Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [49,10,Mod(18,49)] 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("49.18"); S:= CuspForms(chi, 10); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(49, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([4])) N = Newforms(chi, 10, names="a")
 
Level: \( N \) \(=\) \( 49 = 7^{2} \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 49.c (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,6,-86] 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: no
Analytic conductor: \(25.2367559720\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{193})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} + 49x^{2} + 48x + 2304 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 7)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 18.1
Root \(-3.22311 - 5.58259i\) of defining polynomial
Character \(\chi\) \(=\) 49.18
Dual form 49.10.c.b.30.1

$q$-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+(-5.44622 - 9.43313i) q^{2} +(-97.9084 + 169.582i) q^{3} +(196.677 - 340.655i) q^{4} +(100.391 + 173.882i) q^{5} +2132.92 q^{6} -9861.52 q^{8} +(-9330.63 - 16161.1i) q^{9} +(1093.50 - 1894.01i) q^{10} +(-31932.1 + 55308.1i) q^{11} +(38512.7 + 66706.0i) q^{12} +164679. q^{13} -39316.5 q^{15} +(-46990.7 - 81390.3i) q^{16} +(-181455. + 314289. i) q^{17} +(-101633. + 176034. i) q^{18} +(-218249. - 378018. i) q^{19} +78978.6 q^{20} +695638. q^{22} +(-459100. - 795184. i) q^{23} +(965527. - 1.67234e6i) q^{24} +(956406. - 1.65654e6i) q^{25} +(-896877. - 1.55344e6i) q^{26} -200076. q^{27} -3.68643e6 q^{29} +(214127. + 370878. i) q^{30} +(1.73814e6 - 3.01055e6i) q^{31} +(-3.03639e6 + 5.25919e6i) q^{32} +(-6.25285e6 - 1.08303e7i) q^{33} +3.95298e6 q^{34} -7.34049e6 q^{36} +(-9.40745e6 - 1.62942e7i) q^{37} +(-2.37726e6 + 4.11754e6i) q^{38} +(-1.61234e7 + 2.79266e7i) q^{39} +(-990009. - 1.71475e6i) q^{40} -2.40714e6 q^{41} -1.25306e7 q^{43} +(1.25607e7 + 2.17557e7i) q^{44} +(1.87342e6 - 3.24486e6i) q^{45} +(-5.00072e6 + 8.66150e6i) q^{46} +(-2.77255e7 - 4.80219e7i) q^{47} +1.84032e7 q^{48} -2.08352e7 q^{50} +(-3.55320e7 - 6.15432e7i) q^{51} +(3.23886e7 - 5.60987e7i) q^{52} +(4.63444e7 - 8.02709e7i) q^{53} +(1.08966e6 + 1.88734e6i) q^{54} -1.28228e7 q^{55} +8.54737e7 q^{57} +(2.00771e7 + 3.47746e7i) q^{58} +(-1.26300e7 + 2.18758e7i) q^{59} +(-7.73267e6 + 1.33934e7i) q^{60} +(3.46637e7 + 6.00394e7i) q^{61} -3.78653e7 q^{62} +1.80290e7 q^{64} +(1.65323e7 + 2.86347e7i) q^{65} +(-6.81088e7 + 1.17968e8i) q^{66} +(1.16747e7 - 2.02211e7i) q^{67} +(7.13762e7 + 1.23627e8i) q^{68} +1.79799e8 q^{69} -1.06194e8 q^{71} +(9.20142e7 + 1.59373e8i) q^{72} +(-1.05058e8 + 1.81965e8i) q^{73} +(-1.02470e8 + 1.77484e8i) q^{74} +(1.87280e8 + 3.24379e8i) q^{75} -1.71699e8 q^{76} +3.51247e8 q^{78} +(74803.2 + 129563. i) q^{79} +(9.43490e6 - 1.63417e7i) q^{80} +(2.03244e8 - 3.52029e8i) q^{81} +(1.31098e7 + 2.27069e7i) q^{82} -5.21565e8 q^{83} -7.28659e7 q^{85} +(6.82444e7 + 1.18203e8i) q^{86} +(3.60932e8 - 6.25153e8i) q^{87} +(3.14900e8 - 5.45422e8i) q^{88} +(1.49294e8 + 2.58584e8i) q^{89} -4.08123e7 q^{90} -3.61178e8 q^{92} +(3.40358e8 + 5.89517e8i) q^{93} +(-3.01998e8 + 5.23076e8i) q^{94} +(4.38205e7 - 7.58993e7i) q^{95} +(-5.94577e8 - 1.02984e9i) q^{96} +8.95983e8 q^{97} +1.19179e9 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 6 q^{2} - 86 q^{3} + 620 q^{4} - 2238 q^{5} + 7976 q^{6} + 5232 q^{8} - 11038 q^{9} + 43384 q^{10} - 35316 q^{11} + 52136 q^{12} + 53060 q^{13} - 614272 q^{15} + 752 q^{16} - 463920 q^{17} - 332042 q^{18}+ \cdots + 2818835720 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/49\mathbb{Z}\right)^\times\).

\(n\) \(3\)
\(\chi(n)\) \(e\left(\frac{2}{3}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −5.44622 − 9.43313i −0.240691 − 0.416890i 0.720220 − 0.693746i \(-0.244041\pi\)
−0.960911 + 0.276856i \(0.910707\pi\)
\(3\) −97.9084 + 169.582i −0.697870 + 1.20875i 0.271334 + 0.962485i \(0.412535\pi\)
−0.969204 + 0.246261i \(0.920798\pi\)
\(4\) 196.677 − 340.655i 0.384135 − 0.665342i
\(5\) 100.391 + 173.882i 0.0718340 + 0.124420i 0.899705 − 0.436498i \(-0.143781\pi\)
−0.827871 + 0.560918i \(0.810448\pi\)
\(6\) 2132.92 0.671885
\(7\) 0 0
\(8\) −9861.52 −0.851215
\(9\) −9330.63 − 16161.1i −0.474045 − 0.821070i
\(10\) 1093.50 − 1894.01i 0.0345796 − 0.0598937i
\(11\) −31932.1 + 55308.1i −0.657599 + 1.13899i 0.323637 + 0.946181i \(0.395094\pi\)
−0.981236 + 0.192813i \(0.938239\pi\)
\(12\) 38512.7 + 66706.0i 0.536153 + 0.928644i
\(13\) 164679. 1.59916 0.799581 − 0.600558i \(-0.205055\pi\)
0.799581 + 0.600558i \(0.205055\pi\)
\(14\) 0 0
\(15\) −39316.5 −0.200523
\(16\) −46990.7 − 81390.3i −0.179255 − 0.310480i
\(17\) −181455. + 314289.i −0.526925 + 0.912661i 0.472582 + 0.881286i \(0.343322\pi\)
−0.999508 + 0.0313748i \(0.990011\pi\)
\(18\) −101633. + 176034.i −0.228197 + 0.395249i
\(19\) −218249. − 378018.i −0.384203 − 0.665460i 0.607455 − 0.794354i \(-0.292190\pi\)
−0.991658 + 0.128895i \(0.958857\pi\)
\(20\) 78978.6 0.110376
\(21\) 0 0
\(22\) 695638. 0.633113
\(23\) −459100. − 795184.i −0.342083 − 0.592505i 0.642736 − 0.766088i \(-0.277799\pi\)
−0.984819 + 0.173582i \(0.944466\pi\)
\(24\) 965527. − 1.67234e6i 0.594037 − 1.02890i
\(25\) 956406. − 1.65654e6i 0.489680 − 0.848150i
\(26\) −896877. − 1.55344e6i −0.384904 − 0.666674i
\(27\) −200076. −0.0724531
\(28\) 0 0
\(29\) −3.68643e6 −0.967865 −0.483932 − 0.875105i \(-0.660792\pi\)
−0.483932 + 0.875105i \(0.660792\pi\)
\(30\) 214127. + 370878.i 0.0482642 + 0.0835960i
\(31\) 1.73814e6 − 3.01055e6i 0.338032 − 0.585489i −0.646030 − 0.763312i \(-0.723572\pi\)
0.984063 + 0.177823i \(0.0569054\pi\)
\(32\) −3.03639e6 + 5.25919e6i −0.511898 + 0.886633i
\(33\) −6.25285e6 − 1.08303e7i −0.917837 − 1.58974i
\(34\) 3.95298e6 0.507305
\(35\) 0 0
\(36\) −7.34049e6 −0.728390
\(37\) −9.40745e6 − 1.62942e7i −0.825210 − 1.42931i −0.901759 − 0.432239i \(-0.857724\pi\)
0.0765491 − 0.997066i \(-0.475610\pi\)
\(38\) −2.37726e6 + 4.11754e6i −0.184949 + 0.320341i
\(39\) −1.61234e7 + 2.79266e7i −1.11601 + 1.93298i
\(40\) −990009. − 1.71475e6i −0.0611462 − 0.105908i
\(41\) −2.40714e6 −0.133038 −0.0665188 − 0.997785i \(-0.521189\pi\)
−0.0665188 + 0.997785i \(0.521189\pi\)
\(42\) 0 0
\(43\) −1.25306e7 −0.558938 −0.279469 − 0.960155i \(-0.590158\pi\)
−0.279469 + 0.960155i \(0.590158\pi\)
\(44\) 1.25607e7 + 2.17557e7i 0.505214 + 0.875056i
\(45\) 1.87342e6 − 3.24486e6i 0.0681051 − 0.117961i
\(46\) −5.00072e6 + 8.66150e6i −0.164673 + 0.285222i
\(47\) −2.77255e7 − 4.80219e7i −0.828779 − 1.43549i −0.898997 − 0.437955i \(-0.855703\pi\)
0.0702184 − 0.997532i \(-0.477630\pi\)
\(48\) 1.84032e7 0.500388
\(49\) 0 0
\(50\) −2.08352e7 −0.471447
\(51\) −3.55320e7 − 6.15432e7i −0.735451 − 1.27384i
\(52\) 3.23886e7 − 5.60987e7i 0.614295 − 1.06399i
\(53\) 4.63444e7 − 8.02709e7i 0.806782 − 1.39739i −0.108299 − 0.994118i \(-0.534540\pi\)
0.915081 − 0.403269i \(-0.132126\pi\)
\(54\) 1.08966e6 + 1.88734e6i 0.0174388 + 0.0302049i
\(55\) −1.28228e7 −0.188952
\(56\) 0 0
\(57\) 8.54737e7 1.07250
\(58\) 2.00771e7 + 3.47746e7i 0.232957 + 0.403493i
\(59\) −1.26300e7 + 2.18758e7i −0.135696 + 0.235033i −0.925863 − 0.377859i \(-0.876661\pi\)
0.790167 + 0.612892i \(0.209994\pi\)
\(60\) −7.73267e6 + 1.33934e7i −0.0770281 + 0.133417i
\(61\) 3.46637e7 + 6.00394e7i 0.320547 + 0.555203i 0.980601 − 0.196015i \(-0.0628001\pi\)
−0.660054 + 0.751218i \(0.729467\pi\)
\(62\) −3.78653e7 −0.325446
\(63\) 0 0
\(64\) 1.80290e7 0.134326
\(65\) 1.65323e7 + 2.86347e7i 0.114874 + 0.198968i
\(66\) −6.81088e7 + 1.17968e8i −0.441831 + 0.765273i
\(67\) 1.16747e7 − 2.02211e7i 0.0707796 − 0.122594i −0.828464 − 0.560043i \(-0.810785\pi\)
0.899243 + 0.437449i \(0.144118\pi\)
\(68\) 7.13762e7 + 1.23627e8i 0.404821 + 0.701171i
\(69\) 1.79799e8 0.954918
\(70\) 0 0
\(71\) −1.06194e8 −0.495950 −0.247975 − 0.968766i \(-0.579765\pi\)
−0.247975 + 0.968766i \(0.579765\pi\)
\(72\) 9.20142e7 + 1.59373e8i 0.403514 + 0.698907i
\(73\) −1.05058e8 + 1.81965e8i −0.432987 + 0.749956i −0.997129 − 0.0757223i \(-0.975874\pi\)
0.564142 + 0.825678i \(0.309207\pi\)
\(74\) −1.02470e8 + 1.77484e8i −0.397242 + 0.688043i
\(75\) 1.87280e8 + 3.24379e8i 0.683466 + 1.18380i
\(76\) −1.71699e8 −0.590344
\(77\) 0 0
\(78\) 3.51247e8 1.07445
\(79\) 74803.2 + 129563.i 0.000216072 + 0.000374248i 0.866133 − 0.499813i \(-0.166598\pi\)
−0.865917 + 0.500187i \(0.833265\pi\)
\(80\) 9.43490e6 − 1.63417e7i 0.0257533 − 0.0446060i
\(81\) 2.03244e8 − 3.52029e8i 0.524608 − 0.908647i
\(82\) 1.31098e7 + 2.27069e7i 0.0320210 + 0.0554620i
\(83\) −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
\(86\) 6.82444e7 + 1.18203e8i 0.134531 + 0.233015i
\(87\) 3.60932e8 − 6.25153e8i 0.675444 − 1.16990i
\(88\) 3.14900e8 − 5.45422e8i 0.559758 − 0.969529i
\(89\) 1.49294e8 + 2.58584e8i 0.252224 + 0.436865i 0.964138 − 0.265402i \(-0.0855047\pi\)
−0.711914 + 0.702267i \(0.752171\pi\)
\(90\) −4.08123e7 −0.0655692
\(91\) 0 0
\(92\) −3.61178e8 −0.525625
\(93\) 3.40358e8 + 5.89517e8i 0.471805 + 0.817190i
\(94\) −3.01998e8 + 5.23076e8i −0.398960 + 0.691018i
\(95\) 4.38205e7 − 7.58993e7i 0.0551977 − 0.0956053i
\(96\) −5.94577e8 − 1.02984e9i −0.714476 − 1.23751i
\(97\) 8.95983e8 1.02761 0.513803 − 0.857908i \(-0.328236\pi\)
0.513803 + 0.857908i \(0.328236\pi\)
\(98\) 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 49.10.c.b.18.1 4
7.2 even 3 inner 49.10.c.b.30.1 4
7.3 odd 6 7.10.a.a.1.2 ✓ 2
7.4 even 3 49.10.a.b.1.2 2
7.5 odd 6 49.10.c.c.30.1 4
7.6 odd 2 49.10.c.c.18.1 4
21.17 even 6 63.10.a.d.1.1 2
28.3 even 6 112.10.a.e.1.2 2
35.3 even 12 175.10.b.b.99.2 4
35.17 even 12 175.10.b.b.99.3 4
35.24 odd 6 175.10.a.b.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
7.10.a.a.1.2 ✓ 2 7.3 odd 6
49.10.a.b.1.2 2 7.4 even 3
49.10.c.b.18.1 4 1.1 even 1 trivial
49.10.c.b.30.1 4 7.2 even 3 inner
49.10.c.c.18.1 4 7.6 odd 2
49.10.c.c.30.1 4 7.5 odd 6
63.10.a.d.1.1 2 21.17 even 6
112.10.a.e.1.2 2 28.3 even 6
175.10.a.b.1.1 2 35.24 odd 6
175.10.b.b.99.2 4 35.3 even 12
175.10.b.b.99.3 4 35.17 even 12