↑

Properties

Label 900.2.h.d.899.3
Level $900$
Weight $2$
Character 900.899
Analytic conductor $7.187$
Analytic rank $0$
Dimension $16$
Inner twists $8$

Related objects

Downloads

Learn more

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 = [16,0] 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: \(7.18653618192\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\Q(\zeta_{40})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - x^{12} + x^{8} - x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{16} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 899.3
Root \(0.156434 + 0.987688i\) of defining polynomial
Character \(\chi\) \(=\) 900.899
Dual form 900.2.h.d.899.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-1.14412 + 0.831254i) q^{2} +(0.618034 - 1.90211i) q^{4} -0.726543 q^{7} +(0.874032 + 2.68999i) q^{8} +4.35250 q^{11} -3.47214i q^{13} +(0.831254 - 0.603941i) q^{14} +(-3.23607 - 2.35114i) q^{16} -3.16228 q^{17} -6.88191i q^{19} +(-4.97980 + 3.61803i) q^{22} +6.40747i q^{23} +(2.88623 + 3.97255i) q^{26} +(-0.449028 + 1.38197i) q^{28} -0.333851i q^{29} +3.97574i q^{31} +5.65685 q^{32} +(3.61803 - 2.62866i) q^{34} -6.47214i q^{37} +(5.72061 + 7.87375i) q^{38} -10.9010i q^{41} +11.5842 q^{43} +(2.68999 - 8.27895i) q^{44} +(-5.32624 - 7.33094i) q^{46} -7.67752i q^{47} -6.47214 q^{49} +(-6.60440 - 2.14590i) q^{52} -3.90879 q^{53} +(-0.635021 - 1.95440i) q^{56} +(0.277515 + 0.381966i) q^{58} +3.08246 q^{59} +13.9443 q^{61} +(-3.30485 - 4.54873i) q^{62} +(-6.47214 + 4.70228i) q^{64} +3.97574 q^{67} +(-1.95440 + 6.01501i) q^{68} +12.0300 q^{71} -10.4721i q^{73} +(5.37999 + 7.40492i) q^{74} +(-13.0902 - 4.25325i) q^{76} -3.16228 q^{77} +2.90617i q^{79} +(9.06154 + 12.4721i) q^{82} -7.67752i q^{83} +(-13.2537 + 9.62940i) q^{86} +(3.80423 + 11.7082i) q^{88} +3.49613i q^{89} +2.52265i q^{91} +(12.1877 + 3.96004i) q^{92} +(6.38197 + 8.78402i) q^{94} -3.00000i q^{97} +(7.40492 - 5.37999i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 8 q^{4} - 16 q^{16} + 40 q^{34} + 40 q^{46} - 32 q^{49} + 80 q^{61} - 32 q^{64} - 120 q^{76} + 120 q^{94}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.14412 + 0.831254i −0.809017 + 0.587785i
\(3\) 0 0
\(4\) 0.618034 − 1.90211i 0.309017 − 0.951057i
\(5\) 0 0
\(6\) 0 0
\(7\) −0.726543 −0.274607 −0.137304 − 0.990529i \(-0.543844\pi\)
−0.137304 + 0.990529i \(0.543844\pi\)
\(8\) 0.874032 + 2.68999i 0.309017 + 0.951057i
\(9\) 0 0
\(10\) 0 0
\(11\) 4.35250 1.31233 0.656164 − 0.754618i \(-0.272178\pi\)
0.656164 + 0.754618i \(0.272178\pi\)
\(12\) 0 0
\(13\) − 3.47214i − 0.962997i −0.876447 − 0.481499i \(-0.840093\pi\)
0.876447 − 0.481499i \(-0.159907\pi\)
\(14\) 0.831254 − 0.603941i 0.222162 − 0.161410i
\(15\) 0 0
\(16\) −3.23607 − 2.35114i −0.809017 − 0.587785i
\(17\) −3.16228 −0.766965 −0.383482 − 0.923548i \(-0.625275\pi\)
−0.383482 + 0.923548i \(0.625275\pi\)
\(18\) 0 0
\(19\) − 6.88191i − 1.57882i −0.613867 − 0.789409i \(-0.710387\pi\)
0.613867 − 0.789409i \(-0.289613\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −4.97980 + 3.61803i −1.06170 + 0.771367i
\(23\) 6.40747i 1.33605i 0.744138 + 0.668025i \(0.232860\pi\)
−0.744138 + 0.668025i \(0.767140\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 2.88623 + 3.97255i 0.566036 + 0.779081i
\(27\) 0 0
\(28\) −0.449028 + 1.38197i −0.0848583 + 0.261167i
\(29\) − 0.333851i − 0.0619945i −0.999519 − 0.0309972i \(-0.990132\pi\)
0.999519 − 0.0309972i \(-0.00986831\pi\)
\(30\) 0 0
\(31\) 3.97574i 0.714064i 0.934092 + 0.357032i \(0.116211\pi\)
−0.934092 + 0.357032i \(0.883789\pi\)
\(32\) 5.65685 1.00000
\(33\) 0 0
\(34\) 3.61803 − 2.62866i 0.620488 − 0.450811i
\(35\) 0 0
\(36\) 0 0
\(37\) − 6.47214i − 1.06401i −0.846740 − 0.532006i \(-0.821438\pi\)
0.846740 − 0.532006i \(-0.178562\pi\)
\(38\) 5.72061 + 7.87375i 0.928006 + 1.27729i
\(39\) 0 0
\(40\) 0 0
\(41\) − 10.9010i − 1.70246i −0.524795 − 0.851229i \(-0.675858\pi\)
0.524795 − 0.851229i \(-0.324142\pi\)
\(42\) 0 0
\(43\) 11.5842 1.76657 0.883286 − 0.468834i \(-0.155326\pi\)
0.883286 + 0.468834i \(0.155326\pi\)
\(44\) 2.68999 − 8.27895i 0.405532 − 1.24810i
\(45\) 0 0
\(46\) −5.32624 − 7.33094i −0.785311 − 1.08089i
\(47\) − 7.67752i − 1.11988i −0.828533 − 0.559940i \(-0.810824\pi\)
0.828533 − 0.559940i \(-0.189176\pi\)
\(48\) 0 0
\(49\) −6.47214 −0.924591
\(50\) 0 0
\(51\) 0 0
\(52\) −6.60440 − 2.14590i −0.915865 − 0.297583i
\(53\) −3.90879 −0.536914 −0.268457 − 0.963292i \(-0.586514\pi\)
−0.268457 + 0.963292i \(0.586514\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −0.635021 − 1.95440i −0.0848583 − 0.261167i
\(57\) 0 0
\(58\) 0.277515 + 0.381966i 0.0364394 + 0.0501546i
\(59\) 3.08246 0.401302 0.200651 − 0.979663i \(-0.435694\pi\)
0.200651 + 0.979663i \(0.435694\pi\)
\(60\) 0 0
\(61\) 13.9443 1.78538 0.892691 − 0.450670i \(-0.148815\pi\)
0.892691 + 0.450670i \(0.148815\pi\)
\(62\) −3.30485 − 4.54873i −0.419716 − 0.577690i
\(63\) 0 0
\(64\) −6.47214 + 4.70228i −0.809017 + 0.587785i
\(65\) 0 0
\(66\) 0 0
\(67\) 3.97574 0.485714 0.242857 − 0.970062i \(-0.421915\pi\)
0.242857 + 0.970062i \(0.421915\pi\)
\(68\) −1.95440 + 6.01501i −0.237005 + 0.729427i
\(69\) 0 0
\(70\) 0 0
\(71\) 12.0300 1.42770 0.713850 − 0.700298i \(-0.246950\pi\)
0.713850 + 0.700298i \(0.246950\pi\)
\(72\) 0 0
\(73\) − 10.4721i − 1.22567i −0.790211 − 0.612835i \(-0.790029\pi\)
0.790211 − 0.612835i \(-0.209971\pi\)
\(74\) 5.37999 + 7.40492i 0.625411 + 0.860804i
\(75\) 0 0
\(76\) −13.0902 − 4.25325i −1.50155 − 0.487882i
\(77\) −3.16228 −0.360375
\(78\) 0 0
\(79\) 2.90617i 0.326970i 0.986546 + 0.163485i \(0.0522735\pi\)
−0.986546 + 0.163485i \(0.947727\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 9.06154 + 12.4721i 1.00068 + 1.37732i
\(83\) − 7.67752i − 0.842717i −0.906894 − 0.421359i \(-0.861553\pi\)
0.906894 − 0.421359i \(-0.138447\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −13.2537 + 9.62940i −1.42919 + 1.03837i
\(87\) 0 0
\(88\) 3.80423 + 11.7082i 0.405532 + 1.24810i
\(89\) 3.49613i 0.370589i 0.982683 + 0.185294i \(0.0593239\pi\)
−0.982683 + 0.185294i \(0.940676\pi\)
\(90\) 0 0
\(91\) 2.52265i 0.264446i
\(92\) 12.1877 + 3.96004i 1.27066 + 0.412862i
\(93\) 0 0
\(94\) 6.38197 + 8.78402i 0.658250 + 0.906003i
\(95\) 0 0
\(96\) 0 0
\(97\) − 3.00000i − 0.304604i −0.988334 − 0.152302i \(-0.951331\pi\)
0.988334 − 0.152302i \(-0.0486686\pi\)
\(98\) 7.40492 − 5.37999i 0.748010 − 0.543461i
\(99\) 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.d.899.3 16
3.2 odd 2 inner 900.2.h.d.899.13 16
4.3 odd 2 inner 900.2.h.d.899.2 16
5.2 odd 4 900.2.e.g.251.3 yes 8
5.3 odd 4 900.2.e.e.251.6 yes 8
5.4 even 2 inner 900.2.h.d.899.14 16
12.11 even 2 inner 900.2.h.d.899.16 16
15.2 even 4 900.2.e.g.251.6 yes 8
15.8 even 4 900.2.e.e.251.3 ✓ 8
15.14 odd 2 inner 900.2.h.d.899.4 16
20.3 even 4 900.2.e.e.251.4 yes 8
20.7 even 4 900.2.e.g.251.5 yes 8
20.19 odd 2 inner 900.2.h.d.899.15 16
60.23 odd 4 900.2.e.e.251.5 yes 8
60.47 odd 4 900.2.e.g.251.4 yes 8
60.59 even 2 inner 900.2.h.d.899.1 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
900.2.e.e.251.3 ✓ 8 15.8 even 4
900.2.e.e.251.4 yes 8 20.3 even 4
900.2.e.e.251.5 yes 8 60.23 odd 4
900.2.e.e.251.6 yes 8 5.3 odd 4
900.2.e.g.251.3 yes 8 5.2 odd 4
900.2.e.g.251.4 yes 8 60.47 odd 4
900.2.e.g.251.5 yes 8 20.7 even 4
900.2.e.g.251.6 yes 8 15.2 even 4
900.2.h.d.899.1 16 60.59 even 2 inner
900.2.h.d.899.2 16 4.3 odd 2 inner
900.2.h.d.899.3 16 1.1 even 1 trivial
900.2.h.d.899.4 16 15.14 odd 2 inner
900.2.h.d.899.13 16 3.2 odd 2 inner
900.2.h.d.899.14 16 5.4 even 2 inner
900.2.h.d.899.15 16 20.19 odd 2 inner
900.2.h.d.899.16 16 12.11 even 2 inner