↑

Properties

Label 790.2.p.a.719.18
Level $790$
Weight $2$
Character 790.719
Analytic conductor $6.308$
Analytic rank $0$
Dimension $480$
Inner twists $4$

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] = [790,2,Mod(89,790)] 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("790.89"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(790, base_ring=CyclotomicField(26)) chi = DirichletCharacter(H, H._module([13, 22])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 790 = 2 \cdot 5 \cdot 79 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 790.p (of order \(26\), degree \(12\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(6.30818175968\)
Analytic rank: \(0\)
Dimension: \(480\)
Relative dimension: \(40\) over \(\Q(\zeta_{26})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{26}]$

Embedding invariants

Embedding label 719.18
Character \(\chi\) \(=\) 790.719
Dual form 790.2.p.a.89.18

$q$-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+(-0.935016 - 0.354605i) q^{2} +(1.05380 - 2.00785i) q^{3} +(0.748511 + 0.663123i) q^{4} +(-1.04203 + 1.97842i) q^{5} +(-1.69731 + 1.50369i) q^{6} +(0.156322 - 0.297847i) q^{7} +(-0.464723 - 0.885456i) q^{8} +(-1.21676 - 1.76279i) q^{9} +(1.67588 - 1.48035i) q^{10} +(0.210085 + 1.73020i) q^{11} +(2.12023 - 0.804097i) q^{12} +(-1.87781 + 2.11961i) q^{13} +(-0.251782 + 0.223059i) q^{14} +(2.87428 + 4.17711i) q^{15} +(0.120537 + 0.992709i) q^{16} +(5.26528 - 5.94328i) q^{17} +(0.512601 + 2.07970i) q^{18} +(5.95805 - 1.46853i) q^{19} +(-2.09191 + 0.789876i) q^{20} +(-0.433299 - 0.627742i) q^{21} +(0.417106 - 1.69227i) q^{22} -1.33290i q^{23} -2.26759 q^{24} +(-2.82833 - 4.12317i) q^{25} +(2.50741 - 1.31599i) q^{26} +(1.93153 - 0.234530i) q^{27} +(0.314518 - 0.119281i) q^{28} +(1.00783 - 1.46009i) q^{29} +(-1.20627 - 4.92490i) q^{30} +(3.73676 - 9.85303i) q^{31} +(0.239316 - 0.970942i) q^{32} +(3.69537 + 1.40147i) q^{33} +(-7.03064 + 3.68996i) q^{34} +(0.426375 + 0.619639i) q^{35} +(0.258183 - 2.12633i) q^{36} +(1.95911 + 7.94843i) q^{37} +(-6.09162 - 0.739657i) q^{38} +(2.27702 + 6.00400i) q^{39} +(2.23607 + 0.00325578i) q^{40} +(-0.0431886 + 0.355690i) q^{41} +(0.182541 + 0.740599i) q^{42} +(11.2364 + 1.36435i) q^{43} +(-0.990086 + 1.43439i) q^{44} +(4.75545 - 0.570391i) q^{45} +(-0.472654 + 1.24629i) q^{46} +(-2.01135 + 8.16037i) q^{47} +(2.12023 + 0.804097i) q^{48} +(3.91218 + 5.66777i) q^{49} +(1.18244 + 4.85817i) q^{50} +(-6.38464 - 16.8349i) q^{51} +(-2.81112 + 0.341332i) q^{52} +(-5.66643 - 10.7965i) q^{53} +(-1.88918 - 0.465640i) q^{54} +(-3.64199 - 1.38729i) q^{55} -0.336377 q^{56} +(3.33001 - 13.5104i) q^{57} +(-1.46009 + 1.00783i) q^{58} +(-8.71230 + 7.71843i) q^{59} +(-0.618507 + 5.03261i) q^{60} +(8.10944 - 1.99880i) q^{61} +(-6.98787 + 7.88767i) q^{62} +(-0.715248 + 0.0868468i) q^{63} +(-0.568065 + 0.822984i) q^{64} +(-2.23674 - 5.92381i) q^{65} +(-2.95826 - 2.62079i) q^{66} +(-1.89573 + 0.718954i) q^{67} +(7.88224 - 0.957078i) q^{68} +(-2.67627 - 1.40461i) q^{69} +(-0.178941 - 0.730567i) q^{70} +(-1.08990 + 0.572026i) q^{71} +(-0.995411 + 1.89660i) q^{72} +(-8.67999 - 9.79768i) q^{73} +(0.986751 - 8.12663i) q^{74} +(-11.2592 + 1.33385i) q^{75} +(5.43348 + 2.85171i) q^{76} +(0.548177 + 0.207896i) q^{77} -6.42128i q^{78} +(7.30931 + 5.05707i) q^{79} +(-2.08960 - 0.795964i) q^{80} +(3.84318 - 10.1336i) q^{81} +(0.166511 - 0.317261i) q^{82} +(-4.24645 + 4.79325i) q^{83} +(0.0919410 - 0.757202i) q^{84} +(6.27172 + 16.6101i) q^{85} +(-10.0224 - 5.26018i) q^{86} +(-1.86959 - 3.56220i) q^{87} +(1.43439 - 0.990086i) q^{88} +(3.23367 + 1.69716i) q^{89} +(-4.64868 - 1.15298i) q^{90} +(0.337776 + 0.890642i) q^{91} +(0.883879 - 0.997693i) q^{92} +(-15.8456 - 17.8860i) q^{93} +(4.77435 - 6.91685i) q^{94} +(-3.30312 + 13.3178i) q^{95} +(-1.69731 - 1.50369i) q^{96} +(-4.35954 - 17.6874i) q^{97} +(-1.64813 - 6.68673i) q^{98} +(2.79435 - 2.47558i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 480 q + 40 q^{4} + 4 q^{6} + 48 q^{9} - 4 q^{10} - 8 q^{11} - 8 q^{14} - 8 q^{15} - 40 q^{16} + 16 q^{19} + 52 q^{21} + 48 q^{24} - 14 q^{25} - 20 q^{29} + 20 q^{30} - 68 q^{31} + 70 q^{35} - 48 q^{36}+ \cdots + 16 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/790\mathbb{Z}\right)^\times\).

\(n\) \(161\) \(317\)
\(\chi(n)\) \(e\left(\frac{2}{13}\right)\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.935016 − 0.354605i −0.661156 − 0.250744i
\(3\) 1.05380 − 2.00785i 0.608411 − 1.15923i −0.365181 − 0.930937i \(-0.618993\pi\)
0.973592 − 0.228294i \(-0.0733149\pi\)
\(4\) 0.748511 + 0.663123i 0.374255 + 0.331561i
\(5\) −1.04203 + 1.97842i −0.466012 + 0.884778i
\(6\) −1.69731 + 1.50369i −0.692925 + 0.613878i
\(7\) 0.156322 − 0.297847i 0.0590843 − 0.112576i −0.854123 − 0.520071i \(-0.825905\pi\)
0.913207 + 0.407495i \(0.133598\pi\)
\(8\) −0.464723 − 0.885456i −0.164304 − 0.313056i
\(9\) −1.21676 − 1.76279i −0.405588 − 0.587595i
\(10\) 1.67588 − 1.48035i 0.529959 − 0.468127i
\(11\) 0.210085 + 1.73020i 0.0633429 + 0.521676i 0.989499 + 0.144537i \(0.0461691\pi\)
−0.926157 + 0.377139i \(0.876908\pi\)
\(12\) 2.12023 − 0.804097i 0.612057 − 0.232123i
\(13\) −1.87781 + 2.11961i −0.520811 + 0.587874i −0.948787 − 0.315916i \(-0.897688\pi\)
0.427976 + 0.903790i \(0.359227\pi\)
\(14\) −0.251782 + 0.223059i −0.0672915 + 0.0596151i
\(15\) 2.87428 + 4.17711i 0.742136 + 1.07852i
\(16\) 0.120537 + 0.992709i 0.0301342 + 0.248177i
\(17\) 5.26528 − 5.94328i 1.27702 − 1.44146i 0.436471 − 0.899718i \(-0.356228\pi\)
0.840547 − 0.541738i \(-0.182234\pi\)
\(18\) 0.512601 + 2.07970i 0.120821 + 0.490191i
\(19\) 5.95805 − 1.46853i 1.36687 − 0.336904i 0.513447 − 0.858121i \(-0.328368\pi\)
0.853424 + 0.521218i \(0.174522\pi\)
\(20\) −2.09191 + 0.789876i −0.467766 + 0.176622i
\(21\) −0.433299 − 0.627742i −0.0945537 − 0.136985i
\(22\) 0.417106 − 1.69227i 0.0889273 − 0.360792i
\(23\) − 1.33290i − 0.277930i −0.990297 − 0.138965i \(-0.955622\pi\)
0.990297 − 0.138965i \(-0.0443775\pi\)
\(24\) −2.26759 −0.462869
\(25\) −2.82833 − 4.12317i −0.565666 − 0.824635i
\(26\) 2.50741 − 1.31599i 0.491743 − 0.258087i
\(27\) 1.93153 − 0.234530i 0.371723 − 0.0451353i
\(28\) 0.314518 − 0.119281i 0.0594383 − 0.0225420i
\(29\) 1.00783 − 1.46009i 0.187149 − 0.271132i −0.718167 − 0.695871i \(-0.755019\pi\)
0.905316 + 0.424739i \(0.139634\pi\)
\(30\) −1.20627 − 4.92490i −0.220235 − 0.899159i
\(31\) 3.73676 − 9.85303i 0.671142 − 1.76966i 0.0333555 − 0.999444i \(-0.489381\pi\)
0.637787 − 0.770213i \(-0.279850\pi\)
\(32\) 0.239316 − 0.970942i 0.0423054 − 0.171640i
\(33\) 3.69537 + 1.40147i 0.643281 + 0.243964i
\(34\) −7.03064 + 3.68996i −1.20574 + 0.632824i
\(35\) 0.426375 + 0.619639i 0.0720705 + 0.104738i
\(36\) 0.258183 − 2.12633i 0.0430305 − 0.354388i
\(37\) 1.95911 + 7.94843i 0.322076 + 1.30671i 0.879267 + 0.476330i \(0.158033\pi\)
−0.557191 + 0.830385i \(0.688121\pi\)
\(38\) −6.09162 − 0.739657i −0.988192 − 0.119988i
\(39\) 2.27702 + 6.00400i 0.364614 + 0.961409i
\(40\) 2.23607 + 0.00325578i 0.353553 + 0.000514784i
\(41\) −0.0431886 + 0.355690i −0.00674492 + 0.0555494i −0.995693 − 0.0927148i \(-0.970446\pi\)
0.988948 + 0.148264i \(0.0473686\pi\)
\(42\) 0.182541 + 0.740599i 0.0281667 + 0.114277i
\(43\) 11.2364 + 1.36435i 1.71354 + 0.208061i 0.917652 − 0.397385i \(-0.130082\pi\)
0.795886 + 0.605446i \(0.207005\pi\)
\(44\) −0.990086 + 1.43439i −0.149261 + 0.216242i
\(45\) 4.75545 − 0.570391i 0.708900 − 0.0850288i
\(46\) −0.472654 + 1.24629i −0.0696891 + 0.183755i
\(47\) −2.01135 + 8.16037i −0.293386 + 1.19031i 0.620456 + 0.784241i \(0.286948\pi\)
−0.913841 + 0.406071i \(0.866899\pi\)
\(48\) 2.12023 + 0.804097i 0.306029 + 0.116061i
\(49\) 3.91218 + 5.66777i 0.558882 + 0.809681i
\(50\) 1.18244 + 4.85817i 0.167222 + 0.687049i
\(51\) −6.38464 − 16.8349i −0.894028 − 2.35736i
\(52\) −2.81112 + 0.341332i −0.389832 + 0.0473342i
\(53\) −5.66643 − 10.7965i −0.778345 − 1.48301i −0.873116 − 0.487513i \(-0.837904\pi\)
0.0947715 − 0.995499i \(-0.469788\pi\)
\(54\) −1.88918 − 0.465640i −0.257084 − 0.0633656i
\(55\) −3.64199 − 1.38729i −0.491086 − 0.187063i
\(56\) −0.336377 −0.0449503
\(57\) 3.33001 − 13.5104i 0.441071 − 1.78950i
\(58\) −1.46009 + 1.00783i −0.191719 + 0.132334i
\(59\) −8.71230 + 7.71843i −1.13425 + 1.00485i −0.134342 + 0.990935i \(0.542892\pi\)
−0.999903 + 0.0139185i \(0.995569\pi\)
\(60\) −0.618507 + 5.03261i −0.0798489 + 0.649707i
\(61\) 8.10944 − 1.99880i 1.03831 − 0.255920i 0.316920 − 0.948452i \(-0.397351\pi\)
0.721387 + 0.692532i \(0.243505\pi\)
\(62\) −6.98787 + 7.88767i −0.887460 + 1.00174i
\(63\) −0.715248 + 0.0868468i −0.0901127 + 0.0109417i
\(64\) −0.568065 + 0.822984i −0.0710081 + 0.102873i
\(65\) −2.23674 − 5.92381i −0.277434 − 0.734758i
\(66\) −2.95826 − 2.62079i −0.364137 − 0.322597i
\(67\) −1.89573 + 0.718954i −0.231600 + 0.0878343i −0.467675 − 0.883901i \(-0.654908\pi\)
0.236075 + 0.971735i \(0.424139\pi\)
\(68\) 7.88224 − 0.957078i 0.955862 − 0.116063i
\(69\) −2.67627 − 1.40461i −0.322185 − 0.169096i
\(70\) −0.178941 − 0.730567i −0.0213875 − 0.0873195i
\(71\) −1.08990 + 0.572026i −0.129348 + 0.0678869i −0.528156 − 0.849147i \(-0.677116\pi\)
0.398808 + 0.917034i \(0.369424\pi\)
\(72\) −0.995411 + 1.89660i −0.117310 + 0.223516i
\(73\) −8.67999 − 9.79768i −1.01592 − 1.14673i −0.989123 − 0.147090i \(-0.953009\pi\)
−0.0267924 − 0.999641i \(-0.508529\pi\)
\(74\) 0.986751 − 8.12663i 0.114707 − 0.944701i
\(75\) −11.2592 + 1.33385i −1.30010 + 0.154020i
\(76\) 5.43348 + 2.85171i 0.623263 + 0.327114i
\(77\) 0.548177 + 0.207896i 0.0624706 + 0.0236920i
\(78\) − 6.42128i − 0.727066i
\(79\) 7.30931 + 5.05707i 0.822362 + 0.568965i
\(80\) −2.08960 − 0.795964i −0.233625 − 0.0889915i
\(81\) 3.84318 − 10.1336i 0.427019 − 1.12596i
\(82\) 0.166511 − 0.317261i 0.0183881 − 0.0350356i
\(83\) −4.24645 + 4.79325i −0.466109 + 0.526128i −0.933815 − 0.357756i \(-0.883542\pi\)
0.467706 + 0.883884i \(0.345080\pi\)
\(84\) 0.0919410 − 0.757202i 0.0100316 − 0.0826176i
\(85\) 6.27172 + 16.6101i 0.680263 + 1.80161i
\(86\) −10.0224 − 5.26018i −1.08075 − 0.567220i
\(87\) −1.86959 − 3.56220i −0.200441 − 0.381908i
\(88\) 1.43439 − 0.990086i 0.152906 − 0.105544i
\(89\) 3.23367 + 1.69716i 0.342769 + 0.179899i 0.627319 − 0.778762i \(-0.284152\pi\)
−0.284551 + 0.958661i \(0.591844\pi\)
\(90\) −4.64868 − 1.15298i −0.490014 − 0.121535i
\(91\) 0.337776 + 0.890642i 0.0354086 + 0.0933647i
\(92\) 0.883879 − 0.997693i 0.0921507 − 0.104017i
\(93\) −15.8456 − 17.8860i −1.64311 − 1.85469i
\(94\) 4.77435 − 6.91685i 0.492437 − 0.713418i
\(95\) −3.30312 + 13.3178i −0.338893 + 1.36638i
\(96\) −1.69731 − 1.50369i −0.173231 − 0.153469i
\(97\) −4.35954 − 17.6874i −0.442645 − 1.79588i −0.590689 − 0.806899i \(-0.701144\pi\)
0.148044 − 0.988981i \(-0.452702\pi\)
\(98\) −1.64813 − 6.68673i −0.166486 − 0.675462i
\(99\) 2.79435 − 2.47558i 0.280843 − 0.248805i
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 790.2.p.a.719.18 yes 480
5.4 even 2 inner 790.2.p.a.719.23 yes 480
79.10 even 13 inner 790.2.p.a.89.23 yes 480
395.89 even 26 inner 790.2.p.a.89.18 ✓ 480
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
790.2.p.a.89.18 ✓ 480 395.89 even 26 inner
790.2.p.a.89.23 yes 480 79.10 even 13 inner
790.2.p.a.719.18 yes 480 1.1 even 1 trivial
790.2.p.a.719.23 yes 480 5.4 even 2 inner