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Properties

Label 1694.2.a.z.1.4
Level $1694$
Weight $2$
Character 1694.1
Self dual yes
Analytic conductor $13.527$
Analytic rank $1$
Dimension $4$
CM no
Inner twists $1$

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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] = [1694,2,Mod(1,1694)] 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("1694.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1694, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1694 = 2 \cdot 7 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1694.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,4,-3,4,-5] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(13.5266581024\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{70 +2 \sqrt{5}})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 9x^{2} + 4x + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 154)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(2.96645\) of defining polynomial
Character \(\chi\) \(=\) 1694.1

$q$-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)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.96645 q^{3} +1.00000 q^{4} -4.18178 q^{5} +1.96645 q^{6} -1.00000 q^{7} +1.00000 q^{8} +0.866918 q^{9} -4.18178 q^{10} +1.96645 q^{12} -4.40270 q^{13} -1.00000 q^{14} -8.22325 q^{15} +1.00000 q^{16} +1.10299 q^{17} +0.866918 q^{18} -1.43625 q^{19} -4.18178 q^{20} -1.96645 q^{21} -1.16663 q^{23} +1.96645 q^{24} +12.4873 q^{25} -4.40270 q^{26} -4.19460 q^{27} -1.00000 q^{28} -7.30550 q^{29} -8.22325 q^{30} -2.83337 q^{31} +1.00000 q^{32} +1.10299 q^{34} +4.18178 q^{35} +0.866918 q^{36} -7.08458 q^{37} -1.43625 q^{38} -8.65769 q^{39} -4.18178 q^{40} -1.13308 q^{41} -1.96645 q^{42} -4.55093 q^{43} -3.62526 q^{45} -1.16663 q^{46} -7.19316 q^{47} +1.96645 q^{48} +1.00000 q^{49} +12.4873 q^{50} +2.16896 q^{51} -4.40270 q^{52} +10.0023 q^{53} -4.19460 q^{54} -1.00000 q^{56} -2.82432 q^{57} -7.30550 q^{58} +7.41439 q^{59} -8.22325 q^{60} -11.0151 q^{61} -2.83337 q^{62} -0.866918 q^{63} +1.00000 q^{64} +18.4111 q^{65} +11.7327 q^{67} +1.10299 q^{68} -2.29413 q^{69} +4.18178 q^{70} +8.47447 q^{71} +0.866918 q^{72} -13.6090 q^{73} -7.08458 q^{74} +24.5556 q^{75} -1.43625 q^{76} -8.65769 q^{78} -7.26403 q^{79} -4.18178 q^{80} -10.8492 q^{81} -1.13308 q^{82} +6.44235 q^{83} -1.96645 q^{84} -4.61244 q^{85} -4.55093 q^{86} -14.3659 q^{87} +7.52083 q^{89} -3.62526 q^{90} +4.40270 q^{91} -1.16663 q^{92} -5.57167 q^{93} -7.19316 q^{94} +6.00610 q^{95} +1.96645 q^{96} +1.40616 q^{97} +1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{2} - 3 q^{3} + 4 q^{4} - 5 q^{5} - 3 q^{6} - 4 q^{7} + 4 q^{8} + 9 q^{9} - 5 q^{10} - 3 q^{12} - 14 q^{13} - 4 q^{14} - 5 q^{15} + 4 q^{16} + q^{17} + 9 q^{18} - 13 q^{19} - 5 q^{20} + 3 q^{21}+ \cdots + 4 q^{98}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) 1.96645 1.13533 0.567665 − 0.823260i \(-0.307847\pi\)
0.567665 + 0.823260i \(0.307847\pi\)
\(4\) 1.00000 0.500000
\(5\) −4.18178 −1.87015 −0.935074 − 0.354452i \(-0.884668\pi\)
−0.935074 + 0.354452i \(0.884668\pi\)
\(6\) 1.96645 0.802799
\(7\) −1.00000 −0.377964
\(8\) 1.00000 0.353553
\(9\) 0.866918 0.288973
\(10\) −4.18178 −1.32239
\(11\) 0 0
\(12\) 1.96645 0.567665
\(13\) −4.40270 −1.22109 −0.610545 − 0.791982i \(-0.709050\pi\)
−0.610545 + 0.791982i \(0.709050\pi\)
\(14\) −1.00000 −0.267261
\(15\) −8.22325 −2.12323
\(16\) 1.00000 0.250000
\(17\) 1.10299 0.267513 0.133757 − 0.991014i \(-0.457296\pi\)
0.133757 + 0.991014i \(0.457296\pi\)
\(18\) 0.866918 0.204334
\(19\) −1.43625 −0.329499 −0.164750 − 0.986335i \(-0.552682\pi\)
−0.164750 + 0.986335i \(0.552682\pi\)
\(20\) −4.18178 −0.935074
\(21\) −1.96645 −0.429114
\(22\) 0 0
\(23\) −1.16663 −0.243260 −0.121630 − 0.992576i \(-0.538812\pi\)
−0.121630 + 0.992576i \(0.538812\pi\)
\(24\) 1.96645 0.401400
\(25\) 12.4873 2.49746
\(26\) −4.40270 −0.863441
\(27\) −4.19460 −0.807250
\(28\) −1.00000 −0.188982
\(29\) −7.30550 −1.35660 −0.678299 − 0.734786i \(-0.737282\pi\)
−0.678299 + 0.734786i \(0.737282\pi\)
\(30\) −8.22325 −1.50135
\(31\) −2.83337 −0.508887 −0.254444 − 0.967088i \(-0.581892\pi\)
−0.254444 + 0.967088i \(0.581892\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) 1.10299 0.189160
\(35\) 4.18178 0.706850
\(36\) 0.866918 0.144486
\(37\) −7.08458 −1.16470 −0.582349 − 0.812939i \(-0.697866\pi\)
−0.582349 + 0.812939i \(0.697866\pi\)
\(38\) −1.43625 −0.232991
\(39\) −8.65769 −1.38634
\(40\) −4.18178 −0.661197
\(41\) −1.13308 −0.176958 −0.0884789 − 0.996078i \(-0.528201\pi\)
−0.0884789 + 0.996078i \(0.528201\pi\)
\(42\) −1.96645 −0.303430
\(43\) −4.55093 −0.694010 −0.347005 − 0.937863i \(-0.612801\pi\)
−0.347005 + 0.937863i \(0.612801\pi\)
\(44\) 0 0
\(45\) −3.62526 −0.540422
\(46\) −1.16663 −0.172011
\(47\) −7.19316 −1.04923 −0.524615 − 0.851340i \(-0.675791\pi\)
−0.524615 + 0.851340i \(0.675791\pi\)
\(48\) 1.96645 0.283832
\(49\) 1.00000 0.142857
\(50\) 12.4873 1.76597
\(51\) 2.16896 0.303716
\(52\) −4.40270 −0.610545
\(53\) 10.0023 1.37393 0.686963 − 0.726693i \(-0.258944\pi\)
0.686963 + 0.726693i \(0.258944\pi\)
\(54\) −4.19460 −0.570812
\(55\) 0 0
\(56\) −1.00000 −0.133631
\(57\) −2.82432 −0.374090
\(58\) −7.30550 −0.959259
\(59\) 7.41439 0.965272 0.482636 − 0.875821i \(-0.339679\pi\)
0.482636 + 0.875821i \(0.339679\pi\)
\(60\) −8.22325 −1.06162
\(61\) −11.0151 −1.41034 −0.705172 − 0.709036i \(-0.749130\pi\)
−0.705172 + 0.709036i \(0.749130\pi\)
\(62\) −2.83337 −0.359838
\(63\) −0.866918 −0.109221
\(64\) 1.00000 0.125000
\(65\) 18.4111 2.28362
\(66\) 0 0
\(67\) 11.7327 1.43338 0.716689 − 0.697393i \(-0.245657\pi\)
0.716689 + 0.697393i \(0.245657\pi\)
\(68\) 1.10299 0.133757
\(69\) −2.29413 −0.276180
\(70\) 4.18178 0.499818
\(71\) 8.47447 1.00573 0.502867 − 0.864364i \(-0.332279\pi\)
0.502867 + 0.864364i \(0.332279\pi\)
\(72\) 0.866918 0.102167
\(73\) −13.6090 −1.59281 −0.796406 − 0.604763i \(-0.793268\pi\)
−0.796406 + 0.604763i \(0.793268\pi\)
\(74\) −7.08458 −0.823566
\(75\) 24.5556 2.83544
\(76\) −1.43625 −0.164750
\(77\) 0 0
\(78\) −8.65769 −0.980290
\(79\) −7.26403 −0.817267 −0.408634 − 0.912699i \(-0.633995\pi\)
−0.408634 + 0.912699i \(0.633995\pi\)
\(80\) −4.18178 −0.467537
\(81\) −10.8492 −1.20547
\(82\) −1.13308 −0.125128
\(83\) 6.44235 0.707140 0.353570 − 0.935408i \(-0.384968\pi\)
0.353570 + 0.935408i \(0.384968\pi\)
\(84\) −1.96645 −0.214557
\(85\) −4.61244 −0.500290
\(86\) −4.55093 −0.490739
\(87\) −14.3659 −1.54018
\(88\) 0 0
\(89\) 7.52083 0.797207 0.398603 − 0.917123i \(-0.369495\pi\)
0.398603 + 0.917123i \(0.369495\pi\)
\(90\) −3.62526 −0.382136
\(91\) 4.40270 0.461529
\(92\) −1.16663 −0.121630
\(93\) −5.57167 −0.577755
\(94\) −7.19316 −0.741917
\(95\) 6.00610 0.616213
\(96\) 1.96645 0.200700
\(97\) 1.40616 0.142774 0.0713868 − 0.997449i \(-0.477258\pi\)
0.0713868 + 0.997449i \(0.477258\pi\)
\(98\) 1.00000 0.101015
\(99\) 0 0
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 1694.2.a.z.1.4 4
11.7 odd 10 154.2.f.e.71.1 ✓ 8
11.8 odd 10 154.2.f.e.141.1 yes 8
11.10 odd 2 1694.2.a.x.1.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
154.2.f.e.71.1 ✓ 8 11.7 odd 10
154.2.f.e.141.1 yes 8 11.8 odd 10
1694.2.a.x.1.4 4 11.10 odd 2
1694.2.a.z.1.4 4 1.1 even 1 trivial