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Properties

Label 605.6.a.b.1.3
Level $605$
Weight $6$
Character 605.1
Self dual yes
Analytic conductor $97.032$
Analytic rank $1$
Dimension $3$
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 = [3,7] 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: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.21865.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 30x + 40 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 55)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(5.25849\) 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)
 
\(f(q)\) \(=\) \(q+8.45512 q^{2} -26.7755 q^{3} +39.4891 q^{4} +25.0000 q^{5} -226.390 q^{6} -22.2927 q^{7} +63.3212 q^{8} +473.926 q^{9} +211.378 q^{10} -1057.34 q^{12} +225.547 q^{13} -188.488 q^{14} -669.387 q^{15} -728.262 q^{16} +1059.59 q^{17} +4007.11 q^{18} -2525.94 q^{19} +987.227 q^{20} +596.899 q^{21} -337.423 q^{23} -1695.46 q^{24} +625.000 q^{25} +1907.03 q^{26} -6183.16 q^{27} -880.320 q^{28} +7644.60 q^{29} -5659.75 q^{30} +6754.80 q^{31} -8183.83 q^{32} +8958.95 q^{34} -557.318 q^{35} +18714.9 q^{36} -5663.43 q^{37} -21357.2 q^{38} -6039.13 q^{39} +1583.03 q^{40} +13317.4 q^{41} +5046.85 q^{42} -9007.03 q^{43} +11848.2 q^{45} -2852.96 q^{46} -16644.2 q^{47} +19499.6 q^{48} -16310.0 q^{49} +5284.45 q^{50} -28371.0 q^{51} +8906.65 q^{52} +21261.8 q^{53} -52279.4 q^{54} -1411.60 q^{56} +67633.4 q^{57} +64636.1 q^{58} -44329.0 q^{59} -26433.5 q^{60} +36591.1 q^{61} +57112.7 q^{62} -10565.1 q^{63} -45890.9 q^{64} +5638.68 q^{65} -45840.6 q^{67} +41842.2 q^{68} +9034.68 q^{69} -4712.19 q^{70} -31877.6 q^{71} +30009.6 q^{72} -49936.7 q^{73} -47885.0 q^{74} -16734.7 q^{75} -99747.3 q^{76} -51061.6 q^{78} -48257.1 q^{79} -18206.6 q^{80} +50393.1 q^{81} +112600. q^{82} -66052.0 q^{83} +23571.0 q^{84} +26489.7 q^{85} -76155.6 q^{86} -204688. q^{87} -124047. q^{89} +100178. q^{90} -5028.06 q^{91} -13324.5 q^{92} -180863. q^{93} -140729. q^{94} -63148.6 q^{95} +219126. q^{96} +73854.3 q^{97} -137903. q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 7 q^{2} - 36 q^{3} + 41 q^{4} + 75 q^{5} - 101 q^{6} + 102 q^{7} + 15 q^{8} + 249 q^{9} + 175 q^{10} - 1237 q^{12} + 1646 q^{13} - 963 q^{14} - 900 q^{15} - 2687 q^{16} + 1742 q^{17} + 3076 q^{18}+ \cdots - 209376 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 8.45512 1.49467 0.747334 − 0.664448i \(-0.231333\pi\)
0.747334 + 0.664448i \(0.231333\pi\)
\(3\) −26.7755 −1.71765 −0.858824 − 0.512271i \(-0.828804\pi\)
−0.858824 + 0.512271i \(0.828804\pi\)
\(4\) 39.4891 1.23403
\(5\) 25.0000 0.447214
\(6\) −226.390 −2.56731
\(7\) −22.2927 −0.171956 −0.0859782 − 0.996297i \(-0.527402\pi\)
−0.0859782 + 0.996297i \(0.527402\pi\)
\(8\) 63.3212 0.349804
\(9\) 473.926 1.95031
\(10\) 211.378 0.668436
\(11\) 0 0
\(12\) −1057.34 −2.11964
\(13\) 225.547 0.370151 0.185075 − 0.982724i \(-0.440747\pi\)
0.185075 + 0.982724i \(0.440747\pi\)
\(14\) −188.488 −0.257018
\(15\) −669.387 −0.768155
\(16\) −728.262 −0.711194
\(17\) 1059.59 0.889232 0.444616 − 0.895721i \(-0.353340\pi\)
0.444616 + 0.895721i \(0.353340\pi\)
\(18\) 4007.11 2.91507
\(19\) −2525.94 −1.60524 −0.802620 − 0.596491i \(-0.796561\pi\)
−0.802620 + 0.596491i \(0.796561\pi\)
\(20\) 987.227 0.551877
\(21\) 596.899 0.295360
\(22\) 0 0
\(23\) −337.423 −0.133001 −0.0665006 − 0.997786i \(-0.521183\pi\)
−0.0665006 + 0.997786i \(0.521183\pi\)
\(24\) −1695.46 −0.600839
\(25\) 625.000 0.200000
\(26\) 1907.03 0.553253
\(27\) −6183.16 −1.63230
\(28\) −880.320 −0.212200
\(29\) 7644.60 1.68795 0.843976 − 0.536381i \(-0.180209\pi\)
0.843976 + 0.536381i \(0.180209\pi\)
\(30\) −5659.75 −1.14814
\(31\) 6754.80 1.26243 0.631217 − 0.775607i \(-0.282556\pi\)
0.631217 + 0.775607i \(0.282556\pi\)
\(32\) −8183.83 −1.41280
\(33\) 0 0
\(34\) 8958.95 1.32911
\(35\) −557.318 −0.0769012
\(36\) 18714.9 2.40675
\(37\) −5663.43 −0.680104 −0.340052 − 0.940407i \(-0.610445\pi\)
−0.340052 + 0.940407i \(0.610445\pi\)
\(38\) −21357.2 −2.39930
\(39\) −6039.13 −0.635789
\(40\) 1583.03 0.156437
\(41\) 13317.4 1.23725 0.618627 − 0.785685i \(-0.287689\pi\)
0.618627 + 0.785685i \(0.287689\pi\)
\(42\) 5046.85 0.441466
\(43\) −9007.03 −0.742866 −0.371433 − 0.928460i \(-0.621133\pi\)
−0.371433 + 0.928460i \(0.621133\pi\)
\(44\) 0 0
\(45\) 11848.2 0.872207
\(46\) −2852.96 −0.198793
\(47\) −16644.2 −1.09905 −0.549526 − 0.835477i \(-0.685192\pi\)
−0.549526 + 0.835477i \(0.685192\pi\)
\(48\) 19499.6 1.22158
\(49\) −16310.0 −0.970431
\(50\) 5284.45 0.298934
\(51\) −28371.0 −1.52739
\(52\) 8906.65 0.456779
\(53\) 21261.8 1.03971 0.519853 − 0.854256i \(-0.325987\pi\)
0.519853 + 0.854256i \(0.325987\pi\)
\(54\) −52279.4 −2.43975
\(55\) 0 0
\(56\) −1411.60 −0.0601509
\(57\) 67633.4 2.75724
\(58\) 64636.1 2.52293
\(59\) −44329.0 −1.65790 −0.828948 − 0.559325i \(-0.811060\pi\)
−0.828948 + 0.559325i \(0.811060\pi\)
\(60\) −26433.5 −0.947930
\(61\) 36591.1 1.25907 0.629537 − 0.776970i \(-0.283245\pi\)
0.629537 + 0.776970i \(0.283245\pi\)
\(62\) 57112.7 1.88692
\(63\) −10565.1 −0.335369
\(64\) −45890.9 −1.40048
\(65\) 5638.68 0.165537
\(66\) 0 0
\(67\) −45840.6 −1.24757 −0.623783 − 0.781598i \(-0.714405\pi\)
−0.623783 + 0.781598i \(0.714405\pi\)
\(68\) 41842.2 1.09734
\(69\) 9034.68 0.228449
\(70\) −4712.19 −0.114942
\(71\) −31877.6 −0.750481 −0.375240 − 0.926928i \(-0.622440\pi\)
−0.375240 + 0.926928i \(0.622440\pi\)
\(72\) 30009.6 0.682227
\(73\) −49936.7 −1.09676 −0.548382 − 0.836228i \(-0.684756\pi\)
−0.548382 + 0.836228i \(0.684756\pi\)
\(74\) −47885.0 −1.01653
\(75\) −16734.7 −0.343530
\(76\) −99747.3 −1.98092
\(77\) 0 0
\(78\) −51061.6 −0.950294
\(79\) −48257.1 −0.869949 −0.434974 − 0.900443i \(-0.643243\pi\)
−0.434974 + 0.900443i \(0.643243\pi\)
\(80\) −18206.6 −0.318056
\(81\) 50393.1 0.853411
\(82\) 112600. 1.84928
\(83\) −66052.0 −1.05242 −0.526212 − 0.850353i \(-0.676388\pi\)
−0.526212 + 0.850353i \(0.676388\pi\)
\(84\) 23571.0 0.364485
\(85\) 26489.7 0.397677
\(86\) −76155.6 −1.11034
\(87\) −204688. −2.89931
\(88\) 0 0
\(89\) −124047. −1.66002 −0.830009 − 0.557750i \(-0.811665\pi\)
−0.830009 + 0.557750i \(0.811665\pi\)
\(90\) 100178. 1.30366
\(91\) −5028.06 −0.0636498
\(92\) −13324.5 −0.164128
\(93\) −180863. −2.16842
\(94\) −140729. −1.64272
\(95\) −63148.6 −0.717885
\(96\) 219126. 2.42670
\(97\) 73854.3 0.796978 0.398489 − 0.917173i \(-0.369535\pi\)
0.398489 + 0.917173i \(0.369535\pi\)
\(98\) −137903. −1.45047
\(99\) 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.b.1.3 3
11.10 odd 2 55.6.a.a.1.1 ✓ 3
33.32 even 2 495.6.a.f.1.3 3
44.43 even 2 880.6.a.l.1.3 3
55.32 even 4 275.6.b.c.199.1 6
55.43 even 4 275.6.b.c.199.6 6
55.54 odd 2 275.6.a.c.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
55.6.a.a.1.1 ✓ 3 11.10 odd 2
275.6.a.c.1.3 3 55.54 odd 2
275.6.b.c.199.1 6 55.32 even 4
275.6.b.c.199.6 6 55.43 even 4
495.6.a.f.1.3 3 33.32 even 2
605.6.a.b.1.3 3 1.1 even 1 trivial
880.6.a.l.1.3 3 44.43 even 2