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Properties

Label 343.4.a.d.1.18
Level $343$
Weight $4$
Character 343.1
Self dual yes
Analytic conductor $20.238$
Analytic rank $1$
Dimension $18$
CM no
Inner twists $1$

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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] = [343,4,Mod(1,343)] 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("343.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(343, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 343 = 7^{3} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 343.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [18,2,-22] 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: yes
Analytic conductor: \(20.2376551320\)
Analytic rank: \(1\)
Dimension: \(18\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{18} - 2 x^{17} - 101 x^{16} + 200 x^{15} + 4071 x^{14} - 7805 x^{13} - 84126 x^{12} + 151605 x^{11} + \cdots + 1016000 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{3}\cdot 7^{10} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.18
Root \(5.04500\) of defining polynomial
Character \(\chi\) \(=\) 343.1

$q$-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)
 
\(f(q)\) \(=\) \(q+5.04500 q^{2} -6.94713 q^{3} +17.4520 q^{4} -6.41867 q^{5} -35.0483 q^{6} +47.6854 q^{8} +21.2627 q^{9} -32.3822 q^{10} -56.2974 q^{11} -121.241 q^{12} +60.0682 q^{13} +44.5914 q^{15} +100.957 q^{16} -99.0407 q^{17} +107.270 q^{18} -108.566 q^{19} -112.019 q^{20} -284.020 q^{22} -24.9424 q^{23} -331.277 q^{24} -83.8007 q^{25} +303.044 q^{26} +39.8579 q^{27} -202.942 q^{29} +224.963 q^{30} +96.6846 q^{31} +127.843 q^{32} +391.106 q^{33} -499.660 q^{34} +371.077 q^{36} -89.5962 q^{37} -547.716 q^{38} -417.302 q^{39} -306.077 q^{40} -383.626 q^{41} +455.487 q^{43} -982.504 q^{44} -136.478 q^{45} -125.835 q^{46} +351.972 q^{47} -701.359 q^{48} -422.774 q^{50} +688.049 q^{51} +1048.31 q^{52} +307.972 q^{53} +201.083 q^{54} +361.355 q^{55} +754.224 q^{57} -1023.84 q^{58} -281.036 q^{59} +778.209 q^{60} +53.6060 q^{61} +487.774 q^{62} -162.685 q^{64} -385.558 q^{65} +1973.13 q^{66} +566.231 q^{67} -1728.46 q^{68} +173.278 q^{69} +59.5211 q^{71} +1013.92 q^{72} -233.678 q^{73} -452.013 q^{74} +582.175 q^{75} -1894.70 q^{76} -2105.29 q^{78} -9.50122 q^{79} -648.007 q^{80} -850.991 q^{81} -1935.39 q^{82} -251.011 q^{83} +635.709 q^{85} +2297.93 q^{86} +1409.87 q^{87} -2684.57 q^{88} +463.981 q^{89} -688.532 q^{90} -435.296 q^{92} -671.681 q^{93} +1775.70 q^{94} +696.851 q^{95} -888.143 q^{96} -606.353 q^{97} -1197.03 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q + 2 q^{2} - 22 q^{3} + 62 q^{4} - 62 q^{5} - 41 q^{6} - 18 q^{8} + 138 q^{9} - 78 q^{10} - 26 q^{11} - 245 q^{12} - 168 q^{13} + 118 q^{15} + 358 q^{16} - 568 q^{17} + 106 q^{18} - 188 q^{19} - 756 q^{20}+ \cdots + 512 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 5.04500 1.78368 0.891838 − 0.452355i \(-0.149416\pi\)
0.891838 + 0.452355i \(0.149416\pi\)
\(3\) −6.94713 −1.33698 −0.668488 − 0.743723i \(-0.733058\pi\)
−0.668488 + 0.743723i \(0.733058\pi\)
\(4\) 17.4520 2.18150
\(5\) −6.41867 −0.574103 −0.287052 − 0.957915i \(-0.592675\pi\)
−0.287052 + 0.957915i \(0.592675\pi\)
\(6\) −35.0483 −2.38473
\(7\) 0 0
\(8\) 47.6854 2.10742
\(9\) 21.2627 0.787507
\(10\) −32.3822 −1.02401
\(11\) −56.2974 −1.54312 −0.771560 − 0.636157i \(-0.780523\pi\)
−0.771560 + 0.636157i \(0.780523\pi\)
\(12\) −121.241 −2.91662
\(13\) 60.0682 1.28153 0.640767 − 0.767736i \(-0.278617\pi\)
0.640767 + 0.767736i \(0.278617\pi\)
\(14\) 0 0
\(15\) 44.5914 0.767563
\(16\) 100.957 1.57745
\(17\) −99.0407 −1.41299 −0.706497 − 0.707716i \(-0.749725\pi\)
−0.706497 + 0.707716i \(0.749725\pi\)
\(18\) 107.270 1.40466
\(19\) −108.566 −1.31088 −0.655442 − 0.755246i \(-0.727518\pi\)
−0.655442 + 0.755246i \(0.727518\pi\)
\(20\) −112.019 −1.25241
\(21\) 0 0
\(22\) −284.020 −2.75243
\(23\) −24.9424 −0.226124 −0.113062 − 0.993588i \(-0.536066\pi\)
−0.113062 + 0.993588i \(0.536066\pi\)
\(24\) −331.277 −2.81757
\(25\) −83.8007 −0.670405
\(26\) 303.044 2.28584
\(27\) 39.8579 0.284098
\(28\) 0 0
\(29\) −202.942 −1.29950 −0.649749 − 0.760148i \(-0.725126\pi\)
−0.649749 + 0.760148i \(0.725126\pi\)
\(30\) 224.963 1.36908
\(31\) 96.6846 0.560163 0.280082 − 0.959976i \(-0.409638\pi\)
0.280082 + 0.959976i \(0.409638\pi\)
\(32\) 127.843 0.706239
\(33\) 391.106 2.06311
\(34\) −499.660 −2.52032
\(35\) 0 0
\(36\) 371.077 1.71795
\(37\) −89.5962 −0.398095 −0.199048 − 0.979990i \(-0.563785\pi\)
−0.199048 + 0.979990i \(0.563785\pi\)
\(38\) −547.716 −2.33819
\(39\) −417.302 −1.71338
\(40\) −306.077 −1.20987
\(41\) −383.626 −1.46128 −0.730639 − 0.682764i \(-0.760778\pi\)
−0.730639 + 0.682764i \(0.760778\pi\)
\(42\) 0 0
\(43\) 455.487 1.61538 0.807688 − 0.589610i \(-0.200719\pi\)
0.807688 + 0.589610i \(0.200719\pi\)
\(44\) −982.504 −3.36632
\(45\) −136.478 −0.452110
\(46\) −125.835 −0.403333
\(47\) 351.972 1.09235 0.546175 − 0.837671i \(-0.316084\pi\)
0.546175 + 0.837671i \(0.316084\pi\)
\(48\) −701.359 −2.10901
\(49\) 0 0
\(50\) −422.774 −1.19579
\(51\) 688.049 1.88914
\(52\) 1048.31 2.79567
\(53\) 307.972 0.798173 0.399086 − 0.916913i \(-0.369327\pi\)
0.399086 + 0.916913i \(0.369327\pi\)
\(54\) 201.083 0.506740
\(55\) 361.355 0.885910
\(56\) 0 0
\(57\) 754.224 1.75262
\(58\) −1023.84 −2.31789
\(59\) −281.036 −0.620132 −0.310066 − 0.950715i \(-0.600351\pi\)
−0.310066 + 0.950715i \(0.600351\pi\)
\(60\) 778.209 1.67444
\(61\) 53.6060 0.112517 0.0562585 − 0.998416i \(-0.482083\pi\)
0.0562585 + 0.998416i \(0.482083\pi\)
\(62\) 487.774 0.999150
\(63\) 0 0
\(64\) −162.685 −0.317745
\(65\) −385.558 −0.735733
\(66\) 1973.13 3.67993
\(67\) 566.231 1.03248 0.516240 − 0.856444i \(-0.327331\pi\)
0.516240 + 0.856444i \(0.327331\pi\)
\(68\) −1728.46 −3.08245
\(69\) 173.278 0.302323
\(70\) 0 0
\(71\) 59.5211 0.0994909 0.0497455 − 0.998762i \(-0.484159\pi\)
0.0497455 + 0.998762i \(0.484159\pi\)
\(72\) 1013.92 1.65960
\(73\) −233.678 −0.374657 −0.187329 − 0.982297i \(-0.559983\pi\)
−0.187329 + 0.982297i \(0.559983\pi\)
\(74\) −452.013 −0.710073
\(75\) 582.175 0.896316
\(76\) −1894.70 −2.85969
\(77\) 0 0
\(78\) −2105.29 −3.05612
\(79\) −9.50122 −0.0135313 −0.00676563 − 0.999977i \(-0.502154\pi\)
−0.00676563 + 0.999977i \(0.502154\pi\)
\(80\) −648.007 −0.905618
\(81\) −850.991 −1.16734
\(82\) −1935.39 −2.60645
\(83\) −251.011 −0.331952 −0.165976 − 0.986130i \(-0.553077\pi\)
−0.165976 + 0.986130i \(0.553077\pi\)
\(84\) 0 0
\(85\) 635.709 0.811204
\(86\) 2297.93 2.88131
\(87\) 1409.87 1.73740
\(88\) −2684.57 −3.25200
\(89\) 463.981 0.552606 0.276303 − 0.961071i \(-0.410891\pi\)
0.276303 + 0.961071i \(0.410891\pi\)
\(90\) −688.532 −0.806418
\(91\) 0 0
\(92\) −435.296 −0.493290
\(93\) −671.681 −0.748925
\(94\) 1775.70 1.94840
\(95\) 696.851 0.752583
\(96\) −888.143 −0.944226
\(97\) −606.353 −0.634699 −0.317350 − 0.948309i \(-0.602793\pi\)
−0.317350 + 0.948309i \(0.602793\pi\)
\(98\) 0 0
\(99\) −1197.03 −1.21522
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 343.4.a.d.1.18 ✓ 18
7.6 odd 2 343.4.a.e.1.18 yes 18
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
343.4.a.d.1.18 ✓ 18 1.1 even 1 trivial
343.4.a.e.1.18 yes 18 7.6 odd 2