Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [4600,2,Mod(4049,4600)] 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("4600.4049"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4600, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 4600 = 2^{3} \cdot 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4600.e (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [10,0,0,0,0,0,0,0,-6,0,0,0,0,0,0,0,0,0,0] 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: \(36.7311849298\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: 10.0.278379347567616.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} + 18x^{8} + 117x^{6} + 333x^{4} + 396x^{2} + 144 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 4049.7
Root \(1.36629i\) of defining polynomial
Character \(\chi\) \(=\) 4600.4049
Dual form 4600.2.e.w.4049.4

$q$-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)
 
\(f(q)\) \(=\) \(q+1.36629i q^{3} +3.28093i q^{7} +1.13327 q^{9} +3.49709 q^{11} +3.41420i q^{13} +7.46023i q^{17} -3.49709 q^{19} -4.48269 q^{21} +1.00000i q^{23} +5.64722i q^{27} +3.46268 q^{29} +2.01105 q^{31} +4.77803i q^{33} -0.511497i q^{37} -4.66477 q^{39} -7.07954 q^{41} -2.76452i q^{43} +0.889198i q^{47} -3.76452 q^{49} -10.1928 q^{51} -14.2383i q^{53} -4.77803i q^{57} +4.71325 q^{59} +13.4769 q^{61} +3.71817i q^{63} +2.30025i q^{67} -1.36629 q^{69} -10.6214 q^{71} +3.70765i q^{73} +11.4737i q^{77} +7.97978 q^{79} -4.31592 q^{81} +9.42336i q^{83} +4.73101i q^{87} -0.801390 q^{89} -11.2018 q^{91} +2.74766i q^{93} -11.7722i q^{97} +3.96313 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 6 q^{9} - 24 q^{29} - 36 q^{31} - 18 q^{39} - 12 q^{41} - 30 q^{49} - 12 q^{51} + 2 q^{59} + 20 q^{61} + 16 q^{71} - 54 q^{81} - 28 q^{89} - 92 q^{91} + 12 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4600\mathbb{Z}\right)^\times\).

\(n\) \(1151\) \(1201\) \(2301\) \(2577\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 1.36629i 0.788825i 0.918933 + 0.394413i \(0.129052\pi\)
−0.918933 + 0.394413i \(0.870948\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 3.28093i 1.24008i 0.784572 + 0.620038i \(0.212883\pi\)
−0.784572 + 0.620038i \(0.787117\pi\)
\(8\) 0 0
\(9\) 1.13327 0.377755
\(10\) 0 0
\(11\) 3.49709 1.05441 0.527207 − 0.849737i \(-0.323239\pi\)
0.527207 + 0.849737i \(0.323239\pi\)
\(12\) 0 0
\(13\) 3.41420i 0.946928i 0.880813 + 0.473464i \(0.156997\pi\)
−0.880813 + 0.473464i \(0.843003\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 7.46023i 1.80937i 0.426080 + 0.904685i \(0.359894\pi\)
−0.426080 + 0.904685i \(0.640106\pi\)
\(18\) 0 0
\(19\) −3.49709 −0.802288 −0.401144 − 0.916015i \(-0.631387\pi\)
−0.401144 + 0.916015i \(0.631387\pi\)
\(20\) 0 0
\(21\) −4.48269 −0.978203
\(22\) 0 0
\(23\) 1.00000i 0.208514i
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 5.64722i 1.08681i
\(28\) 0 0
\(29\) 3.46268 0.643004 0.321502 − 0.946909i \(-0.395812\pi\)
0.321502 + 0.946909i \(0.395812\pi\)
\(30\) 0 0
\(31\) 2.01105 0.361195 0.180597 − 0.983557i \(-0.442197\pi\)
0.180597 + 0.983557i \(0.442197\pi\)
\(32\) 0 0
\(33\) 4.77803i 0.831748i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) − 0.511497i − 0.0840896i −0.999116 − 0.0420448i \(-0.986613\pi\)
0.999116 − 0.0420448i \(-0.0133872\pi\)
\(38\) 0 0
\(39\) −4.66477 −0.746961
\(40\) 0 0
\(41\) −7.07954 −1.10564 −0.552819 − 0.833301i \(-0.686448\pi\)
−0.552819 + 0.833301i \(0.686448\pi\)
\(42\) 0 0
\(43\) − 2.76452i − 0.421586i −0.977531 − 0.210793i \(-0.932395\pi\)
0.977531 − 0.210793i \(-0.0676046\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0.889198i 0.129703i 0.997895 + 0.0648514i \(0.0206573\pi\)
−0.997895 + 0.0648514i \(0.979343\pi\)
\(48\) 0 0
\(49\) −3.76452 −0.537789
\(50\) 0 0
\(51\) −10.1928 −1.42728
\(52\) 0 0
\(53\) − 14.2383i − 1.95577i −0.209132 − 0.977887i \(-0.567064\pi\)
0.209132 − 0.977887i \(-0.432936\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) − 4.77803i − 0.632865i
\(58\) 0 0
\(59\) 4.71325 0.613613 0.306807 − 0.951772i \(-0.400739\pi\)
0.306807 + 0.951772i \(0.400739\pi\)
\(60\) 0 0
\(61\) 13.4769 1.72554 0.862769 − 0.505599i \(-0.168728\pi\)
0.862769 + 0.505599i \(0.168728\pi\)
\(62\) 0 0
\(63\) 3.71817i 0.468445i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 2.30025i 0.281020i 0.990079 + 0.140510i \(0.0448743\pi\)
−0.990079 + 0.140510i \(0.955126\pi\)
\(68\) 0 0
\(69\) −1.36629 −0.164481
\(70\) 0 0
\(71\) −10.6214 −1.26053 −0.630264 − 0.776381i \(-0.717053\pi\)
−0.630264 + 0.776381i \(0.717053\pi\)
\(72\) 0 0
\(73\) 3.70765i 0.433948i 0.976177 + 0.216974i \(0.0696186\pi\)
−0.976177 + 0.216974i \(0.930381\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 11.4737i 1.30755i
\(78\) 0 0
\(79\) 7.97978 0.897796 0.448898 − 0.893583i \(-0.351817\pi\)
0.448898 + 0.893583i \(0.351817\pi\)
\(80\) 0 0
\(81\) −4.31592 −0.479546
\(82\) 0 0
\(83\) 9.42336i 1.03435i 0.855880 + 0.517174i \(0.173016\pi\)
−0.855880 + 0.517174i \(0.826984\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 4.73101i 0.507218i
\(88\) 0 0
\(89\) −0.801390 −0.0849471 −0.0424736 − 0.999098i \(-0.513524\pi\)
−0.0424736 + 0.999098i \(0.513524\pi\)
\(90\) 0 0
\(91\) −11.2018 −1.17426
\(92\) 0 0
\(93\) 2.74766i 0.284919i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) − 11.7722i − 1.19529i −0.801762 − 0.597644i \(-0.796104\pi\)
0.801762 − 0.597644i \(-0.203896\pi\)
\(98\) 0 0
\(99\) 3.96313 0.398310
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 4600.2.e.w.4049.7 10
5.2 odd 4 4600.2.a.bd.1.4 ✓ 5
5.3 odd 4 4600.2.a.bf.1.2 yes 5
5.4 even 2 inner 4600.2.e.w.4049.4 10
20.3 even 4 9200.2.a.ct.1.4 5
20.7 even 4 9200.2.a.cv.1.2 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4600.2.a.bd.1.4 ✓ 5 5.2 odd 4
4600.2.a.bf.1.2 yes 5 5.3 odd 4
4600.2.e.w.4049.4 10 5.4 even 2 inner
4600.2.e.w.4049.7 10 1.1 even 1 trivial
9200.2.a.ct.1.4 5 20.3 even 4
9200.2.a.cv.1.2 5 20.7 even 4