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Properties

Label 546.4.l.b.295.1
Level $546$
Weight $4$
Character 546.295
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,-4,44] 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 295.1
Root \(0.500000 + 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 546.295
Dual form 546.4.l.b.211.1

$q$-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} +22.0000 q^{5} +(3.00000 + 5.19615i) q^{6} +(-3.50000 - 6.06218i) q^{7} +8.00000 q^{8} +(-4.50000 - 7.79423i) q^{9} +(-22.0000 + 38.1051i) q^{10} +(-8.00000 + 13.8564i) q^{11} -12.0000 q^{12} +(45.5000 + 11.2583i) q^{13} +14.0000 q^{14} +(33.0000 - 57.1577i) q^{15} +(-8.00000 + 13.8564i) q^{16} +(49.5000 + 85.7365i) q^{17} +18.0000 q^{18} +(11.0000 + 19.0526i) q^{19} +(-44.0000 - 76.2102i) q^{20} -21.0000 q^{21} +(-16.0000 - 27.7128i) q^{22} +(-76.5000 + 132.502i) q^{23} +(12.0000 - 20.7846i) q^{24} +359.000 q^{25} +(-65.0000 + 67.5500i) q^{26} -27.0000 q^{27} +(-14.0000 + 24.2487i) q^{28} +(-111.000 + 192.258i) q^{29} +(66.0000 + 114.315i) q^{30} +91.0000 q^{31} +(-16.0000 - 27.7128i) q^{32} +(24.0000 + 41.5692i) q^{33} -198.000 q^{34} +(-77.0000 - 133.368i) q^{35} +(-18.0000 + 31.1769i) q^{36} +(133.000 - 230.363i) q^{37} -44.0000 q^{38} +(97.5000 - 101.325i) q^{39} +176.000 q^{40} +(189.000 - 327.358i) q^{41} +(21.0000 - 36.3731i) q^{42} +(-42.5000 - 73.6122i) q^{43} +64.0000 q^{44} +(-99.0000 - 171.473i) q^{45} +(-153.000 - 265.004i) q^{46} -262.000 q^{47} +(24.0000 + 41.5692i) q^{48} +(-24.5000 + 42.4352i) q^{49} +(-359.000 + 621.806i) q^{50} +297.000 q^{51} +(-52.0000 - 180.133i) q^{52} +371.000 q^{53} +(27.0000 - 46.7654i) q^{54} +(-176.000 + 304.841i) q^{55} +(-28.0000 - 48.4974i) q^{56} +66.0000 q^{57} +(-222.000 - 384.515i) q^{58} +(-257.500 - 446.003i) q^{59} -264.000 q^{60} +(-241.500 - 418.290i) q^{61} +(-91.0000 + 157.617i) q^{62} +(-31.5000 + 54.5596i) q^{63} +64.0000 q^{64} +(1001.00 + 247.683i) q^{65} -96.0000 q^{66} +(77.5000 - 134.234i) q^{67} +(198.000 - 342.946i) q^{68} +(229.500 + 397.506i) q^{69} +308.000 q^{70} +(424.500 + 735.256i) q^{71} +(-36.0000 - 62.3538i) q^{72} +284.000 q^{73} +(266.000 + 460.726i) q^{74} +(538.500 - 932.709i) q^{75} +(44.0000 - 76.2102i) q^{76} +112.000 q^{77} +(78.0000 + 270.200i) q^{78} -116.000 q^{79} +(-176.000 + 304.841i) q^{80} +(-40.5000 + 70.1481i) q^{81} +(378.000 + 654.715i) q^{82} +323.000 q^{83} +(42.0000 + 72.7461i) q^{84} +(1089.00 + 1886.20i) q^{85} +170.000 q^{86} +(333.000 + 576.773i) q^{87} +(-64.0000 + 110.851i) q^{88} +(-268.500 + 465.056i) q^{89} +396.000 q^{90} +(-91.0000 - 315.233i) q^{91} +612.000 q^{92} +(136.500 - 236.425i) q^{93} +(262.000 - 453.797i) q^{94} +(242.000 + 419.156i) q^{95} -96.0000 q^{96} +(-446.000 - 772.495i) q^{97} +(-49.0000 - 84.8705i) q^{98} +144.000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} + 3 q^{3} - 4 q^{4} + 44 q^{5} + 6 q^{6} - 7 q^{7} + 16 q^{8} - 9 q^{9} - 44 q^{10} - 16 q^{11} - 24 q^{12} + 91 q^{13} + 28 q^{14} + 66 q^{15} - 16 q^{16} + 99 q^{17} + 36 q^{18} + 22 q^{19}+ \cdots + 288 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(157\) \(365\) \(379\)
\(\chi(n)\) \(1\) \(1\) \(e\left(\frac{2}{3}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −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
\(5\) 22.0000 1.96774 0.983870 − 0.178885i \(-0.0572491\pi\)
0.983870 + 0.178885i \(0.0572491\pi\)
\(6\) 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
\(10\) −22.0000 + 38.1051i −0.695701 + 1.20499i
\(11\) −8.00000 + 13.8564i −0.219281 + 0.379806i −0.954588 − 0.297928i \(-0.903704\pi\)
0.735307 + 0.677734i \(0.237038\pi\)
\(12\) −12.0000 −0.288675
\(13\) 45.5000 + 11.2583i 0.970725 + 0.240192i
\(14\) 14.0000 0.267261
\(15\) 33.0000 − 57.1577i 0.568038 − 0.983870i
\(16\) −8.00000 + 13.8564i −0.125000 + 0.216506i
\(17\) 49.5000 + 85.7365i 0.706207 + 1.22319i 0.966254 + 0.257590i \(0.0829284\pi\)
−0.260048 + 0.965596i \(0.583738\pi\)
\(18\) 18.0000 0.235702
\(19\) 11.0000 + 19.0526i 0.132820 + 0.230050i 0.924762 − 0.380545i \(-0.124264\pi\)
−0.791943 + 0.610595i \(0.790930\pi\)
\(20\) −44.0000 − 76.2102i −0.491935 − 0.852056i
\(21\) −21.0000 −0.218218
\(22\) −16.0000 − 27.7128i −0.155055 − 0.268563i
\(23\) −76.5000 + 132.502i −0.693537 + 1.20124i 0.277134 + 0.960831i \(0.410615\pi\)
−0.970671 + 0.240410i \(0.922718\pi\)
\(24\) 12.0000 − 20.7846i 0.102062 − 0.176777i
\(25\) 359.000 2.87200
\(26\) −65.0000 + 67.5500i −0.490290 + 0.509525i
\(27\) −27.0000 −0.192450
\(28\) −14.0000 + 24.2487i −0.0944911 + 0.163663i
\(29\) −111.000 + 192.258i −0.710765 + 1.23108i 0.253805 + 0.967255i \(0.418318\pi\)
−0.964570 + 0.263826i \(0.915016\pi\)
\(30\) 66.0000 + 114.315i 0.401663 + 0.695701i
\(31\) 91.0000 0.527228 0.263614 − 0.964628i \(-0.415085\pi\)
0.263614 + 0.964628i \(0.415085\pi\)
\(32\) −16.0000 − 27.7128i −0.0883883 − 0.153093i
\(33\) 24.0000 + 41.5692i 0.126602 + 0.219281i
\(34\) −198.000 −0.998727
\(35\) −77.0000 − 133.368i −0.371868 − 0.644094i
\(36\) −18.0000 + 31.1769i −0.0833333 + 0.144338i
\(37\) 133.000 − 230.363i 0.590948 − 1.02355i −0.403157 − 0.915131i \(-0.632087\pi\)
0.994105 − 0.108421i \(-0.0345794\pi\)
\(38\) −44.0000 −0.187835
\(39\) 97.5000 − 101.325i 0.400320 − 0.416025i
\(40\) 176.000 0.695701
\(41\) 189.000 − 327.358i 0.719923 − 1.24694i −0.241107 − 0.970499i \(-0.577510\pi\)
0.961030 − 0.276445i \(-0.0891562\pi\)
\(42\) 21.0000 − 36.3731i 0.0771517 − 0.133631i
\(43\) −42.5000 − 73.6122i −0.150725 − 0.261064i 0.780769 − 0.624820i \(-0.214828\pi\)
−0.931494 + 0.363756i \(0.881494\pi\)
\(44\) 64.0000 0.219281
\(45\) −99.0000 − 171.473i −0.327957 − 0.568038i
\(46\) −153.000 − 265.004i −0.490405 − 0.849406i
\(47\) −262.000 −0.813120 −0.406560 − 0.913624i \(-0.633272\pi\)
−0.406560 + 0.913624i \(0.633272\pi\)
\(48\) 24.0000 + 41.5692i 0.0721688 + 0.125000i
\(49\) −24.5000 + 42.4352i −0.0714286 + 0.123718i
\(50\) −359.000 + 621.806i −1.01541 + 1.75873i
\(51\) 297.000 0.815457
\(52\) −52.0000 − 180.133i −0.138675 − 0.480384i
\(53\) 371.000 0.961524 0.480762 − 0.876851i \(-0.340360\pi\)
0.480762 + 0.876851i \(0.340360\pi\)
\(54\) 27.0000 − 46.7654i 0.0680414 − 0.117851i
\(55\) −176.000 + 304.841i −0.431488 + 0.747359i
\(56\) −28.0000 − 48.4974i −0.0668153 − 0.115728i
\(57\) 66.0000 0.153367
\(58\) −222.000 − 384.515i −0.502587 − 0.870506i
\(59\) −257.500 − 446.003i −0.568197 − 0.984147i −0.996744 − 0.0806272i \(-0.974308\pi\)
0.428547 − 0.903520i \(-0.359026\pi\)
\(60\) −264.000 −0.568038
\(61\) −241.500 − 418.290i −0.506900 − 0.877977i −0.999968 − 0.00798597i \(-0.997458\pi\)
0.493068 − 0.869991i \(-0.335875\pi\)
\(62\) −91.0000 + 157.617i −0.186403 + 0.322860i
\(63\) −31.5000 + 54.5596i −0.0629941 + 0.109109i
\(64\) 64.0000 0.125000
\(65\) 1001.00 + 247.683i 1.91013 + 0.472636i
\(66\) −96.0000 −0.179042
\(67\) 77.5000 − 134.234i 0.141315 − 0.244765i −0.786677 − 0.617365i \(-0.788200\pi\)
0.927992 + 0.372600i \(0.121533\pi\)
\(68\) 198.000 − 342.946i 0.353103 − 0.611593i
\(69\) 229.500 + 397.506i 0.400414 + 0.693537i
\(70\) 308.000 0.525901
\(71\) 424.500 + 735.256i 0.709562 + 1.22900i 0.965020 + 0.262177i \(0.0844404\pi\)
−0.255458 + 0.966820i \(0.582226\pi\)
\(72\) −36.0000 − 62.3538i −0.0589256 − 0.102062i
\(73\) 284.000 0.455338 0.227669 − 0.973739i \(-0.426890\pi\)
0.227669 + 0.973739i \(0.426890\pi\)
\(74\) 266.000 + 460.726i 0.417863 + 0.723760i
\(75\) 538.500 − 932.709i 0.829075 − 1.43600i
\(76\) 44.0000 − 76.2102i 0.0664098 − 0.115025i
\(77\) 112.000 0.165761
\(78\) 78.0000 + 270.200i 0.113228 + 0.392232i
\(79\) −116.000 −0.165203 −0.0826014 − 0.996583i \(-0.526323\pi\)
−0.0826014 + 0.996583i \(0.526323\pi\)
\(80\) −176.000 + 304.841i −0.245967 + 0.426028i
\(81\) −40.5000 + 70.1481i −0.0555556 + 0.0962250i
\(82\) 378.000 + 654.715i 0.509062 + 0.881722i
\(83\) 323.000 0.427155 0.213577 − 0.976926i \(-0.431488\pi\)
0.213577 + 0.976926i \(0.431488\pi\)
\(84\) 42.0000 + 72.7461i 0.0545545 + 0.0944911i
\(85\) 1089.00 + 1886.20i 1.38963 + 2.40691i
\(86\) 170.000 0.213158
\(87\) 333.000 + 576.773i 0.410360 + 0.710765i
\(88\) −64.0000 + 110.851i −0.0775275 + 0.134282i
\(89\) −268.500 + 465.056i −0.319786 + 0.553885i −0.980443 − 0.196802i \(-0.936944\pi\)
0.660657 + 0.750688i \(0.270278\pi\)
\(90\) 396.000 0.463801
\(91\) −91.0000 − 315.233i −0.104828 − 0.363137i
\(92\) 612.000 0.693537
\(93\) 136.500 − 236.425i 0.152198 − 0.263614i
\(94\) 262.000 − 453.797i 0.287481 − 0.497932i
\(95\) 242.000 + 419.156i 0.261354 + 0.452679i
\(96\) −96.0000 −0.102062
\(97\) −446.000 − 772.495i −0.466850 − 0.808608i 0.532433 − 0.846472i \(-0.321278\pi\)
−0.999283 + 0.0378644i \(0.987944\pi\)
\(98\) −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.b.295.1 yes 2
13.3 even 3 inner 546.4.l.b.211.1 ✓ 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
546.4.l.b.211.1 ✓ 2 13.3 even 3 inner
546.4.l.b.295.1 yes 2 1.1 even 1 trivial