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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [864,2,Mod(109,864)] 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("864.109"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(864, base_ring=CyclotomicField(8)) chi = DirichletCharacter(H, H._module([0, 7, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 864 = 2^{5} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 864.v (of order \(8\), degree \(4\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [128,0,0,0,0,0,0,0,0,16] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(10)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(6.89907473464\)
Analytic rank: \(0\)
Dimension: \(128\)
Relative dimension: \(32\) over \(\Q(\zeta_{8})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{8}]$

Embedding invariants

Embedding label 109.16
Character \(\chi\) \(=\) 864.109
Dual form 864.2.v.b.325.16

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.239074 + 1.39386i) q^{2} +(-1.88569 - 0.666470i) q^{4} +(0.369546 - 0.153071i) q^{5} +(-1.24156 + 1.24156i) q^{7} +(1.37978 - 2.46905i) q^{8} +(0.125011 + 0.551690i) q^{10} +(-0.490552 - 1.18430i) q^{11} +(4.60261 + 1.90646i) q^{13} +(-1.43373 - 2.02738i) q^{14} +(3.11164 + 2.51351i) q^{16} +4.73651i q^{17} +(2.25236 + 0.932959i) q^{19} +(-0.798865 + 0.0423528i) q^{20} +(1.76802 - 0.400626i) q^{22} +(-4.41161 - 4.41161i) q^{23} +(-3.42240 + 3.42240i) q^{25} +(-3.75771 + 5.95961i) q^{26} +(3.16865 - 1.51373i) q^{28} +(-3.72693 + 8.99760i) q^{29} +5.82890 q^{31} +(-4.24739 + 3.73627i) q^{32} +(-6.60203 - 1.13238i) q^{34} +(-0.268766 + 0.648859i) q^{35} +(-1.91286 + 0.792332i) q^{37} +(-1.83889 + 2.91643i) q^{38} +(0.131954 - 1.12363i) q^{40} +(-3.14074 - 3.14074i) q^{41} +(1.99088 + 4.80640i) q^{43} +(0.135729 + 2.56015i) q^{44} +(7.20386 - 5.09446i) q^{46} +3.99894i q^{47} +3.91707i q^{49} +(-3.95214 - 5.58855i) q^{50} +(-7.40849 - 6.66250i) q^{52} +(0.855802 + 2.06609i) q^{53} +(-0.362563 - 0.362563i) q^{55} +(1.35239 + 4.77854i) q^{56} +(-11.6504 - 7.34590i) q^{58} +(-1.65001 + 0.683456i) q^{59} +(0.408897 - 0.987165i) q^{61} +(-1.39354 + 8.12467i) q^{62} +(-4.19240 - 6.81350i) q^{64} +1.99270 q^{65} +(-4.01800 + 9.70032i) q^{67} +(3.15674 - 8.93159i) q^{68} +(-0.840163 - 0.529747i) q^{70} +(2.17183 - 2.17183i) q^{71} +(7.91821 + 7.91821i) q^{73} +(-0.647085 - 2.85568i) q^{74} +(-3.62546 - 3.26040i) q^{76} +(2.07942 + 0.861324i) q^{77} -0.868785i q^{79} +(1.53464 + 0.452555i) q^{80} +(5.12862 - 3.62688i) q^{82} +(-5.47101 - 2.26617i) q^{83} +(0.725022 + 1.75036i) q^{85} +(-7.17541 + 1.62592i) q^{86} +(-3.60094 - 0.422876i) q^{88} +(-4.82488 + 4.82488i) q^{89} +(-8.08139 + 3.34742i) q^{91} +(5.37871 + 11.2591i) q^{92} +(-5.57396 - 0.956040i) q^{94} +0.975160 q^{95} +6.97696 q^{97} +(-5.45985 - 0.936468i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 128 q + 16 q^{10} - 32 q^{16} - 16 q^{22} - 32 q^{40} - 32 q^{46} - 80 q^{52} + 32 q^{55} - 32 q^{58} + 64 q^{61} + 48 q^{64} + 64 q^{67} - 96 q^{70} + 32 q^{76} - 80 q^{82} - 80 q^{88} + 96 q^{91} - 48 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/864\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(353\) \(703\)
\(\chi(n)\) \(e\left(\frac{7}{8}\right)\) \(1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.239074 + 1.39386i −0.169051 + 0.985607i
\(3\) 0 0
\(4\) −1.88569 − 0.666470i −0.942844 − 0.333235i
\(5\) 0.369546 − 0.153071i 0.165266 − 0.0684554i −0.298517 − 0.954404i \(-0.596492\pi\)
0.463783 + 0.885949i \(0.346492\pi\)
\(6\) 0 0
\(7\) −1.24156 + 1.24156i −0.469265 + 0.469265i −0.901676 − 0.432412i \(-0.857663\pi\)
0.432412 + 0.901676i \(0.357663\pi\)
\(8\) 1.37978 − 2.46905i 0.487827 − 0.872940i
\(9\) 0 0
\(10\) 0.125011 + 0.551690i 0.0395318 + 0.174460i
\(11\) −0.490552 − 1.18430i −0.147907 − 0.357079i 0.832511 − 0.554009i \(-0.186903\pi\)
−0.980417 + 0.196930i \(0.936903\pi\)
\(12\) 0 0
\(13\) 4.60261 + 1.90646i 1.27654 + 0.528758i 0.914945 − 0.403579i \(-0.132234\pi\)
0.361590 + 0.932337i \(0.382234\pi\)
\(14\) −1.43373 − 2.02738i −0.383181 − 0.541840i
\(15\) 0 0
\(16\) 3.11164 + 2.51351i 0.777909 + 0.628377i
\(17\) 4.73651i 1.14877i 0.818584 + 0.574387i \(0.194759\pi\)
−0.818584 + 0.574387i \(0.805241\pi\)
\(18\) 0 0
\(19\) 2.25236 + 0.932959i 0.516727 + 0.214035i 0.625779 − 0.780001i \(-0.284781\pi\)
−0.109051 + 0.994036i \(0.534781\pi\)
\(20\) −0.798865 + 0.0423528i −0.178632 + 0.00947038i
\(21\) 0 0
\(22\) 1.76802 − 0.400626i 0.376943 − 0.0854137i
\(23\) −4.41161 − 4.41161i −0.919884 − 0.919884i 0.0771364 − 0.997021i \(-0.475422\pi\)
−0.997021 + 0.0771364i \(0.975422\pi\)
\(24\) 0 0
\(25\) −3.42240 + 3.42240i −0.684480 + 0.684480i
\(26\) −3.75771 + 5.95961i −0.736947 + 1.16878i
\(27\) 0 0
\(28\) 3.16865 − 1.51373i 0.598819 − 0.286068i
\(29\) −3.72693 + 8.99760i −0.692073 + 1.67081i 0.0484889 + 0.998824i \(0.484559\pi\)
−0.740562 + 0.671988i \(0.765441\pi\)
\(30\) 0 0
\(31\) 5.82890 1.04690 0.523451 − 0.852056i \(-0.324644\pi\)
0.523451 + 0.852056i \(0.324644\pi\)
\(32\) −4.24739 + 3.73627i −0.750839 + 0.660485i
\(33\) 0 0
\(34\) −6.60203 − 1.13238i −1.13224 − 0.194201i
\(35\) −0.268766 + 0.648859i −0.0454298 + 0.109677i
\(36\) 0 0
\(37\) −1.91286 + 0.792332i −0.314472 + 0.130259i −0.534336 − 0.845272i \(-0.679438\pi\)
0.219865 + 0.975530i \(0.429438\pi\)
\(38\) −1.83889 + 2.91643i −0.298308 + 0.473108i
\(39\) 0 0
\(40\) 0.131954 − 1.12363i 0.0208637 − 0.177662i
\(41\) −3.14074 − 3.14074i −0.490501 − 0.490501i 0.417963 − 0.908464i \(-0.362744\pi\)
−0.908464 + 0.417963i \(0.862744\pi\)
\(42\) 0 0
\(43\) 1.99088 + 4.80640i 0.303606 + 0.732969i 0.999885 + 0.0151975i \(0.00483770\pi\)
−0.696279 + 0.717771i \(0.745162\pi\)
\(44\) 0.135729 + 2.56015i 0.0204620 + 0.385957i
\(45\) 0 0
\(46\) 7.20386 − 5.09446i 1.06215 − 0.751138i
\(47\) 3.99894i 0.583305i 0.956524 + 0.291652i \(0.0942051\pi\)
−0.956524 + 0.291652i \(0.905795\pi\)
\(48\) 0 0
\(49\) 3.91707i 0.559581i
\(50\) −3.95214 − 5.58855i −0.558917 − 0.790340i
\(51\) 0 0
\(52\) −7.40849 − 6.66250i −1.02737 − 0.923922i
\(53\) 0.855802 + 2.06609i 0.117553 + 0.283799i 0.971694 − 0.236243i \(-0.0759162\pi\)
−0.854141 + 0.520042i \(0.825916\pi\)
\(54\) 0 0
\(55\) −0.362563 − 0.362563i −0.0488879 − 0.0488879i
\(56\) 1.35239 + 4.77854i 0.180720 + 0.638560i
\(57\) 0 0
\(58\) −11.6504 − 7.34590i −1.52977 − 0.964564i
\(59\) −1.65001 + 0.683456i −0.214813 + 0.0889783i −0.487495 − 0.873126i \(-0.662089\pi\)
0.272682 + 0.962104i \(0.412089\pi\)
\(60\) 0 0
\(61\) 0.408897 − 0.987165i 0.0523539 − 0.126394i −0.895539 − 0.444984i \(-0.853209\pi\)
0.947893 + 0.318590i \(0.103209\pi\)
\(62\) −1.39354 + 8.12467i −0.176979 + 1.03183i
\(63\) 0 0
\(64\) −4.19240 − 6.81350i −0.524050 − 0.851688i
\(65\) 1.99270 0.247164
\(66\) 0 0
\(67\) −4.01800 + 9.70032i −0.490877 + 1.18508i 0.463397 + 0.886151i \(0.346630\pi\)
−0.954274 + 0.298932i \(0.903370\pi\)
\(68\) 3.15674 − 8.93159i 0.382811 − 1.08311i
\(69\) 0 0
\(70\) −0.840163 − 0.529747i −0.100419 − 0.0633169i
\(71\) 2.17183 − 2.17183i 0.257748 − 0.257748i −0.566389 − 0.824138i \(-0.691660\pi\)
0.824138 + 0.566389i \(0.191660\pi\)
\(72\) 0 0
\(73\) 7.91821 + 7.91821i 0.926757 + 0.926757i 0.997495 − 0.0707380i \(-0.0225354\pi\)
−0.0707380 + 0.997495i \(0.522535\pi\)
\(74\) −0.647085 − 2.85568i −0.0752222 − 0.331966i
\(75\) 0 0
\(76\) −3.62546 − 3.26040i −0.415869 − 0.373994i
\(77\) 2.07942 + 0.861324i 0.236972 + 0.0981570i
\(78\) 0 0
\(79\) − 0.868785i − 0.0977460i −0.998805 − 0.0488730i \(-0.984437\pi\)
0.998805 − 0.0488730i \(-0.0155630\pi\)
\(80\) 1.53464 + 0.452555i 0.171578 + 0.0505972i
\(81\) 0 0
\(82\) 5.12862 − 3.62688i 0.566361 − 0.400522i
\(83\) −5.47101 − 2.26617i −0.600521 − 0.248744i 0.0616485 − 0.998098i \(-0.480364\pi\)
−0.662170 + 0.749354i \(0.730364\pi\)
\(84\) 0 0
\(85\) 0.725022 + 1.75036i 0.0786397 + 0.189853i
\(86\) −7.17541 + 1.62592i −0.773744 + 0.175327i
\(87\) 0 0
\(88\) −3.60094 − 0.422876i −0.383861 − 0.0450788i
\(89\) −4.82488 + 4.82488i −0.511437 + 0.511437i −0.914966 − 0.403530i \(-0.867783\pi\)
0.403530 + 0.914966i \(0.367783\pi\)
\(90\) 0 0
\(91\) −8.08139 + 3.34742i −0.847160 + 0.350905i
\(92\) 5.37871 + 11.2591i 0.560770 + 1.17384i
\(93\) 0 0
\(94\) −5.57396 − 0.956040i −0.574910 − 0.0986080i
\(95\) 0.975160 0.100049
\(96\) 0 0
\(97\) 6.97696 0.708403 0.354201 − 0.935169i \(-0.384753\pi\)
0.354201 + 0.935169i \(0.384753\pi\)
\(98\) −5.45985 − 0.936468i −0.551528 − 0.0945975i
\(99\) 0 0
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 864.2.v.b.109.16 ✓ 128
3.2 odd 2 inner 864.2.v.b.109.17 yes 128
32.5 even 8 inner 864.2.v.b.325.16 yes 128
96.5 odd 8 inner 864.2.v.b.325.17 yes 128
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
864.2.v.b.109.16 ✓ 128 1.1 even 1 trivial
864.2.v.b.109.17 yes 128 3.2 odd 2 inner
864.2.v.b.325.16 yes 128 32.5 even 8 inner
864.2.v.b.325.17 yes 128 96.5 odd 8 inner