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Properties

Label 285.2.k.d.77.1
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 77.1
Character \(\chi\) \(=\) 285.77
Dual form 285.2.k.d.248.1

$q$-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+(-1.91639 + 1.91639i) q^{2} +(-0.520717 - 1.65192i) q^{3} -5.34506i q^{4} +(0.478119 - 2.18435i) q^{5} +(4.16362 + 2.16783i) q^{6} +(-1.54943 - 1.54943i) q^{7} +(6.41043 + 6.41043i) q^{8} +(-2.45771 + 1.72037i) q^{9} +(3.26980 + 5.10232i) q^{10} -2.77537i q^{11} +(-8.82964 + 2.78327i) q^{12} +(-3.18746 + 3.18746i) q^{13} +5.93863 q^{14} +(-3.85735 + 0.347614i) q^{15} -13.8796 q^{16} +(-3.06903 + 3.06903i) q^{17} +(1.41302 - 8.00681i) q^{18} +1.00000i q^{19} +(-11.6755 - 2.55558i) q^{20} +(-1.75273 + 3.36636i) q^{21} +(5.31867 + 5.31867i) q^{22} +(1.05802 + 1.05802i) q^{23} +(7.25153 - 13.9276i) q^{24} +(-4.54280 - 2.08876i) q^{25} -12.2168i q^{26} +(4.12169 + 3.16412i) q^{27} +(-8.28182 + 8.28182i) q^{28} +0.341074 q^{29} +(6.72601 - 8.05833i) q^{30} +6.05973 q^{31} +(13.7778 - 13.7778i) q^{32} +(-4.58469 + 1.44518i) q^{33} -11.7629i q^{34} +(-4.12533 + 2.64370i) q^{35} +(9.19549 + 13.1366i) q^{36} +(-5.33202 - 5.33202i) q^{37} +(-1.91639 - 1.91639i) q^{38} +(6.92521 + 3.60568i) q^{39} +(17.0676 - 10.9377i) q^{40} -0.460773i q^{41} +(-3.09234 - 9.81016i) q^{42} +(-3.43474 + 3.43474i) q^{43} -14.8345 q^{44} +(2.58282 + 6.19105i) q^{45} -4.05513 q^{46} +(1.93023 - 1.93023i) q^{47} +(7.22733 + 22.9280i) q^{48} -2.19851i q^{49} +(12.7086 - 4.70289i) q^{50} +(6.66790 + 3.47171i) q^{51} +(17.0372 + 17.0372i) q^{52} +(-4.19597 - 4.19597i) q^{53} +(-13.9624 + 1.83507i) q^{54} +(-6.06238 - 1.32695i) q^{55} -19.8651i q^{56} +(1.65192 - 0.520717i) q^{57} +(-0.653630 + 0.653630i) q^{58} -12.0249 q^{59} +(1.85802 + 20.6178i) q^{60} -12.4849 q^{61} +(-11.6128 + 11.6128i) q^{62} +(6.47366 + 1.14246i) q^{63} +25.0478i q^{64} +(5.43856 + 8.48653i) q^{65} +(6.01652 - 11.5556i) q^{66} +(-2.96882 - 2.96882i) q^{67} +(16.4042 + 16.4042i) q^{68} +(1.19683 - 2.29869i) q^{69} +(2.83937 - 12.9721i) q^{70} +9.44996i q^{71} +(-26.7833 - 4.72666i) q^{72} +(11.0638 - 11.0638i) q^{73} +20.4364 q^{74} +(-1.08496 + 8.59202i) q^{75} +5.34506 q^{76} +(-4.30025 + 4.30025i) q^{77} +(-20.1812 + 6.36150i) q^{78} +1.95283i q^{79} +(-6.63609 + 30.3179i) q^{80} +(3.08066 - 8.45633i) q^{81} +(0.883018 + 0.883018i) q^{82} +(-1.98775 - 1.98775i) q^{83} +(17.9934 + 9.36846i) q^{84} +(5.23649 + 8.17121i) q^{85} -13.1646i q^{86} +(-0.177603 - 0.563429i) q^{87} +(17.7913 - 17.7913i) q^{88} +2.63417 q^{89} +(-16.8141 - 6.91475i) q^{90} +9.87752 q^{91} +(5.65516 - 5.65516i) q^{92} +(-3.15540 - 10.0102i) q^{93} +7.39814i q^{94} +(2.18435 + 0.478119i) q^{95} +(-29.9341 - 15.5855i) q^{96} +(-7.00016 - 7.00016i) q^{97} +(4.21319 + 4.21319i) q^{98} +(4.77465 + 6.82104i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.91639 + 1.91639i −1.35509 + 1.35509i −0.475224 + 0.879865i \(0.657633\pi\)
−0.879865 + 0.475224i \(0.842367\pi\)
\(3\) −0.520717 − 1.65192i −0.300636 − 0.953739i
\(4\) − 5.34506i − 2.67253i
\(5\) 0.478119 − 2.18435i 0.213821 − 0.976873i
\(6\) 4.16362 + 2.16783i 1.69979 + 0.885013i
\(7\) −1.54943 − 1.54943i −0.585631 − 0.585631i 0.350814 − 0.936445i \(-0.385905\pi\)
−0.936445 + 0.350814i \(0.885905\pi\)
\(8\) 6.41043 + 6.41043i 2.26643 + 2.26643i
\(9\) −2.45771 + 1.72037i −0.819236 + 0.573457i
\(10\) 3.26980 + 5.10232i 1.03400 + 1.61350i
\(11\) − 2.77537i − 0.836804i −0.908262 − 0.418402i \(-0.862590\pi\)
0.908262 − 0.418402i \(-0.137410\pi\)
\(12\) −8.82964 + 2.78327i −2.54890 + 0.803459i
\(13\) −3.18746 + 3.18746i −0.884043 + 0.884043i −0.993943 − 0.109900i \(-0.964947\pi\)
0.109900 + 0.993943i \(0.464947\pi\)
\(14\) 5.93863 1.58716
\(15\) −3.85735 + 0.347614i −0.995964 + 0.0897535i
\(16\) −13.8796 −3.46990
\(17\) −3.06903 + 3.06903i −0.744349 + 0.744349i −0.973412 − 0.229063i \(-0.926434\pi\)
0.229063 + 0.973412i \(0.426434\pi\)
\(18\) 1.41302 − 8.00681i 0.333053 − 1.88722i
\(19\) 1.00000i 0.229416i
\(20\) −11.6755 − 2.55558i −2.61072 − 0.571444i
\(21\) −1.75273 + 3.36636i −0.382477 + 0.734601i
\(22\) 5.31867 + 5.31867i 1.13394 + 1.13394i
\(23\) 1.05802 + 1.05802i 0.220611 + 0.220611i 0.808756 − 0.588144i \(-0.200141\pi\)
−0.588144 + 0.808756i \(0.700141\pi\)
\(24\) 7.25153 − 13.9276i 1.48021 − 2.84295i
\(25\) −4.54280 − 2.08876i −0.908561 − 0.417752i
\(26\) − 12.2168i − 2.39591i
\(27\) 4.12169 + 3.16412i 0.793220 + 0.608936i
\(28\) −8.28182 + 8.28182i −1.56512 + 1.56512i
\(29\) 0.341074 0.0633359 0.0316680 − 0.999498i \(-0.489918\pi\)
0.0316680 + 0.999498i \(0.489918\pi\)
\(30\) 6.72601 − 8.05833i 1.22800 − 1.47124i
\(31\) 6.05973 1.08836 0.544180 − 0.838969i \(-0.316841\pi\)
0.544180 + 0.838969i \(0.316841\pi\)
\(32\) 13.7778 − 13.7778i 2.43559 − 2.43559i
\(33\) −4.58469 + 1.44518i −0.798093 + 0.251573i
\(34\) − 11.7629i − 2.01732i
\(35\) −4.12533 + 2.64370i −0.697307 + 0.446867i
\(36\) 9.19549 + 13.1366i 1.53258 + 2.18943i
\(37\) −5.33202 − 5.33202i −0.876579 − 0.876579i 0.116600 − 0.993179i \(-0.462801\pi\)
−0.993179 + 0.116600i \(0.962801\pi\)
\(38\) −1.91639 − 1.91639i −0.310879 − 0.310879i
\(39\) 6.92521 + 3.60568i 1.10892 + 0.577371i
\(40\) 17.0676 − 10.9377i 2.69862 − 1.72940i
\(41\) − 0.460773i − 0.0719606i −0.999352 − 0.0359803i \(-0.988545\pi\)
0.999352 − 0.0359803i \(-0.0114554\pi\)
\(42\) −3.09234 − 9.81016i −0.477159 − 1.51374i
\(43\) −3.43474 + 3.43474i −0.523794 + 0.523794i −0.918715 − 0.394921i \(-0.870772\pi\)
0.394921 + 0.918715i \(0.370772\pi\)
\(44\) −14.8345 −2.23639
\(45\) 2.58282 + 6.19105i 0.385024 + 0.922907i
\(46\) −4.05513 −0.597896
\(47\) 1.93023 − 1.93023i 0.281553 − 0.281553i −0.552175 − 0.833728i \(-0.686202\pi\)
0.833728 + 0.552175i \(0.186202\pi\)
\(48\) 7.22733 + 22.9280i 1.04318 + 3.30937i
\(49\) − 2.19851i − 0.314073i
\(50\) 12.7086 − 4.70289i 1.79727 − 0.665089i
\(51\) 6.66790 + 3.47171i 0.933693 + 0.486137i
\(52\) 17.0372 + 17.0372i 2.36263 + 2.36263i
\(53\) −4.19597 − 4.19597i −0.576360 − 0.576360i 0.357538 − 0.933899i \(-0.383616\pi\)
−0.933899 + 0.357538i \(0.883616\pi\)
\(54\) −13.9624 + 1.83507i −1.90005 + 0.249722i
\(55\) −6.06238 − 1.32695i −0.817451 − 0.178926i
\(56\) − 19.8651i − 2.65458i
\(57\) 1.65192 − 0.520717i 0.218803 − 0.0689706i
\(58\) −0.653630 + 0.653630i −0.0858258 + 0.0858258i
\(59\) −12.0249 −1.56551 −0.782757 − 0.622327i \(-0.786187\pi\)
−0.782757 + 0.622327i \(0.786187\pi\)
\(60\) 1.85802 + 20.6178i 0.239869 + 2.66175i
\(61\) −12.4849 −1.59853 −0.799263 − 0.600981i \(-0.794777\pi\)
−0.799263 + 0.600981i \(0.794777\pi\)
\(62\) −11.6128 + 11.6128i −1.47482 + 1.47482i
\(63\) 6.47366 + 1.14246i 0.815604 + 0.143936i
\(64\) 25.0478i 3.13098i
\(65\) 5.43856 + 8.48653i 0.674570 + 1.05262i
\(66\) 6.01652 − 11.5556i 0.740582 − 1.42239i
\(67\) −2.96882 − 2.96882i −0.362699 − 0.362699i 0.502107 − 0.864806i \(-0.332558\pi\)
−0.864806 + 0.502107i \(0.832558\pi\)
\(68\) 16.4042 + 16.4042i 1.98930 + 1.98930i
\(69\) 1.19683 − 2.29869i 0.144082 − 0.276730i
\(70\) 2.83937 − 12.9721i 0.339369 − 1.55046i
\(71\) 9.44996i 1.12150i 0.827984 + 0.560752i \(0.189488\pi\)
−0.827984 + 0.560752i \(0.810512\pi\)
\(72\) −26.7833 − 4.72666i −3.15644 − 0.557042i
\(73\) 11.0638 − 11.0638i 1.29492 − 1.29492i 0.363214 − 0.931706i \(-0.381679\pi\)
0.931706 − 0.363214i \(-0.118321\pi\)
\(74\) 20.4364 2.37569
\(75\) −1.08496 + 8.59202i −0.125280 + 0.992121i
\(76\) 5.34506 0.613121
\(77\) −4.30025 + 4.30025i −0.490058 + 0.490058i
\(78\) −20.1812 + 6.36150i −2.28508 + 0.720298i
\(79\) 1.95283i 0.219710i 0.993948 + 0.109855i \(0.0350387\pi\)
−0.993948 + 0.109855i \(0.964961\pi\)
\(80\) −6.63609 + 30.3179i −0.741937 + 3.38965i
\(81\) 3.08066 − 8.45633i 0.342295 − 0.939593i
\(82\) 0.883018 + 0.883018i 0.0975130 + 0.0975130i
\(83\) −1.98775 − 1.98775i −0.218184 − 0.218184i 0.589549 − 0.807733i \(-0.299306\pi\)
−0.807733 + 0.589549i \(0.799306\pi\)
\(84\) 17.9934 + 9.36846i 1.96324 + 1.02218i
\(85\) 5.23649 + 8.17121i 0.567977 + 0.886292i
\(86\) − 13.1646i − 1.41957i
\(87\) −0.177603 − 0.563429i −0.0190411 − 0.0604059i
\(88\) 17.7913 − 17.7913i 1.89656 − 1.89656i
\(89\) 2.63417 0.279222 0.139611 − 0.990206i \(-0.455415\pi\)
0.139611 + 0.990206i \(0.455415\pi\)
\(90\) −16.8141 − 6.91475i −1.77236 − 0.728878i
\(91\) 9.87752 1.03545
\(92\) 5.65516 − 5.65516i 0.589591 − 0.589591i
\(93\) −3.15540 − 10.0102i −0.327200 − 1.03801i
\(94\) 7.39814i 0.763060i
\(95\) 2.18435 + 0.478119i 0.224110 + 0.0490540i
\(96\) −29.9341 − 15.5855i −3.05514 − 1.59069i
\(97\) −7.00016 − 7.00016i −0.710759 − 0.710759i 0.255935 − 0.966694i \(-0.417617\pi\)
−0.966694 + 0.255935i \(0.917617\pi\)
\(98\) 4.21319 + 4.21319i 0.425596 + 0.425596i
\(99\) 4.77465 + 6.82104i 0.479871 + 0.685540i
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.77.1 ✓ 36
3.2 odd 2 inner 285.2.k.d.77.18 yes 36
5.3 odd 4 inner 285.2.k.d.248.18 yes 36
15.8 even 4 inner 285.2.k.d.248.1 yes 36
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
285.2.k.d.77.1 ✓ 36 1.1 even 1 trivial
285.2.k.d.77.18 yes 36 3.2 odd 2 inner
285.2.k.d.248.1 yes 36 15.8 even 4 inner
285.2.k.d.248.18 yes 36 5.3 odd 4 inner