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Properties

Label 900.2.h.c.899.2
Level $900$
Weight $2$
Character 900.899
Analytic conductor $7.187$
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] = [900,2,Mod(899,900)] 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("900.899"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(900, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 900 = 2^{2} \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 900.h (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,4] 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: \(7.18653618192\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.18939904.2
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 4x^{7} + 14x^{6} - 28x^{5} + 43x^{4} - 44x^{3} + 30x^{2} - 12x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{5} \)
Twist minimal: no (minimal twist has level 180)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 899.2
Root \(0.500000 - 0.691860i\) of defining polynomial
Character \(\chi\) \(=\) 900.899
Dual form 900.2.h.c.899.4

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.207107 - 1.39897i) q^{2} +(-1.91421 + 0.579471i) q^{4} +1.63899 q^{7} +(1.20711 + 2.55791i) q^{8} -3.95687 q^{11} +0.585786i q^{13} +(-0.339446 - 2.29289i) q^{14} +(3.32843 - 2.21846i) q^{16} -4.00000 q^{17} -7.91375i q^{19} +(0.819496 + 5.53553i) q^{22} -5.59587i q^{23} +(0.819496 - 0.121320i) q^{26} +(-3.13738 + 0.949747i) q^{28} -7.65685i q^{29} -5.59587i q^{31} +(-3.79289 - 4.19690i) q^{32} +(0.828427 + 5.59587i) q^{34} +9.07107i q^{37} +(-11.0711 + 1.63899i) q^{38} -1.41421i q^{41} +7.91375 q^{43} +(7.57430 - 2.29289i) q^{44} +(-7.82843 + 1.15894i) q^{46} -3.27798i q^{47} -4.31371 q^{49} +(-0.339446 - 1.12132i) q^{52} -5.17157 q^{53} +(1.97844 + 4.19239i) q^{56} +(-10.7117 + 1.58579i) q^{58} -11.8706 q^{59} -0.828427 q^{61} +(-7.82843 + 1.15894i) q^{62} +(-5.08579 + 6.17534i) q^{64} -11.1917 q^{67} +(7.65685 - 2.31788i) q^{68} +4.63577 q^{71} +8.82843i q^{73} +(12.6901 - 1.87868i) q^{74} +(4.58579 + 15.1486i) q^{76} -6.48528 q^{77} -10.2316i q^{79} +(-1.97844 + 0.292893i) q^{82} -3.27798i q^{83} +(-1.63899 - 11.0711i) q^{86} +(-4.77637 - 10.1213i) q^{88} +5.41421i q^{89} +0.960099i q^{91} +(3.24264 + 10.7117i) q^{92} +(-4.58579 + 0.678892i) q^{94} -14.4853i q^{97} +(0.893398 + 6.03473i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 4 q^{2} - 4 q^{4} + 4 q^{8} + 4 q^{16} - 32 q^{17} - 36 q^{32} - 16 q^{34} - 32 q^{38} - 40 q^{46} + 56 q^{49} - 64 q^{53} + 16 q^{61} - 40 q^{62} - 52 q^{64} + 16 q^{68} + 48 q^{76} + 16 q^{77} - 8 q^{92}+ \cdots + 92 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/900\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(451\) \(577\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.207107 − 1.39897i −0.146447 − 0.989219i
\(3\) 0 0
\(4\) −1.91421 + 0.579471i −0.957107 + 0.289735i
\(5\) 0 0
\(6\) 0 0
\(7\) 1.63899 0.619480 0.309740 − 0.950821i \(-0.399758\pi\)
0.309740 + 0.950821i \(0.399758\pi\)
\(8\) 1.20711 + 2.55791i 0.426777 + 0.904357i
\(9\) 0 0
\(10\) 0 0
\(11\) −3.95687 −1.19304 −0.596521 − 0.802597i \(-0.703451\pi\)
−0.596521 + 0.802597i \(0.703451\pi\)
\(12\) 0 0
\(13\) 0.585786i 0.162468i 0.996695 + 0.0812340i \(0.0258861\pi\)
−0.996695 + 0.0812340i \(0.974114\pi\)
\(14\) −0.339446 − 2.29289i −0.0907208 − 0.612801i
\(15\) 0 0
\(16\) 3.32843 − 2.21846i 0.832107 − 0.554615i
\(17\) −4.00000 −0.970143 −0.485071 − 0.874475i \(-0.661206\pi\)
−0.485071 + 0.874475i \(0.661206\pi\)
\(18\) 0 0
\(19\) − 7.91375i − 1.81554i −0.419470 − 0.907769i \(-0.637784\pi\)
0.419470 − 0.907769i \(-0.362216\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0.819496 + 5.53553i 0.174717 + 1.18018i
\(23\) − 5.59587i − 1.16682i −0.812178 − 0.583409i \(-0.801718\pi\)
0.812178 − 0.583409i \(-0.198282\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0.819496 − 0.121320i 0.160716 − 0.0237929i
\(27\) 0 0
\(28\) −3.13738 + 0.949747i −0.592909 + 0.179485i
\(29\) − 7.65685i − 1.42184i −0.703272 − 0.710921i \(-0.748278\pi\)
0.703272 − 0.710921i \(-0.251722\pi\)
\(30\) 0 0
\(31\) − 5.59587i − 1.00505i −0.864564 − 0.502524i \(-0.832405\pi\)
0.864564 − 0.502524i \(-0.167595\pi\)
\(32\) −3.79289 − 4.19690i −0.670495 − 0.741914i
\(33\) 0 0
\(34\) 0.828427 + 5.59587i 0.142074 + 0.959683i
\(35\) 0 0
\(36\) 0 0
\(37\) 9.07107i 1.49127i 0.666352 + 0.745637i \(0.267855\pi\)
−0.666352 + 0.745637i \(0.732145\pi\)
\(38\) −11.0711 + 1.63899i −1.79596 + 0.265879i
\(39\) 0 0
\(40\) 0 0
\(41\) − 1.41421i − 0.220863i −0.993884 − 0.110432i \(-0.964777\pi\)
0.993884 − 0.110432i \(-0.0352233\pi\)
\(42\) 0 0
\(43\) 7.91375 1.20684 0.603418 − 0.797425i \(-0.293805\pi\)
0.603418 + 0.797425i \(0.293805\pi\)
\(44\) 7.57430 − 2.29289i 1.14187 − 0.345667i
\(45\) 0 0
\(46\) −7.82843 + 1.15894i −1.15424 + 0.170877i
\(47\) − 3.27798i − 0.478143i −0.971002 − 0.239071i \(-0.923157\pi\)
0.971002 − 0.239071i \(-0.0768430\pi\)
\(48\) 0 0
\(49\) −4.31371 −0.616244
\(50\) 0 0
\(51\) 0 0
\(52\) −0.339446 − 1.12132i −0.0470727 − 0.155499i
\(53\) −5.17157 −0.710370 −0.355185 − 0.934796i \(-0.615582\pi\)
−0.355185 + 0.934796i \(0.615582\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 1.97844 + 4.19239i 0.264380 + 0.560231i
\(57\) 0 0
\(58\) −10.7117 + 1.58579i −1.40651 + 0.208224i
\(59\) −11.8706 −1.54542 −0.772712 − 0.634757i \(-0.781100\pi\)
−0.772712 + 0.634757i \(0.781100\pi\)
\(60\) 0 0
\(61\) −0.828427 −0.106069 −0.0530346 − 0.998593i \(-0.516889\pi\)
−0.0530346 + 0.998593i \(0.516889\pi\)
\(62\) −7.82843 + 1.15894i −0.994211 + 0.147186i
\(63\) 0 0
\(64\) −5.08579 + 6.17534i −0.635723 + 0.771917i
\(65\) 0 0
\(66\) 0 0
\(67\) −11.1917 −1.36729 −0.683644 − 0.729816i \(-0.739606\pi\)
−0.683644 + 0.729816i \(0.739606\pi\)
\(68\) 7.65685 − 2.31788i 0.928530 − 0.281085i
\(69\) 0 0
\(70\) 0 0
\(71\) 4.63577 0.550164 0.275082 − 0.961421i \(-0.411295\pi\)
0.275082 + 0.961421i \(0.411295\pi\)
\(72\) 0 0
\(73\) 8.82843i 1.03329i 0.856200 + 0.516645i \(0.172819\pi\)
−0.856200 + 0.516645i \(0.827181\pi\)
\(74\) 12.6901 − 1.87868i 1.47520 − 0.218392i
\(75\) 0 0
\(76\) 4.58579 + 15.1486i 0.526026 + 1.73766i
\(77\) −6.48528 −0.739066
\(78\) 0 0
\(79\) − 10.2316i − 1.15115i −0.817750 − 0.575574i \(-0.804779\pi\)
0.817750 − 0.575574i \(-0.195221\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) −1.97844 + 0.292893i −0.218482 + 0.0323446i
\(83\) − 3.27798i − 0.359805i −0.983684 − 0.179903i \(-0.942422\pi\)
0.983684 − 0.179903i \(-0.0575783\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −1.63899 − 11.0711i −0.176737 − 1.19382i
\(87\) 0 0
\(88\) −4.77637 − 10.1213i −0.509163 − 1.07894i
\(89\) 5.41421i 0.573905i 0.957945 + 0.286953i \(0.0926423\pi\)
−0.957945 + 0.286953i \(0.907358\pi\)
\(90\) 0 0
\(91\) 0.960099i 0.100646i
\(92\) 3.24264 + 10.7117i 0.338069 + 1.11677i
\(93\) 0 0
\(94\) −4.58579 + 0.678892i −0.472988 + 0.0700224i
\(95\) 0 0
\(96\) 0 0
\(97\) − 14.4853i − 1.47076i −0.677656 − 0.735379i \(-0.737004\pi\)
0.677656 − 0.735379i \(-0.262996\pi\)
\(98\) 0.893398 + 6.03473i 0.0902469 + 0.609600i
\(99\) 0 0
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 900.2.h.c.899.2 8
3.2 odd 2 900.2.h.b.899.8 8
4.3 odd 2 inner 900.2.h.c.899.3 8
5.2 odd 4 180.2.e.a.71.7 yes 8
5.3 odd 4 900.2.e.d.251.2 8
5.4 even 2 900.2.h.b.899.7 8
12.11 even 2 900.2.h.b.899.5 8
15.2 even 4 180.2.e.a.71.2 yes 8
15.8 even 4 900.2.e.d.251.7 8
15.14 odd 2 inner 900.2.h.c.899.1 8
20.3 even 4 900.2.e.d.251.8 8
20.7 even 4 180.2.e.a.71.1 ✓ 8
20.19 odd 2 900.2.h.b.899.6 8
40.27 even 4 2880.2.h.e.1151.6 8
40.37 odd 4 2880.2.h.e.1151.7 8
60.23 odd 4 900.2.e.d.251.1 8
60.47 odd 4 180.2.e.a.71.8 yes 8
60.59 even 2 inner 900.2.h.c.899.4 8
120.77 even 4 2880.2.h.e.1151.3 8
120.107 odd 4 2880.2.h.e.1151.2 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
180.2.e.a.71.1 ✓ 8 20.7 even 4
180.2.e.a.71.2 yes 8 15.2 even 4
180.2.e.a.71.7 yes 8 5.2 odd 4
180.2.e.a.71.8 yes 8 60.47 odd 4
900.2.e.d.251.1 8 60.23 odd 4
900.2.e.d.251.2 8 5.3 odd 4
900.2.e.d.251.7 8 15.8 even 4
900.2.e.d.251.8 8 20.3 even 4
900.2.h.b.899.5 8 12.11 even 2
900.2.h.b.899.6 8 20.19 odd 2
900.2.h.b.899.7 8 5.4 even 2
900.2.h.b.899.8 8 3.2 odd 2
900.2.h.c.899.1 8 15.14 odd 2 inner
900.2.h.c.899.2 8 1.1 even 1 trivial
900.2.h.c.899.3 8 4.3 odd 2 inner
900.2.h.c.899.4 8 60.59 even 2 inner
2880.2.h.e.1151.2 8 120.107 odd 4
2880.2.h.e.1151.3 8 120.77 even 4
2880.2.h.e.1151.6 8 40.27 even 4
2880.2.h.e.1151.7 8 40.37 odd 4