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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 109.10
Character \(\chi\) \(=\) 864.109
Dual form 864.2.v.b.325.10

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.877362 + 1.10916i) q^{2} +(-0.460473 - 1.94627i) q^{4} +(3.16129 - 1.30945i) q^{5} +(3.16733 - 3.16733i) q^{7} +(2.56273 + 1.19684i) q^{8} +(-1.32120 + 4.65523i) q^{10} +(1.03307 + 2.49405i) q^{11} +(2.98219 + 1.23526i) q^{13} +(0.734183 + 6.29197i) q^{14} +(-3.57593 + 1.79241i) q^{16} -2.14766i q^{17} +(-6.92299 - 2.86759i) q^{19} +(-4.00422 - 5.54975i) q^{20} +(-3.67267 - 1.04234i) q^{22} +(5.16599 + 5.16599i) q^{23} +(4.74354 - 4.74354i) q^{25} +(-3.98656 + 2.22395i) q^{26} +(-7.62295 - 4.70601i) q^{28} +(0.875489 - 2.11362i) q^{29} -4.47936 q^{31} +(1.14931 - 5.53887i) q^{32} +(2.38210 + 1.88427i) q^{34} +(5.86539 - 14.1603i) q^{35} +(-2.25327 + 0.933333i) q^{37} +(9.25458 - 5.16278i) q^{38} +(9.66871 + 0.427809i) q^{40} +(-1.06719 - 1.06719i) q^{41} +(-1.57823 - 3.81019i) q^{43} +(4.37839 - 3.15907i) q^{44} +(-10.2623 + 1.19747i) q^{46} +5.62964i q^{47} -13.0640i q^{49} +(1.09955 + 9.42315i) q^{50} +(1.03094 - 6.37295i) q^{52} +(1.31226 + 3.16807i) q^{53} +(6.53164 + 6.53164i) q^{55} +(11.9078 - 4.32620i) q^{56} +(1.57622 + 2.82546i) q^{58} +(-10.5601 + 4.37413i) q^{59} +(-1.34023 + 3.23561i) q^{61} +(3.93002 - 4.96833i) q^{62} +(5.13513 + 6.13436i) q^{64} +11.0451 q^{65} +(-2.09986 + 5.06951i) q^{67} +(-4.17992 + 0.988940i) q^{68} +(10.5600 + 18.9294i) q^{70} +(-4.98493 + 4.98493i) q^{71} +(8.05547 + 8.05547i) q^{73} +(0.941713 - 3.31810i) q^{74} +(-2.39326 + 14.7944i) q^{76} +(11.1715 + 4.62740i) q^{77} -14.7761i q^{79} +(-8.95747 + 10.3488i) q^{80} +(2.11999 - 0.247373i) q^{82} +(-4.09240 - 1.69513i) q^{83} +(-2.81225 - 6.78937i) q^{85} +(5.61078 + 1.59240i) q^{86} +(-0.337513 + 7.62798i) q^{88} +(-10.3239 + 10.3239i) q^{89} +(13.3581 - 5.53310i) q^{91} +(7.67560 - 12.4332i) q^{92} +(-6.24417 - 4.93923i) q^{94} -25.6405 q^{95} +14.7875 q^{97} +(14.4901 + 11.4618i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 128 q + 16 q^{10} - 32 q^{16} - 16 q^{22} - 32 q^{40} - 32 q^{46} - 80 q^{52} + 32 q^{55} - 32 q^{58} + 64 q^{61} + 48 q^{64} + 64 q^{67} - 96 q^{70} + 32 q^{76} - 80 q^{82} - 80 q^{88} + 96 q^{91} - 48 q^{94}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.877362 + 1.10916i −0.620388 + 0.784295i
\(3\) 0 0
\(4\) −0.460473 − 1.94627i −0.230237 − 0.973135i
\(5\) 3.16129 − 1.30945i 1.41377 − 0.585603i 0.460483 − 0.887669i \(-0.347676\pi\)
0.953287 + 0.302066i \(0.0976763\pi\)
\(6\) 0 0
\(7\) 3.16733 − 3.16733i 1.19714 − 1.19714i 0.222120 − 0.975019i \(-0.428702\pi\)
0.975019 − 0.222120i \(-0.0712975\pi\)
\(8\) 2.56273 + 1.19684i 0.906061 + 0.423148i
\(9\) 0 0
\(10\) −1.32120 + 4.65523i −0.417801 + 1.47211i
\(11\) 1.03307 + 2.49405i 0.311482 + 0.751983i 0.999651 + 0.0264327i \(0.00841477\pi\)
−0.688169 + 0.725550i \(0.741585\pi\)
\(12\) 0 0
\(13\) 2.98219 + 1.23526i 0.827111 + 0.342600i 0.755758 − 0.654851i \(-0.227268\pi\)
0.0713524 + 0.997451i \(0.477268\pi\)
\(14\) 0.734183 + 6.29197i 0.196219 + 1.68160i
\(15\) 0 0
\(16\) −3.57593 + 1.79241i −0.893982 + 0.448102i
\(17\) − 2.14766i − 0.520884i −0.965489 − 0.260442i \(-0.916132\pi\)
0.965489 − 0.260442i \(-0.0838683\pi\)
\(18\) 0 0
\(19\) −6.92299 − 2.86759i −1.58824 − 0.657871i −0.598550 − 0.801085i \(-0.704256\pi\)
−0.989692 + 0.143214i \(0.954256\pi\)
\(20\) −4.00422 − 5.54975i −0.895372 − 1.24096i
\(21\) 0 0
\(22\) −3.67267 − 1.04234i −0.783016 − 0.222228i
\(23\) 5.16599 + 5.16599i 1.07718 + 1.07718i 0.996761 + 0.0804218i \(0.0256267\pi\)
0.0804218 + 0.996761i \(0.474373\pi\)
\(24\) 0 0
\(25\) 4.74354 − 4.74354i 0.948708 − 0.948708i
\(26\) −3.98656 + 2.22395i −0.781830 + 0.436153i
\(27\) 0 0
\(28\) −7.62295 − 4.70601i −1.44060 − 0.889352i
\(29\) 0.875489 − 2.11362i 0.162574 − 0.392489i −0.821509 − 0.570195i \(-0.806868\pi\)
0.984084 + 0.177706i \(0.0568676\pi\)
\(30\) 0 0
\(31\) −4.47936 −0.804517 −0.402259 − 0.915526i \(-0.631775\pi\)
−0.402259 + 0.915526i \(0.631775\pi\)
\(32\) 1.14931 − 5.53887i 0.203172 − 0.979143i
\(33\) 0 0
\(34\) 2.38210 + 1.88427i 0.408527 + 0.323150i
\(35\) 5.86539 − 14.1603i 0.991431 − 2.39353i
\(36\) 0 0
\(37\) −2.25327 + 0.933333i −0.370435 + 0.153439i −0.560132 − 0.828404i \(-0.689249\pi\)
0.189697 + 0.981843i \(0.439249\pi\)
\(38\) 9.25458 − 5.16278i 1.50129 − 0.837514i
\(39\) 0 0
\(40\) 9.66871 + 0.427809i 1.52876 + 0.0676426i
\(41\) −1.06719 − 1.06719i −0.166667 − 0.166667i 0.618846 − 0.785513i \(-0.287601\pi\)
−0.785513 + 0.618846i \(0.787601\pi\)
\(42\) 0 0
\(43\) −1.57823 − 3.81019i −0.240678 − 0.581048i 0.756673 − 0.653794i \(-0.226824\pi\)
−0.997350 + 0.0727462i \(0.976824\pi\)
\(44\) 4.37839 − 3.15907i 0.660066 − 0.476248i
\(45\) 0 0
\(46\) −10.2623 + 1.19747i −1.51310 + 0.176557i
\(47\) 5.62964i 0.821167i 0.911823 + 0.410584i \(0.134675\pi\)
−0.911823 + 0.410584i \(0.865325\pi\)
\(48\) 0 0
\(49\) − 13.0640i − 1.86628i
\(50\) 1.09955 + 9.42315i 0.155499 + 1.33263i
\(51\) 0 0
\(52\) 1.03094 − 6.37295i 0.142965 − 0.883769i
\(53\) 1.31226 + 3.16807i 0.180252 + 0.435168i 0.988019 − 0.154335i \(-0.0493236\pi\)
−0.807766 + 0.589503i \(0.799324\pi\)
\(54\) 0 0
\(55\) 6.53164 + 6.53164i 0.880727 + 0.880727i
\(56\) 11.9078 − 4.32620i 1.59125 − 0.578113i
\(57\) 0 0
\(58\) 1.57622 + 2.82546i 0.206968 + 0.371002i
\(59\) −10.5601 + 4.37413i −1.37481 + 0.569464i −0.943088 − 0.332544i \(-0.892093\pi\)
−0.431719 + 0.902008i \(0.642093\pi\)
\(60\) 0 0
\(61\) −1.34023 + 3.23561i −0.171600 + 0.414278i −0.986159 − 0.165802i \(-0.946979\pi\)
0.814560 + 0.580080i \(0.196979\pi\)
\(62\) 3.93002 − 4.96833i 0.499113 − 0.630979i
\(63\) 0 0
\(64\) 5.13513 + 6.13436i 0.641891 + 0.766796i
\(65\) 11.0451 1.36997
\(66\) 0 0
\(67\) −2.09986 + 5.06951i −0.256539 + 0.619340i −0.998705 − 0.0508765i \(-0.983799\pi\)
0.742166 + 0.670216i \(0.233799\pi\)
\(68\) −4.17992 + 0.988940i −0.506890 + 0.119927i
\(69\) 0 0
\(70\) 10.5600 + 18.9294i 1.26216 + 2.26249i
\(71\) −4.98493 + 4.98493i −0.591603 + 0.591603i −0.938064 − 0.346462i \(-0.887383\pi\)
0.346462 + 0.938064i \(0.387383\pi\)
\(72\) 0 0
\(73\) 8.05547 + 8.05547i 0.942821 + 0.942821i 0.998451 − 0.0556300i \(-0.0177167\pi\)
−0.0556300 + 0.998451i \(0.517717\pi\)
\(74\) 0.941713 − 3.31810i 0.109472 − 0.385722i
\(75\) 0 0
\(76\) −2.39326 + 14.7944i −0.274526 + 1.69704i
\(77\) 11.1715 + 4.62740i 1.27312 + 0.527342i
\(78\) 0 0
\(79\) − 14.7761i − 1.66244i −0.555945 − 0.831219i \(-0.687643\pi\)
0.555945 − 0.831219i \(-0.312357\pi\)
\(80\) −8.95747 + 10.3488i −1.00148 + 1.15703i
\(81\) 0 0
\(82\) 2.11999 − 0.247373i 0.234114 − 0.0273177i
\(83\) −4.09240 − 1.69513i −0.449200 − 0.186065i 0.146603 − 0.989195i \(-0.453166\pi\)
−0.595803 + 0.803131i \(0.703166\pi\)
\(84\) 0 0
\(85\) −2.81225 − 6.78937i −0.305031 − 0.736410i
\(86\) 5.61078 + 1.59240i 0.605027 + 0.171713i
\(87\) 0 0
\(88\) −0.337513 + 7.62798i −0.0359790 + 0.813145i
\(89\) −10.3239 + 10.3239i −1.09433 + 1.09433i −0.0992728 + 0.995060i \(0.531652\pi\)
−0.995060 + 0.0992728i \(0.968348\pi\)
\(90\) 0 0
\(91\) 13.3581 − 5.53310i 1.40031 − 0.580026i
\(92\) 7.67560 − 12.4332i 0.800237 − 1.29625i
\(93\) 0 0
\(94\) −6.24417 − 4.93923i −0.644037 − 0.509442i
\(95\) −25.6405 −2.63066
\(96\) 0 0
\(97\) 14.7875 1.50145 0.750724 − 0.660616i \(-0.229705\pi\)
0.750724 + 0.660616i \(0.229705\pi\)
\(98\) 14.4901 + 11.4618i 1.46372 + 1.15782i
\(99\) 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.109.10 ✓ 128
3.2 odd 2 inner 864.2.v.b.109.23 yes 128
32.5 even 8 inner 864.2.v.b.325.10 yes 128
96.5 odd 8 inner 864.2.v.b.325.23 yes 128
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
864.2.v.b.109.10 ✓ 128 1.1 even 1 trivial
864.2.v.b.109.23 yes 128 3.2 odd 2 inner
864.2.v.b.325.10 yes 128 32.5 even 8 inner
864.2.v.b.325.23 yes 128 96.5 odd 8 inner