Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [15,11,Mod(7,15)] 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("15.7"); S:= CuspForms(chi, 11); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(15, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 11, names="a")
 
Level: \( N \) \(=\) \( 15 = 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 11 \)
Character orbit: \([\chi]\) \(=\) 15.f (of order \(4\), degree \(2\), 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: \(9.53035879011\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(10\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} - 4 x^{19} + 8 x^{18} - 49316 x^{17} + 18276332 x^{16} - 230627572 x^{15} + 1992333560 x^{14} + \cdots + 14\!\cdots\!04 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{21}\cdot 3^{34}\cdot 5^{14} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 13.3
Root \(23.8698 + 23.8698i\) of defining polynomial
Character \(\chi\) \(=\) 15.13
Dual form 15.11.f.a.7.3

$q$-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+(-28.0945 + 28.0945i) q^{2} +(99.2043 + 99.2043i) q^{3} -554.607i q^{4} +(2550.39 - 1805.86i) q^{5} -5574.20 q^{6} +(20119.5 - 20119.5i) q^{7} +(-13187.4 - 13187.4i) q^{8} +19683.0i q^{9} +(-20917.3 + 122387. i) q^{10} +109313. q^{11} +(55019.5 - 55019.5i) q^{12} +(255406. + 255406. i) q^{13} +1.13049e6i q^{14} +(432159. + 73860.9i) q^{15} +1.30890e6 q^{16} +(-1.32676e6 + 1.32676e6i) q^{17} +(-552985. - 552985. i) q^{18} -3.95975e6i q^{19} +(-1.00154e6 - 1.41447e6i) q^{20} +3.99188e6 q^{21} +(-3.07110e6 + 3.07110e6i) q^{22} +(1.52260e6 + 1.52260e6i) q^{23} -2.61649e6i q^{24} +(3.24337e6 - 9.21130e6i) q^{25} -1.43511e7 q^{26} +(-1.95264e6 + 1.95264e6i) q^{27} +(-1.11584e7 - 1.11584e7i) q^{28} +1.53114e7i q^{29} +(-1.42164e7 + 1.00662e7i) q^{30} +2.75563e7 q^{31} +(-2.32692e7 + 2.32692e7i) q^{32} +(1.08443e7 + 1.08443e7i) q^{33} -7.45492e7i q^{34} +(1.49796e7 - 8.76454e7i) q^{35} +1.09163e7 q^{36} +(-1.67469e7 + 1.67469e7i) q^{37} +(1.11247e8 + 1.11247e8i) q^{38} +5.06749e7i q^{39} +(-5.74475e7 - 9.81843e6i) q^{40} +1.16167e8 q^{41} +(-1.12150e8 + 1.12150e8i) q^{42} +(7.69664e7 + 7.69664e7i) q^{43} -6.06259e7i q^{44} +(3.55447e7 + 5.01994e7i) q^{45} -8.55535e7 q^{46} +(-1.29633e8 + 1.29633e8i) q^{47} +(1.29849e8 + 1.29849e8i) q^{48} -5.27111e8i q^{49} +(1.67666e8 + 3.49908e8i) q^{50} -2.63240e8 q^{51} +(1.41650e8 - 1.41650e8i) q^{52} +(-3.99810e8 - 3.99810e8i) q^{53} -1.09717e8i q^{54} +(2.78791e8 - 1.97404e8i) q^{55} -5.30646e8 q^{56} +(3.92825e8 - 3.92825e8i) q^{57} +(-4.30166e8 - 4.30166e8i) q^{58} -1.93818e8i q^{59} +(4.09638e7 - 2.39679e8i) q^{60} -4.41801e7 q^{61} +(-7.74183e8 + 7.74183e8i) q^{62} +(3.96011e8 + 3.96011e8i) q^{63} +3.28421e7i q^{64} +(1.11261e9 + 1.90158e8i) q^{65} -6.09333e8 q^{66} +(-5.17381e8 + 5.17381e8i) q^{67} +(7.35828e8 + 7.35828e8i) q^{68} +3.02097e8i q^{69} +(2.04151e9 + 2.88320e9i) q^{70} -1.06591e9 q^{71} +(2.59567e8 - 2.59567e8i) q^{72} +(1.19631e9 + 1.19631e9i) q^{73} -9.40993e8i q^{74} +(1.23556e9 - 5.92044e8i) q^{75} -2.19611e9 q^{76} +(2.19932e9 - 2.19932e9i) q^{77} +(-1.42369e9 - 1.42369e9i) q^{78} -1.89420e9i q^{79} +(3.33822e9 - 2.36370e9i) q^{80} -3.87420e8 q^{81} +(-3.26365e9 + 3.26365e9i) q^{82} +(-3.98203e8 - 3.98203e8i) q^{83} -2.21392e9i q^{84} +(-9.87813e8 + 5.77968e9i) q^{85} -4.32467e9 q^{86} +(-1.51895e9 + 1.51895e9i) q^{87} +(-1.44155e9 - 1.44155e9i) q^{88} -3.69253e9i q^{89} +(-2.40894e9 - 4.11715e8i) q^{90} +1.02773e10 q^{91} +(8.44445e8 - 8.44445e8i) q^{92} +(2.73371e9 + 2.73371e9i) q^{93} -7.28397e9i q^{94} +(-7.15076e9 - 1.00989e10i) q^{95} -4.61681e9 q^{96} +(-1.18887e10 + 1.18887e10i) q^{97} +(1.48089e10 + 1.48089e10i) q^{98} +2.15161e9i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - 64 q^{2} + 10676 q^{5} - 4860 q^{6} + 10604 q^{7} - 39948 q^{8} + 153704 q^{10} + 32080 q^{11} - 620136 q^{12} - 69352 q^{13} + 662904 q^{15} - 3111700 q^{16} + 347360 q^{17} - 1259712 q^{18} + 24132564 q^{20}+ \cdots + 54432471592 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/15\mathbb{Z}\right)^\times\).

\(n\) \(7\) \(11\)
\(\chi(n)\) \(e\left(\frac{3}{4}\right)\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −28.0945 + 28.0945i −0.877955 + 0.877955i −0.993323 − 0.115368i \(-0.963195\pi\)
0.115368 + 0.993323i \(0.463195\pi\)
\(3\) 99.2043 + 99.2043i 0.408248 + 0.408248i
\(4\) − 554.607i − 0.541609i
\(5\) 2550.39 − 1805.86i 0.816125 − 0.577875i
\(6\) −5574.20 −0.716847
\(7\) 20119.5 − 20119.5i 1.19709 − 1.19709i 0.222054 − 0.975034i \(-0.428724\pi\)
0.975034 − 0.222054i \(-0.0712762\pi\)
\(8\) −13187.4 − 13187.4i −0.402447 − 0.402447i
\(9\) 19683.0i 0.333333i
\(10\) −20917.3 + 122387.i −0.209173 + 1.22387i
\(11\) 109313. 0.678748 0.339374 − 0.940651i \(-0.389785\pi\)
0.339374 + 0.940651i \(0.389785\pi\)
\(12\) 55019.5 − 55019.5i 0.221111 − 0.221111i
\(13\) 255406. + 255406.i 0.687884 + 0.687884i 0.961764 − 0.273880i \(-0.0883071\pi\)
−0.273880 + 0.961764i \(0.588307\pi\)
\(14\) 1.13049e6i 2.10198i
\(15\) 432159. + 73860.9i 0.569098 + 0.0972653i
\(16\) 1.30890e6 1.24827
\(17\) −1.32676e6 + 1.32676e6i −0.934429 + 0.934429i −0.997979 − 0.0635497i \(-0.979758\pi\)
0.0635497 + 0.997979i \(0.479758\pi\)
\(18\) −552985. − 552985.i −0.292652 − 0.292652i
\(19\) − 3.95975e6i − 1.59919i −0.600540 − 0.799595i \(-0.705047\pi\)
0.600540 − 0.799595i \(-0.294953\pi\)
\(20\) −1.00154e6 − 1.41447e6i −0.312982 − 0.442021i
\(21\) 3.99188e6 0.977419
\(22\) −3.07110e6 + 3.07110e6i −0.595910 + 0.595910i
\(23\) 1.52260e6 + 1.52260e6i 0.236563 + 0.236563i 0.815425 − 0.578862i \(-0.196503\pi\)
−0.578862 + 0.815425i \(0.696503\pi\)
\(24\) − 2.61649e6i − 0.328596i
\(25\) 3.24337e6 − 9.21130e6i 0.332121 − 0.943237i
\(26\) −1.43511e7 −1.20786
\(27\) −1.95264e6 + 1.95264e6i −0.136083 + 0.136083i
\(28\) −1.11584e7 − 1.11584e7i −0.648354 − 0.648354i
\(29\) 1.53114e7i 0.746490i 0.927733 + 0.373245i \(0.121755\pi\)
−0.927733 + 0.373245i \(0.878245\pi\)
\(30\) −1.42164e7 + 1.00662e7i −0.585037 + 0.414248i
\(31\) 2.75563e7 0.962527 0.481264 − 0.876576i \(-0.340178\pi\)
0.481264 + 0.876576i \(0.340178\pi\)
\(32\) −2.32692e7 + 2.32692e7i −0.693477 + 0.693477i
\(33\) 1.08443e7 + 1.08443e7i 0.277098 + 0.277098i
\(34\) − 7.45492e7i − 1.64077i
\(35\) 1.49796e7 − 8.76454e7i 0.285207 − 1.66874i
\(36\) 1.09163e7 0.180536
\(37\) −1.67469e7 + 1.67469e7i −0.241505 + 0.241505i −0.817472 − 0.575968i \(-0.804625\pi\)
0.575968 + 0.817472i \(0.304625\pi\)
\(38\) 1.11247e8 + 1.11247e8i 1.40402 + 1.40402i
\(39\) 5.06749e7i 0.561655i
\(40\) −5.74475e7 − 9.81843e6i −0.561011 − 0.0958831i
\(41\) 1.16167e8 1.00268 0.501340 − 0.865250i \(-0.332841\pi\)
0.501340 + 0.865250i \(0.332841\pi\)
\(42\) −1.12150e8 + 1.12150e8i −0.858129 + 0.858129i
\(43\) 7.69664e7 + 7.69664e7i 0.523551 + 0.523551i 0.918642 − 0.395091i \(-0.129287\pi\)
−0.395091 + 0.918642i \(0.629287\pi\)
\(44\) − 6.06259e7i − 0.367616i
\(45\) 3.55447e7 + 5.01994e7i 0.192625 + 0.272042i
\(46\) −8.55535e7 −0.415383
\(47\) −1.29633e8 + 1.29633e8i −0.565232 + 0.565232i −0.930789 − 0.365557i \(-0.880879\pi\)
0.365557 + 0.930789i \(0.380879\pi\)
\(48\) 1.29849e8 + 1.29849e8i 0.509604 + 0.509604i
\(49\) − 5.27111e8i − 1.86604i
\(50\) 1.67666e8 + 3.49908e8i 0.536532 + 1.11971i
\(51\) −2.63240e8 −0.762958
\(52\) 1.41650e8 − 1.41650e8i 0.372564 − 0.372564i
\(53\) −3.99810e8 − 3.99810e8i −0.956036 − 0.956036i 0.0430371 − 0.999073i \(-0.486297\pi\)
−0.999073 + 0.0430371i \(0.986297\pi\)
\(54\) − 1.09717e8i − 0.238949i
\(55\) 2.78791e8 − 1.97404e8i 0.553944 − 0.392232i
\(56\) −5.30646e8 −0.963529
\(57\) 3.92825e8 − 3.92825e8i 0.652867 − 0.652867i
\(58\) −4.30166e8 − 4.30166e8i −0.655384 − 0.655384i
\(59\) − 1.93818e8i − 0.271103i −0.990770 − 0.135551i \(-0.956719\pi\)
0.990770 − 0.135551i \(-0.0432806\pi\)
\(60\) 4.09638e7 − 2.39679e8i 0.0526798 − 0.308229i
\(61\) −4.41801e7 −0.0523091 −0.0261546 − 0.999658i \(-0.508326\pi\)
−0.0261546 + 0.999658i \(0.508326\pi\)
\(62\) −7.74183e8 + 7.74183e8i −0.845055 + 0.845055i
\(63\) 3.96011e8 + 3.96011e8i 0.399029 + 0.399029i
\(64\) 3.28421e7i 0.0305866i
\(65\) 1.11261e9 + 1.90158e8i 0.958910 + 0.163889i
\(66\) −6.09333e8 −0.486559
\(67\) −5.17381e8 + 5.17381e8i −0.383210 + 0.383210i −0.872257 − 0.489048i \(-0.837344\pi\)
0.489048 + 0.872257i \(0.337344\pi\)
\(68\) 7.35828e8 + 7.35828e8i 0.506095 + 0.506095i
\(69\) 3.02097e8i 0.193153i
\(70\) 2.04151e9 + 2.88320e9i 1.21468 + 1.71548i
\(71\) −1.06591e9 −0.590782 −0.295391 − 0.955376i \(-0.595450\pi\)
−0.295391 + 0.955376i \(0.595450\pi\)
\(72\) 2.59567e8 − 2.59567e8i 0.134149 − 0.134149i
\(73\) 1.19631e9 + 1.19631e9i 0.577071 + 0.577071i 0.934095 − 0.357024i \(-0.116209\pi\)
−0.357024 + 0.934095i \(0.616209\pi\)
\(74\) − 9.40993e8i − 0.424060i
\(75\) 1.23556e9 − 5.92044e8i 0.520663 − 0.249487i
\(76\) −2.19611e9 −0.866135
\(77\) 2.19932e9 − 2.19932e9i 0.812522 − 0.812522i
\(78\) −1.42369e9 − 1.42369e9i −0.493107 − 0.493107i
\(79\) − 1.89420e9i − 0.615588i −0.951453 − 0.307794i \(-0.900409\pi\)
0.951453 − 0.307794i \(-0.0995908\pi\)
\(80\) 3.33822e9 − 2.36370e9i 1.01874 − 0.721343i
\(81\) −3.87420e8 −0.111111
\(82\) −3.26365e9 + 3.26365e9i −0.880308 + 0.880308i
\(83\) −3.98203e8 − 3.98203e8i −0.101091 − 0.101091i 0.654752 − 0.755844i \(-0.272773\pi\)
−0.755844 + 0.654752i \(0.772773\pi\)
\(84\) − 2.21392e9i − 0.529379i
\(85\) −9.87813e8 + 5.77968e9i −0.222628 + 1.30259i
\(86\) −4.32467e9 −0.919308
\(87\) −1.51895e9 + 1.51895e9i −0.304753 + 0.304753i
\(88\) −1.44155e9 − 1.44155e9i −0.273160 − 0.273160i
\(89\) − 3.69253e9i − 0.661263i −0.943760 − 0.330632i \(-0.892738\pi\)
0.943760 − 0.330632i \(-0.107262\pi\)
\(90\) −2.40894e9 − 4.11715e8i −0.407956 − 0.0697244i
\(91\) 1.02773e10 1.64692
\(92\) 8.44445e8 − 8.44445e8i 0.128125 − 0.128125i
\(93\) 2.73371e9 + 2.73371e9i 0.392950 + 0.392950i
\(94\) − 7.28397e9i − 0.992496i
\(95\) −7.15076e9 − 1.00989e10i −0.924132 − 1.30514i
\(96\) −4.61681e9 −0.566221
\(97\) −1.18887e10 + 1.18887e10i −1.38445 + 1.38445i −0.547904 + 0.836541i \(0.684574\pi\)
−0.836541 + 0.547904i \(0.815426\pi\)
\(98\) 1.48089e10 + 1.48089e10i 1.63830 + 1.63830i
\(99\) 2.15161e9i 0.226249i
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 15.11.f.a.13.3 yes 20
3.2 odd 2 45.11.g.c.28.8 20
5.2 odd 4 inner 15.11.f.a.7.3 ✓ 20
5.3 odd 4 75.11.f.d.7.8 20
5.4 even 2 75.11.f.d.43.8 20
15.2 even 4 45.11.g.c.37.8 20
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
15.11.f.a.7.3 ✓ 20 5.2 odd 4 inner
15.11.f.a.13.3 yes 20 1.1 even 1 trivial
45.11.g.c.28.8 20 3.2 odd 2
45.11.g.c.37.8 20 15.2 even 4
75.11.f.d.7.8 20 5.3 odd 4
75.11.f.d.43.8 20 5.4 even 2