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Properties

Label 68.2.b.a.33.1
Level $68$
Weight $2$
Character 68.33
Analytic conductor $0.543$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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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] = [68,2,Mod(33,68)] 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("68.33"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(68, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 68 = 2^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 68.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(0.542982733745\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-2}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 33.1
Root \(-1.41421i\) of defining polynomial
Character \(\chi\) \(=\) 68.33
Dual form 68.2.b.a.33.2

$q$-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)
 
\(f(q)\) \(=\) \(q-1.41421i q^{3} -2.82843i q^{5} +4.24264i q^{7} +1.00000 q^{9} +1.41421i q^{11} -4.00000 q^{13} -4.00000 q^{15} +(3.00000 + 2.82843i) q^{17} -4.00000 q^{19} +6.00000 q^{21} +1.41421i q^{23} -3.00000 q^{25} -5.65685i q^{27} -2.82843i q^{29} -4.24264i q^{31} +2.00000 q^{33} +12.0000 q^{35} +8.48528i q^{37} +5.65685i q^{39} -11.3137i q^{41} +8.00000 q^{43} -2.82843i q^{45} -12.0000 q^{47} -11.0000 q^{49} +(4.00000 - 4.24264i) q^{51} -6.00000 q^{53} +4.00000 q^{55} +5.65685i q^{57} +8.48528i q^{61} +4.24264i q^{63} +11.3137i q^{65} -4.00000 q^{67} +2.00000 q^{69} -7.07107i q^{71} +4.24264i q^{75} -6.00000 q^{77} -4.24264i q^{79} -5.00000 q^{81} +(8.00000 - 8.48528i) q^{85} -4.00000 q^{87} +12.0000 q^{89} -16.9706i q^{91} -6.00000 q^{93} +11.3137i q^{95} +1.41421i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{9} - 8 q^{13} - 8 q^{15} + 6 q^{17} - 8 q^{19} + 12 q^{21} - 6 q^{25} + 4 q^{33} + 24 q^{35} + 16 q^{43} - 24 q^{47} - 22 q^{49} + 8 q^{51} - 12 q^{53} + 8 q^{55} - 8 q^{67} + 4 q^{69} - 12 q^{77}+ \cdots - 12 q^{93}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/68\mathbb{Z}\right)^\times\).

\(n\) \(35\) \(37\)
\(\chi(n)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) − 1.41421i − 0.816497i −0.912871 − 0.408248i \(-0.866140\pi\)
0.912871 − 0.408248i \(-0.133860\pi\)
\(4\) 0 0
\(5\) − 2.82843i − 1.26491i −0.774597 − 0.632456i \(-0.782047\pi\)
0.774597 − 0.632456i \(-0.217953\pi\)
\(6\) 0 0
\(7\) 4.24264i 1.60357i 0.597614 + 0.801784i \(0.296115\pi\)
−0.597614 + 0.801784i \(0.703885\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 1.41421i 0.426401i 0.977008 + 0.213201i \(0.0683888\pi\)
−0.977008 + 0.213201i \(0.931611\pi\)
\(12\) 0 0
\(13\) −4.00000 −1.10940 −0.554700 − 0.832050i \(-0.687167\pi\)
−0.554700 + 0.832050i \(0.687167\pi\)
\(14\) 0 0
\(15\) −4.00000 −1.03280
\(16\) 0 0
\(17\) 3.00000 + 2.82843i 0.727607 + 0.685994i
\(18\) 0 0
\(19\) −4.00000 −0.917663 −0.458831 − 0.888523i \(-0.651732\pi\)
−0.458831 + 0.888523i \(0.651732\pi\)
\(20\) 0 0
\(21\) 6.00000 1.30931
\(22\) 0 0
\(23\) 1.41421i 0.294884i 0.989071 + 0.147442i \(0.0471040\pi\)
−0.989071 + 0.147442i \(0.952896\pi\)
\(24\) 0 0
\(25\) −3.00000 −0.600000
\(26\) 0 0
\(27\) − 5.65685i − 1.08866i
\(28\) 0 0
\(29\) − 2.82843i − 0.525226i −0.964901 − 0.262613i \(-0.915416\pi\)
0.964901 − 0.262613i \(-0.0845842\pi\)
\(30\) 0 0
\(31\) − 4.24264i − 0.762001i −0.924575 − 0.381000i \(-0.875580\pi\)
0.924575 − 0.381000i \(-0.124420\pi\)
\(32\) 0 0
\(33\) 2.00000 0.348155
\(34\) 0 0
\(35\) 12.0000 2.02837
\(36\) 0 0
\(37\) 8.48528i 1.39497i 0.716599 + 0.697486i \(0.245698\pi\)
−0.716599 + 0.697486i \(0.754302\pi\)
\(38\) 0 0
\(39\) 5.65685i 0.905822i
\(40\) 0 0
\(41\) − 11.3137i − 1.76690i −0.468521 − 0.883452i \(-0.655213\pi\)
0.468521 − 0.883452i \(-0.344787\pi\)
\(42\) 0 0
\(43\) 8.00000 1.21999 0.609994 − 0.792406i \(-0.291172\pi\)
0.609994 + 0.792406i \(0.291172\pi\)
\(44\) 0 0
\(45\) − 2.82843i − 0.421637i
\(46\) 0 0
\(47\) −12.0000 −1.75038 −0.875190 − 0.483779i \(-0.839264\pi\)
−0.875190 + 0.483779i \(0.839264\pi\)
\(48\) 0 0
\(49\) −11.0000 −1.57143
\(50\) 0 0
\(51\) 4.00000 − 4.24264i 0.560112 − 0.594089i
\(52\) 0 0
\(53\) −6.00000 −0.824163 −0.412082 − 0.911147i \(-0.635198\pi\)
−0.412082 + 0.911147i \(0.635198\pi\)
\(54\) 0 0
\(55\) 4.00000 0.539360
\(56\) 0 0
\(57\) 5.65685i 0.749269i
\(58\) 0 0
\(59\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) 8.48528i 1.08643i 0.839594 + 0.543214i \(0.182793\pi\)
−0.839594 + 0.543214i \(0.817207\pi\)
\(62\) 0 0
\(63\) 4.24264i 0.534522i
\(64\) 0 0
\(65\) 11.3137i 1.40329i
\(66\) 0 0
\(67\) −4.00000 −0.488678 −0.244339 − 0.969690i \(-0.578571\pi\)
−0.244339 + 0.969690i \(0.578571\pi\)
\(68\) 0 0
\(69\) 2.00000 0.240772
\(70\) 0 0
\(71\) − 7.07107i − 0.839181i −0.907713 − 0.419591i \(-0.862174\pi\)
0.907713 − 0.419591i \(-0.137826\pi\)
\(72\) 0 0
\(73\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(74\) 0 0
\(75\) 4.24264i 0.489898i
\(76\) 0 0
\(77\) −6.00000 −0.683763
\(78\) 0 0
\(79\) − 4.24264i − 0.477334i −0.971101 − 0.238667i \(-0.923290\pi\)
0.971101 − 0.238667i \(-0.0767105\pi\)
\(80\) 0 0
\(81\) −5.00000 −0.555556
\(82\) 0 0
\(83\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 0 0
\(85\) 8.00000 − 8.48528i 0.867722 − 0.920358i
\(86\) 0 0
\(87\) −4.00000 −0.428845
\(88\) 0 0
\(89\) 12.0000 1.27200 0.635999 − 0.771690i \(-0.280588\pi\)
0.635999 + 0.771690i \(0.280588\pi\)
\(90\) 0 0
\(91\) − 16.9706i − 1.77900i
\(92\) 0 0
\(93\) −6.00000 −0.622171
\(94\) 0 0
\(95\) 11.3137i 1.16076i
\(96\) 0 0
\(97\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(98\) 0 0
\(99\) 1.41421i 0.142134i
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 68.2.b.a.33.1 ✓ 2
3.2 odd 2 612.2.b.a.577.2 2
4.3 odd 2 272.2.b.c.33.2 2
5.2 odd 4 1700.2.g.a.849.2 4
5.3 odd 4 1700.2.g.a.849.4 4
5.4 even 2 1700.2.c.a.101.2 2
7.6 odd 2 3332.2.b.a.2549.2 2
8.3 odd 2 1088.2.b.f.577.1 2
8.5 even 2 1088.2.b.e.577.2 2
12.11 even 2 2448.2.c.d.577.2 2
17.2 even 8 1156.2.e.a.829.1 2
17.3 odd 16 1156.2.h.d.977.2 8
17.4 even 4 1156.2.a.c.1.1 2
17.5 odd 16 1156.2.h.d.757.2 8
17.6 odd 16 1156.2.h.d.1001.1 8
17.7 odd 16 1156.2.h.d.733.1 8
17.8 even 8 1156.2.e.a.905.1 2
17.9 even 8 1156.2.e.b.905.1 2
17.10 odd 16 1156.2.h.d.733.2 8
17.11 odd 16 1156.2.h.d.1001.2 8
17.12 odd 16 1156.2.h.d.757.1 8
17.13 even 4 1156.2.a.c.1.2 2
17.14 odd 16 1156.2.h.d.977.1 8
17.15 even 8 1156.2.e.b.829.1 2
17.16 even 2 inner 68.2.b.a.33.2 yes 2
51.50 odd 2 612.2.b.a.577.1 2
68.47 odd 4 4624.2.a.n.1.1 2
68.55 odd 4 4624.2.a.n.1.2 2
68.67 odd 2 272.2.b.c.33.1 2
85.33 odd 4 1700.2.g.a.849.1 4
85.67 odd 4 1700.2.g.a.849.3 4
85.84 even 2 1700.2.c.a.101.1 2
119.118 odd 2 3332.2.b.a.2549.1 2
136.67 odd 2 1088.2.b.f.577.2 2
136.101 even 2 1088.2.b.e.577.1 2
204.203 even 2 2448.2.c.d.577.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
68.2.b.a.33.1 ✓ 2 1.1 even 1 trivial
68.2.b.a.33.2 yes 2 17.16 even 2 inner
272.2.b.c.33.1 2 68.67 odd 2
272.2.b.c.33.2 2 4.3 odd 2
612.2.b.a.577.1 2 51.50 odd 2
612.2.b.a.577.2 2 3.2 odd 2
1088.2.b.e.577.1 2 136.101 even 2
1088.2.b.e.577.2 2 8.5 even 2
1088.2.b.f.577.1 2 8.3 odd 2
1088.2.b.f.577.2 2 136.67 odd 2
1156.2.a.c.1.1 2 17.4 even 4
1156.2.a.c.1.2 2 17.13 even 4
1156.2.e.a.829.1 2 17.2 even 8
1156.2.e.a.905.1 2 17.8 even 8
1156.2.e.b.829.1 2 17.15 even 8
1156.2.e.b.905.1 2 17.9 even 8
1156.2.h.d.733.1 8 17.7 odd 16
1156.2.h.d.733.2 8 17.10 odd 16
1156.2.h.d.757.1 8 17.12 odd 16
1156.2.h.d.757.2 8 17.5 odd 16
1156.2.h.d.977.1 8 17.14 odd 16
1156.2.h.d.977.2 8 17.3 odd 16
1156.2.h.d.1001.1 8 17.6 odd 16
1156.2.h.d.1001.2 8 17.11 odd 16
1700.2.c.a.101.1 2 85.84 even 2
1700.2.c.a.101.2 2 5.4 even 2
1700.2.g.a.849.1 4 85.33 odd 4
1700.2.g.a.849.2 4 5.2 odd 4
1700.2.g.a.849.3 4 85.67 odd 4
1700.2.g.a.849.4 4 5.3 odd 4
2448.2.c.d.577.1 2 204.203 even 2
2448.2.c.d.577.2 2 12.11 even 2
3332.2.b.a.2549.1 2 119.118 odd 2
3332.2.b.a.2549.2 2 7.6 odd 2
4624.2.a.n.1.1 2 68.47 odd 4
4624.2.a.n.1.2 2 68.55 odd 4