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Properties

Label 3600.2.f.h.2449.1
Level $3600$
Weight $2$
Character 3600.2449
Analytic conductor $28.746$
Analytic rank $1$
Dimension $2$
CM discriminant -3
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] = [3600,2,Mod(2449,3600)] 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("3600.2449"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3600, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 3600 = 2^{4} \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3600.f (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-14,0,0,0,0,0,0,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(29)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(28.7461447277\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 900)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 2449.1
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 3600.2449
Dual form 3600.2.f.h.2449.2

$q$-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)
 
\(f(q)\) \(=\) \(q-1.00000i q^{7} -7.00000i q^{13} -7.00000 q^{19} -11.0000 q^{31} +10.0000i q^{37} +13.0000i q^{43} +6.00000 q^{49} -1.00000 q^{61} +11.0000i q^{67} -10.0000i q^{73} -4.00000 q^{79} -7.00000 q^{91} +19.0000i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 14 q^{19} - 22 q^{31} + 12 q^{49} - 2 q^{61} - 8 q^{79} - 14 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3600\mathbb{Z}\right)^\times\).

\(n\) \(577\) \(901\) \(2801\) \(3151\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(1\)

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 0
\(3\) 0 0
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) − 1.00000i − 0.377964i −0.981981 − 0.188982i \(-0.939481\pi\)
0.981981 − 0.188982i \(-0.0605189\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 0 0
\(13\) − 7.00000i − 1.94145i −0.240192 − 0.970725i \(-0.577210\pi\)
0.240192 − 0.970725i \(-0.422790\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(18\) 0 0
\(19\) −7.00000 −1.60591 −0.802955 − 0.596040i \(-0.796740\pi\)
−0.802955 + 0.596040i \(0.796740\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) 0 0
\(31\) −11.0000 −1.97566 −0.987829 − 0.155543i \(-0.950287\pi\)
−0.987829 + 0.155543i \(0.950287\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 10.0000i 1.64399i 0.569495 + 0.821995i \(0.307139\pi\)
−0.569495 + 0.821995i \(0.692861\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) 0 0
\(43\) 13.0000i 1.98248i 0.132068 + 0.991241i \(0.457838\pi\)
−0.132068 + 0.991241i \(0.542162\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(48\) 0 0
\(49\) 6.00000 0.857143
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) −1.00000 −0.128037 −0.0640184 − 0.997949i \(-0.520392\pi\)
−0.0640184 + 0.997949i \(0.520392\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 11.0000i 1.34386i 0.740613 + 0.671932i \(0.234535\pi\)
−0.740613 + 0.671932i \(0.765465\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) − 10.0000i − 1.17041i −0.810885 − 0.585206i \(-0.801014\pi\)
0.810885 − 0.585206i \(-0.198986\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −4.00000 −0.450035 −0.225018 − 0.974355i \(-0.572244\pi\)
−0.225018 + 0.974355i \(0.572244\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) 0 0
\(91\) −7.00000 −0.733799
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 19.0000i 1.92916i 0.263795 + 0.964579i \(0.415026\pi\)
−0.263795 + 0.964579i \(0.584974\pi\)
\(98\) 0 0
\(99\) 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 3600.2.f.h.2449.1 2
3.2 odd 2 CM 3600.2.f.h.2449.1 2
4.3 odd 2 900.2.d.d.649.2 2
5.2 odd 4 3600.2.a.x.1.1 1
5.3 odd 4 3600.2.a.q.1.1 1
5.4 even 2 inner 3600.2.f.h.2449.2 2
12.11 even 2 900.2.d.d.649.2 2
15.2 even 4 3600.2.a.x.1.1 1
15.8 even 4 3600.2.a.q.1.1 1
15.14 odd 2 inner 3600.2.f.h.2449.2 2
20.3 even 4 900.2.a.f.1.1 yes 1
20.7 even 4 900.2.a.d.1.1 ✓ 1
20.19 odd 2 900.2.d.d.649.1 2
60.23 odd 4 900.2.a.f.1.1 yes 1
60.47 odd 4 900.2.a.d.1.1 ✓ 1
60.59 even 2 900.2.d.d.649.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
900.2.a.d.1.1 ✓ 1 20.7 even 4
900.2.a.d.1.1 ✓ 1 60.47 odd 4
900.2.a.f.1.1 yes 1 20.3 even 4
900.2.a.f.1.1 yes 1 60.23 odd 4
900.2.d.d.649.1 2 20.19 odd 2
900.2.d.d.649.1 2 60.59 even 2
900.2.d.d.649.2 2 4.3 odd 2
900.2.d.d.649.2 2 12.11 even 2
3600.2.a.q.1.1 1 5.3 odd 4
3600.2.a.q.1.1 1 15.8 even 4
3600.2.a.x.1.1 1 5.2 odd 4
3600.2.a.x.1.1 1 15.2 even 4
3600.2.f.h.2449.1 2 1.1 even 1 trivial
3600.2.f.h.2449.1 2 3.2 odd 2 CM
3600.2.f.h.2449.2 2 5.4 even 2 inner
3600.2.f.h.2449.2 2 15.14 odd 2 inner