Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [465,2,Mod(119,465)] 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("465.119"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(465, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([3, 3, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 465 = 3 \cdot 5 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 465.t (of order \(6\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [104,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(3.71304369399\)
Analytic rank: \(0\)
Dimension: \(104\)
Relative dimension: \(52\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 119.1
Character \(\chi\) \(=\) 465.119
Dual form 465.2.t.d.254.1

$q$-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-2.66243 q^{2} +(0.290440 + 1.70753i) q^{3} +5.08855 q^{4} +(0.356449 + 2.20747i) q^{5} +(-0.773277 - 4.54617i) q^{6} +(0.379861 + 0.219313i) q^{7} -8.22305 q^{8} +(-2.83129 + 0.991867i) q^{9} +(-0.949020 - 5.87725i) q^{10} +(-2.47410 - 4.28527i) q^{11} +(1.47792 + 8.68883i) q^{12} +(-3.16669 - 5.48487i) q^{13} +(-1.01135 - 0.583906i) q^{14} +(-3.66579 + 1.24978i) q^{15} +11.7162 q^{16} +(-4.37419 - 2.52544i) q^{17} +(7.53812 - 2.64078i) q^{18} +(-0.872311 + 1.51089i) q^{19} +(1.81381 + 11.2328i) q^{20} +(-0.264156 + 0.712320i) q^{21} +(6.58713 + 11.4092i) q^{22} +0.000828308i q^{23} +(-2.38830 - 14.0411i) q^{24} +(-4.74589 + 1.57370i) q^{25} +(8.43111 + 14.6031i) q^{26} +(-2.51596 - 4.54642i) q^{27} +(1.93294 + 1.11598i) q^{28} +7.21938 q^{29} +(9.75993 - 3.32747i) q^{30} +(-5.53613 + 0.592659i) q^{31} -14.7475 q^{32} +(6.59863 - 5.46921i) q^{33} +(11.6460 + 6.72382i) q^{34} +(-0.348727 + 0.916708i) q^{35} +(-14.4071 + 5.04716i) q^{36} +(-1.82699 + 3.16444i) q^{37} +(2.32247 - 4.02263i) q^{38} +(8.44583 - 7.00024i) q^{39} +(-2.93109 - 18.1522i) q^{40} +(-4.57160 + 2.63942i) q^{41} +(0.703297 - 1.89650i) q^{42} +(-0.557302 + 0.965275i) q^{43} +(-12.5896 - 21.8058i) q^{44} +(-3.19873 - 5.89645i) q^{45} -0.00220531i q^{46} +4.32268 q^{47} +(3.40286 + 20.0057i) q^{48} +(-3.40380 - 5.89556i) q^{49} +(12.6356 - 4.18988i) q^{50} +(3.04182 - 8.20254i) q^{51} +(-16.1139 - 27.9100i) q^{52} +(-0.205473 + 0.118630i) q^{53} +(6.69857 + 12.1045i) q^{54} +(8.57774 - 6.98900i) q^{55} +(-3.12362 - 1.80342i) q^{56} +(-2.83323 - 1.05067i) q^{57} -19.2211 q^{58} +(-3.46839 - 2.00247i) q^{59} +(-18.6536 + 6.35958i) q^{60} +6.12726i q^{61} +(14.7396 - 1.57792i) q^{62} +(-1.29303 - 0.244166i) q^{63} +15.8319 q^{64} +(10.9790 - 8.94547i) q^{65} +(-17.5684 + 14.5614i) q^{66} +(-6.76334 + 3.90481i) q^{67} +(-22.2583 - 12.8508i) q^{68} +(-0.00141436 + 0.000240574i) q^{69} +(0.928461 - 2.44067i) q^{70} +(-12.0026 + 6.92969i) q^{71} +(23.2818 - 8.15617i) q^{72} +(1.51325 + 2.62103i) q^{73} +(4.86424 - 8.42510i) q^{74} +(-4.06553 - 7.64666i) q^{75} +(-4.43879 + 7.68822i) q^{76} -2.17041i q^{77} +(-22.4864 + 18.6377i) q^{78} +(0.0399067 + 0.0230401i) q^{79} +(4.17623 + 25.8632i) q^{80} +(7.03240 - 5.61653i) q^{81} +(12.1716 - 7.02727i) q^{82} +(9.70239 - 5.60168i) q^{83} +(-1.34417 + 3.62467i) q^{84} +(4.01567 - 10.5561i) q^{85} +(1.48378 - 2.56998i) q^{86} +(2.09680 + 12.3273i) q^{87} +(20.3447 + 35.2380i) q^{88} +3.39765 q^{89} +(8.51641 + 15.6989i) q^{90} -2.77799i q^{91} +0.00421488i q^{92} +(-2.61989 - 9.28096i) q^{93} -11.5089 q^{94} +(-3.64618 - 1.38705i) q^{95} +(-4.28327 - 25.1818i) q^{96} +8.40856i q^{97} +(9.06240 + 15.6965i) q^{98} +(11.2553 + 9.67886i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 104 q + 56 q^{4} - 12 q^{6} - 2 q^{9} - 20 q^{10} - 24 q^{16} + 12 q^{19} - 6 q^{21} - 48 q^{24} + 8 q^{25} - 40 q^{31} + 108 q^{34} + 36 q^{36} + 52 q^{39} - 76 q^{40} - 4 q^{45} + 80 q^{49} - 20 q^{51}+ \cdots + 66 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/465\mathbb{Z}\right)^\times\).

\(n\) \(187\) \(311\) \(406\)
\(\chi(n)\) \(-1\) \(-1\) \(e\left(\frac{1}{6}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.66243 −1.88262 −0.941312 − 0.337538i \(-0.890406\pi\)
−0.941312 + 0.337538i \(0.890406\pi\)
\(3\) 0.290440 + 1.70753i 0.167686 + 0.985841i
\(4\) 5.08855 2.54427
\(5\) 0.356449 + 2.20747i 0.159409 + 0.987213i
\(6\) −0.773277 − 4.54617i −0.315689 − 1.85597i
\(7\) 0.379861 + 0.219313i 0.143574 + 0.0828925i 0.570066 − 0.821599i \(-0.306918\pi\)
−0.426492 + 0.904491i \(0.640251\pi\)
\(8\) −8.22305 −2.90729
\(9\) −2.83129 + 0.991867i −0.943763 + 0.330622i
\(10\) −0.949020 − 5.87725i −0.300107 − 1.85855i
\(11\) −2.47410 − 4.28527i −0.745970 − 1.29206i −0.949740 − 0.313039i \(-0.898653\pi\)
0.203770 − 0.979019i \(-0.434680\pi\)
\(12\) 1.47792 + 8.68883i 0.426638 + 2.50825i
\(13\) −3.16669 − 5.48487i −0.878283 − 1.52123i −0.853224 − 0.521544i \(-0.825356\pi\)
−0.0250583 − 0.999686i \(-0.507977\pi\)
\(14\) −1.01135 − 0.583906i −0.270296 − 0.156055i
\(15\) −3.66579 + 1.24978i −0.946504 + 0.322693i
\(16\) 11.7162 2.92905
\(17\) −4.37419 − 2.52544i −1.06090 − 0.612510i −0.135217 − 0.990816i \(-0.543173\pi\)
−0.925681 + 0.378306i \(0.876507\pi\)
\(18\) 7.53812 − 2.64078i 1.77675 − 0.622438i
\(19\) −0.872311 + 1.51089i −0.200122 + 0.346621i −0.948568 − 0.316575i \(-0.897467\pi\)
0.748446 + 0.663196i \(0.230800\pi\)
\(20\) 1.81381 + 11.2328i 0.405579 + 2.51174i
\(21\) −0.264156 + 0.712320i −0.0576435 + 0.155441i
\(22\) 6.58713 + 11.4092i 1.40438 + 2.43246i
\(23\) 0 0.000828308i 0 0.000172714i 1.00000 8.63571e-5i \(2.74883e-5\pi\)
−1.00000 8.63571e-5i \(0.999973\pi\)
\(24\) −2.38830 − 14.0411i −0.487510 − 2.86612i
\(25\) −4.74589 + 1.57370i −0.949178 + 0.314741i
\(26\) 8.43111 + 14.6031i 1.65348 + 2.86390i
\(27\) −2.51596 − 4.54642i −0.484196 − 0.874959i
\(28\) 1.93294 + 1.11598i 0.365292 + 0.210901i
\(29\) 7.21938 1.34061 0.670303 − 0.742088i \(-0.266164\pi\)
0.670303 + 0.742088i \(0.266164\pi\)
\(30\) 9.75993 − 3.32747i 1.78191 − 0.607509i
\(31\) −5.53613 + 0.592659i −0.994319 + 0.106445i
\(32\) −14.7475 −2.60702
\(33\) 6.59863 − 5.46921i 1.14867 − 0.952067i
\(34\) 11.6460 + 6.72382i 1.99727 + 1.15313i
\(35\) −0.348727 + 0.916708i −0.0589456 + 0.154952i
\(36\) −14.4071 + 5.04716i −2.40119 + 0.841194i
\(37\) −1.82699 + 3.16444i −0.300355 + 0.520230i −0.976216 − 0.216798i \(-0.930439\pi\)
0.675861 + 0.737029i \(0.263772\pi\)
\(38\) 2.32247 − 4.02263i 0.376754 − 0.652557i
\(39\) 8.44583 − 7.00024i 1.35241 − 1.12093i
\(40\) −2.93109 − 18.1522i −0.463447 − 2.87011i
\(41\) −4.57160 + 2.63942i −0.713965 + 0.412208i −0.812527 − 0.582923i \(-0.801909\pi\)
0.0985627 + 0.995131i \(0.468576\pi\)
\(42\) 0.703297 − 1.89650i 0.108521 − 0.292637i
\(43\) −0.557302 + 0.965275i −0.0849877 + 0.147203i −0.905386 − 0.424589i \(-0.860418\pi\)
0.820398 + 0.571793i \(0.193752\pi\)
\(44\) −12.5896 − 21.8058i −1.89795 − 3.28735i
\(45\) −3.19873 − 5.89645i −0.476839 − 0.878991i
\(46\) − 0.00220531i − 0.000325156i
\(47\) 4.32268 0.630528 0.315264 − 0.949004i \(-0.397907\pi\)
0.315264 + 0.949004i \(0.397907\pi\)
\(48\) 3.40286 + 20.0057i 0.491160 + 2.88758i
\(49\) −3.40380 − 5.89556i −0.486258 − 0.842223i
\(50\) 12.6356 − 4.18988i 1.78694 − 0.592538i
\(51\) 3.04182 − 8.20254i 0.425940 − 1.14859i
\(52\) −16.1139 − 27.9100i −2.23459 − 3.87043i
\(53\) −0.205473 + 0.118630i −0.0282239 + 0.0162951i −0.514046 − 0.857763i \(-0.671854\pi\)
0.485822 + 0.874058i \(0.338520\pi\)
\(54\) 6.69857 + 12.1045i 0.911560 + 1.64722i
\(55\) 8.57774 − 6.98900i 1.15662 − 0.942396i
\(56\) −3.12362 − 1.80342i −0.417411 − 0.240992i
\(57\) −2.83323 − 1.05067i −0.375271 − 0.139165i
\(58\) −19.2211 −2.52386
\(59\) −3.46839 − 2.00247i −0.451546 − 0.260700i 0.256937 − 0.966428i \(-0.417287\pi\)
−0.708483 + 0.705728i \(0.750620\pi\)
\(60\) −18.6536 + 6.35958i −2.40816 + 0.821019i
\(61\) 6.12726i 0.784516i 0.919855 + 0.392258i \(0.128306\pi\)
−0.919855 + 0.392258i \(0.871694\pi\)
\(62\) 14.7396 − 1.57792i 1.87193 − 0.200395i
\(63\) −1.29303 − 0.244166i −0.162906 − 0.0307621i
\(64\) 15.8319 1.97899
\(65\) 10.9790 − 8.94547i 1.36177 − 1.10955i
\(66\) −17.5684 + 14.5614i −2.16252 + 1.79238i
\(67\) −6.76334 + 3.90481i −0.826273 + 0.477049i −0.852575 − 0.522605i \(-0.824960\pi\)
0.0263018 + 0.999654i \(0.491627\pi\)
\(68\) −22.2583 − 12.8508i −2.69921 − 1.55839i
\(69\) −0.00141436 0.000240574i −0.000170269 2.89617e-5i
\(70\) 0.928461 − 2.44067i 0.110972 − 0.291716i
\(71\) −12.0026 + 6.92969i −1.42444 + 0.822403i −0.996675 − 0.0814841i \(-0.974034\pi\)
−0.427770 + 0.903888i \(0.640701\pi\)
\(72\) 23.2818 − 8.15617i 2.74379 − 0.961214i
\(73\) 1.51325 + 2.62103i 0.177113 + 0.306769i 0.940890 − 0.338711i \(-0.109991\pi\)
−0.763777 + 0.645480i \(0.776658\pi\)
\(74\) 4.86424 − 8.42510i 0.565456 − 0.979398i
\(75\) −4.06553 − 7.64666i −0.469447 − 0.882960i
\(76\) −4.43879 + 7.68822i −0.509165 + 0.881899i
\(77\) − 2.17041i − 0.247341i
\(78\) −22.4864 + 18.6377i −2.54609 + 2.11030i
\(79\) 0.0399067 + 0.0230401i 0.00448985 + 0.00259222i 0.502243 − 0.864726i \(-0.332508\pi\)
−0.497753 + 0.867319i \(0.665842\pi\)
\(80\) 4.17623 + 25.8632i 0.466916 + 2.89160i
\(81\) 7.03240 − 5.61653i 0.781378 − 0.624059i
\(82\) 12.1716 − 7.02727i 1.34413 − 0.776032i
\(83\) 9.70239 − 5.60168i 1.06498 − 0.614864i 0.138171 − 0.990408i \(-0.455877\pi\)
0.926805 + 0.375544i \(0.122544\pi\)
\(84\) −1.34417 + 3.62467i −0.146661 + 0.395484i
\(85\) 4.01567 − 10.5561i 0.435561 − 1.14497i
\(86\) 1.48378 − 2.56998i 0.160000 − 0.277128i
\(87\) 2.09680 + 12.3273i 0.224800 + 1.32162i
\(88\) 20.3447 + 35.2380i 2.16875 + 3.75638i
\(89\) 3.39765 0.360150 0.180075 − 0.983653i \(-0.442366\pi\)
0.180075 + 0.983653i \(0.442366\pi\)
\(90\) 8.51641 + 15.6989i 0.897708 + 1.65481i
\(91\) − 2.77799i − 0.291212i
\(92\) 0.00421488i 0.000439432i
\(93\) −2.61989 − 9.28096i −0.271670 − 0.962390i
\(94\) −11.5089 −1.18705
\(95\) −3.64618 − 1.38705i −0.374090 − 0.142308i
\(96\) −4.28327 − 25.1818i −0.437160 − 2.57011i
\(97\) 8.40856i 0.853760i 0.904308 + 0.426880i \(0.140387\pi\)
−0.904308 + 0.426880i \(0.859613\pi\)
\(98\) 9.06240 + 15.6965i 0.915440 + 1.58559i
\(99\) 11.2553 + 9.67886i 1.13120 + 0.972762i
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 465.2.t.d.119.1 ✓ 104
3.2 odd 2 inner 465.2.t.d.119.51 yes 104
5.4 even 2 inner 465.2.t.d.119.52 yes 104
15.14 odd 2 inner 465.2.t.d.119.2 yes 104
31.6 odd 6 inner 465.2.t.d.254.2 yes 104
93.68 even 6 inner 465.2.t.d.254.52 yes 104
155.99 odd 6 inner 465.2.t.d.254.51 yes 104
465.254 even 6 inner 465.2.t.d.254.1 yes 104
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
465.2.t.d.119.1 ✓ 104 1.1 even 1 trivial
465.2.t.d.119.2 yes 104 15.14 odd 2 inner
465.2.t.d.119.51 yes 104 3.2 odd 2 inner
465.2.t.d.119.52 yes 104 5.4 even 2 inner
465.2.t.d.254.1 yes 104 465.254 even 6 inner
465.2.t.d.254.2 yes 104 31.6 odd 6 inner
465.2.t.d.254.51 yes 104 155.99 odd 6 inner
465.2.t.d.254.52 yes 104 93.68 even 6 inner