↑

Properties

Label 392.6.i.a.361.1
Level $392$
Weight $6$
Character 392.361
Analytic conductor $62.870$
Analytic rank $0$
Dimension $2$
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [392,6,Mod(177,392)] 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("392.177"); S:= CuspForms(chi, 6); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(392, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 2])) N = Newforms(chi, 6, names="a")
 
Level: \( N \) \(=\) \( 392 = 2^{3} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 392.i (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,-30,0,-32,0,0,0,-657] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(62.8704573667\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 56)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 361.1
Root \(0.500000 + 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 392.361
Dual form 392.6.i.a.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)
 
\(f(q)\) \(=\) \(q+(-15.0000 + 25.9808i) q^{3} +(-16.0000 - 27.7128i) q^{5} +(-328.500 - 568.979i) q^{9} +(312.000 - 540.400i) q^{11} -708.000 q^{13} +960.000 q^{15} +(-467.000 + 808.868i) q^{17} +(-929.000 - 1609.08i) q^{19} +(560.000 + 969.948i) q^{23} +(1050.50 - 1819.52i) q^{25} +12420.0 q^{27} -1174.00 q^{29} +(-1454.00 + 2518.40i) q^{31} +(9360.00 + 16212.0i) q^{33} +(6231.00 + 10792.4i) q^{37} +(10620.0 - 18394.4i) q^{39} +2662.00 q^{41} -7144.00 q^{43} +(-10512.0 + 18207.3i) q^{45} +(3734.00 + 6467.48i) q^{47} +(-14010.0 - 24266.0i) q^{51} +(13637.0 - 23620.0i) q^{53} -19968.0 q^{55} +55740.0 q^{57} +(-1245.00 + 2156.40i) q^{59} +(5548.00 + 9609.42i) q^{61} +(11328.0 + 19620.7i) q^{65} +(-19878.0 + 34429.7i) q^{67} -33600.0 q^{69} -69888.0 q^{71} +(-8225.00 + 14246.1i) q^{73} +(31515.0 + 54585.6i) q^{75} +(-39188.0 - 67875.6i) q^{79} +(-106474. + 184419. i) q^{81} +109818. q^{83} +29888.0 q^{85} +(17610.0 - 30501.4i) q^{87} +(28483.0 + 49334.0i) q^{89} +(-43620.0 - 75552.1i) q^{93} +(-29728.0 + 51490.4i) q^{95} -115946. q^{97} -409968. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 30 q^{3} - 32 q^{5} - 657 q^{9} + 624 q^{11} - 1416 q^{13} + 1920 q^{15} - 934 q^{17} - 1858 q^{19} + 1120 q^{23} + 2101 q^{25} + 24840 q^{27} - 2348 q^{29} - 2908 q^{31} + 18720 q^{33} + 12462 q^{37}+ \cdots - 819936 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/392\mathbb{Z}\right)^\times\).

\(n\) \(197\) \(295\) \(297\)
\(\chi(n)\) \(1\) \(1\) \(e\left(\frac{2}{3}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −15.0000 + 25.9808i −0.962250 + 1.66667i −0.245423 + 0.969416i \(0.578927\pi\)
−0.716827 + 0.697251i \(0.754406\pi\)
\(4\) 0 0
\(5\) −16.0000 − 27.7128i −0.286217 − 0.495742i 0.686687 − 0.726953i \(-0.259064\pi\)
−0.972904 + 0.231212i \(0.925731\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) −328.500 − 568.979i −1.35185 − 2.34148i
\(10\) 0 0
\(11\) 312.000 − 540.400i 0.777451 − 1.34658i −0.155956 − 0.987764i \(-0.549846\pi\)
0.933407 − 0.358820i \(-0.116821\pi\)
\(12\) 0 0
\(13\) −708.000 −1.16192 −0.580958 − 0.813933i \(-0.697322\pi\)
−0.580958 + 0.813933i \(0.697322\pi\)
\(14\) 0 0
\(15\) 960.000 1.10165
\(16\) 0 0
\(17\) −467.000 + 808.868i −0.391917 + 0.678821i −0.992702 − 0.120590i \(-0.961521\pi\)
0.600785 + 0.799411i \(0.294855\pi\)
\(18\) 0 0
\(19\) −929.000 − 1609.08i −0.590380 − 1.02257i −0.994181 − 0.107721i \(-0.965645\pi\)
0.403801 − 0.914847i \(-0.367689\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 560.000 + 969.948i 0.220734 + 0.382322i 0.955031 − 0.296506i \(-0.0958215\pi\)
−0.734297 + 0.678828i \(0.762488\pi\)
\(24\) 0 0
\(25\) 1050.50 − 1819.52i 0.336160 − 0.582246i
\(26\) 0 0
\(27\) 12420.0 3.27878
\(28\) 0 0
\(29\) −1174.00 −0.259223 −0.129611 − 0.991565i \(-0.541373\pi\)
−0.129611 + 0.991565i \(0.541373\pi\)
\(30\) 0 0
\(31\) −1454.00 + 2518.40i −0.271744 + 0.470675i −0.969309 − 0.245847i \(-0.920934\pi\)
0.697564 + 0.716522i \(0.254267\pi\)
\(32\) 0 0
\(33\) 9360.00 + 16212.0i 1.49620 + 2.59150i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 6231.00 + 10792.4i 0.748262 + 1.29603i 0.948655 + 0.316312i \(0.102445\pi\)
−0.200394 + 0.979715i \(0.564222\pi\)
\(38\) 0 0
\(39\) 10620.0 − 18394.4i 1.11805 − 1.93653i
\(40\) 0 0
\(41\) 2662.00 0.247314 0.123657 − 0.992325i \(-0.460538\pi\)
0.123657 + 0.992325i \(0.460538\pi\)
\(42\) 0 0
\(43\) −7144.00 −0.589210 −0.294605 − 0.955619i \(-0.595188\pi\)
−0.294605 + 0.955619i \(0.595188\pi\)
\(44\) 0 0
\(45\) −10512.0 + 18207.3i −0.773845 + 1.34034i
\(46\) 0 0
\(47\) 3734.00 + 6467.48i 0.246564 + 0.427061i 0.962570 − 0.271033i \(-0.0873651\pi\)
−0.716006 + 0.698094i \(0.754032\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) −14010.0 − 24266.0i −0.754245 − 1.30639i
\(52\) 0 0
\(53\) 13637.0 − 23620.0i 0.666852 − 1.15502i −0.311928 − 0.950106i \(-0.600975\pi\)
0.978780 − 0.204915i \(-0.0656918\pi\)
\(54\) 0 0
\(55\) −19968.0 −0.890078
\(56\) 0 0
\(57\) 55740.0 2.27237
\(58\) 0 0
\(59\) −1245.00 + 2156.40i −0.0465628 + 0.0806492i −0.888367 − 0.459133i \(-0.848160\pi\)
0.841805 + 0.539782i \(0.181493\pi\)
\(60\) 0 0
\(61\) 5548.00 + 9609.42i 0.190903 + 0.330653i 0.945550 − 0.325478i \(-0.105525\pi\)
−0.754647 + 0.656131i \(0.772192\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 11328.0 + 19620.7i 0.332560 + 0.576011i
\(66\) 0 0
\(67\) −19878.0 + 34429.7i −0.540986 + 0.937014i 0.457862 + 0.889023i \(0.348615\pi\)
−0.998848 + 0.0479913i \(0.984718\pi\)
\(68\) 0 0
\(69\) −33600.0 −0.849604
\(70\) 0 0
\(71\) −69888.0 −1.64534 −0.822672 − 0.568516i \(-0.807518\pi\)
−0.822672 + 0.568516i \(0.807518\pi\)
\(72\) 0 0
\(73\) −8225.00 + 14246.1i −0.180646 + 0.312888i −0.942101 − 0.335330i \(-0.891152\pi\)
0.761455 + 0.648218i \(0.224486\pi\)
\(74\) 0 0
\(75\) 31515.0 + 54585.6i 0.646940 + 1.12053i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −39188.0 − 67875.6i −0.706456 − 1.22362i −0.966163 − 0.257931i \(-0.916959\pi\)
0.259707 − 0.965687i \(-0.416374\pi\)
\(80\) 0 0
\(81\) −106474. + 184419.i −1.80316 + 3.12316i
\(82\) 0 0
\(83\) 109818. 1.74976 0.874880 − 0.484340i \(-0.160940\pi\)
0.874880 + 0.484340i \(0.160940\pi\)
\(84\) 0 0
\(85\) 29888.0 0.448693
\(86\) 0 0
\(87\) 17610.0 − 30501.4i 0.249437 − 0.432038i
\(88\) 0 0
\(89\) 28483.0 + 49334.0i 0.381163 + 0.660194i 0.991229 − 0.132157i \(-0.0421903\pi\)
−0.610066 + 0.792351i \(0.708857\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) −43620.0 − 75552.1i −0.522972 − 0.905814i
\(94\) 0 0
\(95\) −29728.0 + 51490.4i −0.337953 + 0.585352i
\(96\) 0 0
\(97\) −115946. −1.25120 −0.625600 − 0.780144i \(-0.715146\pi\)
−0.625600 + 0.780144i \(0.715146\pi\)
\(98\) 0 0
\(99\) −409968. −4.20399
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 392.6.i.a.361.1 2
7.2 even 3 inner 392.6.i.a.177.1 2
7.3 odd 6 392.6.a.a.1.1 1
7.4 even 3 56.6.a.b.1.1 ✓ 1
7.5 odd 6 392.6.i.f.177.1 2
7.6 odd 2 392.6.i.f.361.1 2
21.11 odd 6 504.6.a.b.1.1 1
28.3 even 6 784.6.a.n.1.1 1
28.11 odd 6 112.6.a.a.1.1 1
56.11 odd 6 448.6.a.p.1.1 1
56.53 even 6 448.6.a.a.1.1 1
84.11 even 6 1008.6.a.h.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
56.6.a.b.1.1 ✓ 1 7.4 even 3
112.6.a.a.1.1 1 28.11 odd 6
392.6.a.a.1.1 1 7.3 odd 6
392.6.i.a.177.1 2 7.2 even 3 inner
392.6.i.a.361.1 2 1.1 even 1 trivial
392.6.i.f.177.1 2 7.5 odd 6
392.6.i.f.361.1 2 7.6 odd 2
448.6.a.a.1.1 1 56.53 even 6
448.6.a.p.1.1 1 56.11 odd 6
504.6.a.b.1.1 1 21.11 odd 6
784.6.a.n.1.1 1 28.3 even 6
1008.6.a.h.1.1 1 84.11 even 6