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Properties

Label 1680.2.ba.a.911.3
Level $1680$
Weight $2$
Character 1680.911
Analytic conductor $13.415$
Analytic rank $0$
Dimension $8$
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] = [1680,2,Mod(911,1680)] 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("1680.911"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1680, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0, 1, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1680 = 2^{4} \cdot 3 \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1680.ba (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,0,0,0,0,0,0,0,-2,0,-12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(13.4148675396\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.7278137344.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 2x^{7} + x^{6} + 6x^{5} - 20x^{4} + 18x^{3} + 9x^{2} - 54x + 81 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 911.3
Root \(1.67936 + 0.423958i\) of defining polynomial
Character \(\chi\) \(=\) 1680.911
Dual form 1680.2.ba.a.911.4

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.423958 - 1.67936i) q^{3} +1.00000i q^{5} -1.00000i q^{7} +(-2.64052 + 1.42396i) q^{9} -4.47197 q^{11} +4.98278 q^{13} +(1.67936 - 0.423958i) q^{15} +1.35873i q^{17} -0.809072i q^{19} +(-1.67936 + 0.423958i) q^{21} -8.63977 q^{23} -1.00000 q^{25} +(3.51081 + 3.83069i) q^{27} -1.07440i q^{29} -0.886759i q^{31} +(1.89592 + 7.51006i) q^{33} +1.00000 q^{35} -6.41328 q^{37} +(-2.11249 - 8.36789i) q^{39} +6.63977i q^{41} +8.63977i q^{43} +(-1.42396 - 2.64052i) q^{45} -0.130464 q^{47} -1.00000 q^{49} +(2.28179 - 0.576042i) q^{51} +12.5481i q^{53} -4.47197i q^{55} +(-1.35873 + 0.343012i) q^{57} -5.02162 q^{59} -10.3356 q^{61} +(1.42396 + 2.64052i) q^{63} +4.98278i q^{65} +2.97838i q^{67} +(3.66290 + 14.5093i) q^{69} +4.85231 q^{71} +1.19093 q^{73} +(0.423958 + 1.67936i) q^{75} +4.47197i q^{77} -2.77023i q^{79} +(4.94469 - 7.51998i) q^{81} +7.58370 q^{83} -1.35873 q^{85} +(-1.80430 + 0.455499i) q^{87} -1.32579i q^{89} -4.98278i q^{91} +(-1.48919 + 0.375948i) q^{93} +0.809072 q^{95} -14.6786 q^{97} +(11.8083 - 6.36789i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 2 q^{9} - 12 q^{11} + 2 q^{15} - 2 q^{21} - 8 q^{23} - 8 q^{25} + 12 q^{27} - 12 q^{33} + 8 q^{35} + 8 q^{37} + 26 q^{39} - 8 q^{45} - 40 q^{47} - 8 q^{49} + 22 q^{51} + 12 q^{57} - 8 q^{59} - 8 q^{61}+ \cdots + 38 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1680\mathbb{Z}\right)^\times\).

\(n\) \(241\) \(337\) \(421\) \(1121\) \(1471\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-1\) \(-1\)

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 0
\(3\) −0.423958 − 1.67936i −0.244772 − 0.969581i
\(4\) 0 0
\(5\) 1.00000i 0.447214i
\(6\) 0 0
\(7\) − 1.00000i − 0.377964i
\(8\) 0 0
\(9\) −2.64052 + 1.42396i −0.880173 + 0.474653i
\(10\) 0 0
\(11\) −4.47197 −1.34835 −0.674174 − 0.738572i \(-0.735500\pi\)
−0.674174 + 0.738572i \(0.735500\pi\)
\(12\) 0 0
\(13\) 4.98278 1.38197 0.690987 − 0.722867i \(-0.257176\pi\)
0.690987 + 0.722867i \(0.257176\pi\)
\(14\) 0 0
\(15\) 1.67936 − 0.423958i 0.433610 − 0.109465i
\(16\) 0 0
\(17\) 1.35873i 0.329539i 0.986332 + 0.164770i \(0.0526881\pi\)
−0.986332 + 0.164770i \(0.947312\pi\)
\(18\) 0 0
\(19\) − 0.809072i − 0.185614i −0.995684 − 0.0928069i \(-0.970416\pi\)
0.995684 − 0.0928069i \(-0.0295839\pi\)
\(20\) 0 0
\(21\) −1.67936 + 0.423958i −0.366467 + 0.0925152i
\(22\) 0 0
\(23\) −8.63977 −1.80152 −0.900758 − 0.434322i \(-0.856988\pi\)
−0.900758 + 0.434322i \(0.856988\pi\)
\(24\) 0 0
\(25\) −1.00000 −0.200000
\(26\) 0 0
\(27\) 3.51081 + 3.83069i 0.675656 + 0.737217i
\(28\) 0 0
\(29\) − 1.07440i − 0.199511i −0.995012 − 0.0997554i \(-0.968194\pi\)
0.995012 − 0.0997554i \(-0.0318060\pi\)
\(30\) 0 0
\(31\) − 0.886759i − 0.159267i −0.996824 − 0.0796333i \(-0.974625\pi\)
0.996824 − 0.0796333i \(-0.0253749\pi\)
\(32\) 0 0
\(33\) 1.89592 + 7.51006i 0.330038 + 1.30733i
\(34\) 0 0
\(35\) 1.00000 0.169031
\(36\) 0 0
\(37\) −6.41328 −1.05434 −0.527169 − 0.849761i \(-0.676746\pi\)
−0.527169 + 0.849761i \(0.676746\pi\)
\(38\) 0 0
\(39\) −2.11249 − 8.36789i −0.338269 − 1.33994i
\(40\) 0 0
\(41\) 6.63977i 1.03696i 0.855091 + 0.518479i \(0.173501\pi\)
−0.855091 + 0.518479i \(0.826499\pi\)
\(42\) 0 0
\(43\) 8.63977i 1.31755i 0.752339 + 0.658776i \(0.228925\pi\)
−0.752339 + 0.658776i \(0.771075\pi\)
\(44\) 0 0
\(45\) −1.42396 − 2.64052i −0.212271 − 0.393625i
\(46\) 0 0
\(47\) −0.130464 −0.0190301 −0.00951504 − 0.999955i \(-0.503029\pi\)
−0.00951504 + 0.999955i \(0.503029\pi\)
\(48\) 0 0
\(49\) −1.00000 −0.142857
\(50\) 0 0
\(51\) 2.28179 − 0.576042i 0.319515 − 0.0806621i
\(52\) 0 0
\(53\) 12.5481i 1.72362i 0.507231 + 0.861810i \(0.330669\pi\)
−0.507231 + 0.861810i \(0.669331\pi\)
\(54\) 0 0
\(55\) − 4.47197i − 0.603000i
\(56\) 0 0
\(57\) −1.35873 + 0.343012i −0.179968 + 0.0454331i
\(58\) 0 0
\(59\) −5.02162 −0.653759 −0.326880 − 0.945066i \(-0.605997\pi\)
−0.326880 + 0.945066i \(0.605997\pi\)
\(60\) 0 0
\(61\) −10.3356 −1.32334 −0.661669 − 0.749796i \(-0.730151\pi\)
−0.661669 + 0.749796i \(0.730151\pi\)
\(62\) 0 0
\(63\) 1.42396 + 2.64052i 0.179402 + 0.332674i
\(64\) 0 0
\(65\) 4.98278i 0.618038i
\(66\) 0 0
\(67\) 2.97838i 0.363867i 0.983311 + 0.181933i \(0.0582355\pi\)
−0.983311 + 0.181933i \(0.941764\pi\)
\(68\) 0 0
\(69\) 3.66290 + 14.5093i 0.440961 + 1.74671i
\(70\) 0 0
\(71\) 4.85231 0.575864 0.287932 − 0.957651i \(-0.407032\pi\)
0.287932 + 0.957651i \(0.407032\pi\)
\(72\) 0 0
\(73\) 1.19093 0.139388 0.0696938 − 0.997568i \(-0.477798\pi\)
0.0696938 + 0.997568i \(0.477798\pi\)
\(74\) 0 0
\(75\) 0.423958 + 1.67936i 0.0489544 + 0.193916i
\(76\) 0 0
\(77\) 4.47197i 0.509628i
\(78\) 0 0
\(79\) − 2.77023i − 0.311675i −0.987783 − 0.155838i \(-0.950192\pi\)
0.987783 − 0.155838i \(-0.0498076\pi\)
\(80\) 0 0
\(81\) 4.94469 − 7.51998i 0.549410 − 0.835553i
\(82\) 0 0
\(83\) 7.58370 0.832419 0.416210 − 0.909269i \(-0.363358\pi\)
0.416210 + 0.909269i \(0.363358\pi\)
\(84\) 0 0
\(85\) −1.35873 −0.147375
\(86\) 0 0
\(87\) −1.80430 + 0.455499i −0.193442 + 0.0488347i
\(88\) 0 0
\(89\) − 1.32579i − 0.140533i −0.997528 − 0.0702667i \(-0.977615\pi\)
0.997528 − 0.0702667i \(-0.0223850\pi\)
\(90\) 0 0
\(91\) − 4.98278i − 0.522337i
\(92\) 0 0
\(93\) −1.48919 + 0.375948i −0.154422 + 0.0389840i
\(94\) 0 0
\(95\) 0.809072 0.0830091
\(96\) 0 0
\(97\) −14.6786 −1.49039 −0.745193 − 0.666848i \(-0.767643\pi\)
−0.745193 + 0.666848i \(0.767643\pi\)
\(98\) 0 0
\(99\) 11.8083 − 6.36789i 1.18678 − 0.639997i
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 1680.2.ba.a.911.3 ✓ 8
3.2 odd 2 1680.2.ba.b.911.5 yes 8
4.3 odd 2 1680.2.ba.b.911.6 yes 8
12.11 even 2 inner 1680.2.ba.a.911.4 yes 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1680.2.ba.a.911.3 ✓ 8 1.1 even 1 trivial
1680.2.ba.a.911.4 yes 8 12.11 even 2 inner
1680.2.ba.b.911.5 yes 8 3.2 odd 2
1680.2.ba.b.911.6 yes 8 4.3 odd 2