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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [275,6,Mod(199,275)] 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("275.199"); 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([1, 0])) N = Newforms(chi, 6, names="a")
 
Level: \( N \) \(=\) \( 275 = 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 275.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,0,0,-122] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(4)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(44.1055504486\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} + \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 70x^{6} + 1541x^{4} + 10660x^{2} + 5476 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{5}\cdot 3^{2}\cdot 5^{4} \)
Twist minimal: no (minimal twist has level 55)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 199.4
Root \(-0.746657i\) of defining polynomial
Character \(\chi\) \(=\) 275.199
Dual form 275.6.b.d.199.5

$q$-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)
 
\(f(q)\) \(=\) \(q-2.64192i q^{2} +6.66534i q^{3} +25.0202 q^{4} +17.6093 q^{6} +12.8802i q^{7} -150.643i q^{8} +198.573 q^{9} -121.000 q^{11} +166.769i q^{12} -485.167i q^{13} +34.0284 q^{14} +402.660 q^{16} -266.661i q^{17} -524.615i q^{18} +149.702 q^{19} -85.8507 q^{21} +319.673i q^{22} -3213.11i q^{23} +1004.09 q^{24} -1281.77 q^{26} +2943.24i q^{27} +322.265i q^{28} -2948.81 q^{29} +2145.87 q^{31} -5884.38i q^{32} -806.507i q^{33} -704.498 q^{34} +4968.35 q^{36} +808.357i q^{37} -395.500i q^{38} +3233.80 q^{39} +10105.2 q^{41} +226.811i q^{42} +2763.15i q^{43} -3027.45 q^{44} -8488.79 q^{46} -9973.36i q^{47} +2683.87i q^{48} +16641.1 q^{49} +1777.39 q^{51} -12139.0i q^{52} +7126.92i q^{53} +7775.81 q^{54} +1940.31 q^{56} +997.814i q^{57} +7790.52i q^{58} +33337.2 q^{59} -11871.1 q^{61} -5669.22i q^{62} +2557.66i q^{63} -2660.94 q^{64} -2130.73 q^{66} -4500.58i q^{67} -6671.93i q^{68} +21416.5 q^{69} -45977.8 q^{71} -29913.7i q^{72} -62039.1i q^{73} +2135.62 q^{74} +3745.57 q^{76} -1558.50i q^{77} -8543.46i q^{78} +57486.6 q^{79} +28635.6 q^{81} -26697.2i q^{82} -90511.7i q^{83} -2148.01 q^{84} +7300.03 q^{86} -19654.8i q^{87} +18227.8i q^{88} +127861. q^{89} +6249.03 q^{91} -80392.8i q^{92} +14303.0i q^{93} -26348.9 q^{94} +39221.4 q^{96} -132338. i q^{97} -43964.5i q^{98} -24027.4 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 122 q^{4} - 314 q^{6} - 44 q^{9} - 968 q^{11} + 3374 q^{14} - 5342 q^{16} + 6788 q^{19} - 9416 q^{21} - 5442 q^{24} - 7300 q^{26} + 10496 q^{29} + 9464 q^{31} - 19518 q^{34} + 14300 q^{36} + 42160 q^{39}+ \cdots + 5324 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/275\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(177\)
\(\chi(n)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 2.64192i − 0.467030i −0.972353 − 0.233515i \(-0.924977\pi\)
0.972353 − 0.233515i \(-0.0750229\pi\)
\(3\) 6.66534i 0.427582i 0.976879 + 0.213791i \(0.0685812\pi\)
−0.976879 + 0.213791i \(0.931419\pi\)
\(4\) 25.0202 0.781883
\(5\) 0 0
\(6\) 17.6093 0.199694
\(7\) 12.8802i 0.0993520i 0.998765 + 0.0496760i \(0.0158189\pi\)
−0.998765 + 0.0496760i \(0.984181\pi\)
\(8\) − 150.643i − 0.832193i
\(9\) 198.573 0.817174
\(10\) 0 0
\(11\) −121.000 −0.301511
\(12\) 166.769i 0.334319i
\(13\) − 485.167i − 0.796219i −0.917338 − 0.398110i \(-0.869666\pi\)
0.917338 − 0.398110i \(-0.130334\pi\)
\(14\) 34.0284 0.0464004
\(15\) 0 0
\(16\) 402.660 0.393223
\(17\) − 266.661i − 0.223788i −0.993720 − 0.111894i \(-0.964308\pi\)
0.993720 − 0.111894i \(-0.0356918\pi\)
\(18\) − 524.615i − 0.381645i
\(19\) 149.702 0.0951356 0.0475678 − 0.998868i \(-0.484853\pi\)
0.0475678 + 0.998868i \(0.484853\pi\)
\(20\) 0 0
\(21\) −85.8507 −0.0424811
\(22\) 319.673i 0.140815i
\(23\) − 3213.11i − 1.26650i −0.773946 − 0.633251i \(-0.781720\pi\)
0.773946 − 0.633251i \(-0.218280\pi\)
\(24\) 1004.09 0.355831
\(25\) 0 0
\(26\) −1281.77 −0.371859
\(27\) 2943.24i 0.776991i
\(28\) 322.265i 0.0776816i
\(29\) −2948.81 −0.651106 −0.325553 − 0.945524i \(-0.605550\pi\)
−0.325553 + 0.945524i \(0.605550\pi\)
\(30\) 0 0
\(31\) 2145.87 0.401051 0.200525 − 0.979689i \(-0.435735\pi\)
0.200525 + 0.979689i \(0.435735\pi\)
\(32\) − 5884.38i − 1.01584i
\(33\) − 806.507i − 0.128921i
\(34\) −704.498 −0.104516
\(35\) 0 0
\(36\) 4968.35 0.638934
\(37\) 808.357i 0.0970731i 0.998821 + 0.0485366i \(0.0154557\pi\)
−0.998821 + 0.0485366i \(0.984544\pi\)
\(38\) − 395.500i − 0.0444312i
\(39\) 3233.80 0.340449
\(40\) 0 0
\(41\) 10105.2 0.938829 0.469414 − 0.882978i \(-0.344465\pi\)
0.469414 + 0.882978i \(0.344465\pi\)
\(42\) 226.811i 0.0198400i
\(43\) 2763.15i 0.227894i 0.993487 + 0.113947i \(0.0363494\pi\)
−0.993487 + 0.113947i \(0.963651\pi\)
\(44\) −3027.45 −0.235746
\(45\) 0 0
\(46\) −8488.79 −0.591495
\(47\) − 9973.36i − 0.658562i −0.944232 − 0.329281i \(-0.893194\pi\)
0.944232 − 0.329281i \(-0.106806\pi\)
\(48\) 2683.87i 0.168135i
\(49\) 16641.1 0.990129
\(50\) 0 0
\(51\) 1777.39 0.0956878
\(52\) − 12139.0i − 0.622550i
\(53\) 7126.92i 0.348508i 0.984701 + 0.174254i \(0.0557513\pi\)
−0.984701 + 0.174254i \(0.944249\pi\)
\(54\) 7775.81 0.362878
\(55\) 0 0
\(56\) 1940.31 0.0826800
\(57\) 997.814i 0.0406783i
\(58\) 7790.52i 0.304086i
\(59\) 33337.2 1.24681 0.623403 − 0.781901i \(-0.285750\pi\)
0.623403 + 0.781901i \(0.285750\pi\)
\(60\) 0 0
\(61\) −11871.1 −0.408476 −0.204238 − 0.978921i \(-0.565472\pi\)
−0.204238 + 0.978921i \(0.565472\pi\)
\(62\) − 5669.22i − 0.187303i
\(63\) 2557.66i 0.0811878i
\(64\) −2660.94 −0.0812053
\(65\) 0 0
\(66\) −2130.73 −0.0602099
\(67\) − 4500.58i − 0.122485i −0.998123 − 0.0612423i \(-0.980494\pi\)
0.998123 − 0.0612423i \(-0.0195062\pi\)
\(68\) − 6671.93i − 0.174976i
\(69\) 21416.5 0.541533
\(70\) 0 0
\(71\) −45977.8 −1.08244 −0.541218 − 0.840882i \(-0.682037\pi\)
−0.541218 + 0.840882i \(0.682037\pi\)
\(72\) − 29913.7i − 0.680046i
\(73\) − 62039.1i − 1.36257i −0.732019 − 0.681284i \(-0.761422\pi\)
0.732019 − 0.681284i \(-0.238578\pi\)
\(74\) 2135.62 0.0453361
\(75\) 0 0
\(76\) 3745.57 0.0743848
\(77\) − 1558.50i − 0.0299557i
\(78\) − 8543.46i − 0.159000i
\(79\) 57486.6 1.03633 0.518166 − 0.855280i \(-0.326615\pi\)
0.518166 + 0.855280i \(0.326615\pi\)
\(80\) 0 0
\(81\) 28635.6 0.484946
\(82\) − 26697.2i − 0.438461i
\(83\) − 90511.7i − 1.44215i −0.692858 − 0.721074i \(-0.743649\pi\)
0.692858 − 0.721074i \(-0.256351\pi\)
\(84\) −2148.01 −0.0332152
\(85\) 0 0
\(86\) 7300.03 0.106434
\(87\) − 19654.8i − 0.278401i
\(88\) 18227.8i 0.250916i
\(89\) 127861. 1.71105 0.855524 − 0.517764i \(-0.173235\pi\)
0.855524 + 0.517764i \(0.173235\pi\)
\(90\) 0 0
\(91\) 6249.03 0.0791059
\(92\) − 80392.8i − 0.990256i
\(93\) 14303.0i 0.171482i
\(94\) −26348.9 −0.307569
\(95\) 0 0
\(96\) 39221.4 0.434355
\(97\) − 132338.i − 1.42809i −0.700099 − 0.714046i \(-0.746861\pi\)
0.700099 − 0.714046i \(-0.253139\pi\)
\(98\) − 43964.5i − 0.462420i
\(99\) −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.b.d.199.4 8
5.2 odd 4 55.6.a.b.1.3 ✓ 4
5.3 odd 4 275.6.a.d.1.2 4
5.4 even 2 inner 275.6.b.d.199.5 8
15.2 even 4 495.6.a.g.1.2 4
20.7 even 4 880.6.a.n.1.2 4
55.32 even 4 605.6.a.c.1.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
55.6.a.b.1.3 ✓ 4 5.2 odd 4
275.6.a.d.1.2 4 5.3 odd 4
275.6.b.d.199.4 8 1.1 even 1 trivial
275.6.b.d.199.5 8 5.4 even 2 inner
495.6.a.g.1.2 4 15.2 even 4
605.6.a.c.1.2 4 55.32 even 4
880.6.a.n.1.2 4 20.7 even 4