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Properties

Label 420.2.bf.a.11.12
Level $420$
Weight $2$
Character 420.11
Analytic conductor $3.354$
Analytic rank $0$
Dimension $128$
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(11,420)] 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("420.11"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(420, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([3, 3, 0, 4])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 420 = 2^{2} \cdot 3 \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 420.bf (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: \(3.35371688489\)
Analytic rank: \(0\)
Dimension: \(128\)
Relative dimension: \(64\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 11.12
Character \(\chi\) \(=\) 420.11
Dual form 420.2.bf.a.191.12

$q$-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+(-1.19058 + 0.763225i) q^{2} +(-0.457845 - 1.67044i) q^{3} +(0.834976 - 1.81736i) q^{4} +(0.866025 - 0.500000i) q^{5} +(1.82002 + 1.63936i) q^{6} +(0.775771 + 2.52946i) q^{7} +(0.392949 + 2.80100i) q^{8} +(-2.58076 + 1.52961i) q^{9} +(-0.649463 + 1.25626i) q^{10} +(-3.10364 + 5.37566i) q^{11} +(-3.41809 - 0.562710i) q^{12} +5.38039 q^{13} +(-2.85417 - 2.41945i) q^{14} +(-1.23173 - 1.21772i) q^{15} +(-2.60563 - 3.03491i) q^{16} +(2.76539 + 1.59660i) q^{17} +(1.90517 - 3.79082i) q^{18} +(2.56104 - 1.47862i) q^{19} +(-0.185572 - 1.99137i) q^{20} +(3.87014 - 2.45398i) q^{21} +(-0.407696 - 8.76895i) q^{22} +(0.955531 + 1.65503i) q^{23} +(4.49900 - 1.93882i) q^{24} +(0.500000 - 0.866025i) q^{25} +(-6.40580 + 4.10644i) q^{26} +(3.73670 + 3.61068i) q^{27} +(5.24471 + 0.702184i) q^{28} -6.85121i q^{29} +(2.39587 + 0.509717i) q^{30} +(-1.18808 - 0.685936i) q^{31} +(5.41854 + 1.62464i) q^{32} +(10.4007 + 2.72324i) q^{33} +(-4.51100 + 0.209730i) q^{34} +(1.93657 + 1.80269i) q^{35} +(0.624981 + 5.96736i) q^{36} +(4.48674 + 7.77126i) q^{37} +(-1.92061 + 3.71507i) q^{38} +(-2.46338 - 8.98763i) q^{39} +(1.74080 + 2.22926i) q^{40} +3.80179i q^{41} +(-2.73478 + 5.87545i) q^{42} -3.69756i q^{43} +(7.17808 + 10.1290i) q^{44} +(-1.47020 + 2.61506i) q^{45} +(-2.40080 - 1.24116i) q^{46} +(-3.61736 - 6.26545i) q^{47} +(-3.87668 + 5.74207i) q^{48} +(-5.79636 + 3.92456i) q^{49} +(0.0656803 + 1.41269i) q^{50} +(1.40091 - 5.35043i) q^{51} +(4.49250 - 9.77812i) q^{52} +(2.45478 + 1.41727i) q^{53} +(-7.20462 - 1.44688i) q^{54} +6.20728i q^{55} +(-6.78018 + 3.16688i) q^{56} +(-3.64250 - 3.60109i) q^{57} +(5.22901 + 8.15694i) q^{58} +(-2.47349 + 4.28421i) q^{59} +(-3.24151 + 1.22173i) q^{60} +(-0.574593 - 0.995225i) q^{61} +(1.93803 - 0.0901049i) q^{62} +(-5.87116 - 5.34130i) q^{63} +(-7.69118 + 2.20130i) q^{64} +(4.65955 - 2.69019i) q^{65} +(-14.4614 + 4.69585i) q^{66} +(7.35911 + 4.24878i) q^{67} +(5.21064 - 3.69260i) q^{68} +(2.32714 - 2.35390i) q^{69} +(-3.68151 - 0.668220i) q^{70} +6.39472 q^{71} +(-5.29853 - 6.62764i) q^{72} +(0.778221 - 1.34792i) q^{73} +(-11.2731 - 5.82794i) q^{74} +(-1.67557 - 0.438716i) q^{75} +(-0.548779 - 5.88896i) q^{76} +(-16.0053 - 3.68026i) q^{77} +(9.79244 + 8.82040i) q^{78} +(-1.70113 + 0.982146i) q^{79} +(-3.77400 - 1.32550i) q^{80} +(4.32061 - 7.89508i) q^{81} +(-2.90162 - 4.52634i) q^{82} +8.02951 q^{83} +(-1.22830 - 9.08247i) q^{84} +3.19320 q^{85} +(2.82207 + 4.40225i) q^{86} +(-11.4446 + 3.13679i) q^{87} +(-16.2768 - 6.58093i) q^{88} +(5.16092 - 2.97966i) q^{89} +(-0.245482 - 4.23553i) q^{90} +(4.17395 + 13.6095i) q^{91} +(3.80563 - 0.354639i) q^{92} +(-0.601862 + 2.29866i) q^{93} +(9.08872 + 4.69868i) q^{94} +(1.47862 - 2.56104i) q^{95} +(0.233015 - 9.79519i) q^{96} -14.0167 q^{97} +(3.90572 - 9.09645i) q^{98} +(-0.212905 - 18.6206i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 128 q + 10 q^{12} + 16 q^{13} + 16 q^{16} + 10 q^{18} - 4 q^{21} - 56 q^{22} + 20 q^{24} + 64 q^{25} - 12 q^{28} - 48 q^{34} - 40 q^{36} + 8 q^{37} - 54 q^{42} - 4 q^{45} - 16 q^{46} - 36 q^{48} - 24 q^{49}+ \cdots - 96 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(211\) \(241\) \(281\) \(337\)
\(\chi(n)\) \(-1\) \(e\left(\frac{2}{3}\right)\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.19058 + 0.763225i −0.841869 + 0.539681i
\(3\) −0.457845 − 1.67044i −0.264337 − 0.964430i
\(4\) 0.834976 − 1.81736i 0.417488 − 0.908682i
\(5\) 0.866025 − 0.500000i 0.387298 − 0.223607i
\(6\) 1.82002 + 1.63936i 0.743022 + 0.669267i
\(7\) 0.775771 + 2.52946i 0.293214 + 0.956047i
\(8\) 0.392949 + 2.80100i 0.138928 + 0.990302i
\(9\) −2.58076 + 1.52961i −0.860252 + 0.509869i
\(10\) −0.649463 + 1.25626i −0.205378 + 0.397265i
\(11\) −3.10364 + 5.37566i −0.935783 + 1.62082i −0.162551 + 0.986700i \(0.551972\pi\)
−0.773232 + 0.634124i \(0.781361\pi\)
\(12\) −3.41809 − 0.562710i −0.986718 − 0.162440i
\(13\) 5.38039 1.49225 0.746125 − 0.665805i \(-0.231912\pi\)
0.746125 + 0.665805i \(0.231912\pi\)
\(14\) −2.85417 − 2.41945i −0.762808 − 0.646625i
\(15\) −1.23173 − 1.21772i −0.318030 − 0.314415i
\(16\) −2.60563 − 3.03491i −0.651407 − 0.758728i
\(17\) 2.76539 + 1.59660i 0.670706 + 0.387233i 0.796344 − 0.604844i \(-0.206764\pi\)
−0.125638 + 0.992076i \(0.540098\pi\)
\(18\) 1.90517 − 3.79082i 0.449054 − 0.893505i
\(19\) 2.56104 − 1.47862i 0.587543 − 0.339218i −0.176582 − 0.984286i \(-0.556504\pi\)
0.764125 + 0.645068i \(0.223171\pi\)
\(20\) −0.185572 − 1.99137i −0.0414951 − 0.445284i
\(21\) 3.87014 − 2.45398i 0.844534 − 0.535502i
\(22\) −0.407696 − 8.76895i −0.0869211 − 1.86955i
\(23\) 0.955531 + 1.65503i 0.199242 + 0.345097i 0.948283 − 0.317427i \(-0.102819\pi\)
−0.749041 + 0.662524i \(0.769485\pi\)
\(24\) 4.49900 − 1.93882i 0.918354 − 0.395760i
\(25\) 0.500000 − 0.866025i 0.100000 − 0.173205i
\(26\) −6.40580 + 4.10644i −1.25628 + 0.805340i
\(27\) 3.73670 + 3.61068i 0.719129 + 0.694877i
\(28\) 5.24471 + 0.702184i 0.991156 + 0.132700i
\(29\) − 6.85121i − 1.27224i −0.771591 − 0.636119i \(-0.780539\pi\)
0.771591 − 0.636119i \(-0.219461\pi\)
\(30\) 2.39587 + 0.509717i 0.437424 + 0.0930612i
\(31\) −1.18808 − 0.685936i −0.213385 − 0.123198i 0.389499 − 0.921027i \(-0.372648\pi\)
−0.602883 + 0.797829i \(0.705982\pi\)
\(32\) 5.41854 + 1.62464i 0.957871 + 0.287198i
\(33\) 10.4007 + 2.72324i 1.81053 + 0.474055i
\(34\) −4.51100 + 0.209730i −0.773629 + 0.0359685i
\(35\) 1.93657 + 1.80269i 0.327340 + 0.304711i
\(36\) 0.624981 + 5.96736i 0.104163 + 0.994560i
\(37\) 4.48674 + 7.77126i 0.737615 + 1.27759i 0.953566 + 0.301183i \(0.0973815\pi\)
−0.215951 + 0.976404i \(0.569285\pi\)
\(38\) −1.92061 + 3.71507i −0.311565 + 0.602663i
\(39\) −2.46338 − 8.98763i −0.394457 − 1.43917i
\(40\) 1.74080 + 2.22926i 0.275245 + 0.352477i
\(41\) 3.80179i 0.593740i 0.954918 + 0.296870i \(0.0959427\pi\)
−0.954918 + 0.296870i \(0.904057\pi\)
\(42\) −2.73478 + 5.87545i −0.421986 + 0.906602i
\(43\) − 3.69756i − 0.563872i −0.959433 − 0.281936i \(-0.909023\pi\)
0.959433 − 0.281936i \(-0.0909766\pi\)
\(44\) 7.17808 + 10.1290i 1.08214 + 1.52700i
\(45\) −1.47020 + 2.61506i −0.219164 + 0.389830i
\(46\) −2.40080 − 1.24116i −0.353978 − 0.183000i
\(47\) −3.61736 − 6.26545i −0.527646 − 0.913910i −0.999481 − 0.0322231i \(-0.989741\pi\)
0.471834 − 0.881687i \(-0.343592\pi\)
\(48\) −3.87668 + 5.74207i −0.559550 + 0.828797i
\(49\) −5.79636 + 3.92456i −0.828051 + 0.560652i
\(50\) 0.0656803 + 1.41269i 0.00928860 + 0.199784i
\(51\) 1.40091 − 5.35043i 0.196167 − 0.749210i
\(52\) 4.49250 − 9.77812i 0.622997 − 1.35598i
\(53\) 2.45478 + 1.41727i 0.337191 + 0.194677i 0.659029 − 0.752118i \(-0.270967\pi\)
−0.321838 + 0.946795i \(0.604301\pi\)
\(54\) −7.20462 − 1.44688i −0.980425 − 0.196895i
\(55\) 6.20728i 0.836990i
\(56\) −6.78018 + 3.16688i −0.906040 + 0.423192i
\(57\) −3.64250 − 3.60109i −0.482462 − 0.476977i
\(58\) 5.22901 + 8.15694i 0.686603 + 1.07106i
\(59\) −2.47349 + 4.28421i −0.322021 + 0.557757i −0.980905 − 0.194488i \(-0.937695\pi\)
0.658884 + 0.752245i \(0.271029\pi\)
\(60\) −3.24151 + 1.22173i −0.418477 + 0.157724i
\(61\) −0.574593 − 0.995225i −0.0735692 − 0.127426i 0.826894 − 0.562358i \(-0.190106\pi\)
−0.900463 + 0.434932i \(0.856772\pi\)
\(62\) 1.93803 − 0.0901049i 0.246129 − 0.0114433i
\(63\) −5.87116 − 5.34130i −0.739696 − 0.672941i
\(64\) −7.69118 + 2.20130i −0.961398 + 0.275162i
\(65\) 4.65955 − 2.69019i 0.577946 − 0.333677i
\(66\) −14.4614 + 4.69585i −1.78007 + 0.578019i
\(67\) 7.35911 + 4.24878i 0.899058 + 0.519072i 0.876895 − 0.480683i \(-0.159611\pi\)
0.0221639 + 0.999754i \(0.492944\pi\)
\(68\) 5.21064 − 3.69260i 0.631883 − 0.447794i
\(69\) 2.32714 − 2.35390i 0.280155 − 0.283377i
\(70\) −3.68151 − 0.668220i −0.440024 − 0.0798676i
\(71\) 6.39472 0.758914 0.379457 − 0.925209i \(-0.376111\pi\)
0.379457 + 0.925209i \(0.376111\pi\)
\(72\) −5.29853 − 6.62764i −0.624438 − 0.781075i
\(73\) 0.778221 − 1.34792i 0.0910839 − 0.157762i −0.816884 − 0.576803i \(-0.804300\pi\)
0.907967 + 0.419041i \(0.137634\pi\)
\(74\) −11.2731 − 5.82794i −1.31047 − 0.677484i
\(75\) −1.67557 − 0.438716i −0.193478 − 0.0506586i
\(76\) −0.548779 − 5.88896i −0.0629492 − 0.675510i
\(77\) −16.0053 − 3.68026i −1.82397 − 0.419405i
\(78\) 9.79244 + 8.82040i 1.10878 + 0.998714i
\(79\) −1.70113 + 0.982146i −0.191392 + 0.110500i −0.592634 − 0.805472i \(-0.701912\pi\)
0.401242 + 0.915972i \(0.368579\pi\)
\(80\) −3.77400 − 1.32550i −0.421946 − 0.148195i
\(81\) 4.32061 − 7.89508i 0.480068 − 0.877231i
\(82\) −2.90162 − 4.52634i −0.320430 − 0.499851i
\(83\) 8.02951 0.881354 0.440677 − 0.897666i \(-0.354738\pi\)
0.440677 + 0.897666i \(0.354738\pi\)
\(84\) −1.22830 − 9.08247i −0.134019 − 0.990979i
\(85\) 3.19320 0.346351
\(86\) 2.82207 + 4.40225i 0.304311 + 0.474707i
\(87\) −11.4446 + 3.13679i −1.22698 + 0.336299i
\(88\) −16.2768 − 6.58093i −1.73511 − 0.701530i
\(89\) 5.16092 − 2.97966i 0.547057 − 0.315843i −0.200877 − 0.979616i \(-0.564379\pi\)
0.747934 + 0.663773i \(0.231046\pi\)
\(90\) −0.245482 − 4.23553i −0.0258761 − 0.446464i
\(91\) 4.17395 + 13.6095i 0.437548 + 1.42666i
\(92\) 3.80563 − 0.354639i 0.396765 − 0.0369736i
\(93\) −0.601862 + 2.29866i −0.0624102 + 0.238360i
\(94\) 9.08872 + 4.69868i 0.937430 + 0.484632i
\(95\) 1.47862 − 2.56104i 0.151703 − 0.262757i
\(96\) 0.233015 − 9.79519i 0.0237819 − 0.999717i
\(97\) −14.0167 −1.42318 −0.711589 − 0.702596i \(-0.752024\pi\)
−0.711589 + 0.702596i \(0.752024\pi\)
\(98\) 3.90572 − 9.09645i 0.394538 − 0.918880i
\(99\) −0.212905 − 18.6206i −0.0213977 − 1.87144i
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.bf.a.11.12 yes 128
3.2 odd 2 inner 420.2.bf.a.11.53 yes 128
4.3 odd 2 inner 420.2.bf.a.11.10 ✓ 128
7.2 even 3 inner 420.2.bf.a.191.55 yes 128
12.11 even 2 inner 420.2.bf.a.11.55 yes 128
21.2 odd 6 inner 420.2.bf.a.191.10 yes 128
28.23 odd 6 inner 420.2.bf.a.191.53 yes 128
84.23 even 6 inner 420.2.bf.a.191.12 yes 128
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
420.2.bf.a.11.10 ✓ 128 4.3 odd 2 inner
420.2.bf.a.11.12 yes 128 1.1 even 1 trivial
420.2.bf.a.11.53 yes 128 3.2 odd 2 inner
420.2.bf.a.11.55 yes 128 12.11 even 2 inner
420.2.bf.a.191.10 yes 128 21.2 odd 6 inner
420.2.bf.a.191.12 yes 128 84.23 even 6 inner
420.2.bf.a.191.53 yes 128 28.23 odd 6 inner
420.2.bf.a.191.55 yes 128 7.2 even 3 inner