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Properties

Label 684.2.f.a.379.1
Level $684$
Weight $2$
Character 684.379
Analytic conductor $5.462$
Analytic rank $0$
Dimension $4$
CM discriminant -228
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] = [684,2,Mod(379,684)] 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("684.379"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(684, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 684 = 2^{2} \cdot 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 684.f (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(5.46176749826\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{19})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 20x^{2} + 81 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 379.1
Root \(2.37510i\) of defining polynomial
Character \(\chi\) \(=\) 684.379
Dual form 684.2.f.a.379.4

$q$-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)
 
\(f(q)\) \(=\) \(q-1.41421i q^{2} -2.00000 q^{4} +2.82843i q^{8} -6.16441i q^{11} +4.00000 q^{16} -4.35890 q^{19} -8.71780 q^{22} -6.16441i q^{23} -5.00000 q^{25} +1.41421i q^{29} -8.71780 q^{31} -5.65685i q^{32} +6.16441i q^{38} -7.07107i q^{41} +12.3288i q^{44} -8.71780 q^{46} -6.16441i q^{47} +7.00000 q^{49} +7.07107i q^{50} +9.89949i q^{53} +2.00000 q^{58} -4.00000 q^{61} +12.3288i q^{62} -8.00000 q^{64} -8.71780 q^{67} +8.00000 q^{73} +8.71780 q^{76} +17.4356 q^{79} -10.0000 q^{82} -6.16441i q^{83} +17.4356 q^{88} -15.5563i q^{89} +12.3288i q^{92} -8.71780 q^{94} -9.89949i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 8 q^{4} + 16 q^{16} - 20 q^{25} + 28 q^{49} + 8 q^{58} - 16 q^{61} - 32 q^{64} + 32 q^{73} - 40 q^{82}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/684\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(343\) \(533\)
\(\chi(n)\) \(-1\) \(-1\) \(1\)

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
\(7\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(8\) 2.82843i 1.00000i
\(9\) 0 0
\(10\) 0 0
\(11\) − 6.16441i − 1.85864i −0.369274 − 0.929320i \(-0.620394\pi\)
0.369274 − 0.929320i \(-0.379606\pi\)
\(12\) 0 0
\(13\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 4.00000 1.00000
\(17\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) 0 0
\(19\) −4.35890 −1.00000
\(20\) 0 0
\(21\) 0 0
\(22\) −8.71780 −1.85864
\(23\) − 6.16441i − 1.28537i −0.766131 − 0.642685i \(-0.777821\pi\)
0.766131 − 0.642685i \(-0.222179\pi\)
\(24\) 0 0
\(25\) −5.00000 −1.00000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 1.41421i 0.262613i 0.991342 + 0.131306i \(0.0419172\pi\)
−0.991342 + 0.131306i \(0.958083\pi\)
\(30\) 0 0
\(31\) −8.71780 −1.56576 −0.782881 − 0.622171i \(-0.786251\pi\)
−0.782881 + 0.622171i \(0.786251\pi\)
\(32\) − 5.65685i − 1.00000i
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(38\) 6.16441i 1.00000i
\(39\) 0 0
\(40\) 0 0
\(41\) − 7.07107i − 1.10432i −0.833740 − 0.552158i \(-0.813805\pi\)
0.833740 − 0.552158i \(-0.186195\pi\)
\(42\) 0 0
\(43\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(44\) 12.3288i 1.85864i
\(45\) 0 0
\(46\) −8.71780 −1.28537
\(47\) − 6.16441i − 0.899172i −0.893237 − 0.449586i \(-0.851571\pi\)
0.893237 − 0.449586i \(-0.148429\pi\)
\(48\) 0 0
\(49\) 7.00000 1.00000
\(50\) 7.07107i 1.00000i
\(51\) 0 0
\(52\) 0 0
\(53\) 9.89949i 1.35980i 0.733305 + 0.679900i \(0.237977\pi\)
−0.733305 + 0.679900i \(0.762023\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 2.00000 0.262613
\(59\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) −4.00000 −0.512148 −0.256074 − 0.966657i \(-0.582429\pi\)
−0.256074 + 0.966657i \(0.582429\pi\)
\(62\) 12.3288i 1.56576i
\(63\) 0 0
\(64\) −8.00000 −1.00000
\(65\) 0 0
\(66\) 0 0
\(67\) −8.71780 −1.06505 −0.532524 − 0.846415i \(-0.678756\pi\)
−0.532524 + 0.846415i \(0.678756\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) 8.00000 0.936329 0.468165 − 0.883641i \(-0.344915\pi\)
0.468165 + 0.883641i \(0.344915\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 8.71780 1.00000
\(77\) 0 0
\(78\) 0 0
\(79\) 17.4356 1.96166 0.980829 − 0.194871i \(-0.0624288\pi\)
0.980829 + 0.194871i \(0.0624288\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) −10.0000 −1.10432
\(83\) − 6.16441i − 0.676632i −0.941033 − 0.338316i \(-0.890143\pi\)
0.941033 − 0.338316i \(-0.109857\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 17.4356 1.85864
\(89\) − 15.5563i − 1.64897i −0.565884 − 0.824485i \(-0.691465\pi\)
0.565884 − 0.824485i \(-0.308535\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 12.3288i 1.28537i
\(93\) 0 0
\(94\) −8.71780 −0.899172
\(95\) 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 684.2.f.a.379.1 ✓ 4
3.2 odd 2 inner 684.2.f.a.379.4 yes 4
4.3 odd 2 inner 684.2.f.a.379.2 yes 4
12.11 even 2 inner 684.2.f.a.379.3 yes 4
19.18 odd 2 inner 684.2.f.a.379.3 yes 4
57.56 even 2 inner 684.2.f.a.379.2 yes 4
76.75 even 2 inner 684.2.f.a.379.4 yes 4
228.227 odd 2 CM 684.2.f.a.379.1 ✓ 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
684.2.f.a.379.1 ✓ 4 1.1 even 1 trivial
684.2.f.a.379.1 ✓ 4 228.227 odd 2 CM
684.2.f.a.379.2 yes 4 4.3 odd 2 inner
684.2.f.a.379.2 yes 4 57.56 even 2 inner
684.2.f.a.379.3 yes 4 12.11 even 2 inner
684.2.f.a.379.3 yes 4 19.18 odd 2 inner
684.2.f.a.379.4 yes 4 3.2 odd 2 inner
684.2.f.a.379.4 yes 4 76.75 even 2 inner