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Properties

Label 152.16.a.d.1.10
Level $152$
Weight $16$
Character 152.1
Self dual yes
Analytic conductor $216.894$
Analytic rank $0$
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] = [152,16,Mod(1,152)] 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("152.1"); S:= CuspForms(chi, 16); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(152, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 16, names="a")
 
Level: \( N \) \(=\) \( 152 = 2^{3} \cdot 19 \)
Weight: \( k \) \(=\) \( 16 \)
Character orbit: \([\chi]\) \(=\) 152.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [18,0,-208] 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: \(216.894127752\)
Analytic rank: \(0\)
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: \( x^{18} - 8 x^{17} - 183998036 x^{16} - 16776585512 x^{15} + \cdots + 19\!\cdots\!00 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: multiple of \( 2^{104}\cdot 3^{5}\cdot 5^{6} \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.10
Root \(1003.27\) of defining polynomial
Character \(\chi\) \(=\) 152.1

$q$-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)
 
\(f(q)\) \(=\) \(q+991.271 q^{3} -246518. q^{5} -1.67649e6 q^{7} -1.33663e7 q^{9} +2.48183e7 q^{11} -3.33893e7 q^{13} -2.44366e8 q^{15} +7.78056e8 q^{17} +8.93872e8 q^{19} -1.66186e9 q^{21} -2.05625e10 q^{23} +3.02537e10 q^{25} -2.74733e10 q^{27} -4.54910e10 q^{29} -2.28185e11 q^{31} +2.46017e10 q^{33} +4.13285e11 q^{35} -1.07156e12 q^{37} -3.30979e10 q^{39} -1.79048e12 q^{41} -5.56561e11 q^{43} +3.29503e12 q^{45} -5.63637e12 q^{47} -1.93694e12 q^{49} +7.71264e11 q^{51} -1.45405e13 q^{53} -6.11818e12 q^{55} +8.86069e11 q^{57} +2.68287e13 q^{59} -8.93028e12 q^{61} +2.24084e13 q^{63} +8.23108e12 q^{65} -1.28306e13 q^{67} -2.03831e13 q^{69} -1.10931e14 q^{71} -7.79296e13 q^{73} +2.99896e13 q^{75} -4.16077e13 q^{77} +2.79499e13 q^{79} +1.64558e14 q^{81} +2.93399e14 q^{83} -1.91805e14 q^{85} -4.50939e13 q^{87} +5.80680e14 q^{89} +5.59769e13 q^{91} -2.26193e14 q^{93} -2.20356e14 q^{95} +1.10050e15 q^{97} -3.31729e14 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q - 208 q^{3} - 61468 q^{5} + 602848 q^{7} + 109718210 q^{9} - 64761792 q^{11} + 705753236 q^{13} - 219662688 q^{15} + 257457716 q^{17} + 16089691302 q^{19} - 17795696968 q^{21} + 8636005896 q^{23} + 187142654182 q^{25}+ \cdots - 20\!\cdots\!64 q^{99}+O(q^{100}) \) 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 0
\(3\) 991.271 0.261687 0.130844 − 0.991403i \(-0.458231\pi\)
0.130844 + 0.991403i \(0.458231\pi\)
\(4\) 0 0
\(5\) −246518. −1.41115 −0.705577 − 0.708634i \(-0.749312\pi\)
−0.705577 + 0.708634i \(0.749312\pi\)
\(6\) 0 0
\(7\) −1.67649e6 −0.769424 −0.384712 − 0.923037i \(-0.625699\pi\)
−0.384712 + 0.923037i \(0.625699\pi\)
\(8\) 0 0
\(9\) −1.33663e7 −0.931520
\(10\) 0 0
\(11\) 2.48183e7 0.383997 0.191998 − 0.981395i \(-0.438503\pi\)
0.191998 + 0.981395i \(0.438503\pi\)
\(12\) 0 0
\(13\) −3.33893e7 −0.147582 −0.0737909 − 0.997274i \(-0.523510\pi\)
−0.0737909 + 0.997274i \(0.523510\pi\)
\(14\) 0 0
\(15\) −2.44366e8 −0.369281
\(16\) 0 0
\(17\) 7.78056e8 0.459879 0.229940 − 0.973205i \(-0.426147\pi\)
0.229940 + 0.973205i \(0.426147\pi\)
\(18\) 0 0
\(19\) 8.93872e8 0.229416
\(20\) 0 0
\(21\) −1.66186e9 −0.201349
\(22\) 0 0
\(23\) −2.05625e10 −1.25927 −0.629634 − 0.776892i \(-0.716795\pi\)
−0.629634 + 0.776892i \(0.716795\pi\)
\(24\) 0 0
\(25\) 3.02537e10 0.991353
\(26\) 0 0
\(27\) −2.74733e10 −0.505454
\(28\) 0 0
\(29\) −4.54910e10 −0.489712 −0.244856 − 0.969559i \(-0.578741\pi\)
−0.244856 + 0.969559i \(0.578741\pi\)
\(30\) 0 0
\(31\) −2.28185e11 −1.48962 −0.744809 − 0.667278i \(-0.767459\pi\)
−0.744809 + 0.667278i \(0.767459\pi\)
\(32\) 0 0
\(33\) 2.46017e10 0.100487
\(34\) 0 0
\(35\) 4.13285e11 1.08577
\(36\) 0 0
\(37\) −1.07156e12 −1.85567 −0.927837 − 0.372986i \(-0.878334\pi\)
−0.927837 + 0.372986i \(0.878334\pi\)
\(38\) 0 0
\(39\) −3.30979e10 −0.0386203
\(40\) 0 0
\(41\) −1.79048e12 −1.43579 −0.717895 − 0.696151i \(-0.754894\pi\)
−0.717895 + 0.696151i \(0.754894\pi\)
\(42\) 0 0
\(43\) −5.56561e11 −0.312248 −0.156124 − 0.987738i \(-0.549900\pi\)
−0.156124 + 0.987738i \(0.549900\pi\)
\(44\) 0 0
\(45\) 3.29503e12 1.31452
\(46\) 0 0
\(47\) −5.63637e12 −1.62280 −0.811401 − 0.584490i \(-0.801295\pi\)
−0.811401 + 0.584490i \(0.801295\pi\)
\(48\) 0 0
\(49\) −1.93694e12 −0.407987
\(50\) 0 0
\(51\) 7.71264e11 0.120345
\(52\) 0 0
\(53\) −1.45405e13 −1.70024 −0.850121 − 0.526588i \(-0.823471\pi\)
−0.850121 + 0.526588i \(0.823471\pi\)
\(54\) 0 0
\(55\) −6.11818e12 −0.541878
\(56\) 0 0
\(57\) 8.86069e11 0.0600352
\(58\) 0 0
\(59\) 2.68287e13 1.40349 0.701745 − 0.712428i \(-0.252405\pi\)
0.701745 + 0.712428i \(0.252405\pi\)
\(60\) 0 0
\(61\) −8.93028e12 −0.363824 −0.181912 − 0.983315i \(-0.558229\pi\)
−0.181912 + 0.983315i \(0.558229\pi\)
\(62\) 0 0
\(63\) 2.24084e13 0.716733
\(64\) 0 0
\(65\) 8.23108e12 0.208260
\(66\) 0 0
\(67\) −1.28306e13 −0.258635 −0.129317 − 0.991603i \(-0.541279\pi\)
−0.129317 + 0.991603i \(0.541279\pi\)
\(68\) 0 0
\(69\) −2.03831e13 −0.329535
\(70\) 0 0
\(71\) −1.10931e14 −1.44749 −0.723744 − 0.690069i \(-0.757580\pi\)
−0.723744 + 0.690069i \(0.757580\pi\)
\(72\) 0 0
\(73\) −7.79296e13 −0.825622 −0.412811 − 0.910817i \(-0.635453\pi\)
−0.412811 + 0.910817i \(0.635453\pi\)
\(74\) 0 0
\(75\) 2.99896e13 0.259425
\(76\) 0 0
\(77\) −4.16077e13 −0.295456
\(78\) 0 0
\(79\) 2.79499e13 0.163748 0.0818742 − 0.996643i \(-0.473909\pi\)
0.0818742 + 0.996643i \(0.473909\pi\)
\(80\) 0 0
\(81\) 1.64558e14 0.799249
\(82\) 0 0
\(83\) 2.93399e14 1.18679 0.593394 − 0.804912i \(-0.297788\pi\)
0.593394 + 0.804912i \(0.297788\pi\)
\(84\) 0 0
\(85\) −1.91805e14 −0.648960
\(86\) 0 0
\(87\) −4.50939e13 −0.128151
\(88\) 0 0
\(89\) 5.80680e14 1.39159 0.695796 − 0.718240i \(-0.255052\pi\)
0.695796 + 0.718240i \(0.255052\pi\)
\(90\) 0 0
\(91\) 5.59769e13 0.113553
\(92\) 0 0
\(93\) −2.26193e14 −0.389814
\(94\) 0 0
\(95\) −2.20356e14 −0.323741
\(96\) 0 0
\(97\) 1.10050e15 1.38294 0.691469 − 0.722406i \(-0.256964\pi\)
0.691469 + 0.722406i \(0.256964\pi\)
\(98\) 0 0
\(99\) −3.31729e14 −0.357701
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 152.16.a.d.1.10 ✓ 18
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
152.16.a.d.1.10 ✓ 18 1.1 even 1 trivial