Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [690,2,Mod(139,690)] 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("690.139"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(690, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 690 = 2 \cdot 3 \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 690.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,-4,0,4,0,0,-4,0,0,0,0,-16] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(14)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(5.50967773947\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{5})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 3x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 139.3
Root \(1.61803i\) of defining polynomial
Character \(\chi\) \(=\) 690.139
Dual form 690.2.d.a.139.1

$q$-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)
 
\(f(q)\) \(=\) \(q+1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} -2.23607 q^{5} +1.00000 q^{6} +4.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} -2.23607i q^{10} +1.00000i q^{12} -6.47214i q^{13} -4.00000 q^{14} +2.23607i q^{15} +1.00000 q^{16} -6.47214i q^{17} -1.00000i q^{18} -2.00000 q^{19} +2.23607 q^{20} +4.00000 q^{21} +1.00000i q^{23} -1.00000 q^{24} +5.00000 q^{25} +6.47214 q^{26} +1.00000i q^{27} -4.00000i q^{28} -8.47214 q^{29} -2.23607 q^{30} +1.00000i q^{32} +6.47214 q^{34} -8.94427i q^{35} +1.00000 q^{36} -8.47214i q^{37} -2.00000i q^{38} -6.47214 q^{39} +2.23607i q^{40} -6.94427 q^{41} +4.00000i q^{42} +2.23607 q^{45} -1.00000 q^{46} -12.9443i q^{47} -1.00000i q^{48} -9.00000 q^{49} +5.00000i q^{50} -6.47214 q^{51} +6.47214i q^{52} -8.94427i q^{53} -1.00000 q^{54} +4.00000 q^{56} +2.00000i q^{57} -8.47214i q^{58} +6.00000 q^{59} -2.23607i q^{60} +8.47214 q^{61} -4.00000i q^{63} -1.00000 q^{64} +14.4721i q^{65} +12.9443i q^{67} +6.47214i q^{68} +1.00000 q^{69} +8.94427 q^{70} -16.4721 q^{71} +1.00000i q^{72} +12.9443i q^{73} +8.47214 q^{74} -5.00000i q^{75} +2.00000 q^{76} -6.47214i q^{78} -3.52786 q^{79} -2.23607 q^{80} +1.00000 q^{81} -6.94427i q^{82} -10.4721i q^{83} -4.00000 q^{84} +14.4721i q^{85} +8.47214i q^{87} +7.52786 q^{89} +2.23607i q^{90} +25.8885 q^{91} -1.00000i q^{92} +12.9443 q^{94} +4.47214 q^{95} +1.00000 q^{96} +13.4164i q^{97} -9.00000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{4} + 4 q^{6} - 4 q^{9} - 16 q^{14} + 4 q^{16} - 8 q^{19} + 16 q^{21} - 4 q^{24} + 20 q^{25} + 8 q^{26} - 16 q^{29} + 8 q^{34} + 4 q^{36} - 8 q^{39} + 8 q^{41} - 4 q^{46} - 36 q^{49} - 8 q^{51}+ \cdots + 4 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/690\mathbb{Z}\right)^\times\).

\(n\) \(277\) \(461\) \(511\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000i 0.707107i
\(3\) − 1.00000i − 0.577350i
\(4\) −1.00000 −0.500000
\(5\) −2.23607 −1.00000
\(6\) 1.00000 0.408248
\(7\) 4.00000i 1.51186i 0.654654 + 0.755929i \(0.272814\pi\)
−0.654654 + 0.755929i \(0.727186\pi\)
\(8\) − 1.00000i − 0.353553i
\(9\) −1.00000 −0.333333
\(10\) − 2.23607i − 0.707107i
\(11\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 1.00000i 0.288675i
\(13\) − 6.47214i − 1.79505i −0.440966 − 0.897524i \(-0.645364\pi\)
0.440966 − 0.897524i \(-0.354636\pi\)
\(14\) −4.00000 −1.06904
\(15\) 2.23607i 0.577350i
\(16\) 1.00000 0.250000
\(17\) − 6.47214i − 1.56972i −0.619671 − 0.784862i \(-0.712734\pi\)
0.619671 − 0.784862i \(-0.287266\pi\)
\(18\) − 1.00000i − 0.235702i
\(19\) −2.00000 −0.458831 −0.229416 − 0.973329i \(-0.573682\pi\)
−0.229416 + 0.973329i \(0.573682\pi\)
\(20\) 2.23607 0.500000
\(21\) 4.00000 0.872872
\(22\) 0 0
\(23\) 1.00000i 0.208514i
\(24\) −1.00000 −0.204124
\(25\) 5.00000 1.00000
\(26\) 6.47214 1.26929
\(27\) 1.00000i 0.192450i
\(28\) − 4.00000i − 0.755929i
\(29\) −8.47214 −1.57324 −0.786618 − 0.617440i \(-0.788170\pi\)
−0.786618 + 0.617440i \(0.788170\pi\)
\(30\) −2.23607 −0.408248
\(31\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) 1.00000i 0.176777i
\(33\) 0 0
\(34\) 6.47214 1.10996
\(35\) − 8.94427i − 1.51186i
\(36\) 1.00000 0.166667
\(37\) − 8.47214i − 1.39281i −0.717649 − 0.696405i \(-0.754782\pi\)
0.717649 − 0.696405i \(-0.245218\pi\)
\(38\) − 2.00000i − 0.324443i
\(39\) −6.47214 −1.03637
\(40\) 2.23607i 0.353553i
\(41\) −6.94427 −1.08451 −0.542257 − 0.840213i \(-0.682430\pi\)
−0.542257 + 0.840213i \(0.682430\pi\)
\(42\) 4.00000i 0.617213i
\(43\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(44\) 0 0
\(45\) 2.23607 0.333333
\(46\) −1.00000 −0.147442
\(47\) − 12.9443i − 1.88812i −0.329779 − 0.944058i \(-0.606974\pi\)
0.329779 − 0.944058i \(-0.393026\pi\)
\(48\) − 1.00000i − 0.144338i
\(49\) −9.00000 −1.28571
\(50\) 5.00000i 0.707107i
\(51\) −6.47214 −0.906280
\(52\) 6.47214i 0.897524i
\(53\) − 8.94427i − 1.22859i −0.789076 − 0.614295i \(-0.789440\pi\)
0.789076 − 0.614295i \(-0.210560\pi\)
\(54\) −1.00000 −0.136083
\(55\) 0 0
\(56\) 4.00000 0.534522
\(57\) 2.00000i 0.264906i
\(58\) − 8.47214i − 1.11245i
\(59\) 6.00000 0.781133 0.390567 − 0.920575i \(-0.372279\pi\)
0.390567 + 0.920575i \(0.372279\pi\)
\(60\) − 2.23607i − 0.288675i
\(61\) 8.47214 1.08475 0.542373 − 0.840138i \(-0.317526\pi\)
0.542373 + 0.840138i \(0.317526\pi\)
\(62\) 0 0
\(63\) − 4.00000i − 0.503953i
\(64\) −1.00000 −0.125000
\(65\) 14.4721i 1.79505i
\(66\) 0 0
\(67\) 12.9443i 1.58139i 0.612207 + 0.790697i \(0.290282\pi\)
−0.612207 + 0.790697i \(0.709718\pi\)
\(68\) 6.47214i 0.784862i
\(69\) 1.00000 0.120386
\(70\) 8.94427 1.06904
\(71\) −16.4721 −1.95488 −0.977441 − 0.211207i \(-0.932261\pi\)
−0.977441 + 0.211207i \(0.932261\pi\)
\(72\) 1.00000i 0.117851i
\(73\) 12.9443i 1.51501i 0.652828 + 0.757506i \(0.273582\pi\)
−0.652828 + 0.757506i \(0.726418\pi\)
\(74\) 8.47214 0.984866
\(75\) − 5.00000i − 0.577350i
\(76\) 2.00000 0.229416
\(77\) 0 0
\(78\) − 6.47214i − 0.732825i
\(79\) −3.52786 −0.396916 −0.198458 − 0.980109i \(-0.563593\pi\)
−0.198458 + 0.980109i \(0.563593\pi\)
\(80\) −2.23607 −0.250000
\(81\) 1.00000 0.111111
\(82\) − 6.94427i − 0.766867i
\(83\) − 10.4721i − 1.14947i −0.818341 − 0.574733i \(-0.805106\pi\)
0.818341 − 0.574733i \(-0.194894\pi\)
\(84\) −4.00000 −0.436436
\(85\) 14.4721i 1.56972i
\(86\) 0 0
\(87\) 8.47214i 0.908308i
\(88\) 0 0
\(89\) 7.52786 0.797952 0.398976 − 0.916961i \(-0.369366\pi\)
0.398976 + 0.916961i \(0.369366\pi\)
\(90\) 2.23607i 0.235702i
\(91\) 25.8885 2.71386
\(92\) − 1.00000i − 0.104257i
\(93\) 0 0
\(94\) 12.9443 1.33510
\(95\) 4.47214 0.458831
\(96\) 1.00000 0.102062
\(97\) 13.4164i 1.36223i 0.732177 + 0.681115i \(0.238505\pi\)
−0.732177 + 0.681115i \(0.761495\pi\)
\(98\) − 9.00000i − 0.909137i
\(99\) 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 690.2.d.a.139.3 yes 4
3.2 odd 2 2070.2.d.b.829.2 4
5.2 odd 4 3450.2.a.bc.1.1 2
5.3 odd 4 3450.2.a.bn.1.2 2
5.4 even 2 inner 690.2.d.a.139.1 ✓ 4
15.14 odd 2 2070.2.d.b.829.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
690.2.d.a.139.1 ✓ 4 5.4 even 2 inner
690.2.d.a.139.3 yes 4 1.1 even 1 trivial
2070.2.d.b.829.2 4 3.2 odd 2
2070.2.d.b.829.4 4 15.14 odd 2
3450.2.a.bc.1.1 2 5.2 odd 4
3450.2.a.bn.1.2 2 5.3 odd 4