Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [380,2,Mod(49,380)] 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("380.49"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(380, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 3, 4])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 380 = 2^{2} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 380.r (of order \(6\), 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: \(3.03431527681\)
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: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 349.1
Root \(-2.48777 - 1.43632i\) of defining polynomial
Character \(\chi\) \(=\) 380.349
Dual form 380.2.r.a.49.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-2.48777 + 1.43632i) q^{3} +(1.38776 - 1.75332i) q^{5} +3.54568i q^{7} +(2.62601 - 4.54838i) q^{9} -1.81575 q^{11} +(2.78308 + 1.60681i) q^{13} +(-0.934123 + 6.35512i) q^{15} +(-6.92193 + 3.99638i) q^{17} +(0.863760 + 4.27246i) q^{19} +(-5.09271 - 8.82084i) q^{21} +(-7.30026 - 4.21480i) q^{23} +(-1.14823 - 4.86637i) q^{25} +6.46921i q^{27} +(-4.29124 + 7.43265i) q^{29} -1.70874 q^{31} +(4.51718 - 2.60799i) q^{33} +(6.21669 + 4.92056i) q^{35} +5.50608i q^{37} -9.23155 q^{39} +(4.05694 + 7.02683i) q^{41} +(4.35373 - 2.51363i) q^{43} +(-4.33047 - 10.9163i) q^{45} +(1.16834 + 0.674543i) q^{47} -5.57183 q^{49} +(11.4801 - 19.8842i) q^{51} +(-1.92201 - 1.10967i) q^{53} +(-2.51983 + 3.18359i) q^{55} +(-8.28544 - 9.38828i) q^{57} +(0.960774 + 1.66411i) q^{59} +(2.83047 - 4.90251i) q^{61} +(16.1271 + 9.31098i) q^{63} +(6.67950 - 2.64974i) q^{65} +(8.04360 + 4.64397i) q^{67} +24.2152 q^{69} +(-2.94365 - 5.09854i) q^{71} +(-2.82716 + 1.63226i) q^{73} +(9.84618 + 10.4572i) q^{75} -6.43807i q^{77} +(2.08739 + 3.61546i) q^{79} +(-1.41381 - 2.44879i) q^{81} -6.30268i q^{83} +(-2.59909 + 17.6824i) q^{85} -24.6543i q^{87} +(2.73646 - 4.73968i) q^{89} +(-5.69723 + 9.86789i) q^{91} +(4.25096 - 2.45429i) q^{93} +(8.68966 + 4.41472i) q^{95} +(6.91255 - 3.99096i) q^{97} +(-4.76818 + 8.25873i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + q^{5} + 10 q^{9} - 5 q^{15} + 14 q^{19} - 8 q^{21} + 9 q^{25} - 16 q^{29} + 8 q^{31} - 2 q^{35} - 8 q^{39} + 26 q^{41} - 32 q^{45} - 44 q^{49} + 26 q^{51} - 12 q^{55} + 4 q^{59} + 2 q^{61} - 18 q^{65}+ \cdots - 20 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/380\mathbb{Z}\right)^\times\).

\(n\) \(21\) \(77\) \(191\)
\(\chi(n)\) \(e\left(\frac{1}{3}\right)\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −2.48777 + 1.43632i −1.43632 + 0.829257i −0.997591 − 0.0693641i \(-0.977903\pi\)
−0.438725 + 0.898622i \(0.644570\pi\)
\(4\) 0 0
\(5\) 1.38776 − 1.75332i 0.620626 − 0.784106i
\(6\) 0 0
\(7\) 3.54568i 1.34014i 0.742298 + 0.670070i \(0.233736\pi\)
−0.742298 + 0.670070i \(0.766264\pi\)
\(8\) 0 0
\(9\) 2.62601 − 4.54838i 0.875336 − 1.51613i
\(10\) 0 0
\(11\) −1.81575 −0.547470 −0.273735 − 0.961805i \(-0.588259\pi\)
−0.273735 + 0.961805i \(0.588259\pi\)
\(12\) 0 0
\(13\) 2.78308 + 1.60681i 0.771887 + 0.445649i 0.833547 − 0.552448i \(-0.186306\pi\)
−0.0616606 + 0.998097i \(0.519640\pi\)
\(14\) 0 0
\(15\) −0.934123 + 6.35512i −0.241190 + 1.64088i
\(16\) 0 0
\(17\) −6.92193 + 3.99638i −1.67881 + 0.969264i −0.716396 + 0.697694i \(0.754210\pi\)
−0.962419 + 0.271570i \(0.912457\pi\)
\(18\) 0 0
\(19\) 0.863760 + 4.27246i 0.198160 + 0.980170i
\(20\) 0 0
\(21\) −5.09271 − 8.82084i −1.11132 − 1.92486i
\(22\) 0 0
\(23\) −7.30026 − 4.21480i −1.52221 − 0.878848i −0.999656 − 0.0262406i \(-0.991646\pi\)
−0.522553 − 0.852607i \(-0.675020\pi\)
\(24\) 0 0
\(25\) −1.14823 − 4.86637i −0.229646 − 0.973274i
\(26\) 0 0
\(27\) 6.46921i 1.24500i
\(28\) 0 0
\(29\) −4.29124 + 7.43265i −0.796863 + 1.38021i 0.124786 + 0.992184i \(0.460176\pi\)
−0.921649 + 0.388024i \(0.873158\pi\)
\(30\) 0 0
\(31\) −1.70874 −0.306899 −0.153450 − 0.988156i \(-0.549038\pi\)
−0.153450 + 0.988156i \(0.549038\pi\)
\(32\) 0 0
\(33\) 4.51718 − 2.60799i 0.786340 − 0.453994i
\(34\) 0 0
\(35\) 6.21669 + 4.92056i 1.05081 + 0.831726i
\(36\) 0 0
\(37\) 5.50608i 0.905193i 0.891715 + 0.452597i \(0.149502\pi\)
−0.891715 + 0.452597i \(0.850498\pi\)
\(38\) 0 0
\(39\) −9.23155 −1.47823
\(40\) 0 0
\(41\) 4.05694 + 7.02683i 0.633588 + 1.09741i 0.986812 + 0.161868i \(0.0517520\pi\)
−0.353224 + 0.935539i \(0.614915\pi\)
\(42\) 0 0
\(43\) 4.35373 − 2.51363i 0.663938 − 0.383325i −0.129838 − 0.991535i \(-0.541446\pi\)
0.793776 + 0.608211i \(0.208112\pi\)
\(44\) 0 0
\(45\) −4.33047 − 10.9163i −0.645548 − 1.62730i
\(46\) 0 0
\(47\) 1.16834 + 0.674543i 0.170420 + 0.0983922i 0.582784 − 0.812627i \(-0.301963\pi\)
−0.412364 + 0.911019i \(0.635297\pi\)
\(48\) 0 0
\(49\) −5.57183 −0.795976
\(50\) 0 0
\(51\) 11.4801 − 19.8842i 1.60754 − 2.78434i
\(52\) 0 0
\(53\) −1.92201 − 1.10967i −0.264009 − 0.152426i 0.362153 − 0.932119i \(-0.382042\pi\)
−0.626162 + 0.779693i \(0.715375\pi\)
\(54\) 0 0
\(55\) −2.51983 + 3.18359i −0.339774 + 0.429275i
\(56\) 0 0
\(57\) −8.28544 − 9.38828i −1.09743 − 1.24351i
\(58\) 0 0
\(59\) 0.960774 + 1.66411i 0.125082 + 0.216649i 0.921765 − 0.387749i \(-0.126747\pi\)
−0.796683 + 0.604398i \(0.793414\pi\)
\(60\) 0 0
\(61\) 2.83047 − 4.90251i 0.362404 − 0.627702i −0.625952 − 0.779862i \(-0.715289\pi\)
0.988356 + 0.152159i \(0.0486227\pi\)
\(62\) 0 0
\(63\) 16.1271 + 9.31098i 2.03182 + 1.17307i
\(64\) 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 \(-0.141301\pi\)
0.0796032 + 0.996827i \(0.474635\pi\)
\(68\) 0 0
\(69\) 24.2152 2.91516
\(70\) 0 0
\(71\) −2.94365 − 5.09854i −0.349346 − 0.605086i 0.636787 − 0.771040i \(-0.280263\pi\)
−0.986134 + 0.165954i \(0.946930\pi\)
\(72\) 0 0
\(73\) −2.82716 + 1.63226i −0.330894 + 0.191042i −0.656238 − 0.754554i \(-0.727853\pi\)
0.325344 + 0.945596i \(0.394520\pi\)
\(74\) 0 0
\(75\) 9.84618 + 10.4572i 1.13694 + 1.20749i
\(76\) 0 0
\(77\) − 6.43807i − 0.733687i
\(78\) 0 0
\(79\) 2.08739 + 3.61546i 0.234850 + 0.406771i 0.959229 − 0.282630i \(-0.0912069\pi\)
−0.724379 + 0.689402i \(0.757874\pi\)
\(80\) 0 0
\(81\) −1.41381 − 2.44879i −0.157090 − 0.272087i
\(82\) 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
\(87\) − 24.6543i − 2.64322i
\(88\) 0 0
\(89\) 2.73646 − 4.73968i 0.290064 − 0.502405i −0.683761 − 0.729706i \(-0.739657\pi\)
0.973825 + 0.227301i \(0.0729901\pi\)
\(90\) 0 0
\(91\) −5.69723 + 9.86789i −0.597232 + 1.03444i
\(92\) 0 0
\(93\) 4.25096 − 2.45429i 0.440804 − 0.254499i
\(94\) 0 0
\(95\) 8.68966 + 4.41472i 0.891541 + 0.452941i
\(96\) 0 0
\(97\) 6.91255 − 3.99096i 0.701863 − 0.405221i −0.106178 − 0.994347i \(-0.533861\pi\)
0.808041 + 0.589126i \(0.200528\pi\)
\(98\) 0 0
\(99\) −4.76818 + 8.25873i −0.479220 + 0.830034i
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 380.2.r.a.349.1 yes 20
3.2 odd 2 3420.2.bj.c.2629.4 20
5.2 odd 4 1900.2.i.g.501.10 20
5.3 odd 4 1900.2.i.g.501.1 20
5.4 even 2 inner 380.2.r.a.349.10 yes 20
15.14 odd 2 3420.2.bj.c.2629.10 20
19.11 even 3 inner 380.2.r.a.49.10 yes 20
57.11 odd 6 3420.2.bj.c.1189.10 20
95.49 even 6 inner 380.2.r.a.49.1 ✓ 20
95.68 odd 12 1900.2.i.g.201.1 20
95.87 odd 12 1900.2.i.g.201.10 20
285.239 odd 6 3420.2.bj.c.1189.4 20
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
380.2.r.a.49.1 ✓ 20 95.49 even 6 inner
380.2.r.a.49.10 yes 20 19.11 even 3 inner
380.2.r.a.349.1 yes 20 1.1 even 1 trivial
380.2.r.a.349.10 yes 20 5.4 even 2 inner
1900.2.i.g.201.1 20 95.68 odd 12
1900.2.i.g.201.10 20 95.87 odd 12
1900.2.i.g.501.1 20 5.3 odd 4
1900.2.i.g.501.10 20 5.2 odd 4
3420.2.bj.c.1189.4 20 285.239 odd 6
3420.2.bj.c.1189.10 20 57.11 odd 6
3420.2.bj.c.2629.4 20 3.2 odd 2
3420.2.bj.c.2629.10 20 15.14 odd 2