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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1936,2,Mod(1935,1936)] 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("1936.1935"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1936, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1936 = 2^{4} \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1936.e (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,0,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(15.4590378313\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{-3})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1935.1
Root \(1.22474 + 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 1936.1935
Dual form 1936.2.e.d.1935.3

$q$-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)
 
\(f(q)\) \(=\) \(q-1.73205i q^{3} +1.00000 q^{5} -2.44949 q^{7} +5.65685i q^{13} -1.73205i q^{15} -4.24264i q^{17} +4.24264i q^{21} -8.66025i q^{23} -4.00000 q^{25} -5.19615i q^{27} -5.65685i q^{29} +1.73205i q^{31} -2.44949 q^{35} +5.00000 q^{37} +9.79796 q^{39} -5.65685i q^{41} -12.2474 q^{43} -6.92820i q^{47} -1.00000 q^{49} -7.34847 q^{51} +4.00000 q^{53} -12.1244i q^{59} -5.65685i q^{61} +5.65685i q^{65} -5.19615i q^{67} -15.0000 q^{69} +12.1244i q^{71} +11.3137i q^{73} +6.92820i q^{75} -7.34847 q^{79} -9.00000 q^{81} +2.44949 q^{83} -4.24264i q^{85} -9.79796 q^{87} -17.0000 q^{89} -13.8564i q^{91} +3.00000 q^{93} +7.00000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{5} - 16 q^{25} + 20 q^{37} - 4 q^{49} + 16 q^{53} - 60 q^{69} - 36 q^{81} - 68 q^{89} + 12 q^{93} + 28 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1936\mathbb{Z}\right)^\times\).

\(n\) \(485\) \(849\) \(1695\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

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
\(3\) − 1.73205i − 1.00000i −0.866025 − 0.500000i \(-0.833333\pi\)
0.866025 − 0.500000i \(-0.166667\pi\)
\(4\) 0 0
\(5\) 1.00000 0.447214 0.223607 − 0.974679i \(-0.428217\pi\)
0.223607 + 0.974679i \(0.428217\pi\)
\(6\) 0 0
\(7\) −2.44949 −0.925820 −0.462910 − 0.886405i \(-0.653195\pi\)
−0.462910 + 0.886405i \(0.653195\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 0 0
\(12\) 0 0
\(13\) 5.65685i 1.56893i 0.620174 + 0.784465i \(0.287062\pi\)
−0.620174 + 0.784465i \(0.712938\pi\)
\(14\) 0 0
\(15\) − 1.73205i − 0.447214i
\(16\) 0 0
\(17\) − 4.24264i − 1.02899i −0.857493 − 0.514496i \(-0.827979\pi\)
0.857493 − 0.514496i \(-0.172021\pi\)
\(18\) 0 0
\(19\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) 0 0
\(21\) 4.24264i 0.925820i
\(22\) 0 0
\(23\) − 8.66025i − 1.80579i −0.429863 − 0.902894i \(-0.641438\pi\)
0.429863 − 0.902894i \(-0.358562\pi\)
\(24\) 0 0
\(25\) −4.00000 −0.800000
\(26\) 0 0
\(27\) − 5.19615i − 1.00000i
\(28\) 0 0
\(29\) − 5.65685i − 1.05045i −0.850963 − 0.525226i \(-0.823981\pi\)
0.850963 − 0.525226i \(-0.176019\pi\)
\(30\) 0 0
\(31\) 1.73205i 0.311086i 0.987829 + 0.155543i \(0.0497126\pi\)
−0.987829 + 0.155543i \(0.950287\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −2.44949 −0.414039
\(36\) 0 0
\(37\) 5.00000 0.821995 0.410997 − 0.911636i \(-0.365181\pi\)
0.410997 + 0.911636i \(0.365181\pi\)
\(38\) 0 0
\(39\) 9.79796 1.56893
\(40\) 0 0
\(41\) − 5.65685i − 0.883452i −0.897150 − 0.441726i \(-0.854366\pi\)
0.897150 − 0.441726i \(-0.145634\pi\)
\(42\) 0 0
\(43\) −12.2474 −1.86772 −0.933859 − 0.357641i \(-0.883581\pi\)
−0.933859 + 0.357641i \(0.883581\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) − 6.92820i − 1.01058i −0.862949 − 0.505291i \(-0.831385\pi\)
0.862949 − 0.505291i \(-0.168615\pi\)
\(48\) 0 0
\(49\) −1.00000 −0.142857
\(50\) 0 0
\(51\) −7.34847 −1.02899
\(52\) 0 0
\(53\) 4.00000 0.549442 0.274721 − 0.961524i \(-0.411414\pi\)
0.274721 + 0.961524i \(0.411414\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) − 12.1244i − 1.57846i −0.614100 − 0.789228i \(-0.710481\pi\)
0.614100 − 0.789228i \(-0.289519\pi\)
\(60\) 0 0
\(61\) − 5.65685i − 0.724286i −0.932123 − 0.362143i \(-0.882045\pi\)
0.932123 − 0.362143i \(-0.117955\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 5.65685i 0.701646i
\(66\) 0 0
\(67\) − 5.19615i − 0.634811i −0.948290 − 0.317406i \(-0.897188\pi\)
0.948290 − 0.317406i \(-0.102812\pi\)
\(68\) 0 0
\(69\) −15.0000 −1.80579
\(70\) 0 0
\(71\) 12.1244i 1.43890i 0.694546 + 0.719448i \(0.255605\pi\)
−0.694546 + 0.719448i \(0.744395\pi\)
\(72\) 0 0
\(73\) 11.3137i 1.32417i 0.749429 + 0.662085i \(0.230328\pi\)
−0.749429 + 0.662085i \(0.769672\pi\)
\(74\) 0 0
\(75\) 6.92820i 0.800000i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −7.34847 −0.826767 −0.413384 − 0.910557i \(-0.635653\pi\)
−0.413384 + 0.910557i \(0.635653\pi\)
\(80\) 0 0
\(81\) −9.00000 −1.00000
\(82\) 0 0
\(83\) 2.44949 0.268866 0.134433 − 0.990923i \(-0.457079\pi\)
0.134433 + 0.990923i \(0.457079\pi\)
\(84\) 0 0
\(85\) − 4.24264i − 0.460179i
\(86\) 0 0
\(87\) −9.79796 −1.05045
\(88\) 0 0
\(89\) −17.0000 −1.80200 −0.900998 − 0.433823i \(-0.857164\pi\)
−0.900998 + 0.433823i \(0.857164\pi\)
\(90\) 0 0
\(91\) − 13.8564i − 1.45255i
\(92\) 0 0
\(93\) 3.00000 0.311086
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 7.00000 0.710742 0.355371 − 0.934725i \(-0.384354\pi\)
0.355371 + 0.934725i \(0.384354\pi\)
\(98\) 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 1936.2.e.d.1935.1 ✓ 4
4.3 odd 2 inner 1936.2.e.d.1935.4 yes 4
11.10 odd 2 inner 1936.2.e.d.1935.2 yes 4
44.43 even 2 inner 1936.2.e.d.1935.3 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1936.2.e.d.1935.1 ✓ 4 1.1 even 1 trivial
1936.2.e.d.1935.2 yes 4 11.10 odd 2 inner
1936.2.e.d.1935.3 yes 4 44.43 even 2 inner
1936.2.e.d.1935.4 yes 4 4.3 odd 2 inner