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 11.36
Character \(\chi\) \(=\) 380.11
Dual form 380.3.q.a.311.36

$q$-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+(-0.281421 + 1.98010i) q^{2} +(-3.64654 - 2.10533i) q^{3} +(-3.84160 - 1.11448i) q^{4} +(-1.11803 + 1.93649i) q^{5} +(5.19498 - 6.62804i) q^{6} +11.0873i q^{7} +(3.28790 - 7.29313i) q^{8} +(4.36484 + 7.56012i) q^{9} +(-3.51981 - 2.75879i) q^{10} +15.1236i q^{11} +(11.6622 + 12.1519i) q^{12} +(1.44060 + 2.49519i) q^{13} +(-21.9539 - 3.12018i) q^{14} +(8.15391 - 4.70766i) q^{15} +(13.5159 + 8.56281i) 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} +(-29.9463 - 4.25611i) q^{22} +(-25.0290 + 14.4505i) q^{23} +(-27.3439 + 19.6726i) q^{24} +(-2.50000 - 4.33013i) q^{25} +(-5.34615 + 2.15033i) q^{26} +1.13823i q^{27} +(12.3566 - 42.5928i) q^{28} +(-8.55435 - 14.8166i) q^{29} +(7.02697 + 17.4704i) q^{30} -56.4381i q^{31} +(-20.7589 + 24.3530i) q^{32} +(31.8403 - 55.1490i) q^{33} +(23.6883 + 18.5667i) q^{34} +(-21.4704 - 12.3959i) q^{35} +(-8.34235 - 33.9075i) q^{36} -49.1749 q^{37} +(20.6225 - 31.9172i) q^{38} -12.1318i q^{39} +(10.4471 + 14.5210i) q^{40} +(34.0978 - 59.0591i) q^{41} +(73.4867 + 57.5981i) q^{42} +(36.3654 + 20.9956i) q^{43} +(16.8551 - 58.0991i) q^{44} -19.5202 q^{45} +(-21.5698 - 53.6266i) q^{46} +(9.81283 - 5.66544i) q^{47} +(-31.2586 - 59.6800i) q^{48} -73.9272 q^{49} +(9.27764 - 3.73167i) q^{50} +(-54.8758 + 31.6826i) q^{51} +(-2.75336 - 11.1911i) q^{52} +(33.6392 + 58.2648i) q^{53} +(-2.25381 - 0.320322i) 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} +(-36.5707 + 8.99758i) q^{60} +(28.9650 + 50.1688i) q^{61} +(111.753 + 15.8829i) q^{62} +(-83.8210 + 48.3941i) q^{63} +(-42.3794 - 47.9581i) q^{64} -6.44255 q^{65} +(100.240 + 78.5671i) q^{66} +(20.5217 - 11.8482i) q^{67} +(-43.4303 + 41.6803i) q^{68} +121.692 q^{69} +(30.5874 - 39.0250i) q^{70} +(-37.6226 - 21.7214i) q^{71} +(69.4881 - 6.97641i) q^{72} +(-2.71972 + 4.71070i) q^{73} +(13.8388 - 97.3713i) q^{74} +21.0533i q^{75} +(57.3958 + 49.8169i) q^{76} -167.680 q^{77} +(24.0221 + 3.41413i) q^{78} +(-1.47604 - 0.852193i) q^{79} +(-31.6930 + 16.5998i) q^{80} +(41.6799 - 72.1917i) q^{81} +(107.347 + 84.1375i) q^{82} +25.5889i q^{83} +(-134.731 + 129.302i) q^{84} +(16.8250 + 29.1418i) q^{85} +(-51.8073 + 66.0986i) q^{86} +72.0390i q^{87} +(110.299 + 49.7250i) q^{88} +(6.56207 + 11.3658i) q^{89} +(5.49338 - 38.6519i) q^{90} +(-27.6648 + 15.9723i) q^{91} +(112.256 - 27.6187i) 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} +(20.8046 - 146.383i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.281421 + 1.98010i −0.140710 + 0.990051i
\(3\) −3.64654 − 2.10533i −1.21551 − 0.701777i −0.251558 − 0.967842i \(-0.580943\pi\)
−0.963955 + 0.266065i \(0.914276\pi\)
\(4\) −3.84160 − 1.11448i −0.960401 − 0.278621i
\(5\) −1.11803 + 1.93649i −0.223607 + 0.387298i
\(6\) 5.19498 − 6.62804i 0.865830 − 1.10467i
\(7\) 11.0873i 1.58389i 0.610590 + 0.791947i \(0.290932\pi\)
−0.610590 + 0.791947i \(0.709068\pi\)
\(8\) 3.28790 − 7.29313i 0.410987 − 0.911641i
\(9\) 4.36484 + 7.56012i 0.484982 + 0.840014i
\(10\) −3.51981 − 2.75879i −0.351981 − 0.275879i
\(11\) 15.1236i 1.37488i 0.726243 + 0.687438i \(0.241265\pi\)
−0.726243 + 0.687438i \(0.758735\pi\)
\(12\) 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 \(0.797987\pi\)
\(14\) −21.9539 − 3.12018i −1.56813 − 0.222870i
\(15\) 8.15391 − 4.70766i 0.543594 − 0.313844i
\(16\) 13.5159 + 8.56281i 0.844741 + 0.535176i
\(17\) 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
\(22\) −29.9463 − 4.25611i −1.36120 − 0.193459i
\(23\) −25.0290 + 14.4505i −1.08822 + 0.628282i −0.933101 − 0.359614i \(-0.882908\pi\)
−0.155116 + 0.987896i \(0.549575\pi\)
\(24\) −27.3439 + 19.6726i −1.13933 + 0.819691i
\(25\) −2.50000 − 4.33013i −0.100000 − 0.173205i
\(26\) −5.34615 + 2.15033i −0.205621 + 0.0827051i
\(27\) 1.13823i 0.0421567i
\(28\) 12.3566 − 42.5928i 0.441306 − 1.52117i
\(29\) −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 \(-0.635857\pi\)
0.413968 − 0.910292i \(-0.364143\pi\)
\(32\) −20.7589 + 24.3530i −0.648715 + 0.761031i
\(33\) 31.8403 − 55.1490i 0.964857 − 1.67118i
\(34\) 23.6883 + 18.5667i 0.696716 + 0.546078i
\(35\) −21.4704 − 12.3959i −0.613439 − 0.354169i
\(36\) −8.34235 − 33.9075i −0.231732 − 0.941876i
\(37\) −49.1749 −1.32905 −0.664526 − 0.747266i \(-0.731366\pi\)
−0.664526 + 0.747266i \(0.731366\pi\)
\(38\) 20.6225 − 31.9172i 0.542699 − 0.839928i
\(39\) − 12.1318i − 0.311071i
\(40\) 10.4471 + 14.5210i 0.261178 + 0.363024i
\(41\) 34.0978 − 59.0591i 0.831653 − 1.44047i −0.0650737 − 0.997880i \(-0.520728\pi\)
0.896727 − 0.442585i \(-0.145938\pi\)
\(42\) 73.4867 + 57.5981i 1.74968 + 1.37138i
\(43\) 36.3654 + 20.9956i 0.845707 + 0.488269i 0.859200 − 0.511640i \(-0.170962\pi\)
−0.0134933 + 0.999909i \(0.504295\pi\)
\(44\) 16.8551 − 58.0991i 0.383069 − 1.32043i
\(45\) −19.5202 −0.433781
\(46\) −21.5698 − 53.6266i −0.468908 − 1.16580i
\(47\) 9.81283 − 5.66544i 0.208784 − 0.120541i −0.391962 − 0.919981i \(-0.628204\pi\)
0.600746 + 0.799440i \(0.294870\pi\)
\(48\) −31.2586 − 59.6800i −0.651220 − 1.24333i
\(49\) −73.9272 −1.50872
\(50\) 9.27764 − 3.73167i 0.185553 − 0.0746333i
\(51\) −54.8758 + 31.6826i −1.07600 + 0.621227i
\(52\) −2.75336 − 11.1911i −0.0529493 − 0.215213i
\(53\) 33.6392 + 58.2648i 0.634702 + 1.09934i 0.986578 + 0.163290i \(0.0522105\pi\)
−0.351876 + 0.936047i \(0.614456\pi\)
\(54\) −2.25381 − 0.320322i −0.0417373 − 0.00593189i
\(55\) −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 −0.432952 − 0.901417i \(-0.642528\pi\)
−0.997126 + 0.0757608i \(0.975861\pi\)
\(60\) −36.5707 + 8.99758i −0.609512 + 0.149960i
\(61\) 28.9650 + 50.1688i 0.474836 + 0.822440i 0.999585 − 0.0288173i \(-0.00917409\pi\)
−0.524749 + 0.851257i \(0.675841\pi\)
\(62\) 111.753 + 15.8829i 1.80247 + 0.256175i
\(63\) −83.8210 + 48.3941i −1.33049 + 0.768160i
\(64\) −42.3794 − 47.9581i −0.662179 − 0.749346i
\(65\) −6.44255 −0.0991162
\(66\) 100.240 + 78.5671i 1.51879 + 1.19041i
\(67\) 20.5217 − 11.8482i 0.306294 − 0.176839i −0.338973 − 0.940796i \(-0.610080\pi\)
0.645267 + 0.763957i \(0.276746\pi\)
\(68\) −43.4303 + 41.6803i −0.638680 + 0.612945i
\(69\) 121.692 1.76366
\(70\) 30.5874 − 39.0250i 0.436963 − 0.557501i
\(71\) −37.6226 − 21.7214i −0.529895 − 0.305935i 0.211078 − 0.977469i \(-0.432302\pi\)
−0.740974 + 0.671534i \(0.765636\pi\)
\(72\) 69.4881 − 6.97641i 0.965113 − 0.0968947i
\(73\) −2.71972 + 4.71070i −0.0372565 + 0.0645301i −0.884052 − 0.467388i \(-0.845195\pi\)
0.846796 + 0.531918i \(0.178529\pi\)
\(74\) 13.8388 − 97.3713i 0.187011 − 1.31583i
\(75\) 21.0533i 0.280711i
\(76\) 57.3958 + 49.8169i 0.755208 + 0.655486i
\(77\) −167.680 −2.17766
\(78\) 24.0221 + 3.41413i 0.307976 + 0.0437709i
\(79\) −1.47604 − 0.852193i −0.0186841 − 0.0107873i 0.490629 − 0.871369i \(-0.336767\pi\)
−0.509313 + 0.860581i \(0.670100\pi\)
\(80\) −31.6930 + 16.5998i −0.396162 + 0.207498i
\(81\) 41.6799 − 72.1917i 0.514567 − 0.891256i
\(82\) 107.347 + 84.1375i 1.30911 + 1.02607i
\(83\) 25.5889i 0.308300i 0.988047 + 0.154150i \(0.0492638\pi\)
−0.988047 + 0.154150i \(0.950736\pi\)
\(84\) −134.731 + 129.302i −1.60394 + 1.53931i
\(85\) 16.8250 + 29.1418i 0.197941 + 0.342844i
\(86\) −51.8073 + 66.0986i −0.602411 + 0.768588i
\(87\) 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 \(0.809843\pi\)
\(90\) 5.49338 − 38.6519i 0.0610375 − 0.429465i
\(91\) −27.6648 + 15.9723i −0.304009 + 0.175520i
\(92\) 112.256 − 27.6187i 1.22018 − 0.300203i
\(93\) −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 \(0.0214718\pi\)
\(98\) 20.8046 − 146.383i 0.212292 − 1.49371i
\(99\) −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.11.36 yes 160
4.3 odd 2 inner 380.3.q.a.11.17 ✓ 160
19.7 even 3 inner 380.3.q.a.311.17 yes 160
76.7 odd 6 inner 380.3.q.a.311.36 yes 160
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
380.3.q.a.11.17 ✓ 160 4.3 odd 2 inner
380.3.q.a.11.36 yes 160 1.1 even 1 trivial
380.3.q.a.311.17 yes 160 19.7 even 3 inner
380.3.q.a.311.36 yes 160 76.7 odd 6 inner