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Properties

Label 920.2.a.f.1.1
Level $920$
Weight $2$
Character 920.1
Self dual yes
Analytic conductor $7.346$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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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] = [920,2,Mod(1,920)] 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("920.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(920, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 920 = 2^{3} \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 920.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(7.34623698596\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.56155\) of defining polynomial
Character \(\chi\) \(=\) 920.1

$q$-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)
 
\(f(q)\) \(=\) \(q-1.56155 q^{3} +1.00000 q^{5} -3.12311 q^{7} -0.561553 q^{9} -4.00000 q^{11} +3.56155 q^{13} -1.56155 q^{15} +5.12311 q^{17} +4.00000 q^{19} +4.87689 q^{21} +1.00000 q^{23} +1.00000 q^{25} +5.56155 q^{27} -4.43845 q^{29} +5.56155 q^{31} +6.24621 q^{33} -3.12311 q^{35} +1.12311 q^{37} -5.56155 q^{39} -3.56155 q^{41} -0.876894 q^{43} -0.561553 q^{45} +8.68466 q^{47} +2.75379 q^{49} -8.00000 q^{51} +12.2462 q^{53} -4.00000 q^{55} -6.24621 q^{57} +10.2462 q^{59} +2.87689 q^{61} +1.75379 q^{63} +3.56155 q^{65} -10.2462 q^{67} -1.56155 q^{69} -8.68466 q^{71} +12.4384 q^{73} -1.56155 q^{75} +12.4924 q^{77} +6.24621 q^{79} -7.00000 q^{81} +12.0000 q^{83} +5.12311 q^{85} +6.93087 q^{87} +10.0000 q^{89} -11.1231 q^{91} -8.68466 q^{93} +4.00000 q^{95} +0.246211 q^{97} +2.24621 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{3} + 2 q^{5} + 2 q^{7} + 3 q^{9} - 8 q^{11} + 3 q^{13} + q^{15} + 2 q^{17} + 8 q^{19} + 18 q^{21} + 2 q^{23} + 2 q^{25} + 7 q^{27} - 13 q^{29} + 7 q^{31} - 4 q^{33} + 2 q^{35} - 6 q^{37} - 7 q^{39}+ \cdots - 12 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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
\(3\) −1.56155 −0.901563 −0.450781 − 0.892634i \(-0.648855\pi\)
−0.450781 + 0.892634i \(0.648855\pi\)
\(4\) 0 0
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) −3.12311 −1.18042 −0.590211 − 0.807249i \(-0.700956\pi\)
−0.590211 + 0.807249i \(0.700956\pi\)
\(8\) 0 0
\(9\) −0.561553 −0.187184
\(10\) 0 0
\(11\) −4.00000 −1.20605 −0.603023 − 0.797724i \(-0.706037\pi\)
−0.603023 + 0.797724i \(0.706037\pi\)
\(12\) 0 0
\(13\) 3.56155 0.987797 0.493899 − 0.869520i \(-0.335571\pi\)
0.493899 + 0.869520i \(0.335571\pi\)
\(14\) 0 0
\(15\) −1.56155 −0.403191
\(16\) 0 0
\(17\) 5.12311 1.24254 0.621268 − 0.783598i \(-0.286618\pi\)
0.621268 + 0.783598i \(0.286618\pi\)
\(18\) 0 0
\(19\) 4.00000 0.917663 0.458831 − 0.888523i \(-0.348268\pi\)
0.458831 + 0.888523i \(0.348268\pi\)
\(20\) 0 0
\(21\) 4.87689 1.06423
\(22\) 0 0
\(23\) 1.00000 0.208514
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 5.56155 1.07032
\(28\) 0 0
\(29\) −4.43845 −0.824199 −0.412099 − 0.911139i \(-0.635204\pi\)
−0.412099 + 0.911139i \(0.635204\pi\)
\(30\) 0 0
\(31\) 5.56155 0.998884 0.499442 − 0.866347i \(-0.333538\pi\)
0.499442 + 0.866347i \(0.333538\pi\)
\(32\) 0 0
\(33\) 6.24621 1.08733
\(34\) 0 0
\(35\) −3.12311 −0.527901
\(36\) 0 0
\(37\) 1.12311 0.184637 0.0923187 − 0.995730i \(-0.470572\pi\)
0.0923187 + 0.995730i \(0.470572\pi\)
\(38\) 0 0
\(39\) −5.56155 −0.890561
\(40\) 0 0
\(41\) −3.56155 −0.556221 −0.278111 − 0.960549i \(-0.589708\pi\)
−0.278111 + 0.960549i \(0.589708\pi\)
\(42\) 0 0
\(43\) −0.876894 −0.133725 −0.0668626 − 0.997762i \(-0.521299\pi\)
−0.0668626 + 0.997762i \(0.521299\pi\)
\(44\) 0 0
\(45\) −0.561553 −0.0837114
\(46\) 0 0
\(47\) 8.68466 1.26679 0.633394 − 0.773830i \(-0.281661\pi\)
0.633394 + 0.773830i \(0.281661\pi\)
\(48\) 0 0
\(49\) 2.75379 0.393398
\(50\) 0 0
\(51\) −8.00000 −1.12022
\(52\) 0 0
\(53\) 12.2462 1.68215 0.841073 − 0.540921i \(-0.181924\pi\)
0.841073 + 0.540921i \(0.181924\pi\)
\(54\) 0 0
\(55\) −4.00000 −0.539360
\(56\) 0 0
\(57\) −6.24621 −0.827331
\(58\) 0 0
\(59\) 10.2462 1.33394 0.666972 − 0.745083i \(-0.267590\pi\)
0.666972 + 0.745083i \(0.267590\pi\)
\(60\) 0 0
\(61\) 2.87689 0.368349 0.184174 − 0.982894i \(-0.441039\pi\)
0.184174 + 0.982894i \(0.441039\pi\)
\(62\) 0 0
\(63\) 1.75379 0.220957
\(64\) 0 0
\(65\) 3.56155 0.441756
\(66\) 0 0
\(67\) −10.2462 −1.25177 −0.625887 − 0.779914i \(-0.715263\pi\)
−0.625887 + 0.779914i \(0.715263\pi\)
\(68\) 0 0
\(69\) −1.56155 −0.187989
\(70\) 0 0
\(71\) −8.68466 −1.03068 −0.515340 − 0.856986i \(-0.672334\pi\)
−0.515340 + 0.856986i \(0.672334\pi\)
\(72\) 0 0
\(73\) 12.4384 1.45581 0.727905 − 0.685678i \(-0.240494\pi\)
0.727905 + 0.685678i \(0.240494\pi\)
\(74\) 0 0
\(75\) −1.56155 −0.180313
\(76\) 0 0
\(77\) 12.4924 1.42364
\(78\) 0 0
\(79\) 6.24621 0.702754 0.351377 − 0.936234i \(-0.385714\pi\)
0.351377 + 0.936234i \(0.385714\pi\)
\(80\) 0 0
\(81\) −7.00000 −0.777778
\(82\) 0 0
\(83\) 12.0000 1.31717 0.658586 − 0.752506i \(-0.271155\pi\)
0.658586 + 0.752506i \(0.271155\pi\)
\(84\) 0 0
\(85\) 5.12311 0.555679
\(86\) 0 0
\(87\) 6.93087 0.743067
\(88\) 0 0
\(89\) 10.0000 1.06000 0.529999 − 0.847998i \(-0.322192\pi\)
0.529999 + 0.847998i \(0.322192\pi\)
\(90\) 0 0
\(91\) −11.1231 −1.16602
\(92\) 0 0
\(93\) −8.68466 −0.900557
\(94\) 0 0
\(95\) 4.00000 0.410391
\(96\) 0 0
\(97\) 0.246211 0.0249990 0.0124995 − 0.999922i \(-0.496021\pi\)
0.0124995 + 0.999922i \(0.496021\pi\)
\(98\) 0 0
\(99\) 2.24621 0.225753
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 920.2.a.f.1.1 ✓ 2
3.2 odd 2 8280.2.a.bb.1.1 2
4.3 odd 2 1840.2.a.k.1.2 2
5.2 odd 4 4600.2.e.m.4049.3 4
5.3 odd 4 4600.2.e.m.4049.2 4
5.4 even 2 4600.2.a.r.1.2 2
8.3 odd 2 7360.2.a.bm.1.1 2
8.5 even 2 7360.2.a.bj.1.2 2
20.19 odd 2 9200.2.a.bx.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
920.2.a.f.1.1 ✓ 2 1.1 even 1 trivial
1840.2.a.k.1.2 2 4.3 odd 2
4600.2.a.r.1.2 2 5.4 even 2
4600.2.e.m.4049.2 4 5.3 odd 4
4600.2.e.m.4049.3 4 5.2 odd 4
7360.2.a.bj.1.2 2 8.5 even 2
7360.2.a.bm.1.1 2 8.3 odd 2
8280.2.a.bb.1.1 2 3.2 odd 2
9200.2.a.bx.1.1 2 20.19 odd 2