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Properties

Label 300.2.e.a.251.4
Level $300$
Weight $2$
Character 300.251
Analytic conductor $2.396$
Analytic rank $0$
Dimension $4$
CM discriminant -15
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] = [300,2,Mod(251,300)] 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("300.251"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(300, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 300 = 2^{2} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 300.e (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,-2,0,6,0,0,12,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(12)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(2.39551206064\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{3}, \sqrt{-5})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 60)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 251.4
Root \(0.866025 + 1.11803i\) of defining polynomial
Character \(\chi\) \(=\) 300.251
Dual form 300.2.e.a.251.3

$q$-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)
 
\(f(q)\) \(=\) \(q+(0.866025 + 1.11803i) q^{2} +1.73205 q^{3} +(-0.500000 + 1.93649i) q^{4} +(1.50000 + 1.93649i) q^{6} +(-2.59808 + 1.11803i) q^{8} +3.00000 q^{9} +(-0.866025 + 3.35410i) q^{12} +(-3.50000 - 1.93649i) q^{16} +4.47214i q^{17} +(2.59808 + 3.35410i) q^{18} -7.74597i q^{19} -3.46410 q^{23} +(-4.50000 + 1.93649i) q^{24} +5.19615 q^{27} -7.74597i q^{31} +(-0.866025 - 5.59017i) q^{32} +(-5.00000 + 3.87298i) q^{34} +(-1.50000 + 5.80948i) q^{36} +(8.66025 - 6.70820i) q^{38} +(-3.00000 - 3.87298i) q^{46} -10.3923 q^{47} +(-6.06218 - 3.35410i) q^{48} +7.00000 q^{49} +7.74597i q^{51} +4.47214i q^{53} +(4.50000 + 5.80948i) q^{54} -13.4164i q^{57} -2.00000 q^{61} +(8.66025 - 6.70820i) q^{62} +(5.50000 - 5.80948i) q^{64} +(-8.66025 - 2.23607i) q^{68} -6.00000 q^{69} +(-7.79423 + 3.35410i) q^{72} +(15.0000 + 3.87298i) q^{76} +7.74597i q^{79} +9.00000 q^{81} +3.46410 q^{83} +(1.73205 - 6.70820i) q^{92} -13.4164i q^{93} +(-9.00000 - 11.6190i) q^{94} +(-1.50000 - 9.68246i) q^{96} +(6.06218 + 7.82624i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{4} + 6 q^{6} + 12 q^{9} - 14 q^{16} - 18 q^{24} - 20 q^{34} - 6 q^{36} - 12 q^{46} + 28 q^{49} + 18 q^{54} - 8 q^{61} + 22 q^{64} - 24 q^{69} + 60 q^{76} + 36 q^{81} - 36 q^{94} - 6 q^{96}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.866025 + 1.11803i 0.612372 + 0.790569i
\(3\) 1.73205 1.00000
\(4\) −0.500000 + 1.93649i −0.250000 + 0.968246i
\(5\) 0 0
\(6\) 1.50000 + 1.93649i 0.612372 + 0.790569i
\(7\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(8\) −2.59808 + 1.11803i −0.918559 + 0.395285i
\(9\) 3.00000 1.00000
\(10\) 0 0
\(11\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) −0.866025 + 3.35410i −0.250000 + 0.968246i
\(13\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −3.50000 − 1.93649i −0.875000 − 0.484123i
\(17\) 4.47214i 1.08465i 0.840168 + 0.542326i \(0.182456\pi\)
−0.840168 + 0.542326i \(0.817544\pi\)
\(18\) 2.59808 + 3.35410i 0.612372 + 0.790569i
\(19\) − 7.74597i − 1.77705i −0.458831 − 0.888523i \(-0.651732\pi\)
0.458831 − 0.888523i \(-0.348268\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −3.46410 −0.722315 −0.361158 − 0.932505i \(-0.617618\pi\)
−0.361158 + 0.932505i \(0.617618\pi\)
\(24\) −4.50000 + 1.93649i −0.918559 + 0.395285i
\(25\) 0 0
\(26\) 0 0
\(27\) 5.19615 1.00000
\(28\) 0 0
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) 0 0
\(31\) − 7.74597i − 1.39122i −0.718421 − 0.695608i \(-0.755135\pi\)
0.718421 − 0.695608i \(-0.244865\pi\)
\(32\) −0.866025 − 5.59017i −0.153093 − 0.988212i
\(33\) 0 0
\(34\) −5.00000 + 3.87298i −0.857493 + 0.664211i
\(35\) 0 0
\(36\) −1.50000 + 5.80948i −0.250000 + 0.968246i
\(37\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(38\) 8.66025 − 6.70820i 1.40488 − 1.08821i
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(42\) 0 0
\(43\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) −3.00000 − 3.87298i −0.442326 − 0.571040i
\(47\) −10.3923 −1.51587 −0.757937 − 0.652328i \(-0.773792\pi\)
−0.757937 + 0.652328i \(0.773792\pi\)
\(48\) −6.06218 − 3.35410i −0.875000 − 0.484123i
\(49\) 7.00000 1.00000
\(50\) 0 0
\(51\) 7.74597i 1.08465i
\(52\) 0 0
\(53\) 4.47214i 0.614295i 0.951662 + 0.307148i \(0.0993745\pi\)
−0.951662 + 0.307148i \(0.900625\pi\)
\(54\) 4.50000 + 5.80948i 0.612372 + 0.790569i
\(55\) 0 0
\(56\) 0 0
\(57\) − 13.4164i − 1.77705i
\(58\) 0 0
\(59\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) −2.00000 −0.256074 −0.128037 − 0.991769i \(-0.540868\pi\)
−0.128037 + 0.991769i \(0.540868\pi\)
\(62\) 8.66025 − 6.70820i 1.09985 − 0.851943i
\(63\) 0 0
\(64\) 5.50000 − 5.80948i 0.687500 − 0.726184i
\(65\) 0 0
\(66\) 0 0
\(67\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(68\) −8.66025 − 2.23607i −1.05021 − 0.271163i
\(69\) −6.00000 −0.722315
\(70\) 0 0
\(71\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) −7.79423 + 3.35410i −0.918559 + 0.395285i
\(73\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 15.0000 + 3.87298i 1.72062 + 0.444262i
\(77\) 0 0
\(78\) 0 0
\(79\) 7.74597i 0.871489i 0.900070 + 0.435745i \(0.143515\pi\)
−0.900070 + 0.435745i \(0.856485\pi\)
\(80\) 0 0
\(81\) 9.00000 1.00000
\(82\) 0 0
\(83\) 3.46410 0.380235 0.190117 − 0.981761i \(-0.439113\pi\)
0.190117 + 0.981761i \(0.439113\pi\)
\(84\) 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
\(91\) 0 0
\(92\) 1.73205 − 6.70820i 0.180579 − 0.699379i
\(93\) − 13.4164i − 1.39122i
\(94\) −9.00000 − 11.6190i −0.928279 − 1.19840i
\(95\) 0 0
\(96\) −1.50000 − 9.68246i −0.153093 − 0.988212i
\(97\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(98\) 6.06218 + 7.82624i 0.612372 + 0.790569i
\(99\) 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 300.2.e.a.251.4 4
3.2 odd 2 inner 300.2.e.a.251.1 4
4.3 odd 2 inner 300.2.e.a.251.2 4
5.2 odd 4 60.2.h.b.59.2 yes 4
5.3 odd 4 60.2.h.b.59.3 yes 4
5.4 even 2 inner 300.2.e.a.251.1 4
12.11 even 2 inner 300.2.e.a.251.3 4
15.2 even 4 60.2.h.b.59.3 yes 4
15.8 even 4 60.2.h.b.59.2 yes 4
15.14 odd 2 CM 300.2.e.a.251.4 4
20.3 even 4 60.2.h.b.59.4 yes 4
20.7 even 4 60.2.h.b.59.1 ✓ 4
20.19 odd 2 inner 300.2.e.a.251.3 4
40.3 even 4 960.2.o.a.959.4 4
40.13 odd 4 960.2.o.a.959.2 4
40.27 even 4 960.2.o.a.959.1 4
40.37 odd 4 960.2.o.a.959.3 4
60.23 odd 4 60.2.h.b.59.1 ✓ 4
60.47 odd 4 60.2.h.b.59.4 yes 4
60.59 even 2 inner 300.2.e.a.251.2 4
120.53 even 4 960.2.o.a.959.3 4
120.77 even 4 960.2.o.a.959.2 4
120.83 odd 4 960.2.o.a.959.1 4
120.107 odd 4 960.2.o.a.959.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
60.2.h.b.59.1 ✓ 4 20.7 even 4
60.2.h.b.59.1 ✓ 4 60.23 odd 4
60.2.h.b.59.2 yes 4 5.2 odd 4
60.2.h.b.59.2 yes 4 15.8 even 4
60.2.h.b.59.3 yes 4 5.3 odd 4
60.2.h.b.59.3 yes 4 15.2 even 4
60.2.h.b.59.4 yes 4 20.3 even 4
60.2.h.b.59.4 yes 4 60.47 odd 4
300.2.e.a.251.1 4 3.2 odd 2 inner
300.2.e.a.251.1 4 5.4 even 2 inner
300.2.e.a.251.2 4 4.3 odd 2 inner
300.2.e.a.251.2 4 60.59 even 2 inner
300.2.e.a.251.3 4 12.11 even 2 inner
300.2.e.a.251.3 4 20.19 odd 2 inner
300.2.e.a.251.4 4 1.1 even 1 trivial
300.2.e.a.251.4 4 15.14 odd 2 CM
960.2.o.a.959.1 4 40.27 even 4
960.2.o.a.959.1 4 120.83 odd 4
960.2.o.a.959.2 4 40.13 odd 4
960.2.o.a.959.2 4 120.77 even 4
960.2.o.a.959.3 4 40.37 odd 4
960.2.o.a.959.3 4 120.53 even 4
960.2.o.a.959.4 4 40.3 even 4
960.2.o.a.959.4 4 120.107 odd 4