Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [49,6,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, 6); 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, 6, names="a")
 
Level: \( N \) \(=\) \( 49 = 7^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 49.c (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,10,0] 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: \(7.85880717084\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{-3}, \sqrt{113})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 4x^{7} - 102x^{6} + 320x^{5} + 4283x^{4} - 9104x^{3} - 85298x^{2} + 89904x + 714364 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 7^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 18.2
Root \(-5.52218 - 1.22474i\) of defining polynomial
Character \(\chi\) \(=\) 49.18
Dual form 49.6.c.h.30.2

$q$-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+(-1.40754 - 2.43792i) q^{2} +(3.27401 - 5.67075i) q^{3} +(12.0377 - 20.8499i) q^{4} +(-22.9955 - 39.8294i) q^{5} -18.4331 q^{6} -157.856 q^{8} +(100.062 + 173.312i) q^{9} +(-64.7341 + 112.123i) q^{10} +(275.890 - 477.856i) q^{11} +(-78.8229 - 136.525i) q^{12} -1094.10 q^{13} -301.150 q^{15} +(-163.017 - 282.354i) q^{16} +(-590.357 + 1022.53i) q^{17} +(281.681 - 487.886i) q^{18} +(-583.063 - 1009.89i) q^{19} -1107.25 q^{20} -1553.30 q^{22} +(-22.1925 - 38.4386i) q^{23} +(-516.823 + 895.163i) q^{24} +(504.912 - 874.533i) q^{25} +(1539.98 + 2667.33i) q^{26} +2901.58 q^{27} +3329.02 q^{29} +(423.880 + 734.181i) q^{30} +(4392.01 - 7607.18i) q^{31} +(-2984.61 + 5169.49i) q^{32} +(-1806.53 - 3129.01i) q^{33} +3323.80 q^{34} +4818.05 q^{36} +(1278.56 + 2214.53i) q^{37} +(-1641.37 + 2842.93i) q^{38} +(-3582.08 + 6204.35i) q^{39} +(3629.99 + 6287.32i) q^{40} +12761.3 q^{41} -96.7714 q^{43} +(-6642.16 - 11504.6i) q^{44} +(4601.94 - 7970.80i) q^{45} +(-62.4736 + 108.207i) q^{46} +(3839.58 + 6650.34i) q^{47} -2134.88 q^{48} -2842.73 q^{50} +(3865.67 + 6695.53i) q^{51} +(-13170.4 + 22811.8i) q^{52} +(5976.67 - 10351.9i) q^{53} +(-4084.08 - 7073.83i) q^{54} -25377.0 q^{55} -7635.81 q^{57} +(-4685.71 - 8115.89i) q^{58} +(4928.62 - 8536.62i) q^{59} +(-3625.15 + 6278.94i) q^{60} +(-19258.9 - 33357.5i) q^{61} -24727.6 q^{62} +6370.65 q^{64} +(25159.4 + 43577.3i) q^{65} +(-5085.53 + 8808.39i) q^{66} +(33774.5 - 58499.1i) q^{67} +(14213.1 + 24617.7i) q^{68} -290.634 q^{69} -61374.6 q^{71} +(-15795.4 - 27358.4i) q^{72} +(925.199 - 1602.49i) q^{73} +(3599.24 - 6234.07i) q^{74} +(-3306.17 - 5726.45i) q^{75} -28074.9 q^{76} +20167.7 q^{78} +(4.26456 + 7.38644i) q^{79} +(-7497.34 + 12985.8i) q^{80} +(-14815.2 + 25660.7i) q^{81} +(-17961.9 - 31111.0i) q^{82} +95039.3 q^{83} +54302.3 q^{85} +(136.209 + 235.921i) q^{86} +(10899.2 - 18878.0i) q^{87} +(-43551.0 + 75432.6i) q^{88} +(26802.8 + 46423.8i) q^{89} -25909.6 q^{90} -1068.59 q^{92} +(-28758.9 - 49811.9i) q^{93} +(10808.7 - 18721.2i) q^{94} +(-26815.7 + 46446.1i) q^{95} +(19543.2 + 33849.9i) q^{96} -3110.79 q^{97} +110424. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 10 q^{2} - 10 q^{4} - 540 q^{8} - 220 q^{9} + 1952 q^{11} - 8192 q^{15} + 1566 q^{16} + 5974 q^{18} + 7048 q^{22} + 7136 q^{23} - 2764 q^{25} - 6704 q^{29} - 25608 q^{30} - 27810 q^{32} + 55340 q^{36}+ \cdots - 84544 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.40754 − 2.43792i −0.248820 − 0.430968i 0.714379 − 0.699759i \(-0.246709\pi\)
−0.963199 + 0.268791i \(0.913376\pi\)
\(3\) 3.27401 − 5.67075i 0.210028 − 0.363779i −0.741695 − 0.670737i \(-0.765978\pi\)
0.951723 + 0.306958i \(0.0993113\pi\)
\(4\) 12.0377 − 20.8499i 0.376178 − 0.651559i
\(5\) −22.9955 − 39.8294i −0.411356 − 0.712490i 0.583682 − 0.811982i \(-0.301612\pi\)
−0.995038 + 0.0994921i \(0.968278\pi\)
\(6\) −18.4331 −0.209036
\(7\) 0 0
\(8\) −157.856 −0.872041
\(9\) 100.062 + 173.312i 0.411777 + 0.713218i
\(10\) −64.7341 + 112.123i −0.204707 + 0.354563i
\(11\) 275.890 − 477.856i 0.687472 − 1.19074i −0.285181 − 0.958474i \(-0.592054\pi\)
0.972653 − 0.232263i \(-0.0746129\pi\)
\(12\) −78.8229 − 136.525i −0.158015 − 0.273691i
\(13\) −1094.10 −1.79555 −0.897776 − 0.440453i \(-0.854818\pi\)
−0.897776 + 0.440453i \(0.854818\pi\)
\(14\) 0 0
\(15\) −301.150 −0.345585
\(16\) −163.017 − 282.354i −0.159197 − 0.275737i
\(17\) −590.357 + 1022.53i −0.495442 + 0.858130i −0.999986 − 0.00525555i \(-0.998327\pi\)
0.504545 + 0.863386i \(0.331660\pi\)
\(18\) 281.681 − 487.886i 0.204916 − 0.354925i
\(19\) −583.063 − 1009.89i −0.370537 − 0.641789i 0.619111 − 0.785303i \(-0.287493\pi\)
−0.989648 + 0.143514i \(0.954160\pi\)
\(20\) −1107.25 −0.618972
\(21\) 0 0
\(22\) −1553.30 −0.684226
\(23\) −22.1925 − 38.4386i −0.00874757 − 0.0151512i 0.861618 − 0.507556i \(-0.169451\pi\)
−0.870366 + 0.492405i \(0.836118\pi\)
\(24\) −516.823 + 895.163i −0.183153 + 0.317230i
\(25\) 504.912 − 874.533i 0.161572 − 0.279850i
\(26\) 1539.98 + 2667.33i 0.446768 + 0.773826i
\(27\) 2901.58 0.765993
\(28\) 0 0
\(29\) 3329.02 0.735057 0.367529 − 0.930012i \(-0.380204\pi\)
0.367529 + 0.930012i \(0.380204\pi\)
\(30\) 423.880 + 734.181i 0.0859883 + 0.148936i
\(31\) 4392.01 − 7607.18i 0.820841 − 1.42174i −0.0842165 − 0.996447i \(-0.526839\pi\)
0.905057 − 0.425290i \(-0.139828\pi\)
\(32\) −2984.61 + 5169.49i −0.515243 + 0.892427i
\(33\) −1806.53 − 3129.01i −0.288776 − 0.500175i
\(34\) 3323.80 0.493102
\(35\) 0 0
\(36\) 4818.05 0.619605
\(37\) 1278.56 + 2214.53i 0.153539 + 0.265937i 0.932526 − 0.361103i \(-0.117600\pi\)
−0.778987 + 0.627040i \(0.784266\pi\)
\(38\) −1641.37 + 2842.93i −0.184394 + 0.319379i
\(39\) −3582.08 + 6204.35i −0.377115 + 0.653183i
\(40\) 3629.99 + 6287.32i 0.358720 + 0.621321i
\(41\) 12761.3 1.18559 0.592794 − 0.805354i \(-0.298025\pi\)
0.592794 + 0.805354i \(0.298025\pi\)
\(42\) 0 0
\(43\) −96.7714 −0.00798135 −0.00399067 − 0.999992i \(-0.501270\pi\)
−0.00399067 + 0.999992i \(0.501270\pi\)
\(44\) −6642.16 − 11504.6i −0.517223 − 0.895857i
\(45\) 4601.94 − 7970.80i 0.338774 − 0.586774i
\(46\) −62.4736 + 108.207i −0.00435313 + 0.00753985i
\(47\) 3839.58 + 6650.34i 0.253535 + 0.439136i 0.964497 − 0.264095i \(-0.0850732\pi\)
−0.710961 + 0.703231i \(0.751740\pi\)
\(48\) −2134.88 −0.133743
\(49\) 0 0
\(50\) −2842.73 −0.160809
\(51\) 3865.67 + 6695.53i 0.208113 + 0.360462i
\(52\) −13170.4 + 22811.8i −0.675446 + 1.16991i
\(53\) 5976.67 − 10351.9i 0.292260 − 0.506209i −0.682084 − 0.731274i \(-0.738926\pi\)
0.974344 + 0.225065i \(0.0722594\pi\)
\(54\) −4084.08 − 7073.83i −0.190594 − 0.330119i
\(55\) −25377.0 −1.13118
\(56\) 0 0
\(57\) −7635.81 −0.311292
\(58\) −4685.71 − 8115.89i −0.182897 − 0.316786i
\(59\) 4928.62 − 8536.62i 0.184330 − 0.319268i −0.759021 − 0.651066i \(-0.774322\pi\)
0.943350 + 0.331798i \(0.107655\pi\)
\(60\) −3625.15 + 6278.94i −0.130001 + 0.225169i
\(61\) −19258.9 − 33357.5i −0.662686 − 1.14781i −0.979907 − 0.199454i \(-0.936083\pi\)
0.317221 − 0.948352i \(-0.397250\pi\)
\(62\) −24727.6 −0.816965
\(63\) 0 0
\(64\) 6370.65 0.194417
\(65\) 25159.4 + 43577.3i 0.738612 + 1.27931i
\(66\) −5085.53 + 8808.39i −0.143706 + 0.248907i
\(67\) 33774.5 − 58499.1i 0.919182 − 1.59207i 0.118521 − 0.992952i \(-0.462185\pi\)
0.800661 − 0.599118i \(-0.204482\pi\)
\(68\) 14213.1 + 24617.7i 0.372748 + 0.645619i
\(69\) −290.634 −0.00734893
\(70\) 0 0
\(71\) −61374.6 −1.44492 −0.722458 − 0.691415i \(-0.756988\pi\)
−0.722458 + 0.691415i \(0.756988\pi\)
\(72\) −15795.4 − 27358.4i −0.359086 − 0.621955i
\(73\) 925.199 − 1602.49i 0.0203202 − 0.0351956i −0.855686 − 0.517495i \(-0.826865\pi\)
0.876007 + 0.482299i \(0.160198\pi\)
\(74\) 3599.24 − 6234.07i 0.0764068 − 0.132340i
\(75\) −3306.17 − 5726.45i −0.0678691 − 0.117553i
\(76\) −28074.9 −0.557551
\(77\) 0 0
\(78\) 20167.7 0.375335
\(79\) 4.26456 + 7.38644i 7.68788e−5 + 0.000133158i 0.866064 − 0.499933i \(-0.166642\pi\)
−0.865987 + 0.500067i \(0.833309\pi\)
\(80\) −7497.34 + 12985.8i −0.130973 + 0.226852i
\(81\) −14815.2 + 25660.7i −0.250897 + 0.434566i
\(82\) −17961.9 − 31111.0i −0.294998 − 0.510951i
\(83\) 95039.3 1.51429 0.757143 − 0.653249i \(-0.226595\pi\)
0.757143 + 0.653249i \(0.226595\pi\)
\(84\) 0 0
\(85\) 54302.3 0.815212
\(86\) 136.209 + 235.921i 0.00198592 + 0.00343971i
\(87\) 10899.2 − 18878.0i 0.154382 − 0.267398i
\(88\) −43551.0 + 75432.6i −0.599503 + 1.03837i
\(89\) 26802.8 + 46423.8i 0.358678 + 0.621249i 0.987740 − 0.156106i \(-0.0498942\pi\)
−0.629062 + 0.777355i \(0.716561\pi\)
\(90\) −25909.6 −0.337175
\(91\) 0 0
\(92\) −1068.59 −0.0131626
\(93\) −28758.9 − 49811.9i −0.344798 − 0.597208i
\(94\) 10808.7 − 18721.2i 0.126169 − 0.218531i
\(95\) −26815.7 + 46446.1i −0.304846 + 0.528008i
\(96\) 19543.2 + 33849.9i 0.216431 + 0.374869i
\(97\) −3110.79 −0.0335693 −0.0167846 − 0.999859i \(-0.505343\pi\)
−0.0167846 + 0.999859i \(0.505343\pi\)
\(98\) 0 0
\(99\) 110424. 1.13234
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 49.6.c.h.18.2 8
7.2 even 3 inner 49.6.c.h.30.2 8
7.3 odd 6 49.6.a.g.1.4 yes 4
7.4 even 3 49.6.a.g.1.3 ✓ 4
7.5 odd 6 inner 49.6.c.h.30.1 8
7.6 odd 2 inner 49.6.c.h.18.1 8
21.11 odd 6 441.6.a.z.1.1 4
21.17 even 6 441.6.a.z.1.2 4
28.3 even 6 784.6.a.bf.1.2 4
28.11 odd 6 784.6.a.bf.1.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
49.6.a.g.1.3 ✓ 4 7.4 even 3
49.6.a.g.1.4 yes 4 7.3 odd 6
49.6.c.h.18.1 8 7.6 odd 2 inner
49.6.c.h.18.2 8 1.1 even 1 trivial
49.6.c.h.30.1 8 7.5 odd 6 inner
49.6.c.h.30.2 8 7.2 even 3 inner
441.6.a.z.1.1 4 21.11 odd 6
441.6.a.z.1.2 4 21.17 even 6
784.6.a.bf.1.2 4 28.3 even 6
784.6.a.bf.1.3 4 28.11 odd 6