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Properties

Label 1100.4.b.h.749.5
Level $1100$
Weight $4$
Character 1100.749
Analytic conductor $64.902$
Analytic rank $0$
Dimension $6$
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] = [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 = [6,0,0,0,0,0,0,0,-48,0,66,0,0,0,0,0,0,0,-342] 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: \(6\)
Coefficient field: 6.0.1351885824.3
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} + 37x^{4} + 384x^{2} + 900 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{5} \)
Twist minimal: no (minimal twist has level 220)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 749.5
Root \(3.67648i\) of defining polynomial
Character \(\chi\) \(=\) 1100.749
Dual form 1100.4.b.h.749.2

$q$-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)
 
\(f(q)\) \(=\) \(q+6.86946i q^{3} -15.2554i q^{7} -20.1894 q^{9} +11.0000 q^{11} +30.1909i q^{13} -74.4793i q^{17} -30.6126 q^{19} +104.796 q^{21} +111.339i q^{23} +46.7850i q^{27} -23.5336 q^{29} -272.083 q^{31} +75.5640i q^{33} -292.415i q^{37} -207.395 q^{39} -127.493 q^{41} -466.210i q^{43} -430.418i q^{47} +110.273 q^{49} +511.632 q^{51} +235.761i q^{53} -210.292i q^{57} -167.162 q^{59} -363.291 q^{61} +307.998i q^{63} -611.297i q^{67} -764.836 q^{69} -315.466 q^{71} -372.861i q^{73} -167.809i q^{77} +300.829 q^{79} -866.502 q^{81} -1218.18i q^{83} -161.663i q^{87} -73.6581 q^{89} +460.574 q^{91} -1869.06i q^{93} +1389.82i q^{97} -222.084 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 48 q^{9} + 66 q^{11} - 342 q^{19} + 518 q^{21} - 110 q^{29} + 362 q^{31} - 1108 q^{39} + 604 q^{41} + 536 q^{49} + 1666 q^{51} + 84 q^{59} + 698 q^{61} + 516 q^{69} + 1854 q^{71} + 2292 q^{79} - 3066 q^{81}+ \cdots - 528 q^{99}+O(q^{100}) \) 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
\(3\) 6.86946i 1.32203i 0.750374 + 0.661014i \(0.229873\pi\)
−0.750374 + 0.661014i \(0.770127\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) − 15.2554i − 0.823715i −0.911248 − 0.411857i \(-0.864880\pi\)
0.911248 − 0.411857i \(-0.135120\pi\)
\(8\) 0 0
\(9\) −20.1894 −0.747756
\(10\) 0 0
\(11\) 11.0000 0.301511
\(12\) 0 0
\(13\) 30.1909i 0.644110i 0.946721 + 0.322055i \(0.104374\pi\)
−0.946721 + 0.322055i \(0.895626\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) − 74.4793i − 1.06258i −0.847190 − 0.531290i \(-0.821707\pi\)
0.847190 − 0.531290i \(-0.178293\pi\)
\(18\) 0 0
\(19\) −30.6126 −0.369632 −0.184816 − 0.982773i \(-0.559169\pi\)
−0.184816 + 0.982773i \(0.559169\pi\)
\(20\) 0 0
\(21\) 104.796 1.08897
\(22\) 0 0
\(23\) 111.339i 1.00938i 0.863301 + 0.504690i \(0.168393\pi\)
−0.863301 + 0.504690i \(0.831607\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 46.7850i 0.333473i
\(28\) 0 0
\(29\) −23.5336 −0.150692 −0.0753462 − 0.997157i \(-0.524006\pi\)
−0.0753462 + 0.997157i \(0.524006\pi\)
\(30\) 0 0
\(31\) −272.083 −1.57637 −0.788185 − 0.615439i \(-0.788979\pi\)
−0.788185 + 0.615439i \(0.788979\pi\)
\(32\) 0 0
\(33\) 75.5640i 0.398606i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) − 292.415i − 1.29926i −0.760249 − 0.649632i \(-0.774923\pi\)
0.760249 − 0.649632i \(-0.225077\pi\)
\(38\) 0 0
\(39\) −207.395 −0.851532
\(40\) 0 0
\(41\) −127.493 −0.485634 −0.242817 − 0.970072i \(-0.578071\pi\)
−0.242817 + 0.970072i \(0.578071\pi\)
\(42\) 0 0
\(43\) − 466.210i − 1.65341i −0.562639 − 0.826703i \(-0.690214\pi\)
0.562639 − 0.826703i \(-0.309786\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) − 430.418i − 1.33581i −0.744248 − 0.667903i \(-0.767192\pi\)
0.744248 − 0.667903i \(-0.232808\pi\)
\(48\) 0 0
\(49\) 110.273 0.321494
\(50\) 0 0
\(51\) 511.632 1.40476
\(52\) 0 0
\(53\) 235.761i 0.611024i 0.952188 + 0.305512i \(0.0988276\pi\)
−0.952188 + 0.305512i \(0.901172\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) − 210.292i − 0.488664i
\(58\) 0 0
\(59\) −167.162 −0.368859 −0.184429 − 0.982846i \(-0.559044\pi\)
−0.184429 + 0.982846i \(0.559044\pi\)
\(60\) 0 0
\(61\) −363.291 −0.762535 −0.381267 − 0.924465i \(-0.624512\pi\)
−0.381267 + 0.924465i \(0.624512\pi\)
\(62\) 0 0
\(63\) 307.998i 0.615938i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) − 611.297i − 1.11465i −0.830293 − 0.557327i \(-0.811827\pi\)
0.830293 − 0.557327i \(-0.188173\pi\)
\(68\) 0 0
\(69\) −764.836 −1.33443
\(70\) 0 0
\(71\) −315.466 −0.527309 −0.263654 − 0.964617i \(-0.584928\pi\)
−0.263654 + 0.964617i \(0.584928\pi\)
\(72\) 0 0
\(73\) − 372.861i − 0.597810i −0.954283 − 0.298905i \(-0.903379\pi\)
0.954283 − 0.298905i \(-0.0966214\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) − 167.809i − 0.248359i
\(78\) 0 0
\(79\) 300.829 0.428430 0.214215 − 0.976787i \(-0.431281\pi\)
0.214215 + 0.976787i \(0.431281\pi\)
\(80\) 0 0
\(81\) −866.502 −1.18862
\(82\) 0 0
\(83\) − 1218.18i − 1.61100i −0.592599 − 0.805498i \(-0.701898\pi\)
0.592599 − 0.805498i \(-0.298102\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) − 161.663i − 0.199219i
\(88\) 0 0
\(89\) −73.6581 −0.0877275 −0.0438637 − 0.999038i \(-0.513967\pi\)
−0.0438637 + 0.999038i \(0.513967\pi\)
\(90\) 0 0
\(91\) 460.574 0.530563
\(92\) 0 0
\(93\) − 1869.06i − 2.08400i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 1389.82i 1.45480i 0.686216 + 0.727398i \(0.259271\pi\)
−0.686216 + 0.727398i \(0.740729\pi\)
\(98\) 0 0
\(99\) −222.084 −0.225457
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.h.749.5 6
5.2 odd 4 220.4.a.f.1.3 ✓ 3
5.3 odd 4 1100.4.a.i.1.1 3
5.4 even 2 inner 1100.4.b.h.749.2 6
15.2 even 4 1980.4.a.l.1.3 3
20.7 even 4 880.4.a.w.1.1 3
55.32 even 4 2420.4.a.i.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
220.4.a.f.1.3 ✓ 3 5.2 odd 4
880.4.a.w.1.1 3 20.7 even 4
1100.4.a.i.1.1 3 5.3 odd 4
1100.4.b.h.749.2 6 5.4 even 2 inner
1100.4.b.h.749.5 6 1.1 even 1 trivial
1980.4.a.l.1.3 3 15.2 even 4
2420.4.a.i.1.3 3 55.32 even 4