Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [112,10,Mod(1,112)] 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("112.1"); S:= CuspForms(chi, 10); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(112, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 10, names="a")
 
Level: \( N \) \(=\) \( 112 = 2^{4} \cdot 7 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 112.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [3,0,92,0,274] 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: yes
Analytic conductor: \(57.6840136504\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: \(\mathbb{Q}[x]/(x^{3} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - 823x - 4578 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{6}\cdot 3 \)
Twist minimal: no (minimal twist has level 56)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-5.79960\) of defining polynomial
Character \(\chi\) \(=\) 112.1

$q$-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)
 
\(f(q)\) \(=\) \(q-189.265 q^{3} -729.944 q^{5} +2401.00 q^{7} +16138.2 q^{9} +33434.1 q^{11} -131907. q^{13} +138153. q^{15} -504408. q^{17} -280047. q^{19} -454425. q^{21} -1.37786e6 q^{23} -1.42031e6 q^{25} +670906. q^{27} +2.14540e6 q^{29} -2.31562e6 q^{31} -6.32790e6 q^{33} -1.75259e6 q^{35} -2.44526e6 q^{37} +2.49653e7 q^{39} -3.40357e7 q^{41} +1.28493e7 q^{43} -1.17800e7 q^{45} +2.22094e7 q^{47} +5.76480e6 q^{49} +9.54666e7 q^{51} +4.76786e7 q^{53} -2.44050e7 q^{55} +5.30030e7 q^{57} +1.04479e7 q^{59} -5.53164e7 q^{61} +3.87478e7 q^{63} +9.62845e7 q^{65} +2.10177e8 q^{67} +2.60780e8 q^{69} -2.00438e8 q^{71} +3.73407e8 q^{73} +2.68814e8 q^{75} +8.02753e7 q^{77} +1.59503e8 q^{79} -4.44627e8 q^{81} -2.02178e8 q^{83} +3.68189e8 q^{85} -4.06049e8 q^{87} -2.91432e8 q^{89} -3.16708e8 q^{91} +4.38265e8 q^{93} +2.04418e8 q^{95} +7.97606e8 q^{97} +5.39566e8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 92 q^{3} + 274 q^{5} + 7203 q^{7} + 51871 q^{9} + 45364 q^{11} - 11158 q^{13} + 95584 q^{15} - 55866 q^{17} - 488772 q^{19} + 220892 q^{21} + 253888 q^{23} - 3872619 q^{25} + 10157192 q^{27} - 765318 q^{29}+ \cdots - 168732636 q^{99}+O(q^{100}) \) 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
\(3\) −189.265 −1.34904 −0.674519 − 0.738257i \(-0.735649\pi\)
−0.674519 + 0.738257i \(0.735649\pi\)
\(4\) 0 0
\(5\) −729.944 −0.522305 −0.261153 − 0.965298i \(-0.584103\pi\)
−0.261153 + 0.965298i \(0.584103\pi\)
\(6\) 0 0
\(7\) 2401.00 0.377964
\(8\) 0 0
\(9\) 16138.2 0.819905
\(10\) 0 0
\(11\) 33434.1 0.688530 0.344265 − 0.938873i \(-0.388128\pi\)
0.344265 + 0.938873i \(0.388128\pi\)
\(12\) 0 0
\(13\) −131907. −1.28092 −0.640460 − 0.767991i \(-0.721256\pi\)
−0.640460 + 0.767991i \(0.721256\pi\)
\(14\) 0 0
\(15\) 138153. 0.704610
\(16\) 0 0
\(17\) −504408. −1.46474 −0.732371 − 0.680905i \(-0.761587\pi\)
−0.732371 + 0.680905i \(0.761587\pi\)
\(18\) 0 0
\(19\) −280047. −0.492992 −0.246496 − 0.969144i \(-0.579279\pi\)
−0.246496 + 0.969144i \(0.579279\pi\)
\(20\) 0 0
\(21\) −454425. −0.509889
\(22\) 0 0
\(23\) −1.37786e6 −1.02666 −0.513332 − 0.858190i \(-0.671589\pi\)
−0.513332 + 0.858190i \(0.671589\pi\)
\(24\) 0 0
\(25\) −1.42031e6 −0.727197
\(26\) 0 0
\(27\) 670906. 0.242954
\(28\) 0 0
\(29\) 2.14540e6 0.563271 0.281635 − 0.959521i \(-0.409123\pi\)
0.281635 + 0.959521i \(0.409123\pi\)
\(30\) 0 0
\(31\) −2.31562e6 −0.450338 −0.225169 − 0.974320i \(-0.572293\pi\)
−0.225169 + 0.974320i \(0.572293\pi\)
\(32\) 0 0
\(33\) −6.32790e6 −0.928853
\(34\) 0 0
\(35\) −1.75259e6 −0.197413
\(36\) 0 0
\(37\) −2.44526e6 −0.214495 −0.107247 − 0.994232i \(-0.534204\pi\)
−0.107247 + 0.994232i \(0.534204\pi\)
\(38\) 0 0
\(39\) 2.49653e7 1.72801
\(40\) 0 0
\(41\) −3.40357e7 −1.88108 −0.940541 − 0.339681i \(-0.889681\pi\)
−0.940541 + 0.339681i \(0.889681\pi\)
\(42\) 0 0
\(43\) 1.28493e7 0.573153 0.286576 − 0.958057i \(-0.407483\pi\)
0.286576 + 0.958057i \(0.407483\pi\)
\(44\) 0 0
\(45\) −1.17800e7 −0.428241
\(46\) 0 0
\(47\) 2.22094e7 0.663891 0.331946 − 0.943299i \(-0.392295\pi\)
0.331946 + 0.943299i \(0.392295\pi\)
\(48\) 0 0
\(49\) 5.76480e6 0.142857
\(50\) 0 0
\(51\) 9.54666e7 1.97599
\(52\) 0 0
\(53\) 4.76786e7 0.830007 0.415003 − 0.909820i \(-0.363780\pi\)
0.415003 + 0.909820i \(0.363780\pi\)
\(54\) 0 0
\(55\) −2.44050e7 −0.359623
\(56\) 0 0
\(57\) 5.30030e7 0.665065
\(58\) 0 0
\(59\) 1.04479e7 0.112252 0.0561259 − 0.998424i \(-0.482125\pi\)
0.0561259 + 0.998424i \(0.482125\pi\)
\(60\) 0 0
\(61\) −5.53164e7 −0.511529 −0.255764 − 0.966739i \(-0.582327\pi\)
−0.255764 + 0.966739i \(0.582327\pi\)
\(62\) 0 0
\(63\) 3.87478e7 0.309895
\(64\) 0 0
\(65\) 9.62845e7 0.669031
\(66\) 0 0
\(67\) 2.10177e8 1.27423 0.637115 − 0.770769i \(-0.280128\pi\)
0.637115 + 0.770769i \(0.280128\pi\)
\(68\) 0 0
\(69\) 2.60780e8 1.38501
\(70\) 0 0
\(71\) −2.00438e8 −0.936089 −0.468045 − 0.883705i \(-0.655041\pi\)
−0.468045 + 0.883705i \(0.655041\pi\)
\(72\) 0 0
\(73\) 3.73407e8 1.53897 0.769485 − 0.638665i \(-0.220513\pi\)
0.769485 + 0.638665i \(0.220513\pi\)
\(74\) 0 0
\(75\) 2.68814e8 0.981017
\(76\) 0 0
\(77\) 8.02753e7 0.260240
\(78\) 0 0
\(79\) 1.59503e8 0.460732 0.230366 − 0.973104i \(-0.426008\pi\)
0.230366 + 0.973104i \(0.426008\pi\)
\(80\) 0 0
\(81\) −4.44627e8 −1.14766
\(82\) 0 0
\(83\) −2.02178e8 −0.467608 −0.233804 − 0.972284i \(-0.575117\pi\)
−0.233804 + 0.972284i \(0.575117\pi\)
\(84\) 0 0
\(85\) 3.68189e8 0.765043
\(86\) 0 0
\(87\) −4.06049e8 −0.759874
\(88\) 0 0
\(89\) −2.91432e8 −0.492360 −0.246180 − 0.969224i \(-0.579175\pi\)
−0.246180 + 0.969224i \(0.579175\pi\)
\(90\) 0 0
\(91\) −3.16708e8 −0.484142
\(92\) 0 0
\(93\) 4.38265e8 0.607524
\(94\) 0 0
\(95\) 2.04418e8 0.257492
\(96\) 0 0
\(97\) 7.97606e8 0.914778 0.457389 − 0.889267i \(-0.348785\pi\)
0.457389 + 0.889267i \(0.348785\pi\)
\(98\) 0 0
\(99\) 5.39566e8 0.564529
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 112.10.a.i.1.1 3
4.3 odd 2 56.10.a.a.1.3 ✓ 3
28.27 even 2 392.10.a.d.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
56.10.a.a.1.3 ✓ 3 4.3 odd 2
112.10.a.i.1.1 3 1.1 even 1 trivial
392.10.a.d.1.1 3 28.27 even 2