Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [5,22,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, 22); 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, 22, names="a")
 
Level: \( N \) \(=\) \( 5 \)
Weight: \( k \) \(=\) \( 22 \)
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: \(13.9738672144\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} + \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} + 3780655 x^{8} + 4653816871660 x^{6} + \cdots + 14\!\cdots\!76 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{30}\cdot 3^{10}\cdot 5^{19}\cdot 7^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 4.2
Root \(-934.563i\) of defining polynomial
Character \(\chi\) \(=\) 5.4
Dual form 5.22.b.a.4.9

$q$-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-1869.13i q^{2} -163448. i q^{3} -1.39648e6 q^{4} +(-1.04751e7 + 1.91601e7i) q^{5} -3.05505e8 q^{6} +1.05403e9i q^{7} -1.30964e9i q^{8} -1.62549e10 q^{9} +(3.58126e10 + 1.95794e10i) q^{10} -7.20273e10 q^{11} +2.28252e11i q^{12} -3.87633e11i q^{13} +1.97011e12 q^{14} +(3.13168e12 + 1.71214e12i) q^{15} -5.37652e12 q^{16} +9.31350e12i q^{17} +3.03825e13i q^{18} -2.10301e13 q^{19} +(1.46283e13 - 2.67567e13i) q^{20} +1.72279e14 q^{21} +1.34628e14i q^{22} -2.30102e14i q^{23} -2.14059e14 q^{24} +(-2.57380e14 - 4.01409e14i) q^{25} -7.24535e14 q^{26} +9.47115e14i q^{27} -1.47193e15i q^{28} -4.35009e14 q^{29} +(3.20021e15 - 5.85350e15i) q^{30} -2.68314e15 q^{31} +7.30288e15i q^{32} +1.17727e16i q^{33} +1.74081e16 q^{34} +(-2.01952e16 - 1.10411e16i) q^{35} +2.26997e16 q^{36} +1.95520e16i q^{37} +3.93080e16i q^{38} -6.33579e16 q^{39} +(2.50929e16 + 1.37187e16i) q^{40} +9.77147e15 q^{41} -3.22011e17i q^{42} +2.07501e17i q^{43} +1.00585e17 q^{44} +(1.70273e17 - 3.11446e17i) q^{45} -4.30089e17 q^{46} -4.68939e17i q^{47} +8.78782e17i q^{48} -5.52427e17 q^{49} +(-7.50284e17 + 4.81076e17i) q^{50} +1.52227e18 q^{51} +5.41322e17i q^{52} -4.34960e17i q^{53} +1.77028e18 q^{54} +(7.54496e17 - 1.38005e18i) q^{55} +1.38040e18 q^{56} +3.43734e18i q^{57} +8.13086e17i q^{58} -2.45200e18 q^{59} +(-4.37333e18 - 2.39097e18i) q^{60} -9.09214e18 q^{61} +5.01512e18i q^{62} -1.71331e19i q^{63} +2.37462e18 q^{64} +(7.42708e18 + 4.06051e18i) q^{65} +2.20047e19 q^{66} -1.15773e19i q^{67} -1.30061e19i q^{68} -3.76097e19 q^{69} +(-2.06372e19 + 3.77474e19i) q^{70} +4.36629e19 q^{71} +2.12882e19i q^{72} +2.22424e18i q^{73} +3.65451e19 q^{74} +(-6.56096e19 + 4.20683e19i) q^{75} +2.93682e19 q^{76} -7.59187e19i q^{77} +1.18424e20i q^{78} +2.53872e19 q^{79} +(5.63198e19 - 1.03015e20i) q^{80} -1.52283e19 q^{81} -1.82641e19i q^{82} -1.37569e20i q^{83} -2.40584e20 q^{84} +(-1.78447e20 - 9.75603e19i) q^{85} +3.87845e20 q^{86} +7.11014e19i q^{87} +9.43300e19i q^{88} +4.75737e19 q^{89} +(-5.82132e20 - 3.18261e20i) q^{90} +4.08576e20 q^{91} +3.21332e20i q^{92} +4.38554e20i q^{93} -8.76507e20 q^{94} +(2.20294e20 - 4.02939e20i) q^{95} +1.19364e21 q^{96} +4.39922e20i q^{97} +1.03256e21i q^{98} +1.17080e21 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 9273720 q^{4} - 25175970 q^{5} + 190183320 q^{6} - 46796905530 q^{9} + 42469996280 q^{10} + 150626450520 q^{11} + 1196972791560 q^{14} + 28918735560 q^{15} + 13236859984160 q^{16} - 111339219544600 q^{19}+ \cdots - 22\!\cdots\!60 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 1869.13i − 1.29070i −0.763889 − 0.645348i \(-0.776713\pi\)
0.763889 − 0.645348i \(-0.223287\pi\)
\(3\) − 163448.i − 1.59811i −0.601258 − 0.799055i \(-0.705333\pi\)
0.601258 − 0.799055i \(-0.294667\pi\)
\(4\) −1.39648e6 −0.665894
\(5\) −1.04751e7 + 1.91601e7i −0.479706 + 0.877429i
\(6\) −3.05505e8 −2.06267
\(7\) 1.05403e9i 1.41033i 0.709041 + 0.705167i \(0.249128\pi\)
−0.709041 + 0.705167i \(0.750872\pi\)
\(8\) − 1.30964e9i − 0.431229i
\(9\) −1.62549e10 −1.55396
\(10\) 3.58126e10 + 1.95794e10i 1.13249 + 0.619154i
\(11\) −7.20273e10 −0.837286 −0.418643 − 0.908151i \(-0.637494\pi\)
−0.418643 + 0.908151i \(0.637494\pi\)
\(12\) 2.28252e11i 1.06417i
\(13\) − 3.87633e11i − 0.779858i −0.920845 − 0.389929i \(-0.872500\pi\)
0.920845 − 0.389929i \(-0.127500\pi\)
\(14\) 1.97011e12 1.82031
\(15\) 3.13168e12 + 1.71214e12i 1.40223 + 0.766623i
\(16\) −5.37652e12 −1.22248
\(17\) 9.31350e12i 1.12047i 0.828335 + 0.560234i \(0.189289\pi\)
−0.828335 + 0.560234i \(0.810711\pi\)
\(18\) 3.03825e13i 2.00569i
\(19\) −2.10301e13 −0.786918 −0.393459 − 0.919342i \(-0.628722\pi\)
−0.393459 + 0.919342i \(0.628722\pi\)
\(20\) 1.46283e13 − 2.67567e13i 0.319433 − 0.584275i
\(21\) 1.72279e14 2.25387
\(22\) 1.34628e14i 1.08068i
\(23\) − 2.30102e14i − 1.15818i −0.815262 − 0.579092i \(-0.803407\pi\)
0.815262 − 0.579092i \(-0.196593\pi\)
\(24\) −2.14059e14 −0.689152
\(25\) −2.57380e14 − 4.01409e14i −0.539765 − 0.841816i
\(26\) −7.24535e14 −1.00656
\(27\) 9.47115e14i 0.885285i
\(28\) − 1.47193e15i − 0.939134i
\(29\) −4.35009e14 −0.192008 −0.0960039 − 0.995381i \(-0.530606\pi\)
−0.0960039 + 0.995381i \(0.530606\pi\)
\(30\) 3.20021e15 − 5.85350e15i 0.989476 − 1.80985i
\(31\) −2.68314e15 −0.587957 −0.293978 − 0.955812i \(-0.594979\pi\)
−0.293978 + 0.955812i \(0.594979\pi\)
\(32\) 7.30288e15i 1.14662i
\(33\) 1.17727e16i 1.33808i
\(34\) 1.74081e16 1.44618
\(35\) −2.01952e16 − 1.10411e16i −1.23747 − 0.676546i
\(36\) 2.26997e16 1.03477
\(37\) 1.95520e16i 0.668455i 0.942492 + 0.334228i \(0.108475\pi\)
−0.942492 + 0.334228i \(0.891525\pi\)
\(38\) 3.93080e16i 1.01567i
\(39\) −6.33579e16 −1.24630
\(40\) 2.50929e16 + 1.37187e16i 0.378373 + 0.206863i
\(41\) 9.77147e15 0.113692 0.0568459 − 0.998383i \(-0.481896\pi\)
0.0568459 + 0.998383i \(0.481896\pi\)
\(42\) − 3.22011e17i − 2.90906i
\(43\) 2.07501e17i 1.46420i 0.681196 + 0.732101i \(0.261460\pi\)
−0.681196 + 0.732101i \(0.738540\pi\)
\(44\) 1.00585e17 0.557544
\(45\) 1.70273e17 − 3.11446e17i 0.745442 − 1.36349i
\(46\) −4.30089e17 −1.49486
\(47\) − 4.68939e17i − 1.30044i −0.759747 − 0.650218i \(-0.774677\pi\)
0.759747 − 0.650218i \(-0.225323\pi\)
\(48\) 8.78782e17i 1.95366i
\(49\) −5.52427e17 −0.989045
\(50\) −7.50284e17 + 4.81076e17i −1.08653 + 0.696672i
\(51\) 1.52227e18 1.79063
\(52\) 5.41322e17i 0.519303i
\(53\) − 4.34960e17i − 0.341627i −0.985303 − 0.170814i \(-0.945360\pi\)
0.985303 − 0.170814i \(-0.0546396\pi\)
\(54\) 1.77028e18 1.14263
\(55\) 7.54496e17 − 1.38005e18i 0.401651 − 0.734659i
\(56\) 1.38040e18 0.608177
\(57\) 3.43734e18i 1.25758i
\(58\) 8.13086e17i 0.247824i
\(59\) −2.45200e18 −0.624561 −0.312280 − 0.949990i \(-0.601093\pi\)
−0.312280 + 0.949990i \(0.601093\pi\)
\(60\) −4.37333e18 − 2.39097e18i −0.933736 − 0.510490i
\(61\) −9.09214e18 −1.63193 −0.815967 − 0.578098i \(-0.803795\pi\)
−0.815967 + 0.578098i \(0.803795\pi\)
\(62\) 5.01512e18i 0.758873i
\(63\) − 1.71331e19i − 2.19160i
\(64\) 2.37462e18 0.257456
\(65\) 7.42708e18 + 4.06051e18i 0.684271 + 0.374102i
\(66\) 2.20047e19 1.72705
\(67\) − 1.15773e19i − 0.775926i −0.921675 − 0.387963i \(-0.873179\pi\)
0.921675 − 0.387963i \(-0.126821\pi\)
\(68\) − 1.30061e19i − 0.746113i
\(69\) −3.76097e19 −1.85090
\(70\) −2.06372e19 + 3.77474e19i −0.873214 + 1.59720i
\(71\) 4.36629e19 1.59184 0.795920 − 0.605402i \(-0.206988\pi\)
0.795920 + 0.605402i \(0.206988\pi\)
\(72\) 2.12882e19i 0.670111i
\(73\) 2.22424e18i 0.0605747i 0.999541 + 0.0302874i \(0.00964224\pi\)
−0.999541 + 0.0302874i \(0.990358\pi\)
\(74\) 3.65451e19 0.862772
\(75\) −6.56096e19 + 4.20683e19i −1.34531 + 0.862604i
\(76\) 2.93682e19 0.524004
\(77\) − 7.59187e19i − 1.18085i
\(78\) 1.18424e20i 1.60859i
\(79\) 2.53872e19 0.301669 0.150835 − 0.988559i \(-0.451804\pi\)
0.150835 + 0.988559i \(0.451804\pi\)
\(80\) 5.63198e19 − 1.03015e20i 0.586430 − 1.07264i
\(81\) −1.52283e19 −0.139174
\(82\) − 1.82641e19i − 0.146742i
\(83\) − 1.37569e20i − 0.973195i −0.873626 − 0.486597i \(-0.838238\pi\)
0.873626 − 0.486597i \(-0.161762\pi\)
\(84\) −2.40584e20 −1.50084
\(85\) −1.78447e20 − 9.75603e19i −0.983131 − 0.537495i
\(86\) 3.87845e20 1.88984
\(87\) 7.11014e19i 0.306850i
\(88\) 9.43300e19i 0.361062i
\(89\) 4.75737e19 0.161723 0.0808615 − 0.996725i \(-0.474233\pi\)
0.0808615 + 0.996725i \(0.474233\pi\)
\(90\) −5.82132e20 − 3.18261e20i −1.75985 − 0.962139i
\(91\) 4.08576e20 1.09986
\(92\) 3.21332e20i 0.771227i
\(93\) 4.38554e20i 0.939620i
\(94\) −8.76507e20 −1.67847
\(95\) 2.20294e20 − 4.02939e20i 0.377489 − 0.690465i
\(96\) 1.19364e21 1.83242
\(97\) 4.39922e20i 0.605721i 0.953035 + 0.302860i \(0.0979416\pi\)
−0.953035 + 0.302860i \(0.902058\pi\)
\(98\) 1.03256e21i 1.27656i
\(99\) 1.17080e21 1.30111
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 5.22.b.a.4.2 ✓ 10
3.2 odd 2 45.22.b.b.19.9 10
4.3 odd 2 80.22.c.a.49.9 10
5.2 odd 4 25.22.a.f.1.9 10
5.3 odd 4 25.22.a.f.1.2 10
5.4 even 2 inner 5.22.b.a.4.9 yes 10
15.14 odd 2 45.22.b.b.19.2 10
20.19 odd 2 80.22.c.a.49.2 10
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
5.22.b.a.4.2 ✓ 10 1.1 even 1 trivial
5.22.b.a.4.9 yes 10 5.4 even 2 inner
25.22.a.f.1.2 10 5.3 odd 4
25.22.a.f.1.9 10 5.2 odd 4
45.22.b.b.19.2 10 15.14 odd 2
45.22.b.b.19.9 10 3.2 odd 2
80.22.c.a.49.2 10 20.19 odd 2
80.22.c.a.49.9 10 4.3 odd 2