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Properties

Label 900.2.h.d.899.8
Level $900$
Weight $2$
Character 900.899
Analytic conductor $7.187$
Analytic rank $0$
Dimension $16$
Inner twists $8$

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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 = [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 899.8
Root \(0.891007 + 0.453990i\) of defining polynomial
Character \(\chi\) \(=\) 900.899
Dual form 900.2.h.d.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)
 
\(f(q)\) \(=\) \(q+(-0.437016 + 1.34500i) q^{2} +(-1.61803 - 1.17557i) q^{4} +3.07768 q^{7} +(2.28825 - 1.66251i) q^{8} -1.02749 q^{11} -5.47214i q^{13} +(-1.34500 + 4.13948i) q^{14} +(1.23607 + 3.80423i) q^{16} -3.16228 q^{17} -1.62460i q^{19} +(0.449028 - 1.38197i) q^{22} -7.67752i q^{23} +(7.36001 + 2.39141i) q^{26} +(-4.97980 - 3.61803i) q^{28} +5.99070i q^{29} -10.6861i q^{31} -5.65685 q^{32} +(1.38197 - 4.25325i) q^{34} -2.47214i q^{37} +(2.18508 + 0.709976i) q^{38} +8.07262i q^{41} +5.98385 q^{43} +(1.66251 + 1.20788i) q^{44} +(10.3262 + 3.35520i) q^{46} -6.40747i q^{47} +2.47214 q^{49} +(-6.43288 + 8.85410i) q^{52} +10.2333 q^{53} +(7.04250 - 5.11667i) q^{56} +(-8.05748 - 2.61803i) q^{58} +13.0575 q^{59} -3.94427 q^{61} +(14.3728 + 4.67001i) q^{62} +(2.47214 - 7.60845i) q^{64} +10.6861 q^{67} +(5.11667 + 3.71748i) q^{68} -7.43496 q^{71} +1.52786i q^{73} +(3.32502 + 1.08036i) q^{74} +(-1.90983 + 2.62866i) q^{76} -3.16228 q^{77} +12.3107i q^{79} +(-10.8576 - 3.52786i) q^{82} -6.40747i q^{83} +(-2.61504 + 8.04827i) q^{86} +(-2.35114 + 1.70820i) q^{88} -9.15298i q^{89} -16.8415i q^{91} +(-9.02546 + 12.4225i) q^{92} +(8.61803 + 2.80017i) q^{94} +3.00000i q^{97} +(-1.08036 + 3.32502i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 8 q^{4} - 16 q^{16} + 40 q^{34} + 40 q^{46} - 32 q^{49} + 80 q^{61} - 32 q^{64} - 120 q^{76} + 120 q^{94}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.437016 + 1.34500i −0.309017 + 0.951057i
\(3\) 0 0
\(4\) −1.61803 − 1.17557i −0.809017 − 0.587785i
\(5\) 0 0
\(6\) 0 0
\(7\) 3.07768 1.16326 0.581628 − 0.813455i \(-0.302416\pi\)
0.581628 + 0.813455i \(0.302416\pi\)
\(8\) 2.28825 − 1.66251i 0.809017 − 0.587785i
\(9\) 0 0
\(10\) 0 0
\(11\) −1.02749 −0.309799 −0.154899 − 0.987930i \(-0.549505\pi\)
−0.154899 + 0.987930i \(0.549505\pi\)
\(12\) 0 0
\(13\) − 5.47214i − 1.51770i −0.651267 − 0.758849i \(-0.725762\pi\)
0.651267 − 0.758849i \(-0.274238\pi\)
\(14\) −1.34500 + 4.13948i −0.359466 + 1.10632i
\(15\) 0 0
\(16\) 1.23607 + 3.80423i 0.309017 + 0.951057i
\(17\) −3.16228 −0.766965 −0.383482 − 0.923548i \(-0.625275\pi\)
−0.383482 + 0.923548i \(0.625275\pi\)
\(18\) 0 0
\(19\) − 1.62460i − 0.372708i −0.982483 − 0.186354i \(-0.940333\pi\)
0.982483 − 0.186354i \(-0.0596672\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0.449028 − 1.38197i 0.0957331 − 0.294636i
\(23\) − 7.67752i − 1.60087i −0.599418 − 0.800437i \(-0.704601\pi\)
0.599418 − 0.800437i \(-0.295399\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 7.36001 + 2.39141i 1.44342 + 0.468994i
\(27\) 0 0
\(28\) −4.97980 − 3.61803i −0.941093 − 0.683744i
\(29\) 5.99070i 1.11245i 0.831033 + 0.556223i \(0.187750\pi\)
−0.831033 + 0.556223i \(0.812250\pi\)
\(30\) 0 0
\(31\) − 10.6861i − 1.91929i −0.281220 − 0.959643i \(-0.590739\pi\)
0.281220 − 0.959643i \(-0.409261\pi\)
\(32\) −5.65685 −1.00000
\(33\) 0 0
\(34\) 1.38197 − 4.25325i 0.237005 − 0.729427i
\(35\) 0 0
\(36\) 0 0
\(37\) − 2.47214i − 0.406417i −0.979136 − 0.203208i \(-0.934863\pi\)
0.979136 − 0.203208i \(-0.0651369\pi\)
\(38\) 2.18508 + 0.709976i 0.354467 + 0.115173i
\(39\) 0 0
\(40\) 0 0
\(41\) 8.07262i 1.26073i 0.776298 + 0.630366i \(0.217095\pi\)
−0.776298 + 0.630366i \(0.782905\pi\)
\(42\) 0 0
\(43\) 5.98385 0.912529 0.456265 − 0.889844i \(-0.349187\pi\)
0.456265 + 0.889844i \(0.349187\pi\)
\(44\) 1.66251 + 1.20788i 0.250632 + 0.182095i
\(45\) 0 0
\(46\) 10.3262 + 3.35520i 1.52252 + 0.494697i
\(47\) − 6.40747i − 0.934626i −0.884092 − 0.467313i \(-0.845222\pi\)
0.884092 − 0.467313i \(-0.154778\pi\)
\(48\) 0 0
\(49\) 2.47214 0.353162
\(50\) 0 0
\(51\) 0 0
\(52\) −6.43288 + 8.85410i −0.892080 + 1.22784i
\(53\) 10.2333 1.40566 0.702829 − 0.711359i \(-0.251920\pi\)
0.702829 + 0.711359i \(0.251920\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 7.04250 − 5.11667i 0.941093 − 0.683744i
\(57\) 0 0
\(58\) −8.05748 − 2.61803i −1.05800 − 0.343765i
\(59\) 13.0575 1.69994 0.849971 − 0.526829i \(-0.176619\pi\)
0.849971 + 0.526829i \(0.176619\pi\)
\(60\) 0 0
\(61\) −3.94427 −0.505012 −0.252506 − 0.967595i \(-0.581255\pi\)
−0.252506 + 0.967595i \(0.581255\pi\)
\(62\) 14.3728 + 4.67001i 1.82535 + 0.593092i
\(63\) 0 0
\(64\) 2.47214 − 7.60845i 0.309017 − 0.951057i
\(65\) 0 0
\(66\) 0 0
\(67\) 10.6861 1.30552 0.652760 − 0.757565i \(-0.273611\pi\)
0.652760 + 0.757565i \(0.273611\pi\)
\(68\) 5.11667 + 3.71748i 0.620488 + 0.450811i
\(69\) 0 0
\(70\) 0 0
\(71\) −7.43496 −0.882367 −0.441184 − 0.897417i \(-0.645441\pi\)
−0.441184 + 0.897417i \(0.645441\pi\)
\(72\) 0 0
\(73\) 1.52786i 0.178823i 0.995995 + 0.0894115i \(0.0284986\pi\)
−0.995995 + 0.0894115i \(0.971501\pi\)
\(74\) 3.32502 + 1.08036i 0.386525 + 0.125590i
\(75\) 0 0
\(76\) −1.90983 + 2.62866i −0.219073 + 0.301527i
\(77\) −3.16228 −0.360375
\(78\) 0 0
\(79\) 12.3107i 1.38507i 0.721386 + 0.692533i \(0.243505\pi\)
−0.721386 + 0.692533i \(0.756495\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) −10.8576 − 3.52786i −1.19903 − 0.389587i
\(83\) − 6.40747i − 0.703312i −0.936129 − 0.351656i \(-0.885619\pi\)
0.936129 − 0.351656i \(-0.114381\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −2.61504 + 8.04827i −0.281987 + 0.867867i
\(87\) 0 0
\(88\) −2.35114 + 1.70820i −0.250632 + 0.182095i
\(89\) − 9.15298i − 0.970214i −0.874455 − 0.485107i \(-0.838781\pi\)
0.874455 − 0.485107i \(-0.161219\pi\)
\(90\) 0 0
\(91\) − 16.8415i − 1.76547i
\(92\) −9.02546 + 12.4225i −0.940970 + 1.29513i
\(93\) 0 0
\(94\) 8.61803 + 2.80017i 0.888882 + 0.288815i
\(95\) 0 0
\(96\) 0 0
\(97\) 3.00000i 0.304604i 0.988334 + 0.152302i \(0.0486686\pi\)
−0.988334 + 0.152302i \(0.951331\pi\)
\(98\) −1.08036 + 3.32502i −0.109133 + 0.335877i
\(99\) 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.899.8 16
3.2 odd 2 inner 900.2.h.d.899.10 16
4.3 odd 2 inner 900.2.h.d.899.5 16
5.2 odd 4 900.2.e.e.251.1 ✓ 8
5.3 odd 4 900.2.e.g.251.8 yes 8
5.4 even 2 inner 900.2.h.d.899.9 16
12.11 even 2 inner 900.2.h.d.899.11 16
15.2 even 4 900.2.e.e.251.8 yes 8
15.8 even 4 900.2.e.g.251.1 yes 8
15.14 odd 2 inner 900.2.h.d.899.7 16
20.3 even 4 900.2.e.g.251.2 yes 8
20.7 even 4 900.2.e.e.251.7 yes 8
20.19 odd 2 inner 900.2.h.d.899.12 16
60.23 odd 4 900.2.e.g.251.7 yes 8
60.47 odd 4 900.2.e.e.251.2 yes 8
60.59 even 2 inner 900.2.h.d.899.6 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
900.2.e.e.251.1 ✓ 8 5.2 odd 4
900.2.e.e.251.2 yes 8 60.47 odd 4
900.2.e.e.251.7 yes 8 20.7 even 4
900.2.e.e.251.8 yes 8 15.2 even 4
900.2.e.g.251.1 yes 8 15.8 even 4
900.2.e.g.251.2 yes 8 20.3 even 4
900.2.e.g.251.7 yes 8 60.23 odd 4
900.2.e.g.251.8 yes 8 5.3 odd 4
900.2.h.d.899.5 16 4.3 odd 2 inner
900.2.h.d.899.6 16 60.59 even 2 inner
900.2.h.d.899.7 16 15.14 odd 2 inner
900.2.h.d.899.8 16 1.1 even 1 trivial
900.2.h.d.899.9 16 5.4 even 2 inner
900.2.h.d.899.10 16 3.2 odd 2 inner
900.2.h.d.899.11 16 12.11 even 2 inner
900.2.h.d.899.12 16 20.19 odd 2 inner