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Properties

Label 147.3.h.b.116.2
Level $147$
Weight $3$
Character 147.116
Analytic conductor $4.005$
Analytic rank $0$
Dimension $4$
Inner twists $4$

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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] = [147,3,Mod(116,147)] 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("147.116"); S:= CuspForms(chi, 3); 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, 4])) N = Newforms(chi, 3, names="a")
 
Level: \( N \) \(=\) \( 147 = 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 147.h (of order \(6\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,4] 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: \(4.00545988610\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{-5})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 5x^{2} + 25 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 21)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 116.2
Root \(1.93649 - 1.11803i\) of defining polynomial
Character \(\chi\) \(=\) 147.116
Dual form 147.3.h.b.128.2

$q$-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+(1.93649 - 1.11803i) q^{2} +(2.93649 - 0.614017i) q^{3} +(0.500000 - 0.866025i) q^{4} +(1.93649 - 1.11803i) q^{5} +(5.00000 - 4.47214i) q^{6} +6.70820i q^{8} +(8.24597 - 3.60611i) q^{9} +(2.50000 - 4.33013i) q^{10} +(-9.68246 - 5.59017i) q^{11} +(0.936492 - 2.85008i) q^{12} +2.00000 q^{13} +(5.00000 - 4.47214i) q^{15} +(9.50000 + 16.4545i) q^{16} +(-23.2379 - 13.4164i) q^{17} +(11.9365 - 16.2025i) q^{18} +(8.00000 + 13.8564i) q^{19} -2.23607i q^{20} -25.0000 q^{22} +(-11.6190 + 6.70820i) q^{23} +(4.11895 + 19.6986i) q^{24} +(-10.0000 + 17.3205i) q^{25} +(3.87298 - 2.23607i) q^{26} +(22.0000 - 15.6525i) q^{27} -15.6525i q^{29} +(4.68246 - 14.2504i) q^{30} +(-1.50000 + 2.59808i) q^{31} +(13.5554 + 7.82624i) q^{32} +(-31.8649 - 10.4703i) q^{33} -60.0000 q^{34} +(1.00000 - 8.94427i) q^{36} +(-6.00000 - 10.3923i) q^{37} +(30.9839 + 17.8885i) q^{38} +(5.87298 - 1.22803i) q^{39} +(7.50000 + 12.9904i) q^{40} +31.3050i q^{41} +44.0000 q^{43} +(-9.68246 + 5.59017i) q^{44} +(11.9365 - 16.2025i) q^{45} +(-15.0000 + 25.9808i) q^{46} +(-11.6190 + 6.70820i) q^{47} +(38.0000 + 42.4853i) q^{48} +44.7214i q^{50} +(-76.4758 - 25.1287i) q^{51} +(1.00000 - 1.73205i) q^{52} +(17.4284 + 10.0623i) q^{53} +(25.1028 - 54.9076i) q^{54} -25.0000 q^{55} +(32.0000 + 35.7771i) q^{57} +(-17.5000 - 30.3109i) q^{58} +(17.4284 + 10.0623i) q^{59} +(-1.37298 - 6.56619i) q^{60} +(-13.0000 - 22.5167i) q^{61} +6.70820i q^{62} -41.0000 q^{64} +(3.87298 - 2.23607i) q^{65} +(-73.4123 + 15.3504i) q^{66} +(-26.0000 + 45.0333i) q^{67} +(-23.2379 + 13.4164i) q^{68} +(-30.0000 + 26.8328i) q^{69} -93.9149i q^{71} +(24.1905 + 55.3156i) q^{72} +(9.00000 - 15.5885i) q^{73} +(-23.2379 - 13.4164i) q^{74} +(-18.7298 + 57.0017i) q^{75} +16.0000 q^{76} +(10.0000 - 8.94427i) q^{78} +(39.5000 + 68.4160i) q^{79} +(36.7933 + 21.2426i) q^{80} +(54.9919 - 59.4717i) q^{81} +(35.0000 + 60.6218i) q^{82} -140.872i q^{83} -60.0000 q^{85} +(85.2056 - 49.1935i) q^{86} +(-9.61088 - 45.9634i) q^{87} +(37.5000 - 64.9519i) q^{88} +(42.6028 - 24.5967i) q^{89} +(5.00000 - 44.7214i) q^{90} +13.4164i q^{92} +(-2.80948 + 8.55025i) q^{93} +(-15.0000 + 25.9808i) q^{94} +(30.9839 + 17.8885i) q^{95} +(44.6109 + 14.6584i) q^{96} +93.0000 q^{97} +(-100.000 - 11.1803i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{3} + 2 q^{4} + 20 q^{6} + 2 q^{9} + 10 q^{10} - 4 q^{12} + 8 q^{13} + 20 q^{15} + 38 q^{16} + 40 q^{18} + 32 q^{19} - 100 q^{22} - 30 q^{24} - 40 q^{25} + 88 q^{27} - 20 q^{30} - 6 q^{31} - 50 q^{33}+ \cdots - 400 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(50\) \(52\)
\(\chi(n)\) \(-1\) \(e\left(\frac{2}{3}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.93649 − 1.11803i 0.968246 − 0.559017i 0.0695448 − 0.997579i \(-0.477845\pi\)
0.898701 + 0.438562i \(0.144512\pi\)
\(3\) 2.93649 − 0.614017i 0.978831 − 0.204672i
\(4\) 0.500000 − 0.866025i 0.125000 − 0.216506i
\(5\) 1.93649 − 1.11803i 0.387298 − 0.223607i −0.293691 − 0.955901i \(-0.594884\pi\)
0.680989 + 0.732294i \(0.261550\pi\)
\(6\) 5.00000 − 4.47214i 0.833333 − 0.745356i
\(7\) 0 0
\(8\) 6.70820i 0.838525i
\(9\) 8.24597 − 3.60611i 0.916219 − 0.400679i
\(10\) 2.50000 − 4.33013i 0.250000 − 0.433013i
\(11\) −9.68246 − 5.59017i −0.880223 − 0.508197i −0.00949140 − 0.999955i \(-0.503021\pi\)
−0.870732 + 0.491758i \(0.836355\pi\)
\(12\) 0.936492 − 2.85008i 0.0780410 − 0.237507i
\(13\) 2.00000 0.153846 0.0769231 − 0.997037i \(-0.475490\pi\)
0.0769231 + 0.997037i \(0.475490\pi\)
\(14\) 0 0
\(15\) 5.00000 − 4.47214i 0.333333 − 0.298142i
\(16\) 9.50000 + 16.4545i 0.593750 + 1.02841i
\(17\) −23.2379 − 13.4164i −1.36694 − 0.789200i −0.376400 − 0.926457i \(-0.622838\pi\)
−0.990535 + 0.137257i \(0.956171\pi\)
\(18\) 11.9365 − 16.2025i 0.663138 − 0.900137i
\(19\) 8.00000 + 13.8564i 0.421053 + 0.729285i 0.996043 − 0.0888758i \(-0.0283274\pi\)
−0.574990 + 0.818160i \(0.694994\pi\)
\(20\) − 2.23607i − 0.111803i
\(21\) 0 0
\(22\) −25.0000 −1.13636
\(23\) −11.6190 + 6.70820i −0.505172 + 0.291661i −0.730847 − 0.682542i \(-0.760875\pi\)
0.225675 + 0.974203i \(0.427541\pi\)
\(24\) 4.11895 + 19.6986i 0.171623 + 0.820774i
\(25\) −10.0000 + 17.3205i −0.400000 + 0.692820i
\(26\) 3.87298 − 2.23607i 0.148961 − 0.0860026i
\(27\) 22.0000 − 15.6525i 0.814815 − 0.579721i
\(28\) 0 0
\(29\) − 15.6525i − 0.539741i −0.962897 − 0.269870i \(-0.913019\pi\)
0.962897 − 0.269870i \(-0.0869808\pi\)
\(30\) 4.68246 − 14.2504i 0.156082 − 0.475014i
\(31\) −1.50000 + 2.59808i −0.0483871 + 0.0838089i −0.889205 − 0.457510i \(-0.848741\pi\)
0.840817 + 0.541319i \(0.182075\pi\)
\(32\) 13.5554 + 7.82624i 0.423608 + 0.244570i
\(33\) −31.8649 − 10.4703i −0.965604 − 0.317282i
\(34\) −60.0000 −1.76471
\(35\) 0 0
\(36\) 1.00000 − 8.94427i 0.0277778 − 0.248452i
\(37\) −6.00000 − 10.3923i −0.162162 − 0.280873i 0.773482 − 0.633819i \(-0.218513\pi\)
−0.935644 + 0.352946i \(0.885180\pi\)
\(38\) 30.9839 + 17.8885i 0.815365 + 0.470751i
\(39\) 5.87298 − 1.22803i 0.150589 − 0.0314880i
\(40\) 7.50000 + 12.9904i 0.187500 + 0.324760i
\(41\) 31.3050i 0.763535i 0.924258 + 0.381768i \(0.124685\pi\)
−0.924258 + 0.381768i \(0.875315\pi\)
\(42\) 0 0
\(43\) 44.0000 1.02326 0.511628 − 0.859207i \(-0.329043\pi\)
0.511628 + 0.859207i \(0.329043\pi\)
\(44\) −9.68246 + 5.59017i −0.220056 + 0.127049i
\(45\) 11.9365 − 16.2025i 0.265255 − 0.360055i
\(46\) −15.0000 + 25.9808i −0.326087 + 0.564799i
\(47\) −11.6190 + 6.70820i −0.247212 + 0.142728i −0.618487 − 0.785795i \(-0.712254\pi\)
0.371275 + 0.928523i \(0.378921\pi\)
\(48\) 38.0000 + 42.4853i 0.791667 + 0.885110i
\(49\) 0 0
\(50\) 44.7214i 0.894427i
\(51\) −76.4758 − 25.1287i −1.49953 − 0.492720i
\(52\) 1.00000 − 1.73205i 0.0192308 − 0.0333087i
\(53\) 17.4284 + 10.0623i 0.328838 + 0.189855i 0.655325 − 0.755347i \(-0.272532\pi\)
−0.326487 + 0.945202i \(0.605865\pi\)
\(54\) 25.1028 − 54.9076i 0.464867 − 1.01681i
\(55\) −25.0000 −0.454545
\(56\) 0 0
\(57\) 32.0000 + 35.7771i 0.561404 + 0.627668i
\(58\) −17.5000 − 30.3109i −0.301724 − 0.522602i
\(59\) 17.4284 + 10.0623i 0.295397 + 0.170548i 0.640373 − 0.768064i \(-0.278780\pi\)
−0.344976 + 0.938611i \(0.612113\pi\)
\(60\) −1.37298 − 6.56619i −0.0228831 − 0.109437i
\(61\) −13.0000 − 22.5167i −0.213115 − 0.369126i 0.739573 − 0.673076i \(-0.235027\pi\)
−0.952688 + 0.303951i \(0.901694\pi\)
\(62\) 6.70820i 0.108197i
\(63\) 0 0
\(64\) −41.0000 −0.640625
\(65\) 3.87298 − 2.23607i 0.0595844 − 0.0344010i
\(66\) −73.4123 + 15.3504i −1.11231 + 0.232582i
\(67\) −26.0000 + 45.0333i −0.388060 + 0.672139i −0.992189 − 0.124748i \(-0.960188\pi\)
0.604129 + 0.796887i \(0.293521\pi\)
\(68\) −23.2379 + 13.4164i −0.341734 + 0.197300i
\(69\) −30.0000 + 26.8328i −0.434783 + 0.388881i
\(70\) 0 0
\(71\) − 93.9149i − 1.32274i −0.750058 − 0.661372i \(-0.769974\pi\)
0.750058 − 0.661372i \(-0.230026\pi\)
\(72\) 24.1905 + 55.3156i 0.335980 + 0.768273i
\(73\) 9.00000 − 15.5885i 0.123288 − 0.213541i −0.797775 − 0.602956i \(-0.793990\pi\)
0.921062 + 0.389415i \(0.127323\pi\)
\(74\) −23.2379 − 13.4164i −0.314026 − 0.181303i
\(75\) −18.7298 + 57.0017i −0.249731 + 0.760023i
\(76\) 16.0000 0.210526
\(77\) 0 0
\(78\) 10.0000 − 8.94427i 0.128205 − 0.114670i
\(79\) 39.5000 + 68.4160i 0.500000 + 0.866025i 1.00000 \(0\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(80\) 36.7933 + 21.2426i 0.459917 + 0.265533i
\(81\) 54.9919 − 59.4717i 0.678913 − 0.734219i
\(82\) 35.0000 + 60.6218i 0.426829 + 0.739290i
\(83\) − 140.872i − 1.69726i −0.528990 − 0.848628i \(-0.677429\pi\)
0.528990 − 0.848628i \(-0.322571\pi\)
\(84\) 0 0
\(85\) −60.0000 −0.705882
\(86\) 85.2056 − 49.1935i 0.990763 − 0.572017i
\(87\) −9.61088 − 45.9634i −0.110470 − 0.528315i
\(88\) 37.5000 − 64.9519i 0.426136 − 0.738090i
\(89\) 42.6028 − 24.5967i 0.478683 − 0.276368i −0.241184 − 0.970479i \(-0.577536\pi\)
0.719868 + 0.694111i \(0.244202\pi\)
\(90\) 5.00000 − 44.7214i 0.0555556 − 0.496904i
\(91\) 0 0
\(92\) 13.4164i 0.145831i
\(93\) −2.80948 + 8.55025i −0.0302094 + 0.0919382i
\(94\) −15.0000 + 25.9808i −0.159574 + 0.276391i
\(95\) 30.9839 + 17.8885i 0.326146 + 0.188300i
\(96\) 44.6109 + 14.6584i 0.464697 + 0.152692i
\(97\) 93.0000 0.958763 0.479381 − 0.877607i \(-0.340861\pi\)
0.479381 + 0.877607i \(0.340861\pi\)
\(98\) 0 0
\(99\) −100.000 − 11.1803i −1.01010 − 0.112933i
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 147.3.h.b.116.2 4
3.2 odd 2 inner 147.3.h.b.116.1 4
7.2 even 3 inner 147.3.h.b.128.1 4
7.3 odd 6 147.3.b.d.50.2 2
7.4 even 3 147.3.b.c.50.2 2
7.5 odd 6 21.3.h.b.2.1 ✓ 4
7.6 odd 2 21.3.h.b.11.2 yes 4
21.2 odd 6 inner 147.3.h.b.128.2 4
21.5 even 6 21.3.h.b.2.2 yes 4
21.11 odd 6 147.3.b.c.50.1 2
21.17 even 6 147.3.b.d.50.1 2
21.20 even 2 21.3.h.b.11.1 yes 4
28.19 even 6 336.3.bn.f.65.1 4
28.27 even 2 336.3.bn.f.305.2 4
84.47 odd 6 336.3.bn.f.65.2 4
84.83 odd 2 336.3.bn.f.305.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
21.3.h.b.2.1 ✓ 4 7.5 odd 6
21.3.h.b.2.2 yes 4 21.5 even 6
21.3.h.b.11.1 yes 4 21.20 even 2
21.3.h.b.11.2 yes 4 7.6 odd 2
147.3.b.c.50.1 2 21.11 odd 6
147.3.b.c.50.2 2 7.4 even 3
147.3.b.d.50.1 2 21.17 even 6
147.3.b.d.50.2 2 7.3 odd 6
147.3.h.b.116.1 4 3.2 odd 2 inner
147.3.h.b.116.2 4 1.1 even 1 trivial
147.3.h.b.128.1 4 7.2 even 3 inner
147.3.h.b.128.2 4 21.2 odd 6 inner
336.3.bn.f.65.1 4 28.19 even 6
336.3.bn.f.65.2 4 84.47 odd 6
336.3.bn.f.305.1 4 84.83 odd 2
336.3.bn.f.305.2 4 28.27 even 2