↑

Properties

Label 784.2.x.n.765.8
Level $784$
Weight $2$
Character 784.765
Analytic conductor $6.260$
Analytic rank $0$
Dimension $40$
Inner twists $8$

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [784,2,Mod(165,784)] 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("784.165"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(784, base_ring=CyclotomicField(12)) chi = DirichletCharacter(H, H._module([0, 3, 8])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 784 = 2^{4} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 784.x (of order \(12\), degree \(4\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [40,0,0,8,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(6.26027151847\)
Analytic rank: \(0\)
Dimension: \(40\)
Relative dimension: \(10\) over \(\Q(\zeta_{12})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label 765.8
Character \(\chi\) \(=\) 784.765
Dual form 784.2.x.n.165.8

$q$-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+(0.674626 + 1.24293i) q^{2} +(0.839505 - 3.13308i) q^{3} +(-1.08976 + 1.67703i) q^{4} +(-0.734404 - 2.74083i) q^{5} +(4.46055 - 1.07021i) q^{6} +(-2.81961 - 0.223131i) q^{8} +(-6.51332 - 3.76047i) q^{9} +(2.91122 - 2.76185i) q^{10} +(4.32748 + 1.15955i) q^{11} +(4.33940 + 4.82218i) q^{12} +(-0.558438 - 0.558438i) q^{13} -9.20377 q^{15} +(-1.62485 - 3.65512i) q^{16} +(-1.09678 - 1.89967i) q^{17} +(0.279952 - 10.6325i) q^{18} +(-2.55326 + 0.684144i) q^{19} +(5.39678 + 1.75523i) q^{20} +(1.47820 + 6.16103i) q^{22} +(0.647926 + 0.374080i) q^{23} +(-3.06617 + 8.64674i) q^{24} +(-2.64268 + 1.52575i) q^{25} +(0.317364 - 1.07084i) q^{26} +(-10.3691 + 10.3691i) q^{27} +(-3.07468 - 3.07468i) q^{29} +(-6.20910 - 11.4397i) q^{30} +(-4.43398 - 7.67989i) q^{31} +(3.44690 - 4.48541i) q^{32} +(7.26589 - 12.5849i) q^{33} +(1.62125 - 2.64479i) q^{34} +(13.4044 - 6.82502i) q^{36} +(1.69807 + 6.33730i) q^{37} +(-2.57284 - 2.71199i) q^{38} +(-2.21844 + 1.28082i) q^{39} +(1.45917 + 7.89195i) q^{40} +0.267969i q^{41} +(4.53661 - 4.53661i) q^{43} +(-6.66051 + 5.99369i) q^{44} +(-5.52341 + 20.6136i) q^{45} +(-0.0278487 + 1.05769i) q^{46} +(-3.74156 + 6.48057i) q^{47} +(-12.8158 + 2.02228i) q^{48} +(-3.67923 - 2.25536i) q^{50} +(-6.87257 + 1.84150i) q^{51} +(1.54508 - 0.327953i) q^{52} +(4.36564 + 1.16977i) q^{53} +(-19.8833 - 5.89282i) q^{54} -12.7125i q^{55} +8.57391i q^{57} +(1.74736 - 5.89587i) q^{58} +(7.07539 + 1.89585i) q^{59} +(10.0299 - 15.4350i) q^{60} +(-5.73728 + 1.53730i) q^{61} +(6.55430 - 10.6922i) q^{62} +(7.90042 + 1.25829i) q^{64} +(-1.12047 + 1.94070i) q^{65} +(20.5439 + 0.540916i) q^{66} +(1.48228 - 5.53193i) q^{67} +(4.38103 + 0.230862i) q^{68} +(1.71596 - 1.71596i) q^{69} -9.37949i q^{71} +(17.5260 + 12.0564i) q^{72} +(9.08221 - 5.24362i) q^{73} +(-6.73127 + 6.38590i) q^{74} +(2.56176 + 9.56061i) q^{75} +(1.63511 - 5.02744i) q^{76} +(-3.08859 - 1.89330i) q^{78} +(5.10952 - 8.84995i) q^{79} +(-8.82477 + 7.13776i) q^{80} +(12.5009 + 21.6521i) q^{81} +(-0.333067 + 0.180778i) q^{82} +(0.474267 + 0.474267i) q^{83} +(-4.40120 + 4.40120i) q^{85} +(8.69922 + 2.57819i) q^{86} +(-12.2144 + 7.05199i) q^{87} +(-11.9431 - 4.23507i) q^{88} +(12.5849 + 7.26589i) q^{89} +(-29.3476 + 7.04127i) q^{90} +(-1.33343 + 0.678932i) q^{92} +(-27.7840 + 7.44471i) q^{93} +(-10.5791 - 0.278544i) q^{94} +(3.75025 + 6.49562i) q^{95} +(-11.1594 - 14.5649i) q^{96} +16.4976 q^{97} +(-23.8259 - 23.8259i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 40 q + 8 q^{4} - 4 q^{11} - 32 q^{15} - 16 q^{18} - 8 q^{29} - 8 q^{30} + 40 q^{32} + 80 q^{36} + 20 q^{37} + 120 q^{43} - 56 q^{44} + 64 q^{46} - 112 q^{50} + 16 q^{51} - 28 q^{53} + 72 q^{58} + 24 q^{60}+ \cdots - 216 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/784\mathbb{Z}\right)^\times\).

\(n\) \(197\) \(687\) \(689\)
\(\chi(n)\) \(e\left(\frac{3}{4}\right)\) \(1\) \(e\left(\frac{1}{3}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.674626 + 1.24293i 0.477032 + 0.878886i
\(3\) 0.839505 − 3.13308i 0.484689 − 1.80888i −0.0967700 − 0.995307i \(-0.530851\pi\)
0.581459 − 0.813576i \(-0.302482\pi\)
\(4\) −1.08976 + 1.67703i −0.544880 + 0.838514i
\(5\) −0.734404 − 2.74083i −0.328435 − 1.22574i −0.910813 − 0.412819i \(-0.864544\pi\)
0.582378 − 0.812918i \(-0.302122\pi\)
\(6\) 4.46055 − 1.07021i 1.82101 − 0.436910i
\(7\) 0 0
\(8\) −2.81961 − 0.223131i −0.996883 − 0.0788889i
\(9\) −6.51332 − 3.76047i −2.17111 − 1.25349i
\(10\) 2.91122 − 2.76185i 0.920609 − 0.873374i
\(11\) 4.32748 + 1.15955i 1.30479 + 0.349616i 0.843258 − 0.537509i \(-0.180635\pi\)
0.461528 + 0.887126i \(0.347301\pi\)
\(12\) 4.33940 + 4.82218i 1.25268 + 1.39204i
\(13\) −0.558438 − 0.558438i −0.154883 − 0.154883i 0.625412 − 0.780295i \(-0.284931\pi\)
−0.780295 + 0.625412i \(0.784931\pi\)
\(14\) 0 0
\(15\) −9.20377 −2.37640
\(16\) −1.62485 − 3.65512i −0.406211 − 0.913779i
\(17\) −1.09678 − 1.89967i −0.266007 − 0.460738i 0.701820 − 0.712355i \(-0.252371\pi\)
−0.967827 + 0.251616i \(0.919038\pi\)
\(18\) 0.279952 − 10.6325i 0.0659852 − 2.50611i
\(19\) −2.55326 + 0.684144i −0.585758 + 0.156954i −0.539514 − 0.841977i \(-0.681392\pi\)
−0.0462445 + 0.998930i \(0.514725\pi\)
\(20\) 5.39678 + 1.75523i 1.20676 + 0.392482i
\(21\) 0 0
\(22\) 1.47820 + 6.16103i 0.315152 + 1.31354i
\(23\) 0.647926 + 0.374080i 0.135102 + 0.0780011i 0.566028 − 0.824386i \(-0.308480\pi\)
−0.430926 + 0.902387i \(0.641813\pi\)
\(24\) −3.06617 + 8.64674i −0.625879 + 1.76501i
\(25\) −2.64268 + 1.52575i −0.528537 + 0.305151i
\(26\) 0.317364 − 1.07084i 0.0622402 − 0.210009i
\(27\) −10.3691 + 10.3691i −1.99553 + 1.99553i
\(28\) 0 0
\(29\) −3.07468 − 3.07468i −0.570953 − 0.570953i 0.361442 − 0.932395i \(-0.382285\pi\)
−0.932395 + 0.361442i \(0.882285\pi\)
\(30\) −6.20910 − 11.4397i −1.13362 − 2.08859i
\(31\) −4.43398 − 7.67989i −0.796367 − 1.37935i −0.921968 − 0.387267i \(-0.873419\pi\)
0.125601 − 0.992081i \(-0.459914\pi\)
\(32\) 3.44690 − 4.48541i 0.609331 − 0.792916i
\(33\) 7.26589 − 12.5849i 1.26483 − 2.19075i
\(34\) 1.62125 − 2.64479i 0.278042 − 0.453577i
\(35\) 0 0
\(36\) 13.4044 − 6.82502i 2.23406 − 1.13750i
\(37\) 1.69807 + 6.33730i 0.279162 + 1.04185i 0.953001 + 0.302967i \(0.0979773\pi\)
−0.673839 + 0.738878i \(0.735356\pi\)
\(38\) −2.57284 − 2.71199i −0.417370 − 0.439943i
\(39\) −2.21844 + 1.28082i −0.355235 + 0.205095i
\(40\) 1.45917 + 7.89195i 0.230715 + 1.24783i
\(41\) 0.267969i 0.0418497i 0.999781 + 0.0209248i \(0.00666107\pi\)
−0.999781 + 0.0209248i \(0.993339\pi\)
\(42\) 0 0
\(43\) 4.53661 − 4.53661i 0.691827 − 0.691827i −0.270807 − 0.962634i \(-0.587290\pi\)
0.962634 + 0.270807i \(0.0872904\pi\)
\(44\) −6.66051 + 5.99369i −1.00411 + 0.903582i
\(45\) −5.52341 + 20.6136i −0.823381 + 3.07290i
\(46\) −0.0278487 + 1.05769i −0.00410607 + 0.155948i
\(47\) −3.74156 + 6.48057i −0.545763 + 0.945289i 0.452796 + 0.891614i \(0.350427\pi\)
−0.998558 + 0.0536746i \(0.982907\pi\)
\(48\) −12.8158 + 2.02228i −1.84981 + 0.291890i
\(49\) 0 0
\(50\) −3.67923 − 2.25536i −0.520322 − 0.318956i
\(51\) −6.87257 + 1.84150i −0.962352 + 0.257861i
\(52\) 1.54508 − 0.327953i 0.214264 − 0.0454789i
\(53\) 4.36564 + 1.16977i 0.599667 + 0.160680i 0.545868 − 0.837871i \(-0.316200\pi\)
0.0537996 + 0.998552i \(0.482867\pi\)
\(54\) −19.8833 − 5.89282i −2.70578 − 0.801912i
\(55\) − 12.7125i − 1.71415i
\(56\) 0 0
\(57\) 8.57391i 1.13564i
\(58\) 1.74736 − 5.89587i 0.229439 − 0.774166i
\(59\) 7.07539 + 1.89585i 0.921137 + 0.246818i 0.688071 − 0.725643i \(-0.258458\pi\)
0.233066 + 0.972461i \(0.425124\pi\)
\(60\) 10.0299 − 15.4350i 1.29486 − 1.99265i
\(61\) −5.73728 + 1.53730i −0.734583 + 0.196831i −0.606669 − 0.794954i \(-0.707495\pi\)
−0.127914 + 0.991785i \(0.540828\pi\)
\(62\) 6.55430 − 10.6922i 0.832396 − 1.35791i
\(63\) 0 0
\(64\) 7.90042 + 1.25829i 0.987553 + 0.157286i
\(65\) −1.12047 + 1.94070i −0.138977 + 0.240715i
\(66\) 20.5439 + 0.540916i 2.52878 + 0.0665822i
\(67\) 1.48228 − 5.53193i 0.181089 − 0.675833i −0.814345 − 0.580381i \(-0.802904\pi\)
0.995434 − 0.0954519i \(-0.0304296\pi\)
\(68\) 4.38103 + 0.230862i 0.531277 + 0.0279962i
\(69\) 1.71596 − 1.71596i 0.206577 − 0.206577i
\(70\) 0 0
\(71\) − 9.37949i − 1.11314i −0.830801 − 0.556570i \(-0.812117\pi\)
0.830801 − 0.556570i \(-0.187883\pi\)
\(72\) 17.5260 + 12.0564i 2.06546 + 1.42086i
\(73\) 9.08221 − 5.24362i 1.06299 − 0.613719i 0.136734 − 0.990608i \(-0.456339\pi\)
0.926258 + 0.376889i \(0.123006\pi\)
\(74\) −6.73127 + 6.38590i −0.782494 + 0.742345i
\(75\) 2.56176 + 9.56061i 0.295806 + 1.10396i
\(76\) 1.63511 − 5.02744i 0.187560 − 0.576687i
\(77\) 0 0
\(78\) −3.08859 − 1.89330i −0.349714 − 0.214374i
\(79\) 5.10952 − 8.84995i 0.574866 − 0.995697i −0.421190 − 0.906972i \(-0.638387\pi\)
0.996056 − 0.0887248i \(-0.0282792\pi\)
\(80\) −8.82477 + 7.13776i −0.986639 + 0.798026i
\(81\) 12.5009 + 21.6521i 1.38898 + 2.40579i
\(82\) −0.333067 + 0.180778i −0.0367811 + 0.0199636i
\(83\) 0.474267 + 0.474267i 0.0520576 + 0.0520576i 0.732656 − 0.680599i \(-0.238280\pi\)
−0.680599 + 0.732656i \(0.738280\pi\)
\(84\) 0 0
\(85\) −4.40120 + 4.40120i −0.477378 + 0.477378i
\(86\) 8.69922 + 2.57819i 0.938061 + 0.278013i
\(87\) −12.2144 + 7.05199i −1.30952 + 0.756053i
\(88\) −11.9431 − 4.23507i −1.27314 − 0.451460i
\(89\) 12.5849 + 7.26589i 1.33400 + 0.770183i 0.985910 − 0.167279i \(-0.0534981\pi\)
0.348087 + 0.937462i \(0.386831\pi\)
\(90\) −29.3476 + 7.04127i −3.09351 + 0.742215i
\(91\) 0 0
\(92\) −1.33343 + 0.678932i −0.139019 + 0.0707835i
\(93\) −27.7840 + 7.44471i −2.88107 + 0.771980i
\(94\) −10.5791 − 0.278544i −1.09115 − 0.0287296i
\(95\) 3.75025 + 6.49562i 0.384768 + 0.666437i
\(96\) −11.1594 − 14.5649i −1.13896 − 1.48653i
\(97\) 16.4976 1.67507 0.837537 − 0.546380i \(-0.183995\pi\)
0.837537 + 0.546380i \(0.183995\pi\)
\(98\) 0 0
\(99\) −23.8259 − 23.8259i −2.39459 − 2.39459i
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 784.2.x.n.765.8 40
7.2 even 3 784.2.m.i.589.8 yes 20
7.3 odd 6 inner 784.2.x.n.557.2 40
7.4 even 3 inner 784.2.x.n.557.1 40
7.5 odd 6 784.2.m.i.589.7 yes 20
7.6 odd 2 inner 784.2.x.n.765.7 40
16.5 even 4 inner 784.2.x.n.373.1 40
112.5 odd 12 784.2.m.i.197.7 ✓ 20
112.37 even 12 784.2.m.i.197.8 yes 20
112.53 even 12 inner 784.2.x.n.165.8 40
112.69 odd 4 inner 784.2.x.n.373.2 40
112.101 odd 12 inner 784.2.x.n.165.7 40
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
784.2.m.i.197.7 ✓ 20 112.5 odd 12
784.2.m.i.197.8 yes 20 112.37 even 12
784.2.m.i.589.7 yes 20 7.5 odd 6
784.2.m.i.589.8 yes 20 7.2 even 3
784.2.x.n.165.7 40 112.101 odd 12 inner
784.2.x.n.165.8 40 112.53 even 12 inner
784.2.x.n.373.1 40 16.5 even 4 inner
784.2.x.n.373.2 40 112.69 odd 4 inner
784.2.x.n.557.1 40 7.4 even 3 inner
784.2.x.n.557.2 40 7.3 odd 6 inner
784.2.x.n.765.7 40 7.6 odd 2 inner
784.2.x.n.765.8 40 1.1 even 1 trivial