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 4049.3
Root \(-1.83957i\) of defining polynomial
Character \(\chi\) \(=\) 4600.4049
Dual form 4600.2.e.w.4049.8

$q$-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)
 
\(f(q)\) \(=\) \(q-1.83957i q^{3} -3.97272i q^{7} -0.384010 q^{9} +2.10339 q^{11} -5.35673i q^{13} +1.29567i q^{17} -2.10339 q^{19} -7.30809 q^{21} +1.00000i q^{23} -4.81229i q^{27} -6.03130 q^{29} -8.32489 q^{31} -3.86933i q^{33} +5.10131i q^{37} -9.85407 q^{39} -8.33994 q^{41} -7.78253i q^{43} +11.3964i q^{47} -8.78253 q^{49} +2.38347 q^{51} +0.573664i q^{53} +3.86933i q^{57} +9.17951 q^{59} +13.5149 q^{61} +1.52557i q^{63} -15.8314i q^{67} +1.83957 q^{69} +14.9449 q^{71} -8.36459i q^{73} -8.35619i q^{77} +9.41149 q^{79} -10.0046 q^{81} -1.51206i q^{83} +11.0950i q^{87} -10.5903 q^{89} -21.2808 q^{91} +15.3142i q^{93} -0.337451i q^{97} -0.807724 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 6 q^{9} - 24 q^{29} - 36 q^{31} - 18 q^{39} - 12 q^{41} - 30 q^{49} - 12 q^{51} + 2 q^{59} + 20 q^{61} + 16 q^{71} - 54 q^{81} - 28 q^{89} - 92 q^{91} + 12 q^{99}+O(q^{100}) \) 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
\(3\) − 1.83957i − 1.06208i −0.847348 − 0.531038i \(-0.821802\pi\)
0.847348 − 0.531038i \(-0.178198\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) − 3.97272i − 1.50155i −0.660559 − 0.750774i \(-0.729681\pi\)
0.660559 − 0.750774i \(-0.270319\pi\)
\(8\) 0 0
\(9\) −0.384010 −0.128003
\(10\) 0 0
\(11\) 2.10339 0.634196 0.317098 − 0.948393i \(-0.397292\pi\)
0.317098 + 0.948393i \(0.397292\pi\)
\(12\) 0 0
\(13\) − 5.35673i − 1.48569i −0.669463 − 0.742845i \(-0.733476\pi\)
0.669463 − 0.742845i \(-0.266524\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 1.29567i 0.314246i 0.987579 + 0.157123i \(0.0502218\pi\)
−0.987579 + 0.157123i \(0.949778\pi\)
\(18\) 0 0
\(19\) −2.10339 −0.482551 −0.241276 − 0.970457i \(-0.577566\pi\)
−0.241276 + 0.970457i \(0.577566\pi\)
\(20\) 0 0
\(21\) −7.30809 −1.59476
\(22\) 0 0
\(23\) 1.00000i 0.208514i
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) − 4.81229i − 0.926126i
\(28\) 0 0
\(29\) −6.03130 −1.11998 −0.559992 − 0.828498i \(-0.689196\pi\)
−0.559992 + 0.828498i \(0.689196\pi\)
\(30\) 0 0
\(31\) −8.32489 −1.49519 −0.747597 − 0.664153i \(-0.768793\pi\)
−0.747597 + 0.664153i \(0.768793\pi\)
\(32\) 0 0
\(33\) − 3.86933i − 0.673564i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 5.10131i 0.838650i 0.907836 + 0.419325i \(0.137733\pi\)
−0.907836 + 0.419325i \(0.862267\pi\)
\(38\) 0 0
\(39\) −9.85407 −1.57791
\(40\) 0 0
\(41\) −8.33994 −1.30248 −0.651240 − 0.758872i \(-0.725751\pi\)
−0.651240 + 0.758872i \(0.725751\pi\)
\(42\) 0 0
\(43\) − 7.78253i − 1.18682i −0.804899 − 0.593412i \(-0.797780\pi\)
0.804899 − 0.593412i \(-0.202220\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 11.3964i 1.66234i 0.556018 + 0.831171i \(0.312329\pi\)
−0.556018 + 0.831171i \(0.687671\pi\)
\(48\) 0 0
\(49\) −8.78253 −1.25465
\(50\) 0 0
\(51\) 2.38347 0.333752
\(52\) 0 0
\(53\) 0.573664i 0.0787988i 0.999224 + 0.0393994i \(0.0125445\pi\)
−0.999224 + 0.0393994i \(0.987456\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 3.86933i 0.512505i
\(58\) 0 0
\(59\) 9.17951 1.19507 0.597535 − 0.801843i \(-0.296147\pi\)
0.597535 + 0.801843i \(0.296147\pi\)
\(60\) 0 0
\(61\) 13.5149 1.73040 0.865201 − 0.501425i \(-0.167191\pi\)
0.865201 + 0.501425i \(0.167191\pi\)
\(62\) 0 0
\(63\) 1.52557i 0.192203i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) − 15.8314i − 1.93411i −0.254569 − 0.967055i \(-0.581934\pi\)
0.254569 − 0.967055i \(-0.418066\pi\)
\(68\) 0 0
\(69\) 1.83957 0.221458
\(70\) 0 0
\(71\) 14.9449 1.77363 0.886817 − 0.462121i \(-0.152911\pi\)
0.886817 + 0.462121i \(0.152911\pi\)
\(72\) 0 0
\(73\) − 8.36459i − 0.979002i −0.872003 − 0.489501i \(-0.837179\pi\)
0.872003 − 0.489501i \(-0.162821\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) − 8.35619i − 0.952276i
\(78\) 0 0
\(79\) 9.41149 1.05887 0.529437 − 0.848349i \(-0.322403\pi\)
0.529437 + 0.848349i \(0.322403\pi\)
\(80\) 0 0
\(81\) −10.0046 −1.11162
\(82\) 0 0
\(83\) − 1.51206i − 0.165970i −0.996551 − 0.0829849i \(-0.973555\pi\)
0.996551 − 0.0829849i \(-0.0264453\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 11.0950i 1.18951i
\(88\) 0 0
\(89\) −10.5903 −1.12256 −0.561282 − 0.827624i \(-0.689692\pi\)
−0.561282 + 0.827624i \(0.689692\pi\)
\(90\) 0 0
\(91\) −21.2808 −2.23084
\(92\) 0 0
\(93\) 15.3142i 1.58801i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) − 0.337451i − 0.0342630i −0.999853 − 0.0171315i \(-0.994547\pi\)
0.999853 − 0.0171315i \(-0.00545339\pi\)
\(98\) 0 0
\(99\) −0.807724 −0.0811793
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.4049.3 10
5.2 odd 4 4600.2.a.bd.1.2 ✓ 5
5.3 odd 4 4600.2.a.bf.1.4 yes 5
5.4 even 2 inner 4600.2.e.w.4049.8 10
20.3 even 4 9200.2.a.ct.1.2 5
20.7 even 4 9200.2.a.cv.1.4 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4600.2.a.bd.1.2 ✓ 5 5.2 odd 4
4600.2.a.bf.1.4 yes 5 5.3 odd 4
4600.2.e.w.4049.3 10 1.1 even 1 trivial
4600.2.e.w.4049.8 10 5.4 even 2 inner
9200.2.a.ct.1.2 5 20.3 even 4
9200.2.a.cv.1.4 5 20.7 even 4