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Properties

Label 672.2.bd.a.431.16
Level $672$
Weight $2$
Character 672.431
Analytic conductor $5.366$
Analytic rank $0$
Dimension $56$
Inner twists $8$

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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] = [672,2,Mod(431,672)] 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("672.431"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(672, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([3, 3, 3, 4])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 672 = 2^{5} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 672.bd (of order \(6\), degree \(2\), 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: \(5.36594701583\)
Analytic rank: \(0\)
Dimension: \(56\)
Relative dimension: \(28\) over \(\Q(\zeta_{6})\)
Twist minimal: no (minimal twist has level 168)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 431.16
Character \(\chi\) \(=\) 672.431
Dual form 672.2.bd.a.527.16

$q$-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)
 
\(f(q)\) \(=\) \(q+(0.316681 - 1.70285i) q^{3} +(1.86088 + 3.22314i) q^{5} +(-2.46429 + 0.962962i) q^{7} +(-2.79943 - 1.07852i) q^{9} +(2.26958 + 1.31034i) q^{11} +3.57182i q^{13} +(6.07784 - 2.14810i) q^{15} +(0.186192 + 0.107498i) q^{17} +(1.14103 + 1.97631i) q^{19} +(0.859391 + 4.50127i) q^{21} +(2.33586 + 4.04583i) q^{23} +(-4.42574 + 7.66560i) q^{25} +(-2.72309 + 4.42547i) q^{27} +2.57415 q^{29} +(-4.26196 - 2.46064i) q^{31} +(2.95006 - 3.44980i) q^{33} +(-7.68949 - 6.15077i) q^{35} +(7.31104 - 4.22103i) q^{37} +(6.08229 + 1.13113i) q^{39} +0.909442i q^{41} -3.73541 q^{43} +(-1.73317 - 11.0299i) q^{45} +(0.586728 + 1.01624i) q^{47} +(5.14541 - 4.74603i) q^{49} +(0.242017 - 0.283015i) q^{51} +(-1.06171 + 1.83894i) q^{53} +9.75355i q^{55} +(3.72672 - 1.31714i) q^{57} +(6.79199 + 3.92136i) q^{59} +(-0.301301 + 0.173956i) q^{61} +(7.93716 - 0.0379510i) q^{63} +(-11.5125 + 6.64673i) q^{65} +(4.98736 - 8.63836i) q^{67} +(7.62919 - 2.69640i) q^{69} -10.0808 q^{71} +(4.45173 - 7.71062i) q^{73} +(11.6519 + 9.96394i) q^{75} +(-6.85470 - 1.04354i) q^{77} +(9.70950 - 5.60578i) q^{79} +(6.67358 + 6.03849i) q^{81} +1.73937i q^{83} +0.800163i q^{85} +(0.815185 - 4.38341i) q^{87} +(15.4694 - 8.93127i) q^{89} +(-3.43953 - 8.80199i) q^{91} +(-5.53980 + 6.47826i) q^{93} +(-4.24662 + 7.35536i) q^{95} -8.88553 q^{97} +(-4.94029 - 6.11600i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 56 q + 2 q^{3} - 2 q^{9} + 4 q^{19} - 16 q^{25} + 8 q^{27} - 14 q^{33} + 16 q^{43} - 16 q^{49} + 34 q^{51} + 4 q^{57} + 36 q^{67} + 4 q^{73} - 10 q^{81} - 72 q^{91} - 32 q^{97} + 44 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/672\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(421\) \(449\) \(577\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\) \(e\left(\frac{2}{3}\right)\)

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
\(3\) 0.316681 − 1.70285i 0.182836 − 0.983143i
\(4\) 0 0
\(5\) 1.86088 + 3.22314i 0.832210 + 1.44143i 0.896282 + 0.443485i \(0.146258\pi\)
−0.0640716 + 0.997945i \(0.520409\pi\)
\(6\) 0 0
\(7\) −2.46429 + 0.962962i −0.931412 + 0.363965i
\(8\) 0 0
\(9\) −2.79943 − 1.07852i −0.933142 − 0.359508i
\(10\) 0 0
\(11\) 2.26958 + 1.31034i 0.684304 + 0.395083i 0.801475 − 0.598029i \(-0.204049\pi\)
−0.117171 + 0.993112i \(0.537382\pi\)
\(12\) 0 0
\(13\) 3.57182i 0.990645i 0.868709 + 0.495323i \(0.164950\pi\)
−0.868709 + 0.495323i \(0.835050\pi\)
\(14\) 0 0
\(15\) 6.07784 − 2.14810i 1.56929 − 0.554637i
\(16\) 0 0
\(17\) 0.186192 + 0.107498i 0.0451582 + 0.0260721i 0.522409 − 0.852695i \(-0.325033\pi\)
−0.477251 + 0.878767i \(0.658367\pi\)
\(18\) 0 0
\(19\) 1.14103 + 1.97631i 0.261769 + 0.453398i 0.966712 − 0.255866i \(-0.0823607\pi\)
−0.704943 + 0.709264i \(0.749027\pi\)
\(20\) 0 0
\(21\) 0.859391 + 4.50127i 0.187535 + 0.982258i
\(22\) 0 0
\(23\) 2.33586 + 4.04583i 0.487061 + 0.843615i 0.999889 − 0.0148767i \(-0.00473557\pi\)
−0.512828 + 0.858491i \(0.671402\pi\)
\(24\) 0 0
\(25\) −4.42574 + 7.66560i −0.885148 + 1.53312i
\(26\) 0 0
\(27\) −2.72309 + 4.42547i −0.524060 + 0.851682i
\(28\) 0 0
\(29\) 2.57415 0.478008 0.239004 − 0.971019i \(-0.423179\pi\)
0.239004 + 0.971019i \(0.423179\pi\)
\(30\) 0 0
\(31\) −4.26196 − 2.46064i −0.765471 − 0.441945i 0.0657857 − 0.997834i \(-0.479045\pi\)
−0.831257 + 0.555889i \(0.812378\pi\)
\(32\) 0 0
\(33\) 2.95006 − 3.44980i 0.513539 − 0.600534i
\(34\) 0 0
\(35\) −7.68949 − 6.15077i −1.29976 − 1.03967i
\(36\) 0 0
\(37\) 7.31104 − 4.22103i 1.20193 − 0.693933i 0.240944 − 0.970539i \(-0.422543\pi\)
0.960983 + 0.276606i \(0.0892098\pi\)
\(38\) 0 0
\(39\) 6.08229 + 1.13113i 0.973946 + 0.181125i
\(40\) 0 0
\(41\) 0.909442i 0.142031i 0.997475 + 0.0710155i \(0.0226240\pi\)
−0.997475 + 0.0710155i \(0.977376\pi\)
\(42\) 0 0
\(43\) −3.73541 −0.569644 −0.284822 − 0.958580i \(-0.591935\pi\)
−0.284822 + 0.958580i \(0.591935\pi\)
\(44\) 0 0
\(45\) −1.73317 − 11.0299i −0.258365 − 1.64425i
\(46\) 0 0
\(47\) 0.586728 + 1.01624i 0.0855831 + 0.148234i 0.905640 − 0.424048i \(-0.139391\pi\)
−0.820056 + 0.572283i \(0.806058\pi\)
\(48\) 0 0
\(49\) 5.14541 − 4.74603i 0.735058 − 0.678004i
\(50\) 0 0
\(51\) 0.242017 − 0.283015i 0.0338891 − 0.0396301i
\(52\) 0 0
\(53\) −1.06171 + 1.83894i −0.145837 + 0.252598i −0.929685 − 0.368355i \(-0.879921\pi\)
0.783848 + 0.620953i \(0.213254\pi\)
\(54\) 0 0
\(55\) 9.75355i 1.31517i
\(56\) 0 0
\(57\) 3.72672 − 1.31714i 0.493616 − 0.174459i
\(58\) 0 0
\(59\) 6.79199 + 3.92136i 0.884242 + 0.510517i 0.872055 − 0.489408i \(-0.162787\pi\)
0.0121872 + 0.999926i \(0.496121\pi\)
\(60\) 0 0
\(61\) −0.301301 + 0.173956i −0.0385776 + 0.0222728i −0.519165 − 0.854674i \(-0.673757\pi\)
0.480587 + 0.876947i \(0.340424\pi\)
\(62\) 0 0
\(63\) 7.93716 − 0.0379510i 0.999989 − 0.00478137i
\(64\) 0 0
\(65\) −11.5125 + 6.64673i −1.42795 + 0.824425i
\(66\) 0 0
\(67\) 4.98736 − 8.63836i 0.609303 − 1.05534i −0.382053 − 0.924140i \(-0.624783\pi\)
0.991356 − 0.131203i \(-0.0418839\pi\)
\(68\) 0 0
\(69\) 7.62919 − 2.69640i 0.918446 − 0.324608i
\(70\) 0 0
\(71\) −10.0808 −1.19637 −0.598183 − 0.801359i \(-0.704111\pi\)
−0.598183 + 0.801359i \(0.704111\pi\)
\(72\) 0 0
\(73\) 4.45173 − 7.71062i 0.521036 − 0.902460i −0.478665 − 0.877998i \(-0.658879\pi\)
0.999701 − 0.0244626i \(-0.00778746\pi\)
\(74\) 0 0
\(75\) 11.6519 + 9.96394i 1.34544 + 1.15054i
\(76\) 0 0
\(77\) −6.85470 − 1.04354i −0.781166 − 0.118922i
\(78\) 0 0
\(79\) 9.70950 − 5.60578i 1.09240 − 0.630700i 0.158189 − 0.987409i \(-0.449434\pi\)
0.934216 + 0.356709i \(0.116101\pi\)
\(80\) 0 0
\(81\) 6.67358 + 6.03849i 0.741508 + 0.670944i
\(82\) 0 0
\(83\) 1.73937i 0.190921i 0.995433 + 0.0954603i \(0.0304323\pi\)
−0.995433 + 0.0954603i \(0.969568\pi\)
\(84\) 0 0
\(85\) 0.800163i 0.0867898i
\(86\) 0 0
\(87\) 0.815185 − 4.38341i 0.0873970 − 0.469950i
\(88\) 0 0
\(89\) 15.4694 − 8.93127i 1.63975 − 0.946712i 0.658837 − 0.752286i \(-0.271049\pi\)
0.980917 − 0.194426i \(-0.0622845\pi\)
\(90\) 0 0
\(91\) −3.43953 − 8.80199i −0.360560 − 0.922699i
\(92\) 0 0
\(93\) −5.53980 + 6.47826i −0.574451 + 0.671764i
\(94\) 0 0
\(95\) −4.24662 + 7.35536i −0.435694 + 0.754644i
\(96\) 0 0
\(97\) −8.88553 −0.902189 −0.451094 − 0.892476i \(-0.648966\pi\)
−0.451094 + 0.892476i \(0.648966\pi\)
\(98\) 0 0
\(99\) −4.94029 − 6.11600i −0.496518 − 0.614681i
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 672.2.bd.a.431.16 56
3.2 odd 2 inner 672.2.bd.a.431.21 56
4.3 odd 2 168.2.v.a.11.6 yes 56
7.2 even 3 inner 672.2.bd.a.527.22 56
8.3 odd 2 inner 672.2.bd.a.431.15 56
8.5 even 2 168.2.v.a.11.3 ✓ 56
12.11 even 2 168.2.v.a.11.23 yes 56
21.2 odd 6 inner 672.2.bd.a.527.15 56
24.5 odd 2 168.2.v.a.11.26 yes 56
24.11 even 2 inner 672.2.bd.a.431.22 56
28.23 odd 6 168.2.v.a.107.26 yes 56
56.37 even 6 168.2.v.a.107.23 yes 56
56.51 odd 6 inner 672.2.bd.a.527.21 56
84.23 even 6 168.2.v.a.107.3 yes 56
168.107 even 6 inner 672.2.bd.a.527.16 56
168.149 odd 6 168.2.v.a.107.6 yes 56
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
168.2.v.a.11.3 ✓ 56 8.5 even 2
168.2.v.a.11.6 yes 56 4.3 odd 2
168.2.v.a.11.23 yes 56 12.11 even 2
168.2.v.a.11.26 yes 56 24.5 odd 2
168.2.v.a.107.3 yes 56 84.23 even 6
168.2.v.a.107.6 yes 56 168.149 odd 6
168.2.v.a.107.23 yes 56 56.37 even 6
168.2.v.a.107.26 yes 56 28.23 odd 6
672.2.bd.a.431.15 56 8.3 odd 2 inner
672.2.bd.a.431.16 56 1.1 even 1 trivial
672.2.bd.a.431.21 56 3.2 odd 2 inner
672.2.bd.a.431.22 56 24.11 even 2 inner
672.2.bd.a.527.15 56 21.2 odd 6 inner
672.2.bd.a.527.16 56 168.107 even 6 inner
672.2.bd.a.527.21 56 56.51 odd 6 inner
672.2.bd.a.527.22 56 7.2 even 3 inner