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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 109.20
Character \(\chi\) \(=\) 864.109
Dual form 864.2.v.b.325.20

$q$-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)
 
\(f(q)\) \(=\) \(q+(0.463875 + 1.33597i) q^{2} +(-1.56964 + 1.23945i) q^{4} +(0.137547 - 0.0569738i) q^{5} +(0.385875 - 0.385875i) q^{7} +(-2.38398 - 1.52205i) q^{8} +(0.139920 + 0.157330i) q^{10} +(-1.66259 - 4.01386i) q^{11} +(-5.30371 - 2.19687i) q^{13} +(0.694516 + 0.336521i) q^{14} +(0.927543 - 3.89097i) q^{16} -4.41571i q^{17} +(-7.14906 - 2.96124i) q^{19} +(-0.145283 + 0.259911i) q^{20} +(4.59117 - 4.08311i) q^{22} +(5.04601 + 5.04601i) q^{23} +(-3.51986 + 3.51986i) q^{25} +(0.474698 - 8.10468i) q^{26} +(-0.127414 + 1.08396i) q^{28} +(-1.66821 + 4.02742i) q^{29} +0.274792 q^{31} +(5.62849 - 0.565751i) q^{32} +(5.89927 - 2.04834i) q^{34} +(0.0310912 - 0.0750608i) q^{35} +(-1.49115 + 0.617653i) q^{37} +(0.639862 - 10.9246i) q^{38} +(-0.414626 - 0.0735285i) q^{40} +(-0.482993 - 0.482993i) q^{41} +(-0.554390 - 1.33842i) q^{43} +(7.58464 + 4.23962i) q^{44} +(-4.40061 + 9.08204i) q^{46} -12.2231i q^{47} +6.70220i q^{49} +(-6.33521 - 3.06966i) q^{50} +(11.0478 - 3.12537i) q^{52} +(-2.13483 - 5.15392i) q^{53} +(-0.457370 - 0.457370i) q^{55} +(-1.50724 + 0.332599i) q^{56} +(-6.15436 - 0.360466i) q^{58} +(8.90946 - 3.69042i) q^{59} +(4.31381 - 10.4145i) q^{61} +(0.127469 + 0.367114i) q^{62} +(3.36674 + 7.25707i) q^{64} -0.854674 q^{65} +(-4.38272 + 10.5808i) q^{67} +(5.47304 + 6.93108i) q^{68} +(0.114702 + 0.00671816i) q^{70} +(6.62943 - 6.62943i) q^{71} +(-4.79216 - 4.79216i) q^{73} +(-1.51687 - 1.70562i) q^{74} +(14.8918 - 4.21280i) q^{76} +(-2.19040 - 0.907295i) q^{77} +3.60122i q^{79} +(-0.0941028 - 0.588037i) q^{80} +(0.421217 - 0.869313i) q^{82} +(9.62080 + 3.98506i) q^{83} +(-0.251580 - 0.607368i) q^{85} +(1.53092 - 1.36151i) q^{86} +(-2.14569 + 12.0995i) q^{88} +(-10.0686 + 10.0686i) q^{89} +(-2.89429 + 1.19885i) q^{91} +(-14.1747 - 1.66616i) q^{92} +(16.3297 - 5.66998i) q^{94} -1.15205 q^{95} -6.06867 q^{97} +(-8.95395 + 3.10898i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 128 q + 16 q^{10} - 32 q^{16} - 16 q^{22} - 32 q^{40} - 32 q^{46} - 80 q^{52} + 32 q^{55} - 32 q^{58} + 64 q^{61} + 48 q^{64} + 64 q^{67} - 96 q^{70} + 32 q^{76} - 80 q^{82} - 80 q^{88} + 96 q^{91} - 48 q^{94}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.463875 + 1.33597i 0.328009 + 0.944675i
\(3\) 0 0
\(4\) −1.56964 + 1.23945i −0.784820 + 0.619723i
\(5\) 0.137547 − 0.0569738i 0.0615129 − 0.0254795i −0.351715 − 0.936107i \(-0.614401\pi\)
0.413228 + 0.910628i \(0.364401\pi\)
\(6\) 0 0
\(7\) 0.385875 − 0.385875i 0.145847 − 0.145847i −0.630413 − 0.776260i \(-0.717114\pi\)
0.776260 + 0.630413i \(0.217114\pi\)
\(8\) −2.38398 − 1.52205i −0.842865 − 0.538125i
\(9\) 0 0
\(10\) 0.139920 + 0.157330i 0.0442466 + 0.0497522i
\(11\) −1.66259 − 4.01386i −0.501291 − 1.21022i −0.948781 − 0.315935i \(-0.897682\pi\)
0.447490 − 0.894289i \(-0.352318\pi\)
\(12\) 0 0
\(13\) −5.30371 − 2.19687i −1.47099 − 0.609302i −0.503902 − 0.863761i \(-0.668103\pi\)
−0.967084 + 0.254459i \(0.918103\pi\)
\(14\) 0.694516 + 0.336521i 0.185617 + 0.0899390i
\(15\) 0 0
\(16\) 0.927543 − 3.89097i 0.231886 − 0.972743i
\(17\) − 4.41571i − 1.07097i −0.844545 − 0.535484i \(-0.820129\pi\)
0.844545 − 0.535484i \(-0.179871\pi\)
\(18\) 0 0
\(19\) −7.14906 − 2.96124i −1.64011 − 0.679355i −0.643798 − 0.765195i \(-0.722642\pi\)
−0.996309 + 0.0858404i \(0.972642\pi\)
\(20\) −0.145283 + 0.259911i −0.0324863 + 0.0581178i
\(21\) 0 0
\(22\) 4.59117 − 4.08311i 0.978840 − 0.870521i
\(23\) 5.04601 + 5.04601i 1.05217 + 1.05217i 0.998562 + 0.0536038i \(0.0170708\pi\)
0.0536038 + 0.998562i \(0.482929\pi\)
\(24\) 0 0
\(25\) −3.51986 + 3.51986i −0.703972 + 0.703972i
\(26\) 0.474698 − 8.10468i 0.0930960 − 1.58946i
\(27\) 0 0
\(28\) −0.127414 + 1.08396i −0.0240789 + 0.204849i
\(29\) −1.66821 + 4.02742i −0.309779 + 0.747873i 0.689933 + 0.723874i \(0.257640\pi\)
−0.999712 + 0.0239999i \(0.992360\pi\)
\(30\) 0 0
\(31\) 0.274792 0.0493541 0.0246770 − 0.999695i \(-0.492144\pi\)
0.0246770 + 0.999695i \(0.492144\pi\)
\(32\) 5.62849 − 0.565751i 0.994986 − 0.100012i
\(33\) 0 0
\(34\) 5.89927 − 2.04834i 1.01172 − 0.351287i
\(35\) 0.0310912 − 0.0750608i 0.00525537 − 0.0126876i
\(36\) 0 0
\(37\) −1.49115 + 0.617653i −0.245143 + 0.101542i −0.501872 − 0.864942i \(-0.667355\pi\)
0.256729 + 0.966483i \(0.417355\pi\)
\(38\) 0.639862 − 10.9246i 0.103799 − 1.77220i
\(39\) 0 0
\(40\) −0.414626 − 0.0735285i −0.0655582 − 0.0116259i
\(41\) −0.482993 − 0.482993i −0.0754308 − 0.0754308i 0.668385 − 0.743816i \(-0.266986\pi\)
−0.743816 + 0.668385i \(0.766986\pi\)
\(42\) 0 0
\(43\) −0.554390 − 1.33842i −0.0845437 − 0.204107i 0.875954 − 0.482395i \(-0.160233\pi\)
−0.960498 + 0.278288i \(0.910233\pi\)
\(44\) 7.58464 + 4.23962i 1.14343 + 0.639147i
\(45\) 0 0
\(46\) −4.40061 + 9.08204i −0.648835 + 1.33907i
\(47\) − 12.2231i − 1.78292i −0.453100 − 0.891460i \(-0.649682\pi\)
0.453100 − 0.891460i \(-0.350318\pi\)
\(48\) 0 0
\(49\) 6.70220i 0.957457i
\(50\) −6.33521 − 3.06966i −0.895934 − 0.434116i
\(51\) 0 0
\(52\) 11.0478 − 3.12537i 1.53206 − 0.433411i
\(53\) −2.13483 − 5.15392i −0.293241 − 0.707946i −1.00000 0.000491566i \(-0.999844\pi\)
0.706759 − 0.707454i \(-0.250156\pi\)
\(54\) 0 0
\(55\) −0.457370 − 0.457370i −0.0616717 − 0.0616717i
\(56\) −1.50724 + 0.332599i −0.201414 + 0.0444455i
\(57\) 0 0
\(58\) −6.15436 − 0.360466i −0.808107 − 0.0473315i
\(59\) 8.90946 − 3.69042i 1.15991 − 0.480451i 0.282067 − 0.959395i \(-0.408980\pi\)
0.877846 + 0.478943i \(0.158980\pi\)
\(60\) 0 0
\(61\) 4.31381 − 10.4145i 0.552327 − 1.33344i −0.363399 − 0.931634i \(-0.618384\pi\)
0.915726 − 0.401803i \(-0.131616\pi\)
\(62\) 0.127469 + 0.367114i 0.0161886 + 0.0466235i
\(63\) 0 0
\(64\) 3.36674 + 7.25707i 0.420843 + 0.907134i
\(65\) −0.854674 −0.106009
\(66\) 0 0
\(67\) −4.38272 + 10.5808i −0.535434 + 1.29265i 0.392447 + 0.919775i \(0.371629\pi\)
−0.927881 + 0.372877i \(0.878371\pi\)
\(68\) 5.47304 + 6.93108i 0.663704 + 0.840517i
\(69\) 0 0
\(70\) 0.114702 + 0.00671816i 0.0137095 + 0.000802974i
\(71\) 6.62943 − 6.62943i 0.786768 − 0.786768i −0.194195 − 0.980963i \(-0.562209\pi\)
0.980963 + 0.194195i \(0.0622094\pi\)
\(72\) 0 0
\(73\) −4.79216 − 4.79216i −0.560879 − 0.560879i 0.368678 − 0.929557i \(-0.379811\pi\)
−0.929557 + 0.368678i \(0.879811\pi\)
\(74\) −1.51687 − 1.70562i −0.176333 − 0.198274i
\(75\) 0 0
\(76\) 14.8918 − 4.21280i 1.70820 − 0.483241i
\(77\) −2.19040 − 0.907295i −0.249620 − 0.103396i
\(78\) 0 0
\(79\) 3.60122i 0.405169i 0.979265 + 0.202584i \(0.0649340\pi\)
−0.979265 + 0.202584i \(0.935066\pi\)
\(80\) −0.0941028 − 0.588037i −0.0105210 − 0.0657446i
\(81\) 0 0
\(82\) 0.421217 − 0.869313i 0.0465156 − 0.0959996i
\(83\) 9.62080 + 3.98506i 1.05602 + 0.437418i 0.842037 − 0.539420i \(-0.181356\pi\)
0.213983 + 0.976837i \(0.431356\pi\)
\(84\) 0 0
\(85\) −0.251580 − 0.607368i −0.0272877 − 0.0658783i
\(86\) 1.53092 − 1.36151i 0.165083 − 0.146815i
\(87\) 0 0
\(88\) −2.14569 + 12.0995i −0.228731 + 1.28981i
\(89\) −10.0686 + 10.0686i −1.06727 + 1.06727i −0.0697058 + 0.997568i \(0.522206\pi\)
−0.997568 + 0.0697058i \(0.977794\pi\)
\(90\) 0 0
\(91\) −2.89429 + 1.19885i −0.303404 + 0.125674i
\(92\) −14.1747 − 1.66616i −1.47781 − 0.173709i
\(93\) 0 0
\(94\) 16.3297 − 5.66998i 1.68428 − 0.584813i
\(95\) −1.15205 −0.118197
\(96\) 0 0
\(97\) −6.06867 −0.616180 −0.308090 − 0.951357i \(-0.599690\pi\)
−0.308090 + 0.951357i \(0.599690\pi\)
\(98\) −8.95395 + 3.10898i −0.904486 + 0.314054i
\(99\) 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.109.20 yes 128
3.2 odd 2 inner 864.2.v.b.109.13 ✓ 128
32.5 even 8 inner 864.2.v.b.325.20 yes 128
96.5 odd 8 inner 864.2.v.b.325.13 yes 128
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
864.2.v.b.109.13 ✓ 128 3.2 odd 2 inner
864.2.v.b.109.20 yes 128 1.1 even 1 trivial
864.2.v.b.325.13 yes 128 96.5 odd 8 inner
864.2.v.b.325.20 yes 128 32.5 even 8 inner