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Properties

Label 147.4.g.c.80.2
Level $147$
Weight $4$
Character 147.80
Analytic conductor $8.673$
Analytic rank $0$
Dimension $8$
Inner twists $8$

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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] = [147,4,Mod(68,147)] 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("147.68"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(147, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([3, 5])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 147 = 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 147.g (of order \(6\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,0,0] 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: \(8.67328077084\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{-3}, \sqrt{-17})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 16x^{6} + 175x^{4} - 1296x^{2} + 6561 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 3^{2} \)
Twist minimal: no (minimal twist has level 21)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 80.2
Root \(-2.17132 + 2.07011i\) of defining polynomial
Character \(\chi\) \(=\) 147.80
Dual form 147.4.g.c.68.2

$q$-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+(-3.57071 + 2.06155i) q^{2} +(3.58554 - 3.76084i) q^{3} +(4.50000 - 7.79423i) q^{4} +(-5.04975 - 8.74643i) q^{5} +(-5.04975 + 20.8207i) q^{6} +4.12311i q^{8} +(-1.28786 - 26.9693i) q^{9} +(36.0624 + 20.8207i) q^{10} +(-28.5657 - 16.4924i) q^{11} +(-13.1779 - 44.8703i) q^{12} +56.3383i q^{13} +(-51.0000 - 12.3693i) q^{15} +(27.5000 + 47.6314i) q^{16} +(-30.2985 + 52.4786i) q^{17} +(60.1971 + 93.6446i) q^{18} +(-31.8198 + 18.3712i) q^{19} -90.8955 q^{20} +136.000 q^{22} +(-78.5557 + 45.3542i) q^{23} +(15.5063 + 14.7835i) q^{24} +(11.5000 - 19.9186i) q^{25} +(-116.144 - 201.168i) q^{26} +(-106.045 - 91.8559i) q^{27} +57.7235i q^{29} +(207.606 - 60.9719i) q^{30} +(-220.617 - 127.373i) q^{31} +(-224.955 - 129.878i) q^{32} +(-164.449 + 48.2969i) q^{33} -249.848i q^{34} +(-216.000 - 111.324i) q^{36} +(-115.000 - 199.186i) q^{37} +(75.7463 - 131.196i) q^{38} +(211.879 + 202.003i) q^{39} +(36.0624 - 20.8207i) q^{40} -141.393 q^{41} +44.0000 q^{43} +(-257.091 + 148.432i) q^{44} +(-229.381 + 147.452i) q^{45} +(187.000 - 323.894i) q^{46} +(171.692 + 297.379i) q^{47} +(277.736 + 67.3610i) q^{48} +94.8314i q^{50} +(88.7271 + 302.112i) q^{51} +(439.113 + 253.522i) q^{52} +(-178.536 - 103.078i) q^{53} +(568.021 + 109.374i) q^{54} +333.131i q^{55} +(-45.0000 + 185.540i) q^{57} +(-119.000 - 206.114i) q^{58} +(-65.6468 + 113.704i) q^{59} +(-325.909 + 341.844i) q^{60} +(-61.5183 + 35.5176i) q^{61} +1050.35 q^{62} +631.000 q^{64} +(492.759 - 284.494i) q^{65} +(487.633 - 511.474i) q^{66} +(32.0000 - 55.4256i) q^{67} +(272.687 + 472.307i) q^{68} +(-111.095 + 458.055i) q^{69} -461.788i q^{71} +(111.197 - 5.30997i) q^{72} +(76.3675 + 44.0908i) q^{73} +(821.264 + 474.157i) q^{74} +(-33.6770 - 114.668i) q^{75} +330.681i q^{76} +(-1173.00 - 284.494i) q^{78} +(221.000 + 382.783i) q^{79} +(277.736 - 481.054i) q^{80} +(-725.683 + 69.4651i) q^{81} +(504.874 - 291.489i) q^{82} -494.876 q^{83} +612.000 q^{85} +(-157.111 + 90.7083i) q^{86} +(217.089 + 206.970i) q^{87} +(68.0000 - 117.779i) q^{88} +(242.388 + 419.829i) q^{89} +(515.075 - 999.392i) q^{90} +816.375i q^{92} +(-1270.06 + 373.005i) q^{93} +(-1226.12 - 707.903i) q^{94} +(321.364 + 185.540i) q^{95} +(-1295.03 + 380.338i) q^{96} -1092.47i q^{97} +(-408.000 + 791.636i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 36 q^{4} - 96 q^{9} - 408 q^{15} + 220 q^{16} - 204 q^{18} + 1088 q^{22} + 92 q^{25} + 204 q^{30} - 1728 q^{36} - 920 q^{37} - 276 q^{39} + 352 q^{43} + 1496 q^{46} + 1224 q^{51} - 360 q^{57} - 952 q^{58}+ \cdots - 3264 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/147\mathbb{Z}\right)^\times\).

\(n\) \(50\) \(52\)
\(\chi(n)\) \(-1\) \(e\left(\frac{1}{6}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −3.57071 + 2.06155i −1.26244 + 0.728869i −0.973546 − 0.228493i \(-0.926620\pi\)
−0.288892 + 0.957362i \(0.593287\pi\)
\(3\) 3.58554 − 3.76084i 0.690037 − 0.723774i
\(4\) 4.50000 − 7.79423i 0.562500 − 0.974279i
\(5\) −5.04975 − 8.74643i −0.451664 − 0.782304i 0.546826 − 0.837246i \(-0.315836\pi\)
−0.998490 + 0.0549420i \(0.982503\pi\)
\(6\) −5.04975 + 20.8207i −0.343592 + 1.41667i
\(7\) 0 0
\(8\) 4.12311i 0.182217i
\(9\) −1.28786 − 26.9693i −0.0476984 − 0.998862i
\(10\) 36.0624 + 20.8207i 1.14039 + 0.658407i
\(11\) −28.5657 − 16.4924i −0.782990 − 0.452059i 0.0544991 − 0.998514i \(-0.482644\pi\)
−0.837489 + 0.546455i \(0.815977\pi\)
\(12\) −13.1779 − 44.8703i −0.317012 − 1.07941i
\(13\) 56.3383i 1.20196i 0.799266 + 0.600978i \(0.205222\pi\)
−0.799266 + 0.600978i \(0.794778\pi\)
\(14\) 0 0
\(15\) −51.0000 − 12.3693i −0.877876 − 0.212916i
\(16\) 27.5000 + 47.6314i 0.429688 + 0.744241i
\(17\) −30.2985 + 52.4786i −0.432263 + 0.748701i −0.997068 − 0.0765232i \(-0.975618\pi\)
0.564805 + 0.825224i \(0.308951\pi\)
\(18\) 60.1971 + 93.6446i 0.788256 + 1.22624i
\(19\) −31.8198 + 18.3712i −0.384209 + 0.221823i −0.679648 − 0.733539i \(-0.737867\pi\)
0.295439 + 0.955362i \(0.404534\pi\)
\(20\) −90.8955 −1.01624
\(21\) 0 0
\(22\) 136.000 1.31797
\(23\) −78.5557 + 45.3542i −0.712174 + 0.411174i −0.811865 − 0.583845i \(-0.801548\pi\)
0.0996916 + 0.995018i \(0.468214\pi\)
\(24\) 15.5063 + 14.7835i 0.131884 + 0.125737i
\(25\) 11.5000 − 19.9186i 0.0920000 − 0.159349i
\(26\) −116.144 − 201.168i −0.876068 − 1.51739i
\(27\) −106.045 − 91.8559i −0.755864 − 0.654729i
\(28\) 0 0
\(29\) 57.7235i 0.369620i 0.982774 + 0.184810i \(0.0591670\pi\)
−0.982774 + 0.184810i \(0.940833\pi\)
\(30\) 207.606 − 60.9719i 1.26345 − 0.371063i
\(31\) −220.617 − 127.373i −1.27819 − 0.737966i −0.301679 − 0.953410i \(-0.597547\pi\)
−0.976516 + 0.215444i \(0.930880\pi\)
\(32\) −224.955 − 129.878i −1.24271 − 0.717480i
\(33\) −164.449 + 48.2969i −0.867481 + 0.254770i
\(34\) − 249.848i − 1.26025i
\(35\) 0 0
\(36\) −216.000 − 111.324i −1.00000 − 0.515388i
\(37\) −115.000 − 199.186i −0.510970 − 0.885026i −0.999919 − 0.0127135i \(-0.995953\pi\)
0.488949 − 0.872312i \(-0.337380\pi\)
\(38\) 75.7463 − 131.196i 0.323360 − 0.560076i
\(39\) 211.879 + 202.003i 0.869945 + 0.829394i
\(40\) 36.0624 − 20.8207i 0.142549 − 0.0823009i
\(41\) −141.393 −0.538583 −0.269291 − 0.963059i \(-0.586789\pi\)
−0.269291 + 0.963059i \(0.586789\pi\)
\(42\) 0 0
\(43\) 44.0000 0.156045 0.0780225 − 0.996952i \(-0.475139\pi\)
0.0780225 + 0.996952i \(0.475139\pi\)
\(44\) −257.091 + 148.432i −0.880863 + 0.508567i
\(45\) −229.381 + 147.452i −0.759870 + 0.488464i
\(46\) 187.000 − 323.894i 0.599384 − 1.03816i
\(47\) 171.692 + 297.379i 0.532847 + 0.922917i 0.999264 + 0.0383528i \(0.0122111\pi\)
−0.466418 + 0.884565i \(0.654456\pi\)
\(48\) 277.736 + 67.3610i 0.835162 + 0.202557i
\(49\) 0 0
\(50\) 94.8314i 0.268224i
\(51\) 88.7271 + 302.112i 0.243613 + 0.829492i
\(52\) 439.113 + 253.522i 1.17104 + 0.676100i
\(53\) −178.536 − 103.078i −0.462713 − 0.267147i 0.250472 − 0.968124i \(-0.419414\pi\)
−0.713184 + 0.700977i \(0.752748\pi\)
\(54\) 568.021 + 109.374i 1.43144 + 0.275628i
\(55\) 333.131i 0.816715i
\(56\) 0 0
\(57\) −45.0000 + 185.540i −0.104568 + 0.431146i
\(58\) −119.000 − 206.114i −0.269405 − 0.466622i
\(59\) −65.6468 + 113.704i −0.144856 + 0.250897i −0.929319 − 0.369278i \(-0.879605\pi\)
0.784463 + 0.620175i \(0.212938\pi\)
\(60\) −325.909 + 341.844i −0.701245 + 0.735531i
\(61\) −61.5183 + 35.5176i −0.129125 + 0.0745502i −0.563171 − 0.826340i \(-0.690419\pi\)
0.434046 + 0.900891i \(0.357085\pi\)
\(62\) 1050.35 2.15152
\(63\) 0 0
\(64\) 631.000 1.23242
\(65\) 492.759 − 284.494i 0.940295 − 0.542880i
\(66\) 487.633 − 511.474i 0.909446 − 0.953911i
\(67\) 32.0000 − 55.4256i 0.0583496 − 0.101064i −0.835375 − 0.549680i \(-0.814750\pi\)
0.893725 + 0.448616i \(0.148083\pi\)
\(68\) 272.687 + 472.307i 0.486296 + 0.842289i
\(69\) −111.095 + 458.055i −0.193829 + 0.799178i
\(70\) 0 0
\(71\) − 461.788i − 0.771889i −0.922522 − 0.385945i \(-0.873876\pi\)
0.922522 − 0.385945i \(-0.126124\pi\)
\(72\) 111.197 − 5.30997i 0.182010 − 0.00869147i
\(73\) 76.3675 + 44.0908i 0.122440 + 0.0706910i 0.559969 − 0.828513i \(-0.310813\pi\)
−0.437529 + 0.899204i \(0.644146\pi\)
\(74\) 821.264 + 474.157i 1.29014 + 0.744860i
\(75\) −33.6770 − 114.668i −0.0518491 − 0.176544i
\(76\) 330.681i 0.499102i
\(77\) 0 0
\(78\) −1173.00 − 284.494i −1.70277 − 0.412982i
\(79\) 221.000 + 382.783i 0.314740 + 0.545145i 0.979382 − 0.202016i \(-0.0647494\pi\)
−0.664642 + 0.747162i \(0.731416\pi\)
\(80\) 277.736 − 481.054i 0.388148 − 0.672293i
\(81\) −725.683 + 69.4651i −0.995450 + 0.0952883i
\(82\) 504.874 − 291.489i 0.679927 − 0.392556i
\(83\) −494.876 −0.654454 −0.327227 − 0.944946i \(-0.606114\pi\)
−0.327227 + 0.944946i \(0.606114\pi\)
\(84\) 0 0
\(85\) 612.000 0.780950
\(86\) −157.111 + 90.7083i −0.196997 + 0.113736i
\(87\) 217.089 + 206.970i 0.267521 + 0.255051i
\(88\) 68.0000 − 117.779i 0.0823730 − 0.142674i
\(89\) 242.388 + 419.829i 0.288686 + 0.500020i 0.973496 − 0.228702i \(-0.0734482\pi\)
−0.684810 + 0.728722i \(0.740115\pi\)
\(90\) 515.075 − 999.392i 0.603263 − 1.17050i
\(91\) 0 0
\(92\) 816.375i 0.925141i
\(93\) −1270.06 + 373.005i −1.41612 + 0.415901i
\(94\) −1226.12 − 707.903i −1.34537 − 0.776751i
\(95\) 321.364 + 185.540i 0.347066 + 0.200379i
\(96\) −1295.03 + 380.338i −1.37681 + 0.404355i
\(97\) − 1092.47i − 1.14354i −0.820413 − 0.571772i \(-0.806256\pi\)
0.820413 − 0.571772i \(-0.193744\pi\)
\(98\) 0 0
\(99\) −408.000 + 791.636i −0.414197 + 0.803661i
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 147.4.g.c.80.2 8
3.2 odd 2 inner 147.4.g.c.80.3 8
7.2 even 3 inner 147.4.g.c.68.4 8
7.3 odd 6 21.4.c.b.20.2 yes 4
7.4 even 3 21.4.c.b.20.1 ✓ 4
7.5 odd 6 inner 147.4.g.c.68.3 8
7.6 odd 2 inner 147.4.g.c.80.1 8
21.2 odd 6 inner 147.4.g.c.68.1 8
21.5 even 6 inner 147.4.g.c.68.2 8
21.11 odd 6 21.4.c.b.20.4 yes 4
21.17 even 6 21.4.c.b.20.3 yes 4
21.20 even 2 inner 147.4.g.c.80.4 8
28.3 even 6 336.4.k.b.209.1 4
28.11 odd 6 336.4.k.b.209.4 4
84.11 even 6 336.4.k.b.209.2 4
84.59 odd 6 336.4.k.b.209.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
21.4.c.b.20.1 ✓ 4 7.4 even 3
21.4.c.b.20.2 yes 4 7.3 odd 6
21.4.c.b.20.3 yes 4 21.17 even 6
21.4.c.b.20.4 yes 4 21.11 odd 6
147.4.g.c.68.1 8 21.2 odd 6 inner
147.4.g.c.68.2 8 21.5 even 6 inner
147.4.g.c.68.3 8 7.5 odd 6 inner
147.4.g.c.68.4 8 7.2 even 3 inner
147.4.g.c.80.1 8 7.6 odd 2 inner
147.4.g.c.80.2 8 1.1 even 1 trivial
147.4.g.c.80.3 8 3.2 odd 2 inner
147.4.g.c.80.4 8 21.20 even 2 inner
336.4.k.b.209.1 4 28.3 even 6
336.4.k.b.209.2 4 84.11 even 6
336.4.k.b.209.3 4 84.59 odd 6
336.4.k.b.209.4 4 28.11 odd 6