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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 49.4
Root \(-0.917374i\) of defining polynomial
Character \(\chi\) \(=\) 624.49
Dual form 624.4.bv.h.433.2

$q$-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 \(\left(\mathbb{Z}/624\mathbb{Z}\right)^\times\).

\(n\) \(79\) \(145\) \(209\) \(469\)
\(\chi(n)\) \(1\) \(e\left(\frac{5}{6}\right)\) \(1\) \(1\)

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 \(0.243208\pi\)
−0.722034 + 0.691858i \(0.756792\pi\)
\(6\) 0 0
\(7\) −17.8257 − 10.2917i −0.962497 − 0.555698i −0.0655563 − 0.997849i \(-0.520882\pi\)
−0.896941 + 0.442151i \(0.854216\pi\)
\(8\) 0 0
\(9\) −4.50000 + 7.79423i −0.166667 + 0.288675i
\(10\) 0 0
\(11\) 57.0209 − 32.9210i 1.56295 − 0.902369i 0.565992 − 0.824411i \(-0.308493\pi\)
0.996957 − 0.0779583i \(-0.0248401\pi\)
\(12\) 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 −0.663666 − 0.748029i \(-0.731001\pi\)
0.979645 + 0.200738i \(0.0643339\pi\)
\(18\) 0 0
\(19\) −127.352 − 73.5266i −1.53771 − 0.887798i −0.998972 − 0.0453247i \(-0.985568\pi\)
−0.538738 − 0.842473i \(-0.681099\pi\)
\(20\) 0 0
\(21\) − 61.7500i − 0.641665i
\(22\) 0 0
\(23\) −26.5793 − 46.0367i −0.240964 − 0.417362i 0.720025 − 0.693948i \(-0.244130\pi\)
−0.960989 + 0.276586i \(0.910797\pi\)
\(24\) 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 0.921211 − 0.389064i \(-0.127202\pi\)
−0.797545 + 0.603260i \(0.793868\pi\)
\(30\) 0 0
\(31\) 88.3894i 0.512104i 0.966663 + 0.256052i \(0.0824218\pi\)
−0.966663 + 0.256052i \(0.917578\pi\)
\(32\) 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 −0.340333 − 0.940305i \(-0.610540\pi\)
0.644161 + 0.764890i \(0.277206\pi\)
\(38\) 0 0
\(39\) 139.906 − 14.1155i 0.574434 − 0.0579561i
\(40\) 0 0
\(41\) 307.410 − 177.483i 1.17096 − 0.676054i 0.217053 − 0.976160i \(-0.430355\pi\)
0.953906 + 0.300106i \(0.0970221\pi\)
\(42\) 0 0
\(43\) 203.923 − 353.205i 0.723208 − 1.25263i −0.236499 − 0.971632i \(-0.576000\pi\)
0.959707 − 0.281002i \(-0.0906666\pi\)
\(44\) 0 0
\(45\) −120.580 − 69.6169i −0.399444 − 0.230619i
\(46\) 0 0
\(47\) 67.9674i 0.210938i 0.994423 + 0.105469i \(0.0336343\pi\)
−0.994423 + 0.105469i \(0.966366\pi\)
\(48\) 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 0.629531 − 0.776976i \(-0.283247\pi\)
−0.358115 + 0.933677i \(0.616580\pi\)
\(60\) 0 0
\(61\) −133.416 + 231.083i −0.280035 + 0.485034i −0.971393 − 0.237478i \(-0.923679\pi\)
0.691358 + 0.722512i \(0.257013\pi\)
\(62\) 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 −0.788195 − 0.615426i \(-0.788984\pi\)
0.138877 + 0.990310i \(0.455651\pi\)
\(68\) 0 0
\(69\) 79.7379 − 138.110i 0.139121 − 0.240964i
\(70\) 0 0
\(71\) 79.2458 + 45.7526i 0.132461 + 0.0764765i 0.564766 − 0.825251i \(-0.308966\pi\)
−0.432305 + 0.901727i \(0.642300\pi\)
\(72\) 0 0
\(73\) 63.1328i 0.101221i 0.998718 + 0.0506105i \(0.0161167\pi\)
−0.998718 + 0.0506105i \(0.983883\pi\)
\(74\) 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 \(-0.920497\pi\)
0.968971 − 0.247176i \(-0.0795026\pi\)
\(84\) 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 −0.437156 − 0.899386i \(-0.644014\pi\)
0.560313 + 0.828281i \(0.310681\pi\)
\(90\) 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 0.729868 − 0.683589i \(-0.239582\pi\)
−0.227071 + 0.973878i \(0.572915\pi\)
\(98\) 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.49.4 10
4.3 odd 2 39.4.j.c.10.3 yes 10
12.11 even 2 117.4.q.e.10.3 10
13.4 even 6 inner 624.4.bv.h.433.2 10
52.3 odd 6 507.4.b.i.337.5 10
52.11 even 12 507.4.a.r.1.6 10
52.15 even 12 507.4.a.r.1.5 10
52.23 odd 6 507.4.b.i.337.6 10
52.43 odd 6 39.4.j.c.4.3 ✓ 10
156.11 odd 12 1521.4.a.bk.1.5 10
156.95 even 6 117.4.q.e.82.3 10
156.119 odd 12 1521.4.a.bk.1.6 10
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
39.4.j.c.4.3 ✓ 10 52.43 odd 6
39.4.j.c.10.3 yes 10 4.3 odd 2
117.4.q.e.10.3 10 12.11 even 2
117.4.q.e.82.3 10 156.95 even 6
507.4.a.r.1.5 10 52.15 even 12
507.4.a.r.1.6 10 52.11 even 12
507.4.b.i.337.5 10 52.3 odd 6
507.4.b.i.337.6 10 52.23 odd 6
624.4.bv.h.49.4 10 1.1 even 1 trivial
624.4.bv.h.433.2 10 13.4 even 6 inner
1521.4.a.bk.1.5 10 156.11 odd 12
1521.4.a.bk.1.6 10 156.119 odd 12