Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [42,12,Mod(25,42)] 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("42.25"); S:= CuspForms(chi, 12); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(42, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 4])) N = Newforms(chi, 12, names="a")
 
Level: \( N \) \(=\) \( 42 = 2 \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 12 \)
Character orbit: \([\chi]\) \(=\) 42.e (of order \(3\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,128] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(32.2704135835\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 2 x^{7} + 131491 x^{6} + 44838722 x^{5} + 18152831051 x^{4} + 2926931386118 x^{3} + \cdots + 82\!\cdots\!00 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{8}\cdot 3^{3}\cdot 7^{6} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 37.1
Root \(217.088 + 376.008i\) of defining polynomial
Character \(\chi\) \(=\) 42.37
Dual form 42.12.e.d.25.1

$q$-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+(16.0000 - 27.7128i) q^{2} +(121.500 + 210.444i) q^{3} +(-512.000 - 886.810i) q^{4} +(-4438.44 + 7687.60i) q^{5} +7776.00 q^{6} +(-14427.2 + 42061.6i) q^{7} -32768.0 q^{8} +(-29524.5 + 51137.9i) q^{9} +(142030. + 246003. i) q^{10} +(-334697. - 579713. i) q^{11} +(124416. - 215495. i) q^{12} +813042. q^{13} +(934811. + 1.07281e6i) q^{14} -2.15708e6 q^{15} +(-524288. + 908093. i) q^{16} +(-4.92429e6 - 8.52912e6i) q^{17} +(944784. + 1.63641e6i) q^{18} +(6.00270e6 - 1.03970e7i) q^{19} +9.08992e6 q^{20} +(-1.06045e7 + 2.07437e6i) q^{21} -2.14206e7 q^{22} +(-142538. + 246882. i) q^{23} +(-3.98131e6 - 6.89583e6i) q^{24} +(-1.49854e7 - 2.59555e7i) q^{25} +(1.30087e7 - 2.25317e7i) q^{26} -1.43489e7 q^{27} +(4.46874e7 - 8.74137e6i) q^{28} +1.90584e8 q^{29} +(-3.45133e7 + 5.97788e7i) q^{30} +(-9.50052e7 - 1.64554e8i) q^{31} +(1.67772e7 + 2.90590e7i) q^{32} +(8.13314e7 - 1.40870e8i) q^{33} -3.15155e8 q^{34} +(-2.59319e8 - 2.97599e8i) q^{35} +6.04662e7 q^{36} +(-8.41168e7 + 1.45695e8i) q^{37} +(-1.92086e8 - 3.32703e8i) q^{38} +(9.87846e7 + 1.71100e8i) q^{39} +(1.45439e8 - 2.51907e8i) q^{40} -5.04071e8 q^{41} +(-1.12186e8 + 3.27071e8i) q^{42} -9.47416e8 q^{43} +(-3.42730e8 + 5.93626e8i) q^{44} +(-2.62085e8 - 4.53945e8i) q^{45} +(4.56120e6 + 7.90023e6i) q^{46} +(-1.17680e9 + 2.03829e9i) q^{47} -2.54804e8 q^{48} +(-1.56104e9 - 1.21366e9i) q^{49} -9.59068e8 q^{50} +(1.19660e9 - 2.07258e9i) q^{51} +(-4.16278e8 - 7.21014e8i) q^{52} +(-2.19625e9 - 3.80402e9i) q^{53} +(-2.29583e8 + 3.97649e8i) q^{54} +5.94213e9 q^{55} +(4.72751e8 - 1.37828e9i) q^{56} +2.91731e9 q^{57} +(3.04934e9 - 5.28161e9i) q^{58} +(-4.59414e9 - 7.95729e9i) q^{59} +(1.10443e9 + 1.91292e9i) q^{60} +(-4.72985e8 + 8.19235e8i) q^{61} -6.08033e9 q^{62} +(-1.72499e9 - 1.97963e9i) q^{63} +1.07374e9 q^{64} +(-3.60864e9 + 6.25035e9i) q^{65} +(-2.60261e9 - 4.50785e9i) q^{66} +(7.47404e9 + 1.29454e10i) q^{67} +(-5.04247e9 + 8.73382e9i) q^{68} -6.92733e7 q^{69} +(-1.23964e10 + 2.42488e9i) q^{70} +1.74281e10 q^{71} +(9.67459e8 - 1.67569e9i) q^{72} +(-2.73831e9 - 4.74289e9i) q^{73} +(2.69174e9 + 4.66223e9i) q^{74} +(3.64146e9 - 6.30719e9i) q^{75} -1.22935e10 q^{76} +(2.92124e10 - 5.71428e9i) q^{77} +6.32222e9 q^{78} +(-1.11997e10 + 1.93985e10i) q^{79} +(-4.65404e9 - 8.06104e9i) q^{80} +(-1.74339e9 - 3.01964e9i) q^{81} +(-8.06514e9 + 1.39692e10i) q^{82} -7.41530e9 q^{83} +(7.26909e9 + 8.34213e9i) q^{84} +8.74246e10 q^{85} +(-1.51587e10 + 2.62556e10i) q^{86} +(2.31559e10 + 4.01072e10i) q^{87} +(1.09674e10 + 1.89960e10i) q^{88} +(1.96276e9 - 3.39961e9i) q^{89} -1.67735e10 q^{90} +(-1.17299e10 + 3.41979e10i) q^{91} +2.91917e8 q^{92} +(2.30863e10 - 3.99866e10i) q^{93} +(3.76577e10 + 6.52251e10i) q^{94} +(5.32852e10 + 9.22927e10i) q^{95} +(-4.07686e9 + 7.06133e9i) q^{96} -1.53114e10 q^{97} +(-5.86107e10 + 2.38421e10i) q^{98} +3.95271e10 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 128 q^{2} + 972 q^{3} - 4096 q^{4} - 1420 q^{5} + 62208 q^{6} - 66362 q^{7} - 262144 q^{8} - 236196 q^{9} + 45440 q^{10} - 861962 q^{11} + 995328 q^{12} + 836748 q^{13} - 1361216 q^{14} - 690120 q^{15}+ \cdots + 101795988276 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/42\mathbb{Z}\right)^\times\).

\(n\) \(29\) \(31\)
\(\chi(n)\) \(1\) \(e\left(\frac{1}{3}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 16.0000 − 27.7128i 0.353553 − 0.612372i
\(3\) 121.500 + 210.444i 0.288675 + 0.500000i
\(4\) −512.000 − 886.810i −0.250000 − 0.433013i
\(5\) −4438.44 + 7687.60i −0.635178 + 1.10016i 0.351300 + 0.936263i \(0.385740\pi\)
−0.986477 + 0.163897i \(0.947593\pi\)
\(6\) 7776.00 0.408248
\(7\) −14427.2 + 42061.6i −0.324447 + 0.945904i
\(8\) −32768.0 −0.353553
\(9\) −29524.5 + 51137.9i −0.166667 + 0.288675i
\(10\) 142030. + 246003.i 0.449139 + 0.777931i
\(11\) −334697. − 579713.i −0.626603 − 1.08531i −0.988229 − 0.152985i \(-0.951112\pi\)
0.361626 − 0.932323i \(-0.382222\pi\)
\(12\) 124416. − 215495.i 0.144338 − 0.250000i
\(13\) 813042. 0.607330 0.303665 − 0.952779i \(-0.401790\pi\)
0.303665 + 0.952779i \(0.401790\pi\)
\(14\) 934811. + 1.07281e6i 0.464536 + 0.533110i
\(15\) −2.15708e6 −0.733440
\(16\) −524288. + 908093.i −0.125000 + 0.216506i
\(17\) −4.92429e6 − 8.52912e6i −0.841152 − 1.45692i −0.888921 − 0.458060i \(-0.848545\pi\)
0.0477695 − 0.998858i \(-0.484789\pi\)
\(18\) 944784. + 1.63641e6i 0.117851 + 0.204124i
\(19\) 6.00270e6 − 1.03970e7i 0.556163 − 0.963302i −0.441649 − 0.897188i \(-0.645607\pi\)
0.997812 − 0.0661142i \(-0.0210602\pi\)
\(20\) 9.08992e6 0.635178
\(21\) −1.06045e7 + 2.07437e6i −0.566612 + 0.110836i
\(22\) −2.14206e7 −0.886150
\(23\) −142538. + 246882.i −0.00461771 + 0.00799810i −0.868325 − 0.495996i \(-0.834803\pi\)
0.863707 + 0.503994i \(0.168137\pi\)
\(24\) −3.98131e6 − 6.89583e6i −0.102062 − 0.176777i
\(25\) −1.49854e7 − 2.59555e7i −0.306902 − 0.531569i
\(26\) 1.30087e7 − 2.25317e7i 0.214724 − 0.371912i
\(27\) −1.43489e7 −0.192450
\(28\) 4.46874e7 − 8.74137e6i 0.490700 − 0.0959865i
\(29\) 1.90584e8 1.72543 0.862714 − 0.505693i \(-0.168763\pi\)
0.862714 + 0.505693i \(0.168763\pi\)
\(30\) −3.45133e7 + 5.97788e7i −0.259310 + 0.449139i
\(31\) −9.50052e7 − 1.64554e8i −0.596016 − 1.03233i −0.993403 − 0.114679i \(-0.963416\pi\)
0.397386 − 0.917651i \(-0.369917\pi\)
\(32\) 1.67772e7 + 2.90590e7i 0.0883883 + 0.153093i
\(33\) 8.13314e7 − 1.40870e8i 0.361769 − 0.626603i
\(34\) −3.15155e8 −1.18957
\(35\) −2.59319e8 − 2.97599e8i −0.834565 − 0.957761i
\(36\) 6.04662e7 0.166667
\(37\) −8.41168e7 + 1.45695e8i −0.199422 + 0.345409i −0.948341 − 0.317252i \(-0.897240\pi\)
0.748919 + 0.662661i \(0.230573\pi\)
\(38\) −1.92086e8 − 3.32703e8i −0.393266 − 0.681157i
\(39\) 9.87846e7 + 1.71100e8i 0.175321 + 0.303665i
\(40\) 1.45439e8 − 2.51907e8i 0.224569 − 0.388965i
\(41\) −5.04071e8 −0.679486 −0.339743 − 0.940518i \(-0.610340\pi\)
−0.339743 + 0.940518i \(0.610340\pi\)
\(42\) −1.12186e8 + 3.27071e8i −0.132455 + 0.386164i
\(43\) −9.47416e8 −0.982798 −0.491399 − 0.870935i \(-0.663514\pi\)
−0.491399 + 0.870935i \(0.663514\pi\)
\(44\) −3.42730e8 + 5.93626e8i −0.313301 + 0.542654i
\(45\) −2.62085e8 − 4.53945e8i −0.211726 − 0.366720i
\(46\) 4.56120e6 + 7.90023e6i 0.00326521 + 0.00565551i
\(47\) −1.17680e9 + 2.03829e9i −0.748456 + 1.29636i 0.200107 + 0.979774i \(0.435871\pi\)
−0.948563 + 0.316589i \(0.897462\pi\)
\(48\) −2.54804e8 −0.144338
\(49\) −1.56104e9 − 1.21366e9i −0.789469 − 0.613791i
\(50\) −9.59068e8 −0.434025
\(51\) 1.19660e9 − 2.07258e9i 0.485639 − 0.841152i
\(52\) −4.16278e8 − 7.21014e8i −0.151832 − 0.262982i
\(53\) −2.19625e9 − 3.80402e9i −0.721381 − 1.24947i −0.960446 − 0.278465i \(-0.910174\pi\)
0.239065 − 0.971004i \(-0.423159\pi\)
\(54\) −2.29583e8 + 3.97649e8i −0.0680414 + 0.117851i
\(55\) 5.94213e9 1.59202
\(56\) 4.72751e8 − 1.37828e9i 0.114709 − 0.334428i
\(57\) 2.91731e9 0.642201
\(58\) 3.04934e9 − 5.28161e9i 0.610031 − 1.05660i
\(59\) −4.59414e9 − 7.95729e9i −0.836601 − 1.44904i −0.892720 − 0.450611i \(-0.851206\pi\)
0.0561191 − 0.998424i \(-0.482127\pi\)
\(60\) 1.10443e9 + 1.91292e9i 0.183360 + 0.317589i
\(61\) −4.72985e8 + 8.19235e8i −0.0717024 + 0.124192i −0.899648 − 0.436617i \(-0.856177\pi\)
0.827945 + 0.560809i \(0.189510\pi\)
\(62\) −6.08033e9 −0.842894
\(63\) −1.72499e9 − 1.97963e9i −0.218985 − 0.251310i
\(64\) 1.07374e9 0.125000
\(65\) −3.60864e9 + 6.25035e9i −0.385762 + 0.668160i
\(66\) −2.60261e9 − 4.50785e9i −0.255810 − 0.443075i
\(67\) 7.47404e9 + 1.29454e10i 0.676308 + 1.17140i 0.976085 + 0.217390i \(0.0697543\pi\)
−0.299777 + 0.954009i \(0.596912\pi\)
\(68\) −5.04247e9 + 8.73382e9i −0.420576 + 0.728459i
\(69\) −6.92733e7 −0.00533207
\(70\) −1.23964e10 + 2.42488e9i −0.881569 + 0.172445i
\(71\) 1.74281e10 1.14638 0.573191 − 0.819422i \(-0.305705\pi\)
0.573191 + 0.819422i \(0.305705\pi\)
\(72\) 9.67459e8 − 1.67569e9i 0.0589256 − 0.102062i
\(73\) −2.73831e9 − 4.74289e9i −0.154599 − 0.267774i 0.778314 − 0.627875i \(-0.216075\pi\)
−0.932913 + 0.360102i \(0.882742\pi\)
\(74\) 2.69174e9 + 4.66223e9i 0.141013 + 0.244241i
\(75\) 3.64146e9 − 6.30719e9i 0.177190 − 0.306902i
\(76\) −1.22935e10 −0.556163
\(77\) 2.92124e10 − 5.71428e9i 1.22990 − 0.240582i
\(78\) 6.32222e9 0.247941
\(79\) −1.11997e10 + 1.93985e10i −0.409503 + 0.709281i −0.994834 − 0.101514i \(-0.967631\pi\)
0.585331 + 0.810795i \(0.300965\pi\)
\(80\) −4.65404e9 − 8.06104e9i −0.158794 − 0.275040i
\(81\) −1.74339e9 − 3.01964e9i −0.0555556 − 0.0962250i
\(82\) −8.06514e9 + 1.39692e10i −0.240235 + 0.416099i
\(83\) −7.41530e9 −0.206633 −0.103316 − 0.994649i \(-0.532945\pi\)
−0.103316 + 0.994649i \(0.532945\pi\)
\(84\) 7.26909e9 + 8.34213e9i 0.189646 + 0.217641i
\(85\) 8.74246e10 2.13712
\(86\) −1.51587e10 + 2.62556e10i −0.347471 + 0.601838i
\(87\) 2.31559e10 + 4.01072e10i 0.498088 + 0.862714i
\(88\) 1.09674e10 + 1.89960e10i 0.221538 + 0.383714i
\(89\) 1.96276e9 − 3.39961e9i 0.0372583 − 0.0645333i −0.846795 − 0.531919i \(-0.821471\pi\)
0.884053 + 0.467386i \(0.154804\pi\)
\(90\) −1.67735e10 −0.299426
\(91\) −1.17299e10 + 3.41979e10i −0.197046 + 0.574476i
\(92\) 2.91917e8 0.00461771
\(93\) 2.30863e10 − 3.99866e10i 0.344110 − 0.596016i
\(94\) 3.76577e10 + 6.52251e10i 0.529238 + 0.916667i
\(95\) 5.32852e10 + 9.22927e10i 0.706524 + 1.22374i
\(96\) −4.07686e9 + 7.06133e9i −0.0510310 + 0.0883883i
\(97\) −1.53114e10 −0.181038 −0.0905189 − 0.995895i \(-0.528853\pi\)
−0.0905189 + 0.995895i \(0.528853\pi\)
\(98\) −5.86107e10 + 2.38421e10i −0.654988 + 0.266441i
\(99\) 3.95271e10 0.417735
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 42.12.e.d.37.1 yes 8
3.2 odd 2 126.12.g.d.37.4 8
7.4 even 3 inner 42.12.e.d.25.1 ✓ 8
21.11 odd 6 126.12.g.d.109.4 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
42.12.e.d.25.1 ✓ 8 7.4 even 3 inner
42.12.e.d.37.1 yes 8 1.1 even 1 trivial
126.12.g.d.37.4 8 3.2 odd 2
126.12.g.d.109.4 8 21.11 odd 6