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 30.1
Root \(-4.10797 - 1.22474i\) of defining polynomial
Character \(\chi\) \(=\) 49.30
Dual form 49.6.c.h.18.1

$q$-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+(-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 \(\left(\mathbb{Z}/49\mathbb{Z}\right)^\times\).

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

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.40754 + 2.43792i −0.248820 + 0.430968i −0.963199 − 0.268791i \(-0.913376\pi\)
0.714379 + 0.699759i \(0.246709\pi\)
\(3\) −3.27401 − 5.67075i −0.210028 − 0.363779i 0.741695 − 0.670737i \(-0.234022\pi\)
−0.951723 + 0.306958i \(0.900689\pi\)
\(4\) 12.0377 + 20.8499i 0.376178 + 0.651559i
\(5\) 22.9955 − 39.8294i 0.411356 − 0.712490i −0.583682 − 0.811982i \(-0.698388\pi\)
0.995038 + 0.0994921i \(0.0317218\pi\)
\(6\) 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 0.972653 + 0.232263i \(0.0746129\pi\)
−0.285181 + 0.958474i \(0.592054\pi\)
\(12\) 78.8229 − 136.525i 0.158015 − 0.273691i
\(13\) 1094.10 1.79555 0.897776 − 0.440453i \(-0.145182\pi\)
0.897776 + 0.440453i \(0.145182\pi\)
\(14\) 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 \(-0.00167290\pi\)
−0.504545 + 0.863386i \(0.668340\pi\)
\(18\) 281.681 + 487.886i 0.204916 + 0.354925i
\(19\) 583.063 − 1009.89i 0.370537 − 0.641789i −0.619111 − 0.785303i \(-0.712507\pi\)
0.989648 + 0.143514i \(0.0458404\pi\)
\(20\) 1107.25 0.618972
\(21\) 0 0
\(22\) −1553.30 −0.684226
\(23\) −22.1925 + 38.4386i −0.00874757 + 0.0151512i −0.870366 − 0.492405i \(-0.836118\pi\)
0.861618 + 0.507556i \(0.169451\pi\)
\(24\) 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 −0.905057 − 0.425290i \(-0.860172\pi\)
0.0842165 − 0.996447i \(-0.473161\pi\)
\(32\) −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 −0.778987 − 0.627040i \(-0.784266\pi\)
0.932526 + 0.361103i \(0.117600\pi\)
\(38\) 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 \(-0.701975\pi\)
−0.592794 + 0.805354i \(0.701975\pi\)
\(42\) 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 \(-0.914927\pi\)
0.710961 + 0.703231i \(0.248260\pi\)
\(48\) 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 0.974344 − 0.225065i \(-0.0722594\pi\)
−0.682084 + 0.731274i \(0.738926\pi\)
\(54\) 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 \(-0.225678\pi\)
−0.943350 + 0.331798i \(0.892345\pi\)
\(60\) −3625.15 − 6278.94i −0.130001 − 0.225169i
\(61\) 19258.9 − 33357.5i 0.662686 − 1.14781i −0.317221 − 0.948352i \(-0.602750\pi\)
0.979907 − 0.199454i \(-0.0639169\pi\)
\(62\) 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 0.800661 + 0.599118i \(0.204482\pi\)
0.118521 + 0.992952i \(0.462185\pi\)
\(68\) −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 \(-0.173135\pi\)
−0.876007 + 0.482299i \(0.839802\pi\)
\(74\) 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 −0.865987 − 0.500067i \(-0.833309\pi\)
0.866064 + 0.499933i \(0.166642\pi\)
\(80\) 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 \(-0.773405\pi\)
−0.757143 + 0.653249i \(0.773405\pi\)
\(84\) 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 \(-0.950106\pi\)
0.629062 + 0.777355i \(0.283439\pi\)
\(90\) 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 \(-0.494657\pi\)
0.0167846 + 0.999859i \(0.494657\pi\)
\(98\) 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.30.1 8
7.2 even 3 49.6.a.g.1.4 yes 4
7.3 odd 6 inner 49.6.c.h.18.2 8
7.4 even 3 inner 49.6.c.h.18.1 8
7.5 odd 6 49.6.a.g.1.3 ✓ 4
7.6 odd 2 inner 49.6.c.h.30.2 8
21.2 odd 6 441.6.a.z.1.2 4
21.5 even 6 441.6.a.z.1.1 4
28.19 even 6 784.6.a.bf.1.3 4
28.23 odd 6 784.6.a.bf.1.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
49.6.a.g.1.3 ✓ 4 7.5 odd 6
49.6.a.g.1.4 yes 4 7.2 even 3
49.6.c.h.18.1 8 7.4 even 3 inner
49.6.c.h.18.2 8 7.3 odd 6 inner
49.6.c.h.30.1 8 1.1 even 1 trivial
49.6.c.h.30.2 8 7.6 odd 2 inner
441.6.a.z.1.1 4 21.5 even 6
441.6.a.z.1.2 4 21.2 odd 6
784.6.a.bf.1.2 4 28.23 odd 6
784.6.a.bf.1.3 4 28.19 even 6