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Properties

Label 285.2.k.c.77.7
Level $285$
Weight $2$
Character 285.77
Analytic conductor $2.276$
Analytic rank $0$
Dimension $28$
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 = [28,0] 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: \(28\)
Relative dimension: \(14\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 77.7
Character \(\chi\) \(=\) 285.77
Dual form 285.2.k.c.248.7

$q$-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+(-0.0307908 + 0.0307908i) q^{2} +(-1.45499 - 0.939687i) q^{3} +1.99810i q^{4} +(-1.20318 - 1.88477i) q^{5} +(0.0737341 - 0.0158665i) q^{6} +(-0.715376 - 0.715376i) q^{7} +(-0.123105 - 0.123105i) q^{8} +(1.23398 + 2.73447i) q^{9} +(0.0950807 + 0.0209865i) q^{10} +5.75603i q^{11} +(1.87759 - 2.90722i) q^{12} +(-1.65056 + 1.65056i) q^{13} +0.0440541 q^{14} +(-0.0204736 + 3.87293i) q^{15} -3.98863 q^{16} +(-4.22727 + 4.22727i) q^{17} +(-0.122192 - 0.0462014i) q^{18} -1.00000i q^{19} +(3.76596 - 2.40409i) q^{20} +(0.368633 + 1.71309i) q^{21} +(-0.177233 - 0.177233i) q^{22} +(-1.96284 - 1.96284i) q^{23} +(0.0634360 + 0.294796i) q^{24} +(-2.10469 + 4.53545i) q^{25} -0.101644i q^{26} +(0.774125 - 5.13816i) q^{27} +(1.42939 - 1.42939i) q^{28} -5.38666 q^{29} +(-0.118620 - 0.119881i) q^{30} +4.85064 q^{31} +(0.369023 - 0.369023i) q^{32} +(5.40887 - 8.37495i) q^{33} -0.260323i q^{34} +(-0.487588 + 2.20905i) q^{35} +(-5.46375 + 2.46561i) q^{36} +(6.33347 + 6.33347i) q^{37} +(0.0307908 + 0.0307908i) q^{38} +(3.95255 - 0.850533i) q^{39} +(-0.0839062 + 0.380142i) q^{40} +1.84462i q^{41} +(-0.0640981 - 0.0413970i) q^{42} +(-1.35872 + 1.35872i) q^{43} -11.5011 q^{44} +(3.66913 - 5.61582i) q^{45} +0.120875 q^{46} +(5.93582 - 5.93582i) q^{47} +(5.80340 + 3.74806i) q^{48} -5.97648i q^{49} +(-0.0748449 - 0.204456i) q^{50} +(10.1229 - 2.17831i) q^{51} +(-3.29799 - 3.29799i) q^{52} +(-0.615292 - 0.615292i) q^{53} +(0.134372 + 0.182044i) q^{54} +(10.8488 - 6.92556i) q^{55} +0.176133i q^{56} +(-0.939687 + 1.45499i) q^{57} +(0.165860 - 0.165860i) q^{58} -2.01172 q^{59} +(-7.73851 - 0.0409085i) q^{60} -5.22784 q^{61} +(-0.149355 + 0.149355i) q^{62} +(1.07341 - 2.83893i) q^{63} -7.95453i q^{64} +(5.09685 + 1.12499i) q^{65} +(0.0913281 + 0.424415i) q^{66} +(-8.61810 - 8.61810i) q^{67} +(-8.44653 - 8.44653i) q^{68} +(1.01145 + 4.70035i) q^{69} +(-0.0530052 - 0.0830316i) q^{70} +8.55888i q^{71} +(0.184718 - 0.488535i) q^{72} +(-8.09254 + 8.09254i) q^{73} -0.390026 q^{74} +(7.32420 - 4.62126i) q^{75} +1.99810 q^{76} +(4.11772 - 4.11772i) q^{77} +(-0.0955138 + 0.147891i) q^{78} -13.6363i q^{79} +(4.79905 + 7.51763i) q^{80} +(-5.95461 + 6.74853i) q^{81} +(-0.0567975 - 0.0567975i) q^{82} +(-1.52493 - 1.52493i) q^{83} +(-3.42294 + 0.736567i) q^{84} +(13.0536 + 2.88124i) q^{85} -0.0836722i q^{86} +(7.83751 + 5.06177i) q^{87} +(0.708596 - 0.708596i) q^{88} +7.80034 q^{89} +(0.0599403 + 0.285892i) q^{90} +2.36154 q^{91} +(3.92195 - 3.92195i) q^{92} +(-7.05761 - 4.55808i) q^{93} +0.365538i q^{94} +(-1.88477 + 1.20318i) q^{95} +(-0.883691 + 0.190158i) q^{96} +(9.06784 + 9.06784i) q^{97} +(0.184021 + 0.184021i) q^{98} +(-15.7397 + 7.10280i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 28 q - 2 q^{3} - 12 q^{6} - 8 q^{10} + 34 q^{12} + 8 q^{13} - 14 q^{15} - 20 q^{16} - 24 q^{18} - 4 q^{21} - 32 q^{22} + 8 q^{25} + 22 q^{27} - 28 q^{28} + 12 q^{30} + 72 q^{31} - 84 q^{36} - 12 q^{37}+ \cdots - 80 q^{96}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.0307908 + 0.0307908i −0.0217724 + 0.0217724i −0.717909 − 0.696137i \(-0.754901\pi\)
0.696137 + 0.717909i \(0.254901\pi\)
\(3\) −1.45499 − 0.939687i −0.840037 − 0.542529i
\(4\) 1.99810i 0.999052i
\(5\) −1.20318 − 1.88477i −0.538080 − 0.842893i
\(6\) 0.0737341 − 0.0158665i 0.0301018 − 0.00647748i
\(7\) −0.715376 − 0.715376i −0.270387 − 0.270387i 0.558869 − 0.829256i \(-0.311235\pi\)
−0.829256 + 0.558869i \(0.811235\pi\)
\(8\) −0.123105 − 0.123105i −0.0435242 − 0.0435242i
\(9\) 1.23398 + 2.73447i 0.411325 + 0.911489i
\(10\) 0.0950807 + 0.0209865i 0.0300671 + 0.00663652i
\(11\) 5.75603i 1.73551i 0.496994 + 0.867754i \(0.334437\pi\)
−0.496994 + 0.867754i \(0.665563\pi\)
\(12\) 1.87759 − 2.90722i 0.542014 − 0.839241i
\(13\) −1.65056 + 1.65056i −0.457783 + 0.457783i −0.897927 − 0.440144i \(-0.854927\pi\)
0.440144 + 0.897927i \(0.354927\pi\)
\(14\) 0.0440541 0.0117739
\(15\) −0.0204736 + 3.87293i −0.00528627 + 0.999986i
\(16\) −3.98863 −0.997157
\(17\) −4.22727 + 4.22727i −1.02526 + 1.02526i −0.0255918 + 0.999672i \(0.508147\pi\)
−0.999672 + 0.0255918i \(0.991853\pi\)
\(18\) −0.122192 − 0.0462014i −0.0288009 − 0.0108898i
\(19\) − 1.00000i − 0.229416i
\(20\) 3.76596 − 2.40409i 0.842094 − 0.537570i
\(21\) 0.368633 + 1.71309i 0.0804423 + 0.373827i
\(22\) −0.177233 − 0.177233i −0.0377862 − 0.0377862i
\(23\) −1.96284 − 1.96284i −0.409280 − 0.409280i 0.472208 − 0.881487i \(-0.343457\pi\)
−0.881487 + 0.472208i \(0.843457\pi\)
\(24\) 0.0634360 + 0.294796i 0.0129488 + 0.0601751i
\(25\) −2.10469 + 4.53545i −0.420939 + 0.907089i
\(26\) − 0.101644i − 0.0199341i
\(27\) 0.774125 − 5.13816i 0.148980 − 0.988840i
\(28\) 1.42939 − 1.42939i 0.270130 − 0.270130i
\(29\) −5.38666 −1.00028 −0.500138 − 0.865945i \(-0.666718\pi\)
−0.500138 + 0.865945i \(0.666718\pi\)
\(30\) −0.118620 − 0.119881i −0.0216570 − 0.0218872i
\(31\) 4.85064 0.871200 0.435600 − 0.900140i \(-0.356536\pi\)
0.435600 + 0.900140i \(0.356536\pi\)
\(32\) 0.369023 − 0.369023i 0.0652347 − 0.0652347i
\(33\) 5.40887 − 8.37495i 0.941563 − 1.45789i
\(34\) − 0.260323i − 0.0446450i
\(35\) −0.487588 + 2.20905i −0.0824173 + 0.373397i
\(36\) −5.46375 + 2.46561i −0.910625 + 0.410935i
\(37\) 6.33347 + 6.33347i 1.04122 + 1.04122i 0.999113 + 0.0421033i \(0.0134059\pi\)
0.0421033 + 0.999113i \(0.486594\pi\)
\(38\) 0.0307908 + 0.0307908i 0.00499494 + 0.00499494i
\(39\) 3.95255 − 0.850533i 0.632915 − 0.136194i
\(40\) −0.0839062 + 0.380142i −0.0132667 + 0.0601058i
\(41\) 1.84462i 0.288081i 0.989572 + 0.144041i \(0.0460096\pi\)
−0.989572 + 0.144041i \(0.953990\pi\)
\(42\) −0.0640981 − 0.0413970i −0.00989055 − 0.00638770i
\(43\) −1.35872 + 1.35872i −0.207203 + 0.207203i −0.803077 − 0.595875i \(-0.796805\pi\)
0.595875 + 0.803077i \(0.296805\pi\)
\(44\) −11.5011 −1.73386
\(45\) 3.66913 − 5.61582i 0.546962 − 0.837158i
\(46\) 0.120875 0.0178220
\(47\) 5.93582 − 5.93582i 0.865829 − 0.865829i −0.126179 − 0.992008i \(-0.540271\pi\)
0.992008 + 0.126179i \(0.0402712\pi\)
\(48\) 5.80340 + 3.74806i 0.837649 + 0.540986i
\(49\) − 5.97648i − 0.853782i
\(50\) −0.0748449 − 0.204456i −0.0105847 − 0.0289144i
\(51\) 10.1229 − 2.17831i 1.41750 − 0.305025i
\(52\) −3.29799 − 3.29799i −0.457349 − 0.457349i
\(53\) −0.615292 − 0.615292i −0.0845168 − 0.0845168i 0.663585 − 0.748101i \(-0.269034\pi\)
−0.748101 + 0.663585i \(0.769034\pi\)
\(54\) 0.134372 + 0.182044i 0.0182858 + 0.0247731i
\(55\) 10.8488 − 6.92556i 1.46285 − 0.933843i
\(56\) 0.176133i 0.0235367i
\(57\) −0.939687 + 1.45499i −0.124465 + 0.192718i
\(58\) 0.165860 − 0.165860i 0.0217784 − 0.0217784i
\(59\) −2.01172 −0.261904 −0.130952 − 0.991389i \(-0.541803\pi\)
−0.130952 + 0.991389i \(0.541803\pi\)
\(60\) −7.73851 − 0.0409085i −0.999038 − 0.00528126i
\(61\) −5.22784 −0.669356 −0.334678 − 0.942333i \(-0.608628\pi\)
−0.334678 + 0.942333i \(0.608628\pi\)
\(62\) −0.149355 + 0.149355i −0.0189681 + 0.0189681i
\(63\) 1.07341 − 2.83893i 0.135238 − 0.357671i
\(64\) − 7.95453i − 0.994316i
\(65\) 5.09685 + 1.12499i 0.632186 + 0.139538i
\(66\) 0.0913281 + 0.424415i 0.0112417 + 0.0522419i
\(67\) −8.61810 − 8.61810i −1.05287 − 1.05287i −0.998522 − 0.0543460i \(-0.982693\pi\)
−0.0543460 − 0.998522i \(-0.517307\pi\)
\(68\) −8.44653 − 8.44653i −1.02429 − 1.02429i
\(69\) 1.01145 + 4.70035i 0.121764 + 0.565856i
\(70\) −0.0530052 − 0.0830316i −0.00633533 − 0.00992418i
\(71\) 8.55888i 1.01575i 0.861430 + 0.507876i \(0.169569\pi\)
−0.861430 + 0.507876i \(0.830431\pi\)
\(72\) 0.184718 − 0.488535i 0.0217692 − 0.0575744i
\(73\) −8.09254 + 8.09254i −0.947160 + 0.947160i −0.998672 − 0.0515119i \(-0.983596\pi\)
0.0515119 + 0.998672i \(0.483596\pi\)
\(74\) −0.390026 −0.0453396
\(75\) 7.32420 − 4.62126i 0.845726 − 0.533617i
\(76\) 1.99810 0.229198
\(77\) 4.11772 − 4.11772i 0.469258 − 0.469258i
\(78\) −0.0955138 + 0.147891i −0.0108148 + 0.0167454i
\(79\) − 13.6363i − 1.53421i −0.641522 − 0.767104i \(-0.721697\pi\)
0.641522 − 0.767104i \(-0.278303\pi\)
\(80\) 4.79905 + 7.51763i 0.536551 + 0.840497i
\(81\) −5.95461 + 6.74853i −0.661623 + 0.749836i
\(82\) −0.0567975 − 0.0567975i −0.00627223 − 0.00627223i
\(83\) −1.52493 − 1.52493i −0.167383 − 0.167383i 0.618445 − 0.785828i \(-0.287763\pi\)
−0.785828 + 0.618445i \(0.787763\pi\)
\(84\) −3.42294 + 0.736567i −0.373473 + 0.0803660i
\(85\) 13.0536 + 2.88124i 1.41586 + 0.312514i
\(86\) − 0.0836722i − 0.00902260i
\(87\) 7.83751 + 5.06177i 0.840270 + 0.542679i
\(88\) 0.708596 − 0.708596i 0.0755366 − 0.0755366i
\(89\) 7.80034 0.826834 0.413417 − 0.910542i \(-0.364335\pi\)
0.413417 + 0.910542i \(0.364335\pi\)
\(90\) 0.0599403 + 0.285892i 0.00631826 + 0.0301356i
\(91\) 2.36154 0.247557
\(92\) 3.92195 − 3.92195i 0.408891 − 0.408891i
\(93\) −7.05761 − 4.55808i −0.731840 − 0.472651i
\(94\) 0.365538i 0.0377024i
\(95\) −1.88477 + 1.20318i −0.193373 + 0.123444i
\(96\) −0.883691 + 0.190158i −0.0901913 + 0.0194079i
\(97\) 9.06784 + 9.06784i 0.920699 + 0.920699i 0.997079 − 0.0763795i \(-0.0243360\pi\)
−0.0763795 + 0.997079i \(0.524336\pi\)
\(98\) 0.184021 + 0.184021i 0.0185889 + 0.0185889i
\(99\) −15.7397 + 7.10280i −1.58190 + 0.713858i
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.c.77.7 ✓ 28
3.2 odd 2 inner 285.2.k.c.77.8 yes 28
5.3 odd 4 inner 285.2.k.c.248.8 yes 28
15.8 even 4 inner 285.2.k.c.248.7 yes 28
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
285.2.k.c.77.7 ✓ 28 1.1 even 1 trivial
285.2.k.c.77.8 yes 28 3.2 odd 2 inner
285.2.k.c.248.7 yes 28 15.8 even 4 inner
285.2.k.c.248.8 yes 28 5.3 odd 4 inner