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Properties

Label 420.2.i.a.139.13
Level $420$
Weight $2$
Character 420.139
Analytic conductor $3.354$
Analytic rank $0$
Dimension $48$
Inner twists $8$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [420,2,Mod(139,420)] 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("420.139"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(420, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0, 1, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 420 = 2^{2} \cdot 3 \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 420.i (of order \(2\), degree \(1\), 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: \(3.35371688489\)
Analytic rank: \(0\)
Dimension: \(48\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 139.13
Character \(\chi\) \(=\) 420.139
Dual form 420.2.i.a.139.16

$q$-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+(-0.846354 - 1.13300i) q^{2} -1.00000i q^{3} +(-0.567368 + 1.91784i) q^{4} +(-2.22361 + 0.235752i) q^{5} +(-1.13300 + 0.846354i) q^{6} +(2.54904 + 0.708793i) q^{7} +(2.65310 - 0.980342i) q^{8} -1.00000 q^{9} +(2.14906 + 2.31981i) q^{10} +5.27789i q^{11} +(1.91784 + 0.567368i) q^{12} +2.60538 q^{13} +(-1.35433 - 3.48795i) q^{14} +(0.235752 + 2.22361i) q^{15} +(-3.35619 - 2.17624i) q^{16} -3.66896 q^{17} +(0.846354 + 1.13300i) q^{18} +6.51939 q^{19} +(0.809470 - 4.39827i) q^{20} +(0.708793 - 2.54904i) q^{21} +(5.97984 - 4.46697i) q^{22} +3.54996 q^{23} +(-0.980342 - 2.65310i) q^{24} +(4.88884 - 1.04844i) q^{25} +(-2.20508 - 2.95189i) q^{26} +1.00000i q^{27} +(-2.80559 + 4.48650i) q^{28} +6.64523 q^{29} +(2.31981 - 2.14906i) q^{30} -4.51142 q^{31} +(0.374851 + 5.64442i) q^{32} +5.27789 q^{33} +(3.10524 + 4.15692i) q^{34} +(-5.83516 - 0.975136i) q^{35} +(0.567368 - 1.91784i) q^{36} +0.593655i q^{37} +(-5.51771 - 7.38645i) q^{38} -2.60538i q^{39} +(-5.66833 + 2.80537i) q^{40} -8.01200i q^{41} +(-3.48795 + 1.35433i) q^{42} -0.678893 q^{43} +(-10.1221 - 2.99451i) q^{44} +(2.22361 - 0.235752i) q^{45} +(-3.00452 - 4.02210i) q^{46} +5.79334i q^{47} +(-2.17624 + 3.35619i) q^{48} +(5.99522 + 3.61349i) q^{49} +(-5.32557 - 4.65170i) q^{50} +3.66896i q^{51} +(-1.47821 + 4.99669i) q^{52} +1.21027i q^{53} +(1.13300 - 0.846354i) q^{54} +(-1.24427 - 11.7360i) q^{55} +(7.45772 - 0.618433i) q^{56} -6.51939i q^{57} +(-5.62422 - 7.52903i) q^{58} +10.4372 q^{59} +(-4.39827 - 0.809470i) q^{60} +11.9597i q^{61} +(3.81826 + 5.11143i) q^{62} +(-2.54904 - 0.708793i) q^{63} +(6.07786 - 5.20189i) q^{64} +(-5.79334 + 0.614223i) q^{65} +(-4.46697 - 5.97984i) q^{66} -8.59277 q^{67} +(2.08165 - 7.03646i) q^{68} -3.54996i q^{69} +(3.83379 + 7.43654i) q^{70} -0.618749i q^{71} +(-2.65310 + 0.980342i) q^{72} +1.27551 q^{73} +(0.672610 - 0.502443i) q^{74} +(-1.04844 - 4.88884i) q^{75} +(-3.69889 + 12.5031i) q^{76} +(-3.74094 + 13.4536i) q^{77} +(-2.95189 + 2.20508i) q^{78} +6.11484i q^{79} +(7.97589 + 4.04787i) q^{80} +1.00000 q^{81} +(-9.07758 + 6.78099i) q^{82} +12.0472i q^{83} +(4.48650 + 2.80559i) q^{84} +(8.15831 - 0.864963i) q^{85} +(0.574584 + 0.769185i) q^{86} -6.64523i q^{87} +(5.17414 + 14.0028i) q^{88} -15.9259i q^{89} +(-2.14906 - 2.31981i) q^{90} +(6.64123 + 1.84668i) q^{91} +(-2.01413 + 6.80824i) q^{92} +4.51142i q^{93} +(6.56384 - 4.90322i) q^{94} +(-14.4965 + 1.53696i) q^{95} +(5.64442 - 0.374851i) q^{96} -0.241900 q^{97} +(-0.980011 - 9.85087i) q^{98} -5.27789i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 48 q - 48 q^{9} + 20 q^{14} - 16 q^{16} + 8 q^{25} - 16 q^{30} - 40 q^{44} + 16 q^{46} - 16 q^{49} + 48 q^{50} + 28 q^{56} - 32 q^{60} - 112 q^{74} + 48 q^{81} - 28 q^{84} + 56 q^{85} + 8 q^{86}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/420\mathbb{Z}\right)^\times\).

\(n\) \(211\) \(241\) \(281\) \(337\)
\(\chi(n)\) \(-1\) \(-1\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.846354 − 1.13300i −0.598463 − 0.801150i
\(3\) − 1.00000i − 0.577350i
\(4\) −0.567368 + 1.91784i −0.283684 + 0.958918i
\(5\) −2.22361 + 0.235752i −0.994427 + 0.105431i
\(6\) −1.13300 + 0.846354i −0.462544 + 0.345523i
\(7\) 2.54904 + 0.708793i 0.963447 + 0.267899i
\(8\) 2.65310 − 0.980342i 0.938012 − 0.346603i
\(9\) −1.00000 −0.333333
\(10\) 2.14906 + 2.31981i 0.679594 + 0.733589i
\(11\) 5.27789i 1.59134i 0.605727 + 0.795672i \(0.292882\pi\)
−0.605727 + 0.795672i \(0.707118\pi\)
\(12\) 1.91784 + 0.567368i 0.553631 + 0.163785i
\(13\) 2.60538 0.722603 0.361302 − 0.932449i \(-0.382333\pi\)
0.361302 + 0.932449i \(0.382333\pi\)
\(14\) −1.35433 − 3.48795i −0.361960 − 0.932194i
\(15\) 0.235752 + 2.22361i 0.0608708 + 0.574132i
\(16\) −3.35619 − 2.17624i −0.839047 − 0.544059i
\(17\) −3.66896 −0.889853 −0.444927 − 0.895567i \(-0.646770\pi\)
−0.444927 + 0.895567i \(0.646770\pi\)
\(18\) 0.846354 + 1.13300i 0.199488 + 0.267050i
\(19\) 6.51939 1.49565 0.747825 − 0.663896i \(-0.231098\pi\)
0.747825 + 0.663896i \(0.231098\pi\)
\(20\) 0.809470 − 4.39827i 0.181003 − 0.983483i
\(21\) 0.708793 − 2.54904i 0.154671 − 0.556246i
\(22\) 5.97984 − 4.46697i 1.27491 − 0.952361i
\(23\) 3.54996 0.740218 0.370109 − 0.928988i \(-0.379320\pi\)
0.370109 + 0.928988i \(0.379320\pi\)
\(24\) −0.980342 − 2.65310i −0.200111 − 0.541561i
\(25\) 4.88884 − 1.04844i 0.977768 − 0.209688i
\(26\) −2.20508 − 2.95189i −0.432451 − 0.578914i
\(27\) 1.00000i 0.192450i
\(28\) −2.80559 + 4.48650i −0.530207 + 0.847868i
\(29\) 6.64523 1.23399 0.616994 − 0.786968i \(-0.288350\pi\)
0.616994 + 0.786968i \(0.288350\pi\)
\(30\) 2.31981 − 2.14906i 0.423538 − 0.392364i
\(31\) −4.51142 −0.810275 −0.405138 − 0.914256i \(-0.632776\pi\)
−0.405138 + 0.914256i \(0.632776\pi\)
\(32\) 0.374851 + 5.64442i 0.0662649 + 0.997802i
\(33\) 5.27789 0.918763
\(34\) 3.10524 + 4.15692i 0.532544 + 0.712906i
\(35\) −5.83516 − 0.975136i −0.986322 − 0.164828i
\(36\) 0.567368 − 1.91784i 0.0945614 − 0.319639i
\(37\) 0.593655i 0.0975963i 0.998809 + 0.0487981i \(0.0155391\pi\)
−0.998809 + 0.0487981i \(0.984461\pi\)
\(38\) −5.51771 − 7.38645i −0.895091 − 1.19824i
\(39\) − 2.60538i − 0.417195i
\(40\) −5.66833 + 2.80537i −0.896241 + 0.443567i
\(41\) − 8.01200i − 1.25126i −0.780118 − 0.625632i \(-0.784841\pi\)
0.780118 − 0.625632i \(-0.215159\pi\)
\(42\) −3.48795 + 1.35433i −0.538202 + 0.208978i
\(43\) −0.678893 −0.103530 −0.0517651 − 0.998659i \(-0.516485\pi\)
−0.0517651 + 0.998659i \(0.516485\pi\)
\(44\) −10.1221 − 2.99451i −1.52597 − 0.451439i
\(45\) 2.22361 − 0.235752i 0.331476 − 0.0351438i
\(46\) −3.00452 − 4.02210i −0.442993 − 0.593026i
\(47\) 5.79334i 0.845046i 0.906352 + 0.422523i \(0.138855\pi\)
−0.906352 + 0.422523i \(0.861145\pi\)
\(48\) −2.17624 + 3.35619i −0.314113 + 0.484424i
\(49\) 5.99522 + 3.61349i 0.856461 + 0.516212i
\(50\) −5.32557 − 4.65170i −0.753149 − 0.657849i
\(51\) 3.66896i 0.513757i
\(52\) −1.47821 + 4.99669i −0.204991 + 0.692917i
\(53\) 1.21027i 0.166243i 0.996539 + 0.0831216i \(0.0264890\pi\)
−0.996539 + 0.0831216i \(0.973511\pi\)
\(54\) 1.13300 − 0.846354i 0.154181 − 0.115174i
\(55\) −1.24427 − 11.7360i −0.167778 − 1.58248i
\(56\) 7.45772 − 0.618433i 0.996579 − 0.0826416i
\(57\) − 6.51939i − 0.863514i
\(58\) −5.62422 − 7.52903i −0.738496 − 0.988610i
\(59\) 10.4372 1.35880 0.679401 − 0.733767i \(-0.262240\pi\)
0.679401 + 0.733767i \(0.262240\pi\)
\(60\) −4.39827 − 0.809470i −0.567814 − 0.104502i
\(61\) 11.9597i 1.53128i 0.643269 + 0.765640i \(0.277578\pi\)
−0.643269 + 0.765640i \(0.722422\pi\)
\(62\) 3.81826 + 5.11143i 0.484920 + 0.649153i
\(63\) −2.54904 − 0.708793i −0.321149 − 0.0892996i
\(64\) 6.07786 − 5.20189i 0.759732 − 0.650236i
\(65\) −5.79334 + 0.614223i −0.718576 + 0.0761850i
\(66\) −4.46697 − 5.97984i −0.549846 − 0.736068i
\(67\) −8.59277 −1.04977 −0.524887 − 0.851172i \(-0.675892\pi\)
−0.524887 + 0.851172i \(0.675892\pi\)
\(68\) 2.08165 − 7.03646i 0.252437 − 0.853296i
\(69\) − 3.54996i − 0.427365i
\(70\) 3.83379 + 7.43654i 0.458225 + 0.888836i
\(71\) − 0.618749i − 0.0734320i −0.999326 − 0.0367160i \(-0.988310\pi\)
0.999326 − 0.0367160i \(-0.0116897\pi\)
\(72\) −2.65310 + 0.980342i −0.312671 + 0.115534i
\(73\) 1.27551 0.149287 0.0746435 − 0.997210i \(-0.476218\pi\)
0.0746435 + 0.997210i \(0.476218\pi\)
\(74\) 0.672610 − 0.502443i 0.0781893 − 0.0584078i
\(75\) −1.04844 − 4.88884i −0.121063 − 0.564515i
\(76\) −3.69889 + 12.5031i −0.424292 + 1.43421i
\(77\) −3.74094 + 13.4536i −0.426319 + 1.53318i
\(78\) −2.95189 + 2.20508i −0.334236 + 0.249676i
\(79\) 6.11484i 0.687973i 0.938975 + 0.343986i \(0.111777\pi\)
−0.938975 + 0.343986i \(0.888223\pi\)
\(80\) 7.97589 + 4.04787i 0.891731 + 0.452565i
\(81\) 1.00000 0.111111
\(82\) −9.07758 + 6.78099i −1.00245 + 0.748835i
\(83\) 12.0472i 1.32235i 0.750233 + 0.661174i \(0.229941\pi\)
−0.750233 + 0.661174i \(0.770059\pi\)
\(84\) 4.48650 + 2.80559i 0.489517 + 0.306115i
\(85\) 8.15831 − 0.864963i 0.884893 − 0.0938184i
\(86\) 0.574584 + 0.769185i 0.0619590 + 0.0829433i
\(87\) − 6.64523i − 0.712443i
\(88\) 5.17414 + 14.0028i 0.551565 + 1.49270i
\(89\) − 15.9259i − 1.68814i −0.536234 − 0.844069i \(-0.680154\pi\)
0.536234 − 0.844069i \(-0.319846\pi\)
\(90\) −2.14906 − 2.31981i −0.226531 − 0.244530i
\(91\) 6.64123 + 1.84668i 0.696190 + 0.193584i
\(92\) −2.01413 + 6.80824i −0.209988 + 0.709808i
\(93\) 4.51142i 0.467813i
\(94\) 6.56384 − 4.90322i 0.677009 − 0.505729i
\(95\) −14.4965 + 1.53696i −1.48731 + 0.157689i
\(96\) 5.64442 − 0.374851i 0.576081 − 0.0382581i
\(97\) −0.241900 −0.0245612 −0.0122806 − 0.999925i \(-0.503909\pi\)
−0.0122806 + 0.999925i \(0.503909\pi\)
\(98\) −0.980011 − 9.85087i −0.0989961 − 0.995088i
\(99\) − 5.27789i − 0.530448i
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 420.2.i.a.139.13 ✓ 48
4.3 odd 2 inner 420.2.i.a.139.34 yes 48
5.4 even 2 inner 420.2.i.a.139.36 yes 48
7.6 odd 2 inner 420.2.i.a.139.14 yes 48
20.19 odd 2 inner 420.2.i.a.139.15 yes 48
28.27 even 2 inner 420.2.i.a.139.33 yes 48
35.34 odd 2 inner 420.2.i.a.139.35 yes 48
140.139 even 2 inner 420.2.i.a.139.16 yes 48
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
420.2.i.a.139.13 ✓ 48 1.1 even 1 trivial
420.2.i.a.139.14 yes 48 7.6 odd 2 inner
420.2.i.a.139.15 yes 48 20.19 odd 2 inner
420.2.i.a.139.16 yes 48 140.139 even 2 inner
420.2.i.a.139.33 yes 48 28.27 even 2 inner
420.2.i.a.139.34 yes 48 4.3 odd 2 inner
420.2.i.a.139.35 yes 48 35.34 odd 2 inner
420.2.i.a.139.36 yes 48 5.4 even 2 inner