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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1425,2,Mod(799,1425)] 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("1425.799"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1425, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1425 = 3 \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1425.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,2,0,2,0,0,-2,0,10,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(14)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(11.3786822880\)
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: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 799.1
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 1425.799
Dual form 1425.2.c.h.799.2

$q$-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)
 
\(f(q)\) \(=\) \(q-1.00000i q^{2} +1.00000i q^{3} +1.00000 q^{4} +1.00000 q^{6} -3.00000i q^{8} -1.00000 q^{9} +5.00000 q^{11} +1.00000i q^{12} -4.00000i q^{13} -1.00000 q^{16} -4.00000i q^{17} +1.00000i q^{18} +1.00000 q^{19} -5.00000i q^{22} +9.00000i q^{23} +3.00000 q^{24} -4.00000 q^{26} -1.00000i q^{27} -7.00000 q^{29} +3.00000 q^{31} -5.00000i q^{32} +5.00000i q^{33} -4.00000 q^{34} -1.00000 q^{36} -10.0000i q^{37} -1.00000i q^{38} +4.00000 q^{39} -2.00000 q^{41} -4.00000i q^{43} +5.00000 q^{44} +9.00000 q^{46} +8.00000i q^{47} -1.00000i q^{48} +7.00000 q^{49} +4.00000 q^{51} -4.00000i q^{52} -11.0000i q^{53} -1.00000 q^{54} +1.00000i q^{57} +7.00000i q^{58} -8.00000 q^{59} +13.0000 q^{61} -3.00000i q^{62} -7.00000 q^{64} +5.00000 q^{66} +9.00000i q^{67} -4.00000i q^{68} -9.00000 q^{69} +10.0000 q^{71} +3.00000i q^{72} +5.00000i q^{73} -10.0000 q^{74} +1.00000 q^{76} -4.00000i q^{78} +15.0000 q^{79} +1.00000 q^{81} +2.00000i q^{82} -9.00000i q^{83} -4.00000 q^{86} -7.00000i q^{87} -15.0000i q^{88} -3.00000 q^{89} +9.00000i q^{92} +3.00000i q^{93} +8.00000 q^{94} +5.00000 q^{96} -10.0000i q^{97} -7.00000i q^{98} -5.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{4} + 2 q^{6} - 2 q^{9} + 10 q^{11} - 2 q^{16} + 2 q^{19} + 6 q^{24} - 8 q^{26} - 14 q^{29} + 6 q^{31} - 8 q^{34} - 2 q^{36} + 8 q^{39} - 4 q^{41} + 10 q^{44} + 18 q^{46} + 14 q^{49} + 8 q^{51}+ \cdots - 10 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1425\mathbb{Z}\right)^\times\).

\(n\) \(476\) \(1027\) \(1351\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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 − 0.707107i −0.935414 − 0.353553i \(-0.884973\pi\)
0.935414 − 0.353553i \(-0.115027\pi\)
\(3\) 1.00000i 0.577350i
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) 1.00000 0.408248
\(7\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(8\) − 3.00000i − 1.06066i
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) 5.00000 1.50756 0.753778 − 0.657129i \(-0.228229\pi\)
0.753778 + 0.657129i \(0.228229\pi\)
\(12\) 1.00000i 0.288675i
\(13\) − 4.00000i − 1.10940i −0.832050 − 0.554700i \(-0.812833\pi\)
0.832050 − 0.554700i \(-0.187167\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −1.00000 −0.250000
\(17\) − 4.00000i − 0.970143i −0.874475 − 0.485071i \(-0.838794\pi\)
0.874475 − 0.485071i \(-0.161206\pi\)
\(18\) 1.00000i 0.235702i
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) 0 0
\(22\) − 5.00000i − 1.06600i
\(23\) 9.00000i 1.87663i 0.345782 + 0.938315i \(0.387614\pi\)
−0.345782 + 0.938315i \(0.612386\pi\)
\(24\) 3.00000 0.612372
\(25\) 0 0
\(26\) −4.00000 −0.784465
\(27\) − 1.00000i − 0.192450i
\(28\) 0 0
\(29\) −7.00000 −1.29987 −0.649934 − 0.759991i \(-0.725203\pi\)
−0.649934 + 0.759991i \(0.725203\pi\)
\(30\) 0 0
\(31\) 3.00000 0.538816 0.269408 − 0.963026i \(-0.413172\pi\)
0.269408 + 0.963026i \(0.413172\pi\)
\(32\) − 5.00000i − 0.883883i
\(33\) 5.00000i 0.870388i
\(34\) −4.00000 −0.685994
\(35\) 0 0
\(36\) −1.00000 −0.166667
\(37\) − 10.0000i − 1.64399i −0.569495 − 0.821995i \(-0.692861\pi\)
0.569495 − 0.821995i \(-0.307139\pi\)
\(38\) − 1.00000i − 0.162221i
\(39\) 4.00000 0.640513
\(40\) 0 0
\(41\) −2.00000 −0.312348 −0.156174 − 0.987730i \(-0.549916\pi\)
−0.156174 + 0.987730i \(0.549916\pi\)
\(42\) 0 0
\(43\) − 4.00000i − 0.609994i −0.952353 − 0.304997i \(-0.901344\pi\)
0.952353 − 0.304997i \(-0.0986555\pi\)
\(44\) 5.00000 0.753778
\(45\) 0 0
\(46\) 9.00000 1.32698
\(47\) 8.00000i 1.16692i 0.812142 + 0.583460i \(0.198301\pi\)
−0.812142 + 0.583460i \(0.801699\pi\)
\(48\) − 1.00000i − 0.144338i
\(49\) 7.00000 1.00000
\(50\) 0 0
\(51\) 4.00000 0.560112
\(52\) − 4.00000i − 0.554700i
\(53\) − 11.0000i − 1.51097i −0.655168 − 0.755483i \(-0.727402\pi\)
0.655168 − 0.755483i \(-0.272598\pi\)
\(54\) −1.00000 −0.136083
\(55\) 0 0
\(56\) 0 0
\(57\) 1.00000i 0.132453i
\(58\) 7.00000i 0.919145i
\(59\) −8.00000 −1.04151 −0.520756 − 0.853706i \(-0.674350\pi\)
−0.520756 + 0.853706i \(0.674350\pi\)
\(60\) 0 0
\(61\) 13.0000 1.66448 0.832240 − 0.554416i \(-0.187058\pi\)
0.832240 + 0.554416i \(0.187058\pi\)
\(62\) − 3.00000i − 0.381000i
\(63\) 0 0
\(64\) −7.00000 −0.875000
\(65\) 0 0
\(66\) 5.00000 0.615457
\(67\) 9.00000i 1.09952i 0.835321 + 0.549762i \(0.185282\pi\)
−0.835321 + 0.549762i \(0.814718\pi\)
\(68\) − 4.00000i − 0.485071i
\(69\) −9.00000 −1.08347
\(70\) 0 0
\(71\) 10.0000 1.18678 0.593391 − 0.804914i \(-0.297789\pi\)
0.593391 + 0.804914i \(0.297789\pi\)
\(72\) 3.00000i 0.353553i
\(73\) 5.00000i 0.585206i 0.956234 + 0.292603i \(0.0945214\pi\)
−0.956234 + 0.292603i \(0.905479\pi\)
\(74\) −10.0000 −1.16248
\(75\) 0 0
\(76\) 1.00000 0.114708
\(77\) 0 0
\(78\) − 4.00000i − 0.452911i
\(79\) 15.0000 1.68763 0.843816 − 0.536633i \(-0.180304\pi\)
0.843816 + 0.536633i \(0.180304\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 2.00000i 0.220863i
\(83\) − 9.00000i − 0.987878i −0.869496 − 0.493939i \(-0.835557\pi\)
0.869496 − 0.493939i \(-0.164443\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −4.00000 −0.431331
\(87\) − 7.00000i − 0.750479i
\(88\) − 15.0000i − 1.59901i
\(89\) −3.00000 −0.317999 −0.159000 − 0.987279i \(-0.550827\pi\)
−0.159000 + 0.987279i \(0.550827\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 9.00000i 0.938315i
\(93\) 3.00000i 0.311086i
\(94\) 8.00000 0.825137
\(95\) 0 0
\(96\) 5.00000 0.510310
\(97\) − 10.0000i − 1.01535i −0.861550 − 0.507673i \(-0.830506\pi\)
0.861550 − 0.507673i \(-0.169494\pi\)
\(98\) − 7.00000i − 0.707107i
\(99\) −5.00000 −0.502519
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 1425.2.c.h.799.1 2
5.2 odd 4 1425.2.a.h.1.1 yes 1
5.3 odd 4 1425.2.a.b.1.1 ✓ 1
5.4 even 2 inner 1425.2.c.h.799.2 2
15.2 even 4 4275.2.a.f.1.1 1
15.8 even 4 4275.2.a.l.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1425.2.a.b.1.1 ✓ 1 5.3 odd 4
1425.2.a.h.1.1 yes 1 5.2 odd 4
1425.2.c.h.799.1 2 1.1 even 1 trivial
1425.2.c.h.799.2 2 5.4 even 2 inner
4275.2.a.f.1.1 1 15.2 even 4
4275.2.a.l.1.1 1 15.8 even 4