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Properties

Label 1232.2.q.h.529.1
Level $1232$
Weight $2$
Character 1232.529
Analytic conductor $9.838$
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] = [1232,2,Mod(177,1232)] 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("1232.177"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1232, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 2, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1232 = 2^{4} \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1232.q (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,2,0,0,0,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(9.83756952902\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{-3})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 2x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 616)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 529.1
Root \(-0.707107 + 1.22474i\) of defining polynomial
Character \(\chi\) \(=\) 1232.529
Dual form 1232.2.q.h.177.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.207107 + 0.358719i) q^{3} +(-0.707107 - 1.22474i) q^{5} +(-2.62132 + 0.358719i) q^{7} +(1.41421 + 2.44949i) q^{9} +(0.500000 - 0.866025i) q^{11} +1.82843 q^{13} +0.585786 q^{15} +(1.00000 - 1.73205i) q^{17} +(-1.29289 - 2.23936i) q^{19} +(0.414214 - 1.01461i) q^{21} +(2.70711 + 4.68885i) q^{23} +(1.50000 - 2.59808i) q^{25} -2.41421 q^{27} +1.00000 q^{29} +(-4.82843 + 8.36308i) q^{31} +(0.207107 + 0.358719i) q^{33} +(2.29289 + 2.95680i) q^{35} +(4.53553 + 7.85578i) q^{37} +(-0.378680 + 0.655892i) q^{39} +6.24264 q^{41} +8.00000 q^{43} +(2.00000 - 3.46410i) q^{45} +(2.41421 + 4.18154i) q^{47} +(6.74264 - 1.88064i) q^{49} +(0.414214 + 0.717439i) q^{51} +(-1.29289 + 2.23936i) q^{53} -1.41421 q^{55} +1.07107 q^{57} +(2.20711 - 3.82282i) q^{59} +(4.08579 + 7.07679i) q^{61} +(-4.58579 - 5.91359i) q^{63} +(-1.29289 - 2.23936i) q^{65} +(4.03553 - 6.98975i) q^{67} -2.24264 q^{69} +5.75736 q^{71} +(1.70711 - 2.95680i) q^{73} +(0.621320 + 1.07616i) q^{75} +(-1.00000 + 2.44949i) q^{77} +(4.03553 + 6.98975i) q^{79} +(-3.74264 + 6.48244i) q^{81} +10.4853 q^{83} -2.82843 q^{85} +(-0.207107 + 0.358719i) q^{87} +(4.58579 + 7.94282i) q^{89} +(-4.79289 + 0.655892i) q^{91} +(-2.00000 - 3.46410i) q^{93} +(-1.82843 + 3.16693i) q^{95} -15.8284 q^{97} +2.82843 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{3} - 2 q^{7} + 2 q^{11} - 4 q^{13} + 8 q^{15} + 4 q^{17} - 8 q^{19} - 4 q^{21} + 8 q^{23} + 6 q^{25} - 4 q^{27} + 4 q^{29} - 8 q^{31} - 2 q^{33} + 12 q^{35} + 4 q^{37} - 10 q^{39} + 8 q^{41}+ \cdots - 52 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1232\mathbb{Z}\right)^\times\).

\(n\) \(309\) \(353\) \(463\) \(673\)
\(\chi(n)\) \(1\) \(e\left(\frac{2}{3}\right)\) \(1\) \(1\)

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
\(3\) −0.207107 + 0.358719i −0.119573 + 0.207107i −0.919599 − 0.392859i \(-0.871486\pi\)
0.800025 + 0.599966i \(0.204819\pi\)
\(4\) 0 0
\(5\) −0.707107 − 1.22474i −0.316228 − 0.547723i 0.663470 − 0.748203i \(-0.269083\pi\)
−0.979698 + 0.200480i \(0.935750\pi\)
\(6\) 0 0
\(7\) −2.62132 + 0.358719i −0.990766 + 0.135583i
\(8\) 0 0
\(9\) 1.41421 + 2.44949i 0.471405 + 0.816497i
\(10\) 0 0
\(11\) 0.500000 − 0.866025i 0.150756 − 0.261116i
\(12\) 0 0
\(13\) 1.82843 0.507114 0.253557 − 0.967320i \(-0.418399\pi\)
0.253557 + 0.967320i \(0.418399\pi\)
\(14\) 0 0
\(15\) 0.585786 0.151249
\(16\) 0 0
\(17\) 1.00000 − 1.73205i 0.242536 − 0.420084i −0.718900 − 0.695113i \(-0.755354\pi\)
0.961436 + 0.275029i \(0.0886875\pi\)
\(18\) 0 0
\(19\) −1.29289 − 2.23936i −0.296610 − 0.513744i 0.678748 − 0.734371i \(-0.262523\pi\)
−0.975358 + 0.220628i \(0.929189\pi\)
\(20\) 0 0
\(21\) 0.414214 − 1.01461i 0.0903888 − 0.221406i
\(22\) 0 0
\(23\) 2.70711 + 4.68885i 0.564471 + 0.977692i 0.997099 + 0.0761195i \(0.0242531\pi\)
−0.432628 + 0.901573i \(0.642414\pi\)
\(24\) 0 0
\(25\) 1.50000 − 2.59808i 0.300000 − 0.519615i
\(26\) 0 0
\(27\) −2.41421 −0.464616
\(28\) 0 0
\(29\) 1.00000 0.185695 0.0928477 − 0.995680i \(-0.470403\pi\)
0.0928477 + 0.995680i \(0.470403\pi\)
\(30\) 0 0
\(31\) −4.82843 + 8.36308i −0.867211 + 1.50205i −0.00237631 + 0.999997i \(0.500756\pi\)
−0.864835 + 0.502057i \(0.832577\pi\)
\(32\) 0 0
\(33\) 0.207107 + 0.358719i 0.0360527 + 0.0624450i
\(34\) 0 0
\(35\) 2.29289 + 2.95680i 0.387570 + 0.499790i
\(36\) 0 0
\(37\) 4.53553 + 7.85578i 0.745637 + 1.29148i 0.949896 + 0.312565i \(0.101188\pi\)
−0.204259 + 0.978917i \(0.565479\pi\)
\(38\) 0 0
\(39\) −0.378680 + 0.655892i −0.0606373 + 0.105027i
\(40\) 0 0
\(41\) 6.24264 0.974937 0.487468 − 0.873141i \(-0.337920\pi\)
0.487468 + 0.873141i \(0.337920\pi\)
\(42\) 0 0
\(43\) 8.00000 1.21999 0.609994 − 0.792406i \(-0.291172\pi\)
0.609994 + 0.792406i \(0.291172\pi\)
\(44\) 0 0
\(45\) 2.00000 − 3.46410i 0.298142 − 0.516398i
\(46\) 0 0
\(47\) 2.41421 + 4.18154i 0.352149 + 0.609940i 0.986626 − 0.163002i \(-0.0521176\pi\)
−0.634477 + 0.772942i \(0.718784\pi\)
\(48\) 0 0
\(49\) 6.74264 − 1.88064i 0.963234 − 0.268662i
\(50\) 0 0
\(51\) 0.414214 + 0.717439i 0.0580015 + 0.100462i
\(52\) 0 0
\(53\) −1.29289 + 2.23936i −0.177593 + 0.307599i −0.941055 − 0.338252i \(-0.890164\pi\)
0.763463 + 0.645852i \(0.223498\pi\)
\(54\) 0 0
\(55\) −1.41421 −0.190693
\(56\) 0 0
\(57\) 1.07107 0.141866
\(58\) 0 0
\(59\) 2.20711 − 3.82282i 0.287341 − 0.497689i −0.685833 − 0.727759i \(-0.740562\pi\)
0.973174 + 0.230070i \(0.0738954\pi\)
\(60\) 0 0
\(61\) 4.08579 + 7.07679i 0.523131 + 0.906090i 0.999638 + 0.0269190i \(0.00856962\pi\)
−0.476506 + 0.879171i \(0.658097\pi\)
\(62\) 0 0
\(63\) −4.58579 − 5.91359i −0.577755 − 0.745042i
\(64\) 0 0
\(65\) −1.29289 − 2.23936i −0.160364 − 0.277758i
\(66\) 0 0
\(67\) 4.03553 − 6.98975i 0.493019 − 0.853934i −0.506949 − 0.861976i \(-0.669227\pi\)
0.999968 + 0.00804237i \(0.00255999\pi\)
\(68\) 0 0
\(69\) −2.24264 −0.269982
\(70\) 0 0
\(71\) 5.75736 0.683273 0.341636 − 0.939832i \(-0.389019\pi\)
0.341636 + 0.939832i \(0.389019\pi\)
\(72\) 0 0
\(73\) 1.70711 − 2.95680i 0.199802 − 0.346067i −0.748662 − 0.662952i \(-0.769304\pi\)
0.948464 + 0.316885i \(0.102637\pi\)
\(74\) 0 0
\(75\) 0.621320 + 1.07616i 0.0717439 + 0.124264i
\(76\) 0 0
\(77\) −1.00000 + 2.44949i −0.113961 + 0.279145i
\(78\) 0 0
\(79\) 4.03553 + 6.98975i 0.454033 + 0.786408i 0.998632 − 0.0522883i \(-0.0166515\pi\)
−0.544599 + 0.838697i \(0.683318\pi\)
\(80\) 0 0
\(81\) −3.74264 + 6.48244i −0.415849 + 0.720272i
\(82\) 0 0
\(83\) 10.4853 1.15091 0.575455 − 0.817834i \(-0.304825\pi\)
0.575455 + 0.817834i \(0.304825\pi\)
\(84\) 0 0
\(85\) −2.82843 −0.306786
\(86\) 0 0
\(87\) −0.207107 + 0.358719i −0.0222042 + 0.0384588i
\(88\) 0 0
\(89\) 4.58579 + 7.94282i 0.486092 + 0.841937i 0.999872 − 0.0159854i \(-0.00508851\pi\)
−0.513780 + 0.857922i \(0.671755\pi\)
\(90\) 0 0
\(91\) −4.79289 + 0.655892i −0.502432 + 0.0687562i
\(92\) 0 0
\(93\) −2.00000 − 3.46410i −0.207390 − 0.359211i
\(94\) 0 0
\(95\) −1.82843 + 3.16693i −0.187593 + 0.324920i
\(96\) 0 0
\(97\) −15.8284 −1.60713 −0.803567 − 0.595215i \(-0.797067\pi\)
−0.803567 + 0.595215i \(0.797067\pi\)
\(98\) 0 0
\(99\) 2.82843 0.284268
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 1232.2.q.h.529.1 4
4.3 odd 2 616.2.q.b.529.2 yes 4
7.2 even 3 inner 1232.2.q.h.177.1 4
7.3 odd 6 8624.2.a.cd.1.1 2
7.4 even 3 8624.2.a.bg.1.2 2
28.3 even 6 4312.2.a.m.1.2 2
28.11 odd 6 4312.2.a.u.1.1 2
28.23 odd 6 616.2.q.b.177.2 ✓ 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
616.2.q.b.177.2 ✓ 4 28.23 odd 6
616.2.q.b.529.2 yes 4 4.3 odd 2
1232.2.q.h.177.1 4 7.2 even 3 inner
1232.2.q.h.529.1 4 1.1 even 1 trivial
4312.2.a.m.1.2 2 28.3 even 6
4312.2.a.u.1.1 2 28.11 odd 6
8624.2.a.bg.1.2 2 7.4 even 3
8624.2.a.cd.1.1 2 7.3 odd 6