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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,-8] 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 325.6
Character \(\chi\) \(=\) 864.325
Dual form 864.2.v.a.109.6

$q$-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)
 
\(f(q)\) \(=\) \(q+(-1.18499 - 0.771885i) q^{2} +(0.808388 + 1.82935i) q^{4} +(-0.796479 - 0.329912i) q^{5} +(3.41414 + 3.41414i) q^{7} +(0.454114 - 2.79173i) q^{8} +(0.689163 + 1.00573i) q^{10} +(-0.811278 + 1.95860i) q^{11} +(-3.44869 + 1.42849i) q^{13} +(-1.41039 - 6.68103i) q^{14} +(-2.69302 + 2.95764i) q^{16} +5.12913i q^{17} +(1.84398 - 0.763801i) q^{19} +(-0.0403403 - 1.72373i) q^{20} +(2.47316 - 1.69470i) q^{22} +(-0.655049 + 0.655049i) q^{23} +(-3.01000 - 3.01000i) q^{25} +(5.18928 + 0.969242i) q^{26} +(-3.48569 + 9.00559i) q^{28} +(-3.09893 - 7.48148i) q^{29} -7.46197 q^{31} +(5.47415 - 1.42607i) q^{32} +(3.95910 - 6.07796i) q^{34} +(-1.59292 - 3.84565i) q^{35} +(1.04147 + 0.431390i) q^{37} +(-2.77466 - 0.518245i) q^{38} +(-1.28272 + 2.07374i) q^{40} +(-7.85358 + 7.85358i) q^{41} +(1.55013 - 3.74235i) q^{43} +(-4.23878 + 0.0991997i) q^{44} +(1.28185 - 0.270602i) q^{46} +8.53070i q^{47} +16.3127i q^{49} +(1.24344 + 5.89018i) q^{50} +(-5.40108 - 5.15406i) q^{52} +(2.21888 - 5.35684i) q^{53} +(1.29233 - 1.29233i) q^{55} +(11.0818 - 7.98095i) q^{56} +(-2.10265 + 11.2575i) q^{58} +(-3.55676 - 1.47326i) q^{59} +(5.53850 + 13.3711i) q^{61} +(8.84233 + 5.75978i) q^{62} +(-7.58756 - 2.53553i) q^{64} +3.21808 q^{65} +(1.49796 + 3.61639i) q^{67} +(-9.38296 + 4.14633i) q^{68} +(-1.08081 + 5.78660i) q^{70} +(3.65387 + 3.65387i) q^{71} +(-1.36736 + 1.36736i) q^{73} +(-0.901143 - 1.31508i) q^{74} +(2.88791 + 2.75583i) q^{76} +(-9.45673 + 3.91711i) q^{77} -9.69095i q^{79} +(3.12069 - 1.46724i) q^{80} +(15.3684 - 3.24433i) q^{82} +(-1.86485 + 0.772448i) q^{83} +(1.69216 - 4.08525i) q^{85} +(-4.72555 + 3.23811i) q^{86} +(5.09947 + 3.15430i) q^{88} +(9.11861 + 9.11861i) q^{89} +(-16.6514 - 6.89722i) q^{91} +(-1.72785 - 0.668777i) q^{92} +(6.58472 - 10.1088i) q^{94} -1.72068 q^{95} +15.7002 q^{97} +(12.5915 - 19.3303i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 128 q - 8 q^{10} - 32 q^{16} + 32 q^{22} + 64 q^{40} + 64 q^{46} + 88 q^{52} - 64 q^{55} + 64 q^{58} - 32 q^{61} - 96 q^{64} + 64 q^{67} + 48 q^{70} + 32 q^{76} + 40 q^{82} + 40 q^{88} - 48 q^{91} + 24 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.18499 − 0.771885i −0.837912 − 0.545805i
\(3\) 0 0
\(4\) 0.808388 + 1.82935i 0.404194 + 0.914673i
\(5\) −0.796479 − 0.329912i −0.356196 − 0.147541i 0.197408 − 0.980321i \(-0.436748\pi\)
−0.553604 + 0.832780i \(0.686748\pi\)
\(6\) 0 0
\(7\) 3.41414 + 3.41414i 1.29042 + 1.29042i 0.934525 + 0.355897i \(0.115825\pi\)
0.355897 + 0.934525i \(0.384175\pi\)
\(8\) 0.454114 − 2.79173i 0.160554 − 0.987027i
\(9\) 0 0
\(10\) 0.689163 + 1.00573i 0.217932 + 0.318040i
\(11\) −0.811278 + 1.95860i −0.244609 + 0.590539i −0.997730 − 0.0673433i \(-0.978548\pi\)
0.753120 + 0.657883i \(0.228548\pi\)
\(12\) 0 0
\(13\) −3.44869 + 1.42849i −0.956493 + 0.396193i −0.805668 − 0.592368i \(-0.798193\pi\)
−0.150826 + 0.988560i \(0.548193\pi\)
\(14\) −1.41039 − 6.68103i −0.376942 − 1.78558i
\(15\) 0 0
\(16\) −2.69302 + 2.95764i −0.673254 + 0.739411i
\(17\) 5.12913i 1.24400i 0.783018 + 0.621999i \(0.213679\pi\)
−0.783018 + 0.621999i \(0.786321\pi\)
\(18\) 0 0
\(19\) 1.84398 − 0.763801i 0.423038 − 0.175228i −0.161000 − 0.986954i \(-0.551472\pi\)
0.584038 + 0.811726i \(0.301472\pi\)
\(20\) −0.0403403 − 1.72373i −0.00902037 − 0.385438i
\(21\) 0 0
\(22\) 2.47316 − 1.69470i 0.527280 − 0.361311i
\(23\) −0.655049 + 0.655049i −0.136587 + 0.136587i −0.772095 − 0.635508i \(-0.780791\pi\)
0.635508 + 0.772095i \(0.280791\pi\)
\(24\) 0 0
\(25\) −3.01000 − 3.01000i −0.602000 − 0.602000i
\(26\) 5.18928 + 0.969242i 1.01770 + 0.190084i
\(27\) 0 0
\(28\) −3.48569 + 9.00559i −0.658734 + 1.70190i
\(29\) −3.09893 − 7.48148i −0.575457 − 1.38928i −0.896852 − 0.442331i \(-0.854152\pi\)
0.321394 − 0.946945i \(-0.395848\pi\)
\(30\) 0 0
\(31\) −7.46197 −1.34021 −0.670104 − 0.742267i \(-0.733751\pi\)
−0.670104 + 0.742267i \(0.733751\pi\)
\(32\) 5.47415 − 1.42607i 0.967702 − 0.252096i
\(33\) 0 0
\(34\) 3.95910 − 6.07796i 0.678980 − 1.04236i
\(35\) −1.59292 − 3.84565i −0.269253 − 0.650034i
\(36\) 0 0
\(37\) 1.04147 + 0.431390i 0.171216 + 0.0709201i 0.466645 − 0.884445i \(-0.345463\pi\)
−0.295429 + 0.955365i \(0.595463\pi\)
\(38\) −2.77466 − 0.518245i −0.450109 − 0.0840704i
\(39\) 0 0
\(40\) −1.28272 + 2.07374i −0.202816 + 0.327887i
\(41\) −7.85358 + 7.85358i −1.22652 + 1.22652i −0.261253 + 0.965270i \(0.584136\pi\)
−0.965270 + 0.261253i \(0.915864\pi\)
\(42\) 0 0
\(43\) 1.55013 − 3.74235i 0.236393 − 0.570703i −0.760512 − 0.649324i \(-0.775052\pi\)
0.996905 + 0.0786213i \(0.0250518\pi\)
\(44\) −4.23878 + 0.0991997i −0.639020 + 0.0149549i
\(45\) 0 0
\(46\) 1.28185 − 0.270602i 0.188998 − 0.0398981i
\(47\) 8.53070i 1.24433i 0.782886 + 0.622165i \(0.213747\pi\)
−0.782886 + 0.622165i \(0.786253\pi\)
\(48\) 0 0
\(49\) 16.3127i 2.33038i
\(50\) 1.24344 + 5.89018i 0.175849 + 0.832997i
\(51\) 0 0
\(52\) −5.40108 − 5.15406i −0.748996 − 0.714740i
\(53\) 2.21888 − 5.35684i 0.304786 − 0.735818i −0.695072 − 0.718940i \(-0.744627\pi\)
0.999858 − 0.0168778i \(-0.00537263\pi\)
\(54\) 0 0
\(55\) 1.29233 − 1.29233i 0.174258 − 0.174258i
\(56\) 11.0818 − 7.98095i 1.48086 − 1.06650i
\(57\) 0 0
\(58\) −2.10265 + 11.2575i −0.276091 + 1.47818i
\(59\) −3.55676 − 1.47326i −0.463050 − 0.191802i 0.138947 − 0.990300i \(-0.455628\pi\)
−0.601997 + 0.798498i \(0.705628\pi\)
\(60\) 0 0
\(61\) 5.53850 + 13.3711i 0.709132 + 1.71200i 0.702160 + 0.712020i \(0.252219\pi\)
0.00697220 + 0.999976i \(0.497781\pi\)
\(62\) 8.84233 + 5.75978i 1.12298 + 0.731492i
\(63\) 0 0
\(64\) −7.58756 − 2.53553i −0.948445 − 0.316942i
\(65\) 3.21808 0.399154
\(66\) 0 0
\(67\) 1.49796 + 3.61639i 0.183005 + 0.441813i 0.988583 − 0.150677i \(-0.0481453\pi\)
−0.805578 + 0.592489i \(0.798145\pi\)
\(68\) −9.38296 + 4.14633i −1.13785 + 0.502817i
\(69\) 0 0
\(70\) −1.08081 + 5.78660i −0.129181 + 0.691631i
\(71\) 3.65387 + 3.65387i 0.433635 + 0.433635i 0.889863 − 0.456228i \(-0.150800\pi\)
−0.456228 + 0.889863i \(0.650800\pi\)
\(72\) 0 0
\(73\) −1.36736 + 1.36736i −0.160038 + 0.160038i −0.782583 − 0.622546i \(-0.786098\pi\)
0.622546 + 0.782583i \(0.286098\pi\)
\(74\) −0.901143 − 1.31508i −0.104756 − 0.152875i
\(75\) 0 0
\(76\) 2.88791 + 2.75583i 0.331266 + 0.316115i
\(77\) −9.45673 + 3.91711i −1.07769 + 0.446396i
\(78\) 0 0
\(79\) − 9.69095i − 1.09032i −0.838333 − 0.545158i \(-0.816470\pi\)
0.838333 − 0.545158i \(-0.183530\pi\)
\(80\) 3.12069 − 1.46724i 0.348904 − 0.164043i
\(81\) 0 0
\(82\) 15.3684 − 3.24433i 1.69716 − 0.358277i
\(83\) −1.86485 + 0.772448i −0.204694 + 0.0847871i −0.482675 − 0.875799i \(-0.660335\pi\)
0.277981 + 0.960587i \(0.410335\pi\)
\(84\) 0 0
\(85\) 1.69216 − 4.08525i 0.183541 − 0.443107i
\(86\) −4.72555 + 3.23811i −0.509569 + 0.349175i
\(87\) 0 0
\(88\) 5.09947 + 3.15430i 0.543605 + 0.336249i
\(89\) 9.11861 + 9.11861i 0.966571 + 0.966571i 0.999459 − 0.0328878i \(-0.0104704\pi\)
−0.0328878 + 0.999459i \(0.510470\pi\)
\(90\) 0 0
\(91\) −16.6514 − 6.89722i −1.74554 − 0.723025i
\(92\) −1.72785 − 0.668777i −0.180140 − 0.0697249i
\(93\) 0 0
\(94\) 6.58472 − 10.1088i 0.679162 − 1.04264i
\(95\) −1.72068 −0.176538
\(96\) 0 0
\(97\) 15.7002 1.59412 0.797059 − 0.603901i \(-0.206388\pi\)
0.797059 + 0.603901i \(0.206388\pi\)
\(98\) 12.5915 − 19.3303i 1.27193 − 1.95265i
\(99\) 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.a.325.6 yes 128
3.2 odd 2 inner 864.2.v.a.325.27 yes 128
32.13 even 8 inner 864.2.v.a.109.6 ✓ 128
96.77 odd 8 inner 864.2.v.a.109.27 yes 128
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
864.2.v.a.109.6 ✓ 128 32.13 even 8 inner
864.2.v.a.109.27 yes 128 96.77 odd 8 inner
864.2.v.a.325.6 yes 128 1.1 even 1 trivial
864.2.v.a.325.27 yes 128 3.2 odd 2 inner