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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [4650,2,Mod(3349,4650)] 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("4650.3349"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4650, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 4650 = 2 \cdot 3 \cdot 5^{2} \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4650.d (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,-4,0,-4,0,0,-4,0,2,0,0,-2,0,4,0,0,6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(19)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(37.1304369399\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{17})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 9x^{2} + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 930)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 3349.4
Root \(2.56155i\) of defining polynomial
Character \(\chi\) \(=\) 4650.3349
Dual form 4650.2.d.bd.3349.1

$q$-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)
 
\(f(q)\) \(=\) \(q+1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} +2.56155i q^{7} -1.00000i q^{8} -1.00000 q^{9} +2.56155 q^{11} -1.00000i q^{12} -2.00000i q^{13} -2.56155 q^{14} +1.00000 q^{16} -3.12311i q^{17} -1.00000i q^{18} +7.68466 q^{19} -2.56155 q^{21} +2.56155i q^{22} -1.43845i q^{23} +1.00000 q^{24} +2.00000 q^{26} -1.00000i q^{27} -2.56155i q^{28} -7.12311 q^{29} +1.00000 q^{31} +1.00000i q^{32} +2.56155i q^{33} +3.12311 q^{34} +1.00000 q^{36} -3.12311i q^{37} +7.68466i q^{38} +2.00000 q^{39} +7.12311 q^{41} -2.56155i q^{42} -12.8078i q^{43} -2.56155 q^{44} +1.43845 q^{46} +5.12311i q^{47} +1.00000i q^{48} +0.438447 q^{49} +3.12311 q^{51} +2.00000i q^{52} -7.43845i q^{53} +1.00000 q^{54} +2.56155 q^{56} +7.68466i q^{57} -7.12311i q^{58} +13.1231 q^{59} +6.00000 q^{61} +1.00000i q^{62} -2.56155i q^{63} -1.00000 q^{64} -2.56155 q^{66} -15.3693i q^{67} +3.12311i q^{68} +1.43845 q^{69} -7.68466 q^{71} +1.00000i q^{72} +10.8078i q^{73} +3.12311 q^{74} -7.68466 q^{76} +6.56155i q^{77} +2.00000i q^{78} +4.31534 q^{79} +1.00000 q^{81} +7.12311i q^{82} -14.2462i q^{83} +2.56155 q^{84} +12.8078 q^{86} -7.12311i q^{87} -2.56155i q^{88} +13.6847 q^{89} +5.12311 q^{91} +1.43845i q^{92} +1.00000i q^{93} -5.12311 q^{94} -1.00000 q^{96} -6.00000i q^{97} +0.438447i q^{98} -2.56155 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{4} - 4 q^{6} - 4 q^{9} + 2 q^{11} - 2 q^{14} + 4 q^{16} + 6 q^{19} - 2 q^{21} + 4 q^{24} + 8 q^{26} - 12 q^{29} + 4 q^{31} - 4 q^{34} + 4 q^{36} + 8 q^{39} + 12 q^{41} - 2 q^{44} + 14 q^{46}+ \cdots - 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4650\mathbb{Z}\right)^\times\).

\(n\) \(1801\) \(2977\) \(3101\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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
\(3\) 1.00000i 0.577350i
\(4\) −1.00000 −0.500000
\(5\) 0 0
\(6\) −1.00000 −0.408248
\(7\) 2.56155i 0.968176i 0.875019 + 0.484088i \(0.160849\pi\)
−0.875019 + 0.484088i \(0.839151\pi\)
\(8\) − 1.00000i − 0.353553i
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) 2.56155 0.772337 0.386169 − 0.922428i \(-0.373798\pi\)
0.386169 + 0.922428i \(0.373798\pi\)
\(12\) − 1.00000i − 0.288675i
\(13\) − 2.00000i − 0.554700i −0.960769 − 0.277350i \(-0.910544\pi\)
0.960769 − 0.277350i \(-0.0894562\pi\)
\(14\) −2.56155 −0.684604
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) − 3.12311i − 0.757464i −0.925506 − 0.378732i \(-0.876360\pi\)
0.925506 − 0.378732i \(-0.123640\pi\)
\(18\) − 1.00000i − 0.235702i
\(19\) 7.68466 1.76298 0.881491 − 0.472201i \(-0.156540\pi\)
0.881491 + 0.472201i \(0.156540\pi\)
\(20\) 0 0
\(21\) −2.56155 −0.558977
\(22\) 2.56155i 0.546125i
\(23\) − 1.43845i − 0.299937i −0.988691 − 0.149968i \(-0.952083\pi\)
0.988691 − 0.149968i \(-0.0479172\pi\)
\(24\) 1.00000 0.204124
\(25\) 0 0
\(26\) 2.00000 0.392232
\(27\) − 1.00000i − 0.192450i
\(28\) − 2.56155i − 0.484088i
\(29\) −7.12311 −1.32273 −0.661364 − 0.750065i \(-0.730022\pi\)
−0.661364 + 0.750065i \(0.730022\pi\)
\(30\) 0 0
\(31\) 1.00000 0.179605
\(32\) 1.00000i 0.176777i
\(33\) 2.56155i 0.445909i
\(34\) 3.12311 0.535608
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) − 3.12311i − 0.513435i −0.966486 − 0.256718i \(-0.917359\pi\)
0.966486 − 0.256718i \(-0.0826411\pi\)
\(38\) 7.68466i 1.24662i
\(39\) 2.00000 0.320256
\(40\) 0 0
\(41\) 7.12311 1.11244 0.556221 − 0.831034i \(-0.312251\pi\)
0.556221 + 0.831034i \(0.312251\pi\)
\(42\) − 2.56155i − 0.395256i
\(43\) − 12.8078i − 1.95317i −0.215142 − 0.976583i \(-0.569021\pi\)
0.215142 − 0.976583i \(-0.430979\pi\)
\(44\) −2.56155 −0.386169
\(45\) 0 0
\(46\) 1.43845 0.212087
\(47\) 5.12311i 0.747282i 0.927573 + 0.373641i \(0.121891\pi\)
−0.927573 + 0.373641i \(0.878109\pi\)
\(48\) 1.00000i 0.144338i
\(49\) 0.438447 0.0626353
\(50\) 0 0
\(51\) 3.12311 0.437322
\(52\) 2.00000i 0.277350i
\(53\) − 7.43845i − 1.02175i −0.859655 − 0.510875i \(-0.829322\pi\)
0.859655 − 0.510875i \(-0.170678\pi\)
\(54\) 1.00000 0.136083
\(55\) 0 0
\(56\) 2.56155 0.342302
\(57\) 7.68466i 1.01786i
\(58\) − 7.12311i − 0.935310i
\(59\) 13.1231 1.70848 0.854241 − 0.519877i \(-0.174022\pi\)
0.854241 + 0.519877i \(0.174022\pi\)
\(60\) 0 0
\(61\) 6.00000 0.768221 0.384111 − 0.923287i \(-0.374508\pi\)
0.384111 + 0.923287i \(0.374508\pi\)
\(62\) 1.00000i 0.127000i
\(63\) − 2.56155i − 0.322725i
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) −2.56155 −0.315305
\(67\) − 15.3693i − 1.87766i −0.344380 − 0.938830i \(-0.611911\pi\)
0.344380 − 0.938830i \(-0.388089\pi\)
\(68\) 3.12311i 0.378732i
\(69\) 1.43845 0.173169
\(70\) 0 0
\(71\) −7.68466 −0.912001 −0.456001 − 0.889979i \(-0.650719\pi\)
−0.456001 + 0.889979i \(0.650719\pi\)
\(72\) 1.00000i 0.117851i
\(73\) 10.8078i 1.26495i 0.774580 + 0.632477i \(0.217962\pi\)
−0.774580 + 0.632477i \(0.782038\pi\)
\(74\) 3.12311 0.363054
\(75\) 0 0
\(76\) −7.68466 −0.881491
\(77\) 6.56155i 0.747758i
\(78\) 2.00000i 0.226455i
\(79\) 4.31534 0.485514 0.242757 − 0.970087i \(-0.421948\pi\)
0.242757 + 0.970087i \(0.421948\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 7.12311i 0.786615i
\(83\) − 14.2462i − 1.56372i −0.623451 − 0.781862i \(-0.714270\pi\)
0.623451 − 0.781862i \(-0.285730\pi\)
\(84\) 2.56155 0.279488
\(85\) 0 0
\(86\) 12.8078 1.38110
\(87\) − 7.12311i − 0.763677i
\(88\) − 2.56155i − 0.273062i
\(89\) 13.6847 1.45057 0.725285 − 0.688448i \(-0.241708\pi\)
0.725285 + 0.688448i \(0.241708\pi\)
\(90\) 0 0
\(91\) 5.12311 0.537047
\(92\) 1.43845i 0.149968i
\(93\) 1.00000i 0.103695i
\(94\) −5.12311 −0.528408
\(95\) 0 0
\(96\) −1.00000 −0.102062
\(97\) − 6.00000i − 0.609208i −0.952479 − 0.304604i \(-0.901476\pi\)
0.952479 − 0.304604i \(-0.0985241\pi\)
\(98\) 0.438447i 0.0442899i
\(99\) −2.56155 −0.257446
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 4650.2.d.bd.3349.4 4
5.2 odd 4 4650.2.a.ce.1.1 2
5.3 odd 4 930.2.a.p.1.2 ✓ 2
5.4 even 2 inner 4650.2.d.bd.3349.1 4
15.8 even 4 2790.2.a.be.1.2 2
20.3 even 4 7440.2.a.bl.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
930.2.a.p.1.2 ✓ 2 5.3 odd 4
2790.2.a.be.1.2 2 15.8 even 4
4650.2.a.ce.1.1 2 5.2 odd 4
4650.2.d.bd.3349.1 4 5.4 even 2 inner
4650.2.d.bd.3349.4 4 1.1 even 1 trivial
7440.2.a.bl.1.1 2 20.3 even 4