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Properties

Label 605.6.a.o.1.2
Level $605$
Weight $6$
Character 605.1
Self dual yes
Analytic conductor $97.032$
Analytic rank $0$
Dimension $20$
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] = [605,6,Mod(1,605)] 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("605.1"); S:= CuspForms(chi, 6); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(605, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 6, names="a")
 
Level: \( N \) \(=\) \( 605 = 5 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 605.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [20,-1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(97.0322109869\)
Analytic rank: \(0\)
Dimension: \(20\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} - x^{19} - 523 x^{18} + 521 x^{17} + 115018 x^{16} - 115347 x^{15} - 13821739 x^{14} + \cdots - 32708279373824 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4}\cdot 11^{8} \)
Twist minimal: no (minimal twist has level 55)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(10.1332\) of defining polynomial
Character \(\chi\) \(=\) 605.1

$q$-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)
 
\(f(q)\) \(=\) \(q-10.1332 q^{2} -21.6453 q^{3} +70.6821 q^{4} -25.0000 q^{5} +219.336 q^{6} -136.893 q^{7} -391.974 q^{8} +225.517 q^{9} +253.330 q^{10} -1529.93 q^{12} +1057.89 q^{13} +1387.16 q^{14} +541.132 q^{15} +1710.13 q^{16} -1182.79 q^{17} -2285.22 q^{18} +791.506 q^{19} -1767.05 q^{20} +2963.08 q^{21} -2023.89 q^{23} +8484.38 q^{24} +625.000 q^{25} -10719.8 q^{26} +378.416 q^{27} -9675.86 q^{28} +2805.74 q^{29} -5483.40 q^{30} +8876.39 q^{31} -4785.97 q^{32} +11985.5 q^{34} +3422.32 q^{35} +15940.0 q^{36} -9.45154 q^{37} -8020.51 q^{38} -22898.3 q^{39} +9799.35 q^{40} +12340.6 q^{41} -30025.5 q^{42} +13002.0 q^{43} -5637.93 q^{45} +20508.5 q^{46} +18164.1 q^{47} -37016.2 q^{48} +1932.60 q^{49} -6333.26 q^{50} +25601.8 q^{51} +74773.7 q^{52} -27840.3 q^{53} -3834.57 q^{54} +53658.4 q^{56} -17132.4 q^{57} -28431.2 q^{58} +7850.12 q^{59} +38248.3 q^{60} +28115.6 q^{61} -89946.4 q^{62} -30871.7 q^{63} -6226.99 q^{64} -26447.2 q^{65} -68413.2 q^{67} -83602.2 q^{68} +43807.6 q^{69} -34679.1 q^{70} -12120.0 q^{71} -88397.0 q^{72} -22815.6 q^{73} +95.7745 q^{74} -13528.3 q^{75} +55945.3 q^{76} +232033. q^{78} +17254.6 q^{79} -42753.3 q^{80} -62991.6 q^{81} -125050. q^{82} -52119.1 q^{83} +209437. q^{84} +29569.8 q^{85} -131752. q^{86} -60731.0 q^{87} -140392. q^{89} +57130.4 q^{90} -144817. q^{91} -143053. q^{92} -192132. q^{93} -184061. q^{94} -19787.7 q^{95} +103593. q^{96} -6629.26 q^{97} -19583.5 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - q^{2} + 407 q^{4} - 500 q^{5} + 264 q^{6} + 167 q^{7} + 57 q^{8} + 1598 q^{9} + 25 q^{10} - 253 q^{12} + 769 q^{13} - 1045 q^{14} + 6963 q^{16} - 2989 q^{17} + 3775 q^{18} + 5828 q^{19} - 10175 q^{20}+ \cdots - 420757 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −10.1332 −1.79132 −0.895658 − 0.444743i \(-0.853295\pi\)
−0.895658 + 0.444743i \(0.853295\pi\)
\(3\) −21.6453 −1.38854 −0.694272 − 0.719713i \(-0.744274\pi\)
−0.694272 + 0.719713i \(0.744274\pi\)
\(4\) 70.6821 2.20882
\(5\) −25.0000 −0.447214
\(6\) 219.336 2.48732
\(7\) −136.893 −1.05593 −0.527965 − 0.849266i \(-0.677045\pi\)
−0.527965 + 0.849266i \(0.677045\pi\)
\(8\) −391.974 −2.16537
\(9\) 225.517 0.928055
\(10\) 253.330 0.801101
\(11\) 0 0
\(12\) −1529.93 −3.06704
\(13\) 1057.89 1.73613 0.868063 − 0.496455i \(-0.165365\pi\)
0.868063 + 0.496455i \(0.165365\pi\)
\(14\) 1387.16 1.89150
\(15\) 541.132 0.620976
\(16\) 1710.13 1.67005
\(17\) −1182.79 −0.992627 −0.496313 − 0.868143i \(-0.665313\pi\)
−0.496313 + 0.868143i \(0.665313\pi\)
\(18\) −2285.22 −1.66244
\(19\) 791.506 0.503003 0.251501 − 0.967857i \(-0.419076\pi\)
0.251501 + 0.967857i \(0.419076\pi\)
\(20\) −1767.05 −0.987812
\(21\) 2963.08 1.46621
\(22\) 0 0
\(23\) −2023.89 −0.797750 −0.398875 − 0.917005i \(-0.630599\pi\)
−0.398875 + 0.917005i \(0.630599\pi\)
\(24\) 8484.38 3.00671
\(25\) 625.000 0.200000
\(26\) −10719.8 −3.10995
\(27\) 378.416 0.0998988
\(28\) −9675.86 −2.33235
\(29\) 2805.74 0.619516 0.309758 − 0.950815i \(-0.399752\pi\)
0.309758 + 0.950815i \(0.399752\pi\)
\(30\) −5483.40 −1.11236
\(31\) 8876.39 1.65895 0.829473 − 0.558547i \(-0.188641\pi\)
0.829473 + 0.558547i \(0.188641\pi\)
\(32\) −4785.97 −0.826218
\(33\) 0 0
\(34\) 11985.5 1.77811
\(35\) 3422.32 0.472226
\(36\) 15940.0 2.04990
\(37\) −9.45154 −0.00113501 −0.000567503 − 1.00000i \(-0.500181\pi\)
−0.000567503 1.00000i \(0.500181\pi\)
\(38\) −8020.51 −0.901037
\(39\) −22898.3 −2.41069
\(40\) 9799.35 0.968384
\(41\) 12340.6 1.14650 0.573252 − 0.819379i \(-0.305682\pi\)
0.573252 + 0.819379i \(0.305682\pi\)
\(42\) −30025.5 −2.62644
\(43\) 13002.0 1.07235 0.536176 − 0.844106i \(-0.319868\pi\)
0.536176 + 0.844106i \(0.319868\pi\)
\(44\) 0 0
\(45\) −5637.93 −0.415039
\(46\) 20508.5 1.42902
\(47\) 18164.1 1.19941 0.599707 − 0.800220i \(-0.295284\pi\)
0.599707 + 0.800220i \(0.295284\pi\)
\(48\) −37016.2 −2.31894
\(49\) 1932.60 0.114988
\(50\) −6333.26 −0.358263
\(51\) 25601.8 1.37831
\(52\) 74773.7 3.83478
\(53\) −27840.3 −1.36140 −0.680698 − 0.732565i \(-0.738323\pi\)
−0.680698 + 0.732565i \(0.738323\pi\)
\(54\) −3834.57 −0.178950
\(55\) 0 0
\(56\) 53658.4 2.28648
\(57\) −17132.4 −0.698442
\(58\) −28431.2 −1.10975
\(59\) 7850.12 0.293593 0.146797 − 0.989167i \(-0.453104\pi\)
0.146797 + 0.989167i \(0.453104\pi\)
\(60\) 38248.3 1.37162
\(61\) 28115.6 0.967436 0.483718 − 0.875224i \(-0.339286\pi\)
0.483718 + 0.875224i \(0.339286\pi\)
\(62\) −89946.4 −2.97170
\(63\) −30871.7 −0.979961
\(64\) −6226.99 −0.190033
\(65\) −26447.2 −0.776419
\(66\) 0 0
\(67\) −68413.2 −1.86188 −0.930942 − 0.365167i \(-0.881012\pi\)
−0.930942 + 0.365167i \(0.881012\pi\)
\(68\) −83602.2 −2.19253
\(69\) 43807.6 1.10771
\(70\) −34679.1 −0.845907
\(71\) −12120.0 −0.285337 −0.142669 − 0.989771i \(-0.545568\pi\)
−0.142669 + 0.989771i \(0.545568\pi\)
\(72\) −88397.0 −2.00958
\(73\) −22815.6 −0.501101 −0.250550 − 0.968104i \(-0.580612\pi\)
−0.250550 + 0.968104i \(0.580612\pi\)
\(74\) 95.7745 0.00203316
\(75\) −13528.3 −0.277709
\(76\) 55945.3 1.11104
\(77\) 0 0
\(78\) 232033. 4.31830
\(79\) 17254.6 0.311055 0.155527 − 0.987832i \(-0.450292\pi\)
0.155527 + 0.987832i \(0.450292\pi\)
\(80\) −42753.3 −0.746869
\(81\) −62991.6 −1.06677
\(82\) −125050. −2.05375
\(83\) −52119.1 −0.830427 −0.415214 − 0.909724i \(-0.636293\pi\)
−0.415214 + 0.909724i \(0.636293\pi\)
\(84\) 209437. 3.23858
\(85\) 29569.8 0.443916
\(86\) −131752. −1.92092
\(87\) −60731.0 −0.860225
\(88\) 0 0
\(89\) −140392. −1.87874 −0.939370 − 0.342905i \(-0.888589\pi\)
−0.939370 + 0.342905i \(0.888589\pi\)
\(90\) 57130.4 0.743466
\(91\) −144817. −1.83323
\(92\) −143053. −1.76208
\(93\) −192132. −2.30352
\(94\) −184061. −2.14853
\(95\) −19787.7 −0.224950
\(96\) 103593. 1.14724
\(97\) −6629.26 −0.0715378 −0.0357689 − 0.999360i \(-0.511388\pi\)
−0.0357689 + 0.999360i \(0.511388\pi\)
\(98\) −19583.5 −0.205980
\(99\) 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 605.6.a.o.1.2 20
11.2 odd 10 55.6.g.b.26.1 ✓ 40
11.6 odd 10 55.6.g.b.36.1 yes 40
11.10 odd 2 605.6.a.p.1.19 20
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
55.6.g.b.26.1 ✓ 40 11.2 odd 10
55.6.g.b.36.1 yes 40 11.6 odd 10
605.6.a.o.1.2 20 1.1 even 1 trivial
605.6.a.p.1.19 20 11.10 odd 2