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Properties

Label 420.2.bf.a.11.51
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 11.51
Character \(\chi\) \(=\) 420.11
Dual form 420.2.bf.a.191.51

$q$-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+(1.10993 - 0.876385i) q^{2} +(-0.737304 + 1.56729i) q^{3} +(0.463900 - 1.94546i) q^{4} +(0.866025 - 0.500000i) q^{5} +(0.555187 + 2.38574i) q^{6} +(1.77239 - 1.96434i) q^{7} +(-1.19007 - 2.56588i) q^{8} +(-1.91277 - 2.31113i) q^{9} +(0.523038 - 1.31394i) q^{10} +(-0.0979164 + 0.169596i) q^{11} +(2.70705 + 2.16146i) q^{12} +1.04433 q^{13} +(0.245723 - 3.73358i) q^{14} +(0.145119 + 1.72596i) q^{15} +(-3.56959 - 1.80500i) q^{16} +(3.55032 + 2.04978i) q^{17} +(-4.14848 - 0.888882i) q^{18} +(3.36056 - 1.94022i) q^{19} +(-0.570978 - 1.91676i) q^{20} +(1.77188 + 4.22616i) q^{21} +(0.0399509 + 0.274053i) q^{22} +(-0.636494 - 1.10244i) q^{23} +(4.89891 + 0.0266560i) q^{24} +(0.500000 - 0.866025i) q^{25} +(1.15914 - 0.915237i) q^{26} +(5.03249 - 1.29384i) q^{27} +(-2.99932 - 4.35937i) q^{28} +6.29611i q^{29} +(1.67368 + 1.78852i) q^{30} +(-1.62626 - 0.938920i) q^{31} +(-5.54388 + 1.12491i) q^{32} +(-0.193611 - 0.278507i) q^{33} +(5.73700 - 0.836329i) q^{34} +(0.552770 - 2.58736i) q^{35} +(-5.38354 + 2.64907i) q^{36} +(-2.36278 - 4.09246i) q^{37} +(2.02962 - 5.09866i) q^{38} +(-0.769991 + 1.63677i) q^{39} +(-2.31357 - 1.62708i) q^{40} +8.06002i q^{41} +(5.67041 + 3.13790i) q^{42} +5.76882i q^{43} +(0.284518 + 0.269168i) q^{44} +(-2.81207 - 1.04512i) q^{45} +(-1.67263 - 0.665820i) q^{46} +(-4.77067 - 8.26304i) q^{47} +(5.46082 - 4.26374i) q^{48} +(-0.717240 - 6.96316i) q^{49} +(-0.204005 - 1.39942i) q^{50} +(-5.83025 + 4.05305i) q^{51} +(0.484466 - 2.03170i) q^{52} +(-5.12381 - 2.95823i) q^{53} +(4.45182 - 5.84648i) q^{54} +0.195833i q^{55} +(-7.14952 - 2.21005i) q^{56} +(0.563124 + 6.69749i) q^{57} +(5.51781 + 6.98826i) q^{58} +(-6.40739 + 11.0979i) q^{59} +(3.42510 + 0.518352i) q^{60} +(5.83873 + 10.1130i) q^{61} +(-2.62789 + 0.383089i) q^{62} +(-7.93001 - 0.338919i) q^{63} +(-5.16747 + 6.10715i) q^{64} +(0.904419 - 0.522166i) q^{65} +(-0.458975 - 0.139446i) q^{66} +(7.97596 + 4.60492i) q^{67} +(5.63474 - 5.95609i) q^{68} +(2.19713 - 0.184734i) q^{69} +(-1.65399 - 3.35624i) q^{70} -12.0523 q^{71} +(-3.65376 + 7.65833i) q^{72} +(-6.54572 + 11.3375i) q^{73} +(-6.20910 - 2.47165i) q^{74} +(0.988657 + 1.42217i) q^{75} +(-2.21565 - 7.43789i) q^{76} +(0.159598 + 0.492932i) q^{77} +(0.579800 + 2.49151i) q^{78} +(-4.84428 + 2.79685i) q^{79} +(-3.99386 + 0.221625i) q^{80} +(-1.68266 + 8.84130i) q^{81} +(7.06368 + 8.94608i) q^{82} +12.0808 q^{83} +(9.04378 - 1.48660i) q^{84} +4.09955 q^{85} +(5.05571 + 6.40300i) q^{86} +(-9.86780 - 4.64215i) q^{87} +(0.551691 + 0.0494105i) q^{88} +(6.86736 - 3.96487i) q^{89} +(-4.03713 + 1.30445i) q^{90} +(1.85097 - 2.05142i) q^{91} +(-2.44002 + 0.726848i) q^{92} +(2.67060 - 1.85654i) q^{93} +(-12.5367 - 4.99048i) q^{94} +(1.94022 - 3.36056i) q^{95} +(2.32446 - 9.51824i) q^{96} -17.6878 q^{97} +(-6.89849 - 7.10006i) q^{98} +(0.579250 - 0.0981001i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.10993 − 0.876385i 0.784841 − 0.619697i
\(3\) −0.737304 + 1.56729i −0.425683 + 0.904873i
\(4\) 0.463900 − 1.94546i 0.231950 − 0.972728i
\(5\) 0.866025 − 0.500000i 0.387298 − 0.223607i
\(6\) 0.555187 + 2.38574i 0.226654 + 0.973975i
\(7\) 1.77239 − 1.96434i 0.669902 − 0.742450i
\(8\) −1.19007 − 2.56588i −0.420753 − 0.907175i
\(9\) −1.91277 − 2.31113i −0.637589 − 0.770377i
\(10\) 0.523038 − 1.31394i 0.165399 − 0.415504i
\(11\) −0.0979164 + 0.169596i −0.0295229 + 0.0511352i −0.880409 − 0.474214i \(-0.842732\pi\)
0.850886 + 0.525350i \(0.176065\pi\)
\(12\) 2.70705 + 2.16146i 0.781457 + 0.623959i
\(13\) 1.04433 0.289646 0.144823 − 0.989458i \(-0.453739\pi\)
0.144823 + 0.989458i \(0.453739\pi\)
\(14\) 0.245723 − 3.73358i 0.0656723 − 0.997841i
\(15\) 0.145119 + 1.72596i 0.0374695 + 0.445641i
\(16\) −3.56959 − 1.80500i −0.892398 − 0.451249i
\(17\) 3.55032 + 2.04978i 0.861078 + 0.497144i 0.864373 − 0.502851i \(-0.167715\pi\)
−0.00329491 + 0.999995i \(0.501049\pi\)
\(18\) −4.14848 − 0.888882i −0.977806 − 0.209511i
\(19\) 3.36056 − 1.94022i 0.770966 − 0.445117i −0.0622534 − 0.998060i \(-0.519829\pi\)
0.833219 + 0.552943i \(0.186495\pi\)
\(20\) −0.570978 − 1.91676i −0.127675 − 0.428601i
\(21\) 1.77188 + 4.22616i 0.386657 + 0.922224i
\(22\) 0.0399509 + 0.274053i 0.00851755 + 0.0584282i
\(23\) −0.636494 − 1.10244i −0.132718 − 0.229875i 0.792005 − 0.610514i \(-0.209037\pi\)
−0.924723 + 0.380640i \(0.875704\pi\)
\(24\) 4.89891 + 0.0266560i 0.999985 + 0.00544113i
\(25\) 0.500000 − 0.866025i 0.100000 − 0.173205i
\(26\) 1.15914 − 0.915237i 0.227326 − 0.179493i
\(27\) 5.03249 − 1.29384i 0.968503 − 0.249000i
\(28\) −2.99932 − 4.35937i −0.566817 − 0.823843i
\(29\) 6.29611i 1.16916i 0.811337 + 0.584579i \(0.198740\pi\)
−0.811337 + 0.584579i \(0.801260\pi\)
\(30\) 1.67368 + 1.78852i 0.305570 + 0.326538i
\(31\) −1.62626 − 0.938920i −0.292084 − 0.168635i 0.346797 − 0.937940i \(-0.387269\pi\)
−0.638881 + 0.769305i \(0.720603\pi\)
\(32\) −5.54388 + 1.12491i −0.980028 + 0.198858i
\(33\) −0.193611 − 0.278507i −0.0337034 − 0.0484818i
\(34\) 5.73700 − 0.836329i 0.983888 − 0.143429i
\(35\) 0.552770 − 2.58736i 0.0934351 − 0.437344i
\(36\) −5.38354 + 2.64907i −0.897256 + 0.441511i
\(37\) −2.36278 − 4.09246i −0.388439 − 0.672797i 0.603800 − 0.797135i \(-0.293652\pi\)
−0.992240 + 0.124339i \(0.960319\pi\)
\(38\) 2.02962 − 5.09866i 0.329247 − 0.827111i
\(39\) −0.769991 + 1.63677i −0.123297 + 0.262092i
\(40\) −2.31357 − 1.62708i −0.365807 − 0.257264i
\(41\) 8.06002i 1.25876i 0.777096 + 0.629382i \(0.216692\pi\)
−0.777096 + 0.629382i \(0.783308\pi\)
\(42\) 5.67041 + 3.13790i 0.874964 + 0.484189i
\(43\) 5.76882i 0.879737i 0.898062 + 0.439869i \(0.144975\pi\)
−0.898062 + 0.439869i \(0.855025\pi\)
\(44\) 0.284518 + 0.269168i 0.0428927 + 0.0405786i
\(45\) −2.81207 − 1.04512i −0.419199 − 0.155797i
\(46\) −1.67263 − 0.665820i −0.246615 − 0.0981699i
\(47\) −4.77067 − 8.26304i −0.695873 − 1.20529i −0.969885 − 0.243562i \(-0.921684\pi\)
0.274012 − 0.961726i \(-0.411649\pi\)
\(48\) 5.46082 − 4.26374i 0.788201 − 0.615418i
\(49\) −0.717240 − 6.96316i −0.102463 − 0.994737i
\(50\) −0.204005 − 1.39942i −0.0288507 − 0.197908i
\(51\) −5.83025 + 4.05305i −0.816398 + 0.567541i
\(52\) 0.484466 − 2.03170i 0.0671834 − 0.281746i
\(53\) −5.12381 − 2.95823i −0.703809 − 0.406345i 0.104955 − 0.994477i \(-0.466530\pi\)
−0.808765 + 0.588132i \(0.799863\pi\)
\(54\) 4.45182 − 5.84648i 0.605816 − 0.795605i
\(55\) 0.195833i 0.0264061i
\(56\) −7.14952 − 2.21005i −0.955395 − 0.295331i
\(57\) 0.563124 + 6.69749i 0.0745876 + 0.887104i
\(58\) 5.51781 + 6.98826i 0.724525 + 0.917603i
\(59\) −6.40739 + 11.0979i −0.834172 + 1.44483i 0.0605312 + 0.998166i \(0.480721\pi\)
−0.894703 + 0.446662i \(0.852613\pi\)
\(60\) 3.42510 + 0.518352i 0.442179 + 0.0669190i
\(61\) 5.83873 + 10.1130i 0.747573 + 1.29483i 0.948983 + 0.315327i \(0.102114\pi\)
−0.201410 + 0.979507i \(0.564552\pi\)
\(62\) −2.62789 + 0.383089i −0.333742 + 0.0486523i
\(63\) −7.93001 − 0.338919i −0.999088 − 0.0426997i
\(64\) −5.16747 + 6.10715i −0.645934 + 0.763393i
\(65\) 0.904419 − 0.522166i 0.112179 − 0.0647668i
\(66\) −0.458975 − 0.139446i −0.0564959 − 0.0171646i
\(67\) 7.97596 + 4.60492i 0.974419 + 0.562581i 0.900581 − 0.434689i \(-0.143142\pi\)
0.0738383 + 0.997270i \(0.476475\pi\)
\(68\) 5.63474 − 5.95609i 0.683313 − 0.722282i
\(69\) 2.19713 − 0.184734i 0.264503 − 0.0222394i
\(70\) −1.65399 − 3.35624i −0.197689 − 0.401147i
\(71\) −12.0523 −1.43034 −0.715172 − 0.698949i \(-0.753651\pi\)
−0.715172 + 0.698949i \(0.753651\pi\)
\(72\) −3.65376 + 7.65833i −0.430600 + 0.902543i
\(73\) −6.54572 + 11.3375i −0.766119 + 1.32696i 0.173535 + 0.984828i \(0.444481\pi\)
−0.939653 + 0.342129i \(0.888852\pi\)
\(74\) −6.20910 − 2.47165i −0.721793 − 0.287323i
\(75\) 0.988657 + 1.42217i 0.114160 + 0.164218i
\(76\) −2.21565 − 7.43789i −0.254152 − 0.853185i
\(77\) 0.159598 + 0.492932i 0.0181878 + 0.0561748i
\(78\) 0.579800 + 2.49151i 0.0656494 + 0.282108i
\(79\) −4.84428 + 2.79685i −0.545024 + 0.314670i −0.747113 − 0.664698i \(-0.768560\pi\)
0.202089 + 0.979367i \(0.435227\pi\)
\(80\) −3.99386 + 0.221625i −0.446527 + 0.0247784i
\(81\) −1.68266 + 8.84130i −0.186962 + 0.982367i
\(82\) 7.06368 + 8.94608i 0.780053 + 0.987929i
\(83\) 12.0808 1.32604 0.663019 − 0.748602i \(-0.269275\pi\)
0.663019 + 0.748602i \(0.269275\pi\)
\(84\) 9.04378 − 1.48660i 0.986758 − 0.162202i
\(85\) 4.09955 0.444659
\(86\) 5.05571 + 6.40300i 0.545171 + 0.690454i
\(87\) −9.86780 − 4.64215i −1.05794 − 0.497691i
\(88\) 0.551691 + 0.0494105i 0.0588104 + 0.00526718i
\(89\) 6.86736 − 3.96487i 0.727939 − 0.420276i −0.0897288 − 0.995966i \(-0.528600\pi\)
0.817668 + 0.575691i \(0.195267\pi\)
\(90\) −4.03713 + 1.30445i −0.425551 + 0.137501i
\(91\) 1.85097 − 2.05142i 0.194034 − 0.215047i
\(92\) −2.44002 + 0.726848i −0.254389 + 0.0757792i
\(93\) 2.67060 − 1.85654i 0.276928 − 0.192514i
\(94\) −12.5367 − 4.99048i −1.29306 − 0.514728i
\(95\) 1.94022 − 3.36056i 0.199062 − 0.344786i
\(96\) 2.32446 − 9.51824i 0.237240 − 0.971451i
\(97\) −17.6878 −1.79592 −0.897962 − 0.440072i \(-0.854953\pi\)
−0.897962 + 0.440072i \(0.854953\pi\)
\(98\) −6.89849 − 7.10006i −0.696853 − 0.717214i
\(99\) 0.579250 − 0.0981001i 0.0582168 − 0.00985943i
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.11.51 yes 128
3.2 odd 2 inner 420.2.bf.a.11.14 yes 128
4.3 odd 2 inner 420.2.bf.a.11.56 yes 128
7.2 even 3 inner 420.2.bf.a.191.9 yes 128
12.11 even 2 inner 420.2.bf.a.11.9 ✓ 128
21.2 odd 6 inner 420.2.bf.a.191.56 yes 128
28.23 odd 6 inner 420.2.bf.a.191.14 yes 128
84.23 even 6 inner 420.2.bf.a.191.51 yes 128
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
420.2.bf.a.11.9 ✓ 128 12.11 even 2 inner
420.2.bf.a.11.14 yes 128 3.2 odd 2 inner
420.2.bf.a.11.51 yes 128 1.1 even 1 trivial
420.2.bf.a.11.56 yes 128 4.3 odd 2 inner
420.2.bf.a.191.9 yes 128 7.2 even 3 inner
420.2.bf.a.191.14 yes 128 28.23 odd 6 inner
420.2.bf.a.191.51 yes 128 84.23 even 6 inner
420.2.bf.a.191.56 yes 128 21.2 odd 6 inner