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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1100,2,Mod(749,1100)] 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("1100.749"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1100, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1100 = 2^{2} \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1100.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,-10,0,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(8.78354422234\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{21})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 11x^{2} + 25 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 749.1
Root \(-2.79129i\) of defining polynomial
Character \(\chi\) \(=\) 1100.749
Dual form 1100.2.b.d.749.4

$q$-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)
 
\(f(q)\) \(=\) \(q-2.79129i q^{3} +0.208712i q^{7} -4.79129 q^{9} -1.00000 q^{11} -1.00000i q^{13} -0.791288i q^{17} -6.58258 q^{19} +0.582576 q^{21} -3.79129i q^{23} +5.00000i q^{27} -6.79129 q^{29} -8.58258 q^{31} +2.79129i q^{33} +2.58258i q^{37} -2.79129 q^{39} -1.41742 q^{41} -10.0000i q^{43} +1.41742i q^{47} +6.95644 q^{49} -2.20871 q^{51} +11.3739i q^{53} +18.3739i q^{57} +10.5826 q^{59} +4.20871 q^{61} -1.00000i q^{63} +4.00000i q^{67} -10.5826 q^{69} -10.7477 q^{71} -7.79129i q^{73} -0.208712i q^{77} +15.5390 q^{79} -0.417424 q^{81} -9.95644i q^{83} +18.9564i q^{87} +0.791288 q^{89} +0.208712 q^{91} +23.9564i q^{93} +6.20871i q^{97} +4.79129 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 10 q^{9} - 4 q^{11} - 8 q^{19} - 16 q^{21} - 18 q^{29} - 16 q^{31} - 2 q^{39} - 24 q^{41} - 18 q^{49} - 18 q^{51} + 24 q^{59} + 26 q^{61} - 24 q^{69} + 12 q^{71} - 2 q^{79} - 20 q^{81} - 6 q^{89}+ \cdots + 10 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1100\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(177\) \(551\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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
\(3\) − 2.79129i − 1.61155i −0.592221 − 0.805775i \(-0.701749\pi\)
0.592221 − 0.805775i \(-0.298251\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 0.208712i 0.0788858i 0.999222 + 0.0394429i \(0.0125583\pi\)
−0.999222 + 0.0394429i \(0.987442\pi\)
\(8\) 0 0
\(9\) −4.79129 −1.59710
\(10\) 0 0
\(11\) −1.00000 −0.301511
\(12\) 0 0
\(13\) − 1.00000i − 0.277350i −0.990338 − 0.138675i \(-0.955716\pi\)
0.990338 − 0.138675i \(-0.0442844\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) − 0.791288i − 0.191915i −0.995385 − 0.0959577i \(-0.969409\pi\)
0.995385 − 0.0959577i \(-0.0305914\pi\)
\(18\) 0 0
\(19\) −6.58258 −1.51015 −0.755073 − 0.655640i \(-0.772399\pi\)
−0.755073 + 0.655640i \(0.772399\pi\)
\(20\) 0 0
\(21\) 0.582576 0.127128
\(22\) 0 0
\(23\) − 3.79129i − 0.790538i −0.918565 − 0.395269i \(-0.870651\pi\)
0.918565 − 0.395269i \(-0.129349\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 5.00000i 0.962250i
\(28\) 0 0
\(29\) −6.79129 −1.26111 −0.630555 − 0.776144i \(-0.717173\pi\)
−0.630555 + 0.776144i \(0.717173\pi\)
\(30\) 0 0
\(31\) −8.58258 −1.54148 −0.770738 − 0.637152i \(-0.780112\pi\)
−0.770738 + 0.637152i \(0.780112\pi\)
\(32\) 0 0
\(33\) 2.79129i 0.485901i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 2.58258i 0.424573i 0.977207 + 0.212286i \(0.0680910\pi\)
−0.977207 + 0.212286i \(0.931909\pi\)
\(38\) 0 0
\(39\) −2.79129 −0.446964
\(40\) 0 0
\(41\) −1.41742 −0.221364 −0.110682 − 0.993856i \(-0.535304\pi\)
−0.110682 + 0.993856i \(0.535304\pi\)
\(42\) 0 0
\(43\) − 10.0000i − 1.52499i −0.646997 − 0.762493i \(-0.723975\pi\)
0.646997 − 0.762493i \(-0.276025\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 1.41742i 0.206753i 0.994642 + 0.103376i \(0.0329646\pi\)
−0.994642 + 0.103376i \(0.967035\pi\)
\(48\) 0 0
\(49\) 6.95644 0.993777
\(50\) 0 0
\(51\) −2.20871 −0.309282
\(52\) 0 0
\(53\) 11.3739i 1.56232i 0.624331 + 0.781160i \(0.285372\pi\)
−0.624331 + 0.781160i \(0.714628\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 18.3739i 2.43368i
\(58\) 0 0
\(59\) 10.5826 1.37773 0.688867 − 0.724888i \(-0.258108\pi\)
0.688867 + 0.724888i \(0.258108\pi\)
\(60\) 0 0
\(61\) 4.20871 0.538870 0.269435 − 0.963019i \(-0.413163\pi\)
0.269435 + 0.963019i \(0.413163\pi\)
\(62\) 0 0
\(63\) − 1.00000i − 0.125988i
\(64\) 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
\(69\) −10.5826 −1.27399
\(70\) 0 0
\(71\) −10.7477 −1.27552 −0.637760 − 0.770235i \(-0.720139\pi\)
−0.637760 + 0.770235i \(0.720139\pi\)
\(72\) 0 0
\(73\) − 7.79129i − 0.911901i −0.890005 − 0.455951i \(-0.849299\pi\)
0.890005 − 0.455951i \(-0.150701\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) − 0.208712i − 0.0237850i
\(78\) 0 0
\(79\) 15.5390 1.74828 0.874138 − 0.485678i \(-0.161427\pi\)
0.874138 + 0.485678i \(0.161427\pi\)
\(80\) 0 0
\(81\) −0.417424 −0.0463805
\(82\) 0 0
\(83\) − 9.95644i − 1.09286i −0.837504 − 0.546431i \(-0.815986\pi\)
0.837504 − 0.546431i \(-0.184014\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 18.9564i 2.03234i
\(88\) 0 0
\(89\) 0.791288 0.0838763 0.0419382 − 0.999120i \(-0.486647\pi\)
0.0419382 + 0.999120i \(0.486647\pi\)
\(90\) 0 0
\(91\) 0.208712 0.0218790
\(92\) 0 0
\(93\) 23.9564i 2.48417i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 6.20871i 0.630399i 0.949025 + 0.315200i \(0.102071\pi\)
−0.949025 + 0.315200i \(0.897929\pi\)
\(98\) 0 0
\(99\) 4.79129 0.481543
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 1100.2.b.d.749.1 4
3.2 odd 2 9900.2.c.x.5149.3 4
4.3 odd 2 4400.2.b.s.4049.4 4
5.2 odd 4 1100.2.a.g.1.1 ✓ 2
5.3 odd 4 1100.2.a.h.1.2 yes 2
5.4 even 2 inner 1100.2.b.d.749.4 4
15.2 even 4 9900.2.a.bh.1.2 2
15.8 even 4 9900.2.a.bz.1.1 2
15.14 odd 2 9900.2.c.x.5149.2 4
20.3 even 4 4400.2.a.bi.1.1 2
20.7 even 4 4400.2.a.bu.1.2 2
20.19 odd 2 4400.2.b.s.4049.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1100.2.a.g.1.1 ✓ 2 5.2 odd 4
1100.2.a.h.1.2 yes 2 5.3 odd 4
1100.2.b.d.749.1 4 1.1 even 1 trivial
1100.2.b.d.749.4 4 5.4 even 2 inner
4400.2.a.bi.1.1 2 20.3 even 4
4400.2.a.bu.1.2 2 20.7 even 4
4400.2.b.s.4049.1 4 20.19 odd 2
4400.2.b.s.4049.4 4 4.3 odd 2
9900.2.a.bh.1.2 2 15.2 even 4
9900.2.a.bz.1.1 2 15.8 even 4
9900.2.c.x.5149.2 4 15.14 odd 2
9900.2.c.x.5149.3 4 3.2 odd 2