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Properties

Label 1694.2.a.r.1.2
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 1.2
Root \(-0.618034\) 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)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} +0.618034 q^{3} +1.00000 q^{4} +2.00000 q^{5} +0.618034 q^{6} +1.00000 q^{7} +1.00000 q^{8} -2.61803 q^{9} +2.00000 q^{10} +0.618034 q^{12} +1.23607 q^{13} +1.00000 q^{14} +1.23607 q^{15} +1.00000 q^{16} -0.854102 q^{17} -2.61803 q^{18} +6.85410 q^{19} +2.00000 q^{20} +0.618034 q^{21} -0.472136 q^{23} +0.618034 q^{24} -1.00000 q^{25} +1.23607 q^{26} -3.47214 q^{27} +1.00000 q^{28} +8.00000 q^{29} +1.23607 q^{30} +4.47214 q^{31} +1.00000 q^{32} -0.854102 q^{34} +2.00000 q^{35} -2.61803 q^{36} +5.70820 q^{37} +6.85410 q^{38} +0.763932 q^{39} +2.00000 q^{40} +5.85410 q^{41} +0.618034 q^{42} -1.85410 q^{43} -5.23607 q^{45} -0.472136 q^{46} -11.7082 q^{47} +0.618034 q^{48} +1.00000 q^{49} -1.00000 q^{50} -0.527864 q^{51} +1.23607 q^{52} -12.9443 q^{53} -3.47214 q^{54} +1.00000 q^{56} +4.23607 q^{57} +8.00000 q^{58} +10.8541 q^{59} +1.23607 q^{60} +2.47214 q^{61} +4.47214 q^{62} -2.61803 q^{63} +1.00000 q^{64} +2.47214 q^{65} -10.0902 q^{67} -0.854102 q^{68} -0.291796 q^{69} +2.00000 q^{70} +1.52786 q^{71} -2.61803 q^{72} +4.61803 q^{73} +5.70820 q^{74} -0.618034 q^{75} +6.85410 q^{76} +0.763932 q^{78} -3.23607 q^{79} +2.00000 q^{80} +5.70820 q^{81} +5.85410 q^{82} -16.0344 q^{83} +0.618034 q^{84} -1.70820 q^{85} -1.85410 q^{86} +4.94427 q^{87} +0.326238 q^{89} -5.23607 q^{90} +1.23607 q^{91} -0.472136 q^{92} +2.76393 q^{93} -11.7082 q^{94} +13.7082 q^{95} +0.618034 q^{96} -9.85410 q^{97} +1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} - q^{3} + 2 q^{4} + 4 q^{5} - q^{6} + 2 q^{7} + 2 q^{8} - 3 q^{9} + 4 q^{10} - q^{12} - 2 q^{13} + 2 q^{14} - 2 q^{15} + 2 q^{16} + 5 q^{17} - 3 q^{18} + 7 q^{19} + 4 q^{20} - q^{21} + 8 q^{23}+ \cdots + 2 q^{98}+O(q^{100}) \) 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
\(3\) 0.618034 0.356822 0.178411 − 0.983956i \(-0.442904\pi\)
0.178411 + 0.983956i \(0.442904\pi\)
\(4\) 1.00000 0.500000
\(5\) 2.00000 0.894427 0.447214 − 0.894427i \(-0.352416\pi\)
0.447214 + 0.894427i \(0.352416\pi\)
\(6\) 0.618034 0.252311
\(7\) 1.00000 0.377964
\(8\) 1.00000 0.353553
\(9\) −2.61803 −0.872678
\(10\) 2.00000 0.632456
\(11\) 0 0
\(12\) 0.618034 0.178411
\(13\) 1.23607 0.342824 0.171412 − 0.985199i \(-0.445167\pi\)
0.171412 + 0.985199i \(0.445167\pi\)
\(14\) 1.00000 0.267261
\(15\) 1.23607 0.319151
\(16\) 1.00000 0.250000
\(17\) −0.854102 −0.207150 −0.103575 − 0.994622i \(-0.533028\pi\)
−0.103575 + 0.994622i \(0.533028\pi\)
\(18\) −2.61803 −0.617077
\(19\) 6.85410 1.57244 0.786219 − 0.617947i \(-0.212036\pi\)
0.786219 + 0.617947i \(0.212036\pi\)
\(20\) 2.00000 0.447214
\(21\) 0.618034 0.134866
\(22\) 0 0
\(23\) −0.472136 −0.0984472 −0.0492236 − 0.998788i \(-0.515675\pi\)
−0.0492236 + 0.998788i \(0.515675\pi\)
\(24\) 0.618034 0.126156
\(25\) −1.00000 −0.200000
\(26\) 1.23607 0.242413
\(27\) −3.47214 −0.668213
\(28\) 1.00000 0.188982
\(29\) 8.00000 1.48556 0.742781 − 0.669534i \(-0.233506\pi\)
0.742781 + 0.669534i \(0.233506\pi\)
\(30\) 1.23607 0.225674
\(31\) 4.47214 0.803219 0.401610 − 0.915811i \(-0.368451\pi\)
0.401610 + 0.915811i \(0.368451\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) −0.854102 −0.146477
\(35\) 2.00000 0.338062
\(36\) −2.61803 −0.436339
\(37\) 5.70820 0.938423 0.469211 − 0.883086i \(-0.344538\pi\)
0.469211 + 0.883086i \(0.344538\pi\)
\(38\) 6.85410 1.11188
\(39\) 0.763932 0.122327
\(40\) 2.00000 0.316228
\(41\) 5.85410 0.914257 0.457129 − 0.889401i \(-0.348878\pi\)
0.457129 + 0.889401i \(0.348878\pi\)
\(42\) 0.618034 0.0953647
\(43\) −1.85410 −0.282748 −0.141374 − 0.989956i \(-0.545152\pi\)
−0.141374 + 0.989956i \(0.545152\pi\)
\(44\) 0 0
\(45\) −5.23607 −0.780547
\(46\) −0.472136 −0.0696126
\(47\) −11.7082 −1.70782 −0.853909 − 0.520423i \(-0.825774\pi\)
−0.853909 + 0.520423i \(0.825774\pi\)
\(48\) 0.618034 0.0892055
\(49\) 1.00000 0.142857
\(50\) −1.00000 −0.141421
\(51\) −0.527864 −0.0739158
\(52\) 1.23607 0.171412
\(53\) −12.9443 −1.77803 −0.889016 − 0.457876i \(-0.848610\pi\)
−0.889016 + 0.457876i \(0.848610\pi\)
\(54\) −3.47214 −0.472498
\(55\) 0 0
\(56\) 1.00000 0.133631
\(57\) 4.23607 0.561081
\(58\) 8.00000 1.05045
\(59\) 10.8541 1.41308 0.706542 − 0.707671i \(-0.250254\pi\)
0.706542 + 0.707671i \(0.250254\pi\)
\(60\) 1.23607 0.159576
\(61\) 2.47214 0.316525 0.158262 − 0.987397i \(-0.449411\pi\)
0.158262 + 0.987397i \(0.449411\pi\)
\(62\) 4.47214 0.567962
\(63\) −2.61803 −0.329841
\(64\) 1.00000 0.125000
\(65\) 2.47214 0.306631
\(66\) 0 0
\(67\) −10.0902 −1.23271 −0.616355 − 0.787468i \(-0.711391\pi\)
−0.616355 + 0.787468i \(0.711391\pi\)
\(68\) −0.854102 −0.103575
\(69\) −0.291796 −0.0351281
\(70\) 2.00000 0.239046
\(71\) 1.52786 0.181324 0.0906621 − 0.995882i \(-0.471102\pi\)
0.0906621 + 0.995882i \(0.471102\pi\)
\(72\) −2.61803 −0.308538
\(73\) 4.61803 0.540500 0.270250 − 0.962790i \(-0.412894\pi\)
0.270250 + 0.962790i \(0.412894\pi\)
\(74\) 5.70820 0.663565
\(75\) −0.618034 −0.0713644
\(76\) 6.85410 0.786219
\(77\) 0 0
\(78\) 0.763932 0.0864983
\(79\) −3.23607 −0.364086 −0.182043 − 0.983291i \(-0.558271\pi\)
−0.182043 + 0.983291i \(0.558271\pi\)
\(80\) 2.00000 0.223607
\(81\) 5.70820 0.634245
\(82\) 5.85410 0.646477
\(83\) −16.0344 −1.76001 −0.880004 − 0.474966i \(-0.842460\pi\)
−0.880004 + 0.474966i \(0.842460\pi\)
\(84\) 0.618034 0.0674330
\(85\) −1.70820 −0.185281
\(86\) −1.85410 −0.199933
\(87\) 4.94427 0.530082
\(88\) 0 0
\(89\) 0.326238 0.0345812 0.0172906 − 0.999851i \(-0.494496\pi\)
0.0172906 + 0.999851i \(0.494496\pi\)
\(90\) −5.23607 −0.551930
\(91\) 1.23607 0.129575
\(92\) −0.472136 −0.0492236
\(93\) 2.76393 0.286606
\(94\) −11.7082 −1.20761
\(95\) 13.7082 1.40643
\(96\) 0.618034 0.0630778
\(97\) −9.85410 −1.00053 −0.500266 − 0.865872i \(-0.666765\pi\)
−0.500266 + 0.865872i \(0.666765\pi\)
\(98\) 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.1.2 2
11.7 odd 10 154.2.f.c.71.1 ✓ 4
11.8 odd 10 154.2.f.c.141.1 yes 4
11.10 odd 2 1694.2.a.m.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
154.2.f.c.71.1 ✓ 4 11.7 odd 10
154.2.f.c.141.1 yes 4 11.8 odd 10
1694.2.a.m.1.2 2 11.10 odd 2
1694.2.a.r.1.2 2 1.1 even 1 trivial