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Properties

Label 9.8.c.a.7.1
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 7.1
Root \(0.500000 + 11.4952i\) of defining polynomial
Character \(\chi\) \(=\) 9.7
Dual form 9.8.c.a.4.1

$q$-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+(-10.7051 + 18.5418i) q^{2} +(21.6493 + 41.4525i) q^{3} +(-165.199 - 286.133i) q^{4} +(32.6274 + 56.5123i) q^{5} +(-1000.36 - 42.3372i) q^{6} +(-118.194 + 204.717i) q^{7} +4333.37 q^{8} +(-1249.62 + 1794.83i) q^{9} -1397.12 q^{10} +(-2305.59 + 3993.39i) q^{11} +(8284.48 - 13042.5i) q^{12} +(68.1364 + 118.016i) q^{13} +(-2530.55 - 4383.04i) q^{14} +(-1636.22 + 2575.94i) q^{15} +(-25243.8 + 43723.5i) q^{16} +5319.46 q^{17} +(-19902.1 - 42384.0i) q^{18} +38228.5 q^{19} +(10780.0 - 18671.5i) q^{20} +(-11044.8 - 467.439i) q^{21} +(-49363.1 - 85499.4i) q^{22} +(36480.5 + 63186.1i) q^{23} +(93814.3 + 179629. i) q^{24} +(36933.4 - 63970.5i) q^{25} -2917.63 q^{26} +(-101454. - 12943.0i) q^{27} +78101.7 q^{28} +(-35682.4 + 61803.7i) q^{29} +(-30246.6 - 57914.1i) q^{30} +(28810.3 + 49900.8i) q^{31} +(-263139. - 455770. i) q^{32} +(-215450. - 9118.28i) q^{33} +(-56945.4 + 98632.4i) q^{34} -15425.4 q^{35} +(719995. + 61052.5i) q^{36} +88485.6 q^{37} +(-409240. + 708824. i) q^{38} +(-3416.94 + 5379.38i) q^{39} +(141387. + 244889. i) q^{40} +(-127791. - 221341. i) q^{41} +(126903. - 199787. i) q^{42} +(324806. - 562580. i) q^{43} +1.52352e6 q^{44} +(-142202. - 12058.1i) q^{45} -1.56211e6 q^{46} +(522539. - 905063. i) q^{47} +(-2.35896e6 - 99835.7i) q^{48} +(383832. + 664817. i) q^{49} +(790752. + 1.36962e6i) q^{50} +(115162. + 220505. i) q^{51} +(22512.1 - 38992.1i) q^{52} -777845. q^{53} +(1.32606e6 - 1.74258e6i) q^{54} -300901. q^{55} +(-512177. + 887116. i) q^{56} +(827618. + 1.58467e6i) q^{57} +(-763967. - 1.32323e6i) q^{58} +(-1.01959e6 - 1.76598e6i) q^{59} +(1.00736e6 + 42633.5i) q^{60} +(-927927. + 1.60722e6i) q^{61} -1.23367e6 q^{62} +(-219736. - 467956. i) q^{63} +4.80532e6 q^{64} +(-4446.23 + 7701.10i) q^{65} +(2.47549e6 - 3.89722e6i) q^{66} +(1.81717e6 + 3.14743e6i) q^{67} +(-878768. - 1.52207e6i) q^{68} +(-1.82945e6 + 2.88014e6i) q^{69} +(165131. - 286015. i) q^{70} +1.93117e6 q^{71} +(-5.41506e6 + 7.77768e6i) q^{72} -2.06016e6 q^{73} +(-947248. + 1.64068e6i) q^{74} +(3.45132e6 + 146067. i) q^{75} +(-6.31529e6 - 1.09384e7i) q^{76} +(-545011. - 943987. i) q^{77} +(-63164.6 - 120943. i) q^{78} +(3.13421e6 - 5.42860e6i) q^{79} -3.29456e6 q^{80} +(-1.65988e6 - 4.48571e6i) q^{81} +5.47209e6 q^{82} +(568792. - 985177. i) q^{83} +(1.69085e6 + 3.23751e6i) q^{84} +(173560. + 300615. i) q^{85} +(6.95416e6 + 1.20450e7i) q^{86} +(-3.33441e6 - 141119. i) q^{87} +(-9.99096e6 + 1.73049e7i) q^{88} -428904. q^{89} +(1.74587e6 - 2.50760e6i) q^{90} -32213.2 q^{91} +(1.20531e7 - 2.08765e7i) q^{92} +(-1.44479e6 + 2.27457e6i) q^{93} +(1.11877e7 + 1.93776e7i) q^{94} +(1.24730e6 + 2.16038e6i) q^{95} +(1.31960e7 - 2.07749e7i) q^{96} +(5.85362e6 - 1.01388e7i) q^{97} -1.64359e7 q^{98} +(-4.28637e6 - 9.12836e6i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −10.7051 + 18.5418i −0.946207 + 1.63888i −0.192890 + 0.981220i \(0.561786\pi\)
−0.753317 + 0.657658i \(0.771547\pi\)
\(3\) 21.6493 + 41.4525i 0.462934 + 0.886393i
\(4\) −165.199 − 286.133i −1.29061 − 2.23541i
\(5\) 32.6274 + 56.5123i 0.116731 + 0.202185i 0.918471 − 0.395489i \(-0.129425\pi\)
−0.801739 + 0.597674i \(0.796092\pi\)
\(6\) −1000.36 − 42.3372i −1.89072 − 0.0800190i
\(7\) −118.194 + 204.717i −0.130242 + 0.225586i −0.923770 − 0.382948i \(-0.874909\pi\)
0.793528 + 0.608534i \(0.208242\pi\)
\(8\) 4333.37 2.99234
\(9\) −1249.62 + 1794.83i −0.571385 + 0.820682i
\(10\) −1397.12 −0.441808
\(11\) −2305.59 + 3993.39i −0.522284 + 0.904623i 0.477379 + 0.878697i \(0.341587\pi\)
−0.999664 + 0.0259259i \(0.991747\pi\)
\(12\) 8284.48 − 13042.5i 1.38398 − 2.17884i
\(13\) 68.1364 + 118.016i 0.00860157 + 0.0148984i 0.870294 − 0.492532i \(-0.163929\pi\)
−0.861693 + 0.507431i \(0.830595\pi\)
\(14\) −2530.55 − 4383.04i −0.246472 − 0.426901i
\(15\) −1636.22 + 2575.94i −0.125176 + 0.197068i
\(16\) −25243.8 + 43723.5i −1.54076 + 2.66867i
\(17\) 5319.46 0.262601 0.131301 − 0.991343i \(-0.458085\pi\)
0.131301 + 0.991343i \(0.458085\pi\)
\(18\) −19902.1 − 42384.0i −0.804350 − 1.71297i
\(19\) 38228.5 1.27864 0.639321 − 0.768940i \(-0.279215\pi\)
0.639321 + 0.768940i \(0.279215\pi\)
\(20\) 10780.0 − 18671.5i 0.301310 − 0.521885i
\(21\) −11044.8 − 467.439i −0.260251 − 0.0110143i
\(22\) −49363.1 − 85499.4i −0.988378 − 1.71192i
\(23\) 36480.5 + 63186.1i 0.625192 + 1.08286i 0.988504 + 0.151197i \(0.0483128\pi\)
−0.363311 + 0.931668i \(0.618354\pi\)
\(24\) 93814.3 + 179629.i 1.38526 + 2.65239i
\(25\) 36933.4 − 63970.5i 0.472748 − 0.818823i
\(26\) −2917.63 −0.0325554
\(27\) −101454. − 12943.0i −0.991960 − 0.126550i
\(28\) 78101.7 0.672369
\(29\) −35682.4 + 61803.7i −0.271682 + 0.470567i −0.969293 − 0.245910i \(-0.920913\pi\)
0.697611 + 0.716477i \(0.254246\pi\)
\(30\) −30246.6 − 57914.1i −0.204528 − 0.391616i
\(31\) 28810.3 + 49900.8i 0.173693 + 0.300844i 0.939708 − 0.341978i \(-0.111097\pi\)
−0.766015 + 0.642822i \(0.777763\pi\)
\(32\) −263139. − 455770.i −1.41958 − 2.45879i
\(33\) −215450. − 9118.28i −1.04363 − 0.0441686i
\(34\) −56945.4 + 98632.4i −0.248475 + 0.430371i
\(35\) −15425.4 −0.0608133
\(36\) 719995. + 61052.5i 2.57200 + 0.218095i
\(37\) 88485.6 0.287188 0.143594 − 0.989637i \(-0.454134\pi\)
0.143594 + 0.989637i \(0.454134\pi\)
\(38\) −409240. + 708824.i −1.20986 + 2.09554i
\(39\) −3416.94 + 5379.38i −0.00922384 + 0.0145213i
\(40\) 141387. + 244889.i 0.349300 + 0.605005i
\(41\) −127791. − 221341.i −0.289573 − 0.501555i 0.684135 − 0.729356i \(-0.260180\pi\)
−0.973708 + 0.227800i \(0.926847\pi\)
\(42\) 126903. − 199787.i 0.264302 − 0.416098i
\(43\) 324806. − 562580.i 0.622994 − 1.07906i −0.365931 − 0.930642i \(-0.619249\pi\)
0.988925 − 0.148415i \(-0.0474172\pi\)
\(44\) 1.52352e6 2.69627
\(45\) −142202. − 12058.1i −0.232628 − 0.0197259i
\(46\) −1.56211e6 −2.36624
\(47\) 522539. − 905063.i 0.734135 − 1.27156i −0.220967 − 0.975281i \(-0.570921\pi\)
0.955102 − 0.296277i \(-0.0957453\pi\)
\(48\) −2.35896e6 − 99835.7i −3.07876 − 0.130299i
\(49\) 383832. + 664817.i 0.466074 + 0.807264i
\(50\) 790752. + 1.36962e6i 0.894634 + 1.54955i
\(51\) 115162. + 220505.i 0.121567 + 0.232768i
\(52\) 22512.1 − 38992.1i 0.0222026 − 0.0384561i
\(53\) −777845. −0.717675 −0.358837 − 0.933400i \(-0.616827\pi\)
−0.358837 + 0.933400i \(0.616827\pi\)
\(54\) 1.32606e6 − 1.74258e6i 1.14600 − 1.50596i
\(55\) −300901. −0.243868
\(56\) −512177. + 887116.i −0.389728 + 0.675029i
\(57\) 827618. + 1.58467e6i 0.591927 + 1.13338i
\(58\) −763967. − 1.32323e6i −0.514134 − 0.890507i
\(59\) −1.01959e6 − 1.76598e6i −0.646314 − 1.11945i −0.983996 − 0.178189i \(-0.942976\pi\)
0.337682 − 0.941260i \(-0.390357\pi\)
\(60\) 1.00736e6 + 42633.5i 0.602082 + 0.0254813i
\(61\) −927927. + 1.60722e6i −0.523431 + 0.906609i 0.476197 + 0.879339i \(0.342015\pi\)
−0.999628 + 0.0272705i \(0.991318\pi\)
\(62\) −1.23367e6 −0.657396
\(63\) −219736. − 467956.i −0.110716 − 0.235783i
\(64\) 4.80532e6 2.29135
\(65\) −4446.23 + 7701.10i −0.00200815 + 0.00347821i
\(66\) 2.47549e6 − 3.89722e6i 1.05988 − 1.66860i
\(67\) 1.81717e6 + 3.14743e6i 0.738131 + 1.27848i 0.953336 + 0.301911i \(0.0976247\pi\)
−0.215205 + 0.976569i \(0.569042\pi\)
\(68\) −878768. − 1.52207e6i −0.338917 − 0.587021i
\(69\) −1.82945e6 + 2.88014e6i −0.670421 + 1.05546i
\(70\) 165131. − 286015.i 0.0575419 − 0.0996656i
\(71\) 1.93117e6 0.640348 0.320174 − 0.947359i \(-0.396259\pi\)
0.320174 + 0.947359i \(0.396259\pi\)
\(72\) −5.41506e6 + 7.77768e6i −1.70978 + 2.45576i
\(73\) −2.06016e6 −0.619826 −0.309913 − 0.950765i \(-0.600300\pi\)
−0.309913 + 0.950765i \(0.600300\pi\)
\(74\) −947248. + 1.64068e6i −0.271740 + 0.470667i
\(75\) 3.45132e6 + 146067.i 0.944650 + 0.0399794i
\(76\) −6.31529e6 − 1.09384e7i −1.65024 − 2.85829i
\(77\) −545011. − 943987.i −0.136047 − 0.235640i
\(78\) −63164.6 − 120943.i −0.0150710 − 0.0288569i
\(79\) 3.13421e6 − 5.42860e6i 0.715208 − 1.23878i −0.247671 − 0.968844i \(-0.579665\pi\)
0.962879 − 0.269933i \(-0.0870016\pi\)
\(80\) −3.29456e6 −0.719419
\(81\) −1.65988e6 − 4.48571e6i −0.347039 − 0.937851i
\(82\) 5.47209e6 1.09598
\(83\) 568792. − 985177.i 0.109189 − 0.189122i −0.806253 − 0.591571i \(-0.798508\pi\)
0.915442 + 0.402450i \(0.131841\pi\)
\(84\) 1.69085e6 + 3.23751e6i 0.311262 + 0.595983i
\(85\) 173560. + 300615.i 0.0306538 + 0.0530939i
\(86\) 6.95416e6 + 1.20450e7i 1.17896 + 2.04202i
\(87\) −3.33441e6 − 141119.i −0.542878 − 0.0229756i
\(88\) −9.99096e6 + 1.73049e7i −1.56285 + 2.70694i
\(89\) −428904. −0.0644904 −0.0322452 − 0.999480i \(-0.510266\pi\)
−0.0322452 + 0.999480i \(0.510266\pi\)
\(90\) 1.74587e6 − 2.50760e6i 0.252442 − 0.362584i
\(91\) −32213.2 −0.00448114
\(92\) 1.20531e7 − 2.08765e7i 1.61376 − 2.79512i
\(93\) −1.44479e6 + 2.27457e6i −0.186258 + 0.293231i
\(94\) 1.11877e7 + 1.93776e7i 1.38929 + 2.40632i
\(95\) 1.24730e6 + 2.16038e6i 0.149258 + 0.258522i
\(96\) 1.31960e7 − 2.07749e7i 1.52228 − 2.39656i
\(97\) 5.85362e6 − 1.01388e7i 0.651214 − 1.12793i −0.331615 − 0.943415i \(-0.607594\pi\)
0.982829 − 0.184520i \(-0.0590731\pi\)
\(98\) −1.64359e7 −1.76401
\(99\) −4.28637e6 − 9.12836e6i −0.443983 − 0.945517i
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.7.1 yes 12
3.2 odd 2 27.8.c.a.19.6 12
4.3 odd 2 144.8.i.c.97.3 12
9.2 odd 6 81.8.a.c.1.1 6
9.4 even 3 inner 9.8.c.a.4.1 ✓ 12
9.5 odd 6 27.8.c.a.10.6 12
9.7 even 3 81.8.a.e.1.6 6
12.11 even 2 432.8.i.c.289.4 12
36.23 even 6 432.8.i.c.145.4 12
36.31 odd 6 144.8.i.c.49.3 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
9.8.c.a.4.1 ✓ 12 9.4 even 3 inner
9.8.c.a.7.1 yes 12 1.1 even 1 trivial
27.8.c.a.10.6 12 9.5 odd 6
27.8.c.a.19.6 12 3.2 odd 2
81.8.a.c.1.1 6 9.2 odd 6
81.8.a.e.1.6 6 9.7 even 3
144.8.i.c.49.3 12 36.31 odd 6
144.8.i.c.97.3 12 4.3 odd 2
432.8.i.c.145.4 12 36.23 even 6
432.8.i.c.289.4 12 12.11 even 2