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Properties

Label 444.2.e.a.73.3
Level $444$
Weight $2$
Character 444.73
Analytic conductor $3.545$
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] = [444,2,Mod(73,444)] 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("444.73"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(444, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 444 = 2^{2} \cdot 3 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 444.e (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,-4] 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: no
Analytic conductor: \(3.54535784974\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{5})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 6x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 73.3
Root \(-0.874032i\) of defining polynomial
Character \(\chi\) \(=\) 444.73
Dual form 444.2.e.a.73.2

$q$-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)
 
\(f(q)\) \(=\) \(q-1.00000 q^{3} +0.540182i q^{5} -1.23607 q^{7} +1.00000 q^{9} +2.47214 q^{11} +4.57649i q^{13} -0.540182i q^{15} +3.36861i q^{17} -1.74806i q^{19} +1.23607 q^{21} +8.61280i q^{23} +4.70820 q^{25} -1.00000 q^{27} +7.94510i q^{29} -6.32456i q^{31} -2.47214 q^{33} -0.667701i q^{35} +(-2.23607 + 5.65685i) q^{37} -4.57649i q^{39} +2.00000 q^{41} -2.82843i q^{43} +0.540182i q^{45} +4.00000 q^{47} -5.47214 q^{49} -3.36861i q^{51} +10.9443 q^{53} +1.33540i q^{55} +1.74806i q^{57} +2.28825i q^{59} -5.65685i q^{61} -1.23607 q^{63} -2.47214 q^{65} -7.70820 q^{67} -8.61280i q^{69} -8.94427 q^{71} +3.23607 q^{73} -4.70820 q^{75} -3.05573 q^{77} -9.56564i q^{79} +1.00000 q^{81} +1.52786 q^{83} -1.81966 q^{85} -7.94510i q^{87} -8.61280i q^{89} -5.65685i q^{91} +6.32456i q^{93} +0.944272 q^{95} -13.7295i q^{97} +2.47214 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{3} + 4 q^{7} + 4 q^{9} - 8 q^{11} - 4 q^{21} - 8 q^{25} - 4 q^{27} + 8 q^{33} + 8 q^{41} + 16 q^{47} - 4 q^{49} + 8 q^{53} + 4 q^{63} + 8 q^{65} - 4 q^{67} + 4 q^{73} + 8 q^{75} - 48 q^{77} + 4 q^{81}+ \cdots - 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/444\mathbb{Z}\right)^\times\).

\(n\) \(149\) \(223\) \(409\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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
\(3\) −1.00000 −0.577350
\(4\) 0 0
\(5\) 0.540182i 0.241577i 0.992678 + 0.120788i \(0.0385422\pi\)
−0.992678 + 0.120788i \(0.961458\pi\)
\(6\) 0 0
\(7\) −1.23607 −0.467190 −0.233595 − 0.972334i \(-0.575049\pi\)
−0.233595 + 0.972334i \(0.575049\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 2.47214 0.745377 0.372689 − 0.927957i \(-0.378436\pi\)
0.372689 + 0.927957i \(0.378436\pi\)
\(12\) 0 0
\(13\) 4.57649i 1.26929i 0.772804 + 0.634645i \(0.218854\pi\)
−0.772804 + 0.634645i \(0.781146\pi\)
\(14\) 0 0
\(15\) − 0.540182i − 0.139474i
\(16\) 0 0
\(17\) 3.36861i 0.817008i 0.912757 + 0.408504i \(0.133949\pi\)
−0.912757 + 0.408504i \(0.866051\pi\)
\(18\) 0 0
\(19\) − 1.74806i − 0.401033i −0.979690 − 0.200517i \(-0.935738\pi\)
0.979690 − 0.200517i \(-0.0642621\pi\)
\(20\) 0 0
\(21\) 1.23607 0.269732
\(22\) 0 0
\(23\) 8.61280i 1.79589i 0.440104 + 0.897947i \(0.354941\pi\)
−0.440104 + 0.897947i \(0.645059\pi\)
\(24\) 0 0
\(25\) 4.70820 0.941641
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) 7.94510i 1.47537i 0.675146 + 0.737684i \(0.264081\pi\)
−0.675146 + 0.737684i \(0.735919\pi\)
\(30\) 0 0
\(31\) − 6.32456i − 1.13592i −0.823055 − 0.567962i \(-0.807732\pi\)
0.823055 − 0.567962i \(-0.192268\pi\)
\(32\) 0 0
\(33\) −2.47214 −0.430344
\(34\) 0 0
\(35\) − 0.667701i − 0.112862i
\(36\) 0 0
\(37\) −2.23607 + 5.65685i −0.367607 + 0.929981i
\(38\) 0 0
\(39\) − 4.57649i − 0.732825i
\(40\) 0 0
\(41\) 2.00000 0.312348 0.156174 − 0.987730i \(-0.450084\pi\)
0.156174 + 0.987730i \(0.450084\pi\)
\(42\) 0 0
\(43\) − 2.82843i − 0.431331i −0.976467 − 0.215666i \(-0.930808\pi\)
0.976467 − 0.215666i \(-0.0691921\pi\)
\(44\) 0 0
\(45\) 0.540182i 0.0805255i
\(46\) 0 0
\(47\) 4.00000 0.583460 0.291730 − 0.956501i \(-0.405769\pi\)
0.291730 + 0.956501i \(0.405769\pi\)
\(48\) 0 0
\(49\) −5.47214 −0.781734
\(50\) 0 0
\(51\) − 3.36861i − 0.471700i
\(52\) 0 0
\(53\) 10.9443 1.50331 0.751656 − 0.659556i \(-0.229256\pi\)
0.751656 + 0.659556i \(0.229256\pi\)
\(54\) 0 0
\(55\) 1.33540i 0.180066i
\(56\) 0 0
\(57\) 1.74806i 0.231537i
\(58\) 0 0
\(59\) 2.28825i 0.297904i 0.988844 + 0.148952i \(0.0475900\pi\)
−0.988844 + 0.148952i \(0.952410\pi\)
\(60\) 0 0
\(61\) − 5.65685i − 0.724286i −0.932123 − 0.362143i \(-0.882045\pi\)
0.932123 − 0.362143i \(-0.117955\pi\)
\(62\) 0 0
\(63\) −1.23607 −0.155730
\(64\) 0 0
\(65\) −2.47214 −0.306631
\(66\) 0 0
\(67\) −7.70820 −0.941707 −0.470853 − 0.882211i \(-0.656054\pi\)
−0.470853 + 0.882211i \(0.656054\pi\)
\(68\) 0 0
\(69\) − 8.61280i − 1.03686i
\(70\) 0 0
\(71\) −8.94427 −1.06149 −0.530745 − 0.847532i \(-0.678088\pi\)
−0.530745 + 0.847532i \(0.678088\pi\)
\(72\) 0 0
\(73\) 3.23607 0.378753 0.189377 − 0.981905i \(-0.439353\pi\)
0.189377 + 0.981905i \(0.439353\pi\)
\(74\) 0 0
\(75\) −4.70820 −0.543657
\(76\) 0 0
\(77\) −3.05573 −0.348233
\(78\) 0 0
\(79\) − 9.56564i − 1.07622i −0.842875 − 0.538110i \(-0.819139\pi\)
0.842875 − 0.538110i \(-0.180861\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 1.52786 0.167705 0.0838524 − 0.996478i \(-0.473278\pi\)
0.0838524 + 0.996478i \(0.473278\pi\)
\(84\) 0 0
\(85\) −1.81966 −0.197370
\(86\) 0 0
\(87\) − 7.94510i − 0.851804i
\(88\) 0 0
\(89\) − 8.61280i − 0.912955i −0.889735 − 0.456478i \(-0.849111\pi\)
0.889735 − 0.456478i \(-0.150889\pi\)
\(90\) 0 0
\(91\) − 5.65685i − 0.592999i
\(92\) 0 0
\(93\) 6.32456i 0.655826i
\(94\) 0 0
\(95\) 0.944272 0.0968803
\(96\) 0 0
\(97\) − 13.7295i − 1.39402i −0.717063 − 0.697008i \(-0.754514\pi\)
0.717063 − 0.697008i \(-0.245486\pi\)
\(98\) 0 0
\(99\) 2.47214 0.248459
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 444.2.e.a.73.3 yes 4
3.2 odd 2 1332.2.e.e.73.2 4
4.3 odd 2 1776.2.h.f.961.3 4
12.11 even 2 5328.2.h.g.2737.2 4
37.36 even 2 inner 444.2.e.a.73.2 ✓ 4
111.110 odd 2 1332.2.e.e.73.3 4
148.147 odd 2 1776.2.h.f.961.2 4
444.443 even 2 5328.2.h.g.2737.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
444.2.e.a.73.2 ✓ 4 37.36 even 2 inner
444.2.e.a.73.3 yes 4 1.1 even 1 trivial
1332.2.e.e.73.2 4 3.2 odd 2
1332.2.e.e.73.3 4 111.110 odd 2
1776.2.h.f.961.2 4 148.147 odd 2
1776.2.h.f.961.3 4 4.3 odd 2
5328.2.h.g.2737.2 4 12.11 even 2
5328.2.h.g.2737.3 4 444.443 even 2