Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [736,2,Mod(1,736)] 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("736.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(736, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 736 = 2^{5} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 736.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [3,0,4,0,-2,0,-2] 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: \(5.87698958877\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.316.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 4x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.81361\) of defining polynomial
Character \(\chi\) \(=\) 736.1

$q$-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)
 
\(f(q)\) \(=\) \(q-0.813607 q^{3} -3.10278 q^{5} -3.10278 q^{7} -2.33804 q^{9} +5.10278 q^{11} +0.289169 q^{13} +2.52444 q^{15} -1.10278 q^{17} +7.04888 q^{19} +2.52444 q^{21} +1.00000 q^{23} +4.62721 q^{25} +4.34307 q^{27} +7.54359 q^{29} -0.235269 q^{31} -4.15165 q^{33} +9.62721 q^{35} -1.94610 q^{37} -0.235269 q^{39} -4.28917 q^{41} +1.42166 q^{43} +7.25443 q^{45} +11.0192 q^{47} +2.62721 q^{49} +0.897225 q^{51} -11.8328 q^{53} -15.8328 q^{55} -5.73501 q^{57} +1.79445 q^{59} +6.67609 q^{61} +7.25443 q^{63} -0.897225 q^{65} -14.5628 q^{67} -0.813607 q^{69} -4.81361 q^{71} +8.12193 q^{73} -3.76473 q^{75} -15.8328 q^{77} +8.88164 q^{79} +3.48059 q^{81} +16.9355 q^{83} +3.42166 q^{85} -6.13752 q^{87} +13.0872 q^{89} -0.897225 q^{91} +0.191417 q^{93} -21.8711 q^{95} -4.05390 q^{97} -11.9305 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 4 q^{3} - 2 q^{5} - 2 q^{7} + 5 q^{9} + 8 q^{11} + 2 q^{15} + 4 q^{17} + 10 q^{19} + 2 q^{21} + 3 q^{23} + q^{25} + 16 q^{27} - 4 q^{29} + 4 q^{31} + 6 q^{33} + 16 q^{35} - 2 q^{37} + 4 q^{39} - 12 q^{41}+ \cdots + 14 q^{99}+O(q^{100}) \) 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
\(3\) −0.813607 −0.469736 −0.234868 − 0.972027i \(-0.575466\pi\)
−0.234868 + 0.972027i \(0.575466\pi\)
\(4\) 0 0
\(5\) −3.10278 −1.38760 −0.693802 − 0.720166i \(-0.744066\pi\)
−0.693802 + 0.720166i \(0.744066\pi\)
\(6\) 0 0
\(7\) −3.10278 −1.17274 −0.586369 − 0.810044i \(-0.699443\pi\)
−0.586369 + 0.810044i \(0.699443\pi\)
\(8\) 0 0
\(9\) −2.33804 −0.779348
\(10\) 0 0
\(11\) 5.10278 1.53854 0.769272 − 0.638921i \(-0.220619\pi\)
0.769272 + 0.638921i \(0.220619\pi\)
\(12\) 0 0
\(13\) 0.289169 0.0802009 0.0401005 − 0.999196i \(-0.487232\pi\)
0.0401005 + 0.999196i \(0.487232\pi\)
\(14\) 0 0
\(15\) 2.52444 0.651807
\(16\) 0 0
\(17\) −1.10278 −0.267462 −0.133731 − 0.991018i \(-0.542696\pi\)
−0.133731 + 0.991018i \(0.542696\pi\)
\(18\) 0 0
\(19\) 7.04888 1.61712 0.808562 − 0.588412i \(-0.200247\pi\)
0.808562 + 0.588412i \(0.200247\pi\)
\(20\) 0 0
\(21\) 2.52444 0.550878
\(22\) 0 0
\(23\) 1.00000 0.208514
\(24\) 0 0
\(25\) 4.62721 0.925443
\(26\) 0 0
\(27\) 4.34307 0.835824
\(28\) 0 0
\(29\) 7.54359 1.40081 0.700405 − 0.713745i \(-0.253003\pi\)
0.700405 + 0.713745i \(0.253003\pi\)
\(30\) 0 0
\(31\) −0.235269 −0.0422556 −0.0211278 − 0.999777i \(-0.506726\pi\)
−0.0211278 + 0.999777i \(0.506726\pi\)
\(32\) 0 0
\(33\) −4.15165 −0.722710
\(34\) 0 0
\(35\) 9.62721 1.62730
\(36\) 0 0
\(37\) −1.94610 −0.319937 −0.159969 − 0.987122i \(-0.551139\pi\)
−0.159969 + 0.987122i \(0.551139\pi\)
\(38\) 0 0
\(39\) −0.235269 −0.0376733
\(40\) 0 0
\(41\) −4.28917 −0.669856 −0.334928 − 0.942244i \(-0.608712\pi\)
−0.334928 + 0.942244i \(0.608712\pi\)
\(42\) 0 0
\(43\) 1.42166 0.216802 0.108401 − 0.994107i \(-0.465427\pi\)
0.108401 + 0.994107i \(0.465427\pi\)
\(44\) 0 0
\(45\) 7.25443 1.08143
\(46\) 0 0
\(47\) 11.0192 1.60731 0.803655 − 0.595096i \(-0.202886\pi\)
0.803655 + 0.595096i \(0.202886\pi\)
\(48\) 0 0
\(49\) 2.62721 0.375316
\(50\) 0 0
\(51\) 0.897225 0.125637
\(52\) 0 0
\(53\) −11.8328 −1.62536 −0.812678 − 0.582714i \(-0.801991\pi\)
−0.812678 + 0.582714i \(0.801991\pi\)
\(54\) 0 0
\(55\) −15.8328 −2.13489
\(56\) 0 0
\(57\) −5.73501 −0.759621
\(58\) 0 0
\(59\) 1.79445 0.233617 0.116809 − 0.993154i \(-0.462734\pi\)
0.116809 + 0.993154i \(0.462734\pi\)
\(60\) 0 0
\(61\) 6.67609 0.854786 0.427393 − 0.904066i \(-0.359432\pi\)
0.427393 + 0.904066i \(0.359432\pi\)
\(62\) 0 0
\(63\) 7.25443 0.913972
\(64\) 0 0
\(65\) −0.897225 −0.111287
\(66\) 0 0
\(67\) −14.5628 −1.77912 −0.889562 − 0.456815i \(-0.848990\pi\)
−0.889562 + 0.456815i \(0.848990\pi\)
\(68\) 0 0
\(69\) −0.813607 −0.0979467
\(70\) 0 0
\(71\) −4.81361 −0.571270 −0.285635 − 0.958338i \(-0.592204\pi\)
−0.285635 + 0.958338i \(0.592204\pi\)
\(72\) 0 0
\(73\) 8.12193 0.950600 0.475300 − 0.879824i \(-0.342339\pi\)
0.475300 + 0.879824i \(0.342339\pi\)
\(74\) 0 0
\(75\) −3.76473 −0.434714
\(76\) 0 0
\(77\) −15.8328 −1.80431
\(78\) 0 0
\(79\) 8.88164 0.999262 0.499631 − 0.866238i \(-0.333469\pi\)
0.499631 + 0.866238i \(0.333469\pi\)
\(80\) 0 0
\(81\) 3.48059 0.386732
\(82\) 0 0
\(83\) 16.9355 1.85892 0.929458 − 0.368927i \(-0.120275\pi\)
0.929458 + 0.368927i \(0.120275\pi\)
\(84\) 0 0
\(85\) 3.42166 0.371131
\(86\) 0 0
\(87\) −6.13752 −0.658011
\(88\) 0 0
\(89\) 13.0872 1.38724 0.693620 − 0.720341i \(-0.256015\pi\)
0.693620 + 0.720341i \(0.256015\pi\)
\(90\) 0 0
\(91\) −0.897225 −0.0940547
\(92\) 0 0
\(93\) 0.191417 0.0198490
\(94\) 0 0
\(95\) −21.8711 −2.24393
\(96\) 0 0
\(97\) −4.05390 −0.411611 −0.205806 − 0.978593i \(-0.565981\pi\)
−0.205806 + 0.978593i \(0.565981\pi\)
\(98\) 0 0
\(99\) −11.9305 −1.19906
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 736.2.a.f.1.1 yes 3
3.2 odd 2 6624.2.a.y.1.3 3
4.3 odd 2 736.2.a.e.1.3 ✓ 3
8.3 odd 2 1472.2.a.x.1.1 3
8.5 even 2 1472.2.a.w.1.3 3
12.11 even 2 6624.2.a.z.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
736.2.a.e.1.3 ✓ 3 4.3 odd 2
736.2.a.f.1.1 yes 3 1.1 even 1 trivial
1472.2.a.w.1.3 3 8.5 even 2
1472.2.a.x.1.1 3 8.3 odd 2
6624.2.a.y.1.3 3 3.2 odd 2
6624.2.a.z.1.3 3 12.11 even 2