Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [5,18,Mod(4,5)] 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("5.4"); S:= CuspForms(chi, 18); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(5, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1])) N = Newforms(chi, 18, names="a")
 
Level: \( N \) \(=\) \( 5 \)
Weight: \( k \) \(=\) \( 18 \)
Character orbit: \([\chi]\) \(=\) 5.b (of order \(2\), degree \(1\), 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: \(9.16110436723\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} + \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 203459x^{6} + 12362849196x^{4} + 237701205446144x^{2} + 1320400799499206656 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{21}\cdot 3^{8}\cdot 5^{12}\cdot 11 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 4.2
Root \(-256.320i\) of defining polynomial
Character \(\chi\) \(=\) 5.4
Dual form 5.18.b.a.4.7

$q$-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-512.640i q^{2} +18245.5i q^{3} -131728. q^{4} +(346646. - 801733. i) q^{5} +9.35337e6 q^{6} -1.34806e7i q^{7} +336370. i q^{8} -2.03757e8 q^{9} +(-4.11001e8 - 1.77705e8i) q^{10} +8.20239e8 q^{11} -2.40344e9i q^{12} -2.92679e9i q^{13} -6.91067e9 q^{14} +(1.46280e10 + 6.32472e9i) q^{15} -1.70934e10 q^{16} -3.73919e10i q^{17} +1.04454e11i q^{18} +3.00949e10 q^{19} +(-4.56630e10 + 1.05611e11i) q^{20} +2.45959e11 q^{21} -4.20488e11i q^{22} +4.40887e11i q^{23} -6.13723e9 q^{24} +(-5.22613e11 - 5.55835e11i) q^{25} -1.50039e12 q^{26} -1.36143e12i q^{27} +1.77577e12i q^{28} +1.80846e12 q^{29} +(3.24231e12 - 7.49891e12i) q^{30} +2.90970e12 q^{31} +8.80687e12i q^{32} +1.49657e13i q^{33} -1.91686e13 q^{34} +(-1.08078e13 - 4.67297e12i) q^{35} +2.68406e13 q^{36} +3.31096e12i q^{37} -1.54279e13i q^{38} +5.34006e13 q^{39} +(2.69679e11 + 1.16601e11i) q^{40} -7.87183e13 q^{41} -1.26089e14i q^{42} +4.61355e12i q^{43} -1.08049e14 q^{44} +(-7.06316e13 + 1.63359e14i) q^{45} +2.26016e14 q^{46} +1.39204e14i q^{47} -3.11878e14i q^{48} +5.09053e13 q^{49} +(-2.84943e14 + 2.67913e14i) q^{50} +6.82233e14 q^{51} +3.85540e14i q^{52} +4.39747e13i q^{53} -6.97923e14 q^{54} +(2.84332e14 - 6.57613e14i) q^{55} +4.53445e12 q^{56} +5.49096e14i q^{57} -9.27089e14i q^{58} -3.18887e14 q^{59} +(-1.92692e15 - 8.33143e14i) q^{60} +7.17560e14 q^{61} -1.49163e15i q^{62} +2.74676e15i q^{63} +2.27429e15 q^{64} +(-2.34650e15 - 1.01456e15i) q^{65} +7.67200e15 q^{66} -2.87655e15i q^{67} +4.92556e15i q^{68} -8.04420e15 q^{69} +(-2.39556e15 + 5.54052e15i) q^{70} -3.39253e14 q^{71} -6.85378e13i q^{72} +8.49295e15i q^{73} +1.69733e15 q^{74} +(1.01415e16 - 9.53533e15i) q^{75} -3.96434e15 q^{76} -1.10573e16i q^{77} -2.73753e16i q^{78} +1.05825e16 q^{79} +(-5.92536e15 + 1.37044e16i) q^{80} -1.47335e15 q^{81} +4.03542e16i q^{82} -2.86866e16i q^{83} -3.23997e16 q^{84} +(-2.99783e16 - 1.29617e16i) q^{85} +2.36509e15 q^{86} +3.29962e16i q^{87} +2.75904e14i q^{88} +6.62578e16 q^{89} +(8.37445e16 + 3.62086e16i) q^{90} -3.94547e16 q^{91} -5.80772e16i q^{92} +5.30888e16i q^{93} +7.13616e16 q^{94} +(1.04323e16 - 2.41281e16i) q^{95} -1.60686e17 q^{96} +2.94405e16i q^{97} -2.60961e16i q^{98} -1.67130e17 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 579096 q^{4} + 379200 q^{5} + 357816 q^{6} - 234916344 q^{9} + 329570200 q^{10} + 463296576 q^{11} - 29937907992 q^{14} + 30646226400 q^{15} + 30848001568 q^{16} - 20615713280 q^{19} - 47558579400 q^{20}+ \cdots - 56\!\cdots\!68 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/5\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 512.640i − 1.41598i −0.706221 − 0.707991i \(-0.749602\pi\)
0.706221 − 0.707991i \(-0.250398\pi\)
\(3\) 18245.5i 1.60555i 0.596280 + 0.802777i \(0.296645\pi\)
−0.596280 + 0.802777i \(0.703355\pi\)
\(4\) −131728. −1.00501
\(5\) 346646. − 801733.i 0.396863 − 0.917878i
\(6\) 9.35337e6 2.27343
\(7\) − 1.34806e7i − 0.883841i −0.897054 − 0.441921i \(-0.854297\pi\)
0.897054 − 0.441921i \(-0.145703\pi\)
\(8\) 336370.i 0.00708846i
\(9\) −2.03757e8 −1.57780
\(10\) −4.11001e8 − 1.77705e8i −1.29970 − 0.561951i
\(11\) 8.20239e8 1.15373 0.576863 − 0.816841i \(-0.304277\pi\)
0.576863 + 0.816841i \(0.304277\pi\)
\(12\) − 2.40344e9i − 1.61359i
\(13\) − 2.92679e9i − 0.995114i −0.867431 − 0.497557i \(-0.834231\pi\)
0.867431 − 0.497557i \(-0.165769\pi\)
\(14\) −6.91067e9 −1.25150
\(15\) 1.46280e10 + 6.32472e9i 1.47370 + 0.637185i
\(16\) −1.70934e10 −0.994969
\(17\) − 3.73919e10i − 1.30005i −0.759911 − 0.650027i \(-0.774758\pi\)
0.759911 − 0.650027i \(-0.225242\pi\)
\(18\) 1.04454e11i 2.23414i
\(19\) 3.00949e10 0.406525 0.203262 − 0.979124i \(-0.434846\pi\)
0.203262 + 0.979124i \(0.434846\pi\)
\(20\) −4.56630e10 + 1.05611e11i −0.398850 + 0.922473i
\(21\) 2.45959e11 1.41905
\(22\) − 4.20488e11i − 1.63365i
\(23\) 4.40887e11i 1.17393i 0.809614 + 0.586963i \(0.199677\pi\)
−0.809614 + 0.586963i \(0.800323\pi\)
\(24\) −6.13723e9 −0.0113809
\(25\) −5.22613e11 − 5.55835e11i −0.684999 − 0.728544i
\(26\) −1.50039e12 −1.40906
\(27\) − 1.36143e12i − 0.927690i
\(28\) 1.77577e12i 0.888266i
\(29\) 1.80846e12 0.671314 0.335657 − 0.941984i \(-0.391042\pi\)
0.335657 + 0.941984i \(0.391042\pi\)
\(30\) 3.24231e12 − 7.49891e12i 0.902242 − 2.08674i
\(31\) 2.90970e12 0.612736 0.306368 − 0.951913i \(-0.400886\pi\)
0.306368 + 0.951913i \(0.400886\pi\)
\(32\) 8.80687e12i 1.41595i
\(33\) 1.49657e13i 1.85237i
\(34\) −1.91686e13 −1.84085
\(35\) −1.08078e13 − 4.67297e12i −0.811258 − 0.350764i
\(36\) 2.68406e13 1.58570
\(37\) 3.31096e12i 0.154967i 0.996994 + 0.0774834i \(0.0246885\pi\)
−0.996994 + 0.0774834i \(0.975312\pi\)
\(38\) − 1.54279e13i − 0.575632i
\(39\) 5.34006e13 1.59771
\(40\) 2.69679e11 + 1.16601e11i 0.00650634 + 0.00281315i
\(41\) −7.87183e13 −1.53962 −0.769809 − 0.638275i \(-0.779648\pi\)
−0.769809 + 0.638275i \(0.779648\pi\)
\(42\) − 1.26089e14i − 2.00936i
\(43\) 4.61355e12i 0.0601940i 0.999547 + 0.0300970i \(0.00958162\pi\)
−0.999547 + 0.0300970i \(0.990418\pi\)
\(44\) −1.08049e14 −1.15950
\(45\) −7.06316e13 + 1.63359e14i −0.626171 + 1.44823i
\(46\) 2.26016e14 1.66226
\(47\) 1.39204e14i 0.852746i 0.904547 + 0.426373i \(0.140209\pi\)
−0.904547 + 0.426373i \(0.859791\pi\)
\(48\) − 3.11878e14i − 1.59748i
\(49\) 5.09053e13 0.218824
\(50\) −2.84943e14 + 2.67913e14i −1.03160 + 0.969947i
\(51\) 6.82233e14 2.08731
\(52\) 3.85540e14i 1.00010i
\(53\) 4.39747e13i 0.0970194i 0.998823 + 0.0485097i \(0.0154472\pi\)
−0.998823 + 0.0485097i \(0.984553\pi\)
\(54\) −6.97923e14 −1.31359
\(55\) 2.84332e14 − 6.57613e14i 0.457871 − 1.05898i
\(56\) 4.53445e12 0.00626508
\(57\) 5.49096e14i 0.652697i
\(58\) − 9.27089e14i − 0.950569i
\(59\) −3.18887e14 −0.282745 −0.141372 − 0.989957i \(-0.545151\pi\)
−0.141372 + 0.989957i \(0.545151\pi\)
\(60\) −1.92692e15 − 8.33143e14i −1.48108 − 0.640375i
\(61\) 7.17560e14 0.479242 0.239621 − 0.970867i \(-0.422977\pi\)
0.239621 + 0.970867i \(0.422977\pi\)
\(62\) − 1.49163e15i − 0.867623i
\(63\) 2.74676e15i 1.39453i
\(64\) 2.27429e15 1.00999
\(65\) −2.34650e15 − 1.01456e15i −0.913393 − 0.394924i
\(66\) 7.67200e15 2.62292
\(67\) − 2.87655e15i − 0.865438i −0.901529 − 0.432719i \(-0.857554\pi\)
0.901529 − 0.432719i \(-0.142446\pi\)
\(68\) 4.92556e15i 1.30656i
\(69\) −8.04420e15 −1.88480
\(70\) −2.39556e15 + 5.54052e15i −0.496676 + 1.14873i
\(71\) −3.39253e14 −0.0623488 −0.0311744 − 0.999514i \(-0.509925\pi\)
−0.0311744 + 0.999514i \(0.509925\pi\)
\(72\) − 6.85378e13i − 0.0111842i
\(73\) 8.49295e15i 1.23258i 0.787520 + 0.616289i \(0.211365\pi\)
−0.787520 + 0.616289i \(0.788635\pi\)
\(74\) 1.69733e15 0.219430
\(75\) 1.01415e16 − 9.53533e15i 1.16972 − 1.09980i
\(76\) −3.96434e15 −0.408560
\(77\) − 1.10573e16i − 1.01971i
\(78\) − 2.73753e16i − 2.26233i
\(79\) 1.05825e16 0.784800 0.392400 − 0.919795i \(-0.371645\pi\)
0.392400 + 0.919795i \(0.371645\pi\)
\(80\) −5.92536e15 + 1.37044e16i −0.394866 + 0.913260i
\(81\) −1.47335e15 −0.0883454
\(82\) 4.03542e16i 2.18007i
\(83\) − 2.86866e16i − 1.39803i −0.715108 − 0.699014i \(-0.753623\pi\)
0.715108 − 0.699014i \(-0.246377\pi\)
\(84\) −3.23997e16 −1.42616
\(85\) −2.99783e16 − 1.29617e16i −1.19329 − 0.515943i
\(86\) 2.36509e15 0.0852337
\(87\) 3.29962e16i 1.07783i
\(88\) 2.75904e14i 0.00817814i
\(89\) 6.62578e16 1.78411 0.892055 − 0.451926i \(-0.149263\pi\)
0.892055 + 0.451926i \(0.149263\pi\)
\(90\) 8.37445e16 + 3.62086e16i 2.05067 + 0.886647i
\(91\) −3.94547e16 −0.879523
\(92\) − 5.80772e16i − 1.17980i
\(93\) 5.30888e16i 0.983780i
\(94\) 7.13616e16 1.20747
\(95\) 1.04323e16 − 2.41281e16i 0.161335 − 0.373140i
\(96\) −1.60686e17 −2.27338
\(97\) 2.94405e16i 0.381405i 0.981648 + 0.190702i \(0.0610765\pi\)
−0.981648 + 0.190702i \(0.938923\pi\)
\(98\) − 2.60961e16i − 0.309852i
\(99\) −1.67130e17 −1.82035
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 5.18.b.a.4.2 ✓ 8
3.2 odd 2 45.18.b.b.19.7 8
4.3 odd 2 80.18.c.b.49.1 8
5.2 odd 4 25.18.a.f.1.7 8
5.3 odd 4 25.18.a.f.1.2 8
5.4 even 2 inner 5.18.b.a.4.7 yes 8
15.14 odd 2 45.18.b.b.19.2 8
20.19 odd 2 80.18.c.b.49.8 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
5.18.b.a.4.2 ✓ 8 1.1 even 1 trivial
5.18.b.a.4.7 yes 8 5.4 even 2 inner
25.18.a.f.1.2 8 5.3 odd 4
25.18.a.f.1.7 8 5.2 odd 4
45.18.b.b.19.2 8 15.14 odd 2
45.18.b.b.19.7 8 3.2 odd 2
80.18.c.b.49.1 8 4.3 odd 2
80.18.c.b.49.8 8 20.19 odd 2