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Properties

Label 444.2.m.b.401.1
Level $444$
Weight $2$
Character 444.401
Analytic conductor $3.545$
Analytic rank $0$
Dimension $4$
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] = [444,2,Mod(401,444)] 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("444.401"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(444, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 2, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 444 = 2^{2} \cdot 3 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 444.m (of order \(4\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(3.54535784974\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
Coefficient field: \(\Q(\zeta_{12})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{4}]$

Embedding invariants

Embedding label 401.1
Root \(0.866025 + 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 444.401
Dual form 444.2.m.b.413.2

$q$-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)
 
\(f(q)\) \(=\) \(q-1.73205i q^{3} +3.46410 q^{7} -3.00000 q^{9} +(4.46410 - 4.46410i) q^{13} +(-2.26795 + 2.26795i) q^{19} -6.00000i q^{21} -5.00000i q^{25} +5.19615i q^{27} +(-7.19615 - 7.19615i) q^{31} +(5.00000 + 3.46410i) q^{37} +(-7.73205 - 7.73205i) q^{39} +(1.19615 - 1.19615i) q^{43} +5.00000 q^{49} +(3.92820 + 3.92820i) q^{57} +(10.4641 + 10.4641i) q^{61} -10.3923 q^{63} +16.0000i q^{67} +10.0000i q^{73} -8.66025 q^{75} +(-6.66025 + 6.66025i) q^{79} +9.00000 q^{81} +(15.4641 - 15.4641i) q^{91} +(-12.4641 + 12.4641i) q^{93} +(13.9282 - 13.9282i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 12 q^{9} + 4 q^{13} - 16 q^{19} - 8 q^{31} + 20 q^{37} - 24 q^{39} - 16 q^{43} + 20 q^{49} - 12 q^{57} + 28 q^{61} + 8 q^{79} + 36 q^{81} + 48 q^{91} - 36 q^{93} + 28 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/444\mathbb{Z}\right)^\times\).

\(n\) \(149\) \(223\) \(409\)
\(\chi(n)\) \(-1\) \(1\) \(e\left(\frac{1}{4}\right)\)

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
\(3\) − 1.73205i − 1.00000i
\(4\) 0 0
\(5\) 0 0 0.707107 − 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(6\) 0 0
\(7\) 3.46410 1.30931 0.654654 − 0.755929i \(-0.272814\pi\)
0.654654 + 0.755929i \(0.272814\pi\)
\(8\) 0 0
\(9\) −3.00000 −1.00000
\(10\) 0 0
\(11\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 0 0
\(13\) 4.46410 − 4.46410i 1.23812 − 1.23812i 0.277350 − 0.960769i \(-0.410544\pi\)
0.960769 − 0.277350i \(-0.0894562\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 0 0 0.707107 − 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(18\) 0 0
\(19\) −2.26795 + 2.26795i −0.520303 + 0.520303i −0.917663 − 0.397360i \(-0.869927\pi\)
0.397360 + 0.917663i \(0.369927\pi\)
\(20\) 0 0
\(21\) − 6.00000i − 1.30931i
\(22\) 0 0
\(23\) 0 0 0.707107 − 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(24\) 0 0
\(25\) − 5.00000i − 1.00000i
\(26\) 0 0
\(27\) 5.19615i 1.00000i
\(28\) 0 0
\(29\) 0 0 −0.707107 − 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(30\) 0 0
\(31\) −7.19615 − 7.19615i −1.29247 − 1.29247i −0.933257 − 0.359211i \(-0.883046\pi\)
−0.359211 − 0.933257i \(-0.616954\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 5.00000 + 3.46410i 0.821995 + 0.569495i
\(38\) 0 0
\(39\) −7.73205 − 7.73205i −1.23812 − 1.23812i
\(40\) 0 0
\(41\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) 0 0
\(43\) 1.19615 − 1.19615i 0.182412 − 0.182412i −0.609994 − 0.792406i \(-0.708828\pi\)
0.792406 + 0.609994i \(0.208828\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(48\) 0 0
\(49\) 5.00000 0.714286
\(50\) 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
\(57\) 3.92820 + 3.92820i 0.520303 + 0.520303i
\(58\) 0 0
\(59\) 0 0 0.707107 − 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(60\) 0 0
\(61\) 10.4641 + 10.4641i 1.33979 + 1.33979i 0.896258 + 0.443533i \(0.146275\pi\)
0.443533 + 0.896258i \(0.353725\pi\)
\(62\) 0 0
\(63\) −10.3923 −1.30931
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 16.0000i 1.95471i 0.211604 + 0.977356i \(0.432131\pi\)
−0.211604 + 0.977356i \(0.567869\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 0 0
\(73\) 10.0000i 1.17041i 0.810885 + 0.585206i \(0.198986\pi\)
−0.810885 + 0.585206i \(0.801014\pi\)
\(74\) 0 0
\(75\) −8.66025 −1.00000
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −6.66025 + 6.66025i −0.749337 + 0.749337i −0.974355 − 0.225018i \(-0.927756\pi\)
0.225018 + 0.974355i \(0.427756\pi\)
\(80\) 0 0
\(81\) 9.00000 1.00000
\(82\) 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 −0.707107 − 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(90\) 0 0
\(91\) 15.4641 − 15.4641i 1.62108 − 1.62108i
\(92\) 0 0
\(93\) −12.4641 + 12.4641i −1.29247 + 1.29247i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 13.9282 − 13.9282i 1.41419 − 1.41419i 0.703452 − 0.710742i \(-0.251641\pi\)
0.710742 − 0.703452i \(-0.248359\pi\)
\(98\) 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 444.2.m.b.401.1 ✓ 4
3.2 odd 2 CM 444.2.m.b.401.1 ✓ 4
37.6 odd 4 inner 444.2.m.b.413.2 yes 4
111.80 even 4 inner 444.2.m.b.413.2 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
444.2.m.b.401.1 ✓ 4 1.1 even 1 trivial
444.2.m.b.401.1 ✓ 4 3.2 odd 2 CM
444.2.m.b.413.2 yes 4 37.6 odd 4 inner
444.2.m.b.413.2 yes 4 111.80 even 4 inner