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Properties

Label 49.14.c.g.18.1
Level $49$
Weight $14$
Character 49.18
Analytic conductor $52.543$
Analytic rank $0$
Dimension $16$
Inner twists $2$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [49,14,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, 14); 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, 14, names="a")
 
Level: \( N \) \(=\) \( 49 = 7^{2} \)
Weight: \( k \) \(=\) \( 14 \)
Character orbit: \([\chi]\) \(=\) 49.c (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [16,-2,-728] 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: \(52.5431551864\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 6 x^{15} - 24508 x^{14} + 79772 x^{13} + 237177405 x^{12} + 118141132 x^{11} + \cdots + 13\!\cdots\!00 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{34}\cdot 3^{8}\cdot 7^{12} \)
Twist minimal: no (minimal twist has level 7)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 18.1
Root \(-84.9382 + 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 49.18
Dual form 49.14.c.g.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+(-85.4382 - 147.983i) q^{2} +(-275.475 + 477.137i) q^{3} +(-10503.4 + 18192.4i) q^{4} +(-18457.8 - 31969.8i) q^{5} +94144.4 q^{6} +2.18974e6 q^{8} +(645388. + 1.11785e6i) q^{9} +(-3.15400e6 + 5.46289e6i) q^{10} +(-417437. + 723022. i) q^{11} +(-5.78684e6 - 1.00231e7i) q^{12} +1.14265e7 q^{13} +2.03387e7 q^{15} +(-1.01044e8 - 1.75013e8i) q^{16} +(1.80296e7 - 3.12283e7i) q^{17} +(1.10282e8 - 1.91013e8i) q^{18} +(-2.00069e8 - 3.46530e8i) q^{19} +7.75476e8 q^{20} +1.42660e8 q^{22} +(2.53935e8 + 4.39828e8i) q^{23} +(-6.03218e8 + 1.04480e9i) q^{24} +(-7.10293e7 + 1.23026e8i) q^{25} +(-9.76261e8 - 1.69093e9i) q^{26} -1.58955e9 q^{27} -4.26139e9 q^{29} +(-1.73770e9 - 3.00978e9i) q^{30} +(1.81984e9 - 3.15206e9i) q^{31} +(-8.29679e9 + 1.43705e10i) q^{32} +(-2.29987e8 - 3.98350e8i) q^{33} -6.16168e9 q^{34} -2.71150e10 q^{36} +(2.77238e8 + 4.80190e8i) q^{37} +(-3.41871e10 + 5.92138e10i) q^{38} +(-3.14772e9 + 5.45202e9i) q^{39} +(-4.04177e10 - 7.00056e10i) q^{40} +3.81654e10 q^{41} -7.34797e8 q^{43} +(-8.76899e9 - 1.51883e10i) q^{44} +(2.38249e10 - 4.12659e10i) q^{45} +(4.33915e10 - 7.51563e10i) q^{46} +(-1.79468e10 - 3.10848e10i) q^{47} +1.11340e11 q^{48} +2.42744e10 q^{50} +(9.93344e9 + 1.72052e10i) q^{51} +(-1.20017e11 + 2.07875e11i) q^{52} +(-7.30982e10 + 1.26610e11i) q^{53} +(1.35808e11 + 2.35226e11i) q^{54} +3.08199e10 q^{55} +2.20457e11 q^{57} +(3.64086e11 + 6.30615e11i) q^{58} +(1.41841e11 - 2.45676e11i) q^{59} +(-2.13625e11 + 3.70009e11i) q^{60} +(1.37245e11 + 2.37715e11i) q^{61} -6.21936e11 q^{62} +1.17996e12 q^{64} +(-2.10908e11 - 3.65304e11i) q^{65} +(-3.92994e10 + 6.80685e10i) q^{66} +(4.52479e11 - 7.83716e11i) q^{67} +(3.78744e11 + 6.56004e11i) q^{68} -2.79811e11 q^{69} +1.65485e11 q^{71} +(1.41323e12 + 2.44779e12i) q^{72} +(-3.13378e11 + 5.42787e11i) q^{73} +(4.73734e10 - 8.20532e10i) q^{74} +(-3.91336e10 - 6.77814e10i) q^{75} +8.40561e12 q^{76} +1.07574e12 q^{78} +(-3.45341e11 - 5.98148e11i) q^{79} +(-3.73008e12 + 6.46069e12i) q^{80} +(-5.91076e11 + 1.02377e12i) q^{81} +(-3.26079e12 - 5.64785e12i) q^{82} -3.52783e12 q^{83} -1.33115e12 q^{85} +(6.27797e10 + 1.08738e11i) q^{86} +(1.17391e12 - 2.03327e12i) q^{87} +(-9.14077e11 + 1.58323e12i) q^{88} +(6.89592e11 + 1.19441e12i) q^{89} -8.14222e12 q^{90} -1.06687e13 q^{92} +(1.00264e12 + 1.73663e12i) q^{93} +(-3.06669e12 + 5.31166e12i) q^{94} +(-7.38568e12 + 1.27924e13i) q^{95} +(-4.57112e12 - 7.91742e12i) q^{96} -3.29185e10 q^{97} -1.07764e12 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 2 q^{2} - 728 q^{3} - 32588 q^{4} - 46928 q^{5} + 30212 q^{6} - 1093248 q^{8} - 2910808 q^{9} - 879018 q^{10} + 3746000 q^{11} - 13786276 q^{12} + 63556528 q^{13} + 179729504 q^{15} - 158020976 q^{16}+ \cdots + 1803523183088 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −85.4382 − 147.983i −0.943968 − 1.63500i −0.757805 − 0.652481i \(-0.773728\pi\)
−0.186162 − 0.982519i \(-0.559605\pi\)
\(3\) −275.475 + 477.137i −0.218170 + 0.377881i −0.954248 − 0.299015i \(-0.903342\pi\)
0.736079 + 0.676896i \(0.236675\pi\)
\(4\) −10503.4 + 18192.4i −1.28215 + 2.22075i
\(5\) −18457.8 − 31969.8i −0.528293 − 0.915030i −0.999456 − 0.0329842i \(-0.989499\pi\)
0.471163 − 0.882046i \(-0.343834\pi\)
\(6\) 94144.4 0.823781
\(7\) 0 0
\(8\) 2.18974e6 2.95330
\(9\) 645388. + 1.11785e6i 0.404804 + 0.701141i
\(10\) −3.15400e6 + 5.46289e6i −0.997383 + 1.72752i
\(11\) −417437. + 723022.i −0.0710458 + 0.123055i −0.899360 − 0.437209i \(-0.855967\pi\)
0.828314 + 0.560264i \(0.189300\pi\)
\(12\) −5.78684e6 − 1.00231e7i −0.559452 − 0.969000i
\(13\) 1.14265e7 0.656572 0.328286 − 0.944578i \(-0.393529\pi\)
0.328286 + 0.944578i \(0.393529\pi\)
\(14\) 0 0
\(15\) 2.03387e7 0.461030
\(16\) −1.01044e8 − 1.75013e8i −1.50567 − 2.60789i
\(17\) 1.80296e7 − 3.12283e7i 0.181163 − 0.313783i −0.761114 − 0.648618i \(-0.775347\pi\)
0.942277 + 0.334835i \(0.108681\pi\)
\(18\) 1.10282e8 − 1.91013e8i 0.764244 − 1.32371i
\(19\) −2.00069e8 − 3.46530e8i −0.975623 − 1.68983i −0.677863 − 0.735188i \(-0.737094\pi\)
−0.297760 − 0.954641i \(-0.596240\pi\)
\(20\) 7.75476e8 2.70940
\(21\) 0 0
\(22\) 1.42660e8 0.268260
\(23\) 2.53935e8 + 4.39828e8i 0.357677 + 0.619516i 0.987572 − 0.157165i \(-0.0502354\pi\)
−0.629895 + 0.776680i \(0.716902\pi\)
\(24\) −6.03218e8 + 1.04480e9i −0.644320 + 1.11599i
\(25\) −7.10293e7 + 1.23026e8i −0.0581872 + 0.100783i
\(26\) −9.76261e8 − 1.69093e9i −0.619782 − 1.07349i
\(27\) −1.58955e9 −0.789603
\(28\) 0 0
\(29\) −4.26139e9 −1.33035 −0.665173 − 0.746689i \(-0.731642\pi\)
−0.665173 + 0.746689i \(0.731642\pi\)
\(30\) −1.73770e9 − 3.00978e9i −0.435198 − 0.753784i
\(31\) 1.81984e9 − 3.15206e9i 0.368284 − 0.637886i −0.621013 − 0.783800i \(-0.713279\pi\)
0.989297 + 0.145913i \(0.0466121\pi\)
\(32\) −8.29679e9 + 1.43705e10i −1.36595 + 2.36590i
\(33\) −2.29987e8 − 3.98350e8i −0.0310001 − 0.0536937i
\(34\) −6.16168e9 −0.684047
\(35\) 0 0
\(36\) −2.71150e10 −2.07608
\(37\) 2.77238e8 + 4.80190e8i 0.0177640 + 0.0307682i 0.874771 − 0.484537i \(-0.161012\pi\)
−0.857007 + 0.515305i \(0.827679\pi\)
\(38\) −3.41871e10 + 5.92138e10i −1.84191 + 3.19029i
\(39\) −3.14772e9 + 5.45202e9i −0.143244 + 0.248106i
\(40\) −4.04177e10 − 7.00056e10i −1.56021 − 2.70236i
\(41\) 3.81654e10 1.25480 0.627401 − 0.778696i \(-0.284119\pi\)
0.627401 + 0.778696i \(0.284119\pi\)
\(42\) 0 0
\(43\) −7.34797e8 −0.0177265 −0.00886324 − 0.999961i \(-0.502821\pi\)
−0.00886324 + 0.999961i \(0.502821\pi\)
\(44\) −8.76899e9 − 1.51883e10i −0.182183 − 0.315550i
\(45\) 2.38249e10 − 4.12659e10i 0.427710 − 0.740816i
\(46\) 4.33915e10 − 7.51563e10i 0.675272 − 1.16961i
\(47\) −1.79468e10 − 3.10848e10i −0.242857 − 0.420641i 0.718670 − 0.695352i \(-0.244751\pi\)
−0.961527 + 0.274710i \(0.911418\pi\)
\(48\) 1.11340e11 1.31396
\(49\) 0 0
\(50\) 2.42744e10 0.219707
\(51\) 9.93344e9 + 1.72052e10i 0.0790485 + 0.136916i
\(52\) −1.20017e11 + 2.07875e11i −0.841823 + 1.45808i
\(53\) −7.30982e10 + 1.26610e11i −0.453016 + 0.784647i −0.998572 − 0.0534279i \(-0.982985\pi\)
0.545556 + 0.838075i \(0.316319\pi\)
\(54\) 1.35808e11 + 2.35226e11i 0.745360 + 1.29100i
\(55\) 3.08199e10 0.150132
\(56\) 0 0
\(57\) 2.20457e11 0.851406
\(58\) 3.64086e11 + 6.30615e11i 1.25580 + 2.17512i
\(59\) 1.41841e11 − 2.45676e11i 0.437788 − 0.758271i −0.559731 − 0.828674i \(-0.689095\pi\)
0.997519 + 0.0704038i \(0.0224288\pi\)
\(60\) −2.13625e11 + 3.70009e11i −0.591110 + 1.02383i
\(61\) 1.37245e11 + 2.37715e11i 0.341076 + 0.590762i 0.984633 − 0.174637i \(-0.0558751\pi\)
−0.643556 + 0.765399i \(0.722542\pi\)
\(62\) −6.21936e11 −1.39059
\(63\) 0 0
\(64\) 1.17996e12 2.14633
\(65\) −2.10908e11 − 3.65304e11i −0.346862 − 0.600783i
\(66\) −3.92994e10 + 6.80685e10i −0.0585262 + 0.101370i
\(67\) 4.52479e11 − 7.83716e11i 0.611100 − 1.05846i −0.379956 − 0.925005i \(-0.624061\pi\)
0.991055 − 0.133451i \(-0.0426059\pi\)
\(68\) 3.78744e11 + 6.56004e11i 0.464556 + 0.804634i
\(69\) −2.79811e11 −0.312138
\(70\) 0 0
\(71\) 1.65485e11 0.153313 0.0766566 − 0.997058i \(-0.475575\pi\)
0.0766566 + 0.997058i \(0.475575\pi\)
\(72\) 1.41323e12 + 2.44779e12i 1.19551 + 2.07068i
\(73\) −3.13378e11 + 5.42787e11i −0.242365 + 0.419789i −0.961388 − 0.275198i \(-0.911257\pi\)
0.719022 + 0.694987i \(0.244590\pi\)
\(74\) 4.73734e10 − 8.20532e10i 0.0335374 − 0.0580884i
\(75\) −3.91336e10 − 6.77814e10i −0.0253894 − 0.0439756i
\(76\) 8.40561e12 5.00358
\(77\) 0 0
\(78\) 1.07574e12 0.540871
\(79\) −3.45341e11 − 5.98148e11i −0.159835 − 0.276843i 0.774974 − 0.631993i \(-0.217763\pi\)
−0.934809 + 0.355151i \(0.884430\pi\)
\(80\) −3.73008e12 + 6.46069e12i −1.59087 + 2.75546i
\(81\) −5.91076e11 + 1.02377e12i −0.232536 + 0.402765i
\(82\) −3.26079e12 − 5.64785e12i −1.18449 − 2.05160i
\(83\) −3.52783e12 −1.18441 −0.592203 − 0.805789i \(-0.701742\pi\)
−0.592203 + 0.805789i \(0.701742\pi\)
\(84\) 0 0
\(85\) −1.33115e12 −0.382828
\(86\) 6.27797e10 + 1.08738e11i 0.0167332 + 0.0289828i
\(87\) 1.17391e12 − 2.03327e12i 0.290241 − 0.502713i
\(88\) −9.14077e11 + 1.58323e12i −0.209819 + 0.363418i
\(89\) 6.89592e11 + 1.19441e12i 0.147081 + 0.254752i 0.930147 − 0.367186i \(-0.119679\pi\)
−0.783066 + 0.621938i \(0.786345\pi\)
\(90\) −8.14222e12 −1.61498
\(91\) 0 0
\(92\) −1.06687e13 −1.83438
\(93\) 1.00264e12 + 1.73663e12i 0.160697 + 0.278335i
\(94\) −3.06669e12 + 5.31166e12i −0.458499 + 0.794144i
\(95\) −7.38568e12 + 1.27924e13i −1.03083 + 1.78545i
\(96\) −4.57112e12 − 7.91742e12i −0.596019 − 1.03233i
\(97\) −3.29185e10 −0.00401258 −0.00200629 − 0.999998i \(-0.500639\pi\)
−0.00200629 + 0.999998i \(0.500639\pi\)
\(98\) 0 0
\(99\) −1.07764e12 −0.115039
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 49.14.c.g.18.1 16
7.2 even 3 inner 49.14.c.g.30.1 16
7.3 odd 6 49.14.a.e.1.8 8
7.4 even 3 49.14.a.f.1.8 8
7.5 odd 6 7.14.c.a.2.1 ✓ 16
7.6 odd 2 7.14.c.a.4.1 yes 16
21.5 even 6 63.14.e.c.37.8 16
21.20 even 2 63.14.e.c.46.8 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
7.14.c.a.2.1 ✓ 16 7.5 odd 6
7.14.c.a.4.1 yes 16 7.6 odd 2
49.14.a.e.1.8 8 7.3 odd 6
49.14.a.f.1.8 8 7.4 even 3
49.14.c.g.18.1 16 1.1 even 1 trivial
49.14.c.g.30.1 16 7.2 even 3 inner
63.14.e.c.37.8 16 21.5 even 6
63.14.e.c.46.8 16 21.20 even 2