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Properties

Label 285.2.k.d.77.3
Level $285$
Weight $2$
Character 285.77
Analytic conductor $2.276$
Analytic rank $0$
Dimension $36$
Inner twists $4$

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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] = [285,2,Mod(77,285)] 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("285.77"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(285, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([2, 1, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 285 = 3 \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 285.k (of order \(4\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [36,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(2.27573645761\)
Analytic rank: \(0\)
Dimension: \(36\)
Relative dimension: \(18\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 77.3
Character \(\chi\) \(=\) 285.77
Dual form 285.2.k.d.248.3

$q$-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+(-1.53257 + 1.53257i) q^{2} +(1.34142 - 1.09572i) q^{3} -2.69753i q^{4} +(2.05989 + 0.869965i) q^{5} +(-0.376555 + 3.73508i) q^{6} +(1.55058 + 1.55058i) q^{7} +(1.06901 + 1.06901i) q^{8} +(0.598810 - 2.93963i) q^{9} +(-4.49021 + 1.82365i) q^{10} -2.52140i q^{11} +(-2.95573 - 3.61852i) q^{12} +(-0.427081 + 0.427081i) q^{13} -4.75273 q^{14} +(3.71642 - 1.09007i) q^{15} +2.11839 q^{16} +(1.45764 - 1.45764i) q^{17} +(3.58747 + 5.42290i) q^{18} +1.00000i q^{19} +(2.34676 - 5.55663i) q^{20} +(3.77897 + 0.380980i) q^{21} +(3.86422 + 3.86422i) q^{22} +(3.56822 + 3.56822i) q^{23} +(2.60533 + 0.262659i) q^{24} +(3.48632 + 3.58407i) q^{25} -1.30906i q^{26} +(-2.41775 - 4.59940i) q^{27} +(4.18273 - 4.18273i) q^{28} -6.37067 q^{29} +(-4.02505 + 7.36627i) q^{30} -3.46288 q^{31} +(-5.38460 + 5.38460i) q^{32} +(-2.76274 - 3.38225i) q^{33} +4.46787i q^{34} +(1.84508 + 4.54297i) q^{35} +(-7.92975 - 1.61531i) q^{36} +(6.17292 + 6.17292i) q^{37} +(-1.53257 - 1.53257i) q^{38} +(-0.104935 + 1.04085i) q^{39} +(1.27205 + 3.13206i) q^{40} -4.29278i q^{41} +(-6.37541 + 5.20765i) q^{42} +(-2.42991 + 2.42991i) q^{43} -6.80155 q^{44} +(3.79086 - 5.53438i) q^{45} -10.9371 q^{46} +(-2.49615 + 2.49615i) q^{47} +(2.84164 - 2.32115i) q^{48} -2.19142i q^{49} +(-10.8359 - 0.149808i) q^{50} +(0.358146 - 3.55247i) q^{51} +(1.15206 + 1.15206i) q^{52} +(8.40313 + 8.40313i) q^{53} +(10.7543 + 3.34353i) q^{54} +(2.19353 - 5.19381i) q^{55} +3.31518i q^{56} +(1.09572 + 1.34142i) q^{57} +(9.76349 - 9.76349i) q^{58} -0.181303 q^{59} +(-2.94050 - 10.0251i) q^{60} -14.9722 q^{61} +(5.30710 - 5.30710i) q^{62} +(5.48663 - 3.62963i) q^{63} -12.2678i q^{64} +(-1.25129 + 0.508196i) q^{65} +(9.41762 + 0.949446i) q^{66} +(-6.58649 - 6.58649i) q^{67} +(-3.93204 - 3.93204i) q^{68} +(8.69623 + 0.876719i) q^{69} +(-9.79012 - 4.13471i) q^{70} -14.8059i q^{71} +(3.78265 - 2.50237i) q^{72} +(4.76047 - 4.76047i) q^{73} -18.9208 q^{74} +(8.60374 + 0.987720i) q^{75} +2.69753 q^{76} +(3.90963 - 3.90963i) q^{77} +(-1.43436 - 1.75600i) q^{78} +7.82253i q^{79} +(4.36365 + 1.84292i) q^{80} +(-8.28285 - 3.52056i) q^{81} +(6.57898 + 6.57898i) q^{82} +(-10.7308 - 10.7308i) q^{83} +(1.02771 - 10.1939i) q^{84} +(4.27069 - 1.73449i) q^{85} -7.44800i q^{86} +(-8.54574 + 6.98045i) q^{87} +(2.69541 - 2.69541i) q^{88} +4.99091 q^{89} +(2.67207 + 14.2916i) q^{90} -1.32444 q^{91} +(9.62538 - 9.62538i) q^{92} +(-4.64518 + 3.79434i) q^{93} -7.65103i q^{94} +(-0.869965 + 2.05989i) q^{95} +(-1.32301 + 13.1230i) q^{96} +(-4.73002 - 4.73002i) q^{97} +(3.35849 + 3.35849i) q^{98} +(-7.41198 - 1.50984i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 36 q - 2 q^{3} + 4 q^{6} + 4 q^{7} - 4 q^{10} - 18 q^{12} - 8 q^{13} - 8 q^{15} - 84 q^{16} + 8 q^{21} + 40 q^{22} - 20 q^{25} - 14 q^{27} + 36 q^{28} + 28 q^{30} - 28 q^{33} + 92 q^{36} - 4 q^{37} - 20 q^{40}+ \cdots + 32 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/285\mathbb{Z}\right)^\times\).

\(n\) \(172\) \(191\) \(211\)
\(\chi(n)\) \(e\left(\frac{1}{4}\right)\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.53257 + 1.53257i −1.08369 + 1.08369i −0.0875274 + 0.996162i \(0.527897\pi\)
−0.996162 + 0.0875274i \(0.972103\pi\)
\(3\) 1.34142 − 1.09572i 0.774469 − 0.632612i
\(4\) − 2.69753i − 1.34877i
\(5\) 2.05989 + 0.869965i 0.921212 + 0.389060i
\(6\) −0.376555 + 3.73508i −0.153728 + 1.52484i
\(7\) 1.55058 + 1.55058i 0.586063 + 0.586063i 0.936563 − 0.350499i \(-0.113988\pi\)
−0.350499 + 0.936563i \(0.613988\pi\)
\(8\) 1.06901 + 1.06901i 0.377954 + 0.377954i
\(9\) 0.598810 − 2.93963i 0.199603 − 0.979877i
\(10\) −4.49021 + 1.82365i −1.41993 + 0.576688i
\(11\) − 2.52140i − 0.760230i −0.924939 − 0.380115i \(-0.875884\pi\)
0.924939 − 0.380115i \(-0.124116\pi\)
\(12\) −2.95573 − 3.61852i −0.853246 − 1.04458i
\(13\) −0.427081 + 0.427081i −0.118451 + 0.118451i −0.763848 − 0.645397i \(-0.776692\pi\)
0.645397 + 0.763848i \(0.276692\pi\)
\(14\) −4.75273 −1.27022
\(15\) 3.71642 − 1.09007i 0.959574 − 0.281455i
\(16\) 2.11839 0.529597
\(17\) 1.45764 − 1.45764i 0.353530 − 0.353530i −0.507891 − 0.861421i \(-0.669575\pi\)
0.861421 + 0.507891i \(0.169575\pi\)
\(18\) 3.58747 + 5.42290i 0.845574 + 1.27819i
\(19\) 1.00000i 0.229416i
\(20\) 2.34676 − 5.55663i 0.524751 − 1.24250i
\(21\) 3.77897 + 0.380980i 0.824639 + 0.0831367i
\(22\) 3.86422 + 3.86422i 0.823854 + 0.823854i
\(23\) 3.56822 + 3.56822i 0.744025 + 0.744025i 0.973350 − 0.229325i \(-0.0736518\pi\)
−0.229325 + 0.973350i \(0.573652\pi\)
\(24\) 2.60533 + 0.262659i 0.531812 + 0.0536151i
\(25\) 3.48632 + 3.58407i 0.697264 + 0.716814i
\(26\) − 1.30906i − 0.256728i
\(27\) −2.41775 − 4.59940i −0.465296 − 0.885155i
\(28\) 4.18273 − 4.18273i 0.790462 − 0.790462i
\(29\) −6.37067 −1.18300 −0.591502 − 0.806303i \(-0.701465\pi\)
−0.591502 + 0.806303i \(0.701465\pi\)
\(30\) −4.02505 + 7.36627i −0.734870 + 1.34489i
\(31\) −3.46288 −0.621952 −0.310976 − 0.950418i \(-0.600656\pi\)
−0.310976 + 0.950418i \(0.600656\pi\)
\(32\) −5.38460 + 5.38460i −0.951872 + 0.951872i
\(33\) −2.76274 − 3.38225i −0.480931 − 0.588775i
\(34\) 4.46787i 0.766234i
\(35\) 1.84508 + 4.54297i 0.311875 + 0.767903i
\(36\) −7.92975 − 1.61531i −1.32162 − 0.269218i
\(37\) 6.17292 + 6.17292i 1.01482 + 1.01482i 0.999888 + 0.0149328i \(0.00475344\pi\)
0.0149328 + 0.999888i \(0.495247\pi\)
\(38\) −1.53257 − 1.53257i −0.248615 − 0.248615i
\(39\) −0.104935 + 1.04085i −0.0168030 + 0.166670i
\(40\) 1.27205 + 3.13206i 0.201129 + 0.495223i
\(41\) − 4.29278i − 0.670420i −0.942144 − 0.335210i \(-0.891193\pi\)
0.942144 − 0.335210i \(-0.108807\pi\)
\(42\) −6.37541 + 5.20765i −0.983747 + 0.803558i
\(43\) −2.42991 + 2.42991i −0.370557 + 0.370557i −0.867680 − 0.497123i \(-0.834390\pi\)
0.497123 + 0.867680i \(0.334390\pi\)
\(44\) −6.80155 −1.02537
\(45\) 3.79086 − 5.53438i 0.565108 − 0.825017i
\(46\) −10.9371 −1.61258
\(47\) −2.49615 + 2.49615i −0.364101 + 0.364101i −0.865320 − 0.501220i \(-0.832885\pi\)
0.501220 + 0.865320i \(0.332885\pi\)
\(48\) 2.84164 − 2.32115i 0.410156 − 0.335029i
\(49\) − 2.19142i − 0.313059i
\(50\) −10.8359 − 0.149808i −1.53242 − 0.0211861i
\(51\) 0.358146 − 3.55247i 0.0501504 − 0.497446i
\(52\) 1.15206 + 1.15206i 0.159763 + 0.159763i
\(53\) 8.40313 + 8.40313i 1.15426 + 1.15426i 0.985690 + 0.168568i \(0.0539144\pi\)
0.168568 + 0.985690i \(0.446086\pi\)
\(54\) 10.7543 + 3.34353i 1.46347 + 0.454997i
\(55\) 2.19353 − 5.19381i 0.295775 − 0.700334i
\(56\) 3.31518i 0.443010i
\(57\) 1.09572 + 1.34142i 0.145131 + 0.177675i
\(58\) 9.76349 − 9.76349i 1.28201 − 1.28201i
\(59\) −0.181303 −0.0236036 −0.0118018 − 0.999930i \(-0.503757\pi\)
−0.0118018 + 0.999930i \(0.503757\pi\)
\(60\) −2.94050 − 10.0251i −0.379617 − 1.29424i
\(61\) −14.9722 −1.91699 −0.958497 − 0.285102i \(-0.907973\pi\)
−0.958497 + 0.285102i \(0.907973\pi\)
\(62\) 5.30710 − 5.30710i 0.674003 − 0.674003i
\(63\) 5.48663 − 3.62963i 0.691250 − 0.457290i
\(64\) − 12.2678i − 1.53347i
\(65\) −1.25129 + 0.508196i −0.155203 + 0.0630339i
\(66\) 9.41762 + 0.949446i 1.15923 + 0.116869i
\(67\) −6.58649 − 6.58649i −0.804668 − 0.804668i 0.179153 − 0.983821i \(-0.442664\pi\)
−0.983821 + 0.179153i \(0.942664\pi\)
\(68\) −3.93204 − 3.93204i −0.476829 − 0.476829i
\(69\) 8.69623 + 0.876719i 1.04690 + 0.105545i
\(70\) −9.79012 − 4.13471i −1.17014 − 0.494193i
\(71\) − 14.8059i − 1.75713i −0.477618 − 0.878567i \(-0.658500\pi\)
0.477618 − 0.878567i \(-0.341500\pi\)
\(72\) 3.78265 − 2.50237i 0.445789 − 0.294907i
\(73\) 4.76047 − 4.76047i 0.557171 − 0.557171i −0.371330 − 0.928501i \(-0.621098\pi\)
0.928501 + 0.371330i \(0.121098\pi\)
\(74\) −18.9208 −2.19950
\(75\) 8.60374 + 0.987720i 0.993475 + 0.114052i
\(76\) 2.69753 0.309428
\(77\) 3.90963 − 3.90963i 0.445543 − 0.445543i
\(78\) −1.43436 − 1.75600i −0.162409 − 0.198828i
\(79\) 7.82253i 0.880104i 0.897972 + 0.440052i \(0.145040\pi\)
−0.897972 + 0.440052i \(0.854960\pi\)
\(80\) 4.36365 + 1.84292i 0.487871 + 0.206045i
\(81\) −8.28285 − 3.52056i −0.920317 − 0.391173i
\(82\) 6.57898 + 6.57898i 0.726527 + 0.726527i
\(83\) −10.7308 − 10.7308i −1.17786 − 1.17786i −0.980288 − 0.197574i \(-0.936694\pi\)
−0.197574 − 0.980288i \(-0.563306\pi\)
\(84\) 1.02771 − 10.1939i 0.112132 − 1.11224i
\(85\) 4.27069 − 1.73449i 0.463221 − 0.188132i
\(86\) − 7.44800i − 0.803138i
\(87\) −8.54574 + 6.98045i −0.916200 + 0.748383i
\(88\) 2.69541 − 2.69541i 0.287332 − 0.287332i
\(89\) 4.99091 0.529036 0.264518 − 0.964381i \(-0.414787\pi\)
0.264518 + 0.964381i \(0.414787\pi\)
\(90\) 2.67207 + 14.2916i 0.281661 + 1.50646i
\(91\) −1.32444 −0.138840
\(92\) 9.62538 − 9.62538i 1.00352 − 1.00352i
\(93\) −4.64518 + 3.79434i −0.481682 + 0.393455i
\(94\) − 7.65103i − 0.789144i
\(95\) −0.869965 + 2.05989i −0.0892565 + 0.211341i
\(96\) −1.32301 + 13.1230i −0.135029 + 1.33936i
\(97\) −4.73002 − 4.73002i −0.480261 − 0.480261i 0.424954 − 0.905215i \(-0.360290\pi\)
−0.905215 + 0.424954i \(0.860290\pi\)
\(98\) 3.35849 + 3.35849i 0.339259 + 0.339259i
\(99\) −7.41198 − 1.50984i −0.744932 − 0.151744i
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 285.2.k.d.77.3 ✓ 36
3.2 odd 2 inner 285.2.k.d.77.16 yes 36
5.3 odd 4 inner 285.2.k.d.248.16 yes 36
15.8 even 4 inner 285.2.k.d.248.3 yes 36
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
285.2.k.d.77.3 ✓ 36 1.1 even 1 trivial
285.2.k.d.77.16 yes 36 3.2 odd 2 inner
285.2.k.d.248.3 yes 36 15.8 even 4 inner
285.2.k.d.248.16 yes 36 5.3 odd 4 inner