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Properties

Label 420.2.bf.a.11.14
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 11.14
Character \(\chi\) \(=\) 420.11
Dual form 420.2.bf.a.191.14

$q$-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} +(0.988657 - 1.42217i) q^{3} +(0.463900 - 1.94546i) q^{4} +(-0.866025 + 0.500000i) q^{5} +(0.149022 + 2.44495i) q^{6} +(1.77239 - 1.96434i) q^{7} +(1.19007 + 2.56588i) q^{8} +(-1.04512 - 2.81207i) q^{9} +(0.523038 - 1.31394i) q^{10} +(0.0979164 - 0.169596i) q^{11} +(-2.30812 - 2.58313i) 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} +(3.62446 + 2.20528i) q^{18} +(3.36056 - 1.94022i) q^{19} +(0.570978 + 1.91676i) q^{20} +(-1.04133 - 4.46269i) q^{21} +(0.0399509 + 0.274053i) q^{22} +(0.636494 + 1.10244i) q^{23} +(4.82568 + 0.844298i) 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} +(-1.35153 - 2.04288i) q^{30} +(-1.62626 - 0.938920i) q^{31} +(5.54388 - 1.12491i) q^{32} +(-0.144388 - 0.306926i) q^{33} +(5.73700 - 0.836329i) q^{34} +(-0.552770 + 2.58736i) q^{35} +(-5.95558 + 0.728706i) q^{36} +(-2.36278 - 4.09246i) q^{37} +(-2.02962 + 5.09866i) q^{38} +(1.03249 - 1.48521i) q^{39} +(-2.31357 - 1.62708i) q^{40} -8.06002i q^{41} +(5.06684 + 4.04069i) q^{42} +5.76882i q^{43} +(-0.284518 - 0.269168i) q^{44} +(2.31113 + 1.91277i) q^{45} +(-1.67263 - 0.665820i) q^{46} +(4.77067 + 8.26304i) q^{47} +(-6.09611 + 3.29204i) q^{48} +(-0.717240 - 6.96316i) q^{49} +(0.204005 + 1.39942i) q^{50} +(-6.42517 + 3.02262i) q^{51} +(0.484466 - 2.03170i) q^{52} +(5.12381 + 2.95823i) q^{53} +(6.71963 - 2.97432i) 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} +(3.29046 + 1.08300i) q^{60} +(5.83873 + 10.1130i) q^{61} +(2.62789 - 0.383089i) q^{62} +(-7.37621 - 2.93113i) q^{63} +(-5.16747 + 6.10715i) q^{64} +(-0.904419 + 0.522166i) q^{65} +(0.429246 + 0.214127i) 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} +(5.97167 - 6.02820i) q^{72} +(-6.54572 + 11.3375i) q^{73} +(6.20910 + 2.47165i) q^{74} +(-0.737304 - 1.56729i) q^{75} +(-2.21565 - 7.43789i) q^{76} +(-0.159598 - 0.492932i) q^{77} +(0.155629 + 2.55334i) q^{78} +(-4.84428 + 2.79685i) q^{79} +(3.99386 - 0.221625i) q^{80} +(-6.81547 + 5.87787i) q^{81} +(7.06368 + 8.94608i) q^{82} -12.0808 q^{83} +(-9.16504 - 0.0443939i) q^{84} +4.09955 q^{85} +(-5.05571 - 6.40300i) q^{86} +(-8.95412 - 6.22469i) q^{87} +(0.551691 + 0.0494105i) q^{88} +(-6.86736 + 3.96487i) q^{89} +(-4.24152 - 0.0976011i) q^{90} +(1.85097 - 2.05142i) q^{91} +(2.44002 - 0.726848i) q^{92} +(-2.94311 + 1.38454i) q^{93} +(-12.5367 - 4.99048i) q^{94} +(-1.94022 + 3.36056i) q^{95} +(3.88118 - 8.99647i) 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
\(3\) 0.988657 − 1.42217i 0.570801 − 0.821088i
\(4\) 0.463900 − 1.94546i 0.231950 − 0.972728i
\(5\) −0.866025 + 0.500000i −0.387298 + 0.223607i
\(6\) 0.149022 + 2.44495i 0.0608382 + 0.998148i
\(7\) 1.77239 − 1.96434i 0.669902 − 0.742450i
\(8\) 1.19007 + 2.56588i 0.420753 + 0.907175i
\(9\) −1.04512 − 2.81207i −0.348372 − 0.937356i
\(10\) 0.523038 − 1.31394i 0.165399 − 0.415504i
\(11\) 0.0979164 − 0.169596i 0.0295229 − 0.0511352i −0.850886 − 0.525350i \(-0.823935\pi\)
0.880409 + 0.474214i \(0.157268\pi\)
\(12\) −2.30812 − 2.58313i −0.666298 − 0.745686i
\(13\) 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 0.00329491 − 0.999995i \(-0.498951\pi\)
−0.864373 + 0.502851i \(0.832285\pi\)
\(18\) 3.62446 + 2.20528i 0.854294 + 0.519790i
\(19\) 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
\(21\) −1.04133 − 4.46269i −0.227236 − 0.973840i
\(22\) 0.0399509 + 0.274053i 0.00851755 + 0.0584282i
\(23\) 0.636494 + 1.10244i 0.132718 + 0.229875i 0.924723 − 0.380640i \(-0.124296\pi\)
−0.792005 + 0.610514i \(0.790963\pi\)
\(24\) 4.82568 + 0.844298i 0.985037 + 0.172342i
\(25\) 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 \(-0.801260\pi\)
0.811337 − 0.584579i \(-0.198740\pi\)
\(30\) −1.35153 − 2.04288i −0.246755 − 0.372977i
\(31\) −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
\(33\) −0.144388 − 0.306926i −0.0251348 − 0.0534289i
\(34\) 5.73700 − 0.836329i 0.983888 − 0.143429i
\(35\) −0.552770 + 2.58736i −0.0934351 + 0.437344i
\(36\) −5.95558 + 0.728706i −0.992597 + 0.121451i
\(37\) −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
\(39\) 1.03249 − 1.48521i 0.165330 − 0.237825i
\(40\) −2.31357 − 1.62708i −0.365807 − 0.257264i
\(41\) − 8.06002i − 1.25876i −0.777096 − 0.629382i \(-0.783308\pi\)
0.777096 − 0.629382i \(-0.216692\pi\)
\(42\) 5.06684 + 4.04069i 0.781830 + 0.623492i
\(43\) 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
\(45\) 2.31113 + 1.91277i 0.344523 + 0.285138i
\(46\) −1.67263 − 0.665820i −0.246615 − 0.0981699i
\(47\) 4.77067 + 8.26304i 0.695873 + 1.20529i 0.969885 + 0.243562i \(0.0783159\pi\)
−0.274012 + 0.961726i \(0.588351\pi\)
\(48\) −6.09611 + 3.29204i −0.879897 + 0.475164i
\(49\) −0.717240 − 6.96316i −0.102463 − 0.994737i
\(50\) 0.204005 + 1.39942i 0.0288507 + 0.197908i
\(51\) −6.42517 + 3.02262i −0.899704 + 0.423251i
\(52\) 0.484466 − 2.03170i 0.0671834 − 0.281746i
\(53\) 5.12381 + 2.95823i 0.703809 + 0.406345i 0.808765 − 0.588132i \(-0.200137\pi\)
−0.104955 + 0.994477i \(0.533470\pi\)
\(54\) 6.71963 − 2.97432i 0.914426 − 0.404754i
\(55\) 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 \(-0.519279\pi\)
0.894703 − 0.446662i \(-0.147387\pi\)
\(60\) 3.29046 + 1.08300i 0.424796 + 0.139814i
\(61\) 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
\(63\) −7.37621 − 2.93113i −0.929315 − 0.369288i
\(64\) −5.16747 + 6.10715i −0.645934 + 0.763393i
\(65\) −0.904419 + 0.522166i −0.112179 + 0.0647668i
\(66\) 0.429246 + 0.214127i 0.0528366 + 0.0263572i
\(67\) 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 \(-0.246349\pi\)
0.715172 + 0.698949i \(0.246349\pi\)
\(72\) 5.97167 − 6.02820i 0.703768 − 0.710430i
\(73\) −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
\(75\) −0.737304 − 1.56729i −0.0851365 − 0.180975i
\(76\) −2.21565 − 7.43789i −0.254152 − 0.853185i
\(77\) −0.159598 − 0.492932i −0.0181878 − 0.0561748i
\(78\) 0.155629 + 2.55334i 0.0176215 + 0.289109i
\(79\) −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
\(81\) −6.81547 + 5.87787i −0.757274 + 0.653097i
\(82\) 7.06368 + 8.94608i 0.780053 + 0.987929i
\(83\) −12.0808 −1.32604 −0.663019 − 0.748602i \(-0.730725\pi\)
−0.663019 + 0.748602i \(0.730725\pi\)
\(84\) −9.16504 − 0.0443939i −0.999988 − 0.00484378i
\(85\) 4.09955 0.444659
\(86\) −5.05571 − 6.40300i −0.545171 − 0.690454i
\(87\) −8.95412 − 6.22469i −0.959982 − 0.667357i
\(88\) 0.551691 + 0.0494105i 0.0588104 + 0.00526718i
\(89\) −6.86736 + 3.96487i −0.727939 + 0.420276i −0.817668 − 0.575691i \(-0.804733\pi\)
0.0897288 + 0.995966i \(0.471400\pi\)
\(90\) −4.24152 − 0.0976011i −0.447095 − 0.0102881i
\(91\) 1.85097 − 2.05142i 0.194034 − 0.215047i
\(92\) 2.44002 − 0.726848i 0.254389 − 0.0757792i
\(93\) −2.94311 + 1.38454i −0.305186 + 0.143570i
\(94\) −12.5367 − 4.99048i −1.29306 − 0.514728i
\(95\) −1.94022 + 3.36056i −0.199062 + 0.344786i
\(96\) 3.88118 − 8.99647i 0.396121 − 0.918198i
\(97\) −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.11.14 yes 128
3.2 odd 2 inner 420.2.bf.a.11.51 yes 128
4.3 odd 2 inner 420.2.bf.a.11.9 ✓ 128
7.2 even 3 inner 420.2.bf.a.191.56 yes 128
12.11 even 2 inner 420.2.bf.a.11.56 yes 128
21.2 odd 6 inner 420.2.bf.a.191.9 yes 128
28.23 odd 6 inner 420.2.bf.a.191.51 yes 128
84.23 even 6 inner 420.2.bf.a.191.14 yes 128
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
420.2.bf.a.11.9 ✓ 128 4.3 odd 2 inner
420.2.bf.a.11.14 yes 128 1.1 even 1 trivial
420.2.bf.a.11.51 yes 128 3.2 odd 2 inner
420.2.bf.a.11.56 yes 128 12.11 even 2 inner
420.2.bf.a.191.9 yes 128 21.2 odd 6 inner
420.2.bf.a.191.14 yes 128 84.23 even 6 inner
420.2.bf.a.191.51 yes 128 28.23 odd 6 inner
420.2.bf.a.191.56 yes 128 7.2 even 3 inner