↑

Properties

Label 285.2.k.d.248.9
Level $285$
Weight $2$
Character 285.248
Analytic conductor $2.276$
Analytic rank $0$
Dimension $36$
Inner twists $4$

Related objects

Downloads

Learn more

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 248.9
Character \(\chi\) \(=\) 285.248
Dual form 285.2.k.d.77.9

$q$-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+(-0.0580413 - 0.0580413i) q^{2} +(0.605290 + 1.62284i) q^{3} -1.99326i q^{4} +(-1.89399 + 1.18861i) q^{5} +(0.0590602 - 0.129324i) q^{6} +(-2.14350 + 2.14350i) q^{7} +(-0.231774 + 0.231774i) q^{8} +(-2.26725 + 1.96458i) q^{9} +(0.178918 + 0.0409414i) q^{10} +4.16350i q^{11} +(3.23476 - 1.20650i) q^{12} +(2.98648 + 2.98648i) q^{13} +0.248823 q^{14} +(-3.07534 - 2.35420i) q^{15} -3.95962 q^{16} +(1.86072 + 1.86072i) q^{17} +(0.245621 + 0.0175672i) q^{18} -1.00000i q^{19} +(2.36921 + 3.77523i) q^{20} +(-4.77601 - 2.18113i) q^{21} +(0.241655 - 0.241655i) q^{22} +(5.35400 - 5.35400i) q^{23} +(-0.516424 - 0.235843i) q^{24} +(2.17442 - 4.50243i) q^{25} -0.346678i q^{26} +(-4.56055 - 2.49025i) q^{27} +(4.27256 + 4.27256i) q^{28} -5.48965 q^{29} +(0.0418557 + 0.315138i) q^{30} -0.743102 q^{31} +(0.693369 + 0.693369i) q^{32} +(-6.75672 + 2.52013i) q^{33} -0.215997i q^{34} +(1.51199 - 6.60756i) q^{35} +(3.91593 + 4.51922i) q^{36} +(-3.93981 + 3.93981i) q^{37} +(-0.0580413 + 0.0580413i) q^{38} +(-3.03891 + 6.65428i) q^{39} +(0.163490 - 0.714467i) q^{40} -7.30907i q^{41} +(0.150610 + 0.403801i) q^{42} +(2.54619 + 2.54619i) q^{43} +8.29896 q^{44} +(1.95904 - 6.41578i) q^{45} -0.621506 q^{46} +(4.88333 + 4.88333i) q^{47} +(-2.39672 - 6.42585i) q^{48} -2.18918i q^{49} +(-0.387533 + 0.135121i) q^{50} +(-1.89338 + 4.14593i) q^{51} +(5.95284 - 5.95284i) q^{52} +(1.58196 - 1.58196i) q^{53} +(0.120163 + 0.409238i) q^{54} +(-4.94878 - 7.88565i) q^{55} -0.993615i q^{56} +(1.62284 - 0.605290i) q^{57} +(0.318626 + 0.318626i) q^{58} +12.7007 q^{59} +(-4.69255 + 6.12996i) q^{60} -5.26300 q^{61} +(0.0431306 + 0.0431306i) q^{62} +(0.648767 - 9.07093i) q^{63} +7.83875i q^{64} +(-9.20613 - 2.10662i) q^{65} +(0.538440 + 0.245897i) q^{66} +(1.58677 - 1.58677i) q^{67} +(3.70890 - 3.70890i) q^{68} +(11.9294 + 5.44799i) q^{69} +(-0.471269 + 0.295753i) q^{70} +7.97592i q^{71} +(0.0701504 - 0.980828i) q^{72} +(3.09062 + 3.09062i) q^{73} +0.457343 q^{74} +(8.62290 + 0.803471i) q^{75} -1.99326 q^{76} +(-8.92447 - 8.92447i) q^{77} +(0.562605 - 0.209841i) q^{78} -4.68054i q^{79} +(7.49949 - 4.70644i) q^{80} +(1.28084 - 8.90839i) q^{81} +(-0.424228 + 0.424228i) q^{82} +(-0.803758 + 0.803758i) q^{83} +(-4.34756 + 9.51983i) q^{84} +(-5.73586 - 1.31252i) q^{85} -0.295569i q^{86} +(-3.32283 - 8.90884i) q^{87} +(-0.964992 - 0.964992i) q^{88} +7.94955 q^{89} +(-0.486085 + 0.258675i) q^{90} -12.8030 q^{91} +(-10.6719 - 10.6719i) q^{92} +(-0.449792 - 1.20594i) q^{93} -0.566869i q^{94} +(1.18861 + 1.89399i) q^{95} +(-0.705541 + 1.54492i) q^{96} +(11.4514 - 11.4514i) q^{97} +(-0.127063 + 0.127063i) q^{98} +(-8.17954 - 9.43970i) 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 \(\left(\mathbb{Z}/285\mathbb{Z}\right)^\times\).

\(n\) \(172\) \(191\) \(211\)
\(\chi(n)\) \(e\left(\frac{3}{4}\right)\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.0580413 − 0.0580413i −0.0410414 − 0.0410414i 0.686288 − 0.727330i \(-0.259239\pi\)
−0.727330 + 0.686288i \(0.759239\pi\)
\(3\) 0.605290 + 1.62284i 0.349464 + 0.936950i
\(4\) − 1.99326i − 0.996631i
\(5\) −1.89399 + 1.18861i −0.847020 + 0.531562i
\(6\) 0.0590602 − 0.129324i 0.0241112 − 0.0527962i
\(7\) −2.14350 + 2.14350i −0.810167 + 0.810167i −0.984659 − 0.174492i \(-0.944172\pi\)
0.174492 + 0.984659i \(0.444172\pi\)
\(8\) −0.231774 + 0.231774i −0.0819445 + 0.0819445i
\(9\) −2.26725 + 1.96458i −0.755750 + 0.654861i
\(10\) 0.178918 + 0.0409414i 0.0565789 + 0.0129468i
\(11\) 4.16350i 1.25534i 0.778478 + 0.627672i \(0.215992\pi\)
−0.778478 + 0.627672i \(0.784008\pi\)
\(12\) 3.23476 − 1.20650i 0.933793 − 0.348287i
\(13\) 2.98648 + 2.98648i 0.828300 + 0.828300i 0.987282 − 0.158981i \(-0.0508210\pi\)
−0.158981 + 0.987282i \(0.550821\pi\)
\(14\) 0.248823 0.0665007
\(15\) −3.07534 − 2.35420i −0.794050 − 0.607853i
\(16\) −3.95962 −0.989905
\(17\) 1.86072 + 1.86072i 0.451291 + 0.451291i 0.895783 − 0.444492i \(-0.146616\pi\)
−0.444492 + 0.895783i \(0.646616\pi\)
\(18\) 0.245621 + 0.0175672i 0.0578934 + 0.00414063i
\(19\) − 1.00000i − 0.229416i
\(20\) 2.36921 + 3.77523i 0.529771 + 0.844166i
\(21\) −4.77601 − 2.18113i −1.04221 − 0.475961i
\(22\) 0.241655 − 0.241655i 0.0515210 − 0.0515210i
\(23\) 5.35400 − 5.35400i 1.11639 − 1.11639i 0.124120 − 0.992267i \(-0.460389\pi\)
0.992267 − 0.124120i \(-0.0396107\pi\)
\(24\) −0.516424 − 0.235843i −0.105415 − 0.0481412i
\(25\) 2.17442 − 4.50243i 0.434884 − 0.900486i
\(26\) − 0.346678i − 0.0679892i
\(27\) −4.56055 − 2.49025i −0.877679 − 0.479249i
\(28\) 4.27256 + 4.27256i 0.807438 + 0.807438i
\(29\) −5.48965 −1.01940 −0.509701 − 0.860352i \(-0.670244\pi\)
−0.509701 + 0.860352i \(0.670244\pi\)
\(30\) 0.0418557 + 0.315138i 0.00764177 + 0.0575360i
\(31\) −0.743102 −0.133465 −0.0667325 − 0.997771i \(-0.521257\pi\)
−0.0667325 + 0.997771i \(0.521257\pi\)
\(32\) 0.693369 + 0.693369i 0.122572 + 0.122572i
\(33\) −6.75672 + 2.52013i −1.17619 + 0.438698i
\(34\) − 0.215997i − 0.0370432i
\(35\) 1.51199 − 6.60756i 0.255573 − 1.11688i
\(36\) 3.91593 + 4.51922i 0.652655 + 0.753204i
\(37\) −3.93981 + 3.93981i −0.647701 + 0.647701i −0.952437 − 0.304736i \(-0.901432\pi\)
0.304736 + 0.952437i \(0.401432\pi\)
\(38\) −0.0580413 + 0.0580413i −0.00941554 + 0.00941554i
\(39\) −3.03891 + 6.65428i −0.486615 + 1.06554i
\(40\) 0.163490 − 0.714467i 0.0258500 − 0.112967i
\(41\) − 7.30907i − 1.14148i −0.821129 − 0.570742i \(-0.806655\pi\)
0.821129 − 0.570742i \(-0.193345\pi\)
\(42\) 0.150610 + 0.403801i 0.0232396 + 0.0623078i
\(43\) 2.54619 + 2.54619i 0.388291 + 0.388291i 0.874078 − 0.485786i \(-0.161467\pi\)
−0.485786 + 0.874078i \(0.661467\pi\)
\(44\) 8.29896 1.25111
\(45\) 1.95904 − 6.41578i 0.292036 − 0.956407i
\(46\) −0.621506 −0.0916361
\(47\) 4.88333 + 4.88333i 0.712306 + 0.712306i 0.967017 − 0.254711i \(-0.0819803\pi\)
−0.254711 + 0.967017i \(0.581980\pi\)
\(48\) −2.39672 − 6.42585i −0.345936 − 0.927491i
\(49\) − 2.18918i − 0.312741i
\(50\) −0.387533 + 0.135121i −0.0548054 + 0.0191090i
\(51\) −1.89338 + 4.14593i −0.265127 + 0.580547i
\(52\) 5.95284 − 5.95284i 0.825510 − 0.825510i
\(53\) 1.58196 − 1.58196i 0.217299 − 0.217299i −0.590060 − 0.807359i \(-0.700896\pi\)
0.807359 + 0.590060i \(0.200896\pi\)
\(54\) 0.120163 + 0.409238i 0.0163521 + 0.0556902i
\(55\) −4.94878 − 7.88565i −0.667293 − 1.06330i
\(56\) − 0.993615i − 0.132777i
\(57\) 1.62284 − 0.605290i 0.214951 − 0.0801726i
\(58\) 0.318626 + 0.318626i 0.0418376 + 0.0418376i
\(59\) 12.7007 1.65349 0.826746 − 0.562575i \(-0.190189\pi\)
0.826746 + 0.562575i \(0.190189\pi\)
\(60\) −4.69255 + 6.12996i −0.605805 + 0.791375i
\(61\) −5.26300 −0.673858 −0.336929 − 0.941530i \(-0.609388\pi\)
−0.336929 + 0.941530i \(0.609388\pi\)
\(62\) 0.0431306 + 0.0431306i 0.00547759 + 0.00547759i
\(63\) 0.648767 − 9.07093i 0.0817370 − 1.14283i
\(64\) 7.83875i 0.979844i
\(65\) −9.20613 − 2.10662i −1.14188 − 0.261294i
\(66\) 0.538440 + 0.245897i 0.0662774 + 0.0302679i
\(67\) 1.58677 − 1.58677i 0.193855 − 0.193855i −0.603505 − 0.797360i \(-0.706229\pi\)
0.797360 + 0.603505i \(0.206229\pi\)
\(68\) 3.70890 − 3.70890i 0.449771 − 0.449771i
\(69\) 11.9294 + 5.44799i 1.43614 + 0.655861i
\(70\) −0.471269 + 0.295753i −0.0563274 + 0.0353492i
\(71\) 7.97592i 0.946568i 0.880910 + 0.473284i \(0.156932\pi\)
−0.880910 + 0.473284i \(0.843068\pi\)
\(72\) 0.0701504 − 0.980828i 0.00826730 − 0.115592i
\(73\) 3.09062 + 3.09062i 0.361729 + 0.361729i 0.864449 − 0.502720i \(-0.167667\pi\)
−0.502720 + 0.864449i \(0.667667\pi\)
\(74\) 0.457343 0.0531650
\(75\) 8.62290 + 0.803471i 0.995687 + 0.0927769i
\(76\) −1.99326 −0.228643
\(77\) −8.92447 − 8.92447i −1.01704 − 1.01704i
\(78\) 0.562605 − 0.209841i 0.0637024 − 0.0237598i
\(79\) − 4.68054i − 0.526602i −0.964714 − 0.263301i \(-0.915189\pi\)
0.964714 − 0.263301i \(-0.0848113\pi\)
\(80\) 7.49949 − 4.70644i 0.838469 − 0.526196i
\(81\) 1.28084 − 8.90839i 0.142315 − 0.989821i
\(82\) −0.424228 + 0.424228i −0.0468481 + 0.0468481i
\(83\) −0.803758 + 0.803758i −0.0882239 + 0.0882239i −0.749841 − 0.661618i \(-0.769870\pi\)
0.661618 + 0.749841i \(0.269870\pi\)
\(84\) −4.34756 + 9.51983i −0.474358 + 1.03870i
\(85\) −5.73586 − 1.31252i −0.622141 − 0.142363i
\(86\) − 0.295569i − 0.0318720i
\(87\) −3.32283 − 8.90884i −0.356244 − 0.955128i
\(88\) −0.964992 − 0.964992i −0.102868 − 0.102868i
\(89\) 7.94955 0.842650 0.421325 − 0.906910i \(-0.361565\pi\)
0.421325 + 0.906910i \(0.361565\pi\)
\(90\) −0.486085 + 0.258675i −0.0512378 + 0.0272667i
\(91\) −12.8030 −1.34212
\(92\) −10.6719 − 10.6719i −1.11263 − 1.11263i
\(93\) −0.449792 − 1.20594i −0.0466412 − 0.125050i
\(94\) − 0.566869i − 0.0584681i
\(95\) 1.18861 + 1.89399i 0.121949 + 0.194320i
\(96\) −0.705541 + 1.54492i −0.0720090 + 0.157678i
\(97\) 11.4514 − 11.4514i 1.16271 − 1.16271i 0.178830 − 0.983880i \(-0.442769\pi\)
0.983880 − 0.178830i \(-0.0572313\pi\)
\(98\) −0.127063 + 0.127063i −0.0128353 + 0.0128353i
\(99\) −8.17954 − 9.43970i −0.822075 − 0.948726i
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.248.9 yes 36
3.2 odd 2 inner 285.2.k.d.248.10 yes 36
5.2 odd 4 inner 285.2.k.d.77.10 yes 36
15.2 even 4 inner 285.2.k.d.77.9 ✓ 36
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
285.2.k.d.77.9 ✓ 36 15.2 even 4 inner
285.2.k.d.77.10 yes 36 5.2 odd 4 inner
285.2.k.d.248.9 yes 36 1.1 even 1 trivial
285.2.k.d.248.10 yes 36 3.2 odd 2 inner