Properties

Label 504.2.p.c.307.1
Level $504$
Weight $2$
Character 504.307
Analytic conductor $4.024$
Analytic rank $0$
Dimension $4$
CM discriminant -168
Inner twists $8$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [504,2,Mod(307,504)] 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("504.307"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(504, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1, 0, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 504 = 2^{3} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 504.p (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,-8,0,0,0,0,0,0,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(14)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(4.02446026187\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{7})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 8x^{2} + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 307.1
Root \(-1.16372i\) of defining polynomial
Character \(\chi\) \(=\) 504.307
Dual form 504.2.p.c.307.3

$q$-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)
 
\(f(q)\) \(=\) \(q-1.41421i q^{2} -2.00000 q^{4} -2.64575 q^{7} +2.82843i q^{8} -5.29150 q^{13} +3.74166i q^{14} +4.00000 q^{16} +7.48331i q^{17} -2.82843i q^{23} -5.00000 q^{25} +7.48331i q^{26} +5.29150 q^{28} +5.65685i q^{29} -5.29150 q^{31} -5.65685i q^{32} +10.5830 q^{34} +7.48331i q^{41} +2.00000 q^{43} -4.00000 q^{46} +7.00000 q^{49} +7.07107i q^{50} +10.5830 q^{52} -11.3137i q^{53} -7.48331i q^{56} +8.00000 q^{58} -14.9666i q^{59} -5.29150 q^{61} +7.48331i q^{62} -8.00000 q^{64} -10.0000 q^{67} -14.9666i q^{68} +14.1421i q^{71} +10.5830 q^{82} -14.9666i q^{83} -2.82843i q^{86} +7.48331i q^{89} +14.0000 q^{91} +5.65685i q^{92} -9.89949i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 8 q^{4} + 16 q^{16} - 20 q^{25} + 8 q^{43} - 16 q^{46} + 28 q^{49} + 32 q^{58} - 32 q^{64} - 40 q^{67} + 56 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/504\mathbb{Z}\right)^\times\).

\(n\) \(73\) \(127\) \(253\) \(281\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 1.41421i − 1.00000i
\(3\) 0 0
\(4\) −2.00000 −1.00000
\(5\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(6\) 0 0
\(7\) −2.64575 −1.00000
\(8\) 2.82843i 1.00000i
\(9\) 0 0
\(10\) 0 0
\(11\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 0 0
\(13\) −5.29150 −1.46760 −0.733799 − 0.679366i \(-0.762255\pi\)
−0.733799 + 0.679366i \(0.762255\pi\)
\(14\) 3.74166i 1.00000i
\(15\) 0 0
\(16\) 4.00000 1.00000
\(17\) 7.48331i 1.81497i 0.420084 + 0.907485i \(0.362001\pi\)
−0.420084 + 0.907485i \(0.637999\pi\)
\(18\) 0 0
\(19\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) − 2.82843i − 0.589768i −0.955533 − 0.294884i \(-0.904719\pi\)
0.955533 − 0.294884i \(-0.0952810\pi\)
\(24\) 0 0
\(25\) −5.00000 −1.00000
\(26\) 7.48331i 1.46760i
\(27\) 0 0
\(28\) 5.29150 1.00000
\(29\) 5.65685i 1.05045i 0.850963 + 0.525226i \(0.176019\pi\)
−0.850963 + 0.525226i \(0.823981\pi\)
\(30\) 0 0
\(31\) −5.29150 −0.950382 −0.475191 − 0.879883i \(-0.657621\pi\)
−0.475191 + 0.879883i \(0.657621\pi\)
\(32\) − 5.65685i − 1.00000i
\(33\) 0 0
\(34\) 10.5830 1.81497
\(35\) 0 0
\(36\) 0 0
\(37\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 7.48331i 1.16870i 0.811503 + 0.584349i \(0.198650\pi\)
−0.811503 + 0.584349i \(0.801350\pi\)
\(42\) 0 0
\(43\) 2.00000 0.304997 0.152499 − 0.988304i \(-0.451268\pi\)
0.152499 + 0.988304i \(0.451268\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) −4.00000 −0.589768
\(47\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 0 0
\(49\) 7.00000 1.00000
\(50\) 7.07107i 1.00000i
\(51\) 0 0
\(52\) 10.5830 1.46760
\(53\) − 11.3137i − 1.55406i −0.629465 − 0.777029i \(-0.716726\pi\)
0.629465 − 0.777029i \(-0.283274\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) − 7.48331i − 1.00000i
\(57\) 0 0
\(58\) 8.00000 1.05045
\(59\) − 14.9666i − 1.94849i −0.225494 − 0.974245i \(-0.572400\pi\)
0.225494 − 0.974245i \(-0.427600\pi\)
\(60\) 0 0
\(61\) −5.29150 −0.677507 −0.338754 − 0.940875i \(-0.610005\pi\)
−0.338754 + 0.940875i \(0.610005\pi\)
\(62\) 7.48331i 0.950382i
\(63\) 0 0
\(64\) −8.00000 −1.00000
\(65\) 0 0
\(66\) 0 0
\(67\) −10.0000 −1.22169 −0.610847 − 0.791748i \(-0.709171\pi\)
−0.610847 + 0.791748i \(0.709171\pi\)
\(68\) − 14.9666i − 1.81497i
\(69\) 0 0
\(70\) 0 0
\(71\) 14.1421i 1.67836i 0.543852 + 0.839181i \(0.316965\pi\)
−0.543852 + 0.839181i \(0.683035\pi\)
\(72\) 0 0
\(73\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 10.5830 1.16870
\(83\) − 14.9666i − 1.64280i −0.570352 − 0.821401i \(-0.693193\pi\)
0.570352 − 0.821401i \(-0.306807\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) − 2.82843i − 0.304997i
\(87\) 0 0
\(88\) 0 0
\(89\) 7.48331i 0.793230i 0.917985 + 0.396615i \(0.129815\pi\)
−0.917985 + 0.396615i \(0.870185\pi\)
\(90\) 0 0
\(91\) 14.0000 1.46760
\(92\) 5.65685i 0.589768i
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(98\) − 9.89949i − 1.00000i
\(99\) 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 504.2.p.c.307.1 ✓ 4
3.2 odd 2 inner 504.2.p.c.307.3 yes 4
4.3 odd 2 2016.2.p.c.559.4 4
7.6 odd 2 inner 504.2.p.c.307.2 yes 4
8.3 odd 2 inner 504.2.p.c.307.4 yes 4
8.5 even 2 2016.2.p.c.559.2 4
12.11 even 2 2016.2.p.c.559.3 4
21.20 even 2 inner 504.2.p.c.307.4 yes 4
24.5 odd 2 2016.2.p.c.559.1 4
24.11 even 2 inner 504.2.p.c.307.2 yes 4
28.27 even 2 2016.2.p.c.559.1 4
56.13 odd 2 2016.2.p.c.559.3 4
56.27 even 2 inner 504.2.p.c.307.3 yes 4
84.83 odd 2 2016.2.p.c.559.2 4
168.83 odd 2 CM 504.2.p.c.307.1 ✓ 4
168.125 even 2 2016.2.p.c.559.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
504.2.p.c.307.1 ✓ 4 1.1 even 1 trivial
504.2.p.c.307.1 ✓ 4 168.83 odd 2 CM
504.2.p.c.307.2 yes 4 7.6 odd 2 inner
504.2.p.c.307.2 yes 4 24.11 even 2 inner
504.2.p.c.307.3 yes 4 3.2 odd 2 inner
504.2.p.c.307.3 yes 4 56.27 even 2 inner
504.2.p.c.307.4 yes 4 8.3 odd 2 inner
504.2.p.c.307.4 yes 4 21.20 even 2 inner
2016.2.p.c.559.1 4 24.5 odd 2
2016.2.p.c.559.1 4 28.27 even 2
2016.2.p.c.559.2 4 8.5 even 2
2016.2.p.c.559.2 4 84.83 odd 2
2016.2.p.c.559.3 4 12.11 even 2
2016.2.p.c.559.3 4 56.13 odd 2
2016.2.p.c.559.4 4 4.3 odd 2
2016.2.p.c.559.4 4 168.125 even 2