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Properties

Label 4400.2.b.w.4049.3
Level $4400$
Weight $2$
Character 4400.4049
Analytic conductor $35.134$
Analytic rank $0$
Dimension $4$
Inner twists $2$

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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 4049.3
Root \(1.56155i\) of defining polynomial
Character \(\chi\) \(=\) 4400.4049
Dual form 4400.2.b.w.4049.2

$q$-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)
 
\(f(q)\) \(=\) \(q+1.56155i q^{3} +0.438447i q^{7} +0.561553 q^{9} +1.00000 q^{11} +7.12311i q^{13} -4.68466i q^{17} +5.56155 q^{19} -0.684658 q^{21} +7.12311i q^{23} +5.56155i q^{27} -4.43845 q^{29} +5.56155 q^{31} +1.56155i q^{33} -11.5616i q^{37} -11.1231 q^{39} +4.24621 q^{41} -5.12311i q^{43} +13.3693i q^{47} +6.80776 q^{49} +7.31534 q^{51} -2.68466i q^{53} +8.68466i q^{57} -7.12311 q^{59} -8.43845 q^{61} +0.246211i q^{63} -11.1231 q^{69} +8.68466 q^{71} -7.12311i q^{73} +0.438447i q^{77} +13.3693 q^{79} -7.00000 q^{81} +6.00000i q^{83} -6.93087i q^{87} +2.68466 q^{89} -3.12311 q^{91} +8.68466i q^{93} +13.1231i q^{97} +0.561553 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 6 q^{9} + 4 q^{11} + 14 q^{19} + 22 q^{21} - 26 q^{29} + 14 q^{31} - 28 q^{39} - 16 q^{41} - 14 q^{49} + 54 q^{51} - 12 q^{59} - 42 q^{61} - 28 q^{69} + 10 q^{71} + 4 q^{79} - 28 q^{81} - 14 q^{89}+ \cdots - 6 q^{99}+O(q^{100}) \) 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
\(3\) 1.56155i 0.901563i 0.892634 + 0.450781i \(0.148855\pi\)
−0.892634 + 0.450781i \(0.851145\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 0.438447i 0.165717i 0.996561 + 0.0828587i \(0.0264050\pi\)
−0.996561 + 0.0828587i \(0.973595\pi\)
\(8\) 0 0
\(9\) 0.561553 0.187184
\(10\) 0 0
\(11\) 1.00000 0.301511
\(12\) 0 0
\(13\) 7.12311i 1.97559i 0.155747 + 0.987797i \(0.450222\pi\)
−0.155747 + 0.987797i \(0.549778\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) − 4.68466i − 1.13620i −0.822961 − 0.568098i \(-0.807679\pi\)
0.822961 − 0.568098i \(-0.192321\pi\)
\(18\) 0 0
\(19\) 5.56155 1.27591 0.637954 − 0.770075i \(-0.279781\pi\)
0.637954 + 0.770075i \(0.279781\pi\)
\(20\) 0 0
\(21\) −0.684658 −0.149405
\(22\) 0 0
\(23\) 7.12311i 1.48527i 0.669696 + 0.742635i \(0.266424\pi\)
−0.669696 + 0.742635i \(0.733576\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 5.56155i 1.07032i
\(28\) 0 0
\(29\) −4.43845 −0.824199 −0.412099 − 0.911139i \(-0.635204\pi\)
−0.412099 + 0.911139i \(0.635204\pi\)
\(30\) 0 0
\(31\) 5.56155 0.998884 0.499442 − 0.866347i \(-0.333538\pi\)
0.499442 + 0.866347i \(0.333538\pi\)
\(32\) 0 0
\(33\) 1.56155i 0.271831i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) − 11.5616i − 1.90071i −0.311171 − 0.950354i \(-0.600721\pi\)
0.311171 − 0.950354i \(-0.399279\pi\)
\(38\) 0 0
\(39\) −11.1231 −1.78112
\(40\) 0 0
\(41\) 4.24621 0.663147 0.331573 − 0.943429i \(-0.392421\pi\)
0.331573 + 0.943429i \(0.392421\pi\)
\(42\) 0 0
\(43\) − 5.12311i − 0.781266i −0.920546 − 0.390633i \(-0.872256\pi\)
0.920546 − 0.390633i \(-0.127744\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 13.3693i 1.95012i 0.221952 + 0.975058i \(0.428757\pi\)
−0.221952 + 0.975058i \(0.571243\pi\)
\(48\) 0 0
\(49\) 6.80776 0.972538
\(50\) 0 0
\(51\) 7.31534 1.02435
\(52\) 0 0
\(53\) − 2.68466i − 0.368766i −0.982854 − 0.184383i \(-0.940971\pi\)
0.982854 − 0.184383i \(-0.0590287\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 8.68466i 1.15031i
\(58\) 0 0
\(59\) −7.12311 −0.927349 −0.463675 − 0.886006i \(-0.653469\pi\)
−0.463675 + 0.886006i \(0.653469\pi\)
\(60\) 0 0
\(61\) −8.43845 −1.08043 −0.540216 − 0.841526i \(-0.681658\pi\)
−0.540216 + 0.841526i \(0.681658\pi\)
\(62\) 0 0
\(63\) 0.246211i 0.0310197i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(68\) 0 0
\(69\) −11.1231 −1.33906
\(70\) 0 0
\(71\) 8.68466 1.03068 0.515340 − 0.856986i \(-0.327666\pi\)
0.515340 + 0.856986i \(0.327666\pi\)
\(72\) 0 0
\(73\) − 7.12311i − 0.833696i −0.908976 − 0.416848i \(-0.863135\pi\)
0.908976 − 0.416848i \(-0.136865\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0.438447i 0.0499657i
\(78\) 0 0
\(79\) 13.3693 1.50417 0.752083 − 0.659069i \(-0.229049\pi\)
0.752083 + 0.659069i \(0.229049\pi\)
\(80\) 0 0
\(81\) −7.00000 −0.777778
\(82\) 0 0
\(83\) 6.00000i 0.658586i 0.944228 + 0.329293i \(0.106810\pi\)
−0.944228 + 0.329293i \(0.893190\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) − 6.93087i − 0.743067i
\(88\) 0 0
\(89\) 2.68466 0.284573 0.142287 − 0.989825i \(-0.454555\pi\)
0.142287 + 0.989825i \(0.454555\pi\)
\(90\) 0 0
\(91\) −3.12311 −0.327390
\(92\) 0 0
\(93\) 8.68466i 0.900557i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 13.1231i 1.33245i 0.745751 + 0.666225i \(0.232091\pi\)
−0.745751 + 0.666225i \(0.767909\pi\)
\(98\) 0 0
\(99\) 0.561553 0.0564382
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.4049.3 4
4.3 odd 2 2200.2.b.f.1849.2 4
5.2 odd 4 880.2.a.k.1.2 2
5.3 odd 4 4400.2.a.bt.1.1 2
5.4 even 2 inner 4400.2.b.w.4049.2 4
15.2 even 4 7920.2.a.by.1.2 2
20.3 even 4 2200.2.a.l.1.2 2
20.7 even 4 440.2.a.g.1.1 ✓ 2
20.19 odd 2 2200.2.b.f.1849.3 4
40.27 even 4 3520.2.a.bm.1.2 2
40.37 odd 4 3520.2.a.br.1.1 2
55.32 even 4 9680.2.a.bm.1.2 2
60.47 odd 4 3960.2.a.bf.1.1 2
220.87 odd 4 4840.2.a.m.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
440.2.a.g.1.1 ✓ 2 20.7 even 4
880.2.a.k.1.2 2 5.2 odd 4
2200.2.a.l.1.2 2 20.3 even 4
2200.2.b.f.1849.2 4 4.3 odd 2
2200.2.b.f.1849.3 4 20.19 odd 2
3520.2.a.bm.1.2 2 40.27 even 4
3520.2.a.br.1.1 2 40.37 odd 4
3960.2.a.bf.1.1 2 60.47 odd 4
4400.2.a.bt.1.1 2 5.3 odd 4
4400.2.b.w.4049.2 4 5.4 even 2 inner
4400.2.b.w.4049.3 4 1.1 even 1 trivial
4840.2.a.m.1.1 2 220.87 odd 4
7920.2.a.by.1.2 2 15.2 even 4
9680.2.a.bm.1.2 2 55.32 even 4