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Properties

Label 3174.2.a.i.1.2
Level $3174$
Weight $2$
Character 3174.1
Self dual yes
Analytic conductor $25.345$
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] = [3174,2,Mod(1,3174)] 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("3174.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3174, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 3174 = 2 \cdot 3 \cdot 23^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3174.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,-2,-2,2,0,2,0,-2,2,0,0,-2,12,0,0,2,0,-2,0,0,0,0,0,2,6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(25)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(25.3445176016\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.41421\) of defining polynomial
Character \(\chi\) \(=\) 3174.1

$q$-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)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +2.82843 q^{5} +1.00000 q^{6} -2.82843 q^{7} -1.00000 q^{8} +1.00000 q^{9} -2.82843 q^{10} -5.65685 q^{11} -1.00000 q^{12} +6.00000 q^{13} +2.82843 q^{14} -2.82843 q^{15} +1.00000 q^{16} -2.82843 q^{17} -1.00000 q^{18} -8.48528 q^{19} +2.82843 q^{20} +2.82843 q^{21} +5.65685 q^{22} +1.00000 q^{24} +3.00000 q^{25} -6.00000 q^{26} -1.00000 q^{27} -2.82843 q^{28} +2.00000 q^{29} +2.82843 q^{30} +8.00000 q^{31} -1.00000 q^{32} +5.65685 q^{33} +2.82843 q^{34} -8.00000 q^{35} +1.00000 q^{36} +8.48528 q^{38} -6.00000 q^{39} -2.82843 q^{40} -6.00000 q^{41} -2.82843 q^{42} +2.82843 q^{43} -5.65685 q^{44} +2.82843 q^{45} +8.00000 q^{47} -1.00000 q^{48} +1.00000 q^{49} -3.00000 q^{50} +2.82843 q^{51} +6.00000 q^{52} +8.48528 q^{53} +1.00000 q^{54} -16.0000 q^{55} +2.82843 q^{56} +8.48528 q^{57} -2.00000 q^{58} +4.00000 q^{59} -2.82843 q^{60} -8.00000 q^{62} -2.82843 q^{63} +1.00000 q^{64} +16.9706 q^{65} -5.65685 q^{66} +8.48528 q^{67} -2.82843 q^{68} +8.00000 q^{70} +8.00000 q^{71} -1.00000 q^{72} +6.00000 q^{73} -3.00000 q^{75} -8.48528 q^{76} +16.0000 q^{77} +6.00000 q^{78} +2.82843 q^{79} +2.82843 q^{80} +1.00000 q^{81} +6.00000 q^{82} +5.65685 q^{83} +2.82843 q^{84} -8.00000 q^{85} -2.82843 q^{86} -2.00000 q^{87} +5.65685 q^{88} +2.82843 q^{89} -2.82843 q^{90} -16.9706 q^{91} -8.00000 q^{93} -8.00000 q^{94} -24.0000 q^{95} +1.00000 q^{96} +5.65685 q^{97} -1.00000 q^{98} -5.65685 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} + 2 q^{6} - 2 q^{8} + 2 q^{9} - 2 q^{12} + 12 q^{13} + 2 q^{16} - 2 q^{18} + 2 q^{24} + 6 q^{25} - 12 q^{26} - 2 q^{27} + 4 q^{29} + 16 q^{31} - 2 q^{32} - 16 q^{35} + 2 q^{36}+ \cdots - 2 q^{98}+O(q^{100}) \) 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
\(3\) −1.00000 −0.577350
\(4\) 1.00000 0.500000
\(5\) 2.82843 1.26491 0.632456 − 0.774597i \(-0.282047\pi\)
0.632456 + 0.774597i \(0.282047\pi\)
\(6\) 1.00000 0.408248
\(7\) −2.82843 −1.06904 −0.534522 − 0.845154i \(-0.679509\pi\)
−0.534522 + 0.845154i \(0.679509\pi\)
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) −2.82843 −0.894427
\(11\) −5.65685 −1.70561 −0.852803 − 0.522233i \(-0.825099\pi\)
−0.852803 + 0.522233i \(0.825099\pi\)
\(12\) −1.00000 −0.288675
\(13\) 6.00000 1.66410 0.832050 − 0.554700i \(-0.187167\pi\)
0.832050 + 0.554700i \(0.187167\pi\)
\(14\) 2.82843 0.755929
\(15\) −2.82843 −0.730297
\(16\) 1.00000 0.250000
\(17\) −2.82843 −0.685994 −0.342997 − 0.939336i \(-0.611442\pi\)
−0.342997 + 0.939336i \(0.611442\pi\)
\(18\) −1.00000 −0.235702
\(19\) −8.48528 −1.94666 −0.973329 − 0.229416i \(-0.926318\pi\)
−0.973329 + 0.229416i \(0.926318\pi\)
\(20\) 2.82843 0.632456
\(21\) 2.82843 0.617213
\(22\) 5.65685 1.20605
\(23\) 0 0
\(24\) 1.00000 0.204124
\(25\) 3.00000 0.600000
\(26\) −6.00000 −1.17670
\(27\) −1.00000 −0.192450
\(28\) −2.82843 −0.534522
\(29\) 2.00000 0.371391 0.185695 − 0.982607i \(-0.440546\pi\)
0.185695 + 0.982607i \(0.440546\pi\)
\(30\) 2.82843 0.516398
\(31\) 8.00000 1.43684 0.718421 − 0.695608i \(-0.244865\pi\)
0.718421 + 0.695608i \(0.244865\pi\)
\(32\) −1.00000 −0.176777
\(33\) 5.65685 0.984732
\(34\) 2.82843 0.485071
\(35\) −8.00000 −1.35225
\(36\) 1.00000 0.166667
\(37\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(38\) 8.48528 1.37649
\(39\) −6.00000 −0.960769
\(40\) −2.82843 −0.447214
\(41\) −6.00000 −0.937043 −0.468521 − 0.883452i \(-0.655213\pi\)
−0.468521 + 0.883452i \(0.655213\pi\)
\(42\) −2.82843 −0.436436
\(43\) 2.82843 0.431331 0.215666 − 0.976467i \(-0.430808\pi\)
0.215666 + 0.976467i \(0.430808\pi\)
\(44\) −5.65685 −0.852803
\(45\) 2.82843 0.421637
\(46\) 0 0
\(47\) 8.00000 1.16692 0.583460 − 0.812142i \(-0.301699\pi\)
0.583460 + 0.812142i \(0.301699\pi\)
\(48\) −1.00000 −0.144338
\(49\) 1.00000 0.142857
\(50\) −3.00000 −0.424264
\(51\) 2.82843 0.396059
\(52\) 6.00000 0.832050
\(53\) 8.48528 1.16554 0.582772 − 0.812636i \(-0.301968\pi\)
0.582772 + 0.812636i \(0.301968\pi\)
\(54\) 1.00000 0.136083
\(55\) −16.0000 −2.15744
\(56\) 2.82843 0.377964
\(57\) 8.48528 1.12390
\(58\) −2.00000 −0.262613
\(59\) 4.00000 0.520756 0.260378 − 0.965507i \(-0.416153\pi\)
0.260378 + 0.965507i \(0.416153\pi\)
\(60\) −2.82843 −0.365148
\(61\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(62\) −8.00000 −1.01600
\(63\) −2.82843 −0.356348
\(64\) 1.00000 0.125000
\(65\) 16.9706 2.10494
\(66\) −5.65685 −0.696311
\(67\) 8.48528 1.03664 0.518321 − 0.855186i \(-0.326557\pi\)
0.518321 + 0.855186i \(0.326557\pi\)
\(68\) −2.82843 −0.342997
\(69\) 0 0
\(70\) 8.00000 0.956183
\(71\) 8.00000 0.949425 0.474713 − 0.880141i \(-0.342552\pi\)
0.474713 + 0.880141i \(0.342552\pi\)
\(72\) −1.00000 −0.117851
\(73\) 6.00000 0.702247 0.351123 − 0.936329i \(-0.385800\pi\)
0.351123 + 0.936329i \(0.385800\pi\)
\(74\) 0 0
\(75\) −3.00000 −0.346410
\(76\) −8.48528 −0.973329
\(77\) 16.0000 1.82337
\(78\) 6.00000 0.679366
\(79\) 2.82843 0.318223 0.159111 − 0.987261i \(-0.449137\pi\)
0.159111 + 0.987261i \(0.449137\pi\)
\(80\) 2.82843 0.316228
\(81\) 1.00000 0.111111
\(82\) 6.00000 0.662589
\(83\) 5.65685 0.620920 0.310460 − 0.950586i \(-0.399517\pi\)
0.310460 + 0.950586i \(0.399517\pi\)
\(84\) 2.82843 0.308607
\(85\) −8.00000 −0.867722
\(86\) −2.82843 −0.304997
\(87\) −2.00000 −0.214423
\(88\) 5.65685 0.603023
\(89\) 2.82843 0.299813 0.149906 − 0.988700i \(-0.452103\pi\)
0.149906 + 0.988700i \(0.452103\pi\)
\(90\) −2.82843 −0.298142
\(91\) −16.9706 −1.77900
\(92\) 0 0
\(93\) −8.00000 −0.829561
\(94\) −8.00000 −0.825137
\(95\) −24.0000 −2.46235
\(96\) 1.00000 0.102062
\(97\) 5.65685 0.574367 0.287183 − 0.957876i \(-0.407281\pi\)
0.287183 + 0.957876i \(0.407281\pi\)
\(98\) −1.00000 −0.101015
\(99\) −5.65685 −0.568535
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 3174.2.a.i.1.2 yes 2
3.2 odd 2 9522.2.a.bk.1.1 2
23.22 odd 2 inner 3174.2.a.i.1.1 ✓ 2
69.68 even 2 9522.2.a.bk.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3174.2.a.i.1.1 ✓ 2 23.22 odd 2 inner
3174.2.a.i.1.2 yes 2 1.1 even 1 trivial
9522.2.a.bk.1.1 2 3.2 odd 2
9522.2.a.bk.1.2 2 69.68 even 2