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 = [4,0,2,0,6,0,-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: \(5.87698958877\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.13768.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 5x^{2} + 2x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(0.849256\) 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)
 
\(f(q)\) \(=\) \(q+3.12802 q^{3} +4.35500 q^{5} -2.65649 q^{7} +6.78451 q^{9} -2.35500 q^{11} -3.78451 q^{13} +13.6225 q^{15} -0.656490 q^{17} +0.301488 q^{19} -8.30955 q^{21} -1.00000 q^{23} +13.9660 q^{25} +11.8380 q^{27} -8.49451 q^{29} -6.52504 q^{31} -7.36649 q^{33} -11.5690 q^{35} +4.35500 q^{37} -11.8380 q^{39} +3.07451 q^{41} +1.44247 q^{43} +29.5466 q^{45} -3.73100 q^{47} +0.0569399 q^{49} -2.05351 q^{51} -3.95455 q^{53} -10.2560 q^{55} +0.943060 q^{57} -13.8251 q^{59} +7.95455 q^{61} -18.0230 q^{63} -16.4816 q^{65} +7.66798 q^{67} -3.12802 q^{69} +13.1510 q^{71} -5.33055 q^{73} +43.6861 q^{75} +6.25604 q^{77} -6.85902 q^{79} +16.6760 q^{81} +8.61104 q^{83} -2.85902 q^{85} -26.5710 q^{87} +13.5690 q^{89} +10.0535 q^{91} -20.4105 q^{93} +1.31298 q^{95} +7.28245 q^{97} -15.9775 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{3} + 6 q^{5} - 4 q^{7} + 10 q^{9} + 2 q^{11} + 2 q^{13} + 4 q^{15} + 4 q^{17} + 6 q^{19} + 4 q^{21} - 4 q^{23} + 12 q^{25} + 14 q^{27} + 6 q^{29} - 6 q^{31} - 12 q^{35} + 6 q^{37} - 14 q^{39}+ \cdots - 2 q^{99}+O(q^{100}) \) 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
\(3\) 3.12802 1.80596 0.902982 − 0.429679i \(-0.141373\pi\)
0.902982 + 0.429679i \(0.141373\pi\)
\(4\) 0 0
\(5\) 4.35500 1.94762 0.973808 − 0.227371i \(-0.0730131\pi\)
0.973808 + 0.227371i \(0.0730131\pi\)
\(6\) 0 0
\(7\) −2.65649 −1.00406 −0.502029 − 0.864851i \(-0.667413\pi\)
−0.502029 + 0.864851i \(0.667413\pi\)
\(8\) 0 0
\(9\) 6.78451 2.26150
\(10\) 0 0
\(11\) −2.35500 −0.710060 −0.355030 − 0.934855i \(-0.615529\pi\)
−0.355030 + 0.934855i \(0.615529\pi\)
\(12\) 0 0
\(13\) −3.78451 −1.04963 −0.524817 − 0.851215i \(-0.675866\pi\)
−0.524817 + 0.851215i \(0.675866\pi\)
\(14\) 0 0
\(15\) 13.6225 3.51732
\(16\) 0 0
\(17\) −0.656490 −0.159222 −0.0796111 − 0.996826i \(-0.525368\pi\)
−0.0796111 + 0.996826i \(0.525368\pi\)
\(18\) 0 0
\(19\) 0.301488 0.0691661 0.0345830 − 0.999402i \(-0.488990\pi\)
0.0345830 + 0.999402i \(0.488990\pi\)
\(20\) 0 0
\(21\) −8.30955 −1.81329
\(22\) 0 0
\(23\) −1.00000 −0.208514
\(24\) 0 0
\(25\) 13.9660 2.79321
\(26\) 0 0
\(27\) 11.8380 2.27823
\(28\) 0 0
\(29\) −8.49451 −1.57739 −0.788696 − 0.614784i \(-0.789243\pi\)
−0.788696 + 0.614784i \(0.789243\pi\)
\(30\) 0 0
\(31\) −6.52504 −1.17193 −0.585966 − 0.810335i \(-0.699285\pi\)
−0.585966 + 0.810335i \(0.699285\pi\)
\(32\) 0 0
\(33\) −7.36649 −1.28234
\(34\) 0 0
\(35\) −11.5690 −1.95552
\(36\) 0 0
\(37\) 4.35500 0.715958 0.357979 − 0.933730i \(-0.383466\pi\)
0.357979 + 0.933730i \(0.383466\pi\)
\(38\) 0 0
\(39\) −11.8380 −1.89560
\(40\) 0 0
\(41\) 3.07451 0.480157 0.240079 − 0.970753i \(-0.422827\pi\)
0.240079 + 0.970753i \(0.422827\pi\)
\(42\) 0 0
\(43\) 1.44247 0.219975 0.109987 − 0.993933i \(-0.464919\pi\)
0.109987 + 0.993933i \(0.464919\pi\)
\(44\) 0 0
\(45\) 29.5466 4.40454
\(46\) 0 0
\(47\) −3.73100 −0.544222 −0.272111 − 0.962266i \(-0.587722\pi\)
−0.272111 + 0.962266i \(0.587722\pi\)
\(48\) 0 0
\(49\) 0.0569399 0.00813427
\(50\) 0 0
\(51\) −2.05351 −0.287550
\(52\) 0 0
\(53\) −3.95455 −0.543200 −0.271600 − 0.962410i \(-0.587553\pi\)
−0.271600 + 0.962410i \(0.587553\pi\)
\(54\) 0 0
\(55\) −10.2560 −1.38292
\(56\) 0 0
\(57\) 0.943060 0.124911
\(58\) 0 0
\(59\) −13.8251 −1.79987 −0.899935 − 0.436024i \(-0.856386\pi\)
−0.899935 + 0.436024i \(0.856386\pi\)
\(60\) 0 0
\(61\) 7.95455 1.01848 0.509238 − 0.860626i \(-0.329927\pi\)
0.509238 + 0.860626i \(0.329927\pi\)
\(62\) 0 0
\(63\) −18.0230 −2.27068
\(64\) 0 0
\(65\) −16.4816 −2.04428
\(66\) 0 0
\(67\) 7.66798 0.936793 0.468397 − 0.883518i \(-0.344832\pi\)
0.468397 + 0.883518i \(0.344832\pi\)
\(68\) 0 0
\(69\) −3.12802 −0.376569
\(70\) 0 0
\(71\) 13.1510 1.56074 0.780369 − 0.625320i \(-0.215031\pi\)
0.780369 + 0.625320i \(0.215031\pi\)
\(72\) 0 0
\(73\) −5.33055 −0.623893 −0.311947 − 0.950100i \(-0.600981\pi\)
−0.311947 + 0.950100i \(0.600981\pi\)
\(74\) 0 0
\(75\) 43.6861 5.04443
\(76\) 0 0
\(77\) 6.25604 0.712942
\(78\) 0 0
\(79\) −6.85902 −0.771700 −0.385850 − 0.922562i \(-0.626092\pi\)
−0.385850 + 0.922562i \(0.626092\pi\)
\(80\) 0 0
\(81\) 16.6760 1.85289
\(82\) 0 0
\(83\) 8.61104 0.945185 0.472592 − 0.881281i \(-0.343318\pi\)
0.472592 + 0.881281i \(0.343318\pi\)
\(84\) 0 0
\(85\) −2.85902 −0.310104
\(86\) 0 0
\(87\) −26.5710 −2.84871
\(88\) 0 0
\(89\) 13.5690 1.43831 0.719157 − 0.694848i \(-0.244528\pi\)
0.719157 + 0.694848i \(0.244528\pi\)
\(90\) 0 0
\(91\) 10.0535 1.05389
\(92\) 0 0
\(93\) −20.4105 −2.11647
\(94\) 0 0
\(95\) 1.31298 0.134709
\(96\) 0 0
\(97\) 7.28245 0.739421 0.369710 − 0.929147i \(-0.379457\pi\)
0.369710 + 0.929147i \(0.379457\pi\)
\(98\) 0 0
\(99\) −15.9775 −1.60580
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.h.1.4 yes 4
3.2 odd 2 6624.2.a.be.1.1 4
4.3 odd 2 736.2.a.g.1.1 ✓ 4
8.3 odd 2 1472.2.a.z.1.4 4
8.5 even 2 1472.2.a.y.1.1 4
12.11 even 2 6624.2.a.bf.1.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
736.2.a.g.1.1 ✓ 4 4.3 odd 2
736.2.a.h.1.4 yes 4 1.1 even 1 trivial
1472.2.a.y.1.1 4 8.5 even 2
1472.2.a.z.1.4 4 8.3 odd 2
6624.2.a.be.1.1 4 3.2 odd 2
6624.2.a.bf.1.1 4 12.11 even 2