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Properties

Label 285.2.k.d.77.10
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 77.10
Character \(\chi\) \(=\) 285.77
Dual form 285.2.k.d.248.10

$q$-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} +(1.62284 - 0.605290i) 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} +(1.20650 + 3.23476i) q^{12} +(2.98648 - 2.98648i) q^{13} -0.248823 q^{14} +(3.79311 + 0.782512i) q^{15} -3.95962 q^{16} +(-1.86072 + 1.86072i) q^{17} +(0.0175672 - 0.245621i) 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} +(2.49025 - 4.56055i) q^{27} +(4.27256 - 4.27256i) q^{28} +5.48965 q^{29} +(0.265575 - 0.174739i) q^{30} -0.743102 q^{31} +(-0.693369 + 0.693369i) q^{32} +(2.52013 + 6.75672i) 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} +(-0.403801 + 0.150610i) q^{42} +(2.54619 - 2.54619i) q^{43} -8.29896 q^{44} +(6.62927 - 1.02603i) q^{45} -0.621506 q^{46} +(-4.88333 + 4.88333i) q^{47} +(-6.42585 + 2.39672i) 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} +(0.605290 + 1.62284i) q^{57} +(0.318626 - 0.318626i) q^{58} -12.7007 q^{59} +(-1.55975 + 7.56066i) q^{60} -5.26300 q^{61} +(-0.0431306 + 0.0431306i) q^{62} +(-9.07093 - 0.648767i) 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} +(0.980828 + 0.0701504i) q^{72} +(3.09062 - 3.09062i) q^{73} -0.457343 q^{74} +(6.25402 + 5.99059i) q^{75} -1.99326 q^{76} +(8.92447 - 8.92447i) q^{77} +(-0.209841 - 0.562605i) 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} +(8.90884 - 3.32283i) q^{87} +(-0.964992 + 0.964992i) q^{88} -7.94955 q^{89} +(0.325219 - 0.444324i) q^{90} -12.8030 q^{91} +(10.6719 - 10.6719i) q^{92} +(-1.20594 + 0.449792i) 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 \(\left(\mathbb{Z}/285\mathbb{Z}\right)^\times\).

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

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.0580413 − 0.0580413i 0.0410414 − 0.0410414i −0.686288 − 0.727330i \(-0.740761\pi\)
0.727330 + 0.686288i \(0.240761\pi\)
\(3\) 1.62284 − 0.605290i 0.936950 − 0.349464i
\(4\) 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 0.174492 − 0.984659i \(-0.444172\pi\)
−0.984659 + 0.174492i \(0.944172\pi\)
\(8\) 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 \(0.784008\pi\)
\(12\) 1.20650 + 3.23476i 0.348287 + 0.933793i
\(13\) 2.98648 − 2.98648i 0.828300 − 0.828300i −0.158981 − 0.987282i \(-0.550821\pi\)
0.987282 + 0.158981i \(0.0508210\pi\)
\(14\) −0.248823 −0.0665007
\(15\) 3.79311 + 0.782512i 0.979376 + 0.202044i
\(16\) −3.95962 −0.989905
\(17\) −1.86072 + 1.86072i −0.451291 + 0.451291i −0.895783 − 0.444492i \(-0.853384\pi\)
0.444492 + 0.895783i \(0.353384\pi\)
\(18\) 0.0175672 − 0.245621i 0.00414063 − 0.0578934i
\(19\) 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 −0.992267 − 0.124120i \(-0.960389\pi\)
−0.124120 − 0.992267i \(-0.539611\pi\)
\(24\) 0.516424 + 0.235843i 0.105415 + 0.0481412i
\(25\) 2.17442 + 4.50243i 0.434884 + 0.900486i
\(26\) − 0.346678i − 0.0679892i
\(27\) 2.49025 − 4.56055i 0.479249 − 0.877679i
\(28\) 4.27256 − 4.27256i 0.807438 − 0.807438i
\(29\) 5.48965 1.01940 0.509701 − 0.860352i \(-0.329756\pi\)
0.509701 + 0.860352i \(0.329756\pi\)
\(30\) 0.265575 − 0.174739i 0.0484871 − 0.0319028i
\(31\) −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
\(33\) 2.52013 + 6.75672i 0.438698 + 1.17619i
\(34\) 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 0.304736 − 0.952437i \(-0.401432\pi\)
−0.952437 + 0.304736i \(0.901432\pi\)
\(38\) 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 \(-0.193345\pi\)
\(42\) −0.403801 + 0.150610i −0.0623078 + 0.0232396i
\(43\) 2.54619 − 2.54619i 0.388291 − 0.388291i −0.485786 − 0.874078i \(-0.661467\pi\)
0.874078 + 0.485786i \(0.161467\pi\)
\(44\) −8.29896 −1.25111
\(45\) 6.62927 − 1.02603i 0.988234 − 0.152952i
\(46\) −0.621506 −0.0916361
\(47\) −4.88333 + 4.88333i −0.712306 + 0.712306i −0.967017 − 0.254711i \(-0.918020\pi\)
0.254711 + 0.967017i \(0.418020\pi\)
\(48\) −6.42585 + 2.39672i −0.927491 + 0.345936i
\(49\) 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 \(-0.299104\pi\)
−0.807359 + 0.590060i \(0.799104\pi\)
\(54\) −0.120163 − 0.409238i −0.0163521 − 0.0556902i
\(55\) −4.94878 + 7.88565i −0.667293 + 1.06330i
\(56\) − 0.993615i − 0.132777i
\(57\) 0.605290 + 1.62284i 0.0801726 + 0.214951i
\(58\) 0.318626 − 0.318626i 0.0418376 − 0.0418376i
\(59\) −12.7007 −1.65349 −0.826746 − 0.562575i \(-0.809811\pi\)
−0.826746 + 0.562575i \(0.809811\pi\)
\(60\) −1.55975 + 7.56066i −0.201363 + 0.976077i
\(61\) −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
\(63\) −9.07093 − 0.648767i −1.14283 − 0.0817370i
\(64\) − 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 0.797360 − 0.603505i \(-0.206229\pi\)
−0.603505 + 0.797360i \(0.706229\pi\)
\(68\) −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 \(0.843068\pi\)
\(72\) 0.980828 + 0.0701504i 0.115592 + 0.00826730i
\(73\) 3.09062 − 3.09062i 0.361729 − 0.361729i −0.502720 − 0.864449i \(-0.667667\pi\)
0.864449 + 0.502720i \(0.167667\pi\)
\(74\) −0.457343 −0.0531650
\(75\) 6.25402 + 5.99059i 0.722152 + 0.691734i
\(76\) −1.99326 −0.228643
\(77\) 8.92447 − 8.92447i 1.01704 − 1.01704i
\(78\) −0.209841 − 0.562605i −0.0237598 − 0.0637024i
\(79\) 4.68054i 0.526602i 0.964714 + 0.263301i \(0.0848113\pi\)
−0.964714 + 0.263301i \(0.915189\pi\)
\(80\) −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 \(-0.230130\pi\)
−0.661618 + 0.749841i \(0.730130\pi\)
\(84\) 4.34756 − 9.51983i 0.474358 − 1.03870i
\(85\) −5.73586 + 1.31252i −0.622141 + 0.142363i
\(86\) − 0.295569i − 0.0318720i
\(87\) 8.90884 − 3.32283i 0.955128 − 0.356244i
\(88\) −0.964992 + 0.964992i −0.102868 + 0.102868i
\(89\) −7.94955 −0.842650 −0.421325 − 0.906910i \(-0.638435\pi\)
−0.421325 + 0.906910i \(0.638435\pi\)
\(90\) 0.325219 − 0.444324i 0.0342811 − 0.0468358i
\(91\) −12.8030 −1.34212
\(92\) 10.6719 − 10.6719i 1.11263 − 1.11263i
\(93\) −1.20594 + 0.449792i −0.125050 + 0.0466412i
\(94\) 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 0.983880 + 0.178830i \(0.0572313\pi\)
0.178830 + 0.983880i \(0.442769\pi\)
\(98\) 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.77.10 yes 36
3.2 odd 2 inner 285.2.k.d.77.9 ✓ 36
5.3 odd 4 inner 285.2.k.d.248.9 yes 36
15.8 even 4 inner 285.2.k.d.248.10 yes 36
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
285.2.k.d.77.9 ✓ 36 3.2 odd 2 inner
285.2.k.d.77.10 yes 36 1.1 even 1 trivial
285.2.k.d.248.9 yes 36 5.3 odd 4 inner
285.2.k.d.248.10 yes 36 15.8 even 4 inner