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Properties

Label 605.6.a.c.1.2
Level $605$
Weight $6$
Character 605.1
Self dual yes
Analytic conductor $97.032$
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] = [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 = [4,5] 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: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{3} - 33x^{2} - 8x + 116 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2\cdot 3 \)
Twist minimal: no (minimal twist has level 55)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.74666\) 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)
 
\(f(q)\) \(=\) \(q-2.64192 q^{2} +6.66534 q^{3} -25.0202 q^{4} -25.0000 q^{5} -17.6093 q^{6} +12.8802 q^{7} +150.643 q^{8} -198.573 q^{9} +66.0481 q^{10} -166.769 q^{12} +485.167 q^{13} -34.0284 q^{14} -166.634 q^{15} +402.660 q^{16} -266.661 q^{17} +524.615 q^{18} +149.702 q^{19} +625.506 q^{20} +85.8507 q^{21} -3213.11 q^{23} +1004.09 q^{24} +625.000 q^{25} -1281.77 q^{26} -2943.24 q^{27} -322.265 q^{28} -2948.81 q^{29} +440.233 q^{30} +2145.87 q^{31} -5884.38 q^{32} +704.498 q^{34} -322.004 q^{35} +4968.35 q^{36} -808.357 q^{37} -395.500 q^{38} +3233.80 q^{39} -3766.08 q^{40} -10105.2 q^{41} -226.811 q^{42} -2763.15 q^{43} +4964.33 q^{45} +8488.79 q^{46} +9973.36 q^{47} +2683.87 q^{48} -16641.1 q^{49} -1651.20 q^{50} -1777.39 q^{51} -12139.0 q^{52} +7126.92 q^{53} +7775.81 q^{54} +1940.31 q^{56} +997.814 q^{57} +7790.52 q^{58} -33337.2 q^{59} +4169.21 q^{60} +11871.1 q^{61} -5669.22 q^{62} -2557.66 q^{63} +2660.94 q^{64} -12129.2 q^{65} +4500.58 q^{67} +6671.93 q^{68} -21416.5 q^{69} +850.710 q^{70} -45977.8 q^{71} -29913.7 q^{72} +62039.1 q^{73} +2135.62 q^{74} +4165.84 q^{75} -3745.57 q^{76} -8543.46 q^{78} +57486.6 q^{79} -10066.5 q^{80} +28635.6 q^{81} +26697.2 q^{82} +90511.7 q^{83} -2148.01 q^{84} +6666.53 q^{85} +7300.03 q^{86} -19654.8 q^{87} -127861. q^{89} -13115.4 q^{90} +6249.03 q^{91} +80392.8 q^{92} +14303.0 q^{93} -26348.9 q^{94} -3742.54 q^{95} -39221.4 q^{96} +132338. q^{97} +43964.5 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 5 q^{2} + 61 q^{4} - 100 q^{5} + 157 q^{6} + 90 q^{7} + 135 q^{8} + 22 q^{9} - 125 q^{10} - 795 q^{12} - 820 q^{13} - 1687 q^{14} - 2671 q^{16} + 3800 q^{17} + 1610 q^{18} + 3394 q^{19} - 1525 q^{20}+ \cdots + 393590 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.64192 −0.467030 −0.233515 − 0.972353i \(-0.575023\pi\)
−0.233515 + 0.972353i \(0.575023\pi\)
\(3\) 6.66534 0.427582 0.213791 − 0.976879i \(-0.431419\pi\)
0.213791 + 0.976879i \(0.431419\pi\)
\(4\) −25.0202 −0.781883
\(5\) −25.0000 −0.447214
\(6\) −17.6093 −0.199694
\(7\) 12.8802 0.0993520 0.0496760 − 0.998765i \(-0.484181\pi\)
0.0496760 + 0.998765i \(0.484181\pi\)
\(8\) 150.643 0.832193
\(9\) −198.573 −0.817174
\(10\) 66.0481 0.208862
\(11\) 0 0
\(12\) −166.769 −0.334319
\(13\) 485.167 0.796219 0.398110 − 0.917338i \(-0.369666\pi\)
0.398110 + 0.917338i \(0.369666\pi\)
\(14\) −34.0284 −0.0464004
\(15\) −166.634 −0.191220
\(16\) 402.660 0.393223
\(17\) −266.661 −0.223788 −0.111894 − 0.993720i \(-0.535692\pi\)
−0.111894 + 0.993720i \(0.535692\pi\)
\(18\) 524.615 0.381645
\(19\) 149.702 0.0951356 0.0475678 − 0.998868i \(-0.484853\pi\)
0.0475678 + 0.998868i \(0.484853\pi\)
\(20\) 625.506 0.349669
\(21\) 85.8507 0.0424811
\(22\) 0 0
\(23\) −3213.11 −1.26650 −0.633251 − 0.773946i \(-0.718280\pi\)
−0.633251 + 0.773946i \(0.718280\pi\)
\(24\) 1004.09 0.355831
\(25\) 625.000 0.200000
\(26\) −1281.77 −0.371859
\(27\) −2943.24 −0.776991
\(28\) −322.265 −0.0776816
\(29\) −2948.81 −0.651106 −0.325553 − 0.945524i \(-0.605550\pi\)
−0.325553 + 0.945524i \(0.605550\pi\)
\(30\) 440.233 0.0893058
\(31\) 2145.87 0.401051 0.200525 − 0.979689i \(-0.435735\pi\)
0.200525 + 0.979689i \(0.435735\pi\)
\(32\) −5884.38 −1.01584
\(33\) 0 0
\(34\) 704.498 0.104516
\(35\) −322.004 −0.0444315
\(36\) 4968.35 0.638934
\(37\) −808.357 −0.0970731 −0.0485366 − 0.998821i \(-0.515456\pi\)
−0.0485366 + 0.998821i \(0.515456\pi\)
\(38\) −395.500 −0.0444312
\(39\) 3233.80 0.340449
\(40\) −3766.08 −0.372168
\(41\) −10105.2 −0.938829 −0.469414 − 0.882978i \(-0.655535\pi\)
−0.469414 + 0.882978i \(0.655535\pi\)
\(42\) −226.811 −0.0198400
\(43\) −2763.15 −0.227894 −0.113947 − 0.993487i \(-0.536349\pi\)
−0.113947 + 0.993487i \(0.536349\pi\)
\(44\) 0 0
\(45\) 4964.33 0.365451
\(46\) 8488.79 0.591495
\(47\) 9973.36 0.658562 0.329281 − 0.944232i \(-0.393194\pi\)
0.329281 + 0.944232i \(0.393194\pi\)
\(48\) 2683.87 0.168135
\(49\) −16641.1 −0.990129
\(50\) −1651.20 −0.0934061
\(51\) −1777.39 −0.0956878
\(52\) −12139.0 −0.622550
\(53\) 7126.92 0.348508 0.174254 − 0.984701i \(-0.444249\pi\)
0.174254 + 0.984701i \(0.444249\pi\)
\(54\) 7775.81 0.362878
\(55\) 0 0
\(56\) 1940.31 0.0826800
\(57\) 997.814 0.0406783
\(58\) 7790.52 0.304086
\(59\) −33337.2 −1.24681 −0.623403 − 0.781901i \(-0.714250\pi\)
−0.623403 + 0.781901i \(0.714250\pi\)
\(60\) 4169.21 0.149512
\(61\) 11871.1 0.408476 0.204238 − 0.978921i \(-0.434528\pi\)
0.204238 + 0.978921i \(0.434528\pi\)
\(62\) −5669.22 −0.187303
\(63\) −2557.66 −0.0811878
\(64\) 2660.94 0.0812053
\(65\) −12129.2 −0.356080
\(66\) 0 0
\(67\) 4500.58 0.122485 0.0612423 − 0.998123i \(-0.480494\pi\)
0.0612423 + 0.998123i \(0.480494\pi\)
\(68\) 6671.93 0.174976
\(69\) −21416.5 −0.541533
\(70\) 850.710 0.0207509
\(71\) −45977.8 −1.08244 −0.541218 − 0.840882i \(-0.682037\pi\)
−0.541218 + 0.840882i \(0.682037\pi\)
\(72\) −29913.7 −0.680046
\(73\) 62039.1 1.36257 0.681284 − 0.732019i \(-0.261422\pi\)
0.681284 + 0.732019i \(0.261422\pi\)
\(74\) 2135.62 0.0453361
\(75\) 4165.84 0.0855164
\(76\) −3745.57 −0.0743848
\(77\) 0 0
\(78\) −8543.46 −0.159000
\(79\) 57486.6 1.03633 0.518166 − 0.855280i \(-0.326615\pi\)
0.518166 + 0.855280i \(0.326615\pi\)
\(80\) −10066.5 −0.175855
\(81\) 28635.6 0.484946
\(82\) 26697.2 0.438461
\(83\) 90511.7 1.44215 0.721074 − 0.692858i \(-0.243649\pi\)
0.721074 + 0.692858i \(0.243649\pi\)
\(84\) −2148.01 −0.0332152
\(85\) 6666.53 0.100081
\(86\) 7300.03 0.106434
\(87\) −19654.8 −0.278401
\(88\) 0 0
\(89\) −127861. −1.71105 −0.855524 − 0.517764i \(-0.826765\pi\)
−0.855524 + 0.517764i \(0.826765\pi\)
\(90\) −13115.4 −0.170677
\(91\) 6249.03 0.0791059
\(92\) 80392.8 0.990256
\(93\) 14303.0 0.171482
\(94\) −26348.9 −0.307569
\(95\) −3742.54 −0.0425459
\(96\) −39221.4 −0.434355
\(97\) 132338. 1.42809 0.714046 − 0.700099i \(-0.246861\pi\)
0.714046 + 0.700099i \(0.246861\pi\)
\(98\) 43964.5 0.462420
\(99\) 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.c.1.2 4
11.10 odd 2 55.6.a.b.1.3 ✓ 4
33.32 even 2 495.6.a.g.1.2 4
44.43 even 2 880.6.a.n.1.2 4
55.32 even 4 275.6.b.d.199.5 8
55.43 even 4 275.6.b.d.199.4 8
55.54 odd 2 275.6.a.d.1.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
55.6.a.b.1.3 ✓ 4 11.10 odd 2
275.6.a.d.1.2 4 55.54 odd 2
275.6.b.d.199.4 8 55.43 even 4
275.6.b.d.199.5 8 55.32 even 4
495.6.a.g.1.2 4 33.32 even 2
605.6.a.c.1.2 4 1.1 even 1 trivial
880.6.a.n.1.2 4 44.43 even 2