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Properties

Label 790.2.a.g.1.3
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 1.3
Root \(-0.751024\) 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)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.58049 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.58049 q^{6} +4.76748 q^{7} +1.00000 q^{8} -0.502048 q^{9} -1.00000 q^{10} +2.00000 q^{11} +1.58049 q^{12} -6.03291 q^{13} +4.76748 q^{14} -1.58049 q^{15} +1.00000 q^{16} +6.76748 q^{17} -0.502048 q^{18} -1.36988 q^{19} -1.00000 q^{20} +7.53496 q^{21} +2.00000 q^{22} -2.66303 q^{23} +1.58049 q^{24} +1.00000 q^{25} -6.03291 q^{26} -5.53496 q^{27} +4.76748 q^{28} +1.81711 q^{29} -1.58049 q^{30} -1.79110 q^{31} +1.00000 q^{32} +3.16098 q^{33} +6.76748 q^{34} -4.76748 q^{35} -0.502048 q^{36} +3.78939 q^{37} -1.36988 q^{38} -9.53496 q^{39} -1.00000 q^{40} -4.53086 q^{41} +7.53496 q^{42} -1.21471 q^{43} +2.00000 q^{44} +0.502048 q^{45} -2.66303 q^{46} -10.4593 q^{47} +1.58049 q^{48} +15.7288 q^{49} +1.00000 q^{50} +10.6959 q^{51} -6.03291 q^{52} -0.419509 q^{53} -5.53496 q^{54} -2.00000 q^{55} +4.76748 q^{56} -2.16508 q^{57} +1.81711 q^{58} +8.87883 q^{59} -1.58049 q^{60} -2.81301 q^{61} -1.79110 q^{62} -2.39350 q^{63} +1.00000 q^{64} +6.03291 q^{65} +3.16098 q^{66} -10.5638 q^{67} +6.76748 q^{68} -4.20890 q^{69} -4.76748 q^{70} +6.37397 q^{71} -0.502048 q^{72} +9.56786 q^{73} +3.78939 q^{74} +1.58049 q^{75} -1.36988 q^{76} +9.53496 q^{77} -9.53496 q^{78} -1.00000 q^{79} -1.00000 q^{80} -7.24180 q^{81} -4.53086 q^{82} -8.39869 q^{83} +7.53496 q^{84} -6.76748 q^{85} -1.21471 q^{86} +2.87193 q^{87} +2.00000 q^{88} +3.23771 q^{89} +0.502048 q^{90} -28.7618 q^{91} -2.66303 q^{92} -2.83082 q^{93} -10.4593 q^{94} +1.36988 q^{95} +1.58049 q^{96} -14.2747 q^{97} +15.7288 q^{98} -1.00410 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{2} + 2 q^{3} + 4 q^{4} - 4 q^{5} + 2 q^{6} + 4 q^{7} + 4 q^{8} + 4 q^{9} - 4 q^{10} + 8 q^{11} + 2 q^{12} + 4 q^{14} - 2 q^{15} + 4 q^{16} + 12 q^{17} + 4 q^{18} + 4 q^{19} - 4 q^{20} + 8 q^{22}+ \cdots + 8 q^{99}+O(q^{100}) \) 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
\(3\) 1.58049 0.912497 0.456248 − 0.889852i \(-0.349193\pi\)
0.456248 + 0.889852i \(0.349193\pi\)
\(4\) 1.00000 0.500000
\(5\) −1.00000 −0.447214
\(6\) 1.58049 0.645233
\(7\) 4.76748 1.80194 0.900969 − 0.433884i \(-0.142857\pi\)
0.900969 + 0.433884i \(0.142857\pi\)
\(8\) 1.00000 0.353553
\(9\) −0.502048 −0.167349
\(10\) −1.00000 −0.316228
\(11\) 2.00000 0.603023 0.301511 − 0.953463i \(-0.402509\pi\)
0.301511 + 0.953463i \(0.402509\pi\)
\(12\) 1.58049 0.456248
\(13\) −6.03291 −1.67323 −0.836614 − 0.547793i \(-0.815468\pi\)
−0.836614 + 0.547793i \(0.815468\pi\)
\(14\) 4.76748 1.27416
\(15\) −1.58049 −0.408081
\(16\) 1.00000 0.250000
\(17\) 6.76748 1.64135 0.820677 − 0.571392i \(-0.193596\pi\)
0.820677 + 0.571392i \(0.193596\pi\)
\(18\) −0.502048 −0.118334
\(19\) −1.36988 −0.314271 −0.157136 − 0.987577i \(-0.550226\pi\)
−0.157136 + 0.987577i \(0.550226\pi\)
\(20\) −1.00000 −0.223607
\(21\) 7.53496 1.64426
\(22\) 2.00000 0.426401
\(23\) −2.66303 −0.555280 −0.277640 − 0.960685i \(-0.589552\pi\)
−0.277640 + 0.960685i \(0.589552\pi\)
\(24\) 1.58049 0.322616
\(25\) 1.00000 0.200000
\(26\) −6.03291 −1.18315
\(27\) −5.53496 −1.06520
\(28\) 4.76748 0.900969
\(29\) 1.81711 0.337429 0.168714 − 0.985665i \(-0.446038\pi\)
0.168714 + 0.985665i \(0.446038\pi\)
\(30\) −1.58049 −0.288557
\(31\) −1.79110 −0.321692 −0.160846 − 0.986980i \(-0.551422\pi\)
−0.160846 + 0.986980i \(0.551422\pi\)
\(32\) 1.00000 0.176777
\(33\) 3.16098 0.550256
\(34\) 6.76748 1.16061
\(35\) −4.76748 −0.805851
\(36\) −0.502048 −0.0836747
\(37\) 3.78939 0.622971 0.311486 − 0.950251i \(-0.399173\pi\)
0.311486 + 0.950251i \(0.399173\pi\)
\(38\) −1.36988 −0.222223
\(39\) −9.53496 −1.52681
\(40\) −1.00000 −0.158114
\(41\) −4.53086 −0.707601 −0.353801 − 0.935321i \(-0.615111\pi\)
−0.353801 + 0.935321i \(0.615111\pi\)
\(42\) 7.53496 1.16267
\(43\) −1.21471 −0.185242 −0.0926208 − 0.995701i \(-0.529524\pi\)
−0.0926208 + 0.995701i \(0.529524\pi\)
\(44\) 2.00000 0.301511
\(45\) 0.502048 0.0748409
\(46\) −2.66303 −0.392642
\(47\) −10.4593 −1.52565 −0.762824 − 0.646606i \(-0.776188\pi\)
−0.762824 + 0.646606i \(0.776188\pi\)
\(48\) 1.58049 0.228124
\(49\) 15.7288 2.24698
\(50\) 1.00000 0.141421
\(51\) 10.6959 1.49773
\(52\) −6.03291 −0.836614
\(53\) −0.419509 −0.0576240 −0.0288120 − 0.999585i \(-0.509172\pi\)
−0.0288120 + 0.999585i \(0.509172\pi\)
\(54\) −5.53496 −0.753212
\(55\) −2.00000 −0.269680
\(56\) 4.76748 0.637081
\(57\) −2.16508 −0.286772
\(58\) 1.81711 0.238598
\(59\) 8.87883 1.15592 0.577962 − 0.816063i \(-0.303848\pi\)
0.577962 + 0.816063i \(0.303848\pi\)
\(60\) −1.58049 −0.204041
\(61\) −2.81301 −0.360169 −0.180085 − 0.983651i \(-0.557637\pi\)
−0.180085 + 0.983651i \(0.557637\pi\)
\(62\) −1.79110 −0.227471
\(63\) −2.39350 −0.301553
\(64\) 1.00000 0.125000
\(65\) 6.03291 0.748290
\(66\) 3.16098 0.389090
\(67\) −10.5638 −1.29057 −0.645285 − 0.763942i \(-0.723261\pi\)
−0.645285 + 0.763942i \(0.723261\pi\)
\(68\) 6.76748 0.820677
\(69\) −4.20890 −0.506691
\(70\) −4.76748 −0.569823
\(71\) 6.37397 0.756452 0.378226 − 0.925713i \(-0.376534\pi\)
0.378226 + 0.925713i \(0.376534\pi\)
\(72\) −0.502048 −0.0591670
\(73\) 9.56786 1.11983 0.559917 − 0.828549i \(-0.310833\pi\)
0.559917 + 0.828549i \(0.310833\pi\)
\(74\) 3.78939 0.440507
\(75\) 1.58049 0.182499
\(76\) −1.36988 −0.157136
\(77\) 9.53496 1.08661
\(78\) −9.53496 −1.07962
\(79\) −1.00000 −0.112509
\(80\) −1.00000 −0.111803
\(81\) −7.24180 −0.804645
\(82\) −4.53086 −0.500350
\(83\) −8.39869 −0.921876 −0.460938 − 0.887432i \(-0.652487\pi\)
−0.460938 + 0.887432i \(0.652487\pi\)
\(84\) 7.53496 0.822131
\(85\) −6.76748 −0.734036
\(86\) −1.21471 −0.130986
\(87\) 2.87193 0.307903
\(88\) 2.00000 0.213201
\(89\) 3.23771 0.343196 0.171598 − 0.985167i \(-0.445107\pi\)
0.171598 + 0.985167i \(0.445107\pi\)
\(90\) 0.502048 0.0529205
\(91\) −28.7618 −3.01505
\(92\) −2.66303 −0.277640
\(93\) −2.83082 −0.293543
\(94\) −10.4593 −1.07880
\(95\) 1.36988 0.140546
\(96\) 1.58049 0.161308
\(97\) −14.2747 −1.44938 −0.724689 − 0.689077i \(-0.758016\pi\)
−0.724689 + 0.689077i \(0.758016\pi\)
\(98\) 15.7288 1.58885
\(99\) −1.00410 −0.100916
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.g.1.3 ✓ 4
3.2 odd 2 7110.2.a.bp.1.4 4
4.3 odd 2 6320.2.a.t.1.2 4
5.4 even 2 3950.2.a.o.1.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
790.2.a.g.1.3 ✓ 4 1.1 even 1 trivial
3950.2.a.o.1.2 4 5.4 even 2
6320.2.a.t.1.2 4 4.3 odd 2
7110.2.a.bp.1.4 4 3.2 odd 2