Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [380,3,Mod(11,380)] 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("380.11"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(380, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([3, 0, 4])) N = Newforms(chi, 3, names="a")
 
Level: \( N \) \(=\) \( 380 = 2^{2} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 380.q (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: \(10.3542500457\)
Analytic rank: \(0\)
Dimension: \(160\)
Relative dimension: \(80\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 11.17
Character \(\chi\) \(=\) 380.11
Dual form 380.3.q.a.311.17

$q$-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+(-1.57411 + 1.23377i) q^{2} +(3.64654 + 2.10533i) q^{3} +(0.955631 - 3.88417i) q^{4} +(-1.11803 + 1.93649i) q^{5} +(-8.33754 + 1.18497i) q^{6} -11.0873i q^{7} +(3.28790 + 7.29313i) q^{8} +(4.36484 + 7.56012i) q^{9} +(-0.629276 - 4.42764i) q^{10} -15.1236i q^{11} +(11.6622 - 12.1519i) q^{12} +(1.44060 + 2.49519i) q^{13} +(13.6791 + 17.4525i) q^{14} +(-8.15391 + 4.70766i) q^{15} +(-14.1735 - 7.42367i) q^{16} +(7.52437 - 13.0326i) q^{17} +(-16.1982 - 6.51525i) q^{18} +(17.2508 + 7.96314i) q^{19} +(6.45323 + 6.19320i) q^{20} +(23.3423 - 40.4301i) q^{21} +(18.6591 + 23.8062i) q^{22} +(25.0290 - 14.4505i) q^{23} +(-3.36500 + 33.5168i) q^{24} +(-2.50000 - 4.33013i) q^{25} +(-5.34615 - 2.15033i) q^{26} -1.13823i q^{27} +(-43.0648 - 10.5953i) q^{28} +(-8.55435 - 14.8166i) q^{29} +(7.02697 - 17.4704i) q^{30} +56.4381i q^{31} +(31.4698 - 5.80121i) q^{32} +(31.8403 - 55.1490i) q^{33} +(4.23503 + 29.7980i) q^{34} +(21.4704 + 12.3959i) q^{35} +(33.5360 - 9.72908i) q^{36} -49.1749 q^{37} +(-36.9792 + 8.74859i) q^{38} +12.1318i q^{39} +(-17.7991 - 1.78698i) q^{40} +(34.0978 - 59.0591i) q^{41} +(13.1380 + 92.4404i) q^{42} +(-36.3654 - 20.9956i) q^{43} +(-58.7428 - 14.4526i) q^{44} -19.5202 q^{45} +(-21.5698 + 53.6266i) q^{46} +(-9.81283 + 5.66544i) q^{47} +(-36.0551 - 56.9107i) q^{48} -73.9272 q^{49} +(9.27764 + 3.73167i) q^{50} +(54.8758 - 31.6826i) q^{51} +(11.0684 - 3.21105i) q^{52} +(33.6392 + 58.2648i) q^{53} +(1.40431 + 1.79170i) q^{54} +(29.2868 + 16.9087i) q^{55} +(80.8608 - 36.4538i) q^{56} +(46.1405 + 65.3565i) q^{57} +(31.7457 + 12.7688i) q^{58} +(84.3746 + 48.7137i) q^{59} +(10.4932 + 36.1700i) q^{60} +(28.9650 + 50.1688i) q^{61} +(-69.6315 - 88.8396i) q^{62} +(83.8210 - 48.3941i) q^{63} +(-42.3794 + 47.9581i) q^{64} -6.44255 q^{65} +(17.9210 + 126.094i) q^{66} +(-20.5217 + 11.8482i) q^{67} +(-43.4303 - 41.6803i) q^{68} +121.692 q^{69} +(-49.0904 + 6.97694i) q^{70} +(37.6226 + 21.7214i) q^{71} +(-40.7858 + 56.6903i) q^{72} +(-2.71972 + 4.71070i) q^{73} +(77.4066 - 60.6704i) q^{74} -21.0533i q^{75} +(47.4155 - 59.3950i) q^{76} -167.680 q^{77} +(-14.9678 - 19.0967i) q^{78} +(1.47604 + 0.852193i) q^{79} +(30.2224 - 19.1470i) q^{80} +(41.6799 - 72.1917i) q^{81} +(19.1916 + 135.034i) q^{82} -25.5889i q^{83} +(-134.731 - 129.302i) q^{84} +(16.8250 + 29.1418i) q^{85} +(83.1467 - 11.8172i) q^{86} -72.0390i q^{87} +(110.299 - 49.7250i) q^{88} +(6.56207 + 11.3658i) q^{89} +(30.7268 - 24.0834i) q^{90} +(27.6648 - 15.9723i) q^{91} +(-32.2097 - 111.026i) q^{92} +(-118.821 + 205.804i) q^{93} +(8.45661 - 21.0248i) q^{94} +(-34.7075 + 24.5029i) q^{95} +(126.969 + 45.0999i) q^{96} +(42.7274 - 74.0060i) q^{97} +(116.369 - 91.2090i) q^{98} +(114.337 - 66.0123i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 160 q + 2 q^{4} + 6 q^{6} + 248 q^{9} - 10 q^{10} - 16 q^{13} - 14 q^{16} + 48 q^{17} + 48 q^{21} - 44 q^{24} - 400 q^{25} + 68 q^{26} + 60 q^{28} - 80 q^{30} + 30 q^{32} - 40 q^{33} - 22 q^{34} + 52 q^{36}+ \cdots - 226 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(21\) \(77\) \(191\)
\(\chi(n)\) \(e\left(\frac{2}{3}\right)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.57411 + 1.23377i −0.787054 + 0.616884i
\(3\) 3.64654 + 2.10533i 1.21551 + 0.701777i 0.963955 − 0.266065i \(-0.0857236\pi\)
0.251558 + 0.967842i \(0.419057\pi\)
\(4\) 0.955631 − 3.88417i 0.238908 − 0.971042i
\(5\) −1.11803 + 1.93649i −0.223607 + 0.387298i
\(6\) −8.33754 + 1.18497i −1.38959 + 0.197495i
\(7\) − 11.0873i − 1.58389i −0.610590 − 0.791947i \(-0.709068\pi\)
0.610590 − 0.791947i \(-0.290932\pi\)
\(8\) 3.28790 + 7.29313i 0.410987 + 0.911641i
\(9\) 4.36484 + 7.56012i 0.484982 + 0.840014i
\(10\) −0.629276 − 4.42764i −0.0629276 − 0.442764i
\(11\) − 15.1236i − 1.37488i −0.726243 − 0.687438i \(-0.758735\pi\)
0.726243 − 0.687438i \(-0.241265\pi\)
\(12\) 11.6622 − 12.1519i 0.971851 − 1.01266i
\(13\) 1.44060 + 2.49519i 0.110815 + 0.191938i 0.916099 − 0.400952i \(-0.131320\pi\)
−0.805284 + 0.592889i \(0.797987\pi\)
\(14\) 13.6791 + 17.4525i 0.977079 + 1.24661i
\(15\) −8.15391 + 4.70766i −0.543594 + 0.313844i
\(16\) −14.1735 − 7.42367i −0.885846 − 0.463979i
\(17\) 7.52437 − 13.0326i 0.442610 − 0.766623i −0.555272 − 0.831669i \(-0.687386\pi\)
0.997882 + 0.0650457i \(0.0207193\pi\)
\(18\) −16.1982 − 6.51525i −0.899898 − 0.361958i
\(19\) 17.2508 + 7.96314i 0.907934 + 0.419113i
\(20\) 6.45323 + 6.19320i 0.322662 + 0.309660i
\(21\) 23.3423 − 40.4301i 1.11154 − 1.92524i
\(22\) 18.6591 + 23.8062i 0.848140 + 1.08210i
\(23\) 25.0290 − 14.4505i 1.08822 − 0.628282i 0.155116 − 0.987896i \(-0.450425\pi\)
0.933101 + 0.359614i \(0.117092\pi\)
\(24\) −3.36500 + 33.5168i −0.140208 + 1.39653i
\(25\) −2.50000 − 4.33013i −0.100000 − 0.173205i
\(26\) −5.34615 − 2.15033i −0.205621 − 0.0827051i
\(27\) − 1.13823i − 0.0421567i
\(28\) −43.0648 − 10.5953i −1.53803 − 0.378404i
\(29\) −8.55435 − 14.8166i −0.294978 − 0.510916i 0.680002 − 0.733210i \(-0.261979\pi\)
−0.974980 + 0.222294i \(0.928646\pi\)
\(30\) 7.02697 − 17.4704i 0.234232 − 0.582347i
\(31\) 56.4381i 1.82058i 0.413968 + 0.910292i \(0.364143\pi\)
−0.413968 + 0.910292i \(0.635857\pi\)
\(32\) 31.4698 − 5.80121i 0.983430 − 0.181288i
\(33\) 31.8403 − 55.1490i 0.964857 − 1.67118i
\(34\) 4.23503 + 29.7980i 0.124560 + 0.876413i
\(35\) 21.4704 + 12.3959i 0.613439 + 0.354169i
\(36\) 33.5360 − 9.72908i 0.931555 − 0.270252i
\(37\) −49.1749 −1.32905 −0.664526 − 0.747266i \(-0.731366\pi\)
−0.664526 + 0.747266i \(0.731366\pi\)
\(38\) −36.9792 + 8.74859i −0.973137 + 0.230226i
\(39\) 12.1318i 0.311071i
\(40\) −17.7991 − 1.78698i −0.444977 − 0.0446744i
\(41\) 34.0978 − 59.0591i 0.831653 − 1.44047i −0.0650737 − 0.997880i \(-0.520728\pi\)
0.896727 − 0.442585i \(-0.145938\pi\)
\(42\) 13.1380 + 92.4404i 0.312811 + 2.20096i
\(43\) −36.3654 − 20.9956i −0.845707 − 0.488269i 0.0134933 − 0.999909i \(-0.495705\pi\)
−0.859200 + 0.511640i \(0.829038\pi\)
\(44\) −58.7428 − 14.4526i −1.33506 − 0.328469i
\(45\) −19.5202 −0.433781
\(46\) −21.5698 + 53.6266i −0.468908 + 1.16580i
\(47\) −9.81283 + 5.66544i −0.208784 + 0.120541i −0.600746 − 0.799440i \(-0.705130\pi\)
0.391962 + 0.919981i \(0.371796\pi\)
\(48\) −36.0551 − 56.9107i −0.751148 − 1.18564i
\(49\) −73.9272 −1.50872
\(50\) 9.27764 + 3.73167i 0.185553 + 0.0746333i
\(51\) 54.8758 − 31.6826i 1.07600 − 0.621227i
\(52\) 11.0684 − 3.21105i 0.212854 − 0.0617509i
\(53\) 33.6392 + 58.2648i 0.634702 + 1.09934i 0.986578 + 0.163290i \(0.0522105\pi\)
−0.351876 + 0.936047i \(0.614456\pi\)
\(54\) 1.40431 + 1.79170i 0.0260058 + 0.0331796i
\(55\) 29.2868 + 16.9087i 0.532487 + 0.307432i
\(56\) 80.8608 − 36.4538i 1.44394 − 0.650960i
\(57\) 46.1405 + 65.3565i 0.809483 + 1.14660i
\(58\) 31.7457 + 12.7688i 0.547339 + 0.220152i
\(59\) 84.3746 + 48.7137i 1.43008 + 0.825656i 0.997126 − 0.0757608i \(-0.0241385\pi\)
0.432952 + 0.901417i \(0.357472\pi\)
\(60\) 10.4932 + 36.1700i 0.174887 + 0.602833i
\(61\) 28.9650 + 50.1688i 0.474836 + 0.822440i 0.999585 − 0.0288173i \(-0.00917409\pi\)
−0.524749 + 0.851257i \(0.675841\pi\)
\(62\) −69.6315 − 88.8396i −1.12309 − 1.43290i
\(63\) 83.8210 − 48.3941i 1.33049 − 0.768160i
\(64\) −42.3794 + 47.9581i −0.662179 + 0.749346i
\(65\) −6.44255 −0.0991162
\(66\) 17.9210 + 126.094i 0.271531 + 1.91051i
\(67\) −20.5217 + 11.8482i −0.306294 + 0.176839i −0.645267 − 0.763957i \(-0.723254\pi\)
0.338973 + 0.940796i \(0.389920\pi\)
\(68\) −43.4303 − 41.6803i −0.638680 − 0.612945i
\(69\) 121.692 1.76366
\(70\) −49.0904 + 6.97694i −0.701291 + 0.0996706i
\(71\) 37.6226 + 21.7214i 0.529895 + 0.305935i 0.740974 − 0.671534i \(-0.234364\pi\)
−0.211078 + 0.977469i \(0.567698\pi\)
\(72\) −40.7858 + 56.6903i −0.566469 + 0.787365i
\(73\) −2.71972 + 4.71070i −0.0372565 + 0.0645301i −0.884052 − 0.467388i \(-0.845195\pi\)
0.846796 + 0.531918i \(0.178529\pi\)
\(74\) 77.4066 − 60.6704i 1.04603 − 0.819871i
\(75\) − 21.0533i − 0.280711i
\(76\) 47.4155 − 59.3950i 0.623889 − 0.781513i
\(77\) −167.680 −2.17766
\(78\) −14.9678 − 19.0967i −0.191895 − 0.244829i
\(79\) 1.47604 + 0.852193i 0.0186841 + 0.0107873i 0.509313 − 0.860581i \(-0.329900\pi\)
−0.490629 + 0.871369i \(0.663233\pi\)
\(80\) 30.2224 − 19.1470i 0.377780 − 0.239338i
\(81\) 41.6799 − 72.1917i 0.514567 − 0.891256i
\(82\) 19.1916 + 135.034i 0.234044 + 1.64676i
\(83\) − 25.5889i − 0.308300i −0.988047 − 0.154150i \(-0.950736\pi\)
0.988047 − 0.154150i \(-0.0492638\pi\)
\(84\) −134.731 − 129.302i −1.60394 − 1.53931i
\(85\) 16.8250 + 29.1418i 0.197941 + 0.342844i
\(86\) 83.1467 − 11.8172i 0.966822 − 0.137409i
\(87\) − 72.0390i − 0.828034i
\(88\) 110.299 − 49.7250i 1.25339 − 0.565057i
\(89\) 6.56207 + 11.3658i 0.0737311 + 0.127706i 0.900534 − 0.434786i \(-0.143176\pi\)
−0.826803 + 0.562492i \(0.809843\pi\)
\(90\) 30.7268 − 24.0834i 0.341409 − 0.267593i
\(91\) 27.6648 − 15.9723i 0.304009 − 0.175520i
\(92\) −32.2097 − 111.026i −0.350105 − 1.20681i
\(93\) −118.821 + 205.804i −1.27764 + 2.21294i
\(94\) 8.45661 − 21.0248i 0.0899639 − 0.223668i
\(95\) −34.7075 + 24.5029i −0.365342 + 0.257925i
\(96\) 126.969 + 45.0999i 1.32260 + 0.469791i
\(97\) 42.7274 − 74.0060i 0.440489 − 0.762949i −0.557237 − 0.830354i \(-0.688139\pi\)
0.997726 + 0.0674046i \(0.0214718\pi\)
\(98\) 116.369 − 91.2090i 1.18744 − 0.930704i
\(99\) 114.337 − 66.0123i 1.15492 − 0.666791i
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 380.3.q.a.11.17 ✓ 160
4.3 odd 2 inner 380.3.q.a.11.36 yes 160
19.7 even 3 inner 380.3.q.a.311.36 yes 160
76.7 odd 6 inner 380.3.q.a.311.17 yes 160
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
380.3.q.a.11.17 ✓ 160 1.1 even 1 trivial
380.3.q.a.11.36 yes 160 4.3 odd 2 inner
380.3.q.a.311.17 yes 160 76.7 odd 6 inner
380.3.q.a.311.36 yes 160 19.7 even 3 inner