\pi\)
\(80\) 1.91903e6 0.419051
\(81\) −4.50725e6 − 1.60047e6i −0.942354 − 0.334619i
\(82\) −2.12479e6 −0.425567
\(83\) −192939. − 334181.i −0.0370380 − 0.0641517i 0.846912 − 0.531733i \(-0.178459\pi\)
−0.883950 + 0.467581i \(0.845126\pi\)
\(84\) 6.09491e6 − 5.13469e6i 1.12199 − 0.945229i
\(85\) −1.08913e6 + 1.88643e6i −0.192360 + 0.333177i
\(86\) 94958.4 − 164473.i 0.0160986 − 0.0278836i
\(87\) 7.16623e6 + 2.59235e6i 1.16674 + 0.422062i
\(88\) −260519. − 451232.i −0.0407521 − 0.0705846i
\(89\) −9.20167e6 −1.38357 −0.691786 − 0.722102i \(-0.743176\pi\)
−0.691786 + 0.722102i \(0.743176\pi\)
\(90\) 606888. − 3.52279e6i 0.0877525 − 0.509375i
\(91\) 1.42170e7 1.97771
\(92\) 4.33022e6 + 7.50017e6i 0.579766 + 1.00418i
\(93\) −161433. − 905119.i −0.0208114 − 0.116685i
\(94\) 4.18282e6 − 7.24486e6i 0.519424 − 0.899668i
\(95\) −494350. + 856240.i −0.0591565 + 0.102462i
\(96\) −2.05083e6 − 1.14986e7i −0.236581 − 1.32646i
\(97\) −1.90585e6 − 3.30104e6i −0.212026 − 0.367239i 0.740323 − 0.672252i \(-0.234673\pi\)
−0.952348 + 0.305012i \(0.901339\pi\)
\(98\) −2.31636e7 −2.48607
\(99\) −4.28934e6 + 1.58032e6i −0.444291 + 0.163690i
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.4.6 ✓ 12
3.2 odd 2 27.8.c.a.10.1 12
4.3 odd 2 144.8.i.c.49.6 12
9.2 odd 6 27.8.c.a.19.1 12
9.4 even 3 81.8.a.e.1.1 6
9.5 odd 6 81.8.a.c.1.6 6
9.7 even 3 inner 9.8.c.a.7.6 yes 12
12.11 even 2 432.8.i.c.145.2 12
36.7 odd 6 144.8.i.c.97.6 12
36.11 even 6 432.8.i.c.289.2 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
9.8.c.a.4.6 ✓ 12 1.1 even 1 trivial
9.8.c.a.7.6 yes 12 9.7 even 3 inner
27.8.c.a.10.1 12 3.2 odd 2
27.8.c.a.19.1 12 9.2 odd 6
81.8.a.c.1.6 6 9.5 odd 6
81.8.a.e.1.1 6 9.4 even 3
144.8.i.c.49.6 12 4.3 odd 2
144.8.i.c.97.6 12 36.7 odd 6
432.8.i.c.145.2 12 12.11 even 2
432.8.i.c.289.2 12 36.11 even 6