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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 1189.4
Root \(-2.48777 - 1.43632i\) of defining polynomial
Character \(\chi\) \(=\) 3420.1189
Dual form 3420.2.bj.c.2629.4

$q$-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)
 
\(f(q)\) \(=\) \(q+(-1.38776 - 1.75332i) q^{5} -3.54568i q^{7} +1.81575 q^{11} +(2.78308 - 1.60681i) q^{13} +(6.92193 + 3.99638i) q^{17} +(0.863760 - 4.27246i) q^{19} +(7.30026 - 4.21480i) q^{23} +(-1.14823 + 4.86637i) q^{25} +(4.29124 + 7.43265i) q^{29} -1.70874 q^{31} +(-6.21669 + 4.92056i) q^{35} -5.50608i q^{37} +(-4.05694 + 7.02683i) q^{41} +(4.35373 + 2.51363i) q^{43} +(-1.16834 + 0.674543i) q^{47} -5.57183 q^{49} +(1.92201 - 1.10967i) q^{53} +(-2.51983 - 3.18359i) q^{55} +(-0.960774 + 1.66411i) q^{59} +(2.83047 + 4.90251i) q^{61} +(-6.67950 - 2.64974i) q^{65} +(8.04360 - 4.64397i) q^{67} +(2.94365 - 5.09854i) q^{71} +(-2.82716 - 1.63226i) q^{73} -6.43807i q^{77} +(2.08739 - 3.61546i) q^{79} -6.30268i q^{83} +(-2.59909 - 17.6824i) q^{85} +(-2.73646 - 4.73968i) q^{89} +(-5.69723 - 9.86789i) q^{91} +(-8.68966 + 4.41472i) q^{95} +(6.91255 + 3.99096i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - q^{5} + 14 q^{19} + 9 q^{25} + 16 q^{29} + 8 q^{31} + 2 q^{35} - 26 q^{41} - 44 q^{49} - 12 q^{55} - 4 q^{59} + 2 q^{61} + 18 q^{65} + 2 q^{71} - 16 q^{79} - 39 q^{85} + 40 q^{89} - 4 q^{91} + 43 q^{95}+O(q^{100}) \) 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
\(5\) −1.38776 − 1.75332i −0.620626 − 0.784106i
\(6\) 0 0
\(7\) − 3.54568i − 1.34014i −0.742298 − 0.670070i \(-0.766264\pi\)
0.742298 − 0.670070i \(-0.233736\pi\)
\(8\) 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 −0.0616606 − 0.998097i \(-0.519640\pi\)
0.833547 + 0.552448i \(0.186306\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 6.92193 + 3.99638i 1.67881 + 0.969264i 0.962419 + 0.271570i \(0.0875429\pi\)
0.716396 + 0.697694i \(0.245790\pi\)
\(18\) 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 \(-0.324980\pi\)
0.999656 − 0.0262406i \(-0.00835362\pi\)
\(24\) 0 0
\(25\) −1.14823 + 4.86637i −0.229646 + 0.973274i
\(26\) 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
\(35\) −6.21669 + 4.92056i −1.05081 + 0.831726i
\(36\) 0 0
\(37\) − 5.50608i − 0.905193i −0.891715 − 0.452597i \(-0.850498\pi\)
0.891715 − 0.452597i \(-0.149502\pi\)
\(38\) 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 0.793776 − 0.608211i \(-0.208112\pi\)
−0.129838 + 0.991535i \(0.541446\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −1.16834 + 0.674543i −0.170420 + 0.0983922i −0.582784 − 0.812627i \(-0.698037\pi\)
0.412364 + 0.911019i \(0.364703\pi\)
\(48\) 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 −0.362153 − 0.932119i \(-0.617958\pi\)
0.626162 + 0.779693i \(0.284625\pi\)
\(54\) 0 0
\(55\) −2.51983 − 3.18359i −0.339774 − 0.429275i
\(56\) 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 0.0796032 − 0.996827i \(-0.474635\pi\)
0.903079 + 0.429475i \(0.141301\pi\)
\(68\) 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 0.325344 − 0.945596i \(-0.394520\pi\)
−0.656238 + 0.754554i \(0.727853\pi\)
\(74\) 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 \(-0.887572\pi\)
0.938270 − 0.345905i \(-0.112428\pi\)
\(84\) 0 0
\(85\) −2.59909 − 17.6824i −0.281910 − 1.91792i
\(86\) 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
\(95\) −8.68966 + 4.41472i −0.891541 + 0.452941i
\(96\) 0 0
\(97\) 6.91255 + 3.99096i 0.701863 + 0.405221i 0.808041 − 0.589126i \(-0.200528\pi\)
−0.106178 + 0.994347i \(0.533861\pi\)
\(98\) 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.1189.4 20
3.2 odd 2 380.2.r.a.49.1 ✓ 20
5.4 even 2 inner 3420.2.bj.c.1189.10 20
15.2 even 4 1900.2.i.g.201.1 20
15.8 even 4 1900.2.i.g.201.10 20
15.14 odd 2 380.2.r.a.49.10 yes 20
19.7 even 3 inner 3420.2.bj.c.2629.10 20
57.26 odd 6 380.2.r.a.349.10 yes 20
95.64 even 6 inner 3420.2.bj.c.2629.4 20
285.83 even 12 1900.2.i.g.501.10 20
285.197 even 12 1900.2.i.g.501.1 20
285.254 odd 6 380.2.r.a.349.1 yes 20
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
380.2.r.a.49.1 ✓ 20 3.2 odd 2
380.2.r.a.49.10 yes 20 15.14 odd 2
380.2.r.a.349.1 yes 20 285.254 odd 6
380.2.r.a.349.10 yes 20 57.26 odd 6
1900.2.i.g.201.1 20 15.2 even 4
1900.2.i.g.201.10 20 15.8 even 4
1900.2.i.g.501.1 20 285.197 even 12
1900.2.i.g.501.10 20 285.83 even 12
3420.2.bj.c.1189.4 20 1.1 even 1 trivial
3420.2.bj.c.1189.10 20 5.4 even 2 inner
3420.2.bj.c.2629.4 20 95.64 even 6 inner
3420.2.bj.c.2629.10 20 19.7 even 3 inner