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Properties

Label 300.3.b.a.149.2
Level $300$
Weight $3$
Character 300.149
Analytic conductor $8.174$
Analytic rank $0$
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] = [300,3,Mod(149,300)] 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("300.149"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(300, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 1])) N = Newforms(chi, 3, names="a")
 
Level: \( N \) \(=\) \( 300 = 2^{2} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 300.b (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,-18,0,0,0,0,0,0,0,0,0,-52] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(19)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(8.17440793081\)
Analytic rank: \(0\)
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 12)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 149.2
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 300.149
Dual form 300.3.b.a.149.1

$q$-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)
 
\(f(q)\) \(=\) \(q+3.00000i q^{3} +2.00000i q^{7} -9.00000 q^{9} +22.0000i q^{13} -26.0000 q^{19} -6.00000 q^{21} -27.0000i q^{27} -46.0000 q^{31} +26.0000i q^{37} -66.0000 q^{39} +22.0000i q^{43} +45.0000 q^{49} -78.0000i q^{57} +74.0000 q^{61} -18.0000i q^{63} +122.000i q^{67} +46.0000i q^{73} +142.000 q^{79} +81.0000 q^{81} -44.0000 q^{91} -138.000i q^{93} +2.00000i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 18 q^{9} - 52 q^{19} - 12 q^{21} - 92 q^{31} - 132 q^{39} + 90 q^{49} + 148 q^{61} + 284 q^{79} + 162 q^{81} - 88 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/300\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(151\) \(277\)
\(\chi(n)\) \(-1\) \(1\) \(-1\)

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 0
\(3\) 3.00000i 1.00000i
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 2.00000i 0.285714i 0.989743 + 0.142857i \(0.0456289\pi\)
−0.989743 + 0.142857i \(0.954371\pi\)
\(8\) 0 0
\(9\) −9.00000 −1.00000
\(10\) 0 0
\(11\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(12\) 0 0
\(13\) 22.0000i 1.69231i 0.532939 + 0.846154i \(0.321088\pi\)
−0.532939 + 0.846154i \(0.678912\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) 0 0
\(19\) −26.0000 −1.36842 −0.684211 − 0.729285i \(-0.739853\pi\)
−0.684211 + 0.729285i \(0.739853\pi\)
\(20\) 0 0
\(21\) −6.00000 −0.285714
\(22\) 0 0
\(23\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) − 27.0000i − 1.00000i
\(28\) 0 0
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) 0 0
\(31\) −46.0000 −1.48387 −0.741935 − 0.670471i \(-0.766092\pi\)
−0.741935 + 0.670471i \(0.766092\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 26.0000i 0.702703i 0.936244 + 0.351351i \(0.114278\pi\)
−0.936244 + 0.351351i \(0.885722\pi\)
\(38\) 0 0
\(39\) −66.0000 −1.69231
\(40\) 0 0
\(41\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(42\) 0 0
\(43\) 22.0000i 0.511628i 0.966726 + 0.255814i \(0.0823435\pi\)
−0.966726 + 0.255814i \(0.917657\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 0 0
\(49\) 45.0000 0.918367
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) − 78.0000i − 1.36842i
\(58\) 0 0
\(59\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(60\) 0 0
\(61\) 74.0000 1.21311 0.606557 − 0.795040i \(-0.292550\pi\)
0.606557 + 0.795040i \(0.292550\pi\)
\(62\) 0 0
\(63\) − 18.0000i − 0.285714i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 122.000i 1.82090i 0.413624 + 0.910448i \(0.364263\pi\)
−0.413624 + 0.910448i \(0.635737\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 0 0
\(73\) 46.0000i 0.630137i 0.949069 + 0.315068i \(0.102027\pi\)
−0.949069 + 0.315068i \(0.897973\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 142.000 1.79747 0.898734 − 0.438494i \(-0.144488\pi\)
0.898734 + 0.438494i \(0.144488\pi\)
\(80\) 0 0
\(81\) 81.0000 1.00000
\(82\) 0 0
\(83\) 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
\(89\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(90\) 0 0
\(91\) −44.0000 −0.483516
\(92\) 0 0
\(93\) − 138.000i − 1.48387i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 2.00000i 0.0206186i 0.999947 + 0.0103093i \(0.00328160\pi\)
−0.999947 + 0.0103093i \(0.996718\pi\)
\(98\) 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 300.3.b.a.149.2 2
3.2 odd 2 CM 300.3.b.a.149.2 2
4.3 odd 2 1200.3.c.c.449.1 2
5.2 odd 4 300.3.g.b.101.1 1
5.3 odd 4 12.3.c.a.5.1 ✓ 1
5.4 even 2 inner 300.3.b.a.149.1 2
12.11 even 2 1200.3.c.c.449.1 2
15.2 even 4 300.3.g.b.101.1 1
15.8 even 4 12.3.c.a.5.1 ✓ 1
15.14 odd 2 inner 300.3.b.a.149.1 2
20.3 even 4 48.3.e.a.17.1 1
20.7 even 4 1200.3.l.b.401.1 1
20.19 odd 2 1200.3.c.c.449.2 2
35.3 even 12 588.3.p.b.569.1 2
35.13 even 4 588.3.c.c.197.1 1
35.18 odd 12 588.3.p.c.569.1 2
35.23 odd 12 588.3.p.c.557.1 2
35.33 even 12 588.3.p.b.557.1 2
40.3 even 4 192.3.e.a.65.1 1
40.13 odd 4 192.3.e.b.65.1 1
45.13 odd 12 324.3.g.b.269.1 2
45.23 even 12 324.3.g.b.269.1 2
45.38 even 12 324.3.g.b.53.1 2
45.43 odd 12 324.3.g.b.53.1 2
55.43 even 4 1452.3.e.b.485.1 1
60.23 odd 4 48.3.e.a.17.1 1
60.47 odd 4 1200.3.l.b.401.1 1
60.59 even 2 1200.3.c.c.449.2 2
80.3 even 4 768.3.h.b.641.2 2
80.13 odd 4 768.3.h.a.641.1 2
80.43 even 4 768.3.h.b.641.1 2
80.53 odd 4 768.3.h.a.641.2 2
105.23 even 12 588.3.p.c.557.1 2
105.38 odd 12 588.3.p.b.569.1 2
105.53 even 12 588.3.p.c.569.1 2
105.68 odd 12 588.3.p.b.557.1 2
105.83 odd 4 588.3.c.c.197.1 1
120.53 even 4 192.3.e.b.65.1 1
120.83 odd 4 192.3.e.a.65.1 1
165.98 odd 4 1452.3.e.b.485.1 1
180.23 odd 12 1296.3.q.b.593.1 2
180.43 even 12 1296.3.q.b.1025.1 2
180.83 odd 12 1296.3.q.b.1025.1 2
180.103 even 12 1296.3.q.b.593.1 2
240.53 even 4 768.3.h.a.641.2 2
240.83 odd 4 768.3.h.b.641.2 2
240.173 even 4 768.3.h.a.641.1 2
240.203 odd 4 768.3.h.b.641.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
12.3.c.a.5.1 ✓ 1 5.3 odd 4
12.3.c.a.5.1 ✓ 1 15.8 even 4
48.3.e.a.17.1 1 20.3 even 4
48.3.e.a.17.1 1 60.23 odd 4
192.3.e.a.65.1 1 40.3 even 4
192.3.e.a.65.1 1 120.83 odd 4
192.3.e.b.65.1 1 40.13 odd 4
192.3.e.b.65.1 1 120.53 even 4
300.3.b.a.149.1 2 5.4 even 2 inner
300.3.b.a.149.1 2 15.14 odd 2 inner
300.3.b.a.149.2 2 1.1 even 1 trivial
300.3.b.a.149.2 2 3.2 odd 2 CM
300.3.g.b.101.1 1 5.2 odd 4
300.3.g.b.101.1 1 15.2 even 4
324.3.g.b.53.1 2 45.38 even 12
324.3.g.b.53.1 2 45.43 odd 12
324.3.g.b.269.1 2 45.13 odd 12
324.3.g.b.269.1 2 45.23 even 12
588.3.c.c.197.1 1 35.13 even 4
588.3.c.c.197.1 1 105.83 odd 4
588.3.p.b.557.1 2 35.33 even 12
588.3.p.b.557.1 2 105.68 odd 12
588.3.p.b.569.1 2 35.3 even 12
588.3.p.b.569.1 2 105.38 odd 12
588.3.p.c.557.1 2 35.23 odd 12
588.3.p.c.557.1 2 105.23 even 12
588.3.p.c.569.1 2 35.18 odd 12
588.3.p.c.569.1 2 105.53 even 12
768.3.h.a.641.1 2 80.13 odd 4
768.3.h.a.641.1 2 240.173 even 4
768.3.h.a.641.2 2 80.53 odd 4
768.3.h.a.641.2 2 240.53 even 4
768.3.h.b.641.1 2 80.43 even 4
768.3.h.b.641.1 2 240.203 odd 4
768.3.h.b.641.2 2 80.3 even 4
768.3.h.b.641.2 2 240.83 odd 4
1200.3.c.c.449.1 2 4.3 odd 2
1200.3.c.c.449.1 2 12.11 even 2
1200.3.c.c.449.2 2 20.19 odd 2
1200.3.c.c.449.2 2 60.59 even 2
1200.3.l.b.401.1 1 20.7 even 4
1200.3.l.b.401.1 1 60.47 odd 4
1296.3.q.b.593.1 2 180.23 odd 12
1296.3.q.b.593.1 2 180.103 even 12
1296.3.q.b.1025.1 2 180.43 even 12
1296.3.q.b.1025.1 2 180.83 odd 12
1452.3.e.b.485.1 1 55.43 even 4
1452.3.e.b.485.1 1 165.98 odd 4