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Properties

Label 450.2.q.a.241.1
Level $450$
Weight $2$
Character 450.241
Analytic conductor $3.593$
Analytic rank $0$
Dimension $8$
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] = [450,2,Mod(31,450)] 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("450.31"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(450, base_ring=CyclotomicField(30)) chi = DirichletCharacter(H, H._module([10, 12])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 450 = 2 \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 450.q (of order \(15\), degree \(8\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(3.59326809096\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\Q(\zeta_{15})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{7} + x^{5} - x^{4} + x^{3} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{15}]$

Embedding invariants

Embedding label 241.1
Root \(-0.104528 + 0.994522i\) of defining polynomial
Character \(\chi\) \(=\) 450.241
Dual form 450.2.q.a.211.1

$q$-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+(0.913545 - 0.406737i) q^{2} +(-1.64728 + 0.535233i) q^{3} +(0.669131 - 0.743145i) q^{4} +(1.49622 - 1.66172i) q^{5} +(-1.28716 + 1.15897i) q^{6} +(-2.11803 + 3.66854i) q^{7} +(0.309017 - 0.951057i) q^{8} +(2.42705 - 1.76336i) q^{9} +(0.690983 - 2.12663i) q^{10} +(3.30524 - 1.47159i) q^{11} +(-0.704489 + 1.58231i) q^{12} +(4.21878 + 1.87832i) q^{13} +(-0.442790 + 4.21286i) q^{14} +(-1.57528 + 3.53815i) q^{15} +(-0.104528 - 0.994522i) q^{16} +(1.92705 - 5.93085i) q^{17} +(1.50000 - 2.59808i) q^{18} +(0.0729490 - 0.224514i) q^{19} +(-0.233733 - 2.22382i) q^{20} +(1.52547 - 7.17675i) q^{21} +(2.42094 - 2.68872i) q^{22} +(-0.233733 + 2.22382i) q^{23} +1.73205i q^{24} +(-0.522642 - 4.97261i) q^{25} +4.61803 q^{26} +(-3.05422 + 4.20378i) q^{27} +(1.30902 + 4.02874i) q^{28} +(2.32991 + 0.495239i) q^{29} +3.87298i q^{30} +(6.56161 - 1.39471i) q^{31} +(-0.500000 - 0.866025i) q^{32} +(-4.65701 + 4.19319i) q^{33} +(-0.651847 - 6.20191i) q^{34} +(2.92705 + 9.00854i) q^{35} +(0.313585 - 2.98357i) q^{36} +(-7.47214 + 5.42882i) q^{37} +(-0.0246758 - 0.234775i) q^{38} +(-7.95485 - 0.836089i) q^{39} +(-1.11803 - 1.93649i) q^{40} +(2.39169 + 1.06485i) q^{41} +(-1.52547 - 7.17675i) q^{42} +(2.73607 - 4.73901i) q^{43} +(1.11803 - 3.44095i) q^{44} +(0.701198 - 6.67146i) q^{45} +(0.690983 + 2.12663i) q^{46} +(-4.00079 - 0.850394i) q^{47} +(0.704489 + 1.58231i) q^{48} +(-5.47214 - 9.47802i) q^{49} +(-2.50000 - 4.33013i) q^{50} +10.8012i q^{51} +(4.21878 - 1.87832i) q^{52} +(3.42705 + 10.5474i) q^{53} +(-1.08034 + 5.08260i) q^{54} +(2.50000 - 7.69421i) q^{55} +(2.83448 + 3.14801i) q^{56} +0.408882i q^{57} +(2.32991 - 0.495239i) q^{58} +(-7.73968 - 3.44593i) q^{59} +(1.57528 + 3.53815i) q^{60} +(-11.3430 + 5.05021i) q^{61} +(5.42705 - 3.94298i) q^{62} +(1.32837 + 12.6386i) q^{63} +(-0.809017 - 0.587785i) q^{64} +(9.43349 - 4.20006i) q^{65} +(-2.54886 + 5.72484i) q^{66} +(3.76988 - 0.801313i) q^{67} +(-3.11803 - 5.40059i) q^{68} +(-0.805239 - 3.78835i) q^{69} +(6.33810 + 7.03917i) q^{70} +(4.09017 + 12.5882i) q^{71} +(-0.927051 - 2.85317i) q^{72} +(4.85410 + 3.52671i) q^{73} +(-4.61803 + 7.99867i) q^{74} +(3.52244 + 7.91154i) q^{75} +(-0.118034 - 0.204441i) q^{76} +(-1.60203 + 15.2423i) q^{77} +(-7.60719 + 2.47172i) q^{78} +(-14.2441 - 3.02767i) q^{79} +(-1.80902 - 1.31433i) q^{80} +(2.78115 - 8.55951i) q^{81} +2.61803 q^{82} +(-9.42816 - 10.4710i) q^{83} +(-4.31263 - 5.93583i) q^{84} +(-6.97214 - 12.0761i) q^{85} +(0.571994 - 5.44216i) q^{86} +(-4.10309 + 0.431252i) q^{87} +(-0.378188 - 3.59821i) q^{88} +(-5.42705 - 3.94298i) q^{89} +(-2.07295 - 6.37988i) q^{90} +(-15.8262 + 11.4984i) q^{91} +(1.49622 + 1.66172i) q^{92} +(-10.0623 + 5.80948i) q^{93} +(-4.00079 + 0.850394i) q^{94} +(-0.263932 - 0.457144i) q^{95} +(1.28716 + 1.15897i) q^{96} +(-6.33070 - 1.34563i) q^{97} +(-8.85410 - 6.43288i) q^{98} +(5.42705 - 9.39993i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + q^{2} + q^{4} - 5 q^{5} + 3 q^{6} - 8 q^{7} - 2 q^{8} + 6 q^{9} + 10 q^{10} + 5 q^{11} - 3 q^{12} + 6 q^{13} + 7 q^{14} - 15 q^{15} + q^{16} + 2 q^{17} + 12 q^{18} + 14 q^{19} + 5 q^{20} - 9 q^{21}+ \cdots + 30 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/450\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(127\)
\(\chi(n)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{1}{5}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.913545 − 0.406737i 0.645974 − 0.287606i
\(3\) −1.64728 + 0.535233i −0.951057 + 0.309017i
\(4\) 0.669131 − 0.743145i 0.334565 − 0.371572i
\(5\) 1.49622 − 1.66172i 0.669131 − 0.743145i
\(6\) −1.28716 + 1.15897i −0.525483 + 0.473147i
\(7\) −2.11803 + 3.66854i −0.800542 + 1.38658i 0.118718 + 0.992928i \(0.462121\pi\)
−0.919260 + 0.393651i \(0.871212\pi\)
\(8\) 0.309017 − 0.951057i 0.109254 − 0.336249i
\(9\) 2.42705 − 1.76336i 0.809017 − 0.587785i
\(10\) 0.690983 − 2.12663i 0.218508 − 0.672499i
\(11\) 3.30524 − 1.47159i 0.996567 − 0.443700i 0.157377 − 0.987539i \(-0.449696\pi\)
0.839190 + 0.543838i \(0.183030\pi\)
\(12\) −0.704489 + 1.58231i −0.203368 + 0.456773i
\(13\) 4.21878 + 1.87832i 1.17008 + 0.520953i 0.897429 − 0.441160i \(-0.145433\pi\)
0.272651 + 0.962113i \(0.412099\pi\)
\(14\) −0.442790 + 4.21286i −0.118341 + 1.12593i
\(15\) −1.57528 + 3.53815i −0.406737 + 0.913545i
\(16\) −0.104528 − 0.994522i −0.0261321 − 0.248630i
\(17\) 1.92705 − 5.93085i 0.467379 − 1.43844i −0.388588 − 0.921412i \(-0.627037\pi\)
0.855966 − 0.517031i \(-0.172963\pi\)
\(18\) 1.50000 − 2.59808i 0.353553 − 0.612372i
\(19\) 0.0729490 − 0.224514i 0.0167357 − 0.0515070i −0.942340 − 0.334657i \(-0.891379\pi\)
0.959076 + 0.283150i \(0.0913795\pi\)
\(20\) −0.233733 − 2.22382i −0.0522642 − 0.497261i
\(21\) 1.52547 − 7.17675i 0.332884 − 1.56610i
\(22\) 2.42094 − 2.68872i 0.516146 − 0.573238i
\(23\) −0.233733 + 2.22382i −0.0487366 + 0.463698i 0.942751 + 0.333497i \(0.108229\pi\)
−0.991488 + 0.130201i \(0.958438\pi\)
\(24\) 1.73205i 0.353553i
\(25\) −0.522642 − 4.97261i −0.104528 − 0.994522i
\(26\) 4.61803 0.905671
\(27\) −3.05422 + 4.20378i −0.587785 + 0.809017i
\(28\) 1.30902 + 4.02874i 0.247381 + 0.761360i
\(29\) 2.32991 + 0.495239i 0.432654 + 0.0919635i 0.419090 − 0.907945i \(-0.362349\pi\)
0.0135638 + 0.999908i \(0.495682\pi\)
\(30\) 3.87298i 0.707107i
\(31\) 6.56161 − 1.39471i 1.17850 − 0.250498i 0.423299 − 0.905990i \(-0.360872\pi\)
0.755202 + 0.655492i \(0.227539\pi\)
\(32\) −0.500000 − 0.866025i −0.0883883 − 0.153093i
\(33\) −4.65701 + 4.19319i −0.810681 + 0.729940i
\(34\) −0.651847 − 6.20191i −0.111791 − 1.06362i
\(35\) 2.92705 + 9.00854i 0.494762 + 1.52272i
\(36\) 0.313585 − 2.98357i 0.0522642 − 0.497261i
\(37\) −7.47214 + 5.42882i −1.22841 + 0.892493i −0.996770 − 0.0803080i \(-0.974410\pi\)
−0.231641 + 0.972801i \(0.574410\pi\)
\(38\) −0.0246758 − 0.234775i −0.00400295 − 0.0380855i
\(39\) −7.95485 − 0.836089i −1.27380 − 0.133881i
\(40\) −1.11803 − 1.93649i −0.176777 − 0.306186i
\(41\) 2.39169 + 1.06485i 0.373520 + 0.166302i 0.584906 − 0.811101i \(-0.301131\pi\)
−0.211387 + 0.977403i \(0.567798\pi\)
\(42\) −1.52547 − 7.17675i −0.235384 − 1.10740i
\(43\) 2.73607 − 4.73901i 0.417246 − 0.722692i −0.578415 − 0.815743i \(-0.696329\pi\)
0.995661 + 0.0930507i \(0.0296619\pi\)
\(44\) 1.11803 − 3.44095i 0.168550 − 0.518743i
\(45\) 0.701198 − 6.67146i 0.104528 − 0.994522i
\(46\) 0.690983 + 2.12663i 0.101880 + 0.313554i
\(47\) −4.00079 − 0.850394i −0.583575 − 0.124043i −0.0933421 − 0.995634i \(-0.529755\pi\)
−0.490233 + 0.871591i \(0.663088\pi\)
\(48\) 0.704489 + 1.58231i 0.101684 + 0.228386i
\(49\) −5.47214 − 9.47802i −0.781734 − 1.35400i
\(50\) −2.50000 − 4.33013i −0.353553 − 0.612372i
\(51\) 10.8012i 1.51247i
\(52\) 4.21878 − 1.87832i 0.585040 − 0.260477i
\(53\) 3.42705 + 10.5474i 0.470742 + 1.44879i 0.851616 + 0.524167i \(0.175623\pi\)
−0.380874 + 0.924627i \(0.624377\pi\)
\(54\) −1.08034 + 5.08260i −0.147016 + 0.691655i
\(55\) 2.50000 − 7.69421i 0.337100 − 1.03749i
\(56\) 2.83448 + 3.14801i 0.378774 + 0.420671i
\(57\) 0.408882i 0.0541577i
\(58\) 2.32991 − 0.495239i 0.305933 − 0.0650280i
\(59\) −7.73968 − 3.44593i −1.00762 − 0.448622i −0.164518 − 0.986374i \(-0.552607\pi\)
−0.843103 + 0.537752i \(0.819273\pi\)
\(60\) 1.57528 + 3.53815i 0.203368 + 0.456773i
\(61\) −11.3430 + 5.05021i −1.45232 + 0.646613i −0.972953 − 0.231004i \(-0.925799\pi\)
−0.479363 + 0.877616i \(0.659132\pi\)
\(62\) 5.42705 − 3.94298i 0.689236 − 0.500759i
\(63\) 1.32837 + 12.6386i 0.167359 + 1.59231i
\(64\) −0.809017 − 0.587785i −0.101127 − 0.0734732i
\(65\) 9.43349 − 4.20006i 1.17008 − 0.520953i
\(66\) −2.54886 + 5.72484i −0.313743 + 0.704679i
\(67\) 3.76988 − 0.801313i 0.460564 − 0.0978959i 0.0282144 − 0.999602i \(-0.491018\pi\)
0.432350 + 0.901706i \(0.357685\pi\)
\(68\) −3.11803 − 5.40059i −0.378117 − 0.654918i
\(69\) −0.805239 − 3.78835i −0.0969393 − 0.456064i
\(70\) 6.33810 + 7.03917i 0.757547 + 0.841342i
\(71\) 4.09017 + 12.5882i 0.485414 + 1.49395i 0.831381 + 0.555703i \(0.187551\pi\)
−0.345967 + 0.938247i \(0.612449\pi\)
\(72\) −0.927051 − 2.85317i −0.109254 − 0.336249i
\(73\) 4.85410 + 3.52671i 0.568130 + 0.412770i 0.834425 − 0.551121i \(-0.185800\pi\)
−0.266296 + 0.963891i \(0.585800\pi\)
\(74\) −4.61803 + 7.99867i −0.536836 + 0.929826i
\(75\) 3.52244 + 7.91154i 0.406737 + 0.913545i
\(76\) −0.118034 − 0.204441i −0.0135394 − 0.0234510i
\(77\) −1.60203 + 15.2423i −0.182568 + 1.73702i
\(78\) −7.60719 + 2.47172i −0.861344 + 0.279868i
\(79\) −14.2441 − 3.02767i −1.60258 − 0.340640i −0.682051 − 0.731304i \(-0.738912\pi\)
−0.920533 + 0.390664i \(0.872245\pi\)
\(80\) −1.80902 − 1.31433i −0.202254 − 0.146946i
\(81\) 2.78115 − 8.55951i 0.309017 − 0.951057i
\(82\) 2.61803 0.289113
\(83\) −9.42816 − 10.4710i −1.03488 − 1.14935i −0.988623 − 0.150416i \(-0.951939\pi\)
−0.0462526 − 0.998930i \(-0.514728\pi\)
\(84\) −4.31263 − 5.93583i −0.470547 − 0.647652i
\(85\) −6.97214 − 12.0761i −0.756234 − 1.30984i
\(86\) 0.571994 − 5.44216i 0.0616797 − 0.586843i
\(87\) −4.10309 + 0.431252i −0.439897 + 0.0462350i
\(88\) −0.378188 − 3.59821i −0.0403149 − 0.383571i
\(89\) −5.42705 − 3.94298i −0.575266 − 0.417955i 0.261748 − 0.965136i \(-0.415701\pi\)
−0.837014 + 0.547181i \(0.815701\pi\)
\(90\) −2.07295 − 6.37988i −0.218508 − 0.672499i
\(91\) −15.8262 + 11.4984i −1.65904 + 1.20536i
\(92\) 1.49622 + 1.66172i 0.155992 + 0.173247i
\(93\) −10.0623 + 5.80948i −1.04341 + 0.602414i
\(94\) −4.00079 + 0.850394i −0.412650 + 0.0877115i
\(95\) −0.263932 − 0.457144i −0.0270789 − 0.0469020i
\(96\) 1.28716 + 1.15897i 0.131371 + 0.118287i
\(97\) −6.33070 − 1.34563i −0.642786 − 0.136628i −0.125025 − 0.992154i \(-0.539901\pi\)
−0.517761 + 0.855525i \(0.673234\pi\)
\(98\) −8.85410 − 6.43288i −0.894399 − 0.649819i
\(99\) 5.42705 − 9.39993i 0.545439 − 0.944728i
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 450.2.q.a.241.1 yes 8
9.4 even 3 inner 450.2.q.a.391.1 yes 8
25.11 even 5 inner 450.2.q.a.61.1 ✓ 8
225.211 even 15 inner 450.2.q.a.211.1 yes 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
450.2.q.a.61.1 ✓ 8 25.11 even 5 inner
450.2.q.a.211.1 yes 8 225.211 even 15 inner
450.2.q.a.241.1 yes 8 1.1 even 1 trivial
450.2.q.a.391.1 yes 8 9.4 even 3 inner