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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 433.2
Root \(0.917374i\) of defining polynomial
Character \(\chi\) \(=\) 624.433
Dual form 624.4.bv.h.49.4

$q$-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)
 
\(f(q)\) \(=\) \(q+(1.50000 - 2.59808i) q^{3} -15.4704i q^{5} +(-17.8257 + 10.2917i) q^{7} +(-4.50000 - 7.79423i) q^{9} +(57.0209 + 32.9210i) q^{11} +(19.2429 + 42.7400i) q^{13} +(-40.1933 - 23.2056i) q^{15} +(22.1478 + 38.3611i) q^{17} +(-127.352 + 73.5266i) q^{19} +61.7500i q^{21} +(-26.5793 + 46.0367i) q^{23} -114.334 q^{25} -27.0000 q^{27} +(19.3128 - 33.4508i) q^{29} -88.3894i q^{31} +(171.063 - 98.7630i) q^{33} +(159.216 + 275.771i) q^{35} +(68.3803 + 39.4794i) q^{37} +(139.906 + 14.1155i) q^{39} +(307.410 + 177.483i) q^{41} +(203.923 + 353.205i) q^{43} +(-120.580 + 69.6169i) q^{45} -67.9674i q^{47} +(40.3369 - 69.8656i) q^{49} +132.887 q^{51} +226.572 q^{53} +(509.302 - 882.136i) q^{55} +441.160i q^{57} +(123.002 - 71.0154i) q^{59} +(-133.416 - 231.083i) q^{61} +(160.431 + 92.6250i) q^{63} +(661.206 - 297.696i) q^{65} +(-356.098 - 205.593i) q^{67} +(79.7379 + 138.110i) q^{69} +(79.2458 - 45.7526i) q^{71} -63.1328i q^{73} +(-171.500 + 297.047i) q^{75} -1355.25 q^{77} +287.115 q^{79} +(-40.5000 + 70.1481i) q^{81} +373.812i q^{83} +(593.463 - 342.636i) q^{85} +(-57.9385 - 100.352i) q^{87} +(103.406 + 59.7013i) q^{89} +(-782.885 - 563.829i) q^{91} +(-229.643 - 132.584i) q^{93} +(1137.49 + 1970.18i) q^{95} +(480.341 - 277.325i) q^{97} -592.578i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + 15 q^{3} - 30 q^{7} - 45 q^{9} - 60 q^{11} + 25 q^{13} - 45 q^{15} + 105 q^{17} - 180 q^{19} + 60 q^{23} - 960 q^{25} - 270 q^{27} - 495 q^{29} - 180 q^{33} - 60 q^{35} - 405 q^{37} - 345 q^{39}+ \cdots - 3750 q^{97}+O(q^{100}) \) 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
\(5\) − 15.4704i − 1.38372i −0.722034 − 0.691858i \(-0.756792\pi\)
0.722034 − 0.691858i \(-0.243208\pi\)
\(6\) 0 0
\(7\) −17.8257 + 10.2917i −0.962497 + 0.555698i −0.896941 − 0.442151i \(-0.854216\pi\)
−0.0655563 + 0.997849i \(0.520882\pi\)
\(8\) 0 0
\(9\) −4.50000 − 7.79423i −0.166667 − 0.288675i
\(10\) 0 0
\(11\) 57.0209 + 32.9210i 1.56295 + 0.902369i 0.996957 + 0.0779583i \(0.0248401\pi\)
0.565992 + 0.824411i \(0.308493\pi\)
\(12\) 0 0
\(13\) 19.2429 + 42.7400i 0.410540 + 0.911842i
\(14\) 0 0
\(15\) −40.1933 − 23.2056i −0.691858 − 0.399444i
\(16\) 0 0
\(17\) 22.1478 + 38.3611i 0.315979 + 0.547291i 0.979645 − 0.200738i \(-0.0643339\pi\)
−0.663666 + 0.748029i \(0.731001\pi\)
\(18\) 0 0
\(19\) −127.352 + 73.5266i −1.53771 + 0.887798i −0.538738 + 0.842473i \(0.681099\pi\)
−0.998972 + 0.0453247i \(0.985568\pi\)
\(20\) 0 0
\(21\) 61.7500i 0.641665i
\(22\) 0 0
\(23\) −26.5793 + 46.0367i −0.240964 + 0.417362i −0.960989 − 0.276586i \(-0.910797\pi\)
0.720025 + 0.693948i \(0.244130\pi\)
\(24\) 0 0
\(25\) −114.334 −0.914669
\(26\) 0 0
\(27\) −27.0000 −0.192450
\(28\) 0 0
\(29\) 19.3128 − 33.4508i 0.123666 − 0.214195i −0.797545 − 0.603260i \(-0.793868\pi\)
0.921211 + 0.389064i \(0.127202\pi\)
\(30\) 0 0
\(31\) − 88.3894i − 0.512104i −0.966663 − 0.256052i \(-0.917578\pi\)
0.966663 − 0.256052i \(-0.0824218\pi\)
\(32\) 0 0
\(33\) 171.063 − 98.7630i 0.902369 − 0.520983i
\(34\) 0 0
\(35\) 159.216 + 275.771i 0.768928 + 1.33182i
\(36\) 0 0
\(37\) 68.3803 + 39.4794i 0.303828 + 0.175415i 0.644161 − 0.764890i \(-0.277206\pi\)
−0.340333 + 0.940305i \(0.610540\pi\)
\(38\) 0 0
\(39\) 139.906 + 14.1155i 0.574434 + 0.0579561i
\(40\) 0 0
\(41\) 307.410 + 177.483i 1.17096 + 0.676054i 0.953906 − 0.300106i \(-0.0970221\pi\)
0.217053 + 0.976160i \(0.430355\pi\)
\(42\) 0 0
\(43\) 203.923 + 353.205i 0.723208 + 1.25263i 0.959707 + 0.281002i \(0.0906666\pi\)
−0.236499 + 0.971632i \(0.576000\pi\)
\(44\) 0 0
\(45\) −120.580 + 69.6169i −0.399444 + 0.230619i
\(46\) 0 0
\(47\) − 67.9674i − 0.210938i −0.994423 − 0.105469i \(-0.966366\pi\)
0.994423 − 0.105469i \(-0.0336343\pi\)
\(48\) 0 0
\(49\) 40.3369 − 69.8656i 0.117600 − 0.203690i
\(50\) 0 0
\(51\) 132.887 0.364861
\(52\) 0 0
\(53\) 226.572 0.587209 0.293604 − 0.955927i \(-0.405145\pi\)
0.293604 + 0.955927i \(0.405145\pi\)
\(54\) 0 0
\(55\) 509.302 − 882.136i 1.24862 − 2.16268i
\(56\) 0 0
\(57\) 441.160i 1.02514i
\(58\) 0 0
\(59\) 123.002 − 71.0154i 0.271416 − 0.156702i −0.358115 − 0.933677i \(-0.616580\pi\)
0.629531 + 0.776976i \(0.283247\pi\)
\(60\) 0 0
\(61\) −133.416 − 231.083i −0.280035 − 0.485034i 0.691358 − 0.722512i \(-0.257013\pi\)
−0.971393 + 0.237478i \(0.923679\pi\)
\(62\) 0 0
\(63\) 160.431 + 92.6250i 0.320832 + 0.185233i
\(64\) 0 0
\(65\) 661.206 − 297.696i 1.26173 − 0.568071i
\(66\) 0 0
\(67\) −356.098 − 205.593i −0.649318 − 0.374884i 0.138877 − 0.990310i \(-0.455651\pi\)
−0.788195 + 0.615426i \(0.788984\pi\)
\(68\) 0 0
\(69\) 79.7379 + 138.110i 0.139121 + 0.240964i
\(70\) 0 0
\(71\) 79.2458 − 45.7526i 0.132461 − 0.0764765i −0.432305 − 0.901727i \(-0.642300\pi\)
0.564766 + 0.825251i \(0.308966\pi\)
\(72\) 0 0
\(73\) − 63.1328i − 0.101221i −0.998718 − 0.0506105i \(-0.983883\pi\)
0.998718 − 0.0506105i \(-0.0161167\pi\)
\(74\) 0 0
\(75\) −171.500 + 297.047i −0.264042 + 0.457335i
\(76\) 0 0
\(77\) −1355.25 −2.00578
\(78\) 0 0
\(79\) 287.115 0.408899 0.204449 − 0.978877i \(-0.434460\pi\)
0.204449 + 0.978877i \(0.434460\pi\)
\(80\) 0 0
\(81\) −40.5000 + 70.1481i −0.0555556 + 0.0962250i
\(82\) 0 0
\(83\) 373.812i 0.494352i 0.968971 + 0.247176i \(0.0795026\pi\)
−0.968971 + 0.247176i \(0.920497\pi\)
\(84\) 0 0
\(85\) 593.463 − 342.636i 0.757295 − 0.437224i
\(86\) 0 0
\(87\) −57.9385 − 100.352i −0.0713984 − 0.123666i
\(88\) 0 0
\(89\) 103.406 + 59.7013i 0.123157 + 0.0711047i 0.560313 − 0.828281i \(-0.310681\pi\)
−0.437156 + 0.899386i \(0.644014\pi\)
\(90\) 0 0
\(91\) −782.885 − 563.829i −0.901853 − 0.649509i
\(92\) 0 0
\(93\) −229.643 − 132.584i −0.256052 − 0.147832i
\(94\) 0 0
\(95\) 1137.49 + 1970.18i 1.22846 + 2.12775i
\(96\) 0 0
\(97\) 480.341 − 277.325i 0.502796 − 0.290290i −0.227071 − 0.973878i \(-0.572915\pi\)
0.729868 + 0.683589i \(0.239582\pi\)
\(98\) 0 0
\(99\) − 592.578i − 0.601579i
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.433.2 10
4.3 odd 2 39.4.j.c.4.3 ✓ 10
12.11 even 2 117.4.q.e.82.3 10
13.10 even 6 inner 624.4.bv.h.49.4 10
52.7 even 12 507.4.a.r.1.5 10
52.19 even 12 507.4.a.r.1.6 10
52.23 odd 6 39.4.j.c.10.3 yes 10
52.35 odd 6 507.4.b.i.337.6 10
52.43 odd 6 507.4.b.i.337.5 10
156.23 even 6 117.4.q.e.10.3 10
156.59 odd 12 1521.4.a.bk.1.6 10
156.71 odd 12 1521.4.a.bk.1.5 10
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
39.4.j.c.4.3 ✓ 10 4.3 odd 2
39.4.j.c.10.3 yes 10 52.23 odd 6
117.4.q.e.10.3 10 156.23 even 6
117.4.q.e.82.3 10 12.11 even 2
507.4.a.r.1.5 10 52.7 even 12
507.4.a.r.1.6 10 52.19 even 12
507.4.b.i.337.5 10 52.43 odd 6
507.4.b.i.337.6 10 52.35 odd 6
624.4.bv.h.49.4 10 13.10 even 6 inner
624.4.bv.h.433.2 10 1.1 even 1 trivial
1521.4.a.bk.1.5 10 156.71 odd 12
1521.4.a.bk.1.6 10 156.59 odd 12