Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [3150,2,Mod(899,3150)] 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("3150.899"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3150, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([3, 3, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 3150 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3150.bp (of order \(6\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,-4,0,-4,0,0,0,8,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(25.1528766367\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{24})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 126)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 899.4
Root \(-0.258819 - 0.965926i\) of defining polynomial
Character \(\chi\) \(=\) 3150.899
Dual form 3150.2.bp.b.1349.4

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.500000 - 0.866025i) q^{2} +(-0.500000 + 0.866025i) q^{4} +(2.09077 - 1.62132i) q^{7} +1.00000 q^{8} +(2.59808 + 1.50000i) q^{11} -2.44949 q^{13} +(-2.44949 - 1.00000i) q^{14} +(-0.500000 - 0.866025i) q^{16} +(0.878680 + 0.507306i) q^{17} +(0.878680 - 0.507306i) q^{19} -3.00000i q^{22} +(-2.12132 - 3.67423i) q^{23} +(1.22474 + 2.12132i) q^{26} +(0.358719 + 2.62132i) q^{28} -1.24264i q^{29} +(4.86396 + 2.80821i) q^{31} +(-0.500000 + 0.866025i) q^{32} -1.01461i q^{34} +(7.13834 - 4.12132i) q^{37} +(-0.878680 - 0.507306i) q^{38} -2.02922 q^{41} -8.24264i q^{43} +(-2.59808 + 1.50000i) q^{44} +(-2.12132 + 3.67423i) q^{46} +(-0.878680 + 0.507306i) q^{47} +(1.74264 - 6.77962i) q^{49} +(1.22474 - 2.12132i) q^{52} +(-0.621320 + 1.07616i) q^{53} +(2.09077 - 1.62132i) q^{56} +(-1.07616 + 0.621320i) q^{58} +(-5.76500 + 9.98528i) q^{59} +(5.12132 - 2.95680i) q^{61} -5.61642i q^{62} +1.00000 q^{64} +(8.66025 + 5.00000i) q^{67} +(-0.878680 + 0.507306i) q^{68} +10.2426i q^{71} +(4.18154 - 7.24264i) q^{73} +(-7.13834 - 4.12132i) q^{74} +1.01461i q^{76} +(7.86396 - 1.07616i) q^{77} +(-5.62132 - 9.73641i) q^{79} +(1.01461 + 1.75736i) q^{82} -3.16693i q^{83} +(-7.13834 + 4.12132i) q^{86} +(2.59808 + 1.50000i) q^{88} +(5.19615 + 9.00000i) q^{89} +(-5.12132 + 3.97141i) q^{91} +4.24264 q^{92} +(0.878680 + 0.507306i) q^{94} +3.76127 q^{97} +(-6.74264 + 1.88064i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 4 q^{2} - 4 q^{4} + 8 q^{8} - 4 q^{16} + 24 q^{17} + 24 q^{19} - 12 q^{31} - 4 q^{32} - 24 q^{38} - 24 q^{47} - 20 q^{49} + 12 q^{53} + 24 q^{61} + 8 q^{64} - 24 q^{68} + 12 q^{77} - 28 q^{79} - 24 q^{91}+ \cdots - 20 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3150\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(451\) \(2801\)
\(\chi(n)\) \(-1\) \(e\left(\frac{1}{6}\right)\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.500000 − 0.866025i −0.353553 − 0.612372i
\(3\) 0 0
\(4\) −0.500000 + 0.866025i −0.250000 + 0.433013i
\(5\) 0 0
\(6\) 0 0
\(7\) 2.09077 − 1.62132i 0.790237 − 0.612801i
\(8\) 1.00000 0.353553
\(9\) 0 0
\(10\) 0 0
\(11\) 2.59808 + 1.50000i 0.783349 + 0.452267i 0.837616 − 0.546259i \(-0.183949\pi\)
−0.0542666 + 0.998526i \(0.517282\pi\)
\(12\) 0 0
\(13\) −2.44949 −0.679366 −0.339683 − 0.940540i \(-0.610320\pi\)
−0.339683 + 0.940540i \(0.610320\pi\)
\(14\) −2.44949 − 1.00000i −0.654654 − 0.267261i
\(15\) 0 0
\(16\) −0.500000 − 0.866025i −0.125000 − 0.216506i
\(17\) 0.878680 + 0.507306i 0.213111 + 0.123040i 0.602756 − 0.797925i \(-0.294069\pi\)
−0.389645 + 0.920965i \(0.627402\pi\)
\(18\) 0 0
\(19\) 0.878680 − 0.507306i 0.201583 − 0.116384i −0.395811 − 0.918332i \(-0.629536\pi\)
0.597394 + 0.801948i \(0.296203\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) − 3.00000i − 0.639602i
\(23\) −2.12132 − 3.67423i −0.442326 − 0.766131i 0.555536 − 0.831493i \(-0.312513\pi\)
−0.997862 + 0.0653618i \(0.979180\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 1.22474 + 2.12132i 0.240192 + 0.416025i
\(27\) 0 0
\(28\) 0.358719 + 2.62132i 0.0677916 + 0.495383i
\(29\) − 1.24264i − 0.230753i −0.993322 − 0.115376i \(-0.963193\pi\)
0.993322 − 0.115376i \(-0.0368074\pi\)
\(30\) 0 0
\(31\) 4.86396 + 2.80821i 0.873593 + 0.504369i 0.868541 − 0.495618i \(-0.165058\pi\)
0.00505256 + 0.999987i \(0.498392\pi\)
\(32\) −0.500000 + 0.866025i −0.0883883 + 0.153093i
\(33\) 0 0
\(34\) − 1.01461i − 0.174005i
\(35\) 0 0
\(36\) 0 0
\(37\) 7.13834 − 4.12132i 1.17354 − 0.677541i 0.219025 − 0.975719i \(-0.429712\pi\)
0.954510 + 0.298178i \(0.0963790\pi\)
\(38\) −0.878680 − 0.507306i −0.142541 − 0.0822959i
\(39\) 0 0
\(40\) 0 0
\(41\) −2.02922 −0.316912 −0.158456 − 0.987366i \(-0.550652\pi\)
−0.158456 + 0.987366i \(0.550652\pi\)
\(42\) 0 0
\(43\) − 8.24264i − 1.25699i −0.777813 − 0.628495i \(-0.783671\pi\)
0.777813 − 0.628495i \(-0.216329\pi\)
\(44\) −2.59808 + 1.50000i −0.391675 + 0.226134i
\(45\) 0 0
\(46\) −2.12132 + 3.67423i −0.312772 + 0.541736i
\(47\) −0.878680 + 0.507306i −0.128169 + 0.0739982i −0.562713 − 0.826652i \(-0.690243\pi\)
0.434545 + 0.900650i \(0.356909\pi\)
\(48\) 0 0
\(49\) 1.74264 − 6.77962i 0.248949 − 0.968517i
\(50\) 0 0
\(51\) 0 0
\(52\) 1.22474 − 2.12132i 0.169842 − 0.294174i
\(53\) −0.621320 + 1.07616i −0.0853449 + 0.147822i −0.905538 − 0.424265i \(-0.860533\pi\)
0.820193 + 0.572087i \(0.193866\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 2.09077 − 1.62132i 0.279391 − 0.216658i
\(57\) 0 0
\(58\) −1.07616 + 0.621320i −0.141307 + 0.0815834i
\(59\) −5.76500 + 9.98528i −0.750540 + 1.29997i 0.197022 + 0.980399i \(0.436873\pi\)
−0.947561 + 0.319574i \(0.896460\pi\)
\(60\) 0 0
\(61\) 5.12132 − 2.95680i 0.655718 − 0.378579i −0.134926 − 0.990856i \(-0.543080\pi\)
0.790643 + 0.612277i \(0.209746\pi\)
\(62\) − 5.61642i − 0.713286i
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) 8.66025 + 5.00000i 1.05802 + 0.610847i 0.924883 − 0.380251i \(-0.124162\pi\)
0.133135 + 0.991098i \(0.457496\pi\)
\(68\) −0.878680 + 0.507306i −0.106556 + 0.0615199i
\(69\) 0 0
\(70\) 0 0
\(71\) 10.2426i 1.21558i 0.794099 + 0.607789i \(0.207943\pi\)
−0.794099 + 0.607789i \(0.792057\pi\)
\(72\) 0 0
\(73\) 4.18154 − 7.24264i 0.489412 − 0.847687i −0.510513 − 0.859870i \(-0.670545\pi\)
0.999926 + 0.0121828i \(0.00387799\pi\)
\(74\) −7.13834 − 4.12132i −0.829815 − 0.479094i
\(75\) 0 0
\(76\) 1.01461i 0.116384i
\(77\) 7.86396 − 1.07616i 0.896182 − 0.122640i
\(78\) 0 0
\(79\) −5.62132 − 9.73641i −0.632448 − 1.09543i −0.987050 − 0.160415i \(-0.948717\pi\)
0.354602 − 0.935017i \(-0.384616\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 1.01461 + 1.75736i 0.112045 + 0.194068i
\(83\) − 3.16693i − 0.347616i −0.984780 − 0.173808i \(-0.944393\pi\)
0.984780 − 0.173808i \(-0.0556071\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −7.13834 + 4.12132i −0.769747 + 0.444413i
\(87\) 0 0
\(88\) 2.59808 + 1.50000i 0.276956 + 0.159901i
\(89\) 5.19615 + 9.00000i 0.550791 + 0.953998i 0.998218 + 0.0596775i \(0.0190072\pi\)
−0.447427 + 0.894321i \(0.647659\pi\)
\(90\) 0 0
\(91\) −5.12132 + 3.97141i −0.536860 + 0.416317i
\(92\) 4.24264 0.442326
\(93\) 0 0
\(94\) 0.878680 + 0.507306i 0.0906289 + 0.0523246i
\(95\) 0 0
\(96\) 0 0
\(97\) 3.76127 0.381900 0.190950 − 0.981600i \(-0.438843\pi\)
0.190950 + 0.981600i \(0.438843\pi\)
\(98\) −6.74264 + 1.88064i −0.681110 + 0.189973i
\(99\) 0 0
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 3150.2.bp.b.899.4 8
3.2 odd 2 3150.2.bp.e.899.4 8
5.2 odd 4 3150.2.bf.a.1151.4 8
5.3 odd 4 126.2.k.a.17.1 ✓ 8
5.4 even 2 3150.2.bp.e.899.1 8
7.5 odd 6 inner 3150.2.bp.b.1349.1 8
15.2 even 4 3150.2.bf.a.1151.2 8
15.8 even 4 126.2.k.a.17.4 yes 8
15.14 odd 2 inner 3150.2.bp.b.899.1 8
20.3 even 4 1008.2.bt.c.17.1 8
21.5 even 6 3150.2.bp.e.1349.1 8
35.3 even 12 882.2.d.a.881.1 8
35.12 even 12 3150.2.bf.a.1601.2 8
35.13 even 4 882.2.k.a.521.2 8
35.18 odd 12 882.2.d.a.881.4 8
35.19 odd 6 3150.2.bp.e.1349.4 8
35.23 odd 12 882.2.k.a.215.3 8
35.33 even 12 126.2.k.a.89.4 yes 8
45.13 odd 12 1134.2.l.f.269.1 8
45.23 even 12 1134.2.l.f.269.4 8
45.38 even 12 1134.2.t.e.1025.1 8
45.43 odd 12 1134.2.t.e.1025.4 8
60.23 odd 4 1008.2.bt.c.17.4 8
105.23 even 12 882.2.k.a.215.2 8
105.38 odd 12 882.2.d.a.881.8 8
105.47 odd 12 3150.2.bf.a.1601.4 8
105.53 even 12 882.2.d.a.881.5 8
105.68 odd 12 126.2.k.a.89.1 yes 8
105.83 odd 4 882.2.k.a.521.3 8
105.89 even 6 inner 3150.2.bp.b.1349.4 8
140.3 odd 12 7056.2.k.f.881.2 8
140.103 odd 12 1008.2.bt.c.593.4 8
140.123 even 12 7056.2.k.f.881.8 8
315.68 odd 12 1134.2.t.e.593.4 8
315.103 even 12 1134.2.t.e.593.1 8
315.173 odd 12 1134.2.l.f.215.3 8
315.313 even 12 1134.2.l.f.215.2 8
420.143 even 12 7056.2.k.f.881.7 8
420.263 odd 12 7056.2.k.f.881.1 8
420.383 even 12 1008.2.bt.c.593.1 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
126.2.k.a.17.1 ✓ 8 5.3 odd 4
126.2.k.a.17.4 yes 8 15.8 even 4
126.2.k.a.89.1 yes 8 105.68 odd 12
126.2.k.a.89.4 yes 8 35.33 even 12
882.2.d.a.881.1 8 35.3 even 12
882.2.d.a.881.4 8 35.18 odd 12
882.2.d.a.881.5 8 105.53 even 12
882.2.d.a.881.8 8 105.38 odd 12
882.2.k.a.215.2 8 105.23 even 12
882.2.k.a.215.3 8 35.23 odd 12
882.2.k.a.521.2 8 35.13 even 4
882.2.k.a.521.3 8 105.83 odd 4
1008.2.bt.c.17.1 8 20.3 even 4
1008.2.bt.c.17.4 8 60.23 odd 4
1008.2.bt.c.593.1 8 420.383 even 12
1008.2.bt.c.593.4 8 140.103 odd 12
1134.2.l.f.215.2 8 315.313 even 12
1134.2.l.f.215.3 8 315.173 odd 12
1134.2.l.f.269.1 8 45.13 odd 12
1134.2.l.f.269.4 8 45.23 even 12
1134.2.t.e.593.1 8 315.103 even 12
1134.2.t.e.593.4 8 315.68 odd 12
1134.2.t.e.1025.1 8 45.38 even 12
1134.2.t.e.1025.4 8 45.43 odd 12
3150.2.bf.a.1151.2 8 15.2 even 4
3150.2.bf.a.1151.4 8 5.2 odd 4
3150.2.bf.a.1601.2 8 35.12 even 12
3150.2.bf.a.1601.4 8 105.47 odd 12
3150.2.bp.b.899.1 8 15.14 odd 2 inner
3150.2.bp.b.899.4 8 1.1 even 1 trivial
3150.2.bp.b.1349.1 8 7.5 odd 6 inner
3150.2.bp.b.1349.4 8 105.89 even 6 inner
3150.2.bp.e.899.1 8 5.4 even 2
3150.2.bp.e.899.4 8 3.2 odd 2
3150.2.bp.e.1349.1 8 21.5 even 6
3150.2.bp.e.1349.4 8 35.19 odd 6
7056.2.k.f.881.1 8 420.263 odd 12
7056.2.k.f.881.2 8 140.3 odd 12
7056.2.k.f.881.7 8 420.143 even 12
7056.2.k.f.881.8 8 140.123 even 12