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Properties

Label 147.4.g.d.80.3
Level $147$
Weight $4$
Character 147.80
Analytic conductor $8.673$
Analytic rank $0$
Dimension $12$
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] = [147,4,Mod(68,147)] 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("147.68"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(147, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([3, 5])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 147 = 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 147.g (of order \(6\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [12,0,3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(8.67328077084\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{6})\)
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: \( x^{12} - x^{11} - 29x^{9} + 6x^{8} - 49x^{7} + 1564x^{6} - 441x^{5} + 486x^{4} - 21141x^{3} - 59049x + 531441 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 3^{3} \)
Twist minimal: no (minimal twist has level 21)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 80.3
Root \(-2.23014 - 2.00661i\) of defining polynomial
Character \(\chi\) \(=\) 147.80
Dual form 147.4.g.d.68.3

$q$-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+(-1.65310 + 0.954416i) q^{2} +(3.47555 - 3.86271i) q^{3} +(-2.17818 + 3.77272i) q^{4} +(-0.623706 - 1.08029i) q^{5} +(-2.05878 + 9.70256i) q^{6} -23.5862i q^{8} +(-2.84113 - 26.8501i) q^{9} +(2.06209 + 1.19055i) q^{10} +(35.2392 + 20.3453i) q^{11} +(7.00257 + 21.5260i) q^{12} -19.5973i q^{13} +(-6.34057 - 1.34540i) q^{15} +(5.08559 + 8.80850i) q^{16} +(52.3592 - 90.6889i) q^{17} +(30.3228 + 41.6742i) q^{18} +(-35.0345 + 20.2272i) q^{19} +5.43418 q^{20} -77.6716 q^{22} +(69.6324 - 40.2023i) q^{23} +(-91.1068 - 81.9750i) q^{24} +(61.7220 - 106.906i) q^{25} +(18.7040 + 32.3962i) q^{26} +(-113.589 - 82.3444i) q^{27} -211.712i q^{29} +(11.7656 - 3.82746i) q^{30} +(86.6242 + 50.0125i) q^{31} +(146.596 + 84.6373i) q^{32} +(201.064 - 65.4076i) q^{33} +199.890i q^{34} +(107.486 + 47.7656i) q^{36} +(94.9875 + 164.523i) q^{37} +(38.6103 - 66.8750i) q^{38} +(-75.6987 - 68.1113i) q^{39} +(-25.4799 + 14.7109i) q^{40} +186.753 q^{41} +158.618 q^{43} +(-153.515 + 88.6317i) q^{44} +(-27.2339 + 19.8158i) q^{45} +(-76.7393 + 132.916i) q^{46} +(-179.034 - 310.097i) q^{47} +(51.6999 + 10.9702i) q^{48} +235.634i q^{50} +(-168.328 - 517.442i) q^{51} +(73.9351 + 42.6865i) q^{52} +(-366.460 - 211.576i) q^{53} +(266.364 + 27.7123i) q^{54} -50.7580i q^{55} +(-43.6323 + 205.629i) q^{57} +(202.061 + 349.980i) q^{58} +(-312.781 + 541.753i) q^{59} +(18.8867 - 20.9907i) q^{60} +(-699.575 + 403.900i) q^{61} -190.931 q^{62} -404.486 q^{64} +(-21.1708 + 12.2229i) q^{65} +(-269.952 + 300.023i) q^{66} +(-149.272 + 258.547i) q^{67} +(228.096 + 395.074i) q^{68} +(86.7208 - 408.695i) q^{69} +455.386i q^{71} +(-633.292 + 67.0114i) q^{72} +(434.467 + 250.840i) q^{73} +(-314.047 - 181.315i) q^{74} +(-198.428 - 609.970i) q^{75} -176.234i q^{76} +(190.144 + 40.3465i) q^{78} +(30.9561 + 53.6176i) q^{79} +(6.34382 - 10.9878i) q^{80} +(-712.856 + 152.569i) q^{81} +(-308.720 + 178.240i) q^{82} +73.1180 q^{83} -130.627 q^{85} +(-262.211 + 151.388i) q^{86} +(-817.783 - 735.816i) q^{87} +(479.870 - 831.158i) q^{88} +(-57.3723 - 99.3717i) q^{89} +(26.1077 - 58.7498i) q^{90} +350.271i q^{92} +(494.251 - 160.784i) q^{93} +(591.922 + 341.746i) q^{94} +(43.7025 + 25.2316i) q^{95} +(836.432 - 272.098i) q^{96} +1416.51i q^{97} +(446.156 - 1003.98i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 3 q^{3} + 14 q^{4} - 3 q^{9} - 30 q^{10} + 192 q^{12} + 6 q^{15} + 134 q^{16} + 66 q^{18} - 300 q^{19} - 268 q^{22} - 414 q^{24} - 42 q^{25} - 822 q^{30} + 930 q^{31} + 855 q^{33} + 852 q^{36} + 764 q^{37}+ \cdots - 3354 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/147\mathbb{Z}\right)^\times\).

\(n\) \(50\) \(52\)
\(\chi(n)\) \(-1\) \(e\left(\frac{1}{6}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.65310 + 0.954416i −0.584458 + 0.337437i −0.762903 − 0.646513i \(-0.776227\pi\)
0.178445 + 0.983950i \(0.442893\pi\)
\(3\) 3.47555 − 3.86271i 0.668870 − 0.743380i
\(4\) −2.17818 + 3.77272i −0.272273 + 0.471590i
\(5\) −0.623706 − 1.08029i −0.0557859 − 0.0966240i 0.836784 − 0.547533i \(-0.184433\pi\)
−0.892570 + 0.450909i \(0.851100\pi\)
\(6\) −2.05878 + 9.70256i −0.140082 + 0.660175i
\(7\) 0 0
\(8\) − 23.5862i − 1.04237i
\(9\) −2.84113 − 26.8501i −0.105227 − 0.994448i
\(10\) 2.06209 + 1.19055i 0.0652090 + 0.0376484i
\(11\) 35.2392 + 20.3453i 0.965910 + 0.557668i 0.897987 − 0.440022i \(-0.145029\pi\)
0.0679230 + 0.997691i \(0.478363\pi\)
\(12\) 7.00257 + 21.5260i 0.168456 + 0.517834i
\(13\) − 19.5973i − 0.418101i −0.977905 − 0.209050i \(-0.932963\pi\)
0.977905 − 0.209050i \(-0.0670373\pi\)
\(14\) 0 0
\(15\) −6.34057 − 1.34540i −0.109142 − 0.0231588i
\(16\) 5.08559 + 8.80850i 0.0794623 + 0.137633i
\(17\) 52.3592 − 90.6889i 0.746999 − 1.29384i −0.202256 − 0.979333i \(-0.564827\pi\)
0.949255 − 0.314507i \(-0.101839\pi\)
\(18\) 30.3228 + 41.6742i 0.397064 + 0.545706i
\(19\) −35.0345 + 20.2272i −0.423025 + 0.244234i −0.696371 − 0.717682i \(-0.745203\pi\)
0.273346 + 0.961916i \(0.411870\pi\)
\(20\) 5.43418 0.0607559
\(21\) 0 0
\(22\) −77.6716 −0.752711
\(23\) 69.6324 − 40.2023i 0.631276 − 0.364467i −0.149970 − 0.988691i \(-0.547918\pi\)
0.781246 + 0.624223i \(0.214584\pi\)
\(24\) −91.1068 − 81.9750i −0.774879 − 0.697212i
\(25\) 61.7220 − 106.906i 0.493776 − 0.855245i
\(26\) 18.7040 + 32.3962i 0.141083 + 0.244362i
\(27\) −113.589 − 82.3444i −0.809636 − 0.586933i
\(28\) 0 0
\(29\) − 211.712i − 1.35565i −0.735222 − 0.677827i \(-0.762922\pi\)
0.735222 − 0.677827i \(-0.237078\pi\)
\(30\) 11.7656 − 3.82746i 0.0716034 − 0.0232932i
\(31\) 86.6242 + 50.0125i 0.501876 + 0.289758i 0.729488 − 0.683994i \(-0.239758\pi\)
−0.227612 + 0.973752i \(0.573092\pi\)
\(32\) 146.596 + 84.6373i 0.809837 + 0.467560i
\(33\) 201.064 − 65.4076i 1.06063 − 0.345030i
\(34\) 199.890i 1.00826i
\(35\) 0 0
\(36\) 107.486 + 47.7656i 0.497622 + 0.221137i
\(37\) 94.9875 + 164.523i 0.422050 + 0.731012i 0.996140 − 0.0877801i \(-0.0279773\pi\)
−0.574090 + 0.818792i \(0.694644\pi\)
\(38\) 38.6103 − 66.8750i 0.164827 − 0.285488i
\(39\) −75.6987 − 68.1113i −0.310808 − 0.279655i
\(40\) −25.4799 + 14.7109i −0.100718 + 0.0581497i
\(41\) 186.753 0.711362 0.355681 − 0.934607i \(-0.384249\pi\)
0.355681 + 0.934607i \(0.384249\pi\)
\(42\) 0 0
\(43\) 158.618 0.562536 0.281268 − 0.959629i \(-0.409245\pi\)
0.281268 + 0.959629i \(0.409245\pi\)
\(44\) −153.515 + 88.6317i −0.525982 + 0.303676i
\(45\) −27.2339 + 19.8158i −0.0902174 + 0.0656437i
\(46\) −76.7393 + 132.916i −0.245969 + 0.426032i
\(47\) −179.034 − 310.097i −0.555635 − 0.962388i −0.997854 − 0.0654808i \(-0.979142\pi\)
0.442219 − 0.896907i \(-0.354191\pi\)
\(48\) 51.6999 + 10.9702i 0.155463 + 0.0329877i
\(49\) 0 0
\(50\) 235.634i 0.666473i
\(51\) −168.328 − 517.442i −0.462170 − 1.42071i
\(52\) 73.9351 + 42.6865i 0.197172 + 0.113837i
\(53\) −366.460 − 211.576i −0.949758 − 0.548343i −0.0567521 − 0.998388i \(-0.518074\pi\)
−0.893006 + 0.450045i \(0.851408\pi\)
\(54\) 266.364 + 27.7123i 0.671251 + 0.0698364i
\(55\) − 50.7580i − 0.124440i
\(56\) 0 0
\(57\) −43.6323 + 205.629i −0.101390 + 0.477829i
\(58\) 202.061 + 349.980i 0.457447 + 0.792322i
\(59\) −312.781 + 541.753i −0.690180 + 1.19543i 0.281599 + 0.959532i \(0.409135\pi\)
−0.971779 + 0.235895i \(0.924198\pi\)
\(60\) 18.8867 − 20.9907i 0.0406378 − 0.0451647i
\(61\) −699.575 + 403.900i −1.46838 + 0.847772i −0.999372 − 0.0354209i \(-0.988723\pi\)
−0.469011 + 0.883192i \(0.655389\pi\)
\(62\) −190.931 −0.391101
\(63\) 0 0
\(64\) −404.486 −0.790012
\(65\) −21.1708 + 12.2229i −0.0403986 + 0.0233241i
\(66\) −269.952 + 300.023i −0.503466 + 0.559550i
\(67\) −149.272 + 258.547i −0.272187 + 0.471441i −0.969421 − 0.245402i \(-0.921080\pi\)
0.697235 + 0.716843i \(0.254413\pi\)
\(68\) 228.096 + 395.074i 0.406775 + 0.704555i
\(69\) 86.7208 − 408.695i 0.151304 − 0.713059i
\(70\) 0 0
\(71\) 455.386i 0.761189i 0.924742 + 0.380594i \(0.124281\pi\)
−0.924742 + 0.380594i \(0.875719\pi\)
\(72\) −633.292 + 67.0114i −1.03659 + 0.109686i
\(73\) 434.467 + 250.840i 0.696582 + 0.402172i 0.806073 − 0.591816i \(-0.201589\pi\)
−0.109491 + 0.993988i \(0.534922\pi\)
\(74\) −314.047 − 181.315i −0.493341 − 0.284831i
\(75\) −198.428 − 609.970i −0.305500 − 0.939110i
\(76\) − 176.234i − 0.265993i
\(77\) 0 0
\(78\) 190.144 + 40.3465i 0.276020 + 0.0585685i
\(79\) 30.9561 + 53.6176i 0.0440865 + 0.0763601i 0.887227 − 0.461334i \(-0.152629\pi\)
−0.843140 + 0.537694i \(0.819296\pi\)
\(80\) 6.34382 − 10.9878i 0.00886576 − 0.0153559i
\(81\) −712.856 + 152.569i −0.977855 + 0.209285i
\(82\) −308.720 + 178.240i −0.415761 + 0.240040i
\(83\) 73.1180 0.0966957 0.0483478 − 0.998831i \(-0.484604\pi\)
0.0483478 + 0.998831i \(0.484604\pi\)
\(84\) 0 0
\(85\) −130.627 −0.166688
\(86\) −262.211 + 151.388i −0.328779 + 0.189820i
\(87\) −817.783 − 735.816i −1.00777 − 0.906755i
\(88\) 479.870 − 831.158i 0.581298 − 1.00684i
\(89\) −57.3723 − 99.3717i −0.0683309 − 0.118353i 0.829836 − 0.558008i \(-0.188434\pi\)
−0.898167 + 0.439655i \(0.855101\pi\)
\(90\) 26.1077 − 58.7498i 0.0305777 − 0.0688086i
\(91\) 0 0
\(92\) 350.271i 0.396938i
\(93\) 494.251 − 160.784i 0.551090 − 0.179274i
\(94\) 591.922 + 341.746i 0.649490 + 0.374983i
\(95\) 43.7025 + 25.2316i 0.0471977 + 0.0272496i
\(96\) 836.432 − 272.098i 0.889250 − 0.289280i
\(97\) 1416.51i 1.48273i 0.671101 + 0.741366i \(0.265822\pi\)
−0.671101 + 0.741366i \(0.734178\pi\)
\(98\) 0 0
\(99\) 446.156 − 1003.98i 0.452933 − 1.01923i
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 147.4.g.d.80.3 12
3.2 odd 2 inner 147.4.g.d.80.4 12
7.2 even 3 21.4.g.a.5.4 yes 12
7.3 odd 6 147.4.c.a.146.6 12
7.4 even 3 147.4.c.a.146.5 12
7.5 odd 6 inner 147.4.g.d.68.4 12
7.6 odd 2 21.4.g.a.17.3 yes 12
21.2 odd 6 21.4.g.a.5.3 ✓ 12
21.5 even 6 inner 147.4.g.d.68.3 12
21.11 odd 6 147.4.c.a.146.8 12
21.17 even 6 147.4.c.a.146.7 12
21.20 even 2 21.4.g.a.17.4 yes 12
28.23 odd 6 336.4.bc.d.257.3 12
28.27 even 2 336.4.bc.d.17.5 12
84.23 even 6 336.4.bc.d.257.5 12
84.83 odd 2 336.4.bc.d.17.3 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
21.4.g.a.5.3 ✓ 12 21.2 odd 6
21.4.g.a.5.4 yes 12 7.2 even 3
21.4.g.a.17.3 yes 12 7.6 odd 2
21.4.g.a.17.4 yes 12 21.20 even 2
147.4.c.a.146.5 12 7.4 even 3
147.4.c.a.146.6 12 7.3 odd 6
147.4.c.a.146.7 12 21.17 even 6
147.4.c.a.146.8 12 21.11 odd 6
147.4.g.d.68.3 12 21.5 even 6 inner
147.4.g.d.68.4 12 7.5 odd 6 inner
147.4.g.d.80.3 12 1.1 even 1 trivial
147.4.g.d.80.4 12 3.2 odd 2 inner
336.4.bc.d.17.3 12 84.83 odd 2
336.4.bc.d.17.5 12 28.27 even 2
336.4.bc.d.257.3 12 28.23 odd 6
336.4.bc.d.257.5 12 84.23 even 6