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Properties

Label 21.4.g.a.17.3
Level $21$
Weight $4$
Character 21.17
Analytic conductor $1.239$
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] = [21,4,Mod(5,21)] 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("21.5"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(21, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([3, 5])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 21 = 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 21.g (of order \(6\), 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: \(1.23904011012\)
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: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 17.3
Root \(-2.23014 - 2.00661i\) of defining polynomial
Character \(\chi\) \(=\) 21.17
Dual form 21.4.g.a.5.3

$q$-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} +(10.0808 + 15.5363i) q^{7} -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} +(-31.4927 - 16.0617i) q^{14} +(-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} +(-95.0487 - 15.0578i) q^{21} -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} +(-80.5720 + 4.19132i) q^{28} -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} +(-10.4962 + 20.5803i) q^{35} +(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} +(171.496 - 65.8241i) q^{42} +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} +(-139.753 + 313.238i) q^{49} +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} +(366.442 - 237.769i) q^{56} +(-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} +(388.510 - 314.812i) q^{63} -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} +(-2.29089 - 44.0390i) q^{70} +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} +(39.1491 + 752.584i) q^{77} +(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} +(263.842 - 325.794i) q^{84} -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} +(-304.469 + 197.557i) q^{91} +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} +(-67.9336 - 651.195i) q^{98} +(446.156 - 1003.98i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 3 q^{3} + 14 q^{4} - 56 q^{7} - 3 q^{9} + 30 q^{10} - 192 q^{12} + 6 q^{15} + 134 q^{16} + 66 q^{18} + 300 q^{19} + 357 q^{21} - 268 q^{22} + 414 q^{24} - 42 q^{25} - 602 q^{28} - 822 q^{30} - 930 q^{31}+ \cdots - 3354 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/21\mathbb{Z}\right)^\times\).

\(n\) \(8\) \(10\)
\(\chi(n)\) \(-1\) \(e\left(\frac{1}{6}\right)\)

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 0.892570 − 0.450909i \(-0.148900\pi\)
−0.836784 + 0.547533i \(0.815567\pi\)
\(6\) 2.05878 − 9.70256i 0.140082 − 0.660175i
\(7\) 10.0808 + 15.5363i 0.544314 + 0.838881i
\(8\) − 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 \(0.0670373\pi\)
−0.977905 + 0.209050i \(0.932963\pi\)
\(14\) −31.4927 − 16.0617i −0.601198 − 0.306619i
\(15\) −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 \(0.435173\pi\)
−0.949255 + 0.314507i \(0.898161\pi\)
\(18\) 30.3228 + 41.6742i 0.397064 + 0.545706i
\(19\) 35.0345 − 20.2272i 0.423025 − 0.244234i −0.273346 − 0.961916i \(-0.588130\pi\)
0.696371 + 0.717682i \(0.254797\pi\)
\(20\) −5.43418 −0.0607559
\(21\) −95.0487 − 15.0578i −0.987683 − 0.156470i
\(22\) −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
\(28\) −80.5720 + 4.19132i −0.543810 + 0.0282888i
\(29\) − 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 0.227612 − 0.973752i \(-0.426908\pi\)
−0.729488 + 0.683994i \(0.760242\pi\)
\(32\) 146.596 + 84.6373i 0.809837 + 0.467560i
\(33\) −201.064 + 65.4076i −1.06063 + 0.345030i
\(34\) − 199.890i − 1.00826i
\(35\) −10.4962 + 20.5803i −0.0506910 + 0.0993916i
\(36\) 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 \(-0.615751\pi\)
−0.355681 + 0.934607i \(0.615751\pi\)
\(42\) 171.496 − 65.8241i 0.630058 − 0.241830i
\(43\) 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 \(0.0208581\pi\)
−0.442219 + 0.896907i \(0.645809\pi\)
\(48\) −51.6999 − 10.9702i −0.155463 − 0.0329877i
\(49\) −139.753 + 313.238i −0.407444 + 0.913230i
\(50\) 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
\(56\) 366.442 − 237.769i 0.874427 − 0.567379i
\(57\) −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 \(-0.590865\pi\)
0.971779 − 0.235895i \(-0.0758020\pi\)
\(60\) 18.8867 − 20.9907i 0.0406378 − 0.0451647i
\(61\) 699.575 − 403.900i 1.46838 − 0.847772i 0.469011 − 0.883192i \(-0.344611\pi\)
0.999372 + 0.0354209i \(0.0112772\pi\)
\(62\) 190.931 0.391101
\(63\) 388.510 − 314.812i 0.776948 − 0.629565i
\(64\) −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
\(70\) −2.29089 − 44.0390i −0.00391162 − 0.0751952i
\(71\) 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 0.109491 − 0.993988i \(-0.465078\pi\)
−0.806073 + 0.591816i \(0.798411\pi\)
\(74\) −314.047 − 181.315i −0.493341 − 0.284831i
\(75\) 198.428 + 609.970i 0.305500 + 0.939110i
\(76\) 176.234i 0.265993i
\(77\) 39.1491 + 752.584i 0.0579410 + 1.11383i
\(78\) 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 \(-0.515396\pi\)
−0.0483478 + 0.998831i \(0.515396\pi\)
\(84\) 263.842 − 325.794i 0.342709 − 0.423179i
\(85\) −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 0.898167 − 0.439655i \(-0.144899\pi\)
−0.829836 + 0.558008i \(0.811566\pi\)
\(90\) −26.1077 + 58.7498i −0.0305777 + 0.0688086i
\(91\) −304.469 + 197.557i −0.350737 + 0.227578i
\(92\) 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 \(-0.734178\pi\)
0.671101 − 0.741366i \(-0.265822\pi\)
\(98\) −67.9336 − 651.195i −0.0700238 − 0.671231i
\(99\) 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 21.4.g.a.17.3 yes 12
3.2 odd 2 inner 21.4.g.a.17.4 yes 12
4.3 odd 2 336.4.bc.d.17.5 12
7.2 even 3 147.4.g.d.68.4 12
7.3 odd 6 147.4.c.a.146.5 12
7.4 even 3 147.4.c.a.146.6 12
7.5 odd 6 inner 21.4.g.a.5.4 yes 12
7.6 odd 2 147.4.g.d.80.3 12
12.11 even 2 336.4.bc.d.17.3 12
21.2 odd 6 147.4.g.d.68.3 12
21.5 even 6 inner 21.4.g.a.5.3 ✓ 12
21.11 odd 6 147.4.c.a.146.7 12
21.17 even 6 147.4.c.a.146.8 12
21.20 even 2 147.4.g.d.80.4 12
28.19 even 6 336.4.bc.d.257.3 12
84.47 odd 6 336.4.bc.d.257.5 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
21.4.g.a.5.3 ✓ 12 21.5 even 6 inner
21.4.g.a.5.4 yes 12 7.5 odd 6 inner
21.4.g.a.17.3 yes 12 1.1 even 1 trivial
21.4.g.a.17.4 yes 12 3.2 odd 2 inner
147.4.c.a.146.5 12 7.3 odd 6
147.4.c.a.146.6 12 7.4 even 3
147.4.c.a.146.7 12 21.11 odd 6
147.4.c.a.146.8 12 21.17 even 6
147.4.g.d.68.3 12 21.2 odd 6
147.4.g.d.68.4 12 7.2 even 3
147.4.g.d.80.3 12 7.6 odd 2
147.4.g.d.80.4 12 21.20 even 2
336.4.bc.d.17.3 12 12.11 even 2
336.4.bc.d.17.5 12 4.3 odd 2
336.4.bc.d.257.3 12 28.19 even 6
336.4.bc.d.257.5 12 84.47 odd 6