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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [275,4,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, 4); 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, 4, names="a")
 
Level: \( N \) \(=\) \( 275 = 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 275.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,14] 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: \(16.2255252516\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 55)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 199.1
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 275.199
Dual form 275.4.b.a.199.2

$q$-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)
 
\(f(q)\) \(=\) \(q-1.00000i q^{2} -3.00000i q^{3} +7.00000 q^{4} -3.00000 q^{6} +9.00000i q^{7} -15.0000i q^{8} +18.0000 q^{9} +11.0000 q^{11} -21.0000i q^{12} +2.00000i q^{13} +9.00000 q^{14} +41.0000 q^{16} -21.0000i q^{17} -18.0000i q^{18} +85.0000 q^{19} +27.0000 q^{21} -11.0000i q^{22} +22.0000i q^{23} -45.0000 q^{24} +2.00000 q^{26} -135.000i q^{27} +63.0000i q^{28} +165.000 q^{29} -83.0000 q^{31} -161.000i q^{32} -33.0000i q^{33} -21.0000 q^{34} +126.000 q^{36} -1.00000i q^{37} -85.0000i q^{38} +6.00000 q^{39} -478.000 q^{41} -27.0000i q^{42} -8.00000i q^{43} +77.0000 q^{44} +22.0000 q^{46} -126.000i q^{47} -123.000i q^{48} +262.000 q^{49} -63.0000 q^{51} +14.0000i q^{52} -683.000i q^{53} -135.000 q^{54} +135.000 q^{56} -255.000i q^{57} -165.000i q^{58} +290.000 q^{59} +257.000 q^{61} +83.0000i q^{62} +162.000i q^{63} +167.000 q^{64} -33.0000 q^{66} -776.000i q^{67} -147.000i q^{68} +66.0000 q^{69} -313.000 q^{71} -270.000i q^{72} +902.000i q^{73} -1.00000 q^{74} +595.000 q^{76} +99.0000i q^{77} -6.00000i q^{78} -830.000 q^{79} +81.0000 q^{81} +478.000i q^{82} +842.000i q^{83} +189.000 q^{84} -8.00000 q^{86} -495.000i q^{87} -165.000i q^{88} -25.0000 q^{89} -18.0000 q^{91} +154.000i q^{92} +249.000i q^{93} -126.000 q^{94} -483.000 q^{96} +1784.00i q^{97} -262.000i q^{98} +198.000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 14 q^{4} - 6 q^{6} + 36 q^{9} + 22 q^{11} + 18 q^{14} + 82 q^{16} + 170 q^{19} + 54 q^{21} - 90 q^{24} + 4 q^{26} + 330 q^{29} - 166 q^{31} - 42 q^{34} + 252 q^{36} + 12 q^{39} - 956 q^{41} + 154 q^{44}+ \cdots + 396 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 1.00000i − 0.353553i −0.984251 − 0.176777i \(-0.943433\pi\)
0.984251 − 0.176777i \(-0.0565670\pi\)
\(3\) − 3.00000i − 0.577350i −0.957427 − 0.288675i \(-0.906785\pi\)
0.957427 − 0.288675i \(-0.0932147\pi\)
\(4\) 7.00000 0.875000
\(5\) 0 0
\(6\) −3.00000 −0.204124
\(7\) 9.00000i 0.485954i 0.970032 + 0.242977i \(0.0781240\pi\)
−0.970032 + 0.242977i \(0.921876\pi\)
\(8\) − 15.0000i − 0.662913i
\(9\) 18.0000 0.666667
\(10\) 0 0
\(11\) 11.0000 0.301511
\(12\) − 21.0000i − 0.505181i
\(13\) 2.00000i 0.0426692i 0.999772 + 0.0213346i \(0.00679154\pi\)
−0.999772 + 0.0213346i \(0.993208\pi\)
\(14\) 9.00000 0.171811
\(15\) 0 0
\(16\) 41.0000 0.640625
\(17\) − 21.0000i − 0.299603i −0.988716 − 0.149801i \(-0.952137\pi\)
0.988716 − 0.149801i \(-0.0478634\pi\)
\(18\) − 18.0000i − 0.235702i
\(19\) 85.0000 1.02633 0.513167 − 0.858289i \(-0.328472\pi\)
0.513167 + 0.858289i \(0.328472\pi\)
\(20\) 0 0
\(21\) 27.0000 0.280566
\(22\) − 11.0000i − 0.106600i
\(23\) 22.0000i 0.199449i 0.995015 + 0.0997243i \(0.0317961\pi\)
−0.995015 + 0.0997243i \(0.968204\pi\)
\(24\) −45.0000 −0.382733
\(25\) 0 0
\(26\) 2.00000 0.0150859
\(27\) − 135.000i − 0.962250i
\(28\) 63.0000i 0.425210i
\(29\) 165.000 1.05654 0.528271 − 0.849076i \(-0.322840\pi\)
0.528271 + 0.849076i \(0.322840\pi\)
\(30\) 0 0
\(31\) −83.0000 −0.480879 −0.240439 − 0.970664i \(-0.577292\pi\)
−0.240439 + 0.970664i \(0.577292\pi\)
\(32\) − 161.000i − 0.889408i
\(33\) − 33.0000i − 0.174078i
\(34\) −21.0000 −0.105926
\(35\) 0 0
\(36\) 126.000 0.583333
\(37\) − 1.00000i − 0.00444322i −0.999998 − 0.00222161i \(-0.999293\pi\)
0.999998 − 0.00222161i \(-0.000707160\pi\)
\(38\) − 85.0000i − 0.362864i
\(39\) 6.00000 0.0246351
\(40\) 0 0
\(41\) −478.000 −1.82076 −0.910379 − 0.413776i \(-0.864210\pi\)
−0.910379 + 0.413776i \(0.864210\pi\)
\(42\) − 27.0000i − 0.0991950i
\(43\) − 8.00000i − 0.0283718i −0.999899 − 0.0141859i \(-0.995484\pi\)
0.999899 − 0.0141859i \(-0.00451567\pi\)
\(44\) 77.0000 0.263822
\(45\) 0 0
\(46\) 22.0000 0.0705157
\(47\) − 126.000i − 0.391042i −0.980699 − 0.195521i \(-0.937360\pi\)
0.980699 − 0.195521i \(-0.0626398\pi\)
\(48\) − 123.000i − 0.369865i
\(49\) 262.000 0.763848
\(50\) 0 0
\(51\) −63.0000 −0.172976
\(52\) 14.0000i 0.0373356i
\(53\) − 683.000i − 1.77014i −0.465461 − 0.885069i \(-0.654111\pi\)
0.465461 − 0.885069i \(-0.345889\pi\)
\(54\) −135.000 −0.340207
\(55\) 0 0
\(56\) 135.000 0.322145
\(57\) − 255.000i − 0.592554i
\(58\) − 165.000i − 0.373544i
\(59\) 290.000 0.639912 0.319956 − 0.947432i \(-0.396332\pi\)
0.319956 + 0.947432i \(0.396332\pi\)
\(60\) 0 0
\(61\) 257.000 0.539434 0.269717 − 0.962940i \(-0.413070\pi\)
0.269717 + 0.962940i \(0.413070\pi\)
\(62\) 83.0000i 0.170016i
\(63\) 162.000i 0.323970i
\(64\) 167.000 0.326172
\(65\) 0 0
\(66\) −33.0000 −0.0615457
\(67\) − 776.000i − 1.41498i −0.706725 − 0.707489i \(-0.749828\pi\)
0.706725 − 0.707489i \(-0.250172\pi\)
\(68\) − 147.000i − 0.262152i
\(69\) 66.0000 0.115152
\(70\) 0 0
\(71\) −313.000 −0.523187 −0.261593 − 0.965178i \(-0.584248\pi\)
−0.261593 + 0.965178i \(0.584248\pi\)
\(72\) − 270.000i − 0.441942i
\(73\) 902.000i 1.44618i 0.690754 + 0.723090i \(0.257279\pi\)
−0.690754 + 0.723090i \(0.742721\pi\)
\(74\) −1.00000 −0.00157091
\(75\) 0 0
\(76\) 595.000 0.898042
\(77\) 99.0000i 0.146521i
\(78\) − 6.00000i − 0.00870982i
\(79\) −830.000 −1.18205 −0.591027 − 0.806652i \(-0.701277\pi\)
−0.591027 + 0.806652i \(0.701277\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 478.000i 0.643735i
\(83\) 842.000i 1.11351i 0.830676 + 0.556756i \(0.187954\pi\)
−0.830676 + 0.556756i \(0.812046\pi\)
\(84\) 189.000 0.245495
\(85\) 0 0
\(86\) −8.00000 −0.0100310
\(87\) − 495.000i − 0.609995i
\(88\) − 165.000i − 0.199876i
\(89\) −25.0000 −0.0297752 −0.0148876 − 0.999889i \(-0.504739\pi\)
−0.0148876 + 0.999889i \(0.504739\pi\)
\(90\) 0 0
\(91\) −18.0000 −0.0207353
\(92\) 154.000i 0.174517i
\(93\) 249.000i 0.277635i
\(94\) −126.000 −0.138254
\(95\) 0 0
\(96\) −483.000 −0.513500
\(97\) 1784.00i 1.86740i 0.358057 + 0.933700i \(0.383439\pi\)
−0.358057 + 0.933700i \(0.616561\pi\)
\(98\) − 262.000i − 0.270061i
\(99\) 198.000 0.201008
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 275.4.b.a.199.1 2
5.2 odd 4 55.4.a.a.1.1 ✓ 1
5.3 odd 4 275.4.a.a.1.1 1
5.4 even 2 inner 275.4.b.a.199.2 2
15.2 even 4 495.4.a.a.1.1 1
15.8 even 4 2475.4.a.h.1.1 1
20.7 even 4 880.4.a.j.1.1 1
55.32 even 4 605.4.a.b.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
55.4.a.a.1.1 ✓ 1 5.2 odd 4
275.4.a.a.1.1 1 5.3 odd 4
275.4.b.a.199.1 2 1.1 even 1 trivial
275.4.b.a.199.2 2 5.4 even 2 inner
495.4.a.a.1.1 1 15.2 even 4
605.4.a.b.1.1 1 55.32 even 4
880.4.a.j.1.1 1 20.7 even 4
2475.4.a.h.1.1 1 15.8 even 4