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Properties

Label 84.6.i.c.37.4
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 37.4
Root \(-11.2416 - 19.4709i\) of defining polynomial
Character \(\chi\) \(=\) 84.37
Dual form 84.6.i.c.25.4

$q$-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)
 
\(f(q)\) \(=\) \(q+(4.50000 + 7.79423i) q^{3} +(46.4128 - 80.3893i) q^{5} +(-118.369 - 52.8745i) q^{7} +(-40.5000 + 70.1481i) q^{9} +(-70.3812 - 121.904i) q^{11} -1111.24 q^{13} +835.430 q^{15} +(-27.4435 - 47.5335i) q^{17} +(855.929 - 1482.51i) q^{19} +(-120.546 - 1160.53i) q^{21} +(1643.95 - 2847.41i) q^{23} +(-2745.79 - 4755.85i) q^{25} -729.000 q^{27} -3790.72 q^{29} +(2423.66 + 4197.90i) q^{31} +(633.431 - 1097.13i) q^{33} +(-9744.39 + 7061.57i) q^{35} +(5683.35 - 9843.86i) q^{37} +(-5000.59 - 8661.28i) q^{39} -10385.6 q^{41} +7137.16 q^{43} +(3759.43 + 6511.53i) q^{45} +(8207.53 - 14215.9i) q^{47} +(11215.6 + 12517.4i) q^{49} +(246.991 - 427.801i) q^{51} +(10487.2 + 18164.3i) q^{53} -13066.3 q^{55} +15406.7 q^{57} +(18106.0 + 31360.5i) q^{59} +(-2474.35 + 4285.70i) q^{61} +(8503.00 - 6161.96i) q^{63} +(-51575.9 + 89332.0i) q^{65} +(11482.8 + 19888.8i) q^{67} +29591.2 q^{69} -26341.8 q^{71} +(-27693.7 - 47966.9i) q^{73} +(24712.1 - 42802.6i) q^{75} +(1885.36 + 18151.0i) q^{77} +(24978.3 - 43263.7i) q^{79} +(-3280.50 - 5681.99i) q^{81} +44858.9 q^{83} -5094.91 q^{85} +(-17058.2 - 29545.8i) q^{87} +(-63972.5 + 110804. i) q^{89} +(131537. + 58756.4i) q^{91} +(-21812.9 + 37781.1i) q^{93} +(-79452.1 - 137615. i) q^{95} -65685.9 q^{97} +11401.8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 36 q^{3} - 42 q^{7} - 324 q^{9} - 462 q^{11} - 1204 q^{13} + 228 q^{17} + 358 q^{19} + 1404 q^{21} - 2148 q^{23} - 5454 q^{25} - 5832 q^{27} - 11064 q^{29} + 830 q^{31} + 4158 q^{33} + 7692 q^{35}+ \cdots + 74844 q^{99}+O(q^{100}) \) 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
\(5\) 46.4128 − 80.3893i 0.830257 − 1.43805i −0.0675775 − 0.997714i \(-0.521527\pi\)
0.897834 − 0.440333i \(-0.145140\pi\)
\(6\) 0 0
\(7\) −118.369 − 52.8745i −0.913049 − 0.407851i
\(8\) 0 0
\(9\) −40.5000 + 70.1481i −0.166667 + 0.288675i
\(10\) 0 0
\(11\) −70.3812 − 121.904i −0.175378 − 0.303763i 0.764914 − 0.644132i \(-0.222781\pi\)
−0.940292 + 0.340369i \(0.889448\pi\)
\(12\) 0 0
\(13\) −1111.24 −1.82369 −0.911844 − 0.410537i \(-0.865341\pi\)
−0.911844 + 0.410537i \(0.865341\pi\)
\(14\) 0 0
\(15\) 835.430 0.958698
\(16\) 0 0
\(17\) −27.4435 − 47.5335i −0.0230312 − 0.0398912i 0.854280 − 0.519813i \(-0.173998\pi\)
−0.877311 + 0.479922i \(0.840665\pi\)
\(18\) 0 0
\(19\) 855.929 − 1482.51i 0.543943 − 0.942138i −0.454729 − 0.890630i \(-0.650264\pi\)
0.998673 − 0.0515079i \(-0.0164027\pi\)
\(20\) 0 0
\(21\) −120.546 − 1160.53i −0.0596490 − 0.574261i
\(22\) 0 0
\(23\) 1643.95 − 2847.41i 0.647993 − 1.12236i −0.335609 − 0.942001i \(-0.608942\pi\)
0.983602 − 0.180355i \(-0.0577247\pi\)
\(24\) 0 0
\(25\) −2745.79 − 4755.85i −0.878653 − 1.52187i
\(26\) 0 0
\(27\) −729.000 −0.192450
\(28\) 0 0
\(29\) −3790.72 −0.837003 −0.418501 − 0.908216i \(-0.637445\pi\)
−0.418501 + 0.908216i \(0.637445\pi\)
\(30\) 0 0
\(31\) 2423.66 + 4197.90i 0.452968 + 0.784563i 0.998569 − 0.0534817i \(-0.0170319\pi\)
−0.545601 + 0.838045i \(0.683699\pi\)
\(32\) 0 0
\(33\) 633.431 − 1097.13i 0.101254 − 0.175378i
\(34\) 0 0
\(35\) −9744.39 + 7061.57i −1.34457 + 0.974386i
\(36\) 0 0
\(37\) 5683.35 − 9843.86i 0.682496 − 1.18212i −0.291720 − 0.956504i \(-0.594228\pi\)
0.974217 − 0.225615i \(-0.0724391\pi\)
\(38\) 0 0
\(39\) −5000.59 − 8661.28i −0.526453 − 0.911844i
\(40\) 0 0
\(41\) −10385.6 −0.964881 −0.482440 − 0.875929i \(-0.660249\pi\)
−0.482440 + 0.875929i \(0.660249\pi\)
\(42\) 0 0
\(43\) 7137.16 0.588646 0.294323 − 0.955706i \(-0.404906\pi\)
0.294323 + 0.955706i \(0.404906\pi\)
\(44\) 0 0
\(45\) 3759.43 + 6511.53i 0.276752 + 0.479349i
\(46\) 0 0
\(47\) 8207.53 − 14215.9i 0.541961 − 0.938704i −0.456831 − 0.889554i \(-0.651015\pi\)
0.998791 − 0.0491499i \(-0.0156512\pi\)
\(48\) 0 0
\(49\) 11215.6 + 12517.4i 0.667315 + 0.744775i
\(50\) 0 0
\(51\) 246.991 − 427.801i 0.0132971 − 0.0230312i
\(52\) 0 0
\(53\) 10487.2 + 18164.3i 0.512824 + 0.888238i 0.999889 + 0.0148720i \(0.00473407\pi\)
−0.487065 + 0.873366i \(0.661933\pi\)
\(54\) 0 0
\(55\) −13066.3 −0.582435
\(56\) 0 0
\(57\) 15406.7 0.628092
\(58\) 0 0
\(59\) 18106.0 + 31360.5i 0.677161 + 1.17288i 0.975832 + 0.218521i \(0.0701231\pi\)
−0.298672 + 0.954356i \(0.596544\pi\)
\(60\) 0 0
\(61\) −2474.35 + 4285.70i −0.0851405 + 0.147468i −0.905451 − 0.424451i \(-0.860467\pi\)
0.820311 + 0.571918i \(0.193801\pi\)
\(62\) 0 0
\(63\) 8503.00 − 6161.96i 0.269911 − 0.195599i
\(64\) 0 0
\(65\) −51575.9 + 89332.0i −1.51413 + 2.62255i
\(66\) 0 0
\(67\) 11482.8 + 19888.8i 0.312507 + 0.541279i 0.978905 − 0.204318i \(-0.0654978\pi\)
−0.666397 + 0.745597i \(0.732164\pi\)
\(68\) 0 0
\(69\) 29591.2 0.748238
\(70\) 0 0
\(71\) −26341.8 −0.620154 −0.310077 − 0.950711i \(-0.600355\pi\)
−0.310077 + 0.950711i \(0.600355\pi\)
\(72\) 0 0
\(73\) −27693.7 − 47966.9i −0.608238 − 1.05350i −0.991531 − 0.129872i \(-0.958543\pi\)
0.383293 − 0.923627i \(-0.374790\pi\)
\(74\) 0 0
\(75\) 24712.1 − 42802.6i 0.507291 − 0.878653i
\(76\) 0 0
\(77\) 1885.36 + 18151.0i 0.0362383 + 0.348879i
\(78\) 0 0
\(79\) 24978.3 − 43263.7i 0.450293 − 0.779930i −0.548111 − 0.836406i \(-0.684653\pi\)
0.998404 + 0.0564752i \(0.0179862\pi\)
\(80\) 0 0
\(81\) −3280.50 − 5681.99i −0.0555556 − 0.0962250i
\(82\) 0 0
\(83\) 44858.9 0.714749 0.357374 − 0.933961i \(-0.383672\pi\)
0.357374 + 0.933961i \(0.383672\pi\)
\(84\) 0 0
\(85\) −5094.91 −0.0764873
\(86\) 0 0
\(87\) −17058.2 − 29545.8i −0.241622 − 0.418501i
\(88\) 0 0
\(89\) −63972.5 + 110804.i −0.856087 + 1.48279i 0.0195454 + 0.999809i \(0.493778\pi\)
−0.875633 + 0.482978i \(0.839555\pi\)
\(90\) 0 0
\(91\) 131537. + 58756.4i 1.66512 + 0.743793i
\(92\) 0 0
\(93\) −21812.9 + 37781.1i −0.261521 + 0.452968i
\(94\) 0 0
\(95\) −79452.1 − 137615.i −0.903226 − 1.56443i
\(96\) 0 0
\(97\) −65685.9 −0.708831 −0.354415 − 0.935088i \(-0.615320\pi\)
−0.354415 + 0.935088i \(0.615320\pi\)
\(98\) 0 0
\(99\) 11401.8 0.116919
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.37.4 yes 8
3.2 odd 2 252.6.k.f.37.1 8
4.3 odd 2 336.6.q.i.289.4 8
7.2 even 3 588.6.a.n.1.1 4
7.3 odd 6 588.6.i.o.361.1 8
7.4 even 3 inner 84.6.i.c.25.4 ✓ 8
7.5 odd 6 588.6.a.p.1.4 4
7.6 odd 2 588.6.i.o.373.1 8
21.11 odd 6 252.6.k.f.109.1 8
28.11 odd 6 336.6.q.i.193.4 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
84.6.i.c.25.4 ✓ 8 7.4 even 3 inner
84.6.i.c.37.4 yes 8 1.1 even 1 trivial
252.6.k.f.37.1 8 3.2 odd 2
252.6.k.f.109.1 8 21.11 odd 6
336.6.q.i.193.4 8 28.11 odd 6
336.6.q.i.289.4 8 4.3 odd 2
588.6.a.n.1.1 4 7.2 even 3
588.6.a.p.1.4 4 7.5 odd 6
588.6.i.o.361.1 8 7.3 odd 6
588.6.i.o.373.1 8 7.6 odd 2