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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [2200,2,Mod(1849,2200)] 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("2200.1849"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2200, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 2200 = 2^{3} \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2200.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,-6,0,-4,0,0,0,0,0,0,0,-14] 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: \(17.5670884447\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{17})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 9x^{2} + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 440)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1849.4
Root \(2.56155i\) of defining polynomial
Character \(\chi\) \(=\) 2200.1849
Dual form 2200.2.b.f.1849.1

$q$-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)
 
\(f(q)\) \(=\) \(q+2.56155i q^{3} -4.56155i q^{7} -3.56155 q^{9} -1.00000 q^{11} -1.12311i q^{13} +7.68466i q^{17} -1.43845 q^{19} +11.6847 q^{21} +1.12311i q^{23} -1.43845i q^{27} -8.56155 q^{29} -1.43845 q^{31} -2.56155i q^{33} -7.43845i q^{37} +2.87689 q^{39} -12.2462 q^{41} -3.12311i q^{43} +11.3693i q^{47} -13.8078 q^{49} -19.6847 q^{51} +9.68466i q^{53} -3.68466i q^{57} -1.12311 q^{59} -12.5616 q^{61} +16.2462i q^{63} -2.87689 q^{69} +3.68466 q^{71} +1.12311i q^{73} +4.56155i q^{77} +11.3693 q^{79} -7.00000 q^{81} -6.00000i q^{83} -21.9309i q^{87} -9.68466 q^{89} -5.12311 q^{91} -3.68466i q^{93} +4.87689i q^{97} +3.56155 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 6 q^{9} - 4 q^{11} - 14 q^{19} + 22 q^{21} - 26 q^{29} - 14 q^{31} + 28 q^{39} - 16 q^{41} - 14 q^{49} - 54 q^{51} + 12 q^{59} - 42 q^{61} - 28 q^{69} - 10 q^{71} - 4 q^{79} - 28 q^{81} - 14 q^{89}+ \cdots + 6 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2200\mathbb{Z}\right)^\times\).

\(n\) \(177\) \(551\) \(1101\) \(1201\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(1\)

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
\(3\) 2.56155i 1.47891i 0.673204 + 0.739457i \(0.264917\pi\)
−0.673204 + 0.739457i \(0.735083\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) − 4.56155i − 1.72410i −0.506819 − 0.862052i \(-0.669179\pi\)
0.506819 − 0.862052i \(-0.330821\pi\)
\(8\) 0 0
\(9\) −3.56155 −1.18718
\(10\) 0 0
\(11\) −1.00000 −0.301511
\(12\) 0 0
\(13\) − 1.12311i − 0.311493i −0.987797 − 0.155747i \(-0.950222\pi\)
0.987797 − 0.155747i \(-0.0497784\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 7.68466i 1.86380i 0.362711 + 0.931902i \(0.381851\pi\)
−0.362711 + 0.931902i \(0.618149\pi\)
\(18\) 0 0
\(19\) −1.43845 −0.330002 −0.165001 − 0.986293i \(-0.552763\pi\)
−0.165001 + 0.986293i \(0.552763\pi\)
\(20\) 0 0
\(21\) 11.6847 2.54980
\(22\) 0 0
\(23\) 1.12311i 0.234184i 0.993121 + 0.117092i \(0.0373572\pi\)
−0.993121 + 0.117092i \(0.962643\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) − 1.43845i − 0.276829i
\(28\) 0 0
\(29\) −8.56155 −1.58984 −0.794920 − 0.606714i \(-0.792487\pi\)
−0.794920 + 0.606714i \(0.792487\pi\)
\(30\) 0 0
\(31\) −1.43845 −0.258353 −0.129176 − 0.991622i \(-0.541233\pi\)
−0.129176 + 0.991622i \(0.541233\pi\)
\(32\) 0 0
\(33\) − 2.56155i − 0.445909i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) − 7.43845i − 1.22287i −0.791293 − 0.611437i \(-0.790592\pi\)
0.791293 − 0.611437i \(-0.209408\pi\)
\(38\) 0 0
\(39\) 2.87689 0.460672
\(40\) 0 0
\(41\) −12.2462 −1.91254 −0.956268 − 0.292490i \(-0.905516\pi\)
−0.956268 + 0.292490i \(0.905516\pi\)
\(42\) 0 0
\(43\) − 3.12311i − 0.476269i −0.971232 − 0.238135i \(-0.923464\pi\)
0.971232 − 0.238135i \(-0.0765359\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 11.3693i 1.65839i 0.558963 + 0.829193i \(0.311199\pi\)
−0.558963 + 0.829193i \(0.688801\pi\)
\(48\) 0 0
\(49\) −13.8078 −1.97254
\(50\) 0 0
\(51\) −19.6847 −2.75640
\(52\) 0 0
\(53\) 9.68466i 1.33029i 0.746714 + 0.665145i \(0.231630\pi\)
−0.746714 + 0.665145i \(0.768370\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) − 3.68466i − 0.488045i
\(58\) 0 0
\(59\) −1.12311 −0.146216 −0.0731079 − 0.997324i \(-0.523292\pi\)
−0.0731079 + 0.997324i \(0.523292\pi\)
\(60\) 0 0
\(61\) −12.5616 −1.60834 −0.804171 − 0.594398i \(-0.797390\pi\)
−0.804171 + 0.594398i \(0.797390\pi\)
\(62\) 0 0
\(63\) 16.2462i 2.04683i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(68\) 0 0
\(69\) −2.87689 −0.346337
\(70\) 0 0
\(71\) 3.68466 0.437289 0.218644 − 0.975805i \(-0.429837\pi\)
0.218644 + 0.975805i \(0.429837\pi\)
\(72\) 0 0
\(73\) 1.12311i 0.131450i 0.997838 + 0.0657248i \(0.0209359\pi\)
−0.997838 + 0.0657248i \(0.979064\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 4.56155i 0.519837i
\(78\) 0 0
\(79\) 11.3693 1.27915 0.639574 − 0.768729i \(-0.279111\pi\)
0.639574 + 0.768729i \(0.279111\pi\)
\(80\) 0 0
\(81\) −7.00000 −0.777778
\(82\) 0 0
\(83\) − 6.00000i − 0.658586i −0.944228 − 0.329293i \(-0.893190\pi\)
0.944228 − 0.329293i \(-0.106810\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) − 21.9309i − 2.35124i
\(88\) 0 0
\(89\) −9.68466 −1.02657 −0.513286 − 0.858218i \(-0.671572\pi\)
−0.513286 + 0.858218i \(0.671572\pi\)
\(90\) 0 0
\(91\) −5.12311 −0.537047
\(92\) 0 0
\(93\) − 3.68466i − 0.382081i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 4.87689i 0.495174i 0.968866 + 0.247587i \(0.0796375\pi\)
−0.968866 + 0.247587i \(0.920362\pi\)
\(98\) 0 0
\(99\) 3.56155 0.357950
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 2200.2.b.f.1849.4 4
4.3 odd 2 4400.2.b.w.4049.1 4
5.2 odd 4 440.2.a.g.1.2 ✓ 2
5.3 odd 4 2200.2.a.l.1.1 2
5.4 even 2 inner 2200.2.b.f.1849.1 4
15.2 even 4 3960.2.a.bf.1.2 2
20.3 even 4 4400.2.a.bt.1.2 2
20.7 even 4 880.2.a.k.1.1 2
20.19 odd 2 4400.2.b.w.4049.4 4
40.27 even 4 3520.2.a.br.1.2 2
40.37 odd 4 3520.2.a.bm.1.1 2
55.32 even 4 4840.2.a.m.1.2 2
60.47 odd 4 7920.2.a.by.1.1 2
220.87 odd 4 9680.2.a.bm.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
440.2.a.g.1.2 ✓ 2 5.2 odd 4
880.2.a.k.1.1 2 20.7 even 4
2200.2.a.l.1.1 2 5.3 odd 4
2200.2.b.f.1849.1 4 5.4 even 2 inner
2200.2.b.f.1849.4 4 1.1 even 1 trivial
3520.2.a.bm.1.1 2 40.37 odd 4
3520.2.a.br.1.2 2 40.27 even 4
3960.2.a.bf.1.2 2 15.2 even 4
4400.2.a.bt.1.2 2 20.3 even 4
4400.2.b.w.4049.1 4 4.3 odd 2
4400.2.b.w.4049.4 4 20.19 odd 2
4840.2.a.m.1.2 2 55.32 even 4
7920.2.a.by.1.1 2 60.47 odd 4
9680.2.a.bm.1.1 2 220.87 odd 4