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 = [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: \(11.7539791775\)
Analytic rank: \(1\)
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: no (minimal twist has level 736)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(0.849256\) 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)
 
\(f(q)\) \(=\) \(q-3.12802 q^{3} -4.35500 q^{5} -2.65649 q^{7} +6.78451 q^{9} +2.35500 q^{11} +3.78451 q^{13} +13.6225 q^{15} -0.656490 q^{17} -0.301488 q^{19} +8.30955 q^{21} -1.00000 q^{23} +13.9660 q^{25} -11.8380 q^{27} +8.49451 q^{29} -6.52504 q^{31} -7.36649 q^{33} +11.5690 q^{35} -4.35500 q^{37} -11.8380 q^{39} +3.07451 q^{41} -1.44247 q^{43} -29.5466 q^{45} -3.73100 q^{47} +0.0569399 q^{49} +2.05351 q^{51} +3.95455 q^{53} -10.2560 q^{55} +0.943060 q^{57} +13.8251 q^{59} -7.95455 q^{61} -18.0230 q^{63} -16.4816 q^{65} -7.66798 q^{67} +3.12802 q^{69} +13.1510 q^{71} -5.33055 q^{73} -43.6861 q^{75} -6.25604 q^{77} -6.85902 q^{79} +16.6760 q^{81} -8.61104 q^{83} +2.85902 q^{85} -26.5710 q^{87} +13.5690 q^{89} -10.0535 q^{91} +20.4105 q^{93} +1.31298 q^{95} +7.28245 q^{97} +15.9775 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{3} - 6 q^{5} - 4 q^{7} + 10 q^{9} - 2 q^{11} - 2 q^{13} + 4 q^{15} + 4 q^{17} - 6 q^{19} - 4 q^{21} - 4 q^{23} + 12 q^{25} - 14 q^{27} - 6 q^{29} - 6 q^{31} + 12 q^{35} - 6 q^{37} - 14 q^{39}+ \cdots + 2 q^{99}+O(q^{100}) \) 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 \(-0.858627\pi\)
−0.902982 + 0.429679i \(0.858627\pi\)
\(4\) 0 0
\(5\) −4.35500 −1.94762 −0.973808 − 0.227371i \(-0.926987\pi\)
−0.973808 + 0.227371i \(0.926987\pi\)
\(6\) 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 \(-0.384471\pi\)
0.355030 + 0.934855i \(0.384471\pi\)
\(12\) 0 0
\(13\) 3.78451 1.04963 0.524817 − 0.851215i \(-0.324134\pi\)
0.524817 + 0.851215i \(0.324134\pi\)
\(14\) 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 \(-0.511010\pi\)
−0.0345830 + 0.999402i \(0.511010\pi\)
\(20\) 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 \(-0.210757\pi\)
0.788696 + 0.614784i \(0.210757\pi\)
\(30\) 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 \(-0.616534\pi\)
−0.357979 + 0.933730i \(0.616534\pi\)
\(38\) 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 \(-0.535081\pi\)
−0.109987 + 0.993933i \(0.535081\pi\)
\(44\) 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 \(-0.412447\pi\)
0.271600 + 0.962410i \(0.412447\pi\)
\(54\) 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 \(-0.143614\pi\)
0.899935 + 0.436024i \(0.143614\pi\)
\(60\) 0 0
\(61\) −7.95455 −1.01848 −0.509238 − 0.860626i \(-0.670073\pi\)
−0.509238 + 0.860626i \(0.670073\pi\)
\(62\) 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 \(-0.655168\pi\)
−0.468397 + 0.883518i \(0.655168\pi\)
\(68\) 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 \(-0.656682\pi\)
−0.472592 + 0.881281i \(0.656682\pi\)
\(84\) 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 1472.2.a.y.1.1 4
4.3 odd 2 1472.2.a.z.1.4 4
8.3 odd 2 736.2.a.g.1.1 ✓ 4
8.5 even 2 736.2.a.h.1.4 yes 4
24.5 odd 2 6624.2.a.be.1.1 4
24.11 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 8.3 odd 2
736.2.a.h.1.4 yes 4 8.5 even 2
1472.2.a.y.1.1 4 1.1 even 1 trivial
1472.2.a.z.1.4 4 4.3 odd 2
6624.2.a.be.1.1 4 24.5 odd 2
6624.2.a.bf.1.1 4 24.11 even 2