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Properties

Label 1694.2.a.r.1.1
Level $1694$
Weight $2$
Character 1694.1
Self dual yes
Analytic conductor $13.527$
Analytic rank $0$
Dimension $2$
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 = [2,2,-1,2,4] 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: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{10})^+\)
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: \(\Z[a_1, a_2, a_3]\)
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 1.1
Root \(1.61803\) 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)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.61803 q^{3} +1.00000 q^{4} +2.00000 q^{5} -1.61803 q^{6} +1.00000 q^{7} +1.00000 q^{8} -0.381966 q^{9} +2.00000 q^{10} -1.61803 q^{12} -3.23607 q^{13} +1.00000 q^{14} -3.23607 q^{15} +1.00000 q^{16} +5.85410 q^{17} -0.381966 q^{18} +0.145898 q^{19} +2.00000 q^{20} -1.61803 q^{21} +8.47214 q^{23} -1.61803 q^{24} -1.00000 q^{25} -3.23607 q^{26} +5.47214 q^{27} +1.00000 q^{28} +8.00000 q^{29} -3.23607 q^{30} -4.47214 q^{31} +1.00000 q^{32} +5.85410 q^{34} +2.00000 q^{35} -0.381966 q^{36} -7.70820 q^{37} +0.145898 q^{38} +5.23607 q^{39} +2.00000 q^{40} -0.854102 q^{41} -1.61803 q^{42} +4.85410 q^{43} -0.763932 q^{45} +8.47214 q^{46} +1.70820 q^{47} -1.61803 q^{48} +1.00000 q^{49} -1.00000 q^{50} -9.47214 q^{51} -3.23607 q^{52} +4.94427 q^{53} +5.47214 q^{54} +1.00000 q^{56} -0.236068 q^{57} +8.00000 q^{58} +4.14590 q^{59} -3.23607 q^{60} -6.47214 q^{61} -4.47214 q^{62} -0.381966 q^{63} +1.00000 q^{64} -6.47214 q^{65} +1.09017 q^{67} +5.85410 q^{68} -13.7082 q^{69} +2.00000 q^{70} +10.4721 q^{71} -0.381966 q^{72} +2.38197 q^{73} -7.70820 q^{74} +1.61803 q^{75} +0.145898 q^{76} +5.23607 q^{78} +1.23607 q^{79} +2.00000 q^{80} -7.70820 q^{81} -0.854102 q^{82} +13.0344 q^{83} -1.61803 q^{84} +11.7082 q^{85} +4.85410 q^{86} -12.9443 q^{87} -15.3262 q^{89} -0.763932 q^{90} -3.23607 q^{91} +8.47214 q^{92} +7.23607 q^{93} +1.70820 q^{94} +0.291796 q^{95} -1.61803 q^{96} -3.14590 q^{97} +1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} - q^{3} + 2 q^{4} + 4 q^{5} - q^{6} + 2 q^{7} + 2 q^{8} - 3 q^{9} + 4 q^{10} - q^{12} - 2 q^{13} + 2 q^{14} - 2 q^{15} + 2 q^{16} + 5 q^{17} - 3 q^{18} + 7 q^{19} + 4 q^{20} - q^{21} + 8 q^{23}+ \cdots + 2 q^{98}+O(q^{100}) \) 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
\(3\) −1.61803 −0.934172 −0.467086 − 0.884212i \(-0.654696\pi\)
−0.467086 + 0.884212i \(0.654696\pi\)
\(4\) 1.00000 0.500000
\(5\) 2.00000 0.894427 0.447214 − 0.894427i \(-0.352416\pi\)
0.447214 + 0.894427i \(0.352416\pi\)
\(6\) −1.61803 −0.660560
\(7\) 1.00000 0.377964
\(8\) 1.00000 0.353553
\(9\) −0.381966 −0.127322
\(10\) 2.00000 0.632456
\(11\) 0 0
\(12\) −1.61803 −0.467086
\(13\) −3.23607 −0.897524 −0.448762 − 0.893651i \(-0.648135\pi\)
−0.448762 + 0.893651i \(0.648135\pi\)
\(14\) 1.00000 0.267261
\(15\) −3.23607 −0.835549
\(16\) 1.00000 0.250000
\(17\) 5.85410 1.41983 0.709914 − 0.704288i \(-0.248734\pi\)
0.709914 + 0.704288i \(0.248734\pi\)
\(18\) −0.381966 −0.0900303
\(19\) 0.145898 0.0334713 0.0167357 − 0.999860i \(-0.494673\pi\)
0.0167357 + 0.999860i \(0.494673\pi\)
\(20\) 2.00000 0.447214
\(21\) −1.61803 −0.353084
\(22\) 0 0
\(23\) 8.47214 1.76656 0.883281 − 0.468844i \(-0.155329\pi\)
0.883281 + 0.468844i \(0.155329\pi\)
\(24\) −1.61803 −0.330280
\(25\) −1.00000 −0.200000
\(26\) −3.23607 −0.634645
\(27\) 5.47214 1.05311
\(28\) 1.00000 0.188982
\(29\) 8.00000 1.48556 0.742781 − 0.669534i \(-0.233506\pi\)
0.742781 + 0.669534i \(0.233506\pi\)
\(30\) −3.23607 −0.590822
\(31\) −4.47214 −0.803219 −0.401610 − 0.915811i \(-0.631549\pi\)
−0.401610 + 0.915811i \(0.631549\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) 5.85410 1.00397
\(35\) 2.00000 0.338062
\(36\) −0.381966 −0.0636610
\(37\) −7.70820 −1.26722 −0.633610 − 0.773652i \(-0.718428\pi\)
−0.633610 + 0.773652i \(0.718428\pi\)
\(38\) 0.145898 0.0236678
\(39\) 5.23607 0.838442
\(40\) 2.00000 0.316228
\(41\) −0.854102 −0.133388 −0.0666942 − 0.997773i \(-0.521245\pi\)
−0.0666942 + 0.997773i \(0.521245\pi\)
\(42\) −1.61803 −0.249668
\(43\) 4.85410 0.740244 0.370122 − 0.928983i \(-0.379316\pi\)
0.370122 + 0.928983i \(0.379316\pi\)
\(44\) 0 0
\(45\) −0.763932 −0.113880
\(46\) 8.47214 1.24915
\(47\) 1.70820 0.249167 0.124584 − 0.992209i \(-0.460241\pi\)
0.124584 + 0.992209i \(0.460241\pi\)
\(48\) −1.61803 −0.233543
\(49\) 1.00000 0.142857
\(50\) −1.00000 −0.141421
\(51\) −9.47214 −1.32636
\(52\) −3.23607 −0.448762
\(53\) 4.94427 0.679148 0.339574 − 0.940579i \(-0.389717\pi\)
0.339574 + 0.940579i \(0.389717\pi\)
\(54\) 5.47214 0.744663
\(55\) 0 0
\(56\) 1.00000 0.133631
\(57\) −0.236068 −0.0312680
\(58\) 8.00000 1.05045
\(59\) 4.14590 0.539750 0.269875 − 0.962895i \(-0.413018\pi\)
0.269875 + 0.962895i \(0.413018\pi\)
\(60\) −3.23607 −0.417775
\(61\) −6.47214 −0.828672 −0.414336 − 0.910124i \(-0.635986\pi\)
−0.414336 + 0.910124i \(0.635986\pi\)
\(62\) −4.47214 −0.567962
\(63\) −0.381966 −0.0481232
\(64\) 1.00000 0.125000
\(65\) −6.47214 −0.802770
\(66\) 0 0
\(67\) 1.09017 0.133185 0.0665927 − 0.997780i \(-0.478787\pi\)
0.0665927 + 0.997780i \(0.478787\pi\)
\(68\) 5.85410 0.709914
\(69\) −13.7082 −1.65027
\(70\) 2.00000 0.239046
\(71\) 10.4721 1.24281 0.621407 − 0.783488i \(-0.286561\pi\)
0.621407 + 0.783488i \(0.286561\pi\)
\(72\) −0.381966 −0.0450151
\(73\) 2.38197 0.278788 0.139394 − 0.990237i \(-0.455485\pi\)
0.139394 + 0.990237i \(0.455485\pi\)
\(74\) −7.70820 −0.896061
\(75\) 1.61803 0.186834
\(76\) 0.145898 0.0167357
\(77\) 0 0
\(78\) 5.23607 0.592868
\(79\) 1.23607 0.139069 0.0695343 − 0.997580i \(-0.477849\pi\)
0.0695343 + 0.997580i \(0.477849\pi\)
\(80\) 2.00000 0.223607
\(81\) −7.70820 −0.856467
\(82\) −0.854102 −0.0943198
\(83\) 13.0344 1.43072 0.715358 − 0.698758i \(-0.246264\pi\)
0.715358 + 0.698758i \(0.246264\pi\)
\(84\) −1.61803 −0.176542
\(85\) 11.7082 1.26993
\(86\) 4.85410 0.523431
\(87\) −12.9443 −1.38777
\(88\) 0 0
\(89\) −15.3262 −1.62458 −0.812289 − 0.583255i \(-0.801779\pi\)
−0.812289 + 0.583255i \(0.801779\pi\)
\(90\) −0.763932 −0.0805255
\(91\) −3.23607 −0.339232
\(92\) 8.47214 0.883281
\(93\) 7.23607 0.750345
\(94\) 1.70820 0.176188
\(95\) 0.291796 0.0299376
\(96\) −1.61803 −0.165140
\(97\) −3.14590 −0.319418 −0.159709 − 0.987164i \(-0.551056\pi\)
−0.159709 + 0.987164i \(0.551056\pi\)
\(98\) 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.r.1.1 2
11.2 odd 10 154.2.f.c.15.1 ✓ 4
11.6 odd 10 154.2.f.c.113.1 yes 4
11.10 odd 2 1694.2.a.m.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
154.2.f.c.15.1 ✓ 4 11.2 odd 10
154.2.f.c.113.1 yes 4 11.6 odd 10
1694.2.a.m.1.1 2 11.10 odd 2
1694.2.a.r.1.1 2 1.1 even 1 trivial