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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [3420,2,Mod(1189,3420)] 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("3420.1189"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3420, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 3, 4])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 3420 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3420.bj (of order \(6\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [20,0,0,0,-1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(27.3088374913\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(10\) over \(\Q(\zeta_{6})\)
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: \( x^{20} - 20 x^{18} + 261 x^{16} - 1994 x^{14} + 11074 x^{12} - 39211 x^{10} + 99376 x^{8} - 134299 x^{6} + \cdots + 4096 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 380)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 1189.10
Root \(2.48777 + 1.43632i\) of defining polynomial
Character \(\chi\) \(=\) 3420.1189
Dual form 3420.2.bj.c.2629.10

$q$-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)
 
\(f(q)\) \(=\) \(q+(2.21230 + 0.325180i) q^{5} +3.54568i q^{7} +1.81575 q^{11} +(-2.78308 + 1.60681i) q^{13} +(-6.92193 - 3.99638i) q^{17} +(0.863760 - 4.27246i) q^{19} +(-7.30026 + 4.21480i) q^{23} +(4.78852 + 1.43879i) q^{25} +(4.29124 + 7.43265i) q^{29} -1.70874 q^{31} +(-1.15298 + 7.84409i) q^{35} +5.50608i q^{37} +(-4.05694 + 7.02683i) q^{41} +(-4.35373 - 2.51363i) q^{43} +(1.16834 - 0.674543i) q^{47} -5.57183 q^{49} +(-1.92201 + 1.10967i) q^{53} +(4.01698 + 0.590447i) q^{55} +(-0.960774 + 1.66411i) q^{59} +(2.83047 + 4.90251i) q^{61} +(-6.67950 + 2.64974i) q^{65} +(-8.04360 + 4.64397i) q^{67} +(2.94365 - 5.09854i) q^{71} +(2.82716 + 1.63226i) q^{73} +6.43807i q^{77} +(2.08739 - 3.61546i) q^{79} +6.30268i q^{83} +(-14.0138 - 11.0920i) q^{85} +(-2.73646 - 4.73968i) q^{89} +(-5.69723 - 9.86789i) q^{91} +(3.30021 - 9.17107i) q^{95} +(-6.91255 - 3.99096i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - q^{5} + 14 q^{19} + 9 q^{25} + 16 q^{29} + 8 q^{31} + 2 q^{35} - 26 q^{41} - 44 q^{49} - 12 q^{55} - 4 q^{59} + 2 q^{61} + 18 q^{65} + 2 q^{71} - 16 q^{79} - 39 q^{85} + 40 q^{89} - 4 q^{91} + 43 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3420\mathbb{Z}\right)^\times\).

\(n\) \(1711\) \(1901\) \(2737\) \(3061\)
\(\chi(n)\) \(1\) \(1\) \(-1\) \(e\left(\frac{2}{3}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0 0
\(4\) 0 0
\(5\) 2.21230 + 0.325180i 0.989369 + 0.145425i
\(6\) 0 0
\(7\) 3.54568i 1.34014i 0.742298 + 0.670070i \(0.233736\pi\)
−0.742298 + 0.670070i \(0.766264\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 1.81575 0.547470 0.273735 − 0.961805i \(-0.411741\pi\)
0.273735 + 0.961805i \(0.411741\pi\)
\(12\) 0 0
\(13\) −2.78308 + 1.60681i −0.771887 + 0.445649i −0.833547 − 0.552448i \(-0.813694\pi\)
0.0616606 + 0.998097i \(0.480360\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −6.92193 − 3.99638i −1.67881 − 0.969264i −0.962419 − 0.271570i \(-0.912457\pi\)
−0.716396 − 0.697694i \(-0.754210\pi\)
\(18\) 0 0
\(19\) 0.863760 − 4.27246i 0.198160 − 0.980170i
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −7.30026 + 4.21480i −1.52221 + 0.878848i −0.522553 + 0.852607i \(0.675020\pi\)
−0.999656 + 0.0262406i \(0.991646\pi\)
\(24\) 0 0
\(25\) 4.78852 + 1.43879i 0.957703 + 0.287758i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 4.29124 + 7.43265i 0.796863 + 1.38021i 0.921649 + 0.388024i \(0.126842\pi\)
−0.124786 + 0.992184i \(0.539824\pi\)
\(30\) 0 0
\(31\) −1.70874 −0.306899 −0.153450 − 0.988156i \(-0.549038\pi\)
−0.153450 + 0.988156i \(0.549038\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −1.15298 + 7.84409i −0.194890 + 1.32589i
\(36\) 0 0
\(37\) 5.50608i 0.905193i 0.891715 + 0.452597i \(0.149502\pi\)
−0.891715 + 0.452597i \(0.850498\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −4.05694 + 7.02683i −0.633588 + 1.09741i 0.353224 + 0.935539i \(0.385085\pi\)
−0.986812 + 0.161868i \(0.948248\pi\)
\(42\) 0 0
\(43\) −4.35373 − 2.51363i −0.663938 − 0.383325i 0.129838 − 0.991535i \(-0.458554\pi\)
−0.793776 + 0.608211i \(0.791888\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 1.16834 − 0.674543i 0.170420 − 0.0983922i −0.412364 − 0.911019i \(-0.635297\pi\)
0.582784 + 0.812627i \(0.301963\pi\)
\(48\) 0 0
\(49\) −5.57183 −0.795976
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −1.92201 + 1.10967i −0.264009 + 0.152426i −0.626162 − 0.779693i \(-0.715375\pi\)
0.362153 + 0.932119i \(0.382042\pi\)
\(54\) 0 0
\(55\) 4.01698 + 0.590447i 0.541650 + 0.0796158i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −0.960774 + 1.66411i −0.125082 + 0.216649i −0.921765 − 0.387749i \(-0.873253\pi\)
0.796683 + 0.604398i \(0.206586\pi\)
\(60\) 0 0
\(61\) 2.83047 + 4.90251i 0.362404 + 0.627702i 0.988356 − 0.152159i \(-0.0486227\pi\)
−0.625952 + 0.779862i \(0.715289\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −6.67950 + 2.64974i −0.828489 + 0.328660i
\(66\) 0 0
\(67\) −8.04360 + 4.64397i −0.982682 + 0.567352i −0.903079 − 0.429475i \(-0.858699\pi\)
−0.0796032 + 0.996827i \(0.525365\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 2.94365 − 5.09854i 0.349346 − 0.605086i −0.636787 − 0.771040i \(-0.719737\pi\)
0.986134 + 0.165954i \(0.0530703\pi\)
\(72\) 0 0
\(73\) 2.82716 + 1.63226i 0.330894 + 0.191042i 0.656238 − 0.754554i \(-0.272147\pi\)
−0.325344 + 0.945596i \(0.605480\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 6.43807i 0.733687i
\(78\) 0 0
\(79\) 2.08739 − 3.61546i 0.234850 − 0.406771i −0.724379 − 0.689402i \(-0.757874\pi\)
0.959229 + 0.282630i \(0.0912069\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 6.30268i 0.691809i 0.938270 + 0.345905i \(0.112428\pi\)
−0.938270 + 0.345905i \(0.887572\pi\)
\(84\) 0 0
\(85\) −14.0138 − 11.0920i −1.52001 − 1.20310i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −2.73646 − 4.73968i −0.290064 − 0.502405i 0.683761 − 0.729706i \(-0.260343\pi\)
−0.973825 + 0.227301i \(0.927010\pi\)
\(90\) 0 0
\(91\) −5.69723 − 9.86789i −0.597232 − 1.03444i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 3.30021 − 9.17107i 0.338595 − 0.940932i
\(96\) 0 0
\(97\) −6.91255 − 3.99096i −0.701863 − 0.405221i 0.106178 − 0.994347i \(-0.466139\pi\)
−0.808041 + 0.589126i \(0.799472\pi\)
\(98\) 0 0
\(99\) 0 0
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 3420.2.bj.c.1189.10 20
3.2 odd 2 380.2.r.a.49.10 yes 20
5.4 even 2 inner 3420.2.bj.c.1189.4 20
15.2 even 4 1900.2.i.g.201.10 20
15.8 even 4 1900.2.i.g.201.1 20
15.14 odd 2 380.2.r.a.49.1 ✓ 20
19.7 even 3 inner 3420.2.bj.c.2629.4 20
57.26 odd 6 380.2.r.a.349.1 yes 20
95.64 even 6 inner 3420.2.bj.c.2629.10 20
285.83 even 12 1900.2.i.g.501.1 20
285.197 even 12 1900.2.i.g.501.10 20
285.254 odd 6 380.2.r.a.349.10 yes 20
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
380.2.r.a.49.1 ✓ 20 15.14 odd 2
380.2.r.a.49.10 yes 20 3.2 odd 2
380.2.r.a.349.1 yes 20 57.26 odd 6
380.2.r.a.349.10 yes 20 285.254 odd 6
1900.2.i.g.201.1 20 15.8 even 4
1900.2.i.g.201.10 20 15.2 even 4
1900.2.i.g.501.1 20 285.83 even 12
1900.2.i.g.501.10 20 285.197 even 12
3420.2.bj.c.1189.4 20 5.4 even 2 inner
3420.2.bj.c.1189.10 20 1.1 even 1 trivial
3420.2.bj.c.2629.4 20 19.7 even 3 inner
3420.2.bj.c.2629.10 20 95.64 even 6 inner