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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 109.4
Character \(\chi\) \(=\) 864.109
Dual form 864.2.v.b.325.4

$q$-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)
 
\(f(q)\) \(=\) \(q+(-1.23997 + 0.680049i) q^{2} +(1.07507 - 1.68649i) q^{4} +(1.02237 - 0.423478i) q^{5} +(0.485592 - 0.485592i) q^{7} +(-0.186159 + 2.82229i) q^{8} +(-0.979722 + 1.22036i) q^{10} +(1.70905 + 4.12601i) q^{11} +(-2.03122 - 0.841358i) q^{13} +(-0.271894 + 0.932347i) q^{14} +(-1.68847 - 3.62617i) q^{16} +4.47838i q^{17} +(2.51262 + 1.04076i) q^{19} +(0.384922 - 2.17947i) q^{20} +(-4.92507 - 3.95390i) q^{22} +(-1.15719 - 1.15719i) q^{23} +(-2.66963 + 2.66963i) q^{25} +(3.09082 - 0.338067i) q^{26} +(-0.296901 - 1.34099i) q^{28} +(-1.62983 + 3.93476i) q^{29} +6.62389 q^{31} +(4.55963 + 3.34811i) q^{32} +(-3.04552 - 5.55306i) q^{34} +(0.290816 - 0.702091i) q^{35} +(9.60845 - 3.97995i) q^{37} +(-3.82335 + 0.418190i) q^{38} +(1.00486 + 2.96426i) q^{40} +(3.06238 + 3.06238i) q^{41} +(-2.15582 - 5.20462i) q^{43} +(8.79580 + 1.55345i) q^{44} +(2.22184 + 0.647940i) q^{46} -8.66512i q^{47} +6.52840i q^{49} +(1.49479 - 5.12575i) q^{50} +(-3.60263 + 2.52110i) q^{52} +(2.26817 + 5.47584i) q^{53} +(3.49455 + 3.49455i) q^{55} +(1.28009 + 1.46088i) q^{56} +(-0.654885 - 5.98736i) q^{58} +(7.36470 - 3.05056i) q^{59} +(-1.16848 + 2.82095i) q^{61} +(-8.21344 + 4.50457i) q^{62} +(-7.93069 - 1.05079i) q^{64} -2.43295 q^{65} +(-1.63065 + 3.93673i) q^{67} +(7.55272 + 4.81455i) q^{68} +(0.116853 + 1.06834i) q^{70} +(-2.11730 + 2.11730i) q^{71} +(6.06373 + 6.06373i) q^{73} +(-9.20765 + 11.4692i) q^{74} +(4.45646 - 3.11861i) q^{76} +(2.83346 + 1.17366i) q^{77} +5.65785i q^{79} +(-3.26184 - 2.99224i) q^{80} +(-5.87983 - 1.71470i) q^{82} +(-1.59099 - 0.659009i) q^{83} +(1.89650 + 4.57854i) q^{85} +(6.21256 + 4.98752i) q^{86} +(-11.9630 + 4.05535i) q^{88} +(0.998730 - 0.998730i) q^{89} +(-1.39490 + 0.577786i) q^{91} +(-3.19565 + 0.707531i) q^{92} +(5.89271 + 10.7445i) q^{94} +3.00956 q^{95} +9.78679 q^{97} +(-4.43963 - 8.09504i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 128 q + 16 q^{10} - 32 q^{16} - 16 q^{22} - 32 q^{40} - 32 q^{46} - 80 q^{52} + 32 q^{55} - 32 q^{58} + 64 q^{61} + 48 q^{64} + 64 q^{67} - 96 q^{70} + 32 q^{76} - 80 q^{82} - 80 q^{88} + 96 q^{91} - 48 q^{94}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.23997 + 0.680049i −0.876793 + 0.480867i
\(3\) 0 0
\(4\) 1.07507 − 1.68649i 0.537533 − 0.843243i
\(5\) 1.02237 − 0.423478i 0.457217 − 0.189385i −0.142175 − 0.989842i \(-0.545409\pi\)
0.599391 + 0.800456i \(0.295409\pi\)
\(6\) 0 0
\(7\) 0.485592 − 0.485592i 0.183537 − 0.183537i −0.609358 − 0.792895i \(-0.708573\pi\)
0.792895 + 0.609358i \(0.208573\pi\)
\(8\) −0.186159 + 2.82229i −0.0658172 + 0.997832i
\(9\) 0 0
\(10\) −0.979722 + 1.22036i −0.309815 + 0.385912i
\(11\) 1.70905 + 4.12601i 0.515298 + 1.24404i 0.940763 + 0.339064i \(0.110110\pi\)
−0.425465 + 0.904975i \(0.639890\pi\)
\(12\) 0 0
\(13\) −2.03122 − 0.841358i −0.563358 − 0.233351i 0.0827840 − 0.996568i \(-0.473619\pi\)
−0.646142 + 0.763217i \(0.723619\pi\)
\(14\) −0.271894 + 0.932347i −0.0726668 + 0.249180i
\(15\) 0 0
\(16\) −1.68847 − 3.62617i −0.422117 − 0.906541i
\(17\) 4.47838i 1.08617i 0.839679 + 0.543083i \(0.182743\pi\)
−0.839679 + 0.543083i \(0.817257\pi\)
\(18\) 0 0
\(19\) 2.51262 + 1.04076i 0.576434 + 0.238767i 0.651802 − 0.758389i \(-0.274013\pi\)
−0.0753681 + 0.997156i \(0.524013\pi\)
\(20\) 0.384922 − 2.17947i 0.0860711 − 0.487345i
\(21\) 0 0
\(22\) −4.92507 − 3.95390i −1.05003 − 0.842975i
\(23\) −1.15719 − 1.15719i −0.241291 − 0.241291i 0.576093 − 0.817384i \(-0.304577\pi\)
−0.817384 + 0.576093i \(0.804577\pi\)
\(24\) 0 0
\(25\) −2.66963 + 2.66963i −0.533927 + 0.533927i
\(26\) 3.09082 − 0.338067i 0.606160 − 0.0663004i
\(27\) 0 0
\(28\) −0.296901 − 1.34099i −0.0561089 − 0.253423i
\(29\) −1.62983 + 3.93476i −0.302652 + 0.730667i 0.697252 + 0.716826i \(0.254406\pi\)
−0.999904 + 0.0138407i \(0.995594\pi\)
\(30\) 0 0
\(31\) 6.62389 1.18969 0.594843 − 0.803842i \(-0.297214\pi\)
0.594843 + 0.803842i \(0.297214\pi\)
\(32\) 4.55963 + 3.34811i 0.806035 + 0.591867i
\(33\) 0 0
\(34\) −3.04552 − 5.55306i −0.522302 − 0.952343i
\(35\) 0.290816 − 0.702091i 0.0491568 − 0.118675i
\(36\) 0 0
\(37\) 9.60845 − 3.97995i 1.57962 − 0.654300i 0.591266 − 0.806477i \(-0.298628\pi\)
0.988353 + 0.152177i \(0.0486284\pi\)
\(38\) −3.82335 + 0.418190i −0.620229 + 0.0678393i
\(39\) 0 0
\(40\) 1.00486 + 2.96426i 0.158882 + 0.468690i
\(41\) 3.06238 + 3.06238i 0.478263 + 0.478263i 0.904576 − 0.426313i \(-0.140188\pi\)
−0.426313 + 0.904576i \(0.640188\pi\)
\(42\) 0 0
\(43\) −2.15582 − 5.20462i −0.328760 − 0.793696i −0.998685 − 0.0512678i \(-0.983674\pi\)
0.669925 − 0.742429i \(-0.266326\pi\)
\(44\) 8.79580 + 1.55345i 1.32602 + 0.234191i
\(45\) 0 0
\(46\) 2.22184 + 0.647940i 0.327592 + 0.0955335i
\(47\) − 8.66512i − 1.26394i −0.774994 − 0.631969i \(-0.782247\pi\)
0.774994 − 0.631969i \(-0.217753\pi\)
\(48\) 0 0
\(49\) 6.52840i 0.932629i
\(50\) 1.49479 − 5.12575i 0.211395 − 0.724891i
\(51\) 0 0
\(52\) −3.60263 + 2.52110i −0.499595 + 0.349614i
\(53\) 2.26817 + 5.47584i 0.311557 + 0.752165i 0.999648 + 0.0265403i \(0.00844902\pi\)
−0.688091 + 0.725625i \(0.741551\pi\)
\(54\) 0 0
\(55\) 3.49455 + 3.49455i 0.471205 + 0.471205i
\(56\) 1.28009 + 1.46088i 0.171059 + 0.195218i
\(57\) 0 0
\(58\) −0.654885 − 5.98736i −0.0859906 − 0.786179i
\(59\) 7.36470 − 3.05056i 0.958802 − 0.397149i 0.152270 − 0.988339i \(-0.451342\pi\)
0.806532 + 0.591190i \(0.201342\pi\)
\(60\) 0 0
\(61\) −1.16848 + 2.82095i −0.149608 + 0.361186i −0.980861 − 0.194708i \(-0.937624\pi\)
0.831253 + 0.555894i \(0.187624\pi\)
\(62\) −8.21344 + 4.50457i −1.04311 + 0.572081i
\(63\) 0 0
\(64\) −7.93069 − 1.05079i −0.991336 − 0.131349i
\(65\) −2.43295 −0.301770
\(66\) 0 0
\(67\) −1.63065 + 3.93673i −0.199215 + 0.480948i −0.991642 − 0.129019i \(-0.958817\pi\)
0.792427 + 0.609967i \(0.208817\pi\)
\(68\) 7.55272 + 4.81455i 0.915901 + 0.583850i
\(69\) 0 0
\(70\) 0.116853 + 1.06834i 0.0139666 + 0.127691i
\(71\) −2.11730 + 2.11730i −0.251277 + 0.251277i −0.821494 − 0.570217i \(-0.806859\pi\)
0.570217 + 0.821494i \(0.306859\pi\)
\(72\) 0 0
\(73\) 6.06373 + 6.06373i 0.709706 + 0.709706i 0.966473 − 0.256767i \(-0.0826573\pi\)
−0.256767 + 0.966473i \(0.582657\pi\)
\(74\) −9.20765 + 11.4692i −1.07037 + 1.33327i
\(75\) 0 0
\(76\) 4.45646 − 3.11861i 0.511191 − 0.357729i
\(77\) 2.83346 + 1.17366i 0.322903 + 0.133751i
\(78\) 0 0
\(79\) 5.65785i 0.636558i 0.947997 + 0.318279i \(0.103105\pi\)
−0.947997 + 0.318279i \(0.896895\pi\)
\(80\) −3.26184 − 2.99224i −0.364684 − 0.334543i
\(81\) 0 0
\(82\) −5.87983 − 1.71470i −0.649319 − 0.189357i
\(83\) −1.59099 − 0.659009i −0.174634 − 0.0723356i 0.293654 − 0.955912i \(-0.405129\pi\)
−0.468287 + 0.883576i \(0.655129\pi\)
\(84\) 0 0
\(85\) 1.89650 + 4.57854i 0.205704 + 0.496613i
\(86\) 6.21256 + 4.98752i 0.669917 + 0.537818i
\(87\) 0 0
\(88\) −11.9630 + 4.05535i −1.27526 + 0.432301i
\(89\) 0.998730 − 0.998730i 0.105865 − 0.105865i −0.652190 − 0.758055i \(-0.726150\pi\)
0.758055 + 0.652190i \(0.226150\pi\)
\(90\) 0 0
\(91\) −1.39490 + 0.577786i −0.146225 + 0.0605685i
\(92\) −3.19565 + 0.707531i −0.333169 + 0.0737652i
\(93\) 0 0
\(94\) 5.89271 + 10.7445i 0.607787 + 1.10821i
\(95\) 3.00956 0.308774
\(96\) 0 0
\(97\) 9.78679 0.993698 0.496849 − 0.867837i \(-0.334490\pi\)
0.496849 + 0.867837i \(0.334490\pi\)
\(98\) −4.43963 − 8.09504i −0.448471 − 0.817723i
\(99\) 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.109.4 ✓ 128
3.2 odd 2 inner 864.2.v.b.109.29 yes 128
32.5 even 8 inner 864.2.v.b.325.4 yes 128
96.5 odd 8 inner 864.2.v.b.325.29 yes 128
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
864.2.v.b.109.4 ✓ 128 1.1 even 1 trivial
864.2.v.b.109.29 yes 128 3.2 odd 2 inner
864.2.v.b.325.4 yes 128 32.5 even 8 inner
864.2.v.b.325.29 yes 128 96.5 odd 8 inner