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Properties

Label 343.4.a.d.1.10
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 1.10
Root \(0.636124\) 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)
 
\(f(q)\) \(=\) \(q+0.636124 q^{2} +1.52022 q^{3} -7.59535 q^{4} +18.5469 q^{5} +0.967051 q^{6} -9.92057 q^{8} -24.6889 q^{9} +11.7981 q^{10} -57.3509 q^{11} -11.5466 q^{12} -13.6359 q^{13} +28.1954 q^{15} +54.4521 q^{16} -95.0423 q^{17} -15.7052 q^{18} +72.5147 q^{19} -140.870 q^{20} -36.4823 q^{22} -112.522 q^{23} -15.0815 q^{24} +218.986 q^{25} -8.67413 q^{26} -78.5788 q^{27} -12.2203 q^{29} +17.9358 q^{30} +164.712 q^{31} +114.003 q^{32} -87.1863 q^{33} -60.4587 q^{34} +187.521 q^{36} -69.4323 q^{37} +46.1283 q^{38} -20.7297 q^{39} -183.995 q^{40} -506.162 q^{41} -324.214 q^{43} +435.600 q^{44} -457.902 q^{45} -71.5780 q^{46} -380.673 q^{47} +82.7794 q^{48} +139.302 q^{50} -144.486 q^{51} +103.569 q^{52} -303.650 q^{53} -49.9858 q^{54} -1063.68 q^{55} +110.239 q^{57} -7.77364 q^{58} +354.369 q^{59} -214.154 q^{60} +727.617 q^{61} +104.777 q^{62} -363.097 q^{64} -252.903 q^{65} -55.4613 q^{66} -618.806 q^{67} +721.879 q^{68} -171.059 q^{69} +779.719 q^{71} +244.928 q^{72} -302.145 q^{73} -44.1675 q^{74} +332.908 q^{75} -550.774 q^{76} -13.1866 q^{78} +170.131 q^{79} +1009.91 q^{80} +547.143 q^{81} -321.982 q^{82} +12.9130 q^{83} -1762.74 q^{85} -206.240 q^{86} -18.5776 q^{87} +568.954 q^{88} -1125.76 q^{89} -291.282 q^{90} +854.644 q^{92} +250.399 q^{93} -242.155 q^{94} +1344.92 q^{95} +173.310 q^{96} +1365.31 q^{97} +1415.93 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q + 2 q^{2} - 22 q^{3} + 62 q^{4} - 62 q^{5} - 41 q^{6} - 18 q^{8} + 138 q^{9} - 78 q^{10} - 26 q^{11} - 245 q^{12} - 168 q^{13} + 118 q^{15} + 358 q^{16} - 568 q^{17} + 106 q^{18} - 188 q^{19} - 756 q^{20}+ \cdots + 512 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.636124 0.224904 0.112452 − 0.993657i \(-0.464130\pi\)
0.112452 + 0.993657i \(0.464130\pi\)
\(3\) 1.52022 0.292567 0.146284 − 0.989243i \(-0.453269\pi\)
0.146284 + 0.989243i \(0.453269\pi\)
\(4\) −7.59535 −0.949418
\(5\) 18.5469 1.65888 0.829440 − 0.558595i \(-0.188659\pi\)
0.829440 + 0.558595i \(0.188659\pi\)
\(6\) 0.967051 0.0657995
\(7\) 0 0
\(8\) −9.92057 −0.438431
\(9\) −24.6889 −0.914404
\(10\) 11.7981 0.373089
\(11\) −57.3509 −1.57200 −0.785998 − 0.618229i \(-0.787850\pi\)
−0.785998 + 0.618229i \(0.787850\pi\)
\(12\) −11.5466 −0.277769
\(13\) −13.6359 −0.290917 −0.145459 − 0.989364i \(-0.546466\pi\)
−0.145459 + 0.989364i \(0.546466\pi\)
\(14\) 0 0
\(15\) 28.1954 0.485334
\(16\) 54.4521 0.850813
\(17\) −95.0423 −1.35595 −0.677975 − 0.735085i \(-0.737142\pi\)
−0.677975 + 0.735085i \(0.737142\pi\)
\(18\) −15.7052 −0.205653
\(19\) 72.5147 0.875580 0.437790 − 0.899077i \(-0.355761\pi\)
0.437790 + 0.899077i \(0.355761\pi\)
\(20\) −140.870 −1.57497
\(21\) 0 0
\(22\) −36.4823 −0.353548
\(23\) −112.522 −1.02011 −0.510054 − 0.860142i \(-0.670375\pi\)
−0.510054 + 0.860142i \(0.670375\pi\)
\(24\) −15.0815 −0.128271
\(25\) 218.986 1.75189
\(26\) −8.67413 −0.0654283
\(27\) −78.5788 −0.560092
\(28\) 0 0
\(29\) −12.2203 −0.0782503 −0.0391251 − 0.999234i \(-0.512457\pi\)
−0.0391251 + 0.999234i \(0.512457\pi\)
\(30\) 17.9358 0.109154
\(31\) 164.712 0.954295 0.477148 − 0.878823i \(-0.341671\pi\)
0.477148 + 0.878823i \(0.341671\pi\)
\(32\) 114.003 0.629783
\(33\) −87.1863 −0.459915
\(34\) −60.4587 −0.304958
\(35\) 0 0
\(36\) 187.521 0.868152
\(37\) −69.4323 −0.308503 −0.154251 − 0.988032i \(-0.549297\pi\)
−0.154251 + 0.988032i \(0.549297\pi\)
\(38\) 46.1283 0.196921
\(39\) −20.7297 −0.0851128
\(40\) −183.995 −0.727306
\(41\) −506.162 −1.92803 −0.964015 − 0.265848i \(-0.914348\pi\)
−0.964015 + 0.265848i \(0.914348\pi\)
\(42\) 0 0
\(43\) −324.214 −1.14982 −0.574909 − 0.818217i \(-0.694963\pi\)
−0.574909 + 0.818217i \(0.694963\pi\)
\(44\) 435.600 1.49248
\(45\) −457.902 −1.51689
\(46\) −71.5780 −0.229426
\(47\) −380.673 −1.18142 −0.590711 − 0.806883i \(-0.701153\pi\)
−0.590711 + 0.806883i \(0.701153\pi\)
\(48\) 82.7794 0.248920
\(49\) 0 0
\(50\) 139.302 0.394006
\(51\) −144.486 −0.396707
\(52\) 103.569 0.276202
\(53\) −303.650 −0.786973 −0.393487 − 0.919330i \(-0.628731\pi\)
−0.393487 + 0.919330i \(0.628731\pi\)
\(54\) −49.9858 −0.125967
\(55\) −1063.68 −2.60775
\(56\) 0 0
\(57\) 110.239 0.256166
\(58\) −7.77364 −0.0175988
\(59\) 354.369 0.781947 0.390974 − 0.920402i \(-0.372138\pi\)
0.390974 + 0.920402i \(0.372138\pi\)
\(60\) −214.154 −0.460785
\(61\) 727.617 1.52724 0.763622 − 0.645664i \(-0.223419\pi\)
0.763622 + 0.645664i \(0.223419\pi\)
\(62\) 104.777 0.214625
\(63\) 0 0
\(64\) −363.097 −0.709173
\(65\) −252.903 −0.482597
\(66\) −55.4613 −0.103437
\(67\) −618.806 −1.12835 −0.564173 − 0.825656i \(-0.690805\pi\)
−0.564173 + 0.825656i \(0.690805\pi\)
\(68\) 721.879 1.28736
\(69\) −171.059 −0.298450
\(70\) 0 0
\(71\) 779.719 1.30332 0.651660 − 0.758511i \(-0.274073\pi\)
0.651660 + 0.758511i \(0.274073\pi\)
\(72\) 244.928 0.400904
\(73\) −302.145 −0.484429 −0.242215 − 0.970223i \(-0.577874\pi\)
−0.242215 + 0.970223i \(0.577874\pi\)
\(74\) −44.1675 −0.0693834
\(75\) 332.908 0.512545
\(76\) −550.774 −0.831291
\(77\) 0 0
\(78\) −13.1866 −0.0191422
\(79\) 170.131 0.242294 0.121147 − 0.992635i \(-0.461343\pi\)
0.121147 + 0.992635i \(0.461343\pi\)
\(80\) 1009.91 1.41140
\(81\) 547.143 0.750540
\(82\) −321.982 −0.433621
\(83\) 12.9130 0.0170770 0.00853849 − 0.999964i \(-0.497282\pi\)
0.00853849 + 0.999964i \(0.497282\pi\)
\(84\) 0 0
\(85\) −1762.74 −2.24936
\(86\) −206.240 −0.258598
\(87\) −18.5776 −0.0228935
\(88\) 568.954 0.689212
\(89\) −1125.76 −1.34079 −0.670396 − 0.742004i \(-0.733876\pi\)
−0.670396 + 0.742004i \(0.733876\pi\)
\(90\) −291.282 −0.341154
\(91\) 0 0
\(92\) 854.644 0.968509
\(93\) 250.399 0.279196
\(94\) −242.155 −0.265706
\(95\) 1344.92 1.45248
\(96\) 173.310 0.184254
\(97\) 1365.31 1.42913 0.714567 − 0.699567i \(-0.246624\pi\)
0.714567 + 0.699567i \(0.246624\pi\)
\(98\) 0 0
\(99\) 1415.93 1.43744
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.1.10 ✓ 18
7.6 odd 2 343.4.a.e.1.10 yes 18
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
343.4.a.d.1.10 ✓ 18 1.1 even 1 trivial
343.4.a.e.1.10 yes 18 7.6 odd 2