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,-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: \(35.1341768894\)
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 4049.4
Root \(2.56155i\) of defining polynomial
Character \(\chi\) \(=\) 4400.4049
Dual form 4400.2.b.w.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)
 
\(f(q)\) \(=\) \(q+2.56155i q^{3} -4.56155i q^{7} -3.56155 q^{9} +1.00000 q^{11} +1.12311i q^{13} -7.68466i q^{17} +1.43845 q^{19} +11.6847 q^{21} +1.12311i q^{23} -1.43845i q^{27} -8.56155 q^{29} +1.43845 q^{31} +2.56155i q^{33} +7.43845i q^{37} -2.87689 q^{39} -12.2462 q^{41} -3.12311i q^{43} +11.3693i q^{47} -13.8078 q^{49} +19.6847 q^{51} -9.68466i q^{53} +3.68466i q^{57} +1.12311 q^{59} -12.5616 q^{61} +16.2462i q^{63} -2.87689 q^{69} -3.68466 q^{71} -1.12311i q^{73} -4.56155i q^{77} -11.3693 q^{79} -7.00000 q^{81} -6.00000i q^{83} -21.9309i q^{87} -9.68466 q^{89} +5.12311 q^{91} +3.68466i q^{93} -4.87689i q^{97} -3.56155 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 6 q^{9} + 4 q^{11} + 14 q^{19} + 22 q^{21} - 26 q^{29} + 14 q^{31} - 28 q^{39} - 16 q^{41} - 14 q^{49} + 54 q^{51} - 12 q^{59} - 42 q^{61} - 28 q^{69} + 10 q^{71} + 4 q^{79} - 28 q^{81} - 14 q^{89}+ \cdots - 6 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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
\(3\) 2.56155i 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 \(0.0497784\pi\)
−0.987797 + 0.155747i \(0.950222\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) − 7.68466i − 1.86380i −0.362711 − 0.931902i \(-0.618149\pi\)
0.362711 − 0.931902i \(-0.381851\pi\)
\(18\) 0 0
\(19\) 1.43845 0.330002 0.165001 − 0.986293i \(-0.447237\pi\)
0.165001 + 0.986293i \(0.447237\pi\)
\(20\) 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 \(-0.458767\pi\)
0.129176 + 0.991622i \(0.458767\pi\)
\(32\) 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 \(0.209408\pi\)
−0.791293 + 0.611437i \(0.790592\pi\)
\(38\) 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 \(-0.768370\pi\)
0.746714 − 0.665145i \(-0.231630\pi\)
\(54\) 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 \(-0.476708\pi\)
0.0731079 + 0.997324i \(0.476708\pi\)
\(60\) 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 \(-0.570163\pi\)
−0.218644 + 0.975805i \(0.570163\pi\)
\(72\) 0 0
\(73\) − 1.12311i − 0.131450i −0.997838 − 0.0657248i \(-0.979064\pi\)
0.997838 − 0.0657248i \(-0.0209359\pi\)
\(74\) 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 \(-0.720889\pi\)
−0.639574 + 0.768729i \(0.720889\pi\)
\(80\) 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 \(-0.920362\pi\)
0.968866 − 0.247587i \(-0.0796375\pi\)
\(98\) 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 4400.2.b.w.4049.4 4
4.3 odd 2 2200.2.b.f.1849.1 4
5.2 odd 4 4400.2.a.bt.1.2 2
5.3 odd 4 880.2.a.k.1.1 2
5.4 even 2 inner 4400.2.b.w.4049.1 4
15.8 even 4 7920.2.a.by.1.1 2
20.3 even 4 440.2.a.g.1.2 ✓ 2
20.7 even 4 2200.2.a.l.1.1 2
20.19 odd 2 2200.2.b.f.1849.4 4
40.3 even 4 3520.2.a.bm.1.1 2
40.13 odd 4 3520.2.a.br.1.2 2
55.43 even 4 9680.2.a.bm.1.1 2
60.23 odd 4 3960.2.a.bf.1.2 2
220.43 odd 4 4840.2.a.m.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
440.2.a.g.1.2 ✓ 2 20.3 even 4
880.2.a.k.1.1 2 5.3 odd 4
2200.2.a.l.1.1 2 20.7 even 4
2200.2.b.f.1849.1 4 4.3 odd 2
2200.2.b.f.1849.4 4 20.19 odd 2
3520.2.a.bm.1.1 2 40.3 even 4
3520.2.a.br.1.2 2 40.13 odd 4
3960.2.a.bf.1.2 2 60.23 odd 4
4400.2.a.bt.1.2 2 5.2 odd 4
4400.2.b.w.4049.1 4 5.4 even 2 inner
4400.2.b.w.4049.4 4 1.1 even 1 trivial
4840.2.a.m.1.2 2 220.43 odd 4
7920.2.a.by.1.1 2 15.8 even 4
9680.2.a.bm.1.1 2 55.43 even 4