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Properties

Label 900.2.h.c.899.8
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 899.8
Root \(0.500000 + 1.44392i\) of defining polynomial
Character \(\chi\) \(=\) 900.899
Dual form 900.2.h.c.899.6

$q$-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)
 
\(f(q)\) \(=\) \(q+(1.20711 + 0.736813i) q^{2} +(0.914214 + 1.77882i) q^{4} +5.03127 q^{7} +(-0.207107 + 2.82083i) q^{8} +2.08402 q^{11} -3.41421i q^{13} +(6.07328 + 3.70711i) q^{14} +(-2.32843 + 3.25245i) q^{16} -4.00000 q^{17} -4.16804i q^{19} +(2.51564 + 1.53553i) q^{22} +2.94725i q^{23} +(2.51564 - 4.12132i) q^{26} +(4.59966 + 8.94975i) q^{28} -3.65685i q^{29} +2.94725i q^{31} +(-5.20711 + 2.21044i) q^{32} +(-4.82843 - 2.94725i) q^{34} +5.07107i q^{37} +(3.07107 - 5.03127i) q^{38} -1.41421i q^{41} -4.16804 q^{43} +(1.90524 + 3.70711i) q^{44} +(-2.17157 + 3.55765i) q^{46} +10.0625i q^{47} +18.3137 q^{49} +(6.07328 - 3.12132i) q^{52} -10.8284 q^{53} +(-1.04201 + 14.1924i) q^{56} +(2.69442 - 4.41421i) q^{58} +6.25206 q^{59} +4.82843 q^{61} +(-2.17157 + 3.55765i) q^{62} +(-7.91421 - 1.16843i) q^{64} -5.89450 q^{67} +(-3.65685 - 7.11529i) q^{68} -14.2306 q^{71} -3.17157i q^{73} +(-3.73643 + 6.12132i) q^{74} +(7.41421 - 3.81048i) q^{76} +10.4853 q^{77} -11.2833i q^{79} +(1.04201 - 1.70711i) q^{82} +10.0625i q^{83} +(-5.03127 - 3.07107i) q^{86} +(-0.431615 + 5.87868i) q^{88} -2.58579i q^{89} -17.1778i q^{91} +(-5.24264 + 2.69442i) q^{92} +(-7.41421 + 12.1466i) q^{94} -2.48528i q^{97} +(22.1066 + 13.4938i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 4 q^{2} - 4 q^{4} + 4 q^{8} + 4 q^{16} - 32 q^{17} - 36 q^{32} - 16 q^{34} - 32 q^{38} - 40 q^{46} + 56 q^{49} - 64 q^{53} + 16 q^{61} - 40 q^{62} - 52 q^{64} + 16 q^{68} + 48 q^{76} + 16 q^{77} - 8 q^{92}+ \cdots + 92 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.20711 + 0.736813i 0.853553 + 0.521005i
\(3\) 0 0
\(4\) 0.914214 + 1.77882i 0.457107 + 0.889412i
\(5\) 0 0
\(6\) 0 0
\(7\) 5.03127 1.90164 0.950821 − 0.309740i \(-0.100242\pi\)
0.950821 + 0.309740i \(0.100242\pi\)
\(8\) −0.207107 + 2.82083i −0.0732233 + 0.997316i
\(9\) 0 0
\(10\) 0 0
\(11\) 2.08402 0.628356 0.314178 − 0.949364i \(-0.398271\pi\)
0.314178 + 0.949364i \(0.398271\pi\)
\(12\) 0 0
\(13\) − 3.41421i − 0.946932i −0.880812 − 0.473466i \(-0.843003\pi\)
0.880812 − 0.473466i \(-0.156997\pi\)
\(14\) 6.07328 + 3.70711i 1.62315 + 0.990766i
\(15\) 0 0
\(16\) −2.32843 + 3.25245i −0.582107 + 0.813112i
\(17\) −4.00000 −0.970143 −0.485071 − 0.874475i \(-0.661206\pi\)
−0.485071 + 0.874475i \(0.661206\pi\)
\(18\) 0 0
\(19\) − 4.16804i − 0.956215i −0.878301 − 0.478107i \(-0.841323\pi\)
0.878301 − 0.478107i \(-0.158677\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 2.51564 + 1.53553i 0.536336 + 0.327377i
\(23\) 2.94725i 0.614544i 0.951622 + 0.307272i \(0.0994162\pi\)
−0.951622 + 0.307272i \(0.900584\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 2.51564 − 4.12132i 0.493357 − 0.808257i
\(27\) 0 0
\(28\) 4.59966 + 8.94975i 0.869254 + 1.69134i
\(29\) − 3.65685i − 0.679061i −0.940595 − 0.339530i \(-0.889732\pi\)
0.940595 − 0.339530i \(-0.110268\pi\)
\(30\) 0 0
\(31\) 2.94725i 0.529342i 0.964339 + 0.264671i \(0.0852634\pi\)
−0.964339 + 0.264671i \(0.914737\pi\)
\(32\) −5.20711 + 2.21044i −0.920495 + 0.390754i
\(33\) 0 0
\(34\) −4.82843 − 2.94725i −0.828068 − 0.505449i
\(35\) 0 0
\(36\) 0 0
\(37\) 5.07107i 0.833678i 0.908980 + 0.416839i \(0.136862\pi\)
−0.908980 + 0.416839i \(0.863138\pi\)
\(38\) 3.07107 − 5.03127i 0.498193 − 0.816180i
\(39\) 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
\(43\) −4.16804 −0.635621 −0.317810 − 0.948154i \(-0.602948\pi\)
−0.317810 + 0.948154i \(0.602948\pi\)
\(44\) 1.90524 + 3.70711i 0.287226 + 0.558867i
\(45\) 0 0
\(46\) −2.17157 + 3.55765i −0.320181 + 0.524546i
\(47\) 10.0625i 1.46777i 0.679272 + 0.733887i \(0.262296\pi\)
−0.679272 + 0.733887i \(0.737704\pi\)
\(48\) 0 0
\(49\) 18.3137 2.61624
\(50\) 0 0
\(51\) 0 0
\(52\) 6.07328 − 3.12132i 0.842213 − 0.432849i
\(53\) −10.8284 −1.48740 −0.743699 − 0.668514i \(-0.766931\pi\)
−0.743699 + 0.668514i \(0.766931\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −1.04201 + 14.1924i −0.139245 + 1.89654i
\(57\) 0 0
\(58\) 2.69442 − 4.41421i 0.353794 − 0.579615i
\(59\) 6.25206 0.813949 0.406975 − 0.913439i \(-0.366584\pi\)
0.406975 + 0.913439i \(0.366584\pi\)
\(60\) 0 0
\(61\) 4.82843 0.618217 0.309108 − 0.951027i \(-0.399969\pi\)
0.309108 + 0.951027i \(0.399969\pi\)
\(62\) −2.17157 + 3.55765i −0.275790 + 0.451822i
\(63\) 0 0
\(64\) −7.91421 − 1.16843i −0.989277 − 0.146053i
\(65\) 0 0
\(66\) 0 0
\(67\) −5.89450 −0.720128 −0.360064 − 0.932928i \(-0.617245\pi\)
−0.360064 + 0.932928i \(0.617245\pi\)
\(68\) −3.65685 − 7.11529i −0.443459 − 0.862856i
\(69\) 0 0
\(70\) 0 0
\(71\) −14.2306 −1.68886 −0.844430 − 0.535666i \(-0.820061\pi\)
−0.844430 + 0.535666i \(0.820061\pi\)
\(72\) 0 0
\(73\) − 3.17157i − 0.371205i −0.982625 − 0.185602i \(-0.940576\pi\)
0.982625 − 0.185602i \(-0.0594236\pi\)
\(74\) −3.73643 + 6.12132i −0.434351 + 0.711589i
\(75\) 0 0
\(76\) 7.41421 − 3.81048i 0.850469 − 0.437092i
\(77\) 10.4853 1.19491
\(78\) 0 0
\(79\) − 11.2833i − 1.26947i −0.772728 − 0.634737i \(-0.781108\pi\)
0.772728 − 0.634737i \(-0.218892\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 1.04201 − 1.70711i 0.115071 − 0.188518i
\(83\) 10.0625i 1.10451i 0.833676 + 0.552254i \(0.186232\pi\)
−0.833676 + 0.552254i \(0.813768\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −5.03127 − 3.07107i −0.542536 − 0.331162i
\(87\) 0 0
\(88\) −0.431615 + 5.87868i −0.0460103 + 0.626669i
\(89\) − 2.58579i − 0.274093i −0.990565 − 0.137046i \(-0.956239\pi\)
0.990565 − 0.137046i \(-0.0437609\pi\)
\(90\) 0 0
\(91\) − 17.1778i − 1.80073i
\(92\) −5.24264 + 2.69442i −0.546583 + 0.280912i
\(93\) 0 0
\(94\) −7.41421 + 12.1466i −0.764718 + 1.25282i
\(95\) 0 0
\(96\) 0 0
\(97\) − 2.48528i − 0.252342i −0.992009 − 0.126171i \(-0.959731\pi\)
0.992009 − 0.126171i \(-0.0402688\pi\)
\(98\) 22.1066 + 13.4938i 2.23310 + 1.36308i
\(99\) 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.899.8 8
3.2 odd 2 900.2.h.b.899.2 8
4.3 odd 2 inner 900.2.h.c.899.5 8
5.2 odd 4 900.2.e.d.251.4 8
5.3 odd 4 180.2.e.a.71.5 yes 8
5.4 even 2 900.2.h.b.899.1 8
12.11 even 2 900.2.h.b.899.3 8
15.2 even 4 900.2.e.d.251.5 8
15.8 even 4 180.2.e.a.71.4 yes 8
15.14 odd 2 inner 900.2.h.c.899.7 8
20.3 even 4 180.2.e.a.71.3 ✓ 8
20.7 even 4 900.2.e.d.251.6 8
20.19 odd 2 900.2.h.b.899.4 8
40.3 even 4 2880.2.h.e.1151.4 8
40.13 odd 4 2880.2.h.e.1151.1 8
60.23 odd 4 180.2.e.a.71.6 yes 8
60.47 odd 4 900.2.e.d.251.3 8
60.59 even 2 inner 900.2.h.c.899.6 8
120.53 even 4 2880.2.h.e.1151.5 8
120.83 odd 4 2880.2.h.e.1151.8 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
180.2.e.a.71.3 ✓ 8 20.3 even 4
180.2.e.a.71.4 yes 8 15.8 even 4
180.2.e.a.71.5 yes 8 5.3 odd 4
180.2.e.a.71.6 yes 8 60.23 odd 4
900.2.e.d.251.3 8 60.47 odd 4
900.2.e.d.251.4 8 5.2 odd 4
900.2.e.d.251.5 8 15.2 even 4
900.2.e.d.251.6 8 20.7 even 4
900.2.h.b.899.1 8 5.4 even 2
900.2.h.b.899.2 8 3.2 odd 2
900.2.h.b.899.3 8 12.11 even 2
900.2.h.b.899.4 8 20.19 odd 2
900.2.h.c.899.5 8 4.3 odd 2 inner
900.2.h.c.899.6 8 60.59 even 2 inner
900.2.h.c.899.7 8 15.14 odd 2 inner
900.2.h.c.899.8 8 1.1 even 1 trivial
2880.2.h.e.1151.1 8 40.13 odd 4
2880.2.h.e.1151.4 8 40.3 even 4
2880.2.h.e.1151.5 8 120.53 even 4
2880.2.h.e.1151.8 8 120.83 odd 4