Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [4400,2,Mod(1,4400)] 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("4400.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4400, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 4400 = 2^{4} \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4400.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,1,0,0,0,5,0,1,0,2,0,-10] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(35.1341768894\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{13}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 3 \) 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 275)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(2.30278\) of defining polynomial
Character \(\chi\) \(=\) 4400.1

$q$-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)
 
\(f(q)\) \(=\) \(q+2.30278 q^{3} +0.697224 q^{7} +2.30278 q^{9} +1.00000 q^{11} -5.00000 q^{13} -6.90833 q^{17} +1.00000 q^{19} +1.60555 q^{21} -7.30278 q^{23} -1.60555 q^{27} +0.908327 q^{29} -10.2111 q^{31} +2.30278 q^{33} -2.39445 q^{37} -11.5139 q^{39} -5.60555 q^{41} +7.21110 q^{43} -3.00000 q^{47} -6.51388 q^{49} -15.9083 q^{51} +1.30278 q^{53} +2.30278 q^{57} +14.2111 q^{59} -7.90833 q^{61} +1.60555 q^{63} -4.00000 q^{67} -16.8167 q^{69} +2.60555 q^{71} +7.90833 q^{73} +0.697224 q^{77} +10.9083 q^{79} -10.6056 q^{81} -3.51388 q^{83} +2.09167 q^{87} +1.69722 q^{89} -3.48612 q^{91} -23.5139 q^{93} -15.3028 q^{97} +2.30278 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{3} + 5 q^{7} + q^{9} + 2 q^{11} - 10 q^{13} - 3 q^{17} + 2 q^{19} - 4 q^{21} - 11 q^{23} + 4 q^{27} - 9 q^{29} - 6 q^{31} + q^{33} - 12 q^{37} - 5 q^{39} - 4 q^{41} - 6 q^{47} + 5 q^{49} - 21 q^{51}+ \cdots + q^{99}+O(q^{100}) \) 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
\(3\) 2.30278 1.32951 0.664754 − 0.747062i \(-0.268536\pi\)
0.664754 + 0.747062i \(0.268536\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 0.697224 0.263526 0.131763 − 0.991281i \(-0.457936\pi\)
0.131763 + 0.991281i \(0.457936\pi\)
\(8\) 0 0
\(9\) 2.30278 0.767592
\(10\) 0 0
\(11\) 1.00000 0.301511
\(12\) 0 0
\(13\) −5.00000 −1.38675 −0.693375 − 0.720577i \(-0.743877\pi\)
−0.693375 + 0.720577i \(0.743877\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −6.90833 −1.67552 −0.837758 − 0.546042i \(-0.816134\pi\)
−0.837758 + 0.546042i \(0.816134\pi\)
\(18\) 0 0
\(19\) 1.00000 0.229416 0.114708 − 0.993399i \(-0.463407\pi\)
0.114708 + 0.993399i \(0.463407\pi\)
\(20\) 0 0
\(21\) 1.60555 0.350360
\(22\) 0 0
\(23\) −7.30278 −1.52273 −0.761367 − 0.648321i \(-0.775471\pi\)
−0.761367 + 0.648321i \(0.775471\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) −1.60555 −0.308988
\(28\) 0 0
\(29\) 0.908327 0.168672 0.0843360 − 0.996437i \(-0.473123\pi\)
0.0843360 + 0.996437i \(0.473123\pi\)
\(30\) 0 0
\(31\) −10.2111 −1.83397 −0.916984 − 0.398924i \(-0.869384\pi\)
−0.916984 + 0.398924i \(0.869384\pi\)
\(32\) 0 0
\(33\) 2.30278 0.400862
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −2.39445 −0.393645 −0.196822 − 0.980439i \(-0.563062\pi\)
−0.196822 + 0.980439i \(0.563062\pi\)
\(38\) 0 0
\(39\) −11.5139 −1.84370
\(40\) 0 0
\(41\) −5.60555 −0.875440 −0.437720 − 0.899111i \(-0.644214\pi\)
−0.437720 + 0.899111i \(0.644214\pi\)
\(42\) 0 0
\(43\) 7.21110 1.09968 0.549841 − 0.835269i \(-0.314688\pi\)
0.549841 + 0.835269i \(0.314688\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −3.00000 −0.437595 −0.218797 − 0.975770i \(-0.570213\pi\)
−0.218797 + 0.975770i \(0.570213\pi\)
\(48\) 0 0
\(49\) −6.51388 −0.930554
\(50\) 0 0
\(51\) −15.9083 −2.22761
\(52\) 0 0
\(53\) 1.30278 0.178950 0.0894750 − 0.995989i \(-0.471481\pi\)
0.0894750 + 0.995989i \(0.471481\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 2.30278 0.305010
\(58\) 0 0
\(59\) 14.2111 1.85013 0.925064 − 0.379811i \(-0.124011\pi\)
0.925064 + 0.379811i \(0.124011\pi\)
\(60\) 0 0
\(61\) −7.90833 −1.01256 −0.506279 − 0.862370i \(-0.668979\pi\)
−0.506279 + 0.862370i \(0.668979\pi\)
\(62\) 0 0
\(63\) 1.60555 0.202280
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −4.00000 −0.488678 −0.244339 − 0.969690i \(-0.578571\pi\)
−0.244339 + 0.969690i \(0.578571\pi\)
\(68\) 0 0
\(69\) −16.8167 −2.02449
\(70\) 0 0
\(71\) 2.60555 0.309222 0.154611 − 0.987975i \(-0.450588\pi\)
0.154611 + 0.987975i \(0.450588\pi\)
\(72\) 0 0
\(73\) 7.90833 0.925600 0.462800 − 0.886463i \(-0.346845\pi\)
0.462800 + 0.886463i \(0.346845\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0.697224 0.0794561
\(78\) 0 0
\(79\) 10.9083 1.22728 0.613641 − 0.789585i \(-0.289704\pi\)
0.613641 + 0.789585i \(0.289704\pi\)
\(80\) 0 0
\(81\) −10.6056 −1.17839
\(82\) 0 0
\(83\) −3.51388 −0.385698 −0.192849 − 0.981228i \(-0.561773\pi\)
−0.192849 + 0.981228i \(0.561773\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 2.09167 0.224251
\(88\) 0 0
\(89\) 1.69722 0.179905 0.0899527 − 0.995946i \(-0.471328\pi\)
0.0899527 + 0.995946i \(0.471328\pi\)
\(90\) 0 0
\(91\) −3.48612 −0.365445
\(92\) 0 0
\(93\) −23.5139 −2.43828
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −15.3028 −1.55376 −0.776881 − 0.629648i \(-0.783199\pi\)
−0.776881 + 0.629648i \(0.783199\pi\)
\(98\) 0 0
\(99\) 2.30278 0.231438
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 4400.2.a.bs.1.2 2
4.3 odd 2 275.2.a.e.1.2 ✓ 2
5.2 odd 4 4400.2.b.y.4049.1 4
5.3 odd 4 4400.2.b.y.4049.4 4
5.4 even 2 4400.2.a.bh.1.1 2
12.11 even 2 2475.2.a.t.1.1 2
20.3 even 4 275.2.b.c.199.2 4
20.7 even 4 275.2.b.c.199.3 4
20.19 odd 2 275.2.a.f.1.1 yes 2
44.43 even 2 3025.2.a.n.1.1 2
60.23 odd 4 2475.2.c.k.199.3 4
60.47 odd 4 2475.2.c.k.199.2 4
60.59 even 2 2475.2.a.o.1.2 2
220.219 even 2 3025.2.a.h.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
275.2.a.e.1.2 ✓ 2 4.3 odd 2
275.2.a.f.1.1 yes 2 20.19 odd 2
275.2.b.c.199.2 4 20.3 even 4
275.2.b.c.199.3 4 20.7 even 4
2475.2.a.o.1.2 2 60.59 even 2
2475.2.a.t.1.1 2 12.11 even 2
2475.2.c.k.199.2 4 60.47 odd 4
2475.2.c.k.199.3 4 60.23 odd 4
3025.2.a.h.1.2 2 220.219 even 2
3025.2.a.n.1.1 2 44.43 even 2
4400.2.a.bh.1.1 2 5.4 even 2
4400.2.a.bs.1.2 2 1.1 even 1 trivial
4400.2.b.y.4049.1 4 5.2 odd 4
4400.2.b.y.4049.4 4 5.3 odd 4