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Properties

Label 49.5.d.c.19.2
Level $49$
Weight $5$
Character 49.19
Analytic conductor $5.065$
Analytic rank $0$
Dimension $16$
Inner twists $4$

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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] = [49,5,Mod(19,49)] 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("49.19"); S:= CuspForms(chi, 5); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(49, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([5])) N = Newforms(chi, 5, names="a")
 
Level: \( N \) \(=\) \( 49 = 7^{2} \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 49.d (of order \(6\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [16] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(5.06512819111\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
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: \( x^{16} - 8 x^{15} - 140 x^{14} + 1120 x^{13} + 8358 x^{12} - 65072 x^{11} - 277636 x^{10} + \cdots + 4890937156 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 7^{12} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 19.2
Root \(-4.32803 - 0.382683i\) of defining polynomial
Character \(\chi\) \(=\) 49.19
Dual form 49.5.d.c.31.2

$q$-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+(-1.99543 + 3.45618i) q^{2} +(3.81864 - 2.20470i) q^{3} +(0.0365359 + 0.0632820i) q^{4} +(-37.9221 - 21.8943i) q^{5} +17.5972i q^{6} -64.1453 q^{8} +(-30.7786 + 53.3102i) q^{9} +(151.342 - 87.3772i) q^{10} +(-66.8625 - 115.809i) q^{11} +(0.279035 + 0.161101i) q^{12} -114.216i q^{13} -193.081 q^{15} +(127.413 - 220.685i) q^{16} +(-34.2113 + 19.7519i) q^{17} +(-122.833 - 212.753i) q^{18} +(-52.5127 - 30.3182i) q^{19} -3.19972i q^{20} +533.677 q^{22} +(-298.231 + 516.551i) q^{23} +(-244.948 + 141.421i) q^{24} +(646.225 + 1119.29i) q^{25} +(394.750 + 227.909i) q^{26} +628.591i q^{27} +307.119 q^{29} +(385.280 - 667.325i) q^{30} +(-1171.94 + 676.618i) q^{31} +(-4.67656 - 8.10004i) q^{32} +(-510.648 - 294.823i) q^{33} -157.654i q^{34} -4.49810 q^{36} +(27.6799 - 47.9429i) q^{37} +(209.571 - 120.996i) q^{38} +(-251.811 - 436.149i) q^{39} +(2432.53 + 1404.42i) q^{40} -1310.24i q^{41} +1094.10 q^{43} +(4.88576 - 8.46238i) q^{44} +(2334.38 - 1347.76i) q^{45} +(-1190.20 - 2061.48i) q^{46} +(-2419.04 - 1396.64i) q^{47} -1123.63i q^{48} -5157.98 q^{50} +(-87.0939 + 150.851i) q^{51} +(7.22779 - 4.17297i) q^{52} +(340.913 + 590.478i) q^{53} +(-2172.52 - 1254.31i) q^{54} +5855.64i q^{55} -267.370 q^{57} +(-612.833 + 1061.46i) q^{58} +(1815.47 - 1048.16i) q^{59} +(-7.05440 - 12.2186i) q^{60} +(-1499.39 - 865.675i) q^{61} -5400.57i q^{62} +4114.54 q^{64} +(-2500.68 + 4331.30i) q^{65} +(2037.92 - 1176.59i) q^{66} +(528.898 + 916.078i) q^{67} +(-2.49988 - 1.44331i) q^{68} +2630.03i q^{69} -712.465 q^{71} +(1974.31 - 3419.60i) q^{72} +(-4626.30 + 2670.99i) q^{73} +(110.466 + 191.333i) q^{74} +(4935.41 + 2849.46i) q^{75} -4.43081i q^{76} +2009.88 q^{78} +(5912.37 - 10240.5i) q^{79} +(-9663.52 + 5579.24i) q^{80} +(-1107.22 - 1917.76i) q^{81} +(4528.44 + 2614.50i) q^{82} -7632.44i q^{83} +1729.82 q^{85} +(-2183.20 + 3781.41i) q^{86} +(1172.78 - 677.103i) q^{87} +(4288.91 + 7428.62i) q^{88} +(6221.17 + 3591.79i) q^{89} +10757.4i q^{90} -43.5845 q^{92} +(-2983.47 + 5167.52i) q^{93} +(9654.06 - 5573.77i) q^{94} +(1327.59 + 2299.46i) q^{95} +(-35.7162 - 20.6208i) q^{96} -6917.85i q^{97} +8231.74 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 12 q^{2} - 52 q^{4} - 744 q^{8} + 352 q^{9} - 120 q^{11} - 1264 q^{15} + 300 q^{16} - 340 q^{18} + 3904 q^{22} - 1752 q^{23} + 2192 q^{25} + 2496 q^{29} - 456 q^{30} - 3156 q^{32} + 9880 q^{36} - 2368 q^{37}+ \cdots + 60288 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(3\)
\(\chi(n)\) \(e\left(\frac{5}{6}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.99543 + 3.45618i −0.498857 + 0.864046i −0.999999 − 0.00131938i \(-0.999580\pi\)
0.501142 + 0.865365i \(0.332913\pi\)
\(3\) 3.81864 − 2.20470i 0.424294 − 0.244966i −0.272619 − 0.962122i \(-0.587890\pi\)
0.696913 + 0.717156i \(0.254556\pi\)
\(4\) 0.0365359 + 0.0632820i 0.00228349 + 0.00395513i
\(5\) −37.9221 − 21.8943i −1.51688 − 0.875774i −0.999803 − 0.0198378i \(-0.993685\pi\)
−0.517082 − 0.855936i \(-0.672982\pi\)
\(6\) 17.5972i 0.488812i
\(7\) 0 0
\(8\) −64.1453 −1.00227
\(9\) −30.7786 + 53.3102i −0.379983 + 0.658150i
\(10\) 151.342 − 87.3772i 1.51342 − 0.873772i
\(11\) −66.8625 − 115.809i −0.552582 − 0.957101i −0.998087 − 0.0618211i \(-0.980309\pi\)
0.445505 − 0.895279i \(-0.353024\pi\)
\(12\) 0.279035 + 0.161101i 0.00193774 + 0.00111876i
\(13\) − 114.216i − 0.675832i −0.941176 − 0.337916i \(-0.890278\pi\)
0.941176 − 0.337916i \(-0.109722\pi\)
\(14\) 0 0
\(15\) −193.081 −0.858140
\(16\) 127.413 − 220.685i 0.497706 − 0.862052i
\(17\) −34.2113 + 19.7519i −0.118378 + 0.0683457i −0.558020 − 0.829828i \(-0.688439\pi\)
0.439642 + 0.898173i \(0.355105\pi\)
\(18\) −122.833 − 212.753i −0.379114 − 0.656646i
\(19\) −52.5127 − 30.3182i −0.145464 − 0.0839840i 0.425501 − 0.904958i \(-0.360098\pi\)
−0.570966 + 0.820974i \(0.693431\pi\)
\(20\) − 3.19972i − 0.00799930i
\(21\) 0 0
\(22\) 533.677 1.10264
\(23\) −298.231 + 516.551i −0.563763 + 0.976467i 0.433400 + 0.901202i \(0.357314\pi\)
−0.997164 + 0.0752650i \(0.976020\pi\)
\(24\) −244.948 + 141.421i −0.425257 + 0.245522i
\(25\) 646.225 + 1119.29i 1.03396 + 1.79087i
\(26\) 394.750 + 227.909i 0.583949 + 0.337143i
\(27\) 628.591i 0.862264i
\(28\) 0 0
\(29\) 307.119 0.365183 0.182591 − 0.983189i \(-0.441551\pi\)
0.182591 + 0.983189i \(0.441551\pi\)
\(30\) 385.280 − 667.325i 0.428089 − 0.741472i
\(31\) −1171.94 + 676.618i −1.21950 + 0.704077i −0.964810 − 0.262947i \(-0.915306\pi\)
−0.254686 + 0.967024i \(0.581972\pi\)
\(32\) −4.67656 − 8.10004i −0.00456695 − 0.00791019i
\(33\) −510.648 − 294.823i −0.468915 − 0.270728i
\(34\) − 157.654i − 0.136379i
\(35\) 0 0
\(36\) −4.49810 −0.00347076
\(37\) 27.6799 − 47.9429i 0.0202190 − 0.0350204i −0.855739 − 0.517408i \(-0.826897\pi\)
0.875958 + 0.482388i \(0.160230\pi\)
\(38\) 209.571 − 120.996i 0.145132 − 0.0837920i
\(39\) −251.811 − 436.149i −0.165556 − 0.286751i
\(40\) 2432.53 + 1404.42i 1.52033 + 0.877762i
\(41\) − 1310.24i − 0.779443i −0.920933 − 0.389722i \(-0.872571\pi\)
0.920933 − 0.389722i \(-0.127429\pi\)
\(42\) 0 0
\(43\) 1094.10 0.591725 0.295863 − 0.955231i \(-0.404393\pi\)
0.295863 + 0.955231i \(0.404393\pi\)
\(44\) 4.88576 − 8.46238i 0.00252364 − 0.00437107i
\(45\) 2334.38 − 1347.76i 1.15278 − 0.665559i
\(46\) −1190.20 − 2061.48i −0.562474 − 0.974234i
\(47\) −2419.04 − 1396.64i −1.09509 − 0.632248i −0.160160 − 0.987091i \(-0.551201\pi\)
−0.934926 + 0.354843i \(0.884534\pi\)
\(48\) − 1123.63i − 0.487685i
\(49\) 0 0
\(50\) −5157.98 −2.06319
\(51\) −87.0939 + 150.851i −0.0334848 + 0.0579973i
\(52\) 7.22779 − 4.17297i 0.00267300 − 0.00154326i
\(53\) 340.913 + 590.478i 0.121364 + 0.210209i 0.920306 − 0.391199i \(-0.127940\pi\)
−0.798942 + 0.601409i \(0.794606\pi\)
\(54\) −2172.52 − 1254.31i −0.745036 − 0.430147i
\(55\) 5855.64i 1.93575i
\(56\) 0 0
\(57\) −267.370 −0.0822929
\(58\) −612.833 + 1061.46i −0.182174 + 0.315534i
\(59\) 1815.47 − 1048.16i 0.521538 − 0.301110i −0.216026 − 0.976388i \(-0.569310\pi\)
0.737563 + 0.675278i \(0.235976\pi\)
\(60\) −7.05440 − 12.2186i −0.00195956 − 0.00339405i
\(61\) −1499.39 − 865.675i −0.402955 − 0.232646i 0.284803 − 0.958586i \(-0.408072\pi\)
−0.687758 + 0.725940i \(0.741405\pi\)
\(62\) − 5400.57i − 1.40493i
\(63\) 0 0
\(64\) 4114.54 1.00453
\(65\) −2500.68 + 4331.30i −0.591876 + 1.02516i
\(66\) 2037.92 − 1176.59i 0.467843 − 0.270109i
\(67\) 528.898 + 916.078i 0.117821 + 0.204072i 0.918904 − 0.394482i \(-0.129076\pi\)
−0.801083 + 0.598553i \(0.795742\pi\)
\(68\) −2.49988 − 1.44331i −0.000540632 − 0.000312134i
\(69\) 2630.03i 0.552412i
\(70\) 0 0
\(71\) −712.465 −0.141334 −0.0706670 − 0.997500i \(-0.522513\pi\)
−0.0706670 + 0.997500i \(0.522513\pi\)
\(72\) 1974.31 − 3419.60i 0.380846 − 0.659644i
\(73\) −4626.30 + 2670.99i −0.868136 + 0.501218i −0.866728 − 0.498781i \(-0.833781\pi\)
−0.00140745 + 0.999999i \(0.500448\pi\)
\(74\) 110.466 + 191.333i 0.0201728 + 0.0349403i
\(75\) 4935.41 + 2849.46i 0.877406 + 0.506570i
\(76\) − 4.43081i − 0.000767107i
\(77\) 0 0
\(78\) 2009.88 0.330355
\(79\) 5912.37 − 10240.5i 0.947343 − 1.64085i 0.196352 − 0.980534i \(-0.437091\pi\)
0.750991 − 0.660312i \(-0.229576\pi\)
\(80\) −9663.52 + 5579.24i −1.50993 + 0.871756i
\(81\) −1107.22 − 1917.76i −0.168758 − 0.292297i
\(82\) 4528.44 + 2614.50i 0.673475 + 0.388831i
\(83\) − 7632.44i − 1.10792i −0.832544 − 0.553959i \(-0.813117\pi\)
0.832544 − 0.553959i \(-0.186883\pi\)
\(84\) 0 0
\(85\) 1729.82 0.239422
\(86\) −2183.20 + 3781.41i −0.295186 + 0.511277i
\(87\) 1172.78 − 677.103i 0.154945 − 0.0894574i
\(88\) 4288.91 + 7428.62i 0.553837 + 0.959274i
\(89\) 6221.17 + 3591.79i 0.785402 + 0.453452i 0.838341 − 0.545146i \(-0.183526\pi\)
−0.0529394 + 0.998598i \(0.516859\pi\)
\(90\) 10757.4i 1.32807i
\(91\) 0 0
\(92\) −43.5845 −0.00514940
\(93\) −2983.47 + 5167.52i −0.344950 + 0.597471i
\(94\) 9654.06 − 5573.77i 1.09258 − 0.630803i
\(95\) 1327.59 + 2299.46i 0.147102 + 0.254788i
\(96\) −35.7162 − 20.6208i −0.00387546 − 0.00223750i
\(97\) − 6917.85i − 0.735237i −0.929977 − 0.367619i \(-0.880173\pi\)
0.929977 − 0.367619i \(-0.119827\pi\)
\(98\) 0 0
\(99\) 8231.74 0.839888
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 49.5.d.c.19.2 16
7.2 even 3 49.5.b.b.48.8 yes 8
7.3 odd 6 inner 49.5.d.c.31.2 16
7.4 even 3 inner 49.5.d.c.31.1 16
7.5 odd 6 49.5.b.b.48.7 ✓ 8
7.6 odd 2 inner 49.5.d.c.19.1 16
21.2 odd 6 441.5.d.g.244.1 8
21.5 even 6 441.5.d.g.244.2 8
28.19 even 6 784.5.c.g.97.5 8
28.23 odd 6 784.5.c.g.97.4 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
49.5.b.b.48.7 ✓ 8 7.5 odd 6
49.5.b.b.48.8 yes 8 7.2 even 3
49.5.d.c.19.1 16 7.6 odd 2 inner
49.5.d.c.19.2 16 1.1 even 1 trivial
49.5.d.c.31.1 16 7.4 even 3 inner
49.5.d.c.31.2 16 7.3 odd 6 inner
441.5.d.g.244.1 8 21.2 odd 6
441.5.d.g.244.2 8 21.5 even 6
784.5.c.g.97.4 8 28.23 odd 6
784.5.c.g.97.5 8 28.19 even 6