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Properties

Label 900.2.e.c.251.2
Level $900$
Weight $2$
Character 900.251
Analytic conductor $7.187$
Analytic rank $0$
Dimension $2$
CM discriminant -4
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(251,900)] 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("900.251"); 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, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 900 = 2^{2} \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 900.e (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,-4,0,0,0,0,0,0,0,0,12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(7.18653618192\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-2}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 180)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 251.2
Root \(-1.41421i\) of defining polynomial
Character \(\chi\) \(=\) 900.251
Dual form 900.2.e.c.251.1

$q$-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)
 
\(f(q)\) \(=\) \(q+1.41421i q^{2} -2.00000 q^{4} -2.82843i q^{8} +6.00000 q^{13} +4.00000 q^{16} -7.07107i q^{17} +8.48528i q^{26} +4.24264i q^{29} +5.65685i q^{32} +10.0000 q^{34} +12.0000 q^{37} +12.7279i q^{41} +7.00000 q^{49} -12.0000 q^{52} -7.07107i q^{53} -6.00000 q^{58} +10.0000 q^{61} -8.00000 q^{64} +14.1421i q^{68} -6.00000 q^{73} +16.9706i q^{74} -18.0000 q^{82} -4.24264i q^{89} +18.0000 q^{97} +9.89949i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 4 q^{4} + 12 q^{13} + 8 q^{16} + 20 q^{34} + 24 q^{37} + 14 q^{49} - 24 q^{52} - 12 q^{58} + 20 q^{61} - 16 q^{64} - 12 q^{73} - 36 q^{82} + 36 q^{97}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.41421i 1.00000i
\(3\) 0 0
\(4\) −2.00000 −1.00000
\(5\) 0 0
\(6\) 0 0
\(7\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(8\) − 2.82843i − 1.00000i
\(9\) 0 0
\(10\) 0 0
\(11\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 0 0
\(13\) 6.00000 1.66410 0.832050 − 0.554700i \(-0.187167\pi\)
0.832050 + 0.554700i \(0.187167\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 4.00000 1.00000
\(17\) − 7.07107i − 1.71499i −0.514496 − 0.857493i \(-0.672021\pi\)
0.514496 − 0.857493i \(-0.327979\pi\)
\(18\) 0 0
\(19\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 8.48528i 1.66410i
\(27\) 0 0
\(28\) 0 0
\(29\) 4.24264i 0.787839i 0.919145 + 0.393919i \(0.128881\pi\)
−0.919145 + 0.393919i \(0.871119\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(32\) 5.65685i 1.00000i
\(33\) 0 0
\(34\) 10.0000 1.71499
\(35\) 0 0
\(36\) 0 0
\(37\) 12.0000 1.97279 0.986394 − 0.164399i \(-0.0525685\pi\)
0.986394 + 0.164399i \(0.0525685\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 12.7279i 1.98777i 0.110432 + 0.993884i \(0.464777\pi\)
−0.110432 + 0.993884i \(0.535223\pi\)
\(42\) 0 0
\(43\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 0 0
\(49\) 7.00000 1.00000
\(50\) 0 0
\(51\) 0 0
\(52\) −12.0000 −1.66410
\(53\) − 7.07107i − 0.971286i −0.874157 − 0.485643i \(-0.838586\pi\)
0.874157 − 0.485643i \(-0.161414\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) −6.00000 −0.787839
\(59\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) 10.0000 1.28037 0.640184 − 0.768221i \(-0.278858\pi\)
0.640184 + 0.768221i \(0.278858\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) −8.00000 −1.00000
\(65\) 0 0
\(66\) 0 0
\(67\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(68\) 14.1421i 1.71499i
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) −6.00000 −0.702247 −0.351123 − 0.936329i \(-0.614200\pi\)
−0.351123 + 0.936329i \(0.614200\pi\)
\(74\) 16.9706i 1.97279i
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) −18.0000 −1.98777
\(83\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) − 4.24264i − 0.449719i −0.974391 − 0.224860i \(-0.927808\pi\)
0.974391 − 0.224860i \(-0.0721923\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 18.0000 1.82762 0.913812 − 0.406138i \(-0.133125\pi\)
0.913812 + 0.406138i \(0.133125\pi\)
\(98\) 9.89949i 1.00000i
\(99\) 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.e.c.251.2 2
3.2 odd 2 inner 900.2.e.c.251.1 2
4.3 odd 2 CM 900.2.e.c.251.2 2
5.2 odd 4 180.2.h.a.179.1 ✓ 4
5.3 odd 4 180.2.h.a.179.3 yes 4
5.4 even 2 900.2.e.a.251.1 2
12.11 even 2 inner 900.2.e.c.251.1 2
15.2 even 4 180.2.h.a.179.4 yes 4
15.8 even 4 180.2.h.a.179.2 yes 4
15.14 odd 2 900.2.e.a.251.2 2
20.3 even 4 180.2.h.a.179.3 yes 4
20.7 even 4 180.2.h.a.179.1 ✓ 4
20.19 odd 2 900.2.e.a.251.1 2
40.3 even 4 2880.2.o.a.2879.4 4
40.13 odd 4 2880.2.o.a.2879.4 4
40.27 even 4 2880.2.o.a.2879.2 4
40.37 odd 4 2880.2.o.a.2879.2 4
60.23 odd 4 180.2.h.a.179.2 yes 4
60.47 odd 4 180.2.h.a.179.4 yes 4
60.59 even 2 900.2.e.a.251.2 2
120.53 even 4 2880.2.o.a.2879.1 4
120.77 even 4 2880.2.o.a.2879.3 4
120.83 odd 4 2880.2.o.a.2879.1 4
120.107 odd 4 2880.2.o.a.2879.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
180.2.h.a.179.1 ✓ 4 5.2 odd 4
180.2.h.a.179.1 ✓ 4 20.7 even 4
180.2.h.a.179.2 yes 4 15.8 even 4
180.2.h.a.179.2 yes 4 60.23 odd 4
180.2.h.a.179.3 yes 4 5.3 odd 4
180.2.h.a.179.3 yes 4 20.3 even 4
180.2.h.a.179.4 yes 4 15.2 even 4
180.2.h.a.179.4 yes 4 60.47 odd 4
900.2.e.a.251.1 2 5.4 even 2
900.2.e.a.251.1 2 20.19 odd 2
900.2.e.a.251.2 2 15.14 odd 2
900.2.e.a.251.2 2 60.59 even 2
900.2.e.c.251.1 2 3.2 odd 2 inner
900.2.e.c.251.1 2 12.11 even 2 inner
900.2.e.c.251.2 2 1.1 even 1 trivial
900.2.e.c.251.2 2 4.3 odd 2 CM
2880.2.o.a.2879.1 4 120.53 even 4
2880.2.o.a.2879.1 4 120.83 odd 4
2880.2.o.a.2879.2 4 40.27 even 4
2880.2.o.a.2879.2 4 40.37 odd 4
2880.2.o.a.2879.3 4 120.77 even 4
2880.2.o.a.2879.3 4 120.107 odd 4
2880.2.o.a.2879.4 4 40.3 even 4
2880.2.o.a.2879.4 4 40.13 odd 4