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Properties

Label 1045.2.a.g.1.4
Level $1045$
Weight $2$
Character 1045.1
Self dual yes
Analytic conductor $8.344$
Analytic rank $0$
Dimension $6$
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] = [1045,2,Mod(1,1045)] 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("1045.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1045, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1045 = 5 \cdot 11 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1045.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6,0,3,8,6] 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: \(8.34436701122\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.6.131947641.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 10x^{4} + 25x^{2} - 3x - 9 \) 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 1.4
Root \(0.759131\) of defining polynomial
Character \(\chi\) \(=\) 1045.1

$q$-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)
 
\(f(q)\) \(=\) \(q+0.759131 q^{2} -2.25829 q^{3} -1.42372 q^{4} +1.00000 q^{5} -1.71434 q^{6} +4.95189 q^{7} -2.59905 q^{8} +2.09989 q^{9} +0.759131 q^{10} -1.00000 q^{11} +3.21518 q^{12} +0.499163 q^{13} +3.75913 q^{14} -2.25829 q^{15} +0.874419 q^{16} -4.34828 q^{17} +1.59409 q^{18} -1.00000 q^{19} -1.42372 q^{20} -11.1828 q^{21} -0.759131 q^{22} -2.78646 q^{23} +5.86942 q^{24} +1.00000 q^{25} +0.378930 q^{26} +2.03271 q^{27} -7.05010 q^{28} +8.84908 q^{29} -1.71434 q^{30} +1.67449 q^{31} +5.86190 q^{32} +2.25829 q^{33} -3.30091 q^{34} +4.95189 q^{35} -2.98966 q^{36} -1.81215 q^{37} -0.759131 q^{38} -1.12726 q^{39} -2.59905 q^{40} +10.5406 q^{41} -8.48922 q^{42} +2.65924 q^{43} +1.42372 q^{44} +2.09989 q^{45} -2.11529 q^{46} +11.6115 q^{47} -1.97470 q^{48} +17.5212 q^{49} +0.759131 q^{50} +9.81969 q^{51} -0.710668 q^{52} +10.6541 q^{53} +1.54310 q^{54} -1.00000 q^{55} -12.8702 q^{56} +2.25829 q^{57} +6.71761 q^{58} +10.0174 q^{59} +3.21518 q^{60} -12.5990 q^{61} +1.27116 q^{62} +10.3984 q^{63} +2.70111 q^{64} +0.499163 q^{65} +1.71434 q^{66} +2.84997 q^{67} +6.19073 q^{68} +6.29265 q^{69} +3.75913 q^{70} -13.3351 q^{71} -5.45772 q^{72} -13.2131 q^{73} -1.37566 q^{74} -2.25829 q^{75} +1.42372 q^{76} -4.95189 q^{77} -0.855735 q^{78} -7.08995 q^{79} +0.874419 q^{80} -10.8901 q^{81} +8.00173 q^{82} +15.1160 q^{83} +15.9212 q^{84} -4.34828 q^{85} +2.01871 q^{86} -19.9838 q^{87} +2.59905 q^{88} -7.25826 q^{89} +1.59409 q^{90} +2.47180 q^{91} +3.96714 q^{92} -3.78150 q^{93} +8.81467 q^{94} -1.00000 q^{95} -13.2379 q^{96} +0.0194069 q^{97} +13.3009 q^{98} -2.09989 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 3 q^{3} + 8 q^{4} + 6 q^{5} + 2 q^{6} + 5 q^{7} + 9 q^{9} - 6 q^{11} + 19 q^{12} - 9 q^{13} + 18 q^{14} + 3 q^{15} + 4 q^{16} - 5 q^{17} + 2 q^{18} - 6 q^{19} + 8 q^{20} + 3 q^{21} + 8 q^{23} - 7 q^{24}+ \cdots - 9 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.759131 0.536787 0.268393 − 0.963309i \(-0.413507\pi\)
0.268393 + 0.963309i \(0.413507\pi\)
\(3\) −2.25829 −1.30383 −0.651913 − 0.758294i \(-0.726033\pi\)
−0.651913 + 0.758294i \(0.726033\pi\)
\(4\) −1.42372 −0.711860
\(5\) 1.00000 0.447214
\(6\) −1.71434 −0.699877
\(7\) 4.95189 1.87164 0.935819 − 0.352482i \(-0.114662\pi\)
0.935819 + 0.352482i \(0.114662\pi\)
\(8\) −2.59905 −0.918904
\(9\) 2.09989 0.699963
\(10\) 0.759131 0.240058
\(11\) −1.00000 −0.301511
\(12\) 3.21518 0.928142
\(13\) 0.499163 0.138443 0.0692214 − 0.997601i \(-0.477949\pi\)
0.0692214 + 0.997601i \(0.477949\pi\)
\(14\) 3.75913 1.00467
\(15\) −2.25829 −0.583089
\(16\) 0.874419 0.218605
\(17\) −4.34828 −1.05461 −0.527306 − 0.849675i \(-0.676798\pi\)
−0.527306 + 0.849675i \(0.676798\pi\)
\(18\) 1.59409 0.375731
\(19\) −1.00000 −0.229416
\(20\) −1.42372 −0.318354
\(21\) −11.1828 −2.44029
\(22\) −0.759131 −0.161847
\(23\) −2.78646 −0.581017 −0.290509 − 0.956872i \(-0.593824\pi\)
−0.290509 + 0.956872i \(0.593824\pi\)
\(24\) 5.86942 1.19809
\(25\) 1.00000 0.200000
\(26\) 0.378930 0.0743142
\(27\) 2.03271 0.391196
\(28\) −7.05010 −1.33234
\(29\) 8.84908 1.64323 0.821616 − 0.570041i \(-0.193073\pi\)
0.821616 + 0.570041i \(0.193073\pi\)
\(30\) −1.71434 −0.312994
\(31\) 1.67449 0.300748 0.150374 − 0.988629i \(-0.451952\pi\)
0.150374 + 0.988629i \(0.451952\pi\)
\(32\) 5.86190 1.03625
\(33\) 2.25829 0.393118
\(34\) −3.30091 −0.566102
\(35\) 4.95189 0.837022
\(36\) −2.98966 −0.498276
\(37\) −1.81215 −0.297916 −0.148958 − 0.988844i \(-0.547592\pi\)
−0.148958 + 0.988844i \(0.547592\pi\)
\(38\) −0.759131 −0.123147
\(39\) −1.12726 −0.180505
\(40\) −2.59905 −0.410946
\(41\) 10.5406 1.64617 0.823086 − 0.567916i \(-0.192250\pi\)
0.823086 + 0.567916i \(0.192250\pi\)
\(42\) −8.48922 −1.30992
\(43\) 2.65924 0.405531 0.202765 − 0.979227i \(-0.435007\pi\)
0.202765 + 0.979227i \(0.435007\pi\)
\(44\) 1.42372 0.214634
\(45\) 2.09989 0.313033
\(46\) −2.11529 −0.311882
\(47\) 11.6115 1.69372 0.846858 − 0.531819i \(-0.178491\pi\)
0.846858 + 0.531819i \(0.178491\pi\)
\(48\) −1.97470 −0.285023
\(49\) 17.5212 2.50303
\(50\) 0.759131 0.107357
\(51\) 9.81969 1.37503
\(52\) −0.710668 −0.0985519
\(53\) 10.6541 1.46345 0.731726 − 0.681598i \(-0.238715\pi\)
0.731726 + 0.681598i \(0.238715\pi\)
\(54\) 1.54310 0.209989
\(55\) −1.00000 −0.134840
\(56\) −12.8702 −1.71985
\(57\) 2.25829 0.299118
\(58\) 6.71761 0.882065
\(59\) 10.0174 1.30416 0.652079 − 0.758151i \(-0.273897\pi\)
0.652079 + 0.758151i \(0.273897\pi\)
\(60\) 3.21518 0.415078
\(61\) −12.5990 −1.61314 −0.806569 − 0.591140i \(-0.798678\pi\)
−0.806569 + 0.591140i \(0.798678\pi\)
\(62\) 1.27116 0.161438
\(63\) 10.3984 1.31008
\(64\) 2.70111 0.337639
\(65\) 0.499163 0.0619135
\(66\) 1.71434 0.211021
\(67\) 2.84997 0.348180 0.174090 − 0.984730i \(-0.444302\pi\)
0.174090 + 0.984730i \(0.444302\pi\)
\(68\) 6.19073 0.750736
\(69\) 6.29265 0.757546
\(70\) 3.75913 0.449302
\(71\) −13.3351 −1.58258 −0.791291 − 0.611440i \(-0.790590\pi\)
−0.791291 + 0.611440i \(0.790590\pi\)
\(72\) −5.45772 −0.643199
\(73\) −13.2131 −1.54647 −0.773236 − 0.634118i \(-0.781363\pi\)
−0.773236 + 0.634118i \(0.781363\pi\)
\(74\) −1.37566 −0.159917
\(75\) −2.25829 −0.260765
\(76\) 1.42372 0.163312
\(77\) −4.95189 −0.564320
\(78\) −0.855735 −0.0968929
\(79\) −7.08995 −0.797681 −0.398841 − 0.917020i \(-0.630587\pi\)
−0.398841 + 0.917020i \(0.630587\pi\)
\(80\) 0.874419 0.0977631
\(81\) −10.8901 −1.21001
\(82\) 8.00173 0.883643
\(83\) 15.1160 1.65920 0.829600 − 0.558359i \(-0.188569\pi\)
0.829600 + 0.558359i \(0.188569\pi\)
\(84\) 15.9212 1.73715
\(85\) −4.34828 −0.471637
\(86\) 2.01871 0.217683
\(87\) −19.9838 −2.14249
\(88\) 2.59905 0.277060
\(89\) −7.25826 −0.769374 −0.384687 − 0.923047i \(-0.625691\pi\)
−0.384687 + 0.923047i \(0.625691\pi\)
\(90\) 1.59409 0.168032
\(91\) 2.47180 0.259115
\(92\) 3.96714 0.413603
\(93\) −3.78150 −0.392123
\(94\) 8.81467 0.909164
\(95\) −1.00000 −0.102598
\(96\) −13.2379 −1.35109
\(97\) 0.0194069 0.00197047 0.000985234 − 1.00000i \(-0.499686\pi\)
0.000985234 1.00000i \(0.499686\pi\)
\(98\) 13.3009 1.34359
\(99\) −2.09989 −0.211047
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 1045.2.a.g.1.4 ✓ 6
3.2 odd 2 9405.2.a.w.1.3 6
5.4 even 2 5225.2.a.k.1.3 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1045.2.a.g.1.4 ✓ 6 1.1 even 1 trivial
5225.2.a.k.1.3 6 5.4 even 2
9405.2.a.w.1.3 6 3.2 odd 2