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Properties

Label 84.6.i.c.37.3
Level $84$
Weight $6$
Character 84.37
Analytic conductor $13.472$
Analytic rank $0$
Dimension $8$
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] = [84,6,Mod(25,84)] 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("84.25"); S:= CuspForms(chi, 6); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(84, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 4])) N = Newforms(chi, 6, names="a")
 
Level: \( N \) \(=\) \( 84 = 2^{2} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 84.i (of order \(3\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(13.4722408643\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 2x^{7} + 703x^{6} + 2770x^{5} + 427565x^{4} + 718170x^{3} + 42175732x^{2} - 40929504x + 3559792896 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{6}\cdot 3^{3}\cdot 7 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 37.3
Root \(13.1471 + 22.7714i\) of defining polynomial
Character \(\chi\) \(=\) 84.37
Dual form 84.6.i.c.25.3

$q$-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)
 
\(f(q)\) \(=\) \(q+(4.50000 + 7.79423i) q^{3} +(15.9808 - 27.6796i) q^{5} +(85.7043 + 97.2716i) q^{7} +(-40.5000 + 70.1481i) q^{9} +(-130.442 - 225.932i) q^{11} +769.735 q^{13} +287.654 q^{15} +(776.659 + 1345.21i) q^{17} +(-375.024 + 649.561i) q^{19} +(-372.488 + 1105.72i) q^{21} +(-377.427 + 653.723i) q^{23} +(1051.73 + 1821.65i) q^{25} -729.000 q^{27} +6008.93 q^{29} +(3210.02 + 5559.92i) q^{31} +(1173.98 - 2033.39i) q^{33} +(4062.06 - 817.779i) q^{35} +(2387.86 - 4135.90i) q^{37} +(3463.81 + 5999.49i) q^{39} -5423.27 q^{41} -11896.4 q^{43} +(1294.44 + 2242.04i) q^{45} +(8714.00 - 15093.1i) q^{47} +(-2116.54 + 16673.2i) q^{49} +(-6989.93 + 12106.9i) q^{51} +(-18825.3 - 32606.4i) q^{53} -8338.26 q^{55} -6750.44 q^{57} +(-11039.0 - 19120.2i) q^{59} +(4086.69 - 7078.36i) q^{61} +(-10294.4 + 2072.49i) q^{63} +(12301.0 - 21305.9i) q^{65} +(-6500.87 - 11259.8i) q^{67} -6793.69 q^{69} -12349.6 q^{71} +(-21800.2 - 37759.0i) q^{73} +(-9465.55 + 16394.8i) q^{75} +(10797.3 - 32051.6i) q^{77} +(-38374.7 + 66467.0i) q^{79} +(-3280.50 - 5681.99i) q^{81} +21893.6 q^{83} +49646.5 q^{85} +(27040.2 + 46835.0i) q^{87} +(68483.5 - 118617. i) q^{89} +(65969.7 + 74873.4i) q^{91} +(-28890.2 + 50039.3i) q^{93} +(11986.4 + 20761.0i) q^{95} -93050.1 q^{97} +21131.6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 36 q^{3} - 42 q^{7} - 324 q^{9} - 462 q^{11} - 1204 q^{13} + 228 q^{17} + 358 q^{19} + 1404 q^{21} - 2148 q^{23} - 5454 q^{25} - 5832 q^{27} - 11064 q^{29} + 830 q^{31} + 4158 q^{33} + 7692 q^{35}+ \cdots + 74844 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/84\mathbb{Z}\right)^\times\).

\(n\) \(29\) \(43\) \(73\)
\(\chi(n)\) \(1\) \(1\) \(e\left(\frac{1}{3}\right)\)

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
\(3\) 4.50000 + 7.79423i 0.288675 + 0.500000i
\(4\) 0 0
\(5\) 15.9808 − 27.6796i 0.285873 − 0.495147i −0.686947 − 0.726707i \(-0.741050\pi\)
0.972821 + 0.231560i \(0.0743831\pi\)
\(6\) 0 0
\(7\) 85.7043 + 97.2716i 0.661086 + 0.750311i
\(8\) 0 0
\(9\) −40.5000 + 70.1481i −0.166667 + 0.288675i
\(10\) 0 0
\(11\) −130.442 − 225.932i −0.325039 − 0.562984i 0.656481 − 0.754342i \(-0.272044\pi\)
−0.981520 + 0.191359i \(0.938711\pi\)
\(12\) 0 0
\(13\) 769.735 1.26323 0.631616 − 0.775281i \(-0.282392\pi\)
0.631616 + 0.775281i \(0.282392\pi\)
\(14\) 0 0
\(15\) 287.654 0.330098
\(16\) 0 0
\(17\) 776.659 + 1345.21i 0.651791 + 1.12893i 0.982688 + 0.185268i \(0.0593152\pi\)
−0.330898 + 0.943667i \(0.607351\pi\)
\(18\) 0 0
\(19\) −375.024 + 649.561i −0.238328 + 0.412797i −0.960235 − 0.279194i \(-0.909933\pi\)
0.721906 + 0.691991i \(0.243266\pi\)
\(20\) 0 0
\(21\) −372.488 + 1105.72i −0.184316 + 0.547139i
\(22\) 0 0
\(23\) −377.427 + 653.723i −0.148769 + 0.257676i −0.930773 − 0.365598i \(-0.880865\pi\)
0.782004 + 0.623274i \(0.214198\pi\)
\(24\) 0 0
\(25\) 1051.73 + 1821.65i 0.336553 + 0.582927i
\(26\) 0 0
\(27\) −729.000 −0.192450
\(28\) 0 0
\(29\) 6008.93 1.32679 0.663395 − 0.748269i \(-0.269115\pi\)
0.663395 + 0.748269i \(0.269115\pi\)
\(30\) 0 0
\(31\) 3210.02 + 5559.92i 0.599934 + 1.03912i 0.992830 + 0.119533i \(0.0381399\pi\)
−0.392896 + 0.919583i \(0.628527\pi\)
\(32\) 0 0
\(33\) 1173.98 − 2033.39i 0.187661 − 0.325039i
\(34\) 0 0
\(35\) 4062.06 − 817.779i 0.560501 − 0.112841i
\(36\) 0 0
\(37\) 2387.86 − 4135.90i 0.286751 − 0.496668i −0.686281 − 0.727336i \(-0.740758\pi\)
0.973032 + 0.230669i \(0.0740914\pi\)
\(38\) 0 0
\(39\) 3463.81 + 5999.49i 0.364664 + 0.631616i
\(40\) 0 0
\(41\) −5423.27 −0.503850 −0.251925 − 0.967747i \(-0.581064\pi\)
−0.251925 + 0.967747i \(0.581064\pi\)
\(42\) 0 0
\(43\) −11896.4 −0.981171 −0.490585 − 0.871393i \(-0.663217\pi\)
−0.490585 + 0.871393i \(0.663217\pi\)
\(44\) 0 0
\(45\) 1294.44 + 2242.04i 0.0952911 + 0.165049i
\(46\) 0 0
\(47\) 8714.00 − 15093.1i 0.575404 − 0.996629i −0.420594 − 0.907249i \(-0.638178\pi\)
0.995998 − 0.0893798i \(-0.0284885\pi\)
\(48\) 0 0
\(49\) −2116.54 + 16673.2i −0.125932 + 0.992039i
\(50\) 0 0
\(51\) −6989.93 + 12106.9i −0.376311 + 0.651791i
\(52\) 0 0
\(53\) −18825.3 − 32606.4i −0.920561 − 1.59446i −0.798549 − 0.601930i \(-0.794399\pi\)
−0.122012 − 0.992529i \(-0.538935\pi\)
\(54\) 0 0
\(55\) −8338.26 −0.371679
\(56\) 0 0
\(57\) −6750.44 −0.275198
\(58\) 0 0
\(59\) −11039.0 − 19120.2i −0.412859 − 0.715092i 0.582342 − 0.812944i \(-0.302136\pi\)
−0.995201 + 0.0978516i \(0.968803\pi\)
\(60\) 0 0
\(61\) 4086.69 − 7078.36i 0.140620 − 0.243561i −0.787110 − 0.616812i \(-0.788424\pi\)
0.927730 + 0.373251i \(0.121757\pi\)
\(62\) 0 0
\(63\) −10294.4 + 2072.49i −0.326777 + 0.0657872i
\(64\) 0 0
\(65\) 12301.0 − 21305.9i 0.361124 − 0.625485i
\(66\) 0 0
\(67\) −6500.87 − 11259.8i −0.176923 − 0.306440i 0.763902 − 0.645332i \(-0.223281\pi\)
−0.940825 + 0.338892i \(0.889948\pi\)
\(68\) 0 0
\(69\) −6793.69 −0.171784
\(70\) 0 0
\(71\) −12349.6 −0.290742 −0.145371 − 0.989377i \(-0.546438\pi\)
−0.145371 + 0.989377i \(0.546438\pi\)
\(72\) 0 0
\(73\) −21800.2 − 37759.0i −0.478798 − 0.829303i 0.520906 − 0.853614i \(-0.325594\pi\)
−0.999704 + 0.0243110i \(0.992261\pi\)
\(74\) 0 0
\(75\) −9465.55 + 16394.8i −0.194309 + 0.336553i
\(76\) 0 0
\(77\) 10797.3 − 32051.6i 0.207534 − 0.616060i
\(78\) 0 0
\(79\) −38374.7 + 66467.0i −0.691796 + 1.19823i 0.279453 + 0.960159i \(0.409847\pi\)
−0.971249 + 0.238066i \(0.923487\pi\)
\(80\) 0 0
\(81\) −3280.50 − 5681.99i −0.0555556 − 0.0962250i
\(82\) 0 0
\(83\) 21893.6 0.348836 0.174418 − 0.984672i \(-0.444196\pi\)
0.174418 + 0.984672i \(0.444196\pi\)
\(84\) 0 0
\(85\) 49646.5 0.745318
\(86\) 0 0
\(87\) 27040.2 + 46835.0i 0.383011 + 0.663395i
\(88\) 0 0
\(89\) 68483.5 − 118617.i 0.916454 − 1.58735i 0.111696 − 0.993742i \(-0.464372\pi\)
0.804758 − 0.593603i \(-0.202295\pi\)
\(90\) 0 0
\(91\) 65969.7 + 74873.4i 0.835104 + 0.947816i
\(92\) 0 0
\(93\) −28890.2 + 50039.3i −0.346372 + 0.599934i
\(94\) 0 0
\(95\) 11986.4 + 20761.0i 0.136263 + 0.236015i
\(96\) 0 0
\(97\) −93050.1 −1.00412 −0.502062 − 0.864832i \(-0.667425\pi\)
−0.502062 + 0.864832i \(0.667425\pi\)
\(98\) 0 0
\(99\) 21131.6 0.216692
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 84.6.i.c.37.3 yes 8
3.2 odd 2 252.6.k.f.37.2 8
4.3 odd 2 336.6.q.i.289.3 8
7.2 even 3 588.6.a.n.1.2 4
7.3 odd 6 588.6.i.o.361.2 8
7.4 even 3 inner 84.6.i.c.25.3 ✓ 8
7.5 odd 6 588.6.a.p.1.3 4
7.6 odd 2 588.6.i.o.373.2 8
21.11 odd 6 252.6.k.f.109.2 8
28.11 odd 6 336.6.q.i.193.3 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
84.6.i.c.25.3 ✓ 8 7.4 even 3 inner
84.6.i.c.37.3 yes 8 1.1 even 1 trivial
252.6.k.f.37.2 8 3.2 odd 2
252.6.k.f.109.2 8 21.11 odd 6
336.6.q.i.193.3 8 28.11 odd 6
336.6.q.i.289.3 8 4.3 odd 2
588.6.a.n.1.2 4 7.2 even 3
588.6.a.p.1.3 4 7.5 odd 6
588.6.i.o.361.2 8 7.3 odd 6
588.6.i.o.373.2 8 7.6 odd 2