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(4049,4400)] 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("4400.4049"); 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, 1, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 4400 = 2^{4} \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4400.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,0,0,0,0,0,-2,0,4,0,0,0,0,0,0,0,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(19)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(35.1341768894\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{13})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 7x^{2} + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 275)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 4049.4
Root \(2.30278i\) of defining polynomial
Character \(\chi\) \(=\) 4400.4049
Dual form 4400.2.b.y.4049.1

$q$-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)
 
\(f(q)\) \(=\) \(q+2.30278i q^{3} -0.697224i q^{7} -2.30278 q^{9} +1.00000 q^{11} -5.00000i q^{13} +6.90833i q^{17} -1.00000 q^{19} +1.60555 q^{21} -7.30278i q^{23} +1.60555i q^{27} -0.908327 q^{29} -10.2111 q^{31} +2.30278i q^{33} +2.39445i q^{37} +11.5139 q^{39} -5.60555 q^{41} +7.21110i q^{43} +3.00000i q^{47} +6.51388 q^{49} -15.9083 q^{51} +1.30278i q^{53} -2.30278i q^{57} -14.2111 q^{59} -7.90833 q^{61} +1.60555i q^{63} +4.00000i q^{67} +16.8167 q^{69} +2.60555 q^{71} +7.90833i q^{73} -0.697224i q^{77} -10.9083 q^{79} -10.6056 q^{81} -3.51388i q^{83} -2.09167i q^{87} -1.69722 q^{89} -3.48612 q^{91} -23.5139i q^{93} +15.3028i q^{97} -2.30278 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{9} + 4 q^{11} - 4 q^{19} - 8 q^{21} + 18 q^{29} - 12 q^{31} + 10 q^{39} - 8 q^{41} - 10 q^{49} - 42 q^{51} - 28 q^{59} - 10 q^{61} + 24 q^{69} - 4 q^{71} - 22 q^{79} - 28 q^{81} - 14 q^{89} - 50 q^{91}+ \cdots - 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4400\mathbb{Z}\right)^\times\).

\(n\) \(177\) \(1201\) \(2751\) \(3301\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(1\)

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
\(3\) 2.30278i 1.32951i 0.747062 + 0.664754i \(0.231464\pi\)
−0.747062 + 0.664754i \(0.768536\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) − 0.697224i − 0.263526i −0.991281 − 0.131763i \(-0.957936\pi\)
0.991281 − 0.131763i \(-0.0420638\pi\)
\(8\) 0 0
\(9\) −2.30278 −0.767592
\(10\) 0 0
\(11\) 1.00000 0.301511
\(12\) 0 0
\(13\) − 5.00000i − 1.38675i −0.720577 − 0.693375i \(-0.756123\pi\)
0.720577 − 0.693375i \(-0.243877\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 6.90833i 1.67552i 0.546042 + 0.837758i \(0.316134\pi\)
−0.546042 + 0.837758i \(0.683866\pi\)
\(18\) 0 0
\(19\) −1.00000 −0.229416 −0.114708 − 0.993399i \(-0.536593\pi\)
−0.114708 + 0.993399i \(0.536593\pi\)
\(20\) 0 0
\(21\) 1.60555 0.350360
\(22\) 0 0
\(23\) − 7.30278i − 1.52273i −0.648321 − 0.761367i \(-0.724529\pi\)
0.648321 − 0.761367i \(-0.275471\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 1.60555i 0.308988i
\(28\) 0 0
\(29\) −0.908327 −0.168672 −0.0843360 − 0.996437i \(-0.526877\pi\)
−0.0843360 + 0.996437i \(0.526877\pi\)
\(30\) 0 0
\(31\) −10.2111 −1.83397 −0.916984 − 0.398924i \(-0.869384\pi\)
−0.916984 + 0.398924i \(0.869384\pi\)
\(32\) 0 0
\(33\) 2.30278i 0.400862i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 2.39445i 0.393645i 0.980439 + 0.196822i \(0.0630623\pi\)
−0.980439 + 0.196822i \(0.936938\pi\)
\(38\) 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
\(43\) 7.21110i 1.09968i 0.835269 + 0.549841i \(0.185312\pi\)
−0.835269 + 0.549841i \(0.814688\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 3.00000i 0.437595i 0.975770 + 0.218797i \(0.0702134\pi\)
−0.975770 + 0.218797i \(0.929787\pi\)
\(48\) 0 0
\(49\) 6.51388 0.930554
\(50\) 0 0
\(51\) −15.9083 −2.22761
\(52\) 0 0
\(53\) 1.30278i 0.178950i 0.995989 + 0.0894750i \(0.0285189\pi\)
−0.995989 + 0.0894750i \(0.971481\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) − 2.30278i − 0.305010i
\(58\) 0 0
\(59\) −14.2111 −1.85013 −0.925064 − 0.379811i \(-0.875989\pi\)
−0.925064 + 0.379811i \(0.875989\pi\)
\(60\) 0 0
\(61\) −7.90833 −1.01256 −0.506279 − 0.862370i \(-0.668979\pi\)
−0.506279 + 0.862370i \(0.668979\pi\)
\(62\) 0 0
\(63\) 1.60555i 0.202280i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 4.00000i 0.488678i 0.969690 + 0.244339i \(0.0785709\pi\)
−0.969690 + 0.244339i \(0.921429\pi\)
\(68\) 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
\(73\) 7.90833i 0.925600i 0.886463 + 0.462800i \(0.153155\pi\)
−0.886463 + 0.462800i \(0.846845\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) − 0.697224i − 0.0794561i
\(78\) 0 0
\(79\) −10.9083 −1.22728 −0.613641 − 0.789585i \(-0.710296\pi\)
−0.613641 + 0.789585i \(0.710296\pi\)
\(80\) 0 0
\(81\) −10.6056 −1.17839
\(82\) 0 0
\(83\) − 3.51388i − 0.385698i −0.981228 − 0.192849i \(-0.938227\pi\)
0.981228 − 0.192849i \(-0.0617728\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) − 2.09167i − 0.224251i
\(88\) 0 0
\(89\) −1.69722 −0.179905 −0.0899527 − 0.995946i \(-0.528672\pi\)
−0.0899527 + 0.995946i \(0.528672\pi\)
\(90\) 0 0
\(91\) −3.48612 −0.365445
\(92\) 0 0
\(93\) − 23.5139i − 2.43828i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 15.3028i 1.55376i 0.629648 + 0.776881i \(0.283199\pi\)
−0.629648 + 0.776881i \(0.716801\pi\)
\(98\) 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.b.y.4049.4 4
4.3 odd 2 275.2.b.c.199.2 4
5.2 odd 4 4400.2.a.bs.1.2 2
5.3 odd 4 4400.2.a.bh.1.1 2
5.4 even 2 inner 4400.2.b.y.4049.1 4
12.11 even 2 2475.2.c.k.199.3 4
20.3 even 4 275.2.a.f.1.1 yes 2
20.7 even 4 275.2.a.e.1.2 ✓ 2
20.19 odd 2 275.2.b.c.199.3 4
60.23 odd 4 2475.2.a.o.1.2 2
60.47 odd 4 2475.2.a.t.1.1 2
60.59 even 2 2475.2.c.k.199.2 4
220.43 odd 4 3025.2.a.h.1.2 2
220.87 odd 4 3025.2.a.n.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
275.2.a.e.1.2 ✓ 2 20.7 even 4
275.2.a.f.1.1 yes 2 20.3 even 4
275.2.b.c.199.2 4 4.3 odd 2
275.2.b.c.199.3 4 20.19 odd 2
2475.2.a.o.1.2 2 60.23 odd 4
2475.2.a.t.1.1 2 60.47 odd 4
2475.2.c.k.199.2 4 60.59 even 2
2475.2.c.k.199.3 4 12.11 even 2
3025.2.a.h.1.2 2 220.43 odd 4
3025.2.a.n.1.1 2 220.87 odd 4
4400.2.a.bh.1.1 2 5.3 odd 4
4400.2.a.bs.1.2 2 5.2 odd 4
4400.2.b.y.4049.1 4 5.4 even 2 inner
4400.2.b.y.4049.4 4 1.1 even 1 trivial