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Properties

Label 392.2.i.a.361.1
Level $392$
Weight $2$
Character 392.361
Analytic conductor $3.130$
Analytic rank $0$
Dimension $2$
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] = [392,2,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, 2); 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, 2, names="a")
 
Level: \( N \) \(=\) \( 392 = 2^{3} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
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,-2,0,4,0,0,0,-1] 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: \(3.13013575923\)
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 392.2.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)
 
\(f(q)\) \(=\) \(q+(-1.00000 + 1.73205i) q^{3} +(2.00000 + 3.46410i) q^{5} +(-0.500000 - 0.866025i) q^{9} -8.00000 q^{15} +(1.00000 - 1.73205i) q^{17} +(1.00000 + 1.73205i) q^{19} +(-4.00000 - 6.92820i) q^{23} +(-5.50000 + 9.52628i) q^{25} -4.00000 q^{27} +2.00000 q^{29} +(-2.00000 + 3.46410i) q^{31} +(3.00000 + 5.19615i) q^{37} -2.00000 q^{41} +8.00000 q^{43} +(2.00000 - 3.46410i) q^{45} +(2.00000 + 3.46410i) q^{47} +(2.00000 + 3.46410i) q^{51} +(5.00000 - 8.66025i) q^{53} -4.00000 q^{57} +(-3.00000 + 5.19615i) q^{59} +(-2.00000 - 3.46410i) q^{61} +(6.00000 - 10.3923i) q^{67} +16.0000 q^{69} +(7.00000 - 12.1244i) q^{73} +(-11.0000 - 19.0526i) q^{75} +(4.00000 + 6.92820i) q^{79} +(5.50000 - 9.52628i) q^{81} +6.00000 q^{83} +8.00000 q^{85} +(-2.00000 + 3.46410i) q^{87} +(-5.00000 - 8.66025i) q^{89} +(-4.00000 - 6.92820i) q^{93} +(-4.00000 + 6.92820i) q^{95} -2.00000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{3} + 4 q^{5} - q^{9} - 16 q^{15} + 2 q^{17} + 2 q^{19} - 8 q^{23} - 11 q^{25} - 8 q^{27} + 4 q^{29} - 4 q^{31} + 6 q^{37} - 4 q^{41} + 16 q^{43} + 4 q^{45} + 4 q^{47} + 4 q^{51} + 10 q^{53}+ \cdots - 4 q^{97}+O(q^{100}) \) 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
\(3\) −1.00000 + 1.73205i −0.577350 + 1.00000i 0.418432 + 0.908248i \(0.362580\pi\)
−0.995782 + 0.0917517i \(0.970753\pi\)
\(4\) 0 0
\(5\) 2.00000 + 3.46410i 0.894427 + 1.54919i 0.834512 + 0.550990i \(0.185750\pi\)
0.0599153 + 0.998203i \(0.480917\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) −0.500000 − 0.866025i −0.166667 − 0.288675i
\(10\) 0 0
\(11\) 0 0 −0.866025 − 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(12\) 0 0
\(13\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(14\) 0 0
\(15\) −8.00000 −2.06559
\(16\) 0 0
\(17\) 1.00000 − 1.73205i 0.242536 − 0.420084i −0.718900 − 0.695113i \(-0.755354\pi\)
0.961436 + 0.275029i \(0.0886875\pi\)
\(18\) 0 0
\(19\) 1.00000 + 1.73205i 0.229416 + 0.397360i 0.957635 − 0.287984i \(-0.0929851\pi\)
−0.728219 + 0.685344i \(0.759652\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −4.00000 − 6.92820i −0.834058 − 1.44463i −0.894795 − 0.446476i \(-0.852679\pi\)
0.0607377 − 0.998154i \(-0.480655\pi\)
\(24\) 0 0
\(25\) −5.50000 + 9.52628i −1.10000 + 1.90526i
\(26\) 0 0
\(27\) −4.00000 −0.769800
\(28\) 0 0
\(29\) 2.00000 0.371391 0.185695 − 0.982607i \(-0.440546\pi\)
0.185695 + 0.982607i \(0.440546\pi\)
\(30\) 0 0
\(31\) −2.00000 + 3.46410i −0.359211 + 0.622171i −0.987829 − 0.155543i \(-0.950287\pi\)
0.628619 + 0.777714i \(0.283621\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 3.00000 + 5.19615i 0.493197 + 0.854242i 0.999969 − 0.00783774i \(-0.00249486\pi\)
−0.506772 + 0.862080i \(0.669162\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −2.00000 −0.312348 −0.156174 − 0.987730i \(-0.549916\pi\)
−0.156174 + 0.987730i \(0.549916\pi\)
\(42\) 0 0
\(43\) 8.00000 1.21999 0.609994 − 0.792406i \(-0.291172\pi\)
0.609994 + 0.792406i \(0.291172\pi\)
\(44\) 0 0
\(45\) 2.00000 − 3.46410i 0.298142 − 0.516398i
\(46\) 0 0
\(47\) 2.00000 + 3.46410i 0.291730 + 0.505291i 0.974219 − 0.225605i \(-0.0724358\pi\)
−0.682489 + 0.730896i \(0.739102\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 2.00000 + 3.46410i 0.280056 + 0.485071i
\(52\) 0 0
\(53\) 5.00000 − 8.66025i 0.686803 − 1.18958i −0.286064 − 0.958211i \(-0.592347\pi\)
0.972867 − 0.231367i \(-0.0743197\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −4.00000 −0.529813
\(58\) 0 0
\(59\) −3.00000 + 5.19615i −0.390567 + 0.676481i −0.992524 − 0.122047i \(-0.961054\pi\)
0.601958 + 0.798528i \(0.294388\pi\)
\(60\) 0 0
\(61\) −2.00000 − 3.46410i −0.256074 − 0.443533i 0.709113 − 0.705095i \(-0.249096\pi\)
−0.965187 + 0.261562i \(0.915762\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 6.00000 − 10.3923i 0.733017 − 1.26962i −0.222571 − 0.974916i \(-0.571445\pi\)
0.955588 − 0.294706i \(-0.0952216\pi\)
\(68\) 0 0
\(69\) 16.0000 1.92617
\(70\) 0 0
\(71\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) 7.00000 − 12.1244i 0.819288 − 1.41905i −0.0869195 − 0.996215i \(-0.527702\pi\)
0.906208 − 0.422833i \(-0.138964\pi\)
\(74\) 0 0
\(75\) −11.0000 − 19.0526i −1.27017 − 2.20000i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 4.00000 + 6.92820i 0.450035 + 0.779484i 0.998388 − 0.0567635i \(-0.0180781\pi\)
−0.548352 + 0.836247i \(0.684745\pi\)
\(80\) 0 0
\(81\) 5.50000 − 9.52628i 0.611111 − 1.05848i
\(82\) 0 0
\(83\) 6.00000 0.658586 0.329293 − 0.944228i \(-0.393190\pi\)
0.329293 + 0.944228i \(0.393190\pi\)
\(84\) 0 0
\(85\) 8.00000 0.867722
\(86\) 0 0
\(87\) −2.00000 + 3.46410i −0.214423 + 0.371391i
\(88\) 0 0
\(89\) −5.00000 − 8.66025i −0.529999 − 0.917985i −0.999388 − 0.0349934i \(-0.988859\pi\)
0.469389 − 0.882992i \(-0.344474\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) −4.00000 − 6.92820i −0.414781 − 0.718421i
\(94\) 0 0
\(95\) −4.00000 + 6.92820i −0.410391 + 0.710819i
\(96\) 0 0
\(97\) −2.00000 −0.203069 −0.101535 − 0.994832i \(-0.532375\pi\)
−0.101535 + 0.994832i \(0.532375\pi\)
\(98\) 0 0
\(99\) 0 0
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 392.2.i.a.361.1 2
3.2 odd 2 3528.2.s.a.361.1 2
4.3 odd 2 784.2.i.j.753.1 2
7.2 even 3 inner 392.2.i.a.177.1 2
7.3 odd 6 392.2.a.b.1.1 1
7.4 even 3 56.2.a.b.1.1 ✓ 1
7.5 odd 6 392.2.i.e.177.1 2
7.6 odd 2 392.2.i.e.361.1 2
21.2 odd 6 3528.2.s.a.3313.1 2
21.5 even 6 3528.2.s.ba.3313.1 2
21.11 odd 6 504.2.a.h.1.1 1
21.17 even 6 3528.2.a.b.1.1 1
21.20 even 2 3528.2.s.ba.361.1 2
28.3 even 6 784.2.a.i.1.1 1
28.11 odd 6 112.2.a.a.1.1 1
28.19 even 6 784.2.i.b.177.1 2
28.23 odd 6 784.2.i.j.177.1 2
28.27 even 2 784.2.i.b.753.1 2
35.4 even 6 1400.2.a.a.1.1 1
35.18 odd 12 1400.2.g.b.449.2 2
35.24 odd 6 9800.2.a.bj.1.1 1
35.32 odd 12 1400.2.g.b.449.1 2
56.3 even 6 3136.2.a.c.1.1 1
56.11 odd 6 448.2.a.h.1.1 1
56.45 odd 6 3136.2.a.w.1.1 1
56.53 even 6 448.2.a.c.1.1 1
77.32 odd 6 6776.2.a.h.1.1 1
84.11 even 6 1008.2.a.m.1.1 1
84.59 odd 6 7056.2.a.c.1.1 1
91.25 even 6 9464.2.a.h.1.1 1
112.11 odd 12 1792.2.b.h.897.2 2
112.53 even 12 1792.2.b.a.897.1 2
112.67 odd 12 1792.2.b.h.897.1 2
112.109 even 12 1792.2.b.a.897.2 2
140.39 odd 6 2800.2.a.bd.1.1 1
140.67 even 12 2800.2.g.g.449.2 2
140.123 even 12 2800.2.g.g.449.1 2
168.11 even 6 4032.2.a.a.1.1 1
168.53 odd 6 4032.2.a.d.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
56.2.a.b.1.1 ✓ 1 7.4 even 3
112.2.a.a.1.1 1 28.11 odd 6
392.2.a.b.1.1 1 7.3 odd 6
392.2.i.a.177.1 2 7.2 even 3 inner
392.2.i.a.361.1 2 1.1 even 1 trivial
392.2.i.e.177.1 2 7.5 odd 6
392.2.i.e.361.1 2 7.6 odd 2
448.2.a.c.1.1 1 56.53 even 6
448.2.a.h.1.1 1 56.11 odd 6
504.2.a.h.1.1 1 21.11 odd 6
784.2.a.i.1.1 1 28.3 even 6
784.2.i.b.177.1 2 28.19 even 6
784.2.i.b.753.1 2 28.27 even 2
784.2.i.j.177.1 2 28.23 odd 6
784.2.i.j.753.1 2 4.3 odd 2
1008.2.a.m.1.1 1 84.11 even 6
1400.2.a.a.1.1 1 35.4 even 6
1400.2.g.b.449.1 2 35.32 odd 12
1400.2.g.b.449.2 2 35.18 odd 12
1792.2.b.a.897.1 2 112.53 even 12
1792.2.b.a.897.2 2 112.109 even 12
1792.2.b.h.897.1 2 112.67 odd 12
1792.2.b.h.897.2 2 112.11 odd 12
2800.2.a.bd.1.1 1 140.39 odd 6
2800.2.g.g.449.1 2 140.123 even 12
2800.2.g.g.449.2 2 140.67 even 12
3136.2.a.c.1.1 1 56.3 even 6
3136.2.a.w.1.1 1 56.45 odd 6
3528.2.a.b.1.1 1 21.17 even 6
3528.2.s.a.361.1 2 3.2 odd 2
3528.2.s.a.3313.1 2 21.2 odd 6
3528.2.s.ba.361.1 2 21.20 even 2
3528.2.s.ba.3313.1 2 21.5 even 6
4032.2.a.a.1.1 1 168.11 even 6
4032.2.a.d.1.1 1 168.53 odd 6
6776.2.a.h.1.1 1 77.32 odd 6
7056.2.a.c.1.1 1 84.59 odd 6
9464.2.a.h.1.1 1 91.25 even 6
9800.2.a.bj.1.1 1 35.24 odd 6