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Properties

Label 392.4.a.i.1.3
Level $392$
Weight $4$
Character 392.1
Self dual yes
Analytic conductor $23.129$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

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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] = [392,4,Mod(1,392)] 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("392.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(392, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 392 = 2^{3} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 392.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [3,0,-7,0,-3,0,0,0,18] 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: yes
Analytic conductor: \(23.1287487223\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.1929.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 10x + 13 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 56)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(2.90222\) of defining polynomial
Character \(\chi\) \(=\) 392.1

$q$-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)
 
\(f(q)\) \(=\) \(q+2.80445 q^{3} -6.69159 q^{5} -19.1351 q^{9} +31.4881 q^{11} +18.6837 q^{13} -18.7662 q^{15} -87.9538 q^{17} +13.0970 q^{19} -9.37681 q^{23} -80.2226 q^{25} -129.383 q^{27} -5.11923 q^{29} -257.412 q^{31} +88.3067 q^{33} -380.215 q^{37} +52.3973 q^{39} -217.959 q^{41} +377.049 q^{43} +128.044 q^{45} -357.710 q^{47} -246.662 q^{51} +764.390 q^{53} -210.706 q^{55} +36.7298 q^{57} -450.671 q^{59} -174.078 q^{61} -125.023 q^{65} -497.234 q^{67} -26.2967 q^{69} +350.238 q^{71} -1062.69 q^{73} -224.980 q^{75} +560.449 q^{79} +153.799 q^{81} -1105.27 q^{83} +588.551 q^{85} -14.3566 q^{87} +1206.71 q^{89} -721.897 q^{93} -87.6398 q^{95} +1442.99 q^{97} -602.528 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 7 q^{3} - 3 q^{5} + 18 q^{9} - 3 q^{11} - 26 q^{13} + 127 q^{15} - 31 q^{17} - 89 q^{19} - 201 q^{23} + 300 q^{25} - 469 q^{27} + 190 q^{29} - 339 q^{31} - 105 q^{33} - 535 q^{37} - 134 q^{39} + 58 q^{41}+ \cdots + 2310 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 2.80445 0.539716 0.269858 − 0.962900i \(-0.413023\pi\)
0.269858 + 0.962900i \(0.413023\pi\)
\(4\) 0 0
\(5\) −6.69159 −0.598514 −0.299257 − 0.954173i \(-0.596739\pi\)
−0.299257 + 0.954173i \(0.596739\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) −19.1351 −0.708707
\(10\) 0 0
\(11\) 31.4881 0.863093 0.431546 − 0.902091i \(-0.357968\pi\)
0.431546 + 0.902091i \(0.357968\pi\)
\(12\) 0 0
\(13\) 18.6837 0.398609 0.199304 − 0.979938i \(-0.436132\pi\)
0.199304 + 0.979938i \(0.436132\pi\)
\(14\) 0 0
\(15\) −18.7662 −0.323028
\(16\) 0 0
\(17\) −87.9538 −1.25482 −0.627410 − 0.778689i \(-0.715885\pi\)
−0.627410 + 0.778689i \(0.715885\pi\)
\(18\) 0 0
\(19\) 13.0970 0.158140 0.0790700 − 0.996869i \(-0.474805\pi\)
0.0790700 + 0.996869i \(0.474805\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −9.37681 −0.0850087 −0.0425043 − 0.999096i \(-0.513534\pi\)
−0.0425043 + 0.999096i \(0.513534\pi\)
\(24\) 0 0
\(25\) −80.2226 −0.641781
\(26\) 0 0
\(27\) −129.383 −0.922216
\(28\) 0 0
\(29\) −5.11923 −0.0327799 −0.0163900 − 0.999866i \(-0.505217\pi\)
−0.0163900 + 0.999866i \(0.505217\pi\)
\(30\) 0 0
\(31\) −257.412 −1.49137 −0.745685 − 0.666298i \(-0.767878\pi\)
−0.745685 + 0.666298i \(0.767878\pi\)
\(32\) 0 0
\(33\) 88.3067 0.465825
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −380.215 −1.68938 −0.844688 − 0.535259i \(-0.820214\pi\)
−0.844688 + 0.535259i \(0.820214\pi\)
\(38\) 0 0
\(39\) 52.3973 0.215136
\(40\) 0 0
\(41\) −217.959 −0.830230 −0.415115 − 0.909769i \(-0.636259\pi\)
−0.415115 + 0.909769i \(0.636259\pi\)
\(42\) 0 0
\(43\) 377.049 1.33720 0.668598 − 0.743624i \(-0.266895\pi\)
0.668598 + 0.743624i \(0.266895\pi\)
\(44\) 0 0
\(45\) 128.044 0.424171
\(46\) 0 0
\(47\) −357.710 −1.11016 −0.555079 − 0.831798i \(-0.687312\pi\)
−0.555079 + 0.831798i \(0.687312\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) −246.662 −0.677246
\(52\) 0 0
\(53\) 764.390 1.98108 0.990538 − 0.137240i \(-0.0438233\pi\)
0.990538 + 0.137240i \(0.0438233\pi\)
\(54\) 0 0
\(55\) −210.706 −0.516573
\(56\) 0 0
\(57\) 36.7298 0.0853506
\(58\) 0 0
\(59\) −450.671 −0.994447 −0.497224 − 0.867622i \(-0.665647\pi\)
−0.497224 + 0.867622i \(0.665647\pi\)
\(60\) 0 0
\(61\) −174.078 −0.365383 −0.182691 − 0.983170i \(-0.558481\pi\)
−0.182691 + 0.983170i \(0.558481\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −125.023 −0.238573
\(66\) 0 0
\(67\) −497.234 −0.906669 −0.453334 − 0.891340i \(-0.649766\pi\)
−0.453334 + 0.891340i \(0.649766\pi\)
\(68\) 0 0
\(69\) −26.2967 −0.0458805
\(70\) 0 0
\(71\) 350.238 0.585432 0.292716 − 0.956199i \(-0.405441\pi\)
0.292716 + 0.956199i \(0.405441\pi\)
\(72\) 0 0
\(73\) −1062.69 −1.70381 −0.851903 − 0.523699i \(-0.824552\pi\)
−0.851903 + 0.523699i \(0.824552\pi\)
\(74\) 0 0
\(75\) −224.980 −0.346379
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 560.449 0.798170 0.399085 − 0.916914i \(-0.369328\pi\)
0.399085 + 0.916914i \(0.369328\pi\)
\(80\) 0 0
\(81\) 153.799 0.210972
\(82\) 0 0
\(83\) −1105.27 −1.46168 −0.730840 − 0.682549i \(-0.760871\pi\)
−0.730840 + 0.682549i \(0.760871\pi\)
\(84\) 0 0
\(85\) 588.551 0.751027
\(86\) 0 0
\(87\) −14.3566 −0.0176918
\(88\) 0 0
\(89\) 1206.71 1.43721 0.718604 − 0.695420i \(-0.244782\pi\)
0.718604 + 0.695420i \(0.244782\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) −721.897 −0.804916
\(94\) 0 0
\(95\) −87.6398 −0.0946490
\(96\) 0 0
\(97\) 1442.99 1.51045 0.755226 − 0.655465i \(-0.227527\pi\)
0.755226 + 0.655465i \(0.227527\pi\)
\(98\) 0 0
\(99\) −602.528 −0.611680
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 392.4.a.i.1.3 3
4.3 odd 2 784.4.a.be.1.1 3
7.2 even 3 56.4.i.b.25.1 yes 6
7.3 odd 6 392.4.i.m.177.3 6
7.4 even 3 56.4.i.b.9.1 ✓ 6
7.5 odd 6 392.4.i.m.361.3 6
7.6 odd 2 392.4.a.l.1.1 3
21.2 odd 6 504.4.s.h.361.2 6
21.11 odd 6 504.4.s.h.289.2 6
28.11 odd 6 112.4.i.e.65.3 6
28.23 odd 6 112.4.i.e.81.3 6
28.27 even 2 784.4.a.bb.1.3 3
56.11 odd 6 448.4.i.m.65.1 6
56.37 even 6 448.4.i.j.193.3 6
56.51 odd 6 448.4.i.m.193.1 6
56.53 even 6 448.4.i.j.65.3 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
56.4.i.b.9.1 ✓ 6 7.4 even 3
56.4.i.b.25.1 yes 6 7.2 even 3
112.4.i.e.65.3 6 28.11 odd 6
112.4.i.e.81.3 6 28.23 odd 6
392.4.a.i.1.3 3 1.1 even 1 trivial
392.4.a.l.1.1 3 7.6 odd 2
392.4.i.m.177.3 6 7.3 odd 6
392.4.i.m.361.3 6 7.5 odd 6
448.4.i.j.65.3 6 56.53 even 6
448.4.i.j.193.3 6 56.37 even 6
448.4.i.m.65.1 6 56.11 odd 6
448.4.i.m.193.1 6 56.51 odd 6
504.4.s.h.289.2 6 21.11 odd 6
504.4.s.h.361.2 6 21.2 odd 6
784.4.a.bb.1.3 3 28.27 even 2
784.4.a.be.1.1 3 4.3 odd 2