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Properties

Label 285.2.i.f.106.3
Level $285$
Weight $2$
Character 285.106
Analytic conductor $2.276$
Analytic rank $0$
Dimension $10$
Inner twists $2$

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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(106,285)] 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("285.106"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(285, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 4])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 285 = 3 \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 285.i (of order \(3\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [10,1] 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: \(10\)
Relative dimension: \(5\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - x^{9} + 9x^{8} - 2x^{7} + 56x^{6} - 18x^{5} + 125x^{4} + x^{3} + 189x^{2} - 52x + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 3 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 106.3
Root \(0.145349 + 0.251751i\) of defining polynomial
Character \(\chi\) \(=\) 285.106
Dual form 285.2.i.f.121.3

$q$-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+(0.145349 - 0.251751i) q^{2} +(0.500000 - 0.866025i) q^{3} +(0.957748 + 1.65887i) q^{4} +(-0.500000 + 0.866025i) q^{5} +(-0.145349 - 0.251751i) q^{6} -0.486575 q^{7} +1.13822 q^{8} +(-0.500000 - 0.866025i) q^{9} +(0.145349 + 0.251751i) q^{10} +5.34764 q^{11} +1.91550 q^{12} +(1.24329 + 2.15344i) q^{13} +(-0.0707229 + 0.122496i) q^{14} +(0.500000 + 0.866025i) q^{15} +(-1.75006 + 3.03119i) q^{16} +(1.70780 - 2.95800i) q^{17} -0.290697 q^{18} +(-2.46291 + 3.59640i) q^{19} -1.91550 q^{20} +(-0.243287 + 0.421386i) q^{21} +(0.777272 - 1.34627i) q^{22} +(-3.20619 - 5.55329i) q^{23} +(0.569112 - 0.985730i) q^{24} +(-0.500000 - 0.866025i) q^{25} +0.722840 q^{26} -1.00000 q^{27} +(-0.466016 - 0.807163i) q^{28} +(-1.38312 - 2.39564i) q^{29} +0.290697 q^{30} +4.83099 q^{31} +(1.64696 + 2.85262i) q^{32} +(2.67382 - 4.63119i) q^{33} +(-0.496454 - 0.859883i) q^{34} +(0.243287 - 0.421386i) q^{35} +(0.957748 - 1.65887i) q^{36} -6.31756 q^{37} +(0.547418 + 1.14277i) q^{38} +2.48657 q^{39} +(-0.569112 + 0.985730i) q^{40} +(-1.29070 + 2.23555i) q^{41} +(0.0707229 + 0.122496i) q^{42} +(1.24329 - 2.15344i) q^{43} +(5.12169 + 8.87102i) q^{44} +1.00000 q^{45} -1.86406 q^{46} +(-5.55533 - 9.62211i) q^{47} +(1.75006 + 3.03119i) q^{48} -6.76325 q^{49} -0.290697 q^{50} +(-1.70780 - 2.95800i) q^{51} +(-2.38151 + 4.12490i) q^{52} +(-1.49839 - 2.59529i) q^{53} +(-0.145349 + 0.251751i) q^{54} +(-2.67382 + 4.63119i) q^{55} -0.553831 q^{56} +(1.88312 + 3.93114i) q^{57} -0.804139 q^{58} +(-6.36659 + 11.0273i) q^{59} +(-0.957748 + 1.65887i) q^{60} +(-5.92742 - 10.2666i) q^{61} +(0.702178 - 1.21621i) q^{62} +(0.243287 + 0.421386i) q^{63} -6.04269 q^{64} -2.48657 q^{65} +(-0.777272 - 1.34627i) q^{66} +(7.58770 + 13.1423i) q^{67} +6.54258 q^{68} -6.41238 q^{69} +(-0.0707229 - 0.122496i) q^{70} +(4.96452 - 8.59879i) q^{71} +(-0.569112 - 0.985730i) q^{72} +(5.50642 - 9.53740i) q^{73} +(-0.918249 + 1.59045i) q^{74} -1.00000 q^{75} +(-8.32479 - 0.641189i) q^{76} -2.60202 q^{77} +(0.361420 - 0.625998i) q^{78} +(-4.06636 + 7.04314i) q^{79} +(-1.75006 - 3.03119i) q^{80} +(-0.500000 + 0.866025i) q^{81} +(0.375202 + 0.649869i) q^{82} -5.86106 q^{83} -0.932031 q^{84} +(1.70780 + 2.95800i) q^{85} +(-0.361420 - 0.625998i) q^{86} -2.76624 q^{87} +6.08681 q^{88} +(3.25521 + 5.63820i) q^{89} +(0.145349 - 0.251751i) q^{90} +(-0.604952 - 1.04781i) q^{91} +(6.14145 - 10.6373i) q^{92} +(2.41550 - 4.18376i) q^{93} -3.22984 q^{94} +(-1.88312 - 3.93114i) q^{95} +3.29392 q^{96} +(-1.00000 + 1.73205i) q^{97} +(-0.983028 + 1.70265i) q^{98} +(-2.67382 - 4.63119i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + q^{2} + 5 q^{3} - 7 q^{4} - 5 q^{5} - q^{6} + 4 q^{7} - 12 q^{8} - 5 q^{9} + q^{10} + 10 q^{11} - 14 q^{12} + 8 q^{13} + 4 q^{14} + 5 q^{15} - 7 q^{16} - 10 q^{17} - 2 q^{18} + 5 q^{19} + 14 q^{20}+ \cdots - 5 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/285\mathbb{Z}\right)^\times\).

\(n\) \(172\) \(191\) \(211\)
\(\chi(n)\) \(1\) \(1\) \(e\left(\frac{2}{3}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.145349 − 0.251751i 0.102777 − 0.178015i −0.810051 − 0.586360i \(-0.800561\pi\)
0.912828 + 0.408345i \(0.133894\pi\)
\(3\) 0.500000 − 0.866025i 0.288675 − 0.500000i
\(4\) 0.957748 + 1.65887i 0.478874 + 0.829434i
\(5\) −0.500000 + 0.866025i −0.223607 + 0.387298i
\(6\) −0.145349 − 0.251751i −0.0593383 − 0.102777i
\(7\) −0.486575 −0.183908 −0.0919539 − 0.995763i \(-0.529311\pi\)
−0.0919539 + 0.995763i \(0.529311\pi\)
\(8\) 1.13822 0.402423
\(9\) −0.500000 − 0.866025i −0.166667 − 0.288675i
\(10\) 0.145349 + 0.251751i 0.0459633 + 0.0796107i
\(11\) 5.34764 1.61237 0.806187 − 0.591661i \(-0.201528\pi\)
0.806187 + 0.591661i \(0.201528\pi\)
\(12\) 1.91550 0.552956
\(13\) 1.24329 + 2.15344i 0.344826 + 0.597256i 0.985322 − 0.170705i \(-0.0546046\pi\)
−0.640496 + 0.767961i \(0.721271\pi\)
\(14\) −0.0707229 + 0.122496i −0.0189015 + 0.0327384i
\(15\) 0.500000 + 0.866025i 0.129099 + 0.223607i
\(16\) −1.75006 + 3.03119i −0.437514 + 0.757796i
\(17\) 1.70780 − 2.95800i 0.414203 − 0.717421i −0.581141 − 0.813803i \(-0.697394\pi\)
0.995344 + 0.0963817i \(0.0307270\pi\)
\(18\) −0.290697 −0.0685180
\(19\) −2.46291 + 3.59640i −0.565029 + 0.825071i
\(20\) −1.91550 −0.428318
\(21\) −0.243287 + 0.421386i −0.0530896 + 0.0919539i
\(22\) 0.777272 − 1.34627i 0.165715 − 0.287027i
\(23\) −3.20619 − 5.55329i −0.668537 − 1.15794i −0.978313 − 0.207131i \(-0.933587\pi\)
0.309776 − 0.950810i \(-0.399746\pi\)
\(24\) 0.569112 − 0.985730i 0.116169 − 0.201211i
\(25\) −0.500000 − 0.866025i −0.100000 − 0.173205i
\(26\) 0.722840 0.141761
\(27\) −1.00000 −0.192450
\(28\) −0.466016 − 0.807163i −0.0880687 − 0.152539i
\(29\) −1.38312 − 2.39564i −0.256839 − 0.444859i 0.708554 − 0.705656i \(-0.249348\pi\)
−0.965393 + 0.260798i \(0.916014\pi\)
\(30\) 0.290697 0.0530738
\(31\) 4.83099 0.867671 0.433836 − 0.900992i \(-0.357160\pi\)
0.433836 + 0.900992i \(0.357160\pi\)
\(32\) 1.64696 + 2.85262i 0.291144 + 0.504276i
\(33\) 2.67382 − 4.63119i 0.465452 − 0.806187i
\(34\) −0.496454 − 0.859883i −0.0851411 − 0.147469i
\(35\) 0.243287 − 0.421386i 0.0411231 − 0.0712272i
\(36\) 0.957748 − 1.65887i 0.159625 − 0.276478i
\(37\) −6.31756 −1.03860 −0.519301 − 0.854592i \(-0.673808\pi\)
−0.519301 + 0.854592i \(0.673808\pi\)
\(38\) 0.547418 + 1.14277i 0.0888030 + 0.185382i
\(39\) 2.48657 0.398171
\(40\) −0.569112 + 0.985730i −0.0899845 + 0.155858i
\(41\) −1.29070 + 2.23555i −0.201573 + 0.349135i −0.949035 − 0.315169i \(-0.897939\pi\)
0.747462 + 0.664304i \(0.231272\pi\)
\(42\) 0.0707229 + 0.122496i 0.0109128 + 0.0189015i
\(43\) 1.24329 − 2.15344i 0.189600 − 0.328396i −0.755517 − 0.655129i \(-0.772614\pi\)
0.945117 + 0.326733i \(0.105948\pi\)
\(44\) 5.12169 + 8.87102i 0.772123 + 1.33736i
\(45\) 1.00000 0.149071
\(46\) −1.86406 −0.274841
\(47\) −5.55533 − 9.62211i −0.810328 − 1.40353i −0.912634 − 0.408777i \(-0.865956\pi\)
0.102306 − 0.994753i \(-0.467378\pi\)
\(48\) 1.75006 + 3.03119i 0.252599 + 0.437514i
\(49\) −6.76325 −0.966178
\(50\) −0.290697 −0.0411108
\(51\) −1.70780 − 2.95800i −0.239140 − 0.414203i
\(52\) −2.38151 + 4.12490i −0.330256 + 0.572020i
\(53\) −1.49839 − 2.59529i −0.205820 − 0.356490i 0.744574 − 0.667540i \(-0.232653\pi\)
−0.950394 + 0.311050i \(0.899319\pi\)
\(54\) −0.145349 + 0.251751i −0.0197794 + 0.0342590i
\(55\) −2.67382 + 4.63119i −0.360538 + 0.624470i
\(56\) −0.553831 −0.0740087
\(57\) 1.88312 + 3.93114i 0.249426 + 0.520692i
\(58\) −0.804139 −0.105589
\(59\) −6.36659 + 11.0273i −0.828859 + 1.43563i 0.0700752 + 0.997542i \(0.477676\pi\)
−0.898934 + 0.438084i \(0.855657\pi\)
\(60\) −0.957748 + 1.65887i −0.123645 + 0.214159i
\(61\) −5.92742 − 10.2666i −0.758929 − 1.31450i −0.943397 − 0.331664i \(-0.892390\pi\)
0.184469 − 0.982838i \(-0.440944\pi\)
\(62\) 0.702178 − 1.21621i 0.0891767 − 0.154458i
\(63\) 0.243287 + 0.421386i 0.0306513 + 0.0530896i
\(64\) −6.04269 −0.755336
\(65\) −2.48657 −0.308422
\(66\) −0.777272 − 1.34627i −0.0956755 − 0.165715i
\(67\) 7.58770 + 13.1423i 0.926985 + 1.60559i 0.788336 + 0.615244i \(0.210943\pi\)
0.138649 + 0.990342i \(0.455724\pi\)
\(68\) 6.54258 0.793404
\(69\) −6.41238 −0.771960
\(70\) −0.0707229 − 0.122496i −0.00845301 − 0.0146410i
\(71\) 4.96452 − 8.59879i 0.589180 − 1.02049i −0.405160 − 0.914246i \(-0.632784\pi\)
0.994340 − 0.106244i \(-0.0338823\pi\)
\(72\) −0.569112 − 0.985730i −0.0670705 − 0.116169i
\(73\) 5.50642 − 9.53740i 0.644478 − 1.11627i −0.339944 − 0.940446i \(-0.610408\pi\)
0.984422 − 0.175823i \(-0.0562585\pi\)
\(74\) −0.918249 + 1.59045i −0.106744 + 0.184887i
\(75\) −1.00000 −0.115470
\(76\) −8.32479 − 0.641189i −0.954919 − 0.0735494i
\(77\) −2.60202 −0.296528
\(78\) 0.361420 − 0.625998i 0.0409228 − 0.0708803i
\(79\) −4.06636 + 7.04314i −0.457501 + 0.792415i −0.998828 − 0.0483971i \(-0.984589\pi\)
0.541327 + 0.840812i \(0.317922\pi\)
\(80\) −1.75006 − 3.03119i −0.195662 − 0.338897i
\(81\) −0.500000 + 0.866025i −0.0555556 + 0.0962250i
\(82\) 0.375202 + 0.649869i 0.0414341 + 0.0717660i
\(83\) −5.86106 −0.643335 −0.321668 − 0.946853i \(-0.604243\pi\)
−0.321668 + 0.946853i \(0.604243\pi\)
\(84\) −0.932031 −0.101693
\(85\) 1.70780 + 2.95800i 0.185237 + 0.320840i
\(86\) −0.361420 − 0.625998i −0.0389729 − 0.0675031i
\(87\) −2.76624 −0.296572
\(88\) 6.08681 0.648856
\(89\) 3.25521 + 5.63820i 0.345052 + 0.597647i 0.985363 − 0.170468i \(-0.0545280\pi\)
−0.640311 + 0.768116i \(0.721195\pi\)
\(90\) 0.145349 − 0.251751i 0.0153211 − 0.0265369i
\(91\) −0.604952 − 1.04781i −0.0634162 − 0.109840i
\(92\) 6.14145 − 10.6373i 0.640290 − 1.10901i
\(93\) 2.41550 − 4.18376i 0.250475 − 0.433836i
\(94\) −3.22984 −0.333132
\(95\) −1.88312 − 3.93114i −0.193204 − 0.403326i
\(96\) 3.29392 0.336184
\(97\) −1.00000 + 1.73205i −0.101535 + 0.175863i −0.912317 − 0.409484i \(-0.865709\pi\)
0.810782 + 0.585348i \(0.199042\pi\)
\(98\) −0.983028 + 1.70265i −0.0993008 + 0.171994i
\(99\) −2.67382 − 4.63119i −0.268729 − 0.465452i
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.i.f.106.3 ✓ 10
3.2 odd 2 855.2.k.i.676.3 10
19.7 even 3 inner 285.2.i.f.121.3 yes 10
19.8 odd 6 5415.2.a.z.1.3 5
19.11 even 3 5415.2.a.y.1.3 5
57.26 odd 6 855.2.k.i.406.3 10
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
285.2.i.f.106.3 ✓ 10 1.1 even 1 trivial
285.2.i.f.121.3 yes 10 19.7 even 3 inner
855.2.k.i.406.3 10 57.26 odd 6
855.2.k.i.676.3 10 3.2 odd 2
5415.2.a.y.1.3 5 19.11 even 3
5415.2.a.z.1.3 5 19.8 odd 6