Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [275,6,Mod(1,275)] 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("275.1"); S:= CuspForms(chi, 6); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(275, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 6, names="a")
 
Level: \( N \) \(=\) \( 275 = 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 275.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,5,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(44.1055504486\)
Analytic rank: \(1\)
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 \(\chi\) \(=\) 275.1

$q$-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)
 
\(f(q)\) \(=\) \(q-2.64192 q^{2} -6.66534 q^{3} -25.0202 q^{4} +17.6093 q^{6} +12.8802 q^{7} +150.643 q^{8} -198.573 q^{9} -121.000 q^{11} +166.769 q^{12} +485.167 q^{13} -34.0284 q^{14} +402.660 q^{16} -266.661 q^{17} +524.615 q^{18} -149.702 q^{19} -85.8507 q^{21} +319.673 q^{22} +3213.11 q^{23} -1004.09 q^{24} -1281.77 q^{26} +2943.24 q^{27} -322.265 q^{28} +2948.81 q^{29} +2145.87 q^{31} -5884.38 q^{32} +806.507 q^{33} +704.498 q^{34} +4968.35 q^{36} +808.357 q^{37} +395.500 q^{38} -3233.80 q^{39} +10105.2 q^{41} +226.811 q^{42} -2763.15 q^{43} +3027.45 q^{44} -8488.79 q^{46} -9973.36 q^{47} -2683.87 q^{48} -16641.1 q^{49} +1777.39 q^{51} -12139.0 q^{52} -7126.92 q^{53} -7775.81 q^{54} +1940.31 q^{56} +997.814 q^{57} -7790.52 q^{58} -33337.2 q^{59} -11871.1 q^{61} -5669.22 q^{62} -2557.66 q^{63} +2660.94 q^{64} -2130.73 q^{66} -4500.58 q^{67} +6671.93 q^{68} -21416.5 q^{69} -45977.8 q^{71} -29913.7 q^{72} +62039.1 q^{73} -2135.62 q^{74} +3745.57 q^{76} -1558.50 q^{77} +8543.46 q^{78} -57486.6 q^{79} +28635.6 q^{81} -26697.2 q^{82} +90511.7 q^{83} +2148.01 q^{84} +7300.03 q^{86} -19654.8 q^{87} -18227.8 q^{88} -127861. q^{89} +6249.03 q^{91} -80392.8 q^{92} -14303.0 q^{93} +26348.9 q^{94} +39221.4 q^{96} -132338. q^{97} +43964.5 q^{98} +24027.4 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 5 q^{2} + 61 q^{4} - 157 q^{6} + 90 q^{7} + 135 q^{8} + 22 q^{9} - 484 q^{11} + 795 q^{12} - 820 q^{13} - 1687 q^{14} - 2671 q^{16} + 3800 q^{17} + 1610 q^{18} - 3394 q^{19} - 4708 q^{21} - 605 q^{22}+ \cdots - 2662 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −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 \(-0.568581\pi\)
−0.213791 + 0.976879i \(0.568581\pi\)
\(4\) −25.0202 −0.781883
\(5\) 0 0
\(6\) 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
\(10\) 0 0
\(11\) −121.000 −0.301511
\(12\) 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
\(15\) 0 0
\(16\) 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 \(-0.515147\pi\)
−0.0475678 + 0.998868i \(0.515147\pi\)
\(20\) 0 0
\(21\) −85.8507 −0.0424811
\(22\) 319.673 0.140815
\(23\) 3213.11 1.26650 0.633251 − 0.773946i \(-0.281720\pi\)
0.633251 + 0.773946i \(0.281720\pi\)
\(24\) −1004.09 −0.355831
\(25\) 0 0
\(26\) −1281.77 −0.371859
\(27\) 2943.24 0.776991
\(28\) −322.265 −0.0776816
\(29\) 2948.81 0.651106 0.325553 − 0.945524i \(-0.394450\pi\)
0.325553 + 0.945524i \(0.394450\pi\)
\(30\) 0 0
\(31\) 2145.87 0.401051 0.200525 − 0.979689i \(-0.435735\pi\)
0.200525 + 0.979689i \(0.435735\pi\)
\(32\) −5884.38 −1.01584
\(33\) 806.507 0.128921
\(34\) 704.498 0.104516
\(35\) 0 0
\(36\) 4968.35 0.638934
\(37\) 808.357 0.0970731 0.0485366 − 0.998821i \(-0.484544\pi\)
0.0485366 + 0.998821i \(0.484544\pi\)
\(38\) 395.500 0.0444312
\(39\) −3233.80 −0.340449
\(40\) 0 0
\(41\) 10105.2 0.938829 0.469414 − 0.882978i \(-0.344465\pi\)
0.469414 + 0.882978i \(0.344465\pi\)
\(42\) 226.811 0.0198400
\(43\) −2763.15 −0.227894 −0.113947 − 0.993487i \(-0.536349\pi\)
−0.113947 + 0.993487i \(0.536349\pi\)
\(44\) 3027.45 0.235746
\(45\) 0 0
\(46\) −8488.79 −0.591495
\(47\) −9973.36 −0.658562 −0.329281 − 0.944232i \(-0.606806\pi\)
−0.329281 + 0.944232i \(0.606806\pi\)
\(48\) −2683.87 −0.168135
\(49\) −16641.1 −0.990129
\(50\) 0 0
\(51\) 1777.39 0.0956878
\(52\) −12139.0 −0.622550
\(53\) −7126.92 −0.348508 −0.174254 − 0.984701i \(-0.555751\pi\)
−0.174254 + 0.984701i \(0.555751\pi\)
\(54\) −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 \(0.714250\pi\)
\(60\) 0 0
\(61\) −11871.1 −0.408476 −0.204238 − 0.978921i \(-0.565472\pi\)
−0.204238 + 0.978921i \(0.565472\pi\)
\(62\) −5669.22 −0.187303
\(63\) −2557.66 −0.0811878
\(64\) 2660.94 0.0812053
\(65\) 0 0
\(66\) −2130.73 −0.0602099
\(67\) −4500.58 −0.122485 −0.0612423 − 0.998123i \(-0.519506\pi\)
−0.0612423 + 0.998123i \(0.519506\pi\)
\(68\) 6671.93 0.174976
\(69\) −21416.5 −0.541533
\(70\) 0 0
\(71\) −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
\(75\) 0 0
\(76\) 3745.57 0.0743848
\(77\) −1558.50 −0.0299557
\(78\) 8543.46 0.159000
\(79\) −57486.6 −1.03633 −0.518166 − 0.855280i \(-0.673385\pi\)
−0.518166 + 0.855280i \(0.673385\pi\)
\(80\) 0 0
\(81\) 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
\(85\) 0 0
\(86\) 7300.03 0.106434
\(87\) −19654.8 −0.278401
\(88\) −18227.8 −0.250916
\(89\) −127861. −1.71105 −0.855524 − 0.517764i \(-0.826765\pi\)
−0.855524 + 0.517764i \(0.826765\pi\)
\(90\) 0 0
\(91\) 6249.03 0.0791059
\(92\) −80392.8 −0.990256
\(93\) −14303.0 −0.171482
\(94\) 26348.9 0.307569
\(95\) 0 0
\(96\) 39221.4 0.434355
\(97\) −132338. −1.42809 −0.714046 − 0.700099i \(-0.753139\pi\)
−0.714046 + 0.700099i \(0.753139\pi\)
\(98\) 43964.5 0.462420
\(99\) 24027.4 0.246387
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 275.6.a.d.1.2 4
5.2 odd 4 275.6.b.d.199.4 8
5.3 odd 4 275.6.b.d.199.5 8
5.4 even 2 55.6.a.b.1.3 ✓ 4
15.14 odd 2 495.6.a.g.1.2 4
20.19 odd 2 880.6.a.n.1.2 4
55.54 odd 2 605.6.a.c.1.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
55.6.a.b.1.3 ✓ 4 5.4 even 2
275.6.a.d.1.2 4 1.1 even 1 trivial
275.6.b.d.199.4 8 5.2 odd 4
275.6.b.d.199.5 8 5.3 odd 4
495.6.a.g.1.2 4 15.14 odd 2
605.6.a.c.1.2 4 55.54 odd 2
880.6.a.n.1.2 4 20.19 odd 2