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Properties

Label 32.4.a.c.1.1
Level $32$
Weight $4$
Character 32.1
Self dual yes
Analytic conductor $1.888$
Analytic rank $0$
Dimension $1$
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] = [32,4,Mod(1,32)] 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("32.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(32, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 32 = 2^{5} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 32.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,0,8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(1.88806112018\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 32.1

$q$-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)
 
\(f(q)\) \(=\) \(q+8.00000 q^{3} -10.0000 q^{5} +16.0000 q^{7} +37.0000 q^{9} -40.0000 q^{11} -50.0000 q^{13} -80.0000 q^{15} -30.0000 q^{17} +40.0000 q^{19} +128.000 q^{21} +48.0000 q^{23} -25.0000 q^{25} +80.0000 q^{27} -34.0000 q^{29} +320.000 q^{31} -320.000 q^{33} -160.000 q^{35} +310.000 q^{37} -400.000 q^{39} +410.000 q^{41} +152.000 q^{43} -370.000 q^{45} -416.000 q^{47} -87.0000 q^{49} -240.000 q^{51} -410.000 q^{53} +400.000 q^{55} +320.000 q^{57} -200.000 q^{59} +30.0000 q^{61} +592.000 q^{63} +500.000 q^{65} +776.000 q^{67} +384.000 q^{69} +400.000 q^{71} -630.000 q^{73} -200.000 q^{75} -640.000 q^{77} -1120.00 q^{79} -359.000 q^{81} +552.000 q^{83} +300.000 q^{85} -272.000 q^{87} -326.000 q^{89} -800.000 q^{91} +2560.00 q^{93} -400.000 q^{95} -110.000 q^{97} -1480.00 q^{99} +O(q^{100})\)

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
\(3\) 8.00000 1.53960 0.769800 − 0.638285i \(-0.220356\pi\)
0.769800 + 0.638285i \(0.220356\pi\)
\(4\) 0 0
\(5\) −10.0000 −0.894427 −0.447214 − 0.894427i \(-0.647584\pi\)
−0.447214 + 0.894427i \(0.647584\pi\)
\(6\) 0 0
\(7\) 16.0000 0.863919 0.431959 − 0.901893i \(-0.357822\pi\)
0.431959 + 0.901893i \(0.357822\pi\)
\(8\) 0 0
\(9\) 37.0000 1.37037
\(10\) 0 0
\(11\) −40.0000 −1.09640 −0.548202 − 0.836346i \(-0.684688\pi\)
−0.548202 + 0.836346i \(0.684688\pi\)
\(12\) 0 0
\(13\) −50.0000 −1.06673 −0.533366 − 0.845885i \(-0.679073\pi\)
−0.533366 + 0.845885i \(0.679073\pi\)
\(14\) 0 0
\(15\) −80.0000 −1.37706
\(16\) 0 0
\(17\) −30.0000 −0.428004 −0.214002 − 0.976833i \(-0.568650\pi\)
−0.214002 + 0.976833i \(0.568650\pi\)
\(18\) 0 0
\(19\) 40.0000 0.482980 0.241490 − 0.970403i \(-0.422364\pi\)
0.241490 + 0.970403i \(0.422364\pi\)
\(20\) 0 0
\(21\) 128.000 1.33009
\(22\) 0 0
\(23\) 48.0000 0.435161 0.217580 − 0.976042i \(-0.430184\pi\)
0.217580 + 0.976042i \(0.430184\pi\)
\(24\) 0 0
\(25\) −25.0000 −0.200000
\(26\) 0 0
\(27\) 80.0000 0.570222
\(28\) 0 0
\(29\) −34.0000 −0.217712 −0.108856 − 0.994058i \(-0.534719\pi\)
−0.108856 + 0.994058i \(0.534719\pi\)
\(30\) 0 0
\(31\) 320.000 1.85399 0.926995 − 0.375073i \(-0.122383\pi\)
0.926995 + 0.375073i \(0.122383\pi\)
\(32\) 0 0
\(33\) −320.000 −1.68803
\(34\) 0 0
\(35\) −160.000 −0.772712
\(36\) 0 0
\(37\) 310.000 1.37740 0.688698 − 0.725048i \(-0.258182\pi\)
0.688698 + 0.725048i \(0.258182\pi\)
\(38\) 0 0
\(39\) −400.000 −1.64234
\(40\) 0 0
\(41\) 410.000 1.56174 0.780869 − 0.624695i \(-0.214777\pi\)
0.780869 + 0.624695i \(0.214777\pi\)
\(42\) 0 0
\(43\) 152.000 0.539065 0.269532 − 0.962991i \(-0.413131\pi\)
0.269532 + 0.962991i \(0.413131\pi\)
\(44\) 0 0
\(45\) −370.000 −1.22570
\(46\) 0 0
\(47\) −416.000 −1.29106 −0.645530 − 0.763735i \(-0.723364\pi\)
−0.645530 + 0.763735i \(0.723364\pi\)
\(48\) 0 0
\(49\) −87.0000 −0.253644
\(50\) 0 0
\(51\) −240.000 −0.658955
\(52\) 0 0
\(53\) −410.000 −1.06260 −0.531300 − 0.847184i \(-0.678296\pi\)
−0.531300 + 0.847184i \(0.678296\pi\)
\(54\) 0 0
\(55\) 400.000 0.980654
\(56\) 0 0
\(57\) 320.000 0.743597
\(58\) 0 0
\(59\) −200.000 −0.441318 −0.220659 − 0.975351i \(-0.570821\pi\)
−0.220659 + 0.975351i \(0.570821\pi\)
\(60\) 0 0
\(61\) 30.0000 0.0629690 0.0314845 − 0.999504i \(-0.489977\pi\)
0.0314845 + 0.999504i \(0.489977\pi\)
\(62\) 0 0
\(63\) 592.000 1.18389
\(64\) 0 0
\(65\) 500.000 0.954113
\(66\) 0 0
\(67\) 776.000 1.41498 0.707489 − 0.706725i \(-0.249828\pi\)
0.707489 + 0.706725i \(0.249828\pi\)
\(68\) 0 0
\(69\) 384.000 0.669973
\(70\) 0 0
\(71\) 400.000 0.668609 0.334305 − 0.942465i \(-0.391499\pi\)
0.334305 + 0.942465i \(0.391499\pi\)
\(72\) 0 0
\(73\) −630.000 −1.01008 −0.505041 − 0.863096i \(-0.668522\pi\)
−0.505041 + 0.863096i \(0.668522\pi\)
\(74\) 0 0
\(75\) −200.000 −0.307920
\(76\) 0 0
\(77\) −640.000 −0.947205
\(78\) 0 0
\(79\) −1120.00 −1.59506 −0.797531 − 0.603278i \(-0.793861\pi\)
−0.797531 + 0.603278i \(0.793861\pi\)
\(80\) 0 0
\(81\) −359.000 −0.492455
\(82\) 0 0
\(83\) 552.000 0.729998 0.364999 − 0.931008i \(-0.381069\pi\)
0.364999 + 0.931008i \(0.381069\pi\)
\(84\) 0 0
\(85\) 300.000 0.382818
\(86\) 0 0
\(87\) −272.000 −0.335189
\(88\) 0 0
\(89\) −326.000 −0.388269 −0.194134 − 0.980975i \(-0.562190\pi\)
−0.194134 + 0.980975i \(0.562190\pi\)
\(90\) 0 0
\(91\) −800.000 −0.921569
\(92\) 0 0
\(93\) 2560.00 2.85440
\(94\) 0 0
\(95\) −400.000 −0.431991
\(96\) 0 0
\(97\) −110.000 −0.115142 −0.0575712 − 0.998341i \(-0.518336\pi\)
−0.0575712 + 0.998341i \(0.518336\pi\)
\(98\) 0 0
\(99\) −1480.00 −1.50248
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 32.4.a.c.1.1 yes 1
3.2 odd 2 288.4.a.i.1.1 1
4.3 odd 2 32.4.a.a.1.1 ✓ 1
5.2 odd 4 800.4.c.a.449.1 2
5.3 odd 4 800.4.c.a.449.2 2
5.4 even 2 800.4.a.a.1.1 1
7.6 odd 2 1568.4.a.c.1.1 1
8.3 odd 2 64.4.a.e.1.1 1
8.5 even 2 64.4.a.a.1.1 1
12.11 even 2 288.4.a.h.1.1 1
16.3 odd 4 256.4.b.e.129.1 2
16.5 even 4 256.4.b.c.129.1 2
16.11 odd 4 256.4.b.e.129.2 2
16.13 even 4 256.4.b.c.129.2 2
20.3 even 4 800.4.c.b.449.1 2
20.7 even 4 800.4.c.b.449.2 2
20.19 odd 2 800.4.a.k.1.1 1
24.5 odd 2 576.4.a.h.1.1 1
24.11 even 2 576.4.a.g.1.1 1
28.27 even 2 1568.4.a.o.1.1 1
40.19 odd 2 1600.4.a.e.1.1 1
40.29 even 2 1600.4.a.bw.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
32.4.a.a.1.1 ✓ 1 4.3 odd 2
32.4.a.c.1.1 yes 1 1.1 even 1 trivial
64.4.a.a.1.1 1 8.5 even 2
64.4.a.e.1.1 1 8.3 odd 2
256.4.b.c.129.1 2 16.5 even 4
256.4.b.c.129.2 2 16.13 even 4
256.4.b.e.129.1 2 16.3 odd 4
256.4.b.e.129.2 2 16.11 odd 4
288.4.a.h.1.1 1 12.11 even 2
288.4.a.i.1.1 1 3.2 odd 2
576.4.a.g.1.1 1 24.11 even 2
576.4.a.h.1.1 1 24.5 odd 2
800.4.a.a.1.1 1 5.4 even 2
800.4.a.k.1.1 1 20.19 odd 2
800.4.c.a.449.1 2 5.2 odd 4
800.4.c.a.449.2 2 5.3 odd 4
800.4.c.b.449.1 2 20.3 even 4
800.4.c.b.449.2 2 20.7 even 4
1568.4.a.c.1.1 1 7.6 odd 2
1568.4.a.o.1.1 1 28.27 even 2
1600.4.a.e.1.1 1 40.19 odd 2
1600.4.a.bw.1.1 1 40.29 even 2