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Properties

Label 420.2.bf.a.11.26
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 11.26
Character \(\chi\) \(=\) 420.11
Dual form 420.2.bf.a.191.26

$q$-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+(-0.459814 - 1.33737i) q^{2} +(-0.823062 - 1.52400i) q^{3} +(-1.57714 + 1.22989i) q^{4} +(0.866025 - 0.500000i) q^{5} +(-1.65970 + 1.80150i) q^{6} +(-2.44888 - 1.00148i) q^{7} +(2.37001 + 1.54371i) q^{8} +(-1.64514 + 2.50869i) q^{9} +(-1.06690 - 0.928294i) q^{10} +(-2.46066 + 4.26200i) q^{11} +(3.17243 + 1.39129i) q^{12} -1.25975 q^{13} +(-0.213328 + 3.73557i) q^{14} +(-1.47479 - 0.908290i) q^{15} +(0.974758 - 3.87941i) q^{16} +(-3.19672 - 1.84563i) q^{17} +(4.11151 + 1.04664i) q^{18} +(-1.02899 + 0.594085i) q^{19} +(-0.750902 + 1.85368i) q^{20} +(0.489326 + 4.55638i) q^{21} +(6.83133 + 1.33111i) q^{22} +(2.09464 + 3.62803i) q^{23} +(0.401947 - 4.88246i) q^{24} +(0.500000 - 0.866025i) q^{25} +(0.579250 + 1.68476i) q^{26} +(5.17729 + 0.442379i) q^{27} +(5.09395 - 1.43237i) q^{28} +5.06278i q^{29} +(-0.536594 + 2.38999i) q^{30} +(-3.50538 - 2.02383i) q^{31} +(-5.63644 + 0.480191i) q^{32} +(8.52055 + 0.242160i) q^{33} +(-0.998401 + 5.12386i) q^{34} +(-2.62154 + 0.357132i) q^{35} +(-0.490786 - 5.97989i) q^{36} +(-4.23839 - 7.34110i) q^{37} +(1.26766 + 1.10297i) q^{38} +(1.03685 + 1.91985i) q^{39} +(2.82435 + 0.151888i) q^{40} +2.96112i q^{41} +(5.86858 - 2.74950i) q^{42} +2.21961i q^{43} +(-1.36095 - 9.74812i) q^{44} +(-0.170386 + 2.99516i) q^{45} +(3.88889 - 4.46954i) q^{46} +(1.66702 + 2.88736i) q^{47} +(-6.71450 + 1.70747i) q^{48} +(4.99406 + 4.90503i) q^{49} +(-1.38811 - 0.270477i) q^{50} +(-0.181633 + 6.39087i) q^{51} +(1.98680 - 1.54935i) q^{52} +(3.21902 + 1.85850i) q^{53} +(-1.78896 - 7.12739i) q^{54} +4.92133i q^{55} +(-4.25788 - 6.15390i) q^{56} +(1.75230 + 1.07920i) q^{57} +(6.77084 - 2.32794i) q^{58} +(3.78628 - 6.55802i) q^{59} +(3.44305 - 0.381324i) q^{60} +(-6.61367 - 11.4552i) q^{61} +(-1.09480 + 5.61859i) q^{62} +(6.54116 - 4.49591i) q^{63} +(3.23391 + 7.31723i) q^{64} +(-1.09097 + 0.629875i) q^{65} +(-3.59401 - 11.5065i) q^{66} +(-9.45467 - 5.45866i) q^{67} +(7.31160 - 1.02079i) q^{68} +(3.80508 - 6.17832i) q^{69} +(1.68304 + 3.34176i) q^{70} -15.2344 q^{71} +(-7.77169 + 3.40600i) q^{72} +(-6.26696 + 10.8547i) q^{73} +(-7.86894 + 9.04385i) q^{74} +(-1.73135 - 0.0492062i) q^{75} +(0.892200 - 2.20249i) q^{76} +(10.2942 - 7.97282i) q^{77} +(2.09081 - 2.26943i) q^{78} +(-13.1608 + 7.59837i) q^{79} +(-1.09554 - 3.84705i) q^{80} +(-3.58704 - 8.25428i) q^{81} +(3.96012 - 1.36156i) q^{82} -7.40993 q^{83} +(-6.37556 - 6.58424i) q^{84} -3.69126 q^{85} +(2.96846 - 1.02061i) q^{86} +(7.71567 - 4.16699i) q^{87} +(-12.4111 + 6.30242i) q^{88} +(-3.54523 + 2.04684i) q^{89} +(4.08399 - 1.14934i) q^{90} +(3.08498 + 1.26162i) q^{91} +(-7.76561 - 3.14574i) q^{92} +(-0.199170 + 7.00792i) q^{93} +(3.09497 - 3.55708i) q^{94} +(-0.594085 + 1.02899i) q^{95} +(5.37095 + 8.19469i) q^{96} -12.2546 q^{97} +(4.26352 - 8.93434i) q^{98} +(-6.64389 - 13.1846i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.459814 − 1.33737i −0.325137 − 0.945667i
\(3\) −0.823062 − 1.52400i −0.475195 − 0.879880i
\(4\) −1.57714 + 1.22989i −0.788571 + 0.614943i
\(5\) 0.866025 − 0.500000i 0.387298 − 0.223607i
\(6\) −1.65970 + 1.80150i −0.677570 + 0.735458i
\(7\) −2.44888 − 1.00148i −0.925591 − 0.378525i
\(8\) 2.37001 + 1.54371i 0.837925 + 0.545785i
\(9\) −1.64514 + 2.50869i −0.548379 + 0.836230i
\(10\) −1.06690 − 0.928294i −0.337383 − 0.293552i
\(11\) −2.46066 + 4.26200i −0.741918 + 1.28504i 0.209702 + 0.977765i \(0.432750\pi\)
−0.951621 + 0.307275i \(0.900583\pi\)
\(12\) 3.17243 + 1.39129i 0.915802 + 0.401630i
\(13\) −1.25975 −0.349392 −0.174696 − 0.984622i \(-0.555894\pi\)
−0.174696 + 0.984622i \(0.555894\pi\)
\(14\) −0.213328 + 3.73557i −0.0570142 + 0.998373i
\(15\) −1.47479 − 0.908290i −0.380790 − 0.234519i
\(16\) 0.974758 − 3.87941i 0.243689 − 0.969853i
\(17\) −3.19672 − 1.84563i −0.775319 − 0.447631i 0.0594495 − 0.998231i \(-0.481065\pi\)
−0.834769 + 0.550600i \(0.814399\pi\)
\(18\) 4.11151 + 1.04664i 0.969093 + 0.246694i
\(19\) −1.02899 + 0.594085i −0.236065 + 0.136292i −0.613367 − 0.789798i \(-0.710185\pi\)
0.377302 + 0.926090i \(0.376852\pi\)
\(20\) −0.750902 + 1.85368i −0.167907 + 0.414496i
\(21\) 0.489326 + 4.55638i 0.106780 + 0.994283i
\(22\) 6.83133 + 1.33111i 1.45645 + 0.283793i
\(23\) 2.09464 + 3.62803i 0.436763 + 0.756496i 0.997438 − 0.0715405i \(-0.0227915\pi\)
−0.560675 + 0.828036i \(0.689458\pi\)
\(24\) 0.401947 − 4.88246i 0.0820472 − 0.996628i
\(25\) 0.500000 − 0.866025i 0.100000 − 0.173205i
\(26\) 0.579250 + 1.68476i 0.113600 + 0.330408i
\(27\) 5.17729 + 0.442379i 0.996369 + 0.0851359i
\(28\) 5.09395 − 1.43237i 0.962666 − 0.270692i
\(29\) 5.06278i 0.940135i 0.882630 + 0.470068i \(0.155771\pi\)
−0.882630 + 0.470068i \(0.844229\pi\)
\(30\) −0.536594 + 2.38999i −0.0979683 + 0.436351i
\(31\) −3.50538 − 2.02383i −0.629584 − 0.363491i 0.151007 − 0.988533i \(-0.451748\pi\)
−0.780591 + 0.625042i \(0.785082\pi\)
\(32\) −5.63644 + 0.480191i −0.996391 + 0.0848865i
\(33\) 8.52055 + 0.242160i 1.48324 + 0.0421546i
\(34\) −0.998401 + 5.12386i −0.171224 + 0.878735i
\(35\) −2.62154 + 0.357132i −0.443121 + 0.0603664i
\(36\) −0.490786 − 5.97989i −0.0817977 − 0.996649i
\(37\) −4.23839 − 7.34110i −0.696787 − 1.20687i −0.969575 − 0.244796i \(-0.921279\pi\)
0.272788 − 0.962074i \(-0.412054\pi\)
\(38\) 1.26766 + 1.10297i 0.205641 + 0.178925i
\(39\) 1.03685 + 1.91985i 0.166029 + 0.307423i
\(40\) 2.82435 + 0.151888i 0.446568 + 0.0240156i
\(41\) 2.96112i 0.462449i 0.972900 + 0.231224i \(0.0742732\pi\)
−0.972900 + 0.231224i \(0.925727\pi\)
\(42\) 5.86858 − 2.74950i 0.905542 − 0.424256i
\(43\) 2.21961i 0.338488i 0.985574 + 0.169244i \(0.0541326\pi\)
−0.985574 + 0.169244i \(0.945867\pi\)
\(44\) −1.36095 − 9.74812i −0.205171 − 1.46958i
\(45\) −0.170386 + 2.99516i −0.0253997 + 0.446492i
\(46\) 3.88889 − 4.46954i 0.573385 − 0.658997i
\(47\) 1.66702 + 2.88736i 0.243160 + 0.421165i 0.961613 − 0.274411i \(-0.0884828\pi\)
−0.718453 + 0.695576i \(0.755149\pi\)
\(48\) −6.71450 + 1.70747i −0.969155 + 0.246452i
\(49\) 4.99406 + 4.90503i 0.713438 + 0.700719i
\(50\) −1.38811 − 0.270477i −0.196308 − 0.0382512i
\(51\) −0.181633 + 6.39087i −0.0254337 + 0.894900i
\(52\) 1.98680 − 1.54935i 0.275520 − 0.214856i
\(53\) 3.21902 + 1.85850i 0.442167 + 0.255285i 0.704516 − 0.709688i \(-0.251164\pi\)
−0.262349 + 0.964973i \(0.584497\pi\)
\(54\) −1.78896 − 7.12739i −0.243447 − 0.969914i
\(55\) 4.92133i 0.663592i
\(56\) −4.25788 − 6.15390i −0.568983 − 0.822349i
\(57\) 1.75230 + 1.07920i 0.232098 + 0.142944i
\(58\) 6.77084 − 2.32794i 0.889055 − 0.305673i
\(59\) 3.78628 − 6.55802i 0.492931 − 0.853782i −0.507036 − 0.861925i \(-0.669259\pi\)
0.999967 + 0.00814325i \(0.00259211\pi\)
\(60\) 3.44305 − 0.381324i 0.444496 − 0.0492287i
\(61\) −6.61367 − 11.4552i −0.846794 − 1.46669i −0.884054 − 0.467384i \(-0.845196\pi\)
0.0372605 − 0.999306i \(-0.488137\pi\)
\(62\) −1.09480 + 5.61859i −0.139040 + 0.713561i
\(63\) 6.54116 − 4.49591i 0.824109 − 0.566432i
\(64\) 3.23391 + 7.31723i 0.404238 + 0.914654i
\(65\) −1.09097 + 0.629875i −0.135319 + 0.0781263i
\(66\) −3.59401 − 11.5065i −0.442392 − 1.41635i
\(67\) −9.45467 − 5.45866i −1.15507 − 0.666881i −0.204954 − 0.978772i \(-0.565704\pi\)
−0.950118 + 0.311891i \(0.899038\pi\)
\(68\) 7.31160 − 1.02079i 0.886662 − 0.123789i
\(69\) 3.80508 − 6.17832i 0.458078 − 0.743782i
\(70\) 1.68304 + 3.34176i 0.201162 + 0.399417i
\(71\) −15.2344 −1.80799 −0.903996 − 0.427541i \(-0.859380\pi\)
−0.903996 + 0.427541i \(0.859380\pi\)
\(72\) −7.77169 + 3.40600i −0.915902 + 0.401401i
\(73\) −6.26696 + 10.8547i −0.733492 + 1.27045i 0.221890 + 0.975072i \(0.428777\pi\)
−0.955382 + 0.295374i \(0.904556\pi\)
\(74\) −7.86894 + 9.04385i −0.914745 + 1.05133i
\(75\) −1.73135 − 0.0492062i −0.199919 − 0.00568184i
\(76\) 0.892200 − 2.20249i 0.102342 − 0.252643i
\(77\) 10.2942 − 7.97282i 1.17313 − 0.908587i
\(78\) 2.09081 − 2.26943i 0.236737 − 0.256963i
\(79\) −13.1608 + 7.59837i −1.48070 + 0.854883i −0.999761 − 0.0218705i \(-0.993038\pi\)
−0.480940 + 0.876754i \(0.659705\pi\)
\(80\) −1.09554 − 3.84705i −0.122485 − 0.430113i
\(81\) −3.58704 − 8.25428i −0.398560 − 0.917142i
\(82\) 3.96012 − 1.36156i 0.437323 − 0.150359i
\(83\) −7.40993 −0.813346 −0.406673 − 0.913574i \(-0.633311\pi\)
−0.406673 + 0.913574i \(0.633311\pi\)
\(84\) −6.37556 − 6.58424i −0.695631 − 0.718399i
\(85\) −3.69126 −0.400373
\(86\) 2.96846 − 1.02061i 0.320097 − 0.110055i
\(87\) 7.71567 − 4.16699i 0.827207 − 0.446748i
\(88\) −12.4111 + 6.30242i −1.32303 + 0.671840i
\(89\) −3.54523 + 2.04684i −0.375794 + 0.216965i −0.675987 − 0.736914i \(-0.736282\pi\)
0.300193 + 0.953879i \(0.402949\pi\)
\(90\) 4.08399 − 1.14934i 0.430491 − 0.121152i
\(91\) 3.08498 + 1.26162i 0.323394 + 0.132253i
\(92\) −7.76561 − 3.14574i −0.809621 − 0.327966i
\(93\) −0.199170 + 7.00792i −0.0206530 + 0.726688i
\(94\) 3.09497 − 3.55708i 0.319221 − 0.366884i
\(95\) −0.594085 + 1.02899i −0.0609518 + 0.105572i
\(96\) 5.37095 + 8.19469i 0.548170 + 0.836367i
\(97\) −12.2546 −1.24427 −0.622133 − 0.782911i \(-0.713734\pi\)
−0.622133 + 0.782911i \(0.713734\pi\)
\(98\) 4.26352 − 8.93434i 0.430681 − 0.902504i
\(99\) −6.64389 − 13.1846i −0.667736 − 1.32510i
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.11.26 yes 128
3.2 odd 2 inner 420.2.bf.a.11.39 yes 128
4.3 odd 2 inner 420.2.bf.a.11.47 yes 128
7.2 even 3 inner 420.2.bf.a.191.18 yes 128
12.11 even 2 inner 420.2.bf.a.11.18 ✓ 128
21.2 odd 6 inner 420.2.bf.a.191.47 yes 128
28.23 odd 6 inner 420.2.bf.a.191.39 yes 128
84.23 even 6 inner 420.2.bf.a.191.26 yes 128
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
420.2.bf.a.11.18 ✓ 128 12.11 even 2 inner
420.2.bf.a.11.26 yes 128 1.1 even 1 trivial
420.2.bf.a.11.39 yes 128 3.2 odd 2 inner
420.2.bf.a.11.47 yes 128 4.3 odd 2 inner
420.2.bf.a.191.18 yes 128 7.2 even 3 inner
420.2.bf.a.191.26 yes 128 84.23 even 6 inner
420.2.bf.a.191.39 yes 128 28.23 odd 6 inner
420.2.bf.a.191.47 yes 128 21.2 odd 6 inner