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Properties

Label 1058.2.a.g.1.2
Level $1058$
Weight $2$
Character 1058.1
Self dual yes
Analytic conductor $8.448$
Analytic rank $1$
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] = [1058,2,Mod(1,1058)] 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("1058.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1058, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1058 = 2 \cdot 23^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1058.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,2,-4,2,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(8.44817253385\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{12})^+\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-1.73205\) of defining polynomial
Character \(\chi\) \(=\) 1058.1

$q$-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)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} -2.00000 q^{3} +1.00000 q^{4} +1.73205 q^{5} -2.00000 q^{6} -3.46410 q^{7} +1.00000 q^{8} +1.00000 q^{9} +1.73205 q^{10} +3.46410 q^{11} -2.00000 q^{12} -5.00000 q^{13} -3.46410 q^{14} -3.46410 q^{15} +1.00000 q^{16} -6.92820 q^{17} +1.00000 q^{18} +3.46410 q^{19} +1.73205 q^{20} +6.92820 q^{21} +3.46410 q^{22} -2.00000 q^{24} -2.00000 q^{25} -5.00000 q^{26} +4.00000 q^{27} -3.46410 q^{28} -3.00000 q^{29} -3.46410 q^{30} -8.00000 q^{31} +1.00000 q^{32} -6.92820 q^{33} -6.92820 q^{34} -6.00000 q^{35} +1.00000 q^{36} +3.46410 q^{38} +10.0000 q^{39} +1.73205 q^{40} -9.00000 q^{41} +6.92820 q^{42} +6.92820 q^{43} +3.46410 q^{44} +1.73205 q^{45} -6.00000 q^{47} -2.00000 q^{48} +5.00000 q^{49} -2.00000 q^{50} +13.8564 q^{51} -5.00000 q^{52} -1.73205 q^{53} +4.00000 q^{54} +6.00000 q^{55} -3.46410 q^{56} -6.92820 q^{57} -3.00000 q^{58} -6.00000 q^{59} -3.46410 q^{60} +5.19615 q^{61} -8.00000 q^{62} -3.46410 q^{63} +1.00000 q^{64} -8.66025 q^{65} -6.92820 q^{66} -6.92820 q^{67} -6.92820 q^{68} -6.00000 q^{70} +6.00000 q^{71} +1.00000 q^{72} -11.0000 q^{73} +4.00000 q^{75} +3.46410 q^{76} -12.0000 q^{77} +10.0000 q^{78} -6.92820 q^{79} +1.73205 q^{80} -11.0000 q^{81} -9.00000 q^{82} +6.92820 q^{84} -12.0000 q^{85} +6.92820 q^{86} +6.00000 q^{87} +3.46410 q^{88} -1.73205 q^{89} +1.73205 q^{90} +17.3205 q^{91} +16.0000 q^{93} -6.00000 q^{94} +6.00000 q^{95} -2.00000 q^{96} +12.1244 q^{97} +5.00000 q^{98} +3.46410 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} - 4 q^{3} + 2 q^{4} - 4 q^{6} + 2 q^{8} + 2 q^{9} - 4 q^{12} - 10 q^{13} + 2 q^{16} + 2 q^{18} - 4 q^{24} - 4 q^{25} - 10 q^{26} + 8 q^{27} - 6 q^{29} - 16 q^{31} + 2 q^{32} - 12 q^{35} + 2 q^{36}+ \cdots + 10 q^{98}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) −2.00000 −1.15470 −0.577350 − 0.816497i \(-0.695913\pi\)
−0.577350 + 0.816497i \(0.695913\pi\)
\(4\) 1.00000 0.500000
\(5\) 1.73205 0.774597 0.387298 − 0.921954i \(-0.373408\pi\)
0.387298 + 0.921954i \(0.373408\pi\)
\(6\) −2.00000 −0.816497
\(7\) −3.46410 −1.30931 −0.654654 − 0.755929i \(-0.727186\pi\)
−0.654654 + 0.755929i \(0.727186\pi\)
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) 1.73205 0.547723
\(11\) 3.46410 1.04447 0.522233 − 0.852803i \(-0.325099\pi\)
0.522233 + 0.852803i \(0.325099\pi\)
\(12\) −2.00000 −0.577350
\(13\) −5.00000 −1.38675 −0.693375 − 0.720577i \(-0.743877\pi\)
−0.693375 + 0.720577i \(0.743877\pi\)
\(14\) −3.46410 −0.925820
\(15\) −3.46410 −0.894427
\(16\) 1.00000 0.250000
\(17\) −6.92820 −1.68034 −0.840168 − 0.542326i \(-0.817544\pi\)
−0.840168 + 0.542326i \(0.817544\pi\)
\(18\) 1.00000 0.235702
\(19\) 3.46410 0.794719 0.397360 − 0.917663i \(-0.369927\pi\)
0.397360 + 0.917663i \(0.369927\pi\)
\(20\) 1.73205 0.387298
\(21\) 6.92820 1.51186
\(22\) 3.46410 0.738549
\(23\) 0 0
\(24\) −2.00000 −0.408248
\(25\) −2.00000 −0.400000
\(26\) −5.00000 −0.980581
\(27\) 4.00000 0.769800
\(28\) −3.46410 −0.654654
\(29\) −3.00000 −0.557086 −0.278543 − 0.960424i \(-0.589851\pi\)
−0.278543 + 0.960424i \(0.589851\pi\)
\(30\) −3.46410 −0.632456
\(31\) −8.00000 −1.43684 −0.718421 − 0.695608i \(-0.755135\pi\)
−0.718421 + 0.695608i \(0.755135\pi\)
\(32\) 1.00000 0.176777
\(33\) −6.92820 −1.20605
\(34\) −6.92820 −1.18818
\(35\) −6.00000 −1.01419
\(36\) 1.00000 0.166667
\(37\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(38\) 3.46410 0.561951
\(39\) 10.0000 1.60128
\(40\) 1.73205 0.273861
\(41\) −9.00000 −1.40556 −0.702782 − 0.711405i \(-0.748059\pi\)
−0.702782 + 0.711405i \(0.748059\pi\)
\(42\) 6.92820 1.06904
\(43\) 6.92820 1.05654 0.528271 − 0.849076i \(-0.322841\pi\)
0.528271 + 0.849076i \(0.322841\pi\)
\(44\) 3.46410 0.522233
\(45\) 1.73205 0.258199
\(46\) 0 0
\(47\) −6.00000 −0.875190 −0.437595 − 0.899172i \(-0.644170\pi\)
−0.437595 + 0.899172i \(0.644170\pi\)
\(48\) −2.00000 −0.288675
\(49\) 5.00000 0.714286
\(50\) −2.00000 −0.282843
\(51\) 13.8564 1.94029
\(52\) −5.00000 −0.693375
\(53\) −1.73205 −0.237915 −0.118958 − 0.992899i \(-0.537955\pi\)
−0.118958 + 0.992899i \(0.537955\pi\)
\(54\) 4.00000 0.544331
\(55\) 6.00000 0.809040
\(56\) −3.46410 −0.462910
\(57\) −6.92820 −0.917663
\(58\) −3.00000 −0.393919
\(59\) −6.00000 −0.781133 −0.390567 − 0.920575i \(-0.627721\pi\)
−0.390567 + 0.920575i \(0.627721\pi\)
\(60\) −3.46410 −0.447214
\(61\) 5.19615 0.665299 0.332650 − 0.943051i \(-0.392057\pi\)
0.332650 + 0.943051i \(0.392057\pi\)
\(62\) −8.00000 −1.01600
\(63\) −3.46410 −0.436436
\(64\) 1.00000 0.125000
\(65\) −8.66025 −1.07417
\(66\) −6.92820 −0.852803
\(67\) −6.92820 −0.846415 −0.423207 − 0.906033i \(-0.639096\pi\)
−0.423207 + 0.906033i \(0.639096\pi\)
\(68\) −6.92820 −0.840168
\(69\) 0 0
\(70\) −6.00000 −0.717137
\(71\) 6.00000 0.712069 0.356034 − 0.934473i \(-0.384129\pi\)
0.356034 + 0.934473i \(0.384129\pi\)
\(72\) 1.00000 0.117851
\(73\) −11.0000 −1.28745 −0.643726 − 0.765256i \(-0.722612\pi\)
−0.643726 + 0.765256i \(0.722612\pi\)
\(74\) 0 0
\(75\) 4.00000 0.461880
\(76\) 3.46410 0.397360
\(77\) −12.0000 −1.36753
\(78\) 10.0000 1.13228
\(79\) −6.92820 −0.779484 −0.389742 − 0.920924i \(-0.627436\pi\)
−0.389742 + 0.920924i \(0.627436\pi\)
\(80\) 1.73205 0.193649
\(81\) −11.0000 −1.22222
\(82\) −9.00000 −0.993884
\(83\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 6.92820 0.755929
\(85\) −12.0000 −1.30158
\(86\) 6.92820 0.747087
\(87\) 6.00000 0.643268
\(88\) 3.46410 0.369274
\(89\) −1.73205 −0.183597 −0.0917985 − 0.995778i \(-0.529262\pi\)
−0.0917985 + 0.995778i \(0.529262\pi\)
\(90\) 1.73205 0.182574
\(91\) 17.3205 1.81568
\(92\) 0 0
\(93\) 16.0000 1.65912
\(94\) −6.00000 −0.618853
\(95\) 6.00000 0.615587
\(96\) −2.00000 −0.204124
\(97\) 12.1244 1.23104 0.615521 − 0.788121i \(-0.288946\pi\)
0.615521 + 0.788121i \(0.288946\pi\)
\(98\) 5.00000 0.505076
\(99\) 3.46410 0.348155
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 1058.2.a.g.1.2 yes 2
3.2 odd 2 9522.2.a.t.1.1 2
4.3 odd 2 8464.2.a.bh.1.2 2
23.22 odd 2 inner 1058.2.a.g.1.1 ✓ 2
69.68 even 2 9522.2.a.t.1.2 2
92.91 even 2 8464.2.a.bh.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1058.2.a.g.1.1 ✓ 2 23.22 odd 2 inner
1058.2.a.g.1.2 yes 2 1.1 even 1 trivial
8464.2.a.bh.1.1 2 92.91 even 2
8464.2.a.bh.1.2 2 4.3 odd 2
9522.2.a.t.1.1 2 3.2 odd 2
9522.2.a.t.1.2 2 69.68 even 2