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Properties

Label 546.4.l.a.211.1
Level $546$
Weight $4$
Character 546.211
Analytic conductor $32.215$
Analytic rank $0$
Dimension $2$
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] = [546,4,Mod(211,546)] 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("546.211"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(546, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 2])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 546 = 2 \cdot 3 \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 546.l (of order \(3\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,-2,-3,-4,14] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(32.2150428631\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 211.1
Root \(0.500000 - 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 546.211
Dual form 546.4.l.a.295.1

$q$-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+(-1.00000 - 1.73205i) q^{2} +(-1.50000 - 2.59808i) q^{3} +(-2.00000 + 3.46410i) q^{4} +7.00000 q^{5} +(-3.00000 + 5.19615i) q^{6} +(3.50000 - 6.06218i) q^{7} +8.00000 q^{8} +(-4.50000 + 7.79423i) q^{9} +(-7.00000 - 12.1244i) q^{10} +(8.00000 + 13.8564i) q^{11} +12.0000 q^{12} +(45.5000 - 11.2583i) q^{13} -14.0000 q^{14} +(-10.5000 - 18.1865i) q^{15} +(-8.00000 - 13.8564i) q^{16} +(16.5000 - 28.5788i) q^{17} +18.0000 q^{18} +(-72.0000 + 124.708i) q^{19} +(-14.0000 + 24.2487i) q^{20} -21.0000 q^{21} +(16.0000 - 27.7128i) q^{22} +(22.0000 + 38.1051i) q^{23} +(-12.0000 - 20.7846i) q^{24} -76.0000 q^{25} +(-65.0000 - 67.5500i) q^{26} +27.0000 q^{27} +(14.0000 + 24.2487i) q^{28} +(92.5000 + 160.215i) q^{29} +(-21.0000 + 36.3731i) q^{30} +184.000 q^{31} +(-16.0000 + 27.7128i) q^{32} +(24.0000 - 41.5692i) q^{33} -66.0000 q^{34} +(24.5000 - 42.4352i) q^{35} +(-18.0000 - 31.1769i) q^{36} +(112.500 + 194.856i) q^{37} +288.000 q^{38} +(-97.5000 - 101.325i) q^{39} +56.0000 q^{40} +(82.5000 + 142.894i) q^{41} +(21.0000 + 36.3731i) q^{42} +(10.0000 - 17.3205i) q^{43} -64.0000 q^{44} +(-31.5000 + 54.5596i) q^{45} +(44.0000 - 76.2102i) q^{46} -88.0000 q^{47} +(-24.0000 + 41.5692i) q^{48} +(-24.5000 - 42.4352i) q^{49} +(76.0000 + 131.636i) q^{50} -99.0000 q^{51} +(-52.0000 + 180.133i) q^{52} +111.000 q^{53} +(-27.0000 - 46.7654i) q^{54} +(56.0000 + 96.9948i) q^{55} +(28.0000 - 48.4974i) q^{56} +432.000 q^{57} +(185.000 - 320.429i) q^{58} +(-110.000 + 190.526i) q^{59} +84.0000 q^{60} +(134.500 - 232.961i) q^{61} +(-184.000 - 318.697i) q^{62} +(31.5000 + 54.5596i) q^{63} +64.0000 q^{64} +(318.500 - 78.8083i) q^{65} -96.0000 q^{66} +(-350.000 - 606.218i) q^{67} +(66.0000 + 114.315i) q^{68} +(66.0000 - 114.315i) q^{69} -98.0000 q^{70} +(-508.000 + 879.882i) q^{71} +(-36.0000 + 62.3538i) q^{72} +947.000 q^{73} +(225.000 - 389.711i) q^{74} +(114.000 + 197.454i) q^{75} +(-288.000 - 498.831i) q^{76} +112.000 q^{77} +(-78.0000 + 270.200i) q^{78} +1244.00 q^{79} +(-56.0000 - 96.9948i) q^{80} +(-40.5000 - 70.1481i) q^{81} +(165.000 - 285.788i) q^{82} +1140.00 q^{83} +(42.0000 - 72.7461i) q^{84} +(115.500 - 200.052i) q^{85} -40.0000 q^{86} +(277.500 - 480.644i) q^{87} +(64.0000 + 110.851i) q^{88} +(-429.000 - 743.050i) q^{89} +126.000 q^{90} +(91.0000 - 315.233i) q^{91} -176.000 q^{92} +(-276.000 - 478.046i) q^{93} +(88.0000 + 152.420i) q^{94} +(-504.000 + 872.954i) q^{95} +96.0000 q^{96} +(431.000 - 746.514i) q^{97} +(-49.0000 + 84.8705i) q^{98} -144.000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} - 3 q^{3} - 4 q^{4} + 14 q^{5} - 6 q^{6} + 7 q^{7} + 16 q^{8} - 9 q^{9} - 14 q^{10} + 16 q^{11} + 24 q^{12} + 91 q^{13} - 28 q^{14} - 21 q^{15} - 16 q^{16} + 33 q^{17} + 36 q^{18} - 144 q^{19}+ \cdots - 288 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/546\mathbb{Z}\right)^\times\).

\(n\) \(157\) \(365\) \(379\)
\(\chi(n)\) \(1\) \(1\) \(e\left(\frac{1}{3}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 − 1.73205i −0.353553 − 0.612372i
\(3\) −1.50000 − 2.59808i −0.288675 − 0.500000i
\(4\) −2.00000 + 3.46410i −0.250000 + 0.433013i
\(5\) 7.00000 0.626099 0.313050 − 0.949737i \(-0.398649\pi\)
0.313050 + 0.949737i \(0.398649\pi\)
\(6\) −3.00000 + 5.19615i −0.204124 + 0.353553i
\(7\) 3.50000 − 6.06218i 0.188982 − 0.327327i
\(8\) 8.00000 0.353553
\(9\) −4.50000 + 7.79423i −0.166667 + 0.288675i
\(10\) −7.00000 − 12.1244i −0.221359 − 0.383406i
\(11\) 8.00000 + 13.8564i 0.219281 + 0.379806i 0.954588 − 0.297928i \(-0.0962955\pi\)
−0.735307 + 0.677734i \(0.762962\pi\)
\(12\) 12.0000 0.288675
\(13\) 45.5000 − 11.2583i 0.970725 − 0.240192i
\(14\) −14.0000 −0.267261
\(15\) −10.5000 − 18.1865i −0.180739 − 0.313050i
\(16\) −8.00000 − 13.8564i −0.125000 − 0.216506i
\(17\) 16.5000 − 28.5788i 0.235402 − 0.407729i −0.723987 − 0.689813i \(-0.757693\pi\)
0.959390 + 0.282085i \(0.0910259\pi\)
\(18\) 18.0000 0.235702
\(19\) −72.0000 + 124.708i −0.869365 + 1.50578i −0.00671805 + 0.999977i \(0.502138\pi\)
−0.862647 + 0.505807i \(0.831195\pi\)
\(20\) −14.0000 + 24.2487i −0.156525 + 0.271109i
\(21\) −21.0000 −0.218218
\(22\) 16.0000 − 27.7128i 0.155055 − 0.268563i
\(23\) 22.0000 + 38.1051i 0.199449 + 0.345455i 0.948350 − 0.317227i \(-0.102752\pi\)
−0.748901 + 0.662682i \(0.769418\pi\)
\(24\) −12.0000 − 20.7846i −0.102062 − 0.176777i
\(25\) −76.0000 −0.608000
\(26\) −65.0000 − 67.5500i −0.490290 − 0.509525i
\(27\) 27.0000 0.192450
\(28\) 14.0000 + 24.2487i 0.0944911 + 0.163663i
\(29\) 92.5000 + 160.215i 0.592304 + 1.02590i 0.993921 + 0.110093i \(0.0351149\pi\)
−0.401617 + 0.915808i \(0.631552\pi\)
\(30\) −21.0000 + 36.3731i −0.127802 + 0.221359i
\(31\) 184.000 1.06604 0.533022 − 0.846101i \(-0.321056\pi\)
0.533022 + 0.846101i \(0.321056\pi\)
\(32\) −16.0000 + 27.7128i −0.0883883 + 0.153093i
\(33\) 24.0000 − 41.5692i 0.126602 − 0.219281i
\(34\) −66.0000 −0.332909
\(35\) 24.5000 − 42.4352i 0.118322 − 0.204939i
\(36\) −18.0000 − 31.1769i −0.0833333 − 0.144338i
\(37\) 112.500 + 194.856i 0.499862 + 0.865786i 1.00000 0.000159589i \(-5.07987e-5\pi\)
−0.500138 + 0.865946i \(0.666717\pi\)
\(38\) 288.000 1.22947
\(39\) −97.5000 − 101.325i −0.400320 − 0.416025i
\(40\) 56.0000 0.221359
\(41\) 82.5000 + 142.894i 0.314252 + 0.544301i 0.979278 − 0.202520i \(-0.0649130\pi\)
−0.665026 + 0.746820i \(0.731580\pi\)
\(42\) 21.0000 + 36.3731i 0.0771517 + 0.133631i
\(43\) 10.0000 − 17.3205i 0.0354648 − 0.0614268i −0.847748 − 0.530399i \(-0.822042\pi\)
0.883213 + 0.468972i \(0.155376\pi\)
\(44\) −64.0000 −0.219281
\(45\) −31.5000 + 54.5596i −0.104350 + 0.180739i
\(46\) 44.0000 − 76.2102i 0.141031 − 0.244274i
\(47\) −88.0000 −0.273109 −0.136554 − 0.990633i \(-0.543603\pi\)
−0.136554 + 0.990633i \(0.543603\pi\)
\(48\) −24.0000 + 41.5692i −0.0721688 + 0.125000i
\(49\) −24.5000 − 42.4352i −0.0714286 − 0.123718i
\(50\) 76.0000 + 131.636i 0.214960 + 0.372322i
\(51\) −99.0000 −0.271819
\(52\) −52.0000 + 180.133i −0.138675 + 0.480384i
\(53\) 111.000 0.287680 0.143840 − 0.989601i \(-0.454055\pi\)
0.143840 + 0.989601i \(0.454055\pi\)
\(54\) −27.0000 − 46.7654i −0.0680414 − 0.117851i
\(55\) 56.0000 + 96.9948i 0.137292 + 0.237796i
\(56\) 28.0000 − 48.4974i 0.0668153 − 0.115728i
\(57\) 432.000 1.00386
\(58\) 185.000 − 320.429i 0.418822 − 0.725421i
\(59\) −110.000 + 190.526i −0.242725 + 0.420412i −0.961490 − 0.274841i \(-0.911375\pi\)
0.718764 + 0.695254i \(0.244708\pi\)
\(60\) 84.0000 0.180739
\(61\) 134.500 − 232.961i 0.282311 − 0.488977i −0.689643 − 0.724150i \(-0.742232\pi\)
0.971953 + 0.235173i \(0.0755657\pi\)
\(62\) −184.000 − 318.697i −0.376904 − 0.652816i
\(63\) 31.5000 + 54.5596i 0.0629941 + 0.109109i
\(64\) 64.0000 0.125000
\(65\) 318.500 − 78.8083i 0.607770 − 0.150384i
\(66\) −96.0000 −0.179042
\(67\) −350.000 − 606.218i −0.638199 − 1.10539i −0.985828 − 0.167760i \(-0.946347\pi\)
0.347629 − 0.937632i \(-0.386987\pi\)
\(68\) 66.0000 + 114.315i 0.117701 + 0.203864i
\(69\) 66.0000 − 114.315i 0.115152 − 0.199449i
\(70\) −98.0000 −0.167332
\(71\) −508.000 + 879.882i −0.849134 + 1.47074i 0.0328483 + 0.999460i \(0.489542\pi\)
−0.881982 + 0.471283i \(0.843791\pi\)
\(72\) −36.0000 + 62.3538i −0.0589256 + 0.102062i
\(73\) 947.000 1.51833 0.759164 − 0.650899i \(-0.225608\pi\)
0.759164 + 0.650899i \(0.225608\pi\)
\(74\) 225.000 − 389.711i 0.353456 − 0.612203i
\(75\) 114.000 + 197.454i 0.175514 + 0.304000i
\(76\) −288.000 − 498.831i −0.434682 − 0.752892i
\(77\) 112.000 0.165761
\(78\) −78.0000 + 270.200i −0.113228 + 0.392232i
\(79\) 1244.00 1.77166 0.885829 − 0.464012i \(-0.153591\pi\)
0.885829 + 0.464012i \(0.153591\pi\)
\(80\) −56.0000 − 96.9948i −0.0782624 − 0.135554i
\(81\) −40.5000 − 70.1481i −0.0555556 − 0.0962250i
\(82\) 165.000 − 285.788i 0.222210 − 0.384879i
\(83\) 1140.00 1.50761 0.753803 − 0.657101i \(-0.228217\pi\)
0.753803 + 0.657101i \(0.228217\pi\)
\(84\) 42.0000 − 72.7461i 0.0545545 − 0.0944911i
\(85\) 115.500 − 200.052i 0.147385 − 0.255278i
\(86\) −40.0000 −0.0501548
\(87\) 277.500 − 480.644i 0.341967 − 0.592304i
\(88\) 64.0000 + 110.851i 0.0775275 + 0.134282i
\(89\) −429.000 − 743.050i −0.510943 − 0.884979i −0.999920 − 0.0126821i \(-0.995963\pi\)
0.488977 − 0.872297i \(-0.337370\pi\)
\(90\) 126.000 0.147573
\(91\) 91.0000 − 315.233i 0.104828 − 0.363137i
\(92\) −176.000 −0.199449
\(93\) −276.000 − 478.046i −0.307741 − 0.533022i
\(94\) 88.0000 + 152.420i 0.0965586 + 0.167244i
\(95\) −504.000 + 872.954i −0.544309 + 0.942770i
\(96\) 96.0000 0.102062
\(97\) 431.000 − 746.514i 0.451149 − 0.781412i −0.547309 − 0.836931i \(-0.684348\pi\)
0.998458 + 0.0555182i \(0.0176811\pi\)
\(98\) −49.0000 + 84.8705i −0.0505076 + 0.0874818i
\(99\) −144.000 −0.146187
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 546.4.l.a.211.1 ✓ 2
13.9 even 3 inner 546.4.l.a.295.1 yes 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
546.4.l.a.211.1 ✓ 2 1.1 even 1 trivial
546.4.l.a.295.1 yes 2 13.9 even 3 inner