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Properties

Label 1694.2.a.n.1.1
Level $1694$
Weight $2$
Character 1694.1
Self dual yes
Analytic conductor $13.527$
Analytic rank $1$
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,-2,0,2,-3] 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: \(2\)
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: \(\Z[a_1, \ldots, a_{5}]\)
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 1.1
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)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} -2.23607 q^{3} +1.00000 q^{4} -2.61803 q^{5} +2.23607 q^{6} -1.00000 q^{7} -1.00000 q^{8} +2.00000 q^{9} +2.61803 q^{10} -2.23607 q^{12} -3.00000 q^{13} +1.00000 q^{14} +5.85410 q^{15} +1.00000 q^{16} +5.47214 q^{17} -2.00000 q^{18} +3.47214 q^{19} -2.61803 q^{20} +2.23607 q^{21} +3.47214 q^{23} +2.23607 q^{24} +1.85410 q^{25} +3.00000 q^{26} +2.23607 q^{27} -1.00000 q^{28} +3.00000 q^{29} -5.85410 q^{30} -8.70820 q^{31} -1.00000 q^{32} -5.47214 q^{34} +2.61803 q^{35} +2.00000 q^{36} +10.3262 q^{37} -3.47214 q^{38} +6.70820 q^{39} +2.61803 q^{40} -10.4721 q^{41} -2.23607 q^{42} -3.85410 q^{43} -5.23607 q^{45} -3.47214 q^{46} +7.85410 q^{47} -2.23607 q^{48} +1.00000 q^{49} -1.85410 q^{50} -12.2361 q^{51} -3.00000 q^{52} +3.76393 q^{53} -2.23607 q^{54} +1.00000 q^{56} -7.76393 q^{57} -3.00000 q^{58} +5.56231 q^{59} +5.85410 q^{60} -3.14590 q^{61} +8.70820 q^{62} -2.00000 q^{63} +1.00000 q^{64} +7.85410 q^{65} +13.0000 q^{67} +5.47214 q^{68} -7.76393 q^{69} -2.61803 q^{70} -0.236068 q^{71} -2.00000 q^{72} -8.56231 q^{73} -10.3262 q^{74} -4.14590 q^{75} +3.47214 q^{76} -6.70820 q^{78} -7.76393 q^{79} -2.61803 q^{80} -11.0000 q^{81} +10.4721 q^{82} -4.61803 q^{83} +2.23607 q^{84} -14.3262 q^{85} +3.85410 q^{86} -6.70820 q^{87} -1.85410 q^{89} +5.23607 q^{90} +3.00000 q^{91} +3.47214 q^{92} +19.4721 q^{93} -7.85410 q^{94} -9.09017 q^{95} +2.23607 q^{96} +0.472136 q^{97} -1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} + 2 q^{4} - 3 q^{5} - 2 q^{7} - 2 q^{8} + 4 q^{9} + 3 q^{10} - 6 q^{13} + 2 q^{14} + 5 q^{15} + 2 q^{16} + 2 q^{17} - 4 q^{18} - 2 q^{19} - 3 q^{20} - 2 q^{23} - 3 q^{25} + 6 q^{26} - 2 q^{28}+ \cdots - 2 q^{98}+O(q^{100}) \) 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
\(3\) −2.23607 −1.29099 −0.645497 − 0.763763i \(-0.723350\pi\)
−0.645497 + 0.763763i \(0.723350\pi\)
\(4\) 1.00000 0.500000
\(5\) −2.61803 −1.17082 −0.585410 − 0.810737i \(-0.699067\pi\)
−0.585410 + 0.810737i \(0.699067\pi\)
\(6\) 2.23607 0.912871
\(7\) −1.00000 −0.377964
\(8\) −1.00000 −0.353553
\(9\) 2.00000 0.666667
\(10\) 2.61803 0.827895
\(11\) 0 0
\(12\) −2.23607 −0.645497
\(13\) −3.00000 −0.832050 −0.416025 − 0.909353i \(-0.636577\pi\)
−0.416025 + 0.909353i \(0.636577\pi\)
\(14\) 1.00000 0.267261
\(15\) 5.85410 1.51152
\(16\) 1.00000 0.250000
\(17\) 5.47214 1.32719 0.663594 − 0.748093i \(-0.269030\pi\)
0.663594 + 0.748093i \(0.269030\pi\)
\(18\) −2.00000 −0.471405
\(19\) 3.47214 0.796563 0.398281 − 0.917263i \(-0.369607\pi\)
0.398281 + 0.917263i \(0.369607\pi\)
\(20\) −2.61803 −0.585410
\(21\) 2.23607 0.487950
\(22\) 0 0
\(23\) 3.47214 0.723990 0.361995 − 0.932180i \(-0.382096\pi\)
0.361995 + 0.932180i \(0.382096\pi\)
\(24\) 2.23607 0.456435
\(25\) 1.85410 0.370820
\(26\) 3.00000 0.588348
\(27\) 2.23607 0.430331
\(28\) −1.00000 −0.188982
\(29\) 3.00000 0.557086 0.278543 − 0.960424i \(-0.410149\pi\)
0.278543 + 0.960424i \(0.410149\pi\)
\(30\) −5.85410 −1.06881
\(31\) −8.70820 −1.56404 −0.782020 − 0.623254i \(-0.785810\pi\)
−0.782020 + 0.623254i \(0.785810\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) −5.47214 −0.938464
\(35\) 2.61803 0.442529
\(36\) 2.00000 0.333333
\(37\) 10.3262 1.69762 0.848812 − 0.528696i \(-0.177319\pi\)
0.848812 + 0.528696i \(0.177319\pi\)
\(38\) −3.47214 −0.563255
\(39\) 6.70820 1.07417
\(40\) 2.61803 0.413948
\(41\) −10.4721 −1.63547 −0.817736 − 0.575593i \(-0.804771\pi\)
−0.817736 + 0.575593i \(0.804771\pi\)
\(42\) −2.23607 −0.345033
\(43\) −3.85410 −0.587745 −0.293873 − 0.955845i \(-0.594944\pi\)
−0.293873 + 0.955845i \(0.594944\pi\)
\(44\) 0 0
\(45\) −5.23607 −0.780547
\(46\) −3.47214 −0.511939
\(47\) 7.85410 1.14564 0.572819 − 0.819682i \(-0.305850\pi\)
0.572819 + 0.819682i \(0.305850\pi\)
\(48\) −2.23607 −0.322749
\(49\) 1.00000 0.142857
\(50\) −1.85410 −0.262210
\(51\) −12.2361 −1.71339
\(52\) −3.00000 −0.416025
\(53\) 3.76393 0.517016 0.258508 − 0.966009i \(-0.416769\pi\)
0.258508 + 0.966009i \(0.416769\pi\)
\(54\) −2.23607 −0.304290
\(55\) 0 0
\(56\) 1.00000 0.133631
\(57\) −7.76393 −1.02836
\(58\) −3.00000 −0.393919
\(59\) 5.56231 0.724151 0.362075 − 0.932149i \(-0.382068\pi\)
0.362075 + 0.932149i \(0.382068\pi\)
\(60\) 5.85410 0.755761
\(61\) −3.14590 −0.402791 −0.201395 − 0.979510i \(-0.564548\pi\)
−0.201395 + 0.979510i \(0.564548\pi\)
\(62\) 8.70820 1.10594
\(63\) −2.00000 −0.251976
\(64\) 1.00000 0.125000
\(65\) 7.85410 0.974181
\(66\) 0 0
\(67\) 13.0000 1.58820 0.794101 − 0.607785i \(-0.207942\pi\)
0.794101 + 0.607785i \(0.207942\pi\)
\(68\) 5.47214 0.663594
\(69\) −7.76393 −0.934668
\(70\) −2.61803 −0.312915
\(71\) −0.236068 −0.0280161 −0.0140081 − 0.999902i \(-0.504459\pi\)
−0.0140081 + 0.999902i \(0.504459\pi\)
\(72\) −2.00000 −0.235702
\(73\) −8.56231 −1.00214 −0.501071 − 0.865406i \(-0.667060\pi\)
−0.501071 + 0.865406i \(0.667060\pi\)
\(74\) −10.3262 −1.20040
\(75\) −4.14590 −0.478727
\(76\) 3.47214 0.398281
\(77\) 0 0
\(78\) −6.70820 −0.759555
\(79\) −7.76393 −0.873511 −0.436755 − 0.899580i \(-0.643872\pi\)
−0.436755 + 0.899580i \(0.643872\pi\)
\(80\) −2.61803 −0.292705
\(81\) −11.0000 −1.22222
\(82\) 10.4721 1.15645
\(83\) −4.61803 −0.506895 −0.253448 − 0.967349i \(-0.581565\pi\)
−0.253448 + 0.967349i \(0.581565\pi\)
\(84\) 2.23607 0.243975
\(85\) −14.3262 −1.55390
\(86\) 3.85410 0.415599
\(87\) −6.70820 −0.719195
\(88\) 0 0
\(89\) −1.85410 −0.196534 −0.0982672 − 0.995160i \(-0.531330\pi\)
−0.0982672 + 0.995160i \(0.531330\pi\)
\(90\) 5.23607 0.551930
\(91\) 3.00000 0.314485
\(92\) 3.47214 0.361995
\(93\) 19.4721 2.01917
\(94\) −7.85410 −0.810089
\(95\) −9.09017 −0.932632
\(96\) 2.23607 0.228218
\(97\) 0.472136 0.0479381 0.0239691 − 0.999713i \(-0.492370\pi\)
0.0239691 + 0.999713i \(0.492370\pi\)
\(98\) −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.n.1.1 2
11.3 even 5 154.2.f.d.141.1 yes 4
11.4 even 5 154.2.f.d.71.1 ✓ 4
11.10 odd 2 1694.2.a.s.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
154.2.f.d.71.1 ✓ 4 11.4 even 5
154.2.f.d.141.1 yes 4 11.3 even 5
1694.2.a.n.1.1 2 1.1 even 1 trivial
1694.2.a.s.1.1 2 11.10 odd 2