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Properties

Label 420.2.bf.a.11.18
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 11.18
Character \(\chi\) \(=\) 420.11
Dual form 420.2.bf.a.191.18

$q$-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+(-0.928294 - 1.06690i) q^{2} +(1.73135 - 0.0492062i) q^{3} +(-0.276542 + 1.98079i) 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} +(2.99516 - 0.170386i) q^{9} +(1.33737 + 0.459814i) q^{10} +(-2.46066 + 4.26200i) q^{11} +(-0.381324 + 3.44305i) q^{12} -1.25975 q^{13} +(-1.20480 - 3.54238i) q^{14} +(-1.47479 + 0.908290i) q^{15} +(-3.84705 - 1.09554i) q^{16} +(3.19672 + 1.84563i) q^{17} +(-2.96217 - 3.03736i) q^{18} +(1.02899 - 0.594085i) q^{19} +(-0.750902 - 1.85368i) q^{20} +(4.28916 + 1.61342i) q^{21} +(6.83133 - 1.33111i) q^{22} +(2.09464 + 3.62803i) q^{23} +(4.02736 - 2.78933i) q^{24} +(0.500000 - 0.866025i) q^{25} +(1.16942 + 1.34402i) q^{26} +(5.17729 - 0.442379i) q^{27} +(-2.66094 + 4.57377i) q^{28} -5.06278i q^{29} +(2.33809 + 0.730292i) q^{30} +(3.50538 + 2.02383i) q^{31} +(2.40236 + 5.12139i) q^{32} +(-4.05056 + 7.50010i) 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} +(-1.58903 - 0.546337i) q^{38} +(-2.18107 + 0.0619874i) q^{39} +(-1.28063 + 2.52190i) q^{40} -2.96112i q^{41} +(-2.26025 - 6.07382i) q^{42} -2.21961i q^{43} +(-7.76164 - 6.05268i) q^{44} +(-2.50869 + 1.64514i) q^{45} +(1.92629 - 5.60264i) 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} +(5.62547 + 3.03813i) q^{51} +(0.348373 - 2.49530i) q^{52} +(-3.21902 - 1.85850i) q^{53} +(-5.27802 - 5.11298i) q^{54} -4.92133i q^{55} +(7.34988 - 1.40685i) q^{56} +(1.75230 - 1.07920i) q^{57} +(-5.40147 + 4.69975i) q^{58} +(3.78628 - 6.55802i) q^{59} +(-1.39129 - 3.17243i) q^{60} +(-6.61367 - 11.4552i) q^{61} +(-1.09480 - 5.61859i) q^{62} +(7.50543 + 2.58234i) q^{63} +(3.23391 - 7.31723i) q^{64} +(1.09097 - 0.629875i) q^{65} +(11.7619 - 2.64076i) q^{66} +(9.45467 + 5.45866i) q^{67} +(-4.53983 + 5.82164i) q^{68} +(3.80508 + 6.17832i) q^{69} +(2.81458 + 2.46539i) q^{70} -15.2344 q^{71} +(6.83553 - 5.02748i) q^{72} +(-6.26696 + 10.8547i) q^{73} +(-3.89774 + 11.3366i) q^{74} +(0.823062 - 1.52400i) 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} +(3.87941 - 0.974758i) q^{80} +(8.94194 - 1.02067i) q^{81} +(-3.15921 + 2.74879i) q^{82} -7.40993 q^{83} +(-4.38197 + 8.04974i) q^{84} -3.69126 q^{85} +(-2.36810 + 2.06045i) q^{86} +(-0.249120 - 8.76546i) q^{87} +(0.747492 + 13.8995i) 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} +(6.16862 + 3.33148i) q^{93} +(1.53304 - 4.45886i) q^{94} +(-0.594085 + 1.02899i) q^{95} +(4.41134 + 8.74872i) q^{96} -12.2546 q^{97} +(0.597208 - 9.88146i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.928294 − 1.06690i −0.656403 − 0.754411i
\(3\) 1.73135 − 0.0492062i 0.999596 − 0.0284092i
\(4\) −0.276542 + 1.98079i −0.138271 + 0.990394i
\(5\) −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
\(9\) 2.99516 − 0.170386i 0.998386 − 0.0567955i
\(10\) 1.33737 + 0.459814i 0.422915 + 0.145406i
\(11\) −2.46066 + 4.26200i −0.741918 + 1.28504i 0.209702 + 0.977765i \(0.432750\pi\)
−0.951621 + 0.307275i \(0.900583\pi\)
\(12\) −0.381324 + 3.44305i −0.110079 + 0.993923i
\(13\) −1.25975 −0.349392 −0.174696 − 0.984622i \(-0.555894\pi\)
−0.174696 + 0.984622i \(0.555894\pi\)
\(14\) −1.20480 − 3.54238i −0.321997 − 0.946741i
\(15\) −1.47479 + 0.908290i −0.380790 + 0.234519i
\(16\) −3.84705 − 1.09554i −0.961762 − 0.273885i
\(17\) 3.19672 + 1.84563i 0.775319 + 0.447631i 0.834769 − 0.550600i \(-0.185601\pi\)
−0.0594495 + 0.998231i \(0.518935\pi\)
\(18\) −2.96217 − 3.03736i −0.698190 − 0.715912i
\(19\) 1.02899 − 0.594085i 0.236065 − 0.136292i −0.377302 − 0.926090i \(-0.623148\pi\)
0.613367 + 0.789798i \(0.289815\pi\)
\(20\) −0.750902 − 1.85368i −0.167907 − 0.414496i
\(21\) 4.28916 + 1.61342i 0.935971 + 0.352077i
\(22\) 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 \(0.689458\pi\)
\(24\) 4.02736 − 2.78933i 0.822082 − 0.569369i
\(25\) 0.500000 − 0.866025i 0.100000 − 0.173205i
\(26\) 1.16942 + 1.34402i 0.229342 + 0.263585i
\(27\) 5.17729 − 0.442379i 0.996369 − 0.0851359i
\(28\) −2.66094 + 4.57377i −0.502871 + 0.864361i
\(29\) − 5.06278i − 0.940135i −0.882630 − 0.470068i \(-0.844229\pi\)
0.882630 − 0.470068i \(-0.155771\pi\)
\(30\) 2.33809 + 0.730292i 0.426875 + 0.133333i
\(31\) 3.50538 + 2.02383i 0.629584 + 0.363491i 0.780591 − 0.625042i \(-0.214918\pi\)
−0.151007 + 0.988533i \(0.548252\pi\)
\(32\) 2.40236 + 5.12139i 0.424681 + 0.905343i
\(33\) −4.05056 + 7.50010i −0.705112 + 1.30560i
\(34\) −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 \(-0.412054\pi\)
\(38\) −1.58903 − 0.546337i −0.257774 − 0.0886275i
\(39\) −2.18107 + 0.0619874i −0.349251 + 0.00992594i
\(40\) −1.28063 + 2.52190i −0.202486 + 0.398747i
\(41\) − 2.96112i − 0.462449i −0.972900 − 0.231224i \(-0.925727\pi\)
0.972900 − 0.231224i \(-0.0742732\pi\)
\(42\) −2.26025 − 6.07382i −0.348763 − 0.937211i
\(43\) − 2.21961i − 0.338488i −0.985574 − 0.169244i \(-0.945867\pi\)
0.985574 − 0.169244i \(-0.0541326\pi\)
\(44\) −7.76164 − 6.05268i −1.17011 − 0.912475i
\(45\) −2.50869 + 1.64514i −0.373973 + 0.245243i
\(46\) 1.92629 − 5.60264i 0.284016 − 0.826065i
\(47\) 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
\(51\) 5.62547 + 3.03813i 0.787723 + 0.425424i
\(52\) 0.348373 − 2.49530i 0.0483107 − 0.346035i
\(53\) −3.21902 − 1.85850i −0.442167 − 0.255285i 0.262349 − 0.964973i \(-0.415503\pi\)
−0.704516 + 0.709688i \(0.748836\pi\)
\(54\) −5.27802 − 5.11298i −0.718247 − 0.695788i
\(55\) − 4.92133i − 0.663592i
\(56\) 7.34988 − 1.40685i 0.982169 − 0.187998i
\(57\) 1.75230 − 1.07920i 0.232098 − 0.142944i
\(58\) −5.40147 + 4.69975i −0.709248 + 0.617107i
\(59\) 3.78628 − 6.55802i 0.492931 − 0.853782i −0.507036 − 0.861925i \(-0.669259\pi\)
0.999967 + 0.00814325i \(0.00259211\pi\)
\(60\) −1.39129 − 3.17243i −0.179615 − 0.409559i
\(61\) −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
\(63\) 7.50543 + 2.58234i 0.945596 + 0.325345i
\(64\) 3.23391 − 7.31723i 0.404238 − 0.914654i
\(65\) 1.09097 − 0.629875i 0.135319 − 0.0781263i
\(66\) 11.7619 − 2.64076i 1.44780 − 0.325055i
\(67\) 9.45467 + 5.45866i 1.15507 + 0.666881i 0.950118 − 0.311891i \(-0.100962\pi\)
0.204954 + 0.978772i \(0.434296\pi\)
\(68\) −4.53983 + 5.82164i −0.550535 + 0.705978i
\(69\) 3.80508 + 6.17832i 0.458078 + 0.743782i
\(70\) 2.81458 + 2.46539i 0.336407 + 0.294670i
\(71\) −15.2344 −1.80799 −0.903996 − 0.427541i \(-0.859380\pi\)
−0.903996 + 0.427541i \(0.859380\pi\)
\(72\) 6.83553 − 5.02748i 0.805575 − 0.592494i
\(73\) −6.26696 + 10.8547i −0.733492 + 1.27045i 0.221890 + 0.975072i \(0.428777\pi\)
−0.955382 + 0.295374i \(0.904556\pi\)
\(74\) −3.89774 + 11.3366i −0.453103 + 1.31786i
\(75\) 0.823062 − 1.52400i 0.0950390 − 0.175976i
\(76\) 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 0.480940 − 0.876754i \(-0.340295\pi\)
0.999761 + 0.0218705i \(0.00696215\pi\)
\(80\) 3.87941 − 0.974758i 0.433732 − 0.108981i
\(81\) 8.94194 − 1.02067i 0.993549 − 0.113408i
\(82\) −3.15921 + 2.74879i −0.348876 + 0.303553i
\(83\) −7.40993 −0.813346 −0.406673 − 0.913574i \(-0.633311\pi\)
−0.406673 + 0.913574i \(0.633311\pi\)
\(84\) −4.38197 + 8.04974i −0.478112 + 0.878299i
\(85\) −3.69126 −0.400373
\(86\) −2.36810 + 2.06045i −0.255359 + 0.222184i
\(87\) −0.249120 − 8.76546i −0.0267085 − 0.939756i
\(88\) 0.747492 + 13.8995i 0.0796830 + 1.48170i
\(89\) 3.54523 − 2.04684i 0.375794 − 0.216965i −0.300193 − 0.953879i \(-0.597051\pi\)
0.675987 + 0.736914i \(0.263718\pi\)
\(90\) 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
\(93\) 6.16862 + 3.33148i 0.639657 + 0.345458i
\(94\) 1.53304 − 4.45886i 0.158121 − 0.459896i
\(95\) −0.594085 + 1.02899i −0.0609518 + 0.105572i
\(96\) 4.41134 + 8.74872i 0.450230 + 0.892913i
\(97\) −12.2546 −1.24427 −0.622133 − 0.782911i \(-0.713734\pi\)
−0.622133 + 0.782911i \(0.713734\pi\)
\(98\) 0.597208 − 9.88146i 0.0603271 − 0.998179i
\(99\) −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.11.18 ✓ 128
3.2 odd 2 inner 420.2.bf.a.11.47 yes 128
4.3 odd 2 inner 420.2.bf.a.11.39 yes 128
7.2 even 3 inner 420.2.bf.a.191.26 yes 128
12.11 even 2 inner 420.2.bf.a.11.26 yes 128
21.2 odd 6 inner 420.2.bf.a.191.39 yes 128
28.23 odd 6 inner 420.2.bf.a.191.47 yes 128
84.23 even 6 inner 420.2.bf.a.191.18 yes 128
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
420.2.bf.a.11.18 ✓ 128 1.1 even 1 trivial
420.2.bf.a.11.26 yes 128 12.11 even 2 inner
420.2.bf.a.11.39 yes 128 4.3 odd 2 inner
420.2.bf.a.11.47 yes 128 3.2 odd 2 inner
420.2.bf.a.191.18 yes 128 84.23 even 6 inner
420.2.bf.a.191.26 yes 128 7.2 even 3 inner
420.2.bf.a.191.39 yes 128 21.2 odd 6 inner
420.2.bf.a.191.47 yes 128 28.23 odd 6 inner