Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1472,2,Mod(1,1472)] 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("1472.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1472, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1472 = 2^{6} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1472.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,2,0,4,0,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(11.7539791775\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
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: no (minimal twist has level 736)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.41421\) of defining polynomial
Character \(\chi\) \(=\) 1472.1

$q$-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)
 
\(f(q)\) \(=\) \(q+2.41421 q^{3} +3.41421 q^{5} +3.41421 q^{7} +2.82843 q^{9} -4.24264 q^{11} -1.82843 q^{13} +8.24264 q^{15} -5.41421 q^{17} +4.82843 q^{19} +8.24264 q^{21} +1.00000 q^{23} +6.65685 q^{25} -0.414214 q^{27} +5.00000 q^{29} +0.757359 q^{31} -10.2426 q^{33} +11.6569 q^{35} +9.07107 q^{37} -4.41421 q^{39} -12.6569 q^{41} +4.00000 q^{43} +9.65685 q^{45} -3.24264 q^{47} +4.65685 q^{49} -13.0711 q^{51} -13.3137 q^{53} -14.4853 q^{55} +11.6569 q^{57} +6.82843 q^{59} -7.65685 q^{61} +9.65685 q^{63} -6.24264 q^{65} -5.89949 q^{67} +2.41421 q^{69} -6.41421 q^{71} +2.17157 q^{73} +16.0711 q^{75} -14.4853 q^{77} -2.48528 q^{79} -9.48528 q^{81} +14.7279 q^{83} -18.4853 q^{85} +12.0711 q^{87} -8.00000 q^{89} -6.24264 q^{91} +1.82843 q^{93} +16.4853 q^{95} -9.89949 q^{97} -12.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{3} + 4 q^{5} + 4 q^{7} + 2 q^{13} + 8 q^{15} - 8 q^{17} + 4 q^{19} + 8 q^{21} + 2 q^{23} + 2 q^{25} + 2 q^{27} + 10 q^{29} + 10 q^{31} - 12 q^{33} + 12 q^{35} + 4 q^{37} - 6 q^{39} - 14 q^{41}+ \cdots - 24 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 2.41421 1.39385 0.696923 − 0.717146i \(-0.254552\pi\)
0.696923 + 0.717146i \(0.254552\pi\)
\(4\) 0 0
\(5\) 3.41421 1.52688 0.763441 − 0.645877i \(-0.223508\pi\)
0.763441 + 0.645877i \(0.223508\pi\)
\(6\) 0 0
\(7\) 3.41421 1.29045 0.645226 − 0.763992i \(-0.276763\pi\)
0.645226 + 0.763992i \(0.276763\pi\)
\(8\) 0 0
\(9\) 2.82843 0.942809
\(10\) 0 0
\(11\) −4.24264 −1.27920 −0.639602 − 0.768706i \(-0.720901\pi\)
−0.639602 + 0.768706i \(0.720901\pi\)
\(12\) 0 0
\(13\) −1.82843 −0.507114 −0.253557 − 0.967320i \(-0.581601\pi\)
−0.253557 + 0.967320i \(0.581601\pi\)
\(14\) 0 0
\(15\) 8.24264 2.12824
\(16\) 0 0
\(17\) −5.41421 −1.31314 −0.656570 − 0.754265i \(-0.727993\pi\)
−0.656570 + 0.754265i \(0.727993\pi\)
\(18\) 0 0
\(19\) 4.82843 1.10772 0.553859 − 0.832611i \(-0.313155\pi\)
0.553859 + 0.832611i \(0.313155\pi\)
\(20\) 0 0
\(21\) 8.24264 1.79869
\(22\) 0 0
\(23\) 1.00000 0.208514
\(24\) 0 0
\(25\) 6.65685 1.33137
\(26\) 0 0
\(27\) −0.414214 −0.0797154
\(28\) 0 0
\(29\) 5.00000 0.928477 0.464238 − 0.885710i \(-0.346328\pi\)
0.464238 + 0.885710i \(0.346328\pi\)
\(30\) 0 0
\(31\) 0.757359 0.136026 0.0680129 − 0.997684i \(-0.478334\pi\)
0.0680129 + 0.997684i \(0.478334\pi\)
\(32\) 0 0
\(33\) −10.2426 −1.78301
\(34\) 0 0
\(35\) 11.6569 1.97037
\(36\) 0 0
\(37\) 9.07107 1.49127 0.745637 − 0.666352i \(-0.232145\pi\)
0.745637 + 0.666352i \(0.232145\pi\)
\(38\) 0 0
\(39\) −4.41421 −0.706840
\(40\) 0 0
\(41\) −12.6569 −1.97667 −0.988334 − 0.152300i \(-0.951332\pi\)
−0.988334 + 0.152300i \(0.951332\pi\)
\(42\) 0 0
\(43\) 4.00000 0.609994 0.304997 − 0.952353i \(-0.401344\pi\)
0.304997 + 0.952353i \(0.401344\pi\)
\(44\) 0 0
\(45\) 9.65685 1.43956
\(46\) 0 0
\(47\) −3.24264 −0.472988 −0.236494 − 0.971633i \(-0.575998\pi\)
−0.236494 + 0.971633i \(0.575998\pi\)
\(48\) 0 0
\(49\) 4.65685 0.665265
\(50\) 0 0
\(51\) −13.0711 −1.83032
\(52\) 0 0
\(53\) −13.3137 −1.82878 −0.914389 − 0.404836i \(-0.867329\pi\)
−0.914389 + 0.404836i \(0.867329\pi\)
\(54\) 0 0
\(55\) −14.4853 −1.95319
\(56\) 0 0
\(57\) 11.6569 1.54399
\(58\) 0 0
\(59\) 6.82843 0.888985 0.444493 − 0.895782i \(-0.353384\pi\)
0.444493 + 0.895782i \(0.353384\pi\)
\(60\) 0 0
\(61\) −7.65685 −0.980360 −0.490180 − 0.871621i \(-0.663069\pi\)
−0.490180 + 0.871621i \(0.663069\pi\)
\(62\) 0 0
\(63\) 9.65685 1.21665
\(64\) 0 0
\(65\) −6.24264 −0.774304
\(66\) 0 0
\(67\) −5.89949 −0.720738 −0.360369 − 0.932810i \(-0.617349\pi\)
−0.360369 + 0.932810i \(0.617349\pi\)
\(68\) 0 0
\(69\) 2.41421 0.290637
\(70\) 0 0
\(71\) −6.41421 −0.761227 −0.380614 − 0.924734i \(-0.624287\pi\)
−0.380614 + 0.924734i \(0.624287\pi\)
\(72\) 0 0
\(73\) 2.17157 0.254163 0.127082 − 0.991892i \(-0.459439\pi\)
0.127082 + 0.991892i \(0.459439\pi\)
\(74\) 0 0
\(75\) 16.0711 1.85573
\(76\) 0 0
\(77\) −14.4853 −1.65075
\(78\) 0 0
\(79\) −2.48528 −0.279616 −0.139808 − 0.990179i \(-0.544649\pi\)
−0.139808 + 0.990179i \(0.544649\pi\)
\(80\) 0 0
\(81\) −9.48528 −1.05392
\(82\) 0 0
\(83\) 14.7279 1.61660 0.808300 − 0.588771i \(-0.200388\pi\)
0.808300 + 0.588771i \(0.200388\pi\)
\(84\) 0 0
\(85\) −18.4853 −2.00501
\(86\) 0 0
\(87\) 12.0711 1.29415
\(88\) 0 0
\(89\) −8.00000 −0.847998 −0.423999 − 0.905663i \(-0.639374\pi\)
−0.423999 + 0.905663i \(0.639374\pi\)
\(90\) 0 0
\(91\) −6.24264 −0.654407
\(92\) 0 0
\(93\) 1.82843 0.189599
\(94\) 0 0
\(95\) 16.4853 1.69135
\(96\) 0 0
\(97\) −9.89949 −1.00514 −0.502571 − 0.864536i \(-0.667612\pi\)
−0.502571 + 0.864536i \(0.667612\pi\)
\(98\) 0 0
\(99\) −12.0000 −1.20605
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 1472.2.a.v.1.2 2
4.3 odd 2 1472.2.a.o.1.1 2
8.3 odd 2 736.2.a.d.1.2 yes 2
8.5 even 2 736.2.a.a.1.1 ✓ 2
24.5 odd 2 6624.2.a.t.1.2 2
24.11 even 2 6624.2.a.s.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
736.2.a.a.1.1 ✓ 2 8.5 even 2
736.2.a.d.1.2 yes 2 8.3 odd 2
1472.2.a.o.1.1 2 4.3 odd 2
1472.2.a.v.1.2 2 1.1 even 1 trivial
6624.2.a.s.1.2 2 24.11 even 2
6624.2.a.t.1.2 2 24.5 odd 2