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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1100,4,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, 4); 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, 4, names="a")
 
Level: \( N \) \(=\) \( 1100 = 2^{2} \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
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,-70,0,44,0,0,0,0,0,0,0,-24] 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: \(64.9021010063\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{97})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 49x^{2} + 576 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 44)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 749.1
Root \(-5.42443i\) of defining polynomial
Character \(\chi\) \(=\) 1100.749
Dual form 1100.4.b.e.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)
 
\(f(q)\) \(=\) \(q-9.42443i q^{3} -24.5466i q^{7} -61.8199 q^{9} +11.0000 q^{11} +20.8489i q^{13} -99.6977i q^{17} -104.489 q^{19} -231.337 q^{21} +172.762i q^{23} +328.157i q^{27} -137.826 q^{29} +205.424 q^{31} -103.669i q^{33} +57.2381i q^{37} +196.489 q^{39} -247.640 q^{41} -353.035i q^{43} +198.559i q^{47} -259.534 q^{49} -939.594 q^{51} +101.511i q^{53} +984.745i q^{57} -240.646 q^{59} -43.8717 q^{61} +1517.47i q^{63} +631.321i q^{67} +1628.18 q^{69} -38.2856 q^{71} -388.617i q^{73} -270.012i q^{77} -1219.83 q^{79} +1423.56 q^{81} +119.197i q^{83} +1298.93i q^{87} +520.973 q^{89} +511.768 q^{91} -1936.01i q^{93} -335.101i q^{97} -680.018 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 70 q^{9} + 44 q^{11} - 24 q^{19} - 492 q^{21} + 276 q^{29} + 802 q^{31} + 392 q^{39} - 636 q^{41} - 2220 q^{49} - 1828 q^{51} + 42 q^{59} - 924 q^{61} + 3302 q^{69} + 1206 q^{71} - 4052 q^{79} + 4276 q^{81}+ \cdots - 770 q^{99}+O(q^{100}) \) 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
\(3\) − 9.42443i − 1.81373i −0.421419 − 0.906866i \(-0.638468\pi\)
0.421419 − 0.906866i \(-0.361532\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) − 24.5466i − 1.32539i −0.748889 − 0.662695i \(-0.769413\pi\)
0.748889 − 0.662695i \(-0.230587\pi\)
\(8\) 0 0
\(9\) −61.8199 −2.28962
\(10\) 0 0
\(11\) 11.0000 0.301511
\(12\) 0 0
\(13\) 20.8489i 0.444803i 0.974955 + 0.222401i \(0.0713895\pi\)
−0.974955 + 0.222401i \(0.928611\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) − 99.6977i − 1.42237i −0.703006 − 0.711184i \(-0.748159\pi\)
0.703006 − 0.711184i \(-0.251841\pi\)
\(18\) 0 0
\(19\) −104.489 −1.26165 −0.630824 − 0.775926i \(-0.717283\pi\)
−0.630824 + 0.775926i \(0.717283\pi\)
\(20\) 0 0
\(21\) −231.337 −2.40390
\(22\) 0 0
\(23\) 172.762i 1.56623i 0.621876 + 0.783116i \(0.286371\pi\)
−0.621876 + 0.783116i \(0.713629\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 328.157i 2.33903i
\(28\) 0 0
\(29\) −137.826 −0.882540 −0.441270 − 0.897374i \(-0.645472\pi\)
−0.441270 + 0.897374i \(0.645472\pi\)
\(30\) 0 0
\(31\) 205.424 1.19017 0.595086 − 0.803662i \(-0.297118\pi\)
0.595086 + 0.803662i \(0.297118\pi\)
\(32\) 0 0
\(33\) − 103.669i − 0.546861i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 57.2381i 0.254321i 0.991882 + 0.127161i \(0.0405864\pi\)
−0.991882 + 0.127161i \(0.959414\pi\)
\(38\) 0 0
\(39\) 196.489 0.806753
\(40\) 0 0
\(41\) −247.640 −0.943288 −0.471644 − 0.881789i \(-0.656339\pi\)
−0.471644 + 0.881789i \(0.656339\pi\)
\(42\) 0 0
\(43\) − 353.035i − 1.25203i −0.779810 − 0.626016i \(-0.784685\pi\)
0.779810 − 0.626016i \(-0.215315\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 198.559i 0.616230i 0.951349 + 0.308115i \(0.0996981\pi\)
−0.951349 + 0.308115i \(0.900302\pi\)
\(48\) 0 0
\(49\) −259.534 −0.756660
\(50\) 0 0
\(51\) −939.594 −2.57979
\(52\) 0 0
\(53\) 101.511i 0.263088i 0.991310 + 0.131544i \(0.0419935\pi\)
−0.991310 + 0.131544i \(0.958007\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 984.745i 2.28829i
\(58\) 0 0
\(59\) −240.646 −0.531007 −0.265504 − 0.964110i \(-0.585538\pi\)
−0.265504 + 0.964110i \(0.585538\pi\)
\(60\) 0 0
\(61\) −43.8717 −0.0920852 −0.0460426 − 0.998939i \(-0.514661\pi\)
−0.0460426 + 0.998939i \(0.514661\pi\)
\(62\) 0 0
\(63\) 1517.47i 3.03465i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 631.321i 1.15117i 0.817743 + 0.575583i \(0.195225\pi\)
−0.817743 + 0.575583i \(0.804775\pi\)
\(68\) 0 0
\(69\) 1628.18 2.84073
\(70\) 0 0
\(71\) −38.2856 −0.0639953 −0.0319976 − 0.999488i \(-0.510187\pi\)
−0.0319976 + 0.999488i \(0.510187\pi\)
\(72\) 0 0
\(73\) − 388.617i − 0.623071i −0.950235 − 0.311535i \(-0.899157\pi\)
0.950235 − 0.311535i \(-0.100843\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) − 270.012i − 0.399620i
\(78\) 0 0
\(79\) −1219.83 −1.73723 −0.868615 − 0.495488i \(-0.834989\pi\)
−0.868615 + 0.495488i \(0.834989\pi\)
\(80\) 0 0
\(81\) 1423.56 1.95276
\(82\) 0 0
\(83\) 119.197i 0.157633i 0.996889 + 0.0788166i \(0.0251141\pi\)
−0.996889 + 0.0788166i \(0.974886\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 1298.93i 1.60069i
\(88\) 0 0
\(89\) 520.973 0.620483 0.310242 − 0.950658i \(-0.399590\pi\)
0.310242 + 0.950658i \(0.399590\pi\)
\(90\) 0 0
\(91\) 511.768 0.589537
\(92\) 0 0
\(93\) − 1936.01i − 2.15865i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) − 335.101i − 0.350767i −0.984500 − 0.175383i \(-0.943884\pi\)
0.984500 − 0.175383i \(-0.0561165\pi\)
\(98\) 0 0
\(99\) −680.018 −0.690348
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 1100.4.b.e.749.1 4
5.2 odd 4 1100.4.a.e.1.1 2
5.3 odd 4 44.4.a.b.1.2 ✓ 2
5.4 even 2 inner 1100.4.b.e.749.4 4
15.8 even 4 396.4.a.i.1.2 2
20.3 even 4 176.4.a.g.1.1 2
35.13 even 4 2156.4.a.d.1.1 2
40.3 even 4 704.4.a.r.1.2 2
40.13 odd 4 704.4.a.m.1.1 2
55.43 even 4 484.4.a.e.1.2 2
60.23 odd 4 1584.4.a.z.1.2 2
220.43 odd 4 1936.4.a.o.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
44.4.a.b.1.2 ✓ 2 5.3 odd 4
176.4.a.g.1.1 2 20.3 even 4
396.4.a.i.1.2 2 15.8 even 4
484.4.a.e.1.2 2 55.43 even 4
704.4.a.m.1.1 2 40.13 odd 4
704.4.a.r.1.2 2 40.3 even 4
1100.4.a.e.1.1 2 5.2 odd 4
1100.4.b.e.749.1 4 1.1 even 1 trivial
1100.4.b.e.749.4 4 5.4 even 2 inner
1584.4.a.z.1.2 2 60.23 odd 4
1936.4.a.o.1.1 2 220.43 odd 4
2156.4.a.d.1.1 2 35.13 even 4