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Properties

Label 285.2.k.c.77.1
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 77.1
Character \(\chi\) \(=\) 285.77
Dual form 285.2.k.c.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+(-1.82223 + 1.82223i) q^{2} +(-1.28293 + 1.16366i) q^{3} -4.64103i q^{4} +(1.53042 + 1.63028i) q^{5} +(0.217337 - 4.45823i) q^{6} +(-1.83721 - 1.83721i) q^{7} +(4.81255 + 4.81255i) q^{8} +(0.291804 - 2.98577i) q^{9} +(-5.75952 - 0.181957i) q^{10} -4.54215i q^{11} +(5.40056 + 5.95410i) q^{12} +(4.46633 - 4.46633i) q^{13} +6.69563 q^{14} +(-3.86051 - 0.310639i) q^{15} -8.25708 q^{16} +(-1.82800 + 1.82800i) q^{17} +(4.90903 + 5.97250i) q^{18} -1.00000i q^{19} +(7.56616 - 7.10274i) q^{20} +(4.49489 + 0.219124i) q^{21} +(8.27683 + 8.27683i) q^{22} +(-3.81824 - 3.81824i) q^{23} +(-11.7743 - 0.573994i) q^{24} +(-0.315609 + 4.99003i) q^{25} +16.2773i q^{26} +(3.10005 + 4.17009i) q^{27} +(-8.52654 + 8.52654i) q^{28} +2.20595 q^{29} +(7.60078 - 6.46866i) q^{30} -2.07570 q^{31} +(5.42118 - 5.42118i) q^{32} +(5.28550 + 5.82724i) q^{33} -6.66208i q^{34} +(0.183453 - 5.80687i) q^{35} +(-13.8571 - 1.35427i) q^{36} +(0.958376 + 0.958376i) q^{37} +(1.82223 + 1.82223i) q^{38} +(-0.532700 + 10.9272i) q^{39} +(-0.480553 + 15.2110i) q^{40} +1.66549i q^{41} +(-8.59001 + 7.79142i) q^{42} +(5.41477 - 5.41477i) q^{43} -21.0802 q^{44} +(5.31423 - 4.09378i) q^{45} +13.9154 q^{46} +(1.25505 - 1.25505i) q^{47} +(10.5932 - 9.60841i) q^{48} -0.249319i q^{49} +(-8.51786 - 9.66808i) q^{50} +(0.218026 - 4.47237i) q^{51} +(-20.7284 - 20.7284i) q^{52} +(-7.89753 - 7.89753i) q^{53} +(-13.2479 - 1.94985i) q^{54} +(7.40496 - 6.95141i) q^{55} -17.6833i q^{56} +(1.16366 + 1.28293i) q^{57} +(-4.01974 + 4.01974i) q^{58} +8.55290 q^{59} +(-1.44169 + 17.9167i) q^{60} +4.48338 q^{61} +(3.78240 - 3.78240i) q^{62} +(-6.02160 + 4.94939i) q^{63} +3.24307i q^{64} +(14.1167 + 0.445981i) q^{65} +(-20.2500 - 0.987178i) q^{66} +(-2.12117 - 2.12117i) q^{67} +(8.48382 + 8.48382i) q^{68} +(9.34164 + 0.455402i) q^{69} +(10.2471 + 10.9157i) q^{70} -3.02201i q^{71} +(15.7735 - 12.9649i) q^{72} +(3.69215 - 3.69215i) q^{73} -3.49276 q^{74} +(-5.40178 - 6.76910i) q^{75} -4.64103 q^{76} +(-8.34488 + 8.34488i) q^{77} +(-18.9412 - 20.8826i) q^{78} +0.976417i q^{79} +(-12.6368 - 13.4613i) q^{80} +(-8.82970 - 1.74252i) q^{81} +(-3.03491 - 3.03491i) q^{82} +(-0.264467 - 0.264467i) q^{83} +(1.01696 - 20.8609i) q^{84} +(-5.77777 - 0.182534i) q^{85} +19.7339i q^{86} +(-2.83007 + 2.56697i) q^{87} +(21.8593 - 21.8593i) q^{88} +6.33181 q^{89} +(-2.22393 + 17.1435i) q^{90} -16.4112 q^{91} +(-17.7205 + 17.7205i) q^{92} +(2.66297 - 2.41541i) q^{93} +4.57398i q^{94} +(1.63028 - 1.53042i) q^{95} +(-0.646584 + 13.2634i) q^{96} +(-6.87894 - 6.87894i) q^{97} +(0.454315 + 0.454315i) q^{98} +(-13.5618 - 1.32542i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.82223 + 1.82223i −1.28851 + 1.28851i −0.352817 + 0.935692i \(0.614776\pi\)
−0.935692 + 0.352817i \(0.885224\pi\)
\(3\) −1.28293 + 1.16366i −0.740698 + 0.671838i
\(4\) − 4.64103i − 2.32051i
\(5\) 1.53042 + 1.63028i 0.684426 + 0.729082i
\(6\) 0.217337 − 4.45823i 0.0887276 − 1.82007i
\(7\) −1.83721 − 1.83721i −0.694400 − 0.694400i 0.268797 − 0.963197i \(-0.413374\pi\)
−0.963197 + 0.268797i \(0.913374\pi\)
\(8\) 4.81255 + 4.81255i 1.70149 + 1.70149i
\(9\) 0.291804 − 2.98577i 0.0972681 − 0.995258i
\(10\) −5.75952 − 0.181957i −1.82132 − 0.0575399i
\(11\) − 4.54215i − 1.36951i −0.728774 − 0.684754i \(-0.759909\pi\)
0.728774 − 0.684754i \(-0.240091\pi\)
\(12\) 5.40056 + 5.95410i 1.55901 + 1.71880i
\(13\) 4.46633 − 4.46633i 1.23874 − 1.23874i 0.278219 − 0.960518i \(-0.410256\pi\)
0.960518 − 0.278219i \(-0.0897440\pi\)
\(14\) 6.69563 1.78948
\(15\) −3.86051 − 0.310639i −0.996778 − 0.0802067i
\(16\) −8.25708 −2.06427
\(17\) −1.82800 + 1.82800i −0.443356 + 0.443356i −0.893138 − 0.449782i \(-0.851502\pi\)
0.449782 + 0.893138i \(0.351502\pi\)
\(18\) 4.90903 + 5.97250i 1.15707 + 1.40773i
\(19\) − 1.00000i − 0.229416i
\(20\) 7.56616 − 7.10274i 1.69185 − 1.58822i
\(21\) 4.49489 + 0.219124i 0.980865 + 0.0478168i
\(22\) 8.27683 + 8.27683i 1.76463 + 1.76463i
\(23\) −3.81824 − 3.81824i −0.796157 − 0.796157i 0.186330 − 0.982487i \(-0.440341\pi\)
−0.982487 + 0.186330i \(0.940341\pi\)
\(24\) −11.7743 − 0.573994i −2.40342 − 0.117166i
\(25\) −0.315609 + 4.99003i −0.0631218 + 0.998006i
\(26\) 16.2773i 3.19225i
\(27\) 3.10005 + 4.17009i 0.596606 + 0.802535i
\(28\) −8.52654 + 8.52654i −1.61137 + 1.61137i
\(29\) 2.20595 0.409635 0.204817 − 0.978800i \(-0.434340\pi\)
0.204817 + 0.978800i \(0.434340\pi\)
\(30\) 7.60078 − 6.46866i 1.38771 − 1.18101i
\(31\) −2.07570 −0.372807 −0.186404 − 0.982473i \(-0.559683\pi\)
−0.186404 + 0.982473i \(0.559683\pi\)
\(32\) 5.42118 − 5.42118i 0.958338 − 0.958338i
\(33\) 5.28550 + 5.82724i 0.920088 + 1.01439i
\(34\) − 6.66208i − 1.14254i
\(35\) 0.183453 − 5.80687i 0.0310092 − 0.981540i
\(36\) −13.8571 − 1.35427i −2.30951 − 0.225712i
\(37\) 0.958376 + 0.958376i 0.157556 + 0.157556i 0.781483 − 0.623927i \(-0.214464\pi\)
−0.623927 + 0.781483i \(0.714464\pi\)
\(38\) 1.82223 + 1.82223i 0.295604 + 0.295604i
\(39\) −0.532700 + 10.9272i −0.0853002 + 1.74976i
\(40\) −0.480553 + 15.2110i −0.0759822 + 2.40508i
\(41\) 1.66549i 0.260106i 0.991507 + 0.130053i \(0.0415148\pi\)
−0.991507 + 0.130053i \(0.958485\pi\)
\(42\) −8.59001 + 7.79142i −1.32547 + 1.20224i
\(43\) 5.41477 − 5.41477i 0.825745 − 0.825745i −0.161180 − 0.986925i \(-0.551530\pi\)
0.986925 + 0.161180i \(0.0515301\pi\)
\(44\) −21.0802 −3.17796
\(45\) 5.31423 − 4.09378i 0.792198 − 0.610264i
\(46\) 13.9154 2.05171
\(47\) 1.25505 − 1.25505i 0.183068 − 0.183068i −0.609623 − 0.792691i \(-0.708679\pi\)
0.792691 + 0.609623i \(0.208679\pi\)
\(48\) 10.5932 − 9.60841i 1.52900 − 1.38685i
\(49\) − 0.249319i − 0.0356169i
\(50\) −8.51786 − 9.66808i −1.20461 − 1.36727i
\(51\) 0.218026 − 4.47237i 0.0305298 − 0.626256i
\(52\) −20.7284 − 20.7284i −2.87451 − 2.87451i
\(53\) −7.89753 − 7.89753i −1.08481 − 1.08481i −0.996053 − 0.0887564i \(-0.971711\pi\)
−0.0887564 − 0.996053i \(-0.528289\pi\)
\(54\) −13.2479 − 1.94985i −1.80281 − 0.265341i
\(55\) 7.40496 − 6.95141i 0.998485 − 0.937328i
\(56\) − 17.6833i − 2.36304i
\(57\) 1.16366 + 1.28293i 0.154130 + 0.169928i
\(58\) −4.01974 + 4.01974i −0.527818 + 0.527818i
\(59\) 8.55290 1.11349 0.556746 − 0.830683i \(-0.312050\pi\)
0.556746 + 0.830683i \(0.312050\pi\)
\(60\) −1.44169 + 17.9167i −0.186121 + 2.31304i
\(61\) 4.48338 0.574038 0.287019 − 0.957925i \(-0.407336\pi\)
0.287019 + 0.957925i \(0.407336\pi\)
\(62\) 3.78240 − 3.78240i 0.480366 − 0.480366i
\(63\) −6.02160 + 4.94939i −0.758650 + 0.623564i
\(64\) 3.24307i 0.405384i
\(65\) 14.1167 + 0.445981i 1.75096 + 0.0553172i
\(66\) −20.2500 − 0.987178i −2.49260 − 0.121513i
\(67\) −2.12117 − 2.12117i −0.259142 − 0.259142i 0.565563 − 0.824705i \(-0.308659\pi\)
−0.824705 + 0.565563i \(0.808659\pi\)
\(68\) 8.48382 + 8.48382i 1.02881 + 1.02881i
\(69\) 9.34164 + 0.455402i 1.12460 + 0.0548239i
\(70\) 10.2471 + 10.9157i 1.22477 + 1.30468i
\(71\) − 3.02201i − 0.358647i −0.983790 − 0.179324i \(-0.942609\pi\)
0.983790 − 0.179324i \(-0.0573909\pi\)
\(72\) 15.7735 − 12.9649i 1.85893 − 1.52793i
\(73\) 3.69215 − 3.69215i 0.432134 − 0.432134i −0.457220 − 0.889354i \(-0.651155\pi\)
0.889354 + 0.457220i \(0.151155\pi\)
\(74\) −3.49276 −0.406025
\(75\) −5.40178 − 6.76910i −0.623744 − 0.781629i
\(76\) −4.64103 −0.532362
\(77\) −8.34488 + 8.34488i −0.950987 + 0.950987i
\(78\) −18.9412 − 20.8826i −2.14467 − 2.36449i
\(79\) 0.976417i 0.109856i 0.998490 + 0.0549278i \(0.0174929\pi\)
−0.998490 + 0.0549278i \(0.982507\pi\)
\(80\) −12.6368 − 13.4613i −1.41284 − 1.50502i
\(81\) −8.82970 − 1.74252i −0.981078 − 0.193614i
\(82\) −3.03491 − 3.03491i −0.335149 − 0.335149i
\(83\) −0.264467 − 0.264467i −0.0290291 − 0.0290291i 0.692443 − 0.721472i \(-0.256534\pi\)
−0.721472 + 0.692443i \(0.756534\pi\)
\(84\) 1.01696 − 20.8609i 0.110960 − 2.27611i
\(85\) −5.77777 − 0.182534i −0.626688 − 0.0197986i
\(86\) 19.7339i 2.12796i
\(87\) −2.83007 + 2.56697i −0.303416 + 0.275208i
\(88\) 21.8593 − 21.8593i 2.33021 − 2.33021i
\(89\) 6.33181 0.671170 0.335585 − 0.942010i \(-0.391066\pi\)
0.335585 + 0.942010i \(0.391066\pi\)
\(90\) −2.22393 + 17.1435i −0.234423 + 1.80709i
\(91\) −16.4112 −1.72036
\(92\) −17.7205 + 17.7205i −1.84749 + 1.84749i
\(93\) 2.66297 − 2.41541i 0.276138 − 0.250466i
\(94\) 4.57398i 0.471770i
\(95\) 1.63028 − 1.53042i 0.167263 − 0.157018i
\(96\) −0.646584 + 13.2634i −0.0659918 + 1.35369i
\(97\) −6.87894 − 6.87894i −0.698451 − 0.698451i 0.265625 − 0.964076i \(-0.414422\pi\)
−0.964076 + 0.265625i \(0.914422\pi\)
\(98\) 0.454315 + 0.454315i 0.0458928 + 0.0458928i
\(99\) −13.5618 − 1.32542i −1.36301 − 0.133210i
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.77.1 ✓ 28
3.2 odd 2 inner 285.2.k.c.77.14 yes 28
5.3 odd 4 inner 285.2.k.c.248.14 yes 28
15.8 even 4 inner 285.2.k.c.248.1 yes 28
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
285.2.k.c.77.1 ✓ 28 1.1 even 1 trivial
285.2.k.c.77.14 yes 28 3.2 odd 2 inner
285.2.k.c.248.1 yes 28 15.8 even 4 inner
285.2.k.c.248.14 yes 28 5.3 odd 4 inner