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(5,42)] 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("42.5"); 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([3, 5])) N = Newforms(chi, 12, names="a")
 
Level: \( N \) \(=\) \( 42 = 2 \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 12 \)
Character orbit: \([\chi]\) \(=\) 42.f (of order \(6\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(32.2704135835\)
Analytic rank: \(0\)
Dimension: \(60\)
Relative dimension: \(30\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 5.8
Character \(\chi\) \(=\) 42.5
Dual form 42.12.f.a.17.8

$q$-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+(-27.7128 - 16.0000i) q^{2} +(-13.7199 - 420.665i) q^{3} +(512.000 + 886.810i) q^{4} +(5294.36 - 9170.11i) q^{5} +(-6350.42 + 11877.3i) q^{6} +(18339.4 + 40509.2i) q^{7} -32768.0i q^{8} +(-176771. + 11543.0i) q^{9} +(-293443. + 169420. i) q^{10} +(-264480. + 152698. i) q^{11} +(366025. - 227547. i) q^{12} -608159. i q^{13} +(139912. - 1.41605e6i) q^{14} +(-3.93018e6 - 2.10134e6i) q^{15} +(-524288. + 908093. i) q^{16} +(1.67187e6 + 2.89576e6i) q^{17} +(5.08350e6 + 2.50844e6i) q^{18} +(-1.49324e7 - 8.62124e6i) q^{19} +1.08429e7 q^{20} +(1.67892e7 - 8.27050e6i) q^{21} +9.77265e6 q^{22} +(-4.23940e7 - 2.44762e7i) q^{23} +(-1.37843e7 + 449575. i) q^{24} +(-3.16465e7 - 5.48134e7i) q^{25} +(-9.73055e6 + 1.68538e7i) q^{26} +(7.28100e6 + 7.42027e7i) q^{27} +(-2.65342e7 + 3.70042e7i) q^{28} +3.91221e7i q^{29} +(7.52949e7 + 1.21117e8i) q^{30} +(9.40293e7 - 5.42878e7i) q^{31} +(2.90590e7 - 1.67772e7i) q^{32} +(6.78632e7 + 1.09162e8i) q^{33} -1.07000e8i q^{34} +(4.68569e8 + 4.62965e7i) q^{35} +(-1.00743e8 - 1.50852e8i) q^{36} +(-3.92833e8 + 6.80407e8i) q^{37} +(2.75880e8 + 4.77838e8i) q^{38} +(-2.55831e8 + 8.34390e6i) q^{39} +(-3.00486e8 - 1.73486e8i) q^{40} -9.46696e8 q^{41} +(-5.97603e8 - 3.94278e7i) q^{42} -4.57397e8 q^{43} +(-2.70828e8 - 1.56362e8i) q^{44} +(-8.30037e8 + 1.68212e9i) q^{45} +(7.83238e8 + 1.35661e9i) q^{46} +(-6.01188e8 + 1.04129e9i) q^{47} +(3.89196e8 + 2.08090e8i) q^{48} +(-1.30466e9 + 1.48583e9i) q^{49} +2.02538e9i q^{50} +(1.19521e9 - 7.43025e8i) q^{51} +(5.39322e8 - 3.11378e8i) q^{52} +(3.86732e9 - 2.23280e9i) q^{53} +(9.85467e8 - 2.17286e9i) q^{54} +3.23375e9i q^{55} +(1.32741e9 - 6.00944e8i) q^{56} +(-3.42178e9 + 6.39983e9i) q^{57} +(6.25954e8 - 1.08418e9i) q^{58} +(2.02250e9 + 3.50308e9i) q^{59} +(-1.48763e8 - 4.56121e9i) q^{60} +(6.17240e9 + 3.56363e9i) q^{61} -3.47442e9 q^{62} +(-3.70946e9 - 6.94914e9i) q^{63} -1.07374e9 q^{64} +(-5.57689e9 - 3.21982e9i) q^{65} +(-1.34080e8 - 4.11101e9i) q^{66} +(-1.50560e9 - 2.60778e9i) q^{67} +(-1.71199e9 + 2.96526e9i) q^{68} +(-9.71462e9 + 1.81695e10i) q^{69} +(-1.22446e10 - 8.78011e9i) q^{70} -1.79670e10i q^{71} +(3.78240e8 + 5.79242e9i) q^{72} +(7.94868e9 - 4.58917e9i) q^{73} +(2.17730e10 - 1.25707e10i) q^{74} +(-2.26239e10 + 1.40646e10i) q^{75} -1.76563e10i q^{76} +(-1.10361e10 - 7.91350e9i) q^{77} +(7.22330e9 + 3.86207e9i) q^{78} +(2.41511e10 - 4.18309e10i) q^{79} +(5.55154e9 + 9.61555e9i) q^{80} +(3.11146e10 - 4.08092e9i) q^{81} +(2.62356e10 + 1.51471e10i) q^{82} -6.33073e10 q^{83} +(1.59304e10 + 1.06543e10i) q^{84} +3.54059e10 q^{85} +(1.26758e10 + 7.31836e9i) q^{86} +(1.64573e10 - 5.36753e8i) q^{87} +(5.00360e9 + 8.66649e9i) q^{88} +(-2.33441e10 + 4.04332e10i) q^{89} +(4.99165e10 - 3.33356e10i) q^{90} +(2.46360e10 - 1.11533e10i) q^{91} -5.01272e10i q^{92} +(-2.41271e10 - 3.88100e10i) q^{93} +(3.33212e10 - 1.92380e10i) q^{94} +(-1.58115e11 + 9.12879e10i) q^{95} +(-7.45627e9 - 1.19939e10i) q^{96} +6.38508e10i q^{97} +(5.99291e10 - 2.03018e10i) q^{98} +(4.49897e10 - 3.00453e10i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 60 q + 30720 q^{4} - 95298 q^{7} - 91434 q^{9} + 245184 q^{10} - 8480820 q^{15} - 31457280 q^{16} + 12170496 q^{18} + 1649724 q^{19} + 43254216 q^{21} + 20705664 q^{22} + 4521984 q^{24} - 336132444 q^{25}+ \cdots + 288758042496 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(29\) \(31\)
\(\chi(n)\) \(-1\) \(e\left(\frac{5}{6}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −27.7128 − 16.0000i −0.612372 − 0.353553i
\(3\) −13.7199 − 420.665i −0.0325976 − 0.999469i
\(4\) 512.000 + 886.810i 0.250000 + 0.433013i
\(5\) 5294.36 − 9170.11i 0.757668 − 1.31232i −0.186370 − 0.982480i \(-0.559672\pi\)
0.944037 − 0.329839i \(-0.106994\pi\)
\(6\) −6350.42 + 11877.3i −0.333404 + 0.623572i
\(7\) 18339.4 + 40509.2i 0.412425 + 0.910992i
\(8\) − 32768.0i − 0.353553i
\(9\) −176771. + 11543.0i −0.997875 + 0.0651605i
\(10\) −293443. + 169420.i −0.927950 + 0.535752i
\(11\) −264480. + 152698.i −0.495146 + 0.285873i −0.726707 − 0.686948i \(-0.758950\pi\)
0.231561 + 0.972820i \(0.425617\pi\)
\(12\) 366025. − 227547.i 0.424633 − 0.263982i
\(13\) − 608159.i − 0.454286i −0.973861 − 0.227143i \(-0.927062\pi\)
0.973861 − 0.227143i \(-0.0729384\pi\)
\(14\) 139912. − 1.41605e6i 0.0695264 − 0.703680i
\(15\) −3.93018e6 − 2.10134e6i −1.33632 − 0.714487i
\(16\) −524288. + 908093.i −0.125000 + 0.216506i
\(17\) 1.67187e6 + 2.89576e6i 0.285583 + 0.494645i 0.972750 − 0.231854i \(-0.0744793\pi\)
−0.687167 + 0.726499i \(0.741146\pi\)
\(18\) 5.08350e6 + 2.50844e6i 0.634109 + 0.312900i
\(19\) −1.49324e7 − 8.62124e6i −1.38352 − 0.798776i −0.390946 − 0.920414i \(-0.627852\pi\)
−0.992575 + 0.121638i \(0.961185\pi\)
\(20\) 1.08429e7 0.757668
\(21\) 1.67892e7 − 8.27050e6i 0.897063 − 0.441902i
\(22\) 9.77265e6 0.404285
\(23\) −4.23940e7 − 2.44762e7i −1.37341 − 0.792940i −0.382057 − 0.924139i \(-0.624784\pi\)
−0.991356 + 0.131198i \(0.958118\pi\)
\(24\) −1.37843e7 + 449575.i −0.353365 + 0.0115250i
\(25\) −3.16465e7 − 5.48134e7i −0.648120 − 1.12258i
\(26\) −9.73055e6 + 1.68538e7i −0.160614 + 0.278192i
\(27\) 7.28100e6 + 7.42027e7i 0.0976541 + 0.995220i
\(28\) −2.65342e7 + 3.70042e7i −0.291365 + 0.406333i
\(29\) 3.91221e7i 0.354188i 0.984194 + 0.177094i \(0.0566696\pi\)
−0.984194 + 0.177094i \(0.943330\pi\)
\(30\) 7.52949e7 + 1.21117e8i 0.565716 + 0.909992i
\(31\) 9.40293e7 − 5.42878e7i 0.589894 − 0.340575i −0.175162 − 0.984540i \(-0.556045\pi\)
0.765056 + 0.643964i \(0.222711\pi\)
\(32\) 2.90590e7 − 1.67772e7i 0.153093 − 0.0883883i
\(33\) 6.78632e7 + 1.09162e8i 0.301861 + 0.485564i
\(34\) − 1.07000e8i − 0.403876i
\(35\) 4.68569e8 + 4.62965e7i 1.50799 + 0.148996i
\(36\) −1.00743e8 − 1.50852e8i −0.277684 − 0.415802i
\(37\) −3.92833e8 + 6.80407e8i −0.931319 + 1.61309i −0.150250 + 0.988648i \(0.548008\pi\)
−0.781069 + 0.624444i \(0.785326\pi\)
\(38\) 2.75880e8 + 4.77838e8i 0.564820 + 0.978296i
\(39\) −2.55831e8 + 8.34390e6i −0.454044 + 0.0148086i
\(40\) −3.00486e8 − 1.73486e8i −0.463975 − 0.267876i
\(41\) −9.46696e8 −1.27614 −0.638071 − 0.769977i \(-0.720268\pi\)
−0.638071 + 0.769977i \(0.720268\pi\)
\(42\) −5.97603e8 − 3.94278e7i −0.705573 − 0.0465512i
\(43\) −4.57397e8 −0.474479 −0.237240 − 0.971451i \(-0.576243\pi\)
−0.237240 + 0.971451i \(0.576243\pi\)
\(44\) −2.70828e8 − 1.56362e8i −0.247573 − 0.142936i
\(45\) −8.30037e8 + 1.68212e9i −0.670546 + 1.35890i
\(46\) 7.83238e8 + 1.35661e9i 0.560694 + 0.971150i
\(47\) −6.01188e8 + 1.04129e9i −0.382359 + 0.662266i −0.991399 − 0.130874i \(-0.958222\pi\)
0.609040 + 0.793140i \(0.291555\pi\)
\(48\) 3.89196e8 + 2.08090e8i 0.220466 + 0.117876i
\(49\) −1.30466e9 + 1.48583e9i −0.659811 + 0.751431i
\(50\) 2.02538e9i 0.916581i
\(51\) 1.19521e9 − 7.43025e8i 0.485073 − 0.301556i
\(52\) 5.39322e8 − 3.11378e8i 0.196711 − 0.113571i
\(53\) 3.86732e9 − 2.23280e9i 1.27026 − 0.733386i 0.295224 − 0.955428i \(-0.404606\pi\)
0.975037 + 0.222042i \(0.0712722\pi\)
\(54\) 9.85467e8 − 2.17286e9i 0.292063 − 0.643971i
\(55\) 3.23375e9i 0.866386i
\(56\) 1.32741e9 − 6.00944e8i 0.322084 − 0.145814i
\(57\) −3.42178e9 + 6.39983e9i −0.753252 + 1.40882i
\(58\) 6.25954e8 − 1.08418e9i 0.125224 − 0.216895i
\(59\) 2.02250e9 + 3.50308e9i 0.368301 + 0.637916i 0.989300 − 0.145895i \(-0.0466063\pi\)
−0.620999 + 0.783811i \(0.713273\pi\)
\(60\) −1.48763e8 − 4.56121e9i −0.0246981 − 0.757265i
\(61\) 6.17240e9 + 3.56363e9i 0.935706 + 0.540230i 0.888612 − 0.458660i \(-0.151670\pi\)
0.0470946 + 0.998890i \(0.485004\pi\)
\(62\) −3.47442e9 −0.481646
\(63\) −3.70946e9 − 6.94914e9i −0.470909 − 0.882182i
\(64\) −1.07374e9 −0.125000
\(65\) −5.57689e9 − 3.21982e9i −0.596168 − 0.344197i
\(66\) −1.34080e8 − 4.11101e9i −0.0131787 − 0.404070i
\(67\) −1.50560e9 − 2.60778e9i −0.136238 − 0.235971i 0.789832 − 0.613324i \(-0.210168\pi\)
−0.926070 + 0.377353i \(0.876835\pi\)
\(68\) −1.71199e9 + 2.96526e9i −0.142792 + 0.247322i
\(69\) −9.71462e9 + 1.81695e10i −0.747749 + 1.39853i
\(70\) −1.22446e10 − 8.78011e9i −0.870775 − 0.624397i
\(71\) − 1.79670e10i − 1.18183i −0.806734 − 0.590914i \(-0.798767\pi\)
0.806734 − 0.590914i \(-0.201233\pi\)
\(72\) 3.78240e8 + 5.79242e9i 0.0230377 + 0.352802i
\(73\) 7.94868e9 − 4.58917e9i 0.448766 − 0.259095i −0.258543 − 0.966000i \(-0.583242\pi\)
0.707309 + 0.706905i \(0.249909\pi\)
\(74\) 2.17730e10 − 1.25707e10i 1.14063 − 0.658542i
\(75\) −2.26239e10 + 1.40646e10i −1.10085 + 0.684369i
\(76\) − 1.76563e10i − 0.798776i
\(77\) −1.10361e10 − 7.91350e9i −0.464638 − 0.333173i
\(78\) 7.22330e9 + 3.86207e9i 0.283280 + 0.151460i
\(79\) 2.41511e10 − 4.18309e10i 0.883054 − 1.52949i 0.0351252 − 0.999383i \(-0.488817\pi\)
0.847928 − 0.530111i \(-0.177850\pi\)
\(80\) 5.55154e9 + 9.61555e9i 0.189417 + 0.328080i
\(81\) 3.11146e10 − 4.08092e9i 0.991508 − 0.130044i
\(82\) 2.62356e10 + 1.51471e10i 0.781475 + 0.451185i
\(83\) −6.33073e10 −1.76410 −0.882052 − 0.471153i \(-0.843838\pi\)
−0.882052 + 0.471153i \(0.843838\pi\)
\(84\) 1.59304e10 + 1.06543e10i 0.415615 + 0.277964i
\(85\) 3.54059e10 0.865509
\(86\) 1.26758e10 + 7.31836e9i 0.290558 + 0.167754i
\(87\) 1.64573e10 − 5.36753e8i 0.353999 − 0.0115456i
\(88\) 5.00360e9 + 8.66649e9i 0.101071 + 0.175061i
\(89\) −2.33441e10 + 4.04332e10i −0.443131 + 0.767526i −0.997920 − 0.0644654i \(-0.979466\pi\)
0.554789 + 0.831991i \(0.312799\pi\)
\(90\) 4.99165e10 − 3.33356e10i 0.891068 − 0.595079i
\(91\) 2.46360e10 − 1.11533e10i 0.413850 − 0.187359i
\(92\) − 5.01272e10i − 0.792940i
\(93\) −2.41271e10 − 3.88100e10i −0.359624 − 0.578479i
\(94\) 3.33212e10 − 1.92380e10i 0.468293 − 0.270369i
\(95\) −1.58115e11 + 9.12879e10i −2.09650 + 1.21041i
\(96\) −7.45627e9 − 1.19939e10i −0.0933318 − 0.150131i
\(97\) 6.38508e10i 0.754956i 0.926019 + 0.377478i \(0.123209\pi\)
−0.926019 + 0.377478i \(0.876791\pi\)
\(98\) 5.99291e10 − 2.03018e10i 0.669721 − 0.226877i
\(99\) 4.49897e10 − 3.00453e10i 0.475466 − 0.317529i
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.f.a.5.8 ✓ 60
3.2 odd 2 inner 42.12.f.a.5.28 yes 60
7.3 odd 6 inner 42.12.f.a.17.28 yes 60
21.17 even 6 inner 42.12.f.a.17.8 yes 60
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
42.12.f.a.5.8 ✓ 60 1.1 even 1 trivial
42.12.f.a.5.28 yes 60 3.2 odd 2 inner
42.12.f.a.17.8 yes 60 21.17 even 6 inner
42.12.f.a.17.28 yes 60 7.3 odd 6 inner