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Properties

Label 49.16.c.b.30.1
Level $49$
Weight $16$
Character 49.30
Analytic conductor $69.920$
Analytic rank $0$
Dimension $2$
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] = [49,16,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, 16); 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, 16, names="a")
 
Level: \( N \) \(=\) \( 49 = 7^{2} \)
Weight: \( k \) \(=\) \( 16 \)
Character orbit: \([\chi]\) \(=\) 49.c (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,-216,-3348] 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: \(69.9198174990\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 30.1
Root \(0.500000 - 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 49.30
Dual form 49.16.c.b.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+(-108.000 + 187.061i) q^{2} +(-1674.00 - 2899.45i) q^{3} +(-6944.00 - 12027.4i) q^{4} +(26055.0 - 45128.6i) q^{5} +723168. q^{6} -4.07808e6 q^{8} +(1.56990e6 - 2.71915e6i) q^{9} +(5.62788e6 + 9.74777e6i) q^{10} +(-1.02934e7 - 1.78287e7i) q^{11} +(-2.32485e7 + 4.02676e7i) q^{12} +1.90073e8 q^{13} -1.74464e8 q^{15} +(6.67974e8 - 1.15696e9i) q^{16} +(8.23264e8 + 1.42594e9i) q^{17} +(3.39099e8 + 5.87336e8i) q^{18} +(7.81629e8 - 1.35382e9i) q^{19} -7.23704e8 q^{20} +4.44676e9 q^{22} +(-4.72556e9 + 8.18491e9i) q^{23} +(6.82671e9 + 1.18242e10i) q^{24} +(1.39011e10 + 2.40773e10i) q^{25} +(-2.05279e10 + 3.55554e10i) q^{26} -5.85522e10 q^{27} -3.69026e10 q^{29} +(1.88421e10 - 3.26355e10i) q^{30} +(3.57942e10 + 6.19974e10i) q^{31} +(7.74670e10 + 1.34177e11i) q^{32} +(-3.44624e10 + 5.96906e10i) q^{33} -3.55650e11 q^{34} -4.36056e10 q^{36} +(5.16826e11 - 8.95169e11i) q^{37} +(1.68832e11 + 2.92425e11i) q^{38} +(-3.18183e11 - 5.51109e11i) q^{39} +(-1.06254e11 + 1.84038e11i) q^{40} -1.64197e12 q^{41} -4.92403e11 q^{43} +(-1.42955e11 + 2.47605e11i) q^{44} +(-8.18076e10 - 1.41695e11i) q^{45} +(-1.02072e12 - 1.76794e12i) q^{46} +(-1.70534e12 + 2.95374e12i) q^{47} -4.47275e12 q^{48} -6.00526e12 q^{50} +(2.75629e12 - 4.77403e12i) q^{51} +(-1.31987e12 - 2.28608e12i) q^{52} +(-3.39858e12 - 5.88651e12i) q^{53} +(6.32364e12 - 1.09529e13i) q^{54} -1.07278e12 q^{55} -5.23379e12 q^{57} +(3.98548e12 - 6.90305e12i) q^{58} +(4.92943e12 + 8.53802e12i) q^{59} +(1.21148e12 + 2.09834e12i) q^{60} +(2.46592e12 - 4.27110e12i) q^{61} -1.54631e13 q^{62} +1.03106e13 q^{64} +(4.95236e12 - 8.57774e12i) q^{65} +(-7.44388e12 - 1.28932e13i) q^{66} +(1.44189e13 + 2.49743e13i) q^{67} +(1.14335e13 - 1.98034e13i) q^{68} +3.16423e13 q^{69} +1.25050e14 q^{71} +(-6.40218e12 + 1.10889e13i) q^{72} +(-4.10857e13 - 7.11626e13i) q^{73} +(1.11634e14 + 1.93356e14i) q^{74} +(4.65408e13 - 8.06110e13i) q^{75} -2.17105e13 q^{76} +1.37455e14 q^{78} +(1.27065e13 - 2.20084e13i) q^{79} +(-3.48081e13 - 6.02894e13i) q^{80} +(7.54900e13 + 1.30753e14i) q^{81} +(1.77333e14 - 3.07150e14i) q^{82} +2.81737e14 q^{83} +8.58006e13 q^{85} +(5.31795e13 - 9.21097e13i) q^{86} +(6.17749e13 + 1.06997e14i) q^{87} +(4.19774e13 + 7.27070e13i) q^{88} +(3.57809e14 - 6.19744e14i) q^{89} +3.53409e13 q^{90} +1.31257e14 q^{92} +(1.19839e14 - 2.07567e14i) q^{93} +(-3.68354e14 - 6.38008e14i) q^{94} +(-4.07307e13 - 7.05476e13i) q^{95} +(2.59360e14 - 4.49224e14i) q^{96} -6.12786e14 q^{97} -6.46387e13 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 216 q^{2} - 3348 q^{3} - 13888 q^{4} + 52110 q^{5} + 1446336 q^{6} - 8156160 q^{8} + 3139803 q^{9} + 11255760 q^{10} - 20586852 q^{11} - 46497024 q^{12} + 380146676 q^{13} - 348928560 q^{15} + 1335947264 q^{16}+ \cdots - 129277319340312 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −108.000 + 187.061i −0.596621 + 1.03338i 0.396695 + 0.917951i \(0.370157\pi\)
−0.993316 + 0.115428i \(0.963176\pi\)
\(3\) −1674.00 − 2899.45i −0.441922 − 0.765432i 0.555910 − 0.831243i \(-0.312370\pi\)
−0.997832 + 0.0658106i \(0.979037\pi\)
\(4\) −6944.00 − 12027.4i −0.211914 − 0.367046i
\(5\) 26055.0 − 45128.6i 0.149148 − 0.258331i −0.781765 − 0.623573i \(-0.785680\pi\)
0.930913 + 0.365242i \(0.119014\pi\)
\(6\) 723168. 1.05464
\(7\) 0 0
\(8\) −4.07808e6 −0.687513
\(9\) 1.56990e6 − 2.71915e6i 0.109409 − 0.189502i
\(10\) 5.62788e6 + 9.74777e6i 0.177969 + 0.308252i
\(11\) −1.02934e7 − 1.78287e7i −0.159263 − 0.275852i 0.775340 − 0.631544i \(-0.217578\pi\)
−0.934603 + 0.355692i \(0.884245\pi\)
\(12\) −2.32485e7 + 4.02676e7i −0.187299 + 0.324412i
\(13\) 1.90073e8 0.840129 0.420065 − 0.907494i \(-0.362007\pi\)
0.420065 + 0.907494i \(0.362007\pi\)
\(14\) 0 0
\(15\) −1.74464e8 −0.263647
\(16\) 6.67974e8 − 1.15696e9i 0.622099 − 1.07751i
\(17\) 8.23264e8 + 1.42594e9i 0.486600 + 0.842816i 0.999881 − 0.0154043i \(-0.00490355\pi\)
−0.513281 + 0.858220i \(0.671570\pi\)
\(18\) 3.39099e8 + 5.87336e8i 0.130552 + 0.226122i
\(19\) 7.81629e8 − 1.35382e9i 0.200608 − 0.347463i −0.748116 − 0.663567i \(-0.769042\pi\)
0.948725 + 0.316104i \(0.102375\pi\)
\(20\) −7.23704e8 −0.126426
\(21\) 0 0
\(22\) 4.44676e9 0.380079
\(23\) −4.72556e9 + 8.18491e9i −0.289397 + 0.501251i −0.973666 − 0.227979i \(-0.926788\pi\)
0.684269 + 0.729230i \(0.260122\pi\)
\(24\) 6.82671e9 + 1.18242e10i 0.303827 + 0.526244i
\(25\) 1.39011e10 + 2.40773e10i 0.455510 + 0.788967i
\(26\) −2.05279e10 + 3.55554e10i −0.501239 + 0.868171i
\(27\) −5.85522e10 −1.07725
\(28\) 0 0
\(29\) −3.69026e10 −0.397257 −0.198629 − 0.980075i \(-0.563649\pi\)
−0.198629 + 0.980075i \(0.563649\pi\)
\(30\) 1.88421e10 − 3.26355e10i 0.157297 − 0.272447i
\(31\) 3.57942e10 + 6.19974e10i 0.233669 + 0.404726i 0.958885 − 0.283795i \(-0.0915936\pi\)
−0.725216 + 0.688521i \(0.758260\pi\)
\(32\) 7.74670e10 + 1.34177e11i 0.398559 + 0.690324i
\(33\) −3.44624e10 + 5.96906e10i −0.140764 + 0.243810i
\(34\) −3.55650e11 −1.16126
\(35\) 0 0
\(36\) −4.36056e10 −0.0927413
\(37\) 5.16826e11 − 8.95169e11i 0.895017 − 1.55022i 0.0612342 − 0.998123i \(-0.480496\pi\)
0.833783 − 0.552092i \(-0.186170\pi\)
\(38\) 1.68832e11 + 2.92425e11i 0.239374 + 0.414608i
\(39\) −3.18183e11 − 5.51109e11i −0.371272 − 0.643062i
\(40\) −1.06254e11 + 1.84038e11i −0.102541 + 0.177606i
\(41\) −1.64197e12 −1.31670 −0.658351 − 0.752711i \(-0.728746\pi\)
−0.658351 + 0.752711i \(0.728746\pi\)
\(42\) 0 0
\(43\) −4.92403e11 −0.276253 −0.138127 − 0.990415i \(-0.544108\pi\)
−0.138127 + 0.990415i \(0.544108\pi\)
\(44\) −1.42955e11 + 2.47605e11i −0.0675001 + 0.116914i
\(45\) −8.18076e10 − 1.41695e11i −0.0326362 − 0.0565276i
\(46\) −1.02072e12 − 1.76794e12i −0.345321 − 0.598114i
\(47\) −1.70534e12 + 2.95374e12i −0.490996 + 0.850429i −0.999946 − 0.0103663i \(-0.996700\pi\)
0.508951 + 0.860796i \(0.330034\pi\)
\(48\) −4.47275e12 −1.09968
\(49\) 0 0
\(50\) −6.00526e12 −1.08707
\(51\) 2.75629e12 − 4.77403e12i 0.430079 − 0.744919i
\(52\) −1.31987e12 − 2.28608e12i −0.178035 − 0.308366i
\(53\) −3.39858e12 − 5.88651e12i −0.397400 − 0.688317i 0.596004 − 0.802981i \(-0.296754\pi\)
−0.993404 + 0.114664i \(0.963421\pi\)
\(54\) 6.32364e12 − 1.09529e13i 0.642708 − 1.11320i
\(55\) −1.07278e12 −0.0950147
\(56\) 0 0
\(57\) −5.23379e12 −0.354613
\(58\) 3.98548e12 − 6.90305e12i 0.237012 − 0.410517i
\(59\) 4.92943e12 + 8.53802e12i 0.257873 + 0.446650i 0.965672 − 0.259764i \(-0.0836449\pi\)
−0.707799 + 0.706414i \(0.750312\pi\)
\(60\) 1.21148e12 + 2.09834e12i 0.0558704 + 0.0967704i
\(61\) 2.46592e12 − 4.27110e12i 0.100463 − 0.174007i −0.811413 − 0.584474i \(-0.801301\pi\)
0.911875 + 0.410467i \(0.134634\pi\)
\(62\) −1.54631e13 −0.557647
\(63\) 0 0
\(64\) 1.03106e13 0.293044
\(65\) 4.95236e12 − 8.57774e12i 0.125303 − 0.217031i
\(66\) −7.44388e12 − 1.28932e13i −0.167965 − 0.290924i
\(67\) 1.44189e13 + 2.49743e13i 0.290651 + 0.503422i 0.973964 − 0.226704i \(-0.0727949\pi\)
−0.683313 + 0.730126i \(0.739462\pi\)
\(68\) 1.14335e13 − 1.98034e13i 0.206235 − 0.357209i
\(69\) 3.16423e13 0.511564
\(70\) 0 0
\(71\) 1.25050e14 1.63172 0.815862 − 0.578247i \(-0.196263\pi\)
0.815862 + 0.578247i \(0.196263\pi\)
\(72\) −6.40218e12 + 1.10889e13i −0.0752202 + 0.130285i
\(73\) −4.10857e13 − 7.11626e13i −0.435281 − 0.753928i 0.562038 − 0.827112i \(-0.310018\pi\)
−0.997319 + 0.0731831i \(0.976684\pi\)
\(74\) 1.11634e14 + 1.93356e14i 1.06797 + 1.84978i
\(75\) 4.65408e13 − 8.06110e13i 0.402600 − 0.697324i
\(76\) −2.17105e13 −0.170047
\(77\) 0 0
\(78\) 1.37455e14 0.886035
\(79\) 1.27065e13 − 2.20084e13i 0.0744430 − 0.128939i −0.826401 − 0.563082i \(-0.809615\pi\)
0.900844 + 0.434143i \(0.142949\pi\)
\(80\) −3.48081e13 − 6.02894e13i −0.185569 − 0.321415i
\(81\) 7.54900e13 + 1.30753e14i 0.366650 + 0.635057i
\(82\) 1.77333e14 − 3.07150e14i 0.785572 − 1.36065i
\(83\) 2.81737e14 1.13961 0.569807 − 0.821779i \(-0.307018\pi\)
0.569807 + 0.821779i \(0.307018\pi\)
\(84\) 0 0
\(85\) 8.58006e13 0.290301
\(86\) 5.31795e13 − 9.21097e13i 0.164819 − 0.285474i
\(87\) 6.17749e13 + 1.06997e14i 0.175557 + 0.304073i
\(88\) 4.19774e13 + 7.27070e13i 0.109495 + 0.189651i
\(89\) 3.57809e14 − 6.19744e14i 0.857485 − 1.48521i −0.0168353 − 0.999858i \(-0.505359\pi\)
0.874320 − 0.485349i \(-0.161308\pi\)
\(90\) 3.53409e13 0.0778858
\(91\) 0 0
\(92\) 1.31257e14 0.245309
\(93\) 1.19839e14 − 2.07567e14i 0.206527 − 0.357715i
\(94\) −3.68354e14 − 6.38008e14i −0.585877 − 1.01477i
\(95\) −4.07307e13 − 7.05476e13i −0.0598404 − 0.103647i
\(96\) 2.59360e14 − 4.49224e14i 0.352264 − 0.610139i
\(97\) −6.12786e14 −0.770054 −0.385027 − 0.922905i \(-0.625808\pi\)
−0.385027 + 0.922905i \(0.625808\pi\)
\(98\) 0 0
\(99\) −6.46387e13 −0.0696993
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 49.16.c.b.30.1 2
7.2 even 3 49.16.a.a.1.1 1
7.3 odd 6 49.16.c.c.18.1 2
7.4 even 3 inner 49.16.c.b.18.1 2
7.5 odd 6 1.16.a.a.1.1 ✓ 1
7.6 odd 2 49.16.c.c.30.1 2
21.5 even 6 9.16.a.a.1.1 1
28.19 even 6 16.16.a.d.1.1 1
35.12 even 12 25.16.b.a.24.2 2
35.19 odd 6 25.16.a.a.1.1 1
35.33 even 12 25.16.b.a.24.1 2
56.5 odd 6 64.16.a.i.1.1 1
56.19 even 6 64.16.a.c.1.1 1
77.54 even 6 121.16.a.a.1.1 1
84.47 odd 6 144.16.a.f.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1.16.a.a.1.1 ✓ 1 7.5 odd 6
9.16.a.a.1.1 1 21.5 even 6
16.16.a.d.1.1 1 28.19 even 6
25.16.a.a.1.1 1 35.19 odd 6
25.16.b.a.24.1 2 35.33 even 12
25.16.b.a.24.2 2 35.12 even 12
49.16.a.a.1.1 1 7.2 even 3
49.16.c.b.18.1 2 7.4 even 3 inner
49.16.c.b.30.1 2 1.1 even 1 trivial
49.16.c.c.18.1 2 7.3 odd 6
49.16.c.c.30.1 2 7.6 odd 2
64.16.a.c.1.1 1 56.19 even 6
64.16.a.i.1.1 1 56.5 odd 6
121.16.a.a.1.1 1 77.54 even 6
144.16.a.f.1.1 1 84.47 odd 6