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Properties

Label 396.2.h.a.307.1
Level $396$
Weight $2$
Character 396.307
Analytic conductor $3.162$
Analytic rank $0$
Dimension $4$
CM discriminant -132
Inner twists $8$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [396,2,Mod(307,396)] 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("396.307"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(396, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 396 = 2^{2} \cdot 3^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 396.h (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,-8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(4)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(3.16207592004\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{-11})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{3} + 11x^{2} - 10x + 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 307.1
Root \(0.500000 - 0.244099i\) of defining polynomial
Character \(\chi\) \(=\) 396.307
Dual form 396.2.h.a.307.3

$q$-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)
 
\(f(q)\) \(=\) \(q-1.41421i q^{2} -2.00000 q^{4} -4.69042 q^{7} +2.82843i q^{8} +3.31662i q^{11} +6.63325i q^{14} +4.00000 q^{16} +1.41421i q^{17} -4.69042 q^{19} +4.69042 q^{22} +6.63325i q^{23} -5.00000 q^{25} +9.38083 q^{28} -7.07107i q^{29} -5.65685i q^{32} +2.00000 q^{34} -4.00000 q^{37} +6.63325i q^{38} +9.89949i q^{41} -4.69042 q^{43} -6.63325i q^{44} +9.38083 q^{46} -13.2665i q^{47} +15.0000 q^{49} +7.07107i q^{50} -13.2665i q^{56} -10.0000 q^{58} +6.63325i q^{59} -8.00000 q^{64} -2.82843i q^{68} -13.2665i q^{71} +5.65685i q^{74} +9.38083 q^{76} -15.5563i q^{77} -4.69042 q^{79} +14.0000 q^{82} +6.63325i q^{86} -9.38083 q^{88} -13.2665i q^{92} -18.7617 q^{94} -16.0000 q^{97} -21.2132i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 8 q^{4} + 16 q^{16} - 20 q^{25} + 8 q^{34} - 16 q^{37} + 60 q^{49} - 40 q^{58} - 32 q^{64} + 56 q^{82} - 64 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/396\mathbb{Z}\right)^\times\).

\(n\) \(145\) \(199\) \(353\)
\(\chi(n)\) \(-1\) \(-1\) \(1\)

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 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(6\) 0 0
\(7\) −4.69042 −1.77281 −0.886405 − 0.462910i \(-0.846805\pi\)
−0.886405 + 0.462910i \(0.846805\pi\)
\(8\) 2.82843i 1.00000i
\(9\) 0 0
\(10\) 0 0
\(11\) 3.31662i 1.00000i
\(12\) 0 0
\(13\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(14\) 6.63325i 1.77281i
\(15\) 0 0
\(16\) 4.00000 1.00000
\(17\) 1.41421i 0.342997i 0.985184 + 0.171499i \(0.0548609\pi\)
−0.985184 + 0.171499i \(0.945139\pi\)
\(18\) 0 0
\(19\) −4.69042 −1.07606 −0.538028 − 0.842927i \(-0.680830\pi\)
−0.538028 + 0.842927i \(0.680830\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 4.69042 1.00000
\(23\) 6.63325i 1.38313i 0.722315 + 0.691564i \(0.243078\pi\)
−0.722315 + 0.691564i \(0.756922\pi\)
\(24\) 0 0
\(25\) −5.00000 −1.00000
\(26\) 0 0
\(27\) 0 0
\(28\) 9.38083 1.77281
\(29\) − 7.07107i − 1.31306i −0.754298 − 0.656532i \(-0.772023\pi\)
0.754298 − 0.656532i \(-0.227977\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(32\) − 5.65685i − 1.00000i
\(33\) 0 0
\(34\) 2.00000 0.342997
\(35\) 0 0
\(36\) 0 0
\(37\) −4.00000 −0.657596 −0.328798 − 0.944400i \(-0.606644\pi\)
−0.328798 + 0.944400i \(0.606644\pi\)
\(38\) 6.63325i 1.07606i
\(39\) 0 0
\(40\) 0 0
\(41\) 9.89949i 1.54604i 0.634381 + 0.773021i \(0.281255\pi\)
−0.634381 + 0.773021i \(0.718745\pi\)
\(42\) 0 0
\(43\) −4.69042 −0.715282 −0.357641 − 0.933859i \(-0.616419\pi\)
−0.357641 + 0.933859i \(0.616419\pi\)
\(44\) − 6.63325i − 1.00000i
\(45\) 0 0
\(46\) 9.38083 1.38313
\(47\) − 13.2665i − 1.93512i −0.252646 − 0.967559i \(-0.581301\pi\)
0.252646 − 0.967559i \(-0.418699\pi\)
\(48\) 0 0
\(49\) 15.0000 2.14286
\(50\) 7.07107i 1.00000i
\(51\) 0 0
\(52\) 0 0
\(53\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) − 13.2665i − 1.77281i
\(57\) 0 0
\(58\) −10.0000 −1.31306
\(59\) 6.63325i 0.863576i 0.901975 + 0.431788i \(0.142117\pi\)
−0.901975 + 0.431788i \(0.857883\pi\)
\(60\) 0 0
\(61\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) −8.00000 −1.00000
\(65\) 0 0
\(66\) 0 0
\(67\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(68\) − 2.82843i − 0.342997i
\(69\) 0 0
\(70\) 0 0
\(71\) − 13.2665i − 1.57444i −0.616670 − 0.787222i \(-0.711519\pi\)
0.616670 − 0.787222i \(-0.288481\pi\)
\(72\) 0 0
\(73\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(74\) 5.65685i 0.657596i
\(75\) 0 0
\(76\) 9.38083 1.07606
\(77\) − 15.5563i − 1.77281i
\(78\) 0 0
\(79\) −4.69042 −0.527713 −0.263857 − 0.964562i \(-0.584995\pi\)
−0.263857 + 0.964562i \(0.584995\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 14.0000 1.54604
\(83\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 6.63325i 0.715282i
\(87\) 0 0
\(88\) −9.38083 −1.00000
\(89\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) − 13.2665i − 1.38313i
\(93\) 0 0
\(94\) −18.7617 −1.93512
\(95\) 0 0
\(96\) 0 0
\(97\) −16.0000 −1.62455 −0.812277 − 0.583272i \(-0.801772\pi\)
−0.812277 + 0.583272i \(0.801772\pi\)
\(98\) − 21.2132i − 2.14286i
\(99\) 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 396.2.h.a.307.1 ✓ 4
3.2 odd 2 inner 396.2.h.a.307.3 yes 4
4.3 odd 2 inner 396.2.h.a.307.2 yes 4
11.10 odd 2 inner 396.2.h.a.307.4 yes 4
12.11 even 2 inner 396.2.h.a.307.4 yes 4
33.32 even 2 inner 396.2.h.a.307.2 yes 4
44.43 even 2 inner 396.2.h.a.307.3 yes 4
132.131 odd 2 CM 396.2.h.a.307.1 ✓ 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
396.2.h.a.307.1 ✓ 4 1.1 even 1 trivial
396.2.h.a.307.1 ✓ 4 132.131 odd 2 CM
396.2.h.a.307.2 yes 4 4.3 odd 2 inner
396.2.h.a.307.2 yes 4 33.32 even 2 inner
396.2.h.a.307.3 yes 4 3.2 odd 2 inner
396.2.h.a.307.3 yes 4 44.43 even 2 inner
396.2.h.a.307.4 yes 4 11.10 odd 2 inner
396.2.h.a.307.4 yes 4 12.11 even 2 inner