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Properties

Label 9405.2.a.w.1.3
Level $9405$
Weight $2$
Character 9405.1
Self dual yes
Analytic conductor $75.099$
Analytic rank $1$
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] = [9405,2,Mod(1,9405)] 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("9405.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(9405, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 9405 = 3^{2} \cdot 5 \cdot 11 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9405.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6,0,0,8,-6,0,5,0,0,0,6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(75.0993031010\)
Analytic rank: \(1\)
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: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1045)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(0.759131\) of defining polynomial
Character \(\chi\) \(=\) 9405.1

$q$-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)
 
\(f(q)\) \(=\) \(q-0.759131 q^{2} -1.42372 q^{4} -1.00000 q^{5} +4.95189 q^{7} +2.59905 q^{8} +0.759131 q^{10} +1.00000 q^{11} +0.499163 q^{13} -3.75913 q^{14} +0.874419 q^{16} +4.34828 q^{17} -1.00000 q^{19} +1.42372 q^{20} -0.759131 q^{22} +2.78646 q^{23} +1.00000 q^{25} -0.378930 q^{26} -7.05010 q^{28} -8.84908 q^{29} +1.67449 q^{31} -5.86190 q^{32} -3.30091 q^{34} -4.95189 q^{35} -1.81215 q^{37} +0.759131 q^{38} -2.59905 q^{40} -10.5406 q^{41} +2.65924 q^{43} -1.42372 q^{44} -2.11529 q^{46} -11.6115 q^{47} +17.5212 q^{49} -0.759131 q^{50} -0.710668 q^{52} -10.6541 q^{53} -1.00000 q^{55} +12.8702 q^{56} +6.71761 q^{58} -10.0174 q^{59} -12.5990 q^{61} -1.27116 q^{62} +2.70111 q^{64} -0.499163 q^{65} +2.84997 q^{67} -6.19073 q^{68} +3.75913 q^{70} +13.3351 q^{71} -13.2131 q^{73} +1.37566 q^{74} +1.42372 q^{76} +4.95189 q^{77} -7.08995 q^{79} -0.874419 q^{80} +8.00173 q^{82} -15.1160 q^{83} -4.34828 q^{85} -2.01871 q^{86} +2.59905 q^{88} +7.25826 q^{89} +2.47180 q^{91} -3.96714 q^{92} +8.81467 q^{94} +1.00000 q^{95} +0.0194069 q^{97} -13.3009 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 8 q^{4} - 6 q^{5} + 5 q^{7} + 6 q^{11} - 9 q^{13} - 18 q^{14} + 4 q^{16} + 5 q^{17} - 6 q^{19} - 8 q^{20} - 8 q^{23} + 6 q^{25} + 22 q^{26} + 10 q^{28} + 5 q^{29} - q^{31} - 15 q^{32} - 22 q^{34}+ \cdots - 33 q^{98}+O(q^{100}) \) 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 \(-0.586493\pi\)
−0.268393 + 0.963309i \(0.586493\pi\)
\(3\) 0 0
\(4\) −1.42372 −0.711860
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) 4.95189 1.87164 0.935819 − 0.352482i \(-0.114662\pi\)
0.935819 + 0.352482i \(0.114662\pi\)
\(8\) 2.59905 0.918904
\(9\) 0 0
\(10\) 0.759131 0.240058
\(11\) 1.00000 0.301511
\(12\) 0 0
\(13\) 0.499163 0.138443 0.0692214 − 0.997601i \(-0.477949\pi\)
0.0692214 + 0.997601i \(0.477949\pi\)
\(14\) −3.75913 −1.00467
\(15\) 0 0
\(16\) 0.874419 0.218605
\(17\) 4.34828 1.05461 0.527306 − 0.849675i \(-0.323202\pi\)
0.527306 + 0.849675i \(0.323202\pi\)
\(18\) 0 0
\(19\) −1.00000 −0.229416
\(20\) 1.42372 0.318354
\(21\) 0 0
\(22\) −0.759131 −0.161847
\(23\) 2.78646 0.581017 0.290509 − 0.956872i \(-0.406176\pi\)
0.290509 + 0.956872i \(0.406176\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) −0.378930 −0.0743142
\(27\) 0 0
\(28\) −7.05010 −1.33234
\(29\) −8.84908 −1.64323 −0.821616 − 0.570041i \(-0.806927\pi\)
−0.821616 + 0.570041i \(0.806927\pi\)
\(30\) 0 0
\(31\) 1.67449 0.300748 0.150374 − 0.988629i \(-0.451952\pi\)
0.150374 + 0.988629i \(0.451952\pi\)
\(32\) −5.86190 −1.03625
\(33\) 0 0
\(34\) −3.30091 −0.566102
\(35\) −4.95189 −0.837022
\(36\) 0 0
\(37\) −1.81215 −0.297916 −0.148958 − 0.988844i \(-0.547592\pi\)
−0.148958 + 0.988844i \(0.547592\pi\)
\(38\) 0.759131 0.123147
\(39\) 0 0
\(40\) −2.59905 −0.410946
\(41\) −10.5406 −1.64617 −0.823086 − 0.567916i \(-0.807750\pi\)
−0.823086 + 0.567916i \(0.807750\pi\)
\(42\) 0 0
\(43\) 2.65924 0.405531 0.202765 − 0.979227i \(-0.435007\pi\)
0.202765 + 0.979227i \(0.435007\pi\)
\(44\) −1.42372 −0.214634
\(45\) 0 0
\(46\) −2.11529 −0.311882
\(47\) −11.6115 −1.69372 −0.846858 − 0.531819i \(-0.821509\pi\)
−0.846858 + 0.531819i \(0.821509\pi\)
\(48\) 0 0
\(49\) 17.5212 2.50303
\(50\) −0.759131 −0.107357
\(51\) 0 0
\(52\) −0.710668 −0.0985519
\(53\) −10.6541 −1.46345 −0.731726 − 0.681598i \(-0.761285\pi\)
−0.731726 + 0.681598i \(0.761285\pi\)
\(54\) 0 0
\(55\) −1.00000 −0.134840
\(56\) 12.8702 1.71985
\(57\) 0 0
\(58\) 6.71761 0.882065
\(59\) −10.0174 −1.30416 −0.652079 − 0.758151i \(-0.726103\pi\)
−0.652079 + 0.758151i \(0.726103\pi\)
\(60\) 0 0
\(61\) −12.5990 −1.61314 −0.806569 − 0.591140i \(-0.798678\pi\)
−0.806569 + 0.591140i \(0.798678\pi\)
\(62\) −1.27116 −0.161438
\(63\) 0 0
\(64\) 2.70111 0.337639
\(65\) −0.499163 −0.0619135
\(66\) 0 0
\(67\) 2.84997 0.348180 0.174090 − 0.984730i \(-0.444302\pi\)
0.174090 + 0.984730i \(0.444302\pi\)
\(68\) −6.19073 −0.750736
\(69\) 0 0
\(70\) 3.75913 0.449302
\(71\) 13.3351 1.58258 0.791291 − 0.611440i \(-0.209410\pi\)
0.791291 + 0.611440i \(0.209410\pi\)
\(72\) 0 0
\(73\) −13.2131 −1.54647 −0.773236 − 0.634118i \(-0.781363\pi\)
−0.773236 + 0.634118i \(0.781363\pi\)
\(74\) 1.37566 0.159917
\(75\) 0 0
\(76\) 1.42372 0.163312
\(77\) 4.95189 0.564320
\(78\) 0 0
\(79\) −7.08995 −0.797681 −0.398841 − 0.917020i \(-0.630587\pi\)
−0.398841 + 0.917020i \(0.630587\pi\)
\(80\) −0.874419 −0.0977631
\(81\) 0 0
\(82\) 8.00173 0.883643
\(83\) −15.1160 −1.65920 −0.829600 − 0.558359i \(-0.811431\pi\)
−0.829600 + 0.558359i \(0.811431\pi\)
\(84\) 0 0
\(85\) −4.34828 −0.471637
\(86\) −2.01871 −0.217683
\(87\) 0 0
\(88\) 2.59905 0.277060
\(89\) 7.25826 0.769374 0.384687 − 0.923047i \(-0.374309\pi\)
0.384687 + 0.923047i \(0.374309\pi\)
\(90\) 0 0
\(91\) 2.47180 0.259115
\(92\) −3.96714 −0.413603
\(93\) 0 0
\(94\) 8.81467 0.909164
\(95\) 1.00000 0.102598
\(96\) 0 0
\(97\) 0.0194069 0.00197047 0.000985234 − 1.00000i \(-0.499686\pi\)
0.000985234 1.00000i \(0.499686\pi\)
\(98\) −13.3009 −1.34359
\(99\) 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 9405.2.a.w.1.3 6
3.2 odd 2 1045.2.a.g.1.4 ✓ 6
15.14 odd 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 3.2 odd 2
5225.2.a.k.1.3 6 15.14 odd 2
9405.2.a.w.1.3 6 1.1 even 1 trivial