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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1368,2,Mod(1025,1368)] 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("1368.1025"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1368, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1368 = 2^{3} \cdot 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1368.f (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,-2,0,0,0,0,0,6,0,0,0,-20] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(29)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(10.9235349965\)
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: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1025.1
Root \(-1.41421i\) of defining polynomial
Character \(\chi\) \(=\) 1368.1025
Dual form 1368.2.f.a.1025.2

$q$-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)
 
\(f(q)\) \(=\) \(q-1.41421i q^{5} +2.00000 q^{7} +4.24264i q^{11} +2.82843i q^{13} +7.07107i q^{17} +(-1.00000 - 4.24264i) q^{19} +1.41421i q^{23} +3.00000 q^{25} -10.0000 q^{29} +2.82843i q^{31} -2.82843i q^{35} +5.65685i q^{37} +10.0000 q^{41} +12.0000 q^{43} -1.41421i q^{47} -3.00000 q^{49} +10.0000 q^{53} +6.00000 q^{55} -12.0000 q^{59} +8.00000 q^{61} +4.00000 q^{65} -14.1421i q^{67} -8.00000 q^{71} +6.00000 q^{73} +8.48528i q^{77} +11.3137i q^{79} +4.24264i q^{83} +10.0000 q^{85} +6.00000 q^{89} +5.65685i q^{91} +(-6.00000 + 1.41421i) q^{95} +8.48528i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 4 q^{7} - 2 q^{19} + 6 q^{25} - 20 q^{29} + 20 q^{41} + 24 q^{43} - 6 q^{49} + 20 q^{53} + 12 q^{55} - 24 q^{59} + 16 q^{61} + 8 q^{65} - 16 q^{71} + 12 q^{73} + 20 q^{85} + 12 q^{89} - 12 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1368\mathbb{Z}\right)^\times\).

\(n\) \(343\) \(685\) \(1009\) \(1217\)
\(\chi(n)\) \(1\) \(1\) \(-1\) \(-1\)

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
\(3\) 0 0
\(4\) 0 0
\(5\) − 1.41421i − 0.632456i −0.948683 − 0.316228i \(-0.897584\pi\)
0.948683 − 0.316228i \(-0.102416\pi\)
\(6\) 0 0
\(7\) 2.00000 0.755929 0.377964 − 0.925820i \(-0.376624\pi\)
0.377964 + 0.925820i \(0.376624\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 4.24264i 1.27920i 0.768706 + 0.639602i \(0.220901\pi\)
−0.768706 + 0.639602i \(0.779099\pi\)
\(12\) 0 0
\(13\) 2.82843i 0.784465i 0.919866 + 0.392232i \(0.128297\pi\)
−0.919866 + 0.392232i \(0.871703\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 7.07107i 1.71499i 0.514496 + 0.857493i \(0.327979\pi\)
−0.514496 + 0.857493i \(0.672021\pi\)
\(18\) 0 0
\(19\) −1.00000 − 4.24264i −0.229416 − 0.973329i
\(20\) 0 0
\(21\) 0 0
\(22\) 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
\(27\) 0 0
\(28\) 0 0
\(29\) −10.0000 −1.85695 −0.928477 − 0.371391i \(-0.878881\pi\)
−0.928477 + 0.371391i \(0.878881\pi\)
\(30\) 0 0
\(31\) 2.82843i 0.508001i 0.967204 + 0.254000i \(0.0817464\pi\)
−0.967204 + 0.254000i \(0.918254\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) − 2.82843i − 0.478091i
\(36\) 0 0
\(37\) 5.65685i 0.929981i 0.885316 + 0.464991i \(0.153942\pi\)
−0.885316 + 0.464991i \(0.846058\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 10.0000 1.56174 0.780869 − 0.624695i \(-0.214777\pi\)
0.780869 + 0.624695i \(0.214777\pi\)
\(42\) 0 0
\(43\) 12.0000 1.82998 0.914991 − 0.403473i \(-0.132197\pi\)
0.914991 + 0.403473i \(0.132197\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) − 1.41421i − 0.206284i −0.994667 − 0.103142i \(-0.967110\pi\)
0.994667 − 0.103142i \(-0.0328896\pi\)
\(48\) 0 0
\(49\) −3.00000 −0.428571
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 10.0000 1.37361 0.686803 − 0.726844i \(-0.259014\pi\)
0.686803 + 0.726844i \(0.259014\pi\)
\(54\) 0 0
\(55\) 6.00000 0.809040
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −12.0000 −1.56227 −0.781133 − 0.624364i \(-0.785358\pi\)
−0.781133 + 0.624364i \(0.785358\pi\)
\(60\) 0 0
\(61\) 8.00000 1.02430 0.512148 − 0.858898i \(-0.328850\pi\)
0.512148 + 0.858898i \(0.328850\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 4.00000 0.496139
\(66\) 0 0
\(67\) − 14.1421i − 1.72774i −0.503718 − 0.863868i \(-0.668035\pi\)
0.503718 − 0.863868i \(-0.331965\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −8.00000 −0.949425 −0.474713 − 0.880141i \(-0.657448\pi\)
−0.474713 + 0.880141i \(0.657448\pi\)
\(72\) 0 0
\(73\) 6.00000 0.702247 0.351123 − 0.936329i \(-0.385800\pi\)
0.351123 + 0.936329i \(0.385800\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 8.48528i 0.966988i
\(78\) 0 0
\(79\) 11.3137i 1.27289i 0.771321 + 0.636446i \(0.219596\pi\)
−0.771321 + 0.636446i \(0.780404\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 4.24264i 0.465690i 0.972514 + 0.232845i \(0.0748035\pi\)
−0.972514 + 0.232845i \(0.925196\pi\)
\(84\) 0 0
\(85\) 10.0000 1.08465
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 6.00000 0.635999 0.317999 − 0.948091i \(-0.396989\pi\)
0.317999 + 0.948091i \(0.396989\pi\)
\(90\) 0 0
\(91\) 5.65685i 0.592999i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −6.00000 + 1.41421i −0.615587 + 0.145095i
\(96\) 0 0
\(97\) 8.48528i 0.861550i 0.902459 + 0.430775i \(0.141760\pi\)
−0.902459 + 0.430775i \(0.858240\pi\)
\(98\) 0 0
\(99\) 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 1368.2.f.a.1025.1 ✓ 2
3.2 odd 2 1368.2.f.b.1025.2 yes 2
4.3 odd 2 2736.2.f.c.1025.1 2
12.11 even 2 2736.2.f.d.1025.2 2
19.18 odd 2 1368.2.f.b.1025.1 yes 2
57.56 even 2 inner 1368.2.f.a.1025.2 yes 2
76.75 even 2 2736.2.f.d.1025.1 2
228.227 odd 2 2736.2.f.c.1025.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1368.2.f.a.1025.1 ✓ 2 1.1 even 1 trivial
1368.2.f.a.1025.2 yes 2 57.56 even 2 inner
1368.2.f.b.1025.1 yes 2 19.18 odd 2
1368.2.f.b.1025.2 yes 2 3.2 odd 2
2736.2.f.c.1025.1 2 4.3 odd 2
2736.2.f.c.1025.2 2 228.227 odd 2
2736.2.f.d.1025.1 2 76.75 even 2
2736.2.f.d.1025.2 2 12.11 even 2