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Properties

Label 790.2.a.f.1.4
Level $790$
Weight $2$
Character 790.1
Self dual yes
Analytic conductor $6.308$
Analytic rank $0$
Dimension $4$
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] = [790,2,Mod(1,790)] 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("790.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(790, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 790 = 2 \cdot 5 \cdot 79 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 790.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,-4,-2,4,4] 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: yes
Analytic conductor: \(6.30818175968\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.5744.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 5x^{2} - 2x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(-1.92022\) of defining polynomial
Character \(\chi\) \(=\) 790.1

$q$-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)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +2.60747 q^{3} +1.00000 q^{4} +1.00000 q^{5} -2.60747 q^{6} -0.833525 q^{7} -1.00000 q^{8} +3.79889 q^{9} -1.00000 q^{10} -3.84044 q^{11} +2.60747 q^{12} +0.833525 q^{14} +2.60747 q^{15} +1.00000 q^{16} +8.04846 q^{17} -3.79889 q^{18} +3.04155 q^{19} +1.00000 q^{20} -2.17339 q^{21} +3.84044 q^{22} +4.17339 q^{23} -2.60747 q^{24} +1.00000 q^{25} +2.08309 q^{27} -0.833525 q^{28} +8.57284 q^{29} -2.60747 q^{30} +3.04155 q^{31} -1.00000 q^{32} -10.0138 q^{33} -8.04846 q^{34} -0.833525 q^{35} +3.79889 q^{36} +5.40636 q^{37} -3.04155 q^{38} -1.00000 q^{40} +6.00000 q^{41} +2.17339 q^{42} -8.69056 q^{43} -3.84044 q^{44} +3.79889 q^{45} -4.17339 q^{46} -9.97920 q^{47} +2.60747 q^{48} -6.30524 q^{49} -1.00000 q^{50} +20.9861 q^{51} +1.05958 q^{53} -2.08309 q^{54} -3.84044 q^{55} +0.833525 q^{56} +7.93074 q^{57} -8.57284 q^{58} -10.3230 q^{59} +2.60747 q^{60} -10.5867 q^{61} -3.04155 q^{62} -3.16647 q^{63} +1.00000 q^{64} +10.0138 q^{66} +0.332950 q^{67} +8.04846 q^{68} +10.8820 q^{69} +0.833525 q^{70} +2.78506 q^{71} -3.79889 q^{72} -11.3883 q^{73} -5.40636 q^{74} +2.60747 q^{75} +3.04155 q^{76} +3.20111 q^{77} -1.00000 q^{79} +1.00000 q^{80} -5.96508 q^{81} -6.00000 q^{82} -5.54789 q^{83} -2.17339 q^{84} +8.04846 q^{85} +8.69056 q^{86} +22.3534 q^{87} +3.84044 q^{88} -8.47978 q^{89} -3.79889 q^{90} +4.17339 q^{92} +7.93074 q^{93} +9.97920 q^{94} +3.04155 q^{95} -2.60747 q^{96} -5.75014 q^{97} +6.30524 q^{98} -14.5894 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{2} - 2 q^{3} + 4 q^{4} + 4 q^{5} + 2 q^{6} - 4 q^{7} - 4 q^{8} + 8 q^{9} - 4 q^{10} - 2 q^{12} + 4 q^{14} - 2 q^{15} + 4 q^{16} + 8 q^{17} - 8 q^{18} + 4 q^{19} + 4 q^{20} + 8 q^{21} + 2 q^{24}+ \cdots - 24 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) 2.60747 1.50542 0.752711 − 0.658351i \(-0.228746\pi\)
0.752711 + 0.658351i \(0.228746\pi\)
\(4\) 1.00000 0.500000
\(5\) 1.00000 0.447214
\(6\) −2.60747 −1.06449
\(7\) −0.833525 −0.315043 −0.157521 − 0.987516i \(-0.550350\pi\)
−0.157521 + 0.987516i \(0.550350\pi\)
\(8\) −1.00000 −0.353553
\(9\) 3.79889 1.26630
\(10\) −1.00000 −0.316228
\(11\) −3.84044 −1.15794 −0.578968 − 0.815350i \(-0.696545\pi\)
−0.578968 + 0.815350i \(0.696545\pi\)
\(12\) 2.60747 0.752711
\(13\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(14\) 0.833525 0.222769
\(15\) 2.60747 0.673246
\(16\) 1.00000 0.250000
\(17\) 8.04846 1.95204 0.976020 − 0.217683i \(-0.0698499\pi\)
0.976020 + 0.217683i \(0.0698499\pi\)
\(18\) −3.79889 −0.895408
\(19\) 3.04155 0.697779 0.348889 − 0.937164i \(-0.386559\pi\)
0.348889 + 0.937164i \(0.386559\pi\)
\(20\) 1.00000 0.223607
\(21\) −2.17339 −0.474273
\(22\) 3.84044 0.818785
\(23\) 4.17339 0.870212 0.435106 − 0.900379i \(-0.356711\pi\)
0.435106 + 0.900379i \(0.356711\pi\)
\(24\) −2.60747 −0.532247
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 2.08309 0.400892
\(28\) −0.833525 −0.157521
\(29\) 8.57284 1.59194 0.795968 − 0.605339i \(-0.206962\pi\)
0.795968 + 0.605339i \(0.206962\pi\)
\(30\) −2.60747 −0.476057
\(31\) 3.04155 0.546278 0.273139 − 0.961975i \(-0.411938\pi\)
0.273139 + 0.961975i \(0.411938\pi\)
\(32\) −1.00000 −0.176777
\(33\) −10.0138 −1.74318
\(34\) −8.04846 −1.38030
\(35\) −0.833525 −0.140891
\(36\) 3.79889 0.633149
\(37\) 5.40636 0.888801 0.444400 − 0.895828i \(-0.353417\pi\)
0.444400 + 0.895828i \(0.353417\pi\)
\(38\) −3.04155 −0.493404
\(39\) 0 0
\(40\) −1.00000 −0.158114
\(41\) 6.00000 0.937043 0.468521 − 0.883452i \(-0.344787\pi\)
0.468521 + 0.883452i \(0.344787\pi\)
\(42\) 2.17339 0.335362
\(43\) −8.69056 −1.32530 −0.662649 − 0.748930i \(-0.730568\pi\)
−0.662649 + 0.748930i \(0.730568\pi\)
\(44\) −3.84044 −0.578968
\(45\) 3.79889 0.566306
\(46\) −4.17339 −0.615333
\(47\) −9.97920 −1.45562 −0.727808 − 0.685781i \(-0.759461\pi\)
−0.727808 + 0.685781i \(0.759461\pi\)
\(48\) 2.60747 0.376356
\(49\) −6.30524 −0.900748
\(50\) −1.00000 −0.141421
\(51\) 20.9861 2.93864
\(52\) 0 0
\(53\) 1.05958 0.145545 0.0727724 − 0.997349i \(-0.476815\pi\)
0.0727724 + 0.997349i \(0.476815\pi\)
\(54\) −2.08309 −0.283473
\(55\) −3.84044 −0.517845
\(56\) 0.833525 0.111384
\(57\) 7.93074 1.05045
\(58\) −8.57284 −1.12567
\(59\) −10.3230 −1.34394 −0.671969 − 0.740579i \(-0.734551\pi\)
−0.671969 + 0.740579i \(0.734551\pi\)
\(60\) 2.60747 0.336623
\(61\) −10.5867 −1.35548 −0.677742 − 0.735300i \(-0.737041\pi\)
−0.677742 + 0.735300i \(0.737041\pi\)
\(62\) −3.04155 −0.386277
\(63\) −3.16647 −0.398938
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 10.0138 1.23262
\(67\) 0.332950 0.0406763 0.0203381 − 0.999793i \(-0.493526\pi\)
0.0203381 + 0.999793i \(0.493526\pi\)
\(68\) 8.04846 0.976020
\(69\) 10.8820 1.31004
\(70\) 0.833525 0.0996253
\(71\) 2.78506 0.330526 0.165263 − 0.986250i \(-0.447153\pi\)
0.165263 + 0.986250i \(0.447153\pi\)
\(72\) −3.79889 −0.447704
\(73\) −11.3883 −1.33290 −0.666452 − 0.745548i \(-0.732188\pi\)
−0.666452 + 0.745548i \(0.732188\pi\)
\(74\) −5.40636 −0.628477
\(75\) 2.60747 0.301085
\(76\) 3.04155 0.348889
\(77\) 3.20111 0.364800
\(78\) 0 0
\(79\) −1.00000 −0.112509
\(80\) 1.00000 0.111803
\(81\) −5.96508 −0.662787
\(82\) −6.00000 −0.662589
\(83\) −5.54789 −0.608960 −0.304480 − 0.952519i \(-0.598483\pi\)
−0.304480 + 0.952519i \(0.598483\pi\)
\(84\) −2.17339 −0.237136
\(85\) 8.04846 0.872978
\(86\) 8.69056 0.937127
\(87\) 22.3534 2.39654
\(88\) 3.84044 0.409392
\(89\) −8.47978 −0.898855 −0.449427 − 0.893317i \(-0.648372\pi\)
−0.449427 + 0.893317i \(0.648372\pi\)
\(90\) −3.79889 −0.400439
\(91\) 0 0
\(92\) 4.17339 0.435106
\(93\) 7.93074 0.822379
\(94\) 9.97920 1.02928
\(95\) 3.04155 0.312056
\(96\) −2.60747 −0.266124
\(97\) −5.75014 −0.583839 −0.291919 − 0.956443i \(-0.594294\pi\)
−0.291919 + 0.956443i \(0.594294\pi\)
\(98\) 6.30524 0.636925
\(99\) −14.5894 −1.46629
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 790.2.a.f.1.4 ✓ 4
3.2 odd 2 7110.2.a.bq.1.2 4
4.3 odd 2 6320.2.a.x.1.1 4
5.4 even 2 3950.2.a.u.1.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
790.2.a.f.1.4 ✓ 4 1.1 even 1 trivial
3950.2.a.u.1.1 4 5.4 even 2
6320.2.a.x.1.1 4 4.3 odd 2
7110.2.a.bq.1.2 4 3.2 odd 2