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Properties

Label 285.2.k.c.248.2
Level $285$
Weight $2$
Character 285.248
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 248.2
Character \(\chi\) \(=\) 285.248
Dual form 285.2.k.c.77.2

$q$-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+(-1.60748 - 1.60748i) q^{2} +(-1.46845 - 0.918505i) q^{3} +3.16797i q^{4} +(0.294779 + 2.21655i) q^{5} +(0.884027 + 3.83698i) q^{6} +(3.56650 - 3.56650i) q^{7} +(1.87749 - 1.87749i) q^{8} +(1.31270 + 2.69756i) q^{9} +(3.08921 - 4.03691i) q^{10} -3.54860i q^{11} +(2.90979 - 4.65201i) q^{12} +(-1.01743 - 1.01743i) q^{13} -11.4661 q^{14} +(1.60305 - 3.52566i) q^{15} +0.299907 q^{16} +(0.0645525 + 0.0645525i) q^{17} +(2.22613 - 6.44640i) q^{18} +1.00000i q^{19} +(-7.02197 + 0.933850i) q^{20} +(-8.51307 + 1.96138i) q^{21} +(-5.70429 + 5.70429i) q^{22} +(3.09903 - 3.09903i) q^{23} +(-4.48147 + 1.03252i) q^{24} +(-4.82621 + 1.30679i) q^{25} +3.27099i q^{26} +(0.550085 - 5.16695i) q^{27} +(11.2986 + 11.2986i) q^{28} -0.873658 q^{29} +(-8.24427 + 3.09055i) q^{30} -1.93602 q^{31} +(-4.23707 - 4.23707i) q^{32} +(-3.25940 + 5.21094i) q^{33} -0.207533i q^{34} +(8.95666 + 6.85400i) q^{35} +(-8.54578 + 4.15859i) q^{36} +(1.47870 - 1.47870i) q^{37} +(1.60748 - 1.60748i) q^{38} +(0.559532 + 2.42856i) q^{39} +(4.71499 + 3.60810i) q^{40} -8.53257i q^{41} +(16.8375 + 10.5317i) q^{42} +(-4.61880 - 4.61880i) q^{43} +11.2419 q^{44} +(-5.59232 + 3.70485i) q^{45} -9.96325 q^{46} +(-4.86805 - 4.86805i) q^{47} +(-0.440399 - 0.275466i) q^{48} -18.4398i q^{49} +(9.85866 + 5.65740i) q^{50} +(-0.0355004 - 0.154084i) q^{51} +(3.22319 - 3.22319i) q^{52} +(-0.315361 + 0.315361i) q^{53} +(-9.19001 + 7.42151i) q^{54} +(7.86565 - 1.04605i) q^{55} -13.3921i q^{56} +(0.918505 - 1.46845i) q^{57} +(1.40439 + 1.40439i) q^{58} +13.1184 q^{59} +(11.1692 + 5.07840i) q^{60} +5.71358 q^{61} +(3.11212 + 3.11212i) q^{62} +(14.3026 + 4.93910i) q^{63} +13.0222i q^{64} +(1.95527 - 2.55510i) q^{65} +(13.6159 - 3.13706i) q^{66} +(-8.04520 + 8.04520i) q^{67} +(-0.204500 + 0.204500i) q^{68} +(-7.39725 + 1.70430i) q^{69} +(-3.37997 - 25.4153i) q^{70} -5.53300i q^{71} +(7.52920 + 2.60005i) q^{72} +(9.31370 + 9.31370i) q^{73} -4.75395 q^{74} +(8.28734 + 2.51395i) q^{75} -3.16797 q^{76} +(-12.6561 - 12.6561i) q^{77} +(3.00442 - 4.80329i) q^{78} +0.183486i q^{79} +(0.0884063 + 0.664760i) q^{80} +(-5.55364 + 7.08216i) q^{81} +(-13.7159 + 13.7159i) q^{82} +(-2.39122 + 2.39122i) q^{83} +(-6.21361 - 26.9692i) q^{84} +(-0.124055 + 0.162113i) q^{85} +14.8492i q^{86} +(1.28292 + 0.802459i) q^{87} +(-6.66244 - 6.66244i) q^{88} +10.4502 q^{89} +(14.9450 + 3.03407i) q^{90} -7.25732 q^{91} +(9.81764 + 9.81764i) q^{92} +(2.84296 + 1.77825i) q^{93} +15.6506i q^{94} +(-2.21655 + 0.294779i) q^{95} +(2.33016 + 10.1137i) q^{96} +(7.14572 - 7.14572i) q^{97} +(-29.6416 + 29.6416i) q^{98} +(9.57255 - 4.65824i) 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 \(\left(\mathbb{Z}/285\mathbb{Z}\right)^\times\).

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

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.60748 − 1.60748i −1.13666 − 1.13666i −0.989045 − 0.147613i \(-0.952841\pi\)
−0.147613 − 0.989045i \(-0.547159\pi\)
\(3\) −1.46845 − 0.918505i −0.847811 − 0.530299i
\(4\) 3.16797i 1.58398i
\(5\) 0.294779 + 2.21655i 0.131829 + 0.991272i
\(6\) 0.884027 + 3.83698i 0.360903 + 1.56644i
\(7\) 3.56650 − 3.56650i 1.34801 − 1.34801i 0.460188 − 0.887821i \(-0.347782\pi\)
0.887821 − 0.460188i \(-0.152218\pi\)
\(8\) 1.87749 − 1.87749i 0.663791 − 0.663791i
\(9\) 1.31270 + 2.69756i 0.437566 + 0.899186i
\(10\) 3.08921 − 4.03691i 0.976894 − 1.27658i
\(11\) − 3.54860i − 1.06994i −0.844870 − 0.534971i \(-0.820322\pi\)
0.844870 − 0.534971i \(-0.179678\pi\)
\(12\) 2.90979 − 4.65201i 0.839985 − 1.34292i
\(13\) −1.01743 − 1.01743i −0.282184 − 0.282184i 0.551795 − 0.833980i \(-0.313943\pi\)
−0.833980 + 0.551795i \(0.813943\pi\)
\(14\) −11.4661 −3.06445
\(15\) 1.60305 − 3.52566i 0.413904 − 0.910320i
\(16\) 0.299907 0.0749768
\(17\) 0.0645525 + 0.0645525i 0.0156563 + 0.0156563i 0.714892 − 0.699235i \(-0.246476\pi\)
−0.699235 + 0.714892i \(0.746476\pi\)
\(18\) 2.22613 − 6.44640i 0.524704 − 1.51943i
\(19\) 1.00000i 0.229416i
\(20\) −7.02197 + 0.933850i −1.57016 + 0.208815i
\(21\) −8.51307 + 1.96138i −1.85771 + 0.428009i
\(22\) −5.70429 + 5.70429i −1.21616 + 1.21616i
\(23\) 3.09903 − 3.09903i 0.646193 − 0.646193i −0.305878 − 0.952071i \(-0.598950\pi\)
0.952071 + 0.305878i \(0.0989499\pi\)
\(24\) −4.48147 + 1.03252i −0.914777 + 0.210762i
\(25\) −4.82621 + 1.30679i −0.965242 + 0.261357i
\(26\) 3.27099i 0.641494i
\(27\) 0.550085 − 5.16695i 0.105864 − 0.994381i
\(28\) 11.2986 + 11.2986i 2.13523 + 2.13523i
\(29\) −0.873658 −0.162234 −0.0811171 − 0.996705i \(-0.525849\pi\)
−0.0811171 + 0.996705i \(0.525849\pi\)
\(30\) −8.24427 + 3.09055i −1.50519 + 0.564255i
\(31\) −1.93602 −0.347720 −0.173860 − 0.984770i \(-0.555624\pi\)
−0.173860 + 0.984770i \(0.555624\pi\)
\(32\) −4.23707 − 4.23707i −0.749014 − 0.749014i
\(33\) −3.25940 + 5.21094i −0.567389 + 0.907109i
\(34\) − 0.207533i − 0.0355917i
\(35\) 8.95666 + 6.85400i 1.51395 + 1.15854i
\(36\) −8.54578 + 4.15859i −1.42430 + 0.693098i
\(37\) 1.47870 − 1.47870i 0.243097 − 0.243097i −0.575033 − 0.818130i \(-0.695011\pi\)
0.818130 + 0.575033i \(0.195011\pi\)
\(38\) 1.60748 − 1.60748i 0.260767 − 0.260767i
\(39\) 0.559532 + 2.42856i 0.0895969 + 0.388881i
\(40\) 4.71499 + 3.60810i 0.745505 + 0.570491i
\(41\) − 8.53257i − 1.33256i −0.745700 − 0.666282i \(-0.767885\pi\)
0.745700 − 0.666282i \(-0.232115\pi\)
\(42\) 16.8375 + 10.5317i 2.59808 + 1.62508i
\(43\) −4.61880 − 4.61880i −0.704360 − 0.704360i 0.260983 − 0.965343i \(-0.415953\pi\)
−0.965343 + 0.260983i \(0.915953\pi\)
\(44\) 11.2419 1.69477
\(45\) −5.59232 + 3.70485i −0.833654 + 0.552286i
\(46\) −9.96325 −1.46900
\(47\) −4.86805 − 4.86805i −0.710077 − 0.710077i 0.256474 − 0.966551i \(-0.417439\pi\)
−0.966551 + 0.256474i \(0.917439\pi\)
\(48\) −0.440399 − 0.275466i −0.0635662 − 0.0397601i
\(49\) − 18.4398i − 2.63426i
\(50\) 9.85866 + 5.65740i 1.39422 + 0.800077i
\(51\) −0.0355004 − 0.154084i −0.00497106 − 0.0215761i
\(52\) 3.22319 − 3.22319i 0.446976 − 0.446976i
\(53\) −0.315361 + 0.315361i −0.0433181 + 0.0433181i −0.728434 − 0.685116i \(-0.759752\pi\)
0.685116 + 0.728434i \(0.259752\pi\)
\(54\) −9.19001 + 7.42151i −1.25060 + 1.00994i
\(55\) 7.86565 − 1.04605i 1.06060 − 0.141050i
\(56\) − 13.3921i − 1.78959i
\(57\) 0.918505 − 1.46845i 0.121659 − 0.194501i
\(58\) 1.40439 + 1.40439i 0.184405 + 0.184405i
\(59\) 13.1184 1.70788 0.853938 − 0.520374i \(-0.174208\pi\)
0.853938 + 0.520374i \(0.174208\pi\)
\(60\) 11.1692 + 5.07840i 1.44193 + 0.655618i
\(61\) 5.71358 0.731549 0.365774 − 0.930704i \(-0.380804\pi\)
0.365774 + 0.930704i \(0.380804\pi\)
\(62\) 3.11212 + 3.11212i 0.395239 + 0.395239i
\(63\) 14.3026 + 4.93910i 1.80196 + 0.622268i
\(64\) 13.0222i 1.62777i
\(65\) 1.95527 − 2.55510i 0.242521 − 0.316922i
\(66\) 13.6159 − 3.13706i 1.67600 − 0.386145i
\(67\) −8.04520 + 8.04520i −0.982877 + 0.982877i −0.999856 − 0.0169786i \(-0.994595\pi\)
0.0169786 + 0.999856i \(0.494595\pi\)
\(68\) −0.204500 + 0.204500i −0.0247993 + 0.0247993i
\(69\) −7.39725 + 1.70430i −0.890525 + 0.205174i
\(70\) −3.37997 − 25.4153i −0.403984 − 3.03771i
\(71\) − 5.53300i − 0.656646i −0.944565 − 0.328323i \(-0.893517\pi\)
0.944565 − 0.328323i \(-0.106483\pi\)
\(72\) 7.52920 + 2.60005i 0.887325 + 0.306419i
\(73\) 9.31370 + 9.31370i 1.09009 + 1.09009i 0.995518 + 0.0945679i \(0.0301469\pi\)
0.0945679 + 0.995518i \(0.469853\pi\)
\(74\) −4.75395 −0.552636
\(75\) 8.28734 + 2.51395i 0.956940 + 0.290285i
\(76\) −3.16797 −0.363391
\(77\) −12.6561 − 12.6561i −1.44229 − 1.44229i
\(78\) 3.00442 − 4.80329i 0.340184 − 0.543866i
\(79\) 0.183486i 0.0206438i 0.999947 + 0.0103219i \(0.00328561\pi\)
−0.999947 + 0.0103219i \(0.996714\pi\)
\(80\) 0.0884063 + 0.664760i 0.00988413 + 0.0743225i
\(81\) −5.55364 + 7.08216i −0.617071 + 0.786907i
\(82\) −13.7159 + 13.7159i −1.51467 + 1.51467i
\(83\) −2.39122 + 2.39122i −0.262470 + 0.262470i −0.826057 − 0.563587i \(-0.809421\pi\)
0.563587 + 0.826057i \(0.309421\pi\)
\(84\) −6.21361 − 26.9692i −0.677960 − 2.94258i
\(85\) −0.124055 + 0.162113i −0.0134557 + 0.0175836i
\(86\) 14.8492i 1.60123i
\(87\) 1.28292 + 0.802459i 0.137544 + 0.0860326i
\(88\) −6.66244 − 6.66244i −0.710219 − 0.710219i
\(89\) 10.4502 1.10771 0.553857 − 0.832612i \(-0.313155\pi\)
0.553857 + 0.832612i \(0.313155\pi\)
\(90\) 14.9450 + 3.03407i 1.57534 + 0.319819i
\(91\) −7.25732 −0.760774
\(92\) 9.81764 + 9.81764i 1.02356 + 1.02356i
\(93\) 2.84296 + 1.77825i 0.294801 + 0.184396i
\(94\) 15.6506i 1.61423i
\(95\) −2.21655 + 0.294779i −0.227413 + 0.0302437i
\(96\) 2.33016 + 10.1137i 0.237821 + 1.03222i
\(97\) 7.14572 − 7.14572i 0.725538 − 0.725538i −0.244189 − 0.969728i \(-0.578522\pi\)
0.969728 + 0.244189i \(0.0785218\pi\)
\(98\) −29.6416 + 29.6416i −2.99425 + 2.99425i
\(99\) 9.57255 − 4.65824i 0.962078 − 0.468171i
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.248.2 yes 28
3.2 odd 2 inner 285.2.k.c.248.13 yes 28
5.2 odd 4 inner 285.2.k.c.77.13 yes 28
15.2 even 4 inner 285.2.k.c.77.2 ✓ 28
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
285.2.k.c.77.2 ✓ 28 15.2 even 4 inner
285.2.k.c.77.13 yes 28 5.2 odd 4 inner
285.2.k.c.248.2 yes 28 1.1 even 1 trivial
285.2.k.c.248.13 yes 28 3.2 odd 2 inner