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Properties

Label 608.2.bf.a.81.6
Level $608$
Weight $2$
Character 608.81
Analytic conductor $4.855$
Analytic rank $0$
Dimension $108$
Inner twists $4$

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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] = [608,2,Mod(17,608)] 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("608.17"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(608, base_ring=CyclotomicField(18)) chi = DirichletCharacter(H, H._module([0, 9, 10])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 608 = 2^{5} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 608.bf (of order \(18\), degree \(6\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(4.85490444289\)
Analytic rank: \(0\)
Dimension: \(108\)
Relative dimension: \(18\) over \(\Q(\zeta_{18})\)
Twist minimal: no (minimal twist has level 152)
Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label 81.6
Character \(\chi\) \(=\) 608.81
Dual form 608.2.bf.a.593.6

$q$-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)
 
\(f(q)\) \(=\) \(q+(-1.33117 - 0.234720i) q^{3} +(-2.47882 + 2.95414i) q^{5} +(0.0505133 + 0.0874916i) q^{7} +(-1.10217 - 0.401157i) q^{9} +(-4.94957 - 2.85764i) q^{11} +(3.59690 - 0.634230i) q^{13} +(3.99312 - 3.35063i) q^{15} +(5.84283 - 2.12662i) q^{17} +(4.29207 - 0.760365i) q^{19} +(-0.0467055 - 0.128322i) q^{21} +(0.535457 - 0.449302i) q^{23} +(-1.71417 - 9.72156i) q^{25} +(4.88483 + 2.82026i) q^{27} +(-0.348958 + 0.958756i) q^{29} +(-0.103784 - 0.179759i) q^{31} +(5.91795 + 4.96575i) q^{33} +(-0.383677 - 0.0676525i) q^{35} -4.44091i q^{37} -4.93693 q^{39} +(-0.304992 + 1.72970i) q^{41} +(1.85588 - 2.21175i) q^{43} +(3.91716 - 2.26157i) q^{45} +(-7.59618 - 2.76478i) q^{47} +(3.49490 - 6.05334i) q^{49} +(-8.27693 + 1.45945i) q^{51} +(-5.97088 - 7.11582i) q^{53} +(20.7110 - 7.53818i) q^{55} +(-5.89192 + 0.00473651i) q^{57} +(-1.29623 - 3.56137i) q^{59} +(-3.77642 - 4.50056i) q^{61} +(-0.0205764 - 0.116694i) q^{63} +(-7.04246 + 12.1979i) q^{65} +(0.305920 - 0.840509i) q^{67} +(-0.818242 + 0.472412i) q^{69} +(0.175451 + 0.147221i) q^{71} +(-1.15815 + 6.56822i) q^{73} +13.3434i q^{75} -0.577395i q^{77} +(-1.80071 + 10.2123i) q^{79} +(-3.14505 - 2.63901i) q^{81} +(3.67936 - 2.12428i) q^{83} +(-8.20100 + 22.5321i) q^{85} +(0.689561 - 1.19435i) q^{87} +(-1.98775 - 11.2731i) q^{89} +(0.237181 + 0.282661i) q^{91} +(0.0959604 + 0.263649i) q^{93} +(-8.39304 + 14.5642i) q^{95} +(-6.86128 + 2.49730i) q^{97} +(4.30891 + 5.13516i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 108 q + 6 q^{7} - 12 q^{9} + 12 q^{15} - 12 q^{17} + 12 q^{23} - 12 q^{25} - 30 q^{31} - 30 q^{33} + 24 q^{39} - 24 q^{41} + 48 q^{47} - 24 q^{49} + 42 q^{55} - 12 q^{57} - 30 q^{63} - 6 q^{65} + 12 q^{71}+ \cdots - 12 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/608\mathbb{Z}\right)^\times\).

\(n\) \(97\) \(191\) \(229\)
\(\chi(n)\) \(e\left(\frac{8}{9}\right)\) \(1\) \(-1\)

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 0
\(3\) −1.33117 − 0.234720i −0.768549 − 0.135516i −0.224391 − 0.974499i \(-0.572039\pi\)
−0.544157 + 0.838983i \(0.683151\pi\)
\(4\) 0 0
\(5\) −2.47882 + 2.95414i −1.10856 + 1.32113i −0.166370 + 0.986063i \(0.553205\pi\)
−0.942193 + 0.335070i \(0.891240\pi\)
\(6\) 0 0
\(7\) 0.0505133 + 0.0874916i 0.0190922 + 0.0330687i 0.875414 − 0.483374i \(-0.160589\pi\)
−0.856321 + 0.516443i \(0.827256\pi\)
\(8\) 0 0
\(9\) −1.10217 − 0.401157i −0.367390 − 0.133719i
\(10\) 0 0
\(11\) −4.94957 − 2.85764i −1.49235 − 0.861609i −0.492390 − 0.870375i \(-0.663877\pi\)
−0.999962 + 0.00876534i \(0.997210\pi\)
\(12\) 0 0
\(13\) 3.59690 − 0.634230i 0.997600 − 0.175904i 0.349073 − 0.937095i \(-0.386496\pi\)
0.648527 + 0.761192i \(0.275385\pi\)
\(14\) 0 0
\(15\) 3.99312 − 3.35063i 1.03102 − 0.865128i
\(16\) 0 0
\(17\) 5.84283 − 2.12662i 1.41709 − 0.515780i 0.483891 − 0.875128i \(-0.339223\pi\)
0.933203 + 0.359348i \(0.117001\pi\)
\(18\) 0 0
\(19\) 4.29207 − 0.760365i 0.984668 − 0.174440i
\(20\) 0 0
\(21\) −0.0467055 − 0.128322i −0.0101920 − 0.0280022i
\(22\) 0 0
\(23\) 0.535457 − 0.449302i 0.111650 − 0.0936859i −0.585253 − 0.810851i \(-0.699005\pi\)
0.696904 + 0.717165i \(0.254560\pi\)
\(24\) 0 0
\(25\) −1.71417 − 9.72156i −0.342835 − 1.94431i
\(26\) 0 0
\(27\) 4.88483 + 2.82026i 0.940086 + 0.542759i
\(28\) 0 0
\(29\) −0.348958 + 0.958756i −0.0648000 + 0.178036i −0.967867 − 0.251464i \(-0.919088\pi\)
0.903067 + 0.429500i \(0.141310\pi\)
\(30\) 0 0
\(31\) −0.103784 − 0.179759i −0.0186401 − 0.0322857i 0.856555 − 0.516056i \(-0.172600\pi\)
−0.875195 + 0.483770i \(0.839267\pi\)
\(32\) 0 0
\(33\) 5.91795 + 4.96575i 1.03018 + 0.864426i
\(34\) 0 0
\(35\) −0.383677 − 0.0676525i −0.0648532 − 0.0114354i
\(36\) 0 0
\(37\) − 4.44091i − 0.730081i −0.930992 − 0.365040i \(-0.881055\pi\)
0.930992 − 0.365040i \(-0.118945\pi\)
\(38\) 0 0
\(39\) −4.93693 −0.790542
\(40\) 0 0
\(41\) −0.304992 + 1.72970i −0.0476318 + 0.270133i −0.999317 − 0.0369407i \(-0.988239\pi\)
0.951686 + 0.307074i \(0.0993499\pi\)
\(42\) 0 0
\(43\) 1.85588 − 2.21175i 0.283019 − 0.337289i −0.605741 − 0.795662i \(-0.707123\pi\)
0.888760 + 0.458373i \(0.151568\pi\)
\(44\) 0 0
\(45\) 3.91716 − 2.26157i 0.583936 − 0.337136i
\(46\) 0 0
\(47\) −7.59618 − 2.76478i −1.10802 − 0.403285i −0.277751 − 0.960653i \(-0.589589\pi\)
−0.830266 + 0.557368i \(0.811811\pi\)
\(48\) 0 0
\(49\) 3.49490 − 6.05334i 0.499271 − 0.864763i
\(50\) 0 0
\(51\) −8.27693 + 1.45945i −1.15900 + 0.204363i
\(52\) 0 0
\(53\) −5.97088 − 7.11582i −0.820164 − 0.977433i 0.179817 − 0.983700i \(-0.442450\pi\)
−0.999980 + 0.00626717i \(0.998005\pi\)
\(54\) 0 0
\(55\) 20.7110 − 7.53818i 2.79267 − 1.01645i
\(56\) 0 0
\(57\) −5.89192 + 0.00473651i −0.780405 + 0.000627367i
\(58\) 0 0
\(59\) −1.29623 − 3.56137i −0.168755 − 0.463651i 0.826270 − 0.563274i \(-0.190459\pi\)
−0.995025 + 0.0996231i \(0.968236\pi\)
\(60\) 0 0
\(61\) −3.77642 − 4.50056i −0.483521 − 0.576238i 0.468036 − 0.883709i \(-0.344962\pi\)
−0.951557 + 0.307471i \(0.900517\pi\)
\(62\) 0 0
\(63\) −0.0205764 − 0.116694i −0.00259238 − 0.0147021i
\(64\) 0 0
\(65\) −7.04246 + 12.1979i −0.873510 + 1.51296i
\(66\) 0 0
\(67\) 0.305920 − 0.840509i 0.0373741 − 0.102684i −0.919602 − 0.392851i \(-0.871489\pi\)
0.956976 + 0.290167i \(0.0937109\pi\)
\(68\) 0 0
\(69\) −0.818242 + 0.472412i −0.0985048 + 0.0568717i
\(70\) 0 0
\(71\) 0.175451 + 0.147221i 0.0208222 + 0.0174719i 0.653139 − 0.757238i \(-0.273452\pi\)
−0.632317 + 0.774710i \(0.717896\pi\)
\(72\) 0 0
\(73\) −1.15815 + 6.56822i −0.135552 + 0.768752i 0.838922 + 0.544251i \(0.183186\pi\)
−0.974474 + 0.224501i \(0.927925\pi\)
\(74\) 0 0
\(75\) 13.3434i 1.54076i
\(76\) 0 0
\(77\) − 0.577395i − 0.0658002i
\(78\) 0 0
\(79\) −1.80071 + 10.2123i −0.202596 + 1.14898i 0.698583 + 0.715529i \(0.253814\pi\)
−0.901179 + 0.433448i \(0.857297\pi\)
\(80\) 0 0
\(81\) −3.14505 − 2.63901i −0.349451 − 0.293224i
\(82\) 0 0
\(83\) 3.67936 − 2.12428i 0.403862 − 0.233170i −0.284287 − 0.958739i \(-0.591757\pi\)
0.688149 + 0.725569i \(0.258424\pi\)
\(84\) 0 0
\(85\) −8.20100 + 22.5321i −0.889523 + 2.44395i
\(86\) 0 0
\(87\) 0.689561 − 1.19435i 0.0739287 − 0.128048i
\(88\) 0 0
\(89\) −1.98775 − 11.2731i −0.210701 − 1.19494i −0.888213 − 0.459432i \(-0.848053\pi\)
0.677512 − 0.735512i \(-0.263058\pi\)
\(90\) 0 0
\(91\) 0.237181 + 0.282661i 0.0248633 + 0.0296310i
\(92\) 0 0
\(93\) 0.0959604 + 0.263649i 0.00995063 + 0.0273391i
\(94\) 0 0
\(95\) −8.39304 + 14.5642i −0.861108 + 1.49426i
\(96\) 0 0
\(97\) −6.86128 + 2.49730i −0.696657 + 0.253563i −0.665983 − 0.745967i \(-0.731988\pi\)
−0.0306743 + 0.999529i \(0.509765\pi\)
\(98\) 0 0
\(99\) 4.30891 + 5.13516i 0.433062 + 0.516103i
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 608.2.bf.a.81.6 108
4.3 odd 2 152.2.t.a.5.10 yes 108
8.3 odd 2 152.2.t.a.5.6 ✓ 108
8.5 even 2 inner 608.2.bf.a.81.13 108
19.4 even 9 inner 608.2.bf.a.593.13 108
76.23 odd 18 152.2.t.a.61.6 yes 108
152.61 even 18 inner 608.2.bf.a.593.6 108
152.99 odd 18 152.2.t.a.61.10 yes 108
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
152.2.t.a.5.6 ✓ 108 8.3 odd 2
152.2.t.a.5.10 yes 108 4.3 odd 2
152.2.t.a.61.6 yes 108 76.23 odd 18
152.2.t.a.61.10 yes 108 152.99 odd 18
608.2.bf.a.81.6 108 1.1 even 1 trivial
608.2.bf.a.81.13 108 8.5 even 2 inner
608.2.bf.a.593.6 108 152.61 even 18 inner
608.2.bf.a.593.13 108 19.4 even 9 inner