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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 109.17
Character \(\chi\) \(=\) 864.109
Dual form 864.2.v.b.325.17

$q$-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 −0.463783 − 0.885949i \(-0.653508\pi\)
0.298517 + 0.954404i \(0.403508\pi\)
\(6\) 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 0.980417 − 0.196930i \(-0.0630973\pi\)
−0.832511 + 0.554009i \(0.813097\pi\)
\(12\) 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 \(-0.805241\pi\)
0.818584 − 0.574387i \(-0.194759\pi\)
\(18\) 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 0.997021 − 0.0771364i \(-0.0245777\pi\)
−0.0771364 + 0.997021i \(0.524578\pi\)
\(24\) 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 \(-0.515441\pi\)
0.740562 − 0.671988i \(-0.234559\pi\)
\(30\) 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 0.908464 − 0.417963i \(-0.137256\pi\)
−0.417963 + 0.908464i \(0.637256\pi\)
\(42\) 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 \(-0.905795\pi\)
0.956524 − 0.291652i \(-0.0942051\pi\)
\(48\) 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 0.854141 − 0.520042i \(-0.174084\pi\)
−0.971694 + 0.236243i \(0.924084\pi\)
\(54\) 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 −0.272682 − 0.962104i \(-0.587911\pi\)
0.487495 + 0.873126i \(0.337911\pi\)
\(60\) 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 −0.824138 − 0.566389i \(-0.808340\pi\)
0.566389 + 0.824138i \(0.308340\pi\)
\(72\) 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 0.662170 − 0.749354i \(-0.269636\pi\)
−0.0616485 + 0.998098i \(0.519636\pi\)
\(84\) 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 −0.403530 − 0.914966i \(-0.632217\pi\)
0.914966 + 0.403530i \(0.132217\pi\)
\(90\) 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.109.17 yes 128
3.2 odd 2 inner 864.2.v.b.109.16 ✓ 128
32.5 even 8 inner 864.2.v.b.325.17 yes 128
96.5 odd 8 inner 864.2.v.b.325.16 yes 128
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
864.2.v.b.109.16 ✓ 128 3.2 odd 2 inner
864.2.v.b.109.17 yes 128 1.1 even 1 trivial
864.2.v.b.325.16 yes 128 96.5 odd 8 inner
864.2.v.b.325.17 yes 128 32.5 even 8 inner