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Properties

Label 672.2.h.a.575.4
Level $672$
Weight $2$
Character 672.575
Analytic conductor $5.366$
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] = [672,2,Mod(575,672)] 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("672.575"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(672, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0, 1, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 672 = 2^{5} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 672.h (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,-4,0,0,0,0,0,0,0,-8,0,8] 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: no
Analytic conductor: \(5.36594701583\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 575.4
Root \(0.707107 - 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 672.575
Dual form 672.2.h.a.575.3

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.292893 + 1.70711i) q^{3} -3.41421i q^{5} +1.00000i q^{7} +(-2.82843 - 1.00000i) q^{9} -2.00000 q^{11} +3.41421 q^{13} +(5.82843 + 1.00000i) q^{15} -4.82843i q^{17} -6.24264i q^{19} +(-1.70711 - 0.292893i) q^{21} +4.00000 q^{23} -6.65685 q^{25} +(2.53553 - 4.53553i) q^{27} -4.82843i q^{29} +1.17157i q^{31} +(0.585786 - 3.41421i) q^{33} +3.41421 q^{35} -0.828427 q^{37} +(-1.00000 + 5.82843i) q^{39} +10.0000i q^{41} -10.4853i q^{43} +(-3.41421 + 9.65685i) q^{45} -1.65685 q^{47} -1.00000 q^{49} +(8.24264 + 1.41421i) q^{51} -13.3137i q^{53} +6.82843i q^{55} +(10.6569 + 1.82843i) q^{57} +6.24264 q^{59} +14.2426 q^{61} +(1.00000 - 2.82843i) q^{63} -11.6569i q^{65} +9.31371i q^{67} +(-1.17157 + 6.82843i) q^{69} +0.343146 q^{71} -3.17157 q^{73} +(1.94975 - 11.3640i) q^{75} -2.00000i q^{77} +4.00000i q^{79} +(7.00000 + 5.65685i) q^{81} -11.4142 q^{83} -16.4853 q^{85} +(8.24264 + 1.41421i) q^{87} -7.65685i q^{89} +3.41421i q^{91} +(-2.00000 - 0.343146i) q^{93} -21.3137 q^{95} +5.31371 q^{97} +(5.65685 + 2.00000i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{3} - 8 q^{11} + 8 q^{13} + 12 q^{15} - 4 q^{21} + 16 q^{23} - 4 q^{25} - 4 q^{27} + 8 q^{33} + 8 q^{35} + 8 q^{37} - 4 q^{39} - 8 q^{45} + 16 q^{47} - 4 q^{49} + 16 q^{51} + 20 q^{57} + 8 q^{59}+ \cdots - 24 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/672\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(421\) \(449\) \(577\)
\(\chi(n)\) \(-1\) \(1\) \(-1\) \(1\)

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
\(3\) −0.292893 + 1.70711i −0.169102 + 0.985599i
\(4\) 0 0
\(5\) − 3.41421i − 1.52688i −0.645877 − 0.763441i \(-0.723508\pi\)
0.645877 − 0.763441i \(-0.276492\pi\)
\(6\) 0 0
\(7\) 1.00000i 0.377964i
\(8\) 0 0
\(9\) −2.82843 − 1.00000i −0.942809 − 0.333333i
\(10\) 0 0
\(11\) −2.00000 −0.603023 −0.301511 − 0.953463i \(-0.597491\pi\)
−0.301511 + 0.953463i \(0.597491\pi\)
\(12\) 0 0
\(13\) 3.41421 0.946932 0.473466 − 0.880812i \(-0.343003\pi\)
0.473466 + 0.880812i \(0.343003\pi\)
\(14\) 0 0
\(15\) 5.82843 + 1.00000i 1.50489 + 0.258199i
\(16\) 0 0
\(17\) − 4.82843i − 1.17107i −0.810649 − 0.585533i \(-0.800885\pi\)
0.810649 − 0.585533i \(-0.199115\pi\)
\(18\) 0 0
\(19\) − 6.24264i − 1.43216i −0.698018 − 0.716080i \(-0.745935\pi\)
0.698018 − 0.716080i \(-0.254065\pi\)
\(20\) 0 0
\(21\) −1.70711 − 0.292893i −0.372521 − 0.0639145i
\(22\) 0 0
\(23\) 4.00000 0.834058 0.417029 − 0.908893i \(-0.363071\pi\)
0.417029 + 0.908893i \(0.363071\pi\)
\(24\) 0 0
\(25\) −6.65685 −1.33137
\(26\) 0 0
\(27\) 2.53553 − 4.53553i 0.487964 − 0.872864i
\(28\) 0 0
\(29\) − 4.82843i − 0.896616i −0.893879 − 0.448308i \(-0.852027\pi\)
0.893879 − 0.448308i \(-0.147973\pi\)
\(30\) 0 0
\(31\) 1.17157i 0.210421i 0.994450 + 0.105210i \(0.0335516\pi\)
−0.994450 + 0.105210i \(0.966448\pi\)
\(32\) 0 0
\(33\) 0.585786 − 3.41421i 0.101972 − 0.594338i
\(34\) 0 0
\(35\) 3.41421 0.577107
\(36\) 0 0
\(37\) −0.828427 −0.136193 −0.0680963 − 0.997679i \(-0.521693\pi\)
−0.0680963 + 0.997679i \(0.521693\pi\)
\(38\) 0 0
\(39\) −1.00000 + 5.82843i −0.160128 + 0.933295i
\(40\) 0 0
\(41\) 10.0000i 1.56174i 0.624695 + 0.780869i \(0.285223\pi\)
−0.624695 + 0.780869i \(0.714777\pi\)
\(42\) 0 0
\(43\) − 10.4853i − 1.59899i −0.600672 − 0.799495i \(-0.705100\pi\)
0.600672 − 0.799495i \(-0.294900\pi\)
\(44\) 0 0
\(45\) −3.41421 + 9.65685i −0.508961 + 1.43956i
\(46\) 0 0
\(47\) −1.65685 −0.241677 −0.120839 − 0.992672i \(-0.538558\pi\)
−0.120839 + 0.992672i \(0.538558\pi\)
\(48\) 0 0
\(49\) −1.00000 −0.142857
\(50\) 0 0
\(51\) 8.24264 + 1.41421i 1.15420 + 0.198030i
\(52\) 0 0
\(53\) − 13.3137i − 1.82878i −0.404836 − 0.914389i \(-0.632671\pi\)
0.404836 − 0.914389i \(-0.367329\pi\)
\(54\) 0 0
\(55\) 6.82843i 0.920745i
\(56\) 0 0
\(57\) 10.6569 + 1.82843i 1.41153 + 0.242181i
\(58\) 0 0
\(59\) 6.24264 0.812723 0.406361 − 0.913712i \(-0.366797\pi\)
0.406361 + 0.913712i \(0.366797\pi\)
\(60\) 0 0
\(61\) 14.2426 1.82358 0.911792 − 0.410653i \(-0.134699\pi\)
0.911792 + 0.410653i \(0.134699\pi\)
\(62\) 0 0
\(63\) 1.00000 − 2.82843i 0.125988 − 0.356348i
\(64\) 0 0
\(65\) − 11.6569i − 1.44585i
\(66\) 0 0
\(67\) 9.31371i 1.13785i 0.822389 + 0.568925i \(0.192641\pi\)
−0.822389 + 0.568925i \(0.807359\pi\)
\(68\) 0 0
\(69\) −1.17157 + 6.82843i −0.141041 + 0.822046i
\(70\) 0 0
\(71\) 0.343146 0.0407239 0.0203620 − 0.999793i \(-0.493518\pi\)
0.0203620 + 0.999793i \(0.493518\pi\)
\(72\) 0 0
\(73\) −3.17157 −0.371205 −0.185602 − 0.982625i \(-0.559424\pi\)
−0.185602 + 0.982625i \(0.559424\pi\)
\(74\) 0 0
\(75\) 1.94975 − 11.3640i 0.225137 − 1.31220i
\(76\) 0 0
\(77\) − 2.00000i − 0.227921i
\(78\) 0 0
\(79\) 4.00000i 0.450035i 0.974355 + 0.225018i \(0.0722440\pi\)
−0.974355 + 0.225018i \(0.927756\pi\)
\(80\) 0 0
\(81\) 7.00000 + 5.65685i 0.777778 + 0.628539i
\(82\) 0 0
\(83\) −11.4142 −1.25287 −0.626436 − 0.779473i \(-0.715487\pi\)
−0.626436 + 0.779473i \(0.715487\pi\)
\(84\) 0 0
\(85\) −16.4853 −1.78808
\(86\) 0 0
\(87\) 8.24264 + 1.41421i 0.883704 + 0.151620i
\(88\) 0 0
\(89\) − 7.65685i − 0.811625i −0.913956 − 0.405812i \(-0.866989\pi\)
0.913956 − 0.405812i \(-0.133011\pi\)
\(90\) 0 0
\(91\) 3.41421i 0.357907i
\(92\) 0 0
\(93\) −2.00000 − 0.343146i −0.207390 − 0.0355826i
\(94\) 0 0
\(95\) −21.3137 −2.18674
\(96\) 0 0
\(97\) 5.31371 0.539525 0.269763 − 0.962927i \(-0.413055\pi\)
0.269763 + 0.962927i \(0.413055\pi\)
\(98\) 0 0
\(99\) 5.65685 + 2.00000i 0.568535 + 0.201008i
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 672.2.h.a.575.4 yes 4
3.2 odd 2 672.2.h.d.575.2 yes 4
4.3 odd 2 672.2.h.d.575.1 yes 4
8.3 odd 2 1344.2.h.a.575.4 4
8.5 even 2 1344.2.h.e.575.1 4
12.11 even 2 inner 672.2.h.a.575.3 ✓ 4
24.5 odd 2 1344.2.h.a.575.3 4
24.11 even 2 1344.2.h.e.575.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
672.2.h.a.575.3 ✓ 4 12.11 even 2 inner
672.2.h.a.575.4 yes 4 1.1 even 1 trivial
672.2.h.d.575.1 yes 4 4.3 odd 2
672.2.h.d.575.2 yes 4 3.2 odd 2
1344.2.h.a.575.3 4 24.5 odd 2
1344.2.h.a.575.4 4 8.3 odd 2
1344.2.h.e.575.1 4 8.5 even 2
1344.2.h.e.575.2 4 24.11 even 2