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

$q$-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 \(0.0750229\pi\)
−0.972353 + 0.233515i \(0.924977\pi\)
\(3\) − 6.66534i − 0.427582i −0.976879 − 0.213791i \(-0.931419\pi\)
0.976879 − 0.213791i \(-0.0685812\pi\)
\(4\) 25.0202 0.781883
\(5\) 0 0
\(6\) 17.6093 0.199694
\(7\) − 12.8802i − 0.0993520i −0.998765 − 0.0496760i \(-0.984181\pi\)
0.998765 − 0.0496760i \(-0.0158189\pi\)
\(8\) 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 \(0.130334\pi\)
−0.917338 + 0.398110i \(0.869666\pi\)
\(14\) 34.0284 0.0464004
\(15\) 0 0
\(16\) 402.660 0.393223
\(17\) 266.661i 0.223788i 0.993720 + 0.111894i \(0.0356918\pi\)
−0.993720 + 0.111894i \(0.964308\pi\)
\(18\) 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 \(0.218280\pi\)
−0.773946 + 0.633251i \(0.781720\pi\)
\(24\) 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 \(-0.984544\pi\)
0.998821 − 0.0485366i \(-0.0154557\pi\)
\(38\) 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 \(-0.963651\pi\)
0.993487 − 0.113947i \(-0.0363494\pi\)
\(44\) −3027.45 −0.235746
\(45\) 0 0
\(46\) −8488.79 −0.591495
\(47\) 9973.36i 0.658562i 0.944232 + 0.329281i \(0.106806\pi\)
−0.944232 + 0.329281i \(0.893194\pi\)
\(48\) − 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 \(-0.944249\pi\)
0.984701 − 0.174254i \(-0.0557513\pi\)
\(54\) 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 \(0.0195062\pi\)
−0.998123 + 0.0612423i \(0.980494\pi\)
\(68\) 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 \(0.238578\pi\)
−0.732019 + 0.681284i \(0.761422\pi\)
\(74\) 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 \(0.256351\pi\)
−0.692858 + 0.721074i \(0.743649\pi\)
\(84\) −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 \(0.253139\pi\)
−0.700099 + 0.714046i \(0.746861\pi\)
\(98\) 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.199.5 8
5.2 odd 4 275.6.a.d.1.2 4
5.3 odd 4 55.6.a.b.1.3 ✓ 4
5.4 even 2 inner 275.6.b.d.199.4 8
15.8 even 4 495.6.a.g.1.2 4
20.3 even 4 880.6.a.n.1.2 4
55.43 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 5.3 odd 4
275.6.a.d.1.2 4 5.2 odd 4
275.6.b.d.199.4 8 5.4 even 2 inner
275.6.b.d.199.5 8 1.1 even 1 trivial
495.6.a.g.1.2 4 15.8 even 4
605.6.a.c.1.2 4 55.43 even 4
880.6.a.n.1.2 4 20.3 even 4