Properties

Label 5610.2.a.ci.1.3
Level $5610$
Weight $2$
Character 5610.1
Self dual yes
Analytic conductor $44.796$
Analytic rank $0$
Dimension $5$
CM no
Inner twists $1$

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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] = [5610,2,Mod(1,5610)] 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("5610.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(5610, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 5610 = 2 \cdot 3 \cdot 5 \cdot 11 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5610.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [5,5,-5,5,5,-5,2,5,5,5,5,-5,-1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(44.7960755339\)
Analytic rank: \(0\)
Dimension: \(5\)
Coefficient field: 5.5.18569692.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - 23x^{3} - 32x^{2} + 26x - 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-3.25711\) of defining polynomial
Character \(\chi\) \(=\) 5610.1

$q$-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)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} +1.89495 q^{7} +1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} +1.00000 q^{11} -1.00000 q^{12} +2.77639 q^{13} +1.89495 q^{14} -1.00000 q^{15} +1.00000 q^{16} -1.00000 q^{17} +1.00000 q^{18} +2.77639 q^{19} +1.00000 q^{20} -1.89495 q^{21} +1.00000 q^{22} +4.37168 q^{23} -1.00000 q^{24} +1.00000 q^{25} +2.77639 q^{26} -1.00000 q^{27} +1.89495 q^{28} +4.88144 q^{29} -1.00000 q^{30} -8.88591 q^{31} +1.00000 q^{32} -1.00000 q^{33} -1.00000 q^{34} +1.89495 q^{35} +1.00000 q^{36} +3.73784 q^{37} +2.77639 q^{38} -2.77639 q^{39} +1.00000 q^{40} -0.476734 q^{41} -1.89495 q^{42} +6.34664 q^{43} +1.00000 q^{44} +1.00000 q^{45} +4.37168 q^{46} -0.476734 q^{47} -1.00000 q^{48} -3.40918 q^{49} +1.00000 q^{50} +1.00000 q^{51} +2.77639 q^{52} +4.75135 q^{53} -1.00000 q^{54} +1.00000 q^{55} +1.89495 q^{56} -2.77639 q^{57} +4.88144 q^{58} -12.0670 q^{59} -1.00000 q^{60} +8.21458 q^{61} -8.88591 q^{62} +1.89495 q^{63} +1.00000 q^{64} +2.77639 q^{65} -1.00000 q^{66} -9.08051 q^{67} -1.00000 q^{68} -4.37168 q^{69} +1.89495 q^{70} -2.51423 q^{71} +1.00000 q^{72} -2.75135 q^{73} +3.73784 q^{74} -1.00000 q^{75} +2.77639 q^{76} +1.89495 q^{77} -2.77639 q^{78} +8.99096 q^{79} +1.00000 q^{80} +1.00000 q^{81} -0.476734 q^{82} +6.81388 q^{83} -1.89495 q^{84} -1.00000 q^{85} +6.34664 q^{86} -4.88144 q^{87} +1.00000 q^{88} +3.76288 q^{89} +1.00000 q^{90} +5.26111 q^{91} +4.37168 q^{92} +8.88591 q^{93} -0.476734 q^{94} +2.77639 q^{95} -1.00000 q^{96} +3.33313 q^{97} -3.40918 q^{98} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q + 5 q^{2} - 5 q^{3} + 5 q^{4} + 5 q^{5} - 5 q^{6} + 2 q^{7} + 5 q^{8} + 5 q^{9} + 5 q^{10} + 5 q^{11} - 5 q^{12} - q^{13} + 2 q^{14} - 5 q^{15} + 5 q^{16} - 5 q^{17} + 5 q^{18} - q^{19} + 5 q^{20}+ \cdots + 5 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) −1.00000 −0.577350
\(4\) 1.00000 0.500000
\(5\) 1.00000 0.447214
\(6\) −1.00000 −0.408248
\(7\) 1.89495 0.716223 0.358111 − 0.933679i \(-0.383421\pi\)
0.358111 + 0.933679i \(0.383421\pi\)
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) 1.00000 0.316228
\(11\) 1.00000 0.301511
\(12\) −1.00000 −0.288675
\(13\) 2.77639 0.770032 0.385016 − 0.922910i \(-0.374196\pi\)
0.385016 + 0.922910i \(0.374196\pi\)
\(14\) 1.89495 0.506446
\(15\) −1.00000 −0.258199
\(16\) 1.00000 0.250000
\(17\) −1.00000 −0.242536
\(18\) 1.00000 0.235702
\(19\) 2.77639 0.636947 0.318474 − 0.947932i \(-0.396830\pi\)
0.318474 + 0.947932i \(0.396830\pi\)
\(20\) 1.00000 0.223607
\(21\) −1.89495 −0.413511
\(22\) 1.00000 0.213201
\(23\) 4.37168 0.911558 0.455779 − 0.890093i \(-0.349361\pi\)
0.455779 + 0.890093i \(0.349361\pi\)
\(24\) −1.00000 −0.204124
\(25\) 1.00000 0.200000
\(26\) 2.77639 0.544495
\(27\) −1.00000 −0.192450
\(28\) 1.89495 0.358111
\(29\) 4.88144 0.906461 0.453230 − 0.891393i \(-0.350271\pi\)
0.453230 + 0.891393i \(0.350271\pi\)
\(30\) −1.00000 −0.182574
\(31\) −8.88591 −1.59596 −0.797978 − 0.602686i \(-0.794097\pi\)
−0.797978 + 0.602686i \(0.794097\pi\)
\(32\) 1.00000 0.176777
\(33\) −1.00000 −0.174078
\(34\) −1.00000 −0.171499
\(35\) 1.89495 0.320304
\(36\) 1.00000 0.166667
\(37\) 3.73784 0.614497 0.307249 − 0.951629i \(-0.400592\pi\)
0.307249 + 0.951629i \(0.400592\pi\)
\(38\) 2.77639 0.450390
\(39\) −2.77639 −0.444578
\(40\) 1.00000 0.158114
\(41\) −0.476734 −0.0744533 −0.0372266 − 0.999307i \(-0.511852\pi\)
−0.0372266 + 0.999307i \(0.511852\pi\)
\(42\) −1.89495 −0.292397
\(43\) 6.34664 0.967853 0.483927 − 0.875109i \(-0.339210\pi\)
0.483927 + 0.875109i \(0.339210\pi\)
\(44\) 1.00000 0.150756
\(45\) 1.00000 0.149071
\(46\) 4.37168 0.644569
\(47\) −0.476734 −0.0695388 −0.0347694 − 0.999395i \(-0.511070\pi\)
−0.0347694 + 0.999395i \(0.511070\pi\)
\(48\) −1.00000 −0.144338
\(49\) −3.40918 −0.487025
\(50\) 1.00000 0.141421
\(51\) 1.00000 0.140028
\(52\) 2.77639 0.385016
\(53\) 4.75135 0.652648 0.326324 − 0.945258i \(-0.394190\pi\)
0.326324 + 0.945258i \(0.394190\pi\)
\(54\) −1.00000 −0.136083
\(55\) 1.00000 0.134840
\(56\) 1.89495 0.253223
\(57\) −2.77639 −0.367742
\(58\) 4.88144 0.640965
\(59\) −12.0670 −1.57099 −0.785495 − 0.618868i \(-0.787592\pi\)
−0.785495 + 0.618868i \(0.787592\pi\)
\(60\) −1.00000 −0.129099
\(61\) 8.21458 1.05177 0.525884 − 0.850556i \(-0.323734\pi\)
0.525884 + 0.850556i \(0.323734\pi\)
\(62\) −8.88591 −1.12851
\(63\) 1.89495 0.238741
\(64\) 1.00000 0.125000
\(65\) 2.77639 0.344369
\(66\) −1.00000 −0.123091
\(67\) −9.08051 −1.10936 −0.554681 − 0.832063i \(-0.687160\pi\)
−0.554681 + 0.832063i \(0.687160\pi\)
\(68\) −1.00000 −0.121268
\(69\) −4.37168 −0.526288
\(70\) 1.89495 0.226489
\(71\) −2.51423 −0.298384 −0.149192 − 0.988808i \(-0.547667\pi\)
−0.149192 + 0.988808i \(0.547667\pi\)
\(72\) 1.00000 0.117851
\(73\) −2.75135 −0.322021 −0.161010 − 0.986953i \(-0.551475\pi\)
−0.161010 + 0.986953i \(0.551475\pi\)
\(74\) 3.73784 0.434515
\(75\) −1.00000 −0.115470
\(76\) 2.77639 0.318474
\(77\) 1.89495 0.215949
\(78\) −2.77639 −0.314364
\(79\) 8.99096 1.01156 0.505781 − 0.862662i \(-0.331204\pi\)
0.505781 + 0.862662i \(0.331204\pi\)
\(80\) 1.00000 0.111803
\(81\) 1.00000 0.111111
\(82\) −0.476734 −0.0526464
\(83\) 6.81388 0.747921 0.373960 − 0.927445i \(-0.378000\pi\)
0.373960 + 0.927445i \(0.378000\pi\)
\(84\) −1.89495 −0.206756
\(85\) −1.00000 −0.108465
\(86\) 6.34664 0.684376
\(87\) −4.88144 −0.523345
\(88\) 1.00000 0.106600
\(89\) 3.76288 0.398865 0.199432 − 0.979912i \(-0.436090\pi\)
0.199432 + 0.979912i \(0.436090\pi\)
\(90\) 1.00000 0.105409
\(91\) 5.26111 0.551514
\(92\) 4.37168 0.455779
\(93\) 8.88591 0.921426
\(94\) −0.476734 −0.0491713
\(95\) 2.77639 0.284851
\(96\) −1.00000 −0.102062
\(97\) 3.33313 0.338428 0.169214 − 0.985579i \(-0.445877\pi\)
0.169214 + 0.985579i \(0.445877\pi\)
\(98\) −3.40918 −0.344379
\(99\) 1.00000 0.100504
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 5610.2.a.ci.1.3 ✓ 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
5610.2.a.ci.1.3 ✓ 5 1.1 even 1 trivial