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 = [2,0,86,0,-2238] 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: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{193}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 48 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 7)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(7.44622\) 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)
 
\(f(q)\) \(=\) \(q+195.817 q^{3} +200.782 q^{5} +2401.00 q^{7} +18661.3 q^{9} -63864.3 q^{11} -164679. q^{13} +39316.5 q^{15} -362910. q^{17} +436498. q^{19} +470156. q^{21} -918199. q^{23} -1.91281e6 q^{25} -200076. q^{27} -3.68643e6 q^{29} -3.47629e6 q^{31} -1.25057e7 q^{33} +482078. q^{35} +1.88149e7 q^{37} -3.22469e7 q^{39} +2.40714e6 q^{41} +1.25306e7 q^{43} +3.74685e6 q^{45} +5.54509e7 q^{47} +5.76480e6 q^{49} -7.10639e7 q^{51} -9.26889e7 q^{53} -1.28228e7 q^{55} +8.54737e7 q^{57} +2.52600e7 q^{59} +6.93275e7 q^{61} +4.48057e7 q^{63} -3.30646e7 q^{65} +2.33494e7 q^{67} -1.79799e8 q^{69} +1.06194e8 q^{71} -2.10115e8 q^{73} -3.74561e8 q^{75} -1.53338e8 q^{77} +149606. q^{79} -4.06488e8 q^{81} -5.21565e8 q^{83} -7.28659e7 q^{85} -7.21865e8 q^{87} +2.98587e8 q^{89} -3.95394e8 q^{91} -6.80716e8 q^{93} +8.76410e7 q^{95} -8.95983e8 q^{97} -1.19179e9 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 86 q^{3} - 2238 q^{5} + 4802 q^{7} + 11038 q^{9} - 35316 q^{11} - 26530 q^{13} + 307136 q^{15} - 463920 q^{17} + 925426 q^{19} + 206486 q^{21} - 778128 q^{23} + 2081722 q^{25} + 2798612 q^{27} - 10003584 q^{29}+ \cdots - 1409417860 q^{99}+O(q^{100}) \) 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
\(3\) 195.817 1.39574 0.697870 − 0.716225i \(-0.254131\pi\)
0.697870 + 0.716225i \(0.254131\pi\)
\(4\) 0 0
\(5\) 200.782 0.143668 0.0718340 − 0.997417i \(-0.477115\pi\)
0.0718340 + 0.997417i \(0.477115\pi\)
\(6\) 0 0
\(7\) 2401.00 0.377964
\(8\) 0 0
\(9\) 18661.3 0.948090
\(10\) 0 0
\(11\) −63864.3 −1.31520 −0.657599 − 0.753369i \(-0.728428\pi\)
−0.657599 + 0.753369i \(0.728428\pi\)
\(12\) 0 0
\(13\) −164679. −1.59916 −0.799581 − 0.600558i \(-0.794945\pi\)
−0.799581 + 0.600558i \(0.794945\pi\)
\(14\) 0 0
\(15\) 39316.5 0.200523
\(16\) 0 0
\(17\) −362910. −1.05385 −0.526925 − 0.849912i \(-0.676655\pi\)
−0.526925 + 0.849912i \(0.676655\pi\)
\(18\) 0 0
\(19\) 436498. 0.768406 0.384203 − 0.923249i \(-0.374476\pi\)
0.384203 + 0.923249i \(0.374476\pi\)
\(20\) 0 0
\(21\) 470156. 0.527540
\(22\) 0 0
\(23\) −918199. −0.684166 −0.342083 − 0.939670i \(-0.611132\pi\)
−0.342083 + 0.939670i \(0.611132\pi\)
\(24\) 0 0
\(25\) −1.91281e6 −0.979359
\(26\) 0 0
\(27\) −200076. −0.0724531
\(28\) 0 0
\(29\) −3.68643e6 −0.967865 −0.483932 − 0.875105i \(-0.660792\pi\)
−0.483932 + 0.875105i \(0.660792\pi\)
\(30\) 0 0
\(31\) −3.47629e6 −0.676064 −0.338032 − 0.941135i \(-0.609761\pi\)
−0.338032 + 0.941135i \(0.609761\pi\)
\(32\) 0 0
\(33\) −1.25057e7 −1.83567
\(34\) 0 0
\(35\) 482078. 0.0543014
\(36\) 0 0
\(37\) 1.88149e7 1.65042 0.825210 − 0.564826i \(-0.191057\pi\)
0.825210 + 0.564826i \(0.191057\pi\)
\(38\) 0 0
\(39\) −3.22469e7 −2.23201
\(40\) 0 0
\(41\) 2.40714e6 0.133038 0.0665188 − 0.997785i \(-0.478811\pi\)
0.0665188 + 0.997785i \(0.478811\pi\)
\(42\) 0 0
\(43\) 1.25306e7 0.558938 0.279469 − 0.960155i \(-0.409842\pi\)
0.279469 + 0.960155i \(0.409842\pi\)
\(44\) 0 0
\(45\) 3.74685e6 0.136210
\(46\) 0 0
\(47\) 5.54509e7 1.65756 0.828779 − 0.559577i \(-0.189036\pi\)
0.828779 + 0.559577i \(0.189036\pi\)
\(48\) 0 0
\(49\) 5.76480e6 0.142857
\(50\) 0 0
\(51\) −7.10639e7 −1.47090
\(52\) 0 0
\(53\) −9.26889e7 −1.61356 −0.806782 − 0.590849i \(-0.798793\pi\)
−0.806782 + 0.590849i \(0.798793\pi\)
\(54\) 0 0
\(55\) −1.28228e7 −0.188952
\(56\) 0 0
\(57\) 8.54737e7 1.07250
\(58\) 0 0
\(59\) 2.52600e7 0.271393 0.135696 − 0.990750i \(-0.456673\pi\)
0.135696 + 0.990750i \(0.456673\pi\)
\(60\) 0 0
\(61\) 6.93275e7 0.641093 0.320547 − 0.947233i \(-0.396133\pi\)
0.320547 + 0.947233i \(0.396133\pi\)
\(62\) 0 0
\(63\) 4.48057e7 0.358344
\(64\) 0 0
\(65\) −3.30646e7 −0.229748
\(66\) 0 0
\(67\) 2.33494e7 0.141559 0.0707796 − 0.997492i \(-0.477451\pi\)
0.0707796 + 0.997492i \(0.477451\pi\)
\(68\) 0 0
\(69\) −1.79799e8 −0.954918
\(70\) 0 0
\(71\) 1.06194e8 0.495950 0.247975 − 0.968766i \(-0.420235\pi\)
0.247975 + 0.968766i \(0.420235\pi\)
\(72\) 0 0
\(73\) −2.10115e8 −0.865974 −0.432987 − 0.901400i \(-0.642540\pi\)
−0.432987 + 0.901400i \(0.642540\pi\)
\(74\) 0 0
\(75\) −3.74561e8 −1.36693
\(76\) 0 0
\(77\) −1.53338e8 −0.497098
\(78\) 0 0
\(79\) 149606. 0.000432144 0 0.000216072 − 1.00000i \(-0.499931\pi\)
0.000216072 1.00000i \(0.499931\pi\)
\(80\) 0 0
\(81\) −4.06488e8 −1.04922
\(82\) 0 0
\(83\) −5.21565e8 −1.20630 −0.603152 − 0.797626i \(-0.706089\pi\)
−0.603152 + 0.797626i \(0.706089\pi\)
\(84\) 0 0
\(85\) −7.28659e7 −0.151405
\(86\) 0 0
\(87\) −7.21865e8 −1.35089
\(88\) 0 0
\(89\) 2.98587e8 0.504448 0.252224 − 0.967669i \(-0.418838\pi\)
0.252224 + 0.967669i \(0.418838\pi\)
\(90\) 0 0
\(91\) −3.95394e8 −0.604426
\(92\) 0 0
\(93\) −6.80716e8 −0.943610
\(94\) 0 0
\(95\) 8.76410e7 0.110395
\(96\) 0 0
\(97\) −8.95983e8 −1.02761 −0.513803 − 0.857908i \(-0.671764\pi\)
−0.513803 + 0.857908i \(0.671764\pi\)
\(98\) 0 0
\(99\) −1.19179e9 −1.24693
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.e.1.2 2
4.3 odd 2 7.10.a.a.1.2 ✓ 2
12.11 even 2 63.10.a.d.1.1 2
20.3 even 4 175.10.b.b.99.2 4
20.7 even 4 175.10.b.b.99.3 4
20.19 odd 2 175.10.a.b.1.1 2
28.3 even 6 49.10.c.b.30.1 4
28.11 odd 6 49.10.c.c.30.1 4
28.19 even 6 49.10.c.b.18.1 4
28.23 odd 6 49.10.c.c.18.1 4
28.27 even 2 49.10.a.b.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
7.10.a.a.1.2 ✓ 2 4.3 odd 2
49.10.a.b.1.2 2 28.27 even 2
49.10.c.b.18.1 4 28.19 even 6
49.10.c.b.30.1 4 28.3 even 6
49.10.c.c.18.1 4 28.23 odd 6
49.10.c.c.30.1 4 28.11 odd 6
63.10.a.d.1.1 2 12.11 even 2
112.10.a.e.1.2 2 1.1 even 1 trivial
175.10.a.b.1.1 2 20.19 odd 2
175.10.b.b.99.2 4 20.3 even 4
175.10.b.b.99.3 4 20.7 even 4