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Properties

Label 104.2.b.b.53.2
Level $104$
Weight $2$
Character 104.53
Analytic conductor $0.830$
Analytic rank $0$
Dimension $4$
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] = [104,2,Mod(53,104)] 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("104.53"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(104, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 104 = 2^{3} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 104.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(0.830444181021\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 53.2
Root \(-0.866025 - 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 104.53
Dual form 104.2.b.b.53.1

$q$-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+(-0.366025 + 1.36603i) q^{2} -2.00000i q^{3} +(-1.73205 - 1.00000i) q^{4} -3.46410i q^{5} +(2.73205 + 0.732051i) q^{6} -1.26795 q^{7} +(2.00000 - 2.00000i) q^{8} -1.00000 q^{9} +(4.73205 + 1.26795i) q^{10} +4.73205i q^{11} +(-2.00000 + 3.46410i) q^{12} -1.00000i q^{13} +(0.464102 - 1.73205i) q^{14} -6.92820 q^{15} +(2.00000 + 3.46410i) q^{16} +5.46410 q^{17} +(0.366025 - 1.36603i) q^{18} -0.732051i q^{19} +(-3.46410 + 6.00000i) q^{20} +2.53590i q^{21} +(-6.46410 - 1.73205i) q^{22} +4.00000 q^{23} +(-4.00000 - 4.00000i) q^{24} -7.00000 q^{25} +(1.36603 + 0.366025i) q^{26} -4.00000i q^{27} +(2.19615 + 1.26795i) q^{28} -2.00000i q^{29} +(2.53590 - 9.46410i) q^{30} -6.73205 q^{31} +(-5.46410 + 1.46410i) q^{32} +9.46410 q^{33} +(-2.00000 + 7.46410i) q^{34} +4.39230i q^{35} +(1.73205 + 1.00000i) q^{36} +8.92820i q^{37} +(1.00000 + 0.267949i) q^{38} -2.00000 q^{39} +(-6.92820 - 6.92820i) q^{40} +8.92820 q^{41} +(-3.46410 - 0.928203i) q^{42} +0.535898i q^{43} +(4.73205 - 8.19615i) q^{44} +3.46410i q^{45} +(-1.46410 + 5.46410i) q^{46} +6.73205 q^{47} +(6.92820 - 4.00000i) q^{48} -5.39230 q^{49} +(2.56218 - 9.56218i) q^{50} -10.9282i q^{51} +(-1.00000 + 1.73205i) q^{52} +2.92820i q^{53} +(5.46410 + 1.46410i) q^{54} +16.3923 q^{55} +(-2.53590 + 2.53590i) q^{56} -1.46410 q^{57} +(2.73205 + 0.732051i) q^{58} +10.1962i q^{59} +(12.0000 + 6.92820i) q^{60} -2.92820i q^{61} +(2.46410 - 9.19615i) q^{62} +1.26795 q^{63} -8.00000i q^{64} -3.46410 q^{65} +(-3.46410 + 12.9282i) q^{66} -0.732051i q^{67} +(-9.46410 - 5.46410i) q^{68} -8.00000i q^{69} +(-6.00000 - 1.60770i) q^{70} -8.19615 q^{71} +(-2.00000 + 2.00000i) q^{72} -7.46410 q^{73} +(-12.1962 - 3.26795i) q^{74} +14.0000i q^{75} +(-0.732051 + 1.26795i) q^{76} -6.00000i q^{77} +(0.732051 - 2.73205i) q^{78} +5.46410 q^{79} +(12.0000 - 6.92820i) q^{80} -11.0000 q^{81} +(-3.26795 + 12.1962i) q^{82} -3.26795i q^{83} +(2.53590 - 4.39230i) q^{84} -18.9282i q^{85} +(-0.732051 - 0.196152i) q^{86} -4.00000 q^{87} +(9.46410 + 9.46410i) q^{88} -17.3205 q^{89} +(-4.73205 - 1.26795i) q^{90} +1.26795i q^{91} +(-6.92820 - 4.00000i) q^{92} +13.4641i q^{93} +(-2.46410 + 9.19615i) q^{94} -2.53590 q^{95} +(2.92820 + 10.9282i) q^{96} +6.39230 q^{97} +(1.97372 - 7.36603i) q^{98} -4.73205i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{2} + 4 q^{6} - 12 q^{7} + 8 q^{8} - 4 q^{9} + 12 q^{10} - 8 q^{12} - 12 q^{14} + 8 q^{16} + 8 q^{17} - 2 q^{18} - 12 q^{22} + 16 q^{23} - 16 q^{24} - 28 q^{25} + 2 q^{26} - 12 q^{28} + 24 q^{30}+ \cdots + 46 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/104\mathbb{Z}\right)^\times\).

\(n\) \(41\) \(53\) \(79\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.366025 + 1.36603i −0.258819 + 0.965926i
\(3\) − 2.00000i − 1.15470i −0.816497 − 0.577350i \(-0.804087\pi\)
0.816497 − 0.577350i \(-0.195913\pi\)
\(4\) −1.73205 − 1.00000i −0.866025 − 0.500000i
\(5\) − 3.46410i − 1.54919i −0.632456 − 0.774597i \(-0.717953\pi\)
0.632456 − 0.774597i \(-0.282047\pi\)
\(6\) 2.73205 + 0.732051i 1.11536 + 0.298858i
\(7\) −1.26795 −0.479240 −0.239620 − 0.970867i \(-0.577023\pi\)
−0.239620 + 0.970867i \(0.577023\pi\)
\(8\) 2.00000 − 2.00000i 0.707107 − 0.707107i
\(9\) −1.00000 −0.333333
\(10\) 4.73205 + 1.26795i 1.49641 + 0.400961i
\(11\) 4.73205i 1.42677i 0.700774 + 0.713384i \(0.252838\pi\)
−0.700774 + 0.713384i \(0.747162\pi\)
\(12\) −2.00000 + 3.46410i −0.577350 + 1.00000i
\(13\) − 1.00000i − 0.277350i
\(14\) 0.464102 − 1.73205i 0.124036 − 0.462910i
\(15\) −6.92820 −1.78885
\(16\) 2.00000 + 3.46410i 0.500000 + 0.866025i
\(17\) 5.46410 1.32524 0.662620 − 0.748956i \(-0.269445\pi\)
0.662620 + 0.748956i \(0.269445\pi\)
\(18\) 0.366025 − 1.36603i 0.0862730 − 0.321975i
\(19\) − 0.732051i − 0.167944i −0.996468 − 0.0839720i \(-0.973239\pi\)
0.996468 − 0.0839720i \(-0.0267606\pi\)
\(20\) −3.46410 + 6.00000i −0.774597 + 1.34164i
\(21\) 2.53590i 0.553378i
\(22\) −6.46410 − 1.73205i −1.37815 − 0.369274i
\(23\) 4.00000 0.834058 0.417029 − 0.908893i \(-0.363071\pi\)
0.417029 + 0.908893i \(0.363071\pi\)
\(24\) −4.00000 − 4.00000i −0.816497 − 0.816497i
\(25\) −7.00000 −1.40000
\(26\) 1.36603 + 0.366025i 0.267900 + 0.0717835i
\(27\) − 4.00000i − 0.769800i
\(28\) 2.19615 + 1.26795i 0.415034 + 0.239620i
\(29\) − 2.00000i − 0.371391i −0.982607 − 0.185695i \(-0.940546\pi\)
0.982607 − 0.185695i \(-0.0594537\pi\)
\(30\) 2.53590 − 9.46410i 0.462990 − 1.72790i
\(31\) −6.73205 −1.20911 −0.604556 − 0.796563i \(-0.706649\pi\)
−0.604556 + 0.796563i \(0.706649\pi\)
\(32\) −5.46410 + 1.46410i −0.965926 + 0.258819i
\(33\) 9.46410 1.64749
\(34\) −2.00000 + 7.46410i −0.342997 + 1.28008i
\(35\) 4.39230i 0.742435i
\(36\) 1.73205 + 1.00000i 0.288675 + 0.166667i
\(37\) 8.92820i 1.46779i 0.679264 + 0.733894i \(0.262299\pi\)
−0.679264 + 0.733894i \(0.737701\pi\)
\(38\) 1.00000 + 0.267949i 0.162221 + 0.0434671i
\(39\) −2.00000 −0.320256
\(40\) −6.92820 − 6.92820i −1.09545 − 1.09545i
\(41\) 8.92820 1.39435 0.697176 − 0.716900i \(-0.254440\pi\)
0.697176 + 0.716900i \(0.254440\pi\)
\(42\) −3.46410 − 0.928203i −0.534522 − 0.143225i
\(43\) 0.535898i 0.0817237i 0.999165 + 0.0408619i \(0.0130104\pi\)
−0.999165 + 0.0408619i \(0.986990\pi\)
\(44\) 4.73205 − 8.19615i 0.713384 − 1.23562i
\(45\) 3.46410i 0.516398i
\(46\) −1.46410 + 5.46410i −0.215870 + 0.805638i
\(47\) 6.73205 0.981971 0.490985 − 0.871168i \(-0.336637\pi\)
0.490985 + 0.871168i \(0.336637\pi\)
\(48\) 6.92820 − 4.00000i 1.00000 − 0.577350i
\(49\) −5.39230 −0.770329
\(50\) 2.56218 − 9.56218i 0.362347 − 1.35230i
\(51\) − 10.9282i − 1.53025i
\(52\) −1.00000 + 1.73205i −0.138675 + 0.240192i
\(53\) 2.92820i 0.402220i 0.979569 + 0.201110i \(0.0644548\pi\)
−0.979569 + 0.201110i \(0.935545\pi\)
\(54\) 5.46410 + 1.46410i 0.743570 + 0.199239i
\(55\) 16.3923 2.21034
\(56\) −2.53590 + 2.53590i −0.338874 + 0.338874i
\(57\) −1.46410 −0.193925
\(58\) 2.73205 + 0.732051i 0.358736 + 0.0961230i
\(59\) 10.1962i 1.32743i 0.747987 + 0.663713i \(0.231020\pi\)
−0.747987 + 0.663713i \(0.768980\pi\)
\(60\) 12.0000 + 6.92820i 1.54919 + 0.894427i
\(61\) − 2.92820i − 0.374918i −0.982272 − 0.187459i \(-0.939975\pi\)
0.982272 − 0.187459i \(-0.0600252\pi\)
\(62\) 2.46410 − 9.19615i 0.312941 − 1.16791i
\(63\) 1.26795 0.159747
\(64\) − 8.00000i − 1.00000i
\(65\) −3.46410 −0.429669
\(66\) −3.46410 + 12.9282i −0.426401 + 1.59135i
\(67\) − 0.732051i − 0.0894342i −0.999000 − 0.0447171i \(-0.985761\pi\)
0.999000 − 0.0447171i \(-0.0142386\pi\)
\(68\) −9.46410 − 5.46410i −1.14769 − 0.662620i
\(69\) − 8.00000i − 0.963087i
\(70\) −6.00000 − 1.60770i −0.717137 − 0.192156i
\(71\) −8.19615 −0.972704 −0.486352 − 0.873763i \(-0.661673\pi\)
−0.486352 + 0.873763i \(0.661673\pi\)
\(72\) −2.00000 + 2.00000i −0.235702 + 0.235702i
\(73\) −7.46410 −0.873607 −0.436804 − 0.899557i \(-0.643889\pi\)
−0.436804 + 0.899557i \(0.643889\pi\)
\(74\) −12.1962 − 3.26795i −1.41777 − 0.379891i
\(75\) 14.0000i 1.61658i
\(76\) −0.732051 + 1.26795i −0.0839720 + 0.145444i
\(77\) − 6.00000i − 0.683763i
\(78\) 0.732051 − 2.73205i 0.0828884 − 0.309344i
\(79\) 5.46410 0.614759 0.307380 − 0.951587i \(-0.400548\pi\)
0.307380 + 0.951587i \(0.400548\pi\)
\(80\) 12.0000 − 6.92820i 1.34164 − 0.774597i
\(81\) −11.0000 −1.22222
\(82\) −3.26795 + 12.1962i −0.360885 + 1.34684i
\(83\) − 3.26795i − 0.358704i −0.983785 − 0.179352i \(-0.942600\pi\)
0.983785 − 0.179352i \(-0.0574001\pi\)
\(84\) 2.53590 − 4.39230i 0.276689 − 0.479240i
\(85\) − 18.9282i − 2.05305i
\(86\) −0.732051 − 0.196152i −0.0789391 − 0.0211517i
\(87\) −4.00000 −0.428845
\(88\) 9.46410 + 9.46410i 1.00888 + 1.00888i
\(89\) −17.3205 −1.83597 −0.917985 − 0.396615i \(-0.870185\pi\)
−0.917985 + 0.396615i \(0.870185\pi\)
\(90\) −4.73205 − 1.26795i −0.498802 − 0.133654i
\(91\) 1.26795i 0.132917i
\(92\) −6.92820 − 4.00000i −0.722315 − 0.417029i
\(93\) 13.4641i 1.39616i
\(94\) −2.46410 + 9.19615i −0.254153 + 0.948511i
\(95\) −2.53590 −0.260178
\(96\) 2.92820 + 10.9282i 0.298858 + 1.11536i
\(97\) 6.39230 0.649040 0.324520 − 0.945879i \(-0.394797\pi\)
0.324520 + 0.945879i \(0.394797\pi\)
\(98\) 1.97372 − 7.36603i 0.199376 − 0.744081i
\(99\) − 4.73205i − 0.475589i
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 104.2.b.b.53.2 yes 4
3.2 odd 2 936.2.g.b.469.3 4
4.3 odd 2 416.2.b.b.209.3 4
8.3 odd 2 416.2.b.b.209.2 4
8.5 even 2 inner 104.2.b.b.53.1 ✓ 4
12.11 even 2 3744.2.g.b.1873.3 4
16.3 odd 4 3328.2.a.m.1.1 2
16.5 even 4 3328.2.a.n.1.2 2
16.11 odd 4 3328.2.a.bc.1.2 2
16.13 even 4 3328.2.a.bd.1.1 2
24.5 odd 2 936.2.g.b.469.4 4
24.11 even 2 3744.2.g.b.1873.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
104.2.b.b.53.1 ✓ 4 8.5 even 2 inner
104.2.b.b.53.2 yes 4 1.1 even 1 trivial
416.2.b.b.209.2 4 8.3 odd 2
416.2.b.b.209.3 4 4.3 odd 2
936.2.g.b.469.3 4 3.2 odd 2
936.2.g.b.469.4 4 24.5 odd 2
3328.2.a.m.1.1 2 16.3 odd 4
3328.2.a.n.1.2 2 16.5 even 4
3328.2.a.bc.1.2 2 16.11 odd 4
3328.2.a.bd.1.1 2 16.13 even 4
3744.2.g.b.1873.1 4 24.11 even 2
3744.2.g.b.1873.3 4 12.11 even 2