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Properties

Label 624.4.bv.h.433.1
Level $624$
Weight $4$
Character 624.433
Analytic conductor $36.817$
Analytic rank $0$
Dimension $10$
Inner twists $2$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [624,4,Mod(49,624)] 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("624.49"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(624, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 0, 5])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 624 = 2^{4} \cdot 3 \cdot 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 624.bv (of order \(6\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [10,0,15,0,0,0,-30] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(36.8171918436\)
Analytic rank: \(0\)
Dimension: \(10\)
Relative dimension: \(5\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} + \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} + 70x^{8} + 1645x^{6} + 14700x^{4} + 44100x^{2} + 27648 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{6}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 39)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 433.1
Root \(-5.04537i\) of defining polynomial
Character \(\chi\) \(=\) 624.433
Dual form 624.4.bv.h.49.5

$q$-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)
 
\(f(q)\) \(=\) \(q+(1.50000 - 2.59808i) q^{3} -20.1174i q^{5} +(13.3609 - 7.71395i) q^{7} +(-4.50000 - 7.79423i) q^{9} +(-23.3283 - 13.4686i) q^{11} +(-3.96071 - 46.7045i) q^{13} +(-52.2665 - 30.1761i) q^{15} +(11.6167 + 20.1207i) q^{17} +(39.0399 - 22.5397i) q^{19} -46.2837i q^{21} +(71.0050 - 122.984i) q^{23} -279.710 q^{25} -27.0000 q^{27} +(-1.14534 + 1.98379i) q^{29} +37.7740i q^{31} +(-69.9849 + 40.4058i) q^{33} +(-155.185 - 268.787i) q^{35} +(271.793 + 156.920i) q^{37} +(-127.283 - 59.7666i) q^{39} +(5.08201 + 2.93410i) q^{41} +(180.449 + 312.547i) q^{43} +(-156.800 + 90.5283i) q^{45} -209.748i q^{47} +(-52.4900 + 90.9154i) q^{49} +69.7003 q^{51} +276.886 q^{53} +(-270.953 + 469.305i) q^{55} -135.238i q^{57} +(-470.415 + 271.594i) q^{59} +(-102.894 - 178.218i) q^{61} +(-120.249 - 69.4255i) q^{63} +(-939.573 + 79.6791i) q^{65} +(426.585 + 246.289i) q^{67} +(-213.015 - 368.953i) q^{69} +(-716.081 + 413.430i) q^{71} -66.1205i q^{73} +(-419.564 + 726.707i) q^{75} -415.584 q^{77} -317.642 q^{79} +(-40.5000 + 70.1481i) q^{81} -141.450i q^{83} +(404.777 - 233.698i) q^{85} +(3.43602 + 5.95136i) q^{87} +(555.399 + 320.660i) q^{89} +(-413.195 - 593.464i) q^{91} +(98.1396 + 56.6609i) q^{93} +(-453.440 - 785.381i) q^{95} +(-965.551 + 557.461i) q^{97} +242.435i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + 15 q^{3} - 30 q^{7} - 45 q^{9} - 60 q^{11} + 25 q^{13} - 45 q^{15} + 105 q^{17} - 180 q^{19} + 60 q^{23} - 960 q^{25} - 270 q^{27} - 495 q^{29} - 180 q^{33} - 60 q^{35} - 405 q^{37} - 345 q^{39}+ \cdots - 3750 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/624\mathbb{Z}\right)^\times\).

\(n\) \(79\) \(145\) \(209\) \(469\)
\(\chi(n)\) \(1\) \(e\left(\frac{1}{6}\right)\) \(1\) \(1\)

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
\(3\) 1.50000 − 2.59808i 0.288675 − 0.500000i
\(4\) 0 0
\(5\) − 20.1174i − 1.79935i −0.436556 − 0.899677i \(-0.643802\pi\)
0.436556 − 0.899677i \(-0.356198\pi\)
\(6\) 0 0
\(7\) 13.3609 − 7.71395i 0.721423 − 0.416514i −0.0938530 − 0.995586i \(-0.529918\pi\)
0.815276 + 0.579072i \(0.196585\pi\)
\(8\) 0 0
\(9\) −4.50000 − 7.79423i −0.166667 − 0.288675i
\(10\) 0 0
\(11\) −23.3283 − 13.4686i −0.639432 − 0.369176i 0.144964 − 0.989437i \(-0.453693\pi\)
−0.784396 + 0.620261i \(0.787027\pi\)
\(12\) 0 0
\(13\) −3.96071 − 46.7045i −0.0845002 − 0.996423i
\(14\) 0 0
\(15\) −52.2665 − 30.1761i −0.899677 − 0.519429i
\(16\) 0 0
\(17\) 11.6167 + 20.1207i 0.165733 + 0.287059i 0.936915 − 0.349556i \(-0.113668\pi\)
−0.771182 + 0.636615i \(0.780334\pi\)
\(18\) 0 0
\(19\) 39.0399 − 22.5397i 0.471388 − 0.272156i −0.245433 − 0.969414i \(-0.578930\pi\)
0.716821 + 0.697258i \(0.245597\pi\)
\(20\) 0 0
\(21\) − 46.2837i − 0.480949i
\(22\) 0 0
\(23\) 71.0050 − 122.984i 0.643720 − 1.11496i −0.340875 − 0.940109i \(-0.610723\pi\)
0.984595 − 0.174848i \(-0.0559434\pi\)
\(24\) 0 0
\(25\) −279.710 −2.23768
\(26\) 0 0
\(27\) −27.0000 −0.192450
\(28\) 0 0
\(29\) −1.14534 + 1.98379i −0.00733394 + 0.0127028i −0.869669 − 0.493635i \(-0.835668\pi\)
0.862335 + 0.506338i \(0.169001\pi\)
\(30\) 0 0
\(31\) 37.7740i 0.218852i 0.993995 + 0.109426i \(0.0349012\pi\)
−0.993995 + 0.109426i \(0.965099\pi\)
\(32\) 0 0
\(33\) −69.9849 + 40.4058i −0.369176 + 0.213144i
\(34\) 0 0
\(35\) −155.185 − 268.787i −0.749456 − 1.29810i
\(36\) 0 0
\(37\) 271.793 + 156.920i 1.20764 + 0.697228i 0.962242 − 0.272195i \(-0.0877497\pi\)
0.245393 + 0.969424i \(0.421083\pi\)
\(38\) 0 0
\(39\) −127.283 − 59.7666i −0.522605 − 0.245393i
\(40\) 0 0
\(41\) 5.08201 + 2.93410i 0.0193580 + 0.0111763i 0.509648 − 0.860383i \(-0.329776\pi\)
−0.490290 + 0.871559i \(0.663109\pi\)
\(42\) 0 0
\(43\) 180.449 + 312.547i 0.639958 + 1.10844i 0.985441 + 0.170015i \(0.0543817\pi\)
−0.345483 + 0.938425i \(0.612285\pi\)
\(44\) 0 0
\(45\) −156.800 + 90.5283i −0.519429 + 0.299892i
\(46\) 0 0
\(47\) − 209.748i − 0.650956i −0.945550 − 0.325478i \(-0.894475\pi\)
0.945550 − 0.325478i \(-0.105525\pi\)
\(48\) 0 0
\(49\) −52.4900 + 90.9154i −0.153032 + 0.265060i
\(50\) 0 0
\(51\) 69.7003 0.191372
\(52\) 0 0
\(53\) 276.886 0.717609 0.358804 − 0.933413i \(-0.383185\pi\)
0.358804 + 0.933413i \(0.383185\pi\)
\(54\) 0 0
\(55\) −270.953 + 469.305i −0.664278 + 1.15056i
\(56\) 0 0
\(57\) − 135.238i − 0.314259i
\(58\) 0 0
\(59\) −470.415 + 271.594i −1.03801 + 0.599298i −0.919270 − 0.393627i \(-0.871220\pi\)
−0.118744 + 0.992925i \(0.537887\pi\)
\(60\) 0 0
\(61\) −102.894 − 178.218i −0.215971 − 0.374073i 0.737601 − 0.675236i \(-0.235958\pi\)
−0.953573 + 0.301163i \(0.902625\pi\)
\(62\) 0 0
\(63\) −120.249 − 69.4255i −0.240474 − 0.138838i
\(64\) 0 0
\(65\) −939.573 + 79.6791i −1.79292 + 0.152046i
\(66\) 0 0
\(67\) 426.585 + 246.289i 0.777846 + 0.449090i 0.835666 − 0.549237i \(-0.185082\pi\)
−0.0578203 + 0.998327i \(0.518415\pi\)
\(68\) 0 0
\(69\) −213.015 − 368.953i −0.371652 − 0.643720i
\(70\) 0 0
\(71\) −716.081 + 413.430i −1.19695 + 0.691057i −0.959873 − 0.280435i \(-0.909521\pi\)
−0.237073 + 0.971492i \(0.576188\pi\)
\(72\) 0 0
\(73\) − 66.1205i − 0.106011i −0.998594 − 0.0530056i \(-0.983120\pi\)
0.998594 − 0.0530056i \(-0.0168801\pi\)
\(74\) 0 0
\(75\) −419.564 + 726.707i −0.645962 + 1.11884i
\(76\) 0 0
\(77\) −415.584 −0.615068
\(78\) 0 0
\(79\) −317.642 −0.452374 −0.226187 − 0.974084i \(-0.572626\pi\)
−0.226187 + 0.974084i \(0.572626\pi\)
\(80\) 0 0
\(81\) −40.5000 + 70.1481i −0.0555556 + 0.0962250i
\(82\) 0 0
\(83\) − 141.450i − 0.187063i −0.995616 − 0.0935313i \(-0.970184\pi\)
0.995616 − 0.0935313i \(-0.0298155\pi\)
\(84\) 0 0
\(85\) 404.777 − 233.698i 0.516520 − 0.298213i
\(86\) 0 0
\(87\) 3.43602 + 5.95136i 0.00423425 + 0.00733394i
\(88\) 0 0
\(89\) 555.399 + 320.660i 0.661486 + 0.381909i 0.792843 − 0.609426i \(-0.208600\pi\)
−0.131357 + 0.991335i \(0.541933\pi\)
\(90\) 0 0
\(91\) −413.195 − 593.464i −0.475985 − 0.683648i
\(92\) 0 0
\(93\) 98.1396 + 56.6609i 0.109426 + 0.0631771i
\(94\) 0 0
\(95\) −453.440 − 785.381i −0.489705 − 0.848194i
\(96\) 0 0
\(97\) −965.551 + 557.461i −1.01069 + 0.583522i −0.911394 − 0.411536i \(-0.864992\pi\)
−0.0992962 + 0.995058i \(0.531659\pi\)
\(98\) 0 0
\(99\) 242.435i 0.246117i
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 624.4.bv.h.433.1 10
4.3 odd 2 39.4.j.c.4.5 ✓ 10
12.11 even 2 117.4.q.e.82.1 10
13.10 even 6 inner 624.4.bv.h.49.5 10
52.7 even 12 507.4.a.r.1.9 10
52.19 even 12 507.4.a.r.1.2 10
52.23 odd 6 39.4.j.c.10.5 yes 10
52.35 odd 6 507.4.b.i.337.2 10
52.43 odd 6 507.4.b.i.337.9 10
156.23 even 6 117.4.q.e.10.1 10
156.59 odd 12 1521.4.a.bk.1.2 10
156.71 odd 12 1521.4.a.bk.1.9 10
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
39.4.j.c.4.5 ✓ 10 4.3 odd 2
39.4.j.c.10.5 yes 10 52.23 odd 6
117.4.q.e.10.1 10 156.23 even 6
117.4.q.e.82.1 10 12.11 even 2
507.4.a.r.1.2 10 52.19 even 12
507.4.a.r.1.9 10 52.7 even 12
507.4.b.i.337.2 10 52.35 odd 6
507.4.b.i.337.9 10 52.43 odd 6
624.4.bv.h.49.5 10 13.10 even 6 inner
624.4.bv.h.433.1 10 1.1 even 1 trivial
1521.4.a.bk.1.2 10 156.59 odd 12
1521.4.a.bk.1.9 10 156.71 odd 12